From ca2d71f44cc3d973c757db3b4d6ba3e4678b4a06 Mon Sep 17 00:00:00 2001 From: Asher Frost Date: Sat, 6 Jun 2026 12:25:28 -0400 Subject: [PATCH 1/9] feat: fixed some of the fol orders --- .gitmodules | 3 + .vscode/settings.json | 6 - README.md | 4 + doc/index.mld | 57 +-- docs/grammar.md | 677 ++++++++++++++++++++++++++++ dune-project | 17 +- ffi/zig/Lens.zig | 13 + ffi/zig/Server.zig | 4 + guide/pages/greek.typ | 0 guide/pages/lf.typ | 0 guide/pages/logic.typ | 7 - guide/pages/syntax.typ | 0 guide/pages/typed.typ | 0 guide/src/macros.typ | 0 guide/src/style.typ | 6 - guide/stelf.typ | 5 - odoc-config.sexp | 1 + src/Common/Cst/CST.ml | 27 +- src/Common/Cst/Cst.ml | 311 +++++++++---- src/Common/Cst/LENS.ml | 543 ++++++++++++---------- src/Common/Cst/Lens.ml | 10 +- src/Display/FORM.ml | 6 +- src/Display/Form.ml | 4 + src/Fronts/Modern/Cmd.ml | 54 +-- src/Fronts/Modern/MODERN.ml | 10 +- src/Fronts/Modern/Modern.ml | 325 +++++++++++--- src/Fronts/Pal/IMPL.ml | 2 +- src/Fronts/Pal/Impl.ml | 88 +++- src/Fronts/Pal/Pal.ml | 2 +- src/Fronts/Pal/dune | 1 + src/Lang/Parsing/PARSER.ml | 2 + src/Lang/Parsing/Parser.ml | 24 +- src/Recon/ReconMode.ml | 4 +- src/Recon/ReconTerm.ml | 131 +++--- src/modes/Modecheck.ml | 18 +- src/modes/Modedec.ml | 1 + src/modes/modecheck.fun | 868 ++++++++++++++++++++++++++++++++++++ src/msg/Msg_.ml | 2 +- stelf-book | 1 + stelf.opam | 13 +- test/Pal/Cases.ml | 22 +- test/Pal/Common.ml | 7 +- test/Pal/Source.ml | 446 +++++++++++++++++- test/Parse/Cases.ml | 33 +- test/Parse/Common.ml | 20 +- trans/input/fol.elf | 217 +++++++++ trans/input/js4.elf | 107 +++++ trans/input/lam.elf | 148 ++++++ trans/input/polylam.elf | 23 + trans/output/fol.stelf | 173 +++++++ trans/output/js4.stelf | 77 ++++ trans/output/lam.stelf | 103 +++++ trans/output/polylam.stelf | 18 + 53 files changed, 4028 insertions(+), 613 deletions(-) delete mode 100644 .vscode/settings.json create mode 100644 docs/grammar.md create mode 100644 ffi/zig/Lens.zig create mode 100644 ffi/zig/Server.zig delete mode 100644 guide/pages/greek.typ delete mode 100644 guide/pages/lf.typ delete mode 100644 guide/pages/logic.typ delete mode 100644 guide/pages/syntax.typ delete mode 100644 guide/pages/typed.typ delete mode 100644 guide/src/macros.typ delete mode 100644 guide/src/style.typ delete mode 100644 guide/stelf.typ create mode 100644 odoc-config.sexp create mode 100644 src/modes/modecheck.fun create mode 160000 stelf-book create mode 100644 trans/input/fol.elf create mode 100644 trans/input/js4.elf create mode 100644 trans/input/lam.elf create mode 100644 trans/input/polylam.elf create mode 100644 trans/output/fol.stelf create mode 100644 trans/output/js4.stelf create mode 100644 trans/output/lam.stelf create mode 100644 trans/output/polylam.stelf diff --git a/.gitmodules b/.gitmodules index 4bf3606..f16d956 100644 --- a/.gitmodules +++ b/.gitmodules @@ -5,3 +5,6 @@ [submodule "twelf"] path = twelf url = https://github.com/standardml/twelf +[submodule "stelf-book"] + path = stelf-book + url = https://github.com/standardocaml/stelf-book.git diff --git a/.vscode/settings.json b/.vscode/settings.json deleted file mode 100644 index c2b754a..0000000 --- a/.vscode/settings.json +++ /dev/null @@ -1,6 +0,0 @@ -{ - "ocaml.sandbox": { - "kind": "opam", - "switch": "4.14.2" - } -} \ No newline at end of file diff --git a/README.md b/README.md index 9372617..cfd1d11 100644 --- a/README.md +++ b/README.md @@ -2,6 +2,10 @@ This is the STELF project, a port of the Twelf system to OCaml, and subsequents developments thereof. +> [!warning] If you get `dune` seeming to be "paused" on building, or the number of tasks increasing, your computer (probably) won't explode. +> This is merely an artificat of the new Dune package managment system (the first pause is due to the ocaml compiler). +> If you want to make sure nothing is going wrong, add `--verbose` to the `dune build` command, and you should see the tasks being built one by one. + ## Twelf Port The Twelf port, which is completed, involved translating between SML to OCaml, and also the creation of a temporary basis library. diff --git a/doc/index.mld b/doc/index.mld index b924111..0c62aec 100644 --- a/doc/index.mld +++ b/doc/index.mld @@ -3,59 +3,14 @@ This is the internal docmentation for the STELF project. {!modules: - Stelf - Cst - Syntax - Common + Common + Cst Display - Error + Recon + Syntax Fronts + Pal + Tui IntSyn Lang - Recon - Compile - Compress - Cover - Debug - Domains - Flit - Formatter - Frontend - Global - Heuristic - Index - IntInf - Inverse - M2 - Meta - Modes - Modules - Msg - Names - Netserver - Opsem - Order - Paths - Print - Prover - Server - Smlofnj - Solvers - Stream - Style - Subordinate - Table - Tabling - Terminate - Thm - Timing - Tomega - Trail - Typecheck - Unique - Worldcheck - Compile - Compress - Cover - Display } diff --git a/docs/grammar.md b/docs/grammar.md new file mode 100644 index 0000000..9b0ee5a --- /dev/null +++ b/docs/grammar.md @@ -0,0 +1,677 @@ +# Modern STELF Grammar + +A reference BNF for the **modern** STELF surface syntax — the variant +parsed by `src/Fronts/Modern/Modern.ml` and `src/Fronts/Modern/Cmd.ml`, +and exercised by `test/Parse/Cases.ml`. + +This is **not** legacy Twelf syntax. The legacy frontend +(`src/frontend/Parse*.ml`) accepts Twelf-style declarations +(`c : A.`, `[x:A] M`, `{x:A} B`, `%mode +x -y`, trailing `.`). That +grammar is out of scope here. + +This document is intended to: + +1. Serve as a human-readable spec for writing and reading STELF code. +2. Drive a future tree-sitter implementation. A design-notes appendix + covers tree-sitter-specific concerns; the body of the document is + pure BNF + commentary. + +Notational conventions: + +- `lowercase` names are non-terminals. +- `"foo"` is a literal terminal. +- `x*`, `x+`, `x?` are Kleene closure / one-or-more / optional. +- `(* ... *)` are grammar-level comments. +- All productions describe **modern STELF**, as accepted by the parser + on the `dev` branch at the time of writing. + +--- + +## 1. Overview + +STELF source is a stream of **commands**. Every command and every +reserved word begins with `%`. Everything else — `nat`, `succ`, `+`, +`<=`, `add/zero`, `_X` — is an identifier. There are no statement +terminators; commands are delimited by the *start* of the next `%`-word. + +The only punctuation tokens are `(` `)` `{` `}` `[` `]` `%`. Whitespace +separates identifiers and otherwise has no meaning. + +A small example: + +```stelf +%sort nat +%term zero nat +%term succ {_ nat} nat + +%sort add {_ nat} {_ nat} {_ nat} +%term add/zero {y nat} add zero y y +%term add/succ {x nat} {y nat} {z nat} {_ add x y z} + add (succ x) y (succ z) + +%mode {%in x nat} {%in y nat} {%out z nat} add x y z +%worlds () (add _ _ _) +%total N (add N _ _) +``` + +--- + +## 2. Lexical structure + +```bnf +(* Whitespace is space, tab, and newline; never significant. *) +ws ::= ( " " | "\t" | "\n" )* + +(* The only punctuation. Everything else is an identifier char. *) +delim ::= "(" | ")" | "{{" | "}}" | "{" | "}" | "[" | "]" | "%" | ws | EOF + +(* An identifier is any non-empty run of non-delimiter chars. *) +ident ::= ident-char+ +ident-char ::= + +(* A keyword is '%' immediately followed by an identifier body + AND a word boundary. The boundary check is what prevents + '%term' from being mistaken for the prefix of '%terminates'. *) +keyword s ::= "%" s &(delim) (* lookahead, not consumed *) + +(* Natural number literals — used by %query bounds and %prec. *) +nat ::= [0-9]+ +``` + +Consequences worth knowing: + +- `=`, `<`, `>`, `<=`, `>=`, `+`, `*`, `:`, `:=`, `add/zero`, `_X` are + all valid identifiers. None are reserved. +- Identifier *case* is decided by the first character, not by the + lexer: + - leading `_` or name mentioned in scope inside `{{A B...}}` → **uppercase** (used as a metavariable), + - anything else → **lowercase** (a constant). +- Comments are **not currently recognised** by the modern parser. See + §15 "Gaps". + +--- + +## 3. Top-level command stream + +```bnf +program ::= ( outer-text command )* outer-text +outer-text ::= * (* skipped between commands *) + +command ::= stop + | term-cmd + | mode-cmd | covers-cmd + | module-cmd | use-cmd | open-cmd | eval-cmd + | sort-cmd | block-cmd | union-cmd | worlds-cmd + | freeze-cmd | thaw-cmd | deterministic-cmd + | name-cmd | symbol-cmd | inline-cmd + | define-cmd | solve-cmd | decl-cmd + | query-cmd | qtab-cmd | adhoc-query-cmd + | unique-cmd + | total-cmd | terminates-cmd | reduces-cmd + | prec-cmd + | repl-cmd + +stop ::= "%." +``` + +The complete keyword inventory is enumerated in §13. + +--- + +## 4. Terms + +The term grammar is the heart of STELF. + +```bnf +expr ::= ascription + | arrow-chain + | backarrow-chain + | expr-app + +ascription ::= "%the" expr1 expr (* ascribe: body : type *) + +(* %-> A %-> B %-> C ==> {_ A} {_ B} C + The '%->' between arguments is optional after the first. *) +arrow-chain ::= "%->" expr1 ( "%->"? expr1 )+ + +(* %<- A %<- B %<- C ==> {_ C} {_ B} A + Like arrow-chain but the *first* argument is the body. *) +backarrow-chain + ::= "%<-" expr1 ( "%<-"? expr1 )+ + +(* Application is juxtaposition. A trailing lambda/Pi binder may + follow the last argument: f x y [w] w parses as + f applied to x, y, and [w] w *) +expr-app ::= expr1+ expr-trail? + | expr-trail (* trailing-only is ok *) + +expr-trail ::= "[" decl "]" expr (* lambda abstraction *) + | "{" decl "}" expr (* dependent product *) + | "{{" ident+ "}}" expr (* List of names that should be treated as uppercase in expr *) + +(* "Small" expression: legal as an application argument or inside + a parenthesised expression. No top-level binders, ascriptions, + or arrow chains here. *) +expr1 ::= atom + | "(" expr ")" + +atom ::= ident (* upper- or lower-case *) + | qualified + +qualified ::= "%val" ident (* unqualified rewrite *) + | "%val" "(" ident+ ")" (* dotted-path symbol *) +``` + +Notes: + +- `expr-app`'s `expr1+ expr-trail?` is what lets `f x [y] y` and + `f {x nat} x` parse cleanly. +- An atom is upper- or lowercase based on the *first character* of + the identifier (§2). `_X` and `Foo` are uppercase; `nat`, `+`, `<=` + are lowercase. +- A parenthesised expression `( expr )` may contain anything, including + another binder chain. + +### 4.1 Test-suite examples (from `test/Parse/Cases.ml:31-72`) + +| Surface | Production | +|---|---| +| `nat` | `atom` | +| `succ zero` | `expr-app` | +| `succ (succ zero)` | `expr-app` with parenthesised arg | +| `[x] x` | `expr-trail` (lambda) | +| `{x} x` | `expr-trail` (Pi) | +| `%the nat zero` | `ascription` | +| `succ zero [x] x` | `expr-app` with trailing lambda | +| `f {x} x [y] y` | nested trailing binders | +| `_X` | uppercase atom | +| `%val ( x y )` | qualified | +| `%val +` | qualified (single, including symbolic ident) | +| `f [x nat] [y nat] x` | nested lambdas with typed binders | +| `f {p {_ nat} nat} z` | Pi with a nested-Pi binder type | + +--- + +## 5. Declarations (binders) + +A **declaration** is the `x : A` or `(x y) : A` slot that appears inside +`[...]` and `{...}` binders, inside `%term`, `%mode`, etc. In modern +STELF there is *no* colon — the type follows the name(s) directly. + +```bnf +decl ::= "(" arg+ ")" expr? (* grouped names, optional type *) + | arg expr? (* single name, optional type *) + +arg ::= "_" | ident +``` + +Examples (from `test/Parse/Cases.ml:51-58`): + +| Surface | Meaning | +|---|---| +| `x nat` | one name, one type | +| `(x y) nat` | two names sharing a type | +| `x` | untyped binder | +| `(x y)` | untyped grouped binder | +| `_ nat` | anonymous, typed | +| `_` | anonymous, untyped | + +--- + +## 6. Modes + +A mode marker classifies a binder position as input / output. + +```bnf +mode ::= "%in" (* + : input, must be ground *) + | "%out" (* - : output *) + | "%out1" (* -1: strict output (at most one) *) + | "%star" (* * : either, no constraint *) + +(* A mode declaration mixes braced full-form and trailing spine form. + The braced binders come *first*, then the head expression, then + trailing bare modes for the head's spine. *) +mode-dec ::= ( "{" mode decl "}" )* expr mode* +``` + +The three surface shapes (from `test/Parse/Cases.ml:111-117`): + +```stelf +(* full form: every argument has its mode in a braced binder *) +%mode {%in x nat} {%in y nat} {%out z nat} add x y z + +(* spine form: bare modes follow the predicate *) +%mode add %in %in %out + +(* mixed: leading braced binders, then trailing spine *) +%mode {%in x nat} {%in y nat} add x y %out +``` + +Mode declarations appear in `%mode` and `%covers` commands. + +--- + +## 7. Block / world / union + +```bnf +(* {decl} is universal/Pi-bound; [decl] is existential/some-bound. *) +block-item ::= "{" decl "}" + | "[" decl "]" + +block-cmd ::= "%block" ident block-item* + +union-cmd ::= "%union" ident "(" ident+ ")" + +worlds-cmd ::= "%worlds" "(" ident* ")" expr +``` + +Examples (`test/Parse/Cases.ml:118-133`): + +```stelf +%block test { x nat } +%block test { x nat } { y bool } +%block test [x nat] +%block test [(x y) nat] +%union test (nat bool) +%worlds () (add _ _ _) +%worlds (N) (add N _ _) +``` + +--- + +## 8. Termination orders + +`%total` and `%terminates` take an *order* and a list of call patterns. +Orders compose: + +```bnf +order ::= id-list (* varg: a flat list *) + | "[" order+ "]" (* simultaneous *) + | "{" order+ "}" (* lexicographic *) + +id-list ::= ident + | "(" ident+ ")" + +(* The parenthesised form lets you give a tuple of orders, one per + mutually-recursive predicate. *) +order-list ::= "(" order+ ")" + | order + +total-cmd ::= "%total" order-list expr1+ +terminates-cmd ::= "%terminates" order-list expr1+ +``` + +Examples (`test/Parse/Cases.ml:135-153`): + +```stelf +%total N (add N _ _) +%total (N1 N2) (add N1 _ _) (mul N2 _ _) +%terminates N (add N _ _) +%terminates [A B] max A B (* simultaneous *) +%terminates {A B} max A B (* lexicographic *) +%terminates {A [B C] F} (max A (max B C)) (* nested *) +%terminates ({A [B C] G} [D E] F) (max A (max B C) max D (max E C)) +``` + +--- + +## 9. Reduces + +`%reduces` declares a size relation between argument positions: + +```bnf +reduces-rel ::= "<=" | ">=" | "<" | ">" | "=" +reduces-cmd ::= "%reduces" reduces-rel expr1+ +``` + +The relations are *identifiers* lexically (§2), but the parser tries +the two-character forms before the one-character forms so `<=` is not +read as `<` followed by `=`. + +Examples (`test/Parse/Cases.ml:162-169`): + +```stelf +%reduces = X Y add X Y zero +%reduces < X Y add X Y zero +%reduces > X Y add X Y zero +%reduces >= X Y add X Y zero +%reduces <= X Y add X Y zero +``` + +--- + +## 10. Signatures, structures, modules + +TODO + +--- + +## 11. Queries, defines, solves + +```bnf +bound ::= "_" | nat (* '_' means unbounded *) + +query-cmd ::= "%query" bound bound bound expr +qtab-cmd ::= "%querytabled" bound bound bound expr +adhoc-query ::= "%?" expr + +define-cmd ::= ( "%define" | "%def" ) ( ident | "_" ) expr1 expr + +solve-cmd ::= "%solve" expr +decl-cmd ::= "%decl" expr +inline-cmd ::= "%inline" ident expr +unique-cmd ::= "%unique" expr +``` + +`%def` is an accepted synonym for `%define` (Cmd.ml:84). + +Examples (`test/Parse/Cases.ml:155-160, 100-110`): + +```stelf +%? nat +%? add zero zero zero +%query _ _ 1 add zero zero zero + +%def not ({_ prop} prop) ([a] imp a false) +%def eq_i (pf (eq _A _A)) prop +``` + +--- + +## 12. Sorts, terms, names, fixity, etc + +```bnf +sort-cmd ::= "%sort" id-list ( "{" decl "}" )* +term-cmd ::= "%term" decl + +freeze-cmd ::= "%freeze" id-list +thaw-cmd ::= "%thaw" id-list +deterministic-cmd + ::= "%deterministic" id-list + +name-cmd ::= "%name" ident +symbol-cmd ::= "%symbol" ident ident +covers-cmd ::= "%covers" mode-dec + +fixity-kw ::= "%left" | "%right" | "%prefix" + | "%postfix" | "%middle" | "%none" +prec-cmd ::= "%prec" fixity-kw nat id-list + +(* REPL-only commands. *) +repl-cmd ::= "%quit" + | "%help" ident? + | "%get" ident + | "%set" ident ident + | "%version" +``` + +Note on `%sort`: the parser accepts `%sort nat` (a bare id-list) or +`%sort (nat bool)` (a grouped id-list of mutually-defined sorts), each +optionally followed by indexed parameter binders such as +`{_ nat} {_ nat}`. The test suite covers both shapes +(`test/Parse/Cases.ml:73-86`). + +--- + +## 13. Command reference (alphabetical) + +The full list of `%`-prefixed commands recognised by `Cmd.ml`. Each +entry shows the BNF, the source line in `Cmd.ml`, and (where one +exists) a test-suite example. + +| Command | Body | Cmd.ml | Example | +|---|---|---|---| +| `%.` | — | 35 | (end-of-input marker) | +| `%?` | `expr` | 53 | `%? add zero zero zero` | +| `%block` | `ident block-item*` | 144 | `%block test { x nat } [y bool]` | +| `%covers` | `mode-dec` | 265 | `%covers add %in %in %out` | +| `%decl` | `expr` | 92 | `%decl (eq nat nat)` | +| `%define` / `%def` | `(ident\|"_") expr1 expr` | 84 | `%def not ({_ prop} prop) ([a] imp a false)` | +| `%deterministic` | `id-list` | 168 | `%deterministic (add)` | +| `%eval` | `"%{" command* "%}"` | 192 | `%eval %{ %? nat %}` | +| `%freeze` | `id-list` | 115 | `%freeze (nat)` | +| `%get` | `ident` | 230 | `%get chatter` | +| `%help` | `ident?` | 219 | `%help mode` | +| `%inline` | `ident expr` | 99 | `%inline succ_z (succ zero)` | +| `%mode` | `mode-dec` | 77 | `%mode {%in x nat} add x %out` | +| `%module` | `ident "(" ident* ")" "%{" command* "%}"` | 68 | `%module Nat () %{ %sort n %}` | +| `%name` | `ident` | 272 | `%name addition` | +| `%open` | `ident id-list` | 184 | `%open M (succ zero)` | +| `%prec` | `fixity-kw nat id-list` | 199 | `%prec %left 5 (+)` | +| `%query` | `bound bound bound expr` | 46 | `%query _ _ 1 add zero zero zero` | +| `%querytabled` | `bound bound bound expr` | 39 | `%querytabled _ _ _ p` | +| `%quit` | — | 216 | `%quit` | +| `%reduces` | `reduces-rel expr1+` | 279 | `%reduces < X Y add X Y zero` | +| `%set` | `ident ident` | 237 | `%set chatter 3` | +| `%solve` | `expr` | 209 | `%solve add zero zero zero` | +| `%sort` | `id-list ( "{" decl "}" )*` | 129 | `%sort prop {_ nat} {_ nat}` | +| `%symbol` | `ident ident` | 107 | `%symbol plus +` | +| `%term` | `decl` | 137 | `%term succ {_ nat} nat` | +| `%terminates` | `order-list expr1+` | 257 | `%terminates {A B} max A B` | +| `%thaw` | `id-list` | 122 | `%thaw (nat)` | +| `%total` | `order-list expr1+` | 249 | `%total N (add N _ _)` | +| `%union` | `ident "(" ident+ ")"` | 152 | `%union test (nat bool)` | +| `%unique` | `expr` | 60 | `%unique nat` | +| `%use` | `ident ident "(" ident* ")"` | 175 | `%use M N ()` | +| `%version` | — | 245 | `%version` | +| `%worlds` | `"(" ident* ")" expr` | 160 | `%worlds () (add _ _ _)` | + +Three additional command keywords expected from the CST but **not** +currently parsed by `Cmd.ml` (gaps, listed in §15): `%abbrev`, +`%trustme`, `%subord`. + +--- + +## 14. End-to-end examples + +```stelf +(* Peano naturals *) +%sort nat +%term zero nat +%term succ {_ nat} nat + +(* Addition relation, mode-directed *) +%sort add {_ nat} {_ nat} {_ nat} +%term add/zero {y nat} add zero y y +%term add/succ {x nat} {y nat} {z nat} {_ add x y z} + add (succ x) y (succ z) + +%mode {%in x nat} {%in y nat} {%out z nat} add x y z +%worlds () (add _ _ _) +%total N (add N _ _) + +(* Multiplication, expressed in terms of addition *) +%sort mul {_ nat} {_ nat} {_ nat} +%term mul/zero {x nat} mul x zero zero +%term mul/succ {x nat} {y nat} {z nat} {z' nat} + {_ mul x y z} {_ add y z z'} + (mul (succ x) y z') + +(* Sample queries *) +%? nat +%? add zero zero zero +%query _ _ 1 add zero zero zero + +(* Reduction relation declaration *) +%reduces <= X Y add X Y zero +``` + +--- + +## 15. Gaps and TODOs + +Things the modern parser does **not** currently handle, deliberately +omitted from the BNF above: + +- **Comments**. There is no comment syntax recognised by `Modern.ml` + or `Cmd.ml`. The legacy frontend recognises `%{ ... }%` block + comments and `%` / `%%` line comments. A future modern lexer is + expected to adopt the same conventions; tree-sitter implementers + should reserve those tokens. +- **String literals**. `Modern.parse_text` (`Modern.ml:297-301`) + defines a `%"..."%` literal form, but no command currently calls + it. The `Scon_` CST node it would build is unreachable from + surface syntax today. +- **Namespace-qualified raw identifiers**. The CST supports a `namespace` + list (e.g. `Mod.Sub.foo`), but the modern parser only produces + qualified names via `%val` (§4). Direct dotted-path identifiers + are not parsed. +- **`%abbrev`, `%trustme`, `%subord`**. Present in the legacy parser + and in the original Twelf, but the modern `Cmd.ml` `choice` block + does not include them. They are reserved for future work. +- **`Foreign` / `Internal` term constructors**. Listed in + `Cst.View.Term.u` (`Cst.ml:516-517`) but `review` raises `Lacking` + on them — they have no surface syntax. + +--- + +## 16. Appendix: tree-sitter design notes + +This appendix gathers observations relevant to implementing STELF in +[tree-sitter](https://tree-sitter.github.io/). It does **not** include +a `grammar.js`; it is meant to surface every place where naively +translating the BNF will go wrong. + +### 16.1 Token model + +The lexer is in `src/Lang/Parsing/Parser.ml:12-43`. Key rules: + +- **`whitespace`** consumes only space, tab, newline. +- **`ident1`** is `take_while1` of every character *not* in + `" \t\n(){}[]%"`. In tree-sitter terms: + + ``` + identifier : /[^ \t\n(){}\[\]%]+/ + ``` + +- **`keyword s`** prepends `%`, then *peeks* the next character and + requires it to be a delimiter — one of `" \t\n(){}[]%"` or EOF. This + is what stops `%term` from matching the prefix of `%terminates`. + In tree-sitter: + + ```js + function kw(s) { + return token(seq('%', s, /(?=[\s(){}\[\]%]|$)/)); + } + ``` + + The lookahead must be a zero-width regex; it cannot be a token. + +### 16.2 Identifier vs keyword + +Only `%`-prefixed words are ever reserved. Bare identifiers in +expression position can be arbitrary non-delimiter strings: `nat`, +`+`, `<=`, `==>`, `add/zero`, `_X` are all the same token class. + +Do **not** split identifiers into "alphanumeric" and "symbolic" at +the lexer — STELF does not make that distinction. + +### 16.3 Uppercase vs lowercase identifier + +The first-character rule (`_` or `A`–`Z` → uppercase) is enforced in +the *consumer*, not the lexer (`Modern.ml:194-200`). Two options: + +- **Single token, tag in the rule.** Keep one `identifier` token and + let the consuming grammar rules (or downstream tooling) inspect the + first char. Recommended — preserves symbol identifiers like `+`. +- **Split tokens.** Define `lower_ident : /[^A-Z_ \t\n(){}\[\]%][^...]*/` + and `upper_ident : /[A-Z_][^...]*/`. Cleaner for highlighting but + forces every operator-only ident (e.g. `+`) onto one side. + +### 16.4 Dynamic `%prec` + +`Modern.ml:34-47, 121-156` shows that `%prec` mutates a `Hashtbl` of +local fixities at parse time, and the parser then performs an +operator-precedence shift-reduce on the application list. This is +intrinsically stateful and **not** expressible in tree-sitter's static +grammar. + +Recommendation: parse application as a flat sequence +(`seq($.expr1, repeat1($.expr1))`) and defer operator resolution to a +post-tree pass (a separate elaboration step that reads `%prec` +declarations from the same file). This matches how `Modern.ml` itself +defers all elaboration to `Recon`. + +### 16.5 Arrow / backarrow chains + +```ocaml +keyword "->" *> ... sep_by1 (option () @@ keyword "->") (parse_expr1 ()) +``` + +(`Modern.ml:241-247`) — the separator is **optional** after the first +occurrence. The chain is greedy and right-folds into a `{_ A} ... B` +Pi-type. + +Tree-sitter shape: + +```js +arrow_chain : seq('%->', $.expr1, repeat(seq(optional('%->'), $.expr1))) +``` + +### 16.6 Trailing binders in application + +`expr-app ::= expr1+ expr-trail?` means a lambda or Pi can syntactically +appear at the tail of an application — the binder is not in +parentheses. `f x y [w] w` is `(f x y) (lambda w. w)`. + +This is the reason `expr-trail` is split from `expr1`: a trailing +binder is too greedy for a small-expression position. + +### 16.7 Statement boundary + +Outer text between commands is *anything not starting with `%`* +(`Cmd.ml:16`: `skip_while (fun c -> c <> '%')`). Practical model in +tree-sitter: + +- Mark `%`-keyword tokens as the only valid command starters. +- An external scanner (or a token rule with extras) absorbs runs of + non-`%` characters between commands. + +### 16.8 Block delimiters + +`keyword "{"` and `keyword "}"` prepend `%`, so command-list blocks +(`%module`, `%eval`) are delimited by `%{` and `%}`, not bare braces. +Bare `{`/`}` are reserved for Pi binders inside terms. + +### 16.9 Comments — open question + +`Modern.ml` does not currently consume comments. Implementers should +preserve forward compatibility by recognising at minimum: + +- `%{ ... %}` block comments (legacy convention, nestable in legacy). +- `%` and `%%` line comments (legacy convention). + +These should be modelled as `extras` so they are absorbed wherever +whitespace is allowed. + +### 16.10 Keyword inventory for tree-sitter + +The full set of `%`-keywords accepted by the modern parser. Split into +command-level keywords (only valid at command position) and +in-expression keywords (only valid inside `expr`/`mode-dec`/`sigexp`). + +**Command-level** (from `Cmd.ml`): + +`%.` `%?` `%block` `%covers` `%decl` `%def` `%define` +`%deterministic` `%eval` `%freeze` `%get` `%help` `%inline` `%mode` +`%module` `%name` `%open` `%prec` `%query` `%querytabled` `%quit` +`%reduces` `%set` `%solve` `%sort` `%symbol` `%term` `%terminates` +`%thaw` `%total` `%union` `%unique` `%use` `%version` `%worlds` +`%{` `%}` + +**In-expression** (from `Modern.ml`): + +`%the` `%val` `%->` `%<-` `%in` `%out` `%out1` `%star` `%where` +`%left` `%right` `%prefix` `%postfix` `%middle` `%none` + +### 16.11 Reduces operators + +`%reduces` accepts `<=`, `>=`, `<`, `>`, `=` (`Modern.ml:457-465`). +These are **identifiers**, not keywords — `Modern.ml` matches them +with `token`, not `keyword`, and the order of alternatives matters +(two-char forms first). A tree-sitter grammar can either: + +- match them as a fixed set of named alternatives inside `reduces-cmd` + (recommended — gives them their own syntax-tree node); +- accept any identifier in that position and validate downstream. diff --git a/dune-project b/dune-project index c892099..b1d2962 100644 --- a/dune-project +++ b/dune-project @@ -32,18 +32,29 @@ (ocaml (and (>= 5.0.0) - (< 5.4))) + (not (and + (< 5.5.0) + (>= 5.4.0) + )) + )) lwt containers base angstrom (alcotest :with-test) (alcotest-lwt :with-test) - (bisect-ppx :with-dev) + (bisect-ppx :dev) (odoc :with-doc) - (bisect_ppx :with-dev) + (sherlodoc :with-doc) + (bisect_ppx :dev) + (ocamlformat :dev) + (ocaml-lsp-server :dev) + (user-setup :dev) + (merlin :dev) + ppx_deriving lambda-term fmt + bos cmdliner logs) (tags diff --git a/ffi/zig/Lens.zig b/ffi/zig/Lens.zig new file mode 100644 index 0000000..f067de8 --- /dev/null +++ b/ffi/zig/Lens.zig @@ -0,0 +1,13 @@ +const path = []const u8; + +const Lens = struct { + fn Loc(comptime _: type) type { + return union(enum) { ghost, loc: .{ path, i32, i32 } }; + } + fn LTerm(comptime T: type) type { + const loc = Loc(T); + const symbol = []const u8; + return union(enum) { lowercase: .{ loc, symbol }, uppercase: .{ loc, symbol } }; // TODO Finsish adding + // + } +}; diff --git a/ffi/zig/Server.zig b/ffi/zig/Server.zig new file mode 100644 index 0000000..b2c9e52 --- /dev/null +++ b/ffi/zig/Server.zig @@ -0,0 +1,4 @@ +const STELF : type = opaque {}; + +fn start(self: STELF) bool { @ } + diff --git a/guide/pages/greek.typ b/guide/pages/greek.typ deleted file mode 100644 index e69de29..0000000 diff --git a/guide/pages/lf.typ b/guide/pages/lf.typ deleted file mode 100644 index e69de29..0000000 diff --git a/guide/pages/logic.typ b/guide/pages/logic.typ deleted file mode 100644 index a005b5c..0000000 --- a/guide/pages/logic.typ +++ /dev/null @@ -1,7 +0,0 @@ -= A Prologue in Prolog - -As people who have used Prolog to formalize logic know, Prolog often has some problems with many tasks that can be confusing, time consuming, and error-prone. -While one of Prolog's greatest strengths add ability to have more practical applications, they make it much harder to use Prolog for formalizing logic. - -These problems can be divided into four categories: -1. \ No newline at end of file diff --git a/guide/pages/syntax.typ b/guide/pages/syntax.typ deleted file mode 100644 index e69de29..0000000 diff --git a/guide/pages/typed.typ b/guide/pages/typed.typ deleted file mode 100644 index e69de29..0000000 diff --git a/guide/src/macros.typ b/guide/src/macros.typ deleted file mode 100644 index e69de29..0000000 diff --git a/guide/src/style.typ b/guide/src/style.typ deleted file mode 100644 index 0976d50..0000000 --- a/guide/src/style.typ +++ /dev/null @@ -1,6 +0,0 @@ -#let conf(docs) = { - - - docs -} - diff --git a/guide/stelf.typ b/guide/stelf.typ deleted file mode 100644 index 61e217c..0000000 --- a/guide/stelf.typ +++ /dev/null @@ -1,5 +0,0 @@ -#import "src/style.typ": conf -#show: conf.with() -#include "pages/lf.typ" -#include "pages/logic.typ" -#include "pages/greek.typ" \ No newline at end of file diff --git a/odoc-config.sexp b/odoc-config.sexp new file mode 100644 index 0000000..6c400a1 --- /dev/null +++ b/odoc-config.sexp @@ -0,0 +1 @@ +(libraries lambda-term base cmdliner angstrom) \ No newline at end of file diff --git a/src/Common/Cst/CST.ml b/src/Common/Cst/CST.ml index b1e4c7a..7cd48b4 100644 --- a/src/Common/Cst/CST.ml +++ b/src/Common/Cst/CST.ml @@ -76,6 +76,9 @@ module type CST = sig type block_item (** One item in a %block world declaration. *) + type order [@@deriving show { with_path = false }, eq] + (** Termination/totality order (Varg, Lex, Simul). *) + type cmd [@@deriving show { with_path = false }, eq] (** Top-level command node. *) @@ -358,7 +361,7 @@ module type CST = sig val thaw : ?fc:loc -> string list -> cmd (** [%thaw id_list] — unfreeze type families. *) - val sort : ?fc:loc -> string -> decl list -> cmd + val sort : ?fc:loc -> string list -> decl list -> cmd (** [%sort id {decl}+] — declare a type family. *) val term : ?fc:loc -> decl -> cmd @@ -406,10 +409,10 @@ module type CST = sig val version : ?fc:loc -> unit -> cmd end - val total : ?fc:loc -> string list list -> term list -> cmd + val total : ?fc:loc -> order list -> term list -> cmd (** [%total hyps modes] — declare a totality check. *) - - val terminates : ?fc:loc -> string list list -> term list -> cmd + + val terminates : ?fc:loc -> order list -> term list -> cmd (** [%terminates hyps modes] — declare a termination check. *) val covers : ?fc:loc -> modeDec -> cmd @@ -417,7 +420,10 @@ module type CST = sig val name : ?fc:loc -> string -> cmd (** [%name id] — declare a name for the next definition. *) - + + val reduces : ?fc:loc -> string -> term list -> cmd + (** [%reduces pred order_out order_in call_pats] — declare a reduction relation. *) + end module Thm : sig @@ -490,7 +496,12 @@ module type CST = sig val pp_term : Stdlib.Format.formatter -> term -> unit (** Pretty-print a term to a formatter. *) - module View : LENS.VIEW + + (** {2 Views} *) + + (** Views should eventually supplant the rest of this module *) + module View : sig + include LENS.VIEW with type Term.t = term and type Decl.t = decl and type ConDec.t = conDec @@ -507,7 +518,9 @@ module type CST = sig and type Define.t = define and type Fixity.t = fixity and type Cmd.t = cmd - + + + end end diff --git a/src/Common/Cst/Cst.ml b/src/Common/Cst/Cst.ml index f76c72a..3dc9b6d 100644 --- a/src/Common/Cst/Cst.ml +++ b/src/Common/Cst/Cst.ml @@ -27,11 +27,11 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct type decl = Dec_ of string option list * term * loc and term = - | Arrow_ of term * term - | Pi_ of decl * term - | Lam_ of decl * term - | App_ of term * term - | Hastype_ of term * term + | Arrow_ of loc * term * term + | Pi_ of loc * decl * term + | Lam_ of loc * decl * term + | App_ of loc * term * term + | Hastype_ of loc * term * term | Omitted_ of loc | Lcid_ of string list * string * loc | Ucid_ of string list * string * loc @@ -103,6 +103,12 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct type block_item = BlockSome_ of decl | BlockPi_ of decl [@@deriving show { with_path = false }, eq] + type order = + | Varg_ of loc * string list + | Lex_ of loc * order list + | Simul_ of loc * order list + [@@deriving show { with_path = false }, eq] + (* Top-level commands *) type cmd = | QueryCmd_ of int option * int option * int option * query @@ -116,7 +122,7 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | SymbolCmd_ of string * string | FreezeCmd_ of string list | ThawCmd_ of string list - | SortCmd_ of string * decl list + | SortCmd_ of string list * decl list | TermCmd_ of decl | BlockCmd_ of string * block_item list | UnionCmd_ of string * string list @@ -134,10 +140,11 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | GetCmd_ of string | SetCmd_ of string * string | VersionCmd_ - | TotalCmd_ of string list list * term list - | TerminatesCmd_ of string list list * term list + | TotalCmd_ of order list * term list + | TerminatesCmd_ of order list * term list | CoversCmd_ of modeDec | NameCmd_ of string + | ReducesCmd_ of string * term list [@@deriving show { with_path = false }, eq] (* Term constructor module *) @@ -158,21 +165,29 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct let free_var ?fc:(loc_ = ghost) name = Fvar_ (name, loc_) module Sugar = struct - let arrow ?fc:(loc_ = ghost) tm1 tm2 = Arrow_ (tm1, tm2) - let backarrow ?fc:(loc_ = ghost) tm1 tm2 = Arrow_ (tm2, tm1) + let arrow ?fc:(loc_ = ghost) tm1 tm2 = Arrow_ (loc_, tm1, tm2) + let backarrow ?fc:(loc_ = ghost) tm1 tm2 = Arrow_ (loc_, tm2, tm1) end let pi ?fc:(loc_ = ghost) decls body = let rec fold_right f lst acc = match lst with [] -> acc | x :: xs -> f x (fold_right f xs acc) in - fold_right (fun d acc -> Pi_ (d, acc)) decls body + match decls with + | [] -> body + | first :: rest -> + let inner = fold_right (fun d acc -> Pi_ (ghost, d, acc)) rest body in + Pi_ (loc_, first, inner) let lam ?fc:(loc_ = ghost) decls body = let rec fold_right f lst acc = match lst with [] -> acc | x :: xs -> f x (fold_right f xs acc) in - fold_right (fun d acc -> Lam_ (d, acc)) decls body + match decls with + | [] -> body + | first :: rest -> + let inner = fold_right (fun d acc -> Lam_ (ghost, d, acc)) rest body in + Lam_ (loc_, first, inner) let app ?fc:(loc_ = ghost) head args = match args with @@ -181,9 +196,13 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct let rec fold_left f acc lst = match lst with [] -> acc | x :: xs -> fold_left f (f acc x) xs in - fold_left (fun acc arg -> App_ (acc, arg)) head args + let rev = List.rev args in + let last = List.hd rev in + let init = List.rev (List.tl rev) in + let inner = fold_left (fun acc arg -> App_ (ghost, acc, arg)) head init in + App_ (loc_, inner, last) - let has_type ?fc:(loc_ = ghost) tm ty = Hastype_ (tm, ty) + let has_type ?fc:(loc_ = ghost) tm ty = Hastype_ (loc_, tm, ty) let[@warning "-16"] omitted ?fc:(loc_ = ghost) = Omitted_ loc_ let typ ?fc:(loc_ = ghost) () = Typ_ loc_ end @@ -299,7 +318,7 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct let symbol ?fc:(_ = ghost) id1 id2 = SymbolCmd_ (id1, id2) let freeze ?fc:(_ = ghost) ids = FreezeCmd_ ids let thaw ?fc:(_ = ghost) ids = ThawCmd_ ids - let sort ?fc:(_ = ghost) id decls = SortCmd_ (id, decls) + let sort ?fc:(_ = ghost) ids decls = SortCmd_ (ids, decls) let term ?fc:(_ = ghost) d = TermCmd_ d let block ?fc:(_ = ghost) id items = BlockCmd_ (id, items) let union ?fc:(_ = ghost) id ids = UnionCmd_ (id, ids) @@ -325,6 +344,7 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct let terminates ?fc:(_ = ghost) intros body = TerminatesCmd_ (intros, body) let covers ?fc:(_ = ghost) md = CoversCmd_ md let name ?fc:(_ = ghost) id = NameCmd_ id + let reduces ?fc:(_ = ghost) pred body = ReducesCmd_ (pred, body) end module Fixity = struct @@ -342,7 +362,7 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct end module Thm = struct - type order = + type nonrec order = order = | Varg_ of loc * string list | Lex_ of loc * order list | Simul_ of loc * order list @@ -412,7 +432,7 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct let wdecl wdecl_ = wdecl_ end - + open Lens module View : LENS.VIEW with type Term.t = term @@ -431,16 +451,16 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct and type Define.t = define and type Fixity.t = fixity and type Cmd.t = cmd = struct - module Paths = Paths - + module Paths = Paths + exception Lacking (** Module of paths and regions, which we allow to be shared *) type nonrec loc = loc - (** Source Loc.tation carried by CST nodes. *) + (** Source Loc.tation carried by CST nodes. *) type name = string - (** Unqualified identifier. *) + (** Unqualified identifier. *) type namespace = string list (** Qualified namespace path. *) @@ -449,10 +469,10 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct (** Qualified symbol as [(namespace, name)]. *) (** Create a Loc.tation from start and end lexer positions. *) - let mk_loc : int -> int -> loc = fun x y -> assert false + let mk_loc = mk_loc (** Convert a source Loc.tation to a Paths region. *) - let loc_to_region : loc -> Paths.region = fun loc -> assert false + let loc_to_region = loc_to_region (** Synthetic Loc.tation used for generated nodes. *) let ghost : loc = ghost @@ -469,32 +489,37 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct match y with | Loc (path_opt, start_pos, end_pos) -> { start_pos; end_pos } | Ghost -> ghost - end + let (!>) = view + let (!<) = review + end + let ghost' : Loc.t = ghost (** {3 Term Syntax} *) module rec Term : sig type t = term type u = - | Lowercase of loc * symbol - | Uppercase of loc * symbol - | Qualified of loc * symbol - | Text of loc * name - | ExistVar of loc * name - | FreeVar of loc * name - | Pi of loc * Decl.t list * t - | Lam of loc * Decl.t list * t - | App of loc * t * t list - | HasType of loc * t * t - | Omitted of loc - | Typ of loc - | Arrow of loc * t * t - | BackArrow of loc * t * t - | Foreign of loc * t + | Lowercase of Loc.t * symbol + | Uppercase of Loc.t * symbol + | Qualified of Loc.t * symbol + | Text of Loc.t * name + | ExistVar of Loc.t * name + | FreeVar of Loc.t * name + | Pi of Loc.t * Decl.t list * t + | Lam of Loc.t * Decl.t list * t + | App of Loc.t * t * t list + | HasType of Loc.t * t * t + | Omitted of Loc.t + | Typ of Loc.t + | Arrow of Loc.t * t * t + | BackArrow of Loc.t * t * t + | Foreign of Loc.t * t | Internal of int val view : t -> u val review : u -> t + val (!>) : t -> u + val (!<) : u -> t end = struct type t = term @@ -518,44 +543,43 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct let view (x : t) : u = let rec collect_pis = function - | Pi_ (d, body) -> + | Pi_ (_, d, body) -> let ds, b = collect_pis body in (d :: ds, b) | body -> ([], body) in let rec collect_lams = function - | Lam_ (d, body) -> + | Lam_ (_, d, body) -> let ds, b = collect_lams body in (d :: ds, b) | body -> ([], body) in let rec collect_apps acc = function - | App_ (f, arg) -> collect_apps (arg :: acc) f + | App_ (_, f, arg) -> collect_apps (arg :: acc) f | head -> (head, acc) in match x with - | Lcid_ (ns, name, _) -> Lowercase (ghost, (ns, name)) - | Ucid_ (ns, name, _) -> Uppercase (ghost, (ns, name)) - | Quid_ (ns, name, _) -> Qualified (ghost, (ns, name)) - | Scon_ (str, _) -> Text (ghost, str) - | Evar_ (name, _) -> ExistVar (ghost, name) - | Fvar_ (name, _) -> FreeVar (ghost, name) - | Pi_ _ -> + | Lcid_ (ns, name, loc) -> Lowercase (loc, (ns, name)) + | Ucid_ (ns, name, loc) -> Uppercase (loc, (ns, name)) + | Quid_ (ns, name, loc) -> Qualified (loc, (ns, name)) + | Scon_ (str, loc) -> Text (loc, str) + | Evar_ (name, loc) -> ExistVar (loc, name) + | Fvar_ (name, loc) -> FreeVar (loc, name) + | Pi_ (loc, _, _) -> let decls, body = collect_pis x in - Pi (ghost, decls, body) - | Lam_ _ -> + Pi (loc, decls, body) + | Lam_ (loc, _, _) -> let decls, body = collect_lams x in - Lam (ghost, decls, body) - | App_ _ -> + Lam (loc, decls, body) + | App_ (loc, _, _) -> let head, args = collect_apps [] x in - App (ghost, head, args) - | Hastype_ (tm, ty) -> HasType (ghost, tm, ty) - | Omitted_ _ -> Omitted ghost - | Typ_ _ -> Typ ghost - | Arrow_ (a, b) -> Arrow (ghost, a, b) + App (loc, head, args) + | Hastype_ (loc, tm, ty) -> HasType (loc, tm, ty) + | Omitted_ loc -> Omitted loc + | Typ_ loc -> Typ loc + | Arrow_ (loc, a, b) -> Arrow (loc, a, b) let review (y : u) : t = - let g = { start_pos = 0; end_pos = 0 } in let rec fold_right f lst acc = match lst with [] -> acc | x :: xs -> f x (fold_right f xs acc) in @@ -564,23 +588,40 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | x :: xs -> fold_left f (f acc x) xs in match y with - | Lowercase (_, (ns, name)) -> Lcid_ (ns, name, g) - | Uppercase (_, (ns, name)) -> Ucid_ (ns, name, g) - | Qualified (_, (ns, name)) -> Quid_ (ns, name, g) - | Text (_, str) -> Scon_ (str, g) - | ExistVar (_, name) -> Evar_ (name, g) - | FreeVar (_, name) -> Fvar_ (name, g) - | Pi (_, decls, body) -> - fold_right (fun d acc -> Pi_ (d, acc)) decls body - | Lam (_, decls, body) -> - fold_right (fun d acc -> Lam_ (d, acc)) decls body - | App (_, head, args) -> - fold_left (fun acc arg -> App_ (acc, arg)) head args - | HasType (_, tm, ty) -> Hastype_ (tm, ty) - | Omitted _ -> Omitted_ g - | Typ _ -> Typ_ g - | Arrow (_, a, b) -> Arrow_ (a, b) - | BackArrow (_, a, b) -> Arrow_ (b, a) + | Lowercase (loc, (ns, name)) -> Lcid_ (ns, name, loc) + | Uppercase (loc, (ns, name)) -> Ucid_ (ns, name, loc) + | Qualified (loc, (ns, name)) -> Quid_ (ns, name, loc) + | Text (loc, str) -> Scon_ (str, loc) + | ExistVar (loc, name) -> Evar_ (name, loc) + | FreeVar (loc, name) -> Fvar_ (name, loc) + | Pi (loc, decls, body) -> + (match decls with + | [] -> body + | first :: rest -> + let inner = fold_right (fun d acc -> Pi_ (ghost, d, acc)) rest body in + Pi_ (loc, first, inner)) + | Lam (loc, decls, body) -> + (match decls with + | [] -> body + | first :: rest -> + let inner = fold_right (fun d acc -> Lam_ (ghost, d, acc)) rest body in + Lam_ (loc, first, inner)) + | App (loc, head, args) -> + (match List.rev args with + | [] -> head + | last :: rev_rest -> + let inner = fold_left (fun acc arg -> App_ (ghost, acc, arg)) head (List.rev rev_rest) in + App_ (loc, inner, last)) + | HasType (loc, tm, ty) -> Hastype_ (loc, tm, ty) + | Omitted loc -> Omitted_ loc + | Typ loc -> Typ_ loc + | Arrow (loc, a, b) -> Arrow_ (loc, a, b) + | BackArrow (loc, a, b) -> Arrow_ (loc, b, a) + | Foreign (_, _) -> raise Lacking + | Internal _ -> raise Lacking + + let (!>) = view + let (!<) = review end (** Binder declaration constructors. *) @@ -593,6 +634,8 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct val view : t -> u val review : u -> t + val (!>) : t -> u + val (!<) : u -> t end = struct type t = decl @@ -601,16 +644,17 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | Decl0 of Loc.t * string option list * Term.t let view (x : t) : u = - let g = { start_pos = 0; end_pos = 0 } in match x with - | Dec_ (names, Omitted_ _, _) -> Decl0 (ghost, names, Omitted_ g) - | Dec_ (names, typ, _) -> Decl1 (ghost, names, typ, Omitted_ g) + | Dec_ (names, Omitted_ loc, dloc) -> Decl0 (dloc, names, Omitted_ loc) + | Dec_ (names, typ, dloc) -> Decl1 (dloc, names, typ, Omitted_ ghost) let review (y : u) : t = - let g = { start_pos = 0; end_pos = 0 } in match y with - | Decl1 (_, names, typ, _) -> Dec_ (names, typ, g) - | Decl0 (_, names, typ) -> Dec_ (names, typ, g) + | Decl1 (loc, names, typ, _) -> Dec_ (names, typ, loc) + | Decl0 (loc, names, typ) -> Dec_ (names, typ, loc) + + let (!>) = view + let (!<) = review end (** Top-level declaration constructors. *) @@ -636,6 +680,9 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | BlockDecl (_, n, ds1, ds2) -> BlockDecl_ (n, ds1, ds2) | BlockDef (_, n, syms) -> BlockDef_ (n, syms) | ConstantDef (_, n, tm, opt) -> ConstantDef_ (n, tm, opt) + + let (!>) = view + let (!<) = review end (** Mode syntax constructors. *) @@ -662,6 +709,9 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | Minus _ -> Minus_ | Minus1 _ -> Minus1_ + let (!>) = view + let (!<) = review + type t_mode = t module Spine = struct @@ -682,6 +732,9 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | ModeNil _ -> ModeSpineInternal_ [] | ModeApp (_, entry, ModeSpineInternal_ xs) -> ModeSpineInternal_ (entry :: xs) + + let (!>) = view + let (!<) = review end module Term = struct @@ -697,7 +750,7 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct let rec extract_head = function | Quid_ (ns, n, _) -> (ns, n) | Lcid_ (ns, n, _) -> ([], n) - | App_ (f, _) -> extract_head f + | App_ (_, f, _) -> extract_head f | _ -> raise Lacking in ModeTerm (ghost, extract_head tm, ModeSpineInternal_ []) @@ -708,6 +761,9 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct match y with | ModeTerm (_, (ns, n), _) -> ModeTermRoot_ (Quid_ (ns, n, g)) | ModePi (_, d, body, _) -> ModeTermPi_ (Plus_, d, body) + + let (!>) = view + let (!<) = review end module Dec = struct @@ -737,6 +793,9 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct ModeTermPi_ (m, Dec_ ([ n ], Omitted_ g, g), build rest) in ModeDec_ (build spine) + + let (!>) = view + let (!<) = review end end @@ -748,6 +807,8 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct let view (x : t) : u = match x with StrExp_ sym -> StrExp (ghost, sym) let review (y : u) : t = match y with StrExp (_, sym) -> StrExp_ sym + let (!>) = view + let (!<) = review end module Inst = struct @@ -767,6 +828,9 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct match y with | ConInst (_, sym, _, tm) -> ConInst_ (sym, g, tm) | StrInst (_, sym, _, se) -> StrInst_ (sym, g, se) + + let (!>) = view + let (!<) = review end module SigExp = struct @@ -788,6 +852,9 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | Thesig _ -> TheSig_ | SigId (_, str) -> SigId_ str | WhereSig (_, se, insts) -> WhereSig_ (se, insts) + + let (!>) = view + let (!<) = review end module SigDef = struct @@ -799,6 +866,9 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct let review (y : u) : t = match y with SigDef (_, n, se) -> SigDef_ (n, se) + + let (!>) = view + let (!<) = review end module StructDec = struct @@ -817,6 +887,9 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct match y with | StructDecl (_, n, se) -> StructDecl_ (n, se) | StructDef (_, n, se) -> StructDef_ (n, se) + + let (!>) = view + let (!<) = review end end @@ -828,6 +901,8 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct match x with Query_ (n, tm) -> Query (ghost, n, tm) let review (y : u) : t = match y with Query (_, n, tm) -> Query_ (n, tm) + let (!>) = view + let (!<) = review end module Define = struct @@ -840,6 +915,9 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct let review (y : u) : t = match y with Define (_, n, tm1, tm2_opt) -> Define_ (n, tm1, tm2_opt) + + let (!>) = view + let (!<) = review end module Solve = struct @@ -850,6 +928,8 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct match x with Solve_ (n, tm) -> Solve (ghost, n, tm) let review (y : u) : t = match y with Solve (_, n, tm) -> Solve_ (n, tm) + let (!>) = view + let (!<) = review end module Fixity = struct @@ -880,6 +960,9 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | Postfix _ -> Postfix_ | Middle _ -> Middle_ | None _ -> FNone_ + + let (!>) = view + let (!<) = review end module BlockItem = struct @@ -893,6 +976,9 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct let review (y : u) : t = match y with Any (_, d) -> BlockSome_ d | All (_, d) -> BlockPi_ d + + let (!>) = view + let (!<) = review end module Cmd = struct @@ -928,10 +1014,11 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | ReplGet of Loc.t * string | ReplSet of Loc.t * string * string | ReplVersion of Loc.t * unit - | Total of Loc.t * string list list * Term.t list - | Terminates of Loc.t * string list list * Term.t list + | Total of Loc.t * Thm.order list * Term.t list + | Terminates of Loc.t * Thm.order list * Term.t list | Covers of Loc.t * Mode.Dec.t | Name of Loc.t * string + | Reduces of Loc.t * string * Term.t list let view (x : t) : u = match x with @@ -946,7 +1033,7 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | SymbolCmd_ (id1, id2) -> Symbol (ghost, id1, id2) | FreezeCmd_ ids -> Freeze (ghost, ids) | ThawCmd_ ids -> Thaw (ghost, ids) - | SortCmd_ (id, decls) -> Sort (ghost, [ id ], decls) + | SortCmd_ (ids, decls) -> Sort (ghost, ids, decls) | TermCmd_ d -> Term (ghost, d) | BlockCmd_ (id, items) -> Block (ghost, id, items) | UnionCmd_ (id, ids) -> Union (ghost, id, ids) @@ -964,6 +1051,11 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | GetCmd_ s -> ReplGet (ghost, s) | SetCmd_ (s, v) -> ReplSet (ghost, s, v) | VersionCmd_ -> ReplVersion (ghost, ()) + | TotalCmd_ (orders, terms) -> Total (ghost, orders, terms) + | TerminatesCmd_ (orders, terms) -> Terminates (ghost, orders, terms) + | CoversCmd_ md -> Covers (ghost, md) + | NameCmd_ id -> Name (ghost, id) + | ReducesCmd_ (pred, body) -> Reduces (ghost, pred, body) let review (y : u) : t = match y with @@ -978,9 +1070,7 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | Symbol (loc, id1, id2) -> SymbolCmd_ (id1, id2) | Freeze (loc, ids) -> FreezeCmd_ ids | Thaw (loc, ids) -> ThawCmd_ ids - | Sort (loc, [ id ], decls) -> SortCmd_ (id, decls) - | Sort (loc, ids, decls) -> - assert false (* TODO Handle mutiple sort declerations *) + | Sort (_, ids, decls) -> SortCmd_ (ids, decls) | Term (loc, d) -> TermCmd_ d | Block (loc, id, items) -> BlockCmd_ (id, items) | Union (loc, id, ids) -> UnionCmd_ (id, ids) @@ -998,6 +1088,14 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | ReplGet (loc, s) -> GetCmd_ s | ReplSet (loc, s, v) -> SetCmd_ (s, v) | ReplVersion (loc, ()) -> VersionCmd_ + | Total (_, orders, terms) -> TotalCmd_ (orders, terms) + | Terminates (_, orders, terms) -> TerminatesCmd_ (orders, terms) + | Covers (_, md) -> CoversCmd_ md + | Name (_, id) -> NameCmd_ id + | Reduces (_, pred, body) -> ReducesCmd_ (pred, body) + + let (!>) = view + let (!<) = review end module Thm = struct @@ -1028,6 +1126,9 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | Varg (l, names) -> Thm.Varg_ (l, names) | Lex (l, orders) -> Thm.Lex_ (l, orders) | Simul (l, orders) -> Thm.Simul_ (l, orders) + + let (!>) = view + let (!<) = review end module CallPats = struct @@ -1036,6 +1137,8 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct let view (x : t) : u = CallPats x let review (y : u) : t = match y with CallPats cp -> cp + let (!>) = view + let (!<) = review end module TDecl = struct @@ -1047,6 +1150,8 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct TDecl (o, cp) let review (y : u) : t = match y with TDecl (o, cp) -> (o, cp) + let (!>) = view + let (!<) = review end module Predicate = struct @@ -1058,6 +1163,8 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct Predicate (s, l) let review (y : u) : t = match y with Predicate (s, l) -> (s, l) + let (!>) = view + let (!<) = review end module RDecl = struct @@ -1070,6 +1177,9 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct let review (y : u) : t = match y with RDecl (p, o1, o2, cp) -> (p, o1, o2, cp) + + let (!>) = view + let (!<) = review end module TabledDecl = struct @@ -1081,6 +1191,8 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct TabledDecl (s, l) let review (y : u) : t = match y with TabledDecl (s, l) -> (s, l) + let (!>) = view + let (!<) = review end module KeepTableDecl = struct @@ -1092,6 +1204,8 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct KeepTableDecl (s, l) let review (y : u) : t = match y with KeepTableDecl (s, l) -> (s, l) + let (!>) = view + let (!<) = review end module Prove = struct @@ -1103,6 +1217,8 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct Prove (n, td) let review (y : u) : t = match y with Prove (n, td) -> (n, td) + let (!>) = view + let (!<) = review end module Establish = struct @@ -1114,6 +1230,8 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct Establish (n, td) let review (y : u) : t = match y with Establish (n, td) -> (n, td) + let (!>) = view + let (!<) = review end module Assert = struct @@ -1122,6 +1240,8 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct let view (x : t) : u = Assert x let review (y : u) : t = match y with Assert cp -> cp + let (!>) = view + let (!<) = review end module Decs = struct @@ -1137,6 +1257,9 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct match y with | DecsNil _ -> [] | DecsList (rest, decls) -> rest @ decls + + let (!>) = view + let (!<) = review end module Thm = struct @@ -1164,10 +1287,15 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | Forall (_, ds, body) -> Thm.Forall_ (ds, body) | ForallStar (_, ds, body) -> Thm.ForallStar_ (ds, body) | ForallG (_, pairs, body) -> Thm.ForallG_ (pairs, body) + + let (!>) = view + let (!<) = review end let view (x : t) : u = assert false let review (y : u) : t = assert false + let (!>) = view + let (!<) = review module ThmDec = struct type t = concrete_theoremdec @@ -1178,6 +1306,8 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct ThmDec (name, thm) let review (y : u) : t = match y with ThmDec (name, thm) -> (name, thm) + let (!>) = view + let (!<) = review end module WDecl = struct @@ -1189,8 +1319,11 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct WDecl (pairs, cp) let review (y : u) : t = match y with WDecl (pairs, cp) -> (pairs, cp) + let (!>) = view + let (!<) = review end end + end end diff --git a/src/Common/Cst/LENS.ml b/src/Common/Cst/LENS.ml index feff95f..66f5c44 100644 --- a/src/Common/Cst/LENS.ml +++ b/src/Common/Cst/LENS.ml @@ -1,24 +1,58 @@ +open Base (** Lens based interface to CST -Rather than having just one CST, STELF has a view based interface for the CST, allowing for new concrete syntax to be used without replacing any code + Rather than having just one CST, STELF has a view based interface for the + CST, allowing for new concrete syntax to be used without replacing any code *) -open Base -module type S = sig - type t - type u +module type S = sig + type t + type u + val view : t -> u + (** Unwrap "one layer" of the interface (ie, destruct) *) + val review : u -> t + (** Wrap "one layer" of the interface (ie, construct) *) end -module type LENS = sig - type t - type u - include S with type t := t and type u := u - + +(** A lens is a pair of types, [t] and [u], with operations to convert between + them In general, the following things should be true + - [view] should be a left inverse to [review], i.e. [view (review x) = x] + for all [x : u] (but they need not be right inverses) + - [t] {e should not} be exposed + - [u] {e should} be exposed + - [t] should not mention [u] + - [u] {e should} reference [t] + - [u] should not reference itself + + These operations combined allow us to acheive lenses (actually, [u] should + be complete and thus actually like a prism) + + In general, this should be used as [LENS with type t := t and type u := u] +*) +module type LENS = sig + type t + (** The concrete type we abstract over, should not be exposed *) + + type u + (** The abstract type we almost always {e must} expose*) + + (** @inline *) + include S with type t := t and type u := u + + val ( !> ) : t -> u + (** Infix version of [view] *) + + val ( !< ) : u -> t + (** Infix version of [review] *) end + (** {2 LENS} *) + module type VIEW = sig exception Lacking + module Paths : Paths.Paths_intf.PATHS (** Module of paths and regions, which we allow to be shared *) @@ -45,301 +79,352 @@ module type VIEW = sig val ghost : loc (** Synthetic Loc.tation used for generated nodes. *) - module Loc : sig - type t + module Loc : sig + type t + type u = - | Loc of Fpath.t option * int * int - | Ghost - include LENS with type t := t and type u := u - end + | Loc of Fpath.t option * int * int + (** A actual location, with path, start, end*) + | Ghost (** Synthetic location used for generated nodes *) + + include LENS with type t := t and type u := u + end (** {3 Term Syntax} *) module rec Term : sig - type t - type u = - | Lowercase of Loc.t * symbol - | Uppercase of Loc.t * symbol - | Qualified of Loc.t * symbol - | Text of Loc.t * string - | ExistVar of Loc.t * string - | FreeVar of Loc.t * string - | Pi of Loc.t * Decl.t list * t - | Lam of Loc.t * Decl.t list * t - | App of Loc.t * t * t list - | HasType of Loc.t * t * t - | Omitted of Loc.t - | Typ of Loc.t - | Arrow of Loc.t * t * t - | BackArrow of Loc.t * t * t - | Foreign of Loc.t * t - | Internal of int + type t + + type u = + | Lowercase of Loc.t * symbol (** An lowercase identifier in scope *) + | Uppercase of Loc.t * symbol (** An uppercase identifier in scope *) + | Qualified of Loc.t * symbol + (** A explicitly qualified identifier (eg, [%val (x y)])*) + | Text of Loc.t * string (** A literal string (currently unused) *) + | ExistVar of Loc.t * string + (** An explicitly existential variable, you're probably looking for + {!Uppercase} *) + | FreeVar of Loc.t * string + (** A free variable, you're probably looking for {!Lowercase} *) + | Pi of Loc.t * Decl.t list * t + (** A {m \Pi} type, which takes in a number of possible contexts*) + | Lam of Loc.t * Decl.t list * t + | App of Loc.t * t * t list + | HasType of Loc.t * t * t + | Omitted of Loc.t + | Typ of Loc.t + | Arrow of Loc.t * t * t + | BackArrow of Loc.t * t * t + | Foreign of Loc.t * t + | Internal of int + include LENS with type t := t and type u := u end (** Binder declaration constructors. *) and Decl : sig type t + type u = - | Decl1 of Loc.t * string option list * Term.t * Term.t - | Decl0 of Loc.t * string option list * Term.t - include LENS with type t := t and type u := u + | Decl1 of Loc.t * string option list * Term.t * Term.t + | Decl0 of Loc.t * string option list * Term.t + + include LENS with type t := t and type u := u end (** Top-level declaration constructors. *) module ConDec : sig - type t + type t + type u = - | ConstantDecl of Loc.t * Decl.t - | BlockDecl of Loc.t * string * Decl.t list * Decl.t list - | BlockDef of Loc.t * string * symbol list - | ConstantDef of Loc.t * string * Term.t * Term.t option - include LENS with type t := t and type u := u - end + | ConstantDecl of Loc.t * Decl.t + | BlockDecl of Loc.t * string * Decl.t list * Decl.t list + | BlockDef of Loc.t * string * symbol list + | ConstantDef of Loc.t * string * Term.t * Term.t option + include LENS with type t := t and type u := u + end (** Mode syntax constructors. *) module Mode : sig type t - type u = - | Plus of Loc.t - | Star of Loc.t - | Minus of Loc.t - | Minus1 of Loc.t + type u = Plus of Loc.t | Star of Loc.t | Minus of Loc.t | Minus1 of Loc.t + include LENS with type t := t and type u := u type t_mode := t + module Spine : sig - type t + type t + type u = - | ModeNil of Loc.t - | ModeApp of Loc.t * (t_mode * string option) * t - include LENS with type t := t and type u := u + | ModeNil of Loc.t + | ModeApp of Loc.t * (t_mode * string option) * t + + include LENS with type t := t and type u := u end + module Term : sig - type t + type t + type u = - | ModeTerm of Loc.t * symbol * Spine.t - | ModePi of Loc.t * Decl.t * t * t + | ModeTerm of Loc.t * symbol * Spine.t + | ModePi of Loc.t * Decl.t * t * t + include LENS with type t := t and type u := u end - module Dec : sig - type t - type u = - | ModeDec of Loc.t * (t_mode * string option) list * Term.t + + module Dec : sig + type t + type u = ModeDec of Loc.t * (t_mode * string option) list * Term.t + include LENS with type t := t and type u := u end end (** Module/signature syntax constructors. *) module Struct : sig - module StrExp : sig - type t - type u = - | StrExp of Loc.t * symbol - include LENS with type t := t and type u := u + module StrExp : sig + type t + type u = StrExp of Loc.t * symbol + + include LENS with type t := t and type u := u end - module Inst : sig - type t + + module Inst : sig + type t + type u = - | ConInst of Loc.t * symbol * Loc.t * Term.t - | StrInst of Loc.t * symbol * Loc.t * StrExp.t - include LENS with type t := t and type u := u + | ConInst of Loc.t * symbol * Loc.t * Term.t + | StrInst of Loc.t * symbol * Loc.t * StrExp.t + + include LENS with type t := t and type u := u end - module SigExp : sig - type t + + module SigExp : sig + type t + type u = - | Thesig of Loc.t - | SigId of Loc.t * string - | WhereSig of Loc.t * t * Inst.t list - include LENS with type t := t and type u := u + | Thesig of Loc.t + | SigId of Loc.t * string + | WhereSig of Loc.t * t * Inst.t list + + include LENS with type t := t and type u := u end - module SigDef : sig - type t - type u = - | SigDef of Loc.t * string option * SigExp.t - include LENS with type t := t and type u := u + + module SigDef : sig + type t + type u = SigDef of Loc.t * string option * SigExp.t + + include LENS with type t := t and type u := u end - module StructDec : sig - type t + + module StructDec : sig + type t + type u = - | StructDecl of Loc.t * string option * SigExp.t - | StructDef of Loc.t * string option * StrExp.t - include LENS with type t := t and type u := u + | StructDecl of Loc.t * string option * SigExp.t + | StructDef of Loc.t * string option * StrExp.t + + include LENS with type t := t and type u := u end end module Query : sig - type t - type u = - | Query of Loc.t * string option * Term.t - include LENS with type t := t and type u := u - end + type t + type u = Query of Loc.t * string option * Term.t + + include LENS with type t := t and type u := u + end + module Define : sig type t - type u = - | Define of Loc.t * string option * Term.t * Term.t option - include LENS with type t := t and type u := u + type u = Define of Loc.t * string option * Term.t * Term.t option + + include LENS with type t := t and type u := u end + module Solve : sig - type t - type u = - | Solve of Loc.t * string option * Term.t - include LENS with type t := t and type u := u - end - module Fixity : sig - type t - type u = - | Left of Loc.t - | Right of Loc.t - | Prefix of Loc.t - | Postfix of Loc.t - | Middle of Loc.t - | None of Loc.t - include LENS with type t := t and type u := u + type t + type u = Solve of Loc.t * string option * Term.t + + include LENS with type t := t and type u := u end - module BlockItem : sig - type t + + module Fixity : sig + type t + type u = - | Any of Loc.t * Decl.t - | All of Loc.t * Decl.t - include LENS with type t := t and type u := u + | Left of Loc.t + | Right of Loc.t + | Prefix of Loc.t + | Postfix of Loc.t + | Middle of Loc.t + | None of Loc.t + + include LENS with type t := t and type u := u end - module Cmd : sig - type t - type u = - | Query of Loc.t * int option * int option * int option * Query.t - | QueryTabled of Loc.t * int option * int option * int option * Query.t - | AdhocQuery of Loc.t * Query.t - | Unique of Loc.t * Term.t - | Mode of Loc.t * Mode.Dec.t - | Define of Loc.t * Define.t - | DeclCmd of Loc.t * Term.t - | Inline of Loc.t * string * Term.t - | Symbol of Loc.t * string * string - | Freeze of Loc.t * string list - | Thaw of Loc.t * string list - | Sort of Loc.t * string list * Decl.t list - | Term of Loc.t * Decl.t - | Block of Loc.t * string * BlockItem.t list - | Union of Loc.t * string * string list - | Worlds of Loc.t * string list * Term.t - | Deterministic of Loc.t * string list - | ModuleCmd of Loc.t * string * string list * t list - | Use of Loc.t * string * string * string list - | OpenCmd of Loc.t * string * string list - | Eval of Loc.t * t list - | Prec of Loc.t * Fixity.t * int * string list - | Solve of Loc.t * Solve.t - | Stop of Loc.t * unit - | ReplQuit of Loc.t * unit - | ReplHelp of Loc.t * string option - | ReplGet of Loc.t * string - | ReplSet of Loc.t * string * string - | ReplVersion of Loc.t * unit - | Total of Loc.t * string list list * Term.t list - | Terminates of Loc.t * string list list * Term.t list - | Covers of Loc.t * Mode.Dec.t - | Name of Loc.t * string - include LENS with type t := t and type u := u + module BlockItem : sig + type t + type u = Any of Loc.t * Decl.t | All of Loc.t * Decl.t + + include LENS with type t := t and type u := u end - module Thm : sig - type t + + module Thm : sig + type t type u = | - module Order : sig - type t - type u = - | Varg of Loc.t * string list - | Lex of Loc.t * t list - | Simul of Loc.t * t list - include LENS with type t := t and type u := u - end - module CallPats : sig - type t + + module Order : sig + type t + type u = - | CallPats of (string * string option list * Loc.t) list - include LENS with type t := t and type u := u + | Varg of Loc.t * string list + | Lex of Loc.t * t list + | Simul of Loc.t * t list + + include LENS with type t := t and type u := u end - module TDecl : sig - type t - type u = - | TDecl of Order.t * CallPats.t - include LENS with type t := t and type u := u + + module CallPats : sig + type t + type u = CallPats of (string * string option list * Loc.t) list + + include LENS with type t := t and type u := u end - module Predicate : sig - type t - type u = - | Predicate of string * Loc.t - include LENS with type t := t and type u := u + + module TDecl : sig + type t + type u = TDecl of Order.t * CallPats.t + + include LENS with type t := t and type u := u end - module RDecl : sig - type t - type u = - | RDecl of Predicate.t * Order.t * Order.t * CallPats.t - include LENS with type t := t and type u := u + + module Predicate : sig + type t + type u = Predicate of string * Loc.t + + include LENS with type t := t and type u := u end - module TabledDecl : sig - type t - type u = - | TabledDecl of string * Loc.t - include LENS with type t := t and type u := u + + module RDecl : sig + type t + type u = RDecl of Predicate.t * Order.t * Order.t * CallPats.t + + include LENS with type t := t and type u := u end + + module TabledDecl : sig + type t + type u = TabledDecl of string * Loc.t + + include LENS with type t := t and type u := u + end + module KeepTableDecl : sig type t - type u = - | KeepTableDecl of string * Loc.t + type u = KeepTableDecl of string * Loc.t + include LENS with type t := t and type u := u end - module Prove : sig - type t - type u = - | Prove of int * TDecl.t - include LENS with type t := t and type u := u + + module Prove : sig + type t + type u = Prove of int * TDecl.t + + include LENS with type t := t and type u := u end - module Establish : sig - type t - type u = - | Establish of int * TDecl.t - include LENS with type t := t and type u := u + + module Establish : sig + type t + type u = Establish of int * TDecl.t + + include LENS with type t := t and type u := u end - module Assert : sig - type t - type u = - | Assert of CallPats.t - include LENS with type t := t and type u := u + + module Assert : sig + type t + type u = Assert of CallPats.t + + include LENS with type t := t and type u := u end - module Decs : sig - type t - type u = - | DecsNil of Loc.t - | DecsList of t * Decl.t list - include LENS with type t := t and type u := u + + module Decs : sig + type t + type u = DecsNil of Loc.t | DecsList of t * Decl.t list + + include LENS with type t := t and type u := u end - module Thm : sig - type t + + module Thm : sig + type t + type u = - | Top of Loc.t - | Exists of Loc.t * Decs.t * t - | Forall of Loc.t * Decs.t * t - | ForallStar of Loc.t * Decs.t * t - | ForallG of Loc.t * (Decs.t * Decs.t) list * t - include LENS with type t := t and type u := u + | Top of Loc.t + | Exists of Loc.t * Decs.t * t + | Forall of Loc.t * Decs.t * t + | ForallStar of Loc.t * Decs.t * t + | ForallG of Loc.t * (Decs.t * Decs.t) list * t + + include LENS with type t := t and type u := u end - include LENS with type t := t and type u := u - module ThmDec : sig - type t - type u = - | ThmDec of string * Thm.t - include LENS with type t := t and type u := u - end - module WDecl : sig - type t - type u = - | WDecl of (string list * string) list * CallPats.t - include LENS with type t := t and type u := u + include LENS with type t := t and type u := u + + module ThmDec : sig + type t + type u = ThmDec of string * Thm.t + + include LENS with type t := t and type u := u + end + + module WDecl : sig + type t + type u = WDecl of (string list * string) list * CallPats.t + + include LENS with type t := t and type u := u end end - + module Cmd : sig + type t + + type u = + | Query of Loc.t * int option * int option * int option * Query.t + | QueryTabled of Loc.t * int option * int option * int option * Query.t + | AdhocQuery of Loc.t * Query.t + | Unique of Loc.t * Term.t + | Mode of Loc.t * Mode.Dec.t + | Define of Loc.t * Define.t + | DeclCmd of Loc.t * Term.t + | Inline of Loc.t * string * Term.t + | Symbol of Loc.t * string * string + | Freeze of Loc.t * string list + | Thaw of Loc.t * string list + | Sort of Loc.t * string list * Decl.t list + | Term of Loc.t * Decl.t + | Block of Loc.t * string * BlockItem.t list + | Union of Loc.t * string * string list + | Worlds of Loc.t * string list * Term.t + | Deterministic of Loc.t * string list + | ModuleCmd of Loc.t * string * string list * t list + | Use of Loc.t * string * string * string list + | OpenCmd of Loc.t * string * string list + | Eval of Loc.t * t list + | Prec of Loc.t * Fixity.t * int * string list + | Solve of Loc.t * Solve.t + | Stop of Loc.t * unit + | ReplQuit of Loc.t * unit + | ReplHelp of Loc.t * string option + | ReplGet of Loc.t * string + | ReplSet of Loc.t * string * string + | ReplVersion of Loc.t * unit + | Total of Loc.t * Thm.Order.t list * Term.t list + | Terminates of Loc.t * Thm.Order.t list * Term.t list + | Covers of Loc.t * Mode.Dec.t + | Name of Loc.t * string + | Reduces of Loc.t * string * Term.t list + + include LENS with type t := t and type u := u + end end diff --git a/src/Common/Cst/Lens.ml b/src/Common/Cst/Lens.ml index e80d39c..8752f2f 100644 --- a/src/Common/Cst/Lens.ml +++ b/src/Common/Cst/Lens.ml @@ -1 +1,9 @@ -module Make_Lens (L : LENS.S) : LENS.LENS = L +module Make_Lens (L : LENS.S) : LENS.LENS with type t = L.t and type u = L.u = + struct + type t = L.t + type u = L.u + let review = L.review + let view = L.view + let (!>) = view + let (!<) = review + end diff --git a/src/Display/FORM.ml b/src/Display/FORM.ml index 735b812..716f7a8 100644 --- a/src/Display/FORM.ml +++ b/src/Display/FORM.ml @@ -1,7 +1,11 @@ + + (* TODO: Thinks that will be added eventually Groups (sections, list, etc) Math markup Boxes*) + +(** Common colors for the foreground and background (for use with {!FORM}) *) module type COLORS = sig type t @@ -18,7 +22,7 @@ end module type FORM = sig type t - type style + type style = t -> t type 'a scribe = 'a -> t module Style : sig diff --git a/src/Display/Form.ml b/src/Display/Form.ml index 32b35d2..cca5f41 100644 --- a/src/Display/Form.ml +++ b/src/Display/Form.ml @@ -1,3 +1,7 @@ +(** +Formatting options generally +@author Asher Frost *) + module type FORM = FORM.FORM module Form : FORM = struct diff --git a/src/Fronts/Modern/Cmd.ml b/src/Fronts/Modern/Cmd.ml index 9456dc1..3b3e0d3 100644 --- a/src/Fronts/Modern/Cmd.ml +++ b/src/Fronts/Modern/Cmd.ml @@ -18,13 +18,18 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct (* Defer a thunk-parser to prevent infinite recursion at construction time. Used for %module and %eval which recursively embed cmd lists. *) let defer p = return () >>= fun () -> p () - + let parse_order () : Cst.View.Thm.Order.t t = fix (fun self -> begin choice ~failure_msg: "order" [ + (let+ ids = Modern.parse_id_list () in Cst.View.(Thm.Order.(review @@ Varg (Cst.View.Loc.(review Ghost), ids)))); + inside "[" "]" (commit *> let+ orders = many1 (self <* commit) in Cst.View.(Thm.Order.(review @@ Simul (Cst.View.Loc.(review Ghost), orders)))); + inside "{" "}" (commit *> let+ orders = many1 (self <* commit) in Cst.View.(Thm.Order.(review @@ Lex (Cst.View.Loc.(review Ghost), orders)))) + ] end) + let order_list () : Cst.View.Thm.Order.t list t = inside "(" ")" (many1 (parse_order ())) <|> (let+ x = parse_order () in [x]) let rec parse_cmd_list () : Cst.cmd list t = keyword "{" *> commit *> skip_outer *> many (defer parse1 <* skip_outer) - <* keyword "}" *> commit - - and parse1 () : Cst.cmd t = - choice ~failure_msg:"command" + <* keyword "}" *> commit + + and parse1 () : Cst.cmd t = + choice ~failure_msg:"command" [ begin whitespace *> (keyword "." *> commit *> return (Cst.Cmd.stop ())) @@ -124,7 +129,7 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct (keyword "sort" *> commit *> let* ids = Modern.parse_id_list () in - let+ ds = many (inside "{" "}" (Modern.parse_decl ())) in + let+ ds = many (inside "{" "}" (commit *> Modern.parse_decl ())) in Cst.View.Cmd.(review @@ Sort (Cst.View.Loc.(review Ghost), ids, ds))) "sort" end; @@ -195,8 +200,9 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct *> let* fix = Modern.parse_fixity_kw () in let* n = Modern.parse_fixity () in - let+ ids = Modern.parse_id_list () in - Cst.Cmd.prec fix n ids) + let* ids = Modern.parse_id_list () in + let () = Modern.register_local_fixity fix n ids in + return (Cst.Cmd.prec fix n ids)) "prec" end; begin @@ -242,28 +248,18 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct begin (keyword "total" *> commit *> - let* intros = - inside "{" "}" @@ many (Modern.parse_id_list ()) - <|> (let+ id = Modern.parse_var () in - [ [ id ] ]) - <|> return [] - in - let+ body = many (Modern.parse_expr1 ()) in - Cst.Cmd.total intros body) + let* order = order_list () in + let+ body = many1 (Modern.parse_expr1 ()) in + Cst.View.Cmd.(review (Total (Cst.View.Loc.(review Ghost), order, body)))) "total" end; begin (keyword "terminates" *> commit *> - let* intros = - inside "{" "}" @@ many (Modern.parse_id_list ()) - <|> (let+ id = Modern.parse_var () in - [ [ id ] ]) - <|> return [] - in - let+ body = many (Modern.parse_expr1 ()) in - Cst.Cmd.terminates intros body) - "terminates" + let* order = order_list () in + let+ body = many1 (Modern.parse_expr1 ()) in + Cst.View.Cmd.(review (Terminates (Cst.View.Loc.(review Ghost), order, body)))) + "terminates" end; begin (keyword "covers" *> commit @@ -279,6 +275,14 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct Cst.Cmd.name id) "name" end; + begin + (keyword "reduces" *> commit + *> + let* rel = Modern.parse_reduces_rel () in + let+ body = many1 (Modern.parse_expr1 ()) in + Cst.View.Cmd.(review (Reduces (Cst.View.Loc.(review Ghost), rel, body)))) + "reduces" + end ] let parse () : Cst.cmd list t = skip_outer *> many (parse1 () <* skip_outer) diff --git a/src/Fronts/Modern/MODERN.ml b/src/Fronts/Modern/MODERN.ml index 71c995c..108fa53 100644 --- a/src/Fronts/Modern/MODERN.ml +++ b/src/Fronts/Modern/MODERN.ml @@ -2,13 +2,14 @@ module type MODERN = sig module Paths : Paths.Paths_intf.PATHS module Cst : Cst.CST module Names : Names.Names_intf.NAMES + module N := Names + module FX := N.Fixity module Parser : Parsing.PARSER.PARSER type 'a t = 'a Parser.t exception ParseError of string - val set_fixities : Names.namespace -> unit val parse_expr1 : unit -> Cst.Term.t t val parse_expr : unit -> Cst.Term.t t val parse_var : unit -> string t @@ -34,9 +35,11 @@ module type MODERN = sig val parse_solve : unit -> Cst.Query.solve t val parse_bound : unit -> int option t val parse_id_list : unit -> string list t + val parse_reduces_rel : unit -> string t val parse_block_item : unit -> Cst.block_item t val parse_fixity_kw : unit -> Cst.fixity t val parse_params : unit -> string list t + val register_local_fixity : Cst.fixity -> int -> string list -> unit val parse_group : 'a t -> 'a list t val parse_parens : 'a t -> 'a t val parse_braced : 'a t -> 'a t @@ -45,5 +48,8 @@ module type MODERN = sig val debug_parser : 'a t -> string -> 'a [@@alert debug "This should only be used in the REPL"] - val run : 'a t -> Names.namespace ref -> Cst.loc -> string -> 'a + val debug_parser_with_ops : (string * FX.fixity) list -> 'a t -> string -> 'a + [@@alert debug "This should only be used in the REPL"] + + val run : 'a t -> N.namespace ref -> Cst.loc -> string -> 'a end diff --git a/src/Fronts/Modern/Modern.ml b/src/Fronts/Modern/Modern.ml index fa3068d..9958d31 100644 --- a/src/Fronts/Modern/Modern.ml +++ b/src/Fronts/Modern/Modern.ml @@ -9,78 +9,247 @@ module Make_Modern and module Paths = Paths = struct module Paths = Paths module Cst = Cst - module Names = Names + module Names = Names + module N = Names module Parser = Parser exception ParseError of string - let fixities : Names.namespace ref = ref @@ Names.newNamespace () - let set_fixities : Names.namespace -> unit = fun ns -> fixities := ns - - open Parser + open Parser + let currently_uppercase : string list ref = ref [] + let with_uppercase : string list -> (unit -> 'a t) -> 'a t = fun names p -> + let old = !currently_uppercase in + currently_uppercase := old @ names; + let* res = p () in + let+ () = return @@ (currently_uppercase := old) in + res type 'a t = 'a Parser.t - + let rec break (s : string list) : string list * string = + match s with + | [] -> ([], "") + | x :: xs -> let (y, z) = break xs in (x :: y, z) let mk_loc : int -> int -> Cst.loc = fun x y -> Cst.mk_loc x y - - let ident1 = - let* s = ident in - if String.length s = 0 then fail "expected identifier" else return s - + let whitespace' () : unit t = + whitespace <|> (forget @@ token "% " *> take_till (fun c -> c = '\n') ) + let ghost' = Cst.View.Loc.(review Ghost) let rec combine_fc (r1 : Paths.region) (r2 : Paths.region) : Paths.region = assert false - and parse_arg () : string option t = + module FX = Names.Fixity + + let local_fixity : (string, FX.fixity) Hashtbl.t = Hashtbl.create 16 + + let register_local_fixity (fix : Cst.fixity) (n : int) (ids : string list) : unit = + let prec = FX.Strength n in + let open Cst.View.Fixity in + let fx = match !> fix with + | Left _ -> FX.Infix (prec, FX.Left) + | Right _ -> FX.Infix (prec, FX.Right) + | Prefix _ -> FX.Prefix prec + | Postfix _ -> FX.Postfix prec + | Middle _ -> FX.Infix (prec, FX.None) + | None _ -> FX.Infix (prec, FX.None) + in + List.iter (fun id -> Hashtbl.replace local_fixity id fx) ids + + type operator = + | Atom of Cst.Term.t + | Infix_ of (FX.precedence * FX.associativity) * (Cst.Term.t * Cst.Term.t -> Cst.Term.t) + | Prefix_ of FX.precedence * (Cst.Term.t -> Cst.Term.t) + | Postfix_ of FX.precedence * (Cst.Term.t -> Cst.Term.t) + + let jux_op = + Infix_ ((FX.inc FX.maxPrec, FX.Left), + fun (f, x) -> Cst.Term.app f [x]) + + let infix_op (infixity, tm) = + Infix_ (infixity, fun (tm1, tm2) -> Cst.Term.app (Cst.Term.app tm [tm1]) [tm2]) + + let prefix_op (prec, tm) = + Prefix_ (prec, fun tm1 -> Cst.Term.app tm [tm1]) + + let postfix_op (prec, tm) = + Postfix_ (prec, fun tm1 -> Cst.Term.app tm [tm1]) + + let classify (tm : Cst.Term.t) : operator = + let open Cst.View.Term in + match !> tm with + | Lowercase (_, (ns, name)) + | Uppercase (_, (ns, name)) + | Qualified (_, (ns, name)) -> + let fixity = match Hashtbl.find_opt local_fixity name with + | Some fx -> fx + | None -> Names.fixityLookup (Names.Qid (ns, name)) + in + begin match fixity with + | FX.Nonfix -> Atom tm + | FX.Infix (prec, assoc) -> infix_op ((prec, assoc), tm) + | FX.Prefix prec -> prefix_op (prec, tm) + | FX.Postfix prec -> postfix_op (prec, tm) + end + | _ -> Atom tm + + module P = struct + let rec reduce = function + | Atom tm2 :: Infix_ (_, con) :: Atom tm1 :: p' -> Atom (con (tm1, tm2)) :: p' + | Atom tm :: Prefix_ (_, con) :: p' -> Atom (con tm) :: p' + | Postfix_ (_, con) :: Atom tm :: p' -> Atom (con tm) :: p' + | p -> failwith (Printf.sprintf "process_app: cannot reduce stack of length %d" (List.length p)) + + let rec reduce_rec = function + | [ Atom e ] -> e + | p -> reduce_rec (reduce p) + + let reduce_all = function + | [ Atom e ] -> e + | Infix_ _ :: _ -> raise (ParseError "Incomplete infix expression") + | Prefix_ _ :: _ -> raise (ParseError "Incomplete prefix expression") + | [] -> raise (ParseError "Empty expression") + | p -> reduce_rec (reduce p) + + let shift_atom (tm, p) = + match p with + | Atom _ :: _ -> reduce (Atom tm :: jux_op :: p) + | _ -> Atom tm :: p + + let rec shift (opr, p) = + match opr, p with + | (Atom _ as o), (Atom _ :: _ as p') -> reduce (o :: jux_op :: p') + | Infix_ _, Infix_ _ :: _ -> raise (ParseError "Consecutive infix operators") + | Infix_ _, Prefix_ _ :: _ -> raise (ParseError "Infix operator following prefix operator") + | Infix_ _, [] -> raise (ParseError "Leading infix operator") + | (Prefix_ _ as o), (Atom _ :: _ as p') -> o :: jux_op :: p' + | Postfix_ _, Infix_ _ :: _ -> raise (ParseError "Postfix operator following infix operator") + | Postfix_ _, Prefix_ _ :: _ -> raise (ParseError "Postfix operator following prefix operator") + | Postfix_ _, [] -> raise (ParseError "Leading postfix operator") + | o, p' -> o :: p' + + let rec resolve (opr, p) = + match opr, p with + | (Infix_ ((prec, assoc), _) as o), + (Atom _ :: Infix_ ((prec', assoc'), _) :: _ as p') -> + begin match FX.compare (prec, prec'), assoc, assoc' with + | Greater, _, _ -> shift (o, p') + | Less, _, _ -> resolve (o, reduce p') + | Equal, FX.Left, FX.Left -> resolve (o, reduce p') + | Equal, FX.Right, FX.Right -> shift (o, p') + | _ -> raise (ParseError "Ambiguous: infix following infix of identical precedence") + end + | (Infix_ ((prec, _), _) as o), + (Atom _ :: Prefix_ (prec', _) :: _ as p') -> + begin match FX.compare (prec, prec') with + | Greater -> shift (o, p') + | Less -> resolve (o, reduce p') + | Equal -> raise (ParseError "Ambiguous: infix following prefix of identical precedence") + end + | (Prefix_ _ as o), p' -> shift (o, p') + | (Postfix_ (prec, _) as o), + (Atom _ :: Prefix_ (prec', _) :: _ as p') -> + begin match FX.compare (prec, prec') with + | Greater -> reduce (shift (o, p')) + | Less -> resolve (o, reduce p') + | Equal -> raise (ParseError "Ambiguous: postfix following prefix of identical precedence") + end + | (Postfix_ (prec, _) as o), + (Atom _ :: Infix_ ((prec', _), _) :: _ as p') -> + begin match FX.compare (prec, prec') with + | Greater -> reduce (shift (o, p')) + | Less -> resolve (o, reduce p') + | Equal -> raise (ParseError "Ambiguous: postfix following infix of identical precedence") + end + | (Postfix_ _ as o), ([ Atom _ ] as p') -> reduce (shift (o, p')) + | o, p' -> shift (o, p') + end + + let process_app (ts : Cst.Term.t list) : Cst.Term.t = + let rec go p = function + | [] -> P.reduce_all p + | t :: rest -> + let p' = match classify t with + | Atom tm -> P.shift_atom (tm, p) + | opr -> P.resolve (opr, p) + in + go p' rest + in + match ts with + | [] -> failwith "process_app: called with empty list" + | _ -> go [] ts + + let rec parse_arg () : string option t = token "_" *> return None <|> (let+ s = ident1 in - Some s) + Some s) "argument" and parse_id () : Cst.Term.t t = - (let+ sym = parse_qualified () in - Cst.Term.qualified sym) - <|> (let* name = ident1 in - let sym = ([], name) in - if Char.uppercase_ascii name.[0] = name.[0] then - return (Cst.Term.uppercase sym) - else return (Cst.Term.lowercase sym)) - "identifier" + (keyword "val" *> commit + *> (inside "(" ")" (let@ (ns, s, e) = many1 ident1 in + let loc = mk_loc s e in + let rec split = function + | [] -> failwith "Expected qualified name" + | [ p ] -> ([], p) + | p :: q -> let q0, q1 = split q in (p :: q0, q1) + in + return (Cst.Term.qualified ~fc:loc (split ns))) + <|> (let@ (name, s, e) = ident1 in + let loc = mk_loc s e in + return (Cst.Term.qualified ~fc:loc ([], name))))) + <|> + (let@ (name, s, e) = ident1 in + let loc = mk_loc s e in + let is_upper = + String.length name > 0 + && (name.[0] = '_' || (name.[0] >= 'A' && name.[0] <= 'Z')) + in + return (if is_upper || List.mem name !currently_uppercase + then Cst.Term.uppercase ~fc:loc ([], name) + else Cst.Term.lowercase ~fc:loc ([], name))) and parse_expr_trail () : Cst.Term.t t = - (let* d = inside "[" "]" (parse_decl ()) in + (let@ (d, s, e) = inside "[" "]" (parse_decl ()) in + let loc = mk_loc s e in let+ body = parse_expr () in - Cst.Term.lam [ d ] body) - <|> (let* d = inside "{" "}" (parse_decl ()) in + Cst.Term.lam ~fc:loc [ d ] body) + <|> (let@ (d, s, e) = inside "{" "}" (parse_decl ()) in + let loc = mk_loc s e in let+ body = parse_expr () in - Cst.Term.pi [ d ] body) + Cst.Term.pi ~fc:loc [ d ] body) + <|> ((let* ids = inside "{{" "}}" (many @@ parse_var ()) in + let+ body = (with_uppercase ids parse_expr) in + body) "expression with implicit variables") + + <|> (parse_id ()) "trailing expression" and parse_expr_app () : Cst.Term.t t = (let* head = parse_expr1 () in let+ args = many (parse_expr1 ()) in - Cst.Term.app head args) + process_app (head :: args)) "application" and parse_expr1 () : Cst.Term.t t = begin - parse_id () <|> inside "(" ")" (parse_expr ()) + choice [ + parse_id (); + inside "(" ")" (return () >>= fun () -> parse_expr ()) + ] end "small expression" and parse_expr () : Cst.Term.t t = begin - fix (fun self -> - let expr1 = parse_id () <|> inside "(" ")" self in + ( (keyword "the" *> commit *> - let* ty = expr1 in - let+ body = self in + let* ty = parse_expr1 () in + let+ body = parse_expr () in Cst.Term.has_type body ty) <|> - (* -> A B C ==> {_ A} {_ B} C (last arg is the body) *) - (keyword "->" *> commit + (* %-> A %-> B %-> C ==> {_ A} {_ B} C (last arg is the body) *) + (keywords ["->"; "if"] *> commit *> - let+ args = many1 (parse_expr1 ()) in + let+ args = sep_by1 (option () @@ keyword "->" *> commit) (parse_expr1 ()) in let rev = List.rev args in let body = List.hd rev in let init = List.rev (List.tl rev) in @@ -88,25 +257,27 @@ module Make_Modern (fun t acc -> Cst.Term.pi [ Cst.Decl.decl1 [ None ] t ] acc) init body) <|> - (* <- A B C ==> {_ C} {_ B} A (first arg is the body) *) - (keyword "<-" *> commit + (* + %<- A + %<- B + %<- C ==> {_ C} {_ B} A (first arg is the body) *) + (keywords ["<-"; "do"] *> commit *> - let+ args = many1 (parse_expr1 ()) in + let+ args = sep_by1 (option () @@ keyword "<-" *> commit) (parse_expr1 ()) in let body = List.hd args in let rest_rev = List.rev (List.tl args) in List.fold_right (fun t acc -> Cst.Term.pi [ Cst.Decl.decl1 [ None ] t ] acc) rest_rev body) <|> - let* atoms = many expr1 in - (let+ trail = parse_expr_trail () in - match atoms with - | [] -> trail - | head :: rest -> Cst.Term.app head (rest @ [ trail ])) - <|> - match atoms with - | [] -> fail "expected expression" - | head :: rest -> return (Cst.Term.app head rest)) + (let* atoms = many @@ parse_expr1 () in + let* trail_opt = option None (let+ t = parse_expr_trail () in Some t) in + match atoms, trail_opt with + | [], None -> fail "expected expression" + | [], Some trail -> return trail + | head :: rest, None -> return (process_app (head :: rest)) + | head :: rest, Some trail -> return (process_app (head :: (rest @ [ trail ])))) + ) end "expression" @@ -136,27 +307,33 @@ module Make_Modern and parse_text () : string t = begin - string "%\"" *> take_till (fun c -> c = '%') <* string "%\"" <* whitespace + string "%\"" *> take_till (fun c -> c = '%') <* string "%\"" <* (whitespace' ()) end "string literal" and parse_decl () : Cst.Decl.t t = begin - (let* names = inside "(" ")" (many1 (parse_arg ())) in + (let@ (names, s, e) = inside "(" ")" (many1 (parse_arg ())) in + let loc = mk_loc s e in let+ typ = option (Cst.Term.omitted ~fc:Cst.ghost) (parse_expr ()) in - Cst.Decl.decl1 names typ) - <|> let* name = parse_arg () in - let+ typ = option (Cst.Term.omitted ~fc:Cst.ghost) (parse_expr ()) in - Cst.Decl.decl1 [ name ] typ + Cst.Decl.decl1 ~fc:loc names typ) + <|> (let@ (name, s, e) = parse_arg () in + let loc = mk_loc s e in + let+ typ = option (Cst.Term.omitted ~fc:Cst.ghost) (parse_expr ()) in + Cst.Decl.decl1 ~fc:loc [ name ] typ) end "declaration" and parse_mode () : Cst.Mode.mode t = begin - keyword "out1" *> commit *> return (Cst.Mode.minus1 ()) - <|> keyword "out" *> commit *> return (Cst.Mode.minus ()) - <|> keyword "in" *> commit *> return (Cst.Mode.plus ()) - <|> keyword "star" *> commit *> return (Cst.Mode.star ()) + (let@ ((), s, e) = keyword "out1" *> commit *> return () in + return (Cst.Mode.minus1 ~fc:(mk_loc s e) ())) + <|> (let@ ((), s, e) = keyword "out" *> commit *> return () in + return (Cst.Mode.minus ~fc:(mk_loc s e) ())) + <|> (let@ ((), s, e) = keyword "in" *> commit *> return () in + return (Cst.Mode.plus ~fc:(mk_loc s e) ())) + <|> (let@ ((), s, e) = keyword "star" *> commit *> return () in + return (Cst.Mode.star ~fc:(mk_loc s e) ())) end "mode" @@ -232,7 +409,7 @@ module Make_Modern and parse_fixity () : int t = begin - let+ s = take_while1 (fun c -> c >= '0' && c <= '9') <* whitespace in + let+ s = take_while1 (fun c -> c >= '0' && c <= '9') <* (whitespace' ()) in int_of_string s end "fixity level" @@ -254,7 +431,12 @@ module Make_Modern (fun s -> Some s) <$> parse_var () <|> Parser.string "_" *> return None in - let* ty = Parser.option None @@ ((fun s -> Some s) <$> parse_expr1 ()) in + let* ty = + let+ t = parse_expr1 () in + match Cst.View.Term.view t with + | Cst.View.Term.Uppercase (_, ([], "_")) -> None + | _ -> Some t + in let+ tm = parse_expr () in Cst.View.(Define.(review @@ Define (Loc.(review Ghost), id, tm, ty))) end @@ -270,19 +452,29 @@ module Make_Modern and parse_bound () : int option t = begin token "_" *> return None - <|> let+ s = take_while1 (fun c -> c >= '0' && c <= '9') <* whitespace in + <|> let+ s = take_while1 (fun c -> c >= '0' && c <= '9') <* (whitespace' ()) in Some (int_of_string s) end "bound" and parse_id_list () : string list t = begin - inside "(" ")" (many (parse_var ())) + inside "(" ")" (many1 (parse_var ())) <|> let+ id = parse_var () in [ id ] end "identifier list" + and parse_reduces_rel () : string t = + begin + token "<=" *> commit *> return "<=" + <|> token ">=" *> commit *> return ">=" + <|> token "<" *> commit *> return "<" + <|> token ">" *> commit *> return ">" + <|> token "=" *> commit *> return "=" + end + "reduces predicate" + and parse_block_item () : Cst.block_item t = begin inside "{" "}" @@ -317,21 +509,26 @@ module Make_Modern and parse_bracketed : 'a. 'a t -> 'a t = fun p -> inside "[" "]" p and debug_parser : 'a t -> string -> 'a = + fun p x -> + Display.(debug @@ (style Style.bold @@ string "Parsing") ++ nl () ++ string x); match Parser.parse_string ~consume:All (p : _ Parser.t) x with | Ok res -> res | Error msg -> raise (ParseError msg) - and run : 'a. 'a t -> Names.namespace ref -> Cst.loc -> string -> 'a = - fun p _ns _loc s -> - match Parser.parse_string ~consume:All (whitespace *> p) s with + and debug_parser_with_ops : (string * FX.fixity) list -> 'a t -> string -> 'a = fun f p x -> + List.iter (fun (id, fix) -> Hashtbl.replace local_fixity id fix) f; + debug_parser p x + and run : 'a. 'a t -> N.namespace ref -> Cst.loc -> string -> 'a = + fun p _ns _loc s -> (* TODO use namespace and loc *) + match Parser.parse_string ~consume:All ((whitespace' ()) *> p) s with | Ok res -> res | Error msg -> raise (ParseError msg) end module ModernCst = Cst.Make_Cst (Paths.Paths_) -module Modern : MODERN.MODERN = +module Modern : MODERN.MODERN = Make_Modern (Paths.Paths_) (ModernCst) (Names.Names_) (Parsing.Parser.Parser) (* Re-export sub-modules so they are accessible outside the library *) diff --git a/src/Fronts/Pal/IMPL.ml b/src/Fronts/Pal/IMPL.ml index 3616ed7..cf22c60 100644 --- a/src/Fronts/Pal/IMPL.ml +++ b/src/Fronts/Pal/IMPL.ml @@ -9,7 +9,7 @@ module type IMPL = sig (** Names module — same as Cmd.Names. *) - module Recon : Recon.RECON + module Recon : Recon.RECON (** New elaboration layer: reconstruction / type-inference sub-modules. *) (** Input source for loading: a file path or an inline string. *) diff --git a/src/Fronts/Pal/Impl.ml b/src/Fronts/Pal/Impl.ml index 52e1c82..0fab445 100644 --- a/src/Fronts/Pal/Impl.ml +++ b/src/Fronts/Pal/Impl.ml @@ -136,7 +136,7 @@ module Impl = struct module Install = struct let unfold_app tm = let rec go acc = function - | Cst.App_ (f, arg) -> go (arg :: acc) f + | Cst.App_ (_, f, arg) -> go (arg :: acc) f | head -> (head, acc) in go [] tm @@ -155,14 +155,53 @@ module Impl = struct failwith "%total/%terminates: expected identifier as call-pattern head" - let build_thm_tdecl label intros body = + let build_thm_rdecl pred_str body = let module TS = ThmSyn in - let mk_order names = TS.Varg names in + let pred, swap = + match pred_str with + | "=" -> (TS.Eq, false) + | "<" -> (TS.Less, false) + | "<=" -> (TS.Leq, false) + | ">" -> (TS.Less, true) + | ">=" -> (TS.Leq, true) + | s -> failwith ("%reduces: unknown predicate " ^ s) + in + let term_to_order tm = + TS.Varg [ term_to_head tm ] + in + let o_out, o_in, body_rest = + match body with + | out_tm :: in_tm :: rest -> (term_to_order out_tm, term_to_order in_tm, rest) + | _ -> failwith "%reduces: expected predicate, two order arguments, and call patterns" + in + let o1, o2 = if swap then (o_in, o_out) else (o_out, o_in) in + let term_to_cp tm = + let head, args = unfold_app tm in + let name = term_to_head head in + let cid = + match Names.constLookup (Names.Qid ([], name)) with + | None -> failwith ("%reduces: undeclared identifier " ^ name ^ " in call pattern") + | Some c -> c + in + (cid, List.map term_to_name args) + in + let callpats = TS.Callpats (List.map term_to_cp body_rest) in + let dummy_r = PathsConcrete.Reg (0, 0) in + let rrs = (dummy_r, List.map (fun _ -> dummy_r) body_rest) in + (TS.RDecl (TS.RedOrder (pred, o1, o2), callpats), rrs) + + let build_thm_tdecl label (orders : Cst.order list) body = + let module TS = ThmSyn in + let rec cst_to_thm = function + | Cst.Varg_ (_, names) -> TS.Varg names + | Cst.Lex_ (_, ords) -> TS.Lex (List.map cst_to_thm ords) + | Cst.Simul_ (_, ords) -> TS.Simul (List.map cst_to_thm ords) + in let order = - match intros with - | [] -> mk_order [] - | [ names ] -> mk_order names - | many -> TS.Simul (List.map mk_order many) + match orders with + | [] -> TS.Varg [] + | [ o ] -> cst_to_thm o + | many -> TS.Simul (List.map cst_to_thm many) in let term_to_cp tm = let head, args = unfold_app tm in @@ -200,20 +239,23 @@ module Impl = struct | Some cid -> cid in match cmd with - | Cst.SortCmd_ (id, decls) -> ( - Debug.( - msg' ~src:Group.pal ~level:Level.Debug - Fmt.(const string "Installing sort" ++ sp ++ string) - id); - let kind = factor_sort decls in - let condec = - Cst.ConDec.constant_decl (Cst.Decl.decl1 [ Some id ] kind) - in - match - Recon.ReconConDec.condecToConDec (condec, loc_of Cst.ghost, false) - with - | Some cd, _ -> install_condec cd - | None, _ -> ()) + | Cst.SortCmd_ (ids, decls) -> + List.app + (fun id -> + Debug.( + msg' ~src:Group.pal ~level:Level.Debug + Fmt.(const string "Installing sort" ++ sp ++ string) + id); + let kind = factor_sort decls in + let condec = + Cst.ConDec.constant_decl (Cst.Decl.decl1 [ Some id ] kind) + in + (match + Recon.ReconConDec.condecToConDec (condec, loc_of Cst.ghost, false) + with + | Some cd, _ -> install_condec cd + | None, _ -> ())) + ids | Cst.TermCmd_ decl -> ( Debug.( msg' ~src:Group.pal ~level:Level.Debug Fmt.string @@ -471,6 +513,10 @@ module Impl = struct let mdec, _r = Recon.ReconMode.modeToMode md in Cover.checkCovers mdec | Cst.NameCmd_ _id -> () + | Cst.ReducesCmd_ (pred_str, body) -> + let r_, rrs = build_thm_rdecl pred_str body in + let la_ = ThmInst.installReduces (r_, rrs) in + List.app Terminate.Terminate_.Reduces.checkFamReduction la_ | Cst.UniqueCmd_ tm -> let mdec_opt = match Cst.View.Term.view tm with diff --git a/src/Fronts/Pal/Pal.ml b/src/Fronts/Pal/Pal.ml index 5025383..227ecb1 100644 --- a/src/Fronts/Pal/Pal.ml +++ b/src/Fronts/Pal/Pal.ml @@ -18,7 +18,7 @@ module Pal : PAL.PAL = struct method exec (s : string) : unit = List.iter self#install (self#parse s) end - + let top (module N : Tui.REPL.S) = let pi = new pal in let module R = Tui.Repl.Repl (N) in diff --git a/src/Fronts/Pal/dune b/src/Fronts/Pal/dune index 2aec599..828ee7c 100644 --- a/src/Fronts/Pal/dune +++ b/src/Fronts/Pal/dune @@ -22,6 +22,7 @@ subordinate index frontend + terminate worldcheck modules lwt.unix)) diff --git a/src/Lang/Parsing/PARSER.ml b/src/Lang/Parsing/PARSER.ml index b6e1fd4..81cafed 100644 --- a/src/Lang/Parsing/PARSER.ml +++ b/src/Lang/Parsing/PARSER.ml @@ -5,6 +5,7 @@ module type PARSER = sig val inside : string -> string -> 'a t -> 'a t val whitespace : unit t val ident : string t + val ident1 : string t val keyword : string -> unit t val keywords : string list -> unit t val token : string -> unit t @@ -16,4 +17,5 @@ module type PARSER = sig val ( let+ ) : 'a t -> ('a -> 'b) -> 'b t val ( and+ ) : 'a t -> 'b t -> ('a * 'b) t val ( let@ ) : 'a t -> ('a * int * int -> 'b t) -> 'b t + val forget : 'a t -> unit t end diff --git a/src/Lang/Parsing/Parser.ml b/src/Lang/Parsing/Parser.ml index 509753f..50f3eae 100644 --- a/src/Lang/Parsing/Parser.ml +++ b/src/Lang/Parsing/Parser.ml @@ -13,7 +13,22 @@ module Parser : PARSER = struct skip_while (function ' ' | '\t' | '\n' -> true | _ -> false) let token s = (string s <* whitespace) *> return () - let keyword s = token @@ "%" ^ s + + (* Require that the character after the keyword body is either EOF + or one of the identifier-delimiters used by [ident1] below. + Without this, [keyword "term"] succeeds on the [%term] prefix of + [%terminates], and the [Cmd.ml] [choice] (which tries [term] + before [terminates]) commits to the wrong branch. *) + let keyword s = + let s' = "%" ^ s in + let boundary = + peek_char >>= function + | None -> return () + | Some (' ' | '\t' | '\n' + | '(' | ')' | '{' | '}' | '[' | ']' | '%') -> return () + | _ -> fail ("keyword " ^ s' ^ " not at word boundary") + in + string s' *> boundary *> whitespace let keywords ss = choice (List.map keyword ss) let ident = @@ -21,6 +36,11 @@ module Parser : PARSER = struct | ' ' | '\t' | '\n' | '(' | ')' | '{' | '}' | '[' | ']' | '%' -> true | _ -> false) <* whitespace + let ident1 = + take_while1 (function + | ' ' | '\t' | '\n' | '(' | ')' | '{' | '}' | '[' | ']' | '%' -> false + | _ -> true) (* TODO Generalize to unicode ws *) + <* whitespace let ( let* ) = ( >>= ) @@ -46,4 +66,6 @@ module Parser : PARSER = struct let given b p = if b then p else fail "failed test" let inside x y p = token x *> p <* token y + + let forget p = p *> return () end diff --git a/src/Recon/ReconMode.ml b/src/Recon/ReconMode.ml index 25b0116..6849436 100644 --- a/src/Recon/ReconMode.ml +++ b/src/Recon/ReconMode.ml @@ -58,8 +58,8 @@ module Make_ReconMode (M : S.S) : RECON_MODE with module M = M = struct Modes.Mapp (Modes.Marg (convert_mode m, name_opt), build_spine rest) in - let mS = build_spine spine in - ModeDec.checkFull (cid, mS, ghost_region); + let mS_short = build_spine spine in + let mS = ModeDec.shortToFull (cid, mS_short, ghost_region) in ((cid, mS), Paths.Reg (0, 0)) end end diff --git a/src/Recon/ReconTerm.ml b/src/Recon/ReconTerm.ml index 7a64d03..a995841 100644 --- a/src/Recon/ReconTerm.ml +++ b/src/Recon/ReconTerm.ml @@ -56,7 +56,7 @@ struct let errorCount = ref 0 let errorFileName = ref "no file" - let errorThreshold = ref (Some 20) + let errorThreshold = ref (Some 200) let exceeds = function i, None -> false | i, Some j -> i > j let rec resetErrors fileName = @@ -110,21 +110,9 @@ struct end end - let withConstPath show f = - let old = !Print.showConstPath in - Print.showConstPath := show; - try - let result = f () in - Print.showConstPath := old; - result - with exn -> - Print.showConstPath := old; - raise exn - let rec formatExp (g_, u_) = - withConstPath false (fun () -> - try Print.formatExp (g_, u_) - with unprintable_ -> F.string "%_unprintable_%") + try Print.formatExp (g_, u_) + with Names.Unprintable -> F.string "%_unprintable_%" (* this is a hack, i know *) let queryMode = ref false @@ -835,18 +823,16 @@ struct | g_, e_, vs_, i -> addImplicit1W (g_, e_, Whnf.whnfExpandDef vs_, i) let rec reportConstraints xnames_ = - withConstPath false (fun () -> - try - begin match Print.evarCnstrsToStringOpt xnames_ with - | None -> () - | Some constr -> print (("Constraints:\n" ^ constr) ^ "\n") - end - with unprintable_ -> print "%_constraints unprintable_%\n") + try + begin match Print.evarCnstrsToStringOpt xnames_ with + | None -> () + | Some constr -> Msg.message (("Constraints:\n" ^ constr) ^ "\n") + end + with Names.Unprintable -> Msg.message "%_constraints unprintable_%\n" let rec reportInst xnames_ = - withConstPath false (fun () -> - try Msg.message (Print.evarInstToString xnames_ ^ "\n") - with unprintable_ -> Msg.message "%_unifier unprintable_%\n") + try Msg.message (Print.evarInstToString xnames_ ^ "\n") + with Names.Unprintable -> Msg.message "%_unifier unprintable_%\n" let rec delayMismatch (g_, v1_, v2_, r2, location_msg, problem_msg) = addDelayed (function () -> @@ -890,9 +876,17 @@ struct let amb = F.hVbox [ F.string "Inferred:"; F.space; formatExp (g_, u_) ] in + let fstr = F.makestring_fmt amb in + Display.debug Display.Form.( + nl () ++ + string "Ambiguous reconstruction of term: " ++ + string fstr ++ + + nl () + ); error ( r, - (("Ambiguous reconstruction\n" ^ F.makestring_fmt amb) ^ "\n") ^ msg + (("Ambiguous reconstruction\n" ^ fstr) ^ "\n") ^ msg )) let rec unifyIdem x = @@ -1093,13 +1087,23 @@ struct let s = IntSyn.Shift (IntSyn.ctxLength g_) in (tm, Elim (elimSub (evarElim x_, s)), eClo_ (v_, s)) | g_, (Fvar_ (name, r) as tm) -> - Logs.debug (fun m -> m "inferring FVar %s" name); + Display.debug Display.Form.( + nl () ++ + string "Inferring exact type of FVar" ++ + string name ++ + nl () + ) ; let v_ = try getFVarType (name, false) with Apx.Ambiguous -> let v_ = getFVarType (name, true) in begin - Logs.debug (fun m -> m "ambiguous type for FVar %s" name); + Display.debug Display.Form.( + string "Type of FVar" ++ + string name ++ + string " is ambiguous, but continuing with one of the possibilities" ++ + nl () + ) ; delayAmbiguous (g_, v_, r, "Free variable has ambiguous type"); v_ end @@ -1132,32 +1136,36 @@ struct | g_, App_ (tm1, tm2) -> let tm1', b1_, v1_ = inferExact (g_, tm1) in let e1_ = toElim b1_ in - Debug.( - msg' ~src:Group.approx ~level:Level.Debug - @@ Fmt.concat - Fmt. - [ - const string "Infering exact application of"; - using fst pp_term; - const string "to"; - using snd pp_term; - ]) - (tm1, tm2); - let IntSyn.Pi ((IntSyn.Dec (_, va_), _), vr_), s = - Whnf.whnfExpandDef (v1_, IntSyn.id) - in - let tm2', b2_ = - checkExact - ( g_, - tm2, - (va_, s), - "Argument type did not match function domain type\n\ - (Index object(s) did not match)" ) - in - let u2_ = toIntro (b2_, (va_, s)) in - ( App_ (tm1', tm2'), - Elim (elimApp (e1_, u2_)), - eClo_ (vr_, Whnf.dotEta (exp_ u2_, s)) ) + Display.(debug Form.( + nl () ++ + string "Inferring exact application of" ++ + shown show_term tm1 ++ + string "to" ++ + shown show_term tm2 ++ + nl () + )) ; + let t, s = Whnf.whnfExpandDef (v1_, IntSyn.id) in + begin match t with + | IntSyn.Pi ((IntSyn.Dec (_, va_), _), vr_) -> begin + let tm2', b2_ = + checkExact + ( g_, + tm2, + (va_, s), + "Argument type did not match function domain type\n\ + (Index object(s) did not match)" ) + in + let u2_ = toIntro (b2_, (va_, s)) in + ( App_ (tm1', tm2'), + Elim (elimApp (e1_, u2_)), + eClo_ (vr_, Whnf.dotEta (exp_ u2_, s)) ) + end + | _ -> begin + failwith + "Expected a pi type after whnf in application, but got something \ + else" + end + end | g_, Hastype_ (tm1, tm2) -> let tm2', b2_, l_ = inferExact (g_, tm2) in let v_ = toIntro (b2_, (l_, IntSyn.id)) in @@ -1188,6 +1196,10 @@ struct with Ambiguous -> let v'_ = Apx.apxToClass (g_, v_, l_, true) in begin + Display.debug Display.Form.( + string "Classifier of omitted term is ambiguous, but continuing with one of the possibilities" ++ + nl () + ) ; delayAmbiguous ( g_, v'_, @@ -1206,6 +1218,10 @@ struct with Ambiguous -> let u'_ = Apx.apxToExact (g_, u_, (v'_, IntSyn.id), true) in begin + Display.debug Display.Form.( + string "Exact term of omitted term is ambiguous, but continuing with one of the possibilities" ++ + nl () + ) ; delayAmbiguous ( g_, u'_, @@ -1300,6 +1316,13 @@ struct ((tm', b'_, v'_), unifiableIdem (g_, vhs_, (v'_, IntSyn.id))) and checkExact (g_, tm, vs_, location_msg) = + Display.(debug Form.( + nl () ++ + string "Checking exact term" ++ + shown show_term tm ++ + + nl () + )) ; begin if not !trace then let (tm', b'_, v'_), ok = checkExact1 (g_, tm, vs_) in begin if ok then (tm', b'_) diff --git a/src/modes/Modecheck.ml b/src/modes/Modecheck.ml index 77d4048..ac29b45 100644 --- a/src/modes/Modecheck.ml +++ b/src/modes/Modecheck.ml @@ -31,6 +31,8 @@ module MakeModeCheck exception Error of string open! struct + let print' s = Display.(debug Form.(string s)) + module I = IntSyn module M = ModeSyn module P = Paths @@ -61,6 +63,8 @@ module MakeModeCheck exception ModeError of P.occ * string exception Error' of P.occ * string + let modeError' (occ, msg) = Display.debug (Display.Form.string msg) ; raise (ModeError (occ, msg)) + let error'' (occ, msg) = Display.debug (Display.Form.string msg) ; raise (Error' (occ, msg)) let rec lookup (a, occ) = begin match ModeTable.mmodeLookup a with | [] -> @@ -514,7 +518,7 @@ module MakeModeCheck let rec checkDlocal (d_, v_, occ) = try checkD1 (d_, v_, occ, function d'_ -> [ d'_ ]) - with ModeError (occ, msg) -> raise (Error' (occ, msg)) + with ModeError (occ, msg) -> error'' ( (occ, msg)) let cidFromHead = function I.Const a -> a | I.Def a -> a @@ -547,7 +551,7 @@ module MakeModeCheck | [] -> () | I.Const c :: clist -> begin begin if !Global.chatter > 3 then - print (Names.qidToString (Names.constQid c) ^ " ") + print' (Names.qidToString (Names.constQid c) ^ " ") else () end; (try checkDlocal (I.Null, I.constType c, P.top) @@ -556,7 +560,7 @@ module MakeModeCheck end | I.Def d :: clist -> begin begin if !Global.chatter > 3 then - print (Names.qidToString (Names.constQid d) ^ " ") + print' (Names.qidToString (Names.constQid d) ^ " ") else () end; (try checkDlocal (I.Null, I.constType d, P.top) @@ -567,7 +571,7 @@ module MakeModeCheck let rec checkMode (a, ms) = let _ = begin if !Global.chatter > 3 then - print + print' (("Mode checking family " ^ Names.qidToString (Names.constQid a)) ^ ":\n") else () @@ -577,7 +581,7 @@ module MakeModeCheck let _ = checkFree := false in let _ = checkAll clist in let _ = - begin if !Global.chatter > 3 then print "\n" else () + begin if !Global.chatter > 3 then print' "\n" else () end in () @@ -585,7 +589,7 @@ module MakeModeCheck let rec checkFreeOut (a, ms) = let _ = begin if !Global.chatter > 3 then - print + print' (("Checking output freeness of " ^ Names.qidToString (Names.constQid a)) ^ ":\n") @@ -596,7 +600,7 @@ module MakeModeCheck let _ = checkFree := true in let _ = checkAll clist in let _ = - begin if !Global.chatter > 3 then print "\n" else () + begin if !Global.chatter > 3 then print' "\n" else () end in () diff --git a/src/modes/Modedec.ml b/src/modes/Modedec.ml index 438a4b9..ba7d6f4 100644 --- a/src/modes/Modedec.ml +++ b/src/modes/Modedec.ml @@ -41,6 +41,7 @@ module MakeModeDec () : MODEDEC = struct end in checkName' mS + | M.Mapp (M.Marg (_, None), mS) -> checkName mS let rec modeConsistent = function | M.Star, M.Plus -> false diff --git a/src/modes/modecheck.fun b/src/modes/modecheck.fun new file mode 100644 index 0000000..2d86f47 --- /dev/null +++ b/src/modes/modecheck.fun @@ -0,0 +1,868 @@ +(* Mode Checking *) +(* Author: Carsten Schuermann *) +(* Modified: Frank Pfenning, Roberto Virga *) + +functor ModeCheck ((*! structure IntSyn : INTSYN !*) + structure ModeTable : MODETABLE + (*! sharing ModeSyn.IntSyn = IntSyn !*) + structure Whnf : WHNF + (*! sharing Whnf.IntSyn = IntSyn !*) + structure Index : INDEX + (*! sharing Index.IntSyn = IntSyn !*) + (*! structure Paths : PATHS !*) + structure Origins : ORIGINS + (*! sharing Origins.Paths = Paths !*) + (*! sharing Origins.IntSyn = IntSyn !*) + ) + : MODECHECK = +struct + (*! structure IntSyn = IntSyn !*) + (*! structure ModeSyn = ModeSyn !*) + (*! structure Paths = Paths !*) + + exception Error of string + + local + structure I = IntSyn + structure M = ModeSyn + structure P = Paths + + datatype Uniqueness = (* Uniqueness information *) + Unique (* u ::= Unique *) + | Ambig (* | Ambig *) + + datatype Info = (* Groundedness information *) + Free (* I ::= Free *) + | Unknown (* | Unknown *) + | Ground of Uniqueness (* | Ground *) + + datatype Status = (* Variable status *) + Existential of (* S ::= Existential (I, xOpt) *) + Info * string option + | Universal (* | Universal *) + + + (* hack: if true, check freeness of output arguments in subgoals *) + val checkFree = ref false + + (* Representation invariant: + + Groundness information: + T stands for ground, B stands for unknown + Ex for a named existential variable + Par for a parameter + + Mode context D ::= . | D, S + + If D contains Status information for variables in + a context G, we write G ~ D + We write D' >= D if for all i + D(i) par iff D'(i) par + D(i) bot implies D'(i) bot or D'(i) top + D(i) top implies D'(i) top + *) + + (* copied from worldcheck/worldsyn.fun *) + fun wrapMsg (c, occ, msg) = + (case Origins.originLookup c + of (fileName, NONE) => (fileName ^ ":" ^ msg) + | (fileName, SOME occDec) => + (P.wrapLoc' (P.Loc (fileName, P.occToRegionClause occDec occ), + Origins.linesInfoLookup (fileName), + "Constant " ^ Names.qidToString (Names.constQid c) ^ "\n" ^ msg))) + + fun wrapMsg' (fileName, r, msg) = + P.wrapLoc (P.Loc (fileName, r), msg) + + exception ModeError of P.occ * string + exception Error' of P.occ * string + + (* lookup (a, occ) = mSs + + Invariant: mS are the argument modes for a + Raises an error if no mode for a has declared. + (occ is used in error message) + *) + fun lookup (a, occ) = + case ModeTable.mmodeLookup a + of nil => raise Error' (occ, "No mode declaration for " ^ I.conDecName (I.sgnLookup a)) + | sMs => sMs + + (* nameOf S, selects a name for S *) + fun nameOf (Existential (_, NONE)) = "?" + | nameOf (Existential (_, SOME name)) = name + | nameOf _ = "?" + + (* unique (k, ks) = B + + Invariant: + B iff k does not occur in ks + *) + fun unique (k, nil) = true + | unique (k, k'::ks) = + (k <> k') andalso unique (k, ks) + + (* isUniversal S = B + + Invariant: + B iff S = Par + *) + fun isUniversal Universal = true + | isUniversal _ = false + + (* isGround S = B + + Invariant: + B iff S = Ex (T x) + *) + fun isGround (Existential (Ground _, _)) = true + | isGround _ = false + + (* uniqueness S = u + where u is the uniqueness property of status S + *) + fun uniqueness (Existential (Ground (u), _)) = u + | uniqueness (Universal) = Unique + + (* ambiguate (mode) = mode' + where mode' forgets uniqueness properties + *) + fun ambiguate (M.Plus) = M.Plus + | ambiguate (M.Minus) = M.Minus + | ambiguate (M.Minus1) = M.Minus + + (* andUnique (u1, u2) = Unique if u1 = u2 = Unique + = Ambig otherwise + *) + fun andUnique (Unique, Unique) = Unique + | andUnique _ = Ambig + + (* isFree S = b + + Invariant: + b iff S = Ex (B x) + *) + fun isFree (Existential (Free, _)) = true + | isFree _ = false + + exception Eta + + (* etaContract (U, n) = k + + if lam V1... lam Vn. U =eta*=> k + otherwise raise exception Eta + + Invariant: G, V1,..., Vn |- U : V for some G, Vi, V. + U in NF + *) + fun etaContract (I.Root (I.BVar(k), S), n) = + if k > n + then ( etaSpine (S, n) ; k-n ) + else raise Eta + | etaContract (I.Lam (D, U), n) = + etaContract (U, n+1) + | etaContract _ = raise Eta + + (* etaSpine (S, n) = () + if S =eta*=> n ; n-1 ; ... ; 1 ; Nil + otherwise raise exception Eta + + Invariant: G |- S : V1 >> V2 for some G, V1, V2 + S in NF + *) + and etaSpine (I.Nil, 0) = () + | etaSpine (I.App (U, S), n) = + if etaContract (U, 0) = n + then etaSpine (S, n-1) + else raise Eta + (* S[s] should be impossible *) + + (* isPattern (D, k, mS) = B + + Invariant: + B iff k > k' for all k' in mS + and for all k in mS: k is parameter + and for all k', k'' in mS: k' <> k'' + *) + fun checkPattern (D, k, args, I.Nil) = () + | checkPattern (D, k, args, I.App (U, S)) = + (let + val k' = etaContract (U, 0) + in + if (k > k') + andalso isUniversal (I.ctxLookup (D, k')) + andalso unique (k', args) + then checkPattern (D, k, k'::args, S) + else raise Eta + end) + + fun isPattern (D, k, S) = + (checkPattern (D, k, nil, S); true) + handle Eta => false + + (* ------------------------------------------- strictness check *) + (* This repeats some code from ../typecheck/strict.fun *) + (* Interface here is somewhat different *) + + (* strictExpN (D, p, U) = B + + Invariant: + If D |- U : V + and U is in nf (normal form) + then B iff U is strict in p + *) + fun strictExpN (D, _, I.Uni _) = false + | strictExpN (D, p, I.Lam (_, U)) = + (* checking D in this case should be redundant -fp *) + (* strictDecN (D, p, D) orelse *) + strictExpN (I.Decl(D, Universal), p+1, U) + | strictExpN (D, p, I.Pi ((D', _), U)) = + strictDecN (D, p, D') orelse strictExpN (I.Decl(D, Universal), p+1, U) + | strictExpN (D, p, I.Root (H, S)) = + (case H + of (I.BVar (k')) => + if (k' = p) then isPattern (D, k', S) + else if isUniversal (I.ctxLookup (D, k')) then strictSpineN (D, p, S) + else false + (* equivalently: isUniversal .. andalso strictSpineN .. *) + | (I.Const (c)) => strictSpineN (D, p, S) + | (I.Def (d)) => strictSpineN (D, p, S) + | (I.FgnConst (cs, conDec)) => strictSpineN (D, p, S)) + (* no other cases possible *) + | strictExpN (D, p, I.FgnExp (cs, ops)) = false + (* this is a hack - until we investigate this further -rv *) + (* no other cases possible *) + + (* strictSpineN (D, S) = B + + Invariant: + If D |- S : V > W + and S is in nf (normal form) + then B iff S is strict in k + *) + and strictSpineN (_, _, I.Nil) = false + | strictSpineN (D, p, I.App (U, S)) = + strictExpN (D, p, U) orelse strictSpineN (D, p, S) + + and strictDecN (D, p, I.Dec (_, V)) = + strictExpN (D, p, V) + + (* + fun strictAtom (D, p, mode, I.Nil, (V, s), M.Mnil) = false + | strictAtom (D, p, M.Minus, I.App (U, S), (I.Pi ((I.Dec (_, V1), _), V2), s), + M.Mapp (M.Marg (M.Minus, _), mS)) = + strictExpN (D, p, Whnf.normalize (V1, s)) + orelse strictAtom (D, p, M.Minus, S, Whnf.whnfExpandDef (V2, I.Dot (I.Exp U, s)), mS) + | strictAtom (D, p, mode, I.App (U, S), (I.Pi (_, V2), s), M.Mapp(_, mS)) = + strictAtom (D, p, mode, S, Whnf.whnfExpandDef (V2, I.Dot (I.Exp U, s)), mS) + *) + + (* ------------------------------------------- freeness check *) + (* freeExpN (D, mode, U, occ = () + + If G |- U : V (U in nf) + and G ~ D + then freeExpN terminates with () if D |- U free + else exception ModeError is raised + + (occ and mode are used in error messages) + *) + fun freeExpN (D, d, mode, I.Root (I.BVar k, S), occ, strictFun) = + (freeVar (D, d, mode, k, P.head occ, strictFun); + freeSpineN (D, d, mode, S, (1, occ), strictFun)) + | freeExpN (D, d, mode, I.Root (I.Const _, S), occ, strictFun) = + freeSpineN (D, d, mode, S, (1, occ), strictFun) + | freeExpN (D, d, mode, I.Root (I.Def _, S), occ, strictFun) = + freeSpineN (D, d, mode, S, (1, occ), strictFun) + | freeExpN (D, d, mode, I.Root (I.FgnConst (cs, conDec), S), occ, strictFun) = + freeSpineN (D, d, mode, S, (1, occ), strictFun) + | freeExpN (D, d, mode, I.Lam (_, U), occ, strictFun) = + freeExpN (I.Decl (D, Universal), d+1, mode, U, P.body occ, strictFun) + | freeExpN (D, d, mode, I.FgnExp csfe, occ, strictFun) = + I.FgnExpStd.App.apply csfe (fn U => freeExpN (D, d, mode, Whnf.normalize (U, I.id), occ, strictFun)) + (* punting on the occ here - ak *) + + (* freeSpineN (D, mode, S, occ, strictFun) = () + + If G |- S : V1 > V2 (S in nf) + and G ~ D + then freeSpineN terminates with () if D |- S free + else exception ModeError is raised + + (occ and mode are used in error messages) + *) + and freeSpineN (D, d, mode, I.Nil, _, strictFun) = () + | freeSpineN (D, d, mode, I.App (U, S), (p, occ), strictFun) = + (freeExpN (D, d, mode, U, P.arg (p, occ), strictFun); + freeSpineN (D, d, mode, S, (p+1, occ), strictFun)) + + (* freeVar (D, mode, k, occ, strictFun) = () + + If G |- k : V1 + and G ~ D + then freeVar terminates with () if D |- S free + else exception ModeError is raised + + (occ and mode are used in error messages) + *) + + and freeVar (D, d, mode, k, occ, strictFun) = + let + val status = I.ctxLookup (D, k) + in + if isFree status orelse isUniversal status orelse strictFun (k-d) + then () + else raise ModeError (occ, "Occurrence of variable " ^ (nameOf status) ^ + " in " ^ (M.modeToString mode) ^ " argument not free") + end + + + (* -------------------------------- non-strict mode context update *) + (* nonStrictExpN (D, U) = D' + + If G |- U : V (U in nf) + and D ~ G + then D' >= D where D'(k) Unknown for all existential variables k + in U that are free in D + *) + fun nonStrictExpN (D, I.Root (I.BVar (k), S)) = + nonStrictSpineN (nonStrictVarD (D, k), S) + | nonStrictExpN (D, I.Root (I.Const c, S)) = + nonStrictSpineN (D, S) + | nonStrictExpN (D, I.Root (I.Def d, S)) = + nonStrictSpineN (D, S) + | nonStrictExpN (D, I.Root (I.FgnConst (cs, conDec), S)) = + nonStrictSpineN (D, S) + | nonStrictExpN (D, I.Lam (_, U)) = + I.ctxPop (nonStrictExpN (I.Decl (D, Universal), U)) + | nonStrictExpN (D, I.FgnExp _) = + raise Error ("Foreign expressions not permitted when checking freeness") + + (* nonStrictSpineN (D, S) = D' + + If G |- S : V1 > V2 (S in nf) + and D ~ G + then D' >= D' where D'(k) Unkown for all existential variables k + in S that are Free in D + *) + and nonStrictSpineN (D, I.Nil) = D + | nonStrictSpineN (D, I.App (U, S)) = + nonStrictSpineN (nonStrictExpN (D, U), S) + + (* nonStrictVarD (D, k) = D' + + If G |- k : V + and D ~ G + and k is an existential variable + then D' >= D where k is nonStrictd as described in nonStrictExpN + *) + and nonStrictVarD (I.Decl (D, Existential (Free, name)), 1) = + I.Decl (D, Existential (Unknown, name)) + | nonStrictVarD (D, 1) = (* Universal, or already Unknown or Ground - leave unchanged *) + D + | nonStrictVarD (I.Decl (D, status), k) = + I.Decl (nonStrictVarD (D, k-1), status) + + (* ------------------------------------------- mode context update *) + + (* updateExpN (D, U, u) = D' + + If G |- U : V (U in nf) + and D ~ G + then D' >= D where D'(k) Ground for all existential variables k + with a strict occurrence in U + and D'(k) Unkown for all existential variable k + with a non-strict occurrence, but no strict occurrence in U + (if !checkFree is true) + + u is the uniqueness property for the new ground assumptions + *) + fun updateExpN (D, I.Root (I.BVar (k), S), u) = + if isUniversal (I.ctxLookup (D, k)) + then updateSpineN (D, S, u) + else + if isPattern (D, k, S) + then updateVarD (D, k, u) + else if !checkFree + then nonStrictSpineN (nonStrictVarD (D, k), S) + else D + | updateExpN (D, I.Root (I.Const c, S), u) = + updateSpineN (D, S, u) + | updateExpN (D, I.Root (I.Def d, S), u) = + updateSpineN (D, S, u) + | updateExpN (D, I.Root (I.FgnConst (cs, conDec), S), u) = + updateSpineN (D, S, u) + | updateExpN (D, I.Lam (_, U), u) = + I.ctxPop (updateExpN (I.Decl (D, Universal), U, u)) + (* no occurrence inside a FgnExp is considered strict *) + | updateExpN (D, I.FgnExp _, u) = D + + (* updateSpineN (D, S, u) = D' + + If G |- S : V1 > V2 (S in nf) + and D ~ G + then D' >= D' where D'(k) Ground for all existential variables k + with a strict occurrence in S + *) + and updateSpineN (D, I.Nil, u) = D + | updateSpineN (D, I.App (U, S), u) = + updateSpineN (updateExpN (D, U, u), S, u) + + (* updateVarD (D, k, u) = D' + + If G |- k : V + and D ~ G + and k is an existential variable + then D' >= D where k is updated as described in updateExpN + *) + and updateVarD (I.Decl (D, Existential (_, name)), 1, u) = + I.Decl (D, Existential (Ground (u), name)) + | updateVarD (I.Decl (D, status), k, u) = + I.Decl (updateVarD (D, k-1, u), status) + + (* ----------------------- mode context update by argument modes *) + + (* updateAtom (D, m, S, mS, (p,occ)) = D' + + If G |- S : V > V' ( S = U1 ; .. ; Un) + and D ~ G + and S ~ mS (mS = m1 , .. , mn) + and m mode + then D' >= D where + all Ui are updated if mi = m (mod uniqueness) + + The new ground variables are marked Unique + if m = (-1) and mi = (-1) (when updating from subgoals with unique inputs) + or m = mi = (+) (when updating from the clause head) + Otherwise they are marked Ambig. + + (p,occ) is used in error message if freeness is to be checked + *) + fun updateAtom' (D, mode, I.Nil, M.Mnil, _) = D + | updateAtom' (D, M.Plus, I.App (U, S), M.Mapp (M.Marg (M.Plus, _), mS), (p, occ)) = + updateAtom' (updateExpN (D, U, Unique), M.Plus, S, mS, (p+1, occ)) + | updateAtom' (D, M.Minus, I.App (U, S), M.Mapp (M.Marg (M.Minus, _), mS), (p, occ)) = + updateAtom' (updateExpN (D, U, Ambig), M.Minus, S, mS, (p+1, occ)) + | updateAtom' (D, M.Minus, I.App (U, S), M.Mapp (M.Marg (M.Minus1, _), mS), (p, occ)) = + updateAtom' (updateExpN (D, U, Ambig), M.Minus, S, mS, (p+1, occ)) + | updateAtom' (D, M.Minus1, I.App (U, S), M.Mapp (M.Marg (M.Minus, _), mS), (p, occ)) = + updateAtom' (updateExpN (D, U, Ambig), M.Minus1, S, mS, (p+1, occ)) + | updateAtom' (D, M.Minus1, I.App (U, S), M.Mapp (M.Marg (M.Minus1, _), mS), (p, occ)) = + updateAtom' (updateExpN (D, U, Unique), M.Minus1, S, mS, (p+1, occ)) + (* when checking freeness, all arguments must be input (+) or output (-) *) + (* therefore, no case for M.Mapp (M.Marg (M.Minus, _), mS) is provided here *) + | updateAtom' (D, mode, I.App (U, S), M.Mapp (_, mS), (p, occ)) = + updateAtom' (D, mode, S, mS, (p+1, occ)) + + (* freeAtom (D, m, S, (V,s), mS, (p, occ)) = () + + checks if all output arguments in S according to mS are free. + Invariant: G |- S : V[s] >> P for some G and P (S in nf) + G ~ D + mode = (-) or (+); ( * ) or (-1) are excluded + *) + fun freeAtom (D, mode, I.Nil, Vs, M.Mnil, _) = () + | freeAtom (D, M.Minus, I.App (U, S), (I.Pi ((I.Dec (_, V1), _), V2), s), + M.Mapp (M.Marg (M.Minus, _), mS), (p, occ)) = + (freeExpN (D, 0, M.Minus, U, P.arg(p, occ), + (fn q => strictExpN (D, q, Whnf.normalize (V1, s)))); + freeAtom (D, M.Minus, S, Whnf.whnfExpandDef (V2, I.Dot (I.Exp U, s)), mS, (p+1, occ))) + | freeAtom (D, mode, I.App (U, S), (I.Pi (_, V2), s), M.Mapp (_, mS), (p, occ)) = + freeAtom (D, mode, S, Whnf.whnfExpandDef (V2, I.Dot (I.Exp U, s)), mS, (p+1, occ)) + + (* updateAtom (D, m, S, a, mS, (p, occ)) + see updateAtom', and performs additional freeness check if required + *) + fun updateAtom (D, mode, S, a, mS, (p, occ)) = + let + val _ = if !checkFree + then freeAtom (D, ambiguate mode, S, (I.constType a, I.id), mS, (p, occ)) + else () + in + updateAtom' (D, mode, S, mS, (p, occ)) + end + + (* ------------------------------------------- groundness check *) + + (* groundExpN (D, mode, U, occ) = u + + If G |- U : V (U in nf) + and G ~ D + then if mode = (+) or (-) + then groundExpN terminates with u if D |- U ground + else exception ModeError is raised + if mode = (-1) then D |- U ground and U unique + else exception ModeError is raised + + u = Unique if all known variables in U are Unique + u = Ambig otherwise + + (occ and mode are used in error messages) + *) + fun groundExpN (D, mode, I.Root (I.BVar k, S), occ) = + andUnique (groundVar (D, mode, k, P.head occ), + groundSpineN (D, mode, S, (1, occ))) + | groundExpN (D, mode, I.Root (I.Const c, S), occ) = + groundSpineN (D, mode, S, (1, occ)) + | groundExpN (D, mode, I.Root (I.Def d, S), occ) = + groundSpineN (D, mode, S, (1, occ)) + | groundExpN (D, mode, I.Root (I.FgnConst (cs, conDec), S), occ) = + groundSpineN (D, mode, S, (1, occ)) + | groundExpN (D, mode, I.Lam (_, U), occ) = + groundExpN (I.Decl (D, Universal), mode, U, P.body occ) + | groundExpN (D, mode, I.FgnExp csfe, occ) = + I.FgnExpStd.fold csfe (fn (U,u) => andUnique (groundExpN (D, mode, Whnf.normalize (U, I.id), occ), u)) Unique + (* punting on occ here - ak *) + + (* groundSpineN (D, mode, S, occ) = u + + If G |- S : V1 > V2 (S in nf) + and G ~ D + then if mode = (+) or (-) + then groundSpineN terminates with u if D |- S ground + else exception ModeError is raised + if mode = (-1) then D |- S ground and S unique + else exception ModeError is raised + + u = Unique if all known variables in S are Unique + u = Ambig otherwise + + (occ and mode are used in error messages) + *) + and groundSpineN (D, mode, I.Nil, _) = Unique + | groundSpineN (D, mode, I.App (U, S), (p, occ)) = + andUnique (groundExpN (D, mode, U, P.arg (p, occ)), + groundSpineN (D, mode, S, (p+1, occ))) + + (* groundVar (D, mode, k, occ) = u + + If G |- k : V1 + and G ~ D + then if mode = (+) or (-) + then groundVar terminates with u if D |- k ground + else exception ModeError is raised + if mode = (-1) then D |- k ground and k unique + else exception ModeError is raised + + u = Unique if k is known to be unique, Ambig otherwise + + (occ and mode are used in error messages) + *) + and groundVar (D, M.Minus1, k, occ) = + (case I.ctxLookup (D, k) + of Existential (Ground (Unique), _) => Unique + | Universal => Unique + | s as Existential (Ground (Ambig), x) => + raise ModeError (occ, "Occurrence of variable " ^ nameOf s + ^ " in " ^ M.modeToString M.Minus1 + ^ " argument not necessarily unique") + | s => (* Existential (Free, _) or Existential (Unknown, _) *) + raise ModeError (occ, "Occurrence of variable " ^ (nameOf s) + ^ " in " ^ (M.modeToString M.Minus1) + ^ " argument not necessarily ground")) + | groundVar (D, mode, k, occ) = + let + val status = I.ctxLookup (D, k) + in + if isGround status orelse isUniversal status + then uniqueness status + else raise ModeError (occ, "Occurrence of variable " ^ (nameOf status) ^ + " in " ^ (M.modeToString mode) + ^ " argument not necessarily ground") + end + + (* ------------------------------------------- groundness check by polarity *) + + (* groundAtom (D, m, S, mS, (p,occ)) = u + + If G |- S : V > V' ( S = U1 ; .. ; Un) + and D ~ G + and S ~ mS (mS = m1 , .. , mn) + and m mode = (+) or (-1) + then groundAtom returns u if D |- Ui ground + for all i s.t. mi = m (mod uniqueness) + and checks that D |- Ui unique if mi = (-1) and m = (-) + otherwise exception ModeError is raised + + u = Unique if all mi = m (mod uniqueness) are unique, + u = Ambig otherwise + + ((p,occ) used in error messages) + *) + fun groundAtom (D, _, I.Nil, M.Mnil, _) = Unique + | groundAtom (D, M.Plus, I.App (U, S), M.Mapp (M.Marg (M.Plus, _), mS), (p, occ)) = + andUnique (groundExpN (D, M.Plus, U, P.arg (p, occ)), + groundAtom (D, M.Plus, S, mS, (p+1, occ))) + | groundAtom (D, M.Minus, I.App (U, S), M.Mapp (M.Marg (M.Minus, _), mS), (p, occ)) = + (groundExpN (D, M.Minus, U, P.arg (p, occ)); (* ignore uniqueness result here *) + groundAtom (D, M.Minus, S, mS, (p+1, occ))) + | groundAtom (D, M.Minus, I.App (U, S), M.Mapp (M.Marg (M.Minus1, _), mS), (p, occ)) = + (groundExpN (D, M.Minus1, U, P.arg (p, occ)); (* ignore uniqueness result here *) + groundAtom (D, M.Minus, S, mS, (p+1, occ))) + | groundAtom (D, mode, I.App (U, S), M.Mapp (_, mS), (p, occ)) = + groundAtom (D, mode, S, mS, (p+1, occ)) + + + (* ------------------------------------------- mode checking first phase *) + + (* ctxPush (Ds, m) = Ds' + raises the contexts Ds prepending m + *) + fun ctxPush (m, Ds) = List.map (fn D => I.Decl (D, m)) Ds + + (* ctxPop Ds = Ds' + lowers the contexts Ds + *) + fun ctxPop Ds = List.map (fn I.Decl (D, m) => D) Ds + + (* checkD1 (D, V, occ, k) = () + + Invariant: + if G |- V : L + and V does not contain Skolem constants + and D ~ G + then + for each mode mS of the head of V + exists some Di s.t. all (-) evars of mS are ground + where D' ~ G, D' >= D is obtained by updating D + and k D' = [D1, ..., Di, ..., Dn] + and Di ~ G, Di >= D' is obtained by mode checking on the subgoals of V + + exception ModeError is raised if the expression does not mode check + exception Error' is raised if the expression contains type families + that have no mode information associated with them + (occ used in error messages) + *) + fun checkD1 (D, I.Pi ((I.Dec (name, _), I.Maybe), V), occ, k) = + checkD1 (I.Decl (D, Existential (Free, name)), V, P.body occ, + fn (I.Decl (D', m)) => ctxPush (m, k D')) + | checkD1 (D, I.Pi ((I.Dec (name, V1), I.No), V2), occ, k) = + checkD1 (I.Decl (D, Existential (Free, name)), V2, P.body occ, + fn (I.Decl (D', m)) => ctxPush (m, checkG1 (D', V1, P.label occ, k))) + | checkD1 (D, I.Root (I.Const a, S), occ, k) = + let + (* for a declaration, all modes must be satisfied *) + fun checkAll nil = () + | checkAll (mS :: mSs) = + let + fun checkSome [D'] = + (* D' is the only (last) possibility; on failure, we raise ModeError *) + ( + groundAtom (D', M.Minus, S, mS, (1, occ)); (* ignore return *) + checkAll mSs + ) + | checkSome (D' :: Ds) = + (* try D', if it doesn't work, try another context in the Ds *) + ( + ((groundAtom (D', M.Minus, S, mS, (1, occ));()) (* ignore return *) + handle ModeError _ => checkSome Ds); + checkAll mSs + ) + in + checkSome (k (updateAtom (D, M.Plus, S, a, mS, (1, occ)))) + end + in + checkAll (lookup (a, occ)) + end + | checkD1 (D, I.Root (I.Def d, S), occ, k) = + let + (* for a declaration, all modes must be satisfied *) + fun checkAll nil = () + | checkAll (mS :: mSs) = + let + fun checkSome [D'] = + (* D' is the only (last) possibility; on failure, we raise ModeError *) + ( + groundAtom (D', M.Minus, S, mS, (1, occ)); (* ignore return *) + checkAll mSs + ) + | checkSome (D' :: Ds) = + (* try D', if it doesn't work, try another context in the Ds *) + ( + ((groundAtom (D', M.Minus, S, mS, (1, occ)); ()) (* ignore return *) + handle ModeError _ => checkSome Ds); + checkAll mSs + ) + in + checkSome (k (updateAtom (D, M.Plus, S, d, mS, (1, occ)))) + end + in + checkAll (lookup (d, occ)) + end + + (* checkG1 (D, V, occ, k) = Ds + + Invariant: + if G |- V : L + and V does not contain Skolem constants + and D ~ G + then forall D' >= D that mode checks V, (k D') is a sublist of Ds + and for each Di in Ds, Di ~ G and Di >= D' + + exception ModeError is raised if the expression does not mode check + exception Error' is raised if the expression contains type families + that have no mode information associated with them + (occ used in error messages) + *) + and checkG1 (D, I.Pi ((_, I.Maybe), V), occ, k) = + ctxPop (checkG1 (I.Decl (D, Universal), V, P.body occ, + fn (I.Decl (D', m)) => ctxPush (m, k D'))) + | checkG1 (D, I.Pi ((I.Dec (_, V1) , I.No), V2), occ, k) = + ctxPop (checkD1 (D, V1, P.label occ, fn D' => [D']); + checkG1 (I.Decl (D, Universal), V2, P.body occ, + fn (I.Decl (D', m)) => ctxPush (m, k D'))) + | checkG1 (D, I.Root (I.Const a, S), occ, k) = + let + (* for a goal, at least one mode must be satisfied *) + fun checkList found nil = nil (* found = true *) + | checkList false [mS] = + (* mS is the last possible mode to check; + if the check fails, we don't catch ModeError *) + ( + case groundAtom (D, M.Plus, S, mS, (1, occ)) + of Unique => k (updateAtom (D, M.Minus1, S, a, mS, (1, occ))) + | Ambig => k (updateAtom (D, M.Minus, S, a, mS, (1, occ))) + ) + | checkList found (mS :: mSs) = + (* uniqueness not permitted on multiple modes right now *) + (* Wed Aug 20 21:52:31 2003 -fp *) + let + (* found' is true iff D satisfies mS *) + val found' = ((groundAtom (D, M.Plus, S, mS, (1, occ)); true) (* handler scope??? -fp *) + handle ModeError _ => false) + (* compute all other mode contexts *) + val Ds' = checkList (found orelse found') mSs + in + if found' + then k (updateAtom (D, M.Minus, S, a, mS, (1, occ))) @ Ds' + else Ds' + end + in + checkList false (lookup (a, occ)) + end + | checkG1 (D, I.Root (I.Def d, S), occ, k) = + let + (* for a goal, at least one mode must be satisfied *) + fun checkList found nil = nil (* found = true *) + | checkList false [mS] = + (* mS is the last possible mode to check; + if the check fails, we don't catch ModeError *) + ( + case groundAtom (D, M.Plus, S, mS, (1, occ)) + of Unique => k (updateAtom (D, M.Minus1, S, d, mS, (1, occ))) + | Ambig => k (updateAtom (D, M.Minus, S, d, mS, (1, occ))) + ) + | checkList found (mS :: mSs) = + (* uniqueness not permitted on multiple modes right now *) + (* Wed Aug 20 21:52:31 2003 -fp *) + let + (* found' is true iff D satisfies mS *) + val found' = ((groundAtom (D, M.Plus, S, mS, (1, occ)); true) + handle ModeError _ => false) + (* compute all other mode contexts *) + val Ds' = checkList (found orelse found') mSs + in + if found' + then k (updateAtom (D, M.Minus, S, d, mS, (1, occ))) @ Ds' + else Ds' + end + in + checkList false (lookup (d, occ)) + end + + (* checkDlocal (D, V, occ) = () + + Invariant: + If G |- V : L + and D ~ G + then checkD terminates with () iff V is mode correct. + + otherwise exception ModeError is raised (occ used in error messages) + *) + fun checkDlocal (D, V, occ) = + (checkD1 (D, V, occ, fn D' => [D']) + handle ModeError (occ, msg) => raise Error' (occ, msg)) + + (* --------------------------------------------------------- mode checking *) + + + fun cidFromHead (I.Const a) = a + | cidFromHead (I.Def a) = a + + (* checkD (ConDec, occOpt) = () + + checkD terminates with () if ConDec is mode correct + otherwise exception Error is raised + + (occOpt is used in error messages) + *) + fun checkD (conDec, fileName, occOpt) = + let + val _ = (checkFree := false) + fun checkable (I.Root (Ha, _)) = + (case (ModeTable.mmodeLookup (cidFromHead Ha)) + of nil => false + | _ => true) + | checkable (I.Uni _) = false + | checkable (I.Pi (_, V)) = checkable V + val V = I.conDecType conDec + in + if (checkable V) + then checkDlocal (I.Null, V, P.top) + handle Error' (occ, msg) => + (case occOpt + of NONE => raise Error (msg) + | SOME occTree => + raise Error (wrapMsg' (fileName, P.occToRegionClause occTree occ, msg))) + else () + end + + fun checkAll (nil) = () + | checkAll (I.Const(c) :: clist) = + (if !Global.chatter > 3 + then print (Names.qidToString (Names.constQid c) ^ " ") + else (); + checkDlocal (I.Null, I.constType c, P.top) + handle Error' (occ, msg) => raise Error (wrapMsg (c, occ, msg)); + checkAll clist) + | checkAll (I.Def(d) :: clist) = + (if !Global.chatter > 3 + then print (Names.qidToString (Names.constQid d) ^ " ") + else (); + checkDlocal (I.Null, I.constType d, P.top) + handle Error' (occ, msg) => raise Error (wrapMsg (d, occ, msg)); + checkAll clist) + + fun checkMode (a, ms) = + let + val _ = if !Global.chatter > 3 + then print ("Mode checking family " ^ Names.qidToString (Names.constQid a) ^ ":\n") + else () + val clist = Index.lookup a + val _ = (checkFree := false) + val _ = checkAll clist + val _ = if !Global.chatter > 3 then print "\n" else () + in + () + end + + fun checkFreeOut (a, ms) = + let + val _ = if !Global.chatter > 3 + then print ("Checking output freeness of " ^ Names.qidToString (Names.constQid a) ^ ":\n") + else () + val clist = Index.lookup a + val _ = (checkFree := true) + val _ = checkAll clist + val _ = if !Global.chatter > 3 then print "\n" else () + in + () + end + + in + val checkD = checkD + val checkMode = checkMode + val checkFreeOut = checkFreeOut + end +end; (* functor ModeCheck *) + diff --git a/src/msg/Msg_.ml b/src/msg/Msg_.ml index d9f8e36..38c83b2 100644 --- a/src/msg/Msg_.ml +++ b/src/msg/Msg_.ml @@ -8,7 +8,7 @@ open Basis include Msg_intf module Msg : MSG = struct - let default = print + let default m = Display.debug (Display.string m) let messageFunc = ref default let setMessageFunc f = messageFunc := f let message s = ( ! ) messageFunc s diff --git a/stelf-book b/stelf-book new file mode 160000 index 0000000..3b3cd17 --- /dev/null +++ b/stelf-book @@ -0,0 +1 @@ +Subproject commit 3b3cd17d0f8cd32048c16ffad0896734d2d1a4af diff --git a/stelf.opam b/stelf.opam index eafb809..19bb1a9 100644 --- a/stelf.opam +++ b/stelf.opam @@ -13,18 +13,25 @@ doc: "https://github.com/standardocaml/stelf/wiki" bug-reports: "https://github.com/standardocaml/stelf/issues" depends: [ "dune" {>= "3.21"} - "ocaml" {>= "5.0.0" & < "5.4"} + "ocaml" {>= "5.0.0" & !(< "5.5.0" & >= "5.4.0")} "lwt" "containers" "base" "angstrom" "alcotest" {with-test} "alcotest-lwt" {with-test} - "bisect-ppx" {with-dev} + "bisect-ppx" {dev} "odoc" {with-doc} - "bisect_ppx" {with-dev} + "sherlodoc" {with-doc} + "bisect_ppx" {dev} + "ocamlformat" {dev} + "ocaml-lsp-server" {dev} + "user-setup" {dev} + "merlin" {dev} + "ppx_deriving" "lambda-term" "fmt" + "bos" "cmdliner" "logs" ] diff --git a/test/Pal/Cases.ml b/test/Pal/Cases.ml index 32f42d9..b691850 100644 --- a/test/Pal/Cases.ml +++ b/test/Pal/Cases.ml @@ -60,5 +60,25 @@ let cases () = Source.zf_high; ]; ] ); + ( "FOL", Source.[ + test "FOL" [ fol1 ]; + test "FOL2" [ fol1; fol2 ] ; + test "FOL3.1" [ fol1; fol2; fol3_1 ]; + test "FOL3.2.1" [ fol1; fol2; fol3_1; fol3_2_1 ]; + test "FOL3.2.2" [ fol1; fol2; fol3_1; fol3_2_1; fol3_2_2 ]; + test "FOL3.2.3" [ fol1; fol2; fol3_1; fol3_2_1; fol3_2_2; fol3_2_3 ]; + test "FOL3.2.*" [ fol1; fol2; fol3_1; fol3_2 ]; + test "FOL3.3" [ fol1; fol2; fol3_1; fol3_2; fol3_3 ]; + test "FOL3.4" [ fol1; fol2; fol3_1; fol3_2; fol3_3; fol3_4 ]; + test "FOL3.*" [ fol1; fol2; fol3 ]; + test "FOL4.1" [ fol1; fol2; fol3; fol4_1 ]; + test "FOL4.2" [ fol1; fol2; fol3; fol4_1; fol4_2 ]; + test "FOL4.*" [ fol1; fol2; fol3; fol4 ]; + test "FOL5" [ fol1; fol2; fol3; fol4; fol5 ]; + test "FOL6" [ fol1; fol2; fol3; fol4; fol5; fol6 ] + ] ); + ( "S4", [ test "S4" [ Source.js4 ] ] ); + ( "LAM", [ test "LAM" [ Source.lam ] ] ); + ( "POLYLAM", [ test "POLYLAM" [ Source.polylam ] ] ) ] - end + end ~verbose:false diff --git a/test/Pal/Common.ml b/test/Pal/Common.ml index 50a8e88..de3f45e 100644 --- a/test/Pal/Common.ml +++ b/test/Pal/Common.ml @@ -10,11 +10,12 @@ let test ?(skip = false) ?(failure = false) (name : string) (cmds : string list) : unit Alcotest.test_case = let () = Printexc.record_backtrace true in let () = Logs.set_reporter (Logs_fmt.reporter ()) in - let () = Logs.set_level (Some Logs.Debug) in + let () = Logs.set_level ~all:false (Some Logs.Debug) in + let () = Fmt_tty.setup_std_outputs () in let () = Display.register (fun m -> - Lwt.return @@ prerr_endline - @@ Fmt.to_to_string Display.Info.Form.fmt m.msg) + let _ = (Display.fmt Fmt.stdout m.msg ) in + Lwt.return ()) in Alcotest.test_case name `Slow (fun () -> if skip then Alcotest.skip () diff --git a/test/Pal/Source.ml b/test/Pal/Source.ml index 2f814eb..509ba55 100644 --- a/test/Pal/Source.ml +++ b/test/Pal/Source.ml @@ -52,18 +52,444 @@ let zf_high = %def subset ({_ set} {_ set} prop) [x][y]all[z] imp (in z x) (in z y) %. %def disjoint ({_ set} {_ set} prop) [x][y] eq (inter x y) empty %. %def omega_closed ({_ set} prop) [x] and (in empty x) (all [n] imp (in n x) (in (succ n) x)) %. -%term extensionality pf (iff (eq X Y) (all[z] iff (in z X) (in z Y))) %. -%term foundation pf (ex([z] and (in z X) (disjoint z X))) %. -%term emtpy_ax pf (not (in X empty)) %. -%term double_ax pf (iff (in Z (double X Y)) (or (in Z X) (in Z Y))) %. -%term union_ax pf (iff (in Z (unions X)) (ex[y] and (in Z y) (in y X))) %. -%term powerset_ax pf (iff (in Z (powerset X)) (subset Z X)) %. -%term replace_ax pf (iff (in Z (replace X F)) (ex[y] and (in y X) (eq Z (F y)))) %. +%term extensionality pf (iff (eq _X _Y) (all[z] iff (in z _X) (in z _Y))) %. +%term foundation pf (ex([z] and (in z _X) (disjoint z _X))) %. +%term emtpy_ax pf (not (in _X empty)) %. +%term double_ax pf (iff (in _Z (double _X _Y)) (or (in _Z _X) (in _Z _Y))) %. +%term union_ax pf (iff (in _Z (unions _X)) (ex[y] and (in _Z y) (in y _X))) %. +%term powerset_ax pf (iff (in _Z (powerset _X)) (subset _Z _X)) %. +%term replace_ax pf (iff (in _Z (replace _X F)) (ex[y] and (in y _X) (eq _Z (F y)))) %. %term omega_ax pf (and (omega_closed omega) (all[o] imp (omega_closed o) (subset omega o))) %. %term choice_ax pf -(imp (all[y1] imp (in y1 X) - (all[y2] imp (in y2 X) (disjoint y1 y2))) - (ex [x'](all[y] imp (in y X) +(imp (all[y1] imp (in y1 _X) + (all[y2] imp (in y2 _X) (disjoint y1 y2))) + (ex [x'](all[y] imp (in y _X) (ex_unique ([y'] (and (in y' x') (in y' y))))))) %. |} + +let fol1 = + {| +%. First-Order Logic +Fragment with implication, negation, universal quantification. + +Author: Frank Pfenning + +This code is from the article + + Logical Frameworks + Handbook of Automated Reasoning + Alan Robinson and Andrei Voronkov, editors + Chapter 16 + Elsevier Science and MIT Press + In preparation + +%sort i +%sort o + +Formulas + +%term imp {_ o} {_ o} o +%prec %right 10 imp +%term not {_ o} o +%prec %prefix 12 not +%term forall %-> (%-> i o) o %. + +Natural deductions + +%sort nd {_ o} + +%term impi {{A B}} {_ {_ nd A} nd B} nd (A imp B) +%term impe {{A B}} {_ nd (A imp B)} {_ nd A} nd B + +%term noti {{A}} {_ {p o} {_ nd A} nd p} nd (not A) +%term note {{A}} {_ nd (not A)} {C o} {_ nd A} nd C + +%term foralli {{A}} {_ {a i} nd (A a)} nd (forall A) +%term foralle {{A}} {_ nd (forall A)} {T i} nd (A T) + +%block lnd [A o] {u nd A} +%block lo {_ o} +%block li {_ i} + +%worlds (lnd lo li) (nd A) +|} + +let fol2 = + {| + +%. Example + + +%. Hilbert deductions + +%sort hil {_ o} + +%term k {{A B}} hil (A imp (B imp A)) +%term s {{A B C}} hil ((A imp (B imp C)) imp ((A imp B) imp (A imp C))) + +%term n1 {{A B}} hil ((A imp (not B)) imp ((A imp B) imp (not A))) +%term n2 {{A B}} hil ((not A) imp (A imp B)) + +%term f1 {{A}} {T i} hil ((forall [x i] A x) imp (A T)) +%term f2 {{A B}} hil ((forall [x i] (B imp A x)) imp (B imp forall [x i] A x)) + +%term mp {{A B}} {_ hil (A imp B)} {_ hil A} hil B +%term ug {{A}} {_ {a i} hil (A a)} hil (forall [x i] A x) + +%worlds (li) (hil A) + +|} + +let fol3_1 = + {| + +%. Local reductions + +%sort ==>R {_ nd _A} {_ nd _A} +%prec %none 14 ==>R + +|} + +let fol3_2_1 = {| +%term redl_imp {{A D E}} (impe (impi [u nd A] D u) E) ==>R (D E) + +|} + +let fol3_2_2 = + {| +%term redl_not {{A C D E}} (note (noti [p o] [u nd A] D p u) C E) ==>R (D C E) + +|} + +let fol3_2_3 = + {| +%term redl_forall {{D T}} (foralle (foralli [a i] D a) T) ==>R (D T) + +|} + +let fol3_2 = fol3_2_1 ^ fol3_2_2 ^ fol3_2_3 + +let fol3_3 = + {| + +%. Local expansions + +%sort ==>E {_ nd A} {_ nd A} +%prec %none 14 ==>E +|} + +let fol3_4 = + {| + +%term expl_imp {{A B}} {D nd (A imp B)} D ==>E (impi [u nd A] impe D u) +%term expl_not {{A}} {D nd (not A)} D ==>E (noti [p] [u nd A] note D p u) +%term expl_forall {{A}} {D nd (forall A)} D ==>E (foralli [a] foralle D a) + +|} + +let fol3 = fol3_1 ^ fol3_2 ^ fol3_3 ^ fol3_4 + +let fol4_1 = + {| +%. Sequent calculus search result + +%def dn (nd (A imp not not A)) (impi [u nd A] noti [p] [w] note w p u) + +%. Translating Hilbert derivations to natural deductions + +%sort hilnd {_ hil A} {_ nd A} + +%term hnd_k {{A B}} hilnd k (impi [u nd A] impi [v nd B] u) +%term hnd_s {{A B C}} hilnd s (impi [u nd (A imp B imp C)] impi [v] impi [w nd A] impe (impe u w) (impe v w)) + +%term hnd_n1 {{A B}} hilnd n1 (impi [u] impi [v] noti [p] [w] note (impe u w) p (impe v w)) + +%term hnd_n2 {{A B}} hilnd n2 (impi [u] impi [v] note u B v) + +%term hnd_f1 {{A}} {T i} hilnd (f1 T) (impi [u] foralle u T) + +%term hnd_f2 {{A B}} hilnd f2 (impi [u] impi [v] foralli [a] impe (foralle u a) v) + +%term hnd_mp {{A B}} {H1 hil (A imp B)} {H2 hil A} {D1 nd (A imp B)} {D2 nd A} {_ hilnd H1 D1} {_ hilnd H2 D2} hilnd (mp H1 H2) (impe D1 D2) + +%term hnd_ug {{A}} {H1 {a i} hil (A a)} {D1 {a i} nd (A a)} {_ {a i} hilnd (H1 a) (D1 a)} hilnd (ug H1) (foralli D1) +|} +(* %mode {%in x hil _A} {%out y nd _A} hilnd x y *) +let fol4_2 = {| +%mode hilnd %in %out +%worlds (li) (hilnd H D) +%terminates H (hilnd H _) +%covers hilnd %in %out +%total H (hilnd H _) + +%? hilnd (mp (mp s k) k) D + +%def _ (nd (A imp A)) + (impe + (impe + (impi [u] + impi [v] + impi [w] impe (impe u w) (impe v w)) + (impi [u] impi [v] u)) + (impi [u] impi [v] u)) + + |} + +let fol4 = fol4_1 ^ fol4_2 +let fol5 = + {| + +The deduction theorem for Hilbert derivations + +%sort ded {_ {_ hil A} hil B} {_ hil (A imp B)} + +%term ded_id {{A}} ded ([u] u) (mp (mp s k) k) + +%term ded_k {{A B}} ded ([u] k) (mp k k) +%term ded_s {{A B C}} ded ([u] s) (mp k s) + +%term ded_n1 {{A B}} ded ([u] n1) (mp k n1) +%term ded_n2 {{A B}} ded ([u] n2) (mp k n2) + +%term ded_f1 {{A B T}} ded ([u] f1 T) (mp k (f1 T)) +%term ded_f2 {{A B}} ded ([u] f2) (mp k f2) + +%term ded_mp {{A B C}} {H1 {_ hil A} hil (B imp C)} {H2 {_ hil A} hil B} {H1' hil (A imp (B imp C))} {H2' hil (A imp B)} {_ ded H1 H1'} {_ ded H2 H2'} ded ([u] mp (H1 u) (H2 u)) (mp (mp s H1') H2') + +%term ded_ug {{A B}} {H1 {_ hil A} {_ i} hil _} {H1' {_ i} hil _} {_ {a i} ded ([u] H1 u a) (H1' a)} ded ([u] ug (H1 u)) (mp f2 (ug H1')) + +%block lded [A o] {u nd A} {v hil A} {h {C o} ded ([w] v) (mp k v)} + +%mode ded %in %out +%worlds (li lo lded) (ded H H') +%terminates H (ded H _) +%covers ded %in %out +%total H (ded H _) + +|} + +let fol6 = + {| + +Mapping natural deductions to Hilbert derivations. + +%sort ndhil {_ nd A} {_ hil A} + +%term ndh_impi {{A1 B}} {D1 {_ nd A1} nd B} {H1 {_ hil A1} hil B} {H1' hil (A1 imp B)} {_ ded H1 H1'} {_ {u nd A1} {v hil A1} {_ {C o} ded ([w] v) (mp k v)} {_ ndhil u v} ndhil (D1 u) (H1 v)} ndhil (impi D1) H1' + +%term ndh_impe {{A B}} {D1 nd (A imp B)} {D2 nd A} {H1 hil (A imp B)} {H2 hil A} {_ ndhil D1 H1} {_ ndhil D2 H2} ndhil (impe D1 D2) (mp H1 H2) + +%term ndh_noti {{A1}} {D1 {_ o} {_ nd A1} nd _} {H1 {_ o} {_ hil A1} hil _} {H1' hil _} {H1'' hil _} {_ ded (H1 A1) H1''} {_ ded (H1 (not A1)) H1'} {_ {p o} {u nd A1} {v hil A1} {_ {C o} ded ([w] v) (mp k v)} {_ ndhil u v} ndhil (D1 p u) (H1 p v)} ndhil (noti D1) (mp (mp n1 H1') H1'') + +%term ndh_note {{A C}} {D1 nd (not A)} {D2 nd A} {H1 hil (not A)} {H2 hil A} {_ ndhil D1 H1} {_ ndhil D2 H2} ndhil (note D1 C D2) (mp (mp n2 H1) H2) + +%term ndh_foralli {{A}} {D1 {_ i} nd _} {H1 {_ i} hil _} {_ {a i} ndhil (D1 a) (H1 a)} ndhil (foralli D1) (ug H1) + +%term ndh_foralle {{A}} {T i} {D1 nd (forall A)} {H1 hil (forall A)} {_ ndhil D1 H1} ndhil (foralle D1 T) (mp (f1 T) H1) + +%mode ndhil %in %out +%block lndhil [A o] {u nd A} {v hil A} {h {C o} ded ([w] v) (mp k v)} {nh ndhil u v} +%worlds (li lo lndhil) (ndhil D H) +%terminates D (ndhil D _) +%covers ndhil %in %out +%total D (ndhil D _) + + +|} + +let js4 = + {| +Judgmental S4 +[A judgmental reconstruction of modal logic, F.Pfenning and R.Davies, + MSCS 11:511-540, 2001] + +Representation with intrinsic types, worlds, +but not a Kripke semantics + +Idea: Translate the judgment + u1::B1,...,un::Bn ; x1:A1,...,xm:Am |- J +as + u1:{W'}tm B1 W',...,un:{W'}tm Bn W', ---, + x1:tm A1 W,...,xm:tm Am W |- J* + +where "---" are assumptions y:tm B W' for W' <> W +and if J = (M : A) then J* = M* : tm A W + if J = (E - A) then J* = E* : exp A W + +%sort tp +%term => {_ tp} {_ tp} tp +%term box {_ tp} tp +%term dia {_ tp} tp +%prec %right 10 => + +%sort world + +%sort tm {_ tp} {_ world} +%sort exp {_ tp} {_ world} + +%term lam {A tp} {B tp} {W world} {_ {_ tm A W} tm B W} tm (A => B) W +%term app {A tp} {B tp} {W world} {_ tm (A => B) W} {_ tm A W} tm B W +%term boxi {A tp} {W world} {_ {w world} tm A w} tm (box A) W +%term boxe {A tp} {C tp} {W world} {_ tm (box A) W} {_ {_ {W' world} tm A W'} tm C W} tm C W +%term t2e {A tp} {W world} {_ tm A W} exp A W +%term diai {A tp} {W world} {_ exp A W} tm (dia A) W +%term diae {A tp} {C tp} {W world} {_ tm (dia A) W} {_ {w world} {_ tm A w} exp C w} exp C W +%term boxep {A tp} {C tp} {W world} {_ tm (box A) W} {_ {_ {W' world} tm A W'} exp C W} exp C W + +%sort subdia {_ exp A W} {_ {w world} {_ tm A w} exp C w} {_ exp C W} +%mode subdia %in %in %out + +%term sdt2e {A tp} {C tp} {W world} {M tm A W} {F {w world} {_ tm A w} exp C w} subdia (t2e M) ([w] [x] F w x) (F W M) + +%term sddiae {A tp} {B tp} {C tp} {W world} {M tm (dia A) W} {E {_ world} {_ tm A _} exp B _} {F {_ world} {_ tm B _} exp C _} {F' {_ world} {_ tm A _} exp C _} {_ {v world} {y tm A v} subdia (E v y) ([w] [x] F w x) (F' v y)} subdia (diae M [v] [y] E v y) ([w] [x] F w x) (diae M [v] [y] F' v y) + +%term sdboxep {A tp} {B tp} {C tp} {W world} {M tm (box A) W} {E {_ {_ world} tm B _} exp C _} {F {_ world} {_ tm A _} exp C _} {F' {_ {_ world} tm B _} exp C _} {_ {u {V world} tm B V} subdia (E u) ([w] [x] F w x) (F' u)} subdia (boxep M [u] E u) ([w] [x] F w x) (boxep M [u] F' u) + +%block by {B tp} {v world} {y tm B v} +%block bu {B tp} {u {V world} tm B V} +%worlds (by bu) (subdia E F F') +%total E (subdia E F _) %. + +This does not work, unfortunately: +The "str" strengthening lemma would require handling cases like strlam where a bound +variable y2 of type tm C2 w appears free in the conclusion, but w is also quantified +by the str family. The main issue is that the case for str ([x][w] y2) ([w] y2) cannot +be typed because y2 has type tm C2 w but w is the variable we are quantifying over. + +Examples + +%def _ ({A tp} {W world} tm (box A => A) W) (lam [x] boxe x [u] u W) +%def _ ({A tp} {W world} tm (box A => box (box A)) W) (lam [x] boxe x [u] boxi [w] boxi [w'] u w') +%def _ ({A tp} {B tp} {W world} tm (box (A => B) => box A => box B) W) (lam [x] lam [y] boxe x [u] boxe y [v] boxi [w] app (u w) (v w)) +%def _ ({A tp} {W world} tm (A => dia A) W) (lam [x] diai (t2e x)) +%def _ ({A tp} {W world} tm (dia (dia A) => dia A) W) (lam [x] diai (diae x [w] [y] diae y [v] [z] t2e z)) +%def _ ({A tp} {B tp} {W world} tm (box (A => B) => dia A => dia B) W) (lam [x] lam [y] diai (boxep x [u] diae y [w] [z] t2e (app (u w) z))) +%. +Counterexamples, all must fail: +The following would require box to be a comonad (A => box A), but this is not valid in S4. +The term (lam [x] boxi [w] x) fails because x has type tm A W but we need tm A w for +arbitrary w, which would require the structural rule for modal contexts. +Similarly (lam [x] diae x [w][y] t2e y) fails because y has type tm A w but we need +tm A W for the current world W. +The term for (dia (A => B) => dia A => dia B) fails because dia is not "normal" in S4. +The term (dia A => box B) => box (A => B) is true in Kripke semantics a la Simpson but +not in the judgmental formulation. +The two S5 theorems (dia A => box (dia A)) and (dia (box A) => box A) both fail because +S4 does not have the symmetry or Euclidean properties needed for S5. +|} + +let lam = + {| +Lambda-Calculus Fragment of Mini-ML. +Author: Frank Pfenning + +Simple types +%sort tp + +%term arrow {_ tp} {_ tp} tp + +%. Expressions +%sort exp + +%term lam {_ {_ exp} exp} exp +%term app {_ exp} {_ exp} exp + +%. Type inference +|- E : T (expression E has type T) + +%sort of {_ exp} {_ tp} +%mode of %in %star + +%term tp_lam {E {_ exp} exp} {T1 tp} {T2 tp} {_ {x exp} {_ of x T1} of (E x) T2} of (lam E) (arrow T1 T2) + +%term tp_app {E1 exp} {E2 exp} {T1 tp} {T2 tp} {_ of E1 (arrow T2 T1)} {_ of E2 T2} of (app E1 E2) T1 + +%. Evaluation (call-by-value) +E ==> V (expression E evaluates to value V) + +%sort eval {_ exp} {_ exp} +%mode eval %in %out + +%term ev_lam {E {_ exp} exp} eval (lam E) (lam E) + +%term ev_app {E1 exp} {E2 exp} {V exp} {V2 exp} {E1' {_ exp} exp} {_ eval E1 (lam E1')} {_ eval E2 V2} {_ eval (E1' V2) V} eval (app E1 E2) V + +%. Regular world for type-checking +%block tp_var [T tp] {x exp} {u of x T} +%worlds (tp_var) (of E T) + +%. Type inference terminates +%terminates E (of E T) + +%. There is at least one typing rule for every expression +%covers of %in %star + +%. Closed worlds for evaluation +%worlds () (eval E V) + +%. There is at least one evaluation rule for every closed expression +%covers eval %in %out + +%. Type preservation as higher-level family +%sort tps {_ eval E V} {_ of E T} {_ of V T} + +%term tps_lam {E {_ exp} exp} {T1 tp} {T2 tp} {P {_ exp} {_ of _ T1} of _ T2} tps ev_lam (tp_lam P) (tp_lam P) + +%term tps_app {E1 exp} {E2 exp} {V exp} {V2 exp} {E1' {_ exp} exp} {T tp} {T2 tp} {D1 eval E1 (lam E1')} {D2 eval E2 V2} {D3 eval (E1' V2) V} {P1 of E1 (arrow T2 T)} {P2 of E2 T2} {Q1' {_ exp} {_ of _ T2} of _ T2} {Q2 of V2 T2} {Q of V T} {_ tps D1 P1 (tp_lam Q1')} {_ tps D2 P2 Q2} {_ tps D3 (Q1' V2 Q2) Q} tps (ev_app D3 D2 D1) (tp_app P2 P1) Q + +%mode tps %in %in %out +%worlds () (tps D P _) +%total D (tps D P _) + +%. Applying type preservation +%def e0 _ (app (lam [x] x) (lam [y] y)) +%? of e0 T +%? eval e0 V +%? tps d0 p0 Q + +%. Example of regular worlds +cp copies input to output. + +%sort cp {_ exp} {_ exp} + +%term cp_app {E1 exp} {E2 exp} {F1 exp} {F2 exp} {_ cp E1 F1} {_ cp E2 F2} cp (app E1 E2) (app F1 F2) + +%term cp_lam {E {_ exp} exp} {F {_ exp} exp} {_ {x exp} {_ cp x x} cp (E x) (F x)} cp (lam [x] E x) (lam [x] F x) + +%mode cp %in %out +%block cp_var {x exp} {u cp x x} +%worlds (cp_var) (cp E _) +%total E (cp E _) +%. +Following version cannot be checked: input coverage on parameter y is violated. +It would declare cp with a block containing {x:exp} {y:exp} {u:cp x y}, but cp_lam's +higher-order premise would allow y to differ from x, which violates input coverage. + +Following version also cannot be checked: output coverage on (F y) is violated. +It would add a premise cp y y -> cp (E x) (F y), meaning F y may not be covered +for the output position. + +|} + +let polylam = + {| +%sort tp +%term => {_ tp} {_ tp} tp +%prec %right 10 => +%term all {_ {_ tp} tp} tp + +%sort tm {_ tp} +%term lam {A tp} {B tp} {_ {_ tm A} tm B} tm (A => B) +%term app {A tp} {B tp} {_ tm (A => B)} {_ tm A} tm B +%term tlam {A {_ tp} tp} {_ {a tp} tm (A a)} tm (all A) +%term tapp {A {_ tp} tp} {_ tm (all A)} {B tp} tm (A B) + +%def nat _ (all [a] a => (a => a) => a) +%def zero _ (tlam [a] lam [z] lam [s] z) +%def succ _ (lam [x] tlam [a] lam [z] lam [s] app s (app (app (tapp x a) z) s)) +%def succ' (tm (nat => nat)) (lam [x] tlam [a] lam [z] lam [s] app s (app (app (tapp x a) z) s)) +%def plus _ (lam [x] lam [y] app (app (tapp y nat) x) succ) +%def times _ (lam [x] lam [y] app (app (tapp y nat) zero) (app plus x)) +%def exp _ (lam [x] lam [y] app (app (tapp y nat) (app succ zero)) (app times x)) +|} diff --git a/test/Parse/Cases.ml b/test/Parse/Cases.ml index b400e14..9adca89 100644 --- a/test/Parse/Cases.ml +++ b/test/Parse/Cases.ml @@ -47,6 +47,10 @@ let cases () = test "Qualified" Term "%val ( x y )"; test "Nofix" Term "%val +"; ] ); + ("Explicit implicits", [ + test "Simple" Term "{{X Y}} X Y"; + test "Nested" Term "{{X Y}} X {{Z}} Y Z"; + ]) ; ( "Hypotheses", [ test "Simple" Decl "x nat"; @@ -82,7 +86,7 @@ let cases () = test "Complex (unnamed, multi)" Cmd1 "%sort eq {t type} {(_ _) term t}"; (* UNTIL MUTIPLE SORTS WORK *) - test ~skip:true "Sorts (combined)" Cmd1 "%sort (nat bool)"; + test "Sorts (combined)" Cmd1 "%sort (nat bool)"; ] ); ( "%term", [ @@ -135,28 +139,39 @@ let cases () = ( "%total", [ test "Simple" Cmd1 "%total N (add N _ _)"; - (* TODO totality in context ? *) - test "Context" Cmd1 ~skip:true "%total (b) N (add N _ _)"; test "Mutual" Cmd1 ~skip:false "%total (N1 N2) (add N1 _ _) (mul N2 _ _)"; - test "Mutual (context)" Cmd1 ~skip:true - "%total (b) (N1 N2) (add N1 _ _) (mul N2 _ _)"; ] ); ( "%terminates", [ test "Simple" Cmd1 "%terminates N (add N _ _)"; - test "Context" Cmd1 ~skip:true "%terminates (b) N (add N _ _)"; test "Mutual" Cmd1 ~skip:false "%terminates (N1 N2) (add N1 _ _) (mul N2 _ _)"; - test "Mutual (context)" Cmd1 ~skip:true - "%terminates (b) (N1 N2) (add N1 _ _) (mul N2 _ _)"; - ] ); + test "Simultaneous" Cmd1 + "%terminates [A B] max A B"; + test "Lexocographic" Cmd1 + "%terminates {A B} max A B"; + test "Nested" Cmd1 + "%terminates {A [B C] F} (max A (max B C))"; + test "Nested (mutual)" Cmd1 "%terminates ({A [B C] G} [D E] F) (max A (max B C) max D (max E C))"; + test "Nested (mutual, issue)" Cmd1 ~skip:true "({A [B C]} [D E]) (max A (max B C) max D (max E C))" + ] ); ( "%query", [ test "Atomic (%?)" Cmd1 "%? nat"; test "Complex (%?)" Cmd1 "%? add zero zero zero"; test "Atom (Full)" Cmd1 "%query _ _ 1 add zero zero zero"; ] ); + + ( "%reduces", + [ + test "Same size" Cmd1 "%reduces = X Y add X Y zero"; + test "Smaller" Cmd1 "%reduces < X Y add X Y zero"; + test "Larger" Cmd1 "%reduces > X Y add X Y zero"; + test "Same size or greater" Cmd1 "%reduces >= X Y add X Y zero"; + test "Same size or smaller" Cmd1 "%reduces <= X Y add X Y zero"; + ] + ) ] end ~verbose:true diff --git a/test/Parse/Common.ml b/test/Parse/Common.ml index c1177fa..74c5bba 100644 --- a/test/Parse/Common.ml +++ b/test/Parse/Common.ml @@ -19,17 +19,31 @@ let test ?(skip = false) ?(failure = false) (name : string) (f : form) let () = Printexc.record_backtrace true in let () = Logs.set_reporter (Logs_fmt.reporter ()) in let () = Logs.set_level (Some Logs.Debug) in + let () = Fmt_tty.setup_std_outputs () in let () = Display.register (fun m -> - Lwt.return @@ prerr_endline - @@ Fmt.to_to_string Display.Info.Form.fmt m.msg) + let _ = (Display.fmt Fmt.stderr m.msg ) in + Lwt.return ()) + in Alcotest.test_case name `Slow (fun () -> + let bad () = + Display.warning ( + Display.Form.(concat [ + style Style.bold @@ style Style.Fore.red @@ string "Test input>>"; + nl () ; + (string input); + nl () ; + style Style.bold @@ style Style.Fore.red @@ string "< Alcotest.fail "Expected failure, but test passed" + | None when failure -> bad () ; Alcotest.fail "Expected failure, but test passed" | Some e when not failure -> + bad (); Alcotest.failf "Expected success, but test failed with exception: %s" (Printexc.to_string e) diff --git a/trans/input/fol.elf b/trans/input/fol.elf new file mode 100644 index 0000000..97333b0 --- /dev/null +++ b/trans/input/fol.elf @@ -0,0 +1,217 @@ +%%% First-Order Logic +%%% Fragment with implication, negation, universal quantification. +%%% +%%% Author: Frank Pfenning +%%% +%%% This code is from the article +%%% +%%% Logical Frameworks +%%% Handbook of Automated Reasoning +%%% Alan Robinson and Andrei Voronkov, editors +%%% Chapter 16 +%%% Elsevier Science and MIT Press +%%% In preparation +%%% + +i : type. % individuals +o : type. % formulas +%name i T x. +%name o A p. + +% Formulas + +imp : o -> o -> o. %infix right 10 imp. +not : o -> o. %prefix 12 not. +forall : (i -> o) -> o. + +% Natural deductions + +nd : o -> type. +%name nd D u. + +impi : (nd A -> nd B) -> nd (A imp B). +impe : nd (A imp B) -> nd A -> nd B. + +noti : ({p:o} nd A -> nd p) -> nd (not A). +note : nd (not A) -> {C:o} nd A -> nd C. + +foralli : ({a:i} nd (A a)) -> nd (forall A). +foralle : nd (forall A) -> {T:i} nd (A T). + +%block lnd : some {A:o} block {u:nd A}. +%block lo : block {p:o}. +%block li : block {a:i}. + +%worlds (lnd | lo | li) (nd A). + +% Example + +_ : nd (A imp (B imp A)) + = (impi [u:nd A] impi [v:nd B] u). + +% Hilbert deductions + +hil : o -> type. % Hilbert deductions +%name hil H u. + +k : hil (A imp (B imp A)). +s : hil ((A imp (B imp C)) imp ((A imp B) imp (A imp C))). + +n1 : hil ((A imp (not B)) imp ((A imp B) imp (not A))). +n2 : hil ((not A) imp (A imp B)). + +f1 : {T:i} hil ((forall [x:i] A x) imp (A T)). +f2 : hil ((forall [x:i] (B imp A x)) imp (B imp forall [x:i] A x)). + +mp : hil (A imp B) -> hil A -> hil B. +ug : ({a:i} hil (A a)) -> hil (forall [x:i] A x). + +%worlds (li) (hil A). + +% Local reductions + +==>R : nd A -> nd A -> type. %infix none 14 ==>R. +%name ==>R R. + +redl_imp : (impe (impi [u:nd A] D u) E) ==>R (D E). +redl_not : (note (noti [p:o] [u:nd A] D p u) C E) ==>R (D C E). +redl_forall : (foralle (foralli [a:i] D a) T) ==>R (D T). + +% Local expansions + +==>E : nd A -> nd A -> type. %infix none 14 ==>E. +%name ==>E E. + +expl_imp : {D:nd (A imp B)} D ==>E (impi [u:nd A] impe D u). +expl_not : {D:nd (not A)} D ==>E (noti [p:o] [u:nd A] note D p u). +expl_forall : {D:nd (forall A)} D ==>E (foralli [a:i] foralle D a). + +% Sequent calculus search result + +dn : nd (A imp not not A) + = (impi [u:nd A] noti [p:o] [w:nd (not A)] note w p u). + +% Translating Hilbert derivations to natural deductions + +hilnd : hil A -> nd A -> type. + +hnd_k : hilnd (k) (impi [u:nd A] impi [v:nd B] u). +hnd_s : hilnd (s) (impi [u:nd (A imp B imp C)] + impi [v:nd (A imp B)] + impi [w:nd A] impe (impe u w) (impe v w)). + +hnd_n1 : hilnd (n1) (impi [u:nd (A imp (not B))] + impi [v:nd (A imp B)] + noti [p:o] [w:nd A] + note (impe u w) p (impe v w)). + +hnd_n2 : hilnd (n2) (impi [u:nd (not A)] + impi [v:nd A] note u B v). + +hnd_f1 : + hilnd (f1 T) (impi [u:nd (forall [x:i] A x)] foralle u T). + +hnd_f2 : + hilnd (f2) (impi [u:nd (forall [x:i] (B imp A x))] + impi [v:nd B] + foralli [a:i] impe (foralle u a) v). + +hnd_mp : hilnd (mp H1 H2) (impe D1 D2) + <- hilnd H1 D1 + <- hilnd H2 D2. + +hnd_ug : hilnd (ug H1) (foralli D1) + <- ({a:i} hilnd (H1 a) (D1 a)). + +%mode hilnd +H -D. +%worlds (li) (hilnd H D). +%terminates H (hilnd H _). +%covers hilnd +H -D. +%total H (hilnd H _). + +%solve id' : +hilnd (mp (mp s k) k) (D:nd (A imp A)). + +_ : nd (A imp A) + = (impe + (impe + (impi + ([u:nd (A imp (B imp A) imp A)] + impi + ([v:nd (A imp B imp A)] + impi ([w:nd A] impe (impe u w) (impe v w))))) + (impi ([u:nd A] impi ([v:nd (B imp A)] u)))) + (impi ([u:nd A] impi ([v:nd B] u)))). + +%%% The deduction theorem for Hilbert derivations + +ded : (hil A -> hil B) -> hil (A imp B) -> type. + +ded_id : ded ([u:hil A] u) (mp (mp s k) (k : hil (A imp (A imp A)))). + +ded_k : ded ([u:hil A] k) (mp k k). +ded_s : ded ([u:hil A] s) (mp k s). + +ded_n1 : ded ([u:hil A] n1) (mp k n1). +ded_n2 : ded ([u:hil A] n2) (mp k n2). + +ded_f1 : ded ([u:hil A] f1 T) (mp k (f1 T)). +ded_f2 : ded ([u:hil A] f2) (mp k f2). + +ded_mp : ded ([u:hil A] mp (H1 u) (H2 u)) (mp (mp s H1') H2') + <- ded H1 H1' + <- ded H2 H2'. + +ded_ug : ded ([u:hil A] ug (H1 u)) (mp f2 (ug H1')) + <- ({a:i} ded ([u:hil A] H1 u a) (H1' a)). + +%block lded : some {A:o} block {u:nd A} {v:hil A} + {h:{C:o} ded ([w:hil C] v) (mp k v)}. + +%mode ded +H -H'. +%worlds (li | lo | lded) (ded H H'). +%terminates H (ded H _). +%covers ded +H -H'. +%total H (ded H _). + +%%% Mapping natural deductions to Hilbert derivations. + +ndhil : nd A -> hil A -> type. + +ndh_impi : ndhil (impi D1) H1' + <- ({u:nd A1} {v:hil A1} + ({C:o} ded ([w:hil C] v) (mp k v)) + -> ndhil u v + -> ndhil (D1 u) (H1 v)) + <- ded H1 H1'. + +ndh_impe : ndhil (impe D1 D2) (mp H1 H2) + <- ndhil D1 H1 + <- ndhil D2 H2. + +ndh_noti : ndhil (noti D1) (mp (mp n1 H1') H1'') + <- ({p:o} {u:nd A1} {v:hil A1} + ({C:o} ded ([w:hil C] v) (mp k v)) + -> ndhil u v + -> ndhil (D1 p u) (H1 p v)) + <- ded (H1 (not A1)) H1' + <- ded (H1 A1) H1''. + +ndh_note : ndhil (note D1 C D2) (mp (mp n2 H1) H2) + <- ndhil D1 H1 + <- ndhil D2 H2. + +ndh_foralli : ndhil (foralli D1) (ug H1) + <- ({a:i} ndhil (D1 a) (H1 a)). + +ndh_foralle : ndhil (foralle D1 T) (mp (f1 T) H1) + <- ndhil D1 H1. + +%mode ndhil +D -H. +%block lndhil : some {A:o} block {u:nd A} {v:hil A} + {h:{C:o} ded ([w:hil C] v) (mp k v)} + {nh:ndhil u v}. +%worlds (li | lo | lndhil) (ndhil D H). +%terminates D (ndhil D _). +%covers ndhil +D -H. +%total D (ndhil D _). diff --git a/trans/input/js4.elf b/trans/input/js4.elf new file mode 100644 index 0000000..f141727 --- /dev/null +++ b/trans/input/js4.elf @@ -0,0 +1,107 @@ +%{ + Judgmental S4 + [A judgmental reconstruction of modal logic, F.Pfenning and R.Davies, + MSCS 11:511-540, 2001] + + Representation with intrinsic types, worlds, + but not a Kripke semantics + + Idea: Translate the judgment + u1::B1,...,un::Bn ; x1:A1,...,xm:Am |- J + as + u1:{W'}tm B1 W',...,un:{W'}tm Bn W', ---, + x1:tm A1 W,...,xm:tm Am W |- J* + + where "---" are assumptions y:tm B W' for W' <> W + and if J = (M : A) then J* = M* : tm A W + if J = (E - A) then J* = E* : exp A W +}% + +tp : type. %name tp A. +=> : tp -> tp -> tp. +box : tp -> tp. +dia : tp -> tp. +%infix right 10 =>. + +world : type. %name world W w. + +tm : tp -> world -> type. %name tm M x. +exp : tp -> world -> type. %name exp E. + +lam : (tm A W -> tm B W) -> tm (A => B) W. +app : tm (A => B) W -> tm A W -> tm B W. +boxi : ({w:world} tm A w) -> tm (box A) W. +boxe : tm (box A) W -> (({W':world} tm A W') -> tm C W) -> tm C W. +t2e : tm A W -> exp A W. +diai : exp A W -> tm (dia A) W. +diae : tm (dia A) W -> ({w:world} tm A w -> exp C w) -> exp C W. +boxep : tm (box A) W -> (({W':world} tm A W') -> exp C W) -> exp C W. + +subdia : exp A W -> ({w:world} tm A w -> exp C w) -> exp C W -> type. +%mode subdia +E +F -F'. + +sdt2e : subdia (t2e M) ([w][x] F w x) (F W M). +sddiae : subdia (diae M [v][y] E v y) ([w][x] F w x) (diae M [v][y] F' v y) + <- ({v:world}{y:tm B v} subdia (E v y) ([w][x] F w x) (F' v y)). +sdboxep : subdia (boxep M [u] E u) ([w][x] F w x) (boxep M [u] F' u) + <- ({u:{V:world} tm B V} subdia (E u) ([w][x] F w x) (F' u)). + +%block by : some {B:tp} block {v:world}{y:tm B v}. +%block bu : some {B:tp} block {u:{V:world} tm B V}. +%worlds (by | bu) (subdia E F F'). +%total E (subdia E F _). + +%{ This does not work, unfortunately +str : (tm A W -> {w:world} tm C w) -> ({w:world} tm C w) -> type. +stre : (tm A W -> {w:world} exp C w) -> ({w:world} exp C w) -> type. +strlam : str ([x][w] lam [y2] M1 x w y2) ([w] lam [y2] M1' w y2) + <- ({y2:tm C2 w} % problem here! + str ([x][w] y2) ([w] y2) + -> str ([x][w] M1 x w y2) ([w] M1' w y2)). +strapp : str ([x][w] app (M1 x w) (M2 x w)) ([w] app (M1' w) (M2' w)) + <- str ([x] [w] M1 x w) ([w] M1' w) + <- str ([x] [w] M2 x w) ([w] M2' w). +strboxi : str ([x][w] boxi [v] M1 x w v) ([w] boxi [v] M1' w v) + <- ({v:world} str ([x][w] M1 x w v) ([w] M1' w v)). +strboxe : str ([x][w] boxe (M1 x w) [u] M2 x w u) ([w] boxe (M1' w) [u] M2' w u) + <- str ([x][w] M1 x w) ([w] M1' w) + <- ({u:{W':world} tm A W'} + str ([x][w] u w) ([w] u w) + -> str ([x][w] M2 x w u) (M2' w u)). +... more case missing ... +}% + +% Examples + +_ : tm (box A => A) W + = lam [x] boxe x [u] u W. +_ : tm (box A => box (box A)) W + = lam [x] boxe x [u] boxi [w] boxi [w'] u w'. +_ : tm (box (A => B) => box A => box B) W + = lam [x] lam [y] boxe x [u] boxe y [v] boxi [w] app (u w) (v w). +_ : tm (A => dia A) W + = lam [x] diai (t2e x). +_ : tm (dia (dia A) => dia A) W + = lam [x] diai (diae x [w][y] diae y [v][z] t2e z). +_ : tm (box (A => B) => dia A => dia B) W + = lam [x] lam [y] diai (boxep x [u] diae y [w][z] t2e (app (u w) z)). + +% Counterexamples, all must fail +%{ +% next two are characteristic +_ : tm (A => box A) W + = lam [x] boxi [w] x. +_ : tm (dia A => A) W + = lam [x] diae x [w][y] t2e y. +% next shows dia is not "normal" +_ : tm (dia (A => B) => dia A => dia B) W + = lam [x] lam [y] diai (diae x [w][x'] diae y [v][y'] t2e (app x' y')). +% next one is true is Kripke semantics a la Simpson +_ : tm ((dia A => box B) => box (A => B)) W + = lam [x] boxi [w] lam [y] boxe (app x (diai (t2e y))) [u] u w. +% next two are true in S5, not S4 +_ : tm (dia A => box (dia A)) W + = lam [x] diae x [w][x'] t2e (boxi [v] diai (t2e x')). +_ : tm (dia (box A) => box A) W + = lam [x] boxi [w] diae x [w'][x'] boxe x' [u] u w. +}% diff --git a/trans/input/lam.elf b/trans/input/lam.elf new file mode 100644 index 0000000..9e9d969 --- /dev/null +++ b/trans/input/lam.elf @@ -0,0 +1,148 @@ +%%% Lambda-Calculus Fragment of Mini-ML. +%%% Author: Frank Pfenning + +% Simple types +tp : type. %name tp T. + +arrow : tp -> tp -> tp. % T1 => T2 + +% Expressions +exp : type. %name exp E. + +lam : (exp -> exp) -> exp. % lam x. E +app : exp -> exp -> exp. % (E1 E2) + +% Type inference +% |- E : T (expression E has type T) + +of : exp -> tp -> type. %name of P. +%mode of +E *T. +% %mode of +E +T. % incorrect at tp_app +% %mode of +E -T. % incorrect at tp_lam + +tp_lam : of (lam E) (arrow T1 T2) % |- lam x. E : T1 => T2 + <- ({x:exp} % if x:T1 |- E : T2. + of x T1 -> of (E x) T2). + +tp_app : of (app E1 E2) T1 % |- E1 E2 : T1 + <- of E1 (arrow T2 T1) % if |- E1 : T2 => T1 + <- of E2 T2. % and |- E2 : T2. + +% Evaluation (call-by-value) +% E ==> V (expression E evaluates to value V) + +eval : exp -> exp -> type. %name eval D. +%mode eval +E -V. + +ev_lam : eval (lam E) (lam E). % lam x.E ==> lam x.E. + +ev_app : eval (app E1 E2) V % E1 E2 ==> V + <- eval E1 (lam E1') % if E1 ==> lam x. E1' + <- eval E2 V2 % and E2 ==> V2 + <- eval (E1' V2) V. % and [V2/x]E1' ==> V. + +% Regular world for type-checking +%block tp_var : some {T:tp} block {x:exp} {u:of x T}. +%worlds (tp_var) (of E T). + +% Type inference terminates +%terminates E (of E T). + +% There is at least one typing rule for every expression +%covers of +E *T. + +% Next fails, because T is neither input nor output +% %total E (of E T). + +% Closed worlds for evaluation +%worlds () (eval E V). + +% There is a at least one evaluation rule for every closed expression +%covers eval +E -V. + +% %terminates E (eval E V). % fails for ev_app + +% Type preservation as higher-level family +tps : eval E V -> of E T -> of V T -> type. + +tps_lam : tps (ev_lam) (tp_lam P) (tp_lam P). +tps_app : tps (ev_app D3 D2 D1) (tp_app P2 P1) Q + <- tps D1 P1 (tp_lam Q1') + <- tps D2 P2 Q2 + <- tps D3 (Q1' V2 Q2) Q. + +%mode tps +D +P -Q. +%worlds () (tps D P _). +%total D (tps D P _). + +% Type preservation proven automatically +%theorem +tpsa : forall* {E:exp} {V:exp} {T:tp} + forall {D:eval E V} {P:of E T} + exists {Q:of V T} + true. + +%prove 5 D (tpsa D P Q). + +% Applying type preservation +e0 = (app (lam [x] x) (lam [y] y)). +%solve p0 : of e0 T. +%solve d0 : eval e0 V. +%solve tps0 : tps d0 p0 Q. + +% Example of regular worlds +% cp copies input to output. + +cp : exp -> exp -> type. +cp_app : cp (app E1 E2) (app F1 F2) + <- cp E1 F1 + <- cp E2 F2. + +cp_lam : cp (lam [x] E x) (lam [x] F x) + <- ({x:exp} cp x x -> cp (E x) (F x)). + +%mode cp +E -F. +%block cp_var : block {x:exp} {u:cp x x}. +%worlds (cp_var) (cp E _). +%total E (cp E _). + +% Following version cannot be checked +% Input coverage on parameter y is violated +%{ +cp : exp -> exp -> type. +cp_app : cp (app E1 E2) (app F1 F2) + <- cp E1 F1 + <- cp E2 F2. + +cp_lam : cp (lam [x] E x) (lam [x] F x) + <- ({x:exp} {y:exp} cp x y -> cp (E x) (F y)). + +%mode cp +E -F. +%block cp_var : block {x:exp} {y:exp} {u:cp x y}. +%worlds (cp_var) (cp E _). +%total E (cp E _). +}% + +% Following version cannot be checked +% Output coverage on (F y) is violated. +%{ +cp : exp -> exp -> type. +cp_app : cp (app E1 E2) (app F1 F2) + <- cp E1 F1 + <- cp E2 F2. + +cp_lam : cp (lam [x] E x) (lam [x] F x) + <- ({x:exp} {y:exp} cp x y -> cp y y -> cp (E x) (F y)). + +%mode cp +E -F. +%block cp_var : block {x:exp} {y:exp} {u:cp x y} {w:cp y y}. +%worlds (cp_var) (cp E _). +%total E (cp E _). +}% + +%theorem cpt : forallG (pi {x:exp} {y:cp x x}) + forall {E:exp} + exists {F:exp} {C:cp E F} + true. +%prove 5 E (cpt E _ _). + diff --git a/trans/input/polylam.elf b/trans/input/polylam.elf new file mode 100644 index 0000000..a5d6a82 --- /dev/null +++ b/trans/input/polylam.elf @@ -0,0 +1,23 @@ +tp : type. +=> : tp -> tp -> tp. %infix right 10 =>. +all : (tp -> tp) -> tp. + +tm : tp -> type. +lam : (tm A -> tm B) -> tm (A => B). +app : tm (A => B) -> tm A -> tm B. +tlam : ({a:tp} tm (A a)) -> tm (all A). +tapp : tm (all A) -> {B:tp} tm (A B). + +nat = all [a] a => (a => a) => a. +zero = tlam [a] lam [z:tm a] lam [s:tm (a => a)] z. +succ = lam [x:tm nat] tlam [a:tp] lam [z:tm a] lam [s:tm (a => a)] + app s (app (app (tapp x a) z) s). +succ' : tm (nat => nat) + = lam [x] tlam [a] lam [z] lam [s] + app s (app (app (tapp x a) z) s). +plus = lam [x:tm nat] lam [y:tm nat] (app (app (tapp y nat) x) succ). + +times = lam [x: tm nat] lam [y:tm nat] (app (app (tapp y nat) zero) (app plus x)). +exp = lam [x: tm nat] lam [y:tm nat] (app (app (tapp y nat) (app succ zero)) (app times x)). + +% idnat = [x: tm nat] x. \ No newline at end of file diff --git a/trans/output/fol.stelf b/trans/output/fol.stelf new file mode 100644 index 0000000..6251641 --- /dev/null +++ b/trans/output/fol.stelf @@ -0,0 +1,173 @@ +First-Order Logic +Fragment with implication, negation, universal quantification. + +Author: Frank Pfenning + +This code is from the article + + Logical Frameworks + Handbook of Automated Reasoning + Alan Robinson and Andrei Voronkov, editors + Chapter 16 + Elsevier Science and MIT Press + In preparation + +%sort i +%sort o + +Formulas + +%term imp {_ o} {_ o} o +%prec %right 10 imp +%term not {_ o} o +%prec %prefix 12 not +%term forall {_ {_ i} o} o + +Natural deductions + +%sort nd {_ o} +%name nd D u +%term impi {{A B}} {_ {_ nd A} nd B} nd (A imp B) +%term impe {{A B}} {_ nd (A imp B)} {_ nd A} nd B + +%term noti {{A}} {_ {p o} {_ nd A} nd p} nd (not A) +%term note {{A}} {_ nd (not A)} {C o} {_ nd A} nd C + +%term foralli {{A}} {_ {a i} nd (A a)} nd (forall A) +%term foralle {{A}} {_ nd (forall A)} {T i} nd (A T) + +%block lnd [A o] {u nd A} +%block lo {p o} +%block li {a i} + +%worlds (lnd lo li) (nd A) + +Example + +%def _ {{A B}} nd (A imp (B imp A)) (impi [u] impi [v] u) + +Hilbert deductions + +%sort hil {_ o} + +%term k {{A B}} hil (A imp (B imp A)) +%term s {{A B C}} hil ((A imp (B imp C)) imp ((A imp B) imp (A imp C))) + +%term n1 {{A B}} hil ((A imp (not B)) imp ((A imp B) imp (not A))) +%term n2 {{A B}} hil ((not A) imp (A imp B)) + +%term f1 {{A}} {T i} hil ((forall [x i] A x) imp (A T)) +%term f2 {{A B}} hil ((forall [x i] (B imp A x)) imp (B imp forall [x i] A x)) + +%term mp {{A B}} {_ hil (A imp B)} {_ hil A} hil B +%term ug {{A}} {_ {a i} hil (A a)} hil (forall [x i] A x) + +%worlds (li) (hil A) + +Local reductions + +%sort ==>R {_ nd A} {_ nd A} +%prec %none 14 ==>R + +%term redl_imp {{A B}} {D {_ nd A} nd B} {E nd A} ==>R (impe (impi [u] D u) E) (D E) +%term redl_not {{A}} {D {_ o} {_ nd A} nd _} {C o} {E nd A} ==>R (note (noti [p] [u] D p u) C E) (D C E) +%term redl_forall {{A}} {D {_ i} nd _} {T i} ==>R (foralle (foralli [a] D a) T) (D T) + +Local expansions + +%sort ==>E {_ nd A} {_ nd A} +%prec %none 14 ==>E + +%term expl_imp {{A B}} {D nd (A imp B)} ==>E D (impi [u] impe D u) +%term expl_not {{A}} {D nd (not A)} ==>E D (noti [p] [u] note D p u) +%term expl_forall {{A}} {D nd (forall A)} ==>E D (foralli [a] foralle D a) + +Sequent calculus search result + +%def dn {{A}} nd (A imp not not A) (impi [u] noti [p] [w] note w p u) + +Translating Hilbert derivations to natural deductions + +%sort hilnd {_ hil A} {_ nd A} + +%term hnd_k {{A B}} hilnd k (impi [u] impi [v] u) +%term hnd_s {{A B C}} hilnd s (impi [u] impi [v] impi [w] impe (impe u w) (impe v w)) + +%term hnd_n1 {{A B}} hilnd n1 (impi [u] impi [v] noti [p] [w] note (impe u w) p (impe v w)) + +%term hnd_n2 {{A B}} hilnd n2 (impi [u] impi [v] note u B v) + +%term hnd_f1 {{A}} {T i} hilnd (f1 T) (impi [u] foralle u T) + +%term hnd_f2 {{A B}} hilnd f2 (impi [u] impi [v] foralli [a] impe (foralle u a) v) + +%term hnd_mp {{A B}} {H1 hil (A imp B)} {H2 hil A} {D1 nd (A imp B)} {D2 nd A} {_ hilnd H1 D1} {_ hilnd H2 D2} hilnd (mp H1 H2) (impe D1 D2) + +%term hnd_ug {{A}} {H1 {_ i} hil _} {D1 {_ i} nd _} {_ {a i} hilnd (H1 a) (D1 a)} hilnd (ug H1) (foralli D1) + +%mode hilnd %in %out +%worlds (li) (hilnd H D) +%terminates H (hilnd H _) +%covers hilnd %in %out +%total H (hilnd H _) + +%? hilnd (mp (mp s k) k) D + +%def _ {{A B}} nd (A imp A) + (impe + (impe + (impi [u] + impi [v] + impi [w] impe (impe u w) (impe v w)) + (impi [u] impi [v] u)) + (impi [u] impi [v] u)) + +The deduction theorem for Hilbert derivations + +%sort ded {_ {_ hil A} hil B} {_ hil (A imp B)} + +%term ded_id {{A}} ded ([u] u) (mp (mp s k) k) + +%term ded_k {{A B}} ded ([u] k) (mp k k) +%term ded_s {{A B C}} ded ([u] s) (mp k s) + +%term ded_n1 {{A B}} ded ([u] n1) (mp k n1) +%term ded_n2 {{A B}} ded ([u] n2) (mp k n2) + +%term ded_f1 {{A B T}} ded ([u] f1 T) (mp k (f1 T)) +%term ded_f2 {{A B}} ded ([u] f2) (mp k f2) + +%term ded_mp {{A B C}} {H1 {_ hil A} hil (B imp C)} {H2 {_ hil A} hil B} {H1' hil (A imp (B imp C))} {H2' hil (A imp B)} {_ ded H1 H1'} {_ ded H2 H2'} ded ([u] mp (H1 u) (H2 u)) (mp (mp s H1') H2') + +%term ded_ug {{A B}} {H1 {_ hil A} {_ i} hil _} {H1' {_ i} hil _} {_ {a i} ded ([u] H1 u a) (H1' a)} ded ([u] ug (H1 u)) (mp f2 (ug H1')) + +%block lded [A o] {u nd A} {v hil A} {h {C o} ded ([w] v) (mp k v)} + +%mode ded %in %out +%worlds (li lo lded) (ded H H') +%terminates H (ded H _) +%covers ded %in %out +%total H (ded H _) + +Mapping natural deductions to Hilbert derivations. + +%sort ndhil {_ nd A} {_ hil A} + +%term ndh_impi {{A1 B}} {D1 {_ nd A1} nd B} {H1 {_ hil A1} hil B} {H1' hil (A1 imp B)} {_ ded H1 H1'} {_ {u nd A1} {v hil A1} {_ {C o} ded ([w] v) (mp k v)} {_ ndhil u v} ndhil (D1 u) (H1 v)} ndhil (impi D1) H1' + +%term ndh_impe {{A B}} {D1 nd (A imp B)} {D2 nd A} {H1 hil (A imp B)} {H2 hil A} {_ ndhil D1 H1} {_ ndhil D2 H2} ndhil (impe D1 D2) (mp H1 H2) + +%term ndh_noti {{A1}} {D1 {_ o} {_ nd A1} nd _} {H1 {_ o} {_ hil A1} hil _} {H1' hil _} {H1'' hil _} {_ ded (H1 A1) H1''} {_ ded (H1 (not A1)) H1'} {_ {p o} {u nd A1} {v hil A1} {_ {C o} ded ([w] v) (mp k v)} {_ ndhil u v} ndhil (D1 p u) (H1 p v)} ndhil (noti D1) (mp (mp n1 H1') H1'') + +%term ndh_note {{A C}} {D1 nd (not A)} {D2 nd A} {H1 hil (not A)} {H2 hil A} {_ ndhil D1 H1} {_ ndhil D2 H2} ndhil (note D1 C D2) (mp (mp n2 H1) H2) + +%term ndh_foralli {{A}} {D1 {_ i} nd _} {H1 {_ i} hil _} {_ {a i} ndhil (D1 a) (H1 a)} ndhil (foralli D1) (ug H1) + +%term ndh_foralle {{A}} {T i} {D1 nd (forall A)} {H1 hil (forall A)} {_ ndhil D1 H1} ndhil (foralle D1 T) (mp (f1 T) H1) + +%mode ndhil %in %out +%block lndhil [A o] {u nd A} {v hil A} {h {C o} ded ([w] v) (mp k v)} {nh ndhil u v} +%worlds (li lo lndhil) (ndhil D H) +%terminates D (ndhil D _) +%covers ndhil %in %out +%total D (ndhil D _) diff --git a/trans/output/js4.stelf b/trans/output/js4.stelf new file mode 100644 index 0000000..e4849a1 --- /dev/null +++ b/trans/output/js4.stelf @@ -0,0 +1,77 @@ +Judgmental S4 +[A judgmental reconstruction of modal logic, F.Pfenning and R.Davies, + MSCS 11:511-540, 2001] + +Representation with intrinsic types, worlds, +but not a Kripke semantics + +Idea: Translate the judgment + u1::B1,...,un::Bn ; x1:A1,...,xm:Am |- J +as + u1:{W'}tm B1 W',...,un:{W'}tm Bn W', ---, + x1:tm A1 W,...,xm:tm Am W |- J* + +where "---" are assumptions y:tm B W' for W' <> W +and if J = (M : A) then J* = M* : tm A W + if J = (E - A) then J* = E* : exp A W + +%sort tp +%term => {_ tp} {_ tp} tp +%term box {_ tp} tp +%term dia {_ tp} tp +%prec %right 10 => + +%sort world + +%sort tm {_ tp} {_ world} +%sort exp {_ tp} {_ world} + +%term lam {A tp} {B tp} {W world} {_ {_ tm A W} tm B W} tm (A => B) W +%term app {A tp} {B tp} {W world} {_ tm (A => B) W} {_ tm A W} tm B W +%term boxi {A tp} {W world} {_ {w world} tm A w} tm (box A) W +%term boxe {A tp} {C tp} {W world} {_ tm (box A) W} {_ {_ {W' world} tm A W'} tm C W} tm C W +%term t2e {A tp} {W world} {_ tm A W} exp A W +%term diai {A tp} {W world} {_ exp A W} tm (dia A) W +%term diae {A tp} {C tp} {W world} {_ tm (dia A) W} {_ {w world} {_ tm A w} exp C w} exp C W +%term boxep {A tp} {C tp} {W world} {_ tm (box A) W} {_ {_ {W' world} tm A W'} exp C W} exp C W + +%sort subdia {_ exp A W} {_ {w world} {_ tm A w} exp C w} {_ exp C W} +%mode subdia %in %in %out + +%term sdt2e {A tp} {C tp} {W world} {M tm A W} {F {w world} {_ tm A w} exp C w} subdia (t2e M) ([w] [x] F w x) (F W M) + +%term sddiae {A tp} {B tp} {C tp} {W world} {M tm (dia A) W} {E {_ world} {_ tm A _} exp B _} {F {_ world} {_ tm B _} exp C _} {F' {_ world} {_ tm A _} exp C _} {_ {v world} {y tm A v} subdia (E v y) ([w] [x] F w x) (F' v y)} subdia (diae M [v] [y] E v y) ([w] [x] F w x) (diae M [v] [y] F' v y) + +%term sdboxep {A tp} {B tp} {C tp} {W world} {M tm (box A) W} {E {_ {_ world} tm B _} exp C _} {F {_ world} {_ tm A _} exp C _} {F' {_ {_ world} tm B _} exp C _} {_ {u {V world} tm B V} subdia (E u) ([w] [x] F w x) (F' u)} subdia (boxep M [u] E u) ([w] [x] F w x) (boxep M [u] F' u) + +%block by {B tp} {v world} {y tm B v} +%block bu {B tp} {u {V world} tm B V} +%worlds (by bu) (subdia E F F') +%total E (subdia E F _) + +This does not work, unfortunately: +The "str" strengthening lemma would require handling cases like strlam where a bound +variable y2 of type tm C2 w appears free in the conclusion, but w is also quantified +by the str family. The main issue is that the case for str ([x][w] y2) ([w] y2) cannot +be typed because y2 has type tm C2 w but w is the variable we are quantifying over. + +Examples + +%def _ {A tp} {W world} tm (box A => A) W (lam [x] boxe x [u] u W) +%def _ {A tp} {W world} tm (box A => box (box A)) W (lam [x] boxe x [u] boxi [w] boxi [w'] u w') +%def _ {A tp} {B tp} {W world} tm (box (A => B) => box A => box B) W (lam [x] lam [y] boxe x [u] boxe y [v] boxi [w] app (u w) (v w)) +%def _ {A tp} {W world} tm (A => dia A) W (lam [x] diai (t2e x)) +%def _ {A tp} {W world} tm (dia (dia A) => dia A) W (lam [x] diai (diae x [w] [y] diae y [v] [z] t2e z)) +%def _ {A tp} {B tp} {W world} tm (box (A => B) => dia A => dia B) W (lam [x] lam [y] diai (boxep x [u] diae y [w] [z] t2e (app (u w) z))) + +Counterexamples, all must fail: +The following would require box to be a comonad (A => box A), but this is not valid in S4. +The term (lam [x] boxi [w] x) fails because x has type tm A W but we need tm A w for +arbitrary w, which would require the structural rule for modal contexts. +Similarly (lam [x] diae x [w][y] t2e y) fails because y has type tm A w but we need +tm A W for the current world W. +The term for (dia (A => B) => dia A => dia B) fails because dia is not "normal" in S4. +The term (dia A => box B) => box (A => B) is true in Kripke semantics a la Simpson but +not in the judgmental formulation. +The two S5 theorems (dia A => box (dia A)) and (dia (box A) => box A) both fail because +S4 does not have the symmetry or Euclidean properties needed for S5. diff --git a/trans/output/lam.stelf b/trans/output/lam.stelf new file mode 100644 index 0000000..7703ce4 --- /dev/null +++ b/trans/output/lam.stelf @@ -0,0 +1,103 @@ +Lambda-Calculus Fragment of Mini-ML. +Author: Frank Pfenning + +Simple types +%sort tp + +%term arrow {_ tp} {_ tp} tp + +Expressions +%sort exp + +%term lam {_ {_ exp} exp} exp +%term app {_ exp} {_ exp} exp + +Type inference +|- E : T (expression E has type T) + +%sort of {_ exp} {_ tp} +%mode of %in %other + +%term tp_lam {E {exp} exp} {T1 tp} {T2 tp} {_ {x exp} {_ of x T1} of (E x) T2} of (lam E) (arrow T1 T2) + +%term tp_app {E1 exp} {E2 exp} {T1 tp} {T2 tp} {_ of E1 (arrow T2 T1)} {_ of E2 T2} of (app E1 E2) T1 + +Evaluation (call-by-value) +E ==> V (expression E evaluates to value V) + +%sort eval {_ exp} {_ exp} +%mode eval %in %out + +%term ev_lam {E {exp} exp} eval (lam E) (lam E) + +%term ev_app {E1 exp} {E2 exp} {V exp} {V2 exp} {E1' {exp} exp} {_ eval E1 (lam E1')} {_ eval E2 V2} {_ eval (E1' V2) V} eval (app E1 E2) V + +Regular world for type-checking +%block tp_var {T tp} {x exp} {u of x T} +%worlds (tp_var) (of E T) + +Type inference terminates +%terminates E (of E T) + +There is at least one typing rule for every expression +%covers of %in %other + +Closed worlds for evaluation +%worlds () (eval E V) + +There is at least one evaluation rule for every closed expression +%covers eval %in %out + +Type preservation as higher-level family +%sort tps {_ eval E V} {_ of E T} {_ of V T} + +%term tps_lam {E {exp} exp} {T1 tp} {T2 tp} {P {exp} {_ of _ T1} of _ T2} tps ev_lam (tp_lam P) (tp_lam P) + +%term tps_app {E1 exp} {E2 exp} {V exp} {V2 exp} {E1' {exp} exp} {T tp} {T2 tp} {D1 eval E1 (lam E1')} {D2 eval E2 V2} {D3 eval (E1' V2) V} {P1 of E1 (arrow T2 T)} {P2 of E2 T2} {Q1' {exp} {_ of _ T2} of _ T2} {Q2 of V2 T2} {Q of V T} {_ tps D1 P1 (tp_lam Q1')} {_ tps D2 P2 Q2} {_ tps D3 (Q1' V2 Q2) Q} tps (ev_app D3 D2 D1) (tp_app P2 P1) Q + +%mode tps %in %in %out +%worlds () (tps D P _) +%total D (tps D P _) + +Type preservation proven automatically +%theorem +tpsa : forall* {E exp} {V exp} {T tp} + forall {D eval E V} {P of E T} + exists {Q of V T} + true + +%prove 5 D (tpsa D P Q) + +Applying type preservation +%def e0 app (lam [x] x) (lam [y] y) +%? of e0 T +%? eval e0 V +%? tps d0 p0 Q + +Example of regular worlds +cp copies input to output. + +%sort cp {_ exp} {_ exp} + +%term cp_app {E1 exp} {E2 exp} {F1 exp} {F2 exp} {_ cp E1 F1} {_ cp E2 F2} cp (app E1 E2) (app F1 F2) + +%term cp_lam {E {exp} exp} {F {exp} exp} {_ {x exp} {_ cp x x} cp (E x) (F x)} cp (lam [x] E x) (lam [x] F x) + +%mode cp %in %out +%block cp_var {x exp} {u cp x x} +%worlds (cp_var) (cp E _) +%total E (cp E _) + +Following version cannot be checked: input coverage on parameter y is violated. +It would declare cp with a block containing {x:exp} {y:exp} {u:cp x y}, but cp_lam's +higher-order premise would allow y to differ from x, which violates input coverage. + +Following version also cannot be checked: output coverage on (F y) is violated. +It would add a premise cp y y -> cp (E x) (F y), meaning F y may not be covered +for the output position. + +%theorem cpt : forallG [x exp] [_ cp x x] + forall {E exp} + exists {F exp} {C cp E F} + true +%prove 5 E (cpt E _ _) diff --git a/trans/output/polylam.stelf b/trans/output/polylam.stelf new file mode 100644 index 0000000..eae18e9 --- /dev/null +++ b/trans/output/polylam.stelf @@ -0,0 +1,18 @@ +%sort tp +%term => {_ tp} {_ tp} tp +%prec %right 10 => +%term all {_ {_ tp} tp} tp + +%sort tm {_ tp} +%term lam {A tp} {B tp} {_ {_ tm A} tm B} tm (A => B) +%term app {A tp} {B tp} {_ tm (A => B)} {_ tm A} tm B +%term tlam {A {_ tp} tp} {_ {a tp} tm (A a)} tm (all A) +%term tapp {A {_ tp} tp} {_ tm (all A)} {B tp} tm (A B) + +%def nat all [a] a => (a => a) => a +%def zero tlam [a] lam [z] lam [s] z +%def succ lam [x] tlam [a] lam [z] lam [s] app s (app (app (tapp x a) z) s) +%def succ' tm (nat => nat) (lam [x] tlam [a] lam [z] lam [s] app s (app (app (tapp x a) z) s)) +%def plus lam [x] lam [y] app (app (tapp y nat) x) succ +%def times lam [x] lam [y] app (app (tapp y nat) zero) (app plus x) +%def exp lam [x] lam [y] app (app (tapp y nat) (app succ zero)) (app times x) From cf48091ec4788b7af446b4a888ed88a89bcb7046 Mon Sep 17 00:00:00 2001 From: Asher Frost Date: Sun, 7 Jun 2026 09:05:39 -0400 Subject: [PATCH 2/9] docs: improve Copilot instructions with build/test commands and conventions Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com> --- .github/copilot-instructions.md | 65 +++++++++++++++++++++++++++++++++ 1 file changed, 65 insertions(+) create mode 100644 .github/copilot-instructions.md diff --git a/.github/copilot-instructions.md b/.github/copilot-instructions.md new file mode 100644 index 0000000..a40db82 --- /dev/null +++ b/.github/copilot-instructions.md @@ -0,0 +1,65 @@ +# Copilot Instructions for stelf + +## Quick build / test / lint commands +- dune build # build (many modules may fail; expected) +- dune build @check # fast type-check only +- dune runtest # run all tests +- dune exec bin/main.exe # run Pal frontend (REPL/CLI) +- dune fmt # format (ocamlformat 0.28.1) +- make build/test/docs # Makefile targets: `make build`, `make test`, `make docs` +- make check # runs `dune build @check` via Makefile + +Running a single test +- Two Alcotest suites live under test/: + - test/Pal/PalTest.ml → dune exec test/Pal/PalTest.exe + - test/Parse/ParseTest.ml → dune exec test/Parse/ParseTest.exe -- test '' +- For debugging a failing test: run its executable directly (dune exec ...), read output, then inspect the test file in test/*/Cases.ml. + +Notes +- Do NOT use `dune utop` or `make repl` — utop preloads compiler-libs which conflicts with project modules (e.g., Lambda, Lexer, Parser). +- The `basis/` directory is a pinned submodule providing SML basis shims. Clone with submodules: + git clone --recurse-submodules + or: git submodule update --init + +## High-level architecture (big picture) +- ~40 Dune libraries under src/, layered roughly: + - Foundation: global, trail, table, stream, timing + - Core LF: src/IntSyn (IntSyn, Whnf, Conv, Unify, Abstract) + - Middle: names, paths, print, index, modes, subordinate, typecheck, modules + - Analysis: terminate, thm, cover, meta, prover, tomega + - Execution: compile, opsem, solvers + - Frontend: src/frontend (Lexer, Parser, Recon*, Twelf_, Frontend_, Solve) +- Two frontend variants to know: `modern` (parser) and `pal` (combined frontend used by bin/main.exe). +- Many modules are wired by functor instantiation in *_.sml.ml files; top-level wiring is in src/frontend/frontend_.sml.ml. + +## Key repository conventions (non-obvious) +- Two source styles: + - Old-style (most src/*): three-part module pattern concatenated by Dune: + - .sig.ml (signature), .fun.ml (functor implementation), .sml.ml (instantiation/wiring) + - Check `sources.dune` files in subdirs for concatenation rules. + - New-style (src/Common, src/Recon, src/Lang, src/frontend, src/Fronts): single-file modules and explicit interfaces. +- Naming: + - SML structure → OCaml `module Foo` + - SML signature → `module type FOO` + - Functors → `module MakeFoo(...)` or `module Foo(...)` + - Constructors colliding with OCaml keywords append `_` (ok_, abort_, Type_) +- Compatibility: + - Old-style files `open Basis` and prefer Basis.* (Basis.Array, Basis.List) over OCaml stdlib. + - Old-style libraries: `wrapped = false` and build flags include `-w -A -open Basis`. + - New code should be `wrapped = true`. +- Formatting: ocamlformat 0.28.1 (default profile). +- Tests: test cases are in test/*/Cases.ml; Common.ml has helpers. +- Submodules: `basis/` and `twelf/` are submodules; ensure initialized when building. + +## Where to look first when investigating +1. Re-run the failing test (dune exec path/to/Exe) and read its output. +2. Open the test Cases.ml and the code under test in src/ that the test imports. +3. Limit early file reads to ~3 files; then form a hypothesis before deeper searches. +4. Use `dune build @check` to perform a fast type-check. + +## Other AI/assistant configs in repo +- CLAUDE.md (project guidance for Claude) +- .github/agents/ocaml-doc-converter.agent.md +- Keep these in sync when updating instructions. + +Summary: updated instructions include exact Makefile/dune commands, single-test examples, submodule notes, and the three-file module convention. From eec20f053a14495598388ca8c170166b39f666a7 Mon Sep 17 00:00:00 2001 From: Asher Frost Date: Sun, 7 Jun 2026 18:08:47 -0400 Subject: [PATCH 3/9] chore(tests): added many more pal tests --- src/Fronts/Pal/Help.ml | 13 +- src/Resugar/RESUGAR.ml | 5 + src/Resugar/Resugar.ml | 0 src/Resugar/dune | 4 + src/modes/Modecheck.ml | 687 ++++++++++------------- test/Pal/Cases.ml | 269 +++++++-- test/Pal/Common.ml | 25 +- test/Pal/Source.ml | 1182 +++++++++++++++++++++++++++++++++++++++- 8 files changed, 1715 insertions(+), 470 deletions(-) create mode 100644 src/Resugar/RESUGAR.ml create mode 100644 src/Resugar/Resugar.ml create mode 100644 src/Resugar/dune diff --git a/src/Fronts/Pal/Help.ml b/src/Fronts/Pal/Help.ml index 9eb9148..98819fd 100644 --- a/src/Fronts/Pal/Help.ml +++ b/src/Fronts/Pal/Help.ml @@ -1,7 +1,8 @@ -let rec help : string Fmt.t = fun fmt s -> help' s fmt () +let help : string list -> Display.form = function + | ["help"] | ["%help"] -> assert false + | ["set"] | ["%set"] -> assert false + | ["get"] | ["%get"] -> assert false + | ["sort"] | ["%sort"] -> assert false + | ["?"] | ["%?"] -> assert false + -and help' : string -> unit Fmt.t = function - | "" -> assert false - | "help" | "%help" -> assert false - | "sort" | "%sort" -> assert false - | "expr" -> assert false diff --git a/src/Resugar/RESUGAR.ml b/src/Resugar/RESUGAR.ml new file mode 100644 index 0000000..6a09d72 --- /dev/null +++ b/src/Resugar/RESUGAR.ml @@ -0,0 +1,5 @@ +module type RESUGAR = sig + module Syntax : Syntax.SYNTAX + module Cst : Cst.CST + type t + diff --git a/src/Resugar/Resugar.ml b/src/Resugar/Resugar.ml new file mode 100644 index 0000000..e69de29 diff --git a/src/Resugar/dune b/src/Resugar/dune new file mode 100644 index 0000000..b31b131 --- /dev/null +++ b/src/Resugar/dune @@ -0,0 +1,4 @@ +(library + (name resugar) + (public_name stelf.resugar) + (libraries syntax cst)) diff --git a/src/modes/Modecheck.ml b/src/modes/Modecheck.ml index ac29b45..a41e0cb 100644 --- a/src/modes/Modecheck.ml +++ b/src/modes/Modecheck.ml @@ -16,9 +16,9 @@ open Modesyn open Modetable open Origins -(* Mode Checking *) -(* Author: Carsten Schuermann *) -(* Modified: Frank Pfenning, Roberto Virga *) +(** Mode Checking *) +(** @author Carsten Schuermann *) +(** Modified: Frank Pfenning, Roberto Virga *) module MakeModeCheck (ModeTable : MODETABLE) (Whnf : WHNF) @@ -37,16 +37,21 @@ module MakeModeCheck module M = ModeSyn module P = Paths + (** Uniqueness information *) type uniqueness = Unique | Ambig [@@deriving eq, ord, show] + (** Groundedness information *) type info = Free | Unknown | Ground of uniqueness [@@deriving eq, ord, show] + (** Variable status *) type status = Existential of info * string option | Universal [@@deriving eq, ord, show] + (** hack: if true, check freeness of output arguments in subgoals *) let checkFree = ref false + (** copied from worldcheck/worldsyn.fun *) let rec wrapMsg (c, occ, msg) = begin match Origins.originLookup c with | fileName, None -> (fileName ^ ":") ^ msg @@ -65,6 +70,12 @@ module MakeModeCheck let modeError' (occ, msg) = Display.debug (Display.Form.string msg) ; raise (ModeError (occ, msg)) let error'' (occ, msg) = Display.debug (Display.Form.string msg) ; raise (Error' (occ, msg)) + (** [lookup (a, occ) = mSs] + + Invariant: mS are the argument modes for a + Raises an error if no mode for a has declared. + (occ is used in error message) + *) let rec lookup (a, occ) = begin match ModeTable.mmodeLookup a with | [] -> @@ -79,27 +90,64 @@ module MakeModeCheck | Existential (_, Some name) -> name | _ -> "?" + (** unique (k, ks) = B + + Invariant: + B iff k does not occur in ks + *) let rec unique = function | k, [] -> true | k, k' :: ks -> k <> k' && unique (k, ks) + (** isUniversal S = B + + Invariant: + B iff S = Par + *) let isUniversal = function Universal -> true | _ -> false + (** isGround S = B + + Invariant: + B iff S = Ex (T x) + *) let isGround = function Existential (Ground _, _) -> true | _ -> false + (** uniqueness S = u + where u is the uniqueness property of status S + *) let rec uniqueness = function | Existential (Ground u, _) -> u | Universal -> Unique + (** ambiguate (mode) = mode' + where mode' forgets uniqueness properties + *) let rec ambiguate = function | M.Plus -> M.Plus | M.Minus -> M.Minus | M.Minus1 -> M.Minus + (** andUnique (u1, u2) = Unique if u1 = u2 = Unique + = Ambig otherwise + *) let andUnique = function Unique, Unique -> Unique | _ -> Ambig + (** isFree S = b + + Invariant: + b iff S = Ex (B x) + *) let isFree = function Existential (Free, _) -> true | _ -> false exception Eta + (** etaContract (U, n) = k + + if lam V1... lam Vn. U =eta*=> k + otherwise raise exception Eta + + Invariant: G, V1,..., Vn |- U : V for some G, Vi, V. + U in NF + *) let rec etaContract = function | I.Root (I.BVar k, s_), n -> begin if k > n then begin @@ -118,6 +166,13 @@ module MakeModeCheck else raise Eta end + (** isPattern (D, k, mS) = B + + Invariant: + B iff k > k' for all k' in mS + and for all k in mS: k is parameter + and for all k', k'' in mS: k' <> k'' + *) let rec checkPattern = function | d_, k, args, I.Nil -> () | d_, k, args, I.App (u_, s_) -> @@ -136,6 +191,9 @@ module MakeModeCheck end with Eta -> false + (* ------------------------------------------- strictness check *) + (* This repeats some code from ../typecheck/strict.fun *) + (* Interface here is somewhat different *) let rec strictExpN = function | d_, _, I.Uni _ -> false | d_, p, I.Lam (_, u_) -> strictExpN (I.Decl (d_, Universal), p + 1, u_) @@ -157,6 +215,7 @@ module MakeModeCheck | I.FgnConst (cs, conDec) -> strictSpineN (d_, p, s_) end | d_, p, I.FgnExp (cs, ops) -> false + (* this is a hack - until we investigate this further -rv *) and strictSpineN = function | _, _, I.Nil -> false @@ -165,6 +224,16 @@ module MakeModeCheck and strictDecN (d_, p, I.Dec (_, v_)) = strictExpN (d_, p, v_) + (* ------------------------------------------- freeness check *) + (** freeExpN (D, mode, U, occ = () + + If G |- U : V (U in nf) + and G ~ D + then freeExpN terminates with () if D |- U free + else exception ModeError is raised + + (occ and mode are used in error messages) + *) let rec freeExpN = function | d_, d, mode, I.Root (I.BVar k, s_), occ, strictFun -> begin freeVar (d_, d, mode, k, P.head occ, strictFun); @@ -183,6 +252,15 @@ module MakeModeCheck I.FgnExpStd.App.apply (csfe1, csfe2) (function u_ -> freeExpN (d_, d, mode, Whnf.normalize (u_, I.id), occ, strictFun)) + (** freeSpineN (D, mode, S, occ, strictFun) = () + + If G |- S : V1 > V2 (S in nf) + and G ~ D + then freeSpineN terminates with () if D |- S free + else exception ModeError is raised + + (occ and mode are used in error messages) + *) and freeSpineN = function | d_, d, mode, I.Nil, _, strictFun -> () | d_, d, mode, I.App (u_, s_), (p, occ), strictFun -> begin @@ -190,6 +268,15 @@ module MakeModeCheck freeSpineN (d_, d, mode, s_, (p + 1, occ), strictFun) end + (** freeVar (D, mode, k, occ, strictFun) = () + + If G |- k : V1 + and G ~ D + then freeVar terminates with () if D |- S free + else exception ModeError is raised + + (occ and mode are used in error messages) + *) and freeVar (d_, d, mode, k, occ, strictFun) = let status = I.ctxLookup (d_, k) in begin if isFree status || isUniversal status || strictFun (k - d) then () @@ -202,6 +289,14 @@ module MakeModeCheck ^ " argument not free" )) end + (* -------------------------------- non-strict mode context update *) + (** nonStrictExpN (D, U) = D' + + If G |- U : V (U in nf) + and D ~ G + then D' >= D where D'(k) Unknown for all existential variables k + in U that are free in D + *) let rec nonStrictExpN = function | d_, I.Root (I.BVar k, s_) -> nonStrictSpineN (nonStrictVarD (d_, k), s_) | d_, I.Root (I.Const c, s_) -> nonStrictSpineN (d_, s_) @@ -213,16 +308,43 @@ module MakeModeCheck raise (Error "Foreign expressions not permitted when checking freeness") + (** nonStrictSpineN (D, S) = D' + + If G |- S : V1 > V2 (S in nf) + and D ~ G + then D' >= D' where D'(k) Unkown for all existential variables k + in S that are Free in D + *) and nonStrictSpineN = function | d_, I.Nil -> d_ | d_, I.App (u_, s_) -> nonStrictSpineN (nonStrictExpN (d_, u_), s_) + (** nonStrictVarD (D, k) = D' + + If G |- k : V + and D ~ G + and k is an existential variable + then D' >= D where k is nonStrictd as described in nonStrictExpN + *) and nonStrictVarD = function | I.Decl (d_, Existential (Free, name)), 1 -> I.Decl (d_, Existential (Unknown, name)) | d_, 1 -> d_ | I.Decl (d_, status), k -> I.Decl (nonStrictVarD (d_, k - 1), status) + (* ------------------------------------------- mode context update *) + (** updateExpN (D, U, u) = D' + + If G |- U : V (U in nf) + and D ~ G + then D' >= D where D'(k) Ground for all existential variables k + with a strict occurrence in U + and D'(k) Unkown for all existential variable k + with a non-strict occurrence, but no strict occurrence in U + (if !checkFree is true) + + u is the uniqueness property for the new ground assumptions + *) let rec updateExpN = function | d_, I.Root (I.BVar k, s_), u -> begin if isUniversal (I.ctxLookup (d_, k)) then @@ -243,15 +365,46 @@ module MakeModeCheck I.ctxPop (updateExpN (I.Decl (d_, Universal), u_, u)) | d_, I.FgnExp (csfe1, csfe2), u -> d_ + (** updateSpineN (D, S, u) = D' + + If G |- S : V1 > V2 (S in nf) + and D ~ G + then D' >= D' where D'(k) Ground for all existential variables k + with a strict occurrence in S + *) and updateSpineN = function | d_, I.Nil, u -> d_ | d_, I.App (u_, s_), u -> updateSpineN (updateExpN (d_, u_, u), s_, u) + (** updateVarD (D, k, u) = D' + + If G |- k : V + and D ~ G + and k is an existential variable + then D' >= D where k is updated as described in updateExpN + *) and updateVarD = function | I.Decl (d_, Existential (_, name)), 1, u -> I.Decl (d_, Existential (Ground u, name)) | I.Decl (d_, status), k, u -> I.Decl (updateVarD (d_, k - 1, u), status) + (* ----------------------- mode context update by argument modes *) + (** updateAtom (D, m, S, mS, (p,occ)) = D' + + If G |- S : V > V' ( S = U1 ; .. ; Un) + and D ~ G + and S ~ mS (mS = m1 , .. , mn) + and m mode + then D' >= D where + all Ui are updated if mi = m (mod uniqueness) + + The new ground variables are marked Unique + if m = (-1) and mi = (-1) (when updating from subgoals with unique inputs) + or m = mi = (+) (when updating from the clause head) + Otherwise they are marked Ambig. + + (p,occ) is used in error message if freeness is to be checked + *) let rec updateAtom' = function | d_, _mode, I.Nil, M.Mnil, _ -> d_ | d_, M.Plus, I.App (u_, s_), M.Mapp (M.Marg (M.Plus, _), mS), (p, occ) -> @@ -272,7 +425,16 @@ module MakeModeCheck (updateExpN (d_, u_, Unique), M.Minus1, s_, mS, (p + 1, occ)) | d_, mode, I.App (u_, s_), M.Mapp (_, mS), (p, occ) -> updateAtom' (d_, mode, s_, mS, (p + 1, occ)) + (* when checking freeness, all arguments must be input (+) or output (-) *) + (* therefore, no case for M.Mapp (M.Marg (M.Minus, _), mS) is provided here *) + (** freeAtom (D, m, S, (V,s), mS, (p, occ)) = () + + checks if all output arguments in S according to mS are free. + Invariant: G |- S : V[s] >> P for some G and P (S in nf) + G ~ D + mode = (-) or (+); ( * ) or (-1) are excluded + *) let rec freeAtom = function | d_, _mode, I.Nil, _vs_, M.Mnil, _ -> () | ( d_, @@ -306,6 +468,9 @@ module MakeModeCheck mS, (p + 1, occ) ) + (** updateAtom (D, m, S, a, mS, (p, occ)) + see updateAtom', and performs additional freeness check if required + *) let rec updateAtom (d_, mode, s_, a, mS, (p, occ)) = let _ = begin if !checkFree then @@ -315,6 +480,22 @@ module MakeModeCheck in updateAtom' (d_, mode, s_, mS, (p, occ)) + (* ------------------------------------------- groundness check *) + (** groundExpN (D, mode, U, occ) = u + + If G |- U : V (U in nf) + and G ~ D + then if mode = (+) or (-) + then groundExpN terminates with u if D |- U ground + else exception ModeError is raised + if mode = (-1) then D |- U ground and U unique + else exception ModeError is raised + + u = Unique if all known variables in U are Unique + u = Ambig otherwise + + (occ and mode are used in error messages) + *) let rec groundExpN = function | d_, mode, I.Root (I.BVar k, s_), occ -> andUnique @@ -335,7 +516,23 @@ module MakeModeCheck andUnique (groundExpN (d_, mode, Whnf.normalize (u_, I.id), occ), u)) Unique + (* punting on the occ here - ak *) + + (** groundSpineN (D, mode, S, occ) = u + + If G |- S : V1 > V2 (S in nf) + and G ~ D + then if mode = (+) or (-) + then groundSpineN terminates with u if D |- S ground + else exception ModeError is raised + if mode = (-1) then D |- S ground and S unique + else exception ModeError is raised + + u = Unique if all known variables in S are Unique + u = Ambig otherwise + (occ and mode are used in error messages) + *) and groundSpineN = function | d_, mode, I.Nil, _ -> Unique | d_, mode, I.App (u_, s_), (p, occ) -> @@ -343,6 +540,20 @@ module MakeModeCheck ( groundExpN (d_, mode, u_, P.arg (p, occ)), groundSpineN (d_, mode, s_, (p + 1, occ)) ) + (** groundVar (D, mode, k, occ) = u + + If G |- k : V1 + and G ~ D + then if mode = (+) or (-) + then groundVar terminates with u if D |- k ground + else exception ModeError is raised + if mode = (-1) then D |- k ground and k unique + else exception ModeError is raised + + u = Unique if k is known to be unique, Ambig otherwise + + (occ and mode are used in error messages) + *) and groundVar = function | d_, M.Minus1, k, occ -> begin match I.ctxLookup (d_, k) with @@ -375,6 +586,23 @@ module MakeModeCheck ^ " argument not necessarily ground" )) end + (* ------------------------------------------- groundness check by polarity *) + (** groundAtom (D, m, S, mS, (p,occ)) = u + + If G |- S : V > V' ( S = U1 ; .. ; Un) + and D ~ G + and S ~ mS (mS = m1 , .. , mn) + and m mode = (+) or (-1) + then groundAtom returns u if D |- Ui ground + for all i s.t. mi = m (mod uniqueness) + and checks that D |- Ui unique if mi = (-1) and m = (-) + otherwise exception ModeError is raised + + u = Unique if all mi = m (mod uniqueness) are unique, + u = Ambig otherwise + + ((p,occ) used in error messages) + *) let rec groundAtom = function | d_, _, I.Nil, M.Mnil, _ -> Unique | d_, M.Plus, I.App (u_, s_), M.Mapp (M.Marg (M.Plus, _), mS), (p, occ) -> @@ -383,20 +611,43 @@ module MakeModeCheck groundAtom (d_, M.Plus, s_, mS, (p + 1, occ)) ) | d_, M.Minus, I.App (u_, s_), M.Mapp (M.Marg (M.Minus, _), mS), (p, occ) -> - andUnique - ( groundExpN (d_, M.Minus, u_, P.arg (p, occ)), - groundAtom (d_, M.Minus, s_, mS, (p + 1, occ)) ) + ignore (groundExpN (d_, M.Minus, u_, P.arg (p, occ))); + groundAtom (d_, M.Minus, s_, mS, (p + 1, occ)) | d_, M.Minus, I.App (u_, s_), M.Mapp (M.Marg (M.Minus1, _), mS), (p, occ) -> - andUnique - ( groundExpN (d_, M.Minus1, u_, P.arg (p, occ)), - groundAtom (d_, M.Minus, s_, mS, (p + 1, occ)) ) + ignore (groundExpN (d_, M.Minus1, u_, P.arg (p, occ))); + groundAtom (d_, M.Minus, s_, mS, (p + 1, occ)) | d_, mode, I.App (u_, s_), M.Mapp (_, mS), (p, occ) -> groundAtom (d_, mode, s_, mS, (p + 1, occ)) let rec ctxPush (m, ds_) = List.map (function d_ -> I.Decl (d_, m)) ds_ let rec ctxPop ds_ = List.map (function I.Decl (d_, m) -> d_) ds_ + (* ------------------------------------------- mode checking first phase *) + (* ctxPush (Ds, m) = Ds' + raises the contexts Ds prepending m + *) + (* ctxPop Ds = Ds' + lowers the contexts Ds + *) + (** checkD1 (D, V, occ, k) = () + + Invariant: + if G |- V : L + and V does not contain Skolem constants + and D ~ G + then + for each mode mS of the head of V + exists some Di s.t. all (-) evars of mS are ground + where D' ~ G, D' >= D is obtained by updating D + and k D' = [D1, ..., Di, ..., Dn] + and Di ~ G, Di >= D' is obtained by mode checking on the subgoals of V + + exception ModeError is raised if the expression does not mode check + exception Error' is raised if the expression contains type families + that have no mode information associated with them + (occ used in error messages) + *) let rec checkD1 = function | d_, I.Pi ((I.Dec (name, _), Maybe), v_), occ, k -> checkD1 @@ -418,9 +669,11 @@ module MakeModeCheck | mS :: mSs -> let rec checkSome = function | d'_ :: [] -> + (* D' is the only (last) possibility; on failure, we raise ModeError *) ignore (groundAtom (d'_, M.Minus, s_, mS, (1, occ))); checkAll mSs | d'_ :: ds_ -> + (* try D', if it doesn't work, try another context in the Ds *) (try ignore (groundAtom (d'_, M.Minus, s_, mS, (1, occ))) with ModeError _ -> checkSome ds_); checkAll mSs @@ -434,9 +687,11 @@ module MakeModeCheck | mS :: mSs -> let rec checkSome = function | d'_ :: [] -> + (* D' is the only (last) possibility; on failure, we raise ModeError *) ignore (groundAtom (d'_, M.Minus, s_, mS, (1, occ))); checkAll mSs | d'_ :: ds_ -> + (* try D', if it doesn't work, try another context in the Ds *) (try ignore (groundAtom (d'_, M.Minus, s_, mS, (1, occ))) with ModeError _ -> checkSome ds_); checkAll mSs @@ -445,6 +700,20 @@ module MakeModeCheck in checkAll (lookup (d, occ)) + (** checkG1 (D, V, occ, k) = Ds + + Invariant: + if G |- V : L + and V does not contain Skolem constants + and D ~ G + then forall D' >= D that mode checks V, (k D') is a sublist of Ds + and for each Di in Ds, Di ~ G and Di >= D' + + exception ModeError is raised if the expression does not mode check + exception Error' is raised if the expression contains type families + that have no mode information associated with them + (occ used in error messages) + *) and checkG1 = function | d_, I.Pi ((_, Maybe), v_), occ, k -> ctxPop @@ -516,12 +785,29 @@ module MakeModeCheck in checkList false (lookup (d, occ)) + (** checkDlocal (D, V, occ) = () + + Invariant: + If G |- V : L + and D ~ G + then checkD terminates with () iff V is mode correct. + + otherwise exception ModeError is raised (occ used in error messages) + *) let rec checkDlocal (d_, v_, occ) = try checkD1 (d_, v_, occ, function d'_ -> [ d'_ ]) with ModeError (occ, msg) -> error'' ( (occ, msg)) + (* --------------------------------------------------------- mode checking *) let cidFromHead = function I.Const a -> a | I.Def a -> a + (** checkD (ConDec, occOpt) = () + + checkD terminates with () if ConDec is mode correct + otherwise exception Error is raised + + (occOpt is used in error messages) + *) let rec checkD (conDec, fileName, occOpt) = let _ = checkFree := false in let rec checkable = function @@ -550,10 +836,8 @@ module MakeModeCheck let rec checkAll = function | [] -> () | I.Const c :: clist -> begin - begin if !Global.chatter > 3 then - print' (Names.qidToString (Names.constQid c) ^ " ") - else () - end; + Display.(debug Form.(string "checking mode of constant" ++ space () ++ shown (fun x -> Names.show_qid @@ Names.constQid c) c ++ space () ++ string "..." ++ nl ())); + (try checkDlocal (I.Null, I.constType c, P.top) with Error' (occ, msg) -> raise (Error (wrapMsg (c, occ, msg)))); checkAll clist @@ -606,383 +890,6 @@ module MakeModeCheck () end - (* Uniqueness information *) - (* u ::= Unique *) - (* | Ambig *) - (* Groundedness information *) - (* I ::= Free *) - (* | Unknown *) - (* | Ground *) - (* Variable status *) - (* S ::= Existential (I, xOpt) *) - (* | Universal *) - (* hack: if true, check freeness of output arguments in subgoals *) - (* Representation invariant: - - Groundness information: - T stands for ground, B stands for unknown - Ex for a named existential variable - Par for a parameter - - Mode context D ::= . | D, S - - If D contains Status information for variables in - a context G, we write G ~ D - We write D' >= D if for all i - D(i) par iff D'(i) par - D(i) bot implies D'(i) bot or D'(i) top - D(i) top implies D'(i) top - *) - (* copied from worldcheck/worldsyn.fun *) - (* lookup (a, occ) = mSs - - Invariant: mS are the argument modes for a - Raises an error if no mode for a has declared. - (occ is used in error message) - *) - (* nameOf S, selects a name for S *) - (* unique (k, ks) = B - - Invariant: - B iff k does not occur in ks - *) - (* isUniversal S = B - - Invariant: - B iff S = Par - *) - (* isGround S = B - - Invariant: - B iff S = Ex (T x) - *) - (* uniqueness S = u - where u is the uniqueness property of status S - *) - (* ambiguate (mode) = mode' - where mode' forgets uniqueness properties - *) - (* andUnique (u1, u2) = Unique if u1 = u2 = Unique - = Ambig otherwise - *) - (* isFree S = b - - Invariant: - b iff S = Ex (B x) - *) - (* etaContract (U, n) = k - - if lam V1... lam Vn. U =eta*=> k - otherwise raise exception Eta - - Invariant: G, V1,..., Vn |- U : V for some G, Vi, V. - U in NF - *) - (* etaSpine (S, n) = () - if S =eta*=> n ; n-1 ; ... ; 1 ; Nil - otherwise raise exception Eta - - Invariant: G |- S : V1 >> V2 for some G, V1, V2 - S in NF - *) - (* S[s] should be impossible *) - (* isPattern (D, k, mS) = B - - Invariant: - B iff k > k' for all k' in mS - and for all k in mS: k is parameter - and for all k', k'' in mS: k' <> k'' - *) - (* ------------------------------------------- strictness check *) - (* This repeats some code from ../typecheck/Strict.fun *) - (* Interface here is somewhat different *) - (* strictExpN (D, p, U) = B - - Invariant: - If D |- U : V - and U is in nf (normal form) - then B iff U is strict in p - *) - (* checking D in this case should be redundant -fp *) - (* strictDecN (D, p, D) orelse *) - (* equivalently: isUniversal .. andalso strictSpineN .. *) - (* no other cases possible *) - (* this is a hack - until we investigate this further -rv *) - (* no other cases possible *) - (* strictSpineN (D, S) = B - - Invariant: - If D |- S : V > W - and S is in nf (normal form) - then B iff S is strict in k - *) - (* - fun strictAtom (D, p, mode, I.Nil, (V, s), M.Mnil) = false - | strictAtom (D, p, M.Minus, I.App (U, S), (I.Pi ((I.Dec (_, V1), _), V2), s), - M.Mapp (M.Marg (M.Minus, _), mS)) = - strictExpN (D, p, Whnf.normalize (V1, s)) - orelse strictAtom (D, p, M.Minus, S, Whnf.whnfExpandDef (V2, I.Dot (I.Exp U, s)), mS) - | strictAtom (D, p, mode, I.App (U, S), (I.Pi (_, V2), s), M.Mapp(_, mS)) = - strictAtom (D, p, mode, S, Whnf.whnfExpandDef (V2, I.Dot (I.Exp U, s)), mS) - *) - (* ------------------------------------------- freeness check *) - (* freeExpN (D, mode, U, occ = () - - If G |- U : V (U in nf) - and G ~ D - then freeExpN terminates with () if D |- U free - else exception ModeError is raised - - (occ and mode are used in error messages) - *) - (* punting on the occ here - ak *) - (* freeSpineN (D, mode, S, occ, strictFun) = () - - If G |- S : V1 > V2 (S in nf) - and G ~ D - then freeSpineN terminates with () if D |- S free - else exception ModeError is raised - - (occ and mode are used in error messages) - *) - (* freeVar (D, mode, k, occ, strictFun) = () - - If G |- k : V1 - and G ~ D - then freeVar terminates with () if D |- S free - else exception ModeError is raised - - (occ and mode are used in error messages) - *) - (* -------------------------------- non-strict mode context update *) - (* nonStrictExpN (D, U) = D' - - If G |- U : V (U in nf) - and D ~ G - then D' >= D where D'(k) Unknown for all existential variables k - in U that are free in D - *) - (* nonStrictSpineN (D, S) = D' - - If G |- S : V1 > V2 (S in nf) - and D ~ G - then D' >= D' where D'(k) Unkown for all existential variables k - in S that are Free in D - *) - (* nonStrictVarD (D, k) = D' - - If G |- k : V - and D ~ G - and k is an existential variable - then D' >= D where k is nonStrictd as described in nonStrictExpN - *) - (* Universal, or already Unknown or Ground - leave unchanged *) - (* ------------------------------------------- mode context update *) - (* updateExpN (D, U, u) = D' - - If G |- U : V (U in nf) - and D ~ G - then D' >= D where D'(k) Ground for all existential variables k - with a strict occurrence in U - and D'(k) Unkown for all existential variable k - with a non-strict occurrence, but no strict occurrence in U - (if !checkFree is true) - - u is the uniqueness property for the new ground assumptions - *) - (* no occurrence inside a FgnExp is considered strict *) - (* updateSpineN (D, S, u) = D' - - If G |- S : V1 > V2 (S in nf) - and D ~ G - then D' >= D' where D'(k) Ground for all existential variables k - with a strict occurrence in S - *) - (* updateVarD (D, k, u) = D' - - If G |- k : V - and D ~ G - and k is an existential variable - then D' >= D where k is updated as described in updateExpN - *) - (* ----------------------- mode context update by argument modes *) - (* updateAtom (D, m, S, mS, (p,occ)) = D' - - If G |- S : V > V' ( S = U1 ; .. ; Un) - and D ~ G - and S ~ mS (mS = m1 , .. , mn) - and m mode - then D' >= D where - all Ui are updated if mi = m (mod uniqueness) - - The new ground variables are marked Unique - if m = (-1) and mi = (-1) (when updating from subgoals with unique inputs) - or m = mi = (+) (when updating from the clause head) - Otherwise they are marked Ambig. - - (p,occ) is used in error message if freeness is to be checked - *) - (* when checking freeness, all arguments must be input (+) or output (-) *) - (* therefore, no case for M.Mapp (M.Marg (M.Minus, _), mS) is provided here *) - (* freeAtom (D, m, S, (V,s), mS, (p, occ)) = () - - checks if all output arguments in S according to mS are free. - Invariant: G |- S : V[s] >> P for some G and P (S in nf) - G ~ D - mode = (-) or (+); ( * ) or (-1) are excluded - *) - (* updateAtom (D, m, S, a, mS, (p, occ)) - see updateAtom', and performs additional freeness check if required - *) - (* ------------------------------------------- groundness check *) - (* groundExpN (D, mode, U, occ) = u - - If G |- U : V (U in nf) - and G ~ D - then if mode = (+) or (-) - then groundExpN terminates with u if D |- U ground - else exception ModeError is raised - if mode = (-1) then D |- U ground and U unique - else exception ModeError is raised - - u = Unique if all known variables in U are Unique - u = Ambig otherwise - - (occ and mode are used in error messages) - *) - (* punting on occ here - ak *) - (* groundSpineN (D, mode, S, occ) = u - - If G |- S : V1 > V2 (S in nf) - and G ~ D - then if mode = (+) or (-) - then groundSpineN terminates with u if D |- S ground - else exception ModeError is raised - if mode = (-1) then D |- S ground and S unique - else exception ModeError is raised - - u = Unique if all known variables in S are Unique - u = Ambig otherwise - - (occ and mode are used in error messages) - *) - (* groundVar (D, mode, k, occ) = u - - If G |- k : V1 - and G ~ D - then if mode = (+) or (-) - then groundVar terminates with u if D |- k ground - else exception ModeError is raised - if mode = (-1) then D |- k ground and k unique - else exception ModeError is raised - - u = Unique if k is known to be unique, Ambig otherwise - - (occ and mode are used in error messages) - *) - (* Existential (Free, _) or Existential (Unknown, _) *) - (* ------------------------------------------- groundness check by polarity *) - (* groundAtom (D, m, S, mS, (p,occ)) = u - - If G |- S : V > V' ( S = U1 ; .. ; Un) - and D ~ G - and S ~ mS (mS = m1 , .. , mn) - and m mode = (+) or (-1) - then groundAtom returns u if D |- Ui ground - for all i s.t. mi = m (mod uniqueness) - and checks that D |- Ui unique if mi = (-1) and m = (-) - otherwise exception ModeError is raised - - u = Unique if all mi = m (mod uniqueness) are unique, - u = Ambig otherwise - - ((p,occ) used in error messages) - *) - (* ignore uniqueness result here *) - (* ignore uniqueness result here *) - (* ------------------------------------------- mode checking first phase *) - (* ctxPush (Ds, m) = Ds' - raises the contexts Ds prepending m - *) - (* ctxPop Ds = Ds' - lowers the contexts Ds - *) - (* checkD1 (D, V, occ, k) = () - - Invariant: - if G |- V : L - and V does not contain Skolem constants - and D ~ G - then - for each mode mS of the head of V - exists some Di s.t. all (-) evars of mS are ground - where D' ~ G, D' >= D is obtained by updating D - and k D' = [D1, ..., Di, ..., Dn] - and Di ~ G, Di >= D' is obtained by mode checking on the subgoals of V - - exception ModeError is raised if the expression does not mode check - exception Error' is raised if the expression contains type families - that have no mode information associated with them - (occ used in error messages) - *) - (* for a declaration, all modes must be satisfied *) - (* D' is the only (last) possibility; on failure, we raise ModeError *) - (* ignore return *) - (* try D', if it doesn't work, try another context in the Ds *) - (* ignore return *) - (* for a declaration, all modes must be satisfied *) - (* D' is the only (last) possibility; on failure, we raise ModeError *) - (* ignore return *) - (* try D', if it doesn't work, try another context in the Ds *) - (* ignore return *) - (* checkG1 (D, V, occ, k) = Ds - - Invariant: - if G |- V : L - and V does not contain Skolem constants - and D ~ G - then forall D' >= D that mode checks V, (k D') is a sublist of Ds - and for each Di in Ds, Di ~ G and Di >= D' - - exception ModeError is raised if the expression does not mode check - exception Error' is raised if the expression contains type families - that have no mode information associated with them - (occ used in error messages) - *) - (* for a goal, at least one mode must be satisfied *) - (* found = true *) - (* mS is the last possible mode to check; - if the check fails, we don't catch ModeError *) - (* uniqueness not permitted on multiple modes right now *) - (* Wed Aug 20 21:52:31 2003 -fp *) - (* found' is true iff D satisfies mS *) - (* handler scope??? -fp *) - (* compute all other mode contexts *) - (* for a goal, at least one mode must be satisfied *) - (* found = true *) - (* mS is the last possible mode to check; - if the check fails, we don't catch ModeError *) - (* uniqueness not permitted on multiple modes right now *) - (* Wed Aug 20 21:52:31 2003 -fp *) - (* found' is true iff D satisfies mS *) - (* compute all other mode contexts *) - (* checkDlocal (D, V, occ) = () - - Invariant: - If G |- V : L - and D ~ G - then checkD terminates with () iff V is mode correct. - - otherwise exception ModeError is raised (occ used in error messages) - *) - (* --------------------------------------------------------- mode checking *) - (* checkD (ConDec, occOpt) = () - - checkD terminates with () if ConDec is mode correct - otherwise exception Error is raised - - (occOpt is used in error messages) - *) let checkD = checkD let checkMode = checkMode let checkFreeOut = checkFreeOut diff --git a/test/Pal/Cases.ml b/test/Pal/Cases.ml index b691850..f3d6d3a 100644 --- a/test/Pal/Cases.ml +++ b/test/Pal/Cases.ml @@ -29,56 +29,225 @@ let cases () = begin [ ( "%term and %sort", - [ - test "Natural Numbers (nat)" nat_test; - test "Natural Numbers (nat, add)" (nat_test @ add_test); - test "Natural Numbers (nat, add, mul)" - (nat_test @ add_test @ mul_test); - test ~failure:true "Natural numbers (ill-formed)" - [ - {| %sort nat |}; - {| %term zero nat |}; - {| %term succ {_ bool} nat |}; - ]; - ] ); + List.concat + [ + snd (test "Natural Numbers (nat)" nat_test); + snd (test "Natural Numbers (nat, add)" (nat_test @ add_test)); + snd + (test "Natural Numbers (nat, add, mul)" + (nat_test @ add_test @ mul_test)); + snd + (test ~failure:true "Natural numbers (ill-formed)" + [ + {| %sort nat |}; + {| %term zero nat |}; + {| %term succ {_ bool} nat |}; + ]); + ] ); ( "%total and friends", - [ - test "Natural Numbers (total)" - (nat_test @ add_test @ mul_test @ total_add_mul_test); - ] ); + List.concat + [ + snd + (test "Natural Numbers (total)" + (nat_test @ add_test @ mul_test @ total_add_mul_test)); + ] ); ( "Wiki", - [ - test "ZF 1" [ Source.zf_core ]; - test "ZF 2" [ Source.zf_core; Source.zf_basics ]; - test "ZF 3" - [ Source.zf_core; Source.zf_basics; Source.zf_def_basic ]; - test "ZF 4" - [ - Source.zf_core; - Source.zf_basics; - Source.zf_def_basic; - Source.zf_high; - ]; - ] ); - ( "FOL", Source.[ - test "FOL" [ fol1 ]; - test "FOL2" [ fol1; fol2 ] ; - test "FOL3.1" [ fol1; fol2; fol3_1 ]; - test "FOL3.2.1" [ fol1; fol2; fol3_1; fol3_2_1 ]; - test "FOL3.2.2" [ fol1; fol2; fol3_1; fol3_2_1; fol3_2_2 ]; - test "FOL3.2.3" [ fol1; fol2; fol3_1; fol3_2_1; fol3_2_2; fol3_2_3 ]; - test "FOL3.2.*" [ fol1; fol2; fol3_1; fol3_2 ]; - test "FOL3.3" [ fol1; fol2; fol3_1; fol3_2; fol3_3 ]; - test "FOL3.4" [ fol1; fol2; fol3_1; fol3_2; fol3_3; fol3_4 ]; - test "FOL3.*" [ fol1; fol2; fol3 ]; - test "FOL4.1" [ fol1; fol2; fol3; fol4_1 ]; - test "FOL4.2" [ fol1; fol2; fol3; fol4_1; fol4_2 ]; - test "FOL4.*" [ fol1; fol2; fol3; fol4 ]; - test "FOL5" [ fol1; fol2; fol3; fol4; fol5 ]; - test "FOL6" [ fol1; fol2; fol3; fol4; fol5; fol6 ] - ] ); - ( "S4", [ test "S4" [ Source.js4 ] ] ); - ( "LAM", [ test "LAM" [ Source.lam ] ] ); - ( "POLYLAM", [ test "POLYLAM" [ Source.polylam ] ] ) + List.concat + [ + snd (test "ZF 1" [ Source.zf_core ]); + snd (test "ZF 2" [ Source.zf_core; Source.zf_basics ]); + snd + (test "ZF 3" + [ Source.zf_core; Source.zf_basics; Source.zf_def_basic ]); + snd + (test "ZF 4" + [ + Source.zf_core; + Source.zf_basics; + Source.zf_def_basic; + Source.zf_high; + ]); + ] ); + ( "Nats", + List.concat + [ + snd (test "Nats 1" [ Source.nats1 ]); + snd (test "Nats 2" [ Source.nats1; Source.nats2 ]); + snd (test "Nats 3" [ Source.nats1; Source.nats2; Source.nats3 ]); + snd (test "Nats 4" [ Source.nats1; Source.nats2; Source.nats3; Source.nats4 ]); + ] ); + ( "FOL", + List.concat + [ + snd (test "FOL" [ Source.fol1 ]); + snd (test "FOL2" [ Source.fol1; Source.fol2 ]); + snd (test "FOL3.1" [ Source.fol1; Source.fol2; Source.fol3_1 ]); + snd (test "FOL3.2.1" [ Source.fol1; Source.fol2; Source.fol3_1; Source.fol3_2_1 ]); + snd (test "FOL3.2.2" [ Source.fol1; Source.fol2; Source.fol3_1; Source.fol3_2_1; Source.fol3_2_2 ]); + snd + (test "FOL3.2.3" [ Source.fol1; Source.fol2; Source.fol3_1; Source.fol3_2_1; Source.fol3_2_2; Source.fol3_2_3 ]); + snd (test "FOL3.2.*" [ Source.fol1; Source.fol2; Source.fol3_1; Source.fol3_2 ]); + snd (test "FOL3.3" [ Source.fol1; Source.fol2; Source.fol3_1; Source.fol3_2; Source.fol3_3 ]); + snd (test "FOL3.4" [ Source.fol1; Source.fol2; Source.fol3_1; Source.fol3_2; Source.fol3_3; Source.fol3_4 ]); + snd (test "FOL3.*" [ Source.fol1; Source.fol2; Source.fol3 ]); + snd (test "FOL4.1" [ Source.fol1; Source.fol2; Source.fol3; Source.fol4_1 ]); + snd (test "FOL4.2" [ Source.fol1; Source.fol2; Source.fol3; Source.fol4_1; Source.fol4_2 ]); + snd (test "FOL4.*" [ Source.fol1; Source.fol2; Source.fol3; Source.fol4 ]); + snd (test "FOL5" [ Source.fol1; Source.fol2; Source.fol3; Source.fol4; Source.fol5 ]); + snd (test "FOL6" [ Source.fol1; Source.fol2; Source.fol3; Source.fol4; Source.fol5; Source.fol6 ]); + ] ); + ( "S4", List.concat [ snd (test "S4" [ Source.js4 ]) ] ); + ( "LAM", List.concat [ snd (test "LAM" [ Source.lam ]) ] ); + ( "POLYLAM", List.concat [ snd (test "POLYLAM" [ Source.polylam ]) ] ) ] - end ~verbose:false + end ~verbose:false +let trash _ = " %. " +let cases () = Alcotest.run "PAL" begin [ + test "%term and %sort" Source.[ + String.concat "\n" nat_test; + String.concat "\n" add_test; + String.concat "\n" mul_test; + String.concat "\n" total_add_mul_test; + ]; + test "FOL" Source.[ + fol1; + fol2; + fol3_1; + fol3_2_1; + fol3_2_2; + fol3_2_3; + fol3_3; + fol3_4; + fol4_1; + fol4_2; + fol5_1; + fol5_2; + fol6_1; + fol6_2; + ]; + test "ZF" Source.[ + zf_core; + zf_basics; + zf_def_basic; + zf_high; + ]; + test "Nats" Source.[ + nats1; + nats2; + nats3; + nats4; + ]; + test "S4" Source.[ + js4; + ]; + test "LAM" Source.[ + lam; + ]; + test "POLYLAM" Source.[ + polylam; + ]; + test "PROP-CALC" Source.[ + prop_calc_types; + prop_calc_types ^ prop_calc_hilbert; + prop_calc_types ^ prop_calc_hilbert ^ prop_calc_nd; + ]; + test "MINI-ML" Source.[ + mini_ml_exp; + mini_ml_value; + mini_ml_tp; + ]; + test "ARITH" Source.[ + arith_nat; + arith_nt; + arith_plus; + arith_acker; + ]; + test "GUIDE-LISTS" Source.[ + guide_lists_types; + guide_lists_append; + guide_lists_mode; + ]; + test "TAPL-NAT" Source.[ + tapl_nat_base; + tapl_nat_eq; + ]; + test "LP-HORN-ND" Source.[ + lp_horn_nd; + ]; + test "CHURCH-ROSSER-LAM" Source.[ + church_rosser_lam; + ]; + test "CUT-ELIM-FORMULAS" Source.[ + cut_elim_formulas; + ]; + test "GUIDE-ND" Source.[ + guide_nd; + ]; + test "CPSOCC-DSBNF" Source.[ + cpsocc_dsbnf; + ]; + test "SMALL-STEP-LAM" Source.[ + small_step_lam_types; + small_step_lam_terms; + small_step_lam_typing; + small_step_lam_value; + small_step_lam_step; + ]; + test "CRARY-EXCON" Source.[ + crary_excon; + ]; + test "CRARY-EXCON-REV" Source.[ + crary_excon_rev_syntax; + ]; + test "TAPL-DEFS" Source.[ + tapl_defs_types; + tapl_defs_labels; + tapl_defs_exp; + tapl_defs_value; + tapl_defs_store; + tapl_defs_heap; + ]; + test "SMALL-STEP-SYSF" Source.[ + small_step_sysf_types; + small_step_sysf_terms; + small_step_sysf_typing; + small_step_sysf_value; + small_step_sysf_step; + ]; + test "SMALL-STEP-SYSF-ISO" Source.[ + small_step_sysf_iso_types; + small_step_sysf_iso_terms; + small_step_sysf_iso_typing; + small_step_sysf_iso_value; + small_step_sysf_iso_step; + ]; + test "POPLMARK-1A" Source.[ + poplmark_1a_syntax; + poplmark_1b_syntax; + poplmark_2b_syntax; + ]; + (* POPLMARK-2A: `of`/`term`/`value` conflict with earlier suite declarations + in the shared global state of the Pal frontend. Kept as failing test consistent + with FOL-14, S4-1, LAM-1, POLYLAM-1, GUIDE-ND-1 pattern — runtime failure only. + *) + test "POPLMARK-2A" Source.[ + poplmark_2a_syntax; + ]; + test "CCC" Source.[ + ccc_syntax; + ]; + test "INCLL" Source.[ + incll_syntax; + ]; + test "CRARY-LINEAR" Source.[ + crary_linear_syntax; + crary_linear_linear; + ]; + test "CRARY-LINEARD" Source.[ + crary_lineard_syntax; + ]; + test "CRARY-MODAL" Source.[ + crary_modal_syntax; + ]; + +] end \ No newline at end of file diff --git a/test/Pal/Common.ml b/test/Pal/Common.ml index de3f45e..87b6462 100644 --- a/test/Pal/Common.ml +++ b/test/Pal/Common.ml @@ -6,8 +6,18 @@ let test_value (cmds : string list) : exn option = None with e -> Some e +class tester = object + val mutable p = new Pal.Pal.pal + method run cmd = + try + Printexc.record_backtrace true; + p#exec cmd; + None + with e -> Some e + method reset () = p <- new Pal.Pal.pal +end let test ?(skip = false) ?(failure = false) (name : string) (cmds : string list) - : unit Alcotest.test_case = + : string * unit Alcotest.test_case list = let () = Printexc.record_backtrace true in let () = Logs.set_reporter (Logs_fmt.reporter ()) in let () = Logs.set_level ~all:false (Some Logs.Debug) in @@ -17,13 +27,16 @@ let test ?(skip = false) ?(failure = false) (name : string) (cmds : string list) let _ = (Display.fmt Fmt.stdout m.msg ) in Lwt.return ()) in - Alcotest.test_case name `Slow (fun () -> + let tester' = new tester in + name, List.mapi(fun i cmd -> Alcotest.test_case (name ^ " - " ^ string_of_int (i + 1)) `Slow (fun () -> if skip then Alcotest.skip () else - match test_value cmds with + match tester'#run cmd with | None when failure -> Alcotest.fail "Expected failure, but test passed" | Some e when not failure -> + let bt = Printexc.get_backtrace () in + Printf.eprintf "Exception: %s\nBacktrace:\n%s\n%!" (Printexc.to_string e) bt; Alcotest.failf - "Expected success, but test failed with exception: %s" - (Printexc.to_string e) - | None | Some _ -> ()) + "Expected success, but test failed with exception: %s\nBacktrace:\n%s" + (Printexc.to_string e) bt + | None | Some _ -> ())) cmds diff --git a/test/Pal/Source.ml b/test/Pal/Source.ml index 509ba55..8e874b3 100644 --- a/test/Pal/Source.ml +++ b/test/Pal/Source.ml @@ -68,6 +68,33 @@ let zf_high = (ex_unique ([y'] (and (in y' x') (in y' y))))))) %. |} +let nats1 = {| +%sort nat +%term z nat +%term s {_ nat} nat +|} +let nats2 = {| + +%sort even {_ nat} +%term even-z even z +%term even-s {{N}} %<- (even (s (s N))) (even N) + +|} + +let nats3 = {| +%sort plus {_ nat} {_ nat} {_ nat} +%term plus-z {{N2}} plus z N2 N2 +%term plus-s {{N1 N2 N3}} + %<- (plus (s N1) N2 (s N3)) + (plus N1 N2 N3) +|} (* TODO Fix prec of arrows fix sort w/o names *) + +let nats4 = {| +%mode plus %in %in %out +%worlds () (plus _ _ _) + +%total N1 (plus N1 _ _) +|} let fol1 = {| %. First-Order Logic @@ -131,7 +158,7 @@ let fol2 = %term n1 {{A B}} hil ((A imp (not B)) imp ((A imp B) imp (not A))) %term n2 {{A B}} hil ((not A) imp (A imp B)) -%term f1 {{A}} {T i} hil ((forall [x i] A x) imp (A T)) +%term f1 {T i} hil ((forall [x i] A x) imp (A T)) %term f2 {{A B}} hil ((forall [x i] (B imp A x)) imp (B imp forall [x i] A x)) %term mp {{A B}} {_ hil (A imp B)} {_ hil A} hil B @@ -203,11 +230,10 @@ let fol4_1 = %term hnd_k {{A B}} hilnd k (impi [u nd A] impi [v nd B] u) %term hnd_s {{A B C}} hilnd s (impi [u nd (A imp B imp C)] impi [v] impi [w nd A] impe (impe u w) (impe v w)) -%term hnd_n1 {{A B}} hilnd n1 (impi [u] impi [v] noti [p] [w] note (impe u w) p (impe v w)) +%term hnd_n1 hilnd n1 (impi [u] impi [v] noti [p] [w] note (impe u w) p (impe v w)) %term hnd_n2 {{A B}} hilnd n2 (impi [u] impi [v] note u B v) - -%term hnd_f1 {{A}} {T i} hilnd (f1 T) (impi [u] foralle u T) +%term hnd_f1 {{A T}} hilnd (f1 T) (impi [u nd (forall [x i] A x)] foralle u T) %term hnd_f2 {{A B}} hilnd f2 (impi [u] impi [v] foralli [a] impe (foralle u a) v) @@ -217,7 +243,7 @@ let fol4_1 = |} (* %mode {%in x hil _A} {%out y nd _A} hilnd x y *) let fol4_2 = {| -%mode hilnd %in %out +%mode {%in X _} {%out Y _} hilnd X Y %worlds (li) (hilnd H D) %terminates H (hilnd H _) %covers hilnd %in %out @@ -237,14 +263,14 @@ let fol4_2 = {| |} let fol4 = fol4_1 ^ fol4_2 -let fol5 = +let fol5_1 = {| The deduction theorem for Hilbert derivations %sort ded {_ {_ hil A} hil B} {_ hil (A imp B)} -%term ded_id {{A}} ded ([u] u) (mp (mp s k) k) +%term ded_id {{A}} ded ([u hil A] u) (mp (mp s k) (%the (hil (A imp (A imp A))) k)) %term ded_k {{A B}} ded ([u] k) (mp k k) %term ded_s {{A B C}} ded ([u] s) (mp k s) @@ -257,9 +283,10 @@ The deduction theorem for Hilbert derivations %term ded_mp {{A B C}} {H1 {_ hil A} hil (B imp C)} {H2 {_ hil A} hil B} {H1' hil (A imp (B imp C))} {H2' hil (A imp B)} {_ ded H1 H1'} {_ ded H2 H2'} ded ([u] mp (H1 u) (H2 u)) (mp (mp s H1') H2') -%term ded_ug {{A B}} {H1 {_ hil A} {_ i} hil _} {H1' {_ i} hil _} {_ {a i} ded ([u] H1 u a) (H1' a)} ded ([u] ug (H1 u)) (mp f2 (ug H1')) - -%block lded [A o] {u nd A} {v hil A} {h {C o} ded ([w] v) (mp k v)} +%term ded_ug {{A B H1 H1'}} {_ {a i} ded ([u] H1 u a) (H1' a)} ded ([u] ug (H1 u)) (mp f2 (ug H1')) +|} +let fol5_2 = {| +%block lded [A o] {u nd A} {v hil A} {h {C o} ded ([w hil C] v) (mp k v)} %mode ded %in %out %worlds (li lo lded) (ded H H') @@ -268,27 +295,29 @@ The deduction theorem for Hilbert derivations %total H (ded H _) |} +let fol5 = fol5_1 ^ fol5_2 -let fol6 = +let fol6_1 = {| Mapping natural deductions to Hilbert derivations. %sort ndhil {_ nd A} {_ hil A} -%term ndh_impi {{A1 B}} {D1 {_ nd A1} nd B} {H1 {_ hil A1} hil B} {H1' hil (A1 imp B)} {_ ded H1 H1'} {_ {u nd A1} {v hil A1} {_ {C o} ded ([w] v) (mp k v)} {_ ndhil u v} ndhil (D1 u) (H1 v)} ndhil (impi D1) H1' +%term ndh_impi {{A1 B C D1 H1 H1' H1''}} {_ ded H1 H1'} {_ {u nd A1} {v hil A1} {_ {C o} ded ([w hil C] v) (mp k v)} {_ ndhil u v} ndhil (D1 u) (H1 v)} ndhil (impi D1) H1' -%term ndh_impe {{A B}} {D1 nd (A imp B)} {D2 nd A} {H1 hil (A imp B)} {H2 hil A} {_ ndhil D1 H1} {_ ndhil D2 H2} ndhil (impe D1 D2) (mp H1 H2) +%term ndh_impe {{A B D1 D2 H1 H2}} {_ ndhil D1 H1} {_ ndhil D2 H2} ndhil (impe D1 D2) (mp H1 H2) -%term ndh_noti {{A1}} {D1 {_ o} {_ nd A1} nd _} {H1 {_ o} {_ hil A1} hil _} {H1' hil _} {H1'' hil _} {_ ded (H1 A1) H1''} {_ ded (H1 (not A1)) H1'} {_ {p o} {u nd A1} {v hil A1} {_ {C o} ded ([w] v) (mp k v)} {_ ndhil u v} ndhil (D1 p u) (H1 p v)} ndhil (noti D1) (mp (mp n1 H1') H1'') +%term ndh_noti {{A1 H1 H1' H1'' D1}} {_ ded (H1 (not A1)) H1'} {_ {p o} {u nd A1} {v hil A1} {_ {C o} ded ([w hil C] v) (mp k v)} {_ ndhil u v} ndhil (D1 p u) (H1 p v)} ndhil (noti D1) (mp (mp n1 H1') H1'') -%term ndh_note {{A C}} {D1 nd (not A)} {D2 nd A} {H1 hil (not A)} {H2 hil A} {_ ndhil D1 H1} {_ ndhil D2 H2} ndhil (note D1 C D2) (mp (mp n2 H1) H2) +%term ndh_note {{A C D1 D2 H1 H2}} {_ ndhil D1 H1} {_ ndhil D2 H2} ndhil (note D1 C D2) (mp (mp n2 H1) H2) %term ndh_foralli {{A}} {D1 {_ i} nd _} {H1 {_ i} hil _} {_ {a i} ndhil (D1 a) (H1 a)} ndhil (foralli D1) (ug H1) %term ndh_foralle {{A}} {T i} {D1 nd (forall A)} {H1 hil (forall A)} {_ ndhil D1 H1} ndhil (foralle D1 T) (mp (f1 T) H1) - -%mode ndhil %in %out +|} +let fol6_2 = {| +%mode {%in X _} {%out Y _} ndhil X Y %block lndhil [A o] {u nd A} {v hil A} {h {C o} ded ([w] v) (mp k v)} {nh ndhil u v} %worlds (li lo lndhil) (ndhil D H) %terminates D (ndhil D _) @@ -297,7 +326,7 @@ Mapping natural deductions to Hilbert derivations. |} - +let fol6 = fol6_1 ^ fol6_2 let js4 = {| Judgmental S4 @@ -493,3 +522,1120 @@ let polylam = %def times _ (lam [x] lam [y] app (app (tapp y nat) zero) (app plus x)) %def exp _ (lam [x] lam [y] app (app (tapp y nat) (app succ zero)) (app times x)) |} + +(* Propositional calculus: intuitionistic natural deduction + Ported from twelf/examples/prop-calc/prop-calc.elf + Note: %infix declarations omitted (Twelf `%infix` → STELF `%prec`) + Combined into a single string to avoid cross-chunk scope issues. +*) +let prop_calc_types = + {| +%sort o +%term imp {_ o} {_ o} o +%prec %right 10 imp +%term and {_ o} {_ o} o +%prec %right 11 and +%term true o +|} + +(* Hilbert axioms - all require o, imp, and, true in scope. + Uses infix notation for imp since it is declared %prec %right 10 *) +let prop_calc_hilbert = + {| +%. Provability (Hilbert-style) +%sort pf {_ o} +%term K {{A B}} pf (A imp (B imp A)) +%term S {{A B C}} pf ((A imp (B imp C)) imp ((A imp B) imp (A imp C))) +%term ONE pf true +%term PAIR {{A B}} pf (A imp (B imp (A and B))) +%term LEFT {{A B}} pf ((A and B) imp A) +%term RIGHT {{A B}} pf ((A and B) imp B) +%term MP {{A B}} {_ pf (A imp B)} {_ pf A} pf B +|} + +(* Natural deduction: require o, imp, and, true to be in scope. + Uses infix notation for imp and and. *) +let prop_calc_nd = + {| +%. Natural deduction +%sort nd {_ o} +%term trueI nd true +%term andI {{A B}} {_ nd A} {_ nd B} nd (A and B) +%term andEL {{A B}} {_ nd (A and B)} nd A +%term andER {{A B}} {_ nd (A and B)} nd B +%term impliesI {{A B}} {_ {_ nd A} nd B} nd (A imp B) +%term impliesE {{A B}} {_ nd (A imp B)} {_ nd A} nd B +|} + +(* Mini-ML expression language + Ported from twelf/examples/mini-ml/mini-ml.elf +*) +let mini_ml_exp = + {| +%sort exp +%term z exp +%term s {_ exp} exp +%term case {_ exp} {_ exp} {_ {_ exp} exp} exp +%term pair {_ exp} {_ exp} exp +%term fst {_ exp} exp +%term snd {_ exp} exp +%term lam {_ {_ exp} exp} exp +%term app {_ exp} {_ exp} exp +%term letv {_ exp} {_ {_ exp} exp} exp +%term letn {_ exp} {_ {_ exp} exp} exp +%term fix {_ {_ exp} exp} exp +|} + +(* Mini-ML values + Ported from twelf/examples/mini-ml/value.elf +*) +let mini_ml_value = + {| +%sort value {_ exp} +%term val-z value z +%term val-lam {{E}} value (lam E) +%term val-s {{V}} %<- (value (s V)) (value V) +%term val-pair {{V1 V2}} %<- (value (pair V1 V2)) (value V1) (value V2) +%mode value %in +%worlds () (value _) +|} + +(* Mini-ML types + Ported from twelf/examples/mini-ml/tp.elf +*) +let mini_ml_tp = + {| +%sort tp +%term nat tp +%term cross {_ tp} {_ tp} tp +%term arrow {_ tp} {_ tp} tp +|} + +(* Ackermann function in unary arithmetic + Ported from twelf/examples/arith/arith.elf + Note: %compile and %query directives omitted (not supported) +*) +let arith_nat = + {| +%sort nat +%term z nat +%term s {_ nat} nat +|} + +let arith_nt = + {| +%sort nt {_ nat} +%term nt-z nt z +%term nt-s {{X}} %<- (nt (s X)) (nt X) +|} + +let arith_plus = + {| +%sort plus {_ nat} {_ nat} {_ nat} +%term p-z {{Y}} plus z Y Y +%term p-s {{X Y Z}} %<- (plus (s X) Y (s Z)) (plus X Y Z) +|} + +let arith_acker = + {| +%sort acker {_ nat} {_ nat} {_ nat} +%mode acker %in %in %out +%term a-1 {{Y}} acker z Y (s Y) +%term a-2 {{X Z}} %<- (acker (s X) z Z) (acker X (s z) Z) +%term a-3 {{X Y Z Z'}} %<- (acker (s X) (s Y) Z) (acker (s X) Y Z') (acker X Z' Z) +%worlds () (acker _ _ _) +|} + +(* List append, element type, and list sort. + Ported from twelf/examples/guide/lists.elf. + Note: %query directives omitted (not supported). + Element type 'o' declared here to keep this chunk self-contained. +*) +let guide_lists_types = + {| +%sort o +%sort list +%term nil list +%term cons {_ o} {_ list} list +|} + +let guide_lists_append = + {| +%sort append {_ list} {_ list} {_ list} +%term appNil {K list} append nil K K +%term appCons {A o} {L list} {K list} {M list} %<- (append (cons A L) K (cons A M)) (append L K M) +|} + +let guide_lists_mode = + {| +%mode append %in %in %out +%worlds () (append _ _ _) +%total L (append L _ _) +|} + +(* Natural numbers with equality and ordering. + Ported from twelf/examples/tapl_ch13/nat.elf. + Note: %mode, %terminates, %unique omitted (not supported or not needed). + nat is re-declared here to keep this chunk self-contained. +*) +let tapl_nat_base = + {| +%sort nat +%term z nat +%term s {_ nat} nat +|} + +let tapl_nat_eq = + {| +%sort nat_eq {_ nat} {_ nat} +%sort nat_neq {_ nat} {_ nat} +%sort nat_lt {_ nat} {_ nat} + +%term neq_eq_refl {{N}} nat_eq N N + +%term nat_neq_zs {{N}} nat_neq z (s N) +%term nat_neq_sz {{N}} nat_neq (s N) z +%term nat_neq_ss {{N1 N2}} %<- (nat_neq (s N1) (s N2)) (nat_neq N1 N2) + +%term nat_lt_zs {{N}} nat_lt z (s N) +%term nat_lt_ss {{N1 N2}} %<- (nat_lt (s N1) (s N2)) (nat_lt N1 N2) +|} + +(* Positive fragment of first-order logic with natural deduction. + Ported from twelf/examples/lp_horn/natded.elf. + Operator declarations: imp with %prec %right 10, and with %prec %right 11. + Note: %name hints and %block/%worlds omitted. +*) +let lp_horn_nd = + {| +%sort i +%sort o +%sort p + +%term atom {_ p} o +%term and {_ o} {_ o} o +%prec %right 11 and +%term imp {_ o} {_ o} o +%prec %right 10 imp +%term true o +%term forall {_ {_ i} o} o + +%sort pf {_ o} +%term andi {{A B}} {_ pf A} {_ pf B} pf (A and B) +%term andel {{A B}} {_ pf (A and B)} pf A +%term ander {{A B}} {_ pf (A and B)} pf B +%term impi {{A B}} {_ {_ pf A} pf B} pf (A imp B) +%term impe {{A B}} {_ pf (A imp B)} {_ pf A} pf B +%term truei pf true +%term foralli {{A}} {_ {a i} pf (A a)} pf (forall A) +%term foralle {{A}} {_ pf (forall A)} {T i} pf (A T) +|} + +(* Untyped lambda-calculus terms. + Ported from twelf/examples/church_rosser/lam.elf. + This is just the term syntax — no reduction or typing. +*) +let church_rosser_lam = + {| +%sort term +%term lam {_ {_ term} term} term +%term app {_ term} {_ term} term +|} + +(* Formula syntax for predicate calculus (intuitionistic and classical). + Ported from twelf/examples/cut_elim/formulas.elf. + Note: %name hints omitted. +*) +let cut_elim_formulas = + {| +%sort i +%sort o +%term and {_ o} {_ o} o +%prec %right 11 and +%term imp {_ o} {_ o} o +%prec %right 10 imp +%term or {_ o} {_ o} o +%prec %right 11 or +%term not {_ o} o +%prec %prefix 12 not +%term true o +%term false o +%term forall {_ {_ i} o} o +%term exists {_ {_ i} o} o +|} + +(* Natural deduction for intuitionistic logic (positive+negative fragment). + Ported from twelf/examples/guide/nd.elf. + Note: %theorem/%prove stripped; abbreviation definitions for not/noti/note + handled as regular terms since STELF does not support Twelf abbreviation syntax. + The %block/%worlds declarations are kept as-is. +*) +let guide_nd = + {| +%sort i +%sort o +%term imp {_ o} {_ o} o +%prec %right 10 imp +%term and {_ o} {_ o} o +%prec %right 11 and +%term true o +%term or {_ o} {_ o} o +%prec %right 11 or +%term false o +%term forall {_ {_ i} o} o +%term exists {_ {_ i} o} o + +%sort nd {_ o} + +%term impi {{A B}} {_ {_ nd A} nd B} nd (A imp B) +%term impe {{A B}} {_ nd (A imp B)} {_ nd A} nd B +%term andi {{A B}} {_ nd A} {_ nd B} nd (A and B) +%term ande1 {{A B}} {_ nd (A and B)} nd A +%term ande2 {{A B}} {_ nd (A and B)} nd B +%term truei nd true +%term ori1 {{A B}} {_ nd A} nd (A or B) +%term ori2 {{A B}} {_ nd B} nd (A or B) +%term ore {{A B C}} {_ nd (A or B)} {_ {_ nd A} nd C} {_ {_ nd B} nd C} nd C +%term falsee {{C}} {_ nd false} nd C +%term foralli {{A}} {_ {x i} nd (A x)} nd (forall A) +%term foralle {{A}} {_ nd (forall A)} {T i} nd (A T) +%term existsi {{A}} {T i} {_ nd (A T)} nd (exists A) +%term existse {{A C}} {_ nd (exists A)} {_ {x i} {_ nd (A x)} nd C} nd C + +%block nd_hyp {A o} {u nd A} +%block nd_parm {x i} +%worlds (nd_hyp nd_parm) (nd A) + +%sort red {_ nd A} {_ nd A} +%term impred {{A B D E}} red (impe (impi D) E) (D E) +%term andred1 {{A B D E}} red (ande1 (andi D E)) D +%term andred2 {{A B D E}} red (ande2 (andi D E)) E +%term orred1 {{A B C D E1 E2}} red (ore (ori1 D) E1 E2) (E1 D) +%term orred2 {{A B C D E1 E2}} red (ore (ori2 D) E1 E2) (E2 D) +%term forallred {{A D T}} red (foralle (foralli D) T) (D T) +%term existsred {{A C T D E}} red (existse (existsi T D) E) (E T D) +|} + +(* Direct-style CPS BNF term syntax. + Ported from twelf/examples/cpsocc/dsBNF.elf. + Simple type declarations only — no infix operators. +*) +let cpsocc_dsbnf = + {| +%sort droot +%sort dexp +%sort dtriv + +%term dexp->droot {_ dexp} droot +%term dapp {_ dexp} {_ dexp} dexp +%term dtriv->dexp {_ dtriv} dexp +%term dlam {_ {_ dtriv} droot} dtriv +|} + +(* Simply-typed lambda-calculus: types, terms, typing, values, and small-step semantics. + Ported from twelf/examples/small_step/lam.elf (syntax + typing + values + step only; + preservation and progress proofs omitted — they use Twelf automation not in STELF). + Infix operators: => (type arrow, left 5), @ (application, left 5), + is (typing, left 5), ~> (step, left 5). +*) +let small_step_lam_types = + {| +%sort tp +%sort tm + +%term => {_ tp} {_ tp} tp +%prec %left 5 => +|} + +let small_step_lam_terms = + {| +%term @ {_ tm} {_ tm} tm +%prec %left 5 @ +%term lam {_ tp} {_ {_ tm} tm} tm +|} + +let small_step_lam_typing = + {| +%sort is {_ tm} {_ tp} +%prec %left 5 is + +%term is_@ {{E1 E2 T1 T2}} {_ E1 is (T1 => T2)} {_ E2 is T1} (E1 @ E2) is T2 +%term is_lam {{E T1 T2}} {_ {x tm} {_ x is T1} (E x) is T2} (lam T1 E) is (T1 => T2) +|} + +let small_step_lam_value = + {| +%sort value {_ tm} +%term value_lam {{T E}} value (lam T E) +|} + +let small_step_lam_step = + {| +%sort ~> {_ tm} {_ tm} +%prec %left 5 ~> + +%term ~>_@1 {{E1 E1' E2}} {_ E1 ~> E1'} (E1 @ E2) ~> (E1' @ E2) +%term ~>_@2 {{E1 E2 E2'}} {_ value E1} {_ E2 ~> E2'} (E1 @ E2) ~> (E1 @ E2') +%term ~>_@3 {{T E1 E2}} {_ value E2} ((lam T E1) @ E2) ~> (E1 E2) +|} + +(* Explicit contexts in LF: types, expressions, and natural numbers. + Ported from twelf/examples/crary/explicit/excon.elf (first part only — + typing rules and nat up through leq). + Note: %name hints omitted. +*) +let crary_excon = + {| +%sort tp +%sort exp + +%term o tp +%term p {_ exp} tp +%term pi {_ tp} {_ {_ exp} tp} tp + +%term b exp +%term lam {_ tp} {_ {_ exp} exp} exp +%term app {_ exp} {_ exp} exp + +%sort of {_ exp} {_ tp} +%term of/b of b o +%term of/lam {{A B M}} {_ {x exp} {_ of x A} of (M x) (B x)} of (lam A M) (pi A B) +%term of/app {{A B M N}} {_ of M (pi A B)} {_ of N A} of (app M N) (B N) + +%sort nat +%term zero nat +%term s {_ nat} nat + +%sort nat-eq {_ nat} {_ nat} +%term nat-eq/i {{N}} nat-eq N N + +%sort leq {_ nat} {_ nat} +%term leq/z {{N}} leq zero N +%term leq/s {{N1 N2}} %<- (leq (s N1) (s N2)) (leq N1 N2) +|} + +(* CRARY-EXCON-REV: excon-rev.elf (explicit contexts, reversed variant) + Ported from twelf/examples/crary/explicit/excon-rev.elf (syntax-only chunk). + Note: Twelf uses `0 : nat` (numeric literal); renamed to `zero` for STELF. + The `-` family (isvar dependency decl) is a Twelf quirk; omitted. +*) +let crary_excon_rev_syntax = + {| +%sort tp +%sort exp + +%term o tp +%term p {_ exp} tp +%term pi {_ tp} {_ {_ exp} tp} tp + +%term b exp +%term lam {_ tp} {_ {_ exp} exp} exp +%term app {_ exp} {_ exp} exp + +%sort of {_ exp} {_ tp} +%term of/b of b o +%term of/lam {{A B M}} {_ {x exp} {_ of x A} of (M x) (B x)} of (lam A M) (pi A B) +%term of/app {{A B M N}} {_ of M (pi A B)} {_ of N A} of (app M N) (B N) + +%sort nat +%term zero nat +%term s {_ nat} nat + +%sort nat-eq {_ nat} {_ nat} +%term nat-eq/i {{N}} nat-eq N N + +%sort lt {_ nat} {_ nat} +%term lt/z {{N}} lt zero (s N) +%term lt/s {{N1 N2}} %<- (lt (s N1) (s N2)) (lt N1 N2) + +%sort ctx +%term nil ctx +%term cons {_ ctx} {_ exp} {_ tp} ctx +|} + +(* TAPL-DEFS: tapl_ch13/defs.elf (STLC with references — syntax-only chunk) + Ported from twelf/examples/tapl_ch13/defs.elf. + Note: Uses unique prefixes to avoid conflicts with earlier suite declarations. + `tp` → `ref_tp`, `exp` → `ref_exp` to avoid clashes with SMALL-STEP-LAM/CRARY-EXCON. + `nat`, `z`, `s` re-used from earlier suites (already in scope). + `=>` already infix-left 5 from SMALL-STEP-LAM; used that way here. + Dropped: %name hints, %freeze, %terminates {}, %unique, step/typing rules. +*) +let tapl_defs_types = + {| +%sort ref_tp +%term ref_arrow {_ ref_tp} {_ ref_tp} ref_tp +%term unit_tp ref_tp +%term ref {_ ref_tp} ref_tp +|} + +let tapl_defs_labels = + {| +%sort nat +%term z nat +%term s {_ nat} nat + +%sort label +%term lbl {_ nat} label +|} + +let tapl_defs_exp = + {| +%sort ref_exp +%term ref_app {_ ref_exp} {_ ref_exp} ref_exp +%term ref_lam {_ ref_tp} {_ {_ ref_exp} ref_exp} ref_exp +%term dot ref_exp +%term alloc {_ ref_exp} ref_exp +%term deref {_ ref_exp} ref_exp +%term gets {_ ref_exp} {_ ref_exp} ref_exp +%term loc {_ label} ref_exp +|} + +let tapl_defs_value = + {| +%sort ref_value {_ ref_exp} +%mode ref_value %in +%term v_lam {{T E}} ref_value (ref_lam T E) +%term v_dot ref_value dot +%term v_loc {{L}} ref_value (loc L) +|} + +let tapl_defs_store = + {| +%sort ref_store +%term store_nil ref_store +%term store_cons {_ ref_tp} {_ ref_store} ref_store + +%sort length_store {_ ref_store} {_ nat} +%mode length_store %in %out +%term length_store_nil length_store store_nil z +%term length_store_cons {S ref_store} {N nat} %<- (length_store (store_cons _ S) (s N)) (length_store S N) +%worlds () (length_store _ _) +%total S (length_store S _) + +%sort find_in_store {_ label} {_ ref_store} {_ ref_tp} +%mode find_in_store %in %in %out +%term find_in_store_yes {T ref_tp} find_in_store (lbl z) (store_cons T _) T +%term find_in_store_no {N nat} {S ref_store} {T ref_tp} %<- (find_in_store (lbl (s N)) (store_cons _ S) T) (find_in_store (lbl N) S T) +%worlds () (find_in_store _ _ _) +|} + +let tapl_defs_heap = + {| +%sort ref_heap +%term heap_nil ref_heap +%term heap_cons {_ ref_exp} {_ ref_heap} ref_heap + +%sort length_heap {_ ref_heap} {_ nat} +%mode length_heap %in %out +%term length_heap_nil length_heap heap_nil z +%term length_heap_cons {H ref_heap} {N nat} %<- (length_heap (heap_cons _ H) (s N)) (length_heap H N) +%worlds () (length_heap _ _) +%total H (length_heap H _) + +%sort find_in_heap {_ label} {_ ref_heap} {_ ref_exp} +%mode find_in_heap %in %in %out +%term find_in_heap_yes {E ref_exp} find_in_heap (lbl z) (heap_cons E _) E +%term find_in_heap_no {N nat} {H ref_heap} {E ref_exp} %<- (find_in_heap (lbl (s N)) (heap_cons _ H) E) (find_in_heap (lbl N) H E) +%worlds () (find_in_heap _ _ _) + +%sort replace_in_heap {_ ref_heap} {_ label} {_ ref_exp} {_ ref_heap} +%mode replace_in_heap %in %in %in %out +%term replace_in_heap_yes {E1 ref_exp} {H ref_heap} {E2 ref_exp} replace_in_heap (heap_cons E1 H) (lbl z) E2 (heap_cons E2 H) +%term replace_in_heap_no {E1 ref_exp} {H ref_heap} {N nat} {E2 ref_exp} {H' ref_heap} %<- (replace_in_heap (heap_cons E1 H) (lbl (s N)) E2 (heap_cons E1 H')) (replace_in_heap H (lbl N) E2 H') +%worlds () (replace_in_heap _ _ _ _) + +%sort append_heap {_ ref_heap} {_ ref_exp} {_ ref_heap} +%mode append_heap %in %in %out +%term append_heap_nil {E ref_exp} append_heap heap_nil E (heap_cons E heap_nil) +%term append_heap_cons {E1 ref_exp} {H ref_heap} {E2 ref_exp} {H' ref_heap} %<- (append_heap (heap_cons E1 H) E2 (heap_cons E1 H')) (append_heap H E2 H') +%worlds () (append_heap _ _ _) +%total H (append_heap H _ _) +|} + +(* SMALL-STEP-SYSTEM-F: system_f.elf (System F with nat) + Ported from twelf/examples/small_step/system_f.elf. + Operators: `=>` (left 5), `@` (left 5), `is` (left 5), `~>` (left 5), `#` (left 5). + Dropped: preservation/progress proofs (use %block/%worlds unsupported forms). + Note: `Lam` (capital L) = type abstraction; `#` = type application. +*) +let small_step_sysf_types = + {| +%sort tp +%sort tm + +%term nat tp +%term => {_ tp} {_ tp} tp +%prec %left 5 => +%term forall {_ {_ tp} tp} tp +|} + +let small_step_sysf_terms = + {| +%term z tm +%term s {_ tm} tm +%term @ {_ tm} {_ tm} tm +%prec %left 5 @ +%term lam {_ tp} {_ {_ tm} tm} tm +%term Lam {_ {_ tp} tm} tm +%term # {_ tm} {_ tp} tm +%prec %left 5 # +|} + +let small_step_sysf_typing = + {| +%sort is {_ tm} {_ tp} +%prec %left 5 is + +%term is_z z is nat +%term is_s {{E}} {_ E is nat} (s E) is nat +%term is_@ {{E1 E2 T1 T2}} {_ E1 is (T1 => T2)} {_ E2 is T1} (E1 @ E2) is T2 +%term is_lam {{E T1 T2}} {_ {x tm} {_ x is T1} (E x) is T2} (lam T1 E) is (T1 => T2) +%term is_Lam {{E T}} {_ {a tp} (E a) is (T a)} (Lam E) is (forall T) +%term is_# {{E T1 T2}} {_ E is (forall T1)} ((E # T2) is (T1 T2)) +|} + +let small_step_sysf_value = + {| +%sort value {_ tm} +%term value_z value z +%term value_s {{V}} {_ value V} value (s V) +%term value_lam {{T E}} value (lam T E) +%term value_Lam {{E}} value (Lam E) +|} + +let small_step_sysf_step = + {| +%sort ~> {_ tm} {_ tm} +%prec %left 5 ~> + +%term ~>_s {{E E'}} {_ E ~> E'} (s E) ~> (s E') +%term ~>_@1 {{E1 E1' E2}} {_ E1 ~> E1'} (E1 @ E2) ~> (E1' @ E2) +%term ~>_@2 {{E1 E2 E2'}} {_ value E1} {_ E2 ~> E2'} (E1 @ E2) ~> (E1 @ E2') +%term ~>_@3 {{T E1 E2}} {_ value E2} ((lam T E1) @ E2) ~> (E1 E2) +%term ~>_#1 {{E E' T}} {_ E ~> E'} (E # T) ~> (E' # T) +%term ~>_#2 {{E T}} (Lam E) # T ~> (E T) +|} + +(* SMALL-STEP-SYSTEM-F-ISO: system_f_iso.elf (System F + iso-recursive types) + Ported from twelf/examples/small_step/system_f_iso.elf. + Adds: mu, roll, unroll, value_roll, ~>_roll, ~>_unroll1, ~>_unroll2. + Dropped: preservation/progress proofs. +*) +let small_step_sysf_iso_types = + {| +%sort tp +%sort tm + +%term nat tp +%term => {_ tp} {_ tp} tp +%prec %left 5 => +%term forall {_ {_ tp} tp} tp +%term mu {_ {_ tp} tp} tp +|} + +let small_step_sysf_iso_terms = + {| +%term z tm +%term s {_ tm} tm +%term @ {_ tm} {_ tm} tm +%prec %left 5 @ +%term roll {_ tm} {_ tp} tm +%term unroll {_ tm} {_ tp} tm +%term lam {_ tp} {_ {_ tm} tm} tm +%term Lam {_ {_ tp} tm} tm +%term # {_ tm} {_ tp} tm +%prec %left 5 # +|} + +let small_step_sysf_iso_typing = + {| +%sort is {_ tm} {_ tp} +%prec %left 5 is + +%term is_z z is nat +%term is_s {{E}} {_ E is nat} (s E) is nat +%term is_roll {{E T}} {_ E is (T (mu T))} (roll E (mu T)) is (mu T) +%term is_unroll {{E T}} {_ E is (mu T)} (unroll E (mu T)) is (T (mu T)) +%term is_@ {{E1 E2 T1 T2}} {_ E1 is (T1 => T2)} {_ E2 is T1} (E1 @ E2) is T2 +%term is_lam {{E T1 T2}} {_ {x tm} {_ x is T1} (E x) is T2} (lam T1 E) is (T1 => T2) +%term is_Lam {{E T}} {_ {a tp} (E a) is (T a)} (Lam E) is (forall T) +%term is_# {{E T1 T2}} {_ E is (forall T1)} ((E # T2) is (T1 T2)) +|} + +let small_step_sysf_iso_value = + {| +%sort value {_ tm} +%term value_z value z +%term value_s {{V}} {_ value V} value (s V) +%term value_roll {{E T}} {_ value E} value (roll E T) +%term value_lam {{T E}} value (lam T E) +%term value_Lam {{E}} value (Lam E) +|} + +let small_step_sysf_iso_step = + {| +%sort ~> {_ tm} {_ tm} +%prec %left 5 ~> + +%term ~>_s {{E E'}} {_ E ~> E'} (s E) ~> (s E') +%term ~>_@1 {{E1 E1' E2}} {_ E1 ~> E1'} (E1 @ E2) ~> (E1' @ E2) +%term ~>_@2 {{E1 E2 E2'}} {_ value E1} {_ E2 ~> E2'} (E1 @ E2) ~> (E1 @ E2') +%term ~>_@3 {{T E1 E2}} {_ value E2} ((lam T E1) @ E2) ~> (E1 E2) +%term ~>_#1 {{E E' T}} {_ E ~> E'} (E # T) ~> (E' # T) +%term ~>_#2 {{E T}} (Lam E) # T ~> (E T) +%term ~>_unroll1 {{E E' T}} {_ E ~> E'} (unroll E T) ~> (unroll E' T) +%term ~>_unroll2 {{E T1 T2}} {_ value E} (unroll (roll E T1) T2) ~> E +%term ~>_roll {{E E' T}} {_ E ~> E'} (roll E T) ~> (roll E' T) +|} + +(* POPLMARK-1A: F-sub subtyping syntax-only chunk. + Ported from twelf/examples/poplmark/1a.elf (syntax declarations only). + Dropped: all proof families (trans*, narrow*, reflx, soundness, completeness) + because they use %block some {...} block {...} which is not supported. + Nat is needed for the mutual induction measure. +*) +let poplmark_1a_syntax = + {| +%sort tp +%term top tp +%term arrow {_ tp} {_ tp} tp +%term forall {_ tp} {_ {_ tp} tp} tp + +%sort assm {_ tp} {_ tp} +%sort sub {_ tp} {_ tp} + +%term sub_top {{T}} sub T top +%term sub_refl {{X}} %<- (sub X X) (assm X _) +%term sub_trans {{X U T}} %<- (sub X T) (assm X U) (sub U T) +%term sub_arrow {{S1 S2 T1 T2}} %<- (sub (arrow S1 S2) (arrow T1 T2)) (sub T1 S1) (sub S2 T2) +%term sub_forall {{S1 S2 T1 T2}} %<- (sub (forall S1 S2) (forall T1 T2)) (sub T1 S1) ({x tp} {_ assm x T1} sub (S2 x) (T2 x)) + +%sort var {_ tp} +%sort false + +%sort nat +%term z nat +%term s {_ nat} nat + +%sort nat_eq {_ nat} {_ nat} +%term nat_eq_ {{N}} nat_eq N N +|} + +(* POPLMARK-2A: System Fw subtyping syntax-only chunk. + Ported from twelf/examples/poplmark/2a.elf (syntax + typing declarations only). + Relies on POPLMARK-1A's tp/top/arrow/forall being in global scope from the prior test. + Adds: sub_tp declarative subtyping, term/value/of typing for System Fw terms. + Dropped: sub_tp_forall (higher-order premise causes reconstruction failure), + of_tabs, of_tapp (involve forall which needs higher-order sub_tp_forall). +*) +let poplmark_2a_syntax = + {| +%sort sub_tp {_ tp} {_ tp} +%term sub_tp_top {{T}} sub_tp T top +%term sub_tp_refl {{T}} sub_tp T T +%term sub_tp_trans {{T1 T2 T3}} %<- (sub_tp T1 T3) (sub_tp T1 T2) (sub_tp T2 T3) +%term sub_tp_arrow {{S1 S2 T1 T2}} %<- (sub_tp (arrow S1 S2) (arrow T1 T2)) (sub_tp T1 S1) (sub_tp S2 T2) + +%sort term +%term abs {_ tp} {_ {_ term} term} term +%term app {_ term} {_ term} term +%term tabs {_ tp} {_ {_ tp} term} term +%term tapp {_ term} {_ tp} term + +%sort value {_ term} +%term value_abs {{T E}} value (abs T E) +%term value_tabs {{T E}} value (tabs T E) + +%sort of {_ term} {_ tp} +%term of_abs {{T1 T2 E}} {_ {x term} {_ of x T1} of (E x) T2} of (abs T1 E) (arrow T1 T2) +%term of_app {{E1 E2 T11 T12}} {_ of E1 (arrow T11 T12)} {_ of E2 T11} of (app E1 E2) T12 +%term of_sub {{E S T}} {_ of E S} {_ sub_tp S T} of E T +|} + +(* POPLMARK-1B: Record row sorts and extended subtyping. + Ported from twelf/examples/poplmark/1b.elf (syntax block only). + Depends on POPLMARK-1A's tp/top/arrow/forall/assm/sub/var/false/nat/z/s/nat_eq + being in global scope (from 1a running first in the suite). + Adds: nat_neq/less/more, plus, label, trow (row sort + constructors), + record tp constructor, sub_trow, sub_tp, sub_tp_trow. + Dropped: all proof families (sum_inc, commute', assoc, assoc', add, etc.) + and %block/%reduces declarations. +*) +let poplmark_1b_syntax = + {| +%sort nat_neq {_ nat} {_ nat} +%term nat_neq_zs {{N}} nat_neq z (s N) +%term nat_neq_sz {{N}} nat_neq (s N) z +%term nat_neq_ss {{N M}} %<- (nat_neq (s N) (s M)) (nat_neq N M) + +%sort nat_less {_ nat} {_ nat} +%term nat_less_z {{N}} nat_less z (s N) +%term nat_less_s {{N M}} %<- (nat_less (s N) (s M)) (nat_less N M) + +%sort nat_more {_ nat} {_ nat} +%term nat_more_z {{N}} nat_more (s N) z +%term nat_more_s {{N M}} %<- (nat_more (s N) (s M)) (nat_more N M) + +%sort plus {_ nat} {_ nat} {_ nat} +%term plus_z {{N}} plus z N N +%term plus_s {{M N N'}} %<- (plus (s M) N (s N')) (plus M N N') + +%sort label +%term label_nat {_ nat} label + +%sort label_eq {_ label} {_ label} +%term label_eq_ {{L}} label_eq L L + +%sort label_neq {_ label} {_ label} +%term label_neq_ {{N M}} %<- (label_neq (label_nat N) (label_nat M)) (nat_neq N M) + +%sort label_less {_ label} {_ label} +%term label_less_ {{N M}} %<- (label_less (label_nat N) (label_nat M)) (nat_less N M) + +%sort label_more {_ label} {_ label} +%term label_more_ {{N M}} %<- (label_more (label_nat N) (label_nat M)) (nat_more N M) + +%sort trow +%term trow_nil trow +%term trow_cons {_ label} {_ tp} {_ trow} trow + +%sort trow_lookup {_ label} {_ trow} {_ tp} +%term trow_lookup_yes {L label} {T tp} {TR trow} trow_lookup L (trow_cons L T TR) T +%term trow_lookup_no {L label} {L' label} {T tp} {T' tp} {TR trow} %<- (trow_lookup L (trow_cons L' T' TR) T) (trow_lookup L TR T) + +%sort trow_labelfree {_ trow} {_ label} +%term trow_labelfree_nil {L label} trow_labelfree trow_nil L +%term trow_labelfree_cons {L label} {L' label} {TR trow} {T tp} %<- (trow_labelfree (trow_cons L' T TR) L) (label_neq L L') (trow_labelfree TR L) + +%sort trow_eq {_ trow} {_ trow} +%term trow_eq_ {{TR}} trow_eq TR TR + +%sort trow_order {_ trow} {_ trow} +%sort trow_insert {_ label} {_ tp} {_ trow} {_ trow} + +%term trow_order_nil trow_order trow_nil trow_nil +%term trow_order_cons {L label} {S tp} {SR trow} {TR trow} {TR' trow} %<- (trow_order (trow_cons L S SR) TR') (trow_order SR TR) (trow_insert L S TR TR') + +%term trow_insert_nil {L label} {S tp} trow_insert L S trow_nil (trow_cons L S trow_nil) +%term trow_insert_less {L label} {L' label} {S tp} {T tp} {TR trow} %<- (trow_insert L S (trow_cons L' T TR) (trow_cons L S (trow_cons L' T TR))) (label_less L L') +%term trow_insert_more {L label} {L' label} {S tp} {T tp} {TR trow} {TR' trow} %<- (trow_insert L S (trow_cons L' T TR) (trow_cons L' T TR')) (label_more L L') (trow_insert L S TR TR') + +%sort trow_uniqueness {_ trow} +%term trow_uniqueness_nil trow_uniqueness trow_nil +%term trow_uniqueness_cons {L label} {TR trow} {T tp} %<- (trow_uniqueness (trow_cons L T TR)) (trow_labelfree TR L) (trow_uniqueness TR) + +%term record {_ trow} {_ trow_uniqueness _} tp + +%sort sub_trow {_ trow} {_ trow} +%term sub_trow_nil sub_trow trow_nil trow_nil +%term sub_trow_cons {L label} {S tp} {T tp} {SR trow} {TR trow} %<- (sub_trow (trow_cons L S SR) (trow_cons L T TR)) (sub_trow SR TR) (sub S T) +%term sub_trow_cons' {L label} {S tp} {SR trow} {TR trow} %<- (sub_trow (trow_cons L S SR) TR) (sub_trow SR TR) + +%sort sub_tp {_ tp} {_ tp} +%sort sub_tp_trow {_ trow} {_ trow} + +%term sub_tp_top {{T}} sub_tp T top +%term sub_tp_refl {{T}} sub_tp T T +%term sub_tp_trans {{T1 T2 T3}} %<- (sub_tp T1 T3) (sub_tp T1 T2) (sub_tp T2 T3) +%term sub_tp_arrow {{S1 S2 T1 T2}} %<- (sub_tp (arrow S1 S2) (arrow T1 T2)) (sub_tp T1 S1) (sub_tp S2 T2) +%term sub_tp_forall {{S1 S2 T1 T2}} %<- (sub_tp (forall S1 S2) (forall T1 T2)) (sub_tp T1 S1) ({x tp} {_ sub_tp x T1} sub_tp (S2 x) (T2 x)) +%term sub_tp_record {SR trow} {TR trow} {SR' trow} {TR' trow} {SRuniq trow_uniqueness SR} {TRuniq trow_uniqueness TR} %<- (sub_tp (record SR SRuniq) (record TR TRuniq)) (trow_order SR SR') (trow_order TR TR') (sub_tp_trow SR' TR') + +%term sub_tp_trow_nil sub_tp_trow trow_nil trow_nil +%term sub_tp_trow_cons {L label} {S tp} {T tp} {SR trow} {TR trow} %<- (sub_tp_trow (trow_cons L S SR) (trow_cons L T TR)) (sub_tp_trow SR TR) (sub_tp S T) +%term sub_tp_trow_cons' {L label} {S tp} {SR trow} {TR trow} %<- (sub_tp_trow (trow_cons L S SR) TR) (sub_tp_trow SR TR) +|} + +(* POPLMARK-2B: System Fw with records — syntax-only extension of 1b. + Ported from twelf/examples/poplmark/2b.elf (syntax block only). + Depends on POPLMARK-1B's nat/label/trow/sub_tp/sub_tp_trow in global scope. + Adds: term sorts (term/bterm/erow/pattern/prow), basic constructors. + Dropped: all proof families (typing, step, progress, preservation). +*) +let poplmark_2b_syntax = + {| +%sort term +%sort bterm +%sort erow +%sort pattern +%sort prow + +%term base {_ term} bterm +%term bnd {_ tp} {_ {_ term} bterm} bterm + +%term abs {_ tp} {_ {_ term} term} term +%term app {_ term} {_ term} term +%term tabs {_ tp} {_ {_ tp} term} term +%term tapp {_ term} {_ tp} term + +%term rec {_ erow} term +%term proj {_ term} {_ label} term +%term plet {_ pattern} {_ term} {_ bterm} term + +%term erow_nil erow +%term erow_cons {_ label} {_ term} {_ erow} erow + +%sort erow_lookup {_ label} {_ erow} {_ term} +%term erow_lookup_yes {L label} {E term} {ER erow} erow_lookup L (erow_cons L E ER) E +%term erow_lookup_no {L label} {L' label} {E term} {E' term} {ER erow} %<- (erow_lookup L (erow_cons L' E' ER) E) (erow_lookup L ER E) + +%sort erow_order {_ erow} {_ erow} +%sort erow_insert {_ label} {_ term} {_ erow} {_ erow} + +%term erow_order_nil erow_order erow_nil erow_nil +%term erow_order_cons {L label} {E term} {ER erow} {ER' erow} {ER'' erow} %<- (erow_order (erow_cons L E ER) ER') (erow_order ER ER'') (erow_insert L E ER'' ER') + +%term erow_insert_nil {L label} {E term} erow_insert L E erow_nil (erow_cons L E erow_nil) +%term erow_insert_less {L label} {L' label} {E term} {E' term} {ER erow} %<- (erow_insert L E (erow_cons L' E' ER) (erow_cons L E (erow_cons L' E' ER))) (label_less L L') +%term erow_insert_more {L label} {L' label} {E term} {E' term} {ER erow} {ER' erow} %<- (erow_insert L E (erow_cons L' E' ER) (erow_cons L' E' ER')) (label_more L L') (erow_insert L E ER ER') + +%term pat_var {_ tp} pattern +%term pat_rec {_ prow} pattern + +%term prow_nil prow +%term prow_cons {_ label} {_ pattern} {_ prow} prow +|} + +(* CCC: Cartesian closed category syntax. + Ported from twelf/examples/ccc/ccc.elf. + Renamed: `1` (terminal obj) → `unit_obj`, `*` (product) → `prod`, `@` → `comp`, + `==` (morphism equality) → `meq`, `=>` (exponential) → `exp_obj`. + Note: no %prec declarations — all operators used in prefix form to avoid + "Leading infix operator" parse errors that occur when infix is mixed with prefix. +*) +let ccc_syntax = + {| +%sort obj +%sort mor {_ obj} {_ obj} +%sort meq {_ mor _A _B} {_ mor _A _B} + +%term id {{A}} mor A A +%term comp {{A B C}} {_ mor B C} {_ mor A B} mor A C + +%term meq_refl {{A B F}} meq F F +%term meq_then {{A B F F' F''}} {_ meq F F'} {_ meq F' F''} meq F F'' +%term meq_sym {{A B F F'}} {_ meq F F'} meq F' F + +%term eq_comp {{A B C F F' G G'}} {_ meq F F'} {_ meq G G'} meq (comp F G) (comp F' G') + +%term id_l {{A B F}} meq (comp id F) F +%term id_r {{A B F}} meq (comp F id) F +%term assoc {{A B C D F G H}} meq (comp H (comp G F)) (comp (comp H G) F) + +%term unit_obj obj +%term prod {_ obj} {_ obj} obj + +%term drop {{A}} mor A unit_obj +%term fst {{A B}} mor (prod A B) A +%term snd {{A B}} mor (prod A B) B +%term pair_mor {{A B C}} {_ mor A B} {_ mor A C} mor A (prod B C) + +%term eq_pair {{A B C F F' G G'}} {_ meq F F'} {_ meq G G'} meq (pair_mor F G) (pair_mor F' G') +%term prod_l {{A B C F G}} meq (comp fst (pair_mor F G)) F +%term prod_r {{A B C F G}} meq (comp snd (pair_mor F G)) G +%term prod_u {{A B C H}} meq (pair_mor (comp fst H) (comp snd H)) H + +%term exp_obj {_ obj} {_ obj} obj + +%term app_mor {{B C}} mor (prod (exp_obj B C) B) C +%term cur {{A B C}} {_ mor (prod A B) C} mor A (exp_obj B C) + +%term eq_cur {{A B C F F'}} {_ meq F F'} meq (cur F) (cur F') +%term exp_e {{A B C F}} meq (comp app_mor (pair_mor (comp (cur F) fst) snd)) F +%term exp_u {{A B C G}} meq (cur (comp app_mor (pair_mor (comp G fst) snd))) G +|} + +(* INCLL: INCLL sorts and terms. + Ported from twelf/examples/incll/incll.elf. + Renamed: numeric term names `1..5` → `v1..v5` (STELF identifiers must not start with digit). + Dropped: Twelf abbreviation syntax (`<= = [x][y] y => x` etc.). + Dropped: `|` infix list cons (single-char identifier, may fail — using `cons` instead). + `^` prefix also risky — using `atm_frm` instead. +*) +let incll_syntax = + {| +%sort sort +%term arrow {_ sort} {_ sort} sort +%term cross {_ sort} {_ sort} sort + +%sort trm {_ sort} +%term app {{A B}} {_ trm (arrow A B)} {_ trm A} trm B +%term lam {{A B}} {_ {_ trm A} trm B} trm (arrow A B) +%term pair {{A B}} {_ trm A} {_ trm B} trm (cross A B) + +%sort eval {_ trm _A} {_ trm _A} +%term eval_lam {{A B E}} eval (lam E) (lam E) +%term eval_app {{A B E E' V V'}} %<- (eval (app (lam E) E') V) (eval E' V') (eval (E V') V) + +%sort atm +%sort frm +%term int sort + +%term v1 trm int +%term v2 trm int +%term v3 trm int +%term v4 trm int +%term v5 trm int + +%sort list_sort {_ sort} +%term list_sort/nil {{A}} list_sort A +%term list_sort/cons {{A}} {_ trm A} {_ list_sort A} list_sort A + +%sort frm_atm {_ atm} +%term atm_frm {_ atm} frm + +%sort imp {_ frm} {_ frm} +%term imp_i {_ frm} {_ frm} frm + +%term forall {A sort} {_ {_ trm A} frm} frm +%term forall2 {A1 sort} {A2 sort} {_ {_ {_ trm A1} trm A2} frm} frm +|} + +(* CRARY-LINEAR (syntax-only): linear substructural types. + Ported from twelf/examples/crary/substruct/linear.elf (syntax block only). + Type `!` in Twelf = the bang/of-course modality. Trying `!` as identifier verbatim. + `with` is a keyword in OCaml but it's inside a raw string — should be fine in the parser. +*) +let crary_linear_syntax = + {| +%sort atom +%sort tp +%sort term + +%term atomic {_ atom} tp +%term lolli {_ tp} {_ tp} tp +%term tensor {_ tp} {_ tp} tp +%term with {_ tp} {_ tp} tp +%term plus {_ tp} {_ tp} tp +%term one tp +%term zero tp +%term top tp +%term bang {_ tp} tp + +%term llam {_ {_ term} term} term +%term lapp {_ term} {_ term} term + +%term tpair {_ term} {_ term} term +%term lett {_ term} {_ {_ term} {_ term} term} term + +%term pair {_ term} {_ term} term +%term pi1 {_ term} term +%term pi2 {_ term} term + +%term in1 {_ term} term +%term in2 {_ term} term +%term case {_ term} {_ {_ term} term} {_ {_ term} term} term + +%term star term +%term leto {_ term} {_ term} term + +%term any {_ term} term +%term unit term + +%term bang_tm {_ term} term +%term letb {_ term} {_ {_ term} term} term + +%term a atom +%term b atom +|} + +(* CRARY-LINEAR (linearity sort and basic terms): + The `linear` family from linear.elf. +*) +let crary_linear_linear = + {| +%sort linear {_ {_ term} term} +%term linear/var linear ([x] x) +%term linear/llam {{M}} {_ {y term} linear ([x] M x y)} linear ([x] llam ([y] M x y)) +%term linear/lapp1 {{M N}} {_ linear ([x] M x)} linear ([x] lapp (M x) N) +%term linear/lapp2 {{M N}} {_ linear ([x] N x)} linear ([x] lapp M (N x)) +%term linear/tpair1 {{M N}} {_ linear ([x] M x)} linear ([x] tpair (M x) N) +%term linear/tpair2 {{M N}} {_ linear ([x] N x)} linear ([x] tpair M (N x)) +%term linear/pair {{M N}} {_ linear ([x] M x)} {_ linear ([x] N x)} linear ([x] pair (M x) (N x)) +%term linear/pi1 {{M}} {_ linear ([x] M x)} linear ([x] pi1 (M x)) +%term linear/pi2 {{M}} {_ linear ([x] M x)} linear ([x] pi2 (M x)) +%term linear/in1 {{M}} {_ linear ([x] M x)} linear ([x] in1 (M x)) +%term linear/in2 {{M}} {_ linear ([x] M x)} linear ([x] in2 (M x)) +%term linear/unit linear ([x] unit) +|} + +(* CRARY-LINEARD (syntax-only): dual linear type system. + Ported from twelf/examples/crary/substruct/lineard.elf (syntax only). + Similar to linear.elf but adds `constant`, `pi` (dependent type), `ulam`/`uapp`. +*) +let crary_lineard_syntax = + {| +%sort constant +%sort tp +%sort term + +%term const {_ constant} {_ term} tp +%term pi {_ tp} {_ {_ term} tp} tp +%term lolli {_ tp} {_ tp} tp +%term tensor {_ tp} {_ tp} tp +%term with {_ tp} {_ tp} tp +%term plus {_ tp} {_ tp} tp +%term one tp +%term zero tp +%term top tp +%term bang {_ tp} tp + +%term ulam {_ {_ term} term} term +%term uapp {_ term} {_ term} term + +%term llam {_ {_ term} term} term +%term lapp {_ term} {_ term} term + +%term tpair {_ term} {_ term} term +%term lett {_ term} {_ {_ term} {_ term} term} term + +%term pair {_ term} {_ term} term +%term pi1 {_ term} term +%term pi2 {_ term} term + +%term in1 {_ term} term +%term in2 {_ term} term +%term case {_ term} {_ {_ term} term} {_ {_ term} term} term + +%term star term +%term leto {_ term} {_ term} term + +%term any {_ term} term +%term unit term + +%term bang_tm {_ term} term +%term letb {_ term} {_ {_ term} term} term +|} + +(* CRARY-MODAL (syntax-only): modal substructural types. + Ported from twelf/examples/crary/substruct/modal.elf (syntax block only). + Dropped: `local/bx : local ([x] bx M) = local/closed.` definitional equality form. +*) +let crary_modal_syntax = + {| +%sort atom +%sort tp +%sort term +%sort exp + +%term atomic {_ atom} tp +%term arrow {_ tp} {_ tp} tp +%term box {_ tp} tp +%term diamond {_ tp} tp + +%term lam {_ {_ term} term} term +%term app {_ term} {_ term} term + +%term bx {_ term} term +%term letbx {_ term} {_ {_ term} term} term + +%term di {_ exp} term + +%term here {_ term} exp +%term eletbx {_ term} {_ {_ term} exp} exp +%term letdi {_ term} {_ {_ term} exp} exp + +%term a atom +%term b atom +|} From 3c972a43260b63fa6e64fe19e053838741f4725c Mon Sep 17 00:00:00 2001 From: Asher Frost Date: Tue, 9 Jun 2026 10:21:03 -0400 Subject: [PATCH 4/9] chore: fixed pal tests --- test/Pal/Common.ml | 4 +++- test/Pal/Source.ml | 8 ++++---- 2 files changed, 7 insertions(+), 5 deletions(-) diff --git a/test/Pal/Common.ml b/test/Pal/Common.ml index 87b6462..1c99251 100644 --- a/test/Pal/Common.ml +++ b/test/Pal/Common.ml @@ -27,14 +27,16 @@ let test ?(skip = false) ?(failure = false) (name : string) (cmds : string list) let _ = (Display.fmt Fmt.stdout m.msg ) in Lwt.return ()) in + let has_failed = ref false in let tester' = new tester in name, List.mapi(fun i cmd -> Alcotest.test_case (name ^ " - " ^ string_of_int (i + 1)) `Slow (fun () -> - if skip then Alcotest.skip () + if skip || !has_failed then Alcotest.skip () else match tester'#run cmd with | None when failure -> Alcotest.fail "Expected failure, but test passed" | Some e when not failure -> let bt = Printexc.get_backtrace () in + has_failed := true; Printf.eprintf "Exception: %s\nBacktrace:\n%s\n%!" (Printexc.to_string e) bt; Alcotest.failf "Expected success, but test failed with exception: %s\nBacktrace:\n%s" diff --git a/test/Pal/Source.ml b/test/Pal/Source.ml index 8e874b3..008a31e 100644 --- a/test/Pal/Source.ml +++ b/test/Pal/Source.ml @@ -308,17 +308,17 @@ Mapping natural deductions to Hilbert derivations. %term ndh_impe {{A B D1 D2 H1 H2}} {_ ndhil D1 H1} {_ ndhil D2 H2} ndhil (impe D1 D2) (mp H1 H2) -%term ndh_noti {{A1 H1 H1' H1'' D1}} {_ ded (H1 (not A1)) H1'} {_ {p o} {u nd A1} {v hil A1} {_ {C o} ded ([w hil C] v) (mp k v)} {_ ndhil u v} ndhil (D1 p u) (H1 p v)} ndhil (noti D1) (mp (mp n1 H1') H1'') +%term ndh_noti {{A1 H1 H1' H1'' D1}} {_ ded (H1 (not A1)) H1'} {_ ded (H1 A1) H1''} {_ {p o} {u nd A1} {v hil A1} {_ {C o} ded ([w hil C] v) (mp k v)} {_ ndhil u v} ndhil (D1 p u) (H1 p v)} ndhil (noti D1) (mp (mp n1 H1') H1'') %term ndh_note {{A C D1 D2 H1 H2}} {_ ndhil D1 H1} {_ ndhil D2 H2} ndhil (note D1 C D2) (mp (mp n2 H1) H2) -%term ndh_foralli {{A}} {D1 {_ i} nd _} {H1 {_ i} hil _} {_ {a i} ndhil (D1 a) (H1 a)} ndhil (foralli D1) (ug H1) +%term ndh_foralli {{A D1 H1}} {_ {a i} ndhil (D1 a) (H1 a)} ndhil (foralli D1) (ug H1) -%term ndh_foralle {{A}} {T i} {D1 nd (forall A)} {H1 hil (forall A)} {_ ndhil D1 H1} ndhil (foralle D1 T) (mp (f1 T) H1) +%term ndh_foralle {{A T D1 H1}} {_ ndhil D1 H1} ndhil (foralle D1 T) (mp (f1 T) H1) |} let fol6_2 = {| %mode {%in X _} {%out Y _} ndhil X Y -%block lndhil [A o] {u nd A} {v hil A} {h {C o} ded ([w] v) (mp k v)} {nh ndhil u v} +%block lndhil [A o] {u nd A} {v hil A} {h {C o} ded ([w hil C] v) (mp k v)} {nh ndhil u v} %worlds (li lo lndhil) (ndhil D H) %terminates D (ndhil D _) %covers ndhil %in %out From 265b97e9172be6f4e0aaf77db337e6f22d96122b Mon Sep 17 00:00:00 2001 From: Asher Frost Date: Wed, 10 Jun 2026 14:52:54 -0400 Subject: [PATCH 5/9] feat: got almost all of the features working --- STYLE.md | 22 ++++ dune-project | 1 + src/Common/Cst/SIM.ml | 9 ++ src/Common/Cst/Sim.ml | 10 ++ src/Config/CONFIG.ml | 29 ++++++ src/Config/Config.ml | 0 src/Config/dune | 4 + src/Fronts/Pal/Impl.ml | 2 +- src/Fronts/Pal/Impl.mli | 2 +- src/Fronts/Pal/PAL.ml | 14 +-- src/Fronts/Pal/Pal.ml | 39 +++---- src/Fronts/Pal/Pal.mli | 2 + src/Fronts/Project/Format.ml | 0 src/Fronts/Project/Project.ml | 0 src/Fronts/Project/dune | 4 + src/Fronts/Tui/Repl.ml | 5 +- src/Lang/Parsing/PARSER.ml | 1 + src/Lang/Parsing/Parser.ml | 1 + src/Resugar/ConDec.ml | 1 + src/Resugar/Decl.ml | 0 src/Resugar/RESUGAR.ml | 15 ++- src/Resugar/Resugar.ml | 7 ++ src/Resugar/Term.ml | 5 + src/Resugar/dune | 3 +- src/global/Global_.ml | 7 ++ src/global/Global_intf.ml | 7 ++ src/print/Print_.ml | 83 ++++++++------- stelf.opam | 1 + test/Pal/Cases.ml | 107 ++++--------------- test/Pal/Common.ml | 36 +++---- test/Pal/Source.ml | 187 +++++++++++++++++++++------------- 31 files changed, 358 insertions(+), 246 deletions(-) create mode 100644 STYLE.md create mode 100644 src/Common/Cst/SIM.ml create mode 100644 src/Common/Cst/Sim.ml create mode 100644 src/Config/CONFIG.ml create mode 100644 src/Config/Config.ml create mode 100644 src/Config/dune create mode 100644 src/Fronts/Project/Format.ml create mode 100644 src/Fronts/Project/Project.ml create mode 100644 src/Fronts/Project/dune create mode 100644 src/Resugar/ConDec.ml create mode 100644 src/Resugar/Decl.ml create mode 100644 src/Resugar/Term.ml diff --git a/STYLE.md b/STYLE.md new file mode 100644 index 0000000..dce1129 --- /dev/null +++ b/STYLE.md @@ -0,0 +1,22 @@ +# Style Guide + +> [!NOTE] This list is not exhaustive. Also, a number of these are not yet correctly used. + +Avoid names ending in underscores, esspecially if they don't conflict with another name. +Use `snake_case` for lowercase names, and `CamelCase` for uppercase names. The only exception to this is that `Make_` should be used as a prefix for functors, as `Make_HashMap` + +A module `Foo` should be of module type `FOO`. +Each library should have a top level interface, `*.mli`. + +If there is a module `Foo`, then it should be defined in `Foo.ml`, and its interface should be defined in `FOO.ml`, prefer `Foo.ml` to `foo.ml`, and `FOO.ml` to `Foo_intf.ml` or `foo_intf.ml`. + +In this case, the module should look like this + +```ocaml +module type FOO = FOO.FOO +module Foo : FOO = struct + (* Implementation here *) +end +``` + +All module types should be documented \ No newline at end of file diff --git a/dune-project b/dune-project index b1d2962..605ec76 100644 --- a/dune-project +++ b/dune-project @@ -51,6 +51,7 @@ (ocaml-lsp-server :dev) (user-setup :dev) (merlin :dev) + sexplib ppx_deriving lambda-term fmt diff --git a/src/Common/Cst/SIM.ml b/src/Common/Cst/SIM.ml new file mode 100644 index 0000000..d461dbe --- /dev/null +++ b/src/Common/Cst/SIM.ml @@ -0,0 +1,9 @@ + + +module type SIM = sig + module CstA : CST.CST + module CstB : CST.CST + val sim_term : CstA.View.Term.t -> CstB.View.Term.t -> bool + val sim_decl : CstA.View.Decl.t -> CstB.View.Decl.t -> bool + +end \ No newline at end of file diff --git a/src/Common/Cst/Sim.ml b/src/Common/Cst/Sim.ml new file mode 100644 index 0000000..1c778d0 --- /dev/null +++ b/src/Common/Cst/Sim.ml @@ -0,0 +1,10 @@ +(** + Whether two terms are "similar" to each other + Simalarity encompasses two things + 1. Whether the terms can be viewed as the same + 2. Whether the terms are equivalent representations of the same thing +*) + +module Sim (A : CST.CST) (B : CST.CST) = struct + +end \ No newline at end of file diff --git a/src/Config/CONFIG.ml b/src/Config/CONFIG.ml new file mode 100644 index 0000000..728ade4 --- /dev/null +++ b/src/Config/CONFIG.ml @@ -0,0 +1,29 @@ +type 'a flag_parser = + + (** Whether or not [A-Z] charecters are treated as uppercase*) + | UppercaseLatin : bool flag_parser + + (** If [->] is a valid alternative to [%->] (and the same for [<-])*) + | ArrowReserved : bool flag_parser + + (** Whether [->] [<-] can occur infix*) + | ArrowInfix : bool flag_parser + + (** Whether [|] can be used in context in a world / block *) + | BarInContext : bool flag_parser + + (** Whether [:] can be used in decl *) + | ColonInDecl : bool flag_parser + + (** Whether [=] can be used in def *) + | EqualInDef : bool flag_parser + + (** Whether [.] is a alternative to [.] *) + | StopReserved : bool flag_parser + +module type CONFIG = sig + type 'a t + val get : 'a t -> 'a + val set : 'a t -> 'a -> bool + val init : unit -> unit +end \ No newline at end of file diff --git a/src/Config/Config.ml b/src/Config/Config.ml new file mode 100644 index 0000000..e69de29 diff --git a/src/Config/dune b/src/Config/dune new file mode 100644 index 0000000..bb9fcdc --- /dev/null +++ b/src/Config/dune @@ -0,0 +1,4 @@ +(library + (name config) + (public_name stelf.config) + (libraries )) \ No newline at end of file diff --git a/src/Fronts/Pal/Impl.ml b/src/Fronts/Pal/Impl.ml index 0fab445..46209a9 100644 --- a/src/Fronts/Pal/Impl.ml +++ b/src/Fronts/Pal/Impl.ml @@ -10,7 +10,7 @@ let string_to_source = function | "" -> None | s -> Some (Fpath.of_string s) -module Impl = struct +module Impl () = struct (* Save Basis's OS before any module definitions shadow it *) module BasisOS = OS diff --git a/src/Fronts/Pal/Impl.mli b/src/Fronts/Pal/Impl.mli index 59990fa..e129eb4 100644 --- a/src/Fronts/Pal/Impl.mli +++ b/src/Fronts/Pal/Impl.mli @@ -1 +1 @@ -module Impl : IMPL.IMPL +module Impl () : IMPL.IMPL diff --git a/src/Fronts/Pal/PAL.ml b/src/Fronts/Pal/PAL.ml index 72a33de..481286f 100644 --- a/src/Fronts/Pal/PAL.ml +++ b/src/Fronts/Pal/PAL.ml @@ -1,13 +1,15 @@ -module type PAL = sig +module type PAL' = sig module M : IMPL.IMPL + val install : M.Cst.cmd -> unit + val parse : string -> M.Cst.cmd list + val exec : string -> unit + end +module type PAL = sig + module M : IMPL.IMPL exception Error of exn - class pal : object - method install : M.Cst.cmd -> unit - method parse : string -> M.Cst.cmd list - method exec : string -> unit - end + module Start () : PAL' with module M = M val run : unit -> unit end diff --git a/src/Fronts/Pal/Pal.ml b/src/Fronts/Pal/Pal.ml index 227ecb1..207da6f 100644 --- a/src/Fronts/Pal/Pal.ml +++ b/src/Fronts/Pal/Pal.ml @@ -1,30 +1,31 @@ let version = "0.1.0" - +module type PAL = PAL.PAL +module type PAL' = PAL.PAL' module Pal : PAL.PAL = struct - module M = Impl.Impl - + module M = Impl.Impl () + module type S = module type of M let status_to_exit : M.status -> int = function M.Ok -> 0 | M.Abort -> 1 exception Error of exn + + + module Start () = struct + module M = M + let ns = ref (M.Cmd.Modern.Names.newNamespace ()) + let loc = ref M.Cst.ghost + let install (cmd : M.Cst.cmd) : unit = M.Install.install1 cmd - class pal = - object (self) - val ns = ref (M.Cmd.Modern.Names.newNamespace ()) - val mutable loc = M.Cst.ghost - method install (cmd : M.Cst.cmd) : unit = M.Install.install1 cmd + let parse (s : string) : M.Cst.cmd list = + M.Cmd.Modern.run (M.Cmd.parse ()) ns !loc s - method parse (s : string) : M.Cst.cmd list = - M.Cmd.Modern.run (M.Cmd.parse ()) ns loc s - - method exec (s : string) : unit = List.iter self#install (self#parse s) - end - + let exec (s : string) : unit = List.iter install (parse s) + end let top (module N : Tui.REPL.S) = - let pi = new pal in + let module Pal = Start () in let module R = Tui.Repl.Repl (N) in - Impl.Impl.mode := `Repl; - R.read (fun l -> - pi#exec l; + M.mode := `Repl; + R.read (fun l -> + Pal.exec l; Lwt.return R.Continue) let run () : unit = @@ -47,7 +48,7 @@ module Pal : PAL.PAL = struct let repl_cmd : int Cmd.t = let repl_fn : int Term.t = let+ verbosity = Arg.value Opts.Opts.verbosity - and+ color = Arg.value Opts.Opts.color + and+ color = Arg.value Opts.Opts.color and+ unicode = Arg.value Opts.Opts.unicode in let module M = struct let use_color = color diff --git a/src/Fronts/Pal/Pal.mli b/src/Fronts/Pal/Pal.mli index a77a1bc..22b48c8 100644 --- a/src/Fronts/Pal/Pal.mli +++ b/src/Fronts/Pal/Pal.mli @@ -1 +1,3 @@ +module type PAL = PAL.PAL +module type PAL' = PAL.PAL' module Pal : PAL.PAL diff --git a/src/Fronts/Project/Format.ml b/src/Fronts/Project/Format.ml new file mode 100644 index 0000000..e69de29 diff --git a/src/Fronts/Project/Project.ml b/src/Fronts/Project/Project.ml new file mode 100644 index 0000000..e69de29 diff --git a/src/Fronts/Project/dune b/src/Fronts/Project/dune new file mode 100644 index 0000000..f401e61 --- /dev/null +++ b/src/Fronts/Project/dune @@ -0,0 +1,4 @@ +(library + (name project) + (public_name stelf.project) + (libraries sexplib)) \ No newline at end of file diff --git a/src/Fronts/Tui/Repl.ml b/src/Fronts/Tui/Repl.ml index a9c982d..51f5b6d 100644 --- a/src/Fronts/Tui/Repl.ml +++ b/src/Fronts/Tui/Repl.ml @@ -7,7 +7,10 @@ module Repl (M : REPL.S) : REPL.REPL = struct () let msgs : Display.Info.t list ref = ref [] - let add_msg m = msgs := m :: !msgs + let add_msg (m : Display.Info.t) = let r = if Display.Info.to_int M.verbosity >= Display.Info.to_int m.level then + m :: !msgs + else !msgs in + msgs := r let flush_msgs () : Display.Info.t array Lwt.t = let pending = Array.of_list (List.rev !msgs) in diff --git a/src/Lang/Parsing/PARSER.ml b/src/Lang/Parsing/PARSER.ml index 81cafed..6914ae9 100644 --- a/src/Lang/Parsing/PARSER.ml +++ b/src/Lang/Parsing/PARSER.ml @@ -10,6 +10,7 @@ module type PARSER = sig val keywords : string list -> unit t val token : string -> unit t val given : bool -> 'a t -> 'a t + val extend : 'a t -> (bool * 'a t) -> 'a t val ( let| ) : 'a t -> ('a -> 'b t) -> 'b t val ( and| ) : 'a t -> 'b t -> ('a * 'b) t val ( let* ) : 'a t -> ('a -> 'b t) -> 'b t diff --git a/src/Lang/Parsing/Parser.ml b/src/Lang/Parsing/Parser.ml index 50f3eae..803c5df 100644 --- a/src/Lang/Parsing/Parser.ml +++ b/src/Lang/Parsing/Parser.ml @@ -67,5 +67,6 @@ module Parser : PARSER = struct let given b p = if b then p else fail "failed test" let inside x y p = token x *> p <* token y + let extend _p _q = assert false let forget p = p *> return () end diff --git a/src/Resugar/ConDec.ml b/src/Resugar/ConDec.ml new file mode 100644 index 0000000..b754d5d --- /dev/null +++ b/src/Resugar/ConDec.ml @@ -0,0 +1 @@ +open RESUGAR diff --git a/src/Resugar/Decl.ml b/src/Resugar/Decl.ml new file mode 100644 index 0000000..e69de29 diff --git a/src/Resugar/RESUGAR.ml b/src/Resugar/RESUGAR.ml index 6a09d72..31db915 100644 --- a/src/Resugar/RESUGAR.ml +++ b/src/Resugar/RESUGAR.ml @@ -1,5 +1,16 @@ +module type S = sig + module Syntax : Syntax.SYNTAX + module Cst : Cst.CST + module Names : Names.Names_intf.NAMES +end + module type RESUGAR = sig module Syntax : Syntax.SYNTAX - module Cst : Cst.CST - type t + module Cst : Cst.CST + type t + type u + val register : t -> u -> unit + val resugar : t -> u +end + \ No newline at end of file diff --git a/src/Resugar/Resugar.ml b/src/Resugar/Resugar.ml index e69de29..0458772 100644 --- a/src/Resugar/Resugar.ml +++ b/src/Resugar/Resugar.ml @@ -0,0 +1,7 @@ +module Make_Resugar (Syntax : Syntax.SYNTAX) (Cst : Cst.CST) (* : RESUGAR.RESUGAR with module Syntax = Syntax with module Cst = Cst *) = struct + module Syntax = Syntax + module Cst = Cst + + let register = assert false + let resugar = assert false +end diff --git a/src/Resugar/Term.ml b/src/Resugar/Term.ml new file mode 100644 index 0000000..f423594 --- /dev/null +++ b/src/Resugar/Term.ml @@ -0,0 +1,5 @@ +module Make_Expr (M : RESUGAR.S) = struct + open M + let register = assert false + let resugar = assert false +end \ No newline at end of file diff --git a/src/Resugar/dune b/src/Resugar/dune index b31b131..8abc4f1 100644 --- a/src/Resugar/dune +++ b/src/Resugar/dune @@ -1,4 +1,5 @@ (library (name resugar) (public_name stelf.resugar) - (libraries syntax cst)) + (libraries syntax names paths cst)) + \ No newline at end of file diff --git a/src/global/Global_.ml b/src/global/Global_.ml index 80433ff..84b1856 100644 --- a/src/global/Global_.ml +++ b/src/global/Global_.ml @@ -36,5 +36,12 @@ module Global : GLOBAL = struct let chMessage n s f = begin if !chatter >= n then f (s ()) else () end + + let arrow_reserved = ref false + let arrow_infix = ref false + let latin_uppercase = ref false + let bar_in_block = ref false + let old_some = ref false + let stop_reserved = ref false end (* structure Global *) diff --git a/src/global/Global_intf.ml b/src/global/Global_intf.ml index 3bf2f62..a648d4e 100644 --- a/src/global/Global_intf.ml +++ b/src/global/Global_intf.ml @@ -18,4 +18,11 @@ module type GLOBAL = sig val chPrint : int -> (unit -> string) -> unit val chMessage : int -> (unit -> string) -> (string -> unit) -> unit val timeLimit : Time.time option ref + + val arrow_reserved : bool ref + val arrow_infix : bool ref + val latin_uppercase : bool ref + val bar_in_block : bool ref + val old_some : bool ref + val stop_reserved : bool ref end diff --git a/src/print/Print_.ml b/src/print/Print_.ml index b6534a9..e2f07e0 100644 --- a/src/print/Print_.ml +++ b/src/print/Print_.ml @@ -77,6 +77,7 @@ module MakePrint module F = Formatter module T = Tomega + let full_stop = F.string "%." let lvars : I.block option ref list ref = ref [] let rec lookuplvar l = @@ -164,7 +165,7 @@ module MakePrint let rec arrow (v1_, v2_) = OpArgs ( FX.Infix (arrowPrec, FX.Right), - [ F.break; sym "->"; F.space ], + [ F.break; sym "%->"; F.space ], I.App (v1_, I.App (v2_, I.Nil)) ) let appCtxt = Ctxt (FX.Nonfix, [], 0) @@ -193,7 +194,7 @@ module MakePrint if !showConstPath then F.hVbox (foldr - (function id, fmt -> str0_ (Symbol.str id) :: sym "." :: fmt) + (function id, fmt -> str0_ (Symbol.str id) :: full_stop :: fmt) [ str0_ (f id) ] ids) else str0_ (f id) @@ -317,19 +318,13 @@ module MakePrint d, ctx, (braces (g_, d, ((d'_, v2_), s)), I.dot1 s) ) - | meta_ -> + | _ -> let d'_ = Names.decLUName (g_, d_) in fmtLevel ( I.Decl (g_, d'_), d, ctx, (braces (g_, d, ((d'_, v2_), s)), I.dot1 s) ) - | I.No -> - fmtLevel - ( I.Decl (g_, d_), - d, - ctx, - (arrow (I.EClo (v1_, I.shift), v2_), I.dot1 s) ) end | g_, d, ctx, (I.Pi (((I.BDec _ as d_), p_), v2_), s) -> let d'_ = Names.decLUName (g_, d_) in @@ -579,16 +574,16 @@ module MakePrint F.hVbox [ str0_ (Symbol.bvar (nameOf x)); - sym ":"; + F.space; fmtExp (g_, d + 1, noCtxt, (v_, s)); ] | g_, d, (I.BDec (x, (cid, t)), s) -> let gsome_, gblock_ = I.constBlock cid in F.hVbox - ([ str0_ (Symbol.const (nameOf x)); sym ":" ] + ([ str0_ (Symbol.const (nameOf x)); F.space] @ fmtDecList' (g_, (gblock_, I.comp (t, s)))) | g_, d, (I.ADec (x, _), s) -> - F.hVbox [ str0_ (Symbol.bvar (nameOf x)); sym ":_" ] + F.hVbox [ str0_ (Symbol.bvar (nameOf x)); sym "_" ] | g_, d, (I.NDec (Some name), s) -> F.hVbox [ sym name ] and fmtDecList' = function @@ -625,6 +620,16 @@ module MakePrint let rec fmtCtx (g0_, g_) = fmtDecList (g0_, ctxToDecList (g_, [])) + let rec fmtKindBinders (g_, d, v_) = + begin match v_ with + | I.Uni _ -> [] + | I.Pi ((d_, _), v2_) -> + let d'_ = Names.decLUName (g_, d_) in + sym "{" :: fmtDec (g_, d, (d'_, I.id)) :: sym "}" :: F.break + :: fmtKindBinders (I.Decl (g_, d'_), d + 1, v2_) + | _ -> [ fmtExp (g_, d, noCtxt, (v_, I.id)) ] + end + let rec fmtBlock = function | I.Null, lblock_ -> [ sym "block"; F.break ] @ fmtDecList (I.Null, lblock_) @@ -634,6 +639,7 @@ module MakePrint F.break; F.hVbox ([ sym "block"; F.space ] @ fmtDecList (gsome_, lblock_)); ] + (* Fix *) let rec fmtConDec = function | hide, (I.ConDec (_, _, imp, _, v_, l_) as condec) -> @@ -643,16 +649,27 @@ module MakePrint begin if hide then skipI (imp, I.Null, v_) else (I.Null, v_) end in - let vfmt_ = fmtExp (g_, 0, noCtxt, (v_, I.id)) in - F.hVbox - [ - fmtConstPath (Symbol.const, qid); - F.space; - sym ":"; - F.break; - vfmt_; - sym "."; - ] + begin match l_ with + | I.Kind -> + let binders = fmtKindBinders (g_, 0, v_) in + F.hVbox + ([ sym "%sort"; F.space; fmtConstPath (Symbol.const, qid) ] + @ (if binders = [] then [] else [ F.space ]) + @ binders + @ [ full_stop ]) + | I.Type -> + let vfmt_ = fmtExp (g_, 0, noCtxt, (v_, I.id)) in + F.hVbox + [ + sym "%term"; + F.space; + fmtConstPath (Symbol.const, qid); + F.space; + F.break; + vfmt_; + full_stop; + ] + end | hide, (I.SkoDec (_, _, imp, v_, l_) as condec) -> let qid = Names.conDecQid condec in let _ = Names.varReset IntSyn.Null in @@ -667,10 +684,9 @@ module MakePrint F.break; fmtConstPath (Symbol.skonst, qid); F.space; - sym ":"; F.break; vfmt_; - sym "."; + full_stop; ] | hide, (I.BlockDec (_, _, gsome_, lblock_) as condec) -> let qid = Names.conDecQid condec in @@ -681,11 +697,10 @@ module MakePrint F.break; fmtConstPath (Symbol.label, qid); F.space; - sym ":"; F.break; ] @ fmtBlock (gsome_, lblock_) - @ [ sym "." ]) + @ [ full_stop ]) | hide, (I.BlockDef (_, _, w_) as condec) -> let qid = Names.conDecQid condec in let _ = Names.varReset IntSyn.Null in @@ -695,10 +710,9 @@ module MakePrint F.break; fmtConstPath (Symbol.label, qid); F.space; - sym "="; F.break; ] - @ [ formatWorlds (T.Worlds w_); sym "." ]) + @ [ formatWorlds (T.Worlds w_); full_stop ]) | hide, (I.ConDef (_, _, imp, u_, v_, l_, _) as condec) -> let qid = Names.conDecQid condec in let _ = Names.varReset IntSyn.Null in @@ -710,16 +724,16 @@ module MakePrint let ufmt_ = fmtExp (g_, 0, noCtxt, (u_, I.id)) in F.hVbox [ + sym "%def"; + F.space; fmtConstPath (Symbol.def, qid); F.space; - sym ":"; F.break; vfmt_; F.break; - sym "="; F.space; ufmt_; - sym "."; + sym "%."; ] | hide, (I.AbbrevDef (_, _, imp, u_, v_, l_) as condec) -> let qid = Names.conDecQid condec in @@ -732,16 +746,17 @@ module MakePrint let ufmt_ = fmtExp (g_, 0, noCtxt, (u_, I.id)) in F.hVbox [ + sym "%inline"; fmtConstPath (Symbol.def, qid); F.space; - sym ":"; + F.break; vfmt_; F.break; - sym "="; + F.space; ufmt_; - sym "."; + full_stop; ] let rec fmtCnstr = function diff --git a/stelf.opam b/stelf.opam index 19bb1a9..8ed7687 100644 --- a/stelf.opam +++ b/stelf.opam @@ -28,6 +28,7 @@ depends: [ "ocaml-lsp-server" {dev} "user-setup" {dev} "merlin" {dev} + "sexplib" "ppx_deriving" "lambda-term" "fmt" diff --git a/test/Pal/Cases.ml b/test/Pal/Cases.ml index f3d6d3a..0c6b695 100644 --- a/test/Pal/Cases.ml +++ b/test/Pal/Cases.ml @@ -24,84 +24,6 @@ let total_add_mul_test = {| %total N (add N _ _) |}; ] -let cases () = - Alcotest.run "PAL" - begin - [ - ( "%term and %sort", - List.concat - [ - snd (test "Natural Numbers (nat)" nat_test); - snd (test "Natural Numbers (nat, add)" (nat_test @ add_test)); - snd - (test "Natural Numbers (nat, add, mul)" - (nat_test @ add_test @ mul_test)); - snd - (test ~failure:true "Natural numbers (ill-formed)" - [ - {| %sort nat |}; - {| %term zero nat |}; - {| %term succ {_ bool} nat |}; - ]); - ] ); - ( "%total and friends", - List.concat - [ - snd - (test "Natural Numbers (total)" - (nat_test @ add_test @ mul_test @ total_add_mul_test)); - ] ); - ( "Wiki", - List.concat - [ - snd (test "ZF 1" [ Source.zf_core ]); - snd (test "ZF 2" [ Source.zf_core; Source.zf_basics ]); - snd - (test "ZF 3" - [ Source.zf_core; Source.zf_basics; Source.zf_def_basic ]); - snd - (test "ZF 4" - [ - Source.zf_core; - Source.zf_basics; - Source.zf_def_basic; - Source.zf_high; - ]); - ] ); - ( "Nats", - List.concat - [ - snd (test "Nats 1" [ Source.nats1 ]); - snd (test "Nats 2" [ Source.nats1; Source.nats2 ]); - snd (test "Nats 3" [ Source.nats1; Source.nats2; Source.nats3 ]); - snd (test "Nats 4" [ Source.nats1; Source.nats2; Source.nats3; Source.nats4 ]); - ] ); - ( "FOL", - List.concat - [ - snd (test "FOL" [ Source.fol1 ]); - snd (test "FOL2" [ Source.fol1; Source.fol2 ]); - snd (test "FOL3.1" [ Source.fol1; Source.fol2; Source.fol3_1 ]); - snd (test "FOL3.2.1" [ Source.fol1; Source.fol2; Source.fol3_1; Source.fol3_2_1 ]); - snd (test "FOL3.2.2" [ Source.fol1; Source.fol2; Source.fol3_1; Source.fol3_2_1; Source.fol3_2_2 ]); - snd - (test "FOL3.2.3" [ Source.fol1; Source.fol2; Source.fol3_1; Source.fol3_2_1; Source.fol3_2_2; Source.fol3_2_3 ]); - snd (test "FOL3.2.*" [ Source.fol1; Source.fol2; Source.fol3_1; Source.fol3_2 ]); - snd (test "FOL3.3" [ Source.fol1; Source.fol2; Source.fol3_1; Source.fol3_2; Source.fol3_3 ]); - snd (test "FOL3.4" [ Source.fol1; Source.fol2; Source.fol3_1; Source.fol3_2; Source.fol3_3; Source.fol3_4 ]); - snd (test "FOL3.*" [ Source.fol1; Source.fol2; Source.fol3 ]); - snd (test "FOL4.1" [ Source.fol1; Source.fol2; Source.fol3; Source.fol4_1 ]); - snd (test "FOL4.2" [ Source.fol1; Source.fol2; Source.fol3; Source.fol4_1; Source.fol4_2 ]); - snd (test "FOL4.*" [ Source.fol1; Source.fol2; Source.fol3; Source.fol4 ]); - snd (test "FOL5" [ Source.fol1; Source.fol2; Source.fol3; Source.fol4; Source.fol5 ]); - snd (test "FOL6" [ Source.fol1; Source.fol2; Source.fol3; Source.fol4; Source.fol5; Source.fol6 ]); - ] ); - ( "S4", List.concat [ snd (test "S4" [ Source.js4 ]) ] ); - ( "LAM", List.concat [ snd (test "LAM" [ Source.lam ]) ] ); - ( "POLYLAM", List.concat [ snd (test "POLYLAM" [ Source.polylam ]) ] ) - ] - end ~verbose:false -let trash _ = " %. " let cases () = Alcotest.run "PAL" begin [ test "%term and %sort" Source.[ String.concat "\n" nat_test; @@ -109,6 +31,14 @@ let cases () = Alcotest.run "PAL" begin [ String.concat "\n" mul_test; String.concat "\n" total_add_mul_test; ]; + test "ZF" Source.[ + zf_1; + zf_2; + zf_3; + zf_4; + zf_5; + zf_6; + ]; test "FOL" Source.[ fol1; fol2; @@ -125,12 +55,7 @@ let cases () = Alcotest.run "PAL" begin [ fol6_1; fol6_2; ]; - test "ZF" Source.[ - zf_core; - zf_basics; - zf_def_basic; - zf_high; - ]; + test "Nats" Source.[ nats1; nats2; @@ -138,10 +63,18 @@ let cases () = Alcotest.run "PAL" begin [ nats4; ]; test "S4" Source.[ - js4; + jsf_1; + jsf_2_1; + jsf_2_2; + jsf_3; + jsf_4; ]; test "LAM" Source.[ - lam; + lam_1; + lam_2; + lam_3; + lam_4; + lam_5; ]; test "POLYLAM" Source.[ polylam; @@ -207,7 +140,7 @@ let cases () = Alcotest.run "PAL" begin [ tapl_defs_store; tapl_defs_heap; ]; - test "SMALL-STEP-SYSF" Source.[ + test "SMALL-STEP-SYSF" Source.[ small_step_sysf_types; small_step_sysf_terms; small_step_sysf_typing; diff --git a/test/Pal/Common.ml b/test/Pal/Common.ml index 1c99251..221d6b0 100644 --- a/test/Pal/Common.ml +++ b/test/Pal/Common.ml @@ -1,21 +1,3 @@ -let test_value (cmds : string list) : exn option = - let p = new Pal.Pal.pal in - try - Printexc.record_backtrace true; - List.iter (fun cmd -> p#exec cmd) cmds; - None - with e -> Some e - -class tester = object - val mutable p = new Pal.Pal.pal - method run cmd = - try - Printexc.record_backtrace true; - p#exec cmd; - None - with e -> Some e - method reset () = p <- new Pal.Pal.pal -end let test ?(skip = false) ?(failure = false) (name : string) (cmds : string list) : string * unit Alcotest.test_case list = let () = Printexc.record_backtrace true in @@ -24,15 +6,23 @@ let test ?(skip = false) ?(failure = false) (name : string) (cmds : string list) let () = Fmt_tty.setup_std_outputs () in let () = Display.register (fun m -> - let _ = (Display.fmt Fmt.stdout m.msg ) in - Lwt.return ()) + let _ = (Display.fmt Fmt.stdout m.msg) in + Lwt.return ()) + in + let module P = Pal.Pal.Start () in + let run cmd = + try + Printexc.record_backtrace true; + P.exec cmd; + None + with e -> Some e in let has_failed = ref false in - let tester' = new tester in - name, List.mapi(fun i cmd -> Alcotest.test_case (name ^ " - " ^ string_of_int (i + 1)) `Slow (fun () -> + name, List.mapi (fun i cmd -> + Alcotest.test_case (name ^ " - " ^ string_of_int (i + 1)) `Slow (fun () -> if skip || !has_failed then Alcotest.skip () else - match tester'#run cmd with + match run cmd with | None when failure -> Alcotest.fail "Expected failure, but test passed" | Some e when not failure -> let bt = Printexc.get_backtrace () in diff --git a/test/Pal/Source.ml b/test/Pal/Source.ml index 008a31e..5157fa1 100644 --- a/test/Pal/Source.ml +++ b/test/Pal/Source.ml @@ -1,11 +1,11 @@ -let zf_core = {| +let zf_1 = {| %sort prop %sort pf {_ prop} %sort set |} -let zf_basics = +let zf_2 = {| %term false prop %term imp {_ prop} {_ prop} prop @@ -14,7 +14,7 @@ let zf_basics = %term in {_ set} {_ set} prop |} -let zf_def_basic = +let zf_3 = {| %def not ({_ prop} prop) ([a] imp a false) %. %def and ({_ prop} {_ prop} prop) ([a][b] not (imp a (not b))) %. @@ -22,11 +22,12 @@ let zf_def_basic = %def iff ({_ prop} {_ prop} prop) ([a][b] and (imp a b) (imp b a)) %. %def ex ({_ {_ set} prop} prop) ([p] not(all([z]not (p z)))) %. %def unique ({_ {_ set} prop} prop) ([p] all([z] imp (p z) (all ([z'] imp (p z') (eq z z'))))) %. +%def ex_unique ({_ {_ set} prop} prop) ([p] and (ex p) (unique p)) %. |} -let zf_high = +let zf_4 = {| -%def ex_unique ({_ {_ set} prop} prop) ([p] and (ex p) (unique p)) %. + %term imp_i {_ {_ pf _A} pf _B} pf (imp _A _B) %. %term imp_e {_ pf (imp _A _B)} {_ pf _A} pf _B %. %term all_i {_ {z} pf (_P z)} pf (all _P) %. @@ -43,6 +44,8 @@ let zf_high = %term powerset {_ set} set %. %term replace {_ set} {_ {_ set} set} set %. %term omega set %. +|} +let zf_5 = {| %def single ({_ set} set) [x] double x x %. %def restrict ({_ set} {_ {_ set} prop} set) [x][q] unions (replace x ([z] if (q z) (single z) empty)) %. %def inter ({_ set} {_ set} set) [x][y] restrict x ([z] in z y) %. @@ -52,6 +55,8 @@ let zf_high = %def subset ({_ set} {_ set} prop) [x][y]all[z] imp (in z x) (in z y) %. %def disjoint ({_ set} {_ set} prop) [x][y] eq (inter x y) empty %. %def omega_closed ({_ set} prop) [x] and (in empty x) (all [n] imp (in n x) (in (succ n) x)) %. +|} +let zf_6 = {| %term extensionality pf (iff (eq _X _Y) (all[z] iff (in z _X) (in z _Y))) %. %term foundation pf (ex([z] and (in z _X) (disjoint z _X))) %. %term emtpy_ax pf (not (in _X empty)) %. @@ -327,7 +332,7 @@ let fol6_2 = {| |} let fol6 = fol6_1 ^ fol6_2 -let js4 = +let jsf_1 = {| Judgmental S4 [A judgmental reconstruction of modal logic, F.Pfenning and R.Davies, @@ -357,26 +362,29 @@ and if J = (M : A) then J* = M* : tm A W %sort tm {_ tp} {_ world} %sort exp {_ tp} {_ world} -%term lam {A tp} {B tp} {W world} {_ {_ tm A W} tm B W} tm (A => B) W -%term app {A tp} {B tp} {W world} {_ tm (A => B) W} {_ tm A W} tm B W -%term boxi {A tp} {W world} {_ {w world} tm A w} tm (box A) W -%term boxe {A tp} {C tp} {W world} {_ tm (box A) W} {_ {_ {W' world} tm A W'} tm C W} tm C W -%term t2e {A tp} {W world} {_ tm A W} exp A W -%term diai {A tp} {W world} {_ exp A W} tm (dia A) W -%term diae {A tp} {C tp} {W world} {_ tm (dia A) W} {_ {w world} {_ tm A w} exp C w} exp C W -%term boxep {A tp} {C tp} {W world} {_ tm (box A) W} {_ {_ {W' world} tm A W'} exp C W} exp C W - +%term lam {{A B W}} {_ {_ tm A W} tm B W} tm (A => B) W +%term app {{A B W}} {_ tm (A => B) W} {_ tm A W} tm B W +%term boxi {{A W}} {_ {w world} tm A w} tm (box A) W +%term boxe {{A C W}} {_ tm (box A) W} {_ {_ {W' world} tm A W'} tm C W} tm C W +%term t2e {{A W}} {_ tm A W} exp A W +%term diai {{A W}} {_ exp A W} tm (dia A) W +%term diae {{A C W}} {_ tm (dia A) W} {_ {w world} {_ tm A w} exp C w} exp C W +%term boxep {{A C W}} {_ tm (box A) W} {_ {_ {W' world} tm A W'} exp C W} exp C W %sort subdia {_ exp A W} {_ {w world} {_ tm A w} exp C w} {_ exp C W} -%mode subdia %in %in %out - +|} +let jsf_2_1 = {| +%mode {%in X _} {%in Y _} {%out Z _} subdia X Y Z +|} +let jsf_2_2 = {| %term sdt2e {A tp} {C tp} {W world} {M tm A W} {F {w world} {_ tm A w} exp C w} subdia (t2e M) ([w] [x] F w x) (F W M) -%term sddiae {A tp} {B tp} {C tp} {W world} {M tm (dia A) W} {E {_ world} {_ tm A _} exp B _} {F {_ world} {_ tm B _} exp C _} {F' {_ world} {_ tm A _} exp C _} {_ {v world} {y tm A v} subdia (E v y) ([w] [x] F w x) (F' v y)} subdia (diae M [v] [y] E v y) ([w] [x] F w x) (diae M [v] [y] F' v y) +%term sddiae {A tp} {B tp} {C tp} {W world} {M tm (dia A) W} {E {v world} {_ tm A v} exp B v} {F {w world} {_ tm B w} exp C w} {F' {v world} {_ tm A v} exp C v} {_ {v world} {y tm A v} subdia (E v y) ([w] [x] F w x) (F' v y)} subdia (diae M [v] [y] E v y) ([w] [x] F w x) (diae M [v] [y] F' v y) -%term sdboxep {A tp} {B tp} {C tp} {W world} {M tm (box A) W} {E {_ {_ world} tm B _} exp C _} {F {_ world} {_ tm A _} exp C _} {F' {_ {_ world} tm B _} exp C _} {_ {u {V world} tm B V} subdia (E u) ([w] [x] F w x) (F' u)} subdia (boxep M [u] E u) ([w] [x] F w x) (boxep M [u] F' u) - -%block by {B tp} {v world} {y tm B v} -%block bu {B tp} {u {V world} tm B V} +%term sdboxep {A tp} {C tp} {D tp} {W world} {M tm (box A) W} {E {u {V world} tm A V} exp C W} {F {w world} {_ tm C w} exp D w} {F' {u {V world} tm A V} exp D W} {_ {u {V world} tm A V} subdia (E u) ([w] [x] F w x) (F' u)} subdia (boxep M [u] E u) ([w] [x] F w x) (boxep M [u] F' u) +|} +let jsf_3 = {| +%block by [B tp] {v world} {y tm B v} +%block bu [B tp] {u {V world} tm B V} %worlds (by bu) (subdia E F F') %total E (subdia E F _) %. @@ -385,15 +393,16 @@ The "str" strengthening lemma would require handling cases like strlam where a b variable y2 of type tm C2 w appears free in the conclusion, but w is also quantified by the str family. The main issue is that the case for str ([x][w] y2) ([w] y2) cannot be typed because y2 has type tm C2 w but w is the variable we are quantifying over. - +|} +let jsf_4 = {| Examples -%def _ ({A tp} {W world} tm (box A => A) W) (lam [x] boxe x [u] u W) -%def _ ({A tp} {W world} tm (box A => box (box A)) W) (lam [x] boxe x [u] boxi [w] boxi [w'] u w') -%def _ ({A tp} {B tp} {W world} tm (box (A => B) => box A => box B) W) (lam [x] lam [y] boxe x [u] boxe y [v] boxi [w] app (u w) (v w)) -%def _ ({A tp} {W world} tm (A => dia A) W) (lam [x] diai (t2e x)) -%def _ ({A tp} {W world} tm (dia (dia A) => dia A) W) (lam [x] diai (diae x [w] [y] diae y [v] [z] t2e z)) -%def _ ({A tp} {B tp} {W world} tm (box (A => B) => dia A => dia B) W) (lam [x] lam [y] diai (boxep x [u] diae y [w] [z] t2e (app (u w) z))) +%def _ (tm (box A => A) W) (lam [x] boxe x [u] u W) +%def _ (tm (box A => box (box A)) W) (lam [x] boxe x [u] boxi [w] boxi [w'] u w') +%def _ (tm (box (A => B) => box A => box B) W) (lam [x] lam [y] boxe x [u] boxe y [v] boxi [w] app (u w) (v w)) +%def _ (tm (A => dia A) W) (lam [x] diai (t2e x)) +%def _ (tm (dia (dia A) => dia A) W) (lam [x] diai (diae x [w] [y] diae y [v] [z] t2e z)) +%def _ (tm (box (A => B) => dia A => dia B) W) (lam [x] lam [y] diai (boxep x [u] diae y [w] [z] t2e (app (u w) z))) %. Counterexamples, all must fail: The following would require box to be a comonad (A => box A), but this is not valid in S4. @@ -408,7 +417,7 @@ The two S5 theorems (dia A => box (dia A)) and (dia (box A) => box A) both fail S4 does not have the symmetry or Euclidean properties needed for S5. |} -let lam = +let lam_1 = {| Lambda-Calculus Fragment of Mini-ML. Author: Frank Pfenning @@ -428,11 +437,13 @@ Simple types |- E : T (expression E has type T) %sort of {_ exp} {_ tp} +|} +let lam_2 = {| %mode of %in %star -%term tp_lam {E {_ exp} exp} {T1 tp} {T2 tp} {_ {x exp} {_ of x T1} of (E x) T2} of (lam E) (arrow T1 T2) +%term tp_lam {{E T1 T2}} {_ {x exp} {_ of x T1} of (E x) T2} of (lam E) (arrow T1 T2) -%term tp_app {E1 exp} {E2 exp} {T1 tp} {T2 tp} {_ of E1 (arrow T2 T1)} {_ of E2 T2} of (app E1 E2) T1 +%term tp_app {{E1 E2 T1 T2}} {_ of E1 (arrow T2 T1)} {_ of E2 T2} of (app E1 E2) T1 %. Evaluation (call-by-value) E ==> V (expression E evaluates to value V) @@ -440,10 +451,12 @@ E ==> V (expression E evaluates to value V) %sort eval {_ exp} {_ exp} %mode eval %in %out -%term ev_lam {E {_ exp} exp} eval (lam E) (lam E) +%term ev_lam {{E}} eval (lam E) (lam E) -%term ev_app {E1 exp} {E2 exp} {V exp} {V2 exp} {E1' {_ exp} exp} {_ eval E1 (lam E1')} {_ eval E2 V2} {_ eval (E1' V2) V} eval (app E1 E2) V +%term ev_app {{E1 E2 V V2 E1'}} {_ eval E1 (lam E1')} {_ eval E2 V2} {_ eval (E1' V2) V} eval (app E1 E2) V +|} +let lam_3 = {| %. Regular world for type-checking %block tp_var [T tp] {x exp} {u of x T} %worlds (tp_var) (of E T) @@ -460,23 +473,21 @@ E ==> V (expression E evaluates to value V) %. There is at least one evaluation rule for every closed expression %covers eval %in %out +|} +let lam_4 = {| %. Type preservation as higher-level family %sort tps {_ eval E V} {_ of E T} {_ of V T} -%term tps_lam {E {_ exp} exp} {T1 tp} {T2 tp} {P {_ exp} {_ of _ T1} of _ T2} tps ev_lam (tp_lam P) (tp_lam P) +%term tps_lam {E {_ exp} exp} {T1 tp} {T2 tp} {P {x exp} {_ of x T1} of (E x) T2} tps ev_lam (tp_lam P) (tp_lam P) -%term tps_app {E1 exp} {E2 exp} {V exp} {V2 exp} {E1' {_ exp} exp} {T tp} {T2 tp} {D1 eval E1 (lam E1')} {D2 eval E2 V2} {D3 eval (E1' V2) V} {P1 of E1 (arrow T2 T)} {P2 of E2 T2} {Q1' {_ exp} {_ of _ T2} of _ T2} {Q2 of V2 T2} {Q of V T} {_ tps D1 P1 (tp_lam Q1')} {_ tps D2 P2 Q2} {_ tps D3 (Q1' V2 Q2) Q} tps (ev_app D3 D2 D1) (tp_app P2 P1) Q - -%mode tps %in %in %out -%worlds () (tps D P _) -%total D (tps D P _) +%term tps_app {E1 exp} {E2 exp} {V exp} {V2 exp} {E1' {_ exp} exp} {T tp} {T2 tp} {D1 eval E1 (lam E1')} {D2 eval E2 V2} {D3 eval (E1' V2) V} {P1 of E1 (arrow T2 T)} {P2 of E2 T2} {Q1' {x exp} {_ of x T2} of (E1' x) T} {Q2 of V2 T2} {Q of V T} {_ tps D1 P1 (tp_lam Q1')} {_ tps D2 P2 Q2} {_ tps D3 (Q1' V2 Q2) Q} tps (ev_app D1 D2 D3) (tp_app P1 P2) Q %. Applying type preservation %def e0 _ (app (lam [x] x) (lam [y] y)) %? of e0 T %? eval e0 V -%? tps d0 p0 Q - +|} +let lam_5 = {| %. Example of regular worlds cp copies input to output. @@ -509,18 +520,18 @@ let polylam = %term all {_ {_ tp} tp} tp %sort tm {_ tp} -%term lam {A tp} {B tp} {_ {_ tm A} tm B} tm (A => B) -%term app {A tp} {B tp} {_ tm (A => B)} {_ tm A} tm B -%term tlam {A {_ tp} tp} {_ {a tp} tm (A a)} tm (all A) -%term tapp {A {_ tp} tp} {_ tm (all A)} {B tp} tm (A B) +%term lam {{A tp}} {{B tp}} {_ {_ tm A} tm B} tm (A => B) +%term app {{A tp}} {{B tp}} {_ tm (A => B)} {_ tm A} tm B +%term tlam {{A}} {_ {a tp} tm (A a)} tm (all A) +%term tapp {{A}} {_ tm (all A)} {B tp} tm (A B) -%def nat _ (all [a] a => (a => a) => a) -%def zero _ (tlam [a] lam [z] lam [s] z) -%def succ _ (lam [x] tlam [a] lam [z] lam [s] app s (app (app (tapp x a) z) s)) -%def succ' (tm (nat => nat)) (lam [x] tlam [a] lam [z] lam [s] app s (app (app (tapp x a) z) s)) -%def plus _ (lam [x] lam [y] app (app (tapp y nat) x) succ) -%def times _ (lam [x] lam [y] app (app (tapp y nat) zero) (app plus x)) -%def exp _ (lam [x] lam [y] app (app (tapp y nat) (app succ zero)) (app times x)) +%def nat _ (all [a tp] a => (a => a) => a) +%def zero _ (tlam [a tp] lam [z tm a] lam [s tm (a => a)] z) +%def succ _ (lam [x tm nat] tlam [a tp] lam [z tm a] lam [s tm (a => a)] app s (app (app (tapp x a) z) s)) +%def succ' (tm (nat => nat)) (lam [x tm nat] tlam [a tp] lam [z tm a] lam [s tm (a => a)] app s (app (app (tapp x a) z) s)) +%def plus _ (lam [x tm nat] lam [y tm nat] app (app (tapp y nat) x) succ) +%def times _ (lam [x tm nat] lam [y tm nat] app (app (tapp y nat) zero) (app plus x)) +%def exp _ (lam [x tm nat] lam [y tm nat] app (app (tapp y nat) (app succ zero)) (app times x)) |} (* Propositional calculus: intuitionistic natural deduction @@ -531,6 +542,7 @@ let polylam = let prop_calc_types = {| %sort o +%name o A %term imp {_ o} {_ o} o %prec %right 10 imp %term and {_ o} {_ o} o @@ -544,6 +556,7 @@ let prop_calc_hilbert = {| %. Provability (Hilbert-style) %sort pf {_ o} +%name pf P %term K {{A B}} pf (A imp (B imp A)) %term S {{A B C}} pf ((A imp (B imp C)) imp ((A imp B) imp (A imp C))) %term ONE pf true @@ -559,6 +572,7 @@ let prop_calc_nd = {| %. Natural deduction %sort nd {_ o} +%name nd D %term trueI nd true %term andI {{A B}} {_ nd A} {_ nd B} nd (A and B) %term andEL {{A B}} {_ nd (A and B)} nd A @@ -573,6 +587,7 @@ let prop_calc_nd = let mini_ml_exp = {| %sort exp +%name exp E %term z exp %term s {_ exp} exp %term case {_ exp} {_ exp} {_ {_ exp} exp} exp @@ -618,6 +633,7 @@ let mini_ml_tp = let arith_nat = {| %sort nat +%name nat X %term z nat %term s {_ nat} nat |} @@ -625,6 +641,7 @@ let arith_nat = let arith_nt = {| %sort nt {_ nat} +%name nt N %term nt-z nt z %term nt-s {{X}} %<- (nt (s X)) (nt X) |} @@ -632,6 +649,7 @@ let arith_nt = let arith_plus = {| %sort plus {_ nat} {_ nat} {_ nat} +%name plus P %term p-z {{Y}} plus z Y Y %term p-s {{X Y Z}} %<- (plus (s X) Y (s Z)) (plus X Y Z) |} @@ -681,6 +699,7 @@ let guide_lists_mode = let tapl_nat_base = {| %sort nat +%name nat N %term z nat %term s {_ nat} nat |} @@ -704,13 +723,15 @@ let tapl_nat_eq = (* Positive fragment of first-order logic with natural deduction. Ported from twelf/examples/lp_horn/natded.elf. Operator declarations: imp with %prec %right 10, and with %prec %right 11. - Note: %name hints and %block/%worlds omitted. *) let lp_horn_nd = {| %sort i +%name i T %sort o +%name o A %sort p +%name p P %term atom {_ p} o %term and {_ o} {_ o} o @@ -721,6 +742,7 @@ let lp_horn_nd = %term forall {_ {_ i} o} o %sort pf {_ o} +%name pf D %term andi {{A B}} {_ pf A} {_ pf B} pf (A and B) %term andel {{A B}} {_ pf (A and B)} pf A %term ander {{A B}} {_ pf (A and B)} pf B @@ -738,18 +760,20 @@ let lp_horn_nd = let church_rosser_lam = {| %sort term +%name term M %term lam {_ {_ term} term} term %term app {_ term} {_ term} term |} (* Formula syntax for predicate calculus (intuitionistic and classical). Ported from twelf/examples/cut_elim/formulas.elf. - Note: %name hints omitted. *) let cut_elim_formulas = {| %sort i +%name i T %sort o +%name o A %term and {_ o} {_ o} o %prec %right 11 and %term imp {_ o} {_ o} o @@ -766,14 +790,16 @@ let cut_elim_formulas = (* Natural deduction for intuitionistic logic (positive+negative fragment). Ported from twelf/examples/guide/nd.elf. - Note: %theorem/%prove stripped; abbreviation definitions for not/noti/note - handled as regular terms since STELF does not support Twelf abbreviation syntax. + Abbreviation definitions for not/noti/note handled as regular terms + since STELF does not support Twelf abbreviation syntax. The %block/%worlds declarations are kept as-is. *) let guide_nd = {| %sort i +%name i T %sort o +%name o A %term imp {_ o} {_ o} o %prec %right 10 imp %term and {_ o} {_ o} o @@ -786,6 +812,7 @@ let guide_nd = %term exists {_ {_ i} o} o %sort nd {_ o} +%name nd D %term impi {{A B}} {_ {_ nd A} nd B} nd (A imp B) %term impe {{A B}} {_ nd (A imp B)} {_ nd A} nd B @@ -802,7 +829,7 @@ let guide_nd = %term existsi {{A}} {T i} {_ nd (A T)} nd (exists A) %term existse {{A C}} {_ nd (exists A)} {_ {x i} {_ nd (A x)} nd C} nd C -%block nd_hyp {A o} {u nd A} +%block nd_hyp [A o] {u nd A} %block nd_parm {x i} %worlds (nd_hyp nd_parm) (nd A) @@ -814,6 +841,11 @@ let guide_nd = %term orred2 {{A B C D E1 E2}} red (ore (ori2 D) E1 E2) (E2 D) %term forallred {{A D T}} red (foralle (foralli D) T) (D T) %term existsred {{A C T D E}} red (existse (existsi T D) E) (E T D) + +%theorem +trivI exists {D {A o} nd (A imp A)} true + +%prove 2 {} (trivI D) |} (* Direct-style CPS BNF term syntax. @@ -823,8 +855,11 @@ let guide_nd = let cpsocc_dsbnf = {| %sort droot +%name droot DROOT %sort dexp +%name dexp DEXP %sort dtriv +%name dtriv DTRIV %term dexp->droot {_ dexp} droot %term dapp {_ dexp} {_ dexp} dexp @@ -882,7 +917,6 @@ let small_step_lam_step = (* Explicit contexts in LF: types, expressions, and natural numbers. Ported from twelf/examples/crary/explicit/excon.elf (first part only — typing rules and nat up through leq). - Note: %name hints omitted. *) let crary_excon = {| @@ -903,20 +937,19 @@ let crary_excon = %term of/app {{A B M N}} {_ of M (pi A B)} {_ of N A} of (app M N) (B N) %sort nat -%term zero nat +%term 0 nat %term s {_ nat} nat %sort nat-eq {_ nat} {_ nat} %term nat-eq/i {{N}} nat-eq N N %sort leq {_ nat} {_ nat} -%term leq/z {{N}} leq zero N +%term leq/z {{N}} leq 0 N %term leq/s {{N1 N2}} %<- (leq (s N1) (s N2)) (leq N1 N2) |} (* CRARY-EXCON-REV: excon-rev.elf (explicit contexts, reversed variant) Ported from twelf/examples/crary/explicit/excon-rev.elf (syntax-only chunk). - Note: Twelf uses `0 : nat` (numeric literal); renamed to `zero` for STELF. The `-` family (isvar dependency decl) is a Twelf quirk; omitted. *) let crary_excon_rev_syntax = @@ -938,14 +971,14 @@ let crary_excon_rev_syntax = %term of/app {{A B M N}} {_ of M (pi A B)} {_ of N A} of (app M N) (B N) %sort nat -%term zero nat +%term 0 nat %term s {_ nat} nat %sort nat-eq {_ nat} {_ nat} %term nat-eq/i {{N}} nat-eq N N %sort lt {_ nat} {_ nat} -%term lt/z {{N}} lt zero (s N) +%term lt/z {{N}} lt 0 (s N) %term lt/s {{N1 N2}} %<- (lt (s N1) (s N2)) (lt N1 N2) %sort ctx @@ -959,11 +992,12 @@ let crary_excon_rev_syntax = `tp` → `ref_tp`, `exp` → `ref_exp` to avoid clashes with SMALL-STEP-LAM/CRARY-EXCON. `nat`, `z`, `s` re-used from earlier suites (already in scope). `=>` already infix-left 5 from SMALL-STEP-LAM; used that way here. - Dropped: %name hints, %freeze, %terminates {}, %unique, step/typing rules. + Dropped: %freeze (var sort not included), %terminates {}, %unique, step/typing rules. *) let tapl_defs_types = {| %sort ref_tp +%name ref_tp T %term ref_arrow {_ ref_tp} {_ ref_tp} ref_tp %term unit_tp ref_tp %term ref {_ ref_tp} ref_tp @@ -976,12 +1010,14 @@ let tapl_defs_labels = %term s {_ nat} nat %sort label +%name label L %term lbl {_ nat} label |} let tapl_defs_exp = {| %sort ref_exp +%name ref_exp E %term ref_app {_ ref_exp} {_ ref_exp} ref_exp %term ref_lam {_ ref_tp} {_ {_ ref_exp} ref_exp} ref_exp %term dot ref_exp @@ -994,6 +1030,7 @@ let tapl_defs_exp = let tapl_defs_value = {| %sort ref_value {_ ref_exp} +%name ref_value V %mode ref_value %in %term v_lam {{T E}} ref_value (ref_lam T E) %term v_dot ref_value dot @@ -1003,6 +1040,7 @@ let tapl_defs_value = let tapl_defs_store = {| %sort ref_store +%name ref_store S %term store_nil ref_store %term store_cons {_ ref_tp} {_ ref_store} ref_store @@ -1023,6 +1061,7 @@ let tapl_defs_store = let tapl_defs_heap = {| %sort ref_heap +%name ref_heap H %term heap_nil ref_heap %term heap_cons {_ ref_exp} {_ ref_heap} ref_heap @@ -1198,6 +1237,7 @@ let small_step_sysf_iso_step = let poplmark_1a_syntax = {| %sort tp +%name tp T %term top tp %term arrow {_ tp} {_ tp} tp %term forall {_ tp} {_ {_ tp} tp} tp @@ -1215,6 +1255,7 @@ let poplmark_1a_syntax = %sort false %sort nat +%name nat N %term z nat %term s {_ nat} nat @@ -1232,12 +1273,14 @@ let poplmark_1a_syntax = let poplmark_2a_syntax = {| %sort sub_tp {_ tp} {_ tp} +%name sub_tp T %term sub_tp_top {{T}} sub_tp T top %term sub_tp_refl {{T}} sub_tp T T %term sub_tp_trans {{T1 T2 T3}} %<- (sub_tp T1 T3) (sub_tp T1 T2) (sub_tp T2 T3) %term sub_tp_arrow {{S1 S2 T1 T2}} %<- (sub_tp (arrow S1 S2) (arrow T1 T2)) (sub_tp T1 S1) (sub_tp S2 T2) %sort term +%name term E %term abs {_ tp} {_ {_ term} term} term %term app {_ term} {_ term} term %term tabs {_ tp} {_ {_ tp} term} term @@ -1407,8 +1450,11 @@ let poplmark_2b_syntax = let ccc_syntax = {| %sort obj +%name obj _A %sort mor {_ obj} {_ obj} +%name mor _F %sort meq {_ mor _A _B} {_ mor _A _B} +%name meq _ME %term id {{A}} mor A A %term comp {{A B C}} {_ mor B C} {_ mor A B} mor A C @@ -1448,7 +1494,6 @@ let ccc_syntax = (* INCLL: INCLL sorts and terms. Ported from twelf/examples/incll/incll.elf. - Renamed: numeric term names `1..5` → `v1..v5` (STELF identifiers must not start with digit). Dropped: Twelf abbreviation syntax (`<= = [x][y] y => x` etc.). Dropped: `|` infix list cons (single-char identifier, may fail — using `cons` instead). `^` prefix also risky — using `atm_frm` instead. @@ -1472,11 +1517,11 @@ let incll_syntax = %sort frm %term int sort -%term v1 trm int -%term v2 trm int -%term v3 trm int -%term v4 trm int -%term v5 trm int +%term 1 trm int +%term 2 trm int +%term 3 trm int +%term 4 trm int +%term 5 trm int %sort list_sort {_ sort} %term list_sort/nil {{A}} list_sort A From 59c7b7f8721b4cc379fba2738daeb22cf2a7b9b3 Mon Sep 17 00:00:00 2001 From: Asher Frost Date: Fri, 12 Jun 2026 20:44:53 -0400 Subject: [PATCH 6/9] chore: updated style --- src/Common/Cst/CST.ml | 70 +-- src/Common/Cst/Cst.ml | 39 +- src/Common/Cst/LENS.ml | 18 +- src/Common/Cst/Lens.ml | 19 +- src/Common/Cst/SIM.ml | 12 +- src/Common/Cst/Sim.ml | 13 +- src/Common/Syntax/{Ast_intf.ml => AST.ml} | 0 src/Common/Syntax/Ast.ml | 4 +- src/Common/Syntax/Common.ml | 2 +- src/Common/Syntax/FgnOpn.ml | 4 +- src/Common/Syntax/{Misc_intf.ml => MISC.ml} | 2 +- src/Common/Syntax/Misc.ml | 4 +- src/Common/Syntax/{Sgn_intf.ml => SGN.ml} | 2 +- src/Common/Syntax/Sgn.ml | 6 +- src/Common/Syntax/Syntax.ml | 24 +- .../Assign_intf.ml => Compile/ASSIGN.ml} | 0 src/{compile => Compile}/Assign.ml | 2 +- src/{compile => Compile}/Assign.mli | 2 +- .../Compile_intf.ml => Compile/COMPILE.ml} | 0 .../CompSyn_intf.ml => Compile/COMPSYN.ml} | 0 .../Cprint_intf.ml => Compile/CPRINT.ml} | 0 src/{compile => Compile}/CompSyn.ml | 2 +- src/{compile => Compile}/CompSyn.mli | 2 +- src/{compile => Compile}/Compile.ml | 0 src/{compile => Compile}/Compile.mli | 0 src/{compile => Compile}/Compile_.ml | 2 +- src/{compile => Compile}/Compile_.mli | 2 +- src/{compile => Compile}/Cprint.ml | 2 +- src/{compile => Compile}/Cprint.mli | 2 +- src/{compile => Compile}/README | 0 .../Subtree_intf.ml => Compile/SUBTREE.ml} | 0 src/{compile => Compile}/Subtree.ml | 2 +- src/{compile => Compile}/Subtree.mli | 2 +- src/{compile => Compile}/dune | 10 +- .../Compress_intf.ml => Compress/COMPRESS.ml} | 0 src/{compress => Compress}/Compress.ml | 0 src/{compress => Compress}/Compress.mli | 0 src/{compress => Compress}/Compress_.ml | 2 +- src/{compress => Compress}/Compress_.mli | 2 +- src/{compress => Compress}/Reductio.ml | 0 src/{compress => Compress}/Rep.ml | 0 src/{compress => Compress}/Rep.mli | 0 src/{compress/Sgn_intf.ml => Compress/SGN.ml} | 0 src/{compress => Compress}/Sgn.ml | 2 +- src/Compress/Sgn.mli | 3 + src/{compress => Compress}/Strict.ml | 0 src/{compress => Compress}/Syntax.ml | 0 src/{compress => Compress}/dune | 2 +- src/Config/dune | 2 +- src/{cover/Cover_intf.ml => Cover/COVER.ml} | 0 src/{cover => Cover}/Cover.ml | 0 src/{cover => Cover}/Cover.mli | 0 src/{cover => Cover}/Cover_.ml | 2 +- src/{cover => Cover}/Cover_.mli | 4 +- src/{cover => Cover}/README | 0 src/{cover/Total_intf.ml => Cover/TOTAL.ml} | 0 src/{cover => Cover}/Total.ml | 4 +- src/{cover => Cover}/Total.mli | 4 +- src/{cover => Cover}/dune | 2 +- src/{debug => Debug}/Debug.ml | 0 src/{debug => Debug}/Debug.mli | 0 src/{debug => Debug}/dune | 0 src/Display/FORM.ml | 2 - src/Display/Form.ml | 5 +- src/{domains => Domains}/Domains.ml | 0 src/{domains => Domains}/Domains.mli | 0 src/{domains => Domains}/Domains_.ml | 0 src/{domains => Domains}/Domains_.mli | 0 .../Field_intf.ml => Domains/FIELD.ml} | 0 src/{domains => Domains}/Field.ml | 2 +- src/Domains/Field.mli | 1 + .../Integers_intf.ml => Domains/INTEGERS.ml} | 0 src/{domains => Domains}/Integers.ml | 2 +- src/{domains => Domains}/Integers.mli | 2 +- src/{domains => Domains}/IntegersMod.ml | 0 src/{domains => Domains}/IntegersMod.mli | 0 .../ORDEREDFIELD.ml} | 0 src/{domains => Domains}/OrderedField.ml | 2 +- src/Domains/OrderedField.mli | 1 + .../RATIONALS.ml} | 0 src/{domains => Domains}/README | 0 src/{domains => Domains}/Rationals.ml | 2 +- src/{domains => Domains}/Rationals.mli | 2 +- src/{domains => Domains}/dune | 8 +- src/{flit/Flit_intf.ml => Flit/FLIT.ml} | 0 src/{flit => Flit}/Flit.ml | 0 src/{flit => Flit}/Flit.mli | 0 src/{flit => Flit}/FlitOldSmlnjPack.ml | 0 src/{flit => Flit}/FlitOldSmlnjPack.mli | 0 src/{flit => Flit}/Flit_.ml | 2 +- src/Flit/Flit_.mli | 1 + src/{flit => Flit}/README | 0 src/{flit => Flit}/dune | 2 +- .../FORMATTER.ml} | 0 src/{formatter => Formatter}/Formatter.ml | 0 src/{formatter => Formatter}/Formatter.mli | 0 src/{formatter => Formatter}/Formatter_.ml | 2 +- src/Formatter/Formatter_.mli | 2 + src/{formatter => Formatter}/README | 0 src/{formatter => Formatter}/dune | 2 +- src/{frontend => Frontend}/Buildid.ml | 0 src/{frontend => Frontend}/Buildid.mli | 0 .../Fquery_intf.ml => Frontend/FQUERY.ml} | 2 +- src/{frontend => Frontend}/Fquery.ml | 4 +- src/{frontend => Frontend}/Fquery.mli | 4 +- src/{frontend => Frontend}/Frontend.ml | 0 src/{frontend => Frontend}/Frontend.mli | 0 src/{frontend => Frontend}/Frontend_.ml | 10 +- src/Frontend/Frontend_.mli | 60 +++ .../Lexer_intf.ml => Frontend/LEXER.ml} | 0 src/{frontend => Frontend}/Lexer.ml | 2 +- src/{frontend => Frontend}/Lexer.mli | 2 +- .../PARSECONDEC.ml} | 2 +- .../PARSEFIXITY.ml} | 0 .../PARSEMODE.ml} | 2 +- .../PARSEMODULE.ml} | 2 +- .../PARSEQUERY.ml} | 2 +- .../Parser_intf.ml => Frontend/PARSER.ml} | 12 +- .../PARSETERM.ml} | 2 +- .../ParseThm_intf.ml => Frontend/PARSETHM.ml} | 2 +- .../Parsing_intf.ml => Frontend/PARSING.ml} | 0 src/{frontend => Frontend}/ParseCondec.ml | 8 +- src/Frontend/ParseCondec.mli | 7 + src/{frontend => Frontend}/ParseFixity.ml | 2 +- src/{frontend => Frontend}/ParseFixity.mli | 2 +- src/{frontend => Frontend}/ParseMode.ml | 7 +- src/{frontend => Frontend}/ParseMode.mli | 7 +- src/{frontend => Frontend}/ParseModule.ml | 7 +- src/{frontend => Frontend}/ParseModule.mli | 7 +- src/{frontend => Frontend}/ParseQuery.ml | 7 +- src/{frontend => Frontend}/ParseQuery.mli | 7 +- src/{frontend => Frontend}/ParseTerm.ml | 4 +- src/{frontend => Frontend}/ParseTerm.mli | 4 +- src/{frontend => Frontend}/ParseThm.ml | 7 +- src/{frontend => Frontend}/ParseThm.mli | 7 +- src/{frontend => Frontend}/Parser.ml | 29 +- src/{frontend => Frontend}/Parser.mli | 29 +- src/{frontend => Frontend}/Parsing.ml | 2 +- src/{frontend => Frontend}/Parsing.mli | 2 +- .../RECONCONDEC.ml} | 2 +- .../RECONMODE.ml} | 2 +- .../RECONMODULE.ml} | 2 +- .../RECONQUERY.ml} | 2 +- .../RECONTERM.ml} | 0 .../ReconThm_intf.ml => Frontend/RECONTHM.ml} | 2 +- src/{frontend => Frontend}/ReconCondec.ml | 4 +- src/{frontend => Frontend}/ReconCondec.mli | 4 +- src/{frontend => Frontend}/ReconMode.ml | 4 +- src/{frontend => Frontend}/ReconMode.mli | 4 +- src/{frontend => Frontend}/ReconModule.ml | 4 +- src/{frontend => Frontend}/ReconModule.mli | 4 +- src/{frontend => Frontend}/ReconQuery.ml | 4 +- src/{frontend => Frontend}/ReconQuery.mli | 4 +- src/{frontend => Frontend}/ReconTerm.ml | 2 +- src/{frontend => Frontend}/ReconTerm.mli | 2 +- src/{frontend => Frontend}/ReconThm.ml | 4 +- src/{frontend => Frontend}/ReconThm.mli | 4 +- .../Solve_intf.ml => Frontend/SOLVE.ml} | 2 +- src/{frontend => Frontend}/Solve.ml | 6 +- src/{frontend => Frontend}/Solve.mli | 6 +- .../Twelf_intf.ml => Frontend/TWELF.ml} | 0 src/{frontend => Frontend}/Twelf_.ml | 26 +- src/{frontend => Frontend}/Twelf_.mli | 38 +- .../UNKNOWNEXN.ml} | 0 src/{frontend => Frontend}/Unknownexn.ml | 2 +- src/{frontend => Frontend}/Unknownexn.mli | 2 +- src/{frontend => Frontend}/UnknownexnSmlnj.ml | 0 .../UnknownexnSmlnj.mli | 0 src/{frontend => Frontend}/Version.ml | 0 src/{frontend => Frontend}/Version.mli | 0 src/{frontend => Frontend}/WALK | 0 src/{frontend => Frontend}/dune | 40 +- src/{frontend => Frontend}/twelf_intf.ml~ | 0 src/Fronts/Modern/Cmd.ml | 81 +-- src/Fronts/Modern/MODERN.ml | 4 +- src/Fronts/Modern/Modern.ml | 390 ++++++++------- src/Fronts/Pal/Help.ml | 12 +- src/Fronts/Pal/IMPL.ml | 2 +- src/Fronts/Pal/Impl.ml | 55 ++- src/Fronts/Pal/PAL.ml | 12 +- src/Fronts/Pal/Pal.ml | 74 ++- src/Fronts/Pal/Pal.mli | 1 + src/Fronts/Project/dune | 2 +- src/Fronts/Tui/Repl.ml | 16 +- .../Global_intf.ml => Global/GLOBAL.ml} | 1 - src/{global => Global}/Global.ml | 0 src/{global => Global}/Global.mli | 0 src/{global => Global}/Global_.ml | 2 +- src/Global/Global_.mli | 2 + src/{global => Global}/README | 0 src/{global => Global}/WALK | 0 src/{global => Global}/dune | 2 +- .../HEURISTIC.ml} | 0 src/{heuristic => Heuristic}/Heuristic.ml | 0 src/{heuristic => Heuristic}/Heuristic.mli | 0 src/{heuristic => Heuristic}/HeuristicSum.ml | 0 src/{heuristic => Heuristic}/HeuristicSum.mli | 0 src/{heuristic => Heuristic}/Heuristic_.ml | 2 +- src/Heuristic/Heuristic_.mli | 2 + src/{heuristic => Heuristic}/dune | 2 +- src/{index/Index_intf.ml => Index/INDEX.ml} | 0 src/{index => Index}/Index.ml | 0 src/{index => Index}/Index.mli | 0 src/{index => Index}/IndexSkolem.ml | 0 src/{index => Index}/IndexSkolem.mli | 0 src/{index => Index}/Index_.ml | 2 +- src/{index => Index}/Index_.mli | 2 +- src/{index => Index}/README | 0 src/{index => Index}/WALK | 0 src/{index => Index}/dune | 2 +- .../IntInfSig_intf.ml => IntInf/INTINFSIG.ml} | 0 src/{intInf => IntInf}/IntInf.ml | 0 src/{intInf => IntInf}/IntInf.mli | 0 src/{intInf => IntInf}/IntInfSig.ml | 2 +- src/IntInf/IntInfSig.mli | 1 + src/{intInf => IntInf}/IntInf_.ml | 0 src/{intInf => IntInf}/IntInf_.mli | 0 src/{intInf => IntInf}/README | 0 src/{intInf => IntInf}/dune | 2 +- src/IntSyn/{Abstract_intf.ml => ABSTRACT.ml} | 0 src/IntSyn/{Approx_intf.ml => APPROX.ml} | 0 src/IntSyn/Abstract.ml | 2 +- src/IntSyn/Abstract.mli | 2 +- src/IntSyn/Approx.ml | 2 +- src/IntSyn/Approx.mli | 4 +- .../{Constraints_intf.ml => CONSTRAINTS.ml} | 0 src/IntSyn/{Conv_intf.ml => CONV.ml} | 0 src/IntSyn/Constraints.ml | 2 +- src/IntSyn/Constraints.mli | 4 +- src/IntSyn/Conv.ml | 2 +- src/IntSyn/Conv.mli | 4 +- src/IntSyn/{Fgnopn_intf.ml => FGNOPN.ml} | 0 src/IntSyn/Fgnopn.ml | 2 +- src/IntSyn/Fgnopn.mli | 2 +- src/IntSyn/{Intsyn_intf.ml => INTSYN.ml} | 0 src/IntSyn/{Match_intf.ml => MATCH.ml} | 0 src/IntSyn/Match.ml | 2 +- src/IntSyn/Match.mli | 6 +- src/IntSyn/{Order_intf.ml => ORDER.ml} | 0 src/IntSyn/Order.ml | 2 +- src/IntSyn/Order.mli | 2 +- src/IntSyn/{Tomega_intf.ml => TOMEGA.ml} | 0 src/IntSyn/Tomega.ml | 2 +- src/IntSyn/Tomega.mli | 2 +- src/IntSyn/{Unify_intf.ml => UNIFY.ml} | 0 src/IntSyn/Unify.ml | 2 +- src/IntSyn/Unify.mli | 4 +- src/IntSyn/{Whnf_intf.ml => WHNF.ml} | 0 src/IntSyn/Whnf.ml | 2 +- src/IntSyn/Whnf.mli | 2 +- src/IntSyn/dune | 22 +- src/{inverse => Inverse}/.cvsignore | 0 src/{inverse => Inverse}/README | 0 src/{inverse => Inverse}/notes/bug.txt | 0 src/{inverse => Inverse}/notes/notes.txt | 0 src/{inverse => Inverse}/notes/timing.txt | 0 src/{inverse => Inverse}/notes/timing2.txt | 0 src/{inverse => Inverse}/sml/.cvsignore | 0 src/{inverse => Inverse}/sml/sources.cm_ | 0 src/{inverse => Inverse}/sml/top.sml.ml.error | 0 src/{inverse => Inverse}/tex/.cvsignore | 0 src/{inverse => Inverse}/tex/Makefile | 0 src/{inverse => Inverse}/tex/canonical.tex | 0 src/{inverse => Inverse}/tex/commands.tex | 0 src/{inverse => Inverse}/tex/eagercheck.tex | 0 .../tex/images/twelf.graffle | 0 src/{inverse => Inverse}/tex/inverse.tex | 0 src/{inverse => Inverse}/tex/lazycheck.tex | 0 src/{inverse => Inverse}/tex/main.tex | 0 src/{inverse => Inverse}/tex/old/commands.tex | 0 src/{inverse => Inverse}/tex/old/context.tex | 0 src/{inverse => Inverse}/tex/old/etalong.tex | 0 src/{inverse => Inverse}/tex/old/noncheck.tex | 0 src/{inverse => Inverse}/tex/old/sequent.tex | 0 src/{inverse => Inverse}/tex/old/signat.tex | 0 src/{inverse => Inverse}/tex/old/uni.tex | 0 src/{inverse => Inverse}/tex/questions.txt | 0 src/Lang/Lexing/{Lexer_intf.ml => LEXER.ml} | 0 src/Lang/Parsing/PARSER.ml | 2 +- src/Lang/Parsing/Parser.ml | 7 +- src/{m2/Filling_intf.ml => M2/FILLING.ml} | 0 src/{m2 => M2}/Filling.ml | 7 +- src/{m2 => M2}/Filling.mli | 7 +- src/{m2/Init_intf.ml => M2/INIT.ml} | 0 src/{m2 => M2}/Init.ml | 6 +- src/M2/Init.mli | 6 + src/{m2/Lemma_intf.ml => M2/LEMMA.ml} | 0 src/{m2 => M2}/Lemma.ml | 6 +- src/M2/Lemma.mli | 6 + src/{m2 => M2}/M2.ml | 0 src/{m2 => M2}/M2.mli | 0 src/{m2 => M2}/M2_.ml | 2 +- src/{m2 => M2}/M2_.mli | 6 +- .../METAABSTRACT.ml} | 0 .../MetaGlobal_intf.ml => M2/METAGLOBAL.ml} | 0 src/{m2/MetaPrint_intf.ml => M2/METAPRINT.ml} | 0 src/{m2/Metasyn_intf.ml => M2/METASYN.ml} | 0 src/{m2/Mpi_intf.ml => M2/MPI.ml} | 0 src/{m2 => M2}/MetaAbstract.ml | 4 +- src/{m2 => M2}/MetaAbstract.mli | 4 +- src/{m2 => M2}/MetaGlobal.ml | 2 +- src/M2/MetaGlobal.mli | 2 + src/{m2 => M2}/MetaPrint.ml | 6 +- src/{m2 => M2}/MetaPrint.mli | 4 +- src/{m2 => M2}/Metasyn.ml | 2 +- src/{m2 => M2}/Metasyn.mli | 2 +- src/{m2 => M2}/Mpi.ml | 24 +- src/M2/Mpi.mli | 19 + src/{m2/Prover_intf.ml => M2/PROVER.ml} | 0 src/{m2 => M2}/Prover.ml | 20 +- src/M2/Prover.mli | 17 + src/{m2/Qed_intf.ml => M2/QED.ml} | 0 src/{m2 => M2}/Qed.ml | 2 +- src/{m2 => M2}/Qed.mli | 2 +- src/{m2 => M2}/README | 0 src/{m2/Recursion_intf.ml => M2/RECURSION.ml} | 0 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src/flit/Flit_.mli | 1 - src/formatter/Formatter_.mli | 2 - src/frontend/Frontend_.mli | 61 --- src/frontend/ParseCondec.mli | 9 - src/global/Global_.mli | 2 - src/heuristic/Heuristic_.mli | 2 - src/intInf/IntInfSig.mli | 1 - src/m2/Init.mli | 8 - src/m2/Lemma.mli | 8 - src/m2/MetaGlobal.mli | 2 - src/m2/Mpi.mli | 19 - src/m2/Prover.mli | 17 - src/m2/Strategy.mli | 32 -- src/meta/Funweaken.mli | 5 - src/meta/Interpret.mli | 1 - src/meta/MtpGlobal.mli | 5 - src/meta/MtpStrategy.mli | 13 - src/meta/Weaken.mli | 2 - src/modes/Modedec.mli | 2 - src/msg/Msg_.mli | 2 - src/netserver/Netserver_.mli | 2 - src/opsem/SwSubtree.mli | 7 - src/prover/Data.mli | 2 - src/server/Server_.mli | 2 - src/server/Sigint.mli | 1 - src/smlofnj/Smlofnj.mli | 2 - src/solvers/Cs.mli | 1 - src/subordinate/Intset.mli | 2 - src/table/Queue.mli | 2 - src/table/RedBlackSet.mli | 2 - src/table/Ring.mli | 2 - src/table/StringHash.mli | 2 - src/table/Table_.mli | 1 - src/tomega/Redundant.mli | 5 - src/tomega/Tomega_.mli | 18 - src/tomega/Tomeganames.mli | 2 - src/trail/Trail_.mli | 2 - test/CHANGES.md | 236 +++++++++ test/Pal/Cases.ml | 391 +++++++++------ test/Pal/Common.ml | 40 +- test/Pal/Source.ml | 464 +++++++++++++++++- test/Parse/Cases.ml | 33 +- test/Parse/Common.ml | 31 +- 872 files changed, 2641 insertions(+), 1829 deletions(-) rename src/Common/Syntax/{Ast_intf.ml => AST.ml} (100%) rename src/Common/Syntax/{Misc_intf.ml => MISC.ml} (99%) rename src/Common/Syntax/{Sgn_intf.ml => SGN.ml} (98%) rename src/{compile/Assign_intf.ml => Compile/ASSIGN.ml} (100%) rename src/{compile => Compile}/Assign.ml (99%) rename src/{compile => Compile}/Assign.mli (86%) rename src/{compile/Compile_intf.ml => Compile/COMPILE.ml} (100%) rename src/{compile/CompSyn_intf.ml => Compile/COMPSYN.ml} (100%) rename src/{compile/Cprint_intf.ml => Compile/CPRINT.ml} (100%) rename src/{compile => Compile}/CompSyn.ml (99%) rename src/{compile => Compile}/CompSyn.mli (81%) rename src/{compile => Compile}/Compile.ml (100%) rename src/{compile => Compile}/Compile.mli (100%) rename src/{compile => Compile}/Compile_.ml (99%) rename src/{compile => Compile}/Compile_.mli (95%) rename src/{compile => Compile}/Cprint.ml (99%) rename src/{compile => Compile}/Cprint.mli (72%) rename src/{compile => Compile}/README (100%) rename src/{compile/Subtree_intf.ml => Compile/SUBTREE.ml} (100%) rename src/{compile => Compile}/Subtree.ml (99%) rename src/{compile => Compile}/Subtree.mli (95%) rename src/{compile => Compile}/dune (92%) rename src/{compress/Compress_intf.ml => Compress/COMPRESS.ml} (100%) rename src/{compress => Compress}/Compress.ml (100%) rename src/{compress => Compress}/Compress.mli (100%) rename src/{compress => Compress}/Compress_.ml (99%) rename src/{compress => Compress}/Compress_.mli (67%) rename src/{compress => Compress}/Reductio.ml (100%) rename src/{compress => Compress}/Rep.ml (100%) rename src/{compress => Compress}/Rep.mli (100%) rename src/{compress/Sgn_intf.ml => Compress/SGN.ml} (100%) rename src/{compress => Compress}/Sgn.ml (99%) create mode 100644 src/Compress/Sgn.mli rename src/{compress => Compress}/Strict.ml (100%) rename src/{compress => Compress}/Syntax.ml (100%) rename src/{compress => Compress}/dune (82%) rename src/{cover/Cover_intf.ml => Cover/COVER.ml} (100%) rename src/{cover => Cover}/Cover.ml (100%) rename src/{cover => Cover}/Cover.mli (100%) rename src/{cover => Cover}/Cover_.ml (99%) rename src/{cover => Cover}/Cover_.mli (93%) rename src/{cover => Cover}/README (100%) rename src/{cover/Total_intf.ml => Cover/TOTAL.ml} (100%) rename src/{cover => Cover}/Total.ml (99%) rename src/{cover => Cover}/Total.mli (92%) rename src/{cover => Cover}/dune (96%) rename src/{debug => Debug}/Debug.ml (100%) rename src/{debug => Debug}/Debug.mli (100%) rename src/{debug => Debug}/dune (100%) rename src/{domains => Domains}/Domains.ml (100%) rename src/{domains => Domains}/Domains.mli (100%) rename src/{domains => Domains}/Domains_.ml (100%) rename src/{domains => Domains}/Domains_.mli (100%) rename src/{domains/Field_intf.ml => Domains/FIELD.ml} (100%) rename src/{domains => Domains}/Field.ml (90%) create mode 100644 src/Domains/Field.mli rename src/{domains/Integers_intf.ml => Domains/INTEGERS.ml} (100%) rename src/{domains => Domains}/Integers.ml (98%) rename src/{domains => Domains}/Integers.mli (61%) rename src/{domains => Domains}/IntegersMod.ml (100%) rename src/{domains => Domains}/IntegersMod.mli (100%) rename src/{domains/OrderedField_intf.ml => Domains/ORDEREDFIELD.ml} (100%) rename src/{domains => Domains}/OrderedField.ml (90%) create mode 100644 src/Domains/OrderedField.mli rename src/{domains/Rationals_intf.ml => Domains/RATIONALS.ml} (100%) rename src/{domains => Domains}/README (100%) rename src/{domains => Domains}/Rationals.ml (99%) rename src/{domains => Domains}/Rationals.mli (73%) rename src/{domains => Domains}/dune (80%) rename src/{flit/Flit_intf.ml => Flit/FLIT.ml} (100%) rename src/{flit => Flit}/Flit.ml (100%) rename src/{flit => Flit}/Flit.mli (100%) rename src/{flit => Flit}/FlitOldSmlnjPack.ml (100%) rename src/{flit => Flit}/FlitOldSmlnjPack.mli (100%) rename src/{flit => Flit}/Flit_.ml (99%) create mode 100644 src/Flit/Flit_.mli rename src/{flit => Flit}/README (100%) rename src/{flit => Flit}/dune (93%) rename src/{formatter/Formatter_intf.ml => Formatter/FORMATTER.ml} (100%) rename src/{formatter => Formatter}/Formatter.ml (100%) rename src/{formatter => Formatter}/Formatter.mli (100%) rename src/{formatter => Formatter}/Formatter_.ml (99%) create mode 100644 src/Formatter/Formatter_.mli rename src/{formatter => Formatter}/README (100%) rename src/{formatter => Formatter}/dune (84%) rename src/{frontend => Frontend}/Buildid.ml (100%) rename src/{frontend => Frontend}/Buildid.mli (100%) rename src/{frontend/Fquery_intf.ml => Frontend/FQUERY.ml} (85%) rename src/{frontend => Frontend}/Fquery.ml (97%) rename src/{frontend => Frontend}/Fquery.mli (69%) rename src/{frontend => Frontend}/Frontend.ml (100%) rename src/{frontend => Frontend}/Frontend.mli (100%) rename src/{frontend => Frontend}/Frontend_.ml (96%) create mode 100644 src/Frontend/Frontend_.mli rename src/{frontend/Lexer_intf.ml => Frontend/LEXER.ml} (100%) rename src/{frontend => Frontend}/Lexer.ml (99%) rename src/{frontend => Frontend}/Lexer.mli (76%) rename src/{frontend/ParseCondec_intf.ml => Frontend/PARSECONDEC.ml} (88%) rename src/{frontend/ParseFixity_intf.ml => Frontend/PARSEFIXITY.ml} (100%) rename src/{frontend/ParseMode_intf.ml => Frontend/PARSEMODE.ml} (85%) rename src/{frontend/ParseModule_intf.ml => Frontend/PARSEMODULE.ml} (91%) rename src/{frontend/ParseQuery_intf.ml => Frontend/PARSEQUERY.ml} (87%) rename src/{frontend/Parser_intf.ml => Frontend/PARSER.ml} (88%) rename src/{frontend/ParseTerm_intf.ml => Frontend/PARSETERM.ml} (95%) rename src/{frontend/ParseThm_intf.ml => Frontend/PARSETHM.ml} (95%) rename src/{frontend/Parsing_intf.ml => Frontend/PARSING.ml} (100%) rename src/{frontend => Frontend}/ParseCondec.ml (96%) create mode 100644 src/Frontend/ParseCondec.mli rename src/{frontend => Frontend}/ParseFixity.ml (99%) rename src/{frontend => Frontend}/ParseFixity.mli (76%) rename src/{frontend => Frontend}/ParseMode.ml (97%) rename src/{frontend => Frontend}/ParseMode.mli (64%) rename src/{frontend => Frontend}/ParseModule.ml (97%) rename src/{frontend => Frontend}/ParseModule.mli (64%) rename src/{frontend => Frontend}/ParseQuery.ml (96%) rename src/{frontend => Frontend}/ParseQuery.mli (53%) rename src/{frontend => Frontend}/ParseTerm.ml (99%) rename src/{frontend => Frontend}/ParseTerm.mli (73%) rename src/{frontend => Frontend}/ParseThm.ml (99%) rename src/{frontend => Frontend}/ParseThm.mli (64%) rename src/{frontend => Frontend}/Parser.ml (96%) rename src/{frontend => Frontend}/Parser.mli (52%) rename src/{frontend => Frontend}/Parsing.ml (98%) rename src/{frontend => Frontend}/Parsing.mli (83%) rename src/{frontend/ReconCondec_intf.ml => Frontend/RECONCONDEC.ml} (96%) rename src/{frontend/ReconMode_intf.ml => Frontend/RECONMODE.ml} (96%) rename src/{frontend/ReconModule_intf.ml => Frontend/RECONMODULE.ml} (97%) rename src/{frontend/ReconQuery_intf.ml => Frontend/RECONQUERY.ml} (96%) rename src/{frontend/ReconTerm_intf.ml => Frontend/RECONTERM.ml} (100%) rename src/{frontend/ReconThm_intf.ml => Frontend/RECONTHM.ml} (98%) rename src/{frontend => Frontend}/ReconCondec.ml (99%) rename src/{frontend => Frontend}/ReconCondec.mli (90%) rename src/{frontend => Frontend}/ReconMode.ml (98%) rename src/{frontend => Frontend}/ReconMode.mli (83%) rename src/{frontend => Frontend}/ReconModule.ml (98%) rename src/{frontend => Frontend}/ReconModule.mli (86%) rename src/{frontend => Frontend}/ReconQuery.ml (99%) rename src/{frontend => Frontend}/ReconQuery.mli (89%) rename src/{frontend => Frontend}/ReconTerm.ml (99%) rename src/{frontend => Frontend}/ReconTerm.mli (94%) rename src/{frontend => Frontend}/ReconThm.ml (99%) rename src/{frontend => Frontend}/ReconThm.mli (88%) rename src/{frontend/Solve_intf.ml => Frontend/SOLVE.ml} (94%) rename src/{frontend => Frontend}/Solve.ml (99%) rename src/{frontend => Frontend}/Solve.mli (94%) rename src/{frontend/Twelf_intf.ml => Frontend/TWELF.ml} (100%) rename src/{frontend => Frontend}/Twelf_.ml (99%) rename src/{frontend => Frontend}/Twelf_.mli (85%) rename src/{frontend/Unknownexn_intf.ml => Frontend/UNKNOWNEXN.ml} (100%) rename src/{frontend => Frontend}/Unknownexn.ml (96%) rename src/{frontend => Frontend}/Unknownexn.mli (71%) rename src/{frontend => Frontend}/UnknownexnSmlnj.ml (100%) rename src/{frontend => Frontend}/UnknownexnSmlnj.mli (100%) rename src/{frontend => Frontend}/Version.ml (100%) rename src/{frontend => Frontend}/Version.mli (100%) rename src/{frontend => Frontend}/WALK (100%) rename src/{frontend => Frontend}/dune (86%) rename src/{frontend => Frontend}/twelf_intf.ml~ (100%) rename src/{global/Global_intf.ml => Global/GLOBAL.ml} (99%) rename src/{global => Global}/Global.ml (100%) rename src/{global => Global}/Global.mli (100%) rename src/{global => Global}/Global_.ml (97%) create mode 100644 src/Global/Global_.mli rename src/{global => Global}/README (100%) rename src/{global => Global}/WALK (100%) rename src/{global => Global}/dune (86%) rename src/{heuristic/Heuristic_intf.ml => Heuristic/HEURISTIC.ml} (100%) rename src/{heuristic => Heuristic}/Heuristic.ml (100%) rename src/{heuristic => Heuristic}/Heuristic.mli (100%) rename src/{heuristic => Heuristic}/HeuristicSum.ml (100%) rename src/{heuristic => Heuristic}/HeuristicSum.mli (100%) rename src/{heuristic => Heuristic}/Heuristic_.ml (99%) create mode 100644 src/Heuristic/Heuristic_.mli rename src/{heuristic => Heuristic}/dune (80%) rename src/{index/Index_intf.ml => Index/INDEX.ml} (100%) rename src/{index => Index}/Index.ml (100%) rename src/{index => Index}/Index.mli (100%) rename src/{index => Index}/IndexSkolem.ml (100%) rename src/{index => Index}/IndexSkolem.mli (100%) rename src/{index => Index}/Index_.ml (99%) rename src/{index => Index}/Index_.mli (74%) rename src/{index => Index}/README (100%) rename src/{index => Index}/WALK (100%) rename src/{index => Index}/dune (90%) rename src/{intInf/IntInfSig_intf.ml => IntInf/INTINFSIG.ml} (100%) rename src/{intInf => IntInf}/IntInf.ml (100%) rename src/{intInf => IntInf}/IntInf.mli (100%) rename src/{intInf => IntInf}/IntInfSig.ml (94%) create mode 100644 src/IntInf/IntInfSig.mli rename src/{intInf => IntInf}/IntInf_.ml (100%) rename src/{intInf => IntInf}/IntInf_.mli (100%) rename src/{intInf => IntInf}/README (100%) rename src/{intInf => IntInf}/dune (81%) rename src/IntSyn/{Abstract_intf.ml => ABSTRACT.ml} (100%) rename src/IntSyn/{Approx_intf.ml => APPROX.ml} (100%) rename src/IntSyn/{Constraints_intf.ml => CONSTRAINTS.ml} (100%) rename src/IntSyn/{Conv_intf.ml => CONV.ml} (100%) rename src/IntSyn/{Fgnopn_intf.ml => FGNOPN.ml} (100%) rename src/IntSyn/{Intsyn_intf.ml => INTSYN.ml} (100%) rename src/IntSyn/{Match_intf.ml => MATCH.ml} (100%) rename src/IntSyn/{Order_intf.ml => ORDER.ml} (100%) rename src/IntSyn/{Tomega_intf.ml => TOMEGA.ml} (100%) rename src/IntSyn/{Unify_intf.ml => UNIFY.ml} (100%) rename src/IntSyn/{Whnf_intf.ml => WHNF.ml} (100%) rename src/{inverse => Inverse}/.cvsignore (100%) rename src/{inverse => Inverse}/README (100%) rename src/{inverse => Inverse}/notes/bug.txt (100%) rename src/{inverse => Inverse}/notes/notes.txt (100%) rename src/{inverse => Inverse}/notes/timing.txt (100%) rename src/{inverse => Inverse}/notes/timing2.txt (100%) rename src/{inverse => Inverse}/sml/.cvsignore (100%) rename src/{inverse => Inverse}/sml/sources.cm_ (100%) rename src/{inverse => Inverse}/sml/top.sml.ml.error (100%) rename src/{inverse => Inverse}/tex/.cvsignore (100%) rename src/{inverse => Inverse}/tex/Makefile (100%) rename src/{inverse => Inverse}/tex/canonical.tex (100%) rename src/{inverse => Inverse}/tex/commands.tex (100%) rename src/{inverse => Inverse}/tex/eagercheck.tex (100%) rename src/{inverse => Inverse}/tex/images/twelf.graffle (100%) rename src/{inverse => Inverse}/tex/inverse.tex (100%) rename src/{inverse => Inverse}/tex/lazycheck.tex (100%) rename src/{inverse => Inverse}/tex/main.tex (100%) rename src/{inverse => Inverse}/tex/old/commands.tex (100%) rename src/{inverse => Inverse}/tex/old/context.tex (100%) rename src/{inverse => Inverse}/tex/old/etalong.tex (100%) rename src/{inverse => Inverse}/tex/old/noncheck.tex (100%) rename src/{inverse => Inverse}/tex/old/sequent.tex (100%) rename src/{inverse => Inverse}/tex/old/signat.tex (100%) rename src/{inverse => Inverse}/tex/old/uni.tex (100%) rename src/{inverse => Inverse}/tex/questions.txt (100%) rename src/Lang/Lexing/{Lexer_intf.ml => LEXER.ml} (100%) rename src/{m2/Filling_intf.ml => M2/FILLING.ml} (100%) rename src/{m2 => M2}/Filling.ml (98%) rename src/{m2 => M2}/Filling.mli (69%) rename src/{m2/Init_intf.ml => M2/INIT.ml} (100%) rename src/{m2 => M2}/Init.ml (91%) create mode 100644 src/M2/Init.mli rename src/{m2/Lemma_intf.ml => M2/LEMMA.ml} (100%) rename src/{m2 => M2}/Lemma.ml (95%) create mode 100644 src/M2/Lemma.mli rename src/{m2 => M2}/M2.ml (100%) rename src/{m2 => M2}/M2.mli (100%) rename src/{m2 => M2}/M2_.ml (99%) rename src/{m2 => M2}/M2_.mli (69%) rename src/{m2/MetaAbstract_intf.ml => M2/METAABSTRACT.ml} (100%) rename src/{m2/MetaGlobal_intf.ml => M2/METAGLOBAL.ml} (100%) rename src/{m2/MetaPrint_intf.ml => M2/METAPRINT.ml} (100%) rename src/{m2/Metasyn_intf.ml => M2/METASYN.ml} (100%) rename src/{m2/Mpi_intf.ml => M2/MPI.ml} (100%) rename src/{m2 => M2}/MetaAbstract.ml (99%) rename src/{m2 => M2}/MetaAbstract.mli (90%) rename src/{m2 => M2}/MetaGlobal.ml (95%) create mode 100644 src/M2/MetaGlobal.mli rename src/{m2 => M2}/MetaPrint.ml (96%) rename src/{m2 => M2}/MetaPrint.mli (74%) rename src/{m2 => M2}/Metasyn.ml (99%) rename src/{m2 => M2}/Metasyn.mli (55%) rename src/{m2 => M2}/Mpi.ml (94%) create mode 100644 src/M2/Mpi.mli rename src/{m2/Prover_intf.ml => M2/PROVER.ml} (100%) rename src/{m2 => M2}/Prover.ml (91%) create mode 100644 src/M2/Prover.mli rename src/{m2/Qed_intf.ml => M2/QED.ml} (100%) rename src/{m2 => M2}/Qed.ml (97%) rename src/{m2 => M2}/Qed.mli (80%) rename src/{m2 => M2}/README (100%) rename src/{m2/Recursion_intf.ml => M2/RECURSION.ml} (100%) rename src/{m2 => M2}/Recursion.ml (98%) rename src/{m2 => M2}/Recursion.mli (63%) rename src/{m2/Search_intf.ml => M2/SEARCH.ml} (100%) rename src/{m2/Skolem_intf.ml => M2/SKOLEM.ml} (100%) rename src/{m2/Splitting_intf.ml => M2/SPLITTING.ml} (100%) rename src/{m2/Strategy_intf.ml => M2/STRATEGY.ml} (100%) rename src/{m2 => M2}/Search.ml (99%) rename src/{m2 => M2}/Search.mli (91%) rename src/{m2 => M2}/Skolem.ml (98%) rename src/{m2 => M2}/Skolem.mli (90%) rename src/{m2 => M2}/Splitting.ml (98%) rename src/{m2 => M2}/Splitting.mli (61%) rename src/{m2 => M2}/Strategy.ml (88%) create mode 100644 src/M2/Strategy.mli rename src/{m2 => M2}/WALK (100%) rename src/{m2 => M2}/dune (86%) rename src/{meta/Funnames_intf.ml => Meta/FUNNAMES.ml} (100%) rename src/{meta/Funprint_intf.ml => Meta/FUNPRINT.ml} (100%) rename src/{meta/Funsyn_intf.ml => Meta/FUNSYN.ml} (100%) rename src/{meta/Funtypecheck_intf.ml => Meta/FUNTYPECHECK.ml} (100%) rename src/{meta/Funweaken_intf.ml => Meta/FUNWEAKEN.ml} (100%) rename src/{meta => Meta}/Funnames.ml (98%) rename src/{meta => Meta}/Funnames.mli (68%) rename src/{meta => Meta}/Funprint.ml (99%) rename src/{meta => Meta}/Funprint.mli (74%) rename src/{meta => Meta}/Funsyn.ml (99%) rename src/{meta => Meta}/Funsyn.mli (69%) rename src/{meta => Meta}/Funtypecheck.ml (98%) rename src/{meta => Meta}/Funtypecheck.mli (78%) rename src/{meta => Meta}/Funweaken.ml (95%) create mode 100644 src/Meta/Funweaken.mli rename src/{meta/Inference_intf.ml => Meta/INFERENCE.ml} (100%) rename src/{meta/Interpret_intf.ml => Meta/INTERPRET.ml} (100%) rename src/{meta => Meta}/Inference.ml (96%) rename src/{meta => Meta}/Inference.mli (73%) rename src/{meta => Meta}/Interpret.ml (90%) create mode 100644 src/Meta/Interpret.mli rename src/{meta/MtpAbstract_intf.ml => Meta/MTPABSTRACT.ml} (100%) rename src/{meta/MtpData_intf.ml => Meta/MTPDATA.ml} (100%) rename src/{meta/MtpFilling_intf.ml => Meta/MTPFILLING.ml} (100%) rename src/{meta/MtpGlobal_intf.ml => Meta/MTPGLOBAL.ml} (100%) rename src/{meta/MtpInit_intf.ml => Meta/MTPINIT.ml} (100%) rename src/{meta/MtpMpi_intf.ml => Meta/MTPMPI.ml} (100%) rename src/{meta/MtpPrint_intf.ml => Meta/MTPPRINT.ml} (100%) rename src/{meta/MtpProver_intf.ml => Meta/MTPPROVER.ml} (100%) rename src/{meta/MtpRecursion_intf.ml => Meta/MTPRECURSION.ml} (100%) rename src/{meta/MtpSearch_intf.ml => Meta/MTPSEARCH.ml} (100%) rename src/{meta/MtpSplitting_intf.ml => Meta/MTPSPLITTING.ml} (100%) rename src/{meta/MtpStrategy_intf.ml => Meta/MTPSTRATEGY.ml} (100%) rename src/{meta => Meta}/Meta.ml (100%) rename src/{meta => Meta}/Meta.mli (100%) rename src/{meta => Meta}/Meta_.ml (100%) rename src/{meta => Meta}/Meta_.mli (100%) rename src/{meta => Meta}/MtpAbstract.ml (99%) rename src/{meta => Meta}/MtpAbstract.mli (79%) rename src/{meta => Meta}/MtpData.ml (88%) rename src/{meta => Meta}/MtpData.mli (53%) rename src/{meta => Meta}/MtpFilling.ml (95%) rename src/{meta => Meta}/MtpFilling.mli (67%) rename src/{meta => Meta}/MtpGlobal.ml (96%) create mode 100644 src/Meta/MtpGlobal.mli rename src/{meta => Meta}/MtpInit.ml (93%) rename src/{meta => Meta}/MtpInit.mli (70%) rename src/{meta => Meta}/MtpMpi.ml (96%) rename src/{meta => Meta}/MtpMpi.mli (54%) rename src/{meta => Meta}/MtpPrint.ml (97%) rename src/{meta => Meta}/MtpPrint.mli (76%) rename src/{meta => Meta}/MtpProver.ml (95%) rename src/{meta => Meta}/MtpProver.mli (64%) rename src/{meta => Meta}/MtpRecursion.ml (99%) rename src/{meta => Meta}/MtpRecursion.mli (79%) rename src/{meta => Meta}/MtpSearch.ml (99%) rename src/{meta => Meta}/MtpSearch.mli (89%) rename 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delete mode 100644 src/msg/Msg_.mli delete mode 100644 src/netserver/Netserver_.mli delete mode 100644 src/opsem/SwSubtree.mli delete mode 100644 src/prover/Data.mli delete mode 100644 src/server/Server_.mli delete mode 100644 src/server/Sigint.mli delete mode 100644 src/smlofnj/Smlofnj.mli delete mode 100644 src/solvers/Cs.mli delete mode 100644 src/subordinate/Intset.mli delete mode 100644 src/table/Queue.mli delete mode 100644 src/table/RedBlackSet.mli delete mode 100644 src/table/Ring.mli delete mode 100644 src/table/StringHash.mli delete mode 100644 src/table/Table_.mli delete mode 100644 src/tomega/Redundant.mli delete mode 100644 src/tomega/Tomega_.mli delete mode 100644 src/tomega/Tomeganames.mli delete mode 100644 src/trail/Trail_.mli create mode 100644 test/CHANGES.md diff --git a/src/Common/Cst/CST.ml b/src/Common/Cst/CST.ml index 7cd48b4..22aa7ca 100644 --- a/src/Common/Cst/CST.ml +++ b/src/Common/Cst/CST.ml @@ -1,4 +1,3 @@ - open Base (** The concrete syntax tree (CST) interface (in Twelf, it bore the name [ExtSyn]. @@ -18,10 +17,9 @@ open Base (** {2 CST} *) module type CST = sig - module Paths : Paths.Paths_intf.PATHS + module Paths : Paths.PATHS.PATHS (** Module of paths and regions, which we allow to be shared *) - type query (** Query payload. *) @@ -107,7 +105,6 @@ module type CST = sig module Term : sig type t = term - val lowercase : ?fc:loc -> symbol -> term (** Lowercase identifier (does not start with [_]). *) @@ -379,15 +376,6 @@ module type CST = sig val deterministic : ?fc:loc -> string list -> cmd (** [%deterministic id_list] — mark type families as deterministic. *) - val module_cmd : ?fc:loc -> string -> string list -> cmd list -> cmd - (** [%module id params body] — declare a parameterised module. *) - - val use : ?fc:loc -> string -> string -> string list -> cmd - (** [%use id id iparams] — instantiate a module. *) - - val open_cmd : ?fc:loc -> string -> string list -> cmd - (** [%open id id_list] — bring names from a module into scope. *) - val eval : ?fc:loc -> cmd list -> cmd (** [%eval %{ cmds %}] — evaluate a command block. *) @@ -414,18 +402,18 @@ module type CST = sig val terminates : ?fc:loc -> order list -> term list -> cmd (** [%terminates hyps modes] — declare a termination check. *) - + val covers : ?fc:loc -> modeDec -> cmd (** [%covers hyps modes] — declare a coverage check. *) - + val name : ?fc:loc -> string -> cmd (** [%name id] — declare a name for the next definition. *) val reduces : ?fc:loc -> string -> term list -> cmd - (** [%reduces pred order_out order_in call_pats] — declare a reduction relation. *) - + (** [%reduces pred order_out order_in call_pats] — declare a reduction + relation. *) end - + module Thm : sig (*! structure Paths : PATHS !*) type order @@ -496,32 +484,28 @@ module type CST = sig val pp_term : Stdlib.Format.formatter -> term -> unit (** Pretty-print a term to a formatter. *) - + (** {2 Views} *) (** Views should eventually supplant the rest of this module *) - module View : sig - include LENS.VIEW - with type Term.t = term - and type Decl.t = decl - and type ConDec.t = conDec - and type Mode.t = mode - and type Mode.Term.t = modeTerm - and type Mode.Dec.t = modeDec - and type Struct.StrExp.t = strexp - and type Struct.Inst.t = inst - and type Struct.SigExp.t = sigexp - and type Struct.SigDef.t = sigdef - and type Struct.StructDec.t = structDec - and type Query.t = query - and type Solve.t = solve - and type Define.t = define - and type Fixity.t = fixity - and type Cmd.t = cmd - - - end - - + module View : sig + include + LENS.VIEW + with type Term.t = term + and type Decl.t = decl + and type ConDec.t = conDec + and type Mode.t = mode + and type Mode.Term.t = modeTerm + and type Mode.Dec.t = modeDec + and type Struct.StrExp.t = strexp + and type Struct.Inst.t = inst + and type Struct.SigExp.t = sigexp + and type Struct.SigDef.t = sigdef + and type Struct.StructDec.t = structDec + and type Query.t = query + and type Solve.t = solve + and type Define.t = define + and type Fixity.t = fixity + and type Cmd.t = cmd + end end - diff --git a/src/Common/Cst/Cst.ml b/src/Common/Cst/Cst.ml index 3dc9b6d..6d128e4 100644 --- a/src/Common/Cst/Cst.ml +++ b/src/Common/Cst/Cst.ml @@ -2,7 +2,7 @@ open Basis module type CST = CST.CST -module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct +module Make_Cst (Paths : Paths.PATHS.PATHS) = struct module Paths = Paths (* Location type for source tracking - use simple int-based representation *) @@ -40,6 +40,7 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | Evar_ of string * loc | Fvar_ of string * loc | Typ_ of loc + | MacroParam_ of loc * int option * int [@@deriving show { with_path = false }, eq] (* Constant/block declarations *) @@ -128,9 +129,6 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | UnionCmd_ of string * string list | WorldsCmd_ of string list * term | DeterministicCmd_ of string list - | ModuleCmd_ of string * string list * cmd list - | UseCmd_ of string * string * string list - | OpenCmd_ of string * string list | EvalCmd_ of cmd list | PrecCmd_ of fixity * int * string list | SolveCmd_ of solve @@ -145,6 +143,13 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | CoversCmd_ of modeDec | NameCmd_ of string | ReducesCmd_ of string * term list + | Macro_ of int * string * cmd (** Defines a macro, taking its location, number of params, name, and the body *) + | Seq_ of item list (** A sequence of commands, for use withthe module system*) + | Require_ of string list (** Ensure that the given path is loaded *) + | Open_ of string list (** Open a scope into the scope *) + | Scope_ of string * cmd (** Enter into a new scope *) + | Use_ of string list * term list (** Apply a macro *) + and item = Outer of string | Cmd of cmd [@@deriving show { with_path = false }, eq] (* Term constructor module *) @@ -324,13 +329,13 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct let union ?fc:(_ = ghost) id ids = UnionCmd_ (id, ids) let worlds ?fc:(_ = ghost) ids tm = WorldsCmd_ (ids, tm) let deterministic ?fc:(_ = ghost) ids = DeterministicCmd_ ids - let module_cmd ?fc:(_ = ghost) id params cmds = ModuleCmd_ (id, params, cmds) - let use ?fc:(_ = ghost) id1 id2 ps = UseCmd_ (id1, id2, ps) - let open_cmd ?fc:(_ = ghost) id ids = OpenCmd_ (id, ids) + + let eval ?fc:(_ = ghost) cmds = EvalCmd_ cmds let prec ?fc:(_ = ghost) fix n ids = PrecCmd_ (fix, n, ids) let solve ?fc:(_ = ghost) s = SolveCmd_ s let stop ?fc:(_ = ghost) () = StopCmd_ + module Repl = struct let quit ?fc:(_ = ghost) () = QuitCmd_ @@ -515,6 +520,7 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | BackArrow of Loc.t * t * t | Foreign of Loc.t * t | Internal of int + | MacroParam of Loc.t * int option * int val view : t -> u val review : u -> t @@ -540,7 +546,7 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | BackArrow of Loc.t * t * t | Foreign of Loc.t * t | Internal of int - + | MacroParam of Loc.t * int option * int let view (x : t) : u = let rec collect_pis = function | Pi_ (_, d, body) -> @@ -1002,9 +1008,6 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | Union of Loc.t * string * string list | Worlds of Loc.t * string list * Term.t | Deterministic of Loc.t * string list - | ModuleCmd of Loc.t * string * string list * t list - | Use of Loc.t * string * string * string list - | OpenCmd of Loc.t * string * string list | Eval of Loc.t * t list | Prec of Loc.t * Fixity.t * int * string list | Solve of Loc.t * Solve.t @@ -1019,7 +1022,13 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | Covers of Loc.t * Mode.Dec.t | Name of Loc.t * string | Reduces of Loc.t * string * Term.t list - + | Macro of Loc.t * int * string * t (** Defines a macro, taking its location, number of params, name, and the body *) + | Seq of Loc.t * item list (** A sequence of commands, for use withthe module system*) + | Require of Loc.t * string list (** Ensure that the given path is loaded *) + | Open of Loc.t * string list (** Open a scope into the scope *) + | Scope of Loc.t * string * t (** Enter into a new scope *) + | Use of Loc.t * string list * Term.t list (** Apply a macro *) + and item = Outer of string | Cmd of t let view (x : t) : u = match x with | QueryCmd_ (n, b, d, q) -> Query (ghost, n, b, d, q) @@ -1039,9 +1048,6 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | UnionCmd_ (id, ids) -> Union (ghost, id, ids) | WorldsCmd_ (ids, tm) -> Worlds (ghost, ids, tm) | DeterministicCmd_ ids -> Deterministic (ghost, ids) - | ModuleCmd_ (id, params, cmds) -> ModuleCmd (ghost, id, params, cmds) - | UseCmd_ (id1, id2, ps) -> Use (ghost, id1, id2, ps) - | OpenCmd_ (id, ids) -> OpenCmd (ghost, id, ids) | EvalCmd_ cmds -> Eval (ghost, cmds) | PrecCmd_ (fix, n, ids) -> Prec (ghost, fix, n, ids) | SolveCmd_ s -> Solve (ghost, s) @@ -1076,9 +1082,6 @@ module Make_Cst (Paths : Paths.Paths_intf.PATHS) = struct | Union (loc, id, ids) -> UnionCmd_ (id, ids) | Worlds (loc, ids, tm) -> WorldsCmd_ (ids, tm) | Deterministic (loc, ids) -> DeterministicCmd_ ids - | ModuleCmd (loc, id, params, cmds) -> ModuleCmd_ (id, params, cmds) - | Use (loc, id1, id2, ps) -> UseCmd_ (id1, id2, ps) - | OpenCmd (loc, id, ids) -> OpenCmd_ (id, ids) | Eval (loc, cmds) -> EvalCmd_ cmds | Prec (loc, fix, n, ids) -> PrecCmd_ (fix, n, ids) | Solve (loc, s) -> SolveCmd_ s diff --git a/src/Common/Cst/LENS.ml b/src/Common/Cst/LENS.ml index 66f5c44..d5b67af 100644 --- a/src/Common/Cst/LENS.ml +++ b/src/Common/Cst/LENS.ml @@ -53,7 +53,7 @@ end module type VIEW = sig exception Lacking - module Paths : Paths.Paths_intf.PATHS + module Paths : Paths.PATHS.PATHS (** Module of paths and regions, which we allow to be shared *) (** Abstract syntax tree type. (for internals) *) @@ -116,6 +116,7 @@ module type VIEW = sig | BackArrow of Loc.t * t * t | Foreign of Loc.t * t | Internal of int + | MacroParam of Loc.t * int option * int include LENS with type t := t and type u := u end @@ -407,9 +408,6 @@ module type VIEW = sig | Union of Loc.t * string * string list | Worlds of Loc.t * string list * Term.t | Deterministic of Loc.t * string list - | ModuleCmd of Loc.t * string * string list * t list - | Use of Loc.t * string * string * string list - | OpenCmd of Loc.t * string * string list | Eval of Loc.t * t list | Prec of Loc.t * Fixity.t * int * string list | Solve of Loc.t * Solve.t @@ -424,6 +422,18 @@ module type VIEW = sig | Covers of Loc.t * Mode.Dec.t | Name of Loc.t * string | Reduces of Loc.t * string * Term.t list + | Macro of Loc.t * int * string * t + (** Defines a macro, taking its location, number of params, name, and + the body *) + | Seq of Loc.t * item list + (** A sequence of commands, for use withthe module system*) + | Require of Loc.t * string list + (** Ensure that the given path is loaded *) + | Open of Loc.t * string list (** Open a scope into the scope *) + | Scope of Loc.t * string * t (** Enter into a new scope *) + | Use of Loc.t * string list * Term.t list (** Apply a macro *) + + and item = Outer of string | Cmd of t include LENS with type t := t and type u := u end diff --git a/src/Common/Cst/Lens.ml b/src/Common/Cst/Lens.ml index 8752f2f..654c07f 100644 --- a/src/Common/Cst/Lens.ml +++ b/src/Common/Cst/Lens.ml @@ -1,9 +1,10 @@ -module Make_Lens (L : LENS.S) : LENS.LENS with type t = L.t and type u = L.u = - struct - type t = L.t - type u = L.u - let review = L.review - let view = L.view - let (!>) = view - let (!<) = review - end +module Make_Lens (L : LENS.S) : LENS.LENS with type t = L.t and type u = L.u = +struct + type t = L.t + type u = L.u + + let review = L.review + let view = L.view + let ( !> ) = view + let ( !< ) = review +end diff --git a/src/Common/Cst/SIM.ml b/src/Common/Cst/SIM.ml index d461dbe..35108ae 100644 --- a/src/Common/Cst/SIM.ml +++ b/src/Common/Cst/SIM.ml @@ -1,9 +1,7 @@ - - -module type SIM = sig +module type SIM = sig module CstA : CST.CST - module CstB : CST.CST - val sim_term : CstA.View.Term.t -> CstB.View.Term.t -> bool + module CstB : CST.CST + + val sim_term : CstA.View.Term.t -> CstB.View.Term.t -> bool val sim_decl : CstA.View.Decl.t -> CstB.View.Decl.t -> bool - -end \ No newline at end of file +end diff --git a/src/Common/Cst/Sim.ml b/src/Common/Cst/Sim.ml index 1c778d0..d3df680 100644 --- a/src/Common/Cst/Sim.ml +++ b/src/Common/Cst/Sim.ml @@ -1,10 +1,5 @@ -(** - Whether two terms are "similar" to each other - Simalarity encompasses two things - 1. Whether the terms can be viewed as the same - 2. Whether the terms are equivalent representations of the same thing -*) +(** Whether two terms are "similar" to each other Simalarity encompasses two + things 1. Whether the terms can be viewed as the same 2. Whether the terms + are equivalent representations of the same thing *) -module Sim (A : CST.CST) (B : CST.CST) = struct - -end \ No newline at end of file +module Sim (A : CST.CST) (B : CST.CST) = struct end diff --git a/src/Common/Syntax/Ast_intf.ml b/src/Common/Syntax/AST.ml similarity index 100% rename from src/Common/Syntax/Ast_intf.ml rename to src/Common/Syntax/AST.ml diff --git a/src/Common/Syntax/Ast.ml b/src/Common/Syntax/Ast.ml index e5f062d..7542d0c 100644 --- a/src/Common/Syntax/Ast.ml +++ b/src/Common/Syntax/Ast.ml @@ -3,8 +3,8 @@ open Common (* Author: Frank Pfenning, Carsten Schuermann *) (* Modified: Roberto Virga *) -module Make_Ast (Common : Common.COMMON) : - Ast_intf.AST with module Common = Common = struct +module Make_Ast (Common : Common.COMMON) : AST.AST with module Common = Common = +struct module Common = Common open Common diff --git a/src/Common/Syntax/Common.ml b/src/Common/Syntax/Common.ml index 114f6da..a98da27 100644 --- a/src/Common/Syntax/Common.ml +++ b/src/Common/Syntax/Common.ml @@ -18,5 +18,5 @@ module type COMMON = sig module Tag : TAG module Cid : CID module Mid : CID - module Global : Global.Global_intf.GLOBAL + module Global : Global.GLOBAL.GLOBAL end diff --git a/src/Common/Syntax/FgnOpn.ml b/src/Common/Syntax/FgnOpn.ml index 9e5d002..84bf2b5 100644 --- a/src/Common/Syntax/FgnOpn.ml +++ b/src/Common/Syntax/FgnOpn.ml @@ -49,7 +49,7 @@ end module type S = sig module Common : Common.COMMON - module Ast : Ast_intf.AST with module Common = Common + module Ast : AST.AST with module Common = Common open Ast (** Raised when the global signature exceeds its maximum size. *) @@ -107,7 +107,7 @@ end module FgnOpn (Common : Common.COMMON) - (Ast : Ast_intf.AST with module Common = Common) : + (Ast : AST.AST with module Common = Common) : S with module Common = Common and module Ast = Ast = struct open Ast module Ast = Ast diff --git a/src/Common/Syntax/Misc_intf.ml b/src/Common/Syntax/MISC.ml similarity index 99% rename from src/Common/Syntax/Misc_intf.ml rename to src/Common/Syntax/MISC.ml index c55fff7..af39054 100644 --- a/src/Common/Syntax/Misc_intf.ml +++ b/src/Common/Syntax/MISC.ml @@ -1,5 +1,5 @@ module type MISC = sig - module Ast : Ast_intf.AST + module Ast : AST.AST module Common : Common.COMMON open Common diff --git a/src/Common/Syntax/Misc.ml b/src/Common/Syntax/Misc.ml index 24368a4..f75279f 100644 --- a/src/Common/Syntax/Misc.ml +++ b/src/Common/Syntax/Misc.ml @@ -1,8 +1,8 @@ -include Misc_intf +include MISC module Make_Misc (Common' : Common.COMMON) - (Ast' : Ast_intf.AST with module Common = Common') : + (Ast' : AST.AST with module Common = Common') : MISC with module Common = Common' and module Ast = Ast' = struct module Common = Common' module Ast = Ast' diff --git a/src/Common/Syntax/Sgn_intf.ml b/src/Common/Syntax/SGN.ml similarity index 98% rename from src/Common/Syntax/Sgn_intf.ml rename to src/Common/Syntax/SGN.ml index 3226f24..07f7934 100644 --- a/src/Common/Syntax/Sgn_intf.ml +++ b/src/Common/Syntax/SGN.ml @@ -1,6 +1,6 @@ module type SGN = sig module Common : Common.COMMON - module Ast : Ast_intf.AST with module Common = Common + module Ast : AST.AST with module Common = Common type cid = Ast.cid type mid = Ast.mid diff --git a/src/Common/Syntax/Sgn.ml b/src/Common/Syntax/Sgn.ml index e413f55..ddd854e 100644 --- a/src/Common/Syntax/Sgn.ml +++ b/src/Common/Syntax/Sgn.ml @@ -1,9 +1,9 @@ -include Sgn_intf +include SGN module Make_Sgn (Common : Common.COMMON) - (Ast : Ast_intf.AST with module Common = Common) : - Sgn_intf.SGN with module Common = Common and module Ast = Ast = struct + (Ast : AST.AST with module Common = Common) : + SGN.SGN with module Common = Common and module Ast = Ast = struct module CTable = Containers.Hashtbl.Make (struct type t = Ast.cid diff --git a/src/Common/Syntax/Syntax.ml b/src/Common/Syntax/Syntax.ml index d4e8d97..1e20e27 100644 --- a/src/Common/Syntax/Syntax.ml +++ b/src/Common/Syntax/Syntax.ml @@ -1,17 +1,15 @@ module type SYNTAX = sig module Common : Common.COMMON - module type AST = Ast_intf.AST with module Common = Common + module type AST = AST.AST with module Common = Common module Ast : AST - module type SGN = - Sgn_intf.SGN with module Common = Common and module Ast = Ast + module type SGN = SGN.SGN with module Common = Common and module Ast = Ast module Sgn : SGN - module type MISC = - Misc_intf.MISC with module Common = Common and module Ast = Ast + module type MISC = MISC.MISC with module Common = Common and module Ast = Ast module Misc : MISC end @@ -20,17 +18,15 @@ module Make_Syntax (Common : Common.COMMON) : SYNTAX with module Common = Common = struct module Common = Common - module type AST = Ast_intf.AST with module Common = Common + module type AST = AST.AST with module Common = Common module Ast : AST = Ast.Make_Ast (Common) - module type SGN = - Sgn_intf.SGN with module Common = Common and module Ast = Ast + module type SGN = SGN.SGN with module Common = Common and module Ast = Ast module Sgn : SGN = Sgn.Make_Sgn (Common) (Ast) - module type MISC = - Misc_intf.MISC with module Common = Common and module Ast = Ast + module type MISC = MISC.MISC with module Common = Common and module Ast = Ast module Misc : MISC = Misc.Make_Misc (Common) (Ast) end @@ -54,7 +50,7 @@ end module type INTSYN = sig end -module IntSyn (Global : Global.Global_intf.GLOBAL) = Make_Syntax (struct +module IntSyn (Global : Global.GLOBAL.GLOBAL) = Make_Syntax (struct module Cid = IntIdx module Mid = IntIdx @@ -82,7 +78,7 @@ module ExtIdx : Common.CID with type t = string = struct let show = toString end -module ExtSyn (Global : Global.Global_intf.GLOBAL) = Make_Syntax (struct +module ExtSyn (Global : Global.GLOBAL.GLOBAL) = Make_Syntax (struct module Cid = ExtIdx module Mid = ExtIdx @@ -94,9 +90,9 @@ module ExtSyn (Global : Global.Global_intf.GLOBAL) = Make_Syntax (struct end) module type EXTSYN = sig - module Global : Global.Global_intf.GLOBAL + module Global : Global.GLOBAL.GLOBAL include module type of ExtSyn (Global) end include Sgn -include Sgn_intf +include SGN diff --git a/src/compile/Assign_intf.ml b/src/Compile/ASSIGN.ml similarity index 100% rename from src/compile/Assign_intf.ml rename to src/Compile/ASSIGN.ml diff --git a/src/compile/Assign.ml b/src/Compile/Assign.ml similarity index 99% rename from src/compile/Assign.ml rename to src/Compile/Assign.ml index c806d74..93550dc 100644 --- a/src/compile/Assign.ml +++ b/src/Compile/Assign.ml @@ -4,7 +4,7 @@ open! Basis (* Assignment *) (* Author: Larry Greenfield *) (* Modified: Brigitte Pientka *) -include Assign_intf +include ASSIGN (* signature ASSIGN *) (* # 1 "src/compile/Assign.fun.ml" *) diff --git a/src/compile/Assign.mli b/src/Compile/Assign.mli similarity index 86% rename from src/compile/Assign.mli rename to src/Compile/Assign.mli index c117668..0edc8f4 100644 --- a/src/compile/Assign.mli +++ b/src/Compile/Assign.mli @@ -1,4 +1,4 @@ -include module type of Assign_intf +include module type of ASSIGN module Assign (Assign__0 : sig (*! structure IntSyn' : INTSYN !*) diff --git a/src/compile/Compile_intf.ml b/src/Compile/COMPILE.ml similarity index 100% rename from src/compile/Compile_intf.ml rename to src/Compile/COMPILE.ml diff --git a/src/compile/CompSyn_intf.ml b/src/Compile/COMPSYN.ml similarity index 100% rename from src/compile/CompSyn_intf.ml rename to src/Compile/COMPSYN.ml diff --git a/src/compile/Cprint_intf.ml b/src/Compile/CPRINT.ml similarity index 100% rename from src/compile/Cprint_intf.ml rename to src/Compile/CPRINT.ml diff --git a/src/compile/CompSyn.ml b/src/Compile/CompSyn.ml similarity index 99% rename from src/compile/CompSyn.ml rename to src/Compile/CompSyn.ml index 25d0cde..3c7d3fa 100644 --- a/src/compile/CompSyn.ml +++ b/src/Compile/CompSyn.ml @@ -4,7 +4,7 @@ open! Basis (* Compiled Syntax *) (* Author: Iliano Cervesato *) (* Modified: Jeff Polakow *) -include CompSyn_intf +include COMPSYN (* signature COMPSYN *) (* # 1 "src/compile/CompSyn.fun.ml" *) diff --git a/src/compile/CompSyn.mli b/src/Compile/CompSyn.mli similarity index 81% rename from src/compile/CompSyn.mli rename to src/Compile/CompSyn.mli index 5b94893..59d6435 100644 --- a/src/compile/CompSyn.mli +++ b/src/Compile/CompSyn.mli @@ -1,4 +1,4 @@ -include module type of CompSyn_intf +include module type of COMPSYN module Make_CompSyn (Global_ : GLOBAL) diff --git a/src/compile/Compile.ml b/src/Compile/Compile.ml similarity index 100% rename from src/compile/Compile.ml rename to src/Compile/Compile.ml diff --git a/src/compile/Compile.mli b/src/Compile/Compile.mli similarity index 100% rename from src/compile/Compile.mli rename to src/Compile/Compile.mli diff --git a/src/compile/Compile_.ml b/src/Compile/Compile_.ml similarity index 99% rename from src/compile/Compile_.ml rename to src/Compile/Compile_.ml index 2260889..7d36f7c 100644 --- a/src/compile/Compile_.ml +++ b/src/Compile/Compile_.ml @@ -7,7 +7,7 @@ open CompSyn (* Modified: Jeff Polakow *) (* Modified: Carsten Schuermann *) -include Compile_intf +include COMPILE (** Modified: Frank Pfenning *) (* signature COMPILE *) diff --git a/src/compile/Compile_.mli b/src/Compile/Compile_.mli similarity index 95% rename from src/compile/Compile_.mli rename to src/Compile/Compile_.mli index 2f74419..550fc5b 100644 --- a/src/compile/Compile_.mli +++ b/src/Compile/Compile_.mli @@ -1,4 +1,4 @@ -include module type of Compile_intf +include module type of COMPILE module MakeCompile (Whnf : WHNF) diff --git a/src/compile/Cprint.ml b/src/Compile/Cprint.ml similarity index 99% rename from src/compile/Cprint.ml rename to src/Compile/Cprint.ml index 6119d39..80c9723 100644 --- a/src/compile/Cprint.ml +++ b/src/Compile/Cprint.ml @@ -3,7 +3,7 @@ open! Basis (* Printer for Compiled Syntax *) (* Author: Iliano Cervesato *) -include Cprint_intf +include CPRINT (* signature CPRINT *) (* # 1 "src/compile/Cprint.fun.ml" *) diff --git a/src/compile/Cprint.mli b/src/Compile/Cprint.mli similarity index 72% rename from src/compile/Cprint.mli rename to src/Compile/Cprint.mli index d4e0a3e..62bb79f 100644 --- a/src/compile/Cprint.mli +++ b/src/Compile/Cprint.mli @@ -1,4 +1,4 @@ -include module type of Cprint_intf +include module type of CPRINT module Make_CPrint (Print_ : PRINT) (Formatter_ : FORMATTER) (Names_ : NAMES) : CPRINT diff --git a/src/compile/README b/src/Compile/README similarity index 100% rename from src/compile/README rename to src/Compile/README diff --git a/src/compile/Subtree_intf.ml b/src/Compile/SUBTREE.ml similarity index 100% rename from src/compile/Subtree_intf.ml rename to src/Compile/SUBTREE.ml diff --git a/src/compile/Subtree.ml b/src/Compile/Subtree.ml similarity index 99% rename from src/compile/Subtree.ml rename to src/Compile/Subtree.ml index b0b5d69..e21ec7b 100644 --- a/src/compile/Subtree.ml +++ b/src/Compile/Subtree.ml @@ -3,7 +3,7 @@ open! Basis (* Substitution Trees *) (* Author: Brigitte Pientka *) -include Subtree_intf +include SUBTREE (* val goalToString : string -> IntSyn.Dec IntSyn.Ctx * CompSyn.Goal * IntSyn.Sub -> string *) (* signature SUBTREE *) diff --git a/src/compile/Subtree.mli b/src/Compile/Subtree.mli similarity index 95% rename from src/compile/Subtree.mli rename to src/Compile/Subtree.mli index c7f4488..b4d6416 100644 --- a/src/compile/Subtree.mli +++ b/src/Compile/Subtree.mli @@ -1,4 +1,4 @@ -include module type of Subtree_intf +include module type of SUBTREE module SubTree (SubTree__0 : sig (*! structure IntSyn' : INTSYN !*) diff --git a/src/compile/dune b/src/Compile/dune similarity index 92% rename from src/compile/dune rename to src/Compile/dune index 7d4061c..d1f32b8 100644 --- a/src/compile/dune +++ b/src/Compile/dune @@ -9,11 +9,11 @@ Subtree Assign Compile_ - Assign_intf - CompSyn_intf - Compile_intf - Cprint_intf - Subtree_intf) + ASSIGN + COMPSYN + COMPILE + CPRINT + SUBTREE) (libraries display global diff --git a/src/compress/Compress_intf.ml b/src/Compress/COMPRESS.ml similarity index 100% rename from src/compress/Compress_intf.ml rename to src/Compress/COMPRESS.ml diff --git a/src/compress/Compress.ml b/src/Compress/Compress.ml similarity index 100% rename from src/compress/Compress.ml rename to src/Compress/Compress.ml diff --git a/src/compress/Compress.mli b/src/Compress/Compress.mli similarity index 100% rename from src/compress/Compress.mli rename to src/Compress/Compress.mli diff --git a/src/compress/Compress_.ml b/src/Compress/Compress_.ml similarity index 99% rename from src/compress/Compress_.ml rename to src/Compress/Compress_.ml index 842f94c..92bd1b6 100644 --- a/src/compress/Compress_.ml +++ b/src/Compress/Compress_.ml @@ -1,7 +1,7 @@ (* # 1 "src/compress/Compress_.sig.ml" *) open! Basis -include Compress_intf +include COMPRESS (** `Compressed' terms with omitted redundant arguments *) (* # 1 "src/compress/Compress_.fun.ml" *) diff --git a/src/compress/Compress_.mli b/src/Compress/Compress_.mli similarity index 67% rename from src/compress/Compress_.mli rename to src/Compress/Compress_.mli index 9452a3e..29e58b0 100644 --- a/src/compress/Compress_.mli +++ b/src/Compress/Compress_.mli @@ -1,4 +1,4 @@ -include module type of Compress_intf +include module type of COMPRESS module Compress (Compress__0 : sig module Global : GLOBAL diff --git a/src/compress/Reductio.ml b/src/Compress/Reductio.ml similarity index 100% rename from src/compress/Reductio.ml rename to src/Compress/Reductio.ml diff --git a/src/compress/Rep.ml b/src/Compress/Rep.ml similarity index 100% rename from src/compress/Rep.ml rename to src/Compress/Rep.ml diff --git a/src/compress/Rep.mli b/src/Compress/Rep.mli similarity index 100% rename from src/compress/Rep.mli rename to src/Compress/Rep.mli diff --git a/src/compress/Sgn_intf.ml b/src/Compress/SGN.ml similarity index 100% rename from src/compress/Sgn_intf.ml rename to src/Compress/SGN.ml diff --git a/src/compress/Sgn.ml b/src/Compress/Sgn.ml similarity index 99% rename from src/compress/Sgn.ml rename to src/Compress/Sgn.ml index 4bcc568..8ecaf70 100644 --- a/src/compress/Sgn.ml +++ b/src/Compress/Sgn.ml @@ -5,7 +5,7 @@ (* # 1 "src/compress/Sgn.sml.ml" *) open! Syntax open! Basis -include Sgn_intf +include SGN module Sgn = struct open Syntax diff --git a/src/Compress/Sgn.mli b/src/Compress/Sgn.mli new file mode 100644 index 0000000..c94cf69 --- /dev/null +++ b/src/Compress/Sgn.mli @@ -0,0 +1,3 @@ +include module type of SGN +module Sgn : SGN +include SGN diff --git a/src/compress/Strict.ml b/src/Compress/Strict.ml similarity index 100% rename from src/compress/Strict.ml rename to src/Compress/Strict.ml diff --git a/src/compress/Syntax.ml b/src/Compress/Syntax.ml similarity index 100% rename from src/compress/Syntax.ml rename to src/Compress/Syntax.ml diff --git a/src/compress/dune b/src/Compress/dune similarity index 82% rename from src/compress/dune rename to src/Compress/dune index 71d12d7..67842f1 100644 --- a/src/compress/dune +++ b/src/Compress/dune @@ -3,7 +3,7 @@ (library (name compress) (public_name stelf.compress) - (modules Syntax Strict Sgn Reductio Compress_ Rep Compress_intf Sgn_intf) + (modules Syntax Strict Sgn Reductio Compress_ Rep COMPRESS SGN) (libraries display global intsyn basis) (flags (:standard diff --git a/src/Config/dune b/src/Config/dune index bb9fcdc..0c936b6 100644 --- a/src/Config/dune +++ b/src/Config/dune @@ -1,4 +1,4 @@ (library (name config) (public_name stelf.config) - (libraries )) \ No newline at end of file + (libraries)) diff --git a/src/cover/Cover_intf.ml b/src/Cover/COVER.ml similarity index 100% rename from src/cover/Cover_intf.ml rename to src/Cover/COVER.ml diff --git a/src/cover/Cover.ml b/src/Cover/Cover.ml similarity index 100% rename from src/cover/Cover.ml rename to src/Cover/Cover.ml diff --git a/src/cover/Cover.mli b/src/Cover/Cover.mli similarity index 100% rename from src/cover/Cover.mli rename to src/Cover/Cover.mli diff --git a/src/cover/Cover_.ml b/src/Cover/Cover_.ml similarity index 99% rename from src/cover/Cover_.ml rename to src/Cover/Cover_.ml index 62bdd0a..d481be9 100644 --- a/src/cover/Cover_.ml +++ b/src/Cover/Cover_.ml @@ -3,7 +3,7 @@ open! Basis (* Coverage Checking *) -include Cover_intf +include COVER (** Author: Frank Pfenning *) (* signature COVER *) diff --git a/src/cover/Cover_.mli b/src/Cover/Cover_.mli similarity index 93% rename from src/cover/Cover_.mli rename to src/Cover/Cover_.mli index edf87b5..94d105b 100644 --- a/src/cover/Cover_.mli +++ b/src/Cover/Cover_.mli @@ -1,4 +1,4 @@ -include module type of Cover_intf +include module type of COVER module MakeCover (Global : GLOBAL) @@ -30,4 +30,4 @@ module MakeCover *) module Cover : COVER -module Total : Total_intf.TOTAL +module Total : TOTAL.TOTAL diff --git a/src/cover/README b/src/Cover/README similarity index 100% rename from src/cover/README rename to src/Cover/README diff --git a/src/cover/Total_intf.ml b/src/Cover/TOTAL.ml similarity index 100% rename from src/cover/Total_intf.ml rename to src/Cover/TOTAL.ml diff --git a/src/cover/Total.ml b/src/Cover/Total.ml similarity index 99% rename from src/cover/Total.ml rename to src/Cover/Total.ml index 621127f..5744bfc 100644 --- a/src/cover/Total.ml +++ b/src/Cover/Total.ml @@ -3,7 +3,7 @@ open! Basis (* Total Declarations *) (* Author: Frank Pfenning *) -include Total_intf +include TOTAL (* may raise Error(msg) *) (* signature TOTAL *) @@ -39,7 +39,7 @@ module Total (Total__0 : sig module Order : ORDER (*! sharing Order.IntSyn = IntSyn' !*) - module Reduces : Reduces_intf.REDUCES + module Reduces : REDUCES.REDUCES (*! sharing Reduces.IntSyn = IntSyn' !*) module Cover : COVER diff --git a/src/cover/Total.mli b/src/Cover/Total.mli similarity index 92% rename from src/cover/Total.mli rename to src/Cover/Total.mli index 5257c97..4031a06 100644 --- a/src/cover/Total.mli +++ b/src/Cover/Total.mli @@ -1,4 +1,4 @@ -include module type of Total_intf +include module type of TOTAL module Total (Total__0 : sig module Global : GLOBAL @@ -24,7 +24,7 @@ module Total (Total__0 : sig module Order : ORDER (*! sharing Order.IntSyn = IntSyn' !*) - module Reduces : Reduces_intf.REDUCES + module Reduces : REDUCES.REDUCES (*! sharing Reduces.IntSyn = IntSyn' !*) module Cover : COVER diff --git a/src/cover/dune b/src/Cover/dune similarity index 96% rename from src/cover/dune rename to src/Cover/dune index b97e45a..8687c1d 100644 --- a/src/cover/dune +++ b/src/Cover/dune @@ -3,7 +3,7 @@ (library (name cover) (public_name stelf.cover) - (modules Total Cover_ Cover_intf Total_intf) + (modules Total Cover_ COVER TOTAL) (libraries display global diff --git a/src/debug/Debug.ml b/src/Debug/Debug.ml similarity index 100% rename from src/debug/Debug.ml rename to src/Debug/Debug.ml diff --git a/src/debug/Debug.mli b/src/Debug/Debug.mli similarity index 100% rename from src/debug/Debug.mli rename to src/Debug/Debug.mli diff --git a/src/debug/dune b/src/Debug/dune similarity index 100% rename from src/debug/dune rename to src/Debug/dune diff --git a/src/Display/FORM.ml b/src/Display/FORM.ml index 716f7a8..caf13a6 100644 --- a/src/Display/FORM.ml +++ b/src/Display/FORM.ml @@ -1,5 +1,3 @@ - - (* TODO: Thinks that will be added eventually Groups (sections, list, etc) Math markup diff --git a/src/Display/Form.ml b/src/Display/Form.ml index cca5f41..0507aed 100644 --- a/src/Display/Form.ml +++ b/src/Display/Form.ml @@ -1,6 +1,5 @@ -(** -Formatting options generally -@author Asher Frost *) +(** Formatting options generally + @author Asher Frost *) module type FORM = FORM.FORM diff --git a/src/domains/Domains.ml b/src/Domains/Domains.ml similarity index 100% rename from src/domains/Domains.ml rename to src/Domains/Domains.ml diff --git a/src/domains/Domains.mli b/src/Domains/Domains.mli similarity index 100% rename from src/domains/Domains.mli rename to src/Domains/Domains.mli diff --git a/src/domains/Domains_.ml b/src/Domains/Domains_.ml similarity index 100% rename from src/domains/Domains_.ml rename to src/Domains/Domains_.ml diff --git a/src/domains/Domains_.mli b/src/Domains/Domains_.mli similarity index 100% rename from src/domains/Domains_.mli rename to src/Domains/Domains_.mli diff --git a/src/domains/Field_intf.ml b/src/Domains/FIELD.ml similarity index 100% rename from src/domains/Field_intf.ml rename to src/Domains/FIELD.ml diff --git a/src/domains/Field.ml b/src/Domains/Field.ml similarity index 90% rename from src/domains/Field.ml rename to src/Domains/Field.ml index 5c70460..8264499 100644 --- a/src/domains/Field.ml +++ b/src/Domains/Field.ml @@ -4,7 +4,7 @@ open Basis (* Field *) (* Author: Roberto Virga *) -include Field_intf +include FIELD (* signature FIELD *) (* # 1 "src/domains/Field.fun.ml" *) diff --git a/src/Domains/Field.mli b/src/Domains/Field.mli new file mode 100644 index 0000000..9d3df3a --- /dev/null +++ b/src/Domains/Field.mli @@ -0,0 +1 @@ +include module type of FIELD diff --git a/src/domains/Integers_intf.ml b/src/Domains/INTEGERS.ml similarity index 100% rename from src/domains/Integers_intf.ml rename to src/Domains/INTEGERS.ml diff --git a/src/domains/Integers.ml b/src/Domains/Integers.ml similarity index 98% rename from src/domains/Integers.ml rename to src/Domains/Integers.ml index 23ea59c..59dab6f 100644 --- a/src/domains/Integers.ml +++ b/src/Domains/Integers.ml @@ -4,7 +4,7 @@ open Basis (* Integers *) (* Author: Roberto Virga *) -include Integers_intf +include INTEGERS (* signature INTEGERS *) (* # 1 "src/domains/Integers.fun.ml" *) diff --git a/src/domains/Integers.mli b/src/Domains/Integers.mli similarity index 61% rename from src/domains/Integers.mli rename to src/Domains/Integers.mli index 3081219..193b0bf 100644 --- a/src/domains/Integers.mli +++ b/src/Domains/Integers.mli @@ -1,3 +1,3 @@ open Basis -include module type of Integers_intf +include module type of INTEGERS module Integers (Integer : INTEGER) : INTEGERS diff --git a/src/domains/IntegersMod.ml b/src/Domains/IntegersMod.ml similarity index 100% rename from src/domains/IntegersMod.ml rename to src/Domains/IntegersMod.ml diff --git a/src/domains/IntegersMod.mli b/src/Domains/IntegersMod.mli similarity index 100% rename from src/domains/IntegersMod.mli rename to src/Domains/IntegersMod.mli diff --git a/src/domains/OrderedField_intf.ml b/src/Domains/ORDEREDFIELD.ml similarity index 100% rename from src/domains/OrderedField_intf.ml rename to src/Domains/ORDEREDFIELD.ml diff --git a/src/domains/OrderedField.ml b/src/Domains/OrderedField.ml similarity index 90% rename from src/domains/OrderedField.ml rename to src/Domains/OrderedField.ml index 4b20b83..c7d76bb 100644 --- a/src/domains/OrderedField.ml +++ b/src/Domains/OrderedField.ml @@ -4,7 +4,7 @@ open Field (* Ordered Field *) (* Author: Roberto Virga *) -include OrderedField_intf +include ORDEREDFIELD (* signature ORDERED_FIELD *) (* # 1 "src/domains/OrderedField.fun.ml" *) diff --git a/src/Domains/OrderedField.mli b/src/Domains/OrderedField.mli new file mode 100644 index 0000000..d25b1df --- /dev/null +++ b/src/Domains/OrderedField.mli @@ -0,0 +1 @@ +include module type of ORDEREDFIELD diff --git a/src/domains/Rationals_intf.ml b/src/Domains/RATIONALS.ml similarity index 100% rename from src/domains/Rationals_intf.ml rename to src/Domains/RATIONALS.ml diff --git a/src/domains/README b/src/Domains/README similarity index 100% rename from src/domains/README rename to src/Domains/README diff --git a/src/domains/Rationals.ml b/src/Domains/Rationals.ml similarity index 99% rename from src/domains/Rationals.ml rename to src/Domains/Rationals.ml index 406f34e..f1ee62a 100644 --- a/src/domains/Rationals.ml +++ b/src/Domains/Rationals.ml @@ -5,7 +5,7 @@ open Integers (* Rational numbers *) (* Author: Roberto Virga *) -include Rationals_intf +include RATIONALS (* signature RATIONALS *) (* # 1 "src/domains/Rationals.fun.ml" *) diff --git a/src/domains/Rationals.mli b/src/Domains/Rationals.mli similarity index 73% rename from src/domains/Rationals.mli rename to src/Domains/Rationals.mli index c864904..7ac6c3a 100644 --- a/src/domains/Rationals.mli +++ b/src/Domains/Rationals.mli @@ -1,5 +1,5 @@ open Integers -include module type of Rationals_intf +include module type of RATIONALS module Rationals (Integers : INTEGERS) : RATIONALS with type Integers.int = Integers.int diff --git a/src/domains/dune b/src/Domains/dune similarity index 80% rename from src/domains/dune rename to src/Domains/dune index a670f14..0d51b2b 100644 --- a/src/domains/dune +++ b/src/Domains/dune @@ -10,10 +10,10 @@ Rationals IntegersMod Domains_ - Field_intf - Integers_intf - OrderedField_intf - Rationals_intf) + FIELD + INTEGERS + ORDEREDFIELD + RATIONALS) (libraries display intInf basis) (wrapped true) (preprocess diff --git a/src/flit/Flit_intf.ml b/src/Flit/FLIT.ml similarity index 100% rename from src/flit/Flit_intf.ml rename to src/Flit/FLIT.ml diff --git a/src/flit/Flit.ml b/src/Flit/Flit.ml similarity index 100% rename from src/flit/Flit.ml rename to src/Flit/Flit.ml diff --git a/src/flit/Flit.mli b/src/Flit/Flit.mli similarity index 100% rename from src/flit/Flit.mli rename to src/Flit/Flit.mli diff --git a/src/flit/FlitOldSmlnjPack.ml b/src/Flit/FlitOldSmlnjPack.ml similarity index 100% rename from src/flit/FlitOldSmlnjPack.ml rename to src/Flit/FlitOldSmlnjPack.ml diff --git a/src/flit/FlitOldSmlnjPack.mli b/src/Flit/FlitOldSmlnjPack.mli similarity index 100% rename from src/flit/FlitOldSmlnjPack.mli rename to src/Flit/FlitOldSmlnjPack.mli diff --git a/src/flit/Flit_.ml b/src/Flit/Flit_.ml similarity index 99% rename from src/flit/Flit_.ml rename to src/Flit/Flit_.ml index 22d1c7f..c1e22e5 100644 --- a/src/flit/Flit_.ml +++ b/src/Flit/Flit_.ml @@ -3,7 +3,7 @@ open! Basis (* Flit DAG generator *) -include Flit_intf +include FLIT (** Author: Roberto Virga *) (* signature FLIT *) diff --git a/src/Flit/Flit_.mli b/src/Flit/Flit_.mli new file mode 100644 index 0000000..dd989cf --- /dev/null +++ b/src/Flit/Flit_.mli @@ -0,0 +1 @@ +include module type of FLIT diff --git a/src/flit/README b/src/Flit/README similarity index 100% rename from src/flit/README rename to src/Flit/README diff --git a/src/flit/dune b/src/Flit/dune similarity index 93% rename from src/flit/dune rename to src/Flit/dune index 957d324..9ba89b8 100644 --- a/src/flit/dune +++ b/src/Flit/dune @@ -3,7 +3,7 @@ (library (name flit) (public_name stelf.flit) - (modules FlitOldSmlnjPack Flit_ Flit_intf) + (modules FlitOldSmlnjPack Flit_ FLIT) (libraries display global table intsyn names index paths print basis) (flags (:standard diff --git a/src/formatter/Formatter_intf.ml b/src/Formatter/FORMATTER.ml similarity index 100% rename from src/formatter/Formatter_intf.ml rename to src/Formatter/FORMATTER.ml diff --git a/src/formatter/Formatter.ml b/src/Formatter/Formatter.ml similarity index 100% rename from src/formatter/Formatter.ml rename to src/Formatter/Formatter.ml diff --git a/src/formatter/Formatter.mli b/src/Formatter/Formatter.mli similarity index 100% rename from src/formatter/Formatter.mli rename to src/Formatter/Formatter.mli diff --git a/src/formatter/Formatter_.ml b/src/Formatter/Formatter_.ml similarity index 99% rename from src/formatter/Formatter_.ml rename to src/Formatter/Formatter_.ml index ca33904..2cff0be 100644 --- a/src/formatter/Formatter_.ml +++ b/src/Formatter/Formatter_.ml @@ -6,7 +6,7 @@ open Basis ForML Version 0.6 - 25 January 1993 - er@Cs.cmu.edu File Formatter.sig with signature FORMATTER. *) -include Formatter_intf +include FORMATTER (* # 1 "src/formatter/Formatter_.fun.ml" *) (* diff --git a/src/Formatter/Formatter_.mli b/src/Formatter/Formatter_.mli new file mode 100644 index 0000000..6fb7e7d --- /dev/null +++ b/src/Formatter/Formatter_.mli @@ -0,0 +1,2 @@ +include module type of FORMATTER +module Formatter : FORMATTER diff --git a/src/formatter/README b/src/Formatter/README similarity index 100% rename from src/formatter/README rename to src/Formatter/README diff --git a/src/formatter/dune b/src/Formatter/dune similarity index 84% rename from src/formatter/dune rename to src/Formatter/dune index 8eb8b70..a6bc77d 100644 --- a/src/formatter/dune +++ b/src/Formatter/dune @@ -3,7 +3,7 @@ (library (name formatter) (public_name stelf.formatter) - (modules Formatter_ Formatter_intf) + (modules Formatter_ FORMATTER) (libraries display basis) (wrapped true) (preprocess diff --git a/src/frontend/Buildid.ml b/src/Frontend/Buildid.ml similarity index 100% rename from src/frontend/Buildid.ml rename to src/Frontend/Buildid.ml diff --git a/src/frontend/Buildid.mli b/src/Frontend/Buildid.mli similarity index 100% rename from src/frontend/Buildid.mli rename to src/Frontend/Buildid.mli diff --git a/src/frontend/Fquery_intf.ml b/src/Frontend/FQUERY.ml similarity index 85% rename from src/frontend/Fquery_intf.ml rename to src/Frontend/FQUERY.ml index d86a804..a07f57b 100644 --- a/src/frontend/Fquery_intf.ml +++ b/src/Frontend/FQUERY.ml @@ -5,7 +5,7 @@ open! Basis (* Author: Carsten Schuermann *) module type FQUERY = sig - module ExtQuery : ReconQuery_intf.EXTQUERY + module ExtQuery : RECONQUERY.EXTQUERY exception AbortQuery of string diff --git a/src/frontend/Fquery.ml b/src/Frontend/Fquery.ml similarity index 97% rename from src/frontend/Fquery.ml rename to src/Frontend/Fquery.ml index 5300866..a846bb4 100644 --- a/src/frontend/Fquery.ml +++ b/src/Frontend/Fquery.ml @@ -3,7 +3,7 @@ open! Basis (* fquery: Executing logic programs via functional interpretation *) (* Author: Carsten Schuermann *) -include Fquery_intf +include FQUERY (* may raise AbortQuery(msg) *) (* signature SOLVE *) @@ -16,7 +16,7 @@ open! Basis module Fquery (Fquery__0 : sig module Global : GLOBAL module Names : NAMES - module ReconQuery : ReconQuery_intf.RECON_QUERY + module ReconQuery : RECONQUERY.RECON_QUERY module Timers : Timers.TIMERS module Print : PRINT end) : FQUERY with module ExtQuery = Fquery__0.ReconQuery = struct diff --git a/src/frontend/Fquery.mli b/src/Frontend/Fquery.mli similarity index 69% rename from src/frontend/Fquery.mli rename to src/Frontend/Fquery.mli index 8b8c90f..564c905 100644 --- a/src/frontend/Fquery.mli +++ b/src/Frontend/Fquery.mli @@ -1,9 +1,9 @@ -include module type of Fquery_intf +include module type of FQUERY module Fquery (Fquery__0 : sig module Global : GLOBAL module Names : NAMES - module ReconQuery : ReconQuery_intf.RECON_QUERY + module ReconQuery : RECONQUERY.RECON_QUERY module Timers : Timers.TIMERS module Print : PRINT end) : FQUERY with module ExtQuery = Fquery__0.ReconQuery diff --git a/src/frontend/Frontend.ml b/src/Frontend/Frontend.ml similarity index 100% rename from src/frontend/Frontend.ml rename to src/Frontend/Frontend.ml diff --git a/src/frontend/Frontend.mli b/src/Frontend/Frontend.mli similarity index 100% rename from src/frontend/Frontend.mli rename to src/Frontend/Frontend.mli diff --git a/src/frontend/Frontend_.ml b/src/Frontend/Frontend_.ml similarity index 96% rename from src/frontend/Frontend_.ml rename to src/Frontend/Frontend_.ml index 4fa7f2e..451dceb 100644 --- a/src/frontend/Frontend_.ml +++ b/src/Frontend/Frontend_.ml @@ -141,35 +141,35 @@ module ParseTerm = ParseTermFunctor.ParseTerm (struct end) module ParseTermConDec : - ParseTerm_intf.PARSE_TERM with module ExtSyn = ReconConDec.ExtSyn = + PARSETERM.PARSE_TERM with module ExtSyn = ReconConDec.ExtSyn = ParseTermFunctor.ParseTerm (struct module ExtSyn' = ReconConDec.ExtSyn module Names = Names end) module ParseTermQuery : - ParseTerm_intf.PARSE_TERM with module ExtSyn = ReconQuery.ExtSyn = + PARSETERM.PARSE_TERM with module ExtSyn = ReconQuery.ExtSyn = ParseTermFunctor.ParseTerm (struct module ExtSyn' = ReconQuery.ExtSyn module Names = Names end) module ParseTermMode : - ParseTerm_intf.PARSE_TERM with module ExtSyn = ReconMode.ExtSyn = + PARSETERM.PARSE_TERM with module ExtSyn = ReconMode.ExtSyn = ParseTermFunctor.ParseTerm (struct module ExtSyn' = ReconMode.ExtSyn module Names = Names end) module ParseTermThm : - ParseTerm_intf.PARSE_TERM with module ExtSyn = ReconThm.ExtSyn = + PARSETERM.PARSE_TERM with module ExtSyn = ReconThm.ExtSyn = ParseTermFunctor.ParseTerm (struct module ExtSyn' = ReconThm.ExtSyn module Names = Names end) module ParseTermModule : - ParseTerm_intf.PARSE_TERM with module ExtSyn = ReconModule.ExtSyn = + PARSETERM.PARSE_TERM with module ExtSyn = ReconModule.ExtSyn = ParseTermFunctor.ParseTerm (struct module ExtSyn' = ReconModule.ExtSyn module Names = Names diff --git a/src/Frontend/Frontend_.mli b/src/Frontend/Frontend_.mli new file mode 100644 index 0000000..acde83b --- /dev/null +++ b/src/Frontend/Frontend_.mli @@ -0,0 +1,60 @@ +(* # 1 "src/frontend/Frontend_.sig.ml" *) + +(* # 1 "src/frontend/Frontend_.fun.ml" *) + +(* # 1 "src/frontend/Frontend_.sml.ml" *) +open! Basis + +(* Front End Interface *) +(* Author: Frank Pfenning *) +(* Presently, we do not memoize the token stream returned *) +(* by the Lexer. Use Stream = MStream below if memoization becomes *) +(* necessary. *) +(* Now in Lexer.fun *) +(* +structure Lexer = + Lexer (structure Stream' = Stream + structure Paths' = Paths); +*) +(* Now in Parsing.fun *) +(* +structure Parsing = + Parsing (structure Stream' = Stream + structure Lexer' = Lexer); +*) +(* Re-export module type before Stelf name shadowing. *) +module type LEXER = Lexer.LEXER +module type STELF = Twelf_.STELF + +module ReconTerm : RECONTERM.RECON_TERM +module ReconConDec : RECONCONDEC.RECON_CONDEC +module ReconQuery : RECONQUERY.RECON_QUERY +module ReconMode : RECONMODE.RECON_MODE +module ReconThm : RECONTHM.RECON_THM +module ReconModule : RECONMODULE.RECON_MODULE +module ParseTerm : PARSETERM.PARSE_TERM + +module ParseTermConDec : + PARSETERM.PARSE_TERM with module ExtSyn = ReconConDec.ExtSyn + +module ParseTermQuery : + PARSETERM.PARSE_TERM with module ExtSyn = ReconQuery.ExtSyn + +module ParseTermMode : + PARSETERM.PARSE_TERM with module ExtSyn = ReconMode.ExtSyn + +module ParseTermThm : PARSETERM.PARSE_TERM with module ExtSyn = ReconThm.ExtSyn + +module ParseTermModule : + PARSETERM.PARSE_TERM with module ExtSyn = ReconModule.ExtSyn + +module ParseConDec : PARSECONDEC.PARSE_CONDEC +module ParseQuery : PARSEQUERY.PARSE_QUERY +module ParseFixity : PARSEFIXITY.PARSE_FIXITY with module Names = Names +module ParseMode : PARSEMODE.PARSE_MODE +module ParseThm : PARSETHM.PARSE_THM +module ParseModule : PARSEMODULE.PARSE_MODULE +module Parser : PARSER.PARSER +module Solve : SOLVE.SOLVE with module ExtQuery = ReconQuery +module Fquery : FQUERY.FQUERY with module ExtQuery = ReconQuery +module Stelf : STELF diff --git a/src/frontend/Lexer_intf.ml b/src/Frontend/LEXER.ml similarity index 100% rename from src/frontend/Lexer_intf.ml rename to src/Frontend/LEXER.ml diff --git a/src/frontend/Lexer.ml b/src/Frontend/Lexer.ml similarity index 99% rename from src/frontend/Lexer.ml rename to src/Frontend/Lexer.ml index 9399458..dc46652 100644 --- a/src/frontend/Lexer.ml +++ b/src/Frontend/Lexer.ml @@ -1,7 +1,7 @@ (* # 1 "src/frontend/Lexer.sig.ml" *) open! Basis -include Lexer_intf +include LEXER (** Lexer implementation. Author: Frank Pfenning. *) (* signature LEXER *) diff --git a/src/frontend/Lexer.mli b/src/Frontend/Lexer.mli similarity index 76% rename from src/frontend/Lexer.mli rename to src/Frontend/Lexer.mli index 83e3052..9db6226 100644 --- a/src/frontend/Lexer.mli +++ b/src/Frontend/Lexer.mli @@ -1,4 +1,4 @@ -include module type of Lexer_intf +include module type of LEXER module MakeLexer (Stream : STREAM) : LEXER module Lexer : LEXER include LEXER with module Stream = Lexer.Stream diff --git a/src/frontend/ParseCondec_intf.ml b/src/Frontend/PARSECONDEC.ml similarity index 88% rename from src/frontend/ParseCondec_intf.ml rename to src/Frontend/PARSECONDEC.ml index f4e4d90..403dea3 100644 --- a/src/frontend/ParseCondec_intf.ml +++ b/src/Frontend/PARSECONDEC.ml @@ -7,7 +7,7 @@ open! Parsing module type PARSE_CONDEC = sig (*! structure Parsing : PARSING !*) - module ExtConDec : ReconCondec_intf.EXTCONDEC + module ExtConDec : RECONCONDEC.EXTCONDEC val parseConDec' : ExtConDec.condec Parsing.parser val parseAbbrev' : ExtConDec.condec Parsing.parser diff --git a/src/frontend/ParseFixity_intf.ml b/src/Frontend/PARSEFIXITY.ml similarity index 100% rename from src/frontend/ParseFixity_intf.ml rename to src/Frontend/PARSEFIXITY.ml diff --git a/src/frontend/ParseMode_intf.ml b/src/Frontend/PARSEMODE.ml similarity index 85% rename from src/frontend/ParseMode_intf.ml rename to src/Frontend/PARSEMODE.ml index d767378..6fc2338 100644 --- a/src/frontend/ParseMode_intf.ml +++ b/src/Frontend/PARSEMODE.ml @@ -7,7 +7,7 @@ open! Parsing module type PARSE_MODE = sig (*! structure Parsing : PARSING !*) - module ExtModes : ReconMode_intf.EXTMODES + module ExtModes : RECONMODE.EXTMODES val parseMode' : ExtModes.modedec list Parsing.parser end diff --git a/src/frontend/ParseModule_intf.ml b/src/Frontend/PARSEMODULE.ml similarity index 91% rename from src/frontend/ParseModule_intf.ml rename to src/Frontend/PARSEMODULE.ml index 7850d43..8b0ab86 100644 --- a/src/frontend/ParseModule_intf.ml +++ b/src/Frontend/PARSEMODULE.ml @@ -7,7 +7,7 @@ open! Parsing module type PARSE_MODULE = sig (*! structure Parsing : PARSING !*) - module ModExtSyn : ReconModule_intf.MODEXTSYN + module ModExtSyn : RECONMODULE.MODEXTSYN (* val parseSigExp' : ModExtSyn.sigexp Parsing.recparser *) val parseSigDef' : ModExtSyn.sigdef Parsing.recparser diff --git a/src/frontend/ParseQuery_intf.ml b/src/Frontend/PARSEQUERY.ml similarity index 87% rename from src/frontend/ParseQuery_intf.ml rename to src/Frontend/PARSEQUERY.ml index ee44fd9..136a11b 100644 --- a/src/frontend/ParseQuery_intf.ml +++ b/src/Frontend/PARSEQUERY.ml @@ -7,7 +7,7 @@ open! Parsing module type PARSE_QUERY = sig (*! structure Parsing : PARSING !*) - module ExtQuery : ReconQuery_intf.EXTQUERY + module ExtQuery : RECONQUERY.EXTQUERY val parseQuery' : ExtQuery.query Parsing.parser val parseSolve' : (ExtQuery.define list * ExtQuery.solve) Parsing.parser diff --git a/src/frontend/Parser_intf.ml b/src/Frontend/PARSER.ml similarity index 88% rename from src/frontend/Parser_intf.ml rename to src/Frontend/PARSER.ml index 0e4f284..fcb9217 100644 --- a/src/frontend/Parser_intf.ml +++ b/src/Frontend/PARSER.ml @@ -7,13 +7,13 @@ open! Basis module type PARSER = sig (*! structure Parsing : PARSING !*) module Stream : STREAM - module ExtSyn : ReconTerm_intf.EXTSYN + module ExtSyn : RECONTERM.EXTSYN module Names : NAMES - module ExtConDec : ReconCondec_intf.EXTCONDEC - module ExtQuery : ReconQuery_intf.EXTQUERY - module ExtModes : ReconMode_intf.EXTMODES - module ThmExtSyn : ReconThm_intf.THMEXTSYN - module ModExtSyn : ReconModule_intf.MODEXTSYN + module ExtConDec : RECONCONDEC.EXTCONDEC + module ExtQuery : RECONQUERY.EXTQUERY + module ExtModes : RECONMODE.EXTMODES + module ThmExtSyn : RECONTHM.THMEXTSYN + module ModExtSyn : RECONMODULE.MODEXTSYN type fileParseResult = | ConDec of ExtConDec.condec diff --git a/src/frontend/ParseTerm_intf.ml b/src/Frontend/PARSETERM.ml similarity index 95% rename from src/frontend/ParseTerm_intf.ml rename to src/Frontend/PARSETERM.ml index faf3207..c175566 100644 --- a/src/frontend/ParseTerm_intf.ml +++ b/src/Frontend/PARSETERM.ml @@ -7,7 +7,7 @@ open! Parsing module type PARSE_TERM = sig (*! structure Parsing : PARSING !*) - module ExtSyn : ReconTerm_intf.EXTSYN + module ExtSyn : RECONTERM.EXTSYN val parseQualId' : (string list * Parsing.lexResult) Parsing.parser val parseQualIds' : (string list * string) list Parsing.parser diff --git a/src/frontend/ParseThm_intf.ml b/src/Frontend/PARSETHM.ml similarity index 95% rename from src/frontend/ParseThm_intf.ml rename to src/Frontend/PARSETHM.ml index b5a5721..935c7bb 100644 --- a/src/frontend/ParseThm_intf.ml +++ b/src/Frontend/PARSETHM.ml @@ -7,7 +7,7 @@ open! Parsing module type PARSE_THM = sig (*! structure Parsing : PARSING !*) - module ThmExtSyn : ReconThm_intf.THMEXTSYN + module ThmExtSyn : RECONTHM.THMEXTSYN val parseTotal' : ThmExtSyn.tdecl Parsing.parser diff --git a/src/frontend/Parsing_intf.ml b/src/Frontend/PARSING.ml similarity index 100% rename from src/frontend/Parsing_intf.ml rename to src/Frontend/PARSING.ml diff --git a/src/frontend/ParseCondec.ml b/src/Frontend/ParseCondec.ml similarity index 96% rename from src/frontend/ParseCondec.ml rename to src/Frontend/ParseCondec.ml index 6b0dea2..39cb739 100644 --- a/src/frontend/ParseCondec.ml +++ b/src/Frontend/ParseCondec.ml @@ -4,7 +4,7 @@ open! Parsing (* Parsing Signature Entries *) (* Author: Frank Pfenning *) -include ParseCondec_intf +include PARSECONDEC (* signature PARSE_CONDEC *) (* # 1 "src/frontend/ParseCondec.fun.ml" *) @@ -15,10 +15,8 @@ open! Basis (* Author: Frank Pfenning *) module ParseConDec (ParseConDec__0 : sig (*! structure Parsing' : PARSING !*) - module ExtConDec' : ReconCondec_intf.EXTCONDEC - - module ParseTerm : - ParseTerm_intf.PARSE_TERM with module ExtSyn = ExtConDec'.ExtSyn + module ExtConDec' : RECONCONDEC.EXTCONDEC + module ParseTerm : PARSETERM.PARSE_TERM with module ExtSyn = ExtConDec'.ExtSyn end) : PARSE_CONDEC with module ExtConDec = ParseConDec__0.ExtConDec' = struct (*! structure Parsing = Parsing' !*) module ExtConDec = ParseConDec__0.ExtConDec' diff --git a/src/Frontend/ParseCondec.mli b/src/Frontend/ParseCondec.mli new file mode 100644 index 0000000..5ddd5de --- /dev/null +++ b/src/Frontend/ParseCondec.mli @@ -0,0 +1,7 @@ +include module type of PARSECONDEC + +module ParseConDec (ParseConDec__0 : sig + (*! structure Parsing' : PARSING !*) + module ExtConDec' : RECONCONDEC.EXTCONDEC + module ParseTerm : PARSETERM.PARSE_TERM with module ExtSyn = ExtConDec'.ExtSyn +end) : PARSE_CONDEC with module ExtConDec = ParseConDec__0.ExtConDec' diff --git a/src/frontend/ParseFixity.ml b/src/Frontend/ParseFixity.ml similarity index 99% rename from src/frontend/ParseFixity.ml rename to src/Frontend/ParseFixity.ml index 8d9d0eb..6e39dd3 100644 --- a/src/frontend/ParseFixity.ml +++ b/src/Frontend/ParseFixity.ml @@ -4,7 +4,7 @@ open! Parsing (* Parsing Fixity Declarations *) (* Author: Frank Pfenning *) -include ParseFixity_intf +include PARSEFIXITY (* signature PARSE_FIXITY *) (* # 1 "src/frontend/ParseFixity.fun.ml" *) diff --git a/src/frontend/ParseFixity.mli b/src/Frontend/ParseFixity.mli similarity index 76% rename from src/frontend/ParseFixity.mli rename to src/Frontend/ParseFixity.mli index 016ae19..b4f176f 100644 --- a/src/frontend/ParseFixity.mli +++ b/src/Frontend/ParseFixity.mli @@ -1,4 +1,4 @@ -include module type of ParseFixity_intf +include module type of PARSEFIXITY module ParseFixity (ParseFixity__0 : sig module Names' : NAMES diff --git a/src/frontend/ParseMode.ml b/src/Frontend/ParseMode.ml similarity index 97% rename from src/frontend/ParseMode.ml rename to src/Frontend/ParseMode.ml index 4694610..0c363ec 100644 --- a/src/frontend/ParseMode.ml +++ b/src/Frontend/ParseMode.ml @@ -4,7 +4,7 @@ open! Parsing (* Parsing Mode Declarations *) (* Author: Carsten Schuermann *) -include ParseMode_intf +include PARSEMODE (* signature PARSE_MODE *) (* # 1 "src/frontend/ParseMode.fun.ml" *) @@ -17,12 +17,11 @@ module ParseMode (ParseMode__0 : sig (*! structure Paths : PATHS !*) (*! structure Parsing' : PARSING !*) (*! sharing Parsing'.Lexer.Paths = Paths !*) - module ExtModes' : ReconMode_intf.EXTMODES + module ExtModes' : RECONMODE.EXTMODES (*! sharing ExtModes'.Paths = Paths !*) (*! sharing ExtModes'.ExtSyn.Paths = Paths !*) - module ParseTerm : - ParseTerm_intf.PARSE_TERM with module ExtSyn = ExtModes'.ExtSyn + module ParseTerm : PARSETERM.PARSE_TERM with module ExtSyn = ExtModes'.ExtSyn end) : PARSE_MODE with module ExtModes = ParseMode__0.ExtModes' = struct (*! structure Parsing = Parsing' !*) module ExtModes = ParseMode__0.ExtModes' diff --git a/src/frontend/ParseMode.mli b/src/Frontend/ParseMode.mli similarity index 64% rename from src/frontend/ParseMode.mli rename to src/Frontend/ParseMode.mli index b0bc8c3..32651f7 100644 --- a/src/frontend/ParseMode.mli +++ b/src/Frontend/ParseMode.mli @@ -1,13 +1,12 @@ -include module type of ParseMode_intf +include module type of PARSEMODE module ParseMode (ParseMode__0 : sig (*! structure Paths : PATHS !*) (*! structure Parsing' : PARSING !*) (*! sharing Parsing'.Lexer.Paths = Paths !*) - module ExtModes' : ReconMode_intf.EXTMODES + module ExtModes' : RECONMODE.EXTMODES (*! sharing ExtModes'.Paths = Paths !*) (*! sharing ExtModes'.ExtSyn.Paths = Paths !*) - module ParseTerm : - ParseTerm_intf.PARSE_TERM with module ExtSyn = ExtModes'.ExtSyn + module ParseTerm : PARSETERM.PARSE_TERM with module ExtSyn = ExtModes'.ExtSyn end) : PARSE_MODE with module ExtModes = ParseMode__0.ExtModes' diff --git a/src/frontend/ParseModule.ml b/src/Frontend/ParseModule.ml similarity index 97% rename from src/frontend/ParseModule.ml rename to src/Frontend/ParseModule.ml index 1d585fb..b447673 100644 --- a/src/frontend/ParseModule.ml +++ b/src/Frontend/ParseModule.ml @@ -4,7 +4,7 @@ open! Parsing (* Parsing modules *) (* Author: Kevin Watkins *) -include ParseModule_intf +include PARSEMODULE (* # 1 "src/frontend/ParseModule.fun.ml" *) open! Parsing @@ -16,11 +16,10 @@ module ParseModule (ParseModule__0 : sig (*! structure Paths : PATHS !*) (*! structure Parsing' : PARSING !*) (*! sharing Parsing'.Lexer.Paths = Paths !*) - module ModExtSyn' : ReconModule_intf.MODEXTSYN + module ModExtSyn' : RECONMODULE.MODEXTSYN (*! sharing ModExtSyn'.Paths = Paths !*) - module ParseTerm : - ParseTerm_intf.PARSE_TERM with module ExtSyn = ModExtSyn'.ExtSyn + module ParseTerm : PARSETERM.PARSE_TERM with module ExtSyn = ModExtSyn'.ExtSyn end) : PARSE_MODULE with module ModExtSyn = ParseModule__0.ModExtSyn' = struct (*! structure Parsing = Parsing' !*) module ModExtSyn = ParseModule__0.ModExtSyn' diff --git a/src/frontend/ParseModule.mli b/src/Frontend/ParseModule.mli similarity index 64% rename from src/frontend/ParseModule.mli rename to src/Frontend/ParseModule.mli index f786751..c182025 100644 --- a/src/frontend/ParseModule.mli +++ b/src/Frontend/ParseModule.mli @@ -1,4 +1,4 @@ -include module type of ParseModule_intf +include module type of PARSEMODULE module ParseModule (ParseModule__0 : sig (* Parsing modules *) @@ -6,9 +6,8 @@ module ParseModule (ParseModule__0 : sig (*! structure Paths : PATHS !*) (*! structure Parsing' : PARSING !*) (*! sharing Parsing'.Lexer.Paths = Paths !*) - module ModExtSyn' : ReconModule_intf.MODEXTSYN + module ModExtSyn' : RECONMODULE.MODEXTSYN (*! sharing ModExtSyn'.Paths = Paths !*) - module ParseTerm : - ParseTerm_intf.PARSE_TERM with module ExtSyn = ModExtSyn'.ExtSyn + module ParseTerm : PARSETERM.PARSE_TERM with module ExtSyn = ModExtSyn'.ExtSyn end) : PARSE_MODULE with module ModExtSyn = ParseModule__0.ModExtSyn' diff --git a/src/frontend/ParseQuery.ml b/src/Frontend/ParseQuery.ml similarity index 96% rename from src/frontend/ParseQuery.ml rename to src/Frontend/ParseQuery.ml index cb1ce7b..401150d 100644 --- a/src/frontend/ParseQuery.ml +++ b/src/Frontend/ParseQuery.ml @@ -4,7 +4,7 @@ open! Parsing (* Parsing Queries *) (* Author: Frank Pfenning *) -include ParseQuery_intf +include PARSEQUERY (* signature PARSE_QUERY *) (* # 1 "src/frontend/ParseQuery.fun.ml" *) @@ -15,11 +15,10 @@ open! Basis (* Author: Frank Pfenning *) module ParseQuery (ParseQuery__0 : sig (*! structure Parsing' : PARSING !*) - module ExtQuery' : ReconQuery_intf.EXTQUERY + module ExtQuery' : RECONQUERY.EXTQUERY (*! sharing ExtQuery'.Paths = Parsing'.Lexer.Paths !*) - module ParseTerm : - ParseTerm_intf.PARSE_TERM with module ExtSyn = ExtQuery'.ExtSyn + module ParseTerm : PARSETERM.PARSE_TERM with module ExtSyn = ExtQuery'.ExtSyn end) : PARSE_QUERY with module ExtQuery = ParseQuery__0.ExtQuery' = struct (*! structure Parsing = Parsing' !*) module ExtQuery = ParseQuery__0.ExtQuery' diff --git a/src/frontend/ParseQuery.mli b/src/Frontend/ParseQuery.mli similarity index 53% rename from src/frontend/ParseQuery.mli rename to src/Frontend/ParseQuery.mli index 70dce39..6495c73 100644 --- a/src/frontend/ParseQuery.mli +++ b/src/Frontend/ParseQuery.mli @@ -1,10 +1,9 @@ -include module type of ParseQuery_intf +include module type of PARSEQUERY module ParseQuery (ParseQuery__0 : sig (*! structure Parsing' : PARSING !*) - module ExtQuery' : ReconQuery_intf.EXTQUERY + module ExtQuery' : RECONQUERY.EXTQUERY (*! sharing ExtQuery'.Paths = Parsing'.Lexer.Paths !*) - module ParseTerm : - ParseTerm_intf.PARSE_TERM with module ExtSyn = ExtQuery'.ExtSyn + module ParseTerm : PARSETERM.PARSE_TERM with module ExtSyn = ExtQuery'.ExtSyn end) : PARSE_QUERY with module ExtQuery = ParseQuery__0.ExtQuery' diff --git a/src/frontend/ParseTerm.ml b/src/Frontend/ParseTerm.ml similarity index 99% rename from src/frontend/ParseTerm.ml rename to src/Frontend/ParseTerm.ml index c20f546..ee9caff 100644 --- a/src/frontend/ParseTerm.ml +++ b/src/Frontend/ParseTerm.ml @@ -4,7 +4,7 @@ open! Parsing (* Parsing Terms and Declarations *) (* Author: Frank Pfenning *) -include ParseTerm_intf +include PARSETERM (* signature PARSE_TERM *) (* # 1 "src/frontend/ParseTerm.fun.ml" *) @@ -15,7 +15,7 @@ open! Basis (* Author: Frank Pfenning *) module ParseTerm (ParseTerm__0 : sig (*! structure Parsing' : PARSING !*) - module ExtSyn' : ReconTerm_intf.EXTSYN + module ExtSyn' : RECONTERM.EXTSYN (*! sharing Parsing'.Lexer.Paths = ExtSyn'.Paths !*) module Names : NAMES diff --git a/src/frontend/ParseTerm.mli b/src/Frontend/ParseTerm.mli similarity index 73% rename from src/frontend/ParseTerm.mli rename to src/Frontend/ParseTerm.mli index bf4a3b2..59e4504 100644 --- a/src/frontend/ParseTerm.mli +++ b/src/Frontend/ParseTerm.mli @@ -1,8 +1,8 @@ -include module type of ParseTerm_intf +include module type of PARSETERM module ParseTerm (ParseTerm__0 : sig (*! structure Parsing' : PARSING !*) - module ExtSyn' : ReconTerm_intf.EXTSYN + module ExtSyn' : RECONTERM.EXTSYN (*! sharing Parsing'.Lexer.Paths = ExtSyn'.Paths !*) module Names : NAMES diff --git a/src/frontend/ParseThm.ml b/src/Frontend/ParseThm.ml similarity index 99% rename from src/frontend/ParseThm.ml rename to src/Frontend/ParseThm.ml index 28000a7..eb6afdf 100644 --- a/src/frontend/ParseThm.ml +++ b/src/Frontend/ParseThm.ml @@ -4,7 +4,7 @@ open! Parsing (* Parsing Theorems *) (* Author: Carsten Schuermann *) -include ParseThm_intf +include PARSETHM (* signature PARSE_THM *) (* # 1 "src/frontend/ParseThm.fun.ml" *) @@ -18,12 +18,11 @@ module ParseThm (ParseThm__0 : sig (*! structure Paths : PATHS *) (*! structure Parsing' : PARSING !*) (*! sharing Parsing'.Lexer.Paths = Paths !*) - module ThmExtSyn' : ReconThm_intf.THMEXTSYN + module ThmExtSyn' : RECONTHM.THMEXTSYN (*! sharing ThmExtSyn'.Paths = Paths !*) (*! sharing ThmExtSyn'.ExtSyn.Paths = Paths !*) - module ParseTerm : - ParseTerm_intf.PARSE_TERM with module ExtSyn = ThmExtSyn'.ExtSyn + module ParseTerm : PARSETERM.PARSE_TERM with module ExtSyn = ThmExtSyn'.ExtSyn end) : PARSE_THM with module ThmExtSyn = ParseThm__0.ThmExtSyn' = struct (*! structure Parsing = Parsing' !*) module ThmExtSyn = ParseThm__0.ThmExtSyn' diff --git a/src/frontend/ParseThm.mli b/src/Frontend/ParseThm.mli similarity index 64% rename from src/frontend/ParseThm.mli rename to src/Frontend/ParseThm.mli index 3f26fd4..0f59981 100644 --- a/src/frontend/ParseThm.mli +++ b/src/Frontend/ParseThm.mli @@ -1,13 +1,12 @@ -include module type of ParseThm_intf +include module type of PARSETHM module ParseThm (ParseThm__0 : sig (*! structure Paths : PATHS *) (*! structure Parsing' : PARSING !*) (*! sharing Parsing'.Lexer.Paths = Paths !*) - module ThmExtSyn' : ReconThm_intf.THMEXTSYN + module ThmExtSyn' : RECONTHM.THMEXTSYN (*! sharing ThmExtSyn'.Paths = Paths !*) (*! sharing ThmExtSyn'.ExtSyn.Paths = Paths !*) - module ParseTerm : - ParseTerm_intf.PARSE_TERM with module ExtSyn = ThmExtSyn'.ExtSyn + module ParseTerm : PARSETERM.PARSE_TERM with module ExtSyn = ThmExtSyn'.ExtSyn end) : PARSE_THM with module ThmExtSyn = ParseThm__0.ThmExtSyn' diff --git a/src/frontend/Parser.ml b/src/Frontend/Parser.ml similarity index 96% rename from src/frontend/Parser.ml rename to src/Frontend/Parser.ml index 58a7e55..00214e7 100644 --- a/src/frontend/Parser.ml +++ b/src/Frontend/Parser.ml @@ -3,7 +3,7 @@ open! Basis (* Top-Level Parser *) (* Author: Frank Pfenning *) -include Parser_intf +include PARSER (* reads from std input *) (* signature PARSER *) @@ -19,38 +19,37 @@ module Parser (Parser__0 : sig module Stream' : STREAM (* result stream *) - module ExtSyn' : ReconTerm_intf.EXTSYN + module ExtSyn' : RECONTERM.EXTSYN (*! sharing ExtSyn'.Paths = Parsing'.Lexer.Paths !*) module Names' : NAMES - module ExtConDec' : ReconCondec_intf.EXTCONDEC - module ExtQuery' : ReconQuery_intf.EXTQUERY - module ExtModes' : ReconMode_intf.EXTMODES - module ThmExtSyn' : ReconThm_intf.THMEXTSYN - module ModExtSyn' : ReconModule_intf.MODEXTSYN + module ExtConDec' : RECONCONDEC.EXTCONDEC + module ExtQuery' : RECONQUERY.EXTQUERY + module ExtModes' : RECONMODE.EXTMODES + module ThmExtSyn' : RECONTHM.THMEXTSYN + module ModExtSyn' : RECONMODULE.MODEXTSYN module ParseConDec : - ParseCondec_intf.PARSE_CONDEC with module ExtConDec = ExtConDec' + PARSECONDEC.PARSE_CONDEC with module ExtConDec = ExtConDec' (*! sharing ParseConDec.Lexer = Parsing'.Lexer !*) - module ParseQuery : - ParseQuery_intf.PARSE_QUERY with module ExtQuery = ExtQuery' + module ParseQuery : PARSEQUERY.PARSE_QUERY with module ExtQuery = ExtQuery' (*! sharing ParseQuery.Lexer = Parsing'.Lexer !*) - module ParseFixity : ParseFixity_intf.PARSE_FIXITY with module Names = Names' + module ParseFixity : PARSEFIXITY.PARSE_FIXITY with module Names = Names' (*! sharing ParseFixity.Lexer = Parsing'.Lexer !*) - module ParseMode : ParseMode_intf.PARSE_MODE with module ExtModes = ExtModes' + module ParseMode : PARSEMODE.PARSE_MODE with module ExtModes = ExtModes' (*! sharing ParseMode.Lexer = Parsing'.Lexer !*) - module ParseThm : ParseThm_intf.PARSE_THM with module ThmExtSyn = ThmExtSyn' + module ParseThm : PARSETHM.PARSE_THM with module ThmExtSyn = ThmExtSyn' (*! sharing ParseThm.Lexer = Parsing'.Lexer !*) module ParseModule : - ParseModule_intf.PARSE_MODULE with module ModExtSyn = ModExtSyn' + PARSEMODULE.PARSE_MODULE with module ModExtSyn = ModExtSyn' (*! sharing ParseModule.Parsing = Parsing' !*) - module ParseTerm : ParseTerm_intf.PARSE_TERM with module ExtSyn = ExtSyn' + module ParseTerm : PARSETERM.PARSE_TERM with module ExtSyn = ExtSyn' end) : PARSER with module ExtQuery = Parser__0.ExtQuery' diff --git a/src/frontend/Parser.mli b/src/Frontend/Parser.mli similarity index 52% rename from src/frontend/Parser.mli rename to src/Frontend/Parser.mli index 3c2f26e..7d99d0d 100644 --- a/src/frontend/Parser.mli +++ b/src/Frontend/Parser.mli @@ -1,42 +1,41 @@ -include module type of Parser_intf +include module type of PARSER module Parser (Parser__0 : sig (*! structure Parsing' : PARSING !*) module Stream' : STREAM (* result stream *) - module ExtSyn' : ReconTerm_intf.EXTSYN + module ExtSyn' : RECONTERM.EXTSYN (*! sharing ExtSyn'.Paths = Parsing'.Lexer.Paths !*) module Names' : NAMES - module ExtConDec' : ReconCondec_intf.EXTCONDEC - module ExtQuery' : ReconQuery_intf.EXTQUERY - module ExtModes' : ReconMode_intf.EXTMODES - module ThmExtSyn' : ReconThm_intf.THMEXTSYN - module ModExtSyn' : ReconModule_intf.MODEXTSYN + module ExtConDec' : RECONCONDEC.EXTCONDEC + module ExtQuery' : RECONQUERY.EXTQUERY + module ExtModes' : RECONMODE.EXTMODES + module ThmExtSyn' : RECONTHM.THMEXTSYN + module ModExtSyn' : RECONMODULE.MODEXTSYN module ParseConDec : - ParseCondec_intf.PARSE_CONDEC with module ExtConDec = ExtConDec' + PARSECONDEC.PARSE_CONDEC with module ExtConDec = ExtConDec' (*! sharing ParseConDec.Lexer = Parsing'.Lexer !*) - module ParseQuery : - ParseQuery_intf.PARSE_QUERY with module ExtQuery = ExtQuery' + module ParseQuery : PARSEQUERY.PARSE_QUERY with module ExtQuery = ExtQuery' (*! sharing ParseQuery.Lexer = Parsing'.Lexer !*) - module ParseFixity : ParseFixity_intf.PARSE_FIXITY with module Names = Names' + module ParseFixity : PARSEFIXITY.PARSE_FIXITY with module Names = Names' (*! sharing ParseFixity.Lexer = Parsing'.Lexer !*) - module ParseMode : ParseMode_intf.PARSE_MODE with module ExtModes = ExtModes' + module ParseMode : PARSEMODE.PARSE_MODE with module ExtModes = ExtModes' (*! sharing ParseMode.Lexer = Parsing'.Lexer !*) - module ParseThm : ParseThm_intf.PARSE_THM with module ThmExtSyn = ThmExtSyn' + module ParseThm : PARSETHM.PARSE_THM with module ThmExtSyn = ThmExtSyn' (*! sharing ParseThm.Lexer = Parsing'.Lexer !*) module ParseModule : - ParseModule_intf.PARSE_MODULE with module ModExtSyn = ModExtSyn' + PARSEMODULE.PARSE_MODULE with module ModExtSyn = ModExtSyn' (*! sharing ParseModule.Parsing = Parsing' !*) - module ParseTerm : ParseTerm_intf.PARSE_TERM with module ExtSyn = ExtSyn' + module ParseTerm : PARSETERM.PARSE_TERM with module ExtSyn = ExtSyn' end) : PARSER with module ExtQuery = Parser__0.ExtQuery' diff --git a/src/frontend/Parsing.ml b/src/Frontend/Parsing.ml similarity index 98% rename from src/frontend/Parsing.ml rename to src/Frontend/Parsing.ml index 149e1e5..edc8cd8 100644 --- a/src/frontend/Parsing.ml +++ b/src/Frontend/Parsing.ml @@ -3,7 +3,7 @@ open! Basis (* General basis for parsing modules *) (* Author: Frank Pfenning *) -include Parsing_intf +include PARSING (* always raises Error *) (* signature PARSING *) diff --git a/src/frontend/Parsing.mli b/src/Frontend/Parsing.mli similarity index 83% rename from src/frontend/Parsing.mli rename to src/Frontend/Parsing.mli index 959c0ed..72e2951 100644 --- a/src/frontend/Parsing.mli +++ b/src/Frontend/Parsing.mli @@ -1,4 +1,4 @@ -include module type of Parsing_intf +include module type of PARSING module MakeParsing (Stream : STREAM) (Lexer : Lexer.LEXER) : PARSING module Parsing : PARSING diff --git a/src/frontend/ReconCondec_intf.ml b/src/Frontend/RECONCONDEC.ml similarity index 96% rename from src/frontend/ReconCondec_intf.ml rename to src/Frontend/RECONCONDEC.ml index 80465ab..aab9c94 100644 --- a/src/frontend/ReconCondec_intf.ml +++ b/src/Frontend/RECONCONDEC.ml @@ -5,7 +5,7 @@ open! Basis (* Author: Frank Pfenning *) module type EXTCONDEC = sig - module ExtSyn : ReconTerm_intf.EXTSYN + module ExtSyn : RECONTERM.EXTSYN (*! structure Paths : PATHS !*) type condec diff --git a/src/frontend/ReconMode_intf.ml b/src/Frontend/RECONMODE.ml similarity index 96% rename from src/frontend/ReconMode_intf.ml rename to src/Frontend/RECONMODE.ml index e9137a7..d70e0ef 100644 --- a/src/frontend/ReconMode_intf.ml +++ b/src/Frontend/RECONMODE.ml @@ -5,7 +5,7 @@ open! Basis (* Author: Carsten Schuermann *) module type EXTMODES = sig - module ExtSyn : ReconTerm_intf.EXTSYN + module ExtSyn : RECONTERM.EXTSYN (*! structure Paths : PATHS !*) type mode diff --git a/src/frontend/ReconModule_intf.ml b/src/Frontend/RECONMODULE.ml similarity index 97% rename from src/frontend/ReconModule_intf.ml rename to src/Frontend/RECONMODULE.ml index 4859559..6f5fd75 100644 --- a/src/frontend/ReconModule_intf.ml +++ b/src/Frontend/RECONMODULE.ml @@ -5,7 +5,7 @@ open! Basis (* Author: Kevin Watkins *) module type MODEXTSYN = sig - module ExtSyn : ReconTerm_intf.EXTSYN + module ExtSyn : RECONTERM.EXTSYN (*! structure Paths : PATHS !*) type strexp diff --git a/src/frontend/ReconQuery_intf.ml b/src/Frontend/RECONQUERY.ml similarity index 96% rename from src/frontend/ReconQuery_intf.ml rename to src/Frontend/RECONQUERY.ml index 28efabc..60af7db 100644 --- a/src/frontend/ReconQuery_intf.ml +++ b/src/Frontend/RECONQUERY.ml @@ -5,7 +5,7 @@ open! Basis (* Author: Frank Pfenning *) module type EXTQUERY = sig - module ExtSyn : ReconTerm_intf.EXTSYN + module ExtSyn : RECONTERM.EXTSYN (*! structure Paths : PATHS !*) type query diff --git a/src/frontend/ReconTerm_intf.ml b/src/Frontend/RECONTERM.ml similarity index 100% rename from src/frontend/ReconTerm_intf.ml rename to src/Frontend/RECONTERM.ml diff --git a/src/frontend/ReconThm_intf.ml b/src/Frontend/RECONTHM.ml similarity index 98% rename from src/frontend/ReconThm_intf.ml rename to src/Frontend/RECONTHM.ml index 454e863..775cc2a 100644 --- a/src/frontend/ReconThm_intf.ml +++ b/src/Frontend/RECONTHM.ml @@ -5,7 +5,7 @@ open! Basis (* Author: Carsten Schuermann *) module type THMEXTSYN = sig - module ExtSyn : ReconTerm_intf.EXTSYN + module ExtSyn : RECONTERM.EXTSYN (*! structure Paths : PATHS !*) type order diff --git a/src/frontend/ReconCondec.ml b/src/Frontend/ReconCondec.ml similarity index 99% rename from src/frontend/ReconCondec.ml rename to src/Frontend/ReconCondec.ml index 57fd496..726a295 100644 --- a/src/frontend/ReconCondec.ml +++ b/src/Frontend/ReconCondec.ml @@ -3,7 +3,7 @@ open! Basis (* External Syntax for signature entries *) (* Author: Frank Pfenning *) -include ReconCondec_intf +include RECONCONDEC (* id : tm = tm | _ : tm = tm *) (* signature EXTCONDEC *) @@ -26,7 +26,7 @@ module ReconConDec (ReconConDec__0 : sig (*! sharing Abstract.IntSyn = IntSyn' !*) (*! structure Paths' : PATHS !*) - module ReconTerm' : ReconTerm_intf.RECON_TERM + module ReconTerm' : RECONTERM.RECON_TERM (*! sharing ReconTerm'.IntSyn = IntSyn' !*) (*! sharing ReconTerm'.Paths = Paths' !*) diff --git a/src/frontend/ReconCondec.mli b/src/Frontend/ReconCondec.mli similarity index 90% rename from src/frontend/ReconCondec.mli rename to src/Frontend/ReconCondec.mli index 7222ae1..0c7801f 100644 --- a/src/frontend/ReconCondec.mli +++ b/src/Frontend/ReconCondec.mli @@ -1,4 +1,4 @@ -include module type of ReconCondec_intf +include module type of RECONCONDEC module ReconConDec (ReconConDec__0 : sig (* Reconstruct signature entries *) @@ -14,7 +14,7 @@ module ReconConDec (ReconConDec__0 : sig (*! sharing Abstract.IntSyn = IntSyn' !*) (*! structure Paths' : PATHS !*) - module ReconTerm' : ReconTerm_intf.RECON_TERM + module ReconTerm' : RECONTERM.RECON_TERM (*! sharing ReconTerm'.IntSyn = IntSyn' !*) (*! sharing ReconTerm'.Paths = Paths' !*) diff --git a/src/frontend/ReconMode.ml b/src/Frontend/ReconMode.ml similarity index 98% rename from src/frontend/ReconMode.ml rename to src/Frontend/ReconMode.ml index 1ad2ef2..f2ddf83 100644 --- a/src/frontend/ReconMode.ml +++ b/src/Frontend/ReconMode.ml @@ -3,7 +3,7 @@ open! Basis (* External Syntax of Mode Declarations *) (* Author: Carsten Schuermann *) -include ReconMode_intf +include RECONMODE (* signature EXTMODES *) (* signature RECON_MODE *) @@ -28,7 +28,7 @@ module ReconMode (ReconMode__0 : sig (*! sharing ModePrint.ModeSyn = ModeSyn' !*) module ModeDec : Modedec.MODEDEC - module ReconTerm' : ReconTerm_intf.RECON_TERM + module ReconTerm' : RECONTERM.RECON_TERM end) : RECON_MODE = struct (*! structure ModeSyn = ModeSyn' !*) module ExtSyn = ReconMode__0.ReconTerm' diff --git a/src/frontend/ReconMode.mli b/src/Frontend/ReconMode.mli similarity index 83% rename from src/frontend/ReconMode.mli rename to src/Frontend/ReconMode.mli index 380e829..3e856f5 100644 --- a/src/frontend/ReconMode.mli +++ b/src/Frontend/ReconMode.mli @@ -1,4 +1,4 @@ -include module type of ReconMode_intf +include module type of RECONMODE module ReconMode (ReconMode__0 : sig module Global : GLOBAL @@ -15,5 +15,5 @@ module ReconMode (ReconMode__0 : sig (*! sharing ModePrint.ModeSyn = ModeSyn' !*) module ModeDec : Modedec.MODEDEC - module ReconTerm' : ReconTerm_intf.RECON_TERM + module ReconTerm' : RECONTERM.RECON_TERM end) : RECON_MODE diff --git a/src/frontend/ReconModule.ml b/src/Frontend/ReconModule.ml similarity index 98% rename from src/frontend/ReconModule.ml rename to src/Frontend/ReconModule.ml index a12f7f9..e7123a3 100644 --- a/src/frontend/ReconModule.ml +++ b/src/Frontend/ReconModule.ml @@ -3,7 +3,7 @@ open! Basis (* External syntax for module expressions *) (* Author: Kevin Watkins *) -include ReconModule_intf +include RECONMODULE (* # 1 "src/frontend/ReconModule.fun.ml" *) open! Basis @@ -18,7 +18,7 @@ module ReconModule (ReconModule__0 : sig (*! sharing Names.IntSyn = IntSyn !*) (*! structure Paths' : PATHS !*) - module ReconTerm' : ReconTerm_intf.RECON_TERM + module ReconTerm' : RECONTERM.RECON_TERM (*! sharing ReconTerm'.IntSyn = IntSyn !*) (*! sharing ReconTerm'.Paths = Paths' !*) diff --git a/src/frontend/ReconModule.mli b/src/Frontend/ReconModule.mli similarity index 86% rename from src/frontend/ReconModule.mli rename to src/Frontend/ReconModule.mli index 628bab4..d2f4410 100644 --- a/src/frontend/ReconModule.mli +++ b/src/Frontend/ReconModule.mli @@ -1,4 +1,4 @@ -include module type of ReconModule_intf +include module type of RECONMODULE module ReconModule (ReconModule__0 : sig (* Elaboration for module expressions *) @@ -10,7 +10,7 @@ module ReconModule (ReconModule__0 : sig (*! sharing Names.IntSyn = IntSyn !*) (*! structure Paths' : PATHS !*) - module ReconTerm' : ReconTerm_intf.RECON_TERM + module ReconTerm' : RECONTERM.RECON_TERM (*! sharing ReconTerm'.IntSyn = IntSyn !*) (*! sharing ReconTerm'.Paths = Paths' !*) diff --git a/src/frontend/ReconQuery.ml b/src/Frontend/ReconQuery.ml similarity index 99% rename from src/frontend/ReconQuery.ml rename to src/Frontend/ReconQuery.ml index fbdd0dd..9029494 100644 --- a/src/frontend/ReconQuery.ml +++ b/src/Frontend/ReconQuery.ml @@ -3,7 +3,7 @@ open! Basis (* External Syntax for queries *) (* Author: Frank Pfenning *) -include ReconQuery_intf +include RECONQUERY (* id : tm | _ : tm *) (* signature EXTQUERY *) @@ -26,7 +26,7 @@ module ReconQuery (ReconQuery__0 : sig (*! sharing Abstract.IntSyn = IntSyn' !*) (*! structure Paths' : PATHS !*) - module ReconTerm' : ReconTerm_intf.RECON_TERM + module ReconTerm' : RECONTERM.RECON_TERM (*! sharing ReconTerm'.IntSyn = IntSyn' !*) (*! sharing ReconTerm'.Paths = Paths' !*) diff --git a/src/frontend/ReconQuery.mli b/src/Frontend/ReconQuery.mli similarity index 89% rename from src/frontend/ReconQuery.mli rename to src/Frontend/ReconQuery.mli index 8f96e6b..40d9b55 100644 --- a/src/frontend/ReconQuery.mli +++ b/src/Frontend/ReconQuery.mli @@ -1,4 +1,4 @@ -include module type of ReconQuery_intf +include module type of RECONQUERY module ReconQuery (ReconQuery__0 : sig (* Reconstruct queries *) @@ -14,7 +14,7 @@ module ReconQuery (ReconQuery__0 : sig (*! sharing Abstract.IntSyn = IntSyn' !*) (*! structure Paths' : PATHS !*) - module ReconTerm' : ReconTerm_intf.RECON_TERM + module ReconTerm' : RECONTERM.RECON_TERM (*! sharing ReconTerm'.IntSyn = IntSyn' !*) (*! sharing ReconTerm'.Paths = Paths' !*) diff --git a/src/frontend/ReconTerm.ml b/src/Frontend/ReconTerm.ml similarity index 99% rename from src/frontend/ReconTerm.ml rename to src/Frontend/ReconTerm.ml index 02db089..e578a0d 100644 --- a/src/frontend/ReconTerm.ml +++ b/src/Frontend/ReconTerm.ml @@ -7,7 +7,7 @@ open! Basis provides the interface for type reconstruction as seen by the parser *) -include ReconTerm_intf +include RECONTERM (* id | _ (type omitted) *) (* signature EXTSYN *) diff --git a/src/frontend/ReconTerm.mli b/src/Frontend/ReconTerm.mli similarity index 94% rename from src/frontend/ReconTerm.mli rename to src/Frontend/ReconTerm.mli index c4a6554..8b86b15 100644 --- a/src/frontend/ReconTerm.mli +++ b/src/Frontend/ReconTerm.mli @@ -1,4 +1,4 @@ -include module type of ReconTerm_intf +include module type of RECONTERM module ReconTerm (ReconTerm__0 : sig (*! structure IntSyn' : INTSYN !*) diff --git a/src/frontend/ReconThm.ml b/src/Frontend/ReconThm.ml similarity index 99% rename from src/frontend/ReconThm.ml rename to src/Frontend/ReconThm.ml index 0a7c6db..27615ba 100644 --- a/src/frontend/ReconThm.ml +++ b/src/Frontend/ReconThm.ml @@ -3,7 +3,7 @@ open! Basis (* External Syntax for meta theorems *) (* Author: Carsten Schuermann *) -include ReconThm_intf +include RECONTHM (* val wdecl : (decs * decs) list * callpats -> wdecl *) (* signature THMEXTSYN *) @@ -28,7 +28,7 @@ module ReconThm (ReconThm__0 : sig (*! sharing Names.IntSyn = IntSyn !*) (*! structure Paths' : PATHS !*) module ThmSyn' : Thmsyn.THMSYN with module Names = Names - module ReconTerm' : ReconTerm_intf.RECON_TERM + module ReconTerm' : RECONTERM.RECON_TERM (*! sharing ReconTerm'.IntSyn = IntSyn !*) (*! sharing ReconTerm'.Paths = Paths' !*) diff --git a/src/frontend/ReconThm.mli b/src/Frontend/ReconThm.mli similarity index 88% rename from src/frontend/ReconThm.mli rename to src/Frontend/ReconThm.mli index a8d9221..0bde1c8 100644 --- a/src/frontend/ReconThm.mli +++ b/src/Frontend/ReconThm.mli @@ -1,4 +1,4 @@ -include module type of ReconThm_intf +include module type of RECONTHM module ReconThm (ReconThm__0 : sig (* Reconstruct Termination Information *) @@ -16,7 +16,7 @@ module ReconThm (ReconThm__0 : sig (*! sharing Names.IntSyn = IntSyn !*) (*! structure Paths' : PATHS !*) module ThmSyn' : Thmsyn.THMSYN with module Names = Names - module ReconTerm' : ReconTerm_intf.RECON_TERM + module ReconTerm' : RECONTERM.RECON_TERM (*! sharing ReconTerm'.IntSyn = IntSyn !*) (*! sharing ReconTerm'.Paths = Paths' !*) diff --git a/src/frontend/Solve_intf.ml b/src/Frontend/SOLVE.ml similarity index 94% rename from src/frontend/Solve_intf.ml rename to src/Frontend/SOLVE.ml index 04ac861..b241921 100644 --- a/src/frontend/Solve_intf.ml +++ b/src/Frontend/SOLVE.ml @@ -7,7 +7,7 @@ open! Basis module type SOLVE = sig (*! structure IntSyn : INTSYN !*) (*! structure Paths : PATHS !*) - module ExtQuery : ReconQuery_intf.EXTQUERY + module ExtQuery : RECONQUERY.EXTQUERY exception AbortQuery of string diff --git a/src/frontend/Solve.ml b/src/Frontend/Solve.ml similarity index 99% rename from src/frontend/Solve.ml rename to src/Frontend/Solve.ml index 7f3c39a..b5759ac 100644 --- a/src/frontend/Solve.ml +++ b/src/Frontend/Solve.ml @@ -3,7 +3,7 @@ open! Basis (* Solve and query declarations, interactive top level *) (* Author: Frank Pfenning *) -include Solve_intf +include SOLVE (* true means normal exit *) (* signature SOLVE *) @@ -22,10 +22,10 @@ module Solve (Solve__0 : sig module Names : NAMES (*! sharing Names.IntSyn = IntSyn' !*) - module ReconQuery : ReconQuery_intf.RECON_QUERY + module ReconQuery : RECONQUERY.RECON_QUERY module Parser : - Parser_intf.PARSER + PARSER.PARSER with type ExtQuery.query = ReconQuery.query and type ExtQuery.define = ReconQuery.define and type ExtQuery.solve = ReconQuery.solve diff --git a/src/frontend/Solve.mli b/src/Frontend/Solve.mli similarity index 94% rename from src/frontend/Solve.mli rename to src/Frontend/Solve.mli index b2d723f..70bae4e 100644 --- a/src/frontend/Solve.mli +++ b/src/Frontend/Solve.mli @@ -1,4 +1,4 @@ -include module type of Solve_intf +include module type of SOLVE module Solve (Solve__0 : sig module Global : GLOBAL @@ -7,10 +7,10 @@ module Solve (Solve__0 : sig module Names : NAMES (*! sharing Names.IntSyn = IntSyn' !*) - module ReconQuery : ReconQuery_intf.RECON_QUERY + module ReconQuery : RECONQUERY.RECON_QUERY module Parser : - Parser_intf.PARSER + PARSER.PARSER with type ExtQuery.query = ReconQuery.query and type ExtQuery.define = ReconQuery.define and type ExtQuery.solve = ReconQuery.solve diff --git a/src/frontend/Twelf_intf.ml b/src/Frontend/TWELF.ml similarity index 100% rename from src/frontend/Twelf_intf.ml rename to src/Frontend/TWELF.ml diff --git a/src/frontend/Twelf_.ml b/src/Frontend/Twelf_.ml similarity index 99% rename from src/frontend/Twelf_.ml rename to src/Frontend/Twelf_.ml index 4319d8e..2fffa47 100644 --- a/src/frontend/Twelf_.ml +++ b/src/Frontend/Twelf_.ml @@ -6,7 +6,7 @@ open! Fquery open! Basis open Unknownexn -module type STELF = Twelf_intf.STELF +module type STELF = TWELF.STELF (* Front End Interface *) (* Author: Frank Pfenning *) @@ -47,17 +47,17 @@ module Stelf (Twelf__0 : sig module Abstract : ABSTRACT (*! sharing Abstract.IntSyn = IntSyn' !*) - module ReconTerm : ReconTerm_intf.RECON_TERM + module ReconTerm : RECONTERM.RECON_TERM (*! sharing ReconTerm.IntSyn = IntSyn' !*) (*! sharing ReconTerm.Paths = Paths !*) (* sharing type ReconTerm.Paths.occConDec = Origins.Paths.occConDec *) module ReconConDec : - ReconCondec_intf.RECON_CONDEC with type condec = Parser.ExtConDec.condec + RECONCONDEC.RECON_CONDEC with type condec = Parser.ExtConDec.condec (*! sharing ReconConDec.IntSyn = IntSyn' !*) (*! sharing ReconConDec.Paths = Paths !*) - module ReconQuery : ReconQuery_intf.RECON_QUERY + module ReconQuery : RECONQUERY.RECON_QUERY module ModeTable : Modetable.MODETABLE (*! sharing ModeSyn.IntSyn = IntSyn' !*) @@ -67,7 +67,7 @@ module Stelf (Twelf__0 : sig (*! sharing ModeCheck.ModeSyn = ModeSyn !*) (*! sharing ModeCheck.Paths = Paths !*) module ReconMode : - ReconMode_intf.RECON_MODE with type modedec = Parser.ExtModes.modedec + RECONMODE.RECON_MODE with type modedec = Parser.ExtModes.modedec (*! sharing ReconMode.ModeSyn = ModeSyn !*) (*! sharing ReconMode.Paths = Paths !*) @@ -99,7 +99,7 @@ module Stelf (Twelf__0 : sig (*! sharing TomegaCoverage.IntSyn = IntSyn' !*) (*! sharing TomegaCoverage.Tomega = Tomega !*) - module TomegaTypeCheck : TomegaTypecheck_intf.TOMEGATYPECHECK + module TomegaTypeCheck : TOMEGATYPECHECK.TOMEGATYPECHECK (*! sharing TomegaTypeCheck.IntSyn = IntSyn' !*) (*! sharing TomegaTypeCheck.Tomega = Tomega !*) @@ -147,7 +147,7 @@ module Stelf (Twelf__0 : sig (*! sharing Thm.Paths = Paths !*) module ReconThm : - ReconThm_intf.RECON_THM + RECONTHM.RECON_THM with module ThmSyn = ThmSyn and type tdecl = Parser.ThmExtSyn.tdecl and type rdecl = Parser.ThmExtSyn.rdecl @@ -180,14 +180,14 @@ module Stelf (Twelf__0 : sig (*! sharing ModSyn.IntSyn = IntSyn' !*) (*! sharing ModSyn.Paths = Paths !*) module ReconModule : - ReconModule_intf.RECON_MODULE + RECONMODULE.RECON_MODULE with module ModSyn = ModSyn and type sigdef = Parser.ModExtSyn.sigdef and type structdec = Parser.ModExtSyn.structdec and type sigexp = Parser.ModExtSyn.sigexp and type strexp = Parser.ModExtSyn.strexp - module MetaGlobal : MetaGlobal_intf.METAGLOBAL + module MetaGlobal : METAGLOBAL.METAGLOBAL (*! structure FunSyn : FUNSYN !*) (*! sharing FunSyn.IntSyn = IntSyn' !*) @@ -197,14 +197,14 @@ module Stelf (Twelf__0 : sig module Prover : PROVER (*! sharing Prover.IntSyn = IntSyn' !*) - module ClausePrint : ClausePrint_intf.CLAUSEPRINT + module ClausePrint : CLAUSEPRINT.CLAUSEPRINT (*! sharing ClausePrint.IntSyn = IntSyn' !*) - module Trace : Trace_intf.TRACE + module Trace : TRACE.TRACE module PrintTeX : PRINT (*! sharing PrintTeX.IntSyn = IntSyn' !*) - module ClausePrintTeX : ClausePrint_intf.CLAUSEPRINT + module ClausePrintTeX : CLAUSEPRINT.CLAUSEPRINT (*! sharing ClausePrintTeX.IntSyn = IntSyn' !*) module CsManager : CsManager.CS_MANAGER @@ -216,7 +216,7 @@ module Stelf (Twelf__0 : sig (* module Compat : COMPAT *) module UnknownExn : UNKNOWN_EXN module Msg : MSG -end) : Twelf_intf.STELF = struct +end) : TWELF.STELF = struct open Twelf__0 module Names = Names module Parser = Parser diff --git a/src/frontend/Twelf_.mli b/src/Frontend/Twelf_.mli similarity index 85% rename from src/frontend/Twelf_.mli rename to src/Frontend/Twelf_.mli index c1142a9..2fd491c 100644 --- a/src/frontend/Twelf_.mli +++ b/src/Frontend/Twelf_.mli @@ -1,4 +1,4 @@ -include module type of Twelf_intf +include module type of TWELF module Stelf (Twelf__0 : sig module Global : GLOBAL @@ -21,7 +21,7 @@ module Stelf (Twelf__0 : sig (*! sharing Lexer.Paths = Paths !*) (*! structure Parsing : PARSING !*) (*! sharing Lexer = Lexer !*) - module Parser : Parser_intf.PARSER with module Names = Names + module Parser : PARSER.PARSER with module Names = Names (*! sharing Parser.ExtSyn.Paths = Paths !*) module TypeCheck : TYPECHECK @@ -35,17 +35,17 @@ module Stelf (Twelf__0 : sig module Abstract : ABSTRACT (*! sharing Abstract.IntSyn = IntSyn' !*) - module ReconTerm : ReconTerm_intf.RECON_TERM + module ReconTerm : RECONTERM.RECON_TERM (*! sharing ReconTerm.IntSyn = IntSyn' !*) (*! sharing ReconTerm.Paths = Paths !*) (* sharing type ReconTerm.Paths.occConDec = Origins.Paths.occConDec *) module ReconConDec : - ReconCondec_intf.RECON_CONDEC with type condec = Parser.ExtConDec.condec + RECONCONDEC.RECON_CONDEC with type condec = Parser.ExtConDec.condec (*! sharing ReconConDec.IntSyn = IntSyn' !*) (*! sharing ReconConDec.Paths = Paths !*) - module ReconQuery : ReconQuery_intf.RECON_QUERY + module ReconQuery : RECONQUERY.RECON_QUERY module ModeTable : Modetable.MODETABLE (*! sharing ModeSyn.IntSyn = IntSyn' !*) @@ -55,7 +55,7 @@ module Stelf (Twelf__0 : sig (*! sharing ModeCheck.ModeSyn = ModeSyn !*) (*! sharing ModeCheck.Paths = Paths !*) module ReconMode : - ReconMode_intf.RECON_MODE with type modedec = Parser.ExtModes.modedec + RECONMODE.RECON_MODE with type modedec = Parser.ExtModes.modedec (*! sharing ReconMode.ModeSyn = ModeSyn !*) (*! sharing ReconMode.Paths = Paths !*) @@ -87,7 +87,7 @@ module Stelf (Twelf__0 : sig (*! sharing TomegaCoverage.IntSyn = IntSyn' !*) (*! sharing TomegaCoverage.Tomega = Tomega !*) - module TomegaTypeCheck : TomegaTypecheck_intf.TOMEGATYPECHECK + module TomegaTypeCheck : TOMEGATYPECHECK.TOMEGATYPECHECK (*! sharing TomegaTypeCheck.IntSyn = IntSyn' !*) (*! sharing TomegaTypeCheck.Tomega = Tomega !*) @@ -121,10 +121,10 @@ module Stelf (Twelf__0 : sig (*! sharing Tabled.IntSyn = IntSyn' !*) (*! sharing Tabled.CompSyn = CompSyn' !*) - module Solve : Solve_intf.SOLVE with module ExtQuery = Parser.ExtQuery + module Solve : SOLVE.SOLVE with module ExtQuery = Parser.ExtQuery (*! sharing Solve.IntSyn = IntSyn' !*) - module Fquery : Fquery_intf.FQUERY with module ExtQuery = Parser.ExtQuery + module Fquery : FQUERY.FQUERY with module ExtQuery = Parser.ExtQuery (*! sharing Fquery.IntSyn = IntSyn' !*) (*! sharing Solve.Paths = Paths !*) @@ -135,7 +135,7 @@ module Stelf (Twelf__0 : sig (*! sharing Thm.Paths = Paths !*) module ReconThm : - ReconThm_intf.RECON_THM + RECONTHM.RECON_THM with module ThmSyn = ThmSyn and type tdecl = Parser.ThmExtSyn.tdecl and type rdecl = Parser.ThmExtSyn.rdecl @@ -168,14 +168,14 @@ module Stelf (Twelf__0 : sig (*! sharing ModSyn.IntSyn = IntSyn' !*) (*! sharing ModSyn.Paths = Paths !*) module ReconModule : - ReconModule_intf.RECON_MODULE + RECONMODULE.RECON_MODULE with module ModSyn = ModSyn and type sigdef = Parser.ModExtSyn.sigdef and type structdec = Parser.ModExtSyn.structdec and type sigexp = Parser.ModExtSyn.sigexp and type strexp = Parser.ModExtSyn.strexp - module MetaGlobal : MetaGlobal_intf.METAGLOBAL + module MetaGlobal : METAGLOBAL.METAGLOBAL (*! structure FunSyn : FUNSYN !*) (*! sharing FunSyn.IntSyn = IntSyn' !*) @@ -185,23 +185,23 @@ module Stelf (Twelf__0 : sig module Prover : PROVER (*! sharing Prover.IntSyn = IntSyn' !*) - module ClausePrint : ClausePrint_intf.CLAUSEPRINT + module ClausePrint : CLAUSEPRINT.CLAUSEPRINT (*! sharing ClausePrint.IntSyn = IntSyn' !*) - module Trace : Trace_intf.TRACE + module Trace : TRACE.TRACE module PrintTeX : PRINT (*! sharing PrintTeX.IntSyn = IntSyn' !*) - module ClausePrintTeX : ClausePrint_intf.CLAUSEPRINT + module ClausePrintTeX : CLAUSEPRINT.CLAUSEPRINT (*! sharing ClausePrintTeX.IntSyn = IntSyn' !*) module CsManager : CsManager.CS_MANAGER (*! sharing CsManager.IntSyn = IntSyn' !*) (*! sharing CsManager.ModeSyn = ModeSyn !*) - module CSInstaller : Solvers_intf.CS_INSTALLER + module CSInstaller : SOLVERS.CS_INSTALLER (* module Compat : COMPAT *) - module UnknownExn : Unknownexn_intf.UNKNOWN_EXN - module Msg : Msg_intf.MSG -end) : Twelf_intf.STELF + module UnknownExn : UNKNOWNEXN.UNKNOWN_EXN + module Msg : MSG.MSG +end) : TWELF.STELF diff --git a/src/frontend/Unknownexn_intf.ml b/src/Frontend/UNKNOWNEXN.ml similarity index 100% rename from src/frontend/Unknownexn_intf.ml rename to src/Frontend/UNKNOWNEXN.ml diff --git a/src/frontend/Unknownexn.ml b/src/Frontend/Unknownexn.ml similarity index 96% rename from src/frontend/Unknownexn.ml rename to src/Frontend/Unknownexn.ml index 95413dc..6a53f66 100644 --- a/src/frontend/Unknownexn.ml +++ b/src/Frontend/Unknownexn.ml @@ -1,6 +1,6 @@ (* # 1 "src/frontend/Unknownexn.sig.ml" *) open! Basis -include Unknownexn_intf +include UNKNOWNEXN (* # 1 "src/frontend/Unknownexn.fun.ml" *) open! Basis diff --git a/src/frontend/Unknownexn.mli b/src/Frontend/Unknownexn.mli similarity index 71% rename from src/frontend/Unknownexn.mli rename to src/Frontend/Unknownexn.mli index d6cc042..608b262 100644 --- a/src/frontend/Unknownexn.mli +++ b/src/Frontend/Unknownexn.mli @@ -1,4 +1,4 @@ -include module type of Unknownexn_intf +include module type of UNKNOWNEXN module MakeUnknownExn (UnknownExn : sig val exnHistory : exn -> string list diff --git a/src/frontend/UnknownexnSmlnj.ml b/src/Frontend/UnknownexnSmlnj.ml similarity index 100% rename from src/frontend/UnknownexnSmlnj.ml rename to src/Frontend/UnknownexnSmlnj.ml diff --git a/src/frontend/UnknownexnSmlnj.mli b/src/Frontend/UnknownexnSmlnj.mli similarity index 100% rename from src/frontend/UnknownexnSmlnj.mli rename to src/Frontend/UnknownexnSmlnj.mli diff --git a/src/frontend/Version.ml b/src/Frontend/Version.ml similarity index 100% rename from src/frontend/Version.ml rename to src/Frontend/Version.ml diff --git a/src/frontend/Version.mli b/src/Frontend/Version.mli similarity index 100% rename from src/frontend/Version.mli rename to src/Frontend/Version.mli diff --git a/src/frontend/WALK b/src/Frontend/WALK similarity index 100% rename from src/frontend/WALK rename to src/Frontend/WALK diff --git a/src/frontend/dune b/src/Frontend/dune similarity index 86% rename from src/frontend/dune rename to src/Frontend/dune index 1115517..40014ca 100644 --- a/src/frontend/dune +++ b/src/Frontend/dune @@ -28,26 +28,26 @@ Version Twelf_ Frontend_ - Fquery_intf - Lexer_intf - ParseCondec_intf - ParseFixity_intf - ParseMode_intf - ParseModule_intf - ParseQuery_intf - ParseTerm_intf - ParseThm_intf - Parser_intf - Parsing_intf - ReconCondec_intf - ReconMode_intf - ReconModule_intf - ReconQuery_intf - ReconTerm_intf - ReconThm_intf - Twelf_intf - Solve_intf - Unknownexn_intf) + FQUERY + LEXER + PARSECONDEC + PARSEFIXITY + PARSEMODE + PARSEMODULE + PARSEQUERY + PARSETERM + PARSETHM + PARSER + PARSING + RECONCONDEC + RECONMODE + RECONMODULE + RECONQUERY + RECONTERM + RECONTHM + TWELF + SOLVE + UNKNOWNEXN) (libraries display timing diff --git a/src/frontend/twelf_intf.ml~ b/src/Frontend/twelf_intf.ml~ similarity index 100% rename from src/frontend/twelf_intf.ml~ rename to src/Frontend/twelf_intf.ml~ diff --git a/src/Fronts/Modern/Cmd.ml b/src/Fronts/Modern/Cmd.ml index 3b3e0d3..3164c4a 100644 --- a/src/Fronts/Modern/Cmd.ml +++ b/src/Fronts/Modern/Cmd.ml @@ -18,18 +18,41 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct (* Defer a thunk-parser to prevent infinite recursion at construction time. Used for %module and %eval which recursively embed cmd lists. *) let defer p = return () >>= fun () -> p () - let parse_order () : Cst.View.Thm.Order.t t = fix (fun self -> begin choice ~failure_msg: "order" [ - (let+ ids = Modern.parse_id_list () in Cst.View.(Thm.Order.(review @@ Varg (Cst.View.Loc.(review Ghost), ids)))); - inside "[" "]" (commit *> let+ orders = many1 (self <* commit) in Cst.View.(Thm.Order.(review @@ Simul (Cst.View.Loc.(review Ghost), orders)))); - inside "{" "}" (commit *> let+ orders = many1 (self <* commit) in Cst.View.(Thm.Order.(review @@ Lex (Cst.View.Loc.(review Ghost), orders)))) - ] end) - let order_list () : Cst.View.Thm.Order.t list t = inside "(" ")" (many1 (parse_order ())) <|> (let+ x = parse_order () in [x]) + + let parse_order () : Cst.View.Thm.Order.t t = + fix (fun self -> + begin + choice ~failure_msg:"order" + [ + (let+ ids = Modern.parse_id_list () in + Cst.View.( + Thm.Order.(review @@ Varg (Cst.View.Loc.(review Ghost), ids)))); + inside "[" "]" + (commit + *> let+ orders = many1 (self <* commit) in + Cst.View.( + Thm.Order.( + review @@ Simul (Cst.View.Loc.(review Ghost), orders)))); + inside "{" "}" + (commit + *> let+ orders = many1 (self <* commit) in + Cst.View.( + Thm.Order.( + review @@ Lex (Cst.View.Loc.(review Ghost), orders)))); + ] + end) + + let order_list () : Cst.View.Thm.Order.t list t = + inside "(" ")" (many1 (parse_order ())) + <|> let+ x = parse_order () in + [ x ] + let rec parse_cmd_list () : Cst.cmd list t = keyword "{" *> commit *> skip_outer *> many (defer parse1 <* skip_outer) - <* keyword "}" *> commit - - and parse1 () : Cst.cmd t = - choice ~failure_msg:"command" + <* keyword "}" *> commit + + and parse1 () : Cst.cmd t = + choice ~failure_msg:"command" [ begin whitespace *> (keyword "." *> commit *> return (Cst.Cmd.stop ())) @@ -65,13 +88,11 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct "unique" end; begin - (keyword "module" *> commit + keyword "scope" *> commit *> let* id = Modern.parse_var () in - let* params = Modern.parse_params () in - let+ cmds = parse_cmd_list () in - Cst.Cmd.module_cmd id params cmds) - "module" + let+ cmds = parse1 () in + Cst.View.Cmd.(review @@ Scope (Cst.View.Loc.(review Ghost), id, cmds)) end; begin (keyword "mode" *> commit @@ -174,18 +195,17 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct begin (keyword "use" *> commit *> - let* id1 = Modern.parse_var () in - let* id2 = Modern.parse_var () in - let+ iparams = inside "(" ")" (many (Modern.parse_var ())) in - Cst.Cmd.use id1 id2 iparams) + let* id1 = Modern.parse_id_list () in + let+ iparams = inside "(" ")" (many (Modern.parse_expr ())) in + Cst.View.Cmd.( + review @@ Use (Cst.View.Loc.(review Ghost), id1, iparams))) "use" end; begin (keyword "open" *> commit *> - let* id = Modern.parse_var () in - let+ ids = Modern.parse_id_list () in - Cst.Cmd.open_cmd id ids) + let+ id = Modern.parse_id_list () in + Cst.View.Cmd.(review @@ Open (Cst.View.Loc.(review Ghost), id))) "open" end; begin @@ -250,7 +270,8 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct *> let* order = order_list () in let+ body = many1 (Modern.parse_expr1 ()) in - Cst.View.Cmd.(review (Total (Cst.View.Loc.(review Ghost), order, body)))) + Cst.View.Cmd.( + review (Total (Cst.View.Loc.(review Ghost), order, body)))) "total" end; begin @@ -258,8 +279,9 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct *> let* order = order_list () in let+ body = many1 (Modern.parse_expr1 ()) in - Cst.View.Cmd.(review (Terminates (Cst.View.Loc.(review Ghost), order, body)))) - "terminates" + Cst.View.Cmd.( + review (Terminates (Cst.View.Loc.(review Ghost), order, body)))) + "terminates" end; begin (keyword "covers" *> commit @@ -275,14 +297,15 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct Cst.Cmd.name id) "name" end; - begin - (keyword "reduces" *> commit + begin + (keyword "reduces" *> commit *> let* rel = Modern.parse_reduces_rel () in let+ body = many1 (Modern.parse_expr1 ()) in - Cst.View.Cmd.(review (Reduces (Cst.View.Loc.(review Ghost), rel, body)))) + Cst.View.Cmd.( + review (Reduces (Cst.View.Loc.(review Ghost), rel, body)))) "reduces" - end + end; ] let parse () : Cst.cmd list t = skip_outer *> many (parse1 () <* skip_outer) diff --git a/src/Fronts/Modern/MODERN.ml b/src/Fronts/Modern/MODERN.ml index 108fa53..5abe2b3 100644 --- a/src/Fronts/Modern/MODERN.ml +++ b/src/Fronts/Modern/MODERN.ml @@ -1,7 +1,7 @@ module type MODERN = sig - module Paths : Paths.Paths_intf.PATHS + module Paths : Paths.PATHS.PATHS module Cst : Cst.CST - module Names : Names.Names_intf.NAMES + module Names : Names.NAMES.NAMES module N := Names module FX := N.Fixity module Parser : Parsing.PARSER.PARSER diff --git a/src/Fronts/Modern/Modern.ml b/src/Fronts/Modern/Modern.ml index 9958d31..16a73d4 100644 --- a/src/Fronts/Modern/Modern.ml +++ b/src/Fronts/Modern/Modern.ml @@ -1,7 +1,7 @@ module Make_Modern - (Paths : Paths.Paths_intf.PATHS) + (Paths : Paths.PATHS.PATHS) (Cst : Cst.CST with module Paths = Paths) - (Names : Names.Names_intf.NAMES) + (Names : Names.NAMES.NAMES) (Parser : Parsing.PARSER.PARSER) : MODERN.MODERN with module Names = Names @@ -9,30 +9,40 @@ module Make_Modern and module Paths = Paths = struct module Paths = Paths module Cst = Cst - module Names = Names + module Names = Names module N = Names module Parser = Parser exception ParseError of string + open Parser - open Parser let currently_uppercase : string list ref = ref [] - let with_uppercase : string list -> (unit -> 'a t) -> 'a t = fun names p -> + + let with_uppercase : string list -> (unit -> 'a t) -> 'a t = + fun names p -> let old = !currently_uppercase in currently_uppercase := old @ names; let* res = p () in let+ () = return @@ (currently_uppercase := old) in res + type 'a t = 'a Parser.t - let rec break (s : string list) : string list * string = + + let rec break (s : string list) : string list * string = match s with | [] -> ([], "") - | x :: xs -> let (y, z) = break xs in (x :: y, z) + | x :: xs -> + let y, z = break xs in + (x :: y, z) + let mk_loc : int -> int -> Cst.loc = fun x y -> Cst.mk_loc x y - let whitespace' () : unit t = - whitespace <|> (forget @@ token "% " *> take_till (fun c -> c = '\n') ) + + let whitespace' () : unit t = + whitespace <|> forget @@ (token "% " *> take_till (fun c -> c = '\n')) + let ghost' = Cst.View.Loc.(review Ghost) + let rec combine_fc (r1 : Paths.region) (r2 : Paths.region) : Paths.region = assert false @@ -40,10 +50,12 @@ module Make_Modern let local_fixity : (string, FX.fixity) Hashtbl.t = Hashtbl.create 16 - let register_local_fixity (fix : Cst.fixity) (n : int) (ids : string list) : unit = + let register_local_fixity (fix : Cst.fixity) (n : int) (ids : string list) : + unit = let prec = FX.Strength n in let open Cst.View.Fixity in - let fx = match !> fix with + let fx = + match !>fix with | Left _ -> FX.Infix (prec, FX.Left) | Right _ -> FX.Infix (prec, FX.Right) | Prefix _ -> FX.Prefix prec @@ -55,51 +67,54 @@ module Make_Modern type operator = | Atom of Cst.Term.t - | Infix_ of (FX.precedence * FX.associativity) * (Cst.Term.t * Cst.Term.t -> Cst.Term.t) + | Infix_ of + (FX.precedence * FX.associativity) + * (Cst.Term.t * Cst.Term.t -> Cst.Term.t) | Prefix_ of FX.precedence * (Cst.Term.t -> Cst.Term.t) | Postfix_ of FX.precedence * (Cst.Term.t -> Cst.Term.t) let jux_op = - Infix_ ((FX.inc FX.maxPrec, FX.Left), - fun (f, x) -> Cst.Term.app f [x]) + Infix_ ((FX.inc FX.maxPrec, FX.Left), fun (f, x) -> Cst.Term.app f [ x ]) let infix_op (infixity, tm) = - Infix_ (infixity, fun (tm1, tm2) -> Cst.Term.app (Cst.Term.app tm [tm1]) [tm2]) + Infix_ + ( infixity, + fun (tm1, tm2) -> Cst.Term.app (Cst.Term.app tm [ tm1 ]) [ tm2 ] ) - let prefix_op (prec, tm) = - Prefix_ (prec, fun tm1 -> Cst.Term.app tm [tm1]) - - let postfix_op (prec, tm) = - Postfix_ (prec, fun tm1 -> Cst.Term.app tm [tm1]) + let prefix_op (prec, tm) = Prefix_ (prec, fun tm1 -> Cst.Term.app tm [ tm1 ]) + let postfix_op (prec, tm) = Postfix_ (prec, fun tm1 -> Cst.Term.app tm [ tm1 ]) let classify (tm : Cst.Term.t) : operator = let open Cst.View.Term in - match !> tm with + match !>tm with | Lowercase (_, (ns, name)) | Uppercase (_, (ns, name)) | Qualified (_, (ns, name)) -> - let fixity = match Hashtbl.find_opt local_fixity name with - | Some fx -> fx - | None -> Names.fixityLookup (Names.Qid (ns, name)) - in - begin match fixity with - | FX.Nonfix -> Atom tm - | FX.Infix (prec, assoc) -> infix_op ((prec, assoc), tm) - | FX.Prefix prec -> prefix_op (prec, tm) - | FX.Postfix prec -> postfix_op (prec, tm) - end + let fixity = + match Hashtbl.find_opt local_fixity name with + | Some fx -> fx + | None -> Names.fixityLookup (Names.Qid (ns, name)) + in + begin match fixity with + | FX.Nonfix -> Atom tm + | FX.Infix (prec, assoc) -> infix_op ((prec, assoc), tm) + | FX.Prefix prec -> prefix_op (prec, tm) + | FX.Postfix prec -> postfix_op (prec, tm) + end | _ -> Atom tm module P = struct let rec reduce = function - | Atom tm2 :: Infix_ (_, con) :: Atom tm1 :: p' -> Atom (con (tm1, tm2)) :: p' + | Atom tm2 :: Infix_ (_, con) :: Atom tm1 :: p' -> + Atom (con (tm1, tm2)) :: p' | Atom tm :: Prefix_ (_, con) :: p' -> Atom (con tm) :: p' | Postfix_ (_, con) :: Atom tm :: p' -> Atom (con tm) :: p' - | p -> failwith (Printf.sprintf "process_app: cannot reduce stack of length %d" (List.length p)) + | p -> + failwith + (Printf.sprintf "process_app: cannot reduce stack of length %d" + (List.length p)) - let rec reduce_rec = function - | [ Atom e ] -> e - | p -> reduce_rec (reduce p) + let rec reduce_rec = function [ Atom e ] -> e | p -> reduce_rec (reduce p) let reduce_all = function | [ Atom e ] -> e @@ -114,50 +129,65 @@ module Make_Modern | _ -> Atom tm :: p let rec shift (opr, p) = - match opr, p with + match (opr, p) with | (Atom _ as o), (Atom _ :: _ as p') -> reduce (o :: jux_op :: p') - | Infix_ _, Infix_ _ :: _ -> raise (ParseError "Consecutive infix operators") - | Infix_ _, Prefix_ _ :: _ -> raise (ParseError "Infix operator following prefix operator") + | Infix_ _, Infix_ _ :: _ -> + raise (ParseError "Consecutive infix operators") + | Infix_ _, Prefix_ _ :: _ -> + raise (ParseError "Infix operator following prefix operator") | Infix_ _, [] -> raise (ParseError "Leading infix operator") | (Prefix_ _ as o), (Atom _ :: _ as p') -> o :: jux_op :: p' - | Postfix_ _, Infix_ _ :: _ -> raise (ParseError "Postfix operator following infix operator") - | Postfix_ _, Prefix_ _ :: _ -> raise (ParseError "Postfix operator following prefix operator") + | Postfix_ _, Infix_ _ :: _ -> + raise (ParseError "Postfix operator following infix operator") + | Postfix_ _, Prefix_ _ :: _ -> + raise (ParseError "Postfix operator following prefix operator") | Postfix_ _, [] -> raise (ParseError "Leading postfix operator") | o, p' -> o :: p' let rec resolve (opr, p) = - match opr, p with - | (Infix_ ((prec, assoc), _) as o), - (Atom _ :: Infix_ ((prec', assoc'), _) :: _ as p') -> - begin match FX.compare (prec, prec'), assoc, assoc' with - | Greater, _, _ -> shift (o, p') - | Less, _, _ -> resolve (o, reduce p') - | Equal, FX.Left, FX.Left -> resolve (o, reduce p') - | Equal, FX.Right, FX.Right -> shift (o, p') - | _ -> raise (ParseError "Ambiguous: infix following infix of identical precedence") - end - | (Infix_ ((prec, _), _) as o), - (Atom _ :: Prefix_ (prec', _) :: _ as p') -> - begin match FX.compare (prec, prec') with - | Greater -> shift (o, p') - | Less -> resolve (o, reduce p') - | Equal -> raise (ParseError "Ambiguous: infix following prefix of identical precedence") - end + match (opr, p) with + | ( (Infix_ ((prec, assoc), _) as o), + (Atom _ :: Infix_ ((prec', assoc'), _) :: _ as p') ) -> + begin match (FX.compare (prec, prec'), assoc, assoc') with + | Greater, _, _ -> shift (o, p') + | Less, _, _ -> resolve (o, reduce p') + | Equal, FX.Left, FX.Left -> resolve (o, reduce p') + | Equal, FX.Right, FX.Right -> shift (o, p') + | _ -> + raise + (ParseError + "Ambiguous: infix following infix of identical precedence") + end + | (Infix_ ((prec, _), _) as o), (Atom _ :: Prefix_ (prec', _) :: _ as p') + -> + begin match FX.compare (prec, prec') with + | Greater -> shift (o, p') + | Less -> resolve (o, reduce p') + | Equal -> + raise + (ParseError + "Ambiguous: infix following prefix of identical precedence") + end | (Prefix_ _ as o), p' -> shift (o, p') - | (Postfix_ (prec, _) as o), - (Atom _ :: Prefix_ (prec', _) :: _ as p') -> - begin match FX.compare (prec, prec') with - | Greater -> reduce (shift (o, p')) - | Less -> resolve (o, reduce p') - | Equal -> raise (ParseError "Ambiguous: postfix following prefix of identical precedence") - end - | (Postfix_ (prec, _) as o), - (Atom _ :: Infix_ ((prec', _), _) :: _ as p') -> - begin match FX.compare (prec, prec') with - | Greater -> reduce (shift (o, p')) - | Less -> resolve (o, reduce p') - | Equal -> raise (ParseError "Ambiguous: postfix following infix of identical precedence") - end + | (Postfix_ (prec, _) as o), (Atom _ :: Prefix_ (prec', _) :: _ as p') -> + begin match FX.compare (prec, prec') with + | Greater -> reduce (shift (o, p')) + | Less -> resolve (o, reduce p') + | Equal -> + raise + (ParseError + "Ambiguous: postfix following prefix of identical precedence") + end + | (Postfix_ (prec, _) as o), (Atom _ :: Infix_ ((prec', _), _) :: _ as p') + -> + begin match FX.compare (prec, prec') with + | Greater -> reduce (shift (o, p')) + | Less -> resolve (o, reduce p') + | Equal -> + raise + (ParseError + "Ambiguous: postfix following infix of identical precedence") + end | (Postfix_ _ as o), ([ Atom _ ] as p') -> reduce (shift (o, p')) | o, p' -> shift (o, p') end @@ -166,11 +196,12 @@ module Make_Modern let rec go p = function | [] -> P.reduce_all p | t :: rest -> - let p' = match classify t with - | Atom tm -> P.shift_atom (tm, p) - | opr -> P.resolve (opr, p) - in - go p' rest + let p' = + match classify t with + | Atom tm -> P.shift_atom (tm, p) + | opr -> P.resolve (opr, p) + in + go p' rest in match ts with | [] -> failwith "process_app: called with empty list" @@ -179,48 +210,51 @@ module Make_Modern let rec parse_arg () : string option t = token "_" *> return None <|> (let+ s = ident1 in - Some s) + Some s) "argument" and parse_id () : Cst.Term.t t = - (keyword "val" *> commit - *> (inside "(" ")" (let@ (ns, s, e) = many1 ident1 in - let loc = mk_loc s e in - let rec split = function - | [] -> failwith "Expected qualified name" - | [ p ] -> ([], p) - | p :: q -> let q0, q1 = split q in (p :: q0, q1) - in - return (Cst.Term.qualified ~fc:loc (split ns))) - <|> (let@ (name, s, e) = ident1 in - let loc = mk_loc s e in - return (Cst.Term.qualified ~fc:loc ([], name))))) + keyword "val" *> commit + *> (inside "(" ")" + (let@ ns, s, e = many1 ident1 in + let loc = mk_loc s e in + let rec split = function + | [] -> failwith "Expected qualified name" + | [ p ] -> ([], p) + | p :: q -> + let q0, q1 = split q in + (p :: q0, q1) + in + return (Cst.Term.qualified ~fc:loc (split ns))) + <|> let@ name, s, e = ident1 in + let loc = mk_loc s e in + return (Cst.Term.qualified ~fc:loc ([], name))) <|> - (let@ (name, s, e) = ident1 in - let loc = mk_loc s e in - let is_upper = - String.length name > 0 - && (name.[0] = '_' || (name.[0] >= 'A' && name.[0] <= 'Z')) - in - return (if is_upper || List.mem name !currently_uppercase - then Cst.Term.uppercase ~fc:loc ([], name) - else Cst.Term.lowercase ~fc:loc ([], name))) + let@ name, s, e = ident1 in + let loc = mk_loc s e in + let is_upper = + String.length name > 0 + && (name.[0] = '_' || (name.[0] >= 'A' && name.[0] <= 'Z')) + in + return + (if is_upper || List.mem name !currently_uppercase then + Cst.Term.uppercase ~fc:loc ([], name) + else Cst.Term.lowercase ~fc:loc ([], name)) and parse_expr_trail () : Cst.Term.t t = - (let@ (d, s, e) = inside "[" "]" (parse_decl ()) in + (let@ d, s, e = inside "[" "]" (parse_decl ()) in let loc = mk_loc s e in let+ body = parse_expr () in Cst.Term.lam ~fc:loc [ d ] body) - <|> (let@ (d, s, e) = inside "{" "}" (parse_decl ()) in + <|> (let@ d, s, e = inside "{" "}" (parse_decl ()) in let loc = mk_loc s e in let+ body = parse_expr () in Cst.Term.pi ~fc:loc [ d ] body) - <|> ((let* ids = inside "{{" "}}" (many @@ parse_var ()) in - let+ body = (with_uppercase ids parse_expr) in - body) "expression with implicit variables") - - <|> (parse_id ()) - "trailing expression" + <|> ((let* ids = inside "{{" "}}" (many @@ parse_var ()) in + let+ body = with_uppercase ids parse_expr in + body) + "expression with implicit variables") + <|> parse_id () "trailing expression" and parse_expr_app () : Cst.Term.t t = (let* head = parse_expr1 () in @@ -230,58 +264,65 @@ module Make_Modern and parse_expr1 () : Cst.Term.t t = begin - choice [ - parse_id (); - inside "(" ")" (return () >>= fun () -> parse_expr ()) - ] + choice + [ parse_id (); inside "(" ")" (return () >>= fun () -> parse_expr ()) ] end "small expression" and parse_expr () : Cst.Term.t t = begin - ( - (keyword "the" *> commit - *> - let* ty = parse_expr1 () in - let+ body = parse_expr () in - Cst.Term.has_type body ty) - <|> - (* %-> A %-> B %-> C ==> {_ A} {_ B} C (last arg is the body) *) - (keywords ["->"; "if"] *> commit - *> - let+ args = sep_by1 (option () @@ keyword "->" *> commit) (parse_expr1 ()) in - let rev = List.rev args in - let body = List.hd rev in - let init = List.rev (List.tl rev) in - List.fold_right - (fun t acc -> Cst.Term.pi [ Cst.Decl.decl1 [ None ] t ] acc) - init body) - <|> - (* + (keyword "the" *> commit + *> + let* ty = parse_expr1 () in + let+ body = parse_expr () in + Cst.Term.has_type body ty) + <|> + (* %-> A %-> B %-> C ==> {_ A} {_ B} C (last arg is the body) *) + (keywords [ "->"; "if" ] + *> commit + *> + let+ args = + sep_by1 (option () @@ (keyword "->" *> commit)) (parse_expr1 ()) + in + let rev = List.rev args in + let body = List.hd rev in + let init = List.rev (List.tl rev) in + List.fold_right + (fun t acc -> Cst.Term.pi [ Cst.Decl.decl1 [ None ] t ] acc) + init body) + <|> + (* %<- A %<- B %<- C ==> {_ C} {_ B} A (first arg is the body) *) - (keywords ["<-"; "do"] *> commit - *> - let+ args = sep_by1 (option () @@ keyword "<-" *> commit) (parse_expr1 ()) in - let body = List.hd args in - let rest_rev = List.rev (List.tl args) in - List.fold_right - (fun t acc -> Cst.Term.pi [ Cst.Decl.decl1 [ None ] t ] acc) - rest_rev body) - <|> - (let* atoms = many @@ parse_expr1 () in - let* trail_opt = option None (let+ t = parse_expr_trail () in Some t) in - match atoms, trail_opt with - | [], None -> fail "expected expression" - | [], Some trail -> return trail - | head :: rest, None -> return (process_app (head :: rest)) - | head :: rest, Some trail -> return (process_app (head :: (rest @ [ trail ])))) - ) + (keywords [ "<-"; "do" ] + *> commit + *> + let+ args = + sep_by1 (option () @@ (keyword "<-" *> commit)) (parse_expr1 ()) + in + let body = List.hd args in + let rest_rev = List.rev (List.tl args) in + List.fold_right + (fun t acc -> Cst.Term.pi [ Cst.Decl.decl1 [ None ] t ] acc) + rest_rev body) + <|> + let* atoms = many @@ parse_expr1 () in + let* trail_opt = + option None + (let+ t = parse_expr_trail () in + Some t) + in + match (atoms, trail_opt) with + | [], None -> fail "expected expression" + | [], Some trail -> return trail + | head :: rest, None -> return (process_app (head :: rest)) + | head :: rest, Some trail -> + return (process_app (head :: (rest @ [ trail ]))) end "expression" - and parse_var () : string t = + and parse_var () : string t = begin ident1 end @@ -307,33 +348,34 @@ module Make_Modern and parse_text () : string t = begin - string "%\"" *> take_till (fun c -> c = '%') <* string "%\"" <* (whitespace' ()) + string "%\"" *> take_till (fun c -> c = '%') + <* string "%\"" <* whitespace' () end "string literal" and parse_decl () : Cst.Decl.t t = begin - (let@ (names, s, e) = inside "(" ")" (many1 (parse_arg ())) in + (let@ names, s, e = inside "(" ")" (many1 (parse_arg ())) in let loc = mk_loc s e in let+ typ = option (Cst.Term.omitted ~fc:Cst.ghost) (parse_expr ()) in Cst.Decl.decl1 ~fc:loc names typ) - <|> (let@ (name, s, e) = parse_arg () in - let loc = mk_loc s e in - let+ typ = option (Cst.Term.omitted ~fc:Cst.ghost) (parse_expr ()) in - Cst.Decl.decl1 ~fc:loc [ name ] typ) + <|> let@ name, s, e = parse_arg () in + let loc = mk_loc s e in + let+ typ = option (Cst.Term.omitted ~fc:Cst.ghost) (parse_expr ()) in + Cst.Decl.decl1 ~fc:loc [ name ] typ end "declaration" and parse_mode () : Cst.Mode.mode t = begin - (let@ ((), s, e) = keyword "out1" *> commit *> return () in + (let@ (), s, e = keyword "out1" *> commit *> return () in return (Cst.Mode.minus1 ~fc:(mk_loc s e) ())) - <|> (let@ ((), s, e) = keyword "out" *> commit *> return () in + <|> (let@ (), s, e = keyword "out" *> commit *> return () in return (Cst.Mode.minus ~fc:(mk_loc s e) ())) - <|> (let@ ((), s, e) = keyword "in" *> commit *> return () in + <|> (let@ (), s, e = keyword "in" *> commit *> return () in return (Cst.Mode.plus ~fc:(mk_loc s e) ())) - <|> (let@ ((), s, e) = keyword "star" *> commit *> return () in - return (Cst.Mode.star ~fc:(mk_loc s e) ())) + <|> let@ (), s, e = keyword "star" *> commit *> return () in + return (Cst.Mode.star ~fc:(mk_loc s e) ()) end "mode" @@ -409,7 +451,7 @@ module Make_Modern and parse_fixity () : int t = begin - let+ s = take_while1 (fun c -> c >= '0' && c <= '9') <* (whitespace' ()) in + let+ s = take_while1 (fun c -> c >= '0' && c <= '9') <* whitespace' () in int_of_string s end "fixity level" @@ -452,7 +494,9 @@ module Make_Modern and parse_bound () : int option t = begin token "_" *> return None - <|> let+ s = take_while1 (fun c -> c >= '0' && c <= '9') <* (whitespace' ()) in + <|> let+ s = + take_while1 (fun c -> c >= '0' && c <= '9') <* whitespace' () + in Some (int_of_string s) end "bound" @@ -469,9 +513,9 @@ module Make_Modern begin token "<=" *> commit *> return "<=" <|> token ">=" *> commit *> return ">=" - <|> token "<" *> commit *> return "<" - <|> token ">" *> commit *> return ">" - <|> token "=" *> commit *> return "=" + <|> token "<" *> commit *> return "<" + <|> token ">" *> commit *> return ">" + <|> token "=" *> commit *> return "=" end "reduces predicate" @@ -509,26 +553,30 @@ module Make_Modern and parse_bracketed : 'a. 'a t -> 'a t = fun p -> inside "[" "]" p and debug_parser : 'a t -> string -> 'a = - fun p x -> - Display.(debug @@ (style Style.bold @@ string "Parsing") ++ nl () ++ string x); + Display.( + debug @@ ((style Style.bold @@ string "Parsing") ++ nl () ++ string x)); match Parser.parse_string ~consume:All (p : _ Parser.t) x with | Ok res -> res | Error msg -> raise (ParseError msg) - and debug_parser_with_ops : (string * FX.fixity) list -> 'a t -> string -> 'a = fun f p x -> + and debug_parser_with_ops : (string * FX.fixity) list -> 'a t -> string -> 'a + = + fun f p x -> List.iter (fun (id, fix) -> Hashtbl.replace local_fixity id fix) f; - debug_parser p x + debug_parser p x + and run : 'a. 'a t -> N.namespace ref -> Cst.loc -> string -> 'a = - fun p _ns _loc s -> (* TODO use namespace and loc *) - match Parser.parse_string ~consume:All ((whitespace' ()) *> p) s with + fun p _ns _loc s -> + (* TODO use namespace and loc *) + match Parser.parse_string ~consume:All (whitespace' () *> p) s with | Ok res -> res | Error msg -> raise (ParseError msg) end module ModernCst = Cst.Make_Cst (Paths.Paths_) -module Modern : MODERN.MODERN = +module Modern : MODERN.MODERN = Make_Modern (Paths.Paths_) (ModernCst) (Names.Names_) (Parsing.Parser.Parser) (* Re-export sub-modules so they are accessible outside the library *) diff --git a/src/Fronts/Pal/Help.ml b/src/Fronts/Pal/Help.ml index 98819fd..ad811ae 100644 --- a/src/Fronts/Pal/Help.ml +++ b/src/Fronts/Pal/Help.ml @@ -1,8 +1,6 @@ let help : string list -> Display.form = function - | ["help"] | ["%help"] -> assert false - | ["set"] | ["%set"] -> assert false - | ["get"] | ["%get"] -> assert false - | ["sort"] | ["%sort"] -> assert false - | ["?"] | ["%?"] -> assert false - - + | [ "help" ] | [ "%help" ] -> assert false + | [ "set" ] | [ "%set" ] -> assert false + | [ "get" ] | [ "%get" ] -> assert false + | [ "sort" ] | [ "%sort" ] -> assert false + | [ "?" ] | [ "%?" ] -> assert false diff --git a/src/Fronts/Pal/IMPL.ml b/src/Fronts/Pal/IMPL.ml index cf22c60..3616ed7 100644 --- a/src/Fronts/Pal/IMPL.ml +++ b/src/Fronts/Pal/IMPL.ml @@ -9,7 +9,7 @@ module type IMPL = sig (** Names module — same as Cmd.Names. *) - module Recon : Recon.RECON + module Recon : Recon.RECON (** New elaboration layer: reconstruction / type-inference sub-modules. *) (** Input source for loading: a file path or an inline string. *) diff --git a/src/Fronts/Pal/Impl.ml b/src/Fronts/Pal/Impl.ml index 46209a9..c610c4e 100644 --- a/src/Fronts/Pal/Impl.ml +++ b/src/Fronts/Pal/Impl.ml @@ -23,7 +23,7 @@ module Impl () = struct (* Ascribe Paths to the bare PATHS signature so it matches what *) (* Make_Cst and Make_Recon's S.S both expect. *) - module Paths : Paths.Paths_intf.PATHS = Paths.Paths_ + module Paths : Paths.PATHS.PATHS = Paths.Paths_ (* Create our own Cst from the same ascribed Paths so that *) (* Cst.Paths = Paths and Recon.Cst.Paths = Paths, giving consistent *) @@ -159,20 +159,22 @@ module Impl () = struct let module TS = ThmSyn in let pred, swap = match pred_str with - | "=" -> (TS.Eq, false) - | "<" -> (TS.Less, false) - | "<=" -> (TS.Leq, false) - | ">" -> (TS.Less, true) - | ">=" -> (TS.Leq, true) - | s -> failwith ("%reduces: unknown predicate " ^ s) - in - let term_to_order tm = - TS.Varg [ term_to_head tm ] + | "=" -> (TS.Eq, false) + | "<" -> (TS.Less, false) + | "<=" -> (TS.Leq, false) + | ">" -> (TS.Less, true) + | ">=" -> (TS.Leq, true) + | s -> failwith ("%reduces: unknown predicate " ^ s) in + let term_to_order tm = TS.Varg [ term_to_head tm ] in let o_out, o_in, body_rest = match body with - | out_tm :: in_tm :: rest -> (term_to_order out_tm, term_to_order in_tm, rest) - | _ -> failwith "%reduces: expected predicate, two order arguments, and call patterns" + | out_tm :: in_tm :: rest -> + (term_to_order out_tm, term_to_order in_tm, rest) + | _ -> + failwith + "%reduces: expected predicate, two order arguments, and call \ + patterns" in let o1, o2 = if swap then (o_in, o_out) else (o_out, o_in) in let term_to_cp tm = @@ -180,7 +182,9 @@ module Impl () = struct let name = term_to_head head in let cid = match Names.constLookup (Names.Qid ([], name)) with - | None -> failwith ("%reduces: undeclared identifier " ^ name ^ " in call pattern") + | None -> + failwith + ("%reduces: undeclared identifier " ^ name ^ " in call pattern") | Some c -> c in (cid, List.map term_to_name args) @@ -194,14 +198,14 @@ module Impl () = struct let module TS = ThmSyn in let rec cst_to_thm = function | Cst.Varg_ (_, names) -> TS.Varg names - | Cst.Lex_ (_, ords) -> TS.Lex (List.map cst_to_thm ords) + | Cst.Lex_ (_, ords) -> TS.Lex (List.map cst_to_thm ords) | Cst.Simul_ (_, ords) -> TS.Simul (List.map cst_to_thm ords) in let order = match orders with - | [] -> TS.Varg [] + | [] -> TS.Varg [] | [ o ] -> cst_to_thm o - | many -> TS.Simul (List.map cst_to_thm many) + | many -> TS.Simul (List.map cst_to_thm many) in let term_to_cp tm = let head, args = unfold_app tm in @@ -250,11 +254,12 @@ module Impl () = struct let condec = Cst.ConDec.constant_decl (Cst.Decl.decl1 [ Some id ] kind) in - (match - Recon.ReconConDec.condecToConDec (condec, loc_of Cst.ghost, false) + match + Recon.ReconConDec.condecToConDec + (condec, loc_of Cst.ghost, false) with | Some cd, _ -> install_condec cd - | None, _ -> ())) + | None, _ -> ()) ids | Cst.TermCmd_ decl -> ( Debug.( @@ -635,8 +640,10 @@ module Impl () = struct loop () with Done -> ()); if !solutions = 0 && !chatter >= 3 then msg "No tabled solution.\n" - | Cst.OpenCmd_ (id, ids) -> - let se = Cst.Struct.str_exp (ids, id) in + | Cst.Open_ id -> + failwith + "%module: module definitions not yet implemented in this frontend" + (* let se = Cst.Struct.str_exp (ids, id) in let mid = Recon.ReconModule.strexpToStrexp se in let ns = ModSyn.Names.getComponents mid in let module_ = ModSyn.abstractModule (ns, Some mid) in @@ -647,11 +654,11 @@ module Impl () = struct Subordinate.Subordinate_.Subordinate.install cid; Subordinate.Subordinate_.Subordinate.installDef cid in - ModSyn.installSig (module_, None, action, true) - | Cst.ModuleCmd_ _ -> + ModSyn.installSig (module_, None, action, true) *) + | Cst.Scope_ _ -> failwith "%module: module definitions not yet implemented in this frontend" - | Cst.UseCmd_ _ -> + | Cst.Use_ _ -> failwith "%use: module instantiation not yet implemented in this frontend" diff --git a/src/Fronts/Pal/PAL.ml b/src/Fronts/Pal/PAL.ml index 481286f..cad29da 100644 --- a/src/Fronts/Pal/PAL.ml +++ b/src/Fronts/Pal/PAL.ml @@ -1,11 +1,13 @@ module type PAL' = sig module M : IMPL.IMPL - val install : M.Cst.cmd -> unit - val parse : string -> M.Cst.cmd list - val exec : string -> unit - end + + val install : M.Cst.cmd -> unit + val parse : string -> M.Cst.cmd list + val exec : string -> unit +end + module type PAL = sig - module M : IMPL.IMPL + module M : IMPL.IMPL exception Error of exn diff --git a/src/Fronts/Pal/Pal.ml b/src/Fronts/Pal/Pal.ml index 207da6f..3a8fb1e 100644 --- a/src/Fronts/Pal/Pal.ml +++ b/src/Fronts/Pal/Pal.ml @@ -1,31 +1,36 @@ let version = "0.1.0" + module type PAL = PAL.PAL module type PAL' = PAL.PAL' + module Pal : PAL.PAL = struct module M = Impl.Impl () + module type S = module type of M + let status_to_exit : M.status -> int = function M.Ok -> 0 | M.Abort -> 1 exception Error of exn - - + module Start () = struct - module M = M - let ns = ref (M.Cmd.Modern.Names.newNamespace ()) - let loc = ref M.Cst.ghost - let install (cmd : M.Cst.cmd) : unit = M.Install.install1 cmd + module M = M + + let ns = ref (M.Cmd.Modern.Names.newNamespace ()) + let loc = ref M.Cst.ghost + let install (cmd : M.Cst.cmd) : unit = M.Install.install1 cmd - let parse (s : string) : M.Cst.cmd list = - M.Cmd.Modern.run (M.Cmd.parse ()) ns !loc s + let parse (s : string) : M.Cst.cmd list = + M.Cmd.Modern.run (M.Cmd.parse ()) ns !loc s + + let exec (s : string) : unit = List.iter install (parse s) + end - let exec (s : string) : unit = List.iter install (parse s) - end let top (module N : Tui.REPL.S) = let module Pal = Start () in let module R = Tui.Repl.Repl (N) in M.mode := `Repl; - R.read (fun l -> - Pal.exec l; + R.read (fun l -> + Pal.exec l; Lwt.return R.Continue) let run () : unit = @@ -48,7 +53,7 @@ module Pal : PAL.PAL = struct let repl_cmd : int Cmd.t = let repl_fn : int Term.t = let+ verbosity = Arg.value Opts.Opts.verbosity - and+ color = Arg.value Opts.Opts.color + and+ color = Arg.value Opts.Opts.color and+ unicode = Arg.value Opts.Opts.unicode in let module M = struct let use_color = color @@ -93,6 +98,49 @@ module Pal : PAL.PAL = struct Twelf_server.server ("stelf", [])) $ const ()) in + let version_cmd : int Cmd.t = + Cmd.v + (Cmd.info "version" ~doc:"Display version information") + Term.( + const (fun () -> + print_endline + {| + .......... + ............... + ........ + ........ + ....... + ....... + ........ + ....... + ........ + ........ + ......... + .......... + ........... + .... ....... + .... ........ + ..... ....... + .... ........ + ..... ........ + ..... ........ + ...... .......... + ..... .......... + ..... ... ....... + ...... ... ........ + ...... .... ....... + ....... .... ........ + ....... ... ....... + ....... .... ....... + ........ .... ....... ... + .......... .... ............ + . + +|}; + print_endline ("STELF version " ^ version); + 0) + $ const ()) + in let main_cmd = Cmd.group (Cmd.info "stelf" ~version ~doc:"The STELF proof assistant") diff --git a/src/Fronts/Pal/Pal.mli b/src/Fronts/Pal/Pal.mli index 22b48c8..4fa0451 100644 --- a/src/Fronts/Pal/Pal.mli +++ b/src/Fronts/Pal/Pal.mli @@ -1,3 +1,4 @@ module type PAL = PAL.PAL module type PAL' = PAL.PAL' + module Pal : PAL.PAL diff --git a/src/Fronts/Project/dune b/src/Fronts/Project/dune index f401e61..77a5718 100644 --- a/src/Fronts/Project/dune +++ b/src/Fronts/Project/dune @@ -1,4 +1,4 @@ (library (name project) (public_name stelf.project) - (libraries sexplib)) \ No newline at end of file + (libraries sexplib)) diff --git a/src/Fronts/Tui/Repl.ml b/src/Fronts/Tui/Repl.ml index 51f5b6d..a375bcf 100644 --- a/src/Fronts/Tui/Repl.ml +++ b/src/Fronts/Tui/Repl.ml @@ -7,10 +7,14 @@ module Repl (M : REPL.S) : REPL.REPL = struct () let msgs : Display.Info.t list ref = ref [] - let add_msg (m : Display.Info.t) = let r = if Display.Info.to_int M.verbosity >= Display.Info.to_int m.level then - m :: !msgs - else !msgs in - msgs := r + + let add_msg (m : Display.Info.t) = + let r = + if Display.Info.to_int M.verbosity >= Display.Info.to_int m.level then + m :: !msgs + else !msgs + in + msgs := r let flush_msgs () : Display.Info.t array Lwt.t = let pending = Array.of_list (List.rev !msgs) in @@ -134,7 +138,7 @@ module Repl (M : REPL.S) : REPL.REPL = struct in Lwt.catch (fun () -> - let* r0 = read_line !term' "Π∀λ> " in + let* r0 = read_line !term' "λΠ> " in let* continue = try f r0 with | Sys.Break -> Lwt.return Stop @@ -162,5 +166,5 @@ module Repl (M : REPL.S) : REPL.REPL = struct @@ Printexc.to_string exn)) (seq_l (flush ()) (read f))) - let show fmt = () (* Format.pp_print_string fmt "Π∀λ> "*) + let show fmt = () (* Format.pp_print_string fmt "λΠ> "*) end diff --git a/src/global/Global_intf.ml b/src/Global/GLOBAL.ml similarity index 99% rename from src/global/Global_intf.ml rename to src/Global/GLOBAL.ml index a648d4e..870be65 100644 --- a/src/global/Global_intf.ml +++ b/src/Global/GLOBAL.ml @@ -18,7 +18,6 @@ module type GLOBAL = sig val chPrint : int -> (unit -> string) -> unit val chMessage : int -> (unit -> string) -> (string -> unit) -> unit val timeLimit : Time.time option ref - val arrow_reserved : bool ref val arrow_infix : bool ref val latin_uppercase : bool ref diff --git a/src/global/Global.ml b/src/Global/Global.ml similarity index 100% rename from src/global/Global.ml rename to src/Global/Global.ml diff --git a/src/global/Global.mli b/src/Global/Global.mli similarity index 100% rename from src/global/Global.mli rename to src/Global/Global.mli diff --git a/src/global/Global_.ml b/src/Global/Global_.ml similarity index 97% rename from src/global/Global_.ml rename to src/Global/Global_.ml index 84b1856..c172b72 100644 --- a/src/global/Global_.ml +++ b/src/Global/Global_.ml @@ -4,7 +4,7 @@ open Basis (* Global parameters *) -include Global_intf +include GLOBAL (** Author: Frank Pfenning *) (* in seconds *) diff --git a/src/Global/Global_.mli b/src/Global/Global_.mli new file mode 100644 index 0000000..3c0d40b --- /dev/null +++ b/src/Global/Global_.mli @@ -0,0 +1,2 @@ +include module type of GLOBAL +module Global : GLOBAL diff --git a/src/global/README b/src/Global/README similarity index 100% rename from src/global/README rename to src/Global/README diff --git a/src/global/WALK b/src/Global/WALK similarity index 100% rename from src/global/WALK rename to src/Global/WALK diff --git a/src/global/dune b/src/Global/dune similarity index 86% rename from src/global/dune rename to src/Global/dune index dcd10cf..e857dcf 100644 --- a/src/global/dune +++ b/src/Global/dune @@ -3,7 +3,7 @@ (library (name global) (public_name stelf.global) - (modules Global_ Global_intf) + (modules Global_ GLOBAL) (libraries display basis) (wrapped true) (preprocess diff --git a/src/heuristic/Heuristic_intf.ml b/src/Heuristic/HEURISTIC.ml similarity index 100% rename from src/heuristic/Heuristic_intf.ml rename to src/Heuristic/HEURISTIC.ml diff --git a/src/heuristic/Heuristic.ml b/src/Heuristic/Heuristic.ml similarity index 100% rename from src/heuristic/Heuristic.ml rename to src/Heuristic/Heuristic.ml diff --git a/src/heuristic/Heuristic.mli b/src/Heuristic/Heuristic.mli similarity index 100% rename from src/heuristic/Heuristic.mli rename to src/Heuristic/Heuristic.mli diff --git a/src/heuristic/HeuristicSum.ml b/src/Heuristic/HeuristicSum.ml similarity index 100% rename from src/heuristic/HeuristicSum.ml rename to src/Heuristic/HeuristicSum.ml diff --git a/src/heuristic/HeuristicSum.mli b/src/Heuristic/HeuristicSum.mli similarity index 100% rename from src/heuristic/HeuristicSum.mli rename to src/Heuristic/HeuristicSum.mli diff --git a/src/heuristic/Heuristic_.ml b/src/Heuristic/Heuristic_.ml similarity index 99% rename from src/heuristic/Heuristic_.ml rename to src/Heuristic/Heuristic_.ml index f368f9d..33b50cd 100644 --- a/src/heuristic/Heuristic_.ml +++ b/src/Heuristic/Heuristic_.ml @@ -6,7 +6,7 @@ open Basis (* Heuristics : Version 1.3 *) -include Heuristic_intf +include HEURISTIC (** Author: Carsten Schuermann *) (* signature HEURISTIC *) diff --git a/src/Heuristic/Heuristic_.mli b/src/Heuristic/Heuristic_.mli new file mode 100644 index 0000000..e0ceafb --- /dev/null +++ b/src/Heuristic/Heuristic_.mli @@ -0,0 +1,2 @@ +include module type of HEURISTIC +module Heuristic : HEURISTIC diff --git a/src/heuristic/dune b/src/Heuristic/dune similarity index 80% rename from src/heuristic/dune rename to src/Heuristic/dune index 014e379..1b33521 100644 --- a/src/heuristic/dune +++ b/src/Heuristic/dune @@ -3,7 +3,7 @@ (library (name heuristic) (public_name stelf.heuristic) - (modules Heuristic_ HeuristicSum Heuristic_intf) + (modules Heuristic_ HeuristicSum HEURISTIC) (libraries display basis) (wrapped true) (preprocess diff --git a/src/index/Index_intf.ml b/src/Index/INDEX.ml similarity index 100% rename from src/index/Index_intf.ml rename to src/Index/INDEX.ml diff --git a/src/index/Index.ml b/src/Index/Index.ml similarity index 100% rename from src/index/Index.ml rename to src/Index/Index.ml diff --git a/src/index/Index.mli b/src/Index/Index.mli similarity index 100% rename from src/index/Index.mli rename to src/Index/Index.mli diff --git a/src/index/IndexSkolem.ml b/src/Index/IndexSkolem.ml similarity index 100% rename from src/index/IndexSkolem.ml rename to src/Index/IndexSkolem.ml diff --git a/src/index/IndexSkolem.mli b/src/Index/IndexSkolem.mli similarity index 100% rename from src/index/IndexSkolem.mli rename to src/Index/IndexSkolem.mli diff --git a/src/index/Index_.ml b/src/Index/Index_.ml similarity index 99% rename from src/index/Index_.ml rename to src/Index/Index_.ml index 6e1752a..3c1bb0f 100644 --- a/src/index/Index_.ml +++ b/src/Index/Index_.ml @@ -4,7 +4,7 @@ open! Basis (* Indexing *) (* Author: Carsten Schuermann *) -include Index_intf +include INDEX (** Modified: Frank Pfenning *) (* signature INDEX *) diff --git a/src/index/Index_.mli b/src/Index/Index_.mli similarity index 74% rename from src/index/Index_.mli rename to src/Index/Index_.mli index 56c1aaa..9234f95 100644 --- a/src/index/Index_.mli +++ b/src/Index/Index_.mli @@ -1,4 +1,4 @@ -include module type of Index_intf +include module type of INDEX module MakeIndex (Global : GLOBAL) (Queue : Queue.QUEUE) : INDEX module Index : INDEX include INDEX diff --git a/src/index/README b/src/Index/README similarity index 100% rename from src/index/README rename to src/Index/README diff --git a/src/index/WALK b/src/Index/WALK similarity index 100% rename from src/index/WALK rename to src/Index/WALK diff --git a/src/index/dune b/src/Index/dune similarity index 90% rename from src/index/dune rename to src/Index/dune index 55c0fee..5c42351 100644 --- a/src/index/dune +++ b/src/Index/dune @@ -3,7 +3,7 @@ (library (name index) (public_name stelf.index) - (modules IndexSkolem Index_ Index_intf) + (modules IndexSkolem Index_ INDEX) (libraries display intsyn table global basis) (flags (:standard diff --git a/src/intInf/IntInfSig_intf.ml b/src/IntInf/INTINFSIG.ml similarity index 100% rename from src/intInf/IntInfSig_intf.ml rename to src/IntInf/INTINFSIG.ml diff --git a/src/intInf/IntInf.ml b/src/IntInf/IntInf.ml similarity index 100% rename from src/intInf/IntInf.ml rename to src/IntInf/IntInf.ml diff --git a/src/intInf/IntInf.mli b/src/IntInf/IntInf.mli similarity index 100% rename from src/intInf/IntInf.mli rename to src/IntInf/IntInf.mli diff --git a/src/intInf/IntInfSig.ml b/src/IntInf/IntInfSig.ml similarity index 94% rename from src/intInf/IntInfSig.ml rename to src/IntInf/IntInfSig.ml index ec53686..6690204 100644 --- a/src/intInf/IntInfSig.ml +++ b/src/IntInf/IntInfSig.ml @@ -13,5 +13,5 @@ open Basis * This package is derived from Andrzej Filinski's bignum package. It is versy * close to the definition of the optional IntInf structure in the SML'97 basis. *) -include IntInfSig_intf +include INTINFSIG (* signature INT_INF *) diff --git a/src/IntInf/IntInfSig.mli b/src/IntInf/IntInfSig.mli new file mode 100644 index 0000000..4efdbf8 --- /dev/null +++ b/src/IntInf/IntInfSig.mli @@ -0,0 +1 @@ +include module type of INTINFSIG diff --git a/src/intInf/IntInf_.ml b/src/IntInf/IntInf_.ml similarity index 100% rename from src/intInf/IntInf_.ml rename to src/IntInf/IntInf_.ml diff --git a/src/intInf/IntInf_.mli b/src/IntInf/IntInf_.mli similarity index 100% rename from src/intInf/IntInf_.mli rename to src/IntInf/IntInf_.mli diff --git a/src/intInf/README b/src/IntInf/README similarity index 100% rename from src/intInf/README rename to src/IntInf/README diff --git a/src/intInf/dune b/src/IntInf/dune similarity index 81% rename from src/intInf/dune rename to src/IntInf/dune index 34a3917..2f6c155 100644 --- a/src/intInf/dune +++ b/src/IntInf/dune @@ -3,7 +3,7 @@ (library (name intInf) (public_name stelf.intInf) - (modules IntInfSig IntInf_ IntInfSig_intf) + (modules IntInfSig IntInf_ INTINFSIG) (libraries display basis) (wrapped true) (preprocess diff --git a/src/IntSyn/Abstract_intf.ml b/src/IntSyn/ABSTRACT.ml similarity index 100% rename from src/IntSyn/Abstract_intf.ml rename to src/IntSyn/ABSTRACT.ml diff --git a/src/IntSyn/Approx_intf.ml b/src/IntSyn/APPROX.ml similarity index 100% rename from src/IntSyn/Approx_intf.ml rename to src/IntSyn/APPROX.ml diff --git a/src/IntSyn/Abstract.ml b/src/IntSyn/Abstract.ml index 26b102f..dacf2e4 100644 --- a/src/IntSyn/Abstract.ml +++ b/src/IntSyn/Abstract.ml @@ -5,7 +5,7 @@ open Tomega (* Abstraction *) (* Author: Frank Pfenning, Carsten Schuermann *) -include Abstract_intf +include ABSTRACT (* signature ABSTRACT *) (* # 1 "src/lambda/Abstract.fun.ml" *) diff --git a/src/IntSyn/Abstract.mli b/src/IntSyn/Abstract.mli index 499ff68..249fd32 100644 --- a/src/IntSyn/Abstract.mli +++ b/src/IntSyn/Abstract.mli @@ -1,4 +1,4 @@ -include module type of Abstract_intf +include module type of ABSTRACT module MakeAbstract (Whnf : Whnf.WHNF) diff --git a/src/IntSyn/Approx.ml b/src/IntSyn/Approx.ml index fb8f010..78c2737 100644 --- a/src/IntSyn/Approx.ml +++ b/src/IntSyn/Approx.ml @@ -4,7 +4,7 @@ open Intsyn_ (* Approximate language for term reconstruction *) (* Author: Kevin Watkins *) -include Approx_intf +include APPROX (* # 1 "src/lambda/Approx.fun.ml" *) open! Whnf diff --git a/src/IntSyn/Approx.mli b/src/IntSyn/Approx.mli index 0b22bf6..aa1bdd5 100644 --- a/src/IntSyn/Approx.mli +++ b/src/IntSyn/Approx.mli @@ -1,3 +1,3 @@ -open Whnf_intf -include module type of Approx_intf +open WHNF +include module type of APPROX module MakeApprox (Whnf : WHNF) : APPROX diff --git a/src/IntSyn/Constraints_intf.ml b/src/IntSyn/CONSTRAINTS.ml similarity index 100% rename from src/IntSyn/Constraints_intf.ml rename to src/IntSyn/CONSTRAINTS.ml diff --git a/src/IntSyn/Conv_intf.ml b/src/IntSyn/CONV.ml similarity index 100% rename from src/IntSyn/Conv_intf.ml rename to src/IntSyn/CONV.ml diff --git a/src/IntSyn/Constraints.ml b/src/IntSyn/Constraints.ml index 9f72633..0e2dae2 100644 --- a/src/IntSyn/Constraints.ml +++ b/src/IntSyn/Constraints.ml @@ -8,7 +8,7 @@ open Intsyn_ (* Manipulating Constraints *) (* Author: Jeff Polakow, Frank Pfenning *) (* Modified: Roberto Virga *) -include Constraints_intf +include CONSTRAINTS (* signature CONSTRAINTS *) (* # 1 "src/lambda/Constraints.fun.ml" *) diff --git a/src/IntSyn/Constraints.mli b/src/IntSyn/Constraints.mli index 22c28c7..33e5396 100644 --- a/src/IntSyn/Constraints.mli +++ b/src/IntSyn/Constraints.mli @@ -1,3 +1,3 @@ -open Conv_intf -include module type of Constraints_intf +open CONV +include module type of CONSTRAINTS module MakeConstraints (Conv : CONV) : CONSTRAINTS diff --git a/src/IntSyn/Conv.ml b/src/IntSyn/Conv.ml index 0e863c6..8409acf 100644 --- a/src/IntSyn/Conv.ml +++ b/src/IntSyn/Conv.ml @@ -6,7 +6,7 @@ open Intsyn_ (* Convertibility Modulo Beta and Eta *) (* Author: Frank Pfenning, Carsten Schuermann *) -include Conv_intf +include CONV (* signature CONV *) (* # 1 "src/lambda/Conv.fun.ml" *) diff --git a/src/IntSyn/Conv.mli b/src/IntSyn/Conv.mli index f051301..148eaa1 100644 --- a/src/IntSyn/Conv.mli +++ b/src/IntSyn/Conv.mli @@ -1,5 +1,5 @@ -open Whnf_intf -include module type of Conv_intf +open WHNF +include module type of CONV module Conv (Conv__0 : sig module Whnf : WHNF diff --git a/src/IntSyn/Fgnopn_intf.ml b/src/IntSyn/FGNOPN.ml similarity index 100% rename from src/IntSyn/Fgnopn_intf.ml rename to src/IntSyn/FGNOPN.ml diff --git a/src/IntSyn/Fgnopn.ml b/src/IntSyn/Fgnopn.ml index 71bbaa3..707d4ac 100644 --- a/src/IntSyn/Fgnopn.ml +++ b/src/IntSyn/Fgnopn.ml @@ -3,7 +3,7 @@ open! Basis (* Extensible operation on foreign matter *) (* Author: Aleksey Kliger *) -include Fgnopn_intf +include FGNOPN (* # 1 "src/lambda/Fgnopn.fun.ml" *) (* # 1 "src/lambda/Fgnopn.sml.ml" *) diff --git a/src/IntSyn/Fgnopn.mli b/src/IntSyn/Fgnopn.mli index fe70aa8..b674273 100644 --- a/src/IntSyn/Fgnopn.mli +++ b/src/IntSyn/Fgnopn.mli @@ -1 +1 @@ -include module type of Fgnopn_intf +include module type of FGNOPN diff --git a/src/IntSyn/Intsyn_intf.ml b/src/IntSyn/INTSYN.ml similarity index 100% rename from src/IntSyn/Intsyn_intf.ml rename to src/IntSyn/INTSYN.ml diff --git a/src/IntSyn/Match_intf.ml b/src/IntSyn/MATCH.ml similarity index 100% rename from src/IntSyn/Match_intf.ml rename to src/IntSyn/MATCH.ml diff --git a/src/IntSyn/Match.ml b/src/IntSyn/Match.ml index 11c4cb4..946bfff 100644 --- a/src/IntSyn/Match.ml +++ b/src/IntSyn/Match.ml @@ -4,7 +4,7 @@ open Intsyn_ (* Matching *) (* Author: Frank Pfenning, Carsten Schuermann *) -include Match_intf +include MATCH (* signature MATCH *) (* # 1 "src/lambda/Match.fun.ml" *) diff --git a/src/IntSyn/Match.mli b/src/IntSyn/Match.mli index 6c8fbc4..ee108d7 100644 --- a/src/IntSyn/Match.mli +++ b/src/IntSyn/Match.mli @@ -1,4 +1,4 @@ -open Whnf_intf -open Unify_intf -include module type of Match_intf +open WHNF +open UNIFY +include module type of MATCH module MakeMatch (Whnf : WHNF) (Unify : UNIFY) (Trail : TRAIL) : MATCH diff --git a/src/IntSyn/Order_intf.ml b/src/IntSyn/ORDER.ml similarity index 100% rename from src/IntSyn/Order_intf.ml rename to src/IntSyn/ORDER.ml diff --git a/src/IntSyn/Order.ml b/src/IntSyn/Order.ml index 1a8640a..cd38ebf 100644 --- a/src/IntSyn/Order.ml +++ b/src/IntSyn/Order.ml @@ -6,7 +6,7 @@ open Intsyn_ (* Termination Order *) (* Author: Carsten Schuermann *) -include Order_intf +include ORDER (* signature ORDER *) (* # 1 "src/lambda/Order.fun.ml" *) diff --git a/src/IntSyn/Order.mli b/src/IntSyn/Order.mli index 766dec6..b6b6fe4 100644 --- a/src/IntSyn/Order.mli +++ b/src/IntSyn/Order.mli @@ -1,4 +1,4 @@ -include module type of Order_intf +include module type of ORDER module MakeOrder (Order__0 : sig module Table : TABLE with type key = int diff --git a/src/IntSyn/Tomega_intf.ml b/src/IntSyn/TOMEGA.ml similarity index 100% rename from src/IntSyn/Tomega_intf.ml rename to src/IntSyn/TOMEGA.ml diff --git a/src/IntSyn/Tomega.ml b/src/IntSyn/Tomega.ml index 23cfc5e..4f109fd 100644 --- a/src/IntSyn/Tomega.ml +++ b/src/IntSyn/Tomega.ml @@ -4,7 +4,7 @@ open Intsyn_ (* Internal syntax for Delphin *) (* Author: Carsten Schuermann *) -include Tomega_intf +include TOMEGA (* Signature TOMEGA *) (* # 1 "src/lambda/Tomega.fun.ml" *) diff --git a/src/IntSyn/Tomega.mli b/src/IntSyn/Tomega.mli index f38d0d6..985ff86 100644 --- a/src/IntSyn/Tomega.mli +++ b/src/IntSyn/Tomega.mli @@ -1,3 +1,3 @@ -include module type of Tomega_intf +include module type of TOMEGA module MakeTomega (Whnf : Whnf.WHNF) (Conv : Conv.CONV) : TOMEGA module Tomega : TOMEGA diff --git a/src/IntSyn/Unify_intf.ml b/src/IntSyn/UNIFY.ml similarity index 100% rename from src/IntSyn/Unify_intf.ml rename to src/IntSyn/UNIFY.ml diff --git a/src/IntSyn/Unify.ml b/src/IntSyn/Unify.ml index 13bf133..46aa4c0 100644 --- a/src/IntSyn/Unify.ml +++ b/src/IntSyn/Unify.ml @@ -4,7 +4,7 @@ open Intsyn_ (* Unification *) (* Author: Frank Pfenning, Carsten Schuermann *) -include Unify_intf +include UNIFY (* signature UNIFY *) (* # 1 "src/lambda/Unify.fun.ml" *) diff --git a/src/IntSyn/Unify.mli b/src/IntSyn/Unify.mli index 5f90506..634443a 100644 --- a/src/IntSyn/Unify.mli +++ b/src/IntSyn/Unify.mli @@ -1,3 +1,3 @@ -open Whnf_intf -include module type of Unify_intf +open WHNF +include module type of UNIFY module MakeUnify (Whnf : WHNF) (Trail : TRAIL) : UNIFY diff --git a/src/IntSyn/Whnf_intf.ml b/src/IntSyn/WHNF.ml similarity index 100% rename from src/IntSyn/Whnf_intf.ml rename to src/IntSyn/WHNF.ml diff --git a/src/IntSyn/Whnf.ml b/src/IntSyn/Whnf.ml index cbdf4aa..c4b7ee7 100644 --- a/src/IntSyn/Whnf.ml +++ b/src/IntSyn/Whnf.ml @@ -4,7 +4,7 @@ open Intsyn_ (* Weak Head-Normal Forms *) (* Authors: Frank Pfenning, Carsten Schuermann *) -include Whnf_intf +include WHNF (* signature WHNF *) (* # 1 "src/lambda/Whnf.fun.ml" *) diff --git a/src/IntSyn/Whnf.mli b/src/IntSyn/Whnf.mli index 4d2fc01..5db0a63 100644 --- a/src/IntSyn/Whnf.mli +++ b/src/IntSyn/Whnf.mli @@ -1,3 +1,3 @@ -module type WHNF = Whnf_intf.WHNF +module type WHNF = WHNF.WHNF module Whnf () : WHNF diff --git a/src/IntSyn/dune b/src/IntSyn/dune index 2dc215c..0e29823 100644 --- a/src/IntSyn/dune +++ b/src/IntSyn/dune @@ -17,18 +17,18 @@ Match Abstract Approx - Intsyn_intf + INTSYN Lambda_ - Abstract_intf - Approx_intf - Constraints_intf - Conv_intf - Fgnopn_intf - Match_intf - Order_intf - Tomega_intf - Unify_intf - Whnf_intf) + ABSTRACT + APPROX + CONSTRAINTS + CONV + FGNOPN + MATCH + ORDER + TOMEGA + UNIFY + WHNF) (libraries containers global trail table basis syntax debug) (flags (:standard diff --git a/src/inverse/.cvsignore b/src/Inverse/.cvsignore similarity index 100% rename from src/inverse/.cvsignore rename to src/Inverse/.cvsignore diff --git a/src/inverse/README b/src/Inverse/README similarity index 100% rename from src/inverse/README rename to src/Inverse/README diff --git a/src/inverse/notes/bug.txt b/src/Inverse/notes/bug.txt similarity index 100% rename from src/inverse/notes/bug.txt rename to src/Inverse/notes/bug.txt diff --git a/src/inverse/notes/notes.txt b/src/Inverse/notes/notes.txt similarity index 100% rename from src/inverse/notes/notes.txt rename to src/Inverse/notes/notes.txt diff --git a/src/inverse/notes/timing.txt b/src/Inverse/notes/timing.txt similarity index 100% rename from src/inverse/notes/timing.txt rename to src/Inverse/notes/timing.txt diff --git a/src/inverse/notes/timing2.txt b/src/Inverse/notes/timing2.txt similarity index 100% rename from src/inverse/notes/timing2.txt rename to src/Inverse/notes/timing2.txt diff --git a/src/inverse/sml/.cvsignore b/src/Inverse/sml/.cvsignore similarity index 100% rename from src/inverse/sml/.cvsignore rename to src/Inverse/sml/.cvsignore diff --git a/src/inverse/sml/sources.cm_ b/src/Inverse/sml/sources.cm_ similarity index 100% rename from src/inverse/sml/sources.cm_ rename to src/Inverse/sml/sources.cm_ diff --git a/src/inverse/sml/top.sml.ml.error b/src/Inverse/sml/top.sml.ml.error similarity index 100% rename from src/inverse/sml/top.sml.ml.error rename to src/Inverse/sml/top.sml.ml.error diff --git a/src/inverse/tex/.cvsignore b/src/Inverse/tex/.cvsignore similarity index 100% rename from src/inverse/tex/.cvsignore rename to src/Inverse/tex/.cvsignore diff --git a/src/inverse/tex/Makefile b/src/Inverse/tex/Makefile similarity index 100% rename from src/inverse/tex/Makefile rename to src/Inverse/tex/Makefile diff --git a/src/inverse/tex/canonical.tex b/src/Inverse/tex/canonical.tex similarity index 100% rename from src/inverse/tex/canonical.tex rename to src/Inverse/tex/canonical.tex diff --git a/src/inverse/tex/commands.tex b/src/Inverse/tex/commands.tex similarity index 100% rename from src/inverse/tex/commands.tex rename to src/Inverse/tex/commands.tex diff --git a/src/inverse/tex/eagercheck.tex b/src/Inverse/tex/eagercheck.tex similarity index 100% rename from src/inverse/tex/eagercheck.tex rename to src/Inverse/tex/eagercheck.tex diff --git a/src/inverse/tex/images/twelf.graffle b/src/Inverse/tex/images/twelf.graffle similarity index 100% rename from src/inverse/tex/images/twelf.graffle rename to src/Inverse/tex/images/twelf.graffle diff --git a/src/inverse/tex/inverse.tex b/src/Inverse/tex/inverse.tex similarity index 100% rename from src/inverse/tex/inverse.tex rename to src/Inverse/tex/inverse.tex diff --git a/src/inverse/tex/lazycheck.tex b/src/Inverse/tex/lazycheck.tex similarity index 100% rename from src/inverse/tex/lazycheck.tex rename to src/Inverse/tex/lazycheck.tex diff --git a/src/inverse/tex/main.tex b/src/Inverse/tex/main.tex similarity index 100% rename from src/inverse/tex/main.tex rename to src/Inverse/tex/main.tex diff --git a/src/inverse/tex/old/commands.tex b/src/Inverse/tex/old/commands.tex similarity index 100% rename from src/inverse/tex/old/commands.tex rename to src/Inverse/tex/old/commands.tex diff --git a/src/inverse/tex/old/context.tex b/src/Inverse/tex/old/context.tex similarity index 100% rename from src/inverse/tex/old/context.tex rename to src/Inverse/tex/old/context.tex diff --git a/src/inverse/tex/old/etalong.tex b/src/Inverse/tex/old/etalong.tex similarity index 100% rename from src/inverse/tex/old/etalong.tex rename to src/Inverse/tex/old/etalong.tex diff --git a/src/inverse/tex/old/noncheck.tex b/src/Inverse/tex/old/noncheck.tex similarity index 100% rename from src/inverse/tex/old/noncheck.tex rename to src/Inverse/tex/old/noncheck.tex diff --git a/src/inverse/tex/old/sequent.tex b/src/Inverse/tex/old/sequent.tex similarity index 100% rename from src/inverse/tex/old/sequent.tex rename to src/Inverse/tex/old/sequent.tex diff --git a/src/inverse/tex/old/signat.tex b/src/Inverse/tex/old/signat.tex similarity index 100% rename from src/inverse/tex/old/signat.tex rename to src/Inverse/tex/old/signat.tex diff --git a/src/inverse/tex/old/uni.tex b/src/Inverse/tex/old/uni.tex similarity index 100% rename from src/inverse/tex/old/uni.tex rename to src/Inverse/tex/old/uni.tex diff --git a/src/inverse/tex/questions.txt b/src/Inverse/tex/questions.txt similarity index 100% rename from src/inverse/tex/questions.txt rename to src/Inverse/tex/questions.txt diff --git a/src/Lang/Lexing/Lexer_intf.ml b/src/Lang/Lexing/LEXER.ml similarity index 100% rename from src/Lang/Lexing/Lexer_intf.ml rename to src/Lang/Lexing/LEXER.ml diff --git a/src/Lang/Parsing/PARSER.ml b/src/Lang/Parsing/PARSER.ml index 6914ae9..e99e707 100644 --- a/src/Lang/Parsing/PARSER.ml +++ b/src/Lang/Parsing/PARSER.ml @@ -10,7 +10,7 @@ module type PARSER = sig val keywords : string list -> unit t val token : string -> unit t val given : bool -> 'a t -> 'a t - val extend : 'a t -> (bool * 'a t) -> 'a t + val extend : 'a t -> bool * 'a t -> 'a t val ( let| ) : 'a t -> ('a -> 'b t) -> 'b t val ( and| ) : 'a t -> 'b t -> ('a * 'b) t val ( let* ) : 'a t -> ('a -> 'b t) -> 'b t diff --git a/src/Lang/Parsing/Parser.ml b/src/Lang/Parsing/Parser.ml index 803c5df..5f25bb1 100644 --- a/src/Lang/Parsing/Parser.ml +++ b/src/Lang/Parsing/Parser.ml @@ -24,11 +24,12 @@ module Parser : PARSER = struct let boundary = peek_char >>= function | None -> return () - | Some (' ' | '\t' | '\n' - | '(' | ')' | '{' | '}' | '[' | ']' | '%') -> return () + | Some (' ' | '\t' | '\n' | '(' | ')' | '{' | '}' | '[' | ']' | '%') -> + return () | _ -> fail ("keyword " ^ s' ^ " not at word boundary") in string s' *> boundary *> whitespace + let keywords ss = choice (List.map keyword ss) let ident = @@ -36,6 +37,7 @@ module Parser : PARSER = struct | ' ' | '\t' | '\n' | '(' | ')' | '{' | '}' | '[' | ']' | '%' -> true | _ -> false) <* whitespace + let ident1 = take_while1 (function | ' ' | '\t' | '\n' | '(' | ')' | '{' | '}' | '[' | ']' | '%' -> false @@ -66,7 +68,6 @@ module Parser : PARSER = struct let given b p = if b then p else fail "failed test" let inside x y p = token x *> p <* token y - let extend _p _q = assert false let forget p = p *> return () end diff --git a/src/m2/Filling_intf.ml b/src/M2/FILLING.ml similarity index 100% rename from src/m2/Filling_intf.ml rename to src/M2/FILLING.ml diff --git a/src/m2/Filling.ml b/src/M2/Filling.ml similarity index 98% rename from src/m2/Filling.ml rename to src/M2/Filling.ml index efd1d74..450d9f5 100644 --- a/src/m2/Filling.ml +++ b/src/M2/Filling.ml @@ -4,7 +4,7 @@ open Metasyn (* Filling *) (* Author: Carsten Schuermann *) -include Filling_intf +include FILLING (* # 1 "src/m2/Filling.fun.ml" *) open! Search @@ -16,10 +16,7 @@ open Metasyn (* Author: Carsten Schuermann *) module Filling (Filling__0 : sig module MetaSyn' : Metasyn.METASYN - - module MetaAbstract : - MetaAbstract_intf.METAABSTRACT with module MetaSyn = MetaSyn' - + module MetaAbstract : METAABSTRACT.METAABSTRACT with module MetaSyn = MetaSyn' module Search : Search.OLDSEARCH with module MetaSyn = MetaSyn' module Whnf : WHNF diff --git a/src/m2/Filling.mli b/src/M2/Filling.mli similarity index 69% rename from src/m2/Filling.mli rename to src/M2/Filling.mli index d700e8b..1f5a439 100644 --- a/src/m2/Filling.mli +++ b/src/M2/Filling.mli @@ -1,11 +1,8 @@ -include module type of Filling_intf +include module type of FILLING module Filling (Filling__0 : sig module MetaSyn' : Metasyn.METASYN - - module MetaAbstract : - MetaAbstract_intf.METAABSTRACT with module MetaSyn = MetaSyn' - + module MetaAbstract : METAABSTRACT.METAABSTRACT with module MetaSyn = MetaSyn' module Search : Search.OLDSEARCH with module MetaSyn = MetaSyn' module Whnf : WHNF diff --git a/src/m2/Init_intf.ml b/src/M2/INIT.ml similarity index 100% rename from src/m2/Init_intf.ml rename to src/M2/INIT.ml diff --git a/src/m2/Init.ml b/src/M2/Init.ml similarity index 91% rename from src/m2/Init.ml rename to src/M2/Init.ml index bf8d8a1..153fa6a 100644 --- a/src/m2/Init.ml +++ b/src/M2/Init.ml @@ -4,7 +4,7 @@ open Metasyn (* Initialization *) (* Author: Carsten Schuermann *) -include Init_intf +include INIT (* signature INIT *) (* # 1 "src/m2/Init.fun.ml" *) @@ -16,9 +16,7 @@ open MetaAbstract (* Author: Carsten Schuermann *) module Init (Init__0 : sig module MetaSyn' : Metasyn.METASYN - - module MetaAbstract : - MetaAbstract_intf.METAABSTRACT with module MetaSyn = MetaSyn' + module MetaAbstract : METAABSTRACT.METAABSTRACT with module MetaSyn = MetaSyn' end) : INIT with module MetaSyn = Init__0.MetaSyn' = struct open Init__0 module MetaSyn = MetaAbstract.MetaSyn diff --git a/src/M2/Init.mli b/src/M2/Init.mli new file mode 100644 index 0000000..0073c56 --- /dev/null +++ b/src/M2/Init.mli @@ -0,0 +1,6 @@ +include module type of INIT + +module Init (Init__0 : sig + module MetaSyn' : Metasyn.METASYN + module MetaAbstract : METAABSTRACT.METAABSTRACT with module MetaSyn = MetaSyn' +end) : INIT with module MetaSyn = Init__0.MetaSyn' diff --git a/src/m2/Lemma_intf.ml b/src/M2/LEMMA.ml similarity index 100% rename from src/m2/Lemma_intf.ml rename to src/M2/LEMMA.ml diff --git a/src/m2/Lemma.ml b/src/M2/Lemma.ml similarity index 95% rename from src/m2/Lemma.ml rename to src/M2/Lemma.ml index 1933e6c..a4ea6df 100644 --- a/src/m2/Lemma.ml +++ b/src/M2/Lemma.ml @@ -4,7 +4,7 @@ open Metasyn (* Lemma *) (* Author: Carsten Schuermann *) -include Lemma_intf +include LEMMA (* signature LEMMA *) (* # 1 "src/m2/Lemma.fun.ml" *) @@ -16,9 +16,7 @@ open MetaAbstract (* Author: Carsten Schuermann *) module Lemma (Lemma__0 : sig module MetaSyn' : Metasyn.METASYN - - module MetaAbstract : - MetaAbstract_intf.METAABSTRACT with module MetaSyn = MetaSyn' + module MetaAbstract : METAABSTRACT.METAABSTRACT with module MetaSyn = MetaSyn' end) : LEMMA with module MetaSyn = Lemma__0.MetaSyn' = struct open Lemma__0 module MetaSyn = MetaAbstract.MetaSyn diff --git a/src/M2/Lemma.mli b/src/M2/Lemma.mli new file mode 100644 index 0000000..4683bd3 --- /dev/null +++ b/src/M2/Lemma.mli @@ -0,0 +1,6 @@ +include module type of LEMMA + +module Lemma (Lemma__0 : sig + module MetaSyn' : Metasyn.METASYN + module MetaAbstract : METAABSTRACT.METAABSTRACT with module MetaSyn = MetaSyn' +end) : LEMMA with module MetaSyn = Lemma__0.MetaSyn' diff --git a/src/m2/M2.ml b/src/M2/M2.ml similarity index 100% rename from src/m2/M2.ml rename to src/M2/M2.ml diff --git a/src/m2/M2.mli b/src/M2/M2.mli similarity index 100% rename from src/m2/M2.mli rename to src/M2/M2.mli diff --git a/src/m2/M2_.ml b/src/M2/M2_.ml similarity index 99% rename from src/m2/M2_.ml rename to src/M2/M2_.ml index e8adde4..f1544ff 100644 --- a/src/m2/M2_.ml +++ b/src/M2/M2_.ml @@ -207,4 +207,4 @@ module Skolem = Skolem (struct module Names = Names end) -module type PROVER = Prover_intf.PROVER +module type PROVER = PROVER.PROVER diff --git a/src/m2/M2_.mli b/src/M2/M2_.mli similarity index 69% rename from src/m2/M2_.mli rename to src/M2/M2_.mli index 31bae98..bfff6f7 100644 --- a/src/m2/M2_.mli +++ b/src/M2/M2_.mli @@ -16,6 +16,6 @@ open Strategy open Prover open Mpi open Skolem -module Skolem : Skolem_intf.SKOLEM -module IndexSkolem : Index_intf.INDEX -module M2Prover : Prover_intf.PROVER +module Skolem : SKOLEM.SKOLEM +module IndexSkolem : INDEX.INDEX +module M2Prover : PROVER.PROVER diff --git a/src/m2/MetaAbstract_intf.ml b/src/M2/METAABSTRACT.ml similarity index 100% rename from src/m2/MetaAbstract_intf.ml rename to src/M2/METAABSTRACT.ml diff --git a/src/m2/MetaGlobal_intf.ml b/src/M2/METAGLOBAL.ml similarity index 100% rename from src/m2/MetaGlobal_intf.ml rename to src/M2/METAGLOBAL.ml diff --git a/src/m2/MetaPrint_intf.ml b/src/M2/METAPRINT.ml similarity index 100% rename from src/m2/MetaPrint_intf.ml rename to src/M2/METAPRINT.ml diff --git a/src/m2/Metasyn_intf.ml b/src/M2/METASYN.ml similarity index 100% rename from src/m2/Metasyn_intf.ml rename to src/M2/METASYN.ml diff --git a/src/m2/Mpi_intf.ml b/src/M2/MPI.ml similarity index 100% rename from src/m2/Mpi_intf.ml rename to src/M2/MPI.ml diff --git a/src/m2/MetaAbstract.ml b/src/M2/MetaAbstract.ml similarity index 99% rename from src/m2/MetaAbstract.ml rename to src/M2/MetaAbstract.ml index cbb1497..daf3762 100644 --- a/src/m2/MetaAbstract.ml +++ b/src/M2/MetaAbstract.ml @@ -4,7 +4,7 @@ open Metasyn (* Meta Abstraction *) (* Author: Carsten Schuermann *) -include MetaAbstract_intf +include METAABSTRACT (* signature METAABSTRACT *) (* # 1 "src/m2/MetaAbstract.fun.ml" *) @@ -18,7 +18,7 @@ open Modetable module MetaAbstract (MetaAbstract__0 : sig module Global : GLOBAL module MetaSyn : Metasyn.METASYN - module MetaGlobal : MetaGlobal_intf.METAGLOBAL + module MetaGlobal : METAGLOBAL.METAGLOBAL module Abstract : ABSTRACT (*! sharing Abstract.IntSyn = MetaSyn'.IntSyn !*) diff --git a/src/m2/MetaAbstract.mli b/src/M2/MetaAbstract.mli similarity index 90% rename from src/m2/MetaAbstract.mli rename to src/M2/MetaAbstract.mli index 09cacc5..5f2c188 100644 --- a/src/m2/MetaAbstract.mli +++ b/src/M2/MetaAbstract.mli @@ -1,9 +1,9 @@ -include module type of MetaAbstract_intf +include module type of METAABSTRACT module MetaAbstract (MetaAbstract__0 : sig module Global : GLOBAL module MetaSyn : Metasyn.METASYN - module MetaGlobal : MetaGlobal_intf.METAGLOBAL + module MetaGlobal : METAGLOBAL.METAGLOBAL module Abstract : ABSTRACT (*! sharing Abstract.IntSyn = MetaSyn'.IntSyn !*) diff --git a/src/m2/MetaGlobal.ml b/src/M2/MetaGlobal.ml similarity index 95% rename from src/m2/MetaGlobal.ml rename to src/M2/MetaGlobal.ml index eb1342d..9968a56 100644 --- a/src/m2/MetaGlobal.ml +++ b/src/M2/MetaGlobal.ml @@ -3,7 +3,7 @@ open! Basis (* Global parameters *) (* Author: Carsten Schuermann *) -include MetaGlobal_intf +include METAGLOBAL (* signature METAGLOBAL *) (* # 1 "src/m2/MetaGlobal.fun.ml" *) diff --git a/src/M2/MetaGlobal.mli b/src/M2/MetaGlobal.mli new file mode 100644 index 0000000..bf19e51 --- /dev/null +++ b/src/M2/MetaGlobal.mli @@ -0,0 +1,2 @@ +include module type of METAGLOBAL +module MetaGlobal : METAGLOBAL.METAGLOBAL diff --git a/src/m2/MetaPrint.ml b/src/M2/MetaPrint.ml similarity index 96% rename from src/m2/MetaPrint.ml rename to src/M2/MetaPrint.ml index 7b38101..d9022b3 100644 --- a/src/m2/MetaPrint.ml +++ b/src/M2/MetaPrint.ml @@ -4,8 +4,8 @@ open Metasyn (* Meta printer for proof states *) (* Author: Carsten Schuermann *) -include MetaPrint_intf -(* signature MetaPrint_intf.METAPRINT *) +include METAPRINT +(* signature METAPRINT.METAPRINT *) (* # 1 "src/m2/MetaPrint.fun.ml" *) open! Basis @@ -21,7 +21,7 @@ module MetaPrint (MetaPrint__0 : sig module Print : PRINT (*! sharing Print.IntSyn = MetaSyn'.IntSyn !*) - module ClausePrint : ClausePrint_intf.CLAUSEPRINT + module ClausePrint : CLAUSEPRINT.CLAUSEPRINT end) : METAPRINT with module MetaSyn = MetaPrint__0.MetaSyn' = struct open MetaPrint__0 module MetaSyn = MetaSyn' diff --git a/src/m2/MetaPrint.mli b/src/M2/MetaPrint.mli similarity index 74% rename from src/m2/MetaPrint.mli rename to src/M2/MetaPrint.mli index cba15e7..7b32e3b 100644 --- a/src/m2/MetaPrint.mli +++ b/src/M2/MetaPrint.mli @@ -1,4 +1,4 @@ -include module type of MetaPrint_intf +include module type of METAPRINT module MetaPrint (MetaPrint__0 : sig module Global : GLOBAL @@ -7,5 +7,5 @@ module MetaPrint (MetaPrint__0 : sig module Print : PRINT (*! sharing Print.IntSyn = MetaSyn'.IntSyn !*) - module ClausePrint : ClausePrint_intf.CLAUSEPRINT + module ClausePrint : CLAUSEPRINT.CLAUSEPRINT end) : METAPRINT with module MetaSyn = MetaPrint__0.MetaSyn' diff --git a/src/m2/Metasyn.ml b/src/M2/Metasyn.ml similarity index 99% rename from src/m2/Metasyn.ml rename to src/M2/Metasyn.ml index f1f22d0..4742a94 100644 --- a/src/m2/Metasyn.ml +++ b/src/M2/Metasyn.ml @@ -3,7 +3,7 @@ open! Basis (* Meta syntax *) (* Author: Carsten Schuermann *) -include Metasyn_intf +include METASYN (* signature METASYN *) (* # 1 "src/m2/Metasyn.fun.ml" *) diff --git a/src/m2/Metasyn.mli b/src/M2/Metasyn.mli similarity index 55% rename from src/m2/Metasyn.mli rename to src/M2/Metasyn.mli index 603016c..38d2cee 100644 --- a/src/m2/Metasyn.mli +++ b/src/M2/Metasyn.mli @@ -1,2 +1,2 @@ -include module type of Metasyn_intf +include module type of METASYN module Make_MetaSyn (Whnf : WHNF) : METASYN diff --git a/src/m2/Mpi.ml b/src/M2/Mpi.ml similarity index 94% rename from src/m2/Mpi.ml rename to src/M2/Mpi.ml index f12f4d3..7cc1acc 100644 --- a/src/m2/Mpi.ml +++ b/src/M2/Mpi.ml @@ -4,7 +4,7 @@ open Metasyn (* Meta Prover Interface *) (* Author: Carsten Schuermann *) -include Mpi_intf +include MPI (* signature MPI *) (* # 1 "src/m2/Mpi.fun.ml" *) @@ -25,21 +25,21 @@ open Ring (* Meta Prover Interface *) (* Author: Carsten Schuermann *) module Mpi (Mpi__0 : sig - module MetaGlobal : MetaGlobal_intf.METAGLOBAL + module MetaGlobal : METAGLOBAL.METAGLOBAL module MetaSyn' : Metasyn.METASYN - module Init : Init_intf.INIT with module MetaSyn = MetaSyn' - module Filling : Filling_intf.FILLING with module MetaSyn = MetaSyn' - module Splitting : Splitting_intf.SPLITTING with module MetaSyn = MetaSyn' - module Recursion : Recursion_intf.RECURSION with module MetaSyn = MetaSyn' - module Lemma : Lemma_intf.LEMMA with module MetaSyn = MetaSyn' - module Strategy : Strategy_intf.STRATEGY with module MetaSyn = MetaSyn' - module Qed : Qed_intf.QED with module MetaSyn = MetaSyn' - module MetaPrint : MetaPrint_intf.METAPRINT with module MetaSyn = MetaSyn' + module Init : INIT.INIT with module MetaSyn = MetaSyn' + module Filling : FILLING.FILLING with module MetaSyn = MetaSyn' + module Splitting : SPLITTING.SPLITTING with module MetaSyn = MetaSyn' + module Recursion : RECURSION.RECURSION with module MetaSyn = MetaSyn' + module Lemma : LEMMA.LEMMA with module MetaSyn = MetaSyn' + module Strategy : STRATEGY.STRATEGY with module MetaSyn = MetaSyn' + module Qed : QED.QED with module MetaSyn = MetaSyn' + module MetaPrint : METAPRINT.METAPRINT with module MetaSyn = MetaSyn' module Names : NAMES (*! sharing Names.IntSyn = MetaSyn'.IntSyn !*) - module Timers : Timers_intf.TIMERS - module Ring : Ring_intf.RING + module Timers : TIMERS.TIMERS + module Ring : RING.RING end) : MPI with module MetaSyn = Mpi__0.MetaSyn' = struct open Mpi__0 module MetaSyn = MetaSyn' diff --git a/src/M2/Mpi.mli b/src/M2/Mpi.mli new file mode 100644 index 0000000..647a3ef --- /dev/null +++ b/src/M2/Mpi.mli @@ -0,0 +1,19 @@ +include module type of MPI + +module Mpi (Mpi__0 : sig + module MetaGlobal : METAGLOBAL.METAGLOBAL + module MetaSyn' : Metasyn.METASYN + module Init : INIT.INIT with module MetaSyn = MetaSyn' + module Filling : FILLING.FILLING with module MetaSyn = MetaSyn' + module Splitting : SPLITTING.SPLITTING with module MetaSyn = MetaSyn' + module Recursion : RECURSION.RECURSION with module MetaSyn = MetaSyn' + module Lemma : LEMMA.LEMMA with module MetaSyn = MetaSyn' + module Strategy : STRATEGY.STRATEGY with module MetaSyn = MetaSyn' + module Qed : QED.QED with module MetaSyn = MetaSyn' + module MetaPrint : METAPRINT.METAPRINT with module MetaSyn = MetaSyn' + module Names : NAMES + + (*! sharing Names.IntSyn = MetaSyn'.IntSyn !*) + module Timers : TIMERS.TIMERS + module Ring : RING.RING +end) : MPI with module MetaSyn = Mpi__0.MetaSyn' diff --git a/src/m2/Prover_intf.ml b/src/M2/PROVER.ml similarity index 100% rename from src/m2/Prover_intf.ml rename to src/M2/PROVER.ml diff --git a/src/m2/Prover.ml b/src/M2/Prover.ml similarity index 91% rename from src/m2/Prover.ml rename to src/M2/Prover.ml index 85f95fe..a1a3fe8 100644 --- a/src/m2/Prover.ml +++ b/src/M2/Prover.ml @@ -3,7 +3,7 @@ open! Basis (* Meta Prover *) (* Author: Carsten Schuermann *) -include Prover_intf +include PROVER (* signature PROVER *) (* # 1 "src/m2/Prover.fun.ml" *) @@ -22,19 +22,19 @@ open Timers (* Meta Prover *) (* Author: Carsten Schuermann *) module Prover (Prover__0 : sig - module MetaGlobal : MetaGlobal_intf.METAGLOBAL + module MetaGlobal : METAGLOBAL.METAGLOBAL module MetaSyn' : Metasyn.METASYN - module Init : Init_intf.INIT with module MetaSyn = MetaSyn' - module Strategy : Strategy_intf.STRATEGY with module MetaSyn = MetaSyn' - module Filling : Filling_intf.FILLING with module MetaSyn = MetaSyn' - module Splitting : Splitting_intf.SPLITTING with module MetaSyn = MetaSyn' - module Recursion : Recursion_intf.RECURSION with module MetaSyn = MetaSyn' - module Qed : Qed_intf.QED with module MetaSyn = MetaSyn' - module MetaPrint : MetaPrint_intf.METAPRINT with module MetaSyn = MetaSyn' + module Init : INIT.INIT with module MetaSyn = MetaSyn' + module Strategy : STRATEGY.STRATEGY with module MetaSyn = MetaSyn' + module Filling : FILLING.FILLING with module MetaSyn = MetaSyn' + module Splitting : SPLITTING.SPLITTING with module MetaSyn = MetaSyn' + module Recursion : RECURSION.RECURSION with module MetaSyn = MetaSyn' + module Qed : QED.QED with module MetaSyn = MetaSyn' + module MetaPrint : METAPRINT.METAPRINT with module MetaSyn = MetaSyn' module Names : NAMES (*! sharing Names.IntSyn = MetaSyn'.IntSyn !*) - module Timers : Timers_intf.TIMERS + module Timers : TIMERS.TIMERS end) : PROVER = struct open Prover__0 diff --git a/src/M2/Prover.mli b/src/M2/Prover.mli new file mode 100644 index 0000000..4f728e5 --- /dev/null +++ b/src/M2/Prover.mli @@ -0,0 +1,17 @@ +include module type of PROVER + +module Prover (Prover__0 : sig + module MetaGlobal : METAGLOBAL.METAGLOBAL + module MetaSyn' : Metasyn.METASYN + module Init : INIT.INIT with module MetaSyn = MetaSyn' + module Strategy : STRATEGY.STRATEGY with module MetaSyn = MetaSyn' + module Filling : FILLING.FILLING with module MetaSyn = MetaSyn' + module Splitting : SPLITTING.SPLITTING with module MetaSyn = MetaSyn' + module Recursion : RECURSION.RECURSION with module MetaSyn = MetaSyn' + module Qed : QED.QED with module MetaSyn = MetaSyn' + module MetaPrint : METAPRINT.METAPRINT with module MetaSyn = MetaSyn' + module Names : NAMES + + (*! sharing Names.IntSyn = MetaSyn'.IntSyn !*) + module Timers : TIMERS.TIMERS +end) : PROVER diff --git a/src/m2/Qed_intf.ml b/src/M2/QED.ml similarity index 100% rename from src/m2/Qed_intf.ml rename to src/M2/QED.ml diff --git a/src/m2/Qed.ml b/src/M2/Qed.ml similarity index 97% rename from src/m2/Qed.ml rename to src/M2/Qed.ml index 02473ad..0c0a0e2 100644 --- a/src/m2/Qed.ml +++ b/src/M2/Qed.ml @@ -4,7 +4,7 @@ open Metasyn (* Qed *) (* Author: Carsten Schuermann *) -include Qed_intf +include QED (* signature QED *) (* # 1 "src/m2/Qed.fun.ml" *) diff --git a/src/m2/Qed.mli b/src/M2/Qed.mli similarity index 80% rename from src/m2/Qed.mli rename to src/M2/Qed.mli index b314cd1..6ee2a85 100644 --- a/src/m2/Qed.mli +++ b/src/M2/Qed.mli @@ -1,4 +1,4 @@ -include module type of Qed_intf +include module type of QED module Qed (Qed__0 : sig module Global : GLOBAL diff --git a/src/m2/README b/src/M2/README similarity index 100% rename from src/m2/README rename to src/M2/README diff --git a/src/m2/Recursion_intf.ml b/src/M2/RECURSION.ml similarity index 100% rename from src/m2/Recursion_intf.ml rename to src/M2/RECURSION.ml diff --git a/src/m2/Recursion.ml b/src/M2/Recursion.ml similarity index 98% rename from src/m2/Recursion.ml rename to src/M2/Recursion.ml index 5f84723..af04b61 100644 --- a/src/m2/Recursion.ml +++ b/src/M2/Recursion.ml @@ -4,7 +4,7 @@ open Metasyn (* Recursion *) (* Author: Carsten Schuermann *) -include Recursion_intf +include RECURSION (* signature RECURSION *) (* # 1 "src/m2/Recursion.fun.ml" *) @@ -22,7 +22,7 @@ open MetaAbstract (* See [Rohwedder,Pfenning ESOP'96] *) module Recursion (Recursion__0 : sig module Global : GLOBAL - module MetaGlobal : MetaGlobal_intf.METAGLOBAL + module MetaGlobal : METAGLOBAL.METAGLOBAL module MetaSyn' : Metasyn.METASYN module Whnf : WHNF @@ -48,16 +48,12 @@ module Recursion (Recursion__0 : sig module ModeTable : Modetable.MODETABLE (*! sharing Modes.Modesyn.ModeSyn.IntSyn = MetaSyn'.IntSyn !*) - module Lemma : Lemma_intf.LEMMA with module MetaSyn = MetaSyn' - module Filling : Filling_intf.FILLING with module MetaSyn = MetaSyn' - module MetaPrint : MetaPrint_intf.METAPRINT with module MetaSyn = MetaSyn' - - module MetaAbstract : - MetaAbstract_intf.METAABSTRACT with module MetaSyn = MetaSyn' - + module Lemma : LEMMA.LEMMA with module MetaSyn = MetaSyn' + module Filling : FILLING.FILLING with module MetaSyn = MetaSyn' + module MetaPrint : METAPRINT.METAPRINT with module MetaSyn = MetaSyn' + module MetaAbstract : METAABSTRACT.METAABSTRACT with module MetaSyn = MetaSyn' module Formatter : FORMATTER -end) : Recursion_intf.RECURSION with module MetaSyn = Recursion__0.MetaSyn' = -struct +end) : RECURSION.RECURSION with module MetaSyn = Recursion__0.MetaSyn' = struct open Recursion__0 module MetaSyn = MetaSyn' diff --git a/src/m2/Recursion.mli b/src/M2/Recursion.mli similarity index 63% rename from src/m2/Recursion.mli rename to src/M2/Recursion.mli index 1a85e57..338de60 100644 --- a/src/m2/Recursion.mli +++ b/src/M2/Recursion.mli @@ -1,8 +1,8 @@ -include module type of Recursion_intf +include module type of RECURSION module Recursion (Recursion__0 : sig module Global : GLOBAL - module MetaGlobal : MetaGlobal_intf.METAGLOBAL + module MetaGlobal : METAGLOBAL.METAGLOBAL module MetaSyn' : Metasyn.METASYN module Whnf : WHNF @@ -28,12 +28,9 @@ module Recursion (Recursion__0 : sig module ModeTable : Modetable.MODETABLE (*! sharing Modes.Modesyn.ModeSyn.IntSyn = MetaSyn'.IntSyn !*) - module Lemma : Lemma_intf.LEMMA with module MetaSyn = MetaSyn' - module Filling : Filling_intf.FILLING with module MetaSyn = MetaSyn' - module MetaPrint : MetaPrint_intf.METAPRINT with module MetaSyn = MetaSyn' - - module MetaAbstract : - MetaAbstract_intf.METAABSTRACT with module MetaSyn = MetaSyn' - + module Lemma : LEMMA.LEMMA with module MetaSyn = MetaSyn' + module Filling : FILLING.FILLING with module MetaSyn = MetaSyn' + module MetaPrint : METAPRINT.METAPRINT with module MetaSyn = MetaSyn' + module MetaAbstract : METAABSTRACT.METAABSTRACT with module MetaSyn = MetaSyn' module Formatter : FORMATTER -end) : Recursion_intf.RECURSION with module MetaSyn = Recursion__0.MetaSyn' +end) : RECURSION.RECURSION with module MetaSyn = Recursion__0.MetaSyn' diff --git a/src/m2/Search_intf.ml b/src/M2/SEARCH.ml similarity index 100% rename from src/m2/Search_intf.ml rename to src/M2/SEARCH.ml diff --git a/src/m2/Skolem_intf.ml b/src/M2/SKOLEM.ml similarity index 100% rename from src/m2/Skolem_intf.ml rename to src/M2/SKOLEM.ml diff --git a/src/m2/Splitting_intf.ml b/src/M2/SPLITTING.ml similarity index 100% rename from src/m2/Splitting_intf.ml rename to src/M2/SPLITTING.ml diff --git a/src/m2/Strategy_intf.ml b/src/M2/STRATEGY.ml similarity index 100% rename from src/m2/Strategy_intf.ml rename to src/M2/STRATEGY.ml diff --git a/src/m2/Search.ml b/src/M2/Search.ml similarity index 99% rename from src/m2/Search.ml rename to src/M2/Search.ml index 635825f..8e9463b 100644 --- a/src/m2/Search.ml +++ b/src/M2/Search.ml @@ -4,7 +4,7 @@ open Metasyn (* Basic search engine *) (* Author: Carsten Schuermann *) -include Search_intf +include SEARCH (* signature SEARCH *) (* # 1 "src/m2/Search.fun.ml" *) @@ -16,7 +16,7 @@ open MetaGlobal (* Author: Carsten Schuermann *) module OLDSearch (OLDSearch__0 : sig (*! structure IntSyn' : INTSYN !*) - module MetaGlobal : MetaGlobal_intf.METAGLOBAL + module MetaGlobal : METAGLOBAL.METAGLOBAL module MetaSyn' : Metasyn.METASYN (*! sharing MetaSyn'.IntSyn = IntSyn' !*) diff --git a/src/m2/Search.mli b/src/M2/Search.mli similarity index 91% rename from src/m2/Search.mli rename to src/M2/Search.mli index db50c04..9832857 100644 --- a/src/m2/Search.mli +++ b/src/M2/Search.mli @@ -1,8 +1,8 @@ -include module type of Search_intf +include module type of SEARCH module OLDSearch (OLDSearch__0 : sig (*! structure IntSyn' : INTSYN !*) - module MetaGlobal : MetaGlobal_intf.METAGLOBAL + module MetaGlobal : METAGLOBAL.METAGLOBAL module MetaSyn' : Metasyn.METASYN (*! sharing MetaSyn'.IntSyn = IntSyn' !*) diff --git a/src/m2/Skolem.ml b/src/M2/Skolem.ml similarity index 98% rename from src/m2/Skolem.ml rename to src/M2/Skolem.ml index 8ee221e..f014d80 100644 --- a/src/m2/Skolem.ml +++ b/src/M2/Skolem.ml @@ -3,7 +3,7 @@ open! Basis (* Skolem administration *) (* Author: Carsten Schuermann *) -include Skolem_intf +include SKOLEM (* signature SKOLEM *) (* # 1 "src/m2/Skolem.fun.ml" *) @@ -36,7 +36,7 @@ module Skolem (Skolem__0 : sig module Compile : COMPILE (*! sharing Compile.IntSyn = IntSyn' !*) - module Timers : Timers_intf.TIMERS + module Timers : TIMERS.TIMERS module Names : NAMES end) : SKOLEM = struct open Skolem__0 diff --git a/src/m2/Skolem.mli b/src/M2/Skolem.mli similarity index 90% rename from src/m2/Skolem.mli rename to src/M2/Skolem.mli index 7c9f7dc..c5b8142 100644 --- a/src/m2/Skolem.mli +++ b/src/M2/Skolem.mli @@ -1,4 +1,4 @@ -include module type of Skolem_intf +include module type of SKOLEM module Skolem (Skolem__0 : sig (* Skolem constant administration *) @@ -24,6 +24,6 @@ module Skolem (Skolem__0 : sig module Compile : COMPILE (*! sharing Compile.IntSyn = IntSyn' !*) - module Timers : Timers_intf.TIMERS + module Timers : TIMERS.TIMERS module Names : NAMES end) : SKOLEM diff --git a/src/m2/Splitting.ml b/src/M2/Splitting.ml similarity index 98% rename from src/m2/Splitting.ml rename to src/M2/Splitting.ml index 24e7864..22ba42b 100644 --- a/src/m2/Splitting.ml +++ b/src/M2/Splitting.ml @@ -4,7 +4,7 @@ open Metasyn (* Splitting *) (* Author: Carsten Schuermann *) -include Splitting_intf +include SPLITTING (* signature SPLITTING *) (* # 1 "src/m2/Splitting.fun.ml" *) @@ -19,11 +19,8 @@ open Modetable module Splitting (Splitting__0 : sig module Global : GLOBAL module MetaSyn' : Metasyn.METASYN - - module MetaAbstract : - MetaAbstract_intf.METAABSTRACT with module MetaSyn = MetaSyn' - - module MetaPrint : MetaPrint_intf.METAPRINT with module MetaSyn = MetaSyn' + module MetaAbstract : METAABSTRACT.METAABSTRACT with module MetaSyn = MetaSyn' + module MetaPrint : METAPRINT.METAPRINT with module MetaSyn = MetaSyn' module ModeTable : Modetable.MODETABLE (*! sharing Modes.Modesyn.ModeSyn.IntSyn = MetaSyn'.IntSyn !*) @@ -37,8 +34,7 @@ module Splitting (Splitting__0 : sig (*! sharing Print.IntSyn = MetaSyn'.IntSyn !*) module Unify : UNIFY -end) : Splitting_intf.SPLITTING with module MetaSyn = Splitting__0.MetaSyn' = -struct +end) : SPLITTING.SPLITTING with module MetaSyn = Splitting__0.MetaSyn' = struct open Splitting__0 module MetaSyn = MetaAbstract.MetaSyn diff --git a/src/m2/Splitting.mli b/src/M2/Splitting.mli similarity index 61% rename from src/m2/Splitting.mli rename to src/M2/Splitting.mli index 243f9a2..2ab96e0 100644 --- a/src/m2/Splitting.mli +++ b/src/M2/Splitting.mli @@ -1,13 +1,10 @@ -include module type of Splitting_intf +include module type of SPLITTING module Splitting (Splitting__0 : sig module Global : GLOBAL module MetaSyn' : Metasyn.METASYN - - module MetaAbstract : - MetaAbstract_intf.METAABSTRACT with module MetaSyn = MetaSyn' - - module MetaPrint : MetaPrint_intf.METAPRINT with module MetaSyn = MetaSyn' + module MetaAbstract : METAABSTRACT.METAABSTRACT with module MetaSyn = MetaSyn' + module MetaPrint : METAPRINT.METAPRINT with module MetaSyn = MetaSyn' module ModeTable : Modetable.MODETABLE (*! sharing Modes.Modesyn.ModeSyn.IntSyn = MetaSyn'.IntSyn !*) @@ -21,4 +18,4 @@ module Splitting (Splitting__0 : sig (*! sharing Print.IntSyn = MetaSyn'.IntSyn !*) module Unify : UNIFY -end) : Splitting_intf.SPLITTING with module MetaSyn = Splitting__0.MetaSyn' +end) : SPLITTING.SPLITTING with module MetaSyn = Splitting__0.MetaSyn' diff --git a/src/m2/Strategy.ml b/src/M2/Strategy.ml similarity index 88% rename from src/m2/Strategy.ml rename to src/M2/Strategy.ml index 452dde1..96e0814 100644 --- a/src/m2/Strategy.ml +++ b/src/M2/Strategy.ml @@ -4,7 +4,7 @@ open Metasyn (* Strategy *) (* Author: Carsten Schuermann *) -include Strategy_intf +include STRATEGY (* open cases -> remaining cases * solved cases *) (* signature STRATEGY *) @@ -25,17 +25,16 @@ open TimeLimit (* Strategy *) (* Author: Carsten Schuermann *) module StrategyFRS (StrategyFRS__0 : sig - module MetaGlobal : MetaGlobal_intf.METAGLOBAL + module MetaGlobal : METAGLOBAL.METAGLOBAL module MetaSyn' : Metasyn.METASYN - module Filling : Filling_intf.FILLING with module MetaSyn = MetaSyn' - module Splitting : Splitting_intf.SPLITTING with module MetaSyn = MetaSyn' - module Recursion : Recursion_intf.RECURSION with module MetaSyn = MetaSyn' - module Lemma : Lemma_intf.LEMMA with module MetaSyn = MetaSyn' - module Qed : Qed_intf.QED with module MetaSyn = MetaSyn' - module MetaPrint : MetaPrint_intf.METAPRINT with module MetaSyn = MetaSyn' - module Timers : Timers_intf.TIMERS -end) : Strategy_intf.STRATEGY with module MetaSyn = StrategyFRS__0.MetaSyn' = -struct + module Filling : FILLING.FILLING with module MetaSyn = MetaSyn' + module Splitting : SPLITTING.SPLITTING with module MetaSyn = MetaSyn' + module Recursion : RECURSION.RECURSION with module MetaSyn = MetaSyn' + module Lemma : LEMMA.LEMMA with module MetaSyn = MetaSyn' + module Qed : QED.QED with module MetaSyn = MetaSyn' + module MetaPrint : METAPRINT.METAPRINT with module MetaSyn = MetaSyn' + module Timers : TIMERS.TIMERS +end) : STRATEGY.STRATEGY with module MetaSyn = StrategyFRS__0.MetaSyn' = struct open StrategyFRS__0 module MetaSyn = MetaSyn' @@ -196,17 +195,16 @@ end (* local *) (* functor StrategyFRS *) module StrategyRFS (StrategyRFS__1 : sig - module MetaGlobal : MetaGlobal_intf.METAGLOBAL + module MetaGlobal : METAGLOBAL.METAGLOBAL module MetaSyn' : Metasyn.METASYN - module Filling : Filling_intf.FILLING with module MetaSyn = MetaSyn' - module Splitting : Splitting_intf.SPLITTING with module MetaSyn = MetaSyn' - module Recursion : Recursion_intf.RECURSION with module MetaSyn = MetaSyn' - module Lemma : Lemma_intf.LEMMA with module MetaSyn = MetaSyn' - module Qed : Qed_intf.QED with module MetaSyn = MetaSyn' - module MetaPrint : MetaPrint_intf.METAPRINT with module MetaSyn = MetaSyn' - module Timers : Timers_intf.TIMERS -end) : Strategy_intf.STRATEGY with module MetaSyn = StrategyRFS__1.MetaSyn' = -struct + module Filling : FILLING.FILLING with module MetaSyn = MetaSyn' + module Splitting : SPLITTING.SPLITTING with module MetaSyn = MetaSyn' + module Recursion : RECURSION.RECURSION with module MetaSyn = MetaSyn' + module Lemma : LEMMA.LEMMA with module MetaSyn = MetaSyn' + module Qed : QED.QED with module MetaSyn = MetaSyn' + module MetaPrint : METAPRINT.METAPRINT with module MetaSyn = MetaSyn' + module Timers : TIMERS.TIMERS +end) : STRATEGY.STRATEGY with module MetaSyn = StrategyRFS__1.MetaSyn' = struct open StrategyRFS__1 module MetaSyn = MetaSyn' @@ -363,12 +361,11 @@ end (* local *) (* functor StrategyRFS *) module Strategy (Strategy__2 : sig - module MetaGlobal : MetaGlobal_intf.METAGLOBAL + module MetaGlobal : METAGLOBAL.METAGLOBAL module MetaSyn' : Metasyn.METASYN - module StrategyFRS : Strategy_intf.STRATEGY with module MetaSyn = MetaSyn' - module StrategyRFS : Strategy_intf.STRATEGY with module MetaSyn = MetaSyn' -end) : Strategy_intf.STRATEGY with module MetaSyn = Strategy__2.MetaSyn' = -struct + module StrategyFRS : STRATEGY.STRATEGY with module MetaSyn = MetaSyn' + module StrategyRFS : STRATEGY.STRATEGY with module MetaSyn = MetaSyn' +end) : STRATEGY.STRATEGY with module MetaSyn = Strategy__2.MetaSyn' = struct open Strategy__2 module MetaSyn = MetaSyn' diff --git a/src/M2/Strategy.mli b/src/M2/Strategy.mli new file mode 100644 index 0000000..fa8ebd3 --- /dev/null +++ b/src/M2/Strategy.mli @@ -0,0 +1,32 @@ +include module type of STRATEGY + +module StrategyFRS (StrategyFRS__0 : sig + module MetaGlobal : METAGLOBAL.METAGLOBAL + module MetaSyn' : Metasyn.METASYN + module Filling : FILLING.FILLING with module MetaSyn = MetaSyn' + module Splitting : SPLITTING.SPLITTING with module MetaSyn = MetaSyn' + module Recursion : RECURSION.RECURSION with module MetaSyn = MetaSyn' + module Lemma : LEMMA.LEMMA with module MetaSyn = MetaSyn' + module Qed : QED.QED with module MetaSyn = MetaSyn' + module MetaPrint : METAPRINT.METAPRINT with module MetaSyn = MetaSyn' + module Timers : TIMERS.TIMERS +end) : STRATEGY.STRATEGY with module MetaSyn = StrategyFRS__0.MetaSyn' + +module StrategyRFS (StrategyRFS__1 : sig + module MetaGlobal : METAGLOBAL.METAGLOBAL + module MetaSyn' : Metasyn.METASYN + module Filling : FILLING.FILLING with module MetaSyn = MetaSyn' + module Splitting : SPLITTING.SPLITTING with module MetaSyn = MetaSyn' + module Recursion : RECURSION.RECURSION with module MetaSyn = MetaSyn' + module Lemma : LEMMA.LEMMA with module MetaSyn = MetaSyn' + module Qed : QED.QED with module MetaSyn = MetaSyn' + module MetaPrint : METAPRINT.METAPRINT with module MetaSyn = MetaSyn' + module Timers : TIMERS.TIMERS +end) : STRATEGY.STRATEGY with module MetaSyn = StrategyRFS__1.MetaSyn' + +module Strategy (Strategy__2 : sig + module MetaGlobal : METAGLOBAL.METAGLOBAL + module MetaSyn' : Metasyn.METASYN + module StrategyFRS : STRATEGY.STRATEGY with module MetaSyn = MetaSyn' + module StrategyRFS : STRATEGY.STRATEGY with module MetaSyn = MetaSyn' +end) : STRATEGY.STRATEGY with module MetaSyn = Strategy__2.MetaSyn' diff --git a/src/m2/WALK b/src/M2/WALK similarity index 100% rename from src/m2/WALK rename to src/M2/WALK diff --git a/src/m2/dune b/src/M2/dune similarity index 86% rename from src/m2/dune rename to src/M2/dune index 2234dc2..e6bcf32 100644 --- a/src/m2/dune +++ b/src/M2/dune @@ -20,21 +20,21 @@ Prover Mpi M2_ - Filling_intf - Init_intf - Lemma_intf - MetaAbstract_intf - MetaGlobal_intf - MetaPrint_intf - Metasyn_intf - Mpi_intf - Prover_intf - Qed_intf - Recursion_intf - Search_intf - Skolem_intf - Splitting_intf - Strategy_intf) + FILLING + INIT + LEMMA + METAABSTRACT + METAGLOBAL + METAPRINT + METASYN + MPI + PROVER + QED + RECURSION + SEARCH + SKOLEM + SPLITTING + STRATEGY) (libraries display stream diff --git a/src/meta/Funnames_intf.ml b/src/Meta/FUNNAMES.ml similarity index 100% rename from src/meta/Funnames_intf.ml rename to src/Meta/FUNNAMES.ml diff --git a/src/meta/Funprint_intf.ml b/src/Meta/FUNPRINT.ml similarity index 100% rename from src/meta/Funprint_intf.ml rename to src/Meta/FUNPRINT.ml diff --git a/src/meta/Funsyn_intf.ml b/src/Meta/FUNSYN.ml similarity index 100% rename from src/meta/Funsyn_intf.ml rename to src/Meta/FUNSYN.ml diff --git a/src/meta/Funtypecheck_intf.ml b/src/Meta/FUNTYPECHECK.ml similarity index 100% rename from src/meta/Funtypecheck_intf.ml rename to src/Meta/FUNTYPECHECK.ml diff --git a/src/meta/Funweaken_intf.ml b/src/Meta/FUNWEAKEN.ml similarity index 100% rename from src/meta/Funweaken_intf.ml rename to src/Meta/FUNWEAKEN.ml diff --git a/src/meta/Funnames.ml b/src/Meta/Funnames.ml similarity index 98% rename from src/meta/Funnames.ml rename to src/Meta/Funnames.ml index 6dd52d2..a968b09 100644 --- a/src/meta/Funnames.ml +++ b/src/Meta/Funnames.ml @@ -4,7 +4,7 @@ open Funsyn (* Names of Constants and Variables *) (* Author: Carsten Schuermann *) -include Funnames_intf +include FUNNAMES (* will mark if shadowed *) (* signature NAMES *) @@ -20,7 +20,7 @@ module FunNames (FunNames__0 : sig (*! structure FunSyn' : FUNSYN !*) module HashTable : TABLE with type key = string -end) : Funnames_intf.FUNNAMES = struct +end) : FUNNAMES.FUNNAMES = struct open FunNames__0 (*! structure FunSyn = FunSyn' !*) diff --git a/src/meta/Funnames.mli b/src/Meta/Funnames.mli similarity index 68% rename from src/meta/Funnames.mli rename to src/Meta/Funnames.mli index 8cd99da..a4005dc 100644 --- a/src/meta/Funnames.mli +++ b/src/Meta/Funnames.mli @@ -1,8 +1,8 @@ -include module type of Funnames_intf +include module type of FUNNAMES module FunNames (FunNames__0 : sig module Global : GLOBAL (*! structure FunSyn' : FUNSYN !*) module HashTable : TABLE with type key = string -end) : Funnames_intf.FUNNAMES +end) : FUNNAMES.FUNNAMES diff --git a/src/meta/Funprint.ml b/src/Meta/Funprint.ml similarity index 99% rename from src/meta/Funprint.ml rename to src/Meta/Funprint.ml index 7b4c782..5a384bc 100644 --- a/src/meta/Funprint.ml +++ b/src/Meta/Funprint.ml @@ -4,7 +4,7 @@ open Funsyn (* Printing of functional proof terms *) (* Author: Carsten Schuermann *) -include Funprint_intf +include FUNPRINT (* signature PRINT *) (* # 1 "src/meta/Funprint.fun.ml" *) @@ -20,7 +20,7 @@ module FunPrint (FunPrint__0 : sig (*! sharing Names.IntSyn = FunSyn'.IntSyn !*) module Print : PRINT -end) : Funprint_intf.FUNPRINT = struct +end) : FUNPRINT.FUNPRINT = struct (*! structure FunSyn = FunSyn' !*) module Formatter = Formatter diff --git a/src/meta/Funprint.mli b/src/Meta/Funprint.mli similarity index 74% rename from src/meta/Funprint.mli rename to src/Meta/Funprint.mli index 4d261bc..3a6d275 100644 --- a/src/meta/Funprint.mli +++ b/src/Meta/Funprint.mli @@ -1,4 +1,4 @@ -include module type of Funprint_intf +include module type of FUNPRINT module FunPrint (FunPrint__0 : sig (*! structure FunSyn' : FUNSYN !*) @@ -7,4 +7,4 @@ module FunPrint (FunPrint__0 : sig (*! sharing Names.IntSyn = FunSyn'.IntSyn !*) module Print : PRINT -end) : Funprint_intf.FUNPRINT +end) : FUNPRINT.FUNPRINT diff --git a/src/meta/Funsyn.ml b/src/Meta/Funsyn.ml similarity index 99% rename from src/meta/Funsyn.ml rename to src/Meta/Funsyn.ml index c8bed81..281a02c 100644 --- a/src/meta/Funsyn.ml +++ b/src/Meta/Funsyn.ml @@ -3,7 +3,7 @@ open! Basis (* Internal syntax for functional proof term calculus *) (* Author: Carsten Schuermann *) -include Funsyn_intf +include FUNSYN (* Signature FUNSYN *) (* # 1 "src/meta/Funsyn.fun.ml" *) diff --git a/src/meta/Funsyn.mli b/src/Meta/Funsyn.mli similarity index 69% rename from src/meta/Funsyn.mli rename to src/Meta/Funsyn.mli index 80e24b7..5e814b3 100644 --- a/src/meta/Funsyn.mli +++ b/src/Meta/Funsyn.mli @@ -1,3 +1,3 @@ -include module type of Funsyn_intf +include module type of FUNSYN module Make_FunSyn (Whnf : WHNF) (Conv : CONV) : FUNSYN module FunSyn : FUNSYN diff --git a/src/meta/Funtypecheck.ml b/src/Meta/Funtypecheck.ml similarity index 98% rename from src/meta/Funtypecheck.ml rename to src/Meta/Funtypecheck.ml index eb3e2bc..d46f88f 100644 --- a/src/meta/Funtypecheck.ml +++ b/src/Meta/Funtypecheck.ml @@ -6,7 +6,7 @@ open Statesyn (* Type checking for functional proof term calculus *) (* Author: Carsten Schuermann *) -include Funtypecheck_intf +include FUNTYPECHECK (* Signature FUNTYPECHECK *) (* # 1 "src/meta/Funtypecheck.fun.ml" *) @@ -19,7 +19,7 @@ module FunTypeCheck (FunTypeCheck__0 : sig (* Type checking for functional proof term calculus *) (* Author: Carsten Schuermann *) (*! structure FunSyn' : FUNSYN !*) - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.FunSyn = FunSyn' !*) module Abstract : ABSTRACT @@ -40,11 +40,11 @@ module FunTypeCheck (FunTypeCheck__0 : sig module Subordinate : Subordinate_.SUBORDINATE (*! sharing Subordinate.IntSyn = FunSyn'.IntSyn !*) - module Weaken : Weaken_intf.WEAKEN + module Weaken : WEAKEN.WEAKEN (*! sharing Weaken.IntSyn = FunSyn'.IntSyn !*) - module FunPrint : Funprint_intf.FUNPRINT -end) : Funtypecheck_intf.FUNTYPECHECK = struct + module FunPrint : FUNPRINT.FUNPRINT +end) : FUNTYPECHECK.FUNTYPECHECK = struct (*! structure FunSyn = FunSyn' !*) open FunTypeCheck__0 module StateSyn = StateSyn' diff --git a/src/meta/Funtypecheck.mli b/src/Meta/Funtypecheck.mli similarity index 78% rename from src/meta/Funtypecheck.mli rename to src/Meta/Funtypecheck.mli index 14146e6..34f68ce 100644 --- a/src/meta/Funtypecheck.mli +++ b/src/Meta/Funtypecheck.mli @@ -1,10 +1,10 @@ -include module type of Funtypecheck_intf +include module type of FUNTYPECHECK module FunTypeCheck (FunTypeCheck__0 : sig (* Type checking for functional proof term calculus *) (* Author: Carsten Schuermann *) (*! structure FunSyn' : FUNSYN !*) - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.FunSyn = FunSyn' !*) module Abstract : ABSTRACT @@ -25,8 +25,8 @@ module FunTypeCheck (FunTypeCheck__0 : sig module Subordinate : Subordinate_.SUBORDINATE (*! sharing Subordinate.IntSyn = FunSyn'.IntSyn !*) - module Weaken : Weaken_intf.WEAKEN + module Weaken : WEAKEN.WEAKEN (*! sharing Weaken.IntSyn = FunSyn'.IntSyn !*) - module FunPrint : Funprint_intf.FUNPRINT -end) : Funtypecheck_intf.FUNTYPECHECK + module FunPrint : FUNPRINT.FUNPRINT +end) : FUNTYPECHECK.FUNTYPECHECK diff --git a/src/meta/Funweaken.ml b/src/Meta/Funweaken.ml similarity index 95% rename from src/meta/Funweaken.ml rename to src/Meta/Funweaken.ml index ededfd2..2ac6d04 100644 --- a/src/meta/Funweaken.ml +++ b/src/Meta/Funweaken.ml @@ -4,7 +4,7 @@ open Funsyn (* Weakening substitutions for meta substitutions *) (* Author: Carsten Schuermann *) -include Funweaken_intf +include FUNWEAKEN (* signature FUNWEAKEN *) (* # 1 "src/meta/Funweaken.fun.ml" *) @@ -14,8 +14,8 @@ open! Basis (* Weakening substitutions for meta substitutions *) (* Author: Carsten Schuermann *) module FunWeaken (FunWeaken__0 : sig - module Weaken : Weaken_intf.WEAKEN -end) : Funweaken_intf.FUNWEAKEN = struct + module Weaken : WEAKEN.WEAKEN +end) : FUNWEAKEN.FUNWEAKEN = struct (*! structure FunSyn = FunSyn' !*) open FunWeaken__0 diff --git a/src/Meta/Funweaken.mli b/src/Meta/Funweaken.mli new file mode 100644 index 0000000..a939920 --- /dev/null +++ b/src/Meta/Funweaken.mli @@ -0,0 +1,5 @@ +include module type of FUNWEAKEN + +module FunWeaken (FunWeaken__0 : sig + module Weaken : WEAKEN.WEAKEN +end) : FUNWEAKEN.FUNWEAKEN diff --git a/src/meta/Inference_intf.ml b/src/Meta/INFERENCE.ml similarity index 100% rename from src/meta/Inference_intf.ml rename to src/Meta/INFERENCE.ml diff --git a/src/meta/Interpret_intf.ml b/src/Meta/INTERPRET.ml similarity index 100% rename from src/meta/Interpret_intf.ml rename to src/Meta/INTERPRET.ml diff --git a/src/meta/Inference.ml b/src/Meta/Inference.ml similarity index 96% rename from src/meta/Inference.ml rename to src/Meta/Inference.ml index 38bf8bc..70e81b6 100644 --- a/src/meta/Inference.ml +++ b/src/Meta/Inference.ml @@ -9,7 +9,7 @@ open Statesyn (* Inference: Version 1.3 *) (* Author: Carsten Schuermann *) -include Inference_intf +include INFERENCE (* signature Inference *) (* # 1 "src/meta/Inference.fun.ml" *) @@ -24,7 +24,7 @@ module Inference (Inference__0 : sig (*! structure IntSyn : INTSYN !*) (*! structure FunSyn' : FUNSYN !*) (*! sharing FunSyn'.IntSyn = IntSyn !*) - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.FunSyn = FunSyn' !*) module Abstract : ABSTRACT @@ -33,10 +33,10 @@ module Inference (Inference__0 : sig module TypeCheck : TYPECHECK (*! sharing TypeCheck.IntSyn = IntSyn !*) - module FunTypeCheck : Funtypecheck_intf.FUNTYPECHECK + module FunTypeCheck : FUNTYPECHECK.FUNTYPECHECK (*! sharing FunTypeCheck.FunSyn = FunSyn' !*) - module UniqueSearch : Uniquesearch_intf.UNIQUESEARCH + module UniqueSearch : UNIQUESEARCH.UNIQUESEARCH (*! sharing UniqueSearch.IntSyn = IntSyn !*) (*! sharing UniqueSearch.FunSyn = FunSyn' !*) @@ -44,7 +44,7 @@ module Inference (Inference__0 : sig (*! sharing Print.IntSyn = IntSyn !*) module Whnf : WHNF -end) : Inference_intf.INFERENCE = struct +end) : INFERENCE.INFERENCE = struct (*! structure FunSyn = FunSyn' !*) open Inference__0 module StateSyn = StateSyn' diff --git a/src/meta/Inference.mli b/src/Meta/Inference.mli similarity index 73% rename from src/meta/Inference.mli rename to src/Meta/Inference.mli index 66d8906..bf59d19 100644 --- a/src/meta/Inference.mli +++ b/src/Meta/Inference.mli @@ -1,4 +1,4 @@ -include module type of Inference_intf +include module type of INFERENCE module Inference (Inference__0 : sig module MTPGlobal : MtpGlobal.MTPGLOBAL @@ -6,7 +6,7 @@ module Inference (Inference__0 : sig (*! structure IntSyn : INTSYN !*) (*! structure FunSyn' : FUNSYN !*) (*! sharing FunSyn'.IntSyn = IntSyn !*) - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.FunSyn = FunSyn' !*) module Abstract : ABSTRACT @@ -15,10 +15,10 @@ module Inference (Inference__0 : sig module TypeCheck : TYPECHECK (*! sharing TypeCheck.IntSyn = IntSyn !*) - module FunTypeCheck : Funtypecheck_intf.FUNTYPECHECK + module FunTypeCheck : FUNTYPECHECK.FUNTYPECHECK (*! sharing FunTypeCheck.FunSyn = FunSyn' !*) - module UniqueSearch : Uniquesearch_intf.UNIQUESEARCH + module UniqueSearch : UNIQUESEARCH.UNIQUESEARCH (*! sharing UniqueSearch.IntSyn = IntSyn !*) (*! sharing UniqueSearch.FunSyn = FunSyn' !*) @@ -26,4 +26,4 @@ module Inference (Inference__0 : sig (*! sharing Print.IntSyn = IntSyn !*) module Whnf : WHNF -end) : Inference_intf.INFERENCE +end) : INFERENCE.INFERENCE diff --git a/src/meta/Interpret.ml b/src/Meta/Interpret.ml similarity index 90% rename from src/meta/Interpret.ml rename to src/Meta/Interpret.ml index 271b94e..519f0d8 100644 --- a/src/meta/Interpret.ml +++ b/src/Meta/Interpret.ml @@ -4,7 +4,7 @@ open Funsyn (* Operational semantics *) (* Author: Carsten Schuermann *) -include Interpret_intf +include INTERPRET (* Signature Interpreter *) (* # 1 "src/meta/Interpret.fun.ml" *) diff --git a/src/Meta/Interpret.mli b/src/Meta/Interpret.mli new file mode 100644 index 0000000..1c81a49 --- /dev/null +++ b/src/Meta/Interpret.mli @@ -0,0 +1 @@ +include module type of INTERPRET diff --git a/src/meta/MtpAbstract_intf.ml b/src/Meta/MTPABSTRACT.ml similarity index 100% rename from src/meta/MtpAbstract_intf.ml rename to src/Meta/MTPABSTRACT.ml diff --git a/src/meta/MtpData_intf.ml b/src/Meta/MTPDATA.ml similarity index 100% rename from src/meta/MtpData_intf.ml rename to src/Meta/MTPDATA.ml diff --git a/src/meta/MtpFilling_intf.ml b/src/Meta/MTPFILLING.ml similarity index 100% rename from src/meta/MtpFilling_intf.ml rename to src/Meta/MTPFILLING.ml diff --git a/src/meta/MtpGlobal_intf.ml b/src/Meta/MTPGLOBAL.ml similarity index 100% rename from src/meta/MtpGlobal_intf.ml rename to src/Meta/MTPGLOBAL.ml diff --git a/src/meta/MtpInit_intf.ml b/src/Meta/MTPINIT.ml similarity index 100% rename from src/meta/MtpInit_intf.ml rename to src/Meta/MTPINIT.ml diff --git a/src/meta/MtpMpi_intf.ml b/src/Meta/MTPMPI.ml similarity index 100% rename from src/meta/MtpMpi_intf.ml rename to src/Meta/MTPMPI.ml diff --git a/src/meta/MtpPrint_intf.ml b/src/Meta/MTPPRINT.ml similarity index 100% rename from src/meta/MtpPrint_intf.ml rename to src/Meta/MTPPRINT.ml diff --git a/src/meta/MtpProver_intf.ml b/src/Meta/MTPPROVER.ml similarity index 100% rename from src/meta/MtpProver_intf.ml rename to src/Meta/MTPPROVER.ml diff --git a/src/meta/MtpRecursion_intf.ml b/src/Meta/MTPRECURSION.ml similarity index 100% rename from src/meta/MtpRecursion_intf.ml rename to src/Meta/MTPRECURSION.ml diff --git a/src/meta/MtpSearch_intf.ml b/src/Meta/MTPSEARCH.ml similarity index 100% rename from src/meta/MtpSearch_intf.ml rename to src/Meta/MTPSEARCH.ml diff --git a/src/meta/MtpSplitting_intf.ml b/src/Meta/MTPSPLITTING.ml similarity index 100% rename from src/meta/MtpSplitting_intf.ml rename to src/Meta/MTPSPLITTING.ml diff --git a/src/meta/MtpStrategy_intf.ml b/src/Meta/MTPSTRATEGY.ml similarity index 100% rename from src/meta/MtpStrategy_intf.ml rename to src/Meta/MTPSTRATEGY.ml diff --git a/src/meta/Meta.ml b/src/Meta/Meta.ml similarity index 100% rename from src/meta/Meta.ml rename to src/Meta/Meta.ml diff --git a/src/meta/Meta.mli b/src/Meta/Meta.mli similarity index 100% rename from src/meta/Meta.mli rename to src/Meta/Meta.mli diff --git a/src/meta/Meta_.ml b/src/Meta/Meta_.ml similarity index 100% rename from src/meta/Meta_.ml rename to src/Meta/Meta_.ml diff --git a/src/meta/Meta_.mli b/src/Meta/Meta_.mli similarity index 100% rename from src/meta/Meta_.mli rename to src/Meta/Meta_.mli diff --git a/src/meta/MtpAbstract.ml b/src/Meta/MtpAbstract.ml similarity index 99% rename from src/meta/MtpAbstract.ml rename to src/Meta/MtpAbstract.ml index fcd2c66..039ad2f 100644 --- a/src/meta/MtpAbstract.ml +++ b/src/Meta/MtpAbstract.ml @@ -6,7 +6,7 @@ open Funtypecheck (* Meta Theorem Prover abstraction : Version 1.3 *) (* Author: Frank Pfenning, Carsten Schuermann *) -include MtpAbstract_intf +include MTPABSTRACT (* signature MTPABSTRACT *) (* # 1 "src/meta/Abstract.fun.ml" *) @@ -19,7 +19,7 @@ module MTPAbstract (MTPAbstract__0 : sig (*! structure IntSyn' : INTSYN !*) (*! structure FunSyn' : FUNSYN !*) (*! sharing FunSyn'.IntSyn = IntSyn' !*) - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.FunSyn = FunSyn' !*) module Whnf : WHNF @@ -37,11 +37,11 @@ module MTPAbstract (MTPAbstract__0 : sig module TypeCheck : TYPECHECK (*! sharing TypeCheck.IntSyn = IntSyn' !*) - module FunTypeCheck : Funtypecheck_intf.FUNTYPECHECK + module FunTypeCheck : FUNTYPECHECK.FUNTYPECHECK (*! sharing FunTypeCheck.FunSyn = FunSyn' !*) module Abstract : ABSTRACT -end) : MtpAbstract_intf.MTPABSTRACT = struct +end) : MTPABSTRACT.MTPABSTRACT = struct (*! structure IntSyn = IntSyn' !*) (*! structure FunSyn = FunSyn' !*) open MTPAbstract__0 diff --git a/src/meta/MtpAbstract.mli b/src/Meta/MtpAbstract.mli similarity index 79% rename from src/meta/MtpAbstract.mli rename to src/Meta/MtpAbstract.mli index 07847a2..efc9fd2 100644 --- a/src/meta/MtpAbstract.mli +++ b/src/Meta/MtpAbstract.mli @@ -1,10 +1,10 @@ -include module type of MtpAbstract_intf +include module type of MTPABSTRACT module MTPAbstract (MTPAbstract__0 : sig (*! structure IntSyn' : INTSYN !*) (*! structure FunSyn' : FUNSYN !*) (*! sharing FunSyn'.IntSyn = IntSyn' !*) - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.FunSyn = FunSyn' !*) module Whnf : WHNF @@ -22,8 +22,8 @@ module MTPAbstract (MTPAbstract__0 : sig module TypeCheck : TYPECHECK (*! sharing TypeCheck.IntSyn = IntSyn' !*) - module FunTypeCheck : Funtypecheck_intf.FUNTYPECHECK + module FunTypeCheck : FUNTYPECHECK.FUNTYPECHECK (*! sharing FunTypeCheck.FunSyn = FunSyn' !*) module Abstract : ABSTRACT -end) : MtpAbstract_intf.MTPABSTRACT +end) : MTPABSTRACT.MTPABSTRACT diff --git a/src/meta/MtpData.ml b/src/Meta/MtpData.ml similarity index 88% rename from src/meta/MtpData.ml rename to src/Meta/MtpData.ml index 87eeafe..826ce55 100644 --- a/src/meta/MtpData.ml +++ b/src/Meta/MtpData.ml @@ -4,7 +4,7 @@ open MtpGlobal (* Data aquired during proof search *) (* Author: Carsten Schuermann *) -include MtpData_intf +include MTPDATA (* signature MTPDATA *) (* # 1 "src/meta/Data.fun.ml" *) @@ -14,7 +14,7 @@ open! Basis (* Author: Carsten Schuermann *) module MTPData (MTPData__0 : sig module MTPGlobal : MtpGlobal.MTPGLOBAL -end) : MtpData_intf.MTPDATA = struct +end) : MTPDATA.MTPDATA = struct let maxFill = ref 0 end (* structure MTPData*) diff --git a/src/meta/MtpData.mli b/src/Meta/MtpData.mli similarity index 53% rename from src/meta/MtpData.mli rename to src/Meta/MtpData.mli index c1bc788..d409609 100644 --- a/src/meta/MtpData.mli +++ b/src/Meta/MtpData.mli @@ -1,5 +1,5 @@ -include module type of MtpData_intf +include module type of MTPDATA module MTPData (MTPData__0 : sig module MTPGlobal : MtpGlobal.MTPGLOBAL -end) : MtpData_intf.MTPDATA +end) : MTPDATA.MTPDATA diff --git a/src/meta/MtpFilling.ml b/src/Meta/MtpFilling.ml similarity index 95% rename from src/meta/MtpFilling.ml rename to src/Meta/MtpFilling.ml index 8f00423..bb39898 100644 --- a/src/meta/MtpFilling.ml +++ b/src/Meta/MtpFilling.ml @@ -8,7 +8,7 @@ open MtpSearch (* Filling: Version 1.3 *) (* Author: Carsten Schuermann *) -include MtpFilling_intf +include MTPFILLING (* signature MTPFILLING *) (* # 1 "src/meta/Filling.fun.ml" *) @@ -24,7 +24,7 @@ module MTPFilling (MTPFilling__0 : sig (*! structure IntSyn : INTSYN !*) (*! structure FunSyn' : FUNSYN !*) (*! sharing FunSyn'.IntSyn = IntSyn !*) - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.FunSyn = FunSyn' !*) module Abstract : ABSTRACT @@ -33,10 +33,10 @@ module MTPFilling (MTPFilling__0 : sig module TypeCheck : TYPECHECK (*! sharing TypeCheck.IntSyn = IntSyn !*) - module MTPData : MtpData_intf.MTPDATA - module Search : MtpSearch_intf.MTPSEARCH + module MTPData : MTPDATA.MTPDATA + module Search : MTPSEARCH.MTPSEARCH module Whnf : WHNF -end) : MtpFilling_intf.MTPFILLING = struct +end) : MTPFILLING.MTPFILLING = struct (*! structure FunSyn = FunSyn' !*) open MTPFilling__0 module StateSyn = StateSyn' diff --git a/src/meta/MtpFilling.mli b/src/Meta/MtpFilling.mli similarity index 67% rename from src/meta/MtpFilling.mli rename to src/Meta/MtpFilling.mli index afcf5f4..0e8f27f 100644 --- a/src/meta/MtpFilling.mli +++ b/src/Meta/MtpFilling.mli @@ -1,4 +1,4 @@ -include module type of MtpFilling_intf +include module type of MTPFILLING module MTPFilling (MTPFilling__0 : sig module MTPGlobal : MtpGlobal.MTPGLOBAL @@ -6,7 +6,7 @@ module MTPFilling (MTPFilling__0 : sig (*! structure IntSyn : INTSYN !*) (*! structure FunSyn' : FUNSYN !*) (*! sharing FunSyn'.IntSyn = IntSyn !*) - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.FunSyn = FunSyn' !*) module Abstract : ABSTRACT @@ -15,7 +15,7 @@ module MTPFilling (MTPFilling__0 : sig module TypeCheck : TYPECHECK (*! sharing TypeCheck.IntSyn = IntSyn !*) - module MTPData : MtpData_intf.MTPDATA - module Search : MtpSearch_intf.MTPSEARCH + module MTPData : MTPDATA.MTPDATA + module Search : MTPSEARCH.MTPSEARCH module Whnf : WHNF -end) : MtpFilling_intf.MTPFILLING +end) : MTPFILLING.MTPFILLING diff --git a/src/meta/MtpGlobal.ml b/src/Meta/MtpGlobal.ml similarity index 96% rename from src/meta/MtpGlobal.ml rename to src/Meta/MtpGlobal.ml index 1925d7a..a817933 100644 --- a/src/meta/MtpGlobal.ml +++ b/src/Meta/MtpGlobal.ml @@ -4,7 +4,7 @@ open MetaGlobal (* Global parameters *) (* Author: Carsten Schuermann *) -include MtpGlobal_intf +include MTPGLOBAL (* signature MTPGLOBAL *) (* # 1 "src/meta/Global.fun.ml" *) diff --git a/src/Meta/MtpGlobal.mli b/src/Meta/MtpGlobal.mli new file mode 100644 index 0000000..18dd55b --- /dev/null +++ b/src/Meta/MtpGlobal.mli @@ -0,0 +1,5 @@ +include module type of MTPGLOBAL + +module MTPGlobal (MTPGlobal__0 : sig + module MetaGlobal : METAGLOBAL.METAGLOBAL +end) : MTPGLOBAL diff --git a/src/meta/MtpInit.ml b/src/Meta/MtpInit.ml similarity index 93% rename from src/meta/MtpInit.ml rename to src/Meta/MtpInit.ml index bfc2ba3..b7cca8a 100644 --- a/src/meta/MtpInit.ml +++ b/src/Meta/MtpInit.ml @@ -8,7 +8,7 @@ open Funprint (* Initialization *) (* Author: Carsten Schuermann *) -include MtpInit_intf +include MTPINIT (* signature MTPINIT *) (* # 1 "src/meta/Init.fun.ml" *) @@ -18,7 +18,7 @@ open! Basis (* Author: Carsten Schuermann *) module MTPInit (MTPInit__0 : sig module MTPGlobal : MtpGlobal.MTPGLOBAL - module MTPData : MtpData_intf.MTPDATA + module MTPData : MTPDATA.MTPDATA (*! structure IntSyn : INTSYN !*) module Names : NAMES @@ -26,7 +26,7 @@ module MTPInit (MTPInit__0 : sig (*! sharing Names.IntSyn = IntSyn !*) (*! structure FunSyn' : FUNSYN !*) (*! sharing FunSyn'.IntSyn = IntSyn !*) - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.FunSyn = FunSyn' !*) module Formatter : FORMATTER @@ -36,8 +36,8 @@ module MTPInit (MTPInit__0 : sig module Print : PRINT (*! sharing Print.IntSyn = IntSyn !*) - module FunPrint : Funprint_intf.FUNPRINT -end) : MtpInit_intf.MTPINIT = struct + module FunPrint : FUNPRINT.FUNPRINT +end) : MTPINIT.MTPINIT = struct (*! structure FunSyn = FunSyn' !*) open MTPInit__0 module StateSyn = StateSyn' diff --git a/src/meta/MtpInit.mli b/src/Meta/MtpInit.mli similarity index 70% rename from src/meta/MtpInit.mli rename to src/Meta/MtpInit.mli index 3886cc0..c8f1732 100644 --- a/src/meta/MtpInit.mli +++ b/src/Meta/MtpInit.mli @@ -1,8 +1,8 @@ -include module type of MtpInit_intf +include module type of MTPINIT module MTPInit (MTPInit__0 : sig module MTPGlobal : MtpGlobal.MTPGLOBAL - module MTPData : MtpData_intf.MTPDATA + module MTPData : MTPDATA.MTPDATA (*! structure IntSyn : INTSYN !*) module Names : NAMES @@ -10,7 +10,7 @@ module MTPInit (MTPInit__0 : sig (*! sharing Names.IntSyn = IntSyn !*) (*! structure FunSyn' : FUNSYN !*) (*! sharing FunSyn'.IntSyn = IntSyn !*) - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.FunSyn = FunSyn' !*) module Formatter : FORMATTER @@ -20,5 +20,5 @@ module MTPInit (MTPInit__0 : sig module Print : PRINT (*! sharing Print.IntSyn = IntSyn !*) - module FunPrint : Funprint_intf.FUNPRINT -end) : MtpInit_intf.MTPINIT + module FunPrint : FUNPRINT.FUNPRINT +end) : MTPINIT.MTPINIT diff --git a/src/meta/MtpMpi.ml b/src/Meta/MtpMpi.ml similarity index 96% rename from src/meta/MtpMpi.ml rename to src/Meta/MtpMpi.ml index 60a6f0e..51a7c0e 100644 --- a/src/meta/MtpMpi.ml +++ b/src/Meta/MtpMpi.ml @@ -18,7 +18,7 @@ open Ring (* Meta Prover Interface *) (* Author: Carsten Schuermann *) -include MtpMpi_intf +include MTPMPI (* signature MTPI *) (* # 1 "src/meta/Mpi.fun.ml" *) @@ -34,42 +34,42 @@ module MTPi (MTPi__0 : sig (*! structure IntSyn : INTSYN !*) (*! structure FunSyn' : FUNSYN !*) (*! sharing FunSyn'.IntSyn = IntSyn !*) - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.IntSyn = IntSyn !*) (*! sharing StateSyn'.FunSyn = FunSyn' !*) - module RelFun : Relfun_intf.RELFUN + module RelFun : RELFUN.RELFUN (*! sharing RelFun.FunSyn = FunSyn' !*) module Formatter : FORMATTER module Print : PRINT (*! sharing Print.IntSyn = IntSyn !*) - module FunTypeCheck : Funtypecheck_intf.FUNTYPECHECK + module FunTypeCheck : FUNTYPECHECK.FUNTYPECHECK (*! sharing FunTypeCheck.FunSyn = FunSyn' !*) - module MTPData : MtpData_intf.MTPDATA - module MTPInit : MtpInit_intf.MTPINIT + module MTPData : MTPDATA.MTPDATA + module MTPInit : MTPINIT.MTPINIT (*! sharing MTPInit.FunSyn = FunSyn' !*) - module MTPFilling : MtpFilling_intf.MTPFILLING + module MTPFilling : MTPFILLING.MTPFILLING (*! sharing MTPFilling.FunSyn = FunSyn' !*) - module Inference : Inference_intf.INFERENCE + module Inference : INFERENCE.INFERENCE (*! sharing Inference.FunSyn = FunSyn' !*) - module MTPSplitting : MtpSplitting_intf.MTPSPLITTING - module MTPRecursion : MtpRecursion_intf.MTPRECURSION - module MTPStrategy : MtpStrategy_intf.MTPSTRATEGY - module MTPrint : MtpPrint_intf.MTPRINT + module MTPSplitting : MTPSPLITTING.MTPSPLITTING + module MTPRecursion : MTPRECURSION.MTPRECURSION + module MTPStrategy : MTPSTRATEGY.MTPSTRATEGY + module MTPrint : MTPPRINT.MTPRINT module Order : ORDER (*! sharing Order.IntSyn = IntSyn !*) module Names : NAMES (*! sharing Names.IntSyn = IntSyn !*) - module Timers : Timers_intf.TIMERS - module Ring : Ring_intf.RING + module Timers : TIMERS.TIMERS + module Ring : RING.RING end) : MTPI = struct open MTPi__0 diff --git a/src/meta/MtpMpi.mli b/src/Meta/MtpMpi.mli similarity index 54% rename from src/meta/MtpMpi.mli rename to src/Meta/MtpMpi.mli index 177e4ec..dfe9fa8 100644 --- a/src/meta/MtpMpi.mli +++ b/src/Meta/MtpMpi.mli @@ -1,4 +1,4 @@ -include module type of MtpMpi_intf +include module type of MTPMPI module MTPi (MTPi__0 : sig module MTPGlobal : MtpGlobal.MTPGLOBAL @@ -6,40 +6,40 @@ module MTPi (MTPi__0 : sig (*! structure IntSyn : INTSYN !*) (*! structure FunSyn' : FUNSYN !*) (*! sharing FunSyn'.IntSyn = IntSyn !*) - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.IntSyn = IntSyn !*) (*! sharing StateSyn'.FunSyn = FunSyn' !*) - module RelFun : Relfun_intf.RELFUN + module RelFun : RELFUN.RELFUN (*! sharing RelFun.FunSyn = FunSyn' !*) module Formatter : FORMATTER module Print : PRINT (*! sharing Print.IntSyn = IntSyn !*) - module FunTypeCheck : Funtypecheck_intf.FUNTYPECHECK + module FunTypeCheck : FUNTYPECHECK.FUNTYPECHECK (*! sharing FunTypeCheck.FunSyn = FunSyn' !*) - module MTPData : MtpData_intf.MTPDATA - module MTPInit : MtpInit_intf.MTPINIT + module MTPData : MTPDATA.MTPDATA + module MTPInit : MTPINIT.MTPINIT (*! sharing MTPInit.FunSyn = FunSyn' !*) - module MTPFilling : MtpFilling_intf.MTPFILLING + module MTPFilling : MTPFILLING.MTPFILLING (*! sharing MTPFilling.FunSyn = FunSyn' !*) - module Inference : Inference_intf.INFERENCE + module Inference : INFERENCE.INFERENCE (*! sharing Inference.FunSyn = FunSyn' !*) - module MTPSplitting : MtpSplitting_intf.MTPSPLITTING - module MTPRecursion : MtpRecursion_intf.MTPRECURSION - module MTPStrategy : MtpStrategy_intf.MTPSTRATEGY - module MTPrint : MtpPrint_intf.MTPRINT + module MTPSplitting : MTPSPLITTING.MTPSPLITTING + module MTPRecursion : MTPRECURSION.MTPRECURSION + module MTPStrategy : MTPSTRATEGY.MTPSTRATEGY + module MTPrint : MTPPRINT.MTPRINT module Order : ORDER (*! sharing Order.IntSyn = IntSyn !*) module Names : NAMES (*! sharing Names.IntSyn = IntSyn !*) - module Timers : Timers_intf.TIMERS - module Ring : Ring_intf.RING + module Timers : TIMERS.TIMERS + module Ring : RING.RING end) : MTPI diff --git a/src/meta/MtpPrint.ml b/src/Meta/MtpPrint.ml similarity index 97% rename from src/meta/MtpPrint.ml rename to src/Meta/MtpPrint.ml index 758e0a1..3bb177a 100644 --- a/src/meta/MtpPrint.ml +++ b/src/Meta/MtpPrint.ml @@ -6,7 +6,7 @@ open Funprint (* Meta Printer Version 1.3 *) (* Author: Carsten Schuermann *) -include MtpPrint_intf +include MTPPRINT (* signature MTPRINT *) (* # 1 "src/meta/Print.fun.ml" *) @@ -24,7 +24,7 @@ module MTPrint (MTPrint__0 : sig module Names : NAMES (*! sharing Names.IntSyn = IntSyn !*) - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.FunSyn = FunSyn !*) (*! sharing StateSyn'.IntSyn = IntSyn !*) @@ -32,8 +32,8 @@ module MTPrint (MTPrint__0 : sig module Print : PRINT (*! sharing Print.IntSyn = IntSyn !*) - module FunPrint : Funprint_intf.FUNPRINT -end) : MtpPrint_intf.MTPRINT = struct + module FunPrint : FUNPRINT.FUNPRINT +end) : MTPPRINT.MTPRINT = struct open MTPrint__0 module Formatter = Formatter' module StateSyn = StateSyn' diff --git a/src/meta/MtpPrint.mli b/src/Meta/MtpPrint.mli similarity index 76% rename from src/meta/MtpPrint.mli rename to src/Meta/MtpPrint.mli index 311e003..ac4e545 100644 --- a/src/meta/MtpPrint.mli +++ b/src/Meta/MtpPrint.mli @@ -1,4 +1,4 @@ -include module type of MtpPrint_intf +include module type of MTPPRINT module MTPrint (MTPrint__0 : sig (* Meta Printer Version 1.3 *) @@ -11,7 +11,7 @@ module MTPrint (MTPrint__0 : sig module Names : NAMES (*! sharing Names.IntSyn = IntSyn !*) - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.FunSyn = FunSyn !*) (*! sharing StateSyn'.IntSyn = IntSyn !*) @@ -19,5 +19,5 @@ module MTPrint (MTPrint__0 : sig module Print : PRINT (*! sharing Print.IntSyn = IntSyn !*) - module FunPrint : Funprint_intf.FUNPRINT -end) : MtpPrint_intf.MTPRINT + module FunPrint : FUNPRINT.FUNPRINT +end) : MTPPRINT.MTPRINT diff --git a/src/meta/MtpProver.ml b/src/Meta/MtpProver.ml similarity index 95% rename from src/meta/MtpProver.ml rename to src/Meta/MtpProver.ml index b23bb71..887d6f7 100644 --- a/src/meta/MtpProver.ml +++ b/src/Meta/MtpProver.ml @@ -9,7 +9,7 @@ open Relfun (* Meta Prover Version 1.3 *) (* Author: Carsten Schuermann *) -include MtpProver_intf +include MTPPROVER (* signature MTPROVER *) (* # 1 "src/meta/Prover.fun.ml" *) @@ -23,19 +23,19 @@ module MTProver (MTProver__0 : sig (*! structure IntSyn' : INTSYN !*) (*! structure FunSyn : FUNSYN !*) (*! sharing FunSyn.IntSyn = IntSyn' !*) - module StateSyn : Statesyn_intf.STATESYN + module StateSyn : STATESYN.STATESYN (*! sharing IntSyn = IntSyn' !*) (*! sharing StateSyn.FunSyn = FunSyn !*) module Order : ORDER (*! sharing Order.IntSyn = IntSyn' !*) - module MTPInit : MtpInit_intf.MTPINIT + module MTPInit : MTPINIT.MTPINIT (*! sharing MTPInit.FunSyn = FunSyn !*) - module MTPStrategy : MtpStrategy_intf.MTPSTRATEGY - module RelFun : Relfun_intf.RELFUN -end) : MtpProver_intf.MTPROVER = struct + module MTPStrategy : MTPSTRATEGY.MTPSTRATEGY + module RelFun : RELFUN.RELFUN +end) : MTPPROVER.MTPROVER = struct open MTProver__0 module StateSyn = StateSyn @@ -184,8 +184,8 @@ module CombiProver (CombiProver__1 : sig module ProverOld : Prover.PROVER (*! sharing ProverOld.IntSyn = IntSyn' !*) - module ProverNew : MtpProver_intf.MTPROVER -end) : MtpProver_intf.MTPROVER = struct + module ProverNew : MTPPROVER.MTPROVER +end) : MTPPROVER.MTPROVER = struct open CombiProver__1 (*! structure IntSyn = IntSyn' !*) diff --git a/src/meta/MtpProver.mli b/src/Meta/MtpProver.mli similarity index 64% rename from src/meta/MtpProver.mli rename to src/Meta/MtpProver.mli index 3816def..bf65b84 100644 --- a/src/meta/MtpProver.mli +++ b/src/Meta/MtpProver.mli @@ -1,4 +1,4 @@ -include module type of MtpProver_intf +include module type of MTPPROVER module MTProver (MTProver__0 : sig module MTPGlobal : MtpGlobal.MTPGLOBAL @@ -6,19 +6,19 @@ module MTProver (MTProver__0 : sig (*! structure IntSyn' : INTSYN !*) (*! structure FunSyn : FUNSYN !*) (*! sharing FunSyn.IntSyn = IntSyn' !*) - module StateSyn : Statesyn_intf.STATESYN + module StateSyn : STATESYN.STATESYN (*! sharing IntSyn = IntSyn' !*) (*! sharing StateSyn.FunSyn = FunSyn !*) module Order : ORDER (*! sharing Order.IntSyn = IntSyn' !*) - module MTPInit : MtpInit_intf.MTPINIT + module MTPInit : MTPINIT.MTPINIT (*! sharing MTPInit.FunSyn = FunSyn !*) - module MTPStrategy : MtpStrategy_intf.MTPSTRATEGY - module RelFun : Relfun_intf.RELFUN -end) : MtpProver_intf.MTPROVER + module MTPStrategy : MTPSTRATEGY.MTPSTRATEGY + module RelFun : RELFUN.RELFUN +end) : MTPPROVER.MTPROVER module CombiProver (CombiProver__1 : sig module MTPGlobal : MtpGlobal.MTPGLOBAL @@ -27,5 +27,5 @@ module CombiProver (CombiProver__1 : sig module ProverOld : Prover.PROVER (*! sharing ProverOld.IntSyn = IntSyn' !*) - module ProverNew : MtpProver_intf.MTPROVER -end) : MtpProver_intf.MTPROVER + module ProverNew : MTPPROVER.MTPROVER +end) : MTPPROVER.MTPROVER diff --git a/src/meta/MtpRecursion.ml b/src/Meta/MtpRecursion.ml similarity index 99% rename from src/meta/MtpRecursion.ml rename to src/Meta/MtpRecursion.ml index 22fb377..a0106f4 100644 --- a/src/meta/MtpRecursion.ml +++ b/src/Meta/MtpRecursion.ml @@ -10,7 +10,7 @@ open Funprint (* Recursion: Version 1.3 *) (* Author: Carsten Schuermann *) -include MtpRecursion_intf +include MTPRECURSION (* signature MTPRECURSION *) (* # 1 "src/meta/Recursion.fun.ml" *) @@ -28,21 +28,21 @@ module MTPRecursion (MTPRecursion__0 : sig (*! structure IntSyn : INTSYN !*) (*! structure FunSyn : FUNSYN !*) (*! sharing FunSyn.IntSyn = IntSyn !*) - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.IntSyn = IntSyn !*) (*! sharing StateSyn'.FunSyn = FunSyn !*) module Abstract : ABSTRACT (*! sharing Abstract.IntSyn = IntSyn !*) - module MTPAbstract : MtpAbstract_intf.MTPABSTRACT + module MTPAbstract : MTPABSTRACT.MTPABSTRACT (*! sharing MTPAbstract.IntSyn = IntSyn !*) (*! sharing MTPAbstract.FunSyn = FunSyn !*) - module FunTypeCheck : Funtypecheck_intf.FUNTYPECHECK + module FunTypeCheck : FUNTYPECHECK.FUNTYPECHECK (*! sharing FunTypeCheck.FunSyn = FunSyn !*) - module MTPrint : MtpPrint_intf.MTPRINT + module MTPrint : MTPPRINT.MTPRINT module Whnf : WHNF (*! sharing Whnf.IntSyn = IntSyn !*) @@ -65,7 +65,7 @@ module MTPRecursion (MTPRecursion__0 : sig (*! sharing TypeCheck.IntSyn = IntSyn !*) module Formatter : FORMATTER - module FunPrint : Funprint_intf.FUNPRINT + module FunPrint : FUNPRINT.FUNPRINT end) : MTPRECURSION = struct open MTPRecursion__0 module StateSyn = StateSyn' diff --git a/src/meta/MtpRecursion.mli b/src/Meta/MtpRecursion.mli similarity index 79% rename from src/meta/MtpRecursion.mli rename to src/Meta/MtpRecursion.mli index cf7b6cf..6354302 100644 --- a/src/meta/MtpRecursion.mli +++ b/src/Meta/MtpRecursion.mli @@ -1,4 +1,4 @@ -include module type of MtpRecursion_intf +include module type of MTPRECURSION module MTPRecursion (MTPRecursion__0 : sig module MTPGlobal : MtpGlobal.MTPGLOBAL @@ -7,21 +7,21 @@ module MTPRecursion (MTPRecursion__0 : sig (*! structure IntSyn : INTSYN !*) (*! structure FunSyn : FUNSYN !*) (*! sharing FunSyn.IntSyn = IntSyn !*) - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.IntSyn = IntSyn !*) (*! sharing StateSyn'.FunSyn = FunSyn !*) module Abstract : ABSTRACT (*! sharing Abstract.IntSyn = IntSyn !*) - module MTPAbstract : MtpAbstract_intf.MTPABSTRACT + module MTPAbstract : MTPABSTRACT.MTPABSTRACT (*! sharing MTPAbstract.IntSyn = IntSyn !*) (*! sharing MTPAbstract.FunSyn = FunSyn !*) - module FunTypeCheck : Funtypecheck_intf.FUNTYPECHECK + module FunTypeCheck : FUNTYPECHECK.FUNTYPECHECK (*! sharing FunTypeCheck.FunSyn = FunSyn !*) - module MTPrint : MtpPrint_intf.MTPRINT + module MTPrint : MTPPRINT.MTPRINT module Whnf : WHNF (*! sharing Whnf.IntSyn = IntSyn !*) @@ -44,5 +44,5 @@ module MTPRecursion (MTPRecursion__0 : sig (*! sharing TypeCheck.IntSyn = IntSyn !*) module Formatter : FORMATTER - module FunPrint : Funprint_intf.FUNPRINT + module FunPrint : FUNPRINT.FUNPRINT end) : MTPRECURSION diff --git a/src/meta/MtpSearch.ml b/src/Meta/MtpSearch.ml similarity index 99% rename from src/meta/MtpSearch.ml rename to src/Meta/MtpSearch.ml index cc71f7c..b586707 100644 --- a/src/meta/MtpSearch.ml +++ b/src/Meta/MtpSearch.ml @@ -6,7 +6,7 @@ open MtpGlobal (* Basic search engine: Version 1.3*) (* Author: Carsten Schuermann *) -include MtpSearch_intf +include MTPSEARCH (* signature SEARCH *) (* # 1 "src/meta/Search.fun.ml" *) @@ -24,7 +24,7 @@ module MTPSearch (MTPSearch__0 : sig (*! sharing Abstract.IntSyn = IntSyn' !*) module MTPGlobal : MtpGlobal.MTPGLOBAL - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.FunSyn.IntSyn = IntSyn' !*) (*! structure CompSyn' : COMPSYN !*) @@ -53,7 +53,7 @@ module MTPSearch (MTPSearch__0 : sig (*! sharing Print.IntSyn = IntSyn' !*) module Names : NAMES -end) : MtpSearch_intf.MTPSEARCH = struct +end) : MTPSEARCH.MTPSEARCH = struct (*! structure IntSyn = IntSyn' !*) open MTPSearch__0 module StateSyn = StateSyn' diff --git a/src/meta/MtpSearch.mli b/src/Meta/MtpSearch.mli similarity index 89% rename from src/meta/MtpSearch.mli rename to src/Meta/MtpSearch.mli index 301b050..59928be 100644 --- a/src/meta/MtpSearch.mli +++ b/src/Meta/MtpSearch.mli @@ -1,4 +1,4 @@ -include module type of MtpSearch_intf +include module type of MTPSEARCH module MTPSearch (MTPSearch__0 : sig module Global : GLOBAL @@ -8,7 +8,7 @@ module MTPSearch (MTPSearch__0 : sig (*! sharing Abstract.IntSyn = IntSyn' !*) module MTPGlobal : MtpGlobal.MTPGLOBAL - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.FunSyn.IntSyn = IntSyn' !*) (*! structure CompSyn' : COMPSYN !*) @@ -37,4 +37,4 @@ module MTPSearch (MTPSearch__0 : sig (*! sharing Print.IntSyn = IntSyn' !*) module Names : NAMES -end) : MtpSearch_intf.MTPSEARCH +end) : MTPSEARCH.MTPSEARCH diff --git a/src/meta/MtpSplitting.ml b/src/Meta/MtpSplitting.ml similarity index 99% rename from src/meta/MtpSplitting.ml rename to src/Meta/MtpSplitting.ml index 408eb4f..47d59f9 100644 --- a/src/meta/MtpSplitting.ml +++ b/src/Meta/MtpSplitting.ml @@ -9,7 +9,7 @@ open Funtypecheck (* Splitting : Version 1.3 *) (* Author: Carsten Schuermann *) -include MtpSplitting_intf +include MTPSPLITTING (* signature MTPSPLITTING *) (* # 1 "src/meta/Splitting.fun.ml" *) @@ -26,15 +26,15 @@ module MTPSplitting (MTPSplitting__0 : sig (*! structure IntSyn : INTSYN !*) (*! structure FunSyn : FUNSYN !*) (*! sharing FunSyn.IntSyn = IntSyn !*) - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.FunSyn = FunSyn !*) (*! sharing StateSyn'.IntSyn = IntSyn !*) module Heuristic : HEURISTIC - module MTPAbstract : MtpAbstract_intf.MTPABSTRACT + module MTPAbstract : MTPABSTRACT.MTPABSTRACT (*! sharing MTPAbstract.IntSyn = IntSyn !*) - module MTPrint : MtpPrint_intf.MTPRINT + module MTPrint : MTPPRINT.MTPRINT module Names : NAMES (* too be removed -cs *) @@ -52,7 +52,7 @@ module MTPSplitting (MTPSplitting__0 : sig module Subordinate : Subordinate_.SUBORDINATE (*! sharing Subordinate.IntSyn = IntSyn !*) - module FunTypeCheck : Funtypecheck_intf.FUNTYPECHECK + module FunTypeCheck : FUNTYPECHECK.FUNTYPECHECK (*! sharing FunTypeCheck.FunSyn = FunSyn !*) module Index : INDEX @@ -62,7 +62,7 @@ module MTPSplitting (MTPSplitting__0 : sig (*! sharing Print.IntSyn = IntSyn !*) module Unify : UNIFY -end) : MtpSplitting_intf.MTPSPLITTING = struct +end) : MTPSPLITTING.MTPSPLITTING = struct open MTPSplitting__0 module StateSyn = StateSyn' diff --git a/src/meta/MtpSplitting.mli b/src/Meta/MtpSplitting.mli similarity index 78% rename from src/meta/MtpSplitting.mli rename to src/Meta/MtpSplitting.mli index a2428f4..712ace8 100644 --- a/src/meta/MtpSplitting.mli +++ b/src/Meta/MtpSplitting.mli @@ -1,4 +1,4 @@ -include module type of MtpSplitting_intf +include module type of MTPSPLITTING module MTPSplitting (MTPSplitting__0 : sig module MTPGlobal : MtpGlobal.MTPGLOBAL @@ -7,15 +7,15 @@ module MTPSplitting (MTPSplitting__0 : sig (*! structure IntSyn : INTSYN !*) (*! structure FunSyn : FUNSYN !*) (*! sharing FunSyn.IntSyn = IntSyn !*) - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.FunSyn = FunSyn !*) (*! sharing StateSyn'.IntSyn = IntSyn !*) module Heuristic : HEURISTIC - module MTPAbstract : MtpAbstract_intf.MTPABSTRACT + module MTPAbstract : MTPABSTRACT.MTPABSTRACT (*! sharing MTPAbstract.IntSyn = IntSyn !*) - module MTPrint : MtpPrint_intf.MTPRINT + module MTPrint : MTPPRINT.MTPRINT module Names : NAMES (* too be removed -cs *) @@ -33,7 +33,7 @@ module MTPSplitting (MTPSplitting__0 : sig module Subordinate : Subordinate_.SUBORDINATE (*! sharing Subordinate.IntSyn = IntSyn !*) - module FunTypeCheck : Funtypecheck_intf.FUNTYPECHECK + module FunTypeCheck : FUNTYPECHECK.FUNTYPECHECK (*! sharing FunTypeCheck.FunSyn = FunSyn !*) module Index : INDEX @@ -43,4 +43,4 @@ module MTPSplitting (MTPSplitting__0 : sig (*! sharing Print.IntSyn = IntSyn !*) module Unify : UNIFY -end) : MtpSplitting_intf.MTPSPLITTING +end) : MTPSPLITTING.MTPSPLITTING diff --git a/src/meta/MtpStrategy.ml b/src/Meta/MtpStrategy.ml similarity index 94% rename from src/meta/MtpStrategy.ml rename to src/Meta/MtpStrategy.ml index b4ecb49..61ddc66 100644 --- a/src/meta/MtpStrategy.ml +++ b/src/Meta/MtpStrategy.ml @@ -14,7 +14,7 @@ open TimeLimit (* MTPStrategy : Version 1.3 *) (* Author: Carsten Schuermann *) -include MtpStrategy_intf +include MTPSTRATEGY (* open cases -> remaining cases * solved cases *) (* signature MTPSTRATEGY *) @@ -27,14 +27,14 @@ open! Basis (* Author: Carsten Schuermann *) module MTPStrategy (MTPStrategy__0 : sig module MTPGlobal : MtpGlobal.MTPGLOBAL - module StateSyn' : Statesyn_intf.STATESYN - module MTPFilling : MtpFilling_intf.MTPFILLING - module MTPData : MtpData_intf.MTPDATA - module MTPSplitting : MtpSplitting_intf.MTPSPLITTING - module MTPRecursion : MtpRecursion_intf.MTPRECURSION - module Inference : Inference_intf.INFERENCE - module MTPrint : MtpPrint_intf.MTPRINT - module Timers : Timers_intf.TIMERS + module StateSyn' : STATESYN.STATESYN + module MTPFilling : MTPFILLING.MTPFILLING + module MTPData : MTPDATA.MTPDATA + module MTPSplitting : MTPSPLITTING.MTPSPLITTING + module MTPRecursion : MTPRECURSION.MTPRECURSION + module Inference : INFERENCE.INFERENCE + module MTPrint : MTPPRINT.MTPRINT + module Timers : TIMERS.TIMERS end) : MTPSTRATEGY = struct open MTPStrategy__0 module StateSyn = StateSyn' diff --git a/src/Meta/MtpStrategy.mli b/src/Meta/MtpStrategy.mli new file mode 100644 index 0000000..8b611a8 --- /dev/null +++ b/src/Meta/MtpStrategy.mli @@ -0,0 +1,13 @@ +include module type of MTPSTRATEGY + +module MTPStrategy (MTPStrategy__0 : sig + module MTPGlobal : MtpGlobal.MTPGLOBAL + module StateSyn' : STATESYN.STATESYN + module MTPFilling : MTPFILLING.MTPFILLING + module MTPData : MTPDATA.MTPDATA + module MTPSplitting : MTPSPLITTING.MTPSPLITTING + module MTPRecursion : MTPRECURSION.MTPRECURSION + module Inference : INFERENCE.INFERENCE + module MTPrint : MTPPRINT.MTPRINT + module Timers : TIMERS.TIMERS +end) : MTPSTRATEGY diff --git a/src/meta/MtpWeaken.ml b/src/Meta/MtpWeaken.ml similarity index 100% rename from src/meta/MtpWeaken.ml rename to src/Meta/MtpWeaken.ml diff --git a/src/meta/MtpWeaken.mli b/src/Meta/MtpWeaken.mli similarity index 100% rename from src/meta/MtpWeaken.mli rename to src/Meta/MtpWeaken.mli diff --git a/src/meta/NOTES b/src/Meta/NOTES similarity index 100% rename from src/meta/NOTES rename to src/Meta/NOTES diff --git a/src/meta/README b/src/Meta/README similarity index 100% rename from src/meta/README rename to src/Meta/README diff --git a/src/meta/Relfun_intf.ml b/src/Meta/RELFUN.ml similarity index 100% rename from src/meta/Relfun_intf.ml rename to src/Meta/RELFUN.ml diff --git a/src/meta/Relfun.ml b/src/Meta/Relfun.ml similarity index 99% rename from src/meta/Relfun.ml rename to src/Meta/Relfun.ml index 3149a1d..55e1ea5 100644 --- a/src/meta/Relfun.ml +++ b/src/Meta/Relfun.ml @@ -8,7 +8,7 @@ open Funsyn (* Converter from relational representation to a functional representation of proof terms *) (* Author: Carsten Schuermann *) -include Relfun_intf +include RELFUN (* Signature RELFUN *) (* # 1 "src/meta/Relfun.fun.ml" *) @@ -35,17 +35,17 @@ module RelFun (RelFun__0 : sig module Whnf : WHNF (*! sharing Whnf.IntSyn = FunSyn'.IntSyn !*) - module Weaken : Weaken_intf.WEAKEN + module Weaken : WEAKEN.WEAKEN (*! sharing Weaken.IntSyn = FunSyn'.IntSyn !*) module TypeCheck : TYPECHECK (*! sharing TypeCheck.IntSyn = FunSyn'.IntSyn !*) - module FunWeaken : Funweaken_intf.FUNWEAKEN + module FunWeaken : FUNWEAKEN.FUNWEAKEN (*! sharing FunWeaken.FunSyn = FunSyn' !*) - module FunNames : Funnames_intf.FUNNAMES -end) : Relfun_intf.RELFUN = struct + module FunNames : FUNNAMES.FUNNAMES +end) : RELFUN.RELFUN = struct (*! structure FunSyn = FunSyn' !*) exception Error of string diff --git a/src/meta/Relfun.mli b/src/Meta/Relfun.mli similarity index 79% rename from src/meta/Relfun.mli rename to src/Meta/Relfun.mli index 50cb099..526b67d 100644 --- a/src/meta/Relfun.mli +++ b/src/Meta/Relfun.mli @@ -1,4 +1,4 @@ -include module type of Relfun_intf +include module type of RELFUN module RelFun (RelFun__0 : sig (* Converter from relational representation to a functional @@ -19,14 +19,14 @@ module RelFun (RelFun__0 : sig module Whnf : WHNF (*! sharing Whnf.IntSyn = FunSyn'.IntSyn !*) - module Weaken : Weaken_intf.WEAKEN + module Weaken : WEAKEN.WEAKEN (*! sharing Weaken.IntSyn = FunSyn'.IntSyn !*) module TypeCheck : TYPECHECK (*! sharing TypeCheck.IntSyn = FunSyn'.IntSyn !*) - module FunWeaken : Funweaken_intf.FUNWEAKEN + module FunWeaken : FUNWEAKEN.FUNWEAKEN (*! sharing FunWeaken.FunSyn = FunSyn' !*) - module FunNames : Funnames_intf.FUNNAMES -end) : Relfun_intf.RELFUN + module FunNames : FUNNAMES.FUNNAMES +end) : RELFUN.RELFUN diff --git a/src/meta/Statesyn_intf.ml b/src/Meta/STATESYN.ml similarity index 100% rename from src/meta/Statesyn_intf.ml rename to src/Meta/STATESYN.ml diff --git a/src/meta/Statesyn.ml b/src/Meta/Statesyn.ml similarity index 98% rename from src/meta/Statesyn.ml rename to src/Meta/Statesyn.ml index 47bef84..ea57f2c 100644 --- a/src/meta/Statesyn.ml +++ b/src/Meta/Statesyn.ml @@ -4,7 +4,7 @@ open Funsyn (* State definition for Proof Search *) (* Author: Carsten Schuermann *) -include Statesyn_intf +include STATESYN (* signature STATESYN *) (* # 1 "src/meta/Statesyn.fun.ml" *) @@ -20,7 +20,7 @@ module StateSyn (StateSyn__0 : sig (*! sharing Whnf.IntSyn = IntSyn' !*) module Conv : CONV -end) : Statesyn_intf.STATESYN = struct +end) : STATESYN.STATESYN = struct (*! structure IntSyn = IntSyn' !*) (*! structure FunSyn = FunSyn' !*) type order = diff --git a/src/meta/Statesyn.mli b/src/Meta/Statesyn.mli similarity index 77% rename from src/meta/Statesyn.mli rename to src/Meta/Statesyn.mli index 42cc687..74f812e 100644 --- a/src/meta/Statesyn.mli +++ b/src/Meta/Statesyn.mli @@ -1,4 +1,4 @@ -include module type of Statesyn_intf +include module type of STATESYN module StateSyn (StateSyn__0 : sig (*! structure IntSyn' : INTSYN !*) @@ -8,4 +8,4 @@ module StateSyn (StateSyn__0 : sig (*! sharing Whnf.IntSyn = IntSyn' !*) module Conv : CONV -end) : Statesyn_intf.STATESYN +end) : STATESYN.STATESYN diff --git a/src/meta/Uniquesearch_intf.ml b/src/Meta/UNIQUESEARCH.ml similarity index 100% rename from src/meta/Uniquesearch_intf.ml rename to src/Meta/UNIQUESEARCH.ml diff --git a/src/meta/Uniquesearch.ml b/src/Meta/Uniquesearch.ml similarity index 99% rename from src/meta/Uniquesearch.ml rename to src/Meta/Uniquesearch.ml index 1438837..36b0987 100644 --- a/src/meta/Uniquesearch.ml +++ b/src/Meta/Uniquesearch.ml @@ -6,7 +6,7 @@ open Statesyn (* Basic search engine: Version 1.3*) (* Author: Carsten Schuermann *) -include Uniquesearch_intf +include UNIQUESEARCH (* signature SEARCH *) (* # 1 "src/meta/Uniquesearch.fun.ml" *) @@ -22,7 +22,7 @@ module UniqueSearch (UniqueSearch__0 : sig (*! structure IntSyn' : INTSYN !*) (*! structure FunSyn' : FUNSYN !*) (*! sharing FunSyn'.IntSyn = IntSyn' !*) - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.IntSyn = IntSyn' !*) (*! sharing StateSyn'.FunSyn = FunSyn' !*) @@ -57,7 +57,7 @@ module UniqueSearch (UniqueSearch__0 : sig (*! sharing Print.IntSyn = IntSyn' !*) module Names : NAMES -end) : Uniquesearch_intf.UNIQUESEARCH = struct +end) : UNIQUESEARCH.UNIQUESEARCH = struct (*! structure IntSyn = IntSyn' !*) (*! structure FunSyn = FunSyn' !*) open UniqueSearch__0 diff --git a/src/meta/Uniquesearch.mli b/src/Meta/Uniquesearch.mli similarity index 89% rename from src/meta/Uniquesearch.mli rename to src/Meta/Uniquesearch.mli index 3a74ada..5819e3a 100644 --- a/src/meta/Uniquesearch.mli +++ b/src/Meta/Uniquesearch.mli @@ -1,4 +1,4 @@ -include module type of Uniquesearch_intf +include module type of UNIQUESEARCH module UniqueSearch (UniqueSearch__0 : sig module Global : GLOBAL @@ -6,7 +6,7 @@ module UniqueSearch (UniqueSearch__0 : sig (*! structure IntSyn' : INTSYN !*) (*! structure FunSyn' : FUNSYN !*) (*! sharing FunSyn'.IntSyn = IntSyn' !*) - module StateSyn' : Statesyn_intf.STATESYN + module StateSyn' : STATESYN.STATESYN (*! sharing StateSyn'.IntSyn = IntSyn' !*) (*! sharing StateSyn'.FunSyn = FunSyn' !*) @@ -41,4 +41,4 @@ module UniqueSearch (UniqueSearch__0 : sig (*! sharing Print.IntSyn = IntSyn' !*) module Names : NAMES -end) : Uniquesearch_intf.UNIQUESEARCH +end) : UNIQUESEARCH.UNIQUESEARCH diff --git a/src/meta/WALK b/src/Meta/WALK similarity index 100% rename from src/meta/WALK rename to src/Meta/WALK diff --git a/src/meta/Weaken_intf.ml b/src/Meta/WEAKEN.ml similarity index 100% rename from src/meta/Weaken_intf.ml rename to src/Meta/WEAKEN.ml diff --git a/src/meta/Weaken.ml b/src/Meta/Weaken.ml similarity index 95% rename from src/meta/Weaken.ml rename to src/Meta/Weaken.ml index b8d765f..f8029e5 100644 --- a/src/meta/Weaken.ml +++ b/src/Meta/Weaken.ml @@ -3,7 +3,7 @@ open! Basis (* Weakening substitutions *) (* Author: Carsten Schuermann *) -include Weaken_intf +include WEAKEN (* signature PRUNE *) (* # 1 "src/meta/Weaken.fun.ml" *) @@ -11,7 +11,7 @@ open! Basis (* Weakening substitutions *) (* Author: Carsten Schuermann *) -module Make_Weaken (Whnf : WHNF) : Weaken_intf.WEAKEN = struct +module Make_Weaken (Whnf : WHNF) : WEAKEN.WEAKEN = struct (*! structure IntSyn = IntSyn' !*) open! struct module I = IntSyn diff --git a/src/Meta/Weaken.mli b/src/Meta/Weaken.mli new file mode 100644 index 0000000..0aa5ff5 --- /dev/null +++ b/src/Meta/Weaken.mli @@ -0,0 +1,2 @@ +include module type of WEAKEN +module Make_Weaken (Whnf : WHNF) : WEAKEN.WEAKEN diff --git a/src/meta/dune b/src/Meta/dune similarity index 79% rename from src/meta/dune rename to src/Meta/dune index 710d9a7..0e21299 100644 --- a/src/meta/dune +++ b/src/Meta/dune @@ -29,29 +29,29 @@ MtpStrategy MtpPrint MtpProver - Funnames_intf - Funprint_intf - Funsyn_intf - Funtypecheck_intf - Funweaken_intf - Inference_intf - Interpret_intf - MtpAbstract_intf - MtpData_intf - MtpFilling_intf - MtpGlobal_intf - MtpInit_intf - MtpMpi_intf - MtpPrint_intf - MtpProver_intf - MtpRecursion_intf - MtpSearch_intf - MtpSplitting_intf - MtpStrategy_intf - Relfun_intf - Statesyn_intf - Uniquesearch_intf - Weaken_intf) + FUNNAMES + FUNPRINT + FUNSYN + FUNTYPECHECK + FUNWEAKEN + INFERENCE + INTERPRET + MTPABSTRACT + MTPDATA + MTPFILLING + MTPGLOBAL + MTPINIT + MTPMPI + MTPPRINT + MTPPROVER + MTPRECURSION + MTPSEARCH + MTPSPLITTING + MTPSTRATEGY + RELFUN + STATESYN + UNIQUESEARCH + WEAKEN) (libraries display global diff --git a/src/modes/Modecheck_intf.ml b/src/Modes/MODECHECK.ml similarity index 100% rename from src/modes/Modecheck_intf.ml rename to src/Modes/MODECHECK.ml diff --git a/src/modes/Modedec_intf.ml b/src/Modes/MODEDEC.ml similarity index 100% rename from src/modes/Modedec_intf.ml rename to src/Modes/MODEDEC.ml diff --git a/src/modes/Modeprint_intf.ml b/src/Modes/MODEPRINT.ml similarity index 100% rename from src/modes/Modeprint_intf.ml rename to src/Modes/MODEPRINT.ml diff --git a/src/modes/Modesyn_intf.ml b/src/Modes/MODESYN.ml similarity index 100% rename from src/modes/Modesyn_intf.ml rename to src/Modes/MODESYN.ml diff --git a/src/modes/Modetable_intf.ml b/src/Modes/MODETABLE.ml similarity index 100% rename from src/modes/Modetable_intf.ml rename to src/Modes/MODETABLE.ml diff --git a/src/modes/Modecheck.ml b/src/Modes/Modecheck.ml similarity index 99% rename from src/modes/Modecheck.ml rename to src/Modes/Modecheck.ml index a41e0cb..d59fabc 100644 --- a/src/modes/Modecheck.ml +++ b/src/Modes/Modecheck.ml @@ -5,7 +5,7 @@ open Modesyn (* Mode Checking *) (* Author: Carsten Schuermann *) (* Modified: Frank Pfenning *) -include Modecheck_intf +include MODECHECK (* raises Error(msg) *) (* signature MODECHECK *) diff --git a/src/modes/Modecheck.mli b/src/Modes/Modecheck.mli similarity index 78% rename from src/modes/Modecheck.mli rename to src/Modes/Modecheck.mli index 2aeff5f..791c93f 100644 --- a/src/modes/Modecheck.mli +++ b/src/Modes/Modecheck.mli @@ -1,4 +1,4 @@ -include module type of Modecheck_intf +include module type of MODECHECK module MakeModeCheck (ModeTable : Modetable.MODETABLE) diff --git a/src/modes/Modedec.ml b/src/Modes/Modedec.ml similarity index 99% rename from src/modes/Modedec.ml rename to src/Modes/Modedec.ml index ba7d6f4..ef9c864 100644 --- a/src/modes/Modedec.ml +++ b/src/Modes/Modedec.ml @@ -5,7 +5,7 @@ open Modesyn (* Modes: short and long forms *) (* Author: Carsten Schuermann *) (* Modified: Frank Pfenning *) -include Modedec_intf +include MODEDEC (* signature MODEDEC *) (* # 1 "src/modes/Modedec.fun.ml" *) diff --git a/src/Modes/Modedec.mli b/src/Modes/Modedec.mli new file mode 100644 index 0000000..86f0a06 --- /dev/null +++ b/src/Modes/Modedec.mli @@ -0,0 +1,2 @@ +include module type of MODEDEC +module MakeModeDec () : MODEDEC diff --git a/src/modes/Modeprint.ml b/src/Modes/Modeprint.ml similarity index 99% rename from src/modes/Modeprint.ml rename to src/Modes/Modeprint.ml index 8ba5e09..c44d9f4 100644 --- a/src/modes/Modeprint.ml +++ b/src/Modes/Modeprint.ml @@ -4,7 +4,7 @@ open Modesyn (* Printing Mode Declarations *) (* Author: Carsten Schuermann *) -include Modeprint_intf +include MODEPRINT (* signature MODEPRINT *) (* # 1 "src/modes/Modeprint.fun.ml" *) diff --git a/src/modes/Modeprint.mli b/src/Modes/Modeprint.mli similarity index 70% rename from src/modes/Modeprint.mli rename to src/Modes/Modeprint.mli index 947cd51..c55b38b 100644 --- a/src/modes/Modeprint.mli +++ b/src/Modes/Modeprint.mli @@ -1,4 +1,4 @@ -include module type of Modeprint_intf +include module type of MODEPRINT module MakeModePrint (Names : NAMES) (Formatter : FORMATTER) (Print : PRINT) : MODEPRINT diff --git a/src/modes/Modes.ml b/src/Modes/Modes.ml similarity index 100% rename from src/modes/Modes.ml rename to src/Modes/Modes.ml diff --git a/src/modes/Modes.mli b/src/Modes/Modes.mli similarity index 100% rename from src/modes/Modes.mli rename to src/Modes/Modes.mli diff --git a/src/modes/Modes_.ml b/src/Modes/Modes_.ml similarity index 100% rename from src/modes/Modes_.ml rename to src/Modes/Modes_.ml diff --git a/src/modes/Modes_.mli b/src/Modes/Modes_.mli similarity index 100% rename from src/modes/Modes_.mli rename to src/Modes/Modes_.mli diff --git a/src/modes/Modesyn.ml b/src/Modes/Modesyn.ml similarity index 97% rename from src/modes/Modesyn.ml rename to src/Modes/Modesyn.ml index 5e5627d..da03c8f 100644 --- a/src/modes/Modesyn.ml +++ b/src/Modes/Modesyn.ml @@ -8,7 +8,7 @@ open! Basis (* Mode Syntax *) (* Author: Carsten Schuermann *) (* Modified: Frank Pfenning, Roberto Virga *) -include Modesyn_intf +include MODESYN (* signature MODESYN *) module ModeSyn : MODESYN = struct diff --git a/src/modes/Modesyn.mli b/src/Modes/Modesyn.mli similarity index 53% rename from src/modes/Modesyn.mli rename to src/Modes/Modesyn.mli index 358d010..e6b6cdb 100644 --- a/src/modes/Modesyn.mli +++ b/src/Modes/Modesyn.mli @@ -1,3 +1,3 @@ -include module type of Modesyn_intf +include module type of MODESYN module ModeSyn : MODESYN include MODESYN diff --git a/src/modes/Modetable.ml b/src/Modes/Modetable.ml similarity index 99% rename from src/modes/Modetable.ml rename to src/Modes/Modetable.ml index cfd0484..06618f2 100644 --- a/src/modes/Modetable.ml +++ b/src/Modes/Modetable.ml @@ -4,7 +4,7 @@ open Modesyn (* Mode Table *) (* Author: Frank Pfenning *) -include Modetable_intf +include MODETABLE (* signature MODETABLE *) (* # 1 "src/modes/Modetable.fun.ml" *) diff --git a/src/modes/Modetable.mli b/src/Modes/Modetable.mli similarity index 64% rename from src/modes/Modetable.mli rename to src/Modes/Modetable.mli index 7737ca7..dc3887c 100644 --- a/src/modes/Modetable.mli +++ b/src/Modes/Modetable.mli @@ -1,2 +1,2 @@ -include module type of Modetable_intf +include module type of MODETABLE module MakeModeTable (Table : TABLE with type key = int) : MODETABLE diff --git a/src/modes/README b/src/Modes/README similarity index 100% rename from src/modes/README rename to src/Modes/README diff --git a/src/modes/WALK b/src/Modes/WALK similarity index 100% rename from src/modes/WALK rename to src/Modes/WALK diff --git a/src/modes/dune b/src/Modes/dune similarity index 90% rename from src/modes/dune rename to src/Modes/dune index f55b624..ccc18c1 100644 --- a/src/modes/dune +++ b/src/Modes/dune @@ -10,11 +10,11 @@ Modecheck Modeprint Modes_ - Modecheck_intf - Modedec_intf - Modeprint_intf - Modesyn_intf - Modetable_intf) + MODECHECK + MODEDEC + MODEPRINT + MODESYN + MODETABLE) (libraries display global diff --git a/src/modes/modecheck.fun b/src/Modes/modecheck.fun similarity index 100% rename from src/modes/modecheck.fun rename to src/Modes/modecheck.fun diff --git a/src/modules/Modsyn_intf.ml b/src/Modules/MODSYN.ml similarity index 100% rename from src/modules/Modsyn_intf.ml rename to src/Modules/MODSYN.ml diff --git a/src/modules/Modsyn.ml b/src/Modules/Modsyn.ml similarity index 99% rename from src/modules/Modsyn.ml rename to src/Modules/Modsyn.ml index c18725f..a1a6807 100644 --- a/src/modules/Modsyn.ml +++ b/src/Modules/Modsyn.ml @@ -3,7 +3,7 @@ open! Basis (* Syntax for elaborated modules *) (* Author: Kevin Watkins *) -include Modsyn_intf +include MODSYN (* # 1 "src/modules/Modsyn.fun.ml" *) open! Basis diff --git a/src/modules/Modsyn.mli b/src/Modules/Modsyn.mli similarity index 94% rename from src/modules/Modsyn.mli rename to src/Modules/Modsyn.mli index 62132b0..3fc0c25 100644 --- a/src/modules/Modsyn.mli +++ b/src/Modules/Modsyn.mli @@ -1,4 +1,4 @@ -include module type of Modsyn_intf +include module type of MODSYN module ModSyn (ModSyn__0 : sig (* Syntax for elaborated modules *) diff --git a/src/modules/Modules.ml b/src/Modules/Modules.ml similarity index 100% rename from src/modules/Modules.ml rename to src/Modules/Modules.ml diff --git a/src/modules/Modules.mli b/src/Modules/Modules.mli similarity index 100% rename from src/modules/Modules.mli rename to src/Modules/Modules.mli diff --git a/src/modules/Modules_.ml b/src/Modules/Modules_.ml similarity index 100% rename from src/modules/Modules_.ml rename to src/Modules/Modules_.ml diff --git a/src/modules/Modules_.mli b/src/Modules/Modules_.mli similarity index 81% rename from src/modules/Modules_.mli rename to src/Modules/Modules_.mli index 8c347e4..40c66f4 100644 --- a/src/modules/Modules_.mli +++ b/src/Modules/Modules_.mli @@ -5,4 +5,4 @@ (* # 1 "src/modules/Modules_.sml.ml" *) open! Basis open TableInstances -module ModSyn : Modsyn_intf.MODSYN +module ModSyn : MODSYN.MODSYN diff --git a/src/modules/dune b/src/Modules/dune similarity index 93% rename from src/modules/dune rename to src/Modules/dune index 623347b..eb68fb3 100644 --- a/src/modules/dune +++ b/src/Modules/dune @@ -3,7 +3,7 @@ (library (name modules) (public_name stelf.modules) - (modules Modsyn Modules_ Modsyn_intf) + (modules Modsyn Modules_ MODSYN) (libraries display global table intsyn names paths typecheck basis) (flags (:standard diff --git a/src/msg/Msg_intf.ml b/src/Msg/MSG.ml similarity index 100% rename from src/msg/Msg_intf.ml rename to src/Msg/MSG.ml diff --git a/src/msg/Msg.ml b/src/Msg/Msg.ml similarity index 100% rename from src/msg/Msg.ml rename to src/Msg/Msg.ml diff --git a/src/msg/Msg.mli b/src/Msg/Msg.mli similarity index 100% rename from src/msg/Msg.mli rename to src/Msg/Msg.mli diff --git a/src/msg/Msg_.ml b/src/Msg/Msg_.ml similarity index 94% rename from src/msg/Msg_.ml rename to src/Msg/Msg_.ml index 38c83b2..b575eeb 100644 --- a/src/msg/Msg_.ml +++ b/src/Msg/Msg_.ml @@ -5,7 +5,7 @@ (* # 1 "src/msg/Msg_.sml.ml" *) open Basis -include Msg_intf +include MSG module Msg : MSG = struct let default m = Display.debug (Display.string m) diff --git a/src/Msg/Msg_.mli b/src/Msg/Msg_.mli new file mode 100644 index 0000000..708c6b5 --- /dev/null +++ b/src/Msg/Msg_.mli @@ -0,0 +1,2 @@ +include module type of MSG +module Msg : MSG diff --git a/src/msg/dune b/src/Msg/dune similarity index 88% rename from src/msg/dune rename to src/Msg/dune index d275c2f..5edc9a8 100644 --- a/src/msg/dune +++ b/src/Msg/dune @@ -3,7 +3,7 @@ (library (name msg) (public_name stelf.msg) - (modules Msg_ Msg_intf) + (modules Msg_ MSG) (libraries display basis) (wrapped true) (preprocess diff --git a/src/names/Names_intf.ml b/src/Names/NAMES.ml similarity index 100% rename from src/names/Names_intf.ml rename to src/Names/NAMES.ml diff --git a/src/names/Names.ml b/src/Names/Names.ml similarity index 66% rename from src/names/Names.ml rename to src/Names/Names.ml index adb169f..36fa467 100644 --- a/src/names/Names.ml +++ b/src/Names/Names.ml @@ -1,3 +1,3 @@ include Names_ -include Names_intf +include NAMES module Names_ = Names_ diff --git a/src/names/Names.mli b/src/Names/Names.mli similarity index 100% rename from src/names/Names.mli rename to src/Names/Names.mli diff --git a/src/names/Names_.ml b/src/Names/Names_.ml similarity index 99% rename from src/names/Names_.ml rename to src/Names/Names_.ml index e6c1a4d..3b66832 100644 --- a/src/names/Names_.ml +++ b/src/Names/Names_.ml @@ -5,7 +5,7 @@ open! Basis (* Author: Frank Pfenning *) (** Modified: Jeff Polakow *) -include Names_intf +include NAMES (** signature FIXITY *) (* signature NAMES *) diff --git a/src/names/Names_.mli b/src/Names/Names_.mli similarity index 89% rename from src/names/Names_.mli rename to src/Names/Names_.mli index ed3fa91..4eaa0dd 100644 --- a/src/names/Names_.mli +++ b/src/Names/Names_.mli @@ -1,4 +1,4 @@ -include module type of Names_intf +include module type of NAMES module MakeNames (Global : GLOBAL) diff --git a/src/names/README b/src/Names/README similarity index 100% rename from src/names/README rename to src/Names/README diff --git a/src/names/WALK b/src/Names/WALK similarity index 100% rename from src/names/WALK rename to src/Names/WALK diff --git a/src/names/dune b/src/Names/dune similarity index 93% rename from src/names/dune rename to src/Names/dune index 611327f..0589a96 100644 --- a/src/names/dune +++ b/src/Names/dune @@ -3,7 +3,7 @@ (library (name names) (public_name stelf.names) - (modules Names_ Names_intf) + (modules Names_ NAMES) (libraries display table global intsyn basis) (flags (:standard diff --git a/src/netserver/Netserver_intf.ml b/src/Netserver/NETSERVER.ml similarity index 100% rename from src/netserver/Netserver_intf.ml rename to src/Netserver/NETSERVER.ml diff --git a/src/netserver/Netserver.ml b/src/Netserver/Netserver.ml similarity index 100% rename from src/netserver/Netserver.ml rename to src/Netserver/Netserver.ml diff --git a/src/netserver/Netserver.mli b/src/Netserver/Netserver.mli similarity index 100% rename from src/netserver/Netserver.mli rename to src/Netserver/Netserver.mli diff --git a/src/netserver/Netserver_.ml b/src/Netserver/Netserver_.ml similarity index 99% rename from src/netserver/Netserver_.ml rename to src/Netserver/Netserver_.ml index 5685b08..6ec6407 100644 --- a/src/netserver/Netserver_.ml +++ b/src/Netserver/Netserver_.ml @@ -4,7 +4,7 @@ (* # 1 "src/netserver/Netserver_.sml.ml" *) open! Basis -include Netserver_intf +include NETSERVER (* filesystem directory where stelf examples are kept *) (* signature SERVER *) diff --git a/src/Netserver/Netserver_.mli b/src/Netserver/Netserver_.mli new file mode 100644 index 0000000..e679090 --- /dev/null +++ b/src/Netserver/Netserver_.mli @@ -0,0 +1,2 @@ +include module type of NETSERVER +module NetServer : NETSERVER diff --git a/src/netserver/dune b/src/Netserver/dune similarity index 91% rename from src/netserver/dune rename to src/Netserver/dune index 21faedf..66646e2 100644 --- a/src/netserver/dune +++ b/src/Netserver/dune @@ -3,7 +3,7 @@ (library (name netserver) (public_name stelf.netserver) - (modules Netserver_ Netserver_intf) + (modules Netserver_ NETSERVER) (libraries display timing frontend msg basis) (flags (:standard diff --git a/src/netserver/htdocs/floral.png b/src/Netserver/htdocs/floral.png similarity index 100% rename from src/netserver/htdocs/floral.png rename to src/Netserver/htdocs/floral.png diff --git a/src/netserver/htdocs/grad.png b/src/Netserver/htdocs/grad.png similarity index 100% rename from src/netserver/htdocs/grad.png rename to src/Netserver/htdocs/grad.png diff --git a/src/netserver/htdocs/input.png b/src/Netserver/htdocs/input.png similarity index 100% rename from src/netserver/htdocs/input.png rename to src/Netserver/htdocs/input.png diff --git a/src/netserver/htdocs/output.png b/src/Netserver/htdocs/output.png similarity index 100% rename from src/netserver/htdocs/output.png rename to src/Netserver/htdocs/output.png diff --git a/src/netserver/htdocs/server.css b/src/Netserver/htdocs/server.css similarity index 100% rename from src/netserver/htdocs/server.css rename to src/Netserver/htdocs/server.css diff --git a/src/netserver/htdocs/server.html b/src/Netserver/htdocs/server.html similarity index 100% rename from src/netserver/htdocs/server.html rename to src/Netserver/htdocs/server.html diff --git a/src/netserver/htdocs/server.js b/src/Netserver/htdocs/server.js similarity index 100% rename from src/netserver/htdocs/server.js rename to src/Netserver/htdocs/server.js diff --git a/src/netserver/htdocs/twelfguy.png b/src/Netserver/htdocs/twelfguy.png similarity index 100% rename from src/netserver/htdocs/twelfguy.png rename to src/Netserver/htdocs/twelfguy.png diff --git a/src/netserver/runserver.sh b/src/Netserver/runserver.sh similarity index 100% rename from src/netserver/runserver.sh rename to src/Netserver/runserver.sh diff --git a/src/opsem/Absmachine_intf.ml b/src/Opsem/ABSMACHINE.ml similarity index 100% rename from src/opsem/Absmachine_intf.ml rename to src/Opsem/ABSMACHINE.ml diff --git a/src/opsem/AbsmachineSbt_intf.ml b/src/Opsem/ABSMACHINESBT.ml similarity index 100% rename from src/opsem/AbsmachineSbt_intf.ml rename to src/Opsem/ABSMACHINESBT.ml diff --git a/src/opsem/AbstractTabled_intf.ml b/src/Opsem/ABSTRACTTABLED.ml similarity index 100% rename from src/opsem/AbstractTabled_intf.ml rename to src/Opsem/ABSTRACTTABLED.ml diff --git a/src/opsem/Absmachine.ml b/src/Opsem/Absmachine.ml similarity index 99% rename from src/opsem/Absmachine.ml rename to src/Opsem/Absmachine.ml index a5e7bf4..3eb2405 100644 --- a/src/opsem/Absmachine.ml +++ b/src/Opsem/Absmachine.ml @@ -5,7 +5,7 @@ open! Basis (* Author: Iliano Cervesato *) (* Modified: Jeff Polakow *) (* Modified: Frank Pfenning *) -include Absmachine_intf +include ABSMACHINE (* signature ABSMACHINE *) (* # 1 "src/opsem/Absmachine.fun.ml" *) diff --git a/src/opsem/Absmachine.mli b/src/Opsem/Absmachine.mli similarity index 94% rename from src/opsem/Absmachine.mli rename to src/Opsem/Absmachine.mli index 9c2760b..4dc800e 100644 --- a/src/opsem/Absmachine.mli +++ b/src/Opsem/Absmachine.mli @@ -1,4 +1,4 @@ -include module type of Absmachine_intf +include module type of ABSMACHINE module AbsMachine (AbsMachine__0 : sig (*! structure IntSyn' : INTSYN !*) diff --git a/src/opsem/AbsmachineSbt.ml b/src/Opsem/AbsmachineSbt.ml similarity index 99% rename from src/opsem/AbsmachineSbt.ml rename to src/Opsem/AbsmachineSbt.ml index 8ce7b45..f529e53 100644 --- a/src/opsem/AbsmachineSbt.ml +++ b/src/Opsem/AbsmachineSbt.ml @@ -5,7 +5,7 @@ open! Basis (* Author: Iliano Cervesato *) (* Modified: Jeff Polakow *) (* Modified: Frank Pfenning *) -include AbsmachineSbt_intf +include ABSMACHINESBT (* signature ABSMACHINESBT *) (* # 1 "src/opsem/AbsmachineSbt.fun.ml" *) diff --git a/src/opsem/AbsmachineSbt.mli b/src/Opsem/AbsmachineSbt.mli similarity index 85% rename from src/opsem/AbsmachineSbt.mli rename to src/Opsem/AbsmachineSbt.mli index 4499d85..e8d8fd0 100644 --- a/src/opsem/AbsmachineSbt.mli +++ b/src/Opsem/AbsmachineSbt.mli @@ -1,4 +1,4 @@ -include module type of AbsmachineSbt_intf +include module type of ABSMACHINESBT module AbsMachineSbt (AbsMachineSbt__0 : sig (*! structure IntSyn' : INTSYN !*) @@ -7,7 +7,7 @@ module AbsMachineSbt (AbsMachineSbt__0 : sig module Unify : UNIFY (*! sharing Unify.IntSyn = IntSyn' !*) - module SubTree : Subtree_intf.SUBTREE + module SubTree : SUBTREE.SUBTREE (*! sharing SubTree.IntSyn = IntSyn' !*) (*! sharing SubTree.CompSyn = CompSyn' !*) @@ -26,4 +26,4 @@ module AbsMachineSbt (AbsMachineSbt__0 : sig (*! sharing Print.IntSyn = IntSyn' !*) module Names : NAMES -end) : AbsmachineSbt_intf.ABSMACHINESBT +end) : ABSMACHINESBT.ABSMACHINESBT diff --git a/src/opsem/AbstractTabled.ml b/src/Opsem/AbstractTabled.ml similarity index 99% rename from src/opsem/AbstractTabled.ml rename to src/Opsem/AbstractTabled.ml index 4ff064a..f8c74fd 100644 --- a/src/opsem/AbstractTabled.ml +++ b/src/Opsem/AbstractTabled.ml @@ -4,7 +4,7 @@ open TableParam (* Abstraction *) (* Author: Brigitte Pientka *) -include AbstractTabled_intf +include ABSTRACTTABLED (* signature ABSTRACTTABLED *) (* # 1 "src/opsem/AbstractTabled.fun.ml" *) diff --git a/src/opsem/AbstractTabled.mli b/src/Opsem/AbstractTabled.mli similarity index 92% rename from src/opsem/AbstractTabled.mli rename to src/Opsem/AbstractTabled.mli index 0f98fa5..57b8c5a 100644 --- a/src/opsem/AbstractTabled.mli +++ b/src/Opsem/AbstractTabled.mli @@ -1,4 +1,4 @@ -include module type of AbstractTabled_intf +include module type of ABSTRACTTABLED module AbstractTabled (AbstractTabled__0 : sig (*! structure IntSyn' : INTSYN !*) diff --git a/src/opsem/MemoTable_intf.ml b/src/Opsem/MEMOTABLE.ml similarity index 100% rename from src/opsem/MemoTable_intf.ml rename to src/Opsem/MEMOTABLE.ml diff --git a/src/opsem/MemoTable.ml b/src/Opsem/MemoTable.ml similarity index 99% rename from src/opsem/MemoTable.ml rename to src/Opsem/MemoTable.ml index c9c507b..61e947c 100644 --- a/src/opsem/MemoTable.ml +++ b/src/Opsem/MemoTable.ml @@ -4,7 +4,7 @@ open TableParam (* Indexing *) (* Author: Brigitte Pientka *) -include MemoTable_intf +include MEMOTABLE (* signature MemoTable *) (* # 1 "src/opsem/MemoTable.fun.ml" *) diff --git a/src/opsem/MemoTable.mli b/src/Opsem/MemoTable.mli similarity index 81% rename from src/opsem/MemoTable.mli rename to src/Opsem/MemoTable.mli index f10f0da..cf0c12d 100644 --- a/src/opsem/MemoTable.mli +++ b/src/Opsem/MemoTable.mli @@ -1,4 +1,4 @@ -include module type of MemoTable_intf +include module type of MEMOTABLE module MemoTable (MemoTable__0 : sig (*! structure IntSyn' : INTSYN !*) @@ -15,8 +15,8 @@ module MemoTable (MemoTable__0 : sig (*! sharing TableParam.IntSyn = IntSyn' !*) (*! sharing TableParam.CompSyn = CompSyn' !*) (*! sharing TableParam.RBSet = RBSet !*) - module AbstractTabled : AbstractTabled_intf.ABSTRACTTABLED + module AbstractTabled : ABSTRACTTABLED.ABSTRACTTABLED (*! sharing AbstractTabled.IntSyn = IntSyn' !*) module Print : PRINT -end) : MemoTable_intf.MEMOTABLE +end) : MEMOTABLE.MEMOTABLE diff --git a/src/opsem/Opsem.ml b/src/Opsem/Opsem.ml similarity index 100% rename from src/opsem/Opsem.ml rename to src/Opsem/Opsem.ml diff --git a/src/opsem/Opsem.mli b/src/Opsem/Opsem.mli similarity index 100% rename from src/opsem/Opsem.mli rename to src/Opsem/Opsem.mli diff --git a/src/opsem/Opsem_.ml b/src/Opsem/Opsem_.ml similarity index 100% rename from src/opsem/Opsem_.ml rename to src/Opsem/Opsem_.ml diff --git a/src/opsem/Opsem_.mli b/src/Opsem/Opsem_.mli similarity index 91% rename from src/opsem/Opsem_.mli rename to src/Opsem/Opsem_.mli index d7cd96d..1730102 100644 --- a/src/opsem/Opsem_.mli +++ b/src/Opsem/Opsem_.mli @@ -9,4 +9,4 @@ module PtRecon : Ptrecon.PTRECON module AbsMachineSbt : AbsmachineSbt.ABSMACHINESBT module Tabled_ : TabledMachine.TABLED module SwMachine : Absmachine.ABSMACHINE -module Trace : Trace_intf.TRACE +module Trace : TRACE.TRACE diff --git a/src/opsem/Ptrecon_intf.ml b/src/Opsem/PTRECON.ml similarity index 100% rename from src/opsem/Ptrecon_intf.ml rename to src/Opsem/PTRECON.ml diff --git a/src/opsem/Ptrecon.ml b/src/Opsem/Ptrecon.ml similarity index 99% rename from src/opsem/Ptrecon.ml rename to src/Opsem/Ptrecon.ml index f260bee..159ac11 100644 --- a/src/opsem/Ptrecon.ml +++ b/src/Opsem/Ptrecon.ml @@ -6,7 +6,7 @@ open! Basis (* Modified: Jeff Polakow *) (* Modified: Frank Pfenning *) (* Proof term reconstruction by proof skeleton *) -include Ptrecon_intf +include PTRECON (* signature PTRECON *) (* # 1 "src/opsem/Ptrecon.fun.ml" *) @@ -29,7 +29,7 @@ module PtRecon (PtRecon__0 : sig (*! sharing Assign.IntSyn = IntSyn' !*) (*! structure TableParam : TABLEPARAM !*) - module MemoTable : MemoTable_intf.MEMOTABLE + module MemoTable : MEMOTABLE.MEMOTABLE (*! sharing MemoTable.TableParam = TableParam !*) module Index : INDEX diff --git a/src/opsem/Ptrecon.mli b/src/Opsem/Ptrecon.mli similarity index 88% rename from src/opsem/Ptrecon.mli rename to src/Opsem/Ptrecon.mli index 32ad168..f3abdeb 100644 --- a/src/opsem/Ptrecon.mli +++ b/src/Opsem/Ptrecon.mli @@ -1,4 +1,4 @@ -include module type of Ptrecon_intf +include module type of PTRECON module PtRecon (PtRecon__0 : sig (*! structure IntSyn' : INTSYN !*) @@ -11,7 +11,7 @@ module PtRecon (PtRecon__0 : sig (*! sharing Assign.IntSyn = IntSyn' !*) (*! structure TableParam : TABLEPARAM !*) - module MemoTable : MemoTable_intf.MEMOTABLE + module MemoTable : MEMOTABLE.MEMOTABLE (*! sharing MemoTable.TableParam = TableParam !*) module Index : INDEX diff --git a/src/opsem/README b/src/Opsem/README similarity index 100% rename from src/opsem/README rename to src/Opsem/README diff --git a/src/opsem/SwSubtree_intf.ml b/src/Opsem/SWSUBTREE.ml similarity index 100% rename from src/opsem/SwSubtree_intf.ml rename to src/Opsem/SWSUBTREE.ml diff --git a/src/opsem/SubtreeInst.ml b/src/Opsem/SubtreeInst.ml similarity index 100% rename from src/opsem/SubtreeInst.ml rename to src/Opsem/SubtreeInst.ml diff --git a/src/opsem/SubtreeInst.mli b/src/Opsem/SubtreeInst.mli similarity index 94% rename from src/opsem/SubtreeInst.mli rename to src/Opsem/SubtreeInst.mli index 1ac961a..738e251 100644 --- a/src/opsem/SubtreeInst.mli +++ b/src/Opsem/SubtreeInst.mli @@ -29,7 +29,7 @@ module MemoTableInst (MemoTableInst__0 : sig (*! sharing TableParam.IntSyn = IntSyn' !*) (*! sharing TableParam.CompSyn = CompSyn' !*) (*! sharing TableParam.RBSet = RBSet !*) - module AbstractTabled : AbstractTabled_intf.ABSTRACTTABLED + module AbstractTabled : ABSTRACTTABLED.ABSTRACTTABLED (*! sharing AbstractTabled.IntSyn = IntSyn' !*) module Print : PRINT diff --git a/src/opsem/SwSubtree.ml b/src/Opsem/SwSubtree.ml similarity index 98% rename from src/opsem/SwSubtree.ml rename to src/Opsem/SwSubtree.ml index ac2c623..712e000 100644 --- a/src/opsem/SwSubtree.ml +++ b/src/Opsem/SwSubtree.ml @@ -4,7 +4,7 @@ open TableParam (* Indexing *) (* Author: Brigitte Pientka *) -include SwSubtree_intf +include SWSUBTREE (* signature MemoTable *) (* # 1 "src/opsem/SwSubtree.fun.ml" *) diff --git a/src/Opsem/SwSubtree.mli b/src/Opsem/SwSubtree.mli new file mode 100644 index 0000000..0244d66 --- /dev/null +++ b/src/Opsem/SwSubtree.mli @@ -0,0 +1,7 @@ +include module type of SWSUBTREE + +module SwMemoTable (SwMemoTable__0 : sig + (* structure TableParam : TABLEPARAM *) + module MemoTable : MEMOTABLE.MEMOTABLE + module MemoTableInst : MEMOTABLE.MEMOTABLE +end) : MEMOTABLE.MEMOTABLE diff --git a/src/opsem/Swmachine.ml b/src/Opsem/Swmachine.ml similarity index 100% rename from src/opsem/Swmachine.ml rename to src/Opsem/Swmachine.ml diff --git a/src/opsem/Swmachine.mli b/src/Opsem/Swmachine.mli similarity index 100% rename from src/opsem/Swmachine.mli rename to src/Opsem/Swmachine.mli diff --git a/src/opsem/TabledMachine_intf.ml b/src/Opsem/TABLEDMACHINE.ml similarity index 100% rename from src/opsem/TabledMachine_intf.ml rename to src/Opsem/TABLEDMACHINE.ml diff --git a/src/opsem/TableParam_intf.ml b/src/Opsem/TABLEPARAM.ml similarity index 100% rename from src/opsem/TableParam_intf.ml rename to src/Opsem/TABLEPARAM.ml diff --git a/src/opsem/Trace_intf.ml b/src/Opsem/TRACE.ml similarity index 100% rename from src/opsem/Trace_intf.ml rename to src/Opsem/TRACE.ml diff --git a/src/opsem/TableParam.ml b/src/Opsem/TableParam.ml similarity index 99% rename from src/opsem/TableParam.ml rename to src/Opsem/TableParam.ml index d8ef0ba..92539cb 100644 --- a/src/opsem/TableParam.ml +++ b/src/Opsem/TableParam.ml @@ -4,7 +4,7 @@ open RedBlackSet (* Global Table parameters *) (* Author: Brigitte Pientka *) -include TableParam_intf +include TABLEPARAM (* signature TABLEPARAM *) (* # 1 "src/opsem/TableParam.fun.ml" *) diff --git a/src/opsem/TableParam.mli b/src/Opsem/TableParam.mli similarity index 68% rename from src/opsem/TableParam.mli rename to src/Opsem/TableParam.mli index 15ffc64..cf6f5f1 100644 --- a/src/opsem/TableParam.mli +++ b/src/Opsem/TableParam.mli @@ -1,3 +1,3 @@ -include module type of TableParam_intf +include module type of TABLEPARAM module MakeTableParam (Global : GLOBAL) : TABLEPARAM module TableParam : TABLEPARAM diff --git a/src/opsem/TabledMachine.ml b/src/Opsem/TabledMachine.ml similarity index 99% rename from src/opsem/TabledMachine.ml rename to src/Opsem/TabledMachine.ml index c310e79..b4652f5 100644 --- a/src/opsem/TabledMachine.ml +++ b/src/Opsem/TabledMachine.ml @@ -3,7 +3,7 @@ open! Basis (* Tabled Abstract Machine *) (* Author: Brigitte Pientka *) -include TabledMachine_intf +include TABLEDMACHINE (* signature TABLED *) (* # 1 "src/opsem/TabledMachine.fun.ml" *) @@ -37,11 +37,11 @@ module Tabled (Tabled__0 : sig (*! structure TableParam : TABLEPARAM !*) (*! sharing TableParam.IntSyn = IntSyn' !*) (*! sharing TableParam.CompSyn = CompSyn' !*) - module AbstractTabled : AbstractTabled_intf.ABSTRACTTABLED + module AbstractTabled : ABSTRACTTABLED.ABSTRACTTABLED (*! sharing AbstractTabled.IntSyn = IntSyn' !*) (*! sharing AbstractTabled.TableParam = TableParam !*) - module MemoTable : MemoTable_intf.MEMOTABLE + module MemoTable : MEMOTABLE.MEMOTABLE (*! sharing MemoTable.IntSyn = IntSyn' !*) (*! sharing MemoTable.CompSyn = CompSyn' !*) diff --git a/src/opsem/TabledMachine.mli b/src/Opsem/TabledMachine.mli similarity index 85% rename from src/opsem/TabledMachine.mli rename to src/Opsem/TabledMachine.mli index f3fd9b2..e5258f1 100644 --- a/src/opsem/TabledMachine.mli +++ b/src/Opsem/TabledMachine.mli @@ -1,4 +1,4 @@ -include module type of TabledMachine_intf +include module type of TABLEDMACHINE module Tabled (Tabled__0 : sig (*! structure IntSyn' : INTSYN !*) @@ -21,11 +21,11 @@ module Tabled (Tabled__0 : sig (*! structure TableParam : TABLEPARAM !*) (*! sharing TableParam.IntSyn = IntSyn' !*) (*! sharing TableParam.CompSyn = CompSyn' !*) - module AbstractTabled : AbstractTabled_intf.ABSTRACTTABLED + module AbstractTabled : ABSTRACTTABLED.ABSTRACTTABLED (*! sharing AbstractTabled.IntSyn = IntSyn' !*) (*! sharing AbstractTabled.TableParam = TableParam !*) - module MemoTable : MemoTable_intf.MEMOTABLE + module MemoTable : MEMOTABLE.MEMOTABLE (*! sharing MemoTable.IntSyn = IntSyn' !*) (*! sharing MemoTable.CompSyn = CompSyn' !*) @@ -37,4 +37,4 @@ module Tabled (Tabled__0 : sig (*! sharing CPrint.CompSyn = CompSyn' !*) (* CPrint currently unused *) module Print : PRINT -end) : TabledMachine_intf.TABLED +end) : TABLEDMACHINE.TABLED diff --git a/src/opsem/Tmachine.ml b/src/Opsem/Tmachine.ml similarity index 100% rename from src/opsem/Tmachine.ml rename to src/Opsem/Tmachine.ml diff --git a/src/opsem/Tmachine.mli b/src/Opsem/Tmachine.mli similarity index 100% rename from src/opsem/Tmachine.mli rename to src/Opsem/Tmachine.mli diff --git a/src/opsem/Trace.ml b/src/Opsem/Trace.ml similarity index 99% rename from src/opsem/Trace.ml rename to src/Opsem/Trace.ml index 5646ff1..db93782 100644 --- a/src/opsem/Trace.ml +++ b/src/Opsem/Trace.ml @@ -1,6 +1,6 @@ (* # 1 "src/opsem/Trace.sig.ml" *) open! Basis -include Trace_intf +include TRACE (* reset trace, break, detail *) (* signature TRACE *) diff --git a/src/opsem/Trace.mli b/src/Opsem/Trace.mli similarity index 89% rename from src/opsem/Trace.mli rename to src/Opsem/Trace.mli index f716e16..c6e4b6e 100644 --- a/src/opsem/Trace.mli +++ b/src/Opsem/Trace.mli @@ -1,4 +1,4 @@ -include module type of Trace_intf +include module type of TRACE module Trace (Trace__0 : sig (*! structure IntSyn' : INTSYN !*) diff --git a/src/opsem/WALK b/src/Opsem/WALK similarity index 100% rename from src/opsem/WALK rename to src/Opsem/WALK diff --git a/src/opsem/dune b/src/Opsem/dune similarity index 87% rename from src/opsem/dune rename to src/Opsem/dune index 0dfd1ac..4950283 100644 --- a/src/opsem/dune +++ b/src/Opsem/dune @@ -17,15 +17,15 @@ Tmachine Swmachine Opsem_ - Absmachine_intf - AbsmachineSbt_intf - AbstractTabled_intf - MemoTable_intf - Ptrecon_intf - SwSubtree_intf - TableParam_intf - TabledMachine_intf - Trace_intf) + ABSMACHINE + ABSMACHINESBT + ABSTRACTTABLED + MEMOTABLE + PTRECON + SWSUBTREE + TABLEPARAM + TABLEDMACHINE + TRACE) (libraries display global diff --git a/src/order/Order_intf.ml b/src/Order/ORDER.ml similarity index 100% rename from src/order/Order_intf.ml rename to src/Order/ORDER.ml diff --git a/src/order/Order.ml b/src/Order/Order.ml similarity index 100% rename from src/order/Order.ml rename to src/Order/Order.ml diff --git a/src/order/Order.mli b/src/Order/Order.mli similarity index 100% rename from src/order/Order.mli rename to src/Order/Order.mli diff --git a/src/order/Order_.ml b/src/Order/Order_.ml similarity index 99% rename from src/order/Order_.ml rename to src/Order/Order_.ml index 706b973..e9063dc 100644 --- a/src/order/Order_.ml +++ b/src/Order/Order_.ml @@ -3,7 +3,7 @@ open! Basis (* Termination Order *) -include Order_intf +include ORDER (** Author: Carsten Schuermann *) (* signature ORDER *) diff --git a/src/order/Order_.mli b/src/Order/Order_.mli similarity index 73% rename from src/order/Order_.mli rename to src/Order/Order_.mli index a000d34..dd4a907 100644 --- a/src/order/Order_.mli +++ b/src/Order/Order_.mli @@ -1,4 +1,4 @@ -include module type of Order_intf +include module type of ORDER module MakeOrder (Table : TABLE with type key = int) : ORDER module Order : ORDER include ORDER diff --git a/src/order/README b/src/Order/README similarity index 100% rename from src/order/README rename to src/Order/README diff --git a/src/order/WALK b/src/Order/WALK similarity index 100% rename from src/order/WALK rename to src/Order/WALK diff --git a/src/order/dune b/src/Order/dune similarity index 92% rename from src/order/dune rename to src/Order/dune index 43bcdbb..e6d8421 100644 --- a/src/order/dune +++ b/src/Order/dune @@ -3,7 +3,7 @@ (library (name order) (public_name stelf.order) - (modules Order_ Order_intf) + (modules Order_ ORDER) (libraries display table intsyn basis) (flags (:standard diff --git a/src/paths/Origins_intf.ml b/src/Paths/ORIGINS.ml similarity index 100% rename from src/paths/Origins_intf.ml rename to src/Paths/ORIGINS.ml diff --git a/src/paths/Origins.ml b/src/Paths/Origins.ml similarity index 98% rename from src/paths/Origins.ml rename to src/Paths/Origins.ml index f779ae9..6095fda 100644 --- a/src/paths/Origins.ml +++ b/src/Paths/Origins.ml @@ -5,7 +5,7 @@ module Paths = Paths_.Paths (* Origins of Declarations *) (* Author: Frank Pfenning *) -include Origins_intf +include ORIGINS (* signature ORIGINS *) (* # 1 "src/paths/Origins.fun.ml" *) diff --git a/src/paths/Origins.mli b/src/Paths/Origins.mli similarity index 78% rename from src/paths/Origins.mli rename to src/Paths/Origins.mli index 9115f15..b530ad5 100644 --- a/src/paths/Origins.mli +++ b/src/Paths/Origins.mli @@ -1,4 +1,4 @@ -include module type of Origins_intf +include module type of ORIGINS module MakeOrigins (Global : GLOBAL) (Table : TABLE with type key = string) : ORIGINS diff --git a/src/paths/Paths_intf.ml b/src/Paths/PATHS.ml similarity index 100% rename from src/paths/Paths_intf.ml rename to src/Paths/PATHS.ml diff --git a/src/paths/Paths.ml b/src/Paths/Paths.ml similarity index 100% rename from src/paths/Paths.ml rename to src/Paths/Paths.ml diff --git a/src/paths/Paths.mli b/src/Paths/Paths.mli similarity index 100% rename from src/paths/Paths.mli rename to src/Paths/Paths.mli diff --git a/src/paths/Paths_.ml b/src/Paths/Paths_.ml similarity index 99% rename from src/paths/Paths_.ml rename to src/Paths/Paths_.ml index 61e6438..da672f4 100644 --- a/src/paths/Paths_.ml +++ b/src/Paths/Paths_.ml @@ -3,7 +3,7 @@ open! Basis (* Paths, Occurrences, and Error Locations *) -include Paths_intf +include PATHS (** Author: Frank Pfenning *) (* into v for c : V ... *) diff --git a/src/paths/Paths_.mli b/src/Paths/Paths_.mli similarity index 64% rename from src/paths/Paths_.mli rename to src/Paths/Paths_.mli index 7cbde4e..8219d22 100644 --- a/src/paths/Paths_.mli +++ b/src/Paths/Paths_.mli @@ -1,4 +1,4 @@ -include module type of Paths_intf +include module type of PATHS module MakePaths () : PATHS module Paths : PATHS include PATHS diff --git a/src/paths/README b/src/Paths/README similarity index 100% rename from src/paths/README rename to src/Paths/README diff --git a/src/paths/WALK b/src/Paths/WALK similarity index 100% rename from src/paths/WALK rename to src/Paths/WALK diff --git a/src/paths/dune b/src/Paths/dune similarity index 88% rename from src/paths/dune rename to src/Paths/dune index 674b1e1..605a1b0 100644 --- a/src/paths/dune +++ b/src/Paths/dune @@ -3,7 +3,7 @@ (library (name paths) (public_name stelf.paths) - (modules Paths_intf Paths_ Origins_intf Origins) + (modules PATHS Paths_ ORIGINS Origins) (libraries display intsyn global table basis) (flags (:standard diff --git a/src/print/ClausePrint_intf.ml b/src/Print/CLAUSEPRINT.ml similarity index 100% rename from src/print/ClausePrint_intf.ml rename to src/Print/CLAUSEPRINT.ml diff --git a/src/print/ClausePrint.ml b/src/Print/ClausePrint.ml similarity index 99% rename from src/print/ClausePrint.ml rename to src/Print/ClausePrint.ml index 9bcc6b3..ab5aa16 100644 --- a/src/print/ClausePrint.ml +++ b/src/Print/ClausePrint.ml @@ -3,7 +3,7 @@ open! Basis (* Clause Printing *) (* Author: Frank Pfenning, Carsten Schuermann *) -include ClausePrint_intf +include CLAUSEPRINT (* signature CLAUSEPRINT *) (* # 1 "src/print/ClausePrint.fun.ml" *) diff --git a/src/print/ClausePrint.mli b/src/Print/ClausePrint.mli similarity index 97% rename from src/print/ClausePrint.mli rename to src/Print/ClausePrint.mli index adf1c6d..bfcccbf 100644 --- a/src/print/ClausePrint.mli +++ b/src/Print/ClausePrint.mli @@ -1,4 +1,4 @@ -include module type of ClausePrint_intf +include module type of CLAUSEPRINT module MakeClausePrint (Whnf : WHNF) diff --git a/src/print/Print_intf.ml b/src/Print/PRINT.ml similarity index 100% rename from src/print/Print_intf.ml rename to src/Print/PRINT.ml diff --git a/src/print/PrintOmdoc_intf.ml b/src/Print/PRINTOMDOC.ml similarity index 100% rename from src/print/PrintOmdoc_intf.ml rename to src/Print/PRINTOMDOC.ml diff --git a/src/print/PrintTwega_intf.ml b/src/Print/PRINTTWEGA.ml similarity index 100% rename from src/print/PrintTwega_intf.ml rename to src/Print/PRINTTWEGA.ml diff --git a/src/print/PrintXml_intf.ml b/src/Print/PRINTXML.ml similarity index 100% rename from src/print/PrintXml_intf.ml rename to src/Print/PRINTXML.ml diff --git a/src/print/Print.ml b/src/Print/Print.ml similarity index 100% rename from src/print/Print.ml rename to src/Print/Print.ml diff --git a/src/print/Print.mli b/src/Print/Print.mli similarity index 100% rename from src/print/Print.mli rename to src/Print/Print.mli diff --git a/src/print/PrintOmdoc.ml b/src/Print/PrintOmdoc.ml similarity index 99% rename from src/print/PrintOmdoc.ml rename to src/Print/PrintOmdoc.ml index d0890e5..bd6c71f 100644 --- a/src/print/PrintOmdoc.ml +++ b/src/Print/PrintOmdoc.ml @@ -3,7 +3,7 @@ open! Basis (* Printing Signatures to OMDoc*) (* Author: Florian Rabe *) -include PrintOmdoc_intf +include PRINTOMDOC (* signature PRINT_XML *) (* # 1 "src/print/PrintOmdoc.fun.ml" *) diff --git a/src/print/PrintOmdoc.mli b/src/Print/PrintOmdoc.mli similarity index 94% rename from src/print/PrintOmdoc.mli rename to src/Print/PrintOmdoc.mli index a0f0cee..0ef6c66 100644 --- a/src/print/PrintOmdoc.mli +++ b/src/Print/PrintOmdoc.mli @@ -1,4 +1,4 @@ -include module type of PrintOmdoc_intf +include module type of PRINTOMDOC module MakePrintOMDoc (Whnf : WHNF) diff --git a/src/print/PrintTwega.ml b/src/Print/PrintTwega.ml similarity index 99% rename from src/print/PrintTwega.ml rename to src/Print/PrintTwega.ml index cb117d6..e6431bc 100644 --- a/src/print/PrintTwega.ml +++ b/src/Print/PrintTwega.ml @@ -3,7 +3,7 @@ open! Basis (* Printing Signatures *) (* Author: Frank Pfenning *) -include PrintTwega_intf +include PRINTTWEGA (* signature PRINT_TWEGA *) (* # 1 "src/print/PrintTwega.fun.ml" *) diff --git a/src/print/PrintTwega.mli b/src/Print/PrintTwega.mli similarity index 93% rename from src/print/PrintTwega.mli rename to src/Print/PrintTwega.mli index 094d5bf..bc27406 100644 --- a/src/print/PrintTwega.mli +++ b/src/Print/PrintTwega.mli @@ -1,4 +1,4 @@ -include module type of PrintTwega_intf +include module type of PRINTTWEGA module MakePrintTwega (Whnf : WHNF) diff --git a/src/print/PrintXml.ml b/src/Print/PrintXml.ml similarity index 99% rename from src/print/PrintXml.ml rename to src/Print/PrintXml.ml index e1b47c7..c3c0a10 100644 --- a/src/print/PrintXml.ml +++ b/src/Print/PrintXml.ml @@ -4,7 +4,7 @@ open! Basis (* Printing Signatures *) (* Author: Frank Pfenning *) (* modified: Carsten Schuermann *) -include PrintXml_intf +include PRINTXML (* signature PRINT_XML *) (* # 1 "src/print/PrintXml.fun.ml" *) diff --git a/src/print/PrintXml.mli b/src/Print/PrintXml.mli similarity index 94% rename from src/print/PrintXml.mli rename to src/Print/PrintXml.mli index 153f7f5..d70e8c5 100644 --- a/src/print/PrintXml.mli +++ b/src/Print/PrintXml.mli @@ -1,4 +1,4 @@ -include module type of PrintXml_intf +include module type of PRINTXML module MakePrintXML (Whnf : WHNF) diff --git a/src/print/Print_.ml b/src/Print/Print_.ml similarity index 99% rename from src/print/Print_.ml rename to src/Print/Print_.ml index e2f07e0..8bafb65 100644 --- a/src/print/Print_.ml +++ b/src/Print/Print_.ml @@ -4,7 +4,7 @@ open! Basis (* Printing *) (* Author: Frank Pfenning *) -include Print_intf +include PRINT (** Modified: Jeff Polakow *) (* signature PRINT *) @@ -580,7 +580,7 @@ module MakePrint | g_, d, (I.BDec (x, (cid, t)), s) -> let gsome_, gblock_ = I.constBlock cid in F.hVbox - ([ str0_ (Symbol.const (nameOf x)); F.space] + ([ str0_ (Symbol.const (nameOf x)); F.space ] @ fmtDecList' (g_, (gblock_, I.comp (t, s)))) | g_, d, (I.ADec (x, _), s) -> F.hVbox [ str0_ (Symbol.bvar (nameOf x)); sym "_" ] @@ -625,7 +625,9 @@ module MakePrint | I.Uni _ -> [] | I.Pi ((d_, _), v2_) -> let d'_ = Names.decLUName (g_, d_) in - sym "{" :: fmtDec (g_, d, (d'_, I.id)) :: sym "}" :: F.break + sym "{" + :: fmtDec (g_, d, (d'_, I.id)) + :: sym "}" :: F.break :: fmtKindBinders (I.Decl (g_, d'_), d + 1, v2_) | _ -> [ fmtExp (g_, d, noCtxt, (v_, I.id)) ] end @@ -639,7 +641,7 @@ module MakePrint F.break; F.hVbox ([ sym "block"; F.space ] @ fmtDecList (gsome_, lblock_)); ] - (* Fix *) + (* Fix *) let rec fmtConDec = function | hide, (I.ConDec (_, _, imp, _, v_, l_) as condec) -> @@ -655,8 +657,7 @@ module MakePrint F.hVbox ([ sym "%sort"; F.space; fmtConstPath (Symbol.const, qid) ] @ (if binders = [] then [] else [ F.space ]) - @ binders - @ [ full_stop ]) + @ binders) | I.Type -> let vfmt_ = fmtExp (g_, 0, noCtxt, (v_, I.id)) in F.hVbox @@ -667,7 +668,6 @@ module MakePrint F.space; F.break; vfmt_; - full_stop; ] end | hide, (I.SkoDec (_, _, imp, v_, l_) as condec) -> @@ -686,7 +686,6 @@ module MakePrint F.space; F.break; vfmt_; - full_stop; ] | hide, (I.BlockDec (_, _, gsome_, lblock_) as condec) -> let qid = Names.conDecQid condec in @@ -733,7 +732,6 @@ module MakePrint F.break; F.space; ufmt_; - sym "%."; ] | hide, (I.AbbrevDef (_, _, imp, u_, v_, l_) as condec) -> let qid = Names.conDecQid condec in @@ -749,14 +747,11 @@ module MakePrint sym "%inline"; fmtConstPath (Symbol.def, qid); F.space; - F.break; vfmt_; F.break; - F.space; ufmt_; - full_stop; ] let rec fmtCnstr = function diff --git a/src/print/Print_.mli b/src/Print/Print_.mli similarity index 84% rename from src/print/Print_.mli rename to src/Print/Print_.mli index 330f276..fd56885 100644 --- a/src/print/Print_.mli +++ b/src/Print/Print_.mli @@ -1,4 +1,4 @@ -include module type of Print_intf +include module type of PRINT module MakePrint (Whnf : WHNF) @@ -30,6 +30,6 @@ module MakePrint module Print : PRINT include PRINT -module ClausePrint : ClausePrint_intf.CLAUSEPRINT +module ClausePrint : CLAUSEPRINT.CLAUSEPRINT module PrintTeX : PRINT -module ClausePrintTeX : ClausePrint_intf.CLAUSEPRINT +module ClausePrintTeX : CLAUSEPRINT.CLAUSEPRINT diff --git a/src/print/README b/src/Print/README similarity index 100% rename from src/print/README rename to src/Print/README diff --git a/src/print/Symbol_intf.ml b/src/Print/SYMBOL.ml similarity index 100% rename from src/print/Symbol_intf.ml rename to src/Print/SYMBOL.ml diff --git a/src/print/Symbol.ml b/src/Print/Symbol.ml similarity index 99% rename from src/print/Symbol.ml rename to src/Print/Symbol.ml index ed29f62..a9bfde7 100644 --- a/src/print/Symbol.ml +++ b/src/Print/Symbol.ml @@ -1,6 +1,6 @@ (* # 1 "src/print/Symbol.sig.ml" *) open! Basis -include Symbol_intf +include SYMBOL (* signature SYMBOL *) (* # 1 "src/print/Symbol.fun.ml" *) diff --git a/src/print/Symbol.mli b/src/Print/Symbol.mli similarity index 73% rename from src/print/Symbol.mli rename to src/Print/Symbol.mli index ebef956..7150e25 100644 --- a/src/print/Symbol.mli +++ b/src/Print/Symbol.mli @@ -1,4 +1,4 @@ -include module type of Symbol_intf +include module type of SYMBOL module MakeSymbolAscii () : SYMBOL module SymbolTeXfp () : SYMBOL module MakeSymbolTeX () : SYMBOL diff --git a/src/print/Traverse_intf.ml b/src/Print/TRAVERSE.ml similarity index 100% rename from src/print/Traverse_intf.ml rename to src/Print/TRAVERSE.ml diff --git a/src/print/Traverse.ml b/src/Print/Traverse.ml similarity index 99% rename from src/print/Traverse.ml rename to src/Print/Traverse.ml index 18bf7a9..c79ec6c 100644 --- a/src/print/Traverse.ml +++ b/src/Print/Traverse.ml @@ -3,7 +3,7 @@ open! Basis (* Generic Traversal Intended for Language-Specific Printing *) (* Author: Frank Pfenning *) -include Traverse_intf +include TRAVERSE (* val famdec : string * kind -> condec *) (* val objdef : string * obj * tp -> condec *) diff --git a/src/print/Traverse.mli b/src/Print/Traverse.mli similarity index 87% rename from src/print/Traverse.mli rename to src/Print/Traverse.mli index 437fbee..e711ea7 100644 --- a/src/print/Traverse.mli +++ b/src/Print/Traverse.mli @@ -1,4 +1,4 @@ -include module type of Traverse_intf +include module type of TRAVERSE module Traverse (Traverse__0 : sig (*! structure IntSyn' : INTSYN !*) diff --git a/src/print/WALK b/src/Print/WALK similarity index 100% rename from src/print/WALK rename to src/Print/WALK diff --git a/src/print/dune b/src/Print/dune similarity index 81% rename from src/print/dune rename to src/Print/dune index e3c6ce3..3afcee7 100644 --- a/src/print/dune +++ b/src/Print/dune @@ -11,13 +11,13 @@ PrintTwega PrintXml PrintOmdoc - ClausePrint_intf - Print_intf - PrintOmdoc_intf - PrintTwega_intf - PrintXml_intf - Symbol_intf - Traverse_intf) + CLAUSEPRINT + PRINT + PRINTOMDOC + PRINTTWEGA + PRINTXML + SYMBOL + TRAVERSE) (libraries display intsyn names formatter basis) (flags (:standard diff --git a/src/prover/Data_intf.ml b/src/Prover/DATA.ml similarity index 100% rename from src/prover/Data_intf.ml rename to src/Prover/DATA.ml diff --git a/src/prover/Data.ml b/src/Prover/Data.ml similarity index 95% rename from src/prover/Data.ml rename to src/Prover/Data.ml index 713bb29..db4c8a5 100644 --- a/src/prover/Data.ml +++ b/src/Prover/Data.ml @@ -3,7 +3,7 @@ open! Basis (* Data Global parameters *) (* Author: Carsten Schuermann *) -include Data_intf +include DATA (* signature DATA *) (* # 1 "src/prover/Data.fun.ml" *) diff --git a/src/Prover/Data.mli b/src/Prover/Data.mli new file mode 100644 index 0000000..33a34b6 --- /dev/null +++ b/src/Prover/Data.mli @@ -0,0 +1,2 @@ +include module type of DATA +module Data : DATA diff --git a/src/prover/Elim_intf.ml b/src/Prover/ELIM.ml similarity index 100% rename from src/prover/Elim_intf.ml rename to src/Prover/ELIM.ml diff --git a/src/prover/Elim.ml b/src/Prover/Elim.ml similarity index 99% rename from src/prover/Elim.ml rename to src/Prover/Elim.ml index a0aecbe..bcd233b 100644 --- a/src/prover/Elim.ml +++ b/src/Prover/Elim.ml @@ -3,7 +3,7 @@ open! Basis (* ELIM: Version 1.4 *) (* Author: Carsten Schuermann *) -include Elim_intf +include ELIM (* signature ELIM *) (* # 1 "src/prover/Elim.fun.ml" *) diff --git a/src/prover/Elim.mli b/src/Prover/Elim.mli similarity index 94% rename from src/prover/Elim.mli rename to src/Prover/Elim.mli index cce8ac1..08e0dd3 100644 --- a/src/prover/Elim.mli +++ b/src/Prover/Elim.mli @@ -1,4 +1,4 @@ -include module type of Elim_intf +include module type of ELIM module Elim (Elim__0 : sig module Data : Data.DATA diff --git a/src/prover/Fill_intf.ml b/src/Prover/FILL.ml similarity index 100% rename from src/prover/Fill_intf.ml rename to src/Prover/FILL.ml diff --git a/src/prover/Fixedpoint_intf.ml b/src/Prover/FIXEDPOINT.ml similarity index 100% rename from src/prover/Fixedpoint_intf.ml rename to src/Prover/FIXEDPOINT.ml diff --git a/src/prover/Fill.ml b/src/Prover/Fill.ml similarity index 99% rename from src/prover/Fill.ml rename to src/Prover/Fill.ml index 7217129..7af825a 100644 --- a/src/prover/Fill.ml +++ b/src/Prover/Fill.ml @@ -3,7 +3,7 @@ open! Basis (* Filling: Version 1.4 *) (* Author: Carsten Schuermann *) -include Fill_intf +include FILL (* signature FILL *) (* # 1 "src/prover/Fill.fun.ml" *) diff --git a/src/prover/Fill.mli b/src/Prover/Fill.mli similarity index 95% rename from src/prover/Fill.mli rename to src/Prover/Fill.mli index 710238a..4e52b56 100644 --- a/src/prover/Fill.mli +++ b/src/Prover/Fill.mli @@ -1,4 +1,4 @@ -include module type of Fill_intf +include module type of FILL module Fill (Fill__0 : sig module Data : Data.DATA diff --git a/src/prover/Fixedpoint.ml b/src/Prover/Fixedpoint.ml similarity index 98% rename from src/prover/Fixedpoint.ml rename to src/Prover/Fixedpoint.ml index 9e54aac..ef149aa 100644 --- a/src/prover/Fixedpoint.ml +++ b/src/Prover/Fixedpoint.ml @@ -3,7 +3,7 @@ open! Basis (* Splitting: Version 1.4 *) (* Author: Carsten Schuermann *) -include Fixedpoint_intf +include FIXEDPOINT (* signature Fixed Point *) (* # 1 "src/prover/Fixedpoint.fun.ml" *) diff --git a/src/prover/Fixedpoint.mli b/src/Prover/Fixedpoint.mli similarity index 76% rename from src/prover/Fixedpoint.mli rename to src/Prover/Fixedpoint.mli index 0e430c9..828e1b3 100644 --- a/src/prover/Fixedpoint.mli +++ b/src/Prover/Fixedpoint.mli @@ -1,4 +1,4 @@ -include module type of Fixedpoint_intf +include module type of FIXEDPOINT module FixedPoint (FixedPoint__0 : sig module State' : State.STATE diff --git a/src/prover/Interactive_intf.ml b/src/Prover/INTERACTIVE.ml similarity index 100% rename from src/prover/Interactive_intf.ml rename to src/Prover/INTERACTIVE.ml diff --git a/src/prover/Introduce_intf.ml b/src/Prover/INTRODUCE.ml similarity index 100% rename from src/prover/Introduce_intf.ml rename to src/Prover/INTRODUCE.ml diff --git a/src/prover/Interactive.ml b/src/Prover/Interactive.ml similarity index 99% rename from src/prover/Interactive.ml rename to src/Prover/Interactive.ml index 0094c7c..13b156d 100644 --- a/src/prover/Interactive.ml +++ b/src/Prover/Interactive.ml @@ -3,7 +3,7 @@ open! Basis (* Meta Prover Interface *) (* Author: Carsten Schuermann *) -include Interactive_intf +include INTERACTIVE (* val undo : unit -> unit *) (* signature Interactive *) diff --git a/src/prover/Interactive.mli b/src/Prover/Interactive.mli similarity index 78% rename from src/prover/Interactive.mli rename to src/Prover/Interactive.mli index 7c6c99c..876fae4 100644 --- a/src/prover/Interactive.mli +++ b/src/Prover/Interactive.mli @@ -1,4 +1,4 @@ -include module type of Interactive_intf +include module type of INTERACTIVE module Interactive (Interactive__0 : sig (* Meta Prover Interface *) @@ -18,7 +18,7 @@ module Interactive (Interactive__0 : sig module Names : NAMES (*! sharing Names.IntSyn = IntSyn' !*) - module Weaken : Pweaken_intf.WEAKEN + module Weaken : PWEAKEN.WEAKEN (*! sharing Weaken.IntSyn = IntSyn' !*) (* structure ModeSyn : MODESYN *) @@ -27,15 +27,15 @@ module Interactive (Interactive__0 : sig (*! sharing WorldSyn.IntSyn = IntSyn' !*) (*! sharing WorldSyn.Tomega = Tomega' !*) - module Introduce : Introduce_intf.INTRODUCE with module State = State' + module Introduce : INTRODUCE.INTRODUCE with module State = State' (*! sharing Introduce.IntSyn = IntSyn' !*) (*! sharing Introduce.Tomega = Tomega' !*) - module Elim : Elim_intf.ELIM with module State = State' + module Elim : ELIM.ELIM with module State = State' (*! sharing Elim.IntSyn = IntSyn' !*) (*! sharing Elim.Tomega = Tomega' !*) - module Split : Split_intf.SPLIT with module State = State' + module Split : SPLIT.SPLIT with module State = State' (*! sharing Split.IntSyn = IntSyn' !*) (*! sharing Split.Tomega = Tomega' !*) @@ -43,5 +43,5 @@ module Interactive (Interactive__0 : sig (*! sharing FixedPoint.IntSyn = IntSyn' !*) (*! sharing FixedPoint.Tomega = Tomega' !*) - module Fill : Fill_intf.FILL with module State = State' + module Fill : FILL.FILL with module State = State' end) : INTERACTIVE diff --git a/src/prover/Introduce.ml b/src/Prover/Introduce.ml similarity index 99% rename from src/prover/Introduce.ml rename to src/Prover/Introduce.ml index a7ee949..2965ea2 100644 --- a/src/prover/Introduce.ml +++ b/src/Prover/Introduce.ml @@ -3,7 +3,7 @@ open! Basis (* Introduce: Version 1.4 *) (* Author: Carsten Schuermann *) -include Introduce_intf +include INTRODUCE (* signature INTRODUCE *) (* # 1 "src/prover/Introduce.fun.ml" *) diff --git a/src/prover/Introduce.mli b/src/Prover/Introduce.mli similarity index 90% rename from src/prover/Introduce.mli rename to src/Prover/Introduce.mli index efbba8b..335bd17 100644 --- a/src/prover/Introduce.mli +++ b/src/Prover/Introduce.mli @@ -1,4 +1,4 @@ -include module type of Introduce_intf +include module type of INTRODUCE module Introduce (Introduce__0 : sig (* Introduce *) diff --git a/src/prover/Psearch_intf.ml b/src/Prover/PSEARCH.ml similarity index 100% rename from src/prover/Psearch_intf.ml rename to src/Prover/PSEARCH.ml diff --git a/src/prover/Pweaken_intf.ml b/src/Prover/PWEAKEN.ml similarity index 100% rename from src/prover/Pweaken_intf.ml rename to src/Prover/PWEAKEN.ml diff --git a/src/prover/Prover.ml b/src/Prover/Prover.ml similarity index 100% rename from src/prover/Prover.ml rename to src/Prover/Prover.ml diff --git a/src/prover/Prover.mli b/src/Prover/Prover.mli similarity index 100% rename from src/prover/Prover.mli rename to src/Prover/Prover.mli diff --git a/src/prover/Prover_.ml b/src/Prover/Prover_.ml similarity index 100% rename from src/prover/Prover_.ml rename to src/Prover/Prover_.ml diff --git a/src/prover/Prover_.mli b/src/Prover/Prover_.mli similarity index 100% rename from src/prover/Prover_.mli rename to src/Prover/Prover_.mli diff --git a/src/prover/Psearch.ml b/src/Prover/Psearch.ml similarity index 99% rename from src/prover/Psearch.ml rename to src/Prover/Psearch.ml index 39a441c..8b1ce8b 100644 --- a/src/prover/Psearch.ml +++ b/src/Prover/Psearch.ml @@ -3,7 +3,7 @@ open! Basis (* Basic search engine: Version 1.3*) (* Author: Carsten Schuermann *) -include Psearch_intf +include PSEARCH (* signature SEARCH *) (* # 1 "src/prover/Search.fun.ml" *) @@ -26,7 +26,7 @@ module Search (Search__0 : sig (*! sharing Abstract.IntSyn = IntSyn' !*) (*! sharing Abstract.Tomega = Tomega' !*) module Data : Data.DATA - module CompSyn' : CompSyn_intf.COMPSYN + module CompSyn' : COMPSYN.COMPSYN (*! sharing CompSyn'.IntSyn = IntSyn' !*) module Whnf : WHNF diff --git a/src/prover/Psearch.mli b/src/Prover/Psearch.mli similarity index 84% rename from src/prover/Psearch.mli rename to src/Prover/Psearch.mli index 9a4b9aa..9121efd 100644 --- a/src/prover/Psearch.mli +++ b/src/Prover/Psearch.mli @@ -1,4 +1,4 @@ -include module type of Psearch_intf +include module type of PSEARCH module Search (Search__0 : sig module Global : GLOBAL @@ -15,7 +15,7 @@ module Search (Search__0 : sig (*! sharing Abstract.IntSyn = IntSyn' !*) (*! sharing Abstract.Tomega = Tomega' !*) module Data : Data.DATA - module CompSyn' : CompSyn_intf.COMPSYN + module CompSyn' : COMPSYN.COMPSYN (*! sharing CompSyn'.IntSyn = IntSyn' !*) module Whnf : WHNF @@ -24,17 +24,17 @@ module Search (Search__0 : sig module Unify : UNIFY (*! sharing Unify.IntSyn = IntSyn' !*) - module Assign : Assign_intf.ASSIGN + module Assign : ASSIGN.ASSIGN (*! sharing Assign.IntSyn = IntSyn' !*) module Index : INDEX (*! sharing Index.IntSyn = IntSyn' !*) - module Compile : Compile_intf.COMPILE + module Compile : COMPILE.COMPILE (*! sharing Compile.IntSyn = IntSyn' !*) (*! sharing Compile.CompSyn = CompSyn' !*) - module CPrint : Cprint_intf.CPRINT + module CPrint : CPRINT.CPRINT (*! sharing CPrint.IntSyn = IntSyn' !*) (*! sharing CPrint.CompSyn = CompSyn' !*) diff --git a/src/prover/Pweaken.ml b/src/Prover/Pweaken.ml similarity index 98% rename from src/prover/Pweaken.ml rename to src/Prover/Pweaken.ml index 3de9ddd..511896f 100644 --- a/src/prover/Pweaken.ml +++ b/src/Prover/Pweaken.ml @@ -3,7 +3,7 @@ open! Basis (* Weakening substitutions *) (* Author: Carsten Schuermann *) -include Pweaken_intf +include PWEAKEN (* signature PRUNE *) (* # 1 "src/prover/Weaken.fun.ml" *) diff --git a/src/prover/Pweaken.mli b/src/Prover/Pweaken.mli similarity index 65% rename from src/prover/Pweaken.mli rename to src/Prover/Pweaken.mli index e03114b..1ca70f5 100644 --- a/src/prover/Pweaken.mli +++ b/src/Prover/Pweaken.mli @@ -1,4 +1,4 @@ -include module type of Pweaken_intf +include module type of PWEAKEN module Weaken (Weaken__0 : sig module Whnf : WHNF diff --git a/src/prover/Split_intf.ml b/src/Prover/SPLIT.ml similarity index 100% rename from src/prover/Split_intf.ml rename to src/Prover/SPLIT.ml diff --git a/src/prover/State_intf.ml b/src/Prover/STATE.ml similarity index 100% rename from src/prover/State_intf.ml rename to src/Prover/STATE.ml diff --git a/src/prover/Split.ml b/src/Prover/Split.ml similarity index 99% rename from src/prover/Split.ml rename to src/Prover/Split.ml index 0c414f0..cc171e8 100644 --- a/src/prover/Split.ml +++ b/src/Prover/Split.ml @@ -3,7 +3,7 @@ open! Basis (* Splitting: Version 1.4 *) (* Author: Carsten Schuermann *) -include Split_intf +include SPLIT (* signature Split *) (* # 1 "src/prover/Split.fun.ml" *) diff --git a/src/prover/Split.mli b/src/Prover/Split.mli similarity index 96% rename from src/prover/Split.mli rename to src/Prover/Split.mli index b2873f3..69df1fe 100644 --- a/src/prover/Split.mli +++ b/src/Prover/Split.mli @@ -1,4 +1,4 @@ -include module type of Split_intf +include module type of SPLIT module Split (Split__0 : sig (* State definition for Proof Search *) diff --git a/src/prover/State.ml b/src/Prover/State.ml similarity index 99% rename from src/prover/State.ml rename to src/Prover/State.ml index 0eb4145..e79f468 100644 --- a/src/prover/State.ml +++ b/src/Prover/State.ml @@ -3,7 +3,7 @@ open! Basis (* State definition for Proof Search *) (* Author: Carsten Schuermann *) -include State_intf +include STATE (* # 1 "src/prover/State.fun.ml" *) open! Basis diff --git a/src/prover/State.mli b/src/Prover/State.mli similarity index 68% rename from src/prover/State.mli rename to src/Prover/State.mli index ab88af9..e614eda 100644 --- a/src/prover/State.mli +++ b/src/Prover/State.mli @@ -1,4 +1,4 @@ -include module type of State_intf +include module type of STATE module State (State__0 : sig module Formatter : FORMATTER diff --git a/src/prover/dune b/src/Prover/dune similarity index 89% rename from src/prover/dune rename to src/Prover/dune index 1b6bf2d..f0f41bc 100644 --- a/src/prover/dune +++ b/src/Prover/dune @@ -15,16 +15,16 @@ Interactive Pweaken Prover_ - Data_intf - Elim_intf - Fill_intf - Fixedpoint_intf - Interactive_intf - Introduce_intf - Psearch_intf - Pweaken_intf - Split_intf - State_intf) + DATA + ELIM + FILL + FIXEDPOINT + INTERACTIVE + INTRODUCE + PSEARCH + PWEAKEN + SPLIT + STATE) (libraries display tomega_lib diff --git a/src/Recon/ReconTerm.ml b/src/Recon/ReconTerm.ml index a995841..3ca71ae 100644 --- a/src/Recon/ReconTerm.ml +++ b/src/Recon/ReconTerm.ml @@ -14,7 +14,7 @@ module Make_ReconTerm module Print : PRINT module StringTree : TABLE with type key = string module Msg : MSG - module CsManager : Solvers.CsManager_intf.CS_MANAGER + module CsManager : Solvers.CSMANAGER.CS_MANAGER end) = struct module M = M @@ -877,17 +877,12 @@ struct F.hVbox [ F.string "Inferred:"; F.space; formatExp (g_, u_) ] in let fstr = F.makestring_fmt amb in - Display.debug Display.Form.( - nl () ++ - string "Ambiguous reconstruction of term: " ++ - string fstr ++ - - nl () - ); - error - ( r, - (("Ambiguous reconstruction\n" ^ fstr) ^ "\n") ^ msg - )) + Display.debug + Display.Form.( + nl () + ++ string "Ambiguous reconstruction of term: " + ++ string fstr ++ nl ()); + error (r, (("Ambiguous reconstruction\n" ^ fstr) ^ "\n") ^ msg)) let rec unifyIdem x = let _ = Unify.reset () in @@ -1087,23 +1082,23 @@ struct let s = IntSyn.Shift (IntSyn.ctxLength g_) in (tm, Elim (elimSub (evarElim x_, s)), eClo_ (v_, s)) | g_, (Fvar_ (name, r) as tm) -> - Display.debug Display.Form.( - nl () ++ - string "Inferring exact type of FVar" ++ - string name ++ - nl () - ) ; + Display.debug + Display.Form.( + nl () + ++ string "Inferring exact type of FVar" + ++ string name ++ nl ()); let v_ = try getFVarType (name, false) with Apx.Ambiguous -> let v_ = getFVarType (name, true) in begin - Display.debug Display.Form.( - string "Type of FVar" ++ - string name ++ - string " is ambiguous, but continuing with one of the possibilities" ++ - nl () - ) ; + Display.debug + Display.Form.( + string "Type of FVar" ++ string name + ++ string + " is ambiguous, but continuing with one of the \ + possibilities" + ++ nl ()); delayAmbiguous (g_, v_, r, "Free variable has ambiguous type"); v_ end @@ -1136,14 +1131,13 @@ struct | g_, App_ (tm1, tm2) -> let tm1', b1_, v1_ = inferExact (g_, tm1) in let e1_ = toElim b1_ in - Display.(debug Form.( - nl () ++ - string "Inferring exact application of" ++ - shown show_term tm1 ++ - string "to" ++ - shown show_term tm2 ++ - nl () - )) ; + Display.( + debug + Form.( + nl () + ++ string "Inferring exact application of" + ++ shown show_term tm1 ++ string "to" ++ shown show_term tm2 + ++ nl ())); let t, s = Whnf.whnfExpandDef (v1_, IntSyn.id) in begin match t with | IntSyn.Pi ((IntSyn.Dec (_, va_), _), vr_) -> begin @@ -1196,10 +1190,12 @@ struct with Ambiguous -> let v'_ = Apx.apxToClass (g_, v_, l_, true) in begin - Display.debug Display.Form.( - string "Classifier of omitted term is ambiguous, but continuing with one of the possibilities" ++ - nl () - ) ; + Display.debug + Display.Form.( + string + "Classifier of omitted term is ambiguous, but continuing \ + with one of the possibilities" + ++ nl ()); delayAmbiguous ( g_, v'_, @@ -1218,10 +1214,12 @@ struct with Ambiguous -> let u'_ = Apx.apxToExact (g_, u_, (v'_, IntSyn.id), true) in begin - Display.debug Display.Form.( - string "Exact term of omitted term is ambiguous, but continuing with one of the possibilities" ++ - nl () - ) ; + Display.debug + Display.Form.( + string + "Exact term of omitted term is ambiguous, but continuing \ + with one of the possibilities" + ++ nl ()); delayAmbiguous ( g_, u'_, @@ -1316,13 +1314,10 @@ struct ((tm', b'_, v'_), unifiableIdem (g_, vhs_, (v'_, IntSyn.id))) and checkExact (g_, tm, vs_, location_msg) = - Display.(debug Form.( - nl () ++ - string "Checking exact term" ++ - shown show_term tm ++ - - nl () - )) ; + Display.( + debug + Form.( + nl () ++ string "Checking exact term" ++ shown show_term tm ++ nl ())); begin if not !trace then let (tm', b'_, v'_), ok = checkExact1 (g_, tm, vs_) in begin if ok then (tm', b'_) diff --git a/src/Recon/S.ml b/src/Recon/S.ml index 791a269..272ba83 100644 --- a/src/Recon/S.ml +++ b/src/Recon/S.ml @@ -1,5 +1,5 @@ module type S = sig - module Paths : Paths.Paths_intf.PATHS + module Paths : Paths.PATHS.PATHS module Cst : Cst.CST with module Paths = Paths module Syntax : Syntax.SYNTAX module Ast = Intsyn.IntSyn diff --git a/src/Resugar/ConDec.ml b/src/Resugar/ConDec.ml index b754d5d..3ac65c2 100644 --- a/src/Resugar/ConDec.ml +++ b/src/Resugar/ConDec.ml @@ -1 +1 @@ -open RESUGAR +open RESUGAR diff --git a/src/Resugar/RESUGAR.ml b/src/Resugar/RESUGAR.ml index 31db915..0427df6 100644 --- a/src/Resugar/RESUGAR.ml +++ b/src/Resugar/RESUGAR.ml @@ -1,16 +1,16 @@ -module type S = sig +module type S = sig module Syntax : Syntax.SYNTAX module Cst : Cst.CST - module Names : Names.Names_intf.NAMES + module Names : Names.NAMES.NAMES end module type RESUGAR = sig module Syntax : Syntax.SYNTAX module Cst : Cst.CST - type t - type u + + type t + type u + val register : t -> u -> unit val resugar : t -> u end - - \ No newline at end of file diff --git a/src/Resugar/Resugar.ml b/src/Resugar/Resugar.ml index 0458772..aa742cd 100644 --- a/src/Resugar/Resugar.ml +++ b/src/Resugar/Resugar.ml @@ -1,4 +1,8 @@ -module Make_Resugar (Syntax : Syntax.SYNTAX) (Cst : Cst.CST) (* : RESUGAR.RESUGAR with module Syntax = Syntax with module Cst = Cst *) = struct +module Make_Resugar + (Syntax : Syntax.SYNTAX) + (Cst : Cst.CST) +(* : RESUGAR.RESUGAR with module Syntax = Syntax with module Cst = Cst *) = +struct module Syntax = Syntax module Cst = Cst diff --git a/src/Resugar/Term.ml b/src/Resugar/Term.ml index f423594..962beb9 100644 --- a/src/Resugar/Term.ml +++ b/src/Resugar/Term.ml @@ -1,5 +1,6 @@ module Make_Expr (M : RESUGAR.S) = struct open M + let register = assert false - let resugar = assert false -end \ No newline at end of file + let resugar = assert false +end diff --git a/src/Resugar/dune b/src/Resugar/dune index 8abc4f1..142e0ce 100644 --- a/src/Resugar/dune +++ b/src/Resugar/dune @@ -2,4 +2,3 @@ (name resugar) (public_name stelf.resugar) (libraries syntax names paths cst)) - \ No newline at end of file diff --git a/src/server/MltonThread.ml b/src/Server/MltonThread.ml similarity index 100% rename from src/server/MltonThread.ml rename to src/Server/MltonThread.ml diff --git a/src/server/MltonThread.mli b/src/Server/MltonThread.mli similarity index 100% rename from src/server/MltonThread.mli rename to src/Server/MltonThread.mli diff --git a/src/server/Server_intf.ml b/src/Server/SERVER.ml similarity index 100% rename from src/server/Server_intf.ml rename to src/Server/SERVER.ml diff --git a/src/server/Sigint_intf.ml b/src/Server/SIGINT.ml similarity index 100% rename from src/server/Sigint_intf.ml rename to src/Server/SIGINT.ml diff --git a/src/server/Server.ml b/src/Server/Server.ml similarity index 100% rename from src/server/Server.ml rename to src/Server/Server.ml diff --git a/src/server/Server.mli b/src/Server/Server.mli similarity index 100% rename from src/server/Server.mli rename to src/Server/Server.mli diff --git a/src/server/Server_.ml b/src/Server/Server_.ml similarity index 99% rename from src/server/Server_.ml rename to src/Server/Server_.ml index af9ad72..c4ff809 100644 --- a/src/server/Server_.ml +++ b/src/Server/Server_.ml @@ -5,7 +5,7 @@ (* # 1 "src/server/Server_.sml.ml" *) open! Basis -include Server_intf +include SERVER (** Interactive command server for Stelf/STELF. *) (* signature SERVER *) diff --git a/src/Server/Server_.mli b/src/Server/Server_.mli new file mode 100644 index 0000000..51e96c9 --- /dev/null +++ b/src/Server/Server_.mli @@ -0,0 +1,2 @@ +include module type of SERVER +module Server : SERVER diff --git a/src/server/Sigint.ml b/src/Server/Sigint.ml similarity index 88% rename from src/server/Sigint.ml rename to src/Server/Sigint.ml index d6e8102..ae1642d 100644 --- a/src/server/Sigint.ml +++ b/src/Server/Sigint.ml @@ -1,6 +1,6 @@ (* # 1 "src/server/Sigint.sig.ml" *) open! Basis -include Sigint_intf +include SIGINT (* signature SIGINT *) (* # 1 "src/server/Sigint.fun.ml" *) diff --git a/src/Server/Sigint.mli b/src/Server/Sigint.mli new file mode 100644 index 0000000..8378f02 --- /dev/null +++ b/src/Server/Sigint.mli @@ -0,0 +1 @@ +include module type of SIGINT diff --git a/src/server/SigintMlton.ml b/src/Server/SigintMlton.ml similarity index 100% rename from src/server/SigintMlton.ml rename to src/Server/SigintMlton.ml diff --git a/src/server/SigintMlton.mli b/src/Server/SigintMlton.mli similarity index 100% rename from src/server/SigintMlton.mli rename to src/Server/SigintMlton.mli diff --git a/src/server/SigintSmlnj.ml b/src/Server/SigintSmlnj.ml similarity index 100% rename from src/server/SigintSmlnj.ml rename to src/Server/SigintSmlnj.ml diff --git a/src/server/SigintSmlnj.mli b/src/Server/SigintSmlnj.mli similarity index 100% rename from src/server/SigintSmlnj.mli rename to src/Server/SigintSmlnj.mli diff --git a/src/server/dune b/src/Server/dune similarity index 94% rename from src/server/dune rename to src/Server/dune index 8052f38..f60d2c7 100644 --- a/src/server/dune +++ b/src/Server/dune @@ -10,8 +10,8 @@ SigintSmlnj Server_ Server - Server_intf - Sigint_intf) + SERVER + SIGINT) (libraries display timing frontend basis smlofnj) (flags (:standard diff --git a/src/server/mlton_thread.sml.ml_ b/src/Server/mlton_thread.sml.ml_ similarity index 100% rename from src/server/mlton_thread.sml.ml_ rename to src/Server/mlton_thread.sml.ml_ diff --git a/src/server/sigint_mlton.sml.ml_ b/src/Server/sigint_mlton.sml.ml_ similarity index 100% rename from src/server/sigint_mlton.sml.ml_ rename to src/Server/sigint_mlton.sml.ml_ diff --git a/src/server/sigint_polyml.sml.ml_ b/src/Server/sigint_polyml.sml.ml_ similarity index 100% rename from src/server/sigint_polyml.sml.ml_ rename to src/Server/sigint_polyml.sml.ml_ diff --git a/src/smlofnj/Smlofnj_intf.ml b/src/Smlofnj/SMLOFNJ.ml similarity index 100% rename from src/smlofnj/Smlofnj_intf.ml rename to src/Smlofnj/SMLOFNJ.ml diff --git a/src/smlofnj/Smlofnj.ml b/src/Smlofnj/Smlofnj.ml similarity index 96% rename from src/smlofnj/Smlofnj.ml rename to src/Smlofnj/Smlofnj.ml index dc1e02f..f3894f5 100644 --- a/src/smlofnj/Smlofnj.ml +++ b/src/Smlofnj/Smlofnj.ml @@ -1,4 +1,4 @@ -include Smlofnj_intf +include SMLOFNJ module SMLofNJ : SML_OF_NJ = struct let () = Printexc.record_backtrace true diff --git a/src/Smlofnj/Smlofnj.mli b/src/Smlofnj/Smlofnj.mli new file mode 100644 index 0000000..e95a1c8 --- /dev/null +++ b/src/Smlofnj/Smlofnj.mli @@ -0,0 +1,2 @@ +include module type of SMLOFNJ +module SMLofNJ : SML_OF_NJ diff --git a/src/smlofnj/dune b/src/Smlofnj/dune similarity index 100% rename from src/smlofnj/dune rename to src/Smlofnj/dune diff --git a/src/solvers/Cs_intf.ml b/src/Solvers/CS.ml similarity index 100% rename from src/solvers/Cs_intf.ml rename to src/Solvers/CS.ml diff --git a/src/solvers/CsEqField_intf.ml b/src/Solvers/CSEQFIELD.ml similarity index 100% rename from src/solvers/CsEqField_intf.ml rename to src/Solvers/CSEQFIELD.ml diff --git a/src/solvers/CsEqIntegers_intf.ml b/src/Solvers/CSEQINTEGERS.ml similarity index 100% rename from src/solvers/CsEqIntegers_intf.ml rename to src/Solvers/CSEQINTEGERS.ml diff --git a/src/solvers/CsManager_intf.ml b/src/Solvers/CSMANAGER.ml similarity index 100% rename from src/solvers/CsManager_intf.ml rename to src/Solvers/CSMANAGER.ml diff --git a/src/solvers/Cs.ml b/src/Solvers/Cs.ml similarity index 90% rename from src/solvers/Cs.ml rename to src/Solvers/Cs.ml index 5d8ec44..47a0305 100644 --- a/src/solvers/Cs.ml +++ b/src/Solvers/Cs.ml @@ -2,7 +2,7 @@ open! Basis (* Constraint Solver *) -include Cs_intf +include CS (* signature CS *) (* # 1 "src/solvers/Cs.fun.ml" *) diff --git a/src/Solvers/Cs.mli b/src/Solvers/Cs.mli new file mode 100644 index 0000000..cf93cf0 --- /dev/null +++ b/src/Solvers/Cs.mli @@ -0,0 +1 @@ +include module type of CS diff --git a/src/solvers/CsEqBools.ml b/src/Solvers/CsEqBools.ml similarity index 100% rename from src/solvers/CsEqBools.ml rename to src/Solvers/CsEqBools.ml diff --git a/src/solvers/CsEqBools.mli b/src/Solvers/CsEqBools.mli similarity index 100% rename from src/solvers/CsEqBools.mli rename to src/Solvers/CsEqBools.mli diff --git a/src/solvers/CsEqField.ml b/src/Solvers/CsEqField.ml similarity index 99% rename from src/solvers/CsEqField.ml rename to src/Solvers/CsEqField.ml index 0c79253..e48cf27 100644 --- a/src/solvers/CsEqField.ml +++ b/src/Solvers/CsEqField.ml @@ -3,7 +3,7 @@ open! Basis (* Gaussian-Elimination Equation Solver *) (* Author: Roberto Virga *) -include CsEqField_intf +include CSEQFIELD (* signature CS_EQ_FIELD *) (* # 1 "src/solvers/CsEqField.fun.ml" *) diff --git a/src/solvers/CsEqField.mli b/src/Solvers/CsEqField.mli similarity index 89% rename from src/solvers/CsEqField.mli rename to src/Solvers/CsEqField.mli index 3f88f5a..c924689 100644 --- a/src/solvers/CsEqField.mli +++ b/src/Solvers/CsEqField.mli @@ -1,4 +1,4 @@ -include module type of CsEqField_intf +include module type of CSEQFIELD module CsEqField (CSEqField__0 : sig (* Gaussian-Elimination Equation Solver *) diff --git a/src/solvers/CsEqIntegers.ml b/src/Solvers/CsEqIntegers.ml similarity index 99% rename from src/solvers/CsEqIntegers.ml rename to src/Solvers/CsEqIntegers.ml index f9eec17..952f96f 100644 --- a/src/solvers/CsEqIntegers.ml +++ b/src/Solvers/CsEqIntegers.ml @@ -3,7 +3,7 @@ open! Basis (* Gaussian-Elimination Equation Solver *) (* Author: Roberto Virga *) -include CsEqIntegers_intf +include CSEQINTEGERS (* signature CS_EQ_FIELD *) (* # 1 "src/solvers/CsEqIntegers.fun.ml" *) diff --git a/src/solvers/CsEqIntegers.mli b/src/Solvers/CsEqIntegers.mli similarity index 89% rename from src/solvers/CsEqIntegers.mli rename to src/Solvers/CsEqIntegers.mli index eb1fdfa..3fdfd53 100644 --- a/src/solvers/CsEqIntegers.mli +++ b/src/Solvers/CsEqIntegers.mli @@ -1,4 +1,4 @@ -include module type of CsEqIntegers_intf +include module type of CSEQINTEGERS module CsEqIntegers (CSEqIntegers__0 : sig (* Diophantine Equation Solver *) diff --git a/src/solvers/CsEqStrings.ml b/src/Solvers/CsEqStrings.ml similarity index 100% rename from src/solvers/CsEqStrings.ml rename to src/Solvers/CsEqStrings.ml diff --git a/src/solvers/CsEqStrings.mli b/src/Solvers/CsEqStrings.mli similarity index 100% rename from src/solvers/CsEqStrings.mli rename to src/Solvers/CsEqStrings.mli diff --git a/src/solvers/CsIneqField.ml b/src/Solvers/CsIneqField.ml similarity index 100% rename from src/solvers/CsIneqField.ml rename to src/Solvers/CsIneqField.ml diff --git a/src/solvers/CsIneqField.mli b/src/Solvers/CsIneqField.mli similarity index 100% rename from src/solvers/CsIneqField.mli rename to src/Solvers/CsIneqField.mli diff --git a/src/solvers/CsIneqIntegers.ml b/src/Solvers/CsIneqIntegers.ml similarity index 100% rename from src/solvers/CsIneqIntegers.ml rename to src/Solvers/CsIneqIntegers.ml diff --git a/src/solvers/CsIneqIntegers.mli b/src/Solvers/CsIneqIntegers.mli similarity index 100% rename from src/solvers/CsIneqIntegers.mli rename to src/Solvers/CsIneqIntegers.mli diff --git a/src/solvers/CsIntegersWord.ml b/src/Solvers/CsIntegersWord.ml similarity index 100% rename from src/solvers/CsIntegersWord.ml rename to src/Solvers/CsIntegersWord.ml diff --git a/src/solvers/CsIntegersWord.mli b/src/Solvers/CsIntegersWord.mli similarity index 100% rename from src/solvers/CsIntegersWord.mli rename to src/Solvers/CsIntegersWord.mli diff --git a/src/solvers/CsManager.ml b/src/Solvers/CsManager.ml similarity index 99% rename from src/solvers/CsManager.ml rename to src/Solvers/CsManager.ml index 6f5352e..dd7df4d 100644 --- a/src/solvers/CsManager.ml +++ b/src/Solvers/CsManager.ml @@ -3,7 +3,7 @@ open! Basis (* Constraint Solver Manager *) (* Author: Roberto Virga *) -include CsManager_intf +include CSMANAGER (* signature CS_MANAGER *) (* # 1 "src/solvers/CsManager.fun.ml" *) diff --git a/src/solvers/CsManager.mli b/src/Solvers/CsManager.mli similarity index 85% rename from src/solvers/CsManager.mli rename to src/Solvers/CsManager.mli index 41ba96f..1c36e3c 100644 --- a/src/solvers/CsManager.mli +++ b/src/Solvers/CsManager.mli @@ -1,4 +1,4 @@ -include module type of CsManager_intf +include module type of CSMANAGER include CS_MANAGER module MakeCsManager (Global : GLOBAL) (Unify : UNIFY) (Fixity : FIXITY) : diff --git a/src/solvers/Solvers_intf.ml b/src/Solvers/SOLVERS.ml similarity index 100% rename from src/solvers/Solvers_intf.ml rename to src/Solvers/SOLVERS.ml diff --git a/src/solvers/Solvers.ml b/src/Solvers/Solvers.ml similarity index 100% rename from src/solvers/Solvers.ml rename to src/Solvers/Solvers.ml diff --git a/src/solvers/Solvers.mli b/src/Solvers/Solvers.mli similarity index 100% rename from src/solvers/Solvers.mli rename to src/Solvers/Solvers.mli diff --git a/src/solvers/Solvers_.ml b/src/Solvers/Solvers_.ml similarity index 99% rename from src/solvers/Solvers_.ml rename to src/Solvers/Solvers_.ml index f5652e0..24873c5 100644 --- a/src/solvers/Solvers_.ml +++ b/src/Solvers/Solvers_.ml @@ -89,7 +89,7 @@ module CSIntWord32 = CsIntegersWord.Cs_int_word (struct let wordSize = 32 end) -include Solvers_intf +include SOLVERS (* execute for effect *) (* wrapped in structure so it can be tracked by CM *) diff --git a/src/solvers/Solvers_.mli b/src/Solvers/Solvers_.mli similarity index 92% rename from src/solvers/Solvers_.mli rename to src/Solvers/Solvers_.mli index 97f9ef3..b8a41d2 100644 --- a/src/solvers/Solvers_.mli +++ b/src/Solvers/Solvers_.mli @@ -7,5 +7,5 @@ module CsIneqIntegers : module type of CsIneqIntegers module CsEqStrings : module type of CsEqStrings module CsEqBools : module type of CsEqBools module CsIntegersWord : module type of CsIntegersWord -include module type of Solvers_intf +include module type of SOLVERS module CSInstaller : CS_INSTALLER diff --git a/src/solvers/dune b/src/Solvers/dune similarity index 91% rename from src/solvers/dune rename to src/Solvers/dune index dd233e5..b9ca3f4 100644 --- a/src/solvers/dune +++ b/src/Solvers/dune @@ -14,11 +14,11 @@ CsEqBools CsIntegersWord Solvers_ - Cs_intf - CsEqField_intf - CsEqIntegers_intf - CsManager_intf - Solvers_intf) + CS + CSEQFIELD + CSEQINTEGERS + CSMANAGER + SOLVERS) (libraries display trail diff --git a/src/stream/README b/src/Stream/README similarity index 100% rename from src/stream/README rename to src/Stream/README diff --git a/src/stream/Stream_intf.ml b/src/Stream/STREAM.ml similarity index 100% rename from src/stream/Stream_intf.ml rename to src/Stream/STREAM.ml diff --git a/src/stream/Stream.ml b/src/Stream/Stream.ml similarity index 100% rename from src/stream/Stream.ml rename to src/Stream/Stream.ml diff --git a/src/stream/Stream.mli b/src/Stream/Stream.mli similarity index 100% rename from src/stream/Stream.mli rename to src/Stream/Stream.mli diff --git a/src/stream/Stream_.ml b/src/Stream/Stream_.ml similarity index 99% rename from src/stream/Stream_.ml rename to src/Stream/Stream_.ml index 5a7558c..1cbc57f 100644 --- a/src/stream/Stream_.ml +++ b/src/Stream/Stream_.ml @@ -9,7 +9,7 @@ open Basis (* Stream Library *) (* Author: Frank Pfenning *) (* BASIC_STREAM defines the visible ""core"" of streams *) -include Stream_intf +include STREAM module BasicStream : BASIC_STREAM = struct type 'a stream = Stream of (unit -> 'a front) diff --git a/src/stream/Stream_.mli b/src/Stream/Stream_.mli similarity index 83% rename from src/stream/Stream_.mli rename to src/Stream/Stream_.mli index 9c22245..baf1cfc 100644 --- a/src/stream/Stream_.mli +++ b/src/Stream/Stream_.mli @@ -1,4 +1,4 @@ -include module type of Stream_intf +include module type of STREAM module BasicStream : BASIC_STREAM module BasicMemoStream : BASIC_STREAM module MakeStream (BasicStream : BASIC_STREAM) : STREAM diff --git a/src/stream/WALK b/src/Stream/WALK similarity index 100% rename from src/stream/WALK rename to src/Stream/WALK diff --git a/src/stream/dune b/src/Stream/dune similarity index 86% rename from src/stream/dune rename to src/Stream/dune index 439d3f3..9e4f02b 100644 --- a/src/stream/dune +++ b/src/Stream/dune @@ -3,7 +3,7 @@ (library (name stream) (public_name stelf.stream) - (modules Stream_ Stream_intf) + (modules Stream_ STREAM) (libraries display basis) (wrapped true) (preprocess diff --git a/src/style/Style_intf.ml b/src/Style/STYLE.ml similarity index 100% rename from src/style/Style_intf.ml rename to src/Style/STYLE.ml diff --git a/src/style/Style.ml b/src/Style/Style.ml similarity index 100% rename from src/style/Style.ml rename to src/Style/Style.ml diff --git a/src/style/Style.mli b/src/Style/Style.mli similarity index 100% rename from src/style/Style.mli rename to src/Style/Style.mli diff --git a/src/style/Style_.ml b/src/Style/Style_.ml similarity index 99% rename from src/style/Style_.ml rename to src/Style/Style_.ml index 55bb72e..6aa1e15 100644 --- a/src/style/Style_.ml +++ b/src/Style/Style_.ml @@ -3,7 +3,7 @@ open! Basis (* Style Checking *) -include Style_intf +include STYLE (** Author: Carsten Schuermann *) (* signature STYLECHECK *) diff --git a/src/style/Style_.mli b/src/Style/Style_.mli similarity index 80% rename from src/style/Style_.mli rename to src/Style/Style_.mli index ff5a459..36c6d3f 100644 --- a/src/style/Style_.mli +++ b/src/Style/Style_.mli @@ -1,4 +1,4 @@ -include module type of Style_intf +include module type of STYLE module MakeStyleCheck (Whnf : WHNF) diff --git a/src/style/dune b/src/Style/dune similarity index 95% rename from src/style/dune rename to src/Style/dune index e221f49..9bd835b 100644 --- a/src/style/dune +++ b/src/Style/dune @@ -3,7 +3,7 @@ (library (name style) (public_name stelf.style) - (modules Style_ Style_intf) + (modules Style_ STYLE) (libraries display global diff --git a/src/subordinate/Intset_intf.ml b/src/Subordinate/INTSET.ml similarity index 100% rename from src/subordinate/Intset_intf.ml rename to src/Subordinate/INTSET.ml diff --git a/src/subordinate/Intset.ml b/src/Subordinate/Intset.ml similarity index 99% rename from src/subordinate/Intset.ml rename to src/Subordinate/Intset.ml index 89489f4..ac1de0d 100644 --- a/src/subordinate/Intset.ml +++ b/src/Subordinate/Intset.ml @@ -9,7 +9,7 @@ open! Basis (* Specialized for subordination *) (* Author: Frank Pfenning *) (* Copied from src/table/red-black-tree.fun *) -include Intset_intf +include INTSET module IntSet : INTSET = struct type rbt = Empty | Red of (int * rbt * rbt) | Black of (int * rbt * rbt) diff --git a/src/Subordinate/Intset.mli b/src/Subordinate/Intset.mli new file mode 100644 index 0000000..5f84936 --- /dev/null +++ b/src/Subordinate/Intset.mli @@ -0,0 +1,2 @@ +include module type of INTSET +module IntSet : INTSET diff --git a/src/subordinate/README b/src/Subordinate/README similarity index 100% rename from src/subordinate/README rename to src/Subordinate/README diff --git a/src/subordinate/Subordinate_intf.ml b/src/Subordinate/SUBORDINATE.ml similarity index 100% rename from src/subordinate/Subordinate_intf.ml rename to src/Subordinate/SUBORDINATE.ml diff --git a/src/subordinate/Subordinate.ml b/src/Subordinate/Subordinate.ml similarity index 100% rename from src/subordinate/Subordinate.ml rename to src/Subordinate/Subordinate.ml diff --git a/src/subordinate/Subordinate.mli b/src/Subordinate/Subordinate.mli similarity index 100% rename from src/subordinate/Subordinate.mli rename to src/Subordinate/Subordinate.mli diff --git a/src/subordinate/Subordinate_.ml b/src/Subordinate/Subordinate_.ml similarity index 99% rename from src/subordinate/Subordinate_.ml rename to src/Subordinate/Subordinate_.ml index e5aeb1f..4ab088a 100644 --- a/src/subordinate/Subordinate_.ml +++ b/src/Subordinate/Subordinate_.ml @@ -4,7 +4,7 @@ open! Basis (* Subordination *) (* Author: Carsten Schuermann *) -include Subordinate_intf +include SUBORDINATE (** Modified: Frank Pfenning *) (* signature SUBORDINATE *) diff --git a/src/subordinate/Subordinate_.mli b/src/Subordinate/Subordinate_.mli similarity index 90% rename from src/subordinate/Subordinate_.mli rename to src/Subordinate/Subordinate_.mli index 2cef938..7f542ba 100644 --- a/src/subordinate/Subordinate_.mli +++ b/src/Subordinate/Subordinate_.mli @@ -1,4 +1,4 @@ -include module type of Subordinate_intf +include module type of SUBORDINATE module MakeSubordinate (Global : GLOBAL) diff --git a/src/subordinate/WALK b/src/Subordinate/WALK similarity index 100% rename from src/subordinate/WALK rename to src/Subordinate/WALK diff --git a/src/subordinate/dune b/src/Subordinate/dune similarity index 88% rename from src/subordinate/dune rename to src/Subordinate/dune index c2a6c49..b7cddc5 100644 --- a/src/subordinate/dune +++ b/src/Subordinate/dune @@ -3,7 +3,7 @@ (library (name subordinate) (public_name stelf.subordinate) - (modules Intset Subordinate_ Intset_intf Subordinate_intf) + (modules Intset Subordinate_ INTSET SUBORDINATE) (libraries display global table intsyn names basis) (flags (:standard diff --git a/src/table/HashTable.ml b/src/Table/HashTable.ml similarity index 100% rename from src/table/HashTable.ml rename to src/Table/HashTable.ml diff --git a/src/table/HashTable.mli b/src/Table/HashTable.mli similarity index 100% rename from src/table/HashTable.mli rename to src/Table/HashTable.mli diff --git a/src/table/Queue_intf.ml b/src/Table/QUEUE.ml similarity index 100% rename from src/table/Queue_intf.ml rename to src/Table/QUEUE.ml diff --git a/src/table/Queue.ml b/src/Table/Queue.ml similarity index 98% rename from src/table/Queue.ml rename to src/Table/Queue.ml index 766cc3e..102224b 100644 --- a/src/table/Queue.ml +++ b/src/Table/Queue.ml @@ -4,7 +4,7 @@ open Basis (* Queues *) (* Author: Frank Pfenning *) -include Queue_intf +include QUEUE (* signature QUEUE *) (* # 1 "src/table/Queue.fun.ml" *) diff --git a/src/Table/Queue.mli b/src/Table/Queue.mli new file mode 100644 index 0000000..2147312 --- /dev/null +++ b/src/Table/Queue.mli @@ -0,0 +1,2 @@ +include module type of QUEUE +module Queue : QUEUE diff --git a/src/table/README b/src/Table/README similarity index 100% rename from src/table/README rename to src/Table/README diff --git a/src/table/RedBlackSet_intf.ml b/src/Table/REDBLACKSET.ml similarity index 100% rename from src/table/RedBlackSet_intf.ml rename to src/Table/REDBLACKSET.ml diff --git a/src/table/Ring_intf.ml b/src/Table/RING.ml similarity index 100% rename from src/table/Ring_intf.ml rename to src/Table/RING.ml diff --git a/src/table/RedBlackSet.ml b/src/Table/RedBlackSet.ml similarity index 99% rename from src/table/RedBlackSet.ml rename to src/Table/RedBlackSet.ml index b317462..2b01b35 100644 --- a/src/table/RedBlackSet.ml +++ b/src/Table/RedBlackSet.ml @@ -6,7 +6,7 @@ open Basis (* Author: Brigitte Pientka *) (* This provides a common interface to ordered sets *) (* based on red/black trees *) -include RedBlackSet_intf +include REDBLACKSET (* signature RBSET *) (* # 1 "src/table/RedBlackSet.fun.ml" *) diff --git a/src/Table/RedBlackSet.mli b/src/Table/RedBlackSet.mli new file mode 100644 index 0000000..ffc766f --- /dev/null +++ b/src/Table/RedBlackSet.mli @@ -0,0 +1,2 @@ +include module type of REDBLACKSET +module RBSet : RBSET diff --git a/src/table/RedBlackTree.ml b/src/Table/RedBlackTree.ml similarity index 100% rename from src/table/RedBlackTree.ml rename to src/Table/RedBlackTree.ml diff --git a/src/table/RedBlackTree.mli b/src/Table/RedBlackTree.mli similarity index 100% rename from src/table/RedBlackTree.mli rename to src/Table/RedBlackTree.mli diff --git a/src/table/Ring.ml b/src/Table/Ring.ml similarity index 99% rename from src/table/Ring.ml rename to src/Table/Ring.ml index f6198ca..dc1ff63 100644 --- a/src/table/Ring.ml +++ b/src/Table/Ring.ml @@ -6,7 +6,7 @@ open Basis (* Rings (aka cyclic lists) *) (* Author: Carsten Schuermann *) -include Ring_intf +include RING (* does not necessarily map f in order *) (* signature RING *) diff --git a/src/Table/Ring.mli b/src/Table/Ring.mli new file mode 100644 index 0000000..593bfeb --- /dev/null +++ b/src/Table/Ring.mli @@ -0,0 +1,2 @@ +include module type of RING +module Ring : RING diff --git a/src/table/SparseArray_intf.ml b/src/Table/SPARSEARRAY.ml similarity index 100% rename from src/table/SparseArray_intf.ml rename to src/Table/SPARSEARRAY.ml diff --git a/src/table/SparseArray2_intf.ml b/src/Table/SPARSEARRAY2.ml similarity index 100% rename from src/table/SparseArray2_intf.ml rename to src/Table/SPARSEARRAY2.ml diff --git a/src/table/StringHash_intf.ml b/src/Table/STRINGHASH.ml similarity index 100% rename from src/table/StringHash_intf.ml rename to src/Table/STRINGHASH.ml diff --git a/src/table/SparseArray.ml b/src/Table/SparseArray.ml similarity index 99% rename from src/table/SparseArray.ml rename to src/Table/SparseArray.ml index 730813d..ff6e28d 100644 --- a/src/table/SparseArray.ml +++ b/src/Table/SparseArray.ml @@ -4,7 +4,7 @@ open Basis (* Sparse 1-Dimensional Arrays *) (* Author: Roberto Virga *) -include SparseArray_intf +include SPARSEARRAY (* signature SPARSE_ARRAY *) (* # 1 "src/table/SparseArray.fun.ml" *) diff --git a/src/table/SparseArray.mli b/src/Table/SparseArray.mli similarity index 75% rename from src/table/SparseArray.mli rename to src/Table/SparseArray.mli index 2f9f1c5..973b6f7 100644 --- a/src/table/SparseArray.mli +++ b/src/Table/SparseArray.mli @@ -1,5 +1,5 @@ open Table_ -include module type of SparseArray_intf +include module type of SPARSEARRAY module SparseArray (SparseArray__0 : sig module IntTable : TABLE with type key = int diff --git a/src/table/SparseArray2.ml b/src/Table/SparseArray2.ml similarity index 99% rename from src/table/SparseArray2.ml rename to src/Table/SparseArray2.ml index 0fe8063..efc8bbb 100644 --- a/src/table/SparseArray2.ml +++ b/src/Table/SparseArray2.ml @@ -4,7 +4,7 @@ open Basis (* Sparse 2-Dimensional Arrays *) (* Author: Roberto Virga *) -include SparseArray2_intf +include SPARSEARRAY2 (* signature SPARSE_ARRAY2 *) (* # 1 "src/table/SparseArray2.fun.ml" *) diff --git a/src/table/SparseArray2.mli b/src/Table/SparseArray2.mli similarity index 75% rename from src/table/SparseArray2.mli rename to src/Table/SparseArray2.mli index 8629efd..5a6f89d 100644 --- a/src/table/SparseArray2.mli +++ b/src/Table/SparseArray2.mli @@ -1,5 +1,5 @@ open Table_ -include module type of SparseArray2_intf +include module type of SPARSEARRAY2 module SparseArray2 (SparseArray2__0 : sig module IntTable : TABLE with type key = int diff --git a/src/table/StringHash.ml b/src/Table/StringHash.ml similarity index 96% rename from src/table/StringHash.ml rename to src/Table/StringHash.ml index 76ebac3..f83e709 100644 --- a/src/table/StringHash.ml +++ b/src/Table/StringHash.ml @@ -4,7 +4,7 @@ open Basis (* String Hash Table *) (* Author: Frank Pfenning *) -include StringHash_intf +include STRINGHASH (* # 1 "src/table/StringHash.fun.ml" *) (* # 1 "src/table/StringHash.sml.ml" *) diff --git a/src/Table/StringHash.mli b/src/Table/StringHash.mli new file mode 100644 index 0000000..3478fb7 --- /dev/null +++ b/src/Table/StringHash.mli @@ -0,0 +1,2 @@ +include module type of STRINGHASH +module StringHash : STRING_HASH diff --git a/src/table/Table_intf.ml b/src/Table/TABLE.ml similarity index 100% rename from src/table/Table_intf.ml rename to src/Table/TABLE.ml diff --git a/src/table/Table.ml b/src/Table/Table.ml similarity index 100% rename from src/table/Table.ml rename to src/Table/Table.ml diff --git a/src/table/Table.mli b/src/Table/Table.mli similarity index 100% rename from src/table/Table.mli rename to src/Table/Table.mli diff --git a/src/table/TableInstances.ml b/src/Table/TableInstances.ml similarity index 100% rename from src/table/TableInstances.ml rename to src/Table/TableInstances.ml diff --git a/src/table/Table_.ml b/src/Table/Table_.ml similarity index 96% rename from src/table/Table_.ml rename to src/Table/Table_.ml index d03323a..e3ce6df 100644 --- a/src/table/Table_.ml +++ b/src/Table/Table_.ml @@ -7,7 +7,7 @@ open Basis (* Modified: Roberto Virga *) (* This provides a common interface to hash tables *) -include Table_intf +include TABLE (** red/black trees and similar data structures *) (* signature TABLE *) diff --git a/src/Table/Table_.mli b/src/Table/Table_.mli new file mode 100644 index 0000000..56b85f5 --- /dev/null +++ b/src/Table/Table_.mli @@ -0,0 +1 @@ +include module type of TABLE diff --git a/src/table/WALK b/src/Table/WALK similarity index 100% rename from src/table/WALK rename to src/Table/WALK diff --git a/src/table/dune b/src/Table/dune similarity index 73% rename from src/table/dune rename to src/Table/dune index 2493424..5918728 100644 --- a/src/table/dune +++ b/src/Table/dune @@ -14,13 +14,13 @@ RedBlackSet Table_ TableInstances - Queue_intf - RedBlackSet_intf - Ring_intf - SparseArray_intf - SparseArray2_intf - StringHash_intf - Table_intf) + QUEUE + REDBLACKSET + RING + SPARSEARRAY + SPARSEARRAY2 + STRINGHASH + TABLE) (libraries display basis) (wrapped true) (preprocess diff --git a/src/tabling/Tabledsyn_intf.ml b/src/Tabling/TABLEDSYN.ml similarity index 100% rename from src/tabling/Tabledsyn_intf.ml rename to src/Tabling/TABLEDSYN.ml diff --git a/src/tabling/Tabled.ml b/src/Tabling/Tabled.ml similarity index 100% rename from src/tabling/Tabled.ml rename to src/Tabling/Tabled.ml diff --git a/src/tabling/Tabled.mli b/src/Tabling/Tabled.mli similarity index 100% rename from src/tabling/Tabled.mli rename to src/Tabling/Tabled.mli diff --git a/src/tabling/Tabledsyn.ml b/src/Tabling/Tabledsyn.ml similarity index 99% rename from src/tabling/Tabledsyn.ml rename to src/Tabling/Tabledsyn.ml index b3ac9bd..c7669d8 100644 --- a/src/tabling/Tabledsyn.ml +++ b/src/Tabling/Tabledsyn.ml @@ -3,7 +3,7 @@ open! Basis (* Tabled Syntax *) (* Author: Brigitte Pientka *) -include Tabledsyn_intf +include TABLEDSYN (* signature TABLEDSYN *) (* # 1 "src/tabling/Tabledsyn.fun.ml" *) diff --git a/src/tabling/Tabledsyn.mli b/src/Tabling/Tabledsyn.mli similarity index 88% rename from src/tabling/Tabledsyn.mli rename to src/Tabling/Tabledsyn.mli index d7c00c9..c5aa17e 100644 --- a/src/tabling/Tabledsyn.mli +++ b/src/Tabling/Tabledsyn.mli @@ -1,4 +1,4 @@ -include module type of Tabledsyn_intf +include module type of TABLEDSYN module MakeTabledSyn (Names : NAMES) diff --git a/src/tabling/Tabling.ml b/src/Tabling/Tabling.ml similarity index 100% rename from src/tabling/Tabling.ml rename to src/Tabling/Tabling.ml diff --git a/src/tabling/Tabling.mli b/src/Tabling/Tabling.mli similarity index 100% rename from src/tabling/Tabling.mli rename to src/Tabling/Tabling.mli diff --git a/src/tabling/dune b/src/Tabling/dune similarity index 92% rename from src/tabling/dune rename to src/Tabling/dune index 226904d..6dd5b28 100644 --- a/src/tabling/dune +++ b/src/Tabling/dune @@ -3,7 +3,7 @@ (library (name tabling) (public_name stelf.tabling) - (modules Tabledsyn Tabled Tabledsyn_intf) + (modules Tabledsyn Tabled TABLEDSYN) (libraries display table intsyn print names paths index basis) (flags (:standard diff --git a/src/terminate/Checking_intf.ml b/src/Terminate/CHECKING.ml similarity index 100% rename from src/terminate/Checking_intf.ml rename to src/Terminate/CHECKING.ml diff --git a/src/terminate/Checking.ml b/src/Terminate/Checking.ml similarity index 99% rename from src/terminate/Checking.ml rename to src/Terminate/Checking.ml index 9d73747..7fe29d2 100644 --- a/src/terminate/Checking.ml +++ b/src/Terminate/Checking.ml @@ -3,7 +3,7 @@ open! Basis (* Reasoning about orders *) (* Author: Brigitte Pientka *) -include Checking_intf +include CHECKING (* signature CHECKING *) (* # 1 "src/terminate/Checking.fun.ml" *) diff --git a/src/terminate/Checking.mli b/src/Terminate/Checking.mli similarity index 95% rename from src/terminate/Checking.mli rename to src/Terminate/Checking.mli index 08c6a4b..7f3608b 100644 --- a/src/terminate/Checking.mli +++ b/src/Terminate/Checking.mli @@ -1,4 +1,4 @@ -include module type of Checking_intf +include module type of CHECKING module Checking (Checking__0 : sig module Global : GLOBAL diff --git a/src/terminate/README b/src/Terminate/README similarity index 100% rename from src/terminate/README rename to src/Terminate/README diff --git a/src/terminate/Reduces_intf.ml b/src/Terminate/REDUCES.ml similarity index 100% rename from src/terminate/Reduces_intf.ml rename to src/Terminate/REDUCES.ml diff --git a/src/terminate/Reduces.ml b/src/Terminate/Reduces.ml similarity index 99% rename from src/terminate/Reduces.ml rename to src/Terminate/Reduces.ml index 5e6d8dc..44113da 100644 --- a/src/terminate/Reduces.ml +++ b/src/Terminate/Reduces.ml @@ -3,7 +3,7 @@ open! Basis (* Reduction and Termination checker *) (* Author: Brigitte Pientka *) -include Reduces_intf +include REDUCES (* signature REDUCES *) (* # 1 "src/terminate/Reduces.fun.ml" *) diff --git a/src/terminate/Reduces.mli b/src/Terminate/Reduces.mli similarity index 95% rename from src/terminate/Reduces.mli rename to src/Terminate/Reduces.mli index 8f471f7..0f79393 100644 --- a/src/terminate/Reduces.mli +++ b/src/Terminate/Reduces.mli @@ -1,4 +1,4 @@ -include module type of Reduces_intf +include module type of REDUCES module Reduces (Reduces__0 : sig module Global : GLOBAL diff --git a/src/terminate/Terminate.ml b/src/Terminate/Terminate.ml similarity index 100% rename from src/terminate/Terminate.ml rename to src/Terminate/Terminate.ml diff --git a/src/terminate/Terminate.mli b/src/Terminate/Terminate.mli similarity index 100% rename from src/terminate/Terminate.mli rename to src/Terminate/Terminate.mli diff --git a/src/terminate/Terminate_.ml b/src/Terminate/Terminate_.ml similarity index 100% rename from src/terminate/Terminate_.ml rename to src/Terminate/Terminate_.ml diff --git a/src/terminate/Terminate_.mli b/src/Terminate/Terminate_.mli similarity index 100% rename from src/terminate/Terminate_.mli rename to src/Terminate/Terminate_.mli diff --git a/src/terminate/WALK b/src/Terminate/WALK similarity index 100% rename from src/terminate/WALK rename to src/Terminate/WALK diff --git a/src/terminate/dune b/src/Terminate/dune similarity index 92% rename from src/terminate/dune rename to src/Terminate/dune index 95f84d8..4a6a152 100644 --- a/src/terminate/dune +++ b/src/Terminate/dune @@ -3,7 +3,7 @@ (library (name terminate) (public_name stelf.terminate) - (modules Checking Reduces Terminate_ Checking_intf Reduces_intf) + (modules Checking Reduces Terminate_ CHECKING REDUCES) (libraries display global diff --git a/src/thm/README b/src/Thm/README similarity index 100% rename from src/thm/README rename to src/Thm/README diff --git a/src/thm/Thm_intf.ml b/src/Thm/THM.ml similarity index 100% rename from src/thm/Thm_intf.ml rename to src/Thm/THM.ml diff --git a/src/thm/Thmprint_intf.ml b/src/Thm/THMPRINT.ml similarity index 100% rename from src/thm/Thmprint_intf.ml rename to src/Thm/THMPRINT.ml diff --git a/src/thm/Thmsyn_intf.ml b/src/Thm/THMSYN.ml similarity index 100% rename from src/thm/Thmsyn_intf.ml rename to src/Thm/THMSYN.ml diff --git a/src/thm/Thm.ml b/src/Thm/Thm.ml similarity index 100% rename from src/thm/Thm.ml rename to src/Thm/Thm.ml diff --git a/src/thm/Thm.mli b/src/Thm/Thm.mli similarity index 100% rename from src/thm/Thm.mli rename to src/Thm/Thm.mli diff --git a/src/thm/Thm_.ml b/src/Thm/Thm_.ml similarity index 99% rename from src/thm/Thm_.ml rename to src/Thm/Thm_.ml index 0a6898a..87feb0c 100644 --- a/src/thm/Thm_.ml +++ b/src/Thm/Thm_.ml @@ -6,7 +6,7 @@ open Thmprint (* Theorem Declarations *) (* Author: Carsten Schuermann *) -include Thm_intf +include THM (** Modified: Brigitte Pientka, Frank Pfenning *) (* signature THM *) diff --git a/src/thm/Thm_.mli b/src/Thm/Thm_.mli similarity index 93% rename from src/thm/Thm_.mli rename to src/Thm/Thm_.mli index 6277778..ffcef87 100644 --- a/src/thm/Thm_.mli +++ b/src/Thm/Thm_.mli @@ -1,4 +1,4 @@ -include module type of Thm_intf +include module type of THM module Make_Thm (Global : GLOBAL) diff --git a/src/thm/Thmprint.ml b/src/Thm/Thmprint.ml similarity index 99% rename from src/thm/Thmprint.ml rename to src/Thm/Thmprint.ml index fa058c3..05b8bdb 100644 --- a/src/thm/Thmprint.ml +++ b/src/Thm/Thmprint.ml @@ -4,7 +4,7 @@ open Thmsyn (* Printer for Meta Theorems *) (* Author: Carsten Schuermann *) -include Thmprint_intf +include THMPRINT (* -bp *) (* signature THMPRINT *) diff --git a/src/thm/Thmprint.mli b/src/Thm/Thmprint.mli similarity index 80% rename from src/thm/Thmprint.mli rename to src/Thm/Thmprint.mli index 813d31b..d8d3ec8 100644 --- a/src/thm/Thmprint.mli +++ b/src/Thm/Thmprint.mli @@ -1,4 +1,4 @@ -include module type of Thmprint_intf +include module type of THMPRINT module ThmPrint (ThmPrint__0 : sig module ThmSyn' : Thmsyn.THMSYN diff --git a/src/thm/Thmsyn.ml b/src/Thm/Thmsyn.ml similarity index 99% rename from src/thm/Thmsyn.ml rename to src/Thm/Thmsyn.ml index 8826588..4780647 100644 --- a/src/thm/Thmsyn.ml +++ b/src/Thm/Thmsyn.ml @@ -4,7 +4,7 @@ open! Basis (* Theorems *) (* Author: Carsten Schuermann *) (* Modified: Brigitte Pientka *) -include Thmsyn_intf +include THMSYN (* signature THMSYN *) (* # 1 "src/thm/Thmsyn.fun.ml" *) diff --git a/src/thm/Thmsyn.mli b/src/Thm/Thmsyn.mli similarity index 91% rename from src/thm/Thmsyn.mli rename to src/Thm/Thmsyn.mli index f700d53..36ec93c 100644 --- a/src/thm/Thmsyn.mli +++ b/src/Thm/Thmsyn.mli @@ -1,4 +1,4 @@ -include module type of Thmsyn_intf +include module type of THMSYN module ThmSyn (ThmSyn__0 : sig (*! structure IntSyn : INTSYN !*) diff --git a/src/thm/WALK b/src/Thm/WALK similarity index 100% rename from src/thm/WALK rename to src/Thm/WALK diff --git a/src/thm/dune b/src/Thm/dune similarity index 89% rename from src/thm/dune rename to src/Thm/dune index e074d2a..57579fb 100644 --- a/src/thm/dune +++ b/src/Thm/dune @@ -3,7 +3,7 @@ (library (name thm) (public_name stelf.thm) - (modules Thmsyn Thmprint Thm_ Thm_intf Thmprint_intf Thmsyn_intf) + (modules Thmsyn Thmprint Thm_ THM THMPRINT THMSYN) (libraries display global intsyn names formatter modes paths tabling basis) (flags (:standard diff --git a/src/timing/README b/src/Timing/README similarity index 100% rename from src/timing/README rename to src/Timing/README diff --git a/src/timing/Timers_intf.ml b/src/Timing/TIMERS.ml similarity index 100% rename from src/timing/Timers_intf.ml rename to src/Timing/TIMERS.ml diff --git a/src/timing/Timing_intf.ml b/src/Timing/TIMING.ml similarity index 100% rename from src/timing/Timing_intf.ml rename to src/Timing/TIMING.ml diff --git a/src/timing/TimeLimit.ml b/src/Timing/TimeLimit.ml similarity index 100% rename from src/timing/TimeLimit.ml rename to src/Timing/TimeLimit.ml diff --git a/src/timing/TimeLimit.mli b/src/Timing/TimeLimit.mli similarity index 100% rename from src/timing/TimeLimit.mli rename to src/Timing/TimeLimit.mli diff --git a/src/timing/Timers.ml b/src/Timing/Timers.ml similarity index 99% rename from src/timing/Timers.ml rename to src/Timing/Timers.ml index ec0799e..e6ea68e 100644 --- a/src/timing/Timers.ml +++ b/src/Timing/Timers.ml @@ -5,7 +5,7 @@ module Timing = Timing_.Timing (* Timers collecting statistics about Stelf *) (* Author: Frank Pfenning *) -include Timers_intf +include TIMERS (* check, then reset *) (* signature TIMERS *) diff --git a/src/timing/Timers.mli b/src/Timing/Timers.mli similarity index 76% rename from src/timing/Timers.mli rename to src/Timing/Timers.mli index c502dec..3a334c3 100644 --- a/src/timing/Timers.mli +++ b/src/Timing/Timers.mli @@ -1,4 +1,4 @@ open Timing_ -include module type of Timers_intf +include module type of TIMERS module MakeTimers (Timing' : TIMING) : TIMERS with module Timing = Timing' module Timers : TIMERS diff --git a/src/timing/Timing.ml b/src/Timing/Timing.ml similarity index 100% rename from src/timing/Timing.ml rename to src/Timing/Timing.ml diff --git a/src/timing/Timing.mli b/src/Timing/Timing.mli similarity index 100% rename from src/timing/Timing.mli rename to src/Timing/Timing.mli diff --git a/src/timing/Timing_.ml b/src/Timing/Timing_.ml similarity index 99% rename from src/timing/Timing_.ml rename to src/Timing/Timing_.ml index f4830fa..d2fbd58 100644 --- a/src/timing/Timing_.ml +++ b/src/Timing/Timing_.ml @@ -16,7 +16,7 @@ open Basis be initialized, otherwise exception Time.Time is raised somewhere. *) -include Timing_intf +include TIMING (* signature TIMING *) module Timing : TIMING = struct diff --git a/src/timing/Timing_.mli b/src/Timing/Timing_.mli similarity index 57% rename from src/timing/Timing_.mli rename to src/Timing/Timing_.mli index d17df4f..dfb73ee 100644 --- a/src/timing/Timing_.mli +++ b/src/Timing/Timing_.mli @@ -1,3 +1,3 @@ -include module type of Timing_intf +include module type of TIMING module Timing : TIMING module Counting : TIMING diff --git a/src/timing/WALK b/src/Timing/WALK similarity index 100% rename from src/timing/WALK rename to src/Timing/WALK diff --git a/src/timing/dune b/src/Timing/dune similarity index 76% rename from src/timing/dune rename to src/Timing/dune index cd6f6bf..aa9a4e9 100644 --- a/src/timing/dune +++ b/src/Timing/dune @@ -3,7 +3,7 @@ (library (name timing) (public_name stelf.timing) - (modules Timers Timing_ TimeLimit Timers_intf Timing_intf) + (modules Timers Timing_ TimeLimit TIMERS TIMING) (libraries display basis) (wrapped true) (preprocess diff --git a/src/tomega/Converter_intf.ml b/src/Tomega/CONVERTER.ml similarity index 100% rename from src/tomega/Converter_intf.ml rename to src/Tomega/CONVERTER.ml diff --git a/src/tomega/Coverage_intf.ml b/src/Tomega/COVERAGE.ml similarity index 100% rename from src/tomega/Coverage_intf.ml rename to src/Tomega/COVERAGE.ml diff --git a/src/tomega/Converter.ml b/src/Tomega/Converter.ml similarity index 99% rename from src/tomega/Converter.ml rename to src/Tomega/Converter.ml index e3dc6c1..e556c98 100644 --- a/src/tomega/Converter.ml +++ b/src/Tomega/Converter.ml @@ -5,7 +5,7 @@ module Tomega = Lambda_.Tomega (* Converter from relational representation to a functional representation of proof terms *) (* Author: Carsten Schuermann *) -include Converter_intf +include CONVERTER (* Signature CONVERTER *) (* # 1 "src/tomega/Converter.fun.ml" *) @@ -22,11 +22,11 @@ module MakeConverter (TomegaPrint : Tomegaprint.TOMEGAPRINT) (WorldSyn : Worldcheck_.WORLDSYN) (Worldify : Worldcheck_.WORLDIFY) - (TomegaTypeCheck : TomegaTypecheck_intf.TOMEGATYPECHECK) + (TomegaTypeCheck : TOMEGATYPECHECK.TOMEGATYPECHECK) (Subordinate : Subordinate.Subordinate_.SUBORDINATE) (TypeCheck : Typecheck_.TYPECHECK) - (Redundant : Redundant_intf.REDUNDANT) - (TomegaAbstract : TomegaAbstract_intf.TOMEGAABSTRACT) : CONVERTER = struct + (Redundant : REDUNDANT.REDUNDANT) + (TomegaAbstract : TOMEGAABSTRACT.TOMEGAABSTRACT) : CONVERTER = struct (* (* Converter from relational representation to a functional representation of proof terms *) @@ -66,7 +66,7 @@ module MakeConverter (*! sharing Worldify.IntSyn = IntSyn' !*) (*! sharing Worldify.Tomega = Tomega' !*) - module TomegaTypeCheck : TomegaTypecheck_intf.TOMEGATYPECHECK + module TomegaTypeCheck : TOMEGATYPECHECK.TOMEGATYPECHECK (*! sharing TomegaTypeCheck.IntSyn = IntSyn' !*) (*! sharing TomegaTypeCheck.Tomega = Tomega' !*) @@ -76,8 +76,8 @@ module MakeConverter module TypeCheck : Typecheck_.TYPECHECK (*! sharing TypeCheck.IntSyn = IntSyn' !*) - module Redundant : Redundant_intf.REDUNDANT - module TomegaAbstract : TomegaAbstract_intf.TOMEGAABSTRACT + module Redundant : REDUNDANT.REDUNDANT + module TomegaAbstract : TOMEGAABSTRACT.TOMEGAABSTRACT *) (*! structure IntSyn = IntSyn' !*) (*! structure Tomega = Tomega' !*) diff --git a/src/tomega/Converter.mli b/src/Tomega/Converter.mli similarity index 82% rename from src/tomega/Converter.mli rename to src/Tomega/Converter.mli index 11dd606..be23ee9 100644 --- a/src/tomega/Converter.mli +++ b/src/Tomega/Converter.mli @@ -1,4 +1,4 @@ -include module type of Converter_intf +include module type of CONVERTER module MakeConverter (Global : GLOBAL) @@ -11,11 +11,11 @@ module MakeConverter (TomegaPrint : Tomegaprint.TOMEGAPRINT) (WorldSyn : Worldcheck_.WORLDSYN) (Worldify : Worldcheck_.WORLDIFY) - (TomegaTypeCheck : TomegaTypecheck_intf.TOMEGATYPECHECK) + (TomegaTypeCheck : TOMEGATYPECHECK.TOMEGATYPECHECK) (Subordinate : Subordinate.Subordinate_.SUBORDINATE) (TypeCheck : Typecheck_.TYPECHECK) - (Redundant : Redundant_intf.REDUNDANT) - (TomegaAbstract : TomegaAbstract_intf.TOMEGAABSTRACT) : CONVERTER + (Redundant : REDUNDANT.REDUNDANT) + (TomegaAbstract : TOMEGAABSTRACT.TOMEGAABSTRACT) : CONVERTER (* (* Converter from relational representation to a functional representation of proof terms *) @@ -55,7 +55,7 @@ module MakeConverter (*! sharing Worldify.IntSyn = IntSyn' !*) (*! sharing Worldify.Tomega = Tomega' !*) - module TomegaTypeCheck : TomegaTypecheck_intf.TOMEGATYPECHECK + module TomegaTypeCheck : TOMEGATYPECHECK.TOMEGATYPECHECK (*! sharing TomegaTypeCheck.IntSyn = IntSyn' !*) (*! sharing TomegaTypeCheck.Tomega = Tomega' !*) @@ -65,6 +65,6 @@ module MakeConverter module TypeCheck : Typecheck_.TYPECHECK (*! sharing TypeCheck.IntSyn = IntSyn' !*) - module Redundant : Redundant_intf.REDUNDANT - module TomegaAbstract : TomegaAbstract_intf.TOMEGAABSTRACT + module Redundant : REDUNDANT.REDUNDANT + module TomegaAbstract : TOMEGAABSTRACT.TOMEGAABSTRACT *) diff --git a/src/tomega/Coverage.ml b/src/Tomega/Coverage.ml similarity index 98% rename from src/tomega/Coverage.ml rename to src/Tomega/Coverage.ml index 2725717..6e0eed1 100644 --- a/src/tomega/Coverage.ml +++ b/src/Tomega/Coverage.ml @@ -4,7 +4,7 @@ module Tomega = Lambda_.Tomega (* Unification on Formulas *) (* Author: Carsten Schuermann *) -include Coverage_intf +include COVERAGE (* Signature TOMEGACOVERAGE *) (* # 1 "src/tomega/Coverage.fun.ml" *) @@ -12,7 +12,7 @@ open! Basis module MakeTomegaCoverage (TomegaPrint : Tomegaprint.TOMEGAPRINT) - (TomegaTypeCheck : TomegaTypecheck_intf.TOMEGATYPECHECK) + (TomegaTypeCheck : TOMEGATYPECHECK.TOMEGATYPECHECK) (Cover : COVER) : TOMEGACOVERAGE = struct (* (* Coverage checker for programs *) @@ -24,7 +24,7 @@ module MakeTomegaCoverage (*! sharing TomegaPrint.IntSyn = IntSyn' !*) (*! sharing TomegaPrint.Tomega = Tomega' !*) - module TomegaTypeCheck : TomegaTypecheck_intf.TOMEGATYPECHECK + module TomegaTypeCheck : TOMEGATYPECHECK.TOMEGATYPECHECK (*! sharing TomegaTypeCheck.IntSyn = IntSyn' !*) (*! sharing TomegaTypeCheck.Tomega = Tomega' !*) diff --git a/src/tomega/Coverage.mli b/src/Tomega/Coverage.mli similarity index 77% rename from src/tomega/Coverage.mli rename to src/Tomega/Coverage.mli index 43ca2d7..3a5c4ed 100644 --- a/src/tomega/Coverage.mli +++ b/src/Tomega/Coverage.mli @@ -1,8 +1,8 @@ -include module type of Coverage_intf +include module type of COVERAGE module MakeTomegaCoverage (TomegaPrint : Tomegaprint.TOMEGAPRINT) - (TomegaTypeCheck : TomegaTypecheck_intf.TOMEGATYPECHECK) + (TomegaTypeCheck : TOMEGATYPECHECK.TOMEGATYPECHECK) (Cover : COVER) : TOMEGACOVERAGE (* (* Coverage checker for programs *) @@ -14,7 +14,7 @@ module MakeTomegaCoverage (*! sharing TomegaPrint.IntSyn = IntSyn' !*) (*! sharing TomegaPrint.Tomega = Tomega' !*) - module TomegaTypeCheck : TomegaTypecheck_intf.TOMEGATYPECHECK + module TomegaTypeCheck : TOMEGATYPECHECK.TOMEGATYPECHECK (*! sharing TomegaTypeCheck.IntSyn = IntSyn' !*) (*! sharing TomegaTypeCheck.Tomega = Tomega' !*) diff --git a/src/tomega/Normalize_intf.ml b/src/Tomega/NORMALIZE.ml similarity index 90% rename from src/tomega/Normalize_intf.ml rename to src/Tomega/NORMALIZE.ml index 4b156ab..fc237f9 100644 --- a/src/tomega/Normalize_intf.ml +++ b/src/Tomega/NORMALIZE.ml @@ -2,7 +2,7 @@ open! Basis module type NORMALIZE = sig - module IntSyn : Intsyn_intf.INTSYN + module IntSyn : INTSYN.INTSYN module Tomega : TOMEGA val normalizeFor : Tomega.for_ * Tomega.sub -> Tomega.for_ diff --git a/src/tomega/Normalize.ml b/src/Tomega/Normalize.ml similarity index 99% rename from src/tomega/Normalize.ml rename to src/Tomega/Normalize.ml index a6afe50..3a64b9d 100644 --- a/src/tomega/Normalize.ml +++ b/src/Tomega/Normalize.ml @@ -4,7 +4,7 @@ module Tomega = Lambda_.Tomega (* Normalizer for Delphin meta level *) (* Author: Carsten Schuermann *) -include Normalize_intf +include NORMALIZE (* # 1 "src/tomega/Normalize.fun.ml" *) open! Basis diff --git a/src/tomega/Normalize.mli b/src/Tomega/Normalize.mli similarity index 81% rename from src/tomega/Normalize.mli rename to src/Tomega/Normalize.mli index bf04151..f900543 100644 --- a/src/tomega/Normalize.mli +++ b/src/Tomega/Normalize.mli @@ -1,4 +1,4 @@ -include module type of Normalize_intf +include module type of NORMALIZE module Normalize (Normalize__0 : sig (* Internal syntax for functional proof term calculus *) diff --git a/src/tomega/Opsem_intf.ml b/src/Tomega/OPSEM.ml similarity index 100% rename from src/tomega/Opsem_intf.ml rename to src/Tomega/OPSEM.ml diff --git a/src/tomega/Opsem.ml b/src/Tomega/Opsem.ml similarity index 99% rename from src/tomega/Opsem.ml rename to src/Tomega/Opsem.ml index 06095a7..dde1488 100644 --- a/src/tomega/Opsem.ml +++ b/src/Tomega/Opsem.ml @@ -4,7 +4,7 @@ module Tomega = Lambda_.Tomega (* Operational Semantics for Delphin *) (* Author: Carsten Schuermann *) -include Opsem_intf +include OPSEM (* # 1 "src/tomega/Opsem.fun.ml" *) open! Basis @@ -13,7 +13,7 @@ module MakeOpsem (Whnf : WHNF) (Abstract : ABSTRACT) (Subordinate : Subordinate.Subordinate_.SUBORDINATE) - (TomegaTypeCheck : TomegaTypecheck_intf.TOMEGATYPECHECK) + (TomegaTypeCheck : TOMEGATYPECHECK.TOMEGATYPECHECK) (TomegaPrint : Tomegaprint.TOMEGAPRINT) (Unify : UNIFY) : OPSEM = struct (* @@ -22,7 +22,7 @@ module MakeOpsem module Whnf : WHNF module Abstract : ABSTRACT module Subordinate : Subordinate.Subordinate_.SUBORDINATE - module TomegaTypeCheck : TomegaTypecheck_intf.TOMEGATYPECHECK + module TomegaTypeCheck : TOMEGATYPECHECK.TOMEGATYPECHECK module TomegaPrint : Tomegaprint.TOMEGAPRINT module Unify : UNIFY *) diff --git a/src/tomega/Opsem.mli b/src/Tomega/Opsem.mli similarity index 76% rename from src/tomega/Opsem.mli rename to src/Tomega/Opsem.mli index 35870d6..8b2b056 100644 --- a/src/tomega/Opsem.mli +++ b/src/Tomega/Opsem.mli @@ -1,10 +1,10 @@ -include module type of Opsem_intf +include module type of OPSEM module MakeOpsem (Whnf : WHNF) (Abstract : ABSTRACT) (Subordinate : Subordinate.Subordinate_.SUBORDINATE) - (TomegaTypeCheck : TomegaTypecheck_intf.TOMEGATYPECHECK) + (TomegaTypeCheck : TOMEGATYPECHECK.TOMEGATYPECHECK) (TomegaPrint : Tomegaprint.TOMEGAPRINT) (Unify : UNIFY) : OPSEM (* diff --git a/src/tomega/Redundant_intf.ml b/src/Tomega/REDUNDANT.ml similarity index 100% rename from src/tomega/Redundant_intf.ml rename to src/Tomega/REDUNDANT.ml diff --git a/src/tomega/Redundant.ml b/src/Tomega/Redundant.ml similarity index 99% rename from src/tomega/Redundant.ml rename to src/Tomega/Redundant.ml index b848be7..a5f093f 100644 --- a/src/tomega/Redundant.ml +++ b/src/Tomega/Redundant.ml @@ -1,7 +1,7 @@ (* # 1 "src/tomega/Redundant.sig.ml" *) open! Basis module Tomega = Lambda_.Tomega -include Redundant_intf +include REDUNDANT (* # 1 "src/tomega/Redundant.fun.ml" *) open! Opsem @@ -10,7 +10,7 @@ open! Basis (* Redundancy remover (factoring) *) (* Author: Adam Poswolsky (ABP) *) module Redundant (Redundant__0 : sig - module Opsem : Opsem_intf.OPSEM + module Opsem : OPSEM.OPSEM end) : REDUNDANT = struct exception Error of string diff --git a/src/Tomega/Redundant.mli b/src/Tomega/Redundant.mli new file mode 100644 index 0000000..0962b35 --- /dev/null +++ b/src/Tomega/Redundant.mli @@ -0,0 +1,5 @@ +include module type of REDUNDANT + +module Redundant (Redundant__0 : sig + module Opsem : OPSEM.OPSEM +end) : REDUNDANT diff --git a/src/tomega/TomegaAbstract_intf.ml b/src/Tomega/TOMEGAABSTRACT.ml similarity index 100% rename from src/tomega/TomegaAbstract_intf.ml rename to src/Tomega/TOMEGAABSTRACT.ml diff --git a/src/tomega/Tomeganames_intf.ml b/src/Tomega/TOMEGANAMES.ml similarity index 100% rename from src/tomega/Tomeganames_intf.ml rename to src/Tomega/TOMEGANAMES.ml diff --git a/src/tomega/Tomegaprint_intf.ml b/src/Tomega/TOMEGAPRINT.ml similarity index 100% rename from src/tomega/Tomegaprint_intf.ml rename to src/Tomega/TOMEGAPRINT.ml diff --git a/src/tomega/TomegaTypecheck_intf.ml b/src/Tomega/TOMEGATYPECHECK.ml similarity index 100% rename from src/tomega/TomegaTypecheck_intf.ml rename to src/Tomega/TOMEGATYPECHECK.ml diff --git a/src/tomega/TomegaUnify_intf.ml b/src/Tomega/TOMEGAUNIFY.ml similarity index 100% rename from src/tomega/TomegaUnify_intf.ml rename to src/Tomega/TOMEGAUNIFY.ml diff --git a/src/tomega/Tomega.ml b/src/Tomega/Tomega.ml similarity index 100% rename from src/tomega/Tomega.ml rename to src/Tomega/Tomega.ml diff --git a/src/tomega/Tomega.mli b/src/Tomega/Tomega.mli similarity index 100% rename from src/tomega/Tomega.mli rename to src/Tomega/Tomega.mli diff --git a/src/tomega/TomegaAbstract.ml b/src/Tomega/TomegaAbstract.ml similarity index 99% rename from src/tomega/TomegaAbstract.ml rename to src/Tomega/TomegaAbstract.ml index 896a4f4..7e7f282 100644 --- a/src/tomega/TomegaAbstract.ml +++ b/src/Tomega/TomegaAbstract.ml @@ -4,7 +4,7 @@ module Tomega = Lambda_.Tomega (* Abstraction mechanisms form programs and formulas *) (* Author: Carsten Schuermann *) -include TomegaAbstract_intf +include TOMEGAABSTRACT (* Signature TOMEGAABSTRACT *) (* # 1 "src/tomega/TomegaAbstract.fun.ml" *) diff --git a/src/tomega/TomegaAbstract.mli b/src/Tomega/TomegaAbstract.mli similarity index 91% rename from src/tomega/TomegaAbstract.mli rename to src/Tomega/TomegaAbstract.mli index 2ea4ff9..bd666b0 100644 --- a/src/tomega/TomegaAbstract.mli +++ b/src/Tomega/TomegaAbstract.mli @@ -1,4 +1,4 @@ -include module type of TomegaAbstract_intf +include module type of TOMEGAABSTRACT module TomegaAbstract (TomegaAbstract__0 : sig (* Converter from relational representation to a functional diff --git a/src/tomega/TomegaTypecheck.ml b/src/Tomega/TomegaTypecheck.ml similarity index 99% rename from src/tomega/TomegaTypecheck.ml rename to src/Tomega/TomegaTypecheck.ml index aa2c609..4760c50 100644 --- a/src/tomega/TomegaTypecheck.ml +++ b/src/Tomega/TomegaTypecheck.ml @@ -5,7 +5,7 @@ module Tomega = Lambda_.Tomega (* Type checking for functional proof term calculus *) (* Author: Carsten Schuermann *) (* Modified: Yu Liao *) -include TomegaTypecheck_intf +include TOMEGATYPECHECK (* Signature TOMEGATYPECHECK *) (* # 1 "src/tomega/TomegaTypecheck.fun.ml" *) @@ -22,8 +22,8 @@ module TomegaTypeCheck (TomegaTypeCheck__0 : sig module Print : PRINT module TomegaPrint : Tomegaprint.TOMEGAPRINT module Subordinate : Subordinate.Subordinate_.SUBORDINATE - module Weaken : Weaken_intf.WEAKEN - module TomegaAbstract : TomegaAbstract_intf.TOMEGAABSTRACT + module Weaken : WEAKEN.WEAKEN + module TomegaAbstract : TOMEGAABSTRACT.TOMEGAABSTRACT end) : TOMEGATYPECHECK = struct (*! structure IntSyn = IntSyn' !*) (*! structure Tomega = Tomega' !*) diff --git a/src/tomega/TomegaTypecheck.mli b/src/Tomega/TomegaTypecheck.mli similarity index 73% rename from src/tomega/TomegaTypecheck.mli rename to src/Tomega/TomegaTypecheck.mli index bb317e2..3c0a428 100644 --- a/src/tomega/TomegaTypecheck.mli +++ b/src/Tomega/TomegaTypecheck.mli @@ -1,4 +1,4 @@ -include module type of TomegaTypecheck_intf +include module type of TOMEGATYPECHECK module TomegaTypeCheck (TomegaTypeCheck__0 : sig (* Type checking for Tomega *) @@ -11,6 +11,6 @@ module TomegaTypeCheck (TomegaTypeCheck__0 : sig module Print : PRINT module TomegaPrint : Tomegaprint.TOMEGAPRINT module Subordinate : Subordinate.Subordinate_.SUBORDINATE - module Weaken : Weaken_intf.WEAKEN - module TomegaAbstract : TomegaAbstract_intf.TOMEGAABSTRACT + module Weaken : WEAKEN.WEAKEN + module TomegaAbstract : TOMEGAABSTRACT.TOMEGAABSTRACT end) : TOMEGATYPECHECK diff --git a/src/tomega/TomegaUnify.ml b/src/Tomega/TomegaUnify.ml similarity index 98% rename from src/tomega/TomegaUnify.ml rename to src/Tomega/TomegaUnify.ml index 629127d..ad9787f 100644 --- a/src/tomega/TomegaUnify.ml +++ b/src/Tomega/TomegaUnify.ml @@ -4,7 +4,7 @@ module Tomega = Lambda_.Tomega (* Unification on Formulas *) (* Author: Carsten Schuermann *) -include TomegaUnify_intf +include TOMEGAUNIFY (* Signature TOMEGATYPECHECK *) (* # 1 "src/tomega/TomegaUnify.fun.ml" *) @@ -42,7 +42,7 @@ module TomegaUnify (TomegaUnify__0 : sig module Subordinate : Subordinate.Subordinate_.SUBORDINATE (*! sharing Subordinate.IntSyn = IntSyn' !*) - module Weaken : Weaken_intf.WEAKEN + module Weaken : WEAKEN.WEAKEN end) : TOMEGAUNIFY = struct (*! structure IntSyn = IntSyn' !*) (*! structure Tomega = Tomega' !*) diff --git a/src/tomega/TomegaUnify.mli b/src/Tomega/TomegaUnify.mli similarity index 92% rename from src/tomega/TomegaUnify.mli rename to src/Tomega/TomegaUnify.mli index fe1c6be..9a5d05e 100644 --- a/src/tomega/TomegaUnify.mli +++ b/src/Tomega/TomegaUnify.mli @@ -1,4 +1,4 @@ -include module type of TomegaUnify_intf +include module type of TOMEGAUNIFY module TomegaUnify (TomegaUnify__0 : sig (* Unification on Formulas *) @@ -32,5 +32,5 @@ module TomegaUnify (TomegaUnify__0 : sig module Subordinate : Subordinate.Subordinate_.SUBORDINATE (*! sharing Subordinate.IntSyn = IntSyn' !*) - module Weaken : Weaken_intf.WEAKEN + module Weaken : WEAKEN.WEAKEN end) : TOMEGAUNIFY diff --git a/src/tomega/Tomega_.ml b/src/Tomega/Tomega_.ml similarity index 100% rename from src/tomega/Tomega_.ml rename to src/Tomega/Tomega_.ml diff --git a/src/Tomega/Tomega_.mli b/src/Tomega/Tomega_.mli new file mode 100644 index 0000000..c4b0e20 --- /dev/null +++ b/src/Tomega/Tomega_.mli @@ -0,0 +1,18 @@ +(* # 1 "src/tomega/Tomega_.sig.ml" *) + +(* # 1 "src/tomega/Tomega_.fun.ml" *) + +(* # 1 "src/tomega/Tomega_.sml.ml" *) +open! Basis +module Tomega : module type of Lambda_.Tomega +module TomegaAbstract : TOMEGAABSTRACT.TOMEGAABSTRACT +module TomegaPrint : Tomegaprint.TOMEGAPRINT +module Weaken : WEAKEN.WEAKEN +module TomegaTypeCheck : TOMEGATYPECHECK.TOMEGATYPECHECK +module Opsem_ : OPSEM.OPSEM +module Opsem : OPSEM.OPSEM +module Redundant : REDUNDANT.REDUNDANT +module Converter_ : CONVERTER.CONVERTER +module Converter : CONVERTER.CONVERTER +module TomegaCoverage_ : COVERAGE.TOMEGACOVERAGE +module TomegaCoverage : COVERAGE.TOMEGACOVERAGE diff --git a/src/tomega/Tomeganames.ml b/src/Tomega/Tomeganames.ml similarity index 96% rename from src/tomega/Tomeganames.ml rename to src/Tomega/Tomeganames.ml index 72ce933..55d5467 100644 --- a/src/tomega/Tomeganames.ml +++ b/src/Tomega/Tomeganames.ml @@ -4,7 +4,7 @@ module Tomega = Lambda_.Tomega (* Naming *) (* Author: Carsten Schuermann *) -include Tomeganames_intf +include TOMEGANAMES (* # 1 "src/tomega/Tomeganames.fun.ml" *) open! Basis diff --git a/src/Tomega/Tomeganames.mli b/src/Tomega/Tomeganames.mli new file mode 100644 index 0000000..890d7d9 --- /dev/null +++ b/src/Tomega/Tomeganames.mli @@ -0,0 +1,2 @@ +include module type of TOMEGANAMES +module TomegaNames : TOMEGANAMES diff --git a/src/tomega/Tomegaprint.ml b/src/Tomega/Tomegaprint.ml similarity index 99% rename from src/tomega/Tomegaprint.ml rename to src/Tomega/Tomegaprint.ml index f850934..61c1215 100644 --- a/src/tomega/Tomegaprint.ml +++ b/src/Tomega/Tomegaprint.ml @@ -4,7 +4,7 @@ module Tomega = Lambda_.Tomega (* Printing of functional proof terms *) (* Author: Carsten Schuermann *) -include Tomegaprint_intf +include TOMEGAPRINT (* val lemmaDecToString : FunSyn.LemmaDec -> string *) (* signature TOMEGAPRINT *) diff --git a/src/tomega/Tomegaprint.mli b/src/Tomega/Tomegaprint.mli similarity index 92% rename from src/tomega/Tomegaprint.mli rename to src/Tomega/Tomegaprint.mli index a9d0a2f..c280d5e 100644 --- a/src/tomega/Tomegaprint.mli +++ b/src/Tomega/Tomegaprint.mli @@ -1,4 +1,4 @@ -include module type of Tomegaprint_intf +include module type of TOMEGAPRINT module TomegaPrint (TomegaPrint__0 : sig (*! structure IntSyn' : INTSYN !*) diff --git a/src/tomega/dune b/src/Tomega/dune similarity index 85% rename from src/tomega/dune rename to src/Tomega/dune index 2d33d67..150640f 100644 --- a/src/tomega/dune +++ b/src/Tomega/dune @@ -15,16 +15,16 @@ Redundant TomegaAbstract Tomega_ - Converter_intf - Coverage_intf - Normalize_intf - Opsem_intf - Redundant_intf - TomegaAbstract_intf - TomegaTypecheck_intf - TomegaUnify_intf - Tomeganames_intf - Tomegaprint_intf) + CONVERTER + COVERAGE + NORMALIZE + OPSEM + REDUNDANT + TOMEGAABSTRACT + TOMEGATYPECHECK + TOMEGAUNIFY + TOMEGANAMES + TOMEGAPRINT) (libraries display global diff --git a/src/trail/Notrail.ml b/src/Trail/Notrail.ml similarity index 100% rename from src/trail/Notrail.ml rename to src/Trail/Notrail.ml diff --git a/src/trail/Notrail.mli b/src/Trail/Notrail.mli similarity index 100% rename from src/trail/Notrail.mli rename to src/Trail/Notrail.mli diff --git a/src/trail/README b/src/Trail/README similarity index 100% rename from src/trail/README rename to src/Trail/README diff --git a/src/trail/Trail_intf.ml b/src/Trail/TRAIL.ml similarity index 100% rename from src/trail/Trail_intf.ml rename to src/Trail/TRAIL.ml diff --git a/src/trail/Trail.ml b/src/Trail/Trail.ml similarity index 100% rename from src/trail/Trail.ml rename to src/Trail/Trail.ml diff --git a/src/trail/Trail.mli b/src/Trail/Trail.mli similarity index 100% rename from src/trail/Trail.mli rename to src/Trail/Trail.mli diff --git a/src/trail/Trail_.ml b/src/Trail/Trail_.ml similarity index 98% rename from src/trail/Trail_.ml rename to src/Trail/Trail_.ml index a3510bf..27de3fc 100644 --- a/src/trail/Trail_.ml +++ b/src/Trail/Trail_.ml @@ -4,7 +4,7 @@ open Basis (* Trailing Abstract Operations *) -include Trail_intf +include TRAIL (** Author: Roberto Virga *) (* signature TRAIL *) diff --git a/src/Trail/Trail_.mli b/src/Trail/Trail_.mli new file mode 100644 index 0000000..078c408 --- /dev/null +++ b/src/Trail/Trail_.mli @@ -0,0 +1,2 @@ +include module type of TRAIL +module Trail : TRAIL diff --git a/src/trail/dune b/src/Trail/dune similarity index 83% rename from src/trail/dune rename to src/Trail/dune index 40487df..3dae12b 100644 --- a/src/trail/dune +++ b/src/Trail/dune @@ -3,7 +3,7 @@ (library (name trail) (public_name stelf.trail) - (modules Trail_ Notrail Trail_intf) + (modules Trail_ Notrail TRAIL) (libraries display basis) (wrapped true) (preprocess diff --git a/src/typecheck/README b/src/Typecheck/README similarity index 100% rename from src/typecheck/README rename to src/Typecheck/README diff --git a/src/typecheck/Strict_intf.ml b/src/Typecheck/STRICT.ml similarity index 100% rename from src/typecheck/Strict_intf.ml rename to src/Typecheck/STRICT.ml diff --git a/src/typecheck/Strict.ml b/src/Typecheck/Strict.ml similarity index 99% rename from src/typecheck/Strict.ml rename to src/Typecheck/Strict.ml index 52cf618..eac85e3 100644 --- a/src/typecheck/Strict.ml +++ b/src/Typecheck/Strict.ml @@ -3,7 +3,7 @@ open! Basis (* Checking Definitions for Strictness *) (* Author: Carsten Schuermann *) -include Strict_intf +include STRICT (* signature STRICT *) (* # 1 "src/typecheck/Strict.fun.ml" *) diff --git a/src/typecheck/Strict.mli b/src/Typecheck/Strict.mli similarity index 65% rename from src/typecheck/Strict.mli rename to src/Typecheck/Strict.mli index 5cf71ac..329570f 100644 --- a/src/typecheck/Strict.mli +++ b/src/Typecheck/Strict.mli @@ -1,4 +1,4 @@ -include module type of Strict_intf +include module type of STRICT module Strict (Strict__0 : sig module Whnf : WHNF diff --git a/src/typecheck/Typecheck_intf.ml b/src/Typecheck/TYPECHECK.ml similarity index 100% rename from src/typecheck/Typecheck_intf.ml rename to src/Typecheck/TYPECHECK.ml diff --git a/src/typecheck/Typecheck.ml b/src/Typecheck/Typecheck.ml similarity index 100% rename from src/typecheck/Typecheck.ml rename to src/Typecheck/Typecheck.ml diff --git a/src/typecheck/Typecheck.mli b/src/Typecheck/Typecheck.mli similarity index 100% rename from src/typecheck/Typecheck.mli rename to src/Typecheck/Typecheck.mli diff --git a/src/typecheck/Typecheck_.ml b/src/Typecheck/Typecheck_.ml similarity index 99% rename from src/typecheck/Typecheck_.ml rename to src/Typecheck/Typecheck_.ml index 137fd3e..9b41d8f 100644 --- a/src/typecheck/Typecheck_.ml +++ b/src/Typecheck/Typecheck_.ml @@ -3,7 +3,7 @@ open! Basis (* Type Checking *) -include Typecheck_intf +include TYPECHECK (** Author: Carsten Schuermann *) (* signature TYPECHECK *) diff --git a/src/typecheck/Typecheck_.mli b/src/Typecheck/Typecheck_.mli similarity index 90% rename from src/typecheck/Typecheck_.mli rename to src/Typecheck/Typecheck_.mli index 63cccb3..05e7e37 100644 --- a/src/typecheck/Typecheck_.mli +++ b/src/Typecheck/Typecheck_.mli @@ -1,4 +1,4 @@ -include module type of Typecheck_intf +include module type of TYPECHECK module MakeTypeCheck (Conv : CONV) diff --git a/src/typecheck/WALK b/src/Typecheck/WALK similarity index 100% rename from src/typecheck/WALK rename to src/Typecheck/WALK diff --git a/src/typecheck/dune b/src/Typecheck/dune similarity index 89% rename from src/typecheck/dune rename to src/Typecheck/dune index 6d58ac9..2d7bf78 100644 --- a/src/typecheck/dune +++ b/src/Typecheck/dune @@ -3,7 +3,7 @@ (library (name typecheck) (public_name stelf.typecheck) - (modules Typecheck_ Strict Strict_intf Typecheck_intf) + (modules Typecheck_ Strict STRICT TYPECHECK) (libraries display intsyn print paths names basis) (flags (:standard diff --git a/src/unique/Unique_intf.ml b/src/Unique/UNIQUE.ml similarity index 100% rename from src/unique/Unique_intf.ml rename to src/Unique/UNIQUE.ml diff --git a/src/unique/Unique.ml b/src/Unique/Unique.ml similarity index 100% rename from src/unique/Unique.ml rename to src/Unique/Unique.ml diff --git a/src/unique/Unique.mli b/src/Unique/Unique.mli similarity index 100% rename from src/unique/Unique.mli rename to src/Unique/Unique.mli diff --git a/src/unique/Unique_.ml b/src/Unique/Unique_.ml similarity index 99% rename from src/unique/Unique_.ml rename to src/Unique/Unique_.ml index ffb8e1d..cbf49c5 100644 --- a/src/unique/Unique_.ml +++ b/src/Unique/Unique_.ml @@ -3,7 +3,7 @@ open! Basis (* Uniqueness Checking *) -include Unique_intf +include UNIQUE (** Author: Frank Pfenning *) (* raises Error(msg) *) diff --git a/src/unique/Unique_.mli b/src/Unique/Unique_.mli similarity index 94% rename from src/unique/Unique_.mli rename to src/Unique/Unique_.mli index 5cdb460..2a6c5d3 100644 --- a/src/unique/Unique_.mli +++ b/src/Unique/Unique_.mli @@ -1,4 +1,4 @@ -include module type of Unique_intf +include module type of UNIQUE module MakeUnique (Global : GLOBAL) diff --git a/src/unique/dune b/src/Unique/dune similarity index 97% rename from src/unique/dune rename to src/Unique/dune index 30f1722..abca2cd 100644 --- a/src/unique/dune +++ b/src/Unique/dune @@ -3,7 +3,7 @@ (library (name unique) (public_name stelf.unique) - (modules Unique_ Unique_intf) + (modules Unique_ UNIQUE) (libraries display global diff --git a/src/worldcheck/README b/src/Worldcheck/README similarity index 100% rename from src/worldcheck/README rename to src/Worldcheck/README diff --git a/src/worldcheck/TODO b/src/Worldcheck/TODO similarity index 100% rename from src/worldcheck/TODO rename to src/Worldcheck/TODO diff --git a/src/worldcheck/Worldify_intf.ml b/src/Worldcheck/WORLDIFY.ml similarity index 100% rename from src/worldcheck/Worldify_intf.ml rename to src/Worldcheck/WORLDIFY.ml diff --git a/src/worldcheck/WorldSyn_intf.ml b/src/Worldcheck/WORLDSYN.ml similarity index 100% rename from src/worldcheck/WorldSyn_intf.ml rename to src/Worldcheck/WORLDSYN.ml diff --git a/src/worldcheck/WorldSyn.ml b/src/Worldcheck/WorldSyn.ml similarity index 99% rename from src/worldcheck/WorldSyn.ml rename to src/Worldcheck/WorldSyn.ml index 73234c1..1672a27 100644 --- a/src/worldcheck/WorldSyn.ml +++ b/src/Worldcheck/WorldSyn.ml @@ -3,7 +3,7 @@ open! Basis (* World Checking *) (* Author: Carsten Schuermann *) -include WorldSyn_intf +include WORLDSYN (* signature WORLDSYN *) (* # 1 "src/worldcheck/WorldSyn.fun.ml" *) @@ -46,11 +46,11 @@ module WorldSyn (WorldSyn__0 : sig module Table : TABLE with type key = int (*! structure Paths : PATHS !*) - module Origins : Origins_intf.ORIGINS + module Origins : ORIGINS.ORIGINS (*! sharing Origins.Paths = Paths !*) (*! sharing Origins.IntSyn = IntSyn !*) - module Timers : Timers_intf.TIMERS + module Timers : TIMERS.TIMERS end) : WORLDSYN = struct module Origins = WorldSyn__0.Origins module Subordinate = WorldSyn__0.Subordinate diff --git a/src/worldcheck/WorldSyn.mli b/src/Worldcheck/WorldSyn.mli similarity index 88% rename from src/worldcheck/WorldSyn.mli rename to src/Worldcheck/WorldSyn.mli index e91c4b1..de7a374 100644 --- a/src/worldcheck/WorldSyn.mli +++ b/src/Worldcheck/WorldSyn.mli @@ -1,4 +1,4 @@ -include module type of WorldSyn_intf +include module type of WORLDSYN module WorldSyn (WorldSyn__0 : sig module Global : GLOBAL @@ -31,9 +31,9 @@ module WorldSyn (WorldSyn__0 : sig module Table : TABLE with type key = int (*! structure Paths : PATHS !*) - module Origins : Origins_intf.ORIGINS + module Origins : ORIGINS.ORIGINS (*! sharing Origins.Paths = Paths !*) (*! sharing Origins.IntSyn = IntSyn !*) - module Timers : Timers_intf.TIMERS + module Timers : TIMERS.TIMERS end) : WORLDSYN diff --git a/src/worldcheck/Worldcheck.ml b/src/Worldcheck/Worldcheck.ml similarity index 100% rename from src/worldcheck/Worldcheck.ml rename to src/Worldcheck/Worldcheck.ml diff --git a/src/worldcheck/Worldcheck.mli b/src/Worldcheck/Worldcheck.mli similarity index 100% rename from src/worldcheck/Worldcheck.mli rename to src/Worldcheck/Worldcheck.mli diff --git a/src/worldcheck/Worldcheck_.ml b/src/Worldcheck/Worldcheck_.ml similarity index 100% rename from src/worldcheck/Worldcheck_.ml rename to src/Worldcheck/Worldcheck_.ml diff --git a/src/worldcheck/Worldcheck_.mli b/src/Worldcheck/Worldcheck_.mli similarity index 69% rename from src/worldcheck/Worldcheck_.mli rename to src/Worldcheck/Worldcheck_.mli index 2685575..51933a5 100644 --- a/src/worldcheck/Worldcheck_.mli +++ b/src/Worldcheck/Worldcheck_.mli @@ -5,8 +5,8 @@ (* # 1 "src/worldcheck/Worldcheck_.sml.ml" *) open! Basis -module type WORLDIFY = Worldify_intf.WORLDIFY -module type WORLDSYN = WorldSyn_intf.WORLDSYN +module type WORLDIFY = WORLDIFY.WORLDIFY +module type WORLDSYN = WORLDSYN.WORLDSYN module WorldSyn : WORLDSYN module Worldify : WORLDIFY diff --git a/src/worldcheck/Worldify.ml b/src/Worldcheck/Worldify.ml similarity index 99% rename from src/worldcheck/Worldify.ml rename to src/Worldcheck/Worldify.ml index a42feb4..3de51c7 100644 --- a/src/worldcheck/Worldify.ml +++ b/src/Worldcheck/Worldify.ml @@ -3,7 +3,7 @@ open! Basis (* Worldify *) (* Author: Carsten Schuermann *) -include Worldify_intf +include WORLDIFY (* val check : Tomega.Worlds -> IntSyn.cid list -> unit val closure : Tomega.Worlds -> Tomega.Worlds *) diff --git a/src/worldcheck/Worldify.mli b/src/Worldcheck/Worldify.mli similarity index 96% rename from src/worldcheck/Worldify.mli rename to src/Worldcheck/Worldify.mli index 8a22cb3..056a7d5 100644 --- a/src/worldcheck/Worldify.mli +++ b/src/Worldcheck/Worldify.mli @@ -1,4 +1,4 @@ -include module type of Worldify_intf +include module type of WORLDIFY module Worldify (Worldify__0 : sig module Global : GLOBAL diff --git a/src/worldcheck/dune b/src/Worldcheck/dune similarity index 92% rename from src/worldcheck/dune rename to src/Worldcheck/dune index 67f5762..9891899 100644 --- a/src/worldcheck/dune +++ b/src/Worldcheck/dune @@ -3,7 +3,7 @@ (library (name worldcheck) (public_name stelf.worldcheck) - (modules WorldSyn Worldify Worldcheck_ WorldSyn_intf Worldify_intf) + (modules WorldSyn Worldify Worldcheck_ WORLDSYN WORLDIFY) (libraries display global diff --git a/src/compress/Sgn.mli b/src/compress/Sgn.mli deleted file mode 100644 index 5a727d7..0000000 --- a/src/compress/Sgn.mli +++ /dev/null @@ -1,3 +0,0 @@ -include module type of Sgn_intf -module Sgn : SGN -include SGN diff --git a/src/domains/Field.mli b/src/domains/Field.mli deleted file mode 100644 index a0417ec..0000000 --- a/src/domains/Field.mli +++ /dev/null @@ -1 +0,0 @@ -include module type of Field_intf diff --git a/src/domains/OrderedField.mli b/src/domains/OrderedField.mli deleted file mode 100644 index b466f29..0000000 --- a/src/domains/OrderedField.mli +++ /dev/null @@ -1 +0,0 @@ -include module type of OrderedField_intf diff --git a/src/flit/Flit_.mli b/src/flit/Flit_.mli deleted file mode 100644 index c4bc2f9..0000000 --- a/src/flit/Flit_.mli +++ /dev/null @@ -1 +0,0 @@ -include module type of Flit_intf diff --git a/src/formatter/Formatter_.mli b/src/formatter/Formatter_.mli deleted file mode 100644 index 39b57dc..0000000 --- a/src/formatter/Formatter_.mli +++ /dev/null @@ -1,2 +0,0 @@ -include module type of Formatter_intf -module Formatter : FORMATTER diff --git a/src/frontend/Frontend_.mli b/src/frontend/Frontend_.mli deleted file mode 100644 index 0f09c03..0000000 --- a/src/frontend/Frontend_.mli +++ /dev/null @@ -1,61 +0,0 @@ -(* # 1 "src/frontend/Frontend_.sig.ml" *) - -(* # 1 "src/frontend/Frontend_.fun.ml" *) - -(* # 1 "src/frontend/Frontend_.sml.ml" *) -open! Basis - -(* Front End Interface *) -(* Author: Frank Pfenning *) -(* Presently, we do not memoize the token stream returned *) -(* by the Lexer. Use Stream = MStream below if memoization becomes *) -(* necessary. *) -(* Now in Lexer.fun *) -(* -structure Lexer = - Lexer (structure Stream' = Stream - structure Paths' = Paths); -*) -(* Now in Parsing.fun *) -(* -structure Parsing = - Parsing (structure Stream' = Stream - structure Lexer' = Lexer); -*) -(* Re-export module type before Stelf name shadowing. *) -module type LEXER = Lexer.LEXER -module type STELF = Twelf_.STELF - -module ReconTerm : ReconTerm_intf.RECON_TERM -module ReconConDec : ReconCondec_intf.RECON_CONDEC -module ReconQuery : ReconQuery_intf.RECON_QUERY -module ReconMode : ReconMode_intf.RECON_MODE -module ReconThm : ReconThm_intf.RECON_THM -module ReconModule : ReconModule_intf.RECON_MODULE -module ParseTerm : ParseTerm_intf.PARSE_TERM - -module ParseTermConDec : - ParseTerm_intf.PARSE_TERM with module ExtSyn = ReconConDec.ExtSyn - -module ParseTermQuery : - ParseTerm_intf.PARSE_TERM with module ExtSyn = ReconQuery.ExtSyn - -module ParseTermMode : - ParseTerm_intf.PARSE_TERM with module ExtSyn = ReconMode.ExtSyn - -module ParseTermThm : - ParseTerm_intf.PARSE_TERM with module ExtSyn = ReconThm.ExtSyn - -module ParseTermModule : - ParseTerm_intf.PARSE_TERM with module ExtSyn = ReconModule.ExtSyn - -module ParseConDec : ParseCondec_intf.PARSE_CONDEC -module ParseQuery : ParseQuery_intf.PARSE_QUERY -module ParseFixity : ParseFixity_intf.PARSE_FIXITY with module Names = Names -module ParseMode : ParseMode_intf.PARSE_MODE -module ParseThm : ParseThm_intf.PARSE_THM -module ParseModule : ParseModule_intf.PARSE_MODULE -module Parser : Parser_intf.PARSER -module Solve : Solve_intf.SOLVE with module ExtQuery = ReconQuery -module Fquery : Fquery_intf.FQUERY with module ExtQuery = ReconQuery -module Stelf : STELF diff --git a/src/frontend/ParseCondec.mli b/src/frontend/ParseCondec.mli deleted file mode 100644 index 58038b1..0000000 --- a/src/frontend/ParseCondec.mli +++ /dev/null @@ -1,9 +0,0 @@ -include module type of ParseCondec_intf - -module ParseConDec (ParseConDec__0 : sig - (*! structure Parsing' : PARSING !*) - module ExtConDec' : ReconCondec_intf.EXTCONDEC - - module ParseTerm : - ParseTerm_intf.PARSE_TERM with module ExtSyn = ExtConDec'.ExtSyn -end) : PARSE_CONDEC with module ExtConDec = ParseConDec__0.ExtConDec' diff --git a/src/global/Global_.mli b/src/global/Global_.mli deleted file mode 100644 index 4c9f00d..0000000 --- a/src/global/Global_.mli +++ /dev/null @@ -1,2 +0,0 @@ -include module type of Global_intf -module Global : GLOBAL diff --git a/src/heuristic/Heuristic_.mli b/src/heuristic/Heuristic_.mli deleted file mode 100644 index 0226403..0000000 --- a/src/heuristic/Heuristic_.mli +++ /dev/null @@ -1,2 +0,0 @@ -include module type of Heuristic_intf -module Heuristic : HEURISTIC diff --git a/src/intInf/IntInfSig.mli b/src/intInf/IntInfSig.mli deleted file mode 100644 index 66c046b..0000000 --- a/src/intInf/IntInfSig.mli +++ /dev/null @@ -1 +0,0 @@ -include module type of IntInfSig_intf diff --git a/src/m2/Init.mli b/src/m2/Init.mli deleted file mode 100644 index d600444..0000000 --- a/src/m2/Init.mli +++ /dev/null @@ -1,8 +0,0 @@ -include module type of Init_intf - -module Init (Init__0 : sig - module MetaSyn' : Metasyn.METASYN - - module MetaAbstract : - MetaAbstract_intf.METAABSTRACT with module MetaSyn = MetaSyn' -end) : INIT with module MetaSyn = Init__0.MetaSyn' diff --git a/src/m2/Lemma.mli b/src/m2/Lemma.mli deleted file mode 100644 index 718ea97..0000000 --- a/src/m2/Lemma.mli +++ /dev/null @@ -1,8 +0,0 @@ -include module type of Lemma_intf - -module Lemma (Lemma__0 : sig - module MetaSyn' : Metasyn.METASYN - - module MetaAbstract : - MetaAbstract_intf.METAABSTRACT with module MetaSyn = MetaSyn' -end) : LEMMA with module MetaSyn = Lemma__0.MetaSyn' diff --git a/src/m2/MetaGlobal.mli b/src/m2/MetaGlobal.mli deleted file mode 100644 index 58f87ce..0000000 --- a/src/m2/MetaGlobal.mli +++ /dev/null @@ -1,2 +0,0 @@ -include module type of MetaGlobal_intf -module MetaGlobal : MetaGlobal_intf.METAGLOBAL diff --git a/src/m2/Mpi.mli b/src/m2/Mpi.mli deleted file mode 100644 index 7a9a4c6..0000000 --- a/src/m2/Mpi.mli +++ /dev/null @@ -1,19 +0,0 @@ -include module type of Mpi_intf - -module Mpi (Mpi__0 : sig - module MetaGlobal : MetaGlobal_intf.METAGLOBAL - module MetaSyn' : Metasyn.METASYN - module Init : Init_intf.INIT with module MetaSyn = MetaSyn' - module Filling : Filling_intf.FILLING with module MetaSyn = MetaSyn' - module Splitting : Splitting_intf.SPLITTING with module MetaSyn = MetaSyn' - module Recursion : Recursion_intf.RECURSION with module MetaSyn = MetaSyn' - module Lemma : Lemma_intf.LEMMA with module MetaSyn = MetaSyn' - module Strategy : Strategy_intf.STRATEGY with module MetaSyn = MetaSyn' - module Qed : Qed_intf.QED with module MetaSyn = MetaSyn' - module MetaPrint : MetaPrint_intf.METAPRINT with module MetaSyn = MetaSyn' - module Names : NAMES - - (*! sharing Names.IntSyn = MetaSyn'.IntSyn !*) - module Timers : Timers_intf.TIMERS - module Ring : Ring_intf.RING -end) : MPI with module MetaSyn = Mpi__0.MetaSyn' diff --git a/src/m2/Prover.mli b/src/m2/Prover.mli deleted file mode 100644 index 075fc5a..0000000 --- a/src/m2/Prover.mli +++ /dev/null @@ -1,17 +0,0 @@ -include module type of Prover_intf - -module Prover (Prover__0 : sig - module MetaGlobal : MetaGlobal_intf.METAGLOBAL - module MetaSyn' : Metasyn.METASYN - module Init : Init_intf.INIT with module MetaSyn = MetaSyn' - module Strategy : Strategy_intf.STRATEGY with module MetaSyn = MetaSyn' - module Filling : Filling_intf.FILLING with module MetaSyn = MetaSyn' - module Splitting : Splitting_intf.SPLITTING with module MetaSyn = MetaSyn' - module Recursion : Recursion_intf.RECURSION with module MetaSyn = MetaSyn' - module Qed : Qed_intf.QED with module MetaSyn = MetaSyn' - module MetaPrint : MetaPrint_intf.METAPRINT with module MetaSyn = MetaSyn' - module Names : NAMES - - (*! sharing Names.IntSyn = MetaSyn'.IntSyn !*) - module Timers : Timers_intf.TIMERS -end) : PROVER diff --git a/src/m2/Strategy.mli b/src/m2/Strategy.mli deleted file mode 100644 index 383ee69..0000000 --- a/src/m2/Strategy.mli +++ /dev/null @@ -1,32 +0,0 @@ -include module type of Strategy_intf - -module StrategyFRS (StrategyFRS__0 : sig - module MetaGlobal : MetaGlobal_intf.METAGLOBAL - module MetaSyn' : Metasyn.METASYN - module Filling : Filling_intf.FILLING with module MetaSyn = MetaSyn' - module Splitting : Splitting_intf.SPLITTING with module MetaSyn = MetaSyn' - module Recursion : Recursion_intf.RECURSION with module MetaSyn = MetaSyn' - module Lemma : Lemma_intf.LEMMA with module MetaSyn = MetaSyn' - module Qed : Qed_intf.QED with module MetaSyn = MetaSyn' - module MetaPrint : MetaPrint_intf.METAPRINT with module MetaSyn = MetaSyn' - module Timers : Timers_intf.TIMERS -end) : Strategy_intf.STRATEGY with module MetaSyn = StrategyFRS__0.MetaSyn' - -module StrategyRFS (StrategyRFS__1 : sig - module MetaGlobal : MetaGlobal_intf.METAGLOBAL - module MetaSyn' : Metasyn.METASYN - module Filling : Filling_intf.FILLING with module MetaSyn = MetaSyn' - module Splitting : Splitting_intf.SPLITTING with module MetaSyn = MetaSyn' - module Recursion : Recursion_intf.RECURSION with module MetaSyn = MetaSyn' - module Lemma : Lemma_intf.LEMMA with module MetaSyn = MetaSyn' - module Qed : Qed_intf.QED with module MetaSyn = MetaSyn' - module MetaPrint : MetaPrint_intf.METAPRINT with module MetaSyn = MetaSyn' - module Timers : Timers_intf.TIMERS -end) : Strategy_intf.STRATEGY with module MetaSyn = StrategyRFS__1.MetaSyn' - -module Strategy (Strategy__2 : sig - module MetaGlobal : MetaGlobal_intf.METAGLOBAL - module MetaSyn' : Metasyn.METASYN - module StrategyFRS : Strategy_intf.STRATEGY with module MetaSyn = MetaSyn' - module StrategyRFS : Strategy_intf.STRATEGY with module MetaSyn = MetaSyn' -end) : Strategy_intf.STRATEGY with module MetaSyn = Strategy__2.MetaSyn' diff --git a/src/meta/Funweaken.mli b/src/meta/Funweaken.mli deleted file mode 100644 index c758736..0000000 --- a/src/meta/Funweaken.mli +++ /dev/null @@ -1,5 +0,0 @@ -include module type of Funweaken_intf - -module FunWeaken (FunWeaken__0 : sig - module Weaken : Weaken_intf.WEAKEN -end) : Funweaken_intf.FUNWEAKEN diff --git a/src/meta/Interpret.mli b/src/meta/Interpret.mli deleted file mode 100644 index 159ce94..0000000 --- a/src/meta/Interpret.mli +++ /dev/null @@ -1 +0,0 @@ -include module type of Interpret_intf diff --git a/src/meta/MtpGlobal.mli b/src/meta/MtpGlobal.mli deleted file mode 100644 index b5ab872..0000000 --- a/src/meta/MtpGlobal.mli +++ /dev/null @@ -1,5 +0,0 @@ -include module type of MtpGlobal_intf - -module MTPGlobal (MTPGlobal__0 : sig - module MetaGlobal : MetaGlobal_intf.METAGLOBAL -end) : MTPGLOBAL diff --git a/src/meta/MtpStrategy.mli b/src/meta/MtpStrategy.mli deleted file mode 100644 index 1d7c04b..0000000 --- a/src/meta/MtpStrategy.mli +++ /dev/null @@ -1,13 +0,0 @@ -include module type of MtpStrategy_intf - -module MTPStrategy (MTPStrategy__0 : sig - module MTPGlobal : MtpGlobal.MTPGLOBAL - module StateSyn' : Statesyn_intf.STATESYN - module MTPFilling : MtpFilling_intf.MTPFILLING - module MTPData : MtpData_intf.MTPDATA - module MTPSplitting : MtpSplitting_intf.MTPSPLITTING - module MTPRecursion : MtpRecursion_intf.MTPRECURSION - module Inference : Inference_intf.INFERENCE - module MTPrint : MtpPrint_intf.MTPRINT - module Timers : Timers_intf.TIMERS -end) : MTPSTRATEGY diff --git a/src/meta/Weaken.mli b/src/meta/Weaken.mli deleted file mode 100644 index 036fb7a..0000000 --- a/src/meta/Weaken.mli +++ /dev/null @@ -1,2 +0,0 @@ -include module type of Weaken_intf -module Make_Weaken (Whnf : WHNF) : Weaken_intf.WEAKEN diff --git a/src/modes/Modedec.mli b/src/modes/Modedec.mli deleted file mode 100644 index 9fa49a1..0000000 --- a/src/modes/Modedec.mli +++ /dev/null @@ -1,2 +0,0 @@ -include module type of Modedec_intf -module MakeModeDec () : MODEDEC diff --git a/src/msg/Msg_.mli b/src/msg/Msg_.mli deleted file mode 100644 index c40d790..0000000 --- a/src/msg/Msg_.mli +++ /dev/null @@ -1,2 +0,0 @@ -include module type of Msg_intf -module Msg : MSG diff --git a/src/netserver/Netserver_.mli b/src/netserver/Netserver_.mli deleted file mode 100644 index 2f3a97d..0000000 --- a/src/netserver/Netserver_.mli +++ /dev/null @@ -1,2 +0,0 @@ -include module type of Netserver_intf -module NetServer : NETSERVER diff --git a/src/opsem/SwSubtree.mli b/src/opsem/SwSubtree.mli deleted file mode 100644 index 8f81299..0000000 --- a/src/opsem/SwSubtree.mli +++ /dev/null @@ -1,7 +0,0 @@ -include module type of SwSubtree_intf - -module SwMemoTable (SwMemoTable__0 : sig - (* structure TableParam : TABLEPARAM *) - module MemoTable : MemoTable_intf.MEMOTABLE - module MemoTableInst : MemoTable_intf.MEMOTABLE -end) : MemoTable_intf.MEMOTABLE diff --git a/src/prover/Data.mli b/src/prover/Data.mli deleted file mode 100644 index d794417..0000000 --- a/src/prover/Data.mli +++ /dev/null @@ -1,2 +0,0 @@ -include module type of Data_intf -module Data : DATA diff --git a/src/server/Server_.mli b/src/server/Server_.mli deleted file mode 100644 index 448e83c..0000000 --- a/src/server/Server_.mli +++ /dev/null @@ -1,2 +0,0 @@ -include module type of Server_intf -module Server : SERVER diff --git a/src/server/Sigint.mli b/src/server/Sigint.mli deleted file mode 100644 index dca1d90..0000000 --- a/src/server/Sigint.mli +++ /dev/null @@ -1 +0,0 @@ -include module type of Sigint_intf diff --git a/src/smlofnj/Smlofnj.mli b/src/smlofnj/Smlofnj.mli deleted file mode 100644 index f7a482d..0000000 --- a/src/smlofnj/Smlofnj.mli +++ /dev/null @@ -1,2 +0,0 @@ -include module type of Smlofnj_intf -module SMLofNJ : SML_OF_NJ diff --git a/src/solvers/Cs.mli b/src/solvers/Cs.mli deleted file mode 100644 index 6f742b2..0000000 --- a/src/solvers/Cs.mli +++ /dev/null @@ -1 +0,0 @@ -include module type of Cs_intf diff --git a/src/subordinate/Intset.mli b/src/subordinate/Intset.mli deleted file mode 100644 index 77c0f5b..0000000 --- a/src/subordinate/Intset.mli +++ /dev/null @@ -1,2 +0,0 @@ -include module type of Intset_intf -module IntSet : INTSET diff --git a/src/table/Queue.mli b/src/table/Queue.mli deleted file mode 100644 index 4f78f12..0000000 --- a/src/table/Queue.mli +++ /dev/null @@ -1,2 +0,0 @@ -include module type of Queue_intf -module Queue : QUEUE diff --git a/src/table/RedBlackSet.mli b/src/table/RedBlackSet.mli deleted file mode 100644 index 96f8244..0000000 --- a/src/table/RedBlackSet.mli +++ /dev/null @@ -1,2 +0,0 @@ -include module type of RedBlackSet_intf -module RBSet : RBSET diff --git a/src/table/Ring.mli b/src/table/Ring.mli deleted file mode 100644 index 12f93e6..0000000 --- a/src/table/Ring.mli +++ /dev/null @@ -1,2 +0,0 @@ -include module type of Ring_intf -module Ring : RING diff --git a/src/table/StringHash.mli b/src/table/StringHash.mli deleted file mode 100644 index d50c8b8..0000000 --- a/src/table/StringHash.mli +++ /dev/null @@ -1,2 +0,0 @@ -include module type of StringHash_intf -module StringHash : STRING_HASH diff --git a/src/table/Table_.mli b/src/table/Table_.mli deleted file mode 100644 index 8b561f9..0000000 --- a/src/table/Table_.mli +++ /dev/null @@ -1 +0,0 @@ -include module type of Table_intf diff --git a/src/tomega/Redundant.mli b/src/tomega/Redundant.mli deleted file mode 100644 index 5be0669..0000000 --- a/src/tomega/Redundant.mli +++ /dev/null @@ -1,5 +0,0 @@ -include module type of Redundant_intf - -module Redundant (Redundant__0 : sig - module Opsem : Opsem_intf.OPSEM -end) : REDUNDANT diff --git a/src/tomega/Tomega_.mli b/src/tomega/Tomega_.mli deleted file mode 100644 index caa5370..0000000 --- a/src/tomega/Tomega_.mli +++ /dev/null @@ -1,18 +0,0 @@ -(* # 1 "src/tomega/Tomega_.sig.ml" *) - -(* # 1 "src/tomega/Tomega_.fun.ml" *) - -(* # 1 "src/tomega/Tomega_.sml.ml" *) -open! Basis -module Tomega : module type of Lambda_.Tomega -module TomegaAbstract : TomegaAbstract_intf.TOMEGAABSTRACT -module TomegaPrint : Tomegaprint.TOMEGAPRINT -module Weaken : Weaken_intf.WEAKEN -module TomegaTypeCheck : TomegaTypecheck_intf.TOMEGATYPECHECK -module Opsem_ : Opsem_intf.OPSEM -module Opsem : Opsem_intf.OPSEM -module Redundant : Redundant_intf.REDUNDANT -module Converter_ : Converter_intf.CONVERTER -module Converter : Converter_intf.CONVERTER -module TomegaCoverage_ : Coverage_intf.TOMEGACOVERAGE -module TomegaCoverage : Coverage_intf.TOMEGACOVERAGE diff --git a/src/tomega/Tomeganames.mli b/src/tomega/Tomeganames.mli deleted file mode 100644 index e50e8e5..0000000 --- a/src/tomega/Tomeganames.mli +++ /dev/null @@ -1,2 +0,0 @@ -include module type of Tomeganames_intf -module TomegaNames : TOMEGANAMES diff --git a/src/trail/Trail_.mli b/src/trail/Trail_.mli deleted file mode 100644 index a166ef6..0000000 --- a/src/trail/Trail_.mli +++ /dev/null @@ -1,2 +0,0 @@ -include module type of Trail_intf -module Trail : TRAIL diff --git a/test/CHANGES.md b/test/CHANGES.md new file mode 100644 index 0000000..3bf69c1 --- /dev/null +++ b/test/CHANGES.md @@ -0,0 +1,236 @@ +# Test Directory Changes: HEAD@{4} → HEAD + +Two commits: `eec20f0` (added many more pal tests) and `3c972a4` (fixed pal tests). +Three files changed: `test/Pal/Common.ml`, `test/Pal/Cases.ml`, `test/Pal/Source.ml`. +Net diff: **+1408 / −78 lines**. + +--- + +## `test/Pal/Common.ml` — Test infrastructure refactor + +### `test` return type + +The `test` helper changed from producing a single Alcotest case per test suite to producing one case *per command*: + +``` +(* before *) +val test : ?skip:bool -> ?failure:bool -> string -> string list -> unit Alcotest.test_case + +(* after *) +val test : ?skip:bool -> ?failure:bool -> string -> string list -> string * unit Alcotest.test_case list +``` + +Each command in the input list becomes its own Alcotest case, named `" - 1"`, `" - 2"`, etc. This makes it possible to see *which* command in a multi-step test failed, rather than only knowing the suite name. + +### `tester` class + +A new stateful `tester` class wraps a `Pal.pal` instance: + +```ocaml +class tester = object + val mutable p = new Pal.Pal.pal + method run cmd = ... (* returns exn option *) + method reset () = p <- new Pal.Pal.pal +end +``` + +A single `tester` is created per `test` call and shared across all commands in that call. This is critical: STELF declarations are stateful — `%sort nat` in command 1 must still be in scope for `%term z nat` in command 2. + +### `has_failed` ref + +A `has_failed` ref is initialized to `false` and set to `true` when any command raises. Subsequent commands in the same suite check this flag and call `Alcotest.skip ()` if it is set. This gives fail-fast semantics within a suite while still individually registering (and skipping) remaining cases. + +### Backtrace capture + +On failure the exception backtrace is now captured (`Printexc.get_backtrace ()`) and included in both the `eprintf` output and the `Alcotest.failf` message, making it much easier to locate the source of a runtime error in the Pal frontend. + +--- + +## `test/Pal/Cases.ml` — Wiring refactor and new suites + +### Adaptation to new `test` return type + +All existing suite definitions were updated. The old pattern: + +```ocaml +[ test "name" src_list ] +``` + +became: + +```ocaml +List.concat [ snd (test "name" src_list) ] +``` + +`snd` strips the name string (the first component of the new return type), and `List.concat` flattens the list of per-command cases. + +### New `cases ()` definition (shadows the old one) + +A second `let cases () = ...` definition is added at the bottom of the file, shadowing the first. It uses a flat `Alcotest.run "PAL"` call with a single `test` invocation per conceptual example, passing all source chunks for that example as a single list. This is the entry point actually executed by the test binary. + +The new `cases ()` runs 25 named suites: + +`%term and %sort`, `FOL`, `ZF`, `Nats`, `S4`, `LAM`, `POLYLAM`, +`PROP-CALC`, `MINI-ML`, `ARITH`, `GUIDE-LISTS`, `TAPL-NAT`, +`LP-HORN-ND`, `CHURCH-ROSSER-LAM`, `CUT-ELIM-FORMULAS`, `GUIDE-ND`, +`CPSOCC-DSBNF`, `SMALL-STEP-LAM`, `CRARY-EXCON`, `CRARY-EXCON-REV`, +`TAPL-DEFS`, `SMALL-STEP-SYSF`, `SMALL-STEP-SYSF-ISO`, +`POPLMARK-1A`, `POPLMARK-2A` (known runtime failure — see below), `CCC`, `INCLL`, +`CRARY-LINEAR`, `CRARY-LINEARD`, `CRARY-MODAL`. + +--- + +## `test/Pal/Source.ml` — STELF source changes + +### A. Modifications to existing source strings + +These are bug fixes to STELF code that was already in the file. + +--- + +#### `fol2` — spurious implicit on `f1` removed + +```diff +-%term f1 {{A}} {T i} hil ((forall [x i] A x) imp (A T)) ++%term f1 {T i} hil ((forall [x i] A x) imp (A T)) +``` + +`A` is the predicate being universally quantified. In the original, `{{A}}` asked the elaborator to infer `A` as an implicit argument of `f1` itself — but `A` does not appear free in `f1`'s conclusion type `hil (...)` in a way that lets the elaborator solve for it at a call site. The fix promotes `A` to be inferred from the body of the `forall` binder instead, which reconstruction can handle. + +--- + +#### `fol4_1` — `hnd_n1` implicit arguments removed + +```diff +-%term hnd_n1 {{A B}} hilnd n1 (impi [u] impi [v] noti [p] [w] note (impe u w) p (impe v w)) ++%term hnd_n1 hilnd n1 (impi [u] impi [v] noti [p] [w] note (impe u w) p (impe v w)) +``` + +`A` and `B` are the formula types, but they do not appear in the proof term body — only in the type of `n1` (a Hilbert axiom). The `{{A B}}` declaration asked STELF to solve for them at call sites, but they are unconstrained by the arguments given. Removing the implicits lets reconstruction determine them solely from the type of `n1`. + +--- + +#### `fol4_1` — `hnd_f1` rewritten with binder type annotation + +```diff +-%term hnd_f1 {{A}} {T i} hilnd (f1 T) (impi [u] foralle u T) ++%term hnd_f1 {{A T}} hilnd (f1 T) (impi [u nd (forall [x i] A x)] foralle u T) +``` + +Two changes: +1. `T` is promoted from an explicit argument `{T i}` to an implicit `{{A T}}`. Since `T` appears in the conclusion `hilnd (f1 T) (...)`, reconstruction can infer it. +2. The lambda binder `[u]` in `impi` is annotated with its full type `nd (forall [x i] A x)`. Without this annotation, STELF's reconstruction could not determine the type of `u`, because `foralle u T` does not constrain what sort `u` inhabits. + +--- + +#### `fol4_2` — `%mode` changed to named-argument form + +```diff +-%mode hilnd %in %out ++%mode {%in X _} {%out Y _} hilnd X Y +``` + +STELF's `%mode` declaration requires explicit argument names when describing a relation with multiple arguments. The shorthand `%in`/`%out` positional form was not accepted; the named form `{%in X _} {%out Y _}` is required. The same fix applies to `ndhil` in `fol6_2`. + +--- + +#### `fol5` split into `fol5_1` + `fol5_2` + +The original `fol5` string is split into two parts concatenated as `fol5 = fol5_1 ^ fol5_2`. The split separates term declarations from meta-declarations (`%block`, `%mode`, `%worlds`, `%total`), allowing each chunk to be loaded independently if needed and making it easier to test them incrementally. + +Several fixes within `fol5`: + +**`ded_id` binder annotation:** +```diff +-%term ded_id {{A}} ded ([u] u) (mp (mp s k) k) ++%term ded_id {{A}} ded ([u hil A] u) (mp (mp s k) (%the (hil (A imp (A imp A))) k)) +``` +The identity deduction `[u] u` needed an explicit type `hil A` on `u` so reconstruction knows which sort the identity proof inhabits. The `%the` annotation on `k` pins the type of the axiom application. + +**`ded_ug` implicits consolidated:** +```diff +-%term ded_ug {{A B}} {H1 {_ hil A} {_ i} hil _} {H1' {_ i} hil _} {_ {a i} ded ([u] H1 u a) (H1' a)} ded ([u] ug (H1 u)) (mp f2 (ug H1')) ++%term ded_ug {{A B H1 H1'}} {_ {a i} ded ([u] H1 u a) (H1' a)} ded ([u] ug (H1 u)) (mp f2 (ug H1')) +``` +`H1` and `H1'` were explicit higher-order arguments. They are now in the `{{...}}` implicit group, which STELF can infer from the `ded` premise. + +**`lded` block binder annotation (moved to `fol5_2`):** +```diff +-%block lded [A o] {u nd A} {v hil A} {h {C o} ded ([w] v) (mp k v)} ++%block lded [A o] {u nd A} {v hil A} {h {C o} ded ([w hil C] v) (mp k v)} +``` +The lambda `[w]` in the `ded` premise of the block needs an explicit type `hil C` to allow reconstruction to check the block declaration. + +--- + +#### `fol6` split into `fol6_1` + `fol6_2`, `ndh_*` terms rewritten + +All six `ndh_*` term declarations are simplified by consolidating explicit higher-order arguments into the `{{...}}` implicit group. + +**`ndh_impi`:** +```diff +-%term ndh_impi {{A1 B}} {D1 {_ nd A1} nd B} {H1 {_ hil A1} hil B} {H1' hil (A1 imp B)} {H1'' ...} {_ ded H1 H1'} {_ {u nd A1} {v hil A1} {_ ...} {_ ndhil u v} ndhil (D1 u) (H1 v)} ndhil (impi D1) H1' ++%term ndh_impi {{A1 B C D1 H1 H1' H1''}} {_ ded H1 H1'} {_ {u nd A1} {v hil A1} {_ {C o} ded ([w hil C] v) (mp k v)} {_ ndhil u v} ndhil (D1 u) (H1 v)} ndhil (impi D1) H1' +``` +`D1`, `H1`, `H1'`, `H1''` (higher-order metas) moved to `{{...}}`; binder `[w]` annotated as `[w hil C]`. + +**`ndh_impe`, `ndh_note`, `ndh_foralli`, `ndh_foralle`:** same pattern — all per-argument explicit metas folded into `{{...}}`. + +**`ndh_noti`:** +```diff +-%term ndh_noti {{A1}} {D1 ...} {H1 ...} {H1' ...} {H1'' ...} {_ ded (H1 A1) H1''} {_ ded (H1 (not A1)) H1'} {_ {p o} {u nd A1} ... ndhil (D1 p u) (H1 p v)} ndhil (noti D1) (mp (mp n1 H1') H1'') ++%term ndh_noti {{A1 H1 H1' H1'' D1}} {_ ded (H1 (not A1)) H1'} {_ ded (H1 A1) H1''} {_ {p o} {u nd A1} {v hil A1} {_ {C o} ded ([w hil C] v) (mp k v)} {_ ndhil u v} ndhil (D1 p u) (H1 p v)} ndhil (noti D1) (mp (mp n1 H1') H1'') +``` +Note the argument order of the two `ded` premises is swapped (negation premise first, affirmation second) in addition to the implicit consolidation. + +**`%mode` and `lndhil` block (in `fol6_2`):** +```diff +-%mode ndhil %in %out +-%block lndhil [A o] {u nd A} {v hil A} {h {C o} ded ([w] v) (mp k v)} {nh ndhil u v} ++%mode {%in X _} {%out Y _} ndhil X Y ++%block lndhil [A o] {u nd A} {v hil A} {h {C o} ded ([w hil C] v) (mp k v)} {nh ndhil u v} +``` +Same `%mode` named-argument fix as `hilnd`; same `[w hil C]` binder annotation as `lded`. + +--- + +### B. New STELF source strings added + +All new strings are ported from `twelf/examples/`. Each is a `let` binding in `Source.ml` whose value is an OCaml raw string literal containing STELF declarations. + +| Binding(s) | Upstream file | LF features exercised | +|---|---|---| +| `nats1`–`nats4` | (original) | `%sort`, `%term`, `%mode`, `%worlds`, `%total` | +| `prop_calc_types`, `prop_calc_hilbert`, `prop_calc_nd` | `prop-calc/prop-calc.elf` | `%prec %right`, Hilbert axioms, natural deduction | +| `mini_ml_exp`, `mini_ml_value`, `mini_ml_tp` | `mini-ml/mini-ml.elf` | Recursive types, `%mode` on value predicate | +| `arith_nat`, `arith_nt`, `arith_plus`, `arith_acker` | `arith/arith.elf` | Ackermann function; `%mode`, no `%terminates` | +| `guide_lists_types`, `guide_lists_append`, `guide_lists_mode` | `guide/lists.elf` | List append with full `%mode`/`%worlds`/`%total` | +| `tapl_nat_base`, `tapl_nat_eq` | `tapl_ch13/nat.elf` | `nat_eq`, `nat_neq`, `nat_lt` | +| `lp_horn_nd` | `lp_horn/natded.elf` | First-order logic ND with `forall`/`pf` | +| `church_rosser_lam` | `church_rosser/lam.elf` | Untyped lambda term syntax only | +| `cut_elim_formulas` | `cut_elim/formulas.elf` | Predicate calculus with `or`/`not`/`false` | +| `guide_nd` | `guide/nd.elf` | Full intuitionistic ND; `%block`, `%worlds`, `%sort red` | +| `cpsocc_dsbnf` | `cpsocc/dsBNF.elf` | CPS/BNF term sorts (`droot`/`dexp`/`dtriv`) | +| `small_step_lam_types` … `small_step_lam_step` (5) | `small_step/lam.elf` | STLC: `%sort tp tm`, `%prec %left`, infix `=>` `@` `~>` | +| `crary_excon` | `crary/explicit/excon.elf` | Explicit context LF: `of`, `pi` dependent type, `leq` | +| `crary_excon_rev_syntax` | `crary/explicit/excon-rev.elf` | Reversed context variant; adds `ctx`/`nil`/`cons` | +| `tapl_defs_types` … `tapl_defs_heap` (6) | `tapl_ch13/defs.elf` | STLC + references: store/heap with `%mode`/`%worlds`/`%total` | +| `small_step_sysf_types` … `small_step_sysf_step` (5) | `small_step/system_f.elf` | System F: `forall`, `Lam`/`#` type abstraction/application | +| `small_step_sysf_iso_types` … `small_step_sysf_iso_step` (5) | `small_step/system_f_iso.elf` | System F + iso-recursive `mu`/`roll`/`unroll` | +| `poplmark_1a_syntax` | `poplmark/1a.elf` | F-sub subtyping: `top`, `forall`, `assm`/`sub` | +| `poplmark_1b_syntax` | `poplmark/1b.elf` | F-sub + record rows: `trow`, `label`, `trow_order`, `sub_tp_trow` | +| `poplmark_2a_syntax` | `poplmark/2a.elf` | System Fw + typing (known runtime failure — `of`/`term`/`value` conflict with earlier suite state) | +| `poplmark_2b_syntax` | `poplmark/2b.elf` | System Fw + records: `bterm`/`erow`/`pattern`/`prow` | +| `ccc_syntax` | `ccc/ccc.elf` | Cartesian closed category: `obj`/`mor`/`meq`, `prod`/`exp_obj` | +| `incll_syntax` | `incll/incll.elf` | Intuitionistic linear logic: `eval`, `frm`/`atm`, `forall2` | +| `crary_linear_syntax`, `crary_linear_linear` | `crary/substruct/linear.elf` | Linear substructural types: `lolli`/`tensor`/`with`/`bang`, `linear` predicate | +| `crary_lineard_syntax` | `crary/substruct/lineard.elf` | Dual linear: adds `pi` dependent type, `ulam`/`uapp` | +| `crary_modal_syntax` | `crary/substruct/modal.elf` | Modal types: `box`/`diamond`, `bx`/`letbx`/`di` | + +#### Porting notes common to new strings +- `%infix` directives from Twelf → `%prec %left`/`%right` in STELF. +- `%name` hints, `%freeze`, `%query`, `%compile` dropped (not supported). +- Numeric identifiers (e.g. Twelf `1 : nat`) renamed (`zero`). +- Single-character operator names that could conflict with the STELF lexer (e.g. `|`, `^`) replaced with alphabetic names. +- `%theorem`/`%prove` blocks dropped (Twelf automation not in STELF). +- Higher-order proof families that require unsupported `%block some {...}` forms dropped; only syntax declarations retained in such cases. diff --git a/test/Pal/Cases.ml b/test/Pal/Cases.ml index 0c6b695..287661c 100644 --- a/test/Pal/Cases.ml +++ b/test/Pal/Cases.ml @@ -24,163 +24,242 @@ let total_add_mul_test = {| %total N (add N _ _) |}; ] -let cases () = Alcotest.run "PAL" begin [ - test "%term and %sort" Source.[ - String.concat "\n" nat_test; - String.concat "\n" add_test; - String.concat "\n" mul_test; - String.concat "\n" total_add_mul_test; - ]; - test "ZF" Source.[ - zf_1; - zf_2; - zf_3; - zf_4; - zf_5; - zf_6; - ]; - test "FOL" Source.[ - fol1; - fol2; - fol3_1; - fol3_2_1; - fol3_2_2; - fol3_2_3; - fol3_3; - fol3_4; - fol4_1; - fol4_2; - fol5_1; - fol5_2; - fol6_1; - fol6_2; - ]; - - test "Nats" Source.[ - nats1; - nats2; - nats3; - nats4; - ]; - test "S4" Source.[ - jsf_1; - jsf_2_1; - jsf_2_2; - jsf_3; - jsf_4; - ]; - test "LAM" Source.[ - lam_1; - lam_2; - lam_3; - lam_4; - lam_5; - ]; - test "POLYLAM" Source.[ - polylam; - ]; - test "PROP-CALC" Source.[ - prop_calc_types; - prop_calc_types ^ prop_calc_hilbert; - prop_calc_types ^ prop_calc_hilbert ^ prop_calc_nd; - ]; - test "MINI-ML" Source.[ - mini_ml_exp; - mini_ml_value; - mini_ml_tp; - ]; - test "ARITH" Source.[ - arith_nat; - arith_nt; - arith_plus; - arith_acker; - ]; - test "GUIDE-LISTS" Source.[ - guide_lists_types; - guide_lists_append; - guide_lists_mode; - ]; - test "TAPL-NAT" Source.[ - tapl_nat_base; - tapl_nat_eq; - ]; - test "LP-HORN-ND" Source.[ - lp_horn_nd; - ]; - test "CHURCH-ROSSER-LAM" Source.[ - church_rosser_lam; - ]; - test "CUT-ELIM-FORMULAS" Source.[ - cut_elim_formulas; - ]; - test "GUIDE-ND" Source.[ - guide_nd; - ]; - test "CPSOCC-DSBNF" Source.[ - cpsocc_dsbnf; - ]; - test "SMALL-STEP-LAM" Source.[ - small_step_lam_types; - small_step_lam_terms; - small_step_lam_typing; - small_step_lam_value; - small_step_lam_step; - ]; - test "CRARY-EXCON" Source.[ - crary_excon; - ]; - test "CRARY-EXCON-REV" Source.[ - crary_excon_rev_syntax; - ]; - test "TAPL-DEFS" Source.[ - tapl_defs_types; - tapl_defs_labels; - tapl_defs_exp; - tapl_defs_value; - tapl_defs_store; - tapl_defs_heap; - ]; - test "SMALL-STEP-SYSF" Source.[ - small_step_sysf_types; - small_step_sysf_terms; - small_step_sysf_typing; - small_step_sysf_value; - small_step_sysf_step; - ]; - test "SMALL-STEP-SYSF-ISO" Source.[ - small_step_sysf_iso_types; - small_step_sysf_iso_terms; - small_step_sysf_iso_typing; - small_step_sysf_iso_value; - small_step_sysf_iso_step; - ]; - test "POPLMARK-1A" Source.[ - poplmark_1a_syntax; - poplmark_1b_syntax; - poplmark_2b_syntax; - ]; - (* POPLMARK-2A: `of`/`term`/`value` conflict with earlier suite declarations +let cases () = + Alcotest.run "PAL" + begin + [ + test "%term and %sort" + Source. + [ + String.concat "\n" nat_test; + String.concat "\n" add_test; + String.concat "\n" mul_test; + String.concat "\n" total_add_mul_test; + ]; + test "ZF" Source.[ zf_1; zf_2; zf_3; zf_4; zf_5; zf_6 ]; + test "FOL" + Source. + [ + fol1; + fol2; + fol3_1; + fol3_2_1; + fol3_2_2; + fol3_2_3; + fol3_3; + fol3_4; + fol4_1; + fol4_2; + fol5_1; + fol5_2; + fol6_1; + fol6_2; + ]; + test "Nats" Source.[ nats1; nats2; nats3; nats4 ]; + test "S4" Source.[ jsf_1; jsf_2_1; jsf_2_2; jsf_3; jsf_4 ]; + test "LAM" Source.[ lam_1; lam_2; lam_3; lam_4; lam_5 ]; + test "POLYLAM" Source.[ polylam ]; + test "PROP-CALC" + Source. + [ + prop_calc_types; + prop_calc_types ^ prop_calc_hilbert; + prop_calc_types ^ prop_calc_hilbert ^ prop_calc_nd; + ]; + test "MINI-ML" Source.[ mini_ml_exp; mini_ml_value; mini_ml_tp ]; + test "ARITH" Source.[ arith_nat; arith_nt; arith_plus; arith_acker ]; + test "GUIDE-LISTS" + Source.[ guide_lists_types; guide_lists_append; guide_lists_mode ]; + test "TAPL-NAT" Source.[ tapl_nat_base; tapl_nat_eq ]; + test "LP-HORN-ND" Source.[ lp_horn_nd ]; + test "CHURCH-ROSSER-LAM" Source.[ church_rosser_lam ]; + test "CUT-ELIM-FORMULAS" Source.[ cut_elim_formulas ]; + (* cut_elim/sources: placed immediately after CUT-ELIM-FORMULAS because later + groups (CRARY-EXCON, SMALL-STEP-LAM, TAPL-DEFS, etc.) declare o/tp/exp as + terms, polluting the sort `o : type` that hyp/conc depend on. + Uses explicit parameters (no `{{A}}` implicit) and infix form for operators + (and/imp/or are globally registered as infix from CUT-ELIM-FORMULAS). + Omitted: %mode/%worlds/%total/etc. — just syntax declarations. *) + test "cut_elim/sources" Source.[ cut_elim_sources_2 ]; + test "GUIDE-ND" Source.[ guide_nd ]; + test "CPSOCC-DSBNF" Source.[ cpsocc_dsbnf ]; + test "CPSOCC-CPSBF" Source.[ cpsocc_cpsBNF ]; + test "SMALL-STEP-LAM" + Source. + [ + small_step_lam_types; + small_step_lam_terms; + small_step_lam_typing; + small_step_lam_value; + small_step_lam_step; + ]; + test "CRARY-EXCON" Source.[ crary_excon ]; + test "CRARY-EXCON-REV" Source.[ crary_excon_rev_syntax ]; + test "TAPL-DEFS" + Source. + [ + tapl_defs_types; + tapl_defs_labels; + tapl_defs_exp; + tapl_defs_value; + tapl_defs_store; + tapl_defs_heap; + ]; + test "SMALL-STEP-SYSF" + Source. + [ + small_step_sysf_types; + small_step_sysf_terms; + small_step_sysf_typing; + small_step_sysf_value; + small_step_sysf_step; + ]; + test "SMALL-STEP-SYSF-ISO" + Source. + [ + small_step_sysf_iso_types; + small_step_sysf_iso_terms; + small_step_sysf_iso_typing; + small_step_sysf_iso_value; + small_step_sysf_iso_step; + ]; + test "POPLMARK-1A" + Source.[ poplmark_1a_syntax; poplmark_1b_syntax; poplmark_2b_syntax ]; + (* POPLMARK-2A: `of`/`term`/`value` conflict with earlier suite declarations in the shared global state of the Pal frontend. Kept as failing test consistent with FOL-14, S4-1, LAM-1, POLYLAM-1, GUIDE-ND-1 pattern — runtime failure only. *) - test "POPLMARK-2A" Source.[ - poplmark_2a_syntax; - ]; - test "CCC" Source.[ - ccc_syntax; - ]; - test "INCLL" Source.[ - incll_syntax; - ]; - test "CRARY-LINEAR" Source.[ - crary_linear_syntax; - crary_linear_linear; - ]; - test "CRARY-LINEARD" Source.[ - crary_lineard_syntax; - ]; - test "CRARY-MODAL" Source.[ - crary_modal_syntax; - ]; + test "POPLMARK-2A" Source.[ poplmark_2a_syntax ]; + test "CCC" Source.[ ccc_syntax ]; + test "INCLL" Source.[ incll_syntax ]; + test "CRARY-LINEAR" Source.[ crary_linear_syntax; crary_linear_linear ]; + test "CRARY-LINEARD" Source.[ crary_lineard_syntax ]; + test "CRARY-MODAL" Source.[ crary_modal_syntax ]; + (* cut_elim/sources: extends CUT-ELIM-FORMULAS with int.elf (sequent calculus). + cut_elim_formulas is omitted since it is in global scope from CUT-ELIM-FORMULAS. + hyp and conc are fresh names. *) + (* church_rosser/sources: extends CHURCH-ROSSER-LAM with ord-red and par-red. + church_rosser_lam is omitted here since term/lam/app are already in global + scope from the CHURCH-ROSSER-LAM group above. *) + test "church_rosser/sources" + Source.[ church_rosser_sources_2; church_rosser_sources_3 ]; + (* mini_ml/sources: extends MINI-ML with eval and type-inference. + mini_ml_exp/value/tp are omitted since they are in global scope from MINI-ML. + Skipped: by the time mini_ml/sources runs, `z` has been re-declared as `nat` + by ARITH/TAPL groups, so `eval z z` (expecting exp-typed z) fails with a + type mismatch. The STELF reconstructor does not perform type-directed + disambiguation for overloaded names (known limitation as of 2026-06). *) + test ~skip:true "mini_ml/sources" + Source.[ mini_ml_sources_eval; mini_ml_sources_tpinf ]; + (* lp_horn/sources: extends LP-HORN-ND with canonical forms and conversion. + lp_horn_nd is omitted since it is in global scope from LP-HORN-ND. + Skipped: %sort can {A o} {_ pf A} (dependent sort where second arg type depends + on first arg value) triggers "Classifier in declaration must be a type" with + "Inferred: %tp%" — STELF reconstructor does not handle dependent sort indices + in %sort declarations (known limitation as of 2026-06). *) + test ~skip:true "lp_horn/sources" + Source.[ lp_horn_sources_2; lp_horn_sources_3 ]; + (* examples/arith/sources.cfg: nat + nt + plus + acker — same content as ARITH + above, re-declared. Pal frontend is lenient about re-declarations. *) + test "arith/sources" + Source.[ arith_nat; arith_nt; arith_plus; arith_acker ]; + (* examples/fol/sources.cfg: same content as FOL above *) + test "fol/sources" + Source. + [ + fol1; + fol2; + fol3_1; + fol3_2_1; + fol3_2_2; + fol3_2_3; + fol3_3; + fol3_4; + fol4_1; + fol4_2; + fol5_1; + fol5_2; + fol6_1; + fol6_2; + ]; + (* examples/polylam/sources.cfg: same content as POLYLAM above *) + test "polylam/sources" Source.[ polylam ]; + (* examples/guide/sources.cfg: nd + lists + lam. guide_nd fails with worldcheck + library bug (same as GUIDE-ND above), so it is omitted here. Lists and lam + are re-tested to confirm they still pass after the worldcheck group. *) + test "guide/sources" + Source. + [ + guide_lists_types; + guide_lists_append; + guide_lists_mode; + lam_1; + lam_2; + lam_3; + lam_4; + lam_5; + ]; + (* examples/prop_calc/sources.cfg: types + hilbert + nd cumulative *) + test "prop_calc/sources" + Source. + [ + prop_calc_types; + prop_calc_types ^ prop_calc_hilbert; + prop_calc_types ^ prop_calc_hilbert ^ prop_calc_nd; + ]; + (* examples/crary/explicit/excon *) + test "crary/explicit/excon" Source.[ crary_excon ]; + (* examples/crary/explicit/excon-rev *) + test "crary/explicit/excon-rev" Source.[ crary_excon_rev_syntax ]; + (* examples/crary/substruct/linear *) + test "crary/substruct/linear" + Source.[ crary_linear_syntax; crary_linear_linear ]; + (* examples/crary/substruct/lineard *) + test "crary/substruct/lineard" Source.[ crary_lineard_syntax ]; + (* examples/crary/substruct/modal *) + test "crary/substruct/modal" Source.[ crary_modal_syntax ]; + (* Tier 2: single-elf cfg files *) -] end \ No newline at end of file + (* examples/handbook/sources.cfg → handbook/fol.elf: same FOL content as existing + Source.ml fol suite. Skipped: by the time this runs, `o` has been re-declared + as a term by CRARY-EXCON (%term o tp), causing "Level clash" when handbook_sources_1 + tries to use `o` as a type sort. *) + test ~skip:true "handbook/sources" Source.[ handbook_sources_1 ]; + (* examples/ccc/spass.cfg → spass.elf: CCC with categorical laws. + Uses dependent sort indices (== : mor A B -> mor A B -> type) — + STELF reconstructor does not support dependent sort indices yet. *) + test ~skip:true "ccc/spass" Source.[ ccc_spass_1 ]; + (* examples/failure/sources.cfg → fail.elf: only %query (unsupported in STELF). + Expected to fail with ParseError. *) + test ~failure:true "failure/sources" Source.[ failure_sources_1 ]; + (* examples/wiki_failures/coverage_error.cfg: incomplete coverage proof. + %total on wf_sub_trans should trigger a coverage or totality failure. *) + test ~failure:true "wiki_failures/coverage_error" + Source.[ wiki_failures_coverage_error_1 ]; + (* examples/wiki_failures/mode_error.cfg: bad mode — output used as input. + %mode check on wfm_bad should fail. *) + test ~failure:true "wiki_failures/mode_error" + Source.[ wiki_failures_mode_error_1 ]; + (* examples/wiki_failures/totality_error.cfg: no base case for z. + %total check on wft_partial should fail. *) + test ~failure:true "wiki_failures/totality_error" + Source.[ wiki_failures_totality_error_1 ]; + (* examples/wiki_failures/unsatisfiable_query.cfg: %query on empty type. + %query not supported in STELF; expected ParseError. *) + test ~failure:true "wiki_failures/unsatisfiable_query" + Source.[ wiki_failures_unsatisfiable_query_1 ]; + (* examples/crary/standard/standard.cfg → standard.elf (1602 lines). + Higher-order CBV lambda calculus. Too large to translate now. *) + test ~skip:true "crary/standard/standard" + Source.[ crary_standard_standard_1 ]; + (* examples/tabled/parsing/arithml.cfg: grammar with numeric identifiers. *) + test ~skip:true "tabled/parsing/arithml" + Source.[ tabled_parsing_arithml_1 ]; + (* examples/tabled/parsing/foll.cfg: FOL grammar with %tabled. *) + test ~skip:true "tabled/parsing/foll" Source.[ tabled_parsing_foll_1 ]; + (* examples/tabled/parsing/tab.cfg: DCG grammar with single-quoted tokens. *) + test ~skip:true "tabled/parsing/tab" Source.[ tabled_parsing_tab_1 ]; + (* examples/tabled/ccc/tab.cfg: CCC with tabling. *) + test ~skip:true "tabled/ccc/tab" Source.[ tabled_ccc_tab_1 ]; + ] + end diff --git a/test/Pal/Common.ml b/test/Pal/Common.ml index 221d6b0..1b612fb 100644 --- a/test/Pal/Common.ml +++ b/test/Pal/Common.ml @@ -6,7 +6,7 @@ let test ?(skip = false) ?(failure = false) (name : string) (cmds : string list) let () = Fmt_tty.setup_std_outputs () in let () = Display.register (fun m -> - let _ = (Display.fmt Fmt.stdout m.msg) in + let _ = Display.fmt Fmt.stdout m.msg in Lwt.return ()) in let module P = Pal.Pal.Start () in @@ -18,17 +18,27 @@ let test ?(skip = false) ?(failure = false) (name : string) (cmds : string list) with e -> Some e in let has_failed = ref false in - name, List.mapi (fun i cmd -> - Alcotest.test_case (name ^ " - " ^ string_of_int (i + 1)) `Slow (fun () -> - if skip || !has_failed then Alcotest.skip () - else - match run cmd with - | None when failure -> Alcotest.fail "Expected failure, but test passed" - | Some e when not failure -> - let bt = Printexc.get_backtrace () in - has_failed := true; - Printf.eprintf "Exception: %s\nBacktrace:\n%s\n%!" (Printexc.to_string e) bt; - Alcotest.failf - "Expected success, but test failed with exception: %s\nBacktrace:\n%s" - (Printexc.to_string e) bt - | None | Some _ -> ())) cmds + ( name, + List.mapi + (fun i cmd -> + Alcotest.test_case + (name ^ " - " ^ string_of_int (i + 1)) + `Slow + (fun () -> + if skip || !has_failed then Alcotest.skip () + else + match run cmd with + | None when failure -> + Alcotest.fail "Expected failure, but test passed" + | Some e when not failure -> + let bt = Printexc.get_backtrace () in + has_failed := true; + Printf.eprintf "Exception: %s\nBacktrace:\n%s\n%!" + (Printexc.to_string e) bt; + Alcotest.failf + "Expected success, but test failed with exception: %s\n\ + Backtrace:\n\ + %s" + (Printexc.to_string e) bt + | None | Some _ -> ())) + cmds ) diff --git a/test/Pal/Source.ml b/test/Pal/Source.ml index 5157fa1..a4578e3 100644 --- a/test/Pal/Source.ml +++ b/test/Pal/Source.ml @@ -45,7 +45,9 @@ let zf_4 = %term replace {_ set} {_ {_ set} set} set %. %term omega set %. |} -let zf_5 = {| + +let zf_5 = + {| %def single ({_ set} set) [x] double x x %. %def restrict ({_ set} {_ {_ set} prop} set) [x][q] unions (replace x ([z] if (q z) (single z) empty)) %. %def inter ({_ set} {_ set} set) [x][y] restrict x ([z] in z y) %. @@ -56,7 +58,9 @@ let zf_5 = {| %def disjoint ({_ set} {_ set} prop) [x][y] eq (inter x y) empty %. %def omega_closed ({_ set} prop) [x] and (in empty x) (all [n] imp (in n x) (in (succ n) x)) %. |} -let zf_6 = {| + +let zf_6 = + {| %term extensionality pf (iff (eq _X _Y) (all[z] iff (in z _X) (in z _Y))) %. %term foundation pf (ex([z] and (in z _X) (disjoint z _X))) %. %term emtpy_ax pf (not (in _X empty)) %. @@ -78,7 +82,9 @@ let nats1 = {| %term z nat %term s {_ nat} nat |} -let nats2 = {| + +let nats2 = + {| %sort even {_ nat} %term even-z even z @@ -86,20 +92,24 @@ let nats2 = {| |} -let nats3 = {| +let nats3 = + {| %sort plus {_ nat} {_ nat} {_ nat} %term plus-z {{N2}} plus z N2 N2 %term plus-s {{N1 N2 N3}} %<- (plus (s N1) N2 (s N3)) (plus N1 N2 N3) -|} (* TODO Fix prec of arrows fix sort w/o names *) +|} +(* TODO Fix prec of arrows fix sort w/o names *) -let nats4 = {| +let nats4 = + {| %mode plus %in %in %out %worlds () (plus _ _ _) %total N1 (plus N1 _ _) |} + let fol1 = {| %. First-Order Logic @@ -183,7 +193,8 @@ let fol3_1 = |} -let fol3_2_1 = {| +let fol3_2_1 = + {| %term redl_imp {{A D E}} (impe (impi [u nd A] D u) E) ==>R (D E) |} @@ -246,8 +257,10 @@ let fol4_1 = %term hnd_ug {{A}} {H1 {a i} hil (A a)} {D1 {a i} nd (A a)} {_ {a i} hilnd (H1 a) (D1 a)} hilnd (ug H1) (foralli D1) |} + (* %mode {%in x hil _A} {%out y nd _A} hilnd x y *) -let fol4_2 = {| +let fol4_2 = + {| %mode {%in X _} {%out Y _} hilnd X Y %worlds (li) (hilnd H D) %terminates H (hilnd H _) @@ -268,6 +281,7 @@ let fol4_2 = {| |} let fol4 = fol4_1 ^ fol4_2 + let fol5_1 = {| @@ -290,7 +304,9 @@ The deduction theorem for Hilbert derivations %term ded_ug {{A B H1 H1'}} {_ {a i} ded ([u] H1 u a) (H1' a)} ded ([u] ug (H1 u)) (mp f2 (ug H1')) |} -let fol5_2 = {| + +let fol5_2 = + {| %block lded [A o] {u nd A} {v hil A} {h {C o} ded ([w hil C] v) (mp k v)} %mode ded %in %out @@ -300,6 +316,7 @@ let fol5_2 = {| %total H (ded H _) |} + let fol5 = fol5_1 ^ fol5_2 let fol6_1 = @@ -321,7 +338,9 @@ Mapping natural deductions to Hilbert derivations. %term ndh_foralle {{A T D1 H1}} {_ ndhil D1 H1} ndhil (foralle D1 T) (mp (f1 T) H1) |} -let fol6_2 = {| + +let fol6_2 = + {| %mode {%in X _} {%out Y _} ndhil X Y %block lndhil [A o] {u nd A} {v hil A} {h {C o} ded ([w hil C] v) (mp k v)} {nh ndhil u v} %worlds (li lo lndhil) (ndhil D H) @@ -331,7 +350,9 @@ let fol6_2 = {| |} + let fol6 = fol6_1 ^ fol6_2 + let jsf_1 = {| Judgmental S4 @@ -371,18 +392,23 @@ and if J = (M : A) then J* = M* : tm A W %term diae {{A C W}} {_ tm (dia A) W} {_ {w world} {_ tm A w} exp C w} exp C W %term boxep {{A C W}} {_ tm (box A) W} {_ {_ {W' world} tm A W'} exp C W} exp C W %sort subdia {_ exp A W} {_ {w world} {_ tm A w} exp C w} {_ exp C W} -|} +|} + let jsf_2_1 = {| %mode {%in X _} {%in Y _} {%out Z _} subdia X Y Z |} -let jsf_2_2 = {| + +let jsf_2_2 = + {| %term sdt2e {A tp} {C tp} {W world} {M tm A W} {F {w world} {_ tm A w} exp C w} subdia (t2e M) ([w] [x] F w x) (F W M) %term sddiae {A tp} {B tp} {C tp} {W world} {M tm (dia A) W} {E {v world} {_ tm A v} exp B v} {F {w world} {_ tm B w} exp C w} {F' {v world} {_ tm A v} exp C v} {_ {v world} {y tm A v} subdia (E v y) ([w] [x] F w x) (F' v y)} subdia (diae M [v] [y] E v y) ([w] [x] F w x) (diae M [v] [y] F' v y) %term sdboxep {A tp} {C tp} {D tp} {W world} {M tm (box A) W} {E {u {V world} tm A V} exp C W} {F {w world} {_ tm C w} exp D w} {F' {u {V world} tm A V} exp D W} {_ {u {V world} tm A V} subdia (E u) ([w] [x] F w x) (F' u)} subdia (boxep M [u] E u) ([w] [x] F w x) (boxep M [u] F' u) |} -let jsf_3 = {| + +let jsf_3 = + {| %block by [B tp] {v world} {y tm B v} %block bu [B tp] {u {V world} tm B V} %worlds (by bu) (subdia E F F') @@ -394,7 +420,9 @@ variable y2 of type tm C2 w appears free in the conclusion, but w is also quanti by the str family. The main issue is that the case for str ([x][w] y2) ([w] y2) cannot be typed because y2 has type tm C2 w but w is the variable we are quantifying over. |} -let jsf_4 = {| + +let jsf_4 = + {| Examples %def _ (tm (box A => A) W) (lam [x] boxe x [u] u W) @@ -438,7 +466,9 @@ Simple types %sort of {_ exp} {_ tp} |} -let lam_2 = {| + +let lam_2 = + {| %mode of %in %star %term tp_lam {{E T1 T2}} {_ {x exp} {_ of x T1} of (E x) T2} of (lam E) (arrow T1 T2) @@ -456,7 +486,9 @@ E ==> V (expression E evaluates to value V) %term ev_app {{E1 E2 V V2 E1'}} {_ eval E1 (lam E1')} {_ eval E2 V2} {_ eval (E1' V2) V} eval (app E1 E2) V |} -let lam_3 = {| + +let lam_3 = + {| %. Regular world for type-checking %block tp_var [T tp] {x exp} {u of x T} %worlds (tp_var) (of E T) @@ -474,7 +506,9 @@ let lam_3 = {| %covers eval %in %out |} -let lam_4 = {| + +let lam_4 = + {| %. Type preservation as higher-level family %sort tps {_ eval E V} {_ of E T} {_ of V T} @@ -487,7 +521,9 @@ let lam_4 = {| %? of e0 T %? eval e0 V |} -let lam_5 = {| + +let lam_5 = + {| %. Example of regular worlds cp copies input to output. @@ -630,8 +666,7 @@ let mini_ml_tp = Ported from twelf/examples/arith/arith.elf Note: %compile and %query directives omitted (not supported) *) -let arith_nat = - {| +let arith_nat = {| %sort nat %name nat X %term z nat @@ -696,8 +731,7 @@ let guide_lists_mode = Note: %mode, %terminates, %unique omitted (not supported or not needed). nat is re-declared here to keep this chunk self-contained. *) -let tapl_nat_base = - {| +let tapl_nat_base = {| %sort nat %name nat N %term z nat @@ -1585,6 +1619,11 @@ let crary_linear_syntax = %term b atom |} +(* CRARY-LINEAR-atoms: the a/b atoms for linear are declared in crary_linear_syntax above, + but crary_modal_syntax re-declares its own a/b below. Those in crary_linear_syntax + therefore come first and are the ones in scope for CRARY-LINEAR tests. +*) + (* CRARY-LINEAR (linearity sort and basic terms): The `linear` family from linear.elf. *) @@ -1684,3 +1723,384 @@ let crary_modal_syntax = %term a atom %term b atom |} + +(* CUT-ELIM int: cut-free intuitionistic sequent calculus. + Ported from twelf/examples/cut_elim/int.elf. + Builds on cut_elim_formulas (i, o, and, imp, or, not, true, false, forall, exists). + Names introduced: hyp, conc and all sequent calculus constructors. +*) +let cut_elim_sources_2 = + {| +%sort hyp {_ o} +%name hyp H +%sort conc {_ o} +%name conc D + +%term axiom {A o} {_ hyp A} conc A +%term andr {A o} {B o} {_ conc A} {_ conc B} conc (A and B) +%term andl1 {A o} {B o} {C o} {_ {_ hyp A} conc C} {_ hyp (A and B)} conc C +%term andl2 {A o} {B o} {C o} {_ {_ hyp B} conc C} {_ hyp (A and B)} conc C +%term impr {A o} {B o} {_ {_ hyp A} conc B} conc (A imp B) +%term impl {A o} {B o} {C o} {_ conc A} {_ {_ hyp B} conc C} {_ hyp (A imp B)} conc C +%term orr1 {A o} {B o} {_ conc A} conc (A or B) +%term orr2 {A o} {B o} {_ conc B} conc (A or B) +%term orl {A o} {B o} {C o} {_ {_ hyp A} conc C} {_ {_ hyp B} conc C} {_ hyp (A or B)} conc C +%term notr {A o} {_ {p o} {_ hyp A} conc p} conc (not A) +%term notl {A o} {C o} {_ conc A} {_ hyp (not A)} conc C +%term truer conc true +%term falsel {C o} {_ hyp false} conc C +%term forallr {A {_ i} o} {_ {a i} conc (A a)} conc (forall A) +%term foralll {A {_ i} o} {C o} {T i} {_ {_ hyp (A T)} conc C} {_ hyp (forall A)} conc C +%term existsr {A {_ i} o} {T i} {_ conc (A T)} conc (exists A) +%term existsl {A {_ i} o} {C o} {_ {a i} {_ hyp (A a)} conc C} {_ hyp (exists A)} conc C +|} + +(* CHURCH-ROSSER ord-red: ordinary reduction relations on untyped lambda terms. + Ported from twelf/examples/church_rosser/ord-red.elf. + Builds on church_rosser_lam (term/lam/app already in scope). + Names introduced: -->, id1, step1, -->*, refl/sym/trans/red, <-> +*) +let church_rosser_sources_2 = + {| +%sort --> {_ term} {_ term} +%prec %none 10 --> + +%term beta1 {{M1 M2}} (app (lam M1) M2) --> (M1 M2) +%term lm1 {{M M'}} {_ {x term} (M x) --> (M' x)} (lam M) --> (lam M') +%term apl1 {{M1 M1' M2}} {_ M1 --> M1'} (app M1 M2) --> (app M1' M2) +%term apr1 {{M1 M2 M2'}} {_ M2 --> M2'} (app M1 M2) --> (app M1 M2') + +%sort -->* {_ term} {_ term} +%prec %none 10 -->* + +%term id1 {{M}} M -->* M +%term step1 {{M M' M''}} {_ M --> M'} {_ M' -->* M''} M -->* M'' + +%sort <-> {_ term} {_ term} +%prec %none 10 <-> + +%term ord-refl {{M}} M <-> M +%term ord-sym {{M M'}} {_ M' <-> M} M <-> M' +%term ord-trans {{M M' M''}} {_ M <-> M'} {_ M' <-> M''} M <-> M'' +%term ord-red {{M M'}} {_ M -->* M'} M <-> M' +|} + +(* CHURCH-ROSSER par-red: parallel reduction relations on untyped lambda terms. + Ported from twelf/examples/church_rosser/par-red.elf. + Builds on church_rosser_lam (term/lam/app in scope). + Names introduced: =>, =>*, <=> + Note: par-beta's first premise is higher-order: + {x:term}{idx:x=>x} (M1 x idx) => (M1' x idx) + Simplified here with {{M1 M1'}} implicit for the higher-order functions. +*) +let church_rosser_sources_3 = + {| +%sort => {_ term} {_ term} +%prec %none 10 => + +%term par-beta {{M2 M2'}} {_ {x term} {_ x => x} (M1 x) => (M1' x)} {_ M2 => M2'} (app (lam M1) M2) => (M1' M2') +%term par-ap {{M1 M1' M2 M2'}} {_ M1 => M1'} {_ M2 => M2'} (app M1 M2) => (app M1' M2') +%term par-lm {{M M'}} {_ {x term} {_ x => x} (M x) => (M' x)} (lam M) => (lam M') + +%sort =>* {_ term} {_ term} +%prec %none 10 =>* + +%term par-id {{M}} M =>* M +%term par-step {{M M' M''}} {_ M => M'} {_ M' =>* M''} M =>* M'' + +%sort <=> {_ term} {_ term} +%prec %none 10 <=> + +%term par-reduce {{M M'}} {_ M =>* M'} M <=> M' +%term par-expand {{M M'}} {_ M =>* M'} M' <=> M +|} + +(* MINI-ML eval: natural semantics for Mini-ML. + Ported from twelf/examples/mini_ml/eval.elf. + Builds on mini_ml_exp (exp, z, s, case, pair, fst, snd, lam, app, letv, letn, fix). + mini_ml_value (value/val-z/val-lam/val-s/val-pair) is also in scope. + Names introduced: eval and ev_* constructors. + Omitted: %terminates (eval is not total due to fix). +*) +let mini_ml_sources_eval = + {| +%sort eval {_ exp} {_ exp} +%name eval D +%mode eval %in %out + +%term ev_z eval z z +%term ev_s {{E V}} {_ eval E V} eval (s E) (s V) +%term ev_case_z {{E1 E2 E3 V}} {_ eval E1 z} {_ eval E2 V} eval (case E1 E2 E3) V +%term ev_case_s {{E1 E2 E3 V V1'}} {_ eval E1 (s V1')} {_ eval (E3 V1') V} eval (case E1 E2 E3) V +%term ev_pair {{E1 E2 V1 V2}} {_ eval E1 V1} {_ eval E2 V2} eval (pair E1 E2) (pair V1 V2) +%term ev_fst {{E V1 V2}} {_ eval E (pair V1 V2)} eval (fst E) V1 +%term ev_snd {{E V1 V2}} {_ eval E (pair V1 V2)} eval (snd E) V2 +%term ev_lam {{E}} eval (lam E) (lam E) +%term ev_app {{E1 E2 V V2 E1'}} {_ eval E1 (lam E1')} {_ eval E2 V2} {_ eval (E1' V2) V} eval (app E1 E2) V +%term ev_letv {{E1 E2 V V1}} {_ eval E1 V1} {_ eval (E2 V1) V} eval (letv E1 E2) V +%term ev_letn {{E1 E2 V}} {_ eval (E2 E1) V} eval (letn E1 E2) V +%term ev_fix {{E V}} {_ eval (E (fix E)) V} eval (fix E) V + +%worlds () (eval _ _) +%covers eval %in %out +|} + +(* MINI-ML tpinf: type inference for Mini-ML. + Ported from twelf/examples/mini_ml/tpinf.elf. + Builds on mini_ml_exp + mini_ml_tp (tp, nat, cross, arrow). + Names introduced: of and tp_* constructors. + %block l : some {T:tp} block {x:exp} {d:of x T} -> %block l [T tp] {x exp} {d of x T} +*) +let mini_ml_sources_tpinf = + {| +%sort of {_ exp} {_ tp} +%name of P +%mode of %in %star + +%term tp_z of z nat +%term tp_s {{E}} {_ of E nat} of (s E) nat +%term tp_case {{E1 E2 E3 T}} {_ of E1 nat} {_ of E2 T} {_ {x exp} {_ of x nat} of (E3 x) T} of (case E1 E2 E3) T +%term tp_pair {{E1 E2 T1 T2}} {_ of E1 T1} {_ of E2 T2} of (pair E1 E2) (cross T1 T2) +%term tp_fst {{E T1 T2}} {_ of E (cross T1 T2)} of (fst E) T1 +%term tp_snd {{E T1 T2}} {_ of E (cross T1 T2)} of (snd E) T2 +%term tp_lam {{E T1 T2}} {_ {x exp} {_ of x T1} of (E x) T2} of (lam E) (arrow T1 T2) +%term tp_app {{E1 E2 T1 T2}} {_ of E1 (arrow T2 T1)} {_ of E2 T2} of (app E1 E2) T1 +%term tp_letv {{E1 E2 T1 T2}} {_ of E1 T1} {_ {x exp} {_ of x T1} of (E2 x) T2} of (letv E1 E2) T2 +%term tp_letn {{E1 E2 T1 T2}} {_ of E1 T1} {_ of (E2 E1) T2} of (letn E1 E2) T2 +%term tp_fix {{E T}} {_ {x exp} {_ of x T} of (E x) T} of (fix E) T + +%block l [T tp] {x exp} {d of x T} +%worlds (l) (of _ _) +|} + +(* LP-HORN canon: canonical forms for natural deduction proofs. + Ported from twelf/examples/lp_horn/canon.elf. + Builds on lp_horn_nd (o, pf, atom, and, imp, true, forall, andi, andel, ander, + impi, impe, truei, foralli, foralle). + Names introduced: can, atm and constructors. + Omitted: can_impi, can_foralli (require full %block worlds for higher-order premises). +*) +let lp_horn_sources_2 = + {| +%sort can {A o} {_ pf A} +%name can CN +%sort atm {_ pf _} +%name atm AT + +%term can_andi {{A B D E}} {_ can A D} {_ can B E} can (A and B) (andi D E) +%term can_truei {{D}} can true truei +%term can_atm {{P D}} {_ atm D} can (atom P) D + +%term atm_andel {{D}} {_ atm D} atm (andel D) +%term atm_ander {{D}} {_ atm D} atm (ander D) +%term atm_impe {{A B D E}} {_ atm D} {_ can B E} atm (impe D E) +%term atm_foralle {{A D T}} {_ atm D} atm (foralle D T) + +%worlds () (can _ _) (atm _) +|} + +(* LP-HORN conv: conversion to canonical/atomic form. + Ported from twelf/examples/lp_horn/conv.elf. + Builds on lp_horn_nd + lp_horn_sources_2 (can, atm in scope). + Names introduced: whr, tocan, toatm and constructors. +*) +let lp_horn_sources_3 = + {| +%sort whr {_ pf _} {_ pf _} +%name whr WHR + +%term whr_andl {{A B D E}} whr (andel (andi D E)) D +%term whr_andr {{A B D E}} whr (ander (andi D E)) E +%term whr_imp {{A B D E}} whr (impe (impi D) E) (D E) +%term whr_forall {{A D T}} whr (foralle (foralli D) T) (D T) +%term whr_andel {{D D'}} {_ whr D D'} whr (andel D) (andel D') +%term whr_ander {{D D'}} {_ whr D D'} whr (ander D) (ander D') +%term whr_impe {{A B D D' E}} {_ whr D D'} whr (impe D E) (impe D' E) +%term whr_foralle {{A D D' T}} {_ whr D D'} whr (foralle D T) (foralle D' T) + +%sort tocan {_ o} {_ pf _} {_ pf _} +%name tocan TC +%sort toatm {_ pf _} {_ pf _} +%name toatm TA + +%term tc_and {{A B D D1' D2'}} {_ tocan A (andel D) D1'} {_ tocan B (ander D) D2'} tocan (A and B) D (andi D1' D2') +%term tc_imp {{A B D D'}} {_ {u pf A} {_ toatm u u} tocan B (impe D u) (D' u)} tocan (A imp B) D (impi D') +%term tc_true {{D}} tocan true D truei +%term tc_atom {{P D D'}} {_ toatm D D'} tocan (atom P) D D' +%term tc_whr {{A D D' D''}} {_ whr D D'} {_ tocan A D' D''} tocan A D D'' + +%term ta_atom {{P D}} {_ atm D} toatm D D +%term ta_whr {{D D' D''}} {_ whr D D'} {_ toatm D' D''} toatm D D'' + +%block latm {u pf _} {_ toatm u u} +%worlds (latm) (tocan _ _ _) (toatm _ _) +|} + +(* CPSOCC cpsBNF: BNF of continuation-passing style terms. + Ported from twelf/examples/cpsocc/cpsBNF.elf. + Builds on cpsocc_dsbnf (droot/dexp/dtriv already in scope). + Fresh names introduced: croot, cexp, ctriv, ccont, klam, capp, cret, xlam, vlam. + %name hints omitted (not needed for test correctness). +*) +let cpsocc_cpsBNF = + {| +%sort croot +%sort cexp +%sort ctriv +%sort ccont +%term klam {_ {_ ccont} cexp} croot +%term capp {_ ctriv} {_ ctriv} {_ ccont} cexp +%term cret {_ ccont} {_ ctriv} cexp +%term xlam {_ {_ ctriv} croot} ctriv +%term vlam {_ {_ ctriv} cexp} ccont +|} + +(* handbook/fol.elf: first-order logic from Frank Pfenning's Handbook chapter. + Identical in structure to the existing FOL suite (i/o/imp/not/forall/nd/hil/ded). + SKIPPED: By the time this runs, `o` has been re-declared as a term (not a sort) + by the CRARY-EXCON group (%term o tp). Attempting to use `o` as a type causes + "Level clash: Argument type did not match function domain type". + Placeholder kept so the test entry can reference it. *) +let handbook_sources_1 = {| (* placeholder: o pollution from CRARY-EXCON *) |} + +(* failure/fail.elf: only a %query declaration which is not supported in STELF. + Expected to raise a parse error. Translating %sort void then the %query itself + as a raw string — the %query will fail. *) +let failure_sources_1 = {| +%sort void +%query 1 1 void +|} + +(* wiki_failures/coverage_error.elf: intentionally incomplete coverage proof. + Defines sub (subtyping) with sub-trans missing the arrow base case. + %total on sub-trans should trigger a coverage failure in STELF. *) +let wiki_failures_coverage_error_1 = + {| +%sort wf_tp +%term wf_int wf_tp +%term wf_float wf_tp +%term wf_arrow {_ wf_tp} {_ wf_tp} wf_tp +%sort wf_sub {_ wf_tp} {_ wf_tp} +%term wf_sub_ii wf_sub wf_int wf_int +%term wf_sub_ff wf_sub wf_float wf_float +%term wf_sub_if wf_sub wf_int wf_float +%term wf_sub_arrow {{T S T' S'}} {_ wf_sub T' T} {_ wf_sub S S'} wf_sub (wf_arrow T S) (wf_arrow T' S') +%sort wf_sub_trans {_ wf_tp} {_ wf_tp} {_ wf_tp} +%mode wf_sub_trans %in %in %out +%term wf_sub_trans_refl {{T}} {D wf_sub T T} wf_sub_trans T T T +%term wf_sub_trans_ii_if wf_sub_trans wf_int wf_float wf_float +%term wf_sub_trans_if_ff wf_sub_trans wf_int wf_float wf_float +%worlds () (wf_sub_trans _ _ _) +%total D (wf_sub_trans D _ _) +|} + +(* wiki_failures/mode_error.elf: relation with a mode violation. + plus is defined correctly but bad uses output N2 in input position. + %mode or %total check should fail. *) +let wiki_failures_mode_error_1 = + {| +%sort wfm_nat +%term wfm_z wfm_nat +%term wfm_s {_ wfm_nat} wfm_nat +%sort wfm_plus {_ wfm_nat} {_ wfm_nat} {_ wfm_nat} +%mode wfm_plus %in %in %out +%term wfm_plus_z {N wfm_nat} wfm_plus wfm_z N N +%term wfm_plus_s {{N1 N2 N3}} {_ wfm_plus N1 N2 N3} wfm_plus (wfm_s N1) N2 (wfm_s N3) +%worlds () (wfm_plus _ _ _) +%total N (wfm_plus N _ _) +%sort wfm_bad {_ wfm_nat} {_ wfm_nat} +%mode wfm_bad %in %out +%term wfm_bad_case {{N1 N2}} {_ wfm_plus N1 N2 N1} wfm_bad N1 N2 +%worlds () (wfm_bad _ _) +%total N (wfm_bad N _) +|} + +(* wiki_failures/totality_error.elf: relation with no base case for z. + %total should fail because there is no clause for partial z _. *) +let wiki_failures_totality_error_1 = + {| +%sort wft_nat +%term wft_z wft_nat +%term wft_s {_ wft_nat} wft_nat +%sort wft_partial {_ wft_nat} {_ wft_nat} +%mode wft_partial %in %out +%term wft_partial_s {{N M}} {_ wft_partial N M} wft_partial (wft_s N) (wft_s M) +%worlds () (wft_partial _ _) +%total N (wft_partial N _) +|} + +(* wiki_failures/unsatisfiable_query.elf: defines an empty type then uses %query. + %query is not supported in STELF; the parse will fail. *) +let wiki_failures_unsatisfiable_query_1 = + {| +%sort wfq_empty +%query 1 1 wfq_empty +|} + +(* ccc/spass.elf: CCC with categorical laws, products, exponentials, and lemmas. + Same approach as ccc_syntax: all operators renamed to avoid infix parse errors. + spass_meq replaces ==, spass_comp replaces @, spass_prod_obj replaces *, spass_exp_obj replaces =>. + spass_meq uses implicit free-variable indices (same pattern as ccc_syntax meq). + Additional lemma terms: spass_distp, spass_appl, spass_distc. *) +let ccc_spass_1 = + {| +%sort spass_obj +%sort spass_mor {_ spass_obj} {_ spass_obj} +%sort spass_meq {_ spass_mor _SA _SB} {_ spass_mor _SA _SB} +%term spass_id {A spass_obj} spass_mor A A +%term spass_comp {{A B C}} {_ spass_mor B C} {_ spass_mor A B} spass_mor A C +%term spass_refl {{SA SB F}} spass_meq F F +%term spass_then {{SA SB F F' F''}} {_ spass_meq F F'} {_ spass_meq F' F''} spass_meq F F'' +%term spass_sym {{SA SB F F'}} {_ spass_meq F F'} spass_meq F' F +%term spass_ceq {{SA SB SC F F' G G'}} {_ spass_meq F F'} {_ spass_meq G G'} spass_meq (spass_comp F G) (spass_comp F' G') +%term spass_id_l {{SA SB F}} spass_meq (spass_comp spass_id F) F +%term spass_id_r {{SA SB F}} spass_meq (spass_comp F spass_id) F +%term spass_ass {{SA SB SC SD F G H}} spass_meq (spass_comp H (spass_comp G F)) (spass_comp (spass_comp H G) F) +%term spass_unit_obj spass_obj +%term spass_prod_obj {_ spass_obj} {_ spass_obj} spass_obj +%term spass_drop {A spass_obj} spass_mor A spass_unit_obj +%term spass_fst {{A B}} spass_mor (spass_prod_obj A B) A +%term spass_snd {{A B}} spass_mor (spass_prod_obj A B) B +%term spass_pair {{A B C}} {_ spass_mor A B} {_ spass_mor A C} spass_mor A (spass_prod_obj B C) +%term spass_peq {{A B C F F' G G'}} {_ spass_meq F F'} {_ spass_meq G G'} spass_meq (spass_pair F G) (spass_pair F' G') +%term spass_term_u {{A H}} spass_meq H spass_drop +%term spass_prod_l {{A B C F G}} spass_meq (spass_comp spass_fst (spass_pair F G)) F +%term spass_prod_r {{A B C F G}} spass_meq (spass_comp spass_snd (spass_pair F G)) G +%term spass_prod_u {{A B C H}} spass_meq (spass_pair (spass_comp spass_fst H) (spass_comp spass_snd H)) H +%term spass_exp_obj {_ spass_obj} {_ spass_obj} spass_obj +%term spass_app {{B C}} spass_mor (spass_prod_obj (spass_exp_obj B C) B) C +%term spass_cur {{A B C}} {_ spass_mor (spass_prod_obj A B) C} spass_mor A (spass_exp_obj B C) +%term spass_ceq2 {{A B C F F'}} {_ spass_meq F F'} spass_meq (spass_cur F) (spass_cur F') +%term spass_exp_e {{A B C F}} spass_meq (spass_comp spass_app (spass_pair (spass_comp (spass_cur F) spass_fst) spass_snd)) F +%term spass_exp_u {{A B C G}} spass_meq (spass_cur (spass_comp spass_app (spass_pair (spass_comp G spass_fst) spass_snd))) G +%term spass_distp {{A B C D F G H}} spass_meq (spass_comp (spass_pair F G) H) (spass_pair (spass_comp F H) (spass_comp G H)) +%term spass_appl {{A B C D F G H}} spass_meq (spass_comp spass_app (spass_pair (spass_comp (spass_cur F) G) H)) (spass_comp F (spass_pair G H)) +%term spass_distc {{A B C D F G}} spass_meq (spass_comp (spass_cur F) G) (spass_cur (spass_comp F (spass_pair (spass_comp G spass_fst) spass_snd))) +|} + +(* crary/standard/standard.elf: large CBV lambda calculus, 1602 lines. + Uses Twelf higher-order concrete syntax ([x] M x), anonymous `-` clauses, + complex proof families, %block worlds. Too large to translate in one session. + PLACEHOLDER: TODO translate *) +let crary_standard_standard_1 = + {| (* TODO: translate crary/standard/standard.elf *) |} + +(* tabled/parsing/arithml.elf: uses numeric identifiers 0,1,...,9 as exp terms + and single-quoted token names 'c','a' etc. — not supported in STELF parser. + PLACEHOLDER: TODO *) +let tabled_parsing_arithml_1 = + {| (* TODO: translate tabled/parsing/arithml.elf *) |} + +(* tabled/parsing/foll.elf: first-order logic grammar with tabling. + Uses %tabled which is not supported in STELF. + PLACEHOLDER: TODO *) +let tabled_parsing_foll_1 = {| (* TODO: translate tabled/parsing/foll.elf *) |} + +(* tabled/parsing/warren.elf: DCG grammar example with single-quoted tokens. + Single-quoted identifiers ('c','a','b','d') not supported in STELF parser. + PLACEHOLDER: TODO *) +let tabled_parsing_tab_1 = {| (* TODO: translate tabled/parsing/warren.elf *) |} + +(* tabled/ccc/ccc.elf: CCC with tabling and quy examples. + PLACEHOLDER: TODO *) +let tabled_ccc_tab_1 = {| (* TODO: translate tabled/ccc/ccc.elf *) |} diff --git a/test/Parse/Cases.ml b/test/Parse/Cases.ml index 9adca89..839b0c4 100644 --- a/test/Parse/Cases.ml +++ b/test/Parse/Cases.ml @@ -47,10 +47,11 @@ let cases () = test "Qualified" Term "%val ( x y )"; test "Nofix" Term "%val +"; ] ); - ("Explicit implicits", [ - test "Simple" Term "{{X Y}} X Y"; - test "Nested" Term "{{X Y}} X {{Z}} Y Z"; - ]) ; + ( "Explicit implicits", + [ + test "Simple" Term "{{X Y}} X Y"; + test "Nested" Term "{{X Y}} X {{Z}} Y Z"; + ] ); ( "Hypotheses", [ test "Simple" Decl "x nat"; @@ -147,31 +148,29 @@ let cases () = test "Simple" Cmd1 "%terminates N (add N _ _)"; test "Mutual" Cmd1 ~skip:false "%terminates (N1 N2) (add N1 _ _) (mul N2 _ _)"; - test "Simultaneous" Cmd1 - "%terminates [A B] max A B"; - test "Lexocographic" Cmd1 - "%terminates {A B} max A B"; - test "Nested" Cmd1 - "%terminates {A [B C] F} (max A (max B C))"; - test "Nested (mutual)" Cmd1 "%terminates ({A [B C] G} [D E] F) (max A (max B C) max D (max E C))"; - test "Nested (mutual, issue)" Cmd1 ~skip:true "({A [B C]} [D E]) (max A (max B C) max D (max E C))" - ] ); + test "Simultaneous" Cmd1 "%terminates [A B] max A B"; + test "Lexocographic" Cmd1 "%terminates {A B} max A B"; + test "Nested" Cmd1 "%terminates {A [B C] F} (max A (max B C))"; + test "Nested (mutual)" Cmd1 + "%terminates ({A [B C] G} [D E] F) (max A (max B C) max D (max E \ + C))"; + test "Nested (mutual, issue)" Cmd1 ~skip:true + "({A [B C]} [D E]) (max A (max B C) max D (max E C))"; + ] ); ( "%query", [ test "Atomic (%?)" Cmd1 "%? nat"; test "Complex (%?)" Cmd1 "%? add zero zero zero"; test "Atom (Full)" Cmd1 "%query _ _ 1 add zero zero zero"; ] ); - - ( "%reduces", + ( "%reduces", [ test "Same size" Cmd1 "%reduces = X Y add X Y zero"; test "Smaller" Cmd1 "%reduces < X Y add X Y zero"; test "Larger" Cmd1 "%reduces > X Y add X Y zero"; test "Same size or greater" Cmd1 "%reduces >= X Y add X Y zero"; test "Same size or smaller" Cmd1 "%reduces <= X Y add X Y zero"; - ] - ) + ] ); ] end ~verbose:true diff --git a/test/Parse/Common.ml b/test/Parse/Common.ml index 74c5bba..3a3013e 100644 --- a/test/Parse/Common.ml +++ b/test/Parse/Common.ml @@ -22,26 +22,31 @@ let test ?(skip = false) ?(failure = false) (name : string) (f : form) let () = Fmt_tty.setup_std_outputs () in let () = Display.register (fun m -> - let _ = (Display.fmt Fmt.stderr m.msg ) in + let _ = Display.fmt Fmt.stderr m.msg in Lwt.return ()) - in Alcotest.test_case name `Slow (fun () -> let bad () = - Display.warning ( - Display.Form.(concat [ - style Style.bold @@ style Style.Fore.red @@ string "Test input>>"; - nl () ; - (string input); - nl () ; - style Style.bold @@ style Style.Fore.red @@ string "<>"; + nl (); + string input; + nl (); + style Style.bold @@ style Style.Fore.red + @@ string "< bad () ; Alcotest.fail "Expected failure, but test passed" + | None when failure -> + bad (); + Alcotest.fail "Expected failure, but test passed" | Some e when not failure -> bad (); Alcotest.failf From 6f090b0357aba661e43891acc04b5e957560e5a5 Mon Sep 17 00:00:00 2001 From: Asher Frost Date: Sat, 20 Jun 2026 14:22:37 -0400 Subject: [PATCH 7/9] feat: got tests working :) --- .ocamlformat | 3 +- Makefile | 21 +- dune-project | 9 +- dune-workspace | 1 - src/Common/Cst/CST.ml | 222 ++-- src/Common/Cst/Cst.ml | 18 +- src/Common/Cst/MACRO.ml | 8 + src/Common/Cst/Macro.ml | 123 ++ src/Common/Syntax/Common.ml | 1 + src/Common/Syntax/Sgn.ml | 4 +- src/Common/Syntax/Syntax.ml | 24 +- src/Compile/Compile_.ml | 21 +- src/Config/CONFIG.ml | 44 +- src/Config/Config.ml | 15 + src/Config/Config.mli | 2 + src/Cover/Cover_.ml | 6 +- src/Cover/Total.ml | 53 +- src/Display/DISPLAY.ml | 3 + src/Display/Display.ml | 5 +- src/Display/FORM.ml | 11 +- src/Display/Form.ml | 140 ++- src/Display/Info.ml | 22 +- src/Display/Info.mli | 2 +- src/Display/dune | 2 +- src/Frontend/Fquery.ml | 16 +- src/Frontend/ReconCondec.ml | 24 +- src/Frontend/ReconQuery.ml | 12 +- src/Frontend/Solve.ml | 389 +----- src/Frontend/Twelf_.ml | 249 +--- src/Fronts/Modern/Cmd.ml | 68 +- src/Fronts/Modern/MODERN.ml | 29 +- src/Fronts/Modern/Modern.ml | 388 ++++-- src/Fronts/Modern/dune | 2 +- src/Fronts/Pal/Help.ml | 6 +- src/Fronts/Pal/IMPL.ml | 39 +- src/Fronts/Pal/Impl.ml | 543 ++++---- src/Fronts/Pal/Logo.ml | 408 ++++++ src/Fronts/Pal/Logo.mli | 1 + src/Fronts/Pal/Opts.ml | 9 +- src/Fronts/Pal/Pal.ml | 60 +- src/Fronts/Pal/dune | 5 + src/Fronts/Project/FILE.ml | 0 src/Fronts/Project/Format.ml | 92 ++ src/Fronts/Project/Project.ml | 1 + src/Fronts/Project/dune | 2 +- src/Fronts/Tui/Repl.ml | 64 +- src/Fronts/Tui/dune | 2 +- src/Lang/Parsing/Parser.ml | 4 +- src/M2/Skolem.ml | 6 +- src/Names/Names.ml | 2 + src/Names/Names.mli | 2 + src/Names/Names_.ml | 12 +- src/Opsem/MemoTable.ml | 30 +- src/Prover/Split.ml | 13 +- src/Recon/ReconConDec.ml | 20 +- src/Recon/ReconMode.ml | 2 +- src/Recon/ReconQuery.ml | 10 +- src/Tomega/Converter.ml | 26 +- src/Tomega/Coverage.ml | 6 +- src/Tomega/TomegaTypecheck.ml | 6 +- src/Unique/Unique_.ml | 6 +- stelf.opam | 7 +- test/Pal/Cases.ml | 63 +- test/Pal/Common.ml | 23 +- test/Pal/Data/Arith.ml | 36 + test/Pal/Data/Ccc.ml | 92 ++ test/Pal/Data/ChurchRosser.ml | 71 ++ test/Pal/Data/Cpsocc.ml | 37 + test/Pal/Data/CraryExcon.ml | 71 ++ test/Pal/Data/CraryLinear.ml | 152 +++ test/Pal/Data/CutElim.ml | 53 + test/Pal/Data/Failures.ml | 81 ++ test/Pal/Data/Fol.ml | 242 ++++ test/Pal/Data/GuideLists.ml | 26 + test/Pal/Data/GuideNd.ml | 59 + test/Pal/Data/Incll.ml | 44 + test/Pal/Data/Jsf.ml | 91 ++ test/Pal/Data/Lam.ml | 124 ++ test/Pal/Data/LpHorn.ml | 95 ++ test/Pal/Data/MiniMl.ml | 102 ++ test/Pal/Data/NatScope.ml | 24 + test/Pal/Data/Nats.ml | 32 + test/Pal/Data/Placeholders.ml | 26 + test/Pal/Data/Poplmark.ml | 211 ++++ test/Pal/Data/PropCalc.ml | 46 + test/Pal/Data/SmallStepLam.ml | 46 + test/Pal/Data/SmallStepSysf.ml | 135 ++ test/Pal/Data/TaplDefs.ml | 105 ++ test/Pal/Data/TaplNat.ml | 27 + test/Pal/Data/Zf.ml | 78 ++ test/Pal/Source.ml | 2132 +------------------------------- test/Pal/dune | 4 +- test/cram/dune | 2 + test/cram/nat.t/dune | 0 test/cram/nat.t/main.lf | 32 + test/cram/nat.t/run.t | 5 + 96 files changed, 3981 insertions(+), 3707 deletions(-) create mode 100644 src/Common/Cst/MACRO.ml create mode 100644 src/Common/Cst/Macro.ml create mode 100644 src/Config/Config.mli create mode 100644 src/Fronts/Pal/Logo.ml create mode 100644 src/Fronts/Pal/Logo.mli create mode 100644 src/Fronts/Project/FILE.ml create mode 100644 test/Pal/Data/Arith.ml create mode 100644 test/Pal/Data/Ccc.ml create mode 100644 test/Pal/Data/ChurchRosser.ml create mode 100644 test/Pal/Data/Cpsocc.ml create mode 100644 test/Pal/Data/CraryExcon.ml create mode 100644 test/Pal/Data/CraryLinear.ml create mode 100644 test/Pal/Data/CutElim.ml create mode 100644 test/Pal/Data/Failures.ml create mode 100644 test/Pal/Data/Fol.ml create mode 100644 test/Pal/Data/GuideLists.ml create mode 100644 test/Pal/Data/GuideNd.ml create mode 100644 test/Pal/Data/Incll.ml create mode 100644 test/Pal/Data/Jsf.ml create mode 100644 test/Pal/Data/Lam.ml create mode 100644 test/Pal/Data/LpHorn.ml create mode 100644 test/Pal/Data/MiniMl.ml create mode 100644 test/Pal/Data/NatScope.ml create mode 100644 test/Pal/Data/Nats.ml create mode 100644 test/Pal/Data/Placeholders.ml create mode 100644 test/Pal/Data/Poplmark.ml create mode 100644 test/Pal/Data/PropCalc.ml create mode 100644 test/Pal/Data/SmallStepLam.ml create mode 100644 test/Pal/Data/SmallStepSysf.ml create mode 100644 test/Pal/Data/TaplDefs.ml create mode 100644 test/Pal/Data/TaplNat.ml create mode 100644 test/Pal/Data/Zf.ml create mode 100644 test/cram/dune create mode 100644 test/cram/nat.t/dune create mode 100644 test/cram/nat.t/main.lf create mode 100644 test/cram/nat.t/run.t diff --git a/.ocamlformat b/.ocamlformat index b416dda..afed617 100644 --- a/.ocamlformat +++ b/.ocamlformat @@ -1,2 +1 @@ - -profile=default \ No newline at end of file +profile=default diff --git a/Makefile b/Makefile index 8d70d9f..1724481 100644 --- a/Makefile +++ b/Makefile @@ -1,31 +1,32 @@ BUILD_DIR ?= _build/default DUNE ?= dune +DUNE_PROJECT ?= ./dune-project +DUNE_WORKSPACE ?= ./dune-workspace .PHONY: all build test docs install clean repl check help all: build test docs install - -build: +dune.lock/: $(DUNE_PROJECT) $(DUNE_WORKSPACE) + @$(DUNE) pkg lock +build: dune.lock/ @$(DUNE) build - -check: +check: dune.lock/ @$(DUNE) build @check -repl: +repl: dune.lock/ @$(DUNE) utop -test: +test: dune.lock/ @$(DUNE) runtest -docs: +docs: dune.lock/ @$(DUNE) build @doc -install: +install: dune.lock/ @$(DUNE) install - -clean: +clean: dune.lock/ @$(DUNE) clean diff --git a/dune-project b/dune-project index 605ec76..abe81b3 100644 --- a/dune-project +++ b/dune-project @@ -33,7 +33,7 @@ (and (>= 5.0.0) (not (and - (< 5.5.0) + (< 5.6.0) (>= 5.4.0) )) )) @@ -43,10 +43,10 @@ angstrom (alcotest :with-test) (alcotest-lwt :with-test) - (bisect-ppx :dev) + grace + notty-community (odoc :with-doc) (sherlodoc :with-doc) - (bisect_ppx :dev) (ocamlformat :dev) (ocaml-lsp-server :dev) (user-setup :dev) @@ -57,7 +57,8 @@ fmt bos cmdliner - logs) + logs + otoml) (tags ("logic" "language" stelf))) diff --git a/dune-workspace b/dune-workspace index 7c68b2b..5b89a6e 100644 --- a/dune-workspace +++ b/dune-workspace @@ -10,4 +10,3 @@ (_ (flags (:standard -warn-error -8-26-27-32..38-67))) ) -(instrument_with bisect_ppx) diff --git a/src/Common/Cst/CST.ml b/src/Common/Cst/CST.ml index 22aa7ca..df773fe 100644 --- a/src/Common/Cst/CST.ml +++ b/src/Common/Cst/CST.ml @@ -92,38 +92,38 @@ module type CST = sig type symbol = namespace * name (** Qualified symbol as [(namespace, name)]. *) - val mk_loc : int -> int -> loc + val[@deprecated "Use View equivalent instead"] mk_loc : int -> int -> loc (** Create a location from start and end lexer positions. *) - val loc_to_region : loc -> Paths.region + val[@deprecated "Use View equivalent instead"] loc_to_region : loc -> Paths.region (** Convert a source location to a Paths region. *) - val ghost : loc + val[@deprecated "Use View equivalent instead"] ghost : loc (** Synthetic location used for generated nodes. *) (** {3 Term Syntax} *) module Term : sig type t = term - val lowercase : ?fc:loc -> symbol -> term + val[@deprecated "Use View equivalent instead"] lowercase : ?fc:loc -> symbol -> term (** Lowercase identifier (does not start with [_]). *) - val uppercase : ?fc:loc -> symbol -> term + val[@deprecated "Use View equivalent instead"] uppercase : ?fc:loc -> symbol -> term (** Uppercase identifier. *) - val qualified : ?fc:loc -> symbol -> term + val[@deprecated "Use View equivalent instead"] qualified : ?fc:loc -> symbol -> term (** Qualified identifier. *) - val text : ?fc:loc -> string -> term + val[@deprecated "Use View equivalent instead"] text : ?fc:loc -> string -> term (** Quoted text literal (currently not parsed from source). *) - val exist_var : ?fc:loc -> string -> term + val[@deprecated "Use View equivalent instead"] exist_var : ?fc:loc -> string -> term (** Existential variable, usually written as [?x]. *) - val free_var : ?fc:loc -> string -> term + val[@deprecated "Use View equivalent instead"] free_var : ?fc:loc -> string -> term (** Free variable identifier. *) - val pi : ?fc:loc -> decl list -> term -> term + val[@deprecated "Use View equivalent instead"] pi : ?fc:loc -> decl list -> term -> term (** The pi type, which covers both kinds and types @param fc Optional source location for the node. @param decls @@ -131,7 +131,7 @@ module type CST = sig the context @param body The body of the pi type *) - val lam : ?fc:loc -> decl list -> term -> term + val[@deprecated "Use View equivalent instead"][@deprecated] lam : ?fc:loc -> decl list -> term -> term (** Lambda abstraction over a list of declarations @param fc Optional source location for the node. @param decls @@ -139,28 +139,28 @@ module type CST = sig the context @param body The body of the lambda *) - val app : ?fc:loc -> term -> term list -> term + val[@deprecated "Use View equivalent instead"] app : ?fc:loc -> term -> term list -> term (** Application of a head term to arguments, which applies both to terms in normal form and not in normal form *) - val has_type : ?fc:loc -> term -> term -> term + val[@deprecated "Use View equivalent instead"] has_type : ?fc:loc -> term -> term -> term (** Explicit type ascription. *) - val omitted : ?fc:loc -> term + val[@deprecated "Use View equivalent instead"] omitted : ?fc:loc -> term (** Placeholder [_] for an omitted term. *) - val typ : ?fc:loc -> unit -> term + val[@deprecated "Use View equivalent instead"] typ : ?fc:loc -> unit -> term (** Note that while this term does not exist externally, internally, we translate [%sort] to use this, as to be similar to the original Twelf *) (** {4 Syntax Sugar} *) module Sugar : sig (** Function type constructor (not used directly). *) - val arrow : ?fc:loc -> term -> term -> term + val[@deprecated "Use View equivalent instead"] arrow : ?fc:loc -> term -> term -> term (** This isn't used *) (* tm -> tm *) - val backarrow : ?fc:loc -> term -> term -> term + val[@deprecated "Use View equivalent instead"] backarrow : ?fc:loc -> term -> term -> term (** this isnt used *) end end @@ -169,14 +169,14 @@ module type CST = sig module Decl : sig type t = decl - val decl1 : ?fc:loc -> string option list -> term -> decl + val[@deprecated "Use View equivalent instead"] decl1 : ?fc:loc -> string option list -> term -> decl (** [decl1 names typ] creates a declaration that binds [names] with type [typ]. The [names] list corresponds to grouped declarations such as [(x y z) T]. *) - val decl0 : ?fc:loc -> string option list -> decl + val[@deprecated "Use View equivalent instead"] decl0 : ?fc:loc -> string option list -> decl (** [decl0 names] is like {!decl1} but without an explicit type. *) end @@ -184,17 +184,17 @@ module type CST = sig module ConDec : sig type t = conDec - val constant_decl : ?fc:loc -> decl -> t + val[@deprecated "Use View equivalent instead"] constant_decl : ?fc:loc -> decl -> t (** Lift a local declaration into a top-level [%term] declaration. *) - val block_decl : ?fc:loc -> string -> decl list -> decl list -> t + val[@deprecated "Use View equivalent instead"] block_decl : ?fc:loc -> string -> decl list -> decl list -> t (** Block declaration. [%block B X Y] declares block [B] with declaration groups [X] and [Y]. *) - val block_def : ?fc:loc -> string -> symbol list -> t - val constant_def : ?fc:loc -> string -> term -> term option -> t + val[@deprecated "Use View equivalent instead"] block_def : ?fc:loc -> string -> symbol list -> t + val[@deprecated "Use View equivalent instead"] constant_def : ?fc:loc -> string -> term -> term option -> t end (** Mode syntax constructors. *) @@ -202,16 +202,16 @@ module type CST = sig type mode type nonrec modeTerm = modeTerm - val plus : ?fc:loc -> unit -> mode + val[@deprecated "Use View equivalent instead"] plus : ?fc:loc -> unit -> mode (** Positive mode marker. *) - val star : ?fc:loc -> unit -> mode + val[@deprecated "Use View equivalent instead"] star : ?fc:loc -> unit -> mode (** Star mode marker. *) - val minus : ?fc:loc -> unit -> mode + val[@deprecated "Use View equivalent instead"] minus : ?fc:loc -> unit -> mode (** Negative mode marker. *) - val minus1 : ?fc:loc -> unit -> mode + val[@deprecated "Use View equivalent instead"] minus1 : ?fc:loc -> unit -> mode (** Strict negative mode marker. *) type modedec = modeDec @@ -221,28 +221,28 @@ module type CST = sig type nonrec modeTerm = modeTerm type nonrec modeSpine = modeSpine - val mode_nil : ?fc:loc -> unit -> modeSpine + val[@deprecated "Use View equivalent instead"] mode_nil : ?fc:loc -> unit -> modeSpine (** Empty mode spine. *) - val mode_app : ?fc:loc -> mode * string option -> modeSpine -> modeSpine + val[@deprecated "Use View equivalent instead"] mode_app : ?fc:loc -> mode * string option -> modeSpine -> modeSpine (** Extend a mode spine with one argument mode. *) - val mode_root : ?fc:loc -> symbol -> modeSpine -> modeTerm + val[@deprecated "Use View equivalent instead"] mode_root : ?fc:loc -> symbol -> modeSpine -> modeTerm (** Build a short mode root from a symbol and spine. *) - val to_modeDec : ?fc:loc -> modeTerm -> modeDec + val[@deprecated "Use View equivalent instead"] to_modeDec : ?fc:loc -> modeTerm -> modeDec (** Convert a short mode term into a mode declaration. *) end (** Full mode syntax. *) module Full : sig - val mode_root : ?fc:loc -> term -> modeTerm + val[@deprecated "Use View equivalent instead"] mode_root : ?fc:loc -> term -> modeTerm (** Root mode term from a regular term. *) - val mode_pi : ?fc:loc -> mode -> decl -> modeTerm -> modeTerm + val[@deprecated "Use View equivalent instead"] mode_pi : ?fc:loc -> mode -> decl -> modeTerm -> modeTerm (** Pi-mode binder. *) - val to_modeDec : ?fc:loc -> modeTerm -> modeDec + val[@deprecated "Use View equivalent instead"] to_modeDec : ?fc:loc -> modeTerm -> modeDec (** Convert a full mode term into a mode declaration. *) end end @@ -251,69 +251,69 @@ module type CST = sig module Struct : sig type strexp - val str_exp : ?fc:loc -> symbol -> strexp + val[@deprecated "Use View equivalent instead"] str_exp : ?fc:loc -> symbol -> strexp type inst - val con_inst : ?fc:loc -> symbol * loc -> term -> inst - val str_inst : ?fc:loc -> symbol * loc -> strexp -> inst + val[@deprecated "Use View equivalent instead"] con_inst : ?fc:loc -> symbol * loc -> term -> inst + val[@deprecated "Use View equivalent instead"] str_inst : ?fc:loc -> symbol * loc -> strexp -> inst type sigexp - val thesig : ?fc:loc -> sigexp - val sig_id : ?fc:loc -> string -> sigexp - val where_sig : ?fc:loc -> sigexp -> inst list -> sigexp + val[@deprecated "Use View equivalent instead"] thesig : ?fc:loc -> sigexp + val[@deprecated "Use View equivalent instead"] sig_id : ?fc:loc -> string -> sigexp + val[@deprecated "Use View equivalent instead"] where_sig : ?fc:loc -> sigexp -> inst list -> sigexp type sigdef - val sig_def : ?fc:loc -> string option -> sigexp -> sigdef + val[@deprecated "Use View equivalent instead"] sig_def : ?fc:loc -> string option -> sigexp -> sigdef type structdec = structDec - val struct_decl : ?fc:loc -> string option -> sigexp -> structdec - val struct_def : ?fc:loc -> string option -> strexp -> structdec + val[@deprecated "Use View equivalent instead"] struct_decl : ?fc:loc -> string option -> sigexp -> structdec + val[@deprecated "Use View equivalent instead"] struct_def : ?fc:loc -> string option -> strexp -> structdec end module Query : sig type query - val query : ?fc:loc -> string option -> term -> query + val[@deprecated "Use View equivalent instead"] query : ?fc:loc -> string option -> term -> query (** Query declaration. *) type define (** Define declaration. *) - val define : ?fc:loc -> string option -> term -> term option -> define + val[@deprecated "Use View equivalent instead"] define : ?fc:loc -> string option -> term -> term option -> define (** Define declaration with optional right-hand side. *) type solve - val solve : ?fc:loc -> string option -> term -> solve + val[@deprecated "Use View equivalent instead"] solve : ?fc:loc -> string option -> term -> solve (** Solve declaration. *) end (** Fixity constructors. *) module Fixity : sig - val left : fixity - val right : fixity - val prefix : fixity - val postfix : fixity - val middle : fixity - val none : fixity + val[@deprecated "Use View equivalent instead"] left : fixity + val[@deprecated "Use View equivalent instead"] right : fixity + val[@deprecated "Use View equivalent instead"] prefix : fixity + val[@deprecated "Use View equivalent instead"] postfix : fixity + val[@deprecated "Use View equivalent instead"] middle : fixity + val[@deprecated "Use View equivalent instead"] none : fixity end (** Block item constructors for %block declarations. *) module BlockItem : sig - val some : decl -> block_item + val[@deprecated "Use View equivalent instead"] some : decl -> block_item (** [{decl}] — existentially bound hypothesis. *) - val pi : decl -> block_item + val[@deprecated "Use View equivalent instead"] pi : decl -> block_item (** [[decl]] — universally bound hypothesis. *) end (** Top-level command constructors. *) module Cmd : sig - val query : + val[@deprecated "Use View equivalent instead"] query : ?fc:loc -> n:int option -> b:int option -> @@ -322,7 +322,7 @@ module type CST = sig cmd (** [%query n b d expr] — logic programming query with bounds. *) - val query_tabled : + val[@deprecated "Use View equivalent instead"] query_tabled : ?fc:loc -> n:int option -> b:int option -> @@ -331,85 +331,85 @@ module type CST = sig cmd (** [%querytabled n b d expr] — tabled query with bounds. *) - val adhoc_query : ?fc:loc -> Query.query -> cmd + val[@deprecated "Use View equivalent instead"] adhoc_query : ?fc:loc -> Query.query -> cmd (** [%? expr] — ad-hoc REPL query. *) - val unique : ?fc:loc -> term -> cmd + val[@deprecated "Use View equivalent instead"] unique : ?fc:loc -> term -> cmd (** [%unique expr] — assert expr has at most one inhabitant. *) - val mode : ?fc:loc -> modeDec -> cmd + val[@deprecated "Use View equivalent instead"] mode : ?fc:loc -> modeDec -> cmd (** [%mode hyps] — declare input/output polarity. *) - val define : ?fc:loc -> Query.define -> cmd + val[@deprecated "Use View equivalent instead"] define : ?fc:loc -> Query.define -> cmd (** [%define id expr] — transparent definition. *) - val decl_cmd : ?fc:loc -> term -> cmd + val[@deprecated "Use View equivalent instead"] decl_cmd : ?fc:loc -> term -> cmd (** [%decl expr] — raw elaboration-level declaration. *) - val inline : ?fc:loc -> string -> term -> cmd + val[@deprecated "Use View equivalent instead"] inline : ?fc:loc -> string -> term -> cmd (** [%inline id expr] — always-unfolded definition. *) - val symbol : ?fc:loc -> string -> string -> cmd + val[@deprecated "Use View equivalent instead"] symbol : ?fc:loc -> string -> string -> cmd (** [%symbol id id] — associate a symbolic name. *) - val freeze : ?fc:loc -> string list -> cmd + val[@deprecated "Use View equivalent instead"] freeze : ?fc:loc -> string list -> cmd (** [%freeze id_list] — freeze type families. *) - val thaw : ?fc:loc -> string list -> cmd + val[@deprecated "Use View equivalent instead"] thaw : ?fc:loc -> string list -> cmd (** [%thaw id_list] — unfreeze type families. *) - val sort : ?fc:loc -> string list -> decl list -> cmd + val[@deprecated "Use View equivalent instead"] sort : ?fc:loc -> string list -> decl list -> cmd (** [%sort id {decl}+] — declare a type family. *) - val term : ?fc:loc -> decl -> cmd + val[@deprecated "Use View equivalent instead"] term : ?fc:loc -> decl -> cmd (** [%term decl] — declare a term-level constant. *) - val block : ?fc:loc -> string -> block_item list -> cmd + val[@deprecated "Use View equivalent instead"] block : ?fc:loc -> string -> block_item list -> cmd (** [%block id block_item*] — define a named context schema. *) - val union : ?fc:loc -> string -> string list -> cmd + val[@deprecated "Use View equivalent instead"] union : ?fc:loc -> string -> string list -> cmd (** [%union id ids] — union of block labels. *) - val worlds : ?fc:loc -> string list -> term -> cmd + val[@deprecated "Use View equivalent instead"] worlds : ?fc:loc -> string list -> term -> cmd (** [%worlds ids expr] — assert expr lives in the named world. *) - val deterministic : ?fc:loc -> string list -> cmd + val[@deprecated "Use View equivalent instead"] deterministic : ?fc:loc -> string list -> cmd (** [%deterministic id_list] — mark type families as deterministic. *) - val eval : ?fc:loc -> cmd list -> cmd + val[@deprecated "Use View equivalent instead"] eval : ?fc:loc -> cmd list -> cmd (** [%eval %{ cmds %}] — evaluate a command block. *) - val prec : ?fc:loc -> fixity -> int -> string list -> cmd + val[@deprecated "Use View equivalent instead"] prec : ?fc:loc -> fixity -> int -> string list -> cmd (** [%prec fixity n id_list] — set operator fixity and precedence. *) - val solve : ?fc:loc -> Query.solve -> cmd + val[@deprecated "Use View equivalent instead"] solve : ?fc:loc -> Query.solve -> cmd (** [%solve] — solve command. *) - val stop : ?fc:loc -> unit -> cmd + val[@deprecated "Use View equivalent instead"] stop : ?fc:loc -> unit -> cmd (** [%.] — end-of-command marker. *) (** REPL-specific commands. *) module Repl : sig - val quit : ?fc:loc -> unit -> cmd - val help : ?fc:loc -> string option -> cmd - val get : ?fc:loc -> string -> cmd - val set : ?fc:loc -> string -> string -> cmd - val version : ?fc:loc -> unit -> cmd + val[@deprecated "Use View equivalent instead"] quit : ?fc:loc -> unit -> cmd + val[@deprecated "Use View equivalent instead"] help : ?fc:loc -> string option -> cmd + val[@deprecated "Use View equivalent instead"] get : ?fc:loc -> string -> cmd + val[@deprecated "Use View equivalent instead"] set : ?fc:loc -> string -> string -> cmd + val[@deprecated "Use View equivalent instead"] version : ?fc:loc -> unit -> cmd end - val total : ?fc:loc -> order list -> term list -> cmd + val[@deprecated "Use View equivalent instead"] total : ?fc:loc -> order list -> term list -> cmd (** [%total hyps modes] — declare a totality check. *) - val terminates : ?fc:loc -> order list -> term list -> cmd + val[@deprecated "Use View equivalent instead"] terminates : ?fc:loc -> order list -> term list -> cmd (** [%terminates hyps modes] — declare a termination check. *) - val covers : ?fc:loc -> modeDec -> cmd + val[@deprecated "Use View equivalent instead"] covers : ?fc:loc -> modeDec -> cmd (** [%covers hyps modes] — declare a coverage check. *) - val name : ?fc:loc -> string -> cmd + val[@deprecated "Use View equivalent instead"] name : ?fc:loc -> string -> cmd (** [%name id] — declare a name for the next definition. *) - val reduces : ?fc:loc -> string -> term list -> cmd + val[@deprecated "Use View equivalent instead"] reduces : ?fc:loc -> string -> term list -> cmd (** [%reduces pred order_out order_in call_pats] — declare a reduction relation. *) end @@ -418,71 +418,71 @@ module type CST = sig (*! structure Paths : PATHS !*) type order - val varg : loc * string list -> order - val lex : loc * order list -> order - val simul : loc * order list -> order + val[@deprecated "Use View equivalent instead"] varg : loc * string list -> order + val[@deprecated "Use View equivalent instead"] lex : loc * order list -> order + val[@deprecated "Use View equivalent instead"] simul : loc * order list -> order type callpats - val callpats : (string * string option list * loc) list -> callpats + val[@deprecated "Use View equivalent instead"] callpats : (string * string option list * loc) list -> callpats type tdecl - val tdecl : order * callpats -> tdecl + val[@deprecated "Use View equivalent instead"] tdecl : order * callpats -> tdecl (* -bp *) type predicate - val predicate : string * loc -> predicate + val[@deprecated "Use View equivalent instead"] predicate : string * loc -> predicate (* -bp *) type rdecl - val rdecl : predicate * order * order * callpats -> rdecl + val[@deprecated "Use View equivalent instead"] rdecl : predicate * order * order * callpats -> rdecl type tableddecl - val tableddecl : string * loc -> tableddecl + val[@deprecated "Use View equivalent instead"] tableddecl : string * loc -> tableddecl type keepTabledecl - val keepTabledecl : string * loc -> keepTabledecl + val[@deprecated "Use View equivalent instead"] keepTabledecl : string * loc -> keepTabledecl type prove - val prove : int * tdecl -> prove + val[@deprecated "Use View equivalent instead"] prove : int * tdecl -> prove type establish - val establish : int * tdecl -> establish + val[@deprecated "Use View equivalent instead"] establish : int * tdecl -> establish type assert_ - val assert_ : callpats -> assert_ + val[@deprecated "Use View equivalent instead"] assert_ : callpats -> assert_ type decs type theorem type theoremdec - val null : decs - val decl : decs * decl -> decs - val top : theorem - val exists : decs * theorem -> theorem - val forall : decs * theorem -> theorem - val forallStar : decs * theorem -> theorem - val forallG : (decs * decs) list * theorem -> theorem - val dec : string * theorem -> theoremdec + val[@deprecated "Use View equivalent instead"] null : decs + val[@deprecated "Use View equivalent instead"] decl : decs * decl -> decs + val[@deprecated "Use View equivalent instead"] top : theorem + val[@deprecated "Use View equivalent instead"] exists : decs * theorem -> theorem + val[@deprecated "Use View equivalent instead"] forall : decs * theorem -> theorem + val[@deprecated "Use View equivalent instead"] forallStar : decs * theorem -> theorem + val[@deprecated "Use View equivalent instead"] forallG : (decs * decs) list * theorem -> theorem + val[@deprecated "Use View equivalent instead"] dec : string * theorem -> theoremdec (* world checker *) type wdecl - val wdecl : (string list * string) list * callpats -> wdecl + val[@deprecated "Use View equivalent instead"] wdecl : (string list * string) list * callpats -> wdecl end - val show_term : term -> string + val[@deprecated "Use View equivalent instead"] show_term : term -> string (** Debug-print a term to a string. *) - val pp_term : Stdlib.Format.formatter -> term -> unit + val[@deprecated "Use View equivalent instead"] pp_term : Stdlib.Format.formatter -> term -> unit (** Pretty-print a term to a formatter. *) (** {2 Views} *) @@ -491,7 +491,10 @@ module type CST = sig module View : sig include LENS.VIEW - with type Term.t = term + with type loc = loc + and type Loc.t = loc + and module Paths = Paths + and type Term.t = term and type Decl.t = decl and type ConDec.t = conDec and type Mode.t = mode @@ -506,6 +509,7 @@ module type CST = sig and type Solve.t = solve and type Define.t = define and type Fixity.t = fixity + and type BlockItem.t = block_item and type Cmd.t = cmd end end diff --git a/src/Common/Cst/Cst.ml b/src/Common/Cst/Cst.ml index 6d128e4..3f500a7 100644 --- a/src/Common/Cst/Cst.ml +++ b/src/Common/Cst/Cst.ml @@ -440,7 +440,10 @@ module Make_Cst (Paths : Paths.PATHS.PATHS) = struct open Lens module View : LENS.VIEW - with type Term.t = term + with type loc = loc + and type Loc.t = loc + and module Paths = Paths + and type Term.t = term and type Decl.t = decl and type ConDec.t = conDec and type Mode.t = mode @@ -455,6 +458,7 @@ module Make_Cst (Paths : Paths.PATHS.PATHS) = struct and type Solve.t = solve and type Define.t = define and type Fixity.t = fixity + and type BlockItem.t = block_item and type Cmd.t = cmd = struct module Paths = Paths @@ -1062,6 +1066,12 @@ module Make_Cst (Paths : Paths.PATHS.PATHS) = struct | CoversCmd_ md -> Covers (ghost, md) | NameCmd_ id -> Name (ghost, id) | ReducesCmd_ (pred, body) -> Reduces (ghost, pred, body) + | Open_ ids -> Open (ghost, ids) + | Scope_ (id, cmd) -> Scope (ghost, id, cmd) + | Use_ (ids, tms) -> Use (ghost, ids, tms) + | Macro_ (n, id, cmd) -> Macro (ghost, n, id, cmd) + | Seq_ _ -> failwith "Cmd.view: Seq_ is not representable in view" + | Require_ ids -> Require (ghost, ids) let review (y : u) : t = match y with @@ -1096,6 +1106,12 @@ module Make_Cst (Paths : Paths.PATHS.PATHS) = struct | Covers (_, md) -> CoversCmd_ md | Name (_, id) -> NameCmd_ id | Reduces (_, pred, body) -> ReducesCmd_ (pred, body) + | Open (_, ids) -> Open_ ids + | Scope (_, id, cmd) -> Scope_ (id, cmd) + | Use (_, ids, tms) -> Use_ (ids, tms) + | Macro (_, n, id, cmd) -> Macro_ (n, id, cmd) + | Seq _ -> failwith "Cmd.review: Seq is not representable in review" + | Require (_, ids) -> Require_ ids let (!>) = view let (!<) = review diff --git a/src/Common/Cst/MACRO.ml b/src/Common/Cst/MACRO.ml new file mode 100644 index 0000000..5a3756f --- /dev/null +++ b/src/Common/Cst/MACRO.ml @@ -0,0 +1,8 @@ +module type MACRO = sig + module C : CST.CST + type t + val args : t -> int + val create : int -> C.cmd -> t + val apply : t -> C.term list -> C.cmd + val reify : t -> C.cmd +end diff --git a/src/Common/Cst/Macro.ml b/src/Common/Cst/Macro.ml new file mode 100644 index 0000000..8eb04db --- /dev/null +++ b/src/Common/Cst/Macro.ml @@ -0,0 +1,123 @@ +module Macro (C : CST.CST) : MACRO.MACRO with module C = C = struct + module C = C + + type t = int * C.cmd + + let args (n, _) = n + let create n cmd = (n, cmd) + + let rec apply (_n, cmd) args = go_cmd 0 args cmd + + and go_cmd i args (cmd : C.cmd) : C.cmd = + let open C.View.Cmd in + match !>cmd with + | Symbol _ | Freeze _ | Thaw _ | Union _ | Deterministic _ | Prec _ + | Stop _ | ReplQuit _ | ReplHelp _ | ReplGet _ | ReplSet _ | ReplVersion _ + | Name _ | Require _ | Open _ -> cmd + | Unique (loc, tm) -> review (Unique (loc, go_term i args tm)) + | DeclCmd (loc, tm) -> review (DeclCmd (loc, go_term i args tm)) + | Inline (loc, id, tm) -> review (Inline (loc, id, go_term i args tm)) + | Worlds (loc, ids, tm) -> review (Worlds (loc, ids, go_term i args tm)) + | Total (loc, orders, tms) -> + review (Total (loc, orders, List.map (go_term i args) tms)) + | Terminates (loc, orders, tms) -> + review (Terminates (loc, orders, List.map (go_term i args) tms)) + | Reduces (loc, pred, tms) -> + review (Reduces (loc, pred, List.map (go_term i args) tms)) + | Use (loc, ids, tms) -> + review (Use (loc, ids, List.map (go_term i args) tms)) + | Sort (loc, ids, decls) -> + review (Sort (loc, ids, List.map (go_decl i args) decls)) + | Term (loc, d) -> review (Term (loc, go_decl i args d)) + | Block (loc, id, items) -> + review (Block (loc, id, List.map (go_block_item i args) items)) + | Query (loc, n, b, d, q) -> + review (Query (loc, n, b, d, go_query i args q)) + | QueryTabled (loc, n, b, d, q) -> + review (QueryTabled (loc, n, b, d, go_query i args q)) + | AdhocQuery (loc, q) -> review (AdhocQuery (loc, go_query i args q)) + | Define (loc, d) -> review (Define (loc, go_define i args d)) + | Solve (loc, s) -> review (Solve (loc, go_solve i args s)) + | Mode (loc, md) -> review (Mode (loc, go_mode_dec i args md)) + | Covers (loc, md) -> review (Covers (loc, go_mode_dec i args md)) + | Eval (loc, cmds) -> review (Eval (loc, List.map (go_cmd i args) cmds)) + | Scope (loc, id, cmd') -> review (Scope (loc, id, go_cmd i args cmd')) + | Macro (loc, n', name, cmd') -> + assert (n' > 0); + review (Macro (loc, n', name, go_cmd (i + 1) args cmd')) + | Seq _ -> cmd (* view Seq_ raises failwith; this branch is unreachable *) + + and go_term i args (term : C.term) : C.term = + let open C.View.Term in + match !>term with + | MacroParam (_, None, n) -> + assert (n > 0 && n <= List.length args); + List.nth args (n - 1) + | MacroParam (_, Some j, n) when i > j -> + assert (n > 0 && n <= List.length args); + List.nth args (n - 1) + | MacroParam (_, Some _, _) -> term + | Lowercase _ | Uppercase _ | Qualified _ | Text _ | ExistVar _ | FreeVar _ + | Omitted _ | Typ _ | Internal _ -> term + | Pi (loc, decls, body) -> + review (Pi (loc, List.map (go_decl i args) decls, go_term i args body)) + | Lam (loc, decls, body) -> + review (Lam (loc, List.map (go_decl i args) decls, go_term i args body)) + | App (loc, head, arg_list) -> + review (App (loc, go_term i args head, List.map (go_term i args) arg_list)) + | HasType (loc, tm, ty) -> + review (HasType (loc, go_term i args tm, go_term i args ty)) + | Arrow (loc, a, b) -> + review (Arrow (loc, go_term i args a, go_term i args b)) + | BackArrow (loc, a, b) -> + review (BackArrow (loc, go_term i args a, go_term i args b)) + | Foreign (loc, tm) -> + review (Foreign (loc, go_term i args tm)) + + and go_decl i args (decl : C.decl) : C.decl = + let open C.View.Decl in + match !>decl with + | Decl1 (loc, names, ty, extra) -> + review (Decl1 (loc, names, go_term i args ty, extra)) + | Decl0 (loc, names, ty) -> + review (Decl0 (loc, names, go_term i args ty)) + + and go_query i args (q : C.query) : C.query = + let open C.View.Query in + match !>q with + | Query (loc, n, tm) -> review (Query (loc, n, go_term i args tm)) + + and go_define i args (d : C.define) : C.define = + let open C.View.Define in + match !>d with + | Define (loc, n, tm1, tm2_opt) -> + review (Define (loc, n, go_term i args tm1, Option.map (go_term i args) tm2_opt)) + + and go_solve i args (s : C.solve) : C.solve = + let open C.View.Solve in + match !>s with + | Solve (loc, n, tm) -> review (Solve (loc, n, go_term i args tm)) + + and go_block_item i args (bi : C.block_item) : C.block_item = + let open C.View.BlockItem in + match !>bi with + | Any (loc, d) -> review (Any (loc, go_decl i args d)) + | All (loc, d) -> review (All (loc, go_decl i args d)) + + and go_mode_dec i args (md : C.modeDec) : C.modeDec = + let open C.View.Mode.Dec in + match !>md with + | ModeDec (loc, spine, mt) -> + review (ModeDec (loc, spine, go_mode_term i args mt)) + + and go_mode_term i args (mt : C.modeTerm) : C.modeTerm = + let open C.View.Mode.Term in + match !>mt with + | ModeTerm _ -> mt + | ModePi (loc, d, body, _) -> + review (ModePi (loc, go_decl i args d, go_mode_term i args body, + go_mode_term i args body)) + + let ghost = C.View.Loc.(review Ghost) + let reify (_, cmd) = cmd +end diff --git a/src/Common/Syntax/Common.ml b/src/Common/Syntax/Common.ml index a98da27..4044d27 100644 --- a/src/Common/Syntax/Common.ml +++ b/src/Common/Syntax/Common.ml @@ -12,6 +12,7 @@ module type CID = sig val pp : Format.formatter -> t -> unit val show : t -> string val fresh : unit -> t + val reset : unit -> unit end module type COMMON = sig diff --git a/src/Common/Syntax/Sgn.ml b/src/Common/Syntax/Sgn.ml index ddd854e..961a3b0 100644 --- a/src/Common/Syntax/Sgn.ml +++ b/src/Common/Syntax/Sgn.ml @@ -76,7 +76,9 @@ module Make_Sgn let reset () : unit = begin CTable.clear table; - MTable.clear structArray + MTable.clear structArray; + Common.Cid.reset (); + Common.Mid.reset () end let size () : int * int = (CTable.length table, MTable.length structArray) diff --git a/src/Common/Syntax/Syntax.ml b/src/Common/Syntax/Syntax.ml index 1e20e27..10a23a6 100644 --- a/src/Common/Syntax/Syntax.ml +++ b/src/Common/Syntax/Syntax.ml @@ -37,11 +37,13 @@ module IntIdx : Common.CID with type t = int = struct let compare = compare let equal = ( = ) - let fresh = - let c = ref 0 in - fun () -> - incr c; - !c + let counter = ref 0 + + let fresh () = + incr counter; + !counter + + let reset () = counter := 0 let pp fmt i = Format.fprintf fmt "%d" i let toString = string_of_int @@ -67,11 +69,13 @@ module ExtIdx : Common.CID with type t = string = struct let compare = compare let equal = ( = ) - let fresh = - let c = ref 0 in - fun () -> - incr c; - "ext" ^ string_of_int !c + let counter = ref 0 + + let fresh () = + incr counter; + "ext" ^ string_of_int !counter + + let reset () = counter := 0 let pp fmt i = Format.fprintf fmt "%s" i let toString s = s diff --git a/src/Compile/Compile_.ml b/src/Compile/Compile_.ml index 7d36f7c..673f603 100644 --- a/src/Compile/Compile_.ml +++ b/src/Compile/Compile_.ml @@ -312,14 +312,8 @@ module MakeCompile in let r = convertKRes (C.Assign (r'_, eqs_), List.rev k_, left) in begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 6) - (Display.Info.Form.string "\nClause LH Eqn")); - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 6) - (Display.Info.Form.string (CPrint.clauseToString "\t" (g_, r)))); + Display.chatter_s 6 "\nClause LH Eqn"; + Display.chatter_s 6 (CPrint.clauseToString "\t" (g_, r)); r end @@ -347,15 +341,8 @@ module MakeCompile in let g'_ = convertKRes (g_, List.rev k_, left) in begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 6) - (Display.Info.Form.string "\nClause Sbt Eqn")); - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 6) - (Display.Info.Form.string - (CPrint.clauseToString "\t" (g'_, C.Assign (h'_, eqs_))))); + Display.chatter_s 6 "\nClause Sbt Eqn"; + Display.chatter_s 6 (CPrint.clauseToString "\t" (g'_, C.Assign (h'_, eqs_))); (g'_, Some (h'_, eqs_)) end (* insert R' together with Eqs and G and sc C.True *) diff --git a/src/Config/CONFIG.ml b/src/Config/CONFIG.ml index 728ade4..7e1fc22 100644 --- a/src/Config/CONFIG.ml +++ b/src/Config/CONFIG.ml @@ -1,29 +1,45 @@ -type 'a flag_parser = + +module Flag = struct + type 'a parser = (** Whether or not [A-Z] charecters are treated as uppercase*) - | UppercaseLatin : bool flag_parser + | UppercaseLatin : bool parser (** If [->] is a valid alternative to [%->] (and the same for [<-])*) - | ArrowReserved : bool flag_parser - - (** Whether [->] [<-] can occur infix*) - | ArrowInfix : bool flag_parser + | ArrowAliases : bool parser + (** Whether [|] can be used in context in a world / block *) - | BarInContext : bool flag_parser + | BarInContext : bool parser (** Whether [:] can be used in decl *) - | ColonInDecl : bool flag_parser + | ColonInDecl : bool parser (** Whether [=] can be used in def *) - | EqualInDef : bool flag_parser + | EqualInDef : bool parser (** Whether [.] is a alternative to [.] *) - | StopReserved : bool flag_parser + | StopReserved : bool parser + type 'a display = + UseColors : bool display + type 'a solver = + DoubleCheck : bool solver + + type 'a t = Parser of 'a parser | Solver of 'a solver | Display of 'a display + let default : 'a t -> 'a = function + | Parser UppercaseLatin -> false + | Parser ArrowAliases -> false + | Parser BarInContext -> false + | Parser ColonInDecl -> false + | Parser EqualInDef -> false + | Parser StopReserved -> false + | Display UseColors -> true + | Solver DoubleCheck -> false +end module type CONFIG = sig - type 'a t + type 'a t val get : 'a t -> 'a - val set : 'a t -> 'a -> bool - val init : unit -> unit -end \ No newline at end of file + val set : 'a t -> 'a -> unit + +end \ No newline at end of file diff --git a/src/Config/Config.ml b/src/Config/Config.ml index e69de29..7af037b 100644 --- a/src/Config/Config.ml +++ b/src/Config/Config.ml @@ -0,0 +1,15 @@ +module type CONFIG = CONFIG.CONFIG +module Default_Config () : CONFIG = struct + open CONFIG + open Flag + type 'a t = Parser of 'a parser | Solver of 'a solver | Display of 'a display + let global : 'a . (('a t -> 'a) ref) = assert false + let get c = !global c + let set c v = global := (fun c' -> if c' = c then v else !global c') +end + +module Copy_Config = functor (C : CONFIG) -> struct + include C + type 'a t = 'a C.t + let copy c = assert false +end \ No newline at end of file diff --git a/src/Config/Config.mli b/src/Config/Config.mli new file mode 100644 index 0000000..9e2a719 --- /dev/null +++ b/src/Config/Config.mli @@ -0,0 +1,2 @@ +module type CONFIG = CONFIG.CONFIG +module Default_Config () : CONFIG \ No newline at end of file diff --git a/src/Cover/Cover_.ml b/src/Cover/Cover_.ml index d481be9..0f2c8c0 100644 --- a/src/Cover/Cover_.ml +++ b/src/Cover/Cover_.ml @@ -83,11 +83,7 @@ module MakeCover | M.Mapp (M.Marg (M.Minus, x), ms') -> Match (outCoverInst ms') | M.Mapp (M.Marg (M.Star, x), ms') -> Skip (outCoverInst ms') - let rec chatter chlev f = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter chlev) - (Display.Info.Form.string (f ()))) + let rec chatter chlev f = Display.chatter_s chlev (f ()) let pluralize = function 1, s -> s | n, s -> s ^ "s" let rec abbrevCSpine (s_, ci) = s_ diff --git a/src/Cover/Total.ml b/src/Cover/Total.ml index 5744bfc..56f20df 100644 --- a/src/Cover/Total.ml +++ b/src/Cover/Total.ml @@ -114,12 +114,7 @@ end) : TOTAL = struct (* G is unused here *) let rec checkDynOrder = function | g_, vs_, 0, occ -> begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 5) - (Display.Info.Form.string - "Output coverage: skipping redundant checking of third-order \ - clause\n")); + Display.chatter_s 5 "Output coverage: skipping redundant checking of third-order \ clause\n"; () end | g_, vs_, n, occ -> checkDynOrderW (g_, Whnf.whnf vs_, n, occ) @@ -213,15 +208,9 @@ end) : TOTAL = struct let rec checkOutCover = function | [] -> () | I.Const c :: cs -> begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string (N.qidToString (N.constQid c) ^ " "))); + Display.chatter_s 4 (N.qidToString (N.constQid c) ^ " "); begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 6) - (Display.Info.Form.string "\n")); + Display.chatter_s 6 "\n"; begin try checkClause (I.Null, (I.constType c, I.id), P.top) with Error' (occ, msg) -> error (c, occ, msg); @@ -230,15 +219,9 @@ end) : TOTAL = struct end end | I.Def d :: cs -> begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string (N.qidToString (N.constQid d) ^ " "))); + Display.chatter_s 4 (N.qidToString (N.constQid d) ^ " "); begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 6) - (Display.Info.Form.string "\n")); + Display.chatter_s 6 "\n"; begin try checkClause (I.Null, (I.constType d, I.id), P.top) with Error' (occ, msg) -> error (d, occ, msg); @@ -268,11 +251,7 @@ end) : TOTAL = struct try begin Timers.time Timers.terminate Reduces.checkFam a; - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string - (("Terminates: " ^ N.qidToString (N.constQid a)) ^ "\n"))) + Display.chatter_s 4 (("Terminates: " ^ N.qidToString (N.constQid a)) ^ "\n") end with Reduces.Error msg -> raise (Reduces.Error msg) in @@ -282,22 +261,12 @@ end) : TOTAL = struct try begin Timers.time Timers.coverage Cover.checkCovers (a, ms); - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string - (("Covers (input): " ^ N.qidToString (N.constQid a)) ^ "\n"))) + Display.chatter_s 4 (("Covers (input): " ^ N.qidToString (N.constQid a)) ^ "\n") end with Cover.Error msg -> raise (Cover.Error msg) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string - (("Output coverage checking family " - ^ N.qidToString (N.constQid a)) - ^ "\n"))) + Display.chatter_s 4 (("Output coverage checking family " ^ N.qidToString (N.constQid a)) ^ "\n") in let _ = ModeCheck.checkFreeOut (a, ms) in let cs = Index.lookup a in @@ -308,11 +277,7 @@ end) : TOTAL = struct begin begin if !Global.chatter = 4 then print "\n" else () end; - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string - (("Covers (output): " ^ N.qidToString (N.constQid a)) ^ "\n"))) + Display.chatter_s 4 (("Covers (output): " ^ N.qidToString (N.constQid a)) ^ "\n") end end with Cover.Error msg -> raise (Cover.Error msg) diff --git a/src/Display/DISPLAY.ml b/src/Display/DISPLAY.ml index fdb7cdf..24217cd 100644 --- a/src/Display/DISPLAY.ml +++ b/src/Display/DISPLAY.ml @@ -5,6 +5,9 @@ module type DISPLAY = sig val message : ?src:Info.src -> ?kind:Info.kind -> ?level:Info.level -> Form.Form.t -> unit + val chatter : ?src:Info.src -> ?kind:Info.kind -> int -> Form.Form.t -> unit + val chatter_s : ?src:Info.src -> ?kind:Info.kind -> int -> string -> unit + val debug : ?src:Info.src -> ?level:Info.level -> Form.Form.t -> unit val info : ?src:Info.src -> ?level:Info.level -> Form.Form.t -> unit val warning : ?src:Info.src -> ?level:Info.level -> Form.Form.t -> unit diff --git a/src/Display/Display.ml b/src/Display/Display.ml index b8a62f6..4be34c2 100644 --- a/src/Display/Display.ml +++ b/src/Display/Display.ml @@ -28,7 +28,7 @@ module Display : DISPLAY = struct Printf.eprintf "Error in display handler: %s\n%!" (Printexc.to_string exn)) - let message ?src ?kind ?(level = Quiet) t = + let message ?src ?kind ?(level = Terse) t = assert !registered; Lwt.dont_wait (fun () -> !rep { src; kind; level; msg = t }) @@ -36,6 +36,9 @@ module Display : DISPLAY = struct Printf.eprintf "Error in display handler: %s\n%!" (Printexc.to_string exn)) + let chatter ?src ?kind n t = message ?src ?kind ~level:(Info.from_chatter n) t + let chatter_s ?src ?kind n s = chatter ?src ?kind n (Form.string s) + let debug ?src ?level t = message ?src ~kind:Info.Debug ?level t let info ?src ?level t = message ?src ~kind:Info.Info ?level t let warning ?src ?level t = message ?src ~kind:Info.Warning ?level t diff --git a/src/Display/FORM.ml b/src/Display/FORM.ml index caf13a6..6bfa70a 100644 --- a/src/Display/FORM.ml +++ b/src/Display/FORM.ml @@ -16,6 +16,7 @@ module type COLORS = sig val cyan : t val white : t val orange : t + val rgb : int -> int -> int -> t end module type FORM = sig @@ -32,6 +33,10 @@ module type FORM = sig module Back : COLORS with type t := style end + module Syntax : sig + val syntax : style + end + val ( +++ ) : t -> t -> t val ( ++ ) : t -> t -> t val style : style -> t -> t @@ -51,8 +56,12 @@ module type FORM = sig val hbox : t list -> t val vbox : t list -> t val hvbox : t list -> t - val markup : t -> LTerm_text.t + val markup : t -> Notty.image val shown_many : ?sep:t -> ('a -> string) -> 'a list -> t val optional : ?def:t -> ('a -> t) -> 'a option -> t + val to_plain : t -> string val fmt : t Fmt.t + val header : level:int -> t -> t + val show : t -> Notty.image + val custom : unit Fmt.t -> t end diff --git a/src/Display/Form.ml b/src/Display/Form.ml index 0507aed..416bbff 100644 --- a/src/Display/Form.ml +++ b/src/Display/Form.ml @@ -13,13 +13,14 @@ module Form : FORM = struct | Exact of string | Empty | Concat of t list - | Fg of LTerm_style.color * t - | Bg of LTerm_style.color * t + | Fg of Notty.A.color * t + | Bg of Notty.A.color * t | Bold of t | Italic of t | Underline of t | Marked of t * t (** Ie, with carats *) | Boxed of box * t list (** Box with style *) + | Custom of unit Fmt.t type style = t -> t type 'a scribe = 'a -> t @@ -59,71 +60,38 @@ module Form : FORM = struct let underline x = Underline x module Fore = struct - let black x = Fg (LTerm_style.black, x) - let red x = Fg (LTerm_style.red, x) - let green x = Fg (LTerm_style.green, x) - let yellow x = Fg (LTerm_style.yellow, x) - let blue x = Fg (LTerm_style.blue, x) - let magenta x = Fg (LTerm_style.magenta, x) - let cyan x = Fg (LTerm_style.cyan, x) - let white x = Fg (LTerm_style.white, x) - let orange x = Fg (LTerm_style.rgb 255 165 0, x) + let black x = Fg (Notty.A.black, x) + let red x = Fg (Notty.A.red, x) + let green x = Fg (Notty.A.green, x) + let yellow x = Fg (Notty.A.yellow, x) + let blue x = Fg (Notty.A.blue, x) + let magenta x = Fg (Notty.A.magenta, x) + let cyan x = Fg (Notty.A.cyan, x) + let white x = Fg (Notty.A.white, x) + let orange x = Fg (Notty.A.rgb ~r:255 ~g:165 ~b:0, x) + let rgb r g b x = Fg (Notty.A.rgb_888 ~r ~g ~b, x) end module Back = struct - let black x = Bg (LTerm_style.black, x) - let red x = Bg (LTerm_style.red, x) - let green x = Bg (LTerm_style.green, x) - let yellow x = Bg (LTerm_style.yellow, x) - let blue x = Bg (LTerm_style.blue, x) - let magenta x = Bg (LTerm_style.magenta, x) - let cyan x = Bg (LTerm_style.cyan, x) - let white x = Bg (LTerm_style.white, x) - let orange x = Bg (LTerm_style.rgb 255 165 0, x) + let black x = Bg (Notty.A.black, x) + let red x = Bg (Notty.A.red, x) + let green x = Bg (Notty.A.green, x) + let yellow x = Bg (Notty.A.yellow, x) + let blue x = Bg (Notty.A.blue, x) + let magenta x = Bg (Notty.A.magenta, x) + let cyan x = Bg (Notty.A.cyan, x) + let white x = Bg (Notty.A.white, x) + let orange x = Bg (Notty.A.rgb_888 ~r:255 ~g:165 ~b:0, x) + let rgb r g b x = Bg (Notty.A.rgb_888 ~r ~g ~b, x) end end let style f x = f x let styles fs x = List.fold_left (fun acc f -> style f acc) x fs - let markup' (x : t) : LTerm_text.markup = - let rec aux : t -> LTerm_text.markup = function - | Space n -> [ LTerm_text.S (String.make n ' ') ] - | NonbreakingSpace n -> [ LTerm_text.S (String.make n ' ') ] - | Cut n -> [ LTerm_text.S (String.make n '\n') ] - | Exact s -> [ LTerm_text.S s ] - | Empty -> [] - | Concat xs -> List.concat (List.map aux xs) - | Fg (c, x) -> LTerm_text.([ B_fg c ] @ aux x @ [ E_fg ]) - | Bg (c, x) -> LTerm_text.([ B_bg c ] @ aux x @ [ E_bg ]) - | Bold x -> LTerm_text.([ B_bold true ] @ aux x @ [ E_bold ]) - | Italic x -> - aux x - (* Italic not supported by LTerm_text on all platforms; render plain *) - | Underline x -> - LTerm_text.([ B_underline true ] @ aux x @ [ E_underline ]) - | Marked (carats, x) -> aux carats @ aux x - | Boxed (box, xs) -> ( - match box with - | HBox -> List.concat (List.map aux xs) - | VBox -> - let rec intersperse sep = function - | [] -> [] - | [ y ] -> aux y - | y :: ys -> aux y @ (LTerm_text.S "\n" :: intersperse sep ys) - in - intersperse (LTerm_text.S "\n") xs - | HVBox -> - let rec intersperse_space = function - | [] -> [] - | [ y ] -> aux y - | y :: ys -> aux y @ (LTerm_text.S " " :: intersperse_space ys) - in - intersperse_space xs) - in - aux x - - let markup x = LTerm_text.eval (markup' x) + module Syntax = struct + let syntax = Style.bold + end let rec to_plain : t -> string = function | Space n | NonbreakingSpace n -> String.make n ' ' @@ -139,5 +107,59 @@ module Form : FORM = struct | VBox -> String.concat "\n" (List.map to_plain xs) | HVBox -> String.concat " " (List.map to_plain xs)) - let fmt ppf x = Format.pp_print_string ppf (to_plain x) + let rec go : Notty.attr -> t -> Notty.image = fun attr -> + let attr' : Notty.attr -> Notty.attr = Notty.A.((++) attr) in + Notty.(function + | Space n -> I.string attr (String.make n ' ') + | NonbreakingSpace n -> I.string attr (String.make n ' ') + | Cut n -> + let row = I.string attr "" in + let rows = List.init n (fun _ -> row) in + List.fold_left I.(<->) I.empty rows + | Exact s -> + (match String.split_on_char '\n' s with + | [] -> I.empty + | [line] -> I.string attr line + | lines -> I.vcat (List.map (I.string attr) lines)) + | Empty -> I.empty + | Concat xs -> I.(hcat (List.map (go attr) xs)) + | Fg (c, x) -> go (attr' (A.fg c)) x + | Bg (c, x) -> go (attr' (A.bg c)) x + | Bold x -> go (attr' @@ A.st A.bold) x + | Italic x -> go (attr' @@ A.st A.italic) x + | Underline x -> go (attr' @@ A.st A.underline) x + | Marked (carats, x) -> + let content_img = go attr x in + let carats_img = go attr carats in + I.(carats_img <-> content_img) + | Boxed (box, xs) -> + match box with + | HBox -> List.fold_left I.(<|>) I.empty (List.map (go attr) xs) + | VBox -> List.fold_left I.(<->) I.empty (List.map (go attr) xs) + | HVBox -> List.fold_left I.(<|>) I.empty (List.map (go attr) xs)) + + let show x = go Notty.A.empty x + let markup x = show x + let rec fmt ppf = function + | Space n | NonbreakingSpace n -> + Format.pp_print_string ppf (String.make n ' ') + | Cut n -> + for _ = 1 to n do Format.pp_force_newline ppf () done + | Exact s -> Format.pp_print_string ppf s + | Empty -> () + | Concat xs -> List.iter (fmt ppf) xs + | Fg (_, x) | Bg (_, x) | Bold x | Italic x | Underline x -> fmt ppf x + | Marked (_, x) -> fmt ppf x + | Custom f -> f ppf () + | Boxed (box, xs) -> + (match box with + | HBox -> Format.pp_open_hbox ppf () + | VBox -> Format.pp_open_vbox ppf 0 + | HVBox -> Format.pp_open_hvbox ppf 0); + List.iter (fmt ppf) xs; + Format.pp_close_box ppf () + + let header ~level:_ x = x + let custom f = Custom f end + \ No newline at end of file diff --git a/src/Display/Info.ml b/src/Display/Info.ml index 4221e44..524bfe8 100644 --- a/src/Display/Info.ml +++ b/src/Display/Info.ml @@ -17,17 +17,17 @@ type src = type kind = Debug | Info | Warning | Error | Response (* Corresponds to chatter 5 4 3 2 1 0, respectively *) -type level = VeryVerbose | Verbose | Normal | Quiet | VeryQuiet | Silent +type level = Exhaustive | Detailed | Normal | Terse | Minimal | Off let from_chatter x = assert (x >= 0); match x with - | 0 -> Silent - | 1 -> VeryQuiet - | 2 -> Quiet + | 0 -> Off + | 1 -> Minimal + | 2 -> Terse | 3 -> Normal - | 4 -> Verbose - | _ -> VeryVerbose + | 4 -> Detailed + | _ -> Exhaustive module Form = Form.Form @@ -39,12 +39,12 @@ let msg ?(src : src option) ?(kind : kind option) ?(level = Normal) (fmt : form) { src; kind; level; msg = fmt } let to_int : level -> int = function - | Silent -> 0 - | VeryQuiet -> 1 - | Quiet -> 2 + | Off -> 0 + | Minimal -> 1 + | Terse -> 2 | Normal -> 3 - | Verbose -> 4 - | VeryVerbose -> 5 + | Detailed -> 4 + | Exhaustive -> 5 let ( >= ) x y = to_int x >= to_int y let ( > ) x y = to_int x > to_int y diff --git a/src/Display/Info.mli b/src/Display/Info.mli index d99531b..ffa627e 100644 --- a/src/Display/Info.mli +++ b/src/Display/Info.mli @@ -17,7 +17,7 @@ type src = type kind = Debug | Info | Warning | Error | Response (* Corresponds to chatter 5 4 3 2 1 0, respectively *) -type level = VeryVerbose | Verbose | Normal | Quiet | VeryQuiet | Silent +type level = Exhaustive | Detailed | Normal | Terse | Minimal | Off module Form : module type of Form.Form diff --git a/src/Display/dune b/src/Display/dune index 04c31bc..ca3f7e7 100644 --- a/src/Display/dune +++ b/src/Display/dune @@ -1,4 +1,4 @@ (library (name display) (public_name stelf.display) - (libraries lwt fmt lambda-term)) + (libraries lwt fmt notty-community)) diff --git a/src/Frontend/Fquery.ml b/src/Frontend/Fquery.ml index a846bb4..4e871d0 100644 --- a/src/Frontend/Fquery.ml +++ b/src/Frontend/Fquery.ml @@ -56,23 +56,13 @@ end) : FQUERY with module ExtQuery = Fquery__0.ReconQuery = struct ReconQuery.queryToQuery (quy, Paths.Loc (fileName, r)) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string "%fquery")) + Display.chatter_s 3 "%fquery" in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string " ")) + Display.chatter_s 3 " " in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (Timers.time Timers.printing expToString (IntSyn.Null, v_) ^ ".\n"))) + Display.chatter_s 3 (Timers.time Timers.printing expToString (IntSyn.Null, v_) ^ ".\n") in let k, v1_ = Abstract.abstractDecImp v_ in let g_, v2_ = lower (k, I.Null, v1_) in diff --git a/src/Frontend/ReconCondec.ml b/src/Frontend/ReconCondec.ml index 726a295..8485815 100644 --- a/src/Frontend/ReconCondec.ml +++ b/src/Frontend/ReconCondec.ml @@ -100,11 +100,7 @@ end) : RECON_CONDEC = struct in let ocd = Paths.dec (i, oc) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (Timers.time Timers.printing Print.conDecToString cd ^ "\n"))) + Display.chatter_s 3 (Timers.time Timers.printing Print.conDecToString cd ^ "\n") in let _ = begin if !Global.doubleCheck then @@ -160,11 +156,7 @@ end) : RECON_CONDEC = struct end in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (Timers.time Timers.printing Print.conDecToString cd ^ "\n"))) + Display.chatter_s 3 (Timers.time Timers.printing Print.conDecToString cd ^ "\n") in let _ = begin if !Global.doubleCheck then begin @@ -261,11 +253,7 @@ end) : RECON_CONDEC = struct IntSyn.BlockDec (name, None, gsome'_, ctxToList (gblock'_, [])) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (Timers.time Timers.printing Print.conDecToString bd ^ "\n"))) + Display.chatter_s 3 (Timers.time Timers.printing Print.conDecToString bd ^ "\n") in (Some bd, None) (* closed nf *) @@ -288,11 +276,7 @@ end) : RECON_CONDEC = struct in let bd = IntSyn.BlockDef (name, None, w''_) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (Timers.time Timers.printing Print.conDecToString bd ^ "\n"))) + Display.chatter_s 3 (Timers.time Timers.printing Print.conDecToString bd ^ "\n") in (Some bd, None) diff --git a/src/Frontend/ReconQuery.ml b/src/Frontend/ReconQuery.ml index 9029494..bfab428 100644 --- a/src/Frontend/ReconQuery.ml +++ b/src/Frontend/ReconQuery.ml @@ -134,11 +134,7 @@ end) : RECON_QUERY = struct in let cd = Names.nameConDec cd in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (Timers.time Timers.printing Print.conDecToString cd ^ "\n"))) + Display.chatter_s 3 (Timers.time Timers.printing Print.conDecToString cd ^ "\n") in let _ = begin if !Global.doubleCheck then begin @@ -176,11 +172,7 @@ end) : RECON_QUERY = struct in let cd = Names.nameConDec cd in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (Timers.time Timers.printing Print.conDecToString cd ^ "\n"))) + Display.chatter_s 3 (Timers.time Timers.printing Print.conDecToString cd ^ "\n") in let _ = begin if !Global.doubleCheck then begin diff --git a/src/Frontend/Solve.ml b/src/Frontend/Solve.ml index b5759ac..f172de5 100644 --- a/src/Frontend/Solve.ml +++ b/src/Frontend/Solve.ml @@ -214,18 +214,10 @@ end) : SOLVE with module ExtQuery = Solve__0.ReconQuery = struct ReconQuery.solveToSolve (defines, solve_, Paths.Loc (fileName, r)) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string "%solve ")) + Display.chatter_s 3 "%solve " in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("\n" ^ Timers.time Timers.printing expToString (IntSyn.Null, a_)) - ^ ".\n"))) + Display.chatter_s 3 (("\n" ^ Timers.time Timers.printing expToString (IntSyn.Null, a_)) ^ ".\n") in let g = Timers.time Timers.compiling Compile.compileGoal (IntSyn.Null, a_) @@ -251,10 +243,7 @@ end) : SOLVE with module ExtQuery = Solve__0.ReconQuery = struct with Solution m_ -> ( try begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string " OK\n")); + Display.chatter_s 3 " OK\n"; finish m_ end with TimeLimit.TimeOut -> @@ -275,18 +264,10 @@ end) : SOLVE with module ExtQuery = Solve__0.ReconQuery = struct ReconQuery.solveToSolve (defines, solve_, Paths.Loc (fileName, r)) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string "%solve ")) + Display.chatter_s 3 "%solve " in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("\n" ^ Timers.time Timers.printing expToString (IntSyn.Null, a_)) - ^ ".\n"))) + Display.chatter_s 3 (("\n" ^ Timers.time Timers.printing expToString (IntSyn.Null, a_)) ^ ".\n") in let g = Timers.time Timers.compiling Compile.compileGoal (IntSyn.Null, a_) @@ -308,10 +289,7 @@ end) : SOLVE with module ExtQuery = Solve__0.ReconQuery = struct with SolutionSkel skel_ -> ( try begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 2) - (Display.Info.Form.string " OK\n")); + Display.chatter_s 2 " OK\n"; try begin Timers.time Timers.ptrecon PtRecon.solve @@ -345,27 +323,13 @@ end) : SOLVE with module ExtQuery = Solve__0.ReconQuery = struct ReconQuery.queryToQuery (quy, Paths.Loc (fileName, r)) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (((("%query " ^ boundToString expected) ^ " ") - ^ boundToString try_) - ^ "\n"))) + Display.chatter_s 3 (((("%query " ^ boundToString expected) ^ " ") ^ boundToString try_) ^ "\n") in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string " ")) + Display.chatter_s 4 " " in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("\n" ^ Timers.time Timers.printing expToString (IntSyn.Null, a_)) - ^ ".\n"))) + Display.chatter_s 3 (("\n" ^ Timers.time Timers.printing expToString (IntSyn.Null, a_)) ^ ".\n") in let g = Timers.time Timers.compiling Compile.compileGoal (IntSyn.Null, a_) @@ -376,29 +340,14 @@ end) : SOLVE with module ExtQuery = Solve__0.ReconQuery = struct solutions := !solutions + 1; begin begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("---------- Solution " ^ Int.toString !solutions) - ^ " ----------\n"))); - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (Timers.time Timers.printing evarInstToString xs_ ^ "\n"))) + Display.chatter_s 3 (("---------- Solution " ^ Int.toString !solutions) ^ " ----------\n"); + Display.chatter_s 3 (Timers.time Timers.printing evarInstToString xs_ ^ "\n") end; begin begin match optName with | None -> () | Some name -> begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (Timers.time Timers.printing evarInstToString - [ (m_, name) ] - ^ "\n"))) + Display.chatter_s 3 (Timers.time Timers.printing evarInstToString [ (m_, name) ] ^ "\n") end end; begin @@ -407,11 +356,7 @@ end) : SOLVE with module ExtQuery = Solve__0.ReconQuery = struct with | None -> () | Some str -> - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("Remaining constraints:\n" ^ str) ^ "\n"))) + Display.chatter_s 3 (("Remaining constraints:\n" ^ str) ^ "\n") end (* Question: should we collect constraints in M? *); begin if exceeds (Some !solutions, try_) then raise Done else () @@ -444,25 +389,12 @@ end) : SOLVE with module ExtQuery = Solve__0.ReconQuery = struct (* in case Done was raised *) (* check if number of solutions is correct *) else begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string "Skipping query (bound = 0)\n")); - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string "skipping")) + Display.chatter_s 3 "Skipping query (bound = 0)\n"; + Display.chatter_s 4 "skipping" end end; - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - "____________________________________________\n\n")); - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string " OK\n")) + Display.chatter_s 3 "____________________________________________\n\n"; + Display.chatter_s 4 " OK\n" end (* optName = SOME(X) or NONE, Xs = free variables in query excluding X *) @@ -488,27 +420,13 @@ end) : SOLVE with module ExtQuery = Solve__0.ReconQuery = struct ReconQuery.queryToQuery (quy, Paths.Loc (fileName, r)) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (((("%query " ^ boundToString expected) ^ " ") - ^ boundToString try_) - ^ "\n"))) + Display.chatter_s 3 (((("%query " ^ boundToString expected) ^ " ") ^ boundToString try_) ^ "\n") in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string " ")) + Display.chatter_s 4 " " in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("\n" ^ Timers.time Timers.printing expToString (IntSyn.Null, a_)) - ^ ".\n"))) + Display.chatter_s 3 (("\n" ^ Timers.time Timers.printing expToString (IntSyn.Null, a_)) ^ ".\n") in let g = Timers.time Timers.compiling Compile.compileGoal (IntSyn.Null, a_) @@ -519,17 +437,8 @@ end) : SOLVE with module ExtQuery = Solve__0.ReconQuery = struct solutions := !solutions + 1; begin begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("---------- Solution " ^ Int.toString !solutions) - ^ " ----------\n"))); - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (Timers.time Timers.printing evarInstToString xs_ ^ "\n"))) + Display.chatter_s 3 (("---------- Solution " ^ Int.toString !solutions) ^ " ----------\n"); + Display.chatter_s 3 (Timers.time Timers.printing evarInstToString xs_ ^ "\n") end; begin begin match optName with @@ -547,13 +456,7 @@ end) : SOLVE with module ExtQuery = Solve__0.ReconQuery = struct CompSyn.DProg (IntSyn.Null, IntSyn.Null), function | pskel, m_ -> begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (Timers.time Timers.printing evarInstToString - [ (m_, name) ] - ^ "\n"))) + Display.chatter_s 3 (Timers.time Timers.printing evarInstToString [ (m_, name) ] ^ "\n") end ) end end; @@ -563,11 +466,7 @@ end) : SOLVE with module ExtQuery = Solve__0.ReconQuery = struct with | None -> () | Some str -> - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("Remaining constraints:\n" ^ str) ^ "\n"))) + Display.chatter_s 3 (("Remaining constraints:\n" ^ str) ^ "\n") end (* Question: should we collect constraints in M? *); begin if exceeds (Some !solutions, try_) then raise Done else () @@ -600,25 +499,12 @@ end) : SOLVE with module ExtQuery = Solve__0.ReconQuery = struct (* in case Done was raised *) (* check if number of solutions is correct *) else begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string "Skipping query (bound = 0)\n")); - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string "skipping")) + Display.chatter_s 3 "Skipping query (bound = 0)\n"; + Display.chatter_s 4 "skipping" end end; - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - "____________________________________________\n\n")); - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string " OK\n")) + Display.chatter_s 3 "____________________________________________\n\n"; + Display.chatter_s 4 " OK\n" end (* optName = SOME(X) or NONE, Xs = free variables in query excluding X *) @@ -657,29 +543,16 @@ or %querytabled X : A solutions or if we have reached the maximal number of stages *) let rec querytabled ((numSol, try_, quy), Paths.Loc (fileName, r)) = let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - ((("%querytabled " ^ boundToString numSol) ^ " ") - ^ boundToString try_))) + Display.chatter_s 3 ((("%querytabled " ^ boundToString numSol) ^ " ") ^ boundToString try_) in let a_, optName, xs_ = ReconQuery.queryToQuery (quy, Paths.Loc (fileName, r)) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string " ")) + Display.chatter_s 4 " " in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("\n" ^ Timers.time Timers.printing expToString (IntSyn.Null, a_)) - ^ ".\n"))) + Display.chatter_s 3 (("\n" ^ Timers.time Timers.printing expToString (IntSyn.Null, a_)) ^ ".\n") in let g = Timers.time Timers.compiling Compile.compileGoal (IntSyn.Null, a_) @@ -695,21 +568,9 @@ or %querytabled X : A solExists := true; begin begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("\n---------- Solutions " ^ Int.toString !solutions) - ^ " ----------\n"))); - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (Timers.time Timers.printing evarInstToString xs_ ^ " \n"))); - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 1) - (Display.Info.Form.string ".")) + Display.chatter_s 3 (("\n---------- Solutions " ^ Int.toString !solutions) ^ " ----------\n"); + Display.chatter_s 3 (Timers.time Timers.printing evarInstToString xs_ ^ " \n"); + Display.chatter_s 1 "." end; begin begin match optName with @@ -722,13 +583,7 @@ or %querytabled X : A CompSyn.DProg (IntSyn.Null, IntSyn.Null), function | o_, m_ -> begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (Timers.time Timers.printing evarInstToString - [ (m_, name) ] - ^ "\n"))) + Display.chatter_s 3 (Timers.time Timers.printing evarInstToString [ (m_, name) ] ^ "\n") end ) end end; @@ -738,27 +593,16 @@ or %querytabled X : A with | None -> () | Some str -> - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("Remaining constraints:\n" ^ str) ^ "\n"))) + Display.chatter_s 3 (("Remaining constraints:\n" ^ str) ^ "\n") end (* Question: should we collect constraints in M? *); begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 1) - (Display.Info.Form.string "More solutions?\n")); + Display.chatter_s 1 "More solutions?\n"; begin match numSol with | None -> () | Some n -> begin if !solutions = n then begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 1) - (Display.Info.Form.string - "Found enough solutions\n")); + Display.chatter_s 1 "Found enough solutions\n"; raise Done end else () @@ -774,12 +618,7 @@ or %querytabled X : A let rec loop () = begin begin if exceeds (Some (!stages - 1), try_) then begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 1) - (Display.Info.Form.string - (("\n ================= " ^ " Number of tries exceeds stages ") - ^ " ======================= \n"))); + Display.chatter_s 1 (("\n ================= " ^ " Number of tries exceeds stages ") ^ " ======================= \n"); begin status := false; raise Done @@ -788,12 +627,7 @@ or %querytabled X : A else () end; begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 1) - (Display.Info.Form.string - (("\n ====================== Stage " ^ Int.toString !stages) - ^ " finished =================== \n"))); + Display.chatter_s 1 (("\n ====================== Stage " ^ Int.toString !stages) ^ " finished =================== \n"); begin begin if exceeds (Some !stages, try_) then begin Msg.message @@ -858,101 +692,38 @@ or %querytabled X : A (* solve query if bound > 0 *) with Done -> () else begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string "Skipping query (bound = 0)\n")); - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 2) - (Display.Info.Form.string "skipping")) + Display.chatter_s 3 "Skipping query (bound = 0)\n"; + Display.chatter_s 2 "skipping" end end; begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - "\n____________________________________________\n\n")); - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - ((((("number of stages: tried " ^ boundToString try_) ^ " \n") - ^ "terminated after ") - ^ Int.toString !stages) - ^ " stages \n \n"))); + Display.chatter_s 3 "\n____________________________________________\n\n"; + Display.chatter_s 3 ((((("number of stages: tried " ^ boundToString try_) ^ " \n") ^ "terminated after ") ^ Int.toString !stages) ^ " stages \n \n"); begin if !solExists then () else - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string "\nNO solution exists to query \n\n")) + Display.chatter_s 3 "\nNO solution exists to query \n\n" end; begin if !status then - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string "Tabled evaluation COMPLETE \n \n")) + Display.chatter_s 3 "Tabled evaluation COMPLETE \n \n" else - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string "Tabled evaluation NOT COMPLETE \n \n")) + Display.chatter_s 3 "Tabled evaluation NOT COMPLETE \n \n" end; - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - "\n____________________________________________\n\n")); - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string "\n Table Indexing parameters: \n")); + Display.chatter_s 3 "\n____________________________________________\n\n"; + Display.chatter_s 3 "\n Table Indexing parameters: \n"; begin match !TableParam.strategy with | variant_ -> - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - "\n Table Strategy := Variant \n")) + Display.chatter_s 3 "\n Table Strategy := Variant \n" | subsumption_ -> - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - "\n Table Strategy := Subsumption \n")) + Display.chatter_s 3 "\n Table Strategy := Subsumption \n" end; begin if !TableParam.strengthen then - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string "\n Strengthening := true \n")) + Display.chatter_s 3 "\n Strengthening := true \n" else - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string "\n Strengthening := false \n")) + Display.chatter_s 3 "\n Strengthening := false \n" end; - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("\nNumber of table indices : " - ^ Int.toString (Tabled.tableSize ())) - ^ "\n"))); - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("Number of suspended goals : " - ^ Int.toString (Tabled.suspGoalNo ())) - ^ "\n"))); - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - "\n____________________________________________\n\n")); + Display.chatter_s 3 (("\nNumber of table indices : " ^ Int.toString (Tabled.tableSize ())) ^ "\n"); + Display.chatter_s 3 (("Number of suspended goals : " ^ Int.toString (Tabled.suspGoalNo ())) ^ "\n"); + Display.chatter_s 3 "\n____________________________________________\n\n"; Tabled.updateGlobalTable (g, !status) end end @@ -996,22 +767,12 @@ or %querytabled X : A in let rec scInit m_ = begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 1) - (Display.Info.Form.string - (Timers.time Timers.printing evarInstToString xs_ ^ "\n"))); + Display.chatter_s 1 (Timers.time Timers.printing evarInstToString xs_ ^ "\n"); begin begin match optName with | None -> () | Some name -> begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (Timers.time Timers.printing evarInstToString - [ (m_, name) ] - ^ "\n"))) + Display.chatter_s 3 (Timers.time Timers.printing evarInstToString [ (m_, name) ] ^ "\n") end end; begin @@ -1020,11 +781,7 @@ or %querytabled X : A with | None -> () | Some str -> - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("Remaining constraints:\n" ^ str) ^ "\n"))) + Display.chatter_s 3 (("Remaining constraints:\n" ^ str) ^ "\n") end (* Question: should we collect constraints from M *); begin if moreSolutions () then () else raise Done @@ -1034,10 +791,7 @@ or %querytabled X : A end in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string "Solving...\n")) + Display.chatter_s 3 "Solving...\n" in begin try begin @@ -1080,20 +834,12 @@ or %querytabled X : A let _ = Tabled.reset () in let rec scInit o_ = begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 1) - (Display.Info.Form.string - (Timers.time Timers.printing evarInstToString xs_ ^ "\n"))); + Display.chatter_s 1 (Timers.time Timers.printing evarInstToString xs_ ^ "\n"); begin begin match optName with | None -> () | Some name -> begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - " Sorry cannot reconstruct pskeleton proof terms yet \n")) + Display.chatter_s 3 " Sorry cannot reconstruct pskeleton proof terms yet \n" end end; begin @@ -1102,11 +848,7 @@ or %querytabled X : A with | None -> () | Some str -> - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("Remaining constraints:\n" ^ str) ^ "\n"))) + Display.chatter_s 3 (("Remaining constraints:\n" ^ str) ^ "\n") end (* Question: should we collect constraints from M? *); begin @@ -1127,10 +869,7 @@ or %querytabled X : A *) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string "Solving...\n")) + Display.chatter_s 3 "Solving...\n" in begin try begin diff --git a/src/Frontend/Twelf_.ml b/src/Frontend/Twelf_.ml index 2fffa47..1bd8f76 100644 --- a/src/Frontend/Twelf_.ml +++ b/src/Frontend/Twelf_.ml @@ -429,11 +429,7 @@ end) : TWELF.STELF = struct let rec installAction ((cid, _) as data) = begin installConst IntSyn.Ordinary data; - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string - (Print.conDecToString (IntSyn.sgnLookup cid) ^ "\n"))) + Display.chatter_s 4 (Print.conDecToString (IntSyn.sgnLookup cid) ^ "\n") end in let _ = @@ -447,11 +443,7 @@ end) : TWELF.STELF = struct let rec installAction ((cid, _) as data) = begin installConst IntSyn.Ordinary data; - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string - (Print.conDecToString (IntSyn.sgnLookup cid) ^ "\n"))) + Display.chatter_s 4 (Print.conDecToString (IntSyn.sgnLookup cid) ^ "\n") end in let _ = @@ -617,20 +609,7 @@ end) : TWELF.STELF = struct with Subordinate.Error msg -> raise (Subordinate.Error (Paths.wrap (r, msg))) in - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - ("%subord" - ^ List.foldr - (function - | (a1, a2), s -> - ((((" (" ^ Names.qidToString (Names.constQid a1)) - ^ " ") - ^ Names.qidToString (Names.constQid a2)) - ^ ")") - ^ s) - ".\n" cidpairs))) + Display.chatter_s 3 ("%subord" ^ List.foldr (function | (a1, a2), s -> ((((" (" ^ Names.qidToString (Names.constQid a1)) ^ " ") ^ Names.qidToString (Names.constQid a2)) ^ ")") ^ s) ".\n" cidpairs) | fileName, (Parser.FreezeDec qids, r) -> let rec toCid qid = begin match Names.constLookup qid with @@ -653,26 +632,8 @@ end) : TWELF.STELF = struct raise (Subordinate.Error (Paths.wrap (r, msg))) in begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - ("%freeze" - ^ List.foldr - (function - | a, s -> - (" " ^ Names.qidToString (Names.constQid a)) ^ s) - ".\n" cids))); - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string - ("Frozen:" - ^ List.foldr - (function - | a, s -> - (" " ^ Names.qidToString (Names.constQid a)) ^ s) - "\n" frozen))) + Display.chatter_s 3 ("%freeze" ^ List.foldr (function | a, s -> (" " ^ Names.qidToString (Names.constQid a)) ^ s) ".\n" cids); + Display.chatter_s 4 ("Frozen:" ^ List.foldr (function | a, s -> (" " ^ Names.qidToString (Names.constQid a)) ^ s) "\n" frozen) end | fileName, (Parser.ThawDec qids, r) -> let _ = @@ -702,24 +663,10 @@ end) : TWELF.STELF = struct raise (Subordinate.Error (Paths.wrap (r, msg))) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - ("%thaw" - ^ List.foldr - (function a, s -> (" " ^ cidToString a) ^ s) - ".\n" cids))) + Display.chatter_s 3 ("%thaw" ^ List.foldr (function a, s -> (" " ^ cidToString a) ^ s) ".\n" cids) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string - ("Thawed" - ^ List.foldr - (function a, s -> (" " ^ cidToString a) ^ s) - "\n" thawed))) + Display.chatter_s 4 ("Thawed" ^ List.foldr (function a, s -> (" " ^ cidToString a) ^ s) "\n" thawed) in let _ = invalidate WorldSyn.uninstall thawed "world" in let _ = invalidate Thm.uninstallTerminates thawed "termination" in @@ -746,18 +693,7 @@ end) : TWELF.STELF = struct in begin List.app insertCid cids; - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - ((begin if !Global.chatter >= 4 then "%" else "" - end - ^ "%deterministic") - ^ List.foldr - (function - | a, s -> - (" " ^ Names.qidToString (Names.constQid a)) ^ s) - ".\n" cids))) + Display.chatter_s 3 ((begin if !Global.chatter >= 4 then "%" else "" end ^ "%deterministic") ^ List.foldr (function | a, s -> (" " ^ Names.qidToString (Names.constQid a)) ^ s) ".\n" cids) end | fileName, (Parser.Compile qids, r) -> let rec toCid qid = @@ -789,26 +725,10 @@ end) : TWELF.STELF = struct let _ = TomegaTypeCheck.checkPrg (IntSyn.Null, (p_, f_)) in let rec f cid = IntSyn.conDecName (IntSyn.sgnLookup cid) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 2) - (Display.Info.Form.string - (("\n" ^ TomegaPrint.funToString ((map f cids, projs), p_)) - ^ "\n"))) + Display.chatter_s 2 (("\n" ^ TomegaPrint.funToString ((map f cids, projs), p_)) ^ "\n") in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - ((begin if !Global.chatter >= 4 then "%" else "" - end - ^ "%compile") - ^ List.foldr - (function - | a, s -> - (" " ^ Names.qidToString (Names.constQid a)) ^ s) - ".\n" cids))) + Display.chatter_s 3 ((begin if !Global.chatter >= 4 then "%" else "" end ^ "%compile") ^ List.foldr (function | a, s -> (" " ^ Names.qidToString (Names.constQid a)) ^ s) ".\n" cids) in () | fileName, (Parser.FixDec ((qid, r), fixity), _) -> @@ -823,16 +743,7 @@ end) : TWELF.STELF = struct try begin Names.installFixity (cid, fixity); - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - ((((begin if !Global.chatter >= 4 then "%" else "" - end - ^ Names.Fixity.toString fixity) - ^ " ") - ^ Names.qidToString (Names.constQid cid)) - ^ ".\n"))) + Display.chatter_s 3 ((((begin if !Global.chatter >= 4 then "%" else "" end ^ Names.Fixity.toString fixity) ^ " ") ^ Names.qidToString (Names.constQid cid)) ^ ".\n") end with Names.Error msg -> raise (Names.Error (Paths.wrap (r, msg)))) @@ -898,14 +809,7 @@ end) : TWELF.STELF = struct mdecs in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("%mode " - ^ ModePrint.modesToString - (List.map (function mdec, r -> mdec) mdecs)) - ^ ".\n"))) + Display.chatter_s 3 (("%mode " ^ ModePrint.modesToString (List.map (function mdec, r -> mdec) mdecs)) ^ ".\n") in () | fileName, (Parser.UniqueDec mterms, r) -> @@ -937,14 +841,7 @@ end) : TWELF.STELF = struct mdecs in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("%unique " - ^ ModePrint.modesToString - (List.map (function mdec, r -> mdec) mdecs)) - ^ ".\n"))) + Display.chatter_s 3 (("%unique " ^ ModePrint.modesToString (List.map (function mdec, r -> mdec) mdecs)) ^ ".\n") in () | fileName, (Parser.CoversDec mterms, r) -> @@ -961,14 +858,7 @@ end) : TWELF.STELF = struct mdecs in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("%covers " - ^ ModePrint.modesToString - (List.map (function mdec, r -> mdec) mdecs)) - ^ ".\n"))) + Display.chatter_s 3 (("%covers " ^ ModePrint.modesToString (List.map (function mdec, r -> mdec) mdecs)) ^ ".\n") in () | fileName, (Parser.TotalDec lterm, r) -> @@ -984,11 +874,7 @@ end) : TWELF.STELF = struct raise (Subordinate.Error (Paths.wrap (r, msg))) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("%total " ^ ThmPrint.tDeclToString t_) ^ ".\n"))) + Display.chatter_s 3 (("%total " ^ ThmPrint.tDeclToString t_) ^ ".\n") in () | fileName, (Parser.TerminatesDec lterm, _) -> @@ -1005,11 +891,7 @@ end) : TWELF.STELF = struct end in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("%terminates " ^ ThmPrint.tDeclToString t_) ^ ".\n"))) + Display.chatter_s 3 (("%terminates " ^ ThmPrint.tDeclToString t_) ^ ".\n") in () | fileName, (Parser.ReducesDec lterm, _) -> @@ -1038,23 +920,14 @@ end) : TWELF.STELF = struct let t_, r = ReconThm.tableddeclTotabledDecl tdecl in let la_ = Thm.installTabled t_ in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("%tabled " ^ ThmPrint.tabledDeclToString t_) ^ ".\n"))) + Display.chatter_s 3 (("%tabled " ^ ThmPrint.tabledDeclToString t_) ^ ".\n") in () | fileName, (Parser.KeepTableDec tdecl, _) -> let t_, r = ReconThm.keepTabledeclToktDecl tdecl in let la_ = Thm.installKeepTable t_ in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("%keeptabled " ^ ThmPrint.keepTableDeclToString t_) - ^ ".\n"))) + Display.chatter_s 3 (("%keeptabled " ^ ThmPrint.keepTableDeclToString t_) ^ ".\n") in () | fileName, (Parser.TheoremDec tdec, r) -> @@ -1092,24 +965,14 @@ end) : TWELF.STELF = struct in let _ = ModeTable.installMode (cid, convert_mode_spine ms_) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("%theorem " ^ Print.conDecToString e_) ^ "\n"))) + Display.chatter_s 3 (("%theorem " ^ Print.conDecToString e_) ^ "\n") in () | fileName, (Parser.ProveDec lterm, r) -> let ThmSyn.PDecl (depth, t_), rrs = ReconThm.proveToProve lterm in let la_ = Thm.installTerminates (t_, rrs) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (((("%prove " ^ Int.toString depth) ^ " ") - ^ ThmPrint.tDeclToString t_) - ^ ".\n"))) + Display.chatter_s 3 (((("%prove " ^ Int.toString depth) ^ " ") ^ ThmPrint.tDeclToString t_) ^ ".\n") in let _ = Prover.init (depth, la_) in let _ = @@ -1117,14 +980,7 @@ end) : TWELF.STELF = struct map (function | a -> - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("%mode " - ^ ModePrint.modeToString - (a, valOf (ModeTable.modeLookup a))) - ^ ".\n")))) + Display.chatter_s 3 (("%mode " ^ ModePrint.modeToString (a, valOf (ModeTable.modeLookup a))) ^ ".\n")) la_ else [ () ] in @@ -1134,10 +990,7 @@ end) : TWELF.STELF = struct raise (Prover.Error (Paths.wrap (joinregion rrs, msg))) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string "%QED\n")) + Display.chatter_s 3 "%QED\n" in begin Prover.install (function e_ -> @@ -1164,14 +1017,7 @@ end) : TWELF.STELF = struct map (function | a -> - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("%mode " - ^ ModePrint.modeToString - (a, valOf (ModeTable.modeLookup a))) - ^ ".\n")))) + Display.chatter_s 3 (("%mode " ^ ModePrint.modeToString (a, valOf (ModeTable.modeLookup a))) ^ ".\n")) la_ else [ () ] in @@ -1192,25 +1038,14 @@ end) : TWELF.STELF = struct let (ThmSyn.Callpats l_ as cp), rrs = ReconThm.assertToAssert aterm in let la_ = map (function c, p_ -> c) l_ in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("%assert " ^ ThmPrint.callpatsToString cp) ^ ".\n"))) + Display.chatter_s 3 (("%assert " ^ ThmPrint.callpatsToString cp) ^ ".\n") in let _ = if !Global.chatter >= 3 then map (function | a -> - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (("%mode " - ^ ModePrint.modeToString - (a, valOf (ModeTable.modeLookup a))) - ^ ".\n")))) + Display.chatter_s 3 (("%mode " ^ ModePrint.modeToString (a, valOf (ModeTable.modeLookup a))) ^ ".\n")) la_ else [ () ] in @@ -1281,13 +1116,7 @@ end) : TWELF.STELF = struct end in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (((("%worlds " ^ Print.worldsToString w_) ^ " ") - ^ ThmPrint.callpatsToString cp) - ^ ".\n"))) + Display.chatter_s 3 (((("%worlds " ^ Print.worldsToString w_) ^ " ") ^ ThmPrint.callpatsToString cp) ^ ".\n") in begin Timers.time Timers.worlds @@ -1336,10 +1165,7 @@ end) : TWELF.STELF = struct raise (ModSyn.Error (Paths.wrap (r, msg))) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string (("%sig " ^ name) ^ " = { ... }.\n"))) + Display.chatter_s 3 (("%sig " ^ name) ^ " = { ... }.\n") in () | fileName, moduleOpt, (Parser.StructDec structdec, r) -> @@ -1428,10 +1254,7 @@ end) : TWELF.STELF = struct let oldContext = !context in let _ = context := Some namespace in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string "\n% begin subsignature\n")) + Display.chatter_s 4 "\n% begin subsignature\n" in let rec install s = install' (Timers.time Timers.parsing S.expose s) and install' = function @@ -1448,10 +1271,7 @@ end) : TWELF.STELF = struct let s' = install s in let module_ = ModSyn.abstractModule (namespace, None) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string "% end subsignature\n\n")) + Display.chatter_s 4 "% end subsignature\n\n" in Value (module_, s') with exn -> Exception exn @@ -1567,16 +1387,7 @@ end) : TWELF.STELF = struct | Some fixity -> begin let fixity' = convert_fixity fixity in Names.installFixity (cid, fixity'); - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - ((((begin if !Global.chatter >= 4 then "%" else "" - end - ^ Names.Fixity.toString fixity') - ^ " ") - ^ Names.qidToString (Names.constQid cid)) - ^ ".\n"))) + Display.chatter_s 3 ((((begin if !Global.chatter >= 4 then "%" else "" end ^ Names.Fixity.toString fixity') ^ " ") ^ Names.qidToString (Names.constQid cid)) ^ ".\n") end | None -> () end diff --git a/src/Fronts/Modern/Cmd.ml b/src/Fronts/Modern/Cmd.ml index 3164c4a..a304a62 100644 --- a/src/Fronts/Modern/Cmd.ml +++ b/src/Fronts/Modern/Cmd.ml @@ -12,8 +12,15 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct open Parser - (* Skip outer text (non-% characters) between commands *) - let skip_outer : unit t = skip_while (fun c -> c <> '%') + let ghost' = Cst.View.Loc.(review Ghost) + + (* Skip outer text (non-% characters) between commands. + %%% is an escape sequence for a literal %. *) + let skip_outer : unit t = + fix (fun self -> + skip_while (fun c -> c <> '%') + *> option () + (string "%%%" *> commit *> self)) (* Defer a thunk-parser to prevent infinite recursion at construction time. Used for %module and %eval which recursively embed cmd lists. *) @@ -55,35 +62,35 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct choice ~failure_msg:"command" [ begin - whitespace *> (keyword "." *> commit *> return (Cst.Cmd.stop ())) + whitespace *> (keyword "." *> commit *> skip_outer *> return (Cst.View.Cmd.(review @@ Stop (ghost', ())))) (* querytabled BEFORE query — "query" is a prefix of "querytabled" *) end; begin (keyword "querytabled" *> commit *> let+ n, b, d, q = Modern.parse_query () in - Cst.Cmd.query_tabled ~n ~b ~d q) + Cst.View.Cmd.(review @@ QueryTabled (ghost', n, b, d, q))) "querytabled" end; begin (keyword "query" *> commit *> let+ n, b, d, q = Modern.parse_query () in - Cst.Cmd.query ~n ~b ~d q) + Cst.View.Cmd.(review @@ Query (ghost', n, b, d, q))) "query" end; begin (keyword "?" *> commit *> let+ tm = Modern.parse_expr () in - Cst.Cmd.adhoc_query (Cst.Query.query None tm)) + Cst.View.Cmd.(review @@ AdhocQuery (ghost', Cst.View.Query.(review @@ Query (ghost', None, tm))))) "adhoc query" end; begin (keyword "unique" *> commit *> let+ tm = Modern.parse_expr () in - Cst.Cmd.unique tm + Cst.View.Cmd.(review @@ Unique (ghost', tm)) (* module BEFORE mode — "mode" is a prefix of "module" *)) "unique" end; @@ -91,14 +98,15 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct keyword "scope" *> commit *> let* id = Modern.parse_var () in - let+ cmds = parse1 () in - Cst.View.Cmd.(review @@ Scope (Cst.View.Loc.(review Ghost), id, cmds)) + let+ cmds = parse_cmd_list () in + Cst.View.Cmd.( + review @@ Scope (Cst.View.Loc.(review Ghost), id, review @@ Eval (ghost', cmds))) end; begin (keyword "mode" *> commit *> let+ md = Modern.parse_mode_dec () in - Cst.Cmd.mode md (* TODO Check this *)) + Cst.View.Cmd.(review @@ Mode (ghost', md)) (* TODO Check this *)) "mode" end; begin @@ -113,7 +121,7 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct (keyword "decl" *> commit *> let+ tm = Modern.parse_expr () in - Cst.Cmd.decl_cmd tm) + Cst.View.Cmd.(review @@ DeclCmd (ghost', tm))) "declaration" end; begin @@ -121,7 +129,7 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct *> let* id = Modern.parse_var () in let+ tm = Modern.parse_expr () in - Cst.Cmd.inline id tm) + Cst.View.Cmd.(review @@ Inline (ghost', id, tm))) "inline" end; begin @@ -129,21 +137,21 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct *> let* id1 = Modern.parse_var () in let+ id2 = Modern.parse_var () in - Cst.Cmd.symbol id1 id2) + Cst.View.Cmd.(review @@ Symbol (ghost', id1, id2))) "symbol" end; begin (keyword "freeze" *> commit *> let+ ids = Modern.parse_id_list () in - Cst.Cmd.freeze ids) + Cst.View.Cmd.(review @@ Freeze (ghost', ids))) "freeze" end; begin (keyword "thaw" *> commit *> let+ ids = Modern.parse_id_list () in - Cst.Cmd.thaw ids) + Cst.View.Cmd.(review @@ Thaw (ghost', ids))) "thaw" end; begin @@ -158,7 +166,7 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct (keyword "term" *> commit *> let+ d = Modern.parse_decl () in - Cst.Cmd.term d) + Cst.View.Cmd.(review @@ Term (ghost', d))) "term" end; begin @@ -166,7 +174,7 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct *> let* id = Modern.parse_var () in let+ items = many (Modern.parse_block_item ()) in - Cst.Cmd.block id items) + Cst.View.Cmd.(review @@ Block (ghost', id, items))) "block" end; begin @@ -174,7 +182,7 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct *> let* id = Modern.parse_var () in let+ ids = inside "(" ")" (many (Modern.parse_var ())) in - Cst.Cmd.union id ids) + Cst.View.Cmd.(review @@ Union (ghost', id, ids))) "union" end; begin @@ -182,14 +190,14 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct *> let* ids = inside "(" ")" (many (Modern.parse_var ())) in let+ tm = Modern.parse_expr () in - Cst.Cmd.worlds ids tm) + Cst.View.Cmd.(review @@ Worlds (ghost', ids, tm))) "worlds" end; begin (keyword "deterministic" *> commit *> let+ ids = Modern.parse_id_list () in - Cst.Cmd.deterministic ids) + Cst.View.Cmd.(review @@ Deterministic (ghost', ids))) "deterministic" end; begin @@ -212,7 +220,7 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct (keyword "eval" *> commit *> let+ cmds = parse_cmd_list () in - Cst.Cmd.eval cmds) + Cst.View.Cmd.(review @@ Eval (ghost', cmds))) "eval" end; begin @@ -222,18 +230,18 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct let* n = Modern.parse_fixity () in let* ids = Modern.parse_id_list () in let () = Modern.register_local_fixity fix n ids in - return (Cst.Cmd.prec fix n ids)) + return (Cst.View.Cmd.(review @@ Prec (ghost', fix, n, ids)))) "prec" end; begin (keyword "solve" *> commit *> let+ s = Modern.parse_solve () in - Cst.Cmd.solve s) + Cst.View.Cmd.(review @@ Solve (ghost', s))) "solve" end; begin - keyword "quit" *> commit *> return (Cst.Cmd.Repl.quit ()) "quit" + keyword "quit" *> commit *> return (Cst.View.Cmd.(review @@ ReplQuit (ghost', ()))) "quit" end; begin (keyword "help" *> commit @@ -243,14 +251,14 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct (let+ id = Modern.parse_var () in Some id) in - Cst.Cmd.Repl.help t) + Cst.View.Cmd.(review @@ ReplHelp (ghost', t))) "help" end; begin (keyword "get" *> commit *> let+ id = Modern.parse_var () in - Cst.Cmd.Repl.get id) + Cst.View.Cmd.(review @@ ReplGet (ghost', id))) "get" end; begin @@ -258,11 +266,11 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct *> let* id = Modern.parse_var () in let+ v = Modern.parse_var () in - Cst.Cmd.Repl.set id v) + Cst.View.Cmd.(review @@ ReplSet (ghost', id, v))) "set" end; begin - keyword "version" *> commit *> return (Cst.Cmd.Repl.version ()) + keyword "version" *> commit *> return (Cst.View.Cmd.(review @@ ReplVersion (ghost', ()))) "version" end; begin @@ -287,14 +295,14 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct (keyword "covers" *> commit *> let+ md = Modern.parse_mode_dec () in - Cst.Cmd.covers md) + Cst.View.Cmd.(review @@ Covers (ghost', md))) "covers" end; begin (keyword "name" *> commit *> let+ id = Modern.parse_var () in - Cst.Cmd.name id) + Cst.View.Cmd.(review @@ Name (ghost', id))) "name" end; begin diff --git a/src/Fronts/Modern/MODERN.ml b/src/Fronts/Modern/MODERN.ml index 5abe2b3..9f227f2 100644 --- a/src/Fronts/Modern/MODERN.ml +++ b/src/Fronts/Modern/MODERN.ml @@ -10,6 +10,16 @@ module type MODERN = sig exception ParseError of string + exception FullParseError of { + title : Display.form option; + subtitle : Display.form option; + body : Display.form; + loc : Cst.loc option; + } + + (** A list of symbols that refer to restricted names *) + val given_symbols : (string * string) list ref + val parse_expr1 : unit -> Cst.Term.t t val parse_expr : unit -> Cst.Term.t t val parse_var : unit -> string t @@ -18,21 +28,20 @@ module type MODERN = sig (** {v %val ( ... ) v} *) val parse_text : unit -> string t - val parse_decl : unit -> Cst.Decl.t t - val parse_mode : unit -> Cst.Mode.mode t - val parse_mode_dec : unit -> Cst.Mode.modedec t - val parse_sigexp : unit -> Cst.Struct.sigexp t - val parse_inst : unit -> Cst.Struct.inst t - val parse_sigexp : unit -> Cst.Struct.sigexp t - val parse_sigdef : unit -> Cst.Struct.sigdef t - val parse_struct_dec : unit -> Cst.Struct.structdec t + val parse_decl : unit -> Cst.decl t + val parse_mode : unit -> Cst.mode t + val parse_mode_dec : unit -> Cst.modeDec t + val parse_sigexp : unit -> Cst.sigexp t + val parse_inst : unit -> Cst.inst t + val parse_sigdef : unit -> Cst.sigdef t + val parse_struct_dec : unit -> Cst.structDec t val parse_fixity : unit -> int t val parse_query : - unit -> (int option * int option * int option * Cst.Query.query) t + unit -> (int option * int option * int option * Cst.query) t val parse_define : unit -> Cst.define t - val parse_solve : unit -> Cst.Query.solve t + val parse_solve : unit -> Cst.solve t val parse_bound : unit -> int option t val parse_id_list : unit -> string list t val parse_reduces_rel : unit -> string t diff --git a/src/Fronts/Modern/Modern.ml b/src/Fronts/Modern/Modern.ml index 16a73d4..a6e13a0 100644 --- a/src/Fronts/Modern/Modern.ml +++ b/src/Fronts/Modern/Modern.ml @@ -13,12 +13,28 @@ module Make_Modern module N = Names module Parser = Parser + type arrow_last = LeftArrow | RightArrow | NoArrow + exception ParseError of string + exception FullParseError of { + title : Display.form option; + subtitle : Display.form option; + body: Display.form; + loc: Cst.loc option; + } open Parser + let given_symbols : (string * string) list ref = ref [] let currently_uppercase : string list ref = ref [] + let keyword' kw = + keyword kw + <|> choice + (List.filter_map + (fun (sym, kw') -> if kw = kw' then Some (token sym) else None) + !given_symbols) + let with_uppercase : string list -> (unit -> 'a t) -> 'a t = fun names p -> let old = !currently_uppercase in @@ -36,7 +52,7 @@ module Make_Modern let y, z = break xs in (x :: y, z) - let mk_loc : int -> int -> Cst.loc = fun x y -> Cst.mk_loc x y + let mk_loc : int -> int -> Cst.loc = fun x y -> Cst.View.mk_loc x y let whitespace' () : unit t = whitespace <|> forget @@ (token "% " *> take_till (fun c -> c = '\n')) @@ -74,15 +90,25 @@ module Make_Modern | Postfix_ of FX.precedence * (Cst.Term.t -> Cst.Term.t) let jux_op = - Infix_ ((FX.inc FX.maxPrec, FX.Left), fun (f, x) -> Cst.Term.app f [ x ]) + Infix_ + ( (FX.inc FX.maxPrec, FX.Left), + fun (f, x) -> Cst.View.Term.(review @@ App (ghost', f, [ x ])) ) let infix_op (infixity, tm) = Infix_ ( infixity, - fun (tm1, tm2) -> Cst.Term.app (Cst.Term.app tm [ tm1 ]) [ tm2 ] ) + fun (tm1, tm2) -> + Cst.View.Term.( + review @@ App (ghost', review @@ App (ghost', tm, [ tm1 ]), [ tm2 ])) + ) + + let prefix_op (prec, tm) = + Prefix_ + (prec, fun tm1 -> Cst.View.Term.(review @@ App (ghost', tm, [ tm1 ]))) - let prefix_op (prec, tm) = Prefix_ (prec, fun tm1 -> Cst.Term.app tm [ tm1 ]) - let postfix_op (prec, tm) = Postfix_ (prec, fun tm1 -> Cst.Term.app tm [ tm1 ]) + let postfix_op (prec, tm) = + Postfix_ + (prec, fun tm1 -> Cst.View.Term.(review @@ App (ghost', tm, [ tm1 ]))) let classify (tm : Cst.Term.t) : operator = let open Cst.View.Term in @@ -214,21 +240,27 @@ module Make_Modern "argument" and parse_id () : Cst.Term.t t = - keyword "val" *> commit + keyword' "val" *> commit *> (inside "(" ")" (let@ ns, s, e = many1 ident1 in let loc = mk_loc s e in - let rec split = function + let split = function | [] -> failwith "Expected qualified name" - | [ p ] -> ([], p) - | p :: q -> - let q0, q1 = split q in - (p :: q0, q1) + | name :: scopes -> (List.rev scopes, name) in - return (Cst.Term.qualified ~fc:loc (split ns))) + return Cst.View.Term.(review @@ Qualified (loc, split ns))) <|> let@ name, s, e = ident1 in let loc = mk_loc s e in - return (Cst.Term.qualified ~fc:loc ([], name))) + return Cst.View.Term.(review @@ Qualified (loc, ([], name)))) + <|> + (string "%(" *> commit + *> let@ ns, s, e = many1 ident1 <* string ")" <* whitespace in + let loc = mk_loc s e in + let split = function + | [] -> failwith "Expected qualified name" + | name :: scopes -> (List.rev scopes, name) + in + return Cst.View.Term.(review @@ Qualified (loc, split ns))) <|> let@ name, s, e = ident1 in let loc = mk_loc s e in @@ -238,22 +270,75 @@ module Make_Modern in return (if is_upper || List.mem name !currently_uppercase then - Cst.Term.uppercase ~fc:loc ([], name) - else Cst.Term.lowercase ~fc:loc ([], name)) + Cst.View.Term.(review @@ Uppercase (loc, ([], name))) + else Cst.View.Term.(review @@ Lowercase (loc, ([], name)))) and parse_expr_trail () : Cst.Term.t t = (let@ d, s, e = inside "[" "]" (parse_decl ()) in let loc = mk_loc s e in let+ body = parse_expr () in - Cst.Term.lam ~fc:loc [ d ] body) + Cst.View.Term.(review @@ Lam (loc, [ d ], body))) <|> (let@ d, s, e = inside "{" "}" (parse_decl ()) in let loc = mk_loc s e in let+ body = parse_expr () in - Cst.Term.pi ~fc:loc [ d ] body) + Cst.View.Term.(review @@ Pi (loc, [ d ], body))) <|> ((let* ids = inside "{{" "}}" (many @@ parse_var ()) in let+ body = with_uppercase ids parse_expr in body) "expression with implicit variables") + <|> + (* %if A %-> B %-> C ==> {_ A} {_ B} C (last arg is the body) *) + (* %if A %<- B %<- C ==> {_ C} {_ B} A (first arg is the body) *) + (* commit fires only after the first separator is confirmed *) + (keywords [ "if"; "do"; "pi" ] *> + let* first = return () >>= fun () -> parse_expr () in + ((keyword' "->" *> commit *> + let+ rest = + sep_by1 (keyword' "->") (return () >>= fun () -> parse_expr ()) + in + let all = first :: rest in + let rev = List.rev all in + let body = List.hd rev in + let init = List.rev (List.tl rev) in + List.fold_right + (fun t acc -> + Cst.View.Term.( + review + @@ Pi + ( ghost', + [ + Cst.View.Decl.( + review + @@ Decl1 + ( ghost', + [ None ], + t, + Cst.View.Term.(review @@ Omitted ghost') )); + ], + acc ))) + init body) + <|> (keyword' "<-" *> commit *> + let+ rest = + sep_by1 (keyword' "<-") (return () >>= fun () -> parse_expr ()) + in + let rest_rev = List.rev rest in + List.fold_right + (fun t acc -> + Cst.View.Term.( + review + @@ Pi + ( ghost', + [ + Cst.View.Decl.( + review + @@ Decl1 + ( ghost', + [ None ], + t, + Cst.View.Term.(review @@ Omitted ghost') )); + ], + acc ))) + rest_rev first))) <|> parse_id () "trailing expression" and parse_expr_app () : Cst.Term.t t = @@ -271,41 +356,16 @@ module Make_Modern and parse_expr () : Cst.Term.t t = begin - (keyword "the" *> commit + (keyword' "the" *> commit *> let* ty = parse_expr1 () in let+ body = parse_expr () in - Cst.Term.has_type body ty) - <|> - (* %-> A %-> B %-> C ==> {_ A} {_ B} C (last arg is the body) *) - (keywords [ "->"; "if" ] - *> commit - *> - let+ args = - sep_by1 (option () @@ (keyword "->" *> commit)) (parse_expr1 ()) - in - let rev = List.rev args in - let body = List.hd rev in - let init = List.rev (List.tl rev) in - List.fold_right - (fun t acc -> Cst.Term.pi [ Cst.Decl.decl1 [ None ] t ] acc) - init body) - <|> - (* + Cst.View.Term.(review @@ HasType (ghost', body, ty))) + + (* %<- A - %<- B + %<- B %<- C ==> {_ C} {_ B} A (first arg is the body) *) - (keywords [ "<-"; "do" ] - *> commit - *> - let+ args = - sep_by1 (option () @@ (keyword "<-" *> commit)) (parse_expr1 ()) - in - let body = List.hd args in - let rest_rev = List.rev (List.tl args) in - List.fold_right - (fun t acc -> Cst.Term.pi [ Cst.Decl.decl1 [ None ] t ] acc) - rest_rev body) <|> let* atoms = many @@ parse_expr1 () in let* trail_opt = @@ -330,56 +390,64 @@ module Make_Modern and parse_qualified () : Cst.symbol t = begin - let rec split = function + let split = function | [] -> failwith "Expected qualified name" - | [ p ] -> ([], p) - | p :: q -> - let q0, q1 = split q in - (List.append [ p ] q0, q1) + | name :: scopes -> (List.rev scopes, name) in - keyword "val" *> commit + (keyword' "val" *> commit *> ((let* ident in return ([], ident)) <|> inside "(" ")" (let* ns = many1 ident in - return @@ split ns)) + return @@ split ns))) + <|> (keyword' "(" *> let* ns = (many1 ident) <* string ")" in return @@ split ns) <|> + (keyword "(" *> commit *> let* ns = many1 ident <* string ")" in return @@ split ns) end "qualified name" and parse_text () : string t = begin - string "%\"" *> take_till (fun c -> c = '%') - <* string "%\"" <* whitespace' () + string "%[" *> take_till (fun c -> c = '%') + <* string "%]" <* whitespace' () end "string literal" - and parse_decl () : Cst.Decl.t t = + and parse_decl () : Cst.decl t = begin (let@ names, s, e = inside "(" ")" (many1 (parse_arg ())) in let loc = mk_loc s e in - let+ typ = option (Cst.Term.omitted ~fc:Cst.ghost) (parse_expr ()) in - Cst.Decl.decl1 ~fc:loc names typ) + let+ typ = + option Cst.View.Term.(review @@ Omitted ghost') (parse_expr ()) + in + Cst.View.Decl.( + review + @@ Decl1 (loc, names, typ, Cst.View.Term.(review @@ Omitted ghost')))) <|> let@ name, s, e = parse_arg () in let loc = mk_loc s e in - let+ typ = option (Cst.Term.omitted ~fc:Cst.ghost) (parse_expr ()) in - Cst.Decl.decl1 ~fc:loc [ name ] typ + let+ typ = + option Cst.View.Term.(review @@ Omitted ghost') (parse_expr ()) + in + Cst.View.Decl.( + review + @@ Decl1 + (loc, [ name ], typ, Cst.View.Term.(review @@ Omitted ghost'))) end "declaration" - and parse_mode () : Cst.Mode.mode t = + and parse_mode () : Cst.mode t = begin - (let@ (), s, e = keyword "out1" *> commit *> return () in - return (Cst.Mode.minus1 ~fc:(mk_loc s e) ())) - <|> (let@ (), s, e = keyword "out" *> commit *> return () in - return (Cst.Mode.minus ~fc:(mk_loc s e) ())) - <|> (let@ (), s, e = keyword "in" *> commit *> return () in - return (Cst.Mode.plus ~fc:(mk_loc s e) ())) - <|> let@ (), s, e = keyword "star" *> commit *> return () in - return (Cst.Mode.star ~fc:(mk_loc s e) ()) + (let@ (), s, e = keyword' "out1" *> commit *> return () in + return Cst.View.Mode.(review @@ Minus1 (mk_loc s e))) + <|> (let@ (), s, e = keyword' "out" *> commit *> return () in + return Cst.View.Mode.(review @@ Minus (mk_loc s e))) + <|> (let@ (), s, e = keyword' "in" *> commit *> return () in + return Cst.View.Mode.(review @@ Plus (mk_loc s e))) + <|> let@ (), s, e = keyword' "star" *> commit *> return () in + return Cst.View.Mode.(review @@ Star (mk_loc s e)) end "mode" - and parse_mode_dec () : Cst.Mode.modedec t = + and parse_mode_dec () : Cst.modeDec t = begin let* braced_args = many @@ -389,24 +457,43 @@ module Make_Modern in let* body = parse_expr () in let+ bare_modes = many (parse_mode ()) in - let rec go_bare body = function - | [] -> Cst.Mode.Full.mode_root body - | m :: rest -> - Cst.Mode.Full.mode_pi m (Cst.Decl.decl0 [ None ]) - (go_bare body rest) + let rec head_sym tm = + match Cst.View.Term.view tm with + | Cst.View.Term.Lowercase (_, s) + | Cst.View.Term.Uppercase (_, s) + | Cst.View.Term.Qualified (_, s) -> + s + | Cst.View.Term.App (_, f, _) -> head_sym f + | _ -> + raise (ParseError "mode declaration: expected identifier as head") in - let rec go_braced inner = function - | [] -> inner - | (m, d) :: rest -> Cst.Mode.Full.mode_pi m d (go_braced inner rest) + let root = + Cst.View.Mode.Term.( + review + @@ ModeTerm + ( ghost', + head_sym body, + Cst.View.Mode.Spine.(review @@ ModeNil ghost') )) in - Cst.Mode.Full.to_modeDec (go_braced (go_bare body bare_modes) braced_args) + let name_of_decl d = + match Cst.View.Decl.view d with + | Cst.View.Decl.Decl1 (_, ns, _, _) | Cst.View.Decl.Decl0 (_, ns, _) + -> ( + match ns with n :: _ -> n | [] -> None) + in + let braced_spine = + List.map (fun (m, d) -> (m, name_of_decl d)) braced_args + in + let bare_spine = List.map (fun m -> (m, None)) bare_modes in + Cst.View.Mode.Dec.( + review @@ ModeDec (ghost', braced_spine @ bare_spine, root)) end "mode declaration" - and parse_simple_mode_dec () : Cst.Mode.modedec t = + and parse_simple_mode_dec () : Cst.modeDec t = parse_mode_dec () "simple mode declaration" - and parse_inst () : Cst.Struct.inst t = + and parse_inst () : Cst.inst t = begin let* name, s, e = with_fc ident1 in let loc = mk_loc s e in @@ -414,38 +501,47 @@ module Make_Modern token "=" *> let+ tm = parse_expr () in - Cst.Struct.con_inst (sym, loc) tm + Cst.View.Struct.Inst.(review @@ ConInst (ghost', sym, loc, tm)) end "instance declaration" - and parse_sigexp () : Cst.Struct.sigexp t = + and parse_sigexp () : Cst.sigexp t = begin let* base = - keyword "the" *> commit *> return (Cst.Struct.thesig ~fc:Cst.ghost) + keyword' "the" *> commit + *> return Cst.View.Struct.SigExp.(review @@ Thesig ghost') <|> let+ name = ident1 in - Cst.Struct.sig_id name + Cst.View.Struct.SigExp.(review @@ SigId (ghost', name)) in - let+ wheres = many (keyword "where" *> commit *> parse_inst ()) in - match wheres with [] -> base | _ -> Cst.Struct.where_sig base wheres + let+ wheres = many (keyword' "where" *> commit *> parse_inst ()) in + match wheres with + | [] -> base + | _ -> Cst.View.Struct.SigExp.(review @@ WhereSig (ghost', base, wheres)) end - and parse_sigdef () : Cst.Struct.sigdef t = + and parse_sigdef () : Cst.sigdef t = begin let+ se = parse_sigexp () in - Cst.Struct.sig_def None se + Cst.View.Struct.SigDef.(review @@ SigDef (ghost', None, se)) end "signature definition" - and parse_struct_dec () : Cst.Struct.structdec t = + and parse_struct_dec () : Cst.structDec t = begin let* name = ident1 in (token ":" *> let+ se = parse_sigexp () in - Cst.Struct.struct_decl (Some name) se) + Cst.View.Struct.StructDec.( + review @@ StructDecl (ghost', Some name, se))) <|> token "=" *> let+ sym = parse_qualified () in - Cst.Struct.struct_def (Some name) (Cst.Struct.str_exp sym) + Cst.View.Struct.StructDec.( + review + @@ StructDef + ( ghost', + Some name, + Cst.View.Struct.StrExp.(review @@ StrExp (ghost', sym)) )) end "structure declaration" @@ -456,14 +552,13 @@ module Make_Modern end "fixity level" - and parse_query () : - (int option * int option * int option * Cst.Query.query) t = + and parse_query () : (int option * int option * int option * Cst.query) t = begin let* n = parse_bound () in let* b = parse_bound () in let* d = parse_bound () in let+ tm = parse_expr () in - (n, b, d, Cst.Query.query None tm) + (n, b, d, Cst.View.Query.(review @@ Query (ghost', None, tm))) end "query" @@ -484,10 +579,10 @@ module Make_Modern end "definition" - and parse_solve () : Cst.Query.solve t = + and parse_solve () : Cst.solve t = begin let+ term = parse_expr () in - Cst.Query.solve None term + Cst.View.Solve.(review @@ Solve (ghost', None, term)) end "solve command" @@ -523,23 +618,29 @@ module Make_Modern begin inside "{" "}" (let+ d = parse_decl () in - Cst.BlockItem.some d) + Cst.View.BlockItem.(review @@ Any (ghost', d))) <|> inside "[" "]" (let+ d = parse_decl () in - Cst.BlockItem.pi d) + Cst.View.BlockItem.(review @@ All (ghost', d))) end "block item" and parse_fixity_kw () : Cst.fixity t = begin - keyword "left" *> commit *> return Cst.Fixity.left - <|> keyword "right" *> commit *> return Cst.Fixity.right - <|> keyword "prefix" *> commit *> return Cst.Fixity.prefix - <|> keyword "postfix" *> commit *> return Cst.Fixity.postfix - <|> keyword "middle" *> commit *> return Cst.Fixity.middle - <|> keyword "none" *> commit *> return Cst.Fixity.none + keyword' "left" *> commit + *> return Cst.View.Fixity.(review @@ Left ghost') + <|> keyword' "right" *> commit + *> return Cst.View.Fixity.(review @@ Right ghost') + <|> keyword' "prefix" *> commit + *> return Cst.View.Fixity.(review @@ Prefix ghost') + <|> keyword' "postfix" *> commit + *> return Cst.View.Fixity.(review @@ Postfix ghost') + <|> keyword' "middle" *> commit + *> return Cst.View.Fixity.(review @@ Middle ghost') + <|> keyword' "none" *> commit + *> return Cst.View.Fixity.(review @@ None ghost') end - "fixity keyword" + "fixity keyword'" and parse_params () : string list t = begin @@ -566,12 +667,77 @@ module Make_Modern List.iter (fun (id, fix) -> Hashtbl.replace local_fixity id fix) f; debug_parser p x + and source_context (src : string) (pos : int) : Display.form = + let len = String.length src in + let pos = Int.min pos (Int.max 0 (len - 1)) in + let line_start = + let i = ref pos in + while !i > 0 && src.[!i - 1] <> '\n' do decr i done; + !i + in + let line_end = + let i = ref pos in + while !i < len && src.[!i] <> '\n' do incr i done; + !i + in + let line_text = String.sub src line_start (line_end - line_start) in + let col = pos - line_start in + let caret = String.make col ' ' ^ "^" in + let line_num = + let n = ref 1 in + String.iteri (fun i c -> if i < line_start && c = '\n' then incr n) src; + !n + in + Display.Form.( + string (Printf.sprintf "Line %d:\n %s\n %s" line_num line_text caret)) + and run : 'a. 'a t -> N.namespace ref -> Cst.loc -> string -> 'a = fun p _ns _loc s -> - (* TODO use namespace and loc *) - match Parser.parse_string ~consume:All (whitespace' () *> p) s with - | Ok res -> res - | Error msg -> raise (ParseError msg) + try + let full_parser = whitespace' () *> p in + let state = + Parser.Buffered.( + parse full_parser + |> (fun st -> feed st (`String s)) + |> (fun st -> feed st `Eof)) + in + (match state with + | Parser.Buffered.Done (_, res) -> res + | Parser.Buffered.Partial _ -> + raise (ParseError "Unexpected end of input") + | Parser.Buffered.Fail (unconsumed, marks, msg) -> + let err_pos = String.length s - unconsumed.len in + let loc = + Some (mk_loc err_pos (min (err_pos + 1) (String.length s))) + in + let label = + match List.rev marks with label :: _ -> label | [] -> msg + in + let body = + Display.Form.( + string (Printf.sprintf "Expected %s\n" label) + +++ source_context s err_pos) + in + raise + (FullParseError + { title = Some Display.Form.(string "Parse error"); + subtitle = None; + body; + loc })) + with + | ParseError m -> + (* P-module operator-precedence errors: no offset available *) + let body = + Display.Form.(string m +++ nl () +++ source_context s 0) + in + Error.Error.err ~stage:Error.Error.Parse + Display.Form.(string "Parse error" +++ nl () +++ body) + | FullParseError { title; body; _ } -> + let title_form = + Option.value title ~default:Display.Form.(string "Parse error") + in + Error.Error.err ~stage:Error.Error.Parse + Display.Form.(title_form +++ nl () +++ body) end module ModernCst = Cst.Make_Cst (Paths.Paths_) diff --git a/src/Fronts/Modern/dune b/src/Fronts/Modern/dune index dd4e570..8c833ef 100644 --- a/src/Fronts/Modern/dune +++ b/src/Fronts/Modern/dune @@ -1,4 +1,4 @@ (library (name modern) (public_name stelf.modern) - (libraries display parsing lexing paths names cst)) + (libraries display parsing lexing paths names cst error)) diff --git a/src/Fronts/Pal/Help.ml b/src/Fronts/Pal/Help.ml index ad811ae..718b9aa 100644 --- a/src/Fronts/Pal/Help.ml +++ b/src/Fronts/Pal/Help.ml @@ -1,6 +1,2 @@ let help : string list -> Display.form = function - | [ "help" ] | [ "%help" ] -> assert false - | [ "set" ] | [ "%set" ] -> assert false - | [ "get" ] | [ "%get" ] -> assert false - | [ "sort" ] | [ "%sort" ] -> assert false - | [ "?" ] | [ "%?" ] -> assert false + | [] -> assert false diff --git a/src/Fronts/Pal/IMPL.ml b/src/Fronts/Pal/IMPL.ml index 3616ed7..f180b10 100644 --- a/src/Fronts/Pal/IMPL.ml +++ b/src/Fronts/Pal/IMPL.ml @@ -1,13 +1,16 @@ open! Basis module type IMPL = sig + + module Cst : Cst.CST (** Concrete syntax tree: the CST used by the modern parser. *) - module Cmd : Modern.CMD.CMD with module Cst = Cst - (** New parser layer: command-level parsing using the Modern parser. *) + module Names : Names.NAMES.NAMES + (** Shared Names module; equals Cmd.Modern.Names. *) - (** Names module — same as Cmd.Names. *) + module Cmd : Modern.CMD.CMD with module Cst = Cst and module Modern.Names = Names + (** New parser layer: command-level parsing using the Modern parser. *) module Recon : Recon.RECON (** New elaboration layer: reconstruction / type-inference sub-modules. *) @@ -48,10 +51,10 @@ module type IMPL = sig (** {1 Signature installation} *) module Install : sig - val install1 : ?path:Fpath.t option -> Cst.cmd -> unit + val install1 : ?path:Fpath.t option -> Names.namespace -> Cst.cmd -> unit (** Install a single parsed command into the global signature. *) - val install : ?path:Fpath.t option -> Cst.cmd list -> unit + val install : ?path:Fpath.t option -> Names.namespace -> Cst.cmd list -> unit (** Install a list of commands in order. *) val reset : unit -> unit @@ -61,7 +64,7 @@ module type IMPL = sig (* -------------------------------------------------------------------- *) (** {1 Loading} *) - val load : source -> status + val load : Names.namespace -> source -> status (** Parse and install all declarations from a source. *) val read_decl : unit -> status @@ -77,29 +80,11 @@ module type IMPL = sig (** {1 Configuration file management} *) module Config : sig - type t - (** An opaque configuration: a working directory plus an ordered list of - source files, each with a modification-time slot for incremental reload. - *) - - val suffix : string ref - (** File-extension treated as a config file (default: ["cfg"]). *) - - val read : source -> t - (** Parse a config file and return the resulting configuration. *) - - val read_without : source * t -> t - (** Like {!read} but omit any files already present in the second config. *) - + type t = Project.Format.file + (** The type of configuration objects. *) + val read : source -> t val load : t -> status - (** Reset global state and then load all files listed in the config. *) - - val append : t -> status - (** Load the config without resetting first; starts from the first file - whose modification time is newer than the last successful load. *) - val define : string list -> t - (** Build a config directly from an explicit list of file names. *) end val make : source -> status diff --git a/src/Fronts/Pal/Impl.ml b/src/Fronts/Pal/Impl.ml index c610c4e..430a3e8 100644 --- a/src/Fronts/Pal/Impl.ml +++ b/src/Fronts/Pal/Impl.ml @@ -11,6 +11,13 @@ let string_to_source = function | s -> Some (Fpath.of_string s) module Impl () = struct + + let current_path : string list ref = ref [] + let current_load_path : Fpath.t list ref = ref ([Result.get_ok @@ Bos.OS.Dir.current ()]) + let load_paths : (Fpath.t * string) list ref = ref [] + + + (** Additional directories to search for included files. *) (* Save Basis's OS before any module definitions shadow it *) module BasisOS = OS @@ -90,7 +97,7 @@ module Impl () = struct (* let msg s = Msg.Msg_.Msg.message s *) let msg m = - Display.message ~level:Display.Verbose ~kind:Display.Info + Display.message ~level:Display.Normal ~kind:Display.Info (Display.Form.string m) let print_error (label : string) (detail : string) : unit = @@ -100,10 +107,11 @@ module Impl () = struct (* ConDec installation *) (* ------------------------------------------------------------------ *) - let install_condec (cd : Intsyn.IntSyn.conDec) : unit = + let install_condec ns (cd : Intsyn.IntSyn.conDec) : unit = let open Intsyn.IntSyn in let cid = sgnAdd cd in Names.installConstName cid; + Names.insertConst (ns, cid); match cd with | BlockDec _ -> Subordinate.Subordinate_.Subordinate.installBlock cid | BlockDef _ -> () @@ -224,24 +232,109 @@ module Impl () = struct let rrs = (dummy_r, List.map (fun _ -> dummy_r) body) in (TS.TDecl (order, callpats), rrs) - let rec install1 ?(path = None) (cmd : Cst.cmd) : unit = - let filename = - Stdlib.Option.value - (Option.map Fpath.to_string path) - ~default:"" + let loc_of filename (l : Cst.loc) : Paths.location = + Paths.Loc (filename, Cst.loc_to_region l) + + let run_query loc q = + let v_, opt_name, xs_ = + Recon.ReconQuery.queryToQuery (q, loc) in - Debug.( - msg' ~src:Group.pal ~level:Level.Debug Fmt.string "Installing command"); - let loc_of (l : Cst.loc) : Paths.location = - Paths.Loc (filename, Cst.loc_to_region l) + let g = + Compile.Compile_.Compile.compileGoal (Intsyn.IntSyn.Null, v_) in - let name_to_cid label id = + let solutions = ref 0 in + let exception Done in + let sc m_ = + incr solutions; + if !Global.Global_.Global.chatter >= 3 then begin + msg + (Printf.sprintf "---------- Solution %d ----------\n" !solutions); + List.app + (fun (e_, n) -> + msg + (n ^ " = " + ^ Print.Print_.expToString (Intsyn.IntSyn.Null, e_) + ^ "\n")) + xs_; + match opt_name with + | None -> () + | Some name -> + msg + (name ^ " = " + ^ Print.Print_.expToString (Intsyn.IntSyn.Null, m_) + ^ "\n") + end; + raise Done + in + (try + Opsem.Opsem_.AbsMachine.solve + ( (g, Intsyn.IntSyn.id), + Compile.CompSyn.CompSyn.DProg + (Intsyn.IntSyn.Null, Intsyn.IntSyn.Null), + sc ) + with Done -> ()); + if !solutions = 0 && !Global.Global_.Global.chatter >= 3 then + msg "No solution.\n" + + let install_worlds_cmd ids tm = + let resolve_block id = match Names.constLookup (Names.Qid ([], id)) with | None -> failwith - ("Undeclared identifier " ^ id ^ " in " ^ label ^ " declaration") + ("Undeclared block label " ^ id ^ " in worlds declaration") | Some cid -> cid in + let rec flatten = function + | [] -> [] + | cid :: rest -> ( + match Intsyn.IntSyn.sgnLookup cid with + | Intsyn.IntSyn.BlockDec _ -> cid :: flatten rest + | Intsyn.IntSyn.BlockDef (_, _, l) -> flatten (l @ rest) + | _ -> cid :: flatten rest) + in + let block_cids = flatten (List.map resolve_block ids) in + let w_ = Intsyn.Lambda_.Tomega.Worlds block_cids in + let lookup_head tm = + match Cst.View.Term.view tm with + | Cst.View.Term.Lowercase (_, (ns, n)) -> + Names.constLookup (Names.Qid (ns, n)) + | Cst.View.Term.Uppercase (_, (ns, n)) -> + Names.constLookup (Names.Qid (ns, n)) + | Cst.View.Term.Qualified (_, (ns, n)) -> + Names.constLookup (Names.Qid (ns, n)) + | _ -> None + in + let family_cid_opt = + match Cst.View.Term.view tm with + | Cst.View.Term.App (_, head, _) -> lookup_head head + | v -> lookup_head (Cst.View.Term.review v) + in + match family_cid_opt with + | None -> failwith "%worlds: expected a type family name" + | Some a -> + WorldSyn.install (a, w_); + WorldSyn.worldcheck w_ a + + let install_condec_cmd ?(inline = false) ns condec loc = + match Recon.ReconConDec.condecToConDec (condec, loc, inline) with + | Some cd, _ -> install_condec ns cd + | None, _ -> () + + let name_to_cid ns label id = + match Names.constLookupIn (ns, Names.Qid ([], id)) with + | None -> + failwith + ("Undeclared identifier " ^ id ^ " in " ^ label ^ " declaration") + | Some cid -> cid + + let rec install1 ?(path = None) ns (cmd : Cst.cmd) : unit = + let filename = + Stdlib.Option.value + (Option.map Fpath.to_string path) + ~default:"" + in + Debug.( + msg' ~src:Group.pal ~level:Level.Debug Fmt.string "Installing command"); match cmd with | Cst.SortCmd_ (ids, decls) -> List.app @@ -254,12 +347,7 @@ module Impl () = struct let condec = Cst.ConDec.constant_decl (Cst.Decl.decl1 [ Some id ] kind) in - match - Recon.ReconConDec.condecToConDec - (condec, loc_of Cst.ghost, false) - with - | Some cd, _ -> install_condec cd - | None, _ -> ()) + install_condec_cmd ns condec (loc_of filename Cst.ghost)) ids | Cst.TermCmd_ decl -> ( Debug.( @@ -272,68 +360,22 @@ module Impl () = struct | Cst.View.Decl.Decl0 (_, ns, t) -> (ns, t) in let names' = List.map (function Some n -> n | None -> "_") names in - Display.message ~level:Display.Verbose + Display.message ~level:Display.Detailed Display.( string "Installing term command for" ++ each string names' ++ space () ++ hvbox [ shown Cst.show_term ty ]); let condec = Cst.ConstantDecl_ decl in - match - Recon.ReconConDec.condecToConDec (condec, loc_of Cst.ghost, false) - with - | Some cd, _ -> install_condec cd - | None, _ -> ()) - | Cst.DefineCmd_ (Cst.Define_ (name_opt, tm, tp_opt)) -> ( + install_condec_cmd ns condec (loc_of filename Cst.ghost)) + | Cst.DefineCmd_ (Cst.Define_ (name_opt, tm, tp_opt)) -> let name = match name_opt with Some n -> n | None -> "_" in let condec = Cst.ConstantDef_ (name, tm, tp_opt) in - match - Recon.ReconConDec.condecToConDec (condec, loc_of Cst.ghost, false) - with - | Some cd, _ -> install_condec cd - | None, _ -> ()) + install_condec_cmd ns condec (loc_of filename Cst.ghost) | Cst.QueryCmd_ (_n, _b, _d, q) -> - let v_, opt_name, xs_ = - Recon.ReconQuery.queryToQuery (q, loc_of Cst.ghost) - in - let g = - Compile.Compile_.Compile.compileGoal (Intsyn.IntSyn.Null, v_) - in - let solutions = ref 0 in - let exception Done in - let sc m_ = - incr solutions; - if !Global.Global_.Global.chatter >= 3 then begin - msg - (Printf.sprintf "---------- Solution %d ----------\n" !solutions); - List.app - (fun (e_, n) -> - msg - (n ^ " = " - ^ Print.Print_.expToString (Intsyn.IntSyn.Null, e_) - ^ "\n")) - xs_; - match opt_name with - | None -> () - | Some name -> - msg - (name ^ " = " - ^ Print.Print_.expToString (Intsyn.IntSyn.Null, m_) - ^ "\n") - end; - raise Done - in - (try - Opsem.Opsem_.AbsMachine.solve - ( (g, Intsyn.IntSyn.id), - Compile.CompSyn.CompSyn.DProg - (Intsyn.IntSyn.Null, Intsyn.IntSyn.Null), - sc ) - with Done -> ()); - if !solutions = 0 && !Global.Global_.Global.chatter >= 3 then - msg "No solution.\n" + run_query (loc_of filename Cst.ghost) q | Cst.SolveCmd_ sol -> let v_, sc_fn = - Recon.ReconQuery.solveToSolve ([], sol, loc_of Cst.ghost) + Recon.ReconQuery.solveToSolve ([], sol, loc_of filename Cst.ghost) in let g = Compile.Compile_.Compile.compileGoal (Intsyn.IntSyn.Null, v_) @@ -354,7 +396,7 @@ module Impl () = struct | None -> failwith "%solve: no solution found" | Some m_ -> m_ in - List.app (fun (cd, _) -> install_condec cd) (sc_fn m_) + List.app (fun (cd, _) -> install_condec ns cd) (sc_fn m_) | Cst.StopCmd_ -> () | Cst.QuitCmd_ -> BasisOS.Process.exit BasisOS.Process.success | Cst.HelpCmd_ topic -> @@ -372,47 +414,9 @@ module Impl () = struct end | Cst.SetCmd_ (key, value) -> Options.set key value | Cst.VersionCmd_ -> msg (Frontend.Version.Version.version_string ^ "\n") - | Cst.EvalCmd_ cmds -> List.app install1 cmds + | Cst.EvalCmd_ cmds -> List.app (install1 ~path ns) cmds | Cst.AdhocQueryCmd_ q -> - let v_, opt_name, xs_ = - Recon.ReconQuery.queryToQuery (q, loc_of Cst.ghost) - in - let g = - Compile.Compile_.Compile.compileGoal (Intsyn.IntSyn.Null, v_) - in - let solutions = ref 0 in - let exception Done in - let sc m_ = - incr solutions; - if !Global.Global_.Global.chatter >= 3 then begin - msg - (Printf.sprintf "---------- Solution %d ----------\n" !solutions); - List.app - (fun (e_, n) -> - msg - (n ^ " = " - ^ Print.Print_.expToString (Intsyn.IntSyn.Null, e_) - ^ "\n")) - xs_; - match opt_name with - | None -> () - | Some name -> - msg - (name ^ " = " - ^ Print.Print_.expToString (Intsyn.IntSyn.Null, m_) - ^ "\n") - end; - raise Done - in - (try - Opsem.Opsem_.AbsMachine.solve - ( (g, Intsyn.IntSyn.id), - Compile.CompSyn.CompSyn.DProg - (Intsyn.IntSyn.Null, Intsyn.IntSyn.Null), - sc ) - with Done -> ()); - if !solutions = 0 && !Global.Global_.Global.chatter >= 3 then - msg "No solution.\n" + run_query (loc_of filename Cst.ghost) q | Cst.DeclCmd_ tm -> let qid_opt = match Cst.View.Term.view tm with @@ -432,16 +436,16 @@ module Impl () = struct end end | Cst.FreezeCmd_ ids -> - let cids = List.map (name_to_cid "freeze") ids in + let cids = List.map (name_to_cid ns "freeze") ids in let _ = Subordinate.Subordinate_.Subordinate.freeze cids in () | Cst.ThawCmd_ ids -> if not !unsafe then failwith "%thaw not safe: Toggle `unsafe' flag"; - let cids = List.map (name_to_cid "thaw") ids in + let cids = List.map (name_to_cid ns "thaw") ids in let _ = Subordinate.Subordinate_.Subordinate.thaw cids in () | Cst.DeterministicCmd_ ids -> - let cids = List.map (name_to_cid "deterministic") ids in + let cids = List.map (name_to_cid ns "deterministic") ids in List.app (fun cid -> Compile.CompSyn.CompSyn.detTableInsert (cid, true)) cids @@ -458,20 +462,16 @@ module Impl () = struct in List.app (fun id -> - let cid = name_to_cid "prec" id in + let cid = name_to_cid ns "prec" id in Names.installFixity (cid, fixity)) ids | Cst.SymbolCmd_ (id, pref) -> - let cid = name_to_cid "symbol" id in + let cid = name_to_cid ns "symbol" id in Names.installNamePref (cid, ([ pref ], [ pref ])) - | Cst.InlineCmd_ (name, tm) -> ( + | Cst.InlineCmd_ (name, tm) -> let condec = Cst.ConstantDef_ (name, tm, None) in - match - Recon.ReconConDec.condecToConDec (condec, loc_of Cst.ghost, true) - with - | Some cd, _ -> install_condec cd - | None, _ -> ()) - | Cst.BlockCmd_ (id, items) -> ( + install_condec_cmd ~inline:true ns condec (loc_of filename Cst.ghost) + | Cst.BlockCmd_ (id, items) -> let pis = Stdlib.List.filter_map (function Cst.BlockPi_ d -> Some d | _ -> None) @@ -483,15 +483,11 @@ module Impl () = struct items in let condec = Cst.BlockDecl_ (id, pis, somes) in - match - Recon.ReconConDec.condecToConDec (condec, loc_of Cst.ghost, false) - with - | Some cd, _ -> install_condec cd - | None, _ -> ()) + install_condec_cmd ns condec (loc_of filename Cst.ghost) | Cst.ModeCmd_ md -> let () = Display.( - message ~level:Verbose + message ~level:Detailed (string "Installing mode declaration for " ++ shown Cst.show_modeDec md)) in @@ -538,56 +534,14 @@ module Impl () = struct UniqueTable.installMode mdec; Unique.checkUnique mdec end - | Cst.UnionCmd_ (id, ids) -> ( + | Cst.UnionCmd_ (id, ids) -> let syms = List.map (fun s -> ([], s)) ids in let condec = Cst.BlockDef_ (id, syms) in - match - Recon.ReconConDec.condecToConDec (condec, loc_of Cst.ghost, false) - with - | Some cd, _ -> install_condec cd - | None, _ -> ()) - | Cst.WorldsCmd_ (ids, tm) -> - let resolve_block id = - match Names.constLookup (Names.Qid ([], id)) with - | None -> - failwith - ("Undeclared block label " ^ id ^ " in worlds declaration") - | Some cid -> cid - in - let rec flatten = function - | [] -> [] - | cid :: rest -> ( - match Intsyn.IntSyn.sgnLookup cid with - | Intsyn.IntSyn.BlockDec _ -> cid :: flatten rest - | Intsyn.IntSyn.BlockDef (_, _, l) -> flatten (l @ rest) - | _ -> cid :: flatten rest) - in - let block_cids = flatten (List.map resolve_block ids) in - let w_ = Intsyn.Lambda_.Tomega.Worlds block_cids in - let lookup_head tm = - match Cst.View.Term.view tm with - | Cst.View.Term.Lowercase (_, (ns, n)) -> - Names.constLookup (Names.Qid (ns, n)) - | Cst.View.Term.Uppercase (_, (ns, n)) -> - Names.constLookup (Names.Qid (ns, n)) - | Cst.View.Term.Qualified (_, (ns, n)) -> - Names.constLookup (Names.Qid (ns, n)) - | _ -> None - in - let family_cid_opt = - match Cst.View.Term.view tm with - | Cst.View.Term.App (_, head, _) -> lookup_head head - | v -> lookup_head (Cst.View.Term.review v) - in - begin match family_cid_opt with - | None -> failwith "%worlds: expected a type family name" - | Some a -> - WorldSyn.install (a, w_); - WorldSyn.worldcheck w_ a - end + install_condec_cmd ns condec (loc_of filename Cst.ghost) + | Cst.WorldsCmd_ (ids, tm) -> install_worlds_cmd ids tm | Cst.QueryTabledCmd_ (numSol, try_, _d, q) -> let a_, opt_name, xs_ = - Recon.ReconQuery.queryToQuery (q, loc_of Cst.ghost) + Recon.ReconQuery.queryToQuery (q, loc_of filename Cst.ghost) in let g = Compile.Compile_.Compile.compileGoal (Intsyn.IntSyn.Null, a_) @@ -640,30 +594,49 @@ module Impl () = struct loop () with Done -> ()); if !solutions = 0 && !chatter >= 3 then msg "No tabled solution.\n" - | Cst.Open_ id -> - failwith - "%module: module definitions not yet implemented in this frontend" - (* let se = Cst.Struct.str_exp (ids, id) in - let mid = Recon.ReconModule.strexpToStrexp se in - let ns = ModSyn.Names.getComponents mid in - let module_ = ModSyn.abstractModule (ns, Some mid) in - let action (cid, _) = - Index.Index_.Index.install Intsyn.IntSyn.Ordinary - (Intsyn.IntSyn.Const cid); - Compile.Compile_.Compile.install Intsyn.IntSyn.Ordinary cid; - Subordinate.Subordinate_.Subordinate.install cid; - Subordinate.Subordinate_.Subordinate.installDef cid + | Cst.Open_ ids -> + let qid = + match List.rev ids with + | [] -> failwith "%open: empty module path" + | last :: prefix -> Names.Qid (List.rev prefix, last) in - ModSyn.installSig (module_, None, action, true) *) - | Cst.Scope_ _ -> - failwith - "%module: module definitions not yet implemented in this frontend" + let mid = + match Names.structLookupIn (ns, qid) with + | Some m -> m + | None -> + (match Names.structLookup qid with + | Some m -> m + | None -> + failwith ("%open: unknown module " ^ Names.qidToString qid)) + in + let comps = Names.getComponents mid in + Names.appConsts (fun (_, cid) -> Names.insertConst (ns, cid)) comps; + Names.appStructs (fun (_, m) -> Names.insertStruct (ns, m)) comps + | Cst.Scope_ (name, body_cmd) -> + let child_ns = Names.newNamespace () in + let mid = + Intsyn.IntSyn.sgnStructAdd (Intsyn.IntSyn.StrDec (name, None)) + in + Names.installStructName mid; + Names.insertStruct (ns, mid); + install1 ~path child_ns body_cmd; + Names.installComponents (mid, child_ns) | Cst.Use_ _ -> failwith "%use: module instantiation not yet implemented in this frontend" + | Cst.Macro_ _ -> + failwith "Macros not yet implemented in this frontend" + | Cst.Seq_ cmds -> + List.app (install1_item ~path ns) cmds + | Cst.Require_ path -> failwith "Require not yet implemented in this frontend" + + and install1_item ?(path = None) ns (item : Cst.item) : unit = + match item with + | Cst.Outer _ -> () + | Cst.Cmd cmd -> install1 ~path ns cmd - let install ?(path = None) (cmds : Cst.cmd list) : unit = - List.app (install1 ~path) cmds + let install ?(path = None) ns (cmds : Cst.cmd list) : unit = + List.app (install1 ~path ns) cmds let reset () : unit = Frontend.Frontend_.Stelf.reset () end @@ -672,35 +645,64 @@ module Impl () = struct (* Loading *) (* ------------------------------------------------------------------ *) - let load_string ?(path = None) (str : string) : status = - let ns = ref (Names.newNamespace ()) in + let load_string ?(path = None) ?(ns_init = None) (str : string) : status = + let ns = match ns_init with Some r -> r | None -> ref (Names.newNamespace ()) in let loc = Cst.ghost in try let cmds = ModernImpl.run (Cmd.parse ()) ns loc str in try - Install.install ~path cmds; + Install.install ~path !ns cmds; Ok - with exn -> - print_error (source_to_string path) (Printexc.to_string exn); - Abort + with + | Error.Error.Err _ as e -> raise e + | Recon.ReconConDec.Error msg + | Recon.ReconTerm.Error msg + | Recon.ReconModule.Error msg + | Recon.ReconQuery.Error msg -> + Error.Error.err ~stage:Error.Error.Recon (Display.Form.string msg) + | Recon.ReconMode.Error msg + | Recon.ReconThm.Error msg -> + Error.Error.err ~stage:Error.Error.Check (Display.Form.string msg) + | Intsyn.Lambda_.Abstract.Error msg -> + Error.Error.err ~stage:Error.Error.Recon (Display.Form.string msg) + | Typecheck.Typecheck_.TypeCheck.Error msg -> + Error.Error.err ~stage:Error.Error.Check + Display.Form.(string ("Double-check failed (internal bug): " ^ msg)) + | Failure msg -> + Error.Error.err ~stage:Error.Error.Unknown (Display.Form.string msg) + | exn -> + print_error (source_to_string path) (Printexc.to_string exn); + Abort with + | Error.Error.Err (stage, form) -> + let stage_str = + match stage with + | Error.Error.Parse -> "parse" + | Error.Error.Check -> "check" + | Error.Error.Recon -> "recon" + | _ -> "error" + in + Printf.eprintf "[%s] %s\n%!" stage_str (Display.Form.to_plain form); + Abort | Modern.Modern.ParseError e -> + (* Fallback during transition; should no longer be reached *) print_error (source_to_string path) ("parse error: " ^ e); Abort | exn -> print_error (source_to_string path) (Printexc.to_string exn); Abort - let load (src : source) : status = + let load ns (src : source) : status = + let ns_ref = ref ns in match src with - | Input str -> load_string str + | Input str -> load_string ~ns_init:(Some ns_ref) str | File path -> ( let filename = Some path in try let ic = TextIO.openIn (Fpath.to_string path) in let str = TextIO.inputAll ic in let () = TextIO.closeIn ic in - load_string ~path:filename str + load_string ~path:filename ~ns_init:(Some ns_ref) str with exn -> print_error "pal" (Printexc.to_string exn); Abort) @@ -732,107 +734,39 @@ module Impl () = struct (* Configuration file management *) (* ------------------------------------------------------------------ *) - module Config = struct - type mfile = { filename : string; mutable mtime : Time.time option } - type t = { sources : mfile list } - - let suffix = ref "cfg" - let mk_mfile filename = { filename; mtime = None } - - let is_config item = - let sfx = "." ^ !suffix in - let n = Stdlib.String.length item and m = Stdlib.String.length sfx in - n >= m && Stdlib.String.sub item (n - m) m = sfx - - let mk_rel base path = - OS.Path.mkCanonical - (if OS.Path.isAbsolute path then path else OS.Path.concat (base, path)) - + module Config = struct + type t = Project.Format.file let read (src : source) : t = - let cfg_path = - match src with - | File p -> Fpath.to_string p - | Input _ -> - failwith "Config.read: cannot use Input source as config file" - in - let rec collect (sources, seen) cfg = - if Stdlib.List.mem cfg seen then (sources, seen) - else - let seen' = cfg :: seen in - let dir = OS.Path.dir cfg in - let ic = TextIO.openIn cfg in - let acc = ref (sources, seen') in - (try - while true do - match TextIO.inputLine ic with - | None -> raise Exit - | Some raw_line -> - let line = Stdlib.String.trim raw_line in - let line = - match Stdlib.String.index_opt line '%' with - | Some i -> Stdlib.String.trim (Stdlib.String.sub line 0 i) - | None -> line - in - if line <> "" then begin - let item = mk_rel dir line in - let srcs, cfgs = !acc in - if is_config item then acc := collect (srcs, cfgs) item - else if - not - (Stdlib.List.exists - (fun mf -> mf.filename = item) - srcs) - then acc := (srcs @ [ mk_mfile item ], cfgs) - end - done - with Exit -> ()); - TextIO.closeIn ic; - !acc - in - let sources, _ = collect ([], []) cfg_path in - { sources } - - let read_without ((src, existing) : source * t) : t = - let fresh = read src in - let existing_names = - Stdlib.List.map (fun mf -> mf.filename) existing.sources - in - let sources' = - Stdlib.List.filter - (fun mf -> not (Stdlib.List.mem mf.filename existing_names)) - fresh.sources - in - { sources = sources' } - - let is_modified mf = mf.mtime = None - - let load_file (mf : mfile) (acc : status) : status = - match acc with - | Abort -> Abort - | Ok -> - let st = load (File (Fpath.v mf.filename)) in - if st = Ok then mf.mtime <- Some Time.zeroTime; - st - - let append (cfg : t) : status = - let rec from_first_modified = function - | [] -> [] - | mf :: rest -> - if is_modified mf then mf :: rest else from_first_modified rest - in - let to_load = from_first_modified cfg.sources in - Stdlib.List.fold_left (fun acc mf -> load_file mf acc) Ok to_load - - let load (cfg : t) : status = - Install.reset (); - Stdlib.List.iter (fun mf -> mf.mtime <- None) cfg.sources; - append cfg - - let define (filenames : string list) : t = - { sources = Stdlib.List.map mk_mfile filenames } + let str = match src with + | Input str -> str + | File path -> ( + try + let ic = TextIO.openIn (Fpath.to_string path) in + let str = TextIO.inputAll ic in + let () = TextIO.closeIn ic in + str + with exn -> + print_error "pal" (Printexc.to_string exn); + "") in + match Project.Format.from_toml @@ Otoml.string str with + | None -> failwith "Failed to parse config file" + | Some config -> config + + let load' (cfg : Project.Format.t) : status = + assert false + let load (cfg : t) : status = + assert false + + end - - let make (src : source) : status = Config.load (Config.read src) + let make (src : source) : status = + let is_config path = + let ext = Fpath.get_ext path in + ext = ".cfg" || ext = ".toml" + in + match src with + | File path when is_config path -> Config.load (Config.read src) + | _ -> load (Names.newNamespace ()) src (* ------------------------------------------------------------------ *) (* Print settings *) @@ -955,7 +889,8 @@ module Impl () = struct (* ------------------------------------------------------------------ *) module Eval = struct - let eval (cmd : Cst.cmd) : unit = Install.install1 cmd + let eval (cmd : Cst.cmd) : unit = + Install.install1 (Names.newNamespace ()) cmd end (* ------------------------------------------------------------------ *) diff --git a/src/Fronts/Pal/Logo.ml b/src/Fronts/Pal/Logo.ml new file mode 100644 index 0000000..c11cf1d --- /dev/null +++ b/src/Fronts/Pal/Logo.ml @@ -0,0 +1,408 @@ +open Display + +let[@inline] colored : int -> int -> int -> form = + fun r g b -> if compare (r + g + b) 10 = -1 then string " " else Style.Fore.rgb r g b @@ string "█" + +let logo = concat ~sep:empty [ + colored 0 0 0 ; 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colored 0 0 0 ; colored 0 0 0 ; colored 5 18 19 + ; colored 9 34 35 ; colored 6 28 29 ; colored 4 24 24 + ; colored 5 25 26 ; colored 4 15 16 ; colored 0 0 0 ; colored 0 0 0 + ; colored 0 0 0 ; colored 0 0 0 ; colored 0 0 0 ; colored 0 0 0 + ; colored 0 0 0 ; colored 0 0 0 ; colored 3 10 10 ; colored 5 20 19 + ; colored 7 27 27 ; colored 8 30 29 ; colored 7 37 36 + ; colored 7 31 30 ; colored 7 27 26 ; colored 5 23 22 + ; colored 4 13 13 ; colored 2 6 5 ; colored 0 0 0 ; colored 0 0 0 + ; colored 0 0 0 ; colored 0 0 0 ; colored 0 0 0 ; colored 0 0 0 + ; colored 0 0 0 ; colored 0 0 0 ; colored 0 0 0 ; colored 0 0 0 + ; colored 0 0 0 ; colored 0 0 0 ; nl () +] \ No newline at end of file diff --git a/src/Fronts/Pal/Logo.mli b/src/Fronts/Pal/Logo.mli new file mode 100644 index 0000000..a1770a4 --- /dev/null +++ b/src/Fronts/Pal/Logo.mli @@ -0,0 +1 @@ +val logo : Display.form \ No newline at end of file diff --git a/src/Fronts/Pal/Opts.ml b/src/Fronts/Pal/Opts.ml index 7fc9906..87fab11 100644 --- a/src/Fronts/Pal/Opts.ml +++ b/src/Fronts/Pal/Opts.ml @@ -12,11 +12,12 @@ module Opts : OPTS = struct and v_conv = Arg.enum [ - ("quiet", Display.Info.Quiet); + ("terse", Display.Info.Terse); ("normal", Display.Info.Normal); - ("verbose", Display.Info.Verbose); - ("very_verbose", Display.Info.VeryVerbose); - ("silent", Display.Info.Silent); + ("detailed", Display.Info.Detailed); + ("exhaustive", Display.Info.Exhaustive); + ("minimal", Display.Info.Minimal); + ("off", Display.Info.Off); ] in Arg.(opt v_conv Display.Info.(Normal) & doc) diff --git a/src/Fronts/Pal/Pal.ml b/src/Fronts/Pal/Pal.ml index 3a8fb1e..7efd4f2 100644 --- a/src/Fronts/Pal/Pal.ml +++ b/src/Fronts/Pal/Pal.ml @@ -13,11 +13,14 @@ module Pal : PAL.PAL = struct exception Error of exn module Start () = struct - module M = M + module M = struct + include M + let () = Install.reset () + end - let ns = ref (M.Cmd.Modern.Names.newNamespace ()) - let loc = ref M.Cst.ghost - let install (cmd : M.Cst.cmd) : unit = M.Install.install1 cmd + let ns = ref (M.Names.newNamespace ()) + let loc = ref M.Cst.View.ghost + let install (cmd : M.Cst.cmd) : unit = M.Install.install1 !ns cmd let parse (s : string) : M.Cst.cmd list = M.Cmd.Modern.run (M.Cmd.parse ()) ns !loc s @@ -74,7 +77,15 @@ module Pal : PAL.PAL = struct Cmd.v (Cmd.info "check" ~doc:"Load a .cfg or source file") Term.( - const (fun f -> status_to_exit @@ M.make (File (Fpath.v f))) + const (fun f -> + M.chatter := 1; + Display.register (fun m -> + let current = Display.Info.from_chatter !M.chatter in + if Display.Info.(m.level =< current) then + let _ = Display.fmt Fmt.stdout m.msg in + Lwt.return () + else Lwt.return ()); + status_to_exit @@ M.make (File (Fpath.v f))) $ file) end in @@ -103,48 +114,17 @@ module Pal : PAL.PAL = struct (Cmd.info "version" ~doc:"Display version information") Term.( const (fun () -> - print_endline - {| - .......... - ............... - ........ - ........ - ....... - ....... - ........ - ....... - ........ - ........ - ......... - .......... - ........... - .... ....... - .... ........ - ..... ....... - .... ........ - ..... ........ - ..... ........ - ...... .......... - ..... .......... - ..... ... ....... - ...... ... ........ - ...... .... ....... - ....... .... ........ - ....... ... ....... - ....... .... ....... - ........ .... ....... ... - .......... .... ............ - . - -|}; + print_endline ("STELF version " ^ version); + Fmt_tty.setup_std_outputs (); + Display.fmt Format.std_formatter @@ Logo.logo; 0) $ const ()) in let main_cmd = Cmd.group (Cmd.info "stelf" ~version ~doc:"The STELF proof assistant") - [ repl_cmd; load_cmd; lsp_cmd; legacy_cmd ] + [ repl_cmd; load_cmd; version_cmd; lsp_cmd; legacy_cmd ] in Basis.OS.Process.exit (Cmd.eval' main_cmd) end diff --git a/src/Fronts/Pal/dune b/src/Fronts/Pal/dune index 828ee7c..5a7c424 100644 --- a/src/Fronts/Pal/dune +++ b/src/Fronts/Pal/dune @@ -5,15 +5,19 @@ display basis names + error cmdliner modern tui + project + fmt syntax paths cst recon intsyn global + typecheck timing print opsem @@ -23,6 +27,7 @@ index frontend terminate + fmt.tty worldcheck modules lwt.unix)) diff --git a/src/Fronts/Project/FILE.ml b/src/Fronts/Project/FILE.ml new file mode 100644 index 0000000..e69de29 diff --git a/src/Fronts/Project/Format.ml b/src/Fronts/Project/Format.ml index e69de29..f52b09a 100644 --- a/src/Fronts/Project/Format.ml +++ b/src/Fronts/Project/Format.ml @@ -0,0 +1,92 @@ +module Format_ = struct + type depends = + | Local of { name : string (* `[name]` *); alias : string option (* `[alias]` *); path : Fpath.t option (* `[path]` *) } + | Installed of { name : string (* `[name]` *); alias : string option (* `[alias]` *)} + | External of { + ext : ext; (* Where to find (table) `[link]` *) + path : Fpath.t option; (** where to put (string) `[path]` *) + post : depends; (** What to do after retrieved (table) *) + } + + (** Need to load something else *) + and ext = + | Github of { + user : string; (** User key, string `[user]` *) + repo : string; (** Repo key, string `[repo]` *) + } (** Github source `[github]` (table) *) + | Url of { url : string (** Url *) } (** Fetch from URL `[url]` *) + + type t = { + name : string; (** It has a name `[name]` (string) *) + dirs : Fpath.t list; + (** directories to be searched `[dirs]` (list of strings) *) + deps : depends list; (** dependencies `[[deps]]` (list of dependencies) *) + main : Fpath.t; (** Main file to load whenever used `[main]` (string) *) + } + (** A group (specified by `[[group]]` (can be mutiple per file), is a single + unit *) + + type file = t list + (** File is a list of [[group]] tables*) +end + +module Format : sig + include module type of Format_ + + val from_toml : Otoml.t -> file option +end = struct + include Format_ + + let ( let* ) = Option.bind + + let fpath_opt s = match Fpath.of_string s with Ok p -> Some p | Error _ -> None + + let rec depends_of_toml toml = + let* type_ = Otoml.Helpers.find_string_opt toml [ "type" ] in + match type_ with + | "local" -> + let* name = Otoml.Helpers.find_string_opt toml [ "name" ] in + let alias = Otoml.Helpers.find_string_opt toml [ "alias" ] in + let path = Option.bind (Otoml.Helpers.find_string_opt toml [ "path" ]) fpath_opt in + Some (Local { name; alias; path }) + | "installed" -> + let* name = Otoml.Helpers.find_string_opt toml [ "name" ] in + let alias = Otoml.Helpers.find_string_opt toml [ "alias" ] in + Some (Installed { name; alias }) + | "external" -> + let* link = Otoml.find_opt toml Otoml.get_value [ "link" ] in + let* ext = ext_of_toml link in + let path = Option.bind (Otoml.Helpers.find_string_opt toml [ "path" ]) fpath_opt in + let* post_t = Otoml.find_opt toml Otoml.get_value [ "post" ] in + let* post = depends_of_toml post_t in + Some (External { ext; path; post }) + | _ -> None + + and ext_of_toml link = + match Otoml.find_opt link Otoml.get_value [ "github" ] with + | Some gh -> + let* user = Otoml.Helpers.find_string_opt gh [ "user" ] in + let* repo = Otoml.Helpers.find_string_opt gh [ "repo" ] in + Some (Github { user; repo }) + | None -> + let* url = Otoml.Helpers.find_string_opt link [ "url" ] in + Some (Url { url }) + + let group_of_toml toml = + let* name = Otoml.Helpers.find_string_opt toml [ "name" ] in + let* dirs_s = Otoml.Helpers.find_strings_opt toml [ "dirs" ] in + let dirs = List.filter_map fpath_opt dirs_s in + let* main_s = Otoml.Helpers.find_string_opt toml [ "main" ] in + let* main = fpath_opt main_s in + let deps = + match Otoml.find_opt toml (Otoml.get_array ~strict:false Otoml.get_value) [ "deps" ] with + | Some items -> List.filter_map depends_of_toml items + | None -> [] + in + Some { name; dirs; deps; main } + + let from_toml toml = + match Otoml.find_opt toml (Otoml.get_array ~strict:false Otoml.get_value) [ "group" ] with + | Some groups -> Some (List.filter_map group_of_toml groups) + | None -> None +end diff --git a/src/Fronts/Project/Project.ml b/src/Fronts/Project/Project.ml index e69de29..cfc1a1c 100644 --- a/src/Fronts/Project/Project.ml +++ b/src/Fronts/Project/Project.ml @@ -0,0 +1 @@ +module Format = Format.Format \ No newline at end of file diff --git a/src/Fronts/Project/dune b/src/Fronts/Project/dune index 77a5718..5d3e6e8 100644 --- a/src/Fronts/Project/dune +++ b/src/Fronts/Project/dune @@ -1,4 +1,4 @@ (library (name project) (public_name stelf.project) - (libraries sexplib)) + (libraries otoml fpath)) diff --git a/src/Fronts/Tui/Repl.ml b/src/Fronts/Tui/Repl.ml index a375bcf..77f6202 100644 --- a/src/Fronts/Tui/Repl.ml +++ b/src/Fronts/Tui/Repl.ml @@ -23,12 +23,6 @@ module Repl (M : REPL.S) : REPL.REPL = struct let () = Display.register (fun m -> Lwt.return @@ add_msg m) - let seq_l : 'a 'b. 'a Lwt.t -> 'b Lwt.t -> 'b Lwt.t = - fun a b -> - let open Lwt.Syntax in - let* _ = a in - b - type response = Continue | Fail of string | Stop let term : LTerm.t ref Lwt.t = Lwt.map ref @@ Lazy.force LTerm.stdout @@ -75,41 +69,34 @@ module Repl (M : REPL.S) : REPL.REPL = struct exception Interrupted let stop code = exit code - + let rec read : (string -> response Lwt.t) -> int Lwt.t = fun f -> let open Lwt.Syntax in let* term' = term in - + let pp_msg fmt m : unit = Grace_ansi_renderer.pp_diagnostic fmt m in + let display_msg (s : Grace.Diagnostic.Severity.t) (m : Display.Info.t) : unit Lwt.t = + let diag = Grace.Diagnostic.create s (fun ppf -> Display.fmt ppf m.msg) in + let str = Format.asprintf "%a" pp_msg diag in + LTerm.printf "%s" str in + let open Grace.Diagnostic.Severity in let display_err (m : Display.Info.t) : unit Lwt.t = - let open Display in - let arrow = Form.(styles Style.[ Fore.red; bold ] (string "!>")) in - let out = Form.(markup @@ concat [ arrow; space (); m.msg ]) in - LTerm.printls out + display_msg Error m in let display_warn (m : Display.Info.t) : unit Lwt.t = - let open Display in - let arrow = Form.(styles Style.[ Fore.yellow; bold ] (string "!!")) in - let out = Form.(markup @@ concat [ arrow; space (); m.msg ]) in - LTerm.printls out + display_msg Warning m in let display_info (m : Display.Info.t) : unit Lwt.t = - let open Display in - let arrow = Form.(styles Style.[ Fore.blue; bold ] (string "*>")) in - let out = Form.(markup @@ concat [ arrow; space (); m.msg ]) in - LTerm.printls out + display_msg Note m in let display_debug (m : Display.Info.t) : unit Lwt.t = - let open Display in - let arrow = Form.(styles Style.[ Fore.magenta; bold ] (string "?>")) in - let out = Form.(markup @@ concat [ arrow; space (); m.msg ]) in - LTerm.printls out + display_msg Note m in let display_response (m : Display.Info.t) : unit Lwt.t = let open Display in let arrow = Form.(styles Style.[ Fore.orange; bold ] (string "=>")) in let out = Form.(markup @@ concat [ arrow; space (); m.msg ]) in - LTerm.printls out + Notty_lwt.output_image (Notty_lwt.eol out) in let should_display (msg : Display.Info.t) : bool = @@ -151,20 +138,29 @@ module Repl (M : REPL.S) : REPL.REPL = struct in match continue with - | Continue -> seq_l (flush ()) (read f) + | Continue -> + let* () = flush () in + read f | Fail msg -> Printf.eprintf "Error: %s\n%!" msg; - seq_l (flush ()) (Lwt.return 1) - | Stop -> seq_l (flush ()) (Lwt.return 0)) + let* () = flush () in + Lwt.return 1 + | Stop -> + let* () = flush () in + Lwt.return 0) (fun exn -> match exn with - | LTerm_read_line.Interrupt -> seq_l (flush ()) (Lwt.return 0) + | LTerm_read_line.Interrupt -> + let* () = flush () in + Lwt.return 0 | exn -> - seq_l - (display_err - (Display.Info.msg @@ Display.Form.string - @@ Printexc.to_string exn)) - (seq_l (flush ()) (read f))) + let* () = + display_err + (Display.Info.msg @@ Display.Form.string + @@ Printexc.to_string exn) + in + let* () = flush () in + read f) let show fmt = () (* Format.pp_print_string fmt "λΠ> "*) end diff --git a/src/Fronts/Tui/dune b/src/Fronts/Tui/dune index e3e75b1..f4aad3e 100644 --- a/src/Fronts/Tui/dune +++ b/src/Fronts/Tui/dune @@ -1,4 +1,4 @@ (library (name tui) (public_name stelf.tui_fronts) - (libraries frontend server fmt lambda-term lwt parsing display)) + (libraries frontend server fmt grace grace.ansi_renderer lambda-term lwt parsing display notty-community.lwt)) diff --git a/src/Lang/Parsing/Parser.ml b/src/Lang/Parsing/Parser.ml index 5f25bb1..c5ae1a0 100644 --- a/src/Lang/Parsing/Parser.ml +++ b/src/Lang/Parsing/Parser.ml @@ -26,7 +26,9 @@ module Parser : PARSER = struct | None -> return () | Some (' ' | '\t' | '\n' | '(' | ')' | '{' | '}' | '[' | ']' | '%') -> return () - | _ -> fail ("keyword " ^ s' ^ " not at word boundary") + | Some c -> + fail (Printf.sprintf + "expected whitespace or delimiter after '%s', found '%c'" s' c) in string s' *> boundary *> whitespace diff --git a/src/M2/Skolem.ml b/src/M2/Skolem.ml index f014d80..597df4e 100644 --- a/src/M2/Skolem.ml +++ b/src/M2/Skolem.ml @@ -80,11 +80,7 @@ end) : SKOLEM = struct in let s_ = spine d in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string - (Print.conDecToString sd_ ^ "\n"))) + Display.chatter_s 3 (Print.conDecToString sd_ ^ "\n") in installSkolem' (d, (v_, mS'), I.Dot (I.Exp (I.Root (h_, s_)), s), k) diff --git a/src/Names/Names.ml b/src/Names/Names.ml index 36fa467..16045b4 100644 --- a/src/Names/Names.ml +++ b/src/Names/Names.ml @@ -1,3 +1,5 @@ include Names_ include NAMES module Names_ = Names_ +module Names = Names_ +include Names diff --git a/src/Names/Names.mli b/src/Names/Names.mli index 1961e68..730bc5e 100644 --- a/src/Names/Names.mli +++ b/src/Names/Names.mli @@ -1,2 +1,4 @@ +module type NAMES = NAMES include module type of Names_ module Names_ : module type of Names_ +module Names = Names_ diff --git a/src/Names/Names_.ml b/src/Names/Names_.ml index 3b66832..85a31fc 100644 --- a/src/Names/Names_.ml +++ b/src/Names/Names_.ml @@ -649,14 +649,14 @@ module MakeNames let extent = function Exist -> Global | Univ ext -> ext let rec namePrefOf'' = function - | Exist, None -> "X" - | Univ _, None -> "x" + | Exist, None -> "_?" + | Univ _, None -> "_" | Exist, Some (ePref, uPref) -> hd ePref | Univ _, Some (ePref, uPref) -> hd uPref let rec namePrefOf' = function - | Exist, None -> "X" - | Univ _, None -> "x" + | Exist, None -> "_?" + | Univ _, None -> "_" | role, Some (IntSyn.Const cid) -> namePrefOf'' (role, Array.sub (namePrefArray, cid)) | role, Some (IntSyn.Def cid) -> @@ -865,8 +865,8 @@ module MakeNames let rec findNameLocal (g_, base, i) = let name = base - ^ begin if i = 0 then "" else Int.toString i - end + ^ Int.toString i + in begin if varDefined name || conDefined name || ctxDefined (g_, name) then findNameLocal (g_, base, i + 1) diff --git a/src/Opsem/MemoTable.ml b/src/Opsem/MemoTable.ml index 61e947c..c09ca70 100644 --- a/src/Opsem/MemoTable.ml +++ b/src/Opsem/MemoTable.ml @@ -679,29 +679,20 @@ end) : MEMOTABLE = struct begin added := true; begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 5) - (Display.Info.Form.string "\t -- Add goal \n")); + Display.chatter_s 5 "\t -- Add goal \n"; T.NewEntry answRef end end end | _, T.RepeatedEntry (((_, asub) as s), answRef, status) -> begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 5) - (Display.Info.Form.string "\t -- Suspend goal\n")); + Display.chatter_s 5 "\t -- Suspend goal\n"; T.RepeatedEntry ((esub, asub), answRef, status) end | _, T.DivergingEntry (_, answRef) -> begin begin added := true; begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 5) - (Display.Info.Form.string "\t -- Add diverging goal\n")); + Display.chatter_s 5 "\t -- Add diverging goal\n"; T.DivergingEntry (I.id, answRef) end end @@ -727,28 +718,19 @@ end) : MEMOTABLE = struct | _, T.NewEntry answRef -> begin added := true; begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 5) - (Display.Info.Form.string "\t -- Add goal \n")); + Display.chatter_s 5 "\t -- Add goal \n"; T.NewEntry answRef end end | _, T.RepeatedEntry (asub, answRef, status) -> begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 5) - (Display.Info.Form.string "\t -- Suspend goal\n")); + Display.chatter_s 5 "\t -- Suspend goal\n"; T.RepeatedEntry (asub, answRef, status) end | _, T.DivergingEntry (_, answRef) -> begin begin added := true; begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 5) - (Display.Info.Form.string "\t -- Add diverging goal\n")); + Display.chatter_s 5 "\t -- Add diverging goal\n"; T.DivergingEntry (I.id, answRef) end end diff --git a/src/Prover/Split.ml b/src/Prover/Split.ml index cc171e8..21caf97 100644 --- a/src/Prover/Split.ml +++ b/src/Prover/Split.ml @@ -239,11 +239,7 @@ end) : SPLIT with module State = Split__0.State' = struct | (g_, i), ([], _, _, _) -> [] | (g_, i), (x_ :: xs_, f_, w_, sc) -> let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 6) - (Display.Info.Form.string - (("Split " ^ Print.expToString (I.Null, x_)) ^ ".\n"))) + Display.chatter_s 6 (("Split " ^ Print.expToString (I.Null, x_)) ^ ".\n") in let os_ = splitXs (g_, i + 1) (xs_, f_, w_, sc) in let _ = resetCases () in @@ -257,12 +253,7 @@ end) : SPLIT with module State = Split__0.State' = struct end with Constraints.Error constrs -> begin - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 6) - (Display.Info.Form.string - (("Inactive split:\n" ^ Print.cnstrsToString constrs) - ^ "\n"))); + Display.chatter_s 6 (("Inactive split:\n" ^ Print.cnstrsToString constrs) ^ "\n"); os_ end in diff --git a/src/Recon/ReconConDec.ml b/src/Recon/ReconConDec.ml index 95336f1..e2896a7 100644 --- a/src/Recon/ReconConDec.ml +++ b/src/Recon/ReconConDec.ml @@ -97,10 +97,7 @@ module Make_ReconConDec in let ocd = Paths.dec (i, oc) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string (Print.conDecToString cd ^ "\n"))) + Display.chatter_s 3 (Print.conDecToString cd ^ "\n") in let _ = if !Global.doubleCheck then @@ -146,10 +143,7 @@ module Make_ReconConDec end in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string (Print.conDecToString cd ^ "\n"))) + Display.chatter_s 3 (Print.conDecToString cd ^ "\n") in let _ = if !Global.doubleCheck then begin @@ -202,10 +196,7 @@ module Make_ReconConDec (IntSyn.BlockDec (name, None, gsome'_, ctxToList gblock'_)) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string (Print.conDecToString bd ^ "\n"))) + Display.chatter_s 3 (Print.conDecToString bd ^ "\n") in (Some bd, None) | Cst.View.ConDec.BlockDef (_, name, worlds) -> @@ -227,10 +218,7 @@ module Make_ReconConDec in let bd = Names.nameConDec (IntSyn.BlockDef (name, None, cids)) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string (Print.conDecToString bd ^ "\n"))) + Display.chatter_s 3 (Print.conDecToString bd ^ "\n") in (Some bd, None) | _ -> raise (Error "condecToConDec: unrecognised conDec variant") diff --git a/src/Recon/ReconMode.ml b/src/Recon/ReconMode.ml index 6849436..a1b2869 100644 --- a/src/Recon/ReconMode.ml +++ b/src/Recon/ReconMode.ml @@ -67,7 +67,7 @@ module Make_ReconMode (M : S.S) : RECON_MODE with module M = M = struct end with Error m -> Display.( - message ~kind:Warning ~level:Verbose + message ~kind:Warning ~level:Detailed (string "Error processing mode declaration: " ++ string m)); raise' m end diff --git a/src/Recon/ReconQuery.ml b/src/Recon/ReconQuery.ml index 8c4e9f9..c011c89 100644 --- a/src/Recon/ReconQuery.ml +++ b/src/Recon/ReconQuery.ml @@ -58,10 +58,7 @@ module Make_ReconQuery in let cd = Names.nameConDec cd in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string (Print.conDecToString cd ^ "\n"))) + Display.chatter_s 3 (Print.conDecToString cd ^ "\n") in let _ = if !Global.doubleCheck then begin @@ -88,10 +85,7 @@ module Make_ReconQuery in let cd = Names.nameConDec cd in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 3) - (Display.Info.Form.string (Print.conDecToString cd ^ "\n"))) + Display.chatter_s 3 (Print.conDecToString cd ^ "\n") in let _ = if !Global.doubleCheck then begin diff --git a/src/Tomega/Converter.ml b/src/Tomega/Converter.ml index e556c98..686b143 100644 --- a/src/Tomega/Converter.ml +++ b/src/Tomega/Converter.ml @@ -113,11 +113,7 @@ module MakeConverter | _ -> raise (Error "Type Constant declaration expected") end - let rec chatter chlev f = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter chlev) - (Display.Info.Form.string ("[tomega] " ^ f ()))) + let rec chatter chlev f = Display.chatter_s chlev ("[tomega] " ^ f ()) let rec strengthenExp (u_, s) = Whnf.normalize (Whnf.cloInv (u_, s), I.id) let rec strengthenSub (s, t) = Whnf.compInv (s, t) @@ -901,17 +897,11 @@ module MakeConverter let name = I.conDecName (I.sgnLookup cid) in let _ = TomegaTypeCheck.checkPrg (I.Null, (p_, f_)) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string "[Redundancy Checker (factoring) ...")) + Display.chatter_s 4 "[Redundancy Checker (factoring) ..." in let factP = Redundant.convert p_ in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string "done]\n")) + Display.chatter_s 4 "done]\n" in let lemma = T.lemmaAdd (T.ValDec (name, factP, f_)) in (lemma, [], []) @@ -924,17 +914,11 @@ module MakeConverter let s = name cids in let _ = TomegaTypeCheck.checkPrg (I.Null, (p_, f_)) in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string "[Redundancy Checker (factoring) ...")) + Display.chatter_s 4 "[Redundancy Checker (factoring) ..." in let factP = Redundant.convert p_ in let _ = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter 4) - (Display.Info.Form.string "done]\n")) + Display.chatter_s 4 "done]\n" in let lemma = T.lemmaAdd (T.ValDec (s, factP, f_)) in let sels = installSelection (cids, projs, f_, lemma) in diff --git a/src/Tomega/Coverage.ml b/src/Tomega/Coverage.ml index 6e0eed1..8297c3a 100644 --- a/src/Tomega/Coverage.ml +++ b/src/Tomega/Coverage.ml @@ -40,11 +40,7 @@ module MakeTomegaCoverage module Cover = Cover module TomegaTypeCheck = TomegaTypeCheck - let rec chatter chlev f = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter chlev) - (Display.Info.Form.string ("[coverage] " ^ f ()))) + let rec chatter chlev f = Display.chatter_s chlev ("[coverage] " ^ f ()) let rec purifyFor = function | (T.Unit, t), (psi_, T.True), s -> (t, psi_, s) diff --git a/src/Tomega/TomegaTypecheck.ml b/src/Tomega/TomegaTypecheck.ml index 4760c50..8dd6bb4 100644 --- a/src/Tomega/TomegaTypecheck.ml +++ b/src/Tomega/TomegaTypecheck.ml @@ -35,11 +35,7 @@ end) : TOMEGATYPECHECK = struct module S = Subordinate module TA = TomegaTypeCheck__0.TomegaAbstract - let rec chatter chlev f = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter chlev) - (Display.Info.Form.string (f ()))) + let rec chatter chlev f = Display.chatter_s chlev (f ()) let rec normalizeHead = function | T.Const lemma, t -> T.Const lemma diff --git a/src/Unique/Unique_.ml b/src/Unique/Unique_.ml index cbf49c5..4388212 100644 --- a/src/Unique/Unique_.ml +++ b/src/Unique/Unique_.ml @@ -44,11 +44,7 @@ module MakeUnique module N = Names module T = Tomega - let rec chatter chlev f = - Display.display' - (Display.Info.msg - ~level:(Display.Info.from_chatter chlev) - (Display.Info.Form.string (f ()))) + let rec chatter chlev f = Display.chatter_s chlev (f ()) let rec cName cid = N.qidToString (N.constQid cid) diff --git a/stelf.opam b/stelf.opam index 8ed7687..b49a614 100644 --- a/stelf.opam +++ b/stelf.opam @@ -13,17 +13,17 @@ doc: "https://github.com/standardocaml/stelf/wiki" bug-reports: "https://github.com/standardocaml/stelf/issues" depends: [ "dune" {>= "3.21"} - "ocaml" {>= "5.0.0" & !(< "5.5.0" & >= "5.4.0")} + "ocaml" {>= "5.0.0" & !(< "5.6.0" & >= "5.4.0")} "lwt" "containers" "base" "angstrom" "alcotest" {with-test} "alcotest-lwt" {with-test} - "bisect-ppx" {dev} + "grace" + "notty-community" "odoc" {with-doc} "sherlodoc" {with-doc} - "bisect_ppx" {dev} "ocamlformat" {dev} "ocaml-lsp-server" {dev} "user-setup" {dev} @@ -35,6 +35,7 @@ depends: [ "bos" "cmdliner" "logs" + "otoml" ] build: [ ["dune" "subst"] {dev} diff --git a/test/Pal/Cases.ml b/test/Pal/Cases.ml index 287661c..312676d 100644 --- a/test/Pal/Cases.ml +++ b/test/Pal/Cases.ml @@ -74,13 +74,8 @@ let cases () = test "LP-HORN-ND" Source.[ lp_horn_nd ]; test "CHURCH-ROSSER-LAM" Source.[ church_rosser_lam ]; test "CUT-ELIM-FORMULAS" Source.[ cut_elim_formulas ]; - (* cut_elim/sources: placed immediately after CUT-ELIM-FORMULAS because later - groups (CRARY-EXCON, SMALL-STEP-LAM, TAPL-DEFS, etc.) declare o/tp/exp as - terms, polluting the sort `o : type` that hyp/conc depend on. - Uses explicit parameters (no `{{A}}` implicit) and infix form for operators - (and/imp/or are globally registered as infix from CUT-ELIM-FORMULAS). - Omitted: %mode/%worlds/%total/etc. — just syntax declarations. *) - test "cut_elim/sources" Source.[ cut_elim_sources_2 ]; + test "cut_elim/sources" + Source.[ cut_elim_formulas; cut_elim_sources_2 ]; test "GUIDE-ND" Source.[ guide_nd ]; test "CPSOCC-DSBNF" Source.[ cpsocc_dsbnf ]; test "CPSOCC-CPSBF" Source.[ cpsocc_cpsBNF ]; @@ -125,40 +120,32 @@ let cases () = ]; test "POPLMARK-1A" Source.[ poplmark_1a_syntax; poplmark_1b_syntax; poplmark_2b_syntax ]; - (* POPLMARK-2A: `of`/`term`/`value` conflict with earlier suite declarations - in the shared global state of the Pal frontend. Kept as failing test consistent - with FOL-14, S4-1, LAM-1, POLYLAM-1, GUIDE-ND-1 pattern — runtime failure only. - *) - test "POPLMARK-2A" Source.[ poplmark_2a_syntax ]; + test "POPLMARK-2A" + Source.[ poplmark_1a_syntax; poplmark_2a_syntax ]; test "CCC" Source.[ ccc_syntax ]; test "INCLL" Source.[ incll_syntax ]; test "CRARY-LINEAR" Source.[ crary_linear_syntax; crary_linear_linear ]; test "CRARY-LINEARD" Source.[ crary_lineard_syntax ]; test "CRARY-MODAL" Source.[ crary_modal_syntax ]; - (* cut_elim/sources: extends CUT-ELIM-FORMULAS with int.elf (sequent calculus). - cut_elim_formulas is omitted since it is in global scope from CUT-ELIM-FORMULAS. - hyp and conc are fresh names. *) - (* church_rosser/sources: extends CHURCH-ROSSER-LAM with ord-red and par-red. - church_rosser_lam is omitted here since term/lam/app are already in global - scope from the CHURCH-ROSSER-LAM group above. *) test "church_rosser/sources" - Source.[ church_rosser_sources_2; church_rosser_sources_3 ]; - (* mini_ml/sources: extends MINI-ML with eval and type-inference. - mini_ml_exp/value/tp are omitted since they are in global scope from MINI-ML. - Skipped: by the time mini_ml/sources runs, `z` has been re-declared as `nat` - by ARITH/TAPL groups, so `eval z z` (expecting exp-typed z) fails with a - type mismatch. The STELF reconstructor does not perform type-directed - disambiguation for overloaded names (known limitation as of 2026-06). *) + Source. + [ + church_rosser_lam; + church_rosser_sources_2; + church_rosser_sources_3; + ]; test ~skip:true "mini_ml/sources" - Source.[ mini_ml_sources_eval; mini_ml_sources_tpinf ]; - (* lp_horn/sources: extends LP-HORN-ND with canonical forms and conversion. - lp_horn_nd is omitted since it is in global scope from LP-HORN-ND. - Skipped: %sort can {A o} {_ pf A} (dependent sort where second arg type depends - on first arg value) triggers "Classifier in declaration must be a type" with - "Inferred: %tp%" — STELF reconstructor does not handle dependent sort indices - in %sort declarations (known limitation as of 2026-06). *) + Source. + [ + mini_ml_exp; + mini_ml_value; + mini_ml_tp; + mini_ml_sources_eval; + mini_ml_sources_tpinf; + ]; + (* Skipped: dependent sort indices in %sort declarations not yet supported. *) test ~skip:true "lp_horn/sources" - Source.[ lp_horn_sources_2; lp_horn_sources_3 ]; + Source.[ lp_horn_nd; lp_horn_sources_2; lp_horn_sources_3 ]; (* examples/arith/sources.cfg: nat + nt + plus + acker — same content as ARITH above, re-declared. Pal frontend is lenient about re-declarations. *) test "arith/sources" @@ -220,10 +207,6 @@ let cases () = test "crary/substruct/modal" Source.[ crary_modal_syntax ]; (* Tier 2: single-elf cfg files *) - (* examples/handbook/sources.cfg → handbook/fol.elf: same FOL content as existing - Source.ml fol suite. Skipped: by the time this runs, `o` has been re-declared - as a term by CRARY-EXCON (%term o tp), causing "Level clash" when handbook_sources_1 - tries to use `o` as a type sort. *) test ~skip:true "handbook/sources" Source.[ handbook_sources_1 ]; (* examples/ccc/spass.cfg → spass.elf: CCC with categorical laws. Uses dependent sort indices (== : mor A B -> mor A B -> type) — @@ -261,5 +244,7 @@ let cases () = test ~skip:true "tabled/parsing/tab" Source.[ tabled_parsing_tab_1 ]; (* examples/tabled/ccc/tab.cfg: CCC with tabling. *) test ~skip:true "tabled/ccc/tab" Source.[ tabled_ccc_tab_1 ]; - ] - end + test "nat-scope" Source.[ nat_scope ]; + ]; + + end diff --git a/test/Pal/Common.ml b/test/Pal/Common.ml index 1b612fb..9178e54 100644 --- a/test/Pal/Common.ml +++ b/test/Pal/Common.ml @@ -6,14 +6,31 @@ let test ?(skip = false) ?(failure = false) (name : string) (cmds : string list) let () = Fmt_tty.setup_std_outputs () in let () = Display.register (fun m -> - let _ = Display.fmt Fmt.stdout m.msg in + if Display.Info.( >= ) Display.Info.Normal m.level then begin + let severity = + let open Grace.Diagnostic.Severity in + match m.kind with + | Some Display.Info.Error -> Error + | Some Warning -> Warning + | Some Info | Some Debug | Some Response | None -> Note + in + let diag = + Grace.Diagnostic.create severity (fun ppf -> + Display.fmt ppf m.msg) + in + Grace_ansi_renderer.pp_diagnostic Format.std_formatter diag + end; Lwt.return ()) in - let module P = Pal.Pal.Start () in + let exec = + lazy + (let module P = Pal.Pal.Start () in + P.exec) + in let run cmd = try Printexc.record_backtrace true; - P.exec cmd; + (Lazy.force exec) cmd; None with e -> Some e in diff --git a/test/Pal/Data/Arith.ml b/test/Pal/Data/Arith.ml new file mode 100644 index 0000000..7e468e6 --- /dev/null +++ b/test/Pal/Data/Arith.ml @@ -0,0 +1,36 @@ +(* Ackermann function in unary arithmetic + Ported from twelf/examples/arith/arith.elf + Note: %compile and %query directives omitted (not supported) +*) +let arith_nat = {| +%sort nat +%name nat X +%term z nat +%term s {_ nat} nat +|} + +let arith_nt = + {| +%sort nt {_ nat} +%name nt N +%term nt-z nt z +%term nt-s {{X}} %if (nt (s X)) %<- (nt X) +|} + +let arith_plus = + {| +%sort plus {_ nat} {_ nat} {_ nat} +%name plus P +%term p-z {{Y}} plus z Y Y +%term p-s {{X Y Z}} %if (plus (s X) Y (s Z)) %<- (plus X Y Z) +|} + +let arith_acker = + {| +%sort acker {_ nat} {_ nat} {_ nat} +%mode acker %in %in %out +%term a-1 {{Y}} acker z Y (s Y) +%term a-2 {{X Z}} %if (acker (s X) z Z) %<- (acker X (s z) Z) +%term a-3 {{X Y Z Z'}} %if (acker (s X) (s Y) Z) %<- (acker (s X) Y Z') %<- (acker X Z' Z) +%worlds () (acker _ _ _) +|} diff --git a/test/Pal/Data/Ccc.ml b/test/Pal/Data/Ccc.ml new file mode 100644 index 0000000..f835bea --- /dev/null +++ b/test/Pal/Data/Ccc.ml @@ -0,0 +1,92 @@ +(* CCC: Cartesian closed category syntax. + Ported from twelf/examples/ccc/ccc.elf. + Renamed: `1` (terminal obj) → `unit_obj`, `*` (product) → `prod`, `@` → `comp`, + `==` (morphism equality) → `meq`, `=>` (exponential) → `exp_obj`. + Note: no %prec declarations — all operators used in prefix form to avoid + "Leading infix operator" parse errors that occur when infix is mixed with prefix. +*) +let ccc_syntax = + {| +%sort obj +%name obj _A +%sort mor {_ obj} {_ obj} +%name mor _F +%sort meq {_ mor _A _B} {_ mor _A _B} +%name meq _ME + +%term id {{A}} mor A A +%term comp {{A B C}} {_ mor B C} {_ mor A B} mor A C + +%term meq_refl {{A B F}} meq F F +%term meq_then {{A B F F' F''}} {_ meq F F'} {_ meq F' F''} meq F F'' +%term meq_sym {{A B F F'}} {_ meq F F'} meq F' F + +%term eq_comp {{A B C F F' G G'}} {_ meq F F'} {_ meq G G'} meq (comp F G) (comp F' G') + +%term id_l {{A B F}} meq (comp id F) F +%term id_r {{A B F}} meq (comp F id) F +%term assoc {{A B C D F G H}} meq (comp H (comp G F)) (comp (comp H G) F) + +%term unit_obj obj +%term prod {_ obj} {_ obj} obj + +%term drop {{A}} mor A unit_obj +%term fst {{A B}} mor (prod A B) A +%term snd {{A B}} mor (prod A B) B +%term pair_mor {{A B C}} {_ mor A B} {_ mor A C} mor A (prod B C) + +%term eq_pair {{A B C F F' G G'}} {_ meq F F'} {_ meq G G'} meq (pair_mor F G) (pair_mor F' G') +%term prod_l {{A B C F G}} meq (comp fst (pair_mor F G)) F +%term prod_r {{A B C F G}} meq (comp snd (pair_mor F G)) G +%term prod_u {{A B C H}} meq (pair_mor (comp fst H) (comp snd H)) H + +%term exp_obj {_ obj} {_ obj} obj + +%term app_mor {{B C}} mor (prod (exp_obj B C) B) C +%term cur {{A B C}} {_ mor (prod A B) C} mor A (exp_obj B C) + +%term eq_cur {{A B C F F'}} {_ meq F F'} meq (cur F) (cur F') +%term exp_e {{A B C F}} meq (comp app_mor (pair_mor (comp (cur F) fst) snd)) F +%term exp_u {{A B C G}} meq (cur (comp app_mor (pair_mor (comp G fst) snd))) G +|} + +(* ccc/spass.elf: CCC with categorical laws, products, exponentials, and lemmas. + Same approach as ccc_syntax: all operators renamed to avoid infix parse errors. + spass_meq replaces ==, spass_comp replaces @, spass_prod_obj replaces *, spass_exp_obj replaces =>. + spass_meq uses implicit free-variable indices (same pattern as ccc_syntax meq). + Additional lemma terms: spass_distp, spass_appl, spass_distc. *) +let ccc_spass_1 = + {| +%sort spass_obj +%sort spass_mor {_ spass_obj} {_ spass_obj} +%sort spass_meq {_ spass_mor _SA _SB} {_ spass_mor _SA _SB} +%term spass_id {A spass_obj} spass_mor A A +%term spass_comp {{A B C}} {_ spass_mor B C} {_ spass_mor A B} spass_mor A C +%term spass_refl {{SA SB F}} spass_meq F F +%term spass_then {{SA SB F F' F''}} {_ spass_meq F F'} {_ spass_meq F' F''} spass_meq F F'' +%term spass_sym {{SA SB F F'}} {_ spass_meq F F'} spass_meq F' F +%term spass_ceq {{SA SB SC F F' G G'}} {_ spass_meq F F'} {_ spass_meq G G'} spass_meq (spass_comp F G) (spass_comp F' G') +%term spass_id_l {{SA SB F}} spass_meq (spass_comp spass_id F) F +%term spass_id_r {{SA SB F}} spass_meq (spass_comp F spass_id) F +%term spass_ass {{SA SB SC SD F G H}} spass_meq (spass_comp H (spass_comp G F)) (spass_comp (spass_comp H G) F) +%term spass_unit_obj spass_obj +%term spass_prod_obj {_ spass_obj} {_ spass_obj} spass_obj +%term spass_drop {A spass_obj} spass_mor A spass_unit_obj +%term spass_fst {{A B}} spass_mor (spass_prod_obj A B) A +%term spass_snd {{A B}} spass_mor (spass_prod_obj A B) B +%term spass_pair {{A B C}} {_ spass_mor A B} {_ spass_mor A C} spass_mor A (spass_prod_obj B C) +%term spass_peq {{A B C F F' G G'}} {_ spass_meq F F'} {_ spass_meq G G'} spass_meq (spass_pair F G) (spass_pair F' G') +%term spass_term_u {{A H}} spass_meq H spass_drop +%term spass_prod_l {{A B C F G}} spass_meq (spass_comp spass_fst (spass_pair F G)) F +%term spass_prod_r {{A B C F G}} spass_meq (spass_comp spass_snd (spass_pair F G)) G +%term spass_prod_u {{A B C H}} spass_meq (spass_pair (spass_comp spass_fst H) (spass_comp spass_snd H)) H +%term spass_exp_obj {_ spass_obj} {_ spass_obj} spass_obj +%term spass_app {{B C}} spass_mor (spass_prod_obj (spass_exp_obj B C) B) C +%term spass_cur {{A B C}} {_ spass_mor (spass_prod_obj A B) C} spass_mor A (spass_exp_obj B C) +%term spass_ceq2 {{A B C F F'}} {_ spass_meq F F'} spass_meq (spass_cur F) (spass_cur F') +%term spass_exp_e {{A B C F}} spass_meq (spass_comp spass_app (spass_pair (spass_comp (spass_cur F) spass_fst) spass_snd)) F +%term spass_exp_u {{A B C G}} spass_meq (spass_cur (spass_comp spass_app (spass_pair (spass_comp G spass_fst) spass_snd))) G +%term spass_distp {{A B C D F G H}} spass_meq (spass_comp (spass_pair F G) H) (spass_pair (spass_comp F H) (spass_comp G H)) +%term spass_appl {{A B C D F G H}} spass_meq (spass_comp spass_app (spass_pair (spass_comp (spass_cur F) G) H)) (spass_comp F (spass_pair G H)) +%term spass_distc {{A B C D F G}} spass_meq (spass_comp (spass_cur F) G) (spass_cur (spass_comp F (spass_pair (spass_comp G spass_fst) spass_snd))) +|} diff --git a/test/Pal/Data/ChurchRosser.ml b/test/Pal/Data/ChurchRosser.ml new file mode 100644 index 0000000..a28a844 --- /dev/null +++ b/test/Pal/Data/ChurchRosser.ml @@ -0,0 +1,71 @@ +(* Untyped lambda-calculus terms. + Ported from twelf/examples/church_rosser/lam.elf. + This is just the term syntax — no reduction or typing. +*) +let church_rosser_lam = + {| +%sort term +%name term M +%term lam {_ {_ term} term} term +%term app {_ term} {_ term} term +|} + +(* CHURCH-ROSSER ord-red: ordinary reduction relations on untyped lambda terms. + Ported from twelf/examples/church_rosser/ord-red.elf. + Builds on church_rosser_lam (term/lam/app already in scope). + Names introduced: -->, id1, step1, -->*, refl/sym/trans/red, <-> +*) +let church_rosser_sources_2 = + {| +%sort --> {_ term} {_ term} +%prec %none 10 --> + +%term beta1 {{M1 M2}} (app (lam M1) M2) --> (M1 M2) +%term lm1 {{M M'}} {_ {x term} (M x) --> (M' x)} (lam M) --> (lam M') +%term apl1 {{M1 M1' M2}} {_ M1 --> M1'} (app M1 M2) --> (app M1' M2) +%term apr1 {{M1 M2 M2'}} {_ M2 --> M2'} (app M1 M2) --> (app M1 M2') + +%sort -->* {_ term} {_ term} +%prec %none 10 -->* + +%term id1 {{M}} M -->* M +%term step1 {{M M' M''}} {_ M --> M'} {_ M' -->* M''} M -->* M'' + +%sort <-> {_ term} {_ term} +%prec %none 10 <-> + +%term ord-refl {{M}} M <-> M +%term ord-sym {{M M'}} {_ M' <-> M} M <-> M' +%term ord-trans {{M M' M''}} {_ M <-> M'} {_ M' <-> M''} M <-> M'' +%term ord-red {{M M'}} {_ M -->* M'} M <-> M' +|} + +(* CHURCH-ROSSER par-red: parallel reduction relations on untyped lambda terms. + Ported from twelf/examples/church_rosser/par-red.elf. + Builds on church_rosser_lam (term/lam/app in scope). + Names introduced: =>, =>*, <=> + Note: par-beta's first premise is higher-order: + {x:term}{idx:x=>x} (M1 x idx) => (M1' x idx) + Simplified here with {{M1 M1'}} implicit for the higher-order functions. +*) +let church_rosser_sources_3 = + {| +%sort => {_ term} {_ term} +%prec %none 10 => + +%term par-beta {{M2 M2'}} {_ {x term} {_ x => x} (M1 x) => (M1' x)} {_ M2 => M2'} (app (lam M1) M2) => (M1' M2') +%term par-ap {{M1 M1' M2 M2'}} {_ M1 => M1'} {_ M2 => M2'} (app M1 M2) => (app M1' M2') +%term par-lm {{M M'}} {_ {x term} {_ x => x} (M x) => (M' x)} (lam M) => (lam M') + +%sort =>* {_ term} {_ term} +%prec %none 10 =>* + +%term par-id {{M}} M =>* M +%term par-step {{M M' M''}} {_ M => M'} {_ M' =>* M''} M =>* M'' + +%sort <=> {_ term} {_ term} +%prec %none 10 <=> + +%term par-reduce {{M M'}} {_ M =>* M'} M <=> M' +%term par-expand {{M M'}} {_ M =>* M'} M' <=> M +|} diff --git a/test/Pal/Data/Cpsocc.ml b/test/Pal/Data/Cpsocc.ml new file mode 100644 index 0000000..5a6e49f --- /dev/null +++ b/test/Pal/Data/Cpsocc.ml @@ -0,0 +1,37 @@ +(* Direct-style CPS BNF term syntax. + Ported from twelf/examples/cpsocc/dsBNF.elf. + Simple type declarations only — no infix operators. +*) +let cpsocc_dsbnf = + {| +%sort droot +%name droot DROOT +%sort dexp +%name dexp DEXP +%sort dtriv +%name dtriv DTRIV + +%term dexp->droot {_ dexp} droot +%term dapp {_ dexp} {_ dexp} dexp +%term dtriv->dexp {_ dtriv} dexp +%term dlam {_ {_ dtriv} droot} dtriv +|} + +(* CPSOCC cpsBNF: BNF of continuation-passing style terms. + Ported from twelf/examples/cpsocc/cpsBNF.elf. + Builds on cpsocc_dsbnf (droot/dexp/dtriv already in scope). + Fresh names introduced: croot, cexp, ctriv, ccont, klam, capp, cret, xlam, vlam. + %name hints omitted (not needed for test correctness). +*) +let cpsocc_cpsBNF = + {| +%sort croot +%sort cexp +%sort ctriv +%sort ccont +%term klam {_ {_ ccont} cexp} croot +%term capp {_ ctriv} {_ ctriv} {_ ccont} cexp +%term cret {_ ccont} {_ ctriv} cexp +%term xlam {_ {_ ctriv} croot} ctriv +%term vlam {_ {_ ctriv} cexp} ccont +|} diff --git a/test/Pal/Data/CraryExcon.ml b/test/Pal/Data/CraryExcon.ml new file mode 100644 index 0000000..1617ba7 --- /dev/null +++ b/test/Pal/Data/CraryExcon.ml @@ -0,0 +1,71 @@ +(* Explicit contexts in LF: types, expressions, and natural numbers. + Ported from twelf/examples/crary/explicit/excon.elf (first part only — + typing rules and nat up through leq). +*) +let crary_excon = + {| +%sort tp +%sort exp + +%term o tp +%term p {_ exp} tp +%term pi {_ tp} {_ {_ exp} tp} tp + +%term b exp +%term lam {_ tp} {_ {_ exp} exp} exp +%term app {_ exp} {_ exp} exp + +%sort of {_ exp} {_ tp} +%term of/b of b o +%term of/lam {{A B M}} {_ {x exp} {_ of x A} of (M x) (B x)} of (lam A M) (pi A B) +%term of/app {{A B M N}} {_ of M (pi A B)} {_ of N A} of (app M N) (B N) + +%sort nat +%term 0 nat +%term s {_ nat} nat + +%sort nat-eq {_ nat} {_ nat} +%term nat-eq/i {{N}} nat-eq N N + +%sort leq {_ nat} {_ nat} +%term leq/z {{N}} leq 0 N +%term leq/s {{N1 N2}} %if (leq (s N1) (s N2)) %<- (leq N1 N2) +|} + +(* CRARY-EXCON-REV: excon-rev.elf (explicit contexts, reversed variant) + Ported from twelf/examples/crary/explicit/excon-rev.elf (syntax-only chunk). + The `-` family (isvar dependency decl) is a Twelf quirk; omitted. +*) +let crary_excon_rev_syntax = + {| +%sort tp +%sort exp + +%term o tp +%term p {_ exp} tp +%term pi {_ tp} {_ {_ exp} tp} tp + +%term b exp +%term lam {_ tp} {_ {_ exp} exp} exp +%term app {_ exp} {_ exp} exp + +%sort of {_ exp} {_ tp} +%term of/b of b o +%term of/lam {{A B M}} {_ {x exp} {_ of x A} of (M x) (B x)} of (lam A M) (pi A B) +%term of/app {{A B M N}} {_ of M (pi A B)} {_ of N A} of (app M N) (B N) + +%sort nat +%term 0 nat +%term s {_ nat} nat + +%sort nat-eq {_ nat} {_ nat} +%term nat-eq/i {{N}} nat-eq N N + +%sort lt {_ nat} {_ nat} +%term lt/z {{N}} lt 0 (s N) +%term lt/s {{N1 N2}} %if (lt (s N1) (s N2)) %<- (lt N1 N2) + +%sort ctx +%term nil ctx +%term cons {_ ctx} {_ exp} {_ tp} ctx +|} diff --git a/test/Pal/Data/CraryLinear.ml b/test/Pal/Data/CraryLinear.ml new file mode 100644 index 0000000..3ff29fc --- /dev/null +++ b/test/Pal/Data/CraryLinear.ml @@ -0,0 +1,152 @@ +(* CRARY-LINEAR (syntax-only): linear substructural types. + Ported from twelf/examples/crary/substruct/linear.elf (syntax block only). + Type `!` in Twelf = the bang/of-course modality. Trying `!` as identifier verbatim. + `with` is a keyword in OCaml but it's inside a raw string — should be fine in the parser. +*) +let crary_linear_syntax = + {| +%sort atom +%sort tp +%sort term + +%term atomic {_ atom} tp +%term lolli {_ tp} {_ tp} tp +%term tensor {_ tp} {_ tp} tp +%term with {_ tp} {_ tp} tp +%term plus {_ tp} {_ tp} tp +%term one tp +%term zero tp +%term top tp +%term bang {_ tp} tp + +%term llam {_ {_ term} term} term +%term lapp {_ term} {_ term} term + +%term tpair {_ term} {_ term} term +%term lett {_ term} {_ {_ term} {_ term} term} term + +%term pair {_ term} {_ term} term +%term pi1 {_ term} term +%term pi2 {_ term} term + +%term in1 {_ term} term +%term in2 {_ term} term +%term case {_ term} {_ {_ term} term} {_ {_ term} term} term + +%term star term +%term leto {_ term} {_ term} term + +%term any {_ term} term +%term unit term + +%term bang_tm {_ term} term +%term letb {_ term} {_ {_ term} term} term + +%term a atom +%term b atom +|} + +(* CRARY-LINEAR-atoms: the a/b atoms for linear are declared in crary_linear_syntax above, + but crary_modal_syntax re-declares its own a/b below. Those in crary_linear_syntax + therefore come first and are the ones in scope for CRARY-LINEAR tests. +*) + +(* CRARY-LINEAR (linearity sort and basic terms): + The `linear` family from linear.elf. +*) +let crary_linear_linear = + {| +%sort linear {_ {_ term} term} +%term linear/var linear ([x] x) +%term linear/llam {{M}} {_ {y term} linear ([x] M x y)} linear ([x] llam ([y] M x y)) +%term linear/lapp1 {{M N}} {_ linear ([x] M x)} linear ([x] lapp (M x) N) +%term linear/lapp2 {{M N}} {_ linear ([x] N x)} linear ([x] lapp M (N x)) +%term linear/tpair1 {{M N}} {_ linear ([x] M x)} linear ([x] tpair (M x) N) +%term linear/tpair2 {{M N}} {_ linear ([x] N x)} linear ([x] tpair M (N x)) +%term linear/pair {{M N}} {_ linear ([x] M x)} {_ linear ([x] N x)} linear ([x] pair (M x) (N x)) +%term linear/pi1 {{M}} {_ linear ([x] M x)} linear ([x] pi1 (M x)) +%term linear/pi2 {{M}} {_ linear ([x] M x)} linear ([x] pi2 (M x)) +%term linear/in1 {{M}} {_ linear ([x] M x)} linear ([x] in1 (M x)) +%term linear/in2 {{M}} {_ linear ([x] M x)} linear ([x] in2 (M x)) +%term linear/unit linear ([x] unit) +|} + +(* CRARY-LINEARD (syntax-only): dual linear type system. + Ported from twelf/examples/crary/substruct/lineard.elf (syntax only). + Similar to linear.elf but adds `constant`, `pi` (dependent type), `ulam`/`uapp`. +*) +let crary_lineard_syntax = + {| +%sort constant +%sort tp +%sort term + +%term const {_ constant} {_ term} tp +%term pi {_ tp} {_ {_ term} tp} tp +%term lolli {_ tp} {_ tp} tp +%term tensor {_ tp} {_ tp} tp +%term with {_ tp} {_ tp} tp +%term plus {_ tp} {_ tp} tp +%term one tp +%term zero tp +%term top tp +%term bang {_ tp} tp + +%term ulam {_ {_ term} term} term +%term uapp {_ term} {_ term} term + +%term llam {_ {_ term} term} term +%term lapp {_ term} {_ term} term + +%term tpair {_ term} {_ term} term +%term lett {_ term} {_ {_ term} {_ term} term} term + +%term pair {_ term} {_ term} term +%term pi1 {_ term} term +%term pi2 {_ term} term + +%term in1 {_ term} term +%term in2 {_ term} term +%term case {_ term} {_ {_ term} term} {_ {_ term} term} term + +%term star term +%term leto {_ term} {_ term} term + +%term any {_ term} term +%term unit term + +%term bang_tm {_ term} term +%term letb {_ term} {_ {_ term} term} term +|} + +(* CRARY-MODAL (syntax-only): modal substructural types. + Ported from twelf/examples/crary/substruct/modal.elf (syntax block only). + Dropped: `local/bx : local ([x] bx M) = local/closed.` definitional equality form. +*) +let crary_modal_syntax = + {| +%sort atom +%sort tp +%sort term +%sort exp + +%term atomic {_ atom} tp +%term arrow {_ tp} {_ tp} tp +%term box {_ tp} tp +%term diamond {_ tp} tp + +%term lam {_ {_ term} term} term +%term app {_ term} {_ term} term + +%term bx {_ term} term +%term letbx {_ term} {_ {_ term} term} term + +%term di {_ exp} term + +%term here {_ term} exp +%term eletbx {_ term} {_ {_ term} exp} exp +%term letdi {_ term} {_ {_ term} exp} exp + +%term a atom +%term b atom +|} diff --git a/test/Pal/Data/CutElim.ml b/test/Pal/Data/CutElim.ml new file mode 100644 index 0000000..d7538e1 --- /dev/null +++ b/test/Pal/Data/CutElim.ml @@ -0,0 +1,53 @@ +(* Formula syntax for predicate calculus (intuitionistic and classical). + Ported from twelf/examples/cut_elim/formulas.elf. +*) +let cut_elim_formulas = + {| +%sort i +%name i T +%sort o +%name o A +%term and {_ o} {_ o} o +%prec %right 11 and +%term imp {_ o} {_ o} o +%prec %right 10 imp +%term or {_ o} {_ o} o +%prec %right 11 or +%term not {_ o} o +%prec %prefix 12 not +%term true o +%term false o +%term forall {_ {_ i} o} o +%term exists {_ {_ i} o} o +|} + +(* CUT-ELIM int: cut-free intuitionistic sequent calculus. + Ported from twelf/examples/cut_elim/int.elf. + Builds on cut_elim_formulas (i, o, and, imp, or, not, true, false, forall, exists). + Names introduced: hyp, conc and all sequent calculus constructors. +*) +let cut_elim_sources_2 = + {| +%sort hyp {_ o} +%name hyp H +%sort conc {_ o} +%name conc D + +%term axiom {A o} {_ hyp A} conc A +%term andr {A o} {B o} {_ conc A} {_ conc B} conc (A and B) +%term andl1 {A o} {B o} {C o} {_ {_ hyp A} conc C} {_ hyp (A and B)} conc C +%term andl2 {A o} {B o} {C o} {_ {_ hyp B} conc C} {_ hyp (A and B)} conc C +%term impr {A o} {B o} {_ {_ hyp A} conc B} conc (A imp B) +%term impl {A o} {B o} {C o} {_ conc A} {_ {_ hyp B} conc C} {_ hyp (A imp B)} conc C +%term orr1 {A o} {B o} {_ conc A} conc (A or B) +%term orr2 {A o} {B o} {_ conc B} conc (A or B) +%term orl {A o} {B o} {C o} {_ {_ hyp A} conc C} {_ {_ hyp B} conc C} {_ hyp (A or B)} conc C +%term notr {A o} {_ {p o} {_ hyp A} conc p} conc (not A) +%term notl {A o} {C o} {_ conc A} {_ hyp (not A)} conc C +%term truer conc true +%term falsel {C o} {_ hyp false} conc C +%term forallr {A {_ i} o} {_ {a i} conc (A a)} conc (forall A) +%term foralll {A {_ i} o} {C o} {T i} {_ {_ hyp (A T)} conc C} {_ hyp (forall A)} conc C +%term existsr {A {_ i} o} {T i} {_ conc (A T)} conc (exists A) +%term existsl {A {_ i} o} {C o} {_ {a i} {_ hyp (A a)} conc C} {_ hyp (exists A)} conc C +|} diff --git a/test/Pal/Data/Failures.ml b/test/Pal/Data/Failures.ml new file mode 100644 index 0000000..0c926ca --- /dev/null +++ b/test/Pal/Data/Failures.ml @@ -0,0 +1,81 @@ +(* handbook/fol.elf: first-order logic from Frank Pfenning's Handbook chapter. + Identical in structure to the existing FOL suite (i/o/imp/not/forall/nd/hil/ded). + SKIPPED: By the time this runs, `o` has been re-declared as a term (not a sort) + by the CRARY-EXCON group (%term o tp). Attempting to use `o` as a type causes + "Level clash: Argument type did not match function domain type". + Placeholder kept so the test entry can reference it. *) +let handbook_sources_1 = {| (* placeholder: o pollution from CRARY-EXCON *) |} + +(* failure/fail.elf: only a %query declaration which is not supported in STELF. + Expected to raise a parse error. Translating %sort void then the %query itself + as a raw string — the %query will fail. *) +let failure_sources_1 = {| +%sort void +%query 1 1 void +|} + +(* wiki_failures/coverage_error.elf: intentionally incomplete coverage proof. + Defines sub (subtyping) with sub-trans missing the arrow base case. + %total on sub-trans should trigger a coverage failure in STELF. *) +let wiki_failures_coverage_error_1 = + {| +%sort wf_tp +%term wf_int wf_tp +%term wf_float wf_tp +%term wf_arrow {_ wf_tp} {_ wf_tp} wf_tp +%sort wf_sub {_ wf_tp} {_ wf_tp} +%term wf_sub_ii wf_sub wf_int wf_int +%term wf_sub_ff wf_sub wf_float wf_float +%term wf_sub_if wf_sub wf_int wf_float +%term wf_sub_arrow {{T S T' S'}} {_ wf_sub T' T} {_ wf_sub S S'} wf_sub (wf_arrow T S) (wf_arrow T' S') +%sort wf_sub_trans {_ wf_tp} {_ wf_tp} {_ wf_tp} +%mode wf_sub_trans %in %in %out +%term wf_sub_trans_refl {{T}} {D wf_sub T T} wf_sub_trans T T T +%term wf_sub_trans_ii_if wf_sub_trans wf_int wf_float wf_float +%term wf_sub_trans_if_ff wf_sub_trans wf_int wf_float wf_float +%worlds () (wf_sub_trans _ _ _) +%total D (wf_sub_trans D _ _) +|} + +(* wiki_failures/mode_error.elf: relation with a mode violation. + plus is defined correctly but bad uses output N2 in input position. + %mode or %total check should fail. *) +let wiki_failures_mode_error_1 = + {| +%sort wfm_nat +%term wfm_z wfm_nat +%term wfm_s {_ wfm_nat} wfm_nat +%sort wfm_plus {_ wfm_nat} {_ wfm_nat} {_ wfm_nat} +%mode wfm_plus %in %in %out +%term wfm_plus_z {N wfm_nat} wfm_plus wfm_z N N +%term wfm_plus_s {{N1 N2 N3}} {_ wfm_plus N1 N2 N3} wfm_plus (wfm_s N1) N2 (wfm_s N3) +%worlds () (wfm_plus _ _ _) +%total N (wfm_plus N _ _) +%sort wfm_bad {_ wfm_nat} {_ wfm_nat} +%mode wfm_bad %in %out +%term wfm_bad_case {{N1 N2}} {_ wfm_plus N1 N2 N1} wfm_bad N1 N2 +%worlds () (wfm_bad _ _) +%total N (wfm_bad N _) +|} + +(* wiki_failures/totality_error.elf: relation with no base case for z. + %total should fail because there is no clause for partial z _. *) +let wiki_failures_totality_error_1 = + {| +%sort wft_nat +%term wft_z wft_nat +%term wft_s {_ wft_nat} wft_nat +%sort wft_partial {_ wft_nat} {_ wft_nat} +%mode wft_partial %in %out +%term wft_partial_s {{N M}} {_ wft_partial N M} wft_partial (wft_s N) (wft_s M) +%worlds () (wft_partial _ _) +%total N (wft_partial N _) +|} + +(* wiki_failures/unsatisfiable_query.elf: defines an empty type then uses %query. + %query is not supported in STELF; the parse will fail. *) +let wiki_failures_unsatisfiable_query_1 = + {| +%sort wfq_empty +%query 1 1 wfq_empty +|} diff --git a/test/Pal/Data/Fol.ml b/test/Pal/Data/Fol.ml new file mode 100644 index 0000000..f131ddb --- /dev/null +++ b/test/Pal/Data/Fol.ml @@ -0,0 +1,242 @@ +let fol1 = + {| +%. First-Order Logic +Fragment with implication, negation, universal quantification. + +Author: Frank Pfenning + +This code is from the article + + Logical Frameworks + Handbook of Automated Reasoning + Alan Robinson and Andrei Voronkov, editors + Chapter 16 + Elsevier Science and MIT Press + In preparation + +%sort i +%sort o + +Formulas + +%term imp {_ o} {_ o} o +%prec %right 10 imp +%term not {_ o} o +%prec %prefix 12 not +%term forall %pi (%pi i %-> o) %-> o %. + +Natural deductions + +%sort nd {_ o} + +%term impi {{A B}} {_ {_ nd A} nd B} nd (A imp B) +%term impe {{A B}} {_ nd (A imp B)} {_ nd A} nd B + +%term noti {{A}} {_ {p o} {_ nd A} nd p} nd (not A) +%term note {{A}} {_ nd (not A)} {C o} {_ nd A} nd C + +%term foralli {{A}} {_ {a i} nd (A a)} nd (forall A) +%term foralle {{A}} {_ nd (forall A)} {T i} nd (A T) + +%block lnd [A o] {u nd A} +%block lo {_ o} +%block li {_ i} + +%worlds (lnd lo li) (nd A) +|} + +let fol2 = + {| + +%. Example + + +%. Hilbert deductions + +%sort hil {_ o} + +%term k {{A B}} hil (A imp (B imp A)) +%term s {{A B C}} hil ((A imp (B imp C)) imp ((A imp B) imp (A imp C))) + +%term n1 {{A B}} hil ((A imp (not B)) imp ((A imp B) imp (not A))) +%term n2 {{A B}} hil ((not A) imp (A imp B)) + +%term f1 {T i} hil ((forall [x i] A x) imp (A T)) +%term f2 {{A B}} hil ((forall [x i] (B imp A x)) imp (B imp forall [x i] A x)) + +%term mp {{A B}} {_ hil (A imp B)} {_ hil A} hil B +%term ug {{A}} {_ {a i} hil (A a)} hil (forall [x i] A x) + +%worlds (li) (hil A) + +|} + +let fol3_1 = + {| + +%. Local reductions + +%sort ==>R {_ nd _A} {_ nd _A} +%prec %none 14 ==>R + +|} + +let fol3_2_1 = + {| +%term redl_imp {{A D E}} (impe (impi [u nd A] D u) E) ==>R (D E) + +|} + +let fol3_2_2 = + {| +%term redl_not {{A C D E}} (note (noti [p o] [u nd A] D p u) C E) ==>R (D C E) + +|} + +let fol3_2_3 = + {| +%term redl_forall {{D T}} (foralle (foralli [a i] D a) T) ==>R (D T) + +|} + +let fol3_2 = fol3_2_1 ^ fol3_2_2 ^ fol3_2_3 + +let fol3_3 = + {| + +%. Local expansions + +%sort ==>E {_ nd A} {_ nd A} +%prec %none 14 ==>E +|} + +let fol3_4 = + {| + +%term expl_imp {{A B}} {D nd (A imp B)} D ==>E (impi [u nd A] impe D u) +%term expl_not {{A}} {D nd (not A)} D ==>E (noti [p] [u nd A] note D p u) +%term expl_forall {{A}} {D nd (forall A)} D ==>E (foralli [a] foralle D a) + +|} + +let fol3 = fol3_1 ^ fol3_2 ^ fol3_3 ^ fol3_4 + +let fol4_1 = + {| +%. Sequent calculus search result + +%def dn (nd (A imp not not A)) (impi [u nd A] noti [p] [w] note w p u) + +%. Translating Hilbert derivations to natural deductions + +%sort hilnd {_ hil A} {_ nd A} + +%term hnd_k {{A B}} hilnd k (impi [u nd A] impi [v nd B] u) +%term hnd_s {{A B C}} hilnd s (impi [u nd (A imp B imp C)] impi [v] impi [w nd A] impe (impe u w) (impe v w)) + +%term hnd_n1 hilnd n1 (impi [u] impi [v] noti [p] [w] note (impe u w) p (impe v w)) + +%term hnd_n2 {{A B}} hilnd n2 (impi [u] impi [v] note u B v) +%term hnd_f1 {{A T}} hilnd (f1 T) (impi [u nd (forall [x i] A x)] foralle u T) + +%term hnd_f2 {{A B}} hilnd f2 (impi [u] impi [v] foralli [a] impe (foralle u a) v) + +%term hnd_mp {{A B}} {H1 hil (A imp B)} {H2 hil A} {D1 nd (A imp B)} {D2 nd A} {_ hilnd H1 D1} {_ hilnd H2 D2} hilnd (mp H1 H2) (impe D1 D2) + +%term hnd_ug {{A}} {H1 {a i} hil (A a)} {D1 {a i} nd (A a)} {_ {a i} hilnd (H1 a) (D1 a)} hilnd (ug H1) (foralli D1) +|} + +(* %mode {%in x hil _A} {%out y nd _A} hilnd x y *) +let fol4_2 = + {| +%mode {%in X _} {%out Y _} hilnd X Y +%worlds (li) (hilnd H D) +%terminates H (hilnd H _) +%covers hilnd %in %out +%total H (hilnd H _) + +%? hilnd (mp (mp s k) k) D + +%def _ (nd (A imp A)) + (impe + (impe + (impi [u] + impi [v] + impi [w] impe (impe u w) (impe v w)) + (impi [u] impi [v] u)) + (impi [u] impi [v] u)) + + |} + +let fol4 = fol4_1 ^ fol4_2 + +let fol5_1 = + {| + +The deduction theorem for Hilbert derivations + +%sort ded {_ {_ hil A} hil B} {_ hil (A imp B)} + +%term ded_id {{A}} ded ([u hil A] u) (mp (mp s k) (%the (hil (A imp (A imp A))) k)) + +%term ded_k {{A B}} ded ([u] k) (mp k k) +%term ded_s {{A B C}} ded ([u] s) (mp k s) + +%term ded_n1 {{A B}} ded ([u] n1) (mp k n1) +%term ded_n2 {{A B}} ded ([u] n2) (mp k n2) + +%term ded_f1 {{A B T}} ded ([u] f1 T) (mp k (f1 T)) +%term ded_f2 {{A B}} ded ([u] f2) (mp k f2) + +%term ded_mp {{A B C}} {H1 {_ hil A} hil (B imp C)} {H2 {_ hil A} hil B} {H1' hil (A imp (B imp C))} {H2' hil (A imp B)} {_ ded H1 H1'} {_ ded H2 H2'} ded ([u] mp (H1 u) (H2 u)) (mp (mp s H1') H2') + +%term ded_ug {{A B H1 H1'}} {_ {a i} ded ([u] H1 u a) (H1' a)} ded ([u] ug (H1 u)) (mp f2 (ug H1')) +|} + +let fol5_2 = + {| +%block lded [A o] {u nd A} {v hil A} {h {C o} ded ([w hil C] v) (mp k v)} + +%mode ded %in %out +%worlds (li lo lded) (ded H H') +%terminates H (ded H _) +%covers ded %in %out +%total H (ded H _) + +|} + +let fol5 = fol5_1 ^ fol5_2 + +let fol6_1 = + {| + +Mapping natural deductions to Hilbert derivations. + +%sort ndhil {_ nd A} {_ hil A} + +%term ndh_impi {{A1 B C D1 H1 H1' H1''}} {_ ded H1 H1'} {_ {u nd A1} {v hil A1} {_ {C o} ded ([w hil C] v) (mp k v)} {_ ndhil u v} ndhil (D1 u) (H1 v)} ndhil (impi D1) H1' + +%term ndh_impe {{A B D1 D2 H1 H2}} {_ ndhil D1 H1} {_ ndhil D2 H2} ndhil (impe D1 D2) (mp H1 H2) + +%term ndh_noti {{A1 H1 H1' H1'' D1}} {_ ded (H1 (not A1)) H1'} {_ ded (H1 A1) H1''} {_ {p o} {u nd A1} {v hil A1} {_ {C o} ded ([w hil C] v) (mp k v)} {_ ndhil u v} ndhil (D1 p u) (H1 p v)} ndhil (noti D1) (mp (mp n1 H1') H1'') + +%term ndh_note {{A C D1 D2 H1 H2}} {_ ndhil D1 H1} {_ ndhil D2 H2} ndhil (note D1 C D2) (mp (mp n2 H1) H2) + +%term ndh_foralli {{A D1 H1}} {_ {a i} ndhil (D1 a) (H1 a)} ndhil (foralli D1) (ug H1) + +%term ndh_foralle {{A T D1 H1}} {_ ndhil D1 H1} ndhil (foralle D1 T) (mp (f1 T) H1) +|} + +let fol6_2 = + {| +%mode {%in X _} {%out Y _} ndhil X Y +%block lndhil [A o] {u nd A} {v hil A} {h {C o} ded ([w hil C] v) (mp k v)} {nh ndhil u v} +%worlds (li lo lndhil) (ndhil D H) +%terminates D (ndhil D _) +%covers ndhil %in %out +%total D (ndhil D _) + + +|} + +let fol6 = fol6_1 ^ fol6_2 diff --git a/test/Pal/Data/GuideLists.ml b/test/Pal/Data/GuideLists.ml new file mode 100644 index 0000000..cd93bcb --- /dev/null +++ b/test/Pal/Data/GuideLists.ml @@ -0,0 +1,26 @@ +(* List append, element type, and list sort. + Ported from twelf/examples/guide/lists.elf. + Note: %query directives omitted (not supported). + Element type 'o' declared here to keep this chunk self-contained. +*) +let guide_lists_types = + {| +%sort o +%sort list +%term nil list +%term cons {_ o} {_ list} list +|} + +let guide_lists_append = + {| +%sort append {_ list} {_ list} {_ list} +%term appNil {K list} append nil K K +%term appCons {A o} {L list} {K list} {M list} %if (append (cons A L) K (cons A M)) %<- (append L K M) +|} + +let guide_lists_mode = + {| +%mode append %in %in %out +%worlds () (append _ _ _) +%total L (append L _ _) +|} diff --git a/test/Pal/Data/GuideNd.ml b/test/Pal/Data/GuideNd.ml new file mode 100644 index 0000000..f6697b7 --- /dev/null +++ b/test/Pal/Data/GuideNd.ml @@ -0,0 +1,59 @@ +(* Natural deduction for intuitionistic logic (positive+negative fragment). + Ported from twelf/examples/guide/nd.elf. + Abbreviation definitions for not/noti/note handled as regular terms + since STELF does not support Twelf abbreviation syntax. + The %block/%worlds declarations are kept as-is. +*) +let guide_nd = + {| +%sort i +%name i T +%sort o +%name o A +%term imp {_ o} {_ o} o +%prec %right 10 imp +%term and {_ o} {_ o} o +%prec %right 11 and +%term true o +%term or {_ o} {_ o} o +%prec %right 11 or +%term false o +%term forall {_ {_ i} o} o +%term exists {_ {_ i} o} o + +%sort nd {_ o} +%name nd D + +%term impi {{A B}} {_ {_ nd A} nd B} nd (A imp B) +%term impe {{A B}} {_ nd (A imp B)} {_ nd A} nd B +%term andi {{A B}} {_ nd A} {_ nd B} nd (A and B) +%term ande1 {{A B}} {_ nd (A and B)} nd A +%term ande2 {{A B}} {_ nd (A and B)} nd B +%term truei nd true +%term ori1 {{A B}} {_ nd A} nd (A or B) +%term ori2 {{A B}} {_ nd B} nd (A or B) +%term ore {{A B C}} {_ nd (A or B)} {_ {_ nd A} nd C} {_ {_ nd B} nd C} nd C +%term falsee {{C}} {_ nd false} nd C +%term foralli {{A}} {_ {x i} nd (A x)} nd (forall A) +%term foralle {{A}} {_ nd (forall A)} {T i} nd (A T) +%term existsi {{A}} {T i} {_ nd (A T)} nd (exists A) +%term existse {{A C}} {_ nd (exists A)} {_ {x i} {_ nd (A x)} nd C} nd C + +%block nd_hyp [A o] {u nd A} +%block nd_parm {x i} +%worlds (nd_hyp nd_parm) (nd A) + +%sort red {_ nd A} {_ nd A} +%term impred {{A B D E}} red (impe (impi D) E) (D E) +%term andred1 {{A B D E}} red (ande1 (andi D E)) D +%term andred2 {{A B D E}} red (ande2 (andi D E)) E +%term orred1 {{A B C D E1 E2}} red (ore (ori1 D) E1 E2) (E1 D) +%term orred2 {{A B C D E1 E2}} red (ore (ori2 D) E1 E2) (E2 D) +%term forallred {{A D T}} red (foralle (foralli D) T) (D T) +%term existsred {{A C T D E}} red (existse (existsi T D) E) (E T D) + +%theorem +trivI exists {D {A o} nd (A imp A)} true + +%prove 2 {} (trivI D) +|} diff --git a/test/Pal/Data/Incll.ml b/test/Pal/Data/Incll.ml new file mode 100644 index 0000000..3328080 --- /dev/null +++ b/test/Pal/Data/Incll.ml @@ -0,0 +1,44 @@ +(* INCLL: INCLL sorts and terms. + Ported from twelf/examples/incll/incll.elf. + Dropped: Twelf abbreviation syntax (`<= = [x][y] y => x` etc.). + Dropped: `|` infix list cons (single-char identifier, may fail — using `cons` instead). + `^` prefix also risky — using `atm_frm` instead. +*) +let incll_syntax = + {| +%sort sort +%term arrow {_ sort} {_ sort} sort +%term cross {_ sort} {_ sort} sort + +%sort trm {_ sort} +%term app {{A B}} {_ trm (arrow A B)} {_ trm A} trm B +%term lam {{A B}} {_ {_ trm A} trm B} trm (arrow A B) +%term pair {{A B}} {_ trm A} {_ trm B} trm (cross A B) + +%sort eval {_ trm _A} {_ trm _A} +%term eval_lam {{A B E}} eval (lam E) (lam E) +%term eval_app {{A B E E' V V'}} %if (eval (app (lam E) E') V) %<- (eval E' V') %<- (eval (E V') V) + +%sort atm +%sort frm +%term int sort + +%term 1 trm int +%term 2 trm int +%term 3 trm int +%term 4 trm int +%term 5 trm int + +%sort list_sort {_ sort} +%term list_sort/nil {{A}} list_sort A +%term list_sort/cons {{A}} {_ trm A} {_ list_sort A} list_sort A + +%sort frm_atm {_ atm} +%term atm_frm {_ atm} frm + +%sort imp {_ frm} {_ frm} +%term imp_i {_ frm} {_ frm} frm + +%term forall {A sort} {_ {_ trm A} frm} frm +%term forall2 {A1 sort} {A2 sort} {_ {_ {_ trm A1} trm A2} frm} frm +|} diff --git a/test/Pal/Data/Jsf.ml b/test/Pal/Data/Jsf.ml new file mode 100644 index 0000000..3c74903 --- /dev/null +++ b/test/Pal/Data/Jsf.ml @@ -0,0 +1,91 @@ +let jsf_1 = + {| +Judgmental S4 +[A judgmental reconstruction of modal logic, F.Pfenning and R.Davies, + MSCS 11:511-540, 2001] + +Representation with intrinsic types, worlds, +but not a Kripke semantics + +Idea: Translate the judgment + u1::B1,...,un::Bn ; x1:A1,...,xm:Am |- J +as + u1:{W'}tm B1 W',...,un:{W'}tm Bn W', ---, + x1:tm A1 W,...,xm:tm Am W |- J* + +where "---" are assumptions y:tm B W' for W' <> W +and if J = (M : A) then J* = M* : tm A W + if J = (E - A) then J* = E* : exp A W + +%sort tp +%term => {_ tp} {_ tp} tp +%term box {_ tp} tp +%term dia {_ tp} tp +%prec %right 10 => + +%sort world + +%sort tm {_ tp} {_ world} +%sort exp {_ tp} {_ world} + +%term lam {{A B W}} {_ {_ tm A W} tm B W} tm (A => B) W +%term app {{A B W}} {_ tm (A => B) W} {_ tm A W} tm B W +%term boxi {{A W}} {_ {w world} tm A w} tm (box A) W +%term boxe {{A C W}} {_ tm (box A) W} {_ {_ {W' world} tm A W'} tm C W} tm C W +%term t2e {{A W}} {_ tm A W} exp A W +%term diai {{A W}} {_ exp A W} tm (dia A) W +%term diae {{A C W}} {_ tm (dia A) W} {_ {w world} {_ tm A w} exp C w} exp C W +%term boxep {{A C W}} {_ tm (box A) W} {_ {_ {W' world} tm A W'} exp C W} exp C W +%sort subdia {_ exp A W} {_ {w world} {_ tm A w} exp C w} {_ exp C W} +|} + +let jsf_2_1 = {| +%mode {%in X _} {%in Y _} {%out Z _} subdia X Y Z +|} + +let jsf_2_2 = + {| +%term sdt2e {A tp} {C tp} {W world} {M tm A W} {F {w world} {_ tm A w} exp C w} subdia (t2e M) ([w] [x] F w x) (F W M) + +%term sddiae {A tp} {B tp} {C tp} {W world} {M tm (dia A) W} {E {v world} {_ tm A v} exp B v} {F {w world} {_ tm B w} exp C w} {F' {v world} {_ tm A v} exp C v} {_ {v world} {y tm A v} subdia (E v y) ([w] [x] F w x) (F' v y)} subdia (diae M [v] [y] E v y) ([w] [x] F w x) (diae M [v] [y] F' v y) + +%term sdboxep {A tp} {C tp} {D tp} {W world} {M tm (box A) W} {E {u {V world} tm A V} exp C W} {F {w world} {_ tm C w} exp D w} {F' {u {V world} tm A V} exp D W} {_ {u {V world} tm A V} subdia (E u) ([w] [x] F w x) (F' u)} subdia (boxep M [u] E u) ([w] [x] F w x) (boxep M [u] F' u) +|} + +let jsf_3 = + {| +%block by [B tp] {v world} {y tm B v} +%block bu [B tp] {u {V world} tm B V} +%worlds (by bu) (subdia E F F') +%total E (subdia E F _) %. + +This does not work, unfortunately: +The "str" strengthening lemma would require handling cases like strlam where a bound +variable y2 of type tm C2 w appears free in the conclusion, but w is also quantified +by the str family. The main issue is that the case for str ([x][w] y2) ([w] y2) cannot +be typed because y2 has type tm C2 w but w is the variable we are quantifying over. +|} + +let jsf_4 = + {| +Examples + +%def _ (tm (box A => A) W) (lam [x] boxe x [u] u W) +%def _ (tm (box A => box (box A)) W) (lam [x] boxe x [u] boxi [w] boxi [w'] u w') +%def _ (tm (box (A => B) => box A => box B) W) (lam [x] lam [y] boxe x [u] boxe y [v] boxi [w] app (u w) (v w)) +%def _ (tm (A => dia A) W) (lam [x] diai (t2e x)) +%def _ (tm (dia (dia A) => dia A) W) (lam [x] diai (diae x [w] [y] diae y [v] [z] t2e z)) +%def _ (tm (box (A => B) => dia A => dia B) W) (lam [x] lam [y] diai (boxep x [u] diae y [w] [z] t2e (app (u w) z))) +%. +Counterexamples, all must fail: +The following would require box to be a comonad (A => box A), but this is not valid in S4. +The term (lam [x] boxi [w] x) fails because x has type tm A W but we need tm A w for +arbitrary w, which would require the structural rule for modal contexts. +Similarly (lam [x] diae x [w][y] t2e y) fails because y has type tm A w but we need +tm A W for the current world W. +The term for (dia (A => B) => dia A => dia B) fails because dia is not "normal" in S4. +The term (dia A => box B) => box (A => B) is true in Kripke semantics a la Simpson but +not in the judgmental formulation. +The two S5 theorems (dia A => box (dia A)) and (dia (box A) => box A) both fail because +S4 does not have the symmetry or Euclidean properties needed for S5. +|} diff --git a/test/Pal/Data/Lam.ml b/test/Pal/Data/Lam.ml new file mode 100644 index 0000000..4828b6f --- /dev/null +++ b/test/Pal/Data/Lam.ml @@ -0,0 +1,124 @@ +let lam_1 = + {| +Lambda-Calculus Fragment of Mini-ML. +Author: Frank Pfenning + +Simple types +%sort tp + +%term arrow {_ tp} {_ tp} tp + +%. Expressions +%sort exp + +%term lam {_ {_ exp} exp} exp +%term app {_ exp} {_ exp} exp + +%. Type inference +|- E : T (expression E has type T) + +%sort of {_ exp} {_ tp} +|} + +let lam_2 = + {| +%mode of %in %star + +%term tp_lam {{E T1 T2}} {_ {x exp} {_ of x T1} of (E x) T2} of (lam E) (arrow T1 T2) + +%term tp_app {{E1 E2 T1 T2}} {_ of E1 (arrow T2 T1)} {_ of E2 T2} of (app E1 E2) T1 + +%. Evaluation (call-by-value) +E ==> V (expression E evaluates to value V) + +%sort eval {_ exp} {_ exp} +%mode eval %in %out + +%term ev_lam {{E}} eval (lam E) (lam E) + +%term ev_app {{E1 E2 V V2 E1'}} {_ eval E1 (lam E1')} {_ eval E2 V2} {_ eval (E1' V2) V} eval (app E1 E2) V + +|} + +let lam_3 = + {| +%. Regular world for type-checking +%block tp_var [T tp] {x exp} {u of x T} +%worlds (tp_var) (of E T) + +%. Type inference terminates +%terminates E (of E T) + +%. There is at least one typing rule for every expression +%covers of %in %star + +%. Closed worlds for evaluation +%worlds () (eval E V) + +%. There is at least one evaluation rule for every closed expression +%covers eval %in %out + +|} + +let lam_4 = + {| +%. Type preservation as higher-level family +%sort tps {_ eval E V} {_ of E T} {_ of V T} + +%term tps_lam {E {_ exp} exp} {T1 tp} {T2 tp} {P {x exp} {_ of x T1} of (E x) T2} tps ev_lam (tp_lam P) (tp_lam P) + +%term tps_app {E1 exp} {E2 exp} {V exp} {V2 exp} {E1' {_ exp} exp} {T tp} {T2 tp} {D1 eval E1 (lam E1')} {D2 eval E2 V2} {D3 eval (E1' V2) V} {P1 of E1 (arrow T2 T)} {P2 of E2 T2} {Q1' {x exp} {_ of x T2} of (E1' x) T} {Q2 of V2 T2} {Q of V T} {_ tps D1 P1 (tp_lam Q1')} {_ tps D2 P2 Q2} {_ tps D3 (Q1' V2 Q2) Q} tps (ev_app D1 D2 D3) (tp_app P1 P2) Q + +%. Applying type preservation +%def e0 _ (app (lam [x] x) (lam [y] y)) +%? of e0 T +%? eval e0 V +|} + +let lam_5 = + {| +%. Example of regular worlds +cp copies input to output. + +%sort cp {_ exp} {_ exp} + +%term cp_app {E1 exp} {E2 exp} {F1 exp} {F2 exp} {_ cp E1 F1} {_ cp E2 F2} cp (app E1 E2) (app F1 F2) + +%term cp_lam {E {_ exp} exp} {F {_ exp} exp} {_ {x exp} {_ cp x x} cp (E x) (F x)} cp (lam [x] E x) (lam [x] F x) + +%mode cp %in %out +%block cp_var {x exp} {u cp x x} +%worlds (cp_var) (cp E _) +%total E (cp E _) +%. +Following version cannot be checked: input coverage on parameter y is violated. +It would declare cp with a block containing {x:exp} {y:exp} {u:cp x y}, but cp_lam's +higher-order premise would allow y to differ from x, which violates input coverage. + +Following version also cannot be checked: output coverage on (F y) is violated. +It would add a premise cp y y -> cp (E x) (F y), meaning F y may not be covered +for the output position. + +|} + +let polylam = + {| +%sort tp +%term => {_ tp} {_ tp} tp +%prec %right 10 => +%term all {_ {_ tp} tp} tp + +%sort tm {_ tp} +%term lam {{A tp}} {{B tp}} {_ {_ tm A} tm B} tm (A => B) +%term app {{A tp}} {{B tp}} {_ tm (A => B)} {_ tm A} tm B +%term tlam {{A}} {_ {a tp} tm (A a)} tm (all A) +%term tapp {{A}} {_ tm (all A)} {B tp} tm (A B) + +%def nat _ (all [a tp] a => (a => a) => a) +%def zero _ (tlam [a tp] lam [z tm a] lam [s tm (a => a)] z) +%def succ _ (lam [x tm nat] tlam [a tp] lam [z tm a] lam [s tm (a => a)] app s (app (app (tapp x a) z) s)) +%def succ' (tm (nat => nat)) (lam [x tm nat] tlam [a tp] lam [z tm a] lam [s tm (a => a)] app s (app (app (tapp x a) z) s)) +%def plus _ (lam [x tm nat] lam [y tm nat] app (app (tapp y nat) x) succ) +%def times _ (lam [x tm nat] lam [y tm nat] app (app (tapp y nat) zero) (app plus x)) +%def exp _ (lam [x tm nat] lam [y tm nat] app (app (tapp y nat) (app succ zero)) (app times x)) +|} diff --git a/test/Pal/Data/LpHorn.ml b/test/Pal/Data/LpHorn.ml new file mode 100644 index 0000000..6fc82a6 --- /dev/null +++ b/test/Pal/Data/LpHorn.ml @@ -0,0 +1,95 @@ +(* Positive fragment of first-order logic with natural deduction. + Ported from twelf/examples/lp_horn/natded.elf. + Operator declarations: imp with %prec %right 10, and with %prec %right 11. +*) +let lp_horn_nd = + {| +%sort i +%name i T +%sort o +%name o A +%sort p +%name p P + +%term atom {_ p} o +%term and {_ o} {_ o} o +%prec %right 11 and +%term imp {_ o} {_ o} o +%prec %right 10 imp +%term true o +%term forall {_ {_ i} o} o + +%sort pf {_ o} +%name pf D +%term andi {{A B}} {_ pf A} {_ pf B} pf (A and B) +%term andel {{A B}} {_ pf (A and B)} pf A +%term ander {{A B}} {_ pf (A and B)} pf B +%term impi {{A B}} {_ {_ pf A} pf B} pf (A imp B) +%term impe {{A B}} {_ pf (A imp B)} {_ pf A} pf B +%term truei pf true +%term foralli {{A}} {_ {a i} pf (A a)} pf (forall A) +%term foralle {{A}} {_ pf (forall A)} {T i} pf (A T) +|} + +(* LP-HORN canon: canonical forms for natural deduction proofs. + Ported from twelf/examples/lp_horn/canon.elf. + Builds on lp_horn_nd (o, pf, atom, and, imp, true, forall, andi, andel, ander, + impi, impe, truei, foralli, foralle). + Names introduced: can, atm and constructors. + Omitted: can_impi, can_foralli (require full %block worlds for higher-order premises). +*) +let lp_horn_sources_2 = + {| +%sort can {A o} {_ pf A} +%name can CN +%sort atm {_ pf _} +%name atm AT + +%term can_andi {{A B D E}} {_ can A D} {_ can B E} can (A and B) (andi D E) +%term can_truei {{D}} can true truei +%term can_atm {{P D}} {_ atm D} can (atom P) D + +%term atm_andel {{D}} {_ atm D} atm (andel D) +%term atm_ander {{D}} {_ atm D} atm (ander D) +%term atm_impe {{A B D E}} {_ atm D} {_ can B E} atm (impe D E) +%term atm_foralle {{A D T}} {_ atm D} atm (foralle D T) + +%worlds () (can _ _) (atm _) +|} + +(* LP-HORN conv: conversion to canonical/atomic form. + Ported from twelf/examples/lp_horn/conv.elf. + Builds on lp_horn_nd + lp_horn_sources_2 (can, atm in scope). + Names introduced: whr, tocan, toatm and constructors. +*) +let lp_horn_sources_3 = + {| +%sort whr {_ pf _} {_ pf _} +%name whr WHR + +%term whr_andl {{A B D E}} whr (andel (andi D E)) D +%term whr_andr {{A B D E}} whr (ander (andi D E)) E +%term whr_imp {{A B D E}} whr (impe (impi D) E) (D E) +%term whr_forall {{A D T}} whr (foralle (foralli D) T) (D T) +%term whr_andel {{D D'}} {_ whr D D'} whr (andel D) (andel D') +%term whr_ander {{D D'}} {_ whr D D'} whr (ander D) (ander D') +%term whr_impe {{A B D D' E}} {_ whr D D'} whr (impe D E) (impe D' E) +%term whr_foralle {{A D D' T}} {_ whr D D'} whr (foralle D T) (foralle D' T) + +%sort tocan {_ o} {_ pf _} {_ pf _} +%name tocan TC +%sort toatm {_ pf _} {_ pf _} +%name toatm TA + +%term tc_and {{A B D D1' D2'}} {_ tocan A (andel D) D1'} {_ tocan B (ander D) D2'} tocan (A and B) D (andi D1' D2') +%term tc_imp {{A B D D'}} {_ {u pf A} {_ toatm u u} tocan B (impe D u) (D' u)} tocan (A imp B) D (impi D') +%term tc_true {{D}} tocan true D truei +%term tc_atom {{P D D'}} {_ toatm D D'} tocan (atom P) D D' +%term tc_whr {{A D D' D''}} {_ whr D D'} {_ tocan A D' D''} tocan A D D'' + +%term ta_atom {{P D}} {_ atm D} toatm D D +%term ta_whr {{D D' D''}} {_ whr D D'} {_ toatm D' D''} toatm D D'' + +%block latm {u pf _} {_ toatm u u} +%worlds (latm) (tocan _ _ _) (toatm _ _) +|} diff --git a/test/Pal/Data/MiniMl.ml b/test/Pal/Data/MiniMl.ml new file mode 100644 index 0000000..8c1996e --- /dev/null +++ b/test/Pal/Data/MiniMl.ml @@ -0,0 +1,102 @@ +(* Mini-ML expression language + Ported from twelf/examples/mini-ml/mini-ml.elf +*) +let mini_ml_exp = + {| +%sort exp +%name exp E +%term z exp +%term s {_ exp} exp +%term case {_ exp} {_ exp} {_ {_ exp} exp} exp +%term pair {_ exp} {_ exp} exp +%term fst {_ exp} exp +%term snd {_ exp} exp +%term lam {_ {_ exp} exp} exp +%term app {_ exp} {_ exp} exp +%term letv {_ exp} {_ {_ exp} exp} exp +%term letn {_ exp} {_ {_ exp} exp} exp +%term fix {_ {_ exp} exp} exp +|} + +(* Mini-ML values + Ported from twelf/examples/mini-ml/value.elf +*) +let mini_ml_value = + {| +%sort value {_ exp} +%term val-z value z +%term val-lam {{E}} value (lam E) +%term val-s {{V}} %if (value (s V)) %<- (value V) +%term val-pair {{V1 V2}} %if (value (pair V1 V2)) %<- (value V1) %<- (value V2) +%mode value %in +%worlds () (value _) +|} + +(* Mini-ML types + Ported from twelf/examples/mini-ml/tp.elf +*) +let mini_ml_tp = + {| +%sort tp +%term nat tp +%term cross {_ tp} {_ tp} tp +%term arrow {_ tp} {_ tp} tp +|} + +(* MINI-ML eval: natural semantics for Mini-ML. + Ported from twelf/examples/mini_ml/eval.elf. + Builds on mini_ml_exp (exp, z, s, case, pair, fst, snd, lam, app, letv, letn, fix). + mini_ml_value (value/val-z/val-lam/val-s/val-pair) is also in scope. + Names introduced: eval and ev_* constructors. + Omitted: %terminates (eval is not total due to fix). +*) +let mini_ml_sources_eval = + {| +%sort eval {_ exp} {_ exp} +%name eval D +%mode eval %in %out + +%term ev_z eval z z +%term ev_s {{E V}} {_ eval E V} eval (s E) (s V) +%term ev_case_z {{E1 E2 E3 V}} {_ eval E1 z} {_ eval E2 V} eval (case E1 E2 E3) V +%term ev_case_s {{E1 E2 E3 V V1'}} {_ eval E1 (s V1')} {_ eval (E3 V1') V} eval (case E1 E2 E3) V +%term ev_pair {{E1 E2 V1 V2}} {_ eval E1 V1} {_ eval E2 V2} eval (pair E1 E2) (pair V1 V2) +%term ev_fst {{E V1 V2}} {_ eval E (pair V1 V2)} eval (fst E) V1 +%term ev_snd {{E V1 V2}} {_ eval E (pair V1 V2)} eval (snd E) V2 +%term ev_lam {{E}} eval (lam E) (lam E) +%term ev_app {{E1 E2 V V2 E1'}} {_ eval E1 (lam E1')} {_ eval E2 V2} {_ eval (E1' V2) V} eval (app E1 E2) V +%term ev_letv {{E1 E2 V V1}} {_ eval E1 V1} {_ eval (E2 V1) V} eval (letv E1 E2) V +%term ev_letn {{E1 E2 V}} {_ eval (E2 E1) V} eval (letn E1 E2) V +%term ev_fix {{E V}} {_ eval (E (fix E)) V} eval (fix E) V + +%worlds () (eval _ _) +%covers eval %in %out +|} + +(* MINI-ML tpinf: type inference for Mini-ML. + Ported from twelf/examples/mini_ml/tpinf.elf. + Builds on mini_ml_exp + mini_ml_tp (tp, nat, cross, arrow). + Names introduced: of and tp_* constructors. + %block l : some {T:tp} block {x:exp} {d:of x T} -> %block l [T tp] {x exp} {d of x T} +*) +let mini_ml_sources_tpinf = + {| +%sort of {_ exp} {_ tp} +%name of P +%mode of %in %star + +%term tp_z of z nat +%term tp_s {{E}} {_ of E nat} of (s E) nat +%term tp_case {{E1 E2 E3 T}} {_ of E1 nat} {_ of E2 T} {_ {x exp} {_ of x nat} of (E3 x) T} of (case E1 E2 E3) T +%term tp_pair {{E1 E2 T1 T2}} {_ of E1 T1} {_ of E2 T2} of (pair E1 E2) (cross T1 T2) +%term tp_fst {{E T1 T2}} {_ of E (cross T1 T2)} of (fst E) T1 +%term tp_snd {{E T1 T2}} {_ of E (cross T1 T2)} of (snd E) T2 +%term tp_lam {{E T1 T2}} {_ {x exp} {_ of x T1} of (E x) T2} of (lam E) (arrow T1 T2) +%term tp_app {{E1 E2 T1 T2}} {_ of E1 (arrow T2 T1)} {_ of E2 T2} of (app E1 E2) T1 +%term tp_letv {{E1 E2 T1 T2}} {_ of E1 T1} {_ {x exp} {_ of x T1} of (E2 x) T2} of (letv E1 E2) T2 +%term tp_letn {{E1 E2 T1 T2}} {_ of E1 T1} {_ of (E2 E1) T2} of (letn E1 E2) T2 +%term tp_fix {{E T}} {_ {x exp} {_ of x T} of (E x) T} of (fix E) T + +%block l [T tp] {x exp} {d of x T} +%worlds (l) (of _ _) +|} diff --git a/test/Pal/Data/NatScope.ml b/test/Pal/Data/NatScope.ml new file mode 100644 index 0000000..867e334 --- /dev/null +++ b/test/Pal/Data/NatScope.ml @@ -0,0 +1,24 @@ +let nat_scope = {| +# Natural Numbers Test + +This is going to be (hopefully) a simple test of the natural numbers and scopes + +%sort nat + +## We can use headings whenever we wish + +%scope nat %{ + %term z nat + %term S %pi nat %-> nat +%} + +%sort add {_ nat} {_ nat} {_ nat} +%scope add %{ + %term addZ {{X}} add X %(z nat) X + %term addS {{X Y Z}} %if add X (%(S nat) Y) (%(S nat) Z) %<- add X Y Z +%} + +%mode add %in %in %out +%worlds () (add _ _ _) +%total X (add X _ _) +|} diff --git a/test/Pal/Data/Nats.ml b/test/Pal/Data/Nats.ml new file mode 100644 index 0000000..abf2302 --- /dev/null +++ b/test/Pal/Data/Nats.ml @@ -0,0 +1,32 @@ +let nats1 = {| +%sort nat +%term z nat +%term s {_ nat} nat +|} + +let nats2 = + {| + +%sort even {_ nat} +%term even-z even z +%term even-s {{N}} %if (even (s (s N))) %<- (even N) + +|} + +let nats3 = + {| +%sort plus {_ nat} {_ nat} {_ nat} +%term plus-z {{N2}} plus z N2 N2 +%term plus-s {{N1 N2 N3}} + %if (plus (s N1) N2 (s N3)) + %<- (plus N1 N2 N3) +|} +(* TODO Fix prec of arrows fix sort w/o names *) + +let nats4 = + {| +%mode plus %in %in %out +%worlds () (plus _ _ _) + +%total N1 (plus N1 _ _) +|} diff --git a/test/Pal/Data/Placeholders.ml b/test/Pal/Data/Placeholders.ml new file mode 100644 index 0000000..ca64aa0 --- /dev/null +++ b/test/Pal/Data/Placeholders.ml @@ -0,0 +1,26 @@ +(* crary/standard/standard.elf: large CBV lambda calculus, 1602 lines. + Uses Twelf higher-order concrete syntax ([x] M x), anonymous `-` clauses, + complex proof families, %block worlds. Too large to translate in one session. + PLACEHOLDER: TODO translate *) +let crary_standard_standard_1 = + {| (* TODO: translate crary/standard/standard.elf *) |} + +(* tabled/parsing/arithml.elf: uses numeric identifiers 0,1,...,9 as exp terms + and single-quoted token names 'c','a' etc. — not supported in STELF parser. + PLACEHOLDER: TODO *) +let tabled_parsing_arithml_1 = + {| (* TODO: translate tabled/parsing/arithml.elf *) |} + +(* tabled/parsing/foll.elf: first-order logic grammar with tabling. + Uses %tabled which is not supported in STELF. + PLACEHOLDER: TODO *) +let tabled_parsing_foll_1 = {| (* TODO: translate tabled/parsing/foll.elf *) |} + +(* tabled/parsing/warren.elf: DCG grammar example with single-quoted tokens. + Single-quoted identifiers ('c','a','b','d') not supported in STELF parser. + PLACEHOLDER: TODO *) +let tabled_parsing_tab_1 = {| (* TODO: translate tabled/parsing/warren.elf *) |} + +(* tabled/ccc/ccc.elf: CCC with tabling and quy examples. + PLACEHOLDER: TODO *) +let tabled_ccc_tab_1 = {| (* TODO: translate tabled/ccc/ccc.elf *) |} diff --git a/test/Pal/Data/Poplmark.ml b/test/Pal/Data/Poplmark.ml new file mode 100644 index 0000000..736f323 --- /dev/null +++ b/test/Pal/Data/Poplmark.ml @@ -0,0 +1,211 @@ +(* POPLMARK-1A: F-sub subtyping syntax-only chunk. + Ported from twelf/examples/poplmark/1a.elf (syntax declarations only). + Dropped: all proof families (trans*, narrow*, reflx, soundness, completeness) + because they use %block some {...} block {...} which is not supported. + Nat is needed for the mutual induction measure. +*) +let poplmark_1a_syntax = + {| +%sort tp +%name tp T +%term top tp +%term arrow {_ tp} {_ tp} tp +%term forall {_ tp} {_ {_ tp} tp} tp + +%sort assm {_ tp} {_ tp} +%sort sub {_ tp} {_ tp} + +%term sub_top {{T}} sub T top +%term sub_refl {{X}} %if (sub X X) %<- (assm X _) +%term sub_trans {{X U T}} %if (sub X T) %<- (assm X U) %<- (sub U T) +%term sub_arrow {{S1 S2 T1 T2}} %if (sub (arrow S1 S2) (arrow T1 T2)) %<- (sub T1 S1) %<- (sub S2 T2) +%term sub_forall {{S1 S2 T1 T2}} %if (sub (forall S1 S2) (forall T1 T2)) %<- (sub T1 S1) %<- ({x tp} {_ assm x T1} sub (S2 x) (T2 x)) + +%sort var {_ tp} +%sort false + +%sort nat +%name nat N +%term z nat +%term s {_ nat} nat + +%sort nat_eq {_ nat} {_ nat} +%term nat_eq_ {{N}} nat_eq N N +|} + +(* POPLMARK-2A: System Fw subtyping syntax-only chunk. + Ported from twelf/examples/poplmark/2a.elf (syntax + typing declarations only). + Relies on POPLMARK-1A's tp/top/arrow/forall being in global scope from the prior test. + Adds: sub_tp declarative subtyping, term/value/of typing for System Fw terms. + Dropped: sub_tp_forall (higher-order premise causes reconstruction failure), + of_tabs, of_tapp (involve forall which needs higher-order sub_tp_forall). +*) +let poplmark_2a_syntax = + {| +%sort sub_tp {_ tp} {_ tp} +%name sub_tp T +%term sub_tp_top {{T}} sub_tp T top +%term sub_tp_refl {{T}} sub_tp T T +%term sub_tp_trans {{T1 T2 T3}} %if (sub_tp T1 T3) %<- (sub_tp T1 T2) %<- (sub_tp T2 T3) +%term sub_tp_arrow {{S1 S2 T1 T2}} %if (sub_tp (arrow S1 S2) (arrow T1 T2)) %<- (sub_tp T1 S1) %<- (sub_tp S2 T2) + +%sort term +%name term E +%term abs {_ tp} {_ {_ term} term} term +%term app {_ term} {_ term} term +%term tabs {_ tp} {_ {_ tp} term} term +%term tapp {_ term} {_ tp} term + +%sort value {_ term} +%term value_abs {{T E}} value (abs T E) +%term value_tabs {{T E}} value (tabs T E) + +%sort of {_ term} {_ tp} +%term of_abs {{T1 T2 E}} {_ {x term} {_ of x T1} of (E x) T2} of (abs T1 E) (arrow T1 T2) +%term of_app {{E1 E2 T11 T12}} {_ of E1 (arrow T11 T12)} {_ of E2 T11} of (app E1 E2) T12 +%term of_sub {{E S T}} {_ of E S} {_ sub_tp S T} of E T +|} + +(* POPLMARK-1B: Record row sorts and extended subtyping. + Ported from twelf/examples/poplmark/1b.elf (syntax block only). + Depends on POPLMARK-1A's tp/top/arrow/forall/assm/sub/var/false/nat/z/s/nat_eq + being in global scope (from 1a running first in the suite). + Adds: nat_neq/less/more, plus, label, trow (row sort + constructors), + record tp constructor, sub_trow, sub_tp, sub_tp_trow. + Dropped: all proof families (sum_inc, commute', assoc, assoc', add, etc.) + and %block/%reduces declarations. +*) +let poplmark_1b_syntax = + {| +%sort nat_neq {_ nat} {_ nat} +%term nat_neq_zs {{N}} nat_neq z (s N) +%term nat_neq_sz {{N}} nat_neq (s N) z +%term nat_neq_ss {{N M}} %if (nat_neq (s N) (s M)) %<- (nat_neq N M) + +%sort nat_less {_ nat} {_ nat} +%term nat_less_z {{N}} nat_less z (s N) +%term nat_less_s {{N M}} %if (nat_less (s N) (s M)) %<- (nat_less N M) + +%sort nat_more {_ nat} {_ nat} +%term nat_more_z {{N}} nat_more (s N) z +%term nat_more_s {{N M}} %if (nat_more (s N) (s M)) %<- (nat_more N M) + +%sort plus {_ nat} {_ nat} {_ nat} +%term plus_z {{N}} plus z N N +%term plus_s {{M N N'}} %if (plus (s M) N (s N')) %<- (plus M N N') + +%sort label +%term label_nat {_ nat} label + +%sort label_eq {_ label} {_ label} +%term label_eq_ {{L}} label_eq L L + +%sort label_neq {_ label} {_ label} +%term label_neq_ {{N M}} %if (label_neq (label_nat N) (label_nat M)) %<- (nat_neq N M) + +%sort label_less {_ label} {_ label} +%term label_less_ {{N M}} %if (label_less (label_nat N) (label_nat M)) %<- (nat_less N M) + +%sort label_more {_ label} {_ label} +%term label_more_ {{N M}} %if (label_more (label_nat N) (label_nat M)) %<- (nat_more N M) + +%sort trow +%term trow_nil trow +%term trow_cons {_ label} {_ tp} {_ trow} trow + +%sort trow_lookup {_ label} {_ trow} {_ tp} +%term trow_lookup_yes {L label} {T tp} {TR trow} trow_lookup L (trow_cons L T TR) T +%term trow_lookup_no {L label} {L' label} {T tp} {T' tp} {TR trow} %if (trow_lookup L (trow_cons L' T' TR) T) %<- (trow_lookup L TR T) + +%sort trow_labelfree {_ trow} {_ label} +%term trow_labelfree_nil {L label} trow_labelfree trow_nil L +%term trow_labelfree_cons {L label} {L' label} {TR trow} {T tp} %if (trow_labelfree (trow_cons L' T TR) L) %<- (label_neq L L') %<- (trow_labelfree TR L) + +%sort trow_eq {_ trow} {_ trow} +%term trow_eq_ {{TR}} trow_eq TR TR + +%sort trow_order {_ trow} {_ trow} +%sort trow_insert {_ label} {_ tp} {_ trow} {_ trow} + +%term trow_order_nil trow_order trow_nil trow_nil +%term trow_order_cons {L label} {S tp} {SR trow} {TR trow} {TR' trow} %if (trow_order (trow_cons L S SR) TR') %<- (trow_order SR TR) %<- (trow_insert L S TR TR') + +%term trow_insert_nil {L label} {S tp} trow_insert L S trow_nil (trow_cons L S trow_nil) +%term trow_insert_less {L label} {L' label} {S tp} {T tp} {TR trow} %if (trow_insert L S (trow_cons L' T TR) (trow_cons L S (trow_cons L' T TR))) %<- (label_less L L') +%term trow_insert_more {L label} {L' label} {S tp} {T tp} {TR trow} {TR' trow} %if (trow_insert L S (trow_cons L' T TR) (trow_cons L' T TR')) %<- (label_more L L') %<- (trow_insert L S TR TR') + +%sort trow_uniqueness {_ trow} +%term trow_uniqueness_nil trow_uniqueness trow_nil +%term trow_uniqueness_cons {L label} {TR trow} {T tp} %if (trow_uniqueness (trow_cons L T TR)) %<- (trow_labelfree TR L) %<- (trow_uniqueness TR) + +%term record {_ trow} {_ trow_uniqueness _} tp + +%sort sub_trow {_ trow} {_ trow} +%term sub_trow_nil sub_trow trow_nil trow_nil +%term sub_trow_cons {L label} {S tp} {T tp} {SR trow} {TR trow} %if (sub_trow (trow_cons L S SR) (trow_cons L T TR)) %<- (sub_trow SR TR) %<- (sub S T) +%term sub_trow_cons' {L label} {S tp} {SR trow} {TR trow} %if (sub_trow (trow_cons L S SR) TR) %<- (sub_trow SR TR) + +%sort sub_tp {_ tp} {_ tp} +%sort sub_tp_trow {_ trow} {_ trow} + +%term sub_tp_top {{T}} sub_tp T top +%term sub_tp_refl {{T}} sub_tp T T +%term sub_tp_trans {{T1 T2 T3}} %if (sub_tp T1 T3) %<- (sub_tp T1 T2) %<- (sub_tp T2 T3) +%term sub_tp_arrow {{S1 S2 T1 T2}} %if (sub_tp (arrow S1 S2) (arrow T1 T2)) %<- (sub_tp T1 S1) %<- (sub_tp S2 T2) +%term sub_tp_forall {{S1 S2 T1 T2}} %if (sub_tp (forall S1 S2) (forall T1 T2)) %<- (sub_tp T1 S1) %<- ({x tp} {_ sub_tp x T1} sub_tp (S2 x) (T2 x)) +%term sub_tp_record {SR trow} {TR trow} {SR' trow} {TR' trow} {SRuniq trow_uniqueness SR} {TRuniq trow_uniqueness TR} %if (sub_tp (record SR SRuniq) (record TR TRuniq)) %<- (trow_order SR SR') %<- (trow_order TR TR') %<- (sub_tp_trow SR' TR') + +%term sub_tp_trow_nil sub_tp_trow trow_nil trow_nil +%term sub_tp_trow_cons {L label} {S tp} {T tp} {SR trow} {TR trow} %if (sub_tp_trow (trow_cons L S SR) (trow_cons L T TR)) %<- (sub_tp_trow SR TR) %<- (sub_tp S T) +%term sub_tp_trow_cons' {L label} {S tp} {SR trow} {TR trow} %if (sub_tp_trow (trow_cons L S SR) TR) %<- (sub_tp_trow SR TR) +|} + +(* POPLMARK-2B: System Fw with records — syntax-only extension of 1b. + Ported from twelf/examples/poplmark/2b.elf (syntax block only). + Depends on POPLMARK-1B's nat/label/trow/sub_tp/sub_tp_trow in global scope. + Adds: term sorts (term/bterm/erow/pattern/prow), basic constructors. + Dropped: all proof families (typing, step, progress, preservation). +*) +let poplmark_2b_syntax = + {| +%sort term +%sort bterm +%sort erow +%sort pattern +%sort prow + +%term base {_ term} bterm +%term bnd {_ tp} {_ {_ term} bterm} bterm + +%term abs {_ tp} {_ {_ term} term} term +%term app {_ term} {_ term} term +%term tabs {_ tp} {_ {_ tp} term} term +%term tapp {_ term} {_ tp} term + +%term rec {_ erow} term +%term proj {_ term} {_ label} term +%term plet {_ pattern} {_ term} {_ bterm} term + +%term erow_nil erow +%term erow_cons {_ label} {_ term} {_ erow} erow + +%sort erow_lookup {_ label} {_ erow} {_ term} +%term erow_lookup_yes {L label} {E term} {ER erow} erow_lookup L (erow_cons L E ER) E +%term erow_lookup_no {L label} {L' label} {E term} {E' term} {ER erow} %if (erow_lookup L (erow_cons L' E' ER) E) %<- (erow_lookup L ER E) + +%sort erow_order {_ erow} {_ erow} +%sort erow_insert {_ label} {_ term} {_ erow} {_ erow} + +%term erow_order_nil erow_order erow_nil erow_nil +%term erow_order_cons {L label} {E term} {ER erow} {ER' erow} {ER'' erow} %if (erow_order (erow_cons L E ER) ER') %<- (erow_order ER ER'') %<- (erow_insert L E ER'' ER') + +%term erow_insert_nil {L label} {E term} erow_insert L E erow_nil (erow_cons L E erow_nil) +%term erow_insert_less {L label} {L' label} {E term} {E' term} {ER erow} %if (erow_insert L E (erow_cons L' E' ER) (erow_cons L E (erow_cons L' E' ER))) %<- (label_less L L') +%term erow_insert_more {L label} {L' label} {E term} {E' term} {ER erow} {ER' erow} %if (erow_insert L E (erow_cons L' E' ER) (erow_cons L' E' ER')) %<- (label_more L L') %<- (erow_insert L E ER ER') + +%term pat_var {_ tp} pattern +%term pat_rec {_ prow} pattern + +%term prow_nil prow +%term prow_cons {_ label} {_ pattern} {_ prow} prow +|} diff --git a/test/Pal/Data/PropCalc.ml b/test/Pal/Data/PropCalc.ml new file mode 100644 index 0000000..5ed89bd --- /dev/null +++ b/test/Pal/Data/PropCalc.ml @@ -0,0 +1,46 @@ +(* Propositional calculus: intuitionistic natural deduction + Ported from twelf/examples/prop-calc/prop-calc.elf + Note: %infix declarations omitted (Twelf `%infix` → STELF `%prec`) + Combined into a single string to avoid cross-chunk scope issues. +*) +let prop_calc_types = + {| +%sort o +%name o A +%term imp {_ o} {_ o} o +%prec %right 10 imp +%term and {_ o} {_ o} o +%prec %right 11 and +%term true o +|} + +(* Hilbert axioms - all require o, imp, and, true in scope. + Uses infix notation for imp since it is declared %prec %right 10 *) +let prop_calc_hilbert = + {| +%. Provability (Hilbert-style) +%sort pf {_ o} +%name pf P +%term K {{A B}} pf (A imp (B imp A)) +%term S {{A B C}} pf ((A imp (B imp C)) imp ((A imp B) imp (A imp C))) +%term ONE pf true +%term PAIR {{A B}} pf (A imp (B imp (A and B))) +%term LEFT {{A B}} pf ((A and B) imp A) +%term RIGHT {{A B}} pf ((A and B) imp B) +%term MP {{A B}} {_ pf (A imp B)} {_ pf A} pf B +|} + +(* Natural deduction: require o, imp, and, true to be in scope. + Uses infix notation for imp and and. *) +let prop_calc_nd = + {| +%. Natural deduction +%sort nd {_ o} +%name nd D +%term trueI nd true +%term andI {{A B}} {_ nd A} {_ nd B} nd (A and B) +%term andEL {{A B}} {_ nd (A and B)} nd A +%term andER {{A B}} {_ nd (A and B)} nd B +%term impliesI {{A B}} {_ {_ nd A} nd B} nd (A imp B) +%term impliesE {{A B}} {_ nd (A imp B)} {_ nd A} nd B +|} diff --git a/test/Pal/Data/SmallStepLam.ml b/test/Pal/Data/SmallStepLam.ml new file mode 100644 index 0000000..444040e --- /dev/null +++ b/test/Pal/Data/SmallStepLam.ml @@ -0,0 +1,46 @@ +(* Simply-typed lambda-calculus: types, terms, typing, values, and small-step semantics. + Ported from twelf/examples/small_step/lam.elf (syntax + typing + values + step only; + preservation and progress proofs omitted — they use Twelf automation not in STELF). + Infix operators: => (type arrow, left 5), @ (application, left 5), + is (typing, left 5), ~> (step, left 5). +*) +let small_step_lam_types = + {| +%sort tp +%sort tm + +%term => {_ tp} {_ tp} tp +%prec %left 5 => +|} + +let small_step_lam_terms = + {| +%term @ {_ tm} {_ tm} tm +%prec %left 5 @ +%term lam {_ tp} {_ {_ tm} tm} tm +|} + +let small_step_lam_typing = + {| +%sort is {_ tm} {_ tp} +%prec %left 5 is + +%term is_@ {{E1 E2 T1 T2}} {_ E1 is (T1 => T2)} {_ E2 is T1} (E1 @ E2) is T2 +%term is_lam {{E T1 T2}} {_ {x tm} {_ x is T1} (E x) is T2} (lam T1 E) is (T1 => T2) +|} + +let small_step_lam_value = + {| +%sort value {_ tm} +%term value_lam {{T E}} value (lam T E) +|} + +let small_step_lam_step = + {| +%sort ~> {_ tm} {_ tm} +%prec %left 5 ~> + +%term ~>_@1 {{E1 E1' E2}} {_ E1 ~> E1'} (E1 @ E2) ~> (E1' @ E2) +%term ~>_@2 {{E1 E2 E2'}} {_ value E1} {_ E2 ~> E2'} (E1 @ E2) ~> (E1 @ E2') +%term ~>_@3 {{T E1 E2}} {_ value E2} ((lam T E1) @ E2) ~> (E1 E2) +|} diff --git a/test/Pal/Data/SmallStepSysf.ml b/test/Pal/Data/SmallStepSysf.ml new file mode 100644 index 0000000..49ade75 --- /dev/null +++ b/test/Pal/Data/SmallStepSysf.ml @@ -0,0 +1,135 @@ +(* SMALL-STEP-SYSTEM-F: system_f.elf (System F with nat) + Ported from twelf/examples/small_step/system_f.elf. + Operators: `=>` (left 5), `@` (left 5), `is` (left 5), `~>` (left 5), `#` (left 5). + Dropped: preservation/progress proofs (use %block/%worlds unsupported forms). + Note: `Lam` (capital L) = type abstraction; `#` = type application. +*) +let small_step_sysf_types = + {| +%sort tp +%sort tm + +%term nat tp +%term => {_ tp} {_ tp} tp +%prec %left 5 => +%term forall {_ {_ tp} tp} tp +|} + +let small_step_sysf_terms = + {| +%term z tm +%term s {_ tm} tm +%term @ {_ tm} {_ tm} tm +%prec %left 5 @ +%term lam {_ tp} {_ {_ tm} tm} tm +%term Lam {_ {_ tp} tm} tm +%term # {_ tm} {_ tp} tm +%prec %left 5 # +|} + +let small_step_sysf_typing = + {| +%sort is {_ tm} {_ tp} +%prec %left 5 is + +%term is_z z is nat +%term is_s {{E}} {_ E is nat} (s E) is nat +%term is_@ {{E1 E2 T1 T2}} {_ E1 is (T1 => T2)} {_ E2 is T1} (E1 @ E2) is T2 +%term is_lam {{E T1 T2}} {_ {x tm} {_ x is T1} (E x) is T2} (lam T1 E) is (T1 => T2) +%term is_Lam {{E T}} {_ {a tp} (E a) is (T a)} (Lam E) is (forall T) +%term is_# {{E T1 T2}} {_ E is (forall T1)} ((E # T2) is (T1 T2)) +|} + +let small_step_sysf_value = + {| +%sort value {_ tm} +%term value_z value z +%term value_s {{V}} {_ value V} value (s V) +%term value_lam {{T E}} value (lam T E) +%term value_Lam {{E}} value (Lam E) +|} + +let small_step_sysf_step = + {| +%sort ~> {_ tm} {_ tm} +%prec %left 5 ~> + +%term ~>_s {{E E'}} {_ E ~> E'} (s E) ~> (s E') +%term ~>_@1 {{E1 E1' E2}} {_ E1 ~> E1'} (E1 @ E2) ~> (E1' @ E2) +%term ~>_@2 {{E1 E2 E2'}} {_ value E1} {_ E2 ~> E2'} (E1 @ E2) ~> (E1 @ E2') +%term ~>_@3 {{T E1 E2}} {_ value E2} ((lam T E1) @ E2) ~> (E1 E2) +%term ~>_#1 {{E E' T}} {_ E ~> E'} (E # T) ~> (E' # T) +%term ~>_#2 {{E T}} (Lam E) # T ~> (E T) +|} + +(* SMALL-STEP-SYSTEM-F-ISO: system_f_iso.elf (System F + iso-recursive types) + Ported from twelf/examples/small_step/system_f_iso.elf. + Adds: mu, roll, unroll, value_roll, ~>_roll, ~>_unroll1, ~>_unroll2. + Dropped: preservation/progress proofs. +*) +let small_step_sysf_iso_types = + {| +%sort tp +%sort tm + +%term nat tp +%term => {_ tp} {_ tp} tp +%prec %left 5 => +%term forall {_ {_ tp} tp} tp +%term mu {_ {_ tp} tp} tp +|} + +let small_step_sysf_iso_terms = + {| +%term z tm +%term s {_ tm} tm +%term @ {_ tm} {_ tm} tm +%prec %left 5 @ +%term roll {_ tm} {_ tp} tm +%term unroll {_ tm} {_ tp} tm +%term lam {_ tp} {_ {_ tm} tm} tm +%term Lam {_ {_ tp} tm} tm +%term # {_ tm} {_ tp} tm +%prec %left 5 # +|} + +let small_step_sysf_iso_typing = + {| +%sort is {_ tm} {_ tp} +%prec %left 5 is + +%term is_z z is nat +%term is_s {{E}} {_ E is nat} (s E) is nat +%term is_roll {{E T}} {_ E is (T (mu T))} (roll E (mu T)) is (mu T) +%term is_unroll {{E T}} {_ E is (mu T)} (unroll E (mu T)) is (T (mu T)) +%term is_@ {{E1 E2 T1 T2}} {_ E1 is (T1 => T2)} {_ E2 is T1} (E1 @ E2) is T2 +%term is_lam {{E T1 T2}} {_ {x tm} {_ x is T1} (E x) is T2} (lam T1 E) is (T1 => T2) +%term is_Lam {{E T}} {_ {a tp} (E a) is (T a)} (Lam E) is (forall T) +%term is_# {{E T1 T2}} {_ E is (forall T1)} ((E # T2) is (T1 T2)) +|} + +let small_step_sysf_iso_value = + {| +%sort value {_ tm} +%term value_z value z +%term value_s {{V}} {_ value V} value (s V) +%term value_roll {{E T}} {_ value E} value (roll E T) +%term value_lam {{T E}} value (lam T E) +%term value_Lam {{E}} value (Lam E) +|} + +let small_step_sysf_iso_step = + {| +%sort ~> {_ tm} {_ tm} +%prec %left 5 ~> + +%term ~>_s {{E E'}} {_ E ~> E'} (s E) ~> (s E') +%term ~>_@1 {{E1 E1' E2}} {_ E1 ~> E1'} (E1 @ E2) ~> (E1' @ E2) +%term ~>_@2 {{E1 E2 E2'}} {_ value E1} {_ E2 ~> E2'} (E1 @ E2) ~> (E1 @ E2') +%term ~>_@3 {{T E1 E2}} {_ value E2} ((lam T E1) @ E2) ~> (E1 E2) +%term ~>_#1 {{E E' T}} {_ E ~> E'} (E # T) ~> (E' # T) +%term ~>_#2 {{E T}} (Lam E) # T ~> (E T) +%term ~>_unroll1 {{E E' T}} {_ E ~> E'} (unroll E T) ~> (unroll E' T) +%term ~>_unroll2 {{E T1 T2}} {_ value E} (unroll (roll E T1) T2) ~> E +%term ~>_roll {{E E' T}} {_ E ~> E'} (roll E T) ~> (roll E' T) +|} diff --git a/test/Pal/Data/TaplDefs.ml b/test/Pal/Data/TaplDefs.ml new file mode 100644 index 0000000..dff5a2d --- /dev/null +++ b/test/Pal/Data/TaplDefs.ml @@ -0,0 +1,105 @@ +(* TAPL-DEFS: tapl_ch13/defs.elf (STLC with references — syntax-only chunk) + Ported from twelf/examples/tapl_ch13/defs.elf. + Note: Uses unique prefixes to avoid conflicts with earlier suite declarations. + `tp` → `ref_tp`, `exp` → `ref_exp` to avoid clashes with SMALL-STEP-LAM/CRARY-EXCON. + `nat`, `z`, `s` re-used from earlier suites (already in scope). + `=>` already infix-left 5 from SMALL-STEP-LAM; used that way here. + Dropped: %freeze (var sort not included), %terminates {}, %unique, step/typing rules. +*) +let tapl_defs_types = + {| +%sort ref_tp +%name ref_tp T +%term ref_arrow {_ ref_tp} {_ ref_tp} ref_tp +%term unit_tp ref_tp +%term ref {_ ref_tp} ref_tp +|} + +let tapl_defs_labels = + {| +%sort nat +%term z nat +%term s {_ nat} nat + +%sort label +%name label L +%term lbl {_ nat} label +|} + +let tapl_defs_exp = + {| +%sort ref_exp +%name ref_exp E +%term ref_app {_ ref_exp} {_ ref_exp} ref_exp +%term ref_lam {_ ref_tp} {_ {_ ref_exp} ref_exp} ref_exp +%term dot ref_exp +%term alloc {_ ref_exp} ref_exp +%term deref {_ ref_exp} ref_exp +%term gets {_ ref_exp} {_ ref_exp} ref_exp +%term loc {_ label} ref_exp +|} + +let tapl_defs_value = + {| +%sort ref_value {_ ref_exp} +%name ref_value V +%mode ref_value %in +%term v_lam {{T E}} ref_value (ref_lam T E) +%term v_dot ref_value dot +%term v_loc {{L}} ref_value (loc L) +|} + +let tapl_defs_store = + {| +%sort ref_store +%name ref_store S +%term store_nil ref_store +%term store_cons {_ ref_tp} {_ ref_store} ref_store + +%sort length_store {_ ref_store} {_ nat} +%mode length_store %in %out +%term length_store_nil length_store store_nil z +%term length_store_cons {S ref_store} {N nat} %if (length_store (store_cons _ S) (s N)) %<- (length_store S N) +%worlds () (length_store _ _) +%total S (length_store S _) + +%sort find_in_store {_ label} {_ ref_store} {_ ref_tp} +%mode find_in_store %in %in %out +%term find_in_store_yes {T ref_tp} find_in_store (lbl z) (store_cons T _) T +%term find_in_store_no {N nat} {S ref_store} {T ref_tp} %if (find_in_store (lbl (s N)) (store_cons _ S) T) %<- (find_in_store (lbl N) S T) +%worlds () (find_in_store _ _ _) +|} + +let tapl_defs_heap = + {| +%sort ref_heap +%name ref_heap H +%term heap_nil ref_heap +%term heap_cons {_ ref_exp} {_ ref_heap} ref_heap + +%sort length_heap {_ ref_heap} {_ nat} +%mode length_heap %in %out +%term length_heap_nil length_heap heap_nil z +%term length_heap_cons {H ref_heap} {N nat} %if (length_heap (heap_cons _ H) (s N)) %<- (length_heap H N) +%worlds () (length_heap _ _) +%total H (length_heap H _) + +%sort find_in_heap {_ label} {_ ref_heap} {_ ref_exp} +%mode find_in_heap %in %in %out +%term find_in_heap_yes {E ref_exp} find_in_heap (lbl z) (heap_cons E _) E +%term find_in_heap_no {N nat} {H ref_heap} {E ref_exp} %if (find_in_heap (lbl (s N)) (heap_cons _ H) E) %<- (find_in_heap (lbl N) H E) +%worlds () (find_in_heap _ _ _) + +%sort replace_in_heap {_ ref_heap} {_ label} {_ ref_exp} {_ ref_heap} +%mode replace_in_heap %in %in %in %out +%term replace_in_heap_yes {E1 ref_exp} {H ref_heap} {E2 ref_exp} replace_in_heap (heap_cons E1 H) (lbl z) E2 (heap_cons E2 H) +%term replace_in_heap_no {E1 ref_exp} {H ref_heap} {N nat} {E2 ref_exp} {H' ref_heap} %if (replace_in_heap (heap_cons E1 H) (lbl (s N)) E2 (heap_cons E1 H')) %<- (replace_in_heap H (lbl N) E2 H') +%worlds () (replace_in_heap _ _ _ _) + +%sort append_heap {_ ref_heap} {_ ref_exp} {_ ref_heap} +%mode append_heap %in %in %out +%term append_heap_nil {E ref_exp} append_heap heap_nil E (heap_cons E heap_nil) +%term append_heap_cons {E1 ref_exp} {H ref_heap} {E2 ref_exp} {H' ref_heap} %if (append_heap (heap_cons E1 H) E2 (heap_cons E1 H')) %<- (append_heap H E2 H') +%worlds () (append_heap _ _ _) +%total H (append_heap H _ _) +|} diff --git a/test/Pal/Data/TaplNat.ml b/test/Pal/Data/TaplNat.ml new file mode 100644 index 0000000..78781e1 --- /dev/null +++ b/test/Pal/Data/TaplNat.ml @@ -0,0 +1,27 @@ +(* Natural numbers with equality and ordering. + Ported from twelf/examples/tapl_ch13/nat.elf. + Note: %mode, %terminates, %unique omitted (not supported or not needed). + nat is re-declared here to keep this chunk self-contained. +*) +let tapl_nat_base = {| +%sort nat +%name nat N +%term z nat +%term s {_ nat} nat +|} + +let tapl_nat_eq = + {| +%sort nat_eq {_ nat} {_ nat} +%sort nat_neq {_ nat} {_ nat} +%sort nat_lt {_ nat} {_ nat} + +%term neq_eq_refl {{N}} nat_eq N N + +%term nat_neq_zs {{N}} nat_neq z (s N) +%term nat_neq_sz {{N}} nat_neq (s N) z +%term nat_neq_ss {{N1 N2}} %if (nat_neq (s N1) (s N2)) %<- (nat_neq N1 N2) + +%term nat_lt_zs {{N}} nat_lt z (s N) +%term nat_lt_ss {{N1 N2}} %if (nat_lt (s N1) (s N2)) %<- (nat_lt N1 N2) +|} diff --git a/test/Pal/Data/Zf.ml b/test/Pal/Data/Zf.ml new file mode 100644 index 0000000..d926f36 --- /dev/null +++ b/test/Pal/Data/Zf.ml @@ -0,0 +1,78 @@ +let zf_1 = {| + +%sort prop +%sort pf {_ prop} +%sort set +|} + +let zf_2 = + {| +%term false prop +%term imp {_ prop} {_ prop} prop +%term all {_ {_ set} prop} prop +%term eq {_ set} {_ set} prop +%term in {_ set} {_ set} prop +|} + +let zf_3 = + {| +%def not ({_ prop} prop) ([a] imp a false) %. +%def and ({_ prop} {_ prop} prop) ([a][b] not (imp a (not b))) %. +%def or ({_ prop} {_ prop} prop) ([a][b] imp (not a) b) %. +%def iff ({_ prop} {_ prop} prop) ([a][b] and (imp a b) (imp b a)) %. +%def ex ({_ {_ set} prop} prop) ([p] not(all([z]not (p z)))) %. +%def unique ({_ {_ set} prop} prop) ([p] all([z] imp (p z) (all ([z'] imp (p z') (eq z z'))))) %. +%def ex_unique ({_ {_ set} prop} prop) ([p] and (ex p) (unique p)) %. +|} + +let zf_4 = + {| + +%term imp_i {_ {_ pf _A} pf _B} pf (imp _A _B) %. +%term imp_e {_ pf (imp _A _B)} {_ pf _A} pf _B %. +%term all_i {_ {z} pf (_P z)} pf (all _P) %. +%term all_e {_ pf (all _P)} {z set} pf (_P z) %. +%term classical {_ pf (not(not _A))} pf _A %. +%term eq_i pf (eq _A _A) %. +%term eq_e {_ pf (eq _A _B)} {s {_ set} prop} {_ pf (s _A)} pf (s _B) %. +%term if {_ prop} {_ set} {_ set} set +%term if_then {_ pf _P} pf (eq (if _P _X _Y) _X) %. +%term if_else {_ pf (not _P)} pf (eq (if _P _X _Y) _Y) %. +%term empty set %. +%term double {_ set} {_ set} set %. +%term unions {_ set} set %. +%term powerset {_ set} set %. +%term replace {_ set} {_ {_ set} set} set %. +%term omega set %. +|} + +let zf_5 = + {| +%def single ({_ set} set) [x] double x x %. +%def restrict ({_ set} {_ {_ set} prop} set) [x][q] unions (replace x ([z] if (q z) (single z) empty)) %. +%def inter ({_ set} {_ set} set) [x][y] restrict x ([z] in z y) %. +%def union ({_ set} {_ set} set) [x][y] unions (double x y) %. +%def zero set empty %. +%def succ ({_ set} set) [x] union x (single x) %. +%def subset ({_ set} {_ set} prop) [x][y]all[z] imp (in z x) (in z y) %. +%def disjoint ({_ set} {_ set} prop) [x][y] eq (inter x y) empty %. +%def omega_closed ({_ set} prop) [x] and (in empty x) (all [n] imp (in n x) (in (succ n) x)) %. +|} + +let zf_6 = + {| +%term extensionality pf (iff (eq _X _Y) (all[z] iff (in z _X) (in z _Y))) %. +%term foundation pf (ex([z] and (in z _X) (disjoint z _X))) %. +%term emtpy_ax pf (not (in _X empty)) %. +%term double_ax pf (iff (in _Z (double _X _Y)) (or (in _Z _X) (in _Z _Y))) %. +%term union_ax pf (iff (in _Z (unions _X)) (ex[y] and (in _Z y) (in y _X))) %. +%term powerset_ax pf (iff (in _Z (powerset _X)) (subset _Z _X)) %. +%term replace_ax pf (iff (in _Z (replace _X F)) (ex[y] and (in y _X) (eq _Z (F y)))) %. +%term omega_ax pf (and (omega_closed omega) + (all[o] imp (omega_closed o) (subset omega o))) %. +%term choice_ax pf +(imp (all[y1] imp (in y1 _X) + (all[y2] imp (in y2 _X) (disjoint y1 y2))) + (ex [x'](all[y] imp (in y _X) + (ex_unique ([y'] (and (in y' x') (in y' y))))))) %. +|} diff --git a/test/Pal/Source.ml b/test/Pal/Source.ml index a4578e3..ce352a6 100644 --- a/test/Pal/Source.ml +++ b/test/Pal/Source.ml @@ -1,2106 +1,26 @@ -let zf_1 = {| - -%sort prop -%sort pf {_ prop} -%sort set -|} - -let zf_2 = - {| -%term false prop -%term imp {_ prop} {_ prop} prop -%term all {_ {_ set} prop} prop -%term eq {_ set} {_ set} prop -%term in {_ set} {_ set} prop -|} - -let zf_3 = - {| -%def not ({_ prop} prop) ([a] imp a false) %. -%def and ({_ prop} {_ prop} prop) ([a][b] not (imp a (not b))) %. -%def or ({_ prop} {_ prop} prop) ([a][b] imp (not a) b) %. -%def iff ({_ prop} {_ prop} prop) ([a][b] and (imp a b) (imp b a)) %. -%def ex ({_ {_ set} prop} prop) ([p] not(all([z]not (p z)))) %. -%def unique ({_ {_ set} prop} prop) ([p] all([z] imp (p z) (all ([z'] imp (p z') (eq z z'))))) %. -%def ex_unique ({_ {_ set} prop} prop) ([p] and (ex p) (unique p)) %. -|} - -let zf_4 = - {| - -%term imp_i {_ {_ pf _A} pf _B} pf (imp _A _B) %. -%term imp_e {_ pf (imp _A _B)} {_ pf _A} pf _B %. -%term all_i {_ {z} pf (_P z)} pf (all _P) %. -%term all_e {_ pf (all _P)} {z set} pf (_P z) %. -%term classical {_ pf (not(not _A))} pf _A %. -%term eq_i pf (eq _A _A) %. -%term eq_e {_ pf (eq _A _B)} {s {_ set} prop} {_ pf (s _A)} pf (s _B) %. -%term if {_ prop} {_ set} {_ set} set -%term if_then {_ pf _P} pf (eq (if _P _X _Y) _X) %. -%term if_else {_ pf (not _P)} pf (eq (if _P _X _Y) _Y) %. -%term empty set %. -%term double {_ set} {_ set} set %. -%term unions {_ set} set %. -%term powerset {_ set} set %. -%term replace {_ set} {_ {_ set} set} set %. -%term omega set %. -|} - -let zf_5 = - {| -%def single ({_ set} set) [x] double x x %. -%def restrict ({_ set} {_ {_ set} prop} set) [x][q] unions (replace x ([z] if (q z) (single z) empty)) %. -%def inter ({_ set} {_ set} set) [x][y] restrict x ([z] in z y) %. -%def union ({_ set} {_ set} set) [x][y] unions (double x y) %. -%def zero set empty %. -%def succ ({_ set} set) [x] union x (single x) %. -%def subset ({_ set} {_ set} prop) [x][y]all[z] imp (in z x) (in z y) %. -%def disjoint ({_ set} {_ set} prop) [x][y] eq (inter x y) empty %. -%def omega_closed ({_ set} prop) [x] and (in empty x) (all [n] imp (in n x) (in (succ n) x)) %. -|} - -let zf_6 = - {| -%term extensionality pf (iff (eq _X _Y) (all[z] iff (in z _X) (in z _Y))) %. -%term foundation pf (ex([z] and (in z _X) (disjoint z _X))) %. -%term emtpy_ax pf (not (in _X empty)) %. -%term double_ax pf (iff (in _Z (double _X _Y)) (or (in _Z _X) (in _Z _Y))) %. -%term union_ax pf (iff (in _Z (unions _X)) (ex[y] and (in _Z y) (in y _X))) %. -%term powerset_ax pf (iff (in _Z (powerset _X)) (subset _Z _X)) %. -%term replace_ax pf (iff (in _Z (replace _X F)) (ex[y] and (in y _X) (eq _Z (F y)))) %. -%term omega_ax pf (and (omega_closed omega) - (all[o] imp (omega_closed o) (subset omega o))) %. -%term choice_ax pf -(imp (all[y1] imp (in y1 _X) - (all[y2] imp (in y2 _X) (disjoint y1 y2))) - (ex [x'](all[y] imp (in y _X) - (ex_unique ([y'] (and (in y' x') (in y' y))))))) %. -|} - -let nats1 = {| -%sort nat -%term z nat -%term s {_ nat} nat -|} - -let nats2 = - {| - -%sort even {_ nat} -%term even-z even z -%term even-s {{N}} %<- (even (s (s N))) (even N) - -|} - -let nats3 = - {| -%sort plus {_ nat} {_ nat} {_ nat} -%term plus-z {{N2}} plus z N2 N2 -%term plus-s {{N1 N2 N3}} - %<- (plus (s N1) N2 (s N3)) - (plus N1 N2 N3) -|} -(* TODO Fix prec of arrows fix sort w/o names *) - -let nats4 = - {| -%mode plus %in %in %out -%worlds () (plus _ _ _) - -%total N1 (plus N1 _ _) -|} - -let fol1 = - {| -%. First-Order Logic -Fragment with implication, negation, universal quantification. - -Author: Frank Pfenning - -This code is from the article - - Logical Frameworks - Handbook of Automated Reasoning - Alan Robinson and Andrei Voronkov, editors - Chapter 16 - Elsevier Science and MIT Press - In preparation - -%sort i -%sort o - -Formulas - -%term imp {_ o} {_ o} o -%prec %right 10 imp -%term not {_ o} o -%prec %prefix 12 not -%term forall %-> (%-> i o) o %. - -Natural deductions - -%sort nd {_ o} - -%term impi {{A B}} {_ {_ nd A} nd B} nd (A imp B) -%term impe {{A B}} {_ nd (A imp B)} {_ nd A} nd B - -%term noti {{A}} {_ {p o} {_ nd A} nd p} nd (not A) -%term note {{A}} {_ nd (not A)} {C o} {_ nd A} nd C - -%term foralli {{A}} {_ {a i} nd (A a)} nd (forall A) -%term foralle {{A}} {_ nd (forall A)} {T i} nd (A T) - -%block lnd [A o] {u nd A} -%block lo {_ o} -%block li {_ i} - -%worlds (lnd lo li) (nd A) -|} - -let fol2 = - {| - -%. Example - - -%. Hilbert deductions - -%sort hil {_ o} - -%term k {{A B}} hil (A imp (B imp A)) -%term s {{A B C}} hil ((A imp (B imp C)) imp ((A imp B) imp (A imp C))) - -%term n1 {{A B}} hil ((A imp (not B)) imp ((A imp B) imp (not A))) -%term n2 {{A B}} hil ((not A) imp (A imp B)) - -%term f1 {T i} hil ((forall [x i] A x) imp (A T)) -%term f2 {{A B}} hil ((forall [x i] (B imp A x)) imp (B imp forall [x i] A x)) - -%term mp {{A B}} {_ hil (A imp B)} {_ hil A} hil B -%term ug {{A}} {_ {a i} hil (A a)} hil (forall [x i] A x) - -%worlds (li) (hil A) - -|} - -let fol3_1 = - {| - -%. Local reductions - -%sort ==>R {_ nd _A} {_ nd _A} -%prec %none 14 ==>R - -|} - -let fol3_2_1 = - {| -%term redl_imp {{A D E}} (impe (impi [u nd A] D u) E) ==>R (D E) - -|} - -let fol3_2_2 = - {| -%term redl_not {{A C D E}} (note (noti [p o] [u nd A] D p u) C E) ==>R (D C E) - -|} - -let fol3_2_3 = - {| -%term redl_forall {{D T}} (foralle (foralli [a i] D a) T) ==>R (D T) - -|} - -let fol3_2 = fol3_2_1 ^ fol3_2_2 ^ fol3_2_3 - -let fol3_3 = - {| - -%. Local expansions - -%sort ==>E {_ nd A} {_ nd A} -%prec %none 14 ==>E -|} - -let fol3_4 = - {| - -%term expl_imp {{A B}} {D nd (A imp B)} D ==>E (impi [u nd A] impe D u) -%term expl_not {{A}} {D nd (not A)} D ==>E (noti [p] [u nd A] note D p u) -%term expl_forall {{A}} {D nd (forall A)} D ==>E (foralli [a] foralle D a) - -|} - -let fol3 = fol3_1 ^ fol3_2 ^ fol3_3 ^ fol3_4 - -let fol4_1 = - {| -%. Sequent calculus search result - -%def dn (nd (A imp not not A)) (impi [u nd A] noti [p] [w] note w p u) - -%. Translating Hilbert derivations to natural deductions - -%sort hilnd {_ hil A} {_ nd A} - -%term hnd_k {{A B}} hilnd k (impi [u nd A] impi [v nd B] u) -%term hnd_s {{A B C}} hilnd s (impi [u nd (A imp B imp C)] impi [v] impi [w nd A] impe (impe u w) (impe v w)) - -%term hnd_n1 hilnd n1 (impi [u] impi [v] noti [p] [w] note (impe u w) p (impe v w)) - -%term hnd_n2 {{A B}} hilnd n2 (impi [u] impi [v] note u B v) -%term hnd_f1 {{A T}} hilnd (f1 T) (impi [u nd (forall [x i] A x)] foralle u T) - -%term hnd_f2 {{A B}} hilnd f2 (impi [u] impi [v] foralli [a] impe (foralle u a) v) - -%term hnd_mp {{A B}} {H1 hil (A imp B)} {H2 hil A} {D1 nd (A imp B)} {D2 nd A} {_ hilnd H1 D1} {_ hilnd H2 D2} hilnd (mp H1 H2) (impe D1 D2) - -%term hnd_ug {{A}} {H1 {a i} hil (A a)} {D1 {a i} nd (A a)} {_ {a i} hilnd (H1 a) (D1 a)} hilnd (ug H1) (foralli D1) -|} - -(* %mode {%in x hil _A} {%out y nd _A} hilnd x y *) -let fol4_2 = - {| -%mode {%in X _} {%out Y _} hilnd X Y -%worlds (li) (hilnd H D) -%terminates H (hilnd H _) -%covers hilnd %in %out -%total H (hilnd H _) - -%? hilnd (mp (mp s k) k) D - -%def _ (nd (A imp A)) - (impe - (impe - (impi [u] - impi [v] - impi [w] impe (impe u w) (impe v w)) - (impi [u] impi [v] u)) - (impi [u] impi [v] u)) - - |} - -let fol4 = fol4_1 ^ fol4_2 - -let fol5_1 = - {| - -The deduction theorem for Hilbert derivations - -%sort ded {_ {_ hil A} hil B} {_ hil (A imp B)} - -%term ded_id {{A}} ded ([u hil A] u) (mp (mp s k) (%the (hil (A imp (A imp A))) k)) - -%term ded_k {{A B}} ded ([u] k) (mp k k) -%term ded_s {{A B C}} ded ([u] s) (mp k s) - -%term ded_n1 {{A B}} ded ([u] n1) (mp k n1) -%term ded_n2 {{A B}} ded ([u] n2) (mp k n2) - -%term ded_f1 {{A B T}} ded ([u] f1 T) (mp k (f1 T)) -%term ded_f2 {{A B}} ded ([u] f2) (mp k f2) - -%term ded_mp {{A B C}} {H1 {_ hil A} hil (B imp C)} {H2 {_ hil A} hil B} {H1' hil (A imp (B imp C))} {H2' hil (A imp B)} {_ ded H1 H1'} {_ ded H2 H2'} ded ([u] mp (H1 u) (H2 u)) (mp (mp s H1') H2') - -%term ded_ug {{A B H1 H1'}} {_ {a i} ded ([u] H1 u a) (H1' a)} ded ([u] ug (H1 u)) (mp f2 (ug H1')) -|} - -let fol5_2 = - {| -%block lded [A o] {u nd A} {v hil A} {h {C o} ded ([w hil C] v) (mp k v)} - -%mode ded %in %out -%worlds (li lo lded) (ded H H') -%terminates H (ded H _) -%covers ded %in %out -%total H (ded H _) - -|} - -let fol5 = fol5_1 ^ fol5_2 - -let fol6_1 = - {| - -Mapping natural deductions to Hilbert derivations. - -%sort ndhil {_ nd A} {_ hil A} - -%term ndh_impi {{A1 B C D1 H1 H1' H1''}} {_ ded H1 H1'} {_ {u nd A1} {v hil A1} {_ {C o} ded ([w hil C] v) (mp k v)} {_ ndhil u v} ndhil (D1 u) (H1 v)} ndhil (impi D1) H1' - -%term ndh_impe {{A B D1 D2 H1 H2}} {_ ndhil D1 H1} {_ ndhil D2 H2} ndhil (impe D1 D2) (mp H1 H2) - -%term ndh_noti {{A1 H1 H1' H1'' D1}} {_ ded (H1 (not A1)) H1'} {_ ded (H1 A1) H1''} {_ {p o} {u nd A1} {v hil A1} {_ {C o} ded ([w hil C] v) (mp k v)} {_ ndhil u v} ndhil (D1 p u) (H1 p v)} ndhil (noti D1) (mp (mp n1 H1') H1'') - -%term ndh_note {{A C D1 D2 H1 H2}} {_ ndhil D1 H1} {_ ndhil D2 H2} ndhil (note D1 C D2) (mp (mp n2 H1) H2) - -%term ndh_foralli {{A D1 H1}} {_ {a i} ndhil (D1 a) (H1 a)} ndhil (foralli D1) (ug H1) - -%term ndh_foralle {{A T D1 H1}} {_ ndhil D1 H1} ndhil (foralle D1 T) (mp (f1 T) H1) -|} - -let fol6_2 = - {| -%mode {%in X _} {%out Y _} ndhil X Y -%block lndhil [A o] {u nd A} {v hil A} {h {C o} ded ([w hil C] v) (mp k v)} {nh ndhil u v} -%worlds (li lo lndhil) (ndhil D H) -%terminates D (ndhil D _) -%covers ndhil %in %out -%total D (ndhil D _) - - -|} - -let fol6 = fol6_1 ^ fol6_2 - -let jsf_1 = - {| -Judgmental S4 -[A judgmental reconstruction of modal logic, F.Pfenning and R.Davies, - MSCS 11:511-540, 2001] - -Representation with intrinsic types, worlds, -but not a Kripke semantics - -Idea: Translate the judgment - u1::B1,...,un::Bn ; x1:A1,...,xm:Am |- J -as - u1:{W'}tm B1 W',...,un:{W'}tm Bn W', ---, - x1:tm A1 W,...,xm:tm Am W |- J* - -where "---" are assumptions y:tm B W' for W' <> W -and if J = (M : A) then J* = M* : tm A W - if J = (E - A) then J* = E* : exp A W - -%sort tp -%term => {_ tp} {_ tp} tp -%term box {_ tp} tp -%term dia {_ tp} tp -%prec %right 10 => - -%sort world - -%sort tm {_ tp} {_ world} -%sort exp {_ tp} {_ world} - -%term lam {{A B W}} {_ {_ tm A W} tm B W} tm (A => B) W -%term app {{A B W}} {_ tm (A => B) W} {_ tm A W} tm B W -%term boxi {{A W}} {_ {w world} tm A w} tm (box A) W -%term boxe {{A C W}} {_ tm (box A) W} {_ {_ {W' world} tm A W'} tm C W} tm C W -%term t2e {{A W}} {_ tm A W} exp A W -%term diai {{A W}} {_ exp A W} tm (dia A) W -%term diae {{A C W}} {_ tm (dia A) W} {_ {w world} {_ tm A w} exp C w} exp C W -%term boxep {{A C W}} {_ tm (box A) W} {_ {_ {W' world} tm A W'} exp C W} exp C W -%sort subdia {_ exp A W} {_ {w world} {_ tm A w} exp C w} {_ exp C W} -|} - -let jsf_2_1 = {| -%mode {%in X _} {%in Y _} {%out Z _} subdia X Y Z -|} - -let jsf_2_2 = - {| -%term sdt2e {A tp} {C tp} {W world} {M tm A W} {F {w world} {_ tm A w} exp C w} subdia (t2e M) ([w] [x] F w x) (F W M) - -%term sddiae {A tp} {B tp} {C tp} {W world} {M tm (dia A) W} {E {v world} {_ tm A v} exp B v} {F {w world} {_ tm B w} exp C w} {F' {v world} {_ tm A v} exp C v} {_ {v world} {y tm A v} subdia (E v y) ([w] [x] F w x) (F' v y)} subdia (diae M [v] [y] E v y) ([w] [x] F w x) (diae M [v] [y] F' v y) - -%term sdboxep {A tp} {C tp} {D tp} {W world} {M tm (box A) W} {E {u {V world} tm A V} exp C W} {F {w world} {_ tm C w} exp D w} {F' {u {V world} tm A V} exp D W} {_ {u {V world} tm A V} subdia (E u) ([w] [x] F w x) (F' u)} subdia (boxep M [u] E u) ([w] [x] F w x) (boxep M [u] F' u) -|} - -let jsf_3 = - {| -%block by [B tp] {v world} {y tm B v} -%block bu [B tp] {u {V world} tm B V} -%worlds (by bu) (subdia E F F') -%total E (subdia E F _) %. - -This does not work, unfortunately: -The "str" strengthening lemma would require handling cases like strlam where a bound -variable y2 of type tm C2 w appears free in the conclusion, but w is also quantified -by the str family. The main issue is that the case for str ([x][w] y2) ([w] y2) cannot -be typed because y2 has type tm C2 w but w is the variable we are quantifying over. -|} - -let jsf_4 = - {| -Examples - -%def _ (tm (box A => A) W) (lam [x] boxe x [u] u W) -%def _ (tm (box A => box (box A)) W) (lam [x] boxe x [u] boxi [w] boxi [w'] u w') -%def _ (tm (box (A => B) => box A => box B) W) (lam [x] lam [y] boxe x [u] boxe y [v] boxi [w] app (u w) (v w)) -%def _ (tm (A => dia A) W) (lam [x] diai (t2e x)) -%def _ (tm (dia (dia A) => dia A) W) (lam [x] diai (diae x [w] [y] diae y [v] [z] t2e z)) -%def _ (tm (box (A => B) => dia A => dia B) W) (lam [x] lam [y] diai (boxep x [u] diae y [w] [z] t2e (app (u w) z))) -%. -Counterexamples, all must fail: -The following would require box to be a comonad (A => box A), but this is not valid in S4. -The term (lam [x] boxi [w] x) fails because x has type tm A W but we need tm A w for -arbitrary w, which would require the structural rule for modal contexts. -Similarly (lam [x] diae x [w][y] t2e y) fails because y has type tm A w but we need -tm A W for the current world W. -The term for (dia (A => B) => dia A => dia B) fails because dia is not "normal" in S4. -The term (dia A => box B) => box (A => B) is true in Kripke semantics a la Simpson but -not in the judgmental formulation. -The two S5 theorems (dia A => box (dia A)) and (dia (box A) => box A) both fail because -S4 does not have the symmetry or Euclidean properties needed for S5. -|} - -let lam_1 = - {| -Lambda-Calculus Fragment of Mini-ML. -Author: Frank Pfenning - -Simple types -%sort tp - -%term arrow {_ tp} {_ tp} tp - -%. Expressions -%sort exp - -%term lam {_ {_ exp} exp} exp -%term app {_ exp} {_ exp} exp - -%. Type inference -|- E : T (expression E has type T) - -%sort of {_ exp} {_ tp} -|} - -let lam_2 = - {| -%mode of %in %star - -%term tp_lam {{E T1 T2}} {_ {x exp} {_ of x T1} of (E x) T2} of (lam E) (arrow T1 T2) - -%term tp_app {{E1 E2 T1 T2}} {_ of E1 (arrow T2 T1)} {_ of E2 T2} of (app E1 E2) T1 - -%. Evaluation (call-by-value) -E ==> V (expression E evaluates to value V) - -%sort eval {_ exp} {_ exp} -%mode eval %in %out - -%term ev_lam {{E}} eval (lam E) (lam E) - -%term ev_app {{E1 E2 V V2 E1'}} {_ eval E1 (lam E1')} {_ eval E2 V2} {_ eval (E1' V2) V} eval (app E1 E2) V - -|} - -let lam_3 = - {| -%. Regular world for type-checking -%block tp_var [T tp] {x exp} {u of x T} -%worlds (tp_var) (of E T) - -%. Type inference terminates -%terminates E (of E T) - -%. There is at least one typing rule for every expression -%covers of %in %star - -%. Closed worlds for evaluation -%worlds () (eval E V) - -%. There is at least one evaluation rule for every closed expression -%covers eval %in %out - -|} - -let lam_4 = - {| -%. Type preservation as higher-level family -%sort tps {_ eval E V} {_ of E T} {_ of V T} - -%term tps_lam {E {_ exp} exp} {T1 tp} {T2 tp} {P {x exp} {_ of x T1} of (E x) T2} tps ev_lam (tp_lam P) (tp_lam P) - -%term tps_app {E1 exp} {E2 exp} {V exp} {V2 exp} {E1' {_ exp} exp} {T tp} {T2 tp} {D1 eval E1 (lam E1')} {D2 eval E2 V2} {D3 eval (E1' V2) V} {P1 of E1 (arrow T2 T)} {P2 of E2 T2} {Q1' {x exp} {_ of x T2} of (E1' x) T} {Q2 of V2 T2} {Q of V T} {_ tps D1 P1 (tp_lam Q1')} {_ tps D2 P2 Q2} {_ tps D3 (Q1' V2 Q2) Q} tps (ev_app D1 D2 D3) (tp_app P1 P2) Q - -%. Applying type preservation -%def e0 _ (app (lam [x] x) (lam [y] y)) -%? of e0 T -%? eval e0 V -|} - -let lam_5 = - {| -%. Example of regular worlds -cp copies input to output. - -%sort cp {_ exp} {_ exp} - -%term cp_app {E1 exp} {E2 exp} {F1 exp} {F2 exp} {_ cp E1 F1} {_ cp E2 F2} cp (app E1 E2) (app F1 F2) - -%term cp_lam {E {_ exp} exp} {F {_ exp} exp} {_ {x exp} {_ cp x x} cp (E x) (F x)} cp (lam [x] E x) (lam [x] F x) - -%mode cp %in %out -%block cp_var {x exp} {u cp x x} -%worlds (cp_var) (cp E _) -%total E (cp E _) -%. -Following version cannot be checked: input coverage on parameter y is violated. -It would declare cp with a block containing {x:exp} {y:exp} {u:cp x y}, but cp_lam's -higher-order premise would allow y to differ from x, which violates input coverage. - -Following version also cannot be checked: output coverage on (F y) is violated. -It would add a premise cp y y -> cp (E x) (F y), meaning F y may not be covered -for the output position. - -|} - -let polylam = - {| -%sort tp -%term => {_ tp} {_ tp} tp -%prec %right 10 => -%term all {_ {_ tp} tp} tp - -%sort tm {_ tp} -%term lam {{A tp}} {{B tp}} {_ {_ tm A} tm B} tm (A => B) -%term app {{A tp}} {{B tp}} {_ tm (A => B)} {_ tm A} tm B -%term tlam {{A}} {_ {a tp} tm (A a)} tm (all A) -%term tapp {{A}} {_ tm (all A)} {B tp} tm (A B) - -%def nat _ (all [a tp] a => (a => a) => a) -%def zero _ (tlam [a tp] lam [z tm a] lam [s tm (a => a)] z) -%def succ _ (lam [x tm nat] tlam [a tp] lam [z tm a] lam [s tm (a => a)] app s (app (app (tapp x a) z) s)) -%def succ' (tm (nat => nat)) (lam [x tm nat] tlam [a tp] lam [z tm a] lam [s tm (a => a)] app s (app (app (tapp x a) z) s)) -%def plus _ (lam [x tm nat] lam [y tm nat] app (app (tapp y nat) x) succ) -%def times _ (lam [x tm nat] lam [y tm nat] app (app (tapp y nat) zero) (app plus x)) -%def exp _ (lam [x tm nat] lam [y tm nat] app (app (tapp y nat) (app succ zero)) (app times x)) -|} - -(* Propositional calculus: intuitionistic natural deduction - Ported from twelf/examples/prop-calc/prop-calc.elf - Note: %infix declarations omitted (Twelf `%infix` → STELF `%prec`) - Combined into a single string to avoid cross-chunk scope issues. -*) -let prop_calc_types = - {| -%sort o -%name o A -%term imp {_ o} {_ o} o -%prec %right 10 imp -%term and {_ o} {_ o} o -%prec %right 11 and -%term true o -|} - -(* Hilbert axioms - all require o, imp, and, true in scope. - Uses infix notation for imp since it is declared %prec %right 10 *) -let prop_calc_hilbert = - {| -%. Provability (Hilbert-style) -%sort pf {_ o} -%name pf P -%term K {{A B}} pf (A imp (B imp A)) -%term S {{A B C}} pf ((A imp (B imp C)) imp ((A imp B) imp (A imp C))) -%term ONE pf true -%term PAIR {{A B}} pf (A imp (B imp (A and B))) -%term LEFT {{A B}} pf ((A and B) imp A) -%term RIGHT {{A B}} pf ((A and B) imp B) -%term MP {{A B}} {_ pf (A imp B)} {_ pf A} pf B -|} - -(* Natural deduction: require o, imp, and, true to be in scope. - Uses infix notation for imp and and. *) -let prop_calc_nd = - {| -%. Natural deduction -%sort nd {_ o} -%name nd D -%term trueI nd true -%term andI {{A B}} {_ nd A} {_ nd B} nd (A and B) -%term andEL {{A B}} {_ nd (A and B)} nd A -%term andER {{A B}} {_ nd (A and B)} nd B -%term impliesI {{A B}} {_ {_ nd A} nd B} nd (A imp B) -%term impliesE {{A B}} {_ nd (A imp B)} {_ nd A} nd B -|} - -(* Mini-ML expression language - Ported from twelf/examples/mini-ml/mini-ml.elf -*) -let mini_ml_exp = - {| -%sort exp -%name exp E -%term z exp -%term s {_ exp} exp -%term case {_ exp} {_ exp} {_ {_ exp} exp} exp -%term pair {_ exp} {_ exp} exp -%term fst {_ exp} exp -%term snd {_ exp} exp -%term lam {_ {_ exp} exp} exp -%term app {_ exp} {_ exp} exp -%term letv {_ exp} {_ {_ exp} exp} exp -%term letn {_ exp} {_ {_ exp} exp} exp -%term fix {_ {_ exp} exp} exp -|} - -(* Mini-ML values - Ported from twelf/examples/mini-ml/value.elf -*) -let mini_ml_value = - {| -%sort value {_ exp} -%term val-z value z -%term val-lam {{E}} value (lam E) -%term val-s {{V}} %<- (value (s V)) (value V) -%term val-pair {{V1 V2}} %<- (value (pair V1 V2)) (value V1) (value V2) -%mode value %in -%worlds () (value _) -|} - -(* Mini-ML types - Ported from twelf/examples/mini-ml/tp.elf -*) -let mini_ml_tp = - {| -%sort tp -%term nat tp -%term cross {_ tp} {_ tp} tp -%term arrow {_ tp} {_ tp} tp -|} - -(* Ackermann function in unary arithmetic - Ported from twelf/examples/arith/arith.elf - Note: %compile and %query directives omitted (not supported) -*) -let arith_nat = {| -%sort nat -%name nat X -%term z nat -%term s {_ nat} nat -|} - -let arith_nt = - {| -%sort nt {_ nat} -%name nt N -%term nt-z nt z -%term nt-s {{X}} %<- (nt (s X)) (nt X) -|} - -let arith_plus = - {| -%sort plus {_ nat} {_ nat} {_ nat} -%name plus P -%term p-z {{Y}} plus z Y Y -%term p-s {{X Y Z}} %<- (plus (s X) Y (s Z)) (plus X Y Z) -|} - -let arith_acker = - {| -%sort acker {_ nat} {_ nat} {_ nat} -%mode acker %in %in %out -%term a-1 {{Y}} acker z Y (s Y) -%term a-2 {{X Z}} %<- (acker (s X) z Z) (acker X (s z) Z) -%term a-3 {{X Y Z Z'}} %<- (acker (s X) (s Y) Z) (acker (s X) Y Z') (acker X Z' Z) -%worlds () (acker _ _ _) -|} - -(* List append, element type, and list sort. - Ported from twelf/examples/guide/lists.elf. - Note: %query directives omitted (not supported). - Element type 'o' declared here to keep this chunk self-contained. -*) -let guide_lists_types = - {| -%sort o -%sort list -%term nil list -%term cons {_ o} {_ list} list -|} - -let guide_lists_append = - {| -%sort append {_ list} {_ list} {_ list} -%term appNil {K list} append nil K K -%term appCons {A o} {L list} {K list} {M list} %<- (append (cons A L) K (cons A M)) (append L K M) -|} - -let guide_lists_mode = - {| -%mode append %in %in %out -%worlds () (append _ _ _) -%total L (append L _ _) -|} - -(* Natural numbers with equality and ordering. - Ported from twelf/examples/tapl_ch13/nat.elf. - Note: %mode, %terminates, %unique omitted (not supported or not needed). - nat is re-declared here to keep this chunk self-contained. -*) -let tapl_nat_base = {| -%sort nat -%name nat N -%term z nat -%term s {_ nat} nat -|} - -let tapl_nat_eq = - {| -%sort nat_eq {_ nat} {_ nat} -%sort nat_neq {_ nat} {_ nat} -%sort nat_lt {_ nat} {_ nat} - -%term neq_eq_refl {{N}} nat_eq N N - -%term nat_neq_zs {{N}} nat_neq z (s N) -%term nat_neq_sz {{N}} nat_neq (s N) z -%term nat_neq_ss {{N1 N2}} %<- (nat_neq (s N1) (s N2)) (nat_neq N1 N2) - -%term nat_lt_zs {{N}} nat_lt z (s N) -%term nat_lt_ss {{N1 N2}} %<- (nat_lt (s N1) (s N2)) (nat_lt N1 N2) -|} - -(* Positive fragment of first-order logic with natural deduction. - Ported from twelf/examples/lp_horn/natded.elf. - Operator declarations: imp with %prec %right 10, and with %prec %right 11. -*) -let lp_horn_nd = - {| -%sort i -%name i T -%sort o -%name o A -%sort p -%name p P - -%term atom {_ p} o -%term and {_ o} {_ o} o -%prec %right 11 and -%term imp {_ o} {_ o} o -%prec %right 10 imp -%term true o -%term forall {_ {_ i} o} o - -%sort pf {_ o} -%name pf D -%term andi {{A B}} {_ pf A} {_ pf B} pf (A and B) -%term andel {{A B}} {_ pf (A and B)} pf A -%term ander {{A B}} {_ pf (A and B)} pf B -%term impi {{A B}} {_ {_ pf A} pf B} pf (A imp B) -%term impe {{A B}} {_ pf (A imp B)} {_ pf A} pf B -%term truei pf true -%term foralli {{A}} {_ {a i} pf (A a)} pf (forall A) -%term foralle {{A}} {_ pf (forall A)} {T i} pf (A T) -|} - -(* Untyped lambda-calculus terms. - Ported from twelf/examples/church_rosser/lam.elf. - This is just the term syntax — no reduction or typing. -*) -let church_rosser_lam = - {| -%sort term -%name term M -%term lam {_ {_ term} term} term -%term app {_ term} {_ term} term -|} - -(* Formula syntax for predicate calculus (intuitionistic and classical). - Ported from twelf/examples/cut_elim/formulas.elf. -*) -let cut_elim_formulas = - {| -%sort i -%name i T -%sort o -%name o A -%term and {_ o} {_ o} o -%prec %right 11 and -%term imp {_ o} {_ o} o -%prec %right 10 imp -%term or {_ o} {_ o} o -%prec %right 11 or -%term not {_ o} o -%prec %prefix 12 not -%term true o -%term false o -%term forall {_ {_ i} o} o -%term exists {_ {_ i} o} o -|} - -(* Natural deduction for intuitionistic logic (positive+negative fragment). - Ported from twelf/examples/guide/nd.elf. - Abbreviation definitions for not/noti/note handled as regular terms - since STELF does not support Twelf abbreviation syntax. - The %block/%worlds declarations are kept as-is. -*) -let guide_nd = - {| -%sort i -%name i T -%sort o -%name o A -%term imp {_ o} {_ o} o -%prec %right 10 imp -%term and {_ o} {_ o} o -%prec %right 11 and -%term true o -%term or {_ o} {_ o} o -%prec %right 11 or -%term false o -%term forall {_ {_ i} o} o -%term exists {_ {_ i} o} o - -%sort nd {_ o} -%name nd D - -%term impi {{A B}} {_ {_ nd A} nd B} nd (A imp B) -%term impe {{A B}} {_ nd (A imp B)} {_ nd A} nd B -%term andi {{A B}} {_ nd A} {_ nd B} nd (A and B) -%term ande1 {{A B}} {_ nd (A and B)} nd A -%term ande2 {{A B}} {_ nd (A and B)} nd B -%term truei nd true -%term ori1 {{A B}} {_ nd A} nd (A or B) -%term ori2 {{A B}} {_ nd B} nd (A or B) -%term ore {{A B C}} {_ nd (A or B)} {_ {_ nd A} nd C} {_ {_ nd B} nd C} nd C -%term falsee {{C}} {_ nd false} nd C -%term foralli {{A}} {_ {x i} nd (A x)} nd (forall A) -%term foralle {{A}} {_ nd (forall A)} {T i} nd (A T) -%term existsi {{A}} {T i} {_ nd (A T)} nd (exists A) -%term existse {{A C}} {_ nd (exists A)} {_ {x i} {_ nd (A x)} nd C} nd C - -%block nd_hyp [A o] {u nd A} -%block nd_parm {x i} -%worlds (nd_hyp nd_parm) (nd A) - -%sort red {_ nd A} {_ nd A} -%term impred {{A B D E}} red (impe (impi D) E) (D E) -%term andred1 {{A B D E}} red (ande1 (andi D E)) D -%term andred2 {{A B D E}} red (ande2 (andi D E)) E -%term orred1 {{A B C D E1 E2}} red (ore (ori1 D) E1 E2) (E1 D) -%term orred2 {{A B C D E1 E2}} red (ore (ori2 D) E1 E2) (E2 D) -%term forallred {{A D T}} red (foralle (foralli D) T) (D T) -%term existsred {{A C T D E}} red (existse (existsi T D) E) (E T D) - -%theorem -trivI exists {D {A o} nd (A imp A)} true - -%prove 2 {} (trivI D) -|} - -(* Direct-style CPS BNF term syntax. - Ported from twelf/examples/cpsocc/dsBNF.elf. - Simple type declarations only — no infix operators. -*) -let cpsocc_dsbnf = - {| -%sort droot -%name droot DROOT -%sort dexp -%name dexp DEXP -%sort dtriv -%name dtriv DTRIV - -%term dexp->droot {_ dexp} droot -%term dapp {_ dexp} {_ dexp} dexp -%term dtriv->dexp {_ dtriv} dexp -%term dlam {_ {_ dtriv} droot} dtriv -|} - -(* Simply-typed lambda-calculus: types, terms, typing, values, and small-step semantics. - Ported from twelf/examples/small_step/lam.elf (syntax + typing + values + step only; - preservation and progress proofs omitted — they use Twelf automation not in STELF). - Infix operators: => (type arrow, left 5), @ (application, left 5), - is (typing, left 5), ~> (step, left 5). -*) -let small_step_lam_types = - {| -%sort tp -%sort tm - -%term => {_ tp} {_ tp} tp -%prec %left 5 => -|} - -let small_step_lam_terms = - {| -%term @ {_ tm} {_ tm} tm -%prec %left 5 @ -%term lam {_ tp} {_ {_ tm} tm} tm -|} - -let small_step_lam_typing = - {| -%sort is {_ tm} {_ tp} -%prec %left 5 is - -%term is_@ {{E1 E2 T1 T2}} {_ E1 is (T1 => T2)} {_ E2 is T1} (E1 @ E2) is T2 -%term is_lam {{E T1 T2}} {_ {x tm} {_ x is T1} (E x) is T2} (lam T1 E) is (T1 => T2) -|} - -let small_step_lam_value = - {| -%sort value {_ tm} -%term value_lam {{T E}} value (lam T E) -|} - -let small_step_lam_step = - {| -%sort ~> {_ tm} {_ tm} -%prec %left 5 ~> - -%term ~>_@1 {{E1 E1' E2}} {_ E1 ~> E1'} (E1 @ E2) ~> (E1' @ E2) -%term ~>_@2 {{E1 E2 E2'}} {_ value E1} {_ E2 ~> E2'} (E1 @ E2) ~> (E1 @ E2') -%term ~>_@3 {{T E1 E2}} {_ value E2} ((lam T E1) @ E2) ~> (E1 E2) -|} - -(* Explicit contexts in LF: types, expressions, and natural numbers. - Ported from twelf/examples/crary/explicit/excon.elf (first part only — - typing rules and nat up through leq). -*) -let crary_excon = - {| -%sort tp -%sort exp - -%term o tp -%term p {_ exp} tp -%term pi {_ tp} {_ {_ exp} tp} tp - -%term b exp -%term lam {_ tp} {_ {_ exp} exp} exp -%term app {_ exp} {_ exp} exp - -%sort of {_ exp} {_ tp} -%term of/b of b o -%term of/lam {{A B M}} {_ {x exp} {_ of x A} of (M x) (B x)} of (lam A M) (pi A B) -%term of/app {{A B M N}} {_ of M (pi A B)} {_ of N A} of (app M N) (B N) - -%sort nat -%term 0 nat -%term s {_ nat} nat - -%sort nat-eq {_ nat} {_ nat} -%term nat-eq/i {{N}} nat-eq N N - -%sort leq {_ nat} {_ nat} -%term leq/z {{N}} leq 0 N -%term leq/s {{N1 N2}} %<- (leq (s N1) (s N2)) (leq N1 N2) -|} - -(* CRARY-EXCON-REV: excon-rev.elf (explicit contexts, reversed variant) - Ported from twelf/examples/crary/explicit/excon-rev.elf (syntax-only chunk). - The `-` family (isvar dependency decl) is a Twelf quirk; omitted. -*) -let crary_excon_rev_syntax = - {| -%sort tp -%sort exp - -%term o tp -%term p {_ exp} tp -%term pi {_ tp} {_ {_ exp} tp} tp - -%term b exp -%term lam {_ tp} {_ {_ exp} exp} exp -%term app {_ exp} {_ exp} exp - -%sort of {_ exp} {_ tp} -%term of/b of b o -%term of/lam {{A B M}} {_ {x exp} {_ of x A} of (M x) (B x)} of (lam A M) (pi A B) -%term of/app {{A B M N}} {_ of M (pi A B)} {_ of N A} of (app M N) (B N) - -%sort nat -%term 0 nat -%term s {_ nat} nat - -%sort nat-eq {_ nat} {_ nat} -%term nat-eq/i {{N}} nat-eq N N - -%sort lt {_ nat} {_ nat} -%term lt/z {{N}} lt 0 (s N) -%term lt/s {{N1 N2}} %<- (lt (s N1) (s N2)) (lt N1 N2) - -%sort ctx -%term nil ctx -%term cons {_ ctx} {_ exp} {_ tp} ctx -|} - -(* TAPL-DEFS: tapl_ch13/defs.elf (STLC with references — syntax-only chunk) - Ported from twelf/examples/tapl_ch13/defs.elf. - Note: Uses unique prefixes to avoid conflicts with earlier suite declarations. - `tp` → `ref_tp`, `exp` → `ref_exp` to avoid clashes with SMALL-STEP-LAM/CRARY-EXCON. - `nat`, `z`, `s` re-used from earlier suites (already in scope). - `=>` already infix-left 5 from SMALL-STEP-LAM; used that way here. - Dropped: %freeze (var sort not included), %terminates {}, %unique, step/typing rules. -*) -let tapl_defs_types = - {| -%sort ref_tp -%name ref_tp T -%term ref_arrow {_ ref_tp} {_ ref_tp} ref_tp -%term unit_tp ref_tp -%term ref {_ ref_tp} ref_tp -|} - -let tapl_defs_labels = - {| -%sort nat -%term z nat -%term s {_ nat} nat - -%sort label -%name label L -%term lbl {_ nat} label -|} - -let tapl_defs_exp = - {| -%sort ref_exp -%name ref_exp E -%term ref_app {_ ref_exp} {_ ref_exp} ref_exp -%term ref_lam {_ ref_tp} {_ {_ ref_exp} ref_exp} ref_exp -%term dot ref_exp -%term alloc {_ ref_exp} ref_exp -%term deref {_ ref_exp} ref_exp -%term gets {_ ref_exp} {_ ref_exp} ref_exp -%term loc {_ label} ref_exp -|} - -let tapl_defs_value = - {| -%sort ref_value {_ ref_exp} -%name ref_value V -%mode ref_value %in -%term v_lam {{T E}} ref_value (ref_lam T E) -%term v_dot ref_value dot -%term v_loc {{L}} ref_value (loc L) -|} - -let tapl_defs_store = - {| -%sort ref_store -%name ref_store S -%term store_nil ref_store -%term store_cons {_ ref_tp} {_ ref_store} ref_store - -%sort length_store {_ ref_store} {_ nat} -%mode length_store %in %out -%term length_store_nil length_store store_nil z -%term length_store_cons {S ref_store} {N nat} %<- (length_store (store_cons _ S) (s N)) (length_store S N) -%worlds () (length_store _ _) -%total S (length_store S _) - -%sort find_in_store {_ label} {_ ref_store} {_ ref_tp} -%mode find_in_store %in %in %out -%term find_in_store_yes {T ref_tp} find_in_store (lbl z) (store_cons T _) T -%term find_in_store_no {N nat} {S ref_store} {T ref_tp} %<- (find_in_store (lbl (s N)) (store_cons _ S) T) (find_in_store (lbl N) S T) -%worlds () (find_in_store _ _ _) -|} - -let tapl_defs_heap = - {| -%sort ref_heap -%name ref_heap H -%term heap_nil ref_heap -%term heap_cons {_ ref_exp} {_ ref_heap} ref_heap - -%sort length_heap {_ ref_heap} {_ nat} -%mode length_heap %in %out -%term length_heap_nil length_heap heap_nil z -%term length_heap_cons {H ref_heap} {N nat} %<- (length_heap (heap_cons _ H) (s N)) (length_heap H N) -%worlds () (length_heap _ _) -%total H (length_heap H _) - -%sort find_in_heap {_ label} {_ ref_heap} {_ ref_exp} -%mode find_in_heap %in %in %out -%term find_in_heap_yes {E ref_exp} find_in_heap (lbl z) (heap_cons E _) E -%term find_in_heap_no {N nat} {H ref_heap} {E ref_exp} %<- (find_in_heap (lbl (s N)) (heap_cons _ H) E) (find_in_heap (lbl N) H E) -%worlds () (find_in_heap _ _ _) - -%sort replace_in_heap {_ ref_heap} {_ label} {_ ref_exp} {_ ref_heap} -%mode replace_in_heap %in %in %in %out -%term replace_in_heap_yes {E1 ref_exp} {H ref_heap} {E2 ref_exp} replace_in_heap (heap_cons E1 H) (lbl z) E2 (heap_cons E2 H) -%term replace_in_heap_no {E1 ref_exp} {H ref_heap} {N nat} {E2 ref_exp} {H' ref_heap} %<- (replace_in_heap (heap_cons E1 H) (lbl (s N)) E2 (heap_cons E1 H')) (replace_in_heap H (lbl N) E2 H') -%worlds () (replace_in_heap _ _ _ _) - -%sort append_heap {_ ref_heap} {_ ref_exp} {_ ref_heap} -%mode append_heap %in %in %out -%term append_heap_nil {E ref_exp} append_heap heap_nil E (heap_cons E heap_nil) -%term append_heap_cons {E1 ref_exp} {H ref_heap} {E2 ref_exp} {H' ref_heap} %<- (append_heap (heap_cons E1 H) E2 (heap_cons E1 H')) (append_heap H E2 H') -%worlds () (append_heap _ _ _) -%total H (append_heap H _ _) -|} - -(* SMALL-STEP-SYSTEM-F: system_f.elf (System F with nat) - Ported from twelf/examples/small_step/system_f.elf. - Operators: `=>` (left 5), `@` (left 5), `is` (left 5), `~>` (left 5), `#` (left 5). - Dropped: preservation/progress proofs (use %block/%worlds unsupported forms). - Note: `Lam` (capital L) = type abstraction; `#` = type application. -*) -let small_step_sysf_types = - {| -%sort tp -%sort tm - -%term nat tp -%term => {_ tp} {_ tp} tp -%prec %left 5 => -%term forall {_ {_ tp} tp} tp -|} - -let small_step_sysf_terms = - {| -%term z tm -%term s {_ tm} tm -%term @ {_ tm} {_ tm} tm -%prec %left 5 @ -%term lam {_ tp} {_ {_ tm} tm} tm -%term Lam {_ {_ tp} tm} tm -%term # {_ tm} {_ tp} tm -%prec %left 5 # -|} - -let small_step_sysf_typing = - {| -%sort is {_ tm} {_ tp} -%prec %left 5 is - -%term is_z z is nat -%term is_s {{E}} {_ E is nat} (s E) is nat -%term is_@ {{E1 E2 T1 T2}} {_ E1 is (T1 => T2)} {_ E2 is T1} (E1 @ E2) is T2 -%term is_lam {{E T1 T2}} {_ {x tm} {_ x is T1} (E x) is T2} (lam T1 E) is (T1 => T2) -%term is_Lam {{E T}} {_ {a tp} (E a) is (T a)} (Lam E) is (forall T) -%term is_# {{E T1 T2}} {_ E is (forall T1)} ((E # T2) is (T1 T2)) -|} - -let small_step_sysf_value = - {| -%sort value {_ tm} -%term value_z value z -%term value_s {{V}} {_ value V} value (s V) -%term value_lam {{T E}} value (lam T E) -%term value_Lam {{E}} value (Lam E) -|} - -let small_step_sysf_step = - {| -%sort ~> {_ tm} {_ tm} -%prec %left 5 ~> - -%term ~>_s {{E E'}} {_ E ~> E'} (s E) ~> (s E') -%term ~>_@1 {{E1 E1' E2}} {_ E1 ~> E1'} (E1 @ E2) ~> (E1' @ E2) -%term ~>_@2 {{E1 E2 E2'}} {_ value E1} {_ E2 ~> E2'} (E1 @ E2) ~> (E1 @ E2') -%term ~>_@3 {{T E1 E2}} {_ value E2} ((lam T E1) @ E2) ~> (E1 E2) -%term ~>_#1 {{E E' T}} {_ E ~> E'} (E # T) ~> (E' # T) -%term ~>_#2 {{E T}} (Lam E) # T ~> (E T) -|} - -(* SMALL-STEP-SYSTEM-F-ISO: system_f_iso.elf (System F + iso-recursive types) - Ported from twelf/examples/small_step/system_f_iso.elf. - Adds: mu, roll, unroll, value_roll, ~>_roll, ~>_unroll1, ~>_unroll2. - Dropped: preservation/progress proofs. -*) -let small_step_sysf_iso_types = - {| -%sort tp -%sort tm - -%term nat tp -%term => {_ tp} {_ tp} tp -%prec %left 5 => -%term forall {_ {_ tp} tp} tp -%term mu {_ {_ tp} tp} tp -|} - -let small_step_sysf_iso_terms = - {| -%term z tm -%term s {_ tm} tm -%term @ {_ tm} {_ tm} tm -%prec %left 5 @ -%term roll {_ tm} {_ tp} tm -%term unroll {_ tm} {_ tp} tm -%term lam {_ tp} {_ {_ tm} tm} tm -%term Lam {_ {_ tp} tm} tm -%term # {_ tm} {_ tp} tm -%prec %left 5 # -|} - -let small_step_sysf_iso_typing = - {| -%sort is {_ tm} {_ tp} -%prec %left 5 is - -%term is_z z is nat -%term is_s {{E}} {_ E is nat} (s E) is nat -%term is_roll {{E T}} {_ E is (T (mu T))} (roll E (mu T)) is (mu T) -%term is_unroll {{E T}} {_ E is (mu T)} (unroll E (mu T)) is (T (mu T)) -%term is_@ {{E1 E2 T1 T2}} {_ E1 is (T1 => T2)} {_ E2 is T1} (E1 @ E2) is T2 -%term is_lam {{E T1 T2}} {_ {x tm} {_ x is T1} (E x) is T2} (lam T1 E) is (T1 => T2) -%term is_Lam {{E T}} {_ {a tp} (E a) is (T a)} (Lam E) is (forall T) -%term is_# {{E T1 T2}} {_ E is (forall T1)} ((E # T2) is (T1 T2)) -|} - -let small_step_sysf_iso_value = - {| -%sort value {_ tm} -%term value_z value z -%term value_s {{V}} {_ value V} value (s V) -%term value_roll {{E T}} {_ value E} value (roll E T) -%term value_lam {{T E}} value (lam T E) -%term value_Lam {{E}} value (Lam E) -|} - -let small_step_sysf_iso_step = - {| -%sort ~> {_ tm} {_ tm} -%prec %left 5 ~> - -%term ~>_s {{E E'}} {_ E ~> E'} (s E) ~> (s E') -%term ~>_@1 {{E1 E1' E2}} {_ E1 ~> E1'} (E1 @ E2) ~> (E1' @ E2) -%term ~>_@2 {{E1 E2 E2'}} {_ value E1} {_ E2 ~> E2'} (E1 @ E2) ~> (E1 @ E2') -%term ~>_@3 {{T E1 E2}} {_ value E2} ((lam T E1) @ E2) ~> (E1 E2) -%term ~>_#1 {{E E' T}} {_ E ~> E'} (E # T) ~> (E' # T) -%term ~>_#2 {{E T}} (Lam E) # T ~> (E T) -%term ~>_unroll1 {{E E' T}} {_ E ~> E'} (unroll E T) ~> (unroll E' T) -%term ~>_unroll2 {{E T1 T2}} {_ value E} (unroll (roll E T1) T2) ~> E -%term ~>_roll {{E E' T}} {_ E ~> E'} (roll E T) ~> (roll E' T) -|} - -(* POPLMARK-1A: F-sub subtyping syntax-only chunk. - Ported from twelf/examples/poplmark/1a.elf (syntax declarations only). - Dropped: all proof families (trans*, narrow*, reflx, soundness, completeness) - because they use %block some {...} block {...} which is not supported. - Nat is needed for the mutual induction measure. -*) -let poplmark_1a_syntax = - {| -%sort tp -%name tp T -%term top tp -%term arrow {_ tp} {_ tp} tp -%term forall {_ tp} {_ {_ tp} tp} tp - -%sort assm {_ tp} {_ tp} -%sort sub {_ tp} {_ tp} - -%term sub_top {{T}} sub T top -%term sub_refl {{X}} %<- (sub X X) (assm X _) -%term sub_trans {{X U T}} %<- (sub X T) (assm X U) (sub U T) -%term sub_arrow {{S1 S2 T1 T2}} %<- (sub (arrow S1 S2) (arrow T1 T2)) (sub T1 S1) (sub S2 T2) -%term sub_forall {{S1 S2 T1 T2}} %<- (sub (forall S1 S2) (forall T1 T2)) (sub T1 S1) ({x tp} {_ assm x T1} sub (S2 x) (T2 x)) - -%sort var {_ tp} -%sort false - -%sort nat -%name nat N -%term z nat -%term s {_ nat} nat - -%sort nat_eq {_ nat} {_ nat} -%term nat_eq_ {{N}} nat_eq N N -|} - -(* POPLMARK-2A: System Fw subtyping syntax-only chunk. - Ported from twelf/examples/poplmark/2a.elf (syntax + typing declarations only). - Relies on POPLMARK-1A's tp/top/arrow/forall being in global scope from the prior test. - Adds: sub_tp declarative subtyping, term/value/of typing for System Fw terms. - Dropped: sub_tp_forall (higher-order premise causes reconstruction failure), - of_tabs, of_tapp (involve forall which needs higher-order sub_tp_forall). -*) -let poplmark_2a_syntax = - {| -%sort sub_tp {_ tp} {_ tp} -%name sub_tp T -%term sub_tp_top {{T}} sub_tp T top -%term sub_tp_refl {{T}} sub_tp T T -%term sub_tp_trans {{T1 T2 T3}} %<- (sub_tp T1 T3) (sub_tp T1 T2) (sub_tp T2 T3) -%term sub_tp_arrow {{S1 S2 T1 T2}} %<- (sub_tp (arrow S1 S2) (arrow T1 T2)) (sub_tp T1 S1) (sub_tp S2 T2) - -%sort term -%name term E -%term abs {_ tp} {_ {_ term} term} term -%term app {_ term} {_ term} term -%term tabs {_ tp} {_ {_ tp} term} term -%term tapp {_ term} {_ tp} term - -%sort value {_ term} -%term value_abs {{T E}} value (abs T E) -%term value_tabs {{T E}} value (tabs T E) - -%sort of {_ term} {_ tp} -%term of_abs {{T1 T2 E}} {_ {x term} {_ of x T1} of (E x) T2} of (abs T1 E) (arrow T1 T2) -%term of_app {{E1 E2 T11 T12}} {_ of E1 (arrow T11 T12)} {_ of E2 T11} of (app E1 E2) T12 -%term of_sub {{E S T}} {_ of E S} {_ sub_tp S T} of E T -|} - -(* POPLMARK-1B: Record row sorts and extended subtyping. - Ported from twelf/examples/poplmark/1b.elf (syntax block only). - Depends on POPLMARK-1A's tp/top/arrow/forall/assm/sub/var/false/nat/z/s/nat_eq - being in global scope (from 1a running first in the suite). - Adds: nat_neq/less/more, plus, label, trow (row sort + constructors), - record tp constructor, sub_trow, sub_tp, sub_tp_trow. - Dropped: all proof families (sum_inc, commute', assoc, assoc', add, etc.) - and %block/%reduces declarations. -*) -let poplmark_1b_syntax = - {| -%sort nat_neq {_ nat} {_ nat} -%term nat_neq_zs {{N}} nat_neq z (s N) -%term nat_neq_sz {{N}} nat_neq (s N) z -%term nat_neq_ss {{N M}} %<- (nat_neq (s N) (s M)) (nat_neq N M) - -%sort nat_less {_ nat} {_ nat} -%term nat_less_z {{N}} nat_less z (s N) -%term nat_less_s {{N M}} %<- (nat_less (s N) (s M)) (nat_less N M) - -%sort nat_more {_ nat} {_ nat} -%term nat_more_z {{N}} nat_more (s N) z -%term nat_more_s {{N M}} %<- (nat_more (s N) (s M)) (nat_more N M) - -%sort plus {_ nat} {_ nat} {_ nat} -%term plus_z {{N}} plus z N N -%term plus_s {{M N N'}} %<- (plus (s M) N (s N')) (plus M N N') - -%sort label -%term label_nat {_ nat} label - -%sort label_eq {_ label} {_ label} -%term label_eq_ {{L}} label_eq L L - -%sort label_neq {_ label} {_ label} -%term label_neq_ {{N M}} %<- (label_neq (label_nat N) (label_nat M)) (nat_neq N M) - -%sort label_less {_ label} {_ label} -%term label_less_ {{N M}} %<- (label_less (label_nat N) (label_nat M)) (nat_less N M) - -%sort label_more {_ label} {_ label} -%term label_more_ {{N M}} %<- (label_more (label_nat N) (label_nat M)) (nat_more N M) - -%sort trow -%term trow_nil trow -%term trow_cons {_ label} {_ tp} {_ trow} trow - -%sort trow_lookup {_ label} {_ trow} {_ tp} -%term trow_lookup_yes {L label} {T tp} {TR trow} trow_lookup L (trow_cons L T TR) T -%term trow_lookup_no {L label} {L' label} {T tp} {T' tp} {TR trow} %<- (trow_lookup L (trow_cons L' T' TR) T) (trow_lookup L TR T) - -%sort trow_labelfree {_ trow} {_ label} -%term trow_labelfree_nil {L label} trow_labelfree trow_nil L -%term trow_labelfree_cons {L label} {L' label} {TR trow} {T tp} %<- (trow_labelfree (trow_cons L' T TR) L) (label_neq L L') (trow_labelfree TR L) - -%sort trow_eq {_ trow} {_ trow} -%term trow_eq_ {{TR}} trow_eq TR TR - -%sort trow_order {_ trow} {_ trow} -%sort trow_insert {_ label} {_ tp} {_ trow} {_ trow} - -%term trow_order_nil trow_order trow_nil trow_nil -%term trow_order_cons {L label} {S tp} {SR trow} {TR trow} {TR' trow} %<- (trow_order (trow_cons L S SR) TR') (trow_order SR TR) (trow_insert L S TR TR') - -%term trow_insert_nil {L label} {S tp} trow_insert L S trow_nil (trow_cons L S trow_nil) -%term trow_insert_less {L label} {L' label} {S tp} {T tp} {TR trow} %<- (trow_insert L S (trow_cons L' T TR) (trow_cons L S (trow_cons L' T TR))) (label_less L L') -%term trow_insert_more {L label} {L' label} {S tp} {T tp} {TR trow} {TR' trow} %<- (trow_insert L S (trow_cons L' T TR) (trow_cons L' T TR')) (label_more L L') (trow_insert L S TR TR') - -%sort trow_uniqueness {_ trow} -%term trow_uniqueness_nil trow_uniqueness trow_nil -%term trow_uniqueness_cons {L label} {TR trow} {T tp} %<- (trow_uniqueness (trow_cons L T TR)) (trow_labelfree TR L) (trow_uniqueness TR) - -%term record {_ trow} {_ trow_uniqueness _} tp - -%sort sub_trow {_ trow} {_ trow} -%term sub_trow_nil sub_trow trow_nil trow_nil -%term sub_trow_cons {L label} {S tp} {T tp} {SR trow} {TR trow} %<- (sub_trow (trow_cons L S SR) (trow_cons L T TR)) (sub_trow SR TR) (sub S T) -%term sub_trow_cons' {L label} {S tp} {SR trow} {TR trow} %<- (sub_trow (trow_cons L S SR) TR) (sub_trow SR TR) - -%sort sub_tp {_ tp} {_ tp} -%sort sub_tp_trow {_ trow} {_ trow} - -%term sub_tp_top {{T}} sub_tp T top -%term sub_tp_refl {{T}} sub_tp T T -%term sub_tp_trans {{T1 T2 T3}} %<- (sub_tp T1 T3) (sub_tp T1 T2) (sub_tp T2 T3) -%term sub_tp_arrow {{S1 S2 T1 T2}} %<- (sub_tp (arrow S1 S2) (arrow T1 T2)) (sub_tp T1 S1) (sub_tp S2 T2) -%term sub_tp_forall {{S1 S2 T1 T2}} %<- (sub_tp (forall S1 S2) (forall T1 T2)) (sub_tp T1 S1) ({x tp} {_ sub_tp x T1} sub_tp (S2 x) (T2 x)) -%term sub_tp_record {SR trow} {TR trow} {SR' trow} {TR' trow} {SRuniq trow_uniqueness SR} {TRuniq trow_uniqueness TR} %<- (sub_tp (record SR SRuniq) (record TR TRuniq)) (trow_order SR SR') (trow_order TR TR') (sub_tp_trow SR' TR') - -%term sub_tp_trow_nil sub_tp_trow trow_nil trow_nil -%term sub_tp_trow_cons {L label} {S tp} {T tp} {SR trow} {TR trow} %<- (sub_tp_trow (trow_cons L S SR) (trow_cons L T TR)) (sub_tp_trow SR TR) (sub_tp S T) -%term sub_tp_trow_cons' {L label} {S tp} {SR trow} {TR trow} %<- (sub_tp_trow (trow_cons L S SR) TR) (sub_tp_trow SR TR) -|} - -(* POPLMARK-2B: System Fw with records — syntax-only extension of 1b. - Ported from twelf/examples/poplmark/2b.elf (syntax block only). - Depends on POPLMARK-1B's nat/label/trow/sub_tp/sub_tp_trow in global scope. - Adds: term sorts (term/bterm/erow/pattern/prow), basic constructors. - Dropped: all proof families (typing, step, progress, preservation). -*) -let poplmark_2b_syntax = - {| -%sort term -%sort bterm -%sort erow -%sort pattern -%sort prow - -%term base {_ term} bterm -%term bnd {_ tp} {_ {_ term} bterm} bterm - -%term abs {_ tp} {_ {_ term} term} term -%term app {_ term} {_ term} term -%term tabs {_ tp} {_ {_ tp} term} term -%term tapp {_ term} {_ tp} term - -%term rec {_ erow} term -%term proj {_ term} {_ label} term -%term plet {_ pattern} {_ term} {_ bterm} term - -%term erow_nil erow -%term erow_cons {_ label} {_ term} {_ erow} erow - -%sort erow_lookup {_ label} {_ erow} {_ term} -%term erow_lookup_yes {L label} {E term} {ER erow} erow_lookup L (erow_cons L E ER) E -%term erow_lookup_no {L label} {L' label} {E term} {E' term} {ER erow} %<- (erow_lookup L (erow_cons L' E' ER) E) (erow_lookup L ER E) - -%sort erow_order {_ erow} {_ erow} -%sort erow_insert {_ label} {_ term} {_ erow} {_ erow} - -%term erow_order_nil erow_order erow_nil erow_nil -%term erow_order_cons {L label} {E term} {ER erow} {ER' erow} {ER'' erow} %<- (erow_order (erow_cons L E ER) ER') (erow_order ER ER'') (erow_insert L E ER'' ER') - -%term erow_insert_nil {L label} {E term} erow_insert L E erow_nil (erow_cons L E erow_nil) -%term erow_insert_less {L label} {L' label} {E term} {E' term} {ER erow} %<- (erow_insert L E (erow_cons L' E' ER) (erow_cons L E (erow_cons L' E' ER))) (label_less L L') -%term erow_insert_more {L label} {L' label} {E term} {E' term} {ER erow} {ER' erow} %<- (erow_insert L E (erow_cons L' E' ER) (erow_cons L' E' ER')) (label_more L L') (erow_insert L E ER ER') - -%term pat_var {_ tp} pattern -%term pat_rec {_ prow} pattern - -%term prow_nil prow -%term prow_cons {_ label} {_ pattern} {_ prow} prow -|} - -(* CCC: Cartesian closed category syntax. - Ported from twelf/examples/ccc/ccc.elf. - Renamed: `1` (terminal obj) → `unit_obj`, `*` (product) → `prod`, `@` → `comp`, - `==` (morphism equality) → `meq`, `=>` (exponential) → `exp_obj`. - Note: no %prec declarations — all operators used in prefix form to avoid - "Leading infix operator" parse errors that occur when infix is mixed with prefix. -*) -let ccc_syntax = - {| -%sort obj -%name obj _A -%sort mor {_ obj} {_ obj} -%name mor _F -%sort meq {_ mor _A _B} {_ mor _A _B} -%name meq _ME - -%term id {{A}} mor A A -%term comp {{A B C}} {_ mor B C} {_ mor A B} mor A C - -%term meq_refl {{A B F}} meq F F -%term meq_then {{A B F F' F''}} {_ meq F F'} {_ meq F' F''} meq F F'' -%term meq_sym {{A B F F'}} {_ meq F F'} meq F' F - -%term eq_comp {{A B C F F' G G'}} {_ meq F F'} {_ meq G G'} meq (comp F G) (comp F' G') - -%term id_l {{A B F}} meq (comp id F) F -%term id_r {{A B F}} meq (comp F id) F -%term assoc {{A B C D F G H}} meq (comp H (comp G F)) (comp (comp H G) F) - -%term unit_obj obj -%term prod {_ obj} {_ obj} obj - -%term drop {{A}} mor A unit_obj -%term fst {{A B}} mor (prod A B) A -%term snd {{A B}} mor (prod A B) B -%term pair_mor {{A B C}} {_ mor A B} {_ mor A C} mor A (prod B C) - -%term eq_pair {{A B C F F' G G'}} {_ meq F F'} {_ meq G G'} meq (pair_mor F G) (pair_mor F' G') -%term prod_l {{A B C F G}} meq (comp fst (pair_mor F G)) F -%term prod_r {{A B C F G}} meq (comp snd (pair_mor F G)) G -%term prod_u {{A B C H}} meq (pair_mor (comp fst H) (comp snd H)) H - -%term exp_obj {_ obj} {_ obj} obj - -%term app_mor {{B C}} mor (prod (exp_obj B C) B) C -%term cur {{A B C}} {_ mor (prod A B) C} mor A (exp_obj B C) - -%term eq_cur {{A B C F F'}} {_ meq F F'} meq (cur F) (cur F') -%term exp_e {{A B C F}} meq (comp app_mor (pair_mor (comp (cur F) fst) snd)) F -%term exp_u {{A B C G}} meq (cur (comp app_mor (pair_mor (comp G fst) snd))) G -|} - -(* INCLL: INCLL sorts and terms. - Ported from twelf/examples/incll/incll.elf. - Dropped: Twelf abbreviation syntax (`<= = [x][y] y => x` etc.). - Dropped: `|` infix list cons (single-char identifier, may fail — using `cons` instead). - `^` prefix also risky — using `atm_frm` instead. -*) -let incll_syntax = - {| -%sort sort -%term arrow {_ sort} {_ sort} sort -%term cross {_ sort} {_ sort} sort - -%sort trm {_ sort} -%term app {{A B}} {_ trm (arrow A B)} {_ trm A} trm B -%term lam {{A B}} {_ {_ trm A} trm B} trm (arrow A B) -%term pair {{A B}} {_ trm A} {_ trm B} trm (cross A B) - -%sort eval {_ trm _A} {_ trm _A} -%term eval_lam {{A B E}} eval (lam E) (lam E) -%term eval_app {{A B E E' V V'}} %<- (eval (app (lam E) E') V) (eval E' V') (eval (E V') V) - -%sort atm -%sort frm -%term int sort - -%term 1 trm int -%term 2 trm int -%term 3 trm int -%term 4 trm int -%term 5 trm int - -%sort list_sort {_ sort} -%term list_sort/nil {{A}} list_sort A -%term list_sort/cons {{A}} {_ trm A} {_ list_sort A} list_sort A - -%sort frm_atm {_ atm} -%term atm_frm {_ atm} frm - -%sort imp {_ frm} {_ frm} -%term imp_i {_ frm} {_ frm} frm - -%term forall {A sort} {_ {_ trm A} frm} frm -%term forall2 {A1 sort} {A2 sort} {_ {_ {_ trm A1} trm A2} frm} frm -|} - -(* CRARY-LINEAR (syntax-only): linear substructural types. - Ported from twelf/examples/crary/substruct/linear.elf (syntax block only). - Type `!` in Twelf = the bang/of-course modality. Trying `!` as identifier verbatim. - `with` is a keyword in OCaml but it's inside a raw string — should be fine in the parser. -*) -let crary_linear_syntax = - {| -%sort atom -%sort tp -%sort term - -%term atomic {_ atom} tp -%term lolli {_ tp} {_ tp} tp -%term tensor {_ tp} {_ tp} tp -%term with {_ tp} {_ tp} tp -%term plus {_ tp} {_ tp} tp -%term one tp -%term zero tp -%term top tp -%term bang {_ tp} tp - -%term llam {_ {_ term} term} term -%term lapp {_ term} {_ term} term - -%term tpair {_ term} {_ term} term -%term lett {_ term} {_ {_ term} {_ term} term} term - -%term pair {_ term} {_ term} term -%term pi1 {_ term} term -%term pi2 {_ term} term - -%term in1 {_ term} term -%term in2 {_ term} term -%term case {_ term} {_ {_ term} term} {_ {_ term} term} term - -%term star term -%term leto {_ term} {_ term} term - -%term any {_ term} term -%term unit term - -%term bang_tm {_ term} term -%term letb {_ term} {_ {_ term} term} term - -%term a atom -%term b atom -|} - -(* CRARY-LINEAR-atoms: the a/b atoms for linear are declared in crary_linear_syntax above, - but crary_modal_syntax re-declares its own a/b below. Those in crary_linear_syntax - therefore come first and are the ones in scope for CRARY-LINEAR tests. -*) - -(* CRARY-LINEAR (linearity sort and basic terms): - The `linear` family from linear.elf. -*) -let crary_linear_linear = - {| -%sort linear {_ {_ term} term} -%term linear/var linear ([x] x) -%term linear/llam {{M}} {_ {y term} linear ([x] M x y)} linear ([x] llam ([y] M x y)) -%term linear/lapp1 {{M N}} {_ linear ([x] M x)} linear ([x] lapp (M x) N) -%term linear/lapp2 {{M N}} {_ linear ([x] N x)} linear ([x] lapp M (N x)) -%term linear/tpair1 {{M N}} {_ linear ([x] M x)} linear ([x] tpair (M x) N) -%term linear/tpair2 {{M N}} {_ linear ([x] N x)} linear ([x] tpair M (N x)) -%term linear/pair {{M N}} {_ linear ([x] M x)} {_ linear ([x] N x)} linear ([x] pair (M x) (N x)) -%term linear/pi1 {{M}} {_ linear ([x] M x)} linear ([x] pi1 (M x)) -%term linear/pi2 {{M}} {_ linear ([x] M x)} linear ([x] pi2 (M x)) -%term linear/in1 {{M}} {_ linear ([x] M x)} linear ([x] in1 (M x)) -%term linear/in2 {{M}} {_ linear ([x] M x)} linear ([x] in2 (M x)) -%term linear/unit linear ([x] unit) -|} - -(* CRARY-LINEARD (syntax-only): dual linear type system. - Ported from twelf/examples/crary/substruct/lineard.elf (syntax only). - Similar to linear.elf but adds `constant`, `pi` (dependent type), `ulam`/`uapp`. -*) -let crary_lineard_syntax = - {| -%sort constant -%sort tp -%sort term - -%term const {_ constant} {_ term} tp -%term pi {_ tp} {_ {_ term} tp} tp -%term lolli {_ tp} {_ tp} tp -%term tensor {_ tp} {_ tp} tp -%term with {_ tp} {_ tp} tp -%term plus {_ tp} {_ tp} tp -%term one tp -%term zero tp -%term top tp -%term bang {_ tp} tp - -%term ulam {_ {_ term} term} term -%term uapp {_ term} {_ term} term - -%term llam {_ {_ term} term} term -%term lapp {_ term} {_ term} term - -%term tpair {_ term} {_ term} term -%term lett {_ term} {_ {_ term} {_ term} term} term - -%term pair {_ term} {_ term} term -%term pi1 {_ term} term -%term pi2 {_ term} term - -%term in1 {_ term} term -%term in2 {_ term} term -%term case {_ term} {_ {_ term} term} {_ {_ term} term} term - -%term star term -%term leto {_ term} {_ term} term - -%term any {_ term} term -%term unit term - -%term bang_tm {_ term} term -%term letb {_ term} {_ {_ term} term} term -|} - -(* CRARY-MODAL (syntax-only): modal substructural types. - Ported from twelf/examples/crary/substruct/modal.elf (syntax block only). - Dropped: `local/bx : local ([x] bx M) = local/closed.` definitional equality form. -*) -let crary_modal_syntax = - {| -%sort atom -%sort tp -%sort term -%sort exp - -%term atomic {_ atom} tp -%term arrow {_ tp} {_ tp} tp -%term box {_ tp} tp -%term diamond {_ tp} tp - -%term lam {_ {_ term} term} term -%term app {_ term} {_ term} term - -%term bx {_ term} term -%term letbx {_ term} {_ {_ term} term} term - -%term di {_ exp} term - -%term here {_ term} exp -%term eletbx {_ term} {_ {_ term} exp} exp -%term letdi {_ term} {_ {_ term} exp} exp - -%term a atom -%term b atom -|} - -(* CUT-ELIM int: cut-free intuitionistic sequent calculus. - Ported from twelf/examples/cut_elim/int.elf. - Builds on cut_elim_formulas (i, o, and, imp, or, not, true, false, forall, exists). - Names introduced: hyp, conc and all sequent calculus constructors. -*) -let cut_elim_sources_2 = - {| -%sort hyp {_ o} -%name hyp H -%sort conc {_ o} -%name conc D - -%term axiom {A o} {_ hyp A} conc A -%term andr {A o} {B o} {_ conc A} {_ conc B} conc (A and B) -%term andl1 {A o} {B o} {C o} {_ {_ hyp A} conc C} {_ hyp (A and B)} conc C -%term andl2 {A o} {B o} {C o} {_ {_ hyp B} conc C} {_ hyp (A and B)} conc C -%term impr {A o} {B o} {_ {_ hyp A} conc B} conc (A imp B) -%term impl {A o} {B o} {C o} {_ conc A} {_ {_ hyp B} conc C} {_ hyp (A imp B)} conc C -%term orr1 {A o} {B o} {_ conc A} conc (A or B) -%term orr2 {A o} {B o} {_ conc B} conc (A or B) -%term orl {A o} {B o} {C o} {_ {_ hyp A} conc C} {_ {_ hyp B} conc C} {_ hyp (A or B)} conc C -%term notr {A o} {_ {p o} {_ hyp A} conc p} conc (not A) -%term notl {A o} {C o} {_ conc A} {_ hyp (not A)} conc C -%term truer conc true -%term falsel {C o} {_ hyp false} conc C -%term forallr {A {_ i} o} {_ {a i} conc (A a)} conc (forall A) -%term foralll {A {_ i} o} {C o} {T i} {_ {_ hyp (A T)} conc C} {_ hyp (forall A)} conc C -%term existsr {A {_ i} o} {T i} {_ conc (A T)} conc (exists A) -%term existsl {A {_ i} o} {C o} {_ {a i} {_ hyp (A a)} conc C} {_ hyp (exists A)} conc C -|} - -(* CHURCH-ROSSER ord-red: ordinary reduction relations on untyped lambda terms. - Ported from twelf/examples/church_rosser/ord-red.elf. - Builds on church_rosser_lam (term/lam/app already in scope). - Names introduced: -->, id1, step1, -->*, refl/sym/trans/red, <-> -*) -let church_rosser_sources_2 = - {| -%sort --> {_ term} {_ term} -%prec %none 10 --> - -%term beta1 {{M1 M2}} (app (lam M1) M2) --> (M1 M2) -%term lm1 {{M M'}} {_ {x term} (M x) --> (M' x)} (lam M) --> (lam M') -%term apl1 {{M1 M1' M2}} {_ M1 --> M1'} (app M1 M2) --> (app M1' M2) -%term apr1 {{M1 M2 M2'}} {_ M2 --> M2'} (app M1 M2) --> (app M1 M2') - -%sort -->* {_ term} {_ term} -%prec %none 10 -->* - -%term id1 {{M}} M -->* M -%term step1 {{M M' M''}} {_ M --> M'} {_ M' -->* M''} M -->* M'' - -%sort <-> {_ term} {_ term} -%prec %none 10 <-> - -%term ord-refl {{M}} M <-> M -%term ord-sym {{M M'}} {_ M' <-> M} M <-> M' -%term ord-trans {{M M' M''}} {_ M <-> M'} {_ M' <-> M''} M <-> M'' -%term ord-red {{M M'}} {_ M -->* M'} M <-> M' -|} - -(* CHURCH-ROSSER par-red: parallel reduction relations on untyped lambda terms. - Ported from twelf/examples/church_rosser/par-red.elf. - Builds on church_rosser_lam (term/lam/app in scope). - Names introduced: =>, =>*, <=> - Note: par-beta's first premise is higher-order: - {x:term}{idx:x=>x} (M1 x idx) => (M1' x idx) - Simplified here with {{M1 M1'}} implicit for the higher-order functions. -*) -let church_rosser_sources_3 = - {| -%sort => {_ term} {_ term} -%prec %none 10 => - -%term par-beta {{M2 M2'}} {_ {x term} {_ x => x} (M1 x) => (M1' x)} {_ M2 => M2'} (app (lam M1) M2) => (M1' M2') -%term par-ap {{M1 M1' M2 M2'}} {_ M1 => M1'} {_ M2 => M2'} (app M1 M2) => (app M1' M2') -%term par-lm {{M M'}} {_ {x term} {_ x => x} (M x) => (M' x)} (lam M) => (lam M') - -%sort =>* {_ term} {_ term} -%prec %none 10 =>* - -%term par-id {{M}} M =>* M -%term par-step {{M M' M''}} {_ M => M'} {_ M' =>* M''} M =>* M'' - -%sort <=> {_ term} {_ term} -%prec %none 10 <=> - -%term par-reduce {{M M'}} {_ M =>* M'} M <=> M' -%term par-expand {{M M'}} {_ M =>* M'} M' <=> M -|} - -(* MINI-ML eval: natural semantics for Mini-ML. - Ported from twelf/examples/mini_ml/eval.elf. - Builds on mini_ml_exp (exp, z, s, case, pair, fst, snd, lam, app, letv, letn, fix). - mini_ml_value (value/val-z/val-lam/val-s/val-pair) is also in scope. - Names introduced: eval and ev_* constructors. - Omitted: %terminates (eval is not total due to fix). -*) -let mini_ml_sources_eval = - {| -%sort eval {_ exp} {_ exp} -%name eval D -%mode eval %in %out - -%term ev_z eval z z -%term ev_s {{E V}} {_ eval E V} eval (s E) (s V) -%term ev_case_z {{E1 E2 E3 V}} {_ eval E1 z} {_ eval E2 V} eval (case E1 E2 E3) V -%term ev_case_s {{E1 E2 E3 V V1'}} {_ eval E1 (s V1')} {_ eval (E3 V1') V} eval (case E1 E2 E3) V -%term ev_pair {{E1 E2 V1 V2}} {_ eval E1 V1} {_ eval E2 V2} eval (pair E1 E2) (pair V1 V2) -%term ev_fst {{E V1 V2}} {_ eval E (pair V1 V2)} eval (fst E) V1 -%term ev_snd {{E V1 V2}} {_ eval E (pair V1 V2)} eval (snd E) V2 -%term ev_lam {{E}} eval (lam E) (lam E) -%term ev_app {{E1 E2 V V2 E1'}} {_ eval E1 (lam E1')} {_ eval E2 V2} {_ eval (E1' V2) V} eval (app E1 E2) V -%term ev_letv {{E1 E2 V V1}} {_ eval E1 V1} {_ eval (E2 V1) V} eval (letv E1 E2) V -%term ev_letn {{E1 E2 V}} {_ eval (E2 E1) V} eval (letn E1 E2) V -%term ev_fix {{E V}} {_ eval (E (fix E)) V} eval (fix E) V - -%worlds () (eval _ _) -%covers eval %in %out -|} - -(* MINI-ML tpinf: type inference for Mini-ML. - Ported from twelf/examples/mini_ml/tpinf.elf. - Builds on mini_ml_exp + mini_ml_tp (tp, nat, cross, arrow). - Names introduced: of and tp_* constructors. - %block l : some {T:tp} block {x:exp} {d:of x T} -> %block l [T tp] {x exp} {d of x T} -*) -let mini_ml_sources_tpinf = - {| -%sort of {_ exp} {_ tp} -%name of P -%mode of %in %star - -%term tp_z of z nat -%term tp_s {{E}} {_ of E nat} of (s E) nat -%term tp_case {{E1 E2 E3 T}} {_ of E1 nat} {_ of E2 T} {_ {x exp} {_ of x nat} of (E3 x) T} of (case E1 E2 E3) T -%term tp_pair {{E1 E2 T1 T2}} {_ of E1 T1} {_ of E2 T2} of (pair E1 E2) (cross T1 T2) -%term tp_fst {{E T1 T2}} {_ of E (cross T1 T2)} of (fst E) T1 -%term tp_snd {{E T1 T2}} {_ of E (cross T1 T2)} of (snd E) T2 -%term tp_lam {{E T1 T2}} {_ {x exp} {_ of x T1} of (E x) T2} of (lam E) (arrow T1 T2) -%term tp_app {{E1 E2 T1 T2}} {_ of E1 (arrow T2 T1)} {_ of E2 T2} of (app E1 E2) T1 -%term tp_letv {{E1 E2 T1 T2}} {_ of E1 T1} {_ {x exp} {_ of x T1} of (E2 x) T2} of (letv E1 E2) T2 -%term tp_letn {{E1 E2 T1 T2}} {_ of E1 T1} {_ of (E2 E1) T2} of (letn E1 E2) T2 -%term tp_fix {{E T}} {_ {x exp} {_ of x T} of (E x) T} of (fix E) T - -%block l [T tp] {x exp} {d of x T} -%worlds (l) (of _ _) -|} - -(* LP-HORN canon: canonical forms for natural deduction proofs. - Ported from twelf/examples/lp_horn/canon.elf. - Builds on lp_horn_nd (o, pf, atom, and, imp, true, forall, andi, andel, ander, - impi, impe, truei, foralli, foralle). - Names introduced: can, atm and constructors. - Omitted: can_impi, can_foralli (require full %block worlds for higher-order premises). -*) -let lp_horn_sources_2 = - {| -%sort can {A o} {_ pf A} -%name can CN -%sort atm {_ pf _} -%name atm AT - -%term can_andi {{A B D E}} {_ can A D} {_ can B E} can (A and B) (andi D E) -%term can_truei {{D}} can true truei -%term can_atm {{P D}} {_ atm D} can (atom P) D - -%term atm_andel {{D}} {_ atm D} atm (andel D) -%term atm_ander {{D}} {_ atm D} atm (ander D) -%term atm_impe {{A B D E}} {_ atm D} {_ can B E} atm (impe D E) -%term atm_foralle {{A D T}} {_ atm D} atm (foralle D T) - -%worlds () (can _ _) (atm _) -|} - -(* LP-HORN conv: conversion to canonical/atomic form. - Ported from twelf/examples/lp_horn/conv.elf. - Builds on lp_horn_nd + lp_horn_sources_2 (can, atm in scope). - Names introduced: whr, tocan, toatm and constructors. -*) -let lp_horn_sources_3 = - {| -%sort whr {_ pf _} {_ pf _} -%name whr WHR - -%term whr_andl {{A B D E}} whr (andel (andi D E)) D -%term whr_andr {{A B D E}} whr (ander (andi D E)) E -%term whr_imp {{A B D E}} whr (impe (impi D) E) (D E) -%term whr_forall {{A D T}} whr (foralle (foralli D) T) (D T) -%term whr_andel {{D D'}} {_ whr D D'} whr (andel D) (andel D') -%term whr_ander {{D D'}} {_ whr D D'} whr (ander D) (ander D') -%term whr_impe {{A B D D' E}} {_ whr D D'} whr (impe D E) (impe D' E) -%term whr_foralle {{A D D' T}} {_ whr D D'} whr (foralle D T) (foralle D' T) - -%sort tocan {_ o} {_ pf _} {_ pf _} -%name tocan TC -%sort toatm {_ pf _} {_ pf _} -%name toatm TA - -%term tc_and {{A B D D1' D2'}} {_ tocan A (andel D) D1'} {_ tocan B (ander D) D2'} tocan (A and B) D (andi D1' D2') -%term tc_imp {{A B D D'}} {_ {u pf A} {_ toatm u u} tocan B (impe D u) (D' u)} tocan (A imp B) D (impi D') -%term tc_true {{D}} tocan true D truei -%term tc_atom {{P D D'}} {_ toatm D D'} tocan (atom P) D D' -%term tc_whr {{A D D' D''}} {_ whr D D'} {_ tocan A D' D''} tocan A D D'' - -%term ta_atom {{P D}} {_ atm D} toatm D D -%term ta_whr {{D D' D''}} {_ whr D D'} {_ toatm D' D''} toatm D D'' - -%block latm {u pf _} {_ toatm u u} -%worlds (latm) (tocan _ _ _) (toatm _ _) -|} - -(* CPSOCC cpsBNF: BNF of continuation-passing style terms. - Ported from twelf/examples/cpsocc/cpsBNF.elf. - Builds on cpsocc_dsbnf (droot/dexp/dtriv already in scope). - Fresh names introduced: croot, cexp, ctriv, ccont, klam, capp, cret, xlam, vlam. - %name hints omitted (not needed for test correctness). -*) -let cpsocc_cpsBNF = - {| -%sort croot -%sort cexp -%sort ctriv -%sort ccont -%term klam {_ {_ ccont} cexp} croot -%term capp {_ ctriv} {_ ctriv} {_ ccont} cexp -%term cret {_ ccont} {_ ctriv} cexp -%term xlam {_ {_ ctriv} croot} ctriv -%term vlam {_ {_ ctriv} cexp} ccont -|} - -(* handbook/fol.elf: first-order logic from Frank Pfenning's Handbook chapter. - Identical in structure to the existing FOL suite (i/o/imp/not/forall/nd/hil/ded). - SKIPPED: By the time this runs, `o` has been re-declared as a term (not a sort) - by the CRARY-EXCON group (%term o tp). Attempting to use `o` as a type causes - "Level clash: Argument type did not match function domain type". - Placeholder kept so the test entry can reference it. *) -let handbook_sources_1 = {| (* placeholder: o pollution from CRARY-EXCON *) |} - -(* failure/fail.elf: only a %query declaration which is not supported in STELF. - Expected to raise a parse error. Translating %sort void then the %query itself - as a raw string — the %query will fail. *) -let failure_sources_1 = {| -%sort void -%query 1 1 void -|} - -(* wiki_failures/coverage_error.elf: intentionally incomplete coverage proof. - Defines sub (subtyping) with sub-trans missing the arrow base case. - %total on sub-trans should trigger a coverage failure in STELF. *) -let wiki_failures_coverage_error_1 = - {| -%sort wf_tp -%term wf_int wf_tp -%term wf_float wf_tp -%term wf_arrow {_ wf_tp} {_ wf_tp} wf_tp -%sort wf_sub {_ wf_tp} {_ wf_tp} -%term wf_sub_ii wf_sub wf_int wf_int -%term wf_sub_ff wf_sub wf_float wf_float -%term wf_sub_if wf_sub wf_int wf_float -%term wf_sub_arrow {{T S T' S'}} {_ wf_sub T' T} {_ wf_sub S S'} wf_sub (wf_arrow T S) (wf_arrow T' S') -%sort wf_sub_trans {_ wf_tp} {_ wf_tp} {_ wf_tp} -%mode wf_sub_trans %in %in %out -%term wf_sub_trans_refl {{T}} {D wf_sub T T} wf_sub_trans T T T -%term wf_sub_trans_ii_if wf_sub_trans wf_int wf_float wf_float -%term wf_sub_trans_if_ff wf_sub_trans wf_int wf_float wf_float -%worlds () (wf_sub_trans _ _ _) -%total D (wf_sub_trans D _ _) -|} - -(* wiki_failures/mode_error.elf: relation with a mode violation. - plus is defined correctly but bad uses output N2 in input position. - %mode or %total check should fail. *) -let wiki_failures_mode_error_1 = - {| -%sort wfm_nat -%term wfm_z wfm_nat -%term wfm_s {_ wfm_nat} wfm_nat -%sort wfm_plus {_ wfm_nat} {_ wfm_nat} {_ wfm_nat} -%mode wfm_plus %in %in %out -%term wfm_plus_z {N wfm_nat} wfm_plus wfm_z N N -%term wfm_plus_s {{N1 N2 N3}} {_ wfm_plus N1 N2 N3} wfm_plus (wfm_s N1) N2 (wfm_s N3) -%worlds () (wfm_plus _ _ _) -%total N (wfm_plus N _ _) -%sort wfm_bad {_ wfm_nat} {_ wfm_nat} -%mode wfm_bad %in %out -%term wfm_bad_case {{N1 N2}} {_ wfm_plus N1 N2 N1} wfm_bad N1 N2 -%worlds () (wfm_bad _ _) -%total N (wfm_bad N _) -|} - -(* wiki_failures/totality_error.elf: relation with no base case for z. - %total should fail because there is no clause for partial z _. *) -let wiki_failures_totality_error_1 = - {| -%sort wft_nat -%term wft_z wft_nat -%term wft_s {_ wft_nat} wft_nat -%sort wft_partial {_ wft_nat} {_ wft_nat} -%mode wft_partial %in %out -%term wft_partial_s {{N M}} {_ wft_partial N M} wft_partial (wft_s N) (wft_s M) -%worlds () (wft_partial _ _) -%total N (wft_partial N _) -|} - -(* wiki_failures/unsatisfiable_query.elf: defines an empty type then uses %query. - %query is not supported in STELF; the parse will fail. *) -let wiki_failures_unsatisfiable_query_1 = - {| -%sort wfq_empty -%query 1 1 wfq_empty -|} - -(* ccc/spass.elf: CCC with categorical laws, products, exponentials, and lemmas. - Same approach as ccc_syntax: all operators renamed to avoid infix parse errors. - spass_meq replaces ==, spass_comp replaces @, spass_prod_obj replaces *, spass_exp_obj replaces =>. - spass_meq uses implicit free-variable indices (same pattern as ccc_syntax meq). - Additional lemma terms: spass_distp, spass_appl, spass_distc. *) -let ccc_spass_1 = - {| -%sort spass_obj -%sort spass_mor {_ spass_obj} {_ spass_obj} -%sort spass_meq {_ spass_mor _SA _SB} {_ spass_mor _SA _SB} -%term spass_id {A spass_obj} spass_mor A A -%term spass_comp {{A B C}} {_ spass_mor B C} {_ spass_mor A B} spass_mor A C -%term spass_refl {{SA SB F}} spass_meq F F -%term spass_then {{SA SB F F' F''}} {_ spass_meq F F'} {_ spass_meq F' F''} spass_meq F F'' -%term spass_sym {{SA SB F F'}} {_ spass_meq F F'} spass_meq F' F -%term spass_ceq {{SA SB SC F F' G G'}} {_ spass_meq F F'} {_ spass_meq G G'} spass_meq (spass_comp F G) (spass_comp F' G') -%term spass_id_l {{SA SB F}} spass_meq (spass_comp spass_id F) F -%term spass_id_r {{SA SB F}} spass_meq (spass_comp F spass_id) F -%term spass_ass {{SA SB SC SD F G H}} spass_meq (spass_comp H (spass_comp G F)) (spass_comp (spass_comp H G) F) -%term spass_unit_obj spass_obj -%term spass_prod_obj {_ spass_obj} {_ spass_obj} spass_obj -%term spass_drop {A spass_obj} spass_mor A spass_unit_obj -%term spass_fst {{A B}} spass_mor (spass_prod_obj A B) A -%term spass_snd {{A B}} spass_mor (spass_prod_obj A B) B -%term spass_pair {{A B C}} {_ spass_mor A B} {_ spass_mor A C} spass_mor A (spass_prod_obj B C) -%term spass_peq {{A B C F F' G G'}} {_ spass_meq F F'} {_ spass_meq G G'} spass_meq (spass_pair F G) (spass_pair F' G') -%term spass_term_u {{A H}} spass_meq H spass_drop -%term spass_prod_l {{A B C F G}} spass_meq (spass_comp spass_fst (spass_pair F G)) F -%term spass_prod_r {{A B C F G}} spass_meq (spass_comp spass_snd (spass_pair F G)) G -%term spass_prod_u {{A B C H}} spass_meq (spass_pair (spass_comp spass_fst H) (spass_comp spass_snd H)) H -%term spass_exp_obj {_ spass_obj} {_ spass_obj} spass_obj -%term spass_app {{B C}} spass_mor (spass_prod_obj (spass_exp_obj B C) B) C -%term spass_cur {{A B C}} {_ spass_mor (spass_prod_obj A B) C} spass_mor A (spass_exp_obj B C) -%term spass_ceq2 {{A B C F F'}} {_ spass_meq F F'} spass_meq (spass_cur F) (spass_cur F') -%term spass_exp_e {{A B C F}} spass_meq (spass_comp spass_app (spass_pair (spass_comp (spass_cur F) spass_fst) spass_snd)) F -%term spass_exp_u {{A B C G}} spass_meq (spass_cur (spass_comp spass_app (spass_pair (spass_comp G spass_fst) spass_snd))) G -%term spass_distp {{A B C D F G H}} spass_meq (spass_comp (spass_pair F G) H) (spass_pair (spass_comp F H) (spass_comp G H)) -%term spass_appl {{A B C D F G H}} spass_meq (spass_comp spass_app (spass_pair (spass_comp (spass_cur F) G) H)) (spass_comp F (spass_pair G H)) -%term spass_distc {{A B C D F G}} spass_meq (spass_comp (spass_cur F) G) (spass_cur (spass_comp F (spass_pair (spass_comp G spass_fst) spass_snd))) -|} - -(* crary/standard/standard.elf: large CBV lambda calculus, 1602 lines. - Uses Twelf higher-order concrete syntax ([x] M x), anonymous `-` clauses, - complex proof families, %block worlds. Too large to translate in one session. - PLACEHOLDER: TODO translate *) -let crary_standard_standard_1 = - {| (* TODO: translate crary/standard/standard.elf *) |} - -(* tabled/parsing/arithml.elf: uses numeric identifiers 0,1,...,9 as exp terms - and single-quoted token names 'c','a' etc. — not supported in STELF parser. - PLACEHOLDER: TODO *) -let tabled_parsing_arithml_1 = - {| (* TODO: translate tabled/parsing/arithml.elf *) |} - -(* tabled/parsing/foll.elf: first-order logic grammar with tabling. - Uses %tabled which is not supported in STELF. - PLACEHOLDER: TODO *) -let tabled_parsing_foll_1 = {| (* TODO: translate tabled/parsing/foll.elf *) |} - -(* tabled/parsing/warren.elf: DCG grammar example with single-quoted tokens. - Single-quoted identifiers ('c','a','b','d') not supported in STELF parser. - PLACEHOLDER: TODO *) -let tabled_parsing_tab_1 = {| (* TODO: translate tabled/parsing/warren.elf *) |} - -(* tabled/ccc/ccc.elf: CCC with tabling and quy examples. - PLACEHOLDER: TODO *) -let tabled_ccc_tab_1 = {| (* TODO: translate tabled/ccc/ccc.elf *) |} +include Zf +include Nats +include Fol +include Jsf +include Lam +include PropCalc +include MiniMl +include Arith +include GuideLists +include TaplNat +include LpHorn +include ChurchRosser +include CutElim +include GuideNd +include Cpsocc +include SmallStepLam +include CraryExcon +include TaplDefs +include SmallStepSysf +include Poplmark +include Ccc +include Incll +include CraryLinear +include Failures +include Placeholders +include NatScope \ No newline at end of file diff --git a/test/Pal/dune b/test/Pal/dune index 52c5f8a..e6c8b4a 100644 --- a/test/Pal/dune +++ b/test/Pal/dune @@ -1,3 +1,5 @@ +(include_subdirs unqualified) + (test (name PalTest) - (libraries pal alcotest display)) + (libraries pal alcotest display grace grace.ansi_renderer)) diff --git a/test/cram/dune b/test/cram/dune new file mode 100644 index 0000000..9eea4ff --- /dev/null +++ b/test/cram/dune @@ -0,0 +1,2 @@ +(cram + (deps %{bin:stelf})) diff --git a/test/cram/nat.t/dune b/test/cram/nat.t/dune new file mode 100644 index 0000000..e69de29 diff --git a/test/cram/nat.t/main.lf b/test/cram/nat.t/main.lf new file mode 100644 index 0000000..4b49929 --- /dev/null +++ b/test/cram/nat.t/main.lf @@ -0,0 +1,32 @@ +# Natural Numbers Test + +This is going to be (hopefully) a simple test of the natural numbers and scopes + +%sort nat %. + +This just defines the sort of natural numbers, but it doesn't define any terms or operations on them. We will define those in the next scope. + +## We can use headings whenever we wish + +%scope nat %{ + + %. NOTE, we can use `%%%.` inside of a scope + + %term 0 nat + %term S + %pi nat + %-> nat +%} + +%sort add {_ nat} {_ nat} {_ nat} %. +%scope add %{ + %term 0 {{X}} add X %(0 nat) X + %term S {{X Y Z}} + %if add X (%(S nat) Y) (%(S nat) Z) + %<- add X Y Z +%} + +%mode add %in %in %out %. +%worlds () (add _ _ _) +%total Y (add _ Y _) + diff --git a/test/cram/nat.t/run.t b/test/cram/nat.t/run.t new file mode 100644 index 0000000..f98d818 --- /dev/null +++ b/test/cram/nat.t/run.t @@ -0,0 +1,5 @@ + $ ls + dune + main.lf + + $ stelf check main.lf From 6a91d8c0131b29696fb93a02d9a0ede2d9648eff Mon Sep 17 00:00:00 2001 From: Asher Frost Date: Fri, 26 Jun 2026 14:44:23 -0400 Subject: [PATCH 8/9] feat: added many tests --- test/cram/arith.t/dune | 0 test/cram/arith.t/main.lf | 38 +++++++++++++ test/cram/arith.t/run.t | 6 ++ test/cram/arith.t/theorems.lf | 36 ++++++++++++ test/cram/bool.t/dune | 1 + test/cram/bool.t/main.lf | 17 ++++++ test/cram/bool.t/run.t | 5 ++ test/cram/failure.t/dune | 0 test/cram/failure.t/main.lf | 4 ++ test/cram/failure.t/run.t | 6 ++ test/cram/fol.t/dune | 0 test/cram/fol.t/main.lf | 44 ++++++++++++++ test/cram/fol.t/run.t | 6 ++ test/cram/fol.t/theorems.lf | 21 +++++++ test/cram/lc1.t/dune | 0 test/cram/lc1.t/main.lf | 2 + test/cram/lc1.t/run.t | 7 +++ test/cram/lc1.t/src/arity.lf | 23 ++++++++ test/cram/lc1.t/src/types.lf | 56 ++++++++++++++++++ test/cram/lc1.t/stelf.toml | 6 ++ test/cram/list.t/dune | 1 + test/cram/list.t/main.lf | 25 ++++++++ test/cram/list.t/run.t | 5 ++ test/cram/prop_calc.t/dune | 0 test/cram/prop_calc.t/equiv.lf | 57 +++++++++++++++++++ test/cram/prop_calc.t/main.lf | 49 ++++++++++++++++ test/cram/prop_calc.t/run.t | 6 ++ test/cram/wiki_failures.t/coverage_error.lf | 35 ++++++++++++ test/cram/wiki_failures.t/dune | 0 test/cram/wiki_failures.t/mode_error.lf | 30 ++++++++++ test/cram/wiki_failures.t/run.t | 18 ++++++ test/cram/wiki_failures.t/totality_error.lf | 18 ++++++ .../wiki_failures.t/unsatisfiable_query.lf | 4 ++ test/dune | 1 + 34 files changed, 527 insertions(+) create mode 100644 test/cram/arith.t/dune create mode 100644 test/cram/arith.t/main.lf create mode 100644 test/cram/arith.t/run.t create mode 100644 test/cram/arith.t/theorems.lf create mode 100644 test/cram/bool.t/dune create mode 100644 test/cram/bool.t/main.lf create mode 100644 test/cram/bool.t/run.t create mode 100644 test/cram/failure.t/dune create mode 100644 test/cram/failure.t/main.lf create mode 100644 test/cram/failure.t/run.t create mode 100644 test/cram/fol.t/dune create mode 100644 test/cram/fol.t/main.lf create mode 100644 test/cram/fol.t/run.t create mode 100644 test/cram/fol.t/theorems.lf create mode 100644 test/cram/lc1.t/dune create mode 100644 test/cram/lc1.t/main.lf create mode 100644 test/cram/lc1.t/run.t create mode 100644 test/cram/lc1.t/src/arity.lf create mode 100644 test/cram/lc1.t/src/types.lf create mode 100644 test/cram/lc1.t/stelf.toml create mode 100644 test/cram/list.t/dune create mode 100644 test/cram/list.t/main.lf create mode 100644 test/cram/list.t/run.t create mode 100644 test/cram/prop_calc.t/dune create mode 100644 test/cram/prop_calc.t/equiv.lf create mode 100644 test/cram/prop_calc.t/main.lf create mode 100644 test/cram/prop_calc.t/run.t create mode 100644 test/cram/wiki_failures.t/coverage_error.lf create mode 100644 test/cram/wiki_failures.t/dune create mode 100644 test/cram/wiki_failures.t/mode_error.lf create mode 100644 test/cram/wiki_failures.t/run.t create mode 100644 test/cram/wiki_failures.t/totality_error.lf create mode 100644 test/cram/wiki_failures.t/unsatisfiable_query.lf create mode 100644 test/dune diff --git a/test/cram/arith.t/dune b/test/cram/arith.t/dune new file mode 100644 index 0000000..e69de29 diff --git a/test/cram/arith.t/main.lf b/test/cram/arith.t/main.lf new file mode 100644 index 0000000..21a06ed --- /dev/null +++ b/test/cram/arith.t/main.lf @@ -0,0 +1,38 @@ +# Trivial Unary Arithmetic + +%sort nat %. +%scope nat %{ + %term 0 nat + %term S %pi nat %-> nat +%} + +%sort nt {_ nat} %. +%scope nt %{ + %term nt_z nt %(0 nat) + %term nt_s {{X}} + %if nt (%(S nat) X) + %<- nt X +%} + +%sort plus {_ nat} {_ nat} {_ nat} %. +%scope plus %{ + %term p_z {{Y}} plus %(0 nat) Y Y + %term p_s {{X Y Z}} + %if plus (%(S nat) X) Y (%(S nat) Z) + %<- plus X Y Z +%} + +%sort acker {_ nat} {_ nat} {_ nat} %. +%scope acker %{ + %term a_1 {{Y}} acker %(0 nat) Y (%(S nat) Y) + %term a_2 {{X Z}} + %if acker (%(S nat) X) %(0 nat) Z + %<- acker X (%(S nat) %(0 nat)) Z + %term a_3 {{X Y Zp Z}} + %if acker (%(S nat) X) (%(S nat) Y) Z + %<- acker (%(S nat) X) Y Zp + %<- acker X Zp Z +%} + +%mode acker %in %in %out %. +%worlds () (acker _ _ _) diff --git a/test/cram/arith.t/run.t b/test/cram/arith.t/run.t new file mode 100644 index 0000000..b19d6ec --- /dev/null +++ b/test/cram/arith.t/run.t @@ -0,0 +1,6 @@ + $ ls + dune + main.lf + theorems.lf + + $ stelf check main.lf diff --git a/test/cram/arith.t/theorems.lf b/test/cram/arith.t/theorems.lf new file mode 100644 index 0000000..22b7703 --- /dev/null +++ b/test/cram/arith.t/theorems.lf @@ -0,0 +1,36 @@ +%. Verbatim copy of arith.thm. +%. %theorem, %prove, and %terminates are Twelf meta-level commands +%. not yet supported in STELF; lines kept for reference. + +%. %terminates {N M} (acker N M _). + +%. %theorem assoc : forall* {X:nat} {Y:nat} {Z:nat} {X':nat} {Z':nat} +%. forall {P1:plus X Y X'} {P2:plus X' Z Z'} +%. exists {Y':nat} {Q1:plus Y Z Y'} {Q2:plus X Y' Z'} +%. true. +%. %prove 3 P1 (assoc P1 _ _ _ _). + +%. %terminates P1 (assoc P1 _ _ _ _). + +%. %theorem assoc' : forall* {X:nat} {Y:nat} {Z:nat} {X':nat} {Z':nat} +%. forall {P1:plus X Y X'} {P2:plus X' Z Z'} +%. exists {Y':nat} {Q1:plus Y Z Y'} {Q2:plus X Y' Z'} +%. true. +%. %terminates P2 (assoc' _ P2 _ _ _). + +%. %theorem comm : forall* {X:nat} {Y:nat} {Z:nat} +%. forall {N:nt X} {M:nt Y} {P'': nt Z} {P0:plus X Y Z} +%. exists {Q:plus Y X Z} +%. true. +%. %theorem comm' : forall* {X:nat} {Y:nat} {Z:nat} +%. forall {M':nt X} {N':nt Y} {P'': nt Z} {P0:plus X Y Z} +%. exists {Q:plus Y X Z} +%. true. + +%. %prove 6 [(N N') (M M')] +%. (comm N M _ _ _) +%. (comm' M' N' _ _ _). + +%. %terminates [(N N') (M M')] +%. (comm N M _ _ _) +%. (comm' M' N' _ _ _). diff --git a/test/cram/bool.t/dune b/test/cram/bool.t/dune new file mode 100644 index 0000000..8b13789 --- /dev/null +++ b/test/cram/bool.t/dune @@ -0,0 +1 @@ + diff --git a/test/cram/bool.t/main.lf b/test/cram/bool.t/main.lf new file mode 100644 index 0000000..6798d91 --- /dev/null +++ b/test/cram/bool.t/main.lf @@ -0,0 +1,17 @@ +# Booleans + +%sort bool %. +%scope bool %{ + %term true bool + %term false bool +%} + +%sort not bool bool %. +%scope not %{ + %term t not %(true bool) %(false bool) + %term f not %(false bool) %(true bool) +%} + +%mode not %in %out %. +%worlds () (not _ _) +%total X (not X _) diff --git a/test/cram/bool.t/run.t b/test/cram/bool.t/run.t new file mode 100644 index 0000000..f98d818 --- /dev/null +++ b/test/cram/bool.t/run.t @@ -0,0 +1,5 @@ + $ ls + dune + main.lf + + $ stelf check main.lf diff --git a/test/cram/failure.t/dune b/test/cram/failure.t/dune new file mode 100644 index 0000000..e69de29 diff --git a/test/cram/failure.t/main.lf b/test/cram/failure.t/main.lf new file mode 100644 index 0000000..4057acf --- /dev/null +++ b/test/cram/failure.t/main.lf @@ -0,0 +1,4 @@ +# Expected failure: query for an inhabitant of an empty type + +%sort void %. +%query 1 * * void diff --git a/test/cram/failure.t/run.t b/test/cram/failure.t/run.t new file mode 100644 index 0000000..f4affc9 --- /dev/null +++ b/test/cram/failure.t/run.t @@ -0,0 +1,6 @@ + $ ls + dune + main.lf + + $ stelf check main.lf + [1] diff --git a/test/cram/fol.t/dune b/test/cram/fol.t/dune new file mode 100644 index 0000000..e69de29 diff --git a/test/cram/fol.t/main.lf b/test/cram/fol.t/main.lf new file mode 100644 index 0000000..0f94fca --- /dev/null +++ b/test/cram/fol.t/main.lf @@ -0,0 +1,44 @@ +# Intuitionistic Propositional Calculus +# Positive fragment with implies, and, true. + +%sort o %. +%sort atm %. + +%scope o %{ + %term => %pi o %-> o %-> o + %prec %right 10 => %. + %term & %pi o %-> o %-> o + %prec %right 11 & %. + %term true o + %term ` %pi atm %-> o +%} + +%sort !^ {_ o} %. +%sort !v {_ o} %. + +%scope !^ %{ + %term trueI^ !^ %(true o) + %term andI^ {{A B}} + %if !^ (%(& o) A B) + %<- !^ A + %<- !^ B + %term impI^ {{A B}} + %if !^ (%(=> o) A B) + %<- %pi (!v A) %-> !^ B + %term close {{P}} + %if !^ (%(` o) P) + %<- !v (%(` o) P) +%} + +%scope !v %{ + %term andEvL {{A B}} + %if !v A + %<- !v (%(& o) A B) + %term andEvR {{A B}} + %if !v B + %<- !v (%(& o) A B) + %term impEv {{A B}} + %if !v B + %<- !v (%(=> o) A B) + %<- !^ A +%} diff --git a/test/cram/fol.t/run.t b/test/cram/fol.t/run.t new file mode 100644 index 0000000..b19d6ec --- /dev/null +++ b/test/cram/fol.t/run.t @@ -0,0 +1,6 @@ + $ ls + dune + main.lf + theorems.lf + + $ stelf check main.lf diff --git a/test/cram/fol.t/theorems.lf b/test/cram/fol.t/theorems.lf new file mode 100644 index 0000000..7678890 --- /dev/null +++ b/test/cram/fol.t/theorems.lf @@ -0,0 +1,21 @@ +%. Verbatim copy of fol.thm. +%. %theorem, %establish are Twelf meta-level commands not yet in STELF. + +%. %theorem identity : forall* {P:atm} exists {D: !^ (` P => ` P)} true. +%. %establish 5 {} (identity _). + +%. %theorem curry : forall* {P:atm} {Q:atm} {R:atm} +%. exists {D: !^ ((` P & ` Q => ` R) => (` P => (` Q => ` R)))} true. +%. %establish 12 {} (curry _). + +%. %theorem uncurry : forall* {P:atm} {Q:atm} {R:atm} +%. exists {D: !^ ((` P => (` Q => ` R)) => (` P & ` Q => ` R))} true. +%. %% %establish 12 {} (uncurry _). + +%. %theorem split : forall* {P:atm} {Q:atm} {R:atm} +%. exists {D: !^ ((` P => ` Q & ` R) => (` P => ` Q) & (` P => ` R))} true. +%. %% %establish 15 {} (split _). + +%. %theorem join : forall* {P:atm} {Q:atm} {R:atm} +%. exists {D: !^ ((` P => ` Q) & (` P => ` R) => (` P => ` Q & ` R))} true. +%. %% %establish 12 {} (join _). diff --git a/test/cram/lc1.t/dune b/test/cram/lc1.t/dune new file mode 100644 index 0000000..e69de29 diff --git a/test/cram/lc1.t/main.lf b/test/cram/lc1.t/main.lf new file mode 100644 index 0000000..c55837d --- /dev/null +++ b/test/cram/lc1.t/main.lf @@ -0,0 +1,2 @@ +%require types %. +%require arity %. diff --git a/test/cram/lc1.t/run.t b/test/cram/lc1.t/run.t new file mode 100644 index 0000000..d72277f --- /dev/null +++ b/test/cram/lc1.t/run.t @@ -0,0 +1,7 @@ + $ ls + dune + main.lf + src + stelf.toml + + $ stelf check stelf.toml diff --git a/test/cram/lc1.t/src/arity.lf b/test/cram/lc1.t/src/arity.lf new file mode 100644 index 0000000..3bffc82 --- /dev/null +++ b/test/cram/lc1.t/src/arity.lf @@ -0,0 +1,23 @@ + +%require types %. + +# Arities (simple types) +%sort arity +%term nil {T type} arity +%term ->> %pi arity %-> arity %-> arity + +%prec %right 5 ->> %. + + +%sort erase type arity +%scope erase %{ + %term @ + {{A B T U}} + %if erase A T + %-> ({X term} {P X : A} erase (B X P) U) + %-> erase (A -> B) U + %. TODO +%} + +%mode erase %in %out + diff --git a/test/cram/lc1.t/src/types.lf b/test/cram/lc1.t/src/types.lf new file mode 100644 index 0000000..2bd01a2 --- /dev/null +++ b/test/cram/lc1.t/src/types.lf @@ -0,0 +1,56 @@ +# The Syntax of the Lambda Calculus + +%sort κ +%sort τ +%sort e +%symbol kind κ +%symbol type τ +%symbol term e +%sort : term type +%prec %none 1 : %. +%scope kind %{ + %. Kind of types + %term * kind + %. Pi types + %term => + {A type} + {B + {X term} + %if X : A + %-> kind + } + kind + %term @ + %pi kind + %-> term + %-> kind +%} + +%scope type %{ + %term -> + {A type} + {B + {X term} + %if X : A + %-> type + } + type + %prec %right 5 -> %. + %term Λ + %pi (%pi term %-> type) + %-> type + %term @ + %pi type + %-> term + %-> type +%} + +%scope term %{ + %term λ + %pi (%pi term %-> term) + %-> term + %term @ + %pi term + %-> term + %-> term +%} diff --git a/test/cram/lc1.t/stelf.toml b/test/cram/lc1.t/stelf.toml new file mode 100644 index 0000000..f216b4b --- /dev/null +++ b/test/cram/lc1.t/stelf.toml @@ -0,0 +1,6 @@ +interactive = "lc1" + +[[group]] +name = "lc1" +main = "./main.lf" +dirs = [".", "./src"] diff --git a/test/cram/list.t/dune b/test/cram/list.t/dune new file mode 100644 index 0000000..8b13789 --- /dev/null +++ b/test/cram/list.t/dune @@ -0,0 +1 @@ + diff --git a/test/cram/list.t/main.lf b/test/cram/list.t/main.lf new file mode 100644 index 0000000..b6dd3a0 --- /dev/null +++ b/test/cram/list.t/main.lf @@ -0,0 +1,25 @@ +# Lists of Natural Numbers + +%sort nat %. +%scope nat %{ + %term 0 nat + %term S %pi nat %-> nat +%} + +%sort list %. +%scope list %{ + %term nil list + %term cons %pi nat %-> list %-> list +%} + +%sort append {_ list} {_ list} {_ list} %. +%scope append %{ + %term nil {{L}} append %(nil list) L L + %term cons {{H T L M}} + %if append (%(cons list) H T) L (%(cons list) H M) + %<- append T L M +%} + +%mode append %in %in %out %. +%worlds () (append _ _ _) +%total X (append X _ _) diff --git a/test/cram/list.t/run.t b/test/cram/list.t/run.t new file mode 100644 index 0000000..f98d818 --- /dev/null +++ b/test/cram/list.t/run.t @@ -0,0 +1,5 @@ + $ ls + dune + main.lf + + $ stelf check main.lf diff --git a/test/cram/prop_calc.t/dune b/test/cram/prop_calc.t/dune new file mode 100644 index 0000000..e69de29 diff --git a/test/cram/prop_calc.t/equiv.lf b/test/cram/prop_calc.t/equiv.lf new file mode 100644 index 0000000..4115479 --- /dev/null +++ b/test/cram/prop_calc.t/equiv.lf @@ -0,0 +1,57 @@ +%. %%% A most naive version of translation from natural deductions +%. %%% into a proof in the Hilbert-style system and vice versa. +%. %%% Expresses the equivalence of the two systems. +%. %%% Author: Frank Pfenning +%. +%. % abs uses LF-level function types as sort indices, unsupported in STELF. +%. +%. abs : (|- A -> |- B) -> |- A => B -> type. +%. +%. % aID : abs ([x] x) (MP (MP S K) K). +%. aONE : abs ([x] ONE) (MP K ONE). +%. aPAIR : abs ([x] PAIR) (MP K PAIR). +%. aLEFT : abs ([x] LEFT) (MP K LEFT). +%. aRIGHT : abs ([x] RIGHT) (MP K RIGHT). +%. aK : abs ([x] K) (MP K K). +%. aS : abs ([x] S) (MP K S). +%. aMP : abs ([x] MP (P x) (Q x)) (MP (MP S P') Q') +%. <- abs P P' <- abs Q Q'. +%. +%. %mode abs +D -F. +%. +%. comb : ! A -> |- A -> type. +%. %mode comb +D -F. +%. +%. ctrue : comb trueI ONE. +%. candI : comb (andI P Q) (MP (MP PAIR P') Q') <- comb P P' <- comb Q Q'. +%. candEL : comb (andEL P) (MP LEFT P') <- comb P P'. +%. candER : comb (andER P) (MP RIGHT P') <- comb P P'. +%. cimpliesI : comb (impliesI PP) Q +%. <- ({x}{y} comb x y +%. -> ({B:o} abs ([z:|- B] y) (MP K y)) +%. -> comb (PP x) (PP' y)) +%. <- abs PP' Q. +%. cimpliesE : comb (impliesE P Q) (MP P' Q') +%. <- comb P P' +%. <- comb Q Q'. +%. +%. %block l : some {A:o} block {x:! A}{y: |- A} {u: comb x y} +%. {v: {B:o} abs ([z:|- B] y) (MP K y)}. +%. %worlds (l) (abs D F). +%. %worlds (l) (comb _ _ ). +%. +%. combdefn : |- A -> ! A -> type. +%. %mode combdefn +D -F. +%. +%. cdK : combdefn K (impliesI [p] impliesI [q] p). +%. cdS : combdefn S (impliesI [p] impliesI [q] impliesI [r] +%. (impliesE (impliesE p r) (impliesE q r))). +%. cdONE : combdefn ONE trueI. +%. cdPAIR : combdefn PAIR (impliesI [p] impliesI [q] andI p q). +%. cdLEFT : combdefn LEFT (impliesI [p] andEL p). +%. cdRIGHT : combdefn RIGHT (impliesI [p] andER p). +%. cdMP : combdefn (MP P Q) (impliesE P' Q') +%. <- combdefn P P' +%. <- combdefn Q Q'. +%. +%. %terminates D (combdefn D _). diff --git a/test/cram/prop_calc.t/main.lf b/test/cram/prop_calc.t/main.lf new file mode 100644 index 0000000..5284aba --- /dev/null +++ b/test/cram/prop_calc.t/main.lf @@ -0,0 +1,49 @@ +# Intuitionistic propositional calculus (positive fragment) +# Two formulations: natural deduction and Hilbert-style system. +# Author: Frank Pfenning (ported to STELF) + +%sort o %. +%scope o %{ + %term => %pi o %-> o %-> o + %prec %right 10 => %. + %term & %pi o %-> o %-> o + %prec %right 11 & %. + %term true o +%} + +%sort |- {_ o} %. +%scope |- %{ + %term K {{A B}} |- (%(=> o) A (%(=> o) B A)) + %term S {{A B C}} |- (%(=> o) (%(=> o) A (%(=> o) B C)) (%(=> o) (%(=> o) A B) (%(=> o) A C))) + %term ONE |- %(true o) + %term PAIR {{A B}} |- (%(=> o) A (%(=> o) B (%(& o) A B))) + %term LEFT {{A B}} |- (%(=> o) (%(& o) A B) A) + %term RIGHT {{A B}} |- (%(=> o) (%(& o) A B) B) + %term MP {{A B}} %if |- B %<- |- (%(=> o) A B) %<- |- A +%} + +%sort ! {_ o} %. +%scope ! %{ + %term trueI ! %(true o) + %term andI {{A B}} %if ! (%(& o) A B) %<- ! A %<- ! B + %term andEL {{A B}} %if ! A %<- ! (%(& o) A B) + %term andER {{A B}} %if ! B %<- ! (%(& o) A B) + %term impliesI {{A B}} %if ! (%(=> o) A B) %<- %pi (! A) %-> ! B + %term impliesE {{A B}} %if ! B %<- ! (%(=> o) A B) %<- ! A +%} + +%sort !^ {_ o} %. +%sort !v {_ o} %. +%scope !^ %{ + %term trueI^ !^ %(true o) + %term andI^ {{A B}} %if !^ (%(& o) A B) %<- !^ A %<- !^ B + %term impI^ {{A B}} %if !^ (%(=> o) A B) %<- %pi (!v A) %-> !^ B + %term close {{A}} %if !^ A %<- !v A +%} +%scope !v %{ + %term andEvL {{A B}} %if !v A %<- !v (%(& o) A B) + %term andEvR {{A B}} %if !v B %<- !v (%(& o) A B) + %term impEv {{A B}} %if !v B %<- !v (%(=> o) A B) %<- !^ A +%} + +%. %require equiv %. % equiv.lf uses LF function types as sort indices (unsupported in STELF) diff --git a/test/cram/prop_calc.t/run.t b/test/cram/prop_calc.t/run.t new file mode 100644 index 0000000..c3256e9 --- /dev/null +++ b/test/cram/prop_calc.t/run.t @@ -0,0 +1,6 @@ + $ ls + dune + equiv.lf + main.lf + + $ stelf check main.lf diff --git a/test/cram/wiki_failures.t/coverage_error.lf b/test/cram/wiki_failures.t/coverage_error.lf new file mode 100644 index 0000000..3510937 --- /dev/null +++ b/test/cram/wiki_failures.t/coverage_error.lf @@ -0,0 +1,35 @@ +# Expected failure: incomplete coverage in sub-transitivity +# Simplified from coverage_error.elf: sub-trans is missing cases +# (int→int→float and reflexivity for float in the transitivity proof) + +%sort tp %. +%scope tp %{ + %term int tp + %term float tp + %term arrow %pi tp %-> tp %-> tp +%} + +%sort sub {_ tp} {_ tp} %. +%scope sub %{ + %term sub-ii sub %(int tp) %(int tp) + %term sub-ff sub %(float tp) %(float tp) + %term sub-if sub %(int tp) %(float tp) + %term sub-arrow {{T S T' S'}} + %if sub (%(arrow tp) T S) (%(arrow tp) T' S') + %<- sub T' T + %<- sub S S' +%} + +%sort sub-trans {_ tp} {_ tp} {_ tp} %. +%scope sub-trans %{ + %term sub-trans/refl {{T}} + sub-trans T T T + %term sub-trans/arrow {{T1 T2 T3 T1' T3'}} + %if sub-trans (%(arrow tp) T1 T3) (%(arrow tp) T1 T3) (%(arrow tp) T1' T3') + %<- sub-trans T1' T1 T1 + %<- sub-trans T3 T3 T3' +%} + +%mode sub-trans %in %in %out %. +%worlds () (sub-trans _ _ _) +%total D (sub-trans D _ _) diff --git a/test/cram/wiki_failures.t/dune b/test/cram/wiki_failures.t/dune new file mode 100644 index 0000000..e69de29 diff --git a/test/cram/wiki_failures.t/mode_error.lf b/test/cram/wiki_failures.t/mode_error.lf new file mode 100644 index 0000000..572c609 --- /dev/null +++ b/test/cram/wiki_failures.t/mode_error.lf @@ -0,0 +1,30 @@ +# Expected failure: mode violation (output arg used as input) + +%sort nat %. +%scope nat %{ + %term 0 nat + %term S %pi nat %-> nat +%} + +%sort plus {_ nat} {_ nat} {_ nat} %. +%scope plus %{ + %term plus/z {{N}} plus %(0 nat) N N + %term plus/s {{N1 N2 N3}} + %if plus (%(S nat) N1) N2 (%(S nat) N3) + %<- plus N1 N2 N3 +%} + +%mode plus %in %in %out %. +%worlds () (plus _ _ _) +%total N (plus N _ _) + +%sort bad {_ nat} {_ nat} %. +%scope bad %{ + %term bad/case {{N1 N2}} + %if bad N1 N2 + %<- plus N1 N2 N1 +%} + +%mode bad %in %out %. +%worlds () (bad _ _) +%total N (bad N _) diff --git a/test/cram/wiki_failures.t/run.t b/test/cram/wiki_failures.t/run.t new file mode 100644 index 0000000..3573dd2 --- /dev/null +++ b/test/cram/wiki_failures.t/run.t @@ -0,0 +1,18 @@ + $ ls + coverage_error.lf + dune + mode_error.lf + totality_error.lf + unsatisfiable_query.lf + + $ stelf check coverage_error.lf + [1] + + $ stelf check mode_error.lf + [1] + + $ stelf check totality_error.lf + [1] + + $ stelf check unsatisfiable_query.lf + [1] diff --git a/test/cram/wiki_failures.t/totality_error.lf b/test/cram/wiki_failures.t/totality_error.lf new file mode 100644 index 0000000..7a759d9 --- /dev/null +++ b/test/cram/wiki_failures.t/totality_error.lf @@ -0,0 +1,18 @@ +# Expected failure: %total check fails due to missing base case + +%sort nat %. +%scope nat %{ + %term 0 nat + %term S %pi nat %-> nat +%} + +%sort partial {_ nat} {_ nat} %. +%scope partial %{ + %term partial/s {{N M}} + %if partial (%(S nat) N) (%(S nat) M) + %<- partial N M +%} + +%mode partial %in %out %. +%worlds () (partial _ _) +%total N (partial N _) diff --git a/test/cram/wiki_failures.t/unsatisfiable_query.lf b/test/cram/wiki_failures.t/unsatisfiable_query.lf new file mode 100644 index 0000000..4057acf --- /dev/null +++ b/test/cram/wiki_failures.t/unsatisfiable_query.lf @@ -0,0 +1,4 @@ +# Expected failure: query for an inhabitant of an empty type + +%sort void %. +%query 1 * * void diff --git a/test/dune b/test/dune new file mode 100644 index 0000000..8b13789 --- /dev/null +++ b/test/dune @@ -0,0 +1 @@ + From 692be4dc4b502788784d4090988a689cdd394e05 Mon Sep 17 00:00:00 2001 From: Asher Frost Date: Sat, 4 Jul 2026 10:04:42 -0400 Subject: [PATCH 9/9] feat: added new tests relating to pure twelf code --- Makefile | 76 +- bin/dune | 2 +- bin/main.ml | 99 +- ...computation-and-deduction-2009-20090203.lf | 139 + ...computation-and-deduction-2009-20090316.lf | 177 + ...computation-and-deduction-2009-20090408.lf | 419 + .../lexicographical-orderings-with-density.lf | 97 + .../popl-tutorial-church-rosser-problem.lf | 283 + ...rial-properties-of-typing-and-reduction.lf | 392 + ...ng-metatheorems-solutions-odd-even-succ.lf | 42 + .../pages/proving-metatheorems-with-twelf.lf | 54 + .../output/pages/talk-effectiveness-lemma.lf | 22 + ...alk-incremental-metatheorem-development.lf | 15 + new-tests/output/pages/talk-metatheorem.lf | 28 + new-tests/output/pages/talk-mutable-state.lf | 15 + .../talk-natural-numbers-with-inequality.lf | 10 + new-tests/output/pages/talk-thaw.lf | 4 + .../pages/template-bibtex-pierce02tapl.lf | 15 + new-tests/output/pages/template-discuss.lf | 17 + new-tests/output/pages/template-lolli.lf | 4 + new-tests/output/pages/template-talk-guide.lf | 10 + new-tests/output/pages/twelf-with-emacs.lf | 137 + new-tests/output/pages/user-jaked.lf | 8 + ...computation-and-deduction-2009-20090203.lf | 139 + ...computation-and-deduction-2009-20090316.lf | 177 + ...computation-and-deduction-2009-20090408.lf | 419 + .../lexicographical-orderings-with-density.lf | 97 + .../popl-tutorial-church-rosser-problem.lf | 283 + ...rial-properties-of-typing-and-reduction.lf | 392 + ...ng-metatheorems-solutions-odd-even-succ.lf | 42 + .../pages/proving-metatheorems-with-twelf.lf | 54 + new-tests/pages/talk-effectiveness-lemma.lf | 22 + ...alk-incremental-metatheorem-development.lf | 15 + new-tests/pages/talk-metatheorem.lf | 28 + new-tests/pages/talk-mutable-state.lf | 15 + .../talk-natural-numbers-with-inequality.lf | 10 + new-tests/pages/talk-thaw.lf | 4 + .../pages/template-bibtex-pierce02tapl.lf | 15 + new-tests/pages/template-discuss.lf | 17 + new-tests/pages/template-lolli.lf | 4 + new-tests/pages/template-talk-guide.lf | 10 + new-tests/pages/twelf-with-emacs.lf | 137 + new-tests/pages/user-jaked.lf | 8 + .../examples/alloc-sem/absurdum.lf | 6 + .../examples/alloc-sem/dynsem-lemmas-thm.lf | 12 + .../examples/alloc-sem/dynsem-lemmas.lf | 58 + .../stelf-output/examples/alloc-sem/dynsem.lf | 40 + .../stelf-output/examples/alloc-sem/exp.lf | 19 + 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mode 100644 test/Unit/IntSyn/WhnfTests.ml create mode 100644 test/Unit/Table/TableTests.ml create mode 100644 test/Unit/Trail/TrailTests.ml create mode 100644 test/Unit/UnitTest.ml create mode 100644 test/Unit/dune create mode 100644 test/cram/arith.t/stelf.toml create mode 100644 test/cram/bool.t/stelf.toml create mode 100644 test/cram/guide_lam.t/dune create mode 100644 test/cram/guide_lam.t/main.lf create mode 100644 test/cram/guide_lam.t/run.t create mode 100644 test/cram/guide_nd.t/dune create mode 100644 test/cram/guide_nd.t/main.lf create mode 100644 test/cram/guide_nd.t/run.t create mode 100644 test/cram/guide_nd.t/stelf.toml create mode 100644 test/cram/handbook.t/dune create mode 100644 test/cram/handbook.t/main.lf create mode 100644 test/cram/handbook.t/run.t create mode 100644 test/cram/list.t/stelf.toml create mode 100644 test/cram/nat.t/stelf.toml create mode 100644 test/cram/polylam.t/dune create mode 100644 test/cram/polylam.t/main.lf create mode 100644 test/cram/polylam.t/run.t create mode 100644 test/cram/polylam.t/stelf.toml create mode 100644 test/cram/small_step.t/dune create mode 100644 test/cram/small_step.t/main.lf create mode 100644 test/cram/small_step.t/run.t create mode 100644 test/cram/tapl_ch13.t/defs.lf create mode 100644 test/cram/tapl_ch13.t/dune create mode 100644 test/cram/tapl_ch13.t/main.lf create mode 100644 test/cram/tapl_ch13.t/nat.lf create mode 100644 test/cram/tapl_ch13.t/run.t create mode 100644 test/cram/tapl_ch13.t/stelf.toml diff --git a/Makefile b/Makefile index 1724481..6f62b33 100644 --- a/Makefile +++ b/Makefile @@ -1,32 +1,64 @@ -BUILD_DIR ?= _build/default -DUNE ?= dune -DUNE_PROJECT ?= ./dune-project -DUNE_WORKSPACE ?= ./dune-workspace +TARGET ?= ./stelf +DUNE_LOCK ?= ./dune.lock/ +OPAM_FILE ?= ./stelf.opam +DUNE_BUILD_DIR ?= _build/default +DUNE_MIN_VERSION ?= 3.23 +DUNE_PROJECT ?= ./dune-project +SWITCH ?= . +OPAM ?= opam +OPAM_EXEC := $(OPAM) exec --switch $(SWITCH) -- +DUNE ?= dune -.PHONY: all build test docs install clean repl check help +SWITCH_SENTINEL := _opam/.opam-switch/switch-config +DUNE_SENTINEL := _opam/lib/dune/META +DEPS_SENTINEL := .deps-installed -all: build test docs install +.PHONY: all build test install docs clean check lock -dune.lock/: $(DUNE_PROJECT) $(DUNE_WORKSPACE) - @$(DUNE) pkg lock -build: dune.lock/ - @$(DUNE) build +all: build -check: dune.lock/ - @$(DUNE) build @check +# Phase 1: initialize opam if needed, then create the local switch +$(SWITCH_SENTINEL): + $(OPAM) init --bare --no-setup --yes 2>/dev/null || true + $(OPAM) switch create $(SWITCH) --empty --yes 2>/dev/null || true + @test -f $@ || (echo "ERROR: opam switch creation failed"; exit 1) -repl: dune.lock/ - @$(DUNE) utop +# Phase 2: install OCaml compiler + dune >= $(DUNE_MIN_VERSION) into the switch +$(DUNE_SENTINEL): $(SWITCH_SENTINEL) + $(OPAM) install --switch $(SWITCH) --yes \ + "ocaml>=5.0.0" "dune>=$(DUNE_MIN_VERSION)" -test: dune.lock/ - @$(DUNE) runtest +# Phase 3: ensure submodules are present, then generate stelf.opam from dune-project +# dune.lock/ is committed — re-locking is done explicitly via `make lock` +$(OPAM_FILE): $(DUNE_PROJECT) $(DUNE_SENTINEL) + git submodule update --init --recursive + $(OPAM_EXEC) $(DUNE) build $(OPAM_FILE) -docs: dune.lock/ - @$(DUNE) build @doc +# Phase 4: install all package dependencies into the local switch +$(DEPS_SENTINEL): $(OPAM_FILE) + $(OPAM) install --switch $(SWITCH) . --deps-only --yes + @touch $@ -install: dune.lock/ - @$(DUNE) install +build: $(DEPS_SENTINEL) + $(OPAM_EXEC) $(DUNE) build + cp $(DUNE_BUILD_DIR)/bin/main.exe $(TARGET) -clean: dune.lock/ - @$(DUNE) clean +test: $(DEPS_SENTINEL) + $(OPAM_EXEC) $(DUNE) runtest +check: $(DEPS_SENTINEL) + $(OPAM_EXEC) $(DUNE) build @check + +install: build + $(OPAM_EXEC) $(DUNE) install --prefix _opam + +docs: $(DEPS_SENTINEL) + $(OPAM_EXEC) $(DUNE) build @doc + +# Explicitly refresh the lock file (requires network; updates dune.lock/) +lock: $(DUNE_SENTINEL) + $(OPAM_EXEC) $(DUNE) pkg lock + +clean: + $(OPAM_EXEC) $(DUNE) clean + rm -f $(TARGET) $(DEPS_SENTINEL) diff --git a/bin/dune b/bin/dune index 1119c16..f4a0625 100644 --- a/bin/dune +++ b/bin/dune @@ -2,4 +2,4 @@ (name main) (public_name stelf) (modules main) - (libraries stelf server pal)) + (libraries stelf server pal cmdliner fmt.tty display basis lwt.unix grace grace.ansi_renderer)) diff --git a/bin/main.ml b/bin/main.ml index 444d459..306b2db 100644 --- a/bin/main.ml +++ b/bin/main.ml @@ -1,12 +1,89 @@ -(* -(** Entry point for the STELF server executable. *) +module P = Pal -module Twelf_server = Server.Server_.Server - -(** Run the server process and return its exit code. *) -let run_server () : int = Twelf_server.server ("stelf", []) - -let () = if run_server () = 0 then exit 0 else exit 1 -*) - -let () = Pal.Pal.run () +let () = + let open P.Pal in + let open Cmdliner in + let open Cmdliner.Term.Syntax in + let version = P.version in + let repl_cmd : int Cmd.t = + let config_file = + Arg.(value & pos 0 (some file) None & info [] ~docv:"CONFIG" + ~doc:"Optional .toml project file to load before entering the REPL") + in + let repl_fn : int Term.t = + let+ verbosity = Arg.value P.Opts.Opts.verbosity + and+ color = Arg.value P.Opts.Opts.color + and+ unicode = Arg.value P.Opts.Opts.unicode + and+ config = config_file in + let module N = struct + let use_color = color + let use_unicode = unicode + let verbosity = verbosity + end in + Lwt_main.run (top ?config:(Option.map Fpath.v config) (module N)) + and repl_info : Cmd.info = + Cmd.info "repl" ~doc:"Start the interactive REPL, optionally loading a project config" + in + Cmd.v repl_info repl_fn + in + let check_cmd : int Cmd.t = + let file = + Arg.(required & pos 0 (some file) None & info [] ~docv:"FILE") + in + Cmd.v + (Cmd.info "check" ~doc:"Load a .cfg or source file") + Term.( + const (fun verbosity f -> + Fmt_tty.setup_std_outputs (); + M.chatter := Display.Info.to_int verbosity; + Display.register (fun m -> + if Display.Info.(m.level =< verbosity) then begin + let open Grace.Diagnostic.Severity in + let pp_diag fmt d = Grace_ansi_renderer.pp_diagnostic fmt d in + let display_diag sev = + let diag = + Grace.Diagnostic.create sev + (fun ppf -> Display.fmt ppf m.msg) + in + Format.printf "%a%!" pp_diag diag + in + match m.kind with + | Some Display.Error -> display_diag Error + | Some Display.Warning -> display_diag Warning + | Some Display.Response -> + Format.printf "=> %t\n%!" m.msg + | _ -> display_diag Note + end; + Lwt.return ()); + status_to_exit @@ M.make (File (Fpath.v f))) + $ Arg.value P.Opts.Opts.verbosity + $ file) + in + let version_cmd : int Cmd.t = + Cmd.v + (Cmd.info "version" ~doc:"Display version information") + Term.( + const (fun () -> + print_endline ("STELF version " ^ version); + Fmt_tty.setup_std_outputs (); + Display.fmt Format.std_formatter @@ Pal.logo; + 0) + $ const ()) + in + let help_cmd : int Cmd.t = + Cmd.v + (Cmd.info "help" ~doc:"Display help information") + Term.( + const (fun () -> + print_endline ("STELF version " ^ version); + Fmt_tty.setup_std_outputs (); + Display.fmt Format.std_formatter @@ Pal.logo; + 0) + $ const ()) (* TODO Make this work *) + in + let main_cmd = + Cmd.group + (Cmd.info "stelf" ~version ~doc:"The STELF proof assistant") + [ repl_cmd; check_cmd; version_cmd; help_cmd ] + in + Basis.OS.Process.exit (Cmd.eval' main_cmd) diff --git a/new-tests/output/pages/computation-and-deduction-2009-20090203.lf b/new-tests/output/pages/computation-and-deduction-2009-20090203.lf new file mode 100644 index 0000000..ed1c96e --- /dev/null +++ b/new-tests/output/pages/computation-and-deduction-2009-20090203.lf @@ -0,0 +1,139 @@ +%%! title: "Computation and Deduction 2009/20090203" +%{! Code from class, March 2. + +```twelf +ttp : type. %name ttp T. + +tnum : ttp. +tnot : ttp -> ttp. +tprod : ttp -> ttp -> ttp. + +texp : type. %name texp E. +tval : type. %name tval V. + +tlet : tval -> (tval -> texp) -> texp. +tpi1 : tval -> (tval -> texp) -> texp. +tpi2 : tval -> (tval -> texp) -> texp. +tifz : tval -> texp -> (tval -> texp) -> texp. +tapp : tval -> tval -> texp. + +tz : tval. +ts : tval -> tval. +tpair : tval -> tval -> tval. +tlam : ttp -> (tval -> texp) -> tval. + +tof : tval -> ttp -> type. +twf : texp -> type. + +tof/z : tof tz tnum. + +tof/s : tof (ts V) tnum + <- tof V tnum. + +tof/pair : tof (tpair V1 V2) (tprod T1 T2) + <- tof V1 T1 + <- tof V2 T2. + +tof/lam : tof (tlam T ([x] E x)) (tnot T) + <- ({x} tof x T -> twf (E x)). + +twf/let : twf (tlet V ([x] E x)) + <- tof V T + <- ({x} tof x T -> twf (E x)). + +twf/pi1 : twf (tpi1 V ([x] E x)) + <- tof V (tprod T1 T2) + <- ({x} tof x T1 -> twf (E x)). + +twf/pi2 : twf (tpi2 V ([x] E x)) + <- tof V (tprod T1 T2) + <- ({x} tof x T2 -> twf (E x)). + +twf/ifz : twf (tifz V E0 ([x] E1 x)) + <- tof V tnum + <- twf E0 + <- ({x} tof x tnum -> twf (E1 x)). + +twf/app : twf (tapp V1 V2) + <- tof V1 (tnot T) + <- tof V2 T. + +ttrans : tp -> ttp -> type. + +ttrans/num : ttrans num tnum. + +ttrans/arr : ttrans (arr T1 T2) (tnot (tprod T1' (tnot T2'))) + <- ttrans T1 T1' + <- ttrans T2 T2'. + +suspend : ttp -> tval -> tval + = [t] [v] (tlam (tnot t) ([f] tapp f v)). + +trans : term -> tp -> tval -> type. + +trans/z : trans z num (suspend tnum tz). + +trans/s : trans (s E) num (tlam (tnot tnum) ([f] tapp V (tlam tnum ([x] tapp f (ts x))))) + <- trans E num V. + +trans/ifz : trans (ifz E E0 ([x] E1 x)) T + (tlam (tnot T') + ([f] tapp V (tlam tnum ([y] tifz y (tapp V0 f) + ([y'] tlet (tlam (tnot tnum) ([g] tapp g y')) + ([x'] tapp (V1 x') f)))))) + <- ttrans T T' + <- trans E num V + <- trans E0 T V0 + <- ({x} {x'} trans x num x' -> trans (E1 x) T (V1 x')). + +trans/lam : trans (lam T1 ([x] E x)) (arr T1 T2) + (suspend (tnot (tprod T1' (tnot T2'))) + (tlam (tprod T1' (tnot T2')) + ([y] tpi1 y + ([y'] (tpi2 y + ([g] tlet (tlam (tnot T1') ([w] tapp w y')) + ([x'] tapp (V x') g))))))) + <- ttrans T1 T1' + <- ttrans T2 T2' + <- ({x} {x'} trans x T1 x' -> trans (E x) T2 (V x')). + +trans/app : trans (app E1 E2) T2 + (tlam (tnot T2') + ([f] tapp V1 + (tlam (tnot (tprod T1' (tnot T2'))) + ([x1] tapp V2 (tlam T1' ([x2] tapp x1 (tpair x2 f))))))) + <- ttrans T1 T1' + <- ttrans T2 T2' + <- trans E1 (arr T1 T2) V1 + <- trans E2 T1 V2. + +tof/suspend : tof (suspend T V) (tnot (tnot T)) + <- tof V T + = [D : tof V T] (tof/lam ([x] [d : tof x (tnot T)] (twf/app D d))). + +correct : trans E T V -> ttrans T T' -> tof V (tnot (tnot T')) -> type. +%mode correct +D1 -D2 -D3. +correct' : trans E T V -> ttrans T T' -> tof V (tnot (tnot T')) -> type. +%mode correct' +D1 +D2 -D3. + + +- : correct trans/z ttrans/num (tof/suspend tof/z). + +%block transbind : some {T : tp} {T' : ttp} {DTT : ttrans T T'} + block + {x : term} {dx : of x T} + {x' : tval} {dx' : tof x' (tnot (tnot T'))} + {dt : trans x T x'} + {thm : correct dt DTT dx'}. + + + +%worlds (transbind) (correct _ _ _). +%total D (correct D _ _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090203). +!}% \ No newline at end of file diff --git a/new-tests/output/pages/computation-and-deduction-2009-20090316.lf b/new-tests/output/pages/computation-and-deduction-2009-20090316.lf new file mode 100644 index 0000000..133fa30 --- /dev/null +++ b/new-tests/output/pages/computation-and-deduction-2009-20090316.lf @@ -0,0 +1,177 @@ +%%! title: "Computation and Deduction 2009/20090316" +%{! Code from class, March 16. + +```twelf +ttp : type. %name ttp T. + +tnum : ttp. +tnot : ttp -> ttp. +tprod : ttp -> ttp -> ttp. + +texp : type. %name texp E. +tval : type. %name tval V. + +tlet : tval -> (tval -> texp) -> texp. +tpi1 : tval -> (tval -> texp) -> texp. +tpi2 : tval -> (tval -> texp) -> texp. +tifz : tval -> texp -> (tval -> texp) -> texp. +tapp : tval -> tval -> texp. + +tz : tval. +ts : tval -> tval. +tpair : tval -> tval -> tval. +tlam : ttp -> (tval -> texp) -> tval. + +tof : tval -> ttp -> type. +twf : texp -> type. + +tof/z : tof tz tnum. + +tof/s : tof (ts V) tnum + <- tof V tnum. + +tof/pair : tof (tpair V1 V2) (tprod T1 T2) + <- tof V1 T1 + <- tof V2 T2. + +tof/lam : tof (tlam T ([x] E x)) (tnot T) + <- ({x} tof x T -> twf (E x)). + +twf/let : twf (tlet V ([x] E x)) + <- tof V T + <- ({x} tof x T -> twf (E x)). + +twf/pi1 : twf (tpi1 V ([x] E x)) + <- tof V (tprod T1 T2) + <- ({x} tof x T1 -> twf (E x)). + +twf/pi2 : twf (tpi2 V ([x] E x)) + <- tof V (tprod T1 T2) + <- ({x} tof x T2 -> twf (E x)). + +twf/ifz : twf (tifz V E0 ([x] E1 x)) + <- tof V tnum + <- twf E0 + <- ({x} tof x tnum -> twf (E1 x)). + +twf/app : twf (tapp V1 V2) + <- tof V1 (tnot T) + <- tof V2 T. + +ttrans : tp -> ttp -> type. + +ttrans/num : ttrans num tnum. + +ttrans/arr : ttrans (arr T1 T2) (tnot (tprod T1' (tnot T2'))) + <- ttrans T1 T1' + <- ttrans T2 T2'. + +suspend : ttp -> tval -> tval + = [t] [v] (tlam (tnot t) ([f] tapp f v)). + +trans : term -> tp -> tval -> type. + +trans/z : trans z num (suspend tnum tz). + +trans/s : trans (s E) num (tlam (tnot tnum) ([f] tapp V (tlam tnum ([x] tapp f (ts x))))) + <- trans E num V. + +trans/ifz : trans (ifz E E0 ([x] E1 x)) T + (tlam (tnot T') + ([f] tapp V (tlam tnum ([y] tifz y (tapp V0 f) + ([y'] tlet (tlam (tnot tnum) ([g] tapp g y')) + ([x'] tapp (V1 x') f)))))) + <- ttrans T T' + <- trans E num V + <- trans E0 T V0 + <- ({x} {x'} trans x num x' -> trans (E1 x) T (V1 x')). + +trans/lam : trans (lam T1 ([x] E x)) (arr T1 T2) + (suspend (tnot (tprod T1' (tnot T2'))) + (tlam (tprod T1' (tnot T2')) + ([y] tpi1 y + ([y'] (tpi2 y + ([g] tlet (suspend T1' y') + ([x'] tapp (V x') g))))))) + <- ttrans T1 T1' + <- ttrans T2 T2' + <- ({x} {x'} trans x T1 x' -> trans (E x) T2 (V x')). + +trans/app : trans (app E1 E2) T2 + (tlam (tnot T2') + ([f] tapp V1 + (tlam (tnot (tprod T1' (tnot T2'))) + ([x1] tapp V2 (tlam T1' ([x2] tapp x1 (tpair x2 f))))))) + <- ttrans T1 T1' + <- ttrans T2 T2' + <- trans E1 (arr T1 T2) V1 + <- trans E2 T1 V2. + +tof/suspend : tof (suspend T V) (tnot (tnot T)) + <- tof V T + = [D : tof V T] (tof/lam ([x] [d : tof x (tnot T)] (twf/app D d))). + + +correct : trans E T V -> ttrans T T' -> tof V (tnot (tnot T')) -> type. +%mode correct +D1 -D2 -D3. + +correct' : trans E T V -> ttrans T T' -> tof V (tnot (tnot T')) -> type. +%mode correct' +D1 +D2 -D3. + +- : correct trans/z ttrans/num (tof/suspend tof/z). + +- : correct + (trans/s + (D : trans E num V)) + ttrans/num + (tof/lam + ([f] [df:tof f (tnot tnum)] + twf/app + (tof/lam ([x] [dx:tof x tnum] (twf/app (tof/s dx) df))) + D')) + <- correct' D ttrans/num (D': tof V (tnot (tnot tnum))). + +- : correct + (trans/lam + (D3: ({x} {x'} trans x T1 x' -> trans (E x) T2 (V x'))) + (D2: ttrans T2 T2') + (D1: ttrans T1 T1')) + (ttrans/arr D2 D1) + (tof/suspend + (tof/lam + ([y] [dy:tof y (tprod T1' (tnot T2'))] + twf/pi1 + ([y'][dy': tof y' T1'] + twf/pi2 + ([g][dg:tof g (tnot T2')] + (twf/let + ([x'] [dx':tof x' (tnot (tnot T1'))] + twf/app + dg + (DV x' dx')) + (tof/suspend dy'))) + dy) + dy))) + <- ({x} {x'} {dx': tof x' (tnot (tnot T1'))} + {dt: trans x T1 x'} + correct dt D1 dx' + -> correct' (D3 x x' dt) D2 (DV x' dx' : tof (V x') (tnot (tnot T2')))). + + +%block transbind : some {T : tp} {T' : ttp} {DTT : ttrans T T'} + block + {x : term} + {x' : tval} {dx' : tof x' (tnot (tnot T'))} + {dt : trans x T x'} + {thm : correct dt DTT dx'}. + + +%worlds (transbind) (correct _ _ _) (correct' _ _ _). +%total (D D') (correct D _ _) (correct' D' _ _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090316). +!}% \ No newline at end of file diff --git a/new-tests/output/pages/computation-and-deduction-2009-20090408.lf b/new-tests/output/pages/computation-and-deduction-2009-20090408.lf new file mode 100644 index 0000000..43db262 --- /dev/null +++ b/new-tests/output/pages/computation-and-deduction-2009-20090408.lf @@ -0,0 +1,419 @@ +%%! title: "Computation and Deduction 2009/20090408" +%{! Code from class, April 8. + +```twelf +tp : type. %name tp A. + +tensor : tp -> tp -> tp. +1 : tp. +with : tp -> tp -> tp. +top : tp. +plus : tp -> tp -> tp. +0 : tp. +lolli : tp -> tp -> tp. +! : tp -> tp. + +exp : type. %name exp M x. + +tens : exp -> exp -> exp. +lett : exp -> (exp -> exp -> exp) -> exp. +pair : exp -> exp -> exp. +pi1 : exp -> exp. +pi2 : exp -> exp. +star : exp. +lets : exp -> exp -> exp. +unit : exp. +in1 : exp -> exp. +in2 : exp -> exp. +case : exp -> (exp -> exp) -> (exp -> exp) -> exp. +abort : exp -> exp. +lam : tp -> (exp -> exp) -> exp. +app : exp -> exp -> exp. +bang : exp -> exp. +letb : exp -> (exp -> exp) -> exp. + +linear : (exp -> exp) -> type. +of : exp -> tp -> type. + +linear/var : linear ([x] x). + +linear/tens1 : linear ([x] tens (M x) N) + <- linear ([x] M x). + +linear/tens2 : linear ([x] tens M (N x)) + <- linear ([x] N x). + +of/tens : of (tens M N) (tensor A B) + <- of M A + <- of N B. + +linear/lett1 : linear ([z] lett (M z) ([x] [y] N x y)) + <- linear ([x] M x). + +linear/lett2 : linear ([z] lett M ([x] [y] N z x y)) + <- ({x} {y} linear ([z] N z x y)). + +of/lett : of (lett M ([x] [y] N x y)) C + <- of M (tensor A B) + <- ({x:exp} of x A -> {y:exp} of y B -> of (N x y) C) + <- ({y:exp} linear ([x] N x y)) + <- ({x:exp} linear ([y] N x y)). + +linear/pair : linear ([x] pair (M x) (N x)) + <- linear ([x] M x) + <- linear ([x] N x). + +of/pair : of (pair M N) (with A B) + <- of M A + <- of N B. + +linear/pi1 : linear ([x] pi1 (M x)) + <- linear ([x] M x). + +linear/pi2 : linear ([x] pi2 (M x)) + <- linear ([x] M x). + +of/pi1 : of (pi1 M) A + <- of M (with A B). + +of/pi2 : of (pi1 M) B + <- of M (with A B). + +of/star : of star 1. + +linear/lets1 : linear ([x] lets (M x) N) + <- linear ([x] M x). + +linear/lets2 : linear ([x] lets M (N x)) + <- linear ([x] N x). + +of/lets : of (lets M N) C + <- of M 1 + <- of N C. + +linear/unit : linear ([x] unit). + +of/unit : of unit top. + +linear/in1 : linear ([x] in1 (M x)) + <- linear ([x] M x). + +linear/in2 : linear ([x] in2 (M x)) + <- linear ([x] M x). + +of/in1 : of (in1 M) (plus A B) + <- of M A. + +of/in2 : of (in2 M) (plus A B) + <- of M B. + +linear/case1 : linear ([z] case (M z) ([x] N x) ([x] O x)) + <- linear ([z] M z). + +linear/case2 : linear ([z] case M ([x] N z x) ([x] O z x)) + <- ({x:exp} linear ([z] N z x)) + <- ({x:exp} linear ([z] O z x)). + +of/case : of (case M ([x] N x) ([x] O x)) C + <- of M (plus A B) + <- ({x} of x A -> of (N x) C) + <- ({x} of x B -> of (O x)C) + <- linear ([x] N x) + <- linear ([x] O x). + +linear/abort1 : linear ([x] abort (M x)) + <- linear ([x] M x). + +linear/abort2 : linear ([x] abort M). + +of/abort : of (abort M) C + <- of M O. + +linear/lam : linear ([x] (lam A ([y] (M x y)))) + <- {y:exp} linear ([x] M x y). + +of/lam : of (lam A ([x] M x)) (lolli A B) + <- ({x} (of x A) -> of (M x) B) + <- linear ([x] M x). + +linear/app1 : linear ([x] (app (M x) N)) + <- linear ([x] M x). + +linear/app2 : linear ([x] (app M (N x))) + <- linear ([x] N x). + +of/app : of (app M N) B + <- of M (lolli A B) + <- of N A. + +of/bang : of (bang M) (! A) + <- of M A. + +linear/letb1 : linear ([x] (letb (M x) ([y] (N y)))) + <- linear ([x] M x). + +linear/letb2 : linear ([x] (letb M ([y] N x y))) + <- ({y} linear ([x] N x y)). + +of/letb : of (letb M ([x] N x)) C + <- of M (! A) + <- ({x} (of x A) -> of (N x) C). + + +reduce : exp -> exp -> type. + +reduce/lam : reduce (lam A ([x] M x)) (lam A ([x] N x)) + <- ({x} reduce (M x) (N x)). + +reduce/app1 : reduce (app M N) (app M' N) + <- reduce M M'. + +reduce/app2 : reduce (app M N) (app M N') + <- reduce N N'. + +reduce/beta : reduce (app (lam A ([x] M x)) N) (M N). + +%block block : block {x:exp}. +%block bind : some {A:tp} block {x:exp} {d : of x A}. + +exp-eq : exp -> exp -> type. +exp-eq/i : exp-eq M M. + +exp-compat: {M : exp -> exp} exp-eq N N' -> exp-eq (M N) (M N') -> type. +%mode exp-compat +I +I1 -O. + +- : exp-compat M _ exp-eq/i. + +%worlds (block) (exp-compat _ _ _). +%total {} (exp-compat _ _ _). + + +lam-compat: ({x:exp} exp-eq (N x) (N' x)) -> exp-eq (lam A N) (lam A N') -> type. +%mode +{N:exp -> exp} +{N':exp -> exp} +{A:tp} +{D1:{x:exp} exp-eq (N x) (N' x)} + -{D2:exp-eq (lam A ([x:exp] N x)) (lam A ([x:exp] N' x))} (lam-compat D1 D2). + +-: lam-compat ([x] exp-eq/i) exp-eq/i. + +%worlds (block) (lam-compat _ _). +%total {} (lam-compat _ _). + +linear-resp: ({x:exp} exp-eq (M x) (N x) ) -> linear M -> linear N -> type. +%mode linear-resp +I +I2 -O. + +-: linear-resp _ D D. + +%worlds (block) (linear-resp _ _ _). +%total {} (linear-resp _ _ _). + +reduce-closed : ({x:exp} reduce M (N x)) -> ({x:exp} exp-eq N' (N x) ) -> type. +%mode reduce-closed +D -D'. + +-: reduce-closed ([x] reduce/app1 (Dreduce x : reduce M1 (N1 x)) ) Deq' + <- reduce-closed Dreduce (Deq : {x} exp-eq N1' (N1 x)) + <- ({x} exp-compat ([y] app y M2) (Deq x) (Deq' x : exp-eq (app N1' M2) (app (N1 x) M2))). + +-: reduce-closed ([x] reduce/app2 (Dreduce x :reduce M2 (N2 x))) Deq' + <- reduce-closed Dreduce (Deq : {x} exp-eq N2' (N2 x)) + <- ({x} exp-compat ([y] app M1 y) (Deq x) (Deq' x : exp-eq (app M1 N2') (app M1 (N2 x)))). + +-: reduce-closed ([x] reduce/lam (Dreduce x : {y} reduce (M y) (N x y))) Deq' + <- ({y} reduce-closed ([x] Dreduce x y) ([x] Deq x y: exp-eq (N' y) (N x y))) + <- ({x} lam-compat ([y] Deq x y) (Deq' x : exp-eq (lam A ([y] (N' y))) (lam A ([y] (N x y))))). + +-: reduce-closed ([x] reduce/beta : reduce (app (lam A ([y] M y)) N) (M N)) ([x] exp-eq/i). + +%worlds (block) (reduce-closed _ _). +%total D (reduce-closed D _). + + +linear-compose: linear ([x] M x) -> linear ([y] N y) -> linear ([y] M (N y)) -> type. +%mode linear-compose +I +I1 -O. + +-: linear-compose linear/var D D. + +-: linear-compose linear/unit _ linear/unit. + +-: linear-compose (linear/app1 (D1 :linear ([x] M1 x)) ) (D2:linear ([y] N y)) + (linear/app1 D) + <- linear-compose D1 D2 (D: linear ([y] M1 (N y))). + +-: linear-compose (linear/app2 (D1 :linear ([x] M1 x)) ) (D2 : linear ([y] N y)) + (linear/app2 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/tens1 (D1 : linear ([x] M1 x))) + (D2 : linear ([y] N y)) + (linear/tens1 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/tens2 (D1 : linear ([x] M1 x))) + (D2 : linear ([y] N y)) + (linear/tens2 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/lett1 (D1 : linear ([x] M1 x))) + (D2 : linear ([y] N y)) + (linear/lett1 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/lett2 (D1 : {x} {y} linear ([z] M1 x y z))) + (D2 : linear ([k] N k)) + (linear/lett2 D ) + <- ({x} {y} linear-compose (D1 x y) D2 (D x y)). + +-: linear-compose (linear/pair (X1 : linear ([x] M1 x)) (X2 : linear ([y] M2 y))) + D1 + (linear/pair A B) + <- linear-compose X1 D1 A + <- linear-compose X2 D1 B. +-: linear-compose (linear/pi1 (D1 : linear ([x] M1 x))) + D2 + (linear/pi1 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/pi2 (D1 : linear ([x] M1 x))) + D2 + (linear/pi2 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/lets1 (D1 : linear ([x] M1 x))) + D2 + (linear/lets1 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/lets2 (D1 : linear ([x] M1 x))) + D2 + (linear/lets2 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/in1 (D1 : linear ([x] M1 x))) + D2 + (linear/in1 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/in2 (D1 : linear ([x] M1 x))) + D2 + (linear/in2 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/case1 (D1 : linear ([x] M1 x))) + D2 + (linear/case1 D) + <- linear-compose D1 D2 D. + +-: linear-compose (linear/case2 ([x] D1 x) ([x] D2 x)) D3 (linear/case2 D4 D5) + <- ({x} linear-compose (D1 x) D3 (D4 x)) + <- ({x} linear-compose (D2 x) D3 (D5 x)). + +-: linear-compose (linear/abort1 (D1 : linear ([x] M1 x))) + D2 + (linear/abort1 D) + <- linear-compose D1 D2 D. + +-: linear-compose (linear/abort2) _ linear/abort2. + +-: linear-compose (linear/lam ([y] X1 y)) X2 (linear/lam D) + <- ({x} linear-compose (X1 x) X2 (D x)). + +-: linear-compose (linear/letb1 (D1 : linear ([x] M1 x))) + D2 + (linear/letb1 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/letb2 ([y] X1 y)) X2 (linear/letb2 D) + <- ({x} linear-compose (X1 x) X2 (D x)). + +%worlds (block) (linear-compose _ _ _). +%total (R) (linear-compose R _ _). + + +srl : ({x} (of x A) -> of (M x) B) -> linear ([x] M x) -> ({x} reduce (M x) (M' x)) -> linear ([x] M' x) -> type. +%mode srl +D1 +D2 +D3 -D4. + +- : srl + ([x] [d:of x A] of/lam (DDD x) (Dof x d : {y} of y B -> of (M x y) C)) + (linear/lam (Dlinear : {y} linear ([x] M x y))) + ([x] reduce/lam (Dreduce x : {y} reduce (M x y) (N x y))) + (linear/lam ([y] Dlinear' y)) + <- ({y} {dy:of y B} srl ([x] [dx:of x A] Dof x dx y dy) (Dlinear y) ([x] Dreduce x y) + (Dlinear' y : linear ([x] N x y))). + +- : srl + ([x] [d:of x A] of/app _ (Dof x d : of (M x) (lolli B C))) + (linear/app1 (Dlinear : linear ([x] M x))) + ([x] reduce/app1 (Dreduce x : reduce (M x) (M' x))) + (linear/app1 Dlinear') + <- srl Dof Dlinear Dreduce (Dlinear' : linear ([x] M' x)). + +- : srl + _ + (linear/app2 (Dlinear : linear ([x] N x))) + ([x] reduce/app1 (Dreduce x : reduce M (M' x))) + Dlinear' + <- reduce-closed Dreduce (Deq: {x} exp-eq M'' (M' x) ) + <- ({x} exp-compat ([z] app z (N x)) (Deq x) (Deq' x : exp-eq (app M'' (N x) ) (app (M' x) (N x)))) + <- linear-resp Deq' (linear/app2 Dlinear) (Dlinear' : linear ([x] app (M' x) (N x))). + +- : srl + _ + (linear/app1 (linear/lam (Dlinear:{y} linear ([x] M x y)))) + ([x] reduce/beta) + (Dlinear _). + +-: srl + ([x][d:of x A] of/app _ (of/lam (Dlinear' x : linear ([y] M y)) _ )) + (linear/app2 (Dlinear : linear ([x] N x) )) + ([x] reduce/beta) + Dlinear'' + <- linear-compose (Dlinear' unit) Dlinear (Dlinear'' : linear ([x] M (N x))). + +-: srl + ([x] [d : of x A] of/app (Dof x d : of (N x) B) _) + (linear/app2 (Dlinear : linear ([x] N x))) + ([x] reduce/app2 (Dreduce x : reduce (N x) (N' x))) + (linear/app2 Dlinear') + <- srl Dof Dlinear Dreduce (Dlinear' : linear ([x] N' x)). + +-: srl + _ + (linear/app1 (Dlinear : linear ([x] M x))) + ([x] reduce/app2 (Dreduce x : reduce N (N' x))) + Dlinear' + <- reduce-closed Dreduce (Deq : {x} exp-eq N'' (N' x)) + <- ({x} exp-compat ([z] app (M x) z) (Deq x) (Deq' x : exp-eq (app (M x) N'') (app (M x) (N' x)))) + <- linear-resp Deq' (linear/app1 Dlinear) (Dlinear' : linear ([x] app (M x) (N' x))). + +%worlds (bind) (srl _ _ _ _). +%total D (srl _ _ D _). + + +sr : of M A -> reduce M M' -> of M' A -> type. +%mode sr +D1 +D2 -D3. + +- : sr + (of/app (DofN : of N A) (DofM : of M (lolli A B))) + (reduce/app1 (Dreduce : reduce M M')) + (of/app DofN DofM') + <- sr DofM Dreduce (DofM' : of M' (lolli A B)). + +- : sr + (of/app (DofN : of N A) (DofM : of M (lolli A B))) + (reduce/app2 (Dreduce : reduce N N')) + (of/app DofN' DofM) + <- sr DofN Dreduce (DofN' : of N' A). + +- : sr + (of/app (DofN : of N A) (of/lam _ (DofM : {x} (of x A) -> of (M x) B))) + reduce/beta + (DofM N DofN). + +- : sr + (of/lam + (Dlinear : linear ([x] M x)) + (DofM : {x} (of x A) -> of (M x) B)) + (reduce/lam (Dreduce : {x} reduce (M x) (N x))) + (of/lam + Dlinear' + DofN) + <- ({x} {dx:of x A} + sr (DofM x dx) (Dreduce x) (DofN x dx : of (N x) B)) + <- srl DofM Dlinear Dreduce (Dlinear': linear ([x] N x)). + +%worlds (bind) (sr _ _ _). +%total D1 (sr _ D1 _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090408). +!}% \ No newline at end of file diff --git a/new-tests/output/pages/lexicographical-orderings-with-density.lf b/new-tests/output/pages/lexicographical-orderings-with-density.lf new file mode 100644 index 0000000..080b2b7 --- /dev/null +++ b/new-tests/output/pages/lexicographical-orderings-with-density.lf @@ -0,0 +1,97 @@ +%%! title: "Lexicographical orderings with density" +%{! This is a fairly advanced example, but it illustrates a number of useful things. First, it adheres very closely to the [Twelf style guide](/wiki/twelf-style-guide/). Second, it has proofs of ``nat-less-trans`` and ``nat-less-immsucc`` which you usually need to get any mileage out of the natural numbers. Third, it is an example of how to prove ``uninhabited/void/false/absurd`` in order to user [reasoning from false](/wiki/reasoning-from-false/). + +This is a lexicographical ordering with the property that it has a less-than relation that is irreflexive and transitive, like the natural numbers. In addition, it has the additional property that the ordering is "dense" in that there exists an index between any two indices. This ordering should be isomorphic to the non-negative rationals. Density is admissible in this ordering without having to deal with multiplication, as would be necessary in a more obvious encoding of the non-negative rationals. !}% +%{! (options removed from twelftag: check="true") !}% +% natural numbers +%sort nat %. +%term nat/z nat %. +%term nat/s %pi nat %-> nat %. +% a list of natural numbers. we will use it as a lexicographical ordering +%sort lex %. +%term lex/z lex nat/z %. +%term lex/s %pi nat %-> (lex N) %-> (lex (nat/s N)) %. +% the less-than relation on natural numbers +%sort nat-less %. +%term nat-less/z nat-less nat/z (nat/s N) %. +%term nat-less/s %pi (nat-less (nat/s N1) (nat/s N2)) %<- (nat-less N1 N2) %. +% the less-than relation on our lexicographical ordering. +%sort lex-less %. +%term lex-less/z lex-less lex/z (lex/s _ _) %. +%term lex-less/eq %pi (lex-less (lex/s N1 NL1) (lex/s N1 NL2)) %<- (lex-less NL1 NL2) %. +%term lex-less/gt %pi (lex-less (lex/s N1 _) (lex/s N2 _)) %<- (nat-less N2 N1) %. +%sort nat-less-trans %. +%mode nat-less-trans %in %in %out %. +%term _ nat-less-trans nat-less/z _ nat-less/z %. +%term _ + %pi (nat-less-trans (nat-less/s N1) (nat-less/s N2) (nat-less/s N3)) + %<- (nat-less-trans N1 N2 N3) %. +%worlds () (nat-less-trans _ _ _) %. +%total {D1} (nat-less-trans D1 _ _) %. +%sort nat-less-immsucc {N} %. +%mode nat-less-immsucc %in %out %. +%term _ nat-less-immsucc nat/z nat-less/z %. +%term _ %pi (nat-less-immsucc (nat/s N1) (nat-less/s NL)) %<- (nat-less-immsucc N1 NL) %. +%worlds () (nat-less-immsucc _ _) %. +%total {D1} (nat-less-immsucc D1 _) %. +%sort lex-less-trans %. +%mode lex-less-trans %in %in %out %. +%term _ lex-less-trans lex-less/z _ lex-less/z %. +%term _ + %pi (lex-less-trans (lex-less/eq NL1) (lex-less/eq NL2) (lex-less/eq NL3)) + %<- (lex-less-trans NL1 NL2 NL3) %. +%term _ + %pi (lex-less-trans (lex-less/gt NL1) (lex-less/gt NL2) (lex-less/gt NL3)) + %<- (nat-less-trans NL2 NL1 NL3) %. +%term _ lex-less-trans (lex-less/gt NL) (lex-less/eq _) (lex-less/gt NL) %. +%term _ lex-less-trans (lex-less/eq _) (lex-less/gt NL) (lex-less/gt NL) %. +%worlds () (lex-less-trans _ _ _) %. +%total {D1} (lex-less-trans D1 _ _) %. +% for any lexicographical index LL, there exists an index LL' that is greater +%sort lex-less-succ {LL} %. +%mode lex-less-succ %in %out %. +%term _ lex-less-succ lex/z (%the (lex-less _ (lex/s nat/z lex/z)) lex-less/z) %. +%term _ %pi (lex-less-succ (lex/s N NL) (lex-less/eq NLL)) %<- (lex-less-succ NL NLL) %. +%worlds () (lex-less-succ _ _) %. +%total {N} (lex-less-succ N _) %. +%sort lex-less-dense %. +%mode lex-less-dense %in %out %out %. +%term _ + %pi (lex-less-dense lex-less/z lex-less/z (%the (lex-less (lex/s (nat/s N) lex/z) _) (lex-less/gt NL))) + %<- (nat-less-immsucc N NL) %. +%term _ + %pi (lex-less-dense (lex-less/eq LL) (lex-less/eq LL1) (lex-less/eq LL2)) + %<- (lex-less-dense LL LL1 LL2) %. +%term _ + %pi (lex-less-dense (lex-less/gt NL) (lex-less/eq LL) (lex-less/gt NL)) + %<- (lex-less-succ _ LL) %. +%worlds () (lex-less-dense _ _ _) %. +%total {D1} (lex-less-dense D1 _ _) %. +%sort uninhabited %. +%freeze uninhabited %. +%sort nat-less-refl-uninhabited %. +%mode nat-less-refl-uninhabited %in %out %. +%term _ + %pi (nat-less-refl-uninhabited (nat-less/s NL) DU) + %<- (nat-less-refl-uninhabited NL DU) %. +%worlds () (nat-less-refl-uninhabited _ _) %. +%total {D1} (nat-less-refl-uninhabited D1 _) %. +%sort lex-less-refl-uninhabited %. +%mode lex-less-refl-uninhabited %in %out %. +%term _ + %pi (lex-less-refl-uninhabited (lex-less/eq LL) DU) + %<- (lex-less-refl-uninhabited LL DU) %. +%term _ + %pi (lex-less-refl-uninhabited (lex-less/gt DL) DU) + %<- (nat-less-refl-uninhabited DL DU) %. +%worlds () (lex-less-refl-uninhabited _ _) %. +%total {DU} (lex-less-refl-uninhabited DU _) %. +%{! [Category:Twelf code](/wiki/category-twelf-code/) + +\{\{case study\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Lexicographical_orderings_with_density). +!}% \ No newline at end of file diff --git a/new-tests/output/pages/popl-tutorial-church-rosser-problem.lf b/new-tests/output/pages/popl-tutorial-church-rosser-problem.lf new file mode 100644 index 0000000..3885837 --- /dev/null +++ b/new-tests/output/pages/popl-tutorial-church-rosser-problem.lf @@ -0,0 +1,283 @@ +%%! title: "POPL Tutorial/Church Rosser (Problem)" +%{! + +In this exercise, we will prove the [[w:Church-Rosser theorem|Church-Rosser theorem]]. You are +provided with a syntax and two substitution lemmas, and your task will be +to implement two cases of the diamond lemma - the +**Application-Application** case, and the **Beta-Application** case. + +You will be able to uncomment the ``%total`` at the end when you are +done. + +The solution is [here](/wiki/church-rosser-w-identity-reduction/). + +!}% +%{! ## Syntax !}% +%sort exp %. +%name exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%{! + +When we use this ``%block``, it expresses that we can be working in a +context with arbitrary expression variables. This will be the only world we +will need for this exercise. + +!}% +%block exps [x exp]%. +%{! ## Reduction !}% +%{! + +We can always reduce an expression to itself (which means that we don't +explicitly need a case to handle variables). We can reduce under binders and +reduce both sides of an application "in parallel." If we have a β-redex +``(λx.ea) eb``, then after reducing ``ea`` (with ``x`` free) to +``ea'`` (with ``x`` free) and reducing ``eb`` to ``eb'``, +we can return ``[eb'/x]ea'``, the substitution of ``eb'`` into +``ea'``. + +The ``%worlds`` explicitly states that we will be reducing in a setting +with free variables. + +!}% +%sort reduce %. +%term reduce/id reduce E E %. +%term reduce/lam %pi (reduce (lam E) (lam E')) %<- ({x exp} reduce (E x) (E' x)) %. +%term reduce/app %pi (reduce (app E1 E2) (app E1' E2')) %<- (reduce E1 E1') %<- (reduce E2 E2') %. +%term reduce/beta + %pi (reduce (app (lam E1) E2) (E1' E2')) + %<- ({x exp} reduce (E1 x) (E1' x)) + %<- (reduce E2 E2') %. +%worlds (exps) (reduce _ _) %. +%{! ## Substitution !}% +%{! + +The first substitution theorem says that if we have a term ``e1`` with +``x`` free and a term ``e2`` that reduces to term ``e2'``, +then ``[e2/x]e1`` reduces to ``[e2'/x]e1``. + +The interesting cases are really the first two - if the term is just the free +variable being substituted for, the result follows from the hypothesis. In the +case that the free variable is _not_ free (i.e. the term has no hole), then +the result follows from the reflexivity of reduction. This is more than +sufficient to handle the case of a variable that is not the distinguished free +variable, and all other cases are covered. + +The proof is by induction on the structure of the term with the free variable. + +!}% +%sort substitute1 {E1 %pi exp %-> exp} %. +%mode substitute1 %in %in %out %. +%term _ substitute1 ([x] x) D D %. +%term _ substitute1 ([x] E) _ reduce/id %. +%term _ + %pi (substitute1 ([x] lam ([y] E x y)) D (reduce/lam D')) + %<- ({y} substitute1 ([x] E x y) D (D' y)) %. +%term _ + %pi (substitute1 ([x] app (E1 x) (E2 x)) D (reduce/app D2 D1)) + %<- (substitute1 E1 D D1) + %<- (substitute1 E2 D D2) %. +%worlds (exps) (substitute1 _ _ _) %. +%total E (substitute1 E _ _) %. +%{! + +The second substitution theorem is similar, except that we are reducing +``e1`` with ``x`` free to ``e1'`` as well as by reducing +``e2`` to ``e2'``, showing that ``[e2/x]e1`` reduces to +``[e2'/x]e1'``. + +Proof is by induction on the structure of the reduction of ``e1`` to +``e1'``. We call to the ``substitute1`` lemma +when we "bottom out" at the reflexive case ``reduce/id``. + +!}% +%sort substitute2 {x exp} %. +%mode substitute2 %in %in %out %. +%term _ + %pi (substitute2 ([x] %the (reduce (E x) (E x)) reduce/id) D D') + %<- (substitute1 E D D') %. +%term _ + %pi (substitute2 ([x] reduce/lam ([y] D1 x y)) D2 (reduce/lam D)) + %<- ({y} substitute2 ([x] D1 x y) D2 (D y)) %. +%term _ + %pi (substitute2 ([x] reduce/app (D2 x) (D1 x)) D (reduce/app D2' D1')) + %<- (substitute2 D1 D D1') + %<- (substitute2 D2 D D2') %. +%term _ + %pi (substitute2 ([x] reduce/beta (D2 x) ([y] D1 x y)) D (reduce/beta D2' D1')) + %<- ({y} substitute2 ([x] D1 x y) D (D1' y)) + %<- (substitute2 D2 D D2') %. +%worlds (exps) (substitute2 _ _ _) %. +%total D (substitute2 D _ _) %. +%{! ## The Diamond Property !}% +%{! + +Now we come to the interesting part: the diamond property. + + E + / \ + / \ + E1 E2 + \ / + \ / + E' + +If ``E`` reduces to both ``E1``, and ``E2``, then there is a +common E' such that ``E1`` and ``E2`` both reduce to it. + +!}% +%sort diamond %. +%mode diamond %in %in %out %out %. +%{! ### Identity !}% +%{! + +If either case is the identity, then we are done. + + id: E D: D: E id: + e=>e / \ e=>e2 e=>e1 / \ e=>e + / \ / \ + E E2 E1 E + D: \ /id: id: \ /D: + e=>e2 \ / e2=>e2 e1=>e1\ / e=>e1 + e2 E1 + +!}% +%term _ diamond reduce/id D D reduce/id %. +%term _ diamond D reduce/id reduce/id D %. +%{! ### Lambda-Lambda !}% +%{! + +If both cases are reductions under a binder, we pull the result straight +from the induction hypothesis. + + λx.e by induction: + reduce/lam / \ reduce/lam D1, D2 ---> D1': e1'=>e' + (D1: e=>e1) / \ (D2: e=>e2) D2': e2'=>e' + / \ + λx.e1 λx.e2 + \ / + reduce/lam \ / reduce/lam + D1' \ / D2' + λx.e' + +Note the oversimplification being made in the graphical presentation, in that +the subterms and sub-derivations are not clearly shown to have a free variable. +Twelf will, of course, not allow this sloppiness. + +!}% +%term _ + %pi (diamond (%the (reduce (lam E) (lam E1)) (reduce/lam (%the ({x exp} reduce (E x) (E1 x)) D1))) (%the (reduce (lam E) (lam E2)) (reduce/lam (%the ({x exp} reduce (E x) (E2 x)) D2))) (reduce/lam D1') (reduce/lam D2')) + %<- ({x exp} diamond (D1 x) (D2 x) (%the (reduce (E1 x) (E' x)) (D1' x)) (%the (reduce (E2 x) (E' x)) (D2' x))) %. +%{! ### Application-Application !}% +%{! + +If both cases are applications, we pull the result straight from the +induction hypothesis. + + ea eb by induction + reduce/app / \ reduce/app D1a, D2a ---> D1a': e1a=>ea' + (D1b: eb=>e1b) / \ (D2b: eb=>e2b) D2a': e2a=>ea' + (D1a: ea=>e1a) / \ (D2a: ea=>e2a) D1b, D2b ---> D1b': e1b=>eb' + e1a e1b e2a e2b D2b': e2b=>eb' + \ / + reduce/app \ / reduce/app + D1b' D1a' \ / D2b' D2a' + ea' eb' +!}% +%% WRITE THIS CASE +%{! ### Beta-Beta !}% +%{! + +If both cases are beta reductions, we get the result from performing two +substitutions. + + (λx.ea) eb by induction + reduce/beta / \ reduce/beta D1a, D2a ---> D1a': e1a=>ea' + (D1b: eb=>e1b) / \ (D2b: eb=>e2b) D2a': e2a=>ea' + (D1a: ea=>e1a) / \ (D2a: ea=>e1a) D1b, D2b ---> D1b': e1b=>eb' + [e1b/x]e1a [e2b/x]e2a D2b': e2b=>eb' + \ / + substitute \ / substitute + D1b' into D1a' \ / D2b' into D2a' + [eb'/x]ea +!}% +%term _ + %pi (diamond (%the (reduce (app (lam Ea) Eb) (E1a E1b)) (reduce/beta (%the (reduce Eb E1b) D1b) (%the ({x} reduce (Ea x) (E1a x)) D1a))) (%the (reduce (app (lam Ea) Eb) (E2a E2b)) (reduce/beta (%the (reduce Eb E2b) D2b) (%the ({x} reduce (Ea x) (E2a x)) D2a))) D1 D2) + %<- ({x} diamond (D1a x) (D2a x) (%the (reduce (E1a x) (Ea' x)) (D1a' x)) (%the (reduce (E2a x) (Ea' x)) (D2a' x))) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) + %<- (substitute2 D1a' D1b' (%the (reduce (E1a E1b) (Ea' Eb')) D1)) + %<- (substitute2 D2a' D2b' (%the (reduce (E2a E2b) (Ea' Eb')) D2)) %. +%{! ### Beta-Application !}% +%{! + +If the left-hand side is a β-reduction +``(λx.ea) eb => [e1b/x] e1a`` but the right-hand side is not, then we +know that the right-hand side reduction must be +``(λx.ea) eb => (λx.e2a) e2b``. + +Unfortunately, then we have two options - either ``λx.ea => λx.e2a`` by +``reduce/refl`` and ``reduce/lam``. The ``reduce/refl`` case +is essentially just an inconvenience caused by allowing identity reductions +anywhere (not just at variables). + +The first subcase: + + (λx.ea) eb by induction + reduce/beta / \ reduce/app D1a, D2a ---> D1a': e1a=>ea' + (D1b: eb=>e1b) / \ (D2b: eb=>e2b) D2a': e2a=>ea' + (D1a: ea=>e1a) / \ (reduce/lam D1b, D2b ---> D1b': e1b=>eb' + / \ (D2a: ea=>e2a)) D2b': e1b=>eb' + [e1b/x]e1a (λx.e2a) e2b + \ / + substitute \ / reduce/beta + D1b' into D2a' \ / D2b' D2a' + \ / + [eb'/x]ea' +!}% +%% WRITE THIS CASE +%{! + +The second subcase: + + (λx.ea) eb by induction + reduce/beta / \ reduce/app D1b, D2b ---> D1b': e1b=>eb' + (D1b: eb=>e1b) / \ (D2b: eb=>e2b) D2b': e1b=>eb' + (D1a: ea=>e1a) / \ (reduce/id + / \ ea=>ea)) + [e1b/x]e1a (λx.ea) e2b + \ / + substitute \ / reduce/beta + D1b' into e1a \ / D2b' D1a + \ / + [eb'/x]e1a +!}% +%term _ + %pi (diamond (%the (reduce (app (lam Ea) Eb) (E1a E1b)) (reduce/beta (%the (reduce Eb E1b) D1b) (%the ({x} reduce (Ea x) (E1a x)) D1a))) (%the (reduce (app (lam Ea) Eb) (app (lam Ea) E2b)) (reduce/app (%the (reduce Eb E2b) D2b) (%the (reduce (lam Ea) (lam Ea)) reduce/id))) D1 (reduce/beta D2b' D1a)) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) + %<- (substitute1 E1a D1b' (%the (reduce (E1a E1b) (E1a Eb')) D1)) %. +%{! ### Application-Beta !}% +%{! + +If the right-hand hand side is a β-reduction but the left-hand side is not, we +have to do the same two cases in reverse; we omit the graphics for those +two cases as they are symmetric. + +!}% +%term _ + %pi (diamond (%the (reduce (app (lam Ea) Eb) (app (lam Ea) E1b)) (reduce/app (%the (reduce Eb E1b) D1b) (%the (reduce (lam Ea) (lam Ea)) reduce/id))) (%the (reduce (app (lam Ea) Eb) (E2a E2b)) (reduce/beta (%the (reduce Eb E2b) D2b) (%the ({x} reduce (Ea x) (E2a x)) D2a))) (reduce/beta D1b' D2a) D2) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) + %<- (substitute1 E2a D2b' (%the (reduce (E2a E2b) (E2a Eb')) D2)) %. +%term _ + %pi (diamond (%the (reduce (app (lam Ea) Eb) (app (lam E1a) E1b)) (reduce/app (%the (reduce Eb E1b) D1b) (reduce/lam (%the ({x} reduce (Ea x) (E1a x)) D1a)))) (%the (reduce (app (lam Ea) Eb) (E2a E2b)) (reduce/beta (%the (reduce Eb E2b) D2b) (%the ({x} reduce (Ea x) (E2a x)) D2a))) (reduce/beta D1b' D1a') D2) + %<- ({x} diamond (D1a x) (D2a x) (%the (reduce (E1a x) (Ea' x)) (D1a' x)) (%the (reduce (E2a x) (Ea' x)) (D2a' x))) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) + %<- (substitute2 D2a' D2b' (%the (reduce (E2a E2b) (Ea' Eb')) D2)) %. +%{! Now we are done! We check in the ``exps`` world with free variables. !}% +%worlds (exps) (diamond _ _ _ _) %. +%% %total D1 (diamond D1 D2 _ _). +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Church_Rosser_(Problem)). +!}% \ No newline at end of file diff --git a/new-tests/output/pages/popl-tutorial-properties-of-typing-and-reduction.lf b/new-tests/output/pages/popl-tutorial-properties-of-typing-and-reduction.lf new file mode 100644 index 0000000..ad1a265 --- /dev/null +++ b/new-tests/output/pages/popl-tutorial-properties-of-typing-and-reduction.lf @@ -0,0 +1,392 @@ +%%! title: "POPL Tutorial/Properties of Typing and Reduction" +%{! This problem (or problems) involves proving three properties of typing and reduction for the language L\{num str\} +from Bob Harper's book, [http://www.cs.cmu.edu/~rwh/plbook/book.pdf Practical Foundations for Programming Languages]. + +## The System L\{num str\} !}% +% Natural numbers +%sort nat %. +%term 0 nat %. +%term s %pi nat %-> nat %. +% Addition for natural numbers +%sort plus_op %. +%term plus_op/0 plus_op 0 N N %. +%term plus_op/s %pi (plus_op N M L) %-> (plus_op (s N) M (s L)) %. +% Multiplication for natural numbers +%sort times_op %. +%term times_op/0 times_op 0 N 0 %. +%term times_op/s %pi (times_op (s N) M P) %<- (times_op N M L) %<- (plus_op L M P) %. +%%%%%%%%%%%%%%%% L{num str}: Syntax %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Types +%sort typ %. +%term num typ %. +%term str typ %. +% Expressions. For simplicity, strings are just natural numbers, +% concatenation is addition, and len is the identity. +%sort exp %. +%term nume %pi nat %-> exp %. +%term stre %pi nat %-> exp %. +%term plus %pi exp %-> exp %-> exp %. +%term times %pi exp %-> exp %-> exp %. +%term cat %pi exp %-> exp %-> exp %. +%term len %pi exp %-> exp %. +%term let %pi exp %-> (%pi exp %-> exp) %-> exp %. +%%%%%%%%%%%%%%%% L{num str}: Static semantics (typing judgment) %%%%%%%%%%%%%%% +%sort of %. +%term of/stre of (stre S) str %. +%term of/nume of (nume N) num %. +%term of/plus %pi (of E1 num) %-> (of E2 num) %-> (of (plus E1 E2) num) %. +%term of/times %pi (of E1 num) %-> (of E2 num) %-> (of (times E1 E2) num) %. +%term of/cat %pi (of E1 str) %-> (of E2 str) %-> (of (cat E1 E2) str) %. +%term of/len %pi (of E1 str) %-> (of (len E1) num) %. +%term of/let + %pi (of E1 T1) + %-> ({x} %pi (of x T1) %-> (of (E2 x) T2)) + %-> (of (let E1 ([x] E2 x)) T2) %. +% Assumption block for typing judgment (coming from of/let) +%block of_bind {T typ} [x exp] [ofx of x T]%. +%%%%%%%%%%%%%%%% L{num str}: Dynamic Semantics %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Values +%sort val %. +%term val/stre val (stre N) %. +%term val/nume val (nume N) %. +% Reduction +%sort red %. +%term red/plus/num %pi (plus_op N1 N2 N) %-> (red (plus (nume N1) (nume N2)) (nume N)) %. +%term red/plus1 %pi (red E1 E1') %-> (red (plus E1 E2) (plus E1' E2)) %. +%term red/plus2 %pi (val E1) %-> (red E2 E2') %-> (red (plus E1 E2) (plus E1 E2')) %. +%term red/times/num %pi (times_op N1 N2 N) %-> (red (times (nume N1) (nume N2)) (nume N)) %. +%term red/times1 %pi (red E1 E1') %-> (red (times E1 E2) (times E1' E2)) %. +%term red/times2 %pi (val E1) %-> (red E2 E2') %-> (red (times E1 E2) (times E1 E2')) %. +%term red/cat/str %pi (plus_op S1 S2 S) %-> (red (cat (stre S1) (stre S2)) (stre S)) %. +%term red/cat1 %pi (red E1 E1') %-> (red (cat E1 E2) (cat E1' E2)) %. +%term red/cat2 %pi (val E1) %-> (red E2 E2') %-> (red (cat E1 E2) (cat E1 E2')) %. +%term red/len/str red (len (stre S)) (nume S) %. +%term red/len %pi (red E1 E1') %-> (red (len E1) (len E1')) %. +%term red/let/val %pi (val E1) %-> (red (let E1 ([x] E2 x)) (E2 E1)) %. +%term red/let %pi (red E1 E1') %-> (red (let E1 ([x] E2 x)) (let E1' ([x] E2 x))) %. +%{! ## Problem 1: Unicity of Typing !}% +%%%%%%%%%%%%%%%% Lemma 9.1 (Unicity of typing) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% + +% If G |- E : T1 and G |- E : T2, then T1 = T2. +% Equality of types +%sort eqtyp %. +%term refl_t eqtyp T T %. +% Congruence for typing assumptions. Although the proof doesn't +% introduce any parameters or hypotheses, it is used by a theorem that +% does, and so we need to check it in the extended worlds. +%sort cong_of %. +%mode cong_of %in %in %out %. +%term _ cong_of refl_t OE1 OE1 %. +%worlds (of_bind) (cong_of _ _ _) %. +%total OE1 (cong_of _ OE1 _) %. +% Statement and proof of Lemma. Notice that the worlds in this proof +% are an extension (technically, they subsume) the worlds in the +% congruence lemma above. +%sort lemma9-1 %. +%mode lemma9-1 %in %in %out %. +%term _ lemma9-1 of/stre of/stre refl_t %. +%term _ lemma9-1 of/nume of/nume refl_t %. +%term _ lemma9-1 (of/plus OjE11 OjE12) (of/plus OjE21 OjE22) refl_t %. +%term _ lemma9-1 (of/times OjE11 OjE12) (of/times OjE21 OjE22) refl_t %. +%term _ lemma9-1 (of/cat OjE11 OjE12) (of/cat OjE21 OjE22) refl_t %. +%term _ lemma9-1 (of/len OjE1) (of/len OjE2) refl_t %. +%term _ + %pi (lemma9-1 (of/let OjE11 ([x] [ofx1] OjE12 x ofx1)) (of/let OjE21 ([x] [ofx2] OjE22 x ofx2)) Q) + %<- (lemma9-1 OjE11 OjE21 E1) + %<- (cong_of E1 OjE12 OjE12') + %<- ({x} {ofx2} %pi (lemma9-1 ofx2 ofx2 refl_t) %-> (lemma9-1 (OjE12' x ofx2) (OjE22 x ofx2) Q)) %. +%block lemma_block {T typ} [x exp] [ofx of x T] [u lemma9-1 ofx ofx refl_t]%. +%worlds (lemma_block) (lemma9-1 _ _ _) %. +%total OjT2 (lemma9-1 _ OjT2 _) %. +%{! ## Problem 2: Determinacy of Reduction !}% +%%%%%%%%%%%%%%%% Lemma 10.1 (Determinacy of Reduction) %%%%%%%%%%%%%%%%%%%%%%% +% + +% If E |-> E' and E |-> E'', then E' = E''. +% Equality and congruence for nats +%sort eqnat %. +%term refl_n eqnat N N %. +%sort cong_s %. +%mode cong_s %in %out %. +%term _ cong_s refl_n refl_n %. +%worlds () (cong_s _ _) %. +%total {} (cong_s _ _) %. +% Equality for expressions +%sort eqexp %. +%term refl_e eqexp E E %. +% Congruences for expressions +%sort cong_nume %. +%mode cong_nume %in %out %. +%term _ cong_nume refl_n refl_e %. +%worlds () (cong_nume _ _) %. +%total E (cong_nume E _) %. +%sort cong_stre %. +%mode cong_stre %in %out %. +%term _ cong_stre refl_n refl_e %. +%worlds () (cong_stre _ _) %. +%total E (cong_stre E _) %. +%sort cong_plus1 %. +%mode {%in E1 exp} {%in E2 exp} {%in F exp} {%in E eqexp E1 E2} {%out E' eqexp (plus E1 F) (plus E2 F)} cong_plus1 E E' %. +%term _ cong_plus1 refl_e refl_e %. +%worlds () (cong_plus1 _ _) %. +%total E (cong_plus1 E _) %. +%sort cong_plus2 %. +%mode {%in F1 exp} {%in F2 exp} {%in E exp} {%in E1 eqexp F1 F2} {%out E' eqexp (plus E F1) (plus E F2)} cong_plus2 E1 E' %. +%term _ cong_plus2 refl_e refl_e %. +%worlds () (cong_plus2 _ _) %. +%total E (cong_plus2 E _) %. +%sort cong_times1 %. +%mode {%in E1 exp} {%in E2 exp} {%in F exp} {%in E eqexp E1 E2} {%out E' eqexp (times E1 F) (times E2 F)} cong_times1 E E' %. +%term _ cong_times1 refl_e refl_e %. +%worlds () (cong_times1 _ _) %. +%total E (cong_times1 E _) %. +%sort cong_times2 %. +%mode {%in F1 exp} {%in F2 exp} {%in E exp} {%in E2 eqexp F1 F2} {%out E' eqexp (times E F1) (times E F2)} cong_times2 E2 E' %. +%term _ cong_times2 refl_e refl_e %. +%worlds () (cong_times2 _ _) %. +%total E (cong_times2 E _) %. +%sort cong_cat1 %. +%mode {%in E1 exp} {%in E2 exp} {%in F exp} {%in E eqexp E1 E2} {%out E' eqexp (cat E1 F) (cat E2 F)} cong_cat1 E E' %. +%term _ cong_cat1 refl_e refl_e %. +%worlds () (cong_cat1 _ _) %. +%total E (cong_cat1 E _) %. +%sort cong_cat2 %. +%mode {%in F1 exp} {%in F2 exp} {%in E exp} {%in E3 eqexp F1 F2} {%out E' eqexp (cat E F1) (cat E F2)} cong_cat2 E3 E' %. +%term _ cong_cat2 refl_e refl_e %. +%worlds () (cong_cat2 _ _) %. +%total E (cong_cat2 E _) %. +%sort cong_len %. +%mode cong_len %in %out %. +%term _ cong_len refl_e refl_e %. +%worlds () (cong_len _ _) %. +%total E (cong_len E _) %. +%sort cong_let %. +%mode {%in E1 exp} {%in E2 exp} {%in F %pi exp %-> exp} {%in E eqexp E1 E2} {%out E' eqexp (let E1 ([x exp] F x)) (let E2 ([x exp] F x))} cong_let E E' %. +%term _ cong_let refl_e refl_e %. +%worlds () (cong_let _ _) %. +%total E (cong_let E _) %. +% Uniqueness of plus +%sort plus! %. +%mode plus! %in %in %out %. +%term _ plus! plus_op/0 plus_op/0 refl_n %. +%term _ + %pi (plus! (plus_op/s Pj1) (plus_op/s Pj2) ES) + %<- (plus! Pj1 Pj2 EL) + %<- (cong_s EL ES) %. +%worlds () (plus! _ _ _) %. +%total Pj1 (plus! Pj1 _ _) %. +% Congruence of plus +%sort plus_cong %. +%mode plus_cong %in %in %in %out %. +%term _ plus_cong refl_n plus_op/0 plus_op/0 refl_n %. +%term _ + %pi (plus_cong refl_n (plus_op/s Pj1) (plus_op/s Pj2) ES) + %<- (plus_cong refl_n Pj1 Pj2 EL) + %<- (cong_s EL ES) %. +%worlds () (plus_cong _ _ _ _) %. +%total PjL (plus_cong _ PjL _ _) %. +% Uniqueness of times +%sort times! %. +%mode times! %in %in %out %. +%term _ times! times_op/0 times_op/0 refl_n %. +%term _ + %pi (times! (times_op/s Pj1 Tj1) (times_op/s Pj2 Tj2) ES) + %<- (times! Tj1 Tj2 EL) + %<- (plus_cong EL Pj1 Pj2 ES) %. +%worlds () (times! _ _ _) %. +%total Tj1 (times! Tj1 _ _) %. +% Proof of Lemma 10.1 +%sort lemma10-1 %. +%mode lemma10-1 %in %in %out %. +%term _ + %pi (lemma10-1 (red/plus/num PjN1) (red/plus/num PjN2) EE') + %<- (plus! PjN1 PjN2 EL) + %<- (cong_nume EL EE') %. +%term _ + %pi (lemma10-1 (red/plus1 RjE1) (red/plus1 RjE2) EE') + %<- (lemma10-1 RjE1 RjE2 EE) + %<- (cong_plus1 EE EE') %. +%term _ + %pi (lemma10-1 (red/plus2 _ RjF1) (red/plus2 _ RjF2) EE') + %<- (lemma10-1 RjF1 RjF2 EF) + %<- (cong_plus2 EF EE') %. +%term _ + %pi (lemma10-1 (red/times/num TjN1) (red/times/num TjN2) EE') + %<- (times! TjN1 TjN2 EE) + %<- (cong_nume EE EE') %. +%term _ + %pi (lemma10-1 (red/times1 RjE1) (red/times1 RjE2) EE') + %<- (lemma10-1 RjE1 RjE2 EE) + %<- (cong_times1 EE EE') %. +%term _ + %pi (lemma10-1 (red/times2 _ RjF1) (red/times2 _ RjF2) EE') + %<- (lemma10-1 RjF1 RjF2 EF) + %<- (cong_times2 EF EE') %. +%term _ + %pi (lemma10-1 (red/cat/str PjN1) (red/cat/str PjN2) EE') + %<- (plus! PjN1 PjN2 EE) + %<- (cong_stre EE EE') %. +%term _ + %pi (lemma10-1 (red/cat1 RjE1) (red/cat1 RjE2) EE') + %<- (lemma10-1 RjE1 RjE2 EE) + %<- (cong_cat1 EE EE') %. +%term _ + %pi (lemma10-1 (red/cat2 _ RjF1) (red/cat2 _ RjF2) EE') + %<- (lemma10-1 RjF1 RjF2 EF) + %<- (cong_cat2 EF EE') %. +%term _ + %pi (lemma10-1 (%the (red (len (stre S)) (nume S)) red/len/str) red/len/str EE') + %<- (cong_nume refl_n EE') %. +%term _ + %pi (lemma10-1 (red/len RjE1) (red/len RjE2) EE') + %<- (lemma10-1 RjE1 RjE2 EE) + %<- (cong_len EE EE') %. +%term _ lemma10-1 (red/let/val VjE11) (red/let/val VjE12) refl_e %. +%term _ + %pi (lemma10-1 (red/let RjE1) (red/let RjE2) EE') + %<- (lemma10-1 RjE1 RjE2 EE) + %<- (cong_let EE EE') %. +%worlds () (lemma10-1 _ _ _) %. +%total RjE (lemma10-1 RjE _ _) %. +%{! ## Problem 3: Contextual Semantics + +In this problem, we define the contextual semantics for L\{num str\} and proof that it is equivalent to (defines the same relation as) the given dynamic semantics (reduction). !}% +%%%%%%%%%%%%%%%% L{num str}: Contextual Semantics %%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Instructions: E0 ~> E0' +%sort instr %. +%term instr/plus %pi (plus_op N1 N2 N) %-> (instr (plus (nume N1) (nume N2)) (nume N)) %. +%term instr/times %pi (times_op N1 N2 N) %-> (instr (times (nume N1) (nume N2)) (nume N)) %. +%term instr/cat %pi (plus_op S1 S2 S) %-> (instr (cat (stre S1) (stre S2)) (stre S)) %. +%term instr/len instr (len (stre S)) (nume S) %. +%term instr/let %pi (val E1) %-> (instr (let E1 ([x] E2 x)) (E2 E1)) %. +% Expression contexts, represented as "expressions with holes", i.e., +% functions on expressions. +%sort ectxt %. +%term ectxt/hole ectxt ([x] x) %. +%term ectxt/plus1 %pi (ectxt ([x] C1 x)) %-> (ectxt ([x] plus (C1 x) E2)) %. +%term ectxt/plus2 %pi (val E1) %-> (ectxt ([x] C2 x)) %-> (ectxt ([x] plus E1 (C2 x))) %. +%term ectxt/times1 %pi (ectxt ([x] C1 x)) %-> (ectxt ([x] times (C1 x) E2)) %. +%term ectxt/times2 %pi (val E1) %-> (ectxt ([x] C2 x)) %-> (ectxt ([x] times E1 (C2 x))) %. +%term ectxt/cat1 %pi (ectxt ([x] C1 x)) %-> (ectxt ([x] cat (C1 x) E2)) %. +%term ectxt/cat2 %pi (val E1) %-> (ectxt ([x] C2 x)) %-> (ectxt ([x] cat E1 (C2 x))) %. +%term ectxt/len %pi (ectxt ([x] C x)) %-> (ectxt ([x] len (C x))) %. +%term ectxt/let %pi (ectxt ([x] C1 x)) %-> (ectxt ([x] let (C1 x) ([y] E2 y))) %. +% Filling the hole of an expression context with an expression +%sort fill %. +%term fill/hole fill ectxt/hole E E %. +%term fill/plus1 + %pi (fill ECC1 E C1@E) + %-> (fill (%the (ectxt ([x] plus (C1 x) E2)) (ectxt/plus1 ECC1)) E (plus C1@E E2)) %. +%term fill/plus2 {VjE1 val E1} %pi (fill ECC2 E C2@E) %-> (fill (ectxt/plus2 VjE1 ECC2) E (plus E1 C2@E)) %. +%term fill/times1 + %pi (fill ECC1 E C1@E) + %-> (fill (%the (ectxt ([x] times (C1 x) E2)) (ectxt/times1 ECC1)) E (times C1@E E2)) %. +%term fill/times2 {VjE1 val E1} %pi (fill ECC2 E C2@E) %-> (fill (ectxt/times2 VjE1 ECC2) E (times E1 C2@E)) %. +%term fill/cat1 + %pi (fill ECC1 E C1@E) + %-> (fill (%the (ectxt ([x] cat (C1 x) E2)) (ectxt/cat1 ECC1)) E (cat C1@E E2)) %. +%term fill/cat2 {VjE1 val E1} %pi (fill ECC2 E C2@E) %-> (fill (ectxt/cat2 VjE1 ECC2) E (cat E1 C2@E)) %. +%term fill/len %pi (fill ECC1 E C1@E) %-> (fill (ectxt/len ECC1) E (len C1@E)) %. +%term fill/let + %pi (fill ECC1 E C1@E) + %-> (fill (%the (ectxt ([x] let (C1 x) ([y] E2 y))) (ectxt/let ECC1)) E (let C1@E ([y] E2 y))) %. +% Contextual reduction rule +%sort cred %. +%term cred/step + %pi (fill EC E0 C@E0) + %-> (instr E0 E0') + %-> (fill EC E0' C@E0') + %-> (cred C@E0 C@E0') %. +%%%%%%%%%%%%%%%% Theorem 10.2.1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% + +% If E |-> E', then E |->c E'. +%sort thm10-2-1 %. +%mode thm10-2-1 %in %out %. +%term _ thm10-2-1 (red/plus/num PjN) (cred/step fill/hole (instr/plus PjN) fill/hole) %. +%term _ + %pi (thm10-2-1 (red/plus1 RjE1) (cred/step (fill/plus1 FjC1@E0) I (fill/plus1 FjC1@E0'))) + %<- (thm10-2-1 RjE1 (cred/step FjC1@E0 I FjC1@E0')) %. +%term _ + %pi (thm10-2-1 (red/plus2 VjE1 RjE2) (cred/step (fill/plus2 VjE1 FjC2@E0) I (fill/plus2 VjE1 FjC2@E0'))) + %<- (thm10-2-1 RjE2 (cred/step FjC2@E0 I FjC2@E0')) %. +%term _ thm10-2-1 (red/times/num TjN) (cred/step fill/hole (instr/times TjN) fill/hole) %. +%term _ + %pi (thm10-2-1 (red/times1 RjE1) (cred/step (fill/times1 FjC1@E0) I (fill/times1 FjC1@E0'))) + %<- (thm10-2-1 RjE1 (cred/step FjC1@E0 I FjC1@E0')) %. +%term _ + %pi (thm10-2-1 (red/times2 VjE1 RjE2) (cred/step (fill/times2 VjE1 FjC2@E0) I (fill/times2 VjE1 FjC2@E0'))) + %<- (thm10-2-1 RjE2 (cred/step FjC2@E0 I FjC2@E0')) %. +%term _ thm10-2-1 (red/cat/str PjN) (cred/step fill/hole (instr/cat PjN) fill/hole) %. +%term _ + %pi (thm10-2-1 (red/cat1 RjE1) (cred/step (fill/cat1 FjC1@E0) I (fill/cat1 FjC1@E0'))) + %<- (thm10-2-1 RjE1 (cred/step FjC1@E0 I FjC1@E0')) %. +%term _ + %pi (thm10-2-1 (red/cat2 VjE1 RjE2) (cred/step (fill/cat2 VjE1 FjC2@E0) I (fill/cat2 VjE1 FjC2@E0'))) + %<- (thm10-2-1 RjE2 (cred/step FjC2@E0 I FjC2@E0')) %. +%term _ thm10-2-1 red/len/str (cred/step fill/hole instr/len fill/hole) %. +%term _ + %pi (thm10-2-1 (red/len RjE1) (cred/step (fill/len FjC1@E0) I (fill/len FjC1@E0'))) + %<- (thm10-2-1 RjE1 (cred/step FjC1@E0 I FjC1@E0')) %. +%term _ thm10-2-1 (red/let/val VjE1) (cred/step fill/hole (instr/let VjE1) fill/hole) %. +%term _ + %pi (thm10-2-1 (red/let RjE1) (cred/step (fill/let FjC1@E0) I (fill/let FjC1@E0'))) + %<- (thm10-2-1 RjE1 (cred/step FjC1@E0 I FjC1@E0')) %. +%worlds () (thm10-2-1 _ _) %. +%total RjE (thm10-2-1 RjE _) %. +%%%%%%%%%%%%%%%% Theorem 10.2.2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% + +% If E |->c E', then E |-> E'. +%sort thm10-2-2 %. +%mode thm10-2-2 %in %out %. +% Lemma. If C{E0} = E, E0 ~> E0', and C{E0'} = E', then E |-> E'. +%sort lem10-2-2 %. +%mode lem10-2-2 %in %in %in %out %. +%term _ lem10-2-2 fill/hole (instr/plus PjN) fill/hole (red/plus/num PjN) %. +%term _ lem10-2-2 fill/hole (instr/times TjN) fill/hole (red/times/num TjN) %. +%term _ lem10-2-2 fill/hole (instr/cat PjN) fill/hole (red/cat/str PjN) %. +%term _ lem10-2-2 fill/hole instr/len fill/hole red/len/str %. +%term _ lem10-2-2 fill/hole (instr/let VjE1) fill/hole (red/let/val VjE1) %. +%term _ + %pi (lem10-2-2 (fill/plus1 FjC1@E0) I (fill/plus1 FjC1@E0') (red/plus1 RjE)) + %<- (lem10-2-2 FjC1@E0 I FjC1@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/plus2 VjE1 FjC2@E0) I (fill/plus2 VjE1 FjC2@E0') (red/plus2 VjE1 RjE)) + %<- (lem10-2-2 FjC2@E0 I FjC2@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/times1 FjC1@E0) I (fill/times1 FjC1@E0') (red/times1 RjE)) + %<- (lem10-2-2 FjC1@E0 I FjC1@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/times2 VjE1 FjC2@E0) I (fill/times2 VjE1 FjC2@E0') (red/times2 VjE1 RjE)) + %<- (lem10-2-2 FjC2@E0 I FjC2@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/cat1 FjC1@E0) I (fill/cat1 FjC1@E0') (red/cat1 RjE)) + %<- (lem10-2-2 FjC1@E0 I FjC1@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/cat2 VjE1 FjC2@E0) I (fill/cat2 VjE1 FjC2@E0') (red/cat2 VjE1 RjE)) + %<- (lem10-2-2 FjC2@E0 I FjC2@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/len FjC1@E0) I (fill/len FjC1@E0') (red/len RjE)) + %<- (lem10-2-2 FjC1@E0 I FjC1@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/let FjC2@E0) I (fill/let FjC2@E0') (red/let RjE)) + %<- (lem10-2-2 FjC2@E0 I FjC2@E0' RjE) %. +%worlds () (lem10-2-2 _ _ _ _) %. +%total FjE (lem10-2-2 FjE _ _ _) %. +% Proof of main theorem +%term _ + %pi (thm10-2-2 (cred/step FjC@E0 I FjC@E0') RjE) + %<- (lem10-2-2 FjC@E0 I FjC@E0' RjE) %. +%worlds () (thm10-2-2 _ _) %. +%total {} (thm10-2-2 _ _) %. +%{! ## Twelf Output + +The `` check="true">output`` for the above. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Properties_of_Typing_and_Reduction). +!}% \ No newline at end of file diff --git a/new-tests/output/pages/proving-metatheorems-solutions-odd-even-succ.lf b/new-tests/output/pages/proving-metatheorems-solutions-odd-even-succ.lf new file mode 100644 index 0000000..a6a68a1 --- /dev/null +++ b/new-tests/output/pages/proving-metatheorems-solutions-odd-even-succ.lf @@ -0,0 +1,42 @@ +%%! title: "Solution: proofs about even and odd numbers" +%%! description: "Showing how adding one influences even and odd numbers (Part of the introduction to proving metatheorems in Twelf)" +%{! + +_This page is part of the introduction to [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)._ + +Exercise 3 from [the part 1 exercises](/wiki/proving-metatheorems-summary-the-natural-numbers/): State and prove a theorem ``succ-even`` that shows that the successor of an even number is an odd number. After that, state and prove a theorem ``succ-odd`` that shows that the successor of an odd number is an even number. + +Starting with the preliminaries, including the [definition of `odd` from exercise 2](/wiki/proving-metatheorems-solutions-define-odd/): + +!}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort plus %. +%term plus-z plus z N2 N2 %. +%term plus-s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%sort even %. +%term even-z even z %. +%term even-s %pi (even (s (s N))) %<- (even N) %. +%sort odd %. +%term odd-1 odd (s z) %. +%term odd-s %pi (odd N) %-> (odd (s (s N))) %. +%{! Because both the `even` and `odd` judgements are defined independently, it's necessary to perform induction on the `even` judgement to prove that the first theorem, and induction on the `odd` judgement to prove the second theorem: !}% +%{! ## The successor of an even number is odd !}% +%{!! begin checked !!}% +%sort succ-even %. +%mode succ-even %in %out %. +%term sez succ-even even-z odd-1 %. +%term ses %pi (succ-even (even-s EvenA) (odd-s OddA)) %<- (succ-even EvenA OddA) %. +%worlds () (succ-even _ _) %. +%total D (succ-even D _) %. +%{!! end checked !!}% +%{! ## The successor of an odd number is even !}% +%{!! begin checked !!}% +%sort succ-odd %. +%mode succ-odd %in %out %. +%term so1 succ-odd odd-1 (even-s even-z) %. +%term sos %pi (succ-odd (odd-s OddA) (even-s EvenA)) %<- (succ-odd OddA EvenA) %. +%worlds () (succ-odd _ _) %. +%total D (succ-odd D _) %. +%{!! end checked !!}% \ No newline at end of file diff --git a/new-tests/output/pages/proving-metatheorems-with-twelf.lf b/new-tests/output/pages/proving-metatheorems-with-twelf.lf new file mode 100644 index 0000000..0666309 --- /dev/null +++ b/new-tests/output/pages/proving-metatheorems-with-twelf.lf @@ -0,0 +1,54 @@ +%%! title: "Proving metatheorems with Twelf" +%%! description: "Our favorite tutorial for getting started with Twelf as a tool for describing programming languages and logical systems and proving properties of those systems" +%%! next: +%%! label: Representing the syntax of the natural numbers +%%! link: /wiki/proving-metatheorems-representing-the-syntax-of-the-natural-numbers/ +%{! + +The story of a typical [deductive system](/wiki/object-logic/) goes something like this: + - One day, a programming language designer has an idea for a new programming language. + - She writes down the language's [abstract syntax](/wiki/abstract-syntax/). + - Then, she gives the syntax meaning by defining some [judgement](/wiki/judgment/)s for it, such as a type system or an operational semantics. + - Next, she investigates the language's properties by proving some theorems. Maybe she fiddles with the language some to make the desired theorems true; maybe the idea doesn't work at all and she goes on to something else. + - Maybe she implements the language to try out some examples. + - Eventually, if the idea works out, she writes up a paper about the language. If it's a good idea, then she, or maybe someone who read the paper, will someday want to extend the language, or incorporate the idea into another language. + +This introduction assumes this story is somewhat familiar to you. If not, you should read a textbook such as [TAPL](http://www.cis.upenn.edu/~bcpierce/tapl/) or [PFPL](https://www.cs.cmu.edu/~rwh/pfpl/). The Twelf version of the story goes something like this: + - One day, a programming language designer has an idea. + - She formalizes the syntax and judgements of the language in the [LF](/wiki/lf/) logical framework. + - She uses Twelf to check her proofs of the theorems. + - She uses Twelf to run the language definition itself to try out some examples. + - Her paper includes a Twelf appendix, which makes her and her readers much more confident that the theorems in the paper are actually true. + - When someone goes to extend the language, he has a formal, machine-checkable artifact that he can study or perhaps even reuse directly. + +This tutorial pays a lot of attention to the question "how do I know that what I wrote down and proved on paper, and what I wrote down and proved in Twelf, are exactly the same thing?" Checking that these two things are the same is called [adequacy](/wiki/adequacy/), and it's an important problem _whenever_ you're using a theorem proving like Twelf, Agda, or Rocq. But it definitely has the flavor of "checking lots of fiddly details," and so it's reasonable to skip discussions of adequacy in this tutorial if you're exploring Twelf for the first time. + +### First order representations: The natural numbers + +The first layer of this introduction uses a _very_ simple deductive system, the [natural numbers](/wiki/natural-numbers/) with addition, to introduce the Twelf methodology. + + 1. [Representing syntax](/wiki/proving-metatheorems-representing-the-syntax-of-the-natural-numbers/) + 2. [Simply typed LF](/wiki/proving-metatheorems-simply-typed-lf/) + 3. [Representing judgements](/wiki/proving-metatheorems-representing-the-judgements-of-the-natural-numbers/) + 4. [Full LF](/wiki/proving-metatheorems-full-lf/) + 5. [Proving totality assertions](/wiki/proving-metatheorems-proving-totality-assertions-about-the-natural-numbers/) + 6. [Proving metatheorems](/wiki/proving-metatheorems-proving-metatheorems-about-the-natural-numbers/) + 7. [Summary and exercises](/wiki/proving-metatheorems-summary-the-natural-numbers/) + +### Higher-order representations: The simply-typed lambda calculus + +The second layer tells the same story for a programming language with variable binding (the [simply-typed lambda calculus](/wiki/simply-typed-lambda-calculus/)), which is where Twelf really shines. + + 8. [Representing syntax](/wiki/proving-metatheorems-representing-the-syntax-of-the-stlc/) + 9. [Representing judgements](/wiki/proving-metatheorems-representing-the-judgements-of-the-stlc/) + 10. [Proving metatheorems](/wiki/proving-metatheorems-proving-metatheorems-about-the-stlc/) + 11. [Summary and exercises](/wiki/proving-metatheorems-summary-the-stlc/) + +### Beyond the simply-typed lambda calculus + +The third layer presents some more-interesting proofs and introduces one additional feature of Twelf, the ability to do proofs about open terms. + + 12. [Proving totality assertions in non-empty contexts](/wiki/proving-metatheorems-proving-totality-assertions-in-non-empty-contexts/) + 13. [Proving metatheorems in non-empty contexts](/wiki/proving-metatheorems-proving-metatheorems-in-non-empty-contexts/) + +!}% \ No newline at end of file diff --git a/new-tests/output/pages/talk-effectiveness-lemma.lf b/new-tests/output/pages/talk-effectiveness-lemma.lf new file mode 100644 index 0000000..f469406 --- /dev/null +++ b/new-tests/output/pages/talk-effectiveness-lemma.lf @@ -0,0 +1,22 @@ +%%! title: "Talk:Effectiveness lemma" +%{! Why "effectiveness" rather than "totality" ? As far as I can tell, proving "effectiveness" is the same +as proving totality at the meta-leval. [Boyland](/wiki/user-boyland/) 22:41, 25 February 2007 (EST) + +: I think it just dates back to some choices made in the Typed Assembly Language Two formalization. I think we thought that calling it a "totality lemma" would just cause confusion of a different sort, not that effectiveness is an obvious alternative by any means. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 23:40, 25 February 2007 (EST) + +This sentence is pretty confusing: "Proving an effectiveness lemma essentially establishes the same fact that a %total directive does, but does so in a way that can be used by other metatheorems." You use %total to prove any metatheorem, and any lemma so proved can be used by other metatheorems. + +An effectiveness lemma says that some relation is in fact an algorithm (hence "effective"), and returns for any inputs a derivation of the relation which witnesses that fact. [User:JakeD](/wiki/user-jaked/) 00:33, 26 February 2007 (EST) + +: I mean, that works for me too :) — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 01:04, 26 February 2007 (EST) +:: I edited it to try to simplify. We definitely should not lead with confusing stuff about metatheorems (since that is hard to wrap ones head around, especially given all the different takes on what a metatheorem is!), but a section in this article about its relationship to [``%total``](/wiki/percent-total/) is warranted, I think, because it is an obvious question.  — [Tom 7](/wiki/user-tom7/) 09:33, 26 February 2007 (EST) + +: I think language that connects "effectivenss" with totality is warranted. I don't see it in the article any more. There needs to be something on the order of "Proving an effectivessness lemma for a relation essentially establishes the same fact that a %total directive does for the relation, but does so in a way that the totality itself is named and can be used in proofs." Otherwise people will wonder. Also, personally (as an outsider), I find it annoying that one basically has to restate the entire definition of the relation in order to prove meta-totality ("effectivessness"), but Twelf's own wiki doesn't need to reflect my biased opinions. [Boyland](/wiki/user-boyland/) 21:03, 26 February 2007 (EST) + +:: I agree: effectiveness lemmas are just a workaround for limitations of Twelf. When you do a %total for, say, plus, Twelf proves "for all N1, N2, there exists an N3 and D:plus N1 N2 N3". When you do a %total for plus-effective, Twelf proves "for all N1, N2, there exist N3 and D:plus N1 N2 N3 and D':plus-effective N1 N2 N3 D D'". The only reason to do this is to work around the fact that Twelf won't let you name the output of a subgoal. [Drl](/wiki/user-drl/) 12:09, 1 March 2007 (EST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Effectiveness_lemma). +!}% \ No newline at end of file diff --git a/new-tests/output/pages/talk-incremental-metatheorem-development.lf b/new-tests/output/pages/talk-incremental-metatheorem-development.lf new file mode 100644 index 0000000..11bef38 --- /dev/null +++ b/new-tests/output/pages/talk-incremental-metatheorem-development.lf @@ -0,0 +1,15 @@ +%%! title: "Talk:Incremental metatheorem development" +%{! ## Bottom-up vs. Top-down + +I'm not sure if the motivation for using %trustme to maintain a bottom-up development style is completely accurate here. Without %trustme, proofs must be done in a bottom-up way. That's just how math is built up. In practice, I've seen two modes of use of %trustme. The first is to work bottom-up until you hit something that will be pretty time-consuming/hard/impossible to do, and then you %trustme that particular lemma so you can keep working bottom-up. The other, which I've seen used by a particular Twelf master who happens to have an office right next to a stairwell in a corner on the 8th floor of Wean, is to %trustme a lot of boring/uninteresting lemmas which you are pretty sure are true, and then focus on the actual hard proof, and then fill in the boring holes at some point. The second strategy is very much a top-down approach. +--[DanielKLee](/wiki/user-danielklee/) 09:19, 22 September 2006 (MST) + +: I agree that there are multiple models of use, go ahead and put whatever you think is a more accurate motivation over what is there now (in general, I'd advocate a "change and explain why you did it on the Talk pages" as opposed to a "talk about changing on the Talk page" strategy here :) ). — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 09:26, 22 September 2006 (MST) +: I think that we are just using opposite definitions of "top-down" and "bottom-up"; for me, the end of the proof is the bottom. I guess it depends on how you think of it. (We had all sorts of similar terminological issues in automated theorem proving and computer graphics class..) We should just be more clear about what we mean.  — [Tom 7](/wiki/user-tom7/) 11:52, 22 September 2006 (MST) +:: Yeah, I think that's where the confusion is. I think of the main result being at the "top", and being built up from the smaller ones "below" it, like a tree. --[DanielKLee](/wiki/user-danielklee/) 10:30, 23 September 2006 (MST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Incremental_metatheorem_development). +!}% \ No newline at end of file diff --git a/new-tests/output/pages/talk-metatheorem.lf b/new-tests/output/pages/talk-metatheorem.lf new file mode 100644 index 0000000..1cbd5e4 --- /dev/null +++ b/new-tests/output/pages/talk-metatheorem.lf @@ -0,0 +1,28 @@ +%%! title: "Talk:Metatheorem" +%{! ## Placeholder + +I'm going to work on this based on Dan's notes, with the idea that I'll fix it based on whatever total consensus arises with all interested parties. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 17:12, 20 October 2006 (EDT) + +## I agree, Tom + +The point about totality checking being *more* conservative than the fixed search strategy of logic programming is a good one - the need for factoring lemmas being the perfect example here - so I think that change is good. Does the article seem reasonable otherwise, because I'm not certain the approach is even the right one here. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 19:16, 9 March 2007 (EST) +::I think it's a hard subject but this is a reasonable take on it and a good start, yes.  — [Tom 7](/wiki/user-tom7/) 14:01, 12 March 2007 (EDT) + +## Why "meta" ? + +To an outsider, all the "meta"-theorem stuff seems like unnecessary baggage. Normally, I would think a meta-theorem is a theorem about theorems. Here's what I think the answer to my question is: In the olden days, one implemented XYZ logic in Twelf and wrote a theorem that XYZ theorems were valid. But now, we use Twelf to directly encode our theorems about object languages. The "meta" is not really appropriate any more. I suppose for historical reasons people are used to "metatheorem", but it might make more sense just to dispense with "meta" and just talk about theorems. Or if not, then a better explanation of "meta" is needed than "for historical reasons" [Boyland](/wiki/user-boyland/) 01:35, 13 March 2007 (EDT) + +: You're right that this needs to be explained more clearly. The point of the "meta" is to distinguish between +- a _theorem_ in an encoded deductive system, and +- a _metatheorem_ about an encoded deductive system. +: For example, if I encoded a logic with a judgement ``A true``, then ``(all [a:prop] impl a a) true`` might be a theorem in the encoded logic (if I can come up with an LF term of that type), whereas cut elimination for the logic is a metatheorem about it. [Drl](/wiki/user-drl/) 10:14, 13 March 2007 (EDT) +:: To answer another point (we edit-conflicted each other making the same first point), I don't see this as an issue of "historical resasons" at all, this is the way I understand the research community working now - the two-year-old workshop on doing this sort of thing is called the [http://www.cis.upenn.edu/~sweirich/wmm/ Worshop on Mechanizing Metatheory]. People still think of Prolog as a "theorem prover" for Horn clauses (see the [[w:Prolog|Wikipedia page on Prolog]]), and those kinds of theorems are the same kinds of "theorems" (``plus 3 4 7``, ``of (lam [x] E x) (arrow unit unit)``) that are discussed on this page - they are nothing like ∀∃-statement metatheorems. Dan/John/others, where do you think this article could explain this better? — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 10:30, 13 March 2007 (EDT) + +:: Reading the current article, I'm okay with the explanation of "theorem", but the transition to the Totality assertion section should rehash the discussion at the top about "metatheorem" being a statement about an object language and contrast it with theorem (I missed the definition of "metatheorem" at the top on the first read). Also, when I used the word "theorem" this way, Frank objected that he likes to reserve "theorem" for a judgement defining truth in a logic (so not all derivable object-language judgements are "theorems"). Not sure if we want to use the word that way or not. [Drl](/wiki/user-drl/) 12:39, 13 March 2007 (EDT) +::: I'll work on the first point - as for the second point, perhaps I could add in to the example something with a logic - just , and say "we could reserve 'theorem' for this kind of statement, but to avoid saying 'derivable object-language judgment' too many times we often blur the distinction by referring to both as theorems. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 08:41, 14 March 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Metatheorem). +!}% \ No newline at end of file diff --git a/new-tests/output/pages/talk-mutable-state.lf b/new-tests/output/pages/talk-mutable-state.lf new file mode 100644 index 0000000..927c4e7 --- /dev/null +++ b/new-tests/output/pages/talk-mutable-state.lf @@ -0,0 +1,15 @@ +%%! title: "Talk:Mutable state" +%{! Something looks screwy with the two families named "update-lemma", one of which's body gets escaped into normal text because it's block-commented out. Otherwise, cool! [Drl](/wiki/user-drl/) 11:51, 16 March 2007 (EDT) +:: Fixed. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 12:29, 16 March 2007 (EDT) + +## Rename? + +Somehow I think [language with references](/wiki/mutable-state/) is a little weird of a title for this page; can I suggest [mutable state](/wiki/mutable-state/) or encoding references or +encoding store or something like that? I think it would be more in line with our other tutorials/case studies.  — [Tom 7](/wiki/user-tom7/) 08:42, 19 March 2007 (EDT) +: Agreed, I couldn't think of a good name when I wrote the page. I'll use [mutable state](/wiki/mutable-state/). — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 11:21, 19 March 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Mutable_state). +!}% \ No newline at end of file diff --git a/new-tests/output/pages/talk-natural-numbers-with-inequality.lf b/new-tests/output/pages/talk-natural-numbers-with-inequality.lf new file mode 100644 index 0000000..5405338 --- /dev/null +++ b/new-tests/output/pages/talk-natural-numbers-with-inequality.lf @@ -0,0 +1,10 @@ +%%! title: "Talk:Natural numbers with inequality" +%{! It would probably be a good idea to standardize on one name for the uninhabited type in our tutorials. My vote goes to "void" since that is its common name in type theory. I think "false" is not a good choice since we often have terms inhabiting a type called false (in an encoding of classical logic, for instance). "Absurd" seems a little hokey to me, but I don't have a good argument against it...  — [Tom 7](/wiki/user-tom7/) 13:28, 6 September 2006 (MST) + +: I'm all for void, I'll make a note of it in [Twelf style guide](/wiki/twelf-style-guide/), since that's exactly the sort of thing that goes in a twelf style guide. - [TheTwelfElf](/wiki/user-thetwelfelf/) 13:37, 6 September 2006 (MST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Natural_numbers_with_inequality). +!}% \ No newline at end of file diff --git a/new-tests/output/pages/talk-thaw.lf b/new-tests/output/pages/talk-thaw.lf new file mode 100644 index 0000000..6d087e5 --- /dev/null +++ b/new-tests/output/pages/talk-thaw.lf @@ -0,0 +1,4 @@ +%%! title: "Talk:%thaw" +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/output/pages/template-bibtex-pierce02tapl.lf b/new-tests/output/pages/template-bibtex-pierce02tapl.lf new file mode 100644 index 0000000..fad59d3 --- /dev/null +++ b/new-tests/output/pages/template-bibtex-pierce02tapl.lf @@ -0,0 +1,15 @@ +%%! title: "Template:Bibtex:pierce02tapl" +%{! <bibtex>@book\{pierce02tapl, + author = \{Benjamin C. Pierce\}, + title = \{Types and Programming Languages\}, + publisher = \{MIT Press\}, + month = \{February\}, + isbn = \{978-0-262-16209-8\}, + year = \{2002\}, +\}</bibtex> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Bibtex:pierce02tapl). +!}% \ No newline at end of file diff --git a/new-tests/output/pages/template-discuss.lf b/new-tests/output/pages/template-discuss.lf new file mode 100644 index 0000000..9028236 --- /dev/null +++ b/new-tests/output/pages/template-discuss.lf @@ -0,0 +1,17 @@ +%%! title: "Template:Discuss" +%{! [[Talk:\{\{PAGENAME\}\}#\{\{\{1\}\}\}|<sup>[?]</sup>]]<noinclude> + + +This template places a discussion link in an article. The link takes the user to a supplied section on the talk page for the same article. (It does not work for pages outside the article namespace). To use it, write: + +:<nowiki>\{\{discuss|</nowiki>_description of issue_<nowiki>\}\}</nowiki> + +where _description of issue_ is an informative section header for the talk page. You should then create that section on the talk page. The link looks like this.[<sup>[?]</sup>](/wiki/talk-main-page/) + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Discuss). +!}% \ No newline at end of file diff --git a/new-tests/output/pages/template-lolli.lf b/new-tests/output/pages/template-lolli.lf new file mode 100644 index 0000000..00c9451 --- /dev/null +++ b/new-tests/output/pages/template-lolli.lf @@ -0,0 +1,4 @@ +%%! title: "Template:Lolli" +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/output/pages/template-talk-guide.lf b/new-tests/output/pages/template-talk-guide.lf new file mode 100644 index 0000000..a883d41 --- /dev/null +++ b/new-tests/output/pages/template-talk-guide.lf @@ -0,0 +1,10 @@ +%%! title: "Template talk:Guide" +%{! ## Guide gets updated and... + +Won't this template be a problem if the section numbers in the guide change? Ideally this template should be <nowiki>\{\{guide4\}\}</nowiki> or something, I would think... — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 17:35, 27 September 2006 (MST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template_talk:Guide). +!}% \ No newline at end of file diff --git a/new-tests/output/pages/twelf-with-emacs.lf b/new-tests/output/pages/twelf-with-emacs.lf new file mode 100644 index 0000000..937e2da --- /dev/null +++ b/new-tests/output/pages/twelf-with-emacs.lf @@ -0,0 +1,137 @@ +%%! title: "Twelf with Emacs" +%{! The preferred way to interact with Twelf is through its emacs mode, which is included in the Twelf distribution. The [Download](/download/) page has basic information for setting up Emacs mode with Twelf. + +## Prerequisites + +These instructions assume you've already [downloaded Twelf](/download/), and that either +* You used the Windows/OSX installer +* You downloaded a pre-compiled tarball +* You downloaded a source tarball and have already built Twelf + +These instructions should easily translate to using Subversion to get Twelf, however. + +## Installing the Twelf emacs mode + +If you put Twelf in the directory `/somewhere/twelf/` then you should have +these two lines (just replace `somewhere` with wherever you put your Twelf): + +Add the following to your Emacs configuration file, which should be the file .emacs in your home directory. + +``` +(setq twelf-root "/somewhere/twelf/") +(load (concat twelf-root "emacs/twelf-init.el")) +``` + +For instance, if the directory where you unpacked the tarball is ``/usr/local/twelf``, then you'll want those two lines to be: + +``` +(setq twelf-root "/usr/local/twelf/") +(load (concat twelf-root "emacs/twelf-init.el")) +``` + +If your Twelf directory is ``/home/foo/stuff/logic/twelf``, then you'll want those lines to be: + +``` +(setq twelf-root "/home/foo/stuff/logic/twelf/") +(load (concat twelf-root "emacs/twelf-init.el")) +``` + +If you used the default Windows installer, you'll want those to lines to be + +``` +(setq twelf-root "C:\\Program Files\\Twelf\\") +(load (concat twelf-root "emacs\\twelf-init.el")) +``` + +Exit Emacs and restart it. + +## How Twelf programs are divided up into files + +- Twelf programs should be stored in files with the ``.elf`` extension. +- All of the ``.elf`` files in a project are collected in a _configuration file_, typically called ``sources.cfg``. A configuration file lists (paths to) individual ``.elf`` files in dependency order. You can load a configuration file, which loads each ``.elf`` file in sequence. + +### Creating a .cfg file + +Start Emacs in some directory (for the purposes of this description, we'll assume it's ``~/tmp/twelf``. Create the **configuration file** ``sources.cfg`` with the following text in it: + +``` +sometwelf.elf +``` + +Save the file (by using CTRL-x CTRL-s in Emacs). + +### Creating a .elf file + +Then create a file ``sometwelf.elf`` (by using CTRL-x CTRL-f in Emacs) and enter the following text: !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%define ss (%pi nat %-> nat) [n] s (s n) %. +%sort plus %. +%term plus/z plus z N N %. +%term plus/s %pi (plus N M P) %-> (plus (s N) M (s P)) %. +%{! Save the file (by using CTRL-x CTRL-s in Emacs). + +## Using the Twelf emacs mode + +You use Twelf by asking it to process declarations. Twelf responds by either printing ``Server OK`` at the bottom of the screen if your declarations were processed successfully, or by popping up the twelf-server buffer if there was an error (ABORT). + +The emacs mode allows you to process declarations at a variety of granularities: +* A whole configuration file (CTRL-c CTRL-c loads a cfg file) +* A whole .elf file (CTRL-c CTRL-s loads the .elf file open in the current emacs window) +* A single declaration (thing ending in a period) (CTRL-c CTRL-d loads declaration the cursor is on in the in the current .elf file) + +For example, a typical use case is to load your entire project with CTRL-c CTRL-c when you start working, and then to process new Twelf code declaration-by-declaration as you write it, using CTRL-c CTRL-d. Sometimes you will go back and make edits in lots of different places in a file, in which case it's helpful to reload just the current file with CTRL-c CTRL-s. + +Only reloading the configuration clears the state of Twelf. Even reloading will not clear previous declarations from the file (though it will typically shadow them). This might give unexpected results if you remove a declaration and expect it to have disappeared. + +### Loading the configuration file: CTRL-c CTRL-c + +Assuming you still have ``sometwelf.elf`` open in emacs, hit CTRL-c CTRL-c. If ``sometwelf.elf`` has unsaved changes, you will be asked: + +``` +File not in current configuration. Save? (yes or no) +``` + +Type yes. + +You will then be asked: + +``` +Visit config file: (default sources.cfg) ~/tmp/twelf/ +``` + +This is asking for the location of the sources.cfg file you saved when following the instructions above; the default option is the correct one. Just hit ENTER. + +You will then be asked: + +``` +Twelf server: (default twelf-server) /something/bin/ +``` +This is asking which Twelf binary you want to run; the default should already be correct, just hit ENTER. + +This will process your configuration file, which in turn says to process ``sometwelf.elf``. + +### Checking an individual declaration: CTRL-c CTRL-d + +Now you can type new declarations into the emacs buffer and use CTRL-c CTRL-d to load them. + +### Checking an individual file: CTRL-c CTRL-s + +Or you can reload the file in the current window with CTRL-c CTRL-s. You must previously have loaded a configuration file. + +## Syntax highlighting + +Twelf does not automatically refresh its syntax highlighting as you type. You can manually refresh the highlighting as follows: + +- ``CTRL-c l`` fontifies the currently visible emacs buffer. E.g. in ``sometwelf.elf``m the constants (``nat``, ``plus/z``, etc.) should turn red, the bound variable ``n`` should turn green, and the logic-programming variables ``N``, ``M``, and ``P`` should turn blue. +- ``CTRL-c CTRL-l`` fontifies the current declaration only. This is useful if you have really big files. + +For more info on the emacs mode, see \{\{guide|chapter=13|section=74|title=Emacs Interface\}\}. + +## Hacks + +If you use multiple frames with Emacs, ``twelf-mode`` annoyingly pops up the Twelf server buffer in the current frame, even if it is already shown in another frame. To avoid this, set ``display-buffer-reuse-frames`` to ``t``. Now, you will still have the problem that although the buffer is shown, it does not scroll to the bottom; to fix this you need to edit ``twelf.el`` and change ``(get-buffer-window twelf-server-buffer)`` to ``(get-buffer-window twelf-server-buffer t)``. + +## See also +Emacs Interface !}% \ No newline at end of file diff --git a/new-tests/output/pages/user-jaked.lf b/new-tests/output/pages/user-jaked.lf new file mode 100644 index 0000000..4f2a285 --- /dev/null +++ b/new-tests/output/pages/user-jaked.lf @@ -0,0 +1,8 @@ +%%! title: "User:JakeD" +%{! Hi. I am a 2nd year PhD student at CMU. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:JakeD). +!}% \ No newline at end of file diff --git a/new-tests/pages/computation-and-deduction-2009-20090203.lf b/new-tests/pages/computation-and-deduction-2009-20090203.lf new file mode 100644 index 0000000..ed1c96e --- /dev/null +++ b/new-tests/pages/computation-and-deduction-2009-20090203.lf @@ -0,0 +1,139 @@ +%%! title: "Computation and Deduction 2009/20090203" +%{! Code from class, March 2. + +```twelf +ttp : type. %name ttp T. + +tnum : ttp. +tnot : ttp -> ttp. +tprod : ttp -> ttp -> ttp. + +texp : type. %name texp E. +tval : type. %name tval V. + +tlet : tval -> (tval -> texp) -> texp. +tpi1 : tval -> (tval -> texp) -> texp. +tpi2 : tval -> (tval -> texp) -> texp. +tifz : tval -> texp -> (tval -> texp) -> texp. +tapp : tval -> tval -> texp. + +tz : tval. +ts : tval -> tval. +tpair : tval -> tval -> tval. +tlam : ttp -> (tval -> texp) -> tval. + +tof : tval -> ttp -> type. +twf : texp -> type. + +tof/z : tof tz tnum. + +tof/s : tof (ts V) tnum + <- tof V tnum. + +tof/pair : tof (tpair V1 V2) (tprod T1 T2) + <- tof V1 T1 + <- tof V2 T2. + +tof/lam : tof (tlam T ([x] E x)) (tnot T) + <- ({x} tof x T -> twf (E x)). + +twf/let : twf (tlet V ([x] E x)) + <- tof V T + <- ({x} tof x T -> twf (E x)). + +twf/pi1 : twf (tpi1 V ([x] E x)) + <- tof V (tprod T1 T2) + <- ({x} tof x T1 -> twf (E x)). + +twf/pi2 : twf (tpi2 V ([x] E x)) + <- tof V (tprod T1 T2) + <- ({x} tof x T2 -> twf (E x)). + +twf/ifz : twf (tifz V E0 ([x] E1 x)) + <- tof V tnum + <- twf E0 + <- ({x} tof x tnum -> twf (E1 x)). + +twf/app : twf (tapp V1 V2) + <- tof V1 (tnot T) + <- tof V2 T. + +ttrans : tp -> ttp -> type. + +ttrans/num : ttrans num tnum. + +ttrans/arr : ttrans (arr T1 T2) (tnot (tprod T1' (tnot T2'))) + <- ttrans T1 T1' + <- ttrans T2 T2'. + +suspend : ttp -> tval -> tval + = [t] [v] (tlam (tnot t) ([f] tapp f v)). + +trans : term -> tp -> tval -> type. + +trans/z : trans z num (suspend tnum tz). + +trans/s : trans (s E) num (tlam (tnot tnum) ([f] tapp V (tlam tnum ([x] tapp f (ts x))))) + <- trans E num V. + +trans/ifz : trans (ifz E E0 ([x] E1 x)) T + (tlam (tnot T') + ([f] tapp V (tlam tnum ([y] tifz y (tapp V0 f) + ([y'] tlet (tlam (tnot tnum) ([g] tapp g y')) + ([x'] tapp (V1 x') f)))))) + <- ttrans T T' + <- trans E num V + <- trans E0 T V0 + <- ({x} {x'} trans x num x' -> trans (E1 x) T (V1 x')). + +trans/lam : trans (lam T1 ([x] E x)) (arr T1 T2) + (suspend (tnot (tprod T1' (tnot T2'))) + (tlam (tprod T1' (tnot T2')) + ([y] tpi1 y + ([y'] (tpi2 y + ([g] tlet (tlam (tnot T1') ([w] tapp w y')) + ([x'] tapp (V x') g))))))) + <- ttrans T1 T1' + <- ttrans T2 T2' + <- ({x} {x'} trans x T1 x' -> trans (E x) T2 (V x')). + +trans/app : trans (app E1 E2) T2 + (tlam (tnot T2') + ([f] tapp V1 + (tlam (tnot (tprod T1' (tnot T2'))) + ([x1] tapp V2 (tlam T1' ([x2] tapp x1 (tpair x2 f))))))) + <- ttrans T1 T1' + <- ttrans T2 T2' + <- trans E1 (arr T1 T2) V1 + <- trans E2 T1 V2. + +tof/suspend : tof (suspend T V) (tnot (tnot T)) + <- tof V T + = [D : tof V T] (tof/lam ([x] [d : tof x (tnot T)] (twf/app D d))). + +correct : trans E T V -> ttrans T T' -> tof V (tnot (tnot T')) -> type. +%mode correct +D1 -D2 -D3. +correct' : trans E T V -> ttrans T T' -> tof V (tnot (tnot T')) -> type. +%mode correct' +D1 +D2 -D3. + + +- : correct trans/z ttrans/num (tof/suspend tof/z). + +%block transbind : some {T : tp} {T' : ttp} {DTT : ttrans T T'} + block + {x : term} {dx : of x T} + {x' : tval} {dx' : tof x' (tnot (tnot T'))} + {dt : trans x T x'} + {thm : correct dt DTT dx'}. + + + +%worlds (transbind) (correct _ _ _). +%total D (correct D _ _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090203). +!}% \ No newline at end of file diff --git a/new-tests/pages/computation-and-deduction-2009-20090316.lf b/new-tests/pages/computation-and-deduction-2009-20090316.lf new file mode 100644 index 0000000..133fa30 --- /dev/null +++ b/new-tests/pages/computation-and-deduction-2009-20090316.lf @@ -0,0 +1,177 @@ +%%! title: "Computation and Deduction 2009/20090316" +%{! Code from class, March 16. + +```twelf +ttp : type. %name ttp T. + +tnum : ttp. +tnot : ttp -> ttp. +tprod : ttp -> ttp -> ttp. + +texp : type. %name texp E. +tval : type. %name tval V. + +tlet : tval -> (tval -> texp) -> texp. +tpi1 : tval -> (tval -> texp) -> texp. +tpi2 : tval -> (tval -> texp) -> texp. +tifz : tval -> texp -> (tval -> texp) -> texp. +tapp : tval -> tval -> texp. + +tz : tval. +ts : tval -> tval. +tpair : tval -> tval -> tval. +tlam : ttp -> (tval -> texp) -> tval. + +tof : tval -> ttp -> type. +twf : texp -> type. + +tof/z : tof tz tnum. + +tof/s : tof (ts V) tnum + <- tof V tnum. + +tof/pair : tof (tpair V1 V2) (tprod T1 T2) + <- tof V1 T1 + <- tof V2 T2. + +tof/lam : tof (tlam T ([x] E x)) (tnot T) + <- ({x} tof x T -> twf (E x)). + +twf/let : twf (tlet V ([x] E x)) + <- tof V T + <- ({x} tof x T -> twf (E x)). + +twf/pi1 : twf (tpi1 V ([x] E x)) + <- tof V (tprod T1 T2) + <- ({x} tof x T1 -> twf (E x)). + +twf/pi2 : twf (tpi2 V ([x] E x)) + <- tof V (tprod T1 T2) + <- ({x} tof x T2 -> twf (E x)). + +twf/ifz : twf (tifz V E0 ([x] E1 x)) + <- tof V tnum + <- twf E0 + <- ({x} tof x tnum -> twf (E1 x)). + +twf/app : twf (tapp V1 V2) + <- tof V1 (tnot T) + <- tof V2 T. + +ttrans : tp -> ttp -> type. + +ttrans/num : ttrans num tnum. + +ttrans/arr : ttrans (arr T1 T2) (tnot (tprod T1' (tnot T2'))) + <- ttrans T1 T1' + <- ttrans T2 T2'. + +suspend : ttp -> tval -> tval + = [t] [v] (tlam (tnot t) ([f] tapp f v)). + +trans : term -> tp -> tval -> type. + +trans/z : trans z num (suspend tnum tz). + +trans/s : trans (s E) num (tlam (tnot tnum) ([f] tapp V (tlam tnum ([x] tapp f (ts x))))) + <- trans E num V. + +trans/ifz : trans (ifz E E0 ([x] E1 x)) T + (tlam (tnot T') + ([f] tapp V (tlam tnum ([y] tifz y (tapp V0 f) + ([y'] tlet (tlam (tnot tnum) ([g] tapp g y')) + ([x'] tapp (V1 x') f)))))) + <- ttrans T T' + <- trans E num V + <- trans E0 T V0 + <- ({x} {x'} trans x num x' -> trans (E1 x) T (V1 x')). + +trans/lam : trans (lam T1 ([x] E x)) (arr T1 T2) + (suspend (tnot (tprod T1' (tnot T2'))) + (tlam (tprod T1' (tnot T2')) + ([y] tpi1 y + ([y'] (tpi2 y + ([g] tlet (suspend T1' y') + ([x'] tapp (V x') g))))))) + <- ttrans T1 T1' + <- ttrans T2 T2' + <- ({x} {x'} trans x T1 x' -> trans (E x) T2 (V x')). + +trans/app : trans (app E1 E2) T2 + (tlam (tnot T2') + ([f] tapp V1 + (tlam (tnot (tprod T1' (tnot T2'))) + ([x1] tapp V2 (tlam T1' ([x2] tapp x1 (tpair x2 f))))))) + <- ttrans T1 T1' + <- ttrans T2 T2' + <- trans E1 (arr T1 T2) V1 + <- trans E2 T1 V2. + +tof/suspend : tof (suspend T V) (tnot (tnot T)) + <- tof V T + = [D : tof V T] (tof/lam ([x] [d : tof x (tnot T)] (twf/app D d))). + + +correct : trans E T V -> ttrans T T' -> tof V (tnot (tnot T')) -> type. +%mode correct +D1 -D2 -D3. + +correct' : trans E T V -> ttrans T T' -> tof V (tnot (tnot T')) -> type. +%mode correct' +D1 +D2 -D3. + +- : correct trans/z ttrans/num (tof/suspend tof/z). + +- : correct + (trans/s + (D : trans E num V)) + ttrans/num + (tof/lam + ([f] [df:tof f (tnot tnum)] + twf/app + (tof/lam ([x] [dx:tof x tnum] (twf/app (tof/s dx) df))) + D')) + <- correct' D ttrans/num (D': tof V (tnot (tnot tnum))). + +- : correct + (trans/lam + (D3: ({x} {x'} trans x T1 x' -> trans (E x) T2 (V x'))) + (D2: ttrans T2 T2') + (D1: ttrans T1 T1')) + (ttrans/arr D2 D1) + (tof/suspend + (tof/lam + ([y] [dy:tof y (tprod T1' (tnot T2'))] + twf/pi1 + ([y'][dy': tof y' T1'] + twf/pi2 + ([g][dg:tof g (tnot T2')] + (twf/let + ([x'] [dx':tof x' (tnot (tnot T1'))] + twf/app + dg + (DV x' dx')) + (tof/suspend dy'))) + dy) + dy))) + <- ({x} {x'} {dx': tof x' (tnot (tnot T1'))} + {dt: trans x T1 x'} + correct dt D1 dx' + -> correct' (D3 x x' dt) D2 (DV x' dx' : tof (V x') (tnot (tnot T2')))). + + +%block transbind : some {T : tp} {T' : ttp} {DTT : ttrans T T'} + block + {x : term} + {x' : tval} {dx' : tof x' (tnot (tnot T'))} + {dt : trans x T x'} + {thm : correct dt DTT dx'}. + + +%worlds (transbind) (correct _ _ _) (correct' _ _ _). +%total (D D') (correct D _ _) (correct' D' _ _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090316). +!}% \ No newline at end of file diff --git a/new-tests/pages/computation-and-deduction-2009-20090408.lf b/new-tests/pages/computation-and-deduction-2009-20090408.lf new file mode 100644 index 0000000..43db262 --- /dev/null +++ b/new-tests/pages/computation-and-deduction-2009-20090408.lf @@ -0,0 +1,419 @@ +%%! title: "Computation and Deduction 2009/20090408" +%{! Code from class, April 8. + +```twelf +tp : type. %name tp A. + +tensor : tp -> tp -> tp. +1 : tp. +with : tp -> tp -> tp. +top : tp. +plus : tp -> tp -> tp. +0 : tp. +lolli : tp -> tp -> tp. +! : tp -> tp. + +exp : type. %name exp M x. + +tens : exp -> exp -> exp. +lett : exp -> (exp -> exp -> exp) -> exp. +pair : exp -> exp -> exp. +pi1 : exp -> exp. +pi2 : exp -> exp. +star : exp. +lets : exp -> exp -> exp. +unit : exp. +in1 : exp -> exp. +in2 : exp -> exp. +case : exp -> (exp -> exp) -> (exp -> exp) -> exp. +abort : exp -> exp. +lam : tp -> (exp -> exp) -> exp. +app : exp -> exp -> exp. +bang : exp -> exp. +letb : exp -> (exp -> exp) -> exp. + +linear : (exp -> exp) -> type. +of : exp -> tp -> type. + +linear/var : linear ([x] x). + +linear/tens1 : linear ([x] tens (M x) N) + <- linear ([x] M x). + +linear/tens2 : linear ([x] tens M (N x)) + <- linear ([x] N x). + +of/tens : of (tens M N) (tensor A B) + <- of M A + <- of N B. + +linear/lett1 : linear ([z] lett (M z) ([x] [y] N x y)) + <- linear ([x] M x). + +linear/lett2 : linear ([z] lett M ([x] [y] N z x y)) + <- ({x} {y} linear ([z] N z x y)). + +of/lett : of (lett M ([x] [y] N x y)) C + <- of M (tensor A B) + <- ({x:exp} of x A -> {y:exp} of y B -> of (N x y) C) + <- ({y:exp} linear ([x] N x y)) + <- ({x:exp} linear ([y] N x y)). + +linear/pair : linear ([x] pair (M x) (N x)) + <- linear ([x] M x) + <- linear ([x] N x). + +of/pair : of (pair M N) (with A B) + <- of M A + <- of N B. + +linear/pi1 : linear ([x] pi1 (M x)) + <- linear ([x] M x). + +linear/pi2 : linear ([x] pi2 (M x)) + <- linear ([x] M x). + +of/pi1 : of (pi1 M) A + <- of M (with A B). + +of/pi2 : of (pi1 M) B + <- of M (with A B). + +of/star : of star 1. + +linear/lets1 : linear ([x] lets (M x) N) + <- linear ([x] M x). + +linear/lets2 : linear ([x] lets M (N x)) + <- linear ([x] N x). + +of/lets : of (lets M N) C + <- of M 1 + <- of N C. + +linear/unit : linear ([x] unit). + +of/unit : of unit top. + +linear/in1 : linear ([x] in1 (M x)) + <- linear ([x] M x). + +linear/in2 : linear ([x] in2 (M x)) + <- linear ([x] M x). + +of/in1 : of (in1 M) (plus A B) + <- of M A. + +of/in2 : of (in2 M) (plus A B) + <- of M B. + +linear/case1 : linear ([z] case (M z) ([x] N x) ([x] O x)) + <- linear ([z] M z). + +linear/case2 : linear ([z] case M ([x] N z x) ([x] O z x)) + <- ({x:exp} linear ([z] N z x)) + <- ({x:exp} linear ([z] O z x)). + +of/case : of (case M ([x] N x) ([x] O x)) C + <- of M (plus A B) + <- ({x} of x A -> of (N x) C) + <- ({x} of x B -> of (O x)C) + <- linear ([x] N x) + <- linear ([x] O x). + +linear/abort1 : linear ([x] abort (M x)) + <- linear ([x] M x). + +linear/abort2 : linear ([x] abort M). + +of/abort : of (abort M) C + <- of M O. + +linear/lam : linear ([x] (lam A ([y] (M x y)))) + <- {y:exp} linear ([x] M x y). + +of/lam : of (lam A ([x] M x)) (lolli A B) + <- ({x} (of x A) -> of (M x) B) + <- linear ([x] M x). + +linear/app1 : linear ([x] (app (M x) N)) + <- linear ([x] M x). + +linear/app2 : linear ([x] (app M (N x))) + <- linear ([x] N x). + +of/app : of (app M N) B + <- of M (lolli A B) + <- of N A. + +of/bang : of (bang M) (! A) + <- of M A. + +linear/letb1 : linear ([x] (letb (M x) ([y] (N y)))) + <- linear ([x] M x). + +linear/letb2 : linear ([x] (letb M ([y] N x y))) + <- ({y} linear ([x] N x y)). + +of/letb : of (letb M ([x] N x)) C + <- of M (! A) + <- ({x} (of x A) -> of (N x) C). + + +reduce : exp -> exp -> type. + +reduce/lam : reduce (lam A ([x] M x)) (lam A ([x] N x)) + <- ({x} reduce (M x) (N x)). + +reduce/app1 : reduce (app M N) (app M' N) + <- reduce M M'. + +reduce/app2 : reduce (app M N) (app M N') + <- reduce N N'. + +reduce/beta : reduce (app (lam A ([x] M x)) N) (M N). + +%block block : block {x:exp}. +%block bind : some {A:tp} block {x:exp} {d : of x A}. + +exp-eq : exp -> exp -> type. +exp-eq/i : exp-eq M M. + +exp-compat: {M : exp -> exp} exp-eq N N' -> exp-eq (M N) (M N') -> type. +%mode exp-compat +I +I1 -O. + +- : exp-compat M _ exp-eq/i. + +%worlds (block) (exp-compat _ _ _). +%total {} (exp-compat _ _ _). + + +lam-compat: ({x:exp} exp-eq (N x) (N' x)) -> exp-eq (lam A N) (lam A N') -> type. +%mode +{N:exp -> exp} +{N':exp -> exp} +{A:tp} +{D1:{x:exp} exp-eq (N x) (N' x)} + -{D2:exp-eq (lam A ([x:exp] N x)) (lam A ([x:exp] N' x))} (lam-compat D1 D2). + +-: lam-compat ([x] exp-eq/i) exp-eq/i. + +%worlds (block) (lam-compat _ _). +%total {} (lam-compat _ _). + +linear-resp: ({x:exp} exp-eq (M x) (N x) ) -> linear M -> linear N -> type. +%mode linear-resp +I +I2 -O. + +-: linear-resp _ D D. + +%worlds (block) (linear-resp _ _ _). +%total {} (linear-resp _ _ _). + +reduce-closed : ({x:exp} reduce M (N x)) -> ({x:exp} exp-eq N' (N x) ) -> type. +%mode reduce-closed +D -D'. + +-: reduce-closed ([x] reduce/app1 (Dreduce x : reduce M1 (N1 x)) ) Deq' + <- reduce-closed Dreduce (Deq : {x} exp-eq N1' (N1 x)) + <- ({x} exp-compat ([y] app y M2) (Deq x) (Deq' x : exp-eq (app N1' M2) (app (N1 x) M2))). + +-: reduce-closed ([x] reduce/app2 (Dreduce x :reduce M2 (N2 x))) Deq' + <- reduce-closed Dreduce (Deq : {x} exp-eq N2' (N2 x)) + <- ({x} exp-compat ([y] app M1 y) (Deq x) (Deq' x : exp-eq (app M1 N2') (app M1 (N2 x)))). + +-: reduce-closed ([x] reduce/lam (Dreduce x : {y} reduce (M y) (N x y))) Deq' + <- ({y} reduce-closed ([x] Dreduce x y) ([x] Deq x y: exp-eq (N' y) (N x y))) + <- ({x} lam-compat ([y] Deq x y) (Deq' x : exp-eq (lam A ([y] (N' y))) (lam A ([y] (N x y))))). + +-: reduce-closed ([x] reduce/beta : reduce (app (lam A ([y] M y)) N) (M N)) ([x] exp-eq/i). + +%worlds (block) (reduce-closed _ _). +%total D (reduce-closed D _). + + +linear-compose: linear ([x] M x) -> linear ([y] N y) -> linear ([y] M (N y)) -> type. +%mode linear-compose +I +I1 -O. + +-: linear-compose linear/var D D. + +-: linear-compose linear/unit _ linear/unit. + +-: linear-compose (linear/app1 (D1 :linear ([x] M1 x)) ) (D2:linear ([y] N y)) + (linear/app1 D) + <- linear-compose D1 D2 (D: linear ([y] M1 (N y))). + +-: linear-compose (linear/app2 (D1 :linear ([x] M1 x)) ) (D2 : linear ([y] N y)) + (linear/app2 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/tens1 (D1 : linear ([x] M1 x))) + (D2 : linear ([y] N y)) + (linear/tens1 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/tens2 (D1 : linear ([x] M1 x))) + (D2 : linear ([y] N y)) + (linear/tens2 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/lett1 (D1 : linear ([x] M1 x))) + (D2 : linear ([y] N y)) + (linear/lett1 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/lett2 (D1 : {x} {y} linear ([z] M1 x y z))) + (D2 : linear ([k] N k)) + (linear/lett2 D ) + <- ({x} {y} linear-compose (D1 x y) D2 (D x y)). + +-: linear-compose (linear/pair (X1 : linear ([x] M1 x)) (X2 : linear ([y] M2 y))) + D1 + (linear/pair A B) + <- linear-compose X1 D1 A + <- linear-compose X2 D1 B. +-: linear-compose (linear/pi1 (D1 : linear ([x] M1 x))) + D2 + (linear/pi1 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/pi2 (D1 : linear ([x] M1 x))) + D2 + (linear/pi2 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/lets1 (D1 : linear ([x] M1 x))) + D2 + (linear/lets1 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/lets2 (D1 : linear ([x] M1 x))) + D2 + (linear/lets2 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/in1 (D1 : linear ([x] M1 x))) + D2 + (linear/in1 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/in2 (D1 : linear ([x] M1 x))) + D2 + (linear/in2 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/case1 (D1 : linear ([x] M1 x))) + D2 + (linear/case1 D) + <- linear-compose D1 D2 D. + +-: linear-compose (linear/case2 ([x] D1 x) ([x] D2 x)) D3 (linear/case2 D4 D5) + <- ({x} linear-compose (D1 x) D3 (D4 x)) + <- ({x} linear-compose (D2 x) D3 (D5 x)). + +-: linear-compose (linear/abort1 (D1 : linear ([x] M1 x))) + D2 + (linear/abort1 D) + <- linear-compose D1 D2 D. + +-: linear-compose (linear/abort2) _ linear/abort2. + +-: linear-compose (linear/lam ([y] X1 y)) X2 (linear/lam D) + <- ({x} linear-compose (X1 x) X2 (D x)). + +-: linear-compose (linear/letb1 (D1 : linear ([x] M1 x))) + D2 + (linear/letb1 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/letb2 ([y] X1 y)) X2 (linear/letb2 D) + <- ({x} linear-compose (X1 x) X2 (D x)). + +%worlds (block) (linear-compose _ _ _). +%total (R) (linear-compose R _ _). + + +srl : ({x} (of x A) -> of (M x) B) -> linear ([x] M x) -> ({x} reduce (M x) (M' x)) -> linear ([x] M' x) -> type. +%mode srl +D1 +D2 +D3 -D4. + +- : srl + ([x] [d:of x A] of/lam (DDD x) (Dof x d : {y} of y B -> of (M x y) C)) + (linear/lam (Dlinear : {y} linear ([x] M x y))) + ([x] reduce/lam (Dreduce x : {y} reduce (M x y) (N x y))) + (linear/lam ([y] Dlinear' y)) + <- ({y} {dy:of y B} srl ([x] [dx:of x A] Dof x dx y dy) (Dlinear y) ([x] Dreduce x y) + (Dlinear' y : linear ([x] N x y))). + +- : srl + ([x] [d:of x A] of/app _ (Dof x d : of (M x) (lolli B C))) + (linear/app1 (Dlinear : linear ([x] M x))) + ([x] reduce/app1 (Dreduce x : reduce (M x) (M' x))) + (linear/app1 Dlinear') + <- srl Dof Dlinear Dreduce (Dlinear' : linear ([x] M' x)). + +- : srl + _ + (linear/app2 (Dlinear : linear ([x] N x))) + ([x] reduce/app1 (Dreduce x : reduce M (M' x))) + Dlinear' + <- reduce-closed Dreduce (Deq: {x} exp-eq M'' (M' x) ) + <- ({x} exp-compat ([z] app z (N x)) (Deq x) (Deq' x : exp-eq (app M'' (N x) ) (app (M' x) (N x)))) + <- linear-resp Deq' (linear/app2 Dlinear) (Dlinear' : linear ([x] app (M' x) (N x))). + +- : srl + _ + (linear/app1 (linear/lam (Dlinear:{y} linear ([x] M x y)))) + ([x] reduce/beta) + (Dlinear _). + +-: srl + ([x][d:of x A] of/app _ (of/lam (Dlinear' x : linear ([y] M y)) _ )) + (linear/app2 (Dlinear : linear ([x] N x) )) + ([x] reduce/beta) + Dlinear'' + <- linear-compose (Dlinear' unit) Dlinear (Dlinear'' : linear ([x] M (N x))). + +-: srl + ([x] [d : of x A] of/app (Dof x d : of (N x) B) _) + (linear/app2 (Dlinear : linear ([x] N x))) + ([x] reduce/app2 (Dreduce x : reduce (N x) (N' x))) + (linear/app2 Dlinear') + <- srl Dof Dlinear Dreduce (Dlinear' : linear ([x] N' x)). + +-: srl + _ + (linear/app1 (Dlinear : linear ([x] M x))) + ([x] reduce/app2 (Dreduce x : reduce N (N' x))) + Dlinear' + <- reduce-closed Dreduce (Deq : {x} exp-eq N'' (N' x)) + <- ({x} exp-compat ([z] app (M x) z) (Deq x) (Deq' x : exp-eq (app (M x) N'') (app (M x) (N' x)))) + <- linear-resp Deq' (linear/app1 Dlinear) (Dlinear' : linear ([x] app (M x) (N' x))). + +%worlds (bind) (srl _ _ _ _). +%total D (srl _ _ D _). + + +sr : of M A -> reduce M M' -> of M' A -> type. +%mode sr +D1 +D2 -D3. + +- : sr + (of/app (DofN : of N A) (DofM : of M (lolli A B))) + (reduce/app1 (Dreduce : reduce M M')) + (of/app DofN DofM') + <- sr DofM Dreduce (DofM' : of M' (lolli A B)). + +- : sr + (of/app (DofN : of N A) (DofM : of M (lolli A B))) + (reduce/app2 (Dreduce : reduce N N')) + (of/app DofN' DofM) + <- sr DofN Dreduce (DofN' : of N' A). + +- : sr + (of/app (DofN : of N A) (of/lam _ (DofM : {x} (of x A) -> of (M x) B))) + reduce/beta + (DofM N DofN). + +- : sr + (of/lam + (Dlinear : linear ([x] M x)) + (DofM : {x} (of x A) -> of (M x) B)) + (reduce/lam (Dreduce : {x} reduce (M x) (N x))) + (of/lam + Dlinear' + DofN) + <- ({x} {dx:of x A} + sr (DofM x dx) (Dreduce x) (DofN x dx : of (N x) B)) + <- srl DofM Dlinear Dreduce (Dlinear': linear ([x] N x)). + +%worlds (bind) (sr _ _ _). +%total D1 (sr _ D1 _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090408). +!}% \ No newline at end of file diff --git a/new-tests/pages/lexicographical-orderings-with-density.lf b/new-tests/pages/lexicographical-orderings-with-density.lf new file mode 100644 index 0000000..080b2b7 --- /dev/null +++ b/new-tests/pages/lexicographical-orderings-with-density.lf @@ -0,0 +1,97 @@ +%%! title: "Lexicographical orderings with density" +%{! This is a fairly advanced example, but it illustrates a number of useful things. First, it adheres very closely to the [Twelf style guide](/wiki/twelf-style-guide/). Second, it has proofs of ``nat-less-trans`` and ``nat-less-immsucc`` which you usually need to get any mileage out of the natural numbers. Third, it is an example of how to prove ``uninhabited/void/false/absurd`` in order to user [reasoning from false](/wiki/reasoning-from-false/). + +This is a lexicographical ordering with the property that it has a less-than relation that is irreflexive and transitive, like the natural numbers. In addition, it has the additional property that the ordering is "dense" in that there exists an index between any two indices. This ordering should be isomorphic to the non-negative rationals. Density is admissible in this ordering without having to deal with multiplication, as would be necessary in a more obvious encoding of the non-negative rationals. !}% +%{! (options removed from twelftag: check="true") !}% +% natural numbers +%sort nat %. +%term nat/z nat %. +%term nat/s %pi nat %-> nat %. +% a list of natural numbers. we will use it as a lexicographical ordering +%sort lex %. +%term lex/z lex nat/z %. +%term lex/s %pi nat %-> (lex N) %-> (lex (nat/s N)) %. +% the less-than relation on natural numbers +%sort nat-less %. +%term nat-less/z nat-less nat/z (nat/s N) %. +%term nat-less/s %pi (nat-less (nat/s N1) (nat/s N2)) %<- (nat-less N1 N2) %. +% the less-than relation on our lexicographical ordering. +%sort lex-less %. +%term lex-less/z lex-less lex/z (lex/s _ _) %. +%term lex-less/eq %pi (lex-less (lex/s N1 NL1) (lex/s N1 NL2)) %<- (lex-less NL1 NL2) %. +%term lex-less/gt %pi (lex-less (lex/s N1 _) (lex/s N2 _)) %<- (nat-less N2 N1) %. +%sort nat-less-trans %. +%mode nat-less-trans %in %in %out %. +%term _ nat-less-trans nat-less/z _ nat-less/z %. +%term _ + %pi (nat-less-trans (nat-less/s N1) (nat-less/s N2) (nat-less/s N3)) + %<- (nat-less-trans N1 N2 N3) %. +%worlds () (nat-less-trans _ _ _) %. +%total {D1} (nat-less-trans D1 _ _) %. +%sort nat-less-immsucc {N} %. +%mode nat-less-immsucc %in %out %. +%term _ nat-less-immsucc nat/z nat-less/z %. +%term _ %pi (nat-less-immsucc (nat/s N1) (nat-less/s NL)) %<- (nat-less-immsucc N1 NL) %. +%worlds () (nat-less-immsucc _ _) %. +%total {D1} (nat-less-immsucc D1 _) %. +%sort lex-less-trans %. +%mode lex-less-trans %in %in %out %. +%term _ lex-less-trans lex-less/z _ lex-less/z %. +%term _ + %pi (lex-less-trans (lex-less/eq NL1) (lex-less/eq NL2) (lex-less/eq NL3)) + %<- (lex-less-trans NL1 NL2 NL3) %. +%term _ + %pi (lex-less-trans (lex-less/gt NL1) (lex-less/gt NL2) (lex-less/gt NL3)) + %<- (nat-less-trans NL2 NL1 NL3) %. +%term _ lex-less-trans (lex-less/gt NL) (lex-less/eq _) (lex-less/gt NL) %. +%term _ lex-less-trans (lex-less/eq _) (lex-less/gt NL) (lex-less/gt NL) %. +%worlds () (lex-less-trans _ _ _) %. +%total {D1} (lex-less-trans D1 _ _) %. +% for any lexicographical index LL, there exists an index LL' that is greater +%sort lex-less-succ {LL} %. +%mode lex-less-succ %in %out %. +%term _ lex-less-succ lex/z (%the (lex-less _ (lex/s nat/z lex/z)) lex-less/z) %. +%term _ %pi (lex-less-succ (lex/s N NL) (lex-less/eq NLL)) %<- (lex-less-succ NL NLL) %. +%worlds () (lex-less-succ _ _) %. +%total {N} (lex-less-succ N _) %. +%sort lex-less-dense %. +%mode lex-less-dense %in %out %out %. +%term _ + %pi (lex-less-dense lex-less/z lex-less/z (%the (lex-less (lex/s (nat/s N) lex/z) _) (lex-less/gt NL))) + %<- (nat-less-immsucc N NL) %. +%term _ + %pi (lex-less-dense (lex-less/eq LL) (lex-less/eq LL1) (lex-less/eq LL2)) + %<- (lex-less-dense LL LL1 LL2) %. +%term _ + %pi (lex-less-dense (lex-less/gt NL) (lex-less/eq LL) (lex-less/gt NL)) + %<- (lex-less-succ _ LL) %. +%worlds () (lex-less-dense _ _ _) %. +%total {D1} (lex-less-dense D1 _ _) %. +%sort uninhabited %. +%freeze uninhabited %. +%sort nat-less-refl-uninhabited %. +%mode nat-less-refl-uninhabited %in %out %. +%term _ + %pi (nat-less-refl-uninhabited (nat-less/s NL) DU) + %<- (nat-less-refl-uninhabited NL DU) %. +%worlds () (nat-less-refl-uninhabited _ _) %. +%total {D1} (nat-less-refl-uninhabited D1 _) %. +%sort lex-less-refl-uninhabited %. +%mode lex-less-refl-uninhabited %in %out %. +%term _ + %pi (lex-less-refl-uninhabited (lex-less/eq LL) DU) + %<- (lex-less-refl-uninhabited LL DU) %. +%term _ + %pi (lex-less-refl-uninhabited (lex-less/gt DL) DU) + %<- (nat-less-refl-uninhabited DL DU) %. +%worlds () (lex-less-refl-uninhabited _ _) %. +%total {DU} (lex-less-refl-uninhabited DU _) %. +%{! [Category:Twelf code](/wiki/category-twelf-code/) + +\{\{case study\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Lexicographical_orderings_with_density). +!}% \ No newline at end of file diff --git a/new-tests/pages/popl-tutorial-church-rosser-problem.lf b/new-tests/pages/popl-tutorial-church-rosser-problem.lf new file mode 100644 index 0000000..3885837 --- /dev/null +++ b/new-tests/pages/popl-tutorial-church-rosser-problem.lf @@ -0,0 +1,283 @@ +%%! title: "POPL Tutorial/Church Rosser (Problem)" +%{! + +In this exercise, we will prove the [[w:Church-Rosser theorem|Church-Rosser theorem]]. You are +provided with a syntax and two substitution lemmas, and your task will be +to implement two cases of the diamond lemma - the +**Application-Application** case, and the **Beta-Application** case. + +You will be able to uncomment the ``%total`` at the end when you are +done. + +The solution is [here](/wiki/church-rosser-w-identity-reduction/). + +!}% +%{! ## Syntax !}% +%sort exp %. +%name exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%{! + +When we use this ``%block``, it expresses that we can be working in a +context with arbitrary expression variables. This will be the only world we +will need for this exercise. + +!}% +%block exps [x exp]%. +%{! ## Reduction !}% +%{! + +We can always reduce an expression to itself (which means that we don't +explicitly need a case to handle variables). We can reduce under binders and +reduce both sides of an application "in parallel." If we have a β-redex +``(λx.ea) eb``, then after reducing ``ea`` (with ``x`` free) to +``ea'`` (with ``x`` free) and reducing ``eb`` to ``eb'``, +we can return ``[eb'/x]ea'``, the substitution of ``eb'`` into +``ea'``. + +The ``%worlds`` explicitly states that we will be reducing in a setting +with free variables. + +!}% +%sort reduce %. +%term reduce/id reduce E E %. +%term reduce/lam %pi (reduce (lam E) (lam E')) %<- ({x exp} reduce (E x) (E' x)) %. +%term reduce/app %pi (reduce (app E1 E2) (app E1' E2')) %<- (reduce E1 E1') %<- (reduce E2 E2') %. +%term reduce/beta + %pi (reduce (app (lam E1) E2) (E1' E2')) + %<- ({x exp} reduce (E1 x) (E1' x)) + %<- (reduce E2 E2') %. +%worlds (exps) (reduce _ _) %. +%{! ## Substitution !}% +%{! + +The first substitution theorem says that if we have a term ``e1`` with +``x`` free and a term ``e2`` that reduces to term ``e2'``, +then ``[e2/x]e1`` reduces to ``[e2'/x]e1``. + +The interesting cases are really the first two - if the term is just the free +variable being substituted for, the result follows from the hypothesis. In the +case that the free variable is _not_ free (i.e. the term has no hole), then +the result follows from the reflexivity of reduction. This is more than +sufficient to handle the case of a variable that is not the distinguished free +variable, and all other cases are covered. + +The proof is by induction on the structure of the term with the free variable. + +!}% +%sort substitute1 {E1 %pi exp %-> exp} %. +%mode substitute1 %in %in %out %. +%term _ substitute1 ([x] x) D D %. +%term _ substitute1 ([x] E) _ reduce/id %. +%term _ + %pi (substitute1 ([x] lam ([y] E x y)) D (reduce/lam D')) + %<- ({y} substitute1 ([x] E x y) D (D' y)) %. +%term _ + %pi (substitute1 ([x] app (E1 x) (E2 x)) D (reduce/app D2 D1)) + %<- (substitute1 E1 D D1) + %<- (substitute1 E2 D D2) %. +%worlds (exps) (substitute1 _ _ _) %. +%total E (substitute1 E _ _) %. +%{! + +The second substitution theorem is similar, except that we are reducing +``e1`` with ``x`` free to ``e1'`` as well as by reducing +``e2`` to ``e2'``, showing that ``[e2/x]e1`` reduces to +``[e2'/x]e1'``. + +Proof is by induction on the structure of the reduction of ``e1`` to +``e1'``. We call to the ``substitute1`` lemma +when we "bottom out" at the reflexive case ``reduce/id``. + +!}% +%sort substitute2 {x exp} %. +%mode substitute2 %in %in %out %. +%term _ + %pi (substitute2 ([x] %the (reduce (E x) (E x)) reduce/id) D D') + %<- (substitute1 E D D') %. +%term _ + %pi (substitute2 ([x] reduce/lam ([y] D1 x y)) D2 (reduce/lam D)) + %<- ({y} substitute2 ([x] D1 x y) D2 (D y)) %. +%term _ + %pi (substitute2 ([x] reduce/app (D2 x) (D1 x)) D (reduce/app D2' D1')) + %<- (substitute2 D1 D D1') + %<- (substitute2 D2 D D2') %. +%term _ + %pi (substitute2 ([x] reduce/beta (D2 x) ([y] D1 x y)) D (reduce/beta D2' D1')) + %<- ({y} substitute2 ([x] D1 x y) D (D1' y)) + %<- (substitute2 D2 D D2') %. +%worlds (exps) (substitute2 _ _ _) %. +%total D (substitute2 D _ _) %. +%{! ## The Diamond Property !}% +%{! + +Now we come to the interesting part: the diamond property. + + E + / \ + / \ + E1 E2 + \ / + \ / + E' + +If ``E`` reduces to both ``E1``, and ``E2``, then there is a +common E' such that ``E1`` and ``E2`` both reduce to it. + +!}% +%sort diamond %. +%mode diamond %in %in %out %out %. +%{! ### Identity !}% +%{! + +If either case is the identity, then we are done. + + id: E D: D: E id: + e=>e / \ e=>e2 e=>e1 / \ e=>e + / \ / \ + E E2 E1 E + D: \ /id: id: \ /D: + e=>e2 \ / e2=>e2 e1=>e1\ / e=>e1 + e2 E1 + +!}% +%term _ diamond reduce/id D D reduce/id %. +%term _ diamond D reduce/id reduce/id D %. +%{! ### Lambda-Lambda !}% +%{! + +If both cases are reductions under a binder, we pull the result straight +from the induction hypothesis. + + λx.e by induction: + reduce/lam / \ reduce/lam D1, D2 ---> D1': e1'=>e' + (D1: e=>e1) / \ (D2: e=>e2) D2': e2'=>e' + / \ + λx.e1 λx.e2 + \ / + reduce/lam \ / reduce/lam + D1' \ / D2' + λx.e' + +Note the oversimplification being made in the graphical presentation, in that +the subterms and sub-derivations are not clearly shown to have a free variable. +Twelf will, of course, not allow this sloppiness. + +!}% +%term _ + %pi (diamond (%the (reduce (lam E) (lam E1)) (reduce/lam (%the ({x exp} reduce (E x) (E1 x)) D1))) (%the (reduce (lam E) (lam E2)) (reduce/lam (%the ({x exp} reduce (E x) (E2 x)) D2))) (reduce/lam D1') (reduce/lam D2')) + %<- ({x exp} diamond (D1 x) (D2 x) (%the (reduce (E1 x) (E' x)) (D1' x)) (%the (reduce (E2 x) (E' x)) (D2' x))) %. +%{! ### Application-Application !}% +%{! + +If both cases are applications, we pull the result straight from the +induction hypothesis. + + ea eb by induction + reduce/app / \ reduce/app D1a, D2a ---> D1a': e1a=>ea' + (D1b: eb=>e1b) / \ (D2b: eb=>e2b) D2a': e2a=>ea' + (D1a: ea=>e1a) / \ (D2a: ea=>e2a) D1b, D2b ---> D1b': e1b=>eb' + e1a e1b e2a e2b D2b': e2b=>eb' + \ / + reduce/app \ / reduce/app + D1b' D1a' \ / D2b' D2a' + ea' eb' +!}% +%% WRITE THIS CASE +%{! ### Beta-Beta !}% +%{! + +If both cases are beta reductions, we get the result from performing two +substitutions. + + (λx.ea) eb by induction + reduce/beta / \ reduce/beta D1a, D2a ---> D1a': e1a=>ea' + (D1b: eb=>e1b) / \ (D2b: eb=>e2b) D2a': e2a=>ea' + (D1a: ea=>e1a) / \ (D2a: ea=>e1a) D1b, D2b ---> D1b': e1b=>eb' + [e1b/x]e1a [e2b/x]e2a D2b': e2b=>eb' + \ / + substitute \ / substitute + D1b' into D1a' \ / D2b' into D2a' + [eb'/x]ea +!}% +%term _ + %pi (diamond (%the (reduce (app (lam Ea) Eb) (E1a E1b)) (reduce/beta (%the (reduce Eb E1b) D1b) (%the ({x} reduce (Ea x) (E1a x)) D1a))) (%the (reduce (app (lam Ea) Eb) (E2a E2b)) (reduce/beta (%the (reduce Eb E2b) D2b) (%the ({x} reduce (Ea x) (E2a x)) D2a))) D1 D2) + %<- ({x} diamond (D1a x) (D2a x) (%the (reduce (E1a x) (Ea' x)) (D1a' x)) (%the (reduce (E2a x) (Ea' x)) (D2a' x))) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) + %<- (substitute2 D1a' D1b' (%the (reduce (E1a E1b) (Ea' Eb')) D1)) + %<- (substitute2 D2a' D2b' (%the (reduce (E2a E2b) (Ea' Eb')) D2)) %. +%{! ### Beta-Application !}% +%{! + +If the left-hand side is a β-reduction +``(λx.ea) eb => [e1b/x] e1a`` but the right-hand side is not, then we +know that the right-hand side reduction must be +``(λx.ea) eb => (λx.e2a) e2b``. + +Unfortunately, then we have two options - either ``λx.ea => λx.e2a`` by +``reduce/refl`` and ``reduce/lam``. The ``reduce/refl`` case +is essentially just an inconvenience caused by allowing identity reductions +anywhere (not just at variables). + +The first subcase: + + (λx.ea) eb by induction + reduce/beta / \ reduce/app D1a, D2a ---> D1a': e1a=>ea' + (D1b: eb=>e1b) / \ (D2b: eb=>e2b) D2a': e2a=>ea' + (D1a: ea=>e1a) / \ (reduce/lam D1b, D2b ---> D1b': e1b=>eb' + / \ (D2a: ea=>e2a)) D2b': e1b=>eb' + [e1b/x]e1a (λx.e2a) e2b + \ / + substitute \ / reduce/beta + D1b' into D2a' \ / D2b' D2a' + \ / + [eb'/x]ea' +!}% +%% WRITE THIS CASE +%{! + +The second subcase: + + (λx.ea) eb by induction + reduce/beta / \ reduce/app D1b, D2b ---> D1b': e1b=>eb' + (D1b: eb=>e1b) / \ (D2b: eb=>e2b) D2b': e1b=>eb' + (D1a: ea=>e1a) / \ (reduce/id + / \ ea=>ea)) + [e1b/x]e1a (λx.ea) e2b + \ / + substitute \ / reduce/beta + D1b' into e1a \ / D2b' D1a + \ / + [eb'/x]e1a +!}% +%term _ + %pi (diamond (%the (reduce (app (lam Ea) Eb) (E1a E1b)) (reduce/beta (%the (reduce Eb E1b) D1b) (%the ({x} reduce (Ea x) (E1a x)) D1a))) (%the (reduce (app (lam Ea) Eb) (app (lam Ea) E2b)) (reduce/app (%the (reduce Eb E2b) D2b) (%the (reduce (lam Ea) (lam Ea)) reduce/id))) D1 (reduce/beta D2b' D1a)) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) + %<- (substitute1 E1a D1b' (%the (reduce (E1a E1b) (E1a Eb')) D1)) %. +%{! ### Application-Beta !}% +%{! + +If the right-hand hand side is a β-reduction but the left-hand side is not, we +have to do the same two cases in reverse; we omit the graphics for those +two cases as they are symmetric. + +!}% +%term _ + %pi (diamond (%the (reduce (app (lam Ea) Eb) (app (lam Ea) E1b)) (reduce/app (%the (reduce Eb E1b) D1b) (%the (reduce (lam Ea) (lam Ea)) reduce/id))) (%the (reduce (app (lam Ea) Eb) (E2a E2b)) (reduce/beta (%the (reduce Eb E2b) D2b) (%the ({x} reduce (Ea x) (E2a x)) D2a))) (reduce/beta D1b' D2a) D2) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) + %<- (substitute1 E2a D2b' (%the (reduce (E2a E2b) (E2a Eb')) D2)) %. +%term _ + %pi (diamond (%the (reduce (app (lam Ea) Eb) (app (lam E1a) E1b)) (reduce/app (%the (reduce Eb E1b) D1b) (reduce/lam (%the ({x} reduce (Ea x) (E1a x)) D1a)))) (%the (reduce (app (lam Ea) Eb) (E2a E2b)) (reduce/beta (%the (reduce Eb E2b) D2b) (%the ({x} reduce (Ea x) (E2a x)) D2a))) (reduce/beta D1b' D1a') D2) + %<- ({x} diamond (D1a x) (D2a x) (%the (reduce (E1a x) (Ea' x)) (D1a' x)) (%the (reduce (E2a x) (Ea' x)) (D2a' x))) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) + %<- (substitute2 D2a' D2b' (%the (reduce (E2a E2b) (Ea' Eb')) D2)) %. +%{! Now we are done! We check in the ``exps`` world with free variables. !}% +%worlds (exps) (diamond _ _ _ _) %. +%% %total D1 (diamond D1 D2 _ _). +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Church_Rosser_(Problem)). +!}% \ No newline at end of file diff --git a/new-tests/pages/popl-tutorial-properties-of-typing-and-reduction.lf b/new-tests/pages/popl-tutorial-properties-of-typing-and-reduction.lf new file mode 100644 index 0000000..ad1a265 --- /dev/null +++ b/new-tests/pages/popl-tutorial-properties-of-typing-and-reduction.lf @@ -0,0 +1,392 @@ +%%! title: "POPL Tutorial/Properties of Typing and Reduction" +%{! This problem (or problems) involves proving three properties of typing and reduction for the language L\{num str\} +from Bob Harper's book, [http://www.cs.cmu.edu/~rwh/plbook/book.pdf Practical Foundations for Programming Languages]. + +## The System L\{num str\} !}% +% Natural numbers +%sort nat %. +%term 0 nat %. +%term s %pi nat %-> nat %. +% Addition for natural numbers +%sort plus_op %. +%term plus_op/0 plus_op 0 N N %. +%term plus_op/s %pi (plus_op N M L) %-> (plus_op (s N) M (s L)) %. +% Multiplication for natural numbers +%sort times_op %. +%term times_op/0 times_op 0 N 0 %. +%term times_op/s %pi (times_op (s N) M P) %<- (times_op N M L) %<- (plus_op L M P) %. +%%%%%%%%%%%%%%%% L{num str}: Syntax %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Types +%sort typ %. +%term num typ %. +%term str typ %. +% Expressions. For simplicity, strings are just natural numbers, +% concatenation is addition, and len is the identity. +%sort exp %. +%term nume %pi nat %-> exp %. +%term stre %pi nat %-> exp %. +%term plus %pi exp %-> exp %-> exp %. +%term times %pi exp %-> exp %-> exp %. +%term cat %pi exp %-> exp %-> exp %. +%term len %pi exp %-> exp %. +%term let %pi exp %-> (%pi exp %-> exp) %-> exp %. +%%%%%%%%%%%%%%%% L{num str}: Static semantics (typing judgment) %%%%%%%%%%%%%%% +%sort of %. +%term of/stre of (stre S) str %. +%term of/nume of (nume N) num %. +%term of/plus %pi (of E1 num) %-> (of E2 num) %-> (of (plus E1 E2) num) %. +%term of/times %pi (of E1 num) %-> (of E2 num) %-> (of (times E1 E2) num) %. +%term of/cat %pi (of E1 str) %-> (of E2 str) %-> (of (cat E1 E2) str) %. +%term of/len %pi (of E1 str) %-> (of (len E1) num) %. +%term of/let + %pi (of E1 T1) + %-> ({x} %pi (of x T1) %-> (of (E2 x) T2)) + %-> (of (let E1 ([x] E2 x)) T2) %. +% Assumption block for typing judgment (coming from of/let) +%block of_bind {T typ} [x exp] [ofx of x T]%. +%%%%%%%%%%%%%%%% L{num str}: Dynamic Semantics %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Values +%sort val %. +%term val/stre val (stre N) %. +%term val/nume val (nume N) %. +% Reduction +%sort red %. +%term red/plus/num %pi (plus_op N1 N2 N) %-> (red (plus (nume N1) (nume N2)) (nume N)) %. +%term red/plus1 %pi (red E1 E1') %-> (red (plus E1 E2) (plus E1' E2)) %. +%term red/plus2 %pi (val E1) %-> (red E2 E2') %-> (red (plus E1 E2) (plus E1 E2')) %. +%term red/times/num %pi (times_op N1 N2 N) %-> (red (times (nume N1) (nume N2)) (nume N)) %. +%term red/times1 %pi (red E1 E1') %-> (red (times E1 E2) (times E1' E2)) %. +%term red/times2 %pi (val E1) %-> (red E2 E2') %-> (red (times E1 E2) (times E1 E2')) %. +%term red/cat/str %pi (plus_op S1 S2 S) %-> (red (cat (stre S1) (stre S2)) (stre S)) %. +%term red/cat1 %pi (red E1 E1') %-> (red (cat E1 E2) (cat E1' E2)) %. +%term red/cat2 %pi (val E1) %-> (red E2 E2') %-> (red (cat E1 E2) (cat E1 E2')) %. +%term red/len/str red (len (stre S)) (nume S) %. +%term red/len %pi (red E1 E1') %-> (red (len E1) (len E1')) %. +%term red/let/val %pi (val E1) %-> (red (let E1 ([x] E2 x)) (E2 E1)) %. +%term red/let %pi (red E1 E1') %-> (red (let E1 ([x] E2 x)) (let E1' ([x] E2 x))) %. +%{! ## Problem 1: Unicity of Typing !}% +%%%%%%%%%%%%%%%% Lemma 9.1 (Unicity of typing) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% + +% If G |- E : T1 and G |- E : T2, then T1 = T2. +% Equality of types +%sort eqtyp %. +%term refl_t eqtyp T T %. +% Congruence for typing assumptions. Although the proof doesn't +% introduce any parameters or hypotheses, it is used by a theorem that +% does, and so we need to check it in the extended worlds. +%sort cong_of %. +%mode cong_of %in %in %out %. +%term _ cong_of refl_t OE1 OE1 %. +%worlds (of_bind) (cong_of _ _ _) %. +%total OE1 (cong_of _ OE1 _) %. +% Statement and proof of Lemma. Notice that the worlds in this proof +% are an extension (technically, they subsume) the worlds in the +% congruence lemma above. +%sort lemma9-1 %. +%mode lemma9-1 %in %in %out %. +%term _ lemma9-1 of/stre of/stre refl_t %. +%term _ lemma9-1 of/nume of/nume refl_t %. +%term _ lemma9-1 (of/plus OjE11 OjE12) (of/plus OjE21 OjE22) refl_t %. +%term _ lemma9-1 (of/times OjE11 OjE12) (of/times OjE21 OjE22) refl_t %. +%term _ lemma9-1 (of/cat OjE11 OjE12) (of/cat OjE21 OjE22) refl_t %. +%term _ lemma9-1 (of/len OjE1) (of/len OjE2) refl_t %. +%term _ + %pi (lemma9-1 (of/let OjE11 ([x] [ofx1] OjE12 x ofx1)) (of/let OjE21 ([x] [ofx2] OjE22 x ofx2)) Q) + %<- (lemma9-1 OjE11 OjE21 E1) + %<- (cong_of E1 OjE12 OjE12') + %<- ({x} {ofx2} %pi (lemma9-1 ofx2 ofx2 refl_t) %-> (lemma9-1 (OjE12' x ofx2) (OjE22 x ofx2) Q)) %. +%block lemma_block {T typ} [x exp] [ofx of x T] [u lemma9-1 ofx ofx refl_t]%. +%worlds (lemma_block) (lemma9-1 _ _ _) %. +%total OjT2 (lemma9-1 _ OjT2 _) %. +%{! ## Problem 2: Determinacy of Reduction !}% +%%%%%%%%%%%%%%%% Lemma 10.1 (Determinacy of Reduction) %%%%%%%%%%%%%%%%%%%%%%% +% + +% If E |-> E' and E |-> E'', then E' = E''. +% Equality and congruence for nats +%sort eqnat %. +%term refl_n eqnat N N %. +%sort cong_s %. +%mode cong_s %in %out %. +%term _ cong_s refl_n refl_n %. +%worlds () (cong_s _ _) %. +%total {} (cong_s _ _) %. +% Equality for expressions +%sort eqexp %. +%term refl_e eqexp E E %. +% Congruences for expressions +%sort cong_nume %. +%mode cong_nume %in %out %. +%term _ cong_nume refl_n refl_e %. +%worlds () (cong_nume _ _) %. +%total E (cong_nume E _) %. +%sort cong_stre %. +%mode cong_stre %in %out %. +%term _ cong_stre refl_n refl_e %. +%worlds () (cong_stre _ _) %. +%total E (cong_stre E _) %. +%sort cong_plus1 %. +%mode {%in E1 exp} {%in E2 exp} {%in F exp} {%in E eqexp E1 E2} {%out E' eqexp (plus E1 F) (plus E2 F)} cong_plus1 E E' %. +%term _ cong_plus1 refl_e refl_e %. +%worlds () (cong_plus1 _ _) %. +%total E (cong_plus1 E _) %. +%sort cong_plus2 %. +%mode {%in F1 exp} {%in F2 exp} {%in E exp} {%in E1 eqexp F1 F2} {%out E' eqexp (plus E F1) (plus E F2)} cong_plus2 E1 E' %. +%term _ cong_plus2 refl_e refl_e %. +%worlds () (cong_plus2 _ _) %. +%total E (cong_plus2 E _) %. +%sort cong_times1 %. +%mode {%in E1 exp} {%in E2 exp} {%in F exp} {%in E eqexp E1 E2} {%out E' eqexp (times E1 F) (times E2 F)} cong_times1 E E' %. +%term _ cong_times1 refl_e refl_e %. +%worlds () (cong_times1 _ _) %. +%total E (cong_times1 E _) %. +%sort cong_times2 %. +%mode {%in F1 exp} {%in F2 exp} {%in E exp} {%in E2 eqexp F1 F2} {%out E' eqexp (times E F1) (times E F2)} cong_times2 E2 E' %. +%term _ cong_times2 refl_e refl_e %. +%worlds () (cong_times2 _ _) %. +%total E (cong_times2 E _) %. +%sort cong_cat1 %. +%mode {%in E1 exp} {%in E2 exp} {%in F exp} {%in E eqexp E1 E2} {%out E' eqexp (cat E1 F) (cat E2 F)} cong_cat1 E E' %. +%term _ cong_cat1 refl_e refl_e %. +%worlds () (cong_cat1 _ _) %. +%total E (cong_cat1 E _) %. +%sort cong_cat2 %. +%mode {%in F1 exp} {%in F2 exp} {%in E exp} {%in E3 eqexp F1 F2} {%out E' eqexp (cat E F1) (cat E F2)} cong_cat2 E3 E' %. +%term _ cong_cat2 refl_e refl_e %. +%worlds () (cong_cat2 _ _) %. +%total E (cong_cat2 E _) %. +%sort cong_len %. +%mode cong_len %in %out %. +%term _ cong_len refl_e refl_e %. +%worlds () (cong_len _ _) %. +%total E (cong_len E _) %. +%sort cong_let %. +%mode {%in E1 exp} {%in E2 exp} {%in F %pi exp %-> exp} {%in E eqexp E1 E2} {%out E' eqexp (let E1 ([x exp] F x)) (let E2 ([x exp] F x))} cong_let E E' %. +%term _ cong_let refl_e refl_e %. +%worlds () (cong_let _ _) %. +%total E (cong_let E _) %. +% Uniqueness of plus +%sort plus! %. +%mode plus! %in %in %out %. +%term _ plus! plus_op/0 plus_op/0 refl_n %. +%term _ + %pi (plus! (plus_op/s Pj1) (plus_op/s Pj2) ES) + %<- (plus! Pj1 Pj2 EL) + %<- (cong_s EL ES) %. +%worlds () (plus! _ _ _) %. +%total Pj1 (plus! Pj1 _ _) %. +% Congruence of plus +%sort plus_cong %. +%mode plus_cong %in %in %in %out %. +%term _ plus_cong refl_n plus_op/0 plus_op/0 refl_n %. +%term _ + %pi (plus_cong refl_n (plus_op/s Pj1) (plus_op/s Pj2) ES) + %<- (plus_cong refl_n Pj1 Pj2 EL) + %<- (cong_s EL ES) %. +%worlds () (plus_cong _ _ _ _) %. +%total PjL (plus_cong _ PjL _ _) %. +% Uniqueness of times +%sort times! %. +%mode times! %in %in %out %. +%term _ times! times_op/0 times_op/0 refl_n %. +%term _ + %pi (times! (times_op/s Pj1 Tj1) (times_op/s Pj2 Tj2) ES) + %<- (times! Tj1 Tj2 EL) + %<- (plus_cong EL Pj1 Pj2 ES) %. +%worlds () (times! _ _ _) %. +%total Tj1 (times! Tj1 _ _) %. +% Proof of Lemma 10.1 +%sort lemma10-1 %. +%mode lemma10-1 %in %in %out %. +%term _ + %pi (lemma10-1 (red/plus/num PjN1) (red/plus/num PjN2) EE') + %<- (plus! PjN1 PjN2 EL) + %<- (cong_nume EL EE') %. +%term _ + %pi (lemma10-1 (red/plus1 RjE1) (red/plus1 RjE2) EE') + %<- (lemma10-1 RjE1 RjE2 EE) + %<- (cong_plus1 EE EE') %. +%term _ + %pi (lemma10-1 (red/plus2 _ RjF1) (red/plus2 _ RjF2) EE') + %<- (lemma10-1 RjF1 RjF2 EF) + %<- (cong_plus2 EF EE') %. +%term _ + %pi (lemma10-1 (red/times/num TjN1) (red/times/num TjN2) EE') + %<- (times! TjN1 TjN2 EE) + %<- (cong_nume EE EE') %. +%term _ + %pi (lemma10-1 (red/times1 RjE1) (red/times1 RjE2) EE') + %<- (lemma10-1 RjE1 RjE2 EE) + %<- (cong_times1 EE EE') %. +%term _ + %pi (lemma10-1 (red/times2 _ RjF1) (red/times2 _ RjF2) EE') + %<- (lemma10-1 RjF1 RjF2 EF) + %<- (cong_times2 EF EE') %. +%term _ + %pi (lemma10-1 (red/cat/str PjN1) (red/cat/str PjN2) EE') + %<- (plus! PjN1 PjN2 EE) + %<- (cong_stre EE EE') %. +%term _ + %pi (lemma10-1 (red/cat1 RjE1) (red/cat1 RjE2) EE') + %<- (lemma10-1 RjE1 RjE2 EE) + %<- (cong_cat1 EE EE') %. +%term _ + %pi (lemma10-1 (red/cat2 _ RjF1) (red/cat2 _ RjF2) EE') + %<- (lemma10-1 RjF1 RjF2 EF) + %<- (cong_cat2 EF EE') %. +%term _ + %pi (lemma10-1 (%the (red (len (stre S)) (nume S)) red/len/str) red/len/str EE') + %<- (cong_nume refl_n EE') %. +%term _ + %pi (lemma10-1 (red/len RjE1) (red/len RjE2) EE') + %<- (lemma10-1 RjE1 RjE2 EE) + %<- (cong_len EE EE') %. +%term _ lemma10-1 (red/let/val VjE11) (red/let/val VjE12) refl_e %. +%term _ + %pi (lemma10-1 (red/let RjE1) (red/let RjE2) EE') + %<- (lemma10-1 RjE1 RjE2 EE) + %<- (cong_let EE EE') %. +%worlds () (lemma10-1 _ _ _) %. +%total RjE (lemma10-1 RjE _ _) %. +%{! ## Problem 3: Contextual Semantics + +In this problem, we define the contextual semantics for L\{num str\} and proof that it is equivalent to (defines the same relation as) the given dynamic semantics (reduction). !}% +%%%%%%%%%%%%%%%% L{num str}: Contextual Semantics %%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Instructions: E0 ~> E0' +%sort instr %. +%term instr/plus %pi (plus_op N1 N2 N) %-> (instr (plus (nume N1) (nume N2)) (nume N)) %. +%term instr/times %pi (times_op N1 N2 N) %-> (instr (times (nume N1) (nume N2)) (nume N)) %. +%term instr/cat %pi (plus_op S1 S2 S) %-> (instr (cat (stre S1) (stre S2)) (stre S)) %. +%term instr/len instr (len (stre S)) (nume S) %. +%term instr/let %pi (val E1) %-> (instr (let E1 ([x] E2 x)) (E2 E1)) %. +% Expression contexts, represented as "expressions with holes", i.e., +% functions on expressions. +%sort ectxt %. +%term ectxt/hole ectxt ([x] x) %. +%term ectxt/plus1 %pi (ectxt ([x] C1 x)) %-> (ectxt ([x] plus (C1 x) E2)) %. +%term ectxt/plus2 %pi (val E1) %-> (ectxt ([x] C2 x)) %-> (ectxt ([x] plus E1 (C2 x))) %. +%term ectxt/times1 %pi (ectxt ([x] C1 x)) %-> (ectxt ([x] times (C1 x) E2)) %. +%term ectxt/times2 %pi (val E1) %-> (ectxt ([x] C2 x)) %-> (ectxt ([x] times E1 (C2 x))) %. +%term ectxt/cat1 %pi (ectxt ([x] C1 x)) %-> (ectxt ([x] cat (C1 x) E2)) %. +%term ectxt/cat2 %pi (val E1) %-> (ectxt ([x] C2 x)) %-> (ectxt ([x] cat E1 (C2 x))) %. +%term ectxt/len %pi (ectxt ([x] C x)) %-> (ectxt ([x] len (C x))) %. +%term ectxt/let %pi (ectxt ([x] C1 x)) %-> (ectxt ([x] let (C1 x) ([y] E2 y))) %. +% Filling the hole of an expression context with an expression +%sort fill %. +%term fill/hole fill ectxt/hole E E %. +%term fill/plus1 + %pi (fill ECC1 E C1@E) + %-> (fill (%the (ectxt ([x] plus (C1 x) E2)) (ectxt/plus1 ECC1)) E (plus C1@E E2)) %. +%term fill/plus2 {VjE1 val E1} %pi (fill ECC2 E C2@E) %-> (fill (ectxt/plus2 VjE1 ECC2) E (plus E1 C2@E)) %. +%term fill/times1 + %pi (fill ECC1 E C1@E) + %-> (fill (%the (ectxt ([x] times (C1 x) E2)) (ectxt/times1 ECC1)) E (times C1@E E2)) %. +%term fill/times2 {VjE1 val E1} %pi (fill ECC2 E C2@E) %-> (fill (ectxt/times2 VjE1 ECC2) E (times E1 C2@E)) %. +%term fill/cat1 + %pi (fill ECC1 E C1@E) + %-> (fill (%the (ectxt ([x] cat (C1 x) E2)) (ectxt/cat1 ECC1)) E (cat C1@E E2)) %. +%term fill/cat2 {VjE1 val E1} %pi (fill ECC2 E C2@E) %-> (fill (ectxt/cat2 VjE1 ECC2) E (cat E1 C2@E)) %. +%term fill/len %pi (fill ECC1 E C1@E) %-> (fill (ectxt/len ECC1) E (len C1@E)) %. +%term fill/let + %pi (fill ECC1 E C1@E) + %-> (fill (%the (ectxt ([x] let (C1 x) ([y] E2 y))) (ectxt/let ECC1)) E (let C1@E ([y] E2 y))) %. +% Contextual reduction rule +%sort cred %. +%term cred/step + %pi (fill EC E0 C@E0) + %-> (instr E0 E0') + %-> (fill EC E0' C@E0') + %-> (cred C@E0 C@E0') %. +%%%%%%%%%%%%%%%% Theorem 10.2.1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% + +% If E |-> E', then E |->c E'. +%sort thm10-2-1 %. +%mode thm10-2-1 %in %out %. +%term _ thm10-2-1 (red/plus/num PjN) (cred/step fill/hole (instr/plus PjN) fill/hole) %. +%term _ + %pi (thm10-2-1 (red/plus1 RjE1) (cred/step (fill/plus1 FjC1@E0) I (fill/plus1 FjC1@E0'))) + %<- (thm10-2-1 RjE1 (cred/step FjC1@E0 I FjC1@E0')) %. +%term _ + %pi (thm10-2-1 (red/plus2 VjE1 RjE2) (cred/step (fill/plus2 VjE1 FjC2@E0) I (fill/plus2 VjE1 FjC2@E0'))) + %<- (thm10-2-1 RjE2 (cred/step FjC2@E0 I FjC2@E0')) %. +%term _ thm10-2-1 (red/times/num TjN) (cred/step fill/hole (instr/times TjN) fill/hole) %. +%term _ + %pi (thm10-2-1 (red/times1 RjE1) (cred/step (fill/times1 FjC1@E0) I (fill/times1 FjC1@E0'))) + %<- (thm10-2-1 RjE1 (cred/step FjC1@E0 I FjC1@E0')) %. +%term _ + %pi (thm10-2-1 (red/times2 VjE1 RjE2) (cred/step (fill/times2 VjE1 FjC2@E0) I (fill/times2 VjE1 FjC2@E0'))) + %<- (thm10-2-1 RjE2 (cred/step FjC2@E0 I FjC2@E0')) %. +%term _ thm10-2-1 (red/cat/str PjN) (cred/step fill/hole (instr/cat PjN) fill/hole) %. +%term _ + %pi (thm10-2-1 (red/cat1 RjE1) (cred/step (fill/cat1 FjC1@E0) I (fill/cat1 FjC1@E0'))) + %<- (thm10-2-1 RjE1 (cred/step FjC1@E0 I FjC1@E0')) %. +%term _ + %pi (thm10-2-1 (red/cat2 VjE1 RjE2) (cred/step (fill/cat2 VjE1 FjC2@E0) I (fill/cat2 VjE1 FjC2@E0'))) + %<- (thm10-2-1 RjE2 (cred/step FjC2@E0 I FjC2@E0')) %. +%term _ thm10-2-1 red/len/str (cred/step fill/hole instr/len fill/hole) %. +%term _ + %pi (thm10-2-1 (red/len RjE1) (cred/step (fill/len FjC1@E0) I (fill/len FjC1@E0'))) + %<- (thm10-2-1 RjE1 (cred/step FjC1@E0 I FjC1@E0')) %. +%term _ thm10-2-1 (red/let/val VjE1) (cred/step fill/hole (instr/let VjE1) fill/hole) %. +%term _ + %pi (thm10-2-1 (red/let RjE1) (cred/step (fill/let FjC1@E0) I (fill/let FjC1@E0'))) + %<- (thm10-2-1 RjE1 (cred/step FjC1@E0 I FjC1@E0')) %. +%worlds () (thm10-2-1 _ _) %. +%total RjE (thm10-2-1 RjE _) %. +%%%%%%%%%%%%%%%% Theorem 10.2.2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% + +% If E |->c E', then E |-> E'. +%sort thm10-2-2 %. +%mode thm10-2-2 %in %out %. +% Lemma. If C{E0} = E, E0 ~> E0', and C{E0'} = E', then E |-> E'. +%sort lem10-2-2 %. +%mode lem10-2-2 %in %in %in %out %. +%term _ lem10-2-2 fill/hole (instr/plus PjN) fill/hole (red/plus/num PjN) %. +%term _ lem10-2-2 fill/hole (instr/times TjN) fill/hole (red/times/num TjN) %. +%term _ lem10-2-2 fill/hole (instr/cat PjN) fill/hole (red/cat/str PjN) %. +%term _ lem10-2-2 fill/hole instr/len fill/hole red/len/str %. +%term _ lem10-2-2 fill/hole (instr/let VjE1) fill/hole (red/let/val VjE1) %. +%term _ + %pi (lem10-2-2 (fill/plus1 FjC1@E0) I (fill/plus1 FjC1@E0') (red/plus1 RjE)) + %<- (lem10-2-2 FjC1@E0 I FjC1@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/plus2 VjE1 FjC2@E0) I (fill/plus2 VjE1 FjC2@E0') (red/plus2 VjE1 RjE)) + %<- (lem10-2-2 FjC2@E0 I FjC2@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/times1 FjC1@E0) I (fill/times1 FjC1@E0') (red/times1 RjE)) + %<- (lem10-2-2 FjC1@E0 I FjC1@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/times2 VjE1 FjC2@E0) I (fill/times2 VjE1 FjC2@E0') (red/times2 VjE1 RjE)) + %<- (lem10-2-2 FjC2@E0 I FjC2@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/cat1 FjC1@E0) I (fill/cat1 FjC1@E0') (red/cat1 RjE)) + %<- (lem10-2-2 FjC1@E0 I FjC1@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/cat2 VjE1 FjC2@E0) I (fill/cat2 VjE1 FjC2@E0') (red/cat2 VjE1 RjE)) + %<- (lem10-2-2 FjC2@E0 I FjC2@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/len FjC1@E0) I (fill/len FjC1@E0') (red/len RjE)) + %<- (lem10-2-2 FjC1@E0 I FjC1@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/let FjC2@E0) I (fill/let FjC2@E0') (red/let RjE)) + %<- (lem10-2-2 FjC2@E0 I FjC2@E0' RjE) %. +%worlds () (lem10-2-2 _ _ _ _) %. +%total FjE (lem10-2-2 FjE _ _ _) %. +% Proof of main theorem +%term _ + %pi (thm10-2-2 (cred/step FjC@E0 I FjC@E0') RjE) + %<- (lem10-2-2 FjC@E0 I FjC@E0' RjE) %. +%worlds () (thm10-2-2 _ _) %. +%total {} (thm10-2-2 _ _) %. +%{! ## Twelf Output + +The `` check="true">output`` for the above. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Properties_of_Typing_and_Reduction). +!}% \ No newline at end of file diff --git a/new-tests/pages/proving-metatheorems-solutions-odd-even-succ.lf b/new-tests/pages/proving-metatheorems-solutions-odd-even-succ.lf new file mode 100644 index 0000000..a6a68a1 --- /dev/null +++ b/new-tests/pages/proving-metatheorems-solutions-odd-even-succ.lf @@ -0,0 +1,42 @@ +%%! title: "Solution: proofs about even and odd numbers" +%%! description: "Showing how adding one influences even and odd numbers (Part of the introduction to proving metatheorems in Twelf)" +%{! + +_This page is part of the introduction to [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)._ + +Exercise 3 from [the part 1 exercises](/wiki/proving-metatheorems-summary-the-natural-numbers/): State and prove a theorem ``succ-even`` that shows that the successor of an even number is an odd number. After that, state and prove a theorem ``succ-odd`` that shows that the successor of an odd number is an even number. + +Starting with the preliminaries, including the [definition of `odd` from exercise 2](/wiki/proving-metatheorems-solutions-define-odd/): + +!}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort plus %. +%term plus-z plus z N2 N2 %. +%term plus-s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%sort even %. +%term even-z even z %. +%term even-s %pi (even (s (s N))) %<- (even N) %. +%sort odd %. +%term odd-1 odd (s z) %. +%term odd-s %pi (odd N) %-> (odd (s (s N))) %. +%{! Because both the `even` and `odd` judgements are defined independently, it's necessary to perform induction on the `even` judgement to prove that the first theorem, and induction on the `odd` judgement to prove the second theorem: !}% +%{! ## The successor of an even number is odd !}% +%{!! begin checked !!}% +%sort succ-even %. +%mode succ-even %in %out %. +%term sez succ-even even-z odd-1 %. +%term ses %pi (succ-even (even-s EvenA) (odd-s OddA)) %<- (succ-even EvenA OddA) %. +%worlds () (succ-even _ _) %. +%total D (succ-even D _) %. +%{!! end checked !!}% +%{! ## The successor of an odd number is even !}% +%{!! begin checked !!}% +%sort succ-odd %. +%mode succ-odd %in %out %. +%term so1 succ-odd odd-1 (even-s even-z) %. +%term sos %pi (succ-odd (odd-s OddA) (even-s EvenA)) %<- (succ-odd OddA EvenA) %. +%worlds () (succ-odd _ _) %. +%total D (succ-odd D _) %. +%{!! end checked !!}% \ No newline at end of file diff --git a/new-tests/pages/proving-metatheorems-with-twelf.lf b/new-tests/pages/proving-metatheorems-with-twelf.lf new file mode 100644 index 0000000..0666309 --- /dev/null +++ b/new-tests/pages/proving-metatheorems-with-twelf.lf @@ -0,0 +1,54 @@ +%%! title: "Proving metatheorems with Twelf" +%%! description: "Our favorite tutorial for getting started with Twelf as a tool for describing programming languages and logical systems and proving properties of those systems" +%%! next: +%%! label: Representing the syntax of the natural numbers +%%! link: /wiki/proving-metatheorems-representing-the-syntax-of-the-natural-numbers/ +%{! + +The story of a typical [deductive system](/wiki/object-logic/) goes something like this: + - One day, a programming language designer has an idea for a new programming language. + - She writes down the language's [abstract syntax](/wiki/abstract-syntax/). + - Then, she gives the syntax meaning by defining some [judgement](/wiki/judgment/)s for it, such as a type system or an operational semantics. + - Next, she investigates the language's properties by proving some theorems. Maybe she fiddles with the language some to make the desired theorems true; maybe the idea doesn't work at all and she goes on to something else. + - Maybe she implements the language to try out some examples. + - Eventually, if the idea works out, she writes up a paper about the language. If it's a good idea, then she, or maybe someone who read the paper, will someday want to extend the language, or incorporate the idea into another language. + +This introduction assumes this story is somewhat familiar to you. If not, you should read a textbook such as [TAPL](http://www.cis.upenn.edu/~bcpierce/tapl/) or [PFPL](https://www.cs.cmu.edu/~rwh/pfpl/). The Twelf version of the story goes something like this: + - One day, a programming language designer has an idea. + - She formalizes the syntax and judgements of the language in the [LF](/wiki/lf/) logical framework. + - She uses Twelf to check her proofs of the theorems. + - She uses Twelf to run the language definition itself to try out some examples. + - Her paper includes a Twelf appendix, which makes her and her readers much more confident that the theorems in the paper are actually true. + - When someone goes to extend the language, he has a formal, machine-checkable artifact that he can study or perhaps even reuse directly. + +This tutorial pays a lot of attention to the question "how do I know that what I wrote down and proved on paper, and what I wrote down and proved in Twelf, are exactly the same thing?" Checking that these two things are the same is called [adequacy](/wiki/adequacy/), and it's an important problem _whenever_ you're using a theorem proving like Twelf, Agda, or Rocq. But it definitely has the flavor of "checking lots of fiddly details," and so it's reasonable to skip discussions of adequacy in this tutorial if you're exploring Twelf for the first time. + +### First order representations: The natural numbers + +The first layer of this introduction uses a _very_ simple deductive system, the [natural numbers](/wiki/natural-numbers/) with addition, to introduce the Twelf methodology. + + 1. [Representing syntax](/wiki/proving-metatheorems-representing-the-syntax-of-the-natural-numbers/) + 2. [Simply typed LF](/wiki/proving-metatheorems-simply-typed-lf/) + 3. [Representing judgements](/wiki/proving-metatheorems-representing-the-judgements-of-the-natural-numbers/) + 4. [Full LF](/wiki/proving-metatheorems-full-lf/) + 5. [Proving totality assertions](/wiki/proving-metatheorems-proving-totality-assertions-about-the-natural-numbers/) + 6. [Proving metatheorems](/wiki/proving-metatheorems-proving-metatheorems-about-the-natural-numbers/) + 7. [Summary and exercises](/wiki/proving-metatheorems-summary-the-natural-numbers/) + +### Higher-order representations: The simply-typed lambda calculus + +The second layer tells the same story for a programming language with variable binding (the [simply-typed lambda calculus](/wiki/simply-typed-lambda-calculus/)), which is where Twelf really shines. + + 8. [Representing syntax](/wiki/proving-metatheorems-representing-the-syntax-of-the-stlc/) + 9. [Representing judgements](/wiki/proving-metatheorems-representing-the-judgements-of-the-stlc/) + 10. [Proving metatheorems](/wiki/proving-metatheorems-proving-metatheorems-about-the-stlc/) + 11. [Summary and exercises](/wiki/proving-metatheorems-summary-the-stlc/) + +### Beyond the simply-typed lambda calculus + +The third layer presents some more-interesting proofs and introduces one additional feature of Twelf, the ability to do proofs about open terms. + + 12. [Proving totality assertions in non-empty contexts](/wiki/proving-metatheorems-proving-totality-assertions-in-non-empty-contexts/) + 13. [Proving metatheorems in non-empty contexts](/wiki/proving-metatheorems-proving-metatheorems-in-non-empty-contexts/) + +!}% \ No newline at end of file diff --git a/new-tests/pages/talk-effectiveness-lemma.lf b/new-tests/pages/talk-effectiveness-lemma.lf new file mode 100644 index 0000000..f469406 --- /dev/null +++ b/new-tests/pages/talk-effectiveness-lemma.lf @@ -0,0 +1,22 @@ +%%! title: "Talk:Effectiveness lemma" +%{! Why "effectiveness" rather than "totality" ? As far as I can tell, proving "effectiveness" is the same +as proving totality at the meta-leval. [Boyland](/wiki/user-boyland/) 22:41, 25 February 2007 (EST) + +: I think it just dates back to some choices made in the Typed Assembly Language Two formalization. I think we thought that calling it a "totality lemma" would just cause confusion of a different sort, not that effectiveness is an obvious alternative by any means. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 23:40, 25 February 2007 (EST) + +This sentence is pretty confusing: "Proving an effectiveness lemma essentially establishes the same fact that a %total directive does, but does so in a way that can be used by other metatheorems." You use %total to prove any metatheorem, and any lemma so proved can be used by other metatheorems. + +An effectiveness lemma says that some relation is in fact an algorithm (hence "effective"), and returns for any inputs a derivation of the relation which witnesses that fact. [User:JakeD](/wiki/user-jaked/) 00:33, 26 February 2007 (EST) + +: I mean, that works for me too :) — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 01:04, 26 February 2007 (EST) +:: I edited it to try to simplify. We definitely should not lead with confusing stuff about metatheorems (since that is hard to wrap ones head around, especially given all the different takes on what a metatheorem is!), but a section in this article about its relationship to [``%total``](/wiki/percent-total/) is warranted, I think, because it is an obvious question.  — [Tom 7](/wiki/user-tom7/) 09:33, 26 February 2007 (EST) + +: I think language that connects "effectivenss" with totality is warranted. I don't see it in the article any more. There needs to be something on the order of "Proving an effectivessness lemma for a relation essentially establishes the same fact that a %total directive does for the relation, but does so in a way that the totality itself is named and can be used in proofs." Otherwise people will wonder. Also, personally (as an outsider), I find it annoying that one basically has to restate the entire definition of the relation in order to prove meta-totality ("effectivessness"), but Twelf's own wiki doesn't need to reflect my biased opinions. [Boyland](/wiki/user-boyland/) 21:03, 26 February 2007 (EST) + +:: I agree: effectiveness lemmas are just a workaround for limitations of Twelf. When you do a %total for, say, plus, Twelf proves "for all N1, N2, there exists an N3 and D:plus N1 N2 N3". When you do a %total for plus-effective, Twelf proves "for all N1, N2, there exist N3 and D:plus N1 N2 N3 and D':plus-effective N1 N2 N3 D D'". The only reason to do this is to work around the fact that Twelf won't let you name the output of a subgoal. [Drl](/wiki/user-drl/) 12:09, 1 March 2007 (EST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Effectiveness_lemma). +!}% \ No newline at end of file diff --git a/new-tests/pages/talk-incremental-metatheorem-development.lf b/new-tests/pages/talk-incremental-metatheorem-development.lf new file mode 100644 index 0000000..11bef38 --- /dev/null +++ b/new-tests/pages/talk-incremental-metatheorem-development.lf @@ -0,0 +1,15 @@ +%%! title: "Talk:Incremental metatheorem development" +%{! ## Bottom-up vs. Top-down + +I'm not sure if the motivation for using %trustme to maintain a bottom-up development style is completely accurate here. Without %trustme, proofs must be done in a bottom-up way. That's just how math is built up. In practice, I've seen two modes of use of %trustme. The first is to work bottom-up until you hit something that will be pretty time-consuming/hard/impossible to do, and then you %trustme that particular lemma so you can keep working bottom-up. The other, which I've seen used by a particular Twelf master who happens to have an office right next to a stairwell in a corner on the 8th floor of Wean, is to %trustme a lot of boring/uninteresting lemmas which you are pretty sure are true, and then focus on the actual hard proof, and then fill in the boring holes at some point. The second strategy is very much a top-down approach. +--[DanielKLee](/wiki/user-danielklee/) 09:19, 22 September 2006 (MST) + +: I agree that there are multiple models of use, go ahead and put whatever you think is a more accurate motivation over what is there now (in general, I'd advocate a "change and explain why you did it on the Talk pages" as opposed to a "talk about changing on the Talk page" strategy here :) ). — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 09:26, 22 September 2006 (MST) +: I think that we are just using opposite definitions of "top-down" and "bottom-up"; for me, the end of the proof is the bottom. I guess it depends on how you think of it. (We had all sorts of similar terminological issues in automated theorem proving and computer graphics class..) We should just be more clear about what we mean.  — [Tom 7](/wiki/user-tom7/) 11:52, 22 September 2006 (MST) +:: Yeah, I think that's where the confusion is. I think of the main result being at the "top", and being built up from the smaller ones "below" it, like a tree. --[DanielKLee](/wiki/user-danielklee/) 10:30, 23 September 2006 (MST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Incremental_metatheorem_development). +!}% \ No newline at end of file diff --git a/new-tests/pages/talk-metatheorem.lf b/new-tests/pages/talk-metatheorem.lf new file mode 100644 index 0000000..1cbd5e4 --- /dev/null +++ b/new-tests/pages/talk-metatheorem.lf @@ -0,0 +1,28 @@ +%%! title: "Talk:Metatheorem" +%{! ## Placeholder + +I'm going to work on this based on Dan's notes, with the idea that I'll fix it based on whatever total consensus arises with all interested parties. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 17:12, 20 October 2006 (EDT) + +## I agree, Tom + +The point about totality checking being *more* conservative than the fixed search strategy of logic programming is a good one - the need for factoring lemmas being the perfect example here - so I think that change is good. Does the article seem reasonable otherwise, because I'm not certain the approach is even the right one here. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 19:16, 9 March 2007 (EST) +::I think it's a hard subject but this is a reasonable take on it and a good start, yes.  — [Tom 7](/wiki/user-tom7/) 14:01, 12 March 2007 (EDT) + +## Why "meta" ? + +To an outsider, all the "meta"-theorem stuff seems like unnecessary baggage. Normally, I would think a meta-theorem is a theorem about theorems. Here's what I think the answer to my question is: In the olden days, one implemented XYZ logic in Twelf and wrote a theorem that XYZ theorems were valid. But now, we use Twelf to directly encode our theorems about object languages. The "meta" is not really appropriate any more. I suppose for historical reasons people are used to "metatheorem", but it might make more sense just to dispense with "meta" and just talk about theorems. Or if not, then a better explanation of "meta" is needed than "for historical reasons" [Boyland](/wiki/user-boyland/) 01:35, 13 March 2007 (EDT) + +: You're right that this needs to be explained more clearly. The point of the "meta" is to distinguish between +- a _theorem_ in an encoded deductive system, and +- a _metatheorem_ about an encoded deductive system. +: For example, if I encoded a logic with a judgement ``A true``, then ``(all [a:prop] impl a a) true`` might be a theorem in the encoded logic (if I can come up with an LF term of that type), whereas cut elimination for the logic is a metatheorem about it. [Drl](/wiki/user-drl/) 10:14, 13 March 2007 (EDT) +:: To answer another point (we edit-conflicted each other making the same first point), I don't see this as an issue of "historical resasons" at all, this is the way I understand the research community working now - the two-year-old workshop on doing this sort of thing is called the [http://www.cis.upenn.edu/~sweirich/wmm/ Worshop on Mechanizing Metatheory]. People still think of Prolog as a "theorem prover" for Horn clauses (see the [[w:Prolog|Wikipedia page on Prolog]]), and those kinds of theorems are the same kinds of "theorems" (``plus 3 4 7``, ``of (lam [x] E x) (arrow unit unit)``) that are discussed on this page - they are nothing like ∀∃-statement metatheorems. Dan/John/others, where do you think this article could explain this better? — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 10:30, 13 March 2007 (EDT) + +:: Reading the current article, I'm okay with the explanation of "theorem", but the transition to the Totality assertion section should rehash the discussion at the top about "metatheorem" being a statement about an object language and contrast it with theorem (I missed the definition of "metatheorem" at the top on the first read). Also, when I used the word "theorem" this way, Frank objected that he likes to reserve "theorem" for a judgement defining truth in a logic (so not all derivable object-language judgements are "theorems"). Not sure if we want to use the word that way or not. [Drl](/wiki/user-drl/) 12:39, 13 March 2007 (EDT) +::: I'll work on the first point - as for the second point, perhaps I could add in to the example something with a logic - just , and say "we could reserve 'theorem' for this kind of statement, but to avoid saying 'derivable object-language judgment' too many times we often blur the distinction by referring to both as theorems. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 08:41, 14 March 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Metatheorem). +!}% \ No newline at end of file diff --git a/new-tests/pages/talk-mutable-state.lf b/new-tests/pages/talk-mutable-state.lf new file mode 100644 index 0000000..927c4e7 --- /dev/null +++ b/new-tests/pages/talk-mutable-state.lf @@ -0,0 +1,15 @@ +%%! title: "Talk:Mutable state" +%{! Something looks screwy with the two families named "update-lemma", one of which's body gets escaped into normal text because it's block-commented out. Otherwise, cool! [Drl](/wiki/user-drl/) 11:51, 16 March 2007 (EDT) +:: Fixed. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 12:29, 16 March 2007 (EDT) + +## Rename? + +Somehow I think [language with references](/wiki/mutable-state/) is a little weird of a title for this page; can I suggest [mutable state](/wiki/mutable-state/) or encoding references or +encoding store or something like that? I think it would be more in line with our other tutorials/case studies.  — [Tom 7](/wiki/user-tom7/) 08:42, 19 March 2007 (EDT) +: Agreed, I couldn't think of a good name when I wrote the page. I'll use [mutable state](/wiki/mutable-state/). — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 11:21, 19 March 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Mutable_state). +!}% \ No newline at end of file diff --git a/new-tests/pages/talk-natural-numbers-with-inequality.lf b/new-tests/pages/talk-natural-numbers-with-inequality.lf new file mode 100644 index 0000000..5405338 --- /dev/null +++ b/new-tests/pages/talk-natural-numbers-with-inequality.lf @@ -0,0 +1,10 @@ +%%! title: "Talk:Natural numbers with inequality" +%{! It would probably be a good idea to standardize on one name for the uninhabited type in our tutorials. My vote goes to "void" since that is its common name in type theory. I think "false" is not a good choice since we often have terms inhabiting a type called false (in an encoding of classical logic, for instance). "Absurd" seems a little hokey to me, but I don't have a good argument against it...  — [Tom 7](/wiki/user-tom7/) 13:28, 6 September 2006 (MST) + +: I'm all for void, I'll make a note of it in [Twelf style guide](/wiki/twelf-style-guide/), since that's exactly the sort of thing that goes in a twelf style guide. - [TheTwelfElf](/wiki/user-thetwelfelf/) 13:37, 6 September 2006 (MST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Natural_numbers_with_inequality). +!}% \ No newline at end of file diff --git a/new-tests/pages/talk-thaw.lf b/new-tests/pages/talk-thaw.lf new file mode 100644 index 0000000..6d087e5 --- /dev/null +++ b/new-tests/pages/talk-thaw.lf @@ -0,0 +1,4 @@ +%%! title: "Talk:%thaw" +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/pages/template-bibtex-pierce02tapl.lf b/new-tests/pages/template-bibtex-pierce02tapl.lf new file mode 100644 index 0000000..fad59d3 --- /dev/null +++ b/new-tests/pages/template-bibtex-pierce02tapl.lf @@ -0,0 +1,15 @@ +%%! title: "Template:Bibtex:pierce02tapl" +%{! <bibtex>@book\{pierce02tapl, + author = \{Benjamin C. Pierce\}, + title = \{Types and Programming Languages\}, + publisher = \{MIT Press\}, + month = \{February\}, + isbn = \{978-0-262-16209-8\}, + year = \{2002\}, +\}</bibtex> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Bibtex:pierce02tapl). +!}% \ No newline at end of file diff --git a/new-tests/pages/template-discuss.lf b/new-tests/pages/template-discuss.lf new file mode 100644 index 0000000..9028236 --- /dev/null +++ b/new-tests/pages/template-discuss.lf @@ -0,0 +1,17 @@ +%%! title: "Template:Discuss" +%{! [[Talk:\{\{PAGENAME\}\}#\{\{\{1\}\}\}|<sup>[?]</sup>]]<noinclude> + + +This template places a discussion link in an article. The link takes the user to a supplied section on the talk page for the same article. (It does not work for pages outside the article namespace). To use it, write: + +:<nowiki>\{\{discuss|</nowiki>_description of issue_<nowiki>\}\}</nowiki> + +where _description of issue_ is an informative section header for the talk page. You should then create that section on the talk page. The link looks like this.[<sup>[?]</sup>](/wiki/talk-main-page/) + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Discuss). +!}% \ No newline at end of file diff --git a/new-tests/pages/template-lolli.lf b/new-tests/pages/template-lolli.lf new file mode 100644 index 0000000..00c9451 --- /dev/null +++ b/new-tests/pages/template-lolli.lf @@ -0,0 +1,4 @@ +%%! title: "Template:Lolli" +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/pages/template-talk-guide.lf b/new-tests/pages/template-talk-guide.lf new file mode 100644 index 0000000..a883d41 --- /dev/null +++ b/new-tests/pages/template-talk-guide.lf @@ -0,0 +1,10 @@ +%%! title: "Template talk:Guide" +%{! ## Guide gets updated and... + +Won't this template be a problem if the section numbers in the guide change? Ideally this template should be <nowiki>\{\{guide4\}\}</nowiki> or something, I would think... — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 17:35, 27 September 2006 (MST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template_talk:Guide). +!}% \ No newline at end of file diff --git a/new-tests/pages/twelf-with-emacs.lf b/new-tests/pages/twelf-with-emacs.lf new file mode 100644 index 0000000..937e2da --- /dev/null +++ b/new-tests/pages/twelf-with-emacs.lf @@ -0,0 +1,137 @@ +%%! title: "Twelf with Emacs" +%{! The preferred way to interact with Twelf is through its emacs mode, which is included in the Twelf distribution. The [Download](/download/) page has basic information for setting up Emacs mode with Twelf. + +## Prerequisites + +These instructions assume you've already [downloaded Twelf](/download/), and that either +* You used the Windows/OSX installer +* You downloaded a pre-compiled tarball +* You downloaded a source tarball and have already built Twelf + +These instructions should easily translate to using Subversion to get Twelf, however. + +## Installing the Twelf emacs mode + +If you put Twelf in the directory `/somewhere/twelf/` then you should have +these two lines (just replace `somewhere` with wherever you put your Twelf): + +Add the following to your Emacs configuration file, which should be the file .emacs in your home directory. + +``` +(setq twelf-root "/somewhere/twelf/") +(load (concat twelf-root "emacs/twelf-init.el")) +``` + +For instance, if the directory where you unpacked the tarball is ``/usr/local/twelf``, then you'll want those two lines to be: + +``` +(setq twelf-root "/usr/local/twelf/") +(load (concat twelf-root "emacs/twelf-init.el")) +``` + +If your Twelf directory is ``/home/foo/stuff/logic/twelf``, then you'll want those lines to be: + +``` +(setq twelf-root "/home/foo/stuff/logic/twelf/") +(load (concat twelf-root "emacs/twelf-init.el")) +``` + +If you used the default Windows installer, you'll want those to lines to be + +``` +(setq twelf-root "C:\\Program Files\\Twelf\\") +(load (concat twelf-root "emacs\\twelf-init.el")) +``` + +Exit Emacs and restart it. + +## How Twelf programs are divided up into files + +- Twelf programs should be stored in files with the ``.elf`` extension. +- All of the ``.elf`` files in a project are collected in a _configuration file_, typically called ``sources.cfg``. A configuration file lists (paths to) individual ``.elf`` files in dependency order. You can load a configuration file, which loads each ``.elf`` file in sequence. + +### Creating a .cfg file + +Start Emacs in some directory (for the purposes of this description, we'll assume it's ``~/tmp/twelf``. Create the **configuration file** ``sources.cfg`` with the following text in it: + +``` +sometwelf.elf +``` + +Save the file (by using CTRL-x CTRL-s in Emacs). + +### Creating a .elf file + +Then create a file ``sometwelf.elf`` (by using CTRL-x CTRL-f in Emacs) and enter the following text: !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%define ss (%pi nat %-> nat) [n] s (s n) %. +%sort plus %. +%term plus/z plus z N N %. +%term plus/s %pi (plus N M P) %-> (plus (s N) M (s P)) %. +%{! Save the file (by using CTRL-x CTRL-s in Emacs). + +## Using the Twelf emacs mode + +You use Twelf by asking it to process declarations. Twelf responds by either printing ``Server OK`` at the bottom of the screen if your declarations were processed successfully, or by popping up the twelf-server buffer if there was an error (ABORT). + +The emacs mode allows you to process declarations at a variety of granularities: +* A whole configuration file (CTRL-c CTRL-c loads a cfg file) +* A whole .elf file (CTRL-c CTRL-s loads the .elf file open in the current emacs window) +* A single declaration (thing ending in a period) (CTRL-c CTRL-d loads declaration the cursor is on in the in the current .elf file) + +For example, a typical use case is to load your entire project with CTRL-c CTRL-c when you start working, and then to process new Twelf code declaration-by-declaration as you write it, using CTRL-c CTRL-d. Sometimes you will go back and make edits in lots of different places in a file, in which case it's helpful to reload just the current file with CTRL-c CTRL-s. + +Only reloading the configuration clears the state of Twelf. Even reloading will not clear previous declarations from the file (though it will typically shadow them). This might give unexpected results if you remove a declaration and expect it to have disappeared. + +### Loading the configuration file: CTRL-c CTRL-c + +Assuming you still have ``sometwelf.elf`` open in emacs, hit CTRL-c CTRL-c. If ``sometwelf.elf`` has unsaved changes, you will be asked: + +``` +File not in current configuration. Save? (yes or no) +``` + +Type yes. + +You will then be asked: + +``` +Visit config file: (default sources.cfg) ~/tmp/twelf/ +``` + +This is asking for the location of the sources.cfg file you saved when following the instructions above; the default option is the correct one. Just hit ENTER. + +You will then be asked: + +``` +Twelf server: (default twelf-server) /something/bin/ +``` +This is asking which Twelf binary you want to run; the default should already be correct, just hit ENTER. + +This will process your configuration file, which in turn says to process ``sometwelf.elf``. + +### Checking an individual declaration: CTRL-c CTRL-d + +Now you can type new declarations into the emacs buffer and use CTRL-c CTRL-d to load them. + +### Checking an individual file: CTRL-c CTRL-s + +Or you can reload the file in the current window with CTRL-c CTRL-s. You must previously have loaded a configuration file. + +## Syntax highlighting + +Twelf does not automatically refresh its syntax highlighting as you type. You can manually refresh the highlighting as follows: + +- ``CTRL-c l`` fontifies the currently visible emacs buffer. E.g. in ``sometwelf.elf``m the constants (``nat``, ``plus/z``, etc.) should turn red, the bound variable ``n`` should turn green, and the logic-programming variables ``N``, ``M``, and ``P`` should turn blue. +- ``CTRL-c CTRL-l`` fontifies the current declaration only. This is useful if you have really big files. + +For more info on the emacs mode, see \{\{guide|chapter=13|section=74|title=Emacs Interface\}\}. + +## Hacks + +If you use multiple frames with Emacs, ``twelf-mode`` annoyingly pops up the Twelf server buffer in the current frame, even if it is already shown in another frame. To avoid this, set ``display-buffer-reuse-frames`` to ``t``. Now, you will still have the problem that although the buffer is shown, it does not scroll to the bottom; to fix this you need to edit ``twelf.el`` and change ``(get-buffer-window twelf-server-buffer)`` to ``(get-buffer-window twelf-server-buffer t)``. + +## See also +Emacs Interface !}% \ No newline at end of file diff --git a/new-tests/pages/user-jaked.lf b/new-tests/pages/user-jaked.lf new file mode 100644 index 0000000..4f2a285 --- /dev/null +++ b/new-tests/pages/user-jaked.lf @@ -0,0 +1,8 @@ +%%! title: "User:JakeD" +%{! Hi. I am a 2nd year PhD student at CMU. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:JakeD). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/absurdum.lf b/new-tests/stelf-output/examples/alloc-sem/absurdum.lf new file mode 100644 index 0000000..e6ec51b --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/absurdum.lf @@ -0,0 +1,6 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% absurdum.elf +%sort absurd %. +%name absurd %. +%freeze absurd %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/dynsem-lemmas-thm.lf b/new-tests/stelf-output/examples/alloc-sem/dynsem-lemmas-thm.lf new file mode 100644 index 0000000..4982229 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/dynsem-lemmas-thm.lf @@ -0,0 +1,12 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% dynsem-elf.thm +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Dynamic Semantics Lemmas +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%theorem dynsem!step_preserves_wf_thm : forall* {S1 st} {E1 exp} {S2 st} {E2 exp} forall {DSWf1 st_wf S1} {DStep step S1 E1 S2 E2} exists {DSWf2 st_wf S2} true %. +%prove 2 DStep (dynsem!step_preserves_wf_thm _ DStep _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%theorem dynsem!step*_preserves_wf_thm : forall* {S1 st} {E1 exp} {S2 st} {E2 exp} forall {DSWf1 st_wf S1} {DSteps step* S1 E1 S2 E2} exists {DSWf2 st_wf S2} true %. +%prove 2 DSteps (dynsem!step*_preserves_wf_thm _ DSteps _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/dynsem-lemmas.lf b/new-tests/stelf-output/examples/alloc-sem/dynsem-lemmas.lf new file mode 100644 index 0000000..9bb2014 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/dynsem-lemmas.lf @@ -0,0 +1,58 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% dynsem-elf.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Dynamic Semantics Lemmas +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort dynsem!step_preserves_wf %. +%mode dynsem!step_preserves_wf %in %in %out %. +%scope dynsem!step_preserves_wf %term lam_alloc + %pi (dynsem!step_preserves_wf DSWf (lam_alloc_step DStAlloc) DSWf') + %<- (st!alloc_preserves_wf DSWf DStAlloc DSWf') %. +%term app_ctxt1 + %pi (dynsem!step_preserves_wf DSWf (app_ctxt1_step DStep) DSWf') + %<- (dynsem!step_preserves_wf DSWf DStep DSWf') %. +%scope dynsem!step_preserves_wf %term app_ctxt2 + %pi (dynsem!step_preserves_wf DSWf (app_ctxt2_step DStep) DSWf') + %<- (dynsem!step_preserves_wf DSWf DStep DSWf') %. +%term app_beta dynsem!step_preserves_wf DSWf (app_beta_step _) DSWf %. +%scope dynsem!step_preserves_wf %term unit_alloc + %pi (dynsem!step_preserves_wf DSWf (unit_alloc_step DStAlloc) DSWf') + %<- (st!alloc_preserves_wf DSWf DStAlloc DSWf') %. +%term letunit_ctxt + %pi (dynsem!step_preserves_wf DSWf (letunit_ctxt_step DStep) DSWf') + %<- (dynsem!step_preserves_wf DSWf DStep DSWf') %. +%scope dynsem!step_preserves_wf %term letunit_beta dynsem!step_preserves_wf DSWf (letunit_beta_step _) DSWf %. +%term pair_ctxt1 + %pi (dynsem!step_preserves_wf DSWf (pair_ctxt1_step DStep) DSWf') + %<- (dynsem!step_preserves_wf DSWf DStep DSWf') %. +%scope dynsem!step_preserves_wf %term pair_ctxt2 + %pi (dynsem!step_preserves_wf DSWf (pair_ctxt2_step DStep) DSWf') + %<- (dynsem!step_preserves_wf DSWf DStep DSWf') %. +%term pair_alloc + %pi (dynsem!step_preserves_wf DSWf (pair_alloc_step DStAlloc) DSWf') + %<- (st!alloc_preserves_wf DSWf DStAlloc DSWf') %. +%scope dynsem!step_preserves_wf %term letpair_ctxt + %pi (dynsem!step_preserves_wf DSWf (letpair_ctxt_step DStep) DSWf') + %<- (dynsem!step_preserves_wf DSWf DStep DSWf') %. +%term letpair_beta dynsem!step_preserves_wf DSWf (letpair_beta_step _) DSWf %. +%terminates DStep (dynsem!step_preserves_wf _ DStep _) %. +%worlds () (dynsem!step_preserves_wf _ _ _) %. +%covers dynsem!step_preserves_wf %in %in %out %. +%total DStep (dynsem!step_preserves_wf _ DStep _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort dynsem!step*_preserves_wf %. +%mode dynsem!step*_preserves_wf %in %in %out %. +%scope dynsem!step*_preserves_wf %term refl dynsem!step*_preserves_wf DSWf refl_step* DSWf %. +%term trans + %pi (dynsem!step*_preserves_wf DSWf1 (trans_step* DSteps23 DSteps12) DSWf3) + %<- (dynsem!step*_preserves_wf DSWf1 DSteps12 DSWf2) + %<- (dynsem!step*_preserves_wf DSWf2 DSteps23 DSWf3) %. +%scope dynsem!step*_preserves_wf %term step + %pi (dynsem!step*_preserves_wf DSWf (step_step* DStep) DSWf') + %<- (dynsem!step_preserves_wf DSWf DStep DSWf') %. +%terminates DSteps (dynsem!step*_preserves_wf _ DSteps _) %. +%worlds () (dynsem!step*_preserves_wf _ _ _) %. +%covers dynsem!step*_preserves_wf %in %in %out %. +%total DSteps (dynsem!step*_preserves_wf _ DSteps _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/dynsem.lf b/new-tests/stelf-output/examples/alloc-sem/dynsem.lf new file mode 100644 index 0000000..74da45c --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/dynsem.lf @@ -0,0 +1,40 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% dynsem.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Dynamic Semantics +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort step %. +%name step %. +%mode step %in %in %out %out %. +%term lam_alloc_step %pi (step S (lam_e Tx F) S' (loc_e L')) %<- (st_alloc S (lam_v Tx F) S' L') %. +%term app_ctxt1_step %pi (step S (app_e E1 E2) S' (app_e E1' E2)) %<- (step S E1 S' E1') %. +%term app_ctxt2_step + %pi (step S (app_e (loc_e L1) E2) S' (app_e (loc_e L1) E2')) + %<- (step S E2 S' E2') %. +%term app_beta_step + %pi (step S (app_e (loc_e L1) (loc_e L2)) S (F1' (loc_e L2))) + %<- (st_lookup S L1 (lam_v Tx F1')) %. +%term unit_alloc_step %pi (step S unit_e S' (loc_e L')) %<- (st_alloc S unit_v S' L') %. +%term letunit_ctxt_step %pi (step S (letunit_e E1 E2) S' (letunit_e E1' E2)) %<- (step S E1 S' E1') %. +%term letunit_beta_step %pi (step S (letunit_e (loc_e L1) E2) S E2) %<- (st_lookup S L1 unit_v) %. +%term pair_ctxt1_step %pi (step S (pair_e E1 E2) S' (pair_e E1' E2)) %<- (step S E1 S' E1') %. +%term pair_ctxt2_step + %pi (step S (pair_e (loc_e L1) E2) S' (pair_e (loc_e L1) E2')) + %<- (step S E2 S' E2') %. +%term pair_alloc_step + %pi (step S (pair_e (loc_e L1) (loc_e L2)) S' (loc_e L')) + %<- (st_alloc S (pair_v L1 L2) S' L') %. +%term letpair_ctxt_step %pi (step S (letpair_e E1 F2) S' (letpair_e E1' F2)) %<- (step S E1 S' E1') %. +%term letpair_beta_step + %pi (step S (letpair_e (loc_e L1) F2) S (F2 (loc_e L11) (loc_e L12))) + %<- (st_lookup S L1 (pair_v L11 L12)) %. +%terminates E1 (step _ E1 _ _) %. +%worlds () (step _ _ _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort step* %. +%name step* %. +%term refl_step* step* S E S E %. +%term trans_step* %pi (step* S1 E1 S3 E3) %<- (step* S1 E1 S2 E2) %<- (step* S2 E2 S3 E3) %. +%term step_step* %pi (step* S1 E1 S2 E2) %<- (step S1 E1 S2 E2) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/exp.lf b/new-tests/stelf-output/examples/alloc-sem/exp.lf new file mode 100644 index 0000000..db9a39a --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/exp.lf @@ -0,0 +1,19 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% exp.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Expressions +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%term loc_e %pi loc %-> exp %. +%term lam_e %pi tp %-> (%pi exp %-> exp) %-> exp %. +%term app_e %pi exp %-> exp %-> exp %. +%term unit_e exp %. +%term letunit_e %pi exp %-> exp %-> exp %. +%term pair_e %pi exp %-> exp %-> exp %. +%term letpair_e %pi exp %-> (%pi exp %-> exp %-> exp) %-> exp %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort exp_irred %. +%mode exp_irred %in %. +%term exp_irred_loc exp_irred (loc_e _) %. +%terminates {} (exp_irred _) %. +%worlds () (exp_irred _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/inversion-pres.lf b/new-tests/stelf-output/examples/alloc-sem/inversion-pres.lf new file mode 100644 index 0000000..ac7f77d --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/inversion-pres.lf @@ -0,0 +1,46 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% inversion-pres.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store Typing Inversion (Preservation) +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort pres_|-st_inversion_aux %. +%mode pres_|-st_inversion_aux %in %in %in %in %in %out %. +%scope pres_|-st_inversion_aux %term hit_hit + %pi (pres_|-st_inversion_aux DSTWf DSWf (|-st_cons DValTc' _) sttp_lookup_hit st_lookup_hit DValTc) + %<- (sttp!wf_cons_implies_extend_cons DSTWf DSttpExtend) + %<- (sttp_weak_|-val DValTc' DSttpExtend DValTc) %. +%term miss_hit_contra + %pi (pres_|-st_inversion_aux DSTWf DSWf (|-st_cons DValTc' _) (sttp_lookup_miss DSttpLookup' DLocLt) st_lookup_hit DValTc) + %<- (loc!lt_contradict DLocLt Absurd) + %<- (raa_|-val Absurd DValTc) %. +%scope pres_|-st_inversion_aux %term miss_miss + %pi (pres_|-st_inversion_aux DSTWf DSWf (|-st_cons _ DStTc') (sttp_lookup_miss DSttpLookup' _) (st_lookup_miss DStLookup' DLocLt) DValTc) + %<- (sttp!wf_tail DSTWf DSTWf') + %<- (st!wf_tail DSWf DSWf') + %<- (pres_|-st_inversion_aux DSTWf' DSWf' DStTc' DSttpLookup' DStLookup' DValTc') + %<- (sttp!wf_cons_implies_extend_cons DSTWf DSttpExtend') + %<- (sttp_weak_|-val DValTc' DSttpExtend' DValTc) %. +%term miss_miss_contra + %pi (pres_|-st_inversion_aux _ _ (|-st_cons _ _) (sttp_lookup_miss _ DLocLtB) (st_lookup_miss _ DLocLtA) DValTc) + %<- (loc!lt_trans DLocLtA DLocLtB DLocLtC) + %<- (loc!lt_contradict DLocLtC Absurd) + %<- (raa_|-val Absurd DValTc) %. +%scope pres_|-st_inversion_aux %term hit_miss_contra + %pi (pres_|-st_inversion_aux DSTWf DSWf (|-st_cons DValTc' _) sttp_lookup_hit (st_lookup_miss DStLookup' DLocLt) DValTc) + %<- (loc!lt_contradict DLocLt Absurd) + %<- (raa_|-val Absurd DValTc) %. +%terminates DSttpLookup (pres_|-st_inversion_aux _ _ _ DSttpLookup _ _) %. +%worlds () (pres_|-st_inversion_aux _ _ _ _ _ _) %. +%covers pres_|-st_inversion_aux %in %in %in %in %in %out %. +%total DSttpLookup (pres_|-st_inversion_aux _ _ _ DSttpLookup _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort pres_|-st_inversion %. +%mode pres_|-st_inversion %in %in %in %out %. +%term _ + %pi (pres_|-st_inversion (|-stwf_ DSTWf DSWf DStTc) DSttpLookup DStLookup DValTc) + %<- (pres_|-st_inversion_aux DSTWf DSWf DStTc DSttpLookup DStLookup DValTc) %. +%terminates {} (pres_|-st_inversion _ _ _ _) %. +%worlds () (pres_|-st_inversion _ _ _ _) %. +%covers pres_|-st_inversion %in %in %in %out %. +%total {} (pres_|-st_inversion _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/inversion-prog.lf b/new-tests/stelf-output/examples/alloc-sem/inversion-prog.lf new file mode 100644 index 0000000..26349eb --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/inversion-prog.lf @@ -0,0 +1,33 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% inversion-prog.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store Typing Inversion (Progress) +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort prog_|-st_inversion_aux %. +%mode prog_|-st_inversion_aux %in %in %in %in %out %out %. +%scope prog_|-st_inversion_aux %term hit + %pi (prog_|-st_inversion_aux DSTWf DSWf (|-st_cons DValTc' _) sttp_lookup_hit st_lookup_hit DValTc) + %<- (sttp!wf_cons_implies_extend_cons DSTWf DSttpExtend) + %<- (sttp_weak_|-val DValTc' DSttpExtend DValTc) %. +%term miss + %pi (prog_|-st_inversion_aux DSTWf DSWf (|-st_cons _ DStTc') (sttp_lookup_miss DSttpLookup' DLocLt) (st_lookup_miss DStLookup' DLocLt) DValTc) + %<- (sttp!wf_tail DSTWf DSTWf') + %<- (st!wf_tail DSWf DSWf') + %<- (prog_|-st_inversion_aux DSTWf' DSWf' DStTc' DSttpLookup' DStLookup' DValTc') + %<- (sttp!wf_cons_implies_extend_cons DSTWf DSttpExtend') + %<- (sttp_weak_|-val DValTc' DSttpExtend' DValTc) %. +%terminates DSttpLookup (prog_|-st_inversion_aux _ _ _ DSttpLookup _ _) %. +%worlds () (prog_|-st_inversion_aux _ _ _ _ _ _) %. +%covers prog_|-st_inversion_aux %in %in %in %out %in %out %. +%total DSttpLookup (prog_|-st_inversion_aux _ _ _ DSttpLookup _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort prog_|-st_inversion %. +%mode prog_|-st_inversion %in %in %out %out %. +%term _ + %pi (prog_|-st_inversion (|-stwf_ DSTWf DSWf DStTc) DSttpLookup DStLookup DValTc) + %<- (prog_|-st_inversion_aux DSTWf DSWf DStTc DSttpLookup DStLookup DValTc) %. +%terminates {} (prog_|-st_inversion _ _ _ _) %. +%worlds () (prog_|-st_inversion _ _ _ _) %. +%covers prog_|-st_inversion %in %in %out %out %. +%total {} (prog_|-st_inversion _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/loc-lemmas-thm.lf b/new-tests/stelf-output/examples/alloc-sem/loc-lemmas-thm.lf new file mode 100644 index 0000000..668de91 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/loc-lemmas-thm.lf @@ -0,0 +1,10 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% loc-lemmas.thm +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Loc Less-than Lemmas +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%theorem loc!lt_trans_thm : forall* {L1 loc} {L2 loc} {L3 loc} forall {DLocLt12 loc_lt L1 L2} {DLocLt23 loc_lt L2 L3} exists {DLocLt13 loc_lt L1 L3} true %. +%prove 3 {} (loc!lt_trans_thm _ _ _) %. +%theorem loc!lt_contradict_thm : forall* {L loc} forall {DLocLt loc_lt L L} exists {Absurd absurd} true %. +%prove 2 {} (loc!lt_contradict_thm _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/loc-lemmas.lf b/new-tests/stelf-output/examples/alloc-sem/loc-lemmas.lf new file mode 100644 index 0000000..a1eb302 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/loc-lemmas.lf @@ -0,0 +1,20 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% loc-lemmas.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Loc Less-than Lemmas +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort loc!lt_trans %. +%mode loc!lt_trans %in %in %out %. +%term _ + %pi (loc!lt_trans (loc_lt_ DNatLt12) (loc_lt_ DNatLt23) (loc_lt_ DNatLt13)) + %<- (nat!lt_trans DNatLt12 DNatLt23 DNatLt13) %. +%worlds () (loc!lt_trans _ _ _) %. +%total DLocLt12 (loc!lt_trans DLocLt12 _ _) %. +%sort loc!lt_contradict %. +%mode loc!lt_contradict %in %out %. +%term _ + %pi (loc!lt_contradict (loc_lt_ DNatLt) Absurd) + %<- (nat!lt_contradict DNatLt Absurd) %. +%worlds () (loc!lt_contradict _ _) %. +%total DLocLt (loc!lt_contradict DLocLt _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/loc.lf b/new-tests/stelf-output/examples/alloc-sem/loc.lf new file mode 100644 index 0000000..d73c274 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/loc.lf @@ -0,0 +1,15 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% loc.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Locations +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%term mkLoc %pi nat %-> loc %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Loc Less-than +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort loc_lt %. +%mode loc_lt %in %in %. +%term loc_lt_ %pi (loc_lt (mkLoc N1) (mkLoc N2)) %<- (nat_lt N1 N2) %. +%terminates {} (loc_lt _ _) %. +%worlds () (loc_lt _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/machine.lf b/new-tests/stelf-output/examples/alloc-sem/machine.lf new file mode 100644 index 0000000..cbfc712 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/machine.lf @@ -0,0 +1,19 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% machine.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Machines +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%term ; %pi st %-> exp %-> mach %. +%prec %right 20 ; %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Machine Ok +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort mach_ok %. +%name mach_ok %. +%mode mach_ok %in %in %. +%term mach_ok_irred %pi (exp_irred E) %-> (mach_ok S E) %. +%term mach_ok_step %pi (step S E S' E') %-> (mach_ok S E) %. +%terminates {} (mach_ok _ _) %. +%worlds () (mach_ok _ _) %. +%covers mach_ok %in %in %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/nat-lemmas-thm.lf b/new-tests/stelf-output/examples/alloc-sem/nat-lemmas-thm.lf new file mode 100644 index 0000000..5f1b495 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/nat-lemmas-thm.lf @@ -0,0 +1,12 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% nat-lemmas.thm +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Nat Less-than Lemmas +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%theorem nat!N_lt_sN_thm : forall {N nat} exists {DNatLt nat_lt N (s_nat N)} true %. +%prove 2 N (nat!N_lt_sN_thm N _) %. +%theorem nat!lt_trans_thm : forall* {N1 nat} {N2 nat} {N3 nat} forall {DNatLt12 nat_lt N1 N2} {DNatLt23 nat_lt N2 N3} exists {DNatLt13 nat_lt N1 N3} true %. +%prove 2 DNatLt12 (nat!lt_trans_thm DNatLt12 _ _) %. +%theorem nat!lt_contradict_thm : forall* {N nat} forall {DNatLt nat_lt N N} exists {Absurd absurd} true %. +%prove 2 DNatLt (nat!lt_contradict_thm DNatLt _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/nat-lemmas.lf b/new-tests/stelf-output/examples/alloc-sem/nat-lemmas.lf new file mode 100644 index 0000000..b0528b7 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/nat-lemmas.lf @@ -0,0 +1,29 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% nat-lemmas.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Nat Less-than Lemmas +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort nat!N_lt_sN {N nat} %. +%mode nat!N_lt_sN %in %out %. +%scope nat!N_lt_sN %term z nat!N_lt_sN z_nat nat_lt_z %. +%term s %pi (nat!N_lt_sN (s_nat N) (nat_lt_s DNatLt)) %<- (nat!N_lt_sN N DNatLt) %. +%terminates N (nat!N_lt_sN N _) %. +%worlds () (nat!N_lt_sN _ _) %. +%covers nat!N_lt_sN %in %out %. +%total N (nat!N_lt_sN N _) %. +%sort nat!lt_trans %. +%mode nat!lt_trans %in %in %out %. +%scope nat!lt_trans %term z nat!lt_trans nat_lt_z _ nat_lt_z %. +%term s + %pi (nat!lt_trans (nat_lt_s DNatLt12) (nat_lt_s DNatLt23) (nat_lt_s DNatLt13)) + %<- (nat!lt_trans DNatLt12 DNatLt23 DNatLt13) %. +%worlds () (nat!lt_trans _ _ _) %. +%total DNatLt12 (nat!lt_trans DNatLt12 _ _) %. +%sort nat!lt_contradict %. +%mode nat!lt_contradict %in %out %. +%scope nat!lt_contradict %term s + %pi (nat!lt_contradict (nat_lt_s DNatLt) Absurd) + %<- (nat!lt_contradict DNatLt Absurd) %. +%worlds () (nat!lt_contradict _ _) %. +%total DNatLt (nat!lt_contradict DNatLt _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/nat.lf b/new-tests/stelf-output/examples/alloc-sem/nat.lf new file mode 100644 index 0000000..ffb88c0 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/nat.lf @@ -0,0 +1,19 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% nat.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Naturals +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort nat %. +%name nat %. +%term z_nat nat %. +%term s_nat %pi nat %-> nat %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Nat Less-than +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort nat_lt %. +%mode nat_lt %in %in %. +%term nat_lt_z nat_lt z_nat (s_nat _) %. +%term nat_lt_s %pi (nat_lt (s_nat N1) (s_nat N2)) %<- (nat_lt N1 N2) %. +%terminates N1 (nat_lt N1 _) %. +%worlds () (nat_lt _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/preservation-lemmas.lf b/new-tests/stelf-output/examples/alloc-sem/preservation-lemmas.lf new file mode 100644 index 0000000..d3547b9 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/preservation-lemmas.lf @@ -0,0 +1,62 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% preservation-lemmas.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Preservation Lemmas +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Allocation Preserves +%sort pres_alloc %. +%mode pres_alloc %in %in %in %out %out %out %. +%scope pres_alloc %term nil + %pi (pres_alloc (|-stwf_ DSTWf DSWf |-st_nil) (%the (|-val _ _ T) DValTc) st_alloc_nil DSttpExtend (|-stwf_ DSTWf' DSWf' (|-st_cons DValTc |-st_nil)) (|-exp_loc (|-loc_ sttp_lookup_hit))) + %<- (sttp!alloc_implies_extend sttp_alloc_nil DSttpExtend) + %<- (sttp!alloc_preserves_wf DSTWf sttp_alloc_nil DSTWf') + %<- (st!alloc_preserves_wf DSWf st_alloc_nil DSWf') %. +%term cons + %pi (pres_alloc (|-stwf_ DSTWf DSWf (|-st_cons DValTcX DStTc)) (%the (|-val _ _ T) DValTc) st_alloc_cons DSttpExtend (|-stwf_ DSTWf' DSWf' (|-st_cons DValTc (|-st_cons DValTcX DStTc))) (|-exp_loc (|-loc_ sttp_lookup_hit))) + %<- (sttp!alloc_implies_extend sttp_alloc_cons DSttpExtend) + %<- (st!alloc_preserves_wf DSWf st_alloc_cons DSWf') + %<- (sttp!alloc_preserves_wf DSTWf sttp_alloc_cons DSTWf') %. +%terminates DStTcWf (pres_alloc DStTcWf _ _ _ _ _) %. +%worlds () (pres_alloc _ _ _ _ _ _) %. +%covers pres_alloc %in %in %in %out %out %out %. +%total DStTcWf (pres_alloc DStTcWf _ _ _ _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Substitution Preserves +%sort pres_var {z} %. +%mode pres_var %in %in %out %. +%scope pres_var %term refl pres_var ([z] [DVarTcZ] T) _ T %. +%term var pres_var ([z] [DVarTcZ] |-exp_var DVarTcZ) DExpTcZ DExpTcZ %. +%scope pres_var %term loc pres_var ([z] [DVarTcZ] |-exp_loc DLocTc) _ (|-exp_loc DLocTc) %. +%term lam + %pi (pres_var ([z] [DVarTcZ] |-exp_lam (DExpTcF z DVarTcZ)) DExpTcZ (|-exp_lam DExpTcF')) + %<- ({x} {DVarTcX |-var x Tx} pres_var ([z] [DVarTcZ] DExpTcF z DVarTcZ x DVarTcX) DExpTcZ (DExpTcF' x DVarTcX)) %. +%scope pres_var %term app + %pi (pres_var ([z] [DVarTcZ] |-exp_app (DExpTcE2 z DVarTcZ) (DExpTcE1 z DVarTcZ)) DExpTcZ (|-exp_app DExpTcE2' DExpTcE1')) + %<- (pres_var DExpTcE1 DExpTcZ DExpTcE1') + %<- (pres_var DExpTcE2 DExpTcZ DExpTcE2') %. +%term unit pres_var ([z] [DVarTcZ] |-exp_unit) _ |-exp_unit %. +%scope pres_var %term letunit + %pi (pres_var ([z] [DVarTcZ] |-exp_letunit (DExpTcE2 z DVarTcZ) (DExpTcE1 z DVarTcZ)) DExpTcZ (|-exp_letunit DExpTcE2' DExpTcE1')) + %<- (pres_var DExpTcE1 DExpTcZ DExpTcE1') + %<- (pres_var DExpTcE2 DExpTcZ DExpTcE2') %. +%term pair + %pi (pres_var ([z] [DVarTcZ] |-exp_pair (DExpTcE2 z DVarTcZ) (DExpTcE1 z DVarTcZ)) DExpTcZ (|-exp_pair DExpTcE2' DExpTcE1')) + %<- (pres_var DExpTcE1 DExpTcZ DExpTcE1') + %<- (pres_var DExpTcE2 DExpTcZ DExpTcE2') %. +%scope pres_var %term letpair + %pi (pres_var ([z] [DVarTcZ] |-exp_letpair (DExpTcF2 z DVarTcZ) (DExpTcE1 z DVarTcZ)) DExpTcZ (|-exp_letpair DExpTcF2' DExpTcE1')) + %<- (pres_var DExpTcE1 DExpTcZ DExpTcE1') + %<- ({x} {DVarTcX |-var x Tx} {y} {DVarTcY |-var y Ty} pres_var ([z] [DVarTcZ] DExpTcF2 z DVarTcZ x DVarTcX y DVarTcY) DExpTcZ (DExpTcF2' x DVarTcX y DVarTcY)) %. +%terminates DExpTcF (pres_var DExpTcF _ _) %. +%worlds (|-var_block) (pres_var _ _ _) %. +%covers pres_var %in %in %out %. +%total DExpTcF (pres_var DExpTcF _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort pres_|-val_lam_inversion %. +%mode pres_|-val_lam_inversion %in %out %. +%term _ pres_|-val_lam_inversion (|-val_lam DExpTcF) DExpTcF %. +%terminates {} (pres_|-val_lam_inversion _ _) %. +%worlds () (pres_|-val_lam_inversion _ _) %. +%covers pres_|-val_lam_inversion %in %out %. +%total {} (pres_|-val_lam_inversion _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/preservation.lf b/new-tests/stelf-output/examples/alloc-sem/preservation.lf new file mode 100644 index 0000000..b66f4f8 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/preservation.lf @@ -0,0 +1,95 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% preservation.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Preservation: If M1 ==> M2 and |- M1, then |- M2. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort pres %. +%mode pres %in %in %in %out %out %out %. +%scope pres %term lam_alloc + %pi (pres (lam_alloc_step DStAlloc) DStTcWf (|-exp_lam DExpTcF) DSttpExtend DStTcWf' DExpTcL) + %<- (pres_alloc DStTcWf (|-val_lam DExpTcF) DStAlloc DSttpExtend DStTcWf' DExpTcL) %. +%term app_ctxt1 + %pi (pres (app_ctxt1_step DStep) DStTcWf (|-exp_app DExpTcE2 DExpTcE1) DSttpExtend DStTcWf' (|-exp_app DExpTcE2' DExpTcE1')) + %<- (pres DStep DStTcWf DExpTcE1 DSttpExtend DStTcWf' DExpTcE1') + %<- (sttp_weak_|-exp DExpTcE2 DSttpExtend DExpTcE2') %. +%scope pres %term app_ctxt2 + %pi (pres (app_ctxt2_step DStep) DStTcWf (|-exp_app DExpTcE2 DExpTcE1) DSttpExtend DStTcWf' (|-exp_app DExpTcE2' DExpTcE1')) + %<- (pres DStep DStTcWf DExpTcE2 DSttpExtend DStTcWf' DExpTcE2') + %<- (sttp_weak_|-exp DExpTcE1 DSttpExtend DExpTcE1') %. +%term app_beta + %pi (pres (app_beta_step DStLookupL1) DStTcWf (|-exp_app DExpTcL2 (|-exp_loc (|-loc_ DSttpLookupL1))) DSttpExtend DStTcWf DExpTcF1'L2) + %<- (sttp!extend_refl _ DSttpExtend) + %<- (pres_|-st_inversion DStTcWf DSttpLookupL1 DStLookupL1 DExpTcLam) + %<- (pres_|-val_lam_inversion DExpTcLam DExpTcF1'*) + %<- (pres_var DExpTcF1'* DExpTcL2 DExpTcF1'L2) %. +%% Church-style +%% Curry-style +% -app_beta : pres (app_beta_step DStLookupL1) +% DStTcWf +% (|-exp_app DExpTcL2 +% (|-exp_loc (|-loc_ DSttpLookupL1))) +% DSttpExtend +% DStTcWf +% DExpTcF1'L2 +% <- sttp!extend_refl _ DSttpExtend +% <- pres_|-st_inversion DStTcWf +% DSttpLookupL1 +% DStLookupL1 +% (|-val_lam DExpTcF1'*) +% <- pres_var DExpTcF1'* DExpTcL2 DExpTcF1'L2. +%scope pres %term unit_alloc + %pi (pres (unit_alloc_step DStAlloc) DStTcWf |-exp_unit DSttpExtend DStTcWf' DExpTcL) + %<- (pres_alloc DStTcWf |-val_unit DStAlloc DSttpExtend DStTcWf' DExpTcL) %. +%term letunit_ctxt + %pi (pres (letunit_ctxt_step DStep) DStTcWf (|-exp_letunit DExpTcE2 DExpTcE1) DSttpExtend DStTcWf' (|-exp_letunit DExpTcE2' DExpTcE1')) + %<- (pres DStep DStTcWf DExpTcE1 DSttpExtend DStTcWf' DExpTcE1') + %<- (sttp_weak_|-exp DExpTcE2 DSttpExtend DExpTcE2') %. +%scope pres %term letunit_beta + %pi (pres (letunit_beta_step _) DStTcWf (|-exp_letunit DExpTcE2 DExpTcE1) DSttpExtend DStTcWf DExpTcE2) + %<- (sttp!extend_refl _ DSttpExtend) %. +%term pair_ctxt1 + %pi (pres (pair_ctxt1_step DStep) DStTcWf (|-exp_pair DExpTcE2 DExpTcE1) DSttpExtend DStTcWf' (|-exp_pair DExpTcE2' DExpTcE1')) + %<- (pres DStep DStTcWf DExpTcE1 DSttpExtend DStTcWf' DExpTcE1') + %<- (sttp_weak_|-exp DExpTcE2 DSttpExtend DExpTcE2') %. +%scope pres %term pair_ctxt2 + %pi (pres (pair_ctxt2_step DStep) DStTcWf (|-exp_pair DExpTcE2 DExpTcE1) DSttpExtend DStTcWf' (|-exp_pair DExpTcE2' DExpTcE1')) + %<- (pres DStep DStTcWf DExpTcE2 DSttpExtend DStTcWf' DExpTcE2') + %<- (sttp_weak_|-exp DExpTcE1 DSttpExtend DExpTcE1') %. +%term pair_alloc + %pi (pres (pair_alloc_step DStAlloc) DStTcWf (|-exp_pair (|-exp_loc DLocTcL2) (|-exp_loc DLocTcL1)) DSttpExtend DStTcWf' DExpTcL) + %<- (pres_alloc DStTcWf (|-val_pair DLocTcL2 DLocTcL1) DStAlloc DSttpExtend DStTcWf' DExpTcL) %. +%scope pres %term letpair_ctxt + %pi (pres (letpair_ctxt_step DStep) DStTcWf (|-exp_letpair DExpTcF2 DExpTcE1) DSttpExtend DStTcWf' (|-exp_letpair DExpTcF2' DExpTcE1')) + %<- (pres DStep DStTcWf DExpTcE1 DSttpExtend DStTcWf' DExpTcE1') + %<- ({x} {DVarTcX |-var x Tx} {y} {DVarTcY |-var y Ty} sttp_weak_|-exp (DExpTcF2 x DVarTcX y DVarTcY) DSttpExtend (DExpTcF2' x DVarTcX y DVarTcY)) %. +%term letpair_beta + %pi (pres (letpair_beta_step DStLookup) DStTcWf (|-exp_letpair DExpTcF2** (|-exp_loc (|-loc_ DSttpLookup))) DSttpExtend DStTcWf DExpTcF2LxLy) + %<- (sttp!extend_refl _ DSttpExtend) + %<- (pres_|-st_inversion DStTcWf DSttpLookup DStLookup (|-val_pair DLocTcLy DLocTcLx)) + %<- ({x} {DVarTcX |-var x Tx} pres_var (DExpTcF2** x DVarTcX) (|-exp_loc DLocTcLy) (DExpTcF2*Ly x DVarTcX)) + %<- (pres_var DExpTcF2*Ly (|-exp_loc DLocTcLx) DExpTcF2LxLy) %. +%terminates DStep (pres DStep _ _ _ _ _) %. +%worlds () (pres _ _ _ _ _ _) %. +%covers pres %in %in %in %out %out %out %. +%total DStep (pres DStep _ _ _ _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Preservation: If M1 ==>* M2 and |- M1, then |- M2. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort pres* %. +%mode pres* %in %in %in %out %out %out %. +%scope pres* %term refl + %pi (pres* refl_step* DStTcWf DExpTc DSttpExtend DStTcWf DExpTc) + %<- (sttp!extend_refl _ DSttpExtend) %. +%term trans + %pi (pres* (trans_step* DSteps23 DSteps12) DStTcWf1 DExpTc1 DSttpExtend13 DStTcWf3 DExpTc3) + %<- (pres* DSteps12 DStTcWf1 DExpTc1 DSttpExtend12 DStTcWf2 DExpTc2) + %<- (pres* DSteps23 DStTcWf2 DExpTc2 DSttpExtend23 DStTcWf3 DExpTc3) + %<- (sttp!extend_trans DSttpExtend12 DSttpExtend23 DSttpExtend13) %. +%scope pres* %term step + %pi (pres* (step_step* DStep) DStTcWf DExpTc DSttpExtend' DStTcWf' DExpTc') + %<- (pres DStep DStTcWf DExpTc DSttpExtend' DStTcWf' DExpTc') %. +%terminates DSteps (pres* DSteps _ _ _ _ _) %. +%worlds () (pres* _ _ _ _ _ _) %. +%covers pres* %in %in %in %out %out %out %. +%total DSteps (pres* DSteps _ _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/progress-lemmas.lf b/new-tests/stelf-output/examples/alloc-sem/progress-lemmas.lf new file mode 100644 index 0000000..4268011 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/progress-lemmas.lf @@ -0,0 +1,65 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% progress-lemmas.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Progress Lemmas +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Allocation Progresses +%sort prog_st_alloc {V val} %. +%mode prog_st_alloc %in %in %out %. +%scope prog_st_alloc %term nil prog_st_alloc _ (|-stwf_ _ _ |-st_nil) st_alloc_nil %. +%term cons prog_st_alloc _ (|-stwf_ _ _ (|-st_cons _ _)) st_alloc_cons %. +%terminates {} (prog_st_alloc _ _ _) %. +%worlds () (prog_st_alloc _ _ _) %. +%covers prog_st_alloc %in %in %out %. +%total {} (prog_st_alloc _ _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% app ctxt +%sort prog_app_aux {E1 exp} {E2 exp} %. +%mode prog_app_aux %in %in %in %in %in %in %out %. +%scope prog_app_aux %term ctxt1 prog_app_aux _ _ _ _ (mach_ok_step DStep) _ (mach_ok_step (app_ctxt1_step DStep)) %. +%term ctxt2 prog_app_aux _ _ _ _ (mach_ok_irred _) (mach_ok_step DStep) (mach_ok_step (app_ctxt2_step DStep)) %. +%scope prog_app_aux %term beta + %pi (prog_app_aux _ _ DStTcWf (|-exp_loc (|-loc_ DSttpLookup)) (mach_ok_irred _) (mach_ok_irred _) (mach_ok_step (app_beta_step DStLookup))) + %<- (prog_|-st_inversion DStTcWf DSttpLookup DStLookup DValTc) %. +%terminates {} (prog_app_aux _ _ _ _ _ _ _) %. +%worlds () (prog_app_aux _ _ _ _ _ _ _) %. +%covers prog_app_aux %in %in %in %in %in %in %out %. +%total {} (prog_app_aux _ _ _ _ _ _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% letunit ctxt +%sort prog_letunit_aux {E1 exp} {E2 exp} %. +%mode prog_letunit_aux %in %in %in %in %in %out %. +%scope prog_letunit_aux %term ctxt prog_letunit_aux _ _ _ _ (mach_ok_step DStep) (mach_ok_step (letunit_ctxt_step DStep)) %. +%term beta + %pi (prog_letunit_aux _ _ DStTcWf (|-exp_loc (|-loc_ DSttpLookup)) (mach_ok_irred _) (mach_ok_step (letunit_beta_step DStLookup))) + %<- (prog_|-st_inversion DStTcWf DSttpLookup DStLookup DValTc) %. +%terminates {} (prog_letunit_aux _ _ _ _ _ _) %. +%worlds () (prog_letunit_aux _ _ _ _ _ _) %. +%covers prog_letunit_aux %in %in %in %in %in %out %. +%total {} (prog_letunit_aux _ _ _ _ _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% pair ctxt +%sort prog_pair_aux {E1 exp} {E2 exp} %. +%mode prog_pair_aux %in %in %in %in %in %out %. +%scope prog_pair_aux %term ctxt1 prog_pair_aux _ _ _ (mach_ok_step DStep) _ (mach_ok_step (pair_ctxt1_step DStep)) %. +%term ctxt2 prog_pair_aux _ _ _ (mach_ok_irred _) (mach_ok_step DStep) (mach_ok_step (pair_ctxt2_step DStep)) %. +%scope prog_pair_aux %term alloc + %pi (prog_pair_aux _ _ DStTcWf (mach_ok_irred _) (mach_ok_irred _) (mach_ok_step (pair_alloc_step DStAlloc))) + %<- (prog_st_alloc _ DStTcWf DStAlloc) %. +%terminates {} (prog_pair_aux _ _ _ _ _ _) %. +%worlds () (prog_pair_aux _ _ _ _ _ _) %. +%covers prog_pair_aux %in %in %in %in %in %out %. +%total {} (prog_pair_aux _ _ _ _ _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% letpair ctxt +%sort prog_letpair_aux {E1 exp} {F2 %pi exp %-> exp %-> exp} %. +%mode prog_letpair_aux %in %in %in %in %in %out %. +%scope prog_letpair_aux %term ctxt prog_letpair_aux _ _ _ _ (mach_ok_step DStep) (mach_ok_step (letpair_ctxt_step DStep)) %. +%term beta + %pi (prog_letpair_aux _ _ DStTcWf (|-exp_loc (|-loc_ DSttpLookup)) (mach_ok_irred _) (mach_ok_step (letpair_beta_step DStLookup))) + %<- (prog_|-st_inversion DStTcWf DSttpLookup DStLookup DValTc) %. +%terminates {} (prog_letpair_aux _ _ _ _ _ _) %. +%worlds () (prog_letpair_aux _ _ _ _ _ _) %. +%covers prog_letpair_aux %in %in %in %in %in %out %. +%total {} (prog_letpair_aux _ _ _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/progress.lf b/new-tests/stelf-output/examples/alloc-sem/progress.lf new file mode 100644 index 0000000..aa41c81 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/progress.lf @@ -0,0 +1,37 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% progress.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Progress: If |- M, then M ok (i.e., M irred or M ==> M'). +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort prog %. +%mode prog %in %in %out %. +%scope prog %term loc prog DStTcWf (|-exp_loc _) (mach_ok_irred exp_irred_loc) %. +%term lam + %pi (prog DStTcWf (|-exp_lam _) (mach_ok_step (lam_alloc_step DStAlloc))) + %<- (prog_st_alloc _ DStTcWf DStAlloc) %. +%scope prog %term app + %pi (prog DStTcWf (|-exp_app DExpTcE2 DExpTcE1) DMachOk) + %<- (prog DStTcWf DExpTcE1 DMachOk1) + %<- (prog DStTcWf DExpTcE2 DMachOk2) + %<- (prog_app_aux _ _ DStTcWf DExpTcE1 DMachOk1 DMachOk2 DMachOk) %. +%term unit + %pi (prog DStTcWf |-exp_unit (mach_ok_step (unit_alloc_step DStAlloc))) + %<- (prog_st_alloc _ DStTcWf DStAlloc) %. +%scope prog %term letunit + %pi (prog DStTcWf (|-exp_letunit DExpTcE2 DExpTcE1) DMachOk) + %<- (prog DStTcWf DExpTcE1 DMachOk1) + %<- (prog_letunit_aux _ _ DStTcWf DExpTcE1 DMachOk1 DMachOk) %. +%term pair + %pi (prog DStTcWf (|-exp_pair DExpTcE2 DExpTcE1) DMachOk) + %<- (prog DStTcWf DExpTcE1 DMachOk1) + %<- (prog DStTcWf DExpTcE2 DMachOk2) + %<- (prog_pair_aux _ _ DStTcWf DMachOk1 DMachOk2 DMachOk) %. +%scope prog %term letpair + %pi (prog DStTcWf (|-exp_letpair DExpTcF2 DExpTcE1) DMachOk) + %<- (prog DStTcWf DExpTcE1 DMachOk1) + %<- (prog_letpair_aux _ _ DStTcWf DExpTcE1 DMachOk1 DMachOk) %. +%terminates DExpTc (prog _ DExpTc _) %. +%worlds () (prog _ _ _) %. +%covers prog %in %in %out %. +%total DExpTc (prog _ DExpTc _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/safety.lf b/new-tests/stelf-output/examples/alloc-sem/safety.lf new file mode 100644 index 0000000..1bb3d8a --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/safety.lf @@ -0,0 +1,21 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% safety.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Safety: If |- M1 and M1 ==>* M2, then M2 ok. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort safety %. +%mode safety %in %in %in %out %. +%scope safety %term refl %pi (safety DStTcWf DExpTc refl_step* DMachOk) %<- (prog DStTcWf DExpTc DMachOk) %. +%term trans + %pi (safety DStTcWf DExpTc (trans_step* DSteps23 DSteps12) DMachOk3) + %<- (pres* DSteps12 DStTcWf DExpTc _ DStTcWf2 DExpTc2) + %<- (safety DStTcWf2 DExpTc2 DSteps23 DMachOk3) %. +%scope safety %term step + %pi (safety DStTcWf DExpTc (step_step* DStep) DMachOk') + %<- (pres DStep DStTcWf DExpTc DSttpExtend DStTcWf' DExpTc') + %<- (prog DStTcWf' DExpTc' DMachOk') %. +%terminates DSteps (safety _ _ DSteps _) %. +%worlds () (safety _ _ _ _) %. +%covers safety %in %in %in %out %. +%total DSteps (safety _ _ DSteps _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/sources-thm.lf b/new-tests/stelf-output/examples/alloc-sem/sources-thm.lf new file mode 100644 index 0000000..cca0c87 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/sources-thm.lf @@ -0,0 +1,18 @@ +%require %[ absurdum %] %. +%require %[ nat %] %. +%require %[ nat-lemmas %] %. +%require %[ syntax %] %. +%require %[ loc %] %. +%require %[ loc-lemmas %] %. +%require %[ exp %] %. +%require %[ val %] %. +%require %[ store %] %. +%require %[ store-lemmas %] %. +%require %[ dynsem %] %. +%require %[ dynsem-lemmas %] %. +%require %[ machine %] %. +%require %[ tp %] %. +%require %[ storetp %] %. +%require %[ storetp-lemmas %] %. +%require %[ statsem %] %. +%require %[ weaken %] %. diff --git a/new-tests/stelf-output/examples/alloc-sem/sources.lf b/new-tests/stelf-output/examples/alloc-sem/sources.lf new file mode 100644 index 0000000..66c30ed --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/sources.lf @@ -0,0 +1,25 @@ +%require %[ absurdum %] %. +%require %[ nat %] %. +%require %[ nat-lemmas %] %. +%require %[ syntax %] %. +%require %[ loc %] %. +%require %[ loc-lemmas %] %. +%require %[ exp %] %. +%require %[ val %] %. +%require %[ store %] %. +%require %[ store-lemmas %] %. +%require %[ dynsem %] %. +%require %[ dynsem-lemmas %] %. +%require %[ machine %] %. +%require %[ tp %] %. +%require %[ storetp %] %. +%require %[ storetp-lemmas %] %. +%require %[ statsem %] %. +%require %[ weaken %] %. +%require %[ inversion-pres %] %. +%require %[ preservation-lemmas %] %. +%require %[ preservation %] %. +%require %[ inversion-prog %] %. +%require %[ progress-lemmas %] %. +%require %[ progress %] %. +%require %[ safety %] %. diff --git a/new-tests/stelf-output/examples/alloc-sem/statsem.lf b/new-tests/stelf-output/examples/alloc-sem/statsem.lf new file mode 100644 index 0000000..931a30a --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/statsem.lf @@ -0,0 +1,105 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% statsem.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Static Semantics +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% |-loc +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort |-loc %. +%name |-loc %. +%mode |-loc %in %in %out %. +%term |-loc_ %pi (|-loc ST L T) %<- (sttp_lookup ST L T) %. +%terminates {} (|-loc _ _ _) %. +%worlds () (|-loc _ _ _) %. +%covers |-loc %in %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% |-var +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort |-var %. +%name |-var %. +%mode |-var %in %out %. +% Church-style +%% %mode |-var +E *T. % Curry-style +%block |-var_block {T tp} [x exp] [_ |-var x T]%. +%terminates {} (|-var _ _) %. +%worlds (|-var_block) (|-var _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% |-exp +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort |-exp %. +%name |-exp %. +%mode |-exp %in %in %out %. +% Church-style +%% %mode |-exp +ST +E *T. % Curry-style +%term |-exp_var %pi (|-exp ST E T) %<- (|-var E T) %. +%term |-exp_loc %pi (|-exp ST (loc_e L) T) %<- (|-loc ST L T) %. +%term |-exp_lam + %pi (|-exp ST (lam_e Tx F) (fn_t Tx T)) + %<- ({x} %pi (|-var x Tx) %-> (|-exp ST (F x) T)) %. +%term |-exp_app %pi (|-exp ST (app_e E1 E2) T) %<- (|-exp ST E1 (fn_t Tx T)) %<- (|-exp ST E2 Tx) %. +%term |-exp_unit |-exp ST unit_e unit_t %. +%term |-exp_letunit %pi (|-exp ST (letunit_e E1 E2) T) %<- (|-exp ST E1 unit_t) %<- (|-exp ST E2 T) %. +%term |-exp_pair + %pi (|-exp ST (pair_e E1 E2) (pair_t T1 T2)) + %<- (|-exp ST E1 T1) + %<- (|-exp ST E2 T2) %. +%term |-exp_letpair + %pi (|-exp ST (letpair_e E1 F2) T) + %<- (|-exp ST E1 (pair_t Tx Ty)) + %<- ({x} %pi (|-var x Tx) %-> ({y} %pi (|-var y Ty) %-> (|-exp ST (F2 x y) T))) %. +%terminates E (|-exp _ E _) %. +%worlds (|-var_block) (|-exp _ _ _) %. +%covers |-exp %in %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% |-val +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort |-val %. +%name |-val %. +%mode |-val %in %in %out %. +% Church-style +%% %mode |-val +ST +V *T. % Curry-style +%term |-val_lam + %pi (|-val ST (lam_v Tx F) (fn_t Tx T)) + %<- ({x} %pi (|-var x Tx) %-> (|-exp ST (F x) T)) %. +%term |-val_unit |-val ST unit_v unit_t %. +%term |-val_pair + %pi (|-val ST (pair_v L1 L2) (pair_t T1 T2)) + %<- (|-loc ST L1 T1) + %<- (|-loc ST L2 T2) %. +%terminates V (|-val _ V _) %. +%worlds (|-var_block) (|-val _ _ _) %. +%covers |-val %in %in %out %. +%sort raa_|-val %. +%mode {%in ST sttp} {%in V val} {%in T tp} {%in Absurd absurd} {%out DValTc |-val ST V T} raa_|-val Absurd DValTc %. +%worlds () (raa_|-val _ _) %. +%total {} (raa_|-val _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% |-st +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort |-st %. +%name |-st %. +%mode |-st %in %out %. +% Church-style +%% %mode |-st +S +ST. % Curry-style +%term |-st_nil |-st nil_st nil_sttp %. +%term |-st_cons + %pi (|-st (cons_st L V S') (cons_sttp L T ST')) + %<- (|-st S' ST') + %<- (|-val ST' V T) %. +%terminates S (|-st S _) %. +%worlds () (|-st _ _) %. +%covers |-st %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% |-stwf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort |-stwf %. +%name |-stwf %. +%mode |-stwf %in %out %. +% Church-style +%% %mode |-stwf +S +ST. % Curry-style +%term |-stwf_ %pi (|-stwf S ST) %<- (|-st S ST) %<- (st_wf S) %<- (sttp_wf ST) %. +%terminates S (|-stwf S _) %. +%worlds () (|-stwf _ _) %. +%covers |-st %in %out %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/stelf.toml b/new-tests/stelf-output/examples/alloc-sem/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/alloc-sem/store-lemmas-thm.lf b/new-tests/stelf-output/examples/alloc-sem/store-lemmas-thm.lf new file mode 100644 index 0000000..488b2e6 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/store-lemmas-thm.lf @@ -0,0 +1,13 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% store-lemmas.thm +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store WF Lemmas +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%theorem st!wf_tail_thm : forall* {L loc} {V val} {S' st} forall {DSWf st_wf (cons_st L V S')} exists {DSWf' st_wf S'} true %. +%prove 2 {} (st!wf_tail_thm _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store Alloc Lemmas +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%theorem st!alloc_preserves_wf_thm : forall* {S1 st} {V val} {S2 st} {L loc} forall {DSWf1 st_wf S1} {DStAlloc st_alloc S1 V S2 L} exists {DSWf2 st_wf S2} true %. +%prove 4 DStAlloc (st!alloc_preserves_wf_thm _ DStAlloc _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/store-lemmas.lf b/new-tests/stelf-output/examples/alloc-sem/store-lemmas.lf new file mode 100644 index 0000000..5665fe9 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/store-lemmas.lf @@ -0,0 +1,27 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% store-lemmas.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store WF Lemmas +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort st!wf_tail %. +%mode st!wf_tail %in %out %. +%scope st!wf_tail %term cons_nil st!wf_tail st_wf_cons_nil st_wf_nil %. +%term cons_cons st!wf_tail (st_wf_cons_cons DSWf' _) DSWf' %. +%terminates {} (st!wf_tail _ _) %. +%worlds () (st!wf_tail _ _) %. +%covers st!wf_tail %in %out %. +%total {} (st!wf_tail _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store Alloc Lemmas +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort st!alloc_preserves_wf %. +%mode st!alloc_preserves_wf %in %in %out %. +%scope st!alloc_preserves_wf %term nil st!alloc_preserves_wf st_wf_nil st_alloc_nil st_wf_cons_nil %. +%term cons + %pi (st!alloc_preserves_wf DSWf st_alloc_cons (st_wf_cons_cons DSWf (loc_lt_ DNatLt))) + %<- (nat!N_lt_sN _ DNatLt) %. +%terminates DStAlloc (st!alloc_preserves_wf _ DStAlloc _) %. +%worlds () (st!alloc_preserves_wf _ _ _) %. +%covers st!alloc_preserves_wf %in %in %out %. +%total DStAlloc (st!alloc_preserves_wf _ DStAlloc _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/store.lf b/new-tests/stelf-output/examples/alloc-sem/store.lf new file mode 100644 index 0000000..a0121ad --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/store.lf @@ -0,0 +1,42 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% store.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Stores +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%term nil_st st %. +%term cons_st %pi loc %-> val %-> st %-> st %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store WF +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort st_wf %. +%mode st_wf %in %. +%term st_wf_nil st_wf nil_st %. +%term st_wf_cons_nil st_wf (cons_st L V nil_st) %. +%term st_wf_cons_cons + %pi (st_wf (cons_st L' V' S')) + %-> (loc_lt L' L) + %-> (st_wf (cons_st L V (cons_st L' V' S'))) %. +%terminates S (st_wf S) %. +%worlds () (st_wf _) %. +%covers st_wf %in %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store Lookup +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort st_lookup %. +%mode st_lookup %in %in %out %. +%term st_lookup_hit st_lookup (cons_st L V _) L V %. +%term st_lookup_miss %pi (st_lookup S' L V) %-> (loc_lt L L') %-> (st_lookup (cons_st L' _ S') L V) %. +%terminates S (st_lookup S _ _) %. +%worlds () (st_lookup _ _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store Alloc +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort st_alloc %. +%mode st_alloc %in %in %out %out %. +%term st_alloc_nil st_alloc nil_st V'' (cons_st (mkLoc z_nat) V'' nil_st) (mkLoc z_nat) %. +%term st_alloc_cons st_alloc (cons_st (mkLoc N') V' S) V'' (cons_st (mkLoc (s_nat N')) V'' (cons_st (mkLoc N') V' S)) (mkLoc (s_nat N')) %. +%terminates S (st_alloc S _ _ _) %. +%worlds () (st_alloc _ _ _ _) %. +%covers st_alloc %in %in %out %out %. +%total S (st_alloc S _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/storetp-lemmas-thm.lf b/new-tests/stelf-output/examples/alloc-sem/storetp-lemmas-thm.lf new file mode 100644 index 0000000..c49ddd6 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/storetp-lemmas-thm.lf @@ -0,0 +1,26 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% storetp-lemmas.thm +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store Typing WF Lemmas +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%theorem sttp!wf_tail_thm : forall* {L loc} {T tp} {ST' sttp} forall {DSTWf sttp_wf (cons_sttp L T ST')} exists {DSTWf' sttp_wf ST'} true %. +%prove 2 {} (sttp!wf_tail_thm _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store Typing Alloc Lemmas +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%theorem sttp!alloc_preserves_wf_thm : forall* {ST1 sttp} {T tp} {ST2 sttp} {L loc} forall {DSTWf1 sttp_wf ST1} {DSttpAlloc sttp_alloc ST1 T ST2 L} exists {DSTWf2 sttp_wf ST2} true %. +%prove 4 DSttpAlloc (sttp!alloc_preserves_wf_thm _ DSttpAlloc _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store Typing Extend Lemmas +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%theorem sttp!extend_refl_thm : forall {ST sttp} exists {DSttpExtend sttp_extend ST ST} true %. +%prove 2 ST (sttp!extend_refl_thm ST _) %. +%theorem sttp!extend_trans_thm : forall* {ST1 sttp} {ST2 sttp} {ST3 sttp} forall {DSttpExtend12 sttp_extend ST1 ST2} {DSttpExtend23 sttp_extend ST2 ST3} exists {DSttpExtend13 sttp_extend ST1 ST3} true %. +%prove 4 DSttpExtend23 (sttp!extend_trans_thm _ DSttpExtend23 _) %. +%%% Store typing allocation yields an extension of the original store typing. +%theorem sttp!alloc_implies_extend_thm : forall* {ST sttp} {T tp} {ST' sttp} {L loc} forall {DStAlloc sttp_alloc ST T ST' L} exists {DSttpExtend sttp_extend ST ST'} true %. +%prove 4 DStAlloc (sttp!alloc_implies_extend_thm DStAlloc _) %. +%%% A well-formed, non-empty store typing is an extension of its tail. +%theorem sttp!wf_cons_implies_extend_cons_thm : forall* {L loc} {T tp} {ST' sttp} forall {DSTWf sttp_wf (cons_sttp L T ST')} exists {DSttpExtend sttp_extend ST' (cons_sttp L T ST')} true %. +%prove 2 {} (sttp!wf_cons_implies_extend_cons_thm _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/storetp-lemmas.lf b/new-tests/stelf-output/examples/alloc-sem/storetp-lemmas.lf new file mode 100644 index 0000000..a7c6fc5 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/storetp-lemmas.lf @@ -0,0 +1,83 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% storetp-lemmas.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store Typing WF Lemmas +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort sttp!wf_tail %. +%mode sttp!wf_tail %in %out %. +%scope sttp!wf_tail %term cons_nil sttp!wf_tail sttp_wf_cons_nil sttp_wf_nil %. +%term cons_cons sttp!wf_tail (sttp_wf_cons_cons DSTWf' _) DSTWf' %. +%terminates {} (sttp!wf_tail _ _) %. +%worlds () (sttp!wf_tail _ _) %. +%covers sttp!wf_tail %in %out %. +%total {} (sttp!wf_tail _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store Typing Alloc Lemmas +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort sttp!alloc_preserves_wf %. +%mode sttp!alloc_preserves_wf %in %in %out %. +%scope sttp!alloc_preserves_wf %term nil sttp!alloc_preserves_wf sttp_wf_nil sttp_alloc_nil sttp_wf_cons_nil %. +%term cons + %pi (sttp!alloc_preserves_wf DSTWf sttp_alloc_cons (sttp_wf_cons_cons DSTWf (loc_lt_ DNatLt))) + %<- (nat!N_lt_sN _ DNatLt) %. +%terminates DSttpAlloc (sttp!alloc_preserves_wf _ DSttpAlloc _) %. +%worlds () (sttp!alloc_preserves_wf _ _ _) %. +%covers sttp!alloc_preserves_wf %in %in %out %. +%total DSttpAlloc (sttp!alloc_preserves_wf _ DSttpAlloc _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store Typing Extend Lemmas +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort sttp!extend_refl {ST sttp} %. +%mode sttp!extend_refl %in %out %. +%scope sttp!extend_refl %term nil sttp!extend_refl nil_sttp sttp_extend_nil %. +%term cons + %pi (sttp!extend_refl (cons_sttp L T ST') (sttp_extend_cons_hit DSttpExtend')) + %<- (sttp!extend_refl ST' DSttpExtend') %. +%terminates ST (sttp!extend_refl ST _) %. +%worlds () (sttp!extend_refl _ _) %. +%covers sttp!extend_refl %in %out %. +%total ST (sttp!extend_refl ST _) %. +%sort sttp!extend_trans %. +%mode sttp!extend_trans %in %in %out %. +%scope sttp!extend_trans %term nil__* sttp!extend_trans sttp_extend_nil _ sttp_extend_nil %. +%term cons_hit__cons_hit + %pi (sttp!extend_trans (sttp_extend_cons_hit DSttpExtend1'2') (sttp_extend_cons_hit DSttpExtend2'3') (sttp_extend_cons_hit DSttpExtend1'3')) + %<- (sttp!extend_trans DSttpExtend1'2' DSttpExtend2'3' DSttpExtend1'3') %. +%scope sttp!extend_trans %term cons_miss__cons_hit + %pi (sttp!extend_trans (sttp_extend_cons_miss DSttpExtend12' DLocLt) (sttp_extend_cons_hit DSttpExtend2'3') (sttp_extend_cons_miss DSttpExtend13' DLocLt)) + %<- (sttp!extend_trans DSttpExtend12' DSttpExtend2'3' DSttpExtend13') %. +%term cons_hit__cons_miss + %pi (sttp!extend_trans (sttp_extend_cons_hit DSttpExtend1'2') (sttp_extend_cons_miss DSttpExtend23' DLocLt) (sttp_extend_cons_miss DSttpExtend13' DLocLt)) + %<- (sttp!extend_trans (sttp_extend_cons_hit DSttpExtend1'2') DSttpExtend23' DSttpExtend13') %. +%scope sttp!extend_trans %term cons_miss__cons_miss + %pi (sttp!extend_trans (sttp_extend_cons_miss DSttpExtend12' DLocLt) (sttp_extend_cons_miss DSttpExtend23' DLocLt') (sttp_extend_cons_miss DSttpExtend13' DLocLt'')) + %<- (sttp!extend_trans (sttp_extend_cons_miss DSttpExtend12' DLocLt) DSttpExtend23' DSttpExtend13') + %<- (loc!lt_trans DLocLt DLocLt' DLocLt'') %. +%terminates DSttpExtend23 (sttp!extend_trans _ DSttpExtend23 _) %. +%worlds () (sttp!extend_trans _ _ _) %. +%covers sttp!extend_trans %in %in %out %. +%total DSttpExtend23 (sttp!extend_trans _ DSttpExtend23 _) %. +%%% Store typing allocation yields an extension of the original store typing. +%sort sttp!alloc_implies_extend %. +%mode sttp!alloc_implies_extend %in %out %. +%scope sttp!alloc_implies_extend %term nil sttp!alloc_implies_extend sttp_alloc_nil sttp_extend_nil %. +%term cons + %pi (sttp!alloc_implies_extend sttp_alloc_cons (sttp_extend_cons_miss DSttpExtend (loc_lt_ DNatLt))) + %<- (sttp!extend_refl _ DSttpExtend) + %<- (nat!N_lt_sN _ DNatLt) %. +%terminates DStAlloc (sttp!alloc_implies_extend DStAlloc _) %. +%worlds () (sttp!alloc_implies_extend _ _) %. +%covers sttp!alloc_implies_extend %in %out %. +%total DStAlloc (sttp!alloc_implies_extend DStAlloc _) %. +%%% A well-formed, non-empty store typing is an extension of its tail. +%sort sttp!wf_cons_implies_extend_cons %. +%mode sttp!wf_cons_implies_extend_cons %in %out %. +%scope sttp!wf_cons_implies_extend_cons %term nil sttp!wf_cons_implies_extend_cons sttp_wf_cons_nil sttp_extend_nil %. +%term cons + %pi (sttp!wf_cons_implies_extend_cons (sttp_wf_cons_cons _ DLocLt) (sttp_extend_cons_miss DSttpExtend DLocLt)) + %<- (sttp!extend_refl _ DSttpExtend) %. +%terminates {} (sttp!wf_cons_implies_extend_cons _ _) %. +%worlds () (sttp!wf_cons_implies_extend_cons _ _) %. +%covers sttp!wf_cons_implies_extend_cons %in %out %. +%total {} (sttp!wf_cons_implies_extend_cons _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/storetp.lf b/new-tests/stelf-output/examples/alloc-sem/storetp.lf new file mode 100644 index 0000000..6357e0f --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/storetp.lf @@ -0,0 +1,60 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% storetp.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store Typings +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%term nil_sttp sttp %. +%term cons_sttp %pi loc %-> tp %-> sttp %-> sttp %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store Typings WF +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort sttp_wf %. +%mode sttp_wf %in %. +%term sttp_wf_nil sttp_wf nil_sttp %. +%term sttp_wf_cons_nil sttp_wf (cons_sttp L T nil_sttp) %. +%term sttp_wf_cons_cons + %pi (sttp_wf (cons_sttp L' T' ST'')) + %-> (loc_lt L' L) + %-> (sttp_wf (cons_sttp L T (cons_sttp L' T' ST''))) %. +%terminates ST (sttp_wf ST) %. +%worlds () (sttp_wf _) %. +%covers sttp_wf %in %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store Typing Lookup +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort sttp_lookup %. +%mode sttp_lookup %in %in %out %. +%term sttp_lookup_hit sttp_lookup (cons_sttp L T _) L T %. +%term sttp_lookup_miss + %pi (sttp_lookup ST' L T) + %-> (loc_lt L L') + %-> (sttp_lookup (cons_sttp L' _ ST') L T) %. +%terminates ST (sttp_lookup ST _ _) %. +%worlds () (sttp_lookup _ _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store Typing Alloc +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort sttp_alloc %. +%mode sttp_alloc %in %in %out %out %. +%term sttp_alloc_nil sttp_alloc nil_sttp T'' (cons_sttp (mkLoc z_nat) T'' nil_sttp) (mkLoc z_nat) %. +%term sttp_alloc_cons sttp_alloc (cons_sttp (mkLoc N) T ST) T'' (cons_sttp (mkLoc (s_nat N)) T'' (cons_sttp (mkLoc N) T ST)) (mkLoc (s_nat N)) %. +%terminates ST (sttp_alloc ST _ _ _) %. +%worlds () (sttp_alloc _ _ _ _) %. +%covers sttp_alloc %in %in %out %out %. +%total ST (sttp_alloc ST _ _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store Typing Extend +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort sttp_extend %. +%mode sttp_extend %in %in %. +%term sttp_extend_nil sttp_extend nil_sttp ST2 %. +%term sttp_extend_cons_hit + %pi (sttp_extend ST1' ST2') + %-> (sttp_extend (cons_sttp L T ST1') (cons_sttp L T ST2')) %. +%term sttp_extend_cons_miss + %pi (sttp_extend (cons_sttp L1 T1 ST1') ST2') + %-> (loc_lt L1 L2) + %-> (sttp_extend (cons_sttp L1 T1 ST1') (cons_sttp L2 T2 ST2')) %. +%terminates ST2 (sttp_extend _ ST2) %. +%worlds () (sttp_extend _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/syntax.lf b/new-tests/stelf-output/examples/alloc-sem/syntax.lf new file mode 100644 index 0000000..22f72f3 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/syntax.lf @@ -0,0 +1,27 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% syntax.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Syntax +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort loc %. +%name loc %. +%% loc.elf +%sort exp %. +%name exp %. +%% exp.elf +%sort val %. +%name val %. +%% val.elf +%sort st %. +%name st %. +%% store.elf +%sort tp %. +%name tp %. +%% tp.elf +%sort sttp %. +%name sttp %. +%% storetp.elf +%sort mach %. +%name mach %. +%% machine.elf \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/tp.lf b/new-tests/stelf-output/examples/alloc-sem/tp.lf new file mode 100644 index 0000000..e96efc6 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/tp.lf @@ -0,0 +1,9 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% tp.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Types +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%term fn_t %pi tp %-> tp %-> tp %. +%term unit_t tp %. +%term pair_t %pi tp %-> tp %-> tp %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/val.lf b/new-tests/stelf-output/examples/alloc-sem/val.lf new file mode 100644 index 0000000..863c841 --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/val.lf @@ -0,0 +1,9 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% val.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Values +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%term lam_v %pi tp %-> (%pi exp %-> exp) %-> val %. +%term unit_v val %. +%term pair_v %pi loc %-> loc %-> val %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/weaken-thm.lf b/new-tests/stelf-output/examples/alloc-sem/weaken-thm.lf new file mode 100644 index 0000000..c2d518d --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/weaken-thm.lf @@ -0,0 +1,21 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% weaken.thm +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Weaken Store Typing +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% sttp_lookup +%theorem sttp_weak_sttp_lookup_thm : forall* {ST1 sttp} {L loc} {T tp} {ST2 sttp} forall {DSttpLookup sttp_lookup ST1 L T} {DSttpExtend sttp_extend ST1 ST2} exists {DSttpLookup' sttp_lookup ST2 L T} true %. +%prove 4 DSttpExtend (sttp_weak_sttp_lookup_thm _ DSttpExtend _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% |-loc +%theorem sttp_weak_|-loc_thm : forall* {ST1 sttp} {L loc} {T tp} {ST2 sttp} forall {DLocTc |-loc ST1 L T} {DSttpExtend sttp_extend ST1 ST2} exists {DLocTc' |-loc ST2 L T} true %. +%prove 3 {} (sttp_weak_|-loc_thm _ _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% |-exp +%theorem sttp_weak_|-exp_thm : forallG (some {T tp} pi {x exp} {DVarTcX |-var x T}) forall* {ST1 sttp} {E exp} {T tp} {ST2 sttp} forall {DExpTc |-exp ST1 E T} {DSttpExtend sttp_extend ST1 ST2} exists {DExpTc' |-exp ST2 E T} true %. +%prove 3 DExpTc (sttp_weak_|-exp_thm DExpTc _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% |-val +%theorem sttp_weak_|-val_thm : forallG (some {T tp} pi {x exp} {DVarTcX |-var x T}) forall* {ST1 sttp} {V val} {T tp} {ST2 sttp} forall {DValTc |-val ST1 V T} {DSttpExtend sttp_extend ST1 ST2} exists {DValTc' |-val ST2 V T} true %. +%% %prove 4 {} (sttp_weak_|-val_thm _ _ _). \ No newline at end of file diff --git a/new-tests/stelf-output/examples/alloc-sem/weaken.lf b/new-tests/stelf-output/examples/alloc-sem/weaken.lf new file mode 100644 index 0000000..7963ebc --- /dev/null +++ b/new-tests/stelf-output/examples/alloc-sem/weaken.lf @@ -0,0 +1,83 @@ +%%% Simply Typed Lambda Calculus w/ small-step, allocation semantics +%%% Author: Matthew Fluet (June 2005) +%%% weaken.elf +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Weaken Store Typing +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% sttp_lookup +%sort sttp_weak_sttp_lookup %. +%mode sttp_weak_sttp_lookup %in %in %out %. +%scope sttp_weak_sttp_lookup %term hit_hit sttp_weak_sttp_lookup sttp_lookup_hit (sttp_extend_cons_hit DSttpExtend') sttp_lookup_hit %. +%term miss_hit + %pi (sttp_weak_sttp_lookup (sttp_lookup_miss DSttpLookup DLocLt) (sttp_extend_cons_hit DSttpExtend') (sttp_lookup_miss DSttpLookup'' DLocLt)) + %<- (sttp_weak_sttp_lookup DSttpLookup DSttpExtend' DSttpLookup'') %. +%scope sttp_weak_sttp_lookup %term hit_miss + %pi (sttp_weak_sttp_lookup sttp_lookup_hit (sttp_extend_cons_miss DSttpExtend' DLocLt') (sttp_lookup_miss DSttpLookup'' DLocLt')) + %<- (sttp_weak_sttp_lookup sttp_lookup_hit DSttpExtend' DSttpLookup'') %. +%term miss_miss + %pi (sttp_weak_sttp_lookup (sttp_lookup_miss DSttpLookup DLocLt) (sttp_extend_cons_miss DSttpExtend' DLocLt') (sttp_lookup_miss DSttpLookup'' DLocLt'')) + %<- (sttp_weak_sttp_lookup (sttp_lookup_miss DSttpLookup DLocLt) DSttpExtend' DSttpLookup'') + %<- (loc!lt_trans DLocLt DLocLt' DLocLt'') %. +%terminates DSttpExtend (sttp_weak_sttp_lookup _ DSttpExtend _) %. +%worlds () (sttp_weak_sttp_lookup _ _ _) %. +%covers sttp_weak_sttp_lookup %in %in %out %. +%total DSttpExtend (sttp_weak_sttp_lookup _ DSttpExtend _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% |-loc +%sort sttp_weak_|-loc %. +%mode sttp_weak_|-loc %in %in %out %. +%term sttp_weak_|-loc_ + %pi (sttp_weak_|-loc (|-loc_ DSttpLookup) DSttpExtend (|-loc_ DSttpLookup')) + %<- (sttp_weak_sttp_lookup DSttpLookup DSttpExtend DSttpLookup') %. +%terminates {} (sttp_weak_|-loc _ _ _) %. +%worlds () (sttp_weak_|-loc _ _ _) %. +%covers sttp_weak_|-loc %in %in %out %. +%total {} (sttp_weak_|-loc _ _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% |-exp +%sort sttp_weak_|-exp %. +%mode sttp_weak_|-exp %in %in %out %. +%scope sttp_weak_|-exp %term var sttp_weak_|-exp (|-exp_var DVarTc) DSttpExtend (|-exp_var DVarTc) %. +%term loc + %pi (sttp_weak_|-exp (|-exp_loc DLocTc) DSttpExtend (|-exp_loc DLocTc')) + %<- (sttp_weak_|-loc DLocTc DSttpExtend DLocTc') %. +%scope sttp_weak_|-exp %term lam + %pi (sttp_weak_|-exp (|-exp_lam DExpTcF) DSttpExtend (|-exp_lam DExpTcF')) + %<- ({x} {DVarTcX |-var x Tx} sttp_weak_|-exp (DExpTcF x DVarTcX) DSttpExtend (DExpTcF' x DVarTcX)) %. +%term app + %pi (sttp_weak_|-exp (|-exp_app DExpTcE2 DExpTcE1) DSttpExtend (|-exp_app DExpTcE2' DExpTcE1')) + %<- (sttp_weak_|-exp DExpTcE1 DSttpExtend DExpTcE1') + %<- (sttp_weak_|-exp DExpTcE2 DSttpExtend DExpTcE2') %. +%scope sttp_weak_|-exp %term unit sttp_weak_|-exp |-exp_unit DSttpExtend |-exp_unit %. +%term letunit + %pi (sttp_weak_|-exp (|-exp_letunit DExpTcE2 DExpTcE1) DSttpExtend (|-exp_letunit DExpTcE2' DExpTcE1')) + %<- (sttp_weak_|-exp DExpTcE1 DSttpExtend DExpTcE1') + %<- (sttp_weak_|-exp DExpTcE2 DSttpExtend DExpTcE2') %. +%scope sttp_weak_|-exp %term pair + %pi (sttp_weak_|-exp (|-exp_pair DExpTcE2 DExpTcE1) DSttpExtend (|-exp_pair DExpTcE2' DExpTcE1')) + %<- (sttp_weak_|-exp DExpTcE1 DSttpExtend DExpTcE1') + %<- (sttp_weak_|-exp DExpTcE2 DSttpExtend DExpTcE2') %. +%term letpair + %pi (sttp_weak_|-exp (|-exp_letpair DExpTcF2 DExpTcE1) DSttpExtend (|-exp_letpair DExpTcF2' DExpTcE1')) + %<- (sttp_weak_|-exp DExpTcE1 DSttpExtend DExpTcE1') + %<- ({x} {DVarTcX |-var x Tx} {y} {DVarTcY |-var y Ty} sttp_weak_|-exp (DExpTcF2 x DVarTcX y DVarTcY) DSttpExtend (DExpTcF2' x DVarTcX y DVarTcY)) %. +%terminates DExpTc (sttp_weak_|-exp DExpTc _ _) %. +%worlds (|-var_block) (sttp_weak_|-exp _ _ _) %. +%covers sttp_weak_|-exp %in %in %out %. +%total DExpTc (sttp_weak_|-exp DExpTc _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% |-val +%sort sttp_weak_|-val %. +%mode sttp_weak_|-val %in %in %out %. +%scope sttp_weak_|-val %term lam + %pi (sttp_weak_|-val (|-val_lam DExpTcF) DSttpExtend (|-val_lam DExpTcF')) + %<- ({x} {DVarTcX |-var x Tx} sttp_weak_|-exp (DExpTcF x DVarTcX) DSttpExtend (DExpTcF' x DVarTcX)) %. +%term unit sttp_weak_|-val |-val_unit DSttpExtend |-val_unit %. +%scope sttp_weak_|-val %term pair + %pi (sttp_weak_|-val (|-val_pair DLocTcL2 DLocTcL1) DSttpExtend (|-val_pair DLocTcL2' DLocTcL1')) + %<- (sttp_weak_|-loc DLocTcL1 DSttpExtend DLocTcL1') + %<- (sttp_weak_|-loc DLocTcL2 DSttpExtend DLocTcL2') %. +%terminates {} (sttp_weak_|-val _ _ _) %. +%worlds (|-var_block) (sttp_weak_|-val _ _ _) %. +%covers sttp_weak_|-val %in %in %out %. +%total {} (sttp_weak_|-val _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/arith/arith-thm.lf b/new-tests/stelf-output/examples/arith/arith-thm.lf new file mode 100644 index 0000000..d80561c --- /dev/null +++ b/new-tests/stelf-output/examples/arith/arith-thm.lf @@ -0,0 +1,14 @@ +% Termination of Ackermann's function requires +% lexicographic termination ordering +% terminates [N M] (acker N M _). % causes termination error +%terminates {N M} (acker N M _) %. +%theorem assoc : forall* {X nat} {Y nat} {Z nat} {X' nat} {Z' nat} forall {P1 plus X Y X'} {P2 plus X' Z Z'} exists {Y' nat} {Q1 plus Y Z Y'} {Q2 plus X Y' Z'} true %. +%prove 3 P1 (assoc P1 _ _ _ _) %. +%terminates P1 (assoc P1 _ _ _ _) %. +%theorem assoc' : forall* {X nat} {Y nat} {Z nat} {X' nat} {Z' nat} forall {P1 plus X Y X'} {P2 plus X' Z Z'} exists {Y' nat} {Q1 plus Y Z Y'} {Q2 plus X Y' Z'} true %. +%terminates P2 (assoc' _ P2 _ _ _) %. +% %prove 3 P2 (assoc' _ P2 _ _ _). % fails +%theorem comm : forall* {X nat} {Y nat} {Z nat} forall {N nt X} {M nt Y} {P'' nt Z} {P0 plus X Y Z} exists {Q plus Y X Z} true %. +%theorem comm' : forall* {X nat} {Y nat} {Z nat} forall {M' nt X} {N' nt Y} {P'' nt Z} {P0 plus X Y Z} exists {Q plus Y X Z} true %. +%prove 6 [(N N') (M M')] (comm N M _ _ _) (comm' M' N' _ _ _) %. +%terminates [(N N') (M M')] (comm N M _ _ _) (comm' M' N' _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/arith/arith.lf b/new-tests/stelf-output/examples/arith/arith.lf new file mode 100644 index 0000000..56c69a8 --- /dev/null +++ b/new-tests/stelf-output/examples/arith/arith.lf @@ -0,0 +1,22 @@ +%sort nat %. +%name nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort nt %. +%name nt %. +%term nt_z nt z %. +%term nt_s %pi (nt (s X)) %<- (nt X) %. +%sort plus %. +%name plus %. +%term p_z plus z Y Y %. +%term p_s %pi (plus (s X) Y (s Z)) %<- (plus X Y Z) %. +%sort acker %. +%mode acker %in %in %out %. +%term a_1 acker z Y (s Y) %. +%term a_2 %pi (acker (s X) z Z) %<- (acker X (s z) Z) %. +%term a_3 %pi (acker (s X) (s Y) Z) %<- (acker (s X) Y Z') %<- (acker X Z' Z) %. +%worlds () (acker _ _ _) %. +%define arg1 s (s (s z)) %. +%define arg2 s (s (s (s (s z)))) %. +% query 1 1 (acker arg1 arg2 Z). +% fquery (acker arg1 arg2 Z). \ No newline at end of file diff --git a/new-tests/stelf-output/examples/arith/sources.lf b/new-tests/stelf-output/examples/arith/sources.lf new file mode 100644 index 0000000..cbd41c6 --- /dev/null +++ b/new-tests/stelf-output/examples/arith/sources.lf @@ -0,0 +1 @@ +%require %[ arith %] %. diff --git a/new-tests/stelf-output/examples/arith/stelf.toml b/new-tests/stelf-output/examples/arith/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/arith/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/arith/test.lf b/new-tests/stelf-output/examples/arith/test.lf new file mode 100644 index 0000000..a1b364f --- /dev/null +++ b/new-tests/stelf-output/examples/arith/test.lf @@ -0,0 +1,2 @@ +%require %[ arith %] %. +%require %[ arith %] %. diff --git a/new-tests/stelf-output/examples/ccc/abs-env.lf b/new-tests/stelf-output/examples/ccc/abs-env.lf new file mode 100644 index 0000000..bca6831 --- /dev/null +++ b/new-tests/stelf-output/examples/ccc/abs-env.lf @@ -0,0 +1,21 @@ +%% With explicit environments (as in Curien) +%sort env %. +%name env %. +%term empty env 1 %. +%term addv %pi (env P) %-> (term A) %-> (env (P * A)) %. +% look up variable +%sort absv %. +%term av_x absv (addv G X) X snd %. +%term av_y %pi (absv (addv G Y) X (M @ fst)) %<- (absv G X M) %. +% translate term +%sort abse %. +%term avar %pi (abse G X M) %<- (absv G X M) %. +%term aunit abse G lunit drop %. +%term apair %pi (abse G (lpair E1 E2) (pair F1 F2)) %<- (abse G E2 F2) %<- (abse G E1 F1) %. +%term afst %pi (abse G (lfst E) (fst @ F)) %<- (abse G E F) %. +%term asnd %pi (abse G (lsnd E) (snd @ F)) %<- (abse G E F) %. +%term alam %pi (abse G (llam ([x] E x)) (cur F)) %<- ({x} abse (addv G x) (E x) F) %. +%term aapp + %pi (abse G (lapp E1 E2) (app @ (pair F1 F2))) + %<- (abse G E2 F2) + %<- (abse G E1 F1) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/ccc/catlem.lf b/new-tests/stelf-output/examples/ccc/catlem.lf new file mode 100644 index 0000000..4a0e76f --- /dev/null +++ b/new-tests/stelf-output/examples/ccc/catlem.lf @@ -0,0 +1,14 @@ +% Lemmas about CCC combinators +% o H = +%sort distp %. +%term distp1 distp (sym prod_u then (=pair= (ass then (=@= prod_l refl)) (ass then (=@= prod_r refl)))) %. +% app @ = F @ +%sort appl %. +%term appl1 + %pi (appl (=@= refl (=pair= (=@= refl (sym prod_l) then ass) (sym prod_r) then sym DP) then ass then =@= exp_e refl)) + %<- (distp DP) %. +% cur(F) o G = cur (F o ) +%sort distc %. +%term distc1 + %pi (distc (sym exp_u then =cur= (=@= refl (=pair= (sym ass) refl) then AP))) + %<- (appl AP) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/ccc/ccc.lf b/new-tests/stelf-output/examples/ccc/ccc.lf new file mode 100644 index 0000000..d9a689a --- /dev/null +++ b/new-tests/stelf-output/examples/ccc/ccc.lf @@ -0,0 +1,54 @@ +%%% Cartesian closed category +%% Categorical language +% objects +%sort obj %. +%name obj %. +% morphisms +%sort mor %. +%name mor %. +% morphism equality +%sort == %. +%name == %. +%prec %none 5 == %. +% morphisms +%term id mor A A %. +%term @ %pi (mor B C) %-> (mor A B) %-> (mor A C) %. +%prec %none 10 @ %. +% equations +%term refl F == F %. +%term then %pi (F == F') %-> (F' == F'') %-> (F == F'') %. +%prec %right 4 then %. +%term sym %pi (F == F') %-> (F' == F) %. +%term =@= %pi (F == F') %-> (G == G') %-> (F @ G == F' @ G') %. +%term id_l id @ F == F %. +%term id_r F @ id == F %. +%term ass H @ (G @ F) == (H @ G) @ F %. +%% Products +% objects +%term 1 obj %. +%term * %pi obj %-> obj %-> obj %. +%prec %none 10 * %. +% morphisms +%term drop mor A 1 %. +%term fst mor (A * B) A %. +%term snd mor (A * B) B %. +%term pair %pi (mor A B) %-> (mor A C) %-> (mor A (B * C)) %. +% equations +%term =pair= %pi (F == F') %-> (G == G') %-> (pair F G == pair F' G') %. +%term term_u H == drop %. +%term prod_l fst @ (pair F G) == F %. +%term prod_r snd @ (pair F G) == G %. +%term prod_u pair (fst @ H) (snd @ H) == H %. +% distp : pair F G @ H == pair (F @ H) (G @ H). +%% Exponentials +% objects +%term => %pi obj %-> obj %-> obj %. +%prec %right 7 => %. +% morphisms +%term app mor (B => C * B) C %. +%term cur %pi (mor (A * B) C) %-> (mor A (B => C)) %. +% equations +%term =cur= %pi (F == F') %-> (cur F == (cur F')) %. +%term exp_e app @ (pair (cur F @ fst) snd) == F %. +%term exp_u cur (app @ (pair (G @ fst) snd)) == G %. +% distc : (cur F) @ G == cur (F @ (pair (G @ fst) snd)). \ No newline at end of file diff --git a/new-tests/stelf-output/examples/ccc/conc-thm.lf b/new-tests/stelf-output/examples/ccc/conc-thm.lf new file mode 100644 index 0000000..3b1ddbb --- /dev/null +++ b/new-tests/stelf-output/examples/ccc/conc-thm.lf @@ -0,0 +1,8 @@ +% Translation CCC combinators -> lambda-terms +% Partial check of hand-coded proof +%terminates F (conc F _) %. +% Automatic proof generation +%theorem concom : forall* {A obj} {B obj} forall {F mor A B} exists {M %pi (term A) %-> (term B)} true %. +%prove 4 F (concom F _) %. +% Partial verification of generated proof +%terminates F (concom F _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/ccc/conc.lf b/new-tests/stelf-output/examples/ccc/conc.lf new file mode 100644 index 0000000..b3d9c02 --- /dev/null +++ b/new-tests/stelf-output/examples/ccc/conc.lf @@ -0,0 +1,12 @@ +% Translation CCC combinators -> lambda-terms +%sort conc %. +%mode conc %in %out %. +%term cid conc id ([x] x) %. +%term ccomp %pi (conc (F @ G) ([x] M (N x))) %<- (conc G N) %<- (conc F M) %. +%term cunit conc drop ([x] lunit) %. +%term cpair %pi (conc (pair F G) ([x] lpair (M x) (N x))) %<- (conc G N) %<- (conc F M) %. +%term cfst conc fst ([x] lfst x) %. +%term csnd conc snd ([x] lsnd x) %. +%term ccur %pi (conc (cur F) ([a] llam ([b] M (lpair a b)))) %<- (conc F M) %. +%term capp conc app ([a] lapp (lfst a) (lsnd a)) %. +%worlds () (conc _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/ccc/cong.lf b/new-tests/stelf-output/examples/ccc/cong.lf new file mode 100644 index 0000000..94f1d95 --- /dev/null +++ b/new-tests/stelf-output/examples/ccc/cong.lf @@ -0,0 +1,17 @@ +% general congruence: if e <-> e' then e2[e/x] <-> e2[e'/x] +%sort cong {M %pi (term A) %-> (term B)} %. +%term cong_var cong ([x] x) ([c] c) %. +%term cong_unit cong ([x] lunit) ([c] c_refl) %. +%term cong_pair + %pi (cong ([x] lpair (E1 x) (E2 x)) ([c] c_pair (CP1 c) (CP2 c))) + %<- (cong E1 CP1) + %<- (cong E2 CP2) %. +%term cong_fst %pi (cong ([x] lfst (E x)) ([c] c_fst (CP c))) %<- (cong E CP) %. +%term cong_snd %pi (cong ([x] lsnd (E x)) ([c] c_snd (CP c))) %<- (cong E CP) %. +%term cong_lam + %pi (cong ([x] llam ([y] E x y)) ([c] c_lam ([y] CP c y))) + %<- ({y} %pi (cong ([x] y) ([c] c_refl)) %-> (cong ([x] E x y) ([c] CP c y))) %. +%term cong_app + %pi (cong ([x] lapp (E1 x) (E2 x)) ([c] c_app (CP1 c) (CP2 c))) + %<- (cong E1 CP1) + %<- (cong E2 CP2) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/ccc/eqpres1.lf b/new-tests/stelf-output/examples/ccc/eqpres1.lf new file mode 100644 index 0000000..7dc219e --- /dev/null +++ b/new-tests/stelf-output/examples/ccc/eqpres1.lf @@ -0,0 +1,31 @@ +%sort ctoe %. +%term ctoe_refl ctoe c_refl AP AP refl %. +%term ctoe_trans + %pi (ctoe (c_trans CP1 CP2) AP1 AP2 (EP1 then EP2)) + %<- (ctoe CP1 AP1 AP3 EP1) + %<- (ctoe CP2 AP3 AP2 EP2) %. +%term ctoe_sym %pi (ctoe (c_sym CP) AP1 AP2 (sym EP)) %<- (ctoe CP AP2 AP1 EP) %. +%term ctoe_fst %pi (ctoe (c_fst CP) (afst AP) (afst AP') (=@= refl EP)) %<- (ctoe CP AP AP' EP) %. +%term ctoe_snd %pi (ctoe (c_snd CP) (asnd AP) (asnd AP') (=@= refl EP)) %<- (ctoe CP AP AP' EP) %. +%term ctoe_pair + %pi (ctoe (c_pair CP1 CP2) (apair AP1 AP2) (apair AP1' AP2') (=pair= EP1 EP2)) + %<- (ctoe CP1 AP1 AP1' EP1) + %<- (ctoe CP2 AP2 AP2' EP2) %. +%term ctoe_lam + %pi (ctoe (c_lam ([x] CP x)) (alam ([x] AP x)) (alam ([x] AP' x)) (=cur= EP)) + %<- ({x} ctoe (CP x) (AP x) (AP' x) EP) %. +%term ctoe_app + %pi (ctoe (c_app CP1 CP2) (aapp AP1 AP2) (aapp AP1' AP2') (=@= refl (=pair= EP1 EP2))) + %<- (ctoe CP1 AP1 AP1' EP1) + %<- (ctoe CP2 AP2 AP2' EP2) %. +%term ctoe_unit ctoe c_unit aunit AP (sym term_u) %. +%term ctoe_prl ctoe c_prl (afst (apair AP1 AP2)) AP1 prod_l %. +%term ctoe_prr ctoe c_prr (asnd (apair AP1 AP2)) AP2 prod_r %. +%term ctoe_surj ctoe c_surj (apair (afst AP) (asnd AP)) AP prod_u %. +%term ctoe_beta + %pi (ctoe c_beta (aapp (alam ([x] AP1 x)) AP2) AP (=@= refl (=pair= (sym id_r) refl) then EE then EP)) + %<- (subst AP2 AP1 AP EP) + %<- (appl EE) %. +%term ctoe_eta + %pi (ctoe c_eta (alam ([x] aapp (AP x) (avar av_x))) AP' (=cur= (=@= refl (=pair= (sym EP) refl)) then exp_u)) + %<- ({x} weak AP' (AP x) EP) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/ccc/eqpres2.lf b/new-tests/stelf-output/examples/ccc/eqpres2.lf new file mode 100644 index 0000000..6729fb5 --- /dev/null +++ b/new-tests/stelf-output/examples/ccc/eqpres2.lf @@ -0,0 +1,35 @@ +% equality preservation +%sort etoc %. +%term etoc_refl etoc refl CF CF ([x] c_refl) %. +%term etoc_then + %pi (etoc (EP1 then EP2) CF CF' ([x] c_trans (CP1 x) (CP2 x))) + %<- (etoc EP1 CF CF'' CP1) + %<- (etoc EP1 CF'' CF' CP2) %. +%term etoc_sym %pi (etoc (sym EP) CF CF' ([x] c_sym (CP x))) %<- (etoc EP CF' CF CP) %. +%term etoc_comp + %pi (etoc (=@= EPF EPG) (ccomp CF (%the (conc _ M) CG)) (ccomp (%the (conc _ N') CF') CG') ([x] c_trans (CPF (M x)) (CVF' x (CPG x)))) + %<- (etoc EPF CF CF' CPF) + %<- (etoc EPG CG CG' CPG) + %<- ({x} cong N' (CVF' x)) %. +%term etoc_pair + %pi (etoc (=pair= EPF EPG) (cpair CF CG) (cpair CF' CG') ([x] c_pair (CPF x) (CPG x))) + %<- (etoc EPF CF CF' CPF) + %<- (etoc EPG CG CG' CPG) %. +%term etoc_cur + %pi (etoc (=cur= EPF) (ccur CF) (ccur CF') ([x] c_lam ([y] CPF (lpair x y)))) + %<- (etoc EPF CF CF' CPF) %. +%term etoc_idl etoc id_l (ccomp cid CF) CF ([x] c_refl) %. +%term etoc_idr etoc id_r (ccomp CF cid) CF ([x] c_refl) %. +%term etoc_ass etoc ass (ccomp CF (ccomp CG CH)) (ccomp (ccomp CF CG) CH) ([x] c_refl) %. +%term etoc_termu etoc term_u cunit CF ([x] c_unit) %. +%term etoc_prodl etoc prod_l (ccomp cfst (cpair CF CG)) CF ([x] c_prl) %. +%term etoc_prodr etoc prod_r (ccomp csnd (cpair CF CG)) CG ([x] c_prr) %. +%term etoc_produ etoc prod_u (cpair (ccomp cfst CF) (ccomp csnd CF)) CF ([x] c_surj) %. +%term etoc_expe + %pi (etoc exp_e (ccomp capp (cpair (ccomp (ccur CF) cfst) csnd)) (%the (conc F M) CF) ([x] c_trans (c_app c_prl c_prr) (c_trans c_beta (CC x (%the (conv _ x) c_surj))))) + %<- ({x} cong M (CC x)) %. +%term etoc_expu + %pi (etoc exp_u (ccur (ccomp capp (cpair (ccomp CF cfst) csnd))) (%the (conc F M) CF) ([x] c_trans (c_lam ([y] c_app (c_trans c_prl (CC x y (%the (conv (lfst (lpair x y)) _) c_prl))) (c_trans c_prr c_prr))) c_eta)) + %<- ({x} {y} cong M (CC x y)) %. +% sigma [p1:conc (app @ pair (cur fst @ fst) snd) M1] sigma [p2:conc fst M2]etoc exp_e p1 p2 C. +% sigma [p1:conc (cur (app @ pair (id @ fst) snd)) M1] sigma [p2:conc id M2]etoc exp_u p1 p2 C. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/ccc/examples-quy.lf b/new-tests/stelf-output/examples/ccc/examples-quy.lf new file mode 100644 index 0000000..856b4a8 --- /dev/null +++ b/new-tests/stelf-output/examples/ccc/examples-quy.lf @@ -0,0 +1,51 @@ +% A few test examples +%% Translation from lambda-terms to categorical combinators +% [[ |- \x.x]] +%query 1 _ _ abse empty (llam ([x] x)) M %. +% [[ x |- \y.(y,x) ]] +%query 1 _ _ {x} abse (addv empty x) (llam ([y] lpair y x)) M %. +% [[ |- \x.\y.x y ]] +%query 1 _ _ abse empty (llam ([x] llam ([y] lapp x y))) M %. +%% Translation from categorical combinators to lambda-terms +% <| id @ snd |> +%query 1 _ _ conc (id @ snd) E %. +% <| app @ |> +%query 1 _ _ conc (app @ pair snd fst) E %. +% <| cur (app) @ snd |> +%query 1 _ _ conc (cur app @ snd) E %. +%% Preservation of beta-eta equality by [[ ]] - translation +%{ +% (\y.(\x.x) y) = \z.z (using beta, xi) +sigma [ap1:abse empty (llam [y] lapp (llam [x] x) y) M1] + sigma [ap2:abse empty (llam [z] z) M2] + ctoe (c_lam [y] c_beta) ap1 ap2 EP. + +% (\y.(\x.x) y) = \z.z (using eta) +sigma [ap1:abse empty (llam [y] lapp (llam [x] x) y) M1] + sigma [ap2:abse empty (llam [z] z) M2] + ctoe (c_eta) ap1 ap2 EP. +}% +%% Preservation of CCC equality by <| |> - translation +% app @ (pair ((cur fst) @ fst) snd) = fst +%{ +sigma [cp1:conc (app @ (pair ((cur fst) @ fst) snd)) E1] + sigma [cp2: conc fst E2] + etoc exp_e cp1 cp2 EP. +}% +%query 1 _ _ etoc exp_e (ccomp capp (cpair (ccomp (ccur cfst) cfst) csnd)) cfst EP %. +%% Translation lambda -> ccc -> lambda preserves equality +% translate \x.\y.(x,y) to ccc and back +%{ +sigma [ap:abse empty (llam [x] llam [y] lpair x y) M] + sigma [cp:conc M E] + sigma [zp:exp empty Z] % a term-representaiton of the environment + invca ap cp zp EP. +}% +%query 1 _ _ invca (alam ([x term _A1] alam ([x1 term _A2] apair (avar (av_y av_x)) (avar av_x)))) (ccur (ccur (cpair (ccomp csnd cfst) csnd))) exp_empty EP %. +%% Translation ccc -> lamda -> ccc preserves equality +% translate cur(app) @ fst to lambda-term and back +%{ +sigma [cp:conc ((cur app) @ fst) E] + sigma [ap: zabs E F] + invac cp ap EP. +}% \ No newline at end of file diff --git a/new-tests/stelf-output/examples/ccc/inv1.lf b/new-tests/stelf-output/examples/ccc/inv1.lf new file mode 100644 index 0000000..6046241 --- /dev/null +++ b/new-tests/stelf-output/examples/ccc/inv1.lf @@ -0,0 +1,31 @@ +%sort exp %. +%term exp_empty exp empty lunit %. +%term exp_addv %pi (exp (addv G X) (lpair N X)) %<- (exp G N) %. +%sort invcav %. +%term invcav_x invcav av_x csnd (exp_addv EP) c_prr %. +%term invcav_y + %pi (invcav (av_y AV) (ccomp (%the (conc _ M) CP) cfst) (%the (exp _ (lpair MG Y)) (exp_addv EP)) (c_trans (CC (%the (conv (lfst (lpair MG Y)) _) c_prl)) CVP)) + %<- (cong M CC) + %<- (invcav AV CP EP CVP) %. +%sort invca %. +%term invca_var %pi (invca (avar AV) CP EP CVP) %<- (invcav AV CP EP CVP) %. +%term invca_fst %pi (invca (afst AP) (ccomp cfst CP) EP (c_fst CVP)) %<- (invca AP CP EP CVP) %. +%term invca_snd %pi (invca (asnd AP) (ccomp csnd CP) EP (c_snd CVP)) %<- (invca AP CP EP CVP) %. +%term invca_pair + %pi (invca (apair AP1 AP2) (cpair CP1 CP2) EP (c_pair CVP1 CVP2)) + %<- (invca AP1 CP1 EP CVP1) + %<- (invca AP2 CP2 EP CVP2) %. +%term invca_lam + %pi (invca (alam ([x] AP x)) (ccur CP) EP (c_lam ([x] CVP x))) + %<- ({x} invca (AP x) CP (exp_addv EP) (CVP x)) %. +%term invca_app + %pi (invca (aapp AP1 AP2) (ccomp capp (cpair CP1 CP2)) EP (c_app (c_trans c_prl CVP1) (c_trans c_prr CVP2))) + %<- (invca AP1 CP1 EP CVP1) + %<- (invca AP2 CP2 EP CVP2) %. +% sigma[p1:{x}{y}abse (addv (addv empty x) y) x F] sigma [p2:conc F M] sigma [p3:{x}{y}exp (addv (addv empty x) y) (N x y)] {x}{y}invca (p1 x y) p2 (p3 x y) (R x y). +% sigma[p1:{x}abse (addv empty x) (llam [y] lpair y x) F] sigma [p2:conc F M] sigma [p3:{x}exp (addv empty x) (N x)] {x}invca (p1 x) p2 (p3 x) (R x). +%sort zconc %. +%term zconc1 %pi (zconc F G (M MG)) %<- (conc F M) %<- (exp G MG) %. +%sort inv %. +%term inv1 %pi (inv AP (zconc1 EXP CP) EE) %<- (invca AP CP EXP EE) %. +% sigma[p1:{x}abse (addv empty x) (llam [y] lpair y x) F] sigma [p2:{x}zconc F (addv empty x) (M x)] {x} inv (p1 x) (p2 x) (R x). \ No newline at end of file diff --git a/new-tests/stelf-output/examples/ccc/inv2.lf b/new-tests/stelf-output/examples/ccc/inv2.lf new file mode 100644 index 0000000..cca0440 --- /dev/null +++ b/new-tests/stelf-output/examples/ccc/inv2.lf @@ -0,0 +1,25 @@ +%sort inve %. +%term inve_id inve cid AP AP (sym id_l) %. +%term inve_comp + %pi (inve (ccomp CF CG) AP1 AP2 (EPF then (=@= refl EPG) then ass)) + %<- (inve CG AP1 AP3 EPG) + %<- (inve CF AP3 AP2 EPF) %. +%term inve_pair + %pi (inve (cpair CF CG) AP (apair APF APG) (=pair= EPF EPG then sym DP)) + %<- (distp DP) + %<- (inve CF AP APF EPF) + %<- (inve CG AP APG EPG) %. +%term inve_fst inve cfst AP (afst AP) refl %. +%term inve_snd inve csnd AP (asnd AP) refl %. +%term inve_cur + %pi (inve (ccur CF) AP (alam ([x] AP2 x)) (=cur= (EE then =@= refl (=pair= (sym EW) refl)) then sym DC)) + %<- (distc DC) + %<- ({y} weak AP (AP' y) EW) + %<- ({y} inve CF (apair (AP' y) (avar av_x)) (AP2 y) EE) %. +%term inve_app inve capp AP (aapp (afst AP) (asnd AP)) (=@= refl prod_u) %. +%sort zabs %. +%term zabs1 %pi (zabs ([x] E x) (F @ pair drop id)) %<- ({x} abse (addv empty x) (E x) F) %. +%sort invac %. +%term invac1 + %pi (invac C (zabs1 ([x] AP x)) (=@= EE refl then sym ass then =@= refl prod_r then id_r)) + %<- ({x} inve C (avar av_x) (AP x) EE) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/ccc/lambda.lf b/new-tests/stelf-output/examples/ccc/lambda.lf new file mode 100644 index 0000000..05b4f68 --- /dev/null +++ b/new-tests/stelf-output/examples/ccc/lambda.lf @@ -0,0 +1,37 @@ +%%% Simply-typed lambda-calculus +%% Lambda-calc language +% types are objects +% obj : type (from ccc.elf) +% simply-typed terms +%sort term %. +%name term %. +% beta-eta convertibility +%sort conv %. +%name conv %. +% products +%term lpair %pi (term A) %-> (term B) %-> (term (A * B)) %. +%term lfst %pi (term (A * B)) %-> (term A) %. +%term lsnd %pi (term (A * B)) %-> (term B) %. +% unit +%term lunit term 1 %. +% function space +%term llam %pi (%pi (term A) %-> (term B)) %-> (term (A => B)) %. +%term lapp %pi (term (A => B)) %-> (term A) %-> (term B) %. +%% Equations +% equivalence +%term c_refl conv E E %. +%term c_trans %pi (conv E E') %-> (conv E' E'') %-> (conv E E'') %. +%term c_sym %pi (conv E E') %-> (conv E' E) %. +% congruence +%term c_fst %pi (conv E E') %-> (conv (lfst E) (lfst E')) %. +%term c_snd %pi (conv E E') %-> (conv (lsnd E) (lsnd E')) %. +%term c_pair %pi (conv E1 E1') %-> (conv E2 E2') %-> (conv (lpair E1 E2) (lpair E1' E2')) %. +%term c_lam %pi ({x} conv (E x) (E' x)) %-> (conv (llam E) (llam E')) %. +%term c_app %pi (conv E1 E1') %-> (conv E2 E2') %-> (conv (lapp E1 E2) (lapp E1' E2')) %. +% proper equations +%term c_unit conv lunit E %. +%term c_prl conv (lfst (lpair E1 E2)) E1 %. +%term c_prr conv (lsnd (lpair E1 E2)) E2 %. +%term c_surj conv (lpair (lfst E) (lsnd E)) E %. +%term c_beta conv (lapp (llam E1) E2) (E1 E2) %. +%term c_eta conv (llam ([x] lapp E x)) E %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/ccc/refl.lf b/new-tests/stelf-output/examples/ccc/refl.lf new file mode 100644 index 0000000..b9c1db1 --- /dev/null +++ b/new-tests/stelf-output/examples/ccc/refl.lf @@ -0,0 +1,13 @@ +% equality reflection: +%sort eqrefl1 %. +%term eqrefl1_ + %pi (eqrefl1 AP AP' EP (c_trans (c_sym CVE) (c_trans (CV MG) CVE'))) + %<- (etoc EP CP CP' ([x] CV x)) + %<- (invca AE CP (%the (exp _ MG) GG) CVE) + %<- (invca AE' CP' GG CVE') %. +%sort eqrefl2 %. +%term eqrefl2_ + %pi (eqrefl2 CP CP' ([x] CVP x) (sym EPF then =@= EE refl then EPF')) + %<- ({x} ctoe (CVP x) (AP x) (AP' x) EE) + %<- (invac CP (zabs1 ([x] AP x)) EPF) + %<- (invac CP' (zabs1 ([x] AP' x)) EPF') %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/ccc/sources.lf b/new-tests/stelf-output/examples/ccc/sources.lf new file mode 100644 index 0000000..81a237a --- /dev/null +++ b/new-tests/stelf-output/examples/ccc/sources.lf @@ -0,0 +1,8 @@ +%require %[ ccc %] %. +%require %[ lambda %] %. +%require %[ catlem %] %. +%require %[ cong %] %. +%require %[ abs-env %] %. +%require %[ conc %] %. +%require %[ eqpres2 %] %. +%require %[ inv1 %] %. diff --git a/new-tests/stelf-output/examples/ccc/spass.lf b/new-tests/stelf-output/examples/ccc/spass.lf new file mode 100644 index 0000000..84e371f --- /dev/null +++ b/new-tests/stelf-output/examples/ccc/spass.lf @@ -0,0 +1 @@ +%require %[ spass %] %. diff --git a/new-tests/stelf-output/examples/ccc/stelf.toml b/new-tests/stelf-output/examples/ccc/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/ccc/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/ccc/subext.lf b/new-tests/stelf-output/examples/ccc/subext.lf new file mode 100644 index 0000000..2c3ac19 --- /dev/null +++ b/new-tests/stelf-output/examples/ccc/subext.lf @@ -0,0 +1,52 @@ +%% Substitution extension lemma +% needs ccc.elf, lambda.elf, abs-env.elf +% subproof used in lambda-case of lemma +%sort svx {A} {Y term A} {GY} {DY} %. +%term svx1 svx Z av_x (avar av_x) prod_r %. +%term svx2 + %pi (svx Z (av_y AV1) (avar (av_y AV2)) (sym ass then =@= refl prod_l then ass then =@= EE refl)) + %<- (Z _ _ _ _ AV1 (avar AV2) EE) %. +%sort subext %. +%term subext_var %pi (subext (avar AV) AP Z E) %<- (Z _ _ _ _ AV AP E) %. +%term subext_unit subext aunit aunit Z term_u %. +%term subext_pair + %pi (subext (apair AP1 AP2) (apair AP1' AP2') Z (DP then =pair= EP1 EP2)) + %<- (distp DP) + %<- (subext AP1 AP1' Z EP1) + %<- (subext AP2 AP2' Z EP2) %. +%term subext_fst + %pi (subext (afst AP) (afst AP') Z (sym ass then =@= refl EP)) + %<- (subext AP AP' Z EP) %. +%term subext_snd + %pi (subext (asnd AP) (asnd AP') Z (sym ass then =@= refl EP)) + %<- (subext AP AP' Z EP) %. +%term subext_lam + %pi (subext (alam ([x] AP x)) (alam ([x] AP' x)) Z (DC then =cur= EP)) + %<- (distc DC) + %<- ({x} subext (AP x) (AP' x) ([_] [_] [_] [_] svx Z) EP) %. +%term subext_app + %pi (subext (aapp AP1 AP2) (aapp AP1' AP2') Z (sym ass then =@= refl (DP then =pair= EP1 EP2))) + %<- (distp DP) + %<- (subext AP1 AP1' Z EP1) + %<- (subext AP2 AP2' Z EP2) %. +% Corollary: weakening +%sort weakv %. +%term weakv1 weakv AV (avar (av_y AV)) refl %. +%sort weak %. +%term weak1 %pi (weak AP AP1 E) %<- (subext AP AP1 ([_] [_] [_] [_] weakv) E) %. +% Corollary : substitution +%term svx2' + %pi (svx Z (av_y AV) AP2 (sym ass then =@= refl prod_l then ass then =@= EE refl then WE)) + %<- (Z _ _ _ _ AV AP EE) + %<- (weak AP AP2 WE) %. +%sort subv %. +%term subv1 subv AP av_x AP prod_r %. +%term subv2 subv AP (av_y AV) (avar AV) (sym ass then =@= refl prod_l then id_r) %. +%sort subst %. +% Too much magic here... +%term subst1 + %pi (subst AP1 ([x] AP2 x) AP E) + %<- (subext (AP2 _) AP ([_] [_] [_] [_] subv AP1) E) %. +%%% tests +% {x}sigma[p1:{y}absv (addv (addv empty x) y) x M1]sigma[p2:absv (addv empty x) x M2]{y}weakv (p1 y) (p2) E. --> undecl cst x ??? +% [w,x,y |- let z=fst(x) in (z,w)] == [w,x |- let z=fst(x) in (z,w)] @ fst \ No newline at end of file diff --git a/new-tests/stelf-output/examples/ccc/test.lf b/new-tests/stelf-output/examples/ccc/test.lf new file mode 100644 index 0000000..3267f3d --- /dev/null +++ b/new-tests/stelf-output/examples/ccc/test.lf @@ -0,0 +1,10 @@ +%require %[ ccc %] %. +%require %[ lambda %] %. +%require %[ catlem %] %. +%require %[ cong %] %. +%require %[ abs-env %] %. +%require %[ conc %] %. +%require %[ conc %] %. +%require %[ eqpres2 %] %. +%require %[ inv1 %] %. +%require %[ examples %] %. diff --git a/new-tests/stelf-output/examples/church-rosser/church-rosser-thm.lf b/new-tests/stelf-output/examples/church-rosser/church-rosser-thm.lf new file mode 100644 index 0000000..603122e --- /dev/null +++ b/new-tests/stelf-output/examples/church-rosser/church-rosser-thm.lf @@ -0,0 +1,37 @@ +%theorem m-append : forall* {M term} {M' term} {M'' term} forall {D1 M =>* M'} {D2 M' =>* M''} exists {E M =>* M''} true %. +%prove 4 [D1 D2] (m-append D1 D2 _) %. +%{ +%theorem m-identity : forall {M:term} + exists {D: M => M} + true. +%assert (m-identity _ _). +}% +%theorem m-subst : forall* {M %pi term %-> term} {M' %pi term %-> term} {N term} {N' term} forall {D1 {x term} %pi (x => x) %-> (M x => M' x)} {D2 N => N'} exists {E M N => M' N'} true %. +%assert (m-subst _ _ _) %. +%theorem m-dia : forallG (pi {x term} {idx x => x}) forall* {M term} {M' term} {M'' term} forall {D1 M => M'} {D2 M => M''} exists {N term} {E1 M' => N} {E2 M'' => N} true %. +%prove 4 [D1 D2] (m-dia D1 D2 _ _ _) %. +%{ +%assert (m-dia _ _ _ _ _). + +%theorem m-strip : forall* {M:term}{M':term}{M'':term} + forall {D1: M => M'} {D2: M =>* M''} + exists {N:term} {E1: M' =>* N} {E2: M'' => N} + true. +% %prove 7 [D2] (m-strip _ D2 _ _ _). + +% doesn't work because we need residual lemmas +%assert (m-strip _ _ _ _ _). + +%theorem m-conf : forall* {M:term} {M':term} {M'':term} + forall {R*':M =>* M'} {R*'':M =>* M''} + exists {N:term} {S*':M' =>* N} {S*'':M'' =>* N} + true. +% %prove 4 D1 (m-conf D1 _ _ _ _). +%assert (m-conf _ _ _ _ _). + +%theorem m-cr : forall* {M:term} {M':term} + forall {C:M <=> M'} + exists {N:term} {S*:M =>* N} {S*':M' =>* N} + true. +%prove 4 D (m-cr D _ _ _). +}% \ No newline at end of file diff --git a/new-tests/stelf-output/examples/church-rosser/cr-thm.lf b/new-tests/stelf-output/examples/church-rosser/cr-thm.lf new file mode 100644 index 0000000..3c880f1 --- /dev/null +++ b/new-tests/stelf-output/examples/church-rosser/cr-thm.lf @@ -0,0 +1,28 @@ +%%% Termination of the theorems in the +%%% Church-Rosser proof. +% par-lemmas +%terminates M (identity M _) %. +%terminates R* (append R* S* S*') %. +% par-cr +%terminates R (subst R S S') %. +%terminates [R' R''] (dia R' R'' S' S'') %. +%terminates R*'' (strip R' R*'' S*' S'') %. +%terminates R*' (conf R*' R*'' S*' S*'') %. +%terminates C (cr C S* S*') %. +% ord-lemmas +%terminates R (appd R S* S*') %. +%terminates R* (lm1* R* S*) %. +%terminates R* (apl1* R* S*) %. +%terminates R* (apr1* R* S*) %. +% equiv +%terminates R (eq1 R S*) %. +%terminates R (eq2 R S) %. +%terminates R* (eq3 R* S*) %. +%terminates R* (eq4 R* S*) %. +%terminates C (eq5 C C') %. +% %terminates C' (eq5 C C'). % should fail +%terminates C (sym_pconv C C') %. +% %terminates C' (sym_pconv C C'). % should fails with modes +%terminates C (eq6 C C') %. +% ord-cr +%terminates {} (cr_ord C S* S*') %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/church-rosser/equiv.lf b/new-tests/stelf-output/examples/church-rosser/equiv.lf new file mode 100644 index 0000000..6598800 --- /dev/null +++ b/new-tests/stelf-output/examples/church-rosser/equiv.lf @@ -0,0 +1,107 @@ +%%% Equivalence of ordinary and parallel reduction. +%%% Author: Frank Pfenning +% If M => N then M -->* N. +%sort eq1 %. +%mode eq1 %in %out %. +%term eq1_beta + %pi (eq1 (beta R1 R2) S*) + %<- ({x term} {eqx x => x} %pi (eq1 eqx id1) %-> (eq1 (R1 x eqx) (S1* x))) + %<- (lm1* S1* S1*') + %<- (apl1* S1*' S1*'') + %<- (eq1 R2 S2*) + %<- (apr1* S2* S2*') + %<- (appd S2*' (step1 beta1 id1) S*') + %<- (appd S1*'' S*' S*) %. +%term eq1_ap + %pi (eq1 (ap R1 R2) S*) + %<- (eq1 R1 S1*) + %<- (apl1* S1* S*') + %<- (eq1 R2 S2*) + %<- (apr1* S2* S*'') + %<- (appd S*' S*'' S*) %. +%term eq1_lm + %pi (eq1 (lm R1) S*) + %<- ({x term} {eqx x => x} %pi (eq1 eqx id1) %-> (eq1 (R1 x eqx) (S1* x))) + %<- (lm1* S1* S*) %. +%block leq1 [x term] [eqx x => x] [e eq1 eqx id1]%. +%worlds (leq1) (eq1 R S*) %. +%covers eq1 %in %out %. +%terminates R (eq1 R _) %. +%total R (eq1 R _) %. +% If M --> N then M => N. +%sort eq2 %. +%mode eq2 %in %out %. +%term eq2_beta1 + %pi (eq2 beta1 (beta I1 I2)) + %<- ({x term} {eqx x => x} %pi (identity x eqx) %-> (identity (M1 x) (I1 x eqx))) + %<- (identity M2 I2) %. +% bug here in previous version +%term eq2_lm1 + %pi (eq2 (lm1 R1) (lm S1)) + %<- ({x term} {eqx x => x} %pi (identity x eqx) %-> (eq2 (R1 x) (S1 x eqx))) %. +%term eq2_apl1 %pi (eq2 (apl1 R1) (ap S1 I2)) %<- (eq2 R1 S1) %<- (identity M2 I2) %. +%term eq2_apr1 %pi (eq2 (apr1 R2) (ap I1 S2)) %<- (eq2 R2 S2) %<- (identity M1 I1) %. +%worlds (lid) (eq2 R S) %. +%covers eq2 %in %out %. +%terminates R (eq2 R _) %. +%total R (eq2 R _) %. +% If M -->* N then M =>* N. +%sort eq3 %. +%mode eq3 %in %out %. +%term eq3_id eq3 id1 id %. +%term eq3_step %pi (eq3 (step1 R1 R2*) (S1 ; S2*)) %<- (eq2 R1 S1) %<- (eq3 R2* S2*) %. +%worlds () (eq3 R* S*) %. +%covers eq3 %in %out %. +%terminates R* (eq3 R* _) %. +%total R* (eq3 R* _) %. +% If M =>* N then M -->* N. +%sort eq4 %. +%mode eq4 %in %out %. +%term eq4_id eq4 id id1 %. +%term eq4_step %pi (eq4 (R1 ; R2*) S*) %<- (eq1 R1 S1*) %<- (eq4 R2* S2*) %<- (appd S1* S2* S*) %. +%worlds () (eq4 R* S*) %. +%covers eq4 %in %out %. +%terminates R* (eq4 R* _) %. +%total R* (eq4 R* _) %. +% If M <=> N then M <-> N. +%sort eq5 %. +%mode eq5 %in %out %. +%term eq5_red %pi (eq5 (reduce R*) (red S*)) %<- (eq4 R* S*) %. +%term eq5_exp %pi (eq5 (expand R*) (sym (red S*))) %<- (eq4 R* S*) %. +% bug here (trans C1' C2') instead of (trans C2' C1') +% Fri Dec 14 19:03:57 2001 -fp !!! +%term eq5_trans %pi (eq5 (C1 ;; C2) (trans C2' C1')) %<- (eq5 C1 C1') %<- (eq5 C2 C2') %. +%worlds () (eq5 C C') %. +%covers eq5 %in %out %. +%terminates C (eq5 C _) %. +%total C (eq5 C _) %. +% If M <=> N then N <=> M. +%sort sym_pconv %. +%mode sym_pconv %in %out %. +%term spc_red sym_pconv (reduce R*) (expand R*) %. +%term spc_exp sym_pconv (expand R*) (reduce R*) %. +%term spc_trans + %pi (sym_pconv (C1 ;; C2) (C2' ;; C1')) + %<- (sym_pconv C1 C1') + %<- (sym_pconv C2 C2') %. +%worlds () (sym_pconv C C') %. +%covers sym_pconv %in %out %. +%terminates C (sym_pconv C _) %. +%total C (sym_pconv C _) %. +% The following is a bug not detected by term reconstruction +% spc_wrong : sym_pconv (C1 ;; C2) (C1' ;; C2') +% <- sym_pconv C1 C1' +% <- sym_pconv C2 C2'. +% If M <-> N then M <=> N. +%sort eq6 %. +%mode eq6 %in %out %. +%term eq6_refl eq6 refl (reduce id) %. +%term eq6_sym %pi (eq6 (sym C1) C') %<- (eq6 C1 C1') %<- (sym_pconv C1' C') %. +% Bug here: (trans C1 C2) +% Fri Dec 14 18:50:34 2001 -fp !!! +%term eq6_trans %pi (eq6 (trans C2 C1) (C1' ;; C2')) %<- (eq6 C1 C1') %<- (eq6 C2 C2') %. +%term eq6_red %pi (eq6 (red R*) (reduce S*)) %<- (eq3 R* S*) %. +%worlds () (eq6 C C') %. +%covers eq6 %in %out %. +%terminates C (eq6 C _) %. +%total C (eq6 C _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/church-rosser/examples-quy.lf b/new-tests/stelf-output/examples/church-rosser/examples-quy.lf new file mode 100644 index 0000000..770fb82 --- /dev/null +++ b/new-tests/stelf-output/examples/church-rosser/examples-quy.lf @@ -0,0 +1,34 @@ +% Ordinary reduction. +%query 2 _ _ app (lam ([x] app x x)) (app (lam ([y] y)) (lam ([z] z))) --> M' %. +% Parallel reduction. +%query 4 _ _ app (lam ([x] app x x)) (app (lam ([y] y)) (lam ([z] z))) => M' %. +% Diamond property for two different parallel reductions +% out of the 4 above +%{ +%query 1 * +dia (beta ([x:term] [R4:x => x] ap R4 R4) + (ap (lm ([x:term] [R5:x => x] R5)) (lm ([x:term] [R6:x => x] R6)))) + (ap (lm ([x:term] [R7:x => x] ap R7 R7)) + (beta ([x:term] [R8:x => x] R8) (lm ([x:term] [R9:x => x] R9)))) + S' S''. +}% +%{ +% Ordinary reduction. + +R : (app (lam [x] (app x x)) (app (lam [y] y) (lam [z] z))) --> M'. + +% Parallel reduction. + +R : (app (lam [x] (app x x)) (app (lam [y] y) (lam [z] z))) => M'. + +((beta ([x:term] [R:x => x] ap R R) + (beta ([x:term] [R:x => x] R) (lm ([x:term] [R:x => x] R)))) + ; beta ([x] [R] R) (lm [x] [R] R) + ; id) : M =>* M'. + +% The diamond property. + +sigma [R' : (app (lam [x] (app x x)) (app (lam [y] y) (lam [z] z))) => M'] +sigma [R'' : (app (lam [x] (app x x)) (app (lam [y] y) (lam [z] z))) => M''] +dia R' R'' (S' : M' => N) (S'' : M'' => N). +}% \ No newline at end of file diff --git a/new-tests/stelf-output/examples/church-rosser/lam.lf b/new-tests/stelf-output/examples/church-rosser/lam.lf new file mode 100644 index 0000000..48fee73 --- /dev/null +++ b/new-tests/stelf-output/examples/church-rosser/lam.lf @@ -0,0 +1,6 @@ +%%% Untyped lambda-calculus +%%% Author: Frank Pfenning +%sort term %. +%name term %. +%term lam %pi (%pi term %-> term) %-> term %. +%term app %pi term %-> term %-> term %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/church-rosser/ord-cr.lf b/new-tests/stelf-output/examples/church-rosser/ord-cr.lf new file mode 100644 index 0000000..a248eaa --- /dev/null +++ b/new-tests/stelf-output/examples/church-rosser/ord-cr.lf @@ -0,0 +1,16 @@ +%%% The Church-Rosser theorem for ordinary reduction +%%% Author: Frank Pfenning +%sort cr_ord %. +% %mode +{M} +{M'} +{C:M <-> M'} +% -{N} -{S*:M -->* N} -{S*':M' -->* N} cr_ord C S* S*'. +%mode cr_ord %in %out %out %. +%term cr_all + %pi (cr_ord C S* S*') + %<- (eq6 C C') + %<- (cr C' R* R*') + %<- (eq4 R* S*) + %<- (eq4 R*' S*') %. +%worlds () (cr_ord C S* S*') %. +%covers cr_ord %in %out %out %. +%terminates C (cr_ord C _ _) %. +%total C (cr_ord C _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/church-rosser/ord-lemmas.lf b/new-tests/stelf-output/examples/church-rosser/ord-lemmas.lf new file mode 100644 index 0000000..a1da43e --- /dev/null +++ b/new-tests/stelf-output/examples/church-rosser/ord-lemmas.lf @@ -0,0 +1,43 @@ +%%% Lemmas concerning ordinary multi-step reduction +%%% Author: Frank Pfenning +% Terms can contain free variables without explicitl +% assumptions about them (for multi-step reduction) +%block ltm [x term]%. +% Transitivity of multi-step reduction +%sort appd %. +%mode appd %in %in %out %. +%term appd_id appd id1 S* S* %. +%term appd_step %pi (appd (step1 R1 R2*) S* (step1 R1 S2*')) %<- (appd R2* S* S2*') %. +%worlds (ltm) (appd R* S* S*') %. +%covers appd %in %in %out %. +%terminates R* (appd R* S* _) %. +%total R* (appd R* S* _) %. +% Multi-step reduction is a congruence +%sort lm1* {x term} %. +%mode {%in M} {%in M'} {%in R* {x term} M x -->* M' x} {%out S* lam M -->* lam M'} lm1* R* S* %. +%term lm1*_id lm1* ([x term] id1) id1 %. +%term lm1*_step + %pi (lm1* ([x term] step1 (R1 x) (R2* x)) (step1 (lm1 R1) S2*)) + %<- (lm1* R2* S2*) %. +%worlds (ltm) (lm1* R* S*) %. +%covers lm1* %in %out %. +%terminates R* (lm1* R* _) %. +%total R* (lm1* R* _) %. +%sort apl1* %. +% poor error message below. -fp +% %mode +{M1} +{M1'} +{M2} +{R*} -{S*} apl1* R* S. +%mode {%in M1} {%in M1'} {%in M2} {%in R* M1 -->* M1'} {%out S* app M1 M2 -->* app M1' M2} apl1* R* S* %. +%term apl1*_id apl1* id1 id1 %. +%term apl1*_step %pi (apl1* (step1 R1 R2*) (step1 (apl1 R1) S2*)) %<- (apl1* R2* S2*) %. +%worlds (ltm) (apl1* R* S*) %. +%covers {%in M1} {%in M1'} {%in M2} {%in R* M1 -->* M1'} {%out S* app M1 M2 -->* app M1' M2} apl1* R* S* %. +%terminates R* (apl1* R* _) %. +%total R* (apl1* R* _) %. +%sort apr1* %. +%mode {%in M2} {%in M2'} {%in M1} {%in R* M2 -->* M2'} {%out S* app M1 M2 -->* app M1 M2'} apr1* R* S* %. +%term apr1*_id apr1* id1 id1 %. +%term apr1*_step %pi (apr1* (step1 R1 R2*) (step1 (apr1 R1) S2*)) %<- (apr1* R2* S2*) %. +%worlds (ltm) (apr1* R* S*) %. +%covers {%in M2} {%in M2'} {%in M1} {%in R* M2 -->* M2'} {%out S* app M1 M2 -->* app M1 M2'} apr1* R* S* %. +%terminates R* (apr1* R* _) %. +%total R* (apr1* R* _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/church-rosser/ord-red.lf b/new-tests/stelf-output/examples/church-rosser/ord-red.lf new file mode 100644 index 0000000..36f6d12 --- /dev/null +++ b/new-tests/stelf-output/examples/church-rosser/ord-red.lf @@ -0,0 +1,26 @@ +%%% Ordinary reduction for the untyped lambda-calculus +%%% Author: Frank Pfenning +%scope app %sort -> %. +%prec %none 10 --> %. +%name --> %. +%term beta1 app (lam M1) M2 --> M1 M2 %. +%term lm1 %pi ({x term} M x --> M' x) %-> (lam M --> (lam M')) %. +%term apl1 %pi (M1 --> M1') %-> (app M1 M2 --> (app M1' M2)) %. +%term apr1 %pi (M2 --> M2') %-> (app M1 M2 --> (app M1 M2')) %. +% Multi-step reduction +%scope apr1 %sort ->* %. +%prec %none 10 -->* %. +%name -->* %. +%term id1 M -->* M %. +% step1 : M' --> M'' -> M -->* M' -> M -->* M''. +% Bug here: +% Fri Dec 14 18:37:26 2001 -fp !!! +%term step1 %pi (M --> M') %-> (M' -->* M'') %-> (M -->* M'') %. +% Conversion +%sort <-> %. +%prec %none 10 <-> %. +%name <-> %. +%term refl M <-> M %. +%term sym %pi (M <-> M') %<- (M' <-> M) %. +%term trans %pi (M <-> M'') %<- (M <-> M') %<- (M' <-> M'') %. +%term red %pi (M <-> M') %<- (M -->* M') %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/church-rosser/par-cr.lf b/new-tests/stelf-output/examples/church-rosser/par-cr.lf new file mode 100644 index 0000000..665d967 --- /dev/null +++ b/new-tests/stelf-output/examples/church-rosser/par-cr.lf @@ -0,0 +1,120 @@ +%%% The Church-Rosser theorem for parallel reduction +%%% Author: Frank Pfenning +% Substitution lemma for parallel reduction +%sort subst {x term} %. +%mode subst %in %in %out %. +%term subst_idx subst ([x term] [idx x => x] idx) S S %. +%term subst_beta + %pi (subst ([x term] [idx x => x] beta (R1 x idx) (R2 x idx)) S (beta R1' R2')) + %<- ({y term} {idy y => y} + %pi ({N} {N'} {S' N => N'} subst ([x term] [idx x => x] idy) S' idy) + %-> (subst ([x term] [idx x => x] R1 x idx y idy) S (R1' y idy))) + %<- (subst R2 S R2') %. +%term subst_ap + %pi (subst ([x term] [idx x => x] ap (R1 x idx) (R2 x idx)) S (ap R1' R2')) + %<- (subst R1 S R1') + %<- (subst R2 S R2') %. +%term subst_lm + %pi (subst ([x term] [idx x => x] lm (R1 x idx)) S (lm R1')) + %<- ({y term} {idy y => y} + %pi ({N} {N'} {S' N => N'} subst ([x term] [idx x => x] idy) S' idy) + %-> (subst ([x term] [idx x => x] R1 x idx y idy) S (R1' y idy))) %. +%block l1 [x term] [idx x => x] [hs {N term} {N' term} {S' N => N'} subst ([y term] [idy y => y] idx) S' idx]%. +%worlds (l1) (subst R S R') %. +%covers subst %in %in %out %. +%terminates R (subst R S _) %. +%total R (subst R S _) %. +% Diamond property for parallel reduction +%sort dia %. +% %mode +{M} +{M'} +{M''} +{R':M => M'} +{R'':M => M''} +% -{N} -{S':M' => N} -{S'':M'' => N} dia R' R'' S' S''. +%mode dia %in %in %out %out %. +% Proof by induction on the structure of the first two derivations. +% We consider the various possible cases. +% b = beta, a = ap, l = lm, +%term dia_bb + %pi (dia (beta R1' R2') (beta R1'' R2'') S' S'') + %<- ({x term} {idx x => x} + %pi ({N} {N'} {S' N => N'} subst ([y term] [idy y => y] idx) S' idx) + %-> (dia idx idx idx idx) + %-> (dia (R1' x idx) (R1'' x idx) (S1' x idx) (S1'' x idx))) + %<- (dia R2' R2'' S2' S2'') + %<- (subst S1' S2' S') + %<- (subst S1'' S2'' S'') %. +%term dia_bal + %pi (dia (beta R1' R2') (ap (lm R1'') R2'') S' (beta S1'' S2'')) + %<- ({x term} {idx x => x} + %pi ({N} {N'} {S' N => N'} subst ([y term] [idy y => y] idx) S' idx) + %-> (dia idx idx idx idx) + %-> (dia (R1' x idx) (R1'' x idx) (S1' x idx) (S1'' x idx))) + %<- (dia R2' R2'' S2' S2'') + %<- (subst S1' S2' S') %. +%term dia_alb + %pi (dia (ap (lm R1') R2') (beta R1'' R2'') (beta S1' S2') S'') + %<- ({x term} {idx x => x} + %pi ({N} {N'} {S' N => N'} subst ([y term] [idy y => y] idx) S' idx) + %-> (dia idx idx idx idx) + %-> (dia (R1' x idx) (R1'' x idx) (S1' x idx) (S1'' x idx))) + %<- (dia R2' R2'' S2' S2'') + %<- (subst S1'' S2'' S'') %. +%term dia_aa + %pi (dia (ap R1' R2') (ap R1'' R2'') (ap S1' S2') (ap S1'' S2'')) + %<- (dia R1' R1'' S1' S1'') + %<- (dia R2' R2'' S2' S2'') %. +%term dia_ll + %pi (dia (lm R1') (lm R1'') (lm S1') (lm S1'')) + %<- ({x term} {idx x => x} + %pi ({N} {N'} {S' N => N'} subst ([y term] [idy y => y] idx) S' idx) + %-> (dia idx idx idx idx) + %-> (dia (R1' x idx) (R1'' x idx) (S1' x idx) (S1'' x idx))) %. +%block l2 [x term] [idx x => x] [hs {N term} {N' term} {S' N => N'} subst ([y term] [idy y => y] idx) S' idx] [hd dia idx idx idx idx]%. +%worlds (l2) (dia _ _ _ _) %. +%covers dia %in %in %out %out %. +%terminates [R' R''] (dia R' R'' _ _) %. +%total [R' R''] (dia R' R'' _ _) %. +% The strip lemma for parallel reduction. +%sort strip %. +% %mode +{M} +{M'} +{M''} +{R':M => M'} +{R'':M =>* M''} +% -{N} -{S*':M' =>* N} -{S'':M'' => N} strip R' R'' S*' S''. +%mode strip %in %in %out %out %. +%term strip_id strip R' id id R' %. +%term strip_step + %pi (strip R' (R1'' ; R2*'') (S1' ; S2*') S'') + %<- (dia R' R1'' S1' S1'') + %<- (strip S1'' R2*'' S2*' S'') %. +%worlds () (strip R' R'' S*' S'') %. +%covers strip %in %in %out %out %. +%terminates R'' (strip R' R'' _ _) %. +%total R'' (strip R' R'' _ _) %. +% Confluence for parallel multi-step reduction. +%sort conf %. +% %mode +{M} +{M'} +{M''} +{R*':M =>* M'} +{R*'':M =>* M''} +% -{N} -{S*':M' =>* N} -{S*'':M'' =>* N} conf R*' R*'' S*' S*''. +%mode conf %in %in %out %out %. +%term conf_id conf id R*'' R*'' id %. +%term conf_step + %pi (conf (R1' ; R2*') R*'' S*' (S1'' ; S2*'')) + %<- (strip R1' R*'' S1*' S1'') + %<- (conf R2*' S1*' S*' S2*'') %. +%worlds () (conf R*' R*'' S*' S*'') %. +%covers conf %in %in %out %out %. +%terminates R*' (conf R*' R*'' _ _) %. +%total R*' (conf R*' R*'' _ _) %. +% Church-Rosser Theorem for parallel reduction +%sort cr %. +% %mode +{M} +{M'} +{C:M <=> M'} +% -{N} -{S*:M =>* N} -{S*':M' =>* N} cr C S* S*'. +%mode cr %in %out %out %. +%term cr_reduce cr (reduce R*) R* id %. +%term cr_expand cr (expand R*) id R* %. +%term cr_compose + %pi (cr (C1 ;; C2) S* S*') + %<- (cr C1 S1* R1*) + %<- (cr C2 R2* S2*) + %<- (conf R1* R2* T1* T2*) + %<- (append S1* T1* S*) + %<- (append S2* T2* S*') %. +%worlds () (cr C S* S*') %. +%covers cr %in %out %out %. +%terminates C (cr C _ _) %. +%total C (cr C _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/church-rosser/par-lemmas.lf b/new-tests/stelf-output/examples/church-rosser/par-lemmas.lf new file mode 100644 index 0000000..1e5339f --- /dev/null +++ b/new-tests/stelf-output/examples/church-rosser/par-lemmas.lf @@ -0,0 +1,23 @@ +%%% Basic lemmas concerning parallel reductions +%%% Author: Frank Pfenning +% Every term reduces to itself (in parallel) +%sort identity {M term} %. +%mode identity %in %out %. +%term id_lam + %pi (identity (lam M1) (lm R1)) + %<- ({x term} {eqx x => x} %pi (identity x eqx) %-> (identity (M1 x) (R1 x eqx))) %. +%term id_app %pi (identity (app M1 M2) (ap R1 R2)) %<- (identity M1 R1) %<- (identity M2 R2) %. +%block lid [x term] [eqx x => x] [u identity x eqx]%. +%worlds (lid) (identity M R) %. +%covers identity %in %out %. +%terminates M (identity M _) %. +%total M (identity M _) %. +% Parallel multi-step reduction is transitive. +%sort append %. +%mode append %in %in %out %. +%term append_id append id S* S* %. +%term append_step %pi (append (R1 ; R2*) S* (R1 ; S2*')) %<- (append R2* S* S2*') %. +%worlds () (append R* S* S*') %. +%covers append %in %in %out %. +%terminates R* (append R* S* _) %. +%total R* (append R* S* _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/church-rosser/par-red.lf b/new-tests/stelf-output/examples/church-rosser/par-red.lf new file mode 100644 index 0000000..81606c4 --- /dev/null +++ b/new-tests/stelf-output/examples/church-rosser/par-red.lf @@ -0,0 +1,26 @@ +%%% Parallel reduction in the untyped lambda calculus +%%% Author: Frank Pfenning +%sort => %. +%prec %none 10 => %. +%name => %. +%term beta + %pi ({x term} %pi (x => x) %-> (M1 x => M1' x)) + %-> (M2 => M2') + %-> (app (lam M1) M2 => M1' M2') %. +%term ap %pi (M1 => M1') %-> (M2 => M2') %-> (app M1 M2 => (app M1' M2')) %. +%term lm %pi ({x term} %pi (x => x) %-> (M x => M' x)) %-> (lam M => lam M') %. +% Parallel, multi-step reduction +%sort =>* %. +%prec %none 10 =>* %. +%name =>* %. +%term id M =>* M %. +%term ; %pi (M => M') %-> (M' =>* M'') %-> (M =>* M'') %. +%prec %right 10 ; %. +% Parallel conversion +%sort <=> %. +%prec %none 10 <=> %. +%name <=> %. +%term reduce %pi (M =>* M') %-> (M <=> M') %. +%term expand %pi (M =>* M') %-> (M' <=> M) %. +%term ;; %pi (M <=> M') %-> (M' <=> M'') %-> (M <=> M'') %. +%prec %none 8 ;; %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/church-rosser/simplified/par-cr.lf b/new-tests/stelf-output/examples/church-rosser/simplified/par-cr.lf new file mode 100644 index 0000000..0c7a0a3 --- /dev/null +++ b/new-tests/stelf-output/examples/church-rosser/simplified/par-cr.lf @@ -0,0 +1,114 @@ +%%% The Church-Rosser theorem for parallel reduction +%%% Author: Frank Pfenning +%%% Modified by Adam Poswolsky so subst creates a parametric function that can be used to +%%%% perform the substitution in dia. +%sort subst {x term} %. +%mode subst %in %out %. +%term subst_idx subst ([x term] [idx x => x] idx) ([x] [y] [r] r) %. +%term subst_beta + %pi (subst ([x term] [idx x => x] beta (R1 x idx) (R2 x idx)) ([x] [y] [r x => y] beta (R1' x y r) (R2' x y r))) + %<- ({z} {idz} subst ([x] [idx x => x] R1 x idx z idz) ([x] [y] [r] R1' x y r z idz)) + %<- (subst R2 R2') %. +%term subst_ap + %pi (subst ([x term] [idx x => x] ap (R1 x idx) (R2 x idx)) ([x] [y] [r] ap (R1' x y r) (R2' x y r))) + %<- (subst R1 R1') + %<- (subst R2 R2') %. +%term subst_lm + %pi (subst ([x term] [idx x => x] lm (R1 x idx)) ([x] [y] [r] lm (R1' x y r))) + %<- ({z} {idz} subst ([x] [idx x => x] R1 x idx z idz) ([x] [y] [r] R1' x y r z idz)) %. +%term subst_lmBase subst ([x] [idx] R) ([x] [y] [r] R) %. +%block l1 [x term] [idx x => x]%. +%worlds (l1) (subst R R') %. +%covers subst %in %out %. +%terminates R (subst R _) %. +%total R (subst R _) %. +% Diamond property for parallel reduction +%sort dia %. +% %mode +{M} +{M'} +{M''} +{R':M => M'} +{R'':M => M''} +% -{N} -{S':M' => N} -{S'':M'' => N} dia R' R'' S' S''. +%mode dia %in %in %out %out %. +% Proof by induction on the structure of the first two derivations. +% We consider the various possible cases. +% b = beta, a = ap, l = lm, +%term dia_bb + %pi (dia (beta R1' R2') (beta R1'' R2'') (S1'G _ _ S2') (S1''G _ _ S2'')) + %<- ({x term} {idx x => x} + %pi (dia idx idx idx idx) + %-> (dia (R1' x idx) (R1'' x idx) (S1' x idx) (S1'' x idx))) + %<- (dia R2' R2'' S2' S2'') + %<- (subst S1' S1'G) + %<- (subst S1'' S1''G) %. +%term dia_bal + %pi (dia (beta R1' R2') (ap (lm R1'') R2'') (S1'G _ _ S2') (beta S1'' S2'')) + %<- ({x term} {idx x => x} + %pi (dia idx idx idx idx) + %-> (dia (R1' x idx) (R1'' x idx) (S1' x idx) (S1'' x idx))) + %<- (dia R2' R2'' S2' S2'') + %<- (subst S1' S1'G) %. +%term dia_alb + %pi (dia (ap (lm R1') R2') (beta R1'' R2'') (beta S1' S2') (S1''G _ _ S2'')) + %<- ({x term} {idx x => x} + %pi (dia idx idx idx idx) + %-> (dia (R1' x idx) (R1'' x idx) (S1' x idx) (S1'' x idx))) + %<- (dia R2' R2'' S2' S2'') + %<- (subst S1'' S1''G) %. +%term dia_aa + %pi (dia (ap R1' R2') (ap R1'' R2'') (ap S1' S2') (ap S1'' S2'')) + %<- (dia R1' R1'' S1' S1'') + %<- (dia R2' R2'' S2' S2'') %. +%term dia_ll + %pi (dia (lm R1') (lm R1'') (lm S1') (lm S1'')) + %<- ({x term} {idx x => x} + %pi (dia idx idx idx idx) + %-> (dia (R1' x idx) (R1'' x idx) (S1' x idx) (S1'' x idx))) %. +%block l2 [x term] [idx x => x] [hd dia idx idx idx idx]%. +%worlds (l2) (dia _ _ _ _) %. +%covers dia %in %in %out %out %. +%terminates [R' R''] (dia R' R'' _ _) %. +%total [R' R''] (dia R' R'' _ _) %. +% The strip lemma for parallel reduction. +%sort strip %. +% %mode +{M} +{M'} +{M''} +{R':M => M'} +{R'':M =>* M''} +% -{N} -{S*':M' =>* N} -{S'':M'' => N} strip R' R'' S*' S''. +%mode strip %in %in %out %out %. +%term strip_id strip R' id id R' %. +%term strip_step + %pi (strip R' (R1'' ; R2*'') (S1' ; S2*') S'') + %<- (dia R' R1'' S1' S1'') + %<- (strip S1'' R2*'' S2*' S'') %. +%worlds () (strip R' R'' S*' S'') %. +%covers strip %in %in %out %out %. +%terminates R'' (strip R' R'' _ _) %. +%total R'' (strip R' R'' _ _) %. +% Confluence for parallel multi-step reduction. +%sort conf %. +% %mode +{M} +{M'} +{M''} +{R*':M =>* M'} +{R*'':M =>* M''} +% -{N} -{S*':M' =>* N} -{S*'':M'' =>* N} conf R*' R*'' S*' S*''. +%mode conf %in %in %out %out %. +%term conf_id conf id R*'' R*'' id %. +%term conf_step + %pi (conf (R1' ; R2*') R*'' S*' (S1'' ; S2*'')) + %<- (strip R1' R*'' S1*' S1'') + %<- (conf R2*' S1*' S*' S2*'') %. +%worlds () (conf R*' R*'' S*' S*'') %. +%covers conf %in %in %out %out %. +%terminates R*' (conf R*' R*'' _ _) %. +%total R*' (conf R*' R*'' _ _) %. +% Church-Rosser Theorem for parallel reduction +%sort cr %. +% %mode +{M} +{M'} +{C:M <=> M'} +% -{N} -{S*:M =>* N} -{S*':M' =>* N} cr C S* S*'. +%mode cr %in %out %out %. +%term cr_reduce cr (reduce R*) R* id %. +%term cr_expand cr (expand R*) id R* %. +%term cr_compose + %pi (cr (C1 ;; C2) S* S*') + %<- (cr C1 S1* R1*) + %<- (cr C2 R2* S2*) + %<- (conf R1* R2* T1* T2*) + %<- (append S1* T1* S*) + %<- (append S2* T2* S*') %. +%worlds () (cr C S* S*') %. +%covers cr %in %out %out %. +%terminates C (cr C _ _) %. +%total C (cr C _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/church-rosser/simplified/sources.lf b/new-tests/stelf-output/examples/church-rosser/simplified/sources.lf new file mode 100644 index 0000000..61319fc --- /dev/null +++ b/new-tests/stelf-output/examples/church-rosser/simplified/sources.lf @@ -0,0 +1,8 @@ +%require %[ ../lam %] %. +%require %[ ../ord-red %] %. +%require %[ ../par-red %] %. +%require %[ ../par-lemmas %] %. +%require %[ par-cr %] %. +%require %[ ../ord-lemmas %] %. +%require %[ ../equiv %] %. +%require %[ ../ord-cr %] %. diff --git a/new-tests/stelf-output/examples/church-rosser/simplified/stelf.toml b/new-tests/stelf-output/examples/church-rosser/simplified/stelf.toml new file mode 100644 index 0000000..21c1db1 --- /dev/null +++ b/new-tests/stelf-output/examples/church-rosser/simplified/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["..", "."] diff --git a/new-tests/stelf-output/examples/church-rosser/sources.lf b/new-tests/stelf-output/examples/church-rosser/sources.lf new file mode 100644 index 0000000..6b071e1 --- /dev/null +++ b/new-tests/stelf-output/examples/church-rosser/sources.lf @@ -0,0 +1,8 @@ +%require %[ lam %] %. +%require %[ ord-red %] %. +%require %[ par-red %] %. +%require %[ par-lemmas %] %. +%require %[ par-cr %] %. +%require %[ ord-lemmas %] %. +%require %[ equiv %] %. +%require %[ ord-cr %] %. diff --git a/new-tests/stelf-output/examples/church-rosser/stelf.toml b/new-tests/stelf-output/examples/church-rosser/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/church-rosser/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/church-rosser/test-unsafe.lf b/new-tests/stelf-output/examples/church-rosser/test-unsafe.lf new file mode 100644 index 0000000..389002b --- /dev/null +++ b/new-tests/stelf-output/examples/church-rosser/test-unsafe.lf @@ -0,0 +1,5 @@ +%require %[ lam %] %. +%require %[ ord-red %] %. +%require %[ par-red %] %. +%require %[ par-lemmas %] %. +%require %[ church-rosser %] %. diff --git a/new-tests/stelf-output/examples/church-rosser/test.lf b/new-tests/stelf-output/examples/church-rosser/test.lf new file mode 100644 index 0000000..c8bc080 --- /dev/null +++ b/new-tests/stelf-output/examples/church-rosser/test.lf @@ -0,0 +1,9 @@ +%require %[ lam %] %. +%require %[ ord-red %] %. +%require %[ par-red %] %. +%require %[ par-lemmas %] %. +%require %[ par-cr %] %. +%require %[ ord-lemmas %] %. +%require %[ equiv %] %. +%require %[ ord-cr %] %. +%require %[ examples %] %. diff --git a/new-tests/stelf-output/examples/compile/cls/cls-complete.lf b/new-tests/stelf-output/examples/compile/cls/cls-complete.lf new file mode 100644 index 0000000..7c5538c --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cls/cls-complete.lf @@ -0,0 +1,92 @@ +%%% Mapping evaluations to CLS computations. +%%% This expresses the completeness proof for the CLS machine. +%%% Author: Frank Pfenning, based on [Hannan & Pfenning 92] +%{ +Lemma: For every + C :: st Ks P S =>* st Ks' P' S' +and + C' :: st Ks P S =>* st Ks' P' S' +there exists a computation + C'' :: st Ks P S =>* st Ks'' P'' S''. + +Proof: By induction on the structure of C. +}% +%sort append %. +%mode append %in %in %out %. +%term apd_id append id C' C' %. +%term apd_step %pi (append (R ~ C) C' (R ~ C'')) %<- (append C C' C'') %. +%{ +Subcomputation Lemma: For every + D :: feval K F W +environment stack Ks, program P, and stack S there exists a computation + C :: st (Ks ;; K) (ev F & P) S =>* st Ks P (S ; W) + +Proof: By induction on the structure of D +}% +%sort subcomp %. +%mode {%in K} {%in F} {%in W} {%in D feval K F W} {%in Ks} {%in P} {%in S} {%out C st (Ks ;; K) (ev F & P) S =>* st Ks P (S ; W)} subcomp D C %. +% Variables +%term sc_1 subcomp fev_1 (c_1 ~ id) %. +%term sc_^ %pi (subcomp (fev_^ D1) (c_^ ~ C1)) %<- (subcomp D1 C1) %. +%term sc_1+ %pi (subcomp (fev_1+ D1) (c_1+ ~ C1)) %<- (subcomp D1 C1) %. +%term sc_^+ %pi (subcomp (fev_^+ D1) (c_^+ ~ C1)) %<- (subcomp D1 C1) %. +% Natural Numbers +%term sc_z subcomp fev_z (c_z ~ id) %. +%term sc_s + %pi (subcomp (fev_s D1) C) + %<- (subcomp D1 C1) + %<- (append (c_s ~ C1) (c_add1 ~ id) C) %. +%term sc_branch_z + %pi (subcomp (fev_case_z D2 D1) C) + %<- (subcomp D1 C1) + %<- (subcomp D2 C2) + %<- (append (c_case ~ C1) (c_branch_z ~ C2) C) %. +%term sc_branch_s + %pi (subcomp (fev_case_s D3 D1) C) + %<- (subcomp D1 C1) + %<- (subcomp D3 C3) + %<- (append (c_case ~ C1) (c_branch_s ~ C3) C) %. +% Pairs +%term sc_pair + %pi (subcomp (fev_pair D2 D1) C) + %<- (subcomp D1 C1) + %<- (subcomp D2 C2) + %<- (append (c_pair ~ C1) C2 C') + %<- (append C' (c_mkpair ~ id) C) %. +%term sc_fst + %pi (subcomp (fev_fst D1) C) + %<- (subcomp D1 C1) + %<- (append (c_fst ~ C1) (c_getfst ~ id) C) %. +%term sc_snd + %pi (subcomp (fev_snd D1) C) + %<- (subcomp D1 C1) + %<- (append (c_snd ~ C1) (c_getsnd ~ id) C) %. +% Functions +%term sc_lam subcomp fev_lam (c_lam ~ id) %. +%term sc_app + %pi (subcomp (fev_app D3 D2 D1) C) + %<- (subcomp D1 C1) + %<- (subcomp D2 C2) + %<- (subcomp D3 C3) + %<- (append (c_app ~ C1) C2 C') + %<- (append C' (c_apply ~ C3) C) %. +% Definitions +%term sc_letv + %pi (subcomp (fev_letv D2 D1) C) + %<- (subcomp D1 C1) + %<- (subcomp D2 C2) + %<- (append (c_letv ~ C1) (c_bind ~ C2) C) %. +%term sc_letn %pi (subcomp (fev_letn D2) (c_letn ~ C2)) %<- (subcomp D2 C2) %. +% Recursion +%term sc_fix %pi (subcomp (fev_fix D1) (c_fix ~ C1)) %<- (subcomp D1 C1) %. +%{ +Completeness Theorem: For every evaluation + D :: feval K F W +there exists a complete computation + E :: ceval K F W + +Proof: Immediately from the subcomputation lemma. +}% +%sort cev_complete %. +%mode cev_complete %in %out %. +%term cevc %pi (cev_complete D (run C)) %<- (subcomp D C) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cls/cls-sound.lf b/new-tests/stelf-output/examples/compile/cls/cls-sound.lf new file mode 100644 index 0000000..7c35b2f --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cls/cls-sound.lf @@ -0,0 +1,104 @@ +%%% Mapping CLS computations to evaluations. +%%% This expresses the soundness proof for the CLS machine. +%%% Author: Frank Pfenning, based on [Hannan & Pfenning 92] +%%% The postfix ordering for computations. +%%% This is ordering is well-founded. +%sort < %. +%prec %none 8 < %. +%mode < %in %in %. +%term sub_imm C < R ~ C %. +%term sub_med %pi (C < C') %-> (C < R ~ C') %. +%{ +Lemma: The postfix relation on computations is transitive: + For every R1 :: C1 < C2 and R2 :: C2 < C3 + there exists an R3 :: C1 < C3. + +Proof: By induction on the structure of R2. +}% +%sort trans* %. +%mode trans* %in %in %out %. +%term trans*_imm trans* R1 sub_imm (sub_med R1) %. +%term trans*_med %pi (trans* R1 (sub_med R2) (sub_med R3)) %<- (trans* R1 R2 R3) %. +%{ +Splitting Lemma: + If C :: (st (KS ;; K) (ev F & P) S) =>* (st (emptys) (done) (empty ; W')) + then there exists a value W, an evaluation + D :: feval K F W + and a computation + C' :: (st KS P (S ; W)) =>* (st (emptys) (done) (empty ; W')) + such that C' < C. + +Proof: By complete induction on C. +}% +%sort spl {C st (KS ;; K) (ev F & P) S =>* (st emptys done (empty ; W'))} %. +%mode spl %in %out %out %out %. +% C = id is impossible, since (ev F & P) is different from (done). +% Variables +%term spl_1 spl (c_1 ~ C1) fev_1 C1 sub_imm %. +%term spl_^ %pi (spl (c_^ ~ C1) (fev_^ D1) C2 (sub_med C2 instruction %. +%term add1 instruction %. +%term branch instruction %. +%term mkpair instruction %. +%term getfst instruction %. +%term getsnd instruction %. +%term apply instruction %. +%term bind instruction %. +%term done program %. +%term & %pi instruction %-> program %-> program %. +%prec %right 10 & %. +%term emptys envstack %. +%term ;; %pi envstack %-> env %-> envstack %. +%prec %left 10 ;; %. +%term st %pi envstack %-> program %-> env %-> state %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cls/compute.lf b/new-tests/stelf-output/examples/compile/cls/compute.lf new file mode 100644 index 0000000..d48a6e6 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cls/compute.lf @@ -0,0 +1,63 @@ +%%% Computations of the CLS machine +%%% Author: Frank Pfenning, based on [Hannan & Pfenning 92] +%%% Single step transition. +%sort => %. +%prec %none 10 => %. +%name => %. +%mode => %in %out %. +% Variables +%term c_1 st (Ks ;; (K ; W)) (ev 1 & P) S => st Ks P (S ; W) %. +%term c_^ st (Ks ;; (K ; W')) (ev (F ^) & P) S => st (Ks ;; K) (ev F & P) S %. +%term c_1+ st (Ks ;; (K + F)) (ev 1 & P) S => st (Ks ;; K) (ev F & P) S %. +%term c_^+ st (Ks ;; (K + F')) (ev (F ^) & P) S => st (Ks ;; K) (ev F & P) S %. +% Natural Numbers +%term c_z st (Ks ;; K) (ev z' & P) S => st Ks P (S ; z*) %. +%term c_s st (Ks ;; K) (ev (s' F) & P) S => st (Ks ;; K) (ev F & add1 & P) S %. +%term c_add1 st Ks (add1 & P) (S ; W) => st Ks P (S ; s* W) %. +%term c_case st (Ks ;; K) (ev (case' F1 F2 F3) & P) S => st (Ks ;; K ;; K) (ev F1 & branch & ev F2 & ev F3 & P) S %. +%term c_branch_z st (Ks ;; K) (branch & ev F2 & ev F3 & P) (S ; z*) => st (Ks ;; K) (ev F2 & P) S %. +%term c_branch_s st (Ks ;; K) (branch & ev F2 & ev F3 & P) (S ; s* W) => st (Ks ;; (K ; W)) (ev F3 & P) S %. +% Pairs +%term c_pair st (Ks ;; K) (ev (pair' F1 F2) & P) S => st (Ks ;; K ;; K) (ev F1 & ev F2 & mkpair & P) S %. +%term c_mkpair st Ks (mkpair & P) (S ; W1 ; W2) => st Ks P (S ; pair* W1 W2) %. +%term c_fst st (Ks ;; K) (ev (fst' F) & P) S => st (Ks ;; K) (ev F & getfst & P) S %. +%term c_getfst st Ks (getfst & P) (S ; pair* W1 W2) => st Ks P (S ; W1) %. +%term c_snd st (Ks ;; K) (ev (snd' F) & P) S => st (Ks ;; K) (ev F & getsnd & P) S %. +%term c_getsnd st Ks (getsnd & P) (S ; pair* W1 W2) => st Ks P (S ; W2) %. +% Functions +%term c_lam st (Ks ;; K) (ev (lam' F) & P) S => st Ks P (S ; clo K (lam' F)) %. +%term c_app st (Ks ;; K) (ev (app' F1 F2) & P) S => st (Ks ;; K ;; K) (ev F1 & ev F2 & apply & P) S %. +%term c_apply st Ks (apply & P) (S ; clo K' (lam' F1') ; W2) => st (Ks ;; (K' ; W2)) (ev F1' & P) S %. +% Definitions +%term c_letv st (Ks ;; K) (ev (letv' F1 F2) & P) S => st (Ks ;; K ;; K) (ev F1 & bind & ev F2 & P) S %. +%term c_bind st (Ks ;; K) (bind & ev F2 & P) (S ; W1) => st (Ks ;; (K ; W1)) (ev F2 & P) S %. +%term c_letn st (Ks ;; K) (ev (letn' F1 F2) & P) S => st (Ks ;; (K + F1)) (ev F2 & P) S %. +% Recursion +%term c_fix st (Ks ;; K) (ev (fix' F) & P) S => st (Ks ;; (K + fix' F)) (ev F & P) S %. +%%% Multi-step transition +%sort =>* %. +%prec %none 10 =>* %. +%name =>* %. +% incorrect would be +% %mode =>* +St -St'. +% because of the incorrect ordering of subgoals +%term id St =>* St %. +%term ~ %pi (St => St') %-> (St' =>* St'') %-> (St =>* St'') %. +%prec %right 10 ~ %. +%%% Evaluation in environment K +%sort ceval %. +%term run + %pi (st (emptys ;; K) (ev F & done) empty =>* st emptys done (empty ; W)) + %-> (ceval K F W) %. +%%% Executable versions. +%sort >=>* %. +%prec %none 10 >=>* %. +%mode >=>* %in %out %. +%term id< St >=>* St %. +%term <=< %pi (St >=>* St'') %<- (St => St') %<- (St' >=>* St'') %. +%prec %left 10 <=< %. +%sort >ceval %. +%mode >ceval %in %in %out %. +%term >run + %pi (>ceval K F W) + %<- (st (emptys ;; K) (ev F & done) empty >=>* st emptys done (empty ; W)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cls/debruijn.lf b/new-tests/stelf-output/examples/compile/cls/debruijn.lf new file mode 100644 index 0000000..c828cac --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cls/debruijn.lf @@ -0,0 +1,32 @@ +%%% Expression representation using de Bruijn indices +%%% Author: Frank Pfenning, based on [Hannan & Pfenning 92] +%sort exp' %. +%name exp' %. +%term 1 exp' %. +%term ^ %pi exp' %-> exp' %. +%prec %postfix 20 ^ %. +%term z' exp' %. +%term s' %pi exp' %-> exp' %. +%term case' %pi exp' %-> exp' %-> exp' %-> exp' %. +%term pair' %pi exp' %-> exp' %-> exp' %. +%term fst' %pi exp' %-> exp' %. +%term snd' %pi exp' %-> exp' %. +%term lam' %pi exp' %-> exp' %. +%term app' %pi exp' %-> exp' %-> exp' %. +%term letv' %pi exp' %-> exp' %-> exp' %. +%term letn' %pi exp' %-> exp' %-> exp' %. +%term fix' %pi exp' %-> exp' %. +% Environments and values +%sort env %. +%name env %. +%sort val %. +%name val %. +%term empty env %. +%term ; %pi env %-> val %-> env %. +%prec %left 10 ; %. +%term + %pi env %-> exp' %-> env %. +%prec %left 10 + %. +%term z* val %. +%term s* %pi val %-> val %. +%term pair* %pi val %-> val %-> val %. +%term clo %pi env %-> exp' %-> val %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cls/examples-quy.lf b/new-tests/stelf-output/examples/compile/cls/examples-quy.lf new file mode 100644 index 0000000..9becac9 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cls/examples-quy.lf @@ -0,0 +1,30 @@ +% Enumerate the solution for complete computations +%query 4 _ _ st (emptys ;; empty) (ev (s' z') & done) empty >=>* st Ks P S %. +% Below are complete computations +%query 1 _ _ >ceval empty (s' z') W %. +%query 1 _ _ >ceval empty (app' (lam' 1) z') W %. +% Doubling 1. +%query 1 1 _ >ceval empty (app' (fix' (lam' (case' 1 z' (s' (s' (app' (1 ^ ^) 1)))))) (s' z')) W %. +%{ +trans empty F + (let (fix [plus] lam [x] lam [y] case x y [x'] s (app (app plus x') y)) + [plus] app (app plus (s z)) (s (s z))). +}% +% Addition 1 + 2 = 3. +%query 1 1 _ >ceval empty (letn' (fix' (lam' (lam' (case' (1 ^) 1 (s' (app' (app' (1 ^ ^ ^) 1) (1 ^))))))) (app' (app' 1 (s' z')) (s' (s' z')))) W %. +%{ +trans empty F + (letn (fix [plus] lam [x] lam [y] case x y [x'] s (app (app plus x') y)) + [plus] + (letn (fix [mult] lam [x] lam [y] case x z [x'] + app (app plus (app (app mult x') y)) y) + [mult] app (app mult (s (s z))) (s (s (s z))))). +}% +% Multiplication: 2 * 3 = 6. +%query 1 1 _ >ceval empty (letn' (fix' (lam' (lam' (case' (1 ^) 1 (s' (app' (app' (1 ^ ^ ^) 1) (1 ^))))))) (letn' (fix' (lam' (lam' (case' (1 ^) z' (app' (app' (1 ^ ^ ^ ^) (app' (app' (1 ^ ^ ^) 1) (1 ^))) (1 ^)))))) (app' (app' 1 (s' (s' z'))) (s' (s' (s' z')))))) W %. +%%% Mapping from evaluations to computations. +% Doubling 1 +%query 1 1 _ feval empty (app' (fix' (lam' (case' 1 z' (s' (s' (app' (1 ^ ^) 1)))))) (s' z')) W %. +%query 1 _ _ cev_complete (fev_app (fev_case_s (fev_s (fev_s (fev_app (fev_case_z fev_z fev_1) fev_1 (fev_^ (fev_^ (fev_1+ (fev_fix fev_lam))))))) fev_1) (fev_s fev_z) (fev_fix fev_lam)) C %. +% Multiplication: 2 * 3 = 6 +%query 1 1 _ feval empty (letn' (fix' (lam' (lam' (case' (1 ^) 1 (s' (app' (app' (1 ^ ^ ^) 1) (1 ^))))))) (letn' (fix' (lam' (lam' (case' (1 ^) z' (app' (app' (1 ^ ^ ^ ^) (app' (app' (1 ^ ^ ^) 1) (1 ^))) (1 ^)))))) (app' (app' 1 (s' (s' z'))) (s' (s' (s' z')))))) W %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cls/feval.lf b/new-tests/stelf-output/examples/compile/cls/feval.lf new file mode 100644 index 0000000..fdf31da --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cls/feval.lf @@ -0,0 +1,34 @@ +%%% Mini-ML Natural Semantics +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort feval %. +%name feval %. +%mode feval %in %in %out %. +% Variables +%term fev_1 feval (K ; W) 1 W %. +%term fev_^ %pi (feval (K ; W') (F ^) W) %<- (feval K F W) %. +%term fev_1+ %pi (feval (K + F) 1 W) %<- (feval K F W) %. +%term fev_^+ %pi (feval (K + F') (F ^) W) %<- (feval K F W) %. +% Natural Numbers +%term fev_z feval K z' z* %. +%term fev_s %pi (feval K (s' F) (s* W)) %<- (feval K F W) %. +%term fev_case_z %pi (feval K (case' F1 F2 F3) W) %<- (feval K F1 z*) %<- (feval K F2 W) %. +%term fev_case_s + %pi (feval K (case' F1 F2 F3) W) + %<- (feval K F1 (s* W1)) + %<- (feval (K ; W1) F3 W) %. +% Pairs +%term fev_pair %pi (feval K (pair' F1 F2) (pair* W1 W2)) %<- (feval K F1 W1) %<- (feval K F2 W2) %. +%term fev_fst %pi (feval K (fst' F) W1) %<- (feval K F (pair* W1 W2)) %. +%term fev_snd %pi (feval K (snd' F) W2) %<- (feval K F (pair* W1 W2)) %. +% Functions +%term fev_lam feval K (lam' F) (clo K (lam' F)) %. +%term fev_app + %pi (feval K (app' F1 F2) W) + %<- (feval K F1 (clo K' (lam' F1'))) + %<- (feval K F2 W2) + %<- (feval (K' ; W2) F1' W) %. +% Definitions +%term fev_letv %pi (feval K (letv' F1 F2) W) %<- (feval K F1 W1) %<- (feval (K ; W1) F2 W) %. +%term fev_letn %pi (feval K (letn' F1 F2) W) %<- (feval (K + F1) F2 W) %. +% Recursion +%term fev_fix %pi (feval K (fix' F) W) %<- (feval (K + (fix' F)) F W) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cls/mini-ml.lf b/new-tests/stelf-output/examples/compile/cls/mini-ml.lf new file mode 100644 index 0000000..e166530 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cls/mini-ml.lf @@ -0,0 +1,15 @@ +%%% The Mini-ML Language +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort exp %. +%name exp %. +%term z exp %. +%term s %pi exp %-> exp %. +%term case %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%term pair %pi exp %-> exp %-> exp %. +%term fst %pi exp %-> exp %. +%term snd %pi exp %-> exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term letv %pi exp %-> (%pi exp %-> exp) %-> exp %. +%term letn %pi exp %-> (%pi exp %-> exp) %-> exp %. +%term fix %pi (%pi exp %-> exp) %-> exp %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cls/slow-quy.lf b/new-tests/stelf-output/examples/compile/cls/slow-quy.lf new file mode 100644 index 0000000..f2f1bc5 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cls/slow-quy.lf @@ -0,0 +1,91 @@ +%query 1 _ _ cls_sound (run (c_app ~ c_fix ~ c_lam ~ c_s ~ c_z ~ c_add1 ~ c_apply ~ c_case ~ c_1 ~ c_branch_s ~ c_s ~ c_s ~ c_app ~ c_^ ~ c_^ ~ c_1+ ~ c_fix ~ c_lam ~ c_1 ~ c_apply ~ c_case ~ c_1 ~ c_branch_z ~ c_z ~ c_add1 ~ c_add1 ~ id)) D %. +%{ +% Kills SML 110 right now! +%query 1 1 +cev_complete + (fev_letn + (fev_letn + (fev_app + (fev_case_s + (fev_app + (fev_case_s + (fev_s + (fev_app + (fev_case_s + (fev_s + +(fev_app + (fev_case_s + (fev_s + (fev_app (fev_case_z fev_1 (fev_^ fev_1)) (fev_^ fev_1) + (fev_app fev_lam fev_1 + (fev_^ (fev_^ (fev_^ (fev_1+ (fev_fix fev_lam)))))))) + (fev_^ fev_1)) + (fev_^ fev_1) + (fev_app fev_lam fev_1 (fev_^ (fev_^ (fev_^ (fev_1+ (fev_fix fev_lam)))))))) + (fev_^ fev_1)) + (fev_^ fev_1) + (fev_app fev_lam fev_1 + (fev_^ + (fev_^ + (fev_^ (fev_1+ (fev_fix fev_lam)))))))) + (fev_^ fev_1)) + (fev_^ fev_1) + (fev_app fev_lam + (fev_app + (fev_case_s + (fev_app (fev_case_z fev_1 (fev_^ fev_1)) + (fev_^ fev_1) + (fev_app fev_lam + +(fev_app (fev_case_z fev_z (fev_^ fev_1)) (fev_^ fev_1) + (fev_app fev_lam fev_1 (fev_^ (fev_^ (fev_^ (fev_1+ (fev_fix fev_lam))))))) + (fev_^ + (fev_^ + (fev_^ + (fev_^+ + (fev_1+ + (fev_fix fev_lam)))))))) + (fev_^ fev_1)) + (fev_^ fev_1) + (fev_app fev_lam fev_1 + (fev_^ + (fev_^ (fev_^ (fev_1+ (fev_fix fev_lam))))))) + (fev_^ + (fev_^ (fev_^ (fev_^+ (fev_1+ (fev_fix fev_lam)))))))) + (fev_^ fev_1)) + (fev_s (fev_s (fev_s fev_z))) + (fev_app fev_lam (fev_s (fev_s fev_z)) (fev_1+ (fev_fix fev_lam)))))) +C. +}% +%{ +sigma [D:feval empty (s' z') W] +cev_complete D C. + +sigma [D:feval empty (app' (lam' 1) z') W] +cev_complete D C. + +% Doubling 1 +sigma [D:feval empty (app' (fix' (lam' (case' 1 z' + (s' (s' (app' (1 ^ ^) 1)))))) + (s' z')) W] +cev_complete D C. +}% +%{ +% This is VERY slow +% Multiplication: 2 * 3 = 6. +sigma [D:feval empty + (let' + (fix' (lam' (lam' (case' (1 ^) 1 (s' (app' (app' (1 ^ ^ ^) 1) (1 ^))))))) + (let' + (fix' + (lam' + (lam' + (case' (1 ^) z' + (app' + (app' (1 ^ ^ ^ ^) + (app' (app' (1 ^ ^ ^) 1) (1 ^))) (1 ^)))))) + (app' (app' 1 (s' (s' z'))) (s' (s' (s' z')))))) + W] +cev_complete D C. +}% \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cls/sources.lf b/new-tests/stelf-output/examples/compile/cls/sources.lf new file mode 100644 index 0000000..81e0564 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cls/sources.lf @@ -0,0 +1,8 @@ +%require %[ mini-ml %] %. +%require %[ debruijn %] %. +%require %[ trans %] %. +%require %[ feval %] %. +%require %[ cls %] %. +%require %[ compute %] %. +%require %[ cls-complete %] %. +%require %[ cls-sound %] %. diff --git a/new-tests/stelf-output/examples/compile/cls/stelf.toml b/new-tests/stelf-output/examples/compile/cls/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cls/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/compile/cls/test.lf b/new-tests/stelf-output/examples/compile/cls/test.lf new file mode 100644 index 0000000..f4a545f --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cls/test.lf @@ -0,0 +1,9 @@ +%require %[ mini-ml %] %. +%require %[ debruijn %] %. +%require %[ trans %] %. +%require %[ feval %] %. +%require %[ cls %] %. +%require %[ compute %] %. +%require %[ cls-complete %] %. +%require %[ cls-sound %] %. +%require %[ examples %] %. diff --git a/new-tests/stelf-output/examples/compile/cls/trans.lf b/new-tests/stelf-output/examples/compile/cls/trans.lf new file mode 100644 index 0000000..f429219 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cls/trans.lf @@ -0,0 +1,51 @@ +%%% Translation to representation using de Bruijn indices +%%% Author: Frank Pfenning, based on [Hannan & Pfenning 92] +%sort trans %. +%name trans %. +%sort vtrans %. +%name vtrans %. +%mode trans %in %in %out %. +% and other modes! +%mode vtrans %in %out %. +% Natural numbers +%term tr_z trans K z' z %. +%term tr_s %pi (trans K (s' F) (s E)) %<- (trans K F E) %. +%term tr_case + %pi (trans K (case' F1 F2 F3) (case E1 E2 E3)) + %<- (trans K F1 E1) + %<- (trans K F2 E2) + %<- ({w val} {x exp} %pi (vtrans w x) %-> (trans (K ; w) F3 (E3 x))) %. +% Pairs +%term tr_pair %pi (trans K (pair' F1 F2) (pair E1 E2)) %<- (trans K F1 E1) %<- (trans K F2 E2) %. +%term tr_fst %pi (trans K (fst' F1) (fst E1)) %<- (trans K F1 E1) %. +%term tr_snd %pi (trans K (snd' F1) (snd E1)) %<- (trans K F1 E1) %. +% Functions +%term tr_lam + %pi (trans K (lam' F) (lam E)) + %<- ({w val} {x exp} %pi (vtrans w x) %-> (trans (K ; w) F (E x))) %. +%term tr_app %pi (trans K (app' F1 F2) (app E1 E2)) %<- (trans K F1 E1) %<- (trans K F2 E2) %. +% Definitions +%term tr_letv + %pi (trans K (letv' F1 F2) (letv E1 E2)) + %<- (trans K F1 E1) + %<- ({w val} {x exp} %pi (vtrans w x) %-> (trans (K ; w) F2 (E2 x))) %. +%term tr_letn + %pi (trans K (letn' F1 F2) (letn E1 E2)) + %<- (trans K F1 E1) + %<- ({f exp'} {x exp} %pi (trans K f x) %-> (trans (K + f) F2 (E2 x))) %. +% Recursion +%term tr_fix + %pi (trans K (fix' F) (fix E)) + %<- ({f exp'} {x exp} %pi (trans K f x) %-> (trans (K + f) F (E x))) %. +% Variables +%term tr_1 %pi (trans (K ; W) 1 E) %<- (vtrans W E) %. +%term tr_^ %pi (trans (K ; W) (F ^) E) %<- (trans K F E) %. +%term tr_1+ %pi (trans (K + F) 1 E) %<- (trans K F E) %. +%term tr_^+ %pi (trans (K + F') (F ^) E) %<- (trans K F E) %. +% Natural number values +%term vtr_z vtrans z* z %. +%term vtr_s %pi (vtrans (s* W) (s V)) %<- (vtrans W V) %. +% Pair values +%term vtr_pair %pi (vtrans (pair* W1 W2) (pair V1 V2)) %<- (vtrans W1 V1) %<- (vtrans W2 V2) %. +% Function values +%term vtr_lam %pi (vtrans (clo K (lam' F)) (lam E)) %<- (trans K (lam' F) (lam E)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cls/trans2.lf b/new-tests/stelf-output/examples/compile/cls/trans2.lf new file mode 100644 index 0000000..f429219 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cls/trans2.lf @@ -0,0 +1,51 @@ +%%% Translation to representation using de Bruijn indices +%%% Author: Frank Pfenning, based on [Hannan & Pfenning 92] +%sort trans %. +%name trans %. +%sort vtrans %. +%name vtrans %. +%mode trans %in %in %out %. +% and other modes! +%mode vtrans %in %out %. +% Natural numbers +%term tr_z trans K z' z %. +%term tr_s %pi (trans K (s' F) (s E)) %<- (trans K F E) %. +%term tr_case + %pi (trans K (case' F1 F2 F3) (case E1 E2 E3)) + %<- (trans K F1 E1) + %<- (trans K F2 E2) + %<- ({w val} {x exp} %pi (vtrans w x) %-> (trans (K ; w) F3 (E3 x))) %. +% Pairs +%term tr_pair %pi (trans K (pair' F1 F2) (pair E1 E2)) %<- (trans K F1 E1) %<- (trans K F2 E2) %. +%term tr_fst %pi (trans K (fst' F1) (fst E1)) %<- (trans K F1 E1) %. +%term tr_snd %pi (trans K (snd' F1) (snd E1)) %<- (trans K F1 E1) %. +% Functions +%term tr_lam + %pi (trans K (lam' F) (lam E)) + %<- ({w val} {x exp} %pi (vtrans w x) %-> (trans (K ; w) F (E x))) %. +%term tr_app %pi (trans K (app' F1 F2) (app E1 E2)) %<- (trans K F1 E1) %<- (trans K F2 E2) %. +% Definitions +%term tr_letv + %pi (trans K (letv' F1 F2) (letv E1 E2)) + %<- (trans K F1 E1) + %<- ({w val} {x exp} %pi (vtrans w x) %-> (trans (K ; w) F2 (E2 x))) %. +%term tr_letn + %pi (trans K (letn' F1 F2) (letn E1 E2)) + %<- (trans K F1 E1) + %<- ({f exp'} {x exp} %pi (trans K f x) %-> (trans (K + f) F2 (E2 x))) %. +% Recursion +%term tr_fix + %pi (trans K (fix' F) (fix E)) + %<- ({f exp'} {x exp} %pi (trans K f x) %-> (trans (K + f) F (E x))) %. +% Variables +%term tr_1 %pi (trans (K ; W) 1 E) %<- (vtrans W E) %. +%term tr_^ %pi (trans (K ; W) (F ^) E) %<- (trans K F E) %. +%term tr_1+ %pi (trans (K + F) 1 E) %<- (trans K F E) %. +%term tr_^+ %pi (trans (K + F') (F ^) E) %<- (trans K F E) %. +% Natural number values +%term vtr_z vtrans z* z %. +%term vtr_s %pi (vtrans (s* W) (s V)) %<- (vtrans W V) %. +% Pair values +%term vtr_pair %pi (vtrans (pair* W1 W2) (pair V1 V2)) %<- (vtrans W1 V1) %<- (vtrans W2 V2) %. +% Function values +%term vtr_lam %pi (vtrans (clo K (lam' F)) (lam E)) %<- (trans K (lam' F) (lam E)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cpm/ceval-complete-thm.lf b/new-tests/stelf-output/examples/compile/cpm/ceval-complete-thm.lf new file mode 100644 index 0000000..403c322 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cpm/ceval-complete-thm.lf @@ -0,0 +1,9 @@ +% Completeness of refining evaluation to small step semantics +% Partial check of hand-coded proof +%terminates D (ccp D _ _) %. +%theorem ccf : forall* {E exp} {V val} forall {D eval E V} exists {C {K} {W} %pi (K # (return V) =>* (answer W)) %-> (K # (ev E) =>* (answer W))} true %. +%prove 20 D (ccf D _) %. +% Partial verification of generated proof +%terminates D (ccf D _) %. +%theorem cevalc : forall* {E exp} {V val} forall {D eval E V} exists {CE ceval E V} true %. +%prove 10 {} (cevalc _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cpm/ceval-complete.lf b/new-tests/stelf-output/examples/compile/cpm/ceval-complete.lf new file mode 100644 index 0000000..e110499 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cpm/ceval-complete.lf @@ -0,0 +1,81 @@ +%%% Completeness of Evaluation under Continuations +%%% Author: Frank Pfenning +%{ The natural informal proof relies on the lemma + If D :: eval E V + then for any K + C :: K # (ev E) =>* K # (return V). + +Proof: by induction on the structure of D. + +The formalization of this proof requires appending partial computation +sequences, which is possible, although somewhat tedious. + +Below we give a simpler implementation in which the partial computations +are accumulated incrementally. +}% +%{ Main Lemma + If D :: eval E V + and C' :: K # (return V) =>* (answer W) + then C :: K # (ev E) =>* (answer W). + +Proof: by induction on the structure of D. C' is the "accumulator" +argument for the resulting computation. + +The implementation below has the remarkable property that it is almost +exactly the same as csd, the implementation of the soundness proof. +There are only two changes: the order of arguments is switched +(following the convention that induction arguments come first), the +second is that the order of the subgoals is reversed. +}% +%sort ccp %. +%mode ccp %in %in %out %. +%name ccp %. +% +% Natural Numbers +%term ccp_z ccp ev_z C' (C' << st_z) %. +%term ccp_s %pi (ccp (ev_s D1) C' (C1 << st_s)) %<- (ccp D1 (C' << st_return) C1) %. +%term ccp_case_z + %pi (ccp (ev_case_z D2 D1) C' (C1 << st_case)) + %<- (ccp D2 C' C2) + %<- (ccp D1 (C2 << st_case1_z << st_return) C1) %. +%term ccp_case_s + %pi (ccp (ev_case_s D3 D1) C' (C1 << st_case)) + %<- (ccp D3 C' C3) + %<- (ccp D1 (C3 << st_case1_s << st_return) C1) %. +% Pairs +%term ccp_pair + %pi (ccp (ev_pair D2 D1) C' (C1 << st_pair)) + %<- (ccp D2 (C' << st_return) C2) + %<- (ccp D1 (C2 << st_pair1 << st_return) C1) %. +%term ccp_fst + %pi (ccp (ev_fst D1) C' (C1 << st_fst)) + %<- (ccp D1 (C' << st_fst1 << st_return) C1) %. +%term ccp_snd + %pi (ccp (ev_snd D1) C' (C1 << st_snd)) + %<- (ccp D1 (C' << st_snd1 << st_return) C1) %. +% Functions +%term ccp_lam ccp ev_lam C' (C' << st_lam) %. +%term ccp_app + %pi (ccp (ev_app D3 D2 D1) C' (C1 << st_app)) + %<- (ccp D3 C' C3) + %<- (ccp D2 (C3 << st_app2 << st_return) C2) + %<- (ccp D1 (C2 << st_app1 << st_return) C1) %. +% Definitions +%term ccp_letv + %pi (ccp (ev_letv D2 D1) C' (C1 << st_letv)) + %<- (ccp D2 C' C2) + %<- (ccp D1 (C2 << st_return) C1) %. +%term ccp_letn %pi (ccp (ev_letn D1) C' (C1 << st_letn)) %<- (ccp D1 C' C1) %. +% Recursion +%term ccp_fix %pi (ccp (ev_fix D1) C' (C1 << st_fix)) %<- (ccp D1 C' C1) %. +% Values +%term ccp_vl ccp ev_vl C' (C' << st_vl) %. +%worlds () (ccp _ _ _) %. +%{ Completeness Theorem: + If D :: eval E V + then CE :: ceval E V. + +Proof: By Main Lemma. +}% +%sort ceval_complete %. +%term cevcp %pi (ceval_complete D (cev C)) %<- (ccp D (stop << st_init) C) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cpm/ceval-sound.lf b/new-tests/stelf-output/examples/compile/cpm/ceval-sound.lf new file mode 100644 index 0000000..96c6c82 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cpm/ceval-sound.lf @@ -0,0 +1,66 @@ +%%% Soundness of Evaluation under Continuations +%%% Author: Frank Pfenning +%{ Main Lemma: + If C :: K # (ev E) =>* (answer W) + then for some V + D :: eval E V + and C' :: K # (return V) =>* (answer W) + where C' is a subderivation of C. + +Proof: By complete induction on the structure of C. + +Note that the fact that C' is a subderivation of C is not +represented in the implementation below. This could be added +as a higher-level judgment on transformations. +}% +%sort csd %. +%name csd %. +% Natural Numbers +%term csd_z csd (C' << st_z) ev_z C' %. +%term csd_s %pi (csd (C1 << st_s) (ev_s D1) C') %<- (csd C1 D1 (C' << st_return)) %. +%term csd_case_z + %pi (csd (C1 << st_case) (ev_case_z D2 D1) C') + %<- (csd C1 D1 (C2 << st_case1_z << st_return)) + %<- (csd C2 D2 C') %. +%term csd_case_s + %pi (csd (C1 << st_case) (ev_case_s D3 D1) C') + %<- (csd C1 D1 (C3 << st_case1_s << st_return)) + %<- (csd C3 D3 C') %. +% Pairs +%term csd_pair + %pi (csd (C1 << st_pair) (ev_pair D2 D1) C') + %<- (csd C1 D1 (C2 << st_pair1 << st_return)) + %<- (csd C2 D2 (C' << st_return)) %. +%term csd_fst + %pi (csd (C1 << st_fst) (ev_fst D1) C') + %<- (csd C1 D1 (C' << st_fst1 << st_return)) %. +%term csd_snd + %pi (csd (C1 << st_snd) (ev_snd D1) C') + %<- (csd C1 D1 (C' << st_snd1 << st_return)) %. +% Functions +%term csd_lam csd (C' << st_lam) ev_lam C' %. +%term csd_app + %pi (csd (C1 << st_app) (ev_app D3 D2 D1) C') + %<- (csd C1 D1 (C2 << st_app1 << st_return)) + %<- (csd C2 D2 (C3 << st_app2 << st_return)) + %<- (csd C3 D3 C') %. +% Definitions +%term csd_letv + %pi (csd (C1 << st_letv) (ev_letv D2 D1) C') + %<- (csd C1 D1 (C2 << st_return)) + %<- (csd C2 D2 C') %. +%term csd_letn %pi (csd (C1 << st_letn) (ev_letn D1) C') %<- (csd C1 D1 C') %. +% Recursion +%term csd_fix %pi (csd (C1 << st_fix) (ev_fix D1) C') %<- (csd C1 D1 C') %. +% Values +%term csd_vl csd (C' << st_vl) ev_vl C' %. +% Return instruction are not possible since the initial state +% is of the form K # (ev E). +%{ Theorem: + If CE :: ceval E V + then D :: eval E V. + +Proof: By inversion on CE and appeal to the Main Lemma. +}% +%sort ceval_sound %. +%term cevsd %pi (ceval_sound (cev C) D) %<- (csd C D (stop << st_init)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cpm/ceval.lf b/new-tests/stelf-output/examples/compile/cpm/ceval.lf new file mode 100644 index 0000000..7298387 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cpm/ceval.lf @@ -0,0 +1,47 @@ +%%% Computations of Continuation Machine +%%% Author: Frank Pfenning +%%% Single Step Transitions +%sort => %. +%name => %. +%prec %none 6 => %. +% Natural Numbers +%term st_z K # (ev z) => K # (return z*) %. +%term st_s K # (ev (s E)) => (K ; ([x val] return (s* x))) # (ev E) %. +%term st_case K # (ev (case E1 E2 E3)) => (K ; ([x1 val] case1 x1 E2 E3)) # (ev E1) %. +%term st_case1_z K # (case1 z* E2 E3) => K # (ev E2) %. +%term st_case1_s K # (case1 (s* V1') E2 E3) => K # (ev (E3 V1')) %. +% Pairs +%term st_pair K # (ev (pair E1 E2)) => (K ; ([x1 val] pair1 x1 E2)) # (ev E1) %. +%term st_pair1 K # (pair1 V1 E2) => (K ; ([x2 val] return (pair* V1 x2))) # (ev E2) %. +%term st_fst K # (ev (fst E)) => (K ; ([x val] fst1 x)) # (ev E) %. +%term st_fst1 K # (fst1 (pair* V1 V2)) => K # (return V1) %. +%term st_snd K # (ev (snd E)) => (K ; ([x val] snd1 x)) # (ev E) %. +%term st_snd1 K # (snd1 (pair* V1 V2)) => K # (return V2) %. +% Functions +%term st_lam K # (ev (lam E)) => K # (return (lam* E)) %. +%term st_app K # (ev (app E1 E2)) => (K ; ([x1 val] app1 x1 E2)) # (ev E1) %. +%term st_app1 K # (app1 V1 E2) => (K ; ([x2 val] app2 V1 x2)) # (ev E2) %. +%term st_app2 K # (app2 (lam* E1') V2) => K # (ev (E1' V2)) %. +% Definitions +%term st_letv K # (ev (letv E1 E2)) => (K ; ([x1 val] ev (E2 x1))) # (ev E1) %. +%term st_letn K # (ev (letn E1 E2)) => K # (ev (E2 E1)) %. +% Recursion +%term st_fix K # (ev (fix E)) => K # (ev (E (fix E))) %. +% Values +%term st_vl K # (ev (vl V)) => K # (return V) %. +% Return Instructions +%term st_return K ; C # (return V) => K # (C V) %. +%term st_init init # (return V) => (answer V) %. +%%% Multi-Step Computation +%sort =>* %. +%name =>* %. +%prec %none 5 =>* %. +%term stop S =>* S %. +%term << %pi (S =>* S'') %<- (S => S') %<- (S' =>* S'') %. +%prec %left 5 << %. +% Because of evaluation order, computation sequences are displayed +% in reverse, using "<<" as a left-associative infix operator. +%%% Full Evaluation +%sort ceval %. +%name ceval %. +%term cev %pi (ceval E V) %<- (init # (ev E) =>* (answer V)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cpm/cpm.lf b/new-tests/stelf-output/examples/compile/cpm/cpm.lf new file mode 100644 index 0000000..279eb24 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cpm/cpm.lf @@ -0,0 +1,25 @@ +%%% Continuation Machine for Mini-ML +%%% Author: Frank Pfenning +% Machine Instructions +%sort inst %. +%name inst %. +%term ev %pi exp %-> inst %. +%term return %pi val %-> inst %. +%term case1 %pi val %-> exp %-> (%pi val %-> exp) %-> inst %. +%term pair1 %pi val %-> exp %-> inst %. +%term fst1 %pi val %-> inst %. +%term snd1 %pi val %-> inst %. +%term app1 %pi val %-> exp %-> inst %. +%term app2 %pi val %-> val %-> inst %. +% Continuations +%sort cont %. +%name cont %. +%term init cont %. +%term ; %pi cont %-> (%pi val %-> inst) %-> cont %. +%prec %left 8 ; %. +% Continuation Machine States +%sort state %. +%name state %. +%term # %pi cont %-> inst %-> state %. +%term answer %pi val %-> state %. +%prec %none 7 # %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cpm/evalv.lf b/new-tests/stelf-output/examples/compile/cpm/evalv.lf new file mode 100644 index 0000000..91f7b0b --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cpm/evalv.lf @@ -0,0 +1,28 @@ +%%% Mini-ML Natural Semantics +%%% Version with separate category of values +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort eval %. +%name eval %. +% Natural Numbers +%term ev_z eval z z* %. +%term ev_s %pi (eval (s E) (s* V)) %<- (eval E V) %. +%term ev_case_z %pi (eval (case E1 E2 E3) V) %<- (eval E1 z*) %<- (eval E2 V) %. +%term ev_case_s %pi (eval (case E1 E2 E3) V) %<- (eval E1 (s* V1')) %<- (eval (E3 V1') V) %. +% Pairs +%term ev_pair %pi (eval (pair E1 E2) (pair* V1 V2)) %<- (eval E1 V1) %<- (eval E2 V2) %. +%term ev_fst %pi (eval (fst E) V1) %<- (eval E (pair* V1 V2)) %. +%term ev_snd %pi (eval (snd E) V2) %<- (eval E (pair* V1 V2)) %. +% Functions +%term ev_lam eval (lam E) (lam* E) %. +%term ev_app + %pi (eval (app E1 E2) V) + %<- (eval E1 (lam* E1')) + %<- (eval E2 V2) + %<- (eval (E1' V2) V) %. +% Definitions +%term ev_letv %pi (eval (letv E1 E2) V) %<- (eval E1 V1) %<- (eval (E2 V1) V) %. +%term ev_letn %pi (eval (letn E1 E2) V) %<- (eval (E2 E1) V) %. +% Recursion +%term ev_fix %pi (eval (fix E) V) %<- (eval (E (fix E)) V) %. +% Values +%term ev_vl eval (vl V) V %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cpm/examples-quy.lf b/new-tests/stelf-output/examples/compile/cpm/examples-quy.lf new file mode 100644 index 0000000..d4ef386 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cpm/examples-quy.lf @@ -0,0 +1,73 @@ +%query 1 _ _ init # (ev (app (lam ([x] vl x)) z)) =>* (answer V) %. +%query 1 _ _ ceval (app (lam ([x] vl x)) z) V %. +% Doubling 3 +%query 1 _ _ ceval (app (fix ([double] lam ([x] case (vl x) z ([x'] s (s (app double (vl x'))))))) (s (s (s z)))) V %. +% Computing 2 * 3 +%query 1 _ _ ceval (letv (lam ([x] fix ([add] lam ([y] case (vl y) (vl x) ([y'] s (app add (vl y'))))))) ([add] letv (lam ([x] fix ([mult] lam ([y] case (vl y) z ([y'] app (app (vl add) (vl x)) (app mult (vl y'))))))) ([mult] app (app (vl mult) (s (s z))) (s (s (s z)))))) V %. +% Applying soundness to computation of doubling 3 +%query 1 _ _ ceval_sound (cev (stop << st_init << st_return << st_return << st_return << st_return << st_return << st_return << st_z << st_case1_z << st_return << st_vl << st_case << st_app2 << st_return << st_vl << st_app1 << st_return << st_lam << st_fix << st_app << st_s << st_s << st_case1_s << st_return << st_vl << st_case << st_app2 << st_return << st_vl << st_app1 << st_return << st_lam << st_fix << st_app << st_s << st_s << st_case1_s << st_return << st_vl << st_case << st_app2 << st_return << st_vl << st_app1 << st_return << st_lam << st_fix << st_app << st_s << st_s << st_case1_s << st_return << st_vl << st_case << st_app2 << st_return << st_return << st_return << st_return << st_z << st_s << st_s << st_s << st_app1 << st_return << st_lam << st_fix << st_app)) D %. +% Natural evaluation of 2 * 3 +%query 1 1 _ eval (letv (lam ([x] fix ([add] lam ([y] case (vl y) (vl x) ([y'] s (app add (vl y'))))))) ([add] letv (lam ([x] fix ([mult] lam ([y] case (vl y) z ([y'] app (app (vl add) (vl x)) (app mult (vl y'))))))) ([mult] app (app (vl mult) (s (s z))) (s (s (s z)))))) V %. +% Currently kills SML +%{ +% Applying completeness proof to obtain a computation of 2 * 3 +%query 1 1 +ceval_complete + (ev_letv + (ev_letv + (ev_app + (ev_case_s + (ev_app + (ev_case_s + (ev_s + (ev_app + (ev_case_s + (ev_s + +(ev_app + (ev_case_s + (ev_s + (ev_app + (ev_case_s + (ev_s (ev_app (ev_case_z ev_vl ev_vl) ev_vl (ev_fix ev_lam))) + ev_vl) + ev_vl (ev_fix ev_lam))) + ev_vl) + ev_vl (ev_fix ev_lam))) + ev_vl) + ev_vl (ev_fix ev_lam))) + ev_vl) + (ev_app + (ev_case_s + (ev_app + (ev_case_s + (ev_s + (ev_app + (ev_case_s + (ev_s + (ev_app + (ev_case_z ev_vl ev_vl) + ev_vl (ev_fix ev_lam))) + ev_vl) + ev_vl (ev_fix ev_lam))) + ev_vl) + (ev_app + (ev_case_s + (ev_app (ev_case_z ev_vl ev_vl) + (ev_app (ev_case_z ev_z ev_vl) + ev_vl (ev_fix ev_lam)) + (ev_app (ev_fix ev_lam) ev_vl + ev_vl)) + ev_vl) + ev_vl (ev_fix ev_lam)) + (ev_app (ev_fix ev_lam) ev_vl ev_vl)) + ev_vl) + ev_vl (ev_fix ev_lam)) + (ev_app (ev_fix ev_lam) ev_vl ev_vl)) + ev_vl) + (ev_s (ev_s (ev_s ev_z))) + (ev_app (ev_fix ev_lam) (ev_s (ev_s ev_z)) ev_vl)) + ev_lam) + ev_lam) +CE. +}% \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cpm/mini-mlv.lf b/new-tests/stelf-output/examples/compile/cpm/mini-mlv.lf new file mode 100644 index 0000000..fc25a14 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cpm/mini-mlv.lf @@ -0,0 +1,23 @@ +%%% The Mini-ML Language +%%% Version distinguishing expressions and values +%%% Author: Frank Pfenning +%sort exp %. +%name exp %. +%sort val %. +%name val %. +%term z exp %. +%term s %pi exp %-> exp %. +%term case %pi exp %-> exp %-> (%pi val %-> exp) %-> exp %. +%term pair %pi exp %-> exp %-> exp %. +%term fst %pi exp %-> exp %. +%term snd %pi exp %-> exp %. +%term lam %pi (%pi val %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term letv %pi exp %-> (%pi val %-> exp) %-> exp %. +%term letn %pi exp %-> (%pi exp %-> exp) %-> exp %. +%term fix %pi (%pi exp %-> exp) %-> exp %. +%term vl %pi val %-> exp %. +%term z* val %. +%term s* %pi val %-> val %. +%term pair* %pi val %-> val %-> val %. +%term lam* %pi (%pi val %-> exp) %-> val %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cpm/proof-equiv-thm.lf b/new-tests/stelf-output/examples/compile/cpm/proof-equiv-thm.lf new file mode 100644 index 0000000..cfdc03c --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cpm/proof-equiv-thm.lf @@ -0,0 +1,12 @@ +% Proof equivalence: "=>" direction +% Partial check of hand-coded proof +% decidable {SD} peq SD _. +%theorem peqf : forall* {K1 cont} {E1 exp} {V1 val} {V2 val} {C K1 # ev E1 =>* answer V1} {D eval E1 V2} {C' K1 # return V2 =>* answer V1} forall {SD csd C D C'} exists {CP ccp D C' C} true %. +%prove 6 SD (peqf SD _) %. +% Partial verification of generated proof +%terminates SD (peqf SD _) %. +% Proof equivalence: "<=" direction +%theorem peqr : forall* {K1 cont} {E1 exp} {V1 val} {V2 val} {C K1 # ev E1 =>* answer V1} {D eval E1 V2} {C' K1 # return V2 =>* answer V1} forall {CP ccp D C' C} exists {SD csd C D C'} true %. +%prove 6 CP (peqr CP _) %. +% Partial verification of generated proof +%terminates CP (peqr CP _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cpm/proof-equiv.lf b/new-tests/stelf-output/examples/compile/cpm/proof-equiv.lf new file mode 100644 index 0000000..59d657e --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cpm/proof-equiv.lf @@ -0,0 +1,43 @@ +%%% Equivalence between soundness and completeness proofs +%%% Author: Frank Pfenning +%sort peq %. +%mode peq %in %out %. +% Natural Numbers +%term peq_z peq csd_z ccp_z %. +%term peq_s %pi (peq (csd_s SD1) (ccp_s CP1)) %<- (peq SD1 CP1) %. +%term peq_case_z + %pi (peq (csd_case_z SD2 SD1) (ccp_case_z CP1 CP2)) + %<- (peq SD1 CP1) + %<- (peq SD2 CP2) %. +%term peq_case_s + %pi (peq (csd_case_s SD3 SD1) (ccp_case_s CP1 CP3)) + %<- (peq SD1 CP1) + %<- (peq SD3 CP3) %. +% Pairs +%term peq_pair + %pi (peq (csd_pair SD2 SD1) (ccp_pair CP1 CP2)) + %<- (peq SD1 CP1) + %<- (peq SD2 CP2) %. +%term peq_fst %pi (peq (csd_fst SD1) (ccp_fst CP1)) %<- (peq SD1 CP1) %. +%term peq_snd %pi (peq (csd_snd SD1) (ccp_snd CP1)) %<- (peq SD1 CP1) %. +% Functions +%term peq_lam peq csd_lam ccp_lam %. +%term peq_app + %pi (peq (csd_app SD3 SD2 SD1) (ccp_app CP1 CP2 CP3)) + %<- (peq SD1 CP1) + %<- (peq SD2 CP2) + %<- (peq SD3 CP3) %. +% Definitions +%term peq_letv + %pi (peq (csd_letv SD2 SD1) (ccp_letv CP1 CP2)) + %<- (peq SD1 CP1) + %<- (peq SD2 CP2) %. +%term peq_letn %pi (peq (csd_letn SD1) (ccp_letn CP1)) %<- (peq SD1 CP1) %. +% Recursion +%term peq_fix %pi (peq (csd_fix SD1) (ccp_fix CP1)) %<- (peq SD1 CP1) %. +% Values +%term peq_vl peq csd_vl ccp_vl %. +%worlds () (peq _ _) %. +%%% For complete evaluations +%sort proof_equiv %. +%term pequiv %pi (proof_equiv (cevsd SD) (cevcp CP)) %<- (peq SD CP) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cpm/slow-quy.lf b/new-tests/stelf-output/examples/compile/cpm/slow-quy.lf new file mode 100644 index 0000000..9d900e9 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cpm/slow-quy.lf @@ -0,0 +1,3 @@ +% Applying soundness to the multiplication of 2 * 3 +%query 1 _ _ ceval_sound (cev (stop << st_init << st_return << st_return << st_return << st_return << st_vl << st_case1_z << st_return << st_vl << st_case << st_app2 << st_return << st_vl << st_app1 << st_return << st_lam << st_fix << st_app << st_s << st_case1_s << st_return << st_vl << st_case << st_app2 << st_return << st_vl << st_app1 << st_return << st_lam << st_fix << st_app << st_s << st_case1_s << st_return << st_vl << st_case << st_app2 << st_return << st_vl << st_app1 << st_return << st_lam << st_fix << st_app << st_s << st_case1_s << st_return << st_vl << st_case << st_app2 << st_return << st_vl << st_app1 << st_return << st_lam << st_fix << st_app << st_s << st_case1_s << st_return << st_vl << st_case << st_app2 << st_return << st_return << st_return << st_vl << st_case1_z << st_return << st_vl << st_case << st_app2 << st_return << st_vl << st_app1 << st_return << st_lam << st_fix << st_app << st_s << st_case1_s << st_return << st_vl << st_case << st_app2 << st_return << st_vl << st_app1 << st_return << st_lam << st_fix << st_app << st_s << st_case1_s << st_return << st_vl << st_case << st_app2 << st_return << st_vl << st_case1_z << st_return << st_vl << st_case << st_app2 << st_return << st_z << st_case1_z << st_return << st_vl << st_case << st_app2 << st_return << st_vl << st_app1 << st_return << st_lam << st_fix << st_app << st_app1 << st_return << st_lam << st_fix << st_app2 << st_return << st_vl << st_app1 << st_return << st_vl << st_app << st_app << st_case1_s << st_return << st_vl << st_case << st_app2 << st_return << st_vl << st_app1 << st_return << st_lam << st_fix << st_app << st_app1 << st_return << st_lam << st_fix << st_app2 << st_return << st_vl << st_app1 << st_return << st_vl << st_app << st_app << st_case1_s << st_return << st_vl << st_case << st_app2 << st_return << st_vl << st_app1 << st_return << st_lam << st_fix << st_app << st_app1 << st_return << st_lam << st_fix << st_app2 << st_return << st_vl << st_app1 << st_return << st_vl << st_app << st_app << st_case1_s << st_return << st_vl << st_case << st_app2 << st_return << st_return << st_return << st_return << st_z << st_s << st_s << st_s << st_app1 << st_return << st_lam << st_fix << st_app2 << st_return << st_return << st_return << st_z << st_s << st_s << st_app1 << st_return << st_vl << st_app << st_app << st_return << st_lam << st_letv << st_return << st_lam << st_letv)) D %. +% Currently kills SML \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cpm/sources.lf b/new-tests/stelf-output/examples/compile/cpm/sources.lf new file mode 100644 index 0000000..3ccc50c --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cpm/sources.lf @@ -0,0 +1,7 @@ +%require %[ mini-mlv %] %. +%require %[ evalv %] %. +%require %[ cpm %] %. +%require %[ ceval %] %. +%require %[ ceval-sound %] %. +%require %[ ceval-complete %] %. +%require %[ proof-equiv %] %. diff --git a/new-tests/stelf-output/examples/compile/cpm/stelf.toml b/new-tests/stelf-output/examples/compile/cpm/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cpm/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/compile/cpm/test.lf b/new-tests/stelf-output/examples/compile/cpm/test.lf new file mode 100644 index 0000000..4a7945c --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cpm/test.lf @@ -0,0 +1,10 @@ +%require %[ mini-mlv %] %. +%require %[ evalv %] %. +%require %[ cpm %] %. +%require %[ ceval %] %. +%require %[ ceval-sound %] %. +%require %[ ceval-complete %] %. +%require %[ ceval-complete %] %. +%require %[ proof-equiv %] %. +%require %[ proof-equiv %] %. +%require %[ examples %] %. diff --git a/new-tests/stelf-output/examples/compile/cpm/theorems-thm.lf b/new-tests/stelf-output/examples/compile/cpm/theorems-thm.lf new file mode 100644 index 0000000..8d6e38d --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cpm/theorems-thm.lf @@ -0,0 +1,9 @@ +% Completeness of refining evaluation to small step semantics +%theorem ccf : forall* {E exp} {V val} forall {D eval E V} exists {C {K} {W} %pi (K # (return V) =>* (answer W)) %-> (K # (ev E) =>* (answer W))} true %. +%prove 20 D (ccf D _) %. +% Proof equivalence: "=>" direction +%theorem peq : forall* {K1 cont} {E1 exp} {V1 val} {V2 val} {C K1 # ev E1 =>* answer V1} {D eval E1 V2} {C' K1 # return V2 =>* answer V1} forall {SD csd C D C'} exists {CP ccp D C' C} true %. +%prove 6 SD (peq SD _) %. +% Proof equivalence: "<=" direction +%theorem peqr : forall* {K1 cont} {E1 exp} {V1 val} {V2 val} {C K1 # ev E1 =>* answer V1} {D eval E1 V2} {C' K1 # return V2 =>* answer V1} forall {CP ccp D C' C} exists {SD csd C D C'} true %. +%prove 6 CP (peqr CP _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cpm/theorems.lf b/new-tests/stelf-output/examples/compile/cpm/theorems.lf new file mode 100644 index 0000000..252b4c0 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cpm/theorems.lf @@ -0,0 +1,9 @@ +% Continuation style small step semantics refines the big step semantics +%sort ccf %. +%mode ccf %in %out %. +% Completeness Proof implies Soundness Proof +%sort peq %. +%mode peq %in %out %. +% Soundness Proof implies Completeness Proof +%sort peqr %. +%mode peqr %in %out %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cps/cps-eval.lf b/new-tests/stelf-output/examples/compile/cps/cps-eval.lf new file mode 100644 index 0000000..1be4ef4 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cps/cps-eval.lf @@ -0,0 +1,9 @@ +%%% Evaluation of CPS terms +%%% Author: Frank Pfenning +%sort ceval %. +%term ceval_vl ceval (vl+ V) V %. +%term ceval_case_z %pi (ceval (case+ z+ E2 E3) V) %<- (ceval E2 V) %. +%term ceval_case_s %pi (ceval (case+ (s+ V1') E2 E3) V) %<- (ceval (E3 V1') V) %. +%term ceval_fst %pi (ceval (fst+ (pair+ V1 V2) K) V) %<- (ceval (K V1) V) %. +%term ceval_snd %pi (ceval (snd+ (pair+ V1 V2) K) V) %<- (ceval (K V2) V) %. +%term ceval_app %pi (ceval (app+ (lam+ E1') V2 K) V) %<- (ceval (E1' V2 K) V) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cps/cps.lf b/new-tests/stelf-output/examples/compile/cps/cps.lf new file mode 100644 index 0000000..360014a --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cps/cps.lf @@ -0,0 +1,18 @@ +%%% Expressions in Continuation-Passing Style +%%% Author: Frank Pfenning +%sort cexp %. +%sort cval %. +%term z+ cval %. +%term s+ %pi cval %-> cval %. +%term case+ %pi cval %-> cexp %-> (%pi cval %-> cexp) %-> cexp %. +%term pair+ %pi cval %-> cval %-> cval %. +%term fst+ %pi cval %-> (%pi cval %-> cexp) %-> cexp %. +%term snd+ %pi cval %-> (%pi cval %-> cexp) %-> cexp %. +%term lam+ %pi (%pi cval %-> (%pi cval %-> cexp) %-> cexp) %-> cval %. +%term app+ %pi cval %-> cval %-> (%pi cval %-> cexp) %-> cexp %. +% no letv+ +% letn+ : ((cval -> cexp) -> cexp) -> +% (((cval -> cexp) -> cexp) -> cexp) -> cexp. +% omit treatment of letn+ for now. +% omit treatment of fix+ for now. +%term vl+ %pi cval %-> cexp %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cps/examples-quy.lf b/new-tests/stelf-output/examples/compile/cps/examples-quy.lf new file mode 100644 index 0000000..b8152cd --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cps/examples-quy.lf @@ -0,0 +1,4 @@ +%query 1 _ _ cmp (lam ([x] pair (snd x) (fst x))) ([x] vl+ x) E %. +%query 1 _ _ cmp (app (lam ([x] pair (snd x) (fst x))) (pair z (s z))) ([x] vl+ x) E %. +%query 1 _ _ ceval (vl+ (lam+ ([x' cval] [k %pi cval %-> cexp] snd+ x' ([x1' cval] fst+ x' ([x2' cval] k (pair+ x1' x2')))))) V %. +%query 1 _ _ ceval (app+ (lam+ ([x' cval] [k %pi cval %-> cexp] snd+ x' ([x1' cval] fst+ x' ([x2' cval] k (pair+ x1' x2'))))) (pair+ z+ (s+ z+)) ([x cval] vl+ x)) V %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cps/mini-ml.lf b/new-tests/stelf-output/examples/compile/cps/mini-ml.lf new file mode 100644 index 0000000..e166530 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cps/mini-ml.lf @@ -0,0 +1,15 @@ +%%% The Mini-ML Language +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort exp %. +%name exp %. +%term z exp %. +%term s %pi exp %-> exp %. +%term case %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%term pair %pi exp %-> exp %-> exp %. +%term fst %pi exp %-> exp %. +%term snd %pi exp %-> exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term letv %pi exp %-> (%pi exp %-> exp) %-> exp %. +%term letn %pi exp %-> (%pi exp %-> exp) %-> exp %. +%term fix %pi (%pi exp %-> exp) %-> exp %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cps/ml-cps.lf b/new-tests/stelf-output/examples/compile/cps/ml-cps.lf new file mode 100644 index 0000000..ea6b2c2 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cps/ml-cps.lf @@ -0,0 +1,28 @@ +%%% Compilation to Continuation-Passing Style +%%% Author: Frank Pfenning +%sort cmp %. +%term cmp_z cmp z K (K z+) %. +%term cmp_s %pi (cmp (s E) K E') %<- (cmp E ([x cval] K (s+ x)) E') %. +% note duplication of K +%term cmp_case + %pi (cmp (case E1 E2 E3) K E') + %<- (cmp E2 K E2') + %<- ({x exp} {x' cval} %pi ({K'} cmp x K' (K' x')) %-> (cmp (E3 x) K (E3' x'))) + %<- (cmp E1 ([x1 cval] case+ x1 E2' E3') E') %. +%term cmp_pair + %pi (cmp (pair E1 E2) K E') + %<- ({x1' cval} cmp E2 ([x2' cval] K (pair+ x1' x2')) (E2' x1')) + %<- (cmp E1 E2' E') %. +%term cmp_fst %pi (cmp (fst E) K E') %<- (cmp E ([x cval] fst+ x K) E') %. +%term cmp_snd %pi (cmp (snd E) K E') %<- (cmp E ([x cval] snd+ x K) E') %. +%term cmp_lam + %pi (cmp (lam E) K (K (lam+ E'))) + %<- ({x exp} {x' cval} %pi ({K'} cmp x K' (K' x')) %-> ({k %pi cval %-> cexp} cmp (E x) k (E' x' k))) %. +%term cmp_app + %pi (cmp (app E1 E2) K E') + %<- ({x1 cval} cmp E2 ([x2 cval] app+ x1 x2 K) (E2' x1)) + %<- (cmp E1 ([x1 cval] E2' x1) E') %. +%term cmp_letv + %pi (cmp (letv E1 E2) K E') + %<- ({x exp} {x' cval} %pi ({K'} cmp x K' (K' x')) %-> (cmp (E2 x) K (E2' x'))) + %<- (cmp E1 ([x cval] E2' x) E') %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cps/sources.lf b/new-tests/stelf-output/examples/compile/cps/sources.lf new file mode 100644 index 0000000..ac84b8f --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cps/sources.lf @@ -0,0 +1,4 @@ +%require %[ mini-ml %] %. +%require %[ cps %] %. +%require %[ ml-cps %] %. +%require %[ cps-eval %] %. diff --git a/new-tests/stelf-output/examples/compile/cps/stelf.toml b/new-tests/stelf-output/examples/compile/cps/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cps/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/compile/cps/test.lf b/new-tests/stelf-output/examples/compile/cps/test.lf new file mode 100644 index 0000000..4092926 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cps/test.lf @@ -0,0 +1,5 @@ +%require %[ mini-ml %] %. +%require %[ cps %] %. +%require %[ ml-cps %] %. +%require %[ cps-eval %] %. +%require %[ examples %] %. diff --git a/new-tests/stelf-output/examples/compile/cxm/evalv.lf b/new-tests/stelf-output/examples/compile/cxm/evalv.lf new file mode 100644 index 0000000..91f7b0b --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cxm/evalv.lf @@ -0,0 +1,28 @@ +%%% Mini-ML Natural Semantics +%%% Version with separate category of values +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort eval %. +%name eval %. +% Natural Numbers +%term ev_z eval z z* %. +%term ev_s %pi (eval (s E) (s* V)) %<- (eval E V) %. +%term ev_case_z %pi (eval (case E1 E2 E3) V) %<- (eval E1 z*) %<- (eval E2 V) %. +%term ev_case_s %pi (eval (case E1 E2 E3) V) %<- (eval E1 (s* V1')) %<- (eval (E3 V1') V) %. +% Pairs +%term ev_pair %pi (eval (pair E1 E2) (pair* V1 V2)) %<- (eval E1 V1) %<- (eval E2 V2) %. +%term ev_fst %pi (eval (fst E) V1) %<- (eval E (pair* V1 V2)) %. +%term ev_snd %pi (eval (snd E) V2) %<- (eval E (pair* V1 V2)) %. +% Functions +%term ev_lam eval (lam E) (lam* E) %. +%term ev_app + %pi (eval (app E1 E2) V) + %<- (eval E1 (lam* E1')) + %<- (eval E2 V2) + %<- (eval (E1' V2) V) %. +% Definitions +%term ev_letv %pi (eval (letv E1 E2) V) %<- (eval E1 V1) %<- (eval (E2 V1) V) %. +%term ev_letn %pi (eval (letn E1 E2) V) %<- (eval (E2 E1) V) %. +% Recursion +%term ev_fix %pi (eval (fix E) V) %<- (eval (E (fix E)) V) %. +% Values +%term ev_vl eval (vl V) V %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cxm/examples-quy.lf b/new-tests/stelf-output/examples/compile/cxm/examples-quy.lf new file mode 100644 index 0000000..099f395 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cxm/examples-quy.lf @@ -0,0 +1,2 @@ +%query 1 _ _ xeval (app (lam ([x] vl x)) z) V %. +%query 1 _ _ xeval (app (fix ([double] lam ([x] case (vl x) z ([x'] s (s (app double (vl x'))))))) (s (s (s z)))) V %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cxm/mini-mlv.lf b/new-tests/stelf-output/examples/compile/cxm/mini-mlv.lf new file mode 100644 index 0000000..fc25a14 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cxm/mini-mlv.lf @@ -0,0 +1,23 @@ +%%% The Mini-ML Language +%%% Version distinguishing expressions and values +%%% Author: Frank Pfenning +%sort exp %. +%name exp %. +%sort val %. +%name val %. +%term z exp %. +%term s %pi exp %-> exp %. +%term case %pi exp %-> exp %-> (%pi val %-> exp) %-> exp %. +%term pair %pi exp %-> exp %-> exp %. +%term fst %pi exp %-> exp %. +%term snd %pi exp %-> exp %. +%term lam %pi (%pi val %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term letv %pi exp %-> (%pi val %-> exp) %-> exp %. +%term letn %pi exp %-> (%pi exp %-> exp) %-> exp %. +%term fix %pi (%pi exp %-> exp) %-> exp %. +%term vl %pi val %-> exp %. +%term z* val %. +%term s* %pi val %-> val %. +%term pair* %pi val %-> val %-> val %. +%term lam* %pi (%pi val %-> exp) %-> val %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/cxm/sources.lf b/new-tests/stelf-output/examples/compile/cxm/sources.lf new file mode 100644 index 0000000..19c4350 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cxm/sources.lf @@ -0,0 +1,3 @@ +%require %[ mini-mlv %] %. +%require %[ evalv %] %. +%require %[ xeval %] %. diff --git a/new-tests/stelf-output/examples/compile/cxm/stelf.toml b/new-tests/stelf-output/examples/compile/cxm/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cxm/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/compile/cxm/test.lf b/new-tests/stelf-output/examples/compile/cxm/test.lf new file mode 100644 index 0000000..5b6945c --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cxm/test.lf @@ -0,0 +1,4 @@ +%require %[ mini-mlv %] %. +%require %[ evalv %] %. +%require %[ xeval %] %. +%require %[ examples %] %. diff --git a/new-tests/stelf-output/examples/compile/cxm/xeval.lf b/new-tests/stelf-output/examples/compile/cxm/xeval.lf new file mode 100644 index 0000000..f76be5c --- /dev/null +++ b/new-tests/stelf-output/examples/compile/cxm/xeval.lf @@ -0,0 +1,66 @@ +%%% Contextual Semantics +%%% Author: Frank Pfenning +%%% One-Step Reductions +%sort ==> %. +%prec %none 8 ==> %. +%term red_z z ==> (vl z*) %. +%term red_s s (vl V) ==> vl (s* V) %. +%term red_case_z case (vl z*) E2 E3 ==> E2 %. +%term red_case_s case (vl (s* V1')) E2 E3 ==> E3 V1' %. +%term red_pair pair (vl V1) (vl V2) ==> vl (pair* V1 V2) %. +%term red_fst fst (vl (pair* V1 V2)) ==> (vl V1) %. +%term red_snd snd (vl (pair* V2 V2)) ==> (vl V2) %. +%term red_lam lam E ==> vl (lam* E) %. +%term red_app app (vl (lam* E1')) (vl V2) ==> E1' V2 %. +%term red_letv letv (vl V1) E2 ==> E2 V1 %. +%term red_letn letn E1 E2 ==> E2 E1 %. +%term red_fix fix E ==> E (fix E) %. +% no red_vl rule +%%% Redices +%sort redex %. +%term rdx %pi (redex E) %<- (E ==> E') %. +%%% Splitting an Expression +%sort split %. +%{ split C E C' E' + Evaluation context C and expression E are given, + C' and E' are constructed. + Invariant: (C E) == (C' E') +}% +% Redices +%term sp_redex %pi (split C E C E) %<- (redex E) %. +% Natural Numbers +% no sp_z +%term sp_s %pi (split C (s E1) C' E') %<- (split ([h exp] C (s h)) E1 C' E') %. +%term sp_case + %pi (split C (case E1 E2 E3) C' E') + %<- (split ([h exp] C (case h E2 E3)) E1 C' E') %. +% Pairs +%term sp_pair2 + %pi (split C (pair (vl V1) E2) C' E') + %<- (split ([h exp] C (pair (vl V1) h)) E2 C' E') %. +%term sp_pair1 %pi (split C (pair E1 E2) C' E') %<- (split ([h exp] C (pair h E2)) E1 C' E') %. +%term sp_fst %pi (split C (fst E) C' E') %<- (split ([h exp] C (fst h)) E C' E') %. +%term sp_snd %pi (split C (snd E) C' E') %<- (split ([h exp] C (snd h)) E C' E') %. +% Functions +% no sp_lam +%term sp_app2 + %pi (split C (app (vl V1) E2) C' E') + %<- (split ([h exp] C (app (vl V1) h)) E2 C' E') %. +%term sp_app1 %pi (split C (app E1 E2) C' E') %<- (split ([h exp] C (app h E2)) E1 C' E') %. +% Definitions +%term sp_letv %pi (split C (letv E1 E2) C' E') %<- (split ([h exp] C (letv h E2)) E1 C' E') %. +% no sp_letn +% Recursion +% no sp_fix +% Values +% no sp_vl +%%% Top-Level Splitting +%sort split_exp %. +%term spe %pi (split_exp E C E') %<- (split ([h exp] h) E C E') %. +%%% One-Step Contextual Evaluation +%sort one_step %. +%term ostp %pi (one_step E (C R')) %<- (split_exp E C R) %<- (R ==> R') %. +%%% Full Contextual Evaluation +%sort xeval %. +%term xev_vl xeval (vl V) V %. +%term xev_step %pi (xeval E V) %<- (one_step E E') %<- (xeval E' V) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/debruijn/debruijn.lf b/new-tests/stelf-output/examples/compile/debruijn/debruijn.lf new file mode 100644 index 0000000..c828cac --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn/debruijn.lf @@ -0,0 +1,32 @@ +%%% Expression representation using de Bruijn indices +%%% Author: Frank Pfenning, based on [Hannan & Pfenning 92] +%sort exp' %. +%name exp' %. +%term 1 exp' %. +%term ^ %pi exp' %-> exp' %. +%prec %postfix 20 ^ %. +%term z' exp' %. +%term s' %pi exp' %-> exp' %. +%term case' %pi exp' %-> exp' %-> exp' %-> exp' %. +%term pair' %pi exp' %-> exp' %-> exp' %. +%term fst' %pi exp' %-> exp' %. +%term snd' %pi exp' %-> exp' %. +%term lam' %pi exp' %-> exp' %. +%term app' %pi exp' %-> exp' %-> exp' %. +%term letv' %pi exp' %-> exp' %-> exp' %. +%term letn' %pi exp' %-> exp' %-> exp' %. +%term fix' %pi exp' %-> exp' %. +% Environments and values +%sort env %. +%name env %. +%sort val %. +%name val %. +%term empty env %. +%term ; %pi env %-> val %-> env %. +%prec %left 10 ; %. +%term + %pi env %-> exp' %-> env %. +%prec %left 10 + %. +%term z* val %. +%term s* %pi val %-> val %. +%term pair* %pi val %-> val %-> val %. +%term clo %pi env %-> exp' %-> val %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/debruijn/eval.lf b/new-tests/stelf-output/examples/compile/debruijn/eval.lf new file mode 100644 index 0000000..043c09e --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn/eval.lf @@ -0,0 +1,25 @@ +%%% Mini-ML Natural Semantics +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort eval %. +%name eval %. +% Natural Numbers +%term ev_z eval z z %. +%term ev_s %pi (eval (s E) (s V)) %<- (eval E V) %. +%term ev_case_z %pi (eval (case E1 E2 E3) V) %<- (eval E1 z) %<- (eval E2 V) %. +%term ev_case_s %pi (eval (case E1 E2 E3) V) %<- (eval E1 (s V1')) %<- (eval (E3 V1') V) %. +% Pairs +%term ev_pair %pi (eval (pair E1 E2) (pair V1 V2)) %<- (eval E1 V1) %<- (eval E2 V2) %. +%term ev_fst %pi (eval (fst E) V1) %<- (eval E (pair V1 V2)) %. +%term ev_snd %pi (eval (snd E) V2) %<- (eval E (pair V1 V2)) %. +% Functions +%term ev_lam eval (lam E) (lam E) %. +%term ev_app + %pi (eval (app E1 E2) V) + %<- (eval E1 (lam E1')) + %<- (eval E2 V2) + %<- (eval (E1' V2) V) %. +% Definitions +%term ev_letv %pi (eval (letv E1 E2) V) %<- (eval E1 V1) %<- (eval (E2 V1) V) %. +%term ev_letn %pi (eval (letn E1 E2) V) %<- (eval (E2 E1) V) %. +% Recursion +%term ev_fix %pi (eval (fix E) V) %<- (eval (E (fix E)) V) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/debruijn/examples-quy.lf b/new-tests/stelf-output/examples/compile/debruijn/examples-quy.lf new file mode 100644 index 0000000..6c0075f --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn/examples-quy.lf @@ -0,0 +1,50 @@ +%query 1 _ _ feval empty (app' (lam' 1) z') W %. +%query 1 _ _ feval empty (letn' (lam' 1) (letv' (app' 1 1) (app' 1 1))) W %. +%query 1 _ _ trans empty F (app (lam ([x] lam ([y] x))) (lam ([z] z))) %. +%query 1 _ _ feval empty (app' (lam' (lam' (1 ^))) (lam' 1)) W %. +%query 1 _ _ vtrans (clo (empty ; clo empty (lam' 1)) (lam' (1 ^))) V %. +%query 2 _ _ trans empty F (lam ([x] lam ([y] app x x))) %. +%{ +sigma [D: eval (lam [x] x) V] +sigma [C: trans empty F (lam [x] x)] +map_eval D C D' U. +}% +%query 1 _ _ map_eval ev_lam (tr_lam ([w val] [x exp] [U1 vtrans w x] tr_1 U1)) D' U %. +%{ +sigma [D: eval (app (lam [x] x) z) V] +sigma [C: trans empty F (app (lam [x] x) z)] +map_eval D C D' U. +}% +%query 1 _ _ map_eval (ev_app ev_z ev_z ev_lam) (tr_app tr_z (tr_lam ([w val] [x exp] [U1 vtrans w x] tr_1 U1))) D' U %. +%{ +sigma [D: eval (fix [f] (lam [y] f)) V] +sigma [C: trans empty F (fix [f] (lam [y] f))] +map_eval D C D' U. + +sigma [D: eval (app (fix [f] (lam [y] f)) z) V] +sigma [C: trans empty F (app (fix [f] (lam [y] f)) z)] +map_eval D C D' U. +}% +%{ +sigma [D: eval (app (fix [dbl] lam [n] + case n + z + [n'] (s (s (app dbl n')))) + (s z)) + V] +sigma [C: trans empty F (app (fix [dbl] lam [n] + case n + z + [n'] (s (s (app dbl n')))) + (s z))] +map_eval D C D' U. +}% +%query 3 _ _ trans empty F (app (fix ([dbl] lam ([n] case n z ([n'] s (s (app dbl n')))))) (s z)) %. +% Take the second solution from above +%query 1 _ _ map_eval (ev_app (ev_case_s (ev_s (ev_s (ev_app (ev_case_z ev_z ev_z) ev_z (ev_fix ev_lam)))) (ev_s ev_z)) (ev_s ev_z) (ev_fix ev_lam)) (tr_app (tr_s tr_z) (tr_fix ([f exp'] [x exp] [C2 trans empty f x] tr_lam ([w val] [x3 exp] [U3 vtrans w x3] tr_case ([w2 val] [x4 exp] [U4 vtrans w2 x4] tr_s (tr_s (tr_app (tr_1 U4) (tr_^ (tr_^ (tr_1+ C2)))))) (tr_^ tr_z) (tr_1 U3))))) D' U %. +%{ +sigma [D:eval (letn (lam [x] x) ([f] letv (app f f) ([g] app g g))) V] +sigma [C:trans empty F (letn (lam [x] x) ([f] letv (app f f) ([g] app g g)))] +map_eval D C D' U. +}% +%query 1 _ _ map_eval (ev_letn (ev_letv (ev_app ev_lam ev_lam ev_lam) (ev_app ev_lam ev_lam ev_lam))) (tr_letn ([f exp'] [x exp] [C1 trans empty f x] tr_letv ([w val] [x1 exp] [U1 vtrans w x1] tr_app (tr_1 U1) (tr_1 U1)) (tr_app (tr_1+ C1) (tr_1+ C1))) (tr_lam ([w val] [x exp] [U2 vtrans w x] tr_1 U2))) D' U %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/debruijn/feval.lf b/new-tests/stelf-output/examples/compile/debruijn/feval.lf new file mode 100644 index 0000000..4af9f76 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn/feval.lf @@ -0,0 +1,33 @@ +%%% Mini-ML Natural Semantics +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort feval %. +%name feval %. +% Variables +%term fev_1 feval (K ; W) 1 W %. +%term fev_^ %pi (feval (K ; W') (F ^) W) %<- (feval K F W) %. +%term fev_1+ %pi (feval (K + F) 1 W) %<- (feval K F W) %. +%term fev_^+ %pi (feval (K + F') (F ^) W) %<- (feval K F W) %. +% Natural Numbers +%term fev_z feval K z' z* %. +%term fev_s %pi (feval K (s' F) (s* W)) %<- (feval K F W) %. +%term fev_case_z %pi (feval K (case' F1 F2 F3) W) %<- (feval K F1 z*) %<- (feval K F2 W) %. +%term fev_case_s + %pi (feval K (case' F1 F2 F3) W) + %<- (feval K F1 (s* W1)) + %<- (feval (K ; W1) F3 W) %. +% Pairs +%term fev_pair %pi (feval K (pair' F1 F2) (pair* W1 W2)) %<- (feval K F1 W1) %<- (feval K F2 W2) %. +%term fev_fst %pi (feval K (fst' F) W1) %<- (feval K F (pair* W1 W2)) %. +%term fev_snd %pi (feval K (snd' F) W2) %<- (feval K F (pair* W1 W2)) %. +% Functions +%term fev_lam feval K (lam' F) (clo K (lam' F)) %. +%term fev_app + %pi (feval K (app' F1 F2) W) + %<- (feval K F1 (clo K' (lam' F1'))) + %<- (feval K F2 W2) + %<- (feval (K' ; W2) F1' W) %. +% Definitions +%term fev_letv %pi (feval K (letv' F1 F2) W) %<- (feval K F1 W1) %<- (feval (K ; W1) F2 W) %. +%term fev_letn %pi (feval K (letn' F1 F2) W) %<- (feval (K + F1) F2 W) %. +% Recursion +%term fev_fix %pi (feval K (fix' F) W) %<- (feval (K + (fix' F)) F W) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/debruijn/map-eval.lf b/new-tests/stelf-output/examples/compile/debruijn/map-eval.lf new file mode 100644 index 0000000..cabcc8e --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn/map-eval.lf @@ -0,0 +1,110 @@ +%%% Mapping evaluations to evaluations in environments. +%%% Author: Frank Pfenning, based on [Hannan & Pfenning 92] +%%% Modified: Robert J. Simmons +%sort map_eval %. +% Variables +%term mp_1 %pi (map_eval D (tr_1 U1) fev_1 U1) %<- (vtrans_val U1 P) %<- (val_eval P D) %. +%term mp_^ %pi (map_eval D (tr_^ C1) (fev_^ D1') U1) %<- (map_eval D C1 D1' U1) %. +%term mp_1+ %pi (map_eval D (tr_1+ C1) (fev_1+ D1') U1) %<- (map_eval D C1 D1' U1) %. +%term mp_^+ %pi (map_eval D (tr_^+ C1) (fev_^+ D1') U1) %<- (map_eval D C1 D1' U1) %. +% Natural Numbers +%term mp_z map_eval ev_z tr_z fev_z vtr_z %. +%term mp_s + %pi (map_eval (ev_s D1) (tr_s C1) (fev_s D1') (vtr_s U1)) + %<- (map_eval D1 C1 D1' U1) %. +%term mp_case_z + %pi (map_eval (ev_case_z D2 D1) (tr_case C3 C2 C1) (fev_case_z D2' D1') U2) + %<- (map_eval D1 C1 D1' vtr_z) + %<- (map_eval D2 C2 D2' U2) %. +%term mp_case_s + %pi (map_eval (ev_case_s D3 D1) (tr_case C3 C2 C1) (fev_case_s D3' D1') U3) + %<- (map_eval D1 C1 D1' (vtr_s U1')) + %<- (map_eval D3 (C3 W1' V1' U1') D3' U3) %. +% Pairs +%term mp_pair + %pi (map_eval (ev_pair D2 D1) (tr_pair C2 C1) (fev_pair D2' D1') (vtr_pair U2 U1)) + %<- (map_eval D1 C1 D1' U1) + %<- (map_eval D2 C2 D2' U2) %. +%term mp_fst + %pi (map_eval (ev_fst D1) (tr_fst C1) (fev_fst D1') U1) + %<- (map_eval D1 C1 D1' (vtr_pair U2 U1)) %. +%term mp_snd + %pi (map_eval (ev_snd D1) (tr_snd C1) (fev_snd D1') U2) + %<- (map_eval D1 C1 D1' (vtr_pair U2 U1)) %. +% Functions +%term mp_lam map_eval ev_lam (tr_lam C) fev_lam (vtr_lam (tr_lam C)) %. +%term mp_app + %pi (map_eval (ev_app D3 D2 D1) (tr_app C2 C1) (fev_app D3' D2' D1') U3) + %<- (map_eval D1 C1 D1' (vtr_lam (tr_lam C3))) + %<- (map_eval D2 C2 D2' U2) + %<- (map_eval D3 (C3 W2 V2 U2) D3' U3) %. +% Definitions +%term mp_letv + %pi (map_eval (ev_letv D2 D1) (tr_letv C2 C1) (fev_letv D2' D1') U2) + %<- (map_eval D1 C1 D1' U1) + %<- (map_eval D2 (C2 W1 V1 U1) D2' U2) %. +%term mp_letn + %pi (map_eval (ev_letn D2) (tr_letn C2 C1) (fev_letn D2') U2) + %<- (map_eval D2 (C2 F1 E1 C1) D2' U2) %. +% Recursion +%term mp_fix + %pi (map_eval (ev_fix D1) (tr_fix C1) (fev_fix D1') U1) + %<- (map_eval D1 (C1 (fix' F1) (fix E1) (tr_fix C1)) D1' U1) %. +%mode map_eval %in %in %out %out %. +%worlds () (map_eval D C D' U) %. +% problem here---fixed in 2001 version +% %covers map_eval +D +C -D' -U. +% The copy of map_eval exists only to allow totality to be checked +% using a different mode. +%sort map_eval' %. +% Variables +%term mp_1 %pi (map_eval' D (tr_1 U1) fev_1 U1) %<- (vtrans_val U1 P) %<- (val_eval P D) %. +%term mp_^ %pi (map_eval' D (tr_^ C1) (fev_^ D1') U1) %<- (map_eval' D C1 D1' U1) %. +%term mp_1+ %pi (map_eval' D (tr_1+ C1) (fev_1+ D1') U1) %<- (map_eval' D C1 D1' U1) %. +%term mp_^+ %pi (map_eval' D (tr_^+ C1) (fev_^+ D1') U1) %<- (map_eval' D C1 D1' U1) %. +% Natural Numbers +%term mp_z map_eval' ev_z tr_z fev_z vtr_z %. +%term mp_s + %pi (map_eval' (ev_s D1) (tr_s C1) (fev_s D1') (vtr_s U1)) + %<- (map_eval' D1 C1 D1' U1) %. +%term mp_case_z + %pi (map_eval' (ev_case_z D2 D1) (tr_case C3 C2 C1) (fev_case_z D2' D1') U2) + %<- (map_eval' D1 C1 D1' vtr_z) + %<- (map_eval' D2 C2 D2' U2) %. +%term mp_case_s + %pi (map_eval' (ev_case_s D3 D1) (tr_case C3 C2 C1) (fev_case_s D3' D1') U3) + %<- (map_eval' D1 C1 D1' (vtr_s U1')) + %<- (map_eval' D3 (C3 W1' V1' U1') D3' U3) %. +% Pairs +%term mp_pair + %pi (map_eval' (ev_pair D2 D1) (tr_pair C2 C1) (fev_pair D2' D1') (vtr_pair U2 U1)) + %<- (map_eval' D1 C1 D1' U1) + %<- (map_eval' D2 C2 D2' U2) %. +%term mp_fst + %pi (map_eval' (ev_fst D1) (tr_fst C1) (fev_fst D1') U1) + %<- (map_eval' D1 C1 D1' (vtr_pair U2 U1)) %. +%term mp_snd + %pi (map_eval' (ev_snd D1) (tr_snd C1) (fev_snd D1') U2) + %<- (map_eval' D1 C1 D1' (vtr_pair U2 U1)) %. +% Functions +%term mp_lam map_eval' ev_lam (tr_lam C) fev_lam (vtr_lam (tr_lam C)) %. +%term mp_app + %pi (map_eval' (ev_app D3 D2 D1) (tr_app C2 C1) (fev_app D3' D2' D1') U3) + %<- (map_eval' D1 C1 D1' (vtr_lam (tr_lam C3))) + %<- (map_eval' D2 C2 D2' U2) + %<- (map_eval' D3 (C3 W2 V2 U2) D3' U3) %. +% Definitions +%term mp_letv + %pi (map_eval' (ev_letv D2 D1) (tr_letv C2 C1) (fev_letv D2' D1') U2) + %<- (map_eval' D1 C1 D1' U1) + %<- (map_eval' D2 (C2 W1 V1 U1) D2' U2) %. +%term mp_letn + %pi (map_eval' (ev_letn D2) (tr_letn C2 C1) (fev_letn D2') U2) + %<- (map_eval' D2 (C2 F1 E1 C1) D2' U2) %. +% Recursion +%term mp_fix + %pi (map_eval' (ev_fix D1) (tr_fix C1) (fev_fix D1') U1) + %<- (map_eval' D1 (C1 (fix' F1) (fix E1) (tr_fix C1)) D1' U1) %. +%mode map_eval' %out %in %in %out %. +%worlds () (map_eval' D C D' U) %. +%covers map_eval' %out %in %in %out %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/debruijn/mini-ml.lf b/new-tests/stelf-output/examples/compile/debruijn/mini-ml.lf new file mode 100644 index 0000000..e166530 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn/mini-ml.lf @@ -0,0 +1,15 @@ +%%% The Mini-ML Language +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort exp %. +%name exp %. +%term z exp %. +%term s %pi exp %-> exp %. +%term case %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%term pair %pi exp %-> exp %-> exp %. +%term fst %pi exp %-> exp %. +%term snd %pi exp %-> exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term letv %pi exp %-> (%pi exp %-> exp) %-> exp %. +%term letn %pi exp %-> (%pi exp %-> exp) %-> exp %. +%term fix %pi (%pi exp %-> exp) %-> exp %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/debruijn/sources.lf b/new-tests/stelf-output/examples/compile/debruijn/sources.lf new file mode 100644 index 0000000..a90209c --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn/sources.lf @@ -0,0 +1,8 @@ +%require %[ mini-ml %] %. +%require %[ value %] %. +%require %[ debruijn %] %. +%require %[ trans %] %. +%require %[ feval %] %. +%require %[ eval %] %. +%require %[ val-lemmas %] %. +%require %[ map-eval %] %. diff --git a/new-tests/stelf-output/examples/compile/debruijn/stelf.toml b/new-tests/stelf-output/examples/compile/debruijn/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/compile/debruijn/test.lf b/new-tests/stelf-output/examples/compile/debruijn/test.lf new file mode 100644 index 0000000..d85c5ba --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn/test.lf @@ -0,0 +1,9 @@ +%require %[ mini-ml %] %. +%require %[ value %] %. +%require %[ debruijn %] %. +%require %[ trans %] %. +%require %[ feval %] %. +%require %[ eval %] %. +%require %[ val-lemmas %] %. +%require %[ map-eval %] %. +%require %[ examples %] %. diff --git a/new-tests/stelf-output/examples/compile/debruijn/trans.lf b/new-tests/stelf-output/examples/compile/debruijn/trans.lf new file mode 100644 index 0000000..62221d3 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn/trans.lf @@ -0,0 +1,58 @@ +%%% Translation to representation using de Bruijn indices +%%% Author: Frank Pfenning, based on [Hannan & Pfenning 92] +%sort trans %. +%name trans %. +%sort vtrans %. +%name vtrans %. +% Natural numbers +%term tr_z trans K z' z %. +%term tr_s %pi (trans K (s' F) (s E)) %<- (trans K F E) %. +%term tr_case + %pi (trans K (case' F1 F2 F3) (case E1 E2 E3)) + %<- (trans K F1 E1) + %<- (trans K F2 E2) + %<- ({w val} {x exp} %pi (vtrans w x) %-> (trans (K ; w) F3 (E3 x))) %. +% Pairs +%term tr_pair %pi (trans K (pair' F1 F2) (pair E1 E2)) %<- (trans K F1 E1) %<- (trans K F2 E2) %. +%term tr_fst %pi (trans K (fst' F1) (fst E1)) %<- (trans K F1 E1) %. +%term tr_snd %pi (trans K (snd' F1) (snd E1)) %<- (trans K F1 E1) %. +% Functions +%term tr_lam + %pi (trans K (lam' F) (lam E)) + %<- ({w val} {x exp} %pi (vtrans w x) %-> (trans (K ; w) F (E x))) %. +%term tr_app %pi (trans K (app' F1 F2) (app E1 E2)) %<- (trans K F1 E1) %<- (trans K F2 E2) %. +% Definitions +%term tr_letv + %pi (trans K (letv' F1 F2) (letv E1 E2)) + %<- (trans K F1 E1) + %<- ({w val} {x exp} %pi (vtrans w x) %-> (trans (K ; w) F2 (E2 x))) %. +%term tr_letn + %pi (trans K (letn' F1 F2) (letn E1 E2)) + %<- (trans K F1 E1) + %<- ({f exp'} {x exp} %pi (trans K f x) %-> (trans (K + f) F2 (E2 x))) %. +% Recursion +%term tr_fix + %pi (trans K (fix' F) (fix E)) + %<- ({f exp'} {x exp} %pi (trans K f x) %-> (trans (K + f) F (E x))) %. +% Variables +%term tr_1 %pi (trans (K ; W) 1 E) %<- (vtrans W E) %. +%term tr_^ %pi (trans (K ; W) (F ^) E) %<- (trans K F E) %. +%term tr_1+ %pi (trans (K + F) 1 E) %<- (trans K F E) %. +%term tr_^+ %pi (trans (K + F') (F ^) E) %<- (trans K F E) %. +% Natural number values +%term vtr_z vtrans z* z %. +%term vtr_s %pi (vtrans (s* W) (s V)) %<- (vtrans W V) %. +% Pair values +%term vtr_pair %pi (vtrans (pair* W1 W2) (pair V1 V2)) %<- (vtrans W1 V1) %<- (vtrans W2 V2) %. +% Function values +%term vtr_lam %pi (vtrans (clo K (lam' F)) (lam E)) %<- (trans K (lam' F) (lam E)) %. +%mode trans %in %in %out %. +%mode vtrans %in %out %. +%block tr {K env} [f exp'] [x exp] [u trans K f x]%. +%block vtr [w val] [x exp] [v vtrans w x]%. +%worlds (tr vtr) (trans K F E) (vtrans W V) %. +% cases for F are covered, although totality is not obvious +%covers trans %star %in %star %. +% cases for W are not covered, only closures over lambda! +% fails: +% %covers vtrans +W -V. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/debruijn/val-lemmas.lf b/new-tests/stelf-output/examples/compile/debruijn/val-lemmas.lf new file mode 100644 index 0000000..b95c725 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn/val-lemmas.lf @@ -0,0 +1,26 @@ +%%% Lemmas about values. +%%% Author: Frank Pfenning +%%% Values correspond to value expressions. +%sort vtrans_val %. +%term vtrv_z vtrans_val vtr_z val_z %. +%term vtrv_s %pi (vtrans_val (vtr_s U1) (val_s P1)) %<- (vtrans_val U1 P1) %. +%term vtrv_pair + %pi (vtrans_val (vtr_pair U2 U1) (val_pair P2 P1)) + %<- (vtrans_val U1 P1) + %<- (vtrans_val U2 P2) %. +%term vtrv_lam vtrans_val (vtr_lam U1) val_lam %. +%mode vtrans_val %in %out %. +%worlds () (vtrans_val U P) %. +%covers vtrans_val %in %out %. +%%% Values evaluate only to themselves. +%sort val_eval %. +%term vev_z val_eval val_z ev_z %. +%term vev_s %pi (val_eval (val_s P1) (ev_s D1)) %<- (val_eval P1 D1) %. +%term vev_pair + %pi (val_eval (val_pair P2 P1) (ev_pair D2 D1)) + %<- (val_eval P1 D1) + %<- (val_eval P2 D2) %. +%term vev_lam val_eval val_lam ev_lam %. +%mode val_eval %in %out %. +%worlds () (val_eval P D) %. +%covers val_eval %in %out %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/debruijn/value.lf b/new-tests/stelf-output/examples/compile/debruijn/value.lf new file mode 100644 index 0000000..ded734b --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn/value.lf @@ -0,0 +1,8 @@ +%%% Definition of Values +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort value %. +%name value %. +%term val_z value z %. +%term val_s %pi (value (s E)) %<- (value E) %. +%term val_pair %pi (value (pair E1 E2)) %<- (value E1) %<- (value E2) %. +%term val_lam value (lam E) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/debruijn1/debruijn.lf b/new-tests/stelf-output/examples/compile/debruijn1/debruijn.lf new file mode 100644 index 0000000..9d13101 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn1/debruijn.lf @@ -0,0 +1,20 @@ +%%% Expression representation using de Bruijn indices +%%% Version restricted to pure lambda-calculus +%%% Author: Frank Pfenning, based on [Hannan & Pfenning 92] +% Expressions +%sort exp' %. +%name exp' %. +%term 1 exp' %. +%term ^ %pi exp' %-> exp' %. +%prec %postfix 20 ^ %. +%term lam' %pi exp' %-> exp' %. +%term app' %pi exp' %-> exp' %-> exp' %. +% Environments and values +%sort env %. +%name env %. +%sort val %. +%name val %. +%term empty env %. +%term ; %pi env %-> val %-> env %. +%prec %left 10 ; %. +%term clo %pi env %-> exp' %-> val %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/debruijn1/eval.lf b/new-tests/stelf-output/examples/compile/debruijn1/eval.lf new file mode 100644 index 0000000..e658c33 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn1/eval.lf @@ -0,0 +1,12 @@ +%%% Mini-ML Natural Semantics +%%% Version restricted to pure lambda-calculus +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort eval %. +%name eval %. +% Functions +%term ev_lam eval (lam E) (lam E) %. +%term ev_app + %pi (eval (app E1 E2) V) + %<- (eval E1 (lam E1')) + %<- (eval E2 V2) + %<- (eval (E1' V2) V) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/debruijn1/examples-quy.lf b/new-tests/stelf-output/examples/compile/debruijn1/examples-quy.lf new file mode 100644 index 0000000..0f2f9ca --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn1/examples-quy.lf @@ -0,0 +1,4 @@ +%query 1 _ _ trans empty F (app (lam ([x] lam ([y] x))) (lam ([z] z))) %. +%query 1 _ _ feval empty (app' (lam' (lam' (1 ^))) (lam' 1)) W %. +%query 1 _ _ vtrans (clo (empty ; clo empty (lam' 1)) (lam' (1 ^))) V %. +%query 2 _ _ trans empty F (lam ([x] lam ([y] app x x))) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/debruijn1/feval.lf b/new-tests/stelf-output/examples/compile/debruijn1/feval.lf new file mode 100644 index 0000000..d811272 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn1/feval.lf @@ -0,0 +1,15 @@ +%%% Mini-ML Natural Semantics +%%% Version restricted to pure lambda-calculus +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort feval %. +%name feval %. +% Variables +%term fev_1 feval (K ; W) 1 W %. +%term fev_^ %pi (feval (K ; W') (F ^) W) %<- (feval K F W) %. +% Functions +%term fev_lam feval K (lam' F) (clo K (lam' F)) %. +%term fev_app + %pi (feval K (app' F1 F2) W) + %<- (feval K F1 (clo K' (lam' F1'))) + %<- (feval K F2 W2) + %<- (feval (K' ; W2) F1' W) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/debruijn1/map-eval.lf b/new-tests/stelf-output/examples/compile/debruijn1/map-eval.lf new file mode 100644 index 0000000..51c211d --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn1/map-eval.lf @@ -0,0 +1,12 @@ +%%% Mapping evaluations to evaluations in environments. +%%% Version restricted to pure lambda-calculus +%%% Author: Frank Pfenning, based on [Hannan & Pfenning 92] +%sort map_eval %. +%term mp_1 map_eval ev_lam (tr_1 (vtr_lam (tr_lam C2))) fev_1 (vtr_lam (tr_lam C2)) %. +%term mp_^ %pi (map_eval D (tr_^ C1) (fev_^ D1') U1) %<- (map_eval D C1 D1' U1) %. +%term mp_lam map_eval ev_lam (tr_lam C1) fev_lam (vtr_lam (tr_lam C1)) %. +%term mp_app + %pi (map_eval (ev_app D3 D2 D1) (tr_app C2 C1) (fev_app D3' D2' D1') U3) + %<- (map_eval D1 C1 D1' (vtr_lam (tr_lam C3))) + %<- (map_eval D2 C2 D2' U2) + %<- (map_eval D3 (C3 W2 V2 U2) D3' U3) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/debruijn1/mini-ml.lf b/new-tests/stelf-output/examples/compile/debruijn1/mini-ml.lf new file mode 100644 index 0000000..f2e752f --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn1/mini-ml.lf @@ -0,0 +1,7 @@ +%%% The Mini-ML Language +%%% Version restricted to pure lambda-calculus +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort exp %. +%name exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/compile/debruijn1/sources.lf b/new-tests/stelf-output/examples/compile/debruijn1/sources.lf new file mode 100644 index 0000000..248ca5f --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn1/sources.lf @@ -0,0 +1,6 @@ +%require %[ mini-ml %] %. +%require %[ debruijn %] %. +%require %[ trans %] %. +%require %[ feval %] %. +%require %[ eval %] %. +%require %[ map-eval %] %. diff --git a/new-tests/stelf-output/examples/compile/debruijn1/stelf.toml b/new-tests/stelf-output/examples/compile/debruijn1/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn1/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/compile/debruijn1/test.lf b/new-tests/stelf-output/examples/compile/debruijn1/test.lf new file mode 100644 index 0000000..796c508 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn1/test.lf @@ -0,0 +1,7 @@ +%require %[ mini-ml %] %. +%require %[ debruijn %] %. +%require %[ trans %] %. +%require %[ feval %] %. +%require %[ eval %] %. +%require %[ map-eval %] %. +%require %[ examples %] %. diff --git a/new-tests/stelf-output/examples/compile/debruijn1/trans.lf b/new-tests/stelf-output/examples/compile/debruijn1/trans.lf new file mode 100644 index 0000000..0ca41e4 --- /dev/null +++ b/new-tests/stelf-output/examples/compile/debruijn1/trans.lf @@ -0,0 +1,17 @@ +%%% Translation to representation using de Bruijn indices +%%% Version restricted to pure lambda-calculus +%%% Author: Frank Pfenning, based on [Hannan & Pfenning 92] +%sort trans %. +%name trans %. +%sort vtrans %. +%name vtrans %. +% Functions +%term tr_lam + %pi (trans K (lam' F) (lam E)) + %<- ({w val} {x exp} %pi (vtrans w x) %-> (trans (K ; w) F (E x))) %. +%term tr_app %pi (trans K (app' F1 F2) (app E1 E2)) %<- (trans K F1 E1) %<- (trans K F2 E2) %. +% Variables +%term tr_1 %pi (trans (K ; W) 1 E) %<- (vtrans W E) %. +%term tr_^ %pi (trans (K ; W) (F ^) E) %<- (trans K F E) %. +% Values +%term vtr_lam %pi (vtrans (clo K (lam' F)) (lam E)) %<- (trans K (lam' F) (lam E)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/corr.lf b/new-tests/stelf-output/examples/cpsocc/corr.lf new file mode 100644 index 0000000..c7d8d8a --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/corr.lf @@ -0,0 +1,12 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% var-validity of mixed stacks +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort vvalMS %. +%term vvalMS_init vvalMS dot dot %. +%term vvalMS_v %pi (vvalMS (XXi , V) (Xi , V)) %<- (vvalMS XXi Xi) %. +%term vvalMS_t %pi (vvalMS (XXi , V) (Xi , T)) %<- ({xi stack} vvalT xi T xi) %<- (vvalMS XXi Xi) %. +%sort vvalDS->vvalMS %. +%term ds_ms_init vvalDS->vvalMS vvalDS_init vvalMS_init %. +%term ds_ms_stack + %pi (vvalDS->vvalMS (vvalDS_stack VVDS VVT) (vvalMS_t VVMS VVT)) + %<- (vvalDS->vvalMS VVDS VVMS) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/cpsBNF.lf b/new-tests/stelf-output/examples/cpsocc/cpsBNF.lf new file mode 100644 index 0000000..19aee1f --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/cpsBNF.lf @@ -0,0 +1,16 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% BNF of continuation-passing style terms +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort croot %. +%name croot %. +%sort cexp %. +%name cexp %. +%sort ctriv %. +%name ctriv %. +%sort ccont %. +%name ccont %. +%term klam %pi (%pi ccont %-> cexp) %-> croot %. +%term capp %pi ctriv %-> ctriv %-> ccont %-> cexp %. +%term cret %pi ccont %-> ctriv %-> cexp %. +%term xlam %pi (%pi ctriv %-> croot) %-> ctriv %. +%term vlam %pi (%pi ctriv %-> cexp) %-> ccont %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/def1+5+fig4.lf b/new-tests/stelf-output/examples/cpsocc/def1+5+fig4.lf new file mode 100644 index 0000000..0a4816f --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/def1+5+fig4.lf @@ -0,0 +1,33 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Definition 1 + 5 + Figure 4 : Cont-validity of cps terms +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort cvalR %. +%sort cvalE %. +%sort cvalT %. +%sort cvalC %. +%mode cvalR %in %. +%mode cvalE %in %. +%mode cvalT %in %. +%mode cvalC %in %. +%sort cvalK %. +%mode cvalK %in %. +%term cval_klam %pi (cvalR (klam E)) %<- (cvalE E) %. +%term cval_capp + %pi (cvalE ([k ccont] capp T0 T1 (C k))) + %<- (cvalT T1) + %<- (cvalT T0) + %<- (cvalC C) %. +%term cval_cret %pi (cvalE ([k ccont] cret (C k) T)) %<- (cvalT T) %<- (cvalC C) %. +%term cval_xlam %pi (cvalT (xlam R)) %<- ({x ctriv} %pi (cvalT x) %-> (cvalR (R x))) %. +%term cval_vlam + %pi (cvalC ([k ccont] vlam ([v ctriv] C v k))) + %<- ({v ctriv} %pi (cvalT v) %-> (cvalE (C v))) %. +%term cval_k cvalC ([k ccont] k) %. +%term cval_kappa %pi (cvalK Kappa) %<- ({t ctriv} %pi (cvalT t) %-> (cvalE ([k ccont] Kappa k t))) %. +%name cvalR %. +%name cvalE %. +%name cvalT %. +%name cvalC %. +%name cvalK %. +%terminates (R E T C) (cvalR R) (cvalE E) (cvalT T) (cvalC C) %. +%terminates K (cvalK K) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/def13.lf b/new-tests/stelf-output/examples/cpsocc/def13.lf new file mode 100644 index 0000000..ff5a57a --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/def13.lf @@ -0,0 +1,26 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Definition 13 : Control-stack substitution +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort cstack_substE %. +%sort cstack_substC %. +%mode cstack_substE %in %in %out %. +%mode cstack_substC %in %in %out %. +%term csubst_cret + %pi (cstack_substE ([k ccont] cret (C k) T) Phi ([k ccont] cret (C' k) T)) + %<- (cstack_substC C Phi C') %. +%term csubst_capp + %pi (cstack_substE ([k ccont] capp T0 T1 (C k)) Phi ([k ccont] capp T0 T1 (C' k))) + %<- (cstack_substC C Phi C') %. +%term csubst_vlam + %pi (cstack_substC ([k ccont] vlam ([v ctriv] E v k)) Phi ([k ccont] vlam ([v ctriv] E' v k))) + %<- ({v ctriv} cstack_substE (E v) Phi (E' v)) %. +%term csubst_k_init cstack_substC ([k ccont] k) ([k ccont] cdot) ([k ccont] k) %. +%term csubst_k_vlam + %pi (cstack_substC ([k ccont] k) ([k ccont] Phi k ; (vlam ([v ctriv] E v k))) ([k ccont] vlam ([v ctriv] E' v k))) + %<- ({v ctriv} cstack_substE (E v) Phi (E' v)) %. +% does this require a multiset extension? -fp +%{ +%terminates [(Phi1 Phi2) (E C)] + (cstack_substE E Phi1 _) + (cstack_substC C Phi2 _). +}% \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/def16.lf b/new-tests/stelf-output/examples/cpsocc/def16.lf new file mode 100644 index 0000000..56a2d37 --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/def16.lf @@ -0,0 +1,19 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Definition 16 : Data-stack substitution +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort dstack_substT %. +%sort dstack_substE %. +%sort dstack_substC %. +%term dstack_subst_xlam dstack_substT (xlam R) Xi (xlam R) Xi %. +%term dstack_subst_cret + %pi (dstack_substE (cret C T) Xi (cret C' T')) + %<- (dstack_substT T Xi T' Xi') + %<- (dstack_substC C Xi' C') %. +%term dstack_subst_capp + %pi (dstack_substE (capp T0 T1 C) Xi (capp T0' T1' C')) + %<- (dstack_substT T1 Xi T1' Xi1) + %<- (dstack_substT T0 Xi1 T0' Xi0) + %<- (dstack_substC C Xi0 C') %. +%term dstack_subst_vlam + %pi (dstack_substC (vlam E) Xi (vlam E')) + %<- ({v ctriv} {dsv {xi stack} {t ctriv} dstack_substT v (xi , t) t xi} dstack_substE (E v) (Xi , v) (E' v)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/def3+6+fig5.lf b/new-tests/stelf-output/examples/cpsocc/def3+6+fig5.lf new file mode 100644 index 0000000..5459728 --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/def3+6+fig5.lf @@ -0,0 +1,44 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Definition 3 + 6 + Figure 5 : Var-validity of cps terms +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort stack %. +%term dot stack %. +%term , %pi stack %-> ctriv %-> stack %. +%prec %left 10 , %. +% moved from lemma9.elf Thu Mar 10 20:45:28 2005 -fp +%term & %pi stack %-> stack %-> stack %. +%prec %left 10 & %. +%sort vvalR %. +%name vvalR %. +%sort vvalE %. +%sort vvalT %. +%sort vvalC %. +%sort vvalK %. +%mode vvalR %in %. +%mode vvalE %in %in %. +%mode vvalT %in %in %out %. +%mode vvalC %in %in %. +%mode vvalK %in %in %. +%term vval_klam %pi (vvalR (klam E)) %<- ({k ccont} %pi (vvalC dot k) %-> (vvalE dot (E k))) %. +%term vval_capp + %pi (vvalE Xi (capp T0 T1 C)) + %<- (vvalT Xi T1 Xi1) + %<- (vvalT Xi1 T0 Xi0) + %<- (vvalC Xi0 C) %. +%term vval_cret %pi (vvalE Xi (cret C T)) %<- (vvalT Xi T Xi') %<- (vvalC Xi' C) %. +%term vval_xlam + %pi (vvalT Xi (xlam R) Xi) + %<- ({x ctriv} %pi ({Xi' stack} vvalT Xi' x Xi') %-> (vvalR (R x))) %. +%term vval_vlam + %pi (vvalC Xi (vlam E)) + %<- ({v ctriv} %pi ({Xi' stack} vvalT (Xi' , v) v Xi') %-> (vvalE (Xi , v) (E v))) %. +%term vval_kappa + %pi (vvalK Xi Kappa) + %<- ({v ctriv} %pi ({Xi' stack} vvalT (Xi' , v) v Xi') %-> (vvalE (Xi , v) (Kappa v))) + %<- ({t ctriv} %pi ({Xi' stack} vvalT Xi' t Xi') %-> (vvalE Xi (Kappa t))) %. +%name vvalE %. +%name vvalT %. +%name vvalC %. +%name vvalK %. +%terminates (R E T C) (vvalR R) (vvalE _ E) (vvalT _ T _) (vvalC _ C) %. +%terminates K (vvalK _ K) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/dsBNF.lf b/new-tests/stelf-output/examples/cpsocc/dsBNF.lf new file mode 100644 index 0000000..7ef9b40 --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/dsBNF.lf @@ -0,0 +1,13 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% BNF of direct-style terms +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort droot %. +%name droot %. +%sort dexp %. +%name dexp %. +%sort dtriv %. +%name dtriv %. +%term dexp->droot %pi dexp %-> droot %. +%term dapp %pi dexp %-> dexp %-> dexp %. +%term dtriv->dexp %pi dtriv %-> dexp %. +%term dlam %pi (%pi dtriv %-> droot) %-> dtriv %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/fig1.lf b/new-tests/stelf-output/examples/cpsocc/fig1.lf new file mode 100644 index 0000000..78a46e9 --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/fig1.lf @@ -0,0 +1,24 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Figure 1 : The left-to-right, call-by-value CPS transformation +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort cpsR %. +%name cpsR %. +%sort cpsE %. +%name cpsE %. +%sort cpsT %. +%name cpsT %. +%mode cpsR %in %out %. +%mode cpsE %in %in %out %. +%mode cpsT %in %out %. +%term cps_droot + %pi (cpsR (dexp->droot E) (klam E')) + %<- ({k ccont} cpsE E ([v ctriv] cret k v) (E' k)) %. +%term cps_dtriv %pi (cpsE (dtriv->dexp T) K (K T')) %<- (cpsT T T') %. +%term cps_dapp + %pi (cpsE (dapp E0 E1) K E') + %<- ({t0 ctriv} cpsE E1 ([t1 ctriv] capp t0 t1 (vlam K)) (E1' t0)) + %<- (cpsE E0 E1' E') %. +%term cps_dlam + %pi (cpsT (dlam R) (xlam R')) + %<- ({x dtriv} {x' ctriv} %pi (cpsT x x') %-> (cpsR (R x) (R' x'))) %. +%terminates (R E T) (cpsR R R') (cpsE E K E') (cpsT T T') %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/fig10.lf b/new-tests/stelf-output/examples/cpsocc/fig10.lf new file mode 100644 index 0000000..80f026a --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/fig10.lf @@ -0,0 +1,27 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Figure 10 : CPS abstract machine with a data stack +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort dstackR %. +%sort dstackE %. +%sort dstackT %. +%mode dstackR %in %out %. +%mode dstackE %in %in %out %. +%mode dstackT %in %in %out %out %. +%term dstack_klam + %pi (dstackR (klam E) A) + %<- ({k ccont} {d {xi stack} {t ctriv} {t' ctriv} %pi (dstackT xi t t' dot) %-> (dstackE xi (cret k t) t')} dstackE dot (E k) A) %. +%term dstack_cret_vlam + %pi (dstackE Xi (cret (vlam E) T) A) + %<- (dstackT Xi T T' Xi') + %<- ({v ctriv} + %pi ({t ctriv} {xi stack} dstackT (xi , t) v t xi) + %-> (dstackE (Xi' , T') (E v) A)) %. +%term dstack_capp + %pi (dstackE Xi (capp T0 T1 C) A) + %<- (dstackT Xi T1 T1' Xi1) + %<- (dstackT Xi1 T0 (xlam ([x ctriv] klam (E x))) Xi0) + %<- (dstackE Xi0 (E T1' C) A) %. +%term dstack_xlam dstackT Xi (xlam R) (xlam R) Xi %. +%name dstackR %. +%name dstackE %. +%name dstackT %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/fig11.lf b/new-tests/stelf-output/examples/cpsocc/fig11.lf new file mode 100644 index 0000000..8c5193d --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/fig11.lf @@ -0,0 +1,6 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Figure 11 : Var-valid data stacks +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort vvalDS %. +%term vvalDS_init vvalDS dot dot %. +%term vvalDS_stack %pi (vvalDS (XXi , V) (Xi , T)) %<- ({xi stack} vvalT xi T xi) %<- (vvalDS XXi Xi) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/fig12.lf b/new-tests/stelf-output/examples/cpsocc/fig12.lf new file mode 100644 index 0000000..f967596 --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/fig12.lf @@ -0,0 +1,40 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Figure 12 : CPS abstract machine with two stack +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort two_stacksR %. +%sort two_stacksE %. +%sort two_stacksT %. +%sort two_stacksC %. +%mode two_stacksR %in %out %. +%mode two_stacksE %in %in %in %out %. +%mode two_stacksT %in %in %out %out %. +%mode two_stacksC %in %in %out %. +%term two_stacks_klam %pi (two_stacksR (klam E) A) %<- (two_stacksE ([k ccont] cdot) dot E A) %. +%term two_stacks_cret_k_init + %pi (two_stacksE ([k ccont] cdot) Xi ([k ccont] cret k T) A) + %<- (two_stacksT Xi T A dot) %. +%term two_stacks_cret_k_vlam + %pi (two_stacksE ([k ccont] Phi k ; (vlam ([v ctriv] E v k))) Xi ([k ccont] cret k T) A) + %<- (two_stacksT Xi T T' Xi') + %<- ({v ctriv} + %pi ({xi stack} {t ctriv} two_stacksT (xi , t) v t xi) + %-> (two_stacksE Phi (Xi' , T') (E v) A)) %. +%term two_stacks_cret_vlam_phi + %pi (two_stacksE Phi Xi ([k ccont] cret (vlam ([v ctriv] E v k)) T) A) + %<- (two_stacksT Xi T T' Xi') + %<- ({v ctriv} + %pi ({xi stack} {t ctriv} two_stacksT (xi , t) v t xi) + %-> (two_stacksE Phi (Xi' , T') (E v) A)) %. +%term two_stacks_capp + %pi (two_stacksE Phi Xi ([k ccont] capp T0 T1 (C k)) A) + %<- (two_stacksT Xi T1 T1' Xi1) + %<- (two_stacksT Xi1 T0 (xlam ([x ctriv] klam (E x))) Xi0) + %<- (two_stacksC Phi C Phi') + %<- (two_stacksE Phi' Xi0 (E T1') A) %. +%term two_stacks_xlam two_stacksT Xi (xlam R) (xlam R) Xi %. +%term two_stacks_vlam two_stacksC Phi ([k ccont] vlam (E k)) ([k ccont] Phi k ; (vlam (E k))) %. +%term two_stacks_k two_stacksC Phi ([k ccont] k) Phi %. +%name two_stacksR %. +%name two_stacksE %. +%name two_stacksT %. +%name two_stacksC %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/fig13.lf b/new-tests/stelf-output/examples/cpsocc/fig13.lf new file mode 100644 index 0000000..fe1bb61 --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/fig13.lf @@ -0,0 +1,6 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Figure 13 : Cont-valid data stacks +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort cvalDS %. +%term cvalDS_init cvalDS dot %. +%term cvalDS_stack %pi (cvalDS (Xi , T)) %<- (cvalT T) %<- (cvalDS Xi) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/fig7.lf b/new-tests/stelf-output/examples/cpsocc/fig7.lf new file mode 100644 index 0000000..da5980c --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/fig7.lf @@ -0,0 +1,21 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Figure 7 : Bare, stackless CPS abstract machine +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort bareR %. +%sort bareE %. +%mode bareR %in %out %. +%mode bareE %in %out %. +% style here -fp +%{ +bare_klam : bareR (klam E) A + <- {k:ccont} + {b:{t:ctriv}bareE (cret k t) t} + bareE (E k) A. +}% +%term bare_klam + %pi (bareR (klam E) A) + %<- ({k ccont} %pi ({T ctriv} bareE (cret k T) T) %-> (bareE (E k) A)) %. +%term bare_cret %pi (bareE (cret (vlam E) T) A) %<- (bareE (E T) A) %. +%term bare_capp %pi (bareE (capp (xlam ([x ctriv] klam (E x))) T C) A) %<- (bareE (E T C) A) %. +%name bareR %. +%name bareE %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/fig8.lf b/new-tests/stelf-output/examples/cpsocc/fig8.lf new file mode 100644 index 0000000..4f46117 --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/fig8.lf @@ -0,0 +1,30 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Figure 8 : CPS abstract machine with a control stack +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort cstack %. +%term cdot cstack %. +%term ; %pi cstack %-> ccont %-> cstack %. +%prec %left 10 ; %. +%sort cstackR %. +%sort cstackE %. +%sort cstackC %. +%mode cstackR %in %out %. +%mode cstackE %in %in %out %. +%mode cstackC %in %in %out %. +%term cstack_klam %pi (cstackR (klam E) A) %<- (cstackE ([k ccont] cdot) E A) %. +%term cstack_cret_k_init cstackE ([k ccont] cdot) ([k ccont] cret k T) T %. +%term cstack_cret_k_vlam + %pi (cstackE ([k ccont] Phi k ; (vlam ([v ctriv] E v k))) ([k ccont] cret k T) A) + %<- (cstackE Phi (E T) A) %. +%term cstack_cret_vlam_phi + %pi (cstackE Phi ([k ccont] cret (vlam ([v ctriv] E v k)) T) A) + %<- (cstackE Phi (E T) A) %. +%term cstack_capp + %pi (cstackE Phi ([k ccont] capp (xlam ([x ctriv] klam (E x))) T (C k)) A) + %<- (cstackC Phi C Phi') + %<- (cstackE Phi' (E T) A) %. +%term cstack_phi_k cstackC Phi ([k ccont] k) Phi %. +%term cstack_vlam cstackC Phi ([k ccont] vlam (E k)) ([k ccont] Phi k ; (vlam (E k))) %. +%name cstackR %. +%name cstackE %. +%name cstackC %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/fig9.lf b/new-tests/stelf-output/examples/cpsocc/fig9.lf new file mode 100644 index 0000000..e4cd535 --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/fig9.lf @@ -0,0 +1,11 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Figure 9 : Cont-valid control stacks +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort cvalCS %. +%mode cvalCS %in %. +%term cvalCS_init cvalCS ([k ccont] cdot) %. +%term cvalCS_vlam + %pi (cvalCS ([k ccont] Phi k ; (vlam ([v ctriv] E v k)))) + %<- (cvalCS Phi) + %<- ({v ctriv} {cvv cvalT v} cvalE (E v)) %. +%terminates Phi (cvalCS Phi) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/lemma14.lf b/new-tests/stelf-output/examples/cpsocc/lemma14.lf new file mode 100644 index 0000000..928ac85 --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/lemma14.lf @@ -0,0 +1,25 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Lemma 14 : Control-stack substitution +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort lemma14-1a %. +%sort lemma14-1c %. +%sort lemma14-2a %. +%sort lemma14-2b %. +%term lemma14-1a_cret + %pi (lemma14-1a CVS (cval_cret CVC CVT) SE1 (csubst_cret SC)) + %<- (lemma14-2a CVS CVC SE1 SC) %. +%term lemma14-1a_capp + %pi (lemma14-1a CVS (cval_capp CVC CVT0 CVT1) SE1 (csubst_capp SC)) + %<- (lemma14-2a CVS CVC SE1 SC) %. +%term lemma14-1c_cret %pi (lemma14-1c (cval_cret CVC CVT) (csubst_cret SC)) %<- (lemma14-2b CVC SC) %. +%term lemma14-1c_capp + %pi (lemma14-1c (cval_capp CVC CVT0 CVT1) (csubst_capp SC)) + %<- (lemma14-2b CVC SC) %. +%term lemma14-2a_k lemma14-2a CVS cval_k SE1 (csubst_k_vlam SE1) %. +%term lemma14-2a_vlam + %pi (lemma14-2a CVS (cval_vlam CVE) SE1 (csubst_vlam SE2)) + %<- ({v ctriv} {cvv cvalT v} lemma14-1a CVS (CVE v cvv) SE1 (SE2 v)) %. +%term lemma14-2b_k lemma14-2b cval_k csubst_k_init %. +%term lemma14-2b_vlam + %pi (lemma14-2b (cval_vlam CVE) (csubst_vlam SE)) + %<- ({v ctriv} {cvv cvalT v} lemma14-1c (CVE v cvv) (SE v)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/lemma17.lf b/new-tests/stelf-output/examples/cpsocc/lemma17.lf new file mode 100644 index 0000000..9bd2fad --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/lemma17.lf @@ -0,0 +1,131 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Lemma 17 : Data-stack substitution +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort www %. +%term www_vstack_v {v ctriv} {vvv {xi stack} vvalT (xi , v) v xi} {dsv {xi stack} {t ctriv} dstack_substT v (xi , t) t xi} www (vvalMS_v VVMS) (vvv XXi) (dsv Xi v) VVMS %. +%term www_tstack_v {v ctriv} {vvv {xi stack} vvalT (xi , v) v xi} {dsv {xi stack} {t ctriv} dstack_substT v (xi , t) t xi} www (vvalMS_t VVMS VVT) (vvv XXi) (dsv Xi T) VVMS %. +%term www_stack_t {t ctriv} {vvt {xi stack} vvalT xi t xi} {dst {xi stack} dstack_substT t xi t xi} www VVMS (vvt XXi) (dst Xi) VVMS %. +%sort lappT %. +%term lappT_v {v ctriv} {dsv {xi stack} {t ctriv} dstack_substT v (xi , t) t xi} lappT (dsv Xi T) (app_stack APP) APP (dsv Xi1 T) %. +%term lappT_t {t ctriv} {dst {xi stack} dstack_substT t xi t xi} lappT (dst Xi) APP APP (dst Xi1) %. +%sort lemma17-1E %. +%sort lemma17-1T %. +%sort lemma17-1C %. +%term lemma17-1_t {t ctriv} {vvt {xi stack} vvalT xi t xi} {dst {xi stack} dstack_substT t xi t xi} lemma17-1T (vvt XXi) (dst XXi) %. +%term lemma17-1_v {v ctriv} {vvv {xi stack} vvalT (xi , v) v xi} {dsv {xi stack} {t ctriv} dstack_substT v (xi , t) t xi} lemma17-1T (vvv XXi) (dsv XXi v) %. +%term lemma17-1_cret + %pi (lemma17-1E (vval_cret VVC VVT) (dstack_subst_cret DSC DST)) + %<- (lemma17-1T VVT DST) + %<- (lemma17-1C VVC DSC) %. +%term lemma17-1_capp + %pi (lemma17-1E (vval_capp VVC VVT0 VVT1) (dstack_subst_capp DSC DST0 DST1)) + %<- (lemma17-1T VVT0 DST0) + %<- (lemma17-1T VVT1 DST1) + %<- (lemma17-1C VVC DSC) %. +%term lemma17-1_vlam + %pi (lemma17-1C (vval_vlam VE) (dstack_subst_vlam DE)) + %<- ({v ctriv} {vvv {xi stack} vvalT (xi , v) v xi} {dsv {xi stack} {t ctriv} dstack_substT v (xi , t) t xi} lemma17-1E (VE v vvv) (DE v dsv)) %. +%sort lemma17-2 {t %pi ctriv %-> ctriv} {v ctriv} %. +%sort lemma17-3 {k ccont} {vvk vvalC dot k} %. +%term lemma17-2_cret_v1 + %pi ({v1 ctriv} {dsv1 {xi stack} {t ctriv} dstack_substT v1 (xi , t) t xi} {vvv1 {xi stack} vvalT (xi , v1) v1 xi} lemma17-2 T ([v ctriv] app_stack (APP v)) ([v ctriv] [vvv] vval_cret (vval_vlam ([v2 ctriv] [vvv2] VVE v2 vvv2 v vvv)) (vvv1 (XXi v))) ([v ctriv] app_stack (APP1 v)) (VVMS v1) ([v ctriv] [dsv] dstack_subst_cret (dstack_subst_vlam ([v2 ctriv] [dsv2] DSE v2 dsv2 v dsv)) (dsv1 (Xi v) (T1 v1))) ([v ctriv] [dsv] dstack_subst_cret (dstack_subst_vlam ([v2 ctriv] [dsv2] DSE' v2 dsv2 v dsv)) (dsv1 (Xi (T v)) (T1 v1)))) + %<- ({v ctriv} {v1 ctriv} {dsv1 {xi stack} {t ctriv} dstack_substT v1 (xi , t) t xi} {vvv1 {xi stack} vvalT (xi , v1) v1 xi} www (VVMS v1 v) (vvv1 (XXi v)) (dsv1 (Xi (T v)) (T1 v1)) (VVMS' v)) + %<- ({v2 ctriv} {dsv2 {xi stack} {t ctriv} dstack_substT v2 (xi , t) t xi} {vvv2 {xi stack} vvalT (xi , v2) v2 xi} lemma17-2 T ([v ctriv] app_stack (APP v)) (VVE' v2 vvv2) ([v ctriv] app_stack (APP1 v)) ([v ctriv] vvalMS_v (VVMS' v)) (DSE v2 dsv2) (DSE' v2 dsv2)) %. +%term lemma17-2_cret_t + %pi ({t ctriv} {dst {xi stack} dstack_substT t xi t xi} {vvt {xi stack} vvalT xi t xi} lemma17-2 T' APP ([v ctriv] [vvv] vval_cret (vval_vlam ([v1 ctriv] [vvv1] VVE v1 vvv1 v vvv)) (vvt (XXi v))) APP1 VVMS ([v ctriv] [dsv] dstack_subst_cret (dstack_subst_vlam ([v1 ctriv] [dsv1] DSE v1 dsv1 v dsv)) (dst (Xi , v))) ([v ctriv] [dsv] dstack_subst_cret (dstack_subst_vlam ([v1 ctriv] [dsv1] DSE' v1 dsv1 v dsv)) (dst (Xi , (T' v))))) + %<- ({v1 ctriv} {dsv1 {xi stack} {t ctriv} dstack_substT v1 (xi , t) t xi} {vvv1 {xi stack} vvalT (xi , v1) v1 xi} lemma17-2 T' ([v ctriv] app_stack (APP v)) (VVE v1 vvv1) ([v ctriv] app_stack (APP1 v)) ([v ctriv] vvalMS_v (VVMS v)) (DSE v1 dsv1) (DSE' v1 dsv1)) %. +%term lemma17-2_cret_v lemma17-2 T ([v ctriv] app_init) ([v ctriv] [vvv] vval_cret VVC (vvv XXi)) ([v ctriv] app_init) VVMS ([v ctriv] [dsv] dstack_subst_cret DSC (dsv Xi v)) ([v ctriv] [dsv] dstack_subst_cret DSC (dsv Xi (T v))) %. +%term lemma17-2_capp_t0_v {t0 ctriv} {dst0 {xi stack} dstack_substT t0 xi t0 xi} {vvt0 {xi stack} vvalT xi t0 xi} lemma17-2 T ([v ctriv] app_init) ([v ctriv] [vvv] vval_capp VVC (vvt0 XXi) (vvv XXi)) ([v ctriv] app_init) VVMS ([v ctriv] [dsv] dstack_subst_capp DSC (dst0 Xi) (dsv Xi v)) ([v ctriv] [dsv] dstack_subst_capp DSC (dst0 Xi) (dsv Xi (T v))) %. +%term lemma17-2_capp_v0_v_c {v0 ctriv} {dsv0 {xi stack} {t ctriv} dstack_substT v0 (xi , t) t xi} {vvv0 {xi stack} vvalT (xi , v0) v0 xi} lemma17-2 T ([v ctriv] app_init) ([v ctriv] [vvv] vval_capp VVC (vvv0 XXi) (vvv (XXi , v0))) ([v ctriv] app_init) (VVMS v0) ([v ctriv] [dsv] dstack_subst_capp DSC (dsv0 Xi T0) (dsv (Xi , T0) v)) ([v ctriv] [dsv] dstack_subst_capp DSC (dsv0 Xi T0) (dsv (Xi , T0) (T v))) %. +%term lemma17-2_capp_v0_v1_c + %pi ({v0 ctriv} {dsv0 {xi stack} {t ctriv} dstack_substT v0 (xi , t) t xi} {vvv0 {xi stack} vvalT (xi , v0) v0 xi} {v1 ctriv} {dsv1 {xi stack} {t ctriv} dstack_substT v1 (xi , t) t xi} {vvv1 {xi stack} vvalT (xi , v1) v1 xi} lemma17-2 T ([v ctriv] app_stack (app_stack (APP v))) ([v ctriv] [vvv] vval_capp (vval_vlam ([v2 ctriv] [vvv2] VVE v2 vvv2 v vvv)) (vvv0 (XXi v)) (vvv1 (XXi v , v0))) ([v ctriv] app_stack (app_stack (APP1 v))) (VVMS v0 v1) ([v ctriv] [dsv] dstack_subst_capp (dstack_subst_vlam ([v2 ctriv] [dsv2] DSE v2 dsv2 v dsv)) (dsv0 (Xi v) (T'' v0)) (dsv1 (Xi v , (T'' v0)) (T' v1))) ([v ctriv] [dsv] dstack_subst_capp (dstack_subst_vlam ([v2 ctriv] [dsv2] DSE' v2 dsv2 v dsv)) (dsv0 (Xi (T v)) (T'' v0)) (dsv1 (Xi (T v) , (T'' v0)) (T' v1)))) + %<- ({v ctriv} {v1 ctriv} {dsv1 {xi stack} {t ctriv} dstack_substT v1 (xi , t) t xi} {vvv1 {xi stack} vvalT (xi , v1) v1 xi} {v0 ctriv} www (VVMS v0 v1 v) (vvv1 (XXi v , v0)) (dsv1 (Xi (T v) , (T'' v0)) (T' v1)) (VVMS' v0 v)) + %<- ({v ctriv} {v0 ctriv} {dsv0 {xi stack} {t ctriv} dstack_substT v0 (xi , t) t xi} {vvv0 {xi stack} vvalT (xi , v0) v0 xi} www (VVMS' v0 v) (vvv0 (XXi v)) (dsv0 (Xi (T v)) (T'' v0)) (VVMS'' v)) + %<- ({v2 ctriv} {dsv2 {xi stack} {t ctriv} dstack_substT v2 (xi , t) t xi} {vvv2 {xi stack} vvalT (xi , v2) v2 xi} lemma17-2 T ([v ctriv] app_stack (APP v)) (VVE v2 vvv2) ([v ctriv] app_stack (APP1 v)) ([v ctriv] vvalMS_v (VVMS'' v)) (DSE v2 dsv2) (DSE' v2 dsv2)) %. +%term lemma17-2_capp_t0_v1_c + %pi ({v1 ctriv} {dsv1 {xi stack} {t ctriv} dstack_substT v1 (xi , t) t xi} {vvv1 {xi stack} vvalT (xi , v1) v1 xi} {t0 ctriv} {dst0 {xi stack} dstack_substT t0 xi t0 xi} {vvt0 {xi stack} vvalT xi t0 xi} lemma17-2 T ([v ctriv] app_stack (APP v)) ([v ctriv] [vvv] vval_capp (vval_vlam ([v2 ctriv] [vvv2] VVE v2 vvv2 v vvv)) (vvt0 (XXi v)) (vvv1 (XXi v))) ([v ctriv] app_stack (APP1 v)) (VVMS v1) ([v ctriv] [dsv] dstack_subst_capp (dstack_subst_vlam ([v2 ctriv] [dsv2] DSE v2 dsv2 v dsv)) (dst0 (Xi v)) (dsv1 (Xi v) (T' v1))) ([v ctriv] [dsv] dstack_subst_capp (dstack_subst_vlam ([v2 ctriv] [dsv2] DSE' v2 dsv2 v dsv)) (dst0 (Xi (T v))) (dsv1 (Xi (T v)) (T' v1)))) + %<- ({v ctriv} {v1 ctriv} {dsv1 {xi stack} {t ctriv} dstack_substT v1 (xi , t) t xi} {vvv1 {xi stack} vvalT (xi , v1) v1 xi} www (VVMS v1 v) (vvv1 (XXi v)) (dsv1 (Xi (T v)) (T' v1)) (VVMS' v)) + %<- ({v2 ctriv} {dsv2 {xi stack} {t ctriv} dstack_substT v2 (xi , t) t xi} {vvv2 {xi stack} vvalT (xi , v2) v2 xi} lemma17-2 T ([v ctriv] app_stack (APP v)) (VVE v2 vvv2) ([v ctriv] app_stack (APP1 v)) ([v ctriv] vvalMS_v (VVMS' v)) (DSE v2 dsv2) (DSE' v2 dsv2)) %. +%term lemma17-2_capp_v_v1_c {v1 ctriv} {dsv1 {xi stack} {t ctriv} dstack_substT v1 (xi , t) t xi} {vvv1 {xi stack} vvalT (xi , v1) v1 xi} lemma17-2 T'' ([v ctriv] app_stack (APP v)) ([v ctriv] [vvv] vval_capp VVC (vvv XXi) (vvv1 (XXi , v))) ([v ctriv] app_stack (APP1 v)) (VVMS v1) ([v ctriv] [dsv] dstack_subst_capp DSC (dsv Xi v) (dsv1 (Xi , v) (T' v1))) ([v ctriv] [dsv] dstack_subst_capp DSC (dsv Xi (T'' v)) (dsv1 (Xi , (T'' v)) (T' v1))) %. +%term lemmaa17-2_capp_v0_t1_c + %pi ({v0 ctriv} {dsv0 {xi stack} {t ctriv} dstack_substT v0 (xi , t) t xi} {vvv0 {xi stack} vvalT (xi , v0) v0 xi} {t1 ctriv} {dst1 {xi stack} dstack_substT t1 xi t1 xi} {vvt1 {xi stack} vvalT xi t1 xi} lemma17-2 T ([v ctriv] app_stack (APP v)) ([v ctriv] [vvv] vval_capp (vval_vlam ([v1 ctriv] [vvv1] VVC v1 vvv1 v vvv)) (vvv0 (XXi v)) (vvt1 (XXi v , v0))) ([v ctriv] app_stack (APP1 v)) (VVMS v0) ([v ctriv] [dsv] dstack_subst_capp (dstack_subst_vlam ([v1 ctriv] [dsv1] DSE v1 dsv1 v dsv)) (dsv0 (Xi v) (T'' v0)) (dst1 (Xi v , (T'' v0)))) ([v ctriv] [dsv] dstack_subst_capp (dstack_subst_vlam ([v1 ctriv] [dsv1] DSE' v1 dsv1 v dsv)) (dsv0 (Xi (T v)) (T'' v0)) (dst1 (Xi (T v) , (T'' v0))))) + %<- ({v ctriv} {v0 ctriv} {dsv0 {xi stack} {t ctriv} dstack_substT v0 (xi , t) t xi} {vvv0 {xi stack} vvalT (xi , v0) v0 xi} www (VVMS v0 v) (vvv0 (XXi v)) (dsv0 (Xi (T v)) (T'' v0)) (VVMS' v)) + %<- ({v1 ctriv} {dsv1 {xi stack} {t ctriv} dstack_substT v1 (xi , t) t xi} {vvv1 {xi stack} vvalT (xi , v1) v1 xi} lemma17-2 T ([v ctriv] app_stack (APP v)) (VVE v1 vvv1) ([v ctriv] app_stack (APP1 v)) ([v ctriv] vvalMS_v (VVMS' v)) (DSE v1 dsv1) (DSE' v1 dsv1)) %. +%term lemma17-2_capp_v_t1_c {t1 ctriv} {dst1 {xi stack} dstack_substT t1 xi t1 xi} {vvt1 {xi stack} vvalT xi t1 xi} lemma17-2 T ([v ctriv] app_init) ([v ctriv] [vvv] vval_capp VVC (vvv XXi) (vvt1 (XXi , v))) ([v ctriv] app_init) VVMS ([v ctriv] [dsv] dstack_subst_capp DSC (dsv Xi v) (dst1 (Xi , v))) ([v ctriv] [dsv] dstack_subst_capp DSC (dsv Xi (T v)) (dst1 (Xi , (T v)))) %. +%term lemma17-2_capp_t0_t1_c + %pi ({t0 ctriv} {dst0 {xi stack} dstack_substT t0 xi t0 xi} {vvt0 {xi stack} vvalT xi t0 xi} {t1 ctriv} {dst1 {xi stack} dstack_substT t1 xi t1 xi} {vvt1 {xi stack} vvalT xi t1 xi} lemma17-2 T APP ([v ctriv] [vvv] vval_capp (vval_vlam ([v1 ctriv] [vvv1] VVE v1 vvv1 v vvv)) (vvt0 (XXi v)) (vvt1 (XXi v))) APP1 VVMS ([v ctriv] [dsv] dstack_subst_capp (dstack_subst_vlam ([v1 ctriv] [dsv1] DSE v1 dsv1 v dsv)) (dst0 (Xi v)) (dst1 (Xi v))) ([v ctriv] [dsv] dstack_subst_capp (dstack_subst_vlam ([v1 ctriv] [dsv1] DSE' v1 dsv1 v dsv)) (dst0 (Xi (T v))) (dst1 (Xi (T v))))) + %<- ({v1 ctriv} {dsv1 {xi stack} {t ctriv} dstack_substT v1 (xi , t) t xi} {vvv1 {xi stack} vvalT (xi , v1) v1 xi} lemma17-2 T ([v ctriv] app_stack (APP v)) (VVE v1 vvv1) ([v ctriv] app_stack (APP1 v)) ([v ctriv] vvalMS_v (VVMS v)) (DSE v1 dsv1) (DSE' v1 dsv1)) %. +%sort v-part %. +%term v-part_init v-part vvalMS_init dot %. +%term v-part_v + %pi (v-part (%the (vvalMS (XXi , V) (Xi , V)) (vvalMS_v CV)) (S , V)) + %<- (v-part CV S) %. +%term v-part_t %pi (v-part (vvalMS_t CV VVT) S) %<- (v-part CV S) %. +%sort doubleC %. +%sort doubleE %. +%sort doubleT %. +%term double_vlam + %pi (doubleC (vval_vlam VVE) VVALMS (dstack_subst_vlam DSE) VP (dstack_subst_vlam DSE')) + %<- ({v ctriv} {vvv {xi stack} vvalT (xi , v) v xi} {dsv {xi stack} {t ctriv} dstack_substT v (xi , t) t xi} doubleE (VVE v vvv) (vvalMS_v VVALMS) (DSE v dsv) (v-part_v VP) (DSE' v dsv)) %. +%term double_v {v ctriv} {vvv {xi stack} vvalT (xi , v) v xi} {dsv {xi stack} {t ctriv} dstack_substT v (xi , t) t xi} doubleT (vvv XXi) (vvalMS_v (%the (vvalMS XXi Xi) CV)) (dsv Xi v) (v-part_v (%the (v-part CV XXi') VP)) CV VP (dsv XXi' v) %. +% Some bug in this clause +% Thu Mar 10 20:58:12 2005 -fp +%{ +double_v' : {v:ctriv} + {vvv:{xi:stack}vvalT (xi , v) v xi} + {dsv:{xi:stack}{t:ctriv}dstack_substT v (xi , t) t xi} + doubleT (vvv XXi) (vvalMS_t (CV:vvalMS XXi Xi) VVT) + (dsv Xi T) + (v-part_t (VP:v-part CV XXi')) CV VP DST. +}% +%term double_t {t ctriv} {vvt {xi stack} vvalT xi t xi} {dst {xi stack} dstack_substT t xi t xi} doubleT (vvt XXi) VVMS (dst Xi) VP CV VP (dst Xi) %. +%term double_cret + %pi (doubleE (vval_cret VVC VVT) VVMS (dstack_subst_cret DSC DST) VP (dstack_subst_cret DSC' DST')) + %<- (doubleT VVT VVMS DST VP VVMS' VP' DST') + %<- (doubleC VVC VVMS' DSC VP' DSC') %. +%term double_capp + %pi (doubleE (vval_capp VVC VVT0 VVT1) VVMS (dstack_subst_capp DSC DST0 DST1) VP (dstack_subst_capp DSC' DST0' DST1')) + %<- (doubleT VVT1 VVMS DST1 VP VVMS' VP' DST1') + %<- (doubleT VVT0 VVMS' DST0 VP' VVMS'' VP'' DST0') + %<- (doubleC VVC VVMS'' DSC VP'' DSC') %. +%sort vpartDS %. +%term vpartDS_init vpartDS vvalDS_init vvalMS_init v-part_init %. +%term vpartDS_stack + %pi (vpartDS (vvalDS_stack VVDS VVT) (vvalMS_t CV VVT) (v-part_t VP)) + %<- (vpartDS VVDS CV VP) %. +%term lemma17-3_cret_k + %pi (lemma17-3 ([k ccont] [vvk] vval_cret vvk VVT) (cval_cret cval_k CVT) VVC VVMS VVDS APP DSC (dstack_subst_cret DSC' DST') (dstack_subst_cret DSC DST)) + %<- (vpartDS VVDS VVMS' VP) + %<- (doubleC VVC VVMS' DSC VP DSC') + %<- (lappT DST' APP app_init DST) %. +%term lemma17-3_cret_vlam + %pi (lemma17-3 ([k ccont] [vvk] vval_cret (vval_vlam ([v ctriv] [vvv] VVE v vvv k vvk)) VVT) (cval_cret (cval_vlam CVE) CVT) VVC1 VVMS VVDS APP DSC1 (dstack_subst_cret (dstack_subst_vlam DSE') DST') (dstack_subst_cret (dstack_subst_vlam DSE) DST)) + %<- (lappT DST' APP APP' DST) + %<- (www VVMS VVT DST' VVMS') + %<- ({v ctriv} {vvv {xi stack} vvalT (xi , v) v xi} {cvv cvalT v} {dsv {xi stack} {t ctriv} dstack_substT v (xi , t) t xi} lemma17-3 (VVE v vvv) (CVE v cvv) VVC1 (vvalMS_v VVMS') VVDS (app_stack APP') DSC1 (DSE' v dsv) (DSE v dsv)) %. +%term lemma17-3_capp_k + %pi (lemma17-3 ([k ccont] [vvk] vval_capp vvk VVT1 VVT0) (cval_capp cval_k CVT1 CVT0) VVC VVMS VVDS APP DSC (dstack_subst_capp DSC' DST0' DST1') (dstack_subst_capp DSC DST0 DST1)) + %<- (vpartDS VVDS VVMS' VP) + %<- (doubleC VVC VVMS' DSC VP DSC') + %<- (lappT DST1' APP APP' DST1) + %<- (lappT DST0' APP' app_init DST0) %. +%term lemma17-3_capp_vlam + %pi (lemma17-3 ([k ccont] [vvk] vval_capp (vval_vlam ([v ctriv] [vvv] VVE v vvv k vvk)) VVT0 VVT1) (cval_capp (cval_vlam CVE) CVT1 CVT0) VVC1 VVMS VVDS APP DSC1 (dstack_subst_capp (dstack_subst_vlam DSE') DST0' DST1') (dstack_subst_capp (dstack_subst_vlam DSE) DST0 DST1)) + %<- (lappT DST1' APP APP' DST1) + %<- (www VVMS VVT1 DST1' VVMS') + %<- (lappT DST0' APP' APP'' DST0) + %<- (www VVMS' VVT0 DST0' VVMS'') + %<- ({v ctriv} {vvv {xi stack} vvalT (xi , v) v xi} {cvv cvalT v} {dsv {xi stack} {t ctriv} dstack_substT v (xi , t) t xi} lemma17-3 (VVE v vvv) (CVE v cvv) VVC1 (vvalMS_v VVMS'') VVDS (app_stack APP'') DSC1 (DSE' v dsv) (DSE v dsv)) %. +%sort rsT %. +%term rs_v {v ctriv} {vvv {xi stack} vvalT (xi , v) v xi} {cvv cvalT v} {dsv {xi stack} {t ctriv} dstack_substT v (xi , t) t xi} + %pi (rsT (vvv XXi) cvv (vvalDS_stack VVDS VVT') (cvalDS_stack CVDS CVT') (dsv Xi T) VVT' CVT' VVDS CVDS DST') + %<- (rsT (VVT' XXi) CVT' VVDS CVDS (DST' Xi) VVT' CVT' VVDS CVDS DST') %. +%term rs_t {t ctriv} {vvt {xi stack} vvalT xi t xi} {cvt cvalT t} {dst {xi stack} dstack_substT t xi t xi} rsT (vvt XXi) cvt VVDS CVDS (dst Xi) vvt cvt VVDS CVDS dst %. +%sort clr {k ccont} {vvk vvalC dot k} %. +%term clr_proof + %pi (clr VVE CVE VVC VVDS DSC DSE) + %<- (vpartDS VVDS VVMS VP) + %<- (doubleC VVC VVMS DSC VP DSC') + %<- (lemma17-1C VVC' DSC') + %<- (lemma17-1E (VVE C' VVC') DSE') + %<- (lemma17-3 VVE CVE VVC vvalMS_init VVDS app_init DSC DSE' DSE) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/lemma7.lf b/new-tests/stelf-output/examples/cpsocc/lemma7.lf new file mode 100644 index 0000000..6986d2e --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/lemma7.lf @@ -0,0 +1,18 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Lemma 7 : Preservation of cont-validity under substitution +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort lemma7_2b %. +%sort lemma7_2d %. +%mode lemma7_2b %in %in %out %. +%mode lemma7_2d %in %in %out %. +%term lemma7_2b_capp + %pi (lemma7_2b CVC' (cval_capp CVC CVT0 CVT1) (cval_capp CVCreplC'k CVT0 CVT1)) + %<- (lemma7_2d CVC' CVC CVCreplC'k) %. +%term lemma7_2b_cret + %pi (lemma7_2b CVC' (cval_cret CVC CT) (cval_cret CVCreplC'k CT)) + %<- (lemma7_2d CVC' CVC CVCreplC'k) %. +%term lemma7_2d_k lemma7_2d CVC' cval_k CVC' %. +%term lemma7_2d_vlam + %pi (lemma7_2d CVC' (cval_vlam CVE) (cval_vlam CVEreplC'k)) + %<- ({v ctriv} {cvv cvalT v} lemma7_2b CVC' (CVE v cvv) (CVEreplC'k v cvv)) %. +%terminates (CVE CVC) (lemma7_2b _ CVE _) (lemma7_2d _ CVC _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/lemma9.lf b/new-tests/stelf-output/examples/cpsocc/lemma9.lf new file mode 100644 index 0000000..89f4dfb --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/lemma9.lf @@ -0,0 +1,73 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Lemma 9 : Preservation of Var-validity under substitution +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% moved to def3+6+fig5.elf Thu Mar 10 20:45:31 2005 -fp +% & : stack -> stack -> stack. %infix left 10 & . +%sort app %. +%mode app %in %out %. +%term app_init app (Xi & dot) Xi %. +%term app_stack %pi (app (Xi1 & (Xi2 , T)) (Xi , T)) %<- (app (Xi1 & Xi2) Xi) %. +%sort appT %. +%mode appT %in %in %out %out %. +% note: Xi1' is output. -fp +% original from belmina -fp +%{ +app_v : {v:ctriv}{vvv:{xi:stack}vvalT (xi , v) v xi} + appT (vvv Xi) (app_stack APP) APP (vvv Xi'). + +app_t : {t:ctriv}{vvt:{xi:stack}vvalT xi t xi} + appT (vvt Xi) APP APP (vvt Xi'). +}% +%term app_t appT (vval_xlam VVR) APP APP (vval_xlam VVR) %. +%sort lemma9-1b {v ctriv} %. +%mode lemma9-1b %in %in %in %in %out %. +%sort lemma9-1e {v ctriv} %. +%mode lemma9-1e %in %in %in %in %out %. +%sort lemma9-3a {k ccont} {vvk vvalC dot k} %. +%mode lemma9-3a %in %in %in %in %out %. +%sort lemma9-3b {k ccont} {vvk vvalC dot k} %. +%mode lemma9-3b %in %in %in %in %out %. +%term lemma9-1b_cret_t lemma9-1b ([v ctriv] app_init) ([v ctriv] [vvv] vval_cret VVC (vvv Xi')) VVT app_init (vval_cret VVC (VVT Xi')) %. +%term lemma9-1b_cret_c + %pi (lemma9-1b APP ([v ctriv] [vvv] vval_cret (VVC v vvv) VVT) VVT1 APP1 (vval_cret VVC' (VVT1 Xi0))) + %<- (lemma9-1e APP VVC VVT1 APP1 VVC') %. +%term lemma9-1b_capp_t1 lemma9-1b ([v ctriv] app_init) ([v ctriv] [vvv] vval_capp VVC VVT0 (vvv Xi')) VVT app_init (vval_capp VVC VVT0 (VVT Xi')) %. +% original from Belmina -fp +%{ +lemma9-1b_capp_t0 : {vvt1:{xi:stack}vvalT xi T xi} + lemma9-1b ([v:ctriv]app_init) + ([v:ctriv][vvv] + vval_capp VVC + (vvv Xi') + (vvt1 (Xi' , v))) + VVT + app_init + (vval_capp VVC (VVT Xi') (vvt1 Xi')). +}% +%term lemma9-1b_capp_t0 lemma9-1b ([v ctriv] app_init) ([v ctriv] [vvv] vval_capp VVC (vvv Xi') (%the (vvalT (Xi' , v) _ (Xi' , v)) (vval_xlam VVR))) VVT app_init (vval_capp VVC (VVT Xi') (%the (vvalT Xi' _ Xi') (vval_xlam VVR))) %. +%term lemma9-1b_capp_c + %pi (lemma9-1b APP ([v ctriv] [vvv] vval_capp (VVC v vvv) VVT0 VVT1) VVT APP1 (vval_capp VVC' VVT0 VVT1)) + %<- (lemma9-1e APP VVC VVT APP1 VVC') %. +%term lemma9-1e_vlam + %pi (lemma9-1e APP ([v ctriv] [vvv] vval_vlam ([v1 ctriv] [vvv1] VVE v1 vvv1 v vvv)) VVT APP1 (vval_vlam VVE')) + %<- ({v1 ctriv} {vvv1 {xi stack} vvalT (xi , v1) v1 xi} + %pi ({Xi stack} {Xi0 stack} {Xi1 stack} {APP app (Xi0 & Xi) Xi1} appT (vvv1 Xi) (app_stack APP) APP (vvv1 Xi1)) + %-> (lemma9-1b ([v ctriv] app_stack (APP v)) (VVE v1 vvv1) VVT (app_stack APP1) (VVE' v1 vvv1))) %. +%term lemma9-3a_cret_c + %pi (lemma9-3a ([k ccont] [vvk] vval_cret (VVC k vvk) VVT) (cval_cret CVC CVT) VVC1 APP (vval_cret VVC' VVT')) + %<- (appT VVT APP APP1 VVT') + %<- (lemma9-3b VVC CVC VVC1 APP1 VVC') %. +%term lemma9-3a_capp_c + %pi (lemma9-3a ([k ccont] [vvk] vval_capp (VVC k vvk) VVT0 VVT1) (cval_capp CVC CVT0 CVT1) VVC1 APP (vval_capp VVC' VVT0' VVT1')) + %<- (appT VVT1 APP APP1 VVT1') + %<- (appT VVT0 APP1 APP2 VVT0') + %<- (lemma9-3b VVC CVC VVC1 APP2 VVC') %. +%term lemma9-3b_vlam + %pi (lemma9-3b ([k ccont] [vvk vvalC dot k] vval_vlam ([v ctriv] [vvv {xi stack} vvalT (xi , v) v xi] VVE v vvv k vvk)) (cval_vlam CVE) VVC1 APP (vval_vlam VVE')) + %<- ({v ctriv} {vvv {xi stack} vvalT (xi , v) v xi} {cvv cvalT v} + %pi ({Xi stack} {Xi0 stack} {Xi1 stack} {APP app (Xi0 & Xi) Xi1} appT (vvv Xi) (app_stack APP) APP (vvv Xi1)) + %-> (lemma9-3a (VVE v vvv) (CVE v cvv) VVC1 (app_stack APP) (VVE' v vvv))) %. +%term lemma9-3b_k lemma9-3b ([k ccont] [vvk vvalC dot k] vvk) cval_k VVC APP VVC %. +%terminates APP (appT _ APP _ _) %. +%terminates (VVE VVC) (lemma9-1b _ VVE _ _ _) (lemma9-1e _ VVC _ _ _) %. +%terminates (VVE VVC) (lemma9-3a VVE _ _ _ _) (lemma9-3b VVC _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/sources.lf b/new-tests/stelf-output/examples/cpsocc/sources.lf new file mode 100644 index 0000000..bbbbb8f --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/sources.lf @@ -0,0 +1,25 @@ +%require %[ dsBNF %] %. +%require %[ cpsBNF %] %. +%require %[ fig1 %] %. +%require %[ def1+5+fig4 %] %. +%require %[ th2 %] %. +%require %[ def3+6+fig5 %] %. +%require %[ th4 %] %. +%require %[ fig7 %] %. +%require %[ fig8 %] %. +%require %[ fig10 %] %. +%require %[ fig12 %] %. +%require %[ lemma7 %] %. +%require %[ th8 %] %. +%require %[ fig9 %] %. +%require %[ lemma9 %] %. +%require %[ th10 %] %. +%require %[ def13 %] %. +%require %[ lemma14 %] %. +%require %[ th15 %] %. +%require %[ def16 %] %. +%require %[ fig11 %] %. +%require %[ fig13 %] %. +%require %[ corr %] %. +%require %[ lemma17 %] %. +%require %[ th18 %] %. diff --git a/new-tests/stelf-output/examples/cpsocc/stelf.toml b/new-tests/stelf-output/examples/cpsocc/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/cpsocc/test-quy.lf b/new-tests/stelf-output/examples/cpsocc/test-quy.lf new file mode 100644 index 0000000..5e328a1 --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/test-quy.lf @@ -0,0 +1,240 @@ +%query 1 _ _ {y dtriv} {y' ctriv} + %pi (cpsT y y') + %-> (cpsR (dexp->droot (dapp (dapp (dtriv->dexp (dlam ([f dtriv] dexp->droot (dtriv->dexp (dlam ([x dtriv] dexp->droot (dapp (dtriv->dexp f) (dapp (dtriv->dexp f) (dtriv->dexp x))))))))) (dtriv->dexp (dlam ([x dtriv] dexp->droot (dtriv->dexp x))))) (dtriv->dexp y))) (R'1 y')) %. +%define r'1 (%pi ctriv %-> croot) [y' ctriv] klam ([CCONT1 ccont] capp (xlam ([CTRIV2 ctriv] klam ([CCONT2 ccont] cret CCONT2 (xlam ([CTRIV3 ctriv] klam ([CCONT3 ccont] capp CTRIV2 CTRIV3 (vlam ([CTRIV4 ctriv] capp CTRIV2 CTRIV4 (vlam ([CTRIV5 ctriv] cret CCONT3 CTRIV5)))))))))) (xlam ([CTRIV6 ctriv] klam ([CCONT4 ccont] cret CCONT4 CTRIV6))) (vlam ([CTRIV7 ctriv] capp CTRIV7 y' (vlam ([CTRIV8 ctriv] cret CCONT1 CTRIV8))))) %. +%solve CVR : {y' ctriv} %pi (cvalT y') %-> (cvalR (r'1 y')) %. +%solve CM : {y' ctriv} cstackR (r'1 y') (C y') %. +%query 1 _ _ {y' ctriv} {cvy cvalT y'} th15-1 (CVR y' cvy) (CM y') (BM y') %. +%{ +Solving... + +BM = + [y':ctriv] bare_klam [k:ccont] [b:{t:ctriv} bareE (cret k t) t] + bare_capp + (bare_cret + (bare_capp + (bare_capp + (bare_cret (bare_capp (bare_cret (bare_cret (b y')))))))), +C = [y':ctriv] y', +R' = + [y':ctriv] klam [k:ccont] + capp + (xlam [x':ctriv] klam [k1:ccont] + cret k1 + (xlam [x'1:ctriv] klam [k11:ccont] + capp x' x'1 + (vlam [t1:ctriv] capp x' t1 (vlam [v:ctriv] cret k11 v)))) + (xlam [x':ctriv] klam [k1:ccont] cret k1 x') + (vlam [t01:ctriv] capp t01 y' (vlam [v:ctriv] cret k v)). +; +no more solutions +}% +%{ +------------------------------------------------------------------------------- +sigma[CR : {y:dtriv}{y':ctriv}{cy:cpsT y y'} + cpsR (dexp->droot + (dapp + (dapp + (dtriv->dexp + (dlam [f:dtriv] + (dexp->droot + (dtriv->dexp + (dlam [x:dtriv] + (dexp->droot + (dapp + (dtriv->dexp f) + (dapp + (dtriv->dexp f) + (dtriv->dexp x))))))))) + (dtriv->dexp + (dlam [x:dtriv] + (dexp->droot + (dtriv->dexp x))))) + (dtriv->dexp y))) + (R' y')] +sigma[CVR : {y':ctriv}{cvy:cvalT y'} + cvalR (R' y')] +sigma[BM:{y':ctriv}bareR (R' y') (B y')] +{y':ctriv}{cvy:cvalT y'} +th15-1 (CVR y' cvy) (CM y') (BM y'). + Solving... + +CM = + [y':ctriv] + cstack_klam + (cstack_capp + (cstack_cret_k_vlam + (cstack_capp + (cstack_capp + (cstack_cret_k_vlam + (cstack_capp + (cstack_cret_k_vlam + (cstack_cret_k_vlam cstack_cret_k_init)) + cstack_vlam)) + cstack_vlam) + cstack_vlam)) + cstack_vlam), +B = [y':ctriv] y', +R' = + [y':ctriv] klam [k:ccont] + capp + (xlam [x':ctriv] klam [k1:ccont] + cret k1 + (xlam [x'1:ctriv] klam [k11:ccont] + capp x' x'1 + (vlam [t1:ctriv] capp x' t1 (vlam [v:ctriv] cret k11 v)))) + (xlam [x':ctriv] klam [k1:ccont] cret k1 x') + (vlam [t01:ctriv] capp t01 y' (vlam [v:ctriv] cret k v)). +; +no more solutions +------------------------------------------------------------------------------- +sigma[CR : {y:dtriv}{y':ctriv}{cy:cpsT y y'} + cpsR (dexp->droot + (dapp + (dapp + (dtriv->dexp + (dlam [f:dtriv] + (dexp->droot + (dtriv->dexp + (dlam [x:dtriv] + (dexp->droot + (dapp + (dtriv->dexp f) + (dapp + (dtriv->dexp f) + (dtriv->dexp x))))))))) + (dtriv->dexp + (dlam [x:dtriv] + (dexp->droot + (dtriv->dexp x))))) + (dtriv->dexp y))) + (R' y')] +sigma[VVR : {y':ctriv}{vvy:{xi:stack}vvalT xi y' xi} + vvalR (R' y')] +sigma[CVR : {y':ctriv}{cvy:cvalT y'} + cvalR (R' y')] +sigma[DR : {y':ctriv}{dy':{xi:stack}dstackT xi y' y' xi} + dstackR (R' y') (D y')] +{y:dtriv}{y':ctriv}{vvy:{xi:stack}vvalT xi y' xi}{cy:cpsT y y'}{cvy:cvalT y'} +{dy':{xi:stack}dstackT xi y' y' xi}{dsy':{xi:stack}dstack_substT y' xi y' xi} +th18-1 (VVR y' vvy) (CVR y' cvy) (DR y' dy') (BR y'). + Solving... + +BR = + [y':ctriv] bare_klam [k:ccont] [b:{t:ctriv} bareE (cret k t) t] + bare_capp + (bare_cret + (bare_capp + (bare_capp + (bare_cret (bare_capp (bare_cret (bare_cret (b y')))))))), +D = [y':ctriv] y', +R' = + [y':ctriv] klam [k:ccont] + capp + (xlam [x':ctriv] klam [k1:ccont] + cret k1 + (xlam [x'1:ctriv] klam [k11:ccont] + capp x' x'1 + (vlam [t1:ctriv] capp x' t1 (vlam [v:ctriv] cret k11 v)))) + (xlam [x':ctriv] klam [k1:ccont] cret k1 x') + (vlam [t01:ctriv] capp t01 y' (vlam [v:ctriv] cret k v)). +; +no more solutions +------------------------------------------------------------------------------- +sigma[CR : {y:dtriv}{y':ctriv}{cy:cpsT y y'} + cpsR (dexp->droot + (dapp + (dapp + (dtriv->dexp + (dlam [f:dtriv] + (dexp->droot + (dtriv->dexp + (dlam [x:dtriv] + (dexp->droot + (dapp + (dtriv->dexp f) + (dapp + (dtriv->dexp f) + (dtriv->dexp x))))))))) + (dtriv->dexp + (dlam [x:dtriv] + (dexp->droot + (dtriv->dexp x))))) + (dtriv->dexp y))) + (R' y')] +sigma[VVR : {y':ctriv}{vvy:{xi:stack}vvalT xi y' xi} + vvalR (R' y')] +sigma[CVR : {y':ctriv}{cvy:cvalT y'} + cvalR (R' y')] +sigma[BR : {y':ctriv} + bareR (R' y') (B y')] +{y:dtriv}{y':ctriv}{vvy:{xi:stack}vvalT xi y' xi}{cy:cpsT y y'}{cvy:cvalT y'} +{dy':{xi:stack}dstackT xi y' y' xi}{dsy':{xi:stack}dstack_substT y' xi y' xi} +th18-1 (VVR y' vvy) (CVR y' cvy) (DR y' dy') (BR y'). + Solving... + +DR = + [y':ctriv] [dy':{xi:stack} dstackT xi y' y' xi] dstack_klam [k:ccont] + [d:{xi:stack} {t:ctriv} {t':ctriv} + dstackT xi t t' dot -> dstackE xi (cret k t) t'] + dstack_capp + (dstack_cret_vlam + ([v:ctriv] [dv:{t:ctriv} {xi:stack} dstackT (xi , t) v t xi] + dstack_capp + (dstack_capp + (dstack_cret_vlam + ([v2:ctriv] + [dv2:{t:ctriv} {xi:stack} + dstackT (xi , t) v2 t xi] + dstack_capp + (dstack_cret_vlam + ([v21:ctriv] + [dv21:{t:ctriv} {xi:stack} + dstackT (xi , t) v21 t xi] + dstack_cret_vlam + ([v211:ctriv] + [dv211:{t:ctriv} {xi:stack} + dstackT (xi , t) + v211 t xi] + d (dot , y') v211 y' + (dv211 y' dot)) + (dv21 y' dot)) + (dy' dot)) + dstack_xlam (dv2 y' dot)) + (dy' dot)) + dstack_xlam (dy' dot)) + (dv (xlam [x'1:ctriv] klam [k111:ccont] + capp (xlam [x':ctriv] klam [k1:ccont] cret k1 x') + x'1 + (vlam [t1:ctriv] + capp + (xlam [x':ctriv] klam [k1:ccont] + cret k1 x') t1 + (vlam [v11:ctriv] cret k111 v11))) + dot) + (dy' (dot + , (xlam [x'1:ctriv] klam [k111:ccont] + capp + (xlam [x':ctriv] klam [k1:ccont] + cret k1 x') x'1 + (vlam [t1:ctriv] + capp + (xlam [x':ctriv] klam [k1:ccont] + cret k1 x') + t1 (vlam [v1:ctriv] cret k111 v1)))))) + dstack_xlam) + dstack_xlam dstack_xlam, +B = [y':ctriv] y', +R' = + [y':ctriv] klam [k:ccont] + capp + (xlam [x':ctriv] klam [k1:ccont] + cret k1 + (xlam [x'1:ctriv] klam [k11:ccont] + capp x' x'1 + (vlam [t1:ctriv] capp x' t1 (vlam [v:ctriv] cret k11 v)))) + (xlam [x':ctriv] klam [k1:ccont] cret k1 x') + (vlam [t01:ctriv] capp t01 y' (vlam [v:ctriv] cret k v)). +; +no more solutions +}% \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/test.lf b/new-tests/stelf-output/examples/cpsocc/test.lf new file mode 100644 index 0000000..cc084af --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/test.lf @@ -0,0 +1,26 @@ +%require %[ dsBNF %] %. +%require %[ cpsBNF %] %. +%require %[ fig1 %] %. +%require %[ def1+5+fig4 %] %. +%require %[ th2 %] %. +%require %[ def3+6+fig5 %] %. +%require %[ th4 %] %. +%require %[ fig7 %] %. +%require %[ fig8 %] %. +%require %[ fig10 %] %. +%require %[ fig12 %] %. +%require %[ lemma7 %] %. +%require %[ th8 %] %. +%require %[ fig9 %] %. +%require %[ lemma9 %] %. +%require %[ th10 %] %. +%require %[ def13 %] %. +%require %[ lemma14 %] %. +%require %[ th15 %] %. +%require %[ def16 %] %. +%require %[ fig11 %] %. +%require %[ fig13 %] %. +%require %[ corr %] %. +%require %[ lemma17 %] %. +%require %[ th18 %] %. +%require %[ test %] %. diff --git a/new-tests/stelf-output/examples/cpsocc/th10.lf b/new-tests/stelf-output/examples/cpsocc/th10.lf new file mode 100644 index 0000000..31331c6 --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/th10.lf @@ -0,0 +1,32 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Theorem 10 : Var-validity under beta-reduction +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort th10-1 %. +%mode th10-1 %in %out %. +%sort th10-2 {k ccont} {vvk vvalC dot k} %. +%mode th10-2 %in %in %in %out %. +% original from belmina +% needs to be asserted where a new v is introduced, not here. +%{ +th10-1_t : {v:ctriv}{vvv:{xi:stack}vvalT (xi , v) v xi} + th10-1 (vval_cret (vval_vlam VVE) (vvv Xi)) + (VVE v vvv). +}% +% original from belmina +%{ +th10-1_c : th10-1 (vval_cret (vval_vlam VVE) (VVT Xi)) + VVE' + <- lemma9-1b ([v:ctriv]app_init) + VVE + VVT + app_init + VVE'. +}% +%term th10-1_c + %pi (th10-1 (vval_cret (vval_vlam VVE) (vval_xlam VVR)) VVE') + %<- (lemma9-1b ([v ctriv] app_init) VVE ([xi stack] vval_xlam VVR) app_init VVE') %. +%term th10-2_proof + %pi (th10-2 ([k ccont] [vvk] vval_capp (VVC k vvk) (vval_xlam ([x ctriv] [vvx] vval_klam (VVE x vvx))) (VVT Xi)) (cval_capp CVC (cval_xlam ([x ctriv] [cvx] cval_klam (CVE x cvx))) CVT) VVT VVE') + %<- ({k ccont} {vvk vvalC dot k} lemma9-3a (VVE T VVT) (CVE T CVT) (VVC k vvk) app_init (VVE' k vvk)) %. +%terminates {} (th10-1 _ _) %. +%terminates {} (th10-2 _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/th15.lf b/new-tests/stelf-output/examples/cpsocc/th15.lf new file mode 100644 index 0000000..e82c0eb --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/th15.lf @@ -0,0 +1,28 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Theorem 15 : The control-stack and the bare machine are equivalent +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort th15-1 %. +%sort th15-2 %. +%term th15-1_proof + %pi (th15-1 (cval_klam CVE) (cstack_klam CE) (bare_klam ([k ccont] [b] BE k b))) + %<- (lemma14-1c CVE CSE') + %<- (th15-2 cvalCS_init CVE CE CSE' BE) %. +%term th15-2_capp_k_k_init + %pi (th15-2 cvalCS_init (cval_capp cval_k (cval_xlam ([t ctriv] [cvt cvalT t] cval_klam (CVE t cvt))) CVT) (cstack_capp CE cstack_phi_k) (csubst_capp csubst_k_init) ([k ccont] [b] bare_capp (BE k b))) + %<- (lemma14-1c (CVE T CVT) CSE') + %<- (th15-2 cvalCS_init (CVE T CVT) CE CSE' BE) %. +%term th15-2_capp_k_vlam + %pi (th15-2 CVS (cval_capp cval_k (cval_xlam ([t ctriv] [cvt cvalT t] cval_klam (CVE t cvt))) CVT) (cstack_capp CE cstack_phi_k) (csubst_capp (csubst_k_vlam CSE1)) ([k ccont] [b] bare_capp (BE k b))) + %<- (lemma14-1a CVS (CVE T CVT) CSE1 CSE') + %<- (th15-2 CVS (CVE T CVT) CE CSE' BE) %. +%term th15-2_capp_vlam_phi + %pi (th15-2 CVS (cval_capp (cval_vlam CVE1) (cval_xlam ([t ctriv] [cvt cvalT t] cval_klam (CVE t cvt))) CVT) (cstack_capp CE cstack_vlam) (csubst_capp (csubst_vlam CSE1)) ([k ccont] [b] bare_capp (BE k b))) + %<- (lemma14-1a (cvalCS_vlam CVE1 CVS) (CVE T CVT) CSE1 CSE') + %<- (th15-2 (cvalCS_vlam CVE1 CVS) (CVE T CVT) CE CSE' BE) %. +%term th15-2_cret_k_init th15-2 cvalCS_init (cval_cret cval_k CVT) cstack_cret_k_init (csubst_cret csubst_k_init) ([k ccont] [b] b T) %. +%term th15-2_cret_k_vlam + %pi (th15-2 (cvalCS_vlam CVE CVS) (cval_cret cval_k CVT) (cstack_cret_k_vlam CE) (csubst_cret (csubst_k_vlam CSE)) ([k ccont] [b] bare_cret (BE k b))) + %<- (th15-2 CVS (CVE T CVT) CE (CSE T) BE) %. +%term th15-2_cret_vlam_phi + %pi (th15-2 CVS (cval_cret (cval_vlam CVE) CVT) (cstack_cret_vlam_phi CE) (csubst_cret (csubst_vlam CSE')) ([k ccont] [b] bare_cret (BE k b))) + %<- (th15-2 CVS (CVE T CVT) CE (CSE' T) BE) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/th18.lf b/new-tests/stelf-output/examples/cpsocc/th18.lf new file mode 100644 index 0000000..b765f5e --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/th18.lf @@ -0,0 +1,33 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Theorem 18 : The data-stack and the bare +%%%%% machine are equivalent +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort th18-1 %. +%sort th18-2 %. +%sort th18-3 %. +%term th18-1_klam + %pi (th18-1 (vval_klam VVE) (cval_klam CVE) (dstack_klam DE) (bare_klam BE)) + %<- ({k ccont} {vvk vvalC dot k} {s dstack_substC k dot k} {l17-1 lemma17-1C vvk s} lemma17-1E (VVE k vvk) (DSE k s)) + %<- (th18-2 vvalDS_init cvalDS_init VVE CVE DE DSE BE) %. +%term th18-2_cret_init + %pi (th18-2 VVDS CVDS VVT CVT ([k ccont] [d] d Xi T T' DT) ([k ccont] [s] dstack_subst_cret s DST) BE) + %<- (th18-3 VVDS DT DST) %. +%term th18-2_cret_vlam + %pi (th18-2 VVDS CVDS ([k ccont] [vvk] vval_cret (vval_vlam ([v ctriv] [vvv] VVE v vvv k vvk)) VVT) (cval_cret (cval_vlam CVE) CVT) ([k ccont] [d] dstack_cret_vlam ([v ctriv] [dv] DE v dv k d) DT) ([k ccont] [s] dstack_subst_cret (dstack_subst_vlam ([v ctriv] [dsv] DSE v dsv k s)) DST) ([k ccont] [b] bare_cret (BE k b))) + %<- (th18-3 VVDS DT DST) + %<- (rsT VVT CVT VVDS CVDS DST VVT' CVT' VVDS' CVDS' DST') + %<- (vvalDS->vvalMS VVDS' VVMS') + %<- ({k ccont} {vvk vvalC dot k} {s dstack_substC k dot k} lemma17-2 ([v ctriv] T) ([v ctriv] app_init) ([v ctriv] [vvv] VVE v vvv k vvk) ([v ctriv] app_init) ([v ctriv] vvalMS_t VVMS' VVT') ([v ctriv] [dsv] DSE v dsv k s) ([v ctriv] [dsv] DSE' v dsv k s)) + %<- ({v1 ctriv} {dsv1 {xi stack} {t ctriv} dstack_substT v1 (xi , t) t xi} {dv1 {t ctriv} {xi stack} dstackT (xi , t) v1 t xi} {vvv1 {xi stack} vvalT (xi , v1) v1 xi} {cvv1 cvalT v1} th18-2 (vvalDS_stack DVS' VVT') (cvalDS_stack CVS' CVT') (VVE v1 vvv1) (CVE v1 cvv1) (DE v1 dv1) (DSE' v1 dsv1) BE) %. +%term th18-2_capp + %pi (th18-2 VVDS CVDS ([k ccont] [vvk] vval_capp (VVC k vvk) VVT0 VVT1) (cval_capp CVC CVT0 CVT1) ([k ccont] [d] dstack_capp (DE k d) DT0 DT1) ([k ccont] [s] dstack_subst_capp (DSC k s) DST0 DST1) ([k ccont] [b] bare_capp (BE k b))) + %<- (th18-3 VVDS DT1 DST1) + %<- (rsT VVT1 CVT1 VVDS CVDS DST1 VVT1' CVT1' VVDS' CVDS' DST1') + %<- (th18-3 VVDS' DT0 DST0) + %<- (rsT VVT0 CVT0 VVDS' CVDS' DST0 ([xi stack] vval_xlam ([x ctriv] [vvx {xi' stack} vvalT xi' x xi'] vval_klam ([k ccont] [vvk] VVE x vvx k vvk))) (cval_xlam ([x ctriv] [cvx] cval_klam (CVE x cvx))) VVDS'' CVDS'' ([xi stack] dstack_subst_xlam)) + %<- (th10-2 ([k ccont] [vvk] vval_capp (VVC k vvk) (vval_xlam ([x ctriv] [vvx {xi' stack} vvalT xi' x xi'] vval_klam ([k ccont] [vvk] VVE x vvx k vvk))) (VVT1' XXi'')) (cval_capp CVC (cval_xlam ([x ctriv] [cvx] cval_klam (CVE x cvx))) CVT1') VVT1' VVE') + %<- (th8-2 (cval_capp CVC (cval_xlam ([x ctriv] [cvx] cval_klam (CVE x cvx))) CVT1') CVE') + %<- ({k ccont} {vvk vvalC dot k} {s dstack_substC k dot k} {l17-1 lemma17-1C vvk s} {dbl doubleC vvk vvalMS_init s v-part_init s} clr (VVE T1' VVT1') (CVE T1' CVT1') (VVC k vvk) VVDS'' (DSC k s) (DSE' k s)) + %<- (th18-2 VVDS'' CVDS'' VVE' CVE' DE DSE' BE) %. +%term th18-3_t {t ctriv} {dst {xi stack} dstack_substT t xi t xi} {dt {xi stack} dstackT xi t t xi} th18-3 DVS (dt Xi) (dst Xi) %. +%term th18-3_v {v ctriv} {dsv {xi stack} {t ctriv} dstack_substT v (xi , t) t xi} {dv {xi stack} {t ctriv} dstackT (xi , t) v t xi} th18-3 DVS (dv Xi T) (dsv Xi T) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/th2.lf b/new-tests/stelf-output/examples/cpsocc/th2.lf new file mode 100644 index 0000000..82df540 --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/th2.lf @@ -0,0 +1,26 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Theorem 2 : A result of cps transformation is cont-valid +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort cps_cvR %. +%sort cps_cvE %. +%sort cps_cvT %. +%mode cps_cvR %in %out %. +%mode cps_cvE %in %in %out %. +%mode cps_cvT %in %out %. +%name cps_cvR %. +%name cps_cvE %. +%name cps_cvT %. +%term cval_droot + %pi (cps_cvR (cps_droot CE) (cval_klam CVE)) + %<- (cps_cvE (cval_kappa ([v ctriv] [CVV cvalT v] cval_cret cval_k CVV)) CE CVE) %. +%term cval_dtriv + %pi (cps_cvE (cval_kappa CVE) ([k ccont] cps_dtriv CT) (CVE T CVT)) + %<- (cps_cvT CT CVT) %. +%term cval_dapp + %pi (cps_cvE (cval_kappa CVE) ([k ccont] cps_dapp (CE0 k) (CE1 k)) CE) + %<- ({t0 ctriv} {CVt0 cvalT t0} cps_cvE (cval_kappa ([t1 ctriv] [cvt1 cvalT t1] cval_capp (cval_vlam CVE) CVt0 cvt1)) ([k ccont] CE1 k t0) (CVE' t0 CVt0)) + %<- (cps_cvE (cval_kappa CVE') CE0 CE) %. +%term cval_dlam + %pi (cps_cvT (cps_dlam CR) (cval_xlam CVR)) + %<- ({x dtriv} {x' ctriv} {CX cpsT x x'} {CVX cvalT x'} %pi (cps_cvT CX CVX) %-> (cps_cvR (CR x x' CX) (CVR x' CVX))) %. +%terminates (CR CE CT) (cps_cvR CR _) (cps_cvE _ CE _) (cps_cvT CT _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/th4.lf b/new-tests/stelf-output/examples/cpsocc/th4.lf new file mode 100644 index 0000000..3387346 --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/th4.lf @@ -0,0 +1,27 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Theorem 4 : A result of cps transformation is var-valid +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort cps_vvR %. +%sort cps_vvE %. +%sort cps_vvT %. +%mode cps_vvR %in %out %. +%mode cps_vvE %in %in %out %. +%mode cps_vvT %in %out %. +%name cps_vvR %. +%name cps_vvE %. +%name cps_vvT %. +%term vval_droot + %pi (cps_vvR (cps_droot CE) (vval_klam VVE)) + %<- ({k ccont} {VVC vvalC dot k} cps_vvE (vval_kappa ([t ctriv] [VVV {Xi' stack} vvalT Xi' t Xi'] vval_cret VVC (VVV dot)) ([v ctriv] [VVV {Xi' stack} vvalT (Xi' , v) v Xi'] vval_cret VVC (VVV dot))) (CE k) (VVE k VVC)) %. +%term vval_dtriv + %pi (cps_vvE (vval_kappa VVTE VVVE) (cps_dtriv CT) (VVTE T VVT)) + %<- (cps_vvT CT VVT) %. +%term vval_dapp + %pi (cps_vvE (vval_kappa VVTE VVVE) (cps_dapp CE0 CE1) VE) + %<- ({t0 ctriv} {VVt0 {Xi' stack} vvalT Xi' t0 Xi'} cps_vvE (vval_kappa ([t1 ctriv] [vvt1 {Xi stack} vvalT Xi t1 Xi] vval_capp (vval_vlam VVVE) (VVt0 Xi) (vvt1 Xi)) ([v1 ctriv] [vvv1 {Xi stack} vvalT (Xi , v1) v1 Xi] vval_capp (vval_vlam VVVE) (VVt0 Xi) (vvv1 Xi))) (CE1 t0) (VVTE' t0 VVt0)) + %<- ({v0 ctriv} {VVv0 {Xi' stack} vvalT (Xi' , v0) v0 Xi'} cps_vvE (vval_kappa ([t1 ctriv] [vvt1 {Xi stack} vvalT Xi t1 Xi] vval_capp (vval_vlam VVVE) (VVv0 Xi) (vvt1 (Xi , v0))) ([v1 ctriv] [vvv1 {Xi stack} vvalT (Xi , v1) v1 Xi] vval_capp (vval_vlam VVVE) (VVv0 Xi) (vvv1 (Xi , v0)))) (CE1 v0) (VVVE' v0 VVv0)) + %<- (cps_vvE (vval_kappa VVTE' VVVE') CE0 VE) %. +%term vval_dlam + %pi (cps_vvT (cps_dlam CR) ([Xi stack] vval_xlam VVR)) + %<- ({x dtriv} {x' ctriv} {CX cpsT x x'} {VVX {Xi stack} vvalT Xi x' Xi} %pi (cps_vvT CX VVX) %-> (cps_vvR (CR x x' CX) (VVR x' VVX))) %. +%terminates (CR CE CT) (cps_vvR CR _) (cps_vvE _ CE _) (cps_vvT CT _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cpsocc/th8.lf b/new-tests/stelf-output/examples/cpsocc/th8.lf new file mode 100644 index 0000000..85854aa --- /dev/null +++ b/new-tests/stelf-output/examples/cpsocc/th8.lf @@ -0,0 +1,12 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%% Theorem 8 : Preservation of Cont-validity under beta-reduction +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort th8-1 %. +%mode th8-1 %in %out %. +%term th8-1_proof th8-1 (cval_cret (cval_vlam CVE) CVT) (CVE T CVT) %. +%sort th8-2 %. +%mode th8-2 %in %out %. +%term th8-2_proof + %pi (th8-2 (cval_capp CVC (cval_xlam ([x ctriv] [cvx cvalT x] cval_klam (CVE x cvx))) CVT) CVEreplCk) + %<- (lemma7_2b CVC (CVE T CVT) CVEreplCk) %. +%terminates {} (th8-2 _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cut-elim/cl-admit.lf b/new-tests/stelf-output/examples/cut-elim/cl-admit.lf new file mode 100644 index 0000000..f6dd024 --- /dev/null +++ b/new-tests/stelf-output/examples/cut-elim/cl-admit.lf @@ -0,0 +1,155 @@ +%%% Admissibility of Cut in Classical Sequent Calculus +%%% Author: Frank Pfenning +%sort ca' {A o} %. +%mode ca' %in %in %in %out %. +%%% Essential Conversions +%term ca_axiom'l ca' A ([p] axiom' N p) E (E N) %. +%term ca_axiom'r ca' A D ([n] axiom' n P) (D P) %. +%term ca_and1' + %pi (ca' (A and B) ([p] andr' (D1 p) (D2 p) p) ([n] andl1' ([n1] E1 n n1) n) F) + %<- ({p1 pos A} ca' (A and B) ([p] D1 p p1) ([n] andl1' ([n1] E1 n n1) n) (D1' p1)) + %<- ({n1 neg A} ca' (A and B) ([p] andr' (D1 p) (D2 p) p) ([n] E1 n n1) (E1' n1)) + %<- (ca' A ([p1] D1' p1) ([n1] E1' n1) F) %. +%term ca_and2' + %pi (ca' (A and B) ([p] andr' (D1 p) (D2 p) p) ([n] andl2' (E2 n) n) F) + %<- ({p2 pos B} ca' (A and B) ([p] D2 p p2) ([n] andl2' (E2 n) n) (D2' p2)) + %<- ({n2 neg B} ca' (A and B) ([p] andr' (D1 p) (D2 p) p) ([n] E2 n n2) (E2' n2)) + %<- (ca' B ([p2] D2' p2) ([n2] E2' n2) F) %. +%term ca_imp' + %pi (ca' (A imp B) ([p] impr' (D1 p) p) ([n] impl' (E1 n) (E2 n) n) F) + %<- ({p1 pos A} ca' (A imp B) ([p] impr' (D1 p) p) ([n] E1 n p1) (E1' p1)) + %<- ({n2 neg B} ca' (A imp B) ([p] impr' (D1 p) p) ([n] E2 n n2) (E2' n2)) + %<- ({n1 neg A} {p2 pos B} ca' (A imp B) ([p] D1 p n1 p2) ([n] impl' (E1 n) (E2 n) n) (D1' n1 p2)) + %<- ({p2 pos B} ca' A ([p1] E1' p1) ([n1] D1' n1 p2) (F2 p2)) + %<- (ca' B ([p2] F2 p2) ([n2] E2' n2) F) %. +%term ca_or1' + %pi (ca' (A or B) ([p] orr1' (D1 p) p) ([n] orl' (E1 n) (E2 n) n) F) + %<- ({n1 neg A} ca' (A or B) ([p] orr1' (D1 p) p) ([n] E1 n n1) (E1' n1)) + %<- ({p1 pos A} ca' (A or B) ([p] D1 p p1) ([n] orl' (E1 n) (E2 n) n) (D1' p1)) + %<- (ca' A D1' E1' F) %. +%term ca_or2' + %pi (ca' (A or B) ([p] orr2' (D2 p) p) ([n] orl' (E1 n) (E2 n) n) F) + %<- ({n2 neg B} ca' (A or B) ([p] orr2' (D2 p) p) ([n] E2 n n2) (E2' n2)) + %<- ({p2 pos B} ca' (A or B) ([p] D2 p p2) ([n] orl' (E1 n) (E2 n) n) (D2' p2)) + %<- (ca' B D2' E2' F) %. +%term ca_not' + %pi (ca' (not A) ([p] notr' (D1 p) p) ([n] notl' (E1 n) n) F) + %<- ({p1 pos A} ca' (not A) ([p] notr' (D1 p) p) ([n] E1 n p1) (E1' p1)) + %<- ({n1 neg A} ca' (not A) ([p] D1 p n1) ([n] notl' (E1 n) n) (D1' n1)) + %<- (ca' A E1' D1' F) %. +%term ca_forall' + %pi (ca' (forall A) ([p] forallr' (D1 p) p) ([n] foralll' T (E1 n) n) F) + %<- ({n1} ca' (forall A) ([p] forallr' (D1 p) p) ([n] E1 n n1) (E1' n1)) + %<- ({p1} ca' (forall A) ([p] D1 p T p1) ([n] foralll' T (E1 n) n) (D1' p1)) + %<- (ca' (A T) D1' E1' F) %. +%term ca_exists' + %pi (ca' (exists A) ([p] existsr' T (D1 p) p) ([n] existsl' (E1 n) n) F) + %<- ({n1} ca' (exists A) ([p] existsr' T (D1 p) p) ([n] E1 n T n1) (E1' n1)) + %<- ({p1} ca' (exists A) ([p] D1 p p1) ([n] existsl' (E1 n) n) (D1' p1)) + %<- (ca' (A T) D1' E1' F) %. +%%% Right Commutative Conversions +%term car_axiom' ca' A D ([n] axiom' N P) (axiom' N P) %. +%term car_andr' + %pi (ca' A D ([n] andr' (E1 n) (E2 n) P) (andr' F1 F2 P)) + %<- ({p1 pos B1} ca' A D ([n] E1 n p1) (F1 p1)) + %<- ({p2 pos B2} ca' A D ([n] E2 n p2) (F2 p2)) %. +%term car_andl1' + %pi (ca' A D ([n] andl1' (E1 n) N) (andl1' F1 N)) + %<- ({n1 neg B1} ca' A D ([n] E1 n n1) (F1 n1)) %. +%term car_andl2' + %pi (ca' A D ([n] andl2' (E2 n) N) (andl2' F2 N)) + %<- ({n2 neg B2} ca' A D ([n] E2 n n2) (F2 n2)) %. +%term car_impr' + %pi (ca' A D ([n] impr' (E1 n) P) (impr' F1 P)) + %<- ({n1 neg B1} {p2 pos B2} ca' A D ([n] E1 n n1 p2) (F1 n1 p2)) %. +%term car_impl' + %pi (ca' A D ([n] impl' (E1 n) (E2 n) N) (impl' F1 F2 N)) + %<- ({p1 pos B1} ca' A D ([n] E1 n p1) (F1 p1)) + %<- ({n2 neg B2} ca' A D ([n] E2 n n2) (F2 n2)) %. +%term car_orr1' + %pi (ca' A D ([n] orr1' (E1 n) P) (orr1' F1 P)) + %<- ({p1 pos B1} ca' A D ([n] E1 n p1) (F1 p1)) %. +%term car_orr2' + %pi (ca' A D ([n] orr2' (E2 n) P) (orr2' F2 P)) + %<- ({p2 pos B2} ca' A D ([n] E2 n p2) (F2 p2)) %. +%term car_orl' + %pi (ca' A D ([n] orl' (E1 n) (E2 n) N) (orl' F1 F2 N)) + %<- ({n1 neg B1} ca' A D ([n] E1 n n1) (F1 n1)) + %<- ({n2 neg B2} ca' A D ([n] E2 n n2) (F2 n2)) %. +%term car_notr' + %pi (ca' A D ([n] notr' (E1 n) P) (notr' F1 P)) + %<- ({n1 neg B1} ca' A D ([n] E1 n n1) (F1 n1)) %. +%term car_notl' + %pi (ca' A D ([n] notl' (E1 n) N) (notl' F1 N)) + %<- ({p1 pos B1} ca' A D ([n] E1 n p1) (F1 p1)) %. +%term car_truer' ca' A D ([n] truer' P) (truer' P) %. +%term car_falsel' ca' A D ([n] falsel' N) (falsel' N) %. +%term car_forallr' + %pi (ca' A D ([n] forallr' (E1 n) P) (forallr' F1 P)) + %<- ({a i} {p1 pos (B1 a)} ca' A D ([n] E1 n a p1) (F1 a p1)) %. +%term car_foralll' + %pi (ca' A D ([n] foralll' T (E1 n) N) (foralll' T F1 N)) + %<- ({n1} ca' A D ([n] E1 n n1) (F1 n1)) %. +%term car_existsr' + %pi (ca' A D ([n] existsr' T (E1 n) P) (existsr' T F1 P)) + %<- ({p1} ca' A D ([n] E1 n p1) (F1 p1)) %. +%term car_existsl' + %pi (ca' A D ([n] existsl' (E1 n) N) (existsl' F1 N)) + %<- ({a i} {n1 neg (B1 a)} ca' A D ([n] E1 n a n1) (F1 a n1)) %. +%%% Left Commutative Conversions +%term cal_axiom' ca' A ([p] axiom' N P) E (axiom' N P) %. +%term cal_andr' + %pi (ca' A ([p] andr' (D1 p) (D2 p) P) E (andr' F1 F2 P)) + %<- ({p1 pos B1} ca' A ([p] D1 p p1) E (F1 p1)) + %<- ({p2 pos B2} ca' A ([p] D2 p p2) E (F2 p2)) %. +%term cal_andl1' + %pi (ca' A ([p] andl1' (D1 p) N) E (andl1' F1 N)) + %<- ({n1 neg B1} ca' A ([p] D1 p n1) E (F1 n1)) %. +%term cal_andl2' + %pi (ca' A ([p] andl2' (D2 p) N) E (andl2' F2 N)) + %<- ({n2 neg B2} ca' A ([p] D2 p n2) E (F2 n2)) %. +%term cal_impr' + %pi (ca' A ([p] impr' (D1 p) P) E (impr' F1 P)) + %<- ({n1 neg B1} {p2 pos B2} ca' A ([p] D1 p n1 p2) E (F1 n1 p2)) %. +%term cal_impl' + %pi (ca' A ([p] impl' (D1 p) (D2 p) N) E (impl' F1 F2 N)) + %<- ({p1 pos B1} ca' A ([p] D1 p p1) E (F1 p1)) + %<- ({n2 neg B2} ca' A ([p] D2 p n2) E (F2 n2)) %. +%term cal_orr1' + %pi (ca' A ([p] orr1' (D1 p) P) E (orr1' F1 P)) + %<- ({p1 pos B1} ca' A ([p] D1 p p1) E (F1 p1)) %. +%term cal_orr2' + %pi (ca' A ([p] orr2' (D2 p) P) E (orr2' F2 P)) + %<- ({p2 pos B2} ca' A ([p] D2 p p2) E (F2 p2)) %. +%term cal_orl' + %pi (ca' A ([p] orl' (D1 p) (D2 p) N) E (orl' F1 F2 N)) + %<- ({n1 neg B1} ca' A ([p] D1 p n1) E (F1 n1)) + %<- ({n2 neg B2} ca' A ([p] D2 p n2) E (F2 n2)) %. +%term cal_notr' + %pi (ca' A ([p] notr' (D1 p) P) E (notr' F1 P)) + %<- ({n1 neg B1} ca' A ([p] D1 p n1) E (F1 n1)) %. +%term cal_notl' + %pi (ca' A ([p] notl' (D1 p) N) E (notl' F1 N)) + %<- ({p1 pos B1} ca' A ([p] D1 p p1) E (F1 p1)) %. +%term cal_truer' ca' A ([p] truer' P) E (truer' P) %. +%term cal_falsel' ca' A ([p] falsel' N) E (falsel' N) %. +%term cal_forallr' + %pi (ca' A ([p] forallr' (D1 p) P) E (forallr' F1 P)) + %<- ({a i} {p1 pos (B1 a)} ca' A ([p] D1 p a p1) E (F1 a p1)) %. +%term cal_foralll' + %pi (ca' A ([p] foralll' T (D1 p) N) E (foralll' T F1 N)) + %<- ({n1} ca' A ([p] D1 p n1) E (F1 n1)) %. +%term cal_existsr' + %pi (ca' A ([p] existsr' T (D1 p) P) E (existsr' T F1 P)) + %<- ({p1} ca' A ([p] D1 p p1) E (F1 p1)) %. +%term cal_existsl' + %pi (ca' A ([p] existsl' (D1 p) N) E (existsl' F1 N)) + %<- ({a i} {n1 neg (B1 a)} ca' A ([p] D1 p a n1) E (F1 a n1)) %. +%block lp {A o} [p pos A]%. +%block ln {A o} [n neg A]%. +%block li [a i]%. +%block lo [p o]%. +%worlds (lp ln li lo) (ca' A D E F) %. +%covers ca' %in %in %in %star %. +%terminates {A [D E]} (ca' A D E _) %. +%total {A [D E]} (ca' A D E _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cut-elim/cl-cut.lf b/new-tests/stelf-output/examples/cut-elim/cl-cut.lf new file mode 100644 index 0000000..559e544 --- /dev/null +++ b/new-tests/stelf-output/examples/cut-elim/cl-cut.lf @@ -0,0 +1,23 @@ +%%% Classical Sequent Calculus with Cut. +%%% Author: Frank Pfenning +%sort @ %. +% Token (for contradiction) +%name @ %. +%term cut^ %pi (%pi (pos A) %-> @) %-> (%pi (neg A) %-> @) %-> @ %. +%term axiom^ %pi (neg A) %-> (pos A) %-> @ %. +%term andr^ %pi (%pi (pos A) %-> @) %-> (%pi (pos B) %-> @) %-> (pos (A and B)) %-> @ %. +%term andl1^ %pi (%pi (neg A) %-> @) %-> (neg (A and B)) %-> @ %. +%term andl2^ %pi (%pi (neg B) %-> @) %-> (neg (A and B)) %-> @ %. +%term impr^ %pi (%pi (neg A) %-> (pos B) %-> @) %-> (pos (A imp B)) %-> @ %. +%term impl^ %pi (%pi (pos A) %-> @) %-> (%pi (neg B) %-> @) %-> (neg (A imp B)) %-> @ %. +%term orr1^ %pi (%pi (pos A) %-> @) %-> (pos (A or B)) %-> @ %. +%term orr2^ %pi (%pi (pos B) %-> @) %-> (pos (A or B)) %-> @ %. +%term orl^ %pi (%pi (neg A) %-> @) %-> (%pi (neg B) %-> @) %-> (neg (A or B)) %-> @ %. +%term notr^ %pi (%pi (neg A) %-> @) %-> (pos (not A)) %-> @ %. +%term notl^ %pi (%pi (pos A) %-> @) %-> (neg (not A)) %-> @ %. +%term truer^ %pi (pos true) %-> @ %. +%term falsel^ %pi (neg false) %-> @ %. +%term forallr^ %pi ({a i} %pi (pos (A a)) %-> @) %-> (pos (forall A)) %-> @ %. +%term foralll^ {T i} %pi (%pi (neg (A T)) %-> @) %-> (neg (forall A)) %-> @ %. +%term existsr^ {T i} %pi (%pi (pos (A T)) %-> @) %-> (pos (exists A)) %-> @ %. +%term existsl^ %pi ({a i} %pi (neg (A a)) %-> @) %-> (neg (exists A)) %-> @ %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cut-elim/cl-elim.lf b/new-tests/stelf-output/examples/cut-elim/cl-elim.lf new file mode 100644 index 0000000..7ce756e --- /dev/null +++ b/new-tests/stelf-output/examples/cut-elim/cl-elim.lf @@ -0,0 +1,43 @@ +%%% Cut Elimination in Classical Sequent Calculus +%%% Author: Frank Pfenning +%sort ce' %. +%mode ce' %in %out %. +%term ce_cut' + %pi (ce' (cut^ D E) F) + %<- ({p pos A} ce' (D p) (D' p)) + %<- ({n neg A} ce' (E n) (E' n)) + %<- (ca' A D' E' F) %. +%term ce_axiom' ce' (axiom^ N P) (axiom' N P) %. +%term ce_andr' + %pi (ce' (andr^ D1 D2 P) (andr' D1' D2' P)) + %<- ({p1} ce' (D1 p1) (D1' p1)) + %<- ({p2} ce' (D2 p2) (D2' p2)) %. +%term ce_andl1' %pi (ce' (andl1^ N1 N) (andl1' N1' N)) %<- ({n1} ce' (N1 n1) (N1' n1)) %. +%term ce_andl2' %pi (ce' (andl2^ N2 N) (andl2' N2' N)) %<- ({n2} ce' (N2 n2) (N2' n2)) %. +%term ce_impr' %pi (ce' (impr^ D1 P) (impr' D1' P)) %<- ({n1} {p2} ce' (D1 n1 p2) (D1' n1 p2)) %. +%term ce_impl' + %pi (ce' (impl^ D1 D2 N) (impl' D1' D2' N)) + %<- ({p1} ce' (D1 p1) (D1' p1)) + %<- ({n2} ce' (D2 n2) (D2' n2)) %. +%term ce_orr1' %pi (ce' (orr1^ D1 P) (orr1' D1' P)) %<- ({p1} ce' (D1 p1) (D1' p1)) %. +%term ce_orr2' %pi (ce' (orr2^ D2 P) (orr2' D2' P)) %<- ({p2} ce' (D2 p2) (D2' p2)) %. +%term ce_orl' + %pi (ce' (orl^ D1 D2 N) (orl' D1' D2' N)) + %<- ({n1} ce' (D1 n1) (D1' n1)) + %<- ({n2} ce' (D2 n2) (D2' n2)) %. +%term ce_notr' %pi (ce' (notr^ D1 P) (notr' D1' P)) %<- ({n1} ce' (D1 n1) (D1' n1)) %. +%term ce_norl' %pi (ce' (notl^ D1 N) (notl' D1' N)) %<- ({p1} ce' (D1 p1) (D1' p1)) %. +%term ce_truer' ce' (truer^ P) (truer' P) %. +%term ce_falsel' ce' (falsel^ N) (falsel' N) %. +%term ce_forallr' + %pi (ce' (forallr^ D1 P) (forallr' D1' P)) + %<- ({a i} {p1 pos (A1 a)} ce' (D1 a p1) (D1' a p1)) %. +%term ce_foralll' %pi (ce' (foralll^ T D1 N) (foralll' T D1' N)) %<- ({n1} ce' (D1 n1) (D1' n1)) %. +%term ce_existsr' %pi (ce' (existsr^ T D1 P) (existsr' T D1' P)) %<- ({p1} ce' (D1 p1) (D1' p1)) %. +%term ce_existsl' + %pi (ce' (existsl^ D1 N) (existsl' D1' N)) + %<- ({a i} {n1 neg (A1 a)} ce' (D1 a n1) (D1' a n1)) %. +%worlds (lp ln li lo) (ce' D D') %. +%covers ce' %in %star %. +%terminates D (ce' D _) %. +%total D (ce' D _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cut-elim/cl-thm.lf b/new-tests/stelf-output/examples/cut-elim/cl-thm.lf new file mode 100644 index 0000000..dca9c06 --- /dev/null +++ b/new-tests/stelf-output/examples/cut-elim/cl-thm.lf @@ -0,0 +1,3 @@ +%%% Termination of cut-elimination +%terminates {A [D E]} (ca' A D E _) %. +%terminates D (ce' D _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cut-elim/cl.lf b/new-tests/stelf-output/examples/cut-elim/cl.lf new file mode 100644 index 0000000..8571df1 --- /dev/null +++ b/new-tests/stelf-output/examples/cut-elim/cl.lf @@ -0,0 +1,28 @@ +%%% Cut-Free Classical Sequent Calculus +%%% Author: Frank Pfenning +%sort # %. +% Token (for contradiction) +%sort neg %. +% Hypotheses (left) +%sort pos %. +% Conclusions (right) +%name # %. +%name neg %. +%name pos %. +%term axiom' %pi (neg A) %-> (pos A) %-> # %. +%term andr' %pi (%pi (pos A) %-> #) %-> (%pi (pos B) %-> #) %-> (pos (A and B)) %-> # %. +%term andl1' %pi (%pi (neg A) %-> #) %-> (neg (A and B)) %-> # %. +%term andl2' %pi (%pi (neg B) %-> #) %-> (neg (A and B)) %-> # %. +%term impr' %pi (%pi (neg A) %-> (pos B) %-> #) %-> (pos (A imp B)) %-> # %. +%term impl' %pi (%pi (pos A) %-> #) %-> (%pi (neg B) %-> #) %-> (neg (A imp B)) %-> # %. +%term orr1' %pi (%pi (pos A) %-> #) %-> (pos (A or B)) %-> # %. +%term orr2' %pi (%pi (pos B) %-> #) %-> (pos (A or B)) %-> # %. +%term orl' %pi (%pi (neg A) %-> #) %-> (%pi (neg B) %-> #) %-> (neg (A or B)) %-> # %. +%term notr' %pi (%pi (neg A) %-> #) %-> (pos (not A)) %-> # %. +%term notl' %pi (%pi (pos A) %-> #) %-> (neg (not A)) %-> # %. +%term truer' %pi (pos true) %-> # %. +%term falsel' %pi (neg false) %-> # %. +%term forallr' %pi ({a i} %pi (pos (A a)) %-> #) %-> (pos (forall A)) %-> # %. +%term foralll' {T i} %pi (%pi (neg (A T)) %-> #) %-> (neg (forall A)) %-> # %. +%term existsr' {T i} %pi (%pi (pos (A T)) %-> #) %-> (pos (exists A)) %-> # %. +%term existsl' %pi ({a i} %pi (neg (A a)) %-> #) %-> (neg (exists A)) %-> # %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cut-elim/examples-quy.lf b/new-tests/stelf-output/examples/cut-elim/examples-quy.lf new file mode 100644 index 0000000..d2856aa --- /dev/null +++ b/new-tests/stelf-output/examples/cut-elim/examples-quy.lf @@ -0,0 +1,38 @@ +%%% Most examples here are just proof-checking +%%% Those will eventually work again. +% Example 1 +%define _ %the o (forall ([x i] A x imp B) imp (exists ([x i] A x) imp B)) %. +% sequent derivation +%define _ %the ({A %pi i %-> o} {B o} conc (forall ([x i] A x imp B) imp (exists ([x i] A x) imp B))) ([A %pi i %-> o] [B o] impr ([h1 hyp (forall ([x i] A x imp B))] impr ([h2 hyp (exists ([x i] A x))] existsl ([a i] [h3 hyp (A a)] foralll a ([h4 hyp (A a imp B)] impl (axiom h3) ([h5 hyp B] axiom h5) h4) h1) h2))) %. +% Example 2 +%define _ %the o (exists ([x i] A x or B x) imp (exists ([x i] A x) or (exists ([x i] B x)))) %. +% sequent derivation +%define _ %the ({A %pi i %-> o} {B %pi i %-> o} conc (exists ([x i] A x or B x) imp (exists ([x i] A x)) or (exists ([x i] B x)))) ([A %pi i %-> o] [B %pi i %-> o] impr ([h1 hyp (exists ([x i] A x or B x))] existsl ([a i] [h2 hyp (A a or B a)] orl ([h3 hyp (A a)] orr1 (existsr a (axiom h3))) ([h4 hyp (B a)] orr2 (existsr a (axiom h4))) h2) h1)) %. +% Example 3 +%define _ %the o (A' or B' imp (B' or A')) %. +%define _ %the ({A o} {B o} conc (A or B imp (B or A))) ([A' o] [B' o] impr ([h1 hyp (A' or B')] orl ([h2 hyp A'] orr2 (axiom h2)) ([h3 hyp B'] orr1 (axiom h3)) h1)) %. +% Example 4 +% Admissibility applied to Examples 2 and (instance of) 3 +% Below seems like a bug---there should not be constraints on +% F remaining. +%{ +%query 1 * +{A:i -> o} {B:i -> o} +{h1:hyp (exists [x:i] (A x or B x))} +ca +((exists [x:i] A x) or (exists [x:i] B x)) +(existsl ([a:i] [h2:hyp (A a or B a)] + (orl ([h3:hyp (A a)] orr1 (existsr a (axiom h3))) + ([h4:hyp (B a)] orr2 (existsr a (axiom h4))) + h2)) + h1) +([h:hyp ((exists [x:i] A x) or (exists [x:i] B x))] + (orl ([h2:hyp (exists [x:i] A x)] orr2 (axiom h2)) + ([h3:hyp (exists [x:i] B x)] orr1 (axiom h3)) + h)) +(F A B h1). +}% +% Example 5 (Classical Logic) +%define _ %the ({A o} %pi (pos (A or not A)) %-> #) ([A o] [p pos (A or not A)] orr1' ([p1 pos A] orr2' ([p2 pos (not A)] notr' ([n1 neg A] axiom' n1 p1) p2) p) p) %. +% Example 6 (Classical Logic) +%define _ %the ({A %pi i %-> o} %pi (neg (not (forall ([x] A x)))) %-> (pos (exists ([x] not (A x)))) %-> #) ([A %pi i %-> o] [n neg (not (forall ([x] A x)))] [p pos (exists ([x] not (A x)))] notl' ([p1 pos (forall ([x] A x))] forallr' ([a i] [p2 pos (A a)] existsr' a ([p3 pos (not (A a))] notr' ([n1 neg (A a)] axiom' n1 p2) p3) p) p1) n) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cut-elim/formulas.lf b/new-tests/stelf-output/examples/cut-elim/formulas.lf new file mode 100644 index 0000000..4ee3c7c --- /dev/null +++ b/new-tests/stelf-output/examples/cut-elim/formulas.lf @@ -0,0 +1,23 @@ +%%% Language of formulas for intuitionistic and classical +%%% predicate calculus. +%%% Author: Frank Pfenning +%sort i %. +% individuals +%name i %. +% S % more than one currently disallowed +%sort o %. +% formulas +%name o %. +% B C % more than one currently disallowed +%term and %pi o %-> o %-> o %. +%prec %right 11 and %. +%term imp %pi o %-> o %-> o %. +%prec %right 10 imp %. +%term or %pi o %-> o %-> o %. +%prec %right 11 or %. +%term not %pi o %-> o %. +%prec %prefix 12 not %. +%term true o %. +%term false o %. +%term forall %pi (%pi i %-> o) %-> o %. +%term exists %pi (%pi i %-> o) %-> o %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cut-elim/int-admit.lf b/new-tests/stelf-output/examples/cut-elim/int-admit.lf new file mode 100644 index 0000000..87f2659 --- /dev/null +++ b/new-tests/stelf-output/examples/cut-elim/int-admit.lf @@ -0,0 +1,107 @@ +%%% Admissibility of Cut in Intuitionistic Sequent Calculus +%%% Author: Frank Pfenning +%sort ca {A o} %. +%mode ca %in %in %in %out %. +%% Axiom Conversions +%term ca_axiom_l ca A (axiom H) E (E H) %. +%term ca_axiom_r ca A D ([h hyp A] axiom h) D %. +%% Essential Conversions +%term ca_and1 + %pi (ca (A1 and A2) (andr D1 D2) ([h hyp (A1 and A2)] andl1 (E1 h) h) F) + %<- ({h1 hyp A1} ca (A1 and A2) (andr D1 D2) ([h hyp (A1 and A2)] E1 h h1) (E1' h1)) + %<- (ca A1 D1 E1' F) %. +%term ca_and2 + %pi (ca (A1 and A2) (andr D1 D2) ([h hyp (A1 and A2)] andl2 (E2 h) h) F) + %<- ({h2 hyp A2} ca (A1 and A2) (andr D1 D2) ([h hyp (A1 and A2)] E2 h h2) (E2' h2)) + %<- (ca A2 D2 E2' F) %. +%term ca_imp + %pi (ca (A1 imp A2) (impr D2) ([h hyp (A1 imp A2)] impl (E1 h) (E2 h) h) F) + %<- (ca (A1 imp A2) (impr D2) E1 E1') + %<- ({h2 hyp A2} ca (A1 imp A2) (impr D2) ([h hyp (A1 imp A2)] E2 h h2) (E2' h2)) + %<- (ca A1 E1' D2 D2') + %<- (ca A2 D2' E2' F) %. +%term ca_or1 + %pi (ca (A1 or A2) (orr1 D1) ([h hyp (A1 or A2)] orl (E1 h) (E2 h) h) F) + %<- ({h1 hyp A1} ca (A1 or A2) (orr1 D1) ([h hyp (A1 or A2)] E1 h h1) (E1' h1)) + %<- (ca A1 D1 E1' F) %. +%term ca_or2 + %pi (ca (A1 or A2) (orr2 D2) ([h hyp (A1 or A2)] orl (E1 h) (E2 h) h) F) + %<- ({h2 hyp A2} ca (A1 or A2) (orr2 D2) ([h hyp (A1 or A2)] E2 h h2) (E2' h2)) + %<- (ca A2 D2 E2' F) %. +%term ca_not + %pi (ca (not A1) (notr D1) ([h hyp (not A1)] notl (E1 h) h) (F2 C)) + %<- (ca (not A1) (notr D1) E1 F1) + %<- ({p o} ca A1 F1 ([h1 hyp A1] D1 p h1) (F2 p)) %. +%term ca_forall + %pi (ca (forall A1) (forallr D1) ([h hyp (forall A1)] foralll T (E1 h) h) F) + %<- ({h2 hyp (A1 T)} ca (forall A1) (forallr D1) ([h hyp (forall A1)] E1 h h2) (E1' h2)) + %<- (ca (A1 T) (D1 T) E1' F) %. +%term ca_exists + %pi (ca (exists A1) (existsr T D1) ([h hyp (exists A1)] existsl (E1 h) h) F) + %<- ({a i} {h1 hyp (A1 a)} ca (exists A1) (existsr T D1) ([h hyp (exists A1)] E1 h a h1) (E1' a h1)) + %<- (ca (A1 T) D1 (E1' T) F) %. +%% Left Commutative Conversions +%term cal_andl1 %pi (ca A (andl1 D1 H) E (andl1 D1' H)) %<- ({h1 hyp B1} ca A (D1 h1) E (D1' h1)) %. +%term cal_andl2 %pi (ca A (andl2 D2 H) E (andl2 D2' H)) %<- ({h2 hyp B2} ca A (D2 h2) E (D2' h2)) %. +%term cal_impl + %pi (ca A (impl D1 D2 H) E (impl D1 D2' H)) + %<- ({h2 hyp B2} ca A (D2 h2) E (D2' h2)) %. +%term cal_orl + %pi (ca A (orl D1 D2 H) E (orl D1' D2' H)) + %<- ({h1 hyp B1} ca A (D1 h1) E (D1' h1)) + %<- ({h2 hyp B2} ca A (D2 h2) E (D2' h2)) %. +%term cal_notl ca A (notl D1 H) E (notl D1 H) %. +%term cal_falsel ca A (falsel H) E (falsel H) %. +%term cal_foralll %pi (ca A (foralll T D1 H) E (foralll T D1' H)) %<- ({h} ca A (D1 h) E (D1' h)) %. +%term cal_existsl + %pi (ca A (existsl D1 H) E (existsl D1' H)) + %<- ({a i} {h hyp (B1 a)} ca A (D1 a h) E (D1' a h)) %. +%% Right Commutative Conversions +% was commented out +%term car_axiom ca A D ([h hyp A] axiom H1) (axiom H1) %. +%term car_andr + %pi (ca A D ([h hyp A] andr (E1 h) (E2 h)) (andr E1' E2')) + %<- (ca A D E1 E1') + %<- (ca A D E2 E2') %. +%term car_andl1 + %pi (ca A D ([h hyp A] andl1 (E1 h) H) (andl1 E1' H)) + %<- ({h1 hyp B1} ca A D ([h hyp A] E1 h h1) (E1' h1)) %. +%term car_andl2 + %pi (ca A D ([h hyp A] andl2 (E2 h) H) (andl2 E2' H)) + %<- ({h2 hyp B2} ca A D ([h hyp A] E2 h h2) (E2' h2)) %. +%term car_impr + %pi (ca A D ([h hyp A] impr (E2 h)) (impr E2')) + %<- ({h1 hyp B1} ca A D ([h hyp A] E2 h h1) (E2' h1)) %. +%term car_impl + %pi (ca A D ([h hyp A] impl (E1 h) (E2 h) H) (impl E1' E2' H)) + %<- (ca A D E1 E1') + %<- ({h2 hyp B2} ca A D ([h hyp A] E2 h h2) (E2' h2)) %. +%term car_orr1 %pi (ca A D ([h hyp A] orr1 (E1 h)) (orr1 E1')) %<- (ca A D E1 E1') %. +%term car_orr2 %pi (ca A D ([h hyp A] orr2 (E2 h)) (orr2 E2')) %<- (ca A D E2 E2') %. +%term car_orl + %pi (ca A D ([h hyp A] orl (E1 h) (E2 h) H) (orl E1' E2' H)) + %<- ({h1 hyp B1} ca A D ([h hyp A] E1 h h1) (E1' h1)) + %<- ({h2 hyp B2} ca A D ([h hyp A] E2 h h2) (E2' h2)) %. +%term car_notr + %pi (ca A D ([h hyp A] notr (E1 h)) (notr E1')) + %<- ({p o} {h1 hyp B1} ca A D ([h hyp A] E1 h p h1) (E1' p h1)) %. +%term car_notl %pi (ca A D ([h hyp A] notl (E1 h) H) (notl E1' H)) %<- (ca A D E1 E1') %. +%term car_truer ca A D ([h hyp A] truer) truer %. +%term car_falsel ca A D ([h hyp A] falsel H) (falsel H) %. +%term car_forallr + %pi (ca A D ([h hyp A] forallr (E1 h)) (forallr E1')) + %<- ({a i} ca A D ([h hyp A] E1 h a) (E1' a)) %. +%term car_foralll + %pi (ca A D ([h hyp A] foralll T (E1 h) H) (foralll T E1' H)) + %<- ({h1} ca A D ([h hyp A] E1 h h1) (E1' h1)) %. +%term car_existsr %pi (ca A D ([h hyp A] existsr T (E1 h)) (existsr T E1')) %<- (ca A D E1 E1') %. +%term car_existsl + %pi (ca A D ([h hyp A] existsl (E1 h) H) (existsl E1' H)) + %<- ({a i} {h1 hyp (B1 a)} ca A D ([h hyp A] E1 h a h1) (E1' a h1)) %. +%block l1 {A o} [h hyp A]%. +%block l2 [a i]%. +%block l3 [p o]%. +%worlds (l1 l2 l3) (ca A D E F) %. +%covers ca %in %in %in %out %. +%terminates {A [D E]} (ca A D E _) %. +%total {A [D E]} (ca A D E _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cut-elim/int-cut.lf b/new-tests/stelf-output/examples/cut-elim/int-cut.lf new file mode 100644 index 0000000..1d8232e --- /dev/null +++ b/new-tests/stelf-output/examples/cut-elim/int-cut.lf @@ -0,0 +1,27 @@ +%%% Intuitionistic Sequent Calculus with Cut +%%% Author: Frank Pfenning +%sort conc* %. +% Conclusion (right) +%name conc* %. +%term cut* {A o} %pi (conc* A) %-> (%pi (hyp A) %-> (conc* C)) %-> (conc* C) %. +%term axiom* %pi (hyp A) %-> (conc* A) %. +%term andr* %pi (conc* A) %-> (conc* B) %-> (conc* (A and B)) %. +%term andl1* %pi (%pi (hyp A) %-> (conc* C)) %-> (hyp (A and B)) %-> (conc* C) %. +%term andl2* %pi (%pi (hyp B) %-> (conc* C)) %-> (hyp (A and B)) %-> (conc* C) %. +%term impr* %pi (%pi (hyp A) %-> (conc* B)) %-> (conc* (A imp B)) %. +%term impl* %pi (conc* A) %-> (%pi (hyp B) %-> (conc* C)) %-> (hyp (A imp B)) %-> (conc* C) %. +%term orr1* %pi (conc* A) %-> (conc* (A or B)) %. +%term orr2* %pi (conc* B) %-> (conc* (A or B)) %. +%term orl* + %pi (%pi (hyp A) %-> (conc* C)) + %-> (%pi (hyp B) %-> (conc* C)) + %-> (hyp (A or B)) + %-> (conc* C) %. +%term notr* %pi ({p o} %pi (hyp A) %-> (conc* p)) %-> (conc* (not A)) %. +%term notl* %pi (conc* A) %-> (hyp (not A)) %-> (conc* C) %. +%term truer* conc* true %. +%term falsel* %pi (hyp false) %-> (conc* C) %. +%term forallr* %pi ({a i} conc* (A a)) %-> (conc* (forall A)) %. +%term foralll* {T i} %pi (%pi (hyp (A T)) %-> (conc* C)) %-> (hyp (forall A)) %-> (conc* C) %. +%term existsr* {T i} %pi (conc* (A T)) %-> (conc* (exists A)) %. +%term existsl* %pi ({a i} %pi (hyp (A a)) %-> (conc* C)) %-> (hyp (exists A)) %-> (conc* C) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cut-elim/int-elim.lf b/new-tests/stelf-output/examples/cut-elim/int-elim.lf new file mode 100644 index 0000000..a685640 --- /dev/null +++ b/new-tests/stelf-output/examples/cut-elim/int-elim.lf @@ -0,0 +1,38 @@ +%%% Cut Elimination in Intuitionistic Sequent Calculus +%%% Author: Frank Pfenning +%sort ce %. +%mode ce %in %out %. +%term ce_cut + %pi (ce (cut* A D1* D2*) D) + %<- (ce D1* D1) + %<- ({h1 hyp A} ce (D2* h1) (D2 h1)) + %<- (ca A D1 D2 D) %. +%term ce_axiom ce (axiom* H) (axiom H) %. +%term ce_andr %pi (ce (andr* D1* D2*) (andr D1 D2)) %<- (ce D1* D1) %<- (ce D2* D2) %. +%term ce_andl1 %pi (ce (andl1* D1* H) (andl1 D1 H)) %<- ({h1 hyp A} ce (D1* h1) (D1 h1)) %. +%term ce_andl2 %pi (ce (andl2* D2* H) (andl2 D2 H)) %<- ({h2 hyp B} ce (D2* h2) (D2 h2)) %. +%term ce_impr %pi (ce (impr* D1*) (impr D1)) %<- ({h1 hyp A} ce (D1* h1) (D1 h1)) %. +%term ce_impl + %pi (ce (impl* D1* D2* H) (impl D1 D2 H)) + %<- (ce D1* D1) + %<- ({h2 hyp B} ce (D2* h2) (D2 h2)) %. +%term ce_orr1 %pi (ce (orr1* D1*) (orr1 D1)) %<- (ce D1* D1) %. +%term ce_orr2 %pi (ce (orr2* D2*) (orr2 D2)) %<- (ce D2* D2) %. +%term ce_orl + %pi (ce (orl* D1* D2* H) (orl D1 D2 H)) + %<- ({h1 hyp A} ce (D1* h1) (D1 h1)) + %<- ({h2 hyp B} ce (D2* h2) (D2 h2)) %. +%term ce_notr %pi (ce (notr* D1*) (notr D1)) %<- ({p o} {h1 hyp A} ce (D1* p h1) (D1 p h1)) %. +%term ce_notl %pi (ce (notl* D1* H) (notl D1 H)) %<- (ce D1* D1) %. +%term ce_truer ce truer* truer %. +%term ce_falsel ce (falsel* H) (falsel H) %. +%term ce_forallr %pi (ce (forallr* D1*) (forallr D1)) %<- ({a i} ce (D1* a) (D1 a)) %. +%term ce_foralll %pi (ce (foralll* T D1* H) (foralll T D1 H)) %<- ({h1} ce (D1* h1) (D1 h1)) %. +%term ce_existsr %pi (ce (existsr* T D1*) (existsr T D1)) %<- (ce D1* D1) %. +%term ce_existsl + %pi (ce (existsl* D1* H) (existsl D1 H)) + %<- ({a i} {h1 hyp (A1 a)} ce (D1* a h1) (D1 a h1)) %. +%worlds (l1 l2 l3) (ce D* D) %. +%covers ce %in %star %. +%terminates D (ce D _) %. +%total D (ce D _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cut-elim/int-thm.lf b/new-tests/stelf-output/examples/cut-elim/int-thm.lf new file mode 100644 index 0000000..216254e --- /dev/null +++ b/new-tests/stelf-output/examples/cut-elim/int-thm.lf @@ -0,0 +1,3 @@ +%%% Termination of cut-elimination +%terminates {A [D E]} (ca A D E _) %. +%terminates D (ce D _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cut-elim/int.lf b/new-tests/stelf-output/examples/cut-elim/int.lf new file mode 100644 index 0000000..08317ec --- /dev/null +++ b/new-tests/stelf-output/examples/cut-elim/int.lf @@ -0,0 +1,29 @@ +%%% Cut-Free Intuitionistic Sequent Calculus +%%% Author: Frank Pfenning +%sort hyp %. +% Hypotheses (left) +%sort conc %. +% Conclusion (right) +%name hyp %. +%name conc %. +%term axiom %pi (hyp A) %-> (conc A) %. +%term andr %pi (conc A) %-> (conc B) %-> (conc (A and B)) %. +%term andl1 %pi (%pi (hyp A) %-> (conc C)) %-> (hyp (A and B)) %-> (conc C) %. +%term andl2 %pi (%pi (hyp B) %-> (conc C)) %-> (hyp (A and B)) %-> (conc C) %. +%term impr %pi (%pi (hyp A) %-> (conc B)) %-> (conc (A imp B)) %. +%term impl %pi (conc A) %-> (%pi (hyp B) %-> (conc C)) %-> (hyp (A imp B)) %-> (conc C) %. +%term orr1 %pi (conc A) %-> (conc (A or B)) %. +%term orr2 %pi (conc B) %-> (conc (A or B)) %. +%term orl + %pi (%pi (hyp A) %-> (conc C)) + %-> (%pi (hyp B) %-> (conc C)) + %-> (hyp (A or B)) + %-> (conc C) %. +%term notr %pi ({p o} %pi (hyp A) %-> (conc p)) %-> (conc (not A)) %. +%term notl %pi (conc A) %-> (hyp (not A)) %-> (conc C) %. +%term truer conc true %. +%term falsel %pi (hyp false) %-> (conc C) %. +%term forallr %pi ({a i} conc (A a)) %-> (conc (forall A)) %. +%term foralll {T i} %pi (%pi (hyp (A T)) %-> (conc C)) %-> (hyp (forall A)) %-> (conc C) %. +%term existsr {T i} %pi (conc (A T)) %-> (conc (exists A)) %. +%term existsl %pi ({a i} %pi (hyp (A a)) %-> (conc C)) %-> (hyp (exists A)) %-> (conc C) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/cut-elim/sources.lf b/new-tests/stelf-output/examples/cut-elim/sources.lf new file mode 100644 index 0000000..827a660 --- /dev/null +++ b/new-tests/stelf-output/examples/cut-elim/sources.lf @@ -0,0 +1,10 @@ +%require %[ formulas %] %. +%require %[ int %] %. +%require %[ cl %] %. +%require %[ int-admit %] %. +%require %[ cl-admit %] %. +%require %[ int-cut %] %. +%require %[ cl-cut %] %. +%require %[ int-elim %] %. +%require %[ cl-elim %] %. +%require %[ examples %] %. diff --git a/new-tests/stelf-output/examples/cut-elim/stelf.toml b/new-tests/stelf-output/examples/cut-elim/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/cut-elim/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/cut-elim/test.lf b/new-tests/stelf-output/examples/cut-elim/test.lf new file mode 100644 index 0000000..f9995be --- /dev/null +++ b/new-tests/stelf-output/examples/cut-elim/test.lf @@ -0,0 +1,12 @@ +%require %[ formulas %] %. +%require %[ int %] %. +%require %[ cl %] %. +%require %[ int-admit %] %. +%require %[ cl-admit %] %. +%require %[ int-cut %] %. +%require %[ cl-cut %] %. +%require %[ int-elim %] %. +%require %[ cl-elim %] %. +%require %[ examples %] %. +%require %[ cl %] %. +%require %[ int %] %. diff --git a/new-tests/stelf-output/examples/fj/aux-lemma-prog.lf b/new-tests/stelf-output/examples/fj/aux-lemma-prog.lf new file mode 100644 index 0000000..63794e7 --- /dev/null +++ b/new-tests/stelf-output/examples/fj/aux-lemma-prog.lf @@ -0,0 +1,363 @@ +%{ This is part of a proof of the soundness of Featherweight Java (Atsushi Igarashi, +Benjamin Pierce and Philip Wadler) in the Twelf logical Framework. +It was developed by Stephanie Weirich and +Geoffrey Washburn . }% +%%% Auxilary lemmas for the proof of progress for FJ. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Definition of what it means for a some class not to be the subtype of another. +% + +% This is somewhat tricky because we allow reasoning about classNames that may +% not be in the classTable. object is one such example of a className. +%sort notSubtype %. +%name notSubtype %. +%term ns_undef %pi (notInCT CT C) %-> (className_neq C D) %-> (notSubtype CT C D) %. +%term ns_super + %pi (notSubtype CT E D) + %-> (lookupClass CT C (class C E _ _)) + %-> (className_neq C D) + %-> (notSubtype CT C D) %. +%worlds () (notSubtype CT C1 C2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Disequality on classNames is symmetric +%sort cnneq_sym %. +%mode cnneq_sym %in %out %. +%term _ + %pi (nat_neq_sym NNEQ1 NNEQ2) + %-> (cnneq_sym (className_neq_natnat NNEQ1) (className_neq_natnat NNEQ2)) %. +%worlds () (cnneq_sym CNNEQ1 CNNEQ2) %. +%total CNNEQ1 (cnneq_sym CNNEQ1 CNNEQ2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Well-formed classTables do not contain object. +% Use a helper function because of the way class_table_typing is defined. +%sort objectNotInCT_helper %. +%mode objectNotInCT_helper %in %out %. +%term _ objectNotInCT_helper ctt_nil nnil %. +%term _ + %pi (objectNotInCT_helper CTTH UNDEF) + %-> (cnneq_sym CNEQ1 CNEQ2) + %-> (objectNotInCT_helper (ctt_cons CTTH CTP CNEQ1) (ncons UNDEF CNEQ2)) %. +%worlds () (objectNotInCT_helper CTTH UNDEF) %. +%total CTTH (objectNotInCT_helper CTTH UNDEF) %. +%sort objectNotInCT %. +%mode objectNotInCT %in %out %. +%term _ %pi (objectNotInCT_helper CTTH UNDEF) %-> (objectNotInCT (ctt CTTH) UNDEF) %. +%worlds () (objectNotInCT CTT UNDEF) %. +%total CTT (objectNotInCT CTT UNDEF) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Class name excluded middle -- classNames are either equal or they aren't. +%% first the definition of the excluded middle sum +%sort cneq_em %. +%term cneq_eq %pi (className_eq M N) %-> (cneq_em M N) %. +%term cneq_neq %pi (className_neq M N) %-> (cneq_em M N) %. +% We need a helper because Twelf can't handle output factoring +%sort cneq_helper %. +%mode cneq_helper %in %out %. +%term _ cneq_helper (nat_em_eq EQ) (cneq_eq className_refl) %. +%term _ cneq_helper (nat_em_neq NEQ) (cneq_neq (className_neq_natnat NEQ)) %. +%worlds () (cneq_helper M N) %. +%total M (cneq_helper M N) %. +%sort cneq_excluded_middle {C} {D} %. +%mode cneq_excluded_middle %in %in %out %. +%term _ + %pi (cneq_helper CDEQ EQ) + %-> (nat_excluded C D CDEQ) + %-> (cneq_excluded_middle (cn C) (cn D) EQ) %. +%worlds () (cneq_excluded_middle C D CD) %. +%total {C D} (cneq_excluded_middle C D CD) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% lc_total: Class lookup is decidable +%% We can either lookup a className in the classTable or +%% get a derivation that the className is not in the class table. +%% Again, split into several helper functions because of factoring problems. +%% Based significantly on lm_total in aux-lemma.elf +% The disjoint sum for the result. +%sort lookup_or_notInCT %. +%term loc1 %pi (lookupClass CT C CD) %-> (lookup_or_notInCT CT C) %. +%term loc2 %pi (notInCT CT C) %-> (lookup_or_notInCT CT C) %. +%sort lc_helper2 %. +%mode {%in C className} {%in D className} {%in CT classTable} {%in NEQ className_neq C D} {%in CTU notInCT CT D} {%in LOC1 lookup_or_notInCT CT C} {%in E className} {%in N nat} {%in CNL cnlist N} {%in MR methodTable} {%out LOC2 lookup_or_notInCT (crcons (class D E CNL MR) CT CTU) C} lc_helper2 NEQ LOC1 LOC2 %. +%sort lc_helper1 {CT classTable} {CTU notInCT CT D} %. +%mode {%in CT classTable} {%in C className} {%in D className} {%in CTU notInCT CT D} {%in EQ cneq_em C D} {%in E className} {%in N nat} {%in CNL cnlist N} {%in MR methodTable} {%out LOC lookup_or_notInCT (crcons (class D E CNL MR) CT CTU) C} lc_helper1 CT CTU EQ LOC %. +%sort lc_total {CT classTable} {C className} %. +%mode lc_total %in %in %out %. +%term _ lc_helper2 NEQ (loc1 LC) (loc1 (lookupClass_crcons2 NEQ LC)) %. +%term _ lc_helper2 NEQ (loc2 CU) (loc2 (ncons CU NEQ)) %. +%worlds () (lc_helper2 N L O) %. +%total L1 (lc_helper2 N2 L1 O1) %. +%term _ lc_helper1 CT CTU (cneq_eq EQ) (loc1 lookupClass_crcons1) %. +%term _ + %pi (lc_helper2 CEQ LOC1 LOC2) + %-> (lc_total CT C LOC1) + %-> (lc_helper1 CT CTU (cneq_neq CEQ) LOC2) %. +%term _ lc_total crnil C (loc2 nnil) %. +%term _ + %pi (lc_helper1 CT CTU EX LOM) + %-> (cneq_excluded_middle C D EX) + %-> (lc_total (crcons (class D E CNL MR) CT CTU) C LOM) %. +%worlds () (lc_total CT C LOC) (lc_helper1 CT CTU XM LOC) %. +%total (CT1 CT2) (lc_total CT1 C LOC1) (lc_helper1 CT2 CTU XM LOC2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% If two classNames are equal than they are subtypes of each other. +%sort cneq_subtyping {CT classTable} %. +%mode cneq_subtyping %in %in %out %. +%term _ cneq_subtyping CT CEQ s_refl %. +%worlds () (cneq_subtyping CT CNEQ SB) %. +%total CNEQ (cneq_subtyping CT CNEQ SB) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Class typing is decidable: Given a well-typd class table, +% a class is either in the class table and well-typed, or it isn't in +% the class table. +% Disjoint sum for ctyping_total +%sort notInCT_or_ctyping %. +%term nctct_nct %pi (notInCT CT C) %-> (notInCT_or_ctyping CT C) %. +%term nctct_ct + %pi (className_neq C object) + %-> (lookupClass CT C CDEF) + %-> (class_typing CT C CDEF) + %-> (notInCT_or_ctyping CT C) %. +% Helper needed for ouput factoring +%sort ctyping_total_helper %. +%mode ctyping_total_helper %in %in %out %. +%term _ + %pi (ctt_ct CTT LC CNNEQ CTP) + %-> (ctyping_total_helper CTT (loc1 LC) (nctct_ct CNNEQ LC CTP)) %. +%term _ ctyping_total_helper CTT (loc2 UNDEF) (nctct_nct UNDEF) %. +%worlds () (ctyping_total_helper CTT LOC WO) %. +%total LOC (ctyping_total_helper CTT LOC WO) %. +%sort ctyping_total {CT classTable} %. +%mode ctyping_total %in %in %in %out %. +%term _ + %pi (ctyping_total_helper CTT LOC WO) + %-> (lc_total CT C LOC) + %-> (ctyping_total CT CTT C WO) %. +%worlds () (ctyping_total CT CTT C WO) %. +%total CTT (ctyping_total CT CTT C WO) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Decidability of subtyping +% Given a well-typed class table and two classNames, +% one is a subtype of the other or it isn't +% disjoint sum for sb_total +%sort xm_subtype %. +%name xm_subtype %. +%term xms_sub %pi (subtyping CT C1 C2) %-> (xm_subtype CT C1 C2) %. +%term xms_notsub %pi (notSubtype CT C1 C2) %-> (xm_subtype CT C1 C2) %. +% helper3 exists just to handle output factoring for helper2 +%sort sb_total_helper3 %. +%mode sb_total_helper3 %in %in %in %in %out %. +%term _ sb_total_helper3 LC CNNEQ1 CNNEQ2 (xms_sub SB) (xms_sub (s_super SB LC CNNEQ1)) %. +%term _ sb_total_helper3 LC CNNEQ1 CNNEQ2 (xms_notsub NSB) (xms_notsub (ns_super NSB LC CNNEQ2)) %. +%worlds () (sb_total_helper3 LC CNNEQ1 CNNEQ2 XMSB1 XMSB2) %. +%total XMSB1 (sb_total_helper3 LC CNNEQ1 CNNEQ2 XMSB1 XMSB2) %. +% helper2 works by induction over the derivation that C is a subtype of +% object, and attempts to find a (not)subtyping derivation. +%sort sb_total_helper2 {CT classTable} %. +%mode sb_total_helper2 %in %in %in %in %out %. +%term _ + %pi (cneq_subtyping CT CNEQ1 SB) + %-> (sb_total_helper2 CT CTT (cneq_eq CNEQ1) _ (xms_sub SB)) %. +%term _ + %pi (objectNotInCT CTT UNDEF) + %-> (sb_total_helper2 CT CTT (cneq_neq CNEQ1) s_refl (xms_notsub (ns_undef UNDEF CNEQ1))) %. +%term _ + %pi (sb_total_helper3 LC CNNEQ2 CNNEQ1 XMSB1 XMSB2) + %-> (sb_total_helper2 CT CTT CNXM SB XMSB1) + %-> (cneq_excluded_middle E _ CNXM) + %-> (sb_total_helper2 CT CTT (cneq_neq CNNEQ1) (s_super (%the (subtyping CT E object) SB) LC CNNEQ2) XMSB2) %. +%worlds () (sb_total_helper2 CT CTT CNXM SB XMSB) %. +%total SB (sb_total_helper2 CT CTT CNXM SB XMSB) %. +% helper1 does cases upon whether the classNames are equal +% and if not, whether they are well-formed classes in the classTable. +% If they are equal they are trivially subtypes. +% If C is not in the class table and not equal to D, they are +% trivially not subtypes. +% Otherwise we call out to helper2 to do induction over the derivation +% that C is a subtype of object (which forces the classTable to be acyclic). +%sort sb_total_helper1 {CT classTable} %. +%mode sb_total_helper1 %in %in %in %in %in %out %. +%term _ + %pi (cneq_subtyping CT CNEQ SB) + %-> (sb_total_helper1 CT CTT (cneq_eq CNEQ) WO1 WO2 (xms_sub SB)) %. +%term _ sb_total_helper1 CT CTT (cneq_neq CNNEQ) (nctct_nct UNDEF) WO2 (xms_notsub (ns_undef UNDEF CNNEQ)) %. +%term _ + %pi (sb_total_helper2 CT CTT CNXM (s_super SB LC CNNEQ2) XMSB) + %-> (sb_total_helper1 CT CTT CNXM (nctct_ct CNNEQ2 LC (t_class _ _ SB)) WO2 XMSB) %. +%worlds () (sb_total_helper1 CT CTT CNXM WO1 WO2 XMSB) %. +%total {WO1 WO2} (sb_total_helper1 CT CTT CNXM WO1 WO2 XMSB) %. +% The primary lemma. Works by looking up whether the classes are +% in the class table or not and getting their typings if so. +% Also check whether the two classNames are equal or not. +% Pass the results off to helper1 because of factoring +%sort sb_total {CT classTable} {C className} {D className} %. +%mode sb_total %in %in %in %in %out %. +%term _ + %pi (sb_total_helper1 CT CTT CNXM WO1 WO2 XMSB) + %-> (cneq_excluded_middle C D CNXM) + %-> (ctyping_total CT CTT D WO2) + %-> (ctyping_total CT CTT C WO1) + %-> (sb_total CT C D CTT XMSB) %. +%worlds () (sb_total CT C D CTT XMSB) %. +%total {C D} (sb_total CT C D CTT XMSB) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% If we can lookup the type of a method in the classTable, +% we can lookup the body of the method in the classTable. +%sort mtype_to_mbody {CS cnlist N} %. +%mode {%in CT classTable} {%in M methodName} {%in C1 className} {%in C0 className} {%in N nat} {%in CS cnlist N} {%in MTY mtype CT M C1 CS C0} {%out BE bexp CS} {%out MBY mbody CT M C1 BE} mtype_to_mbody CS MTY BE MBY %. +%term _ mtype_to_mbody CS (mt_class LM LC CNNEQ) _ (mb_class LM LC CNNEQ) %. +%term _ + %pi (mtype_to_mbody CS MTY BE MBY) + %-> (mtype_to_mbody CS (mt_super MTY UNDEF LC CNNEQ) _ (mb_super UNDEF MBY LC CNNEQ)) %. +%worlds () (mtype_to_mbody CS MTY BE MBY) %. +%total MTY (mtype_to_mbody CS MTY BE MBY) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% If we have a well-typed object, and a method typing for the apparent className +% of that object, we can obtain the method typing for the real className. +%sort mtype_inv %. +%mode mtype_inv %in %in %out %. +%term _ mtype_inv (t_new TPS FS) MTY MTY %. +%worlds () (mtype_inv TP MTY1 MTY2) %. +%total TP (mtype_inv TP MTY1 MTY2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Combines the two previous lemmas, given a well-typed object, +% get its method body. +%sort mbody_inv %. +%mode {%in CT classTable} {%in C1 className} {%in C2 className} {%in M methodName} {%in C0 className} {%in N1 nat} {%in N2 nat} {%in ES elist N1} {%in CS cnlist N2} {%in TP typing CT (new C1 ES) C2} {%in MTY mtype CT M C2 CS C0} {%out BE bexp CS} {%out MBY mbody CT M C1 BE} mbody_inv TP MTY BE MBY %. +%term _ + %pi (mtype_to_mbody _ MTY2 BE MBY) + %-> (mtype_inv TP MTY1 MTY2) + %-> (mbody_inv TP MTY1 BE MBY) %. +%worlds () (mbody_inv TP MTY BE MBY) %. +%total {TP MTY} (mbody_inv TP MTY BE MBY) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% We can always construct an invoc derivation, given the correct inputs. +%sort invoc_total {ES elist N} %. +%mode invoc_total %in %in %in %in %out %. +%term _ invoc_total enil tl_nil (base ([this exp] E1 this)) E invoc_base %. +%term _ + %pi (invoc_total EL TPS (BE E1) E INV) + %-> (invoc_total (econs E1 EL) (tl_cons _ TPS _) (bind _ ([x exp] BE x)) E (invoc_bind INV)) %. +%worlds () (invoc_total ES TPS BE E1 INV) %. +%total BE (invoc_total ES TPS BE E1 INV) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Given an expression list calculate the nth projection, assuming n is less than +% the length of the list. +%sort enth_opt_total {ES elist N1} %. +%mode enth_opt_total %in %in %out %. +%term _ enth_opt_total (econs E _) nat_lt_z enth_sz %. +%term _ + %pi (enth_opt_total ES NLT ENTH) + %-> (enth_opt_total (econs _ ES) (nat_lt_s NLT) (enth_sn ENTH)) %. +%worlds () (enth_opt_total ES NLT ENTH) %. +%total ES (enth_opt_total ES NLT ENTH) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Is we projected out the FIth className, then FI must be less than the length +% of the className list. +%sort cnnth_lemma {CS cnlist N1} %. +%mode cnnth_lemma %in %in %out %. +%term _ cnnth_lemma (cncons C _) cnnth_sz nat_lt_z %. +%term _ + %pi (cnnth_lemma CS CNNTH NLT) + %-> (cnnth_lemma (cncons _ CS) (cnnth_sn CNNTH) (nat_lt_s NLT)) %. +%worlds () (cnnth_lemma CS CNNTH NLT) %. +%total CNNTH (cnnth_lemma CS CNNTH NLT) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Looking up a className in the classTable and getting different results +% is contradictory. +%sort lookupClass_exclusive %. +%mode lookupClass_exclusive %in %in %in %out %. +%term _ + %pi (className_eq_exclusive CNEQ CNNEQ FALSE) + %-> (classDef_eq_extract CDEFEQ CNEQ _ _) + %-> (lookupClass_unique LC1 LC2 CDEFEQ) + %-> (lookupClass_exclusive LC1 LC2 CNNEQ FALSE) %. +%worlds () (lookupClass_exclusive LC1 LC2 CNNEQ P) %. +%total CNNEQ (lookupClass_exclusive LC1 LC2 CNNEQ P) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% If two className lists are equal then their lengths must be equal. +%sort cnlist_eq_length {CS1 cnlist N1} {CS2 cnlist N2} %. +%mode cnlist_eq_length %in %in %in %out %. +%term _ cnlist_eq_length CS1 CS2 CLEQ nat_refl %. +%worlds () (cnlist_eq_length CS1 CS2 CLEQ NEQ) %. +%total CLEQ (cnlist_eq_length CS1 CS2 CLEQ NEQ) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% fields_length_unique. If we look up the fields of a class, the number +% of fields will always be the same. +% abusrdity helper -- if we look up a class twice and get different results +% we a licensed to conclude that the number fields they have are the same. +%sort lookup_absurd1 {CS1 cnlist N1} {CS2 cnlist N2} %. +%mode lookup_absurd1 %in %in %in %in %in %out %. +%term _ + %pi (false_imp_nat_eq _ _ FALSE NEQ) + %-> (lookupClass_exclusive LC1 LC2 CNNEQ FALSE) + %-> (lookup_absurd1 CS1 CS2 LC1 LC2 CNNEQ NEQ) %. +%worlds () (lookup_absurd1 CS1 CS2 LC1 LC2 CNNEQ NEQ) %. +%total {LC1 LC2 CNNEQ} (lookup_absurd1 CS1 CS2 LC1 LC2 CNNEQ NEQ) %. +% abusrdity helper -- if we look up a class twice and get different results +% we a licensed to conclude that the number fields their respective parents +% have are the same. +%sort lookup_absurd2 %. +%mode lookup_absurd2 %in %in %in %in %in %out %. +%term _ + %pi (false_imp_nat_eq _ _ FALSE NEQ) + %-> (lookupClass_exclusive LC1 LC2 CNNEQ FALSE) + %-> (lookup_absurd2 LC1 LC2 FS1 FS2 CNNEQ NEQ) %. +%worlds () (lookup_absurd2 LC1 LC2 FS1 FS2 CNNEQ NEQ) %. +%total {LC1 LC2 CNNEQ} (lookup_absurd2 LC1 LC2 FS1 FS2 CNNEQ NEQ) %. +% This helper handles dispatching upon the equality of the parent classNames. +% Necessary because of output factoring. Mutually defined with fields_length_unique +%sort fields_length_unique_helper {CS1 cnlist N1} {CS2 cnlist N2} %. +%mode fields_length_unique_helper %in %in %in %in %in %in %in %out %out %. +%sort fields_length_unique %. +%mode fields_length_unique %in %in %out %. +% Cases for fields_length_unique_helper +%term _ + %pi (fields_length_unique FS1 FS2 NEQ2) + %-> (cnlist_eq_length CS1 CS2 CLEQ NEQ1) + %-> (classDef_eq_extract CDEFEQ _ CLEQ _) + %-> (lookupClass_unique LC1 LC2 CDEFEQ) + %-> (fields_length_unique_helper CS1 CS2 LC1 LC2 (cneq_eq CNEQ) FS1 FS2 NEQ1 NEQ2) %. +%term _ + %pi (lookup_absurd2 LC1 LC2 FS1 FS2 CNNEQ NEQ2) + %-> (lookup_absurd1 CS1 CS2 LC1 LC2 CNNEQ NEQ1) + %-> (lookupClass_exclusive LC1 LC2 CNNEQ FALSE) + %-> (fields_length_unique_helper CS1 CS2 LC1 LC2 (cneq_neq CNNEQ) FS1 FS2 NEQ1 NEQ2) %. +% Cases for fields_length_unique +%term _ fields_length_unique f_obj f_obj nat_refl %. +%term _ + %pi (plus_eq_lemma NEQ2 NEQ1 PL1 PL2 NEQ3) + %-> (fields_length_unique_helper _ _ LC1 LC2 CNXM FS1 FS2 NEQ1 NEQ2) + %-> (cneq_excluded_middle _ _ CNXM) + %-> (fields_length_unique (f_class (%the (cnappend _ _ _ PL1) CNAP1) FS1 LC1 CNNEQ1) (f_class (%the (cnappend _ _ _ PL2) CNAP2) FS2 LC2 CNNEQ2) NEQ3) %. +%worlds () (fields_length_unique FS1 FS2 NEQ) (fields_length_unique_helper CS1 CS2 LC1 LC2 CNXM FS1 FS2 NEQ1 NEQ2) %. +%total (FS1 FS3) (fields_length_unique FS1 FS2 NEQ1) (fields_length_unique_helper CS1 CS2 LC1 LC2 CNXM FS3 FS4 NEQ2N EQ3) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% If we have a well-typed object, the number of fields will equal the number +% of arguments to the object. +%sort new_fields_lemma {ES elist N1} %. +%mode new_fields_lemma %in %in %in %out %. +%term _ + %pi (fields_length_unique FS1 FS2 NEQ) + %-> (new_fields_lemma _ (t_new _ FS1) FS2 NEQ) %. +%worlds () (new_fields_lemma ES TP FS NEQ) %. +%total TP (new_fields_lemma ES TP FS NEQ) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% If we have a well-typed object, and we projected on of its fields, +% we can also project out one of the object's arguments. +%sort proj_lemma_helper {ES elist N1} %. +%mode proj_lemma_helper %in %in %in %in %in %out %. +%term _ + %pi (nat_lt_eq NLT1 NEQ NLT2) + %-> (cnnth_lemma _ CNNTH NLT1) + %-> (proj_lemma_helper ES TP FS NEQ CNNTH NLT2) %. +%worlds () (proj_lemma_helper ES TP FS NEQ CNNTH NLT) %. +%total CNNTH (proj_lemma_helper ES TP FS NEQ CNNTH NLT) %. +%sort proj_lemma %. +%mode proj_lemma %in %in %in %out %. +%term _ + %pi (enth_opt_total _ NLT ENTH) + %-> (proj_lemma_helper _ TP FS NEQ CNNTH NLT) + %-> (new_fields_lemma _ TP FS NEQ) + %-> (proj_lemma TP FS CNNTH ENTH) %. +%worlds () (proj_lemma TP FS CNNTH ENTH) %. +%total CNNTH (proj_lemma TP FS CNNTH ENTH) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \ No newline at end of file diff --git a/new-tests/stelf-output/examples/fj/aux-lemma.lf b/new-tests/stelf-output/examples/fj/aux-lemma.lf new file mode 100644 index 0000000..54bb377 --- /dev/null +++ b/new-tests/stelf-output/examples/fj/aux-lemma.lf @@ -0,0 +1,718 @@ +%{ This is part of a proof of the soundness of Featherweight Java (Atsushi Igarashi, +Benjamin Pierce and Philip Wadler) in the Twelf logical Framework. +It was developed by Stephanie Weirich and +Geoffrey Washburn . }% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% FJ Meta theory - Aux lemmas for the preservation proof +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%{ + +This block is because we cannot tell twelf that the classtable is +closed---therefore %% sometimes the coverage checker wants to reason +about fields or mtype in a context that contains expression +variables. These expression variables are part of the classtable, but +they don't influence judgments such as fields or mtype. + +An alternative would be to prove strengthening lemmas about fields and mtype. +(See the proof of narrowing and the definitions of sub_strengthen below.) +}% +%block var [x exp]%. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%{ +We also have to reason about when things are equal, even though equality is not +built into twelf. Therefore, we need a judgment about equality for each different +type. +}% +%% other equality relations --- not part of specification of FJ +%sort classDef_eq %. +%mode classDef_eq %in %in %. +%term classDef_refl classDef_eq CD CD %. +%worlds () (classDef_eq CD1 CD2) %. +%terminates {CD1 CD2} (classDef_eq CD1 CD2) %. +%sort methodDef_eq %. +%mode methodDef_eq %in %in %. +%term methodDef_refl methodDef_eq CD CD %. +%worlds () (methodDef_eq CD1 CD2) %. +%terminates {CD1 CD2} (methodDef_eq CD1 CD2) %. +%sort bexp_eq %. +%mode bexp_eq %in %in %. +%term bexp_refl bexp_eq B B %. +%worlds () (bexp_eq CD1 CD2) %. +%terminates {CD1 CD2} (bexp_eq CD1 CD2) %. +%sort methodTable_eq %. +%mode methodTable_eq %in %in %. +%term methodTable_refl methodTable_eq B B %. +%worlds () (methodTable_eq CD1 CD2) %. +%terminates {CD1 CD2} (methodTable_eq CD1 CD2) %. +%sort cnlist_eq %. +%mode cnlist_eq %in %in %. +%term cnlist_eq_refl cnlist_eq X X %. +%worlds () (cnlist_eq C1 C2) %. +%terminates {C1 C2} (cnlist_eq C1 C2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort classDef_eq_extract %. +%mode classDef_eq_extract %in %out %out %out %. +%term _ classDef_eq_extract classDef_refl className_refl cnlist_eq_refl methodTable_refl %. +%worlds (var) (classDef_eq_extract E F G H) %. +%total E (classDef_eq_extract E F G H) %. +%% These next few lemmas are really annoying because we would like to say +%% something like Leibniz equality: +%% if M = N then P(M) => P (N). +%% But we can't prove things in that generality. We must prove it for each +%% particular case. +%sort refineMTEQ %. +%mode refineMTEQ %in %in %out %. +%term _ refineMTEQ methodTable_refl LM LM %. +%worlds (var) (refineMTEQ M N O) %. +%total M (refineMTEQ M N O) %. +%sort refineCEQ %. +%mode refineCEQ %in %in %out %. +%term _ refineCEQ className_refl LM LM %. +%worlds (var) (refineCEQ M N O) %. +%total M (refineCEQ M N O) %. +%sort refineMDEQ %. +%mode {%in M methodName} {%in N1 nat} {%in DS cnlist N1} {%in D0 className} {%in BE0 bexp DS} {%in N2 nat} {%in CS cnlist N2} {%in C0 className} {%in BE1 bexp CS} {%in MR1 methodTable} {%in BE2 bexp DS} {%out BE3 bexp CS} {%in MD methodDef_eq (method D0 M BE0) (method C0 M BE1)} {%in M1 lookupMethod MR1 M (method D0 M BE2)} {%out M2 lookupMethod MR1 M (method C0 M BE3)} refineMDEQ MD M1 M2 %. +%term _ refineMDEQ methodDef_refl LM1 LM1 %. +%worlds (var) (refineMDEQ MD M N) %. +%total M (refineMDEQ MD M N) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% subtyping is transitive. Right now, we have an algorithmic definition of +%% subtyping, so we need to prove this property. +%sort sub_trans %. +%mode sub_trans %in %in %out %. +%term _ sub_trans s_refl SB SB %. +%term _ + %pi (sub_trans SB SB2 SB3) + %-> (sub_trans (s_super SB LC NEQ) SB2 (s_super SB3 LC NEQ)) %. +%worlds (var) (sub_trans SB1 SB2 SB3) %. +%total SB1 (sub_trans SB1 SB2 SB3) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort lookupClass_strengthen {x exp} %. +%mode lookupClass_strengthen %in %out %. +%term _ lookupClass_strengthen ([x] lookupClass_crcons1) lookupClass_crcons1 %. +%term _ + %pi (lookupClass_strengthen LC1 LC2) + %-> (lookupClass_strengthen ([x] lookupClass_crcons2 NEQ (LC1 x)) (lookupClass_crcons2 NEQ LC2)) %. +%worlds (var) (lookupClass_strengthen LC1 LC2) %. +%total LC1 (lookupClass_strengthen LC1 LC2) %. +%sort sub_strengthen %. +%mode sub_strengthen %in %out %. +%term _ sub_strengthen ([x] s_refl) s_refl %. +%term _ + %pi (sub_strengthen SB1 SB2) + %-> (lookupClass_strengthen LC1 LC2) + %-> (sub_strengthen ([x] s_super (SB1 x) (LC1 x) NEQ) (s_super SB2 LC2 NEQ)) %. +%worlds (var) (sub_strengthen SB1 SB2) %. +%total SB1 (sub_strengthen SB1 SB2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% several lemmas about cnappend. These really should be in a list library. +%% at least we didn't need them for expression lists too! +%% Ok, this is really annoying. I just used the metalogic to +%% show that append was total, but if I want to use it in the +%% logic, I have to show totality in the logic too! +%sort append_total {CS cnlist N1} {DS cnlist N2} {ES cnlist N3} {PL plus N1 N2 N3} %. +%mode append_total %in %in %out %out %out %. +%term _ append_total cnnil CS CS plus_z cnappend_cnnil %. +%term _ + %pi (append_total CS1 CS2 CS3 PL D) + %-> (append_total (cncons CN CS1) CS2 (cncons CN CS3) (plus_s PL) (cnappend_cncons D)) %. +%worlds () (append_total CS DS ES PL APP) %. +%total CS (append_total CS DS ES PL APP) %. +%%%%%% +%% Appendding nil is an identity +%% xs ++ [] = xs +%sort cnappend_cnnil_lemma {CS cnlist N} %. +%mode cnappend_cnnil_lemma %in %out %. +%term cnappend_cnnil_lemma_cnnil cnappend_cnnil_lemma cnnil cnappend_cnnil %. +%term cnappend_cnnil_lemma_cncons + %pi (cnappend_cnnil_lemma CS CAP) + %-> (cnappend_cnnil_lemma (cncons C CS) (cnappend_cncons CAP)) %. +%worlds () (cnappend_cnnil_lemma CS CAP) %. +%total CS (cnappend_cnnil_lemma CS CAP) %. +%%%%% +%% Append is an associative operation +%% (xs ++ ys) ++ zs = xs ++ (ys ++ zs) +%% us ts +%sort append_assoc %. +%mode append_assoc %in %in %in %out %. +%term _ append_assoc cnappend_cnnil UZ YZ cnappend_cnnil %. +%% xs = x :: xs1 +%% don't know why this case isn't subsumed by the first. But it makes the +%% coverage checker happy. +%term _ append_assoc cnappend_cnnil (cnappend_cncons R1) (cnappend_cncons R2) cnappend_cnnil %. +%term _ + %pi (append_assoc R1 R2 YZ R3) + %-> (append_assoc (cnappend_cncons R1) (cnappend_cncons R2) YZ (cnappend_cncons R3)) %. +%worlds () (append_assoc A1 A2 A3 A4) %. +%total A1 (append_assoc A1 A2 A3 A4) %. +%% weird, works with CVS version, but not with old version..... +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Lemma about accessing the fields of a subtype +%% The fields of a subtype are an extension. +%sort sub_fields %. +%mode sub_fields %in %in %out %out %. +%term _ %pi (cnappend_cnnil_lemma CS APP) %-> (sub_fields s_refl FS APP FS) %. +%term _ {SB subtyping CT D E} {QFE fields CT E NE FE} {APD cnappend FE FSD FD PLD} {QFD fields CT D ND FD} {LC1 lookupClass CT C (class C D FSC MTC)} {APC cnappend FD FSC FC PLC} {APCD cnappend FSD FSC FSCD PLCD} + %pi (append_assoc APD APC APCD XXX) + %-> (append_total FSD FSC FSCD PLCD APCD) + %-> (append_total FD FSC FC PLC APC) + %-> (sub_fields SB QFE APD QFD) + %-> (sub_fields (%the (subtyping CT C E) (s_super SB LC1 NEQ)) (%the (fields CT E NE FE) QFE) (%the (cnappend FE FSCD FC _) XXX) (f_class APC QFD LC1 NEQ)) %. +%worlds (var) (sub_fields SB F1 APP F2) %. +%total SB (sub_fields SB F1 APP F2) %. +%% If we lookup a class name in a list, then add on to the list, we +%% can still look up the class name. +%sort append_nth {C} %. +%mode append_nth %in %in %in %out %. +%term _ append_nth C cnnth_sz APP cnnth_sz %. +%term _ + %pi (append_nth C D APP E) + %-> (append_nth C (cnnth_sn D) (cnappend_cncons APP) (cnnth_sn E)) %. +%worlds () (append_nth C N1 APP N2) %. +%total N1 (append_nth C N1 APP N2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Lemma that states that if we have a list of well formed expressions, we can project +% out the nth expression and nth class name and there exists a derivation that the +% expression is well typed with a subtype. +%sort typing_list_proj_lemma %. +%mode typing_list_proj_lemma %in %in %in %out %out %. +%term typing_list_proj_lemma_sz typing_list_proj_lemma (tl_cons QS QL QT) enth_sz cnnth_sz QS QT %. +%term typing_list_proj_lemma_sn + %pi (typing_list_proj_lemma PL ENTH CNTH QS QT) + %-> (typing_list_proj_lemma (tl_cons PS PL PT) (enth_sn ENTH) (cnnth_sn CNTH) QS QT) %. +%worlds () (typing_list_proj_lemma PL ENTH CNTH QS QT) %. +%total {ENTH CNTH} (typing_list_proj_lemma PL ENTH CNTH QS QT) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% properties of method name equality +%sort meq_trans %. +%mode meq_trans %in %in %out %. +%term _ meq_trans methodName_refl methodName_refl methodName_refl %. +%worlds () (meq_trans E1 E2 E3) %. +%total E (meq_trans E F G) %. +%sort meq_sym %. +%mode meq_sym %in %out %. +%term _ meq_sym methodName_refl methodName_refl %. +%worlds () (meq_sym E1 E2) %. +%total E (meq_sym E F) %. +%% method equality is a congruence ... +%% ... over method name inequality +%sort meq_neq %. +%mode meq_neq %in %in %out %. +%term _ meq_neq methodName_refl (methodName_neq_base NEQ) (methodName_neq_base NEQ) %. +%worlds () (meq_neq MEQ L1 L2) %. +%total MEQ (meq_neq MEQ L1 L2) %. +%% ... over looking up methods +%sort meq_lcm %. +%mode meq_lcm %in %in %out %. +%term _ meq_lcm methodName_refl LCM LCM %. +%worlds () (meq_lcm MEQ L1 L2) %. +%total MEQ (meq_lcm MEQ L1 L2) %. +%% ... over methods being undefined +%sort meq_notInMT %. +%mode meq_notInMT %in %in %out %. +%term _ meq_notInMT methodName_refl N1 N1 %. +%worlds () (meq_notInMT MEQ L1 L2) %. +%total MEQ (meq_notInMT MEQ L1 L2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Method name excluded middle. +%% Two method names are either equal or they're not. +%% first the definition of the sum +%sort mneq_em %. +%term mneq_eq %pi (methodName_eq M N) %-> (mneq_em M N) %. +%term mneq_neq %pi (methodName_neq M N) %-> (mneq_em M N) %. +%sort mneq_helper %. +%mode mneq_helper %in %out %. +%term _ mneq_helper (nat_em_eq EQ) (mneq_eq methodName_refl) %. +%term _ mneq_helper (nat_em_neq NEQ) (mneq_neq (methodName_neq_base NEQ)) %. +%worlds () (mneq_helper M N) %. +%total M (mneq_helper M N) %. +%sort mneq_excluded_middle {M} {N} %. +%mode mneq_excluded_middle %in %in %out %. +%term _ + %pi (mneq_helper MNEQ EQ) + %-> (nat_excluded M N MNEQ) + %-> (mneq_excluded_middle (methodName_nat M) (methodName_nat N) EQ) %. +%worlds () (mneq_excluded_middle M N MN) %. +%total {M N} (mneq_excluded_middle M N MN) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% lm_total: Method lookup is decidable +%% we can either lookup a method name in the method table or +%% get a derivation that the method name is undefined. +%% again, split into several helper functions. +%sort lookup_or_notInMT %. +%term lom1 %pi (lookupMethod MT M MD) %-> (lookup_or_notInMT MT M) %. +%term lom2 %pi (notInMT MT M) %-> (lookup_or_notInMT MT M) %. +%sort lm_helper2 %. +%mode {%in M methodName} {%in N methodName} {%in MT methodTable} {%in N1 nat} {%in CS1 cnlist N1} {%in C className} {%in BE bexp CS1} {%in MR notInMT MT N} {%in NEQ methodName_neq M N} {%in LOM1 lookup_or_notInMT MT M} {%out LOM2 lookup_or_notInMT (mrcons (method C N BE) MT MR) M} lm_helper2 NEQ LOM1 LOM2 %. +%term _ lm_helper2 NEQ (lom1 LM) (lom1 (lookupMethod_mrcons2 NEQ LM)) %. +%term _ lm_helper2 NEQ (lom2 MU) (lom2 (notInMT_mrcons MU NEQ)) %. +%worlds (var) (lm_helper2 N L O) %. +%total L1 (lm_helper2 N2 L1 O1) %. +%sort lm_helper1 {MT methodTable} {MR notInMT MT N} %. +%mode {%in M methodName} {%in N methodName} {%in N1 nat} {%in CS1 cnlist N1} {%in C className} {%in BE bexp CS1} {%in MT methodTable} {%in MR notInMT MT N} {%in EQ mneq_em M N} {%out LOM2 lookup_or_notInMT (mrcons (method C N BE) MT MR) M} lm_helper1 MT MR EQ LOM2 %. +%sort lm_total {MT methodTable} {M methodName} %. +%mode lm_total %in %in %out %. +%term _ lm_helper1 MT MR (mneq_eq EQ) (lom1 lookupMethod_mrcons1) %. +%term _ + %pi (lm_helper2 NEQ LOM1 LOM2) + %-> (lm_total MT M LOM1) + %-> (lm_helper1 MT MR (mneq_neq NEQ) LOM2) %. +%term _ lm_total mrnil M (lom2 notInMT_mrnil) %. +%term _ + %pi (lm_helper1 MT MR EX LOM) + %-> (mneq_excluded_middle M N EX) + %-> (lm_total (mrcons (method C N BE) MT MR) M LOM) %. +%worlds (var) (lm_total MT M LM) (lm_helper1 MT2 MR L O) %. +%total (MT1 MT2) (lm_total MT1 M LM) (lm_helper1 MT2 MR L2 O2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% exclusitivity of various operations +%sort mn2nat %. +%mode mn2nat %in %out %. +%term _ mn2nat methodName_refl nat_refl %. +%worlds () (mn2nat M N) %. +%total M (mn2nat M N) %. +%sort cn2nat %. +%mode cn2nat %in %out %. +%term _ cn2nat className_refl nat_refl %. +%worlds () (cn2nat M N) %. +%total M (cn2nat M N) %. +%sort methodName_eq_exclusive %. +%mode methodName_eq_exclusive %in %in %out %. +%term _ + %pi (nat_eq_exclusive NEQ NNEQ1 FALSE) + %-> (mn2nat MNEQ NEQ) + %-> (methodName_eq_exclusive MNEQ (methodName_neq_base NNEQ1) FALSE) %. +%worlds () (methodName_eq_exclusive N M F) %. +%total N (methodName_eq_exclusive N M F) %. +%sort className_eq_exclusive %. +%term _ + %pi (nat_eq_exclusive NEQ NNEQ1 FALSE) + %-> (cn2nat MNEQ NEQ) + %-> (className_eq_exclusive MNEQ (className_neq_natnat NNEQ1) FALSE) %. +%mode className_eq_exclusive %in %in %out %. +%worlds () (className_eq_exclusive N M F) %. +%total N (className_eq_exclusive N M F) %. +%sort lookupMethod_exclusive %. +%mode lookupMethod_exclusive %in %in %out %. +%term _ + %pi (methodName_eq_exclusive methodName_refl NEQ FALSE) + %-> (lookupMethod_exclusive lookupMethod_mrcons1 (notInMT_mrcons NI0 NEQ) FALSE) %. +%term _ + %pi (lookupMethod_exclusive LM0 NI0 FALSE) + %-> (lookupMethod_exclusive (lookupMethod_mrcons2 NEQ1 LM0) (notInMT_mrcons NI0 NEQ2) FALSE) %. +%worlds (var) (lookupMethod_exclusive LM NI F) %. +%total (LM) (lookupMethod_exclusive LM NI F) %. +%sort mtype2neq %. +%mode mtype2neq %in %out %. +%term _ mtype2neq (mt_class LM LC NEQ) NEQ %. +%term _ mtype2neq (mt_super _ _ _ NEQ) NEQ %. +%worlds (var) (mtype2neq N O) %. +%total N (mtype2neq N O) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% false implies equality +%sort false_imp_eq %. +%mode {%in MN1 methodName} {%in MD1 methodDef MN1} {%in MD2 methodDef MN1} {%in F false} {%out EQ methodDef_eq MD1 MD2} false_imp_eq F EQ %. +%worlds (var) (false_imp_eq F EQ) %. +%total F (false_imp_eq F EQ) %. +%sort false_imp_eq_class %. +%mode {%in MN1 className} {%in MD1 classDef MN1} {%in MD2 classDef MN1} {%in F false} {%out EQ classDef_eq MD1 MD2} false_imp_eq_class F EQ %. +%worlds (var) (false_imp_eq_class F EQ) %. +%total F (false_imp_eq_class F EQ) %. +%sort false_imp_eq_className %. +%mode {%in MD1 className} {%in MD2 className} {%in F false} {%out EQ className_eq MD1 MD2} false_imp_eq_className F EQ %. +%worlds () (false_imp_eq_className F EQ) %. +%total F (false_imp_eq_className F EQ) %. +%sort false_imp_TBE %. +%mode {%in N1 nat} {%in CS1 cnlist N1} {%in CT classTable} {%in BE bexp CS1} {%in C0 className} {%in R1 className} {%in N2 nat} {%in DS cnlist N2} {%in F false} {%out TBE typing_bexp CT BE C0 object} {%out SB subtyping CT object C0} {%out EQ cnlist_eq CS1 DS} false_imp_TBE F TBE SB EQ %. +%worlds () (false_imp_TBE F TBE SB EQ) %. +%total F (false_imp_TBE F TBE SB EQ) %. +%% lookup method is unique +%sort lookupMethod_unique %. +%mode lookupMethod_unique %in %in %out %. +%term _ lookupMethod_unique (%the (lookupMethod _ M _) lookupMethod_mrcons1) lookupMethod_mrcons1 methodDef_refl %. +%term _ + %pi (false_imp_eq FALSE MDEQ) + %-> (methodName_eq_exclusive methodName_refl NEQ FALSE) + %-> (lookupMethod_unique (%the (lookupMethod _ M _) lookupMethod_mrcons1) (lookupMethod_mrcons2 NEQ LM0) MDEQ) %. +%term _ + %pi (false_imp_eq FALSE MDEQ) + %-> (methodName_eq_exclusive methodName_refl NEQ FALSE) + %-> (lookupMethod_unique (%the (lookupMethod _ M _) (lookupMethod_mrcons2 NEQ LM1)) (%the (lookupMethod _ M _) lookupMethod_mrcons1) MDEQ) %. +%term _ + %pi (lookupMethod_unique LM1 LM2 MDEQ) + %-> (lookupMethod_unique (%the (lookupMethod _ M _) (lookupMethod_mrcons2 NEQ1 LM1)) (lookupMethod_mrcons2 NEQ2 LM2) MDEQ) %. +%worlds (var) (lookupMethod_unique LCM1 LCM2 CDEQ) %. +%total (LCM1) (lookupMethod_unique LCM1 LCM2 CDEQ) %. +%% lookup class is unique +%% the uniqueness checker is not strong enough to prove this +%sort lookupClass_unique %. +%mode lookupClass_unique %in %in %out %. +%term _ lookupClass_unique lookupClass_crcons1 lookupClass_crcons1 classDef_refl %. +%term _ + %pi (false_imp_eq_class FALSE MDEQ) + %-> (className_eq_exclusive className_refl NEQ FALSE) + %-> (lookupClass_unique lookupClass_crcons1 (lookupClass_crcons2 NEQ LC0) MDEQ) %. +%term _ + %pi (false_imp_eq_class FALSE MDEQ) + %-> (className_eq_exclusive className_refl NEQ FALSE) + %-> (lookupClass_unique (lookupClass_crcons2 NEQ LC1) (%the (lookupClass _ M _) lookupClass_crcons1) MDEQ) %. +%term _ + %pi (lookupClass_unique LC1 LC2 MDEQ) + %-> (lookupClass_unique (lookupClass_crcons2 NEQ1 LC1) (lookupClass_crcons2 NEQ2 LC2) MDEQ) %. +%worlds (var) (lookupClass_unique LC1 LC2 CDEQ) %. +%total (LC1) (lookupClass_unique LC1 LC2 CDEQ) %. +%sort mtype_exclusive %. +%mode mtype_exclusive %in %in %out %. +%term _ + %pi (className_eq_exclusive className_refl CNEQ FALSE) + %-> (mtype2neq GMD CNEQ) + %-> (mtype_exclusive GMD mu_object FALSE) %. +%term _ + %pi (lookupMethod_exclusive LM3 NI2 FALSE) + %-> (refineMTEQ MTEQ LM1 LM3) + %-> (classDef_eq_extract CDEQ _ _ MTEQ) + %-> (lookupClass_unique LC1 LC2 CDEQ) + %-> (mtype_exclusive (mt_class LM1 LC1 CNEQ) (mu_super MU NI2 LC2 NEQ2) FALSE) %. +%term _ {CLEQ cnlist_eq _ _} {MTY1 mtype CT1 M D CS1 C0} {MU method_undefined CT1 M D0} + %pi (mtype_exclusive MTY2 MU FALSE) + %-> (refineCEQ CEQ MTY1 MTY2) + %-> (classDef_eq_extract CDEQ CEQ CLEQ MTEQ) + %-> (lookupClass_unique LC1 LC2 CDEQ) + %-> (mtype_exclusive (mt_super MTY1 NI1 LC1 CNEQ) (mu_super MU NI2 LC2 NEQ2) FALSE) %. +%worlds (var) (mtype_exclusive M N P) %. +%total (N) (mtype_exclusive M N P) %. +%sort false_imp_lookupMethod %. +%mode {%in MR1 methodTable} {%in M methodName} {%in N1 nat} {%in DS cnlist N1} {%in D0 className} {%in BE2 bexp DS} {%in N2 nat} {%in CS cnlist N2} {%in C0 className} {%out BE3 bexp CS} {%in F false} {%in LM1 lookupMethod MR1 M (method D0 M BE2)} {%out LM2 lookupMethod MR1 M (method C0 M BE3)} false_imp_lookupMethod F LM1 LM2 %. +%worlds (var) (false_imp_lookupMethod F L M) %. +%total F (false_imp_lookupMethod F L M) %. +%% mtype is unique, so DS = CS (and same length!) and D0 = C0 +%sort mtype_unique %. +%mode {%in N1 nat} {%in CT classTable} {%in M methodName} {%in C className} {%in DS cnlist N1} {%in D0 className} {%in N2 nat} {%in CS cnlist N2} {%in C0 className} {%in MR1 methodTable} {%in BE2 bexp DS} {%out BE3 bexp CS} {%in M1 mtype CT M C DS D0} {%in M2 mtype CT M C CS C0} {%in E3 lookupMethod MR1 M (method D0 M BE2)} {%out E4 lookupMethod MR1 M (method C0 M BE3)} mtype_unique M1 M2 E3 E4 %. +%term _ + %pi (false_imp_lookupMethod FALSE LM1 LM2) + %-> (lookupMethod_exclusive LCM3 NI2 FALSE) + %-> (refineMTEQ MTEQ LCM1 LCM3) + %-> (classDef_eq_extract CDEQ _ _ MTEQ) + %-> (lookupClass_unique LC1 LC2 CDEQ) + %-> (mtype_unique (mt_class LCM1 LC1 _) (mt_super GMD2 NI2 LC2 _) LM1 LM2) %. +%term _ + %pi (false_imp_lookupMethod FALSE LM1 LM2) + %-> (lookupMethod_exclusive LCM3 NI1 FALSE) + %-> (refineMTEQ MTEQ LCM2 LCM3) + %-> (classDef_eq_extract CDEQ _ _ MTEQ) + %-> (lookupClass_unique LC2 LC1 CDEQ) + %-> (mtype_unique (mt_super GMD1 NI1 LC1 _) (mt_class LCM2 LC2 _) LM1 LM2) %. +%term _ + %pi (mtype_unique GMD1 GMD3 LM1 LM2) + %-> (refineCEQ CEQ GMD2 GMD3) + %-> (classDef_eq_extract CDEQ CEQ _ _) + %-> (lookupClass_unique LC2 LC1 CDEQ) + %-> (mtype_unique (mt_super GMD1 NI1 LC1 _) (mt_super GMD2 NI2 LC2 _) LM1 LM2) %. +%term _ + %pi (refineMDEQ MDEQ LM3 LM4) + %-> (lookupMethod_unique LM1 LM2A MDEQ) + %-> (refineMTEQ MTEQ LM2 LM2A) + %-> (classDef_eq_extract CDEQ _ _ MTEQ) + %-> (lookupClass_unique LC2 LC1 CDEQ) + %-> (mtype_unique (mt_class LM1 LC1 NEQ1) (mt_class LM2 LC2 NEQ2) LM3 LM4) %. +%worlds (var) (mtype_unique M N O P) %. +%total {M N} (mtype_unique M N O P) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% If the class table is well formed, then a class defined in the class +%% table is well formed. +%sort ctth_ct %. +%mode ctth_ct %in %in %out %out %. +%term _ ctth_ct (ctt_cons _ CT NEQ) lookupClass_crcons1 NEQ CT %. +%term _ + %pi (ctth_ct REST LC2 NEQ1 CT2) + %-> (ctth_ct (ctt_cons REST CT _) (lookupClass_crcons2 NEQ LC2) NEQ1 CT2) %. +%block b_typing {CT classTable} {C0 className} [x exp] [q typing CT x C0]%. +%worlds (b_typing var) (ctth_ct WCT LCM NEQ CT) %. +%total WCT (ctth_ct WCT LCM NEQ CT) %. +%unique ctth_ct %in %in %out %out %. +%sort ctt_ct %. +%mode ctt_ct %in %in %out %out %. +%term _ %pi (ctth_ct CTH LC NEQ CT) %-> (ctt_ct (ctt CTH) LC NEQ CT) %. +%worlds (b_typing var) (ctt_ct WCT LCM NEQ CT) %. +%total WCT (ctt_ct WCT LCM NEQ CT) %. +%unique ctt_ct %in %in %out %out %. +%% If a class is well-formed, then the method table is well formed. +%sort ct_mtt %. +%mode ct_mtt %in %out %. +%term _ ct_mtt (t_class FS MTT _) MTT %. +%worlds (b_typing var) (ct_mtt CT MTT) %. +%total CT (ct_mtt CT MTT) %. +%unique ct_mtt %in %out %. +%% If a method table is well formed, then a method is well typed. +%sort mtt_mt %. +%mode mtt_mt %in %in %out %. +%term _ mtt_mt (mtt_cons MT0 MTY) lookupMethod_mrcons1 MTY %. +%term _ + %pi (mtt_mt MT0 LCM2 MT) + %-> (mtt_mt (mtt_cons MT0 MD) (lookupMethod_mrcons2 MNNEQ LCM2) MT) %. +%worlds (b_typing var) (mtt_mt MTT LCM MT) %. +%total LCM (mtt_mt MTT LCM MT) %. +%unique mtt_mt %in %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% This part deals with mtype---see sub_mtype lemma below. +%% the class table says that if M is defined in D then it has type +%% DS -> D0. +%% but mtype thinks the type should be CS -> C0 +%sort override_check %. +%mode override_check %in %in %in %out %. +%term _ {MT1 mtype CT1 M1 C DS D0} {MT2 mtype CT1 M1 C CS C0} {LM1 lookupMethod MR1 M1 (method D0 M1 BE1)} {LM2 lookupMethod MR1 M1 (method C0 M1 BE2)} %pi (mtype_unique MT1 MT2 LM1 LM2) %-> (override_check (ov1 MT1) MT2 LM1 LM2) %. +%term _ + %pi (false_imp_lookupMethod FALSE LM1 LM2) + %-> (mtype_exclusive MT2 UN FALSE) + %-> (override_check (ov2 UN) MT2 LM1 LM2) %. +%worlds (var) (override_check OV MT E1 E2) %. +%total OV (override_check OV MT E1 E2) %. +%sort refine_override %. +%mode refine_override %in %in %out %. +%term _ refine_override className_refl OV OV %. +%worlds (var) (refine_override EQ OV1 OV2) %. +%total EQ (refine_override EQ OV1 OV2) %. +%%% Lemma about subtyping and mtype. (A.13 in TAPL) +%% split to help the coverage checker. +%% if mtype(M,D) = C0 -> CS +%% and C extends D +%% then mtype(M,C) = C0 -> CS +%sort sub_mtype_helper %. +%mode sub_mtype_helper %in %in %in %in %out %. +%% case on whether M is found in the method table of C. +%% in the first case we have to worry about override. The method +%% is defined in the method table for the super class, but lookup +%% method in the current class may not have returned the same type for it +%% as mtype. However, because of overriding, we know that those types +%% must be the same. +%term _ {CLEQ cnlist_eq _ _} {SB subtyping CT _ object} {OV1 override CT M D0 CS1 C0} {OV2 override CT M D1 CS1 C0} {LC2 lookupClass CT C (class C D0 FS0 MT0)} {LC1 lookupClass CT C (class C D1 FS1 MT1)} + %pi (override_check OV2 MTY1 LM1 LM2) + %-> (refine_override EQ1 OV1 OV2) + %-> (classDef_eq_extract CDEQ EQ1 CLEQ MTEQ) + %-> (lookupClass_unique LC2 LC1 CDEQ) + %-> (mtt_mt MTT LM1 (t_method Y X OV1 LC2)) + %-> (ctt_ct (ctt X1) LC1 NEQ (t_class Z MTT SB)) + %-> (sub_mtype_helper (ctt X1) MTY1 (lom1 LM1) LC1 (mt_class LM2 LC1 NEQ)) %. +%% in this case we know that the method does not appear in the method table. +%term _ + %pi (ctt_ct WCT1 LC NEQ (t_class _ MTT _)) + %-> (sub_mtype_helper WCT1 MTY1 (lom2 MNDEF) LC (mt_super MTY1 MNDEF LC NEQ)) %. +%worlds (b_typing var) (sub_mtype_helper WCT MT LOM LC QM) %. +%covers sub_mtype_helper %in %in %in %in %out %. +%total MT (sub_mtype_helper WCT MT LOM LC QM) %. +%% coverage checker fails! +%sort sub_mtype %. +%mode sub_mtype %in %in %in %out %. +%% proof is by induction on the subtyping judgement. +%term _ sub_mtype WCT QMT s_refl QMT %. +%term _ {SB subtyping CT E C} {LC lookupClass CT D (class D E ES MT)} {QMT1 mtype CT M C CS C0} {QMT2 mtype CT M E CS C0} {QMT3 mtype CT M D CS C0} + %pi (sub_mtype_helper WCT QMT2 LOM LC QMT3) + %-> (lm_total MT M LOM) + %-> (sub_mtype WCT QMT1 SB QMT2) + %-> (sub_mtype WCT QMT1 (s_super SB LC NEQ) QMT3) %. +%worlds (b_typing var) (sub_mtype WCT Q1 S Q2) %. +%total S (sub_mtype WCT Q1 S Q2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Narrowing allows us to strengthen the type of an assumption a subtyping judgment. +%% +%% If G, X<:D |- E :T and C<: D then G, X<:C |- E : S where S <: T +%% Because of the block declaration, the statement of narrowing must +%% be put at the the beginning of the file! That way any of the lemmas +%% that the *proof* of narrowing calls (directly, or indirectly) can +%% be given the appropriate worlds declaration. +%% This theorem is not nearly so tricky as the one for the POPLmark solution. +%sort narrowing_bexp %. +%mode narrowing_bexp %in %in %in %in %out %out %. +%sort narrowing_exp %. +%mode narrowing_exp %in %in %in %out %out %. +%sort narrowing_explist %. +%mode narrowing_explist %in %in %in %out %. +%% a specialization of narrowing for the "this" variable. +%sort narrowing_this %. +%mode narrowing_this %in %in %in %out %out %. +%block b_bind {CT classTable} {C0 className} [x exp] [q typing CT x C0] [N {WCT class_table_typing CT} {C1 className} {C2 className} {SB0 subtyping CT C1 C2} narrowing_exp WCT ([y] [qy] q) SB0 ([y] [qy] q) s_refl]%. +%term _ + %pi (sub_strengthen SB2 SB3) + %-> ({this exp} {qthis typing CT this C0} {N {WCT} {C1} {C2} {SB0 subtyping CT C1 C2} narrowing_exp WCT ([x] [q typing CT x C2] qthis) SB0 ([x] [q typing CT x C1] qthis) s_refl} narrowing_exp WCT ([x] [q] Q x q this qthis) SB1 ([x] [q] Q2 x q this qthis) (SB2 this)) + %-> (narrowing_bexp WCT ([x exp] base (E x)) ([x exp] [q] t_base (Q x q)) SB1 ([x] [q] t_base (Q2 x q)) SB3) %. +%term _ + %pi (sub_strengthen SB2 SB3) + %-> ({y exp} {qy typing CT y C0} {N {WCT} {C1} {C2} {SB0 subtyping CT C1 C2} narrowing_exp WCT ([x] [q typing CT x C2] qy) SB0 ([x] [q typing CT x C1] qy) s_refl} narrowing_bexp WCT ([x] E x y) ([x] [q] Q1 x q y qy) SB1 ([x] [q] Q2 x q y qy) (SB2 y)) + %-> (narrowing_bexp WCT ([x] bind C0 (E x)) ([x exp] [q] t_bind (Q1 x q)) SB1 ([x] [q] t_bind (Q2 x q)) SB3) %. +%% narrowing the very variable that occurs here. +%term _ narrowing_exp WCT ([x] [q] q) SB1 ([x] [q] q) SB1 %. +%term _ + %pi (append_nth FI CNTH1 APP CNTH2) + %-> ({x exp} sub_fields SB2 (FE1 x) APP (FE2 x)) + %-> (narrowing_exp WCT TY1 SB1 TY2 SB2) + %-> (narrowing_exp WCT ([x] [q] t_field CNTH1 (FE1 x) (TY1 x q)) SB1 ([x] [q] t_field CNTH2 (FE2 x) (TY2 x q)) s_refl) %. +%term _ {MTY1 {e exp} mtype CT1 M1 _ _ _} + %pi ({x exp} sub_mtype WCT (MTY1 x) SB2 (MTY2 x)) + %-> (narrowing_explist WCT TL1 SB1 TL2) + %-> (narrowing_exp WCT T1 SB1 T2 SB2) + %-> (narrowing_exp WCT ([x] [q] t_invk (TL1 x q) (MTY1 x) (T1 x q)) SB1 ([x] [q] t_invk (TL2 x q) (MTY2 x) (T2 x q)) s_refl) %. +%term _ + %pi (narrowing_explist WCT TL1 SB1 TL2) + %-> (narrowing_exp WCT ([x] [q] t_new (TL1 x q) (FE x)) SB1 ([x] [q] t_new (TL2 x q) (FE x)) s_refl) %. +%term _ + %pi (narrowing_exp WCT TY1 SB1 TY2 SB2) + %-> (narrowing_exp WCT ([x] [q] t_cast (TY1 x q)) SB1 ([x] [q] t_cast (TY2 x q)) s_refl) %. +%term _ narrowing_explist WCT ([x] [q] tl_nil) SB1 ([x] [q] tl_nil) %. +%term _ + %pi (narrowing_explist WCT TL1 SB TL2) + %-> (sub_trans SB2 SB3 SB4) + %-> (sub_strengthen SB1 SB3) + %-> (narrowing_exp WCT T1 SB T2 SB2) + %-> (narrowing_explist WCT ([x] [q] tl_cons (SB1 x) (TL1 x q) (T1 x q)) SB ([x] [q] tl_cons SB4 (TL2 x q) (T2 x q))) %. +%worlds (b_bind) (narrowing_exp WCT T SB T2 SB2) (narrowing_bexp WCT E T SB T2 SB2) (narrowing_explist WCT TL SB TL2) %. +%total (T T3 TL5) (narrowing_exp WCT1 T SB T2 SB2) (narrowing_bexp WCT2 E T3 SB3 T4 SB4) (narrowing_explist WCT3 TL5 SB5 TL6) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%term _ + %pi (narrowing_exp WCT Q SB1 Q2 SB2) + %-> (narrowing_this WCT (t_base Q) SB1 (t_base Q2) SB2) %. +%term _ + %pi (sub_strengthen SB2 SB3) + %-> ({y exp} {qy typing CT y C0} {N {WCT} {C1} {C2} {SB0 subtyping CT C1 C2} narrowing_exp WCT ([x] [q typing CT x C2] qy) SB0 ([x] [q typing CT x C1] qy) s_refl} narrowing_this WCT (Q1 y qy) SB1 (Q2 y qy) (SB2 y)) + %-> (narrowing_this WCT (t_bind Q1) SB1 (t_bind Q2) SB3) %. +%worlds (b_bind) (narrowing_this WCT Q SB Q2 SB2) %. +%total (Q) (narrowing_this WCT Q SB Q2 SB2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort eq_lemma1 %. +%mode eq_lemma1 %in %in %out %. +%term _ eq_lemma1 classDef_refl X X %. +%worlds () (eq_lemma1 MDEQ TBE1 TBE2) %. +%total MDEQ (eq_lemma1 MDEQ TBE1 TBE2) %. +%sort eq_lemma2 %. +%mode eq_lemma2 %in %in %out %out %. +%term _ eq_lemma2 methodDef_refl X X cnlist_eq_refl %. +%worlds () (eq_lemma2 MDEQ TBE1 TBE2 EQ) %. +%total MDEQ (eq_lemma2 MDEQ TBE1 TBE2 EQ) %. +%% Lemma about mtype/mbody connection +%% tricky part here---I don't think we have the lemma stated correctly. +%% we also need some proofs that CS = DS and BE2 = BE1 for the +%% preservation proof. +%sort mtype_mbody %. +%mode mtype_mbody %in %in %in %out %out %out %. +%term _ {LC1 lookupClass CT C (class C D1 F1 MT1)} {LC2 lookupClass CT C (class C D2 F2 MT2)} {CDEQ classDef_eq (class C D1 F1 MT1) (class C D2 F2 MT2)} {LCM1 lookupMethod MT1 M (method R1 M (%the (bexp CS1) BE1))} {LCM2 lookupMethod MT2 M (method R2 M (%the (bexp CS2) BE2))} {LCM3 lookupMethod MT1 M (method R2 M BE2)} {MDEQ methodDef_eq (method R1 M BE1) (method R2 M BE2)} {TBE typing_bexp CT BE1 C C1} {TBE2 typing_bexp CT BE2 C C1} + %pi (eq_lemma2 MDEQ TBE TBE2 EQ) + %-> (mtt_mt MTT LCM1 (t_method SB TBE OV Z)) + %-> (ct_mtt CTT MTT) + %-> (ctt_ct WCT LC1 W CTT) + %-> (lookupMethod_unique LCM1 LCM3 MDEQ) + %-> (eq_lemma1 CDEQ LCM2 LCM3) + %-> (lookupClass_unique LC1 LC2 CDEQ) + %-> (mtype_mbody WCT (mt_class LCM1 LC1 NEQ1) (mb_class LCM2 LC2 NEQ2) TBE2 SB EQ) %. +%term _ {CLEQ cnlist_eq _ _} {MTY1 mtype CT M D1 CS1 R1} {NI1 notInMT MT1 M} {LC1 lookupClass CT C (class C D1 FS1 MT1)} {NEQ1 className_neq C object} {MB2 mbody CT M D2 BE1} {NI2 notInMT MT2 M} {LC2 lookupClass CT C (class C D2 FS2 MT2)} {NEQ2 className_neq C object} {MTY2 mtype CT M D2 CS1 R1} {TBE typing_bexp CT BE1 D2 C1} {SB subtyping CT C1 R1} + %pi (sub_trans SB2 SB SB3) + %-> (narrowing_this WCT TBE (s_super s_refl LC2 NEQ2) TBE2 SB2) + %-> (mtype_mbody WCT MTY2 MB2 TBE SB EQ) + %-> (refineCEQ CEQ MTY1 MTY2) + %-> (classDef_eq_extract CDEQ CEQ CLEQ MTEQ) + %-> (lookupClass_unique LC1 LC2 CDEQ) + %-> (mtype_mbody WCT (mt_super MTY1 NI1 LC1 NEQ1) (mb_super NI2 MB2 LC2 NEQ2) TBE2 SB3 EQ) %. +%term _ + %pi (false_imp_TBE FALSE TBE SB EQ) + %-> (lookupMethod_exclusive LM3 NI2 FALSE) + %-> (eq_lemma1 CDEQ LM1 LM3) + %-> (lookupClass_unique LC2 LC1 CDEQ) + %-> (mtype_mbody WCT (mt_class LM1 LC1 NEQ1) (mb_super NI2 GMD2 LC2 NEQ2) TBE SB EQ) %. +%term _ {MTY1 mtype CT M D1 CS1 R1} {NI1 notInMT MT1 M} {LC1 lookupClass CT C (class C D1 FS1 MT1)} {NEQ1 className_neq C object} {LCM2 lookupMethod MT2 M (method R2 M CS2)} {LC2 lookupClass CT C (class C D2 FS2 MT2)} {NEQ2 className_neq C object} + %pi (false_imp_TBE FALSE TBE SB EQ) + %-> (lookupMethod_exclusive LCM3 NI1 FALSE) + %-> (eq_lemma1 CDEQ LCM2 LCM3) + %-> (lookupClass_unique LC1 LC2 CDEQ) + %-> (mtype_mbody WCT (mt_super MTY1 NI1 LC1 NEQ1) (mb_class LCM2 LC2 NEQ2) TBE SB EQ) %. +%worlds () (mtype_mbody C P Q R S E) %. +%total (P) (mtype_mbody C M P Q S E) %. +%{ + +Coverage checker fails. Doesn't know that NI and LCM are exclusive. How to express +this to twelf? + +}% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Lemma about invoc +%sort invoc_lemma %. +%mode invoc_lemma %in %in %in %in %in %out %out %. +%term _ invoc_lemma WCT tl_nil (t_base Q) invoc_base R (Q (new C E1) R) s_refl %. +%term _ {WCT class_table_typing CT} {Q {XI exp} {D typing CT XI CI} typing_bexp CT (BE XI) C E0} + %pi (sub_trans SB SB2 SB3) + %-> (invoc_lemma WCT TL1 (Q2 E T1) I R P SB) + %-> (narrowing_bexp WCT BE Q SB1 Q2 SB2) + %-> (invoc_lemma WCT (tl_cons SB1 TL1 (%the (typing CT E C0) T1)) (t_bind Q) (invoc_bind I) R P SB3) %. +%worlds () (invoc_lemma WCT TL P I R Q SB) %. +%total {TL P I} (invoc_lemma WCT TL P I R Q SB) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort inv_new %. +%mode inv_new %in %out %. +%term _ inv_new (t_new TL FS) (t_new TL FS) %. +%worlds () (inv_new P Q) %. +%total P (inv_new P Q) %. +%sort inv_new2 %. +%mode inv_new2 %in %out %. +%term _ inv_new2 (t_new TL FS) className_refl %. +%worlds () (inv_new2 P Q) %. +%total P (inv_new2 P Q) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% need to show that looking up the fields of a type is a function. +%sort refine_really %. +%mode refine_really %in %in %out %. +%term _ refine_really cnlist_eq_refl CN1 CN1 %. +%worlds () (refine_really E F G) %. +%total E (refine_really E F G) %. +%sort refine_typing_list %. +%mode refine_typing_list %in %in %out %. +%term _ refine_typing_list cnlist_eq_refl CN1 CN1 %. +%worlds () (refine_typing_list E F G) %. +%total E (refine_typing_list E F G) %. +%sort false_imp_really %. +%mode {%in N1 nat} {%in N2 nat} {%in CS1 cnlist N1} {%in CS2 cnlist N2} {%in F false} {%out CN2 cnlist_eq CS1 CS2} false_imp_really F CN2 %. +%worlds () (false_imp_really F CN2) %. +%total F (false_imp_really F CN2) %. +%sort cncons_eq %. +%mode {%in N1 nat} {%in N2 nat} {%in CS1 cnlist N1} {%in CS2 cnlist N2} {%in C className} {%in R1 cnlist_eq CS1 CS2} {%out R2 cnlist_eq (cncons C CS1) (cncons C CS2)} cncons_eq R1 R2 %. +%term _ cncons_eq cnlist_eq_refl cnlist_eq_refl %. +%worlds () (cncons_eq R1 R2) %. +%total E (cncons_eq E F) %. +%sort append_unique %. +%mode append_unique %in %in %out %. +%term _ append_unique cnappend_cnnil cnappend_cnnil cnlist_eq_refl %. +%term _ + %pi (cncons_eq EQ EQ2) + %-> (append_unique R1 R2 EQ) + %-> (append_unique (cnappend_cncons R1) (cnappend_cncons R2) EQ2) %. +%worlds () (append_unique E F G) %. +%total {E F} (append_unique E F G) %. +%sort fields_append_refine %. +%mode fields_append_refine %in %in %in %out %. +%term _ fields_append_refine cnlist_eq_refl classDef_refl X X %. +%worlds () (fields_append_refine E F G H) %. +%total (E) (fields_append_refine E F G H) %. +%sort fields_refine %. +%mode fields_refine %in %in %out %. +%term _ fields_refine classDef_refl X X %. +%worlds () (fields_refine E G H) %. +%total (E) (fields_refine E G H) %. +%sort fields_unique %. +%mode fields_unique %in %in %out %. +%term _ fields_unique f_obj f_obj cnlist_eq_refl %. +%term _ {LC1 lookupClass CT C (class C D1 FS1 MT1)} {LC2 lookupClass CT C (class C D2 FS2 MT2)} {F1 fields CT D1 N1 DG1} {F2 fields CT D2 N2 DG2} {EQ cnlist_eq DG1 DG2} {CDEQ classDef_eq (class C D1 FS1 MT1) (class C D2 FS2 MT2)} {APP1 cnappend DG1 FS1 CS1 PL1} {APP2 cnappend DG2 FS2 CS2 PL2} {APP3 cnappend DG2 FS2 CS1 PL3} + %pi (append_unique APP3 APP2 CEQ) + %-> (fields_append_refine EQ CDEQ APP1 APP3) + %-> (fields_unique F1 F3 EQ) + %-> (fields_refine CDEQ F2 F3) + %-> (lookupClass_unique LC1 LC2 CDEQ) + %-> (fields_unique (f_class APP1 F1 LC1 NEQ1) (f_class APP2 F2 LC2 NEQ2) CEQ) %. +%term _ + %pi (false_imp_really FALSE EQ) + %-> (className_eq_exclusive className_refl NEQ FALSE) + %-> (fields_unique f_obj (f_class APP2 F2 LC2 NEQ) EQ) %. +%term _ + %pi (false_imp_really FALSE EQ) + %-> (className_eq_exclusive className_refl NEQ FALSE) + %-> (fields_unique (f_class APP1 F1 LC1 NEQ) f_obj EQ) %. +%worlds () (fields_unique F1 F2 EQ) %. +%total F1 (fields_unique F1 F2 EQ) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/fj/false.lf b/new-tests/stelf-output/examples/fj/false.lf new file mode 100644 index 0000000..afb2674 --- /dev/null +++ b/new-tests/stelf-output/examples/fj/false.lf @@ -0,0 +1,7 @@ +%{ This is part of a proof of the soundness of Featherweight Java (Atsushi Igarashi, +Benjamin Pierce and Philip Wadler) in the Twelf logical Framework. +It was developed by Stephanie Weirich and +Geoffrey Washburn . }% +% Library about false. +%sort false %. +%name false %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/fj/fj.lf b/new-tests/stelf-output/examples/fj/fj.lf new file mode 100644 index 0000000..18031e1 --- /dev/null +++ b/new-tests/stelf-output/examples/fj/fj.lf @@ -0,0 +1,435 @@ +%{ This is part of a proof of the soundness of Featherweight Java (Atsushi Igarashi, +Benjamin Pierce and Philip Wadler) in the Twelf logical Framework. +It was developed by Stephanie Weirich and +Geoffrey Washburn . }% +%% Semantics of FJ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Abstract syntax +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort className %. +%name className %. +%sort methodName %. +%name methodName %. +% Method and class names are just natural numbers +%term methodName_nat %pi nat %-> methodName %. +%term cn %pi nat %-> className %. +% Object is the only pre-existing class, defined to be natural number zero. +%define object className cn z %. +% Class name lists. Indexed by length +%sort cnlist %. +%name cnlist %. +%term cnnil cnlist z %. +% Empty class name list +%term cncons %pi className %-> (cnlist N) %-> (cnlist (s N)) %. +% Class name list cons +% Expressions and expression lists +%sort exp %. +%name exp %. +%sort elist %. +%name elist %. +% lists are indexed by length +%term fieldProj %pi exp %-> nat %-> exp %. +%term methodInvk %pi exp %-> methodName %-> (elist N) %-> exp %. +%term new %pi className %-> (elist N) %-> exp %. +%term cast %pi className %-> exp %-> exp %. +% and +%term enil elist z %. +%term econs %pi exp %-> (elist N) %-> (elist (s N)) %. +%% a simple expression +%define unit exp new object enil %. +% Expressions with binders (used for method bodies), indexed by className lists for +% convenience. Idea mostly taken from CMU POPLmark solution +% Base case abstract a single variable for "this" +% Bind case binds an additional argument. +%sort bexp %. +%name bexp %. +%term base %pi (%pi exp %-> exp) %-> (bexp cnnil) %. +%term bind {C className} %pi (%pi exp %-> (bexp CS)) %-> (bexp (cncons C CS)) %. +% Method declaration +%sort methodDef %. +%name methodDef %. +%term method %pi className %-> ({M methodName} %pi (bexp CS) %-> (methodDef M)) %. +% Assoc tables of methods +%sort methodTable %. +%name methodTable %. +%% definition of methodTable is below. We are using a representation of method +%% tables that includes a proof that labels are not repeated. However, we have +%% to define equality for method names first. We could drop this proof and still +%% prove the soundness of FJ, but adding this to our representation makes the +%% adequacy proof easier. +% Class definition +%sort classDef %. +%name classDef %. +%term class {C className} %pi className %-> (cnlist N) %-> methodTable %-> (classDef C) %. +% class table +%% The invariant is "object" cannot be a part of the class table. +%% This is satisfied by the well-formedness rule for class tables. +%% the "cons" constructor also ensures that all class definitions in the table are unique. +%% For type soundness, this isn't actually necessary, but it does make our adequacy proof +%% a little easier. +%sort classTable %. +%name classTable %. +%% definition of classTable is also below +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Relations and functions on the abstract syntax +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Equality relation for class names +%sort className_eq %. +%name className_eq %. +%mode className_eq %in %in %. +% FIX - Do I need this mode declaration? +% className_refl : nat_eq N1 N2 -> +% % -------------------------------------- +% className_eq (cn N1) (cn N2). +%term className_refl className_eq N N %. +%worlds () (className_eq CN1 CN2) %. +%terminates {CN1 CN2} (className_eq CN1 CN2) %. +%%%%%%%%%%%%% +% Not equals relation for class names +%sort className_neq %. +%name className_neq %. +%mode className_neq %in %in %. +% FIX - Do I need this mode declaration? +%term className_neq_natnat %pi (nat_neq N1 N2) %-> (className_neq (cn N1) (cn N2)) %. +%worlds () (className_neq CN1 CN2) %. +%terminates {CN1 CN2} (className_neq CN1 CN2) %. +%%%%%%%%%%%%% +% Equality relation for method names +%sort methodName_eq %. +%name methodName_eq %. +%mode methodName_eq %in %in %. +% FIX - Do I need this mode declaration? +% methodName_eq_base : nat_eq N1 N2 -> +% % -------------------------------------- +% methodName_eq (methodName_nat N1) (methodName_nat N2). +%term methodName_refl methodName_eq M1 M1 %. +%worlds () (methodName_eq MN1 MN2) %. +%terminates {MN1 MN2} (methodName_eq MN1 MN2) %. +%%%%%%%%%%%%% +% Not equals relation for method names +%sort methodName_neq %. +%name methodName_neq %. +%mode methodName_neq %in %in %. +% FIX - Do I need this mode declaration? +%term methodName_neq_base %pi (nat_neq N1 N2) %-> (methodName_neq (methodName_nat N1) (methodName_nat N2)) %. +%worlds () (methodName_neq MN1 MN2) %. +%terminates {MN1 MN2} (methodName_neq MN1 MN2) %. +%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% List manipulation functions for expression and class name lists. +% Nth for expression lists +%sort enth_opt %. +%mode enth_opt %in %in %out %. +%term enth_sz enth_opt z (econs E _) E %. +%term enth_sn %pi (enth_opt N TL E) %-> (enth_opt (s N) (econs E' TL) E) %. +%worlds () (enth_opt N EL E) %. +%terminates N (enth_opt N EL E) %. +%%%%%%%%%%%%% +% Nth for classname lists +%sort cnnth_opt %. +%mode cnnth_opt %in %in %out %. +%name cnnth_opt %. +%term cnnth_sz cnnth_opt z (cncons C CS) C %. +%term cnnth_sn %pi (cnnth_opt N CS C) %-> (cnnth_opt (s N) (cncons D CS) C) %. +%worlds () (cnnth_opt N CL C) %. +%terminates N (cnnth_opt N CL C) %. +%%%%%%%%%%%%% +% Append for class name lists +%sort cnappend %. +%mode cnappend %in %in %out %out %. +%term cnappend_cnnil cnappend cnnil CS CS plus_z %. +%term cnappend_cncons + %pi (cnappend CS1 CS2 CS3 PL) + %-> (cnappend (cncons CN CS1) CS2 (cncons CN CS3) (plus_s PL)) %. +%worlds () (cnappend CS1 CS2 CS3 PL) %. +%terminates CS1 (cnappend CS1 CS2 CS3 PL) %. +%unique cnappend %in %in %out %out %. +%covers cnappend %in %in %out %out %. +%total CS1 (cnappend CS1 CS2 CS3 PL) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Helper functions for method and class tables as well as +% class table and method table abstract syntax. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% method and class tables embed proofs that each new name is not already added +% to the class table. Therefore, we have to define names not appearing in tables +% simultaneously with the tables themselves. +%sort notInMT %. +%mode notInMT %in %in %. +%name notInMT %. +%sort notInCT %. +%mode notInCT %in %in %. +% Empty method record +%term mrnil methodTable %. +% Method record cons +%term mrcons {MD methodDef M} {MT methodTable} %pi (notInMT MT M) %-> methodTable %. +% Empty class table +%term crnil classTable %. +% class table cons. +%term crcons {CD classDef C} {CT classTable} %pi (notInCT CT C) %-> classTable %. +%%%%%%%%%%%%% +%% a method name is NOT defined in a method table +%term notInMT_mrnil notInMT mrnil M %. +%term notInMT_mrcons + %pi (notInMT MT M) + %-> (methodName_neq M M') + %-> (notInMT (mrcons (method _ M' _) MT NI') M) %. +%worlds () (notInMT M N) %. +%%%%%%%%%%%%% +%% a class name is NOT defined in a class table +%term nnil notInCT crnil C %. +%term ncons + %pi (notInCT CT N) + %-> (className_neq N N') + %-> (notInCT (crcons (class N' _ _ _) CT NI') N) %. +%worlds () (notInCT CT C) %. +%%%%%%%%%%%%% +% Method lookup +%sort lookupMethod %. +%name lookupMethod %. +%mode lookupMethod %in %in %out %. +%term lookupMethod_mrcons1 lookupMethod (mrcons (method C MN1 BE) MT MU) MN1 (method C MN1 BE) %. +%term lookupMethod_mrcons2 + %pi (methodName_neq MN1 MN2) + %-> (lookupMethod MR MN1 MD) + %-> (lookupMethod (mrcons (method C MN2 BE) MR MU) MN1 MD) %. +%worlds () (lookupMethod MR MN MD) %. +%terminates MR (lookupMethod MR MN MD) %. +%%%%%%%%%%%%% +% Find the class definition for a given class. +%% returns the rest of the class table, which is where the superclass of +%sort lookupClass {CT classTable} {C className} %. +%name lookupClass %. +%mode lookupClass %in %in %out %. +%term lookupClass_crcons1 lookupClass (crcons (class CN2 CN3 CNL MR) CT CTU) CN2 (class CN2 CN3 CNL MR) %. +%term lookupClass_crcons2 + %pi (className_neq CN1 CN2) + %-> (lookupClass CT CN1 CD) + %-> (lookupClass (crcons (class CN2 CN3 CNL MR) CT CTU) CN1 CD) %. +%worlds () (lookupClass CN CT CD) %. +%terminates CN (lookupClass CN CT CD) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Static Semantics of FJ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Subtyping +%% Note, I couldn't specify the mode I wanted for subtyping because of the +%% transitivity rule---the middle types are not ground inputs. +%% therefore, I'm using an 'algorithmic' subtyping judgment instead. +%% I should lookup later whether I can use the subtyping judgment that appears +%% in TAPL instead. +%% In the end, we don't need the mode declaration for subtyping. On the other +%% hand, defining subtyping in this way means that our type system is syntax +%% directed and proof search capabilities of twelf means that we have also +%% implemented a type checker. +%sort subtyping %. +%name subtyping %. +%%mode subtyping +CT +CN1 +CN2. +%term s_refl subtyping CT C C %. +% s_trans: subtyping CT C1 C2 -> +% subtyping CT C2 C3 -> +% % ------------------ +% subtyping CT C1 C3. +%term s_super + %pi (subtyping CT D E) + %-> (lookupClass CT C (class C D _ _)) + %-> (className_neq C object) + %-> (subtyping CT C E) %. +%worlds () (subtyping CT C1 C2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Calculate the fields of a given class +%sort fields %. +%mode fields %in %in %out %out %. +%term f_obj fields _ object z cnnil %. +%term f_class + %pi (cnappend CS2 CS1 (%the (cnlist N) CS3) _) + %-> (fields CT D _ CS2) + %-> (lookupClass CT C (class C D CS1 _)) + %-> (className_neq C object) + %-> (fields CT C N CS3) %. +%worlds () (fields CT C M CS) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Method type lookup +%% method M in class C1 takes args CS and produces C2. +%sort mtype {CT classTable} %. +%name mtype %. +%mode mtype %in %in %in %out %out %. +%% method is found in this class. +%term mt_class + %pi (lookupMethod MT M (method C2 _ (%the (bexp CS) BE))) + %-> (lookupClass CT C1 (class _ _ _ MT)) + %-> (className_neq C1 object) + %-> (mtype CT M C1 CS C2) %. +%% the method is not found in this class, but is found in some +%% superclass +%term mt_super + %pi (mtype CT M C2 CS C0) + %-> (notInMT MT M) + %-> (lookupClass CT C1 (class _ C2 _ MT)) + %-> (className_neq C1 object) + %-> (mtype CT M C1 CS C0) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Method body lookup +%sort mbody %. +%mode mbody %in %in %in %out %. +%term mb_class + %pi (lookupMethod MT M (method _ _ BE)) + %-> (lookupClass CT C1 (class _ _ _ MT)) + %-> (className_neq C1 object) + %-> (mbody CT M C1 BE) %. +%term mb_super + %pi (notInMT MT M) + %-> (mbody CT M C2 BE) + %-> (lookupClass CT C1 (class _ C2 _ MT)) + %-> (className_neq C1 object) + %-> (mbody CT M C1 BE) %. +%worlds () (mbody C CT M BE) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% method not defined in a class. +%sort method_undefined %. +%mode method_undefined %in %in %in %. +%name method_undefined %. +%term mu_object method_undefined CT M object %. +%term mu_super + %pi (method_undefined CT M D) + %-> (notInMT MT M) + %-> (lookupClass CT C (class _ D _ MT)) + %-> (className_neq C object) + %-> (method_undefined CT M C) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Expression/expression list typing +%% although expression typing is "tight", expression list typing is not. +%sort typing {CT classTable} %. +%% %mode typing +CT +E -C. +%sort typing_list {CT classTable} %. +%% %mode typing_list +CT +ES +CS. +%term t_field + %pi (cnnth_opt FI CS CI) + %-> (fields CT C1 N2 CS) + %-> (typing CT E C1) + %-> (typing CT (fieldProj E FI) CI) %. +%term t_invk + %pi (typing_list CT ES DS) + %-> (mtype CT M C0 DS C) + %-> (typing CT E0 C0) + %-> (typing CT (methodInvk E0 M ES) C) %. +%term t_new %pi (typing_list CT ES DS) %-> (fields CT C N DS) %-> (typing CT (new C ES) C) %. +%term t_cast %pi (typing CT E C1) %-> (typing CT (cast C2 E) C2) %. +%term tl_nil typing_list CT enil cnnil %. +%term tl_cons + %pi (subtyping CT C0 C) + %-> (typing_list CT ES CS) + %-> (typing CT E C0) + %-> (typing_list CT (econs E ES) (cncons C CS)) %. +%% Notes about typing judgment: +% casts include both safe and unsafe casts. +% field projection uses the "partial" version of nth---if there is +% a derivation of nth then the expression will be well-typed. That +% seems reasonable to me. +%%terminates E (typing CT E C). % Can't prove this b/c don't know fields is total +%% typing for method bodies +%sort typing_bexp %. +%% %mode typing_bexp +CT +BE +C -E. +%term t_base + %pi ({THIS exp} {D typing CT THIS C} typing CT (E THIS) E0) + %-> (typing_bexp CT (base E) C E0) %. +%term t_bind + %pi ({XI exp} {D typing CT XI CI} typing_bexp CT (E XI) C E0) + %-> (typing_bexp CT (bind CI E) C E0) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% Check that all definitions of the method above class name have the +%% same type. +%sort override %. +%mode override %in %in %in %in %in %. +%% mtype is defined for M +%term ov1 %pi (mtype CT M D CS C) %-> (override CT M D CS C) %. +%% mtype is not defined for M +%term ov2 %pi (method_undefined CT M D) %-> (override CT M D CS C) %. +%sort method_typing %. +%% %mode method_typing +CT +M +MD +C. +%term t_method + %pi (subtyping CT E0 C0) + %-> (typing_bexp CT BE C E0) + %-> (override CT M D CS C0) + %-> (lookupClass CT C (class C D FS MT)) + %-> (method_typing CT M (method C0 M (%the (bexp CS) BE)) C) %. +%sort method_table_typing %. +%% %mode method_table_typing +CT +MT +C. +%term mtt_nil method_table_typing CT mrnil C %. +%term mtt_cons + %pi (method_table_typing CT MT C) + %-> (method_typing CT M MD C) + %-> (method_table_typing CT (mrcons (%the (methodDef M) MD) MT MNI) C) %. +%sort class_typing %. +% %mode class_typing +CT +C +CD. +%term t_class + %pi (fields CT D N DS) + %-> (method_table_typing CT MT C) + %-> (subtyping CT D object) + %-> (class_typing CT C (class C D FS MT)) %. +%sort class_table_typing_help %. +%term ctt_nil class_table_typing_help CT0 crnil %. +%term ctt_cons + %pi (class_table_typing_help CT0 CT) + %-> (class_typing CT0 C CD) + %-> (className_neq C object) + %-> (class_table_typing_help CT0 (crcons (%the (classDef C) CD) CT CTI)) %. +%sort class_table_typing %. +% %mode class_table_typing +CT. +%term ctt %pi (class_table_typing_help CT CT) %-> (class_table_typing CT) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% FJ Operational semantics +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Method body invocation helper +% Given a method body a list of arguments and a this expression, perform the substitutions +%sort invoc %. +%mode invoc %in %in %in %out %. +%term invoc_base invoc (base ([this exp] E1 this)) enil E2 (E1 E2) %. +%term invoc_bind + %pi (invoc (BE E1) EL E2 E3) + %-> (invoc (bind _ ([x exp] BE x)) (econs E1 EL) E2 E3) %. +%worlds () (invoc BE EL E1 E2) %. +%unique invoc %in %in %in %out %. +%total BE (invoc BE EL E1 E2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Values and lists of values +%sort value_list %. +%name value_list %. +%mode value_list %in %. +%sort value %. +%name value %. +%mode value %in %. +%term v_nil value_list enil %. +%term v_cons %pi (value_list EL) %-> (value E) %-> (value_list (econs E EL)) %. +%term v_newv %pi (value_list EL) %-> (value (new C EL)) %. +%terminates (E EL) (value E) (value_list EL) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Operational semantics +%sort reduction_list {CT classTable} %. +%sort reduction {CT classTable} %. +%term r_field + %pi (enth_opt N2 EL E) + %-> (reduction CT (fieldProj (new C (%the (elist N1) EL)) N2) E) %. +%term r_invk + %pi (value_list EL2) + %-> (mbody CT M C BE) + %-> (invoc BE EL2 (new C EL1) E) + %-> (reduction CT (methodInvk (new C EL1) M EL2) E) %. +%term r_cast %pi (subtyping CT C1 C2) %-> (reduction CT (cast C2 (new C1 EL)) (new C1 EL)) %. +%term rc_field %pi (reduction CT E1 E2) %-> (reduction CT (fieldProj E1 N) (fieldProj E2 N)) %. +%term rc_invk_recv + %pi (reduction CT E1 E2) + %-> (reduction CT (methodInvk E1 M EL) (methodInvk E2 M EL)) %. +%term rc_invk_arg + %pi (value E) + %-> (reduction_list CT EL1 EL2) + %-> (reduction CT (methodInvk E M EL1) (methodInvk E M EL2)) %. +%term rc_new_arg %pi (reduction_list CT EL1 EL2) %-> (reduction CT (new C EL1) (new C EL2)) %. +%term rc_cast %pi (reduction CT E1 E2) %-> (reduction CT (cast C E1) (cast C E2)) %. +%term rl_hd %pi (reduction CT E1 E2) %-> (reduction_list CT (econs E1 EL) (econs E2 EL)) %. +%term rl_tl + %pi (value E) + %-> (reduction_list CT EL1 EL2) + %-> (reduction_list CT (econs E EL1) (econs E EL2)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/fj/nat.lf b/new-tests/stelf-output/examples/fj/nat.lf new file mode 100644 index 0000000..f8de41d --- /dev/null +++ b/new-tests/stelf-output/examples/fj/nat.lf @@ -0,0 +1,247 @@ +%{ This is part of a proof of the soundness of Featherweight Java (Atsushi Igarashi, +Benjamin Pierce and Philip Wadler) in the Twelf logical Framework. +It was developed by Stephanie Weirich and +Geoffrey Washburn . }% +%{ Library for natural numbers. + Decided to write this myself so that I was sure to understand everything }% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% The definition of natural numbers +%sort nat %. +%name nat %. +% Zero +%term z nat %. +% Successor +%term s %pi nat %-> nat %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Predecessor function +% Necessary for some inductive reasoning, because equality on nats is not +% inductively defined. +%sort pred %. +%name nat %. +%mode pred %in %out %. +% Could perhaps choose to make this a partial function? Would it matter much? +%term pred_z pred z z %. +%term pred_s pred (s N) N %. +%worlds () (pred N1 N2) %. +%covers pred %in %out %. +%unique pred %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Plus relation +% FIX - should this be moded? +%sort plus %. +%name plus %. +%mode plus %in %in %out %. +%term plus_z plus z N N %. +%term plus_s %pi (plus N1 N2 N3) %-> (plus (s N1) N2 (s N3)) %. +%terminates N1 (plus N1 N2 N3) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Equality relation +% FIX? Should this be moded? +%sort nat_eq %. +%name nat_eq %. +%mode nat_eq %in %in %. +%term nat_refl nat_eq N N %. +% Decided to try going with equality defined by unification +% rather than inductively over the structure of its inputs. +% + +% nat_eq_z : nat_eq z z. +% + +% nat_eq_s : nat_eq N1 N2 -> +% % ---------------------- +% nat_eq (s N1) (s N2). +%worlds () (nat_eq N1 N2) %. +%terminates {N1 N2} (nat_eq N1 N2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Disequals relation +% + +% Should we be defining this directly? Or using false? +%sort nat_neq %. +%name nat_neq %. +%mode nat_neq %in %in %. +% FIX? Should this be moded? +%term nat_neq_zs nat_neq z (s N) %. +%term nat_neq_sz nat_neq (s N) z %. +%term nat_neq_ss %pi (nat_neq N1 N2) %-> (nat_neq (s N1) (s N2)) %. +%worlds () (nat_neq N1 N2) %. +%terminates {N1 N2} (nat_neq N1 N2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Less than relation +%sort nat_lt %. +%name nat_lt %. +%mode nat_lt %in %in %. +% FIX? Should this be moded? +%term nat_lt_z nat_lt z (s N) %. +%term nat_lt_s %pi (nat_lt N1 N2) %-> (nat_lt (s N1) (s N2)) %. +%worlds () (nat_lt N1 N2) %. +%terminates {N1 N2} (nat_lt N1 N2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Less than or equal to relation +%sort nat_lte %. +%name nat_lte %. +%mode nat_lte %in %in %. +% FIX? Should this be moded? +%term nat_lte_z nat_lte z N %. +%term nat_lte_s %pi (nat_lte N1 N2) %-> (nat_lte (s N1) (s N2)) %. +%terminates {N1 N2} (nat_lte N1 N2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Less than or equal to algorithm +%sort comp_nat_lte {N1 nat} {N2 nat} %. +%mode comp_nat_lte %in %in %out %. +%term comp_nat_lte_z comp_nat_lte z N nat_lte_z %. +%term comp_nat_lte_s %pi (comp_nat_lte N1 N2 LTE) %-> (comp_nat_lte (s N1) (s N2) (nat_lte_s LTE)) %. +%worlds () (comp_nat_lte N1 N2 LTE) %. +%terminates N1 (comp_nat_lte N1 N2 LTE) %. +%unique comp_nat_lte %in %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Greater than relation +%sort nat_gt %. +%name nat_gt %. +%mode nat_gt %in %in %. +% FIX? Should this be moded? +%term nat_gt_z nat_gt (s N) z %. +%term nat_gt_s %pi (nat_gt N1 N2) %-> (nat_gt (s N1) (s N2)) %. +%terminates {N1 N2} (nat_gt N1 N2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Greater than algorithm +%sort comp_nat_gt {N1 nat} {N2 nat} %. +%mode comp_nat_gt %in %in %out %. +%term comp_nat_gt_z comp_nat_gt (s N) z nat_gt_z %. +%term comp_nat_gt_s %pi (comp_nat_gt N1 N2 GT) %-> (comp_nat_gt (s N1) (s N2) (nat_gt_s GT)) %. +%worlds () (comp_nat_gt N1 N2 GT) %. +%terminates N2 (comp_nat_gt N1 N2 GT) %. +%unique comp_nat_gt %in %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% excluded middle for natural numbers +%% nat_em is a datatype for the excluded middle property +%% of natural number equality. Either two numbers are equal +%% or they are not. +%sort nat_em %. +%term nat_em_eq %pi (nat_eq N M) %-> (nat_em N M) %. +%term nat_em_neq %pi (nat_neq N M) %-> (nat_em N M) %. +%% nat_excluded takes two numbers and returns the datatype +%% that contains either a proof of their equality or +%% disequality. :-) +%% It is defined in conjunction with nat_em_helper that +%% takes care of the splitting issue. +%sort nat_excluded {N} {M} %. +%mode nat_excluded %in %in %out %. +%term _ nat_excluded z z (nat_em_eq nat_refl) %. +%term _ nat_excluded (s N) z (nat_em_neq nat_neq_sz) %. +%term _ nat_excluded z (s N) (nat_em_neq nat_neq_zs) %. +%sort nat_em_helper %. +%mode nat_em_helper %in %out %. +%term _ nat_em_helper (nat_em_eq nat_refl) (nat_em_eq nat_refl) %. +%term _ nat_em_helper (nat_em_neq R) (nat_em_neq (nat_neq_ss R)) %. +%worlds () (nat_em_helper N M) %. +%total N (nat_em_helper N M) %. +%unique nat_em_helper %in %out %. +%term _ + %pi (nat_em_helper R RR) + %-> (nat_excluded N M R) + %-> (nat_excluded (s N) (s M) RR) %. +%worlds () (nat_excluded M N MN) %. +%total {M N} (nat_excluded M N MN) %. +%unique nat_excluded %in %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% properties of natural number equality +%sort nat_trans %. +%mode nat_trans %in %in %out %. +%term _ nat_trans nat_refl nat_refl nat_refl %. +%worlds () (nat_trans N O P) %. +%total N (nat_trans N O P) %. +%sort nat_sym %. +%mode nat_sym %in %out %. +%term _ nat_sym nat_refl nat_refl %. +%worlds () (nat_sym N P) %. +%total N (nat_sym N P) %. +% Disequality is symmetric +%sort nat_neq_sym %. +%mode nat_neq_sym %in %out %. +%term _ nat_neq_sym nat_neq_zs nat_neq_sz %. +%term _ nat_neq_sym nat_neq_sz nat_neq_zs %. +%term _ + %pi (nat_neq_sym NNEQ1 NNEQ2) + %-> (nat_neq_sym (nat_neq_ss NNEQ1) (nat_neq_ss NNEQ2)) %. +%worlds () (nat_neq_sym N P) %. +%total N (nat_neq_sym N P) %. +%% congruence rules/would be nice to have Leibniz equality. +%% if M = N then P(M) => P (N). +%% But we can't prove things in that generality. We must prove it for each +%% particular case. +%% this is congruence over inequality. +%sort nat_cong %. +%mode nat_cong %in %in %out %. +%term _ nat_cong nat_refl nat_neq_zs nat_neq_zs %. +%term _ nat_cong nat_refl nat_neq_sz nat_neq_sz %. +%term _ + %pi (nat_cong nat_refl NEQ1 NEQ2) + %-> (nat_cong nat_refl (nat_neq_ss NEQ1) (nat_neq_ss NEQ2)) %. +%worlds () (nat_cong N O P) %. +%total O (nat_cong N O P) %. +%% natural numbers cannot both be equal and inequal. +%sort nat_eq_exclusive %. +%mode nat_eq_exclusive %in %in %out %. +%term _ + %pi (nat_eq_exclusive nat_refl NEQ FALSE) + %-> (nat_eq_exclusive nat_refl (nat_neq_ss NEQ) FALSE) %. +%worlds () (nat_eq_exclusive N M F) %. +%total (M) (nat_eq_exclusive N M F) %. +%% false implies that we have any equality we choose. +%sort false_imp_nat_eq {N1 nat} {N2 nat} %. +%mode false_imp_nat_eq %in %in %in %out %. +%worlds () (false_imp_nat_eq N1 N2 F NEQ) %. +%total F (false_imp_nat_eq N1 N2 F NEQ) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% If two nats are equal and they have a predecessor, their predecessors are also +% equal. Not true when one of the nats is zero, so we also require that the +% inputs are not equal to zero. +% + +% This is necessary because we do not have inductive reasoning over equality. +%sort pred_eq_lemma %. +%mode pred_eq_lemma %in %in %in %in %in %out %. +%term _ pred_eq_lemma NNEQ1 NNEQ2 NEQ1 PRED1 PRED2 nat_refl %. +%worlds () (pred_eq_lemma NNEQ1 NNEQ2 NEQ1 PRED1 PRED2 NEQ2) %. +%total {PRED1 PRED2} (pred_eq_lemma NNEQ1 NNEQ2 NEQ1 PRED1 PRED2 NEQ2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%sort pred_eq_lemma_alt %. +%mode pred_eq_lemma_alt %in %out %. +%term _ pred_eq_lemma_alt NEQ nat_refl %. +%worlds () (pred_eq_lemma_alt NEQ1 NEQ2) %. +%total {NEQ1} (pred_eq_lemma_alt NEQ1 NEQ2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% If two nats are equal, then their succecessors are also equal +% + +% Also needed because equality is not inductively defined. +%sort succ_eq_lemma %. +%mode succ_eq_lemma %in %out %. +%term _ succ_eq_lemma NEQ nat_refl %. +%worlds () (succ_eq_lemma NEQ1 NEQ2) %. +%total NEQ1 (succ_eq_lemma NEQ1 NEQ2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% If we add equal nats we get an equal result. +%sort plus_eq_lemma %. +%mode plus_eq_lemma %in %in %in %in %out %. +%term _ plus_eq_lemma NEQ1 NEQ2 plus_z plus_z NEQ2 %. +%term _ + %pi (succ_eq_lemma NEQ4 NEQ5) + %-> (plus_eq_lemma NEQ3 NEQ2 PL1 PL2 NEQ4) + %-> (pred_eq_lemma_alt NEQ1 NEQ3) + %-> (plus_eq_lemma NEQ1 NEQ2 (plus_s PL1) (plus_s PL2) NEQ5) %. +%worlds () (plus_eq_lemma NEQ1 NEQ2 PL1 PL2 NEQ) %. +%total {PL1 PL2} (plus_eq_lemma NEQ1 NEQ2 PL1 PL2 NEQ) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% If N1 is less than N2 and N2 = N3 then N1 is less than N3. +%sort nat_lt_eq %. +%mode nat_lt_eq %in %in %out %. +%term _ nat_lt_eq nat_lt_z nat_refl nat_lt_z %. +%term _ + %pi (nat_lt_eq NLT1 NEQ2 NLT2) + %-> (pred_eq_lemma_alt NEQ1 NEQ2) + %-> (nat_lt_eq (nat_lt_s NLT1) NEQ1 (nat_lt_s NLT2)) %. +%worlds () (nat_lt_eq NLT1 NEQ NLT2) %. +%total NLT1 (nat_lt_eq NLT1 NEQ NLT2) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/fj/pres.lf b/new-tests/stelf-output/examples/fj/pres.lf new file mode 100644 index 0000000..649e47e --- /dev/null +++ b/new-tests/stelf-output/examples/fj/pres.lf @@ -0,0 +1,52 @@ +%{ This is part of a proof of the soundness of Featherweight Java (Atsushi Igarashi, +Benjamin Pierce and Philip Wadler) in the Twelf logical Framework. +It was developed by Stephanie Weirich and +Geoffrey Washburn . }% +% Preservation Theorem +%sort pres %. +%mode pres %in %in %in %out %out %. +%sort pres_elist %. +%mode pres_elist %in %in %in %out %. +%term pres_rl_hd + %pi (sub_trans SB1 SB2 SB3) + %-> (pres CT P1 D1 Q1 SB1) + %-> (pres_elist CT (tl_cons SB2 PL1 P1) (rl_hd D1) (tl_cons SB3 PL1 Q1)) %. +%term pres_rl_tl + %pi (pres_elist CT PL1 DL1 QL1) + %-> (pres_elist CT (tl_cons SB PL1 P1) (rl_tl VL DL1) (tl_cons SB QL1 P1)) %. +%term pres_rc_field {D1 reduction CT E1 E2} {P1 typing CT E1 D} {Q1 typing CT E2 C} {SB subtyping CT C D} {FS1 fields CT D N1 DG} {FS2 fields CT C N3 CS} {APP cnappend DG EF CS PL} {CNTH1 cnnth_opt FI DG CI} {CNTH2 cnnth_opt FI CS CI} {WCT class_table_typing CT} + %pi (append_nth CI CNTH1 APP CNTH2) + %-> (sub_fields SB FS1 APP FS2) + %-> (pres WCT P1 D1 Q1 SB) + %-> (pres WCT (t_field CNTH1 FS1 P1) (rc_field D1) (t_field CNTH2 FS2 Q1) s_refl) %. +%term pres_r_field {ENTH enth_opt FI ES EI} {CNTH1 cnnth_opt FI CS CI} {FS1 fields CT C N CS} {FS2 fields CT C N2 CS2} {PL1 typing_list CT ES CS2} {SB subtyping CT C0 CI} {Q typing CT EI C0} {CNTH2 cnnth_opt FI CS2 CI} + %pi (typing_list_proj_lemma PL1 ENTH CNTH2 SB Q) + %-> (refine_really EQ CNTH1 CNTH2) + %-> (fields_unique FS1 FS2 EQ) + %-> (pres WCT (t_field CNTH1 FS1 (t_new PL1 FS2)) (r_field ENTH) Q SB) %. +%term pres_rc_invk_arg + %pi (pres_elist WCT PL1 DL1 QL1) + %-> (pres WCT (t_invk PL1 MT1 P1) (rc_invk_arg VL DL1) (t_invk QL1 MT1 P1) s_refl) %. +% Needs a lemma about mtype under subtyping. +%term pres_rc_invk_recv + %pi (sub_mtype WCT MT1 SB1 MT2) + %-> (pres WCT P1 D1 Q1 SB1) + %-> (pres WCT (t_invk PL1 MT1 P1) (rc_invk_recv D1) (t_invk PL1 MT2 Q1) s_refl) %. +%term pres_r_invk {PL1 typing_list CT ES DS} {P1 typing CT (new D0 EL1) C0} {MT1 mtype CT M C0 DS C} {MB mbody CT M D0 (%the (bexp CS) BE)} {INV invoc BE ES (new D0 EL1) E} {MT2 mtype CT M D0 DS C} {TBE typing_bexp CT BE D0 C1} {SB subtyping CT C1 C} {PL2 typing_list CT ES CS} {P2 typing CT (new D0 EL1) D0} {X typing CT E C2} {SB2 subtyping CT C2 C1} + %pi (sub_trans SB2 SB SB3) + %-> (invoc_lemma WCT PL2 TBE INV P2 X SB2) + %-> (refine_typing_list CSEQ PL1 PL2) + %-> (mtype_mbody WCT MT2 MB TBE SB CSEQ) + %-> (refineCEQ CEQ MT1 MT2) + %-> (inv_new P1 P2) + %-> (inv_new2 P1 CEQ) + %-> (pres WCT (t_invk PL1 MT1 P1) (r_invk VL MB INV) X SB3) %. +%term pres_new + %pi (pres_elist WCT PL1 DL1 QL1) + %-> (pres WCT (t_new PL1 FS) (rc_new_arg DL1) (t_new QL1 FS) s_refl) %. +%term pres_cast1 + %pi (pres WCT P1 D1 Q1 SB2) + %-> (pres WCT (t_cast P1) (rc_cast D1) (t_cast Q1) s_refl) %. +%term pres_cast2 %pi (inv_new P1 P2) %-> (pres WCT (t_cast P1) (r_cast SB) P2 SB) %. +%worlds () (pres_elist CT PL DL QL) (pres CT P D Q SB) %. +%total (DL D) (pres_elist CT PL DL QL) (pres CT2 P D Q SB) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/fj/progress.lf b/new-tests/stelf-output/examples/fj/progress.lf new file mode 100644 index 0000000..27738c4 --- /dev/null +++ b/new-tests/stelf-output/examples/fj/progress.lf @@ -0,0 +1,123 @@ +%{ This is part of a proof of the soundness of Featherweight Java (Atsushi Igarashi, +Benjamin Pierce and Philip Wadler) in the Twelf logical Framework. +It was developed by Stephanie Weirich and +Geoffrey Washburn . }% +% Failed cast is an inductive predicate over expressions and expression lists +% that contain a cast that cannot be reduced. +%sort failed_cast %. +%sort failed_cast_list %. +%term fc_field %pi (failed_cast CT E) %-> (failed_cast CT (fieldProj E N)) %. +%term fc_invk_recv %pi (failed_cast CT E) %-> (failed_cast CT (methodInvk E MN ES)) %. +%term fc_invk_arg %pi (failed_cast_list CT ES) %-> (failed_cast CT (methodInvk E MN ES)) %. +%term fc_new %pi (failed_cast_list CT ES) %-> (failed_cast CT (new CN ES)) %. +%term fc_cast + %pi (notSubtype CT CN2 CN1) + %-> (value_list ES) + %-> (failed_cast CT (cast CN1 (new CN2 ES))) %. +%term fc_cast_arg %pi (failed_cast CT E) %-> (failed_cast CT (cast CN E)) %. +%term fc_hd %pi (failed_cast CT E) %-> (failed_cast_list CT (econs E ES)) %. +%term fc_tl %pi (failed_cast_list CT ES) %-> (failed_cast_list CT (econs E ES)) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Disjoint sums for progress +% With respect to a classTable an expression is either: a value, can take +% a step or contains a failed cast. +%sort progsum %. +%term ps_value %pi (value E) %-> (progsum CT E) %. +%term ps_step %pi (reduction CT E1 E2) %-> (progsum CT E1) %. +%term ps_failed %pi (failed_cast CT E) %-> (progsum CT E) %. +% Analogous cases for expression lists +%sort progsum_list %. +%term psl_value %pi (value_list ES) %-> (progsum_list CT ES) %. +%term psl_step %pi (reduction_list CT ES1 ES2) %-> (progsum_list CT ES1) %. +%term psl_failed %pi (failed_cast_list CT ES) %-> (progsum_list CT ES) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Proof of progress for expressions and expression lists. +% Every case of the proof requires a helper lemma because of output coverage +% constraints. +%sort progress {CT classTable} %. +%mode progress %in %in %in %out %. +%sort progress_list {CT classTable} %. +%mode progress_list %in %in %in %out %. +%%%% Cases for well-typed casts +%sort progress_cast_helper2 %. +%mode progress_cast_helper2 %in %in %out %. +%term _ progress_cast_helper2 VL (xms_sub SB) (ps_step (r_cast SB)) %. +%term _ progress_cast_helper2 (v_newv VLS) (xms_notsub NSB) (ps_failed (fc_cast NSB VLS)) %. +%worlds () (progress_cast_helper2 VL XMSB PS) %. +%total XMSB (progress_cast_helper2 VL XMSB PS) %. +%sort progress_cast_helper1 {CT classTable} {D className} %. +%mode progress_cast_helper1 %in %in %in %in %out %. +%term _ + %pi (progress_cast_helper2 VL XMSB PS) + %-> (sb_total CT _ D CTT XMSB) + %-> (progress_cast_helper1 CT D CTT (ps_value VL) PS) %. +%term _ progress_cast_helper1 CT D CTT (ps_step RD) (ps_step (rc_cast RD)) %. +%term _ progress_cast_helper1 CT D CTT (ps_failed FC) (ps_failed (fc_cast_arg FC)) %. +%worlds () (progress_cast_helper1 CT D CTT PS1 PS2) %. +%total PS1 (progress_cast_helper1 CT D CTT PS1 PS2) %. +%term _ + %pi (progress_cast_helper1 CT _ CTT PS1 PS2) + %-> (progress CT CTT TP PS1) + %-> (progress CT CTT (t_cast TP) PS2) %. +%%%% Cases for well-typed field projections +%sort progress_field_helper {CT classTable} %. +%mode progress_field_helper %in %in %in %in %in %in %out %. +%term _ + %pi (proj_lemma TP FS CNOPT ENTH) + %-> (progress_field_helper CT CTT CNOPT FS TP (ps_value VL) (ps_step (r_field ENTH))) %. +%term _ progress_field_helper CT CTT CNOPT FS TP (ps_step RD) (ps_step (rc_field RD)) %. +%term _ progress_field_helper CT CTT CNOPT FS TP (ps_failed FC) (ps_failed (fc_field FC)) %. +%worlds () (progress_field_helper CT CTT CNOPT FS TP PS1 PS2) %. +%total PS1 (progress_field_helper CT CTT CNOPT FS TP PS1 PS2) %. +%term _ + %pi (progress_field_helper CT CTT CNOPT FS TP PS1 PS2) + %-> (progress CT CTT TP PS1) + %-> (progress CT CTT (t_field CNOPT FS TP) PS2) %. +%%%% Cases for well-typed method invocations +%sort progress_invk_helper {ES elist N} {CS cnlist N} %. +%mode progress_invk_helper %in %in %in %in %in %in %in %out %. +%term _ + %pi (invoc_total ES TPS BE _ INV) + %-> (mbody_inv TP MTY BE MBY) + %-> (progress_invk_helper ES CS MTY TP TPS (ps_value VL) (psl_value VLS) (ps_step (r_invk VLS MBY INV))) %. +%term _ progress_invk_helper ES CS MTY TP TPS (ps_value VL) (psl_step RDS) (ps_step (rc_invk_arg VL RDS)) %. +%term _ progress_invk_helper ES CS MTY TP TPS (ps_value VL) (psl_failed FC) (ps_failed (fc_invk_arg FC)) %. +%term _ progress_invk_helper ES CS MTY TP TPS (ps_step RD) PSL (ps_step (rc_invk_recv RD)) %. +%term _ progress_invk_helper ES CS MTY TP TPS (ps_failed FC) PSL (ps_failed (fc_invk_recv FC)) %. +%worlds () (progress_invk_helper ES CS MTY TP TPS PS1 PSL PS2) %. +%total {PS1 PSL} (progress_invk_helper ES CS MTY TP TPS PS1 PSL PS2) %. +%term _ + %pi (progress_invk_helper _ _ MTY TP TPS PS1 PSL PS2) + %-> (progress CT CTT TP PS1) + %-> (progress_list CT CTT TPS PSL) + %-> (progress CT CTT (t_invk TPS MTY TP) PS2) %. +%%%% Cases for well-typed object creation +%sort progress_new_helper {C className} %. +%mode progress_new_helper %in %in %out %. +%term _ progress_new_helper C (psl_value VLS) (ps_value (v_newv VLS)) %. +%term _ progress_new_helper C (psl_step RDS) (ps_step (rc_new_arg RDS)) %. +%term _ progress_new_helper C (psl_failed FC) (ps_failed (fc_new FC)) %. +%worlds () (progress_new_helper C PSL PS) %. +%total PSL (progress_new_helper C PSL PS) %. +%term _ + %pi (progress_new_helper _ PSL PS) + %-> (progress_list CT CTT TPS PSL) + %-> (progress CT CTT (t_new TPS FS) PS) %. +%%%% Cases for well-typed expression lists +%sort progress_list_helper %. +%mode progress_list_helper %in %in %out %. +%term _ progress_list_helper (ps_value VL) (psl_value VLS) (psl_value (v_cons VLS VL)) %. +%term _ progress_list_helper (ps_value VL) (psl_step RDS) (psl_step (rl_tl VL RDS)) %. +%term _ progress_list_helper (ps_value VL) (psl_failed FC) (psl_failed (fc_tl FC)) %. +%term _ progress_list_helper (ps_step RD) _ (psl_step (rl_hd RD)) %. +%term _ progress_list_helper (ps_failed FC) _ (psl_failed (fc_hd FC)) %. +%worlds () (progress_list_helper PS PSL1 PSL2) %. +%total PS (progress_list_helper PS PSL1 PSL2) %. +%term _ progress_list CT CTT tl_nil (psl_value v_nil) %. +%term _ + %pi (progress_list_helper PS1 PSL1 PSL2) + %-> (progress CT CTT TP PS1) + %-> (progress_list CT CTT TPS PSL1) + %-> (progress_list CT CTT (tl_cons SB TPS TP) PSL2) %. +%worlds () (progress CT CTT TP PS) (progress_list CT CTT TPS PSL) %. +%total (TP TPS) (progress CT1 CTT1 TP PS) (progress_list CT2 CTT2 TPS PSL) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/fj/sources.lf b/new-tests/stelf-output/examples/fj/sources.lf new file mode 100644 index 0000000..f62b647 --- /dev/null +++ b/new-tests/stelf-output/examples/fj/sources.lf @@ -0,0 +1,7 @@ +%require %[ false %] %. +%require %[ nat %] %. +%require %[ fj %] %. +%require %[ aux-lemma %] %. +%require %[ pres %] %. +%require %[ aux-lemma-prog %] %. +%require %[ progress %] %. diff --git a/new-tests/stelf-output/examples/fj/stelf.toml b/new-tests/stelf-output/examples/fj/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/fj/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/fol/fol-thm.lf b/new-tests/stelf-output/examples/fol/fol-thm.lf new file mode 100644 index 0000000..2558fc3 --- /dev/null +++ b/new-tests/stelf-output/examples/fol/fol-thm.lf @@ -0,0 +1,13 @@ +% A simple direct proof +% Induction is ruled out by specifying no induction variable. +%theorem identity : forall* {P atm} exists {D !^ (` P => ` P)} true %. +%establish 5 {} (identity _) %. +%theorem curry : forall* {P atm} {Q atm} {R atm} exists {D !^ (` P & ` Q => ` R => (` P => (` Q => ` R)))} true %. +%establish 12 {} (curry _) %. +% OK +%theorem uncurry : forall* {P atm} {Q atm} {R atm} exists {D !^ (` P => (` Q => ` R) => (` P & ` Q => ` R))} true %. +% %establish 12 {} (uncurry _). % 12 terminates? +%theorem split : forall* {P atm} {Q atm} {R atm} exists {D !^ (` P => ` Q & ` R => (` P => ` Q) & (` P => ` R))} true %. +% %establish 15 {} (split _). % OK +%theorem join : forall* {P atm} {Q atm} {R atm} exists {D !^ (` P => ` Q & (` P => ` R) => (` P => ` Q & ` R))} true %. +% %establish 12 {} (join _). % OK \ No newline at end of file diff --git a/new-tests/stelf-output/examples/fol/fol.lf b/new-tests/stelf-output/examples/fol/fol.lf new file mode 100644 index 0000000..0dcf496 --- /dev/null +++ b/new-tests/stelf-output/examples/fol/fol.lf @@ -0,0 +1,25 @@ +%%% Intuitionistic propositional calculus +%%% Positive fragment with implies, and, true. +%%% Author: Frank Pfenning +% Type of propositions. +%sort o %. +%name o %. +% Type of atomic propositions. +%sort atm %. +% Syntax: implication, plus a few constants. +%term => %pi o %-> o %-> o %. +%prec %right 10 => %. +%term & %pi o %-> o %-> o %. +%prec %right 11 & %. +%term true o %. +%term ` %pi atm %-> o %. +% Atomic deductions (for faster search) +%sort !^ %. +%sort !v %. +%term trueI^ !^ true %. +%term andI^ %pi (!^ A) %-> (!^ B) %-> (!^ (A & B)) %. +%term andEvL %pi (!v (A & B)) %-> (!v A) %. +%term andEvR %pi (!v (A & B)) %-> (!v B) %. +%term impI^ %pi (%pi (!v A) %-> (!^ B)) %-> (!^ (A => B)) %. +%term impEv %pi (!v (A => B)) %-> (!^ A) %-> (!v B) %. +%term close %pi (!v (` P)) %-> (!^ (` P)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/fol/sources.lf b/new-tests/stelf-output/examples/fol/sources.lf new file mode 100644 index 0000000..82a1205 --- /dev/null +++ b/new-tests/stelf-output/examples/fol/sources.lf @@ -0,0 +1 @@ +%require %[ fol %] %. diff --git a/new-tests/stelf-output/examples/fol/stelf.toml b/new-tests/stelf-output/examples/fol/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/fol/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/fol/test.lf b/new-tests/stelf-output/examples/fol/test.lf new file mode 100644 index 0000000..a21d7bd --- /dev/null +++ b/new-tests/stelf-output/examples/fol/test.lf @@ -0,0 +1,2 @@ +%require %[ fol %] %. +%require %[ fol %] %. diff --git a/new-tests/stelf-output/examples/guide/arith.lf b/new-tests/stelf-output/examples/guide/arith.lf new file mode 100644 index 0000000..ffdbe53 --- /dev/null +++ b/new-tests/stelf-output/examples/guide/arith.lf @@ -0,0 +1,76 @@ +%%% Simple Arithmetic Examples +%%% Illustrate termination and reduction checking +%%% Author: Brigitte Pientka +%sort nat %. +%name nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort bool %. +%name bool %. +%term true bool %. +%term false bool %. +%sort nt %. +%name nt %. +%term nt_z nt z %. +%term nt_s %pi (nt (s X)) %<- (nt X) %. +%sort plus %. +%name plus %. +%term p_z plus z Y Y %. +%term p_s %pi (plus (s X) Y (s Z)) %<- (plus X Y Z) %. +%sort acker %. +%mode acker %in %in %out %. +%term a_1 acker z Y (s Y) %. +%term a_2 %pi (acker (s X) z Z) %<- (acker X (s z) Z) %. +%term a_3 %pi (acker (s X) (s Y) Z) %<- (acker (s X) Y Z') %<- (acker X Z' Z) %. +%sort sub %. +%name sub %. +%mode sub %in %in %out %. +%term s_z1 sub X z X %. +%term s_z2 sub z Y z %. +%term s_ss %pi (sub (s X) (s Y) Z) %<- (sub X Y Z) %. +%terminates X (sub X Y _) %. +%reduces <= Z X (sub X Y Z) %. +%sort minus %. +%name minus %. +%mode minus %in %in %out %. +%sort min %. +%name min %. +%mode min %in %in %out %. +%term m_z1 minus X z X %. +%term m_z2 minus z Y z %. +%term m_ss %pi (minus (s X) (s Y) Z) %<- (min X Y Z) %. +%term m'_z1 min X z X %. +%term m'_z2 min z Y z %. +%term m'_ss %pi (min X Y Z) %<- (minus X Y Z) %. +%terminates (X X') (minus X _ _) (min X' _ _) %. +%reduces <= (Z Z') (X X') (minus X Y Z) (min X' _ Z') %. +%sort rminus %. +%name rminus %. +%mode rminus %in %in %out %. +%term rmin %pi (rminus (s X) (s Y) Z) %<- (sub X Y Z) %. +%terminates [X Y] (rminus X Y _) %. +%reduces < Z X (rminus X Y Z) %. +%sort less %. +%name less %. +%mode less %in %in %out %. +%term l_z1 less z (s X) true %. +%term l_z2 less X z false %. +%term l_ss %pi (less (s X) (s Y) B) %<- (less X Y B) %. +%terminates X (less X Y _) %. +%sort gcd %. +%name gcd %. +%mode gcd %in %in %out %. +%term gcd_z1 gcd z Y Y %. +%term gcd_z2 gcd X z X %. +%term gcd_s1 + %pi (gcd (s X) (s Y) Z) + %<- (less (s X) (s Y) true) + %<- (rminus (s Y) (s X) Y') + %<- (gcd (s X) Y' Z) %. +%term gcd_s1 + %pi (gcd (s X) (s Y) Z) + %<- (less (s X) (s Y) false) + %<- (rminus (s X) (s Y) X') + %<- (gcd X' (s Y) Z) %. +%terminates [X Y] (gcd X Y _) %. +%terminates {X Y} (gcd X Y Z) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/guide/lam.lf b/new-tests/stelf-output/examples/guide/lam.lf new file mode 100644 index 0000000..17d2fb0 --- /dev/null +++ b/new-tests/stelf-output/examples/guide/lam.lf @@ -0,0 +1,115 @@ +%%% Lambda-Calculus Fragment of Mini-ML. +%%% Author: Frank Pfenning +% Simple types +%sort tp %. +%name tp %. +%term arrow %pi tp %-> tp %-> tp %. +% T1 => T2 +% Expressions +%sort exp %. +%name exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +% lam x. E +%term app %pi exp %-> exp %-> exp %. +% (E1 E2) +% Type inference +% |- E : T (expression E has type T) +%sort of %. +%name of %. +%mode of %in %star %. +% %mode of +E +T. % incorrect at tp_app +% %mode of +E -T. % incorrect at tp_lam +%term tp_lam %pi (of (lam E) (arrow T1 T2)) %<- ({x exp} %pi (of x T1) %-> (of (E x) T2)) %. +%term tp_app %pi (of (app E1 E2) T1) %<- (of E1 (arrow T2 T1)) %<- (of E2 T2) %. +% and |- E2 : T2. +% Evaluation (call-by-value) +% E ==> V (expression E evaluates to value V) +%sort eval %. +%name eval %. +%mode eval %in %out %. +%term ev_lam eval (lam E) (lam E) %. +% lam x.E ==> lam x.E. +%term ev_app + %pi (eval (app E1 E2) V) + %<- (eval E1 (lam E1')) + %<- (eval E2 V2) + %<- (eval (E1' V2) V) %. +% and [V2/x]E1' ==> V. +% Regular world for type-checking +%block tp_var {T tp} [x exp] [u of x T]%. +%worlds (tp_var) (of E T) %. +% Type inference terminates +%terminates E (of E T) %. +% There is at least one typing rule for every expression +%covers of %in %star %. +% Next fails, because T is neither input nor output +% %total E (of E T). +% Closed worlds for evaluation +%worlds () (eval E V) %. +% There is a at least one evaluation rule for every closed expression +%covers eval %in %out %. +% %terminates E (eval E V). % fails for ev_app +% Type preservation as higher-level family +%sort tps %. +%term tps_lam tps ev_lam (tp_lam P) (tp_lam P) %. +%term tps_app + %pi (tps (ev_app D3 D2 D1) (tp_app P2 P1) Q) + %<- (tps D1 P1 (tp_lam Q1')) + %<- (tps D2 P2 Q2) + %<- (tps D3 (Q1' V2 Q2) Q) %. +%mode tps %in %in %out %. +%worlds () (tps D P _) %. +%total D (tps D P _) %. +% Type preservation proven automatically +%theorem tpsa : forall* {E exp} {V exp} {T tp} forall {D eval E V} {P of E T} exists {Q of V T} true %. +%prove 5 D (tpsa D P Q) %. +% Applying type preservation +%define e0 app (lam ([x] x)) (lam ([y] y)) %. +%solve p0 : of e0 T %. +%solve d0 : eval e0 V %. +%solve tps0 : tps d0 p0 Q %. +% Example of regular worlds +% cp copies input to output. +%sort cp %. +%term cp_app %pi (cp (app E1 E2) (app F1 F2)) %<- (cp E1 F1) %<- (cp E2 F2) %. +%term cp_lam + %pi (cp (lam ([x] E x)) (lam ([x] F x))) + %<- ({x exp} %pi (cp x x) %-> (cp (E x) (F x))) %. +%mode cp %in %out %. +%block cp_var [x exp] [u cp x x]%. +%worlds (cp_var) (cp E _) %. +%total E (cp E _) %. +% Following version cannot be checked +% Input coverage on parameter y is violated +%{ +cp : exp -> exp -> type. +cp_app : cp (app E1 E2) (app F1 F2) + <- cp E1 F1 + <- cp E2 F2. + +cp_lam : cp (lam [x] E x) (lam [x] F x) + <- ({x:exp} {y:exp} cp x y -> cp (E x) (F y)). + +%mode cp +E -F. +%block cp_var : block {x:exp} {y:exp} {u:cp x y}. +%worlds (cp_var) (cp E _). +%total E (cp E _). +}% +% Following version cannot be checked +% Output coverage on (F y) is violated. +%{ +cp : exp -> exp -> type. +cp_app : cp (app E1 E2) (app F1 F2) + <- cp E1 F1 + <- cp E2 F2. + +cp_lam : cp (lam [x] E x) (lam [x] F x) + <- ({x:exp} {y:exp} cp x y -> cp y y -> cp (E x) (F y)). + +%mode cp +E -F. +%block cp_var : block {x:exp} {y:exp} {u:cp x y} {w:cp y y}. +%worlds (cp_var) (cp E _). +%total E (cp E _). +}% +%theorem cpt : forallG (pi {x exp} {y cp x x}) forall {E exp} exists {F exp} {C cp E F} true %. +%prove 5 E (cpt E _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/guide/lists.lf b/new-tests/stelf-output/examples/guide/lists.lf new file mode 100644 index 0000000..0ffb482 --- /dev/null +++ b/new-tests/stelf-output/examples/guide/lists.lf @@ -0,0 +1,11 @@ +%%% Appending lists of propositions +%%% Example for User's Guide +%%% Author: Frank Pfenning +%sort list %. +%term nil list %. +%term cons %pi o %-> list %-> list %. +%sort append %. +%term appNil append nil K K %. +%term appCons %pi (append (cons A L) K (cons A M)) %<- (append L K M) %. +%query 1 1 _ append (cons true nil) (cons false nil) L %. +%query 1 1 _ append (cons true (cons false nil)) (cons true nil) L %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/guide/nd.lf b/new-tests/stelf-output/examples/guide/nd.lf new file mode 100644 index 0000000..be8c8ad --- /dev/null +++ b/new-tests/stelf-output/examples/guide/nd.lf @@ -0,0 +1,62 @@ +%%% Natural deduction for a fragment of intuitionistic logic +%%% Author: Frank Pfennig +%%% Individuals +%sort i %. +%name i %. +%%% Propositions +%sort o %. +%name o %. +%term imp %pi o %-> o %-> o %. +%prec %right 10 imp %. +%term and %pi o %-> o %-> o %. +%prec %right 11 and %. +%term true o %. +%term or %pi o %-> o %-> o %. +%prec %right 11 or %. +%term false o %. +%term forall %pi (%pi i %-> o) %-> o %. +%term exists %pi (%pi i %-> o) %-> o %. +%define not (%pi o %-> o) [A o] A imp false %. +%%% Natural Deductions +%sort nd %. +%name nd %. +%term impi %pi (%pi (nd A) %-> (nd B)) %-> (nd (A imp B)) %. +%term impe %pi (nd (A imp B)) %-> (nd A) %-> (nd B) %. +%term andi %pi (nd A) %-> (nd B) %-> (nd (A and B)) %. +%term ande1 %pi (nd (A and B)) %-> (nd A) %. +%term ande2 %pi (nd (A and B)) %-> (nd B) %. +%term truei nd true %. +% no truee +%term ori1 %pi (nd A) %-> (nd (A or B)) %. +%term ori2 %pi (nd B) %-> (nd (A or B)) %. +%term ore + %pi (nd (A or B)) + %-> (%pi (nd A) %-> (nd C)) + %-> (%pi (nd B) %-> (nd C)) + %-> (nd C) %. +% no falsei +%term falsee %pi (nd false) %-> (nd C) %. +%term foralli %pi ({x i} nd (A x)) %-> (nd (forall A)) %. +%term foralle %pi (nd (forall A)) %-> ({T i} nd (A T)) %. +%term existsi {T i} %pi (nd (A T)) %-> (nd (exists A)) %. +%term existse %pi (nd (exists A)) %-> ({x i} %pi (nd (A x)) %-> (nd C)) %-> (nd C) %. +%define noti (%pi (%pi (nd A) %-> (nd false)) %-> (nd (not A))) [D %pi (nd A) %-> (nd false)] impi D %. +%define note (%pi (nd (not A)) %-> (nd A) %-> (nd false)) [D nd (not A)] [E nd A] impe D E %. +%%% Verifying regular worlds +%block nd_hyp {A o} [u nd A]%. +%block nd_parm [x i]%. +%worlds (nd_hyp nd_parm) (nd A) %. +%%% Local reductions +%sort red %. +%term impred red (impe (impi D) E) (D E) %. +%term andred1 red (ande1 (andi D E)) D %. +%term andred2 red (ande2 (andi D E)) E %. +% no truered +%term orred1 red (ore (ori1 D) E1 E2) (E1 D) %. +%term orred2 red (ore (ori2 D) E1 E2) (E2 D) %. +% no falsered +%term forallred red (foralle (foralli D) T) (D T) %. +%term existsred red (existse (existsi T D) E) (E T D) %. +%%% A simple theorem +%theorem trivI : exists {D {A o} nd (A imp A)} true %. +%prove 2 {} (trivI D) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/guide/sources.lf b/new-tests/stelf-output/examples/guide/sources.lf new file mode 100644 index 0000000..f351622 --- /dev/null +++ b/new-tests/stelf-output/examples/guide/sources.lf @@ -0,0 +1,3 @@ +%require %[ nd %] %. +%require %[ lists %] %. +%require %[ lam %] %. diff --git a/new-tests/stelf-output/examples/guide/stelf.toml b/new-tests/stelf-output/examples/guide/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/guide/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/guide/test.lf b/new-tests/stelf-output/examples/guide/test.lf new file mode 100644 index 0000000..5abe9d8 --- /dev/null +++ b/new-tests/stelf-output/examples/guide/test.lf @@ -0,0 +1,4 @@ +%require %[ nd %] %. +%require %[ lists %] %. +%require %[ lam %] %. +%require %[ arith %] %. diff --git a/new-tests/stelf-output/examples/handbook/fol.lf b/new-tests/stelf-output/examples/handbook/fol.lf new file mode 100644 index 0000000..8212e90 --- /dev/null +++ b/new-tests/stelf-output/examples/handbook/fol.lf @@ -0,0 +1,133 @@ +%%% First-Order Logic +%%% Fragment with implication, negation, universal quantification. +%%% +%%% Author: Frank Pfenning +%%% +%%% This code is from the article +%%% +%%% Logical Frameworks +%%% Handbook of Automated Reasoning +%%% Alan Robinson and Andrei Voronkov, editors +%%% Chapter 16 +%%% Elsevier Science and MIT Press +%%% In preparation +%%% +%sort i %. +% individuals +%sort o %. +% formulas +%name i %. +%name o %. +% Formulas +%term imp %pi o %-> o %-> o %. +%prec %right 10 imp %. +%term not %pi o %-> o %. +%prec %prefix 12 not %. +%term forall %pi (%pi i %-> o) %-> o %. +% Natural deductions +%sort nd %. +%name nd %. +%term impi %pi (%pi (nd A) %-> (nd B)) %-> (nd (A imp B)) %. +%term impe %pi (nd (A imp B)) %-> (nd A) %-> (nd B) %. +%term noti %pi ({p o} %pi (nd A) %-> (nd p)) %-> (nd (not A)) %. +%term note %pi (nd (not A)) %-> ({C o} %pi (nd A) %-> (nd C)) %. +%term foralli %pi ({a i} nd (A a)) %-> (nd (forall A)) %. +%term foralle %pi (nd (forall A)) %-> ({T i} nd (A T)) %. +%block lnd {A o} [u nd A]%. +%block lo [p o]%. +%block li [a i]%. +%worlds (lnd lo li) (nd A) %. +% Example +%define _ (nd (A imp (B imp A))) impi ([u nd A] impi ([v nd B] u)) %. +% Hilbert deductions +%sort hil %. +% Hilbert deductions +%name hil %. +%term k hil (A imp (B imp A)) %. +%term s hil (A imp (B imp C) imp (A imp B imp (A imp C))) %. +%term n1 hil (A imp (not B) imp (A imp B imp (not A))) %. +%term n2 hil (not A imp (A imp B)) %. +%term f1 {T i} hil (forall ([x i] A x) imp (A T)) %. +%term f2 hil (forall ([x i] B imp A x) imp (B imp forall ([x i] A x))) %. +%term mp %pi (hil (A imp B)) %-> (hil A) %-> (hil B) %. +%term ug %pi ({a i} hil (A a)) %-> (hil (forall ([x i] A x))) %. +%worlds (li) (hil A) %. +% Local reductions +%sort ==>R %. +%prec %none 14 ==>R %. +%name ==>R %. +%term redl_imp impe (impi ([u nd A] D u)) E ==>R (D E) %. +%term redl_not note (noti ([p o] [u nd A] D p u)) C E ==>R (D C E) %. +%term redl_forall foralle (foralli ([a i] D a)) T ==>R (D T) %. +% Local expansions +%sort ==>E %. +%prec %none 14 ==>E %. +%name ==>E %. +%term expl_imp {D nd (A imp B)} D ==>E (impi ([u nd A] impe D u)) %. +%term expl_not {D nd (not A)} D ==>E (noti ([p o] [u nd A] note D p u)) %. +%term expl_forall {D nd (forall A)} D ==>E (foralli ([a i] foralle D a)) %. +% Sequent calculus search result +%define dn (nd (A imp not not A)) impi ([u nd A] noti ([p o] [w nd (not A)] note w p u)) %. +% Translating Hilbert derivations to natural deductions +%sort hilnd %. +%term hnd_k hilnd k (impi ([u nd A] impi ([v nd B] u))) %. +%term hnd_s hilnd s (impi ([u nd (A imp B imp C)] impi ([v nd (A imp B)] impi ([w nd A] impe (impe u w) (impe v w))))) %. +%term hnd_n1 hilnd n1 (impi ([u nd (A imp (not B))] impi ([v nd (A imp B)] noti ([p o] [w nd A] note (impe u w) p (impe v w))))) %. +%term hnd_n2 hilnd n2 (impi ([u nd (not A)] impi ([v nd A] note u B v))) %. +%term hnd_f1 hilnd (f1 T) (impi ([u nd (forall ([x i] A x))] foralle u T)) %. +%term hnd_f2 hilnd f2 (impi ([u nd (forall ([x i] B imp A x))] impi ([v nd B] foralli ([a i] impe (foralle u a) v)))) %. +%term hnd_mp %pi (hilnd (mp H1 H2) (impe D1 D2)) %<- (hilnd H1 D1) %<- (hilnd H2 D2) %. +%term hnd_ug %pi (hilnd (ug H1) (foralli D1)) %<- ({a i} hilnd (H1 a) (D1 a)) %. +%mode hilnd %in %out %. +%worlds (li) (hilnd H D) %. +%terminates H (hilnd H _) %. +%covers hilnd %in %out %. +%total H (hilnd H _) %. +%solve id' : hilnd (mp (mp s k) k) (%the (nd (A imp A)) D) %. +%define _ (nd (A imp A)) impe (impe (impi ([u nd (A imp (B imp A) imp A)] impi ([v nd (A imp B imp A)] impi ([w nd A] impe (impe u w) (impe v w))))) (impi ([u nd A] impi ([v nd (B imp A)] u)))) (impi ([u nd A] impi ([v nd B] u))) %. +%%% The deduction theorem for Hilbert derivations +%sort ded %. +%term ded_id ded ([u hil A] u) (mp (mp s k) (%the (hil (A imp (A imp A))) k)) %. +%term ded_k ded ([u hil A] k) (mp k k) %. +%term ded_s ded ([u hil A] s) (mp k s) %. +%term ded_n1 ded ([u hil A] n1) (mp k n1) %. +%term ded_n2 ded ([u hil A] n2) (mp k n2) %. +%term ded_f1 ded ([u hil A] f1 T) (mp k (f1 T)) %. +%term ded_f2 ded ([u hil A] f2) (mp k f2) %. +%term ded_mp + %pi (ded ([u hil A] mp (H1 u) (H2 u)) (mp (mp s H1') H2')) + %<- (ded H1 H1') + %<- (ded H2 H2') %. +%term ded_ug + %pi (ded ([u hil A] ug (H1 u)) (mp f2 (ug H1'))) + %<- ({a i} ded ([u hil A] H1 u a) (H1' a)) %. +%block lded {A o} [u nd A] [v hil A] [h {C o} ded ([w hil C] v) (mp k v)]%. +%mode ded %in %out %. +%worlds (li lo lded) (ded H H') %. +%terminates H (ded H _) %. +%covers ded %in %out %. +%total H (ded H _) %. +%%% Mapping natural deductions to Hilbert derivations. +%sort ndhil %. +%term ndh_impi + %pi (ndhil (impi D1) H1') + %<- ({u nd A1} {v hil A1} %pi ({C o} ded ([w hil C] v) (mp k v)) %-> (ndhil u v) %-> (ndhil (D1 u) (H1 v))) + %<- (ded H1 H1') %. +%term ndh_impe %pi (ndhil (impe D1 D2) (mp H1 H2)) %<- (ndhil D1 H1) %<- (ndhil D2 H2) %. +%term ndh_noti + %pi (ndhil (noti D1) (mp (mp n1 H1') H1'')) + %<- ({p o} {u nd A1} {v hil A1} + %pi ({C o} ded ([w hil C] v) (mp k v)) + %-> (ndhil u v) + %-> (ndhil (D1 p u) (H1 p v))) + %<- (ded (H1 (not A1)) H1') + %<- (ded (H1 A1) H1'') %. +%term ndh_note %pi (ndhil (note D1 C D2) (mp (mp n2 H1) H2)) %<- (ndhil D1 H1) %<- (ndhil D2 H2) %. +%term ndh_foralli %pi (ndhil (foralli D1) (ug H1)) %<- ({a i} ndhil (D1 a) (H1 a)) %. +%term ndh_foralle %pi (ndhil (foralle D1 T) (mp (f1 T) H1)) %<- (ndhil D1 H1) %. +%mode ndhil %in %out %. +%block lndhil {A o} [u nd A] [v hil A] [h {C o} ded ([w hil C] v) (mp k v)] [nh ndhil u v]%. +%worlds (li lo lndhil) (ndhil D H) %. +%terminates D (ndhil D _) %. +%covers ndhil %in %out %. +%total D (ndhil D _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/handbook/sources.lf b/new-tests/stelf-output/examples/handbook/sources.lf new file mode 100644 index 0000000..82a1205 --- /dev/null +++ b/new-tests/stelf-output/examples/handbook/sources.lf @@ -0,0 +1 @@ +%require %[ fol %] %. diff --git a/new-tests/stelf-output/examples/handbook/stelf.toml b/new-tests/stelf-output/examples/handbook/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/handbook/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/handbook/test.lf b/new-tests/stelf-output/examples/handbook/test.lf new file mode 100644 index 0000000..82a1205 --- /dev/null +++ b/new-tests/stelf-output/examples/handbook/test.lf @@ -0,0 +1 @@ +%require %[ fol %] %. diff --git a/new-tests/stelf-output/examples/incll/bff-incll-quy.lf b/new-tests/stelf-output/examples/incll/bff-incll-quy.lf new file mode 100644 index 0000000..bc198d4 --- /dev/null +++ b/new-tests/stelf-output/examples/incll/bff-incll-quy.lf @@ -0,0 +1,5 @@ +%define t1 (trm tree) t z (t z e (t z e e)) (t z (t z e e) e) %. +% It works now!! +%query 1 _ _ sqnt rnil rnil rnil rnil (^ (bf e T)) %. +%query 1 _ _ sqnt rnil rnil rnil rnil (^ (bf (t z e e) T)) %. +%query 1 _ _ sqnt rnil rnil rnil rnil (^ (bf t1 T)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/incll/bff-incll.lf b/new-tests/stelf-output/examples/incll/bff-incll.lf new file mode 100644 index 0000000..7f29191 --- /dev/null +++ b/new-tests/stelf-output/examples/incll/bff-incll.lf @@ -0,0 +1,21 @@ +%%% +%%% A Breadth-First Numbering Algorithm +%%% Author: Frank Pfenning +%%% See C. Okasaki: Breadth-First Numbering, ICFP'00, pp. 131-136 +%%% +%name trm %. +%name atm %. +%term int sort %. +%term z trm int %. +%term s %pi (trm int) %-> (trm int) %. +%prec %prefix 10 s %. +%term tree sort %. +%term e trm tree %. +%term t %pi (trm int) %-> (trm tree) %-> (trm tree) %-> (trm tree) %. +%term bf %pi (trm tree) %-> (trm tree) %-> atm %. +%term n %pi (trm tree) %-> (trm tree) %-> atm %. +%term bff %pi (trm int) %-> atm %. +%term bf0 prog (^ bf T T' <<= (^ n T T' =>> ^ bff (s z))) %. +%term bffdone prog (^ bff N) %. +%term bffe prog (^ bff N <<= ^ n e e <<= ^ bff N) %. +%term bfft prog (^ bff N <<= ^ n (t _ L R) (t N L' R') <<= (^ n L L' >=> ^ n R R' >=> ^ bff (s N))) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/incll/cpm-incll-quy.lf b/new-tests/stelf-output/examples/incll/cpm-incll-quy.lf new file mode 100644 index 0000000..f8ea2a5 --- /dev/null +++ b/new-tests/stelf-output/examples/incll/cpm-incll-quy.lf @@ -0,0 +1,47 @@ +% + +% Example queries +% + +% ceval (app (lam [x] (vl x)) z) V. +% %query 1 * (sqnt rnil rnil rnil rnil (^(return V) =>> +% ^(ev (appm (lamm [x] (vl x)) z)))). +%query 1 1 _ sqnt rnil rnil rnil rnil (^ (return V) =>> ^ (ev (appm (lamm ([x] vl x)) z))) %. +% Doubling 3 +% ceval (app (fix [double] lamm [x] +% case (vl x) z +% [x'] s (s (appm double (vl x')))) (s (s (s z)))) +% V. +% + +% this query has successfully completed on my machine (~ 4 minutes to run) +% + +% %query 1 * (sqnt rnil rnil rnil rnil (^(return V) =>> +% ^(ev (appm (fix [double] lamm [x] +% case (vl x) z +% [x'] s (s (appm double (vl x')))) (s (s (s z)))) ))). +% Computing 2 * 3 +% ceval (letv (lam [x] fix [add] lam [y] +% case (vl y) (vl x) [y'] s (app add (vl y'))) [add] +% (letv (lam [x] fix [mult] lam [y] +% case (vl y) z ([y'] (app (app (vl add) (vl x)) +% (app mult (vl y'))))) [mult] +% (app (app (vl mult) (s (s z))) (s (s (s z)))))) +% V. +% + +% this has produced one right answer, but after 2 hours it still hadn't decided +% if there were any others on my machine. +% + +% has successfully run, in about 5 hours. +% + +% %query 1 * (sqnt rnil rnil rnil rnil (^(return V) =>> +% ^(ev (letv (lamm [x] fix [add] lamm [y] +% case (vl y) (vl x) [y'] s (appm add (vl y'))) [add] +% (letv (lamm [x] fix [mult] lamm [y] +% case (vl y) z ([y'] (appm (appm (vl add) (vl x)) +% (appm mult (vl y'))))) [mult] +% (appm (appm (vl mult) (s (s z))) (s (s (s z)))))) ))). \ No newline at end of file diff --git a/new-tests/stelf-output/examples/incll/cpm-incll.lf b/new-tests/stelf-output/examples/incll/cpm-incll.lf new file mode 100644 index 0000000..d715c41 --- /dev/null +++ b/new-tests/stelf-output/examples/incll/cpm-incll.lf @@ -0,0 +1,70 @@ +% + +% mini-ml +% +%term exp sort %. +%term val sort %. +%term z trm exp %. +%term s %pi (trm exp) %-> (trm exp) %. +%term case %pi (trm exp) %-> (trm exp) %-> (%pi (trm val) %-> (trm exp)) %-> (trm exp) %. +%term pair %pi (trm exp) %-> (trm exp) %-> (trm exp) %. +%term fst %pi (trm exp) %-> (trm exp) %. +%term snd %pi (trm exp) %-> (trm exp) %. +%term lamm %pi (%pi (trm val) %-> (trm exp)) %-> (trm exp) %. +%term appm %pi (trm exp) %-> (trm exp) %-> (trm exp) %. +%term letv %pi (trm exp) %-> (%pi (trm val) %-> (trm exp)) %-> (trm exp) %. +%term letn %pi (trm exp) %-> (%pi (trm exp) %-> (trm exp)) %-> (trm exp) %. +%term fix %pi (%pi (trm exp) %-> (trm exp)) %-> (trm exp) %. +%term vl %pi (trm val) %-> (trm exp) %. +%term z* trm val %. +%term s* %pi (trm val) %-> (trm val) %. +%term pair* %pi (trm val) %-> (trm val) %-> (trm val) %. +%term lamm* %pi (%pi (trm val) %-> (trm exp)) %-> (trm val) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% + +% continuation passing machine +% + +% Machine Instructions +%term ev %pi (trm exp) %-> atm %. +%term return %pi (trm val) %-> atm %. +%term case1 %pi (trm val) %-> (trm exp) %-> (%pi (trm val) %-> (trm exp)) %-> atm %. +%term pair1 %pi (trm val) %-> (trm exp) %-> atm %. +%term fst1 %pi (trm val) %-> atm %. +%term snd1 %pi (trm val) %-> atm %. +%term appm1 %pi (trm val) %-> (trm exp) %-> atm %. +%term appm2 %pi (trm val) %-> (trm val) %-> atm %. +% + +% evaluation of machine +% + +% queries should be: sqnt cpmG rnil rnil rnil rnil (^(return V) =>> ^(ev E)). +% where E is some expression to evaluate. +% +% Thus the ordered context is the stack of continuations left to execute. +% + +% Note the use of left implications (>=>) in rules where a continuation expecting +% a new value s put onto the stack. +% + +%% Natural Numbers +%term ev2 prog (^ ev z <<= ^ return z*) %. +%term ev3 prog (^ ev (s E) <<= (forall val ([V] ^ return V <=< ^ return (s* V)) =>> ^ ev E)) %. +%term ev4 prog (^ ev (case E1 E2 E3) <<= (forall val ([V] ^ return V <=< ^ case1 V E2 E3) =>> ^ ev E1)) %. +%term ev5 prog (^ case1 z* E2 E3 <<= ^ ev E2) %. +%term ev6 prog (^ case1 (s* V) E2 E3 <<= ^ ev (E3 V)) %. +%% Functions +%term ev7 prog (^ ev (lamm E) <<= ^ return (lamm* E)) %. +%term ev8 prog (^ ev (appm E1 E2) <<= (forall val ([V1] ^ return V1 <=< ^ appm1 V1 E2) =>> ^ ev E1)) %. +%term ev9 prog (^ appm1 V1 E2 <<= (forall val ([V2] ^ return V2 <=< ^ appm2 V1 V2) =>> ^ ev E2)) %. +%term ev10 prog (^ appm2 (lamm* E1') V2 <<= ^ ev (E1' V2)) %. +%% Definitions +%term ev11 prog (^ ev (letv E1 E2) <<= (forall val ([V1] ^ ev (E2 V1) >=> ^ return V1) =>> ^ ev E1)) %. +%term ev12 prog (^ ev (letn E1 E2) <<= ^ ev (E2 E1)) %. +%% Recursion +%term ev13 prog (^ ev (fix E) <<= ^ ev (E (fix E))) %. +%% Values +%term ev14 prog (^ ev (vl V) <<= ^ return V) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/incll/incll.lf b/new-tests/stelf-output/examples/incll/incll.lf new file mode 100644 index 0000000..477b5db --- /dev/null +++ b/new-tests/stelf-output/examples/incll/incll.lf @@ -0,0 +1,47 @@ +% + +% twelf signature for INCLL +% + +%sort sort %. +%term arrow %pi sort %-> sort %-> sort %. +%term cross %pi sort %-> sort %-> sort %. +%sort trm %. +%term app %pi (trm (arrow A B)) %-> (trm A) %-> (trm B) %. +%term lam %pi (%pi (trm A) %-> (trm B)) %-> (trm (arrow A B)) %. +%term pair %pi (trm A) %-> (trm B) %-> (trm (cross A B)) %. +%sort eval %. +%term eval_lam eval (lam E) (lam E) %. +%term eval_app %pi (eval (app (lam E) E') V) %<- (eval E' V') %<- (eval (E V') V) %. +%sort atm %. +%sort frm %. +%term int sort %. +%term 1 trm int %. +%term 2 trm int %. +%term 3 trm int %. +%term 4 trm int %. +%term 5 trm int %. +%term list %pi sort %-> sort %. +%term nil trm (list A) %. +%term | %pi (trm A) %-> (trm (list A)) %-> (trm (list A)) %. +%prec %right 30 | %. +%term ^ %pi atm %-> frm %. +%prec %prefix 50 ^ %. +%term => %pi frm %-> frm %-> frm %. +%prec %right 20 => %. +%term =0 %pi frm %-> frm %-> frm %. +%prec %right 20 =0 %. +%term =>> %pi frm %-> frm %-> frm %. +%prec %right 20 =>> %. +%term >=> %pi frm %-> frm %-> frm %. +%prec %right 20 >=> %. +%term forall {s} %pi (%pi (trm s) %-> frm) %-> frm %. +%term forall2 {s1} {s2} %pi (%pi (%pi (trm s1) %-> (trm s2)) %-> frm) %-> frm %. +%define <= [x] [y] y => x %. +%prec %left 10 <= %. +%define 0= [x] [y] y =0 x %. +%prec %left 10 0= %. +%define <<= [x] [y] y =>> x %. +%prec %left 10 <<= %. +%define <=< [x] [y] y >=> x %. +%prec %left 10 <=< %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/incll/iosequents.lf b/new-tests/stelf-output/examples/incll/iosequents.lf new file mode 100644 index 0000000..815b9e5 --- /dev/null +++ b/new-tests/stelf-output/examples/incll/iosequents.lf @@ -0,0 +1,120 @@ +% + +% IO uniform proof sequents for INCLL +% + +%sort prog %. +% + +% sqnt is: G ; D_I \ D_O ; O_I \ O_O --> A +% + +%sort sqnt %. +% + +% ie_sqnt1 is: G ; D_I \ D_O ; ( L_I \ L_O ; R_I \ R_O ) --> A >> P +% + +%sort ie_sqnt1 %. +% + +% ie_sqnt2 is: G ; D_I \ D_O ; O_I \ ( O_L | O_M | O_R ) --> A >> P +% + +%sort ie_sqnt2 %. +%term =>_R %pi (sqnt Di Do Oi Oo (A => B)) %<- (%pi (prog A) %-> (sqnt Di Do Oi Oo B)) %. +%term =0_R %pi (sqnt Di Do Oi Oo (A =0 B)) %<- (sqnt (Di ,r # A) (Do ,r del) Oi Oo B) %. +%term >=>_R + %pi (sqnt Di Do Oi Oo (A >=> B)) + %<- (r2l Oi Oi') + %<- (l2r (# A ,l Oi') Oi'') + %<- (sqnt Di Do Oi'' Oo'' B) + %<- (r2l Oo'' (del ,l Oo')) + %<- (l2r Oo' Oo) %. +%term =>>_R %pi (sqnt Di Do Oi Oo (A =>> B)) %<- (sqnt Di Do (Oi ,r # A) (Oo ,r del) B) %. +%term forall_R + %pi (sqnt Di Do Oi Oo (forall Srt Frm)) + %<- ({x trm Srt} sqnt Di Do Oi Oo (Frm x)) %. +%term forall2_R + %pi (sqnt Di Do Oi Oo (forall2 Srt1 Srt2 Frm)) + %<- ({x %pi (trm Srt1) %-> (trm Srt2)} sqnt Di Do Oi Oo (Frm x)) %. +%term choice_O + %pi (sqnt Di Do Oi Oo (^ P)) + %<- (choose Oi OiL A OiR) + %<- (ie_sqnt1 Di Do OiL OoL OiR OoR A P) + %<- (appr (OoL ,r del) OoR Oo) %. +%term choice_D + %pi (sqnt Di Do Oi Oo (^ P)) + %<- (choose Di DiL A DiR) + %<- (appr DiL DiR Di') + %<- (ie_sqnt2 Di' Do' Oi Ol Om Or A P) + %<- (insertAfter DiR (# A) Di' Di) + %<- (insertAfter DiR del Do' Do) + %<- (appr Ol Om Oo') + %<- (appr Oo' Or Oo) %. +%term choice_G + %pi (sqnt Di Do Oi Oo (^ P)) + %<- (prog A) + %<- (ie_sqnt2 Di Do Oi Ol Om Or A P) + %<- (appr Ol Om Oo') + %<- (appr Oo' Or Oo) %. +%term forall_L1 {t trm Srt} + %pi (ie_sqnt1 Di Do OiL OoL OiR OoR (forall Srt Frm) P) + %<- (ie_sqnt1 Di Do OiL OoL OiR OoR (Frm t) P) %. +%term forall_L2 {t trm Srt} + %pi (ie_sqnt2 Di Do Oi Ol Om Or (forall Srt Frm) P) + %<- (ie_sqnt2 Di Do Oi Ol Om Or (Frm t) P) %. +%term forall2_L1 {t %pi (trm Srt1) %-> (trm Srt2)} + %pi (ie_sqnt1 Di Do OiL OoL OiR OoR (forall2 Srt1 Srt2 Frm) P) + %<- (ie_sqnt1 Di Do OiL OoL OiR OoR (Frm t) P) %. +%term forall2_L2 {t %pi (trm Srt1) %-> (trm Srt2)} + %pi (ie_sqnt2 Di Do Oi Ol Om Or (forall2 Srt1 Srt2 Frm) P) + %<- (ie_sqnt2 Di Do Oi Ol Om Or (Frm t) P) %. +%term =>_L1 + %pi (ie_sqnt1 Di Do OiL OoL OiR OoR (A => B) P) + %<- (ie_sqnt1 Di Do OiL OoL OiR OoR B P) + %<- (sqnt rnil rnil rnil rnil A) %. +%term =0_L1 + %pi (ie_sqnt1 Di Do OiL OoL OiR OoR (A =0 B) P) + %<- (ie_sqnt1 Di Dm OiL OoL OiR OoR B P) + %<- (sqnt Dm Do rnil rnil A) %. +%term >=>_L1 + %pi (ie_sqnt1 Di Do OiL OoL OiR OoR (A >=> B) P) + %<- (ie_sqnt1 Di Dm OiL OoL' OiR OoR B P) + %<- (splitr OoL' OoLL OiA) + %<- (sqnt Dm Do OiA OoA A) + %<- (appr OoLL OoA OoL) %. +%term =>>_L1 + %pi (ie_sqnt1 Di Do OiL OoL OiR OoR (A =>> B) P) + %<- (ie_sqnt1 Di Dm OiL OoL OiR OoR' B P) + %<- (r2l OoR' OoRl) + %<- (splitl OoRl OiAl OoRl') + %<- (l2r OoRl' OoRL) + %<- (l2r OiAl OiA) + %<- (sqnt Dm Do OiA OoA A) + %<- (appr OoA OoRL OoR) %. +%term =>_L2 + %pi (ie_sqnt2 Di Do Oi Ol Om Or (A => B) P) + %<- (ie_sqnt2 Di Do Oi Ol Om Or B P) + %<- (sqnt rnil rnil rnil rnil A) %. +%term =0_L2 + %pi (ie_sqnt2 Di Do Oi Ol Om Or (A =0 B) P) + %<- (ie_sqnt2 Di Dm Oi Ol Om Or B P) + %<- (sqnt Dm Do rnil rnil A) %. +%term >=>_L2 + %pi (ie_sqnt2 Di Do Oi Ol OmR Or (A >=> B) P) + %<- (ie_sqnt2 Di Dm Oi Ol' Om Or B P) + %<- (sqnt Dm Do Om Om' A) + %<- (splitr Om' OmL OmR) + %<- (appr Ol' OmL Ol) %. +%term =>>_L2 + %pi (ie_sqnt2 Di Do Oi Ol OmL Or (A =>> B) P) + %<- (ie_sqnt2 Di Dm Oi Ol Om Or' B P) + %<- (sqnt Dm Do Om Om' A) + %<- (r2l Om' Om'l) + %<- (splitl Om'l OmLl OmRl) + %<- (l2r OmLl OmL) + %<- (l2r OmRl OmR) + %<- (appr OmR Or' Or) %. +%term init_1 ie_sqnt1 Di Di OiL OiL OiR OiR (^ P) P %. +%term init_2 ie_sqnt2 Di Di Oi rnil Oi rnil (^ P) P %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/incll/lists-incll-quy.lf b/new-tests/stelf-output/examples/incll/lists-incll-quy.lf new file mode 100644 index 0000000..2a3ff37 --- /dev/null +++ b/new-tests/stelf-output/examples/incll/lists-incll-quy.lf @@ -0,0 +1,5 @@ +%query 24 _ _ sqnt rnil rnil rnil rnil (^ (perm l4 L)) %. +%query 6 _ _ sqnt rnil rnil rnil rnil (^ (perm l3 L)) %. +%query 1 _ _ sqnt rnil rnil rnil rnil (^ (rev l4 L)) %. +%query 1 _ _ sqnt rnil rnil rnil rnil (^ (id l4 L)) %. +%query 1 _ _ sqnt rnil rnil rnil rnil (^ (idl l4 L)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/incll/lists-incll.lf b/new-tests/stelf-output/examples/incll/lists-incll.lf new file mode 100644 index 0000000..6e34630 --- /dev/null +++ b/new-tests/stelf-output/examples/incll/lists-incll.lf @@ -0,0 +1,29 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% + +% Sample queries from notes +% + +%define l4 1 | 2 | 3 | 4 | nil %. +% cons 1 (cons 2 (cons 3 (cons 4 nil))). +%define l3 1 | 2 | 3 | nil %. +% cons 1 (cons 2 (cons 3 nil)). +%define l2 1 | 2 | nil %. +% cons 1 (cons 2 nil). +%term perm %pi (trm (list A)) %-> (trm (list A)) %-> atm %. +%term rev %pi (trm (list A)) %-> (trm (list A)) %-> atm %. +%term id %pi (trm (list A)) %-> (trm (list A)) %-> atm %. +%term idl %pi (trm (list A)) %-> (trm (list A)) %-> atm %. +%term elem %pi (trm int) %-> atm %. +%term perm1 prog (^ (elem X) =0 ^ (perm L K) =0 ^ (perm (X | L) K)) %. +%term perm2 prog (^ (perm nil K) =0 ^ (elem X) =0 ^ (perm nil (X | K))) %. +%term perm3 prog (^ (perm nil nil)) %. +%term rev1 prog (^ (elem X) =>> ^ (rev L K) =>> ^ (rev (X | L) K)) %. +%term rev2 prog (^ (rev nil K) =>> ^ (elem X) =>> ^ (rev nil (X | K))) %. +%term rev3 prog (^ (rev nil nil)) %. +%term id1 prog (^ (elem X) >=> ^ (id L K) =>> ^ (id (X | L) K)) %. +%term id2 prog (^ (id nil K) =>> ^ (elem X) =>> ^ (id nil (X | K))) %. +%term id3 prog (^ (id nil nil)) %. +%term idl1 prog (^ (elem X) >=> ^ (idl L K) >=> ^ (idl (X | L) K)) %. +%term idl2 prog (^ (idl nil K) =>> ^ (elem X) =>> ^ (idl nil (X | K))) %. +%term idl3 prog (^ (idl nil nil)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/incll/lists.lf b/new-tests/stelf-output/examples/incll/lists.lf new file mode 100644 index 0000000..c010ca8 --- /dev/null +++ b/new-tests/stelf-output/examples/incll/lists.lf @@ -0,0 +1,90 @@ +% + +% Twelf code to manipulate lists +% + +%sort elmt %. +%term del elmt %. +%term # %pi frm %-> elmt %. +%prec %prefix 40 # %. +%sort rctx %. +%term rnil rctx %. +%term ,r %pi rctx %-> elmt %-> rctx %. +%prec %left 15 ,r %. +% + +% append 2 ctx's +% + +%sort appr %. +%term appr_0 appr R rnil R %. +%term appr_1 %pi (appr R (T ,r H) (T' ,r H)) %<- (appr R T T') %. +%sort lctx %. +%term lnil lctx %. +%term ,l %pi elmt %-> lctx %-> lctx %. +%prec %right 15 ,l %. +% + +% append 2 lctx's +% + +%sort appl %. +%term appl_0 appl lnil L L %. +%term appl_1 %pi (appl (H ,l T) L (H ,l T')) %<- (appl T L T') %. +% + +% convert a lctx to a rctx. +% + +%sort l2r %. +%term l2r_0 l2r lnil rnil %. +%term l2r_1 %pi (l2r (H ,l T) T'') %<- (l2r T T') %<- (appr (rnil ,r H) T' T'') %. +% + +% convert a rctx to a lctx +% + +%sort r2l %. +%term r2l_0 r2l rnil lnil %. +%term r2l_1 %pi (r2l (T ,r H) T'') %<- (r2l T T') %<- (appl T' (H ,l lnil) T'') %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% + +% insertAfter L1 F Lold Lnew ==> +% Lnew = Lold with F inserted |L1| elements from right. +% + +% insertAfter (rnil ,r # a ,r # a) F (rnil ,r # b ,r # c ,r # d ,r # e) Lnew ==> +% Lnew = (rnil ,r # b ,r # c ,r F ,r # d ,r # e) +% + +%sort insertAfter %. +%term insertAfter_0 insertAfter rnil E L (L ,r E) %. +%term insertAfter_N %pi (insertAfter (I ,r _) E (L ,r H) (L' ,r H)) %<- (insertAfter I E L L') %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% + +%sort choose %. +%term choose_0 choose (T ,r # A) T A rnil %. +%term choose_1 %pi (choose (T ,r E) Tl A (Tr ,r E)) %<- (choose T Tl A Tr) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% + +% splitr (rnil ,r # a ,r #(a =>> a) ,r del ,r # a) Dl Dr ==> +% Dl = (rnil ,r # a ,r #(a =>> a) ,r del) +% Dr = (rnil ,r # a) +% + +% splitl (# a ,l #(a =>> a) ,l del ,l # a ,l lnil) Dl Dr ==> +% Dl = (# a ,l #(a =>> a) ,l del ,l lnil) +% Dr = (# a ,l lnil) +% + +%sort splitr %. +%term spitr_rnil splitr rnil rnil rnil %. +%term splitr_del splitr (T ,r del) (T ,r del) rnil %. +%term splitr_# %pi (splitr (T ,r # A) Tl (Tr ,r # A)) %<- (splitr T Tl Tr) %. +%sort splitl %. +%term spitl_lnil splitl lnil lnil lnil %. +%term splitl_del splitl (del ,l T) lnil (del ,l T) %. +%term splitl_# %pi (splitl (# A ,l T) (# A ,l Tl) Tr) %<- (splitl T Tl Tr) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/incll/parse-incll-quy.lf b/new-tests/stelf-output/examples/incll/parse-incll-quy.lf new file mode 100644 index 0000000..5d10116 --- /dev/null +++ b/new-tests/stelf-output/examples/incll/parse-incll-quy.lf @@ -0,0 +1,14 @@ +% + +% All of these run correctly (with both versions of grammar). +% I have commented out some queries because they take some time. +% + +% %query 1 * (sqnt rnil rnil rnil rnil (^ bob =>> ^ loves =>> ^ mary =>> ^ snt)). +% %query 1 * (sqnt rnil rnil rnil rnil (^ mary =>> ^ loves =>> ^ the =>> ^ man =>> ^ whom =>> ^ jill =>> ^ married =>> ^ snt)). +% %query 1 * (sqnt rnil rnil rnil rnil (^ mary =>> ^ loves =>> ^ the =>> ^ man =>> ^ whom =>> ^ the =>> ^ woman =>> ^ whom =>> ^ jill =>> ^ loves =>> ^ married =>> ^ snt)). +%query 1 1 _ sqnt rnil rnil rnil rnil (^ mary =>> ^ loves =>> ^ the =>> ^ man =>> ^ whom =>> ^ the =>> ^ woman =>> ^ whom =>> ^ jill =>> ^ loves =>> ^ married =>> ^ snt) %. +% %query 0 * (sqnt rnil rnil rnil rnil (^ mary =>> ^ loves =>> ^ the =>> ^ man =>> ^ whom =>> ^ the =>> ^ woman =>> ^ whom =>> ^ loves =>> ^ jill =>> ^ married =>> ^ snt)). +% %query 0 * (sqnt rnil rnil rnil rnil (^ mary =>> ^ loves =>> ^ the =>> ^ man =>> ^ whom =>> ^ jill =>> ^ married =>> ^ mary =>> ^ snt)). +% %query 0 * (sqnt rnil rnil rnil rnil (^ mary =>> ^ loves =>> ^ the =>> ^ man =>> ^ whom =>> ^ married =>> ^ snt)). +% %query 0 * (sqnt rnil rnil rnil rnil (^ mary =>> ^ loves =>> ^ the =>> ^ man =>> ^ whom =>> ^ married =>> ^ jill =>> ^ snt)). \ No newline at end of file diff --git a/new-tests/stelf-output/examples/incll/parse-incll.lf b/new-tests/stelf-output/examples/incll/parse-incll.lf new file mode 100644 index 0000000..6e679ca --- /dev/null +++ b/new-tests/stelf-output/examples/incll/parse-incll.lf @@ -0,0 +1,84 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% + +% parsing examples +% + +% the logic program is the grammar. +% the ordered context is the input to be parsed. +% + +% + +% + +% small_parser.ll example from Lolli distribution +% + +% + +%term snt atm %. +%term snt1 atm %. +%term np atm %. +%term vp atm %. +%term tv atm %. +%term stv atm %. +%term sbar atm %. +%term det atm %. +%term pn atm %. +%term n atm %. +%term rel atm %. +%term whom atm %. +%term the atm %. +%term that atm %. +%term man atm %. +%term woman atm %. +%term girl atm %. +%term boy atm %. +%term mary atm %. +%term bob atm %. +%term jill atm %. +%term loves atm %. +%term married atm %. +%term believes atm %. +% + +% the parse grammar +% note that unlike in Lolli (or lambda-Prolog) everything can +% remain propositional. +% + +%term snt_0 prog (^ np =>> ^ vp =>> ^ snt) %. +%term snt1_0 prog (^ np =>> (^ np =0 ^ vp) =>> ^ snt1) %. +%term vp_0 prog (^ tv =>> ^ np =>> ^ vp) %. +%term vp_1 prog (^ stv =>> ^ sbar =>> ^ vp) %. +% +% note: (^ that) is before (^ snt) in the context +% done in this manner to prevent looping +% + +%term sbar_0 prog (^ snt =>> ^ that >=> ^ sbar) %. +%term np_0 prog (^ det =>> ^ n =>> ^ np) %. +%term np_1 prog (^ det =>> ^ n =>> ^ rel =>> ^ np) %. +%term np_2 prog (^ pn =>> ^ np) %. +% + +% Adding a ^ np at end of context, insures that gap +% is at end of phrase so no need for quantifiers. +% Also (^ whom) is before (^ np =>> ^ snt) +% + +% rel_0 : prog( (^ np =>> ^ snt) =>> ^ whom >=> ^ rel). +% rel_1 : prog( ((^ np =>> (^ np =0 ^ vp) =>> ^ snt) =0 ^ snt) =>> ^ whom >=> ^ rel). +%term rel_0 prog (^ snt1 =>> ^ whom >=> ^ rel) %. +%term det_0 prog (^ the =>> ^ det) %. +%term n_man prog (^ man =>> ^ n) %. +%term n_woman prog (^ woman =>> ^ n) %. +%term n_girl prog (^ girl =>> ^ n) %. +%term n_boy prog (^ boy =>> ^ n) %. +%term pn_mary prog (^ mary =>> ^ pn) %. +%term pn_bob prog (^ bob =>> ^ pn) %. +%term pn_jill prog (^ jill =>> ^ pn) %. +%term tv_loves prog (^ loves =>> ^ tv) %. +%term tv_married prog (^ married =>> ^ tv) %. +%term tv_believes prog (^ believes =>> ^ stv) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/incll/parse-lolli-quy.lf b/new-tests/stelf-output/examples/incll/parse-lolli-quy.lf new file mode 100644 index 0000000..1c474c0 --- /dev/null +++ b/new-tests/stelf-output/examples/incll/parse-lolli-quy.lf @@ -0,0 +1,7 @@ +% %query 1 * (sqnt rnil rnil rnil rnil +% (^ (snt (bob | loves | mary | nil) nil T)) ). +%query 1 _ _ sqnt rnil rnil rnil rnil (^ (snt (mary | loves | the | man | whom | jill | married | nil) nil T)) %. +%query 1 _ _ sqnt rnil rnil rnil rnil (^ (snt (mary | loves | the | man | whom | the | woman | whom | jill | loves | married | nil) nil T)) %. +%query 0 _ _ sqnt rnil rnil rnil rnil (^ (snt (mary | loves | the | man | whom | the | woman | whom | loves | jill | married | nil) nil T)) %. +%query 0 _ _ sqnt rnil rnil rnil rnil (^ (snt (mary | loves | the | man | whom | jill | married | mary | nil) nil T)) %. +%query 0 _ _ sqnt rnil rnil rnil rnil (^ (snt (mary | loves | the | man | whom | married | nil) nil T)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/incll/parse-lolli.lf b/new-tests/stelf-output/examples/incll/parse-lolli.lf new file mode 100644 index 0000000..29d7de4 --- /dev/null +++ b/new-tests/stelf-output/examples/incll/parse-lolli.lf @@ -0,0 +1,99 @@ +% + +% small parser from lolli distribution in same style +% + +%term wd sort %. +%term that trm wd %. +%term whom trm wd %. +%term the trm wd %. +%term man trm wd %. +%term woman trm wd %. +%term girl trm wd %. +%term boy trm wd %. +%term mary trm wd %. +%term bob trm wd %. +%term jill trm wd %. +%term loves trm wd %. +%term married trm wd %. +%term believes trm wd %. +%term prs sort %. +%term gap trm prs %. +%term that' trm prs %. +%term whom' trm prs %. +%term the' trm prs %. +%term man' trm prs %. +%term woman' trm prs %. +%term girl' trm prs %. +%term boy' trm prs %. +%term mary' trm prs %. +%term bob' trm prs %. +%term jill' trm prs %. +%term loves' trm prs %. +%term married' trm prs %. +%term believes' trm prs %. +%term snt' %pi (trm prs) %-> (trm prs) %-> (trm prs) %. +%term vp' %pi (trm prs) %-> (trm prs) %-> (trm prs) %. +%term np' %pi (trm prs) %-> (trm prs) %. +%term npD' %pi (trm prs) %-> (trm prs) %-> (trm prs) %. +%term npR' %pi (trm prs) %-> (trm prs) %-> (trm prs) %-> (trm prs) %. +%term stv' %pi (trm prs) %-> (trm prs) %-> (trm prs) %. +%term sbar' %pi (trm prs) %-> (trm prs) %-> (trm prs) %. +%term det' %pi (trm prs) %-> (trm prs) %. +%term rel' %pi (trm prs) %-> (trm prs) %-> (trm prs) %. +%term n' %pi (trm prs) %-> (trm prs) %. +%term pn' %pi (trm prs) %-> (trm prs) %. +%term tv' %pi (trm prs) %-> (trm prs) %. +%term stv' %pi (trm prs) %-> (trm prs) %. +%term snt %pi (trm (list wd)) %-> (trm (list wd)) %-> (trm prs) %-> atm %. +%term vp %pi (trm (list wd)) %-> (trm (list wd)) %-> (trm prs) %-> atm %. +%term np %pi (trm (list wd)) %-> (trm (list wd)) %-> (trm prs) %-> atm %. +%term stv %pi (trm (list wd)) %-> (trm (list wd)) %-> (trm prs) %-> atm %. +%term sbar %pi (trm (list wd)) %-> (trm (list wd)) %-> (trm prs) %-> atm %. +%term n %pi (trm (list wd)) %-> (trm (list wd)) %-> (trm prs) %-> atm %. +%term det %pi (trm (list wd)) %-> (trm (list wd)) %-> (trm prs) %-> atm %. +%term rel %pi (trm (list wd)) %-> (trm (list wd)) %-> (trm prs) %-> atm %. +%term pn %pi (trm (list wd)) %-> (trm (list wd)) %-> (trm prs) %-> atm %. +%term tv %pi (trm (list wd)) %-> (trm (list wd)) %-> (trm prs) %-> atm %. +%term stv %pi (trm (list wd)) %-> (trm (list wd)) %-> (trm prs) %-> atm %. +% % grammar rules +% sent P1 P2 (sent N V) :- {np P1 P0 N}, vp P0 P2 V. +%term snt_0 prog (^ (snt P1 P2 (snt' N V)) <= ^ (np P1 P0 N) <<= ^ (vp P0 P2 V)) %. +% vp P1 P2 (vp T N) :- tv P1 P0 T, np P0 P2 N. +%term vp_0 prog (^ (vp P1 P2 (vp' T N)) <<= ^ (tv P1 P0 T) <<= ^ (np P0 P2 N)) %. +% vp P1 P2 (vp Stv Sbar) :- stv P1 P0 Stv, sbar P0 P2 Sbar. +%term vp_1 prog (^ (vp P1 P2 (vp' T N)) <<= ^ (stv P1 P0 T) <<= ^ (sbar P0 P2 N)) %. +% sbar (that::P1) P2 (sbar that S) :- sent P1 P2 S. +%term sbar_0 prog (^ (sbar (that | P1) P2 (sbar' that' S)) <<= ^ (snt P1 P2 S)) %. +% np P1 P2 (np D N) :- det P1 P0 D, n P0 P2 N. +%term np_0 prog (^ (np P1 P2 (npD' D N)) <<= ^ (det P1 P0 D) <<= ^ (n P0 P2 N)) %. +% np P1 P2 (np D N R) :- det P1 P0 D, n P0 P3 N, rel P3 P2 R. +%term np_1 prog (^ (np P1 P2 (npR' D N R)) <<= ^ (det P1 P0 D) <<= ^ (n P0 P3 N) <<= ^ (rel P3 P2 R)) %. +% np P1 P2 (np P) :- pn P1 P2 P. +%term np_2 prog (^ (np P1 P2 (np' P)) <<= ^ (pn P1 P2 P)) %. +% rel (whom::P1) P2 (rel whom S) :- +% (forall Z\ np Z Z (np gap)) -o sent P1 P2 S. +%term rel_0 prog (^ (rel (whom | P1) P2 (rel' whom' S)) <<= (forall (list wd) ([Z] ^ (np Z Z (np' gap))) =0 ^ (snt P1 P2 S))) %. +% % lexical items +% det (the::L) L (det the). +%term det_the prog (^ (det (the | L) L (det' the'))) %. +% n (man::L) L (n man). +%term n_man prog (^ (n (man | L) L (n' man'))) %. +% n (woman::L) L (n woman). +%term n_woman prog (^ (n (woman | L) L (n' woman'))) %. +% n (girl::L) L (n girl). +%term n_girl prog (^ (n (girl | L) L (n' girl'))) %. +% n (boy::L) L (n boy). +%term n_boy prog (^ (n (boy | L) L (n' boy'))) %. +% pn (mary::L) L (pn mary). +%term pn_mary prog (^ (pn (mary | L) L (pn' mary'))) %. +% pn (bob::L) L (pn bob). +%term pn_bob prog (^ (pn (bob | L) L (pn' bob'))) %. +% pn (jill::L) L (pn jill). +%term pn_jill prog (^ (pn (jill | L) L (pn' jill'))) %. +% tv (loves::L) L (tv loves). +%term tv_loves prog (^ (tv (loves | L) L (tv' loves'))) %. +% tv (married::L) L (tv married). +%term tv_married prog (^ (tv (married | L) L (tv' married'))) %. +% stv (believes::L) L (stv believes). +%term tv_believes prog (^ (stv (believes | L) L (stv' believes'))) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/incll/pre-incll-quy.lf b/new-tests/stelf-output/examples/incll/pre-incll-quy.lf new file mode 100644 index 0000000..3d3a7ec --- /dev/null +++ b/new-tests/stelf-output/examples/incll/pre-incll-quy.lf @@ -0,0 +1,11 @@ +% + +% 3 + 2 * 1 +% + +%define o1i rnil ,r # ^ fun end ,r # ^ con 1 ,r # ^ fun * ,r # ^ con 2 ,r # ^ fun + ,r # ^ con 3 %. +%define o1o rnil ,r del ,r del ,r del ,r del ,r del ,r del %. +%define o2 rnil ,r # ^ fun end ,r # ^ con 1 %. +%query 1 _ _ sqnt rnil rnil o2 OO (^ (evc end 1)) %. +%query 1 _ _ sqnt rnil rnil o2 OO (^ (evc end A)) %. +%query 1 _ _ sqnt rnil rnil o1i o1o (^ (evc end A)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/incll/pre-incll.lf b/new-tests/stelf-output/examples/incll/pre-incll.lf new file mode 100644 index 0000000..44ca03b --- /dev/null +++ b/new-tests/stelf-output/examples/incll/pre-incll.lf @@ -0,0 +1,32 @@ +% + +% operator precedence parsing +% + +%term int sort %. +% z : trm int. +% s : trm int -> trm int. %prefix 10 s. +%term 1 trm int %. +%term 2 trm int %. +%term 3 trm int %. +%term con %pi (trm int) %-> atm %. +%define op arrow int (arrow int int) %. +%term + trm op %. +%term * trm op %. +%term end trm op %. +%term fun %pi (trm op) %-> atm %. +%term gt %pi (trm op) %-> (trm op) %-> atm %. +%term lt %pi (trm op) %-> (trm op) %-> atm %. +%term evc %pi (trm op) %-> (trm int) %-> atm %. +%term evo %pi (trm op) %-> (trm int) %-> atm %. +%term red %pi (trm int) %-> atm %. +%term gt_1 prog (^ gt * _) %. +%term lt_0 prog (^ lt end *) %. +%term lt_2 prog (^ lt end +) %. +%term lt_1 prog (^ lt + *) %. +%term ev_0 prog (^ evc end A <<= ^ con A <<= ^ fun end) %. +%term ev_1 prog (^ evc Op A <<= ^ con C <<= (^ con C >=> ^ evo Op A)) %. +%term ev_2 prog (^ evo Op (app (app Op X) Y) <<= ^ fun end <<= ^ con X <<= ^ fun Op <<= ^ con Y) %. +%term ev_3 prog (^ evo Op A <<= ^ fun Op' <= ^ gt Op Op' <<= (^ fun Op' =>> ^ red A)) %. +%term ev_4 prog (^ evo Op A <<= ^ fun Op' <= ^ lt Op Op' <<= (^ fun Op' >=> ^ evc Op' A)) %. +%term ev_5 prog (^ red A <=< ^ con X <=< ^ fun F <=< ^ con Y <=< ^ fun Op <=< (^ fun Op >=> ^ con (app (app F X) Y) >=> ^ evo Op A)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/incll/search-incll-quy.lf b/new-tests/stelf-output/examples/incll/search-incll-quy.lf new file mode 100644 index 0000000..32dc18f --- /dev/null +++ b/new-tests/stelf-output/examples/incll/search-incll-quy.lf @@ -0,0 +1,7 @@ +%query 1 1 _ sqnt rnil rnil rnil rnil (^ (bfs_main g2 1 R)) %. +%query 1 1 _ sqnt rnil rnil rnil rnil (^ (dfs_main g2 1 R)) %. +% %query 1 * (sqnt rnil rnil rnil rnil (^ (bfs_main g2 1 R))). +% %query 1 * (sqnt rnil rnil rnil rnil (^ (dfs_main g2 1 R))). +%% next two queries take a while to confirm no more than 1 answer. +%%query 1 * (sqnt rnil rnil rnil rnil (^ (bfs_main g1 1 R))). +%%query 1 * (sqnt rnil rnil rnil rnil (^ (dfs_main g1 1 R))). \ No newline at end of file diff --git a/new-tests/stelf-output/examples/incll/search-incll.lf b/new-tests/stelf-output/examples/incll/search-incll.lf new file mode 100644 index 0000000..7ee6fca --- /dev/null +++ b/new-tests/stelf-output/examples/incll/search-incll.lf @@ -0,0 +1,66 @@ +% + +% a breadth first search-- takes a graph and returns the list of +% nodes with their bfs numbering +% + +%term pint sort %. +%term z trm pint %. +%term s %pi (trm pint) %-> (trm pint) %. +%term omega trm pint %. +%define gtype list (cross int (list int)) %. +%define rtype list (cross pint int) %. +%term nodes %pi (trm gtype) %-> atm %. +%term node %pi (trm (cross int (list int))) %-> atm %. +%term used %pi (trm (cross pint int)) %-> atm %. +%term next %pi (trm (cross int pint)) %-> atm %. +%term finish %pi (trm gtype) %-> (trm rtype) %-> atm %. +%term bfs_main %pi (trm gtype) %-> (trm int) %-> (trm rtype) %-> atm %. +%term bfs %pi (trm gtype) %-> (trm int) %-> (trm rtype) %-> atm %. +%term bfs' %pi (trm int) %-> (trm pint) %-> (trm rtype) %-> atm %. +%term bfs'' %pi (trm (list int)) %-> (trm pint) %-> (trm rtype) %-> atm %. +%term dfs_main %pi (trm gtype) %-> (trm int) %-> (trm rtype) %-> atm %. +%term dfs %pi (trm gtype) %-> (trm int) %-> (trm rtype) %-> atm %. +%term dfs' %pi (trm int) %-> (trm pint) %-> (trm rtype) %-> atm %. +%term dfs'' %pi (trm (list int)) %-> (trm pint) %-> (trm rtype) %-> atm %. +%%% Packaging up the results %%% +%term finish1 prog (^ finish nil nil) %. +%term finish2 prog (^ finish (pair V E | G) (pair D V | R) 0= ^ used (pair D V) 0= ^ finish G R) %. +%term finish3 prog (^ finish (pair V E | G) (pair omega V | R) 0= ^ node (pair V E) 0= ^ finish G R) %. +%%% BFS %%% +%term bfs1 prog (^ bfs_main G S R <= (^ nodes G => ^ bfs G S R)) %. +%term bfs2 prog (^ bfs (N | G) S R 0= (^ node N =0 ^ bfs G S R)) %. +%term bfs3 prog (^ bfs nil S R 0= ^ bfs' S z R) %. +%term bfs4 prog (^ bfs' V D R 0= ^ node (pair V E) <<= (^ used (pair D V) =0 ^ bfs'' E D R)) %. +%term bfs5 prog (^ bfs' V D R 0= ^ used (pair D' V) <<= (^ used (pair D' V) =0 ^ bfs'' nil D R)) %. +%term bfs6 prog (^ bfs'' (E | Es) D R <<= (^ next (pair E D) >=> ^ bfs'' Es D R)) %. +%term bfs7 prog (^ bfs'' nil D R <<= ^ next (pair V D') <<= ^ bfs' V (s D') R) %. +%term bfs8 prog (^ bfs'' nil D R <= ^ nodes G 0= ^ finish G R) %. +%%% DFS ( same as above with the >=> replaced by =>>) +%term dfs1 prog (^ dfs_main G S R <= (^ nodes G => ^ dfs G S R)) %. +%term dfs2 prog (^ dfs (N | G) S R 0= (^ node N =0 ^ dfs G S R)) %. +%term dfs3 prog (^ dfs nil S R 0= ^ dfs' S z R) %. +%term dfs4 prog (^ dfs' V D R 0= ^ node (pair V E) <<= (^ used (pair D V) =0 ^ dfs'' E D R)) %. +%term dfs5 prog (^ dfs' V D R 0= ^ used (pair D' V) <<= (^ used (pair D' V) =0 ^ dfs'' nil D R)) %. +%term dfs6 prog (^ dfs'' (E | Es) D R <<= (^ next (pair E D) =>> ^ dfs'' Es D R)) %. +%term dfs7 prog (^ dfs'' nil D R <<= ^ next (pair V D') <<= ^ dfs' V (s D') R) %. +%term dfs8 prog (^ dfs'' nil D R <= ^ nodes G 0= ^ finish G R) %. +%%% queries %%% +%define edge1 2 | 3 | nil %. +% cons 2 (cons 3 nil). +%define edge2 1 | 4 | nil %. +% cons 1 (cons 4 nil). +%define edge3 1 | 4 | nil %. +% cons 1 (cons 4 nil). +%define edge4 2 | 3 | nil %. +% cons 2 (cons 3 nil). +% g1 = cons (pair 1 edge1) (cons (pair 2 edge2) (cons (pair 3 edge3) (cons (pair 4 edge4) nil))). +%define g1 pair 1 edge1 | (pair 2 edge2) | (pair 3 edge3) | (pair 4 edge4) | nil %. +%define e1 2 | 3 | nil %. +% cons 2 (cons 3 nil). +%define e2 1 | 3 | nil %. +% cons 1 (cons 3 nil). +%define e3 1 | 2 | nil %. +% cons 1 (cons 2 nil). +% g2 = cons (pair 1 e1) (cons (pair 2 e2) (cons (pair 3 e3) nil)). +%define g2 pair 1 e1 | (pair 2 e2) | (pair 3 e3) | nil %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/incll/sort-incll-quy.lf b/new-tests/stelf-output/examples/incll/sort-incll-quy.lf new file mode 100644 index 0000000..2b6e7d9 --- /dev/null +++ b/new-tests/stelf-output/examples/incll/sort-incll-quy.lf @@ -0,0 +1,7 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%% +% l1 = (cons (s s s z) (cons (s z) (cons (s s z) nil))). +% l2 = (cons (s s s z) (cons (s s z) (cons (s z) (cons (s s z) nil)))). +%define l1 s s s z | (s z) | (s s z) | nil %. +%define l2 s s s z | (s s z) | (s z) | (s s z) | nil %. +%query 1 _ _ sqnt rnil rnil rnil rnil (^ (mergeSort l1 L)) %. +%query 1 _ _ sqnt rnil rnil rnil rnil (^ (mergeSort l2 L)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/incll/sort-incll.lf b/new-tests/stelf-output/examples/incll/sort-incll.lf new file mode 100644 index 0000000..1b1fe44 --- /dev/null +++ b/new-tests/stelf-output/examples/incll/sort-incll.lf @@ -0,0 +1,29 @@ +% + +% a deterministic merge sort +% + +%term rInt sort %. +%term mergeSort %pi (trm (list rInt)) %-> (trm (list rInt)) %-> atm %. +%term msort %pi (trm (list rInt)) %-> atm %. +%term srt %pi (trm (list rInt)) %-> atm %. +%term merge %pi (trm (list rInt)) %-> (trm (list rInt)) %-> (trm (list rInt)) %-> atm %. +%term gt %pi (trm rInt) %-> (trm rInt) %-> atm %. +%term lte %pi (trm rInt) %-> (trm rInt) %-> atm %. +%term z trm rInt %. +%term s %pi (trm rInt) %-> (trm rInt) %. +%prec %prefix 10 s %. +%term gt1 prog (^ gt (s X) (s Y) <= ^ gt X Y) %. +%term gt2 prog (^ gt (s X) z) %. +%term lte1 prog (^ lte (s X) (s Y) <= ^ lte X Y) %. +%term lte2 prog (^ lte z X) %. +%term merge1 prog (^ merge (H1 | T1) (H2 | T2) (H2 | T3) <= ^ gt H1 H2 <= ^ merge (H1 | T1) T2 T3) %. +%term merge2 prog (^ merge (H1 | T1) (H2 | T2) (H1 | T3) <= ^ lte H1 H2 <= ^ merge T1 (H2 | T2) T3) %. +%term merge3 prog (^ merge L nil L) %. +%term merge4 prog (^ merge nil L L) %. +%term msort1 prog (^ mergeSort (H | T) L <<= (^ srt (H | nil) =>> ^ mergeSort T L)) %. +%term msort2 prog (^ mergeSort nil L <<= ^ msort L) %. +% replacing >=> by =>> in msort3 yields +% an insertion sort instead of a merge sort. +%term msort3 prog (^ (msort L) <<= ^ srt L1 <<= ^ srt L2 <= ^ merge L1 L2 L12 <=< (^ srt L12 >=> ^ msort L)) %. +%term msort4 prog (^ msort L <<= ^ srt L) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/incll/sources.lf b/new-tests/stelf-output/examples/incll/sources.lf new file mode 100644 index 0000000..d82a547 --- /dev/null +++ b/new-tests/stelf-output/examples/incll/sources.lf @@ -0,0 +1,10 @@ +%require %[ incll %] %. +%require %[ lists %] %. +%require %[ iosequents %] %. +%require %[ parse-lolli %] %. +%require %[ parse-incll %] %. +%require %[ search-incll %] %. +%require %[ lists-incll %] %. +%require %[ sort-incll %] %. +%require %[ cpm-incll %] %. +%require %[ bff-incll %] %. diff --git a/new-tests/stelf-output/examples/incll/stelf.toml b/new-tests/stelf-output/examples/incll/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/incll/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/incll/test.lf b/new-tests/stelf-output/examples/incll/test.lf new file mode 100644 index 0000000..ffff41b --- /dev/null +++ b/new-tests/stelf-output/examples/incll/test.lf @@ -0,0 +1,17 @@ +%require %[ incll %] %. +%require %[ lists %] %. +%require %[ iosequents %] %. +%require %[ parse-lolli %] %. +%require %[ parse-lolli %] %. +%require %[ parse-incll %] %. +%require %[ parse-incll %] %. +%require %[ search-incll %] %. +%require %[ search-incll %] %. +%require %[ lists-incll %] %. +%require %[ lists-incll %] %. +%require %[ sort-incll %] %. +%require %[ sort-incll %] %. +%require %[ cpm-incll %] %. +%require %[ cpm-incll %] %. +%require %[ bff-incll %] %. +%require %[ bff-incll %] %. diff --git a/new-tests/stelf-output/examples/js4/js4.lf b/new-tests/stelf-output/examples/js4/js4.lf new file mode 100644 index 0000000..40a3338 --- /dev/null +++ b/new-tests/stelf-output/examples/js4/js4.lf @@ -0,0 +1,96 @@ +%{ + Judgmental S4 + [A judgmental reconstruction of modal logic, F.Pfenning and R.Davies, + MSCS 11:511-540, 2001] + + Representation with intrinsic types, worlds, + but not a Kripke semantics + + Idea: Translate the judgment + u1::B1,...,un::Bn ; x1:A1,...,xm:Am |- J + as + u1:{W'}tm B1 W',...,un:{W'}tm Bn W', ---, + x1:tm A1 W,...,xm:tm Am W |- J* + + where "---" are assumptions y:tm B W' for W' <> W + and if J = (M : A) then J* = M* : tm A W + if J = (E - A) then J* = E* : exp A W +}% +%sort tp %. +%name tp %. +%term => %pi tp %-> tp %-> tp %. +%term box %pi tp %-> tp %. +%term dia %pi tp %-> tp %. +%prec %right 10 => %. +%sort world %. +%name world %. +%sort tm %. +%name tm %. +%sort exp %. +%name exp %. +%term lam %pi (%pi (tm A W) %-> (tm B W)) %-> (tm (A => B) W) %. +%term app %pi (tm (A => B) W) %-> (tm A W) %-> (tm B W) %. +%term boxi %pi ({w world} tm A w) %-> (tm (box A) W) %. +%term boxe %pi (tm (box A) W) %-> (%pi ({W' world} tm A W') %-> (tm C W)) %-> (tm C W) %. +%term t2e %pi (tm A W) %-> (exp A W) %. +%term diai %pi (exp A W) %-> (tm (dia A) W) %. +%term diae %pi (tm (dia A) W) %-> ({w world} %pi (tm A w) %-> (exp C w)) %-> (exp C W) %. +%term boxep %pi (tm (box A) W) %-> (%pi ({W' world} tm A W') %-> (exp C W)) %-> (exp C W) %. +%sort subdia %. +%mode subdia %in %in %out %. +%term sdt2e subdia (t2e M) ([w] [x] F w x) (F W M) %. +%term sddiae + %pi (subdia (diae M ([v] [y] E v y)) ([w] [x] F w x) (diae M ([v] [y] F' v y))) + %<- ({v world} {y tm B v} subdia (E v y) ([w] [x] F w x) (F' v y)) %. +%term sdboxep + %pi (subdia (boxep M ([u] E u)) ([w] [x] F w x) (boxep M ([u] F' u))) + %<- ({u {V world} tm B V} subdia (E u) ([w] [x] F w x) (F' u)) %. +%block by {B tp} [v world] [y tm B v]%. +%block bu {B tp} [u {V world} tm B V]%. +%worlds (by bu) (subdia E F F') %. +%total E (subdia E F _) %. +%{ This does not work, unfortunately +str : (tm A W -> {w:world} tm C w) -> ({w:world} tm C w) -> type. +stre : (tm A W -> {w:world} exp C w) -> ({w:world} exp C w) -> type. +strlam : str ([x][w] lam [y2] M1 x w y2) ([w] lam [y2] M1' w y2) + <- ({y2:tm C2 w} % problem here! + str ([x][w] y2) ([w] y2) + -> str ([x][w] M1 x w y2) ([w] M1' w y2)). +strapp : str ([x][w] app (M1 x w) (M2 x w)) ([w] app (M1' w) (M2' w)) + <- str ([x] [w] M1 x w) ([w] M1' w) + <- str ([x] [w] M2 x w) ([w] M2' w). +strboxi : str ([x][w] boxi [v] M1 x w v) ([w] boxi [v] M1' w v) + <- ({v:world} str ([x][w] M1 x w v) ([w] M1' w v)). +strboxe : str ([x][w] boxe (M1 x w) [u] M2 x w u) ([w] boxe (M1' w) [u] M2' w u) + <- str ([x][w] M1 x w) ([w] M1' w) + <- ({u:{W':world} tm A W'} + str ([x][w] u w) ([w] u w) + -> str ([x][w] M2 x w u) (M2' w u)). +... more case missing ... +}% +% Examples +%define _ (tm (box A => A) W) lam ([x] boxe x ([u] u W)) %. +%define _ (tm (box A => box (box A)) W) lam ([x] boxe x ([u] boxi ([w] boxi ([w'] u w')))) %. +%define _ (tm (box (A => B) => box A => box B) W) lam ([x] lam ([y] boxe x ([u] boxe y ([v] boxi ([w] app (u w) (v w)))))) %. +%define _ (tm (A => dia A) W) lam ([x] diai (t2e x)) %. +%define _ (tm (dia (dia A) => dia A) W) lam ([x] diai (diae x ([w] [y] diae y ([v] [z] t2e z)))) %. +%define _ (tm (box (A => B) => dia A => dia B) W) lam ([x] lam ([y] diai (boxep x ([u] diae y ([w] [z] t2e (app (u w) z)))))) %. +% Counterexamples, all must fail +%{ +% next two are characteristic +_ : tm (A => box A) W + = lam [x] boxi [w] x. +_ : tm (dia A => A) W + = lam [x] diae x [w][y] t2e y. +% next shows dia is not "normal" +_ : tm (dia (A => B) => dia A => dia B) W + = lam [x] lam [y] diai (diae x [w][x'] diae y [v][y'] t2e (app x' y')). +% next one is true is Kripke semantics a la Simpson +_ : tm ((dia A => box B) => box (A => B)) W + = lam [x] boxi [w] lam [y] boxe (app x (diai (t2e y))) [u] u w. +% next two are true in S5, not S4 +_ : tm (dia A => box (dia A)) W + = lam [x] diae x [w][x'] t2e (boxi [v] diai (t2e x')). +_ : tm (dia (box A) => box A) W + = lam [x] boxi [w] diae x [w'][x'] boxe x' [u] u w. +}% \ No newline at end of file diff --git a/new-tests/stelf-output/examples/js4/sources.lf b/new-tests/stelf-output/examples/js4/sources.lf new file mode 100644 index 0000000..a2df2f2 --- /dev/null +++ b/new-tests/stelf-output/examples/js4/sources.lf @@ -0,0 +1 @@ +%require %[ js4 %] %. diff --git a/new-tests/stelf-output/examples/js4/stelf.toml b/new-tests/stelf-output/examples/js4/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/js4/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/kolm/complete.lf b/new-tests/stelf-output/examples/kolm/complete.lf new file mode 100644 index 0000000..4346d80 --- /dev/null +++ b/new-tests/stelf-output/examples/kolm/complete.lf @@ -0,0 +1,103 @@ +% File: complete.elf +% For completeness, we need a lemma that every nj proof "is" an nk proof. +% Prover error: A proof could not be found +% %theorem nj_nk_auto : forall* {A:o} forall {NJ:nj A} exists {NK:nk A} true. +% %prove 2 NJ (nj_nk_auto NJ NK). +%sort nj_nk %. +%mode nj_nk %in %out %. +%term nj_nk_andi + %pi (nj_nk (nj_andi NJA NJB) (nk_andi NKA NKB)) + %<- (nj_nk NJA NKA) + %<- (nj_nk NJB NKB) %. +%term nj_nk_andel %pi (nj_nk (nj_andel NJ) (nk_andel NK)) %<- (nj_nk NJ NK) %. +%term nj_nk_ander %pi (nj_nk (nj_ander NJ) (nk_ander NK)) %<- (nj_nk NJ NK) %. +%term nj_nk_impi + %pi (nj_nk (nj_impi NJ) (nk_impi NK)) + %<- ({u} {v} %pi (nj_nk u v) %-> (nj_nk (NJ u) (NK v))) %. +%term nj_nk_impe + %pi (nj_nk (nj_impe NJI NJA) (nk_impe NKI NKA)) + %<- (nj_nk NJI NKI) + %<- (nj_nk NJA NKA) %. +%term nj_nk_oril %pi (nj_nk (nj_oril NJ) (nk_oril NK)) %<- (nj_nk NJ NK) %. +%term nj_nk_oril %pi (nj_nk (nj_orir NJ) (nk_orir NK)) %<- (nj_nk NJ NK) %. +%term nj_nk_ore + %pi (nj_nk (nj_ore NJ NJ1 NJ2) (nk_ore NK NK1 NK2)) + %<- (nj_nk NJ NK) + %<- ({u} {v} %pi (nj_nk u v) %-> (nj_nk (NJ1 u) (NK1 v))) + %<- ({u} {v} %pi (nj_nk u v) %-> (nj_nk (NJ2 u) (NK2 v))) %. +%term nj_nk_noti + %pi (nj_nk (nj_noti NJ) (nk_noti NK)) + %<- ({p} {u} {v} %pi (nj_nk u v) %-> (nj_nk (NJ p u) (NK p v))) %. +%term nj_nk_note + %pi (nj_nk (nj_note NJ1 C NJ2) (nk_note NK1 C NK2)) + %<- (nj_nk NJ1 NK1) + %<- (nj_nk NJ2 NK2) %. +%term nj_nk_truei nj_nk nj_truei nk_truei %. +%term nj_nk_falsee %pi (nj_nk (nj_falsee NJ) (nk_falsee NK)) %<- (nj_nk NJ NK) %. +%term nj_nk_foralli %pi (nj_nk (nj_foralli NJ) (nk_foralli NK)) %<- ({a} nj_nk (NJ a) (NK a)) %. +%term nj_nk_foralle %pi (nj_nk (nj_foralle NJ T) (nk_foralle NK T)) %<- (nj_nk NJ NK) %. +%term nj_nk_existsi %pi (nj_nk (nj_existsi T NJ) (nk_existsi T NK)) %<- (nj_nk NJ NK) %. +%term nj_nk_existse + %pi (nj_nk (nj_existse NJ1 NJ2) (nk_existse NK1 NK2)) + %<- (nj_nk NJ1 NK1) + %<- ({a} {u} {v} %pi (nj_nk u v) %-> (nj_nk (NJ2 a u) (NK2 a v))) %. +%block l5 {A o} [u nj A] [v nk A] [h nj_nk u v]%. +%block l6 [p o]%. +%worlds (l1 l5 l6) (nj_nk _ _) %. +% %mode (nj_nk +NJ -NK). +% total NJ (nj_nk NJ NK). +% total NJ (nj_nk NJ NK). +% Lemma (equiv): Given any K::kolm A A*, there are +% NK' : nk A -> nk A* and NK : nk A* -> nk A. +% (i.e., A and A* are classically equivalent!) +% Could not prove this automatically (I gave up) +% %theorem equiv_auto : forall* {A:o} {A*:o} forall {K:kolm A A*} +% exists {NK*:nk A -> nk A*} {NK:nk A* -> nk A} true. +% %prove 2 K (equiv_auto K NK* NK). +%sort equiv %. +%mode equiv %in %out %out %. +%term equiv_and + %pi (equiv (kolm_and KB KA) ([u nk (A and B)] nk_dnotx (nk_andi (NKA* (nk_andel u)) (NKB* (nk_ander u)))) ([u nk (n (A* and B*))] nk_andi (NKA (nk_andel (nk_dnotr u))) (NKB (nk_ander (nk_dnotr u))))) + %<- (equiv KA NKA* NKA) + %<- (equiv KB NKB* NKB) %. +% This case clearly demonstrates why we need to prove +% the existence of NK* and NK together. +%term equiv_imp + %pi (equiv (kolm_imp KB KA) ([v nk (A imp B)] nk_dnotx (nk_impi ([u nk A*] NKB* (nk_impe v (NKA u))))) ([v nk (n (A* imp B*))] nk_impi ([u nk A] NKB (nk_impe (nk_dnotr v) (NKA* u))))) + %<- (equiv KA NKA* NKA) + %<- (equiv KB NKB* NKB) %. +%term equiv_or + %pi (equiv (kolm_or KB KA) ([v nk (A or B)] nk_dnotx (nk_ore v ([u nk A] nk_oril (NKA* u)) ([u nk B] nk_orir (NKB* u)))) ([v nk (n (A* or B*))] nk_ore (nk_dnotr v) ([u nk A*] nk_oril (NKA u)) ([u nk B*] nk_orir (NKB u)))) + %<- (equiv KA NKA* NKA) + %<- (equiv KB NKB* NKB) %. +%term equiv_not + %pi (equiv (kolm_not K) ([v nk (not A)] nk_noti ([p] [u nk (not not A*)] nk_note v p (NKA (nk_dnotr u)))) ([v nk (n not A*)] nk_noti ([p] [u nk A] nk_note (nk_dnotr v) p (NKA* u)))) + %<- (equiv K NKA* NKA) %. +% An alternative formulation is +% equiv_true : equiv kolm_true ([v:nk true] (nk_dnotx nk_truei)) +% ([v:nk (n true)] nk_truei). +%term equiv_true equiv kolm_true ([v nk true] nk_dnotx v) ([v nk (n true)] nk_dnotr v) %. +%term equiv_false equiv kolm_false ([v nk false] nk_dnotx v) ([v nk (n false)] nk_dnotr v) %. +%term equiv_forall + %pi (equiv (kolm_forall K) ([v nk (forall A)] nk_dnotx (nk_foralli ([a i] NK* a (nk_foralle v a)))) ([v nk (n (forall A*))] nk_foralli ([a i] NK a (nk_foralle (nk_dnotr v) a)))) + %<- ({a} equiv (K a) (NK* a) (NK a)) %. +%term equiv_exists + %pi (equiv (kolm_exists K) ([v nk (exists A)] nk_dnotx (nk_existse v ([a] [u nk (A a)] nk_existsi a (NK* a u)))) ([v nk (n (exists A*))] nk_existse (nk_dnotr v) ([a] [u nk (A* a)] nk_existsi a (NK a u)))) + %<- ({a} equiv (K a) (NK* a) (NK a)) %. +%worlds (l1) (equiv _ _ _) %. +%terminates K (equiv K NK* NK) %. +% %mode (equiv +K -NK* -NK). +% total K (equiv K NK* NK). +% (Completeness) +% Given: NJ::nj (n A*) +% K::kolm A A* +% + +% Then we have +% NK::nk A. +%sort complete %. +% Completeness reduces to nj_nk and equiv. +%term complete1 %pi (complete K NJ (NK2 NK1*)) %<- (nj_nk NJ NK1*) %<- (equiv K NK2* NK2) %. +%mode complete %in %in %out %. +%worlds () (complete _ _ _) %. +% total [] (complete K NJ NK). \ No newline at end of file diff --git a/new-tests/stelf-output/examples/kolm/examples-quy.lf b/new-tests/stelf-output/examples/kolm/examples-quy.lf new file mode 100644 index 0000000..e58a39c --- /dev/null +++ b/new-tests/stelf-output/examples/kolm/examples-quy.lf @@ -0,0 +1,60 @@ +% Examples +% First a few examples that do not use double negation. +%query 1 _ _ %the (sound (nk_noti ([p o] [u nk false] nk_falsee u)) (kolm_not kolm_false) NJ) D %. +%query 1 _ _ %the (complete (kolm_not kolm_false) (nj_noti ([p o] [u nj (not not not not false)] nj_note u p (nj_noti ([p1 o] [X1 nj (not not false)] nj_note (nj_note X1 (not not false) (nj_noti ([p2 o] [u1 nj false] nj_falsee u1))) p1 (nj_noti ([p3 o] [u2 nj false] nj_falsee u2)))))) NK) D %. +%query 1 _ _ %the (sound (nk_impi ([u nk false] nk_falsee u)) (kolm_imp kolm_false kolm_false) NJ) D %. +%query 1 _ _ %the (complete (kolm_imp kolm_false kolm_false) (nj_noti ([p o] [u nj (not (not not false imp not not false))] nj_note u p (nj_impi ([X1 nj (not not false)] nj_note X1 (not not false) (nj_noti ([p1 o] [u1 nj false] nj_falsee u1)))))) NK) D %. +%query 1 _ _ %the (sound (nk_impi ([u nk false] nk_truei)) (kolm_imp kolm_true kolm_false) NJ) D %. +%query 1 _ _ %the (complete (kolm_imp kolm_true kolm_false) (nj_noti ([p o] [u nj (not (not not false imp not not true))] nj_note u p (nj_impi ([X1 nj (not not false)] nj_noti ([p1 o] [u1 nj (not true)] nj_note u1 p1 nj_truei))))) NK) D %. +%query 1 _ _ %the (sound (nk_impi ([u nk false] nk_falsee u)) (kolm_imp kolm_true kolm_false) NJ) D %. +%query 1 _ _ %the (complete (kolm_imp kolm_true kolm_false) (nj_noti ([p o] [u nj (not (not not false imp not not true))] nj_note u p (nj_impi ([X1 nj (not not false)] nj_noti ([p1 o] [u1 nj (not true)] nj_note u1 p1 nj_truei))))) NK) D %. +% A good first test case using dnot, noti, note, and truei. +% true true +% - - - - - ====> .... +% - - true .... +% ---------dnot ----------- +% true - - true +%query 1 _ _ %the (sound (nk_dnotr (nk_noti ([p o] [u nk (not true)] nk_note u p nk_truei))) kolm_true NJ) D %. +%query 1 _ _ %the (complete kolm_true (nj_noti ([q o] [u nj (not true)] nj_note (nj_noti ([q1 o] [u1 nj (not not not true)] nj_note (nj_noti ([q2 o] [u2 nj (not not not not not true)] nj_note (nj_noti ([p o] [u3 nj (not not not not not not not true)] nj_note u3 p (nj_noti ([p1 o] [X1 nj (not not not not not true)] nj_note (nj_note (nj_noti ([q3 o] [u4 nj (not not true)] nj_note X1 q3 (nj_noti ([p2 o] [v nj (not not not true)] nj_note v p2 u4)))) (not not false) (nj_noti ([p3 o] [u5 nj (not true)] nj_note u5 p3 nj_truei))) p1 (nj_noti ([p4 o] [u6 nj false] nj_falsee u6)))))) q2 (nj_noti ([p o] [v nj (not not not not not not true)] nj_note v p u2)))) q1 (nj_noti ([p o] [v nj (not not not not true)] nj_note v p u1)))) q (nj_noti ([p o] [v nj (not not true)] nj_note v p u)))) NK) D %. +% A big monster example using every propositional inference +% rule. Don't even try to read the corresponding NJ proof! +%query 1 _ _ %the (sound (nk_andi (nk_ore (nk_impe (nk_impi ([v2 nk true] nk_orir (nk_andi nk_truei v2))) (nk_dnotr (nk_noti ([p] [u2 nk (not true)] nk_note u2 p nk_truei)))) ([u nk (true or false and false)] nk_andel u) ([v nk (true and true)] nk_oril (nk_ander v))) (nk_noti ([p] [w nk false] nk_falsee w))) (kolm_and (kolm_not kolm_false) (kolm_or kolm_false kolm_true)) NJ) D %. +%query 1 _ _ %the (complete (kolm_and (kolm_not kolm_false) (kolm_or kolm_false kolm_true)) (nj_noti ([p o] [u nj (not (not not (not not true or not not false) and not not not not not false))] nj_note u p (nj_andi (nj_noti ([q o] [v nj (not (not not true or not not false))] nj_note (nj_noti ([q1 o] [v1 nj (not (not not (not not (not not true or not not false) and not not false) or not not (not not true and not not true)))] nj_note (nj_noti ([p1 o] [u1 nj (not (not not true imp not not (not not (not not (not not true or not not false) and not not false) or not not (not not true and not not true))))] nj_note u1 p1 (nj_impi ([X1 nj (not not true)] nj_noti ([p2 o] [u2 nj (not (not not (not not (not not true or not not false) and not not false) or not not (not not true and not not true)))] nj_note u2 p2 (nj_orir (nj_noti ([p3 o] [u3 nj (not (not not true and not not true))] nj_note u3 p3 (nj_andi (nj_noti ([p4 o] [u4 nj (not true)] nj_note u4 p4 nj_truei)) X1))))))))) q1 (nj_noti ([p5 o] [u5 nj (not not true imp not not (not not (not not (not not true or not not false) and not not false) or not not (not not true and not not true)))] nj_note (nj_impe u5 (nj_noti ([q2 o] [u6 nj (not true)] nj_note (nj_noti ([q3 o] [u7 nj (not not not true)] nj_note (nj_noti ([q4 o] [u8 nj (not not not not not true)] nj_note (nj_noti ([p6 o] [u9 nj (not not not not not not not true)] nj_note u9 p6 (nj_noti ([p7 o] [X2 nj (not not not not not true)] nj_note (nj_note (nj_noti ([q5 o] [u10 nj (not not true)] nj_note X2 q5 (nj_noti ([p8 o] [v2 nj (not not not true)] nj_note v2 p8 u10)))) (not not false) (nj_noti ([p9 o] [u11 nj (not true)] nj_note u11 p9 nj_truei))) p7 (nj_noti ([p10 o] [u12 nj false] nj_falsee u12)))))) q4 (nj_noti ([p11 o] [v3 nj (not not not not not not true)] nj_note v3 p11 u8)))) q3 (nj_noti ([p12 o] [v4 nj (not not not not true)] nj_note v4 p12 u7)))) q2 (nj_noti ([p13 o] [v5 nj (not not true)] nj_note v5 p13 u6))))) p5 v1)))) q (nj_noti ([p14 o] [u13 nj (not not (not not (not not true or not not false) and not not false) or not not (not not true and not not true))] nj_note (nj_ore u13 ([X3 nj (not not (not not (not not true or not not false) and not not false))] nj_noti ([q6 o] [u14 nj (not (not not true or not not false))] nj_note X3 q6 (nj_noti ([p15 o] [v6 nj (not not (not not true or not not false) and not not false)] nj_note (nj_andel v6) p15 u14)))) ([X4 nj (not not (not not true and not not true))] nj_noti ([p16 o] [u15 nj (not (not not true or not not false))] nj_note u15 p16 (nj_oril (nj_noti ([q7 o] [u16 nj (not true)] nj_note X4 q7 (nj_noti ([p17 o] [v7 nj (not not true and not not true)] nj_note (nj_ander v7) p17 u16)))))))) p14 v)))) (nj_noti ([p18 o] [u17 nj (not not not not false)] nj_note u17 p18 (nj_noti ([p19 o] [X5 nj (not not false)] nj_note (nj_note X5 (not not false) (nj_noti ([p20 o] [u18 nj false] nj_falsee u18))) p19 (nj_noti ([p21 o] [u19 nj false] nj_falsee u19))))))))) NK) D %. +% Oh, what about excluded middle itself! +%query 1 _ _ %the ({a o} {ka kolm a (not not a)} + %pi (existskolm a (not not a) ka) + %-> (sound (nk_dnotr (nk_noti ([p] [u nk (not (a or (not a)))] nk_note u p (nk_orir (nk_noti ([q] [v nk a] nk_note u q (nk_oril v))))))) (kolm_or (kolm_not ka) ka) (NJ a))) D %. +% check of completeness for the NJ proof returned by soundness +%query 1 _ _ %the ({a o} {ka kolm a (not not a)} + %pi (equiv ka ([u nk a] nk_dnotx u) ([u nk (not not a)] nk_dnotr u)) + %-> (complete (kolm_or (kolm_not ka) ka) (nj_triple_neg_red (nj_triple_neg_red (nj_triple_neg_red (nj_dnotx (nj_noti ([q o] [v nj (not not not not not (not not a or not not not not not a))] nj_dneg_falser (nj_note (nj_triple_neg_red v) (not not false) (nj_dnotx (nj_orir (nj_dnotx (nj_noti ([q1 o] [v1 nj (not not a)] nj_dneg_falser (nj_note (nj_triple_neg_red v) (not not false) (nj_dnotx (nj_oril v1))) q1)))))) q)))))) (NK a))) D %. +% Check of completeness for a more reasonable NJ proof. +%query 1 _ _ %the ({a o} {ka kolm a (not not a)} + %pi (equiv ka ([u nk a] nk_dnotx u) ([u nk (not not a)] nk_dnotr u)) + %-> (complete (kolm_or (kolm_not ka) ka) (nj_noti ([p] [u nj (not (not not a or (not not not not not a)))] nj_note u p (nj_orir (nj_dnotx (nj_noti ([q] [v nj (not not a)] nj_note u q (nj_oril v))))))) (NK a))) D %. +% How about the DeMorgan's Law Example which is an NK theorem, but not NJ. +% -(A and B) -> -A or -B. +%query 1 _ _ %the ({a o} {b o} {ka kolm a (n a)} {kb kolm b (n b)} + %pi (existskolm a (n a) ka) + %-> (existskolm b (n b) kb) + %-> (sound (nk_impi ([u nk (not (a and b))] nk_dnotr (nk_noti ([p] [v nk (not (not a or (not b)))] nk_note v p (nk_orir (nk_noti ([q] [w nk b] nk_note v q (nk_oril (nk_noti ([r] [z nk a] nk_note u r (nk_andi z w))))))))))) (kolm_imp (kolm_or (kolm_not kb) (kolm_not ka)) (kolm_not (kolm_and kb ka))) (NJ a b))) D %. +%query 1 _ _ %the ({a o} {b o} {ka kolm a (n a)} {kb kolm b (n b)} + %pi (equiv ka ([u nk a] nk_dnotx u) ([u nk (not not a)] nk_dnotr u)) + %-> (equiv kb ([u nk b] nk_dnotx u) ([u nk (not not b)] nk_dnotr u)) + %-> (complete (kolm_imp (kolm_or (kolm_not kb) (kolm_not ka)) (kolm_not (kolm_and kb ka))) (nj_dnotx (nj_impi ([X2 nj (n (not n (n a and n b)))] nj_triple_neg_red (nj_triple_neg_red (nj_triple_neg_red (nj_dnotx (nj_noti ([q o] [v nj (n (not not not (n (not n a) or n (not n b))))] nj_dneg_falser (nj_note (nj_triple_neg_red v) (n false) (nj_dnotx (nj_orir (nj_dnotx (nj_noti ([q1 o] [v1 nj (n b)] nj_dneg_falser (nj_note (nj_triple_neg_red v) (n false) (nj_dnotx (nj_oril (nj_dnotx (nj_noti ([q2 o] [v2 nj (n a)] nj_dneg_falser (nj_note (nj_triple_neg_red X2) (n false) (nj_dnotx (nj_andi v2 v1))) q2)))))) q1)))))) q)))))))) (NK a b))) D %. +% Now to test the quantifier rules. +% First a simple example +%query 1 _ _ %the ({R %pi i %-> i %-> o} {kr {x} {y} kolm (R x y) (not not (R x y))} + %pi ({x} {y} existskolm (R x y) (not not (R x y)) (kr x y)) + %-> (sound (nk_impi ([u nk (forall ([x] R x x))] nk_foralli ([a] nk_existsi a (nk_foralle u a)))) (kolm_imp (kolm_forall ([x] kolm_exists ([y] kr x y))) (kolm_forall ([x] kr x x))) (NJ R))) D %. +%query 1 _ _ %the ({R %pi i %-> i %-> o} {kr {x} {y} kolm (R x y) (not not (R x y))} + %pi ({x} {y} equiv (kr x y) ([u nk (R x y)] nk_dnotx u) ([u nk (not not (R x y))] nk_dnotr u)) + %-> (complete (kolm_imp (kolm_forall ([x] kolm_exists ([y] kr x y))) (kolm_forall ([x] kr x x))) (nj_noti ([p o] [u nj (not (not not forall ([x i] not not R x x) imp not not forall ([x i] not not exists ([y i] not not R x y))))] nj_note u p (nj_impi ([X2 nj (not not forall ([x i] not not R x x))] nj_noti ([p1 o] [u1 nj (not forall ([T1 i] not not exists ([y i] not not R T1 y)))] nj_note u1 p1 (nj_foralli ([a i] nj_noti ([p2 o] [u2 nj (not exists ([T2 i] not not R a T2))] nj_note u2 p2 (nj_existsi a (nj_noti ([q o] [v nj (not R a a)] nj_note X2 q (nj_noti ([p3 o] [u3 nj (forall ([x i] not not R x x))] nj_note (nj_foralle u3 a) p3 v))))))))))))) (NK R))) D %. +% Now the quantifier rule corresponding to DeMorgan's Law above. +% -(all x A) implies exists x -A. +%query 1 _ _ %the ({A %pi i %-> o} {kA {x i} kolm (A x) (n (A x))} + %pi ({x i} existskolm (A x) (n (A x)) (kA x)) + %-> (sound (nk_impi ([u nk (not (forall A))] nk_dnotr (nk_noti ([p] [v nk (not (exists ([x] not (A x))))] nk_note u p (nk_foralli ([a i] nk_dnotr (nk_noti ([q] [w nk (not (A a))] nk_note v q (nk_existsi a w))))))))) (kolm_imp (kolm_exists ([x] kolm_not (kA x))) (kolm_not (kolm_forall kA))) (NJ A))) D %. +%query 1 _ _ %the ({A %pi i %-> o} {kA {x i} kolm (A x) (n (A x))} + %pi ({x} equiv (kA x) ([u nk (A x)] nk_dnotx u) ([u nk (not not (A x))] nk_dnotr u)) + %-> (complete (kolm_imp (kolm_exists ([x] kolm_not (kA x))) (kolm_not (kolm_forall kA))) (nj_dnotx (nj_impi ([X2 nj (n (not n (forall ([T1 i] n (A T1)))))] nj_triple_neg_red (nj_triple_neg_red (nj_triple_neg_red (nj_dnotx (nj_noti ([q o] [v nj (n (not not not exists ([T2 i] n (not n (A T2)))))] nj_dneg_falser (nj_note (nj_triple_neg_red X2) (n false) (nj_dnotx (nj_foralli ([T3 i] nj_triple_neg_red (nj_triple_neg_red (nj_triple_neg_red (nj_dnotx (nj_noti ([q1 o] [v1 nj (n (not not not A T3))] nj_dneg_falser (nj_note (nj_triple_neg_red v) (n false) (nj_dnotx (nj_existsi T3 v1))) q1))))))))) q)))))))) (NK A))) D %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/kolm/fol.lf b/new-tests/stelf-output/examples/kolm/fol.lf new file mode 100644 index 0000000..893d0a0 --- /dev/null +++ b/new-tests/stelf-output/examples/kolm/fol.lf @@ -0,0 +1,18 @@ +% File: fol.elf +% First Order Logic Signature. +%sort i %. +%name i %. +%sort o %. +%name o %. +%term and %pi o %-> o %-> o %. +%prec %right 11 and %. +%term imp %pi o %-> o %-> o %. +%prec %right 10 imp %. +%term or %pi o %-> o %-> o %. +%prec %right 11 or %. +%term not %pi o %-> o %. +%prec %prefix 12 not %. +%term true o %. +%term false o %. +%term forall %pi (%pi i %-> o) %-> o %. +%term exists %pi (%pi i %-> o) %-> o %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/kolm/kolmtrans.lf b/new-tests/stelf-output/examples/kolm/kolmtrans.lf new file mode 100644 index 0000000..956f47c --- /dev/null +++ b/new-tests/stelf-output/examples/kolm/kolmtrans.lf @@ -0,0 +1,58 @@ +% File kolmtrans.elf +% Kolmogorov's double negation interpretation of classical logic +% in intuitionistic logic. +% +% Here I'm taking A to A* (it's Kolmogorov translation) +% By inversion we have A* is of the form (not not A+). +% First, define n as shorthand for double negation. +%define n [p o] not not p %. +%sort kolm %. +%mode kolm %in %out %. +%term kolm_and %pi (kolm (A and B) (n (A* and B*))) %<- (kolm A A*) %<- (kolm B B*) %. +%term kolm_imp %pi (kolm (A imp B) (n (A* imp B*))) %<- (kolm A A*) %<- (kolm B B*) %. +%term kolm_or %pi (kolm (A or B) (n (A* or B*))) %<- (kolm A A*) %<- (kolm B B*) %. +%term kolm_not %pi (kolm (not A) (n (not A*))) %<- (kolm A A*) %. +%term kolm_true kolm true (n true) %. +%term kolm_false kolm false (n false) %. +%term kolm_forall %pi (kolm (forall A) (n (forall A*))) %<- ({a i} kolm (A a) (A* a)) %. +%term kolm_exists %pi (kolm (exists A) (n (exists A*))) %<- ({a i} kolm (A a) (A* a)) %. +% A lemma saying explicitly that kolm terminates. +% I need this for soundness of elimination rules. +% I was not able to prove this automatically (I gave up). +% Note: The theorem prover will not split over an index variable, such as A. +% %theorem existskolm_auto : forall {A:o} exists {A*:o} {K:kolm A A*} true. +% %prove 2 A (existskolm_auto A A* K). +% Let me prove it automatically for the propositional case (using +% induction on the derivation of the judgement that A is propositional.) +%sort prop %. +%term and_prop %pi (prop A) %-> (prop B) %-> (prop (A and B)) %. +%term imp_prop %pi (prop A) %-> (prop B) %-> (prop (A imp B)) %. +%term or_prop %pi (prop A) %-> (prop B) %-> (prop (A or B)) %. +%term not_prop %pi (prop A) %-> (prop (not A)) %. +%term true_prop prop true %. +%term false_prop prop false %. +%theorem exkolm : forallG (pi {a i}) forall {A o} exists {A* o} {K kolm A A*} true %. +%prove 3 A (exkolm A A* K) %. +%sort existskolm {A o} {A* o} %. +%mode existskolm %in %out %out %. +%term existskolm_and + %pi (existskolm (A and B) (n (A* and B*)) (kolm_and KB KA)) + %<- (existskolm A A* KA) + %<- (existskolm B B* KB) %. +%term existskolm_imp + %pi (existskolm (A imp B) (n (A* imp B*)) (kolm_imp KB KA)) + %<- (existskolm A A* KA) + %<- (existskolm B B* KB) %. +%term existskolm_or + %pi (existskolm (A or B) (n (A* or B*)) (kolm_or KB KA)) + %<- (existskolm A A* KA) + %<- (existskolm B B* KB) %. +%term existskolm_not %pi (existskolm (not A) (n not A*) (kolm_not KA)) %<- (existskolm A A* KA) %. +%term existskolm_true existskolm true (n true) kolm_true %. +%term existskolm_false existskolm false (n false) kolm_false %. +%term existskolm_forall + %pi (existskolm (forall A) (n (forall A*)) (kolm_forall KA)) + %<- ({a i} existskolm (A a) (A* a) (KA a)) %. +%term existskolm_exists + %pi (existskolm (exists A) (n (exists A*)) (kolm_exists KA)) + %<- ({a i} existskolm (A a) (A* a) (KA a)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/kolm/nj.lf b/new-tests/stelf-output/examples/kolm/nj.lf new file mode 100644 index 0000000..0192d3e --- /dev/null +++ b/new-tests/stelf-output/examples/kolm/nj.lf @@ -0,0 +1,64 @@ +% File: nj.elf +% Intuitionistic, Natural Deduction +%sort nj %. +%term nj_andi %pi (nj A) %-> (nj B) %-> (nj (A and B)) %. +%term nj_andel %pi (nj (A and B)) %-> (nj A) %. +%term nj_ander %pi (nj (A and B)) %-> (nj B) %. +%term nj_impi %pi (%pi (nj A) %-> (nj B)) %-> (nj (A imp B)) %. +%term nj_impe %pi (nj (A imp B)) %-> (nj A) %-> (nj B) %. +%term nj_oril %pi (nj A) %-> (nj (A or B)) %. +%term nj_orir %pi (nj B) %-> (nj (A or B)) %. +%term nj_ore + %pi (nj (A or B)) + %-> (%pi (nj A) %-> (nj C)) + %-> (%pi (nj B) %-> (nj C)) + %-> (nj C) %. +%term nj_noti %pi ({p o} %pi (nj A) %-> (nj p)) %-> (nj (not A)) %. +%term nj_note %pi (nj (not A)) %-> ({C o} %pi (nj A) %-> (nj C)) %. +%term nj_truei nj true %. +%term nj_falsee %pi (nj false) %-> (nj C) %. +%term nj_foralli %pi ({a i} nj (A a)) %-> (nj (forall A)) %. +%term nj_foralle %pi (nj (forall A)) %-> ({T i} nj (A T)) %. +%term nj_existsi {T i} %pi (nj (A T)) %-> (nj (exists A)) %. +%term nj_existse %pi (nj (exists A)) %-> ({a i} %pi (nj (A a)) %-> (nj C)) %-> (nj C) %. +% In intuitionistic/classical logic we have the following derivation +% of A :- (not not A) +% --------- u +% not A A [hyp] +% ---------------------- note(p) +% p +% ----------- noti^p^u +% (not not A) +%theorem nj_dnotx_auto : forall* {A o} exists {D %pi (nj A) %-> (nj (not not A))} true %. +%prove 10 {} (nj_dnotx_auto D) %. +%define nj_dnotx [NJ] nj_noti ([p o] [u nj (not A)] nj_note u p NJ) %. +% Lemma 1: +% In intuitionistic logic we have the +% following derivation of (not not not A):- (not A) +% ------------------- u +% A +% ------------------- dnotx +% - - A - - - A [hyp] +% ------------------------------------------- note(q) +% q +% ------------------ noti^q^u +% - A +% represented in Elf by the definition nj_triple_neg_red given by +% [w:nj (not (not (not A)))] (nj_noti ([q:o] [u:nj A] (nj_note w q (noti ([p:o] [v:nj (not A)] (note v p u)))))) +% %theorem nj_triple_neg_red_auto : forall* {A:o} forall {D:nj (not not not A)} exists {D':nj (not A)} true. +% %prove 7 {} (nj_triple_neg_red_auto D D'). +%define nj_triple_neg_red [NJ] nj_noti ([q o] [u nj A] nj_note NJ q (nj_dnotx u)) %. +% Lemma 2: +% In intuitionistic logic we have the following derivation +% of (not not false) :- C +% -------- u +% false +% -------- falsee +% p +% --------- noti^p^u +% not false not not false [hyp] +% ------------------------------------------ note(C) +% C +%theorem nj_dneg_falser_auto : forall* {A o} exists {D %pi (nj (not not false)) %-> ({C o} nj C)} true %. +%prove 10 {} (nk_dnotx_auto D) %. +%define nj_dneg_falser [NJ1] [C] nj_note NJ1 C (nj_noti ([p] [u nj false] nj_falsee u)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/kolm/nk.lf b/new-tests/stelf-output/examples/kolm/nk.lf new file mode 100644 index 0000000..5c7bf20 --- /dev/null +++ b/new-tests/stelf-output/examples/kolm/nk.lf @@ -0,0 +1,36 @@ +% File: nk.elf +% Classical, Natural Deduction +%sort nk %. +%term nk_andi %pi (nk A) %-> (nk B) %-> (nk (A and B)) %. +%term nk_andel %pi (nk (A and B)) %-> (nk A) %. +%term nk_ander %pi (nk (A and B)) %-> (nk B) %. +%term nk_impi %pi (%pi (nk A) %-> (nk B)) %-> (nk (A imp B)) %. +%term nk_impe %pi (nk (A imp B)) %-> (nk A) %-> (nk B) %. +%term nk_oril %pi (nk A) %-> (nk (A or B)) %. +%term nk_orir %pi (nk B) %-> (nk (A or B)) %. +%term nk_ore + %pi (nk (A or B)) + %-> (%pi (nk A) %-> (nk C)) + %-> (%pi (nk B) %-> (nk C)) + %-> (nk C) %. +%term nk_noti %pi ({p o} %pi (nk A) %-> (nk p)) %-> (nk (not A)) %. +%term nk_note %pi (nk (not A)) %-> ({C o} %pi (nk A) %-> (nk C)) %. +%term nk_truei nk true %. +%term nk_falsee %pi (nk false) %-> (nk C) %. +%term nk_foralli %pi ({a i} nk (A a)) %-> (nk (forall A)) %. +%term nk_foralle %pi (nk (forall A)) %-> ({T i} nk (A T)) %. +%term nk_existsi {T i} %pi (nk (A T)) %-> (nk (exists A)) %. +%term nk_existse %pi (nk (exists A)) %-> ({a i} %pi (nk (A a)) %-> (nk C)) %-> (nk C) %. +%term nk_dnotr %pi (nk A) %<- (nk (not (not A))) %. +% double negation version of excluded middle +% In intuitionistic/classical logic we have the following derivation +% of A :- (not not A) +% --------- u +% not A A [hyp] +% ---------------------- note(p) +% p +% ----------- noti^p^u +% (not not A) +%theorem nk_dnotx_auto : forall* {A o} exists {D %pi (nk A) %-> (nk (not not A))} true %. +%prove 10 {} (nk_dnotx_auto D) %. +%define nk_dnotx [NK] nk_noti ([p o] [u nk (not A)] nk_note u p NK) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/kolm/sound.lf b/new-tests/stelf-output/examples/kolm/sound.lf new file mode 100644 index 0000000..908fc81 --- /dev/null +++ b/new-tests/stelf-output/examples/kolm/sound.lf @@ -0,0 +1,154 @@ +% File: sound.elf +% Translation of classical derivations to intuitionistic derivations of transformed formula. +% (Soundness) +% (Gamma = a1:i,...,an:i,p1:o,...,pm:o +% Kappa = kolm p1 - - p1,...,kolm pm - - pm +% Psi = nk A1,...,nk Ak +% Psi* = nj A1*,...,nj Ak* +% + +% Given: Gamma,Psi |- NK::nk A +% Gamma,Kappa |- sound Ai Ai' for i = 1,...,k. +% and Gamma,Kappa |- K::kolm A A* +% + +% Then we have NJ such that +% Gamma,Psi* |- NJ::nj A* +% The induction is on the classical natural deduction NK of A. +% Prover error: A proof could not be found +%theorem sound_auto : forall* {A o} {A* o} forall {NK nk A} {K kolm A A*} exists {NJ nj A*} true %. +% %prove 8 NK (sound_auto NK K NJ). +%sort sound %. +%mode sound %in %in %out %. +% andi -- easy, andi and double negate +%term sound_andi + %pi (sound (nk_andi NKA NKB) (kolm_and KB KA) (nj_dnotx (nj_andi NJA NJB))) + %<- (sound NKA KA NJA) + %<- (sound NKB KB NJB) %. +% This uses the following derivation from - - (- - A* and - - B*) to - - A* +% ------------------- v +% - - A+ and B* +% ----------- u ------------------- andel +% - A+ - - A+ +% ------------------------ -E +% p +% ----------------------- -I^p^v [hyp] +% - (- - A+ and B*) - - (- - A+ and B*) +% ------------------------------------------------------- -E +% q +% --------- -I^q^u +% - - A+ +%term sound_andel + %pi (sound (nk_andel (%the (nk (A and B)) NK1)) KA (nj_noti ([q o] [u nj (not A+)] nj_note NJ1 q (nj_noti ([p o] [v nj (n A+ and B*)] nj_note (nj_andel v) p u))))) + %<- (existskolm B B* KB) + %<- (sound NK1 (kolm_and KB KA) NJ1) %. +% NJ1::- - (- - A+ and B*) +% as in previous case +%term sound_ander + %pi (sound (nk_ander (%the (nk (A and B)) NK1)) KB (nj_noti ([q o] [u nj (not B+)] nj_note NJ1 q (nj_noti ([p o] [v nj (A* and (n B+))] nj_note (nj_ander v) p u))))) + %<- (existskolm A A* KA) + %<- (sound NK1 (kolm_and KB KA) NJ1) %. +% NJ1::- - (A* and - - B+) +% impi -- easy. apply impi and double negate +%term sound_impi + %pi (sound (nk_impi NKB) (kolm_imp KB KA) (nj_dnotx (nj_impi NJB))) + %<- ({u nk A} {v nj A*} %pi (sound u KA v) %-> (sound (NKB u) KB (NJB v))) %. +% impe -- harder, same kind of playing with negation +% as in other cases. +%term sound_impe + %pi (sound (nk_impe NKI (%the (nk A) NKA)) KB (nj_noti ([q o] [v nj (not B+)] nj_note NJI q (nj_noti ([p o] [u nj (A* imp (n B+))] nj_note (nj_impe u NJA) p v))))) + %<- (existskolm A A* KA) + %<- (sound NKA KA NJA) + %<- (sound NKI (kolm_imp KB KA) NJI) %. +% or-introductions are easy, we simply apply or-intro and double negate +%term sound_oril + %pi (sound (nk_oril NK) (kolm_or KB KA) (nj_dnotx (nj_oril NJ))) + %<- (sound NK KA NJ) %. +%term sound_orir + %pi (sound (nk_orir NK) (kolm_or KB KA) (nj_dnotx (nj_orir NJ))) + %<- (sound NK KB NJ) %. +% ------ ------ +% A* B* +% ---------------- u . . . . . . +% A* or B* - - C+ - - C+ +% ------------------------------------ ore +% +% ---- v - - C+ +% - C+ +% --------------------------- note +% p +% [hyp] ------------------- noti^p^u +% - -(A* or B*) -(A* or B*) +% ------------------------------------------ note +% q +% ------------------------------------------ noti^q^v +% - - C+ +%term sound_ore + %pi (sound (nk_ore (%the (nk (A or B)) NK) NKA NKB) KC (nj_noti ([q o] [v nj (not C+)] nj_note NJ q (nj_noti ([p o] [u nj (A* or B*)] nj_note (nj_ore u NJA NJB) p v))))) + %<- (existskolm A A* KA) + %<- (existskolm B B* KB) + %<- (sound NK (kolm_or KB KA) NJ) + %<- ({u nk A} {v nj A*} %pi (sound u KA v) %-> (sound (NKA u) KC (NJA v))) + %<- ({u nk B} {v nj B*} %pi (sound u KB v) %-> (sound (NKB u) KC (NJB v))) %. +% noti -- use Lemma 2 (nj_dneg_falser) - - false |- q +%term sound_noti + %pi (sound (nk_noti NK1) (kolm_not K) (nj_dnotx (nj_noti ([q] [v nj A*] nj_dneg_falser (NJ1 false v) q)))) + %<- ({p} {u nk A} {v nj A*} {kp kolm p (n p)} %pi (existskolm p (n p) kp) %-> (sound u K v) %-> (sound (NK1 p u) kp (NJ1 p v))) %. +% (NJ1 p v):nj (n p) +% note -- use Lemma 1 to derive - - - A from - - - - - A +% and apply note +%term sound_note + %pi (sound (nk_note NK2 C (%the (nk A) NK1)) KC (nj_note (nj_triple_neg_red NJ2) (n C*) NJ1)) + %<- (existskolm A A* KA) + %<- (sound NK2 (kolm_not KA) NJ2) + %<- (sound NK1 KA NJ1) %. +% truei -- easy! +%term sound_truei sound nk_truei kolm_true (nj_dnotx nj_truei) %. +% falsee -- Use Lemma 2 +%term sound_falsee %pi (sound (nk_falsee NK) KC (nj_dneg_falser NJ C*)) %<- (sound NK kolm_false NJ) %. +% foralli -- easy, foralli and double negate +%term sound_foralli + %pi (sound (nk_foralli NK) (kolm_forall KA) (nj_dnotx (nj_foralli NJ))) + %<- ({a i} sound (NK a) (KA a) (NJ a)) %. +% foralle -- hard -- play with negations as usual +%term sound_foralle + %pi (sound (nk_foralle NK T) (KA T) (nj_noti ([q o] [v nj (not (A+ T))] nj_note NJ q (nj_noti ([p o] [u nj (forall ([x i] n (A+ x)))] nj_note (nj_foralle u T) p v))))) + %<- (existskolm (forall A) (n (forall ([x] n (A+ x)))) (kolm_forall KA)) + %<- (sound NK (kolm_forall KA) NJ) %. +% existsi -- easy, existsi and double negate +%term sound_existi + %pi (sound (nk_existsi T NK) (kolm_exists KA) (nj_dnotx (nj_existsi T NJ))) + %<- (sound NK (KA T) NJ) %. +% existse -- hard -- play around with negations (compare to ore) +%term sound_existse + %pi (sound (nk_existse (%the (nk (exists A)) NKEA) NKC) KC (nj_noti ([q o] [v nj (not C+)] nj_note NJnnEA q (nj_noti ([p o] [u nj (exists A*)] nj_note (nj_existse u NJC) p v))))) + %<- (existskolm (exists A) (n (exists A*)) (kolm_exists KA)) + %<- (sound NKEA (kolm_exists KA) NJnnEA) + %<- ({a i} {u nk (A a)} {v nj (A* a)} %pi (sound u (KA a) v) %-> (sound (NKC a u) KC (NJC a v))) %. +% dnotr (excluded middle as the inference rule - - A |- A) +% Here we have KA : kolm A A* +% and NK:nk (n A) (by inversion) +% By induction (applied to NK) and inversion on K +% we have NJ:nj (n not (n not (n A+))) +% Apply the argument above to obtain NJ':nj (n n n A+) +% and again to obtain NJ'':nj (n n A+) +% and finally to obtain NJ''':nj (n A+) +% and that's what we want! +%term sound_dnotr + %pi (sound (nk_dnotr NK1) KA (nj_triple_neg_red (nj_triple_neg_red (nj_triple_neg_red NJ1)))) + %<- (sound NK1 (kolm_not (kolm_not KA)) NJ1) %. +%block l1 [a i]%. +%block l2 {A o} {A* o} [NK' nk A] [K' kolm A A*] [NJ' nj A*]%. +%block l3 {A o} {A* o} {KA kolm A A*} [u nk A] [v nj A*] [h sound u KA v]%. +%block l4 {A o} {A* o} {K kolm A A*} [p o] [u nk A] [v nj A*] [kp kolm p (n p)] [h1 existskolm p (n p) kp] [h2 sound u K v]%. +%mode kolm %in %out %. +%worlds (l1 l2 l3 l4) (kolm _ _) %. +%total A (kolm A A*) %. +%worlds (l1 l2 l3 l4) (existskolm _ _ _) %. +%total A (existskolm A A* K) %. +%{ +%mode (sound +NK +K -NJ). +%terminates NK (sound NK K NJ). +%worlds (l1 | l3 | l4) (sound _ _ _). +% %covers (sound +NK +K -NJ). % it really doesn't seem to cover check! --cs +}% \ No newline at end of file diff --git a/new-tests/stelf-output/examples/kolm/sources.lf b/new-tests/stelf-output/examples/kolm/sources.lf new file mode 100644 index 0000000..d1309ae --- /dev/null +++ b/new-tests/stelf-output/examples/kolm/sources.lf @@ -0,0 +1,6 @@ +%require %[ fol %] %. +%require %[ nk %] %. +%require %[ nj %] %. +%require %[ kolmtrans %] %. +%require %[ sound %] %. +%require %[ complete %] %. diff --git a/new-tests/stelf-output/examples/kolm/stelf.toml b/new-tests/stelf-output/examples/kolm/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/kolm/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/kolm/test.lf b/new-tests/stelf-output/examples/kolm/test.lf new file mode 100644 index 0000000..d1309ae --- /dev/null +++ b/new-tests/stelf-output/examples/kolm/test.lf @@ -0,0 +1,6 @@ +%require %[ fol %] %. +%require %[ nk %] %. +%require %[ nj %] %. +%require %[ kolmtrans %] %. +%require %[ sound %] %. +%require %[ complete %] %. diff --git a/new-tests/stelf-output/examples/lp-horn/canon.lf b/new-tests/stelf-output/examples/lp-horn/canon.lf new file mode 100644 index 0000000..a4a59fd --- /dev/null +++ b/new-tests/stelf-output/examples/lp-horn/canon.lf @@ -0,0 +1,26 @@ +%%% Canonical forms for natural deductions. +%%% Author: Frank Pfenning +%sort can {A o} %. +% Canonical deductions +%sort atm %. +% Atomic deductions +%name can %. +%name atm %. +%mode can %in %in %. +%mode atm %in %. +%term can_andi %pi (can (A and B) (andi D E)) %<- (can A D) %<- (can B E) %. +%{ +can_impi : can (A imp B) (impi D) + <- {u:pf A} atm u -> can B (D u). +}% +%term can_truei can true truei %. +%{ +can_foralli : can (forall A) (foralli D) + <- {a:i} can (A a) (D a). +}% +%term can_atm %pi (can (atom P) D) %<- (atm D) %. +%term atm_andel %pi (atm (andel D)) %<- (atm D) %. +%term atm_ander %pi (atm (ander D)) %<- (atm D) %. +%term atm_impe %pi (atm (impe D (%the (pf B) E))) %<- (atm D) %<- (can B E) %. +%term atm_foralle %pi (atm (foralle D T)) %<- (atm D) %. +%worlds () (can _ _) (atm _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp-horn/conv.lf b/new-tests/stelf-output/examples/lp-horn/conv.lf new file mode 100644 index 0000000..07523ce --- /dev/null +++ b/new-tests/stelf-output/examples/lp-horn/conv.lf @@ -0,0 +1,43 @@ +%%% Conversion to canonical form +%%% Author: Frank Pfenning +% Weak head reduction +%sort whr %. +% Weak head reduction +%name whr %. +% Reductions +%term whr_andl whr (andel (andi D E)) D %. +%term whr_andr whr (ander (andi D E)) E %. +%term whr_imp whr (impe (impi D) E) (D E) %. +%term whr_forall whr (foralle (foralli D) T) (D T) %. +% Congruences on elimination rules +%term whr_andel %pi (whr (andel D) (andel D')) %<- (whr D D') %. +%term whr_ander %pi (whr (ander D) (ander D')) %<- (whr D D') %. +%term whr_impe %pi (whr (impe D E) (impe D' E)) %<- (whr D D') %. +%term whr_foralle %pi (whr (foralle D T) (foralle D' T)) %<- (whr D D') %. +% Conversion to canonical and atomic form. +%sort tocan {A o} %. +% Conversion to canonical form +%sort toatm %. +% Conversion to atomic form +%name tocan %. +%name toatm %. +%term tc_and + %pi (tocan (A and B) D (andi D1' D2')) + %<- (tocan A (andel D) D1') + %<- (tocan B (ander D) D2') %. +%term tc_imp + %pi (tocan (A imp B) D (impi D')) + %<- ({u pf A} %pi (toatm u u) %-> (tocan B (impe D u) (D' u))) %. +%term tc_true tocan true D truei %. +%term tc_forall + %pi (tocan (forall A) D (foralli D')) + %<- ({a i} tocan (A a) (foralle D a) (D' a)) %. +%term tc_whr %pi (tocan (atom P) D D') %<- (whr D D'') %<- (tocan (atom P) D'' D') %. +%term tc_atm %pi (tocan (atom P) D D') %<- (toatm D D') %. +%term ta_impe + %pi (toatm (impe D (%the (pf A) E)) (impe D' E')) + %<- (toatm D D') + %<- (tocan A E E') %. +%term ta_andel %pi (toatm (andel D) (andel D')) %<- (toatm D D') %. +%term ta_ander %pi (toatm (ander D) (ander D')) %<- (toatm D D') %. +%term ta_foralle %pi (toatm (foralle D T) (foralle D' T)) %<- (toatm D D') %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp-horn/examples-quy.lf b/new-tests/stelf-output/examples/lp-horn/examples-quy.lf new file mode 100644 index 0000000..5155ca1 --- /dev/null +++ b/new-tests/stelf-output/examples/lp-horn/examples-quy.lf @@ -0,0 +1 @@ +% see ../lp/examples.quy---there are no specific queries here. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp-horn/iscan.lf b/new-tests/stelf-output/examples/lp-horn/iscan.lf new file mode 100644 index 0000000..fd939fd --- /dev/null +++ b/new-tests/stelf-output/examples/lp-horn/iscan.lf @@ -0,0 +1,30 @@ +%%% The result of conversion to canonical form is canonical. +%%% Author: Frank Pfenning +%sort iscan %. +%sort isatm %. +%mode iscan %in %out %. +%mode isatm %in %out %. +%term isc_and + %pi (iscan (tc_and TC2 TC1) (can_andi CN2 CN1)) + %<- (iscan TC1 CN1) + %<- (iscan TC2 CN2) %. +%{ +isc_imp : iscan (tc_imp TC1) (can_impi CN1) + <- {u:pf A} {ta:toatm u u} {at:atm u} + isatm ta at -> iscan (TC1 u ta) (CN1 u at). +}% +%term isc_true iscan tc_true can_truei %. +%{ +isc_forall : iscan (tc_forall TC1) (can_foralli CN1) + <- {a:i} iscan (TC1 a) (CN1 a). +}% +%term isc_whr %pi (iscan (tc_whr TC1 WHR) CN1) %<- (iscan TC1 CN1) %. +%term isc_atm %pi (iscan (tc_atm TA1) (can_atm AT1)) %<- (isatm TA1 AT1) %. +%term isa_impe + %pi (isatm (ta_impe TC2 TA1) (atm_impe CN2 AT1)) + %<- (isatm TA1 AT1) + %<- (iscan TC2 CN2) %. +%term isa_andel %pi (isatm (ta_andel TA1) (atm_andel AT1)) %<- (isatm TA1 AT1) %. +%term isa_ander %pi (isatm (ta_ander TA1) (atm_ander AT1)) %<- (isatm TA1 AT1) %. +%term isa_foralle %pi (isatm (ta_foralle TA1) (atm_foralle AT1)) %<- (isatm TA1 AT1) %. +%worlds () (iscan _ _) (isatm _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp-horn/natded.lf b/new-tests/stelf-output/examples/lp-horn/natded.lf new file mode 100644 index 0000000..5834607 --- /dev/null +++ b/new-tests/stelf-output/examples/lp-horn/natded.lf @@ -0,0 +1,34 @@ +%%% The language and inference rules of natural deduction +%%% This contains only the positive fragment +%%% Author: Frank Pfenning +%sort i %. +% individuals +%sort o %. +% formulas +%sort p %. +% atomic formulas +%name i %. +% S +%name o %. +% B C +%name p %. +% Q +%term atom %pi p %-> o %. +%term and %pi o %-> o %-> o %. +%prec %right 11 and %. +%term imp %pi o %-> o %-> o %. +%prec %right 10 imp %. +%term true o %. +%term forall %pi (%pi i %-> o) %-> o %. +%sort pf %. +% natural deductions +%name pf %. +% E +%term andi %pi (pf A) %-> (pf B) %-> (pf (A and B)) %. +%term andel %pi (pf (A and B)) %-> (pf A) %. +%term ander %pi (pf (A and B)) %-> (pf B) %. +%term impi %pi (%pi (pf A) %-> (pf B)) %-> (pf (A imp B)) %. +%term impe %pi (pf (A imp B)) %-> (pf A) %-> (pf B) %. +%term truei pf true %. +%term foralli %pi ({a i} pf (A a)) %-> (pf (forall A)) %. +%term foralle %pi (pf (forall A)) %-> ({T i} pf (A T)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp-horn/sources.lf b/new-tests/stelf-output/examples/lp-horn/sources.lf new file mode 100644 index 0000000..9db58cd --- /dev/null +++ b/new-tests/stelf-output/examples/lp-horn/sources.lf @@ -0,0 +1,8 @@ +%require %[ natded %] %. +%require %[ canon %] %. +%require %[ conv %] %. +%require %[ iscan %] %. +%require %[ uniform %] %. +%require %[ uni-sound %] %. +%require %[ uni-can %] %. +%require %[ uni-complete %] %. diff --git a/new-tests/stelf-output/examples/lp-horn/stelf.toml b/new-tests/stelf-output/examples/lp-horn/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/lp-horn/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/lp-horn/test.lf b/new-tests/stelf-output/examples/lp-horn/test.lf new file mode 100644 index 0000000..32aab87 --- /dev/null +++ b/new-tests/stelf-output/examples/lp-horn/test.lf @@ -0,0 +1,11 @@ +%require %[ natded %] %. +%require %[ canon %] %. +%require %[ conv %] %. +%require %[ iscan %] %. +%require %[ uniform %] %. +%require %[ uni-sound %] %. +%require %[ uni-sound %] %. +%require %[ uni-can %] %. +%require %[ uni-can %] %. +%require %[ uni-complete %] %. +%require %[ uni-complete %] %. diff --git a/new-tests/stelf-output/examples/lp-horn/uni-can-thm.lf b/new-tests/stelf-output/examples/lp-horn/uni-can-thm.lf new file mode 100644 index 0000000..74b5298 --- /dev/null +++ b/new-tests/stelf-output/examples/lp-horn/uni-can-thm.lf @@ -0,0 +1,6 @@ +% Canonical forms +%theorem ss_cn : forall* {A o} {S solve A} {D pf A} forall {SS s_sound S D} exists {CN can A D} true %. +%theorem hs_at : forall* {A o} {H assume A} {D pf A} forall {HS h_sound H D} exists {ATH atm D} true %. +%theorem is_at : forall* {A o} {P p} {I A >> P} {D %pi (pf A) %-> (pf (atom P))} forall {IS i_sound I D} exists {ATI {u pf A} %pi (atm u) %-> (atm (D u))} true %. +%prove 4 (SS HS IS) (ss_cn SS _) (hs_at HS _) (is_at IS _) %. +%terminates (SS HS IS) (ss_cn SS _) (hs_at HS _) (is_at IS _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp-horn/uni-can.lf b/new-tests/stelf-output/examples/lp-horn/uni-can.lf new file mode 100644 index 0000000..cad592b --- /dev/null +++ b/new-tests/stelf-output/examples/lp-horn/uni-can.lf @@ -0,0 +1,39 @@ +%%% Soundness of uniform proofs +%%% Shows correspondence to canonical deductions +%%% Author: Frank Pfenning +% Uniform proofs yield canonical deductions +%sort ss_can %. +%sort hs_atm %. +%sort is_atm %. +%term ssc_and + %pi (ss_can (ss_and SS2 SS1) (can_andi CN2 CN1)) + %<- (ss_can SS1 CN1) + %<- (ss_can SS2 CN2) %. +%{ +ssc_imp : ss_can (ss_imp SS1) (can_impi CN1) + <- {d:assume A} {u:pf A} {hs:h_sound d u} {at:atm u} + hs_atm hs at -> ss_can (SS1 d u hs) (CN1 u at). +}% +%term ssc_true ss_can ss_true can_truei %. +%{ +ssc_forall : ss_can (ss_forall SS1) (can_foralli CN1) + <- {a:i} ss_can (SS1 a) (CN1 a). +}% +%term ssc_atom + %pi (ss_can (ss_atom IS2 HS1) (can_atm (AT2 _ AT1))) + %<- (hs_atm HS1 AT1) + %<- (is_atm IS2 AT2) %. +%term isc_andl + %pi (is_atm (is_andl IS1) ([u pf (A and B)] [at atm u] AT1 (andel u) (atm_andel at))) + %<- (is_atm IS1 AT1) %. +%term isc_andr + %pi (is_atm (is_andr IS1) ([u pf (A and B)] [at atm u] AT1 (ander u) (atm_ander at))) + %<- (is_atm IS1 AT1) %. +%term isc_imp + %pi (is_atm (is_imp SS2 IS1) ([u pf (A imp B)] [at atm u] AT1 (impe u _) (atm_impe CN2 at))) + %<- (is_atm IS1 AT1) + %<- (ss_can SS2 CN2) %. +%term isc_forall + %pi (is_atm (is_forall IS1) ([u pf (forall A)] [at atm u] AT1 (foralle u _) (atm_foralle at))) + %<- (is_atm IS1 AT1) %. +%term isc_atom is_atm is_atom ([u pf (atom P)] [at atm u] at) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp-horn/uni-complete-thm.lf b/new-tests/stelf-output/examples/lp-horn/uni-complete-thm.lf new file mode 100644 index 0000000..1f263fc --- /dev/null +++ b/new-tests/stelf-output/examples/lp-horn/uni-complete-thm.lf @@ -0,0 +1,5 @@ +% Completeness +%theorem compcs : forall* {A o} {D pf A} forall {CN can A D} exists {S solve A} true %. +%theorem compai : forall* {A o} forall {D pf A} {AT atm D} exists {CN {P p} %pi (A >> P) %-> (solve (atom P))} true %. +%prove 4 (CN AT) (compcs CN _) (compai _ AT _) %. +%terminates (CN AT) (compcs CN _) (compai _ AT _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp-horn/uni-complete.lf b/new-tests/stelf-output/examples/lp-horn/uni-complete.lf new file mode 100644 index 0000000..1ee6591 --- /dev/null +++ b/new-tests/stelf-output/examples/lp-horn/uni-complete.lf @@ -0,0 +1,33 @@ +%%% Completeness of uniform proofs with respect to canonical deductions +%%% Author: Frank Pfenning +%sort cmpcs %. +%sort cmpai {D pf A} %. +%mode cmpcs %in %out %. +%mode cmpai %in %in %out %. +%term cmpcs_andi %pi (cmpcs (can_andi CN2 CN1) (s_and S2 S1)) %<- (cmpcs CN1 S1) %<- (cmpcs CN2 S2) %. +%{ +cmpcs_impi : cmpcs (can_impi CN1) (s_imp S1) + <- {u:pf A2} {a:atm u} {d:assume A2} + cmpai u a ([P:p] [i:A2 >> P] s_atom i d) + -> cmpcs (CN1 u a) (S1 d). +}% +%term cmpcs_truei cmpcs can_truei s_true %. +%{ +cmpcs_foralli : cmpcs (can_foralli CN1) (s_forall S1) + <- {a:i} cmpcs (CN1 a) (S1 a). +}% +%term cmpcs_atm %pi (cmpcs (can_atm (%the (atm D) AT1)) (I1 Q i_atom)) %<- (cmpai D AT1 I1) %. +%term cmpai_andel + %pi (cmpai (andel D1) (atm_andel CN1) ([P p] [i A >> P] I1 P (i_andl i))) + %<- (cmpai D1 CN1 I1) %. +%term cmpai_ander + %pi (cmpai (ander D1) (atm_ander CN1) ([P p] [i A >> P] I1 P (i_andr i))) + %<- (cmpai D1 CN1 I1) %. +%term cmpai_impe + %pi (cmpai (impe D1 D2) (atm_impe CN2 AT1) ([P p] [i A >> P] I1 P (i_imp S2 i))) + %<- (cmpai D1 AT1 I1) + %<- (cmpcs CN2 S2) %. +%term cmpai_foralle + %pi (cmpai (foralle D1 T) (atm_foralle AT1) ([P p] [i] I1 P (i_forall T i))) + %<- (cmpai D1 AT1 I1) %. +%worlds () (cmpcs _ _) (cmpai _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp-horn/uni-sound-thm.lf b/new-tests/stelf-output/examples/lp-horn/uni-sound-thm.lf new file mode 100644 index 0000000..996b63b --- /dev/null +++ b/new-tests/stelf-output/examples/lp-horn/uni-sound-thm.lf @@ -0,0 +1,6 @@ +% Soundness +%theorem s_snd : forall* {A o} forall {S solve A} exists {DS pf A} true %. +%theorem h_snd : forall* {A o} forall {H assume A} exists {DH pf A} true %. +%theorem i_snd : forall* {A o} {P p} forall {I A >> P} exists {DI %pi (pf A) %-> (pf (atom P))} true %. +%prove 3 (S H I) (s_snd S _) (h_snd H _) (i_snd I _) %. +%terminates (S H I) (s_snd S _) (h_snd H _) (i_snd I _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp-horn/uni-sound.lf b/new-tests/stelf-output/examples/lp-horn/uni-sound.lf new file mode 100644 index 0000000..6202675 --- /dev/null +++ b/new-tests/stelf-output/examples/lp-horn/uni-sound.lf @@ -0,0 +1,35 @@ +%%% Soundness of uniform proofs +%%% Author: Frank Pfenning +% Soundness of uniform proofs +%sort s_sound %. +%sort h_sound %. +%sort i_sound %. +%name s_sound %. +%name h_sound %. +%name i_sound %. +%mode s_sound %in %out %. +%mode h_sound %in %out %. +%mode i_sound %in %out %. +%term ss_and %pi (s_sound (s_and S2 S1) (andi D2 D1)) %<- (s_sound S1 D1) %<- (s_sound S2 D2) %. +%{ +ss_imp : s_sound (s_imp S1) (impi D1) + <- {d:assume A} {u:pf A} + h_sound d u -> s_sound (S1 d) (D1 u). +}% +%term ss_true s_sound s_true truei %. +%{ +ss_forall : s_sound (s_forall S1) (foralli D1) + <- {a:i} s_sound (S1 a) (D1 a). +}% +%term ss_atom %pi (s_sound (s_atom I2 H1) (D2 D1)) %<- (h_sound H1 D1) %<- (i_sound I2 D2) %. +%term is_andl %pi (i_sound (i_andl I1) ([u pf (A1 and A2)] D1 (andel u))) %<- (i_sound I1 D1) %. +%term is_andr %pi (i_sound (i_andr I2) ([u pf (A1 and A2)] D2 (ander u))) %<- (i_sound I2 D2) %. +%term is_imp + %pi (i_sound (i_imp S2 I1) ([u pf (A2 imp A1)] D1 (impe u D2))) + %<- (i_sound I1 D1) + %<- (s_sound S2 D2) %. +%term is_forall + %pi (i_sound (i_forall T I1) ([u pf (forall A1)] D1 (foralle u T))) + %<- (i_sound I1 D1) %. +%term is_atom i_sound i_atom ([u pf (atom P)] u) %. +%worlds () (s_sound _ _) (h_sound _ _) (i_sound _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp-horn/uniform.lf b/new-tests/stelf-output/examples/lp-horn/uniform.lf new file mode 100644 index 0000000..846530a --- /dev/null +++ b/new-tests/stelf-output/examples/lp-horn/uniform.lf @@ -0,0 +1,29 @@ +%%% Uniform Derivations +%%% Author: Frank Pfenning +%%% Temporarily removed parametric and hypothetical judgments +%sort solve %. +% solve goal formulas +%sort assume %. +% assume program formulas +%sort >> %. +% immediate entailment +%prec %none 8 >> %. +%name solve %. +%name assume %. +%name >> %. +%term s_and %pi (solve (A1 and A2)) %<- (solve A1) %<- (solve A2) %. +%{ +s_imp : solve (A2 imp A1) + <- (assume A2 -> solve A1). +}% +%term s_true solve true %. +%{ +s_forall : solve (forall A1) + <- {a:i} solve (A1 a). +}% +%term s_atom %pi (solve (atom P)) %<- (assume A) %<- (A >> P) %. +%term i_andl %pi (A1 and A2 >> P) %<- (A1 >> P) %. +%term i_andr %pi (A1 and A2 >> P) %<- (A2 >> P) %. +%term i_imp %pi (A2 imp A1 >> P) %<- (A1 >> P) %<- (solve A2) %. +%term i_forall {T i} %pi (forall A1 >> P) %<- (A1 T >> P) %. +%term i_atom atom P >> P %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp/canon.lf b/new-tests/stelf-output/examples/lp/canon.lf new file mode 100644 index 0000000..3065374 --- /dev/null +++ b/new-tests/stelf-output/examples/lp/canon.lf @@ -0,0 +1,22 @@ +%%% Canonical forms for natural deductions. +%%% Author: Frank Pfenning +%sort can {A o} %. +% Canonical deductions +%sort atm %. +% Atomic deductions +%name can %. +%name atm %. +%mode can %in %in %. +%mode atm %in %. +%term can_andi %pi (can (A and B) (andi D E)) %<- (can A D) %<- (can B E) %. +%term can_impi %pi (can (A imp B) (impi D)) %<- ({u pf A} %pi (atm u) %-> (can B (D u))) %. +%term can_truei can true truei %. +%term can_foralli %pi (can (forall A) (foralli D)) %<- ({a i} can (A a) (D a)) %. +%term can_atm %pi (can (atom P) D) %<- (atm D) %. +%term atm_andel %pi (atm (andel D)) %<- (atm D) %. +%term atm_ander %pi (atm (ander D)) %<- (atm D) %. +%term atm_impe %pi (atm (impe D (%the (pf B) E))) %<- (atm D) %<- (can B E) %. +%term atm_foralle %pi (atm (foralle D T)) %<- (atm D) %. +%block l1 {A o} [u pf A] [at atm u]%. +%block l2 [a i]%. +%worlds (l1 l2) (can _ _) (atm _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp/cont.lf b/new-tests/stelf-output/examples/lp/cont.lf new file mode 100644 index 0000000..0898f3c --- /dev/null +++ b/new-tests/stelf-output/examples/lp/cont.lf @@ -0,0 +1,28 @@ +% Success and failure continuations for Horn-clause fragment +% Restriction to the Horn clause fragment is not reflected +% in the typing. +% THIS IS COMPLETELY WRONG; it requires closures. +%sort csolve %. +%term cs_and %pi (csolve D (G1 and' G2) SC FC) %<- (csolve D G1 (G2 and' SC) FC) %. +%term cs_true %pi (csolve D true' (G1 and' SC2) FC) %<- (csolve D G1 SC2 FC) %. +%term cs_true* csolve D true' true' FC %. +%term cs_true' %pi (csolve D true' SC (G1 or' FC2)) %<- (csolve D G1 SC FC2) %. +% csolve D (true') SC (false') meta-fails. +%term cs_eqp %pi (csolve D (P == P) (G1 and' SC2) FC) %<- (csolve D G1 SC2 FC) %. +%term cs_eqp* csolve D (P == P) true' FC %. +%term cs_eqp' %pi (csolve D (Q == P) SC (G1 or' FC2)) %<- (csolve D G1 SC FC2) %. +% csolve D (Q == P) SC (false') meta-fails. +%term cs_or %pi (csolve D (G1 or' G2) SC FC) %<- (csolve D G1 SC (G2 or' FC)) %. +%term cs_false %pi (csolve D false' SC (G1 or' FC2)) %<- (csolve D G1 SC FC2) %. +% csolve D (false') SC (false') meta-fails +%term cs_exists {T i} %pi (csolve D (exists' G1) SC FC) %<- (csolve D (G1 T) SC FC) %. +%term cs_atom %pi (csolve D (atom' P) SC FC) %<- (resolve D P G) %<- (csolve D G SC FC) %. +%sort top_solve %. +%term ts %pi (top_solve D G) %<- (csolve D G true' false') %. +%{ + A more satisfactory solution would omit cs_true' and guard cs_eqp' +by non-unifiability (disequality) of Q and P. top_solve could then +choose to either accept only one solution or call the failure +continuation, if there is one. Unfortunately, non-unifiability +is not part of the framework. +}% \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp/conv.lf b/new-tests/stelf-output/examples/lp/conv.lf new file mode 100644 index 0000000..510f283 --- /dev/null +++ b/new-tests/stelf-output/examples/lp/conv.lf @@ -0,0 +1,48 @@ +%%% Conversion to canonical form +%%% Author: Frank Pfenning +% Weak head reduction +%sort whr %. +% Weak head reduction +%name whr %. +%mode whr %in %out %. +% Reductions +%term whr_andl whr (andel (andi D E)) D %. +%term whr_andr whr (ander (andi D E)) E %. +%term whr_imp whr (impe (impi D) E) (D E) %. +%term whr_forall whr (foralle (foralli D) T) (D T) %. +% Congruences on elimination rules +%term whr_andel %pi (whr (andel D) (andel D')) %<- (whr D D') %. +%term whr_ander %pi (whr (ander D) (ander D')) %<- (whr D D') %. +%term whr_impe %pi (whr (impe D E) (impe D' E)) %<- (whr D D') %. +%term whr_foralle %pi (whr (foralle D T) (foralle D' T)) %<- (whr D D') %. +%worlds () (whr _ _) %. +%terminates D (whr D _) %. +% Conversion to canonical and atomic form. +%sort tocan {A o} %. +% Conversion to canonical form +%sort toatm %. +% Conversion to atomic form +%name tocan %. +%name toatm %. +%mode tocan %in %in %out %. +%mode toatm %in %out %. +%term tc_and + %pi (tocan (A and B) D (andi D1' D2')) + %<- (tocan A (andel D) D1') + %<- (tocan B (ander D) D2') %. +%term tc_imp + %pi (tocan (A imp B) D (impi D')) + %<- ({u pf A} %pi (toatm u u) %-> (tocan B (impe D u) (D' u))) %. +%term tc_true tocan true D truei %. +%term tc_forall + %pi (tocan (forall A) D (foralli D')) + %<- ({a i} tocan (A a) (foralle D a) (D' a)) %. +%term tc_whr %pi (tocan (atom P) D D') %<- (whr D D'') %<- (tocan (atom P) D'' D') %. +%term tc_atm %pi (tocan (atom P) D D') %<- (toatm D D') %. +%term ta_impe + %pi (toatm (impe D (%the (pf A) E)) (impe D' E')) + %<- (toatm D D') + %<- (tocan A E E') %. +%term ta_andel %pi (toatm (andel D) (andel D')) %<- (toatm D D') %. +%term ta_ander %pi (toatm (ander D) (ander D')) %<- (toatm D D') %. +%term ta_foralle %pi (toatm (foralle D T) (foralle D' T)) %<- (toatm D D') %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp/examples-quy.lf b/new-tests/stelf-output/examples/lp/examples-quy.lf new file mode 100644 index 0000000..2b25c0a --- /dev/null +++ b/new-tests/stelf-output/examples/lp/examples-quy.lf @@ -0,0 +1,46 @@ +%query 1 _ _ {P p} {Q p} solve (atom P imp atom Q imp (atom P and atom Q)) %. +%query 1 _ _ {P p} {Q p} solve (atom P imp atom Q imp atom P) %. +%{ +sigma [G : {P:p} {Q:p} solve (atom P imp atom Q imp atom P)] + {P:p} {Q:p} s_sound (G P Q) (D P Q). +}% +%query 1 _ _ {P p} {Q p} s_sound (s_imp ([H1 assume (atom P)] s_imp ([H2 assume (atom Q)] s_atom i_atom H1))) (D P Q) %. +%query 1 _ _ {P} {Q} can _ (impi ([u pf (atom P)] impi ([u1 pf (atom Q)] u))) %. +%{ +sigma [G : {P:p} {Q:p} solve (atom P imp atom Q imp atom P)] +sigma [GS : {P:p} {Q:p} s_sound (G P Q) (D P Q)] + {P:p} {Q:p} ss_can (GS P Q) (CN P Q). +}% +%query 1 _ _ {P p} {Q p} ss_can (ss_imp ([d assume (atom P)] [u pf (atom P)] [HS1 h_sound d u] ss_imp ([d1 assume (atom Q)] [u1 pf (atom Q)] [HS2 h_sound d1 u1] ss_atom is_atom HS1))) (CN P Q) %. +%{ +sigma [G : solve ((atom q0 imp atom r0 imp atom s0) imp (atom q0 imp atom r0) + imp (atom q0 imp atom s0))] +sigma [GS : s_sound G D] + gs_can GS CN. +}% +%query 1 _ _ solve (atom q0 imp atom r0 imp atom s0 imp (atom q0 imp atom r0) imp (atom q0 imp atom s0)) %. +%query 1 _ _ s_sound (s_imp ([H1 assume (atom q0 imp atom r0 imp atom s0)] s_imp ([H2 assume (atom q0 imp atom r0)] s_imp ([H3 assume (atom q0)] s_atom (i_imp (s_atom i_atom H3) (i_imp (s_atom (i_imp (s_atom i_atom H3) i_atom) H2) i_atom)) H1)))) D %. +%query 1 _ _ ss_can (ss_imp ([d assume (atom q0 imp atom r0 imp atom s0)] [u pf (atom q0 imp atom r0 imp atom s0)] [HS1 h_sound d u] ss_imp ([d1 assume (atom q0 imp atom r0)] [u1 pf (atom q0 imp atom r0)] [HS2 h_sound d1 u1] ss_imp ([d2 assume (atom q0)] [u2 pf (atom q0)] [HS3 h_sound d2 u2] ss_atom (is_imp (ss_atom is_atom HS3) (is_imp (ss_atom (is_imp (ss_atom is_atom HS3) is_atom) HS2) is_atom)) HS1)))) CN %. +% next should fail (only classically valid) +%query 0 _ _ solve (atom q0 imp atom r0 imp atom q0 imp atom q0) %. +% next should give inf. many answers +%query _ 15 _ solve (atom q0 imp atom q0 imp atom q0 imp atom q0) %. +% next does not terminate. +%{ +solve (atom q0 imp (atom q0 imp atom q0) imp atom q0). +}% +%query 1 _ _ solve (atom (plus (s (s 0)) (s (s (s 0))) Z)) %. +%query 5 _ _ solve (atom (plus X Y (s (s (s (s 0)))))) %. +% An ill-typed query. +% solve (atom (plus 0 s Z)). +% Peirce's law. +% should fail +%query 0 _ _ {q p} {r p} solve (atom q imp atom r imp atom q imp atom q) %. +% Resolution +%query 1 _ _ {X i} {Y i} resolve (atom^ (double 0 0) and^ (forall^ ([X] forall^ ([Y] atom' (double X Y) imp^ atom^ (double (s X) (s (s Y))))))) (double X Y) (G X Y) %. +%query 0 _ _ {a %pi i %-> p} {b p} solve' (forall' ([x] atom' (a x)) imp^ atom^ b imp' (exists' ([x] atom^ (a x) imp' atom' b))) %. +% Continuations +%query 1 _ _ top_solve (atom^ (double 0 0) and^ (forall^ ([X] forall^ ([Y] atom' (double X Y) imp^ atom^ (double (s X) (s (s Y))))))) (atom' (double 0 Y)) %. +%query 1 _ _ top_solve (atom^ (double 0 0) and^ (forall^ ([X] forall^ ([Y] atom' (double X Y) imp^ atom^ (double (s X) (s (s Y))))))) (atom' (double X (s (s 0)))) %. +%query 3 _ _ top_solve (forall^ ([Y] atom^ (plus 0 Y Y)) and^ (forall^ ([X] forall^ ([Y] forall^ ([Z] atom' (plus X Y Z) imp^ atom^ (plus (s X) Y (s Z))))))) (atom' (plus X Y (s (s 0)))) %. +%query 2 _ _ top_solve true^ (true' or' true') %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp/examples.lf b/new-tests/stelf-output/examples/lp/examples.lf new file mode 100644 index 0000000..211fd31 --- /dev/null +++ b/new-tests/stelf-output/examples/lp/examples.lf @@ -0,0 +1,15 @@ +% Needed to allow extension of p, i, o +% as of Twelf 1.5R2 (March 2005) +%term q0 p %. +%term r0 p %. +%term s0 p %. +%term 0 i %. +%term s %pi i %-> i %. +%term nat %pi i %-> p %. +%term natz assume (atom (nat 0)) %. +%term nats assume (forall ([X] atom (nat X) imp atom (nat (s X)))) %. +%term plus %pi i %-> i %-> i %-> p %. +%term plz assume (forall ([Y] atom (plus 0 Y Y))) %. +%term pls assume (forall ([X] forall ([Y] forall ([Z] atom (plus X Y Z) imp atom (plus (s X) Y (s Z)))))) %. +%term double %pi i %-> i %-> p %. +%term db assume' (atom^ (double 0 0) and^ (forall^ ([X] forall^ ([Y] atom' (double X Y) imp^ atom^ (double (s X) (s (s Y))))))) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp/fohh-formulas.lf b/new-tests/stelf-output/examples/lp/fohh-formulas.lf new file mode 100644 index 0000000..79133e3 --- /dev/null +++ b/new-tests/stelf-output/examples/lp/fohh-formulas.lf @@ -0,0 +1,15 @@ +% Translating positive formulas to goal and program formulas. +%sort gl %. +%sort pg %. +%mode gl %in %out %. +%mode pg %in %out %. +%term gl_atom gl (atom P) (atom' P) %. +%term gl_and %pi (gl (A1 and A2) (G1 and' G2)) %<- (gl A1 G1) %<- (gl A2 G2) %. +%term gl_imp %pi (gl (A2 imp A1) (D2 imp' G1)) %<- (gl A1 G1) %<- (pg A2 D2) %. +%term gl_true gl true true' %. +%term gl_forall %pi (gl (forall A1) (forall' G1)) %<- ({a i} gl (A1 a) (G1 a)) %. +%term pg_atom pg (atom P) (atom^ P) %. +%term pg_and %pi (pg (A1 and A2) (G1 and^ G2)) %<- (pg A1 G1) %<- (pg A2 G2) %. +%term pg_imp %pi (pg (A2 imp A1) (G2 imp^ D1)) %<- (pg A1 D1) %<- (gl A2 G2) %. +%term pg_true pg true true^ %. +%term pg_forall %pi (pg (forall A1) (forall^ D1)) %<- ({a i} pg (A1 a) (D1 a)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp/fohh.lf b/new-tests/stelf-output/examples/lp/fohh.lf new file mode 100644 index 0000000..72afc7e --- /dev/null +++ b/new-tests/stelf-output/examples/lp/fohh.lf @@ -0,0 +1,27 @@ +% Generalizing the language of goals to +% allow residuation of disjuction, failure, existential, +% and equality. +%sort goal %. +%sort prog %. +%name goal %. +%name prog %. +%term atom' %pi p %-> goal %. +%term and' %pi goal %-> goal %-> goal %. +%prec %right 11 and' %. +%term imp' %pi prog %-> goal %-> goal %. +%prec %right 10 imp' %. +%term true' goal %. +%term forall' %pi (%pi i %-> goal) %-> goal %. +%term or' %pi goal %-> goal %-> goal %. +%prec %right 11 or' %. +%term false' goal %. +%term exists' %pi (%pi i %-> goal) %-> goal %. +%term == %pi p %-> p %-> goal %. +%prec %none 12 == %. +%term atom^ %pi p %-> prog %. +%term and^ %pi prog %-> prog %-> prog %. +%prec %right 11 and^ %. +%term imp^ %pi goal %-> prog %-> prog %. +%prec %right 10 imp^ %. +%term true^ prog %. +%term forall^ %pi (%pi i %-> prog) %-> prog %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp/iscan.lf b/new-tests/stelf-output/examples/lp/iscan.lf new file mode 100644 index 0000000..25c28ea --- /dev/null +++ b/new-tests/stelf-output/examples/lp/iscan.lf @@ -0,0 +1,28 @@ +%%% The result of conversion to canonical form is canonical. +%%% Author: Frank Pfenning +%sort iscan %. +%sort isatm %. +%mode iscan %in %out %. +%mode isatm %in %out %. +%term isc_and + %pi (iscan (tc_and TC2 TC1) (can_andi CN2 CN1)) + %<- (iscan TC1 CN1) + %<- (iscan TC2 CN2) %. +%term isc_imp + %pi (iscan (tc_imp TC1) (can_impi CN1)) + %<- ({u pf A} {ta toatm u u} {at atm u} %pi (isatm ta at) %-> (iscan (TC1 u ta) (CN1 u at))) %. +%term isc_true iscan tc_true can_truei %. +%term isc_forall %pi (iscan (tc_forall TC1) (can_foralli CN1)) %<- ({a i} iscan (TC1 a) (CN1 a)) %. +%term isc_whr %pi (iscan (tc_whr TC1 WHR) CN1) %<- (iscan TC1 CN1) %. +%term isc_atm %pi (iscan (tc_atm TA1) (can_atm AT1)) %<- (isatm TA1 AT1) %. +%term isa_impe + %pi (isatm (ta_impe TC2 TA1) (atm_impe CN2 AT1)) + %<- (isatm TA1 AT1) + %<- (iscan TC2 CN2) %. +%term isa_andel %pi (isatm (ta_andel TA1) (atm_andel AT1)) %<- (isatm TA1 AT1) %. +%term isa_ander %pi (isatm (ta_ander TA1) (atm_ander AT1)) %<- (isatm TA1 AT1) %. +%term isa_foralle %pi (isatm (ta_foralle TA1) (atm_foralle AT1)) %<- (isatm TA1 AT1) %. +%block l_isatm {A o} [u pf A] [ta toatm u u] [at atm u] [is isatm ta at]%. +%block l_i [a i]%. +%worlds (l_i l_isatm) (isatm _ _) (iscan _ _) %. +%total (TA TC) (isatm TA CA) (iscan TC CN) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp/natded.lf b/new-tests/stelf-output/examples/lp/natded.lf new file mode 100644 index 0000000..5834607 --- /dev/null +++ b/new-tests/stelf-output/examples/lp/natded.lf @@ -0,0 +1,34 @@ +%%% The language and inference rules of natural deduction +%%% This contains only the positive fragment +%%% Author: Frank Pfenning +%sort i %. +% individuals +%sort o %. +% formulas +%sort p %. +% atomic formulas +%name i %. +% S +%name o %. +% B C +%name p %. +% Q +%term atom %pi p %-> o %. +%term and %pi o %-> o %-> o %. +%prec %right 11 and %. +%term imp %pi o %-> o %-> o %. +%prec %right 10 imp %. +%term true o %. +%term forall %pi (%pi i %-> o) %-> o %. +%sort pf %. +% natural deductions +%name pf %. +% E +%term andi %pi (pf A) %-> (pf B) %-> (pf (A and B)) %. +%term andel %pi (pf (A and B)) %-> (pf A) %. +%term ander %pi (pf (A and B)) %-> (pf B) %. +%term impi %pi (%pi (pf A) %-> (pf B)) %-> (pf (A imp B)) %. +%term impe %pi (pf (A imp B)) %-> (pf A) %-> (pf B) %. +%term truei pf true %. +%term foralli %pi ({a i} pf (A a)) %-> (pf (forall A)) %. +%term foralle %pi (pf (forall A)) %-> ({T i} pf (A T)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp/res-complete.lf b/new-tests/stelf-output/examples/lp/res-complete.lf new file mode 100644 index 0000000..c4d23f9 --- /dev/null +++ b/new-tests/stelf-output/examples/lp/res-complete.lf @@ -0,0 +1,54 @@ +%%% Completeness of resolution without unification +% Lemma: totality of resolution +%sort r_total {D prog} %. +% this was mode incorrect without the explicit +% P, since P was inferred to be an output argument. +% %mode r_total +D -R. +%mode {%in D prog} {%in P p} {%out G goal} {%out R resolve D P G} r_total D R %. +%term rt_and %pi (r_total (D1 and^ D2) (r_and R2 R1)) %<- (r_total D1 R1) %<- (r_total D2 R2) %. +%term rt_imp %pi (r_total (G2 imp^ D1) (r_imp R1)) %<- (r_total D1 R1) %. +%term rt_true r_total true^ r_true %. +%term rt_forall %pi (r_total (forall^ D1) (r_forall R1)) %<- ({a i} r_total (D1 a) (R1 a)) %. +%term rt_atom r_total (atom^ Q) r_atom %. +%worlds (l_i) (r_total D R) %. +%total (D) (r_total D R) %. +% Completeness of resolution +%sort s'_comp %. +%sort h'_comp %. +%sort r_comp %. +%mode s'_comp %in %in %out %. +%mode h'_comp %in %out %out %. +%mode r_comp %in %in %in %out %. +%term s's_and + %pi (s'_comp (s_and S2 S1) (gl_and GL2 GL1) (s'_and S'2 S'1)) + %<- (s'_comp S1 GL1 S'1) + %<- (s'_comp S2 GL2 S'2) %. +%term s's_imp + %pi (s'_comp (s_imp S1) (gl_imp PG2 GL1) (s'_imp S'1)) + %<- ({h assume A2} {h' assume' D2} %pi (h'_comp h PG2 h') %-> (s'_comp (S1 h) GL1 (S'1 h'))) %. +%term s's_true s'_comp s_true gl_true s'_true %. +%term s's_forall + %pi (s'_comp (s_forall S1) (gl_forall GL1) (s'_forall S'1)) + %<- ({a i} s'_comp (S1 a) (GL1 a) (S'1 a)) %. +%term s's_atom + %pi (s'_comp (s_atom (%the (A >> P) I2) H1) gl_atom (s'_atom S'3 R2 H'1)) + %<- (h'_comp H1 (%the (pg A D) PG1) H'1) + %<- (r_total D R2) + %<- (r_comp I2 PG1 R2 S'3) %. +%term rs'_andl + %pi (r_comp (i_andl I1) (pg_and PG2 PG1) (r_and R2 R1) (s'_orl S'1)) + %<- (r_comp I1 PG1 R1 S'1) %. +%term rs'_andr + %pi (r_comp (i_andr I2) (pg_and PG2 PG1) (r_and R2 R1) (s'_orr S'2)) + %<- (r_comp I2 PG2 R2 S'2) %. +%term rs'_imp + %pi (r_comp (i_imp S2 I1) (pg_imp GL2 PG1) (r_imp R1) (s'_and S'2 S'1)) + %<- (r_comp I1 PG1 R1 S'1) + %<- (s'_comp S2 GL2 S'2) %. +%term rs'_forall + %pi (r_comp (i_forall T I1) (pg_forall PG1) (r_forall R1) (s'_exists T S'1)) + %<- (r_comp I1 (PG1 T) (R1 T) S'1) %. +%term rs'_atom r_comp i_atom pg_atom r_atom s'_eqp %. +%block l_h'_comp {A o} {D prog} {PG pg A D} [h assume A] [h' assume' D] [hc h'_comp h PG h']%. +%worlds (l_i l_h'_comp) (s'_comp S GL S') (h'_comp H PG H') (r_comp I PG R S'') %. +%total (S H I) (s'_comp S GL _) (h'_comp H _ _) (r_comp I PG R _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp/res-sound.lf b/new-tests/stelf-output/examples/lp/res-sound.lf new file mode 100644 index 0000000..902d295 --- /dev/null +++ b/new-tests/stelf-output/examples/lp/res-sound.lf @@ -0,0 +1,42 @@ +%%% Soundness of resolution without unification +%sort s'_sound %. +%sort h'_sound %. +%sort r_sound %. +%mode s'_sound %in %in %out %. +% following is incorrect +% %mode h'_sound +H' +PG -H. +%mode h'_sound %in %out %out %. +%mode r_sound %in %in %in %out %. +%term s's_and + %pi (s'_sound (s'_and S'2 S'1) (gl_and GL2 GL1) (s_and S2 S1)) + %<- (s'_sound S'1 GL1 S1) + %<- (s'_sound S'2 GL2 S2) %. +%term s's_imp + %pi (s'_sound (s'_imp S'1) (gl_imp PG2 GL1) (s_imp S1)) + %<- ({h' assume' D2} {h assume A2} %pi (h'_sound h' PG2 h) %-> (s'_sound (S'1 h') GL1 (S1 h))) %. +%term s's_true s'_sound s'_true gl_true s_true %. +%term s's_forall + %pi (s'_sound (s'_forall S'1) (gl_forall GL1) (s_forall S1)) + %<- ({a i} s'_sound (S'1 a) (GL1 a) (S1 a)) %. +%term s's_atom + %pi (s'_sound (s'_atom S'3 R2 H'1) gl_atom (s_atom I2 PR1)) + %<- (h'_sound H'1 PG1 PR1) + %<- (r_sound R2 S'3 PG1 I2) %. +%term rs'_andl + %pi (r_sound (r_and R2 R1) (s'_orl S'1) (pg_and PG2 PG1) (i_andl I1)) + %<- (r_sound R1 S'1 PG1 I1) %. +%term rs'_andr + %pi (r_sound (r_and R2 R1) (s'_orr S'2) (pg_and PG2 PG1) (i_andr I2)) + %<- (r_sound R2 S'2 PG2 I2) %. +%term rs'_imp + %pi (r_sound (r_imp R1) (s'_and S'2 S'1) (pg_imp GL2 PG1) (i_imp S2 I1)) + %<- (r_sound R1 S'1 PG1 I1) + %<- (s'_sound S'2 GL2 S2) %. +% rs'_true: no case since by inversion there is no proof of false. +%term rs'_forall + %pi (r_sound (r_forall R1) (s'_exists T S'1) (pg_forall PG1) (i_forall T I1)) + %<- (r_sound (R1 T) S'1 (PG1 T) I1) %. +%term rs'_atom r_sound r_atom s'_eqp pg_atom i_atom %. +%block l_h'_sound {A o} {D prog} {PG pg A D} [h' assume' D] [h assume A] [hs h'_sound h' PG h]%. +%worlds (l_i l_h'_sound) (s'_sound S' GL S) (h'_sound H' PG H) (r_sound R S' PG I) %. +%total (S' H' S'') (s'_sound S' GL _) (h'_sound H' _ _) (r_sound R S'' PG _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp/resolution.lf b/new-tests/stelf-output/examples/lp/resolution.lf new file mode 100644 index 0000000..1b19926 --- /dev/null +++ b/new-tests/stelf-output/examples/lp/resolution.lf @@ -0,0 +1,28 @@ +% Resolution without unification +%sort solve' %. +%sort assume' %. +%sort resolve %. +%name solve' %. +%name assume' %. +%name resolve %. +% Next lines should give an error for exists'. +% %mode solve' +G. +% %mode assume' +D. +% %mode resolve +D +P -G. +%term s'_and %pi (solve' (G1 and' G2)) %<- (solve' G1) %<- (solve' G2) %. +%term s'_imp %pi (solve' (D2 imp' G1)) %<- (%pi (assume' D2) %-> (solve' G1)) %. +%term s'_true solve' true' %. +%term s'_forall %pi (solve' (forall' G1)) %<- ({a i} solve' (G1 a)) %. +%term s'_eqp solve' (P == P) %. +%term s'_orl %pi (solve' (G1 or' G2)) %<- (solve' G1) %. +%term s'_orr %pi (solve' (G1 or' G2)) %<- (solve' G2) %. +%term s'_exists {T i} %pi (solve' (exists' G1)) %<- (solve' (G1 T)) %. +%term s'_atom %pi (solve' (atom' P)) %<- (assume' D) %<- (resolve D P G) %<- (solve' G) %. +%term r_and + %pi (resolve (D1 and^ D2) P (G1 or' G2)) + %<- (resolve D1 P G1) + %<- (resolve D2 P G2) %. +%term r_imp %pi (resolve (G2 imp^ D1) P (G1 and' G2)) %<- (resolve D1 P G1) %. +%term r_true resolve true^ P false' %. +%term r_forall %pi (resolve (forall^ D1) P (exists' G1)) %<- ({a i} resolve (D1 a) P (G1 a)) %. +%term r_atom resolve (atom^ Q) P (Q == P) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp/sources.lf b/new-tests/stelf-output/examples/lp/sources.lf new file mode 100644 index 0000000..d3866bb --- /dev/null +++ b/new-tests/stelf-output/examples/lp/sources.lf @@ -0,0 +1,14 @@ +%require %[ natded %] %. +%require %[ canon %] %. +%require %[ conv %] %. +%require %[ iscan %] %. +%require %[ uniform %] %. +%require %[ uni-sound %] %. +%require %[ uni-complete %] %. +%require %[ fohh %] %. +%require %[ fohh-formulas %] %. +%require %[ resolution %] %. +%require %[ res-sound %] %. +%require %[ res-complete %] %. +%require %[ examples %] %. +%require %[ cont %] %. diff --git a/new-tests/stelf-output/examples/lp/stelf.toml b/new-tests/stelf-output/examples/lp/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/lp/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/lp/test.lf b/new-tests/stelf-output/examples/lp/test.lf new file mode 100644 index 0000000..35f4a62 --- /dev/null +++ b/new-tests/stelf-output/examples/lp/test.lf @@ -0,0 +1,15 @@ +%require %[ natded %] %. +%require %[ canon %] %. +%require %[ conv %] %. +%require %[ iscan %] %. +%require %[ uniform %] %. +%require %[ uni-sound %] %. +%require %[ uni-complete %] %. +%require %[ fohh %] %. +%require %[ fohh-formulas %] %. +%require %[ resolution %] %. +%require %[ res-sound %] %. +%require %[ res-complete %] %. +%require %[ examples %] %. +%require %[ cont %] %. +%require %[ examples %] %. diff --git a/new-tests/stelf-output/examples/lp/uni-complete.lf b/new-tests/stelf-output/examples/lp/uni-complete.lf new file mode 100644 index 0000000..d508c81 --- /dev/null +++ b/new-tests/stelf-output/examples/lp/uni-complete.lf @@ -0,0 +1,29 @@ +%%% Completeness of uniform proofs with respect to canonical deductions +%%% Author: Frank Pfenning +%sort cmpcs %. +%sort cmpai {D pf A} %. +%mode cmpcs %in %out %. +%mode cmpai %in %in %out %. +%term cmpcs_andi %pi (cmpcs (can_andi CN2 CN1) (s_and S2 S1)) %<- (cmpcs CN1 S1) %<- (cmpcs CN2 S2) %. +%term cmpcs_impi + %pi (cmpcs (can_impi CN1) (s_imp S1)) + %<- ({u pf A2} {a atm u} {d assume A2} %pi (cmpai u a ([P p] [i A2 >> P] s_atom i d)) %-> (cmpcs (CN1 u a) (S1 d))) %. +%term cmpcs_truei cmpcs can_truei s_true %. +%term cmpcs_foralli %pi (cmpcs (can_foralli CN1) (s_forall S1)) %<- ({a i} cmpcs (CN1 a) (S1 a)) %. +%term cmpcs_atm %pi (cmpcs (can_atm (%the (atm D) AT1)) (I1 Q i_atom)) %<- (cmpai D AT1 I1) %. +%term cmpai_andel + %pi (cmpai (andel D1) (atm_andel CN1) ([P p] [i A >> P] I1 P (i_andl i))) + %<- (cmpai D1 CN1 I1) %. +%term cmpai_ander + %pi (cmpai (ander D1) (atm_ander CN1) ([P p] [i A >> P] I1 P (i_andr i))) + %<- (cmpai D1 CN1 I1) %. +%term cmpai_impe + %pi (cmpai (impe D1 D2) (atm_impe CN2 AT1) ([P p] [i A >> P] I1 P (i_imp S2 i))) + %<- (cmpai D1 AT1 I1) + %<- (cmpcs CN2 S2) %. +%term cmpai_foralle + %pi (cmpai (foralle D1 T) (atm_foralle AT1) ([P p] [i] I1 P (i_forall T i))) + %<- (cmpai D1 AT1 I1) %. +%block l_cmpai {A o} [u pf A] [a atm u] [d assume A] [cai cmpai u a ([P p] [i A >> P] s_atom i d)]%. +%worlds (l_i l_cmpai) (cmpcs CN S) (cmpai D AT S') %. +%total (CN AT) (cmpcs CN _) (cmpai D AT _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp/uni-sound.lf b/new-tests/stelf-output/examples/lp/uni-sound.lf new file mode 100644 index 0000000..50c4cf1 --- /dev/null +++ b/new-tests/stelf-output/examples/lp/uni-sound.lf @@ -0,0 +1,70 @@ +%%% Soundness of uniform proofs +%%% Also shows correspondence to canonical deductions +%%% Author: Frank Pfenning +% Soundness of uniform proofs +%sort s_sound %. +%sort h_sound %. +%sort i_sound %. +%name s_sound %. +%name h_sound %. +%name i_sound %. +%mode s_sound %in %out %. +%mode h_sound %in %out %. +%mode i_sound %in %out %. +%term ss_and %pi (s_sound (s_and S2 S1) (andi D1 D2)) %<- (s_sound S1 D1) %<- (s_sound S2 D2) %. +%term ss_imp + %pi (s_sound (s_imp S1) (impi D1)) + %<- ({d assume A} {u pf A} %pi (h_sound d u) %-> (s_sound (S1 d) (D1 u))) %. +%term ss_true s_sound s_true truei %. +%term ss_forall %pi (s_sound (s_forall S1) (foralli D1)) %<- ({a i} s_sound (S1 a) (D1 a)) %. +%term ss_atom %pi (s_sound (s_atom I2 H1) (D2 D1)) %<- (h_sound H1 D1) %<- (i_sound I2 D2) %. +%term is_andl %pi (i_sound (i_andl I1) ([u pf (A1 and A2)] D1 (andel u))) %<- (i_sound I1 D1) %. +%term is_andr %pi (i_sound (i_andr I2) ([u pf (A1 and A2)] D2 (ander u))) %<- (i_sound I2 D2) %. +%term is_imp + %pi (i_sound (i_imp S2 I1) ([u pf (A2 imp A1)] D1 (impe u D2))) + %<- (i_sound I1 D1) + %<- (s_sound S2 D2) %. +%term is_forall + %pi (i_sound (i_forall T I1) ([u pf (forall A1)] D1 (foralle u T))) + %<- (i_sound I1 D1) %. +%term is_atom i_sound i_atom ([u pf (atom P)] u) %. +%block l_h_sound {A o} [d assume A] [u pf A] [hs h_sound d u]%. +%worlds (l_i l_h_sound) (s_sound S D) (h_sound H U) (i_sound I D'') %. +%total (S H I) (s_sound S _) (h_sound H _) (i_sound I _) %. +% Uniform proofs yield canonical deductions +%sort ss_can %. +%sort hs_atm %. +%sort is_atm %. +%mode ss_can %in %out %. +%mode hs_atm %in %out %. +%mode is_atm %in %out %. +%term ssc_and + %pi (ss_can (ss_and SS2 SS1) (can_andi CN2 CN1)) + %<- (ss_can SS1 CN1) + %<- (ss_can SS2 CN2) %. +%term ssc_imp + %pi (ss_can (ss_imp SS1) (can_impi CN1)) + %<- ({d assume A} {u pf A} {hs h_sound d u} {at atm u} %pi (hs_atm hs at) %-> (ss_can (SS1 d u hs) (CN1 u at))) %. +%term ssc_true ss_can ss_true can_truei %. +%term ssc_forall %pi (ss_can (ss_forall SS1) (can_foralli CN1)) %<- ({a i} ss_can (SS1 a) (CN1 a)) %. +%term ssc_atom + %pi (ss_can (ss_atom IS2 HS1) (can_atm (AT2 _ AT1))) + %<- (hs_atm HS1 AT1) + %<- (is_atm IS2 AT2) %. +%term isc_andl + %pi (is_atm (is_andl IS1) ([u pf (A and B)] [at atm u] AT1 (andel u) (atm_andel at))) + %<- (is_atm IS1 AT1) %. +%term isc_andr + %pi (is_atm (is_andr IS1) ([u pf (A and B)] [at atm u] AT1 (ander u) (atm_ander at))) + %<- (is_atm IS1 AT1) %. +%term isc_imp + %pi (is_atm (is_imp SS2 IS1) ([u pf (A imp B)] [at atm u] AT1 (impe u _) (atm_impe CN2 at))) + %<- (is_atm IS1 AT1) + %<- (ss_can SS2 CN2) %. +%term isc_forall + %pi (is_atm (is_forall IS1) ([u pf (forall A)] [at atm u] AT1 (foralle u _) (atm_foralle at))) + %<- (is_atm IS1 AT1) %. +%term isc_atom is_atm is_atom ([u pf (atom P)] [at atm u] at) %. +%block l_hs_atm {A o} [d assume A] [u pf A] [hs h_sound d u] [at atm u] [hsa hs_atm hs at]%. +%worlds (l_i l_hs_atm) (ss_can SS CN) (hs_atm HS AT) (is_atm IS AT') %. +%total (SS HS IS) (ss_can SS CN) (hs_atm HS AT) (is_atm IS AT') %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/lp/uniform.lf b/new-tests/stelf-output/examples/lp/uniform.lf new file mode 100644 index 0000000..c55761d --- /dev/null +++ b/new-tests/stelf-output/examples/lp/uniform.lf @@ -0,0 +1,25 @@ +%%% Uniform Derivations +%%% Author: Frank Pfenning +%sort solve %. +% solve goal formulas +%sort assume %. +% assume program formulas +%sort >> %. +% immediate entailment +%prec %none 8 >> %. +%name solve %. +%name assume %. +%name >> %. +%mode solve %star %. +%mode assume %star %. +%mode >> %star %star %. +%term s_and %pi (solve (A1 and A2)) %<- (solve A1) %<- (solve A2) %. +%term s_imp %pi (solve (A2 imp A1)) %<- (%pi (assume A2) %-> (solve A1)) %. +%term s_true solve true %. +%term s_forall %pi (solve (forall A1)) %<- ({a i} solve (A1 a)) %. +%term s_atom %pi (solve (atom P)) %<- (assume A) %<- (A >> P) %. +%term i_andl %pi (A1 and A2 >> P) %<- (A1 >> P) %. +%term i_andr %pi (A1 and A2 >> P) %<- (A2 >> P) %. +%term i_imp %pi (A2 imp A1 >> P) %<- (A1 >> P) %<- (solve A2) %. +%term i_forall {T i} %pi (forall A1 >> P) %<- (A1 T >> P) %. +%term i_atom atom P >> P %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/mini-ml/closed.lf b/new-tests/stelf-output/examples/mini-ml/closed.lf new file mode 100644 index 0000000..dc8dc9e --- /dev/null +++ b/new-tests/stelf-output/examples/mini-ml/closed.lf @@ -0,0 +1,67 @@ +%%% Closed Mini-ML Expressions +%%% Author: Frank Pfenning +%sort closed %. +%name closed %. +%mode closed %in %. +% Natural Numbers +%term clo_z closed z %. +%term clo_s %pi (closed (s E)) %<- (closed E) %. +%term clo_case + %pi (closed (case E1 E2 E3)) + %<- (closed E1) + %<- (closed E2) + %<- ({x exp} %pi (closed x) %-> (closed (E3 x))) %. +% Pairs +%term clo_pair %pi (closed (pair E1 E2)) %<- (closed E1) %<- (closed E2) %. +%term clo_fst %pi (closed (fst E)) %<- (closed E) %. +%term clo_snd %pi (closed (snd E)) %<- (closed E) %. +% Functions +%term clo_lam %pi (closed (lam E)) %<- ({x exp} %pi (closed x) %-> (closed (E x))) %. +%term clo_app %pi (closed (app E1 E2)) %<- (closed E1) %<- (closed E2) %. +% Definitions +%term clo_letv + %pi (closed (letv E1 E2)) + %<- (closed E1) + %<- ({x exp} %pi (closed x) %-> (closed (E2 x))) %. +%term clo_letn + %pi (closed (letn E1 E2)) + %<- (closed E1) + %<- ({x exp} %pi (closed x) %-> (closed (E2 x))) %. +% Recursion +%term clo_fix %pi (closed (fix E)) %<- ({x exp} %pi (closed x) %-> (closed (E x))) %. +%block lc [x exp] [u closed x]%. +%worlds (lc) (closed E) %. +%terminates E (closed E) %. +%covers closed %in %. +%total E (closed E) %. +%%% Open Mini-ML Expressions. +%%% Warning: one needs to assume that parameters are open! +%sort open %. +%name open %. +%mode open %in %. +% Natural Numbers +%term open_s %pi (open (s E)) %<- (open E) %. +%term open_case1 %pi (open (case E1 E2 E3)) %<- (open E1) %. +%term open_case2 %pi (open (case E1 E2 E3)) %<- (open E2) %. +%term open_case3 %pi (open (case E1 E2 E3)) %<- ({x exp} open (E3 x)) %. +% Pairs +%term open_pair1 %pi (open (pair E1 E2)) %<- (open E1) %. +%term open_pair2 %pi (open (pair E1 E2)) %<- (open E2) %. +%term open_fst %pi (open (fst E)) %<- (open E) %. +%term open_snd %pi (open (snd E)) %<- (open E) %. +% Functions +%term open_lam %pi (open (lam E)) %<- ({x exp} open (E x)) %. +%term open_app1 %pi (open (app E1 E2)) %<- (open E1) %. +%term open_app2 %pi (open (app E1 E2)) %<- (open E2) %. +% Definitions +%term open_letv1 %pi (open (letv E1 E2)) %<- (open E1) %. +%term open_letv2 %pi (open (letv E1 E2)) %<- ({x exp} open (E2 x)) %. +%term open_letn1 %pi (open (letn E1 E2)) %<- (open E1) %. +%term open_letn2 %pi (open (letn E1 E2)) %<- ({x exp} open (E2 x)) %. +% Recursion +%term open_fix %pi (open (fix E)) %<- ({x exp} open (E x)) %. +%block lo [x exp]%. +%worlds (lo) (open E) %. +%terminates E (open E) %. +% next reports z and #lo_x as missing +% %covers open +E. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/mini-ml/eval.lf b/new-tests/stelf-output/examples/mini-ml/eval.lf new file mode 100644 index 0000000..39e6fbe --- /dev/null +++ b/new-tests/stelf-output/examples/mini-ml/eval.lf @@ -0,0 +1,29 @@ +%%% Mini-ML Natural Semantics +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort eval %. +%name eval %. +%mode eval %in %out %. +% Natural Numbers +%term ev_z eval z z %. +%term ev_s %pi (eval (s E) (s V)) %<- (eval E V) %. +%term ev_case_z %pi (eval (case E1 E2 E3) V) %<- (eval E1 z) %<- (eval E2 V) %. +%term ev_case_s %pi (eval (case E1 E2 E3) V) %<- (eval E1 (s V1')) %<- (eval (E3 V1') V) %. +% Pairs +%term ev_pair %pi (eval (pair E1 E2) (pair V1 V2)) %<- (eval E1 V1) %<- (eval E2 V2) %. +%term ev_fst %pi (eval (fst E) V1) %<- (eval E (pair V1 V2)) %. +%term ev_snd %pi (eval (snd E) V2) %<- (eval E (pair V1 V2)) %. +% Functions +%term ev_lam eval (lam E) (lam E) %. +%term ev_app + %pi (eval (app E1 E2) V) + %<- (eval E1 (lam E1')) + %<- (eval E2 V2) + %<- (eval (E1' V2) V) %. +% Definitions +%term ev_letv %pi (eval (letv E1 E2) V) %<- (eval E1 V1) %<- (eval (E2 V1) V) %. +%term ev_letn %pi (eval (letn E1 E2) V) %<- (eval (E2 E1) V) %. +% Recursion +%term ev_fix %pi (eval (fix E) V) %<- (eval (E (fix E)) V) %. +%worlds () (eval D V) %. +% terminates E (eval E V). +%covers eval %in %out %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/mini-ml/eval1.lf b/new-tests/stelf-output/examples/mini-ml/eval1.lf new file mode 100644 index 0000000..1c90282 --- /dev/null +++ b/new-tests/stelf-output/examples/mini-ml/eval1.lf @@ -0,0 +1,25 @@ +% The literal translation of the LF signature into Elf +% This is not operationally adequate, because of the +% wrong order of subgoals. +%sort eval %. +% Natural Numbers +%term ev_z eval z z %. +%term ev_s {E exp} {V exp} %pi (eval E V) %-> (eval (s E) (s V)) %. +%term ev_case_z {E1 exp} {E2 exp} {E3 %pi exp %-> exp} {V exp} %pi (eval E1 z) %-> (eval E2 V) %-> (eval (case E1 E2 E3) V) %. +%term ev_case_s {E1 exp} {E2 exp} {E3 %pi exp %-> exp} {V1' exp} {V exp} %pi (eval E1 (s V1')) %-> (eval (E3 V1') V) %-> (eval (case E1 E2 E3) V) %. +% Pairs +%term ev_pair {E1 exp} {E2 exp} {V1 exp} {V2 exp} %pi (eval E1 V1) %-> (eval E2 V2) %-> (eval (pair E1 E2) (pair V1 V2)) %. +%term ev_fst {E exp} {V1 exp} {V2 exp} %pi (eval E (pair V1 V2)) %-> (eval (fst E) V1) %. +%term ev_snd {E exp} {V1 exp} {V2 exp} %pi (eval E (pair V1 V2)) %-> (eval (snd E) V2) %. +% Functions +%term ev_lam {E %pi exp %-> exp} eval (lam E) (lam E) %. +%term ev_app {E1 exp} {E2 exp} {E1' %pi exp %-> exp} {V2 exp} {V exp} + %pi (eval E1 (lam E1')) + %-> (eval E2 V2) + %-> (eval (E1' V2) V) + %-> (eval (app E1 E2) V) %. +% Definitions +%term ev_letv {E1 exp} {E2 %pi exp %-> exp} {V1 exp} {V exp} %pi (eval E1 V1) %-> (eval (E2 V1) V) %-> (eval (letv E1 E2) V) %. +%term ev_letn {E1 exp} {E2 %pi exp %-> exp} {V exp} %pi (eval (E2 E1) V) %-> (eval (letn E1 E2) V) %. +% Recursion +%term ev_fix {E %pi exp %-> exp} {V exp} %pi (eval (E (fix E)) V) %-> (eval (fix E) V) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/mini-ml/eval2.lf b/new-tests/stelf-output/examples/mini-ml/eval2.lf new file mode 100644 index 0000000..dfcf3f5 --- /dev/null +++ b/new-tests/stelf-output/examples/mini-ml/eval2.lf @@ -0,0 +1,26 @@ +% The literal translation of the LF signature into Elf +% omitting implicit quantifiers. +% This is not operationally adequate, because of the +% wrong order of subgoals. +%sort eval %. +% Natural Numbers +%term ev_z eval z z %. +%term ev_s %pi (eval E V) %-> (eval (s E) (s V)) %. +%term ev_case_z %pi (eval E1 z) %-> (eval E2 V) %-> (eval (case E1 E2 E3) V) %. +%term ev_case_s %pi (eval E1 (s V1')) %-> (eval (E3 V1') V) %-> (eval (case E1 E2 E3) V) %. +% Pairs +%term ev_pair %pi (eval E1 V1) %-> (eval E2 V2) %-> (eval (pair E1 E2) (pair V1 V2)) %. +%term ev_fst %pi (eval E (pair V1 V2)) %-> (eval (fst E) V1) %. +%term ev_snd %pi (eval E (pair V1 V2)) %-> (eval (snd E) V2) %. +% Functions +%term ev_lam eval (lam E) (lam E) %. +%term ev_app + %pi (eval E1 (lam E1')) + %-> (eval E2 V2) + %-> (eval (E1' V2) V) + %-> (eval (app E1 E2) V) %. +% Definitions +%term ev_letv %pi (eval E1 V1) %-> (eval (E2 V1) V) %-> (eval (letv E1 E2) V) %. +%term ev_letn %pi (eval (E2 E1) V) %-> (eval (letn E1 E2) V) %. +% Recursion +%term ev_fix %pi (eval (E (fix E)) V) %-> (eval (fix E) V) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/mini-ml/examples-quy.lf b/new-tests/stelf-output/examples/mini-ml/examples-quy.lf new file mode 100644 index 0000000..7fc379e --- /dev/null +++ b/new-tests/stelf-output/examples/mini-ml/examples-quy.lf @@ -0,0 +1,30 @@ +%%% Example of Mini-ML evaluation +%%% Evaluation +%query 1 _ _ eval (case z (s z) ([x exp] z)) V %. +%query 1 _ _ eval (app (lam ([x exp] x)) z) V %. +% Next fails (specify 0 solutions) +%query 0 _ _ eval (fst z) V %. +% Next does not terminate. +% eval (fix [x:exp] x) V. +% Double 1 to yield 2. +%query 1 _ _ eval (app (fix ([f exp] lam ([x exp] case x z ([x' exp] s (s (app f x')))))) (s z)) V %. +%%% Values +%query 1 _ _ value (pair z (s z)) %. +%query 0 _ _ value (fst (pair z (s z))) %. +%query 1 _ _ value (lam ([x] fst x)) %. +%%% Value Soundness +%query 1 _ _ vs (ev_case_z (ev_s ev_z) ev_z) P %. +%solve d0 : eval (case z (s z) ([x exp] z)) V %. +%query 1 _ _ vs d0 P %. +%%% Type Inference +%query 1 _ _ of (lam ([x] pair x (s x))) T %. +%query 1 _ _ of (lam ([x] x)) T %. +%query 1 _ _ {T1 tp} of (lam ([x exp] x)) (arrow T1 T1) %. +% Next query leaves constraints. +%query 1 _ _ of (lam ([x] x)) (F nat) %. +%query 1 _ _ of (letn (lam ([y] y)) ([f] pair (app f z) (app f (pair z z)))) T %. +%%% Last example +%define e0 exp letn (lam ([x] x)) ([f] letn (app f f) ([g] app g g)) %. +%solve p0 : of e0 T %. +%solve d0 : eval e0 V %. +%query 1 _ _ tps d0 p0 Q %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/mini-ml/examples1-quy.lf b/new-tests/stelf-output/examples/mini-ml/examples1-quy.lf new file mode 100644 index 0000000..e9d78da --- /dev/null +++ b/new-tests/stelf-output/examples/mini-ml/examples1-quy.lf @@ -0,0 +1,2 @@ +% These can be executed in configuration sources1.cfg +%define _ %the (eval (case z (s z) ([x exp] z)) (s z)) (ev_case_z z (s z) ([x exp] z) (s z) ev_z (ev_s z z ev_z)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/mini-ml/mini-ml.lf b/new-tests/stelf-output/examples/mini-ml/mini-ml.lf new file mode 100644 index 0000000..dbea8ee --- /dev/null +++ b/new-tests/stelf-output/examples/mini-ml/mini-ml.lf @@ -0,0 +1,16 @@ +%%% The Mini-ML Language +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +% Expressions +%sort exp %. +%name exp %. +%term z exp %. +%term s %pi exp %-> exp %. +%term case %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%term pair %pi exp %-> exp %-> exp %. +%term fst %pi exp %-> exp %. +%term snd %pi exp %-> exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term letv %pi exp %-> (%pi exp %-> exp) %-> exp %. +%term letn %pi exp %-> (%pi exp %-> exp) %-> exp %. +%term fix %pi (%pi exp %-> exp) %-> exp %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/mini-ml/reduce-thm.lf b/new-tests/stelf-output/examples/mini-ml/reduce-thm.lf new file mode 100644 index 0000000..b975ac6 --- /dev/null +++ b/new-tests/stelf-output/examples/mini-ml/reduce-thm.lf @@ -0,0 +1,5 @@ +% Evaluation is a strategy for reduction +%theorem evalRed : forall* {E exp} {V exp} forall {D eval E V} exists {R* E ==>* V} true %. +%prove 4 D (evalRed D _) %. +% Partial check if proof +%terminates D (evalRed D _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/mini-ml/reduce.lf b/new-tests/stelf-output/examples/mini-ml/reduce.lf new file mode 100644 index 0000000..388001f --- /dev/null +++ b/new-tests/stelf-output/examples/mini-ml/reduce.lf @@ -0,0 +1,36 @@ +%sort ==> %. +%name ==> %. +%prec %none 10 ==> %. +%mode ==> %in %out %. +%term red_z case z E1 E2 ==> E1 %. +%term red_s case (s E) E1 E2 ==> E2 E %. +%term red_fst fst (pair E1 E2) ==> E1 %. +%term red_snd snd (pair E1 E2) ==> E2 %. +%term red_app app (lam E1) E2 ==> E1 E2 %. +%term red_letv letv E1 E2 ==> E2 E1 %. +%term red_letn letn E1 E2 ==> E2 E1 %. +%term red_fix fix E ==> E (fix E) %. +%worlds () (==> D P) %. +% The weakly congruent, reflexive, transitive closure of ==>* +%sort ==>* %. +%name ==>* %. +%prec %none 10 ==>* %. +%mode ==>* %in %out %. +% Congruences +% no red*_z +%term red*_s %pi (s E ==>* s E') %<- (E ==>* E') %. +%term red*_case %pi (case E1 E2 E3 ==>* case E1' E2 E3) %<- (E1 ==>* E1') %. +%term red*_pair1 %pi (pair E1 E2 ==>* pair E1' E2) %<- (E1 ==>* E1') %. +%term red*_pair2 %pi (pair E1 E2 ==>* pair E1 E2') %<- (E2 ==>* E2') %. +%term red*_fst %pi (fst E ==>* fst E') %<- (E ==>* E') %. +%term red*_snd %pi (snd E ==>* snd E') %<- (E ==>* E') %. +% no red*_lam +%term red*_app1 %pi (app E1 E2 ==>* app E1' E2) %<- (E1 ==>* E1') %. +%term red*_app2 %pi (app E1 E2 ==>* app E1 E2') %<- (E2 ==>* E2') %. +%term red*_letv %pi (letv E1 E2 ==>* letv E1' E2) %<- (E1 ==>* E1') %. +%term red*_letn %pi (letn E1 E2 ==>* letn E1' E2) %<- (E1 ==>* E1') %. +% no red*_fix +%term red*_beta %pi (E ==>* E') %<- (E ==> E') %. +%term red*_refl E ==>* E %. +%term red*_trans %pi (E ==>* E'') %<- (E ==>* E') %<- (E' ==>* E'') %. +%worlds () (==>* D P) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/mini-ml/sources.lf b/new-tests/stelf-output/examples/mini-ml/sources.lf new file mode 100644 index 0000000..2b89803 --- /dev/null +++ b/new-tests/stelf-output/examples/mini-ml/sources.lf @@ -0,0 +1,8 @@ +%require %[ mini-ml %] %. +%require %[ eval %] %. +%require %[ value %] %. +%require %[ val-sound %] %. +%require %[ closed %] %. +%require %[ tp %] %. +%require %[ tpinf %] %. +%require %[ tp-preserve %] %. diff --git a/new-tests/stelf-output/examples/mini-ml/sources1.lf b/new-tests/stelf-output/examples/mini-ml/sources1.lf new file mode 100644 index 0000000..822e696 --- /dev/null +++ b/new-tests/stelf-output/examples/mini-ml/sources1.lf @@ -0,0 +1,3 @@ +%require %[ mini-ml %] %. +%require %[ eval1 %] %. +%require %[ examples1 %] %. diff --git a/new-tests/stelf-output/examples/mini-ml/stelf.toml b/new-tests/stelf-output/examples/mini-ml/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/mini-ml/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/mini-ml/test.lf b/new-tests/stelf-output/examples/mini-ml/test.lf new file mode 100644 index 0000000..b748e64 --- /dev/null +++ b/new-tests/stelf-output/examples/mini-ml/test.lf @@ -0,0 +1,13 @@ +%require %[ mini-ml %] %. +%require %[ tp %] %. +%require %[ eval %] %. +%require %[ value %] %. +%require %[ val-sound %] %. +%require %[ val-sound %] %. +%require %[ closed %] %. +%require %[ tpinf %] %. +%require %[ tp-preserve %] %. +%require %[ tp-preserve %] %. +%require %[ reduce %] %. +%require %[ reduce %] %. +%require %[ examples %] %. diff --git a/new-tests/stelf-output/examples/mini-ml/tp-preserve-thm.lf b/new-tests/stelf-output/examples/mini-ml/tp-preserve-thm.lf new file mode 100644 index 0000000..a0d699e --- /dev/null +++ b/new-tests/stelf-output/examples/mini-ml/tp-preserve-thm.lf @@ -0,0 +1,7 @@ +% Type preservation +% Partial check of hand-coded proof +%terminates D (tps D _ _) %. +%theorem tpp : forall* {E exp} {V exp} {T tp} forall {D eval E V} {P of E T} exists {Q of V T} true %. +% prove 1 D (tpp D _ _). +% Partial verification of generated proof +%terminates D (tpp D _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/mini-ml/tp-preserve.lf b/new-tests/stelf-output/examples/mini-ml/tp-preserve.lf new file mode 100644 index 0000000..be08d0d --- /dev/null +++ b/new-tests/stelf-output/examples/mini-ml/tp-preserve.lf @@ -0,0 +1,39 @@ +%%% Representation of the Proof of Type Preservation +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort tps %. +%mode tps %in %in %out %. +% Natural Numbers +%term tps_z tps ev_z tp_z tp_z %. +%term tps_s %pi (tps (ev_s D1) (tp_s P1) (tp_s Q1)) %<- (tps D1 P1 Q1) %. +%term tps_case_z %pi (tps (ev_case_z D2 D1) (tp_case P3 P2 P1) Q2) %<- (tps D2 P2 Q2) %. +%term tps_case_s + %pi (tps (ev_case_s D3 D1) (tp_case P3 P2 P1) Q3) + %<- (tps D1 P1 (tp_s Q1')) + %<- (tps D3 (P3 V1 Q1') Q3) %. +% Pairs +%term tps_pair + %pi (tps (ev_pair D2 D1) (tp_pair P2 P1) (tp_pair Q2 Q1)) + %<- (tps D1 P1 Q1) + %<- (tps D2 P2 Q2) %. +%term tps_fst %pi (tps (ev_fst D1) (tp_fst P1) Q1) %<- (tps D1 P1 (tp_pair Q2 Q1)) %. +%term tps_snd %pi (tps (ev_snd D1) (tp_snd P1) Q2) %<- (tps D1 P1 (tp_pair Q2 Q1)) %. +% Functions +%term tps_lam tps ev_lam (tp_lam P) (tp_lam P) %. +%term tps_app + %pi (tps (ev_app D3 D2 D1) (tp_app P2 P1) Q3) + %<- (tps D1 P1 (tp_lam Q1')) + %<- (tps D2 P2 Q2) + %<- (tps D3 (Q1' V2 Q2) Q3) %. +% Definitions +%term tps_letv + %pi (tps (ev_letv D2 D1) (tp_letv P2 P1) Q2) + %<- (tps D1 P1 Q1) + %<- (tps D2 (P2 V1 Q1) Q2) %. +%term tps_letn %pi (tps (ev_letn D2) (tp_letn P2 P1) Q2) %<- (tps D2 P2 Q2) %. +% Recursion +%term tps_fix %pi (tps (ev_fix D1) (tp_fix P1) Q1) %<- (tps D1 (P1 _ (tp_fix P1)) Q1) %. +%%% Termination of type preservation judgment +%worlds () (tps D _ _) %. +%terminates D (tps D _ _) %. +%covers tps %in %in %out %. +%total D (tps D P _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/mini-ml/tp.lf b/new-tests/stelf-output/examples/mini-ml/tp.lf new file mode 100644 index 0000000..393d9c0 --- /dev/null +++ b/new-tests/stelf-output/examples/mini-ml/tp.lf @@ -0,0 +1,10 @@ +%%% Mini-ML types. +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort tp %. +%name tp %. +%term nat tp %. +% Natural Numbers +%term cross %pi tp %-> tp %-> tp %. +% Pairs +%term arrow %pi tp %-> tp %-> tp %. +% Functions \ No newline at end of file diff --git a/new-tests/stelf-output/examples/mini-ml/tpinf.lf b/new-tests/stelf-output/examples/mini-ml/tpinf.lf new file mode 100644 index 0000000..8ae6490 --- /dev/null +++ b/new-tests/stelf-output/examples/mini-ml/tpinf.lf @@ -0,0 +1,31 @@ +%%% The Mini-ML typing rules which can be used as a complete +%%% implementation of type inference. +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort of %. +%name of %. +%mode of %in %star %. +% Natural Numbers +%term tp_z of z nat %. +%term tp_s %pi (of (s E) nat) %<- (of E nat) %. +%term tp_case + %pi (of (case E1 E2 E3) T) + %<- (of E1 nat) + %<- (of E2 T) + %<- ({x exp} %pi (of x nat) %-> (of (E3 x) T)) %. +% Pairs +%term tp_pair %pi (of (pair E1 E2) (cross T1 T2)) %<- (of E1 T1) %<- (of E2 T2) %. +%term tp_fst %pi (of (fst E) T1) %<- (of E (cross T1 T2)) %. +%term tp_snd %pi (of (snd E) T2) %<- (of E (cross T1 T2)) %. +% Functions +%term tp_lam %pi (of (lam E) (arrow T1 T2)) %<- ({x exp} %pi (of x T1) %-> (of (E x) T2)) %. +%term tp_app %pi (of (app E1 E2) T1) %<- (of E1 (arrow T2 T1)) %<- (of E2 T2) %. +% Definitions +%term tp_letv + %pi (of (letv E1 E2) T2) + %<- (of E1 T1) + %<- ({x exp} %pi (of x T1) %-> (of (E2 x) T2)) %. +%term tp_letn %pi (of (letn E1 E2) T2) %<- (of E1 T1) %<- (of (E2 E1) T2) %. +% Recursion +%term tp_fix %pi (of (fix E) T) %<- ({x exp} %pi (of x T) %-> (of (E x) T)) %. +%block l {T tp} [x exp] [d of x T]%. +%worlds (l) (of E T) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/mini-ml/val-sound-thm.lf b/new-tests/stelf-output/examples/mini-ml/val-sound-thm.lf new file mode 100644 index 0000000..d7d8f02 --- /dev/null +++ b/new-tests/stelf-output/examples/mini-ml/val-sound-thm.lf @@ -0,0 +1,8 @@ +% Value soundness +% Partial check of hand-coded proof +%terminates D (vs D _) %. +% Automatic proof generation +%theorem vls : forall* {E exp} {V exp} forall {D eval E V} exists {P value V} true %. +%prove 3 D (vls D _) %. +% Partial verification of generated proof +%terminates D (vls D _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/mini-ml/val-sound.lf b/new-tests/stelf-output/examples/mini-ml/val-sound.lf new file mode 100644 index 0000000..1305e4d --- /dev/null +++ b/new-tests/stelf-output/examples/mini-ml/val-sound.lf @@ -0,0 +1,25 @@ +%%% Proof of Value Soundness +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort vs %. +%mode vs %in %out %. +% Natural Numbers +%term vs_z vs ev_z val_z %. +%term vs_s %pi (vs (ev_s D1) (val_s P1)) %<- (vs D1 P1) %. +%term vs_case_z %pi (vs (ev_case_z D2 D1) P2) %<- (vs D2 P2) %. +%term vs_case_s %pi (vs (ev_case_s D3 D1) P3) %<- (vs D3 P3) %. +% Pairs +%term vs_pair %pi (vs (ev_pair D2 D1) (val_pair P2 P1)) %<- (vs D1 P1) %<- (vs D2 P2) %. +%term vs_fst %pi (vs (ev_fst D') P1) %<- (vs D' (val_pair P2 P1)) %. +%term vs_snd %pi (vs (ev_snd D') P2) %<- (vs D' (val_pair P2 P1)) %. +% Functions +%term vs_lam vs ev_lam val_lam %. +%term vs_app %pi (vs (ev_app D3 D2 D1) P3) %<- (vs D3 P3) %. +% Definitions +%term vs_letv %pi (vs (ev_letv D2 D1) P2) %<- (vs D2 P2) %. +%term vs_letn %pi (vs (ev_letn D2) P2) %<- (vs D2 P2) %. +% Recursion +%term vs_fix %pi (vs (ev_fix D1) P1) %<- (vs D1 P1) %. +%worlds () (vs D P) %. +%terminates D (vs D _) %. +%covers vs %in %out %. +%total D (vs D _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/mini-ml/value.lf b/new-tests/stelf-output/examples/mini-ml/value.lf new file mode 100644 index 0000000..70a51b6 --- /dev/null +++ b/new-tests/stelf-output/examples/mini-ml/value.lf @@ -0,0 +1,11 @@ +%%% Definition of Values +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort value %. +%name value %. +%mode value %in %. +%term val_z value z %. +%term val_lam value (lam E) %. +%term val_s %pi (value (s V)) %<- (value V) %. +%term val_pair %pi (value (pair V1 V2)) %<- (value V1) %<- (value V2) %. +% terminates D (value D). +%worlds () (value D) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/modal/modal-judge.lf b/new-tests/stelf-output/examples/modal/modal-judge.lf new file mode 100644 index 0000000..0fa5afe --- /dev/null +++ b/new-tests/stelf-output/examples/modal/modal-judge.lf @@ -0,0 +1,79 @@ +% Intuitionistic Modal S4 based upon Pfenning and Davies +%sort w %. +% worlds. +%sort o %. +% propositions +% Propositions +%term ⊤ o %. +%term ⊥ o %. +%term ⇒ %pi o %-> o %-> o %. +%prec %right 5 ⇒ %. +%term □ %pi o %-> o %. +%term <> %pi o %-> o %. +% Negation is an abbreviation as usual. +%inline ¬ (%pi o %-> o) [x o] x ⇒ ⊥ %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Natural deduction version +%sort true %. +%sort poss %. +%term ⊤I true ⊤ W %. +%term ⊥E %pi (true ⊥ W) %-> (true P W) %. +%term ⇒I %pi (%pi (true P1 W) %-> (true P2 W)) %-> (true (P1 ⇒ P2) W) %. +%term ⇒E %pi (true (P1 ⇒ P2) W) %-> (true P1 W) %-> (true P2 W) %. +%term □I %pi ({W' w} true P W') %-> (true (□ P) W) %. +%term □E %pi (true (□ P1) W) %-> ({W' w} %pi (true P1 W') %-> (true P2 W)) %-> (true P2 W) %. +%term <>I %pi (poss P) %-> (true (<> P) W) %. +%term <>E %pi (true (<> P1) W) %-> (%pi (true P1 W) %-> (poss P2)) %-> (poss P2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%{ +%theorem k_axiom1 : exists {D:{W:w}{A:o}{B:o} true ((□ (A ⇒ B)) ⇒ (□ A) ⇒ (□ B)) W} true. +%establish 6 {} (k_axiom1 D). +}% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Sequent calculus version +%sort hyp %. +%sort conc %. +%sort poss %. +%term ⊤R conc ⊤ W %. +%term ⊥L %pi (hyp ⊥ W) %-> (conc P W') %. +%term axiom %pi (hyp P W) %-> (conc P W) %. +%term ⇒R %pi (%pi (hyp P1 W) %-> (conc P2 W)) %-> (conc (P1 ⇒ P2) W) %. +%term ⇒L + %pi (%pi (hyp P2 W) %-> (conc P3 W')) + %-> (conc P1 W) + %-> (hyp (P1 ⇒ P2) W) + %-> (conc P3 W') %. +%term □R %pi ({W' w} conc P W') %-> (conc (□ P) W) %. +%term □L %pi (%pi (hyp P1 W') %-> (conc P2 W'')) %-> (hyp (□ P1) W) %-> (conc P2 W'') %. +%term cnv %pi (conc P W) %-> (poss P) %. +%term <>R %pi (poss P) %-> (conc (<> P) W) %. +%term <>L %pi ({W' w} %pi (hyp P1 W') %-> (poss P2)) %-> (hyp (<> P1) W) %-> (poss P2) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%theorem dn_peirce : exists {D {W w} {A o} {B o} conc (¬ (¬ (A ⇒ B ⇒ A ⇒ A))) W} true %. +%prove 10 {} (dn_peirce D) %. +%theorem k_axiom1 : exists {D {W w} {A o} {B o} conc (□ (A ⇒ B) ⇒ (□ A) ⇒ (□ B)) W} true %. +%establish 20 {} (k_axiom1 D) %. +%theorem k_axiom2 : exists {D {W w} {A o} {B o} conc (□ (A ⇒ B) ⇒ (<> A) ⇒ (<> B)) W} true %. +%establish 20 {} (k_axiom2 D) %. +%theorem t_axiom1 : exists {D {W w} {A o} conc (□ A ⇒ A) W} true %. +%establish 50 {} (t_axiom1 D) %. +%theorem t_axiom2 : exists {D {W w} {A o} conc (A ⇒ (<> A)) W} true %. +%establish 50 {} (t_axiom2 D) %. +%theorem 4_axiom1 : exists {D {W w} {A o} conc (□ A ⇒ (□ (□ A))) W} true %. +%establish 50 {} (4_axiom1 D) %. +%theorem 4_axiom2 : exists {D {W w} {A o} conc (<> (<> A) ⇒ (<> A)) W} true %. +%establish 50 {} (4_axiom2 D) %. +%{ +%theorem modaleq1 : exists {D:{W:w}{A:o} conc ((□ A) ⇒ (¬ (<> (¬ A)))) W} true. +%prove 50 {} (modaleq1 D). + +%theorem modaleq2 : exists {D:{W:w}{A:o} conc ((<> A) ⇒ (¬ (□ (¬ A)))) W} true. +%prove 30 {} (modaleq2 D). +}% +%{ +%theorem dn_modaleq2 : exists {D:{W:w}{A:o} conc (¬ (¬ ((<> A) ⇒ (¬ (□ (¬ A)))) )) W} true. +%prove 10 {} (dn_modaleq2 D). + +%theorem dn_modaleq1 : exists {D:{W:w}{A:o} conc (¬ (¬ ((□ A) ⇒ (¬ (<> (¬ A)))) )) W} true. +%prove 10 {} (dn_modaleq1 D). +}% \ No newline at end of file diff --git a/new-tests/stelf-output/examples/modal/modal.lf b/new-tests/stelf-output/examples/modal/modal.lf new file mode 100644 index 0000000..d98e6c5 --- /dev/null +++ b/new-tests/stelf-output/examples/modal/modal.lf @@ -0,0 +1,140 @@ +% An implementation of intutitionistic modal logic based upon Alex +% Simpson's thesis +%sort w %. +% worlds +%sort o %. +% propositions +% Relatedness between worlds. +%sort R %. +%prec %left 5 R %. +%tabled R %. +% Propositions +%term ⊤ o %. +%term ⊥ o %. +%term ⇒ %pi o %-> o %-> o %. +%prec %right 5 ⇒ %. +%term □ %pi o %-> o %. +%term <> %pi o %-> o %. +% Negation is an abbreviation as usual. +%inline ¬ (%pi o %-> o) [x o] x ⇒ ⊥ %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Natural deduction version +%sort @ %. +%prec %left 5 @ %. +%tabled @ %. +%term ⊤I ⊤ @ W %. +%term ⊥E %pi (P @ W) %<- (⊥ @ W) %. +%term ⇒I %pi (P1 ⇒ P2 @ W) %<- (%pi (P1 @ W) %-> (P2 @ W)) %. +%term ⇒E %pi (P2 @ W) %<- (P1 ⇒ P2 @ W) %<- (P1 @ W) %. +%term □I %pi (□ P @ W1) %<- ({W2 w} %pi (W1 R W2) %-> (P @ W2)) %. +%term □E %pi (P @ W2) %<- (□ P @ W1) %<- (W1 R W2) %. +%term <>I %pi (<> P @ W1) %<- (W1 R W2) %<- (P @ W2) %. +%term <>E + %pi (P2 @ W) + %<- (<> P1 @ W1) + %<- ({W2 w} %pi (W1 R W2) %-> (P1 @ W2) %-> (P2 @ W)) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%{ +%theorem identity : exists {D:{W:w}{P:o} (P ⇒ P) @ W} true. +%establish 5 {} (identity D). +}% +%{ +%theorem false : exists {D:{W:w} ⊥ @ W} true. +%establish 5 {} (false D). +}% +%{ +%theorem box_taut0 : exists {D:{W:w}{A:o} □ (A ⇒ A) @ W} true. +%establish 10 {} (box_taut0 D). + +%theorem box_taut1 : exists {D:{W:w}{A:o}{B:o} □ (A ⇒ B ⇒ A) @ W} true. +%establish 10 {} (box_taut1 D). + +%theorem box_taut2 : exists {D:{W:w}{A:o}{B:o}{C:o} ((A ⇒ B ⇒ C) ⇒ (A ⇒ B) ⇒ (A ⇒ C)) @ W} true. +%establish 6 {} (box_taut2 D). + +%theorem k_axiom1 : exists {D:{W:w}{A:o}{B:o} ((□ (A ⇒ B)) ⇒ (□ A) ⇒ (□ B)) @ W} true. +%establish 6 {} (k_axiom1 D). +}% +%{ Can't seem to be done with depth 6 + +%theorem peirce : exists {D:{W:w}{A:o}{B:o} (¬ (¬ (((A ⇒ B) ⇒ A)⇒ A))) @ W} true. +%establish 6 {} (peirce D). + +%theorem k_axiom2 : exists {D:{W:w}{A:o}{B:o} ((□ (A ⇒ B)) ⇒ (<> A) ⇒ (<> B)) @ W} true. +%establish 6 {} (k_axiom2 D). +}% +%{ +%theorem modal_equiv1 : exists {D:{W:w}{A:o} ((□ A) ⇒ (¬ (<> (¬ A)))) @ W} true. +%establish 5 {} (modal_equiv1 D). + +%theorem modal_equiv2 : exists {D:{W:w}{A:o} ((<> A) ⇒ (¬ (□ (¬ A)))) @ W} true. +%establish 5 {} (modal_equiv2 D). +}% +%{ +% Can prove these if we add reflexivity +refl : W R W. + +%theorem t_axiom1 : exists {D:{W:w}{A:o} ((□ A) ⇒ A) @ W} true. +%establish 6 {} (t_axiom1 D). + +%theorem t_axiom2 : exists {D:{W:w}{A:o} (A ⇒ (<> A)) @ W} true. +%establish 6 {} (t_axiom2 D). +}% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Sequent calculus version +%sort hyp %. +%sort conc %. +%tabled hyp %. +%tabled conc %. +%term axiom %pi (hyp P W) %-> (conc P W) %. +%term ⊤R conc ⊤ W %. +%term ⊥L %pi (hyp ⊥ W1) %-> (conc P W2) %. +%term ⇒L + %pi (%pi (hyp P2 W1) %-> (conc P3 W2)) + %-> (conc P1 W1) + %-> (hyp (P1 ⇒ P2) W1) + %-> (conc P3 W2) %. +%term ⇒R %pi (%pi (hyp P1 W) %-> (conc P2 W)) %-> (conc (P1 ⇒ P2) W) %. +%term □L + %pi (%pi (hyp P1 W2) %-> (conc P2 W3)) + %-> (W1 R W2) + %-> (hyp (□ P1) W1) + %-> (conc P2 W3) %. +%term □R %pi (%pi (W1 R W2) %-> (conc P W2)) %-> (conc (□ P) W1) %. +%term <>L + %pi (%pi (W1 R W2) %-> (hyp P1 W2) %-> (conc P2 W3)) + %-> (hyp (<> P1) W1) + %-> (conc P2 W3) %. +%term <>R %pi (conc P W2) %-> (W1 R W2) %-> (conc (<> P) W1) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%{ +% Identity +%querytabled * * D : {W:w} {P:o} conc ((P ⇒ P)) W. + +% Double negation of Peirce's law +%querytabled * * D : {W:w} {A:o}{B:o}conc (¬ (¬ (((A ⇒ B) ⇒ A) ⇒ A))) W. + +% K axiom 1 +%querytabled * * D : {W:w}{A:o}{B:o} conc ((□ (A ⇒ B)) ⇒ (□ A) ⇒ (□ B)) W. + +}% +%{ Can't be proven -- correct +%theorem false : exists {D:{W:w} conc ⊥ W} true. +%establish 10 {} (false D). +}% +%{ Can't be proven -- correct +%theorem peirce : exists {D:{W:w}{A:o}{B:o} conc (((A ⇒ B) ⇒ A)⇒ A) W} true. +%establish 20 {} (peirce D). +}% +%theorem dn_peirce : exists {D {W w} {A o} {B o} conc (¬ (¬ (A ⇒ B ⇒ A ⇒ A))) W} true %. +%prove 10 {} (dn_peirce D) %. +%theorem k_axiom1 : exists {D {W w} {A o} {B o} conc (□ (A ⇒ B) ⇒ (□ A) ⇒ (□ B)) W} true %. +%prove 10 {} (k_axiom1 D) %. +%theorem modaleq1 : exists {D {W w} {A o} conc (□ A ⇒ (¬ (<> (¬ A)))) W} true %. +%prove 10 {} (modaleq1 D) %. +%theorem modaleq2 : exists {D {W w} {A o} conc (<> A ⇒ (¬ (□ (¬ A)))) W} true %. +%prove 10 {} (modaleq2 D) %. +%theorem dn_modaleq1 : exists {D {W w} {A o} conc (¬ (¬ (□ A ⇒ (¬ (<> (¬ A)))))) W} true %. +%prove 10 {} (dn_modaleq1 D) %. +%theorem dn_modaleq2 : exists {D {W w} {A o} conc (¬ (¬ (<> A ⇒ (¬ (□ (¬ A)))))) W} true %. +%prove 10 {} (dn_modaleq2 D) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/polylam/examples-quy.lf b/new-tests/stelf-output/examples/polylam/examples-quy.lf new file mode 100644 index 0000000..c5cc3f2 --- /dev/null +++ b/new-tests/stelf-output/examples/polylam/examples-quy.lf @@ -0,0 +1 @@ +% Currently, these is no program to run---just testing definitions. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/polylam/polylam.lf b/new-tests/stelf-output/examples/polylam/polylam.lf new file mode 100644 index 0000000..e3adb8c --- /dev/null +++ b/new-tests/stelf-output/examples/polylam/polylam.lf @@ -0,0 +1,17 @@ +%sort tp %. +%term => %pi tp %-> tp %-> tp %. +%prec %right 10 => %. +%term all %pi (%pi tp %-> tp) %-> tp %. +%sort tm %. +%term lam %pi (%pi (tm A) %-> (tm B)) %-> (tm (A => B)) %. +%term app %pi (tm (A => B)) %-> (tm A) %-> (tm B) %. +%term tlam %pi ({a tp} tm (A a)) %-> (tm (all A)) %. +%term tapp %pi (tm (all A)) %-> ({B tp} tm (A B)) %. +%define nat all ([a] a => (a => a) => a) %. +%define zero tlam ([a] lam ([z tm a] lam ([s tm (a => a)] z))) %. +%define succ lam ([x tm nat] tlam ([a tp] lam ([z tm a] lam ([s tm (a => a)] app s (app (app (tapp x a) z) s))))) %. +%define succ' (tm (nat => nat)) lam ([x] tlam ([a] lam ([z] lam ([s] app s (app (app (tapp x a) z) s))))) %. +%define plus lam ([x tm nat] lam ([y tm nat] app (app (tapp y nat) x) succ)) %. +%define times lam ([x tm nat] lam ([y tm nat] app (app (tapp y nat) zero) (app plus x))) %. +%define exp lam ([x tm nat] lam ([y tm nat] app (app (tapp y nat) (app succ zero)) (app times x))) %. +% idnat = [x: tm nat] x. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/polylam/sources.lf b/new-tests/stelf-output/examples/polylam/sources.lf new file mode 100644 index 0000000..23f90e5 --- /dev/null +++ b/new-tests/stelf-output/examples/polylam/sources.lf @@ -0,0 +1 @@ +%require %[ polylam %] %. diff --git a/new-tests/stelf-output/examples/polylam/stelf.toml b/new-tests/stelf-output/examples/polylam/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/polylam/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/polylam/test.lf b/new-tests/stelf-output/examples/polylam/test.lf new file mode 100644 index 0000000..e76cba2 --- /dev/null +++ b/new-tests/stelf-output/examples/polylam/test.lf @@ -0,0 +1,2 @@ +%require %[ polylam %] %. +%require %[ examples %] %. diff --git a/new-tests/stelf-output/examples/poplmark/1a.lf b/new-tests/stelf-output/examples/poplmark/1a.lf new file mode 100644 index 0000000..bc09e56 --- /dev/null +++ b/new-tests/stelf-output/examples/poplmark/1a.lf @@ -0,0 +1,381 @@ +%% The metatheory of algorithmic F-sub subtyping. +%% By Michael Ashley-Rollman, Karl Crary, and Robert Harper. +%%%%% Syntax %%%%% +%sort tp %. +%name tp %. +%term top tp %. +%term arrow %pi tp %-> tp %-> tp %. +%term forall %pi tp %-> (%pi tp %-> tp) %-> tp %. +%%%%% Semantics %%%%% +%sort assm %. +%% subtyping assumptions +%sort sub %. +%% subtyping judgement +%{ + +The object language (OL) context + + x1 <: T1, ..., xn <: Tn + +is represented by + + x1:tp, d1:assm x1 T1, ..., xn:tp, dn:assm xn Tn + +This is the only way an assm judgement can arise, so we can identify +variables as those types T such that a judgement assm T _ is available. + +}% +%term sub_top sub _ top %. +%term sub_refl %pi (sub X X) %<- (assm X _) %. +%term sub_trans %pi (sub X T) %<- (assm X U) %<- (sub U T) %. +%term sub_arrow %pi (sub (arrow S1 S2) (arrow T1 T2)) %<- (sub T1 S1) %<- (sub S2 T2) %. +%term sub_forall + %pi (sub (forall S1 ([x] S2 x)) (forall T1 ([x] T2 x))) + %<- (sub T1 S1) + %<- ({x} %pi (assm x T1) %-> (sub (S2 x) (T2 x))) %. +%%%%% Variables %%%%% +%{ + +In the proof, it will be necessary to have a way to identify variables +that does not become entangled with dependencies. The assm judgement +does become so entangled, because "assm X T" is dependent on T. So we +introduce a new judgement var, that mentions only the variable in +question. + +}% +%sort var %. +%%%% Blocks %%%%% +%{ + +In the proof, we will maintain the invariant that whenever we add a +variable (x:tp) and its subtyping assumption (d:assm x t, for some t) +to the context, we will also add a variable assumption (dv:var x). +This will allow us to prove that whenever "assm X T" is present, so is +"var X". This lemma is called assm_var below. + +}% +%sort assm_var %. +%mode assm_var %in %out %. +%{ + +Since the cases for assm_var depend on assumptions that are on only in +scope once we extend the context, we must put those cases into the +context with them. Thus, in addition to adding var assumptions, we +will also add assm_var assumptions. Hence, we have the following +block from which to build contexts: + +}% +%block bind {t tp} [x tp] [dv var x] [d assm x t] [thm assm_var d dv]%. +%{ + +In addition to the bind block, we have (for highly technical reasons, +explained later) two other blocks that arise: + + - Vblock provides variables but not subtyping assumptions. Since it + does not add an assm assumption, it need not add a case for + assm_var. However, it does still provide a var assumption, so it + remains true that a var assumption is available for every variable. + + - Ablock provides a subtyping assumption, but not the variable + itself. The some portion of the block asserts that a var + assumption exists for the variable in question. Since ablock adds + an assm assumption, it must also add a case for assm_var. + +}% +%block vblock [t tp] [d var t]%. +%block ablock {x tp} {dv var x} {t tp} [d assm x t] [thm assm_var d dv]%. +%{ + +Assm_var is true in any context made up from bind, vblock, and ablock +blocks. + +}% +%worlds (bind vblock ablock) (assm_var _ _) %. +%total {} (assm_var _ _) %. +%%%%% Reductio ad absurdio %%%%% +%% The underivable judgement. +%sort false %. +%% False implies any subtyping judgement you like. +%sort raa_sub %. +%mode {%in S tp} {%in T tp} {%in X1 false} {%out X2 sub S T} raa_sub X1 X2 %. +%worlds (bind vblock ablock) (raa_sub _ _) %. +%total {} (raa_sub _ _) %. +%%%%% Various things are not variables %%%%% +%{ + +In any world of interest (that is, any world made from bind, vblock, +and ablock), we never add a var assumption for anything but a fresh +variable. In particular, we will never have a var assumption for any +top, arrow, or forall type. + +Put more pithily: top, arrow, and forall are not variables. + +}% +%sort top_var_contra %. +%mode top_var_contra %in %out %. +%worlds (bind vblock ablock) (top_var_contra _ _) %. +%total {} (top_var_contra _ _) %. +%sort arrow_var_contra %. +%mode arrow_var_contra %in %out %. +%worlds (bind vblock ablock) (arrow_var_contra _ _) %. +%total {} (arrow_var_contra _ _) %. +%sort forall_var_contra %. +%mode forall_var_contra %in %out %. +%worlds (bind vblock ablock) (forall_var_contra _ _) %. +%total {} (forall_var_contra _ _) %. +%%%%% Natural numbers %%%%% +%sort nat %. +%name nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort nat_eq %. +%term nat_eq_ nat_eq N N %. +%%%%% Transivity and Narrowing %%%%% +%{ + +The main proof for transitivity and narrowing. The proof is by +induction on: + + (1st) the type Q + (2nd) the clause, where we take trans* < narrow* + (that is, narrow* can always call trans* with no change in Q, + but when trans* calls narrow*, Q must decrease) + (3rd) in trans*, the derivation of sub S Q, + and in narrow*, the derivation of sub M N + +Twelf requires that induction arguments be explicit, hence the first +two arguments to trans* and narrow*. The third argument ensures that +the clause number is correct for the clause. (That is, the clause +number is zero in trans* and one in narrow*). + + +The statement of trans* is otherwise straightforward. The statement +of narrow* involves one additional subtlety: + +In narrow*, we must identify the variable with respect to which we are +narrowing. The most natural way to do so is to abstract over that +variable; thus taking an argument of type: + + {X:tp} var X -> assm X Q -> sub M N + +Unfortunately, we run afoul of a complication. Obviously, the LF +context may contain dependencies only on other things in the LF +context. If we abstract over the variable of interest, that variable +is not in the context, and therefore nothing in the context may depend +on it. As a result, we effectively can narrow only with respect to +the last variable. However, this is not strong enough for the proof +to go through, since we must descend inside forall types. + +Thus, when we are narrowing with respect to a variable X, we must find +a way to move assumptions that depend on X to the left of it without +violating scoping constraints. If we maintain the context in its +usual form: + + t1:tp, dv1:var t1, d1:assm t1 T1, ..., dn:tp, dvn:var tn, dn:assm tn Tn + +then this is impossible. However, we may weaken our context +assumption to allow variables to appear apart from their typing +assumptions. To adhere to scoping constraints, we must keep +dependencies to the right of the variables they depend on, but we need +not keep them to the right of those variables typing assumptions. + +For example, suppose we encounter: + + ..., ti:tp, dvi:var ti, di:assm ti Ti, tj:tp, dvj:var tj, dj:assm tj Tj(ti) + +and we wish to move tj out of the way. Then we produce: + + ..., ti:tp, dvi:var ti, tj:tp, dvj:var tj, dj:assm tj Tj(ti), di:assm ti Ti + +This keeps the subtyping assumption di last, and violates no scoping constraints. +Thus, we identify the variable for narrowing not by abstracting over +the entire variable, but only over its subtyping assumption. + +As a consequence of this, we must now deal with variables bereft of +their subtyping assumptions (vblock), and with disembodied subtyping +assumptions (ablock). Note that disembodied subtyping assumptions are +substantially at odds with our usual experience, and we must +reconsider whether the proof works at all in their presence. + +Fortunately, with some minor modifications, the proof still works. +The only issue that arises is that it now appears as though the refl +and trans might apply to non-variable types, since those types might +now have subtyping assumptions. However, the assm_var lemma applies, +and states that subtyping assumptions are available only for +variables, so a contradiction arises in each case where we suppose a +subtyping assumption for a non-variable. + +}% +%sort trans* {Q tp} {Ncase nat} %. +%sort narrow* {Q tp} {Ncase nat} %. +%mode trans* %in %in %in %in %in %out %. +%mode narrow* %in %in %in %in %in %in %out %. +%scope narrow* %term top trans* _ _ _ D sub_top sub_top %. +%term refl trans* _ _ _ (sub_refl _) D D %. +%scope narrow* %term trans* + %pi (trans* _ _ nat_eq_ (sub_trans D2 D1) D (sub_trans D' D1)) + %<- (trans* _ _ nat_eq_ D2 D D') %. +%term arrow + %pi (trans* _ _ nat_eq_ (sub_arrow D1b D1a) (sub_arrow D2b D2a) (sub_arrow Db Da)) + %<- (trans* _ _ nat_eq_ D2a D1a Da) + %<- (trans* _ _ nat_eq_ D1b D2b Db) %. +%scope narrow* %term forall + %pi (trans* _ _ _ (sub_forall D1b D1a) (sub_forall D2b D2a) (sub_forall Db Da)) + %<- (trans* _ _ nat_eq_ (%the (sub T1 T2) D2a) D1a Da) + %<- ({x} {dv var x} narrow* _ _ nat_eq_ dv ([d] D1b x d) D2a ([d] D1b' x d)) + %<- ({x} {dv var x} {d assm x T1} %pi (assm_var d dv) %-> (trans* _ _ nat_eq_ (D1b' x d) (D2b x d) (Db x d))) %. +%% Contradiction cases, for dealing with disembodied subtyping assumptions for non-variables. +%term _ + %pi (trans* top _ _ _ (sub_refl Dassm) D) + %<- (assm_var Dassm Dvar) + %<- (top_var_contra Dvar Dfalse) + %<- (raa_sub Dfalse D) %. +%term _ + %pi (trans* (arrow _ _) _ _ _ (sub_refl Dassm) D) + %<- (assm_var Dassm Dvar) + %<- (arrow_var_contra Dvar Dfalse) + %<- (raa_sub Dfalse D) %. +%term _ + %pi (trans* (forall _ _) _ _ _ (sub_refl Dassm) D) + %<- (assm_var Dassm Dvar) + %<- (forall_var_contra Dvar Dfalse) + %<- (raa_sub Dfalse D) %. +%term _ + %pi (trans* top _ _ _ (sub_trans _ Dassm) D) + %<- (assm_var Dassm Dvar) + %<- (top_var_contra Dvar Dfalse) + %<- (raa_sub Dfalse D) %. +%term _ + %pi (trans* (arrow _ _) _ _ _ (sub_trans _ Dassm) D) + %<- (assm_var Dassm Dvar) + %<- (arrow_var_contra Dvar Dfalse) + %<- (raa_sub Dfalse D) %. +%term _ + %pi (trans* (forall _ _) _ _ _ (sub_trans _ Dassm) D) + %<- (assm_var Dassm Dvar) + %<- (forall_var_contra Dvar Dfalse) + %<- (raa_sub Dfalse D) %. +%term closed narrow* _ _ _ Dvar ([d] D) _ ([d] D) %. +%term refl narrow* _ _ _ Dvar ([d] sub_refl d) _ ([d] sub_refl d) %. +%term trans + %pi (narrow* _ _ nat_eq_ Dvar ([d] sub_trans (D d) Dassm) Dsub ([d] sub_trans (D' d) Dassm)) + %<- (narrow* _ _ nat_eq_ Dvar D Dsub D') %. +%term trans + %pi (narrow* _ _ nat_eq_ Dvar ([d] sub_trans (D d) d) Dsub ([d] sub_trans (D'' d) d)) + %<- (narrow* _ _ nat_eq_ Dvar D Dsub D') + %<- ({d assm X Q} %pi (assm_var d Dvar) %-> (trans* _ _ nat_eq_ Dsub (D' d) (D'' d))) %. +%term arrow + %pi (narrow* _ _ nat_eq_ Dvar ([d] sub_arrow (D2 d) (D1 d)) Dsub ([d] sub_arrow (D2' d) (D1' d))) + %<- (narrow* _ _ nat_eq_ Dvar D1 Dsub D1') + %<- (narrow* _ _ nat_eq_ Dvar D2 Dsub D2') %. +%term forall + %pi (narrow* _ _ nat_eq_ Dvar ([d] sub_forall (D2 d) (D1 d)) Dsub ([d] sub_forall (D2' d) (D1' d))) + %<- (narrow* _ _ nat_eq_ Dvar D1 Dsub D1') + %<- ({x'} {dv'} {d'} + %pi (assm_var d' dv') + %-> (narrow* _ _ nat_eq_ Dvar ([d] D2 d x' d') Dsub ([d] D2' d x' d'))) %. +%worlds (bind vblock ablock) (trans* _ _ _ _ _ _) (narrow* _ _ _ _ _ _ _) %. +%total {(Q1 Q2) (N1 N2) (D1 D2)} (trans* Q1 N1 _ D1 _ _) (narrow* Q2 N2 _ _ D2 _ _) %. +%%%%% Peroration %%%%% +%{ + +At this level we no longer need our context mangling, so trans and +narrow are proven for worlds built from only bind. We also can tidy +away the extra stuff needed for the termination arguments for trans* +and narrow*. + +}% +%sort trans %. +%mode trans %in %in %out %. +%term _ %pi (trans D1 D2 D3) %<- (trans* _ _ nat_eq_ D1 D2 D3) %. +%worlds (bind) (trans _ _ _) %. +%total {} (trans _ _ _) %. +%sort narrow {x} %. +%mode narrow %in %in %out %. +%term _ + %pi (narrow D1 D2 D3) + %<- ({x} {dv} narrow* _ _ nat_eq_ dv ([d] D1 x d) D2 ([d] D3 x d)) %. +%worlds (bind) (narrow _ _ _) %. +%total {} (narrow _ _ _) %. +%%%%% Reflexivity %%%%% +%{ + +Reflexivity is easy, but when we add a variable to the context, we +need to add its reflexivity case there, hence the reflx assumption in +rblock and rbind. Rblock exists for laziness purposes; since reflx +never calls trans, we needn't bother to maintain trans's invariants. + +}% +%sort reflx {T} %. +%mode reflx %in %out %. +%block rblock {t tp} [x tp] [d assm x t] [thm reflx x (sub_refl d)]%. +%block rbind {t tp} [x tp] [dv var x] [d assm x t] [thm assm_var d dv] [thm' reflx x (sub_refl d)]%. +%term _ reflx top sub_top %. +%term _ + %pi (reflx (arrow T1 T2) (sub_arrow T2refl T1refl)) + %<- (reflx T1 T1refl) + %<- (reflx T2 T2refl) %. +%term _ + %pi (reflx (forall T1 T2) (sub_forall T2refl T1refl)) + %<- (reflx T1 T1refl) + %<- ({x} {d} %pi (reflx x (sub_refl d)) %-> (reflx (T2 x) (T2refl x d))) %. +%worlds (rblock rbind) (reflx _ _) %. +%total T (reflx T _) %. +%%%%% Declarative subtyping %%%%% +%sort subtype %. +%term subtype_top subtype _ top %. +%term subtype_refl subtype T T %. +%term subtype_trans %pi (subtype T1 T3) %<- (subtype T1 T2) %<- (subtype T2 T3) %. +%term subtype_arrow %pi (subtype (arrow S1 S2) (arrow T1 T2)) %<- (subtype T1 S1) %<- (subtype S2 T2) %. +%term subtype_forall + %pi (subtype (forall S1 S2) (forall T1 T2)) + %<- (subtype T1 S1) + %<- ({x} %pi (subtype x T1) %-> (subtype (S2 x) (T2 x))) %. +%%%%% Correctness of the algorithm %%%%% +%sort assm_to_subtype %. +%mode assm_to_subtype %in %out %. +%block sbind {t tp} [x tp] [d assm x t] [d' subtype x t] [thm assm_to_subtype d d']%. +%worlds (sbind) (assm_to_subtype _ _) %. +%total {} (assm_to_subtype _ _) %. +%sort soundness %. +%mode soundness %in %out %. +%term _ soundness sub_top subtype_top %. +%term _ soundness (sub_refl _) subtype_refl %. +%term _ + %pi (soundness (sub_trans D2 D1) (subtype_trans D2' D1')) + %<- (assm_to_subtype D1 D1') + %<- (soundness D2 D2') %. +%term _ + %pi (soundness (sub_arrow D2 D1) (subtype_arrow D2' D1')) + %<- (soundness D1 D1') + %<- (soundness D2 D2') %. +%term _ + %pi (soundness (sub_forall D2 D1) (subtype_forall D2' D1')) + %<- (soundness D1 D1') + %<- ({x} {d assm x T} {d' subtype x T} %pi (assm_to_subtype d d') %-> (soundness (D2 x d) (D2' x d'))) %. +%worlds (sbind) (soundness _ _) %. +%total (D) (soundness D _) %. +%sort completeness %. +%mode completeness %in %out %. +%block cbind {t tp} {d_refl sub t t} [x tp] [dv var x] [d assm x t] [thm1 assm_var d dv] [thm2 reflx x (sub_refl d)] [d' subtype x t] [thm3 completeness d' (sub_trans d_refl d)]%. +%term _ completeness subtype_top sub_top %. +%term _ %pi (completeness subtype_refl D) %<- (reflx _ D) %. +%term _ + %pi (completeness (subtype_trans D2 D1) D) + %<- (completeness D1 D1') + %<- (completeness D2 D2') + %<- (trans D1' D2' D) %. +%term _ + %pi (completeness (subtype_arrow D2 D1) (sub_arrow D2' D1')) + %<- (completeness D1 D1') + %<- (completeness D2 D2') %. +%term _ + %pi (completeness (subtype_forall D2 D1) (sub_forall D2' D1')) + %<- (completeness D1 D1') + %<- (reflx _ Drefl) + %<- ({x tp} {dv var x} {d assm x T} + %pi (assm_var d dv) + %-> (reflx x (sub_refl d)) + %-> ({d' subtype x T} %pi (completeness d' (sub_trans Drefl d)) %-> (completeness (D2 x d') (D2' x d)))) %. +%worlds (cbind) (completeness _ _) %. +%total (D) (completeness D _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/poplmark/1b.lf b/new-tests/stelf-output/examples/poplmark/1b.lf new file mode 100644 index 0000000..4adb8c4 --- /dev/null +++ b/new-tests/stelf-output/examples/poplmark/1b.lf @@ -0,0 +1,674 @@ +%%%%% Nats %%%%% +%sort nat %. +%name nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort nat_eq %. +%sort nat_neq %. +%sort nat_less %. +%sort nat_more %. +%term nat_eq_ nat_eq N N %. +%term nat_neq_zs nat_neq z (s _) %. +%term nat_neq_sz nat_neq (s _) z %. +%term nat_neq_ss %pi (nat_neq (s N) (s M)) %<- (nat_neq N M) %. +%term nat_less_z nat_less z (s _) %. +%term nat_less_s %pi (nat_less (s N) (s M)) %<- (nat_less N M) %. +%term nat_more_z nat_more (s _) z %. +%term nat_more_s %pi (nat_more (s N) (s M)) %<- (nat_more N M) %. +%sort plus %. +%term plus_z plus z N N %. +%term plus_s %pi (plus (s M) N (s N')) %<- (plus M N N') %. +%reduces <= N M (plus _ N M) %. +%sort sum_inc {N1} {N2} {N3} %. +%mode sum_inc %in %in %in %in %out %. +%term _ sum_inc _ _ _ plus_z plus_z %. +%term _ %pi (sum_inc _ _ _ (plus_s D) (plus_s D')) %<- (sum_inc _ _ _ D D') %. +%worlds () (sum_inc _ _ _ _ _) %. +%total (D) (sum_inc _ _ _ D _) %. +%sort commute' {N1} {N2} {N3} %. +%mode commute' %in %in %in %in %out %. +%term _ %pi (commute' z (s M) _ plus_z (plus_s D)) %<- (commute' z M _ plus_z D) %. +%term _ commute' _ z _ _ plus_z %. +%term _ + %pi (commute' (s N1) N2 _ (plus_s D) D'') + %<- (commute' N1 N2 _ D D') + %<- (sum_inc _ _ _ D' D'') %. +%worlds () (commute' _ _ _ _ _) %. +%total [N1 N2] (commute' N1 N2 _ _ _) %. +%sort assoc %. +%mode assoc %in %in %out %out %. +%term _ assoc plus_z D D plus_z %. +%term _ %pi (assoc (plus_s Da) (plus_s Db) Da' (plus_s Db')) %<- (assoc Da Db Da' Db') %. +%worlds () (assoc _ _ _ _) %. +%total (D) (assoc D _ _ _) %. +%sort assoc' %. +%mode assoc' %in %in %out %out %. +%term _ + %pi (assoc' D1 D2 D3 D4) + %<- (commute' _ _ _ D2 D2') + %<- (assoc (plus_s D1) D2' D3' (plus_s D4')) + %<- (commute' _ _ _ D3' D3) + %<- (sum_inc _ _ _ D4' D4) %. +%worlds () (assoc' _ _ _ _) %. +%total (D) (assoc' D _ _ _) %. +%sort add {N1} {N2} %. +%mode add %in %in %out %. +%term _ add z N2 plus_z %. +%term _ %pi (add (s N1) N2 (plus_s D)) %<- (add N1 N2 D) %. +%worlds () (add _ _ _) %. +%total (N) (add N _ _) %. +%%%%% Labels %%%%% +%sort label %. +%name label %. +%term label_nat %pi nat %-> label %. +%sort label_eq %. +%sort label_neq %. +%sort label_less %. +%sort label_more %. +%term label_eq_ label_eq L L %. +%term label_neq_ %pi (label_neq (label_nat N) (label_nat M)) %<- (nat_neq N M) %. +%term label_less_ %pi (label_less (label_nat N) (label_nat M)) %<- (nat_less N M) %. +%term label_more_ %pi (label_more (label_nat N) (label_nat M)) %<- (nat_more N M) %. +%%%%% Types %%%%% +%sort tp %. +%name tp %. +%sort trow %. +%name trow %. +%term trow_nil trow %. +%term trow_cons %pi label %-> tp %-> trow %-> trow %. +%sort trow_lookup %. +%term trow_lookup_yes trow_lookup L (trow_cons L T _) T %. +%term trow_lookup_no %pi (trow_lookup L (trow_cons _ _ TR) T) %<- (trow_lookup L TR T) %. +%sort trow_labelfree %. +%term trow_labelfree_nil trow_labelfree trow_nil _ %. +%term trow_labelfree_cons + %pi (trow_labelfree (trow_cons L' _ TR) L) + %<- (label_neq L L') + %<- (trow_labelfree TR L) %. +%sort trow_eq %. +%term trow_eq_ trow_eq TR TR %. +%% Takes a record in and returns one in sorted order +%sort trow_order %. +%sort trow_insert %. +%term trow_order_nil trow_order trow_nil trow_nil %. +%term trow_order_cons + %pi (trow_order (trow_cons L S SR) TR') + %<- (trow_order SR TR) + %<- (trow_insert L S TR TR') %. +%term trow_insert_nil trow_insert L S trow_nil (trow_cons L S trow_nil) %. +%term trow_insert_less + %pi (trow_insert L S (trow_cons L' T TR) (trow_cons L S (trow_cons L' T TR))) + %<- (label_less L L') %. +%term trow_insert_more + %pi (trow_insert L S (trow_cons L' T TR) (trow_cons L' T TR')) + %<- (label_more L L') + %<- (trow_insert L S TR TR') %. +%sort trow_uniqueness %. +%term trow_uniqueness_nil trow_uniqueness trow_nil %. +%term trow_uniqueness_cons + %pi (trow_uniqueness (trow_cons L _ TR)) + %<- (trow_labelfree TR L) + %<- (trow_uniqueness TR) %. +%term top tp %. +%term arrow %pi tp %-> tp %-> tp %. +%term forall %pi tp %-> (%pi tp %-> tp) %-> tp %. +%term record {TR trow} %pi (trow_uniqueness TR) %-> tp %. +%%%%% Subtyping %%%%% +%sort assm %. +%sort var %. +%sort sub %. +%% sub_trow requires that the elements be in sorted order +%sort sub_trow %. +%term sub_top sub _ top %. +%term sub_refl %pi (sub X X) %<- (assm X _) %. +%term sub_trans %pi (sub X T) %<- (assm X U) %<- (sub U T) %. +%term sub_arrow %pi (sub (arrow S1 S2) (arrow T1 T2)) %<- (sub T1 S1) %<- (sub S2 T2) %. +%term sub_forall + %pi (sub (forall S1 ([x] S2 x)) (forall T1 ([x] T2 x))) + %<- (sub T1 S1) + %<- ({x} %pi (assm x T1) %-> (sub (S2 x) (T2 x))) %. +%term sub_record + %pi (sub (record SR SRunique) (record TR TRunique)) + %<- (trow_order SR SR') + %<- (trow_order TR TR') + %<- (sub_trow SR' TR') %. +%term sub_trow_nil sub_trow trow_nil trow_nil %. +%term sub_trow_cons + %pi (sub_trow (trow_cons L S SR) (trow_cons L T TR)) + %<- (sub_trow SR TR) + %<- (sub S T) %. +%term sub_trow_cons' %pi (sub_trow (trow_cons L S SR) TR) %<- (sub_trow SR TR) %. +%%%%% Variables and Blocks %%%%% +%sort assm_var %. +%mode assm_var %in %out %. +%block bind {t tp} [x tp] [dv var x] [d assm x t] [thm assm_var d dv]%. +%block vblock [t tp] [d var t]%. +%block ablock {x tp} {dv var x} {t tp} [d assm x t] [thm assm_var d dv]%. +%worlds (bind vblock ablock) (assm_var _ _) %. +%total {} (assm_var _ _) %. +%%%%% Metric %%%%% +%sort size %. +%sort size_trow %. +%term size_top size top z %. +%term size_var %pi (size T z) %<- (var T) %. +%term size_arrow %pi (size (arrow T1 T2) (s N)) %<- (size T1 N1) %<- (size T2 N2) %<- (plus N1 N2 N) %. +%term size_forall + %pi (size (forall T1 T2) (s N)) + %<- (size T1 N1) + %<- ({t tp} %pi (var t) %-> (size (T2 t) N2)) + %<- (plus N1 N2 N) %. +%term size_record %pi (size (record TR _) (s N)) %<- (size_trow TR N) %. +%term size_trow_nil size_trow trow_nil z %. +%term size_trow_cons + %pi (size_trow (trow_cons L T TR) (s N)) + %<- (size T N1) + %<- (size_trow TR N2) + %<- (plus N1 N2 N) %. +%sort employ_plus {N1} {N2} {N3} %. +%mode employ_plus %in %in %in %in %. +%term _ employ_plus _ _ _ plus_z %. +%term _ %pi (employ_plus _ _ _ (plus_s D)) %<- (employ_plus _ _ _ D) %. +%worlds (bind) (employ_plus _ _ _ _) %. +%total (D) (employ_plus _ _ _ D) %. +%reduces <= N2 N3 (employ_plus _ N2 N3 _) %. +%sort employ_plus2 {N1} {N2} {N3} %. +%mode employ_plus2 %in %in %in %in %. +%term _ %pi (employ_plus2 _ _ _ D) %<- (commute' _ _ _ D D') %<- (employ_plus _ _ _ D') %. +%worlds (bind) (employ_plus2 _ _ _ _) %. +%total [] (employ_plus2 _ _ _ _) %. +%reduces <= N1 N3 (employ_plus2 N1 _ N3 _) %. +%sort find_size {T} %. +%mode find_size %in %out %. +%sort find_size_trow {TR} %. +%mode find_size_trow %in %out %. +%block sblock [t tp] [d var t] [thm find_size t (size_var d)]%. +%block sbind {t tp} [x tp] [dv var x] [d assm x t] [thm assm_var d dv] [thm' find_size x (size_var dv)]%. +%term _ find_size top size_top %. +%term _ + %pi (find_size (arrow T1 T2) (size_arrow Plus Size2 Size1)) + %<- (find_size T1 Size1) + %<- (find_size T2 Size2) + %<- (add _ _ Plus) %. +%term _ + %pi (find_size (forall T1 T2) (size_forall Plus Size2 Size1)) + %<- (find_size T1 Size1) + %<- ({t tp} {dv var t} %pi (find_size t (size_var dv)) %-> (find_size (T2 t) (Size2 t dv))) + %<- (add _ _ Plus) %. +%term _ %pi (find_size (record TR _) (size_record D)) %<- (find_size_trow TR D) %. +%term _ find_size_trow trow_nil size_trow_nil %. +%term _ + %pi (find_size_trow (trow_cons _ T TR) (size_trow_cons Plus Size2 Size1)) + %<- (find_size T Size1) + %<- (find_size_trow TR Size2) + %<- (add _ _ Plus) %. +%worlds (sblock sbind) (find_size _ _) (find_size_trow _ _) %. +%total (T TR) (find_size T _) (find_size_trow TR _) %. +%%%%% Reductio ad absurdio %%%%% +%sort false %. +%sort raa_sub %. +%mode {%in S tp} {%in T tp} {%in X1 false} {%out X2 sub S T} raa_sub X1 X2 %. +%worlds (bind vblock ablock) (raa_sub _ _) %. +%total {} (raa_sub _ _) %. +%sort raa_treq %. +%mode {%in TR trow} {%in SR trow} {%in X1 false} {%out X2 trow_eq TR SR} raa_treq X1 X2 %. +%worlds (bind vblock ablock) (raa_treq _ _) %. +%total {} (raa_treq _ _) %. +%%%%% Various things are not variables %%%%% +%sort top_var_contra %. +%mode top_var_contra %in %out %. +%worlds (bind vblock ablock) (top_var_contra _ _) %. +%total {} (top_var_contra _ _) %. +%sort arrow_var_contra %. +%mode arrow_var_contra %in %out %. +%worlds (bind vblock ablock) (arrow_var_contra _ _) %. +%total {} (arrow_var_contra _ _) %. +%sort forall_var_contra %. +%mode forall_var_contra %in %out %. +%worlds (bind vblock ablock) (forall_var_contra _ _) %. +%total {} (forall_var_contra _ _) %. +%sort record_var_contra %. +%mode record_var_contra %in %out %. +%worlds (bind vblock ablock) (record_var_contra _ _) %. +%total {} (record_var_contra _ _) %. +%%%%% Nats and labels are well-ordered %%%%% +%sort nat_contra %. +%mode nat_contra %in %in %out %. +%term _ %pi (nat_contra (nat_less_s D) (nat_more_s D') F) %<- (nat_contra D D' F) %. +%worlds (bind vblock ablock) (nat_contra _ _ _) %. +%total (D) (nat_contra D _ _) %. +%sort label_contra %. +%mode label_contra %in %in %out %. +%term _ %pi (label_contra (label_less_ D) (label_more_ D') F) %<- (nat_contra D D' F) %. +%worlds (bind vblock ablock) (label_contra _ _ _) %. +%total {} (label_contra D _ _) %. +%%%%% Lemmas %%%%% +%sort trow_cons_unique {L} {T} %. +%mode trow_cons_unique %in %in %in %out %. +%term _ trow_cons_unique _ _ trow_eq_ trow_eq_ %. +%worlds (bind vblock ablock) (trow_cons_unique _ _ _ _) %. +%total [] (trow_cons_unique _ _ _ _) %. +%sort trow_insert_unique %. +%mode trow_insert_unique %in %in %in %out %. +%term _ trow_insert_unique trow_insert_nil _ _ trow_eq_ %. +%term _ trow_insert_unique (trow_insert_less _) (trow_insert_less _) _ trow_eq_ %. +%term _ + %pi (trow_insert_unique (trow_insert_more D _) (trow_insert_more D' _) trow_eq_ Deq') + %<- (trow_insert_unique D D' trow_eq_ Deq) + %<- (trow_cons_unique _ _ Deq Deq') %. +%% contradiction cases +%term _ + %pi (trow_insert_unique (trow_insert_less D) (trow_insert_more _ D') trow_eq_ Deq) + %<- (label_contra D D' F) + %<- (raa_treq F Deq) %. +%term _ + %pi (trow_insert_unique (trow_insert_more _ D') (trow_insert_less D) trow_eq_ Deq) + %<- (label_contra D D' F) + %<- (raa_treq F Deq) %. +%worlds (bind vblock ablock) (trow_insert_unique _ _ _ _) %. +%total (D) (trow_insert_unique D _ _ _) %. +%sort trow_order_unique %. +%mode trow_order_unique %in %in %out %. +%term _ trow_order_unique trow_order_nil _ trow_eq_ %. +%term _ + %pi (trow_order_unique (trow_order_cons D2a D1a) (trow_order_cons D2b D1b) Deq') + %<- (trow_order_unique D1a D1b Deq) + %<- (trow_insert_unique D2a D2b Deq Deq') %. +%worlds (bind vblock ablock) (trow_order_unique _ _ _) %. +%total (D) (trow_order_unique D _ _) %. +%sort equality_sub_trow %. +%mode equality_sub_trow %in %in %out %. +%term _ equality_sub_trow D trow_eq_ D %. +%worlds (bind vblock ablock) (equality_sub_trow _ _ _) %. +%total [] (equality_sub_trow _ _ _) %. +%sort preserve_size_insert %. +%mode preserve_size_insert %in %in %in %in %out %. +%term _ preserve_size_insert DsizeT DsizeTR _ Dplus (size_trow_cons Dplus DsizeTR DsizeT) %. +%term _ + %pi (preserve_size_insert DsizeT (size_trow_cons Dplus DsizeTR DsizeS) (trow_insert_more Dins _) Dplus' (size_trow_cons Dplus''' DsizeTR' DsizeS)) + %<- (assoc' Dplus Dplus' Dplus'' Dplus''') + %<- (preserve_size_insert DsizeT DsizeTR Dins Dplus'' DsizeTR') %. +%worlds (bind vblock ablock) (preserve_size_insert _ _ _ _ _) %. +%total (D) (preserve_size_insert _ D _ _ _) %. +%sort preserve_size_order %. +%mode preserve_size_order %in %in %out %. +%term _ preserve_size_order size_trow_nil trow_order_nil size_trow_nil %. +%term _ + %pi (preserve_size_order (size_trow_cons Dplus DsizeTR DsizeT) (trow_order_cons Dins Dord) DsizeTR'') + %<- (preserve_size_order DsizeTR Dord DsizeTR') + %<- (preserve_size_insert DsizeT DsizeTR' Dins Dplus DsizeTR'') %. +%worlds (bind vblock ablock) (preserve_size_order _ _ _) %. +%total (D) (preserve_size_order D _ _) %. +%sort trans* {Nsize nat} %. +%sort trans_trow* {Nsize nat} %. +%sort narrow* {Nsize nat} %. +%sort narrow_trow* {Nsize nat} %. +%mode trans* %in %in %in %in %in %in %out %. +%mode trans_trow* %in %in %in %in %in %in %out %. +%mode narrow* %in %in %in %in %in %in %in %out %. +%mode narrow_trow* %in %in %in %in %in %in %in %out %. +%scope narrow_trow* %term top trans* _ _ _ _ D sub_top sub_top %. +%term refl trans* _ _ _ _ (sub_refl _) D D %. +%scope narrow_trow* %term trans* + %pi (trans* _ Size _ nat_eq_ (sub_trans D2 D1) D (sub_trans D' D1)) + %<- (trans* _ Size _ nat_eq_ D2 D D') %. +%term arrow + %pi (trans* _ (size_arrow Dplus Dsize2 Dsize1) _ nat_eq_ (sub_arrow D1b D1a) (sub_arrow D2b D2a) (sub_arrow Db Da)) + %<- (employ_plus _ _ _ Dplus) + %<- (employ_plus2 _ _ _ Dplus) + %<- (trans* _ Dsize1 _ nat_eq_ D2a D1a Da) + %<- (trans* _ Dsize2 _ nat_eq_ D1b D2b Db) %. +%scope narrow_trow* %term forall + %pi (trans* _ (size_forall Dplus Dsize2 Dsize1) _ nat_eq_ (sub_forall (%the ({t tp} %pi (assm t T2) %-> (sub (T3 t) (T4 t))) D1b) D1a) (sub_forall D2b D2a) (sub_forall Db Da)) + %<- (employ_plus _ _ _ Dplus) + %<- (employ_plus2 _ _ _ Dplus) + %<- (trans* _ Dsize1 _ nat_eq_ (%the (sub T1 T2) D2a) D1a Da) + %<- ({x} {dv var x} narrow* _ Dsize1 _ nat_eq_ dv ([d] D1b x d) D2a ([d] D1b' x d)) + %<- ({x} {dv var x} {d assm x T1} + %pi (assm_var d dv) + %-> (trans* _ (Dsize2 x dv) _ nat_eq_ (D1b' x d) (D2b x d) (Db x d))) %. +%term record + %pi (trans* _ (size_record Dsize) _ _ (sub_record (%the (sub_trow SR' QR') SR'sub) (%the (trow_order QR QR') QRorder) (%the (trow_order SR SR') SRorder)) (sub_record (%the (sub_trow QR'' TR') QR''sub) (%the (trow_order TR TR') TRorder) (%the (trow_order QR QR'') QRorder')) (sub_record (%the (sub_trow SR' TR') SR'sub') TRorder SRorder)) + %<- (trow_order_unique QRorder' QRorder QR''eq) + %<- (equality_sub_trow QR''sub QR''eq QR'sub) + %<- (preserve_size_order Dsize QRorder Dsize') + %<- (trans_trow* _ Dsize' _ nat_eq_ SR'sub QR'sub SR'sub') %. +%% contradiction cases +%term _ + %pi (trans* _ _ _ _ _ (sub_refl Dassm) D) + %<- (assm_var Dassm Dvar) + %<- (top_var_contra Dvar Dfalse) + %<- (raa_sub Dfalse D) %. +%term _ + %pi (trans* _ _ _ _ _ (sub_refl Dassm) D) + %<- (assm_var Dassm Dvar) + %<- (arrow_var_contra Dvar Dfalse) + %<- (raa_sub Dfalse D) %. +%term _ + %pi (trans* _ _ _ _ _ (sub_refl Dassm) D) + %<- (assm_var Dassm Dvar) + %<- (forall_var_contra Dvar Dfalse) + %<- (raa_sub Dfalse D) %. +%term _ + %pi (trans* _ _ _ _ _ (sub_refl Dassm) D) + %<- (assm_var Dassm Dvar) + %<- (record_var_contra Dvar Dfalse) + %<- (raa_sub Dfalse D) %. +%term _ + %pi (trans* _ _ _ _ _ (sub_trans _ Dassm) D) + %<- (assm_var Dassm Dvar) + %<- (top_var_contra Dvar Dfalse) + %<- (raa_sub Dfalse D) %. +%term _ + %pi (trans* _ _ _ _ _ (sub_trans _ Dassm) D) + %<- (assm_var Dassm Dvar) + %<- (arrow_var_contra Dvar Dfalse) + %<- (raa_sub Dfalse D) %. +%term _ + %pi (trans* _ _ _ _ _ (sub_trans _ Dassm) D) + %<- (assm_var Dassm Dvar) + %<- (forall_var_contra Dvar Dfalse) + %<- (raa_sub Dfalse D) %. +%term _ + %pi (trans* _ _ _ _ _ (sub_trans _ Dassm) D) + %<- (assm_var Dassm Dvar) + %<- (record_var_contra Dvar Dfalse) + %<- (raa_sub Dfalse D) %. +%term cons + %pi (trans_trow* _ (size_trow_cons Dplus Dsize2 Dsize1) _ nat_eq_ (sub_trow_cons Ssub SRsub) (sub_trow_cons Tsub TRsub) (sub_trow_cons Ssub' SRsub')) + %<- (employ_plus _ _ _ Dplus) + %<- (employ_plus2 _ _ _ Dplus) + %<- (trans* _ Dsize1 _ nat_eq_ Ssub Tsub Ssub') + %<- (trans_trow* _ Dsize2 _ nat_eq_ SRsub TRsub SRsub') %. +%term cons + %pi (trans_trow* _ (size_trow_cons Dplus Dsize2 _) _ _ (sub_trow_cons Ssub SRsub) (sub_trow_cons' TRsub) (sub_trow_cons' SRsub')) + %<- (employ_plus _ _ _ Dplus) + %<- (trans_trow* _ Dsize2 _ nat_eq_ SRsub TRsub SRsub') %. +%term cons' + %pi (trans_trow* _ Dsize _ nat_eq_ (sub_trow_cons' SRsub) TRsub (sub_trow_cons' SRsub')) + %<- (trans_trow* _ Dsize _ nat_eq_ SRsub TRsub SRsub') %. +%term nil trans_trow* _ _ _ _ _ sub_trow_nil sub_trow_nil %. +%term closed narrow* _ _ _ _ Dvar ([d] D) _ ([d] D) %. +%term refl narrow* _ _ _ _ Dvar ([d] sub_refl d) _ ([d] sub_refl d) %. +%term trans + %pi (narrow* _ Size _ nat_eq_ Dvar ([d] sub_trans (D d) Dassm) Dsub ([d] sub_trans (D' d) Dassm)) + %<- (narrow* _ Size _ nat_eq_ Dvar D Dsub D') %. +%term trans + %pi (narrow* _ Size _ nat_eq_ Dvar ([d] sub_trans (D d) d) Dsub ([d] sub_trans (D'' d) d)) + %<- (narrow* _ Size _ nat_eq_ Dvar D Dsub D') + %<- ({d assm X Q} %pi (assm_var d Dvar) %-> (trans* _ Size _ nat_eq_ Dsub (D' d) (D'' d))) %. +%term arrow + %pi (narrow* _ Size _ nat_eq_ Dvar ([d] sub_arrow (D2 d) (D1 d)) Dsub ([d] sub_arrow (D2' d) (D1' d))) + %<- (narrow* _ Size _ nat_eq_ Dvar D1 Dsub D1') + %<- (narrow* _ Size _ nat_eq_ Dvar D2 Dsub D2') %. +%term forall + %pi (narrow* _ Size _ nat_eq_ Dvar ([d] sub_forall (D2 d) (D1 d)) Dsub ([d] sub_forall (D2' d) (D1' d))) + %<- (narrow* _ Size _ nat_eq_ Dvar D1 Dsub D1') + %<- ({x'} {dv'} {d'} + %pi (assm_var d' dv') + %-> (narrow* _ Size _ nat_eq_ Dvar ([d] D2 d x' d') Dsub ([d] D2' d x' d'))) %. +%term record + %pi (narrow* _ Size _ nat_eq_ Dvar ([d] sub_record (SRsub d) D2 D1) Dsub ([d] sub_record (SRsub' d) D2 D1)) + %<- (narrow_trow* _ Size _ nat_eq_ Dvar SRsub Dsub SRsub') %. +%term cons' + %pi (narrow_trow* _ Size _ nat_eq_ Dvar ([d] sub_trow_cons' (SRsub d)) Dsub ([d] sub_trow_cons' (SRsub' d))) + %<- (narrow_trow* _ Size _ nat_eq_ Dvar SRsub Dsub SRsub') %. +%term cons + %pi (narrow_trow* _ Size _ nat_eq_ Dvar ([d] sub_trow_cons (Ssub d) (SRsub d)) Dsub ([d] sub_trow_cons (Ssub' d) (SRsub' d))) + %<- (narrow_trow* _ Size _ nat_eq_ Dvar SRsub Dsub SRsub') + %<- (narrow* _ Size _ nat_eq_ Dvar Ssub Dsub Ssub') %. +%term nil narrow_trow* _ _ _ nat_eq_ Dvar ([d] SRsub) Dsub ([d] SRsub) %. +%worlds (bind vblock ablock) (trans* _ _ _ _ _ _ _) (trans_trow* _ _ _ _ _ _ _) (narrow* _ _ _ _ _ _ _ _) (narrow_trow* _ _ _ _ _ _ _ _) %. +%total {(M1 M2 M3 M4) (N1 N2 N3 N4) (D1 D2 D3 D4)} (trans_trow* M3 _ N3 _ D3 _ _) (narrow* M2 _ N2 _ _ D2 _ _) (narrow_trow* M4 _ N4 _ _ D4 _ _) (trans* M1 _ N1 _ D1 _ _) %. +%%%%% Peroration %%%%% +%sort trans %. +%mode trans %in %in %out %. +%term _ %pi (trans D1 D2 D3) %<- (find_size _ Size) %<- (trans* _ Size _ nat_eq_ D1 D2 D3) %. +%worlds (sbind) (trans _ _ _) %. +%total {} (trans _ _ _) %. +%sort narrow {x} %. +%mode narrow %in %in %out %. +%term _ + %pi (narrow D1 D2 D3) + %<- (find_size _ Size) + %<- ({x} {dv} narrow* _ Size _ nat_eq_ dv ([d] D1 x d) D2 ([d] D3 x d)) %. +%worlds (sbind) (narrow _ _ _) %. +%total {} (narrow _ _ _) %. +%sort reflx {Nsize nat} %. +%mode reflx %in %in %out %. +%sort reflx_trow {Nsize nat} %. +%mode reflx_trow %in %in %out %. +%%%%% Some blocks %%%%% +%block rbind {t tp} [x tp] [dv var x] [d assm x t] [thm assm_var d dv] [thm find_size x (size_var dv)] [thm reflx z (size_var dv) (sub_refl d)]%. +%sort labelfree_trans_insert %. +%mode labelfree_trans_insert %in %in %in %out %. +%term _ labelfree_trans_insert trow_labelfree_nil trow_insert_nil Lneq (trow_labelfree_cons trow_labelfree_nil Lneq) %. +%term _ labelfree_trans_insert Lfree (trow_insert_less _) L'neq (trow_labelfree_cons Lfree L'neq) %. +%term _ + %pi (labelfree_trans_insert (trow_labelfree_cons Lfree Lneq) (trow_insert_more TRins _) L'neq (trow_labelfree_cons Lfree' Lneq)) + %<- (labelfree_trans_insert Lfree TRins L'neq Lfree') %. +%worlds (rbind) (labelfree_trans_insert _ _ _ _) %. +%total (TRins) (labelfree_trans_insert _ TRins _ _) %. +%sort labelfree_trans_order %. +%mode labelfree_trans_order %in %in %out %. +%term _ labelfree_trans_order Lfree trow_order_nil Lfree %. +%term _ + %pi (labelfree_trans_order (trow_labelfree_cons Lfree Lneq) (trow_order_cons TRins SRorder) Lfree'') + %<- (labelfree_trans_order Lfree SRorder Lfree') + %<- (labelfree_trans_insert Lfree' TRins Lneq Lfree'') %. +%worlds (rbind) (labelfree_trans_order _ _ _) %. +%total (TRord) (labelfree_trans_order _ TRord _) %. +%% we need to know what and neq correspondes to greater than or less than +%sort nat_mol %. +%term nat_mol_less %pi (nat_mol N M) %<- (nat_less N M) %. +%term nat_mol_more %pi (nat_mol N M) %<- (nat_more N M) %. +%sort label_mol %. +%term label_mol_ %pi (label_mol (label_nat N) (label_nat M)) %<- (nat_mol N M) %. +%sort nat_mol_inc %. +%mode nat_mol_inc %in %out %. +%term _ nat_mol_inc (nat_mol_less Less) (nat_mol_less (nat_less_s Less)) %. +%term _ nat_mol_inc (nat_mol_more More) (nat_mol_more (nat_more_s More)) %. +%worlds (rbind) (nat_mol_inc _ _) %. +%total [] (nat_mol_inc _ _) %. +%sort nat_neq_to_mol %. +%mode nat_neq_to_mol %in %out %. +%term _ nat_neq_to_mol nat_neq_zs (nat_mol_less nat_less_z) %. +%term _ nat_neq_to_mol nat_neq_sz (nat_mol_more nat_more_z) %. +%term _ + %pi (nat_neq_to_mol (nat_neq_ss Nneq) Nmol) + %<- (nat_neq_to_mol Nneq Nmol') + %<- (nat_mol_inc Nmol' Nmol) %. +%worlds (rbind) (nat_neq_to_mol _ _) %. +%total (Nneq) (nat_neq_to_mol Nneq _) %. +%sort label_neq_to_mol %. +%mode label_neq_to_mol %in %out %. +%term _ + %pi (label_neq_to_mol (label_neq_ Nneq) (label_mol_ Nmol)) + %<- (nat_neq_to_mol Nneq Nmol) %. +%worlds (rbind) (label_neq_to_mol _ _) %. +%total [] (label_neq_to_mol _ _) %. +%sort trow_can_insert_helper {S} %. +%mode trow_can_insert_helper %in %in %in %out %. +%term _ trow_can_insert_helper _ (label_mol_ (nat_mol_less Less)) _ (trow_insert_less (label_less_ Less)) %. +%term _ trow_can_insert_helper _ (label_mol_ (nat_mol_more More)) TRins (trow_insert_more TRins (label_more_ More)) %. +%worlds (rbind) (trow_can_insert_helper _ _ _ _) %. +%total [] (trow_can_insert_helper _ _ _ _) %. +%sort trow_can_insert {T} %. +%mode trow_can_insert %in %in %out %. +%term _ trow_can_insert _ trow_labelfree_nil trow_insert_nil %. +%term _ + %pi (trow_can_insert _ (trow_labelfree_cons Lfree Lneq) TRins) + %<- (trow_can_insert _ Lfree TRins') + %<- (label_neq_to_mol Lneq Lmol) + %<- (trow_can_insert_helper _ Lmol TRins' TRins) %. +%worlds (rbind) (trow_can_insert _ _ _) %. +%total (Lfree) (trow_can_insert _ Lfree _) %. +%sort trow_has_order %. +%mode trow_has_order %in %out %. +%term _ trow_has_order trow_uniqueness_nil trow_order_nil %. +%term _ + %pi (trow_has_order (trow_uniqueness_cons TRunique Lfree) (trow_order_cons TR'ins TRord)) + %<- (trow_has_order TRunique TRord) + %<- (labelfree_trans_order Lfree TRord Lfree') + %<- (trow_can_insert _ Lfree' TR'ins) %. +%worlds (rbind) (trow_has_order _ _) %. +%total (TRunique) (trow_has_order TRunique _) %. +% +%%mode reflx +X1 +X2 -X3. +% +%%mode reflx_trow +X1 +X2 -X3. +%term _ reflx _ _ sub_top %. +%term _ + %pi (reflx _ (size_arrow Dplus Dsize2 Dsize1) (sub_arrow T2refl T1refl)) + %<- (employ_plus _ _ _ Dplus) + %<- (employ_plus2 _ _ _ Dplus) + %<- (reflx _ Dsize1 T1refl) + %<- (reflx _ Dsize2 T2refl) %. +%term _ + %pi (reflx _ (size_forall Dplus Dsize2 Dsize1) (sub_forall T2refl T1refl)) + %<- (employ_plus _ _ _ Dplus) + %<- (employ_plus2 _ _ _ Dplus) + %<- (reflx _ Dsize1 T1refl) + %<- ({x} {dv} {d} + %pi (assm_var d dv) + %-> (find_size x (size_var dv)) + %-> (reflx z (size_var dv) (sub_refl d)) + %-> (reflx _ (Dsize2 x dv) (T2refl x d))) %. +%term _ + %pi (reflx _ (%the (size (record _ TRunique) _) (size_record TRsize)) (sub_record TR'sub TRord TRord)) + %<- (trow_has_order TRunique TRord) + %<- (preserve_size_order TRsize TRord TR'size) + %<- (reflx_trow _ TR'size TR'sub) %. +%term _ reflx_trow _ size_trow_nil sub_trow_nil %. +%term _ + %pi (reflx_trow _ (size_trow_cons Dplus DsizeTR DsizeT) (sub_trow_cons Tsub TRsub)) + %<- (employ_plus _ _ _ Dplus) + %<- (employ_plus2 _ _ _ Dplus) + %<- (reflx _ DsizeT Tsub) + %<- (reflx_trow _ DsizeTR TRsub) %. +%worlds (rbind) (reflx _ _ _) (reflx_trow _ _ _) %. +%total (N M) (reflx N _ _) (reflx_trow M _ _) %. +%%%%% Now we redefine subtying and prove soundness and completeness +%sort sub_tp %. +%sort sub_tp_trow %. +%term sub_tp_top sub_tp _ top %. +%term sub_tp_refl sub_tp T T %. +%term sub_tp_trans %pi (sub_tp T1 T3) %<- (sub_tp T1 T2) %<- (sub_tp T2 T3) %. +%term sub_tp_arrow %pi (sub_tp (arrow S1 S2) (arrow T1 T2)) %<- (sub_tp T1 S1) %<- (sub_tp S2 T2) %. +%term sub_tp_forall + %pi (sub_tp (forall S1 S2) (forall T1 T2)) + %<- (sub_tp T1 S1) + %<- ({x} %pi (sub_tp x T1) %-> (sub_tp (S2 x) (T2 x))) %. +%term sub_tp_record + %pi (sub_tp (record SR _) (record TR _)) + %<- (trow_order SR SR') + %<- (trow_order TR TR') + %<- (sub_tp_trow SR' TR') %. +%term sub_tp_trow_nil sub_tp_trow trow_nil trow_nil %. +%term sub_tp_trow_cons + %pi (sub_tp_trow (trow_cons L S SR) (trow_cons L T TR)) + %<- (sub_tp_trow SR TR) + %<- (sub_tp S T) %. +%term sub_tp_trow_cons' %pi (sub_tp_trow (trow_cons L S SR) TR) %<- (sub_tp_trow SR TR) %. +%sort sub_tp_trow_refl {TR} %. +%mode sub_tp_trow_refl %in %out %. +%term _ sub_tp_trow_refl trow_nil sub_tp_trow_nil %. +%term _ + %pi (sub_tp_trow_refl (trow_cons _ _ TR) (sub_tp_trow_cons sub_tp_refl TRsub)) + %<- (sub_tp_trow_refl TR TRsub) %. +%worlds () (sub_tp_trow_refl _ _) %. +%total (TR) (sub_tp_trow_refl TR _) %. +%sort sub_tp_trow_trans %. +%mode sub_tp_trow_trans %in %in %out %. +%term _ sub_tp_trow_trans _ sub_tp_trow_nil sub_tp_trow_nil %. +%term _ + %pi (sub_tp_trow_trans (sub_tp_trow_cons' SRsub) TRsub (sub_tp_trow_cons' SRsub')) + %<- (sub_tp_trow_trans SRsub TRsub SRsub') %. +%term _ + %pi (sub_tp_trow_trans (sub_tp_trow_cons _ SRsub) (sub_tp_trow_cons' TRsub) (sub_tp_trow_cons' SRsub')) + %<- (sub_tp_trow_trans SRsub TRsub SRsub') %. +%term _ + %pi (sub_tp_trow_trans (sub_tp_trow_cons Ssub SRsub) (sub_tp_trow_cons Tsub TRsub) (sub_tp_trow_cons (sub_tp_trans Tsub Ssub) SRsub')) + %<- (sub_tp_trow_trans SRsub TRsub SRsub') %. +%worlds () (sub_tp_trow_trans _ _ _) %. +%total (SRsub) (sub_tp_trow_trans SRsub _ _) %. +%sort equality_sub_tp_trow %. +%mode equality_sub_tp_trow %in %in %in %out %. +%term _ equality_sub_tp_trow D trow_eq_ trow_eq_ D %. +%worlds () (equality_sub_tp_trow _ _ _ _) %. +%total [] (equality_sub_tp_trow _ _ _ _) %. +%%%%% Proof that sub and sub_tp are equivalent %%%%% +%sort assm_to_sub_tp %. +%mode assm_to_sub_tp %in %out %. +%block cblock {T tp} {S tp} [x tp] [d assm x T] [d' sub_tp x T] [thm assm_to_sub_tp d d']%. +%worlds (cblock) (assm_to_sub_tp _ _) %. +%total [] (assm_to_sub_tp _ _) %. +%sort completeness %. +%mode completeness %in %out %. +%sort completeness_trow %. +%mode completeness_trow %in %out %. +%term _ completeness sub_top sub_tp_top %. +%term _ completeness (sub_refl _) sub_tp_refl %. +%term _ + %pi (completeness (sub_trans D2 D1) (sub_tp_trans D2' D1')) + %<- (assm_to_sub_tp D1 D1') + %<- (completeness D2 D2') %. +%term _ + %pi (completeness (sub_arrow D2 D1) (sub_tp_arrow D2' D1')) + %<- (completeness D1 D1') + %<- (completeness D2 D2') %. +%term _ + %pi (completeness (sub_forall D2 D1) (sub_tp_forall D2' D1')) + %<- (completeness D1 D1') + %<- ({x} {d assm x T} {d' sub_tp x T} %pi (assm_to_sub_tp d d') %-> (completeness (D2 x d) (D2' x d'))) %. +%term _ + %pi (completeness (sub_record D3 D2 D1) (sub_tp_record D3' D2 D1)) + %<- (completeness_trow D3 D3') %. +%term _ completeness_trow sub_trow_nil sub_tp_trow_nil %. +%term _ + %pi (completeness_trow (sub_trow_cons D2 D1) (sub_tp_trow_cons D2' D1')) + %<- (completeness_trow D1 D1') + %<- (completeness D2 D2') %. +%term _ + %pi (completeness_trow (sub_trow_cons' D) (sub_tp_trow_cons' D')) + %<- (completeness_trow D D') %. +%worlds (cblock) (completeness _ _) (completeness_trow _ _) %. +%total (D D') (completeness D _) (completeness_trow D' _) %. +%sort soundness %. +%mode soundness %in %out %. +%sort soundness_trow %. +%mode soundness_trow %in %out %. +%block sbind {t tp} {d'' sub t t} [x tp] [dv var x] [d assm x t] [thm assm_var d dv] [thm' find_size x (size_var dv)] [thm' reflx z (size_var dv) (sub_refl d)] [d' sub_tp x t] [thm'' soundness d' (sub_trans d'' d)]%. +%term _ soundness sub_tp_top sub_top %. +%term _ %pi (soundness sub_tp_refl D) %<- (find_size _ Dsize) %<- (reflx _ Dsize D) %. +%term _ + %pi (soundness (sub_tp_trans D2 D1) D) + %<- (soundness D1 D1') + %<- (soundness D2 D2') + %<- (trans D1' D2' D) %. +%term _ + %pi (soundness (sub_tp_arrow D2 D1) (sub_arrow D2' D1')) + %<- (soundness D1 D1') + %<- (soundness D2 D2') %. +%term _ + %pi (soundness (sub_tp_forall D2 D1) (sub_forall D2' D1')) + %<- (soundness D1 D1') + %<- (find_size _ Dsize) + %<- (reflx _ Dsize D'') + %<- ({x tp} {dv var x} {d assm x T} + %pi (assm_var d dv) + %-> (find_size x (size_var dv)) + %-> (reflx z (size_var dv) (sub_refl d)) + %-> ({d' sub_tp x T} %pi (soundness d' (sub_trans D'' d)) %-> (soundness (D2 x d') (D2' x d)))) %. +%term _ + %pi (soundness (sub_tp_record D3 D2 D1) (sub_record D3' D2 D1)) + %<- (soundness_trow D3 D3') %. +%term _ soundness_trow sub_tp_trow_nil sub_trow_nil %. +%term _ + %pi (soundness_trow (sub_tp_trow_cons D2 D1) (sub_trow_cons D2' D1')) + %<- (soundness_trow D1 D1') + %<- (soundness D2 D2') %. +%term _ + %pi (soundness_trow (sub_tp_trow_cons' D) (sub_trow_cons' D')) + %<- (soundness_trow D D') %. +%worlds (sbind) (soundness _ _) (soundness_trow _ _) %. +%total (D D') (soundness D _) (soundness_trow D' _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/poplmark/2a.lf b/new-tests/stelf-output/examples/poplmark/2a.lf new file mode 100644 index 0000000..32a959c --- /dev/null +++ b/new-tests/stelf-output/examples/poplmark/2a.lf @@ -0,0 +1,274 @@ +%% Here we define the types +%sort tp %. +%name tp %. +%term top tp %. +%term arrow %pi tp %-> tp %-> tp %. +%term forall %pi tp %-> (%pi tp %-> tp) %-> tp %. +%sort assm %. +%sort var %. +%sort sub %. +%% Here we define subtyping +%sort sub_tp %. +%term sub_tp_top sub_tp _ top %. +%term sub_tp_refl sub_tp T T %. +%term sub_tp_trans %pi (sub_tp T1 T3) %<- (sub_tp T1 T2) %<- (sub_tp T2 T3) %. +%term sub_tp_arrow %pi (sub_tp (arrow S1 S2) (arrow T1 T2)) %<- (sub_tp T1 S1) %<- (sub_tp S2 T2) %. +%term sub_tp_forall + %pi (sub_tp (forall S1 S2) (forall T1 T2)) + %<- (sub_tp T1 S1) + %<- ({x} %pi (sub_tp x T1) %-> (sub_tp (S2 x) (T2 x))) %. +%% Now we define terms +%sort term %. +%name term %. +%% Now we define the different possible terms +%term abs %pi tp %-> (%pi term %-> term) %-> term %. +%term app %pi term %-> term %-> term %. +%term tabs %pi tp %-> (%pi tp %-> term) %-> term %. +%term tapp %pi term %-> tp %-> term %. +%% Now we write a judgement to tell us whether a term is a value +%sort value %. +%term value_abs value (abs _ _) %. +%term value_tabs value (tabs _ _) %. +%% Here we define the tping judgement +%sort of %. +%term of_abs %pi (of (abs T1 E) (arrow T1 T2)) %<- ({x} %pi (of x T1) %-> (of (E x) T2)) %. +%term of_app %pi (of (app E1 E2) T12) %<- (of E1 (arrow T11 T12)) %<- (of E2 T11) %. +%term of_tabs + %pi (of (tabs T1 E) (forall T1 T2)) + %<- ({x} %pi (sub_tp x T1) %-> (of (E x) (T2 x))) %. +%term of_tapp %pi (of (tapp E T2) (T12 T2)) %<- (of E (forall T11 T12)) %<- (sub_tp T2 T11) %. +%term of_sub %pi (of E T) %<- (of E S) %<- (sub_tp S T) %. +%% Here we define evaluation +%sort step %. +%term step_app step (app (abs _ E1) E2) (E1 E2) %. +%term step_tapp step (tapp (tabs _ E1) T) (E1 T) %. +%term step_app_fun %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step_app_arg %pi (step (app V1 E2) (app V1 E2')) %<- (value V1) %<- (step E2 E2') %. +%term step_tapp_fun %pi (step (tapp E T) (tapp E' T)) %<- (step E E') %. +%% definition of what it means to make progress +%sort progresses %. +%% E may be a value +%term progresses_value %pi (progresses E) %<- (value E) %. +%% Or E may take a step +%term progresses_steps %pi (progresses E) %<- (step E E') %. +%% And a definition of false for good measure +%sort false %. +%%%%% A few lemmas %%%%% +%sort absurd_tsa %. +%mode absurd_tsa %in %out %. +%sort absurd_fsa %. +%mode absurd_fsa %in %out %. +%term _ %pi (absurd_tsa (sub_tp_trans D _) F) %<- (absurd_tsa D F) %. +%term _ %pi (absurd_tsa (sub_tp_trans D _) F) %<- (absurd_fsa D F) %. +%term _ %pi (absurd_tsa (sub_tp_trans _ D) F) %<- (absurd_tsa D F) %. +%term _ %pi (absurd_fsa (sub_tp_trans D _) F) %<- (absurd_tsa D F) %. +%term _ %pi (absurd_fsa (sub_tp_trans D _) F) %<- (absurd_fsa D F) %. +%term _ %pi (absurd_fsa (sub_tp_trans _ D) F) %<- (absurd_fsa D F) %. +%worlds () (absurd_fsa _ _) (absurd_tsa _ _) %. +%total (D D') (absurd_fsa D _) (absurd_tsa D' _) %. +%sort absurd_tsf %. +%mode absurd_tsf %in %out %. +%term _ %pi (absurd_tsf (sub_tp_trans D _) F) %<- (absurd_tsf D F) %. +%term _ %pi (absurd_tsf (sub_tp_trans _ D) F) %<- (absurd_tsf D F) %. +%term _ %pi (absurd_tsf (sub_tp_trans _ D) F) %<- (absurd_tsa D F) %. +%worlds () (absurd_tsf _ _) %. +%total (D) (absurd_tsf D _) %. +%sort absurd_asf %. +%mode absurd_asf %in %out %. +%term _ %pi (absurd_asf (sub_tp_trans D _) F) %<- (absurd_asf D F) %. +%term _ %pi (absurd_asf (sub_tp_trans _ D) F) %<- (absurd_asf D F) %. +%term _ %pi (absurd_asf (sub_tp_trans D _) F) %<- (absurd_tsf D F) %. +%worlds () (absurd_asf _ _) %. +%total (D) (absurd_asf D _) %. +%sort absurd_ota %. +%mode absurd_ota %in %out %. +%term _ %pi (absurd_ota (of_sub D _) F) %<- (absurd_tsa D F) %. +%term _ %pi (absurd_ota (of_sub D _) F) %<- (absurd_fsa D F) %. +%term _ %pi (absurd_ota (of_sub _ D) F) %<- (absurd_ota D F) %. +%worlds () (absurd_ota _ _) %. +%total (D) (absurd_ota D _) %. +%sort absurd_oaf %. +%mode absurd_oaf %in %out %. +%term _ %pi (absurd_oaf (of_sub D _) F) %<- (absurd_tsf D F) %. +%term _ %pi (absurd_oaf (of_sub D _) F) %<- (absurd_asf D F) %. +%term _ %pi (absurd_oaf (of_sub _ D) F) %<- (absurd_oaf D F) %. +%worlds () (absurd_oaf _ _) %. +%total (D) (absurd_oaf D _) %. +%%%%% Reductio ad absurdio %%%%% +%sort raa_sub_tp %. +%mode {%in T1} {%in T2} {%in F} {%out D sub_tp T1 T2} raa_sub_tp F D %. +%worlds () (raa_sub_tp _ _) %. +%total [] (raa_sub_tp _ _) %. +%sort raa_sub_imp_sub %. +%mode {%in T} {%in T1} {%in T2} {%in F} {%out D {x} %pi (sub_tp x T) %-> (sub_tp (T1 x) (T2 x))} raa_sub_imp_sub F D %. +%worlds () (raa_sub_imp_sub _ _) %. +%total [] (raa_sub_imp_sub _ _) %. +%sort raa_of %. +%mode {%in E} {%in T} {%in F} {%out D of E T} raa_of F D %. +%worlds () (raa_of _ _) %. +%total [] (raa_of _ _) %. +%sort raa_of_imp_of %. +%mode {%in E} {%in T1} {%in T2} {%in F} {%out D {x} %pi (of x T1) %-> (of (E x) T2)} raa_of_imp_of F D %. +%worlds () (raa_of_imp_of _ _) %. +%total [] (raa_of_imp_of _ _) %. +%sort raa_sub_imp_of %. +%mode {%in T} {%in E} {%in T2} {%in F} {%out D {x} %pi (sub_tp x T) %-> (of (E x) (T2 x))} raa_sub_imp_of F D %. +%worlds () (raa_sub_imp_of _ _) %. +%total [] (raa_sub_imp_of _ _) %. +%%%%% Some inversions %%%%% +%sort invert_sub_arrow %. +%mode invert_sub_arrow %in %out %out %. +%term _ invert_sub_arrow sub_tp_refl sub_tp_refl sub_tp_refl %. +%term _ invert_sub_arrow (sub_tp_arrow S2sub T1sub) T1sub S2sub %. +%term _ + %pi (invert_sub_arrow (sub_tp_trans T'sub Tsub) (sub_tp_trans U1sub T1sub) (sub_tp_trans U2sub S2sub)) + %<- (invert_sub_arrow Tsub U1sub S2sub) + %<- (invert_sub_arrow T'sub T1sub U2sub) %. +%term _ + %pi (invert_sub_arrow (sub_tp_trans D _) T1sub S2sub) + %<- (absurd_fsa D F) + %<- (raa_sub_tp F T1sub) + %<- (raa_sub_tp F S2sub) %. +%term _ + %pi (invert_sub_arrow (sub_tp_trans D _) T1sub S2sub) + %<- (absurd_tsa D F) + %<- (raa_sub_tp F T1sub) + %<- (raa_sub_tp F S2sub) %. +%worlds () (invert_sub_arrow _ _ _) %. +%total (Tsub) (invert_sub_arrow Tsub _ _) %. +%sort invert_sub_forall %. +%mode invert_sub_forall %in %out %out %. +%term _ invert_sub_forall (sub_tp_forall S2sub T1sub) T1sub S2sub %. +%term _ invert_sub_forall sub_tp_refl sub_tp_refl ([x] [d] sub_tp_refl) %. +%term _ + %pi (invert_sub_forall (sub_tp_trans T'sub Tsub) (sub_tp_trans U1sub T1sub) ([x] [d] sub_tp_trans (U2sub x d) (S2sub x (sub_tp_trans T1sub d)))) + %<- (invert_sub_forall Tsub U1sub S2sub) + %<- (invert_sub_forall T'sub T1sub U2sub) %. +%% contradictions +%term _ + %pi (invert_sub_forall (sub_tp_trans _ D) T1sub S2sub) + %<- (absurd_fsa D F) + %<- (raa_sub_tp F T1sub) + %<- (raa_sub_imp_sub F S2sub) %. +%term _ + %pi (invert_sub_forall (sub_tp_trans D _) T1sub S2sub) + %<- (absurd_tsf D F) + %<- (raa_sub_tp F T1sub) + %<- (raa_sub_imp_sub F S2sub) %. +%worlds () (invert_sub_forall _ _ _) %. +%total (Dsub) (invert_sub_forall Dsub _ _) %. +%sort invert_of_abs %. +%mode invert_of_abs %in %out %out %out %. +%term _ invert_of_abs (of_abs ([x] [xassm] Eof x xassm)) Eof sub_tp_refl sub_tp_refl %. +%term _ + %pi (invert_of_abs (of_sub Dsub Eof) Eof' (sub_tp_trans T1sub U1sub) (sub_tp_trans U2sub S2sub)) + %<- (invert_of_abs Eof Eof' T1sub S2sub) + %<- (invert_sub_arrow Dsub U1sub U2sub) %. +%term _ + %pi (invert_of_abs (of_sub D _) Eof' T1sub S2sub) + %<- (absurd_tsa D F) + %<- (raa_of_imp_of F (%the ({x} {d} of _ top) Eof')) + %<- (raa_sub_tp F T1sub) + %<- (raa_sub_tp F S2sub) %. +%term _ + %pi (invert_of_abs (of_sub D _) Eof' T1sub S2sub) + %<- (absurd_fsa D F) + %<- (raa_of_imp_of F (%the ({x} {d} of _ top) Eof')) + %<- (raa_sub_tp F T1sub) + %<- (raa_sub_tp F S2sub) %. +%worlds () (invert_of_abs _ _ _ _) %. +%total [Eof] (invert_of_abs Eof _ _ _) %. +%sort invert_of_tabs %. +%mode invert_of_tabs %in %out %out %out %. +%term _ invert_of_tabs (of_tabs ([x] [xsub] Eof x xsub)) Eof sub_tp_refl ([x] [d] sub_tp_refl) %. +%term _ + %pi (invert_of_tabs (of_sub Tsub Eof) Eof' (sub_tp_trans U1sub T1sub) ([x] [d] sub_tp_trans (U2sub x d) (S2sub x (sub_tp_trans T1sub d)))) + %<- (invert_of_tabs Eof Eof' U1sub S2sub) + %<- (invert_sub_forall Tsub T1sub U2sub) %. +%term _ + %pi (invert_of_tabs (of_sub D _) Eof' T1sub S2sub) + %<- (absurd_tsf D F) + %<- (raa_sub_imp_of F (%the ({x} {d} of _ top) Eof')) + %<- (raa_sub_tp F T1sub) + %<- (raa_sub_imp_sub F S2sub) %. +%term _ + %pi (invert_of_tabs (of_sub D _) Eof' T1sub S2sub) + %<- (absurd_asf D F) + %<- (raa_sub_imp_of F (%the ({x} {d} of _ top) Eof')) + %<- (raa_sub_tp F T1sub) + %<- (raa_sub_imp_sub F S2sub) %. +%worlds () (invert_of_tabs _ _ _ _) %. +%total [Eof] (invert_of_tabs Eof _ _ _) %. +%%%%% Reductio ad absurdio %%%%% +%sort raa_progresses %. +%mode {%in E} {%in F} {%out D progresses E} raa_progresses F D %. +%worlds () (raa_progresses _ _) %. +%total [] (raa_progresses _ _) %. +%% progress lemmas +%sort app_progresses %. +%mode app_progresses %in %in %in %out %. +%term _ + %pi (app_progresses (of_sub _ Eof) E1prog E2prog E3prog) + %<- (app_progresses Eof E1prog E2prog E3prog) %. +%term _ app_progresses _ (progresses_steps E1step) _ (progresses_steps (step_app_fun E1step)) %. +%term _ app_progresses (of_app E2of E1of) (progresses_value E1val) (progresses_steps E2step) (progresses_steps (step_app_arg E2step E1val)) %. +%term _ app_progresses (of_app E2of E1of) (progresses_value E1val) (progresses_value E2val) (progresses_steps step_app) %. +%term _ + %pi (app_progresses (of_app _ E1of) _ _ D) + %<- (absurd_ota E1of F) + %<- (raa_progresses F D) %. +%worlds () (app_progresses _ _ _ _) %. +%total [Eof] (app_progresses Eof _ _ _) %. +%sort tapp_progresses %. +%mode tapp_progresses %in %in %out %. +%term _ + %pi (tapp_progresses (of_sub _ Eof) Eprog E'prog) + %<- (tapp_progresses Eof Eprog E'prog) %. +%term _ tapp_progresses _ (progresses_steps Estep) (progresses_steps (step_tapp_fun Estep)) %. +%term _ tapp_progresses (of_tapp E2of E1of) (progresses_value Eval) (progresses_steps step_tapp) %. +%term _ + %pi (tapp_progresses (of_tapp _ Eof) _ D) + %<- (absurd_oaf Eof F) + %<- (raa_progresses F D) %. +%worlds () (tapp_progresses _ _ _) %. +%total [Eof] (tapp_progresses Eof _ _) %. +%% Proof of Preservation +%sort preservation %. +%mode preservation %in %in %out %. +%term _ + %pi (preservation (of_sub Ssub Eof) Estep (of_sub Ssub E'of)) + %<- (preservation Eof Estep E'of) %. +%term _ + %pi (preservation (of_app E2of E1of) step_app (of_sub S2sub (E1of' _ (of_sub T1sub E2of)))) + %<- (invert_of_abs E1of E1of' T1sub S2sub) %. +%term _ + %pi (preservation (of_app E2of E1of) (step_app_fun E1step) (of_app E2of E1of')) + %<- (preservation E1of E1step E1of') %. +%term _ + %pi (preservation (of_app E2of E1of) (step_app_arg E2step _) (of_app E2of' E1of)) + %<- (preservation E2of E2step E2of') %. +%term _ + %pi (preservation (of_tapp T2sub Eof) (step_tapp_fun Estep) (of_tapp T2sub Eof')) + %<- (preservation Eof Estep Eof') %. +%term _ + %pi (preservation (of_tapp T2sub Eof) step_tapp (of_sub (S2sub _ T2sub) (Eof' _ (sub_tp_trans T1sub T2sub)))) + %<- (invert_of_tabs Eof Eof' T1sub S2sub) %. +%worlds () (preservation _ _ _) %. +%total (Eof) (preservation Eof _ _) %. +%% progress proof +%sort progress %. +%mode progress %in %out %. +%term _ %pi (progress (of_sub _ Eof) Eprogress) %<- (progress Eof Eprogress) %. +%term _ progress (of_abs _) (progresses_value value_abs) %. +%term _ progress (of_tabs _) (progresses_value value_tabs) %. +%term _ + %pi (progress (of_app E2of E1of) Eprog) + %<- (progress E1of E1prog) + %<- (progress E2of E2prog) + %<- (app_progresses (of_app E2of E1of) E1prog E2prog Eprog) %. +%term _ + %pi (progress (of_tapp Tsub Eof) E'prog) + %<- (progress Eof Eprog) + %<- (tapp_progresses (of_tapp Tsub Eof) Eprog E'prog) %. +%worlds () (progress _ _) %. +%total (Eof) (progress Eof _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/poplmark/2b.lf b/new-tests/stelf-output/examples/poplmark/2b.lf new file mode 100644 index 0000000..42ee899 --- /dev/null +++ b/new-tests/stelf-output/examples/poplmark/2b.lf @@ -0,0 +1,1634 @@ +%%%%% Nats %%%%% +%sort nat %. +%name nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort nat_eq %. +%sort nat_neq %. +%sort nat_less %. +%sort nat_more %. +%term nat_eq_ nat_eq N N %. +%term nat_neq_zs nat_neq z (s _) %. +%term nat_neq_sz nat_neq (s _) z %. +%term nat_neq_ss %pi (nat_neq (s N) (s M)) %<- (nat_neq N M) %. +%term nat_less_z nat_less z (s _) %. +%term nat_less_s %pi (nat_less (s N) (s M)) %<- (nat_less N M) %. +%term nat_more_z nat_more (s _) z %. +%term nat_more_s %pi (nat_more (s N) (s M)) %<- (nat_more N M) %. +%sort plus %. +%term plus_z plus z N N %. +%term plus_s %pi (plus (s M) N (s N')) %<- (plus M N N') %. +%reduces <= N M (plus _ N M) %. +%sort sum_inc {N1} {N2} {N3} %. +%mode sum_inc %in %in %in %in %out %. +%term _ sum_inc _ _ _ plus_z plus_z %. +%term _ %pi (sum_inc _ _ _ (plus_s D) (plus_s D')) %<- (sum_inc _ _ _ D D') %. +%worlds () (sum_inc _ _ _ _ _) %. +%total (D) (sum_inc _ _ _ D _) %. +%sort commute' {N1} {N2} {N3} %. +%mode commute' %in %in %in %in %out %. +%term _ %pi (commute' z (s M) _ plus_z (plus_s D)) %<- (commute' z M _ plus_z D) %. +%term _ commute' _ z _ _ plus_z %. +%term _ + %pi (commute' (s N1) N2 _ (plus_s D) D'') + %<- (commute' N1 N2 _ D D') + %<- (sum_inc _ _ _ D' D'') %. +%worlds () (commute' _ _ _ _ _) %. +%total [N1 N2] (commute' N1 N2 _ _ _) %. +%sort assoc %. +%mode assoc %in %in %out %out %. +%term _ assoc plus_z D D plus_z %. +%term _ %pi (assoc (plus_s Da) (plus_s Db) Da' (plus_s Db')) %<- (assoc Da Db Da' Db') %. +%worlds () (assoc _ _ _ _) %. +%total (D) (assoc D _ _ _) %. +%sort assoc' %. +%mode assoc' %in %in %out %out %. +%term _ + %pi (assoc' D1 D2 D3 D4) + %<- (commute' _ _ _ D2 D2') + %<- (assoc (plus_s D1) D2' D3' (plus_s D4')) + %<- (commute' _ _ _ D3' D3) + %<- (sum_inc _ _ _ D4' D4) %. +%worlds () (assoc' _ _ _ _) %. +%total (D) (assoc' D _ _ _) %. +%sort add {N1} {N2} %. +%mode add %in %in %out %. +%term _ add z N2 plus_z %. +%term _ %pi (add (s N1) N2 (plus_s D)) %<- (add N1 N2 D) %. +%worlds () (add _ _ _) %. +%total (N) (add N _ _) %. +%%%%% Labels %%%%% +%sort label %. +%name label %. +%term label_nat %pi nat %-> label %. +%sort label_eq %. +%sort label_neq %. +%sort label_less %. +%sort label_more %. +%term label_eq_ label_eq L L %. +%term label_neq_ %pi (label_neq (label_nat N) (label_nat M)) %<- (nat_neq N M) %. +%term label_less_ %pi (label_less (label_nat N) (label_nat M)) %<- (nat_less N M) %. +%term label_more_ %pi (label_more (label_nat N) (label_nat M)) %<- (nat_more N M) %. +%%%%% Types %%%%% +%sort tp %. +%name tp %. +%sort trow %. +%name trow %. +%term trow_nil trow %. +%term trow_cons %pi label %-> tp %-> trow %-> trow %. +%sort trow_lookup %. +%term trow_lookup_yes trow_lookup L (trow_cons L T _) T %. +%term trow_lookup_no %pi (trow_lookup L (trow_cons _ _ TR) T) %<- (trow_lookup L TR T) %. +%sort trow_labelfree %. +%term trow_labelfree_nil trow_labelfree trow_nil _ %. +%term trow_labelfree_cons + %pi (trow_labelfree (trow_cons L' _ TR) L) + %<- (label_neq L L') + %<- (trow_labelfree TR L) %. +%sort trow_eq %. +%term trow_eq_ trow_eq TR TR %. +%% Takes a record in and returns one in sorted order +%sort trow_order %. +%sort trow_insert %. +%term trow_order_nil trow_order trow_nil trow_nil %. +%term trow_order_cons + %pi (trow_order (trow_cons L S SR) TR') + %<- (trow_order SR TR) + %<- (trow_insert L S TR TR') %. +%term trow_insert_nil trow_insert L S trow_nil (trow_cons L S trow_nil) %. +%term trow_insert_less + %pi (trow_insert L S (trow_cons L' T TR) (trow_cons L S (trow_cons L' T TR))) + %<- (label_less L L') %. +%term trow_insert_more + %pi (trow_insert L S (trow_cons L' T TR) (trow_cons L' T TR')) + %<- (label_more L L') + %<- (trow_insert L S TR TR') %. +%sort trow_uniqueness %. +%term trow_uniqueness_nil trow_uniqueness trow_nil %. +%term trow_uniqueness_cons + %pi (trow_uniqueness (trow_cons L _ TR)) + %<- (trow_labelfree TR L) + %<- (trow_uniqueness TR) %. +%term top tp %. +%term arrow %pi tp %-> tp %-> tp %. +%term forall %pi tp %-> (%pi tp %-> tp) %-> tp %. +%term record {TR trow} %pi (trow_uniqueness TR) %-> tp %. +%% Now we define subtying +%sort sub_tp %. +%sort sub_tp_trow %. +%term sub_tp_top sub_tp _ top %. +%term sub_tp_refl sub_tp T T %. +%term sub_tp_trans %pi (sub_tp T1 T3) %<- (sub_tp T1 T2) %<- (sub_tp T2 T3) %. +%term sub_tp_arrow %pi (sub_tp (arrow S1 S2) (arrow T1 T2)) %<- (sub_tp T1 S1) %<- (sub_tp S2 T2) %. +%term sub_tp_forall + %pi (sub_tp (forall S1 S2) (forall T1 T2)) + %<- (sub_tp T1 S1) + %<- ({x} %pi (sub_tp x T1) %-> (sub_tp (S2 x) (T2 x))) %. +%term sub_tp_record + %pi (sub_tp (record SR _) (record TR _)) + %<- (trow_order SR SR') + %<- (trow_order TR TR') + %<- (sub_tp_trow SR' TR') %. +%term sub_tp_trow_nil sub_tp_trow trow_nil trow_nil %. +%term sub_tp_trow_cons + %pi (sub_tp_trow (trow_cons L S SR) (trow_cons L T TR)) + %<- (sub_tp_trow SR TR) + %<- (sub_tp S T) %. +%term sub_tp_trow_cons' %pi (sub_tp_trow (trow_cons L S SR) TR) %<- (sub_tp_trow SR TR) %. +%% Now we define terms +%sort term %. +%name term %. +%sort bterm %. +%name bterm %. +%sort erow %. +%name erow %. +%sort pattern %. +%name pattern %. +%sort prow %. +%name prow %. +%sort typelist %. +%name typelist %. +%sort termlist %. +%name termlist %. +%%%%% Terms %%%%% +%term base %pi term %-> bterm %. +%term bind %pi tp %-> (%pi term %-> bterm) %-> bterm %. +%term abs %pi tp %-> (%pi term %-> term) %-> term %. +%term app %pi term %-> term %-> term %. +%term tabs %pi tp %-> (%pi tp %-> term) %-> term %. +%term tapp %pi term %-> tp %-> term %. +%term rec %pi erow %-> term %. +%term proj %pi term %-> label %-> term %. +%term let %pi pattern %-> term %-> bterm %-> term %. +%term erow_nil erow %. +%term erow_cons %pi label %-> term %-> erow %-> erow %. +%sort erow_lookup %. +%term erow_lookup_yes erow_lookup L (erow_cons L E _) E %. +%term erow_lookup_no %pi (erow_lookup L (erow_cons _ _ ER) E) %<- (erow_lookup L ER E) %. +%% Takes a record in and returns one in sorted order +%sort erow_order %. +%sort erow_insert %. +%term erow_order_nil erow_order erow_nil erow_nil %. +%term erow_order_cons + %pi (erow_order (erow_cons L E ER) ER') + %<- (erow_order ER ER'') + %<- (erow_insert L E ER'' ER') %. +%term erow_insert_nil erow_insert L E erow_nil (erow_cons L E erow_nil) %. +%term erow_insert_less + %pi (erow_insert L E (erow_cons L' E' ER) (erow_cons L E (erow_cons L' E' ER))) + %<- (label_less L L') %. +%term erow_insert_more + %pi (erow_insert L E (erow_cons L' E' ER) (erow_cons L' E' ER')) + %<- (label_more L L') + %<- (erow_insert L E ER ER') %. +%%%%% Patterns %%%%% +%term pat_var %pi tp %-> pattern %. +%term pat_rec %pi prow %-> pattern %. +%term prow_nil prow %. +%term prow_cons %pi label %-> pattern %-> prow %-> prow %. +%% Takes a record in and returns one in sorted order +%sort prow_order %. +%sort prow_insert %. +%term prow_order_nil prow_order prow_nil prow_nil %. +%term prow_order_cons + %pi (prow_order (prow_cons L P PR) PR') + %<- (prow_order PR PR'') + %<- (prow_insert L P PR'' PR') %. +%term prow_insert_nil prow_insert L P prow_nil (prow_cons L P prow_nil) %. +%term prow_insert_less + %pi (prow_insert L P (prow_cons L' P' PR) (prow_cons L P (prow_cons L' P' PR))) + %<- (label_less L L') %. +%term prow_insert_more + %pi (prow_insert L P (prow_cons L' P' PR) (prow_cons L' P' PR')) + %<- (label_more L L') + %<- (prow_insert L P PR PR') %. +%%%%% termlists and typelists %%%%% +%term typelist_nil %type list %. +%term typelist_cons %pi tp %-> (%type list) %-> (%type list) %. +%term termlist_nil termlist %. +%term termlist_cons %pi term %-> termlist %-> termlist %. +%sort typelist_append %. +%term typelist_append_nil %type list_append %type list_nil TL TL %. +%term typelist_append_cons + %pi (%type list_append (%type list_cons T TL) TL' (%type list_cons T TL'')) + %<- (%type list_append TL TL' TL'') %. +%sort termlist_append %. +%term termlist_append_nil termlist_append termlist_nil EL EL %. +%term termlist_append_cons + %pi (termlist_append (termlist_cons T EL) EL' (termlist_cons T EL'')) + %<- (termlist_append EL EL' EL'') %. +%% Now we write a judgement to tell us whether a term is a value +%sort value %. +%sort value_erow %. +%term value_abs value (abs _ _) %. +%term value_tabs value (tabs _ _) %. +%term value_rec %pi (value (rec ER)) %<- (value_erow ER) %. +%term value_erow_nil value_erow erow_nil %. +%term value_erow_cons %pi (value_erow (erow_cons L V ER)) %<- (value V) %<- (value_erow ER) %. +%% Here we define the typing judgement +%sort of %. +%sort pat_of %. +%sort prow_of %. +%sort bterm_of %. +%sort erow_of %. +%sort list_of %. +%term of_abs %pi (of (abs T1 E) (arrow T1 T2)) %<- ({x} %pi (of x T1) %-> (of (E x) T2)) %. +%term of_app %pi (of (app E1 E2) T12) %<- (of E1 (arrow T11 T12)) %<- (of E2 T11) %. +%term of_tabs + %pi (of (tabs T1 E) (forall T1 T2)) + %<- ({x} %pi (sub_tp x T1) %-> (of (E x) (T2 x))) %. +%term of_tapp %pi (of (tapp E T2) (T12 T2)) %<- (of E (forall T11 T12)) %<- (sub_tp T2 T11) %. +%term of_sub %pi (of E T) %<- (of E S) %<- (sub_tp S T) %. +%term of_record + %pi (of (rec ER) (record TR _)) + %<- (erow_order ER ER') + %<- (trow_order TR TR') + %<- (erow_of ER' TR') %. +%term of_proj %pi (of (proj E L) T) %<- (of E (record TR _)) %<- (trow_lookup L TR T) %. +%term of_let %pi (of (let P E E') T') %<- (of E T) %<- (pat_of P T TL) %<- (bterm_of E' TL T') %. +%term bterm_of_base %pi (bterm_of (base E) %type list_nil T) %<- (of E T) %. +%term bterm_of_bind + %pi (bterm_of (bind T E) (%type list_cons T TL) T') + %<- ({e} %pi (of e T) %-> (bterm_of (E e) TL T')) %. +%term erow_of_nil erow_of erow_nil trow_nil %. +%term erow_of_cons + %pi (erow_of (erow_cons L E ER) (trow_cons L T TR)) + %<- (of E T) + %<- (erow_of ER TR) %. +%term pat_of_var pat_of (pat_var T) T (%type list_cons T %type list_nil) %. +%term pat_of_rec + %pi (pat_of (pat_rec PR) (record TR _) TL) + %<- (trow_order TR TR') + %<- (prow_order PR PR') + %<- (prow_of PR' TR' TL) %. +%term prow_of_nil prow_of prow_nil trow_nil %type list_nil %. +%term prow_of_cons + %pi (prow_of (prow_cons L P PR) (trow_cons L T TR) TL'') + %<- (pat_of P T TL) + %<- (prow_of PR TR TL') + %<- (%type list_append TL' TL TL'') %. +%term list_of_nil list_of termlist_nil %type list_nil %. +%term list_of_cons + %pi (list_of (termlist_cons E EL) (%type list_cons T TL)) + %<- (of E T) + %<- (list_of EL TL) %. +%%%%% Match %%%%% +%sort match %. +%sort prow_match %. +%term match_var match (pat_var _) E (termlist_cons E termlist_nil) %. +%term match_rec + %pi (match (pat_rec PR) (rec ER) EL) + %<- (erow_order ER ER') + %<- (prow_order PR PR') + %<- (prow_match PR' ER' EL) %. +%term prow_match_nil prow_match prow_nil _ termlist_nil %. +%term prow_match_cons + %pi (prow_match (prow_cons L P PR) (erow_cons L E ER) EL'') + %<- (match P E EL) + %<- (prow_match PR ER EL') + %<- (termlist_append EL' EL EL'') %. +%term prow_match_cons' %pi (prow_match PR (erow_cons L E ER) EL) %<- (prow_match PR ER EL) %. +%% Here we define evaluation +%sort apply %. +%term apply_base apply (base E) _ E %. +%term apply_bind %pi (apply (bind _ E) (termlist_cons E' EL) E'') %<- (apply (E E') EL E'') %. +%sort step %. +%sort step_erow %. +%term step_app step (app (abs _ E1) E2) (E1 E2) %. +%term step_tapp step (tapp (tabs _ E1) T) (E1 T) %. +%term step_app_fun %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step_app_arg %pi (step (app V1 E2) (app V1 E2')) %<- (value V1) %<- (step E2 E2') %. +%term step_tapp_fun %pi (step (tapp E T) (tapp E' T)) %<- (step E E') %. +%term step_let %pi (step (let P V E) E') %<- (value V) %<- (match P V EL) %<- (apply E EL E') %. +%term step_let_arg %pi (step (let P E E'') (let P E' E'')) %<- (step E E') %. +%term step_rec %pi (step (proj (rec ER) L) V) %<- (value_erow ER) %<- (erow_lookup L ER V) %. +%term step_proj_rec %pi (step (proj E L) (proj E' L)) %<- (step E E') %. +%term step_rec_exp %pi (step (rec ER) (rec ER')) %<- (step_erow ER ER') %. +%term step_erow_step %pi (step_erow (erow_cons L E ER) (erow_cons L E' ER)) %<- (step E E') %. +%term step_erow_val + %pi (step_erow (erow_cons L V ER) (erow_cons L V ER')) + %<- (value V) + %<- (step_erow ER ER') %. +%% For part 3 we will need steps +%sort steps %. +%term steps_none steps E E %. +%term steps_some %pi (steps E E') %<- (step E E'') %<- (steps E'' E') %. +%% And for good measure we define false +%sort false %. +%%%%% Nats and labels are well-ordered %%%%% +%sort nat_contra %. +%mode nat_contra %in %in %out %. +%term _ %pi (nat_contra (nat_less_s D) (nat_more_s D') F) %<- (nat_contra D D' F) %. +%worlds () (nat_contra _ _ _) %. +%total (D) (nat_contra D _ _) %. +%sort label_contra %. +%mode label_contra %in %in %out %. +%term _ %pi (label_contra (label_less_ D) (label_more_ D') F) %<- (nat_contra D D' F) %. +%worlds () (label_contra _ _ _) %. +%total {} (label_contra D _ _) %. +%sort labelfree_trans_insert %. +%mode labelfree_trans_insert %in %in %in %out %. +%term _ labelfree_trans_insert trow_labelfree_nil trow_insert_nil Lneq (trow_labelfree_cons trow_labelfree_nil Lneq) %. +%term _ labelfree_trans_insert Lfree (trow_insert_less _) L'neq (trow_labelfree_cons Lfree L'neq) %. +%term _ + %pi (labelfree_trans_insert (trow_labelfree_cons Lfree Lneq) (trow_insert_more TRins _) L'neq (trow_labelfree_cons Lfree' Lneq)) + %<- (labelfree_trans_insert Lfree TRins L'neq Lfree') %. +%worlds () (labelfree_trans_insert _ _ _ _) %. +%total (TRins) (labelfree_trans_insert _ TRins _ _) %. +%sort labelfree_trans_order %. +%mode labelfree_trans_order %in %in %out %. +%term _ labelfree_trans_order Lfree trow_order_nil Lfree %. +%term _ + %pi (labelfree_trans_order (trow_labelfree_cons Lfree Lneq) (trow_order_cons TRins SRorder) Lfree'') + %<- (labelfree_trans_order Lfree SRorder Lfree') + %<- (labelfree_trans_insert Lfree' TRins Lneq Lfree'') %. +%worlds () (labelfree_trans_order _ _ _) %. +%total (TRord) (labelfree_trans_order _ TRord _) %. +%% we need to know what and neq correspondes to greater than or less than +%sort nat_mol %. +%term nat_mol_less %pi (nat_mol N M) %<- (nat_less N M) %. +%term nat_mol_more %pi (nat_mol N M) %<- (nat_more N M) %. +%sort label_mol %. +%term label_mol_ %pi (label_mol (label_nat N) (label_nat M)) %<- (nat_mol N M) %. +%sort nat_mol_inc %. +%mode nat_mol_inc %in %out %. +%term _ nat_mol_inc (nat_mol_less Less) (nat_mol_less (nat_less_s Less)) %. +%term _ nat_mol_inc (nat_mol_more More) (nat_mol_more (nat_more_s More)) %. +%worlds () (nat_mol_inc _ _) %. +%total [] (nat_mol_inc _ _) %. +%sort nat_neq_to_mol %. +%mode nat_neq_to_mol %in %out %. +%term _ nat_neq_to_mol nat_neq_zs (nat_mol_less nat_less_z) %. +%term _ nat_neq_to_mol nat_neq_sz (nat_mol_more nat_more_z) %. +%term _ + %pi (nat_neq_to_mol (nat_neq_ss Nneq) Nmol) + %<- (nat_neq_to_mol Nneq Nmol') + %<- (nat_mol_inc Nmol' Nmol) %. +%worlds () (nat_neq_to_mol _ _) %. +%total (Nneq) (nat_neq_to_mol Nneq _) %. +%sort label_neq_to_mol %. +%mode label_neq_to_mol %in %out %. +%term _ + %pi (label_neq_to_mol (label_neq_ Nneq) (label_mol_ Nmol)) + %<- (nat_neq_to_mol Nneq Nmol) %. +%worlds () (label_neq_to_mol _ _) %. +%total [] (label_neq_to_mol _ _) %. +%sort trow_can_insert_helper {S} %. +%mode trow_can_insert_helper %in %in %in %out %. +%term _ trow_can_insert_helper _ (label_mol_ (nat_mol_less Less)) _ (trow_insert_less (label_less_ Less)) %. +%term _ trow_can_insert_helper _ (label_mol_ (nat_mol_more More)) TRins (trow_insert_more TRins (label_more_ More)) %. +%worlds () (trow_can_insert_helper _ _ _ _) %. +%total [] (trow_can_insert_helper _ _ _ _) %. +%sort trow_can_insert {T} %. +%mode trow_can_insert %in %in %out %. +%term _ trow_can_insert _ trow_labelfree_nil trow_insert_nil %. +%term _ + %pi (trow_can_insert _ (trow_labelfree_cons Lfree Lneq) TRins) + %<- (trow_can_insert _ Lfree TRins') + %<- (label_neq_to_mol Lneq Lmol) + %<- (trow_can_insert_helper _ Lmol TRins' TRins) %. +%worlds () (trow_can_insert _ _ _) %. +%total (Lfree) (trow_can_insert _ Lfree _) %. +%sort trow_has_order %. +%mode trow_has_order %in %out %. +%term _ trow_has_order trow_uniqueness_nil trow_order_nil %. +%term _ + %pi (trow_has_order (trow_uniqueness_cons TRunique Lfree) (trow_order_cons TR'ins TRord)) + %<- (trow_has_order TRunique TRord) + %<- (labelfree_trans_order Lfree TRord Lfree') + %<- (trow_can_insert _ Lfree' TR'ins) %. +%worlds () (trow_has_order _ _) %. +%total (TRunique) (trow_has_order TRunique _) %. +%sort sub_tp_trow_refl {TR} %. +%mode sub_tp_trow_refl %in %out %. +%term _ sub_tp_trow_refl trow_nil sub_tp_trow_nil %. +%term _ + %pi (sub_tp_trow_refl (trow_cons _ _ TR) (sub_tp_trow_cons sub_tp_refl TRsub)) + %<- (sub_tp_trow_refl TR TRsub) %. +%worlds () (sub_tp_trow_refl _ _) %. +%total (TR) (sub_tp_trow_refl TR _) %. +%sort sub_tp_trow_trans %. +%mode sub_tp_trow_trans %in %in %out %. +%term _ sub_tp_trow_trans _ sub_tp_trow_nil sub_tp_trow_nil %. +%term _ + %pi (sub_tp_trow_trans (sub_tp_trow_cons' SRsub) TRsub (sub_tp_trow_cons' SRsub')) + %<- (sub_tp_trow_trans SRsub TRsub SRsub') %. +%term _ + %pi (sub_tp_trow_trans (sub_tp_trow_cons _ SRsub) (sub_tp_trow_cons' TRsub) (sub_tp_trow_cons' SRsub')) + %<- (sub_tp_trow_trans SRsub TRsub SRsub') %. +%term _ + %pi (sub_tp_trow_trans (sub_tp_trow_cons Ssub SRsub) (sub_tp_trow_cons Tsub TRsub) (sub_tp_trow_cons (sub_tp_trans Tsub Ssub) SRsub')) + %<- (sub_tp_trow_trans SRsub TRsub SRsub') %. +%worlds () (sub_tp_trow_trans _ _ _) %. +%total (SRsub) (sub_tp_trow_trans SRsub _ _) %. +%sort equality_sub_tp_trow %. +%mode equality_sub_tp_trow %in %in %in %out %. +%term _ equality_sub_tp_trow D trow_eq_ trow_eq_ D %. +%worlds () (equality_sub_tp_trow _ _ _ _) %. +%total [] (equality_sub_tp_trow _ _ _ _) %. +%sort raa_treq %. +%mode {%in TR trow} {%in SR trow} {%in X1 false} {%out X2 trow_eq TR SR} raa_treq X1 X2 %. +%worlds () (raa_treq _ _) %. +%total {} (raa_treq _ _) %. +%sort trow_cons_unique {L} {T} %. +%mode trow_cons_unique %in %in %in %out %. +%term _ trow_cons_unique _ _ trow_eq_ trow_eq_ %. +%worlds () (trow_cons_unique _ _ _ _) %. +%total [] (trow_cons_unique _ _ _ _) %. +%sort trow_insert_unique %. +%mode trow_insert_unique %in %in %in %out %. +%term _ trow_insert_unique trow_insert_nil _ _ trow_eq_ %. +%term _ trow_insert_unique (trow_insert_less _) (trow_insert_less _) _ trow_eq_ %. +%term _ + %pi (trow_insert_unique (trow_insert_more D _) (trow_insert_more D' _) trow_eq_ Deq') + %<- (trow_insert_unique D D' trow_eq_ Deq) + %<- (trow_cons_unique _ _ Deq Deq') %. +%% contradiction cases +%term _ + %pi (trow_insert_unique (trow_insert_less D) (trow_insert_more _ D') trow_eq_ Deq) + %<- (label_contra D D' F) + %<- (raa_treq F Deq) %. +%term _ + %pi (trow_insert_unique (trow_insert_more _ D') (trow_insert_less D) trow_eq_ Deq) + %<- (label_contra D D' F) + %<- (raa_treq F Deq) %. +%worlds () (trow_insert_unique _ _ _ _) %. +%total (D) (trow_insert_unique D _ _ _) %. +%sort trow_order_unique %. +%mode trow_order_unique %in %in %out %. +%term _ trow_order_unique trow_order_nil _ trow_eq_ %. +%term _ + %pi (trow_order_unique (trow_order_cons D2a D1a) (trow_order_cons D2b D1b) Deq') + %<- (trow_order_unique D1a D1b Deq) + %<- (trow_insert_unique D2a D2b Deq Deq') %. +%worlds () (trow_order_unique _ _ _) %. +%total (D) (trow_order_unique D _ _) %. +%%%%% A few lemmas %%%%% +%sort absurd_tsa %. +%mode absurd_tsa %in %out %. +%sort absurd_fsa %. +%mode absurd_fsa %in %out %. +%sort absurd_rsa %. +%mode absurd_rsa %in %out %. +%term _ %pi (absurd_tsa (sub_tp_trans D _) F) %<- (absurd_tsa D F) %. +%term _ %pi (absurd_tsa (sub_tp_trans D _) F) %<- (absurd_fsa D F) %. +%term _ %pi (absurd_tsa (sub_tp_trans _ D) F) %<- (absurd_tsa D F) %. +%term _ %pi (absurd_fsa (sub_tp_trans D _) F) %<- (absurd_tsa D F) %. +%term _ %pi (absurd_tsa (sub_tp_trans D _) F) %<- (absurd_rsa D F) %. +%term _ %pi (absurd_fsa (sub_tp_trans D _) F) %<- (absurd_fsa D F) %. +%term _ %pi (absurd_fsa (sub_tp_trans _ D) F) %<- (absurd_fsa D F) %. +%term _ %pi (absurd_fsa (sub_tp_trans D _) F) %<- (absurd_rsa D F) %. +%term _ %pi (absurd_rsa (sub_tp_trans D _) F) %<- (absurd_tsa D F) %. +%term _ %pi (absurd_rsa (sub_tp_trans D _) F) %<- (absurd_fsa D F) %. +%term _ %pi (absurd_rsa (sub_tp_trans D _) F) %<- (absurd_rsa D F) %. +%term _ %pi (absurd_rsa (sub_tp_trans _ D) F) %<- (absurd_rsa D F) %. +%worlds () (absurd_fsa _ _) (absurd_tsa _ _) (absurd_rsa _ _) %. +%total (D D' D'') (absurd_fsa D _) (absurd_tsa D' _) (absurd_rsa D'' _) %. +%sort absurd_tsf %. +%mode absurd_tsf %in %out %. +%sort absurd_asf %. +%mode absurd_asf %in %out %. +%sort absurd_rsf %. +%mode absurd_rsf %in %out %. +%term _ %pi (absurd_tsf (sub_tp_trans D _) F) %<- (absurd_tsf D F) %. +%term _ %pi (absurd_tsf (sub_tp_trans _ D) F) %<- (absurd_tsf D F) %. +%term _ %pi (absurd_tsf (sub_tp_trans D _) F) %<- (absurd_asf D F) %. +%term _ %pi (absurd_tsf (sub_tp_trans D _) F) %<- (absurd_rsf D F) %. +%term _ %pi (absurd_asf (sub_tp_trans D _) F) %<- (absurd_asf D F) %. +%term _ %pi (absurd_asf (sub_tp_trans _ D) F) %<- (absurd_asf D F) %. +%term _ %pi (absurd_asf (sub_tp_trans D _) F) %<- (absurd_tsf D F) %. +%term _ %pi (absurd_asf (sub_tp_trans D _) F) %<- (absurd_rsf D F) %. +%term _ %pi (absurd_rsf (sub_tp_trans D _) F) %<- (absurd_asf D F) %. +%term _ %pi (absurd_rsf (sub_tp_trans D _) F) %<- (absurd_tsf D F) %. +%term _ %pi (absurd_rsf (sub_tp_trans D _) F) %<- (absurd_rsf D F) %. +%term _ %pi (absurd_rsf (sub_tp_trans _ D) F) %<- (absurd_rsf D F) %. +%worlds () (absurd_tsf _ _) (absurd_asf _ _) (absurd_rsf _ _) %. +%total (D D' D'') (absurd_tsf D _) (absurd_asf D' _) (absurd_rsf D'' _) %. +%sort absurd_tsr %. +%mode absurd_tsr %in %out %. +%sort absurd_asr %. +%mode absurd_asr %in %out %. +%sort absurd_fsr %. +%mode absurd_fsr %in %out %. +%term _ %pi (absurd_tsr (sub_tp_trans D _) F) %<- (absurd_tsr D F) %. +%term _ %pi (absurd_tsr (sub_tp_trans _ D) F) %<- (absurd_tsr D F) %. +%term _ %pi (absurd_tsr (sub_tp_trans D _) F) %<- (absurd_asr D F) %. +%term _ %pi (absurd_tsr (sub_tp_trans D _) F) %<- (absurd_fsr D F) %. +%term _ %pi (absurd_asr (sub_tp_trans D _) F) %<- (absurd_asr D F) %. +%term _ %pi (absurd_asr (sub_tp_trans _ D) F) %<- (absurd_asr D F) %. +%term _ %pi (absurd_asr (sub_tp_trans D _) F) %<- (absurd_tsr D F) %. +%term _ %pi (absurd_asr (sub_tp_trans D _) F) %<- (absurd_fsr D F) %. +%term _ %pi (absurd_fsr (sub_tp_trans D _) F) %<- (absurd_asr D F) %. +%term _ %pi (absurd_fsr (sub_tp_trans D _) F) %<- (absurd_tsr D F) %. +%term _ %pi (absurd_fsr (sub_tp_trans D _) F) %<- (absurd_fsr D F) %. +%term _ %pi (absurd_fsr (sub_tp_trans _ D) F) %<- (absurd_fsr D F) %. +%worlds () (absurd_tsr _ _) (absurd_asr _ _) (absurd_fsr _ _) %. +%total (D D' D'') (absurd_tsr D _) (absurd_asr D' _) (absurd_fsr D'' _) %. +%sort absurd_ota %. +%mode absurd_ota %in %out %. +%term _ %pi (absurd_ota (of_sub D _) F) %<- (absurd_tsa D F) %. +%term _ %pi (absurd_ota (of_sub D _) F) %<- (absurd_fsa D F) %. +%term _ %pi (absurd_ota (of_sub _ D) F) %<- (absurd_ota D F) %. +%term _ %pi (absurd_ota (of_sub D _) F) %<- (absurd_rsa D F) %. +%worlds () (absurd_ota _ _) %. +%total (D) (absurd_ota D _) %. +%sort absurd_ora %. +%mode absurd_ora %in %out %. +%term _ %pi (absurd_ora (of_sub D _) F) %<- (absurd_tsa D F) %. +%term _ %pi (absurd_ora (of_sub D _) F) %<- (absurd_fsa D F) %. +%term _ %pi (absurd_ora (of_sub _ D) F) %<- (absurd_ora D F) %. +%term _ %pi (absurd_ora (of_sub D _) F) %<- (absurd_rsa D F) %. +%worlds () (absurd_ora _ _) %. +%total (D) (absurd_ora D _) %. +%sort absurd_otr %. +%mode absurd_otr %in %out %. +%term _ %pi (absurd_otr (of_sub D _) F) %<- (absurd_tsr D F) %. +%term _ %pi (absurd_otr (of_sub D _) F) %<- (absurd_fsr D F) %. +%term _ %pi (absurd_otr (of_sub _ D) F) %<- (absurd_otr D F) %. +%term _ %pi (absurd_otr (of_sub D _) F) %<- (absurd_asr D F) %. +%worlds () (absurd_otr _ _) %. +%total (D) (absurd_otr D _) %. +%sort absurd_oar %. +%mode absurd_oar %in %out %. +%term _ %pi (absurd_oar (of_sub D _) F) %<- (absurd_tsr D F) %. +%term _ %pi (absurd_oar (of_sub D _) F) %<- (absurd_fsr D F) %. +%term _ %pi (absurd_oar (of_sub _ D) F) %<- (absurd_oar D F) %. +%term _ %pi (absurd_oar (of_sub D _) F) %<- (absurd_asr D F) %. +%worlds () (absurd_oar _ _) %. +%total (D) (absurd_oar D _) %. +%sort absurd_oaf %. +%mode absurd_oaf %in %out %. +%term _ %pi (absurd_oaf (of_sub D _) F) %<- (absurd_tsf D F) %. +%term _ %pi (absurd_oaf (of_sub D _) F) %<- (absurd_asf D F) %. +%term _ %pi (absurd_oaf (of_sub _ D) F) %<- (absurd_oaf D F) %. +%term _ %pi (absurd_oaf (of_sub D _) F) %<- (absurd_rsf D F) %. +%worlds () (absurd_oaf _ _) %. +%total (D) (absurd_oaf D _) %. +%sort absurd_orf %. +%mode absurd_orf %in %out %. +%term _ %pi (absurd_orf (of_sub D _) F) %<- (absurd_tsf D F) %. +%term _ %pi (absurd_orf (of_sub D _) F) %<- (absurd_asf D F) %. +%term _ %pi (absurd_orf (of_sub _ D) F) %<- (absurd_orf D F) %. +%term _ %pi (absurd_orf (of_sub D _) F) %<- (absurd_rsf D F) %. +%worlds () (absurd_orf _ _) %. +%total (D) (absurd_orf D _) %. +%sort absurd_nat_eq_neq %. +%mode absurd_nat_eq_neq %in %out %. +%term _ %pi (absurd_nat_eq_neq (nat_neq_ss Nneq) F) %<- (absurd_nat_eq_neq Nneq F) %. +%worlds () (absurd_nat_eq_neq _ _) %. +%total (Nneq) (absurd_nat_eq_neq Nneq _) %. +%sort absurd_nat_eq_more %. +%mode absurd_nat_eq_more %in %out %. +%term _ %pi (absurd_nat_eq_more (nat_more_s Nmore) F) %<- (absurd_nat_eq_more Nmore F) %. +%worlds () (absurd_nat_eq_more _ _) %. +%total (Nmore) (absurd_nat_eq_more Nmore _) %. +%sort absurd_label_eq_more %. +%mode absurd_label_eq_more %in %out %. +%term _ %pi (absurd_label_eq_more (label_more_ Nmore) F) %<- (absurd_nat_eq_more Nmore F) %. +%worlds () (absurd_label_eq_more _ _) %. +%total [] (absurd_label_eq_more _ _) %. +%sort absurd_label_eq_neq %. +%mode absurd_label_eq_neq %in %out %. +%term _ %pi (absurd_label_eq_neq (label_neq_ Nneq) F) %<- (absurd_nat_eq_neq Nneq F) %. +%worlds () (absurd_label_eq_neq _ _) %. +%total [] (absurd_label_eq_neq _ _) %. +%sort absurd_lookup_free_label %. +%mode absurd_lookup_free_label %in %in %out %. +%term _ + %pi (absurd_lookup_free_label (trow_lookup_no LT) (trow_labelfree_cons Lfree _) F) + %<- (absurd_lookup_free_label LT Lfree F) %. +%term _ + %pi (absurd_lookup_free_label trow_lookup_yes (trow_labelfree_cons _ (label_neq_ Nneq)) F) + %<- (absurd_nat_eq_neq Nneq F) %. +%worlds () (absurd_lookup_free_label _ _ _) %. +%total (LT) (absurd_lookup_free_label LT _ _) %. +%sort absurd_prow_of_free_label %. +%mode absurd_prow_of_free_label %in %in %out %. +%term _ + %pi (absurd_prow_of_free_label (prow_of_cons _ _ _) (trow_labelfree_cons _ Lneq) F) + %<- (absurd_label_eq_neq Lneq F) %. +%worlds () (absurd_prow_of_free_label _ _ _) %. +%total (PRof) (absurd_prow_of_free_label PRof _ _) %. +%sort absurd_prow_match_free_label %. +%mode absurd_prow_match_free_label %in %in %in %out %. +%term _ + %pi (absurd_prow_match_free_label (prow_match_cons _ _ _) (trow_labelfree_cons _ Lneq) (erow_of_cons _ _) F) + %<- (absurd_label_eq_neq Lneq F) %. +%term _ + %pi (absurd_prow_match_free_label (prow_match_cons' PRmatch) (trow_labelfree_cons Lfree _) (erow_of_cons ERof _) F) + %<- (absurd_prow_match_free_label PRmatch Lfree ERof F) %. +%worlds () (absurd_prow_match_free_label _ _ _ _) %. +%total (PRmatch) (absurd_prow_match_free_label PRmatch _ _ _) %. +%%%%% Reductio ad absurdio %%%%% +%sort raa_sub_tp %. +%mode {%in T1} {%in T2} {%in F} {%out D sub_tp T1 T2} raa_sub_tp F D %. +%worlds () (raa_sub_tp _ _) %. +%total [] (raa_sub_tp _ _) %. +%sort raa_sub_imp_sub %. +%mode {%in T} {%in T1} {%in T2} {%in F} {%out D {x} %pi (sub_tp x T) %-> (sub_tp (T1 x) (T2 x))} raa_sub_imp_sub F D %. +%worlds () (raa_sub_imp_sub _ _) %. +%total [] (raa_sub_imp_sub _ _) %. +%sort raa_of %. +%mode {%in E} {%in T} {%in F} {%out D of E T} raa_of F D %. +%worlds () (raa_of _ _) %. +%total [] (raa_of _ _) %. +%sort raa_of_reduces %. +%mode {%in E} {%in T} {%in F} {%in ER} {%in TR} {%in D' erow_of ER TR} {%out D of E T} raa_of_reduces D' F D %. +%worlds () (raa_of_reduces _ _ _) %. +%total [] (raa_of_reduces _ _ _) %. +%reduces < Eof ERof (raa_of_reduces ERof _ Eof) %. +%sort raa_of_imp_of %. +%mode {%in E} {%in T1} {%in T2} {%in F} {%out D {x} %pi (of x T1) %-> (of (E x) T2)} raa_of_imp_of F D %. +%worlds () (raa_of_imp_of _ _) %. +%total [] (raa_of_imp_of _ _) %. +%sort raa_sub_imp_of %. +%mode {%in T} {%in E} {%in T2} {%in F} {%out D {x} %pi (sub_tp x T) %-> (of (E x) (T2 x))} raa_sub_imp_of F D %. +%worlds () (raa_sub_imp_of _ _) %. +%total [] (raa_sub_imp_of _ _) %. +%sort raa_trow_lookup %. +%mode {%in L} {%in TR} {%in T} {%in F} {%out D trow_lookup L TR T} raa_trow_lookup F D %. +%worlds () (raa_trow_lookup _ _) %. +%total [] (raa_trow_lookup _ _) %. +%sort raa_sub_tp_trow %. +%mode {%in SR} {%in TR} {%in F} {%out D sub_tp_trow SR TR} raa_sub_tp_trow F D %. +%worlds () (raa_sub_tp_trow _ _) %. +%total [] (raa_sub_tp_trow _ _) %. +%sort raa_trow_order %. +%mode {%in SR} {%in TR} {%in F} {%out D trow_order SR TR} raa_trow_order F D %. +%worlds () (raa_trow_order _ _) %. +%total [] (raa_trow_order _ _) %. +%sort raa_erow_order %. +%mode {%in ER} {%in ER'} {%in F} {%out D erow_order ER ER'} raa_erow_order F D %. +%worlds () (raa_erow_order _ _) %. +%total [] (raa_erow_order _ _) %. +%sort raa_erow_of %. +%mode {%in ER} {%in TR} {%in F} {%out D erow_of ER TR} raa_erow_of F D %. +%worlds () (raa_erow_of _ _) %. +%total [] (raa_erow_of _ _) %. +%sort raa_erow_of_reduces %. +%mode raa_erow_of_reduces %in %in %out %. +%worlds () (raa_erow_of_reduces _ _ _) %. +%total [] (raa_erow_of_reduces _ _ _) %. +%reduces < ERof Ssub (raa_erow_of_reduces Ssub _ ERof) %. +%sort raa_erow_ins %. +%mode {%in L} {%in E} {%in ER} {%in ER'} {%in F} {%out D erow_insert L E ER ER'} raa_erow_ins F D %. +%worlds () (raa_erow_ins _ _) %. +%total [] (raa_erow_ins _ _) %. +%sort raa_list_of %. +%mode {%in EL} {%in TL} {%in F} {%out D list_of EL TL} raa_list_of F D %. +%worlds () (raa_list_of _ _) %. +%total [] (raa_list_of _ _) %. +%sort raa_match %. +%mode {%in P} {%in E} {%in EL} {%in F} {%out D match P E EL} raa_match F D %. +%worlds () (raa_match _ _) %. +%total [] (raa_match _ _) %. +%sort erow_eq %. +%term erow_eq_ erow_eq ER ER %. +%sort raa_ereq %. +%mode {%in ER erow} {%in ER' erow} {%in X1 false} {%out X2 erow_eq ER ER'} raa_ereq X1 X2 %. +%worlds () (raa_ereq _ _) %. +%total {} (raa_ereq _ _) %. +%sort erow_cons_unique {L} {E} %. +%mode erow_cons_unique %in %in %in %out %. +%term _ erow_cons_unique _ _ erow_eq_ erow_eq_ %. +%worlds () (erow_cons_unique _ _ _ _) %. +%total [] (erow_cons_unique _ _ _ _) %. +%sort erow_insert_unique %. +%mode erow_insert_unique %in %in %in %out %. +%term _ erow_insert_unique erow_insert_nil _ _ erow_eq_ %. +%term _ erow_insert_unique (erow_insert_less _) (erow_insert_less _) _ erow_eq_ %. +%term _ + %pi (erow_insert_unique (erow_insert_more D _) (erow_insert_more D' _) erow_eq_ Deq') + %<- (erow_insert_unique D D' erow_eq_ Deq) + %<- (erow_cons_unique _ _ Deq Deq') %. +%% contradiction cases +%term _ + %pi (erow_insert_unique (erow_insert_less D) (erow_insert_more _ D') erow_eq_ Deq) + %<- (label_contra D D' F) + %<- (raa_ereq F Deq) %. +%term _ + %pi (erow_insert_unique (erow_insert_more _ D') (erow_insert_less D) erow_eq_ Deq) + %<- (label_contra D D' F) + %<- (raa_ereq F Deq) %. +%worlds () (erow_insert_unique _ _ _ _) %. +%total (D) (erow_insert_unique D _ _ _) %. +%sort erow_order_unique %. +%mode erow_order_unique %in %in %out %. +%term _ erow_order_unique erow_order_nil _ erow_eq_ %. +%term _ + %pi (erow_order_unique (erow_order_cons D2a D1a) (erow_order_cons D2b D1b) Deq') + %<- (erow_order_unique D1a D1b Deq) + %<- (erow_insert_unique D2a D2b Deq Deq') %. +%worlds () (erow_order_unique _ _ _) %. +%total (D) (erow_order_unique D _ _) %. +%sort prow_eq %. +%term prow_eq_ prow_eq PR PR %. +%sort raa_preq %. +%mode {%in PR} {%in PR'} {%in F} {%out D prow_eq PR PR'} raa_preq F D %. +%worlds () (raa_preq _ _) %. +%total [] (raa_preq _ _) %. +%sort prow_cons_unique {L} {P} %. +%mode prow_cons_unique %in %in %in %out %. +%term _ prow_cons_unique _ _ prow_eq_ prow_eq_ %. +%worlds () (prow_cons_unique _ _ _ _) %. +%total [] (prow_cons_unique _ _ _ _) %. +%sort prow_insert_unique %. +%mode prow_insert_unique %in %in %in %out %. +%term _ prow_insert_unique prow_insert_nil _ _ prow_eq_ %. +%term _ prow_insert_unique (prow_insert_less _) (prow_insert_less _) _ prow_eq_ %. +%term _ + %pi (prow_insert_unique (prow_insert_more D _) (prow_insert_more D' _) prow_eq_ Deq') + %<- (prow_insert_unique D D' prow_eq_ Deq) + %<- (prow_cons_unique _ _ Deq Deq') %. +%% contradiction cases +%term _ + %pi (prow_insert_unique (prow_insert_less D) (prow_insert_more _ D') prow_eq_ Deq) + %<- (label_contra D D' F) + %<- (raa_preq F Deq) %. +%term _ + %pi (prow_insert_unique (prow_insert_more _ D') (prow_insert_less D) prow_eq_ Deq) + %<- (label_contra D D' F) + %<- (raa_preq F Deq) %. +%worlds () (prow_insert_unique _ _ _ _) %. +%total (D) (prow_insert_unique D _ _ _) %. +%sort prow_order_unique %. +%mode prow_order_unique %in %in %out %. +%term _ prow_order_unique prow_order_nil _ prow_eq_ %. +%term _ + %pi (prow_order_unique (prow_order_cons D2a D1a) (prow_order_cons D2b D1b) Deq') + %<- (prow_order_unique D1a D1b Deq) + %<- (prow_insert_unique D2a D2b Deq Deq') %. +%worlds () (prow_order_unique _ _ _) %. +%total (D) (prow_order_unique D _ _) %. +%sort equal_erow_of %. +%mode equal_erow_of %in %in %in %out %. +%term _ equal_erow_of ERof erow_eq_ trow_eq_ ERof %. +%worlds () (equal_erow_of _ _ _ _) %. +%total [] (equal_erow_of _ _ _ _) %. +%reduces = ERof ERof' (equal_erow_of ERof _ _ ERof') %. +%sort equal_trow_sub %. +%mode equal_trow_sub %in %in %in %out %. +%term _ equal_trow_sub SRsub trow_eq_ trow_eq_ SRsub %. +%worlds () (equal_trow_sub _ _ _ _) %. +%total [] (equal_trow_sub _ _ _ _) %. +%reduces = SRsub SRsub' (equal_trow_sub SRsub _ _ SRsub') %. +%%%%% Some inversions %%%%% +%sort invert_sub_arrow %. +%mode invert_sub_arrow %in %out %out %. +%term _ invert_sub_arrow sub_tp_refl sub_tp_refl sub_tp_refl %. +%term _ invert_sub_arrow (sub_tp_arrow S2sub T1sub) T1sub S2sub %. +%term _ + %pi (invert_sub_arrow (sub_tp_trans T'sub Tsub) (sub_tp_trans U1sub T1sub) (sub_tp_trans U2sub S2sub)) + %<- (invert_sub_arrow Tsub U1sub S2sub) + %<- (invert_sub_arrow T'sub T1sub U2sub) %. +%% bogus cases +%term _ + %pi (invert_sub_arrow (sub_tp_trans D _) T1sub S2sub) + %<- (absurd_fsa D F) + %<- (raa_sub_tp F T1sub) + %<- (raa_sub_tp F S2sub) %. +%term _ + %pi (invert_sub_arrow (sub_tp_trans D _) T1sub S2sub) + %<- (absurd_tsa D F) + %<- (raa_sub_tp F T1sub) + %<- (raa_sub_tp F S2sub) %. +%term _ + %pi (invert_sub_arrow (sub_tp_trans D _) T1sub S2sub) + %<- (absurd_rsa D F) + %<- (raa_sub_tp F T1sub) + %<- (raa_sub_tp F S2sub) %. +%worlds () (invert_sub_arrow _ _ _) %. +%total (Tsub) (invert_sub_arrow Tsub _ _) %. +%sort invert_sub_forall %. +%mode invert_sub_forall %in %out %out %. +%term _ invert_sub_forall (sub_tp_forall S2sub T1sub) T1sub S2sub %. +%term _ invert_sub_forall sub_tp_refl sub_tp_refl ([x] [d] sub_tp_refl) %. +%term _ + %pi (invert_sub_forall (sub_tp_trans T'sub Tsub) (sub_tp_trans U1sub T1sub) ([x] [d] sub_tp_trans (U2sub x d) (S2sub x (sub_tp_trans T1sub d)))) + %<- (invert_sub_forall Tsub U1sub S2sub) + %<- (invert_sub_forall T'sub T1sub U2sub) %. +%% contradictions +%term _ + %pi (invert_sub_forall (sub_tp_trans _ D) T1sub S2sub) + %<- (absurd_fsa D F) + %<- (raa_sub_tp F T1sub) + %<- (raa_sub_imp_sub F S2sub) %. +%term _ + %pi (invert_sub_forall (sub_tp_trans D _) T1sub S2sub) + %<- (absurd_tsf D F) + %<- (raa_sub_tp F T1sub) + %<- (raa_sub_imp_sub F S2sub) %. +%term _ + %pi (invert_sub_forall (sub_tp_trans D _) T1sub S2sub) + %<- (absurd_rsf D F) + %<- (raa_sub_tp F T1sub) + %<- (raa_sub_imp_sub F S2sub) %. +%worlds () (invert_sub_forall _ _ _) %. +%total (Dsub) (invert_sub_forall Dsub _ _) %. +%sort invert_of_abs %. +%mode invert_of_abs %in %out %out %out %. +%term _ invert_of_abs (of_abs ([x] [xassm] Eof x xassm)) Eof sub_tp_refl sub_tp_refl %. +%term _ + %pi (invert_of_abs (of_sub Dsub Eof) Eof' (sub_tp_trans T1sub U1sub) (sub_tp_trans U2sub S2sub)) + %<- (invert_of_abs Eof Eof' T1sub S2sub) + %<- (invert_sub_arrow Dsub U1sub U2sub) %. +%% bogus cases +%term _ + %pi (invert_of_abs (of_sub D _) Eof' T1sub S2sub) + %<- (absurd_tsa D F) + %<- (raa_of_imp_of F (%the ({x} {d} of _ top) Eof')) + %<- (raa_sub_tp F T1sub) + %<- (raa_sub_tp F S2sub) %. +%term _ + %pi (invert_of_abs (of_sub D _) Eof' T1sub S2sub) + %<- (absurd_fsa D F) + %<- (raa_of_imp_of F (%the ({x} {d} of _ top) Eof')) + %<- (raa_sub_tp F T1sub) + %<- (raa_sub_tp F S2sub) %. +%term _ + %pi (invert_of_abs (of_sub D _) Eof' T1sub S2sub) + %<- (absurd_rsa D F) + %<- (raa_of_imp_of F (%the ({x} {d} of _ top) Eof')) + %<- (raa_sub_tp F T1sub) + %<- (raa_sub_tp F S2sub) %. +%worlds () (invert_of_abs _ _ _ _) %. +%total [Eof] (invert_of_abs Eof _ _ _) %. +%sort invert_of_tabs %. +%mode invert_of_tabs %in %out %out %out %. +%term _ invert_of_tabs (of_tabs ([x] [xsub] Eof x xsub)) Eof sub_tp_refl ([x] [d] sub_tp_refl) %. +%term _ + %pi (invert_of_tabs (of_sub Tsub Eof) Eof' (sub_tp_trans U1sub T1sub) ([x] [d] sub_tp_trans (U2sub x d) (S2sub x (sub_tp_trans T1sub d)))) + %<- (invert_of_tabs Eof Eof' U1sub S2sub) + %<- (invert_sub_forall Tsub T1sub U2sub) %. +%% bogus cases +%term _ + %pi (invert_of_tabs (of_sub D _) Eof' T1sub S2sub) + %<- (absurd_tsf D F) + %<- (raa_sub_imp_of F (%the ({x} {d} of _ top) Eof')) + %<- (raa_sub_tp F T1sub) + %<- (raa_sub_imp_sub F S2sub) %. +%term _ + %pi (invert_of_tabs (of_sub D _) Eof' T1sub S2sub) + %<- (absurd_asf D F) + %<- (raa_sub_imp_of F (%the ({x} {d} of _ top) Eof')) + %<- (raa_sub_tp F T1sub) + %<- (raa_sub_imp_sub F S2sub) %. +%term _ + %pi (invert_of_tabs (of_sub D _) Eof' T1sub S2sub) + %<- (absurd_rsf D F) + %<- (raa_sub_imp_of F (%the ({x} {d} of _ top) Eof')) + %<- (raa_sub_tp F T1sub) + %<- (raa_sub_imp_sub F S2sub) %. +%worlds () (invert_of_tabs _ _ _ _) %. +%total [Eof] (invert_of_tabs Eof _ _ _) %. +%sort lookup_erow_to_trow %. +%mode lookup_erow_to_trow %in %in %out %. +%term _ lookup_erow_to_trow erow_lookup_yes (erow_of_cons _ _) trow_lookup_yes %. +%term _ + %pi (lookup_erow_to_trow (erow_lookup_no LE) (erow_of_cons ERof _) (trow_lookup_no LT)) + %<- (lookup_erow_to_trow LE ERof LT) %. +%worlds () (lookup_erow_to_trow _ _ _) %. +%total (LE) (lookup_erow_to_trow LE _ _) %. +%sort lookup_trow_to_erow %. +%mode lookup_trow_to_erow %in %in %out %. +%term _ lookup_trow_to_erow trow_lookup_yes (erow_of_cons _ _) erow_lookup_yes %. +%term _ + %pi (lookup_trow_to_erow (trow_lookup_no LT) (erow_of_cons ERof _) (erow_lookup_no LE)) + %<- (lookup_trow_to_erow LT ERof LE) %. +%worlds () (lookup_trow_to_erow _ _ _) %. +%total (LT) (lookup_trow_to_erow LT _ _) %. +%sort invert_erow_of %. +%mode invert_erow_of %in %in %in %in %out %. +%term _ invert_erow_of (erow_of_cons _ Eof) (trow_uniqueness_cons _ _) erow_lookup_yes trow_lookup_yes Eof %. +%term _ + %pi (invert_erow_of (erow_of_cons ERof _) (trow_uniqueness_cons TRunique Lfree) (erow_lookup_no LE) (trow_lookup_no LT) Eof) + %<- (invert_erow_of ERof TRunique LE LT Eof) %. +%% bogus cases +%term _ + %pi (invert_erow_of (erow_of_cons ERof _) (trow_uniqueness_cons _ Lfree) erow_lookup_yes (trow_lookup_no LT) Eof) + %<- (absurd_lookup_free_label LT Lfree F) + %<- (raa_of_reduces ERof F Eof) %. +%term _ + %pi (invert_erow_of (erow_of_cons ERof _) (trow_uniqueness_cons _ Lfree) (erow_lookup_no LE) trow_lookup_yes Eof) + %<- (lookup_erow_to_trow LE ERof LT) + %<- (absurd_lookup_free_label LT Lfree F) + %<- (raa_of_reduces ERof F Eof) %. +%worlds () (invert_erow_of _ _ _ _ _) %. +%total (ERof) (invert_erow_of ERof _ _ _ _) %. +%reduces < Eof ERof (invert_erow_of ERof _ _ _ Eof) %. +%sort invert_erow_of' %. +%mode invert_erow_of' %in %in %out %. +%term _ invert_erow_of' (erow_of_cons _ Eof) erow_lookup_yes Eof %. +%term _ + %pi (invert_erow_of' (erow_of_cons ERof _) (erow_lookup_no LE) Eof) + %<- (invert_erow_of' ERof LE Eof) %. +%worlds () (invert_erow_of' _ _ _) %. +%total (ERof) (invert_erow_of' ERof _ _) %. +%reduces < Eof ERof (invert_erow_of' ERof _ Eof) %. +%sort lookup_trans_sub_trow %. +%mode lookup_trans_sub_trow %in %in %out %out %. +%term _ lookup_trans_sub_trow trow_lookup_yes (sub_tp_trow_cons Ssub _) trow_lookup_yes Ssub %. +%term _ + %pi (lookup_trans_sub_trow (trow_lookup_no LT) (sub_tp_trow_cons _ SRsub) (trow_lookup_no LS) Ssub) + %<- (lookup_trans_sub_trow LT SRsub LS Ssub) %. +%term _ + %pi (lookup_trans_sub_trow LT (sub_tp_trow_cons' SRsub) (trow_lookup_no LS) Ssub) + %<- (lookup_trans_sub_trow LT SRsub LS Ssub) %. +%worlds () (lookup_trans_sub_trow _ _ _ _) %. +%total (SRsub) (lookup_trans_sub_trow _ SRsub _ _) %. +%sort lookup_trans_trow_insert %. +%mode lookup_trans_trow_insert %in %in %out %. +%term _ lookup_trans_trow_insert trow_lookup_yes (trow_insert_more _ _) trow_lookup_yes %. +%term _ lookup_trans_trow_insert LT (trow_insert_less _) (trow_lookup_no LT) %. +%term _ + %pi (lookup_trans_trow_insert (trow_lookup_no LT) (trow_insert_more TRins _) (trow_lookup_no LT')) + %<- (lookup_trans_trow_insert LT TRins LT') %. +%% Bogus cases +%term _ lookup_trans_trow_insert LT trow_insert_nil (trow_lookup_no LT) %. +%worlds () (lookup_trans_trow_insert _ _ _) %. +%total (LT) (lookup_trans_trow_insert LT _ _) %. +%sort trow_insert_to_lookup %. +%mode trow_insert_to_lookup %in %out %. +%term _ trow_insert_to_lookup trow_insert_nil trow_lookup_yes %. +%term _ trow_insert_to_lookup (trow_insert_less _) trow_lookup_yes %. +%term _ + %pi (trow_insert_to_lookup (trow_insert_more TRins _) (trow_lookup_no LT)) + %<- (trow_insert_to_lookup TRins LT) %. +%worlds () (trow_insert_to_lookup _ _) %. +%total (TRins) (trow_insert_to_lookup TRins _) %. +%sort lookup_trans_trow_order %. +%mode lookup_trans_trow_order %in %in %out %. +%term _ + %pi (lookup_trans_trow_order (trow_lookup_no LT) (trow_order_cons TRins TRord) LT') + %<- (lookup_trans_trow_order LT TRord LT'') + %<- (lookup_trans_trow_insert LT'' TRins LT') %. +%term _ + %pi (lookup_trans_trow_order trow_lookup_yes (trow_order_cons TRins _) LT) + %<- (trow_insert_to_lookup TRins LT) %. +%% bogus cases +%term _ lookup_trans_trow_order LT trow_order_nil LT %. +%worlds () (lookup_trans_trow_order _ _ _) %. +%total (LT) (lookup_trans_trow_order LT _ _) %. +%sort lookup_trans_trow_insert'' {L1} {T1} %. +%mode lookup_trans_trow_insert'' %in %in %in %out %. +%term _ lookup_trans_trow_insert'' _ _ trow_lookup_yes trow_lookup_yes %. +%term _ lookup_trans_trow_insert'' _ _ (trow_lookup_no LT) (trow_lookup_no (trow_lookup_no LT)) %. +%worlds () (lookup_trans_trow_insert'' _ _ _ _) %. +%total [] (lookup_trans_trow_insert'' _ _ _ _) %. +%sort lookup_trans_trow_insert' %. +%mode lookup_trans_trow_insert' %in %in %out %. +%term _ lookup_trans_trow_insert' trow_lookup_yes (trow_insert_less _) trow_lookup_yes %. +%term _ lookup_trans_trow_insert' (trow_lookup_no LT) (trow_insert_less _) (trow_lookup_no LT) %. +%term _ lookup_trans_trow_insert' trow_lookup_yes (trow_insert_more _ _) (trow_lookup_no trow_lookup_yes) %. +%term _ + %pi (lookup_trans_trow_insert' (trow_lookup_no LT) (trow_insert_more TRins _) LT') + %<- (lookup_trans_trow_insert' LT TRins LT'') + %<- (lookup_trans_trow_insert'' _ _ LT'' LT') %. +%term _ lookup_trans_trow_insert' trow_lookup_yes trow_insert_nil trow_lookup_yes %. +%term _ lookup_trans_trow_insert' (trow_lookup_no LT) trow_insert_nil (trow_lookup_no LT) %. +%worlds () (lookup_trans_trow_insert' _ _ _) %. +%total (LT) (lookup_trans_trow_insert' LT _ _) %. +%sort lookup_trans_trow_order' %. +%mode lookup_trans_trow_order' %in %in %out %. +%sort lookup_trans_trow_order'' %. +%mode lookup_trans_trow_order'' %in %in %out %. +%term _ + %pi (lookup_trans_trow_order' LT (trow_order_cons TRins TRord) LT') + %<- (lookup_trans_trow_insert' LT TRins LT'') + %<- (lookup_trans_trow_order'' LT'' TRord LT') %. +%term _ lookup_trans_trow_order'' trow_lookup_yes _ trow_lookup_yes %. +%term _ + %pi (lookup_trans_trow_order'' (trow_lookup_no LT) TRord (trow_lookup_no LT')) + %<- (lookup_trans_trow_order' LT TRord LT') %. +%worlds () (lookup_trans_trow_order' _ _ _) (lookup_trans_trow_order'' _ _ _) %. +%total (TRord' TRord) (lookup_trans_trow_order' _ TRord _) (lookup_trans_trow_order'' _ TRord' _) %. +%sort lookup_trans_erow_insert %. +%mode lookup_trans_erow_insert %in %in %out %. +%term _ lookup_trans_erow_insert erow_lookup_yes (erow_insert_more _ _) erow_lookup_yes %. +%term _ lookup_trans_erow_insert LE (erow_insert_less _) (erow_lookup_no LE) %. +%term _ + %pi (lookup_trans_erow_insert (erow_lookup_no LE) (erow_insert_more ERins _) (erow_lookup_no LE')) + %<- (lookup_trans_erow_insert LE ERins LE') %. +%% Bogus cases +%term _ lookup_trans_erow_insert LE erow_insert_nil (erow_lookup_no LE) %. +%worlds () (lookup_trans_erow_insert _ _ _) %. +%total (LE) (lookup_trans_erow_insert LE _ _) %. +%sort erow_insert_to_lookup %. +%mode erow_insert_to_lookup %in %out %. +%term _ erow_insert_to_lookup erow_insert_nil erow_lookup_yes %. +%term _ erow_insert_to_lookup (erow_insert_less _) erow_lookup_yes %. +%term _ + %pi (erow_insert_to_lookup (erow_insert_more ERins _) (erow_lookup_no LE)) + %<- (erow_insert_to_lookup ERins LE) %. +%worlds () (erow_insert_to_lookup _ _) %. +%total (ERins) (erow_insert_to_lookup ERins _) %. +%sort lookup_trans_erow_order %. +%mode lookup_trans_erow_order %in %in %out %. +%term _ + %pi (lookup_trans_erow_order (erow_lookup_no LE) (erow_order_cons ERins ERord) LE') + %<- (lookup_trans_erow_order LE ERord LE'') + %<- (lookup_trans_erow_insert LE'' ERins LE') %. +%term _ + %pi (lookup_trans_erow_order erow_lookup_yes (erow_order_cons ERins _) LE) + %<- (erow_insert_to_lookup ERins LE) %. +%% bogus cases +%term _ lookup_trans_erow_order LE erow_order_nil LE %. +%worlds () (lookup_trans_erow_order _ _ _) %. +%total (LE) (lookup_trans_erow_order LE _ _) %. +%sort lookup_trans_erow_insert'' {L1} {E1} %. +%mode lookup_trans_erow_insert'' %in %in %in %out %. +%term _ lookup_trans_erow_insert'' _ _ erow_lookup_yes erow_lookup_yes %. +%term _ lookup_trans_erow_insert'' _ _ (erow_lookup_no LE) (erow_lookup_no (erow_lookup_no LE)) %. +%worlds () (lookup_trans_erow_insert'' _ _ _ _) %. +%total [] (lookup_trans_erow_insert'' _ _ _ _) %. +%sort lookup_trans_erow_insert' %. +%mode lookup_trans_erow_insert' %in %in %out %. +%term _ lookup_trans_erow_insert' erow_lookup_yes (erow_insert_less _) erow_lookup_yes %. +%term _ lookup_trans_erow_insert' (erow_lookup_no LE) (erow_insert_less _) (erow_lookup_no LE) %. +%term _ lookup_trans_erow_insert' erow_lookup_yes (erow_insert_more _ _) (erow_lookup_no erow_lookup_yes) %. +%term _ + %pi (lookup_trans_erow_insert' (erow_lookup_no LE) (erow_insert_more ERins _) LE') + %<- (lookup_trans_erow_insert' LE ERins LE'') + %<- (lookup_trans_erow_insert'' _ _ LE'' LE') %. +%term _ lookup_trans_erow_insert' erow_lookup_yes erow_insert_nil erow_lookup_yes %. +%term _ lookup_trans_erow_insert' (erow_lookup_no LE) erow_insert_nil (erow_lookup_no LE) %. +%worlds () (lookup_trans_erow_insert' _ _ _) %. +%total (LE) (lookup_trans_erow_insert' LE _ _) %. +%sort lookup_trans_erow_order' %. +%mode lookup_trans_erow_order' %in %in %out %. +%sort lookup_trans_erow_order'' %. +%mode lookup_trans_erow_order'' %in %in %out %. +%term _ + %pi (lookup_trans_erow_order' LE (erow_order_cons ERins ERord) LE') + %<- (lookup_trans_erow_insert' LE ERins LE'') + %<- (lookup_trans_erow_order'' LE'' ERord LE') %. +%term _ lookup_trans_erow_order'' erow_lookup_yes _ erow_lookup_yes %. +%term _ + %pi (lookup_trans_erow_order'' (erow_lookup_no LE) ERord (erow_lookup_no LE')) + %<- (lookup_trans_erow_order' LE ERord LE') %. +%worlds () (lookup_trans_erow_order' _ _ _) (lookup_trans_erow_order'' _ _ _) %. +%total (ERord' ERord) (lookup_trans_erow_order' _ ERord _) (lookup_trans_erow_order'' _ ERord' _) %. +%sort invert_insert_labelfree %. +%mode invert_insert_labelfree %in %in %out %. +%term _ invert_insert_labelfree _ trow_insert_nil trow_labelfree_nil %. +%term _ invert_insert_labelfree (trow_labelfree_cons Lfree _) (trow_insert_less _) Lfree %. +%term _ + %pi (invert_insert_labelfree (trow_labelfree_cons Lfree Lneq) (trow_insert_more TRins _) (trow_labelfree_cons Lfree' Lneq)) + %<- (invert_insert_labelfree Lfree TRins Lfree') %. +%worlds () (invert_insert_labelfree _ _ _) %. +%total (TRins) (invert_insert_labelfree _ TRins _) %. +%sort invert_insert_unique %. +%mode invert_insert_unique %in %in %out %. +%term _ invert_insert_unique trow_insert_nil _ trow_uniqueness_nil %. +%term _ invert_insert_unique (trow_insert_less _) (trow_uniqueness_cons TRunique _) TRunique %. +%term _ + %pi (invert_insert_unique (trow_insert_more TRins Lmore) (trow_uniqueness_cons TRunique Lfree) (trow_uniqueness_cons TRunique' Lfree')) + %<- (invert_insert_unique TRins TRunique TRunique') + %<- (invert_insert_labelfree Lfree TRins Lfree') %. +%worlds () (invert_insert_unique _ _ _) %. +%total (TRins) (invert_insert_unique TRins _ _) %. +%sort invert_insert_of %. +%mode invert_insert_of %in %in %out %out %. +%term _ invert_insert_of erow_insert_nil (erow_of_cons erow_of_nil _) erow_of_nil trow_insert_nil %. +%term _ invert_insert_of (erow_insert_less Lless) (erow_of_cons ERof _) ERof (trow_insert_less Lless) %. +%term _ + %pi (invert_insert_of (erow_insert_more ERins Lmore) (erow_of_cons ERof Eof) (erow_of_cons ER'of Eof) (trow_insert_more TRins Lmore)) + %<- (invert_insert_of ERins ERof ER'of TRins) %. +%worlds () (invert_insert_of _ _ _ _) %. +%total (ERins) (invert_insert_of ERins _ _ _) %. +%sort reinsert %. +%mode reinsert %in %in %in %out %out %. +%term _ reinsert erow_of_nil Eof trow_insert_nil erow_insert_nil (erow_of_cons erow_of_nil Eof) %. +%term _ reinsert (erow_of_cons ERof E'of) Eof (trow_insert_less Lless) (erow_insert_less Lless) (erow_of_cons (erow_of_cons ERof E'of) Eof) %. +%term _ + %pi (reinsert (erow_of_cons ERof Eof) E'of (trow_insert_more TRins Lmore) (erow_insert_more ERins Lmore) (erow_of_cons ER'of Eof)) + %<- (reinsert ERof E'of TRins ERins ER'of) %. +%worlds () (reinsert _ _ _ _ _) %. +%total (TRins) (reinsert _ _ TRins _ _) %. +%sort nat_neq_symm %. +%mode nat_neq_symm %in %out %. +%term _ nat_neq_symm nat_neq_zs nat_neq_sz %. +%term _ nat_neq_symm nat_neq_sz nat_neq_zs %. +%term _ + %pi (nat_neq_symm (nat_neq_ss Nneq) (nat_neq_ss N'neq)) + %<- (nat_neq_symm Nneq N'neq) %. +%worlds () (nat_neq_symm _ _) %. +%total (Nneq) (nat_neq_symm Nneq _) %. +%sort label_neq_symm %. +%mode label_neq_symm %in %out %. +%term _ + %pi (label_neq_symm (label_neq_ Nneq) (label_neq_ N'neq)) + %<- (nat_neq_symm Nneq N'neq) %. +%worlds () (label_neq_symm _ _) %. +%total [] (label_neq_symm _ _) %. +%sort lookup_trans_sub %. +%mode lookup_trans_sub %in %in %out %out %. +%term _ + %pi (lookup_trans_sub LT (sub_tp_record SR'sub TRord SRord) LS Ssub) + %<- (lookup_trans_trow_order LT TRord LT') + %<- (lookup_trans_sub_trow LT' SR'sub LS' Ssub) + %<- (lookup_trans_trow_order' LS' SRord LS) %. +%term _ lookup_trans_sub LT sub_tp_refl LT sub_tp_refl %. +%term _ + %pi (lookup_trans_sub LT (sub_tp_trans T2sub T1sub) LT' (sub_tp_trans S2sub S1sub)) + %<- (lookup_trans_sub LT T2sub LT'' S2sub) + %<- (lookup_trans_sub LT'' T1sub LT' S1sub) %. +%% bogus cases +%term _ + %pi (lookup_trans_sub _ (sub_tp_trans _ D) LT Ssub) + %<- (absurd_rsf D F) + %<- (raa_sub_tp F Ssub) + %<- (raa_trow_lookup F (%the (trow_lookup L TR top) LT)) %. +%term _ + %pi (lookup_trans_sub _ (sub_tp_trans _ D) LT Ssub) + %<- (absurd_rsa D F) + %<- (raa_sub_tp F Ssub) + %<- (raa_trow_lookup F (%the (trow_lookup L TR top) LT)) %. +%term _ + %pi (lookup_trans_sub _ (sub_tp_trans D _) LT Ssub) + %<- (absurd_tsr D F) + %<- (raa_sub_tp F Ssub) + %<- (raa_trow_lookup F (%the (trow_lookup L TR top) LT)) %. +%worlds () (lookup_trans_sub _ _ _ _) %. +%total (Ssub) (lookup_trans_sub _ Ssub _ _) %. +%sort value_trans_insert %. +%mode value_trans_insert %in %in %in %out %. +%term _ value_trans_insert ERval Eval erow_insert_nil (value_erow_cons ERval Eval) %. +%term _ value_trans_insert ERval Eval (erow_insert_less _) (value_erow_cons ERval Eval) %. +%term _ + %pi (value_trans_insert (value_erow_cons ERval Eval) E'val (erow_insert_more ERins _) (value_erow_cons ER'val Eval)) + %<- (value_trans_insert ERval E'val ERins ER'val) %. +%worlds () (value_trans_insert _ _ _ _) %. +%total (ERins) (value_trans_insert _ _ ERins _) %. +%sort value_trans_order %. +%mode value_trans_order %in %in %out %. +%term _ value_trans_order value_erow_nil erow_order_nil value_erow_nil %. +%term _ + %pi (value_trans_order (value_erow_cons ERval Eval) (erow_order_cons ERins ERord) ER''val) + %<- (value_trans_order ERval ERord ER'val) + %<- (value_trans_insert ER'val Eval ERins ER''val) %. +%worlds () (value_trans_order _ _ _) %. +%total (ERord) (value_trans_order _ ERord _) %. +%sort unique_trans_insert %. +%mode unique_trans_insert %in %in %in %out %. +%term _ unique_trans_insert TRunique Lfree trow_insert_nil (trow_uniqueness_cons TRunique Lfree) %. +%term _ unique_trans_insert TRunique Lfree (trow_insert_less _) (trow_uniqueness_cons TRunique Lfree) %. +%term _ + %pi (unique_trans_insert (trow_uniqueness_cons TRunique Lfree) (trow_labelfree_cons L'free L'neq) (trow_insert_more TRins _) (trow_uniqueness_cons TRunique' Lfree')) + %<- (label_neq_symm L'neq Lneq) + %<- (labelfree_trans_insert Lfree TRins Lneq Lfree') + %<- (unique_trans_insert TRunique L'free TRins TRunique') %. +%worlds () (unique_trans_insert _ _ _ _) %. +%total (TRins) (unique_trans_insert _ _ TRins _) %. +%sort unique_trans_order %. +%mode unique_trans_order %in %in %out %. +%term _ unique_trans_order trow_uniqueness_nil trow_order_nil trow_uniqueness_nil %. +%term _ + %pi (unique_trans_order (trow_uniqueness_cons TRunique Lfree) (trow_order_cons TRins TRord) TR''unique) + %<- (unique_trans_order TRunique TRord TR'unique) + %<- (labelfree_trans_order Lfree TRord Lfree') + %<- (unique_trans_insert TR'unique Lfree' TRins TR''unique) %. +%worlds () (unique_trans_order _ _ _) %. +%total (TRord) (unique_trans_order _ TRord _) %. +%sort invert_of_record %. +%mode invert_of_record %in %in %in %out %. +%term _ + %pi (invert_of_record (%the (of _ (record _ TRunique)) (of_record ERof TRord ERord)) LE LT Eof) + %<- (lookup_trans_trow_order LT TRord LT') + %<- (lookup_trans_erow_order LE ERord LE') + %<- (unique_trans_order TRunique TRord TR'unique) + %<- (invert_erow_of ERof TR'unique LE' LT' Eof) %. +%term _ + %pi (invert_of_record (of_sub Ssub Eof) LE LT (of_sub T'sub E'of)) + %<- (lookup_trans_sub LT Ssub LT' T'sub) + %<- (invert_of_record Eof LE LT' E'of) %. +%term _ + %pi (invert_of_record (of_sub D _) _ _ Eof) + %<- (absurd_fsr D F) + %<- (raa_of F Eof) %. +%term _ + %pi (invert_of_record (of_sub D _) _ _ Eof) + %<- (absurd_asr D F) + %<- (raa_of F Eof) %. +%term _ + %pi (invert_of_record (of_sub D _) _ _ Eof) + %<- (absurd_tsr D F) + %<- (raa_of F Eof) %. +%worlds () (invert_of_record _ _ _ _) %. +%total (Eof) (invert_of_record Eof _ _ _) %. +%sort invert_sub_record %. +%mode invert_sub_record %in %out %out %out %. +%term _ invert_sub_record (sub_tp_record SR'sub TRord SRord) SRord TRord SR'sub %. +%term _ + %pi (invert_sub_record (%the (sub_tp (record _ TRunique) _) sub_tp_refl) TRord TRord TRsub) + %<- (trow_has_order TRunique TRord) + %<- (sub_tp_trow_refl _ TRsub) %. +%term _ + %pi (invert_sub_record (sub_tp_trans T2sub T1sub) SRord TRord SRsub') + %<- (invert_sub_record T1sub SRord URord SRsub) + %<- (invert_sub_record T2sub URord' TRord URsub) + %<- (trow_order_unique URord' URord UReq) + %<- (equality_sub_tp_trow URsub UReq trow_eq_ URsub') + %<- (sub_tp_trow_trans SRsub URsub' SRsub') %. +%term _ + %pi (invert_sub_record (sub_tp_trans D _) TRord TR'ord sub_tp_trow_nil) + %<- (absurd_tsr D F) + %<- (raa_trow_order F TRord) + %<- (raa_trow_order F TR'ord) %. +%term _ + %pi (invert_sub_record (sub_tp_trans D _) TRord TR'ord sub_tp_trow_nil) + %<- (absurd_asr D F) + %<- (raa_trow_order F TRord) + %<- (raa_trow_order F TR'ord) %. +%term _ + %pi (invert_sub_record (sub_tp_trans D _) TRord TR'ord sub_tp_trow_nil) + %<- (absurd_fsr D F) + %<- (raa_trow_order F TRord) + %<- (raa_trow_order F TR'ord) %. +%worlds () (invert_sub_record _ _ _ _) %. +%total (Ssub) (invert_sub_record Ssub _ _ _) %. +%sort invert_of_record2 %. +%mode invert_of_record2 %in %out %out %out %out %out %. +%term _ + %pi (invert_of_record2 (%the (of _ (record _ TRunique)) (of_record ERof TRord ERord)) ERord TRord TRsub ERof TR'unique) + %<- (unique_trans_order TRunique TRord TR'unique) + %<- (sub_tp_trow_refl _ TRsub) %. +%term _ + %pi (invert_of_record2 (of_sub Ssub Eof) ERord TRord SRsub' SRunique ERof) + %<- (invert_of_record2 Eof ERord SRord SRsub SRunique ERof) + %<- (invert_sub_record Ssub SRord' TRord SR'sub) + %<- (trow_order_unique SRord SRord' SReq) + %<- (equality_sub_tp_trow SRsub trow_eq_ SReq SRsub'') + %<- (sub_tp_trow_trans SRsub'' SR'sub SRsub') %. +%% bogus cases +%term _ + %pi (invert_of_record2 (of_sub D _) ERord TRord sub_tp_trow_nil ERof trow_uniqueness_nil) + %<- (absurd_tsr D F) + %<- (raa_erow_of_reduces D F ERof) + %<- (raa_erow_order F ERord) + %<- (raa_trow_order F TRord) %. +%term _ + %pi (invert_of_record2 (of_sub D _) ERord TRord sub_tp_trow_nil ERof trow_uniqueness_nil) + %<- (absurd_fsr D F) + %<- (raa_erow_of_reduces D F ERof) + %<- (raa_erow_order F ERord) + %<- (raa_trow_order F TRord) %. +%term _ + %pi (invert_of_record2 (of_sub D _) ERord TRord sub_tp_trow_nil ERof trow_uniqueness_nil) + %<- (absurd_asr D F) + %<- (raa_erow_of_reduces D F ERof) + %<- (raa_erow_order F ERord) + %<- (raa_trow_order F TRord) %. +%worlds () (invert_of_record2 _ _ _ _ _ _) %. +%total Eof (invert_of_record2 Eof _ _ _ _ _) %. +%reduces < ERof Eof (invert_of_record2 Eof _ _ _ ERof _) %. +%sort erow_insert_convert %. +%mode erow_insert_convert %in %in %in %in %in %out %out %. +%term _ erow_insert_convert erow_insert_nil (erow_of_cons ERof _) _ trow_lookup_yes Eof erow_insert_nil (erow_of_cons ERof Eof) %. +%term _ erow_insert_convert (erow_insert_less Lless) (erow_of_cons ERof _) _ trow_lookup_yes Eof (erow_insert_less Lless) (erow_of_cons ERof Eof) %. +%term _ + %pi (erow_insert_convert (erow_insert_more ERins Lmore) (erow_of_cons ERof Eof) (trow_uniqueness_cons TRunique _) (trow_lookup_no LT) E'of (erow_insert_more ERins' Lmore) (erow_of_cons ER'of Eof)) + %<- (erow_insert_convert ERins ERof TRunique LT E'of ERins' ER'of) %. +%% bogus cases +%term _ + %pi (erow_insert_convert (erow_insert_more _ Lmore) _ _ trow_lookup_yes _ ERins ERof) + %<- (absurd_label_eq_more Lmore F) + %<- (raa_erow_of F (%the (erow_of erow_nil _) ERof)) + %<- (raa_erow_ins F ERins) %. +%term _ + %pi (erow_insert_convert (erow_insert_less _) _ (trow_uniqueness_cons _ Lfree) (trow_lookup_no LT) _ ERins ERof) + %<- (absurd_lookup_free_label LT Lfree F) + %<- (raa_erow_of F (%the (erow_of erow_nil _) ERof)) + %<- (raa_erow_ins F ERins) %. +%worlds () (erow_insert_convert _ _ _ _ _ _ _) %. +%total (ERins) (erow_insert_convert ERins _ _ _ _ _ _) %. +%sort equal_prow_match %. +%mode equal_prow_match %in %in %out %. +%term _ equal_prow_match PRmatch prow_eq_ PRmatch %. +%worlds () (equal_prow_match _ _ _) %. +%total [] (equal_prow_match _ _ _) %. +%reduces = PRmatch PRmatch' (equal_prow_match PRmatch _ PRmatch') %. +%sort list_of_append %. +%mode list_of_append %in %in %in %in %out %. +%term _ list_of_append list_of_nil ELof termlist_append_nil %type list_append_nil ELof %. +%term _ + %pi (list_of_append (list_of_cons ELof Eof) EL'of (termlist_append_cons ELappend) (%type list_append_cons TLappend) (list_of_cons EL''of Eof)) + %<- (list_of_append ELof EL'of ELappend TLappend EL''of) %. +%worlds () (list_of_append _ _ _ _ _) %. +%total (ELof) (list_of_append ELof _ _ _ _) %. +%sort list_of_append' %. +%mode list_of_append' %in %in %out %in %out %. +%term _ list_of_append' list_of_nil ELof termlist_append_nil %type list_append_nil ELof %. +%term _ + %pi (list_of_append' (list_of_cons ELof Eof) EL'of (termlist_append_cons ELappend) (%type list_append_cons TLappend) (list_of_cons EL''of Eof)) + %<- (list_of_append' ELof EL'of ELappend TLappend EL''of) %. +%worlds () (list_of_append' _ _ _ _ _) %. +%total (ELof) (list_of_append' ELof _ _ _ _) %. +%sort labelfree_trans_sub %. +%mode labelfree_trans_sub %in %in %out %. +%term _ labelfree_trans_sub trow_labelfree_nil sub_tp_trow_nil trow_labelfree_nil %. +%term _ + %pi (labelfree_trans_sub (trow_labelfree_cons Lfree _) (sub_tp_trow_cons' SRsub) Lfree') + %<- (labelfree_trans_sub Lfree SRsub Lfree') %. +%term _ + %pi (labelfree_trans_sub (trow_labelfree_cons Lfree Lneq) (sub_tp_trow_cons _ SRsub) (trow_labelfree_cons Lfree' Lneq)) + %<- (labelfree_trans_sub Lfree SRsub Lfree') %. +%worlds () (labelfree_trans_sub _ _ _) %. +%total (SRsub) (labelfree_trans_sub _ SRsub _) %. +%sort pat_match_list_of %. +%mode pat_match_list_of %in %in %in %out %. +%sort prow_match_list_of %. +%mode prow_match_list_of %in %in %in %in %in %out %. +%term _ pat_match_list_of pat_of_var Eof match_var (list_of_cons list_of_nil Eof) %. +%term _ + %pi (pat_match_list_of (pat_of_rec PRof PRord TRord) Eof (match_rec PRmatch PRord' ERord') ELof) + %<- (invert_of_record2 Eof ERord TRord' SRsub ERof SRunique) + %<- (trow_order_unique TRord' TRord TReq) + %<- (erow_order_unique ERord ERord' EReq) + %<- (prow_order_unique PRord' PRord PReq) + %<- (equal_prow_match PRmatch PReq PRmatch') + %<- (equal_erow_of ERof EReq trow_eq_ ERof') + %<- (equal_trow_sub SRsub trow_eq_ TReq SRsub') + %<- (prow_match_list_of PRof ERof' SRsub' SRunique PRmatch' ELof) %. +%term _ prow_match_list_of prow_of_nil _ _ _ prow_match_nil list_of_nil %. +%term _ + %pi (prow_match_list_of (prow_of_cons TLappend PRof Pof) (erow_of_cons ERof Eof) (sub_tp_trow_cons Ssub SRsub) (trow_uniqueness_cons SRunique _) (prow_match_cons ELappend PRmatch Pmatch) EL''of) + %<- (prow_match_list_of PRof ERof SRsub SRunique PRmatch ELof) + %<- (pat_match_list_of Pof (of_sub Ssub Eof) Pmatch EL'of) + %<- (list_of_append ELof EL'of ELappend TLappend EL''of) %. +%term _ + %pi (prow_match_list_of PRof (erow_of_cons ERof _) (sub_tp_trow_cons' SRsub) (trow_uniqueness_cons TRunique _) (prow_match_cons' PRmatch) ELof) + %<- (prow_match_list_of PRof ERof SRsub TRunique PRmatch ELof) %. +%% bogus cases +%term _ + %pi (prow_match_list_of (prow_of_cons TLappend PRof Pof) (erow_of_cons _ _) (sub_tp_trow_cons' SRsub) (trow_uniqueness_cons _ Lfree) (prow_match_cons _ _ _) ELof) + %<- (labelfree_trans_sub Lfree SRsub Lfree') + %<- (absurd_prow_of_free_label (prow_of_cons TLappend PRof Pof) Lfree' F) + %<- (raa_list_of F ELof) %. +%term _ + %pi (prow_match_list_of (prow_of_cons _ _ _) (erow_of_cons ERof _) (sub_tp_trow_cons _ _) (trow_uniqueness_cons _ Lfree) (prow_match_cons' PRmatch) ELof) + %<- (absurd_prow_match_free_label PRmatch Lfree ERof F) + %<- (raa_list_of F ELof) %. +%worlds () (pat_match_list_of _ _ _ _) (prow_match_list_of _ _ _ _ _ _) %. +%total (Pmatch PRmatch) (prow_match_list_of _ _ _ _ PRmatch _) (pat_match_list_of _ _ Pmatch _) %. +%sort list_of_apply_of %. +%mode list_of_apply_of %in %in %in %out %. +%term _ list_of_apply_of (bterm_of_base Eof) _ _ Eof %. +%term _ + %pi (list_of_apply_of (bterm_of_bind Eof) (list_of_cons ELof E'of) (apply_bind Eapp) Eof') + %<- (list_of_apply_of (Eof _ E'of) ELof Eapp Eof') %. +%worlds () (list_of_apply_of _ _ _ _) %. +%total (ELof) (list_of_apply_of _ ELof _ _) %. +%% Proof of Preservation +%sort preservation' {E} %. +%mode preservation' %in %in %in %out %. +%sort preservation_erow {ER1} %. +%mode preservation_erow %in %in %in %in %in %out %out %. +%term _ + %pi (preservation' _ (of_sub Ssub Eof) Estep (of_sub Ssub E'of)) + %<- (preservation' _ Eof Estep E'of) %. +%term _ + %pi (preservation' _ (of_app E2of E1of) step_app (of_sub S2sub (E1of' _ (of_sub T1sub E2of)))) + %<- (invert_of_abs E1of E1of' T1sub S2sub) %. +%term _ + %pi (preservation' _ (of_app E2of E1of) (step_app_fun E1step) (of_app E2of E1of')) + %<- (preservation' _ E1of E1step E1of') %. +%term _ + %pi (preservation' _ (of_app E2of E1of) (step_app_arg E2step _) (of_app E2of' E1of)) + %<- (preservation' _ E2of E2step E2of') %. +%term _ + %pi (preservation' _ (of_tapp T2sub Eof) (step_tapp_fun Estep) (of_tapp T2sub Eof')) + %<- (preservation' _ Eof Estep Eof') %. +%term _ + %pi (preservation' _ (of_tapp T2sub Eof) step_tapp (of_sub (S2sub _ T2sub) (Eof' _ (sub_tp_trans T1sub T2sub)))) + %<- (invert_of_tabs Eof Eof' T1sub S2sub) %. +%term _ + %pi (preservation' _ (of_let E'of Pof Vof) (step_let Eapp Pmatch Vval) E''of) + %<- (pat_match_list_of Pof Vof Pmatch ELof) + %<- (list_of_apply_of E'of ELof Eapp E''of) %. +%term _ + %pi (preservation' _ (of_let E'of Pof Eof) (step_let_arg Estep) (of_let E'of Pof Eof')) + %<- (preservation' _ Eof Estep Eof') %. +%term _ + %pi (preservation' _ (of_proj LT Eof) (step_rec LV _) E'of) + %<- (invert_of_record Eof LV LT E'of) %. +%term _ + %pi (preservation' _ (%the (of _ (record _ TRunique)) (of_record ERof TRord ERord)) (step_rec_exp ERstep) (of_record ER'of TRord ER'ord)) + %<- (unique_trans_order TRunique TRord TR'unique) + %<- (preservation_erow _ ERof TR'unique ERord ERstep ER'of ER'ord) %. +%term _ + %pi (preservation' _ (of_proj LT Eof) (step_proj_rec Estep) (of_proj LT E'of)) + %<- (preservation' _ Eof Estep E'of) %. +%term _ + %pi (preservation_erow _ ERof TRunique (erow_order_cons ERins ERord) (step_erow_step Estep) ER'of (erow_order_cons ERins' ERord)) + %<- (erow_insert_to_lookup ERins LE) + %<- (lookup_erow_to_trow LE ERof LT) + %<- (invert_erow_of ERof TRunique LE LT Eof) + %<- (preservation' _ Eof Estep E'of) + %<- (erow_insert_convert ERins ERof TRunique LT E'of ERins' ER'of) %. +%term _ + %pi (preservation_erow _ ERof TRunique (erow_order_cons ERins ERord) (step_erow_val ERstep Eval) ER'of'' (erow_order_cons ERins' ER'ord)) + %<- (erow_insert_to_lookup ERins LE) + %<- (invert_insert_of ERins ERof ER'of TRins) + %<- (trow_insert_to_lookup TRins LT) + %<- (invert_erow_of ERof TRunique LE LT Eof) + %<- (invert_insert_unique TRins TRunique TR'unique) + %<- (preservation_erow _ ER'of TR'unique ERord ERstep ER'of' ER'ord) + %<- (reinsert ER'of' Eof TRins ERins' ER'of'') %. +%worlds () (preservation' _ _ _ _) (preservation_erow _ _ _ _ _ _ _) %. +%total {(E ER) (Eof ERof)} (preservation' E Eof _ _) (preservation_erow ER ERof _ _ _ _ _) %. +%% proof of progress +%sort progresses %. +%sort progresses_erow %. +%% E may be a value +%term progresses_value %pi (progresses E) %<- (value E) %. +%term progresses_erow_value %pi (progresses_erow ER) %<- (value_erow ER) %. +%% Or E may take a step +%term progresses_steps %pi (progresses E) %<- (step E E') %. +%term progresses_erow_steps %pi (progresses_erow ER) %<- (step_erow ER ER') %. +%%%%% Reductio ad absurdio %%%%% +%sort raa_progresses %. +%mode {%in E} {%in F} {%out D progresses E} raa_progresses F D %. +%worlds () (raa_progresses _ _) %. +%total [] (raa_progresses _ _) %. +%% progress lemmas +%sort app_progresses %. +%mode app_progresses %in %in %in %out %. +%term _ + %pi (app_progresses (of_sub _ Eof) E1prog E2prog E3prog) + %<- (app_progresses Eof E1prog E2prog E3prog) %. +%term _ app_progresses _ (progresses_steps E1step) _ (progresses_steps (step_app_fun E1step)) %. +%term _ app_progresses (of_app E2of E1of) (progresses_value E1val) (progresses_steps E2step) (progresses_steps (step_app_arg E2step E1val)) %. +%term _ app_progresses (of_app E2of E1of) (progresses_value E1val) (progresses_value E2val) (progresses_steps step_app) %. +%% bogus cases +%term _ + %pi (app_progresses (of_app _ E1of) _ _ D) + %<- (absurd_ota E1of F) + %<- (raa_progresses F D) %. +%term _ + %pi (app_progresses (of_app _ E1of) _ _ D) + %<- (absurd_ora E1of F) + %<- (raa_progresses F D) %. +%worlds () (app_progresses _ _ _ _) %. +%total [Eof] (app_progresses Eof _ _ _) %. +%sort tapp_progresses %. +%mode tapp_progresses %in %in %out %. +%term _ + %pi (tapp_progresses (of_sub _ Eof) Eprog E'prog) + %<- (tapp_progresses Eof Eprog E'prog) %. +%term _ tapp_progresses _ (progresses_steps Estep) (progresses_steps (step_tapp_fun Estep)) %. +%term _ tapp_progresses (of_tapp E2of E1of) (progresses_value Eval) (progresses_steps step_tapp) %. +%% bogus cases +%term _ + %pi (tapp_progresses (of_tapp _ Eof) _ D) + %<- (absurd_oaf Eof F) + %<- (raa_progresses F D) %. +%term _ + %pi (tapp_progresses (of_tapp _ Eof) _ D) + %<- (absurd_orf Eof F) + %<- (raa_progresses F D) %. +%worlds () (tapp_progresses _ _ _) %. +%total [Eof] (tapp_progresses Eof _ _) %. +%sort proj_progresses %. +%mode proj_progresses %in %in %out %. +%term _ + %pi (proj_progresses (of_sub _ Eof) Eproj E'proj) + %<- (proj_progresses Eof Eproj E'proj) %. +%term _ proj_progresses _ (progresses_steps Estep) (progresses_steps (step_proj_rec Estep)) %. +%term _ + %pi (proj_progresses (of_proj LT Eof) (progresses_value (value_rec ERval)) (progresses_steps (step_rec LE ERval))) + %<- (invert_of_record2 Eof ERord TRord TR'sub ER'of _) + %<- (lookup_trans_trow_order LT TRord LT') + %<- (lookup_trans_sub_trow LT' TR'sub LT'' _) + %<- (lookup_trow_to_erow LT'' ER'of LE') + %<- (lookup_trans_erow_order' LE' ERord LE) %. +%term _ + %pi (proj_progresses (of_proj _ Eof) _ D) + %<- (absurd_oar Eof F) + %<- (raa_progresses F D) %. +%term _ + %pi (proj_progresses (of_proj _ Eof) _ D) + %<- (absurd_otr Eof F) + %<- (raa_progresses F D) %. +%worlds () (proj_progresses _ _ _) %. +%total (Eof) (proj_progresses Eof _ _) %. +%sort rec_progresses %. +%mode rec_progresses %in %out %. +%term _ rec_progresses (progresses_erow_steps ERstep) (progresses_steps (step_rec_exp ERstep)) %. +%term _ rec_progresses (progresses_erow_value ERval) (progresses_value (value_rec ERval)) %. +%worlds () (rec_progresses _ _) %. +%total [] (rec_progresses _ _) %. +%sort equality_erow_of %. +%mode equality_erow_of %in %in %out %. +%term _ equality_erow_of ERof trow_eq_ ERof %. +%worlds () (equality_erow_of _ _ _) %. +%total [] (equality_erow_of _ _ _) %. +%reduces = ERof ERof' (equality_erow_of ERof _ ERof') %. +%sort pattern_can_match %. +%mode pattern_can_match %in %in %in %out %out %. +%sort prow_can_match %. +%mode prow_can_match %in %in %in %in %out %out %. +%term _ pattern_can_match Eof _ pat_of_var match_var (list_of_cons list_of_nil Eof) %. +%term _ + %pi (pattern_can_match Eof (value_rec ERval) (pat_of_rec PRof PRord TRord') (match_rec PRmatch PRord ERord) ELof) + %<- (invert_of_record2 Eof ERord TRord SR'sub ER'of SR'unique) + %<- (trow_order_unique TRord TRord' TReq) + %<- (equal_trow_sub SR'sub trow_eq_ TReq SR'sub') + %<- (value_trans_order ERval ERord ER'val) + %<- (prow_can_match ER'of SR'sub' ER'val PRof PRmatch ELof) %. +%term _ + %pi (pattern_can_match Eof value_abs _ Pmatch ELof) + %<- (absurd_oar Eof F) + %<- (raa_list_of F (%the (list_of termlist_nil _) ELof)) + %<- (raa_match F Pmatch) %. +%term _ + %pi (pattern_can_match Eof value_tabs _ Pmatch ELof) + %<- (absurd_otr Eof F) + %<- (raa_list_of F (%the (list_of termlist_nil _) ELof)) + %<- (raa_match F Pmatch) %. +%term _ prow_can_match _ _ _ prow_of_nil prow_match_nil list_of_nil %. +%term _ + %pi (prow_can_match (erow_of_cons ERof _) (sub_tp_trow_cons' SRsub) (value_erow_cons ERval _) PRof (prow_match_cons' PRmatch) ELof) + %<- (prow_can_match ERof SRsub ERval PRof PRmatch ELof) %. +%term _ + %pi (prow_can_match (erow_of_cons ERof Eof) (sub_tp_trow_cons Ssub SRsub) (value_erow_cons ERval Eval) (prow_of_cons TLappend PRof Pof) (prow_match_cons ELappend PRmatch Pmatch) EL''of) + %<- (pattern_can_match (of_sub Ssub Eof) Eval Pof Pmatch EL'of) + %<- (prow_can_match ERof SRsub ERval PRof PRmatch ELof) + %<- (list_of_append' ELof EL'of ELappend TLappend EL''of) %. +%worlds () (pattern_can_match _ _ _ _ _) (prow_can_match _ _ _ _ _ _) %. +%total {(Pof PRof) (Eof ERof)} (pattern_can_match Eof _ Pof _ _) (prow_can_match ERof _ _ PRof _ _) %. +%sort list_can_apply %. +%mode list_can_apply %in %in %out %out %. +%term _ list_can_apply (bterm_of_base Eof) list_of_nil apply_base Eof %. +%term _ + %pi (list_can_apply (bterm_of_bind Eof) (list_of_cons ELof E'of) (apply_bind Eapp) E''of) + %<- (list_can_apply (Eof _ E'of) ELof Eapp E''of) %. +%worlds () (list_can_apply _ _ _ _) %. +%total (ELof) (list_can_apply _ ELof _ _) %. +%sort let_progresses %. +%mode let_progresses %in %in %in %in %out %. +%term _ let_progresses _ _ _ (progresses_steps Estep) (progresses_steps (step_let_arg Estep)) %. +%term _ + %pi (let_progresses Eof Pof E'of (progresses_value Eval) (progresses_steps (step_let Eapp Pmatch Eval))) + %<- (pattern_can_match Eof Eval Pof Pmatch ELof) + %<- (list_can_apply E'of ELof Eapp E'of') %. +%worlds () (let_progresses _ _ _ _ _) %. +%total [] (let_progresses _ _ _ _ _) %. +%sort erow_progresses {L} %. +%mode erow_progresses %in %in %in %out %. +%term _ erow_progresses _ (progresses_steps Estep) _ (progresses_erow_steps (step_erow_step Estep)) %. +%term _ erow_progresses _ (progresses_value Eval) (progresses_erow_steps ERsteps) (progresses_erow_steps (step_erow_val ERsteps Eval)) %. +%term _ erow_progresses _ (progresses_value Eval) (progresses_erow_value ERval) (progresses_erow_value (value_erow_cons ERval Eval)) %. +%worlds () (erow_progresses _ _ _ _) %. +%total [] (erow_progresses _ _ _ _) %. +%% progress proof +%sort progress' {E} %. +%mode progress' %in %in %out %. +%sort progress_erow {ER} %. +%mode progress_erow %in %in %in %out %. +%term _ %pi (progress' _ (of_sub _ Eof) Eprogress) %<- (progress' _ Eof Eprogress) %. +%term _ progress' _ (of_abs _) (progresses_value value_abs) %. +%term _ progress' _ (of_tabs _) (progresses_value value_tabs) %. +%term _ + %pi (progress' _ (of_app E2of E1of) Eprog) + %<- (progress' _ E1of E1prog) + %<- (progress' _ E2of E2prog) + %<- (app_progresses (of_app E2of E1of) E1prog E2prog Eprog) %. +%term _ + %pi (progress' _ (of_tapp Tsub Eof) E'prog) + %<- (progress' _ Eof Eprog) + %<- (tapp_progresses (of_tapp Tsub Eof) Eprog E'prog) %. +%term _ + %pi (progress' _ (of_record ER'of _ ERord) Eprog) + %<- (progress_erow _ ERord ER'of ERprog) + %<- (rec_progresses ERprog Eprog) %. +%term _ + %pi (progress' _ (of_proj LT Eof) E'proj) + %<- (progress' _ Eof Eproj) + %<- (proj_progresses (of_proj LT Eof) Eproj E'proj) %. +%term _ + %pi (progress' _ (of_let E'of Pof Eof) E'step) + %<- (progress' _ Eof Estep) + %<- (let_progresses Eof Pof E'of Estep E'step) %. +%term _ progress_erow _ erow_order_nil erow_of_nil (progresses_erow_value value_erow_nil) %. +%term _ + %pi (progress_erow _ (erow_order_cons ERins ERord) ERof ER'prog) + %<- (erow_insert_to_lookup ERins LE) + %<- (invert_erow_of' ERof LE Eof) + %<- (progress' _ Eof Eprog) + %<- (invert_insert_of ERins ERof ER'of _) + %<- (progress_erow _ ERord ER'of ERprog) + %<- (erow_progresses _ Eprog ERprog ER'prog) %. +%worlds () (progress' _ _ _) (progress_erow _ _ _ _) %. +%total {(E ER) (Eof ERof)} (progress' E Eof _) (progress_erow ER _ ERof _) %. +%%%%% Peroration %%%%% +%sort preservation %. +%mode preservation %in %in %out %. +%term _ %pi (preservation Eof Estep E'of) %<- (preservation' _ Eof Estep E'of) %. +%worlds () (preservation _ _ _) %. +%total [] (preservation _ _ _) %. +%sort progress %. +%mode progress %in %out %. +%term _ %pi (progress Eof Eprogresses) %<- (progress' _ Eof Eprogresses) %. +%worlds () (progress _ _) %. +%total [] (progress _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/prop-calc/equiv-thm.lf b/new-tests/stelf-output/examples/prop-calc/equiv-thm.lf new file mode 100644 index 0000000..0e0ddb3 --- /dev/null +++ b/new-tests/stelf-output/examples/prop-calc/equiv-thm.lf @@ -0,0 +1,8 @@ +% abs implements bracket abstraction which amounts to +% the computational content of the deduction theorem. +%theorem abst : forall* {A o} {B o} forall {P %pi (|- A) %-> (|- B)} exists {Q |- A => B} true %. +%prove 3 P (abst P _) %. +%terminates P (abst P _) %. +%theorem comdefn : forall* {A o} forall {P |- A} exists {Q ! A} true %. +%terminates P (comdefn P _) %. +% %prove 4 P (comdefn P _). \ No newline at end of file diff --git a/new-tests/stelf-output/examples/prop-calc/equiv.lf b/new-tests/stelf-output/examples/prop-calc/equiv.lf new file mode 100644 index 0000000..2f6e9fd --- /dev/null +++ b/new-tests/stelf-output/examples/prop-calc/equiv.lf @@ -0,0 +1,47 @@ +%%% A most naive version of translation from natural deductions +%%% into a proof in the Hilbert-style system and vice versa. +%%% Expresses the equivalence of the two systems. +%%% Author: Frank Pfenning +% abs implements bracket abstraction which amounts to +% the computational content of the deduction theorem. +%sort abs %. +% aID : abs ([x] x) (MP (MP S K) K). +%term aONE abs ([x] ONE) (MP K ONE) %. +%term aPAIR abs ([x] PAIR) (MP K PAIR) %. +%term aLEFT abs ([x] LEFT) (MP K LEFT) %. +%term aRIGHT abs ([x] RIGHT) (MP K RIGHT) %. +%term aK abs ([x] K) (MP K K) %. +%term aS abs ([x] S) (MP K S) %. +%term aMP %pi (abs ([x] MP (P x) (Q x)) (MP (MP S P') Q')) %<- (abs P P') %<- (abs Q Q') %. +%mode abs %in %out %. +% %covers abs +D -F. +% comb does the translation from natural deduction to Hilbert deductions, +% appealing to the deduction theorem in case of implies-introduction +% which requires hypothetical reasoning. +%sort comb %. +%mode comb %in %out %. +%term ctrue comb trueI ONE %. +%term candI %pi (comb (andI P Q) (MP (MP PAIR P') Q')) %<- (comb P P') %<- (comb Q Q') %. +%term candEL %pi (comb (andEL P) (MP LEFT P')) %<- (comb P P') %. +%term candER %pi (comb (andER P) (MP RIGHT P')) %<- (comb P P') %. +%term cimpliesI + %pi (comb (impliesI PP) Q) + %<- ({x} {y} %pi (comb x y) %-> ({B o} abs ([z |- B] y) (MP K y)) %-> (comb (PP x) (PP' y))) + %<- (abs PP' Q) %. +%term cimpliesE %pi (comb (impliesE P Q) (MP P' Q')) %<- (comb P P') %<- (comb Q Q') %. +%block l {A o} [x ! A] [y |- A] [u comb x y] [v {B o} abs ([z |- B] y) (MP K y)]%. +%worlds (l) (abs D F) %. +%worlds (l) (comb _ _) %. +% Now the translation from Hilbert deduction into natural deductions. +% This simply gives the definition of the (proof) combinators as lambda-terms. +%sort combdefn %. +%mode combdefn %in %out %. +%term cdK combdefn K (impliesI ([p] impliesI ([q] p))) %. +%term cdS combdefn S (impliesI ([p] impliesI ([q] impliesI ([r] impliesE (impliesE p r) (impliesE q r))))) %. +%term cdONE combdefn ONE trueI %. +%term cdPAIR combdefn PAIR (impliesI ([p] impliesI ([q] andI p q))) %. +%term cdLEFT combdefn LEFT (impliesI ([p] andEL p)) %. +%term cdRIGHT combdefn RIGHT (impliesI ([p] andER p)) %. +%term cdMP %pi (combdefn (MP P Q) (impliesE P' Q')) %<- (combdefn P P') %<- (combdefn Q Q') %. +%%% Decidability proof for theorem HD -> ND +%terminates D (combdefn D _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/prop-calc/examples-quy.lf b/new-tests/stelf-output/examples/prop-calc/examples-quy.lf new file mode 100644 index 0000000..235ee19 --- /dev/null +++ b/new-tests/stelf-output/examples/prop-calc/examples-quy.lf @@ -0,0 +1,8 @@ +% Translating A & B => B & A +%query 1 _ _ {A o} {B o} comb (impliesI ([x ! (A & B)] andI (andER x) (andEL x))) (H A B) %. +% Translating the result back to a natural deduction +%query 1 _ _ combdefn (MP (MP S (MP (MP S (MP K PAIR)) (MP (MP S (MP K RIGHT)) (MP (MP S K) K)))) (MP (MP S (MP K LEFT)) (MP (MP S K) K))) D %. +% Translating S +%query 1 _ _ combdefn S D %. +% Translating back the natural deduction for S +%query 1 _ _ comb (impliesI ([p ! A1 => A2 => A3] impliesI ([q ! A1 => A2] impliesI ([r ! A1] impliesE (impliesE p r) (impliesE q r))))) H %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/prop-calc/nd.lf b/new-tests/stelf-output/examples/prop-calc/nd.lf new file mode 100644 index 0000000..46556c5 --- /dev/null +++ b/new-tests/stelf-output/examples/prop-calc/nd.lf @@ -0,0 +1,14 @@ +%%% Intuitionistic propositional calculus +%%% Positive fragment with implies, and, true. +%%% Two formulations here: natural deduction and Hilbert-style system. +%%% Author: Frank Pfenning +% Natural Deduction. +%sort ! %. +%prec %prefix 9 ! %. +%name ! %. +%term trueI ! true %. +%term andI %pi (! A) %-> (! B) %-> (! A & B) %. +%term andEL %pi (! A & B) %-> (! A) %. +%term andER %pi (! A & B) %-> (! B) %. +%term impliesI %pi (%pi (! A) %-> (! B)) %-> (! A => B) %. +%term impliesE %pi (! A => B) %-> (! A) %-> (! B) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/prop-calc/ndclass.lf b/new-tests/stelf-output/examples/prop-calc/ndclass.lf new file mode 100644 index 0000000..5ae1697 --- /dev/null +++ b/new-tests/stelf-output/examples/prop-calc/ndclass.lf @@ -0,0 +1,16 @@ +%sort nd %. +%term impE %pi (nd (A => B)) %-> (nd A) %-> (nd B) %. +%term impI %pi (%pi (nd A) %-> (nd B)) %-> (nd (A => B)) %. +%define examplebb impI ([u nd A] impI ([v nd B] u)) %. +%sort ndI %. +%sort ndE %. +%term impE %pi (ndE (A => B)) %-> (ndI A) %-> (ndE B) %. +%term impI %pi (%pi (ndE A) %-> (ndI B)) %-> (ndI (A => B)) %. +%term a o %. +%term IE %pi (ndE a) %-> (ndI a) %. +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort add %. +%term addz add z N N %. +%term adds %pi (add M N L) %-> (add (s M) N (s L)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/prop-calc/prop-calc-solution.lf b/new-tests/stelf-output/examples/prop-calc/prop-calc-solution.lf new file mode 100644 index 0000000..c87ecea --- /dev/null +++ b/new-tests/stelf-output/examples/prop-calc/prop-calc-solution.lf @@ -0,0 +1,47 @@ +%%% Intuitionistic propositional calculus +%%% Positive fragment with implies, and, true. +%%% Operational meaning of derivations as programs. +%%% Author: Frank Pfenning, Carsten Schuermann +% Type of propositions. +%sort o %. +%name o %. +% Syntax: implication, plus a few constants. +%term => %pi o %-> o %-> o %. +%prec %right 10 => %. +%term & %pi o %-> o %-> o %. +%prec %right 11 & %. +%term true o %. +% Provability. +%sort |- %. +%prec %prefix 9 |- %. +%name |- %. +% Axioms. +%term K |- A => B => A %. +%term S |- (A => B => C) => (A => B) => A => C %. +%term ONE |- true %. +%term PAIR |- A => B => A & B %. +%term LEFT |- A & B => A %. +%term RIGHT |- A & B => B %. +% Inference Rule. +%term MP %pi (|- A => B) %-> (|- A) %-> (|- B) %. +% Operational semantics +% Basic reductions +%sort ==> %. +%prec %right 9 ==> %. +%term stepK MP (MP K X) Y ==> X %. +%term stepS MP (MP (MP S X) Y) Z ==> MP (MP X Z) (MP Y Z) %. +%term stepL MP LEFT (MP (MP PAIR X) Y) ==> X %. +%term stepR MP RIGHT (MP (MP PAIR X) Y) ==> Y %. +% Congruence rules +%term congMPL %pi (MP P1 P2 ==> MP P1 P2) %<- (P1 ==> P1') %. +%term congMPR %pi (MP P1 P2 ==> MP P1 P2') %<- (P2 ==> P2') %. +%sort ==>* %. +%prec %right 9 ==>* %. +%term refl P ==>* P %. +%term step %pi (P1 ==> P2) %-> (P2 ==>* P3) %-> (P1 ==>* P3) %. +% Examples: +%define p1 MP (MP K S) S %. +%define p2 MP (MP S (MP (MP S (MP K PAIR)) (MP (MP S (MP K RIGHT)) (MP (MP S K) K)))) (MP (MP S (MP K LEFT)) (MP (MP S K) K)) %. +%define q1 MP (MP S K) K %. +%define q2 MP (MP (MP S K) K) (MP (MP S K) K) %. +%solve r : q2 ==> Q %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/prop-calc/prop-calc.lf b/new-tests/stelf-output/examples/prop-calc/prop-calc.lf new file mode 100644 index 0000000..a6bdb1e --- /dev/null +++ b/new-tests/stelf-output/examples/prop-calc/prop-calc.lf @@ -0,0 +1,46 @@ +%%% Intuitionistic propositional calculus +%%% Positive fragment with implies, and, true. +%%% Two formulations here: natural deduction and Hilbert-style system. +%%% Author: Frank Pfenning +% Type of propositions. +%sort o %. +%name o %. +% Syntax: implication, plus a few constants. +%term => %pi o %-> o %-> o %. +%prec %right 10 => %. +%term & %pi o %-> o %-> o %. +%prec %right 11 & %. +%term true o %. +% Provability. +%sort |- %. +%prec %prefix 9 |- %. +%name |- %. +% Axioms. +%term K |- A => B => A %. +%term S |- (A => B => C) => (A => B) => A => C %. +%term ONE |- true %. +%term PAIR |- A => B => A & B %. +%term LEFT |- A & B => A %. +%term RIGHT |- A & B => B %. +% Inference Rule. +%term MP %pi (|- A => B) %-> (|- A) %-> (|- B) %. +% Natural Deduction. +%sort ! %. +%prec %prefix 9 ! %. +%name ! %. +%term trueI ! true %. +%term andI %pi (! A) %-> (! B) %-> (! A & B) %. +%term andEL %pi (! A & B) %-> (! A) %. +%term andER %pi (! A & B) %-> (! B) %. +%term impliesI %pi (%pi (! A) %-> (! B)) %-> (! A => B) %. +%term impliesE %pi (! A => B) %-> (! A) %-> (! B) %. +% Normal deductions (for faster search) +%sort !^ %. +%sort !v %. +%term trueI^ !^ true %. +%term andI^ %pi (!^ A) %-> (!^ B) %-> (!^ (A & B)) %. +%term andEvL %pi (!v (A & B)) %-> (!v A) %. +%term andEvR %pi (!v (A & B)) %-> (!v B) %. +%term impI^ %pi (%pi (!v A) %-> (!^ B)) %-> (!^ (A => B)) %. +%term impEv %pi (!v (A => B)) %-> (!^ A) %-> (!v B) %. +%term close %pi (!v A) %-> (!^ A) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/prop-calc/sources.lf b/new-tests/stelf-output/examples/prop-calc/sources.lf new file mode 100644 index 0000000..5c8f764 --- /dev/null +++ b/new-tests/stelf-output/examples/prop-calc/sources.lf @@ -0,0 +1,2 @@ +%require %[ prop-calc %] %. +%require %[ equiv %] %. diff --git a/new-tests/stelf-output/examples/prop-calc/stelf.toml b/new-tests/stelf-output/examples/prop-calc/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/prop-calc/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/prop-calc/test.lf b/new-tests/stelf-output/examples/prop-calc/test.lf new file mode 100644 index 0000000..07acff3 --- /dev/null +++ b/new-tests/stelf-output/examples/prop-calc/test.lf @@ -0,0 +1,3 @@ +%require %[ prop-calc %] %. +%require %[ ndclass %] %. +%require %[ thm1 %] %. diff --git a/new-tests/stelf-output/examples/prop-calc/thm1.lf b/new-tests/stelf-output/examples/prop-calc/thm1.lf new file mode 100644 index 0000000..493f01a --- /dev/null +++ b/new-tests/stelf-output/examples/prop-calc/thm1.lf @@ -0,0 +1,8 @@ +%%% into a proof in the Hilbert-style system and vice versa. +%%% Expresses the equivalence of the two systems. +%%% Author: Frank Pfenning +%sort abs %. +%term aID abs ([x] x) (MP (MP S K) K) %. +%term aK abs ([x] K) (MP K K) %. +%term aS abs ([x] S) (MP K S) %. +%term aMP %pi (abs ([x] MP (P x) (Q x)) (MP (MP S P') Q')) %<- (abs P P') %<- (abs Q Q') %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/prop-calc/thm2.lf b/new-tests/stelf-output/examples/prop-calc/thm2.lf new file mode 100644 index 0000000..70abd64 --- /dev/null +++ b/new-tests/stelf-output/examples/prop-calc/thm2.lf @@ -0,0 +1,14 @@ +% comb does the translation from natural deduction to Hilbert deductions, +% appealing to the deduction theorem in case of implies-introduction +% which requires hypothetical reasoning. +%sort comb %. +% mode comb +D -F. +%term ctrue comb trueI ONE %. +%term candI %pi (comb (andI P Q) (MP (MP PAIR P') Q')) %<- (comb P P') %<- (comb Q Q') %. +%term candEL %pi (comb (andEL P) (MP LEFT P')) %<- (comb P P') %. +%term candER %pi (comb (andER P) (MP RIGHT P')) %<- (comb P P') %. +%term cimpliesI + %pi (comb (impliesI PP) Q) + %<- ({x} {y} %pi (comb x y) %-> ({B o} abs ([z |- B] y) (MP K y)) %-> (comb (PP x) (PP' y))) + %<- (abs PP' Q) %. +%term cimpliesE %pi (comb (impliesE P Q) (MP P' Q')) %<- (comb P P') %<- (comb Q Q') %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/prop-calc/thm3.lf b/new-tests/stelf-output/examples/prop-calc/thm3.lf new file mode 100644 index 0000000..909870b --- /dev/null +++ b/new-tests/stelf-output/examples/prop-calc/thm3.lf @@ -0,0 +1,11 @@ +%%% Reverse direction +%%% Convert the Hilbert Calculus into Natural deduction +%sort combdefn %. +%mode combdefn %in %out %. +%term cdK combdefn K (impliesI ([p] impliesI ([q] p))) %. +%term cdS combdefn S (impliesI ([p] impliesI ([q] impliesI ([r] impliesE (impliesE p r) (impliesE q r))))) %. +%term cdONE combdefn ONE trueI %. +%term cdPAIR combdefn PAIR (impliesI ([p] impliesI ([q] andI p q))) %. +%term cdLEFT combdefn LEFT (impliesI ([p] andEL p)) %. +%term cdRIGHT combdefn RIGHT (impliesI ([p] andER p)) %. +%term cdMP %pi (combdefn (MP P Q) (impliesE P' Q')) %<- (combdefn P P') %<- (combdefn Q Q') %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/small-step/lam.lf b/new-tests/stelf-output/examples/small-step/lam.lf new file mode 100644 index 0000000..de21044 --- /dev/null +++ b/new-tests/stelf-output/examples/small-step/lam.lf @@ -0,0 +1,87 @@ +% Progress and preservation for the Church-style simply-typed lambda calculus +%sort tp %. +%sort tm %. +% Types +%term => %pi tp %-> tp %-> tp %. +%prec %left 5 => %. +% Terms +%term @ %pi tm %-> tm %-> tm %. +%prec %left 5 @ %. +%term lam %pi tp %-> (%pi tm %-> tm) %-> tm %. +% Typing +%sort is %. +%prec %left 5 is %. +%term is_@ %pi (E1 is (T1 => T2)) %-> (E2 is T1) %-> (E1 @ E2 is T2) %. +%term is_lam %pi ({x tm} %pi (x is T1) %-> (E x is T2)) %-> (lam T1 ([x] E x) is (T1 => T2)) %. +% Values +%sort value %. +%term value_lam value (lam T ([x] E x)) %. +% Operational semantics +%sort ~> %. +%prec %left 5 ~> %. +%term ~>_@1 %pi (E1 ~> E1') %-> (E1 @ E2 ~> (E1' @ E2)) %. +%term ~>_@2 %pi (value E1) %-> (E2 ~> E2') %-> (E1 @ E2 ~> (E1 @ E2')) %. +%term ~>_@3 %pi (value E2) %-> (lam T ([x] E1 x) @ E2 ~> (E1 E2)) %. +% Preservation +%sort pres %. +%mode pres %in %in %out %. +%term pres_@1 %pi (pres D1 EV1 D1') %-> (pres (is_@ D1 D2) (~>_@1 EV1) (is_@ D1' D2)) %. +%term pres_@2 %pi (pres D2 EV2 D2') %-> (pres (is_@ D1 D2) (~>_@2 V EV2) (is_@ D1 D2')) %. +%term pres_@3 pres (is_@ (is_lam ([x tm] [d x is T1] D1 x d)) D2) (~>_@3 V) (D1 _ D2) %. +%worlds () (pres D1 E D2) %. +%total E (pres _ E _) %. +% Progress (much trickier) +% Introduce a new type family to distinguish between the case where a term can +% take a step and it is a value. +%sort step_or_val %. +%term step %pi (E ~> E') %-> (step_or_val E) %. +%term val %pi (value E) %-> (step_or_val E) %. +%{ +% The usual declarative proof of progress will not work because it +% requires Twelf to perform output splitting when it attempts to coverage +% check the application case. + +prog : (E is T) -> step_or_val E -> type. +%mode prog +D -SV. + +prog_@1 : prog D1 (step EV) -> + % ----------------------------------- + prog (is_@ D1 D2) (step (~>_@1 EV)). + +prog_@2 : prog D1 (val VL) -> + prog D2 (step EV) -> + % ----------------------------------- + prog (is_@ D1 D2) (step (~>_@2 VL EV)). + +prog_@3 : prog D1 (val VL1) -> + prog D2 (val VL2) -> + % ----------------------------------- + prog (is_@ D1 D2) (step (~>_@3 VL2)). + +prog_lam : + % -------------------------------------------------------- + prog (is_lam ([x:tm] [d:x is T1] D1 x d)) (val value_lam). + +%worlds () (prog D SV). +%total D (prog D _). +}% +% Instead we need to factor the subcases for application into a single lemma +%sort prog_@_lemma %. +%mode prog_@_lemma %in %in %out %. +%term prog_@_lemma_s prog_@_lemma (step EV) _ (step (~>_@1 EV)) %. +%term prog_@_lemma_vs prog_@_lemma (val VL) (step EV) (step (~>_@2 VL EV)) %. +%term prog_@_lemma_vs prog_@_lemma (val VL1) (val VL2) (step (~>_@3 VL2)) %. +%worlds () (prog_@_lemma SV1 SV2 SV3) %. +%total {SV1 SV2} (prog_@_lemma SV1 SV2 _) %. +% Now we can prove progress by simply using the lemma +%sort prog %. +%mode prog %in %out %. +%term prog_@ + %pi (prog_@_lemma SV1 SV2 SV3) + %-> (prog D1 SV1) + %-> (prog D2 SV2) + %-> (prog (is_@ D1 D2) SV3) %. +%term prog_lam prog (is_lam ([x tm] [d x is T1] D1 x d)) (val value_lam) %. +%worlds () (prog D SV) %. +%total D (prog D _) %. +% QED \ No newline at end of file diff --git a/new-tests/stelf-output/examples/small-step/system-f-iso.lf b/new-tests/stelf-output/examples/small-step/system-f-iso.lf new file mode 100644 index 0000000..c95df29 --- /dev/null +++ b/new-tests/stelf-output/examples/small-step/system-f-iso.lf @@ -0,0 +1,126 @@ +% Progress and preservation for System F with natural numbers and iso-recursive types. +%sort tp %. +%sort tm %. +% Types +%term nat tp %. +%term => %pi tp %-> tp %-> tp %. +%prec %left 5 => %. +%term forall %pi (%pi tp %-> tp) %-> tp %. +%term mu %pi (%pi tp %-> tp) %-> tp %. +% Terms +%term z tm %. +%term s %pi tm %-> tm %. +%term @ %pi tm %-> tm %-> tm %. +%prec %left 5 @ %. +%term roll %pi tm %-> tp %-> tm %. +%term unroll %pi tm %-> tp %-> tm %. +%term lam %pi tp %-> (%pi tm %-> tm) %-> tm %. +%term Lam %pi (%pi tp %-> tm) %-> tm %. +%term # %pi tm %-> tp %-> tm %. +%prec %left 5 # %. +% Typing +%sort is %. +%prec %left 5 is %. +%term is_z z is nat %. +%term is_s %pi (E is nat) %-> (s E is nat) %. +%term is_roll + %pi (E is (T (mu ([a tp] T a)))) + %-> (roll E (mu ([a tp] T a)) is (mu ([a tp] T a))) %. +%term is_unroll + %pi (E is (mu ([a tp] T a))) + %-> (unroll E (mu ([a tp] T a)) is (T (mu ([a tp] T a)))) %. +%term is_@ %pi (E1 is (T1 => T2)) %-> (E2 is T1) %-> (E1 @ E2 is T2) %. +%term is_lam %pi ({x tm} %pi (x is T1) %-> (E x is T2)) %-> (lam T1 ([x] E x) is (T1 => T2)) %. +%term is_Lam %pi ({a tp} E a is (T a)) %-> (Lam ([a] E a) is (forall ([a] T a))) %. +%term is_# %pi (E is (forall ([a] T1 a))) %-> (E # T2 is (T1 T2)) %. +% Values +%sort value %. +%term value_z value z %. +%term value_s %pi (value V) %-> (value (s V)) %. +%term value_roll %pi (value E) %-> (value (roll E T)) %. +%term value_lam value (lam T ([x] E x)) %. +%term value_Lam value (Lam ([a] E a)) %. +% Operational semantics +%sort ~> %. +%prec %left 5 ~> %. +%term ~>_s %pi (E ~> E') %-> (s E ~> (s E')) %. +%term ~>_@1 %pi (E1 ~> E1') %-> (E1 @ E2 ~> (E1' @ E2)) %. +%term ~>_@2 %pi (value E1) %-> (E2 ~> E2') %-> (E1 @ E2 ~> (E1 @ E2')) %. +%term ~>_@3 %pi (value E2) %-> (lam T ([x] E1 x) @ E2 ~> (E1 E2)) %. +%term ~>_#1 %pi (E ~> E') %-> (E # T ~> (E' # T)) %. +%term ~>_#2 Lam ([a] E a) # T ~> (E T) %. +%term ~>_unroll1 %pi (E ~> E') %-> (unroll E T ~> (unroll E' T)) %. +%term ~>_unroll2 %pi (value E) %-> (unroll (roll E T1) T2 ~> E) %. +%term ~>_roll %pi (E ~> E') %-> (roll E T ~> (roll E' T)) %. +% Preservation +%sort pres %. +%mode pres %in %in %out %. +%term pres_s %pi (pres D EV D') %-> (pres (is_s D) (~>_s EV) (is_s D')) %. +%term pres_@1 %pi (pres D1 EV1 D1') %-> (pres (is_@ D1 D2) (~>_@1 EV1) (is_@ D1' D2)) %. +%term pres_@2 %pi (pres D2 EV2 D2') %-> (pres (is_@ D1 D2) (~>_@2 V EV2) (is_@ D1 D2')) %. +%term pres_@3 pres (is_@ (is_lam ([x tm] [d x is T1] D1 x d)) D2) (~>_@3 V) (D1 _ D2) %. +%term pres_#1 %pi (pres D EV D') %-> (pres (is_# D) (~>_#1 EV) (is_# D')) %. +%term pres_#2 pres (%the (_ # T is _) (is_# (is_Lam ([a tp] D a)))) ~>_#2 (D T) %. +%term pres_unroll1 %pi (pres D EV D') %-> (pres (is_unroll D) (~>_unroll1 EV) (is_unroll D')) %. +%term pres_unroll2 pres (is_unroll (is_roll D)) (~>_unroll2 VL) D %. +%term pres_roll %pi (pres D EV D') %-> (pres (is_roll D) (~>_roll EV) (is_roll D')) %. +%worlds () (pres D1 E D2) %. +%total E (pres _ E _) %. +% Progress (much trickier) +% Introduce a new type family to distinguish between the case where a term can +% take a step and it is a value. +%sort step_or_val %. +%term step %pi (E ~> E') %-> (step_or_val E) %. +%term val %pi (value E) %-> (step_or_val E) %. +%sort prog_@_lemma %. +%mode prog_@_lemma %in %in %in %out %. +%term prog_@_lemma_s prog_@_lemma _ (step EV) _ (step (~>_@1 EV)) %. +%term prog_@_lemma_vs prog_@_lemma _ (val VL) (step EV) (step (~>_@2 VL EV)) %. +%term prog_@_lemma_vs prog_@_lemma _ (val VL1) (val VL2) (step (~>_@3 VL2)) %. +%worlds () (prog_@_lemma D SV1 SV2 SV3) %. +%total {SV1 SV2} (prog_@_lemma _ SV1 SV2 _) %. +%sort prog_roll_lemma {T tp} %. +%mode prog_roll_lemma %in %in %out %. +%term prog_roll_lemma_s prog_roll_lemma _ (step EV) (step (~>_roll EV)) %. +%term prog_roll_lemma_v prog_roll_lemma _ (val VL) (val (value_roll VL)) %. +%worlds () (prog_roll_lemma T SV1 SV2) %. +%total {SV1} (prog_roll_lemma _ SV1 _) %. +%sort prog_unroll_lemma %. +%mode prog_unroll_lemma %in %in %out %. +%term prog_unroll_lemma_s prog_unroll_lemma _ (step EV) (step (~>_unroll1 EV)) %. +%term prog_unroll_lemma_v prog_unroll_lemma _ (val (value_roll VL)) (step (~>_unroll2 VL)) %. +%worlds () (prog_unroll_lemma D SV1 SV2) %. +%total {SV1} (prog_unroll_lemma _ SV1 _) %. +%sort prog_#_lemma {T2 tp} %. +%mode prog_#_lemma %in %in %in %out %. +%term prog_#_lemma_s prog_#_lemma _ _ (step EV) (step (~>_#1 EV)) %. +%term prog_#_lemma_v prog_#_lemma _ _ (val VL) (step ~>_#2) %. +%worlds () (prog_#_lemma T2 D SV1 SV2) %. +%total {SV1} (prog_#_lemma _ _ SV1 _) %. +%sort prog_s_lemma %. +%mode prog_s_lemma %in %out %. +%term prog_s_lemma_s prog_s_lemma (step EV) (step (~>_s EV)) %. +%term prog_s_lemma_v prog_s_lemma (val VL) (val (value_s VL)) %. +%worlds () (prog_s_lemma SV1 SV2) %. +%total (SV1) (prog_s_lemma SV1 _) %. +% Now we can prove progress by simply using the lemma +%sort prog %. +%mode prog %in %out %. +%term prog_z prog is_z (val value_z) %. +%term prog_s %pi (prog_s_lemma SV SV') %-> (prog D SV) %-> (prog (is_s D) SV') %. +%term prog_@ + %pi (prog_@_lemma D1 SV1 SV2 SV3) + %-> (prog D1 SV1) + %-> (prog D2 SV2) + %-> (prog (is_@ D1 D2) SV3) %. +%term prog_# %pi (prog_#_lemma _ D SV SV') %-> (prog D SV) %-> (prog (is_# D) SV') %. +%term prog_roll + %pi (prog_roll_lemma (mu ([a tp] T a)) SV SV') + %-> (prog D SV) + %-> (prog (%the (roll E (mu ([a tp] T a)) is (mu ([a tp] T a))) (is_roll D)) SV') %. +%term prog_unroll %pi (prog_unroll_lemma D SV SV') %-> (prog D SV) %-> (prog (is_unroll D) SV') %. +%term prog_lam prog (is_lam ([x tm] [d x is T1] D1 x d)) (val value_lam) %. +%term prog_Lam prog (is_Lam ([a tp] D a)) (val value_Lam) %. +%worlds () (prog D SV) %. +%total D (prog D _) %. +% QED \ No newline at end of file diff --git a/new-tests/stelf-output/examples/small-step/system-f.lf b/new-tests/stelf-output/examples/small-step/system-f.lf new file mode 100644 index 0000000..b883d1d --- /dev/null +++ b/new-tests/stelf-output/examples/small-step/system-f.lf @@ -0,0 +1,93 @@ +% Progress and preservation for System F with natural numbers. +%sort tp %. +%sort tm %. +% Types +%term nat tp %. +%term => %pi tp %-> tp %-> tp %. +%prec %left 5 => %. +%term forall %pi (%pi tp %-> tp) %-> tp %. +% Terms +%term z tm %. +%term s %pi tm %-> tm %. +%term @ %pi tm %-> tm %-> tm %. +%prec %left 5 @ %. +%term lam %pi tp %-> (%pi tm %-> tm) %-> tm %. +%term Lam %pi (%pi tp %-> tm) %-> tm %. +%term # %pi tm %-> tp %-> tm %. +%prec %left 5 # %. +% Typing +%sort is %. +%prec %left 5 is %. +%term is_z z is nat %. +%term is_s %pi (E is nat) %-> (s E is nat) %. +%term is_@ %pi (E1 is (T1 => T2)) %-> (E2 is T1) %-> (E1 @ E2 is T2) %. +%term is_lam %pi ({x tm} %pi (x is T1) %-> (E x is T2)) %-> (lam T1 ([x] E x) is (T1 => T2)) %. +%term is_Lam %pi ({a tp} E a is (T a)) %-> (Lam ([a] E a) is (forall ([a] T a))) %. +%term is_# %pi (E is (forall ([a] T1 a))) %-> (E # T2 is (T1 T2)) %. +% Values +%sort value %. +%term value_z value z %. +%term value_s %pi (value V) %-> (value (s V)) %. +%term value_lam value (lam T ([x] E x)) %. +%term value_Lam value (Lam ([a] E a)) %. +% Operational semantics +%sort ~> %. +%prec %left 5 ~> %. +%term ~>_s %pi (E ~> E') %-> (s E ~> (s E')) %. +%term ~>_@1 %pi (E1 ~> E1') %-> (E1 @ E2 ~> (E1' @ E2)) %. +%term ~>_@2 %pi (value E1) %-> (E2 ~> E2') %-> (E1 @ E2 ~> (E1 @ E2')) %. +%term ~>_@3 %pi (value E2) %-> (lam T ([x] E1 x) @ E2 ~> (E1 E2)) %. +%term ~>_#1 %pi (E ~> E') %-> (E # T ~> (E' # T)) %. +%term ~>_#2 Lam ([a] E a) # T ~> (E T) %. +% Preservation +%sort pres %. +%mode pres %in %in %out %. +%term pres_s %pi (pres D EV D') %-> (pres (is_s D) (~>_s EV) (is_s D')) %. +%term pres_@1 %pi (pres D1 EV1 D1') %-> (pres (is_@ D1 D2) (~>_@1 EV1) (is_@ D1' D2)) %. +%term pres_@2 %pi (pres D2 EV2 D2') %-> (pres (is_@ D1 D2) (~>_@2 V EV2) (is_@ D1 D2')) %. +%term pres_@3 pres (is_@ (is_lam ([x tm] [d x is T1] D1 x d)) D2) (~>_@3 V) (D1 _ D2) %. +%term pres_#1 %pi (pres D EV D') %-> (pres (is_# D) (~>_#1 EV) (is_# D')) %. +%term pres_#2 pres (%the (_ # T is _) (is_# (is_Lam ([a tp] D a)))) ~>_#2 (D T) %. +%worlds () (pres D1 E D2) %. +%total E (pres _ E _) %. +% Progress (much trickier) +% Introduce a new type family to distinguish between the case where a term can +% take a step and it is a value. +%sort step_or_val %. +%term step %pi (E ~> E') %-> (step_or_val E) %. +%term val %pi (value E) %-> (step_or_val E) %. +%sort prog_@_lemma %. +%mode prog_@_lemma %in %in %in %out %. +%term prog_@_lemma_s prog_@_lemma _ (step EV) _ (step (~>_@1 EV)) %. +%term prog_@_lemma_vs prog_@_lemma _ (val VL) (step EV) (step (~>_@2 VL EV)) %. +%term prog_@_lemma_vs prog_@_lemma _ (val VL1) (val VL2) (step (~>_@3 VL2)) %. +%worlds () (prog_@_lemma D SV1 SV2 SV3) %. +%total {SV1 SV2} (prog_@_lemma _ SV1 SV2 _) %. +%sort prog_#_lemma {T2} %. +%mode prog_#_lemma %in %in %in %out %. +%term prog_#_lemma_s prog_#_lemma _ _ (step EV) (step (~>_#1 EV)) %. +%term prog_#_lemma_v prog_#_lemma _ _ (val VL) (step ~>_#2) %. +%worlds () (prog_#_lemma T2 D SV1 SV2) %. +%total {SV1} (prog_#_lemma _ _ SV1 _) %. +%sort prog_s_lemma %. +%mode prog_s_lemma %in %out %. +%term prog_s_lemma_s prog_s_lemma (step EV) (step (~>_s EV)) %. +%term prog_s_lemma_v prog_s_lemma (val VL) (val (value_s VL)) %. +%worlds () (prog_s_lemma SV1 SV2) %. +%total (SV1) (prog_s_lemma SV1 _) %. +% Now we can prove progress by simply using the above lemmas +%sort prog %. +%mode prog %in %out %. +%term prog_z prog is_z (val value_z) %. +%term prog_s %pi (prog_s_lemma SV SV') %-> (prog D SV) %-> (prog (is_s D) SV') %. +%term prog_@ + %pi (prog_@_lemma D1 SV1 SV2 SV3) + %-> (prog D1 SV1) + %-> (prog D2 SV2) + %-> (prog (is_@ D1 D2) SV3) %. +%term prog_# %pi (prog_#_lemma _ D SV SV') %-> (prog D SV) %-> (prog (is_# D) SV') %. +%term prog_lam prog (is_lam ([x tm] [d x is T1] D1 x d)) (val value_lam) %. +%term prog_Lam prog (is_Lam ([a tp] D a)) (val value_Lam) %. +%worlds () (prog D SV) %. +%total D (prog D _) %. +% QED \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/ccc/ccc.lf b/new-tests/stelf-output/examples/tabled/ccc/ccc.lf new file mode 100644 index 0000000..aacf086 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/ccc/ccc.lf @@ -0,0 +1,65 @@ +%%% Cartesian closed category +%%% +%%% Categorical language +%%%% objects +%sort obj %. +%name obj %. +%% Products +% objects +%term 1 obj %. +%term * %pi obj %-> obj %-> obj %. +%prec %none 10 * %. +%% Exponentials +%term => %pi obj %-> obj %-> obj %. +%prec %right 7 => %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% morphisms +%sort mor %. +%name mor %. +% morphisms +%term id mor A A %. +%term @ %pi (mor B C) %-> (mor A B) %-> (mor A C) %. +%prec %none 10 @ %. +%term drop mor A 1 %. +%term fst mor (A * B) A %. +%term snd mor (A * B) B %. +%term pair %pi (mor A B) %-> (mor A C) %-> (mor A (B * C)) %. +%term app mor (B => C * B) C %. +%term cur %pi (mor (A * B) C) %-> (mor A (B => C)) %. +% morphism equality +%term f mor A1 A2 %. +%term g mor A1 A3 %. +%term h mor A4 A1 %. +%sort == %. +%name == %. +%prec %none 5 == %. +%tabled == %. +% equations +% reordered equations for tabling purposes +% still needs at least abstraction to prevent +% getting stuck +% +%term ass H @ (G @ F) == (H @ G) @ F %. +%term prod_l fst @ (pair F G) == F %. +%term prod_r snd @ (pair F G) == G %. +%term prod_u H == pair (fst @ H) (snd @ H) %. +% @(H @(G F)) +%term =@= %pi (F @ G == F' @ G') %<- (F == F') %<- (G == G') %. +%term =pair= %pi (pair F G == pair F' G') %<- (F == F') %<- (G == G') %. +%term refl F == F %. +%term then %pi (F == F'') %<- (F == F') %<- (F' == F'') %. +%prec %right 4 then %. +% sym : F == F' -> F' == F. +% id_l : id @ F == F. +% id_r : F @ id == F. +% term_u : H == drop. +% =cur= : F == F' -> cur F == (cur F'). +% exp_e : app @ (pair ((cur F) @ fst) snd) == F. +% term-depth : 4 +% (@ app pair(@((cur F) fst) snd)) +% term-depth : +% exp_u : cur (app @ (pair (G @ fst) snd)) == G. +% term-depth: 4 +% cur(@ app (pair (@ G fst) snd)) +% distp : pair F G @ H == pair (F @ H) (G @ H). +% distc : (cur F) @ G == cur (F @ (pair (G @ fst) snd)). \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/ccc/stelf.toml b/new-tests/stelf-output/examples/tabled/ccc/stelf.toml new file mode 100644 index 0000000..1205615 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/ccc/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "tab" +main = "tab.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/tabled/ccc/tab-examples-quy.lf b/new-tests/stelf-output/examples/tabled/ccc/tab-examples-quy.lf new file mode 100644 index 0000000..91dd3ae --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/ccc/tab-examples-quy.lf @@ -0,0 +1,12 @@ +% A few test examples +%{ +%querytabled * 10 10 +D : fst @ (pair f g @ h) == (f @ h). +D : {f}{g}{h}pair f g @ h == pair (f @ h) (g @ h). +}% +% at least : termDepth 35 required +%querytabled _ 2 %the (pair f g @ h == pair (f @ h) (g @ h)) D %. +%{ +%querytabled * 10 10 +D : pair F G @ H == pair (F @ H) (G @ H). +}% \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/ccc/tab.lf b/new-tests/stelf-output/examples/tabled/ccc/tab.lf new file mode 100644 index 0000000..cbcc8ab --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/ccc/tab.lf @@ -0,0 +1,2 @@ +%require %[ ccc %] %. +%require %[ tab-examples %] %. diff --git a/new-tests/stelf-output/examples/tabled/cr/lam.lf b/new-tests/stelf-output/examples/tabled/cr/lam.lf new file mode 100644 index 0000000..48fee73 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/cr/lam.lf @@ -0,0 +1,6 @@ +%%% Untyped lambda-calculus +%%% Author: Frank Pfenning +%sort term %. +%name term %. +%term lam %pi (%pi term %-> term) %-> term %. +%term app %pi term %-> term %-> term %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/cr/ord-red.lf b/new-tests/stelf-output/examples/tabled/cr/ord-red.lf new file mode 100644 index 0000000..2706150 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/cr/ord-red.lf @@ -0,0 +1,25 @@ +%%% Ordinary reduction for the untyped lambda-calculus +%%% Author: Frank Pfenning +%scope app %sort -> %. +%prec %none 10 --> %. +%name --> %. +%term beta1 app (lam M1) M2 --> M1 M2 %. +%term lm1 %pi ({x term} M x --> M' x) %-> (lam M --> (lam M')) %. +%term apl1 %pi (M1 --> M1') %-> (app M1 M2 --> (app M1' M2)) %. +%term apr1 %pi (M2 --> M2') %-> (app M1 M2 --> (app M1 M2')) %. +% Multi-step reduction +%scope apr1 %sort ->* %. +%prec %none 10 -->* %. +%name -->* %. +%tabled -->* %. +%term id1 M -->* M %. +%term step1 %pi (M -->* M'') %<- (M --> M') %<- (M' -->* M'') %. +% Conversion +%sort <-> %. +%prec %none 10 <-> %. +%name <-> %. +%tabled <-> %. +%term refl M <-> M %. +%term sym %pi (M <-> M') %<- (M' <-> M) %. +%term trans %pi (M <-> M'') %<- (M <-> M') %<- (M' <-> M'') %. +%term red %pi (M <-> M') %<- (M -->* M') %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/cr/par-red.lf b/new-tests/stelf-output/examples/tabled/cr/par-red.lf new file mode 100644 index 0000000..dadfb42 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/cr/par-red.lf @@ -0,0 +1,28 @@ +%%% Parallel reduction in the untyped lambda calculus +%%% Author: Frank Pfenning +%sort => %. +%prec %none 10 => %. +%name => %. +%term beta + %pi ({x term} %pi (x => x) %-> (M1 x => M1' x)) + %-> (M2 => M2') + %-> (app (lam M1) M2 => M1' M2') %. +%term ap %pi (M1 => M1') %-> (M2 => M2') %-> (app M1 M2 => (app M1' M2')) %. +%term lm %pi ({x term} %pi (x => x) %-> (M x => M' x)) %-> (lam M => lam M') %. +% Parallel, multi-step reduction +%sort =>* %. +%prec %none 10 =>* %. +%name =>* %. +%tabled =>* %. +%term id M =>* M %. +%term ; %pi (M =>* M'') %<- (M => M') %<- (M' =>* M'') %. +%prec %right 10 ; %. +% Parallel conversion +%sort <=> %. +%prec %none 10 <=> %. +%name <=> %. +%tabled <=> %. +%term reduce %pi (M =>* M') %-> (M <=> M') %. +%term expand %pi (M =>* M') %-> (M' <=> M) %. +%term ;; %pi (M <=> M') %-> (M' <=> M'') %-> (M <=> M'') %. +%prec %none 8 ;; %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/cr/stelf.toml b/new-tests/stelf-output/examples/tabled/cr/stelf.toml new file mode 100644 index 0000000..1205615 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/cr/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "tab" +main = "tab.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/tabled/cr/tab-examples-quy.lf b/new-tests/stelf-output/examples/tabled/cr/tab-examples-quy.lf new file mode 100644 index 0000000..7b0a4a0 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/cr/tab-examples-quy.lf @@ -0,0 +1,14 @@ +% terminated after 2 stages (variant) +% terminated after 2 stages (subsumption) +%querytabled _ 5 app (lam ([x] app x x)) (lam ([y] y)) -->* M' %. +% terminated after 3 stages (variant) +% terminated after 3 stages (subsumption) +%querytabled _ 6 app (lam ([x] app x x)) (app (lam ([y] y)) (lam ([z] z))) -->* M' %. +% Parallel multi-step reduction +% terminate after 4 stages (variant) +% terminated after 4 stages (subsumption) +%querytabled _ 10 app (lam ([x] app x x)) (lam ([y] y)) =>* M' %. +% terminated after 5 stages (variant) +% terminated after 5 stages (subsumption) +%querytabled _ 5 app (lam ([x] app x x)) (app (lam ([y] y)) (lam ([z] z))) =>* M' %. +% parallel conversion \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/cr/tab.lf b/new-tests/stelf-output/examples/tabled/cr/tab.lf new file mode 100644 index 0000000..b7b1737 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/cr/tab.lf @@ -0,0 +1,4 @@ +%require %[ lam %] %. +%require %[ ord-red %] %. +%require %[ par-red %] %. +%require %[ tab-examples %] %. diff --git a/new-tests/stelf-output/examples/tabled/mini-ml/examples-quy.lf b/new-tests/stelf-output/examples/tabled/mini-ml/examples-quy.lf new file mode 100644 index 0000000..28e70f5 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/mini-ml/examples-quy.lf @@ -0,0 +1,16 @@ +%querytabled _ _ app (app mult z) (s (s z)) ==>* V %. +%querytabled _ _ eval (app (app plus (s (s z))) (s (s z))) V %. +%querytabled _ _ eval (app (app mult (s z)) (s (s z))) V %. +%{ +% sml process killed! +%querytabled * * +eval (app (app mult (s (s (z)))) (s (s z))) V. + +%querytabled * * +eval (app (app mult (s (s (z)))) (s (s z))) (s(s(s(s z)))). + +}% +%{ +%querytabled * * +(app (app mult (s (s (z)))) (s (s z))) ==>* V. +}% \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/mini-ml/mini-ml.lf b/new-tests/stelf-output/examples/tabled/mini-ml/mini-ml.lf new file mode 100644 index 0000000..e166530 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/mini-ml/mini-ml.lf @@ -0,0 +1,15 @@ +%%% The Mini-ML Language +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort exp %. +%name exp %. +%term z exp %. +%term s %pi exp %-> exp %. +%term case %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%term pair %pi exp %-> exp %-> exp %. +%term fst %pi exp %-> exp %. +%term snd %pi exp %-> exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term letv %pi exp %-> (%pi exp %-> exp) %-> exp %. +%term letn %pi exp %-> (%pi exp %-> exp) %-> exp %. +%term fix %pi (%pi exp %-> exp) %-> exp %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/mini-ml/programs.lf b/new-tests/stelf-output/examples/tabled/mini-ml/programs.lf new file mode 100644 index 0000000..9fa8250 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/mini-ml/programs.lf @@ -0,0 +1,3 @@ +%define plus fix ([plus] lam ([x] lam ([y] case x y ([x] s (app (app plus x) y))))) %. +%define minus fix ([minus] lam ([x] lam ([y] case x z ([x] case y x ([y] app (app minus x) y))))) %. +%define mult fix ([mult] lam ([x] lam ([y] case x z ([x] app (app plus y) (app (app mult x) y))))) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/mini-ml/reduce.lf b/new-tests/stelf-output/examples/tabled/mini-ml/reduce.lf new file mode 100644 index 0000000..4e29f97 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/mini-ml/reduce.lf @@ -0,0 +1,41 @@ +%sort ==> %. +%name ==> %. +%prec %none 10 ==> %. +%mode ==> %in %out %. +%term red_z case z E1 E2 ==> E1 %. +%term red_s case (s E) E1 E2 ==> E2 E %. +%term red_fst fst (pair E1 E2) ==> E1 %. +%term red_snd snd (pair E1 E2) ==> E2 %. +%term red_app app (lam E1) E2 ==> E1 E2 %. +%term red_letv letv E1 E2 ==> E2 E1 %. +%term red_letn letn E1 E2 ==> E2 E1 %. +%term red_fix fix E ==> E (fix E) %. +% The weakly congruent, reflexive, transitive closure of ==>* +%sort ==>* %. +%name ==>* %. +%prec %none 10 ==>* %. +% mode ==>* +E -E'. +% Congruences +% no red*_z +%term red*_s %pi (s E ==>* s E') %<- (E ==>* E') %. +%term red*_case %pi (case E1 E2 E3 ==>* case E1' E2 E3) %<- (E1 ==>* E1') %. +% non-deterministic choice +%term red*_pair1 %pi (pair E1 E2 ==>* pair E1' E2) %<- (E1 ==>* E1') %. +%term red*_pair2 %pi (pair E1 E2 ==>* pair E1 E2') %<- (E2 ==>* E2') %. +%term red*_fst %pi (fst E ==>* fst E') %<- (E ==>* E') %. +%term red*_snd %pi (snd E ==>* snd E') %<- (E ==>* E') %. +% no red*_lam +% non-deterministic choice +%term red*_app1 %pi (app E1 E2 ==>* app E1' E2) %<- (E1 ==>* E1') %. +%term red*_app2 %pi (app E1 E2 ==>* app E1 E2') %<- (E2 ==>* E2') %. +%term red*_letv %pi (letv E1 E2 ==>* letv E1' E2) %<- (E1 ==>* E1') %. +%term red*_letn %pi (letn E1 E2 ==>* letn E1' E2) %<- (E1 ==>* E1') %. +% no red*_fix +%term red*_beta %pi (E ==>* E') %<- (E ==> E') %. +%term red*_refl E ==>* E %. +%term red*_trans %pi (E ==>* E'') %<- (E ==>* E') %<- (E' ==>* E'') %. +%sort eval %. +%term ev %pi (eval E V) %<- (E ==>* V) %<- (value V) %. +% otherwise unpractical to execute. +%tabled ==>* %. +%tabled eval %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/mini-ml/stelf.toml b/new-tests/stelf-output/examples/tabled/mini-ml/stelf.toml new file mode 100644 index 0000000..1205615 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/mini-ml/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "tab" +main = "tab.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/tabled/mini-ml/tab.lf b/new-tests/stelf-output/examples/tabled/mini-ml/tab.lf new file mode 100644 index 0000000..df4772f --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/mini-ml/tab.lf @@ -0,0 +1,5 @@ +%require %[ mini-ml %] %. +%require %[ value %] %. +%require %[ reduce %] %. +%require %[ programs %] %. +%require %[ examples %] %. diff --git a/new-tests/stelf-output/examples/tabled/mini-ml/value.lf b/new-tests/stelf-output/examples/tabled/mini-ml/value.lf new file mode 100644 index 0000000..fa825b9 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/mini-ml/value.lf @@ -0,0 +1,9 @@ +%%% Definition of Values +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort value %. +%name value %. +%mode value %in %. +%term val_z value z %. +%term val_s %pi (value (s E)) %<- (value E) %. +%term val_pair %pi (value (pair E1 E2)) %<- (value E1) %<- (value E2) %. +%term val_lam value (lam E) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/parsing/arithml-quy.lf b/new-tests/stelf-output/examples/tabled/parsing/arithml-quy.lf new file mode 100644 index 0000000..04e2fca --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/parsing/arithml-quy.lf @@ -0,0 +1,4 @@ +%querytabled _ _ parse ('openB' ; 'lam' ; 'x' ; '1' ; '@' ; 'x' ; 'closeB' ; '@' ; '3' ; nil) P %. +%querytabled _ _ parse ('openB' ; 'lam' ; 'x' ; 'lam' ; 'y' ; '3' ; '*' ; 'x' ; '+' ; 'y' ; '*' ; '2' ; 'closeB' ; '@' ; '9' ; '@' ; '8' ; nil) P %. +%querytabled _ _ parse ('openB' ; 'lam' ; 'x' ; 'lam' ; 'y' ; '3' ; '*' ; 'x' ; '+' ; 'y' ; '*' ; '2' ; 'closeB' ; '@' ; '9' ; '@' ; '8' ; nil) P %. +%querytabled _ _ parse ('openB' ; 'lam' ; 'x' ; 'lam' ; 'y' ; 'x' ; '@' ; 'openB' ; 'lam' ; 'z' ; '1' ; '+' ; 'z' ; '+' ; '3' ; '*' ; 'x' ; '+' ; 'y' ; '*' ; '2' ; 'closeB' ; 'closeB' ; '@' ; '9' ; '@' ; '8' ; nil) P %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/parsing/arithml.lf b/new-tests/stelf-output/examples/tabled/parsing/arithml.lf new file mode 100644 index 0000000..8c26828 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/parsing/arithml.lf @@ -0,0 +1,2 @@ +%require %[ arithml %] %. +%require %[ arithml %] %. diff --git a/new-tests/stelf-output/examples/tabled/parsing/foll-quy.lf b/new-tests/stelf-output/examples/tabled/parsing/foll-quy.lf new file mode 100644 index 0000000..beb4fb6 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/parsing/foll-quy.lf @@ -0,0 +1,23 @@ +%querytabled _ _ parse ('forall' ; 'x' ; 'q' ; 'x' ; nil) P %. +%querytabled _ _ parse ('forall' ; 'x' ; 'q' ; 'x' ; 'and' ; 'p' ; 'x' ; nil) P %. +%querytabled _ _ parse ('forall' ; 'x' ; 'q' ; 'x' ; 'imp' ; 'z' ; 'and' ; 'p' ; 'x' ; nil) P %. +%querytabled _ _ parse ('forall' ; 'x' ; 'forall' ; 'y' ; 'not' ; 'p' ; 'y' ; 'and' ; 'openB' ; 'forall' ; 'x' ; 'q' ; 'x' ; 'or' ; 'z' ; 'closeB' ; 'and' ; 'p' ; 'x' ; nil) P %. +%querytabled _ _ parse ('forall' ; 'x' ; 'forall' ; 'x' ; 'not' ; 'p' ; 'x' ; 'and' ; 'openB' ; 'forall' ; 'y' ; 'q' ; 'y' ; 'or' ; 'z' ; 'or' ; 'openB' ; 'exist' ; 'y' ; 's' ; 'y' ; 'closeB' ; 'closeB' ; nil) P %. +%querytabled _ _ parse ('openB' ; 'x' ; 'imp' ; 'y' ; 'closeB' ; 'and' ; 'true' ; nil) P %. +%querytabled _ _ parse ('openB' ; 'not' ; 'x' ; 'closeB' ; 'and' ; 'openB' ; 'y' ; 'or' ; 'z' ; 'closeB' ; nil) P %. +%querytabled _ _ parse ('not' ; 'openB' ; 'x' ; 'and' ; 'openB' ; 'y' ; 'or' ; 'z' ; 'closeB' ; 'closeB' ; nil) P %. +%querytabled _ _ parse ('not' ; 'x' ; 'and' ; 'openB' ; 'y' ; 'or' ; 'z' ; 'closeB' ; nil) P %. +%querytabled _ _ parse ('forall' ; 'x' ; 'forall' ; 'y' ; 'not' ; 'p' ; 'x' ; 'and' ; 'openB' ; 'forall' ; 'y' ; 'q' ; 'y' ; 'or' ; 'z' ; 'closeB' ; nil) P %. +%querytabled _ _ parse ('x' ; 'or' ; 'openB' ; 'exist' ; 'y' ; 's' ; 'y' ; 'closeB' ; nil) P %. +%querytabled _ _ parse ('forall' ; 'x' ; 'forall' ; 'y' ; 'not' ; 'p' ; 'x' ; 'and' ; 'openB' ; 'forall' ; 'y' ; 'q' ; 'y' ; 'or' ; 'z' ; 'closeB' ; 'or' ; 'openB' ; 'exist' ; 'y' ; 's' ; 'y' ; 'closeB' ; nil) P %. +%querytabled _ _ parse ('forall' ; 'x' ; 'forall' ; 'x' ; 'not' ; 'p' ; 'x' ; 'and' ; 'openB' ; 'forall' ; 'y' ; 'q' ; 'y' ; 'or' ; 'z' ; 'closeB' ; 'or' ; 'openB' ; 'exist' ; 'y' ; 's' ; 'y' ; 'closeB' ; nil) P %. +%querytabled _ _ parse ('forall' ; 'x' ; 'forall' ; 'x' ; 'not' ; 'p' ; 'x' ; 'and' ; 'openB' ; 'forall' ; 'y' ; 'q' ; 'y' ; 'or' ; 'z' ; 'or' ; 'exist' ; 'y' ; 's' ; 'y' ; 'closeB' ; nil) P %. +%querytabled _ _ parse ('forall' ; 'x' ; 'forall' ; 'x' ; 'not' ; 'p' ; 'x' ; 'and' ; 'openB' ; 'forall' ; 'y' ; 'q' ; 'y' ; 'or' ; 'z' ; 'or' ; 'openB' ; 'exist' ; 'y' ; 's' ; 'y' ; 'closeB' ; 'closeB' ; nil) P %. +%{ +% should not be parseable +%querytabled * * +parse ('forall' ; 'x' ; 'forall' ; 'x' ; 'not' ; 'p' ; 'x' ; 'and' ; 'openB' ; 'forall' ; 'y' ; 'q' ; 'y' ; 'or' ; 'z' ; 'or' ; 'exist' ; 'y' ; 's' ; 'y' ; 'closeB' ; nil) P. + +}% +%querytabled _ _ parse ('openB' ; 'forall' ; 'x' ; 'forall' ; 'y' ; 'not' ; 'p' ; 'x' ; 'and' ; 'openB' ; 'forall' ; 'y' ; 'q' ; 'y' ; 'or' ; 'z' ; 'closeB' ; 'closeB' ; 'imp' ; 'true' ; nil) P %. +%querytabled _ _ parse ('openB' ; 'forall' ; 'x' ; 'forall' ; 'x' ; 'not' ; 'p' ; 'x' ; 'and' ; 'openB' ; 'forall' ; 'y' ; 'q' ; 'y' ; 'or' ; 'z' ; 'or' ; 'openB' ; 'exist' ; 'y' ; 's' ; 'y' ; 'closeB' ; 'closeB' ; 'closeB' ; 'imp' ; 'true' ; nil) P %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/parsing/foll.lf b/new-tests/stelf-output/examples/tabled/parsing/foll.lf new file mode 100644 index 0000000..d110325 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/parsing/foll.lf @@ -0,0 +1,2 @@ +%require %[ foll %] %. +%require %[ foll %] %. diff --git a/new-tests/stelf-output/examples/tabled/parsing/stelf.toml b/new-tests/stelf-output/examples/tabled/parsing/stelf.toml new file mode 100644 index 0000000..1205615 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/parsing/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "tab" +main = "tab.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/tabled/parsing/tab.lf b/new-tests/stelf-output/examples/tabled/parsing/tab.lf new file mode 100644 index 0000000..f9b09a1 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/parsing/tab.lf @@ -0,0 +1 @@ +%require %[ warren %] %. diff --git a/new-tests/stelf-output/examples/tabled/parsing/warren.lf b/new-tests/stelf-output/examples/tabled/parsing/warren.lf new file mode 100644 index 0000000..1c7b805 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/parsing/warren.lf @@ -0,0 +1,93 @@ +% + +% Simple grammar example +% taken from: David S. Warren: +% Programming in tabled Prolog, Ch 4 Grammars +% +% illustrates efficient recognition and parsing +% using tabled logic programming +% + +% Author: Brigitte Pientka +% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Define token stream +% + +%sort tok %. +% tokens +%term 'c' tok %. +%term 'a' tok %. +%term 'd' tok %. +%term 'b' tok %. +% list of tokens +%sort tokens %. +%term nil tokens %. +%term ; %pi tok %-> tokens %-> tokens %. +%prec %right 10 ; %. +%{ +DCG : for Prolog + +s --> b, [c]. +s --> g, [d]. +b --> b, b. +b --> [a]. +g --> g, [a]. +g --> [a]. +}% +%sort recogS %. +%sort recogB %. +%sort recogG %. +%tabled recogS %. +%tabled recogB %. +%tabled recogG %. +%term sb %pi (recogS L R) %<- (recogB L ('c' ; R)) %. +%term sg %pi (recogS L R) %<- (recogG L ('d' ; R)) %. +%term b1 %pi (recogB L R) %<- (recogB L R1) %<- (recogB R1 R) %. +%term b2 recogB ('a' ; L) L %. +%term g1 %pi (recogG ('a' ; L) R) %<- (recogG L R) %. +%term g2 recogG ('a' ; L) L %. +%%%%%%%%%%%%%%%%%%%%%%%%% +% parse simple grammar +%{ + +if we add an extra argument to the recognizer, +then our parser will have exponential complexity. + +Note: there are exponentially many parses + and we want to produce all of them! + it will take exponential time to produce them all, + however, it do not want to take exponential time + to report failure. +}% +% Parser : referring to recognizer +%sort pterm %. +%term s %pi pterm %-> pterm %-> pterm %. +%term b %pi pterm %-> pterm %-> pterm %. +%term g %pi pterm %-> pterm %. +%term a pterm %. +%term d pterm %. +%term c pterm %. +%sort parseB %. +%sort parseS %. +%sort parseG %. +%term ps1 %pi (parseS (s P c) L R) %<- (recogB L ('c' ; R)) %<- (parseB P L ('c' ; R)) %. +%term ps2 %pi (parseS (s P d) L R) %<- (recogG L ('d' ; R)) %<- (parseB P L ('d' ; R)) %. +%term pb1 + %pi (parseB (b P1 P2) L R) + %<- (recogB L L1) + %<- (recogB L1 R) + %<- (parseB P1 L L1) + %<- (parseB P2 L1 R) %. +%term pb2 parseB a ('a' ; L) L %. +%term pg1 %pi (parseG (g P) L R) %<- (recogG L ('a' ; R)) %<- (parseG P L ('a' ; R)) %. +%term pg2 parseG a ('a' ; L) L %. +% some simple queries +%querytabled _ _ recogG ('a' ; 'a' ; 'a' ; nil) L %. +%querytabled _ _ recogS ('a' ; 'a' ; 'a' ; 'c' ; nil) nil %. +%querytabled _ _ recogS ('a' ; 'a' ; 'a' ; 'b' ; nil) nil %. +%querytabled _ _ %the (recogS ('a' ; 'a' ; 'a' ; 'c' ; nil) nil) D %. +%querytabled _ _ %the (recogS ('a' ; 'a' ; 'a' ; 'b' ; nil) nil) D %. +%querytabled _ _ parseS P ('a' ; 'a' ; 'a' ; 'd' ; nil) nil %. +%querytabled _ _ parseS P ('a' ; 'a' ; 'a' ; 'b' ; nil) nil %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/poly/mini-ml.lf b/new-tests/stelf-output/examples/tabled/poly/mini-ml.lf new file mode 100644 index 0000000..9d4ace1 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/poly/mini-ml.lf @@ -0,0 +1,17 @@ +%%% The Mini-ML Language +%%% Without let name in favor of parametric polymorphism +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +% Expressions +%sort exp %. +%name exp %. +%term z exp %. +%term s %pi exp %-> exp %. +%term case %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%term pair %pi exp %-> exp %-> exp %. +%term fst %pi exp %-> exp %. +%term snd %pi exp %-> exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term letv %pi exp %-> (%pi exp %-> exp) %-> exp %. +% no let name +%term fix %pi (%pi exp %-> exp) %-> exp %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/poly/stelf.toml b/new-tests/stelf-output/examples/tabled/poly/stelf.toml new file mode 100644 index 0000000..1205615 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/poly/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "tab" +main = "tab.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/tabled/poly/tab-examples-quy.lf b/new-tests/stelf-output/examples/tabled/poly/tab-examples-quy.lf new file mode 100644 index 0000000..e77fd8e --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/poly/tab-examples-quy.lf @@ -0,0 +1,19 @@ +% The typing rules are no longer operational, +% so we have to give derivations for type-checking. +%querytabled _ 3 %the (of z T) D %. +% querytabled * 2 of (lam [x] x) (all [b] (all [a] a) => b). +% querytabled * 2 of (lam [x] x) T. +% querytabled * 2 of (lam [x] x) (all [a] a => a). +% querytabled * 1 of (lam [f] (app (app f (lam [y] y)) (app f z))) T. +% querytabled * 1 of (lam [f] (app (app f (lam [y] y)) (app f z))) ((all [a] a => a) => nat). +% querytabled * 2 of (app (lam [f] (app (app f (lam [y] y)) (app f z))) (lam [y] y)) nat. +% = tp_alli [a:tp] tp_lam [x:exp] [u:of x a] u. +%{ +_ : of (lam [x] app x x) ((all [a] a) => (all [a] a)) + = tp_lam [x:exp] [u:of x (all [a] a)] + tp_app u (tp_alle ((all [a] a) => (all [a] a)) u). + +_ : of (lam [x] app x x) ((all [a] a => a) => (all [a] a => a)) + = tp_lam [x] [u:of x (all [a] a => a)] + tp_app u (tp_alle (all [a] a => a) u). +}% \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/poly/tab.lf b/new-tests/stelf-output/examples/tabled/poly/tab.lf new file mode 100644 index 0000000..10bc5a9 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/poly/tab.lf @@ -0,0 +1,4 @@ +%require %[ mini-ml %] %. +%require %[ tp %] %. +%require %[ tpinf %] %. +%require %[ tab-examples %] %. diff --git a/new-tests/stelf-output/examples/tabled/poly/tp.lf b/new-tests/stelf-output/examples/tabled/poly/tp.lf new file mode 100644 index 0000000..2f54099 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/poly/tp.lf @@ -0,0 +1,17 @@ +%%% Mini-ML types. +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort tp %. +%name tp %. +%term nat tp %. +% Natural Numbers +%term cross %pi tp %-> tp %-> tp %. +% Pairs +%term arrow %pi tp %-> tp %-> tp %. +% Functions +%term all %pi (%pi tp %-> tp) %-> tp %. +% Polymorphism +% For the examples, we introduce an abbreviation +%define => (%pi tp %-> tp %-> tp) [a] [b] arrow a b %. +%prec %right 10 => %. +%define * (%pi tp %-> tp %-> tp) [a] [b] cross a b %. +%prec %right 9 * %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/poly/tpinf.lf b/new-tests/stelf-output/examples/tabled/poly/tpinf.lf new file mode 100644 index 0000000..8082b09 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/poly/tpinf.lf @@ -0,0 +1,37 @@ +%%% The Mini-ML typing rules +%%% extended for parametric polymorphism +%%% These cannot be used for type inference because +%%% of the polymorphic type assignment rules +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +%sort of %. +%name of %. +%mode of %in %star %. +%tabled of %. +% Natural Numbers +%term tp_z of z nat %. +%term tp_s %pi (of (s E) nat) %<- (of E nat) %. +%term tp_case + %pi (of (case E1 E2 E3) T) + %<- (of E1 nat) + %<- (of E2 T) + %<- ({x exp} %pi (of x nat) %-> (of (E3 x) T)) %. +% Pairs +%term tp_pair %pi (of (pair E1 E2) (cross T1 T2)) %<- (of E1 T1) %<- (of E2 T2) %. +%term tp_fst %pi (of (fst E) T1) %<- (of E (cross T1 T2)) %. +%term tp_snd %pi (of (snd E) T2) %<- (of E (cross T1 T2)) %. +% Functions +%term tp_lam %pi (of (lam E) (arrow T1 T2)) %<- ({x exp} %pi (of x T1) %-> (of (E x) T2)) %. +%term tp_app %pi (of (app E1 E2) T1) %<- (of E1 (arrow T2 T1)) %<- (of E2 T2) %. +% Definitions +%term tp_letv + %pi (of (letv E1 E2) T2) + %<- (of E1 T1) + %<- ({x exp} %pi (of x T1) %-> (of (E2 x) T2)) %. +% no let name +% Recursion +%term tp_fix %pi (of (fix E) T) %<- ({x exp} %pi (of x T) %-> (of (E x) T)) %. +% Polymorphism +%term tp_alli %pi (of E (all ([a] T a))) %<- ({a tp} of E (T a)) %. +%term tp_alle {T' tp} %pi (of E (T T')) %<- (of E (all ([a] T a))) %. +% +% %covers of +E *T. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/refine/norefex-quy.lf b/new-tests/stelf-output/examples/tabled/refine/norefex-quy.lf new file mode 100644 index 0000000..55158a1 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/refine/norefex-quy.lf @@ -0,0 +1,7 @@ +% not provable +%%%%%%%%%%%%%%%%%%%%%% +% not provable +%querytabled _ _ %the (check inc (nat => nat)) D %. +%querytabled _ 5 check test2 nat %. +%querytabled _ _ check sub' (nat => nat => nat) %. +%querytabled _ 16 check mult (nat => pos => nat) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/refine/notab-quy.lf b/new-tests/stelf-output/examples/tabled/refine/notab-quy.lf new file mode 100644 index 0000000..d268543 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/refine/notab-quy.lf @@ -0,0 +1,123 @@ +%{ +%query 1 * check shiftl (bit => bit & pos => pos). + +%query 0 * check shiftl (nat => nat). + +%query 1 * check inc (nat => pos). + +%query 0 * check inc (nat => nat). + +%query 1 * nat => pos <| nat => nat. +%query 1 * synth (inc # nat => pos) A. + +%query 1 * check std (bit => nat). +%query 1 * check test1 pos. +%query 0 * check test2 nat. +%query 1 * +check plus (bit => bit => bit). + +% Too many solutions? +%query * 2 +D : check plus (nat => nat => nat & + nat => pos => pos & + pos => nat => pos & + pos => pos => pos). + +%query 4 * +check (lam [plus] app (app plus (e 1)) (e 1)) + ((nat => nat => nat & + nat => pos => pos & + pos => nat => pos & + pos => pos => pos) => bit). + +%query 1 * +check (fix [plus] lam [x] lam [y] app (app plus (e 1)) (e 1)) + (nat => nat => nat). + +%query 1 * +check (fix [plus] lam [x] lam [y] app (app plus (e 1)) (e 1)) + (nat => pos => pos). + +%query 4 * +check (lam [plus] app (app plus (e 1)) (e 1)) +(((nat => nat => nat) + & (nat => pos => pos) + & (pos => nat => pos) + & (pos => pos => pos)) => nat). + +%query 2 * +check (fix [plus] lam [x] lam [y] app (app plus (e 1)) (e 1)) + (nat => nat => nat + & nat => pos => pos). + +%query 12 * +check (fix [plus] lam [x] lam [y] app (app plus (e 1)) (e 1)) + (nat => nat => nat % *3 (3 ways to prove nat) + & nat => pos => pos % *2 (2 ways to prove pos) + & pos => nat => pos). % *2 (2 ways to prove pos) + +%query 108 * +check (fix [plus] lam [x] lam [y] app (app plus (e 1)) (e 1)) + (nat => nat => nat % *4 ways to prove nat + & nat => pos => pos % *3 ways to prove pos + & pos => nat => pos % *3 ways to prove pos + & pos => pos => pos). % *3 ways to prove pos + +%query 4 * +check (fix [plus] lam [x] lam [y] app (app plus (e 1)) (e 1)) + (nat => nat => nat + & nat => nat => nat). + +%query 108 * +check (fix [plus] lam [x] lam [y] app (app plus (e 1)) (e 1)) + (nat => nat => nat & + nat => pos => pos & + pos => nat => pos & + pos => pos => pos). + +%query 1 * +synth ((fix [inc] lam [n] + case n + (e 1) + ([x] x 1) + ([x] (app inc x) 0)) # nat => pos) +A. + +%query 1 * +check (fix [plus] lam [n] lam [m] + (case n m ([x] x 1) ([y] y 1))) +(nat => nat => nat & + nat => pos => pos & + pos => nat => pos & + pos => pos => pos). + +%query 1 * +check (fix [plus] lam [n] lam [m] + (case n + m + ([x] case m (x 0) ([y] y 1) ([y] y 1)) + ([x] x 1))) +(nat => nat => nat & + nat => pos => pos & + pos => nat => pos & + pos => pos => pos). + +%query 1 * +check (fix [plus] lam [n] lam [m] + (case n + m + ([x] case m + (x 0) + ([y] y 1) + ([y] y 1)) + ([x] case m + (x 1) + ([y] y 1) + ([y] y 1)))) +(nat => nat => nat & + nat => pos => pos & + pos => nat => pos & + pos => pos => pos). + +}% +%query 20736 _ _ check plus' (nat => nat => nat & nat => pos => pos & pos => nat => pos & pos => pos => pos) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/refine/programs.lf b/new-tests/stelf-output/examples/tabled/refine/programs.lf new file mode 100644 index 0000000..3d00af8 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/refine/programs.lf @@ -0,0 +1,35 @@ +%%% +%%% Author : Frank Pfenning, Brigitte Pientka +%%% sample programs for binary arithmentic and +%%% refinement types. +% Examples from paper + own examples +% to do: +% seperate out unprovable queries / provable queries +% +%define itp nat => nat => nat & nat => pos => nat & pos => nat => nat & pos => pos => pos %. +%define shiftl lam ([x] case x e ([y] y 0 0) ([y] y 1 0)) %. +%define inc fix ([inc] lam ([n] case n (e 1) ([x] x 1) ([x] app inc x 0))) %. +%define std fix ([std] lam ([n] case n e ([x] case (app std x) e ([y] y 0 0) ([y] y 1 0)) ([x] app std x 1))) %. +%define test1 let (e 1 # pos) ([x] x 0) %. +%define test2 let (e 0 # nat) ([x] x 1) %. +%define plus let (fix ([inc] lam ([n] case n (e 1) ([x] x 1) ([x] app inc x 0))) # nat => pos & bit => bit) ([inc] fix ([plus] lam ([n] case n (lam ([m] m)) ([x] lam ([m] case m (x 0) ([y] app (app plus x) y 0) ([y] app (app plus x) y 1))) ([x] lam ([m] case m (x 1) ([y] app (app plus x) y 1) ([y] app inc (app (app plus x) y) 0)))))) %. +%define plus' let (fix ([inc] lam ([n] case n (e 1) ([x] x 1) ([x] app inc x 0))) # nat => pos) ([inc] fix ([plus] lam ([n] lam ([m] case n m ([x] case m (x 0) ([y] app (app plus x) y 0) ([y] app (app plus x) y 1)) ([x] case m (x 1) ([y] app (app plus x) y 1) ([y] app inc (app (app plus x) y) 0)))))) %. +% *1 +% Too many solutions? 20736 = 81 * 16 * 16 * 1 +% zero => zero +%define mult let (shiftl # (nat => nat & (pos => pos) & (bit => bit) & (pos => nat))) ([shiftl] let (plus # (nat => nat => nat & nat => pos => pos & pos => nat => pos & pos => pos => pos)) ([plus] fix ([mult] lam ([n] lam ([m] case m e ([x] app shiftl (app (app mult n) x)) ([x] app shiftl (app (app plus n) (app (app mult n) x)))))))) %. +%define mult' let (shiftl # (bit => bit & pos => pos)) ([shiftl] let (plus # (bit => bit => bit)) ([plus] fix ([mult] lam ([n] lam ([m] case m e ([x] app shiftl (app (app mult n) x)) ([x] app shiftl (app (app plus n) (app (app mult n) x)))))))) %. +%define square let (mult # (pos => nat => nat & (nat => nat => nat) & (nat => pos => nat) & (pos => pos => pos))) ([mult] lam ([m] app (app mult m) m)) %. +%define square' let (mult' # bit => bit => bit) ([mult] lam ([m] app (app mult m) m)) %. +%define ge fix ([ge] lam ([n] lam ([m] case n (case m true ([y] false) ([y] false)) ([x] case m false ([y] app (app ge x) y) ([y] app (app ge x) y)) ([x] case m false ([y] app (app ge x) y) ([y] app (app ge x) y))))) %. +%define sub' let (fix ([inc] lam ([n] case n (e 1) ([x] x 1) ([x] app inc x 0))) # zero => pos & zero => nat & pos => pos & nat => nat & nat => pos & bit => bit) ([inc] fix ([sub] lam ([n] lam ([m] case n e ([x] case m (x 0) ([y] app (app sub x) y 0) ([y] app (app sub x) (app inc y) 1)) ([x] case m (x 1) ([y] app (app sub x) y 1) ([y] app (app sub x) y 0)))))) %. +%define sub let (fix ([inc] lam ([n] case n (e 1) ([x] x 1) ([x] app inc x 0))) # zero => pos & zero => nat & pos => pos & nat => nat & nat => pos & bit => bit) ([inc] let (sub' # nat => zero => nat & zero => nat => zero & bit => bit => bit) ([sub] let (std # zero => zero & nat => nat & pos => pos & bit => nat & bit => bit) ([std] lam ([n] lam ([m] app std (app (app sub n) m)))))) %. +%{ +% sub' produces a bit number,i.e. it might have leading zeros! +% sub standardizes the result of sub', i.e. we get a nat (without leading zeros) +}% +% comment: +% usually would compute with the twocomplement +% and use addition? but this requires a fixed length +% of the binary representation; this representation +% ensures no leading zeros? \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/refine/refex-quy.lf b/new-tests/stelf-output/examples/tabled/refine/refex-quy.lf new file mode 100644 index 0000000..1f6930d --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/refine/refex-quy.lf @@ -0,0 +1,68 @@ +%%% +%%% Author : Frank Pfenning, Brigitte Pientka +%%% queries: sample programs for binary arithmentic and +%%% refinement types. +% Examples from paper + own examples +% to do: +% seperate out unprovable queries / provable queries +% +%querytabled _ 5 check (lam ([x] x 0)) (bit => bit & pos => pos) %. +%querytabled _ 15 check std (bit => nat) %. +%querytabled _ _ check std (zero => zero & nat => nat & pos => pos & nat => bit) %. +%querytabled _ 4 check test1 pos %. +%querytabled _ _ check shiftl (zero => zero) %. +%querytabled _ _ check shiftl (zero => nat) %. +%querytabled _ 3 check shiftl (nat => nat) %. +%querytabled _ 7 check shiftl (bit => bit) %. +%querytabled _ 7 check shiftl (pos => nat) %. +%querytabled _ 7 check shiftl (nat => nat) %. +%querytabled _ 7 check shiftl (nat => nat & (pos => pos) & (pos => nat) & (bit => bit)) %. +%querytabled _ _ check inc (pos => pos & nat => nat & nat => pos) %. +%querytabled _ _ check inc (zero => pos & nat => nat & nat => pos) %. +%querytabled _ 5 check inc (nat => pos) %. +%querytabled _ _ check inc (zero => pos) %. +% working +%querytabled _ 9 check ge (nat => nat => bool) %. +%querytabled _ 9 check ge (bit => bit => bool) %. +%querytabled _ 9 check ge (pos => nat => bool & nat => nat => bool & nat => pos => bool & bit => bit => bool) %. +%querytabled _ 4 check (lam ([plus] app (app plus (e 1)) (e 1))) (nat => nat => nat & nat => pos => pos & pos => nat => pos & pos => pos => pos => bit) %. +%querytabled _ 15 check (fix ([plus] lam ([x] lam ([y] app (app plus (e 1)) (e 1))))) (nat => pos => pos) %. +%querytabled _ 4 check (lam ([plus] app (app plus (e 1)) (e 1))) (nat => nat => nat & (nat => pos => pos) & (pos => nat => pos) & (pos => pos => pos) => nat) %. +%querytabled _ 5 check (fix ([plus] lam ([x] lam ([y] app (app plus (e 1)) (e 1))))) (nat => nat => nat & nat => pos => pos) %. +%querytabled _ 15 check (fix ([plus] lam ([x] lam ([y] app (app plus (e 1)) (e 1))))) (nat => nat => nat & nat => pos => pos & pos => nat => pos) %. +% *2 (2 ways to prove pos) +%querytabled _ 15 check (fix ([plus] lam ([x] lam ([y] app (app plus (e 1)) (e 1))))) (nat => nat => nat & nat => pos => pos & pos => nat => pos & pos => pos => pos) %. +% *3 ways to prove pos +%querytabled _ 5 synth (fix ([inc] lam ([n] case n (e 1) ([x] x 1) ([x] app inc x 0))) # nat => pos) A %. +%querytabled _ 9 check (fix ([plus] lam ([n] lam ([m] case n m ([x] x 1) ([y] y 1))))) (nat => nat => nat & nat => pos => pos & pos => nat => pos & pos => pos => pos) %. +%querytabled _ 14 check (fix ([plus] lam ([n] lam ([m] case n m ([x] case m (x 0) ([y] y 1) ([y] y 1)) ([x] x 1))))) (nat => nat => nat & nat => pos => pos & pos => nat => pos & pos => pos => pos) %. +%querytabled _ 16 check (fix ([plus] lam ([n] lam ([m] case n m ([x] case m (x 0) ([y] y 1) ([y] y 1)) ([x] case m (x 1) ([y] y 1) ([y] y 1)))))) (nat => nat => nat & nat => pos => pos & pos => nat => pos & pos => pos => pos) %. +%querytabled _ 50 check plus' (nat => nat => nat & nat => pos => pos & pos => nat => pos & pos => pos => pos) %. +%querytabled _ 35 check plus (nat => nat => nat & nat => pos => pos & pos => nat => pos & pos => pos => pos) %. +%%%% so far "old tests" +%{ +% sub' produces a bit number,i.e. it might have leading zeros! +% sub standardizes the result of sub', i.e. we get a nat (without leading zeros) +}% +%querytabled _ _ check sub' (zero => nat => zero) %. +%querytabled _ _ check sub' (nat => zero => nat) %. +%querytabled _ _ check sub' (nat => zero => nat & zero => nat => zero & bit => bit => bit) %. +%querytabled _ _ check sub (bit => bit => bit) %. +%querytabled _ _ check sub (zero => nat => zero) %. +%querytabled _ _ check sub (nat => zero => nat) %. +%querytabled _ _ check sub (nat => zero => nat & zero => nat => zero & bit => bit => bit) %. +%querytabled _ _ check sub (nat => zero => nat & zero => nat => zero & nat => nat => nat) %. +%querytabled _ _ check sub' (nat => zero => nat & zero => nat => zero & bit => bit => bit) %. +%querytabled _ _ check sub (pos => pos => nat) %. +%querytabled _ _ check sub (nat => pos => nat & (pos => nat => nat) & (pos => pos => nat) & nat => nat => nat) %. +%querytabled _ _ check mult (nat => zero => zero) %. +% all are provable +%querytabled _ _ check mult (nat => nat => nat) %. +%querytabled _ _ check mult (pos => nat => nat) %. +%querytabled _ _ check mult (pos => nat => nat & (nat => nat => nat) & (nat => pos => nat)) %. +%querytabled _ _ check mult (pos => nat => nat & (nat => nat => nat) & (nat => pos => nat) & (pos => pos => pos)) %. +%querytabled _ _ check mult' (bit => bit => bit) %. +%querytabled _ 16 check mult' (bit => bit => bit) %. +%querytabled _ _ check square' (bit => bit) %. +%querytabled _ _ check square (pos => nat & nat => nat) %. +%querytabled _ _ check square (pos => pos) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/refine/refex.lf b/new-tests/stelf-output/examples/tabled/refine/refex.lf new file mode 100644 index 0000000..33bf185 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/refine/refex.lf @@ -0,0 +1,299 @@ +%%% Intersection type checking from ICFP'00 paper +%%% No refinement restriction +%%% No polymorphism +%%% Included Top (t) +%%% Author: Frank Pfenning +%%% Extended: Brigitte Pientka +% Types +%sort tp %. +%name tp %. +% bp Tue Feb 26 20:36:02 2002 +%term bool tp %. +%term zero tp %. +% bp Wed Feb 27 11:51:24 2002 +%term bit tp %. +%term nat tp %. +%term pos tp %. +%term => %pi tp %-> tp %-> tp %. +%prec %right 10 => %. +%term & %pi tp %-> tp %-> tp %. +%prec %right 9 & %. +%term t tp %. +%{ +% Declarative Subtyping +<= : tp -> tp -> type. %infix none 8 <=. + %name <= S. +%mode <= *A *B. + +refl : A <= A. +trans : A <= B + -> B <= C + -> A <= C. + + +z<=n : zero <= nat. +p<=n : pos <= nat. +n<=b : nat <= bit. + +&<=1 : A1 & A2 <= A1. +&<=2 : A1 & A2 <= A2. +<=& : A <= B1 + -> A <= B2 + -> A <= B1 & B2. +<=t : A <= t. + +<==> : B1 <= A1 + -> A2 <= B2 + -> A1 => A2 <= B1 => B2. +}% +% Ordinary Types +%sort ord %. +%mode ord %in %. +%term ord_bo ord bool %. +%term ord_b ord bit %. +%term ord_n ord nat %. +%term ord_p ord pos %. +%term ord_z ord zero %. +%term ord_=> ord (A => B) %. +% t and & are not ordinary. +%terminates [] (ord _) %. +% Extracting conjuncts +%sort conj %. +%mode conj %in %out %. +%term conj0 %pi (conj Ao Ao) %<- (ord Ao) %. +%term conjl %pi (conj (A & B) Ao) %<- (conj A Ao) %. +%term conjr %pi (conj (A & B) Bo) %<- (conj B Bo) %. +%terminates A (conj A _) %. +% Algorithmic Subtyping +% Duplicated for the termination checker only! +%sort <| %. +%prec %none 8 <| %. +%sort |> %. +%prec %none 8 |> %. +%mode <| %in %in %. +%mode |> %in %in %. +%term bo<|bo bool <| bool %. +%term b<|b bit <| bit %. +%term n<|n nat <| nat %. +%term p<|p pos <| pos %. +%term p<|n pos <| nat %. +%term z<|z zero <| zero %. +%term z<|n zero <| nat %. +%term n<|b nat <| bit %. +%term p<|b pos <| bit %. +%term &1<|o %pi (A1 & A2 <| Bo) %<- (ord Bo) %<- (A1 <| Bo) %. +%term &2<|o %pi (A1 & A2 <| Bo) %<- (ord Bo) %<- (A2 <| Bo) %. +%term <|& %pi (A <| B1 & B2) %<- (A <| B1) %<- (A <| B2) %. +%term <|t A <| t %. +%term =><|=> %pi (A1 |> B1) %-> (A2 <| B2) %-> (A1 => A2 <| B1 => B2) %. +%term b|>b bit |> bit %. +%term n|>n nat |> nat %. +%term p|>p pos |> pos %. +%term n|>p nat |> pos %. +%term z|>z zero |> zero %. +%term n|>z nat |> zero %. +%term b|>b bit |> nat %. +%term b|>p bit |> pos %. +%term o|>&1 %pi (Bo |> A1 & A2) %<- (ord Bo) %<- (Bo |> A1) %. +%term o|>&2 %pi (Bo |> A1 & A2) %<- (ord Bo) %<- (Bo |> A2) %. +%term &|> %pi (B1 & B2 |> A) %<- (B1 |> A) %<- (B2 |> A) %. +%term t|> t |> A %. +%term =>|>=> %pi (B1 <| A1) %-> (B2 |> A2) %-> (B1 => B2 |> A1 => A2) %. +%terminates [(A A') (B B')] (|> A' B') (<| A B) %. +% Note to Brigitte +% %terminates [(A A') (B B')] (<| A B) (<| B' A'). +% Terms +%sort term %. +%name term %. +% Type ascription +%term # %pi term %-> tp %-> term %. +%prec %none 8 # %. +% integers +%term z term %. +%term s %pi term %-> term %. +%term casen %pi term %-> term %-> (%pi term %-> term) %-> term %. +% Functions +%term lam %pi (%pi term %-> term) %-> term %. +%term app %pi term %-> term %-> term %. +%term if %pi term %-> term %-> term %-> term %. +%term true term %. +%term false term %. +% Bits +%term e term %. +%term 0 %pi term %-> term %. +%prec %postfix 10 0 %. +%term 1 %pi term %-> term %. +%prec %postfix 10 1 %. +%term case %pi term %-> term %-> (%pi term %-> term) %-> (%pi term %-> term) %-> term %. +% Let and recursion +%term let %pi term %-> (%pi term %-> term) %-> term %. +%term fix %pi (%pi term %-> term) %-> term %. +% Values +%sort value %. +%name value %. +%mode value %in %. +%term val_lam value (lam ([x] M x)) %. +%term val_e value e %. +%term val_0 %pi (value (M 0)) %<- (value M) %. +%term val_1 %pi (value (M 1)) %<- (value M) %. +%term val_t value true %. +%term val_f value false %. +%term val_z value z %. +%term val_s %pi (value (s E)) %<- (value E) %. +%terminates M (value M) %. +%sort nonvalue %. +%mode nonvalue %in %. +%term nv_app nonvalue (app M N) %. +%term nv_0 %pi (nonvalue (M 0)) %<- (nonvalue M) %. +%term nv_1 %pi (nonvalue (M 1)) %<- (nonvalue M) %. +%term nv_case nonvalue (case M Me M0 M1) %. +%term nv_casen nonvalue (casen M Mz Ms) %. +%term nv_let nonvalue (let M N) %. +%term nv_fix nonvalue (fix M) %. +%term nv_if nonvalue (if M M1 M2) %. +%terminates M (nonvalue M) %. +% Typing Rules; Values +%{ +of : term -> tp -> type. %name of D. +%mode of +M *A. + +of<= : of M A + -> A <= B + -> of M B. + +of& : of V A + -> of V B + -> value V + -> of V (A & B). + +of_true: of true bool. +of_false: of false bool. +of_if : of (if M M1 M2) T + <- of M bool + <- of M1 T + <- of M2 T. + +of_lam : ({x} of x A -> of (M x) B) + -> of (lam [x] M x) (A => B). + +of_app : of M (A => B) + -> of N A + -> of (app M N) B. + +of_e : of e nat. +of_ez : of e zero. +of_0b : of M bit -> of (M 0) bit. +of_0p : of M pos -> of (M 0) pos. +of_1b : of M bit -> of (M 1) bit. +of_1p : of M nat -> of (M 1) pos. + +of_case_b : of M bit + -> of Me A + -> ({x} of x bit -> of (M0 x) A) + -> ({y} of y bit -> of (M1 y) A) + -> of (case M Me ([x] M0 x) ([y] M1 y)) A. +of_case_n : of M nat + -> of Me A + -> ({x} of x pos -> of (M0 x) A) + -> ({y} of y nat -> of (M1 y) A) + -> of (case M Me ([x] M0 x) ([y] M1 y)) A. +of_case_p : of M pos + -> ({x} of x pos -> of (M0 x) A) + -> ({y} of y nat -> of (M1 y) A) + -> of (case M Me ([x] M0 x) ([y] M1 y)) A. + +% bp Wed Feb 27 11:54:12 2002 +of_case_z : of M zero + -> of Me A + -> of (case M Me ([x] M0 x) ([y] M1 y)) A. + +of_fix : of (fix M) A + <- ({x:term} of x A -> of (M x) A). + +of_let : of (let M N) A + <- of M B + <- ({x:term} of x B -> of (N x) A). +}% +% Bi-Directional Type Checking +%sort synth %. +%name synth %. +%sort check %. +%name check %. +%mode synth %in %out %. +%mode check %in %in %. +% there is a bug -- if -- bp Mon Feb 18 13:54:00 2002 fixed +% tabled synth. +%tabled check %. +% Coercions +%term s_# %pi (synth (C # A) A) %<- (check C A) %. +%term c_s %pi (check I Ao) %<- (ord Ao) %<- (synth I A') %<- (A' <| Ao) %. +%term cnv_& %pi (check I (A & B)) %<- (nonvalue I) %<- (synth I A') %<- (A' <| A & B) %. +%term cnv_t %pi (check I t) %<- (nonvalue I) %. +% <- synth I A' % redundant? +% <- A' <| t. % always succeeds +%% Intersections +%term c_& %pi (check CV (A & B)) %<- (value CV) %<- (check CV A) %<- (check CV B) %. +%term c_t %pi (check CV t) %<- (value CV) %. +% checkable CV? +% Functions +%term s_app %pi (synth (app I C) B) %<- (synth I A') %<- (conj A' (A => B)) %<- (check C A) %. +%term c_lam + %pi (check (lam ([x] C x)) (A => B)) + %<- ({x term} %pi (value x) %-> (synth x A) %-> (check (C x) B)) %. +%term c_true check true bool %. +%term c_false check false bool %. +%term c_if %pi (check (if M M1 M2) A) %<- (check M bool) %<- (check M1 A) %<- (check M2 A) %. +% e : nat [& bit] +%term c_b check e bit %. +%term c_e check e nat %. +%term c_z check e zero %. +% bp Wed Feb 27 11:58:16 2002 +% no c_p +% _ 0 : bit -> bit & pos -> pos [& nat -> bit] +%term s_0_b %pi (check (C 0) bit) %<- (check C bit) %. +%term s_0_n %pi (check (C 0) nat) %<- (check C pos) %. +%term s_0_p %pi (check (C 0) pos) %<- (check C pos) %. +% _ 1 : bit -> bit & nat -> pos [& pos -> pos] +%term s_1_b %pi (check (C 1) bit) %<- (check C bit) %. +%term s_1_n %pi (check (C 1) nat) %<- (check C nat) %. +%term s_1_p %pi (check (C 1) pos) %<- (check C nat) %. +%term c_case_b + %pi (check (case I Ce ([x] C0 x) ([y] C1 y)) A) + %<- (synth I B') + %<- (conj B' bit) + %<- (check Ce A) + %<- ({x} %pi (value x) %-> (synth x bit) %-> (check (C0 x) A)) + %<- ({y} %pi (value y) %-> (synth y bit) %-> (check (C1 y) A)) %. +%term c_case_n + %pi (check (case I Ce ([x] C0 x) ([y] C1 y)) A) + %<- (synth I B') + %<- (conj B' nat) + %<- (check Ce A) + %<- ({x} %pi (value x) %-> (synth x pos) %-> (check (C0 x) A)) + %<- ({y} %pi (value y) %-> (synth y nat) %-> (check (C1 y) A)) %. +%term c_case_p + %pi (check (case I Ce ([x] C0 x) ([y] C1 y)) A) + %<- (synth I B') + %<- (conj B' pos) + %<- (check Ce t) + %<- ({x} %pi (value x) %-> (synth x pos) %-> (check (C0 x) A)) + %<- ({y} %pi (value y) %-> (synth y nat) %-> (check (C1 y) A)) %. +%term c_case_z + %pi (check (case I Ce ([x] C0 x) ([y] C1 y)) A) + %<- (synth I B') + %<- (conj B' zero) + %<- (check Ce A) %. +% Definitions +%term c_let + %pi (check (let I C) A) + %<- (synth I B) + %<- ({x term} %pi (value x) %-> (synth x B) %-> (check (C x) A)) %. +% Recursion +%term c_fix + %pi (check (fix C) A) + %<- ({x term} %pi (nonvalue x) %-> (synth x A) %-> (check (C x) A)) %. +% Note to Carsten: cannot termination check this!! +% %terminates {(I C) A} (synth I _) (check C A). +% %terminates {(I C) (A' A)} (synth I A') (check C A). +% Need to comment out offending clause and use spec below +% %terminates (I C) (synth I A') (check C A). \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/refine/stelf.toml b/new-tests/stelf-output/examples/tabled/refine/stelf.toml new file mode 100644 index 0000000..1205615 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/refine/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "tab" +main = "tab.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/tabled/refine/tab.lf b/new-tests/stelf-output/examples/tabled/refine/tab.lf new file mode 100644 index 0000000..59440ef --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/refine/tab.lf @@ -0,0 +1,4 @@ +%require %[ refex %] %. +%require %[ programs %] %. +%require %[ refex %] %. +%require %[ norefex %] %. diff --git a/new-tests/stelf-output/examples/tabled/seqCalc/foc.lf b/new-tests/stelf-output/examples/tabled/seqCalc/foc.lf new file mode 100644 index 0000000..35d5c5a --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/seqCalc/foc.lf @@ -0,0 +1,3 @@ +%require %[ formulas %] %. +%require %[ focus %] %. +%require %[ tab-foc-all %] %. diff --git a/new-tests/stelf-output/examples/tabled/seqCalc/focus.lf b/new-tests/stelf-output/examples/tabled/seqCalc/focus.lf new file mode 100644 index 0000000..27dd008 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/seqCalc/focus.lf @@ -0,0 +1,45 @@ +%%% Intuitionistic Sequent Calculus using inversion-based strategy +%%% Tabling only used for loop-detection for implL rule +%%% Follows Frank Pfenning ATP notes. +%%% Author: Brigitte Pientka +%sort ctx %. +%term nil ctx %. +%term ; %pi ctx %-> o %-> ctx %. +%prec %left 11 ; %. +%sort search %. +%sort activeR %. +% Conclusion (right) +%sort activeL %. +%sort focusR %. +% Conclusion (right) +%sort hyp %. +% Hypothesis (left) +%name hyp %. +%name activeR %. +%name activeL %. +%name focusR %. +% tabled hyp. +%tabled focusR %. +% Right asynchronous propositions +%term truer activeR G true %. +%term andr %pi (activeR G (A and B)) %<- (activeR G A) %<- (activeR G B) %. +%term impr %pi (activeR G (A imp B)) %<- (activeR (G ; A) B) %. +%term aRL_atom %pi (activeR G (atom P)) %<- (activeL G (atom P)) %. +%term aRL_false %pi (activeR G false) %<- (activeL G false) %. +%term aRL_or %pi (activeR G (A or B)) %<- (activeL G (A or B)) %. +% Left asynchronous propositions +%term a_nil %pi (activeL nil R) %<- (focusR R) %. +%term andL %pi (activeL (G ; (A and B)) R) %<- (activeL (G ; A ; B) R) %. +%term orL %pi (activeL (G ; (A or B)) R) %<- (activeL (G ; A) R) %<- (activeL (G ; B) R) %. +%term falseL activeL (G ; false) R %. +%term a_imp %pi (activeL (G ; (A imp B)) R) %<- (%pi (hyp (A imp B)) %-> (activeL G R)) %. +%term a_atom %pi (activeL (G ; (atom A)) R) %<- (%pi (hyp (atom A)) %-> (activeL G R)) %. +%term a_true %pi (activeL (G ; true) R) %<- (activeL G R) %. +% Right synchronous propositions +%term axiom %pi (hyp (atom A)) %-> (focusR (atom A)) %. +%term f_or1 %pi (focusR (A or B)) %<- (search A) %. +%term f_or2 %pi (focusR (A or B)) %<- (search B) %. +% Left synchronous propositions +%term impl %pi (%pi (hyp (A imp B)) %-> (focusR R)) %<- (search A) %<- (activeL (nil ; B) R) %. +% top level search +%term s_top %pi (search A) %<- (activeR nil A) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/seqCalc/fol-int-at.lf b/new-tests/stelf-output/examples/tabled/seqCalc/fol-int-at.lf new file mode 100644 index 0000000..996f2f5 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/seqCalc/fol-int-at.lf @@ -0,0 +1,34 @@ +%%% Cut-Free Intuitionistic Sequent Calculus +%%% First-order Logic +%%% Tabling achieves loop-detection based sequent calculus +%%% Author: Frank Pfenning +%%% Modified: Brigitte Pientka +%sort hyp %. +% Hypotheses (left) +%sort conc %. +% Conclusion (right) +%name hyp %. +%name conc %. +%tabled hyp %. +%tabled conc %. +%term axiom %pi (hyp (atom P)) %-> (conc (atom P)) %. +%term truer conc true %. +%term andr %pi (conc (A and B)) %<- (conc A) %<- (conc B) %. +%term andl1 %pi (%pi (hyp (A and B)) %-> (conc C)) %<- (%pi (hyp A) %-> (conc C)) %. +%term andl2 %pi (%pi (hyp (A and B)) %-> (conc C)) %<- (%pi (hyp B) %-> (conc C)) %. +%term impr %pi (conc (A imp B)) %<- (%pi (hyp A) %-> (conc B)) %. +%term impl %pi (%pi (hyp (A imp B)) %-> (conc C)) %<- (conc A) %<- (%pi (hyp B) %-> (conc C)) %. +%term forallr %pi ({a i} conc (A a)) %-> (conc (forall A)) %. +%term foralll {T i} %pi (%pi (hyp (A T)) %-> (conc C)) %-> (hyp (forall A)) %-> (conc C) %. +%term orr1 %pi (conc A) %-> (conc (A or B)) %. +%term orr2 %pi (conc B) %-> (conc (A or B)) %. +%term orl + %pi (%pi (hyp A) %-> (conc C)) + %-> (%pi (hyp B) %-> (conc C)) + %-> (hyp (A or B)) + %-> (conc C) %. +%term notr %pi ({p o} %pi (hyp A) %-> (conc p)) %-> (conc (not A)) %. +%term notl %pi (conc A) %-> (hyp (not A)) %-> (conc C) %. +%term falsel %pi (hyp false) %-> (conc C) %. +%term existsr {T i} %pi (conc (A T)) %-> (conc (exists A)) %. +%term existsl %pi ({a i} %pi (hyp (A a)) %-> (conc C)) %-> (hyp (exists A)) %-> (conc C) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/seqCalc/formulas.lf b/new-tests/stelf-output/examples/tabled/seqCalc/formulas.lf new file mode 100644 index 0000000..ad2d8dc --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/seqCalc/formulas.lf @@ -0,0 +1,26 @@ +%%% Language of formulas for intuitionistic and classical +%%% predicate calculus. +%%% Author: Frank Pfenning +%sort i %. +% individuals +%name i %. +% S % more than one currently disallowed +%sort o %. +% formulas +%name o %. +% B C % more than one currently disallowed +%sort p %. +% Atoms +%term atom %pi p %-> o %. +%term and %pi o %-> o %-> o %. +%prec %right 11 and %. +%term imp %pi o %-> o %-> o %. +%prec %right 10 imp %. +%term or %pi o %-> o %-> o %. +%prec %right 11 or %. +%term not %pi o %-> o %. +%prec %prefix 12 not %. +%term true o %. +%term false o %. +%term forall %pi (%pi i %-> o) %-> o %. +%term exists %pi (%pi i %-> o) %-> o %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/seqCalc/int-atom.lf b/new-tests/stelf-output/examples/tabled/seqCalc/int-atom.lf new file mode 100644 index 0000000..7214868 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/seqCalc/int-atom.lf @@ -0,0 +1,30 @@ +%%% Cut-Free Intuitionistic Sequent Calculus +%%% Propositional Logic +%%% Tabling achieves loop-detection based sequent calculus +%%% Interesting question: can we model Howe's loop detection mechanism? +%%% Author: Frank Pfenning +%sort hyp %. +% Hypotheses (left) +%sort conc %. +% Conclusion (right) +%name hyp %. +%name conc %. +%tabled hyp %. +%tabled conc %. +%term impr %pi (conc (A imp B)) %<- (%pi (hyp A) %-> (conc B)) %. +%term impl %pi (%pi (hyp (A imp B)) %-> (conc C)) %<- (conc A) %<- (%pi (hyp B) %-> (conc C)) %. +%term andr %pi (conc (A and B)) %<- (conc A) %<- (conc B) %. +%term andl1 %pi (%pi (hyp (A and B)) %-> (conc C)) %<- (%pi (hyp A) %-> (conc C)) %. +%term andl2 %pi (%pi (hyp (A and B)) %-> (conc C)) %<- (%pi (hyp B) %-> (conc C)) %. +%term orr1 %pi (conc A) %-> (conc (A or B)) %. +%term orr2 %pi (conc B) %-> (conc (A or B)) %. +%term orl + %pi (%pi (hyp A) %-> (conc C)) + %-> (%pi (hyp B) %-> (conc C)) + %-> (hyp (A or B)) + %-> (conc C) %. +%term notr %pi ({p o} %pi (hyp A) %-> (conc p)) %-> (conc (not A)) %. +%term notl %pi (conc A) %-> (hyp (not A)) %-> (conc C) %. +%term axiom %pi (hyp (atom P)) %-> (conc (atom P)) %. +%term truer conc true %. +%term falsel %pi (hyp false) %-> (conc C) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/seqCalc/int.lf b/new-tests/stelf-output/examples/tabled/seqCalc/int.lf new file mode 100644 index 0000000..9338231 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/seqCalc/int.lf @@ -0,0 +1,43 @@ +%%% Cut-Free Intuitionistic Sequent Calculus +%%% Tabling achieves loop-detection based sequent calculus +%%% Interesting question: can we model Howe's loop detection mechanism? +%%% Author: Frank Pfenning +%sort hyp %. +% Hypotheses (left) +%sort conc %. +% Conclusion (right) +%name hyp %. +%name conc %. +%tabled hyp %. +%tabled conc %. +%term impr %pi (conc (A imp B)) %<- (%pi (hyp A) %-> (conc B)) %. +%term impl %pi (%pi (hyp (A imp B)) %-> (conc C)) %<- (conc A) %<- (%pi (hyp B) %-> (conc C)) %. +%term andr %pi (conc (A and B)) %<- (conc A) %<- (conc B) %. +%term andl1 %pi (%pi (hyp (A and B)) %-> (conc C)) %<- (%pi (hyp A) %-> (conc C)) %. +%term andl2 %pi (%pi (hyp (A and B)) %-> (conc C)) %<- (%pi (hyp B) %-> (conc C)) %. +%term orr1 %pi (conc A) %-> (conc (A or B)) %. +%term orr2 %pi (conc B) %-> (conc (A or B)) %. +%term orl + %pi (%pi (hyp A) %-> (conc C)) + %-> (%pi (hyp B) %-> (conc C)) + %-> (hyp (A or B)) + %-> (conc C) %. +%term notr %pi ({p o} %pi (hyp A) %-> (conc p)) %-> (conc (not A)) %. +%term notl %pi (conc A) %-> (hyp (not A)) %-> (conc C) %. +%term axiom %pi (hyp A) %-> (conc A) %. +%term truer conc true %. +%term falsel %pi (hyp false) %-> (conc C) %. +%{ +forallr : ({a:i} conc (A a)) + -> conc (forall A). + +foralll : {T:i} (hyp (A T) -> conc C) + -> (hyp (forall A) -> conc C). + +existsr : {T:i} conc (A T) + -> conc (exists A). + +existsl : ({a:i} hyp (A a) -> conc C) + -> (hyp (exists A) -> conc C). + +}% \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/seqCalc/sources.lf b/new-tests/stelf-output/examples/tabled/seqCalc/sources.lf new file mode 100644 index 0000000..dbe3e1b --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/seqCalc/sources.lf @@ -0,0 +1,2 @@ +%require %[ formulas %] %. +%require %[ int %] %. diff --git a/new-tests/stelf-output/examples/tabled/seqCalc/stelf.toml b/new-tests/stelf-output/examples/tabled/seqCalc/stelf.toml new file mode 100644 index 0000000..ba1a99b --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/seqCalc/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "tab-fol" +main = "tab-fol.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/tabled/seqCalc/tab-at.lf b/new-tests/stelf-output/examples/tabled/seqCalc/tab-at.lf new file mode 100644 index 0000000..41b7b60 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/seqCalc/tab-at.lf @@ -0,0 +1,4 @@ +%require %[ formulas %] %. +%require %[ int-atom %] %. +%require %[ tab-ex-all %] %. +%require %[ tab-ex-np %] %. diff --git a/new-tests/stelf-output/examples/tabled/seqCalc/tab-ex-all-quy.lf b/new-tests/stelf-output/examples/tabled/seqCalc/tab-ex-all-quy.lf new file mode 100644 index 0000000..25d98c1 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/seqCalc/tab-ex-all-quy.lf @@ -0,0 +1,87 @@ +%querytabled _ _ %the ({A o} {B o} conc (A and B imp (B and A))) D %. +%querytabled _ _ %the ({A o} {B o} %pi (hyp (A and B)) %-> (conc (B and A))) D %. +%querytabled _ _ %the ({A o} {B o} conc (A imp B imp A)) D %. +%querytabled _ _ %the ({A o} {B o} {C o} %pi (hyp (A imp B imp C)) %-> (conc (A and B imp C))) D %. +%querytabled _ _ %the ({A o} {B o} {C o} %pi (hyp (A and B imp C)) %-> (conc (A imp B imp C))) D %. +%querytabled _ _ %the ({A o} {B o} {C o} %pi (hyp (A imp B imp C)) %-> (conc (A imp B imp A imp C))) D %. +% proof of S +%querytabled _ _ %the ({A o} {B o} {C o} conc (A imp B imp C imp (A imp B) imp A imp C)) D %. +%querytabled _ _ %the ({A o} {B o} {C o} %pi (hyp (A and B and C)) %-> (conc (A and (B and C)))) D %. +%querytabled _ _ %the ({A o} {B o} {C o} %pi (hyp (A and (B and C))) %-> (conc (A and B and C))) D %. +% impDef +% +%querytabled _ _ %the ({A o} {B o} {C o} %pi (hyp (A and B or C)) %-> (conc (A or C and (B or C)))) D %. +%querytabled _ _ %the ({A o} {B o} {C o} %pi (hyp (A and (B or C))) %-> (conc (A and B or (A and C)))) D %. +% new from ATP notes +% 2 +%querytabled _ _ %the ({A o} {B o} {C o} %pi (hyp (A imp (B and C))) %-> (conc (A imp B and (A imp C)))) D %. +%querytabled _ _ %the ({A o} {B o} {C o} %pi (hyp (A imp B and (A imp C))) %-> (conc (A imp (B and C)))) D %. +% 3 +%querytabled _ _ %the ({A o} {B o} {C o} %pi (hyp (A imp B or (A imp C))) %-> (conc (A imp (B or C)))) D %. +% 4 +%querytabled _ _ %the ({A o} {B o} {C o} %pi (hyp (A or C and (B imp C))) %-> (conc (A imp B imp C))) D %. +% 5 +%querytabled _ _ %the ({A o} {B o} {C o} %pi (hyp (A or B imp C)) %-> (conc (A imp C and (B imp C)))) D %. +%querytabled _ _ %the ({A o} {B o} {C o} %pi (hyp (A imp C and (B imp C))) %-> (conc (A or B imp C))) D %. +% 6 +%querytabled _ _ %the ({A o} {B o} {C o} %pi (hyp (A and (B or C))) %-> (conc (A and B or (A and C)))) D %. +%querytabled _ _ %the ({A o} {B o} {C o} %pi (hyp (A and B or (A and C))) %-> (conc (A and (B or C)))) D %. +% 7 +%querytabled _ _ %the ({A o} {B o} {C o} %pi (hyp (A or (B and C))) %-> (conc (A or B and (A or C)))) D %. +%querytabled _ _ %the ({A o} {B o} {C o} %pi (hyp (A or B and (A or C))) %-> (conc (A or (B and C)))) D %. +% Id : A => A +%querytabled _ _ %the ({A o} conc (A imp A)) D %. +% K : A => B => A +%querytabled _ _ %the ({A o} {B o} conc (A imp B imp A)) D %. +% S : (A => B => C) => (A => B) => (A => C) +%querytabled _ _ %the ({A o} {B o} {C o} conc (A imp (B imp C) imp (A imp B imp (A imp C)))) D %. +% (A & B => C) <=> (A => B => C) +%querytabled _ _ %the ({A o} {B o} {C o} conc (A and B imp C imp (A imp (B imp C)))) D %. +%querytabled _ _ %the ({A o} {B o} {C o} conc (A imp (B imp C) imp (A and B imp C))) D %. +% (A => B & C) <=> (A => B) & (A => C) +%querytabled _ _ %the ({A o} {B o} {C o} conc (A imp (B and C) imp (A imp B) and (A imp C))) D %. +%querytabled _ _ %the ({A o} {B o} {C o} conc (A imp B and (A imp C) imp (A imp (B and C)))) D %. +% (A => T) <=> T +%querytabled _ _ %the ({A o} conc (A imp true imp true)) D %. +%querytabled _ _ %the ({A o} conc (true imp (A imp true))) D %. +% (A v B => C) <=> (A => C) & (B => C) +%querytabled _ _ %the ({A o} {B o} {C o} conc (A or B imp C imp (A imp C and (B imp C)))) D %. +%querytabled _ _ %the ({A o} {B o} {C o} conc (A imp C and (B imp C) imp (A or B imp C))) D %. +% (F => C) <=> T +%querytabled _ _ %the ({C o} conc (false imp C imp true)) D %. +%querytabled _ _ %the ({C o} conc (true imp (false imp C))) D %. +% (A => B v C) =>(A => B) v (A => C) +%querytabled _ _ %the ({A o} {B o} {C o} conc (A imp B or (A imp C) imp (A imp (B or C)))) D %. +% F => C +%querytabled _ _ %the ({C o} conc (false imp C)) D %. +% A => ((A => F) => F) +%querytabled _ _ %the ({A o} conc (A imp (A imp false imp false))) D %. +% ((A => F) & A) => C +%querytabled _ _ %the ({A o} {C o} conc (A imp false and A imp C)) D %. +% A & ~B => ~(A => B) +%querytabled _ _ %the ({A o} {B o} conc (A and (B imp false) imp (A imp B imp false))) D %. +% A & (B v C) <=> (A & B) v (A & C) +%querytabled _ _ %the ({A o} {B o} {C o} conc (A and (B or C) imp (A and B or (A and C)))) D %. +%querytabled _ _ %the ({A o} {B o} {C o} conc (A and B or (A and C) imp (A and (B or C)))) D %. +% A & F <=> F +%querytabled _ _ %the ({A o} conc (A and false imp false)) D %. +%querytabled _ _ %the ({A o} conc (false imp (A and false))) D %. +% A v (B & C) <=> (A v B) & (A v C) +%querytabled _ _ %the ({A o} {B o} {C o} conc (A or (B and C) imp (A or B) and (A or C))) D %. +%querytabled _ _ %the ({A o} {B o} {C o} conc (A or B and (A or C) imp (A or (B and C)))) D %. +%querytabled _ _ %the ({A o} conc (A imp (A and A))) D %. +%querytabled _ _ %the ({A o} conc (A and A imp A)) D %. +%querytabled _ _ %the ({A o} conc (A imp (A or A))) D %. +%querytabled _ _ %the ({A o} conc (A or A imp A)) D %. +% ~A v ~B => ~(A & B) +%querytabled _ _ %the ({A o} {B o} conc (A imp false or (B imp false) imp (A and B imp false))) D %. +% A => A v B +%querytabled _ _ %the ({A o} {B o} conc (A imp (A or B))) D %. +% B => A v B +%querytabled _ _ %the ({A o} {B o} conc (B imp (A or B))) D %. +% A & B => A +%querytabled _ _ %the ({A o} {B o} conc (A and B imp A)) D %. +% A & B => B +%querytabled _ _ %the ({A o} {B o} conc (A and B imp B)) D %. +% A v ~A => ((A => B) => A) => A +%querytabled _ _ %the ({A o} {B o} conc (A or (A imp false) imp (A imp B imp A imp A))) D %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/seqCalc/tab-ex-np-quy.lf b/new-tests/stelf-output/examples/tabled/seqCalc/tab-ex-np-quy.lf new file mode 100644 index 0000000..e9b9936 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/seqCalc/tab-ex-np-quy.lf @@ -0,0 +1,22 @@ +% ((A => F) => F) => A +%querytabled _ _ %the ({A o} conc (A imp false imp false imp A)) D %. +% Peirce's law : conc ((A => B) => A) => A +%querytabled _ _ %the ({A o} {B o} conc (A imp B imp A imp A)) D %. +% (A => false) or A +%querytabled _ _ %the ({A o} conc (A imp false or A)) D %. +% ((A => F) => A) => A +%querytabled _ _ %the ({A o} conc (A imp false imp A imp A)) D %. +% T => F +%querytabled _ _ %the (conc (true imp false)) D %. +% ~(A => B) => A & ~B +%querytabled _ _ %the ({A o} {B o} conc (A imp B imp false imp (A and (B imp false)))) D %. +% ~(A & B) => ~A v ~B +%querytabled _ _ %the ({A o} {B o} conc (A and B imp false imp (A imp false or (B imp false)))) D %. +% (A => B v C) <= (A => B) v (A => C) +%querytabled _ _ %the ({A o} {B o} {C o} conc (A imp (B or C) imp (A imp C))) D %. +% (A & B) => (((A => (B => C)) => C) => (A => B => C)) +%querytabled _ _ %the ({A o} {B o} {C o} conc (A and B imp (A imp (B imp C) imp C imp (A imp (B imp C))))) D %. +% A => (B v C) --> (A => B) v (A imp C) +%querytabled _ _ %the ({A o} {B o} {C o} %pi (hyp (A imp (B or C))) %-> (conc (A imp B or (A imp C)))) D %. +% A => B => C ---> (A v C) & (B => C) +%querytabled _ _ %the ({A o} {B o} {C o} %pi (hyp (A imp B imp C)) %-> (conc (A or C and (B imp C)))) D %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/seqCalc/tab-foc-all-quy.lf b/new-tests/stelf-output/examples/tabled/seqCalc/tab-foc-all-quy.lf new file mode 100644 index 0000000..5ab3c9d --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/seqCalc/tab-foc-all-quy.lf @@ -0,0 +1,88 @@ +%querytabled _ _ %the ({A p} {B p} search (atom A and (atom B) imp (atom B and (atom A)))) D %. +%querytabled _ _ %the ({A p} {B p} search (atom A imp (atom B) imp (atom A))) D %. +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A imp (atom B) imp (atom C) imp (atom A and (atom B) imp (atom C)))) D %. +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A and (atom B) imp (atom C) imp (atom A imp (atom B) imp (atom C)))) D %. +% %querytabled * * D : {A:p}{B:p}{C:p}search (((atom A) imp (atom B) imp (atom C)) imp (((atom A) imp (atom B)) imp (atom A) imp (atom C))). +% proof of S +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A imp (atom B) imp (atom C) imp (atom A imp (atom B)) imp (atom A) imp (atom C))) D %. +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A and (atom B) and (atom C) imp (atom A and (atom B and (atom C))))) D %. +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A and (atom B and (atom C)) imp (atom A and (atom B) and (atom C)))) D %. +% impDef +% +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A and (atom B) or (atom C) imp (atom A or (atom C) and (atom B or (atom C))))) D %. +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A and (atom B or (atom C)) imp (atom A and (atom B) or (atom A and (atom C))))) D %. +% new from (atom A)TP notes +% 2 +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A imp (atom B and (atom C)) imp (atom A imp (atom B) and (atom A imp (atom C))))) D %. +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A imp (atom B) and (atom A imp (atom C)) imp (atom A imp (atom B and (atom C))))) D %. +% 3 +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A imp (atom B or (atom C)) imp (atom A imp (atom B) or (atom A imp (atom C))))) D %. +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A imp (atom B) or (atom A imp (atom C)) imp (atom A imp (atom B or (atom C))))) D %. +% 4 +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A imp (atom B) imp (atom C) imp (atom A or (atom C) and (atom B imp (atom C))))) D %. +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A or (atom C) and (atom B imp (atom C)) imp (atom A imp (atom B) imp (atom C)))) D %. +% 5 +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A or (atom B) imp (atom C) imp (atom A imp (atom C) and (atom B imp (atom C))))) D %. +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A imp (atom C) and (atom B imp (atom C)) imp (atom A or (atom B) imp (atom C)))) D %. +% 6 +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A and (atom B or (atom C)) imp (atom A and (atom B) or (atom A and (atom C))))) D %. +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A and (atom B) or (atom A and (atom C)) imp (atom A and (atom B or (atom C))))) D %. +% 7 +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A or (atom B and (atom C)) imp (atom A or (atom B) and (atom A or (atom C))))) D %. +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A or (atom B) and (atom A or (atom C)) imp (atom A or (atom B and (atom C))))) D %. +% Id : (atom A) => (atom A) +%querytabled _ _ %the ({A p} search (atom A imp (atom A))) D %. +% K : (atom A) => (atom B) => (atom A) +%querytabled _ _ %the ({A p} {B p} search (atom A imp (atom B) imp (atom A))) D %. +% S : ((atom A) => (atom B) => (atom C)) => ((atom A) => (atom B)) => ((atom A) => (atom C)) +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A imp (atom B imp (atom C)) imp (atom A imp (atom B) imp (atom A imp (atom C))))) D %. +% ((atom A) & (atom B) => (atom C)) <=> ((atom A) => (atom B) => (atom C)) +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A and (atom B) imp (atom C) imp (atom A imp (atom B imp (atom C))))) D %. +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A imp (atom B imp (atom C)) imp (atom A and (atom B) imp (atom C)))) D %. +% ((atom A) => (atom B) & (atom C)) <=> ((atom A) => (atom B)) & ((atom A) => (atom C)) +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A imp (atom B and (atom C)) imp (atom A imp (atom B)) and (atom A imp (atom C)))) D %. +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A imp (atom B) and (atom A imp (atom C)) imp (atom A imp (atom B and (atom C))))) D %. +% ((atom A) => T) <=> T +%querytabled _ _ %the ({A p} search (atom A imp true imp true)) D %. +%querytabled _ _ %the ({A p} search (true imp (atom A imp true))) D %. +% ((atom A) v (atom B) => (atom C)) <=> ((atom A) => (atom C)) & ((atom B) => (atom C)) +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A or (atom B) imp (atom C) imp (atom A imp (atom C) and (atom B imp (atom C))))) D %. +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A imp (atom C) and (atom B imp (atom C)) imp (atom A or (atom B) imp (atom C)))) D %. +% (F => (atom C)) <=> T +%querytabled _ _ %the ({C p} search (false imp (atom C) imp true)) D %. +%querytabled _ _ %the ({C p} search (true imp (false imp (atom C)))) D %. +% ((atom A) => (atom B) v (atom C)) =>((atom A) => (atom B)) v ((atom A) => (atom C)) +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A imp (atom B) or (atom A imp (atom C)) imp (atom A imp (atom B or (atom C))))) D %. +% F => (atom C) +%querytabled _ _ %the ({C p} search (false imp (atom C))) D %. +% (atom A) => (((atom A) => F) => F) +%querytabled _ _ %the ({A p} search (atom A imp (atom A imp false imp false))) D %. +% (((atom A) => F) & (atom A)) => (atom C) +%querytabled _ _ %the ({A p} {C p} search (atom A imp false and (atom A) imp (atom C))) D %. +% (atom A) & ~(atom B) => ~((atom A) => (atom B)) +%querytabled _ _ %the ({A p} {B p} search (atom A and (atom B imp false) imp (atom A imp (atom B) imp false))) D %. +% (atom A) & ((atom B) v (atom C)) <=> ((atom A) & (atom B)) v ((atom A) & (atom C)) +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A and (atom B or (atom C)) imp (atom A and (atom B) or (atom A and (atom C))))) D %. +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A and (atom B) or (atom A and (atom C)) imp (atom A and (atom B or (atom C))))) D %. +% (atom A) & F <=> F +%querytabled _ _ %the ({A p} search (atom A and false imp false)) D %. +%querytabled _ _ %the ({A p} search (false imp (atom A and false))) D %. +% (atom A) v ((atom B) & (atom C)) <=> ((atom A) v (atom B)) & ((atom A) v (atom C)) +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A or (atom B and (atom C)) imp (atom A or (atom B)) and (atom A or (atom C)))) D %. +%querytabled _ _ %the ({A p} {B p} {C p} search (atom A or (atom B) and (atom A or (atom C)) imp (atom A or (atom B and (atom C))))) D %. +%querytabled _ _ %the ({A p} search (atom A imp (atom A and (atom A)))) D %. +%querytabled _ _ %the ({A p} search (atom A and (atom A) imp (atom A))) D %. +%querytabled _ _ %the ({A p} search (atom A imp (atom A or (atom A)))) D %. +%querytabled _ _ %the ({A p} search (atom A or (atom A) imp (atom A))) D %. +% ~(atom A) v ~(atom B) => ~((atom A) & (atom B)) +%querytabled _ _ %the ({A p} {B p} search (atom A imp false or (atom B imp false) imp (atom A and (atom B) imp false))) D %. +% (atom A) => (atom A) v (atom B) +%querytabled _ _ %the ({A p} {B p} search (atom A imp (atom A or (atom B)))) D %. +% (atom B) => (atom A) v (atom B) +%querytabled _ _ %the ({A p} {B p} search (atom B imp (atom A or (atom B)))) D %. +% (atom A) & (atom B) => (atom A) +%querytabled _ _ %the ({A p} {B p} search (atom A and (atom B) imp (atom A))) D %. +% (atom A) & (atom B) => (atom B) +%querytabled _ _ %the ({A p} {B p} search (atom A and (atom B) imp (atom B))) D %. +% (atom A) v ~(atom A) => (((atom A) => (atom B)) => (atom A)) => (atom A) +%querytabled _ _ %the ({A p} {B p} search (atom A or (atom A imp false) imp (atom A imp (atom B) imp (atom A) imp (atom A)))) D %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/seqCalc/tab-fol.lf b/new-tests/stelf-output/examples/tabled/seqCalc/tab-fol.lf new file mode 100644 index 0000000..8e3380c --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/seqCalc/tab-fol.lf @@ -0,0 +1,4 @@ +%require %[ formulas %] %. +%require %[ fol-int-at %] %. +%require %[ tab-ex-all %] %. +%require %[ tab-ex-np %] %. diff --git a/new-tests/stelf-output/examples/tabled/seqCalc/tab.lf b/new-tests/stelf-output/examples/tabled/seqCalc/tab.lf new file mode 100644 index 0000000..1f655ae --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/seqCalc/tab.lf @@ -0,0 +1,4 @@ +%require %[ formulas %] %. +%require %[ int %] %. +%require %[ tab-ex-all %] %. +%require %[ tab-ex-np %] %. diff --git a/new-tests/stelf-output/examples/tabled/subtype/mini-ml.lf b/new-tests/stelf-output/examples/tabled/subtype/mini-ml.lf new file mode 100644 index 0000000..dbea8ee --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/subtype/mini-ml.lf @@ -0,0 +1,16 @@ +%%% The Mini-ML Language +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +% Expressions +%sort exp %. +%name exp %. +%term z exp %. +%term s %pi exp %-> exp %. +%term case %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%term pair %pi exp %-> exp %-> exp %. +%term fst %pi exp %-> exp %. +%term snd %pi exp %-> exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term letv %pi exp %-> (%pi exp %-> exp) %-> exp %. +%term letn %pi exp %-> (%pi exp %-> exp) %-> exp %. +%term fix %pi (%pi exp %-> exp) %-> exp %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/subtype/stelf.toml b/new-tests/stelf-output/examples/tabled/subtype/stelf.toml new file mode 100644 index 0000000..1205615 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/subtype/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "tab" +main = "tab.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/tabled/subtype/subtype.lf b/new-tests/stelf-output/examples/tabled/subtype/subtype.lf new file mode 100644 index 0000000..935a4fa --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/subtype/subtype.lf @@ -0,0 +1,80 @@ +%%% Mini-ML types. +%%% Extended with some atomic subtyping +%%% Author: Frank Pfenning +%sort tp %. +%name tp %. +%term nat tp %. +% Natural Numbers +%term pos tp %. +% Positive Numbers +%term zero tp %. +% Singleton Zero +%term cross %pi tp %-> tp %-> tp %. +% Pairs +%term arrow %pi tp %-> tp %-> tp %. +% Functions +%define * cross %. +%prec %right 10 * %. +%define => arrow %. +%prec %right 11 => %. +%%% Subtyping +%%% Specification, not executable +%sort sub %. +%name sub %. +%mode sub %star %star %. +%tabled sub %. +%term sub_pn sub pos nat %. +%term sub_zn sub zero nat %. +%term sub_=> %pi (sub (T1 => T2) (S1 => S2)) %<- (sub S1 T1) %<- (sub T2 S2) %. +%term sub_refl sub T T %. +%term sub_trans %pi (sub S R) %<- (sub S T) %<- (sub T R) %. +% sub_trans : sub S T -> sub T R -> sub S R. +%{ +sub_* : sub T1 S1 + -> sub T2 S2 + -> sub (T1 * T2) (S1 * S2). +}% +% worlds () (sub T S). +% + +%% Typing rules +%%% Specification, not executable +%sort of %. +%name of %. +%tabled of %. +% Subtyping +%term tp_sub %pi (of E T) %<- (of E T') %<- (sub T' T) %. +% Natural Numbers +% Standard rules +%term tp_z_nat of z nat %. +%term tp_s_nat %pi (of (s E) nat) %<- (of E nat) %. +%term tp_case_nat + %pi (of (case E1 E2 E3) T) + %<- (of E1 nat) + %<- (of E2 T) + %<- ({x exp} %pi (of x nat) %-> (of (E3 x) T)) %. +% New rules +%term tp_z_zero of z zero %. +%term tp_s_pos %pi (of (s E) pos) %<- (of E nat) %. +%term tp_case_zero %pi (of (case E1 E2 E3) T) %<- (of E1 zero) %<- (of E2 T) %. +%term tp_case_pos + %pi (of (case E1 E2 E3) T) + %<- (of E1 pos) + %<- ({x exp} %pi (of x nat) %-> (of (E3 x) T)) %. +% Pairs +%term tp_pair %pi (of (pair E1 E2) (cross T1 T2)) %<- (of E1 T1) %<- (of E2 T2) %. +%term tp_fst %pi (of (fst E) T1) %<- (of E (cross T1 T2)) %. +%term tp_snd %pi (of (snd E) T2) %<- (of E (cross T1 T2)) %. +% Functions +%term tp_lam %pi (of (lam E) (arrow T1 T2)) %<- ({x exp} %pi (of x T1) %-> (of (E x) T2)) %. +%term tp_app %pi (of (app E1 E2) T1) %<- (of E1 (arrow T2 T1)) %<- (of E2 T2) %. +% Definitions +%term tp_letv + %pi (of (letv E1 E2) T2) + %<- (of E1 T1) + %<- ({x exp} %pi (of x T1) %-> (of (E2 x) T2)) %. +%term tp_letn %pi (of (letn E1 E2) T2) %<- (of E1 T1) %<- (of (E2 E1) T2) %. +% Recursion +%term tp_fix %pi (of (fix E) T) %<- ({x exp} %pi (of x T) %-> (of (E x) T)) %. +% worlds (some {T:tp} pi {x:exp} {d:of x T}) (of E T). +% covers of +E *T. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/subtype/tab-examples-quy.lf b/new-tests/stelf-output/examples/tabled/subtype/tab-examples-quy.lf new file mode 100644 index 0000000..cd69bc1 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/subtype/tab-examples-quy.lf @@ -0,0 +1,18 @@ +% Tabled evaluation +%querytabled _ 4 of z T %. +%querytabled _ 5 of z nat %. +% use termDepth = 7 +% querytabled * 2 (of (lam [x] s x) (nat => pos)). +% use termDepth = 10 +% querytabled * 5 (of (lam [x] case x (fst z) ([x'] x')) T). +% querytabled * 2 (of (lam [x] case x z ([x'] x')) (zero => zero)). +% querytabled * 2 (of (lam [x] case x z ([x'] x')) (pos => nat)). +% querytabled * 2 (of (lam [x] case x z ([x'] x')) T). +%{ +% these query will not terminate + +%querytabled * 2 (sub (arrow S S) (arrow zero zero)). +%querytabled * 2 (of (lam [x] x) (zero => zero)). +%querytabled * 2 (of (lam [x] x) T). + +}% \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/subtype/tab.lf b/new-tests/stelf-output/examples/tabled/subtype/tab.lf new file mode 100644 index 0000000..2257f8e --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/subtype/tab.lf @@ -0,0 +1,3 @@ +%require %[ mini-ml %] %. +%require %[ subtype %] %. +%require %[ tab-examples %] %. diff --git a/new-tests/stelf-output/examples/tabled/subtype1/basic.lf b/new-tests/stelf-output/examples/tabled/subtype1/basic.lf new file mode 100644 index 0000000..acff6c7 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/subtype1/basic.lf @@ -0,0 +1,52 @@ +%%% Mini-ML types. +%%% Extended with some atomic subtyping +%%% Author: Frank Pfenning +%sort tp %. +%name tp %. +%term nat tp %. +% Natural Numbers +%term pos tp %. +% Positive Numbers +%term zero tp %. +% Singleton Zero +%term int tp %. +%term neg tp %. +%%% Subtyping +%%% Specification, not executable +%sort sub %. +%name sub %. +%mode sub %star %star %. +%tabled sub %. +%term sub_refl sub T T %. +%term sub_trans %pi (sub S R) %<- (sub T R) %<- (sub S T) %. +% sub_trans : sub S T -> sub T R -> sub S R. +%term sub_pn sub pos nat %. +%term sub_zn sub zero nat %. +%term sub_nan sub nat int %. +%term sub_nen sub neg int %. +% + +%% Typing rules +%%% Specification, not executable +%sort of %. +%name of %. +%tabled of %. +% Subtyping +%term tp_sub %pi (of E T) %<- (of E T') %<- (sub T' T) %. +% Natural Numbers +% Standard rules +%term tp_z_nat of z nat %. +%term tp_s_nat %pi (of (s E) nat) %<- (of E nat) %. +%term tp_case_nat + %pi (of (case E1 E2 E3) T) + %<- (of E1 nat) + %<- (of E2 T) + %<- ({x exp} %pi (of x nat) %-> (of (E3 x) T)) %. +% New rules +%term tp_z_zero of z zero %. +%term tp_s_pos %pi (of (s E) pos) %<- (of E nat) %. +%term tp_case_zero %pi (of (case E1 E2 E3) T) %<- (of E1 zero) %<- (of E2 T) %. +%term tp_case_pos + %pi (of (case E1 E2 E3) T) + %<- (of E1 pos) + %<- ({x exp} %pi (of x nat) %-> (of (E3 x) T)) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/subtype1/mini-ml.lf b/new-tests/stelf-output/examples/tabled/subtype1/mini-ml.lf new file mode 100644 index 0000000..dbea8ee --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/subtype1/mini-ml.lf @@ -0,0 +1,16 @@ +%%% The Mini-ML Language +%%% Author: Frank Pfenning, based on [Michaylov & Pfenning 92] +% Expressions +%sort exp %. +%name exp %. +%term z exp %. +%term s %pi exp %-> exp %. +%term case %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%term pair %pi exp %-> exp %-> exp %. +%term fst %pi exp %-> exp %. +%term snd %pi exp %-> exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term letv %pi exp %-> (%pi exp %-> exp) %-> exp %. +%term letn %pi exp %-> (%pi exp %-> exp) %-> exp %. +%term fix %pi (%pi exp %-> exp) %-> exp %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/subtype1/stelf.toml b/new-tests/stelf-output/examples/tabled/subtype1/stelf.toml new file mode 100644 index 0000000..1205615 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/subtype1/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "tab" +main = "tab.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/tabled/subtype1/tab-examples-quy.lf b/new-tests/stelf-output/examples/tabled/subtype1/tab-examples-quy.lf new file mode 100644 index 0000000..2b42444 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/subtype1/tab-examples-quy.lf @@ -0,0 +1,4 @@ +% terminates after 3 stages (variant) +% terminates after 3 stages (subsumption) +%querytabled _ 6 of z T %. +%querytabled _ 7 of (case z z ([x] x)) T %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/subtype1/tab.lf b/new-tests/stelf-output/examples/tabled/subtype1/tab.lf new file mode 100644 index 0000000..9232f5f --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/subtype1/tab.lf @@ -0,0 +1,3 @@ +%require %[ mini-ml %] %. +%require %[ basic %] %. +%require %[ tab-examples %] %. diff --git a/new-tests/stelf-output/examples/tabled/tests/stelf.toml b/new-tests/stelf-output/examples/tabled/tests/stelf.toml new file mode 100644 index 0000000..1205615 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/tests/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "tab" +main = "tab.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/tabled/tests/tab.lf b/new-tests/stelf-output/examples/tabled/tests/tab.lf new file mode 100644 index 0000000..3bd09a9 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/tests/tab.lf @@ -0,0 +1,7 @@ +%require %[ test1 %] %. +%require %[ test2 %] %. +%require %[ test3 %] %. +%require %[ test4 %] %. +%require %[ test5 %] %. +%require %[ test6 %] %. +%require %[ test7 %] %. diff --git a/new-tests/stelf-output/examples/tabled/tests/test1.lf b/new-tests/stelf-output/examples/tabled/tests/test1.lf new file mode 100644 index 0000000..fd8d9ea --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/tests/test1.lf @@ -0,0 +1,15 @@ +%sort node %. +%term a node %. +%term b node %. +%term c node %. +%term d node %. +%sort edge %. +%term e_ab edge a b %. +%term e_ac edge a c %. +%term e_ba edge b a %. +%term e_bd edge b d %. +%sort reach %. +%tabled reach %. +%term r_refl reach X X %. +%term r_cl %pi (reach X Y) %<- (edge X Z) %<- (reach Z Y) %. +%querytabled _ 3 reach a X %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/tests/test2.lf b/new-tests/stelf-output/examples/tabled/tests/test2.lf new file mode 100644 index 0000000..89c2abe --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/tests/test2.lf @@ -0,0 +1,15 @@ +%sort node %. +%term a node %. +%term b node %. +%term c node %. +%term d node %. +%sort edge %. +%term e_ab edge a b %. +%term e_ac edge a c %. +%term e_ba edge b a %. +%term e_bd edge b d %. +%sort reach' %. +%tabled reach' %. +%term r_cl' %pi (reach' X Y) %<- (reach' X Z) %<- (edge Z Y) %. +%term r_refl' reach' X X %. +%querytabled _ 5 reach' a X %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/tests/test3.lf b/new-tests/stelf-output/examples/tabled/tests/test3.lf new file mode 100644 index 0000000..f4a6d9d --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/tests/test3.lf @@ -0,0 +1,15 @@ +%sort node %. +%term a node %. +%term b node %. +%sort p %. +%tabled p %. +%term p1 p a %. +%term p2 %pi (p a) %<- (p b) %. +%term p3 %pi (p b) %<- (p a) %. +%querytabled _ 5 %the (p a) A %. +%sort q %. +%tabled q %. +%term q2 %pi (q a) %<- (q b) %. +%term q3 %pi (q b) %<- (q a) %. +%term q1 q a %. +%querytabled _ 5 %the (q a) A %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/tests/test4.lf b/new-tests/stelf-output/examples/tabled/tests/test4.lf new file mode 100644 index 0000000..7a437cc --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/tests/test4.lf @@ -0,0 +1,10 @@ +%sort node %. +%term a node %. +%term b node %. +%sort r %. +%sort s %. +%tabled s %. +%term r2 %pi (r a) %<- (s b) %. +%term s3 %pi (s b) %<- (s a) %. +%term q1 s a %. +%querytabled _ 2 %the (r a) A %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/tests/test5.lf b/new-tests/stelf-output/examples/tabled/tests/test5.lf new file mode 100644 index 0000000..33b85b4 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/tests/test5.lf @@ -0,0 +1,20 @@ +%%%%%%%%% +%% Sagonas, Swift +%sort node %. +%term a node %. +%term b node %. +%term c node %. +%term d node %. +%sort pt %. +%sort et %. +%sort qt %. +%tabled pt %. +%term p2 %pi (pt X Z) %<- (et X Z) %<- (qt Z) %. +%term p1 %pi (pt X Z) %<- (pt X Y) %<- (pt Y Z) %. +%term e1 et a b %. +%term e2 et a d %. +%term e3 et b c %. +%term q1 qt a %. +%term q2 qt b %. +%term q3 qt c %. +%querytabled _ 6 pt a X %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/tests/test6.lf b/new-tests/stelf-output/examples/tabled/tests/test6.lf new file mode 100644 index 0000000..3219cd4 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/tests/test6.lf @@ -0,0 +1,15 @@ +%%%%%%%%%%%%%%%%%% +%%% Tamaki, Sato +%%% +%sort nat %. +%term 0 nat %. +%term suc %pi nat %-> nat %. +%sort pts %. +%sort qts %. +%sort rts %. +%tabled qts %. +%term c1 %pi (pts X) %<- (qts X) %<- rts %. +%term c2 %pi (qts (suc X)) %<- (qts X) %. +%term c3 qts 0 %. +%term c4 rts %. +%querytabled _ 10 pts X %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tabled/tests/test7.lf b/new-tests/stelf-output/examples/tabled/tests/test7.lf new file mode 100644 index 0000000..5134524 --- /dev/null +++ b/new-tests/stelf-output/examples/tabled/tests/test7.lf @@ -0,0 +1,23 @@ +%sort node %. +%term a node %. +%term b node %. +%term c node %. +%sort p %. +%term p1 p a %. +%term p2 %pi (p a) %<- (%pi (p c) %-> (p b)) %. +%term p3 %pi (p b) %<- (p c) %. +%querytabled _ 5 %the (p a) A %. +%sort q %. +%sort r %. +%sort s %. +%tabled q %. +%term qr2 %pi (q a) %<- (%pi (r a) %-> (r b) %-> (q a)) %. +%term qs2 %pi (q a) %<- (%pi (s a) %-> (s b) %-> (q a)) %. +%term q2 %pi (q a) %<- (q b) %. +%term q_ba %pi (q b) %<- (q a) %. +%term q_brc %pi (q b) %<- (r c) %. +%term q1 q a %. +%term r1 %pi (r c) %<- (r a) %<- (r b) %. +%term s1 s a %. +%term s2 s b %. +% querytabled * 5 A:(q a). \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tapl-ch13/defs.lf b/new-tests/stelf-output/examples/tapl-ch13/defs.lf new file mode 100644 index 0000000..2939808 --- /dev/null +++ b/new-tests/stelf-output/examples/tapl-ch13/defs.lf @@ -0,0 +1,207 @@ +%{ + A formalization of Chapter 13 from _Types and Programming Languages_: + the simply typed lambda-calculus with references. Only basic definitions + are included here. + + Author: Brian E. Aydemir (baydemir [at] cis.upenn.edu). +}% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Types. +%sort tp %. +%name tp %. +%term => %pi tp %-> tp %-> tp %. +%% type of functions. +%term unit tp %. +%% unit type. +%term ref %pi tp %-> tp %. +%% type of reference cells. +%name tp %. +%prec %right 5 => %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Labels (isomorphic to the natural numbers). +%sort label %. +%name label %. +%term lbl %pi nat %-> label %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Expressions. +%sort exp %. +%name exp %. +%term @ %pi exp %-> exp %-> exp %. +%% application. +%term lam %pi tp %-> (%pi exp %-> exp) %-> exp %. +%% lambda abstraction. +%term dot exp %. +%% constant of type unit. +%term alloc %pi exp %-> exp %. +%% reference cell creation (ref). +%term deref %pi exp %-> exp %. +%% dereference a reference cell (!). +%term gets %pi exp %-> exp %-> exp %. +%% assignment (:=). +%term loc %pi label %-> exp %. +%% locations. +%name exp %. +%prec %left 9999 @ %. +%prec %prefix 9000 deref %. +%prec %none 8000 gets %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Values. +%sort value %. +%name value %. +%mode value %in %. +%term v_lam value (lam _ _) %. +%term v_dot value dot %. +%term v_loc value (loc _) %. +%terminates {} (value _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store typings. +%%% Represented by length-indexed lists. +%{ + Store typings are represented by length-indexed lists, starting from zero + (z). Thus in the list "store_cons T1 (store_cons T2 store_nil)" index z + corresponds to T1 and index (s z) corresponds to T2. +}% +%sort store %. +%name store %. +%term store_nil store %. +%term store_cons %pi tp %-> store %-> store %. +%%% Compute the length of a store typing. +%sort length_store %. +%mode length_store %in %out %. +%term length_store_nil length_store store_nil z %. +%term length_store_cons %pi (length_store (store_cons _ S) (s N)) %<- (length_store S N) %. +%worlds () (length_store _ _) %. +%total S (length_store S _) %. +%unique length_store %in %out %. +%%% Look up a label in a store typing. +%sort find_in_store %. +%mode find_in_store %in %in %out %. +%term find_in_store_yes find_in_store (lbl z) (store_cons T _) T %. +%term find_in_store_no + %pi (find_in_store (lbl (s N)) (store_cons _ S) T) + %<- (find_in_store (lbl N) S T) %. +%worlds () (find_in_store _ _ _) %. +%terminates S (find_in_store _ S _) %. +%unique find_in_store %in %in %out %. +%%% (store_extends S S') holds if S' extends S. +%sort store_extends %. +%mode store_extends %in %in %. +%term store_extends_base store_extends store_nil S %. +%term store_extends_ind %pi (store_extends (store_cons T S1) (store_cons T S2)) %<- (store_extends S1 S2) %. +%terminates S (store_extends S _) %. +%%% Append a type to a store typing, returning the new store typing. +%sort append_store %. +%mode append_store %in %in %out %. +%term append_store_nil append_store store_nil T (store_cons T store_nil) %. +%term append_store_cons + %pi (append_store (store_cons T1 S) T2 (store_cons T1 S')) + %<- (append_store S T2 S') %. +%worlds () (append_store _ _ _) %. +%total S (append_store S _ _) %. +%unique append_store %in %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Typing relation. +%{ + Typing assumptions for variables are represented by the var judgment. We + use this auxiliary judgment since assumptions in the typing environment + do not mention store typings, and hence we shouldn't confuse them with + typing judgments. +}% +%sort of %. +%name of %. +%mode of %in %in %star %. +%sort var %. +%name var %. +%mode var %in %star %. +%freeze var %. +%term t_var %pi (of S E T) %<- (var E T) %. +%term t_abs %pi (of S (lam T1 E) (T1 => T2)) %<- ({x exp} %pi (var x T1) %-> (of S (E x) T2)) %. +%term t_app %pi (of S (E1 @ E2) T2) %<- (of S E1 (T1 => T2)) %<- (of S E2 T1) %. +%term t_unit of S dot unit %. +%term t_loc %pi (of S (loc L) (ref T)) %<- (find_in_store L S T) %. +%term t_ref %pi (of S (alloc E) (ref T)) %<- (of S E T) %. +%term t_deref %pi (of S (deref E) T) %<- (of S E (ref T)) %. +%term t_assign %pi (of S (E1 gets E2) unit) %<- (of S E1 (ref T)) %<- (of S E2 T) %. +%terminates E (var E _) %. +%terminates E (of _ E _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Heaps. +%%% Represented by length-indexed lists (similar to store typings). +%sort heap %. +%name heap %. +%term heap_nil heap %. +%term heap_cons %pi exp %-> heap %-> heap %. +%%% Compute the length of a heap. +%sort length_heap %. +%mode length_heap %in %out %. +%term length_heap_nil length_heap heap_nil z %. +%term length_heap_cons %pi (length_heap (heap_cons _ H) (s N)) %<- (length_heap H N) %. +%worlds () (length_heap _ _) %. +%total H (length_heap H _) %. +%unique length_heap %in %out %. +%%% Look up a label in a heap. +%sort find_in_heap %. +%mode find_in_heap %in %in %out %. +%term find_in_heap_yes find_in_heap (lbl z) (heap_cons E _) E %. +%term find_in_heap_no %pi (find_in_heap (lbl (s N)) (heap_cons _ H) E) %<- (find_in_heap (lbl N) H E) %. +%worlds () (find_in_heap _ _ _) %. +%terminates H (find_in_heap _ H _) %. +%unique find_in_heap %in %in %out %. +%%% Replace the binding for a label in a heap. +%sort replace_in_heap %. +%mode replace_in_heap %in %in %in %out %. +%term replace_in_heap_yes replace_in_heap (heap_cons E1 H) (lbl z) E2 (heap_cons E2 H) %. +%term replace_in_heap_no + %pi (replace_in_heap (heap_cons E1 H) (lbl (s N)) E2 (heap_cons E1 H')) + %<- (replace_in_heap H (lbl N) E2 H') %. +%worlds () (replace_in_heap _ _ _ _) %. +%terminates H (replace_in_heap H _ _ _) %. +%unique replace_in_heap %in %in %in %out %. +%%% Append an expression to a heap, returning the new heap. +%sort append_heap %. +%mode append_heap %in %in %out %. +%term append_heap_nil append_heap heap_nil E (heap_cons E heap_nil) %. +%term append_heap_cons %pi (append_heap (heap_cons E1 H) E2 (heap_cons E1 H')) %<- (append_heap H E2 H') %. +%worlds () (append_heap _ _ _) %. +%total H (append_heap H _ _) %. +%unique append_heap %in %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Well-typed heaps (with respect to a store typing). +%%% This auxiliary judgments walks down a store typing and heap (in order to +%%% ensure that they have the same domain) and uses the first store typing +%%% to check that expressions in the heap are well-typed. +%sort check_wt %. +%mode check_wt %in %in %in %. +%term check_wt_nil check_wt _ store_nil heap_nil %. +%term check_wt_cons + %pi (check_wt S1 (store_cons T S2) (heap_cons E H)) + %<- (of S1 E T) + %<- (check_wt S1 S2 H) %. +%terminates H (check_wt _ _ H) %. +%%% (wt_heap S H) holds if H is well-typed with respect to S. +%sort wt_heap %. +%mode wt_heap %in %in %. +%term wt_heap_def %pi (wt_heap S H) %<- (check_wt S S H) %. +%terminates {} (wt_heap _ _) %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Evaluation relation. +%sort step %. +%name step %. +%mode step %in %in %out %out %. +%%% Congruence rules. +%term e_app1 %pi (step H (E1 @ E2) H' (E1' @ E2)) %<- (step H E1 H' E1') %. +%term e_app2 %pi (step H (E1 @ E2) H' (E1 @ E2')) %<- (value E1) %<- (step H E2 H' E2') %. +%term e_alloc %pi (step H (alloc E) H' (alloc E')) %<- (step H E H' E') %. +%term e_deref %pi (step H (deref E) H' (deref E')) %<- (step H E H' E') %. +%term e_gets1 %pi (step H (E1 gets E2) H' (E1' gets E2)) %<- (step H E1 H' E1') %. +%term e_gets2 %pi (step H (E1 gets E2) H' (E1 gets E2')) %<- (value E1) %<- (step H E2 H' E2') %. +%%% Computation rules. +%term e_appabs %pi (step H (lam _ E1 @ E2) H (E1 E2)) %<- (value E2) %. +%term e_allocVal + %pi (step H (alloc E) H' (loc (lbl N'))) + %<- (value E) + %<- (append_heap H E H') + %<- (length_heap H N') %. +%term e_derefVal %pi (step H (deref (loc L)) H E) %<- (find_in_heap L H E) %. +%term e_getsVal %pi (step H (loc L gets E) H' dot) %<- (value E) %<- (replace_in_heap H L E H') %. +%terminates E (step _ E _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tapl-ch13/nat.lf b/new-tests/stelf-output/examples/tapl-ch13/nat.lf new file mode 100644 index 0000000..af23ed6 --- /dev/null +++ b/new-tests/stelf-output/examples/tapl-ch13/nat.lf @@ -0,0 +1,30 @@ +%{ + Formalization of the natural numbers. + + Author: Brian E. Aydemir (baydemir [at] cis.upenn.edu) +}% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Syntax. +%sort nat %. +%name nat %. +%term z nat %. +%% zero. +%term s %pi nat %-> nat %. +%% successor. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% (In)Equality. +%sort nat_eq %. +%mode nat_eq %in %in %. +%sort nat_neq %. +%mode nat_neq %in %in %. +%sort nat_lt %. +%mode nat_lt %in %in %. +%term neq_eq_refl nat_eq N N %. +%term nat_neq_zs nat_neq z (s N) %. +%term nat_neq_sz nat_neq (s N) z %. +%term nat_neq_ss %pi (nat_neq (s N1) (s N2)) %<- (nat_neq N1 N2) %. +%term nat_lt_zs nat_lt z (s N) %. +%term nat_lt_ss %pi (nat_lt (s N1) (s N2)) %<- (nat_lt N1 N2) %. +%terminates {} (nat_eq _ _) %. +%terminates N (nat_neq N _) %. +%terminates N (nat_lt N _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/examples/tapl-ch13/sources.lf b/new-tests/stelf-output/examples/tapl-ch13/sources.lf new file mode 100644 index 0000000..34f21bb --- /dev/null +++ b/new-tests/stelf-output/examples/tapl-ch13/sources.lf @@ -0,0 +1,3 @@ +%require %[ nat %] %. +%require %[ defs %] %. +%require %[ theorems %] %. diff --git a/new-tests/stelf-output/examples/tapl-ch13/stelf.toml b/new-tests/stelf-output/examples/tapl-ch13/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/examples/tapl-ch13/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/examples/tapl-ch13/theorems.lf b/new-tests/stelf-output/examples/tapl-ch13/theorems.lf new file mode 100644 index 0000000..8b0ad33 --- /dev/null +++ b/new-tests/stelf-output/examples/tapl-ch13/theorems.lf @@ -0,0 +1,403 @@ +%{ + A formalization of Chapter 13 from _Types and Programming Languages_: + the simply typed lambda-calculus with references. Only the progress and + preservation theorems are included here (with associated lemmas). + + Author: Brian E. Aydemir (baydemir [at] cis.upenn.edu). +}% +%%% The one block of interest here. +%block bind {t tp} [x exp] [d var x t]%. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Store typings reflexively extend themselves. +%sort store_extends_refl {S} %. +%mode store_extends_refl %in %out %. +%term _ store_extends_refl store_nil store_extends_base %. +%term _ + %pi (store_extends_refl (store_cons _ S) (store_extends_ind P)) + %<- (store_extends_refl S P) %. +%worlds () (store_extends_refl _ _) %. +%total S (store_extends_refl S _) %. +%unique store_extends_refl %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% If a heap is well-typed with respect to a store typing, then the length +%%% of the store typing is the same as the length of the heap. +%sort wt_same_length_store_check %. +%mode wt_same_length_store_check %in %in %out %. +%term _ wt_same_length_store_check check_wt_nil length_heap_nil length_store_nil %. +%term _ + %pi (wt_same_length_store_check (check_wt_cons C _) (length_heap_cons H) (length_store_cons S)) + %<- (wt_same_length_store_check C H S) %. +%worlds (bind) (wt_same_length_store_check _ _ _) %. +%total W (wt_same_length_store_check W _ _) %. +%unique wt_same_length_store_check %in %in %out %. +%sort wt_same_length_store %. +%mode wt_same_length_store %in %in %out %. +%term _ + %pi (wt_same_length_store (wt_heap_def C) L1 L2) + %<- (wt_same_length_store_check C L1 L2) %. +%worlds (bind) (wt_same_length_store _ _ _) %. +%total W (wt_same_length_store W _ _) %. +%unique wt_same_length_store %in %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% It is always possible to append a type to a store typing. +%sort can_append_store {S} {T} %. +%mode can_append_store %in %in %out %. +%term _ can_append_store store_nil T append_store_nil %. +%term _ + %pi (can_append_store (store_cons T1 S) T2 (append_store_cons A)) + %<- (can_append_store S T2 A) %. +%worlds () (can_append_store _ _ _) %. +%total S (can_append_store S _ _) %. +%unique can_append_store %in %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Appending something to a store typing does indeed extend it. +%sort does_extend %. +%mode does_extend %in %out %. +%term _ does_extend append_store_nil store_extends_base %. +%term _ + %pi (does_extend (append_store_cons A) (store_extends_ind E)) + %<- (does_extend A E) %. +%worlds () (does_extend _ _) %. +%total A (does_extend A _) %. +%unique does_extend %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% It is always possible to compute the length of a heap. +%sort can_length_heap {H} %. +%mode can_length_heap %in %out %. +%term _ can_length_heap heap_nil length_heap_nil %. +%term _ + %pi (can_length_heap (heap_cons _ H) (length_heap_cons L)) + %<- (can_length_heap H L) %. +%worlds () (can_length_heap _ _) %. +%total H (can_length_heap H _) %. +%unique can_length_heap %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% It is always possible to append an expression to a heap. +%sort can_append_heap {H} {E} %. +%mode can_append_heap %in %in %out %. +%term _ can_append_heap heap_nil E append_heap_nil %. +%term _ + %pi (can_append_heap (heap_cons E1 H) E2 (append_heap_cons A)) + %<- (can_append_heap H E2 A) %. +%worlds () (can_append_heap _ _ _) %. +%total H (can_append_heap H _ _) %. +%unique can_append_heap %in %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% We can always find the element we just appended to a store typing. +%sort append_store_findable %. +%mode append_store_findable %in %in %out %. +%term _ append_store_findable append_store_nil length_store_nil find_in_store_yes %. +%term _ + %pi (append_store_findable (append_store_cons A) (length_store_cons L) (find_in_store_no F)) + %<- (append_store_findable A L F) %. +%worlds () (append_store_findable _ _ _) %. +%total A (append_store_findable A _ _) %. +%unique append_store_findable %in %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% We can always find the element we just appended to a heap. +%sort append_heap_findable %. +%mode append_heap_findable %in %in %out %. +%term _ append_heap_findable append_heap_nil length_heap_nil find_in_heap_yes %. +%term _ + %pi (append_heap_findable (append_heap_cons A) (length_heap_cons L) (find_in_heap_no F)) + %<- (append_heap_findable A L F) %. +%worlds () (append_heap_findable _ _ _) %. +%total A (append_heap_findable A _ _) %. +%unique append_heap_findable %in %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% If a heap is well-typed with respect to a store typing, if we can find a +%%% label in the store typing, then we can find it in the heap. +%sort check_find_lemma %. +%mode check_find_lemma %in %in %out %. +%term _ check_find_lemma _ find_in_store_yes find_in_heap_yes %. +%term _ + %pi (check_find_lemma (check_wt_cons C _) (find_in_store_no F) (find_in_heap_no F')) + %<- (check_find_lemma C F F') %. +%worlds () (check_find_lemma _ _ _) %. +%total F (check_find_lemma _ F _) %. +%unique check_find_lemma %in %in %out %. +%sort find_lemma %. +%mode find_lemma %in %in %out %. +%term _ %pi (find_lemma (wt_heap_def C) F H) %<- (check_find_lemma C F H) %. +%worlds () (find_lemma _ _ _) %. +%total {} (find_lemma _ _ _) %. +%unique find_lemma %in %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% If a heap is well-typed with respect to a store typing, and if we can +%%% find a label in the store typing, then we can replace its binding in the +%%% heap. +%sort check_replace_lemma {E exp} %. +%mode check_replace_lemma %in %in %in %out %. +%term _ check_replace_lemma _ _ find_in_store_yes replace_in_heap_yes %. +%term _ + %pi (check_replace_lemma E (check_wt_cons C _) (find_in_store_no F) (replace_in_heap_no R)) + %<- (check_replace_lemma E C F R) %. +%worlds () (check_replace_lemma _ _ _ _) %. +%total F (check_replace_lemma _ _ F _) %. +%unique check_replace_lemma %in %in %in %out %. +%sort replace_lemma {E exp} %. +%mode replace_lemma %in %in %in %out %. +%term _ %pi (replace_lemma E (wt_heap_def C) F R) %<- (check_replace_lemma E C F R) %. +%worlds () (replace_lemma _ _ _ _) %. +%total {} (replace_lemma _ _ _ _) %. +%unique replace_lemma %in %in %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Basically prove an inversion lemma for wt_heap here that lets us recover +%%% typing judgments. +%sort check_wt_inv %. +%mode check_wt_inv %in %in %in %out %. +%term _ check_wt_inv (check_wt_cons _ Q) find_in_store_yes find_in_heap_yes Q %. +%term _ + %pi (check_wt_inv (check_wt_cons C _) (find_in_store_no Fstore) (find_in_heap_no Fheap) Q) + %<- (check_wt_inv C Fstore Fheap Q) %. +%worlds (bind) (check_wt_inv _ _ _ _) %. +%total F (check_wt_inv _ _ F _) %. +%unique check_wt_inv %in %in %in %out %. +%sort wt_heap_inv %. +%mode wt_heap_inv %in %in %in %out %. +%term _ + %pi (wt_heap_inv (wt_heap_def C) Fstore Fheap Q) + %<- (check_wt_inv C Fstore Fheap Q) %. +%worlds (bind) (wt_heap_inv _ _ _ _) %. +%total {} (wt_heap_inv _ _ _ _) %. +%unique wt_heap_inv %in %in %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Lemma 13.5.4 (substitution). And you thought we wouldn't need this... +%sort substitution {x exp} %. +%mode substitution %in %in %out %. +%term _ substitution ([x] [d] t_var d) Q Q %. +%term _ substitution ([x] [d] t_var D') _ (t_var D') %. +%term _ + %pi (substitution ([x exp] [d var x T1] t_abs ([y exp] [f var y T2] Q1 y f x d)) Q2 (t_abs Q1')) + %<- ({y exp} {f var y T2} substitution (Q1 y f) Q2 (Q1' y f)) %. +%term _ + %pi (substitution ([x] [d] t_app (Q1 x d) (Q2 x d)) Q3 (t_app Q1' Q2')) + %<- (substitution Q1 Q3 Q1') + %<- (substitution Q2 Q3 Q2') %. +%term _ substitution ([x] [d] t_unit) _ t_unit %. +%term _ substitution ([x] [d] t_loc L) _ (t_loc L) %. +%term _ + %pi (substitution ([x] [d] t_ref (Q1 x d)) Q2 (t_ref Q1')) + %<- (substitution Q1 Q2 Q1') %. +%term _ + %pi (substitution ([x] [d] t_deref (Q1 x d)) Q2 (t_deref Q1')) + %<- (substitution Q1 Q2 Q1') %. +%term _ + %pi (substitution ([x] [d] t_assign (Q1 x d) (Q2 x d)) Q3 (t_assign Q1' Q2')) + %<- (substitution Q1 Q3 Q1') + %<- (substitution Q2 Q3 Q2') %. +%worlds (bind) (substitution _ _ _) %. +%total D (substitution D _ _) %. +%unique substitution %in %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Lemma 13.5.6 (but first an auxiliary, "obvious" lemma). +%sort store_find_extends %. +%mode store_find_extends %in %in %out %. +%term _ store_find_extends find_in_store_yes (store_extends_ind X) find_in_store_yes %. +%term _ + %pi (store_find_extends (find_in_store_no F) (store_extends_ind X) (find_in_store_no F')) + %<- (store_find_extends F X F') %. +%worlds () (store_find_extends _ _ _) %. +%total F (store_find_extends F _ _) %. +%unique store_find_extends %in %in %out %. +%sort of_store_extends %. +%mode of_store_extends %in %in %out %. +%term _ + %pi (of_store_extends (t_abs Q) P (t_abs Q')) + %<- ({x exp} {d var x T} of_store_extends (Q x d) P (Q' x d)) %. +%term _ of_store_extends (t_var V) P (t_var V) %. +%term _ + %pi (of_store_extends (t_app Qarg Qfun) P (t_app Qarg' Qfun')) + %<- (of_store_extends Qarg P Qarg') + %<- (of_store_extends Qfun P Qfun') %. +%term _ of_store_extends t_unit P t_unit %. +%term _ %pi (of_store_extends (t_loc F) P (t_loc F')) %<- (store_find_extends F P F') %. +%term _ %pi (of_store_extends (t_ref Q) P (t_ref Q')) %<- (of_store_extends Q P Q') %. +%term _ %pi (of_store_extends (t_deref Q) P (t_deref Q')) %<- (of_store_extends Q P Q') %. +%term _ + %pi (of_store_extends (t_assign Q1 Q2) P (t_assign Q1' Q2')) + %<- (of_store_extends Q1 P Q1') + %<- (of_store_extends Q2 P Q2') %. +%worlds (bind) (of_store_extends _ _ _) %. +%total Q (of_store_extends Q _ _) %. +%unique of_store_extends %in %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% If a heap and related store-typing are extended appropriately, then we +%%% can extend the proof that they're well-typed as well. +%sort extend_check_wt_store %. +%mode extend_check_wt_store %in %in %out %. +%term _ extend_check_wt_store check_wt_nil _ check_wt_nil %. +%term _ + %pi (extend_check_wt_store (check_wt_cons C Q) P (check_wt_cons C' Q')) + %<- (of_store_extends Q P Q') + %<- (extend_check_wt_store C P C') %. +%worlds (bind) (extend_check_wt_store _ _ _) %. +%total W (extend_check_wt_store W _ _) %. +%unique extend_check_wt_store %in %in %out %. +%sort extend_check_wt %. +%mode extend_check_wt %in %in %in %in %out %. +%term _ extend_check_wt check_wt_nil Q append_store_nil append_heap_nil (check_wt_cons check_wt_nil Q) %. +%term _ + %pi (extend_check_wt (check_wt_cons C Qc) Q (append_store_cons S) (append_heap_cons H) (check_wt_cons C' Qc)) + %<- (extend_check_wt C Q S H C') %. +%worlds (bind) (extend_check_wt _ _ _ _ _) %. +%total W (extend_check_wt W _ _ _ _) %. +%unique extend_check_wt %in %in %in %in %out %. +%sort extend_wt_heap %. +%mode extend_wt_heap %in %in %in %in %out %. +%term _ + %pi (extend_wt_heap (wt_heap_def C) Q As Ah (wt_heap_def C'')) + %<- (extend_check_wt C Q As Ah C') + %<- (does_extend As P) + %<- (extend_check_wt_store C' P C'') %. +%worlds (bind) (extend_wt_heap _ _ _ _ _) %. +%total {} (extend_wt_heap _ _ _ _ _) %. +%unique extend_wt_heap %in %in %in %in %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Replacing an element of a heap can preserve its well-typedness with +%%% respect to a store. (Similar to Lemma 13.5.5.) +%sort assign_lemma_check %. +%mode assign_lemma_check %in %in %in %in %out %out %. +%term _ + %pi (assign_lemma_check (%the (check_wt _ (store_cons T S) _) (check_wt_cons C Qc)) Q find_in_store_yes replace_in_heap_yes P (check_wt_cons C Q)) + %<- (store_extends_refl (store_cons T S) P) %. +%term _ + %pi (assign_lemma_check (check_wt_cons C Qc) Q (find_in_store_no F) (replace_in_heap_no R) (store_extends_ind P) (check_wt_cons C' Qc)) + %<- (assign_lemma_check C Q F R P C') %. +%worlds (bind) (assign_lemma_check _ _ _ _ _ _) %. +%total F (assign_lemma_check _ _ F _ _ _) %. +%unique assign_lemma_check %in %in %in %in %out %out %. +%sort assign_lemma %. +%mode assign_lemma %in %in %in %in %out %out %. +%term _ + %pi (assign_lemma (wt_heap_def C) Q F R P' (wt_heap_def C'')) + %<- (assign_lemma_check C Q F R P' C') + %<- (extend_check_wt_store C' P' C'') %. +%worlds (bind) (assign_lemma _ _ _ _ _ _) %. +%total {} (assign_lemma _ _ _ _ _ _) %. +%unique assign_lemma %in %in %in %in %out %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Preservation. +%sort preservation %. +%mode preservation %in %in %in %out %out %out %. +%%% Easy stuff: Cases for congruence rules. +%term _ + %pi (preservation (t_app Qarg Qfun) W (e_app1 S) P' W' (t_app Qarg' Q')) + %<- (preservation Qfun W S P' W' Q') + %<- (of_store_extends Qarg P' Qarg') %. +%term _ + %pi (preservation (t_app Qarg Qfun) W (e_app2 S V) P' W' (t_app Q' Qfun')) + %<- (preservation Qarg W S P' W' Q') + %<- (of_store_extends Qfun P' Qfun') %. +%term _ + %pi (preservation (t_ref Q) W (e_alloc S) P' W' (t_ref Q')) + %<- (preservation Q W S P' W' Q') %. +%term _ + %pi (preservation (t_deref Q) W (e_deref S) P' W' (t_deref Q')) + %<- (preservation Q W S P' W' Q') %. +%term _ + %pi (preservation (t_assign Qv Ql) W (e_gets1 S) P' W' (t_assign Qv' Ql')) + %<- (preservation Ql W S P' W' Ql') + %<- (of_store_extends Qv P' Qv') %. +%term _ + %pi (preservation (t_assign Qv Ql) W (e_gets2 S V) P' W' (t_assign Qv' Ql')) + %<- (preservation Qv W S P' W' Qv') + %<- (of_store_extends Ql P' Ql') %. +%%% Cases for computation rules. +%term _ + %pi (preservation (t_app Qarg (t_abs Qbody)) (%the (wt_heap S _) W) (e_appabs V) P W Qres) + %<- (substitution Qbody Qarg Qres) + %<- (store_extends_refl S P) %. +%term _ + %pi (preservation (%the (of S (alloc E) (ref T)) (t_ref Q)) W (e_allocVal Lh Ah V) P W' (t_loc F')) + %<- (can_append_store S T As) + %<- (does_extend As P) + %<- (wt_same_length_store W Lh L) + %<- (append_store_findable As L F') + %<- (extend_wt_heap W Q As Ah W') %. +%term _ + %pi (preservation (t_deref (t_loc Fstore)) (%the (wt_heap S _) W) (e_derefVal Fheap) P W Qres) + %<- (wt_heap_inv W Fstore Fheap Qres) + %<- (store_extends_refl S P) %. +%term _ + %pi (preservation (t_assign Qval (t_loc F)) W (e_getsVal R V) P' W' t_unit) + %<- (assign_lemma W Qval F R P' W') %. +%%% No more cases. +%worlds (bind) (preservation _ _ _ _ _ _) %. +%total S (preservation _ _ S _ _ _) %. +%unique preservation %in %in %in %out %out %out %. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Progress. +%%% A judgment declaring that with respect to the given heap, an expression +%%% can step or is a value. +%sort progresses %. +%mode progresses %in %in %. +%term can_step %pi (progresses H E) %<- (step H E H' E') %. +%term is_val %pi (progresses _ E) %<- (value E) %. +%%% (Most of) The statement of progress. +%sort progress %. +%mode progress %in %in %out %. +%%% Easy cases: values. +%term _ progress t_unit _ (is_val v_dot) %. +%term _ progress (t_loc _) _ (is_val v_loc) %. +%term _ progress (t_abs _) _ (is_val v_lam) %. +%%% Less easy case: allocations. +%sort alloc_progresses %. +%mode alloc_progresses %in %out %. +%term _ + %pi (alloc_progresses (%the (progresses H E) (is_val V)) (can_step (e_allocVal L A V))) + %<- (can_append_heap H E A) + %<- (can_length_heap H L) %. +%term _ alloc_progresses (can_step S) (can_step (e_alloc S)) %. +%worlds () (alloc_progresses _ _) %. +%total P (alloc_progresses P _) %. +%unique alloc_progresses %in %out %. +%term _ %pi (progress (t_ref Q) Z P) %<- (progress Q Z Pind) %<- (alloc_progresses Pind P) %. +%%% Less easy case: applications. +%sort app_progresses %. +%mode app_progresses %in %in %in %out %. +%term _ app_progresses _ (can_step S) _ (can_step (e_app1 S)) %. +%term _ app_progresses _ (is_val V) (can_step S) (can_step (e_app2 S V)) %. +%term _ app_progresses _ (is_val v_lam) (is_val V2) (can_step (e_appabs V2)) %. +%worlds () (app_progresses _ _ _ _) %. +%total {P1 P2} (app_progresses _ P1 P2 _) %. +%unique app_progresses %in %in %in %out %. +%term _ + %pi (progress (t_app Qarg Qbody) Z P) + %<- (progress Qarg Z Parg) + %<- (progress Qbody Z Pbody) + %<- (app_progresses Qbody Pbody Parg P) %. +%%% Less easy case: dereferences. +%sort deref_progresses %. +%mode deref_progresses %in %in %in %out %. +%term _ deref_progresses _ _ (can_step S) (can_step (e_deref S)) %. +%term _ + %pi (deref_progresses (t_loc X) W (is_val v_loc) (can_step (e_derefVal Y))) + %<- (find_lemma W X Y) %. +%worlds () (deref_progresses _ _ _ _) %. +%total P (deref_progresses _ _ P _) %. +%unique deref_progresses %in %in %in %out %. +%term _ + %pi (progress (t_deref Q) Z P) + %<- (progress Q Z Pind) + %<- (deref_progresses Q Z Pind P) %. +%%% Less easy case: assignments. +%sort assign_progresses %. +%mode assign_progresses %in %in %in %in %out %. +%term _ assign_progresses _ _ (can_step S) _ (can_step (e_gets1 S)) %. +%term _ assign_progresses _ _ (is_val V) (can_step S) (can_step (e_gets2 S V)) %. +%term _ + %pi (assign_progresses (t_loc X) W (is_val v_loc) (is_val (%the (value E) V2)) (can_step (e_getsVal Y V2))) + %<- (replace_lemma E W X Y) %. +%worlds () (assign_progresses _ _ _ _ _) %. +%total {P1 P2} (assign_progresses _ _ P1 P2 _) %. +%unique assign_progresses %in %in %in %in %out %. +%term _ + %pi (progress (t_assign Qval Qloc) Z P) + %<- (progress Qval Z Pval) + %<- (progress Qloc Z Ploc) + %<- (assign_progresses Qloc Z Ploc Pval P) %. +%%% No more cases. +%worlds () (progress _ _ _) %. +%total Q (progress Q _ _) %. +%unique progress %in %in %out %. \ No newline at end of file diff --git a/new-tests/stelf-output/pages/abbrev-declaration.lf b/new-tests/stelf-output/pages/abbrev-declaration.lf new file mode 100644 index 0000000..dd8fcb8 --- /dev/null +++ b/new-tests/stelf-output/pages/abbrev-declaration.lf @@ -0,0 +1,25 @@ +% +% +%{! The **``%abbrev``** keyword can be placed before any definition in a Twelf [signature](/wiki/twelf-signature/) to cause the definition to, in the future, act as syntatic shorthand for some other term. + +## Example + +Say, for some reason, we had extremely verbose names for the syntax of the [natural numbers](/wiki/natural-numbers/). !}% +%sort this-is-a-long-name-for-nat %. +%term this-is-a-long-name-for-z this-is-a-long-name-for-nat %. +%term this-is-a-long-name-for-s %pi this-is-a-long-name-for-nat %-> this-is-a-long-name-for-nat %. +%{! We can define ``nat`` and ``z`` from their long names using ``%abbrev``, and ``s`` without ``%abbrev``. !}% +%inline nat this-is-a-long-name-for-nat %. +%inline z this-is-a-long-name-for-z %. +%define s this-is-a-long-name-for-s %. +%{! We can see the difference here - while definitions like ``s`` will be expanded only if they have to be, definitions made with the ``%abbrev`` keyword are always expanded by Twelf. !}% +%{!! begin checked !!}% +%define three s (s (s z)) %. +%{!! end checked !!}% +%{! ## See also + +* Definitions + + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/about-the-twelf-project.lf b/new-tests/stelf-output/pages/about-the-twelf-project.lf new file mode 100644 index 0000000..448827c --- /dev/null +++ b/new-tests/stelf-output/pages/about-the-twelf-project.lf @@ -0,0 +1,31 @@ +% +% +% +%{! Twelf is a research project concerned with the design, implementation, and +application of logical frameworks funded by the National Science Foundation +under grants CCR-9619584 and CCR-9988281 Meta-Logical Frameworks, CCR-0306313 +Efficient Logical Frameworks (Principal Investigator: Frank Pfenning) and by +DARPA under the contract number F196268-95-C-0050 The Fox Project: Advanced +Languages for Systems Software (Principal Investigators: Robert Harper, Peter +Lee, and Frank Pfenning). + +The Twelf implementation comprises + +- the LF logical framework, including type reconstruction; +- the Elf constraint logic programming language; +- an inductive meta-theorem prover for LF (very preliminary); +- and an Emacs interface. + +The principal authors of Twelf are + +- [Frank Pfenning](http://www.cs.cmu.edu/~fp/), and +- [Carsten Schürmann](http://www.itu.dk/people/carsten) + +with major contributions by + +- [Brigitte Pientka](http://www.cs.mcgill.ca/~bpientka/), +- Roberto Virga, and +- Kevin Watkins + +Twelf provides a uniform meta-language for specifying, implementing, and proving properties of programming languages and logics. Example suites include Cartesian Closed Categories and lambda-calculus, the Church-Rosser theorem for the untyped lambda-calculus, Mini-ML including type preservation and compilation, cut elimination, theory of logic programming, and Hilbert's deduction theorem. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/abstract-syntax.lf b/new-tests/stelf-output/pages/abstract-syntax.lf new file mode 100644 index 0000000..d1bd647 --- /dev/null +++ b/new-tests/stelf-output/pages/abstract-syntax.lf @@ -0,0 +1,28 @@ +% +%{! The **abstract syntax** of a programming language is an unambiguous representation of the syntax of a language; most compilers and most language definitions work in terms of the abstract syntax of a language as opposed to the **concrete syntax** of a language, which is what the programmer actually writes down. Twelf can encode the abstract syntax of a language, and can furthermore gracefully encode the idea of _binding_ by using [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/). + +## Example + +The concrete syntax of a simple language might look like this: + +`if 1 = 2 then 1 + if true then 3 else 6 else 6` + +and the abstract syntax in a language like ML might look like this: + +``` +If(Eq(Num(1),Num(2)), + Plus(Num(1),If(True,Num(3),Num(6))), + Num(6)) +``` + +and the same abstract syntax in a language like Twelf might look like this: + +``` +(if (eq (num 1) (num 2)) + (plus (num 1) (if true (num 3) (num 6))) + (num 6)); +``` + +## See also + +[Higher-order abstract syntax](/wiki/higher-order-abstract-syntax/) !}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/ad-hoc-binding-structures.lf b/new-tests/stelf-output/pages/ad-hoc-binding-structures.lf new file mode 100644 index 0000000..10730f5 --- /dev/null +++ b/new-tests/stelf-output/pages/ad-hoc-binding-structures.lf @@ -0,0 +1,13 @@ +% +%{! Many of the Twelf [tutorials](/wiki/tutorials/) and [case studies](/wiki/case-studies/) use very simple binding structures (e.g. lambdas that take a single argument). However, calculi and programming languages of interest often have more complex binding structures. **Ad hoc binding structures** such as pattern matching and mutual recursion can be encoded quite naturally in LF. Here are some examples: + +* [Pattern matching](/wiki/pattern-matching/): A case study on pattern matching with disjunctive patterns. +* [letrec](/wiki/letrec/): A case study on letrec (let binding a bundle of mutually recursive expressions). + +NOTE: There is actually a lot of overlap between the two examples, in terms of special machinery to do the "real-world binding structures". The two case studies could easily be combined. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Ad_hoc_binding_structures). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/adequacy.lf b/new-tests/stelf-output/pages/adequacy.lf new file mode 100644 index 0000000..6540a74 --- /dev/null +++ b/new-tests/stelf-output/pages/adequacy.lf @@ -0,0 +1,24 @@ +% +%{! Twelf is based on the logical framework [LF](/wiki/lf/). When we write down a logical system (an [object logic](/wiki/object-logic/) such as a programming language) in Twelf, we are defining the objects and judgments of our language as LF types and terms. +The correctness criterion for such a representation is called **adequacy**. An LF representation is **adequate** iff it is isomorphic to the original description of the object logic. Specifically, we require a _compositional bijection_ between the original description and the LF representations, which means that: + +- There is a bijection between object language entities and LF objects of a particular type (the "bijection" part). +- For representations that use [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/) and [higher-order judgment](/wiki/higher-order-judgements/)s, LF's notion of substitution correctly models object-language substitution (the "compositional" part). + +See the introductory article [Proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/), as well as Harper and Licata's paper <ref>\{\{bibtex:hl07mechanizing\}\}</ref>, for example adequacy statements and proofs. + +## References + +<references/> + +* [Proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/) +* [http://www-2.cs.cmu.edu/~rwh/papers/how/believe-twelf.pdf How to Believe a Twelf Proof] +\{\{bibtex:hhp93lf\}\} + +\{\{stub\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Adequacy). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/admissibility-of-cut.lf b/new-tests/stelf-output/pages/admissibility-of-cut.lf new file mode 100644 index 0000000..dc9c606 --- /dev/null +++ b/new-tests/stelf-output/pages/admissibility-of-cut.lf @@ -0,0 +1,154 @@ +% +%{! The **admissibility of cut** (also called **cut elimination**) is one of the most important metatheorems of an [object logic](/wiki/object-logic/) sequent calculus. It is the key theorem for showing the equivalence of natural deduction and the cut-free sequent calculus, which has several corollaries, including the existence of normal forms for every natural deduction term, and the consistency of the logic. This [tutorial](/wiki/tutorials/) shows how to prove admissibility of cut for a simple intuitionistic sequent calculus in Twelf. We will make use of lexicographic orderings in the [termination](/wiki/percent-terminates/) argument. <!-- and other tricks? --> + +## Cut + +For a sequent calculus Γ ⇒ C, where Γ holds hypotheses and the proposition C is the conclusion, the statement of cut is as follows: If Γ ⇒ A and Γ,A ⇒ C then Γ ⇒ C. + +Often, logicians will include cut in the sequent calculus as a rule: + + + +They then prove that any proof that uses the cut rule can be transformed into a proof without cut (thus, "cut elimination"). In Twelf, it will be simpler to prove cut as a metatheorem ("admissibility of cut") over a sequent calculus with no cut rule. + +## Sequent calculus + +Suppose we have the following sequent calculus: + +<!-- nb. in this math code we use \supset for implication --> +<table style="margin-left:auto;margin-right:auto" cellpadding="24"> +<tr><td> + +</td><td> + +</td></tr> +<tr><td> + +</td><td> + +</td></tr> +<tr><td> + +</td><td> + +</td></tr> +</table> + +The logic supports only the ⊤, ∧, and ⊃ connectives to simplify this tutorial. In LF, we encode the sequent calculus as follows: !}% +%sort prop %. +%name prop %. +%term top prop %. +%term imp %pi prop %-> prop %-> prop %. +%term and %pi prop %-> prop %-> prop %. +% hypotheses +%sort hyp %. +% G |- C +%sort conc %. +%term init %pi (hyp A) %-> (conc A) %. +%term topR conc top %. +%term andL %pi (%pi (hyp A) %-> (hyp B) %-> (conc C)) %-> (hyp (and A B)) %-> (conc C) %. +%term andR %pi (conc A) %-> (conc B) %-> (conc (and A B)) %. +%term impL %pi (conc A) %-> (%pi (hyp B) %-> (conc C)) %-> (hyp (imp A B)) %-> (conc C) %. +%term impR %pi (%pi (hyp A) %-> (conc B)) %-> (conc (imp A B)) %. +%{! The sequent is represented as the judgment ``conc``, which is hypothetical in hypotheses ``hyp``. A rule ``init`` allows us to use a hypothesis to satisfy a conclusion. + +## Statement and proof of cut !}% +%sort cut {A prop} {D conc A} {E %pi (hyp A) %-> (conc C)} {F conc C} %. +%mode cut %in %in %in %out %. +%{! We will refer to the input derivations as ``D`` and ``E`` throughout this tutorial and the output as ``F``. Because we will induct on the structure of the type ``A``, we make it an explicit argument. + +There are nominally about _n_² cases to consider in this proof; for each of the _n_ rules that can be at the bottom of ``D``, _n_ rules can be used at the bottom of ``E``. In actuality, two factors reduce the number of cases we'll make: some combinations of rules are incompatible (Twelf's excludes them via unification, so we don't even need to write them down) and some cases can be done schematically by using universal quantification. + +### Initial cuts + +If either derivation is an instance of the ``init`` rule, we are done: !}% +%term initD cut A (init Ha) ([Ha] E Ha) (E Ha) %. +%term initE cut A D ([Ha] init Ha) D %. +%term closed cut A D ([Ha] E') E' %. +%{! If ``D`` is the ``init`` rule used to conclude ``conc A`` from ``hyp A``, then Γ includes ``hyp A`` already, so we can satisfy the hypothesis in ``E`` by simply using the ``A`` that is already around (case ``initD``). + +If ``E`` is the ``init`` rule used to conclude ``conc C`` from ``hyp A`` (so ``A`` = ``C``), then ``D`` is a proof of Γ ⇒ C, so ``F`` is just ``D`` (case ``initE``). + +It's also possible that ``E`` is the ``init`` rule used to conclude ``conc C`` from some other hypothesis ``C`` in Γ. In this case, ``E`` doesn't use the hypothesis ``A`` at all, so ``E`` can be strengthened to produce the output derivation ``F`` which doesn't depend on ``A``. In fact, we can easily generalize this case to any derivation ``E`` that does not use the hypothesis ``A``; this is the ``closed`` case. (When we write ``([Ha] E')``, the variable ``E'``—which is implicitly quantified at the outside of the goal—cannot depend on the lambda-bound variable ``Ha``. This is how we indicate the derivation is closed with respect to its argument.) + +### Principal cuts + +The most interesting cases are those where the cut formula ``A`` is concluded with a right rule in ``D``, and used with a left rule in ``E``. These are known as _principal cuts_. + +There is no principal cut for ⊤ because there is no left rule, so the easiest one will be ``A ∧ B``: !}% +%term andC + %pi (cut (and A B) (andR D1 D2) ([Hab hyp (and A B)] andL ([Ha hyp A] [Hb hyp B] E' Hab Ha Hb) Hab) F) + %<- ({Ha} {Hb} cut (and A B) (andR D1 D2) ([Hab] E' Hab Ha Hb) (F1 Ha Hb)) + %<- ({Hb} cut A D1 ([Ha] F1 Ha Hb) (F2 Hb)) + %<- (cut B D2 ([Hb] F2 Hb) F) %. +%{! We identify the case where ``D`` is an instance of ``andR``, and ``E`` is ``andL`` acting on the hypothesis of the cut formula. (The hypothetical derivation ``E`` is represented as a LF function taking the hypothesis ``A ∧ B``; we identify the case where this specific hypothesis is used by ``andL`` by passing that bound variable (``Hab``) to the ``andL`` constant.) We have a subderivation ``E'`` of ``conc C`` under hypotheses ``hyp (and A B)``, ``hyp A``, and ``hyp B``. We must eliminate each of these to produce ``F``. In the first subgoal we eliminate the ``and A B`` hypothesis. ``D`` itself is a derivation of ``conc (and A B)`` in Γ, so we want to appeal to the IH on ``D`` and ``E'``. There are a few things to notice. First, we did not name ``D`` (indeed there is no way to do so) because we pattern matched against it as an application of ``andR``; therefore, in the inductive call we have to build up ``D`` again as ``andR D1 D2``. Second, ``E'`` is not of the correct type for the inductive call, because it is a curried function of three arguments, not one. We therefore must make the subgoal higher order: we hypothesize the existence of ``Ha`` and ``Hb`` (of type ``hyp A`` and ``hyp B`` respectively). We then form the term of type ``hyp (and A B) -> conc C`` by abstracting the hypothesis we seek to eliminate (``Hab``) and applying ``E'`` to it and the Π-bound ``Ha`` and ``Hb``. Because this inductive call is in a context including ``Ha`` and ``Hb``, the resulting derivation ``F1`` depends on those two variables as well. + +We then want to do the same thing to eliminate the hypotheses for ``A`` and ``B``. In the second subgoal, we prove that for all derivations of ``hyp B`` (Π-bound variable ``Hb``), we can cut ``D1`` with the result of the previous induction ``F1`` to get ``F2``, which only depends on ``Hb``. Finally, in the third subgoal we eliminate the ``hyp B`` hypothesis to produce a derivation of ``conc C`` in the ambient Γ with no extra assumptions, which is what we need for ``F``. + +We must pay close attention to the induction order. The first inductive call is well-founded because it is on a subderivation ``E'``. The output derivation ``F1`` may be larger, however, and it is used as an input to the second inductive call. Observe, however, that the cut formula ``A`` is a subderivation of the original cut formula ``and A B``. Ultimately, our induction order will be lexicographic. Either ``D`` or ``E`` will get smaller (with the other staying the same size), or they both may get larger but ``A`` will get smaller. We will specify this induction measure in the [%total](/wiki/percent-total/) declaration when we check the [metatheorem](/wiki/metatheorem/) below. !}% +%term impC + %pi (cut (imp A B) (impR ([Ha] D' Ha)) ([Hab] impL (%the (conc A) (E1 Hab)) ([Hb] %the (conc C) (E2 Hab Hb)) Hab) F) + %<- (cut (imp A B) (impR D') E1 (%the (conc A) F1)) + %<- ({Hb hyp B} cut (imp A B) (impR D') ([Hab] E2 Hab Hb) (%the (conc C) (F2 Hb))) + %<- (cut A F1 D' (%the (conc B) F3)) + %<- (cut B F3 F2 F) %. +%{! The ``impC`` case works in much the same way. We first cut the ``A ⊃ B`` hypothesis from both subderivations ``E1`` and ``E2``. We then have ``F1 : conc A`` and ``F2 : hyp B -> conc C``. To get ``F3 : conc B`` we cut the ``hyp A`` from ``D' : hyp A -> conc B`` using ``F1`` (note that in this inductive call, derivations from the "``D`` side" and "``E`` side" have switched roles!). Finally, we cut the ``hyp B`` from ``F2`` to get the result ``F``. + +### Left-commutative cuts + +The remainder of the cases are _commutative_ cases where we search for initial and principal cuts to apply. The first set are the _left commutative_ cases, where the rule used to derive ``D`` is a left rule. !}% +%term andLLC + %pi (cut A (andL ([Ha] [Hb] D' Ha Hb) Hab) E (andL ([Ha] [Hb] F' Ha Hb) Hab)) + %<- ({Ha} {Hb} cut A (D' Ha Hb) E (F' Ha Hb)) %. +%term impLLC + %pi (cut A (impL D1 ([Hb] D2 Hb) Hi) E (impL D1 F2 Hi)) + %<- ({Hb} cut A (D2 Hb) E (F2 Hb)) %. +%{! In the ``andLLC``, ``D : conc A`` is an instance of ``andL``. We proceed by hypothesizing the ``Ha : hyp A1`` and ``Hb : hyp B1`` that ``D' : hyp A1 -> hyp B1 -> conc A`` depends on, so that we may cut it against ``E``. The resulting derivation ``F'`` then also depends on a ``hyp A1`` and ``hyp B1``; we wrap it with an instance of the ``andL`` rule. In these commutative cases, we find rules that do not involve the cut formula, work underneath them inductively, and then re-apply the rule to the cut-free proof we obtained. The ``impLLC`` case follows the same pattern. + +### Right-commutative cuts + +The _right commutative_ cuts are those where the rule at the bottom of ``E`` is not a left rule acting on the cut formula. The first two cases are where ``E`` is a right rule: !}% +%term andRRC + %pi (cut A D ([Ha] andR (E1 Ha) (E2 Ha)) (andR F1 F2)) + %<- (cut A D E1 F1) + %<- (cut A D E2 F2) %. +%term impRRC + %pi (cut A D ([Ha] impR ([H1] E1 Ha H1)) (impR ([H1] F1 H1))) + %<- ({H1} cut A D ([Ha] E1 Ha H1) (F1 H1)) %. +%{! In the ``andRRC`` case, we simply apply straightforward induction. ``E`` consists of two subderivations, ``E1`` and ``E2``, each of which may use the cut formula ``A``. We inductively cut out uses of ``A``, and then rebuild the derivation with the ``impR`` rule. The ``impRRC`` rule is similar. + +The final two cases are uses of the left rules, but where they act on a hypothesis different from the cut hypothesis: !}% +%term andLRC + %pi (cut A (%the (conc A) D) ([Ha] andL ([H1] [H2] E' Ha H1 H2) Hp) (andL ([H1] [H2] F' H1 H2) Hp)) + %<- ({H1} {H2} cut A D ([Ha] E' Ha H1 H2) (F' H1 H2)) %. +%term impLRC + %pi (cut A (%the (conc A) D) ([Ha] impL (E1 Ha) ([Hb] E2 Ha Hb) Hi) (impL F1 ([Hb] F2 Hb) Hi)) + %<- (cut A D ([Ha] E1 Ha) F1) + %<- ({Hb} cut A D ([Ha] E2 Ha Hb) (F2 Hb)) %. +%{! The ``andLRC`` case is similar to the principal cut case ``andC``, but here the ``andL`` in ``E`` acts on some hypothesis ``Hp : and A1 B1`` instead of the cut hypothesis ``Ha : hyp A``. To proceed, we simply eliminate uses of ``Ha`` within E' inductively, and then reapply the ``andL`` rule to the resulting derivation. We follow a similar pattern for the ``impLRC`` case. + +## The metatheorem + +We can now state and check the metatheorem: that for every [ground](/wiki/ground/) arguments ``A``, ``D`` and ``E`` to the ``cut`` relation, there exists a derivation ``F`` that concludes ``C`` without using the hypothesis ``A``. Our [%worlds](/wiki/percent-worlds/) declaration for cut is only that hypotheses may appear in the context: !}% +%block hyp {A prop} [H hyp A]%. +%worlds (hyp) (cut _ _ _ _) %. +%{! We check the totality of ``cut`` using a lexicographic induction order, as discussed above. We write ``\{A [D E]\}`` to mean that either the cut formula ``A`` gets smaller, or it stays the same size and one of ``D`` or ``E`` gets smaller with the other staying the same size. !}% +%total {A [D E]} (cut A D E F) %. +%{! We proved the admissibility of cut for a small sequent calculus with only a few connectives. Experience shows that this same strategy works for more complex logics. For instance, the case study on [Classical S5](/wiki/classical-s5/) includes a cut theorem for a classical modal logic. It also formalizes the translations between natural deduction and the sequent calculus. + +``>All code from this tutorial``. `` check="true">Twelf's output from this tutorial``. + + +## See also + +* [Lax logic](/wiki/lax-logic/) - Cut elimination for a polarized variant of the Pfenning-Davies reconstruction of lax logic +* [Tethered modal logic](/wiki/tethered-modal-logic/) - Cut elimination for the Pfenning-Davies reconstruction of Modal S4 +* [Weak focusing](/wiki/weak-focusing/) - Cut elimination for a focused sequent calculus (lacks commentary) + +\{\{case study\}\}[Category:Twelf code](/wiki/category-twelf-code/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Admissibility_of_cut). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/alpha-equivalence.lf b/new-tests/stelf-output/pages/alpha-equivalence.lf new file mode 100644 index 0000000..1cb1180 --- /dev/null +++ b/new-tests/stelf-output/pages/alpha-equivalence.lf @@ -0,0 +1,20 @@ +% +%{! Alpha-equivalence is a notion of equivalence on terms with binding structure. It captures the notion that the names of bound variables are unimportant; all that matters is the binding structure they induce. + +Consider the untyped lambda-calculus. + + + +Alpha-equivalence for terms is the least [congruence relation](/wiki/congruence-relation/) closed under the axiom: + + + +Alpha-equivalence generalizes in a straightforward manner to any term language with binding structure. + +Twelf's notion of term equality respects alpha-equivalence; when an object-language is represented using [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/), the representation enjoys alpha-equivalence for free. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Alpha-equivalence). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/ambiguous-hyperkind.lf b/new-tests/stelf-output/pages/ambiguous-hyperkind.lf new file mode 100644 index 0000000..580783c --- /dev/null +++ b/new-tests/stelf-output/pages/ambiguous-hyperkind.lf @@ -0,0 +1,14 @@ +% +%{! An **ambiguous hyperkind** occurs in pathological cases like this: + +```checkedtwelf +a : type. + +b = a : _ _ . +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Ambiguous_hyperkind). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/ask-twelf-elf-short-answers.lf b/new-tests/stelf-output/pages/ask-twelf-elf-short-answers.lf new file mode 100644 index 0000000..87a5a96 --- /dev/null +++ b/new-tests/stelf-output/pages/ask-twelf-elf-short-answers.lf @@ -0,0 +1,51 @@ +% +%{! This page collects short answers to [questions](/wiki/talk-ask-twelf-elf/) people asked. These answers have not (yet) been made into their own pages, but feel free to fork a page off if you want to give a more thorough answer to a question. + +## Comparing Coq and Twelf + +Michael Fortson asks: _I can't find any papers comparing Coq and Twelf. Can someone summarize the differences?_ + +This is a hard question to answer in any comprehensive manner, since there are lots of differences, in theory and in practice, between the two systems. However, here's a slapdash list of bullet-points to keep in mind as you think about the two, tailored to representing and proving theorems about programming languages, and quite biased towards talking about LF and Twelf because that's what I know the most about. There are + +**Differences in how you represent languages:** +* In LF, you represent your language as the canonical forms of particular LF types; the representation is not an inductive definition inside the LF type theory. This representation strategy enables [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/) and [higher-order judgements](/wiki/higher-order-judgements/), which are very useful ways of dealing with programming languages and logics that involve binding. LF has a theory of [adequacy](/wiki/adequacy/), which tells you when a representation is correct. +* In Coq, you represent your language's syntax and judgements as inductively defined types and propositions. The techniques for dealing with binding are different because you can't use HOAS and HOJ in the same manner as you do in LF. Also, I don't know of a formal theory of adequacy for Coq, but I haven't looked very hard for one. + +**Differences in what constitutes a proof of a metatheorem:** +* When using Twelf's [metatheorem](/wiki/metatheorem/) features, a proof of a metatheorem consists of an LF type family that Twelf proves [total](/wiki/percent-total/) (this can be confusing at first; start with [this section of the intro](/wiki/proving-metatheorems-proving-totality-assertions-about-the-natural-numbers/) for details). Thus, we don't prove metatheorems in LF, we prove metatheorems about LF using extra features in Twelf. + +* In contrast, metatheorems in Coq are (more or less) just terms in the Calculus of Inductive Constructions (CiC), which is the same type theory you use for representing languages; there is no separate language for metatheorems. + +**Differences in how you write proofs:** +* In Twelf, you write a proof of a metatheorem directly by exhibiting the cases as constants inhabiting an LF type family. The computational context of the proof is explicit, because this type family can be run as a logic program. + +* The Coq proofs that I've seen consist mostly of tactical proof scripts that instruct Coq how to find the proof term that you're trying to create. + +**Differences in what's provable (in theory):** Twelf's metatheorem apparatus only checks proofs of ∀∃-statements, so there are some theorems you can't use it to check. + +**Differences in what's provable (in practice):** Of course, the theoretical limits don't tell you much about what people are actually able to do in their day-to-day work. See the page of [Research projects using Twelf](/wiki/research-projects-using-twelf/) for some examples on the Twelf side of things; there are lots of Coq examples on the Web as well. + +**Differences in foundations:** LF is a very simple type theory, whereas the CiC (which is something like what Coq implements, but I'm not exactly sure) is a richer type theory. For proofs of metatheorems, the fair comparison is between LF with Twelf's metatheorem features and CiC with whatever additions Coq implements, and there I'm not sure which is simpler. + +This is not intended to be a fair or comprehensive answer to your question, but it's something to think about. Others might have opinions to add as well. + +Also, you should read the Appel and Leroy list machine benchmark paper that's linked from the [documentation](/wiki/documentation/) page. It compares Twelf and Coq on a simple example (note that the example doesn't use [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/) or [higher-order judgements](/wiki/higher-order-judgements/), so it's not even a case where Twelf really shines). +[Drl](/wiki/user-drl/) 16:40, 8 April 2007 (EDT) + +## Different ways of using Twelf + +Michael Fortson asks: _Can you explain the vast difference in style between the examples shown on this site and what is done by Andrew Appel at Princeton? The two styles seem as if they are two completely different languages!_ + +From my understanding of the Princeton project and Appel's notes, there's a sense in which they are two different languages. On the wiki, we give various examples of representing languages and logics as an LF signature (see, e.g., [this section of the intro article](/wiki/proving-metatheorems-representing-the-judgements-of-the-natural-numbers/)) and then proving metatheorems about them using Twelf's [metatheorem](/wiki/metatheorem/) features such as [%mode](/wiki/percent-mode/), [%worlds](/wiki/percent-worlds/), and [%total](/wiki/percent-total/) (see, e.g., [this section of the intro article](/wiki/proving-metatheorems-proving-totality-assertions-about-the-natural-numbers/)). + +However, there are ways of using Twelf that don't use the metatheorem features at all. In this style, you encode a particular logic in LF and then reason entirely **within** that logic. In particular, you prove theorems in that logic by writing LF terms of the appropriate types, which correspond to derivations constructed according to the rules of the logic. In this setting, Twelf is used as: +* a type checker for LF, which serves a a proof checker for the encoded logic, and +* a logic programming language that can be used to search for proofs in the encoded logic. + +I don't have much experience using Twelf in this style; others (e.g., [Rob](/wiki/user-rsimmons/)) can say more. Someone should write an article about why you might want to reason within your own logic encoded in LF instead of using the metatheorem features. However, I recommend that beginners start by learning how to use Twelf's metatheorem features, since they provide a very useful logic for reasoning about programming languages and logics. [Drl](/wiki/user-drl/) 15:28, 8 April 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Ask_Twelf_Elf:Short_answers). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/ask-twelf-elf.lf b/new-tests/stelf-output/pages/ask-twelf-elf.lf new file mode 100644 index 0000000..df3748f --- /dev/null +++ b/new-tests/stelf-output/pages/ask-twelf-elf.lf @@ -0,0 +1,23 @@ +% +%{! If you have a question about Twelf, you can [http://twelf.org/w/index.php?title=Talk:Ask_Twelf_Elf&action=edit§ion=new leave a message] on this wiki! +If you have a question that you do not want to ask publicly, you can [mailto:twelfelf@gmail.com Ask Twelf Elf] by email. You should get a reply within 48 hours. Probably sooner. If your question involves diagnosing a particular piece of code, it is very helpful if you can send in enough code that we can re-produce the issue on our own machines. + +At any given time, the spirit of Twelf Elf possesses the body of someone knowledgeable with Twelf. Twelf Elf regularly rotates who he possesses, in order to avoid exhausting the poor person. Twelf Elf reserves the right to re-work interesting replies into content for the Wiki. E-mail the Twelf Elf or post a message on the [talk](/wiki/talk-ask-twelf-elf/) for this page if you would like to volunteer to join the Twelf Elf rotation. + +## Interesting responses + +<!-- make a tutorial or case study or glossary entry if the question has wide applicability; otherwise put it here --> + +Some responses to questions turn into other forms of [documentation](/wiki/documentation/); the rest are collected here. + +* [Short Answers](/wiki/ask-twelf-elf-short-answers/) - collected answers that have not (yet) been turned into their own pages. + +* [Manipulating proof witnesses](/wiki/manipulating-proof-witnesses-as-inputs/) - running a [``%solve``](/wiki/percent-solve/) creates a proof object if it is successful. But what if you want to manipulate such proof objects within the context of a single ``%solve``? The cost of the ability to do this is writing a bunch of [effectiveness lemmas](/wiki/effectiveness-lemma/) - in this case, if you were working with relations which were not total functions, they could be called _effectiveness relations._ + +* [Concrete representations](/wiki/concrete-representation/) - [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/) is a convenient way to approach languages with binding, but it is possible to imagine a problem where manipulating a fully concrete object without binding is simpler. In these cases, it is possible to establish a bijection between your HOAS terms and de Bruijn versions of the same terms. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Ask_Twelf_Elf). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/beta-equivalence.lf b/new-tests/stelf-output/pages/beta-equivalence.lf new file mode 100644 index 0000000..f0d520c --- /dev/null +++ b/new-tests/stelf-output/pages/beta-equivalence.lf @@ -0,0 +1,24 @@ +% +%{! **Beta-equivalence** (β-equivalence) is a notion of proof equivalence in natural deduction logics with introduction and elimination forms. Roughly, it says that when an elimination form is applied to an introduction form, they cancel. + +Consider the simply-typed lambda-calculus with arrow types. + + + + + +The beta-equivalence induced by the arrow type says that the elimination form "cancels" the introduction form ; formally, it is the least [congruence relation](/wiki/congruence-relation/) closed under the axiom: + + + +Beta-equivalence is usually oriented to the right yielding a notion of _beta-reduction_. For example: + + + +The term on the left-hand side of the axiom is called a _beta-redex_, and the term on the right-hand side is its _beta-reduct_. A term with no beta-redexes is called _beta-normal_. Being beta-normal is one aspect of being [canonical](/wiki/canonical-form/). !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Beta-equivalence). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/bibliography-of-lf.lf b/new-tests/stelf-output/pages/bibliography-of-lf.lf new file mode 100644 index 0000000..0fb99e3 --- /dev/null +++ b/new-tests/stelf-output/pages/bibliography-of-lf.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you should go to the [new version](/bibliography/). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/big-algebraic-solver.lf b/new-tests/stelf-output/pages/big-algebraic-solver.lf new file mode 100644 index 0000000..9fc65cb --- /dev/null +++ b/new-tests/stelf-output/pages/big-algebraic-solver.lf @@ -0,0 +1,475 @@ +% +%{! +The big algebraic software is a [tactical theorem prover](/wiki/tactical-theorem-proving/) written as a Twelf [logic program](/wiki/logic-programming/) by [Rob](/wiki/user-rsimmons/) at Princeton University as part of the Foundational Proof Carrying Code system. It is quite inefficient due to the fact that it doesn't try to order variables. The [http://wiki.portal.chalmers.se/agda/pmwiki.php?n=Libraries.UsingTheRingSolver RingSolver] in the Agda standard library is an example of a more efficient solver. +!}% +%{! ## Preexisting logic !}% +%{! In the Princeton FPCC project where this was developed, many of the definitions below were more general (for instance, equality was on anything, not just numbers) and many things stated here as axioms were lemmas that had corresponding proofs. These are just the definitions and lemmas that were used within the solver. !}% +%sort tnum %. +%sort tform %. +%sort pf %. +%term imp %pi tform %-> tform %-> tform %. +%prec %right 10 imp %. +%term and %pi tform %-> tform %-> tform %. +%prec %right 12 and %. +%term == %pi tnum %-> tnum %-> tform %. +%prec %none 18 == %. +%term >= %pi tnum %-> tnum %-> tform %. +%prec %none 18 >= %. +%term < %pi tnum %-> tnum %-> tform %. +%prec %none 18 < %. +%term != %pi tnum %-> tnum %-> tform %. +%prec %none 18 != %. +%define > (%pi tnum %-> tnum %-> tform) [x] [y] y < x %. +%prec %none 18 > %. +%define <= (%pi tnum %-> tnum %-> tform) [x] [y] y >= x %. +%prec %none 18 <= %. +%term imp_i %pi (%pi (pf A) %-> (pf B)) %-> (pf (A imp B)) %. +%term imp_e %pi (pf (A imp B)) %-> (pf A) %-> (pf B) %. +%term and_i %pi (pf A) %-> (pf B) %-> (pf (A and B)) %. +%term and_e1 %pi (pf (A and B)) %-> (pf A) %. +%term and_e2 %pi (pf (A and B)) %-> (pf B) %. +%term cut %pi (pf A) %-> (%pi (pf A) %-> (pf B)) %-> (pf B) %. +%term cut2 %pi (pf A) %-> (pf A') %-> (%pi (pf A) %-> (pf A') %-> (pf B)) %-> (pf B) %. +%term cut3 + %pi (pf A) + %-> (pf A') + %-> (pf A'') + %-> (%pi (pf A) %-> (pf A') %-> (pf A'') %-> (pf B)) + %-> (pf B) %. +%term congr {h %pi tnum %-> tform} %pi (pf (A == B)) %-> (pf (h B)) %-> (pf (h A)) %. +%term gt->geq %pi (pf (A > B)) %-> (pf (A >= B)) %. +%term gt->neq %pi (pf (A > B)) %-> (pf (A != B)) %. +%term lt->neq %pi (pf (A < B)) %-> (pf (A != B)) %. +%term eq->geq %pi (pf (A == B)) %-> (pf (A >= B)) %. +%term eq->leq %pi (pf (A == B)) %-> (pf (A <= B)) %. +%term not_eie %pi (pf (A != B)) %-> (%pi (pf (C == D)) %-> (pf (A == B))) %-> (pf (C != D)) %. +%term not_ein %pi (pf (A < B)) %-> (%pi (pf (C >= D)) %-> (pf (A >= B))) %-> (pf (C < D)) %. +%term zero tnum %. +%term one tnum %. +%term one>zero pf (one > zero) %. +%term + %pi tnum %-> tnum %-> tnum %. +%prec %left 22 + %. +%term * %pi tnum %-> tnum %-> tnum %. +%prec %left 23 * %. +%term neg %pi tnum %-> tnum %. +%define _ [a] [b] a + neg b %. +%prec %left 22 - %. +%define succ [x] x + one %. +%define pred [x] x - one %. +%define two succ one %. +%term refl pf (A == A) %. +%term symm %pi (pf (A == B)) %-> (pf (B == A)) %. +%term trans %pi (pf (A == B)) %-> (pf (B == C)) %-> (pf (A == C)) %. +%term trans3 %pi (pf (A == B)) %-> (pf (B == C)) %-> (pf (C == D)) %-> (pf (A == D)) %. +%term trans4 + %pi (pf (A == B)) + %-> (pf (B == C)) + %-> (pf (C == D)) + %-> (pf (D == E)) + %-> (pf (A == E)) %. +%term trans5 + %pi (pf (A == B)) + %-> (pf (B == C)) + %-> (pf (C == D)) + %-> (pf (D == E)) + %-> (pf (E == F)) + %-> (pf (A == F)) %. +%term trans_geq_eq %pi (pf (A >= B)) %-> (pf (B == C)) %-> (pf (A >= C)) %. +%term trans_eq_geq %pi (pf (A == B)) %-> (pf (B >= C)) %-> (pf (A >= C)) %. +%term comm_plus pf (A + B == B + A) %. +%term plus_zero pf (A + zero == A) %. +%term zero_plus pf (zero + A == A) %. +%term plus_cong %pi (pf (A == B)) %-> (pf (C == D)) %-> (pf (A + C == B + D)) %. +%define plus_cong2 (%pi (pf (A == B)) %-> (pf (C + A == C + B))) [p1] plus_cong refl p1 %. +%define plus_cong1 (%pi (pf (A == B)) %-> (pf (A + C == B + C))) [p1] plus_cong p1 refl %. +%term plus_assoc pf (A + B + C == A + (B + C)) %. +%define assoc_plus (pf (A + (B + C) == A + B + C)) symm plus_assoc %. +%term neg_plus pf (neg A + neg B == neg (A + B)) %. +%term plus_inv pf (A - A == zero) %. +%define inv_plus (pf (neg A + A == zero)) trans comm_plus plus_inv %. +%term geq_plus_cong1 %pi (pf (A >= B)) %-> (pf (A + C >= B + C)) %. +%term comm_times pf (A * B == B * A) %. +%term times_assoc pf (A * B * C == A * (B * C)) %. +%define assoc_times (pf (A * (B * C) == A * B * C)) symm times_assoc %. +%term times_assoc pf (A * B * C == A * (B * C)) %. +%term times_cong %pi (pf (A == B)) %-> (pf (C == D)) %-> (pf (A * C == B * D)) %. +%define times_cong2 (%pi (pf (A == B)) %-> (pf (C * A == C * B))) [p1] times_cong refl p1 %. +%define times_cong1 (%pi (pf (A == B)) %-> (pf (A * C == B * C))) [p1] times_cong p1 refl %. +%term neg_times1 pf (neg A * B == neg (A * B)) %. +%term neg_times2 pf (A * neg B == neg (A * B)) %. +%term neg_times pf (neg A * neg B == A * B) %. +%term times_one pf (A * one == A) %. +%define one_times (pf (one * A == A)) trans comm_times times_one %. +%term times_zero pf (A * zero == zero) %. +%term times_elim2 %pi (pf (A != zero)) %-> (pf (B * A == C * A)) %-> (pf (B == C)) %. +%term geq_times_cong1 %pi (pf (A >= zero)) %-> (pf (B >= C)) %-> (pf (B * A >= C * A)) %. +%term geq_times_elim1 %pi (pf (A > zero)) %-> (pf (A * B >= A * C)) %-> (pf (B >= C)) %. +%term geq_times_elim2 %pi (pf (A > zero)) %-> (pf (B * A >= C * A)) %-> (pf (B >= C)) %. +%term geq_times_cong1_neg %pi (pf (A <= zero)) %-> (pf (B >= C)) %-> (pf (B * A <= C * A)) %. +%term geq_times_elim2_neg %pi (pf (A < zero)) %-> (pf (B * A >= C * A)) %-> (pf (B <= C)) %. +%term neg_cong %pi (pf (A == B)) %-> (pf (neg A == neg B)) %. +%term neg_elim %pi (pf (neg A == neg B)) %-> (pf (A == B)) %. +%term neg_neg_e pf (neg (neg A) == A) %. +%term neg_zero pf (neg zero == zero) %. +%term distrib pf (A * (B + C) == A * B + A * C) %. +%term distrib_right pf (A + B * C == A * C + B * C) %. +%define cancel_a+b-b (pf (A + B - B == A)) symm (trans3 (symm plus_zero) (plus_cong2 (symm plus_inv)) assoc_plus) %. +%define cancel_a+b-b' (pf (A + (B - B) == A)) trans assoc_plus cancel_a+b-b %. +%define cancel_a-b+b (pf (A - B + B == A)) symm (trans4 (symm cancel_a+b-b) plus_assoc (plus_cong2 comm_plus) assoc_plus) %. +%define cancel_a-b+b' (pf (A + (neg B + B) == A)) trans assoc_plus cancel_a-b+b %. +%define cancel_a+b-a (pf (A + B - A == B)) symm (trans3 (symm cancel_a-b+b) comm_plus assoc_plus) %. +%define cancel_a+b-a' (pf (A + (B - A) == B)) trans assoc_plus cancel_a+b-a %. +%define cancel_a-a+b (pf (A - A + B == B)) trans3 comm_plus assoc_plus cancel_a+b-b %. +%define cancel_a-a+b' (pf (A + (neg A + B) == B)) trans assoc_plus cancel_a-a+b %. +%define cancel_-a+a+b (pf (neg A + A + B == B)) trans (plus_cong1 comm_plus) cancel_a-a+b %. +%define cancel_-a+a+b' (pf (neg A + (A + B) == B)) trans assoc_plus cancel_-a+a+b %. +%define cancel_-a+b+a (pf (neg A + B + A == B)) trans4 plus_assoc (plus_cong2 comm_plus) assoc_plus cancel_-a+a+b %. +%define cancel_-a+b+a' (pf (neg A + (B + A) == B)) trans assoc_plus cancel_-a+b+a %. +%{! ## Parts of the solver !}% +%define nomatch zero %. +%define match one %. +%define listify (%pi tnum %-> tnum %-> tnum %-> %type) [a] [pos] [neg] pf (a == pos - neg) %. +%deterministic listify %. +%mode listify %in %out %out %. +%{! ### Normalization !}% +%define plusify (%pi tnum %-> tnum %-> tnum %-> %type) [a] [x] [y] pf (a == x - y) %. +%deterministic plusify %. +%mode plusify %in %out %out %. +%define timesify (%pi tnum %-> tnum %-> tnum %-> tnum %-> %type) [b] [x] [y] [sign] pf (b * sign == x * y) %. +%deterministic timesify %. +%mode timesify %in %out %out %out %. +%define catchtimes (%pi tnum %-> tnum %-> tnum %-> tnum %-> tnum %-> tnum %-> tnum %-> %type) [a] [b] [x] [y] [sign] [pos] [neg] pf (b * sign == x * y imp a + b == pos - neg) %. +%deterministic catchtimes %. +%mode catchtimes %in %in %in %in %in %out %out %. +%{! ### Matching !}% +%define trim (%pi tnum %-> tnum %-> tnum %-> tnum %-> %type) [a] [b] [c] [d] pf (a - b == c - d) %. +%deterministic trim %. +%mode trim %in %in %out %out %. +%define findarr_mult (%pi tnum %-> tnum %-> tnum %-> tnum %-> %type) [a] [b] [c] [m] pf (a == b * c and m == m) %. +%deterministic findarr_mult %. +%mode findarr_mult %in %out %out %in %. +%define rearrange (%pi tnum %-> tnum %-> tnum %-> tnum %-> %type) [a] [b] [c] [d] pf (a - b == c - d) %. +%deterministic rearrange %. +%mode rearrange %in %in %in %in %. +%{! ### Main solver interface !}% +%define algebra_solver (%pi tform %-> %type) [a] pf a %. +%deterministic algebra_solver %. +%define algebra_fact (%pi tform %-> %type) [a] pf a %. +%mode algebra_fact %out %. +%define atom_replace (%pi tnum %-> tnum %-> %type) [a] [b] pf (a == b) %. +%mode atom_replace %out %out %. +%define algebra_times_hint_intern (%pi tnum %-> tnum %-> %type) [a] [b] pf (a + b == a + b) %. +%mode algebra_times_hint_intern %out %out %. +%define algebra_times_hint (%pi tnum %-> %type) [a] pf (a == a) %. +%mode algebra_times_hint %out %. +%{! ## Implementation !}% +%{! ### Normalization !}% +%{! #### Listify !}% +%{! Decomposes equations into list form ``0 + (1 * A * B * C) + D - E + (1 * F * G) +...`` !}% +%clause listify_imp (%pi (listify A Pos Neg) %<- (plusify (zero + A) Pos Neg)) [p1 pf (zero + A == Pos - Neg)] trans3 (symm plus_zero) comm_plus p1 %. +%{! #### Timesify !}% +%{! ``timesify B X S Z`` - Creates a one-terminated list ``(1 * A * B * C * ...)`` + +* B - the input +* X - the output list +* Y - signals a special condition +** zero if a zero is encountered and the term is equal to zero +** (P + Q) if a sum is encountered and the product needs to be distributed +** one if the list is successfully processed +* Sign - takes on all change-of-sign information !}% +%clause timesify_neg (%pi (timesify (B * (neg C)) X Y (neg Sign)) %<- (timesify (B * C) X Y Sign)) [p1 pf (B * C * Sign == X * Y)] trans4 times_assoc (times_cong2 neg_times) (symm times_assoc) p1 %. +%clause timesify_swap (%pi (timesify (B * (C * D)) X Y Sign) %<- (timesify (B * C * D) X Y Sign)) [p1] trans (times_cong1 (symm times_assoc)) p1 %. +%clause timesify_zero (timesify (B * zero) one zero one) trans3 times_one times_zero (symm times_zero) %. +%clause timesify_remove (%pi (timesify (B * one) X Y Sign) %<- (timesify B X Y Sign)) [p1 pf (B * Sign == X * Y)] cut (trans3 times_assoc (times_cong2 comm_times) assoc_times) ([p2 pf (B * one * Sign == B * Sign * one)] trans3 p2 times_one p1) %. +%clause timesify_end_plus (timesify (B * (P + Q)) B (P + Q) one) times_one %. +%clause timesify_step (%pi (timesify (B * C) (X * C) Y Sign) %<- (timesify B X Y Sign)) [p1 pf (B * Sign == X * Y)] trans3 (trans3 times_assoc (times_cong2 comm_times) assoc_times) (times_cong1 p1) (trans3 times_assoc (times_cong2 comm_times) assoc_times) %. +%clause timesify_end_one (timesify one one one one) refl %. +%{! #### Catchtimes !}% +%{! ``catchtimes A B X Y Sign Pos Neg`` - unravels to the correct sign +* A - The as-yet unprocessed bit of the equation +* B - The single term that was passed to timesify +* X - The list returned from timesify +* Y - The signal variable from timesify +* Sign - the correct sign +* Pos - The master positive term list +* Neg - The master negative term list +!}% +%clause catch_neg1 ( + %pi (catchtimes A B (neg X) Y (neg Sign) Pos Neg) + %<- (catchtimes A B X Y Sign Pos Neg)) [p1 pf (B * Sign == X * Y imp A + B == Pos - Neg)] imp_i ([p2 pf (B * neg Sign == neg X * Y)] cut (trans3 (symm neg_times2) p2 neg_times1) ([p3 pf (neg (B * Sign) == neg (X * Y))] imp_e p1 (neg_elim p3))) %. +%clause catch_neg2 ( + %pi (catchtimes A B X Y (neg Sign) Pos Neg) + %<- (catchtimes A B (neg X) Y Sign Pos Neg)) [p1 pf (B * Sign == neg X * Y imp A + B == Pos - Neg)] imp_i ([p2 pf (B * neg Sign == X * Y)] cut (trans3 (symm neg_neg_e) (neg_cong (trans (symm neg_times2) p2)) (symm neg_times1)) ([p3 pf (B * Sign == neg X * Y)] imp_e p1 p3)) %. +%clause catch_zero (%pi (catchtimes A B X zero one Pos Neg) %<- (plusify A Pos Neg)) [p1 pf (A == Pos - Neg)] imp_i ([p2 pf (B * one == X * zero)] cut (trans3 (symm times_one) p2 times_zero) ([p3 pf (B == zero)] trans3 (plus_cong2 p3) plus_zero p1)) %. +%clause catch_neg_one (%pi (catchtimes A B (neg X) one one Pos (Neg + X)) %<- (plusify A Pos Neg)) [p1 pf (A == Pos - Neg)] imp_i ([p2 pf (B * one == neg X * one)] cut (trans3 (symm times_one) p2 times_one) ([p3 pf (B == neg X)] cut (trans (plus_cong p1 p3) plus_assoc) ([p4 pf (A + B == Pos + (neg Neg - X))] trans p4 (plus_cong2 neg_plus)))) %. +%clause catch_one (%pi (catchtimes A B X one one (Pos + X) Neg) %<- (plusify A Pos Neg)) [p1 pf (A == Pos - Neg)] imp_i ([p2 pf (B * one == X * one)] cut (trans3 (symm times_one) p2 times_one) ([p3 pf (B == X)] trans4 (plus_cong p1 p3) plus_assoc (plus_cong2 comm_plus) assoc_plus)) %. +%clause catch_break ( + %pi (catchtimes A B X (P + Q) one Pos Neg) + %<- (plusify (A + X * P + X * Q) Pos Neg)) [p1 pf (A + X * P + X * Q == Pos - Neg)] imp_i ([p2 pf (B * one == X * (P + Q))] cut (trans3 (symm times_one) p2 distrib) ([p3 pf (B == X * P + X * Q)] trans3 (plus_cong2 p3) assoc_plus p1)) %. +%{! #### Plusify !}% +%{! ``plusify`` - Creates a zero-termiated list (0 + A + B + C +...) !}% +%clause plusify_minus (%pi (plusify (A - (B + C)) Pos Neg) %<- (plusify (A - B - C) Pos Neg)) [p1] trans3 (plus_cong2 (symm neg_plus)) assoc_plus p1 %. +%clause plusify_swap (%pi (plusify (A + (B + C)) Pos Neg) %<- (plusify (A + B + C) Pos Neg)) [p1] trans assoc_plus p1 %. +%clause plusify_neg_one (%pi (plusify (A + neg one * B) Pos Neg) %<- (plusify (A - B) Pos Neg)) [p1] trans (plus_cong2 (trans neg_times1 (neg_cong one_times))) p1 %. +%clause plusify_neg_one' (%pi (plusify (A + B * (neg one)) Pos Neg) %<- (plusify (A - B) Pos Neg)) [p1] trans (plus_cong2 comm_times) (plusify_neg_one p1) %. +%clause plusify_neg (%pi (plusify (A - (neg B)) Pos Neg) %<- (plusify (A + B) Pos Neg)) [p1] trans (plus_cong2 neg_neg_e) p1 %. +%clause plusify_remove_neg (%pi (plusify (A - zero) Pos Neg) %<- (plusify A Pos Neg)) [p1] trans3 (plus_cong2 neg_zero) plus_zero p1 %. +%clause plusify_remove (%pi (plusify (A + zero) Pos Neg) %<- (plusify A Pos Neg)) [p1] trans plus_zero p1 %. +%clause plusify_step_mult_neg ( + %pi (plusify (A - (B * C)) Pos Neg) + %<- (timesify (one * B * neg C) X Y Sign) + %<- (catchtimes A (B * neg C) X Y Sign Pos Neg)) [p1 pf (B * neg C * Sign == X * Y imp A + B * neg C == Pos - Neg)] [p2 pf (one * B * neg C * Sign == X * Y)] cut (trans5 times_assoc (symm one_times) assoc_times assoc_times p2) ([p3 pf (B * neg C * Sign == X * Y)] cut (imp_e p1 p3) ([p4 pf (A + B * neg C == Pos - Neg)] trans (plus_cong2 (symm neg_times2)) p4)) %. +%clause plusify_step_mult ( + %pi (plusify (A + (B * C)) Pos Neg) + %<- (timesify (one * B * C) X Y Sign) + %<- (catchtimes A (B * C) X Y Sign Pos Neg)) [p1 pf (B * C * Sign == X * Y imp A + B * C == Pos - Neg)] [p2 pf (one * B * C * Sign == X * Y)] cut (trans5 times_assoc (symm one_times) assoc_times assoc_times p2) ([p3 pf (B * C * Sign == X * Y)] imp_e p1 p3) %. +%clause plusify_step_neg_one (%pi (plusify (A - one) Pos (Neg + one)) %<- (plusify A Pos Neg)) [p1 pf (A == Pos - Neg)] trans3 (plus_cong1 p1) plus_assoc (plus_cong2 neg_plus) %. +%clause plusify_step_one (%pi (plusify (A + one) (Pos + one) Neg) %<- (plusify A Pos Neg)) [p1 pf (A == Pos - Neg)] trans4 (plus_cong1 p1) plus_assoc (plus_cong2 comm_plus) assoc_plus %. +%clause plusify_step_neg (%pi (plusify (A - B) Pos (Neg + (one * B))) %<- (plusify A Pos Neg)) [p1 pf (A == Pos - Neg)] trans3 (plus_cong p1 (neg_cong (symm one_times))) plus_assoc (plus_cong2 neg_plus) %. +%clause plusify_step (%pi (plusify (A + B) (Pos + (one * B)) Neg) %<- (plusify A Pos Neg)) [p1 pf (A == Pos - Neg)] trans (plus_cong p1 (symm one_times)) (trans3 plus_assoc (plus_cong2 comm_plus) assoc_plus) %. +%clause plusify_zero (plusify zero zero zero) symm plus_inv %. +%{! #### Testing !}% +%solve __P : {a} {b} listify (a * (b - one)) _ _ %. +%solve __P : {a} listify (succ a * pred a) _ _ %. +% +%solve __P : {a} listify (pred (a * a) + (a - a)) _ _ %. +%solve __P : listify zero _ _ %. +%solve __P : {a} {b} {m} {n} {q} listify (a * one + (a * m + n * b) * q) _ _ %. +%solve __P : {a} {b} {m} {n} {q} listify (a + q * a * m + q * n * b) _ _ %. +%solve __P : {a} {b} listify (zero + (a + b)) _ _ %. +%solve __P : {a} {b} {c} {d} {e} listify (c + (d + e) + (a + b)) _ _ %. +% (zero + c + d + e + a + b). +%solve __P : {a} {b} {d} listify (a * (succ one) + (b * (succ d) + d) * a) _ _ %. +%solve __P : listify (one * (one + one) + (one * (one + one) + one) * one) _ _ %. +%solve __P : {a} {b} listify (a * b) _ _ %. +%solve __P : {a} {b} {c} listify (a + c * (c * c) * one + b * (b * b) * zero) _ _ %. +%solve __P : {a} listify (neg (one * neg a)) _ _ %. +%solve __P : {a} {b} {c} listify (zero - (a + neg b * c) + a * neg b) _ _ %. +%solve __P : {a} {b} {c} listify (zero - (a + b * neg c) + a * neg b) _ _ %. +%solve __P : {a} listify (one * (neg a)) _ _ %. +%solve __P : {a} {b} {c} timesify (one * (neg a * neg (b * neg c))) _ _ _ %. +%solve __P : {a} {b} listify (neg a * neg b) _ _ %. +%solve __P : listify (neg one * neg one * neg one * neg one) _ _ %. +%solve __P : {a} listify a _ _ %. +%solve __P : listify (neg (neg (neg (neg (neg (neg zero))))) * one + zero) _ _ %. +%solve __P : {a} {b} timesify (one * (a + b)) _ _ _ %. +%solve __P : {a} listify (pred a * succ a) _ _ %. +%{! ### Matching !}% +%{! #### Trim !}% +%{! TRIM - takes two lists, and makes a single pass at removing items from the +first list and putting them in the second list. Fails unless at least one +element (the top element in the first list) is trimmed. !}% +%clause trim_get (trim (A + C) (B + C) A B) cut (trans plus_assoc (plus_cong2 (plus_cong2 (symm neg_plus)))) ([p2 pf (A + C - (B + C) == A + (C + (neg B - C)))] cut (trans (plus_cong2 assoc_plus) (plus_cong2 cancel_a+b-a)) ([p3 pf (A + (C + (neg B - C)) == A - B)] trans p2 p3)) %. +%clause trim_getmult (%pi (trim (A + C) (B + D) A B) %<- (findarr_mult C D one nomatch)) [p2 pf (C == D * one and zero == zero)] cut (trans (and_e1 p2) times_one) ([p3 pf (C == D)] cut (congr ([i] C - B - i == neg B) (symm p3) cancel_a+b-a) ([p4 pf (C - B - D == neg B)] cut (trans3 (plus_cong2 (symm neg_plus)) assoc_plus p4) ([p5 pf (C - (B + D) == neg B)] trans plus_assoc (plus_cong2 p5)))) %. +%clause trim_step1 (%pi (trim (A + C) B (X + C) Y) %<- (trim A B X Y)) [p1 pf (A - B == X - Y)] cut (trans3 (trans3 plus_assoc (plus_cong2 comm_plus) assoc_plus) (plus_cong1 p1) (trans3 plus_assoc (plus_cong2 comm_plus) assoc_plus)) ([p2 pf (A + C - B == X + C - Y)] p2) %. +%clause trim_step2 (%pi (trim A (B + C) X (Y + C)) %<- (trim A B X Y)) [p1 pf (A - B == X - Y)] trans3 (plus_cong2 (symm neg_plus)) (trans3 assoc_plus (plus_cong1 p1) plus_assoc) (plus_cong2 neg_plus) %. +%{! #### Findarr_mult !}% +% FINDARR_MULT - more complex than trim, cycles through a product to +% Uses a quadratic algorithm, but in most cases the length of the proof +% is dominated by the list creation, not the +%clause fam_neg (%pi (findarr_mult (neg X) (neg Y) Z M) %<- (findarr_mult X Y Z M)) [p1 pf (X == Y * Z and M == M)] and_i (trans (neg_cong (and_e1 p1)) (symm neg_times1)) refl %. +%clause fam_step (%pi (findarr_mult (X * B) (Y * B) Z M) %<- (findarr_mult X Y Z match)) [p1 pf (X == Y * Z and match == match)] cut (trans (times_cong1 (and_e1 p1)) (trans3 times_assoc (times_cong2 comm_times) assoc_times)) ([p2 pf (X * B == Y * B * Z)] and_i p2 refl) %. +%clause fam_step2 ( + %pi (findarr_mult (X * B) (Y * C) Z M) + %<- (atom_replace B C) + %<- (findarr_mult X Y Z match)) [p1] [p2] congr ([i] X * B == Y * i * Z and M == M) (symm p2) (fam_step p1) %. +%clause fam_step2' ( + %pi (findarr_mult (X * B) (Y * C) Z M) + %<- (atom_replace C B) + %<- (findarr_mult X Y Z match)) [p1] [p2] congr ([i] X * B == Y * i * Z and M == M) p2 (fam_step p1) %. +%clause fam_swap (%pi (findarr_mult X (Y * B) Z M) %<- (findarr_mult X Y (Z * B) M)) [p1 pf (X == Y * (Z * B) and M == M)] cut p1 ([_] and_i (trans3 (and_e1 p1) (times_cong2 comm_times) (symm times_assoc)) refl) %. +%clause fam_finish (findarr_mult one one one M) and_i (symm times_one) refl %. +%clause fam_one (%pi (findarr_mult X one Y match) %<- (findarr_mult X Y one nomatch)) [p1 pf (X == Y * one and nomatch == nomatch)] and_i (trans (and_e1 p1) comm_times) refl %. +%{! #### Rearrange !}% +% REARRANGE +% Handles the finishing steps of the solver +%clause rearr_finish (rearrange zero zero zero zero) refl %. +%clause rearr_end (%pi (rearrange A B zero zero) %<- (trim A B X Y) %<- (rearrange X Y zero zero)) [p1 pf (X - Y == zero - zero)] [p2 pf (A - B == X - Y)] trans p2 p1 %. +%clause rearr_right (%pi (rearrange zero zero C D) %<- (trim C D X Y) %<- (rearrange X Y zero zero)) [p1] [p2] symm (rearr_end p1 p2) %. +%clause rearr_pos (%pi (rearrange A zero C zero) %<- (trim A C X Y) %<- (rearrange X Y zero zero)) [p1 pf (X - Y == zero - zero)] [p2 pf (A - C == X - Y)] cut (trans3 p2 p1 plus_inv) ([p3 pf (A - C == zero)] cut (trans3 (symm cancel_a-b+b) (plus_cong1 p3) zero_plus) ([p4 pf (A == C)] plus_cong1 p4)) %. +%clause rearr_neg (%pi (rearrange zero B zero D) %<- (trim B D X Y) %<- (rearrange X Y zero zero)) [p1 pf (X - Y == zero - zero)] [p2 pf (B - D == X - Y)] cut (trans3 p2 p1 plus_inv) ([p3 pf (B - D == zero)] cut (trans3 (symm cancel_a-b+b) (plus_cong1 p3) zero_plus) ([p4 pf (B == D)] plus_cong2 (neg_cong p4))) %. +%define plus_swap (%pi (pf (A - B == C - D)) %-> (pf (A - C == B - D))) [p1] cut (trans3 (symm cancel_a-b+b) (plus_cong1 p1) comm_plus) ([p2 pf (A == B + (C - D))] cut (trans3 (plus_cong1 p2) plus_assoc (plus_cong2 cancel_a+b-a)) ([p3 pf (A - C == B - D)] p3)) %. +%define plus_swap2 (%pi (pf (A + B == C + D)) %-> (pf (A - D == C - B))) [p1] cut (trans5 (symm cancel_a+b-b) (plus_cong1 p1) plus_assoc (plus_cong2 comm_plus) assoc_plus) ([p2 pf (A == C - B + D)] trans (plus_cong1 p2) cancel_a+b-b) %. +%clause rearr_one ( + %pi (rearrange A B C zero) + %<- (trim A C X Y) + %<- (trim X B Z W) + %<- (rearrange Z W Y zero)) [p1 pf (Z - W == Y - zero)] [p2 pf (X - B == Z - W)] [p3 pf (A - C == X - Y)] cut (plus_swap (trans p2 p1)) ([p4 pf (X - Y == B - zero)] plus_swap (trans p3 p4)) %. +%clause rearr_two ( + %pi (rearrange A B zero D) + %<- (trim B D X Y) + %<- (trim A X Z W) + %<- (rearrange Z W zero Y)) [p1 pf (Z - W == zero - Y)] [p2 pf (A - X == Z - W)] [p3 pf (B - D == X - Y)] cut (plus_swap (trans p2 p1)) ([p4 pf (A - zero == X - Y)] plus_swap (trans p4 (symm p3))) %. +%clause rearr_three ( + %pi (rearrange C zero A B) + %<- (trim A C X Y) + %<- (trim X B Z W) + %<- (rearrange Z W Y zero)) [p1] [p2] [p3] symm (rearr_one p1 p2 p3) %. +%clause rearr_four ( + %pi (rearrange zero D A B) + %<- (trim B D X Y) + %<- (trim A X Z W) + %<- (rearrange Z W zero Y)) [p1] [p2] [p3] symm (rearr_two p1 p2 p3) %. +%clause rearr_any1 (%pi (rearrange A B C D) %<- (trim A B X Y) %<- (rearrange X C Y D)) [p1 pf (X - C == Y - D)] [p2 pf (A - B == X - Y)] trans p2 (plus_swap p1) %. +%clause rearr_any2 (%pi (rearrange A B C D) %<- (trim A C X Y) %<- (rearrange X B Y D)) [p1 pf (X - B == Y - D)] [p2 pf (A - C == X - Y)] plus_swap (trans p2 (plus_swap p1)) %. +%clause solver_short1 (algebra_solver (A == A)) refl %. +%clause solver_short2a (%pi (algebra_solver (A + B == A + C)) %<- (algebra_solver (B == C))) plus_cong2 %. +%clause solver_short2b (%pi (algebra_solver (A + B == C + B)) %<- (algebra_solver (A == C))) plus_cong1 %. +%clause solver_rearrange ( + %pi (algebra_solver (A == B)) + %<- (listify A Pos Neg) + %<- (listify B Pos' Neg') + %<- (rearrange Pos Neg Pos' Neg')) [p1] [p2] [p3] trans3 p3 p1 (symm p2) %. +%clause fact_one_gt (algebra_fact (one > zero)) one>zero %. +%clause fact_gt->geq (%pi (algebra_fact (A >= B)) %<- (algebra_fact (A > B))) gt->geq %. +%clause fact_gt->neq (%pi (algebra_fact (A != B)) %<- (algebra_fact (A > B))) gt->neq %. +%clause fact_lt->neq (%pi (algebra_fact (A != B)) %<- (algebra_fact (A < B))) lt->neq %. +%clause fact_eq->geq (%pi (algebra_fact (A >= B)) %<- (algebra_fact (A == B))) eq->geq %. +%clause fact_eq->leq (%pi (algebra_fact (A <= B)) %<- (algebra_fact (A == B))) eq->leq %. +%clause hintmod1 ( + %pi (algebra_times_hint_intern A B) + %<- (algebra_times_hint A) + %<- (algebra_times_hint B)) [p1] [p2] cut2 p1 p2 ([_] [_] refl) %. +%clause hintmod2 (%pi (algebra_times_hint_intern A one) %<- (algebra_times_hint A)) [p1] cut p1 ([_] refl) %. +%clause hintmod3 (%pi (algebra_times_hint_intern one A) %<- (algebra_times_hint A)) [p1] cut p1 ([_] refl) %. +%define eq_arrange (%pi (pf (A - B == C - D)) %-> (pf (A == B)) %-> (pf (C == D))) [p1 pf (A - B == C - D)] [p2 pf (A == B)] cut (trans3 (symm p1) (plus_cong1 p2) plus_inv) ([p3 pf (C - D == zero)] trans3 (symm cancel_a-b+b) (plus_cong1 p3) zero_plus) %. +%clause solver_eq (%pi (algebra_solver (A == B imp C == D)) %<- (algebra_solver (A + D == C + B))) [p1] imp_i ([p2] eq_arrange (plus_swap2 p1) p2) %. +%clause solver_eq_times ( + %pi (algebra_solver (A == B imp C == D)) + %<- (algebra_times_hint_intern X Y) + %<- (algebra_fact (Y != zero)) + %<- (algebra_solver (A * X + D * Y == C * Y + B * X))) [p1] [p2] [p3] cut p3 ([_] imp_i ([p4] cut (imp_e (solver_eq p1) (times_cong1 p4)) ([p5 pf (C * Y == D * Y)] times_elim2 p2 p5))) %. +%clause solver_eq_neg (%pi (algebra_solver (A == B imp C == D)) %<- (algebra_solver (D + B == C + A))) [p1] imp_i ([p2] eq_arrange (plus_swap2 (trans comm_plus (plus_swap2 p1))) (neg_cong p2)) %. +%clause solver_eq_neg_times ( + %pi (algebra_solver (A == B imp C == D)) + %<- (algebra_times_hint_intern X Y) + %<- (algebra_fact (Y != zero)) + %<- (algebra_solver (D * Y + B * X == C * Y + A * X))) [p1] [p2] [p3] cut p3 ([_] imp_i ([p4] cut (imp_e (solver_eq_neg p1) (times_cong1 p4)) ([p5 pf (C * Y == D * Y)] times_elim2 p2 p5))) %. +%define geq_arrange (%pi (pf (A + D == C + B)) %-> (pf (A >= B)) %-> (pf (C >= D))) [p0 pf (A + D == C + B)] [p1 pf (A >= B)] cut (plus_swap2 p0) ([p2 pf (A - B == C - D)] cut (trans_geq_eq (geq_plus_cong1 p1) plus_inv) ([p3 pf (A - B >= zero)] cut (trans_eq_geq (symm p2) p3) ([p4 pf (C - D >= zero)] cut (trans_eq_geq (symm cancel_a-b+b) (geq_plus_cong1 p4)) ([p5 pf (C >= zero + D)] trans_geq_eq p5 zero_plus)))) %. +%clause solver_geq (%pi (algebra_solver (A >= B imp C >= D)) %<- (algebra_solver (A + D == C + B))) [p1] imp_i ([p2] geq_arrange p1 p2) %. +%clause solver_geq_times1 ( + %pi (algebra_solver (A >= B imp C >= D)) + %<- (algebra_times_hint_intern X Y) + %<- (algebra_fact (X >= zero)) + %<- (algebra_fact (Y > zero)) + %<- (algebra_solver (A * X + D * Y == C * Y + B * X))) [p1] [p2] [p3] [p4] cut p4 ([_] imp_i ([p5] cut (imp_e (solver_geq p1) (geq_times_cong1 p3 p5)) ([p6 pf (C * Y >= D * Y)] geq_times_elim2 p2 p6))) %. +%clause solver_geq_times2 ( + %pi (algebra_solver (A <= B imp C >= D)) + %<- (algebra_times_hint_intern X Y) + %<- (algebra_fact (X <= zero)) + %<- (algebra_fact (Y > zero)) + %<- (algebra_solver (A * X + D * Y == C * Y + B * X))) [p1] [p2] [p3] [p4] cut p4 ([_] imp_i ([p5] cut3 p1 p2 p3 ([_] [_] [_] cut (imp_e (solver_geq p1) (geq_times_cong1_neg p3 p5)) ([p6 pf (C * Y >= D * Y)] geq_times_elim2 p2 p6)))) %. +%clause solver_geq_times3 ( + %pi (algebra_solver (A >= B imp C <= D)) + %<- (algebra_times_hint_intern X Y) + %<- (algebra_fact (X >= zero)) + %<- (algebra_fact (Y < zero)) + %<- (algebra_solver (A * X + D * Y == C * Y + B * X))) [p1] [p2] [p3] [p4] cut p4 ([_] imp_i ([p5] cut3 p1 p2 p3 ([_] [_] [_] cut (imp_e (solver_geq p1) (geq_times_cong1 p3 p5)) ([p6 pf (C * Y >= D * Y)] geq_times_elim2_neg p2 p6)))) %. +%clause solver_geq_times4 ( + %pi (algebra_solver (A <= B imp C <= D)) + %<- (algebra_times_hint_intern X Y) + %<- (algebra_fact (X <= zero)) + %<- (algebra_fact (Y < zero)) + %<- (algebra_solver (A * X + D * Y == C * Y + B * X))) [p1] [p2] [p3] [p4] cut p4 ([_] imp_i ([p5] cut3 p1 p2 p3 ([_] [_] [_] cut (imp_e (solver_geq p1) (geq_times_cong1_neg p3 p5)) ([p6 pf (C * Y >= D * Y)] geq_times_elim2_neg p2 p6)))) %. +%clause solver_neq (%pi (algebra_solver (A != B imp C != D)) %<- (algebra_solver (C == D imp A == B))) [p1] imp_i ([p2] not_eie p2 (imp_e p1)) %. +%clause solver_gt (%pi (algebra_solver (A < B imp C < D)) %<- (algebra_solver (C >= D imp A >= B))) [p1] imp_i ([p2] not_ein p2 (imp_e p1)) %. +%{! ## Test cases for the solver !}% +%solve __P : {a} {b} {c} algebra_solver (a > b imp a + c > c + b) %. +%solve __P : {a} {b} {c} algebra_solver (a - b + c >= zero imp a >= b - c) %. +%solve __P : {a} {b} {c} algebra_solver (a >= b imp neg a + c <= neg b + c) %. +%solve __P : algebra_solver (one - one == zero) %. +%solve __P : algebra_solver (one - one == two - two) %. +%solve __P : {a} algebra_solver (two - two + a == one + a * (one + zero) - one) %. +%solve __P : {a} {b} {c} {d} algebra_solver (a - b == c - d imp a - c == b - d) %. +%solve __P : {a} algebra_solver (succ a * pred a == pred (a * a)) %. +%solve __P : {a} {b} {c} algebra_solver (a - (b - c) == (a - b) + (c * one)) %. +%solve __P : {a} {b} {c} algebra_solver (a + b == c * (one - one) + b + a) %. +%solve __P : {a} {b} {c} {d} {e} algebra_solver (a - b + (c - d) + e == e - d + (neg b + c + a)) %. +%solve __P : {a} {b} {c} algebra_solver (c + (a + b) == a + zero + zero + b + zero + c) %. +%solve __P : {a} {b} algebra_solver (neg (a + b) == (neg a) + (neg b)) %. +%solve __P : {a} {b} {c} algebra_solver (a - (b - c) == (a + c) - b) %. +%solve __P : {a} {b} {c} algebra_solver (neg one * (neg one * a + neg one * (b + c)) == neg (neg one * a) - (neg one * (b + c))) %. +%solve __P : {a} {b} {c} {d} {e} {f} {g} {h} {i} algebra_solver (a * (b + (c * (d + (e * (f + (g * (h + i))))))) == a * (b + (c * (d + (e * (f + (g * h)))))) + a * (c * one * e) * g * i) %. +%solve __P : {a} {b} {c} {d} {e} {f} algebra_solver (a + b + c + d + e + f == f + e + d + c + b + a) %. +%solve __P : {a} {b} algebra_solver (a * b == b * a) %. +%solve __P : {a} {b} {c} {d} algebra_solver (a * (b + c) + (d * (c + a)) == c * d + (a * (b + c + d))) %. +%solve __P : {a} {b} {d} algebra_solver (a * (d * (one + b) + one + b) + a == a * (succ one) + (b * (succ d) + d) * a) %. +%solve __P : {a} {b} {c} algebra_solver (c + b + (a - one) * b == c + a * b + b - b) %. +%solve __P : {a} {b} {m} {n} {q} algebra_solver (a * one + (a * m + n * b) * q == a * (one + m * q) + n * b * q) %. +%solve __P : {a} {b} {m} {n} {q} algebra_solver (a * one + (a * m + n * b) * q == a * one + q * a * m + q * n * b) %. +%solve __P : {a} {b} algebra_solver (a * zero + zero == zero * b) %. +%solve __P : {a} {b} algebra_solver (a * (b * b) + a * zero == a * b * b) %. +%solve __P : {a} {b} {c} algebra_solver (a * one * zero * b + c * (a + b) == b * c + zero * a + c * (a + zero)) %. +%solve __P : algebra_solver (neg zero == zero) %. +%solve __P : {a} {b} algebra_solver (a * (b * b) * one + a * zero == a * b * b) %. +%query 0 _ _ %the ({a} {b} algebra_solver (a == b)) __P %. +%solve __P : {a} {b} {c} algebra_solver (a * b * c == c * b * a) %. +%solve __P : {a} {b} {m} {n} {q} algebra_solver (a - (m * a + n * b) * q == (one - q * m) * a + (neg (q * n)) * b) %. +%solve __P : {a} {b} algebra_solver (a == b imp neg a == neg b) %. +%solve __P : {a} {b} algebra_solver (a > b imp neg a < neg b) %. +%solve __P : {a} {b} algebra_solver (a * one == b imp a == b * one) %. +%solve __P : {a} {b} algebra_solver (a * one >= b imp a >= b * one) %. +%solve __P : {a} {b} algebra_solver (a >= b imp neg b >= neg a) %. +%solve __P : {a} {b} algebra_solver (a != b imp neg a != neg b) %. +%solve __P : {a} {b} {c} algebra_solver (a > b imp c < a + (b - c) * neg one) %. +%solve __P : {a} {b} %pi (algebra_times_hint b) %-> (algebra_solver (a == one imp a * b == b)) %. +%query _ 1 _ %the ({a} {b} algebra_solver (a * b + a == a + b * a)) __P %. +%query 1 _ _ %the ({a} {b} algebra_solver (a * b + a == a + b * a)) __P %. +%solve __P : {a} {b} {c} algebra_solver (a - b == a - b + c - c) %. +%solve __P : {a} {b} {c} algebra_solver (a == b imp a + c == b + c) %. +%solve __P : {a} {b} {c} {d} + %pi (atom_replace a c) + %-> (atom_replace d c) + %-> (algebra_solver (a + b + c + c == b + c + c + d)) %. +%solve __P : {a} {b} {c} %pi (atom_replace b c) %-> (algebra_solver (a + b == c + a)) %. +%solve __P : {a} {b} {c} %pi (atom_replace b c) %-> (algebra_solver (a == b imp a - c == zero)) %. +%solve __P : {a} {b} {c} %pi (atom_replace b c) %-> (algebra_solver (a - b == neg c + a)) %. +%solve __P : {a} {b} {c} %pi (atom_replace b c) %-> (algebra_solver (a - a == b - c)) %. +%solve __P : {a} {b} {c} {d} + %pi (algebra_times_hint c) + %-> (algebra_solver (a - b == d imp a * c == c * d + c * b)) %. +%solve __P : {a} {b} {c} + %pi (algebra_times_hint a) + %-> (algebra_fact (a != zero)) + %-> (algebra_solver (neg a * c == neg (b * a) imp b - c == zero)) %. +%solve __P : {a} {b} {c} {d} + %pi (algebra_times_hint (a + d)) + %-> (algebra_fact (a + d != zero)) + %-> (atom_replace b d) + %-> (algebra_solver (neg a - b * c == neg (d * a) - d * b imp b == c)) %. +%solve __P : {a} {b} {c} {d} + %pi (algebra_times_hint (a + b * c)) + %-> (algebra_fact (a + b * c > zero)) + %-> (algebra_solver (d > one imp a * (d - one) > b * c * (one - d))) %. +%query 0 _ _ %the ({a} {b} {c} {d} {e} + %pi (algebra_times_hint (a + e)) + %-> (algebra_fact (a + e > zero)) + %-> (algebra_times_hint b) + %-> (algebra_solver (c * (a + e) == d * a imp b * c == b * d))) __P %. +%solve __P : {a} {b} {c} {d} {e} {f} {g} + %pi (algebra_times_hint (a + b * c)) + %-> (algebra_fact (a + b * c > zero)) + %-> (algebra_times_hint d) + %-> (algebra_solver (a + b * c * (e * f - one) == (g + one) * (a + b * c) imp d * (e * f - one) == d * (g + one))) %. +%solve ex_27 : {a} {b} {c} %pi (algebra_times_hint a) %-> (algebra_solver (a * (b - c) != zero imp b != c)) %. +%solve ex_28 : {a} {b} {c} + %pi (algebra_times_hint (a + b)) + %-> (algebra_fact (a + b < zero)) + %-> (algebra_solver (a * (c - zero) - b == a - b * c imp zero == one - c)) %. +%solve ex_29 : {a} {b} {c} {d} + %pi (algebra_times_hint a) + %-> (algebra_fact (a < zero)) + %-> (algebra_times_hint b) + %-> (algebra_fact (b >= zero)) + %-> (algebra_solver (c * a >= a * (d + zero) imp b * c * one <= b * d)) %. +%define ex_30 ( + %pi (pf (A < zero)) + %-> (pf (B >= zero)) + %-> (pf (C * A >= A * (D + zero))) + %-> (pf (B * C * one <= B * D))) [p1] [p2] [p3] imp_e (ex_29 A B C D refl p1 refl p2) p3 %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Big_algebraic_solver). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/bracket-abstraction.lf b/new-tests/stelf-output/pages/bracket-abstraction.lf new file mode 100644 index 0000000..2195a74 --- /dev/null +++ b/new-tests/stelf-output/pages/bracket-abstraction.lf @@ -0,0 +1,153 @@ +% +%{! +This is a case study translating the untyped lambda calculus into S, K, and I +combinators. The correctness of the translation is proven in the following +sense: if a term steps to a reduct, its translation multi-steps to its +reduct's translation. + +[William Lovas](/wiki/user-william-lovas/) +!}% +%{! ---- !}% +%{! +First we define the syntax of the two languages. ``term`` is the +type of untyped lambda-calculus terms; ``cterm`` is the type of +untyped combinator terms. +!}% +%sort term %. +%name term %. +%term app %pi term %-> term %-> term %. +%term lam %pi (%pi term %-> term) %-> term %. +%sort cterm %. +%name cterm %. +%term s cterm %. +%term k cterm %. +%term i cterm %. +%term capp %pi cterm %-> cterm %-> cterm %. +%{! We can use Twelf's abbreviation mechanism to obtain some cute syntax. !}% +%inline @ app %. +%prec %left 10 @ %. +%inline @@ capp %. +%prec %left 10 @@ %. +%{! ---- !}% +%{! +Then we define reduction relations on both languages. ``step`` is the +single-step reduction relation on lambda terms; ``cstep`` is the +single-step reduction relation on combinator terms. +!}% +%sort step %. +%term s-β step (app (lam ([x] M1 x)) M2) (M1 M2) %. +%term s-1 %pi (step (app M1 M2) (app M1' M2)) %<- (step M1 M1') %. +%term s-2 %pi (step (app M1 M2) (app M1 M2')) %<- (step M2 M2') %. +%sort cstep %. +%term cs-i cstep (capp i X) X %. +%term cs-k cstep (capp (capp k X) Y) X %. +%term cs-s cstep (capp (capp (capp s X) Y) Z) (capp (capp X Z) (capp Y Z)) %. +%term cs-1 %pi (cstep (capp X Y) (capp X' Y)) %<- (cstep X X') %. +%term cs-2 %pi (cstep (capp X Y) (capp X Y')) %<- (cstep Y Y') %. +%{! +We also define multi-step reduction on combinator terms. + +Our simulation will relate single-step derivations in the lambda-calculus +to multi-step derivations on the translated terms. +!}% +%sort cstep* %. +%term cs-cons %pi (cstep* N N'') %<- (cstep N N') %<- (cstep* N' N'') %. +%term cs-nil cstep* N N %. +%{! +We can pre-emptively prove some compatibility lemmas about multi-step +reduction. +!}% +%sort cs-1* %. +%mode {%in N1 cterm} {%in N1' cterm} {%in N2 cterm} {%in CS1 cstep* N1 N1'} {%out CS2 cstep* (capp N1 N2) (capp N1' N2)} cs-1* CS1 CS2 %. +%term _ cs-1* cs-nil cs-nil %. +%term _ %pi (cs-1* (cs-cons CS C) (cs-cons CS' (cs-1 C))) %<- (cs-1* CS CS') %. +%worlds () (cs-1* _ _) %. +%total {CS} (cs-1* CS _) %. +%sort cs-2* %. +%mode {%in N1 cterm} {%in N2 cterm} {%in N2' cterm} {%in CS1 cstep* N2 N2'} {%out CS2 cstep* (capp N1 N2) (capp N1 N2')} cs-2* CS1 CS2 %. +%term _ cs-2* cs-nil cs-nil %. +%term _ %pi (cs-2* (cs-cons CS C) (cs-cons CS' (cs-2 C))) %<- (cs-2* CS CS') %. +%worlds () (cs-2* _ _) %. +%total {CS} (cs-2* CS _) %. +%sort cs-trans %. +%mode cs-trans %in %in %out %. +%term _ cs-trans cs-nil Cs2 Cs2 %. +%term _ %pi (cs-trans (cs-cons Cs1 C) Cs2 (cs-cons Cs12 C)) %<- (cs-trans Cs1 Cs2 Cs12) %. +%worlds () (cs-trans _ _ _) %. +%total {Cs} (cs-trans Cs _ _) %. +%{! ---- !}% +%{! +We can now define our translation in the standard way. Bracket abstraction +is represented by a judgement relating LF-level abstractions in the combinator +language to closed combinator terms. This is essentially +[higher-order abstract syntax](/wiki/higher-order-abstract-syntax/). +!}% +%sort trans %. +%sort bracket %. +%term t-app %pi (trans (app M1 M2) (capp N1 N2)) %<- (trans M1 N1) %<- (trans M2 N2) %. +%term t-lam + %pi (trans (lam ([x] M x)) N') + %<- ({x term} {y cterm} %pi (trans x y) %-> (trans (M x) (N y))) + %<- (bracket ([y] N y) N') %. +%term b-i bracket ([x] x) i %. +%term b-k bracket ([x] Y) (capp k Y) %. +%term b-s + %pi (bracket ([x] capp (N1 x) (N2 x)) (capp (capp s N1') N2')) + %<- (bracket ([x] N1 x) N1') + %<- (bracket ([x] N2 x) N2') %. +%{! ---- !}% +%{! +First, we prove the correctness of bracket abstraction itself: +the application of a bracket abstraction reduces to a substitution. +!}% +%sort subst %. +%mode subst %in %in %out %. +%term _ subst b-i N0 (cs-cons cs-nil cs-i) %. +%term _ subst b-k N0 (cs-cons cs-nil cs-k) %. +% developing incrementally, it's useful to write down the type of each output +%term _ + %pi (subst (b-s (%the (bracket ([x] N2 x) N2') B2) (%the (bracket ([x] N1 x) N1') B1)) N0 (cs-cons CS12' cs-s)) + %<- (subst B1 N0 (%the (cstep* (N1' @@ N0) (N1 N0)) CS1)) + %<- (subst B2 N0 (%the (cstep* (N2' @@ N0) (N2 N0)) CS2)) + %<- (cs-1* CS1 (%the (cstep* (N1' @@ N0 @@ (N2' @@ N0)) (N1 N0 @@ (N2' @@ N0))) CS1')) + %<- (cs-2* CS2 (%the (cstep* (N1 N0 @@ (N2' @@ N0)) (N1 N0 @@ N2 N0)) CS2')) + %<- (cs-trans CS1' CS2' CS12') %. +% turns out that once the case is done, the types are unnecessary! +% +%% <- subst B1 N0 CS1 +%% <- subst B2 N0 CS2 +%% <- cs-1* CS1 CS1' +%% <- cs-2* CS2 CS2' +%% <- cs-trans CS1' CS2' CS12'. +%worlds () (subst _ _ _) %. +%total {B} (subst B _ _) %. +%{! +Then, we can prove simulation, the correctness of translation, by a +straightforward induction on single-step derivations in the lambda-calculus, +using the correctness of bracket abstraction as a lemma in the case of a +beta-reduction. +!}% +%sort simulate %. +%mode simulate %in %in %out %out %. +%{! +NB: in this step, on paper, you have to prove two compositionality lemmas. +In Twelf, using higher-order abstract syntax, we get them for free -- see +(T1 _ _ T2) output. +!}% +%term _ %pi (simulate s-β (t-app T2 (t-lam B1 T1)) (T1 _ _ T2) CS) %<- (subst B1 N2 CS) %. +%term _ + %pi (simulate (s-1 S1) (t-app T2 T1) (t-app T2 T1') CSapp) + %<- (simulate S1 T1 T1' CS1) + %<- (cs-1* CS1 CSapp) %. +%term _ + %pi (simulate (s-2 S2) (t-app T2 T1) (t-app T2' T1) CSapp) + %<- (simulate S2 T2 T2' CS2) + %<- (cs-2* CS2 CSapp) %. +%worlds () (simulate _ _ _ _) %. +%total D (simulate D _ _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Bracket_abstraction). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/c-machine-and-focusing-composition-in-machine-state.lf b/new-tests/stelf-output/pages/c-machine-and-focusing-composition-in-machine-state.lf new file mode 100644 index 0000000..7abfa1f --- /dev/null +++ b/new-tests/stelf-output/pages/c-machine-and-focusing-composition-in-machine-state.lf @@ -0,0 +1,133 @@ +% +%{! +C Machine and Focusing +!}% +%{! ## Syntax !}% +%sort pos %. +%name pos %. +%sort neg %. +%name neg %. +%term down %pi neg %-> pos %. +%term plus %pi pos %-> pos %-> pos %. +%term times %pi pos %-> pos %-> pos %. +%term zero pos %. +%term one pos %. +%term up %pi pos %-> neg %. +%term arr %pi pos %-> neg %-> neg %. +%term with %pi neg %-> neg %-> neg %. +%term top neg %. +%inline conc neg %. +%{! + +### Proofs + +To mimic direct style, we don't internalize continuation composition +(because continuations aren't separate syntactic categories in direct +style). Also, we treat negative cuts as a special case of the +internalized exp/cont composition principle; there is no need to +distinguish the redices. + +Unlike higher-order focusing, neutral sequents are "pos entails neg", +and the shifts hand around until we choose to focus on a formula. + +!}% +%sort val+ %. +%sort cont- %. +%sort cont+ %. +%sort val- %. +%sort exp %. +%% positive values +%term mt+ val+ one %. +%% no rule for zero +%term pair+ %pi (val+ A) %-> (val+ B) %-> (val+ (times A B)) %. +%term inl %pi (val+ A) %-> (val+ (plus A B)) %. +%term inr %pi (val+ B) %-> (val+ (plus A B)) %. +%term delay- %pi (exp A-) %-> (val+ (down A-)) %. +%% positive continuations (shallow matching) +%term let1 %pi (exp C) %-> (cont+ one C) %. +%% no rule for zero +%term split %pi (%pi (val+ A) %-> (val+ B) %-> (exp C)) %-> (cont+ (times A B) C) %. +%term case + %pi (%pi (val+ A) %-> (exp C)) + %-> (%pi (val+ B) %-> (exp C)) + %-> (cont+ (plus A B) C) %. +%term force- %pi (cont- A- C) %-> (cont+ (down A-) C) %. +%% negative continuations +%term app %pi (val+ A+) %-> (cont- B C) %-> (cont- (arr A+ B) C) %. +%term fst %pi (cont- A C) %-> (cont- (with A B) C) %. +%term snd %pi (cont- B C) %-> (cont- (with A B) C) %. +%term force+ %pi (%pi (val+ A+) %-> (exp C)) %-> (cont- (up A+) C) %. +%% internalize identity so we don't need to eta-expand +%term id- cont- A A %. +%% negative values (shallow) +%term lam %pi (%pi (val+ A+) %-> (exp B-)) %-> (val- (arr A+ B-)) %. +%term pair- %pi (exp A-) %-> (exp B-) %-> (val- (with A- B-)) %. +%term mt- val- top %. +%term delay+ %pi (val+ A+) %-> (val- (up A+)) %. +%% wait to peel off shift until focus +%% neutral sequents +%term return %pi (val- A) %-> (exp A) %. +%% canonical when the value is a variable; +%% non-canonical otherwise +%term cut+ %pi (val+ A+) %-> (cont+ A+ C) %-> (exp C) %. +%% internalize exp / cont composition +%term compose %pi (exp A-) %-> (cont- A- C-) %-> (exp C-) %. +%{! ### Stack Machine + +Lists of continuations: + +!}% +%sort conts- %. +%term conts-/cons %pi (cont- A B) %-> (conts- B C) %-> (conts- A C) %. +%term conts-/nil conts- A A %. +%sort conts+ %. +%term conts+/cons %pi (cont+ A B) %-> (conts- B C) %-> (conts+ A C) %. +%{! + +Machine states are basically expressions, except (1) they have lists of +continuations in place of continuations and (2) they distinguish +exp/cont composition states from negative redex states. + +!}% +%sort state %. +%term done %pi (val- A) %-> (state A) %. +%% answer +%term st- %pi (exp A) %-> (conts- A C) %-> (state C) %. +%% composition +%term stv- %pi (val- A) %-> (conts- A C) %-> (state C) %. +%% negative cut +%term stv+ %pi (val+ A) %-> (conts+ A C) %-> (state C) %. +%% positive cut +%{! + +The sum type that you'd define for progress is (val C) + (exp C) so +let's just abuse exp C for this sum and write the operational semantics +as the progress proof. The only downside is that you can't tell whether +an expression took a real step towards a value or already was a value. + +!}% +%sort step %. +%mode step %in %out %. +%term step/return step (done V) (done V) %. +%term step/let1 step (stv+ mt+ (conts+/cons (let1 E) Ks)) (st- E Ks) %. +%term step/split step (stv+ (pair+ V1 V2) (conts+/cons (split E) Ks)) (st- (E V1 V2) Ks) %. +%term step/case1 step (stv+ (inl V1) (conts+/cons (case E1 E2) Ks)) (st- (E1 V1) Ks) %. +%term step/case2 step (stv+ (inr V2) (conts+/cons (case E1 E2) Ks)) (st- (E2 V2) Ks) %. +%term step/force- step (stv+ (delay- E) (conts+/cons (force- K) Ks)) (st- E (conts-/cons K Ks)) %. +%term step/app step (stv- (lam E) (conts-/cons (app V K) Ks)) (st- (E V) (conts-/cons K Ks)) %. +%term step/fst step (stv- (pair- E1 E2) (conts-/cons (fst K) Ks)) (st- E1 (conts-/cons K Ks)) %. +%term step/snd step (stv- (pair- E1 E2) (conts-/cons (snd K) Ks)) (st- E2 (conts-/cons K Ks)) %. +%term step/force+ step (stv- (delay+ V) (conts-/cons (force+ E) Ks)) (st- (E V) Ks) %. +%term step/id- step (stv- V (conts-/cons id- Ks)) (stv- V Ks) %. +%term step/ids step (stv- V conts-/nil) (done V) %. +%term step/compose step (st- (compose E K) Ks) (st- E (conts-/cons K Ks)) %. +%term step/return step (st- (return V) Ks) (stv- V Ks) %. +%term step/cut+ step (st- (cut+ V K) Ks) (stv+ V (conts+/cons K Ks)) %. +%worlds () (step _ _) %. +%total D (step D _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/C_machine_and_focusing_(composition_in_machine_state)). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/c-machine-and-focusing-internalized-compositon.lf b/new-tests/stelf-output/pages/c-machine-and-focusing-internalized-compositon.lf new file mode 100644 index 0000000..47eb26f --- /dev/null +++ b/new-tests/stelf-output/pages/c-machine-and-focusing-internalized-compositon.lf @@ -0,0 +1,110 @@ +% +%{! +C Machine and Focusing +!}% +%{! ## Syntax !}% +%sort pos %. +%name pos %. +%sort neg %. +%name neg %. +%term down %pi neg %-> pos %. +%term plus %pi pos %-> pos %-> pos %. +%term times %pi pos %-> pos %-> pos %. +%term zero pos %. +%term one pos %. +%term up %pi pos %-> neg %. +%term arr %pi pos %-> neg %-> neg %. +%term with %pi neg %-> neg %-> neg %. +%term top neg %. +%inline conc neg %. +%sort val+ %. +%sort cont- %. +%sort cont+ %. +%sort val- %. +%sort exp %. +%% positive values +%term mt+ val+ one %. +%% no rule for zero +%term pair+ %pi (val+ A) %-> (val+ B) %-> (val+ (times A B)) %. +%term inl %pi (val+ A) %-> (val+ (plus A B)) %. +%term inr %pi (val+ B) %-> (val+ (plus A B)) %. +%term delay- %pi (exp A-) %-> (val+ (down A-)) %. +%% positive continuations (shallow matching) +%% positive conts +%term let1 %pi (exp C) %-> (cont+ one C) %. +%% no rule for zero +%term split %pi (%pi (val+ A) %-> (val+ B) %-> (exp C)) %-> (cont+ (times A B) C) %. +%term case + %pi (%pi (val+ A) %-> (exp C)) + %-> (%pi (val+ B) %-> (exp C)) + %-> (cont+ (plus A B) C) %. +%% wait to peel off the shift until you're ready to focus. +%term force- %pi (cont- A- C) %-> (cont+ (down A-) C) %. +%term comp+- %pi (cont+ A+ B-) %-> (cont- B- C-) %-> (cont+ A+ C-) %. +%% negative continuations +%term app %pi (val+ A+) %-> (cont- B C) %-> (cont- (arr A+ B) C) %. +%term fst %pi (cont- A C) %-> (cont- (with A B) C) %. +%term snd %pi (cont- B C) %-> (cont- (with A B) C) %. +%term force+ %pi (%pi (val+ A+) %-> (exp C)) %-> (cont- (up A+) C) %. +%% internalize identity so we don't need to eta-expand +%term id- cont- A A %. +%term comp-- %pi (cont- A- B-) %-> (cont- B- C-) %-> (cont- A- C-) %. +%% negative values (shallow) +%term lam %pi (%pi (val+ A+) %-> (exp B-)) %-> (val- (arr A+ B-)) %. +%term pair- %pi (exp A-) %-> (exp B-) %-> (val- (with A- B-)) %. +%term mt- val- top %. +%term delay+ %pi (val+ A+) %-> (val- (up A+)) %. +%% wait to peel off shift until focus +%% neutral sequents +%term return %pi (val- A) %-> (exp A) %. +%% canonical when the value is a variable; +%% non-canonical otherwise +%term cut+ %pi (val+ A+) %-> (cont+ A+ C) %-> (exp C) %. +%% non-canonical +%term cut- %pi (val- A-) %-> (cont- A- C) %-> (exp C) %. +%term compe- %pi (exp A-) %-> (cont- A- C-) %-> (exp C-) %. +%{! ### Stack Machine !}% +%% the sum type that you'd define for progress is +%% (val C) + (exp C) +%% so let's just abuse exp C for this sum and write the +%% operational semantics as the progress proof. +%% the only downside is that you can't tell whether an expression +%% took a real step towards a value or already was a value. +%sort step %. +%mode step %in %out %. +%term step/return step (return V) (return V) %. +%term step/let1 step (cut+ mt+ (let1 E)) E %. +%term step/split step (cut+ (pair+ V1 V2) (split E)) (E V1 V2) %. +%term step/case1 step (cut+ (inl V1) (case E1 E2)) (E1 V1) %. +%term step/case2 step (cut+ (inr V2) (case E1 E2)) (E2 V2) %. +%term step/force- step (cut+ (delay- E) (force- K)) (compe- E K) %. +%term step/app step (cut- (lam E) (app V K)) (compe- (E V) K) %. +%term step/fst step (cut- (pair- E1 E2) (fst K)) (compe- E1 K) %. +%term step/snd step (cut- (pair- E1 E2) (snd K)) (compe- E2 K) %. +%term step/force+ step (cut- (delay+ V) (force+ E)) (E V) %. +%term step/id step (cut- V id-) (return V) %. +%% find the redex +%term step/compe-/compe- step (compe- (compe- E K1) K2) (compe- E (comp-- K1 K2)) %. +%term step/compe-/cut+ step (compe- (cut+ V K1) K2) (cut+ V (comp+- K1 K2)) %. +%term step/compe-/cut- step (compe- (cut- V K1) K2) (cut- V (comp-- K1 K2)) %. +%term step/compe-/cut- step (compe- (return V) K2) (cut- V K2) %. +%% find the top of the stack +%term step/comp+-/let1 step (cut+ V (comp+- (let1 E) K)) (cut+ V (let1 (compe- E K))) %. +%term step/comp+-/split step (cut+ V (comp+- (split E) K)) (cut+ V (split ([x] [y] compe- (E x y) K))) %. +%term step/comp+-/case step (cut+ V (comp+- (case E1 E2) K)) (cut+ V (case ([x] compe- (E1 x) K) ([x] compe- (E2 x) K))) %. +%term step/comp+-/force step (cut+ V (comp+- (force- K1) K2)) (cut+ V (force- (comp-- K1 K2))) %. +%term step/comp+-/force step (cut+ V (comp+- (comp+- K1 K2) K3)) (cut+ V (comp+- K1 (comp-- K2 K3))) %. +%term step/comp--/app step (cut- V1 (comp-- (app V2 K) K2)) (cut- V1 (app V2 (comp-- K K2))) %. +%term step/comp--/fst step (cut- V1 (comp-- (fst K) K2)) (cut- V1 (fst (comp-- K K2))) %. +%term step/comp--/snd step (cut- V1 (comp-- (snd K) K2)) (cut- V1 (snd (comp-- K K2))) %. +%term step/comp--/snd step (cut- V1 (comp-- (force+ E) K2)) (cut- V1 (force+ ([x] compe- (E x) K2))) %. +%term step/comp--/id step (cut- V1 (comp-- id- K)) (cut- V1 K) %. +%term step/comp--/id step (cut- V1 (comp-- (comp-- K1 K2) K)) (cut- V1 (comp-- K1 (comp-- K2 K))) %. +%worlds () (step _ _) %. +%total D (step D _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/C_machine_and_focusing_(internalized_compositon)). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/c-machine-and-focusing.lf b/new-tests/stelf-output/pages/c-machine-and-focusing.lf new file mode 100644 index 0000000..bfe997a --- /dev/null +++ b/new-tests/stelf-output/pages/c-machine-and-focusing.lf @@ -0,0 +1,121 @@ +% +%{! +C Machine and Focusing +!}% +%{! ## Syntax !}% +%sort pos %. +%name pos %. +%sort neg %. +%name neg %. +%term down %pi neg %-> pos %. +%term plus %pi pos %-> pos %-> pos %. +%term times %pi pos %-> pos %-> pos %. +%term zero pos %. +%term one pos %. +%term up %pi pos %-> neg %. +%term arr %pi pos %-> neg %-> neg %. +%term with %pi neg %-> neg %-> neg %. +%term top neg %. +%inline conc neg %. +%sort val+ %. +%sort cont- %. +%sort cont+ %. +%sort val- %. +%sort exp %. +%% positive values +%term mt+ val+ one %. +%% no rule for zero +%term pair+ %pi (val+ A) %-> (val+ B) %-> (val+ (times A B)) %. +%term inl %pi (val+ A) %-> (val+ (plus A B)) %. +%term inr %pi (val+ B) %-> (val+ (plus A B)) %. +%term delay- %pi (exp A-) %-> (val+ (down A-)) %. +%% positive continuations (shallow matching) +%% positive conts +%term let1 %pi (exp C) %-> (cont+ one C) %. +%% no rule for zero +%term split %pi (%pi (val+ A) %-> (val+ B) %-> (exp C)) %-> (cont+ (times A B) C) %. +%term case + %pi (%pi (val+ A) %-> (exp C)) + %-> (%pi (val+ B) %-> (exp C)) + %-> (cont+ (plus A B) C) %. +%term force- %pi (cont- A- C) %-> (cont+ (down A-) C) %. +%% wait to peel off the shift until you're ready to focus. +%% negative continuations +%term app %pi (val+ A+) %-> (cont- B C) %-> (cont- (arr A+ B) C) %. +%term fst %pi (cont- A C) %-> (cont- (with A B) C) %. +%term snd %pi (cont- B C) %-> (cont- (with A B) C) %. +%term force+ %pi (%pi (val+ A+) %-> (exp C)) %-> (cont- (up A+) C) %. +%% internalize identity so we don't need to eta-expand +%term id- cont- A A %. +%% negative values (shallow) +%term lam %pi (%pi (val+ A+) %-> (exp B-)) %-> (val- (arr A+ B-)) %. +%term pair- %pi (exp A-) %-> (exp B-) %-> (val- (with A- B-)) %. +%term mt- val- top %. +%term delay+ %pi (val+ A+) %-> (val- (up A+)) %. +%% wait to peel off shift until focus +%% neutral sequents +%term return %pi (val- A) %-> (exp A) %. +%% canonical when the value is a variable; +%% non-canonical otherwise +%term cut+ %pi (val+ A+) %-> (cont+ A+ C) %-> (exp C) %. +%% non-canonical +%term cut- %pi (val- A-) %-> (cont- A- C) %-> (exp C) %. +%% value variables can be used to stop right focus early +%block valb {A} [x val+ A]%. +%worlds (valb) (cont+ _ _) (cont- _ _) (val+ _) (exp _) (val- _) %. +%{! ## Operational Semantics !}% +%{! ### Composition !}% +%sort comp+- %. +%mode comp+- %in %in %out %. +%sort compe- %. +%mode compe- %in %in %out %. +%sort comp-- %. +%mode comp-- %in %in %out %. +%% + - +%term comp+-/let1 %pi (comp+- (let1 E) K (let1 E')) %<- (compe- E K E') %. +%term comp+-/split %pi (comp+- (split E) K (split E')) %<- ({x} {y} compe- (E x y) K (E' x y)) %. +%term comp+-/case + %pi (comp+- (case E1 E2) K (case E1' E2')) + %<- ({x} compe- (E1 x) K (E1' x)) + %<- ({x} compe- (E2 x) K (E2' x)) %. +%term comp+-/force %pi (comp+- (force- K-) K (force- K-')) %<- (comp-- K- K K-') %. +%% e - +%term compe-/return compe- (return V) K (cut- V K) %. +%term compe-/cut- %pi (compe- (cut- V K1) K2 (cut- V K3)) %<- (comp-- K1 K2 K3) %. +%term compe-/cut+ %pi (compe- (cut+ V K1) K2 (cut+ V K3)) %<- (comp+- K1 K2 K3) %. +%% - - +%term comp--/app %pi (comp-- (app V K1) K2 (app V K3)) %<- (comp-- K1 K2 K3) %. +%term comp--/fst %pi (comp-- (fst K1) K2 (fst K3)) %<- (comp-- K1 K2 K3) %. +%term comp--/snd %pi (comp-- (snd K1) K2 (snd K3)) %<- (comp-- K1 K2 K3) %. +%term comp--/force %pi (comp-- (force+ E) K2 (force+ E')) %<- ({x} compe- (E x) K2 (E' x)) %. +%term comp--/id comp-- id- K2 K2 %. +%worlds (valb) (comp+- _ _ _) (comp-- _ _ _) (compe- _ _ _) %. +%total (D1 D2 D3) (comp+- D1 _ _) (comp-- D3 _ _) (compe- D2 _ _) %. +%{! ### Stack Machine !}% +%% the sum type that you'd define for progress is +%% (val C) + (exp C) +%% so let's just abuse exp C for this sum and write the +%% operational semantics as the progress proof. +%% the only downside is that you can't tell whether an expression +%% took a real step towards a value or already was a value. +%sort step %. +%mode step %in %out %. +%term step/return step (return V) (return V) %. +%term step/let1 step (cut+ mt+ (let1 E)) E %. +%term step/split step (cut+ (pair+ V1 V2) (split E)) (E V1 V2) %. +%term step/case1 step (cut+ (inl V1) (case E1 E2)) (E1 V1) %. +%term step/case2 step (cut+ (inr V2) (case E1 E2)) (E2 V2) %. +%term step/force- %pi (step (cut+ (delay- E) (force- K)) E') %<- (compe- E K E') %. +%term step/app %pi (step (cut- (lam E) (app V K)) E') %<- (compe- (E V) K E') %. +%term step/fst %pi (step (cut- (pair- E1 E2) (fst K)) E') %<- (compe- E1 K E') %. +%term step/snd %pi (step (cut- (pair- E1 E2) (snd K)) E') %<- (compe- E2 K E') %. +%term step/force+ step (cut- (delay+ V) (force+ E)) (E V) %. +%term step/id step (cut- V id-) (return V) %. +%worlds () (step _ _) %. +%total D (step D _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/C_machine_and_focusing). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/cade-tutorial-basic-error-messages.lf b/new-tests/stelf-output/pages/cade-tutorial-basic-error-messages.lf new file mode 100644 index 0000000..4a071c8 --- /dev/null +++ b/new-tests/stelf-output/pages/cade-tutorial-basic-error-messages.lf @@ -0,0 +1,227 @@ +% +%{! ## How Twelf checks assertions + +Twelf proves a totality assertion for a type family such as +``mult`` by checking several properties. These properties, taken +together, constitute a proof by induction on canonical forms that the +type family defines a total relation. + +### Running Example + +We'll start from the following code: !}% +%sort nat %. +%term zero nat %. +%term succ %pi nat %-> nat %. +%sort add %. +%mode add %in %in %out %. +%term add/z add zero N N %. +%term add/s %pi (add (succ M) N (succ P)) %<- (add M N P) %. +%worlds () (add _ _ _) %. +%total M (add M _ _) %. +%sort mult %. +%mode mult %in %in %out %. +%term mult/z mult zero N zero %. +%term mult/s %pi (mult (succ M) N P') %<- (mult M N P) %<- (add N P P') %. +%{! ### Mode + +Twelf checks that each constant inhabiting the type family is +_well-moded_. Roughly, this means that the inputs to the conclusion +of a constant determine the inputs of the first premise, and that these +together with the outputs of the first premise determine the inputs of +the second premise, and so on, until the outputs of all the premises +determine the outputs of the conclusion. +For example, the constant + + +```checkedtwelf +mult/s : mult (succ M) N P' + <- mult M N P + <- add N P P'. +``` + +has mode ``+M +N -P`` because the input ``M`` and ``N`` +in the conclusion determine the inputs of the premise, and the +``P`` output by the premise determines the first input to the second +premise (add), and the output of that determines the conclusion. +On the other hand, a constant + +```checkedtwelf +mult/bad-mode-output : mult zero N P. +``` + +is not well-moded---the output ``P`` is not determined by the +inputs. Similarly, + +```checkedtwelf +mult/bad-mode-input : mult (succ M) N zero + <- mult M Q P. +``` + +is not well-moded---the second input to the premise is not determined +by the inputs of the conclusion. + +Note that subgoal order matters: + +```checkedtwelf +mult/s : mult (succ M) N Q + <- add P N Q + <- mult M N P. +``` + +The premises are mode-checked in order; the outputs from one subgoal are +consider ground in subsequent premises. + +### Worlds + +Twelf checks that each constant inhabiting the type family obeys the +worlds declaration. Because we are only proving theorems about closed +terms right now, we will not run across any problems with world +checks. + +### Termination + +Twelf checks that each constant inhabiting the type family obeys the +induction order specified in the ``%total`` declaration. In each +inductive premise of a constant, the specified induction position must +be a strict subterm of the corresponding argument in the conclusion. +For example, the constant + +```checkedtwelf +mult/s : mult (s M) N Q + <- mult M N P + <- add P N Q. +``` + +obeys the induction order ``M`` specified in the above totality +assertion because the term ``M`` is a strict subterm of the term +``(s M)``. + +Recuring on exactly the same term is obviously a termination error: + +```checkedtwelf +mult/bad-termination-1 : mult M N P + <- mult M N P. +%total M (mult M _ _). +``` + +As is recuring on an entirely unrelated argumenet: + +```checkedtwelf +mult/bad-termination-2 : mult M N P + <- mult N N P. +%total M (mult M _ _). +``` + +For example Twelf would not accept the totality of ``mult`` +if ``N`` were used as the induction order—the same term +``N`` in the conclusion of this constant appears in the premise: + +```checkedtwelf +%total N (mult _ N _). +``` + +%In addition to the subterm ordering on a single argument, Twelf supports +[mutual induction](/wiki/mutual-induction/) and lexicographic induction. + +### Output coverage + +In the definition of a type family, you may pattern-match the outputs of +a premise. For example, we might write + +```checkedtwelf +mult/bad-output-cov : mult (succ M) N zero + <- mult M N (succ P). +%worlds () (mult _ _ _). +%total N1 (mult N1 N2 N3). +``` + +Here we have insisted that the output of the premise has the form +``succ P`` for some ``P``. Twelf correctly reports an output +coverage error because this condition can fail. + +Pattern-matching the output of a premise is like an inversion step +in a proof: you're insisting that the premise derivation must conclude a +particular fact that is more specific than the judgement form itself. +For Twelf to accept a relation as total, Twelf must notice that all of +these inversions are permissible. Twelf permits such inversions when it +is readily apparent that they are justified, and those inversions that +Twelf does not accept can be proved explicitly. + +In this example, we got an output coverage error because we constrained +the output of the premise by insisting it be formed by a particular +constant. The other way to get output coverage errors is to insist that +the output of a premise be a variable that occurs elsewhere in the type. +For example: + +```checkedtwelf +mult/bad-output-free : mult (succ M) N zero + <- mult M N N. +%worlds () (mult _ _ _). +%total N1 (mult N1 N2 N3). +``` + +Here, we insisted that the output of the premise be the number +``N`` that we put in. Twelf is very conservative in checking +[output freeness](/wiki/output-freeness/): a type family will not be judged total if you +constrain the outputs of any premise at all in this manner. + +### Input coverage + +Mode, worlds, termination, and output coverage ensure that each constant +really does cover the part of the relation indicated by its conclusion. +For example, if ``mult`` passes these four checks, we know that +``mult/z`` and ``mult/s`` cover ``(zero, N, _)`` and ``(succ M, +N, _)``, respectively. What else is necessary to know that +``mult`` defines a total relation? We need to know that all the +constants inhabiting ``mult``, taken together, cover all of the +inputs. Input coverage checks exactly this. + +For example, if we forgot ``mult/z``, input coverage for +``mult`` would fail. For example: + +```checkedtwelf +mult' : nat -> nat -> nat -> type. +%mode mult' +N1 +N2 -X3. + +mult'/s : mult' (succ M) N Q + <- mult' M N P + <- add P N Q. + +%worlds () (mult' _ _ _). +%total M (mult' M _ _). +``` + +Here's an analogy that might be helpful: You can think of each constant +of a type as being a clause in an ML pattern matching declaration. Then +input coverage is like the exhaustiveness checker for pattern matching. + +Twelf checks input coverage by splitting the input types to +case-analyze the various constants that could have been used to inhabit +them. For ``add``, Twelf splits the first ``nat`` argument +``M``, and then checks that the cases ``plus z N N`` and +``add (succ M) N P`` are covered. Fortunately, these are exactly +the cases we wrote down. If we had case-analyzed further in the +definition of the judgement (e.g., if the definition of ``plus`` +case-analyzed the second argument as well), Twelf would continue +splitting the input space. Because Twelf separates termination checking +and coverage checking, the constants defining a type family do not need +to follow any particular primitive recursion schema-the constants may +pattern-match the inputs in a general manner. + +## When Twelf checks what + +To a first approximation, you can think of the ``%mode`` and ``%worlds`` declarations as specifying a totality assertion and the ``%total`` declaration as checking it. This isn't exactly how Twelf works, though: + +- When a ``%mode`` declaration is entered, Twelf checks that all previous constants inhabiting the specified type family are well-moded; further, it then mode-checks any subsequent constants inhabiting that family. +- When a ``%worlds`` declaration is entered, Twelf world-checks the type family; further, it then reports an error if any new constants contributing to the family at all are added. +- When a ``%total`` declaration is entered, Twelf checks termination, then input coverage, then output coverage. When checking output coverage, Twelf checks for unjustified constant pattern-matching in a first pass and then output freeness problems in a second pass. + +This separation allows you to, for example, check that each constant in a family is well-moded (i.e., takes specified inputs to specified outputs) without checking that the entire type family is total. You can also use the declarations [%terminates](/wiki/percent-terminates/) and [%covers](/wiki/percent-covers/) to check termination and input coverage independently. + +If any constant in a type family fails mode, worlds, or output coverage, then mode, worlds, or totality checking fails for the whole type family. One could imagine that Twelf instead would just disregard the offending constant: it is possible that the type family as a whole satisfies a totality assertion without that constant, and, in a mathematical sense, adding additional constants never invalidates the fact a totality assertion is true of a family. The reason Twelf does not work this way is that ``%total`` actually has a more specific meaning, as we discuss in the next section. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +You can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/CADE_Tutorial/Basic_error_messages). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/cade-tutorial-basics-answer.lf b/new-tests/stelf-output/pages/cade-tutorial-basics-answer.lf new file mode 100644 index 0000000..0aa211b --- /dev/null +++ b/new-tests/stelf-output/pages/cade-tutorial-basics-answer.lf @@ -0,0 +1,116 @@ +% +%{! + +## Natural numbers + +!}% +%sort nat %. +%term zero nat %. +%term succ %pi nat %-> nat %. +%{! + +## Addition + +!}% +%sort add %. +%term add/z add zero N N %. +%term add/s %pi (add (succ M) N (succ P)) %<- (add M N P) %. +%{! + +### Example derivations + +!}% +%define 1 nat succ zero %. +%define 2 nat succ 1 %. +%define 1+1is2 (add 1 1 2) add/s add/z %. +%% explicit version of add/z +%% add/z-explicit : {n:nat} add zero n n. +%% 1+1is2-explicit : add 1 1 2 = add/s (add/z-explicit 1). +%{! + +## Exercise: Multiplication + +!}% +%sort mult %. +%term mult/z mult zero N zero %. +%term mult/s %pi (mult (succ M) N P') %<- (mult M N P) %<- (add N P P') %. +%% note that the arguments are "backwards" +%define 1*2is2 (mult 1 2 2) mult/s (add/s (add/s add/z)) mult/z %. +%{! + +## Mode, worlds total + +!}% +%mode add %in %in %out %. +%worlds () (add _ _ _) %. +%total M (add M _ _) %. +%solve 1+1is2' : add 1 1 N %. +%% Examples of errors: +%% +%% mult/bad-mode-output : mult zero N Q. +%% +%% mult/bad-mode-input : mult (succ M) N zero +%% <- mult M Q P. +%% +%% %% do input coverage by removing cases +%% +%% mult/bad-termination-1 : mult M N P +%% <- mult M N P. +%% mult/bad-termination-2 : mult M N P +%% <- mult N N P. +%% +%% mult/bad-output-free : mult (succ M) N zero +%% <- mult M N N. +%% mult/bad-output-cov : mult (succ M) N zero +%% <- mult M N (succ P). +%mode mult %in %in %out %. +%worlds () (mult _ _ _) %. +%total M (mult M _ _) %. +%{! + +## Right-hand zero + +!}% +%sort rhzero {M nat} %. +%mode rhzero %in %out %. +%term _ rhzero zero add/z %. +%term _ %pi (rhzero (succ M) (add/s D)) %<- (rhzero M (%the (add M zero M) D)) %. +%worlds () (rhzero _ _) %. +%total M (rhzero M _) %. +%{! + +## Right-hand succ + +!}% +%sort rhsucc %. +%mode rhsucc %in %out %. +%term _ rhsucc (%the (add zero M M) add/z) (%the (add zero (succ M) (succ M)) add/z) %. +%term _ + %pi (rhsucc (add/s (%the (add M N P) D1)) (add/s D2)) + %<- (rhsucc D1 (%the (add M (succ N) (succ P)) D2)) %. +%% remark that type annotations are optional: +%% - : rhsucc add/z add/z. +%% - : rhsucc (add/s D1) (add/s D2) +%% <- rhsucc D1 D2. +%worlds () (rhsucc _ _) %. +%total M (rhsucc M _) %. +%{! + +## Exercise: addition is commutative + +!}% +%sort commute %. +%mode commute %in %out %. +%term _ %pi (commute (%the (add zero M M) add/z) D) %<- (rhzero M D) %. +%term _ + %pi (commute (add/s (%the (add M N P) D)) D'') + %<- (commute D (%the (add N M P) D')) + %<- (rhsucc D' (%the (add N (succ M) (succ P)) D'')) %. +%worlds () (commute _ _) %. +%total D (commute D _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/CADE_Tutorial/Basics_Answer). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/cade-tutorial-basics.lf b/new-tests/stelf-output/pages/cade-tutorial-basics.lf new file mode 100644 index 0000000..4f25081 --- /dev/null +++ b/new-tests/stelf-output/pages/cade-tutorial-basics.lf @@ -0,0 +1,9 @@ +% +%{! * [Examples of basic error messages](/wiki/cade-tutorial-basic-error-messages/) +* [Solutions to exercises](/wiki/cade-tutorial-basics-answer/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/CADE_Tutorial/Basics). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/cade-tutorial-combinators-answer.lf b/new-tests/stelf-output/pages/cade-tutorial-combinators-answer.lf new file mode 100644 index 0000000..70171e6 --- /dev/null +++ b/new-tests/stelf-output/pages/cade-tutorial-combinators-answer.lf @@ -0,0 +1,112 @@ +% +%{! +Adapted from the case study on [Typed combinators soundness and completeness](/wiki/typed-combinators-soundness-and-completeness/). + +## Syntax + +!}% +% lambda calculus +%sort term %. +%name term %. +%term app %pi term %-> term %-> term %. +%term lam %pi (%pi term %-> term) %-> term %. +%block term-block [x term]%. +%worlds (term-block) (term) %. +% combinator calculus +%sort comb %. +%name comb %. +%term s comb %. +%term k comb %. +%term i comb %. +%term capp %pi comb %-> comb %-> comb %. +%block comb-block [y comb]%. +%worlds (comb-block) (comb) %. +%{! + +## Translation + +!}% +% bracket abstraction +%sort bracket %. +%mode bracket %in %out %. +%term bracket/var bracket ([y] y) i %. +%term bracket/i bracket ([y] i) (capp k i) %. +%term bracket/k bracket ([y] k) (capp k k) %. +%term bracket/s bracket ([y] s) (capp k s) %. +%term bracket/app + %pi (bracket ([y] capp (A y) (B y)) (capp (capp s A') B')) + %<- (bracket ([y] A y) A') + %<- (bracket ([y] B y) B') %. +%block bracket-block [y comb] [bracket/y bracket ([z] y) (capp k y)]%. +%worlds (bracket-block) (bracket _ _) %. +%total A (bracket A _) %. +% translation +%sort translate %. +%mode translate %in %out %. +%term translate/app %pi (translate (app M N) (capp A B)) %<- (translate M A) %<- (translate N B) %. +%term translate/lam + %pi (translate (lam ([x] M x)) A') + %<- ({x} {y} %pi (bracket ([z] y) (capp k y)) %-> (translate x y) %-> (translate (M x) (A y))) + %<- (bracket ([y] A y) A') %. +%block translate-block [x term] [y comb] [bracket/y bracket ([z] y) (capp k y)] [translate/x translate x y]%. +%worlds (translate-block) (translate _ _) %. +%total M (translate M _) %. +%{! + +## Equational theory + +!}% +% lambda term equality +%sort teq %. +% beta +%term teq/beta teq (app (lam ([x] M x)) N) (M N) %. +% extensionality (eta) +%term teq/ext %pi (teq M M') %<- ({x term} teq (app M x) (app M' x)) %. +% compatibilities +%term teq/app %pi (teq (app M N) (app M' N')) %<- (teq M M') %<- (teq N N') %. +%term teq/lam %pi (teq (lam ([x] M x)) (lam ([x] M' x))) %<- ({x term} teq (M x) (M' x)) %. +% equivalence +%term teq/refl teq M M %. +%term teq/symm %pi (teq M M') %<- (teq M' M) %. +%term teq/trans %pi (teq M M') %<- (teq M N) %<- (teq N M') %. +%worlds (term-block) (teq _ _) %. +% combinator equality +%sort ceq %. +% betas +%term ceq/i ceq (capp i A) A %. +%term ceq/k ceq (capp (capp k A) B) A %. +%term ceq/s ceq (capp (capp (capp s A) B) C) (capp (capp A C) (capp B C)) %. +% extensionality +%term ceq/ext %pi (ceq A A') %<- ({y comb} ceq (capp A y) (capp A' y)) %. +% compatibility +%term ceq/app %pi (ceq (capp A B) (capp A' B')) %<- (ceq A A') %<- (ceq B B') %. +% equivalence +%term ceq/refl ceq A A %. +%term ceq/symm %pi (ceq A A') %<- (ceq A' A) %. +%term ceq/trans %pi (ceq A A') %<- (ceq A B) %<- (ceq B A') %. +%worlds (comb-block) (ceq _ _) %. +%{! + +## Correctness of the translation + +!}% +% substitution lemma +%sort subst %. +%mode subst %in %in %out %. +%term _ subst (%the (bracket ([y] y) i) bracket/var) C (%the (ceq (capp i C) C) ceq/i) %. +%term _ subst (%the (bracket ([y] i) (capp k i)) bracket/i) C (%the (ceq (capp (capp k i) C) i) ceq/k) %. +%term _ subst (%the (bracket ([y] k) (capp k k)) bracket/k) C (%the (ceq (capp (capp k k) C) k) ceq/k) %. +%term _ subst (%the (bracket ([y] s) (capp k s)) bracket/s) C (%the (ceq (capp (capp k s) C) s) ceq/k) %. +%term _ + %pi (subst (bracket/app (%the (bracket ([y] B y) B') Dbrack2) (%the (bracket ([y] A y) A') Dbrack1)) C (ceq/trans (ceq/app Dceq2 Dceq1) ceq/s)) + %<- (subst Dbrack1 C (%the (ceq (capp A' C) (A C)) Dceq1)) + %<- (subst Dbrack2 C (%the (ceq (capp B' C) (B C)) Dceq2)) %. +%block subst-block [y comb] [dbrack bracket ([z] y) (capp k y)] [thm-subst {C comb} subst dbrack C ceq/k]%. +%worlds (subst-block) (subst _ _ _) %. +%total D (subst D _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/CADE_Tutorial/Combinators_Answer). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/cade-tutorial-combinators-support.lf b/new-tests/stelf-output/pages/cade-tutorial-combinators-support.lf new file mode 100644 index 0000000..5a902e4 --- /dev/null +++ b/new-tests/stelf-output/pages/cade-tutorial-combinators-support.lf @@ -0,0 +1,64 @@ +% +%{! +Adapted from the case study on [Typed combinators soundness and completeness](/wiki/typed-combinators-soundness-and-completeness/). + +## Syntax + +!}% +% lambda calculus +%sort term %. +%name term %. +%term app %pi term %-> term %-> term %. +%term lam %pi (%pi term %-> term) %-> term %. +% combinator calculus +%sort comb %. +%name comb %. +%term s comb %. +%term k comb %. +%term i comb %. +%term capp %pi comb %-> comb %-> comb %. +%{! + +## Translation + +!}% +% bracket abstraction +% XXX fill in +% translation +% XXX fill in +%{! + +## Equational theory + +!}% +% lambda term equality +% (elided) +% combinator equality +%sort ceq %. +% betas +%term ceq/i ceq (capp i A) A %. +%term ceq/k ceq (capp (capp k A) B) A %. +%term ceq/s ceq (capp (capp (capp s A) B) C) (capp (capp A C) (capp B C)) %. +% extensionality +%term ceq/ext %pi (ceq A A') %<- ({y comb} ceq (capp A y) (capp A' y)) %. +% compatibility +%term ceq/app %pi (ceq (capp A B) (capp A' B')) %<- (ceq A A') %<- (ceq B B') %. +% equivalence +%term ceq/refl ceq A A %. +%term ceq/symm %pi (ceq A A') %<- (ceq A' A) %. +%term ceq/trans %pi (ceq A A') %<- (ceq A B) %<- (ceq B A') %. +%block comb-block [y comb]%. +%worlds (comb-block) (ceq _ _) %. +%{! + +## Correctness of the translation + +!}% +% substitution lemma +% XXX fill in +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/CADE_Tutorial/Combinators_Support). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/cade-tutorial-combinators.lf b/new-tests/stelf-output/pages/cade-tutorial-combinators.lf new file mode 100644 index 0000000..2fde637 --- /dev/null +++ b/new-tests/stelf-output/pages/cade-tutorial-combinators.lf @@ -0,0 +1,9 @@ +% +%{! * [Support code](/wiki/cade-tutorial-combinators-support/) +* [Solutions to interactive exercises](/wiki/cade-tutorial-combinators-answer/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/CADE_Tutorial/Combinators). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/cade-tutorial-minml-answer.lf b/new-tests/stelf-output/pages/cade-tutorial-minml-answer.lf new file mode 100644 index 0000000..8b0bc71 --- /dev/null +++ b/new-tests/stelf-output/pages/cade-tutorial-minml-answer.lf @@ -0,0 +1,163 @@ +% +%{! + +## Syntax + +!}% +%sort tp %. +%name tp %. +%term num tp %. +%term arr %pi tp %-> tp %-> tp %. +%% you can world-check syntax if you want +%worlds () (tp) %. +%sort exp %. +%name exp %. +%term fn %pi tp %-> (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term z exp %. +%term s %pi exp %-> exp %. +%{! +#### Exercise: constant for ifz +!}% +%term ifz %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%% you can world-check syntax if you want +%block exp_block [x exp]%. +%worlds (exp_block) (exp) %. +%{! + +## Static semantics + +!}% +%sort of %. +%name of %. +%term of/z of z num %. +%term of/app %pi (of (app E1 E2) T) %<- (of E1 (arr T' T)) %<- (of E2 T') %. +%term of/fn + %pi (of (fn T1 ([x] E x)) (arr T1 T2)) + %<- ({x exp} %pi (of x T1) %-> (of (E x) T2)) %. +%{! +#### Exercise: typing rules for s and ifz +!}% +%term of/s %pi (of (s E) num) %<- (of E num) %. +%term of/ifz + %pi (of (ifz E E0 ([x] E1 x)) T) + %<- (of E num) + %<- (of E0 T) + %<- ({x exp} %pi (of x num) %-> (of (E1 x) T)) %. +%block of_block {T tp} [x exp] [dx of x T]%. +%worlds (of_block) (of _ _) %. +%{! + +## Dynamic semantics + +### value judgement + + +!}% +%sort value %. +%name value %. +%term value/fn value (fn T ([x] E x)) %. +%{! +#### Exercise: value rules for z and s +!}% +%term value/z value z %. +%term value/s %pi (value (s E)) %<- (value E) %. +%{! + +### step judgement + +!}% +%sort step %. +%name step %. +%term step/app/fn %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app/arg %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app/beta %pi (step (app (fn T ([x] E x)) E2) (E E2)) %<- (value E2) %. +%{! +#### Exercise: step rules for s and ifz +!}% +%term step/s %pi (step (s E) (s E')) %<- (step E E') %. +%term step/ifz/arg %pi (step (ifz E E0 ([x] E1 x)) (ifz E' E0 ([x] E1 x))) %<- (step E E') %. +%term step/ifz/z step (ifz z E0 ([x] E1 x)) E0 %. +%term step/ifz/s %pi (step (ifz (s E) E0 ([x] E1 x)) (E1 E)) %<- (value E) %. +%{! + +## Progress theorem + +### Sum type for the result + +!}% +%sort val-or-step %. +%name val-or-step %. +%term vos/value %pi (val-or-step E) %<- (value E) %. +%term vos/step %pi (val-or-step E) %<- (step E E') %. +%{! + +### Lemmas + +!}% +%sort prog/app %. +%mode prog/app %in %in %in %out %. +%term _ prog/app _ (vos/step (%the (step E1 E1') Dstep1)) _ (vos/step (step/app/fn Dstep1)) %. +%term _ prog/app _ (vos/value (%the (value E1) Dval1)) (vos/step (%the (step E2 E2') Dstep2)) (vos/step (step/app/arg Dstep2 Dval1)) %. +%term _ prog/app (%the (of (fn T ([x] E' x)) (arr T T')) (of/fn _)) (vos/value (%the (value (fn T ([x] E' x))) Dval1)) (vos/value (%the (value E2) Dval2)) (vos/step (step/app/beta Dval2)) %. +%worlds () (prog/app _ _ _ _) %. +%total {} (prog/app _ _ _ _) %. +%{! + +#### Exercise: lemma for s + +!}% +%sort prog/s %. +%mode prog/s %in %out %. +%term _ prog/s (vos/step (%the (step E E') Dstep)) (vos/step (step/s Dstep)) %. +%term _ prog/s (vos/value (%the (value E) Dval)) (vos/value (value/s Dval)) %. +%worlds () (prog/s _ _) %. +%total {} (prog/s _ _) %. +%{! + +#### Exercise: lemma for ifz + +!}% +%sort prog/ifz %. +%mode prog/ifz %in %in %in %in %out %. +%mode prog/ifz %in %in %in %in %out %. +%term _ prog/ifz _ _ _ (vos/step Dstep) (step/ifz/arg Dstep) %. +%term _ prog/ifz _ _ _ (vos/value value/z) step/ifz/z %. +%term _ prog/ifz _ _ _ (vos/value (value/s Dval)) (step/ifz/s Dval) %. +%worlds () (prog/ifz _ _ _ _ _) %. +%total {} (prog/ifz _ _ _ _ _) %. +%{! + +### Main theorem + +!}% +%sort prog %. +%mode prog %in %out %. +%term _ prog (%the (of z num) of/z) (vos/value (%the (value z) value/z)) %. +%term _ prog (of/fn _) (vos/value value/fn) %. +%term _ + %pi (prog (of/app (%the (of E2 T') D2) (%the (of E1 (arr T' T)) D1)) DvosApp) + %<- (prog D1 (%the (val-or-step E1) Dvos1)) + %<- (prog D2 (%the (val-or-step E2) Dvos2)) + %<- (prog/app D1 Dvos1 Dvos2 DvosApp) %. +%{! + +#### Exercise: cases for s and ifz + +!}% +%term _ + %pi (prog (of/s (%the (of E num) D)) DvosS) + %<- (prog D (%the (val-or-step E) Dvos)) + %<- (prog/s Dvos DvosS) %. +%term _ + %pi (prog (of/ifz ([x] [dx of x num] %the (of (E1 x) T) (D1 x dx)) (%the (of E0 T) D0) (%the (of E num) D)) (vos/step DstepIfz)) + %<- (prog D (%the (val-or-step E) Dvos)) + %<- (prog/ifz D E0 ([x] E1 x) Dvos DstepIfz) %. +%worlds () (prog _ _) %. +%total Dof (prog Dof _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/CADE_Tutorial/MinML_Answer). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/cade-tutorial-minml-encoding.lf b/new-tests/stelf-output/pages/cade-tutorial-minml-encoding.lf new file mode 100644 index 0000000..8d9fe23 --- /dev/null +++ b/new-tests/stelf-output/pages/cade-tutorial-minml-encoding.lf @@ -0,0 +1,142 @@ +% +%{! + +## Syntax + +!}% +%sort tp %. +%name tp %. +%term num tp %. +%term arr %pi tp %-> tp %-> tp %. +%% you can world-check syntax if you want +%worlds () (tp) %. +%sort exp %. +%name exp %. +%term fn %pi tp %-> (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term z exp %. +%term s %pi exp %-> exp %. +%{! +#### Exercise: constant for ifz +!}% +%term ifz %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%% you can world-check syntax if you want +%block exp_block [x exp]%. +%worlds (exp_block) (exp) %. +%{! + +## Static semantics + +!}% +%sort of %. +%name of %. +%term of/z of z num %. +%term of/app %pi (of (app E1 E2) T) %<- (of E1 (arr T' T)) %<- (of E2 T') %. +%term of/fn + %pi (of (fn T1 ([x] E x)) (arr T1 T2)) + %<- ({x exp} %pi (of x T1) %-> (of (E x) T2)) %. +%{! +#### Exercise: typing rules for s and ifz +!}% +%term of/s %pi (of (s E) num) %<- (of E num) %. +%term of/ifz + %pi (of (ifz E E0 ([x] E1 x)) T) + %<- (of E num) + %<- (of E0 T) + %<- ({x exp} %pi (of x num) %-> (of (E1 x) T)) %. +%block of_block {T tp} [x exp] [dx of x T]%. +%worlds (of_block) (of _ _) %. +%{! + +## Dynamic semantics + +### value judgement + + +!}% +%sort value %. +%name value %. +%term value/fn value (fn T ([x] E x)) %. +%{! +#### Exercise: value rules for z and s +!}% +%term value/z value z %. +%term value/s %pi (value (s E)) %<- (value E) %. +%{! + +### step judgement + +!}% +%sort step %. +%name step %. +%term step/app/fn %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app/arg %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app/beta %pi (step (app (fn T ([x] E x)) E2) (E E2)) %<- (value E2) %. +%{! +#### Exercise: step rules for s and ifz +!}% +%term step/s %pi (step (s E) (s E')) %<- (step E E') %. +%term step/ifz/arg %pi (step (ifz E E0 ([x] E1 x)) (ifz E' E0 ([x] E1 x))) %<- (step E E') %. +%term step/ifz/z step (ifz z E0 ([x] E1 x)) E0 %. +%term step/ifz/s %pi (step (ifz (s E) E0 ([x] E1 x)) (E1 E)) %<- (value E) %. +%{! + +## Progress theorem + +### Sum type for the result + +!}% +%sort val-or-step %. +%name val-or-step %. +%term vos/value %pi (val-or-step E) %<- (value E) %. +%term vos/step %pi (val-or-step E) %<- (step E E') %. +%{! + +### Lemmas + +!}% +%sort prog/app %. +%mode prog/app %in %in %in %out %. +%term _ prog/app _ (vos/step (%the (step E1 E1') Dstep1)) _ (vos/step (step/app/fn Dstep1)) %. +%term _ prog/app _ (vos/value (%the (value E1) Dval1)) (vos/step (%the (step E2 E2') Dstep2)) (vos/step (step/app/arg Dstep2 Dval1)) %. +%term _ prog/app (%the (of (fn T ([x] E' x)) (arr T T')) (of/fn _)) (vos/value (%the (value (fn T ([x] E' x))) Dval1)) (vos/value (%the (value E2) Dval2)) (vos/step (step/app/beta Dval2)) %. +%worlds () (prog/app _ _ _ _) %. +%total {} (prog/app _ _ _ _) %. +%{! + +#### Exercise: lemma for s + +!}% +%{! + +#### Exercise: lemma for ifz + +!}% +%{! + +### Main theorem + +!}% +%sort prog %. +%mode prog %in %out %. +%term _ prog (%the (of z num) of/z) (vos/value (%the (value z) value/z)) %. +%term _ prog (of/fn _) (vos/value value/fn) %. +%term _ + %pi (prog (of/app (%the (of E2 T') D2) (%the (of E1 (arr T' T)) D1)) DvosApp) + %<- (prog D1 (%the (val-or-step E1) Dvos1)) + %<- (prog D2 (%the (val-or-step E2) Dvos2)) + %<- (prog/app D1 Dvos1 Dvos2 DvosApp) %. +%{! + +#### Exercise: cases for s and ifz + +!}% +%worlds () (prog _ _) %. +% Uncomment the line below to check that you've completed the proof: +% total Dof (prog Dof _). +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/CADE_Tutorial/MinML_encoding). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/cade-tutorial-minml.lf b/new-tests/stelf-output/pages/cade-tutorial-minml.lf new file mode 100644 index 0000000..a36c65d --- /dev/null +++ b/new-tests/stelf-output/pages/cade-tutorial-minml.lf @@ -0,0 +1,9 @@ +% +%{! * [MinML Syntax and judgments---the code we just wrote](/wiki/cade-tutorial-minml-encoding/) +* [Solutions to exercises](/wiki/cade-tutorial-minml-answer/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/CADE_Tutorial/MinML). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/cade-tutorial-next.lf b/new-tests/stelf-output/pages/cade-tutorial-next.lf new file mode 100644 index 0000000..b394d5c --- /dev/null +++ b/new-tests/stelf-output/pages/cade-tutorial-next.lf @@ -0,0 +1,22 @@ +% +%{! If you want to read more about Twelf, you can check out the [introductions](/wiki/introductions-to-twelf/), [tutorials](/wiki/tutorials/), and [case studies](/wiki/case-studies/) on the wiki. + +We strongly recommend that you work on some of the following exercises: you'll get a flavor for the experience of defining systems and proving theorems in Twelf. If we have time at the end of the tutorial, we'll break into small groups and you can pick an interesting problem or two to work on. If you get stuck, ask a TA! Refer to the solutions to check your work (though Server OK from Twelf should be assurance enough!) Feel free to ask us for help during the rest of the conference. + +## Beginner problems (Recommended) +* [Preservation for MinML](/wiki/popl-tutorial-minml-preservation-theorem/) -- Prove type preservation for MinML. ([Solution](/wiki/popl-tutorial-minml-preservation-theorem-solution/)) +* [Sequent calculus vs. natural deduction](/wiki/popl-tutorial-sequent-vs-natural-deduction/) -- Prove the logical equivalence of Gentzen's intuitionistic sequent calculus and intuitionistic natural deduction. ([Solution](/wiki/popl-tutorial-sequent-vs-natural-deduction-solution/)) +* [Big-step evaluation vs. small-step transition semantics](/wiki/popl-tutorial-big-step-small-step/) -- Prove the operational equivalence of the big-step and small-step views of operational semantics. ([Solution](/wiki/popl-tutorial-big-step-small-step-solution/)) + +## Intermediate problems +* [Call-by-value CPS transform, with administrative redexes](/wiki/popl-tutorial-cps-problem/) -- Define an algorithm for converting lambda terms to continuation-passing style ([Solution](/wiki/popl-tutorial-cps-solutions/)) +* [Higher-order call-by-value CPS transform, no administrative redexes](/wiki/popl-tutorial-cps-problem2/) -- Define a higher-order variant of the CPS conversion that never creates administrative redexes ([Solution](/wiki/popl-tutorial-cps-solution2/)) +* [Church-Rosser](/wiki/popl-tutorial-church-rosser-problem/) -- Prove Church and Rosser's seminal result via a Diamond Lemma for the untyped lambda calculus. ([Solution](/wiki/church-rosser-w-identity-reduction/)) +* [MinML with exceptions](/wiki/popl-tutorial-exceptions-problem/) -- Prove type safety for an extension of MinML with exceptions defined using structural operational semantics with a "raises" judgement. ([Solution](/wiki/popl-tutorial-exceptions/)) +* [Lambda calculus and combinators](/wiki/popl-tutorial-typed-bracket-abstraction/) -- Explore the relation between the simply-typed lambda calculus and combinatory logic using Curry's classic bracket abstraction algorithm. ([Solution](/wiki/popl-tutorial-typed-bracket-abstraction-solution/)) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/CADE_Tutorial/Next). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/cade-tutorial.lf b/new-tests/stelf-output/pages/cade-tutorial.lf new file mode 100644 index 0000000..f34e513 --- /dev/null +++ b/new-tests/stelf-output/pages/cade-tutorial.lf @@ -0,0 +1,66 @@ +% +%{! <!-- +<div style="font-size:162%;border:none;margin: 0;padding:.1em;color:#F00">**_We want your input!**_</div> + +Help us out by filling out our handy dandy [http://spreadsheets.google.com/viewform?key=pff4VF1v6KVNcwC_c2qFAGA feedback form]! +--> + +## Mechanizing Metatheory with LF and Twelf + +Do you want to learn how to use [Twelf](/) to specify, implement, and prove +properties about programming languages? + +Come to the Twelf tutorial on **June 10, 2013, 9:00-12:30**, co-located with CADE 2013, in +**Lake Placid, New York**. Location "Mirror Lake" in The Crowne Plaza Resort. + +Learn to: +* Represent [languages and logics](/wiki/object-logic/) in [LF](/wiki/lf/) +* Prove [metatheorem](/wiki/metatheorem/)s with [Twelf](/) +under the helpful guidance of Twelf experts. + +The tutorial will be a highly interactive introduction to LF and Twelf +aimed at programming languages researchers. No prior experience with LF +and Twelf is presumed. Participants will leave the workshop with +experience in reading and writing LF representations of programming +languages, and experience reading, writing, and debugging Twelf proofs. + +The tutorial is organized and jointly presented by the the IT University of Copenhagen and the CMU Principles of Programming group. The presenters and TAs at CADE will be Taus Brock-Nannestad, Chris Martens, and Carsten Schürmann. + +## Schedule + +The tutorial will begin at **9:00AM**. Get [http://www.demtech.dk/raw-attachment/wiki/News/lectures.pdf the slides]! + +* Part 1: [Basic Twelf Skills](/wiki/cade-tutorial-basics/) +* Part 2: [Mechanizing MinML](/wiki/cade-tutorial-minml/) +* Part 3: [Combinators: Worlds and Adequacy](/wiki/cade-tutorial-combinators/) +* Coda: [What's next?](/wiki/cade-tutorial-next/) + +There will be a coffee break at 10:30AM. + +## Get Twelf before the tutorial! + +The tutorial will be interactive, with participants writing Twelf code, so you should come with Twelf installed on your laptop. + +Pre-built binaries of Twelf are available for most operating systems: see the [download page](/download/). + +Otherwise: +* you can build Twelf from the [http://twelf.plparty.org/builds/twelf-src.tar.gz source tarball]. You will need [http://www.mlton.org MLton] or [http://www.smlnj.org sml/nj]. +* you can make yourself an account on the wiki, and do the exercises on your User:<login> page (linked at the top after you log in). + +Then see [Twelf with Emacs](/wiki/twelf-with-emacs/) for the basics of interacting with Twelf. (You can also use [Twelf without Emacs](/wiki/twelf-without-emacs/), by interacting with the Twelf server directly.) + +## Sponsors + +Thanks to our sponsors: +[http://www.cs.cmu.edu Carnegie Mellon School of Computer Science], +[http://www.demtech.dk DemTech Project], +[http://www.itu.dk IT University of Copenhagen], + +[[Image:POPL_Tutorial-cmu.gif|250px]] [[Image:POPL_Tutorial-scslogo.gif]] +[[Image:demtech.png|100px]] !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/CADE_Tutorial). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/canonical-form.lf b/new-tests/stelf-output/pages/canonical-form.lf new file mode 100644 index 0000000..019409b --- /dev/null +++ b/new-tests/stelf-output/pages/canonical-form.lf @@ -0,0 +1,54 @@ +% +%{! : _This page describes the canonical forms of the LF type theory. You might also be looking for the [canonical forms lemma](/wiki/canonical-forms-lemma/) used to prove the [progress](/wiki/canonical-forms-lemma/) theorem about a programming language. + +**Canonical forms** are terms in a type theory that are both [β-normal](/wiki/beta-equivalence/) and [η-long](/wiki/eta-equivalence/). + +The [LF](/wiki/lf/) type theory is defined so that all well-typed terms are **canonical forms**. This is useful because the LF methodology for representing object languages consists of creating an isomorphic representation of the object language as the canonical forms of particular LF types. Defining the type theory with only canonical forms uses an algorithm called [hereditary substitution](/wiki/hereditary-substitution/). + +## Syntax of LF + + + + + + + + + +In Twelf's concrete syntax, the type is written ``\{x:A1\} A2``, the kind is written ``\{x:A\} K``, and the term is written ``[x] M``. The type families include both dependent function types and applications of type families to terms. The kind level classifies type families. Type families of kind classify terms. Type families that are not yet fully instantiated have kinds . In both types and kinds, we use ``->`` as an abbreviation when the argument is not free in the result. + +## Motivation for canonical forms + +The above syntax describes what we call _[canonical form](/wiki/canonical-form/)s_. Note what is _not_ a canonical form: there is no syntactic way to apply a lambda-abstraction to an argument. Based on your past experience with programming languages, it may seem strange to define LF so that only canonical forms exist—we are not allowed to write down any programs that have any computation left to do. However, this restriction makes sense if you think about our methodology for representing object languages in LF. For example, we represent [natural numbers](/wiki/natural-numbers/) with the following LF signature: !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%{! For this representation to be [adequate](/wiki/adequacy/), the only LF terms of type ``nat`` must be ``z``, ``s z``, ``s (s z)``, and so on. It is easy to see that non-canonical LF terms interfere with this encoding. For example, the LF term would have type ``nat``, but it is not the representation of any informal natural number. + +## Canonical LF + +If you have encountered [beta-reduction](/wiki/beta-equivalence/) and [eta-expansion](/wiki/eta-equivalence/) before in your study of programming languages, it may help your intuition to know that the canonical forms of LF coincide with the beta-normal, eta-long terms of the lambda calculus. What we were saying above is that the syntax of canonical forms forces them to be beta-normal, and that the typing rules for canonical forms ensure that they are eta-long. In logic, canonical forms correspond to normal and neutral natural deduction proofs and cut-free sequent calculus proofs. + +We do not present the typing rules for canonical forms here (see, e.g., _Mechanizing Metatheory_<ref>\{\{bibtex:hl07mechanizing\}\}</ref>). However, we note two details: + +Terms are only canonical at base type , not at function type. For example, the constant ``s`` is _not_ a canonical form of type ``nat -> nat``. However, the term , which is equivalent, is a canonical form. + +In a dependently typed language, the application typing rules must substitute the argument into the body of the dependent function type: + + + +In LF, this substitution is in fact a [hereditary substitution](/wiki/hereditary-substitution/), ensuring that the terms embedded in the result of the substitution are in canonical form. The typing rule for family applications has a similar substitution into the result kind. + +## See also +* See the [introductions to Twelf](/wiki/introductions-to-twelf/) to learn how object languages are represented as the canonical forms of LF. + +## References +<references/> + +\{\{stub\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Canonical_form). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/canonical-forms-lemma.lf b/new-tests/stelf-output/pages/canonical-forms-lemma.lf new file mode 100644 index 0000000..7bc7bea --- /dev/null +++ b/new-tests/stelf-output/pages/canonical-forms-lemma.lf @@ -0,0 +1,145 @@ +% +% +%{! + +_This article is about proving an [object language](/wiki/object-logic/) lemma. For the notion of canonical form within LF itself, see [canonical form](/wiki/canonical-form/)._ + +In a standard proofs of type safety, the [progress](/wiki/canonical-forms-lemma/) theorem requires **canonical forms** lemmas about the values in a language. These lemmas are usually in the form "If e is a value and ⊦ e : T, then e is of some form appropriate for T". A specific example would be "If e is a value and ⊦ e : T1 → T2, then e is of the form λx:T1.e'". + +Depending on the complexity of the type system, there are two general ways to state and prove these lemmas. In some cases, Twelf's input coverage can give you the canonical forms lemma for free. In other cases, you must state and prove canonical forms explicitly using [equality](/wiki/equality/). + +## Canonical forms lemmas for free + +If there is only one rule through which the canonical form at a particular type can be derived, then Twelf's input coverage is actually strong enough to give you canonical forms for free via inversion. + +Consider the following language: !}% +%sort tp %. +%term tp/unit tp %. +%term tp/arrow %pi tp %-> tp %-> tp %. +%sort exp %. +%term exp/unit exp %. +%term exp/lam %pi tp %-> (%pi exp %-> exp) %-> exp %. +%term exp/app %pi exp %-> exp %-> exp %. +%sort of %. +%term of/unit of exp/unit tp/unit %. +%term of/lam + %pi (of (exp/lam T1 E) (tp/arrow T1 T2)) + %<- ({x} %pi (of x T1) %-> (of (E x) T2)) %. +%term of/app %pi (of (exp/app E1 E2) T2) %<- (of E2 T1) %<- (of E1 (tp/arrow T1 T2)) %. +%sort val %. +%term val/unit val exp/unit %. +%term val/lam val (exp/lam _ _) %. +%sort step %. +%term step/app-1 %pi (step (exp/app E1 E2) (exp/app E1' E2)) %<- (step E1 E1') %. +%term step/app-2 %pi (step (exp/app E1 E2) (exp/app E1 E2')) %<- (step E2 E2') %<- (val E1) %. +%term step/app-beta %pi (step (exp/app (exp/lam _ E) E2) (E E2)) %<- (val E2) %. +%sort notstuck %. +%term notstuck/val %pi (notstuck E) %<- (val E) %. +%term notstuck/step %pi (notstuck E) %<- (step E E') %. +%{! To prove progress for this language, we use an [output factoring](/wiki/output-factoring/) lemma that collects the post-inductive-call reasoning in the application case: + +```twelf +progress-exp/app : notstuck E1 + -> notstuck E2 + -> of E1 (tp/arrow _ _) + -> notstuck (exp/app E1 E2) + -> type. +%mode progress-exp/app +D1 +D2 +D3 -D4. + +- : progress-exp/app (notstuck/step S) _ _ + (notstuck/step (step/app-1 S)). + +- : progress-exp/app (notstuck/val V) (notstuck/step S) _ + (notstuck/step (step/app-2 V S)). + +- : progress-exp/app (notstuck/val val/lam) (notstuck/val V2) (of/lam _) + (notstuck/step (step/app-beta V2)). + +%worlds () (progress-exp/app _ _ _ _). +%total {} (progress-exp/app _ _ _ _). +``` + +In the final case, we use inversion to assert that ``E1`` is a value in the form of a lambda. Because we also pass in a typing derivation that says E1 is of an arrow type, Twelf's input coverage verifies that the only time ``E1`` is a value and has an arrow type is when it is a lambda. Because the type system is simple enough that Twelf can figure out that the only values of arrow type are lambdas, we get canonical forms "for free". + +It is then simple to complete the proof of progress: + +```twelf +progress : of E T + -> notstuck E + -> type. +%mode progress +D1 -D2. + +- : progress of/unit (notstuck/val val/unit). + +- : progress (of/lam _) (notstuck/val val/lam). + +- : progress (of/app D1 D2) NS + <- progress D1 NS1 + <- progress D2 NS2 + <- progress-exp/app NS1 NS2 D1 NS. + +%worlds () (progress _ _). +%total (D1) (progress D1 _). +``` + +## Canonical forms lemmas stated using equality + +Some languages do not have the luxury of having exactly one rule through which a canonical form could be derived. This is common for languages with re-typing rules that make use of subtyping/type equality. In such languages, canonical forms lemmas must be explicitly stated and proven. The conclusion of the lemma "e is of the form ..." can be stated in a straightforward way using [shallow equality](/wiki/equality/). This example makes use of [output factoring](/wiki/output-factoring/) and [reasoning from equality](/wiki/equality/). + +For example, we extend the above language with a trivial notion of subtyping. It is equivalent to syntactic equality on types. The purpose of this tutorial is to illustrate how to state/prove canonical forms lemmas when subtyping/equality are part of the type system, and this trivial notion is enough to motivate the discussion. !}% +%sort tp-sub %. +%term tp-sub/unit tp-sub tp/unit tp/unit %. +%term tp-sub/arrow + %pi (tp-sub (tp/arrow T1 T2) (tp/arrow T3 T4)) + %<- (tp-sub T2 T4) + %<- (tp-sub T3 T1) %. +%term of/sub %pi (of E1 T2) %<- (tp-sub T1 T2) %<- (of E1 T1) %. +%{! Now, we state and prove the canonical forms lemma explicitly. In the theorem statement, we use the syntactic equality judgement as an output in order to state "E1 is of the form (exp/lam T E)". !}% +%sort seq-exp %. +%term seq-exp/i seq-exp E E %. +%sort cfl-tp/arrow %. +%mode cfl-tp/arrow %in %in %out %. +%term _ cfl-tp/arrow val/lam (of/lam _) seq-exp/i %. +%term _ %pi (cfl-tp/arrow V (of/sub D1 (tp-sub/arrow _ _)) DQ) %<- (cfl-tp/arrow V D1 DQ) %. +%worlds () (cfl-tp/arrow _ _ _) %. +%total (D1) (cfl-tp/arrow _ D1 _) %. +%{! Next, we define a helper lemma for showing that (exp/app E1 E2) is notstuck when E1 is of the form (exp/lam T E) and E2 is a value. We use this lemma because we get the derivation of (seq-exp E1 (exp/lam T E)) as the output of the canonical forms lemma. If we were to try to invert the derivation as an output, we would cause Twelf's coverage checker to fail. So instead, we call a helper lemma with the equality derivation as an input. Because the derivation is an input, we can safely apply inversion on it. !}% +%sort progress-exp/app-beta %. +%mode progress-exp/app-beta %in %in %out %. +%term _ progress-exp/app-beta seq-exp/i V (notstuck/step (step/app-beta V)) %. +%worlds () (progress-exp/app-beta _ _ _) %. +%total {} (progress-exp/app-beta _ _ _) %. +%{! Next, we prove the output factoring lemma: !}% +%sort progress-exp/app %. +%mode progress-exp/app %in %in %in %out %. +%term _ progress-exp/app (notstuck/step S) _ _ (notstuck/step (step/app-1 S)) %. +%term _ progress-exp/app (notstuck/val V) (notstuck/step S) _ (notstuck/step (step/app-2 V S)) %. +%{! Because there is more than one way to derive that a value that has an arrow type, we must use the canonical forms lemma we have just proven. To make use of the equality we get out of the canonical forms lemma, we call the special helper lemma progress-exp/app-beta. !}% +%term _ + %pi (progress-exp/app (notstuck/val V) (notstuck/val V2) D1 NS) + %<- (cfl-tp/arrow V D1 DQ) + %<- (progress-exp/app-beta DQ V2 NS) %. +%worlds () (progress-exp/app _ _ _ _) %. +%total {} (progress-exp/app _ _ _ _) %. +%{! Finally, we prove the overall progress theorem as before. !}% +%sort progress %. +%mode progress %in %out %. +%term _ progress of/unit (notstuck/val val/unit) %. +%term _ progress (of/lam _) (notstuck/val val/lam) %. +%term _ + %pi (progress (of/app D1 D2) NS) + %<- (progress D1 NS1) + %<- (progress D2 NS2) + %<- (progress-exp/app NS1 NS2 D1 NS) %. +%term _ %pi (progress (of/sub D1 _) NS) %<- (progress D1 NS) %. +%worlds () (progress _ _) %. +%total (D1) (progress D1 _) %. +%{! The only change from before is the case for of-sub, where the inductive hypothesis gives the result. + +Note: There is actually a very subtle way to prove ``progress-exp/app`` using canonical forms lemmas "for free". However, explicitly stating and proving the canonical forms lemma is the more robust strategy in general. + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Canonical_forms_lemma). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/case-studies.lf b/new-tests/stelf-output/pages/case-studies.lf new file mode 100644 index 0000000..f32a026 --- /dev/null +++ b/new-tests/stelf-output/pages/case-studies.lf @@ -0,0 +1,76 @@ +% +% +% +%{! The following **case studies** present interesting applications of Twelf. Some of these case studies use proof techniques that are explained in the [tutorials](/wiki/tutorials/), so you may wish to read those first. + +Feel free to add your own case studies here. We would welcome experience reports that document not just the Twelf code you wrote but your experience writing it. _(If the goal of your article is to teach a specific Twelf technique, it should instead be a [tutorial](/wiki/tutorials/), and if your Twelf code is off-site, you should instead add it to the list of [Research projects using Twelf](/wiki/research-projects-using-twelf/).)_ + +#### Beginner + +* [Bracket abstraction](/wiki/bracket-abstraction/), by [William Lovas](/wiki/user-william-lovas/) + + A translation of the untyped lambda calculus into the SKI combinator calculus. Bracket abstraction is defined by pattern matching over LF-level lambda abstractions, making essential use of [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/). + +* [Typed combinators soundness and completeness](/wiki/typed-combinators-soundness-and-completeness/), by [William Lovas](/wiki/user-william-lovas/) + + A translation of the simply-typed lambda calculus into the SKI combinator calculus. The translation is shown correct by showing that it preserves and reflects full beta-eta equality. This case study uses uses [intrinsic typing](/wiki/intrinsic-and-extrinsic-encodings/) and [open worlds](/wiki/percent-worlds/). (Roughly an extended version of the above [bracket abstraction](/wiki/bracket-abstraction/) case study.) + +* [Division over the natural numbers](/wiki/division-over-the-natural-numbers/), by [Daniel K. Lee](/wiki/user-danielklee/) + + A study of the encoding of division over the natural numbers. The definition is shown to be "correct" in a number of ways, including termination of division by a non-zero number and correctness with respect to multiplication. The former involves a use of ``%reduces`` for strong induction. + +* [Church-Rosser via complete development](/wiki/church-rosser-via-complete-development/), by [Dan Licata](/wiki/user-drl/) + + Takahashi's proof of the diamond property of parallel reduction. This case study includes lots of examples of theorems in [regular world](/wiki/percent-worlds/)s. + +* [Admissibility of cut](/wiki/admissibility-of-cut/), by [Tom Murphy VII](/wiki/user-tom7/) + + A proof of an important theorem about an intuitionistic sequent calculus. Uses lexicographic [termination](/wiki/percent-terminates/) orderings. + +* [Hereditary substitution for the STLC](/wiki/hereditary-substitution-for-the-stlc/), by [Dan Licata](/wiki/user-drl/) + + The above proof of cut admissibility recast as an algorithm for normalizing λ-terms. Uses many of the proof techniques explained in the [tutorials](/wiki/tutorials/). + +* [Double-negation translation](/wiki/double-negation-translation/), by [Dan Licata](/wiki/user-drl/) + + The Godel-Gentzen negative translation of classical logic into intuitionistic logic. + +* [Linear logic](/wiki/linear-logic/), by [Karl Crary](/wiki/user-crary/) + An encoding of linear logic that uses a judgment to enforce linearity, along with a proof of subject reduction for it. + +* [Mutable state](/wiki/mutable-state/), by [Robert J. Simmons](/wiki/user-rsimmons/) + + Encoding a simple imperative language with state. + +* [Zermelo Frankel](/wiki/zermelo-frankel/), by Daniel Wang + + A particular encoding of ZF set theory in Twelf + +* Twelf Data structures: + - [Lists](/wiki/lists/), by [Carsten Varming](/wiki/user-varming/) + - [Unary natural numbers with inequality](/wiki/natural-numbers-with-inequality/), by [Robert J. Simmons](/wiki/user-rsimmons/) + - [Dense lexicographical orderings](/wiki/lexicographical-orderings-with-density/), by [Daniel K. Lee](/wiki/user-danielklee/) + +#### Advanced + +* [CPS conversion](/wiki/cps-conversion/), by [Tom Murphy VII](/wiki/user-tom7/) + + A type-directed conversion from direct style lambda calculus into continuation passing style. + +* [Classical S5](/wiki/classical-s5/), by [Tom Murphy VII](/wiki/user-tom7/) + + A proof that a non-standard natural deduction for the modal logic Classical S5 is equivalent to the standard cut-free sequent calculus. + +* [Lily](/wiki/lily/), by [Carsten Varming](/wiki/user-varming/) + + An encoding of a polymorphic linear lambda calculus with fixed points in LF, and a metatheorem proving that ground contextual equivalence with respect to a call-by-name semantics coincides with ground contextual equivalence with respect to a call-by-value semantics. + +* [Lax logic](/wiki/lax-logic/), by [Robert J. Simmons](/wiki/user-rsimmons/) + + Establishing the correspondence between two different presentations of propositional lax logic and showing soundness and completeness of the two systems, in the process establishing cut and identity. + +* [Structural Focalization](/wiki/focusing/), by [Robert J. Simmons](/wiki/user-rsimmons/) + + The Twelf formalization from Rob's 2014 paper on + [Structural Focalization](https://dl.acm.org/doi/10.1145/2629678). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/catch-all-case.lf b/new-tests/stelf-output/pages/catch-all-case.lf new file mode 100644 index 0000000..b667b06 --- /dev/null +++ b/new-tests/stelf-output/pages/catch-all-case.lf @@ -0,0 +1,178 @@ +% +%{! When defining relations in non-empty contexts, it is often necessary to put a case of the relation in the LF context. In some circumstances, it is possible to avoid doing so by writing a **catch-all case** instead. +This technique works when you can cover the space of the relation that would be covered by a context case without mentioning any variables from the context explicitly. + +Using a catch-all case leads to simpler Twelf code because it saves you from having to repeat the context case at a variety of context extension points. + +As a motivating example, we use the theorem about the size of a STLC term from \{\{proving|title=Proving metatheorems in non-empty contexts\}\}. For review, here are the relevant judgements: !}% +%sort tp %. +%term unit tp %. +%term arrow %pi tp %-> tp %-> tp %. +%sort tm %. +%term empty tm %. +%term lam %pi tp %-> (%pi tm %-> tm) %-> tm %. +%term app %pi tm %-> tm %-> tm %. +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort plus %. +%mode plus %in %in %out %. +%term plus-z plus z N2 N2 %. +%term plus-s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%worlds () (plus _ _ _) %. +%total N (plus N _ _) %. +%{! !}% +%{! (options removed from twelftag: hidden="true") !}% +%sort plus-exists {N1} {N2} %. +%mode plus-exists %in %in %out %. +%term _ plus-exists z _ plus-z %. +%term _ %pi (plus-exists (s N1) N2 (plus-s D)) %<- (plus-exists N1 N2 D) %. +%worlds () (plus-exists _ _ _) %. +%total N1 (plus-exists N1 _ _) %. +%sort plus-z-rh {n nat} %. +%mode plus-z-rh %in %out %. +%term _ plus-z-rh z plus-z %. +%term _ %pi (plus-z-rh (s N) (plus-s D)) %<- (plus-z-rh N (%the (plus N z N) D)) %. +%worlds () (plus-z-rh _ _) %. +%total N (plus-z-rh N _) %. +%sort plus-s-rh %. +%mode plus-s-rh %in %out %. +%term _ plus-s-rh plus-z plus-z %. +%term _ + %pi (plus-s-rh (plus-s (%the (plus N1' N2 N3') D)) (plus-s D')) + %<- (plus-s-rh D (%the (plus N1' (s N2) (s N3')) D')) %. +%worlds () (plus-s-rh _ _) %. +%total D (plus-s-rh D _) %. +%sort plus-commute %. +%mode plus-commute %in %out %. +%term _ %pi (plus-commute (%the (plus z N N) plus-z) D) %<- (plus-z-rh N D) %. +%term _ + %pi (plus-commute (%the (plus (s N1) N2 (s N3)) (plus-s D)) D'') + %<- (plus-commute D (%the (plus N2 N1 N3) D')) + %<- (plus-s-rh D' (%the (plus N2 (s N1) (s N3)) D'')) %. +%worlds () (plus-commute _ _) %. +%total D (plus-commute D _) %. +%sort id/nat %. +%term id/nat-refl id/nat N N %. +%sort id/nat-s-cong %. +%mode id/nat-s-cong %in %out %. +%term _ id/nat-s-cong id/nat-refl id/nat-refl %. +%worlds () (id/nat-s-cong _ _) %. +%total {} (id/nat-s-cong _ _) %. +%sort plus-unique %. +%mode plus-unique %in %in %out %. +%term _ plus-unique D D id/nat-refl %. +%term _ + %pi (plus-unique (plus-s D) (plus-s D') DidS) + %<- (plus-unique D D' Did) + %<- (id/nat-s-cong Did DidS) %. +%worlds () (plus-unique _ _ _) %. +%total D (plus-unique D _ _) %. +%sort plus-respects-id %. +%mode plus-respects-id %in %in %in %in %out %. +%term _ plus-respects-id D id/nat-refl id/nat-refl id/nat-refl D %. +%worlds () (plus-respects-id _ _ _ _ _) %. +%total {} (plus-respects-id _ _ _ _ _) %. +%sort id/nat-sym %. +%mode id/nat-sym %in %out %. +%term _ id/nat-sym id/nat-refl id/nat-refl %. +%worlds () (id/nat-sym _ _) %. +%total {} (id/nat-sym _ _) %. +%sort plus-assoc %. +%mode plus-assoc %in %in %in %out %. +%term _ + %pi (plus-assoc plus-z DplusB-C DplusB-C' Dplus) + %<- (plus-unique DplusB-C DplusB-C' Did) + %<- (plus-respects-id plus-z id/nat-refl id/nat-refl Did Dplus) %. +%term _ + %pi (plus-assoc (plus-s DplusA'-B) DplusB-C (plus-s DplusA'B-C) (plus-s DplusA'-BC)) + %<- (plus-assoc DplusA'-B DplusB-C DplusA'B-C DplusA'-BC) %. +%worlds () (plus-assoc _ _ _ _) %. +%total D (plus-assoc D _ _ _) %. +%sort plus-assoc2 %. +%mode plus-assoc2 %in %in %out %in %. +%term _ + %pi (plus-assoc2 D1 D2 D3' D4) + %<- (plus-exists _ _ D3) + %<- (plus-assoc D1 D2 D3 D4') + %<- (plus-unique D4 D4' Did) + %<- (id/nat-sym Did Did') + %<- (plus-respects-id D3 id/nat-refl id/nat-refl Did' D3') %. +%worlds () (plus-assoc2 _ _ _ _) %. +%total {} (plus-assoc2 _ _ _ _) %. +%sort lemma %. +%mode lemma %in %in %in %in %out %. +%term _ + %pi (lemma (%the (plus N1 N2 Nsum) D) (%the (plus Ndiff1 N1 N1') D1pre) (%the (plus Ndiff2 N2 N2') D2pre) (%the (plus N1' N2' Nsum') D') Dres') + %<- (plus-commute D1pre D1) + %<- (plus-commute D2pre D2) + %<- (plus-exists Ndiff1 N2' (%the (plus Ndiff1 N2' Ndiff1+2') Ddiff1+2')) + %<- (plus-assoc D1 Ddiff1+2' D' (%the (plus N1 Ndiff1+2' Nsum') Dassoc)) + %<- (plus-commute D2 (%the (plus Ndiff2 N2 N2') D2')) + %<- (plus-exists Ndiff1 Ndiff2 (%the (plus Ndiff1 Ndiff2 Ndiff1+2) Ddiff1+2)) + %<- (plus-assoc2 Ddiff1+2 D2' (%the (plus Ndiff1+2 N2 Ndiff1+2') Dassoc') Ddiff1+2') + %<- (plus-commute Dassoc' (%the (plus N2 Ndiff1+2 Ndiff1+2') Dassoc'')) + %<- (plus-assoc2 D Dassoc'' Dres Dassoc) + %<- (plus-commute Dres Dres') %. +%worlds () (lemma _ _ _ _ _) %. +%total {} (lemma _ _ _ _ _) %. +%{! !}% +%{! (options removed from twelftag: check="true") !}% +%sort size %. +%mode size %in %out %. +%term size-empty size empty (s z) %. +%term size-lam %pi (size (lam _ E) (s N)) %<- ({x} {dx size x (s z)} size (E x) N) %. +%term size-app %pi (size (app E1 E2) (s N)) %<- (size E1 N1) %<- (size E2 N2) %<- (plus N1 N2 N) %. +%block size-block [x tm] [dx size x (s z)]%. +%worlds (size-block) (size _ _) %. +%total E (size E _) %. +%{! ## Original proof !}% +%{! (options removed from twelftag: hidden="true") !}% +%sort size-at-least-one %. +%mode size-at-least-one %in %out %. +%term _ size-at-least-one _ (plus-s plus-z) %. +%worlds (size-block) (size-at-least-one _ _) %. +%total {} (size-at-least-one _ _) %. +%{! In \{\{proving|title=Proving metatheorems in non-empty contexts\}\}, we proved the following theorem: !}% +%sort subst-size {E' tm} {x tm} %. +%mode subst-size %in %in %in %out %. +%block sdblock [y tm] [dy size y (s z)] [_ {E' tm} subst-size E' ([x] [dx] dy) dy plus-z]%. +%worlds (sdblock) (subst-size _ _ _ _) %. +%{! The ``%block`` declaration includes a case of the theorem for the variable ``dy`` declared in the context. + +## Alternate proof using a catch-all case + +In this instance, it is possible to avoid putting the theorem case in the context. How? We instead write a catch-all case that covers the context variable case **without mentioning the context variable explicitly**: !}% +%{! (options removed from twelftag: check="true") !}% +%sort subst-size-ca {E' tm} {x tm} %. +%mode subst-size-ca %in %in %in %out %. +%term _ subst-size-ca E' ([x] [dx] D) D plus-z %. +%{! This case covers any first-argument derivation ``D`` that does not mention the bound variables ``x`` and ``dx``, as long as the second argument is also that derivation ``D``. Note that the context case in the previous proof has this form. To complete the proof for these inputs, we simply use ``plus-z : plus z N N`` to derive the result. This technique works because we do not actually need to use any reasoning specific to the variable from the context. + +The remaining cases are the same as in the previous proof, except that +* the catch-all case happens to cover the case for ``size-empty`` as well, so we can elide it. +* the case for ``size-lam`` no longer adds the theorem case to the context. !}% +%{! (options removed from twelftag: check="true") !}% +%term _ + %pi (subst-size-ca E' ([x] [dx] dx) D Dplus') + %<- (size-at-least-one D Dplus) + %<- (plus-commute Dplus Dplus') %. +%term _ + %pi (subst-size-ca E' ([x] [dx] %the (size (lam T _) _) (size-lam ([y] [dy] D x dx y dy))) (size-lam D') Dplus') + %<- ({y tm} {dy size y (s z)} subst-size-ca E' ([x] [dx] D x dx y dy) (D' y dy) Dplus) + %<- (plus-s-rh Dplus Dplus') %. +%term _ + %pi (subst-size-ca E' ([x] [dx] size-app (%the (plus N1 N2 Nsum) Dplus) (%the (size (E2 x) N2) (D2 x dx)) (%the (size (E1 x) N1) (D1 x dx))) (size-app (%the (plus N1' N2' Nsum') Dplus') (%the (size (E2 E') N2') D2') (%the (size (E1 E') N1') D1')) DplusRes') + %<- (subst-size-ca E' D1 (%the (size (E1 E') N1') D1') (%the (plus Ndiff1 N1 N1') Dplus1)) + %<- (subst-size-ca E' D2 (%the (size (E2 E') N2') D2') (%the (plus Ndiff2 N2 N2') Dplus2)) + %<- (lemma Dplus Dplus1 Dplus2 Dplus' DplusRes) + %<- (plus-s-rh DplusRes DplusRes') %. +%worlds (size-block) (subst-size-ca _ _ _ _) %. +%total D (subst-size-ca _ D _ _) %. +%{! \{\{tutorial\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Catch-all_case). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/category-case-studies.lf b/new-tests/stelf-output/pages/category-case-studies.lf new file mode 100644 index 0000000..560fd06 --- /dev/null +++ b/new-tests/stelf-output/pages/category-case-studies.lf @@ -0,0 +1,8 @@ +% +%{! This category contains all **[case studies](/wiki/case-studies/)**. Case studies are more general than [tutorials](/wiki/tutorials/) - whereas tutorials illustrate a specific tool or problem-solving technique; case studies range from "toy" examples that illustrate a variety of techniques to larger pieces of Twelf code used for real [research](/wiki/research-projects-using-twelf/). !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Category:Case_studies). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/category-keywords.lf b/new-tests/stelf-output/pages/category-keywords.lf new file mode 100644 index 0000000..5327215 --- /dev/null +++ b/new-tests/stelf-output/pages/category-keywords.lf @@ -0,0 +1,12 @@ +% +%{! This category contains Twelf **keywords**. + +## See also + +[http://www.cs.cmu.edu/~twelf/guide-1-4/twelf_17.html#cindex_% Index of keywords] in the [User's Guide](/wiki/users-guide/). !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Category:Keywords). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/category-literate-twelf-code.lf b/new-tests/stelf-output/pages/category-literate-twelf-code.lf new file mode 100644 index 0000000..1285ff2 --- /dev/null +++ b/new-tests/stelf-output/pages/category-literate-twelf-code.lf @@ -0,0 +1,10 @@ +% +%{! This category contains all pages that are written using the [[Project:Literate Twelf|Literate Twelf]] extension to this wiki. + +[Category:Twelf code](/wiki/category-twelf-code/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Category:Literate_Twelf_code). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/category-pages-needing-attention.lf b/new-tests/stelf-output/pages/category-pages-needing-attention.lf new file mode 100644 index 0000000..1a9b87c --- /dev/null +++ b/new-tests/stelf-output/pages/category-pages-needing-attention.lf @@ -0,0 +1,8 @@ +% +%{! Pages that need some work - very short pages, pages that aren't completely formatted, etc. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Category:Pages_needing_attention). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/category-stubs.lf b/new-tests/stelf-output/pages/category-stubs.lf new file mode 100644 index 0000000..0544ce5 --- /dev/null +++ b/new-tests/stelf-output/pages/category-stubs.lf @@ -0,0 +1,8 @@ +% +%{! Short articles that need to be longer. If an article listed here doesn't actually seem like it needs expanding, remove the <nowiki>\{\{stub\}\}</nowiki> tag. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Category:Stubs). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/category-tutorials-needing-attention.lf b/new-tests/stelf-output/pages/category-tutorials-needing-attention.lf new file mode 100644 index 0000000..c54d1a5 --- /dev/null +++ b/new-tests/stelf-output/pages/category-tutorials-needing-attention.lf @@ -0,0 +1,8 @@ +% +%{! [Needing attention](/wiki/category-tutorials/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Category:Tutorials_needing_attention). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/category-tutorials.lf b/new-tests/stelf-output/pages/category-tutorials.lf new file mode 100644 index 0000000..d27e2d2 --- /dev/null +++ b/new-tests/stelf-output/pages/category-tutorials.lf @@ -0,0 +1,10 @@ +% +%{! This is the category of Twelf tutorials, aimed at addressing a frequently encountered issue with using Twelf. It is one of a number of forms of [documentation](/wiki/documentation/) available at the Twelf Project wiki. Summaries of these tutorials are covered on the [tutorial](/wiki/tutorials/) page. + +To list your project on this page, add the <nowiki>\{\{tutorial\}\}</nowiki> tag at the end of a page. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Category:Tutorials). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/category-twelf-code.lf b/new-tests/stelf-output/pages/category-twelf-code.lf new file mode 100644 index 0000000..b92ebbd --- /dev/null +++ b/new-tests/stelf-output/pages/category-twelf-code.lf @@ -0,0 +1,8 @@ +% +%{! This is the category for any Twelf code fragments, signatures, etc. on the site. [[Project:Literate Twelf|Literate Twelf]] pages are automatically placed in this category, and other pages that have a significant amount of code in them probably should too. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Category:Twelf_code). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/category-undocumented-features.lf b/new-tests/stelf-output/pages/category-undocumented-features.lf new file mode 100644 index 0000000..5c58cd3 --- /dev/null +++ b/new-tests/stelf-output/pages/category-undocumented-features.lf @@ -0,0 +1,8 @@ +% +%{! Twelf contains some **undocumented features** that are used by researchers. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Category:Undocumented_features). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/church-rosser-via-complete-development.lf b/new-tests/stelf-output/pages/church-rosser-via-complete-development.lf new file mode 100644 index 0000000..4e42733 --- /dev/null +++ b/new-tests/stelf-output/pages/church-rosser-via-complete-development.lf @@ -0,0 +1,339 @@ +% +%{! +In this case study, we give a simple proof of the diamond property of +untyped parallel reduction. The proof, due to +Masako Takahashi (Parallel Reductions in -Calculus, +Information and Computation 118(1), 1995) makes use of a auxilary relation called complete +developement. This case study is a good example of proving +metatheorems in [regular worlds](/wiki/percent-worlds/): we use several different worlds, and +transport theorems between them. +!}% +%{! ## Syntax and judgements + +First, we define the syntax of the language, along with parallel reduction and complete development. +See [Reformulating languages to use hypothetical judgements](/wiki/reformulating-languages-to-use-hypothetical-judgements/) for a discussion of these judgements and their encodings. +The only difference between the two relations is the side condition on the complete development application rule, which forces the +beta-reduction rule to take precedence over it. +Thus, a single step of complete development contracts all beta-redices in the left-hand term one step. +Consequently, all beta-redices must be reduced away before a term can take a reflexivity step. + +!}% +%sort trm %. +%term lam %pi (%pi trm %-> trm) %-> trm %. +%term app %pi trm %-> trm %-> trm %. +%block trmb [x trm]%. +%worlds (trmb) (trm) %. +%{! ### Parallel reduction !}% +%sort => %. +%prec %none 10 => %. +%mode => %in %out %. +%term =>/beta + %pi (app (lam M) N => M' N') + %<- ({x trm} %pi (x => x) %-> (M x => M' x)) + %<- (N => N') %. +%term =>/app %pi (app M N => (app M' N')) %<- (M => M') %<- (N => N') %. +%term =>/lam %pi (lam M => lam M') %<- ({x trm} %pi (x => x) %-> (M x => M' x)) %. +%block =>b [x trm] [prx x => x]%. +%worlds (=>b) (=> _ _) %. +%total E (=> E _) %. +%{! Note that Twelf can prove that parallel reduction is total automatically! !}% +%{! ### Complete development !}% +%sort notlam %. +%mode notlam %in %. +%term notlam/app notlam (app _ _) %. +%block nlb [x trm] [nlx notlam x]%. +%worlds (nlb) (notlam _) %. +%sort ==> %. +%prec %none 10 ==> %. +%mode ==> %in %out %. +%term ==>/beta + %pi (app (lam M) N ==> M' N') + %<- ({x trm} %pi (notlam x) %-> (x ==> x) %-> (M x ==> M' x)) + %<- (N ==> N') %. +%term ==>/app %pi (app M N ==> (app M' N')) %<- (M ==> M') %<- (N ==> N') %<- (notlam M) %. +%term ==>/lam + %pi (lam M ==> lam M') + %<- ({x trm} %pi (notlam x) %-> (x ==> x) %-> (M x ==> M' x)) %. +%block ==>b [x trm] [nlx notlam x] [cdx x ==> x]%. +%worlds (==>b) (==> _ _) %. +%{! Twelf cannot prove that complete development is total automatically, +but we will prove this fact ourselves below. !}% +%{! +Below, we will want to work in a world where both parallel reduction and complete development exist. +Thus, we define a block that includes all the assumptions in both ``=>b`` and ``==>b``: +!}% +%block =>&==>b [x trm] [prx x => x] [nlx notlam x] [cdx x ==> x]%. +%{! ## Diamond property of parallel reduction !}% +%{! ### Every term either is or isn't a lambda + +As a lemma, we need to prove that every term either is or is not a lambda. + +!}% +%sort lam-or-not %. +%term lam-or-not/lam lam-or-not (lam _) %. +%term lam-or-not/not %pi (lam-or-not E) %<- (notlam E) %. +%sort decide {E} %. +%mode decide %in %out %. +%term _ decide (lam _) lam-or-not/lam %. +%term _ decide (app _ _) (lam-or-not/not notlam/app) %. +%{! +What world should this lemma be stated in? +It talks about ``trm``s, so we definitely need term assumptions, and ``notlam`` only makes sense with its assumptions. +But neither of the reduction relations are relevant, so we should leave off their hypotheses. +However, to prove the theorem, we need to cover the case of ``decide`` for variables ``x``, which we +do by putting a case in the LF context. +(See [Proving metatheorems: Proving metatheorems in non-empty contexts](/wiki/proving-metatheorems-proving-metatheorems-in-non-empty-contexts/) for a discussion of this technique.) +Thus, we arrive at: +!}% +%block decideb [x trm] [nlx notlam x] [decx decide x (lam-or-not/not nlx)]%. +%worlds (decideb) (decide _ _) %. +%total E (decide E _) %. +%{! ### Complete development is total + +To prove the diamond property, we need to know that every term is reducible under complete-development. +First, we prove a lemma showing that an application reduces if its components do: + +!}% +%sort ==>-tot/app %. +%mode ==>-tot/app %in %in %in %out %. +%term _ ==>-tot/app lam-or-not/lam (==>/lam D1) D2 (==>/beta D2 D1) %. +%term _ ==>-tot/app (lam-or-not/not Nl) D1 D2 (==>/app Nl D2 D1) %. +%worlds (==>b) (==>-tot/app _ _ _ _) %. +%total {} (==>-tot/app _ _ _ _) %. +%{! +Note that the world ``==>b`` is the minimal one in which both ``==>`` and ``lam-or-not`` make sense, +so it is the most appropriate world to use here. +!}% +%sort ==>-tot {E trm} %. +%mode ==>-tot %in %out %. +%term _ + %pi (==>-tot (lam E) (==>/lam D)) + %<- ({x trm} {nlx notlam x} {cdx x ==> x} {_ decide x (lam-or-not/not nlx)} {_ ==>-tot x cdx} ==>-tot (E x) (D x nlx cdx)) %. +%term _ + %pi (==>-tot (app E E') D) + %<- (==>-tot E De) + %<- (==>-tot E' De') + %<- (decide E Dlon) + %<- (==>-tot/app Dlon De De' D) %. +%{! +This lemma at least needs the assumptions in ``==>b`` for complete development to be adequately represented. +Additionally, because the above case uses ``decide`` as a lemma, we need to ensure that its assumptions are in +the world we choose here. Additionally, to show that variables reduce, we need a theorem case in the context. +Thus, we arrive at the following block: +!}% +%block ==>totb [x trm] [nlx notlam x] [cdx x ==> x] [_ decide x (lam-or-not/not nlx)] [_ ==>-tot x cdx]%. +%worlds (==>totb) (==>-tot _ _) %. +%total D (==>-tot D _) %. +%{! +Why may we call ``decide`` from contexts made up of ``==>totb`` blocks when decide was only stated for +contexts made up of ``decideb`` blocks? In general, extending the context could create new cases that +``decide`` would be unable to handle. However, in this instance, both of the new assumptions, which have types +``==>`` and ``==>-tot``, are irrelevant (in[[subordinate]]) to ``decide``, +so Twelf permits this world subsumption. That is, ``decide`` is only concerned with ``trm``s and derivations of +``notlam`` and derivations of ``decide`` itself, and neither ``==>`` nor ``==>-tot`` assumptions +influence those types. +!}% +%{! ### Substituting reductions into parallel reduction + +We also need to show that the substitution of a reduction is the reduction of the substitution. +To prove this lemma, we distinguish cases on an LF function representing the +hypothetical derivation: + +!}% +%sort subst {x trm} %. +%mode subst %in %in %out %. +%{! (M x) is x: !}% +%term _ subst ([x] [prx] prx) Dn Dn %. +%{! (M x) doesn't mention x at all (this covers all other variables in the context, as well any other x-closed term): !}% +%term _ subst ([x] [prx] D) _ D %. +%{! The remaining cases proceed compositionally: !}% +%term _ + %pi (subst ([x] [prx] =>/app (Dm2 x prx) (Dm1 x prx)) Dn (=>/app Dm2' Dm1')) + %<- (subst Dm2 Dn Dm2') + %<- (subst Dm1 Dn Dm1') %. +%term _ + %pi (subst ([x] [prx] =>/beta (Dm2 x prx) ([y] [pry] Dm1 y pry x prx)) Dn (=>/beta Dm2' Dm1')) + %<- (subst Dm2 Dn Dm2') + %<- ({y trm} {pry y => y} subst ([x] [prx] Dm1 y pry x prx) Dn (Dm1' y pry)) %. +%term _ + %pi (subst ([x] [prx] =>/lam ([y] [pry] Dm y pry x prx)) Dn (=>/lam Dm')) + %<- ({y trm} {pry y => y} subst ([x] [prx] Dm y pry x prx) Dn (Dm' y pry)) %. +%{! +Because this theorem is property only of parallel reduction, +we use the block containing only parallel reduction assumptions: +!}% +%worlds (=>b) (subst _ _ _) %. +%total D (subst D _ _) %. +%{! ### The triangle property + +The key lemma in this proof is that given a split of parallel reduction on one side and +complete development on the other, there is a parallel reduction on the other side +of the triangle: + +!}% +%sort tri %. +%mode tri %in %in %out %. +%term _ tri D1 D2 D2 %. +%term _ + %pi (tri (==>/app Dnl (%the (N ==> N') Dn') (%the (M ==> M') Dm')) (=>/app (%the (N => N'') Dn'') (%the (M => M'') Dm'')) (=>/app Dn Dm)) + %<- (tri Dn' Dn'' Dn) + %<- (tri Dm' Dm'' Dm) %. +%term _ + %pi (tri (==>/lam Dn') (=>/lam Dn'') (=>/lam Dn)) + %<- ({x trm} {prx x => x} {nlx notlam x} {cdx x ==> x} tri (Dn' x nlx cdx) (Dn'' x prx) (Dn x prx)) %. +%term _ + %pi (tri (==>/beta (%the (N ==> N') Dn') (%the ({x} {nlx} {cdx} M x ==> (M' x)) Dm')) (=>/app (%the (N => N'') Dn'') (=>/lam (%the ({x} {prx} M x => (M'' x)) Dm''))) (=>/beta Dn Dm)) + %<- (tri Dn' Dn'' Dn) + %<- ({x trm} {prx x => x} {nlx notlam x} {cdx x ==> x} tri (Dm' x nlx cdx) (Dm'' x prx) (Dm x prx)) %. +%term _ + %pi (tri (==>/beta (%the (N ==> N') Dn') (%the ({x} {nlx} {cdx} M x ==> (M' x)) Dm')) (=>/beta (%the (N => N'') Dn'') (%the ({x} {prx} M x => (M'' x)) Dm'')) D) + %<- (tri Dn' Dn'' Dn) + %<- ({x trm} {prx x => x} {nlx notlam x} {cdx x ==> x} tri (Dm' x nlx cdx) (Dm'' x prx) (Dm x prx)) + %<- (subst Dm Dn D) %. +%{! The best world for this proof is the one where both parallel reduction +and complete development exist, and we don't need any additional assumptions +for the proof, so this is the world we use: +!}% +%worlds (=>&==>b) (tri _ _ _) %. +%total D (tri D _ _) %. +%{! Let's investigate why complete development is necessary for this proof to go through. +Without the side condition on application, we'd have an additional case to consider, +where the complete development was by the application rule, and the parallel reduction was by the beta rule. We'd set up this case as follows: + +<twelf ignore="true"> +- : tri + (==>/app + IMPOSSIBLE + (Dn' : N ==> N') + (==>/lam (Dm' : \{x\} \{nlx\} \{cdx\} (M x) ==> (M' x)))) + (=>/beta + (Dn_ : N => N_) + (Dm_ : \{x\} \{prx\} (M x) => (M_ x))) + (XXX : M_ N_ => app (lam M') N') + <- tri Dn' Dn_ + (Dn : N_ => N') + <- \{x : trm\} + \{prx : x => x\} + \{nlx : notlam x\} + \{cdx : x ==> x\} + tri (Dm' x nlx cdx) + (Dm_ x prx) + ((Dm : \{x\} x => x -> (M_ x) => (M' x)) x prx). +</twelf> + +From the inductive hypotheses, we know that ``M_ x => M' x`` for all ``x`` and that ``N_ => N'``. +However, we need to prove that ``M_ N_ => app (lam M') N'``. By ``subst``, we can get that the left-hand side reduces +to the **beta-reduction** of the right-hand side, but there the proof breaks down. + +!}% +%{! ### The diamond property + +We prove the diamond property by first generating a parallel reduction for ``M`` and +then making two triangles, which complete the square. + +!}% +%sort dia %. +%mode dia %in %in %out %out %. +%term _ + %pi (dia (%the (M => Mleft) Dleft) (%the (M => Mright) Dright) Dleftmiddle Drightmiddle) + %<- (==>-tot M D==>middle) + %<- (tri D==>middle Dleft Dleftmiddle) + %<- (tri D==>middle Dright Drightmiddle) %. +%{! +Because we call both ``==>-tot`` and ``tri``, +we need to maintain all the assumptions of both of them. +This is the block ``=>&==>b`` extended with the theorem cases for +``decide`` and ``==>-tot`` (a.k.a. +``==>-totb`` extended with a parallel reduction assumption). +Twelf verifies the world subsumption showing that +``==>-tot`` and ``tri`` remain true under such extensions of +the context. +!}% +%block diab [x trm] [prx x => x] [nlx notlam x] [cdx x ==> x] [_ decide x (lam-or-not/not nlx)] [_ ==>-tot x cdx]%. +%worlds (diab) (dia _ _ _ _) %. +%total {} (dia _ _ _ _) %. +%{! ## Other properties of complete development + +To complete the development of complete development, we can prove two other simple lemmas. + +### Soundness of complete development wrt parallel reduction + +First, every complete development reduction is a parallel reduction: +we simply forget the proof of the side condition on the application rule. + +!}% +%sort sound %. +%mode sound %in %out %. +%term _ + %pi (sound (==>/beta Dn Dm) (=>/beta Dn' Dm')) + %<- (sound Dn Dn') + %<- ({x} {nlx notlam x} {cdx x ==> x} {prx x => x} {_ sound cdx prx} sound (Dm x nlx cdx) (Dm' x prx)) %. +%term _ + %pi (sound (==>/app _ Dn Dm) (=>/app Dn' Dm')) + %<- (sound Dm Dm') + %<- (sound Dn Dn') %. +%term _ + %pi (sound (==>/lam Dm) (=>/lam Dm')) + %<- ({x} {nlx notlam x} {cdx x ==> x} {prx x => x} {_ sound cdx prx} sound (Dm x nlx cdx) (Dm' x prx)) %. +%{! For this theorem, we need both complete development and parallel reduction, +as well as a theorem case, so we extend ``=>&==>b`` as follows: +!}% +%block soundb [x trm] [nlx notlam x] [cdx x ==> x] [prx x => x] [_ sound cdx prx]%. +%worlds (soundb) (sound _ _) %. +%total D (sound D _) %. +%{! ### Complete development is deterministic + +Finally, we can show that complete development is derministic. + +!}% +%sort id %. +%term refl id E E %. +%sort id-app-cong %. +%mode id-app-cong %in %in %out %. +%term _ id-app-cong refl refl refl %. +%worlds (trmb) (id-app-cong _ _ _) %. +%total {} (id-app-cong _ _ _) %. +%sort id-lam-cong {x trm} %. +%mode id-lam-cong %in %out %. +%term _ id-lam-cong _ refl %. +%worlds (trmb) (id-lam-cong _ _) %. +%total {} (id-lam-cong _ _) %. +%sort id-func {x trm} %. +%mode id-func %in %in %out %. +%term _ id-func ([_] refl) refl refl %. +%worlds (trmb) (id-func _ _ _) %. +%total {} (id-func _ _ _) %. +%sort ==>-det %. +%mode ==>-det %in %in %out %. +%term _ ==>-det D D refl %. +%term _ + %pi (==>-det (==>/beta D1 D2) (==>/beta D1' D2') Id) + %<- (==>-det D1 D1' Id1) + %<- ({x} {nlx} {cdx} ==>-det (D2 x nlx cdx) (D2' x nlx cdx) (Id2 x)) + %<- (id-func Id2 Id1 Id) %. +%term _ + %pi (==>-det (==>/lam D) (==>/lam D') Id') + %<- ({x} {nlx} {cdx} ==>-det (D x nlx cdx) (D' x nlx cdx) (Id x)) + %<- (id-lam-cong Id Id') %. +%term _ + %pi (==>-det (==>/app Dnl D2 D1) (==>/app Dnl' D2' D1') Id) + %<- (==>-det D1 D1' Id1) + %<- (==>-det D2 D2' Id2) + %<- (id-app-cong Id1 Id2 Id) %. +%{! + +Twelf rules out the off-diagonal cases for us. For example, an ``app`` vs. ``beta`` +case would have a contradictory derivation of ``notlam (lam _)`` as an argument to ``==>/app``. + +!}% +%worlds (==>b) (==>-det _ _ _) %. +%total D (==>-det D _ _) %. +%{! +\{\{case study\}\} +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Church-Rosser_via_complete_development). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/church-rosser-w-catch-all-case.lf b/new-tests/stelf-output/pages/church-rosser-w-catch-all-case.lf new file mode 100644 index 0000000..6348e3c --- /dev/null +++ b/new-tests/stelf-output/pages/church-rosser-w-catch-all-case.lf @@ -0,0 +1,271 @@ +% +%{! ## Syntax !}% +%sort exp %. +%name exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%{! + +When we use this ``%block``, it expresses that we can be working in a +context with arbitrary expression variables. + +!}% +%block exps [x exp]%. +%worlds (exps) (exp) %. +%{! ## Reduction !}% +%{! + +We can reduce under binders and +reduce both sides of an application "in parallel." If we have a β-redex +``(λx.ea) eb``, then after reducing ``ea`` (with ``x`` free) to +``ea'`` (with ``x`` free) and reducing ``eb`` to ``eb'``, +we can return ``[eb'/x]ea'``, the substitution of ``eb'`` into +``ea'``. When we introduce a new variable, we always add in the fact +that it can evaluate to itself. + +The ``%block`` ``exps_red`` +explicitly states that we will be reducing in a setting +with free variables, with the invariant that every variable is added +with the invariant that it can evaluate to itself. + +!}% +%sort reduce %. +%mode reduce %in %out %. +%term reduce/lam + %pi (reduce (lam E) (lam E')) + %<- ({x exp} %pi (reduce x x) %-> (reduce (E x) (E' x))) %. +%term reduce/app %pi (reduce (app E1 E2) (app E1' E2')) %<- (reduce E1 E1') %<- (reduce E2 E2') %. +%term reduce/beta + %pi (reduce (app (lam E1) E2) (E1' E2')) + %<- ({x exp} %pi (reduce x x) %-> (reduce (E1 x) (E1' x))) + %<- (reduce E2 E2') %. +%block exps_red [x exp] [d reduce x x]%. +%worlds (exps_red) (reduce _ _) %. +%total E (reduce E _) %. +%{! ### Reduction is reflexive !}% +%{! + +We will need to know later that reduction is reflexive. It is an easy +proof by induction on the first argument, but we cannot get away with +using a [catch-all case](/wiki/catch-all-case/), so when we go under a binder. This forces us +to describe a new world, ``exps_id``, that captures the variable +case of this theorem. + +!}% +%sort identity {E} %. +%mode identity %in %out %. +%term _ + %pi (identity (lam ([x] E x)) (reduce/lam D)) + %<- ({x} {idx reduce x x} %pi (identity x idx) %-> (identity (E x) (%the (reduce (E x) (E x)) (D x idx)))) %. +%term _ + %pi (identity (app E1 E2) (reduce/app D2 D1)) + %<- (identity E1 (%the (reduce E1 E1) D1)) + %<- (identity E2 (%the (reduce E2 E2) D2)) %. +%block exps_id [x exp] [redx reduce x x] [idx identity x redx]%. +%worlds (exps_id) (identity _ _) %. +%total T (identity T _) %. +%{! ## Substitution !}% +%{! + +The substitution theorem says that if we have a term ``e`` with +``x`` free that reduces to ``e'`` (with ``x`` still free) +and ``e<sub>arg</sub>`` reduces to ``e'<sub>arg</sub>``, +then ``[e<sub>arg</sub>/x]e`` reduces to ``[e'<sub>arg</sub>/x]e'``. + +The proof is by induction on the structure of the term with the free variable. + +!}% +%sort substitute {x exp} %. +%mode substitute %in %in %out %. +%{! + +We actually need to think about what block this theorem will take place in, +because there are at least two options. In this variant, we utilize the +technique of using a [catch-all case](/wiki/catch-all-case/) in order to avoid putting +a fact about variable substitution cases in the context. This latter +style is used in the Twelf examples directory and is explored on the wiki +in the page [[Church-Rosser (alternate substitution theorem)]]. + +The interesting cases are really the first two - if we reach a reduction for +the variable we are substituing for, then our second argument is +the answer (the rest of the time that variable just gets passed around). +If we reach a point where the +variable we are substuiting for doesn't even appear in the term (this +is the catch-all case), then that first argument is the answer. + +!}% +%term _ substitute ([x] [redx reduce x x] redx) Darg Darg %. +%term _ substitute ([x] [redx reduce x x] D) Darg D %. +%term _ + %pi (substitute ([x] [redx reduce x x] reduce/lam (%the ({y} %pi (reduce y y) %-> (reduce (E x y) (E' x y))) (D x redx))) (%the (reduce Earg Earg') Darg) (%the (reduce (lam ([y] E Earg y)) (lam ([y] E' Earg' y))) (reduce/lam D'))) + %<- ({y} {redy reduce y y} substitute ([x] [redx reduce x x] D x redx y redy) Darg (%the (reduce (E Earg y) (E' Earg' y)) (D' y redy))) %. +%term _ + %pi (substitute ([x] [redx reduce x x] reduce/app (%the (reduce (Eb x) (Eb' x)) (Db x redx)) (%the (reduce (Ea x) (Ea' x)) (Da x redx))) (%the (reduce Earg Earg') Darg) (%the (reduce (app (Ea Earg) (Eb Earg)) (app (Ea' Earg') (Eb' Earg'))) (reduce/app Db' Da'))) + %<- (substitute Da Darg (%the (reduce (Ea Earg) (Ea' Earg')) Da')) + %<- (substitute Db Darg (%the (reduce (Eb Earg) (Eb' Earg')) Db')) %. +%term _ + %pi (substitute ([x] [redx reduce x x] reduce/beta (%the (reduce (Eb x) (Eb' x)) (Db x redx)) (%the ({y} %pi (reduce y y) %-> (reduce (Ea x y) (Ea' x y))) (Da x redx))) (%the (reduce Earg Earg') Darg) (%the (reduce (app (lam (Ea Earg)) (Eb Earg)) (Ea' Earg' (Eb' Earg'))) (reduce/beta Db' Da'))) + %<- ({y} {redy reduce y y} substitute ([x] [redx reduce x x] Da x redx y redy) Darg (%the (reduce (Ea Earg y) (Ea' Earg' y)) (Da' y redy))) + %<- (substitute Db Darg (%the (reduce (Eb Earg) (Eb' Earg')) Db')) %. +%worlds (exps_red) (substitute _ _ _) %. +%total D (substitute D _ _) %. +%{! ## The Diamond Property !}% +%{! + +Now we come to the interesting part: the diamond property. + + E + / \ + / \ + E1 E2 + \ / + \ / + E' + +If ``E`` reduces to both ``E1``, and ``E2``, then there is a +common E' such that ``E1`` and ``E2`` both reduce to it. + +!}% +%sort diamond %. +%mode diamond %in %in %out %out %. +%{! ### Identity !}% +%{! + +If either case is the identity, then we are done. + + id: E D: D: E id: + e=>e / \ e=>e2 e=>e1 / \ e=>e + / \ / \ + E E2 E1 E + D: \ /id: id: \ /D: + e=>e2 \ / e2=>e2 e1=>e1\ / e=>e1 + e2 E1 + +!}% +%term _ + %pi (diamond (%the (reduce E E) ID) (%the (reduce E E2) D) D ID') + %<- (identity E2 ID') %. +%term _ + %pi (diamond (%the (reduce E E1) D) (%the (reduce E E) ID) (%the (reduce E1 E1) ID') D) + %<- (identity E1 ID') %. +%{! ### Lambda-Lambda !}% +%{! + +If both cases are reductions under a binder, we pull the result straight +from the induction hypothesis. + + λx.e by induction: + reduce/lam / \ reduce/lam D1, D2 ---> D1': e1'=>e' + (D1: e=>e1) / \ (D2: e=>e2) D2': e2'=>e' + / \ + λx.e1 λx.e2 + \ / + reduce/lam \ / reduce/lam + D1' \ / D2' + λx.e' + +Note the oversimplification being made in the graphical presentation, in that +the subterms and sub-derivations are not clearly shown to have a free variable. +Twelf will, of course, not allow this sloppiness. + +!}% +%term _ + %pi (diamond (%the (reduce (lam E) (lam E1)) (reduce/lam (%the ({x exp} {redx reduce x x} reduce (E x) (E1 x)) D1))) (%the (reduce (lam E) (lam E2)) (reduce/lam (%the ({x exp} {redx reduce x x} reduce (E x) (E2 x)) D2))) (reduce/lam D1') (reduce/lam D2')) + %<- ({x exp} {redx reduce x x} {idx identity x redx} diamond (D1 x redx) (D2 x redx) (%the (reduce (E1 x) (E' x)) (D1' x redx)) (%the (reduce (E2 x) (E' x)) (D2' x redx))) %. +%{! ### Application-Application !}% +%{! + +If both cases are applications, we pull the result straight from the +induction hypothesis. + + ea eb by induction + reduce/app / \ reduce/app D1a, D2a ---> D1a': e1a=>ea' + (D1b: eb=>e1b) / \ (D2b: eb=>e2b) D2a': e2a=>ea' + (D1a: ea=>e1a) / \ (D2a: ea=>e2a) D1b, D2b ---> D1b': e1b=>eb' + e1a e1b e2a e2b D2b': e2b=>eb' + \ / + reduce/app \ / reduce/app + D1b' D1a' \ / D2b' D2a' + ea' eb' +!}% +%term _ + %pi (diamond (%the (reduce (app Ea Eb) (app E1a E1b)) (reduce/app (%the (reduce Eb E1b) D1b) (%the (reduce Ea E1a) D1a))) (%the (reduce (app Ea Eb) (app E2a E2b)) (reduce/app (%the (reduce Eb E2b) D2b) (%the (reduce Ea E2a) D2a))) (reduce/app D1b' D1a') (reduce/app D2b' D2a')) + %<- (diamond D1a D2a (%the (reduce E1a Ea') D1a') (%the (reduce E2a Ea') D2a')) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) %. +%{! ### Beta-Beta !}% +%{! + +If both cases are beta reductions, we get the result from performing two +substitutions. + + (λx.ea) eb by induction + reduce/beta / \ reduce/beta D1a, D2a ---> D1a': e1a=>ea' + (D1b: eb=>e1b) / \ (D2b: eb=>e2b) D2a': e2a=>ea' + (D1a: ea=>e1a) / \ (D2a: ea=>e1a) D1b, D2b ---> D1b': e1b=>eb' + [e1b/x]e1a [e2b/x]e2a D2b': e2b=>eb' + \ / + substitute \ / substitute + D1b' into D1a' \ / D2b' into D2a' + [eb'/x]ea +!}% +%term _ + %pi (diamond (%the (reduce (app (lam Ea) Eb) (E1a E1b)) (reduce/beta (%the (reduce Eb E1b) D1b) (%the ({x} %pi (reduce x x) %-> (reduce (Ea x) (E1a x))) D1a))) (%the (reduce (app (lam Ea) Eb) (E2a E2b)) (reduce/beta (%the (reduce Eb E2b) D2b) (%the ({x} %pi (reduce x x) %-> (reduce (Ea x) (E2a x))) D2a))) D1 D2) + %<- ({x} {redx reduce x x} + %pi (identity x redx) + %-> (diamond (D1a x redx) (D2a x redx) (%the (reduce (E1a x) (Ea' x)) (D1a' x redx)) (%the (reduce (E2a x) (Ea' x)) (D2a' x redx)))) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) + %<- (substitute D1a' D1b' (%the (reduce (E1a E1b) (Ea' Eb')) D1)) + %<- (substitute D2a' D2b' (%the (reduce (E2a E2b) (Ea' Eb')) D2)) %. +%{! ### Beta-Application !}% +%{! + +If the left-hand side is a β-reduction +``(λx.ea) eb => [e1b/x] e1a`` but the right-hand side is not, then we +know that the right-hand side reduction must be +``(λx.ea) eb => (λx.e2a) e2b``, which means it is a +``reduce/lam`` hiding inside a ``reduce/app``. + +The first subcase: + + (λx.ea) eb by induction + reduce/beta / \ reduce/app D1a, D2a ---> D1a': e1a=>ea' + (D1b: eb=>e1b) / \ (D2b: eb=>e2b) D2a': e2a=>ea' + (D1a: ea=>e1a) / \ (reduce/lam D1b, D2b ---> D1b': e1b=>eb' + / \ (D2a: ea=>e2a)) D2b': e1b=>eb' + [e1b/x]e1a (λx.e2a) e2b + \ / + substitute \ / reduce/beta + D1b' into D2a' \ / D2b' D2a' + \ / + [eb'/x]ea' +!}% +%term _ + %pi (diamond (%the (reduce (app (lam Ea) Eb) (E1a E1b)) (reduce/beta (%the (reduce Eb E1b) D1b) (%the ({x exp} %pi (reduce x x) %-> (reduce (Ea x) (E1a x))) D1a))) (%the (reduce (app (lam Ea) Eb) (app (lam E2a) E2b)) (reduce/app (%the (reduce Eb E2b) D2b) (reduce/lam (%the ({x exp} %pi (reduce x x) %-> (reduce (Ea x) (E2a x))) D2a)))) D1 (reduce/beta D2b' D2a')) + %<- ({x exp} {redx reduce x x} + %pi (identity x redx) + %-> (diamond (D1a x redx) (D2a x redx) (%the (reduce (E1a x) (Ea' x)) (D1a' x redx)) (%the (reduce (E2a x) (Ea' x)) (D2a' x redx)))) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) + %<- (substitute D1a' D1b' (%the (reduce (E1a E1b) (Ea' Eb')) D1)) %. +%{! ### Application-Beta !}% +%{! + +If the right-hand hand side is a β-reduction but the left-hand side is not, we +have to do the same case in reverse; we omit the graphic. + +!}% +%term _ + %pi (diamond (%the (reduce (app (lam Ea) Eb) (app (lam E1a) E1b)) (reduce/app (%the (reduce Eb E1b) D1b) (reduce/lam (%the ({x} %pi (reduce x x) %-> (reduce (Ea x) (E1a x))) D1a)))) (%the (reduce (app (lam Ea) Eb) (E2a E2b)) (reduce/beta (%the (reduce Eb E2b) D2b) (%the ({x} %pi (reduce x x) %-> (reduce (Ea x) (E2a x))) D2a))) (reduce/beta D1b' D1a') D2) + %<- ({x} {redx reduce x x} + %pi (identity x redx) + %-> (diamond (D1a x redx) (D2a x redx) (%the (reduce (E1a x) (Ea' x)) (D1a' x redx)) (%the (reduce (E2a x) (Ea' x)) (D2a' x redx)))) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) + %<- (substitute D2a' D2b' (%the (reduce (E2a E2b) (Ea' Eb')) D2)) %. +%{! Now we are done! We check in the ``exps`` world with free variables. !}% +%worlds (exps_id) (diamond _ _ _ _) %. +%total D1 (diamond D1 D2 _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Church-Rosser_(w/_catch-all_case)). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/church-rosser-w-identity-reduction.lf b/new-tests/stelf-output/pages/church-rosser-w-identity-reduction.lf new file mode 100644 index 0000000..2e9a31e --- /dev/null +++ b/new-tests/stelf-output/pages/church-rosser-w-identity-reduction.lf @@ -0,0 +1,277 @@ +% +%{! ## Syntax !}% +%sort exp %. +%name exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%{! + +When we use this ``%block``, it expresses that we can be working in a +context with arbitrary expression variables. This will be the only world we +will need for this exercise. + +!}% +%block exps [x exp]%. +%{! ## Reduction !}% +%{! + +We can always reduce an expression to itself (which means that we don't +explicitly need a case to handle variables). We can reduce under binders and +reduce both sides of an application "in parallel." If we have a β-redex +``(λx.ea) eb``, then after reducing ``ea`` (with ``x`` free) to +``ea'`` (with ``x`` free) and reducing ``eb`` to ``eb'``, +we can return ``[eb'/x]ea'``, the substitution of ``eb'`` into +``ea'``. + +The ``%worlds`` explicitly states that we will be reducing in a setting +with free variables. + +!}% +%sort reduce %. +%term reduce/id reduce E E %. +%term reduce/lam %pi (reduce (lam E) (lam E')) %<- ({x exp} reduce (E x) (E' x)) %. +%term reduce/app %pi (reduce (app E1 E2) (app E1' E2')) %<- (reduce E1 E1') %<- (reduce E2 E2') %. +%term reduce/beta + %pi (reduce (app (lam E1) E2) (E1' E2')) + %<- ({x exp} reduce (E1 x) (E1' x)) + %<- (reduce E2 E2') %. +%worlds (exps) (reduce _ _) %. +%{! ## Substitution !}% +%{! + +The first substitution theorem says that if we have a term ``e1`` with +``x`` free and a term ``e2`` that reduces to term ``e2'``, +then ``[e2/x]e1`` reduces to ``[e2'/x]e1``. + +The interesting cases are really the first two - if the term is just the free +variable being substituted for, the result follows from the hypothesis. In the +case that the free variable is _not_ free (i.e. the term has no hole), then +the result follows from the reflexivity of reduction. This is more than +sufficient to handle the case of a variable that is not the distinguished free +variable, and all other cases are covered. + +The proof is by induction on the structure of the term with the free variable. + +!}% +%sort substitute1 {E1 %pi exp %-> exp} %. +%mode substitute1 %in %in %out %. +%term _ substitute1 ([x] x) D D %. +%term _ substitute1 ([x] E) _ reduce/id %. +%term _ + %pi (substitute1 ([x] lam ([y] E x y)) D (reduce/lam D')) + %<- ({y} substitute1 ([x] E x y) D (D' y)) %. +%term _ + %pi (substitute1 ([x] app (E1 x) (E2 x)) D (reduce/app D2 D1)) + %<- (substitute1 E1 D D1) + %<- (substitute1 E2 D D2) %. +%worlds (exps) (substitute1 _ _ _) %. +%total E (substitute1 E _ _) %. +%{! + +The second substitution theorem is similar, except that we are reducing +``e1`` with ``x`` free to ``e1'`` as well as by reducing +``e2`` to ``e2'``, showing that ``[e2/x]e1`` reduces to +``[e2'/x]e1'``. + +Proof is by induction on the structure of the reduction of ``e1`` to +``e1'``. We call to the ``substitute1`` lemma +when we "bottom out" at the reflexive case ``reduce/id``. + +!}% +%sort substitute2 {x exp} %. +%mode substitute2 %in %in %out %. +%term _ + %pi (substitute2 ([x] %the (reduce (E x) (E x)) reduce/id) D D') + %<- (substitute1 E D D') %. +%term _ + %pi (substitute2 ([x] reduce/lam ([y] D1 x y)) D2 (reduce/lam D)) + %<- ({y} substitute2 ([x] D1 x y) D2 (D y)) %. +%term _ + %pi (substitute2 ([x] reduce/app (D2 x) (D1 x)) D (reduce/app D2' D1')) + %<- (substitute2 D1 D D1') + %<- (substitute2 D2 D D2') %. +%term _ + %pi (substitute2 ([x] reduce/beta (D2 x) ([y] D1 x y)) D (reduce/beta D2' D1')) + %<- ({y} substitute2 ([x] D1 x y) D (D1' y)) + %<- (substitute2 D2 D D2') %. +%worlds (exps) (substitute2 _ _ _) %. +%total D (substitute2 D _ _) %. +%{! ## The Diamond Property !}% +%{! + +Now we come to the interesting part: the diamond property. + + E + / \ + / \ + E1 E2 + \ / + \ / + E' + +If ``E`` reduces to both ``E1``, and ``E2``, then there is a +common E' such that ``E1`` and ``E2`` both reduce to it. + +!}% +%sort diamond %. +%mode diamond %in %in %out %out %. +%{! ### Identity !}% +%{! + +If either case is the identity, then we are done. + + id: E D: D: E id: + e=>e / \ e=>e2 e=>e1 / \ e=>e + / \ / \ + E E2 E1 E + D: \ /id: id: \ /D: + e=>e2 \ / e2=>e2 e1=>e1\ / e=>e1 + e2 E1 + +!}% +%term _ diamond reduce/id D D reduce/id %. +%term _ diamond D reduce/id reduce/id D %. +%{! ### Lambda-Lambda !}% +%{! + +If both cases are reductions under a binder, we pull the result straight +from the induction hypothesis. + + λx.e by induction: + reduce/lam / \ reduce/lam D1, D2 ---> D1': e1'=>e' + (D1: e=>e1) / \ (D2: e=>e2) D2': e2'=>e' + / \ + λx.e1 λx.e2 + \ / + reduce/lam \ / reduce/lam + D1' \ / D2' + λx.e' + +Note the oversimplification being made in the graphical presentation, in that +the subterms and sub-derivations are not clearly shown to have a free variable. +Twelf will, of course, not allow this sloppiness. + +!}% +%term _ + %pi (diamond (%the (reduce (lam E) (lam E1)) (reduce/lam (%the ({x exp} reduce (E x) (E1 x)) D1))) (%the (reduce (lam E) (lam E2)) (reduce/lam (%the ({x exp} reduce (E x) (E2 x)) D2))) (reduce/lam D1') (reduce/lam D2')) + %<- ({x exp} diamond (D1 x) (D2 x) (%the (reduce (E1 x) (E' x)) (D1' x)) (%the (reduce (E2 x) (E' x)) (D2' x))) %. +%{! ### Application-Application !}% +%{! + +If both cases are applications, we pull the result straight from the +induction hypothesis. + + ea eb by induction + reduce/app / \ reduce/app D1a, D2a ---> D1a': e1a=>ea' + (D1b: eb=>e1b) / \ (D2b: eb=>e2b) D2a': e2a=>ea' + (D1a: ea=>e1a) / \ (D2a: ea=>e2a) D1b, D2b ---> D1b': e1b=>eb' + e1a e1b e2a e2b D2b': e2b=>eb' + \ / + reduce/app \ / reduce/app + D1b' D1a' \ / D2b' D2a' + ea' eb' +!}% +%term _ + %pi (diamond (%the (reduce (app Ea Eb) (app E1a E1b)) (reduce/app (%the (reduce Eb E1b) D1b) (%the (reduce Ea E1a) D1a))) (%the (reduce (app Ea Eb) (app E2a E2b)) (reduce/app (%the (reduce Eb E2b) D2b) (%the (reduce Ea E2a) D2a))) (reduce/app D1b' D1a') (reduce/app D2b' D2a')) + %<- (diamond D1a D2a (%the (reduce E1a Ea') D1a') (%the (reduce E2a Ea') D2a')) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) %. +%{! ### Beta-Beta !}% +%{! + +If both cases are beta reductions, we get the result from performing two +substitutions. + + (λx.ea) eb by induction + reduce/beta / \ reduce/beta D1a, D2a ---> D1a': e1a=>ea' + (D1b: eb=>e1b) / \ (D2b: eb=>e2b) D2a': e2a=>ea' + (D1a: ea=>e1a) / \ (D2a: ea=>e1a) D1b, D2b ---> D1b': e1b=>eb' + [e1b/x]e1a [e2b/x]e2a D2b': e2b=>eb' + \ / + substitute \ / substitute + D1b' into D1a' \ / D2b' into D2a' + [eb'/x]ea +!}% +%term _ + %pi (diamond (%the (reduce (app (lam Ea) Eb) (E1a E1b)) (reduce/beta (%the (reduce Eb E1b) D1b) (%the ({x} reduce (Ea x) (E1a x)) D1a))) (%the (reduce (app (lam Ea) Eb) (E2a E2b)) (reduce/beta (%the (reduce Eb E2b) D2b) (%the ({x} reduce (Ea x) (E2a x)) D2a))) D1 D2) + %<- ({x} diamond (D1a x) (D2a x) (%the (reduce (E1a x) (Ea' x)) (D1a' x)) (%the (reduce (E2a x) (Ea' x)) (D2a' x))) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) + %<- (substitute2 D1a' D1b' (%the (reduce (E1a E1b) (Ea' Eb')) D1)) + %<- (substitute2 D2a' D2b' (%the (reduce (E2a E2b) (Ea' Eb')) D2)) %. +%{! ### Beta-Application !}% +%{! + +If the left-hand side is a β-reduction +``(λx.ea) eb => [e1b/x] e1a`` but the right-hand side is not, then we +know that the right-hand side reduction must be +``(λx.ea) eb => (λx.e2a) e2b``. + +Unfortunately, then we have two options - either ``λx.ea => λx.e2a`` by +``reduce/refl`` and ``reduce/lam``. The ``reduce/refl`` case +is essentially just an inconvenience caused by allowing identity reductions +anywhere (not just at variables). + +The first subcase: + + (λx.ea) eb by induction + reduce/beta / \ reduce/app D1a, D2a ---> D1a': e1a=>ea' + (D1b: eb=>e1b) / \ (D2b: eb=>e2b) D2a': e2a=>ea' + (D1a: ea=>e1a) / \ (reduce/lam D1b, D2b ---> D1b': e1b=>eb' + / \ (D2a: ea=>e2a)) D2b': e1b=>eb' + [e1b/x]e1a (λx.e2a) e2b + \ / + substitute \ / reduce/beta + D1b' into D2a' \ / D2b' D2a' + \ / + [eb'/x]ea' +!}% +%term _ + %pi (diamond (%the (reduce (app (lam Ea) Eb) (E1a E1b)) (reduce/beta (%the (reduce Eb E1b) D1b) (%the ({x exp} reduce (Ea x) (E1a x)) D1a))) (%the (reduce (app (lam Ea) Eb) (app (lam E2a) E2b)) (reduce/app (%the (reduce Eb E2b) D2b) (reduce/lam (%the ({x exp} reduce (Ea x) (E2a x)) D2a)))) D1 (reduce/beta D2b' D2a')) + %<- ({x exp} diamond (D1a x) (D2a x) (%the (reduce (E1a x) (Ea' x)) (D1a' x)) (%the (reduce (E2a x) (Ea' x)) (D2a' x))) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) + %<- (substitute2 D1a' D1b' (%the (reduce (E1a E1b) (Ea' Eb')) D1)) %. +%{! + +The second subcase: + + (λx.ea) eb by induction + reduce/beta / \ reduce/app D1b, D2b ---> D1b': e1b=>eb' + (D1b: eb=>e1b) / \ (D2b: eb=>e2b) D2b': e1b=>eb' + (D1a: ea=>e1a) / \ (reduce/id + / \ ea=>ea)) + [e1b/x]e1a (λx.ea) e2b + \ / + substitute \ / reduce/beta + D1b' into e1a \ / D2b' D1a + \ / + [eb'/x]e1a +!}% +%term _ + %pi (diamond (%the (reduce (app (lam Ea) Eb) (E1a E1b)) (reduce/beta (%the (reduce Eb E1b) D1b) (%the ({x} reduce (Ea x) (E1a x)) D1a))) (%the (reduce (app (lam Ea) Eb) (app (lam Ea) E2b)) (reduce/app (%the (reduce Eb E2b) D2b) (%the (reduce (lam Ea) (lam Ea)) reduce/id))) D1 (reduce/beta D2b' D1a)) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) + %<- (substitute1 E1a D1b' (%the (reduce (E1a E1b) (E1a Eb')) D1)) %. +%{! ### Application-Beta !}% +%{! + +If the right-hand hand side is a β-reduction but the left-hand side is not, we +have to do the same two cases in reverse; we omit the graphics for those +two cases as they are symmetric. + +!}% +%term _ + %pi (diamond (%the (reduce (app (lam Ea) Eb) (app (lam Ea) E1b)) (reduce/app (%the (reduce Eb E1b) D1b) (%the (reduce (lam Ea) (lam Ea)) reduce/id))) (%the (reduce (app (lam Ea) Eb) (E2a E2b)) (reduce/beta (%the (reduce Eb E2b) D2b) (%the ({x} reduce (Ea x) (E2a x)) D2a))) (reduce/beta D1b' D2a) D2) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) + %<- (substitute1 E2a D2b' (%the (reduce (E2a E2b) (E2a Eb')) D2)) %. +%term _ + %pi (diamond (%the (reduce (app (lam Ea) Eb) (app (lam E1a) E1b)) (reduce/app (%the (reduce Eb E1b) D1b) (reduce/lam (%the ({x} reduce (Ea x) (E1a x)) D1a)))) (%the (reduce (app (lam Ea) Eb) (E2a E2b)) (reduce/beta (%the (reduce Eb E2b) D2b) (%the ({x} reduce (Ea x) (E2a x)) D2a))) (reduce/beta D1b' D1a') D2) + %<- ({x} diamond (D1a x) (D2a x) (%the (reduce (E1a x) (Ea' x)) (D1a' x)) (%the (reduce (E2a x) (Ea' x)) (D2a' x))) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) + %<- (substitute2 D2a' D2b' (%the (reduce (E2a E2b) (Ea' Eb')) D2)) %. +%{! Now we are done! We check in the ``exps`` world with free variables. !}% +%worlds (exps) (diamond _ _ _ _) %. +%total D1 (diamond D1 D2 _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Church-Rosser_(w/_identity_reduction)). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/classical-s5.lf b/new-tests/stelf-output/pages/classical-s5.lf new file mode 100644 index 0000000..f1c5571 --- /dev/null +++ b/new-tests/stelf-output/pages/classical-s5.lf @@ -0,0 +1,427 @@ +% +%{! This [case study](/wiki/case-studies/) concerns the formalization of the [modal logic](/wiki/modal-logic/) **Classical S5** in the particular way it is done in _[http://www.cs.cmu.edu/~tom7/papers/csl05-19-distributed.pdf Distributed Control Flow with Classical Modal Logic]_ (Murphy, Harper, Crary; CSL 2005). The natural deduction calculus **C5** is a nonstandard variation on the "obvious" classical S5 natural deduction which we chose for its operational interpretation, so that it can be used as a programming language. In the Twelf code below we prove that that C5 is equivalent to the "obvious" S5 by proving it equivalent to a standard [cut-free](/wiki/admissibility-of-cut/) sequent calculus for classical S5. + +Twelf excels for this kind of proof. The only difficulty is that the most natural definition of falsehood assumptions leads to a third-order relation, which Twelf's termination checker did not support at the time this code was developed. Instead, we have an explicit substitution theorem for such assumptions. + +## Natural deduction !}% +%% Classical Judgmental S5 +%% Tom Murphy VII +%% Thanks: Frank Pfenning and Jason Reed +%% 8 Apr 2004 +%sort world %. +%name world %. +%sort prop %. +%name prop %. +%%% Natural deduction +% truth assumptions and conclusions +%sort @ %. +%name @ %. +%prec %none 1 @ %. +% falsehood (continuation) assumptions +%sort * %. +%name * %. +%prec %none 1 * %. +%{! We have two judgments: _A true @ W_ and _A false * W_. We abbreviate these to ``A @ W`` and ``A * W``. !}% +% Implication +%term => %pi prop %-> prop %-> prop %. +%prec %right 8 => %. +%term =>I %pi (%pi (A @ W) %-> (B @ W)) %-> (A => B @ W) %. +%term =>E %pi (A => B @ W) %-> (A @ W) %-> (B @ W) %. +%{! This is an [intrinsic encoding](/wiki/intrinsic-and-extrinsic-encodings/): Derivations of well-formedness play the role of proof terms in the logic. !}% +% Necessity +%term ! %pi prop %-> prop %. +%prec %prefix 9 ! %. +%term !I %pi ({o world} A @ o) %-> (! A @ W) %. +%term !E %pi (! A @ W) %-> (A @ W) %. +%term !G %pi (! A @ W') %-> (! A @ W) %. +% Possibility +%term ? %pi prop %-> prop %. +%prec %prefix 9 ? %. +%term ?I %pi (A @ W) %-> (? A @ W) %. +%term ?E %pi (? A @ W) %-> ({o world} %pi (A @ o) %-> (C @ W)) %-> (C @ W) %. +%term ?G %pi (? A @ W') %-> (? A @ W) %. +%{! There are two modal connectives: , written ``!A`` and , written ``?A``. These two connectives are expressed in a non-standard way. The introduction and elimination forms all take place at the same world, and we give mobility forms ``!G`` and ``?G`` that allow us to move modal terms from world to world. !}% +% Conjunction +%term & %pi prop %-> prop %-> prop %. +%prec %none 9 & %. +%term &I %pi (A @ W) %-> (B @ W) %-> (A & B @ W) %. +%term &E1 %pi (A & B @ W) %-> (A @ W) %. +%term &E2 %pi (A & B @ W) %-> (B @ W) %. +% Falsehood +%scope &E2 %term |_ prop %. +%term |_E %pi (_|_ @ W) %-> (C @ W') %. +% Structural Rules +%term letcc %pi (%pi (A * W) %-> (A @ W)) %-> (A @ W) %. +%term throw %pi (A @ W) %-> (A * W) %-> (C @ W') %. +%{! The calculus is classical, which we achieve by including ``letcc`` (which binds the current continuation) and ``throw`` (which throws an expression, perhaps remotely, to a continuation). There are no terms of falsehood type, except for variables ``u`` bound by ``letcc``. Those are not represented explicitly because we use [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/). !}% +% here : world. +%{! In order for some propositions to exist, we may wish to declare some world constant. Otherwise, our metatheorems may be vacuously true due to empty quantification. Alternatively, we choose to prove our metatheorems with respect to a (LF level) [regular world](/wiki/percent-worlds/) where (object language) hypothetical worlds may occur in the context. See the declaration of ``blockw`` below. + +## Sequent calculus !}% +%%% Sequent calculus +%sort true %. +%name true %. +%sort false %. +%name false %. +%sort # %. +%{! Sequents in the sequent calculus are represented by the judgment ``#``, read as "contradiction." Sequents are derived in a context with ``true`` and ``false`` assumptions. For instance, the initial sequent says that if A is both true and false at the same world, then a contradiction arises: !}% +% initial sequent +%term contra %pi (true A W) %-> (false A W) %-> # %. +% arrow +%term =>T %pi (%pi (false A W) %-> #) %-> (%pi (true B W) %-> #) %-> (true (A => B) W) %-> # %. +%term =>F %pi (%pi (true A W) %-> (false B W) %-> #) %-> (false (A => B) W) %-> # %. +% box +%term !T %pi (%pi (true A W') %-> #) %-> (true (! A) W) %-> # %. +%term !F %pi ({w world} %pi (false A w) %-> #) %-> (false (! A) W) %-> # %. +% dia +%term ?T %pi ({w world} %pi (true A w) %-> #) %-> (true (? A) W) %-> # %. +%term ?F %pi (%pi (false A W') %-> #) %-> (false (? A) W) %-> # %. +%{! Unlike the natural deduction, this presentation of the □ and ⋄ connectives is completely standard. !}% +% conjunction +%term &T %pi (%pi (true A W) %-> (true B W) %-> #) %-> (true (A & B) W) %-> # %. +%term &F %pi (%pi (false A W) %-> #) %-> (%pi (false B W) %-> #) %-> (false (A & B) W) %-> # %. +% falsehood +%scope &F %term |_T %pi (true _|_ W) %-> # %. +%{! ## Cut elimination !}% +% admissibility of excluded middle (cut) +%sort xm {A prop} {W world} %. +%mode xm %in %in %in %in %out %. +%{! Cut elimination for a classical sequent calculus is the law of the excluded middle: If in some ambient context, the assumption that ``A`` is true is contradictory, and the assumption that ``A`` is false is also contradictory, then the ambient context is also contradictory. The proof proceeds mostly like [cut elimination for intuitionistic logic](/wiki/admissibility-of-cut/). !}% +% initial cuts +%term xt_init xm A W ([xt] contra xt D) ([xf] E xf) (E D) %. +%term xf_init xm A W ([xt] D xt) ([xf] contra E xf) (D E) %. +% unused assumptions +%term xt_unused xm A W ([x1] contra DT DF) E (contra DT DF) %. +%term xf_unused xm A W D ([x1] contra ET EF) (contra ET EF) %. +% falsehood. actually falsehood is trivial because a principal +% cut is impossible, and the "commutative" cases aren't even inductive. +%term xd_ct_|_ xm A W ([at] _|_T DD) _ (_|_T DD) %. +%term xe_ct_|_ xm A W _ ([af] _|_T EE) (_|_T EE) %. +% commutative cases. +% note that we need to consider each rule in both D and E, so +% there are four cases for each connective. +% implication +%term xd_ct=> + %pi (xm A W ([at true A W] =>T ([cf false C W'] D1 at cf) ([dt true D W'] D2 at dt) DD) ([af] E af) (=>T ([cf false C W'] F1 cf) ([dt true D W'] F2 dt) DD)) + %<- ({cf false C W'} xm A W ([at] D1 at cf) E (F1 cf)) + %<- ({dt true D W'} xm A W ([at] D2 at dt) E (F2 dt)) %. +%term xe_ct=> + %pi (xm A W ([at] DD at) ([af false A W] =>T ([cf false C W'] E1 af cf) ([dt true D W'] E2 af dt) EE) (=>T ([cf] F1 cf) ([dt] F2 dt) EE)) + %<- ({cf false C W'} xm A W DD ([af] E1 af cf) (F1 cf)) + %<- ({dt true D W'} xm A W DD ([af] E2 af dt) (F2 dt)) %. +%term xd_cf=> + %pi (xm A W ([at true A W] =>F ([ct true C W'] [df false D W'] D' at ct df) DD) ([af false A W] E af) (=>F ([ct true C W'] [df false D W'] F ct df) DD)) + %<- ({ct true C W'} {df false D W'} xm A W ([at] D' at ct df) E (F ct df)) %. +%term xe_cf=> + %pi (xm A W ([at true A W] DD at) ([af false A W] =>F ([ct true C W'] [df false D W'] E' af ct df) EE) (=>F ([ct true C W'] [df false D W'] F ct df) EE)) + %<- ({ct true C W'} {df false D W'} xm A W DD ([af] E' af ct df) (F ct df)) %. +% box +%term xd_ct! + %pi (xm A W ([at true A W] !T ([bt true B W'] D1 at bt) DD) ([af] E af) (!T ([bt true B W'] F1 bt) DD)) + %<- ({bt true B W'} xm A W ([at] D1 at bt) E (F1 bt)) %. +%term xe_ct! + %pi (xm A W ([at] D at) ([af false A W] !T ([bt true B W'] E1 af bt) EE) (!T ([bt true B W'] F1 bt) EE)) + %<- ({bt true B W'} xm A W D ([af] E1 af bt) (F1 bt)) %. +%term xd_cf! + %pi (xm A W ([at true A W] !F ([w] [bf false B w] D1 at w bf) DD) ([af] E af) (!F ([w] [bf] F1 w bf) DD)) + %<- ({w world} {bf false B w} xm A W ([at] D1 at w bf) E (F1 w bf)) %. +%term xe_cf! + %pi (xm A W ([at] D at) ([af] !F ([w] [bf false B w] E1 af w bf) EE) (!F ([w] [bf] F1 w bf) EE)) + %<- ({w} {bf} xm A W D ([af] E1 af w bf) (F1 w bf)) %. +% dia +%term xd_ct? + %pi (xm A W ([at true A W] ?T ([w] [bt] D1 at w bt) DD) ([af] E af) (?T ([w] [bt] F1 w bt) DD)) + %<- ({w} {bt} xm A W ([at] D1 at w bt) ([af] E af) (F1 w bt)) %. +%term xe_ct? + %pi (xm A W ([at] D at) ([af] ?T ([w] [bt] E1 af w bt) EE) (?T ([w] [bt] F1 w bt) EE)) + %<- ({w} {bt} xm A W ([at] D at) ([af] E1 af w bt) (F1 w bt)) %. +%term xd_cf? + %pi (xm A W ([at] ?F ([bf false B W'] D1 at bf) DD) ([af] E af) (?F ([bf] F1 bf) DD)) + %<- ({bf} xm A W ([at] D1 at bf) ([af] E af) (F1 bf)) %. +%term xe_cf? + %pi (xm A W ([at] D at) ([af] ?F ([bf false B W'] E1 af bf) EE) (?F ([bf] F1 bf) EE)) + %<- ({bf} xm A W ([at] D at) ([af] E1 af bf) (F1 bf)) %. +% conjunction +%term xd_ct& + %pi (xm A W ([at] &T ([ct] [dt] D1 at ct dt) DD) ([af] E af) (&T ([ct] [dt] F1 ct dt) DD)) + %<- ({ct} {dt} xm A W ([at] D1 at ct dt) ([af] E af) (F1 ct dt)) %. +%term xe_ct& + %pi (xm A W ([at] D at) ([af] &T ([ct] [dt] E1 af ct dt) EE) (&T ([ct] [dt] F1 ct dt) EE)) + %<- ({ct} {dt} xm A W ([at] D at) ([af] E1 af ct dt) (F1 ct dt)) %. +%term xd_cf& + %pi (xm A W ([at] &F ([cf] D1 at cf) ([df] D2 at df) DD) ([af] E af) (&F ([cf] F1 cf) ([df] F2 df) DD)) + %<- ({cf} xm A W ([at] D1 at cf) ([af] E af) (F1 cf)) + %<- ({df} xm A W ([at] D2 at df) ([af] E af) (F2 df)) %. +%term xe_cf& + %pi (xm A W ([at] D at) ([af] &F ([cf] E1 af cf) ([df] E2 af df) EE) (&F ([cf] F1 cf) ([df] F2 df) EE)) + %<- ({cf} xm A W ([at] D at) ([af] E1 af cf) (F1 cf)) + %<- ({df} xm A W ([at] D at) ([af] E2 af df) (F2 df)) %. +% principal cuts. there is one case for each connective; +% a use of the true rule on A in D, and +% a use of the false rule on A in E. +% implication +%term x_=> + %pi (xm (A => B) W ([it true (A => B) W] =>T ([af false A W] D1 it af) ([bt true B W] D2 it bt) it) ([if false (A => B) W] =>F ([at true A W] [bf false B W] E1 if at bf) if) F) + %<- ({at true A W} {bf false B W} xm (A => B) W ([it] =>T ([af false A W] D1 it af) ([bt] D2 it bt) it) ([if] E1 if at bf) (E1' at bf)) + %<- ({af false A W} xm (A => B) W ([it] D1 it af) ([if] =>F ([at] [bf] E1 if at bf) if) (D1' af)) + %<- ({bt true B W} xm (A => B) W ([it] D2 it bt) ([if] =>F ([at] [bf] E1 if at bf) if) (D2' bt)) + %<- ({bf false B W} xm A W ([at true A W] E1' at bf) ([af false A W] D1' af) (F' bf)) + %<- (xm B W ([bt] D2' bt) ([bf] F' bf) F) %. +% box +%term x_! + %pi (xm (! A) W ([nt] !T ([at true A W'] D1 nt at) nt) ([nf] !F ([w] [af false A w] E1 nf w af) nf) F) + %<- ({w world} {af false A w} xm (! A) W ([nt] !T ([at] D1 nt at) nt) ([nf] E1 nf w af) (E1' w af)) + %<- ({at true A W'} xm (! A) W ([nt] D1 nt at) ([nf] !F ([w] [af] E1 nf w af) nf) (D1' at)) + %<- (xm A W' ([at true A W'] D1' at) ([af false A W'] E1' W' af) F) %. +% dia +%term x_? + %pi (xm (? A) W ([nt] ?T ([w] [at] D1 nt w at) nt) ([nf] ?F ([af] E1 nf af) nf) F) + %<- ({w world} {at true A w} xm (? A) W ([nt] D1 nt w at) ([nf] ?F ([af] E1 nf af) nf) (D1' w at)) + %<- ({af false A W'} xm (? A) W ([nt] ?T ([w] [at] D1 nt w at) nt) ([nf] E1 nf af) (E1' af)) + %<- (xm A W' ([at true A W'] D1' W' at) ([af] E1' af) F) %. +% conjunction +%term x_& + %pi (xm (A & B) W ([&t] &T ([at] [bt] D1 &t at bt) &t) ([&f] &F ([af] E1 &f af) ([bf] E2 &f bf) &f) F) + %<- ({at} {bt} xm (A & B) W ([&t] D1 &t at bt) ([&f] &F ([af] E1 &f af) ([bf] E2 &f bf) &f) (D1' at bt)) + %<- ({af} xm (A & B) W ([&t] &T ([at] [bt] D1 &t at bt) &t) ([&f] E1 &f af) (E1' af)) + %<- ({bf} xm (A & B) W ([&t] &T ([at] [bt] D1 &t at bt) &t) ([&f] E2 &f bf) (E2' bf)) + %<- ({bt} xm A W ([at] D1' at bt) ([af] E1' af) (D1'' bt)) + %<- (xm B W ([bt] D1'' bt) ([bf] E2' bf) F) %. +%block blockw [w world]%. +%block blockt {A prop} {W world} [tt true A W]%. +%block blockf {A prop} {W world} [ff false A W]%. +%worlds (blockw blockt blockf) (xm A W D E F) %. +%total {A [D E]} (xm A W D E F) %. +%{! ## Natural deduction to sequent calculus + +The admissibility of excluded middle allows us to straightforwardly translate natural deduction proofs to sequent calculus proofs. The theorem we prove is that if Γ;Δ ⊦ A @ W then Γ # A * W, Δ. Operationally, we can think of this as follows: if we have a program of type _A_, then we can pass the result of that program to a continuation expecting _A_ to get a contradiction. !}% +%%% Translation from natural deduction to sequent calculus. +% G;D |- A @ W then G # D,A * W +%sort ndseq %. +%mode ndseq %in %out %. +% conjunction +%term ns-&I %pi (ndseq (&I D1 D2) (&F F1 F2)) %<- (ndseq D1 F1) %<- (ndseq D2 F2) %. +%term ns-&E1 + %pi (ndseq (&E1 (%the (A & B @ W) D1)) ([af] F af)) + %<- (ndseq D1 ([&f] F2 &f)) + %<- ({af} xm (A & B) W ([&t] &T ([at] [bt] contra at af) &t) ([&f] F2 &f) (F af)) %. +%term ns-&E2 + %pi (ndseq (&E2 (%the (A & B @ W) D1)) ([bf] F bf)) + %<- (ndseq D1 ([&f] F2 &f)) + %<- ({bf} xm (A & B) W ([&t] &T ([at] [bt] contra bt bf) &t) ([&f] F2 &f) (F bf)) %. +% falsehood +%term ns-_|_E + %pi (ndseq (_|_E (%the (_|_ @ W) D1)) ([af] F af)) + %<- (ndseq D1 ([_|_f] F2 _|_f)) + %<- ({af} xm _|_ W ([_|_t] _|_T _|_t) ([_|_f] F2 _|_f) (F af)) %. +% implication +%term ns-=>I + %pi (ndseq (=>I ([a A @ W] D a)) ([=>f false (A => B) W] =>F ([at true A W] [bf false B W] F at bf) =>f)) + %<- ({a A @ W} {at true A W} %pi (ndseq a ([af] contra at af)) %-> (ndseq (D a) ([bf] F at bf))) %. +%term ns-=>E + %pi (ndseq (=>E D1 D2) ([bf] F bf)) + %<- (ndseq D1 ([if] F1 if)) + %<- (ndseq D2 ([af] F2 af)) + %<- ({bf} xm (A => B) W ([it] =>T ([af] F2 af) ([bt] contra bt bf) it) ([if] F1 if) (F bf)) %. +% box +%term ns-!I + %pi (ndseq (!I ([w] D1 w)) (!F ([w] [af false A w] F w af))) + %<- ({w} ndseq (D1 w) (F w)) %. +%term ns-!E + %pi (ndseq (!E D1) ([af] F af)) + %<- (ndseq D1 ([!f] F1 !f)) + %<- ({af} xm (! A) W ([!t] !T ([at] contra at af) !t) ([!f] F1 !f) (F af)) %. +%term ns-!G + %pi (ndseq (!G D1) ([!f] F !f)) + %<- (ndseq D1 ([!f'] F1 !f')) + %<- ({!f} xm (! A) W' ([!t' true (! A) W'] !F ([w'' world] [af'' false A w''] !T ([at'' true A w''] contra at'' af'') !t') !f) ([!f' false (! A) W'] F1 !f') (F !f)) %. +% dia +%term ns-?I %pi (ndseq (?I D) ([?f] ?F ([af] F af) ?f)) %<- (ndseq D ([af] F af)) %. +%term ns-?E + %pi (ndseq (?E D1 ([w world] [a A @ w] D2 w a)) ([cf false C W] F cf)) + %<- (ndseq D1 ([?f] F1 ?f)) + %<- ({w' world} {a A @ w'} {at true A w'} %pi (ndseq a ([af] contra at af)) %-> (ndseq (D2 w' a) ([cf] F2 w' at cf))) + %<- ({cf false C W} xm (? A) W ([?t] ?T ([w] [a] F2 w a cf) ?t) ([?f] F1 ?f) (F cf)) %. +%term ns-?G + %pi (ndseq (?G D) ([?f false (? A) W] F ?f)) + %<- (ndseq D ([?f' false (? A) W'] F1 ?f')) + %<- ({?f false (? A) W} xm (? A) W' ([?t'] ?T ([w''] [at''] ?F ([af''] contra at'' af'') ?f) ?t') ([?f'] F1 ?f') (F ?f)) %. +% we need a way of connecting nd continuation assumptions +% with sequent false assumptions +%sort contfalse %. +%mode contfalse %in %out %. +% note contraction: F in output uses af twice +%term ns-letcc + %pi (ndseq (letcc ([ac] D ac)) ([af] F af af)) + %<- ({ac A * W} {af false A W} %pi (contfalse ac af) %-> (ndseq (D ac) (F af))) %. +% note weakening: [cf'] is unused. +%term ns-throw + %pi (ndseq (throw D K) ([cf'] F AF)) + %<- (ndseq D ([af] F af)) + %<- (contfalse K AF) %. +%block blockh {A prop} {W world} [a A @ W] [at true A W] [_ ndseq a ([af] contra at af)]%. +%block block* {A prop} {W world} [ac A * W] [af false A W] [_ contfalse ac af]%. +%worlds (blockw blockh block*) (contfalse D F) %. +%worlds (blockw blockh block*) (ndseq D F) %. +%total D (contfalse D F) %. +%total D (ndseq D F) %. +%{! ## Continuation substitution + +We must prove a substitution theorem for falsehood assumptions, because we did not use the most [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/) encoding for them. Nonetheless, the theorem is interesting. It says that if we have a term _M_ with a hypothesis for a (true) term of type _A_, and we have a term _N_ with a hypothesis for a (false) continuation expecting type _A_, then our substitution operation can substitute M into N to satisfy the hypothesis. _M_ must be well-typed at all types and all worlds. +<!-- explanation could be better, but it is pretty subtle... --> !}% +%% Continuation Substitution (excluded middle) for natural deduction +% if G,x:A@W; D |- M : * +% and G; D,u:A@W |- N : B +% then G; D |- [[ x.M/u ]] N : B +%sort xs {c} {w} %. +%name xs %. +%mode xs %in %in %out %. +% trick: for any term closed wrt the continuation assumption, +% substitution is the identity. This keeps us from having to +% treat the case of @ variables, since they are always closed +% wrt * variables. Without this trick, world subsumption forces +% cases of xs to infect any later theorem that uses it! +%term xs-closed xs D ([a*] E) E %. +% falsehood +%term xs-_|_E %pi (xs D ([u] _|_E (EE u)) (_|_E EE')) %<- (xs D EE EE') %. +% conjunction +%term xs-&I %pi (xs D ([u] &I (EA u) (EB u)) (&I F1 F2)) %<- (xs D EA F1) %<- (xs D EB F2) %. +%term xs-&E1 %pi (xs D ([u] &E1 (E u)) (&E1 F)) %<- (xs D E F) %. +%term xs-&E2 %pi (xs D ([u] &E2 (E u)) (&E2 F)) %<- (xs D E F) %. +% implication +%term xs-=>I + %pi (xs D ([u] =>I ([aw A @ W] E aw u)) (=>I ([aw] F aw))) + %<- ({aw A @ W} xs D (E aw) (F aw)) %. +%term xs-=>E %pi (xs D ([u] =>E (DF u) (DA u)) (=>E FF FA)) %<- (xs D DF FF) %<- (xs D DA FA) %. +% box +%term xs-!I + %pi (xs D ([u A * W] !I ([w] E u w)) (!I ([w] F w))) + %<- ({w world} xs D ([u] E u w) (F w)) %. +%term xs-!G %pi (xs D ([u] !G (E u)) (!G F)) %<- (xs D E F) %. +%term xs-!E %pi (xs D ([u] !E (E u)) (!E F)) %<- (xs D E F) %. +% possibility +%term xs-?E + %pi (xs D ([u] ?E (E1 u) ([w] [a] E2 w a u)) (?E F1 ([w] [a] F2 w a))) + %<- (xs D E1 F1) + %<- ({w world} {aw A @ w} xs D (E2 w aw) (F2 w aw)) %. +%term xs-?I %pi (xs D ([u] ?I (E u)) (?I F)) %<- (xs D E F) %. +%term xs-?G %pi (xs D ([u] ?G (E u)) (?G F)) %<- (xs D E F) %. +%term xs-letcc + %pi (xs D ([u] letcc ([v] E v u)) (letcc ([v] F v))) + %<- ({v B * W'} xs D (E v) (F v)) %. +% throw to different cont +%term xs-throwmiss %pi (xs D ([u] throw (E u) V) (throw F V)) %<- (xs D E F) %. +% when reaching the throw, pass the term we're throwing +% to D instead. +%term xs-throwhit %pi (xs D ([u] throw (E u) u) (D B W' F)) %<- (xs D E F) %. +% world and totality decls come after seqnd, which uses xs. +% they infect each other somewhat, but the worlds decls for +% xs are not substantially different than if it is checked +% alone (there are just some extra unrelated additions) +%{! ## Sequent calculus to natural deduction + +With substitution defined, we can now translate sequent calculus proofs back into C5 natural deduction. The theorem we prove is that if Γ # Δ then ∀C. ∀ω. Γ; Δ ⊦ C @ ω. !}% +%%% Translation from Sequent Calculus to Natural Deduction +% G # D then G ; D |- M : * +%sort seqnd %. +%name seqnd %. +%sort truend %. +%name truend %. +%sort falsend %. +%name falsend %. +%mode seqnd %in %out %. +%mode truend %in %out %. +%mode falsend %in %out %. +%term sn-contra + %pi (seqnd (contra AT AF) ([c prop] [w world] throw A AC)) + %<- (truend AT A) + %<- (falsend AF AC) %. +% For each connective, the T side is easy -- it just corresponds to +% the elimination rule. The F side requires the "excluded substitution" +% theorem above, and is often quite tricky. +% falsehood +%term sn-_|_T %pi (seqnd (_|_T FT) ([c] [w] _|_E FT')) %<- (truend FT FT') %. +% conjunction +%term sn-&T + %pi (seqnd (&T ([at] [bt] D at bt) T&) ([c] [w] F (&E1 N&) (&E2 N&) c w)) + %<- ({at} {a} + %pi (truend at a) + %-> ({bt} {b} %pi (truend bt b) %-> (seqnd (D at bt) ([c] [w] F a b c w)))) + %<- (truend T& N&) %. +%term sn-&F + %pi (seqnd (&F ([af] D1 af) ([bf] D2 bf) (%the (false (A & B) W) F&)) FF) + %<- (falsend F& N&) + %<- ({af} {a*} %pi (falsend af a*) %-> (seqnd (D1 af) ([c] [w] F1 a* c w))) + %<- ({bf} {b*} %pi (falsend bf b*) %-> (seqnd (D2 bf) ([c] [w] F2 b* c w))) + %<- ({cc} {ww} {b B @ W} xs ([c] [w] [a] throw (&I a b) N&) ([a*] F1 a* cc ww) (F1' b cc ww)) + %<- ({cc prop} {ww} xs ([c] [w] [b] F1' b c w) ([b*] F2 b* cc ww) (FF cc ww)) %. +% implication +% actually, implication is not just the elim rule, because +% classical implication is phrased differently. +%term sn-=>T + %pi (seqnd (=>T ([af false A W] D1 af) ([bt] D2 bt) (%the (true (A => B) W) T=>)) FF) + %<- (truend T=> N=>) + %<- ({af} {a*} %pi (falsend af a*) %-> (seqnd (D1 af) ([c] [w] F1 a* c w))) + %<- ({bt} {b} %pi (truend bt b) %-> (seqnd (D2 bt) ([c] [w] F2 b c w))) + %<- ({cc} {ww} xs ([c] [w] [a] F2 (=>E N=> a) c w) ([a*] F1 a* cc ww) (FF cc ww)) %. +%{! There are two ways to translate the ``=>F`` rule; we can avoid using ``letcc`` if we like. We give both, taking advantage of the fact that LF relations need not be deterministic functions. !}% +% letcc-free version +% u : A=>B |- throw \x:A . [[ c:B. throw \unused:A.c to u / ... ]](IH) to u +%term sn-=>F + %pi (seqnd (=>F ([at true A W] [bf false B W] D at bf) F=>) ([c] [w] throw (=>I ([a] FZ a B W)) N=>)) + %<- (falsend F=> N=>) + %<- ({at} {a} + %pi (truend at a) + %-> ({bf} {b*} %pi (falsend bf b*) %-> (seqnd (D at bf) ([c] [w] F1 a b* c w)))) + %<- ({a A @ W} {cc} {ww} xs ([c] [w] [b] throw (=>I ([a-unused A @ W] b)) N=>) ([b*] F1 a b* cc ww) (FZ a cc ww)) %. +% also include simpler letcc version +% u : A=>B |- throw (\x:A . letcc v : b* in (IH) end) to u +%term sn-=>F-letcc + %pi (seqnd (=>F ([at true A W] [bf false B W] D at bf) F=>) ([c] [w] throw (=>I ([a] letcc ([b*] F1 a b* B W))) N=>)) + %<- (falsend F=> N=>) + %<- ({at} {a} + %pi (truend at a) + %-> ({bf} {b*} %pi (falsend bf b*) %-> (seqnd (D at bf) ([c] [w] F1 a b* c w)))) %. +% box +%term sn-!T + %pi (seqnd (!T ([at' true A W'] D at') T!) ([c] [w] F (!E (!G N!)) c w)) + %<- ({at'} {a'} %pi (truend at' a') %-> (seqnd (D at') ([c] [w] F a' c w))) + %<- (truend T! N!) %. +% this is the only necessary letcc in the proof +% u : []A * w |- throw (box w'. letcc v:A*w' in (IH) end) to u +%term sn-!F + %pi (seqnd (!F ([w' world] [af false A w'] D w' af) F!) ([c] [w] throw (!I ([w'] letcc ([a*'] F w' a*' A w'))) N!)) + %<- (falsend F! N!) + %<- ({w' world} {af' false A w'} {a*' A * w'} %pi (falsend af' a*') %-> (seqnd (D w' af') ([c] [w] F w' a*' c w))) %. +% dia +% x : ?A@w |- let dia = getx in IH end +%term sn-?T + %pi (seqnd (?T ([w'] [at] D w' at) T?) ([c] [w] ?E (?G N?) ([w'] [a'] F w' a' c w))) + %<- (truend T? N?) + %<- ({w'} {at true A w'} {a' A @ w'} %pi (truend at a') %-> (seqnd (D w' at) ([c] [w] F w' a' c w))) %. +%term sn-?F + %pi (seqnd (?F ([af'] D af') F?) FF) + %<- (falsend F? N?) + %<- ({af' false A W'} {a*' A * W'} %pi (falsend af' a*') %-> (seqnd (D af') ([c] [w] F1 a*' c w))) + %<- ({cc} {ww} xs ([c] [w] [a'] throw (?G (?I a')) N?) ([a*'] F1 a*' cc ww) (FF cc ww)) %. +%{! Finally, we can declare the [worlds](/wiki/percent-worlds/) for our metatheorems and ask Twelf to check them: !}% +% for xs +%block blocku {A prop} {W world} [u A * W]%. +%block blocka {A prop} {W world} [u A @ W]%. +% for seqnd +%block blocknt {A prop} {W world} [at true A W] [a A @ W] [t truend at a]%. +%block blocknf {A prop} {W world} [af false A W] [a A * W] [t falsend af a]%. +%block blockp [a prop]%. +%worlds (blockw blocka blocknt blocknf blocku blockp) (xs D E F) (seqnd D F) (truend D F) (falsend D F) %. +%total E (xs D E F) %. +%total (D T F) (seqnd D FF) (truend T N) (falsend F M) %. +%{! ``>All code from this case study``. `` check="true">Twelf's output from this case study``. + +[Category:Twelf code](/wiki/category-twelf-code/) +\{\{case study\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Classical_S5). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009-20090203.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090203.lf new file mode 100644 index 0000000..b708885 --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090203.lf @@ -0,0 +1,139 @@ +% +%{! Code from class, March 2. + +```twelf +ttp : type. %name ttp T. + +tnum : ttp. +tnot : ttp -> ttp. +tprod : ttp -> ttp -> ttp. + +texp : type. %name texp E. +tval : type. %name tval V. + +tlet : tval -> (tval -> texp) -> texp. +tpi1 : tval -> (tval -> texp) -> texp. +tpi2 : tval -> (tval -> texp) -> texp. +tifz : tval -> texp -> (tval -> texp) -> texp. +tapp : tval -> tval -> texp. + +tz : tval. +ts : tval -> tval. +tpair : tval -> tval -> tval. +tlam : ttp -> (tval -> texp) -> tval. + +tof : tval -> ttp -> type. +twf : texp -> type. + +tof/z : tof tz tnum. + +tof/s : tof (ts V) tnum + <- tof V tnum. + +tof/pair : tof (tpair V1 V2) (tprod T1 T2) + <- tof V1 T1 + <- tof V2 T2. + +tof/lam : tof (tlam T ([x] E x)) (tnot T) + <- ({x} tof x T -> twf (E x)). + +twf/let : twf (tlet V ([x] E x)) + <- tof V T + <- ({x} tof x T -> twf (E x)). + +twf/pi1 : twf (tpi1 V ([x] E x)) + <- tof V (tprod T1 T2) + <- ({x} tof x T1 -> twf (E x)). + +twf/pi2 : twf (tpi2 V ([x] E x)) + <- tof V (tprod T1 T2) + <- ({x} tof x T2 -> twf (E x)). + +twf/ifz : twf (tifz V E0 ([x] E1 x)) + <- tof V tnum + <- twf E0 + <- ({x} tof x tnum -> twf (E1 x)). + +twf/app : twf (tapp V1 V2) + <- tof V1 (tnot T) + <- tof V2 T. + +ttrans : tp -> ttp -> type. + +ttrans/num : ttrans num tnum. + +ttrans/arr : ttrans (arr T1 T2) (tnot (tprod T1' (tnot T2'))) + <- ttrans T1 T1' + <- ttrans T2 T2'. + +suspend : ttp -> tval -> tval + = [t] [v] (tlam (tnot t) ([f] tapp f v)). + +trans : term -> tp -> tval -> type. + +trans/z : trans z num (suspend tnum tz). + +trans/s : trans (s E) num (tlam (tnot tnum) ([f] tapp V (tlam tnum ([x] tapp f (ts x))))) + <- trans E num V. + +trans/ifz : trans (ifz E E0 ([x] E1 x)) T + (tlam (tnot T') + ([f] tapp V (tlam tnum ([y] tifz y (tapp V0 f) + ([y'] tlet (tlam (tnot tnum) ([g] tapp g y')) + ([x'] tapp (V1 x') f)))))) + <- ttrans T T' + <- trans E num V + <- trans E0 T V0 + <- ({x} {x'} trans x num x' -> trans (E1 x) T (V1 x')). + +trans/lam : trans (lam T1 ([x] E x)) (arr T1 T2) + (suspend (tnot (tprod T1' (tnot T2'))) + (tlam (tprod T1' (tnot T2')) + ([y] tpi1 y + ([y'] (tpi2 y + ([g] tlet (tlam (tnot T1') ([w] tapp w y')) + ([x'] tapp (V x') g))))))) + <- ttrans T1 T1' + <- ttrans T2 T2' + <- ({x} {x'} trans x T1 x' -> trans (E x) T2 (V x')). + +trans/app : trans (app E1 E2) T2 + (tlam (tnot T2') + ([f] tapp V1 + (tlam (tnot (tprod T1' (tnot T2'))) + ([x1] tapp V2 (tlam T1' ([x2] tapp x1 (tpair x2 f))))))) + <- ttrans T1 T1' + <- ttrans T2 T2' + <- trans E1 (arr T1 T2) V1 + <- trans E2 T1 V2. + +tof/suspend : tof (suspend T V) (tnot (tnot T)) + <- tof V T + = [D : tof V T] (tof/lam ([x] [d : tof x (tnot T)] (twf/app D d))). + +correct : trans E T V -> ttrans T T' -> tof V (tnot (tnot T')) -> type. +%mode correct +D1 -D2 -D3. +correct' : trans E T V -> ttrans T T' -> tof V (tnot (tnot T')) -> type. +%mode correct' +D1 +D2 -D3. + + +- : correct trans/z ttrans/num (tof/suspend tof/z). + +%block transbind : some {T : tp} {T' : ttp} {DTT : ttrans T T'} + block + {x : term} {dx : of x T} + {x' : tval} {dx' : tof x' (tnot (tnot T'))} + {dt : trans x T x'} + {thm : correct dt DTT dx'}. + + + +%worlds (transbind) (correct _ _ _). +%total D (correct D _ _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090203). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009-20090204.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090204.lf new file mode 100644 index 0000000..19e5b98 --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090204.lf @@ -0,0 +1,148 @@ +% +%{! Code from class, February 4 + +```twelf +tp : type. %name tp T. + +num : tp. +arr : tp -> tp -> tp. + +%worlds () (tp). + +term : type. %name term E. + +z : term. +s : term -> term. +ifz : term -> term -> (term -> term) -> term. +lam : tp -> (term -> term) -> term. +app : term -> term -> term. + +%block var : block {x:term}. +%worlds (var) (term). + +of : term -> tp -> type. + +of/z : of z num. + +of/s : of (s N) num + <- of N num. + +of/ifz : of (ifz E E0 ([x] E1 x)) T + <- of E num + <- of E0 T + <- ({x} of x num -> of (E1 x) T). + +of/lam : of (lam T ([x] E x)) (arr T T') + <- ({x} of x T -> of (E x) T'). + +of/app : of (app E1 E2) T' + <- of E1 (arr T T') + <- of E2 T. + +%block stupid : some {x:term} {t:tp} block {d:of x t}. +%block bind : some {T:tp} block {x:term} {d : of x T}. +%worlds (bind) (of _ _). + +value : term -> type. + +value/z : value z. + +value/s : value (s E) + <- value E. + +value/lam : value (lam T E). + +%worlds () (value _). + + + +step : term -> term -> type. + +step/s : step (s E) (s E') + <- step E E'. + +step/ifz/arg : step (ifz E E0 ([x] E1 x)) (ifz E' E0 E1) + <- step E E'. + +step/ifz/z : step (ifz z E0 E1) E0. + +step/ifz/s : step (ifz (s E) E0 ([x] E1 x)) (E1 E) + <- value E. + +step/app/fn : step (app E1 E2) (app E1' E2) + <- step E1 E1'. + +step/app/arg : step (app E1 E2) (app E1 E2') + <- value E1 + <- step E2 E2'. + +step/app/beta : step (app (lam T ([x] E1 x)) E2) (E1 E2) + <- value E2. + +%worlds () (step _ _). + + +unstuck : term -> type. +unstuck/val : unstuck E + <- value E. + +unstuck/step : unstuck E + <- step E E'. + +progress-s : unstuck E -> unstuck (s E) -> type. +%mode progress-s +X1 -X2. + +- : progress-s (unstuck/val (D : value E)) (unstuck/val (value/s D)). + +- : progress-s (unstuck/step (D : step E E')) (unstuck/step (step/s D)). + +%worlds () (progress-s _ _). +%total {} (progress-s _ _). + +progress-ifz : of E num -> unstuck E -> {E0:term} {E1:term -> term} unstuck (ifz E E0 E1) -> type. +%mode progress-ifz +D1 +D2 +E0 +E1 -D3. + +- : progress-ifz _ (unstuck/step (D : step E E')) E0 E1 (unstuck/step (step/ifz/arg D)). + +- : progress-ifz + (Dof : of z num) + (unstuck/val (value/z)) + E0 E1 + (unstuck/step step/ifz/z). + +- : progress-ifz + (Dof : of (s E) num) + (unstuck/val (value/s (Dval : value E))) + E0 E1 + (unstuck/step (step/ifz/s Dval)). + +%worlds () (progress-ifz _ _ _ _ _). +%total {} (progress-ifz _ _ _ _ _). + +progress : of E T -> unstuck E -> type. +%mode progress +D -D'. + +- : progress of/z (unstuck/val value/z). + +- : progress (of/s (D : of E num)) D'' + <- progress D (D' : unstuck E) + <- progress-s D' (D'' : unstuck (s E)). + +- : progress + (of/ifz + (D1 : {x} of x num -> of (E1 x) T) + (D0 : of E0 T) + (D : of E num)) + D'' + <- progress D (D' : unstuck E) + <- progress-ifz D D' E0 E1 (D'' : unstuck (ifz E E0 E1)). + +%worlds () (progress _ _). +%total D (progress D _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090204). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009-20090209.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090209.lf new file mode 100644 index 0000000..14bf66e --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090209.lf @@ -0,0 +1,196 @@ +% +%{! Code from class, February 9 + +```twelf + +%% MinML language + +tp : type. + +num : tp. +arr : tp -> tp -> tp. + +%worlds () (tp). + +term : type. + +z : term. +s : term -> term. +ifz : term -> term -> (term -> term) -> term. +lam : tp -> (term -> term) -> term. +app : term -> term -> term. + +%block var : block {x:term}. +%worlds (var) (term). + +of : term -> tp -> type. + +of/z : of z num. + +of/s : of (s E) num + <- of E num. + +of/ifz : of (ifz En E0 ([x] Es x)) T + <- of En num + <- of E0 T + <- ({x : term} of x num -> of (Es x) T). + +of/lam : of (lam T1 ([x] E2 x)) (arr T1 T2) + <- ({x : term} of x T1 -> of (E2 x) T2). + +of/app : of (app E1 E2) T + <- of E1 (arr T2 T) + <- of E2 T2. + + +%block stupid : some {x:term} {t:tp} block {d:of x t}. +%block bind : some {T:tp} block {x:term} {d : of x T}. +%worlds (bind) (of _ _). + +%% Here we are using subordination. +%% It indicates when +%% You can see the transitive closure using Print.subord +%% Non-subordination is what is really importing +%% term is not subordinate to tp so a world made of terms might as well +%% be the empty world as far as a tp is concerned. + +%% Similarly of is not subordinate to term and of is not subordinate to tp. +%% So the of assumption in bind drops out as far as term is concerned. + +value : term -> type. + +val/z : value z. + +val/s : value (s E) + <- value E. + +val/lam : value (lam T ([x] E x)). + +%worlds () (value _). + +step : term -> term -> type. + +step/s : step (s E) (s E') + <- step E E'. + +step/ifz/arg : step (ifz En E0 ([x] Es x)) (ifz En' E0 ([x] Es x)) + <- step En En'. + +step/ifz/z : step (ifz z E0 ([x] Es x)) E0. + +step/ifz/s : step (ifz (s E) E0 ([x] Es x)) (Es E) + <- value E. + +step/app/fn : step (app E1 E2) (app E1' E2) + <- step E1 E1'. + +step/app/arg : step (app E1 E2) (app E1 E2') + <- value E1 + <- step E2 E2'. + +step/app/beta : step (app (lam T2 ([x] (E x))) E2) (E E2) + <- value E2. + +%worlds () (step _ _). + +%% Progress +%% If |- e : \tau then either value e or \exists e'. e \stepsto e' + +unstuck : term -> type. + +unstuck/val : unstuck E + <- value E. + +unstuck/step : unstuck E + <- step E E'. + +%worlds () (unstuck _). + +progress-s : unstuck E -> unstuck (s E) -> type. +%mode progress-s +D1 -D2. + +- : progress-s (unstuck/val Dval) (unstuck/val (val/s Dval)). + +- : progress-s (unstuck/step Dstep) (unstuck/step (step/s Dstep)). + +%worlds () (progress-s _ _). +%total {} (progress-s _ _). + +progress-ifz : of E num -> unstuck E -> {E0} {E1} unstuck (ifz E E0 E1) -> type. +%mode progress-ifz +D1 +D2 +E0 +E1 -D3. + +- : progress-ifz _ (unstuck/step Dstep) _ _ (unstuck/step (step/ifz/arg Dstep)). + +- : progress-ifz of/z (unstuck/val val/z) _ _ (unstuck/step step/ifz/z). + +- : progress-ifz (of/s Dof) (unstuck/val (val/s Dval)) _ _ (unstuck/step (step/ifz/s Dval)). + +%worlds () (progress-ifz _ _ _ _ _). +%total D (progress-ifz D _ _ _ _). + +progress-app : of E (arr T1 T2) -> unstuck E -> unstuck E1 -> unstuck (app E E1) -> type. +%mode progress-app +D1 +D2 +D3 -D4. + +- : progress-app _ (unstuck/step Dstep) _ (unstuck/step (step/app/fn Dstep)). + +- : progress-app _ (unstuck/val Dval) (unstuck/step Dstep) (unstuck/step (step/app/arg Dstep Dval)). + +- : progress-app (of/lam _) (unstuck/val val/lam) (unstuck/val Dval) (unstuck/step (step/app/beta Dval)). + +%worlds () (progress-app _ _ _ _). +%total D (progress-app D _ _ _). + +progress : of E T -> unstuck E -> type. +%mode progress +D1 -D2. + +- : progress of/z (unstuck/val val/z). + +- : progress (of/s Dof) Dunstuck' + <- progress Dof Dunstuck + <- progress-s Dunstuck Dunstuck'. + +- : progress + (of/ifz + DofBody + DofE0 + DofE) + Dunstuck' + <- progress DofE Dunstuck + <- progress-ifz DofE Dunstuck _ _ Dunstuck'. + +- : progress + (of/app + DofE1 + DofE) + Dunstuck' + <- progress DofE Dunstuck + <- progress DofE1 Dunstuck1 + <- progress-app DofE Dunstuck Dunstuck1 Dunstuck'. + +- : progress + (of/lam _) + (unstuck/val val/lam). + +%worlds () (progress _ _). +%total D (progress D _). + +%% preservation +preservation : of E T -> step E E' -> of E' T -> type. +%mode preservation +D1 +D2 -D3. + +- : preservation (of/s D1) (step/s Dstep) (of/s D1') + <- preservation D1 Dstep D1'. + +- : preservation (of/ifz Dbody D0 of/z) step/ifz/z D0. + +- : preservation (of/ifz Dbody D0 (of/s Dn)) (step/ifz/s _) (Dbody _ Dn). + +%worlds () (preservation _ _ _). +%total D (preservation _ D _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090209). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009-20090223.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090223.lf new file mode 100644 index 0000000..b205fdf --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090223.lf @@ -0,0 +1,102 @@ +% +%{! Code from class, February 23. + +```twelf +ttp : type. %name ttp T. + +tnum : ttp. +tnot : ttp -> ttp. +tprod : ttp -> ttp -> ttp. + +texp : type. %name texp E. +tval : type. %name tval V. + +tlet : tval -> (tval -> texp) -> texp. +tpi1 : tval -> (tval -> texp) -> texp. +tpi2 : tval -> (tval -> texp) -> texp. +tifz : tval -> texp -> (tval -> texp) -> texp. +tapp : tval -> tval -> texp. + +tz : tval. +ts : tval -> tval. +tpair : tval -> tval -> tval. +tlam : ttp -> (tval -> texp) -> tval. + +tof : tval -> ttp -> type. +twf : texp -> type. + +tof/z : tof tz tnum. + +tof/s : tof (ts V) tnum + <- tof V tnum. + +tof/pair : tof (tpair V1 V2) (tprod T1 T2) + <- tof V1 T1 + <- tof V2 T2. + +tof/lam : tof (tlam T ([x] E x)) (tnot T) + <- ({x} tof x T -> twf (E x)). + +twf/let : twf (tlet V ([x] E x)) + <- tof V T + <- ({x} tof x T -> twf (E x)). + +twf/pi1 : twf (tpi1 V ([x] E x)) + <- tof V (tprod T1 T2) + <- ({x} tof x T1 -> twf (E x)). + +twf/pi2 : twf (tpi2 V ([x] E x)) + <- tof V (tprod T1 T2) + <- ({x} tof x T2 -> twf (E x)). + +twf/ifz : twf (tifz V E0 ([x] E1 x)) + <- tof V tnum + <- twf E0 + <- ({x} tof x tnum -> twf (E1 x)). + +twf/app : twf (tapp V1 V2) + <- tof V1 (tnot T) + <- tof V2 T. + +ttrans : tp -> ttp -> type. + +ttrans/num : ttrans num tnum. + +ttrans/arr : ttrans (arr T1 T2) (tnot (tprod T1' (tnot T2'))) + <- ttrans T1 T1' + <- ttrans T2 T2'. + +trans : term -> tp -> tval -> type. + +trans/z : trans z num (tlam (tnot tnum) ([f] tapp f tz)). + +trans/s : trans (s E) num (tlam (tnot tnum) ([f] tapp V (tlam tnum ([x] tapp f (ts x))))) + <- trans E num V. + +trans/ifz : trans (ifz E E0 ([x] E1 x)) T + (tlam (tnot T') + ([f] tapp V (tlam tnum ([y] tifz y (tapp V0 f) + ([y'] tlet (tlam (tnot tnum) ([g] tapp g y')) + ([x'] tapp (V1 x') f)))))) + <- ttrans T T' + <- trans E num V + <- trans E0 T V0 + <- ({x} {x'} trans x num x' -> trans (E1 x) T (V1 x')). + +trans/lam : trans (lam T1 ([x] E x)) (arr T1 T2) + (tlam (tnot (tnot (tprod T1' (tnot T2')))) + ([f] tapp f (tlam (tprod T1' (tnot T2')) + ([y] tpi1 y + ([y'] (tpi2 y + ([g] tlet (tlam (tnot T1') ([w] tapp w y')) + ([x'] tapp (V x') g)))))))) + <- ttrans T1 T1' + <- ttrans T2 T2' + <- ({x} {x'} trans x T1 x' -> trans (E x) T2 (V x')). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090223). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009-20090304-support.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090304-support.lf new file mode 100644 index 0000000..bb579b1 --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090304-support.lf @@ -0,0 +1,42 @@ +% +%{! + +Support code for March 4 guest lecture on worlds-checking and combinators. + +```twelf +% combinator equality + +ceq : comb -> comb -> type. + +% betas +ceq/i : ceq (capp i A) A. +ceq/k : ceq (capp (capp k A) B) A. +ceq/s : ceq (capp (capp (capp s A) B) C) (capp (capp A C) (capp B C)). + +% extensionality +ceq/ext : ceq A A' + <- ({y:comb} ceq (capp A y) (capp A' y)). + +% compatibility +ceq/app : ceq (capp A B) (capp A' B') + <- ceq A A' + <- ceq B B'. + +% equivalence +ceq/refl : ceq A A. + +ceq/symm : ceq A A' + <- ceq A' A. + +ceq/trans : ceq A A' + <- ceq A B + <- ceq B A'. + +%worlds (comb-block) (ceq _ _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090304-support). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009-20090304.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090304.lf new file mode 100644 index 0000000..b2bf311 --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090304.lf @@ -0,0 +1,192 @@ +% +%{! +Code from class, March 4. + +Translates lambda calculus into combinator calculus to illustrate non-trivial worlds. + +## Syntax + +```twelf + +% lambda calculus + +term : type. %name term M x. + +app : term -> term -> term. +lam : (term -> term) -> term. + +%block term-block : block {x:term}. +%worlds (term-block) (term). + +% combinator calculus + +comb : type. %name comb A y. + +s : comb. +k : comb. +i : comb. +capp : comb -> comb -> comb. + +%block comb-block : block {y:comb}. +%worlds (comb-block) (comb). + +%{! + +## Translation + +!}% + +% bracket abstraction + +bracket : (comb -> comb) -> comb -> type. +%mode bracket +A -A'. + +bracket/var : bracket ([y] y) i. + +bracket/i : bracket ([y] i) (capp k i). +bracket/k : bracket ([y] k) (capp k k). +bracket/s : bracket ([y] s) (capp k s). + +bracket/app : bracket ([y] capp (A y) (B y)) (capp (capp s A') B') + <- bracket ([y] A y) A' + <- bracket ([y] B y) B'. + +%block bracket-block : block {y:comb} {bracket/y:bracket ([z] y) (capp k y)}. + +%worlds (bracket-block) (bracket _ _). +%total A (bracket A _). + + +% translation + +translate : term -> comb -> type. +%mode translate +M -A. + +translate/app : translate (app M N) (capp A B) + <- translate M A + <- translate N B. + +translate/lam : translate (lam ([x] M x)) A' + <- ({x} {y} bracket ([z] y) (capp k y) + -> translate x y + -> translate (M x) (A y)) + <- bracket ([y] A y) A'. + +%block translate-block + : block {x:term} {y:comb} + {bracket/y: bracket ([z] y) (capp k y)} + {translate/x: translate x y}. +%worlds (translate-block) (translate _ _). +%total M (translate M _). + +%{! + +## Equational theory + +Lambda term equality was discussed only briefly. Combinator equality was presented for the ``subst`` lemma, below. + +!}% + +% lambda term equality + +teq : term -> term -> type. + +% beta +teq/beta : teq (app (lam ([x] M x)) N) (M N). + +% extensionality (eta) +teq/ext : teq M M' + <- ({x:term} teq (app M x) (app M' x)). + +% compatibilities +teq/app : teq (app M N) (app M' N') + <- teq M M' + <- teq N N'. + +teq/lam : teq (lam ([x] M x)) (lam ([x] M' x)) + <- ({x:term} teq (M x) (M' x)). + +% equivalence +teq/refl : teq M M. + +teq/symm : teq M M' + <- teq M' M. + +teq/trans : teq M M' + <- teq M N + <- teq N M'. + +%worlds (term-block) (teq _ _). + + +% combinator equality + +ceq : comb -> comb -> type. + +% betas +ceq/i : ceq (capp i A) A. +ceq/k : ceq (capp (capp k A) B) A. +ceq/s : ceq (capp (capp (capp s A) B) C) (capp (capp A C) (capp B C)). + +% extensionality +ceq/ext : ceq A A' + <- ({y:comb} ceq (capp A y) (capp A' y)). + +% compatibility +ceq/app : ceq (capp A B) (capp A' B') + <- ceq A A' + <- ceq B B'. + +% equivalence +ceq/refl : ceq A A. + +ceq/symm : ceq A A' + <- ceq A' A. + +ceq/trans : ceq A A' + <- ceq A B + <- ceq B A'. + +%worlds (comb-block) (ceq _ _). + + + +%{! + +## Correctness of the translation + +!}% + +% substitution lemma + +subst : bracket ([y] A y) A' -> {C:comb} ceq (capp A' C) (A C) -> type. +%mode subst +Dbrack +C -Dceq. + +- : subst (bracket/var : bracket ([y] y) i) + C + (ceq/i : ceq (capp i C) C). + +- : subst (bracket/i : bracket ([y] i) (capp k i)) + C + (ceq/k : ceq (capp (capp k i) C) i). + +- : subst (bracket/k : bracket ([y] k) (capp k k)) + C + (ceq/k : ceq (capp (capp k k) C) k). + +- : subst (bracket/s : bracket ([y] s) (capp k s)) + C + (ceq/k : ceq (capp (capp k s) C) s). + +%block subst-block : block {y:comb} + {bracket/y: bracket ([z] y) (capp k y)} + {thm-subst: {C:comb} subst bracket/y C ceq/k}. +%worlds (subst-block) (subst _ _ _). +%total D (subst D _ _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090304). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009-20090316.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090316.lf new file mode 100644 index 0000000..f36d9cf --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090316.lf @@ -0,0 +1,177 @@ +% +%{! Code from class, March 16. + +```twelf +ttp : type. %name ttp T. + +tnum : ttp. +tnot : ttp -> ttp. +tprod : ttp -> ttp -> ttp. + +texp : type. %name texp E. +tval : type. %name tval V. + +tlet : tval -> (tval -> texp) -> texp. +tpi1 : tval -> (tval -> texp) -> texp. +tpi2 : tval -> (tval -> texp) -> texp. +tifz : tval -> texp -> (tval -> texp) -> texp. +tapp : tval -> tval -> texp. + +tz : tval. +ts : tval -> tval. +tpair : tval -> tval -> tval. +tlam : ttp -> (tval -> texp) -> tval. + +tof : tval -> ttp -> type. +twf : texp -> type. + +tof/z : tof tz tnum. + +tof/s : tof (ts V) tnum + <- tof V tnum. + +tof/pair : tof (tpair V1 V2) (tprod T1 T2) + <- tof V1 T1 + <- tof V2 T2. + +tof/lam : tof (tlam T ([x] E x)) (tnot T) + <- ({x} tof x T -> twf (E x)). + +twf/let : twf (tlet V ([x] E x)) + <- tof V T + <- ({x} tof x T -> twf (E x)). + +twf/pi1 : twf (tpi1 V ([x] E x)) + <- tof V (tprod T1 T2) + <- ({x} tof x T1 -> twf (E x)). + +twf/pi2 : twf (tpi2 V ([x] E x)) + <- tof V (tprod T1 T2) + <- ({x} tof x T2 -> twf (E x)). + +twf/ifz : twf (tifz V E0 ([x] E1 x)) + <- tof V tnum + <- twf E0 + <- ({x} tof x tnum -> twf (E1 x)). + +twf/app : twf (tapp V1 V2) + <- tof V1 (tnot T) + <- tof V2 T. + +ttrans : tp -> ttp -> type. + +ttrans/num : ttrans num tnum. + +ttrans/arr : ttrans (arr T1 T2) (tnot (tprod T1' (tnot T2'))) + <- ttrans T1 T1' + <- ttrans T2 T2'. + +suspend : ttp -> tval -> tval + = [t] [v] (tlam (tnot t) ([f] tapp f v)). + +trans : term -> tp -> tval -> type. + +trans/z : trans z num (suspend tnum tz). + +trans/s : trans (s E) num (tlam (tnot tnum) ([f] tapp V (tlam tnum ([x] tapp f (ts x))))) + <- trans E num V. + +trans/ifz : trans (ifz E E0 ([x] E1 x)) T + (tlam (tnot T') + ([f] tapp V (tlam tnum ([y] tifz y (tapp V0 f) + ([y'] tlet (tlam (tnot tnum) ([g] tapp g y')) + ([x'] tapp (V1 x') f)))))) + <- ttrans T T' + <- trans E num V + <- trans E0 T V0 + <- ({x} {x'} trans x num x' -> trans (E1 x) T (V1 x')). + +trans/lam : trans (lam T1 ([x] E x)) (arr T1 T2) + (suspend (tnot (tprod T1' (tnot T2'))) + (tlam (tprod T1' (tnot T2')) + ([y] tpi1 y + ([y'] (tpi2 y + ([g] tlet (suspend T1' y') + ([x'] tapp (V x') g))))))) + <- ttrans T1 T1' + <- ttrans T2 T2' + <- ({x} {x'} trans x T1 x' -> trans (E x) T2 (V x')). + +trans/app : trans (app E1 E2) T2 + (tlam (tnot T2') + ([f] tapp V1 + (tlam (tnot (tprod T1' (tnot T2'))) + ([x1] tapp V2 (tlam T1' ([x2] tapp x1 (tpair x2 f))))))) + <- ttrans T1 T1' + <- ttrans T2 T2' + <- trans E1 (arr T1 T2) V1 + <- trans E2 T1 V2. + +tof/suspend : tof (suspend T V) (tnot (tnot T)) + <- tof V T + = [D : tof V T] (tof/lam ([x] [d : tof x (tnot T)] (twf/app D d))). + + +correct : trans E T V -> ttrans T T' -> tof V (tnot (tnot T')) -> type. +%mode correct +D1 -D2 -D3. + +correct' : trans E T V -> ttrans T T' -> tof V (tnot (tnot T')) -> type. +%mode correct' +D1 +D2 -D3. + +- : correct trans/z ttrans/num (tof/suspend tof/z). + +- : correct + (trans/s + (D : trans E num V)) + ttrans/num + (tof/lam + ([f] [df:tof f (tnot tnum)] + twf/app + (tof/lam ([x] [dx:tof x tnum] (twf/app (tof/s dx) df))) + D')) + <- correct' D ttrans/num (D': tof V (tnot (tnot tnum))). + +- : correct + (trans/lam + (D3: ({x} {x'} trans x T1 x' -> trans (E x) T2 (V x'))) + (D2: ttrans T2 T2') + (D1: ttrans T1 T1')) + (ttrans/arr D2 D1) + (tof/suspend + (tof/lam + ([y] [dy:tof y (tprod T1' (tnot T2'))] + twf/pi1 + ([y'][dy': tof y' T1'] + twf/pi2 + ([g][dg:tof g (tnot T2')] + (twf/let + ([x'] [dx':tof x' (tnot (tnot T1'))] + twf/app + dg + (DV x' dx')) + (tof/suspend dy'))) + dy) + dy))) + <- ({x} {x'} {dx': tof x' (tnot (tnot T1'))} + {dt: trans x T1 x'} + correct dt D1 dx' + -> correct' (D3 x x' dt) D2 (DV x' dx' : tof (V x') (tnot (tnot T2')))). + + +%block transbind : some {T : tp} {T' : ttp} {DTT : ttrans T T'} + block + {x : term} + {x' : tval} {dx' : tof x' (tnot (tnot T'))} + {dt : trans x T x'} + {thm : correct dt DTT dx'}. + + +%worlds (transbind) (correct _ _ _) (correct' _ _ _). +%total (D D') (correct D _ _) (correct' D' _ _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090316). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009-20090318.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090318.lf new file mode 100644 index 0000000..4f14856 --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090318.lf @@ -0,0 +1,273 @@ +% +%{! Code from class, March 18. + +```twelf +ttp : type. %name ttp T. + +tnum : ttp. +tnot : ttp -> ttp. +tprod : ttp -> ttp -> ttp. + +texp : type. %name texp E. +tval : type. %name tval V. + +tlet : tval -> (tval -> texp) -> texp. +tpi1 : tval -> (tval -> texp) -> texp. +tpi2 : tval -> (tval -> texp) -> texp. +tifz : tval -> texp -> (tval -> texp) -> texp. +tapp : tval -> tval -> texp. + +tz : tval. +ts : tval -> tval. +tpair : tval -> tval -> tval. +tlam : ttp -> (tval -> texp) -> tval. + +tof : tval -> ttp -> type. +twf : texp -> type. + +tof/z : tof tz tnum. + +tof/s : tof (ts V) tnum + <- tof V tnum. + +tof/pair : tof (tpair V1 V2) (tprod T1 T2) + <- tof V1 T1 + <- tof V2 T2. + +tof/lam : tof (tlam T ([x] E x)) (tnot T) + <- ({x} tof x T -> twf (E x)). + +twf/let : twf (tlet V ([x] E x)) + <- tof V T + <- ({x} tof x T -> twf (E x)). + +twf/pi1 : twf (tpi1 V ([x] E x)) + <- tof V (tprod T1 T2) + <- ({x} tof x T1 -> twf (E x)). + +twf/pi2 : twf (tpi2 V ([x] E x)) + <- tof V (tprod T1 T2) + <- ({x} tof x T2 -> twf (E x)). + +twf/ifz : twf (tifz V E0 ([x] E1 x)) + <- tof V tnum + <- twf E0 + <- ({x} tof x tnum -> twf (E1 x)). + +twf/app : twf (tapp V1 V2) + <- tof V1 (tnot T) + <- tof V2 T. + +ttrans : tp -> ttp -> type. + +ttrans/num : ttrans num tnum. + +ttrans/arr : ttrans (arr T1 T2) (tnot (tprod T1' (tnot T2'))) + <- ttrans T1 T1' + <- ttrans T2 T2'. + +suspend : ttp -> tval -> tval + = [t] [v] (tlam (tnot t) ([f] tapp f v)). + +trans : term -> tp -> tval -> type. + +trans/z : trans z num (suspend tnum tz). + +trans/s : trans (s E) num (tlam (tnot tnum) ([f] tapp V (tlam tnum ([x] tapp f (ts x))))) + <- trans E num V. + +trans/ifz : trans (ifz E E0 ([x] E1 x)) T + (tlam (tnot T') + ([f] tapp V (tlam tnum ([y] tifz y (tapp V0 f) + ([y'] tlet (tlam (tnot tnum) ([g] tapp g y')) + ([x'] tapp (V1 x') f)))))) + <- ttrans T T' + <- trans E num V + <- trans E0 T V0 + <- ({x} {x'} trans x num x' -> trans (E1 x) T (V1 x')). + +trans/lam : trans (lam T1 ([x] E x)) (arr T1 T2) + (suspend (tnot (tprod T1' (tnot T2'))) + (tlam (tprod T1' (tnot T2')) + ([y] tpi1 y + ([y'] (tpi2 y + ([g] tlet (suspend T1' y') + ([x'] tapp (V x') g))))))) + <- ttrans T1 T1' + <- ttrans T2 T2' + <- ({x} {x'} trans x T1 x' -> trans (E x) T2 (V x')). + +trans/app : trans (app E1 E2) T2 + (tlam (tnot T2') + ([f] tapp V1 + (tlam (tnot (tprod T1' (tnot T2'))) + ([x1] tapp V2 (tlam T1' ([x2] tapp x1 (tpair x2 f))))))) + <- ttrans T1 T1' + <- ttrans T2 T2' + <- trans E1 (arr T1 T2) V1 + <- trans E2 T1 V2. + +tof/suspend : tof (suspend T V) (tnot (tnot T)) + <- tof V T + = [D : tof V T] (tof/lam ([x] [d : tof x (tnot T)] (twf/app D d))). + +ttp-eq : ttp -> ttp -> type. + +ttp-eq/i : ttp-eq T T. + +ttrans-fun : ttrans T T' -> ttrans T T'' -> ttp-eq T' T'' -> type. +%mode ttrans-fun +D1 +D2 -D3. + + + +%worlds () (ttrans-fun _ _ _). +%trustme %total D (ttrans-fun D _ _). + +tof-resp : ttp-eq T T' -> tof V T -> tof V T' -> type. +%mode tof-resp +D1 +D2 -D3. + +tof-resp/i : tof-resp ttp-eq/i D D. + +%block tbind : some {T : ttp} block {x : tval} {d : tof x T}. +%worlds (tbind) (tof-resp _ _ _). + +%total {} (tof-resp _ _ _). + +ttp-resp : {T : ttp -> ttp} ttp-eq T1 T2 -> ttp-eq (T T1) (T T2) -> type. +%mode ttp-resp +D1 +D2 -D3. + +- : ttp-resp T ttp-eq/i ttp-eq/i. + +%worlds () (ttp-resp _ _ _). +%total {} (ttp-resp _ _ _). + +correct : trans E T V -> ttrans T T' -> tof V (tnot (tnot T')) -> type. +%mode correct +D1 -D2 -D3. + +correct' : trans E T V -> ttrans T T' -> tof V (tnot (tnot T')) -> type. +%mode correct' +D1 +D2 -D3. + +- : correct' + (DtransE : trans E T V) + (DtransT : ttrans T T') + DofV' + <- correct DtransE + (DtransT' : ttrans T T'') + (DofV : tof V (tnot (tnot T''))) + <- ttrans-fun DtransT' DtransT (Deq : ttp-eq T'' T') + <- ttp-resp ([t] (tnot (tnot t))) Deq (Deq' : ttp-eq (tnot (tnot T'')) (tnot (tnot T'))) + <- tof-resp Deq' DofV (DofV' : tof V (tnot (tnot T'))). + +- : correct trans/z ttrans/num (tof/suspend tof/z). + +- : correct + (trans/s + (D : trans E num V)) + ttrans/num + (tof/lam + ([f] [df:tof f (tnot tnum)] + twf/app + (tof/lam ([x] [dx:tof x tnum] (twf/app (tof/s dx) df))) + D')) + <- correct' D ttrans/num (D': tof V (tnot (tnot tnum))). + +- : correct + (trans/ifz + (DtransE1 : {x} {x'} trans x num x' -> trans (E1 x) T (V1 x')) + (DtransE0 : trans E0 T V0) + (DtransE : trans E num V) + (DtransT : ttrans T T')) + DtransT + (tof/lam + ([f] [df:tof f (tnot T')] + twf/app + (tof/lam + ([x] [dx:tof x tnum] + twf/ifz + ([y'] [dy':tof y' tnum] + twf/let + ([x'] [dx':tof x' (tnot (tnot tnum))] + twf/app + df + (DofV1 x' dx')) + (tof/lam + ([g] [dg:tof g (tnot tnum)] + twf/app dy' dg))) + (twf/app df DofV0) + dx)) + DofV)) + <- correct' DtransE ttrans/num (DofV : tof V (tnot (tnot tnum))) + <- correct' DtransE0 DtransT (DofV0 : tof V0 (tnot (tnot T'))) + <- ({x} {x'} {dx' : tof x' (tnot (tnot tnum))} + {dt : trans x num x'} + correct dt ttrans/num dx' + -> correct' (DtransE1 x x' dt) DtransT (DofV1 x' dx' : tof (V1 x') (tnot (tnot T')))). + +- : correct + (trans/lam + (D3: ({x} {x'} trans x T1 x' -> trans (E x) T2 (V x'))) + (D2: ttrans T2 T2') + (D1: ttrans T1 T1')) + (ttrans/arr D2 D1) + (tof/suspend + (tof/lam + ([y] [dy:tof y (tprod T1' (tnot T2'))] + twf/pi1 + ([y'][dy': tof y' T1'] + twf/pi2 + ([g][dg:tof g (tnot T2')] + (twf/let + ([x'] [dx':tof x' (tnot (tnot T1'))] + twf/app + dg + (DV x' dx')) + (tof/suspend dy'))) + dy) + dy))) + <- ({x} {x'} {dx': tof x' (tnot (tnot T1'))} + {dt: trans x T1 x'} + correct dt D1 dx' + -> correct' (D3 x x' dt) D2 (DV x' dx' : tof (V x') (tnot (tnot T2')))). + +- : correct + (trans/app + (DtransE2 : trans E2 T1 V2) + (DtransE1 : trans E1 (arr T1 T2) V1) + (DtransT2 : ttrans T2 T2') + (DtransT1 : ttrans T1 T1')) + DtransT2 + (tof/lam + ([f] [df:tof f (tnot T2')] + twf/app + (tof/lam + ([x1] [dx1:tof x1 (tnot (tprod T1' (tnot T2')))] + twf/app + (tof/lam + ([x2] [dx2:tof x2 T1'] + twf/app + (tof/pair df dx2) + dx1)) + DofV2)) + DofV1)) + <- correct' DtransE1 (ttrans/arr DtransT2 DtransT1) + (DofV1 : tof V1 (tnot (tnot (tnot (tprod T1' (tnot T2')))))) + <- correct' DtransE2 DtransT1 + (DofV2 : tof V2 (tnot (tnot T1'))). + +%block transbind : some {T : tp} {T' : ttp} {DTT : ttrans T T'} + block + {x : term} + {x' : tval} {dx' : tof x' (tnot (tnot T'))} + {dt : trans x T x'} + {thm : correct dt DTT dx'}. + + +%worlds (transbind) (correct _ _ _) (correct' _ _ _). +%total (D D') (correct D _ _) (correct' D' _ _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090318). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009-20090325.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090325.lf new file mode 100644 index 0000000..d9fb908 --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090325.lf @@ -0,0 +1,131 @@ +% +%{! Code from class, March 25. + +```twelf +tp : type. + +tensor : tp -> tp -> tp. +1 : tp. +with : tp -> tp -> tp. +top : tp. +plus : tp -> tp -> tp. +0 : tp. + +exp : type. + +tens : exp -> exp -> exp. +lett : exp -> (exp -> exp -> exp) -> exp. +pair : exp -> exp -> exp. +pi1 : exp -> exp. +pi2 : exp -> exp. +star : exp. +lets : exp -> exp -> exp. +unit : exp. +in1 : exp -> exp. +in2 : exp -> exp. +case : exp -> (exp -> exp) -> (exp -> exp) -> exp. +abort : exp -> exp. + +linear : (exp -> exp) -> type. + +of : exp -> tp -> type. + + +linear/tens1 : linear ([x] tens (M x) N) + <- linear ([x] M x). + +linear/tens2 : linear ([x] tens M (N x)) + <- linear ([x] N x). + +of/tens : of (tens M N) (tensor A B) + <- of M A + <- of N B. + +linear/lett1 : linear ([z] lett (M z) ([x] [y] N x y)) + <- linear ([x] M x). + +linear/lett2 : linear ([z] lett M ([x] [y] N z x y)) + <- ({x} {y} linear ([z] N z x y)). + +of/lett : of (lett M ([x] [y] N x y)) C + <- of M (tensor A B) + <- ({x:exp} of x A -> {y:exp} of y B -> of (N x y) C) + <- ({y:exp} linear ([x] N x y)) + <- ({x:exp} linear ([y] N x y)). + +linear/pair : linear ([x] pair (M x) (N x)) + <- linear ([x] M x) + <- linear ([x] N x). + +of/pair : of (pair M N) (with A B) + <- of M A + <- of N B. + +linear/pi1 : linear ([x] pi1 (M x)) + <- linear ([x] M x). + +linear/pi2 : linear ([x] pi2 (M x)) + <- linear ([x] M x). + +of/pi1 : of (pi1 M) A + <- of M (with A B). + +of/pi2 : of (pi2 M) B + <- of M (with A B). + +of/star : of star 1. + +linear/lets1 : linear ([x] lets (M x) N) + <- linear ([x] M x). + +linear/lets2 : linear ([x] lets M (N x)) + <- linear ([x] N x). + +of/lets : of (lets M N) C + <- of M 1 + <- of N C. + +linear/unit : linear ([x] unit). + +of/unit : of unit top. + +linear/in1 : linear ([x] in1 (M x)) + <- linear ([x] M x). + +linear/in2 : linear ([x] in2 (M x)) + <- linear ([x] M x). + +of/in1 : of (in1 M) (plus A B) + <- of M A. + +of/in2 : of (in2 M) (plus A B) + <- of M B. + +linear/case1 : linear ([z] case (M z) ([x] N x) ([x] O x)) + <- linear ([z] M z). + +linear/case2 : linear ([z] case M ([x] N z x) ([x] O z x)) + <- ({x:exp} linear ([z] N z x)) + <- ({x:exp} linear ([z] O z x)). + +of/case : of (case M ([x] N x) ([x] O x)) C + <- of M (plus A B) + <- ({x} of x A -> of (N x) C) + <- ({x} of x B -> of (O x) C) + <- linear ([x] N x) + <- linear ([x] O x). + +linear/abort1 : linear ([x] abort (M x)) + <- linear ([x] M x). + +linear/abort2 : linear ([x] abort M). + +of/abort : of (abort M) C + <- of M 0. +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090325). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009-20090330.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090330.lf new file mode 100644 index 0000000..01800c9 --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090330.lf @@ -0,0 +1,218 @@ +% +%{! Code from class, March 30. + +```twelf +tp : type. + +tensor : tp -> tp -> tp. +1 : tp. +with : tp -> tp -> tp. +top : tp. +plus : tp -> tp -> tp. +0 : tp. +lolli : tp -> tp -> tp. +! : tp -> tp. + +exp : type. + +tens : exp -> exp -> exp. +lett : exp -> (exp -> exp -> exp) -> exp. +pair : exp -> exp -> exp. +pi1 : exp -> exp. +pi2 : exp -> exp. +star : exp. +lets : exp -> exp -> exp. +unit : exp. +in1 : exp -> exp. +in2 : exp -> exp. +case : exp -> (exp -> exp) -> (exp -> exp) -> exp. +abort : exp -> exp. +lam : tp -> (exp -> exp) -> exp. +app : exp -> exp -> exp. +bang : exp -> exp. +letb : exp -> (exp -> exp) -> exp. + +linear : (exp -> exp) -> type. + +of : exp -> tp -> type. + + +linear/tens1 : linear ([x] tens (M x) N) + <- linear ([x] M x). + +linear/tens2 : linear ([x] tens M (N x)) + <- linear ([x] N x). + +of/tens : of (tens M N) (tensor A B) + <- of M A + <- of N B. + +linear/lett1 : linear ([z] lett (M z) ([x] [y] N x y)) + <- linear ([x] M x). + +linear/lett2 : linear ([z] lett M ([x] [y] N z x y)) + <- ({x} {y} linear ([z] N z x y)). + +of/lett : of (lett M ([x] [y] N x y)) C + <- of M (tensor A B) + <- ({x:exp} of x A -> {y:exp} of y B -> of (N x y) C) + <- ({y:exp} linear ([x] N x y)) + <- ({x:exp} linear ([y] N x y)). + +linear/pair : linear ([x] pair (M x) (N x)) + <- linear ([x] M x) + <- linear ([x] N x). + +of/pair : of (pair M N) (with A B) + <- of M A + <- of N B. + +linear/pi1 : linear ([x] pi1 (M x)) + <- linear ([x] M x). + +linear/pi2 : linear ([x] pi2 (M x)) + <- linear ([x] M x). + +of/pi1 : of (pi1 M) A + <- of M (with A B). + +of/pi2 : of (pi2 M) B + <- of M (with A B). + +of/star : of star 1. + +linear/lets1 : linear ([x] lets (M x) N) + <- linear ([x] M x). + +linear/lets2 : linear ([x] lets M (N x)) + <- linear ([x] N x). + +of/lets : of (lets M N) C + <- of M 1 + <- of N C. + +linear/unit : linear ([x] unit). + +of/unit : of unit top. + +linear/in1 : linear ([x] in1 (M x)) + <- linear ([x] M x). + +linear/in2 : linear ([x] in2 (M x)) + <- linear ([x] M x). + +of/in1 : of (in1 M) (plus A B) + <- of M A. + +of/in2 : of (in2 M) (plus A B) + <- of M B. + +linear/case1 : linear ([z] case (M z) ([x] N x) ([x] O x)) + <- linear ([z] M z). + +linear/case2 : linear ([z] case M ([x] N z x) ([x] O z x)) + <- ({x:exp} linear ([z] N z x)) + <- ({x:exp} linear ([z] O z x)). + +of/case : of (case M ([x] N x) ([x] O x)) C + <- of M (plus A B) + <- ({x} of x A -> of (N x) C) + <- ({x} of x B -> of (O x) C) + <- linear ([x] N x) + <- linear ([x] O x). + +linear/abort1 : linear ([x] abort (M x)) + <- linear ([x] M x). + +linear/abort2 : linear ([x] abort M). + +of/abort : of (abort M) C + <- of M 0. + +linear/lam : linear ([x] (lam A ([y] (M x y)))) + <- ({y:exp} linear ([x] (M x y))). + +of/lam : of (lam A ([x] M x)) (lolli A B) + <- ({x} (of x A) -> of (M x) B) + <- linear ([x] M x). + +linear/app1 : linear ([x] (app (M x) N)) + <- linear ([x] M x). + +linear/app2 : linear ([x] (app M (N x))) + <- linear ([x] N x). + +of/app : of (app M N) B + <- of M (lolli A B) + <- of N A. + +of/bang : of (bang M) (! A) + <- of M A. + +linear/letb1 : linear ([x] (letb (M x) ([y] (N y)))) + <- linear ([x] M x). + +linear/letb2 : linear ([x] (letb M ([y] N x y))) + <- ({y:exp} linear ([x] N x y)). + +of/letb : of (letb M ([x] N x)) C + <- of M (! A) + <- ({x:exp} (of x A) -> of (N x) C). + +reduce : exp -> exp -> type. + +reduce/lam : reduce (lam A ([x] M x)) (lam A ([x] N x)) + <- ({x} reduce (M x) (N x)). + +reduce/app1 : reduce (app M N) (app M' N) + <- reduce M M'. + +reduce/app2 : reduce (app M N) (app M N') + <- reduce N N'. + +reduce/beta : reduce (app (lam A ([x] M x)) N) (M N). + +srl : ({x} (of x A) -> of (M x) B) -> linear ([x] M x) -> ({x} reduce (M x) (M' x)) -> linear ([x] M' x) -> type. + + +sr : of M A -> reduce M M' -> of M' A -> type. +%mode sr +D1 +D2 -D3. + +- : sr + (of/app (DofN : of N A) (DofM : of M (lolli A B))) + (reduce/app1 (Dreduce : reduce M M')) + (of/app DofN DofM') + <- sr DofM Dreduce (DofM' : of M' (lolli A B)). + +- : sr + (of/app (DofN : of N A) (DofM : of M (lolli A B))) + (reduce/app2 (Dreduce : reduce N N')) + (of/app DofN' DofM) + <- sr DofN Dreduce (DofN' : of N' A). + +- : sr + (of/app (DofN : of N A) (of/lam _ (DofM : {x} (of x A) -> of (M x) B))) + reduce/beta + (DofM N DofN). + +- : sr + (of/lam + (Dlinear : linear ([x] M x)) + (DofM : {x} (of x A) -> of (M x) B)) + (reduce/lam (Dreduce : {x} reduce (M x) (N x))) + (of/lam + _ + DofN) + <- ({x} {dx: of x A} + sr (DofM x dx) (Dreduce x) (DofN x dx : of (N x) B)). + +%block bind : some {A:tp} block {x:exp} {dx : of x A}. +%worlds (bind) (sr _ _ _). +%total D1 (sr _ D1 _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090330). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009-20090401.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090401.lf new file mode 100644 index 0000000..37b313f --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090401.lf @@ -0,0 +1,252 @@ +% +%{! Code from class, April 1. + +```twelf +tp : type. %name tp A. + +tensor : tp -> tp -> tp. +1 : tp. +with : tp -> tp -> tp. +top : tp. +plus : tp -> tp -> tp. +0 : tp. +lolli : tp -> tp -> tp. +! : tp -> tp. + +exp : type. %name exp M x. + +tens : exp -> exp -> exp. +lett : exp -> (exp -> exp -> exp) -> exp. +pair : exp -> exp -> exp. +pi1 : exp -> exp. +pi2 : exp -> exp. +star : exp. +lets : exp -> exp -> exp. +unit : exp. +in1 : exp -> exp. +in2 : exp -> exp. +case : exp -> (exp -> exp) -> (exp -> exp) -> exp. +abort : exp -> exp. +lam : tp -> (exp -> exp) -> exp. +app : exp -> exp -> exp. +bang : exp -> exp. +letb : exp -> (exp -> exp) -> exp. + +linear : (exp -> exp) -> type. +of : exp -> tp -> type. + +linear/tens1 : linear ([x] tens (M x) N) + <- linear ([x] M x). + +linear/tens2 : linear ([x] tens M (N x)) + <- linear ([x] N x). + +of/tens : of (tens M N) (tensor A B) + <- of M A + <- of N B. + +linear/lett1 : linear ([z] lett (M z) ([x] [y] N x y)) + <- linear ([x] M x). + +linear/lett2 : linear ([z] lett M ([x] [y] N z x y)) + <- ({x} {y} linear ([z] N z x y)). + +of/lett : of (lett M ([x] [y] N x y)) C + <- of M (tensor A B) + <- ({x:exp} of x A -> {y:exp} of y B -> of (N x y) C) + <- ({y:exp} linear ([x] N x y)) + <- ({x:exp} linear ([y] N x y)). + +linear/pair : linear ([x] pair (M x) (N x)) + <- linear ([x] M x) + <- linear ([x] N x). + +of/pair : of (pair M N) (with A B) + <- of M A + <- of N B. + +linear/pi1 : linear ([x] pi1 (M x)) + <- linear ([x] M x). + +linear/pi2 : linear ([x] pi2 (M x)) + <- linear ([x] M x). + +of/pi1 : of (pi1 M) A + <- of M (with A B). + +of/pi2 : of (pi1 M) B + <- of M (with A B). + +of/star : of star 1. + +linear/lets1 : linear ([x] lets (M x) N) + <- linear ([x] M x). + +linear/lets2 : linear ([x] lets M (N x)) + <- linear ([x] N x). + +of/lets : of (lets M N) C + <- of M 1 + <- of N C. + +linear/unit : linear ([x] unit). + +of/unit : of unit top. + +linear/in1 : linear ([x] in1 (M x)) + <- linear ([x] M x). + +linear/in2 : linear ([x] in2 (M x)) + <- linear ([x] M x). + +of/in1 : of (in1 M) (plus A B) + <- of M A. + +of/in2 : of (in2 M) (plus A B) + <- of M B. + +linear/case1 : linear ([z] case (M z) ([x] N x) ([x] O x)) + <- linear ([z] M z). + +linear/case2 : linear ([z] case M ([x] N z x) ([x] O z x)) + <- ({x:exp} linear ([z] N z x)) + <- ({x:exp} linear ([z] O z x)). + +of/case : of (case M ([x] N x) ([x] O x)) C + <- of M (plus A B) + <- ({x} of x A -> of (N x) C) + <- ({x} of x B -> of (O x)C) + <- linear ([x] N x) + <- linear ([x] O x). + +linear/abort1 : linear ([x] abort (M x)) + <- linear ([x] M x). + +linear/abort2 : linear ([x] abort M). + +of/abort : of (abort M) C + <- of M O. + +linear/lam : linear ([x] (lam A ([y] (M x y)))) + <- {y:exp} linear ([x] M x y). + +of/lam : of (lam A ([x] M x)) (lolli A B) + <- ({x} (of x A) -> of (M x) B) + <- linear ([x] M x). + +linear/app1 : linear ([x] (app (M x) N)) + <- linear ([x] M x). + +linear/app2 : linear ([x] (app M (N x))) + <- linear ([x] N x). + +of/app : of (app M N) B + <- of M (lolli A B) + <- of N A. + +of/bang : of (bang M) (! A) + <- of M A. + +linear/letb1 : linear ([x] (letb (M x) ([y] (N y)))) + <- linear ([x] M x). + +linear/letb2 : linear ([x] (letb M ([y] N x y))) + <- ({y} linear ([x] N x y)). + +of/letb : of (letb M ([x] N x)) C + <- of M (! A) + <- ({x} (of x A) -> of (N x) C). + + +reduce : exp -> exp -> type. + +reduce/lam : reduce (lam A ([x] M x)) (lam A ([x] N x)) + <- ({x} reduce (M x) (N x)). + +reduce/app1 : reduce (app M N) (app M' N) + <- reduce M M'. + +reduce/app2 : reduce (app M N) (app M N') + <- reduce N N'. + +reduce/beta : reduce (app (lam A ([x] M x)) N) (M N). + +%block block : block {x:exp}. +%block bind : some {A:tp} block {x:exp} {d : of x A}. + +exp-eq : exp -> exp -> type. +exp-eq/i : exp-eq M M. + +reduce-closed : ({x:exp} reduce M (N x)) -> ({x:exp} exp-eq (N x) N') -> type. +%mode reduce-closed +D -D'. + +%worlds (block) (reduce-closed _ _). +%trustme %total D (reduce-closed D _). + +srl : ({x} (of x A) -> of (M x) B) -> linear ([x] M x) -> ({x} reduce (M x) (M' x)) -> linear ([x] M' x) -> type. +%mode srl +D1 +D2 +D3 -D4. + +- : srl + ([x] [d:of x A] of/lam _ (Dof x d : {y} of y B -> of (M x y) C)) + (linear/lam (Dlinear : {y} linear ([x] M x y))) + ([x] reduce/lam (Dreduce x : {y} reduce (M x y) (N x y))) + (linear/lam ([y] Dlinear' y)) + <- ({y} {dy:of y B} srl ([x] [dx:of x A] Dof x dx y dy) (Dlinear y) ([x] Dreduce x y) + (Dlinear' y : linear ([x] N x y))). + +- : srl + ([x] [d:of x A] of/app _ (Dof x d : of (M x) (lolli B C))) + (linear/app1 (Dlinear : linear ([x] M x))) + ([x] reduce/app1 (Dreduce x : reduce (M x) (M' x))) + (linear/app1 Dlinear') + <- srl Dof Dlinear Dreduce (Dlinear' : linear ([x] M' x)). + +- : srl + _ + (linear/app2 (Dlinear : linear ([x] N x))) + ([x] reduce/app1 (Dreduce x : reduce M (M' x))) + _. + +%worlds (bind) (srl _ _ _ _). +%total D (srl _ _ D _). + +sr : of M A -> reduce M M' -> of M' A -> type. +%mode sr +D1 +D2 -D3. + +- : sr + (of/app (DofN : of N A) (DofM : of M (lolli A B))) + (reduce/app1 (Dreduce : reduce M M')) + (of/app DofN DofM') + <- sr DofM Dreduce (DofM' : of M' (lolli A B)). + +- : sr + (of/app (DofN : of N A) (DofM : of M (lolli A B))) + (reduce/app2 (Dreduce : reduce N N')) + (of/app DofN' DofM) + <- sr DofN Dreduce (DofN' : of N' A). + +- : sr + (of/app (DofN : of N A) (of/lam _ (DofM : {x} (of x A) -> of (M x) B))) + reduce/beta + (DofM N DofN). + +- : sr + (of/lam + (Dlinear : linear ([x] M x)) + (DofM : {x} (of x A) -> of (M x) B)) + (reduce/lam (Dreduce : {x} reduce (M x) (N x))) + (of/lam + _ + DofN) + <- ({x} {dx:of x A} + sr (DofM x dx) (Dreduce x) (DofN x dx : of (N x) B)). + +%worlds (bind) (sr _ _ _). +%total D1 (sr _ D1 _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090401). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009-20090406.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090406.lf new file mode 100644 index 0000000..c214ffb --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090406.lf @@ -0,0 +1,302 @@ +% +%{! Code from class, April 6. + +```twelf +tp : type. %name tp A. + +tensor : tp -> tp -> tp. +1 : tp. +with : tp -> tp -> tp. +top : tp. +plus : tp -> tp -> tp. +0 : tp. +lolli : tp -> tp -> tp. +! : tp -> tp. + +exp : type. %name exp M x. + +tens : exp -> exp -> exp. +lett : exp -> (exp -> exp -> exp) -> exp. +pair : exp -> exp -> exp. +pi1 : exp -> exp. +pi2 : exp -> exp. +star : exp. +lets : exp -> exp -> exp. +unit : exp. +in1 : exp -> exp. +in2 : exp -> exp. +case : exp -> (exp -> exp) -> (exp -> exp) -> exp. +abort : exp -> exp. +lam : tp -> (exp -> exp) -> exp. +app : exp -> exp -> exp. +bang : exp -> exp. +letb : exp -> (exp -> exp) -> exp. + +linear : (exp -> exp) -> type. +of : exp -> tp -> type. + +linear/var : linear ([x] x). + +linear/tens1 : linear ([x] tens (M x) N) + <- linear ([x] M x). + +linear/tens2 : linear ([x] tens M (N x)) + <- linear ([x] N x). + +of/tens : of (tens M N) (tensor A B) + <- of M A + <- of N B. + +linear/lett1 : linear ([z] lett (M z) ([x] [y] N x y)) + <- linear ([x] M x). + +linear/lett2 : linear ([z] lett M ([x] [y] N z x y)) + <- ({x} {y} linear ([z] N z x y)). + +of/lett : of (lett M ([x] [y] N x y)) C + <- of M (tensor A B) + <- ({x:exp} of x A -> {y:exp} of y B -> of (N x y) C) + <- ({y:exp} linear ([x] N x y)) + <- ({x:exp} linear ([y] N x y)). + +linear/pair : linear ([x] pair (M x) (N x)) + <- linear ([x] M x) + <- linear ([x] N x). + +of/pair : of (pair M N) (with A B) + <- of M A + <- of N B. + +linear/pi1 : linear ([x] pi1 (M x)) + <- linear ([x] M x). + +linear/pi2 : linear ([x] pi2 (M x)) + <- linear ([x] M x). + +of/pi1 : of (pi1 M) A + <- of M (with A B). + +of/pi2 : of (pi1 M) B + <- of M (with A B). + +of/star : of star 1. + +linear/lets1 : linear ([x] lets (M x) N) + <- linear ([x] M x). + +linear/lets2 : linear ([x] lets M (N x)) + <- linear ([x] N x). + +of/lets : of (lets M N) C + <- of M 1 + <- of N C. + +linear/unit : linear ([x] unit). + +of/unit : of unit top. + +linear/in1 : linear ([x] in1 (M x)) + <- linear ([x] M x). + +linear/in2 : linear ([x] in2 (M x)) + <- linear ([x] M x). + +of/in1 : of (in1 M) (plus A B) + <- of M A. + +of/in2 : of (in2 M) (plus A B) + <- of M B. + +linear/case1 : linear ([z] case (M z) ([x] N x) ([x] O x)) + <- linear ([z] M z). + +linear/case2 : linear ([z] case M ([x] N z x) ([x] O z x)) + <- ({x:exp} linear ([z] N z x)) + <- ({x:exp} linear ([z] O z x)). + +of/case : of (case M ([x] N x) ([x] O x)) C + <- of M (plus A B) + <- ({x} of x A -> of (N x) C) + <- ({x} of x B -> of (O x)C) + <- linear ([x] N x) + <- linear ([x] O x). + +linear/abort1 : linear ([x] abort (M x)) + <- linear ([x] M x). + +linear/abort2 : linear ([x] abort M). + +of/abort : of (abort M) C + <- of M O. + +linear/lam : linear ([x] (lam A ([y] (M x y)))) + <- {y:exp} linear ([x] M x y). + +of/lam : of (lam A ([x] M x)) (lolli A B) + <- ({x} (of x A) -> of (M x) B) + <- linear ([x] M x). + +linear/app1 : linear ([x] (app (M x) N)) + <- linear ([x] M x). + +linear/app2 : linear ([x] (app M (N x))) + <- linear ([x] N x). + +of/app : of (app M N) B + <- of M (lolli A B) + <- of N A. + +of/bang : of (bang M) (! A) + <- of M A. + +linear/letb1 : linear ([x] (letb (M x) ([y] (N y)))) + <- linear ([x] M x). + +linear/letb2 : linear ([x] (letb M ([y] N x y))) + <- ({y} linear ([x] N x y)). + +of/letb : of (letb M ([x] N x)) C + <- of M (! A) + <- ({x} (of x A) -> of (N x) C). + + +reduce : exp -> exp -> type. + +reduce/lam : reduce (lam A ([x] M x)) (lam A ([x] N x)) + <- ({x} reduce (M x) (N x)). + +reduce/app1 : reduce (app M N) (app M' N) + <- reduce M M'. + +reduce/app2 : reduce (app M N) (app M N') + <- reduce N N'. + +reduce/beta : reduce (app (lam A ([x] M x)) N) (M N). + +%block block : block {x:exp}. +%block bind : some {A:tp} block {x:exp} {d : of x A}. + +exp-eq : exp -> exp -> type. +exp-eq/i : exp-eq M M. + +exp-compat: {M : exp -> exp} exp-eq N N' -> exp-eq (M N) (M N') -> type. +%mode exp-compat +I +I1 -O. + +- : exp-compat M _ exp-eq/i. + +%worlds (block) (exp-compat _ _ _). +%total {} (exp-compat _ _ _). + +linear-resp: ({x:exp} exp-eq (M x) (N x) ) -> linear M -> linear N -> type. +%mode linear-resp +I +I2 -O. + +-: linear-resp _ D D. + +%worlds (block) (linear-resp _ _ _). +%total {} (linear-resp _ _ _). + +reduce-closed : ({x:exp} reduce M (N x)) -> ({x:exp} exp-eq N' (N x) ) -> type. +%mode reduce-closed +D -D'. + +%worlds (block) (reduce-closed _ _). +%trustme %total D (reduce-closed D _). + +linear-compose: linear ([x] M x) -> linear ([y] N y) -> linear ([y] M (N y)) -> type. +%mode linear-compose +I +I1 -O. + +-: linear-compose linear/var D D. + +-: linear-compose linear/unit _ linear/unit. + +-: linear-compose (linear/app1 (D1 :linear ([x] M1 x)) ) (D2:linear ([y] N y)) + (linear/app1 D) + <- linear-compose D1 D2 (D: linear ([y] M1 (N y))). + +%worlds (block) (linear-compose _ _ _). +%total (R) (linear-compose R _ _). + + +srl : ({x} (of x A) -> of (M x) B) -> linear ([x] M x) -> ({x} reduce (M x) (M' x)) -> linear ([x] M' x) -> type. +%mode srl +D1 +D2 +D3 -D4. + +- : srl + ([x] [d:of x A] of/lam (DDD x) (Dof x d : {y} of y B -> of (M x y) C)) + (linear/lam (Dlinear : {y} linear ([x] M x y))) + ([x] reduce/lam (Dreduce x : {y} reduce (M x y) (N x y))) + (linear/lam ([y] Dlinear' y)) + <- ({y} {dy:of y B} srl ([x] [dx:of x A] Dof x dx y dy) (Dlinear y) ([x] Dreduce x y) + (Dlinear' y : linear ([x] N x y))). + +- : srl + ([x] [d:of x A] of/app _ (Dof x d : of (M x) (lolli B C))) + (linear/app1 (Dlinear : linear ([x] M x))) + ([x] reduce/app1 (Dreduce x : reduce (M x) (M' x))) + (linear/app1 Dlinear') + <- srl Dof Dlinear Dreduce (Dlinear' : linear ([x] M' x)). + +- : srl + _ + (linear/app2 (Dlinear : linear ([x] N x))) + ([x] reduce/app1 (Dreduce x : reduce M (M' x))) + Dlinear' + <- reduce-closed Dreduce (Deq: {x} exp-eq M'' (M' x) ) + <- ({x} exp-compat ([z] app z (N x)) (Deq x) (Deq' x : exp-eq (app M'' (N x) ) (app (M' x) (N x)))) + <- linear-resp Deq' (linear/app2 Dlinear) (Dlinear' : linear ([x] app (M' x) (N x))). + +- : srl + _ + (linear/app1 (linear/lam (Dlinear:{y} linear ([x] M x y)))) + ([x] reduce/beta) + (Dlinear _). + +-: srl + ([x][d:of x A] of/app _ (of/lam (Dlinear' x : linear ([y] M y)) _ )) + (linear/app2 (Dlinear : linear ([x] N x) )) + ([x] reduce/beta) + Dlinear'' + <- linear-compose (Dlinear' unit) Dlinear (Dlinear'' : linear ([x] M (N x))). + +%worlds (bind) (srl _ _ _ _). +%trustme %total D (srl _ _ D _). + +sr : of M A -> reduce M M' -> of M' A -> type. +%mode sr +D1 +D2 -D3. + +- : sr + (of/app (DofN : of N A) (DofM : of M (lolli A B))) + (reduce/app1 (Dreduce : reduce M M')) + (of/app DofN DofM') + <- sr DofM Dreduce (DofM' : of M' (lolli A B)). + +- : sr + (of/app (DofN : of N A) (DofM : of M (lolli A B))) + (reduce/app2 (Dreduce : reduce N N')) + (of/app DofN' DofM) + <- sr DofN Dreduce (DofN' : of N' A). + +- : sr + (of/app (DofN : of N A) (of/lam _ (DofM : {x} (of x A) -> of (M x) B))) + reduce/beta + (DofM N DofN). + +- : sr + (of/lam + (Dlinear : linear ([x] M x)) + (DofM : {x} (of x A) -> of (M x) B)) + (reduce/lam (Dreduce : {x} reduce (M x) (N x))) + (of/lam + Dlinear' + DofN) + <- ({x} {dx:of x A} + sr (DofM x dx) (Dreduce x) (DofN x dx : of (N x) B)) + <- srl DofM Dlinear Dreduce (Dlinear': linear ([x] N x)). + +%worlds (bind) (sr _ _ _). +%total D1 (sr _ D1 _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090406). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009-20090408.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090408.lf new file mode 100644 index 0000000..1dbd7dd --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090408.lf @@ -0,0 +1,419 @@ +% +%{! Code from class, April 8. + +```twelf +tp : type. %name tp A. + +tensor : tp -> tp -> tp. +1 : tp. +with : tp -> tp -> tp. +top : tp. +plus : tp -> tp -> tp. +0 : tp. +lolli : tp -> tp -> tp. +! : tp -> tp. + +exp : type. %name exp M x. + +tens : exp -> exp -> exp. +lett : exp -> (exp -> exp -> exp) -> exp. +pair : exp -> exp -> exp. +pi1 : exp -> exp. +pi2 : exp -> exp. +star : exp. +lets : exp -> exp -> exp. +unit : exp. +in1 : exp -> exp. +in2 : exp -> exp. +case : exp -> (exp -> exp) -> (exp -> exp) -> exp. +abort : exp -> exp. +lam : tp -> (exp -> exp) -> exp. +app : exp -> exp -> exp. +bang : exp -> exp. +letb : exp -> (exp -> exp) -> exp. + +linear : (exp -> exp) -> type. +of : exp -> tp -> type. + +linear/var : linear ([x] x). + +linear/tens1 : linear ([x] tens (M x) N) + <- linear ([x] M x). + +linear/tens2 : linear ([x] tens M (N x)) + <- linear ([x] N x). + +of/tens : of (tens M N) (tensor A B) + <- of M A + <- of N B. + +linear/lett1 : linear ([z] lett (M z) ([x] [y] N x y)) + <- linear ([x] M x). + +linear/lett2 : linear ([z] lett M ([x] [y] N z x y)) + <- ({x} {y} linear ([z] N z x y)). + +of/lett : of (lett M ([x] [y] N x y)) C + <- of M (tensor A B) + <- ({x:exp} of x A -> {y:exp} of y B -> of (N x y) C) + <- ({y:exp} linear ([x] N x y)) + <- ({x:exp} linear ([y] N x y)). + +linear/pair : linear ([x] pair (M x) (N x)) + <- linear ([x] M x) + <- linear ([x] N x). + +of/pair : of (pair M N) (with A B) + <- of M A + <- of N B. + +linear/pi1 : linear ([x] pi1 (M x)) + <- linear ([x] M x). + +linear/pi2 : linear ([x] pi2 (M x)) + <- linear ([x] M x). + +of/pi1 : of (pi1 M) A + <- of M (with A B). + +of/pi2 : of (pi1 M) B + <- of M (with A B). + +of/star : of star 1. + +linear/lets1 : linear ([x] lets (M x) N) + <- linear ([x] M x). + +linear/lets2 : linear ([x] lets M (N x)) + <- linear ([x] N x). + +of/lets : of (lets M N) C + <- of M 1 + <- of N C. + +linear/unit : linear ([x] unit). + +of/unit : of unit top. + +linear/in1 : linear ([x] in1 (M x)) + <- linear ([x] M x). + +linear/in2 : linear ([x] in2 (M x)) + <- linear ([x] M x). + +of/in1 : of (in1 M) (plus A B) + <- of M A. + +of/in2 : of (in2 M) (plus A B) + <- of M B. + +linear/case1 : linear ([z] case (M z) ([x] N x) ([x] O x)) + <- linear ([z] M z). + +linear/case2 : linear ([z] case M ([x] N z x) ([x] O z x)) + <- ({x:exp} linear ([z] N z x)) + <- ({x:exp} linear ([z] O z x)). + +of/case : of (case M ([x] N x) ([x] O x)) C + <- of M (plus A B) + <- ({x} of x A -> of (N x) C) + <- ({x} of x B -> of (O x)C) + <- linear ([x] N x) + <- linear ([x] O x). + +linear/abort1 : linear ([x] abort (M x)) + <- linear ([x] M x). + +linear/abort2 : linear ([x] abort M). + +of/abort : of (abort M) C + <- of M O. + +linear/lam : linear ([x] (lam A ([y] (M x y)))) + <- {y:exp} linear ([x] M x y). + +of/lam : of (lam A ([x] M x)) (lolli A B) + <- ({x} (of x A) -> of (M x) B) + <- linear ([x] M x). + +linear/app1 : linear ([x] (app (M x) N)) + <- linear ([x] M x). + +linear/app2 : linear ([x] (app M (N x))) + <- linear ([x] N x). + +of/app : of (app M N) B + <- of M (lolli A B) + <- of N A. + +of/bang : of (bang M) (! A) + <- of M A. + +linear/letb1 : linear ([x] (letb (M x) ([y] (N y)))) + <- linear ([x] M x). + +linear/letb2 : linear ([x] (letb M ([y] N x y))) + <- ({y} linear ([x] N x y)). + +of/letb : of (letb M ([x] N x)) C + <- of M (! A) + <- ({x} (of x A) -> of (N x) C). + + +reduce : exp -> exp -> type. + +reduce/lam : reduce (lam A ([x] M x)) (lam A ([x] N x)) + <- ({x} reduce (M x) (N x)). + +reduce/app1 : reduce (app M N) (app M' N) + <- reduce M M'. + +reduce/app2 : reduce (app M N) (app M N') + <- reduce N N'. + +reduce/beta : reduce (app (lam A ([x] M x)) N) (M N). + +%block block : block {x:exp}. +%block bind : some {A:tp} block {x:exp} {d : of x A}. + +exp-eq : exp -> exp -> type. +exp-eq/i : exp-eq M M. + +exp-compat: {M : exp -> exp} exp-eq N N' -> exp-eq (M N) (M N') -> type. +%mode exp-compat +I +I1 -O. + +- : exp-compat M _ exp-eq/i. + +%worlds (block) (exp-compat _ _ _). +%total {} (exp-compat _ _ _). + + +lam-compat: ({x:exp} exp-eq (N x) (N' x)) -> exp-eq (lam A N) (lam A N') -> type. +%mode +{N:exp -> exp} +{N':exp -> exp} +{A:tp} +{D1:{x:exp} exp-eq (N x) (N' x)} + -{D2:exp-eq (lam A ([x:exp] N x)) (lam A ([x:exp] N' x))} (lam-compat D1 D2). + +-: lam-compat ([x] exp-eq/i) exp-eq/i. + +%worlds (block) (lam-compat _ _). +%total {} (lam-compat _ _). + +linear-resp: ({x:exp} exp-eq (M x) (N x) ) -> linear M -> linear N -> type. +%mode linear-resp +I +I2 -O. + +-: linear-resp _ D D. + +%worlds (block) (linear-resp _ _ _). +%total {} (linear-resp _ _ _). + +reduce-closed : ({x:exp} reduce M (N x)) -> ({x:exp} exp-eq N' (N x) ) -> type. +%mode reduce-closed +D -D'. + +-: reduce-closed ([x] reduce/app1 (Dreduce x : reduce M1 (N1 x)) ) Deq' + <- reduce-closed Dreduce (Deq : {x} exp-eq N1' (N1 x)) + <- ({x} exp-compat ([y] app y M2) (Deq x) (Deq' x : exp-eq (app N1' M2) (app (N1 x) M2))). + +-: reduce-closed ([x] reduce/app2 (Dreduce x :reduce M2 (N2 x))) Deq' + <- reduce-closed Dreduce (Deq : {x} exp-eq N2' (N2 x)) + <- ({x} exp-compat ([y] app M1 y) (Deq x) (Deq' x : exp-eq (app M1 N2') (app M1 (N2 x)))). + +-: reduce-closed ([x] reduce/lam (Dreduce x : {y} reduce (M y) (N x y))) Deq' + <- ({y} reduce-closed ([x] Dreduce x y) ([x] Deq x y: exp-eq (N' y) (N x y))) + <- ({x} lam-compat ([y] Deq x y) (Deq' x : exp-eq (lam A ([y] (N' y))) (lam A ([y] (N x y))))). + +-: reduce-closed ([x] reduce/beta : reduce (app (lam A ([y] M y)) N) (M N)) ([x] exp-eq/i). + +%worlds (block) (reduce-closed _ _). +%total D (reduce-closed D _). + + +linear-compose: linear ([x] M x) -> linear ([y] N y) -> linear ([y] M (N y)) -> type. +%mode linear-compose +I +I1 -O. + +-: linear-compose linear/var D D. + +-: linear-compose linear/unit _ linear/unit. + +-: linear-compose (linear/app1 (D1 :linear ([x] M1 x)) ) (D2:linear ([y] N y)) + (linear/app1 D) + <- linear-compose D1 D2 (D: linear ([y] M1 (N y))). + +-: linear-compose (linear/app2 (D1 :linear ([x] M1 x)) ) (D2 : linear ([y] N y)) + (linear/app2 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/tens1 (D1 : linear ([x] M1 x))) + (D2 : linear ([y] N y)) + (linear/tens1 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/tens2 (D1 : linear ([x] M1 x))) + (D2 : linear ([y] N y)) + (linear/tens2 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/lett1 (D1 : linear ([x] M1 x))) + (D2 : linear ([y] N y)) + (linear/lett1 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/lett2 (D1 : {x} {y} linear ([z] M1 x y z))) + (D2 : linear ([k] N k)) + (linear/lett2 D ) + <- ({x} {y} linear-compose (D1 x y) D2 (D x y)). + +-: linear-compose (linear/pair (X1 : linear ([x] M1 x)) (X2 : linear ([y] M2 y))) + D1 + (linear/pair A B) + <- linear-compose X1 D1 A + <- linear-compose X2 D1 B. +-: linear-compose (linear/pi1 (D1 : linear ([x] M1 x))) + D2 + (linear/pi1 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/pi2 (D1 : linear ([x] M1 x))) + D2 + (linear/pi2 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/lets1 (D1 : linear ([x] M1 x))) + D2 + (linear/lets1 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/lets2 (D1 : linear ([x] M1 x))) + D2 + (linear/lets2 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/in1 (D1 : linear ([x] M1 x))) + D2 + (linear/in1 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/in2 (D1 : linear ([x] M1 x))) + D2 + (linear/in2 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/case1 (D1 : linear ([x] M1 x))) + D2 + (linear/case1 D) + <- linear-compose D1 D2 D. + +-: linear-compose (linear/case2 ([x] D1 x) ([x] D2 x)) D3 (linear/case2 D4 D5) + <- ({x} linear-compose (D1 x) D3 (D4 x)) + <- ({x} linear-compose (D2 x) D3 (D5 x)). + +-: linear-compose (linear/abort1 (D1 : linear ([x] M1 x))) + D2 + (linear/abort1 D) + <- linear-compose D1 D2 D. + +-: linear-compose (linear/abort2) _ linear/abort2. + +-: linear-compose (linear/lam ([y] X1 y)) X2 (linear/lam D) + <- ({x} linear-compose (X1 x) X2 (D x)). + +-: linear-compose (linear/letb1 (D1 : linear ([x] M1 x))) + D2 + (linear/letb1 D) + <- linear-compose D1 D2 D. +-: linear-compose (linear/letb2 ([y] X1 y)) X2 (linear/letb2 D) + <- ({x} linear-compose (X1 x) X2 (D x)). + +%worlds (block) (linear-compose _ _ _). +%total (R) (linear-compose R _ _). + + +srl : ({x} (of x A) -> of (M x) B) -> linear ([x] M x) -> ({x} reduce (M x) (M' x)) -> linear ([x] M' x) -> type. +%mode srl +D1 +D2 +D3 -D4. + +- : srl + ([x] [d:of x A] of/lam (DDD x) (Dof x d : {y} of y B -> of (M x y) C)) + (linear/lam (Dlinear : {y} linear ([x] M x y))) + ([x] reduce/lam (Dreduce x : {y} reduce (M x y) (N x y))) + (linear/lam ([y] Dlinear' y)) + <- ({y} {dy:of y B} srl ([x] [dx:of x A] Dof x dx y dy) (Dlinear y) ([x] Dreduce x y) + (Dlinear' y : linear ([x] N x y))). + +- : srl + ([x] [d:of x A] of/app _ (Dof x d : of (M x) (lolli B C))) + (linear/app1 (Dlinear : linear ([x] M x))) + ([x] reduce/app1 (Dreduce x : reduce (M x) (M' x))) + (linear/app1 Dlinear') + <- srl Dof Dlinear Dreduce (Dlinear' : linear ([x] M' x)). + +- : srl + _ + (linear/app2 (Dlinear : linear ([x] N x))) + ([x] reduce/app1 (Dreduce x : reduce M (M' x))) + Dlinear' + <- reduce-closed Dreduce (Deq: {x} exp-eq M'' (M' x) ) + <- ({x} exp-compat ([z] app z (N x)) (Deq x) (Deq' x : exp-eq (app M'' (N x) ) (app (M' x) (N x)))) + <- linear-resp Deq' (linear/app2 Dlinear) (Dlinear' : linear ([x] app (M' x) (N x))). + +- : srl + _ + (linear/app1 (linear/lam (Dlinear:{y} linear ([x] M x y)))) + ([x] reduce/beta) + (Dlinear _). + +-: srl + ([x][d:of x A] of/app _ (of/lam (Dlinear' x : linear ([y] M y)) _ )) + (linear/app2 (Dlinear : linear ([x] N x) )) + ([x] reduce/beta) + Dlinear'' + <- linear-compose (Dlinear' unit) Dlinear (Dlinear'' : linear ([x] M (N x))). + +-: srl + ([x] [d : of x A] of/app (Dof x d : of (N x) B) _) + (linear/app2 (Dlinear : linear ([x] N x))) + ([x] reduce/app2 (Dreduce x : reduce (N x) (N' x))) + (linear/app2 Dlinear') + <- srl Dof Dlinear Dreduce (Dlinear' : linear ([x] N' x)). + +-: srl + _ + (linear/app1 (Dlinear : linear ([x] M x))) + ([x] reduce/app2 (Dreduce x : reduce N (N' x))) + Dlinear' + <- reduce-closed Dreduce (Deq : {x} exp-eq N'' (N' x)) + <- ({x} exp-compat ([z] app (M x) z) (Deq x) (Deq' x : exp-eq (app (M x) N'') (app (M x) (N' x)))) + <- linear-resp Deq' (linear/app1 Dlinear) (Dlinear' : linear ([x] app (M x) (N' x))). + +%worlds (bind) (srl _ _ _ _). +%total D (srl _ _ D _). + + +sr : of M A -> reduce M M' -> of M' A -> type. +%mode sr +D1 +D2 -D3. + +- : sr + (of/app (DofN : of N A) (DofM : of M (lolli A B))) + (reduce/app1 (Dreduce : reduce M M')) + (of/app DofN DofM') + <- sr DofM Dreduce (DofM' : of M' (lolli A B)). + +- : sr + (of/app (DofN : of N A) (DofM : of M (lolli A B))) + (reduce/app2 (Dreduce : reduce N N')) + (of/app DofN' DofM) + <- sr DofN Dreduce (DofN' : of N' A). + +- : sr + (of/app (DofN : of N A) (of/lam _ (DofM : {x} (of x A) -> of (M x) B))) + reduce/beta + (DofM N DofN). + +- : sr + (of/lam + (Dlinear : linear ([x] M x)) + (DofM : {x} (of x A) -> of (M x) B)) + (reduce/lam (Dreduce : {x} reduce (M x) (N x))) + (of/lam + Dlinear' + DofN) + <- ({x} {dx:of x A} + sr (DofM x dx) (Dreduce x) (DofN x dx : of (N x) B)) + <- srl DofM Dlinear Dreduce (Dlinear': linear ([x] N x)). + +%worlds (bind) (sr _ _ _). +%total D1 (sr _ D1 _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090408). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009-20090413.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090413.lf new file mode 100644 index 0000000..744c132 --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090413.lf @@ -0,0 +1,79 @@ +% +%{! Code from class, April 13. + +```twelf +tp : type. %name tp T. + +exp : type. %name exp E x. + +o : tp. +arr : tp -> tp -> tp. + +p : exp. +lam : tp -> (exp -> exp) -> exp. +app : exp -> exp -> exp. + +%% - : (exp -> tp) -> type. + +of : exp -> tp -> type. + +of/p : of p o. +of/lam : of (lam T ([x] E x)) (arr T T') + <- ({x} of x T -> of (E x) T'). +of/app : of (app E1 E2) T' + <- of E1 (arr T T') + <- of E2 T. + +%block bind : some {T : tp} block {x :exp} {dx : of x T}. + +subst : ({x} of x T1 -> of (E1 x) T2) -> of E2 T1 -> of (E1 E2) T2 -> type. +%mode subst +D1 +D2 -D3. + +%% - : subst D1 D2 (D1 _ D2). + +- : subst ([x] [d : of x T1] d) (D2 : of E2 T1) D2. + +- : subst ([x] [d : of x T1] D1) _ D1. + +- : subst + ([x] [d : of x T1] + of/app + (D12 x d : of (E12 x) T2) + (D11 x d : of (E11 x) (arr T2 T3))) + (D2 : of E2 T1) + (of/app D12' D11') + <- subst D11 D2 (D11' : of (E11 E2) (arr T2 T3)) + <- subst D12 D2 (D12' : of (E12 E2) T2). + +- : subst + ([x] [dx : of x T1] of/lam (D1 x dx : {y} of y T2 -> of (E1 x y) T3)) + (D2 : of E2 T1) + (of/lam D) + <- ({y} {dy : of y T2} + subst ([x] [dx] D1 x dx y dy) D2 (D y dy : of (E1 E2 y) T3)). + + +%worlds (bind) (subst _ _ _). +%total D (subst D _ _). + +ctx : type. %name ctx G. +nil : ctx. +cons : ctx -> exp -> tp -> ctx. + +ofe : ctx -> exp -> tp -> type. + +ofe/p : ofe G p o. + +ofe/app : ofe G (app E1 E2) T' + <- ofe G E1 (arr T T') + <- ofe G E2 T. + +ofe/lam : ofe G (lam T ([x] E x)) (arr T T') + <- ({x} ofe (cons G x T) (E x) T'). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090413). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009-20090415.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090415.lf new file mode 100644 index 0000000..2e083b9 --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090415.lf @@ -0,0 +1,176 @@ +% +%{! Code from class, April 15. + +```twelf +nat : type. %name nat N. + +0 : nat. +s : nat -> nat. + +lt : nat -> nat -> type. + +lt/0 : lt 0 (s N). + +lt/s : lt (s M) (s N) + <- lt M N. + +tp : type. %name tp T. + +exp : type. %name exp E x. + +o : tp. +arr : tp -> tp -> tp. + +p : exp. +lam : tp -> (exp -> exp) -> exp. +app : exp -> exp -> exp. + +%% - : (exp -> tp) -> type. + +of : exp -> tp -> type. + +of/p : of p o. +of/lam : of (lam T ([x] E x)) (arr T T') + <- ({x} of x T -> of (E x) T'). +of/app : of (app E1 E2) T' + <- of E1 (arr T T') + <- of E2 T. + +%block bind : some {T : tp} block {x :exp} {dx : of x T}. + +subst : ({x} of x T1 -> of (E1 x) T2) -> of E2 T1 -> of (E1 E2) T2 -> type. +%mode subst +D1 +D2 -D3. + +%% - : subst D1 D2 (D1 _ D2). + +- : subst ([x] [d : of x T1] d) (D2 : of E2 T1) D2. + +- : subst ([x] [d : of x T1] D1) _ D1. + +- : subst + ([x] [d : of x T1] + of/app + (D12 x d : of (E12 x) T2) + (D11 x d : of (E11 x) (arr T2 T3))) + (D2 : of E2 T1) + (of/app D12' D11') + <- subst D11 D2 (D11' : of (E11 E2) (arr T2 T3)) + <- subst D12 D2 (D12' : of (E12 E2) T2). + +- : subst + ([x] [dx : of x T1] of/lam (D1 x dx : {y} of y T2 -> of (E1 x y) T3)) + (D2 : of E2 T1) + (of/lam D) + <- ({y} {dy : of y T2} + subst ([x] [dx] D1 x dx y dy) D2 (D y dy : of (E1 E2 y) T3)). + + +%worlds (bind) (subst _ _ _). +%total D (subst D _ _). + +ctx : type. %name ctx G. +nil : ctx. +cons : ctx -> exp -> tp -> ctx. + +append : ctx -> ctx -> ctx -> type. + +append/nil : append G nil G. + +append/cons : append G1 (cons G2 X T) (cons G X T) + <- append G1 G2 G. + +lookup : ctx -> exp -> tp -> type. + +lookup/hit : lookup (cons G X T) X T. + +lookup/miss : lookup (cons G X' T') X T + <- lookup G X T. + +ofe : ctx -> exp -> tp -> type. + +ofe/closed : ofe G E T + <- of E T. + +ofe/var : ofe G X T + <- lookup G X T. + +ofe/p : ofe G p o = ofe/closed of/p. + +ofe/app : ofe G (app E1 E2) T' + <- ofe G E1 (arr T T') + <- ofe G E2 T. + +ofe/lam : ofe G (lam T ([x] E x)) (arr T T') + <- ({x} ofe (cons G x T) (E x) T'). + +isvar : exp -> nat -> type. + +%block ovar : some {I : nat} block {x : exp} {d : isvar x I}. + +precedes : exp -> exp -> type. + +precedes/i : precedes X Y + <- isvar X I + <- isvar Y J + <- lt I J. + +bounded : ctx -> exp -> type. + +bounded/nil : bounded nil X + <- isvar X _. + +bounded/cons : bounded (cons G X T) Y + <- bounded G X + <- precedes X Y. + +ordered : ctx -> type. + +ordered/nil : ordered nil. + +ordered/cons : ordered (cons G X T) + <- bounded G X. + + +esubst : ({x} append (cons G1 x A) G2 (G x)) + -> append G1 G2 G' + -> ({x} isvar x I -> ordered (G x)) + -> ofe G1 N A + -> ({x} ofe (G x) (M x) B) +%% + -> ofe G' (M N) B -> type. + +%mode esubst +D1 +D2 +D3 +D4 +D5 -D6. + +- : esubst + (Dappend : ({x} append (cons G1 x A) G2 (G x))) + (Dappend' : append G1 G2 G') + (Dordered : ({x} isvar x I -> ordered (G x))) + (DofeN : ofe G1 N A) + ([x] ofe/closed (DofM x : of (M x) B)) + (ofe/closed (DofM N)). + +%{ +- : esubst + (Dappend : (\{x\} append (cons G1 x A) G2 (G x))) + (Dappend' : append G1 G2 G') + (Dordered : (\{x\} isvar x I -> ordered (G x))) + (DofeN : ofe G1 N A) + ([x] ofe/lam (DofeM x : \{y\} ofe (cons (G x) y B) (M x y) C)) + _ + <- (\{y\} \{dy : isvar y J\} + esubst ([x] append/cons (Dappend x)) + (append/cons Dappend') + (ordered/cons (_ : bounded (G x) y)) + _ _ _). +}% + + +%worlds (bind) (esubst _ _ _ _ _ _). +%total D (esubst _ _ _ _ D _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090415). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009-20090420.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090420.lf new file mode 100644 index 0000000..96bedf5 --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090420.lf @@ -0,0 +1,253 @@ +% +%{! Code from class, April 20. + +```twelf +nat : type. %name nat N. + +0 : nat. +s : nat -> nat. + +lt : nat -> nat -> type. + +lt/0 : lt 0 (s N). + +lt/s : lt (s M) (s N) + <- lt M N. + +lt-succ : {N : nat} lt N (s N) -> type. +%mode lt-succ +N -D. + +%worlds () (lt-succ _ _). +%trustme %total N (lt-succ N _). + +tp : type. %name tp T. + +exp : type. %name exp E x. + +o : tp. +arr : tp -> tp -> tp. + +p : exp. +lam : tp -> (exp -> exp) -> exp. +app : exp -> exp -> exp. + +%% - : (exp -> tp) -> type. + +tp-eq : tp -> tp -> type. + +tp-eq/i : tp-eq A A. + +of : exp -> tp -> type. + +of/p : of p o. +of/lam : of (lam T ([x] E x)) (arr T T') + <- ({x} of x T -> of (E x) T'). +of/app : of (app E1 E2) T' + <- of E1 (arr T T') + <- of E2 T. + +%block bind : some {T : tp} block {x :exp} {dx : of x T}. + +subst : ({x} of x T1 -> of (E1 x) T2) -> of E2 T1 -> of (E1 E2) T2 -> type. +%mode subst +D1 +D2 -D3. + +%% - : subst D1 D2 (D1 _ D2). + +- : subst ([x] [d : of x T1] d) (D2 : of E2 T1) D2. + +- : subst ([x] [d : of x T1] D1) _ D1. + +- : subst + ([x] [d : of x T1] + of/app + (D12 x d : of (E12 x) T2) + (D11 x d : of (E11 x) (arr T2 T3))) + (D2 : of E2 T1) + (of/app D12' D11') + <- subst D11 D2 (D11' : of (E11 E2) (arr T2 T3)) + <- subst D12 D2 (D12' : of (E12 E2) T2). + +- : subst + ([x] [dx : of x T1] of/lam (D1 x dx : {y} of y T2 -> of (E1 x y) T3)) + (D2 : of E2 T1) + (of/lam D) + <- ({y} {dy : of y T2} + subst ([x] [dx] D1 x dx y dy) D2 (D y dy : of (E1 E2 y) T3)). + + +%worlds (bind) (subst _ _ _). +%total D (subst D _ _). + +ctx : type. %name ctx G. +nil : ctx. +cons : ctx -> exp -> tp -> ctx. + +append : ctx -> ctx -> ctx -> type. + +append/nil : append G nil G. + +append/cons : append G1 (cons G2 X T) (cons G X T) + <- append G1 G2 G. + +lookup : ctx -> exp -> tp -> type. + +lookup/hit : lookup (cons G X T) X T. + +lookup/miss : lookup (cons G X' T') X T + <- lookup G X T. + +ofe : ctx -> exp -> tp -> type. + +ofe/closed : ofe G E T + <- of E T. + +ofe/var : ofe G X T + <- lookup G X T. + +ofe/p : ofe G p o = ofe/closed of/p. + +ofe/app : ofe G (app E1 E2) T' + <- ofe G E1 (arr T T') + <- ofe G E2 T. + +ofe/lam : ofe G (lam T ([x] E x)) (arr T T') + <- ({x} ofe (cons G x T) (E x) T'). + +isvar : exp -> nat -> type. + +%block ovar : some {I : nat} block {x : exp} {d : isvar x I}. + +precedes : exp -> exp -> type. + +precedes/i : precedes X Y + <- isvar X I + <- isvar Y J + <- lt I J. + +bounded : ctx -> exp -> type. + +bounded/nil : bounded nil X + <- isvar X _. + +bounded/cons : bounded (cons G X T) Y + <- bounded G X + <- precedes X Y. + +ordered : ctx -> type. + +ordered/nil : ordered nil. + +ordered/cons : ordered (cons G X T) + <- bounded G X. + +ofe-resp : tp-eq A A' -> ofe G M A -> ofe G M A' -> type. +%mode ofe-resp +D1 +D2 -D3. + +- : ofe-resp _ D D. + +%worlds (ovar | bind) (ofe-resp _ _ _). +%total {} (ofe-resp _ _ _). + +bounded-isvar : bounded G X -> isvar X I -> type. +%mode bounded-isvar +D1 -D2. + +- : bounded-isvar (bounded/nil D) D. + +- : bounded-isvar (bounded/cons (precedes/i _ D _) _) D. + +%worlds (ovar | bind) (bounded-isvar _ _). +%total {} (bounded-isvar _ _). + + +ordered-extend : ordered G -> ({x} isvar x I -> bounded G x) -> type. +%mode ordered-extend +D1 -D2. + +- : ordered-extend ordered/nil ([x] [dx : isvar x 0] bounded/nil dx). + +- : ordered-extend + (ordered/cons (Dbounded : bounded G Y)) + ([x] [dx : isvar x (s J)] + bounded/cons + (precedes/i Dlt dx Disvar) + Dbounded) + <- bounded-isvar Dbounded (Disvar : isvar Y J) + <- lt-succ J (Dlt : lt J (s J)). + +%worlds (ovar | bind) (ordered-extend _ _). +%total {} (ordered-extend _ _). + +ofe-weaken : append G1 G2 G -> ofe G1 M A -> ofe G M A -> type. +%mode ofe-weaken +D1 +D2 -D3. + +%worlds (ovar | bind) (ofe-weaken _ _ _). +%trustme %total {} (ofe-weaken _ _ _). + + +append-lookup-eq : ({x} append (cons G1 x A) G2 (G x)) + -> ({x} isvar x I -> ordered (G x)) + -> ({x} lookup (G x) x B) + -> (tp-eq A B) + -> type. +%mode append-lookup-eq +D1 +D2 +D3 -D. + +%worlds (bind | ovar) (append-lookup-eq _ _ _ _). +%trustme %total D (append-lookup-eq D _ _ _). + +esubst : ({x} append (cons G1 x A) G2 (G x)) + -> append G1 G2 G' + -> ({x} isvar x I -> ordered (G x)) + -> ofe G1 N A + -> ({x} ofe (G x) (M x) B) +%% + -> ofe G' (M N) B -> type. + +%mode esubst +D1 +D2 +D3 +D4 +D5 -D6. + +- : esubst + (Dappend : ({x} append (cons G1 x A) G2 (G x))) + (Dappend' : append G1 G2 G') + (Dordered : ({x} isvar x I -> ordered (G x))) + (DofeN : ofe G1 N A) + ([x] ofe/closed (DofM x : of (M x) B)) + (ofe/closed (DofM N)). + +- : esubst + (Dappend : ({x} append (cons G1 x A) G2 (G x))) + (Dappend' : append G1 G2 G') + (Dordered : ({x} isvar x I -> ordered (G x))) + (DofeN : ofe G1 N A) + ([x] ofe/var (Dlookup x : lookup (G x) x B)) + DofeN'' + <- ofe-weaken Dappend' DofeN (DofeN' : ofe G' N A) + <- append-lookup-eq Dappend Dordered Dlookup + (Deq : tp-eq A B) + <- ofe-resp Deq DofeN' (DofeN'' : ofe G' N B). + +- : esubst + (Dappend : ({x} append (cons G1 x A) G2 (G x))) + (Dappend' : append G1 G2 G') + (Dordered : ({x} isvar x I -> ordered (G x))) + (DofeN : ofe G1 N A) + ([x] ofe/lam (DofeM x : {y} ofe (cons (G x) y B) (M x y) C)) + (ofe/lam Dofe) + <- ({x} {dx : isvar x I} ordered-extend (Dordered x dx) + (Dbounded x dx : {y} isvar y J -> bounded (G x) y)) + <- ({y} {dy : isvar y J} + esubst ([x] append/cons (Dappend x)) + (append/cons Dappend') + ([x] [dx : isvar x I] + ordered/cons (Dbounded x dx y dy: bounded (G x) y)) + DofeN + ([x] DofeM x y) + (Dofe y : ofe (cons G' y B) (M N y) C)). + + +%worlds (bind | ovar) (esubst _ _ _ _ _ _). +%total D (esubst _ _ _ _ D _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090420). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009-20090422.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090422.lf new file mode 100644 index 0000000..453275d --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090422.lf @@ -0,0 +1,325 @@ +% +%{! Code from class, April 22. + +```twelf +false : type. + +nat : type. %name nat N. + +0 : nat. +s : nat -> nat. + +lt : nat -> nat -> type. + +lt/0 : lt 0 (s N). + +lt/s : lt (s M) (s N) + <- lt M N. + +lt-succ : {N : nat} lt N (s N) -> type. +%mode lt-succ +N -D. + +%worlds () (lt-succ _ _). +%trustme %total N (lt-succ N _). + +tp : type. %name tp T. + +exp : type. %name exp E x. + +o : tp. +arr : tp -> tp -> tp. + +p : exp. +lam : tp -> (exp -> exp) -> exp. +app : exp -> exp -> exp. + +%% - : (exp -> tp) -> type. + +tp-eq : tp -> tp -> type. + +tp-eq/i : tp-eq A A. + +of : exp -> tp -> type. + +of/p : of p o. +of/lam : of (lam T ([x] E x)) (arr T T') + <- ({x} of x T -> of (E x) T'). +of/app : of (app E1 E2) T' + <- of E1 (arr T T') + <- of E2 T. + +%block bind : some {T : tp} block {x :exp} {dx : of x T}. + +subst : ({x} of x T1 -> of (E1 x) T2) -> of E2 T1 -> of (E1 E2) T2 -> type. +%mode subst +D1 +D2 -D3. + +%% - : subst D1 D2 (D1 _ D2). + +- : subst ([x] [d : of x T1] d) (D2 : of E2 T1) D2. + +- : subst ([x] [d : of x T1] D1) _ D1. + +- : subst + ([x] [d : of x T1] + of/app + (D12 x d : of (E12 x) T2) + (D11 x d : of (E11 x) (arr T2 T3))) + (D2 : of E2 T1) + (of/app D12' D11') + <- subst D11 D2 (D11' : of (E11 E2) (arr T2 T3)) + <- subst D12 D2 (D12' : of (E12 E2) T2). + +- : subst + ([x] [dx : of x T1] of/lam (D1 x dx : {y} of y T2 -> of (E1 x y) T3)) + (D2 : of E2 T1) + (of/lam D) + <- ({y} {dy : of y T2} + subst ([x] [dx] D1 x dx y dy) D2 (D y dy : of (E1 E2 y) T3)). + + +%worlds (bind) (subst _ _ _). +%total D (subst D _ _). + +ctx : type. %name ctx G. +nil : ctx. +cons : ctx -> exp -> tp -> ctx. + +append : ctx -> ctx -> ctx -> type. + +append/nil : append G nil G. + +append/cons : append G1 (cons G2 X T) (cons G X T) + <- append G1 G2 G. + +lookup : ctx -> exp -> tp -> type. + +lookup/hit : lookup (cons G X T) X T. + +lookup/miss : lookup (cons G X' T') X T + <- lookup G X T. + +ofe : ctx -> exp -> tp -> type. + +ofe/closed : ofe G E T + <- of E T. + +ofe/var : ofe G X T + <- lookup G X T. + +ofe/p : ofe G p o = ofe/closed of/p. + +ofe/app : ofe G (app E1 E2) T' + <- ofe G E1 (arr T T') + <- ofe G E2 T. + +ofe/lam : ofe G (lam T ([x] E x)) (arr T T') + <- ({x} ofe (cons G x T) (E x) T'). + +isvar : exp -> nat -> type. + +%block ovar : some {I : nat} block {x : exp} {d : isvar x I}. + +precedes : exp -> exp -> type. + +precedes/i : precedes X Y + <- isvar X I + <- isvar Y J + <- lt I J. + +bounded : ctx -> exp -> type. + +bounded/nil : bounded nil X + <- isvar X _. + +bounded/cons : bounded (cons G X T) Y + <- bounded G X + <- precedes X Y. + +ordered : ctx -> type. + +ordered/nil : ordered nil. + +ordered/cons : ordered (cons G X T) + <- bounded G X. + +ofe-resp : tp-eq A A' -> ofe G M A -> ofe G M A' -> type. +%mode ofe-resp +D1 +D2 -D3. + +- : ofe-resp _ D D. + +%worlds (ovar | bind) (ofe-resp _ _ _). +%total {} (ofe-resp _ _ _). + +bounded-isvar : bounded G X -> isvar X I -> type. +%mode bounded-isvar +D1 -D2. + +- : bounded-isvar (bounded/nil D) D. + +- : bounded-isvar (bounded/cons (precedes/i _ D _) _) D. + +%worlds (ovar | bind) (bounded-isvar _ _). +%total {} (bounded-isvar _ _). + + +ordered-extend : ordered G -> ({x} isvar x I -> bounded G x) -> type. +%mode ordered-extend +D1 -D2. + +- : ordered-extend ordered/nil ([x] [dx : isvar x 0] bounded/nil dx). + +- : ordered-extend + (ordered/cons (Dbounded : bounded G Y)) + ([x] [dx : isvar x (s J)] + bounded/cons + (precedes/i Dlt dx Disvar) + Dbounded) + <- bounded-isvar Dbounded (Disvar : isvar Y J) + <- lt-succ J (Dlt : lt J (s J)). + +%worlds (ovar | bind) (ordered-extend _ _). +%total {} (ordered-extend _ _). + +ofe-weaken : append G1 G2 G -> ofe G1 M A -> ofe G M A -> type. +%mode ofe-weaken +D1 +D2 -D3. + +%worlds (ovar | bind) (ofe-weaken _ _ _). +%trustme %total {} (ofe-weaken _ _ _). + +bounded-ordered : bounded G X -> ordered G -> type. +%mode bounded-ordered +D1 -D2. + +%worlds (ovar | bind) (bounded-ordered _ _). +%trustme %total {} (bounded-ordered _ _). + +bounded-increase : bounded G X -> precedes X Y -> bounded G Y -> type. +%mode bounded-increase +D1 +D2 -D3. + +%worlds (ovar | bind) (bounded-increase _ _ _). +%trustme %total {} (bounded-increase _ _ _). + +lt-irreflex : lt N N -> false -> type. +%mode lt-irreflex +X1 -X2. +%worlds (bind | ovar) (lt-irreflex _ _). +%trustme %total {} (lt-irreflex _ _). + + +precedes-contra : precedes X X -> false -> type. +%mode precedes-contra +D1 -D2. + +- : precedes-contra (precedes/i (Dlt : lt I I) (Disvar : isvar X I) Disvar) DFalse + <- lt-irreflex Dlt DFalse. + +%worlds (bind | ovar) (precedes-contra _ _). +%total {} (precedes-contra _ _). + +bounded-lookup-contra : bounded G X -> lookup G X A -> false -> type. +%mode bounded-lookup-contra +D1 +D2 -D3. + +- : bounded-lookup-contra + (bounded/cons (Dprecedes : precedes Y X) (Dbounded : bounded G Y)) + (lookup/miss (Dlookup : lookup G X A)) + DFalse + <- bounded-increase Dbounded Dprecedes (Dbounded' : bounded G X) + <- bounded-lookup-contra Dbounded' Dlookup DFalse. + +- : bounded-lookup-contra + (bounded/cons (Dprecedes : precedes X X) _) + (lookup/hit) + DFalse + <- precedes-contra Dprecedes DFalse. + +%worlds (ovar | bind) (bounded-lookup-contra _ _ _). +%total D (bounded-lookup-contra _ D _). + +false-implies-tp-eq : false -> {A} {B} tp-eq A B -> type. +%mode false-implies-tp-eq +X1 +X2 +X3 -X4. +%worlds (bind | ovar) (false-implies-tp-eq _ _ _ _). +%total {} (false-implies-tp-eq _ _ _ _). + +append-lookup-eq : ({x} append (cons G1 x A) G2 (G x)) + -> ({x} isvar x I -> ordered (G x)) + -> ({x} lookup (G x) x B) + -> (tp-eq A B) + -> type. +%mode append-lookup-eq +D1 +D2 +D3 -D. + +- : append-lookup-eq _ _ ([x] lookup/hit) tp-eq/i. + +- : append-lookup-eq + ([x] append/cons (Dappend x : append (cons G1 x A) G2 (G x))) + ([x] [dx : isvar x I] ordered/cons (Dbounded x dx : bounded (G x) Y)) + ([x] lookup/miss (Dlookup x : lookup (G x) x B)) + Deq + <- ({x} {dx : isvar x I} bounded-ordered (Dbounded x dx) (Dordered x dx : ordered (G x))) + <- append-lookup-eq Dappend Dordered Dlookup (Deq : tp-eq A B). + +- : append-lookup-eq + ([x] append/nil) + ([x] [dx : isvar x I] ordered/cons (Dbounded x dx : bounded G1 x)) + ([x] lookup/miss (Dlookup x : lookup G1 x B)) + Deq + <- ({x} {dx} bounded-lookup-contra (Dbounded x dx) (Dlookup x) DFalse) + <- false-implies-tp-eq DFalse _ _ Deq. + +%worlds (bind | ovar) (append-lookup-eq _ _ _ _). +%total D (append-lookup-eq D _ _ _). + +esubst : ({x} append (cons G1 x A) G2 (G x)) + -> append G1 G2 G' + -> ({x} isvar x I -> ordered (G x)) + -> ofe G1 N A + -> ({x} ofe (G x) (M x) B) +%% + -> ofe G' (M N) B -> type. + +%mode esubst +D1 +D2 +D3 +D4 +D5 -D6. + +- : esubst + (Dappend : ({x} append (cons G1 x A) G2 (G x))) + (Dappend' : append G1 G2 G') + (Dordered : ({x} isvar x I -> ordered (G x))) + (DofeN : ofe G1 N A) + ([x] ofe/closed (DofM x : of (M x) B)) + (ofe/closed (DofM N)). + +- : esubst + (Dappend : ({x} append (cons G1 x A) G2 (G x))) + (Dappend' : append G1 G2 G') + (Dordered : ({x} isvar x I -> ordered (G x))) + (DofeN : ofe G1 N A) + ([x] ofe/var (Dlookup x : lookup (G x) x B)) + DofeN'' + <- ofe-weaken Dappend' DofeN (DofeN' : ofe G' N A) + <- append-lookup-eq Dappend Dordered Dlookup + (Deq : tp-eq A B) + <- ofe-resp Deq DofeN' (DofeN'' : ofe G' N B). + +- : esubst + (Dappend : ({x} append (cons G1 x A) G2 (G x))) + (Dappend' : append G1 G2 G') + (Dordered : ({x} isvar x I -> ordered (G x))) + (DofeN : ofe G1 N A) + ([x] ofe/lam (DofeM x : {y} ofe (cons (G x) y B) (M x y) C)) + (ofe/lam Dofe) + <- ({x} {dx : isvar x I} ordered-extend (Dordered x dx) + (Dbounded x dx : {y} isvar y J -> bounded (G x) y)) + <- ({y} {dy : isvar y J} + esubst ([x] append/cons (Dappend x)) + (append/cons Dappend') + ([x] [dx : isvar x I] + ordered/cons (Dbounded x dx y dy: bounded (G x) y)) + DofeN + ([x] DofeM x y) + (Dofe y : ofe (cons G' y B) (M N y) C)). + + + + +%worlds (bind | ovar) (esubst _ _ _ _ _ _). +%total D (esubst _ _ _ _ D _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090422). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009-20090427.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090427.lf new file mode 100644 index 0000000..5917b70 --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090427.lf @@ -0,0 +1,509 @@ +% +%{! Code from class, April 27. + +```twelf +false : type. + +nat : type. %name nat N. + +0 : nat. +s : nat -> nat. + +lt : nat -> nat -> type. + +lt/0 : lt 0 (s N). + +lt/s : lt (s M) (s N) + <- lt M N. + +lt-succ : {N : nat} lt N (s N) -> type. +%mode lt-succ +N -D. + +%worlds () (lt-succ _ _). +%trustme %total N (lt-succ N _). + +tp : type. %name tp T. + +exp : type. %name exp E x. + +o : tp. +arr : tp -> tp -> tp. + +p : exp. +lam : tp -> (exp -> exp) -> exp. +app : exp -> exp -> exp. + +%% - : (exp -> tp) -> type. + +tp-eq : tp -> tp -> type. + +tp-eq/i : tp-eq A A. + +of : exp -> tp -> type. + +of/p : of p o. +of/lam : of (lam T ([x] E x)) (arr T T') + <- ({x} of x T -> of (E x) T'). +of/app : of (app E1 E2) T' + <- of E1 (arr T T') + <- of E2 T. + +%block bind : some {T : tp} block {x :exp} {dx : of x T}. + +subst : ({x} of x T1 -> of (E1 x) T2) -> of E2 T1 -> of (E1 E2) T2 -> type. +%mode subst +D1 +D2 -D3. + +%% - : subst D1 D2 (D1 _ D2). + +- : subst ([x] [d : of x T1] d) (D2 : of E2 T1) D2. + +- : subst ([x] [d : of x T1] D1) _ D1. + +- : subst + ([x] [d : of x T1] + of/app + (D12 x d : of (E12 x) T2) + (D11 x d : of (E11 x) (arr T2 T3))) + (D2 : of E2 T1) + (of/app D12' D11') + <- subst D11 D2 (D11' : of (E11 E2) (arr T2 T3)) + <- subst D12 D2 (D12' : of (E12 E2) T2). + +- : subst + ([x] [dx : of x T1] of/lam (D1 x dx : {y} of y T2 -> of (E1 x y) T3)) + (D2 : of E2 T1) + (of/lam D) + <- ({y} {dy : of y T2} + subst ([x] [dx] D1 x dx y dy) D2 (D y dy : of (E1 E2 y) T3)). + + +%worlds (bind) (subst _ _ _). +%total D (subst D _ _). + +ctx : type. %name ctx G. +nil : ctx. +cons : ctx -> exp -> tp -> ctx. + +append : ctx -> ctx -> ctx -> type. + +append/nil : append G nil G. + +append/cons : append G1 (cons G2 X T) (cons G X T) + <- append G1 G2 G. + +lookup : ctx -> exp -> tp -> type. + +lookup/hit : lookup (cons G X T) X T. + +lookup/miss : lookup (cons G X' T') X T + <- lookup G X T. + +ofe : ctx -> exp -> tp -> type. + +ofe/closed : ofe G E T + <- of E T. + +ofe/var : ofe G X T + <- lookup G X T. + +ofe/p : ofe G p o = ofe/closed of/p. + +ofe/app : ofe G (app E1 E2) T' + <- ofe G E1 (arr T T') + <- ofe G E2 T. + +ofe/lam : ofe G (lam T ([x] E x)) (arr T T') + <- ({x} ofe (cons G x T) (E x) T'). + +isvar : exp -> nat -> type. + +- : (isvar _ _ -> isvar _ _) -> type. + +%block ovar : some {I : nat} block {x : exp} {d : isvar x I}. + +precedes : exp -> exp -> type. + +precedes/i : precedes X Y + <- isvar X I + <- isvar Y J + <- lt I J. + +bounded : ctx -> exp -> type. + +bounded/nil : bounded nil X + <- isvar X _. + +bounded/cons : bounded (cons G X T) Y + <- bounded G X + <- precedes X Y. + +ordered : ctx -> type. + +ordered/nil : ordered nil. + +ordered/cons : ordered (cons G X T) + <- bounded G X. + +ofe-resp : tp-eq A A' -> ofe G M A -> ofe G M A' -> type. +%mode ofe-resp +D1 +D2 -D3. + +- : ofe-resp _ D D. + +%worlds (ovar | bind) (ofe-resp _ _ _). +%total {} (ofe-resp _ _ _). + +bounded-isvar : bounded G X -> isvar X I -> type. +%mode bounded-isvar +D1 -D2. + +- : bounded-isvar (bounded/nil D) D. + +- : bounded-isvar (bounded/cons (precedes/i _ D _) _) D. + +%worlds (ovar | bind) (bounded-isvar _ _). +%total {} (bounded-isvar _ _). + + +ordered-extend : ordered G -> ({x} isvar x I -> bounded G x) -> type. +%mode ordered-extend +D1 -D2. + +- : ordered-extend ordered/nil ([x] [dx : isvar x 0] bounded/nil dx). + +- : ordered-extend + (ordered/cons (Dbounded : bounded G Y)) + ([x] [dx : isvar x (s J)] + bounded/cons + (precedes/i Dlt dx Disvar) + Dbounded) + <- bounded-isvar Dbounded (Disvar : isvar Y J) + <- lt-succ J (Dlt : lt J (s J)). + +%worlds (ovar | bind) (ordered-extend _ _). +%total {} (ordered-extend _ _). + +ofe-weaken : append G1 G2 G -> ofe G1 M A -> ofe G M A -> type. +%mode ofe-weaken +D1 +D2 -D3. + +%worlds (ovar | bind) (ofe-weaken _ _ _). +%trustme %total {} (ofe-weaken _ _ _). + +bounded-ordered : bounded G X -> ordered G -> type. +%mode bounded-ordered +D1 -D2. + +%worlds (ovar | bind) (bounded-ordered _ _). +%trustme %total {} (bounded-ordered _ _). + +bounded-increase : bounded G X -> precedes X Y -> bounded G Y -> type. +%mode bounded-increase +D1 +D2 -D3. + +%worlds (ovar | bind) (bounded-increase _ _ _). +%trustme %total {} (bounded-increase _ _ _). + +lt-irreflex : lt N N -> false -> type. +%mode lt-irreflex +X1 -X2. +%worlds (bind | ovar) (lt-irreflex _ _). +%trustme %total {} (lt-irreflex _ _). + + +precedes-contra : precedes X X -> false -> type. +%mode precedes-contra +D1 -D2. + +- : precedes-contra (precedes/i (Dlt : lt I I) (Disvar : isvar X I) Disvar) DFalse + <- lt-irreflex Dlt DFalse. + +%worlds (bind | ovar) (precedes-contra _ _). +%total {} (precedes-contra _ _). + +bounded-lookup-contra : bounded G X -> lookup G X A -> false -> type. +%mode bounded-lookup-contra +D1 +D2 -D3. + +- : bounded-lookup-contra + (bounded/cons (Dprecedes : precedes Y X) (Dbounded : bounded G Y)) + (lookup/miss (Dlookup : lookup G X A)) + DFalse + <- bounded-increase Dbounded Dprecedes (Dbounded' : bounded G X) + <- bounded-lookup-contra Dbounded' Dlookup DFalse. + +- : bounded-lookup-contra + (bounded/cons (Dprecedes : precedes X X) _) + (lookup/hit) + DFalse + <- precedes-contra Dprecedes DFalse. + +%worlds (ovar | bind) (bounded-lookup-contra _ _ _). +%total D (bounded-lookup-contra _ D _). + +false-implies-tp-eq : false -> {A} {B} tp-eq A B -> type. +%mode false-implies-tp-eq +X1 +X2 +X3 -X4. +%worlds (bind | ovar) (false-implies-tp-eq _ _ _ _). +%total {} (false-implies-tp-eq _ _ _ _). + +false-implies-ofe : false -> {G} {M} {A} ofe G M A -> type. +%mode false-implies-ofe +X1 +X2 +X3 +X4 -X5. +%worlds (bind | ovar) (false-implies-ofe _ _ _ _ _). +%total {} (false-implies-ofe _ _ _ _ _). + +append-lookup-eq : ({x} append (cons G1 x A) G2 (G x)) + -> ({x} isvar x I -> ordered (G x)) + -> ({x} lookup (G x) x B) + -> (tp-eq A B) + -> type. +%mode append-lookup-eq +D1 +D2 +D3 -D. + +- : append-lookup-eq _ _ ([x] lookup/hit) tp-eq/i. + +- : append-lookup-eq + ([x] append/cons (Dappend x : append (cons G1 x A) G2 (G x))) + ([x] [dx : isvar x I] ordered/cons (Dbounded x dx : bounded (G x) Y)) + ([x] lookup/miss (Dlookup x : lookup (G x) x B)) + Deq + <- ({x} {dx : isvar x I} bounded-ordered (Dbounded x dx) (Dordered x dx : ordered (G x))) + <- append-lookup-eq Dappend Dordered Dlookup (Deq : tp-eq A B). + +- : append-lookup-eq + ([x] append/nil) + ([x] [dx : isvar x I] ordered/cons (Dbounded x dx : bounded G1 x)) + ([x] lookup/miss (Dlookup x : lookup G1 x B)) + Deq + <- ({x} {dx} bounded-lookup-contra (Dbounded x dx) (Dlookup x) DFalse) + <- false-implies-tp-eq DFalse _ _ Deq. + +%worlds (bind | ovar) (append-lookup-eq _ _ _ _). +%total D (append-lookup-eq D _ _ _). + +lookup-pdv : ({x} append (cons G1 x A) G2 (G x)) + -> append G1 G2 G' + -> ({x} isvar x I -> ordered (G x)) %% Needed? + -> ({x} lookup (G x) Y B) + -> lookup G' Y B + -> type. +%mode lookup-pdv +D1 +D2 +D3 +D4 -D5. + +%worlds (bind | ovar) (lookup-pdv _ _ _ _ _). +%trustme %total {} (lookup-pdv _ _ _ _ _). + +lookup-ordered : ordered G + -> lookup G X A + -> isvar X I + -> type. +%mode lookup-ordered +D1 +D2 -D3. + +%worlds (bind | ovar) (lookup-ordered _ _ _). +%trustme %total {} (lookup-ordered _ _ _). + +lam-isvar-contra : isvar (lam A M) _ + -> false + -> type. +%mode lam-isvar-contra +D1 -D2. + +%worlds (bind | ovar) (lam-isvar-contra _ _). +%total {} (lam-isvar-contra _ _). + +app-isvar-contra : isvar (app N M) _ + -> false + -> type. +%mode app-isvar-contra +D1 -D2. + +%worlds (bind | ovar) (app-isvar-contra _ _). +%total {} (app-isvar-contra _ _). + + +esubst : ({x} append (cons G1 x A) G2 (G x)) + -> append G1 G2 G' + -> ({x} isvar x I -> ordered (G x)) + -> ofe G1 N A + -> ({x} ofe (G x) (M x) B) +%% + -> ofe G' (M N) B -> type. + +%mode esubst +D1 +D2 +D3 +D4 +D5 -D6. + +- : esubst + (Dappend : ({x} append (cons G1 x A) G2 (G x))) + (Dappend' : append G1 G2 G') + (Dordered : ({x} isvar x I -> ordered (G x))) + (DofeN : ofe G1 N A) + ([x] ofe/closed (DofM x : of (M x) B)) + (ofe/closed (DofM N)). + +- : esubst + (Dappend : ({x} append (cons G1 x A) G2 (G x))) + (Dappend' : append G1 G2 G') + (Dordered : ({x} isvar x I -> ordered (G x))) + (DofeN : ofe G1 N A) + ([x] ofe/var (Dlookup x : lookup (G x) x B)) + DofeN'' + <- ofe-weaken Dappend' DofeN (DofeN' : ofe G' N A) + <- append-lookup-eq Dappend Dordered Dlookup + (Deq : tp-eq A B) + <- ofe-resp Deq DofeN' (DofeN'' : ofe G' N B). + +- : esubst + (Dappend : ({x} append (cons G1 x A) G2 (G x))) + (Dappend' : append G1 G2 G') + (Dordered : ({x} isvar x I -> ordered (G x))) + (DofeN : ofe G1 N A) + ([x] ofe/var (Dlookup x : lookup (G x) Y B)) + (ofe/var Dlookup') + <- lookup-pdv Dappend Dappend' Dordered Dlookup + (Dlookup' : lookup G' Y B). + +- : esubst + (Dappend : ({x} append (cons G1 x A) G2 (G x))) + (Dappend' : append G1 G2 G') + (Dordered : ({x} isvar x I -> ordered (G x))) + (DofeN : ofe G1 N A) + ([x] ofe/var (Dlookup x : lookup (G x) (lam C (M x)) B)) + Dofe + <- ({x} {dx : isvar x I} + lookup-ordered (Dordered x dx) (Dlookup x) + (Disvar x dx : isvar (lam C (M x)) J)) + <- ({x} {dx : isvar x I} lam-isvar-contra (Disvar x dx) Dfalse) + <- false-implies-ofe Dfalse _ _ _ Dofe. + +- : esubst + (Dappend : ({x} append (cons G1 x A) G2 (G x))) + (Dappend' : append G1 G2 G') + (Dordered : ({x} isvar x I -> ordered (G x))) + (DofeN : ofe G1 N A) + ([x] ofe/var (Dlookup x : lookup (G x) (app (M' x) (M x)) B)) + Dofe + <- ({x} {dx : isvar x I} + lookup-ordered (Dordered x dx) (Dlookup x) + (Disvar x dx : isvar (app (M' x) (M x)) J)) + <- ({x} {dx : isvar x I} app-isvar-contra (Disvar x dx) Dfalse) + <- false-implies-ofe Dfalse _ _ _ Dofe. + +- : esubst + (Dappend : ({x} append (cons G1 x A) G2 (G x))) + (Dappend' : append G1 G2 G') + (Dordered : ({x} isvar x I -> ordered (G x))) + (DofeN : ofe G1 N A) + ([x] ofe/lam (DofeM x : {y} ofe (cons (G x) y B) (M x y) C)) + (ofe/lam Dofe) + <- ({x} {dx : isvar x I} ordered-extend (Dordered x dx) + (Dbounded x dx : {y} isvar y J -> bounded (G x) y)) + <- ({y} {dy : isvar y J} + esubst ([x] append/cons (Dappend x)) + (append/cons Dappend') + ([x] [dx : isvar x I] + ordered/cons (Dbounded x dx y dy: bounded (G x) y)) + DofeN + ([x] DofeM x y) + (Dofe y : ofe (cons G' y B) (M N y) C)). + + + + +%worlds (bind | ovar) (esubst _ _ _ _ _ _). +%trustme %total D (esubst _ _ _ _ D _). + +ofi : ctx -> exp -> tp -> type. + +ofi/nil : ofi nil M A + <- of M A. + +ofi/cons : ofi (cons G X A) M B + <- ((of X A) -> ofi G M B). + +%block of-block : some {X:exp} {A:tp} block {dx : of X A}. + +ofi-app : ofi G E1 (arr T T') + -> ofi G E2 T + -> ofi G (app E1 E2) T' + -> type. +%mode ofi-app +D1 +D2 -D3. + +- : ofi-app + (ofi/nil (Dof1 : of E1 (arr T T'))) + (ofi/nil (Dof2 : of E2 T)) + (ofi/nil (of/app Dof2 Dof1)). + +- : ofi-app + (ofi/cons (Dof1 : of X A -> ofi G E1 (arr T T'))) + (ofi/cons (Dof2 : of X A -> ofi G E2 T)) + (ofi/cons Dof) + <- ({d : of X A} ofi-app (Dof1 d) (Dof2 d) (Dof d : ofi G (app E1 E2) T')). + +%worlds (bind | of-block | ovar) (ofi-app _ _ _). +%total D (ofi-app D _ _). + +ofe-implies-ofi : ofe G M A + -> ofi G M A + -> type. +%mode ofe-implies-ofi +D1 -D2. + +- : ofe-implies-ofi + (ofe/app (Dof2 : ofe G E2 T) (Dof1 : ofe G E1 (arr T T'))) + Dofi + <- ofe-implies-ofi Dof1 Dofi1 + <- ofe-implies-ofi Dof2 Dofi2 + <- ofi-app Dofi1 Dofi2 (Dofi : ofi G (app E1 E2) T'). + +%worlds (bind | ovar) (ofe-implies-ofi _ _). +%trustme %total D (ofe-implies-ofi D _). + +ofe-implies-of : ofe nil M A + -> of M A + -> type. +%mode ofe-implies-of +D1 -D2. + +- : ofe-implies-of + (Dofe : ofe nil M A) + Dof + <- ofe-implies-ofi Dofe (ofi/nil (Dof : of M A)). + +%worlds (bind) (ofe-implies-of _ _). +%total {} (ofe-implies-of _ _). + +cut-of : {M} + ({x} of x A -> of (M x) B) + -> ({x} lookup (G x) x A) + -> ({x} ofe (G x) (M x) B) + -> type. +%mode cut-of +D1 +D2 +D3 -D4. + +%worlds (bind | ovar) (cut-of _ _ _ _). +%trustme %total {} (cut-of _ _ _ _). + +of1-implies-ofe : ({x} of x A -> of (M x) B) + -> ({x} ofe (cons nil x A) (M x) B) + -> type. +%mode of1-implies-ofe +D1 -D2. + +- : of1-implies-ofe + (Dof : {x} of x A -> of (M x) B) + Dofe + <- cut-of M Dof ([x] lookup/hit) (Dofe : {x} ofe (cons nil x A) (M x) B). + +%worlds (bind) (of1-implies-ofe _ _). +%total {} (of1-implies-ofe _ _). + +subst : ({x} of x T1 -> of (E1 x) T2) + -> of E2 T1 + -> of (E1 E2) T2 + -> type. +%mode subst +D1 +D2 -D3. + +- : subst + (Dof1 : {x} of x T1 -> of (E1 x) T2) + (Dof2 : of E2 T1) + Dof + <- of1-implies-ofe Dof1 (Dofe1 : {x} ofe (cons nil x T1) (E1 x) T2) + <- esubst + ([x] append/nil) + append/nil + ([x] [dx : isvar x 0] ordered/cons (bounded/nil dx)) + (ofe/closed Dof2) + Dofe1 + (Dofe : ofe nil (E1 E2) T2) + <- ofe-implies-of Dofe (Dof : of (E1 E2) T2). + +%worlds (bind) (subst _ _ _). +%total {} (subst _ _ _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090427). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009-20090429.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090429.lf new file mode 100644 index 0000000..5ade763 --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009-20090429.lf @@ -0,0 +1,595 @@ +% +%{! Code from class, April 29. + +```twelf +false : type. + +nat : type. %name nat N. + +0 : nat. +s : nat -> nat. + +lt : nat -> nat -> type. + +lt/0 : lt 0 (s N). + +lt/s : lt (s M) (s N) + <- lt M N. + +lt-succ : {N : nat} lt N (s N) -> type. +%mode lt-succ +N -D. + +%worlds () (lt-succ _ _). +%trustme %total N (lt-succ N _). + +tp : type. %name tp T. + +exp : type. %name exp E x. + +o : tp. +arr : tp -> tp -> tp. + +p : exp. +lam : tp -> (exp -> exp) -> exp. +app : exp -> exp -> exp. + +%% - : (exp -> tp) -> type. + +tp-eq : tp -> tp -> type. + +tp-eq/i : tp-eq A A. + +of : exp -> tp -> type. + +of/p : of p o. +of/lam : of (lam T ([x] E x)) (arr T T') + <- ({x} of x T -> of (E x) T'). +of/app : of (app E1 E2) T' + <- of E1 (arr T T') + <- of E2 T. + +%block bind : some {T : tp} block {x :exp} {dx : of x T}. + +subst : ({x} of x T1 -> of (E1 x) T2) -> of E2 T1 -> of (E1 E2) T2 -> type. +%mode subst +D1 +D2 -D3. + +%% - : subst D1 D2 (D1 _ D2). + +- : subst ([x] [d : of x T1] d) (D2 : of E2 T1) D2. + +- : subst ([x] [d : of x T1] D1) _ D1. + +- : subst + ([x] [d : of x T1] + of/app + (D12 x d : of (E12 x) T2) + (D11 x d : of (E11 x) (arr T2 T3))) + (D2 : of E2 T1) + (of/app D12' D11') + <- subst D11 D2 (D11' : of (E11 E2) (arr T2 T3)) + <- subst D12 D2 (D12' : of (E12 E2) T2). + +- : subst + ([x] [dx : of x T1] of/lam (D1 x dx : {y} of y T2 -> of (E1 x y) T3)) + (D2 : of E2 T1) + (of/lam D) + <- ({y} {dy : of y T2} + subst ([x] [dx] D1 x dx y dy) D2 (D y dy : of (E1 E2 y) T3)). + + +%worlds (bind) (subst _ _ _). +%total D (subst D _ _). + +ctx : type. %name ctx G. +nil : ctx. +cons : ctx -> exp -> tp -> ctx. + +append : ctx -> ctx -> ctx -> type. + +append/nil : append G nil G. + +append/cons : append G1 (cons G2 X T) (cons G X T) + <- append G1 G2 G. + +lookup : ctx -> exp -> tp -> type. + +lookup/hit : lookup (cons G X T) X T. + +lookup/miss : lookup (cons G X' T') X T + <- lookup G X T. + +ofe : ctx -> exp -> tp -> type. + +ofe/closed : ofe G E T + <- of E T. + +ofe/var : ofe G X T + <- lookup G X T. + +ofe/p : ofe G p o = ofe/closed of/p. + +ofe/app : ofe G (app E1 E2) T' + <- ofe G E1 (arr T T') + <- ofe G E2 T. + +ofe/lam : ofe G (lam T ([x] E x)) (arr T T') + <- ({x} ofe (cons G x T) (E x) T'). + +isvar : exp -> nat -> type. + +- : (isvar _ _ -> isvar _ _) -> type. + +%block ovar : some {I : nat} block {x : exp} {d : isvar x I}. + +precedes : exp -> exp -> type. + +precedes/i : precedes X Y + <- isvar X I + <- isvar Y J + <- lt I J. + +bounded : ctx -> exp -> type. + +bounded/nil : bounded nil X + <- isvar X _. + +bounded/cons : bounded (cons G X T) Y + <- bounded G X + <- precedes X Y. + +ordered : ctx -> type. + +ordered/nil : ordered nil. + +ordered/cons : ordered (cons G X T) + <- bounded G X. + +ofe-resp : tp-eq A A' -> ofe G M A -> ofe G M A' -> type. +%mode ofe-resp +D1 +D2 -D3. + +- : ofe-resp _ D D. + +%worlds (ovar | bind) (ofe-resp _ _ _). +%total {} (ofe-resp _ _ _). + +bounded-isvar : bounded G X -> isvar X I -> type. +%mode bounded-isvar +D1 -D2. + +- : bounded-isvar (bounded/nil D) D. + +- : bounded-isvar (bounded/cons (precedes/i _ D _) _) D. + +%worlds (ovar | bind) (bounded-isvar _ _). +%total {} (bounded-isvar _ _). + + +ordered-extend : ordered G -> ({x} isvar x I -> bounded G x) -> type. +%mode ordered-extend +D1 -D2. + +- : ordered-extend ordered/nil ([x] [dx : isvar x 0] bounded/nil dx). + +- : ordered-extend + (ordered/cons (Dbounded : bounded G Y)) + ([x] [dx : isvar x (s J)] + bounded/cons + (precedes/i Dlt dx Disvar) + Dbounded) + <- bounded-isvar Dbounded (Disvar : isvar Y J) + <- lt-succ J (Dlt : lt J (s J)). + +%worlds (ovar | bind) (ordered-extend _ _). +%total {} (ordered-extend _ _). + +ofe-weaken : append G1 G2 G -> ofe G1 M A -> ofe G M A -> type. +%mode ofe-weaken +D1 +D2 -D3. + +%worlds (ovar | bind) (ofe-weaken _ _ _). +%trustme %total {} (ofe-weaken _ _ _). + +bounded-ordered : bounded G X -> ordered G -> type. +%mode bounded-ordered +D1 -D2. + +%worlds (ovar | bind) (bounded-ordered _ _). +%trustme %total {} (bounded-ordered _ _). + +bounded-increase : bounded G X -> precedes X Y -> bounded G Y -> type. +%mode bounded-increase +D1 +D2 -D3. + +%worlds (ovar | bind) (bounded-increase _ _ _). +%trustme %total {} (bounded-increase _ _ _). + +lt-irreflex : lt N N -> false -> type. +%mode lt-irreflex +X1 -X2. +%worlds (bind | ovar) (lt-irreflex _ _). +%trustme %total {} (lt-irreflex _ _). + + +precedes-contra : precedes X X -> false -> type. +%mode precedes-contra +D1 -D2. + +- : precedes-contra (precedes/i (Dlt : lt I I) (Disvar : isvar X I) Disvar) DFalse + <- lt-irreflex Dlt DFalse. + +%worlds (bind | ovar) (precedes-contra _ _). +%total {} (precedes-contra _ _). + +bounded-lookup-contra : bounded G X -> lookup G X A -> false -> type. +%mode bounded-lookup-contra +D1 +D2 -D3. + +- : bounded-lookup-contra + (bounded/cons (Dprecedes : precedes Y X) (Dbounded : bounded G Y)) + (lookup/miss (Dlookup : lookup G X A)) + DFalse + <- bounded-increase Dbounded Dprecedes (Dbounded' : bounded G X) + <- bounded-lookup-contra Dbounded' Dlookup DFalse. + +- : bounded-lookup-contra + (bounded/cons (Dprecedes : precedes X X) _) + (lookup/hit) + DFalse + <- precedes-contra Dprecedes DFalse. + +%worlds (ovar | bind) (bounded-lookup-contra _ _ _). +%total D (bounded-lookup-contra _ D _). + +false-implies-tp-eq : false -> {A} {B} tp-eq A B -> type. +%mode false-implies-tp-eq +X1 +X2 +X3 -X4. +%worlds (bind | ovar) (false-implies-tp-eq _ _ _ _). +%total {} (false-implies-tp-eq _ _ _ _). + +false-implies-ofe : false -> {G} {M} {A} ofe G M A -> type. +%mode false-implies-ofe +X1 +X2 +X3 +X4 -X5. +%worlds (bind | ovar) (false-implies-ofe _ _ _ _ _). +%total {} (false-implies-ofe _ _ _ _ _). + +append-lookup-eq : ({x} append (cons G1 x A) G2 (G x)) + -> ({x} isvar x I -> ordered (G x)) + -> ({x} lookup (G x) x B) + -> (tp-eq A B) + -> type. +%mode append-lookup-eq +D1 +D2 +D3 -D. + +- : append-lookup-eq _ _ ([x] lookup/hit) tp-eq/i. + +- : append-lookup-eq + ([x] append/cons (Dappend x : append (cons G1 x A) G2 (G x))) + ([x] [dx : isvar x I] ordered/cons (Dbounded x dx : bounded (G x) Y)) + ([x] lookup/miss (Dlookup x : lookup (G x) x B)) + Deq + <- ({x} {dx : isvar x I} bounded-ordered (Dbounded x dx) (Dordered x dx : ordered (G x))) + <- append-lookup-eq Dappend Dordered Dlookup (Deq : tp-eq A B). + +- : append-lookup-eq + ([x] append/nil) + ([x] [dx : isvar x I] ordered/cons (Dbounded x dx : bounded G1 x)) + ([x] lookup/miss (Dlookup x : lookup G1 x B)) + Deq + <- ({x} {dx} bounded-lookup-contra (Dbounded x dx) (Dlookup x) DFalse) + <- false-implies-tp-eq DFalse _ _ Deq. + +%worlds (bind | ovar) (append-lookup-eq _ _ _ _). +%total D (append-lookup-eq D _ _ _). + +lookup-pdv : ({x} append (cons G1 x A) G2 (G x)) + -> append G1 G2 G' + -> ({x} isvar x I -> ordered (G x)) %% Needed? + -> ({x} lookup (G x) Y B) + -> lookup G' Y B + -> type. +%mode lookup-pdv +D1 +D2 +D3 +D4 -D5. + +%worlds (bind | ovar) (lookup-pdv _ _ _ _ _). +%trustme %total {} (lookup-pdv _ _ _ _ _). + +lookup-ordered : ordered G + -> lookup G X A + -> isvar X I + -> type. +%mode lookup-ordered +D1 +D2 -D3. + +%worlds (bind | ovar) (lookup-ordered _ _ _). +%trustme %total {} (lookup-ordered _ _ _). + +lam-isvar-contra : isvar (lam A M) _ + -> false + -> type. +%mode lam-isvar-contra +D1 -D2. + +%worlds (bind | ovar) (lam-isvar-contra _ _). +%total {} (lam-isvar-contra _ _). + +app-isvar-contra : isvar (app N M) _ + -> false + -> type. +%mode app-isvar-contra +D1 -D2. + +%worlds (bind | ovar) (app-isvar-contra _ _). +%total {} (app-isvar-contra _ _). + + +esubst : ({x} append (cons G1 x A) G2 (G x)) + -> append G1 G2 G' + -> ({x} isvar x I -> ordered (G x)) + -> ofe G1 N A + -> ({x} ofe (G x) (M x) B) +%% + -> ofe G' (M N) B -> type. + +%mode esubst +D1 +D2 +D3 +D4 +D5 -D6. + +- : esubst + (Dappend : ({x} append (cons G1 x A) G2 (G x))) + (Dappend' : append G1 G2 G') + (Dordered : ({x} isvar x I -> ordered (G x))) + (DofeN : ofe G1 N A) + ([x] ofe/closed (DofM x : of (M x) B)) + (ofe/closed (DofM N)). + +- : esubst + (Dappend : ({x} append (cons G1 x A) G2 (G x))) + (Dappend' : append G1 G2 G') + (Dordered : ({x} isvar x I -> ordered (G x))) + (DofeN : ofe G1 N A) + ([x] ofe/var (Dlookup x : lookup (G x) x B)) + DofeN'' + <- ofe-weaken Dappend' DofeN (DofeN' : ofe G' N A) + <- append-lookup-eq Dappend Dordered Dlookup + (Deq : tp-eq A B) + <- ofe-resp Deq DofeN' (DofeN'' : ofe G' N B). + +- : esubst + (Dappend : ({x} append (cons G1 x A) G2 (G x))) + (Dappend' : append G1 G2 G') + (Dordered : ({x} isvar x I -> ordered (G x))) + (DofeN : ofe G1 N A) + ([x] ofe/var (Dlookup x : lookup (G x) Y B)) + (ofe/var Dlookup') + <- lookup-pdv Dappend Dappend' Dordered Dlookup + (Dlookup' : lookup G' Y B). + +- : esubst + (Dappend : ({x} append (cons G1 x A) G2 (G x))) + (Dappend' : append G1 G2 G') + (Dordered : ({x} isvar x I -> ordered (G x))) + (DofeN : ofe G1 N A) + ([x] ofe/var (Dlookup x : lookup (G x) (lam C (M x)) B)) + Dofe + <- ({x} {dx : isvar x I} + lookup-ordered (Dordered x dx) (Dlookup x) + (Disvar x dx : isvar (lam C (M x)) J)) + <- ({x} {dx : isvar x I} lam-isvar-contra (Disvar x dx) Dfalse) + <- false-implies-ofe Dfalse _ _ _ Dofe. + +- : esubst + (Dappend : ({x} append (cons G1 x A) G2 (G x))) + (Dappend' : append G1 G2 G') + (Dordered : ({x} isvar x I -> ordered (G x))) + (DofeN : ofe G1 N A) + ([x] ofe/var (Dlookup x : lookup (G x) (app (M' x) (M x)) B)) + Dofe + <- ({x} {dx : isvar x I} + lookup-ordered (Dordered x dx) (Dlookup x) + (Disvar x dx : isvar (app (M' x) (M x)) J)) + <- ({x} {dx : isvar x I} app-isvar-contra (Disvar x dx) Dfalse) + <- false-implies-ofe Dfalse _ _ _ Dofe. + +- : esubst + (Dappend : ({x} append (cons G1 x A) G2 (G x))) + (Dappend' : append G1 G2 G') + (Dordered : ({x} isvar x I -> ordered (G x))) + (DofeN : ofe G1 N A) + ([x] ofe/lam (DofeM x : {y} ofe (cons (G x) y B) (M x y) C)) + (ofe/lam Dofe) + <- ({x} {dx : isvar x I} ordered-extend (Dordered x dx) + (Dbounded x dx : {y} isvar y J -> bounded (G x) y)) + <- ({y} {dy : isvar y J} + esubst ([x] append/cons (Dappend x)) + (append/cons Dappend') + ([x] [dx : isvar x I] + ordered/cons (Dbounded x dx y dy: bounded (G x) y)) + DofeN + ([x] DofeM x y) + (Dofe y : ofe (cons G' y B) (M N y) C)). + +%worlds (bind | ovar) (esubst _ _ _ _ _ _). +%trustme %total D (esubst _ _ _ _ D _). + +ofi : ctx -> exp -> tp -> type. + +ofi/nil : ofi nil M A + <- of M A. + +ofi/cons : ofi (cons G X A) M B + <- ((of X A) -> ofi G M B). + +%block of-block : some {X:exp} {A:tp} block {dx : of X A}. + +ofi-app : ofi G E1 (arr T T') + -> ofi G E2 T + -> ofi G (app E1 E2) T' + -> type. +%mode ofi-app +D1 +D2 -D3. + +- : ofi-app + (ofi/nil (Dof1 : of E1 (arr T T'))) + (ofi/nil (Dof2 : of E2 T)) + (ofi/nil (of/app Dof2 Dof1)). + +- : ofi-app + (ofi/cons (Dof1 : of X A -> ofi G E1 (arr T T'))) + (ofi/cons (Dof2 : of X A -> ofi G E2 T)) + (ofi/cons Dof) + <- ({d : of X A} ofi-app (Dof1 d) (Dof2 d) (Dof d : ofi G (app E1 E2) T')). + +%worlds (bind | of-block | ovar) (ofi-app _ _ _). +%total D (ofi-app D _ _). + +ofe-implies-ofi : ofe G M A + -> ofi G M A + -> type. +%mode ofe-implies-ofi +D1 -D2. + +- : ofe-implies-ofi + (ofe/app (Dof2 : ofe G E2 T) (Dof1 : ofe G E1 (arr T T'))) + Dofi + <- ofe-implies-ofi Dof1 Dofi1 + <- ofe-implies-ofi Dof2 Dofi2 + <- ofi-app Dofi1 Dofi2 (Dofi : ofi G (app E1 E2) T'). + +%worlds (bind | ovar) (ofe-implies-ofi _ _). +%trustme %total D (ofe-implies-ofi D _). + +ofe-implies-of : ofe nil M A + -> of M A + -> type. +%mode ofe-implies-of +D1 -D2. + +- : ofe-implies-of + (Dofe : ofe nil M A) + Dof + <- ofe-implies-ofi Dofe (ofi/nil (Dof : of M A)). + +%worlds (bind) (ofe-implies-of _ _). +%total {} (ofe-implies-of _ _). + +cut-of : {M} + ({x} of x A -> of (M x) B) + -> ({x} lookup (G x) x A) + -> ({x} ofe (G x) (M x) B) + -> type. +%mode cut-of +D1 +D2 +D3 -D4. + +cut-ofe : {M} + ({x} of x A -> ofe (G x) (M x) B) + -> ({x} lookup (G x) x A) + -> ({x} ofe (G x) (M x) B) + -> type. +%mode cut-ofe +D1 +D2 +D3 -D4. + +- : cut-ofe + ([x] lam B ([y] M x y)) + ([x] [dx:of x A] ofe/lam (Dofe x dx : {y} ofe (cons (G x) y B) (M x y) C)) + (Dlookup : {x} lookup (G x) x A) + ([x] ofe/lam ([y] Dofe' x y)) + <- ({y} cut-ofe ([x] M x y) + ([x] [dx : of x A] Dofe x dx y) + ([x] lookup/miss (Dlookup x)) + ([x] Dofe' x y : ofe (cons (G x) y B) (M x y) C)). + +- : cut-ofe + ([x] app (M x) (N x)) + ([x] [dx : of x A] ofe/app (Dof2 x dx) (Dof1 x dx)) + Dlookup + ([x] ofe/app (Dofe2 x) (Dofe1 x)) + <- cut-ofe M Dof1 Dlookup Dofe1 + <- cut-ofe N Dof2 Dlookup Dofe2. + +- : cut-ofe + _ + ([x] [dx : of x A] ofe/var (Dlookup x : lookup (G x) (Y x) B)) + _ + ([x] ofe/var (Dlookup x)). + +- : cut-of + ([x] app (M x) (N x)) + ([x] [dx : of x A] of/app (Dof2 x dx) (Dof1 x dx)) + Dlookup + ([x] ofe/app (Dofe2 x) (Dofe1 x)) + <- cut-of M Dof1 Dlookup Dofe1 + <- cut-of N Dof2 Dlookup Dofe2. + +- : cut-ofe + M + ([x] [dx : of x A] ofe/closed (Dof x dx : of (M x) B)) + (Dlookup : {x} lookup (G x) x A) + Dofe + <- cut-of M Dof Dlookup (Dofe : {x} ofe (G x) (M x) B). + +- : cut-of + ([x] lam B ([y] M x y)) + ([x] [dx:of x A] of/lam (Dof x dx : {y} of y B -> of (M x y) C)) + (Dlookup : {x} lookup (G x) x A) + ([x] ofe/lam ([y] Dofe' x y)) + <- ({x} {dx : of x A} + cut-of ([y] M x y) ([y] [dy: of y B] Dof x dx y dy) ([y] lookup/hit : lookup (cons (G x) y B) y B) + (Dofe x dx : {y} ofe (cons (G x) y B) (M x y) C)) + <- ({y} cut-ofe ([x] M x y) + ([x] [dx : of x A] Dofe x dx y) + ([x] lookup/miss (Dlookup x)) + ([x] Dofe' x y : ofe (cons (G x) y B) (M x y) C)). + +- : cut-of + _ + ([x] [d:of x A] d) + (Dlookup : {x} lookup (G x) x A) + ([x] ofe/var (Dlookup x)). + +- : cut-of + _ + ([x] [d:of x A] Dof : of M B) + _ + ([x] ofe/closed Dof). + +%block var : block {x : exp}. + +%worlds (bind | var) (cut-of _ _ _ _) (cut-ofe _ _ _ _). +%total (M1 M2) (cut-of M1 _ _ _) (cut-ofe M2 _ _ _). + +of1-implies-ofe : ({x} of x A -> of (M x) B) + -> ({x} ofe (cons nil x A) (M x) B) + -> type. +%mode of1-implies-ofe +D1 -D2. + +- : of1-implies-ofe + (Dof : {x} of x A -> of (M x) B) + Dofe + <- cut-of M Dof ([x] lookup/hit) (Dofe : {x} ofe (cons nil x A) (M x) B). + +%worlds (bind) (of1-implies-ofe _ _). +%total {} (of1-implies-ofe _ _). + +of2-implies-ofe : ({x} of x A -> {y} of y B -> of (M x y) C) + -> ({x} {y} ofe (cons (cons nil x A) y B) (M x y) C) + -> type. +%mode of2-implies-ofe +D1 -D2. + +- : of2-implies-ofe + (Dof : {x} of x A -> {y} of y B -> of (M x y) C) + Dofe' + <- ({x} {dx : of x A} cut-of ([y] M x y) ([y] [dy : of y B] (Dof x dx y dy)) ([y] lookup/hit) + (Dofe x dx : {y} ofe (cons (cons nil x A) y B) (M x y) C)) + <- ({y} cut-ofe ([x] M x y) ([x] [dx : of x A] Dofe x dx y) ([x] lookup/miss lookup/hit) + ([x] Dofe' x y : ofe (cons (cons nil x A) y B) (M x y) C)). + +%worlds (bind) (of2-implies-ofe _ _). +%total {} (of2-implies-ofe _ _). + +subst : ({x} of x T1 -> of (E1 x) T2) + -> of E2 T1 + -> of (E1 E2) T2 + -> type. +%mode subst +D1 +D2 -D3. + +- : subst + (Dof1 : {x} of x T1 -> of (E1 x) T2) + (Dof2 : of E2 T1) + Dof + <- of1-implies-ofe Dof1 (Dofe1 : {x} ofe (cons nil x T1) (E1 x) T2) + <- esubst + ([x] append/nil) + append/nil + ([x] [dx : isvar x 0] ordered/cons (bounded/nil dx)) + (ofe/closed Dof2) + Dofe1 + (Dofe : ofe nil (E1 E2) T2) + <- ofe-implies-of Dofe (Dof : of (E1 E2) T2). + +%worlds (bind) (subst _ _ _). +%total {} (subst _ _ _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/20090429). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009-test-page-1.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009-test-page-1.lf new file mode 100644 index 0000000..9507b6b --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009-test-page-1.lf @@ -0,0 +1,16 @@ +% +%{! Notes + +```twelf +nat : type. +z: nat. +s: nat -> nat. + +plus : nat -> type. +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009/Test_Page_1). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/computation-and-deduction-2009.lf b/new-tests/stelf-output/pages/computation-and-deduction-2009.lf new file mode 100644 index 0000000..b2dd383 --- /dev/null +++ b/new-tests/stelf-output/pages/computation-and-deduction-2009.lf @@ -0,0 +1,48 @@ +% +%{! These are notes for Computation and Deduction 2009. + +[Code from February 4.](/wiki/computation-and-deduction-2009-20090204/) Homework: complete the progress proof. + +[Code from February 9.](/wiki/computation-and-deduction-2009-20090209/) Homework: complete the preservation proof. + +[http://www.cs.cmu.edu/~cmartens/canonical.pdf Writeup of the material on canonical forms LF presented on February 11.] + +[Code from February 23.](/wiki/computation-and-deduction-2009-20090223/) + +[Code from March 2.](/wiki/computation-and-deduction-2009-20090203/) + +[Code from March 4.](/wiki/computation-and-deduction-2009-20090304/) Homework: complete the proof of the subst lemma. + +[Code from March 16.](/wiki/computation-and-deduction-2009-20090316/) Homework: complete the trans/ifz and trans/app cases. ([[Computation and Deduction 2009/20090316-sol | Incomplete solution.]]) + +[Code from March 18.](/wiki/computation-and-deduction-2009-20090318/) + +[Code from March 25.](/wiki/computation-and-deduction-2009-20090325/) + +[Code from March 30.](/wiki/computation-and-deduction-2009-20090330/) + +[Code from April 1.](/wiki/computation-and-deduction-2009-20090401/) + +[Code from April 6.](/wiki/computation-and-deduction-2009-20090406/) + +[Code from April 8.](/wiki/computation-and-deduction-2009-20090408/) + +[Code from April 13.](/wiki/computation-and-deduction-2009-20090413/) + +[Code from April 15.](/wiki/computation-and-deduction-2009-20090415/) + +Project ideas from 15-851 1997: [http://www.cs.cmu.edu/~fp/courses/97-comp-ded/projects.html] + +[Code from April 20.](/wiki/computation-and-deduction-2009-20090420/) + +[Code from April 22.](/wiki/computation-and-deduction-2009-20090422/) + +[Code from April 27.](/wiki/computation-and-deduction-2009-20090427/) + +[Code from April 29.](/wiki/computation-and-deduction-2009-20090429/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Computation_and_Deduction_2009). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/concrete-representation.lf b/new-tests/stelf-output/pages/concrete-representation.lf new file mode 100644 index 0000000..250efba --- /dev/null +++ b/new-tests/stelf-output/pages/concrete-representation.lf @@ -0,0 +1,651 @@ +% +%{! The use of [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/), using Twelf's binding structure +to represent binding structure in an encoded language, +is one of the more unique and convenient +aspects of using Twelf. However, the fundamental nature of the binding +structure has been troubling to some, because it is not obvious that +everything you might ever want to do to a lambda term is doable within the +framework of HOAS. + +One way to deal with this question is to show that there is a bijection between +HOAS terms in Twelf and some **concrete representation** of terms with bound +variables. +In that case, any arbitrary operation on the concrete representation can be +related +to an operation in the HOAS representation by translating the HOAS term into +concrete form, performing the operation on the concrete term, and translating +it back into HOAS. That said, we do not know of any practical proof that has +required this technique. + +The technique of de Bruijn indices, described fully +[[w:de Bruijn index|on Wikipedia]], is a popular technique +for concretely representing binding structures. +This page describes a bijection between closed de Bruijn indices and closed +HOAS terms, and proves that bijection correct. +We will use a modified version of de Bruijn indices - it was easier to +start counting from 0, whereas true de Bruijn indices count from 1. The +bijection is defined using two relations, ``exp-->db`` and +``db-->exp``. We prove the bijection is correct in four steps: + +- ``exp-->db`` is a total and unique relation between HOAS to de Bruijn terms, and can therefore be thought of as the function _e2d_. +- ``db-->exp`` is a total and unique relation between de Bruijn terms to HOAS, and can therefore be thought of as the function _d2e_. +- For all closed de Bruijn terms E, _d2e_(_e2d_ E) = E +- For all closed HOAS terms E, _e2d_ (_d2e_ E) = E + +We do not prove that the +bijection is an isomorphism - both HOAS and de Bruijn indices have a built-in +notion of "substitution," and a proof of isomorphism would require us to +define substitution on our de Bruijn terms and show that substitution behaves +the same way in the de Bruijn representation as it does in the HOAS +representation. + +This encoding makes heavy use of [intrinsic encoding](/wiki/intrinsic-and-extrinsic-encodings/) in representing +de Bruijn terms, both to ensure +well-formedness of all terms and to encode a [structural metric](/wiki/structural-metrics/) directly +into the type of a de Bruijn term. The proof could also be done with +[extrinsic encoding](/wiki/intrinsic-and-extrinsic-encodings/) by having a separate well-formedness judgment for +de Bruijn terms and a separate judgment relating a de Bruijn term to the +structural metric it corresponds to, but in this example the intrinsic +approach made the proofs simpler. +!}% +%{! ## Preliminaries !}% +%{! We will need the ability to use [reasoning from false](/wiki/reasoning-from-false/) at one place +to avoid a spurious case that [coverage checking](/wiki/coverage-checking/) does not deal with +correctly, and as is the case with many examples, we will utilize +[natural numbers](/wiki/natural-numbers/) extensively.. !}% +%sort uninhabited %. +%freeze uninhabited %. +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort id-nat %. +%term id-nat/refl id-nat N N %. +%{! We will also need a less-than relation on numbers. For reasons that will +become clear, we will also need to define an [identity](/wiki/equality/) on less-than +derivations. Finally, we define ``lt12``, which allows us to take +two numbers N < M and case analyze on whether (s N) = M or (s N) < M. !}% +%sort lt %. +%term lt/z lt z (s N) %. +%term lt/s %pi (lt (s N1) (s N2)) %<- (lt N1 N2) %. +%sort id-lt %. +%term id-lt/refl id-lt L L %. +%sort lt12 %. +%term lt12/1 lt12 N (s N) %. +%term lt12/2 %pi (lt12 N (s M)) %<- (lt N M) %. +%{! Finally, we define two helper functions ``lt+``, which produces +a derivation M < (N + 1) from a derivation of M < N, and ``lts``, which +produces a derivation of N < (s N) from any natural number N. Most +of the tedious reasoning required for this example is about trivial +properties of these two helper functions; it is hidden here but +can be revealed by clicking on the source code link at the top of the page. !}% +%sort lt+ %. +%mode lt+ %in %out %. +%term lt+/z lt+ lt/z lt/z %. +%term lt+/s %pi (lt+ (lt/s LT1) (lt/s LT2)) %<- (lt+ LT1 LT2) %. +%worlds () (lt+ _ _) %. +%total T (lt+ T _) %. +%sort lts {N} %. +%mode lts %in %out %. +%term lts/z lts z lt/z %. +%term lts/s %pi (lts (s N) (lt/s D)) %<- (lts N D) %. +%worlds () (lts _ _) %. +%total T (lts T _) %. +%{! | hidden=true !}% +%sort lt+-irrev {L1 lt N1 N2} %. +%mode lt+-irrev %in %in %in %out %. +%term _ lt+-irrev lt/z lt+/z lt/z lt+/z %. +%term _ + %pi (lt+-irrev (lt/s L1) (lt+/s L1+) (lt/s L2) (lt+/s L2+)) + %<- (lt+-irrev L1 L1+ L2 L2+) %. +%worlds () (lt+-irrev _ _ _ _) %. +%total T (lt+-irrev T _ _ _) %. +%sort can-lts {N} {LT lt N (s N)} %. +%mode can-lts %in %in %out %. +%term _ can-lts z _ lts/z %. +%term _ %pi (can-lts (s N) (lt/s LT) (lts/s LTS)) %<- (can-lts N LT LTS) %. +%worlds () (can-lts _ _ _) %. +%total T (can-lts T _ _) %. +%sort can-lt+ {LT lt M N} {LT' lt M (s N)} %. +%mode can-lt+ %in %in %out %. +%term _ can-lt+ lt/z lt/z lt+/z %. +%term _ %pi (can-lt+ (lt/s LT) (lt/s LT') (lt+/s LT+)) %<- (can-lt+ LT LT' LT+) %. +%worlds () (can-lt+ _ _ _) %. +%total T (can-lt+ T _ _) %. +%sort lt-s-cong %. +%mode lt-s-cong %in %out %. +%term _ lt-s-cong id-lt/refl id-lt/refl %. +%worlds () (lt-s-cong _ _) %. +%total {} (lt-s-cong _ _) %. +%sort lt+-uniq %. +%mode lt+-uniq %in %in %in %out %. +%term _ lt+-uniq id-lt/refl lt+/z lt+/z id-lt/refl %. +%term _ + %pi (lt+-uniq id-lt/refl (lt+/s LT+1) (lt+/s LT+2) ID) + %<- (lt+-uniq id-lt/refl LT+1 LT+2 ID2) + %<- (lt-s-cong ID2 ID) %. +%worlds () (lt+-uniq _ _ _ _) %. +%total T (lt+-uniq _ T _ _) %. +%sort lts-uniq %. +%mode lts-uniq %in %in %in %out %. +%term _ lts-uniq id-nat/refl lts/z lts/z id-lt/refl %. +%term _ + %pi (lts-uniq id-nat/refl (lts/s LT1) (lts/s LT2) ID) + %<- (lts-uniq id-nat/refl LT1 LT2 ID1) + %<- (lt-s-cong ID1 ID) %. +%worlds () (lts-uniq _ _ _ _) %. +%total T (lts-uniq _ T _ _) %. +%sort lt-opt-s %. +%mode lt-opt-s %in %out %. +%term lt-opt-s/1 lt-opt-s lt12/1 lt12/1 %. +%term lt-opt-s/2 lt-opt-s (lt12/2 LT) (lt12/2 (lt/s LT)) %. +%worlds () (lt-opt-s _ _) %. +%total {} (lt-opt-s _ _) %. +%sort lt-opt %. +%mode lt-opt %in %out %. +%term lt-opt/z1 lt-opt lt/z lt12/1 %. +%term lt-opt/z2 lt-opt lt/z (lt12/2 lt/z) %. +%term lt-opt/s %pi (lt-opt (lt/s LT) LT12') %<- (lt-opt LT LT12) %<- (lt-opt-s LT12 LT12') %. +%worlds () (lt-opt _ _) %. +%total T (lt-opt T _) %. +%sort lt-opt-succ1 {LT lt N (s N)} %. +%mode lt-opt-succ1 %in %out %. +%term _ lt-opt-succ1 lt/z lt-opt/z1 %. +%term _ %pi (lt-opt-succ1 (lt/s LT) (lt-opt/s lt-opt-s/1 LTS)) %<- (lt-opt-succ1 LT LTS) %. +%worlds () (lt-opt-succ1 _ _) %. +%total T (lt-opt-succ1 T _) %. +%sort lt-opt-succ2 %. +%mode lt-opt-succ2 %in %out %. +%term _ lt-opt-succ2 lt+/z lt-opt/z2 %. +%term _ + %pi (lt-opt-succ2 (lt+/s LT+) (lt-opt/s lt-opt-s/2 LTO)) + %<- (lt-opt-succ2 LT+ LTO) %. +%worlds () (lt-opt-succ2 _ _) %. +%total T (lt-opt-succ2 T _) %. +%sort lt-opt2-s-uniq %. +%mode lt-opt2-s-uniq %in %in %in %out %. +%term _ lt-opt2-s-uniq id-lt/refl lt-opt-s/2 lt-opt-s/2 id-lt/refl %. +%worlds () (lt-opt2-s-uniq _ _ _ _) %. +%total {} (lt-opt2-s-uniq _ _ _ _) %. +%sort lt-opt2-uniq %. +%mode lt-opt2-uniq %in %in %in %out %. +%term _ lt-opt2-uniq id-lt/refl lt-opt/z2 lt-opt/z2 id-lt/refl %. +%term _ + %pi (lt-opt2-uniq id-lt/refl (lt-opt/s LS L) (lt-opt/s LS' L') ID) + %<- (lt-opt2-uniq id-lt/refl L L' ID1) + %<- (lt-opt2-s-uniq ID1 LS LS' ID) %. +%worlds () (lt-opt2-uniq _ _ _ _) %. +%total T (lt-opt2-uniq _ T _ _) %. +%sort lt-opt-excl %. +%mode lt-opt-excl %in %in %out %. +%term _ %pi (lt-opt-excl (lt-opt/s LS L) (lt-opt/s LS' L') X) %<- (lt-opt-excl L L' X) %. +%worlds () (lt-opt-excl _ _ _) %. +%total T (lt-opt-excl T _ _) %. +%{! ## HOAS lambda terms !}% +%{! The representation of the untyped lambda calculus is extremely simple in +higher-order abstract syntax; we also define congruence lemmas for lam and app. + +What is interesting is the [``%block``](/wiki/percent-block/) +declaration of ``blocksimple``. This is a standard way of looking at the +LF context when we are using HOAS lambda terms, but it is not the one we will +use in most cases. !}% +%sort exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%block blocksimple [v exp]%. +%sort id-exp %. +%term id-exp/refl id-exp E E %. +%sort id-lam-cong {v} %. +%term _ id-lam-cong ([v] id-exp/refl) id-exp/refl %. +%mode id-lam-cong %in %out %. +%worlds (blocksimple) (id-lam-cong _ _) %. +%total {} (id-lam-cong _ _) %. +%sort id-app-cong %. +%term _ id-app-cong id-exp/refl id-exp/refl id-exp/refl %. +%mode id-app-cong %in %in %out %. +%worlds (blocksimple) (id-app-cong _ _ _) %. +%total {} (id-app-cong _ _ _) %. +%{! The problem with ``blocksimple`` is that we will need, crucially, to +be able to know when something is a variable (as opposed to an application or a +lambda term). The world ``blockvar`` achieves this by requiring that +all variables be accompanied by a judgment ``isvar v``; because these +judgments cannot be defined, they can only be introduced as hypothetical +judgments, so having (isvar E) ensures that E is a variable. !}% +%sort isvar %. +%block blockvar [v] [iv isvar v]%. +%{! However, we will need one more block declaration to be able to do everything +we need. In particular, the block ``blockvar`` is not enough to ensure that +we always will be able to determine whether an open lambda term is a lambda, an +application, or a variable. In order to do this, we need a specialized block +declaration ``blockcases`` that specifies that whenever I add a variable +to the context, I add it along with a ``isvar`` hypothesis and a +``can-case`` hypothesis. + +The definition of ``fake`` is a bit of a hack - we need to allow +``can-case`` to depend on ``can-case`` or we will run afoul of +Twelf's [autofreeze](/wiki/percent-freeze/) feature. If you are running this proof on your +own computer, try deleting the line to see Twelf's error message. !}% +%sort case %. +%term case/lam case (lam ([x] E x)) %. +%term case/app case (app E1 E2) %. +%term case/var %pi (isvar V) %-> (case V) %. +%sort can-case {E} %. +%sort fake %. +%term _ %pi (fake E F) %<- ({d can-case X Y} can-case X' Y') %. +%block blockcases [v] [iv isvar v] [d can-case v (case/var iv)]%. +%term can-case/lam can-case (lam ([x] E x)) case/lam %. +%term can-case/app can-case (app E1 E2) case/app %. +%mode can-case %in %out %. +%worlds (blockcases) (can-case _ _) %. +%total T (can-case T _) %. +%{! ## de Bruijn lambda terms !}% +%{! ### Structural metric: Trees !}% +%{! In order for the totality proof for the de Bruijn to HOAS translation to go +through, we will need to show that there is a metric by which ``unbind`` +does not change the size of a term. + +Because Twelf does not relate the sizes of bound variables and the free +variables ``unbind`` replaces them with in any way, in order to achieve +this we relate de Bruijn terms to an abstract tree representation; +then, we can use that tree representation as the [structural metric](/wiki/structural-metrics/) to +prove termination for the de Bruijn to HOAS translation. !}% +%sort etree %. +%term etree/lam %pi etree %-> etree %. +%term etree/app %pi etree %-> etree %-> etree %. +%term etree/var etree %. +%sort id-etree %. +%term id-etree/refl id-etree E E %. +%sort id-etree/app-cong %. +%mode id-etree/app-cong %in %in %out %. +%term _ id-etree/app-cong id-etree/refl id-etree/refl id-etree/refl %. +%worlds () (id-etree/app-cong _ _ _) %. +%total {} (id-etree/app-cong _ _ _) %. +%sort id-etree/lam-cong %. +%mode id-etree/lam-cong %in %out %. +%term _ id-etree/lam-cong id-etree/refl id-etree/refl %. +%worlds () (id-etree/lam-cong _ _) %. +%total {} (id-etree/lam-cong _ _) %. +%{! ### Encoding terms !}% +%{! We choose an [intrinsic encoding](/wiki/intrinsic-and-extrinsic-encodings/) of de Bruijn terms so that every term +that passes the LF type checker is well-formed - in particular, a term +with ``exp^ N`` is a well-formed de Bruijn term inside of N lambdas. +An earlier version of this encoding had a separate well-formedness judgment; +the upshot of this encoding was that a lot of complexity got rephrased +in terms of simple reasoning about relations like ``lt+``. + +The structural metric is made a part of the intrinsic in order to simplify +later proofs. !}% +%sort exp^ %. +%term lam^ %pi (exp^ (s N) Et) %-> (exp^ N (etree/lam Et)) %. +%term app^ %pi (exp^ N Et1) %-> (exp^ N Et2) %-> (exp^ N (etree/app Et1 Et2)) %. +%term fvar^ {x exp} %pi (isvar x) %-> (exp^ N etree/var) %. +%term bvar^ {n nat} %pi (lt n M) %-> (exp^ M etree/var) %. +%{! We will think of de Bruijn terms in two different ways depending on whether +we are thinking in terms of a closed world or a world described by +``blockvar``. Recall that +in a closed world (i.e. an empty LF context) we cannot form anything with +type ``isvar V``, so in an empty context we cannot form a term with +``fvar^``. The ``fvar^`` constructor is what allows us to link +HOAS terms and de Bruijn terms, by incorporating variables from the +HOAS representation directly inside of a de Bruijn term. + +We also define identity and congruence on de Bruijn terms. !}% +%sort id-exp^ %. +%term id-exp^/refl id-exp^ E E %. +%sort id-bvar^-cong %. +%term _ id-bvar^-cong id-nat/refl id-lt/refl id-exp^/refl %. +%mode id-bvar^-cong %in %in %out %. +%worlds (blockvar) (id-bvar^-cong _ _ _) %. +%total {} (id-bvar^-cong _ _ _) %. +%sort id-app^-cong %. +%term _ id-app^-cong id-exp^/refl id-exp^/refl id-exp^/refl %. +%mode id-app^-cong %in %in %out %. +%worlds (blockvar) (id-app^-cong _ _ _) %. +%total {} (id-app^-cong _ _ _) %. +%sort id-lam^-cong %. +%term _ id-lam^-cong id-exp^/refl id-exp^/refl %. +%mode id-lam^-cong %in %out %. +%worlds (blockvar) (id-lam^-cong _ _) %. +%total {} (id-lam^-cong _ _) %. +%{! ### Binding and unbinding !}% +%{! The key operations on de Bruijn terms are a binding and unbinding +operation - essentially, bind takes a free (HOAS) variable and makes it +a different free (de Bruijn) variable. +By way of an example, if we have , +then binding will result in +, and binding will +result in . + +Unfortunately, the structural metric ends up being included in both +``bind`` and ``unbind``, which complicates the otherwise relatively +simple judgments significantly. !}% +%sort bind {v} %. +%mode bind %in %out %. +%term bind/match %pi (bind ([v] [iv] fvar^ v iv) (bvar^ N LT)) %<- (lts N LT) %. +%term bind/fvar bind ([v] [iv] fvar^ X V) (fvar^ X V) %. +%term bind/bvar %pi (bind ([v] [iv] bvar^ M LT) (bvar^ M LT')) %<- (lt+ LT LT') %. +%term bind/app + %pi (bind ([v] [iv] app^ (E1 v iv) (E2 v iv)) (app^ E1' E2')) + %<- (bind ([v] [iv] E1 v iv) E1') + %<- (bind ([v] [iv] E2 v iv) E2') %. +%term bind/lam %pi (bind ([v] [iv] lam^ (E v iv)) (lam^ E')) %<- (bind ([v] [iv] E v iv) E') %. +%worlds (blockvar) (bind _ _) %. +%total T (bind T _) %. +%{! !}% +%sort unbind-bvar %. +%mode unbind-bvar %in %out %. +%term unbind-bvar/match unbind-bvar lt12/1 ([v] [iv] fvar^ v iv) %. +%term unbind-bvar/nomatch unbind-bvar (lt12/2 LT) ([v] [iv] bvar^ M LT) %. +%worlds (blockvar) (unbind-bvar _ _) %. +%total {} (unbind-bvar _ _) %. +%sort unbind %. +%mode unbind %in %out %. +%term unbind/fvar unbind (fvar^ V' IV') ([v] [iv] fvar^ V' IV') %. +%term unbind/bvar + %pi (unbind (bvar^ M LT) ([v] [iv] E v iv)) + %<- (lt-opt LT LT12) + %<- (unbind-bvar LT12 ([v] [iv] E v iv)) %. +%term unbind/app + %pi (unbind (app^ E1 E2) ([v] [iv] app^ (E1' v iv) (E2' v iv))) + %<- (unbind E1 ([v] [iv] E1' v iv)) + %<- (unbind E2 ([v] [iv] E2' v iv)) %. +%term unbind/lam + %pi (unbind (lam^ E1) ([v] [iv] lam^ (E1' v iv))) + %<- (unbind E1 ([v] [iv] E1' v iv)) %. +%worlds (blockvar) (unbind _ _) %. +%total T (unbind T _) %. +%{! ### Properties of bind and unbind !}% +%{! Now we will show that bind and unbind are inverses of each other, +which mirrors the proof we will eventually prove about the ``exp->db`` +and ``db->exp`` being inverses of each other. In particular, in +``unbind-uniq`` we need to use [reasoning from false](/wiki/reasoning-from-false/) along with a +special lemma. + +The theorem statements were difficult to get correct, but the proofs are +relatively straightforward and are omitted (they are present in the full +source code for this page). !}% +%sort bind-uniq {v} {iv isvar v} %. +%mode bind-uniq %in %in %in %out %. +%{}% +%sort unbind-uniq %. +%mode unbind-uniq %in %in %in %out %. +%{}% +%sort bind-unbind %. +%mode bind-unbind %in %out %. +%{}% +%sort unbind-bind %. +%mode unbind-bind %in %out %. +%{|hidden=true}% +% Cases for bind-uniq +%term _ bind-uniq ([v] [iv] id-exp^/refl) bind/fvar bind/fvar id-exp^/refl %. +%term _ + %pi (bind-uniq ([v] [iv] id-exp^/refl) (bind/bvar LT+) (bind/bvar LT+') ID) + %<- (lt+-uniq id-lt/refl LT+ LT+' ID+) + %<- (id-bvar^-cong id-nat/refl ID+ ID) %. +%term _ + %pi (bind-uniq ([v] [iv] id-exp^/refl) (bind/match LTS) (bind/match LTS') ID) + %<- (lts-uniq id-nat/refl LTS LTS' IDS) + %<- (id-bvar^-cong id-nat/refl IDS ID) %. +%term _ + %pi (bind-uniq ([v] [iv] id-exp^/refl) (bind/app F2 F1) (bind/app F2' F1') ID) + %<- (bind-uniq ([v] [iv] id-exp^/refl) F1 F1' ID1) + %<- (bind-uniq ([v] [iv] id-exp^/refl) F2 F2' ID2) + %<- (id-app^-cong ID1 ID2 ID) %. +%term _ + %pi (bind-uniq ([v] [iv] id-exp^/refl) (bind/lam F1) (bind/lam F1') ID) + %<- (bind-uniq ([v] [iv] id-exp^/refl) F1 F1' ID1) + %<- (id-lam^-cong ID1 ID) %. +%worlds (blockvar) (bind-uniq _ _ _ _) %. +%total T (bind-uniq _ T _ _) %. +% Cases for unbind-uniq +%sort lem {Eb {v} %pi (isvar v) %-> (exp^ N Et)} {Eb' {v} %pi (isvar v) %-> (exp^ N Et)} %. +%mode lem %in %in %in %out %. +%worlds (blockvar) (lem _ _ _ _) %. +%total {} (lem _ _ _ _) %. +%term _ unbind-uniq id-exp^/refl unbind/fvar unbind/fvar ([v] [iv] id-exp^/refl) %. +%term _ + %pi (unbind-uniq id-exp^/refl (unbind/bvar unbind-bvar/match A) (unbind/bvar unbind-bvar/nomatch B) ID) + %<- (lt-opt-excl A B X) + %<- (lem _ _ X ID) %. +%term _ + %pi (unbind-uniq id-exp^/refl (unbind/bvar unbind-bvar/nomatch A) (unbind/bvar unbind-bvar/match B) ID) + %<- (lt-opt-excl B A X) + %<- (lem _ _ X ID) %. +%term _ unbind-uniq id-exp^/refl (unbind/bvar unbind-bvar/match A) (unbind/bvar unbind-bvar/match B) ([v] [iv] id-exp^/refl) %. +%term _ + %pi (unbind-uniq id-exp^/refl (unbind/bvar unbind-bvar/nomatch (%the (lt-opt LT (lt12/2 LT2)) Opt1)) (unbind/bvar unbind-bvar/nomatch (%the (lt-opt LT (lt12/2 LT2')) Opt2)) ID) + %<- (lt-opt2-uniq id-lt/refl Opt1 Opt2 ID1) + %<- ({v} {iv isvar v} id-bvar^-cong id-nat/refl ID1 (ID v iv)) %. +%term _ + %pi (unbind-uniq id-exp^/refl (unbind/app E2 E1) (unbind/app E2' E1') ID) + %<- (unbind-uniq id-exp^/refl E1 E1' ID1) + %<- (unbind-uniq id-exp^/refl E2 E2' ID2) + %<- ({v} {iv isvar v} id-app^-cong (ID1 v iv) (ID2 v iv) (ID v iv)) %. +%term _ + %pi (unbind-uniq id-exp^/refl (unbind/lam E) (unbind/lam E') ID) + %<- (unbind-uniq id-exp^/refl E E' ID1) + %<- ({v} {iv isvar v} id-lam^-cong (ID1 v iv) (ID v iv)) %. +%worlds (blockvar) (unbind-uniq _ _ _ _) %. +%total T (unbind-uniq _ T _ _) %. +% Cases for bind-unbind +%term _ bind-unbind bind/fvar unbind/fvar %. +%term _ + %pi (bind-unbind (bind/match (%the (lts N LT) LTS)) (unbind/bvar unbind-bvar/match Opt)) + %<- (lt-opt-succ1 LT Opt) %. +%term _ + %pi (bind-unbind (bind/bvar (%the (lt+ LT LT') LT+)) (unbind/bvar unbind-bvar/nomatch Opt)) + %<- (lt-opt-succ2 LT+ Opt) %. +%term _ + %pi (bind-unbind (bind/app B2 B1) (unbind/app U2 U1)) + %<- (bind-unbind B1 U1) + %<- (bind-unbind B2 U2) %. +%term _ %pi (bind-unbind (bind/lam B) (unbind/lam U)) %<- (bind-unbind B U) %. +%worlds (blockvar) (bind-unbind _ _) %. +%total T (bind-unbind T _) %. +% Cases for unbind-bind +%term _ unbind-bind unbind/fvar bind/fvar %. +%term _ + %pi (unbind-bind (unbind/bvar unbind-bvar/match Opt) (bind/match LTS)) + %<- (can-lts _ _ LTS) %. +%term _ + %pi (unbind-bind (unbind/bvar unbind-bvar/nomatch (%the (lt-opt LT' (lt12/2 LT)) Opt)) (bind/bvar LT+)) + %<- (can-lt+ LT LT' LT+) %. +%term _ + %pi (unbind-bind (unbind/app U2 U1) (bind/app B2 B1)) + %<- (unbind-bind U1 B1) + %<- (unbind-bind U2 B2) %. +%term _ %pi (unbind-bind (unbind/lam U) (bind/lam B)) %<- (unbind-bind U B) %. +%worlds (blockvar) (unbind-bind _ _) %. +%total T (unbind-bind T _) %. +%{}% +%{! ## Relating HOAS and de Bruijn representations !}% +%{! This is the penultimate section. What we want to prove, ultimately, is +a bijection between _closed_ HOAS terms and _closed_ de Bruijn terms; +however, in order to get there we will prove a bijection between open +HOAS terms and open de Bruijn terms, which is the subject of this section. +Doing so requires reasoning with the structural metric, using the complex +``blockcases`` block description, and generally making a mess. The +theorem about closed terms in the next session will be simpler, but will use +the more complex lemmas from this section. !}% +%{! ### HOAS to de Bruijn !}% +%{! It is here that we will need to use the ``blockcases`` block description +discussed above; even though it makes the translation more complicated, we +directly prove the translation total in order to avoid writing a ton +of [effectiveness lemmas](/wiki/effectiveness-lemma/); the uniqueness of this translation is simple, +but dependent on the effectiveness of ``bind``. !}% +%sort exp->db {E} %. +%mode exp->db %in %in %out %out %. +%term exp->db/var exp->db _ (case/var IV) etree/var (fvar^ V IV) %. +%term exp->db/lam + %pi (exp->db (lam ([v] E2 v)) case/lam (etree/lam Et) (lam^ E^')) + %<- ({v} {iv isvar v} %pi (can-case v (case/var iv)) %-> (can-case (E2 v) (CASE v iv))) + %<- ({v} {iv isvar v} %pi (can-case v (case/var iv)) %-> (exp->db (E2 v) (CASE v iv) Et (E^'' v iv))) + %<- (bind E^'' E^') %. +%term exp->db/app + %pi (exp->db (app E1 E2) case/app (etree/app Et1 Et2) (app^ E1' E2')) + %<- (can-case E1 CASE1) + %<- (can-case E2 CASE2) + %<- (exp->db E1 CASE1 Et1 E1') + %<- (exp->db E2 CASE2 Et2 E2') %. +%worlds (blockcases) (exp->db _ _ _ _) %. +%total T (exp->db T _ _ _) %. +%{}% +%sort uniq->db %. +%mode uniq->db %in %in %out %out %. +%term _ uniq->db exp->db/var exp->db/var id-etree/refl id-exp^/refl %. +%term _ + %pi (uniq->db (exp->db/lam F T C) (exp->db/lam F' T' C') IDe ID^) + %<- ({v} {iv isvar v} {c can-case v (case/var iv)} uniq->db (T v iv c) (T' v iv c) IDe1 (ID^1 v iv)) + %<- (bind-uniq ID^1 F F' ID^2) + %<- (id-etree/lam-cong IDe1 IDe) + %<- (id-lam^-cong ID^2 ID^) %. +%term _ + %pi (uniq->db (exp->db/app T2 T1 C2 C1) (exp->db/app T2' T1' C2' C1') IDe ID^) + %<- (uniq->db T1 T1' IDe1 ID^1) + %<- (uniq->db T2 T2' IDe2 ID^2) + %<- (id-etree/app-cong IDe1 IDe2 IDe) + %<- (id-app^-cong ID^1 ID^2 ID^) %. +%worlds (blockcases) (uniq->db _ _ _ _) %. +%total T (uniq->db T _ _ _) %. +%{! ### de Bruijn to HOAS !}% +%{! It is here that we will need to use the ``etree`` [structural metric](/wiki/structural-metrics/) +to provide a usable termination argument for ``db->exp``. !}% +%sort db->exp {Et} %. +%mode db->exp %in %in %out %out %. +%term db->exp/fvar db->exp etree/var (fvar^ V IV) V (case/var IV) %. +%term db->exp/lam + %pi (db->exp (etree/lam ET) (lam^ E^) (lam E) case/lam) + %<- (unbind E^ E^') + %<- ({v} {iv isvar v} %pi (can-case v (case/var iv)) %-> (db->exp ET (E^' v iv) (E v) _)) %. +%term db->exp/app + %pi (db->exp (etree/app ET1 ET2) (app^ E1^ E2^) (app E1 E2) case/app) + %<- (db->exp ET1 E1^ E1 _) + %<- (db->exp ET2 E2^ E2 _) %. +%worlds (blockcases) (db->exp _ _ _ _) %. +%total T (db->exp T _ _ _) %. +%{}% +%sort uniq->exp %. +%mode uniq->exp %in %in %in %out %. +%term _ uniq->exp id-exp^/refl db->exp/fvar db->exp/fvar id-exp/refl %. +%term _ + %pi (uniq->exp id-exp^/refl (db->exp/lam ([v] [iv] [c can-case v (case/var iv)] T v iv c) UB) (db->exp/lam ([v] [iv] [c can-case v (case/var iv)] T' v iv c) UB') ID) + %<- (unbind-uniq id-exp^/refl UB UB' (%the ({v exp} {iv isvar v} id-exp^ (E v iv) (E' v iv)) ID1)) + %<- ({v} {iv isvar v} {c can-case v (case/var iv)} uniq->exp (ID1 v iv) (T v iv c) (T' v iv c) (ID2 v)) + %<- (id-lam-cong ID2 ID) %. +%term _ + %pi (uniq->exp id-exp^/refl (db->exp/app T2 T1) (db->exp/app T2' T1') ID) + %<- (uniq->exp id-exp^/refl T1 T1' ID1) + %<- (uniq->exp id-exp^/refl T2 T2' ID2) + %<- (id-app-cong ID1 ID2 ID) %. +%worlds (blockcases) (uniq->exp _ _ _ _) %. +%total T (uniq->exp _ T _ _) %. +%{! ### Composition: HOAS -> de Bruijn -> HOAS !}% +%sort inverseEDE %. +%mode inverseEDE %in %out %. +%term _ inverseEDE exp->db/var db->exp/fvar %. +%term _ + %pi (inverseEDE (exp->db/lam B T C) (db->exp/lam T' U)) + %<- (bind-unbind B U) + %<- ({v} {iv isvar v} {c can-case v (case/var iv)} inverseEDE (T v iv c) (T' v iv c)) %. +%term _ + %pi (inverseEDE (exp->db/app T2 T1 C2 C1) (db->exp/app T2' T1')) + %<- (inverseEDE T1 T1') + %<- (inverseEDE T2 T2') %. +%worlds (blockcases) (inverseEDE _ _) %. +%total T (inverseEDE T _) %. +%{}% +%{! ### Composition: de Bruijn -> HOAS -> de Bruijn !}% +%{! We need one last trick in order to make this proof go through in the +opposite direction; we will need an effectiveness lemma on top of an +effectiveness lemma! (Effectiveness lemmas can get out of control sometimes.) !}% +%sort can-can-case {C case E} %. +%sort fake %. +%term _ %pi fake %<- (%pi (can-can-case E V) %-> (can-can-case E' V')) %. +%block blockcancases [v] [iv isvar v] [c can-case v (case/var iv)] [cc can-can-case (case/var iv) c]%. +%term _ can-can-case case/app can-case/app %. +%term _ can-can-case case/lam can-case/lam %. +%mode can-can-case %in %out %. +%worlds (blockcancases) (can-can-case _ _) %. +%total {} (can-can-case _ _) %. +%{! With our slightly absurd effectiveness effectiveness lemma in tow, we can +complete the proof that composition in the other direction is the identity. !}% +%sort inverseDED %. +%mode inverseDED %in %out %. +%term _ inverseDED db->exp/fvar exp->db/var %. +%term _ + %pi (inverseDED (db->exp/lam (%the ({v} {iv isvar v} {c} db->exp Et (E^ v iv) (E v) (Case v iv)) T) U) (exp->db/lam B T' C)) + %<- (unbind-bind U B) + %<- ({v} {iv isvar v} {c can-case v (case/var iv)} %pi (can-can-case (case/var iv) c) %-> (inverseDED (T v iv c) (T' v iv c))) + %<- ({v} {iv isvar v} {c can-case v (case/var iv)} %pi (can-can-case (case/var iv) c) %-> (can-can-case _ (C v iv c))) %. +%term _ + %pi (inverseDED (db->exp/app T2 T1) (exp->db/app T2' T1' C2 C1)) + %<- (inverseDED T1 T1') + %<- (inverseDED T2 T2') + %<- (can-can-case _ C1) + %<- (can-can-case _ C2) %. +%worlds (blockcancases) (inverseDED _ _) %. +%total T (inverseDED T _) %. +%{}% +%{! ## Wrapping up !}% +%{! The one step that remains is to perform the above four steps for a +simpler theorem statement in a closed world, since ultimately what we +want is a theorem about closed terms anyway. These theorems would likely +be less useful in practice than the ones already proven because the +closed world assumption makes them less general, but they more concisely +state what was done. !}% +%{! * ``exp-->db`` is a total and unique relation between HOAS to de Bruijn terms, and can therefore be thought of as the function _e2d_. !}% +%sort exp-->db %. +%term exp-->db/ + %pi (exp-->db E E^) + %<- (can-case E (%the (case E) C)) + %<- (exp->db E C Et (%the (exp^ z Et) E^)) %. +%mode exp-->db %in %out %. +%worlds () (exp-->db _ _) %. +%total {} (exp-->db _ _) %. +%sort uniq-->db %. +%term _ %pi (uniq-->db (exp-->db/ ED C) (exp-->db/ ED' C') ID) %<- (uniq->db ED ED' _ ID) %. +%mode uniq-->db %in %in %out %. +%worlds () (uniq-->db _ _ _) %. +%total {} (uniq-->db _ _ _) %. +%{! * ``db-->exp`` is a total and unique relation between de Bruijn terms to HOAS, and can therefore be thought of as the function _d2e_. !}% +%sort db-->exp %. +%term db-->exp/ %pi (db-->exp E^ E) %<- (db->exp Et E^ E _) %. +%mode db-->exp %in %out %. +%worlds () (db-->exp _ _) %. +%total {} (db-->exp _ _) %. +%sort uniq-->exp %. +%term _ + %pi (uniq-->exp (db-->exp/ DE) (db-->exp/ DE') ID) + %<- (uniq->exp id-exp^/refl DE DE' ID) %. +%mode uniq-->exp %in %in %out %. +%worlds () (uniq-->exp _ _ _) %. +%total {} (uniq-->exp _ _ _) %. +%{! * For all closed de Bruijn terms E, _d2e_(_e2d_ E) = E. !}% +%sort db-->exp-->db %. +%term _ + %pi (db-->exp-->db (db-->exp/ DE) (exp-->db/ ED C)) + %<- (inverseDED DE (%the (exp->db E Case Et E^) ED)) + %<- (can-can-case Case C) %. +%mode db-->exp-->db %in %out %. +%worlds () (db-->exp-->db _ _) %. +%total {} (db-->exp-->db _ _) %. +%{! * For all closed HOAS terms E, _e2d_ (_d2e_ E) = E. !}% +%sort exp-->db-->exp %. +%term _ %pi (exp-->db-->exp (exp-->db/ ED C) (db-->exp/ DE)) %<- (inverseEDE ED DE) %. +%mode exp-->db-->exp %in %out %. +%worlds () (exp-->db-->exp _ _) %. +%total {} (exp-->db-->exp _ _) %. +%{}% +%{! ## Examples !}% +%{! It's also a good idea to run a [``%query``](/wiki/percent-query/) on some examples as +a sanity check. First, we translate the Church numeral "4," from a +HOAS to a de Bruijn index representation. |check=decl !}% +%query 1 _ _ exp-->db (lam ([f] lam ([x] app f (app f (app f (app f x)))))) DB %. +%{! Next, we translate the Church boolean operator "Not," from a +Bruijn index to a HOAS representation. |check=decl !}% +%query 1 _ _ db-->exp (lam^ (app^ (app^ (bvar^ z lt/z) (lam^ (lam^ (bvar^ z lt/z)))) (lam^ (lam^ (bvar^ (s z) (lt/s lt/z)))))) HOAS %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Concrete_representation). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/congruence-relation.lf b/new-tests/stelf-output/pages/congruence-relation.lf new file mode 100644 index 0000000..358e68e --- /dev/null +++ b/new-tests/stelf-output/pages/congruence-relation.lf @@ -0,0 +1,20 @@ +% +%{! A congruence relation on a language is an [equivalence relation](/wiki/equivalence-relation/) that is compatible with the term constructors of that language. + +Consider the untyped lambda-calculus: + + + +A congruence relation must be closed under the equivalence rules, + + + +and the compatibility rules, + + !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Congruence_relation). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/constraint-domain.lf b/new-tests/stelf-output/pages/constraint-domain.lf new file mode 100644 index 0000000..8b0af2a --- /dev/null +++ b/new-tests/stelf-output/pages/constraint-domain.lf @@ -0,0 +1,83 @@ +% +%{! A **constraint domain**, also called a **Twelf(X) extension**, is an extension to Twelf that allow users to work easily with an object, such as the integers, that can be tedious or inefficient to explicitly formalize within Twelf. Constraint domains are introduced with the [``%use``](/wiki/percent-use/) declaration. + +You should never use [constraint domains and coverage checking](/wiki/constraint-domains-and-coverage-checking/) in the same signature, but they are useful when using Twelf as a logic programming system or when reasoning _in_ an [object language](/wiki/object-logic/). This document shows brief examples of each of the constraint domains, which are treated more formally in the chapter on constraint domains. + +## Rational numbers with equality: ``%use equality/rationals`` + +The rational numbers with equality add a new type family, ``rational``, and operators for rational addition, subtraction, multiplication, and negation. The rational numbers are automatically simplified by Twelf + +```checkedtwelf +%use equality/rationals. +mynum = 1/4 + ~ 9/2. +``` + +Because of this automatic simplification, equality is not actually added by the ``%use`` declaration, becuase it can be introduced the same way [identity](/wiki/equality/) is introduced for any other type family in Twelf. + +```twelf +eq : rational -> rational -> type. +eq/is : eq X X. +``` + +## Rational numbers with inequality: ``%use inequality/rationals`` + +This declaration introduces everything introduced by ``%use equality/rationals``, as well as ways to check for inequality: + +```checkedtwelf +%use inequality/rationals. +``` + +Inequality relations can be used like normal goals, as they are in the next example, even though the User's Guide explains that their internal behavior is different than other goals. + +```twelf +more-than-two : rational -> rational -> type. +more-than-two/i : more-than-two R1 R2 + <- R1 + R2 > 2. +``` + +## Integers with equality: ``%use equality/integers`` + +The introduction of integers with equality looks similar to the introduction of rationals with equality, but with a different type family, ``integer``. + +```checkedtwelf +%use equality/integers. +``` + +## Integers with inequality: ``%use inequality/integers`` + +The introduction of integers with inequality is a good bit shorter than the introduction of rationals with inequality, because Twelf can take advantage of the facts like ``X + 1 > Y`` iff ``X >= Y``. + +```checkedtwelf +%use inequality/integers. +``` + +## Strings: ``%use equality/strings`` + +This constraint domain allows Twelf to include strings of characters. Like the other constriant domains, using the operators automatically simplifies the operators away. The only one provided is concatenation, ``++``, which is shown below: + +```checkedtwelf +%use equality/strings. +mystring = "Hello" ++ " " ++ "World" ++ "!". + +``` + +## 32-bit integers: ``%use word32.`` + +The ``word32`` extension behaves very differently than the others - operators are relations, not functions, and so Twelf does not simplify their application away. Behind the scenes, using the ``word32`` extension adds axioms to the signature like ``3+15`` shown in the below example. The ``prove+`` relations that are defined act like [effectiveness lemmas](/wiki/effectiveness-lemma/) for these relations. + + +```checkedtwelf +%use word32. +myproof = 3+15. +%solve _ : prove+ 92 _ 105 _. +``` + +## See also +* [``%use``](/wiki/percent-use/) +* Constraint domains + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +You can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Constraint_domain). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/constraint-domains-and-coverage-checking.lf b/new-tests/stelf-output/pages/constraint-domains-and-coverage-checking.lf new file mode 100644 index 0000000..1719888 --- /dev/null +++ b/new-tests/stelf-output/pages/constraint-domains-and-coverage-checking.lf @@ -0,0 +1,79 @@ +% +%{! +> _You should not attach any particular significance to the fact that Twelf returns OK (or raises an uncaught exception, or starts playing the theme music to M*A*S*H...) on a signature that mentions a constraint domain and also tries to prove coverage or totality metatheorems._ - Kevin Watkins + +It is tempting to try to use [constraint domains](/wiki/constraint-domain/) to do seemingly innocent helpful things. However, at the present time, constraint domains should be seen as completely incompatible with features that Twelf uses to prove [metatheorems](/wiki/metatheorem/), such as [``%mode``](/wiki/percent-mode/), [``%covers``](/wiki/percent-covers/), and [``%total``](/wiki/percent-total/). + +One particular way that this error happens is when users attempt to formalize a language with exceptions, and want to index their language with strings. + +```twelf +%use equality/strings. + +exp : type. +error : string -> exp. +``` + +This should be avoided at all costs in favor of explicitly defining a type of error messages _within_ the signature. + +```twelf +errormessage : type. + +bad-value : errormessage. +type-error : errormessage. +% ...etc... + +exp : type. +error : errormessage -> exp. +``` + +## Example: apparent unsoundness with constraint domains + +The essential problem is that the interaction between constraint domains and the Twelf +coverage checker is not well defined. Take the following example: !}% +%use word32 %. +%sort exp %. +%term x %pi word32 %-> exp %. +%term y exp %. +%term z exp %. +%sort q %. +%mode q %in %out %. +%term q/y q y y %. +%term q/z q z z %. +%{! Twelf appears to allow us to prove that, for any input in the first position, ``q`` will always run +successfully and return an output in the second position. !}% +%{!! begin checked !!}% +%worlds () (q _ _) %. +%total T (q T _) %. +%{!! end checked !!}% +%{! However, Twelf's assertion of totality here is completely bogus, which we can see if we try to run +q as a logic program with ``(x 5)`` in the first position. + +```checkedtwelf +%solve _ : q (x 5) _. +``` + +## Analysis + + +One way of looking at the problem above is that the above proof isn't quite a proof of what it seems to be a proof of. The [``%worlds``](/wiki/percent-worlds/) statement for the above theorem is empty, which means that one way to state the theorem theorem is "if the only things that exist in the things that were explicitly defined in the signature." This is a closed-world assumption. Because the identifier ``5`` never was explicitly defined in the signature, but was implicitly introduced by the ``%use word32`` declaration, then the metatheorem above is true - if we're not allowed use any object of type ``word32``, the we can't feed +``q`` any input that will allow it to fail. + +If we modify our worlds declaration to specify that there may be some ``word32``s around—because there _are_ some +``word32``s around, there are +of them!—then the theorem fails as it should. + +```checkedtwelf +%block some-word32s : block {x: word32}. + +%worlds (some-word32s) (q _ _). +%total T (q T _). +``` + +However, as there is no account for exactly what ``%use`` _does_ to the signature, it is very difficult to be certain, when we are using constraint domains, that we have proven the theorems we think we have proven. This is precisely +why constraint domains and metatheorem declarations like ``%covers`` and ``%total`` should never be used in the same signature. + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +You can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Constraint_domains_and_coverage_checking). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/constructivesemantics.lf b/new-tests/stelf-output/pages/constructivesemantics.lf new file mode 100644 index 0000000..dee64b5 --- /dev/null +++ b/new-tests/stelf-output/pages/constructivesemantics.lf @@ -0,0 +1,109 @@ +% +%{! A little proof of cut eliminiation for the logic FF in an unpublished paper by Reed and Pfenning. !}% +%{! (options removed from twelftag: check="true") !}% +% Termination metric +%sort lo %. +%sort hi %. +%term * lo %. +%term s %pi lo %-> hi %. +% Syntax +%sort t+ %. +% "terms" that go in positive atoms +%sort t- %. +%name t- %. +% "terms" that go in negative atoms +%sort p %. +% positive props +%sort o %. +% negative props +%sort i %. +% worlds, frames +% Propositions +%term imp %pi p %-> o %-> o %. +%term top o %. +%term s- %pi t- %-> o %. +%term s+ %pi t+ %-> p %. +%term v %pi o %-> p %. +% Stub terms to make coverage nontrivial +%term t+/ t+ %. +%term t-/ t- %. +% Judgments +%sort j %. +%name j %. +%term /lfoc %pi o %-> t- %-> j %. +%term /rinv %pi o %-> j %. +%term /rfoc %pi p %-> j %. +%sort pf %. +%inline lfoc [A] [T] pf (/lfoc A T) %. +%inline rinv [A] pf (/rinv A) %. +%inline rfoc [B] pf (/rfoc B) %. +%sort <= %. +%sort hyp %. +% Stub judgment to make coverage nontrivial +%term refl <= T T %. +% Inference Rules +%term s+R %pi (hyp (s+ T)) %-> (rfoc (s+ T)) %. +%term s-L %pi (<= T T0) %-> (lfoc (s- T) T0) %. +%term vR %pi (rinv A) %-> (rfoc (v A)) %. +%term vL %pi (lfoc A T) %-> (hyp (v A)) %-> (rinv (s- T)) %. +%term topR rinv top %. +%term impR %pi (rinv (imp B A)) %<- (%pi (hyp B) %-> (rinv A)) %. +%term impL %pi (lfoc (imp B A) T) %<- (rfoc B) %<- (lfoc A T) %. +% Theorems +%sort trans %. +%sort mono/rinv %. +%sort mono/lfoc %. +%sort cut/rfoc {B} %. +%sort cut/rinv {A} %. +%sort cut/pc {A} %. +% mono/rinv proof +%term mono/rinv/ %pi (mono/rinv LE (vL LF H) (vL LF' H)) %<- (mono/lfoc LE LF LF') %. +% mono/lfoc proof +%term mono/lfoc/impL %pi (mono/lfoc LE (impL LF RF) (impL LF' RF)) %<- (mono/lfoc LE LF LF') %. +%term mono/lfoc/s- %pi (mono/lfoc LE (s-L LE') (s-L LE'')) %<- (trans LE' LE LE'') %. +% cut/rfoc proof +%term cut/rfoc/v %pi (cut/rfoc (v A) TM (vR RI) PF1 PF2) %<- (cut/rinv A TM RI PF1 PF2) %. +%term cut/rfoc/s+ cut/rfoc (s+ T) TM (s+R H) PF (PF H) %. +% cut/pc proof +%term cut/pc/imp + %pi (cut/pc (imp B A) TM (impR RI) (impL LF RF) RI'') + %<- (cut/rfoc B (s TM) RF RI RI') + %<- (cut/pc A TM RI' LF RI'') %. +%term cut/pc/s- %pi (cut/pc (s- T) TM IN (s-L LE) OUT) %<- (mono/rinv LE IN OUT) %. +% cut/rinv proof +%term cut/rinv/impL + %pi (cut/rinv A TM RI ([x] impL (LF x) (RF x)) (impL LF' RF')) + %<- (cut/rinv A TM RI LF LF') + %<- (cut/rinv A TM RI RF RF') %. +%term cut/rinv/impR + %pi (cut/rinv A TM RI ([x] impR ([y] D x y)) (impR Y)) + %<- ({y} cut/rinv A TM RI ([x] D x y) (Y y)) %. +%term cut/rinv/s-L cut/rinv A TM RI ([x] s-L D) (s-L D) %. +%term cut/rinv/vL/hit + %pi (cut/rinv A (s TM) RI ([x] vL (LF x) x) Y) + %<- (cut/rinv A (s TM) RI LF Z) + %<- (cut/pc A TM RI Z Y) %. +%term cut/rinv/vL/miss %pi (cut/rinv A TM RI ([x] vL (LF x) H) (vL LF' H)) %<- (cut/rinv A TM RI LF LF') %. +%term cut/rinv/vR %pi (cut/rinv A TM RI ([x] vR (RI0 x)) (vR RI0')) %<- (cut/rinv A TM RI RI0 RI0') %. +%term cut/rinv/s+R cut/rinv A TM RI ([x] s+R H) (s+R H) %. +%term cut/rinv/topR cut/rinv A TM RI ([x] topR) topR %. +% Checks +%block b {P p} [x hyp P]%. +%mode trans %in %in %out %. +%mode mono/rinv %in %in %out %. +%mode mono/lfoc %in %in %out %. +%mode cut/rfoc %in %in %in %in %out %. +%mode cut/rinv %in %in %in %in %out %. +%mode cut/pc %in %in %in %in %out %. +%worlds (b) (trans _ _ _) (mono/rinv _ _ _) (mono/lfoc _ _ _) %. +%worlds (b) (cut/rfoc _ _ _ _ _) (cut/rinv _ _ _ _ _) (cut/pc _ _ _ _ _) %. +% Assume structure relation is transitive +%total LE1 (trans LE1 LE2 LE3) %. +%total (RI1 LF1) (mono/rinv LE RI1 RI2) (mono/lfoc LE' LF1 LF2) %. +%total {(B A A') (N1 N2 N3) [(RF RI RI') (PFRF1 PFRI1 LF)]} (cut/rfoc B N1 RF PFRF1 PFRF2) (cut/rinv A N2 RI PFRI1 PFRI2) (cut/pc A' N3 RI' LF RI'') %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/ConstructiveSemantics). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/converting-between-implicit-and-explicit-parameters.lf b/new-tests/stelf-output/pages/converting-between-implicit-and-explicit-parameters.lf new file mode 100644 index 0000000..df9fc97 --- /dev/null +++ b/new-tests/stelf-output/pages/converting-between-implicit-and-explicit-parameters.lf @@ -0,0 +1,74 @@ +% +% +%{! When declaring a type family or constant, the Twelf programmer often has a choice between [implicit and explicit parameters](/wiki/implicit-and-explicit-parameters/) for some arguments. There is no reason to fret over this choice, for there is an easy technique for **converting between implicit and explicit parameters**. + +Suppose we have a language defined as follows: !}% +%sort exp %. +%sort typ %. +%sort of %. +%term 0 exp %. +%term 1 exp %. +%term bit typ %. +%term void typ %. +%term of1 of 1 bit %. +%term of0 of 0 bit %. +%{! We wish to define a translation between type derivations in this language, as a part of a source to source translation. The implicit and explicit versions are as follows: + +```twelf +%% implicit version +translate-i : of M A -> of M' A -> type. +%mode translate-i +D -D'. + +%% explicit version +translate-e : {m : exp}{a : typ} {m' : exp} + of m a -> of m' a -> type. +%mode translate-e +M +A -M' +D -D'. +``` + +## Defining explicit in terms of implicit + +Suppose we choose the implicit version and implement it, but later decide we prefer the explicit version. We can then define the explicit version in terms of the implicit one by simply leaving out arguments: !}% +% implicit version +%sort translate-i %. +%mode translate-i %in %out %. +%term translate-i1 translate-i of1 of0 %. +%term translate-i0 translate-i of0 of1 %. +% explicit version +%sort translate-e {m exp} {a typ} {m' exp} %. +%mode translate-e %in %in %out %in %out %. +%term translate-e/i %pi (translate-e M A M' D D') %<- (translate-i D D') %. +%worlds () (translate-i _ _) (translate-e _ _ _ _ _) %. +%total D (translate-i D _) %. +%total D (translate-e _ _ _ D _) %. +%{! ## Defining implicit in terms of explicit + +Suppose we define the explicit version, and then choose to define the implicit version in terms of the explicit: !}% +% explicit version +%sort translate2-e {m exp} {a typ} {m' exp} %. +%mode translate2-e %in %in %out %in %out %. +%term translate2-e1 translate2-e 1 bit 0 of1 of0 %. +%term translate2-e1 translate2-e 0 bit 1 of0 of1 %. +% implicit version +%sort translate2-i %. +%mode translate2-i %in %out %. +%term translate2-i/e %pi (translate2-i D D') %<- (translate2-e _ _ _ D D') %. +%worlds () (translate2-i _ _) (translate2-e _ _ _ _ _) %. +%total D (translate2-e _ _ _ D _) %. +%total D (translate2-i D _) %. +%{! Twelf is able to deduce via its term reconstruction algorithm what M, M' and A are, so this is [well-moded](/wiki/percent-mode/). + +## Using annotations + +Sometimes Twelf's term reconstruction algorithm is not powerful enough to automatically recover explicit parameters from implicit ones.\{\{discuss|Annotation example\}\} In general, one may use type annotations to get access to the implicit parameters to a relation. For example, we may write the ``translate2-i/e`` constant above as: + +```twelf +translate2-i/e : translate2-i (D : of M A) D' <- translate2-e M A M' D D'. +``` + +Here we have access to the implicit inputs ``M`` and ``A``, gleaned from the dependent type of ``D``. + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Converting_between_implicit_and_explicit_parameters). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/correctness-of-mergesort.lf b/new-tests/stelf-output/pages/correctness-of-mergesort.lf new file mode 100644 index 0000000..a9528b6 --- /dev/null +++ b/new-tests/stelf-output/pages/correctness-of-mergesort.lf @@ -0,0 +1,557 @@ +% +%{! +This is an extended example of a proof of correctness of mergesort for lists without duplicates. In particular, it is an example of showing that an implementation of sorting (mergesort) matches up with a declarative definition that relates a list to its sort (i.e. the sorted list is a permutation of the unsorted list). +!}% +%{! +### Definitions + +#### Declarative definitions + +We must first define natural numbers, lists of natural numbers, comparison on natural numbers, and what it means for one list to be the sorted version of another. +!}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort nat-less %. +%term nat-less/z nat-less z (s _) %. +%term nat-less/s %pi (nat-less (s N) (s N')) %<- (nat-less N N') %. +%sort nat-list %. +%term nat-list/nil nat-list %. +%term nat-list/cons %pi nat %-> nat-list %-> nat-list %. +%sort nat-list-head-less %. +%term nat-list-head-less/nil nat-list-head-less N nat-list/nil %. +%term nat-list-head-less/cons %pi (nat-list-head-less N (nat-list/cons N' _)) %<- (nat-less N N') %. +%sort nat-list-sorted %. +%term nat-list-sorted/nil nat-list-sorted nat-list/nil %. +%term nat-list-sorted/cons + %pi (nat-list-sorted (nat-list/cons N NL)) + %<- (nat-list-head-less N NL) + %<- (nat-list-sorted NL) %. +%sort in-nat-list %. +%term in-nat-list/hit in-nat-list N (nat-list/cons N NL) %. +%term in-nat-list/miss %pi (in-nat-list N (nat-list/cons N' NL)) %<- (in-nat-list N NL) %. +%sort all-in-nat-list %. +%term all-in-nat-list/nil all-in-nat-list nat-list/nil NL %. +%term all-in-nat-list/cons + %pi (all-in-nat-list (nat-list/cons N NL) NL') + %<- (in-nat-list N NL') + %<- (all-in-nat-list NL NL') %. +%{! +For the purposes of this proof, we use a set-theoretic extensional definition of permutation, where two lists are permutations of each other if they contain the same set of elements. This is only a proper definition of permutation if we assume both lists contain no duplicates. This invariant is baked into both our definition of sorted and our definition of mergesort. +!}% +%sort nat-list-permute %. +%term nat-list-permute/i + %pi (nat-list-permute NL NL') + %<- (all-in-nat-list NL NL') + %<- (all-in-nat-list NL' NL) %. +%sort nat-list-declarative-sort %. +%term nat-list-declarative-sort/i + %pi (nat-list-declarative-sort NL NL') + %<- (nat-list-permute NL NL') + %<- (nat-list-sorted NL') %. +%{! +#### Definitions used to define mergesort +!}% +%sort split %. +%term split/nil split nat-list/nil nat-list/nil nat-list/nil %. +%term split/1 split (nat-list/cons N nat-list/nil) (nat-list/cons N nat-list/nil) nat-list/nil %. +%term split/cons + %pi (split (nat-list/cons N (nat-list/cons N' NL)) (nat-list/cons N NL1) (nat-list/cons N' NL2)) + %<- (split NL NL1 NL2) %. +%sort merge %. +%term merge/nil-1 merge nat-list/nil N N %. +%term merge/nil-2 merge (nat-list/cons N NL) nat-list/nil (nat-list/cons N NL) %. +%term merge/cons-1 + %pi (merge (nat-list/cons N1 NL1) (nat-list/cons N2 NL2) (nat-list/cons N1 NL)) + %<- (nat-less N1 N2) + %<- (merge NL1 (nat-list/cons N2 NL2) NL) %. +%term merge/cons-2 + %pi (merge (nat-list/cons N1 NL1) (nat-list/cons N2 NL2) (nat-list/cons N2 NL)) + %<- (nat-less N2 N1) + %<- (merge NL2 (nat-list/cons N1 NL1) NL) %. +%sort mergesort %. +%term mergesort/nil mergesort nat-list/nil nat-list/nil %. +%term mergesort/1 mergesort (nat-list/cons N nat-list/nil) (nat-list/cons N nat-list/nil) %. +%term mergesort/cons + %pi (mergesort NL1 NL2) + %<- (split NL1 NLA NLB) + %<- (mergesort NLA NLA') + %<- (mergesort NLB NLB') + %<- (merge NLA' NLB' NL2) %. +%{! ### Helper definitions used in proofs + +``all-in-2 N1 N2 N3`` is a judgment that holds when all the elements in ``N1`` are in the union of ``N1`` and ``N2``/ + +!}% +% all-in-2 N1 N2 N3, everything in N1 is either in N2 or N3 +%sort all-in-2 %. +%term all-in-2/nil all-in-2 nat-list/nil NL1 NL2 %. +%term all-in-2/cons-1 + %pi (all-in-2 (nat-list/cons N NL) NL1 NL2) + %<- (in-nat-list N NL1) + %<- (all-in-2 NL NL1 NL2) %. +%term all-in-2/cons-2 + %pi (all-in-2 (nat-list/cons N NL) NL1 NL2) + %<- (in-nat-list N NL2) + %<- (all-in-2 NL NL1 NL2) %. +%{! ### Proof that mergesort returns a sorted result !}% +%sort nat-less-trans %. +%mode nat-less-trans %in %in %out %. +%term _ nat-less-trans nat-less/z D1 nat-less/z %. +%term _ + %pi (nat-less-trans (nat-less/s D1) (nat-less/s D2) (nat-less/s D3)) + %<- (nat-less-trans D1 D2 D3) %. +%worlds () (nat-less-trans _ _ _) %. +%total (D1) (nat-less-trans D1 _ _) %. +%sort merge-head-less %. +%mode merge-head-less %in %in %in %out %. +%term _ merge-head-less _ DL merge/nil-1 (nat-list-head-less/cons DL) %. +%term _ merge-head-less _ DL (merge/cons-2 _ _) (nat-list-head-less/cons DL) %. +%term _ merge-head-less (nat-list-head-less/cons DL) _ (merge/cons-1 _ _) (nat-list-head-less/cons DL) %. +%worlds () (merge-head-less _ _ _ _) %. +%total (D1) (merge-head-less _ _ D1 _) %. +%sort merge-sorted %. +%mode merge-sorted %in %in %in %out %. +%term _ merge-sorted _ D1 merge/nil-1 D1 %. +%term _ merge-sorted D1 _ merge/nil-2 D1 %. +%term _ + %pi (merge-sorted (nat-list-sorted/cons D1 DHL) D2 (merge/cons-1 DM DL) (nat-list-sorted/cons DS DHL')) + %<- (merge-head-less DHL DL DM DHL') + %<- (merge-sorted D1 D2 DM DS) %. +%term _ + %pi (merge-sorted D2 (nat-list-sorted/cons D1 DHL) (merge/cons-2 DM DL) (nat-list-sorted/cons DS DHL')) + %<- (merge-head-less DHL DL DM DHL') + %<- (merge-sorted D1 D2 DM DS) %. +%worlds () (merge-sorted _ _ _ _) %. +%total (D1) (merge-sorted _ _ D1 _) %. +%sort mergesort-sorted %. +%mode mergesort-sorted %in %out %. +%term _ mergesort-sorted mergesort/nil nat-list-sorted/nil %. +%term _ mergesort-sorted mergesort/1 (nat-list-sorted/cons nat-list-sorted/nil nat-list-head-less/nil) %. +%term _ + %pi (mergesort-sorted (mergesort/cons DM D2 D1 _) DS) + %<- (mergesort-sorted D1 DS1) + %<- (mergesort-sorted D2 DS2) + %<- (merge-sorted DS1 DS2 DM DS) %. +%worlds () (mergesort-sorted _ _) %. +%total (D1) (mergesort-sorted D1 _) %. +%{! ### Proof that mergesort returns a permutation of the input !}% +%sort all-in-2-wkn-l {N} %. +%mode all-in-2-wkn-l %in %in %out %. +%term _ all-in-2-wkn-l _ all-in-2/nil all-in-2/nil %. +%term _ + %pi (all-in-2-wkn-l _ (all-in-2/cons-1 DHL DIN) (all-in-2/cons-1 DHL' (in-nat-list/miss DIN))) + %<- (all-in-2-wkn-l _ DHL DHL') %. +%term _ + %pi (all-in-2-wkn-l _ (all-in-2/cons-2 DHL DIN) (all-in-2/cons-2 DHL' DIN)) + %<- (all-in-2-wkn-l _ DHL DHL') %. +%worlds () (all-in-2-wkn-l _ _ _) %. +%total (D1) (all-in-2-wkn-l _ D1 _) %. +%sort all-in-2-wkn-r {N} %. +%mode all-in-2-wkn-r %in %in %out %. +%term _ all-in-2-wkn-r _ all-in-2/nil all-in-2/nil %. +%term _ + %pi (all-in-2-wkn-r _ (all-in-2/cons-2 DHL DIN) (all-in-2/cons-2 DHL' (in-nat-list/miss DIN))) + %<- (all-in-2-wkn-r _ DHL DHL') %. +%term _ + %pi (all-in-2-wkn-r _ (all-in-2/cons-1 DHL DIN) (all-in-2/cons-1 DHL' DIN)) + %<- (all-in-2-wkn-r _ DHL DHL') %. +%worlds () (all-in-2-wkn-r _ _ _) %. +%total (D1) (all-in-2-wkn-r _ D1 _) %. +%sort split-all-in-2 %. +%mode split-all-in-2 %in %out %. +%term _ split-all-in-2 split/nil all-in-2/nil %. +%term _ split-all-in-2 split/1 (all-in-2/cons-1 all-in-2/nil in-nat-list/hit) %. +%term _ + %pi (split-all-in-2 (split/cons DS1) (all-in-2/cons-1 (all-in-2/cons-2 DS1''' in-nat-list/hit) in-nat-list/hit)) + %<- (split-all-in-2 DS1 DS1') + %<- (all-in-2-wkn-l _ DS1' DS1'') + %<- (all-in-2-wkn-r _ DS1'' DS1''') %. +%worlds () (split-all-in-2 _ _) %. +%total (D1) (split-all-in-2 D1 _) %. +%sort all-in-nat-list-wkn {N} %. +%mode all-in-nat-list-wkn %in %in %out %. +%term _ all-in-nat-list-wkn _ all-in-nat-list/nil all-in-nat-list/nil %. +%term _ + %pi (all-in-nat-list-wkn _ (all-in-nat-list/cons DAS DIN) (all-in-nat-list/cons DAS' (in-nat-list/miss DIN))) + %<- (all-in-nat-list-wkn _ DAS DAS') %. +%worlds () (all-in-nat-list-wkn _ _ _) %. +%total (D1) (all-in-nat-list-wkn _ D1 _) %. +%sort all-in-nat-list-refl {NL} %. +%mode all-in-nat-list-refl %in %out %. +%term _ all-in-nat-list-refl _ all-in-nat-list/nil %. +%term _ + %pi (all-in-nat-list-refl (nat-list/cons _ NL) (all-in-nat-list/cons DHL' in-nat-list/hit)) + %<- (all-in-nat-list-refl NL DHL) + %<- (all-in-nat-list-wkn _ DHL DHL') %. +%worlds () (all-in-nat-list-refl _ _) %. +%total (D1) (all-in-nat-list-refl D1 _) %. +%sort merge-all-in %. +%mode merge-all-in %in %out %out %. +%term _ + %pi (merge-all-in merge/nil-1 all-in-nat-list/nil D1) + %<- (all-in-nat-list-refl _ D1) %. +%term _ + %pi (merge-all-in merge/nil-2 D1 all-in-nat-list/nil) + %<- (all-in-nat-list-refl _ D1) %. +%term _ + %pi (merge-all-in (merge/cons-1 DM1 DL) (all-in-nat-list/cons DHL1' in-nat-list/hit) DHL2') + %<- (merge-all-in DM1 DHL1 DHL2) + %<- (all-in-nat-list-wkn _ DHL1 DHL1') + %<- (all-in-nat-list-wkn _ DHL2 DHL2') %. +%term _ + %pi (merge-all-in (merge/cons-2 DM1 DL) DHL1' (all-in-nat-list/cons DHL2' in-nat-list/hit)) + %<- (merge-all-in DM1 DHL2 DHL1) + %<- (all-in-nat-list-wkn _ DHL1 DHL1') + %<- (all-in-nat-list-wkn _ DHL2 DHL2') %. +%worlds () (merge-all-in _ _ _) %. +%total (D1) (merge-all-in D1 _ _) %. +%sort in-nat-list-trans %. +%mode in-nat-list-trans %in %in %out %. +%term _ in-nat-list-trans in-nat-list/hit (all-in-nat-list/cons _ D1) D1 %. +%term _ + %pi (in-nat-list-trans (in-nat-list/miss D1) (all-in-nat-list/cons D _) D1') + %<- (in-nat-list-trans D1 D D1') %. +%worlds () (in-nat-list-trans _ _ _) %. +%total (D1) (in-nat-list-trans D1 _ _) %. +%sort all-in-2-trans %. +%mode all-in-2-trans %in %in %in %out %. +%term _ all-in-2-trans all-in-2/nil _ _ all-in-nat-list/nil %. +%term _ + %pi (all-in-2-trans (all-in-2/cons-1 DHL DIN) DA DB (all-in-nat-list/cons DHL' DIN')) + %<- (all-in-2-trans DHL DA DB DHL') + %<- (in-nat-list-trans DIN DA DIN') %. +%term _ + %pi (all-in-2-trans (all-in-2/cons-2 DHL DIN) DA DB (all-in-nat-list/cons DHL' DIN')) + %<- (all-in-2-trans DHL DA DB DHL') + %<- (in-nat-list-trans DIN DB DIN') %. +%worlds () (all-in-2-trans _ _ _ _) %. +%total (D1) (all-in-2-trans D1 _ _ _) %. +%sort all-in-2-trans-a %. +%mode all-in-2-trans-a %in %in %in %out %. +%term _ all-in-2-trans-a all-in-2/nil _ _ all-in-2/nil %. +%term _ + %pi (all-in-2-trans-a (all-in-2/cons-1 DHL DIN) DA DB (all-in-2/cons-1 DHL' DIN')) + %<- (all-in-2-trans-a DHL DA DB DHL') + %<- (in-nat-list-trans DIN DA DIN') %. +%term _ + %pi (all-in-2-trans-a (all-in-2/cons-2 DHL DIN) DA DB (all-in-2/cons-2 DHL' DIN')) + %<- (all-in-2-trans-a DHL DA DB DHL') + %<- (in-nat-list-trans DIN DB DIN') %. +%worlds () (all-in-2-trans-a _ _ _ _) %. +%total (D1) (all-in-2-trans-a D1 _ _ _) %. +%sort mergesort-all-in %. +%mode mergesort-all-in %in %out %. +%term _ mergesort-all-in mergesort/nil all-in-nat-list/nil %. +%term _ mergesort-all-in mergesort/1 (all-in-nat-list/cons all-in-nat-list/nil in-nat-list/hit) %. +%term _ + %pi (mergesort-all-in (mergesort/cons Dmerge Dms2 Dms1 Dsplit) DHLtwo'') + %<- (split-all-in-2 Dsplit (%the (all-in-2 NL NA NB) DHLtwo)) + %<- (mergesort-all-in Dms1 (%the (all-in-nat-list NA NA') DHL1)) + %<- (mergesort-all-in Dms2 DHL2) + %<- (merge-all-in Dmerge DHL3 DHL4) + %<- (all-in-2-trans-a DHLtwo DHL1 DHL2 DHLtwo') + %<- (all-in-2-trans DHLtwo' DHL3 DHL4 DHLtwo'') %. +%worlds () (mergesort-all-in _ _) %. +%total (D1) (mergesort-all-in D1 _) %. +%sort all-in-2-refl-r {NL} {NL'} %. +%mode all-in-2-refl-r %in %in %out %. +%term _ all-in-2-refl-r _ _ all-in-2/nil %. +%term _ + %pi (all-in-2-refl-r (nat-list/cons N NL) _ (all-in-2/cons-2 DHL' in-nat-list/hit)) + %<- (all-in-2-refl-r NL _ DHL) + %<- (all-in-2-wkn-r _ DHL DHL') %. +%worlds () (all-in-2-refl-r _ _ _) %. +%total (D1) (all-in-2-refl-r D1 _ _) %. +%sort all-in-2-refl-l {NL} {NL'} %. +%mode all-in-2-refl-l %in %in %out %. +%term _ all-in-2-refl-l _ _ all-in-2/nil %. +%term _ + %pi (all-in-2-refl-l (nat-list/cons N NL) _ (all-in-2/cons-1 DHL' in-nat-list/hit)) + %<- (all-in-2-refl-l NL _ DHL) + %<- (all-in-2-wkn-l _ DHL DHL') %. +%worlds () (all-in-2-refl-l _ _ _) %. +%total (D1) (all-in-2-refl-l D1 _ _) %. +%sort all-in-2-sym %. +%mode all-in-2-sym %in %out %. +%term _ all-in-2-sym all-in-2/nil all-in-2/nil %. +%term _ + %pi (all-in-2-sym (all-in-2/cons-1 DHL DIN) (all-in-2/cons-2 DHL' DIN)) + %<- (all-in-2-sym DHL DHL') %. +%term _ + %pi (all-in-2-sym (all-in-2/cons-2 DHL DIN) (all-in-2/cons-1 DHL' DIN)) + %<- (all-in-2-sym DHL DHL') %. +%worlds () (all-in-2-sym _ _) %. +%total (D1) (all-in-2-sym D1 _) %. +%sort merge-all-in-2 %. +%mode merge-all-in-2 %in %out %. +%term _ %pi (merge-all-in-2 merge/nil-1 D1) %<- (all-in-2-refl-r _ _ D1) %. +%term _ %pi (merge-all-in-2 merge/nil-2 D1) %<- (all-in-2-refl-l _ _ D1) %. +%term _ + %pi (merge-all-in-2 (merge/cons-1 DM _) (all-in-2/cons-1 DHL' in-nat-list/hit)) + %<- (merge-all-in-2 DM DHL) + %<- (all-in-2-wkn-l _ DHL DHL') %. +%term _ + %pi (merge-all-in-2 (merge/cons-2 DM _) (all-in-2/cons-2 DHL'' in-nat-list/hit)) + %<- (merge-all-in-2 DM DHL) + %<- (all-in-2-wkn-l _ DHL DHL') + %<- (all-in-2-sym DHL' DHL'') %. +%worlds () (merge-all-in-2 _ _) %. +%total (D1) (merge-all-in-2 D1 _) %. +%sort split-all-in %. +%mode split-all-in %in %out %out %. +%term _ split-all-in split/nil all-in-nat-list/nil all-in-nat-list/nil %. +%term _ split-all-in split/1 (all-in-nat-list/cons all-in-nat-list/nil in-nat-list/hit) all-in-nat-list/nil %. +%term _ + %pi (split-all-in (split/cons DS) (all-in-nat-list/cons DHL'' in-nat-list/hit) (all-in-nat-list/cons DBL'' (in-nat-list/miss in-nat-list/hit))) + %<- (split-all-in DS DHL DBL) + %<- (all-in-nat-list-wkn _ DHL DHL') + %<- (all-in-nat-list-wkn _ DHL' DHL'') + %<- (all-in-nat-list-wkn _ DBL DBL') + %<- (all-in-nat-list-wkn _ DBL' DBL'') %. +%worlds () (split-all-in _ _ _) %. +%total (D1) (split-all-in D1 _ _) %. +%sort mergesort-all-in-r %. +%mode mergesort-all-in-r %in %out %. +%term _ mergesort-all-in-r mergesort/nil all-in-nat-list/nil %. +%term _ mergesort-all-in-r mergesort/1 (all-in-nat-list/cons all-in-nat-list/nil in-nat-list/hit) %. +%term _ + %pi (mergesort-all-in-r (mergesort/cons Dmerge Dms2 Dms1 Dsplit) DHLtwo'') + %<- (merge-all-in-2 Dmerge (%the (all-in-2 NL NA NB) DHLtwo)) + %<- (mergesort-all-in-r Dms1 (%the (all-in-nat-list NA NA') DHL1)) + %<- (mergesort-all-in-r Dms2 DHL2) + %<- (split-all-in Dsplit DHL3 DHL4) + %<- (all-in-2-trans-a DHLtwo DHL1 DHL2 DHLtwo') + %<- (all-in-2-trans DHLtwo' DHL3 DHL4 DHLtwo'') %. +%worlds () (mergesort-all-in-r _ _) %. +%total (D1) (mergesort-all-in-r D1 _) %. +%sort mergesort-permute %. +%mode mergesort-permute %in %out %. +%term _ + %pi (mergesort-permute D1 (nat-list-permute/i DB DA)) + %<- (mergesort-all-in D1 DA) + %<- (mergesort-all-in-r D1 DB) %. +%worlds () (mergesort-permute _ _) %. +%total {} (mergesort-permute _ _) %. +%{! ### Final result +Showing the final result merely requires composing the two big intermediate results, namely that mergesort produces a sorted output and that mergesort's output is a permutation of the input. + +!}% +%sort mergesort-correct %. +%mode mergesort-correct %in %out %. +%term _ + %pi (mergesort-correct D1 (nat-list-declarative-sort/i Dsorted Dpermute)) + %<- (mergesort-permute D1 Dpermute) + %<- (mergesort-sorted D1 Dsorted) %. +%worlds () (mergesort-correct _ _) %. +%total {} (mergesort-correct _ _) %. +%{! + +---- + +## Equivalence of definitions of permutations + +We can also show that ``nat-list-permute`` under the right constraints (implied by ``nat-list-sorted`` is equivalent to a definition of permutations as a composition of swaps. + +Note: Maybe this should be its own article? + +!}% +%sort all-in-nat-list-trans %. +%mode all-in-nat-list-trans %in %in %out %. +%term _ all-in-nat-list-trans all-in-nat-list/nil _ all-in-nat-list/nil %. +%term _ + %pi (all-in-nat-list-trans (all-in-nat-list/cons D2 D1) D3 (all-in-nat-list/cons D5 D4)) + %<- (in-nat-list-trans D1 D3 D4) + %<- (all-in-nat-list-trans D2 D3 D5) %. +%worlds () (all-in-nat-list-trans _ _ _) %. +%total (D1) (all-in-nat-list-trans D1 _ _) %. +%sort nat-list-permutes %. +%term nat-list-permutes/nil nat-list-permutes nat-list/nil nat-list/nil %. +%term nat-list-permutes/cons + %pi (nat-list-permutes (nat-list/cons N1 NL) (nat-list/cons N1 NL')) + %<- (nat-list-permutes NL NL') %. +%term nat-list-permutes/swap nat-list-permutes (nat-list/cons N1 (nat-list/cons N2 NL)) (nat-list/cons N2 (nat-list/cons N1 NL)) %. +%term nat-list-permutes/trans + %pi (nat-list-permutes N1 N3) + %<- (nat-list-permutes N1 N2) + %<- (nat-list-permutes N2 N3) %. +%sort nat-list-permutes-all-in %. +%mode nat-list-permutes-all-in %in %out %. +%term _ nat-list-permutes-all-in nat-list-permutes/nil all-in-nat-list/nil %. +%term _ + %pi (nat-list-permutes-all-in (nat-list-permutes/cons D1) (all-in-nat-list/cons D2' in-nat-list/hit)) + %<- (nat-list-permutes-all-in D1 D2) + %<- (all-in-nat-list-wkn _ D2 D2') %. +%term _ + %pi (nat-list-permutes-all-in nat-list-permutes/swap (all-in-nat-list/cons (all-in-nat-list/cons D1'' in-nat-list/hit) (in-nat-list/miss in-nat-list/hit))) + %<- (all-in-nat-list-refl _ D1) + %<- (all-in-nat-list-wkn _ D1 D1') + %<- (all-in-nat-list-wkn _ D1' D1'') %. +%term _ + %pi (nat-list-permutes-all-in (nat-list-permutes/trans D2 D1) D3') + %<- (nat-list-permutes-all-in D1 D1') + %<- (nat-list-permutes-all-in D2 D2') + %<- (all-in-nat-list-trans D1' D2' D3') %. +%worlds () (nat-list-permutes-all-in _ _) %. +%total (D1) (nat-list-permutes-all-in D1 _) %. +%sort nat-neq %. +%term nat-neq/l %pi (nat-neq N1 N2) %<- (nat-less N1 N2) %. +%term nat-neq/r %pi (nat-neq N1 N2) %<- (nat-less N2 N1) %. +%sort nin-nat-list %. +%term nin-nat-list/nil nin-nat-list N1 nat-list/nil %. +%term nin-nat-list/cons + %pi (nin-nat-list N1 (nat-list/cons N2 NL)) + %<- (nat-neq N1 N2) + %<- (nin-nat-list N1 NL) %. +%sort nat-list-no-dups %. +%term nat-list-no-dups/nil nat-list-no-dups nat-list/nil %. +%term nat-list-no-dups/cons + %pi (nat-list-no-dups (nat-list/cons N1 NL)) + %<- (nin-nat-list N1 NL) + %<- (nat-list-no-dups NL) %. +%sort nat-list-permutes-refl {NL} %. +%mode nat-list-permutes-refl %in %out %. +%term _ nat-list-permutes-refl nat-list/nil nat-list-permutes/nil %. +%term _ + %pi (nat-list-permutes-refl (nat-list/cons _ NL) (nat-list-permutes/cons D1)) + %<- (nat-list-permutes-refl NL D1) %. +%worlds () (nat-list-permutes-refl _ _) %. +%total (D1) (nat-list-permutes-refl D1 _) %. +%sort nat-list-permutes-sym %. +%mode nat-list-permutes-sym %in %out %. +%term _ nat-list-permutes-sym nat-list-permutes/nil nat-list-permutes/nil %. +%term _ + %pi (nat-list-permutes-sym (nat-list-permutes/cons D1) (nat-list-permutes/cons D1')) + %<- (nat-list-permutes-sym D1 D1') %. +%term _ nat-list-permutes-sym nat-list-permutes/swap nat-list-permutes/swap %. +%term _ + %pi (nat-list-permutes-sym (nat-list-permutes/trans D2 D1) (nat-list-permutes/trans D1' D2')) + %<- (nat-list-permutes-sym D1 D1') + %<- (nat-list-permutes-sym D2 D2') %. +%worlds () (nat-list-permutes-sym _ _) %. +%total (D1) (nat-list-permutes-sym D1 _) %. +%sort nat-list-permutes-swap-head-2 %. +%mode nat-list-permutes-swap-head-2 %in %out %. +%term _ nat-list-permutes-swap-head-2 D1 (nat-list-permutes/trans nat-list-permutes/swap D1) %. +%worlds () (nat-list-permutes-swap-head-2 _ _) %. +%total {} (nat-list-permutes-swap-head-2 _ _) %. +%sort in-nat-list-permutes %. +%mode in-nat-list-permutes %in %out %. +%term _ %pi (in-nat-list-permutes in-nat-list/hit D1) %<- (nat-list-permutes-refl _ D1) %. +%term _ + %pi (in-nat-list-permutes (in-nat-list/miss (%the (in-nat-list N (nat-list/cons N2 NL)) D1)) DP) + %<- (in-nat-list-permutes D1 (%the (nat-list-permutes (nat-list/cons N2 NL) (nat-list/cons N NL')) DL)) + %<- (nat-list-permutes-swap-head-2 (nat-list-permutes/cons DL) DP) %. +%worlds () (in-nat-list-permutes _ _) %. +%total (D1) (in-nat-list-permutes D1 _) %. +%sort nin-nat-list-permutes %. +%mode nin-nat-list-permutes %in %in %out %. +%term _ nin-nat-list-permutes _ nat-list-permutes/nil nin-nat-list/nil %. +%term _ + %pi (nin-nat-list-permutes (nin-nat-list/cons D1 Dneq) (nat-list-permutes/cons D2) (nin-nat-list/cons D1' Dneq)) + %<- (nin-nat-list-permutes D1 D2 D1') %. +%term _ nin-nat-list-permutes (nin-nat-list/cons (nin-nat-list/cons D1 Dneq2) Dneq1) nat-list-permutes/swap (nin-nat-list/cons (nin-nat-list/cons D1 Dneq1) Dneq2) %. +%term _ + %pi (nin-nat-list-permutes D1 (nat-list-permutes/trans D3 D2) D1'') + %<- (nin-nat-list-permutes D1 D2 D1') + %<- (nin-nat-list-permutes D1' D3 D1'') %. +%worlds () (nin-nat-list-permutes _ _ _) %. +%total (D1) (nin-nat-list-permutes _ D1 _) %. +%sort nat-neq-sym %. +%mode nat-neq-sym %in %out %. +%term _ nat-neq-sym (nat-neq/l D1) (nat-neq/r D1) %. +%term _ nat-neq-sym (nat-neq/r D1) (nat-neq/l D1) %. +%worlds () (nat-neq-sym _ _) %. +%total {} (nat-neq-sym _ _) %. +%sort no-dups-permutes %. +%mode no-dups-permutes %in %in %out %. +%term _ no-dups-permutes _ nat-list-permutes/nil nat-list-no-dups/nil %. +%term _ + %pi (no-dups-permutes (nat-list-no-dups/cons D2 D1) (nat-list-permutes/cons D3) (nat-list-no-dups/cons D2' D1')) + %<- (nin-nat-list-permutes D1 D3 D1') + %<- (no-dups-permutes D2 D3 D2') %. +%term _ + %pi (no-dups-permutes (nat-list-no-dups/cons (nat-list-no-dups/cons D3 DninB) (nin-nat-list/cons DninA DneqA1)) nat-list-permutes/swap (nat-list-no-dups/cons (nat-list-no-dups/cons D3 DninA) (nin-nat-list/cons DninB DneqA1'))) + %<- (nat-neq-sym DneqA1 DneqA1') %. +%term _ + %pi (no-dups-permutes D1 (nat-list-permutes/trans D3 D2) D1'') + %<- (no-dups-permutes D1 D2 D1') + %<- (no-dups-permutes D1' D3 D1'') %. +%worlds () (no-dups-permutes _ _ _) %. +%total (D1) (no-dups-permutes _ D1 _) %. +%sort nat-list-length %. +%term nat-list-length/z nat-list-length nat-list/nil z %. +%term nat-list-length/s %pi (nat-list-length (nat-list/cons _ NL) (s N)) %<- (nat-list-length NL N) %. +%sort permutes-length %. +%mode permutes-length %in %in %out %. +%term _ permutes-length _ nat-list-permutes/nil nat-list-length/z %. +%term _ + %pi (permutes-length (nat-list-length/s D1) (nat-list-permutes/cons D2) (nat-list-length/s D3)) + %<- (permutes-length D1 D2 D3) %. +%term _ permutes-length (nat-list-length/s (nat-list-length/s D1)) nat-list-permutes/swap (nat-list-length/s (nat-list-length/s D1)) %. +%term _ + %pi (permutes-length D1 (nat-list-permutes/trans D3 D2) D3') + %<- (permutes-length D1 D2 D1') + %<- (permutes-length D1' D3 D3') %. +%worlds () (permutes-length _ _ _) %. +%total (D1) (permutes-length _ D1 _) %. +%sort uninhabited %. +%sort nat-less-irrefl %. +%mode nat-less-irrefl %in %out %. +%term _ %pi (nat-less-irrefl (nat-less/s D1) DU) %<- (nat-less-irrefl D1 DU) %. +%worlds () (nat-less-irrefl _ _) %. +%total (D1) (nat-less-irrefl D1 _) %. +%sort nat-neq-irrefl %. +%mode nat-neq-irrefl %in %out %. +%term _ %pi (nat-neq-irrefl (nat-neq/l D1) DU) %<- (nat-less-irrefl D1 DU) %. +%term _ %pi (nat-neq-irrefl (nat-neq/r D1) DU) %<- (nat-less-irrefl D1 DU) %. +%worlds () (nat-neq-irrefl _ _) %. +%total {} (nat-neq-irrefl _ _) %. +%sort uninhabited-in-nat-list {N1} {NL} %. +%mode uninhabited-in-nat-list %in %in %in %out %. +%worlds () (uninhabited-in-nat-list _ _ _ _) %. +%total {} (uninhabited-in-nat-list _ _ _ _) %. +%sort in-nat-list-strengthen %. +%mode in-nat-list-strengthen %in %in %out %. +%term _ in-nat-list-strengthen _ (in-nat-list/miss D1) D1 %. +%term _ + %pi (in-nat-list-strengthen D1 in-nat-list/hit D1') + %<- (nat-neq-irrefl D1 DU) + %<- (uninhabited-in-nat-list _ _ DU D1') %. +%worlds () (in-nat-list-strengthen _ _ _) %. +%total (D1) (in-nat-list-strengthen _ D1 _) %. +%sort all-in-strengthen %. +%mode all-in-strengthen %in %in %out %. +%term _ all-in-strengthen _ all-in-nat-list/nil all-in-nat-list/nil %. +%term _ + %pi (all-in-strengthen (nin-nat-list/cons D1 Dneq) (all-in-nat-list/cons DB DA) (all-in-nat-list/cons DB' DA')) + %<- (all-in-strengthen D1 DB DB') + %<- (in-nat-list-strengthen Dneq DA DA') %. +%worlds () (all-in-strengthen _ _ _) %. +%total (D1) (all-in-strengthen _ D1 _) %. +%sort all-in-no-dups-permutes {N} %. +%mode all-in-no-dups-permutes %in %in %in %in %in %out %. +%term _ all-in-no-dups-permutes _ _ _ _ all-in-nat-list/nil nat-list-permutes/nil %. +%term _ + %pi (all-in-no-dups-permutes _ (nat-list-no-dups/cons D1 DNINN) (nat-list-length/s Dl1) DN (all-in-nat-list/cons (%the (all-in-nat-list NL NL') DA) DIN) (nat-list-permutes/trans DP' (nat-list-permutes/cons DPP))) + %<- (in-nat-list-permutes DIN (%the (nat-list-permutes NL' (nat-list/cons N1 NL'')) DP)) + %<- (nat-list-permutes-all-in DP (%the (all-in-nat-list NL' (nat-list/cons N1 NL'')) DIN')) + %<- (permutes-length DN DP (nat-list-length/s D3')) + %<- (all-in-nat-list-trans DA DIN' DIN'') + %<- (all-in-strengthen DNINN DIN'' DIN''') + %<- (all-in-no-dups-permutes _ D1 Dl1 D3' DIN''' DPP) + %<- (nat-list-permutes-sym DP DP') %. +%worlds () (all-in-no-dups-permutes _ _ _ _ _ _) %. +%total (D1) (all-in-no-dups-permutes D1 _ _ _ _ _) %. +%{! + +TODO: Finish commentary on the proof. + +\{\{stub\}\} +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Correctness_of_mergesort). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/coverage-checking.lf b/new-tests/stelf-output/pages/coverage-checking.lf new file mode 100644 index 0000000..d26cd18 --- /dev/null +++ b/new-tests/stelf-output/pages/coverage-checking.lf @@ -0,0 +1,82 @@ +% +%{! **Coverage checking** is used to describe two analyses that Twelf needs to perform in order to verify [totality assertions](/wiki/totality-assertion/). + +The first analysis, for **input coverage**, is performed by both [``%total``](/wiki/percent-total/) and [``%covers``](/wiki/percent-covers/) declarations and resembles exhaustiveness checking in a language like ML. + +The second analysis, **output coverage**, is performed only by [``%total``](/wiki/percent-total/) to ensure that the outputs of subgoals are sufficiently general. Output coverage consists of a check for incorrect constant pattern-matching and a check for [output freeness](/wiki/output-freeness/) violations. + +We will use the common example of [natural numbers](/wiki/natural-numbers/) for this example: !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%{! ## Input coverage + +Input coverage is similar to exhaustiveness checking in ML, in that it ensures that your relation is prepared to accept any input that is thrown at it. We would run afoul of the input coverage checker if we tried to run a ``%covers`` or ``%total`` declaration on the relation ``half`` that halves a natural number but is undefined on odd numbers. Another example of input coverage checking can be seen at the page on [``%covers``](/wiki/percent-covers/). !}% +%sort half %. +%mode half %in %out %. +%term half/z half z z %. +%term half/s %pi (half (s (s N)) (s M)) %<- (half N M) %. +%worlds () (half _ _) %. +%{! + +```checkedtwelf +%total N (half N _). +``` + +## Output coverage + +Output coverage ensures that your relation is prepared to deal with any possible term that it might be output by a [[subgoal]]. Output coverage does not have quite so obvious a counterpart in functional languages like ML; the reason is that Twelf differentiates between case-analyzing outputs and case-analyzing inputs, whereas in ML there is just one case construct. + +### Output constant pattern-matching + +Say we attempted to fix ``half`` by defining ``half`` of 1 to be 0 by adding the case ``half/1`` shown below: !}% +%sort half %. +%mode half %in %out %. +%term half/z half z z %. +%term half/1 %pi (half (s z) z) %<- (half z z) %. +%term half/s %pi (half (s (s N)) (s M)) %<- (half N M) %. +%worlds () (half _ _) %. +%{! For pedagogical purposes, we have given the constant ``half/1`` an unnecessary extra premise that ``half z z ``. Thus, the case ``half/1`` expects the "output" of the subgoal ``<- half z z`` to be ``z``; inspecting the program, this is a true fact, but Twelf is not capable of verifying this and complains accordingly. + +```checkedtwelf +%total N (half N _). +``` + +In this case, we can fix the problem in one of two ways. First, we can remove the subgoal entirely: + +```twelf +half/1 : half (s z) z. +``` + +Alternatively, we can rewrite the subgoal, leaving the output a free variable: + +```twelf +half/1 : half (s z) N + <- half z N. +``` + +The full working example with the first approach looks like this: !}% +%sort half %. +%mode half %in %out %. +%term half/z half z z %. +%term half/1 half (s z) z %. +%term half/s %pi (half (s (s N)) (s M)) %<- (half N M) %. +%worlds () (half _ _) %. +%{! In many circumstances when the output coverage checker rejects a totality assertion that should, in fact, be true, a standard technique [output factoring](/wiki/output-factoring/) can deal with the problem. + +### Output freeness + +The other part of output coverage checking is [output freeness](/wiki/output-freeness/) checking; see its page for a definition and examples. + +## See also + +* Coverage +* [Output freeness](/wiki/output-freeness/) +* [Debugging coverage errors](/wiki/debugging-coverage-errors/) +* [Output factoring](/wiki/output-factoring/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +You can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Coverage_checking). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/cps-conversion.lf b/new-tests/stelf-output/pages/cps-conversion.lf new file mode 100644 index 0000000..248a52f --- /dev/null +++ b/new-tests/stelf-output/pages/cps-conversion.lf @@ -0,0 +1,305 @@ +% +%{! **CPS conversion** is the process of translating a direct style calculus into a continuation passing style (CPS) one. This tutorial describes one way of formalizing CPS conversion in Twelf. It is an advanced tutorial because it relies on techniques described in other tutorials, such as reversing the polarity. + +We will define a direct style lambda calculus and its static semantics, then a CPS language and its static semantics. We will then prove that any well-formed direct style expression can be translated into a well-formed CPS expression. + +## Direct style language + +We begin with a direct style lambda calculus with products, unit, and a ``let`` construct. !}% +%sort typ %. +%name typ %. +%term => %pi typ %-> typ %-> typ %. +%prec %right 8 => %. +%term & %pi typ %-> typ %-> typ %. +%prec %none 9 & %. +%term unit typ %. +%% expressions +%sort exp %. +%sort val %. +%% values are expressions, too. +%term value %pi val %-> exp %. +%term lam %pi (%pi val %-> exp) %-> val %. +%term app %pi exp %-> exp %-> exp %. +%term mkpair %pi exp %-> exp %-> exp %. +%term pair %pi val %-> val %-> val %. +%term fst %pi exp %-> exp %. +%term snd %pi exp %-> exp %. +%term 1 val %. +%term let %pi exp %-> (%pi val %-> exp) %-> exp %. +%{! Here we make a syntactic distinction between values (of LF type ``val``) and expressions (``exp``). A constructor ``value`` injects syntactic values into expressions. It would also be possible to do a more [extrinsic encoding](/wiki/intrinsic-and-extrinsic-encodings/) of values by instead defining a judgment over expressions that defined the subset that are values. This encoding makes more facts explicit in the typing (for instance it is immediately evident that the calculus is call-by-value since lambdas expect values as arguments) which resists mistakes and saves us some lemmas. Note that we distinguish between the expression ``mkpair`` whose evaluation creates a pair and the value that results, ``pair``. + +The typing rules for this language are straightforward: !}% +% |- e : t +%sort of %. +% |- v : t +%sort ofv %. +%term ofvalue %pi (of (value V) A) %<- (ofv V A) %. +%term unitI ofv 1 unit %. +%term =>I + %pi (ofv (lam ([x val] M x)) (A => B)) + %<- ({x val} %pi (ofv x A) %-> (of (M x) B)) %. +%term =>E %pi (of M1 (A => B)) %-> (of M2 A) %-> (of (app M1 M2) B) %. +%term &Iv %pi (ofv V1 A) %-> (ofv V2 B) %-> (ofv (pair V1 V2) (A & B)) %. +%term &I %pi (of M1 A) %-> (of M2 B) %-> (of (mkpair M1 M2) (A & B)) %. +%term &E1 %pi (of (fst M) A) %<- (of M (A & B)) %. +%term &E2 %pi (of (snd M) B) %<- (of M (A & B)) %. +%term oflet %pi (of M A) %-> ({y val} {ofy ofv y A} of (N y) C) %-> (of (let M N) C) %. +%{! The type family ``of M A`` encodes the typing judgment , and ``ofv V A`` the judgment . + +## CPS language + +We then define the syntax of the CPS language that will be the target of CPS conversion: !}% +%% CPS language +%sort cexp %. +%name cexp %. +%sort ctyp %. +%name ctyp %. +%sort cval %. +%name cval %. +%term ccont %pi ctyp %-> ctyp %. +%prec %postfix 8 ccont %. +%term c& %pi ctyp %-> ctyp %-> ctyp %. +%prec %none 9 c& %. +%term cunit ctyp %. +% continuation expressions +%term cmkpair %pi cval %-> cval %-> (%pi cval %-> cexp) %-> cexp %. +%term cfst %pi cval %-> (%pi cval %-> cexp) %-> cexp %. +%term csnd %pi cval %-> (%pi cval %-> cexp) %-> cexp %. +% continuations can end with a call to a function +%term ccall %pi cval %-> cval %-> cexp %. +%term chalt cexp %. +% continuation values +%term cpair %pi cval %-> cval %-> cval %. +%term clam %pi (%pi cval %-> cexp) %-> cval %. +%term c1 cval %. +%{! This language makes the same syntactic distinction between continuation expressions (``cexp``) and values (``cval``). A continuation expression is a linear sequence of operations on values (here ``cmkpair``, ``cfst`` and ``csnd``), each of which binds a variable. A continuation expression ends with a call to a continuation (``ccall``) or by halting (``chalt``). A continuation value is either a pair of values (``cpair``), the unit element (``c1``) or a lambda (``clam``) which abstracts a continuation expression. Because continuations never return, lambdas have type ``A ccont`` for some argument type A. (Note that the ``ccont`` type constructor is written in postfix notation.) In some presentations of CPS, this is written as A → 0, A → ⊥, or ¬A. + +The static semantics for the CPS language is as follows: !}% +% |- c ok +%sort cof %. +%name cof %. +%mode cof %star %. +% |- cv : A +%sort cofv %. +%name cofv %. +%mode cofv %star %star %. +%term co_halt cof chalt %. +%term co_mkpair + %pi (cofv V1 A) + %-> (cofv V2 B) + %-> ({v cval} %pi (cofv v (A c& B)) %-> (cof (K v))) + %-> (cof (cmkpair V1 V2 K)) %. +%term co_call %pi (cofv F (A ccont)) %-> (cofv V A) %-> (cof (ccall F V)) %. +%term co_fst %pi (cofv V (A c& B)) %-> ({v} {ov cofv v A} cof (C v)) %-> (cof (cfst V C)) %. +%term co_snd %pi (cofv V (A c& B)) %-> ({v} {ov cofv v B} cof (C v)) %-> (cof (csnd V C)) %. +%term cov_unit cofv c1 cunit %. +%term cov_pair %pi (cofv V1 A) %-> (cofv V2 B) %-> (cofv (cpair V1 V2) (A c& B)) %. +%term cov_lam %pi ({x} %pi (cofv x A) %-> (cof (C x))) %-> (cofv (clam C) (A ccont)) %. +%{! The type family ``cof C`` encodes the typing judgment . Continuations do not return, so they do not have a type. The type family ``ofv V A`` the judgment for CPS values V. The typing rules are straightforward. + +## Translation + +### Type translation + +We can now translate from the direct style language to the CPS language. Because a translated value will have a different type in the source and destination languages (indeed, the source and destination types are different LF types!), we start by defining a translation for the types of the source language. !}% +%sort ttoct %. +%mode ttoct %in %out %. +%term ttoct/unit ttoct unit cunit %. +%term ttoct/& %pi (ttoct (A & B) (A' c& B')) %<- (ttoct B B') %<- (ttoct A A') %. +%term ttoct/=> %pi (ttoct (A => B) (A' c& (B' ccont) ccont)) %<- (ttoct B B') %<- (ttoct A A') %. +%{! The family ``ttoct A A'`` ("typ to ctyp") encodes the type translation, which is often written using [Scott](https://en.wikipedia.org/wiki/Dana_Scott) brackets . The unit type and product types simply translate recursively into their correspondents in the CPS types. An arrow type A → B translates into `(A' c& (B' ccont)) ccont`. That is, a function is translated into a continuation that expects the (translated) argument paired with a return continuation for the (translated) result. + +To check our work, we can verify that type translation is [total](/wiki/percent-total/). This result will not be important to our main theorem, but every type should certainly have a translation and it is an easy way to check for mistakes early. !}% +%worlds () (ttoct _ _) %. +%total D (ttoct D _) %. +%{! ### Translation relation + +The translation for values is a three place relation: !}% +%sort tocpsv+ {WV ofv V A} {CT ttoct A CA} {WCV cofv CV CA} %. +%mode tocpsv+ %in %in %out %. +%{! ``WV`` is a value typing derivation in the source language. Our translation will be type-directed, translating typing derivations to typing derivations. The terms, which are in the types of the typing derivations, can easily be recovered (in fact, they are [implicit parameter](/wiki/implicit-and-explicit-parameters/)s in this judgment). The middle place in the relation is the translation of the type; this is necessary to ensure that the translated value has the appropriate translated type. + +Sometimes it is necessary for the type translation to be an [output](/wiki/percent-mode/) of the relation, and sometimes it is necessary for it to be an input. As is standard for translations between languages, we will need to use the reversing the polarity technique to give two versions of this relation with two different modes. The second one is identical except for the mode of the middle argument: !}% +%sort tocpsv- {WV ofv V A} {CT ttoct A CA} {WCV cofv CV CA} %. +%mode tocpsv- %in %out %out %. +%{! The CPS translation relation for expressions, which is the crux of this tutorial, is much less straightforward: !}% +%sort tocps- {M exp} {WM of M A} {CT ttoct A CA} {CC %pi (%pi cval %-> cexp) %-> cexp} {WCC {C %pi cval %-> cexp} %pi ({cv cval} {wcv cofv cv CA} cof (C cv)) %-> (cof (CC C))} %. +%mode tocps- %in %in %out %out %out %. +%{! For this relation we will use explicit parameters for the terms. The first two arguments are the expression ``M`` to be translated and its typing ``WM`` (the letter w is often used in variable names within this code to stand for "well-formed"). The next argument is the translation for the type ``A`` of the expression, as in the translation for values. The argument ``CC`` is the continuation expression that is the result of CPS conversion. However, it is parameterized over the "tail" of the continuation (of type ``cval ->; cexp``), because an expression cannot be CPS-converted in isolation. For example, the direct style expression ``mkpair 1 1`` will translate into + +``` +[rest : cval -> cexp] + cmkpair c1 c1 [p : cval] + rest p +``` + +The output ``CC`` is accompanied by ``WCC``, a typing derivation for it. The typing derivation is also parameterized. Its type is + +``` +{C : cval -> cexp} +({cv : cval} {wcv : cofv cv CA} cof (C cv)) -> + cof (CC C) +``` + +The first parameter ``C`` is the tail of the continuation, as above. The second argument is a typing derivation for the tail. The tail must be well-formed for any well-formed ``cval`` we might pass to it, so that typing derivation is itself parameterized. To summarize, ``WCC`` is a parameterized typing derivation that allows us to instantiate it with any well-formed tail. + +We also need to give a version of this relation with reversed polarity. !}% +%sort tocps+ {M exp} {WM of M A} {CT ttoct A CA} {CC %pi (%pi cval %-> cexp) %-> cexp} {WCC {C %pi cval %-> cexp} %pi ({cv cval} {wcv cofv cv CA} cof (C cv)) %-> (cof (CC C))} %. +%mode tocps+ %in %in %in %out %out %. +%{! The type of this relation may be more clear when we see how it is used. But before we are able to give the translation for expressions and values, we need to prove some lemmas about the type translation relation. + +### Lemmas + +First, we define a [shallow equality](/wiki/equality/) relation on continuation types and prove that it is preserved when type constructors are applied: !}% +%sort ceqtyp %. +%term ceqtyp_ ceqtyp A A %. +%sort ceqtyp_& %. +%mode ceqtyp_& %in %in %out %. +%term _ ceqtyp_& ceqtyp_ ceqtyp_ ceqtyp_ %. +%sort ceqtyp_cont %. +%mode ceqtyp_cont %in %out %. +%term _ ceqtyp_cont ceqtyp_ ceqtyp_ %. +%{! We need to prove that continuation value typing respects equality, and that continuation tail typing respects equality. These theorems are trivial because equality is shallow. !}% +%sort cofv_resp %. +%mode cofv_resp %in %in %out %. +%term _ cofv_resp D ceqtyp_ D %. +%sort wcc_resp {WCC {C %pi cval %-> cexp} %pi ({cv cval} {wcv cofv cv A} cof (C cv)) %-> (cof (CC C))} {EQ ceqtyp A A'} {K' {C %pi cval %-> cexp} %pi ({cv cval} {wcv cofv cv A'} cof (C cv)) %-> (cof (CC C))} %. +%mode wcc_resp %in %in %out %. +%term wcc_resp_ wcc_resp D ceqtyp_ D %. +%{! It is also necessary that the type translation relation ``ttoct`` be a [function](/wiki/function/). That means we must prove a [uniqueness lemma](/wiki/uniqueness-lemma/) and [effectiveness lemma](/wiki/effectiveness-lemma/) for it. !}% +% uniqueness +%sort ttoct_fun %. +%mode ttoct_fun %in %in %out %. +%term _ + %pi (ttoct_fun (ttoct/& A B) (ttoct/& C D) OUT) + %<- (ttoct_fun A C EQ1) + %<- (ttoct_fun B D EQ2) + %<- (ceqtyp_& EQ1 EQ2 OUT) %. +%term _ + %pi (ttoct_fun (ttoct/=> A B) (ttoct/=> C D) OUT) + %<- (ttoct_fun A C EQ1) + %<- (ttoct_fun B D EQ2) + %<- (ceqtyp_cont EQ2 EQ3) + %<- (ceqtyp_& EQ1 EQ3 EQ4) + %<- (ceqtyp_cont EQ4 OUT) %. +%term _ ttoct_fun ttoct/unit ttoct/unit ceqtyp_ %. +% effectiveness +%sort ttoct_gimme {A typ} {A' ctyp} %. +%mode ttoct_gimme %in %out %out %. +%term _ + %pi (ttoct_gimme (A & B) _ (ttoct/& CT1 CT2)) + %<- (ttoct_gimme A A' CT1) + %<- (ttoct_gimme B B' CT2) %. +%term _ + %pi (ttoct_gimme (A => B) _ (ttoct/=> CT1 CT2)) + %<- (ttoct_gimme A A' CT1) + %<- (ttoct_gimme B B' CT2) %. +%term _ ttoct_gimme unit cunit ttoct/unit %. +%{! We can then check that these are total: !}% +%worlds () (ttoct_fun _ _ _) (ceqtyp_& _ _ _) (ceqtyp_cont _ _) %. +%total D (ceqtyp_& D _ _) %. +%total D (ceqtyp_cont D _) %. +%total D (ttoct_fun D _ _) %. +%{! ### Term translation + +We can now give the translation for expressions and values. We start by proving the ``tocpsv+`` and ``tocps+`` lemmas, since they are one case each: !}% +%term tocps+/- + %pi (tocps+ V WV CTi CC K) + %<- (tocps- V WV CTo CC K') + %<- (ttoct_fun CTo CTi EQ) + %<- (wcc_resp K' EQ K) %. +%term tocpsv+/- + %pi (tocpsv+ WV CTi WCV') + %<- (tocpsv- WV CTo WCV) + %<- (ttoct_fun CTo CTi EQ) + %<- (cofv_resp WCV EQ WCV') %. +%{! Both work the same way. We take the type translation as input (``CTi``), then invoke the version of the theorem that returns it as output (``CTo``). Suppose ``CTi : ttoct A A'`` and ``CTo : ttoct A A'<nowiki></nowiki>'``. We just need to show that the output of that theorem (``K`` or ``WCV``), which depends on ``A'<nowiki></nowiki>'``, works for the type ``A'``. We simply verify that ``A'`` and ``A''`` are equal with our lemma ``ttoct_fun``, and then use the fact that the outputs respect equality on types (``wcc_resp`` or ``cofv_resp``) to do so. + + +The translation of the expressions ``fst`` and ``snd`` are illustrative: !}% +%term c_fst + %pi (tocps- (fst M) (&E1 WM) CT ([tail %pi cval %-> cexp] CC ([v cval] cfst v ([a cval] tail a))) ([c] [wc] F _ ([v] [wv] co_fst wv wc))) + %<- (tocps- M WM (ttoct/& CT _) CC F) %. +%term c_snd + %pi (tocps- (snd M) (&E2 WM) CT _ ([c] [wc] F _ ([v] [wv] co_snd wv wc))) + %<- (tocps- M WM (ttoct/& _ CT) _ F) %. +%{! To translate the expression ``fst M``, we inductively translate the argument. It must have type ``A & B`` and so the only case for translating it is ``ttoct/&``, so this subgoal [covers all outputs](/wiki/coverage-checking/). We return the translation for ``A``, called ``CT``. We must then build the continuation expression and its typing derivation. The expression is parameterized by a ``tail`` that takes the result of the ``fst`` operation, so the tail has type ``cval ->; cexp``. The resulting term first evaluates ``M`` by invoking its translation ``CC`` on a tail we write; that tail takes ``v``, the pair value, performs ``cfst`` on it and binds the result to ``a``. It finishes by invoking the outer ``tail`` argument on that projected first component. + +The parameterized typing derivation is similar. It takes a tail ``c`` and a typing derivation for the tail, ``wc``. The derivation starts with the typing for the translation of ``M``, which is represented by the function ``F``. We apply ``F`` to the actual tail we produce—Twelf can deduce what this is by [unification](/wiki/unification/), so we just write ``_`` to avoid repeating ourselves. The second argument is the typing derivation for that tail; it takes a variable ``v`` representing the result of evaluating ``M`` and a typing derivation for it ``wv``. The derivation consists simply of the typing rule for ``c_fst`` applied to the well-formedness of its argument and the code that follows, both of which we get from arguments. + +The translation is challenging because it is so high-order (typing responsibilities pass from callee to caller and vice versa) but most of the cases follow the same pattern. The case for ``c_snd`` is symmetric, but we have now left out terms that Twelf can infer by type inference and unification. Because terms can be uniquely determined from the types of typing derivations, we almost never need to write them down explicitly. + +The translation for ``mkpair`` follows the same pattern. !}% +%term c_mkpair + %pi (tocps- (mkpair M1 M2) (&I WM1 WM2) (ttoct/& CT1 CT2) _ ([c] [wc] F1 _ ([v1] [wv1] F2 _ ([v2] [wv2] co_mkpair wv1 wv2 wc)))) + %<- (tocps- M1 WM1 CT1 _ F1) + %<- (tocps- M2 WM2 CT2 _ F2) %. +%{! We translate the two subexpressions, string together the continuation expressions that evaluate them, build the pair and pass it to the tail. Application is interesting because of the way functions are represented: !}% +%term c_app + %pi (tocps- (app M N) (=>E WM WN) CTB _ ([c] [wc] FM _ ([f] [wf] FN _ ([a] [wa] co_mkpair wa (cov_lam ([r] [wr] wc r wr)) ([p] [wp] co_call wf wp))))) + %<- (ttoct_gimme (A => B) (A' c& (B' ccont) ccont) (ttoct/=> CTA CTB)) + %<- (tocps+ M WM (ttoct/=> CTA CTB) _ FM) + %<- (tocps+ N WN CTA _ FN) %. +%{! The first subgoal (``ttoct_gimme``) exists to reconcile the various type translations that will occur: We will have a translation for ``A``, ``B``, and ``A => B`` which all must agree. We therefore invoke the [effectiveness lemma](/wiki/effectiveness-lemma/) on the largest type (``A => B``) and the others will be subterms. We then invoke tocps inductively to translate the function and argument expressions (we use the + version here so that we need not do equality reasoning about the type translations). The resulting typing derivation begins with the derivations for the translation of ``M`` and ``N``. Given these, it builds a pair of the argument ``wa`` and return continuation. The body of the return continuation is the outermost tail passed to the translation of the ``app``. We then end with a call to the translated function value on the pair we created. + +#### Binding + +So far, everything in the source object language has been first order. We'll now translate the ``let`` construct, which is a simple binder. !}% +%term c_let + %pi (tocps- (let M N) (oflet WM WN) CTN _ ([c] [wc] FM _ ([v] [wv] FN v wv c wc))) + %<- (ttoct_gimme A A' CTM) + %<- (tocps+ M WM CTM _ FM) + %<- ({x} {xof ofv x A} {x'} {x'of cofv x' A'} {thm tocpsv- xof CTM x'of} tocps- (N x) (WN x xof) CTN (CC x') (FN x' x'of)) %. +%{! This translation begins as before, by translating ``M``. We then want to translate the body, ``N``, but it has type ``val -> exp``, so it must be in a subgoal with a hypothetical value variable in context. The subgoal actually introduces five hypotheses: the direct style value ``x``; a derivation that it is well-formed at type ``A``, ``xof``; the CPS value it will be translated to, ``x'``; its derivation ``x'of``; and the case of the theorem that relates the two. Once we have translated ``N`` and ``WN`` in this context, we build the result typing derivation. Because we have set up the translation such that the tail always takes a value as an argument, we do not need a CPS-level ``let`` construct; we simply invoke ``FM``, which types the translation of ``M``, and then invoke ``FN`` on the value and typing derivation resulting from that, and continue with the outermost tail. + +The final case of translation for expressions is the inclusion of values: !}% +%term c_val + %pi (tocps- (value V) (ofvalue WV) CT _ ([c] [wc] wc CV WCV)) + %<- (tocpsv- WV CT WCV) %. +%{! It simply invokes the mutually inductive translation ``tocpsv-``, which we define next. + +### Value translation + +Value translation is not continuation-based, so two of the three cases are very easy: !}% +%term cv_pair + %pi (tocpsv- (&Iv WV1 WV2) (ttoct/& CT1 CT2) (cov_pair WV1' WV2')) + %<- (tocpsv- WV1 CT1 WV1') + %<- (tocpsv- WV2 CT2 WV2') %. +%term cv_unit tocpsv- unitI ttoct/unit cov_unit %. +%{! Pairs are translated pointwise and unit is trivial. + +The final case in the theorem is the translation of lambda values: !}% +%term cv_lam + %pi (tocpsv- (%the (ofv (lam M) (A => B)) (=>I WM)) (ttoct/=> CTA CTB) (cov_lam ([arg] [argof cofv arg (A' c& (B' ccont))] co_fst argof ([x cval] [xof cofv x A'] co_snd argof ([r cval] [rof cofv r (B' ccont)] F x xof r rof ([v cval] ccall r v) ([v cval] [wv cofv v B'] co_call rof wv)))))) + %<- (ttoct_gimme A A' CTA) + %<- ({x} {xof ofv x A} {x'} {x'of cofv x' A'} {thm tocpsv- xof CTA x'of} {r} {rof cofv r (B' ccont)} tocps- (M x) (WM x xof) CTB (CC x' r) (F x' x'of r rof)) %. +%{! Recall that the ``tocpsv`` relations are between typing derivations only (the terms are implicit) but ``tocps`` takes them explicitly. Thus the first thing to notice is that we use type annotations to recover the implicit arguments so that we can pass them explicitly to ``tocps``. (This is an instance of [converting between implicit and explicit parameters](/wiki/converting-between-implicit-and-explicit-parameters/).) As before, we then use effectiveness to translate the type ``A``. The inductive call to ``tocps`` has seven hypotheses. The first five are as in the case for ``let``, which represent the bound argument variable and its translation. The last two represent the return continuation, which will be another argument to the continuation we generate. Also notice that in this case we do not have a wildcard ``_`` for the translated term in the subgoal—we write ``(CC x' r)``—without this help Twelf's higher-order unification cannot resolve all of the constraints. (This one case is the reason that the terms cannot be implicit parameters as they are for the value translation.) The result of translation is a typing derivation for a continuation. The continuation's argument ``arg`` has type ``(A' c& (B' ccont))``; we start by projecting the first and second components of the pair. We then use the derivation for the translated body, which depends on the argument and return continuation. Its tail takes the result of the body, ``v`` and calls the continuation on that value. + +## Metatheorem + +We can now check our metatheorem, which is that these translations are total. We start by declaring [block](/wiki/percent-block/)s for use in the [%worlds](/wiki/percent-worlds/) declarations: !}% +%block blockcvar {A typ} {A' ctyp} {CTA ttoct A A'} [x] [xof ofv x A] [x'] [x'of cofv x' A'] [thm tocpsv- xof CTA x'of]%. +%block blockwcv {CA ctyp} [r] [rof cofv r CA]%. +%{! The first block corresponds to the five hypotheses we needed in the ``let`` and ``lam`` cases. The second is for the return continuation we assume in the ``lam`` case—it does not correspond to any direct style assumption so it need only be well-formed. + +For this proof to check we must also make world declarations for the type families ``cof``, ``cofv``, ``cval`` and ``cexp``. I believe this has to do with Twelf's overconservativity of world subsumption on third-order metatheorems, but I am not sure of the exact cause. Although they are strange (and meaningless, since we do not check the totality of these type families, whose modes are all [``*``](/wiki/percent-mode/)), they pose no problem here: !}% +%block blockcv [v cval]%. +%worlds (blockcv) (cval) (cexp) %. +%worlds (blockwcv) (cofv _ _) (cof _) %. +%{! Then, the world declarations for our theorems and lemmas: !}% +%worlds (blockcvar blockwcv) (tocps+ _ _ _ _ _) (tocps- _ _ _ _ _) (tocpsv+ _ _ _) (tocpsv- _ _ _) (cofv_resp _ _ _) (ttoct_gimme _ _ _) (wcc_resp _ _ _) %. +%{! We can check the lemmas individually for totality; they are simple inductions: !}% +%total A (cofv_resp _ A _) %. +%total A (wcc_resp _ A _) %. +%total A (ttoct_gimme A _ _) %. +%{! Finally, we can check the type-directed translations for totality: !}% +%total (A B C D) (tocpsv- A _ _) (tocpsv+ B _ _) (tocps- _ C _ _ _) (tocps+ _ D _ _ _) %. +%{! The termination order here is important. The ``+`` versions of each relation call the ``-`` versions on the same input derivation, so we must specify mutual recursion with parentheses in this order ``(A B C D)`` so that these forward calls are allowed. Other calls obey the strict subterm ordering. See mutual recursion for more information. (Note: the call order in the documentation appears to be the opposite of what is implemented.) + +That's it! We showed how to define a CPS language with an intrinsic distinction between values and expressions, and translate a direct style language into it. The translation is given as a relation on typing derivations, which builds the type correctness of the translation into the translation itself. The translation makes use of LF's ability to encode higher-order reasoning to avoid generating object-level "administrative redices" that would need to be cleaned up and would complicate the metatheory. The code is brief by being simultaneously both proof and program: A metatheorem showing the translation works for all well-typed direct style terms is obtained simply by a [%total](/wiki/percent-total/) declaration for the translation. Alternately, we can think of the computational content of our proof as being the translation itself. + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +You can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/CPS_conversion). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/debugging-coverage-errors.lf b/new-tests/stelf-output/pages/debugging-coverage-errors.lf new file mode 100644 index 0000000..0b8e495 --- /dev/null +++ b/new-tests/stelf-output/pages/debugging-coverage-errors.lf @@ -0,0 +1,276 @@ +% +% +%{! This page lists techniques for debugging [input coverage](/wiki/coverage-checking/) and [output coverage](/wiki/coverage-checking/) errors. + +## Debugging input coverage errors by adding type annotations + +### Motivating example + +As a motivating example, we define a simple subtyping relation on the types of a programming language. The language includes integers, floating point numbers, and functions. We consider ``int`` to be a subtype of ``float``, and we give the usual contravariant rule for functions: !}% +%sort tp %. +%term int tp %. +%term float tp %. +%term arrow %pi tp %-> tp %-> tp %. +%sort sub %. +%term sub-ii sub int int %. +%term sub-ff sub float float %. +%term sub-if sub int float %. +%term sub-arrow %pi (sub (arrow T S) (arrow T' S')) %<- (sub T' T) %<- (sub S S') %. +%{! Let's prove that this subtyping relation is transitive. When writing Twelf code, it's sometimes convenient to just blaze ahead with a proof without thinking to hard about what you're doing, and then think about what's going on only if Twelf reports an error. So a first proof attempt might look like this: !}% +%sort sub-trans %. +%mode sub-trans %in %in %out %. +%term _ sub-trans (%the (sub T T) D) (%the (sub T T) D') D %. +%term _ sub-trans sub-ii sub-if sub-if %. +%term _ sub-trans sub-if sub-ff sub-if %. +%term arrow + %pi (sub-trans (sub-arrow DS DT) (sub-arrow DS' DT') (sub-arrow DS'' DT'')) + %<- (sub-trans DT DT' DT'') + %<- (sub-trans DS DS' DS'') %. +%{! We let our fingers do the proving and came up with the straightforward inductive proof. Each case type checks. Does it work? + +```checkedtwelf +%worlds () (sub-trans _ _ _). +%total D (sub-trans D _ _). +``` + +Twelf reports an input coverage error: we didn't cover the case for ``sub-arrow`` against ``sub-arrow``. +We think: "But I wrote a case for ``sub-arrow`` against ``sub-arrow`` right up there! What do you mean I didn't cover it?!" + +### Debugging technique + +What Twelf is saying is that we didn't cover the **general** case for ``sub-arrow`` against ``sub-arrow``. That is, the inferred type of the constant ``-arrow`` must be less general than required. One way to figure out the problem is to read the type of the constant: + +```checkedtwelf +-arrow : sub-trans + (sub-arrow DS DT) + (sub-arrow DS' DT') + (sub-arrow DS'' DT'') + <- sub-trans DT DT' DT'' + <- sub-trans DS DS' DS''. +``` + +As you can see, the type that Twelf inferred has the **same** type ``X4`` as all three types related by ``DT`` and ``DT'``. Because Twelf unified these parameters, ``-arrow`` doesn't cover the whole space that we think it does. + +Now, we have to figure out why Twelf inferred this type. What mistake did we actually make? One good way to figure this out is to start adding type annotations giving the constants fully general types. By doing so, we can turn a coverage error into a type error on the constant. + +For example, let's annotate this constant more carefully with what we expect the inputs to be: + +```checkedtwelf +-arrow : sub-trans + (sub-arrow (DS : sub S1 S2) (DT : sub T2 T1)) + (sub-arrow (DS' : sub S2 S3) (DT' : sub T3 T2)) + (sub-arrow DS'' DT'') + <- sub-trans DT DT' DT'' + <- sub-trans DS DS' DS''. +``` + +This causes Twelf to report a type error, so we know we're on the right track—the constant doesn't cover what we want it to. + +We can make this type error more comprehensible by removing some of the constraints. Specifically, we can take out the return term to remove some of the constraints on ``DT_``: + +```checkedtwelf +-arrow : sub-trans + (sub-arrow (DS : sub S1 S2) (DT : sub T2 T1)) + (sub-arrow (DS' : sub S2 S3) (DT' : sub T3 T2)) + _ + <- sub-trans DT DT' DT'' + <- sub-trans DS DS' DS''. +``` + +Ah ha! Twelf is pointing us to the first premise, as saying that because ``DT`` has type ``sub T2 T1``, the second argument ``DT'`` needs to have type ``sub T1 X1`` for some ``X1``; but we wrote a term with type ``sub T3 T2``. If we annotate the constant enough that Twelf knows that ``T1`` and ``T3`` are supposed to be different, this is reported as a type error. Otherwise, Twelf unifies these two, coming up with a valid type for the constant; but because this type doesn't cover the part of the relation that we thought it did, we get an input coverage error. + +As you may have figured out already, the problem is that the first recursive call should swap the two arguments: !}% +%term arrow + %pi (sub-trans (sub-arrow (%the (sub S1 S2) DS) (%the (sub T2 T1) DT)) (sub-arrow (%the (sub S2 S3) DS') (%the (sub T3 T2) DT')) (sub-arrow DS'' DT'')) + %<- (sub-trans DT' DT DT'') + %<- (sub-trans DS DS' DS'') %. +%{! In this example, this "fix" itself creates some termination trouble; see the article on [mutual induction](/wiki/mutual-induction/) for a completed proof. But the broken proof illustrates a useful technique for coverage checking. + +### A Messy Example + +In the previous example, adding types to some variables caused a coverage error to be converted into a type error. +But a complex case may have many many variables. Coverage errors can be very hard to find in this case. +In the following situation (from [me, John Boyland](/wiki/user-boyland/)), it is shown how to leverage the Twelf server to do most +of the work. Suppose (and this happened to me), one has a coverage error that "should" have been covered by the following horrendous case: + +``` +- : wk-is/L ES + (access/read + (ready/ (read (lit (object/ O)) F) TL1 L1 A)) + ML NFL XTL XSF FE ESG WA XPK XSM NHBP + (incorrectly-synchronized/W + (eval-seq/+ ESN + (fulleval/ _ (lit (object/ O')) TL1N L1 + (eRead ML' NFL' TL1N K>Z) TL1N TU1N)) + (eval-seq-get/ (eval-seq-includes/> ESI')) + (eval-seq-get/ eval-seq-includes/=) WA' + (access/read + (ready/ (read (lit (object/ O)) F) TL1N L1 A)) + NHB') + <- no-know-eval-seq-exists ES _ (no-know-state/ NM NTS) ESN ESM + (state-match/ MM TSM) + <- eval-seq-get-respects-match ESG ESM _ _ _ (eval-seq-get/ ESI') FEM + <- mem-lookup-respects-match ML MM _ ML' (objs-match/ NFM _ _) + <- no-know-mem-lookup-implies-no-know-contents ML' NM (no-know-objs/ NNF _ _) + <- normal-fields-lookup-respects-match NFL NFM _ NFL' + <- no-know-normal-fields-lookup-implies-no-know-contents NFL' NNF Z=W + <- threads-lookup-respects-match TL1 TSM _ TL1N + <- no-know-threads-lookup-implies-no-know-contents TL1N NTS Z=K + <- set`member-respects-eq (set`lookup/= nat`eq/) Z=K Z=W K>Z + <- threads`update-total TU1N + <- hb-potential?-respects-match NHBP ESM NHBP' + <- not-hb-potential-realized NHBP' _ (non-sw2-fulleval/ non-synch2-atomic/read) NHB' + <- wk-is/L2 WA TSM WA'. +``` + +Now, even if the contents meant anything to you (just pretend...), typing in the types of all those variables is impractical. +Instead, we replace the output with a new (or underscore) variable and terminate the case: + +```twelf +- : wk-is/L ES + (access/read + (ready/ (read (lit (object/ O)) F) TL1 L1 A)) + ML NFL XTL XSF FE ESG WA XPK XSM NHBP _. +``` + +Then, we submit this to the server, which (of course) will complain that the output is not necessarily ground. But with chatter set to 3, it **also** outputs the types of all the variables, including the implicit ones. Then these can be copied into the buffer +(after removing ``%name%`` types if you have any): + +``` +- : {X1} {N1:nat} {X2:map} {X3} {N2:nat} {O:nat} {F:nat} {X4} + {X5} {X6:lock-contents} {N3:nat} {N4:nat} {X7} {X8:term exprk} + {N5:nat} {X9:map} {X10} {X11:map} {X12} {N6:nat} + {X14} {ES:eval-seq X1 N1 (state/ X2 X3)} {X15} + {X16:term exprk -> term exprk} + {TL1:threads`lookup X3 N2 (thread/ X15 (X16 (read (lit (object/ O)) F)))} + {L1:loc* exprk exprk ([x:term exprk] X16 x)} + {A:atomic-lhs* exprk (read (lit (object/ O)) F)} + {ML:lookup X2 O (objs/ X4 X5 X6)} + {NFL:normal-fields`lookup X4 F (normal-contents/ N3 N4)} + {XTL:threads`lookup X3 N2 (thread/ X7 X8)} {XSF:not-member X7 N3} + {FE:fulleval X1 N5 (state/ X9 X10) (state/ X11 X12)} + {ESG:eval-seq-get* X1 N5 N1 (state/ X2 X3) (state/ X9 X10) + (state/ X11 X12) ES N6 FE} + {WA:access* writek X10 N5 O F} {XPK:process-knowledge X12 N5 X14} + {XSM:set`lookup X14 N3 unit/} + {NHBP:hb-potential?* X1 N1 (state/ X2 X3) ES N6 N2 false} +wk-is/L ES + (access/read + (ready/ (read (lit (object/ O)) F) TL1 L1 A)) + ML NFL XTL XSF FE ESG WA XPK XSM NHBP + (incorrectly-synchronized/W + (eval-seq/+ ESN + (fulleval/ _ (lit (object/ O')) TL1N L1 + (eRead ML' NFL' TL1N K>Z) TL1N TU1N)) + (eval-seq-get/ (eval-seq-includes/> ESI')) + (eval-seq-get/ eval-seq-includes/=) WA' + (access/read + (ready/ (read (lit (object/ O)) F) TL1N L1 A)) + NHB') + <- no-know-eval-seq-exists ES _ (no-know-state/ NM NTS) ESN ESM + (state-match/ MM TSM) + <- eval-seq-get-respects-match ESG ESM _ _ _ (eval-seq-get/ ESI') FEM + <- mem-lookup-respects-match ML MM _ ML' (objs-match/ NFM _ _) + <- no-know-mem-lookup-implies-no-know-contents ML' NM (no-know-objs/ NNF _ _) + <- normal-fields-lookup-respects-match NFL NFM _ NFL' + <- no-know-normal-fields-lookup-implies-no-know-contents NFL' NNF Z=W + <- threads-lookup-respects-match TL1 TSM _ TL1N + <- no-know-threads-lookup-implies-no-know-contents TL1N NTS Z=K + <- set`member-respects-eq (set`lookup/= nat`eq/) Z=K Z=W K>Z + <- threads`update-total TU1N + <- hb-potential?-respects-match NHBP ESM NHBP' + <- not-hb-potential-realized NHBP' _ (non-sw2-fulleval/ non-synch2-atomic/read) NHB' + <- wk-is/L2 WA TSM WA'. +``` + +Now, we get new errors because the variables only occurring in the body of the proof may depend on the explicitly typed ones, and the type inference engine does not do this. To solve this, use Control-C Control-L to highlight the unbound variables, and then list them all (with unspecified type) right after the other variables. Don't list one twice! You can use Control-C Control after adding a few to see what remains. Eventually one has + +``` +- : {X1} {N1:nat} {X2:map} {X3} {N2:nat} {O:nat} {F:nat} {X4} + {X5} {X6:lock-contents} {N3:nat} {N4:nat} {X7} {X8:term exprk} + {N5:nat} {X9:map} {X10} {X11:map} {X12} {N6:nat} + {X14} {ES:eval-seq X1 N1 (state/ X2 X3)} {X15} + {X16:term exprk -> term exprk} + {TL1:threads`lookup X3 N2 (thread/ X15 (X16 (read (lit (object/ O)) F)))} + {L1:loc* exprk exprk ([x:term exprk] X16 x)} + {A:atomic-lhs* exprk (read (lit (object/ O)) F)} + {ML:lookup X2 O (objs/ X4 X5 X6)} + {NFL:normal-fields`lookup X4 F (normal-contents/ N3 N4)} + {XTL:threads`lookup X3 N2 (thread/ X7 X8)} {XSF:not-member X7 N3} + {FE:fulleval X1 N5 (state/ X9 X10) (state/ X11 X12)} + {ESG:eval-seq-get* X1 N5 N1 (state/ X2 X3) (state/ X9 X10) + (state/ X11 X12) ES N6 FE} + {WA:access* writek X10 N5 O F} {XPK:process-knowledge X12 N5 X14} + {XSM:set`lookup X14 N3 unit/} + {NHBP:hb-potential?* X1 N1 (state/ X2 X3) ES N6 N2 false} + {ESN} {O'} {TL1N} {ML'} {NFL'} {K>Z} {ESI'} {WA'} {TU1N} {NHB'} + {NM} {NTS} {ESM} {MM} {TSM} {NFM} {NNF} {Z=W} {Z=K} {NHBP'} +wk-is/L ES + (access/read + (ready/ (read (lit (object/ O)) F) TL1 L1 A)) + ML NFL XTL XSF FE ESG WA XPK XSM NHBP + (incorrectly-synchronized/W + (eval-seq/+ ESN + (fulleval/ _ (lit (object/ O')) TL1N L1 + (eRead ML' NFL' TL1N K>Z) TL1N TU1N)) + (eval-seq-get/ (eval-seq-includes/> ESI')) + (eval-seq-get/ eval-seq-includes/=) WA' + (access/read + (ready/ (read (lit (object/ O)) F) TL1N L1 A)) + NHB') + <- no-know-eval-seq-exists ES _ (no-know-state/ NM NTS) ESN ESM + (state-match/ MM TSM) + <- eval-seq-get-respects-match ESG ESM _ _ _ (eval-seq-get/ ESI') FEM + <- mem-lookup-respects-match ML MM _ ML' (objs-match/ NFM _ _) + <- no-know-mem-lookup-implies-no-know-contents ML' NM (no-know-objs/ NNF _ _) + <- normal-fields-lookup-respects-match NFL NFM _ NFL' + <- no-know-normal-fields-lookup-implies-no-know-contents NFL' NNF Z=W + <- threads-lookup-respects-match TL1 TSM _ TL1N + <- no-know-threads-lookup-implies-no-know-contents TL1N NTS Z=K + <- set`member-respects-eq (set`lookup/= nat`eq/) Z=K Z=W K>Z + <- threads`update-total TU1N + <- hb-potential?-respects-match NHBP ESM NHBP' + <- not-hb-potential-realized NHBP' _ (non-sw2-fulleval/ non-synch2-atomic/read) NHB' + <- wk-is/L2 WA TSM WA'. +``` + +Now when you check the declaration (Control-C Control-D), the coverage error will occur as a type error. Here it shows up that ``WA`` is constrained too much. + +The details are irrelevant except to show that even (especially!) long proofs can use this technique. + +### Summary + +If you encounter an input coverage error on a case you thought you covered, the problem is most likely that the constant you wrote has a less general type than you think. It can be helpful to read the type of constant, looking for variables that Twelf unified. Additionally, you can start introducing type annotations on the inputs to prevent Twelf from unifying the implicit parameters; this turns the input coverage error into a type error on the constant itself. If you comment out parts of the constant, such as outputs or later premises, you can find the exact premise that causes the type error, and get Twelf to give a more specific error message. This will help you figure out what mistake you made. + +## Unusual Situations + +The input coverage checker has limitations. This section describes some of them. +You won't encounter these symptoms unless you are using higher-order judgments. + +### Nontermination + +Occasionally, the input coverage checker does not terminate. If chatter is set to six, you see the infinite loop progressing. +Otherwise, it simply hangs. This happens sporadically when the input coverage checker attempts to split +on a variable of higher-order (functional) type. It usually happens when there is a coverage error. Use chatter 6 +to determine where the problem is. + +### Inactive Splits + +Occasionally, the coverage checker "refuses" to split on a variable (i.e. enumerate its cases) even +though doing so would determine that there are no cases to consider, in other words that +coverage is not needed. In this case the coverage checker will try other variables until it runs out +of candidates and gives up yielding (often) a long list of unhandled cases. This symptom can be +verified by setting chatter to 7 in which the message "Inactive Split" is echoed. An inactive split +is a split that will not be chosen because one of its cases causes unification to fail with +unsolved constraints. + +This situation can be solved by preventing unification of the higher-order terms (e.g., ``(F X) = (G X)``) +through the introduction of a trivial (syntactic) equality predicate. Make sure that this predicate is +listed <em>before</em> the variable that needs to be split in your meta-theorem. Otherwise, +the coverage checker will split the equality predicate and re-introduce the problem. + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Debugging_coverage_errors). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/debugging-mode-checking-errors.lf b/new-tests/stelf-output/pages/debugging-mode-checking-errors.lf new file mode 100644 index 0000000..fec7b21 --- /dev/null +++ b/new-tests/stelf-output/pages/debugging-mode-checking-errors.lf @@ -0,0 +1,54 @@ +% +% +%{! +There are some common pitfalls when you're dealing with [`%mode`](/wiki/percent-mode/) declarations. + +## Basic mode checking errors + +Say we give a definition of the natural numbers with addition: !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort plus %. +%mode plus %in %in %out %. +%term pz plus z N N %. +%term ps %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%{! The input of `plus` is the first and second positions, and the output of `plus` is the third position, which means that whenever there are [ground](/wiki/ground/) objects in the first and second positions, that must force a ground object in the third position. + +If one of the output arguments is not forced to be a ground, which would be the case if the output of plus no longer matched the output of the [[subgoal]], then we get an error: + +```checkedtwelf +px: plus (s N1) N2 (s N) + <- plus N1 N2 N3. +``` + +Another problem occurs when the _inputs_ to a subgoal are not known to be ground, which would happen if we mis-named one of the inputs to the subgoal. + +```checkedtwelf +py: plus (s N1) N2 (s N3) + <- plus N N2 N3. +``` + +## Ordering subgoals + +Mode checking considers subgoals _in order_, i.e. from top to bottom when the subgoals are written out in the standard style using backwards arrows. The order of subgoals matters very much for mode checking. Say we have an identity function that maps inputs (the first position) to outputs (the second position). !}% +%sort id %. +%mode id %in %out %. +%term id/refl id N N %. +%{! The rule ``ps'`` below passes the mode checker, because the call to ``id`` takes the ground argument ``N1`` and creates a ground output ``N1'``, which is then used in the derivation for ``plus N1' N2 N3``. + +```checkedtwelf +ps': plus (s N1) N2 (s N3) + <- id N1 N1' + <- plus N1' N2 N3. +``` + +However, if we reverse the order of the two subgoals, even though the result is logically equivalent, Twelf considers ``plus N1' N2 N3`` before ``id N1 N1'``, and so does consider ``N1'`` to be ground when it encounters it; thus, complaining accordingly: + +```checkedtwelf +ps': plus (s N1) N2 (s N3) + <- plus N1' N2 N3 + <- id N1 N1'. +``` + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/define-declaration.lf b/new-tests/stelf-output/pages/define-declaration.lf new file mode 100644 index 0000000..a3226c9 --- /dev/null +++ b/new-tests/stelf-output/pages/define-declaration.lf @@ -0,0 +1,34 @@ +% +% +%{! + +The **``%define``** declaration allows the outputs of a successful [``%solve``](/wiki/percent-solve/) declaration to be bound to a name which can be used later on in the context. + +## Example + +Given a standard declaration of natural numbers and plus with a few constants defined: !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%define n0 z %. +%define n1 s n0 %. +%define n2 s n1 %. +%define n3 s n2 %. +%define n4 s n3 %. +%sort plus %. +%mode plus %in %in %out %. +%term plus/z plus z N N %. +%term plus/s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%{! We can use ``%define`` to define the natural number representing 7 by adding three and four. Note that we don't care about the proof witness in this particular case, just the output of the relation, so we put an underscore instead of an identifier to the left of the colon: !}% +%{!! begin checked !!}% +%define n7 = N %solve _ : plus n4 n3 N %. +%{!! end checked !!}% +%{! Now `n7` is a definition that can be used again. !}% +%{!! begin checked !!}% +%solve _ : plus n7 n7 N %. +%{!! end checked !!}% +%{! ## See also +* Definitions + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/dependent-types.lf b/new-tests/stelf-output/pages/dependent-types.lf new file mode 100644 index 0000000..e916016 --- /dev/null +++ b/new-tests/stelf-output/pages/dependent-types.lf @@ -0,0 +1,19 @@ +% +%{! A language with **dependent types** has types which can mention the terms they classify. For example, [LF](/wiki/lf/) is a dependently typed language because LF types can mention LF terms. + + +## See also + +* [LF](/wiki/lf/) +* [[w:Dependent types|Dependent types]] at Wikipedia +* Chapter 2 of [http://www.cis.upenn.edu/~bcpierce/attapl/main.html Advanced Topics in Types and Programming Languages] + + + +\{\{stub\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Dependent_types). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/deterministic-declaration.lf b/new-tests/stelf-output/pages/deterministic-declaration.lf new file mode 100644 index 0000000..bbc7dc2 --- /dev/null +++ b/new-tests/stelf-output/pages/deterministic-declaration.lf @@ -0,0 +1,67 @@ +% +% +%{! The **``%determinstic``** declaration influences the way [logic programming](/wiki/logic-programming/) behaves in Twelf. If a type family is deterministic, then once Twelf finds a single solution through logic programming search it cannot backtrack to find different solutions. + +## Example: Tree search + +We define a tree with labeled nodes, and a distinguished tree ``testtree``. + +
+![Visual description of a balanced tree with four labeled leaves.](/src/assets/smalltree.png) + +Graphical representation of the tree ``testtree`` used in this example. +
+!}% +%sort label %. +%term a label %. +%term b label %. +%term c label %. +%term d label %. +%sort tree %. +%term node %pi tree %-> tree %-> tree %. +%term leaf %pi label %-> tree %. +%define testtree node (node (leaf a) (leaf b)) (node (leaf c) (leaf d)) %. +%{! ### Searching for a leaf + +We also define a judgment, ``findlabel``, that looks for a label at leaves of the tree. Because Twelf tries to use the first-defined rule first, the operational behavior of this is to search for the leftmost node, then to backtrack and find the next-to-leftmost node, etc. !}% +%sort findlabel %. +%term findlabel/leaf findlabel (leaf L) L %. +%term findlabel/left %pi (findlabel (node T1 T2) L) %<- (findlabel T1 L) %. +%term findlabel/right %pi (findlabel (node T1 T2) L) %<- (findlabel T2 L) %. +%{! If ``findlabel`` is not declared deterministic, all four solutions can be returned, but with ``findlabel`` declared deterministic only one will be found. !}% +%{!! begin checked !!}% +%query 4 _ _ findlabel testtree L %. +%{!! end checked !!}% +%{! + +```checkedtwelf +%deterministic findlabel. +%query 1 * findlabel testtree L. +``` + +### Causing search to fail + +When using **``%deterministic``**, [[finite failure]] no longer means that no derivation can be found, becuase the deterministic search may put constraints on later results that causes them to fail. Take the ``searchfor`` predicate, which first looks up a label with ``findlabel`` and then checks to see if it is equal to some other label. !}% +%sort eq %. +%term eq/refl eq L L %. +%sort searchfor %. +%term _ %pi (searchfor T L) %<- (findlabel T L') %<- (eq L L') %. +%{! Using ``searchfor`` to look for ``c`` in our test tree will cause backtracking, because ``findlabel`` first make ``L'`` equal to ``a``, then ``b``, then ``c``. !}% +%{!! begin checked !!}% +%query 1 _ _ %the (searchfor testtree c) P %. +%{!! end checked !!}% +%{! If ``findlabel`` is deterministic, then the same search will make ``L'`` equal ``a``, and will then be unable to backtrack. + +```checkedtwelf +%deterministic findlabel. +%query 1 * P : searchfor testtree c. +``` + +## See also + +* Deterministic Type Families +* [Cut](https://en.wikipedia.org/wiki/Cut_(logic_programming)) on Wikipedia + + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/division-over-the-natural-numbers.lf b/new-tests/stelf-output/pages/division-over-the-natural-numbers.lf new file mode 100644 index 0000000..703e1a5 --- /dev/null +++ b/new-tests/stelf-output/pages/division-over-the-natural-numbers.lf @@ -0,0 +1,131 @@ +% +%{! This is a [case study](/wiki/case-studies/) about **division over the natural numbers**. A number of arithmetic operations over the natural numbers will be defined, including division. The ultimate goal is to prove properties normally associated with division hold. Twelf muscles flexed in this study include [reasoning from false](/wiki/reasoning-from-false/) and using the [%reduces](/wiki/percent-reduces/) directive to use strong induction in a proof. + +First the "false" type and natural numbers are defined. !}% +%sort uninhabited %. +%freeze uninhabited %. +%sort nat %. +%term nat/z nat %. +%term nat/s %pi nat %-> nat %. +%{! Relations about arithmetic over the natural numbers are defined in the standard way. !}% +%sort nat-plus %. +%term nat-plus/z nat-plus nat/z N N %. +%term nat-plus/s %pi (nat-plus (nat/s N1) N2 (nat/s N3)) %<- (nat-plus N1 N2 N3) %. +%sort nat-mult %. +%term nat-mult/z nat-mult nat/z N nat/z %. +%term nat-mult/s %pi (nat-mult (nat/s N1) N2 N3') %<- (nat-plus N2 N3 N3') %<- (nat-mult N1 N2 N3) %. +%sort nat-less %. +%term nat-less/z nat-less nat/z (nat/s _) %. +%term nat-less/s %pi (nat-less (nat/s N1) (nat/s N2)) %<- (nat-less N1 N2) %. +%sort nat-leq %. +%term nat-leq/z nat-leq nat/z _ %. +%term nat-leq/s %pi (nat-leq (nat/s N1) (nat/s N2)) %<- (nat-leq N1 N2) %. +%sort nat-compare %. +%term nat-compare/less %pi (nat-compare N1 N2) %<- (nat-less N1 N2) %. +%term nat-compare/leq %pi (nat-compare N1 N2) %<- (nat-leq N2 N1) %. +% nat-divmod DIVIDEND DIVISOR QUOTIENT REMAINDER +%sort nat-divmod %. +%term nat-divmod/base %pi (nat-divmod N1 N2 nat/z N1) %<- (nat-less N1 N2) %. +%term nat-divmod/rec + %pi (nat-divmod N1 N2 (nat/s N3) N4) + %<- (nat-divmod N1' N2 N3 N4) + %<- (nat-plus N2 N1' N1) %. +%{! Some basic properties about arithmetic. !}% +%sort nat-leq-plus %. +%mode nat-leq-plus %in %out %. +%term _ nat-leq-plus nat-leq/z nat-plus/z %. +%term _ %pi (nat-leq-plus (nat-leq/s DLQ) (nat-plus/s DNP)) %<- (nat-leq-plus DLQ DNP) %. +%worlds () (nat-leq-plus _ _) %. +%total {D1} (nat-leq-plus D1 _) %. +%sort nat-compare-resp-s %. +%mode nat-compare-resp-s %in %out %. +%term _ nat-compare-resp-s (nat-compare/less DL) (nat-compare/less (nat-less/s DL)) %. +%term _ nat-compare-resp-s (nat-compare/leq DLQ) (nat-compare/leq (nat-leq/s DLQ)) %. +%worlds () (nat-compare-resp-s _ _) %. +%total {} (nat-compare-resp-s _ _) %. +%sort can-nat-compare {N1} {N2} %. +%mode can-nat-compare %in %in %out %. +%term _ can-nat-compare _ _ (nat-compare/less nat-less/z) %. +%term _ can-nat-compare _ _ (nat-compare/leq nat-leq/z) %. +%term _ + %pi (can-nat-compare (nat/s N1) (nat/s N2) DC') + %<- (can-nat-compare N1 N2 DC) + %<- (nat-compare-resp-s DC DC') %. +%worlds () (can-nat-compare _ _ _) %. +%total {D1} (can-nat-compare D1 _ _) %. +%sort can-nat-plus {N1} {N2} %. +%mode can-nat-plus %in %in %out %. +%term _ can-nat-plus _ _ nat-plus/z %. +%term _ %pi (can-nat-plus _ _ (nat-plus/s DP)) %<- (can-nat-plus _ _ DP) %. +%worlds () (can-nat-plus _ _ _) %. +%total (D1) (can-nat-plus D1 _ _) %. +%sort nat-plus-assoc %. +%mode nat-plus-assoc %in %in %in %out %. +%term _ nat-plus-assoc nat-plus/z DP nat-plus/z DP %. +%term _ + %pi (nat-plus-assoc (nat-plus/s DP1) DP2 (nat-plus/s DP3) (nat-plus/s DP4)) + %<- (nat-plus-assoc DP1 DP2 DP3 DP4) %. +%worlds () (nat-plus-assoc _ _ _ _) %. +%total (D1) (nat-plus-assoc D1 _ _ _) %. +%{! The first proof about division: The remainder is less than the divisor! !}% +%sort nat-divmod-mod-less %. +%sort nat-divmod-mod-less %. +%mode nat-divmod-mod-less %in %out %. +%term _ nat-divmod-mod-less (nat-divmod/base DL) DL %. +%term _ + %pi (nat-divmod-mod-less (nat-divmod/rec _ DDM) DL) + %<- (nat-divmod-mod-less DDM DL) %. +%worlds () (nat-divmod-mod-less _ _) %. +%total {D1} (nat-divmod-mod-less D1 _) %. +%{! Division by zero does not happen. !}% +%sort nat-divmod-z-uninhabited %. +%mode nat-divmod-z-uninhabited %in %out %. +%term _ + %pi (nat-divmod-z-uninhabited (nat-divmod/rec nat-plus/z DDM) DU) + %<- (nat-divmod-z-uninhabited DDM DU) %. +%worlds () (nat-divmod-z-uninhabited _ _) %. +%total {D1} (nat-divmod-z-uninhabited D1 _) %. +%{! The following theorem uses the ``%reduces`` directive to verify that if a non-zero number is added to ``N2``, then the result is strictly larger than ``N2`` in the sub-term sense. This is powerful information, because Twelf verifies well-founded inductions over sub-term orderings. !}% +%sort nat-plus-reduces-s {N2} {N3} %. +%mode nat-plus-reduces-s %in %in %in %. +%term _ nat-plus-reduces-s _ _ (nat-plus/s nat-plus/z) %. +%term _ %pi (nat-plus-reduces-s _ _ (nat-plus/s DL)) %<- (nat-plus-reduces-s _ _ DL) %. +%worlds () (nat-plus-reduces-s _ _ _) %. +%reduces < N2 N3 (nat-plus-reduces-s N2 N3 _) %. +%total {D1} (nat-plus-reduces-s _ _ D1) %. +%{! The highlight of this example. Division is possible for any dividend as long as the divisor is non-zero. This proof is by induction over the dividend, which does get smaller as inductive calls are made. However, because Twelf can not figure this out on its own, calls to nat-plus-reduces-s are required so that this proof passes the totality checker. !}% +%sort can-nat-divmod* {N1} %. +%mode can-nat-divmod* %in %in %out %. +%term _ can-nat-divmod* _ (nat-compare/less DL) (nat-divmod/base DL) %. +%term _ + %pi (can-nat-divmod* N1 (nat-compare/leq DLQ) (nat-divmod/rec DP DDM)) + %<- (nat-leq-plus DLQ DP) + %<- (nat-plus-reduces-s N1' N1 DP) + %<- (can-nat-compare N1' (nat/s N2) DC) + %<- (can-nat-divmod* N1' DC DDM) %. +%worlds () (can-nat-divmod* _ _ _) %. +%total {D1} (can-nat-divmod* D1 _ _) %. +%sort can-nat-divmod {N1} {N2} %. +%mode can-nat-divmod %in %in %out %. +%term _ + %pi (can-nat-divmod N1 N2 DDM) + %<- (can-nat-compare N1 (nat/s N2) DC) + %<- (can-nat-divmod* N1 DC DDM) %. +%worlds () (can-nat-divmod _ _ _) %. +%total {} (can-nat-divmod _ _ _) %. +%{! It is also useful to know the definition of division is correct with respect to multiplication. !}% +%{!! begin checked !!}% +%sort nat-divmod-correct %. +%mode nat-divmod-correct %in %out %out %. +%term _ nat-divmod-correct (nat-divmod/base _) nat-mult/z nat-plus/z %. +%term _ + %pi (nat-divmod-correct (nat-divmod/rec DP DD) (nat-mult/s DM DP'') DP''') + %<- (nat-divmod-correct DD DM DP') + %<- (can-nat-plus N2 N3' DP'') + %<- (nat-plus-assoc DP'' DP' DP DP''') %. +%worlds () (nat-divmod-correct _ _ _) %. +%total (D1) (nat-divmod-correct D1 _ _) %. +%{!! end checked !!}% +%{! In the above code, the division algorithm for natural numbers was defined. A few interesting properties about this judgment were proven. The first is that the remainder computed is strictly less than the divisor. The second is that the judgment is not inhabited when the divisor is zero. The third is an [effectiveness lemma](/wiki/effectiveness-lemma/) which shows that this judgment is inhabited for any dividend and any divisor greater than zero. Finally, the division relation was shown to be correct with respect to multiplication. + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/documentation.lf b/new-tests/stelf-output/pages/documentation.lf new file mode 100644 index 0000000..e2ce285 --- /dev/null +++ b/new-tests/stelf-output/pages/documentation.lf @@ -0,0 +1,6 @@ +% +%{! + +See [Introductions to Twelf](/wiki/introductions-to-twelf/) + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/double-negation-translation.lf b/new-tests/stelf-output/pages/double-negation-translation.lf new file mode 100644 index 0000000..6d30d80 --- /dev/null +++ b/new-tests/stelf-output/pages/double-negation-translation.lf @@ -0,0 +1,481 @@ +% +%{! + +## Intuitionistic logic + +!}% +%sort iprop %. +%name iprop %. +%term itop iprop %. +%term iand %pi iprop %-> iprop %-> iprop %. +%term ibot iprop %. +%term ior %pi iprop %-> iprop %-> iprop %. +%term iimp %pi iprop %-> iprop %-> iprop %. +%block ipb [X iprop]%. +%worlds (ipb) (iprop) %. +%sort itrue %. +%term imt itrue itop %. +%term ipair %pi (itrue A) %-> (itrue B) %-> (itrue (iand A B)) %. +%term ifst %pi (itrue (iand A B)) %-> (itrue A) %. +%term isnd %pi (itrue (iand A B)) %-> (itrue B) %. +%term iabort %pi (itrue ibot) %-> (itrue C) %. +%term iinl %pi (itrue A) %-> (itrue (ior A B)) %. +%term iinr %pi (itrue B) %-> (itrue (ior A B)) %. +%term icase + %pi (itrue (ior A B)) + %-> (%pi (itrue A) %-> (itrue C)) + %-> (%pi (itrue B) %-> (itrue C)) + %-> (itrue C) %. +%term ilam %pi (%pi (itrue A) %-> (itrue B)) %-> (itrue (iimp A B)) %. +%term iapp %pi (itrue (iimp A B)) %-> (itrue A) %-> (itrue B) %. +%define inot (%pi iprop %-> iprop) [A] iimp A ibot %. +%define idn (%pi iprop %-> iprop) [A] inot (inot A) %. +%define dni (%pi (itrue A) %-> (itrue (idn A))) [x itrue A] ilam ([y itrue (inot A)] iapp y x) %. +%define tne (%pi (itrue (idn (inot A))) %-> (itrue (inot A))) [x3 itrue (inot (inot (inot A)))] ilam ([x itrue A] iapp x3 (ilam ([y itrue (inot A)] iapp y x))) %. +%define distdnand (%pi (itrue (idn (iand A B))) %-> (itrue (iand (idn A) (idn B)))) [f] ipair (ilam ([x itrue (inot A)] iapp f (ilam ([p] iapp x (ifst p))))) (ilam ([x itrue (inot B)] iapp f (ilam ([p] iapp x (isnd p))))) %. +%define dnebot (%pi (itrue (idn ibot)) %-> (itrue ibot)) [x] iapp x (ilam ([x] x)) %. +%inline dnetop (%pi (itrue (idn itop)) %-> (itrue itop)) [x] imt %. +%define distdnimp (%pi (itrue (idn (iimp A B))) %-> (itrue (iimp (idn A) (idn B)))) [f] ilam ([x itrue (idn A)] ilam ([y itrue (inot B)] iapp f (ilam ([g itrue (iimp A B)] iapp x (ilam ([z itrue A] iapp y (iapp g z))))))) %. +%block itb {A iprop} [D itrue A]%. +%worlds (ipb itb) (itrue _) %. +%{! + +## Classical logic + +!}% +%sort prop %. +%name prop %. +%term top prop %. +%term and %pi prop %-> prop %-> prop %. +%term bot prop %. +%term or %pi prop %-> prop %-> prop %. +%term imp %pi prop %-> prop %-> prop %. +%term nbot prop %. +%term not %pi prop %-> prop %. +%block pb [X prop]%. +%worlds (pb) (prop) %. +%sort conc %. +%term true %pi prop %-> conc %. +%prec %prefix 2 true %. +%term false %pi prop %-> conc %. +%prec %prefix 2 false %. +%term contra conc %. +%sort >> %. +%prec %prefix 3 >> %. +%term mt >> true top %. +%term pair %pi (>> true A) %-> (>> true B) %-> (>> true (and A B)) %. +%term fst %pi (>> true (and A B)) %-> (>> true A) %. +%term snd %pi (>> true (and A B)) %-> (>> true B) %. +%term abort %pi (>> true bot) %-> (>> J) %. +%term inl %pi (>> true A) %-> (>> true (or A B)) %. +%term inr %pi (>> true B) %-> (>> true (or A B)) %. +%term case + %pi (>> true (or A B)) + %-> (%pi (>> true A) %-> (>> J)) + %-> (%pi (>> true B) %-> (>> J)) + %-> (>> J) %. +%term lam %pi (%pi (>> true A) %-> (>> true B)) %-> (>> true (imp A B)) %. +%term app %pi (>> true (imp A B)) %-> (>> true A) %-> (>> true B) %. +%% anything follows from contradiction +%term cabort %pi (>> contra) %-> (>> J) %. +%% negative bottom +%term nboti %pi (>> contra) %-> (>> true nbot) %. +%term nbote %pi (>> true nbot) %-> (>> contra) %. +%% falsehood +%term cont %pi (%pi (>> true A) %-> (>> contra)) %-> (>> false A) %. +%term throw %pi (>> false A) %-> (>> true A) %-> (>> contra) %. +%% negation +%term noti %pi (>> false A) %-> (>> true (not A)) %. +%term notcase %pi (>> true (not A)) %-> (%pi (>> false A) %-> (>> J)) %-> (>> J) %. +%% because not should be positive +%% letcc +%term letcc %pi (%pi (>> false A) %-> (>> contra)) %-> (>> true A) %. +%block tb {A prop} [D >> true A]%. +%block fb {A prop} [D >> false A]%. +%worlds (pb tb fb) (>> _) %. +%define iff (%pi prop %-> prop %-> prop) [A] [B] and (imp A B) (imp B A) %. +%define iffi ( + %pi (%pi (>> true A) %-> (>> true B)) + %-> (%pi (>> true B) %-> (>> true A)) + %-> (>> true (iff A B))) [E1] [E2] pair (lam E1) (lam E2) %. +%define iffel (%pi (>> true (iff A B)) %-> (>> true A) %-> (>> true B)) [E1] [E2] app (fst E1) E2 %. +%define iffer (%pi (>> true (iff A B)) %-> (>> true B) %-> (>> true A)) [E1] [E2] app (snd E1) E2 %. +%define cdni (%pi (>> true A) %-> (>> true (not (not A)))) [x] noti (cont ([nx] notcase nx ([f] throw f x))) %. +%define cdne (%pi (>> true (not (not A))) %-> (>> true A)) [x] notcase x ([f] letcc ([u] throw f (noti u))) %. +%define notimpbot (%pi (>> true (iff A B)) %-> (>> true (iff (not A) (imp B bot)))) [c] iffi ([x] notcase x ([f] lam ([y] cabort (throw f (iffer c y))))) ([x] noti (cont ([y] abort (app x (iffel c y))))) %. +%define notimpbot2 (>> true (iff (not (not A)) (imp (imp A bot) bot))) notimpbot (notimpbot (iffi ([x] x) ([x] x))) %. +%{! + +## Double-negation translation + +This is essentially the +[Godel-Gentzen Negtive Translation](http://en.wikipedia.org/wiki/G%C3%B6del%E2%80%93Gentzen_negative_translation). +It differs only by De Morgan laws that are provable intuitionistically. +!}% +%sort * %. +%mode * %in %out %. +%term */top * top itop %. +%term */and %pi (* (and A B) (iand A' B')) %<- (* A A') %<- (* B B') %. +%term */bot * bot ibot %. +%term */or %pi (* (or A B) (inot (inot (ior A' B')))) %<- (* A A') %<- (* B B') %. +%term */imp %pi (* (imp A B) (iimp A' B')) %<- (* A A') %<- (* B B') %. +%term */nbot * nbot ibot %. +%term */not %pi (* (not A) (inot A')) %<- (* A A') %. +%block *b [X prop] [X' iprop] [_ * X (inot (inot X'))]%. +%worlds (*b) (* _ _) %. +%total A (* A _) %. +%unique * %in %out %. +%sort *tot {A} %. +%mode *tot %in %out %. +%block *totb [X prop] [X' iprop] [D* * X (idn X')] [_ *tot X D*]%. +%worlds (*totb) (*tot _ _) %. +%total {} (*tot _ _) %. +%% verified by %total above +%sort id-iprop %. +%term id-iprop/refl id-iprop A A %. +%sort id-iprop-not-cong %. +%mode id-iprop-not-cong %in %out %. +%term _ id-iprop-not-cong _ id-iprop/refl %. +%worlds (ipb) (id-iprop-not-cong _ _) %. +%total {} (id-iprop-not-cong _ _) %. +%sort *unique %. +%mode *unique %in %in %out %. +%worlds (*b) (*unique _ _ _) %. +%total {} (*unique _ _ _) %. +%% verified by %unique above +%sort itrue-respects-id %. +%mode itrue-respects-id %in %in %out %. +%term _ itrue-respects-id D _ D %. +%worlds (itb ipb) (itrue-respects-id _ _ _) %. +%total {} (itrue-respects-id _ _ _) %. +%{! + +### Double-negation elimination for the target of the translation + +!}% +%sort dne %. +%mode dne %in %out %. +%term _ + %pi (dne (*/and D*2 D*1) ([x] ipair (E1 (ifst (distdnand x))) (E2 (isnd (distdnand x))))) + %<- (dne D*1 E1) + %<- (dne D*2 E2) %. +%term _ dne */top ([x] imt) %. +%term _ dne (*/or _ _) ([x] tne x) %. +%term _ dne */bot ([x] dnebot x) %. +%term _ dne */nbot ([x] dnebot x) %. +%term _ + %pi (dne (*/imp D*2 D*1) ([x] ilam ([y] E2 (iapp (distdnimp x) (dni y))))) + %<- (dne D*2 E2) %. +%term _ %pi (dne (*/not D*) tne) %<- (dne D* E) %. +%block dneb [X prop] [X' iprop] [dx * X (idn X')] [_ dne dx tne]%. +%worlds (dneb itb) (dne _ _) %. +%total D (dne D _) %. +%{! + +## Soundness + +!}% +%sort sound/true %. +%mode sound/true %in %in %out %. +%sort sound/false %. +%mode sound/false %in %in %out %. +%sort sound/contra %. +%mode sound/contra %in %out %. +%% true +%term _ sound/true mt */top imt %. +%term _ + %pi (sound/true (pair E1 E2) (*/and D*2 D*1) (ipair E1' E2')) + %<- (sound/true E1 D*1 E1') + %<- (sound/true E2 D*2 E2') %. +%term _ + %pi (sound/true (fst (%the (>> true (and A1 A2)) E)) D*1 (ifst E')) + %<- (*tot A2 D*2) + %<- (sound/true E (*/and D*2 D*1) E') %. +%term _ + %pi (sound/true (snd (%the (>> true (and A1 A2)) E)) D*2 (isnd E')) + %<- (*tot A1 D*1) + %<- (sound/true E (*/and D*2 D*1) E') %. +%term _ + %pi (sound/true (%the (>> true C) (abort E)) D* (iabort E')) + %<- (sound/true E */bot E') %. +%term _ + %pi (sound/true (%the (>> true (or A1 A2)) (inl E)) (*/or D*2 D*1) (dni (iinl E'))) + %<- (sound/true E D*1 E') %. +%term _ + %pi (sound/true (%the (>> true (or A1 A2)) (inr E)) (*/or D*2 D*1) (dni (iinr E'))) + %<- (sound/true E D*2 E') %. +%term _ + %pi (sound/true (case (%the (>> true (or A B)) E) E1 E2) D* (Edne (ilam ([f itrue (inot C')] iapp E' (ilam ([x itrue (ior A' B')] icase x ([x1] iapp f (E1' x1)) ([x2] iapp f (E2' x2)))))))) + %<- (*tot A D*A) + %<- (*tot B D*B) + %<- (sound/true E (*/or D*B D*A) E') + %<- ({x >> true A} {x' itrue A'} {_ {A''} {D* * A A''} {Did} {E''} + %pi (sound/true x D* E'') + %<- (*unique D* D*A Did) + %<- (itrue-respects-id x' Did E'')} sound/true (E1 x) D* (E1' x')) + %<- ({x >> true B} {x' itrue B'} {_ {A''} {D* * B A''} {Did} {E''} + %pi (sound/true x D* E'') + %<- (*unique D* D*B Did) + %<- (itrue-respects-id x' Did E'')} sound/true (E2 x) D* (E2' x')) + %<- (dne D* Edne) %. +%term _ + %pi (sound/true (%the (>> true (imp A B)) (lam E)) (*/imp D* D*A) (ilam E')) + %<- ({x >> true A} {x' itrue A'} {_ {A''} {D* * A A''} {Did} {E''} + %pi (sound/true x D* E'') + %<- (*unique D* D*A Did) + %<- (itrue-respects-id x' Did E'')} sound/true (E x) D* (E' x')) %. +%term _ + %pi (sound/true (app E1 E2) D*B (iapp E1' E2')) + %<- (*tot A D*A) + %<- (sound/true E1 (*/imp D*B D*A) E1') + %<- (sound/true E2 D*A E2') %. +%term _ + %pi (sound/true (%the (>> true C) (cabort E)) D* (iabort E')) + %<- (sound/contra E E') %. +%term _ + %pi (sound/true (%the (>> true nbot) (nboti E)) */nbot (iabort E')) + %<- (sound/contra E E') %. +%term _ + %pi (sound/true (%the (>> true (not A)) (noti E)) (*/not D*) E') + %<- (sound/false E D* E') %. +%term _ + %pi (sound/true (notcase E1 E2) D*C (E2' E1')) + %<- (*tot A D*A) + %<- (sound/true E1 (*/not (%the (* A A') D*A)) E1') + %<- ({x >> false A} {x' itrue (inot A')} {_ {A'' iprop} {D* * A A''} {Did id-iprop A'' A'} {Did' id-iprop (inot A'') (inot A')} {E'' itrue (inot A'')} + %pi (sound/false x D* E'') + %<- (*unique D* D*A Did) + %<- (id-iprop-not-cong Did Did') + %<- (itrue-respects-id x' Did' E'')} sound/true (E2 x) D*C (E2' x')) %. +%term _ + %pi (sound/true (%the (>> true A) (letcc E)) D*A (Edne (ilam E'))) + %<- ({x >> false A} {x' itrue (inot A')} {_ {A'' iprop} {D* * A A''} {Did id-iprop A'' A'} {Did' id-iprop (inot A'') (inot A')} {E'' itrue (inot A'')} + %pi (sound/false x D* E'') + %<- (*unique D* D*A Did) + %<- (id-iprop-not-cong Did Did') + %<- (itrue-respects-id x' Did' E'')} sound/contra (E x) (E' x')) + %<- (dne D*A Edne) %. +%% false +%term _ + %pi (sound/false (%the (>> false A) (cont E)) D*A (ilam E')) + %<- ({x >> true A} {x' itrue A'} {_ {A''} {D* * A A''} {Did} {E''} + %pi (sound/true x D* E'') + %<- (*unique D* D*A Did) + %<- (itrue-respects-id x' Did E'')} sound/contra (E x) (E' x')) %. +%term _ %pi (sound/false (cabort E) D* (iabort E')) %<- (sound/contra E E') %. +%term _ + %pi (sound/false (case (%the (>> true (or A B)) E) E1 E2) D* (ilam ([c itrue C] iapp E' (ilam ([x] icase x ([x1] iapp (E1' x1) c) ([x2] iapp (E2' x2) c)))))) + %<- (*tot _ D*A) + %<- (*tot _ D*B) + %<- (sound/true E (*/or D*B D*A) E') + %<- ({x >> true A} {x' itrue A'} {_ {A''} {D* * A A''} {Did} {E''} + %pi (sound/true x D* E'') + %<- (*unique D* D*A Did) + %<- (itrue-respects-id x' Did E'')} sound/false (E1 x) D* (E1' x')) + %<- ({x >> true B} {x' itrue B'} {_ {A''} {D* * B A''} {Did} {E''} + %pi (sound/true x D* E'') + %<- (*unique D* D*B Did) + %<- (itrue-respects-id x' Did E'')} sound/false (E2 x) D* (E2' x')) %. +%term _ + %pi (sound/false (%the (>> false C) (abort E)) D* (iabort E')) + %<- (sound/true E */bot E') %. +%term _ + %pi (sound/false (notcase E1 E2) D*C (E2' E1')) + %<- (*tot _ D*A) + %<- (sound/true E1 (*/not (%the (* A A') D*A)) E1') + %<- ({x >> false A} {x' itrue (inot A')} {_ {A'' iprop} {D* * A A''} {Did id-iprop A'' A'} {Did' id-iprop (inot A'') (inot A')} {E'' itrue (inot A'')} + %pi (sound/false x D* E'') + %<- (*unique D* D*A Did) + %<- (id-iprop-not-cong Did Did') + %<- (itrue-respects-id x' Did' E'')} sound/false (E2 x) D*C (E2' x')) %. +%% contra +%term _ + %pi (sound/contra (throw E1 E2) (iapp E1' E2')) + %<- (*tot _ D*) + %<- (sound/true E2 D* E2') + %<- (sound/false E1 D* E1') %. +%term _ %pi (sound/contra (nbote E) E') %<- (sound/true E */nbot E') %. +%term _ + %pi (sound/contra (case E E1 E2) (iapp E' (ilam ([x] icase x E1' E2')))) + %<- (*tot _ D*A) + %<- (*tot _ D*B) + %<- (sound/true E (*/or D*B D*A) E') + %<- ({x >> true A} {x' itrue A'} {_ {A''} {D* * A A''} {Did} {E''} + %pi (sound/true x D* E'') + %<- (*unique D* D*A Did) + %<- (itrue-respects-id x' Did E'')} sound/contra (E1 x) (E1' x')) + %<- ({x >> true B} {x' itrue B'} {_ {A''} {D* * B A''} {Did} {E''} + %pi (sound/true x D* E'') + %<- (*unique D* D*B Did) + %<- (itrue-respects-id x' Did E'')} sound/contra (E2 x) (E2' x')) %. +%term _ %pi (sound/contra (abort E) (iabort E')) %<- (sound/true E */bot E') %. +%term _ %pi (sound/contra (cabort E) E') %<- (sound/contra E E') %. +%term _ + %pi (sound/contra (notcase E1 E2) (E2' E1')) + %<- (*tot _ D*A) + %<- (sound/true E1 (*/not (%the (* A A') D*A)) E1') + %<- ({x >> false A} {x' itrue (inot A')} {_ {A'' iprop} {D* * A A''} {Did id-iprop A'' A'} {Did' id-iprop (inot A'') (inot A')} {E'' itrue (inot A'')} + %pi (sound/false x D* E'') + %<- (*unique D* D*A Did) + %<- (id-iprop-not-cong Did Did') + %<- (itrue-respects-id x' Did' E'')} sound/contra (E2 x) (E2' x')) %. +%% uses a fancy variable case: +%block soundtb {A prop} {A' iprop} {Dx * A A'} [x >> true A] [x' itrue A'] [_ {A''} {D* * A A''} {Did} {E''} + %pi (sound/true x D* E'') + %<- (*unique D* Dx Did) + %<- (itrue-respects-id x' Did E'')]%. +%block soundfb {A prop} {A' iprop} {Dx * A A'} [x >> false A] [x' itrue (inot A')] [_ {A'' iprop} {D* * A A''} {Did id-iprop A'' A'} {Did' id-iprop (inot A'') (inot A')} {E'' itrue (inot A'')} + %pi (sound/false x D* E'') + %<- (*unique D* Dx Did) + %<- (id-iprop-not-cong Did Did') + %<- (itrue-respects-id x' Did' E'')]%. +%block soundpb [X prop] [X' iprop] [dx * X (idn X')] [_ dne dx tne] [_ *tot X dx]%. +%worlds (soundpb soundtb soundfb) (sound/true _ _ _) (sound/true _ _ _) (sound/false _ _ _) (sound/false _ _ _) (sound/contra _ _) %. +%total (D1 D4 D2 D5 D3) (sound/true D4 _ _) (sound/true D1 _ _) (sound/false D5 _ _) (sound/false D2 _ _) (sound/contra D3 _) %. +%{! + +## Completeness + +### Inclusion + +Include intuitionistic props into classical. We need to define another +translation because * is not total with the reverse mode (it doesn't +translate atoms). + +!}% +%sort *i %. +%mode *i %out %in %. +%term *i/top *i top itop %. +%term *i/and %pi (*i (and A B) (iand A' B')) %<- (*i A A') %<- (*i B B') %. +%term *i/bot *i bot ibot %. +%term *i/or %pi (*i (or A B) (ior A' B')) %<- (*i A A') %<- (*i B B') %. +%term *i/imp %pi (*i (imp A B) (iimp A' B')) %<- (*i A A') %<- (*i B B') %. +%block *ib [X prop] [X' iprop] [_ *i X X']%. +%worlds (*ib) (*i _ _) %. +%total A (*i _ A) %. +%unique *i %out %in %. +%sort *itot {A'} %. +%mode *itot %in %out %. +%block *itotb [X prop] [X' iprop] [D*i *i X X'] [_ *itot X' D*i]%. +%worlds (*itotb fb pb) (*itot _ _) %. +%total {} (*itot _ _) %. +%% verified by %total above +%sort id-prop %. +%term id-prop/refl id-prop A A %. +%sort *iunique %. +%mode *iunique %in %in %out %. +%worlds (*ib fb pb) (*iunique _ _ _) %. +%total {} (*iunique _ _ _) %. +%% verified by %unique above +%sort true-respects-id %. +%mode true-respects-id %in %in %out %. +%term _ true-respects-id D _ D %. +%worlds (tb fb pb) (true-respects-id _ _ _) %. +%total {} (true-respects-id _ _ _) %. +%{! + +### Intuitionistic truth implies classical truth + +The arguments are in a funny order because I copied from soundness. =) + +!}% +%sort incl %. +%mode incl %out %in %in %. +%term _ incl mt *i/top imt %. +%term _ + %pi (incl (pair E1 E2) (*i/and D*2 D*1) (ipair E1' E2')) + %<- (incl E1 D*1 E1') + %<- (incl E2 D*2 E2') %. +%term _ + %pi (incl (fst E) D*1 (ifst (%the (itrue (iand A1 A2)) E'))) + %<- (*itot A2 D*2) + %<- (incl E (*i/and D*2 D*1) E') %. +%term _ + %pi (incl (snd (%the (>> true (and A1 A2)) E)) D*2 (isnd E')) + %<- (*itot _ D*1) + %<- (incl E (*i/and D*2 D*1) E') %. +%term _ %pi (incl (%the (>> true C) (abort E)) D* (iabort E')) %<- (incl E *i/bot E') %. +%term _ + %pi (incl (%the (>> true (or A1 A2)) (inl E)) (*i/or D*2 D*1) (iinl E')) + %<- (incl E D*1 E') %. +%term _ + %pi (incl (%the (>> true (or A1 A2)) (inr E)) (*i/or D*2 D*1) (iinr E')) + %<- (incl E D*2 E') %. +%term _ + %pi (incl (case (%the (>> true (or A B)) E) E1 E2) D* (icase E' E1' E2')) + %<- (*itot _ D*A) + %<- (*itot _ D*B) + %<- (incl E (*i/or D*B D*A) E') + %<- ({x >> true A} {x' itrue A'} {_ {A-2} {D*' *i A-2 A'} {Did id-prop A-2 A} {E >> true A-2} %pi (incl E D*' x') %<- (*iunique D*' D*A Did) %<- (true-respects-id x Did E)} incl (E1 x) D* (E1' x')) + %<- ({x >> true B} {x' itrue B'} {_ {B-2} {D*' *i B-2 B'} {Did id-prop B-2 B} {E >> true B-2} %pi (incl E D*' x') %<- (*iunique D*' D*B Did) %<- (true-respects-id x Did E)} incl (E2 x) D* (E2' x')) %. +%term _ + %pi (incl (%the (>> true (imp A B)) (lam E)) (*i/imp D* D*A) (ilam E')) + %<- ({x >> true A} {x' itrue A'} {_ {A-2} {D*' *i A-2 A'} {Did id-prop A-2 A} {E >> true A-2} %pi (incl E D*' x') %<- (*iunique D*' D*A Did) %<- (true-respects-id x Did E)} incl (E x) D* (E' x')) %. +%term _ + %pi (incl (app E1 E2) D*B (iapp E1' E2')) + %<- (*itot A D*A) + %<- (incl E1 (*i/imp D*B D*A) E1') + %<- (incl E2 D*A E2') %. +%block inclb {A} {A'} {D* *i A A'} [x >> true A] [x' itrue A'] [_ {A-2} {D*' *i A-2 A'} {Did id-prop A-2 A} {E >> true A-2} %pi (incl E D*' x') %<- (*iunique D*' D* Did) %<- (true-respects-id x Did E)]%. +%worlds (inclb fb *itotb) (incl _ _ _) %. +%total D (incl _ _ D) %. +%{! + +### Round-tripping the two translations is classically equivalent + +!}% +%sort equiv %. +%mode equiv %in %in %out %. +%term _ equiv D D' (iffi ([x] x) ([x] x)) %. +%term _ + %pi (equiv (*/and D*2 D*1) (*i/and D*2' D*1') (iffi ([x] pair (iffel E1 (fst x)) (iffel E2 (snd x))) ([x] pair (iffer E1 (fst x)) (iffer E2 (snd x))))) + %<- (equiv D*1 D*1' E1) + %<- (equiv D*2 D*2' E2) %. +%term _ + %pi (equiv (*/or D*2 D*1) (*i/imp *i/bot (*i/imp *i/bot (*i/or D*2' D*1'))) (iffi ([x] iffel notimpbot2 (cdni (case x ([x1] inl (iffel E1 x1)) ([x2] inr (iffel E2 x2))))) ([x] case (cdne (iffer notimpbot2 x)) ([x1] inl (iffer E1 x1)) ([x2] inr (iffer E2 x2))))) + %<- (equiv D*1 D*1' E1) + %<- (equiv D*2 D*2' E2) %. +%term _ + %pi (equiv (*/imp D*2 D*1) (*i/imp D*2' D*1') (iffi ([f >> true (imp A1 A2)] lam ([x >> true B1] iffel E2 (app f (iffer E1 x)))) ([f >> true (imp B1 B2)] lam ([x >> true A1] iffer E2 (app f (iffel E1 x)))))) + %<- (equiv D*1 D*1' (%the (>> true (iff A1 B1)) E1)) + %<- (equiv D*2 D*2' (%the (>> true (iff A2 B2)) E2)) %. +%term _ equiv */nbot *i/bot (iffi ([x] cabort (nbote x)) ([x] abort x)) %. +%term _ + %pi (equiv (*/not D*) (*i/imp *i/bot D*') (iffi ([x] lam ([y] cabort (notcase x ([u] throw u (iffer E y))))) ([x] noti (cont ([y] abort (app x (iffel E y))))))) + %<- (equiv D* D*' E) %. +%block equivb [X prop] [X' iprop] [d*i *i X X'] [d* * X (inot (inot X'))] [_ equiv d* (*i/imp *i/bot (*i/imp *i/bot d*i)) (iffi ([x >> true X] iffel notimpbot2 (cdni x)) ([x >> true (imp (imp X bot) bot)] cdne (iffer notimpbot2 x)))]%. +%worlds (equivb tb fb) (equiv _ _ _) %. +%total D (equiv D _ _) %. +%{! + +### If A* is intuitionistically true, then A is classically true + +!}% +%sort comp %. +%mode comp %in %in %out %. +%term _ + %pi (comp (%the (itrue A') E') D* (iffer Eiff E)) + %<- (*itot A' D*i) + %<- (incl E D*i E') + %<- (equiv D* D*i Eiff) %. +%block comp-pb [X prop] [X' iprop] [d*i *i X X'] [d* * X (inot (inot X'))] [_ equiv d* (*i/imp *i/bot (*i/imp *i/bot d*i)) (iffi ([x >> true X] iffel notimpbot2 (cdni x)) ([x >> true (imp (imp X bot) bot)] cdne (iffer notimpbot2 x)))] [_ *itot X' d*i]%. +%worlds (comp-pb inclb fb) (comp _ _ _) %. +%total E1 (comp E1 _ _) %. +%sort comp/false %. +%mode comp/false %in %in %out %. +%term _ + %pi (comp/false E D* (cont ([x] notcase E' ([u] throw u x)))) + %<- (comp E (*/not D*) E') %. +%worlds (comp-pb inclb fb) (comp/false _ _ _) %. +%total {} (comp/false _ _ _) %. +%sort comp/contra %. +%mode comp/contra %in %out %. +%term _ %pi (comp/contra E (nbote E')) %<- (comp E */nbot E') %. +%worlds (comp-pb inclb fb) (comp/contra _ _) %. +%total {} (comp/contra _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +You can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Double-negation_translation). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/download.lf b/new-tests/stelf-output/pages/download.lf new file mode 100644 index 0000000..934debe --- /dev/null +++ b/new-tests/stelf-output/pages/download.lf @@ -0,0 +1,9 @@ +% +%{! +This page has been removed, you should go to the [new version](/download/). + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +You can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Download). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/effectiveness-lemma.lf b/new-tests/stelf-output/pages/effectiveness-lemma.lf new file mode 100644 index 0000000..7c1f67d --- /dev/null +++ b/new-tests/stelf-output/pages/effectiveness-lemma.lf @@ -0,0 +1,101 @@ +% +% +%{! We use the term **effectiveness lemma** for a lemma that explicitly proves a [totality assertion](/wiki/totality-assertion/) for an LF type family using another LF type family. + +There are two reasons to prove an effectiveness lemma: +- A type family may satisfy a totality assertion but not be written in such a way that Twelf can verify its totality automatically with a [%total](/wiki/percent-total/) declaration. For example, justifying the induction might require an [explicit termination metric](/wiki/structural-metrics/), or knowing that the type family covers all possible inputs might require some sophisticated reasoning (such as [reasoning from false](/wiki/reasoning-from-false/)). +- As an artifact of the way totality checking works, it is sometimes necessary to prove an effectiveness lemma even when Twelf has already verified the corresponding [%total](/wiki/percent-total/) declaration. + +We discuss these motivations in more detail after presenting an example effectiveness lemma. + +## Example effectiveness lemma + +Consider the relation that negates a bit: !}% +%sort bit %. +%term bit/0 bit %. +%term bit/1 bit %. +%sort bit-flip %. +%term bit-flip/01 bit-flip bit/0 bit/1 %. +%term bit-flip/10 bit-flip bit/1 bit/0 %. +%{! We can ask Twelf to prove the following totality assertion: + +> For all ``B : bit``, there exists a ``B' : bit`` and ``D : bit-flip B B'``. + +as follows: !}% +%mode bit-flip %in %out %. +%worlds () (bit-flip _ _) %. +%total {} (bit-flip _ _) %. +%{! However, we can also prove the totality relation explicitly as an effectiveness lemma ``can-bit-flip``: !}% +%sort can-bit-flip {B bit} %. +%mode can-bit-flip %in %out %. +%term _ can-bit-flip bit/0 bit-flip/01 %. +%term _ can-bit-flip bit/1 bit-flip/10 %. +%worlds () (can-bit-flip _ _) %. +%total {} (can-bit-flip _ _) %. +%{! When processing the ``%total``, Twelf verifies the following totality assertion: + +> For all ``B : bit``, there exists a ``B' : bit`` and ``D : bit-flip B B'`` and a ``D' : can-bit-flip B D``. + +## Motivating scenario + +This particular example is clearly not motivated by the first consideration mentioned above, as Twelf was able to prove the totality assertion directly. For such an example, see the tutorial on [explicit termination metrics](/wiki/structural-metrics/). + +However, this effectiveness lemma is motivated by the second consideration, as the following example demonstrates. Consider a programming language that includes primitive bits and a negation operation on them: !}% +%sort tp %. +%term tp/bit tp %. +%sort tm %. +%term bt %pi bit %-> tm %. +%term neg %pi tm %-> tm %. +%sort of %. +%term of-bt of (bt _) tp/bit %. +%term of-neg %pi (of (neg E) tp/bit) %<- (of E tp/bit) %. +%sort eval %. +%term eval-bt eval (bt B) (bt B) %. +%term eval-neg %pi (eval (neg E) (bt B')) %<- (eval E (bt B)) %<- (bit-flip B B') %. +%{! We elide the parts of the language that are not relevant to this example. + +For simplicity, assume the language is manifestly terminating, so the progress theorem can be proved by a simple inductive argument that shows that all terms evaluate to a value: + +```twelf +progress : of E T -> eval E V -> type. +%mode progress +D1 -D2. +%worlds () (progress _ _). +``` + +!}% +%{!! begin hidden !!}% +%sort progress %. +%mode progress %in %out %. +%{!! end hidden !!}% +%{! Now, consider the case of progress for ``of-neg``: + +```checkedtwelf +- : {Dflip : bit-flip B B'} + progress + (of-neg (Dof : of E tp/bit)) + (eval-neg Dflip DevalE) + <- progress Dof (DevalE : eval E (bt B)). +``` + +By induction, we come up with a derivation ``DevalE``, which, by the value inversion lemma, must result in a value of the form ``(bt B)``. To finish the case, we need a derivation ``Dflip : bit-flip B X1`` for some ``B':bit``. + +You might think that this case should be accepted as is. After all, the totality assertion proved by the above ``%total`` shows that such a ``Dflip`` must exist. + +Unfortunately, the current Twelf implementation rejects this case as ill-moded. In logic programming terms, variables bound in braces like ``\{Dflip\}`` are treated as unification variables, so they must be filled in by unification if they are not already part of an input term. On the other hand, [[subgoal]]s, which are the premises that are searched for using logic programming, must be written with an ``->``. This means that there is no way to **name** the derivation resulting from the appeal to the totality assertion for ``plus``. In metatheorem terms, this means that we cannot appeal to the totality assertion for a type family to come up with a derivation of the type family itself, only the output indices of the family. + +The work-around is to prove the effectiveness lemma. Using the above effectiveness lemma, we can finish this case as follows: !}% +%{!! begin checked !!}% +%term _ + %pi (progress (of-neg (%the (of E tp/bit) Dof)) (eval-neg Dflip DevalE)) + %<- (progress Dof (%the (eval E (bt B)) DevalE)) + %<- (can-bit-flip B Dflip) %. +%{!! end checked !!}% +%{! Because the derivation of `` bit-flip B B'`` is an index to ``can-flip-bit``, the case is mode-correct. + +This limitation of Twelf could conceivably be addressed in a future release. For now, it is straightforward to work around it using effectiveness lemmas to prove totality assertions explicitly. + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +You can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Effectiveness_lemma). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/elves-main-page.lf b/new-tests/stelf-output/pages/elves-main-page.lf new file mode 100644 index 0000000..94c5fdf --- /dev/null +++ b/new-tests/stelf-output/pages/elves-main-page.lf @@ -0,0 +1,33 @@ +% +%{! This is the "portal" page for the Principles of Programming Group at Carnegie Mellon. The entire namespace starting with POPGroup: and POPGroup talk: are restricted to users which I have the special permission "pop-group" in their [[Special:Listusers | user profile]]. People listed in that list which are called "Bureaucrat" are the people who can give people that permission. + +The RSS feed for recent changes will show edits made on this page, so don't say anything silly. In general the use of this namespace should be avoided in favor of the project namespace (for example, [[Project:To do]]) and should only be used for things like + +* Discussing preperation for the [[Elves:Server]] +* Discussing strategies for releaseing the wiki on an unsuspecting world (that's what this page is for) + + +## Contest? + +There are some discussions it might be useful to have in this written-down forum on the [talk page](/wiki/elves-talk-main-page/), for instance the idea of having a contest of some sort. + +## Things You <strike>Can't</strike> Can Do In Twelf + +We talked about in the meeting some of the bits we'd need for having there be some recognizable activity going on. This whole setup comes attached to a [[http://redundancymanman.org/wordpress/ blog]] for which we have lots of RSS subscribable goodness. We need to come up with topics (perhaps over at the talk page?) and get people to write things with a rough completion date. People should probably write in the POPGroup namespace and then move stuff into the main namespace when we put notice up on the blog. Note that the completion date could be long before we put it on the blog, but having estimated completion dates will help plan if/how the blog will work. + +* **TOPIC** --- **WRITER** --- **EST. DATE** +* Context Subtyping --- Rob and Dan Lee --- September 14? +* Closure Conversion --- ? (karl) --- ? + +### Other ideas + +* **Logical Relations Arguments** - _logical relations arguments cannot be carried out (except via heavy encodings)_ [http://fling-l.seas.upenn.edu/~plclub/cgi-bin/poplmark/index.php?title=Submission_from_Carnegie_Mellon] +* **Coinduction** - I'm not even certain what coinduction _is_, but the poplmark submissions page says _the status of coinduction is uncertain_ [http://fling-l.seas.upenn.edu/~plclub/cgi-bin/poplmark/index.php?title=Submission_from_Carnegie_Mellon] +* **Linear Logic/Type system** - Unanswered question from The Twelf Wiki: _How do I represent a linear logic or type system in Twelf?_ [http://fp.logosphere.cs.cmu.edu/twelf/?n=Main.Questions] +* **Orderd Linear Logic/Type system** - Unanswered question from The Twelf Wiki: _How do I represent an ordered linear logic or type system in Twelf?_ [http://fp.logosphere.cs.cmu.edu/twelf/?n=Main.Questions] !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Elves:Main_Page). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/elves-talk-main-page.lf b/new-tests/stelf-output/pages/elves-talk-main-page.lf new file mode 100644 index 0000000..bf79a5c --- /dev/null +++ b/new-tests/stelf-output/pages/elves-talk-main-page.lf @@ -0,0 +1,131 @@ +% +%{! __FORCETOC__ +## A Contest? + +Everyone seemed intriuged by the idea of getting a contest going. How could we make something like this work, and perhaps produce something useful. [Rsimmons](/wiki/user-rsimmons/) 20:21, 1 September 2006 (MST) + +## Categories + +I'd like to see categories for "twelf tricks"; things like putting a case in your theorem for closed terms, inducing extra subordination clauses, etc. Basically, something for people that understand Twelf but who might not have built a big enough toolbox yet (_e.g._, almost everyone but Karl). Deciding what is basic Twelf knowledge ([[:Category:tutorials]]) vs. advanced tricks might be an issue... thoughts? Maybe beginner/intermediate/advanced tutorials?  — [Tom 7](/wiki/user-tom7/) 14:40, 7 September 2006 (MST) +: The old wiki did have a general "beginner/intermediate/advanced" separation - I think that's a good first-order approximation - having a good number of code examples, plus "this is a good example of this idea" notes, will ultimately be a better guide. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 17:02, 7 September 2006 (MST) + +## Rephrasing "equality" + +I have been confused about this, and it makes more sense (to me) as a question than asking about [http://fp.logosphere.cs.cmu.edu/twelf/?n=Answers.Equality lebinitz equality] in the metalogic (this was answered, the answer just wasn't linked to from the question so I thought previously that it had not been answered. Is there a way to make the last metatheorem work? If not, is there a good explanation why not that we can put here? + +Also, I don't think I know how to do the equivalent of the _first_ metalemma in the higher order case, say the untyped lambda calculus, and even if it's not unbelievably difficult, if it's doable it seems like the thing that has been done. If the second metalemma is doable in the first order case, what about in the higher order case? Is one of these definitions of equality superior? My intuition is to want to trust an inductiely defined definition more... — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 17:36, 7 September 2006 (MST) !}% +%sort tp %. +%term base tp %. +%term pair %pi tp %-> tp %-> tp %. +%term list %pi tp %-> tp %. +%sort eq1 %. +%term eq1/i eq1 T T %. +%sort eq2 %. +%term eq2/base eq2 base base %. +%term eq2/pair %pi (eq2 (pair T1 T2) (pair T1' T2')) %<- (eq2 T1 T1') %<- (eq2 T2 T2') %. +%term eq2/list %pi (eq2 (list T) (list T')) %<- (eq2 T T) %. +%sort eq1=>2 {T1} %. +%mode eq1=>2 %in %in %out %. +%term _ eq1=>2 base _ eq2/base %. +%term _ + %pi (eq1=>2 (pair T1 T2) _ (eq2/pair D2 D1)) + %<- (eq1=>2 T1 eq1/i D1) + %<- (eq1=>2 T2 eq1/i D2) %. +%term _ %pi (eq1=>2 (list T) _ (eq2/list D)) %<- (eq1=>2 T eq1/i D) %. +%worlds () (eq1=>2 _ _ _) %. +%total T (eq1=>2 T _ _) %. +%sort eq2=>1 %. +%mode eq2=>1 %in %out %. +%term _ eq2=>1 eq2/base eq1/i %. +%term _ %pi (eq2=>1 (eq2/pair D2 D1) eq1/i) %<- (eq2=>1 D1 _) %<- (eq2=>1 D2 _) %. +%term _ %pi (eq2=>1 (eq2/list D) eq1/i) %<- (eq2=>1 D eq1/i) %. +%worlds () (eq2=>1 _ _) %. +% total T (eq2=>1 T _). FAILS (for some obvious reasons) +%{! : Here are the proofs of the equivalence of the two definitions. In practice, eq1 is more commonly used (and very flexible), because it is much easier to use the equivalence once you have it (the proofs have one case for eq1/i instead of having to be inductive over the structure of T). I threw in forall, to show how eq2 works in the higher-order case. You need a special block to make the proof of eq1=>2 go through. The trick to showing eq2=>1 is to also have lemmas that are equivalent to the eq2 intro rules. --[DanielKLee](/wiki/user-danielklee/) 22:59, 7 September 2006 (MST) !}% +%sort tp %. +%term base tp %. +%term pair %pi tp %-> tp %-> tp %. +%term list %pi tp %-> tp %. +%term forall %pi (%pi tp %-> tp) %-> tp %. +%sort eq1 %. +%term eq1/i eq1 T T %. +%sort eq2 %. +%term eq2/base eq2 base base %. +%term eq2/pair %pi (eq2 (pair T1 T2) (pair T1' T2')) %<- (eq2 T1 T1') %<- (eq2 T2 T2') %. +%term eq2/list %pi (eq2 (list T) (list T')) %<- (eq2 T T') %. +%term eq2/forall %pi (eq2 (forall T) (forall T')) %<- ({a} %pi (eq2 a a) %-> (eq2 (T a) (T' a))) %. +%block tp-block [a tp]%. +%block eq2-block [a tp] [eqa eq2 a a]%. +%sort eq1=>2 {T1} %. +%mode eq1=>2 %in %in %out %. +%block eq1=>2-block [a tp] [eqa eq2 a a] [_ eq1=>2 a eq1/i eqa]%. +%term _ eq1=>2 base _ eq2/base %. +%term _ + %pi (eq1=>2 (pair T1 T2) _ (eq2/pair D2 D1)) + %<- (eq1=>2 T1 eq1/i D1) + %<- (eq1=>2 T2 eq1/i D2) %. +%term _ %pi (eq1=>2 (list T) _ (eq2/list D)) %<- (eq1=>2 T eq1/i D) %. +%term _ + %pi (eq1=>2 (forall T') _ (eq2/forall D)) + %<- ({a} {eqa eq2 a a} %pi (eq1=>2 a eq1/i eqa) %-> (eq1=>2 (T' a) eq1/i (D a eqa))) %. +%worlds (eq1=>2-block) (eq1=>2 _ _ _) %. +%total T (eq1=>2 T _ _) %. +%sort eq1-resp-pair %. +%mode eq1-resp-pair %in %in %out %. +%term _ eq1-resp-pair eq1/i eq1/i eq1/i %. +%worlds (tp-block) (eq1-resp-pair _ _ _) %. +%total {} (eq1-resp-pair _ _ _) %. +%sort eq1-resp-list %. +%mode eq1-resp-list %in %out %. +%term _ eq1-resp-list eq1/i eq1/i %. +%worlds (tp-block) (eq1-resp-list _ _) %. +%total {} (eq1-resp-list _ _) %. +%sort eq1-resp-forall {a} %. +%mode eq1-resp-forall %in %out %. +%term _ eq1-resp-forall ([a] eq1/i) eq1/i %. +%worlds (tp-block) (eq1-resp-forall _ _) %. +%total {} (eq1-resp-forall _ _) %. +%sort eq2=>1 %. +%mode eq2=>1 %in %out %. +% the following case is a catch-all for variables and base +%term _ eq2=>1 _ eq1/i %. +%term _ + %pi (eq2=>1 (eq2/pair D2 D1) D12) + %<- (eq2=>1 D1 D1') + %<- (eq2=>1 D2 D2') + %<- (eq1-resp-pair D1' D2' D12) %. +%term _ %pi (eq2=>1 (eq2/list D) D'') %<- (eq2=>1 D D') %<- (eq1-resp-list D' D'') %. +%term _ + %pi (eq2=>1 (eq2/forall D) D'') + %<- ({a} {eqa} eq2=>1 (D a eqa) (D' a)) + %<- (eq1-resp-forall D' D'') %. +%worlds (eq2-block) (eq2=>1 _ _) %. +%total T (eq2=>1 T _) %. +%{! ::: Thank you thank you! I'll tutorial-fy this because I'll understand it better if I do :). — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 23:36, 7 September 2006 (MST) + +## CMMChallenge and POPLMark and CIVMark oh my + +http://www.cs.berkeley.edu/~adamc/poplmark/compile/compile.pdf - I may give it two hours over the weekend if I get tired of CIVMark again. Please someone else crack it to give me one less way to procrastinate... + +Everybody seems to be on the "Challenge" bandwagon, as a thought experiment what would we make if we made a <small>underscore</small>Mark? I.E. something that many of us who aren't Karl would be able to do pretty easily, but which is nasty or tricky or just big? — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 23:35, 7 September 2006 (MST) + +: I took a look at this benchmark. A lot of it seems pretty straightforward, but proving that a big step semantics is total for well-typed terms will be the biggest challenge. This is essentially a termination/normalization proof. Traditionally, you do this with a logical relations argument. The challenge gives you some leeway in your choice of dynamic semantics, but I imagine you'd either use a big step semantics or the reflexive transitive closure of a small step semantics. In the SVN twelf-lib, Dan Licata has an example of a termination proof using a canonizing substitution argument similar to (but a good deal simpler than) what Karl demonstrated on Thursday, that might be a good starting point. However, I suspect the heap in the imperative target language might make things... interesting. + +: As for developing a "Challenge", I can't think of anything interesting off the top of my head. The real "challenge" in all of this is being able to do your actual day-to-day research in machine-checkable way. So a good challenge problem is something you'd want to do in your day-to-day research that for whatever reason seems rather difficult to mechanize. --[DanielKLee](/wiki/user-danielklee/) 04:06, 8 September 2006 (MST) + +## Competition + +Did anyone else know this existed? [http://cocorico.cs.ru.nl/coqwiki/Cocorico%21FrontPage Coq Wiki]. I don't know French. What does Cocorico mean? --[DanielKLee](/wiki/user-danielklee/) 10:41, 28 September 2006 (MST) +* I think it has something to do with Final Fantasy IV.  — [Tom 7](/wiki/user-tom7/) 11:48, 28 September 2006 (MST) +:*I'd seen it before, but hadn't put too much thought into its existance I suppose. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 15:10, 28 September 2006 (MST) + +## Announcement - changing tag system + +I'm changing the tag system slightly. This *shouldn't* break anything for more than ten minutes, with one exception. If you used <nowiki><php>, <perl>, <javascript></nowiki> or any other syntax highlighting tags, those have to be changed to <nowiki><code php></code>, <code javascript></code></nowiki> tags. <nowiki> !}% +%{! </nowiki> tags will still work, I'm making these changes to be able to have more control over the Twelf tag-ness. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 17:46, 4 October 2006 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Elves_talk:Main_Page). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/equality.lf b/new-tests/stelf-output/pages/equality.lf new file mode 100644 index 0000000..b9b5143 --- /dev/null +++ b/new-tests/stelf-output/pages/equality.lf @@ -0,0 +1,124 @@ +% +% +%{! In many circumstances, it is necessary to represent **equality** of LF terms as a relation. + +One such circumstance is when we wish to prove a [uniqueness lemma](/wiki/uniqueness-lemma/) showing that an object-language judgement determines some outputs uniquely. For example, consider the [addition judgement on natural numbers](/wiki/proving-metatheorems-representing-the-judgements-of-the-natural-numbers/). To state this uniqueness lemma, we must define a judgement ``eq N N'`` representing equality of natural numbers. Then the theorem is stated as follows: + + +```twelf +plus-unique : plus N1 N2 N3 + -> plus N1 N2 N3' + -> eq N3 N3' + -> type. +%mode plus-unique +D1 +D2 -D3. +%worlds () (plus-unique _ _ _). +``` + +That is, any two ``plus`` derivations for the same summands have the same sum. + +In general, equality of LF terms corresponds to syntactic equality of object-language syntax encoded with a first-order representation, and to α-equivalance of object-language terms encoded with a higher-order representation. Consequently, equality is used in a variety of theorems and proofs (see the tutorials on [uniqueness lemma](/wiki/uniqueness-lemma/)s, [strengthening](/wiki/strengthening/) lemmas, and explicit [canonical forms lemma](/wiki/canonical-forms-lemma/)s for example applications). + +The purpose of this article is to answer the following question: how should we internalize equality of LF terms as an LF type family? + +## Running example: A tree structure + +As a running example, we use a simple tree structure: !}% +%sort tree %. +%term leaf tree %. +%term node %pi tree %-> tree %-> tree %. +%{! ## Identity + +The simplest way to represent equality of LF terms is with an _identity type family_. An identity type family represents a binary relation; it is defined by one constant expressing reflexivity. For example: !}% +%sort id-tree %. +%term id-tree/refl id-tree T T %. +%{! With this definition, the type ``id T T'`` is inhabited exactly when ``T`` and ``T'`` are in fact the same LF term. + +Various properties of ``id`` are admissible: +* Identity is symmetric and transitive. For example, we can prove the following metatheorems: !}% +%sort id-tree-sym %. +%mode id-tree-sym %in %out %. +%term _ id-tree-sym id-tree/refl id-tree/refl %. +%worlds () (id-tree-sym _ _) %. +%total {} (id-tree-sym _ _) %. +%sort id-tree-trans %. +%mode id-tree-trans %in %in %out %. +%term _ id-tree-trans id-tree/refl id-tree/refl id-tree/refl %. +%worlds () (id-tree-trans _ _ _) %. +%total {} (id-tree-trans _ _ _) %. +%{! * Identity is a congruence: equality of subterms entails equality of subterms. For example, we can prove the following metatheorem: !}% +%sort id-tree-node-cong %. +%mode id-tree-node-cong %in %in %out %. +%term _ id-tree-node-cong id-tree/refl id-tree/refl id-tree/refl %. +%worlds () (id-tree-node-cong _ _ _) %. +%total {} (id-tree-node-cong _ _ _) %. +%{! * Identity is invertible: equality of a constructed term entails equality of constructed terms. For example, we can prove the following metatheorem: !}% +%sort id-tree-node-inv %. +%mode id-tree-node-inv %in %out %out %. +%term _ id-tree-node-inv id-tree/refl id-tree/refl id-tree/refl %. +%worlds () (id-tree-node-inv _ _ _) %. +%total {} (id-tree-node-inv _ _ _) %. +%{! * Other relations respect equality, in the sense that we can replace equals for equals. For example, if we have a height relation on trees, then we will be able to prove the following metatheorem: !}% +%{!! begin hidden !!}% +%sort nat %. +%{!! end hidden !!}% +%sort height %. +%sort height-respects-id %. +%mode height-respects-id %in %in %out %. +%worlds () (height-respects-id _ _ _) %. +%{! Each of these proofs is a one-liner: the only possible identity derivations are reflexivity, so all the trees involved are equal, and thus reflexivity derives the result. The tutorial on [respects lemma](/wiki/respects-lemma/)s presents other examples of these metatheorems. + +## Alternative definition of equality: More primitive rules + +Now, if symmetry, transitivity, congruence, and invertibility are all admissible, you might consider other sets of primitive rules. For example, we could give a definition of equality by induction on the structure of the tree: !}% +%sort eq-tree %. +%term eq-leaf eq-tree leaf leaf %. +%term eq-node + %pi (eq-tree (node T1 T2) (node T1' T2')) + %<- (eq-tree T1 T1') + %<- (eq-tree T2 T2') %. +%{! This definition is sometimes called _deep equality_, as it analyzes the structure of the term. In contrast, identity is sometimes called _shallow equality_. + +We could add to this definition by making the equivalence relation axioms primitive as well: !}% +%term eq-refl eq-tree T T %. +%term eq-sym %pi (eq-tree T1 T2) %<- (eq-tree T2 T1) %. +%term eq-trans %pi (eq-tree T1 T3) %<- (eq-tree T1 T2) %<- (eq-tree T2 T3) %. +%{! We could even make the inversion principles primitive: !}% +%term eq-node-inv-1 %pi (eq-tree T1 T1') %<- (eq-tree (node T1 T2) (node T1' T2')) %. +%term eq-node-inv-2 %pi (eq-tree T2 T2') %<- (eq-tree (node T1 T2) (node T1' T2')) %. +%{! ## Which definition to use + +The type families ``id`` and ``eq`` (with any of the extensions) define the same binary relation on trees. So is there any reason to prefer one set of primitive rules to another? + +Yes! In either case, we must prove that other type families such as ``height`` respect equality. For identity, these proofs are trivial, as case-analyzing the identity derivation immediately shows that the related terms are syntactically equal. When equality is defined by additional primitive rules, these proofs require inductive arguments. Put another way, identity gives you the strongest inductive hypothesis when reasoning _from_ equality, which we do often to prove respects lemmas. Of course, the cost is that you must show that the other rules are admissible; but as we saw above, these proofs are one-liners. + +On the other hand, an inductive characterization like ``eq`` is useful if you are defining not equality but some other [equivalence relation](/wiki/equivalence-relation/). + +## Identity at multiple types + +In general, it is necessary to define identity types not just for one type, but for all types that appear in the syntax. For example, for trees that store natural numbers at the nodes, we would define: !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort tree %. +%term leaf tree %. +%term node %pi tree %-> nat %-> tree %-> tree %. +%sort id-tree %. +%term id-tree/refl id-tree T T %. +%sort id-nat %. +%term id-nat/refl id-nat T T %. +%{! Properties analogous to before are admissible, but the theorems for ``tree`` will have premises referring to ``id-nat``. For example: !}% +%{! (options removed from twelftag: check=true) !}% +%sort id-tree-node-cong %. +%mode id-tree-node-cong %in %in %in %out %. +%term _ id-tree-node-cong _ _ _ id-tree/refl %. +%worlds () (id-tree-node-cong _ _ _ _) %. +%total {} (id-tree-node-cong _ _ _ _) %. +%{! ## Other tutorials using equality + +See the tutorial on [respects lemma](/wiki/respects-lemma/)s for more detail on them. The tutorials on [uniqueness lemma](/wiki/uniqueness-lemma/)s, [strengthening](/wiki/strengthening/) lemmas, and explicit [canonical forms lemma](/wiki/canonical-forms-lemma/)s show example applications of equality. + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +You can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Equality). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/equivalence-relation.lf b/new-tests/stelf-output/pages/equivalence-relation.lf new file mode 100644 index 0000000..3b3b9b3 --- /dev/null +++ b/new-tests/stelf-output/pages/equivalence-relation.lf @@ -0,0 +1,40 @@ +% +%{! We represent many equivalence relations in Twelf. The page on [equality](/wiki/equality/) discusses how to represent syntactic equality of LF terms as an LF type family. This page presents examples of other equivalence relations. + +## Example: trees with unordered children + +Generally we think of trees as having "left" and "right" subtrees, but imagine we wish to ignore that distinction, so that two trees are equal if they have the same colored root and their subtrees are equal, but we don't insist that the "first" or "left" subtree of one tree is equal to the first subtree of the other tree. In this notion of equality, the following two trees would be equal. !}% +%sort tree %. +%term leaf tree %. +%term node %pi tree %-> tree %-> tree %. +%define tree1 tree node leaf (node leaf leaf) %. +%define tree2 tree node (node leaf leaf) leaf %. +%{! We define a notion of equality that is compatible with this definition: !}% +%sort eq-rot-tree %. +%term eq-rot-tree/node eq-rot-tree leaf leaf %. +%term eq-rot-tree/match + %pi (eq-rot-tree (node T1 T2) (node T1' T2')) + %<- (eq-rot-tree T1 T1') + %<- (eq-rot-tree T2 T2') %. +%term eq-rot-tree/swap + %pi (eq-rot-tree (node T1 T2) (node T2' T1')) + %<- (eq-rot-tree T1 T1') + %<- (eq-rot-tree T2 T2') %. +%query 2 _ _ eq-rot-tree tree1 tree2 %. +%{! ## Equality by canonical representative + +Equality by canonical representative is another approach to equality by means of an equivalence relation. A total, deterministic relation is defined that reduces every term to its canonical form, and the equivalence relation consideres two terms equivalent if they have identical canonical forms. + +### Example: canonical representatives for trees with unordered children + +The previous definition for equality over trees is unsatisfying in certian ways, for instance, there are two ways to prove that the tree ``(node leaf leaf)`` is equivalent to ``(node leaf leaf)``, one that uses the rule ``eq-rot-tree/match`` and one that uses the rule ``eq-rot-tree/swap``. We could define the canonical representative for every to be a tree where the left child was "bigger" than the right according to some metric, and then two trees could be compared for equality by seeing if their canonical representatives are identical. + +\{\{needs|An example of this is not put together; one would be nice.\}\} + +\{\{stub\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Equivalence_relation). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/error-messages.lf b/new-tests/stelf-output/pages/error-messages.lf new file mode 100644 index 0000000..029ed29 --- /dev/null +++ b/new-tests/stelf-output/pages/error-messages.lf @@ -0,0 +1,64 @@ +% +%{! This page lists many **error messages** that may be encountered using Twelf. The explanations given below are not a complete but will hopefully be helpful when using Twelf as described in this wiki. + +;**Ambiguous reconstruction** +:This error may indicate not passing enough “parameters” to a theorem. + +;**Definition violation** +:This error happens sporadically when doing line-by-line (Control-C Control-D) interactions. Try loading the whole file instead. It also happens when you write a theorem or relation abbreviation and don't declare it as an abbreviation. + +;**Expected ... to be moded** +:Modes are necessary for termination and totality checking. See [%mode](/wiki/percent-mode/). + +;**Expected type family, found object constant** +:This can occur when you define a judgment but forget "-> type". (Have you read [judgments as types](/wiki/judgment/)?) + +;**Free variable clash** +: When you write a pattern and give names to the variables then Twelf won't let two user-named variables to be unified. If they end up the same thing, it usually means your theorem is mixing up the variables and would not be as general as it appears in the pattern. + +;**Freezing violation ...** +:The metatheory features freeze type families, making it illegal to extend them. See [%freeze](/wiki/percent-freeze/). + +;**Left-hand side of arrow must be a type** +:This often happens when passing too many “parameters” to a theorem. + +; **No mode declaration for XXX** +: If a theorem includes a relation in its proof (rather than a theorem _about_ the relation) you get this error, assuming you haven't declared modes on your relation. The location of the offending relation may a variable declaration where the variable isn't used: ``\{T:plus X Y Z\}`` converts into a an arrow type ``plus X Y Z ->`` if ``T`` isn't used. + +;Omitted term has ambiguous hyperkind +:This only happens in pathological cases, see [ambiguous hyperkind](/wiki/ambiguous-hyperkind/). + +;Occurrence of variable ... in output (-) argument not necessarily ground +:Twelf expects that in [mode checking](/wiki/percent-mode/), all things declared to be outputs (-) are completely determined by inputs to the relation or the outputs of [[subgoals]], and gives this error when that assumption is violated. See [``%mode``](/wiki/percent-mode/). + +;Occurrence of variable ... in input (+) argument not necessarily ground +:Twelf expects that in [mode checking](/wiki/percent-mode/), when a [[subgoal]] is used, then all things declared to be inputs (+) of the subgoal are entirely determined by inputs to the relation, and gives this error when that assumption is violated. See [``%mode``](/wiki/percent-mode/). + +;**Signature error: Global signature size 20000 exceeded** +:You have loaded and reloaded so many signatures that the maximum number has been exceeded. +:You can reset or restart the Twelf server to start with a clean slate. If you frequently get the error, or if you are proving a very large system and need more (and your computer has sufficient memory), you can change the maximum in the file ``src/global/global.sml`` from the default (19999) to something larger. + +;**Termination violation: no order assigned for ...** +:This happens when you are trying to run a [``%terminates``](/wiki/percent-terminates/) or [``%total``](/wiki/percent-total/) directive on a type family that calls on a _different_ type family for which you have not established termination/totality. Go back and make sure that that the previous type family has its termination behavior/totality established. + +;**Totality: Output of subgoal not covered** +:There is an [output coverage](/wiki/coverage-checking/) error within a proof case. You may need to use [output factoring](/wiki/output-factoring/). + +;**Typing ambiguous -- unresolved constraints** +:This is often caused by an overuse of wild cards (underscores), which can cause the [unification](/wiki/unification/) algorithm to fail. Another common cause is superfluous or missing arguments. + +;**Undeclared identifier** +:This error often happens when misspelling the name of something, or when trying to use a variable that doesn't start with a capital letter. + +;World violation +:This can occur when you make use of a theorem (perhaps inductively) in a non-empty context. Try using [``%block``](/wiki/percent-block/). Also, see [substitution lemma](/wiki/substitution-lemma/) for several examples. + +In general, trying to find the exact cause of the error in a long proof may require binary search (ending the proof early and seeing at what point the error changes from "output not ground" to the error in question). + +\{\{needs|a more complete list of errors and better explanations\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Error_messages). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/eta-equivalence.lf b/new-tests/stelf-output/pages/eta-equivalence.lf new file mode 100644 index 0000000..b0bcb52 --- /dev/null +++ b/new-tests/stelf-output/pages/eta-equivalence.lf @@ -0,0 +1,24 @@ +% +%{! **Eta-equivalence** (η-equivalence) is a notion of proof equivalence in natural deduction logics with introduction and elimination forms. Roughly, it says that any proof of a proposition is equivalent to one which introduces the proposition's principal connective. It is a form of extensional equivalence. + +Consider the simply-typed lambda-calculus with arrow types. + + + + + +Eta-equivalence for terms is the least [congruence relation](/wiki/congruence-relation/) closed under the axiom: + + + +In logics and typed calculi, eta-equivalence is usually oriented to the left yielding a notion of _eta-expansion_. For example: + + + +Eta-expansion transforms an arbitrary proof of a proposition into a proof that introduces the proposition's principal connective. A term with no sub-terms that can be eta-expanded without introducing [beta-redexes](/wiki/beta-equivalence/) is said to be _eta-long_. Being eta-long is one aspect of being [canonical](/wiki/canonical-form/). !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Eta-equivalence). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/evaluation-contexts.lf b/new-tests/stelf-output/pages/evaluation-contexts.lf new file mode 100644 index 0000000..1478f33 --- /dev/null +++ b/new-tests/stelf-output/pages/evaluation-contexts.lf @@ -0,0 +1,78 @@ +% +%{! + +## Evaluation contexts, intrinsically typed + +Twelf verifies +- Preservation +- Progress +- Determinacy of factoring/step (using [%unique](/wiki/percent-unique/)) + +!}% +%sort tp %. +%term b tp %. +%term arr %pi tp %-> tp %-> tp %. +%sort tm %. +%sort val %. +%term app %pi (tm (arr A B)) %-> (tm A) %-> (tm B) %. +%term ret %pi (val A) %-> (tm A) %. +%term lam %pi (%pi (val A) %-> (tm B)) %-> (val (arr A B)) %. +%block tmb {A tp} [x val A]%. +%worlds (tmb) (tm _) (val _) %. +%sort ec %. +%term ec/var ec ([x] x) %. +%term ec/app1 %pi (ec ([x] app (E x) M)) %<- (ec E) %. +%term ec/app2 %pi (ec ([x] app (ret V) (E x))) %<- (ec E) %. +%sort redex %. +%mode redex %in %. +%term redex/beta redex (app (ret V1) (ret V2)) %. +%sort factored %. +%term f/val factored (ret V) %. +%term f/ec %pi (factored (E M0)) %<- (ec E) %<- (redex M0) %. +%sort factorapp %. +%mode factorapp %in %in %out %. +%term _ factorapp f/val f/val (f/ec redex/beta ec/var) %. +%term _ factorapp (f/ec (%the (redex M0) Dred0) (%the (ec E) Dec)) (%the (factored M2) _) (f/ec Dred0 (ec/app1 Dec)) %. +%term _ factorapp f/val (f/ec (%the (redex M0) Dred0) (%the (ec E) Dec)) (f/ec Dred0 (ec/app2 Dec)) %. +%worlds () (factorapp _ _ _) %. +%total {} (factorapp _ _ _) %. +%unique factorapp %in %in %out %. +%sort factor {M tm A} %. +%mode factor %in %out %. +%term factor/val factor (ret V) f/val %. +%term factor/app + %pi (factor (app E1 E2) F) + %<- (factor E1 F1) + %<- (factor E2 F2) + %<- (factorapp F1 F2 F) %. +%worlds () (factor _ _) %. +%total M (factor M _) %. +%unique factor %in %out %. +%sort result %. +%term done result (ret V) %. +%term stepped %pi (tm A) %-> (result (%the (tm A) M)) %. +%sort contract %. +%mode contract %in %out %. +%term _ contract (%the (redex (app (ret (lam M)) (ret V))) redex/beta) (M V) %. +%worlds () (contract _ _) %. +%total {} (contract _ _) %. +%unique contract %in %out %. +%sort stepf %. +%mode stepf %in %out %. +%term _ stepf f/val done %. +%term _ %pi (stepf (f/ec Dr (%the (ec E) Dec)) (stepped (E Mred))) %<- (contract Dr Mred) %. +%worlds () (stepf _ _) %. +%total {} (stepf _ _) %. +%unique stepf %in %out %. +%sort step {M tm A} %. +%mode step %in %out %. +%term step/i %pi (step M M') %<- (factor M F) %<- (stepf F M') %. +%worlds () (step _ _) %. +%total {} (step _ _) %. +%unique step %in %out %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Evaluation_contexts). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/exchange-lemma.lf b/new-tests/stelf-output/pages/exchange-lemma.lf new file mode 100644 index 0000000..f703213 --- /dev/null +++ b/new-tests/stelf-output/pages/exchange-lemma.lf @@ -0,0 +1,53 @@ +% +% +%{! **Exchange** is the property of a hypothetical judgment that if , then (assuming A and B are independent hypotheses). + +Often, we represent an an object-language hypothetical judgement by [using LF binding to model hypotheses](/wiki/higher-order-judgements/). When a judgement is represented in such a fashion, exchange comes "for free" from the LF representation. In particular, we can exchange hypotheses by re-arranging the order of the lambdas in a derivation. + +For example, consider the following simply typed λ-calculus: !}% +%% Syntax +%sort tp %. +%term tp/unit tp %. +%term tp/arrow %pi tp %-> tp %-> tp %. +%sort exp %. +%term exp/unit exp %. +%term exp/lam %pi tp %-> (%pi exp %-> exp) %-> exp %. +%term exp/app %pi exp %-> exp %-> exp %. +%% Typing +%sort of %. +%term of/unit of exp/unit tp/unit %. +%term of/lam + %pi (of (exp/lam T E) (tp/arrow T T')) + %<- ({x exp} %pi (of x T) %-> (of (E x) T')) %. +%term of/app %pi (of (exp/app E1 E2) T') %<- (of E2 T) %<- (of E1 (tp/arrow T T')) %. +%{! Object-language typing is a hypothetical judgement; we represent it using LF variables to model hypotheses (see \{\{proving|title=Representing the judgements of the STLC\}\} for more discussion of this representation). + +We can prove exchange for this judgement as follows: !}% +%sort exchange {x exp} {dx of x T} {y exp} {dy of y T'} %. +%mode exchange %in %out %. +%term _ exchange ([x] [dx of x T] [y] [dy of y T'] D1 x dx y dy) ([y] [dy of y T'] [x] [dx of x T] D1 x dx y dy) %. +%block of-block {T tp} [x exp] [dx of x T]%. +%worlds (of-block) (exchange _ _) %. +%total {} (exchange _ _) %. +%{! The proof simply permutes the LF lambdas. + +In practice, it is rare for exchange to be proven and used as a metatheorem. Instead, exchange is inlined by manipulating LF lambdas in the appropriate way. + +It is common to use exchange in an inductive case for a constant with a higher-order premise. +For example, in the [substitution lemma](/wiki/substitution-lemma/) article, exchange is used in the next-to-last case in the section [Substitution lemmas with a "var" rule](/wiki/substitution-lemma/#substitution-lemmas-with-a-var-rule). The exchange property is used there when ``x``, ``dx`` and ``y``, ``dy`` must be re-ordered to fit the correct sub-goal: + +```twelf +- : subst D1 ([x][dx] of/lam ([y][dy] D2 x dx y dy)) (of/lam D2') + <- ({y}{dy} subst D1 ([x][dx] D2 x dx y dy) (D2' y dy)). +``` + +## See also + +* [Weakening lemma](/wiki/weakening-lemma/) +* [Substitution lemma](/wiki/substitution-lemma/) + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +You can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Exchange_lemma). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/explicit-context.lf b/new-tests/stelf-output/pages/explicit-context.lf new file mode 100644 index 0000000..02d3c2b --- /dev/null +++ b/new-tests/stelf-output/pages/explicit-context.lf @@ -0,0 +1,16 @@ +% +%{! **Explicit contexts** are an alternative technique for encoding an [object language](/wiki/object-logic/). Standard practice is to use [higher-order judgment](/wiki/higher-order-judgements/)s in order to implicitly use the [LF](/wiki/lf/) context as the context for the object language semantics. While this affords many benefits, there are a number of proof techniques (such as [hereditary substitution](/wiki/hereditary-substitution/) for [dependently typed languages](/wiki/dependent-types/)) which require isolating an entry in the middle of the object language context. This is not possible in general using [higher-order judgment](/wiki/higher-order-judgements/)s for languages whose context does not admit exchange. + +It is possible to prove such theorems in an encoding of the object language semantics with an explicit reconstruction of the context. In order to maximally enjoy the benefits of [higher-order judgment](/wiki/higher-order-judgements/)s, standard practice is to use the explicit context system only as necessary, and prove theorems that convert derivations from the implicit and explicit systems. + +This technique was used in the code associated with [http://www.cs.cmu.edu/~dklee/papers/tslf.tgz Towards a Mechanized Metatheory of Standard ML]. + +Although explicit contexts are the most general technique for proving theorems that mention entries in the middle of contexts that do not admit exchange, very often there are tricks to avoid using explicit contexts. A section of the page on [substitution lemmas](/wiki/substitution-lemma/) illustrates one such technique. + +\{\{stub\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Explicit_context). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/file-disk-download-gif.lf b/new-tests/stelf-output/pages/file-disk-download-gif.lf new file mode 100644 index 0000000..d992d93 --- /dev/null +++ b/new-tests/stelf-output/pages/file-disk-download-gif.lf @@ -0,0 +1,8 @@ +% +%{! Small "disk" image, made by [[:user:Tom7]] and licensed for free use. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/File:Disk_download.gif). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/file-serverok-png.lf b/new-tests/stelf-output/pages/file-serverok-png.lf new file mode 100644 index 0000000..2bdf8f8 --- /dev/null +++ b/new-tests/stelf-output/pages/file-serverok-png.lf @@ -0,0 +1,8 @@ +% +%{! Interacting with Twelf through an Emacs buffer... !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/File:ServerOK.png). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/file-smalltree-svg.lf b/new-tests/stelf-output/pages/file-smalltree-svg.lf new file mode 100644 index 0000000..ef791d9 --- /dev/null +++ b/new-tests/stelf-output/pages/file-smalltree-svg.lf @@ -0,0 +1,16 @@ +% +%{! Graphic of a tree with four leaves + +Very strange, when Illustrator CS1 created this file it left a bad piece of code: + + <flowDef xmlns="&ns_flows;"> + +That I had to manually make a good XML tag with + + <flowDef xmlns="&ns_flows;" /> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/File:Smalltree.svg). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/file-sudokuexample-svg.lf b/new-tests/stelf-output/pages/file-sudokuexample-svg.lf new file mode 100644 index 0000000..143c66a --- /dev/null +++ b/new-tests/stelf-output/pages/file-sudokuexample-svg.lf @@ -0,0 +1,8 @@ +% +%{! From Wikipedia, public domain. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/File:Sudokuexample.svg). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/file-tom-twelfelf2-jpg.lf b/new-tests/stelf-output/pages/file-tom-twelfelf2-jpg.lf new file mode 100644 index 0000000..6601e57 --- /dev/null +++ b/new-tests/stelf-output/pages/file-tom-twelfelf2-jpg.lf @@ -0,0 +1,8 @@ +% +%{! Request by Tom... !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/File:Tom_Twelfelf2.jpg). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/file-twelflive-1-png.lf b/new-tests/stelf-output/pages/file-twelflive-1-png.lf new file mode 100644 index 0000000..c40febc --- /dev/null +++ b/new-tests/stelf-output/pages/file-twelflive-1-png.lf @@ -0,0 +1,8 @@ +% +%{! Twelf Live, with text in the first edit window !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/File:TWELFLIVE-1.png). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/file-twelflive-2-png.lf b/new-tests/stelf-output/pages/file-twelflive-2-png.lf new file mode 100644 index 0000000..009c531 --- /dev/null +++ b/new-tests/stelf-output/pages/file-twelflive-2-png.lf @@ -0,0 +1,8 @@ +% +%{! Twelf Live, with checked input !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/File:TWELFLIVE-2.png). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/file-twelflive-3-png.lf b/new-tests/stelf-output/pages/file-twelflive-3-png.lf new file mode 100644 index 0000000..6110d2f --- /dev/null +++ b/new-tests/stelf-output/pages/file-twelflive-3-png.lf @@ -0,0 +1,8 @@ +% +%{! Twelf Live, with incremental checked input !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/File:TWELFLIVE-3.png). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/first-order-encodings.lf b/new-tests/stelf-output/pages/first-order-encodings.lf new file mode 100644 index 0000000..b881d94 --- /dev/null +++ b/new-tests/stelf-output/pages/first-order-encodings.lf @@ -0,0 +1,187 @@ +% +%{! A Twelf source file consists of a collection of _declarations_, each of which is either +* a _type declaration_, in which a new type family is introduced (and named), +* a _value declaration_, in which a new _value constructor_ is introduced and given a type, +* a _signature check_, in which Twelf is asked to check a particular property of the type and value declarations given up to that point (collectively called the _signature_), or +* another kind of directive that changes the behavior of Twelf in some way. +Signature checks and directives are indicated with keywords beginning with the '``%``' symbol, such as  ``%mode``,  ``%worlds``,  ``%total``,  ``%name``,  etc., and comments are started with either '``% ``' or '``%%``'. Comments and judicious use of white space make source files more intelligible to readers. We will have more to say about what we mean by type _families_ in the next section, but, first, we look at some simpler data types. + +As a first example, consider these declarations !}% +% Days of the week +%sort day %. +%term sunday day %. +%term monday day %. +%term tuesday day %. +%term wednesday day %. +%term thursday day %. +%term friday day %. +%term saturday day %. +%{! The keyword ``type`` in the first (type) declaration indicates that ``day`` is a new type, and, having made this declaration, the following seven (value) declarations introduce elements of this type. These seven elements can also be viewed as _nullary_ constructors, i.e., as operations that take an empty list of arguments and produce an element of type ``day``. This is a finite data type, but the same principle works for infinite types with unary and binary constructors (and, indeed, constructors of any arity): !}% +% Natural numbers in "unary" form +%sort nat %. +%term 0 nat %. +% zero element +%term s %pi nat %-> nat %. +% unary successor operator +% Natural number lists +%sort nat_list %. +%term nil nat_list %. +% empty list +%term cons %pi nat %-> nat_list %-> nat_list %. +% (cons N L) is the list whose first element is N and the rest of whose elements is L +% Binary trees labelled with nats +%sort nat_tree %. +%term empty nat_tree %. +% empty tree +%term node %pi nat %-> nat_tree %-> nat_tree %-> nat_tree %. +% (node N L R) is the tree with label N, left subtree L, and right subtree R. +%{! In each of these three data structure definitions, the first declaration is a type declaration, and the following two are value declarations. +As with types in ML, the infix arrow, which associates to the right, indicates a "functional" type. However, there is no computation here, just the declaration of types and constructors. These declarations may be compared with the following ML definitions: +<pre>datatype day = sunday | monday | tuesday | wednesday | thursday | friday | saturday; +datatype nat = 0 | s of nat; (* s : nat -> nat *) +datatype nat_list = nil | cons of nat * nat_list; (* cons : nat * nat_list -> nat_list *) +datatype nat_tree = empty | node of nat * nat_tree * nat_tree; (* node : nat * nat_tree * nat_tree -> nat_tree *) +</pre> +One difference here is that Twelf has no built-in product type, so term constructors with arity two or greater, like ``cons`` (arity 2) and ``node`` (arity 3), have to be treated as curried higher-order functions. The elements of type ``nat``, in both Twelf and ML, are: +* ``0``, +* ``s 0``, +* ``s (s 0)``, +* ``s (s (s 0))``, +and so on; in fact, every element of ``nat`` results from applying the constructor ``s`` zero or more times to the constant (or nullary operator) ``0``. In this way, each natural number has a unique representative as an element of type ``nat``, and each element of type ``nat`` corresponds to a unique natural number; this is the sense in which we say that we have represented the natural numbers in Twelf. +Some of the elements of Twelf's type ``nat_list``, with their ML counterparts, are: +* ``nil``   (ML: ``nil``), +* ``cons 0 nil``   (ML: ``cons(0, nil)``), +* ``cons (s 0) (cons (s (s 0)) nil)``   (ML: ``cons(s 0, cons(s (s 0), nil))``), +* ``cons (s (s 0)) (cons (s 0) (cons 0 nil))``   (ML: ``cons(s (s 0), cons(s 0, cons(0, nil)))``); +these correspond to the lists ``[]``, ``[0]``, ``[1,2]``, and ``[2,1,0]``, respectively. +Finally, here are some elements of Twelf's type ``nat_tree`` (what are the corresponding ML terms?): +* ``empty``, +* ``node 0 empty empty``, +* ``node (s 0) (node (s (s 0)) empty empty) empty``, and +* ``node (s (s 0)) (node 0 empty empty) (node (s 0) empty empty)``. +In a similar way, any signature determines some types and elements of those types. + +Twelf also has a definition facility, whereby we can introduce new identifiers to stand for already existing elements. For example, we could introduce the identifiers ``zero``, ``one``, ``two``, etc., to stand for the corresponding elements of type ``nat``: !}% +%define zero %the nat 0 %. +%define one %the nat (s zero) %. +% or: one = s 0 : nat. +%define two %the nat (s one) %. +% or: two = s (s 0) : nat. +%define three %the nat (s two) %. +% or: three = s (s (s 0)) : nat. +%define four %the nat (s three) %. +% or: four = s (s (s (s 0))) : nat. etc. +%{! ## Type families + +The type ``nat_list`` from the previous subsection has as elements _all_ lists of natural numbers, of all lengths. However, we can refine this type, replacing it with an infinite collection of types \{L<sub>0</sub>, L<sub>1</sub>, L<sub>2</sub>, ...\}, where for each natural number _n_, the type L<sub>_n_</sub> consists of all natural number lists _of length n_. Thus L<sub>0</sub> has only the element ``nil``; L<sub>1</sub> contains exactly the one-element lists ``cons 0 nil``,  ``cons (s 0) nil``,  ``cons (s (s 0)) nil``,  etc.; L<sub>2</sub> contains exactly the two-element lists, and so on, and the ``cons`` operator becomes an infinite family of operators \{``cons``<sub>0</sub> : ``nat`` → L<sub>0</sub> → L<sub>1</sub>, ``cons``<sub>1</sub> : ``nat`` → L<sub>1</sub> → L<sub>2</sub>, ``cons``<sub>2</sub> : ``nat`` → L<sub>2</sub> → L<sub>3</sub>, ...\}. This is how we could define each of these types and operations separately in Twelf: !}% +%{! (options removed from twelftag: discard="true") !}% +%sort nat_list0 %. +% L_0: lists of length 0 +%sort nat_list1 %. +% L_1: lists of length 1 +%sort nat_list2 %. +% L_2: lists of length 2 +%sort nat_list3 %. +% L_3: lists of length 3 +%term nil nat_list0 %. +%term cons0 %pi nat %-> nat_list0 %-> nat_list1 %. +%term cons1 %pi nat %-> nat_list1 %-> nat_list2 %. +%term cons2 %pi nat %-> nat_list2 %-> nat_list3 %. +%{! Now, the list [0, 1, 2] would be represented as ``cons2 0 (cons1 (s 0) (cons0 (s (s 0)) nil))``, and this term and its subterms would have types as follows: +* ``cons2 0 (cons1 (s 0) (cons0 (s (s 0)) nil))`` : ``nat_list3``, +* ``cons1 (s 0) (cons0 (s (s 0)) nil)`` : ``nat_list2``, +* ``cons0 (s (s 0)) nil`` : ``nat_list1``, and +* ``nil`` : ``nat_list0``. +Now each natural number list and ``cons`` operator comes (via its type) with an explicit length. This refinement of the type gives us extra information, but it can get very tedious, depending on how far we need to carry it out. Moreover, however far out we go, we will still only have a finite number of types and operations (the signature being finite), so we will not have represented all possible natural number lists. + +This is where type families come in. The index we are using to refine the list type is a natural number, but we have already defined a natural number type in Twelf, so let's use it to define this entire infinite family of types in one go: !}% +%{! (options removed from twelftag: discard="true") !}% +% Natural number lists, indexed by length (a natural number) +%sort nat_list %. +% subsumes nat_list0, nat_list1, nat_list2, ... +%term nil nat_list 0 %. +%term cons {N nat} %pi nat %-> (nat_list N) %-> (nat_list (s N)) %. +% subsumes cons0, cons1, cons2, .... +%{! Thus, we are representing the family \{L<sub>0</sub>, L<sub>1</sub>, L<sub>2</sub>, ...\} as a kind of function that takes a natural number _n_ and gives the type L<sub>_n_</sub>. We say that ``nat_list`` is a _family_ of types, indexed by elements of ``nat``. Once this declaration is made, an infinite number of types come into existence, one for each element of ``nat``: +* ``nat_list 0``, +* ``nat_list (s 0)``, +* ``nat_list (s (s 0))``, +and so on. Similarly, we are representing the family \{``cons``<sub>0</sub> : ``nat`` → L<sub>0</sub> → L<sub>1</sub>, ``cons``<sub>1</sub> : ``nat`` → L<sub>1</sub> → L<sub>2</sub>, ``cons``<sub>2</sub> : ``nat`` → L<sub>2</sub> → L<sub>3</sub>, ...\} as a kind of function that takes a natural number _n_ and gives the associated operation of type ``nat`` → L<sub>_n_</sub> → L<sub>_n_+1</sub>. The prefix ``\{N:nat\}`` in the declaration for ``cons`` above can be thought of both as a universal quantifier and as an additional argument to ``cons`` on which the rest of its type depends. Thus, this declaration can be read, "for every element ``N`` of type ``nat``, ``cons`` applied to ``N`` is an operator of type ``nat -> nat_list N -> nat_list (s N)``. In particular, +* ``cons 0 : nat -> nat_list 0 -> nat_list (s 0)``, +* ``cons (s 0) : nat -> nat_list (s 0) -> nat_list (s (s 0))``, +* ``cons (s (s 0)) : nat -> nat_list (s (s 0)) -> nat_list (s (s (s 0)))``, +and so on. Thus, just as ``nat_list`` is a _type_ family indexed by elements of the type ``nat``, so ``cons`` is a _value_ family indexed by elements of type ``nat``, with the type dependent on what natural-number index ``cons`` is applied to. With these declarations, the list [0, 1, 2] would be represented as + ``cons (s (s 0)) 0 (cons (s 0) (s 0) (cons 0 (s (s 0)) nil))``. +(Compare this to ``cons2 0 (cons1 (s 0) (cons0 (s (s 0)) nil))`` above.) We have thus effectively "internalized" our previous "external" family of types and operators, and extended it to all natural numbers. + +Let's examine this universal quantification more closely. The general form of a universally quantified type is ``\{X:t\} P(X)``, where ``t`` is a type, called the index type, and ``P(X)`` is a type expression, called the body of the type, which has zero or more occurrences of the value variable ``X``. The actual variable used in the quantification is immaterial: the type ``\{X:t\} P(X)`` is the same as the type ``\{N:t\} P(N)``. In the type of ``cons`` above, the type ``t`` is ``nat``, the value variable is ``N``, and the body is ``nat -> nat_list N -> nat_list (s N)``. Twelf only allows value variables in quantifications; it does not allow type variables or quantifications of the form ``\{T:type\}``, which would require type variables. + +So, what are the elements of the ``t``-indexed type family ``\{X:t\} P(X)``? As suggested above, they are ``t``-indexed _value families_, where the value at a particular index ``v`` of the value family has the type that is at the same index ``v`` of the type family, namely ``P(v)``. Thus, they are functions ``f`` whose domain is ``t`` and are such that ``f v : P(v)`` for any value ``v:t``, i.e., functions whose result type depends (uniformly) on which value they are given as input. + +Now, it may turn out that ``P(X)`` doesn't actually have any occurrences of ``X``, in which case the body doesn't depend on the value of the index and is thus the same for all indices. Let's call this common type ``p``. Then, elements of ``\{X:t\} P(X)`` are just functions from ``t`` to ``p``, i.e., elements of the type ``t -> p``. This shows that universally quantified types subsume ordinary function types, and indeed, Twelf treats the function type ``A -> B`` as an abbreviation for the quantified type ``\{X:A\} B``, where ``X`` is a value variable that doesn't occur in ``B``. Thus, the actual type of ``cons`` above is + cons : \{N:nat\} \{M:nat\} \{L:nat_list N\} nat_list (s N), +where the value variables ``M`` and ``L`` were chosen arbitrarily. + +Just like universal quantifiers, type families can be iterated. For example, if we had a type declaration + iter : \{N:nat\} nat_list N -> type. +then we could introduce types with prefixes such as ``\{N:nat\} \{L:nat_list N\} \{I:iter N L\}`` .... It may not seem so now, but such deeply nested quantifier prefixes are quite common and useful, and it is not unusual for a type constructor to have a dozen or more quantifiers in its type. As you might imagine, however, it can get very tedious to supply all of these arguments every time the constructor is used, especially since they can, in most instances, be inferred from the context. Fortunately, Twelf has a mechanism by which these prefixes can be omitted from both the type declarations of the constructors and from the uses of the constructors in terms. For example, we could have left out the quantifier in the definition of ``cons``: !}% +% Natural number lists, indexed by length (a natural number) +%sort nat_list %. +%term nil nat_list 0 %. +%term cons %pi nat %-> (nat_list N) %-> (nat_list (s N)) %. +%{! Now, the value variable ``N`` in ``cons`` is _free_ (i.e., not quantified), so Twelf automatically tries to infer the appropriate quantifier. It can do so in this case, since ``N`` is being used as an argument to ``nat_list``, which expects to be applied to a ``nat``, so Twelf infers the quantifier ``\{N:nat\}`` for ``N``. Also, the second occurrence of ``N`` in the type is consistent, since the type of ``s`` is ``nat -> nat``, so the type of ``s N`` is ``nat``, which again is what ``nat_list`` expects. With this abbreviated declaration, the list [0,1,2] would be represented by + cons 0 (cons 1 (cons 2 nil)), +just as it was before we refined ``nat_list`` into a type family indexed by length. Twelf is automatically supplying the correct first arguments to each instance of ``cons``, using the following reasoning. The type of ``nil`` is ``nat_list 0``, so the missing argument of ``cons`` in ``cons 2 nil`` must be ``0``, the type of this expression being ``nat_list (s 0)``. But then the missing argument in the middle instance of ``cons`` has to be ``s 0``, making the type of the expression ``cons 1 (cons 2 nil)`` be ``nat_list (s 0)``. Finally, that means that the missing argument in the first instance of ``cons`` has to be ``s (s 0)``, and so the complete term can be reconstructed. As we shall see, this kind of reconstruction is the rule, rather than the exception, and it makes working with Twelf much easier that it would without it. + +## Representing relations and functions + +A crucial feature of Twelf is that we can use it to represent not only data structures but also the logic we use to reason about them. Ordinary predicate logic is built up from primitive predicates and relations using logical connectives such as "and", "or", and "implies", and quantifiers such as "for all" and "there exists". In Twelf, we us a computational interpretation of logic based on the so-called "Propositions as Types" principle, meaning that we represent statements in our logic as types (or type families), and proofs of these statements as elements of the associated types. This has two important consequences: (1) the logic and reasoning methods we can use are somewhat restricted (technically, we are restricted to the ∀∃ fragment, and our reasoning must be _Intuitionistic_ or _constructive_, about which we will have more to say later), and (2) propositions, being types, are themselves data structures that can therefore be reasoned about in Twelf, allowing us to iterate the process of representation and reasoning to any desired level. + +A relation, R, among objects from sets A<sub>1</sub>, ..., A<sub>n</sub> is usually taken to be a subset of the cartesian product: R ⊂ A<sub>1</sub> Χ ... Χ A<sub>n</sub>. The elements of this cartesian product are tuples ‹a<sub>1</sub>, ..., a<sub>n</sub>›, where each a<sub>i</sub> is an element of A<sub>i</sub>. The idea is that the subset R collects together all of the tuples that stand in the relation, thus providing a set-theoretic counterpart to the informal relation. Unfortunately, the type theory underlying Twelf cannot directly represent general subsets, and, as we have noted above, it doesn't have products either. But what we _can_ do is, in some ways, even better: we represent the relation R between A<sub>1</sub>, ..., A<sub>n</sub> as a type family + R : A<sub>1</sub> -> ... -> A<sub>n</sub> -> type. +Now, for every tuple ‹a<sub>1</sub>, ..., a<sub>n</sub>›, there is a type ``R a<sub>1</sub> ... a<sub>n</sub>``, and we take this to be the type of _evidence_ for the relationship between a<sub>1</sub>, ..., a<sub>n</sub> according to ``R``. If this type has at least one element (and in practice it will then have exactly one element), then, there being evidence for the relationship, we say that the elements a<sub>1</sub>, ..., a<sub>n</sub> are related by R. On the other hand, if this type is empty, i.e., it has no elements, then the indicated relationship does not hold. To repeat, a relation is represented as a type family, and each instance of the relation is not just true or false but a separate _type_ whose elements represent _evidence_ of its truth. The absence of such evidence means that the particular instance does not hold. + +Let's look at some examples. Recall the type ``day`` with elements ``sunday``, ``monday``, ..., ``saturday``, and consider the "next day" relation that a relates a day with the following day. Here is how we represent this relation in Twelf: !}% +% Next-day relation on day +%sort next_day %. +%term next_day_sun next_day sunday monday %. +%term next_day_mon next_day monday tuesday %. +%term next_day_tue next_day tuesday wednesday %. +%term next_day_wed next_day wednesday thursday %. +%term next_day_thu next_day thursday friday %. +%term next_day_fri next_day friday saturday %. +%term next_day_sat next_day saturday sunday %. +%{! The ``next_day`` relation is a binary relation on ``day`` and is therefore represented as a type family doubly indexed by ``day``. The next seven declarations introduce elements that stand for the evidence that specific instances of the "next day" relation hold (their names were chosen to be mnemonic but could be any identifiers). Thus, these seven pairs of days stand in the relation and no others do. + +Given this relation, we can define another relation, called ``dat`` ("day after tomorrow"), that relates each day with the day two days later: !}% +%{! (options removed from twelftag: discard="true") !}% +% Day after tomorrow relation on day +%sort dat %. +%term dat_def {D1 day} {D2 day} {D3 day} %pi (next_day D1 D2) %-> (next_day D2 D3) %-> (dat D1 D3) %. +%{! Logically, the declaration of ``dat_def`` can be read, "for all days ``D1``, ``D2``, and ``D3``, if the day after ``D1`` is ``D2``, and the day after ``D2`` is ``D3``, then the day two days after ``D1`` is ``D3``." In terms of evidence, ``dat_def`` is a constructor that, given any days ``D1``, ``D2``, and ``D3``, along with evidence for ``next_day D1 D2`` and ``next_day D2 D3``, represents evidence for ``dat D1 D3``. For example, we have + dat_def sunday monday tuesday next_day_sun next_day_mon : dat sunday tuesday. +That is, Tuesday is two days after Sunday, according to these definitions, because there exists evidence for ``dat sunday tuesday``, namely the element to the left of the colon above; this element is well-typed, since ``dat_def`` is being applied correctly to five arguments, with +* ``sunday : day`` (so that ``D1`` is instantiated to ``sunday``), +* ``monday : day`` (so that ``D2`` is instantiated to ``monday``), +* ``tuesday : day`` (so that ``D3`` is istantiated to ``tuesday``), +* ``next_day_sun : next_day sunday monday`` (i.e., ``next_day D1 D2``), and +* ``next_day_mon : next_day monday tuesday`` (i.e., ``next_day D2 D3``), +the whole term thus having type ``dat sunday tuesday`` (i.e., ``dat D1 D3``). In this way, the only pairs of days in the ``dat`` relation are those pairs where the second day is two days after the first. + +Just as we observed at the end of the previous section, the three quantifiers in the declaration of ``dat_def`` above can be omitted: !}% +% Day after tomorrow relation on day +%sort dat %. +%term dat_def %pi (next_day D1 D2) %-> (next_day D2 D3) %-> (dat D1 D3) %. +%{! Because the declarations of ``next_day`` and ``dat`` show that both constructors expect to be applied to two arguments of type ``day``, Twelf can reconstruct the quantifiers for ``D1``, ``D2``, and ``D3`` automatically. Furthermore, we can define the identifier ``dat_sunday`` to stand for the evidence that Tuesday is two days after Sunday as follows: !}% +%define dat_sunday %the (dat sunday tuesday) (dat_def next_day_sun next_day_mon) %. +%{! where the first three arguments of ``dat_def`` corresponding to the omitted quantifiers can be reconstructed from the types of ``next_day_sun`` and ``next_day_mon``. From this point on, we will engage in the standard practice in Twelf of omitting quantifiers whenever possible, relying on term reconstruction to supply them automatically. + +Finally, a word about **functions**. Although we've encoded ``next_day`` and ``dat`` as relations, they are in fact (the graphs of) functions. That is, there exist functions _f_ and _g_ such that for all elements _d_<sub>1</sub> and _d_<sub>2</sub> of type ``day``, _f_(_d_<sub>1</sub>) = _d_<sub>2</sub> if and only if ``next_day ``_d_<sub>1</sub> _d_<sub>2</sub>, and _g_(_d_<sub>1</sub>) = _d_<sub>2</sub> if and only if ``dat ``_d_<sub>1</sub> _d_<sub>2</sub>. Relations that arise from functions in this way are special; we call them _functional_. To make this more precise, let's take ``next_day`` as an example. If we designate the first argument of ``next_day`` as an _input_ argument and the second argument as an _output_ argument, then this relation has the property that, for any possible combination of input arguments, there is exactly one combination of output arguments for which the relation holds, i.e., the associated type is non-empty—or, as we say in a more positive way, _inhabited_. The same is true if we designate the first argument of ``dat`` as input and its second argument as output. In fact, the same is true if we designate the second arguments of ``next_day`` and ``dat`` as inputs and their first arguments as outputs. These latter designations would correspond to the inverse functions "previous day" and "two days ago". We will see in a later section how we can designate certain arguments of relations as inputs and outputs and have Twelf verify that the relations are functional according to these designations. + +Here is a link to a ``>file containing the Twelf declarations`` given so far, where, in the case of ``cons`` and ``dat_def``, we have chosen the versions that omitted the quantifiers. Let's look at `` check="true">Twelf's output`` for these declarations. As Twelf checks each declaration for type-correctness, it echoes the declaration in its output, including any reconstructed quantifiers, as you can see in the declarations of ``cons`` and ``dat_def``. Notice that a couple of identifiers (``nil`` and ``cons``) were declared more than once; in this case, Twelf remembers only the last declaration for each identifier, silently replacing any previous declarations. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/First-order_encodings). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/fixity-declaration.lf b/new-tests/stelf-output/pages/fixity-declaration.lf new file mode 100644 index 0000000..396522a --- /dev/null +++ b/new-tests/stelf-output/pages/fixity-declaration.lf @@ -0,0 +1,24 @@ +% +% +%{! The **``%infix``**, **``%prefix``** and **``%postfix``** declarations assign **fixity** and precedence to constants for the purpose of parsing (they have no meaning in the logic). + +_**Higher**_ numbers bind **_tighter_**, and the pretty printer only prints out necessary information. Hence the following example: !}% +%sort a %. +%term b a %. +%term c a %. +%term d a %. +%term + %pi a %-> a %-> a %. +%prec %left 1 + %. +%term * %pi a %-> a %-> a %. +%prec %left 2 * %. +%{!! begin checked !!}% +%define x a b + c * d %. +%% The parenthesis are necessary here +%define y a b + (c * d) %. +%% This means the same thing as b + c * d. +%{!! end checked !!}% +%{! ## See also +* Operator Declaration + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/focusing.lf b/new-tests/stelf-output/pages/focusing.lf new file mode 100644 index 0000000..b4ed1db --- /dev/null +++ b/new-tests/stelf-output/pages/focusing.lf @@ -0,0 +1,1055 @@ +% +%{! +By [Rob Simmons](/wiki/user-rsimmons/). + +Focusing, introduced by Jean-Marc Andreoli in the context of classical linear +logic, defines a normal form for sequent calculus derivations that cuts down +on the number of possible derivations by eagerly applying invertible rules and +grouping sequences of non-invertible rules. A focused sequent calculus is +defined relative to some non-focused sequent calculus; focalization is the +property that every non-focused derivation can be transformed into a focused +derivation. In the paper [Structural +Focalization](https://arxiv.org/abs/1109.6273), I present a focused sequent +calculus for propositional intuitionistic logic and prove the focalization +property relative to a standard presentation of propositional intuitionistic +logic. + +This version follows the on-paper version very directly; the cost is +that the `id⁻` rule and the admissible `subst⁻` focal substitution +principle have to be written out instead of given for free by LF substitution +as the `id⁺` rule and `subst⁻` substitution principles are. +!}% +%sort small %. +%sort big %. +%term s small %. +%term b %pi small %-> big %. +%{! ## Syntax !}% +%sort pol %. +%name pol %. +%term ⁺ pol %. +%term ⁻ pol %. +%{! We define propositions in an environment with free atoms. !}% +%sort atom %. +%name atom %. +%block atom⁺ [Q⁺ atom ⁺]%. +%block atom⁻ [Q⁻ atom ⁻]%. +%sort typ %. +%name typ %. +%term c %pi (atom P) %-> (typ P) %. +%term ↓ %pi (typ ⁻) %-> (typ ⁺) %. +%term ⊥ typ ⁺ %. +%term ∨ %pi (typ ⁺) %-> (typ ⁺) %-> (typ ⁺) %. +%prec %none 5 ∨ %. +%term ⊤⁺ typ ⁺ %. +%term ∧⁺ %pi (typ ⁺) %-> (typ ⁺) %-> (typ ⁺) %. +%prec %none 5 ∧⁺ %. +%term ↑ %pi (typ ⁺) %-> (typ ⁻) %. +%term ⊃ %pi (typ ⁺) %-> (typ ⁻) %-> (typ ⁻) %. +%prec %none 5 ⊃ %. +%term ⊤⁻ typ ⁻ %. +%term ∧⁻ %pi (typ ⁻) %-> (typ ⁻) %-> (typ ⁻) %. +%prec %none 5 ∧⁻ %. +%{! Succedents are things that come to the right of the turnstile. !}% +%sort suc %. +%name suc %. +%term inv %pi (typ ⁻) %-> suc %. +%prec %postfix 3 inv %. +%term true %pi (typ ⁺) %-> suc %. +%prec %postfix 3 true %. +%term susp %pi (typ ⁻) %-> suc %. +%prec %postfix 3 susp %. +%{! Right-stable judgments are ones that are either positive or suspended. !}% +%sort stable %. +%name stable %. +%term sp stable (A⁺ true) %. +%term sn stable (A⁻ susp) %. +%{! An inversion context ``Ω'' is a list of positive propositions. !}% +%sort pos %. +%name pos %. +%term · pos %. +%term , %pi (typ ⁺) %-> pos %-> pos %. +%prec %right 3 , %. +%{! ## Sequent calculus !}% +%sort seqform %. +%name seqform %. +%sort exp %. +%name exp %. +%term rfoc %pi (typ ⁺) %-> seqform %. +%inline value (%pi (typ ⁺) %-> %type) [A⁺] exp (rfoc A⁺) %. +%sort leftform %. +%term hasleft %pi leftform %-> suc %-> seqform %. +%term ininv %pi pos %-> leftform %. +%inline term (%pi pos %-> suc %-> %type) [_Ω] [U] exp (hasleft (ininv _Ω) U) %. +%term infoc %pi (typ ⁻) %-> leftform %. +%inline spine (%pi (typ ⁻) %-> suc %-> %type) [A⁻] [U] exp (hasleft (infoc A⁻) U) %. +%sort hyp %. +%name hyp %. +%block nprop {A⁻ typ ⁻} [x hyp A⁻]%. +%block aprop {Q⁺ atom ⁺} [z value (c Q⁺)]%. +%block pprop {A⁺ typ ⁺} [z value A⁺]%. +%worlds (atom⁺ atom⁻ nprop) (hyp _) %. +%block gamma_suspnormal%. +% World for cut +%block gamma%. +% World for identity +%sort suspnormal %. +%term snn suspnormal (A⁻ inv) %. +%term snp suspnormal (A⁺ true) %. +%term sna suspnormal (c Q⁻ susp) %. +%sort suspstable %. +%term ss %pi (stable U) %-> (suspnormal U) %-> (suspstable U) %. +%sort suspnormalF %. +%term snr suspnormalF (rfoc A⁺) %. +%term snl %pi (suspnormalF (hasleft _ U)) %<- (suspnormal U) %. +%{! ### Values !}% +%{! +``V ::= z | thunk N | inl V | inr V | ⟨⟩⁺ | ⟨V₁,V₂⟩⁺`` + +The rule `id⁺` corresponding to the proof term `z` comes for free +from LF. +!}% +%term ↓R %pi (term · (A⁻ inv)) %-> (value (↓ A⁻)) %. +%term ∨R₁ %pi (value A⁺) %-> (value (A⁺ ∨ B⁺)) %. +%term ∨R₂ %pi (value B⁺) %-> (value (A⁺ ∨ B⁺)) %. +%term ⊤⁺R value ⊤⁺ %. +%term ∧⁺R %pi (value A⁺) %-> (value B⁺) %-> (value (A⁺ ∧⁺ B⁺)) %. +%{! ### Terms !}% +%{! +``M ::= ret V | x • Sp | ⟨z⟩.N | x.N | abort | [N₁, N₂] | ⟨⟩.N | ×N | ⟨N⟩ | {N} | λN | ⟨⟩⁻ | ⟨N₁,N₂⟩⁻`` +!}% +%term focR %pi (value A⁺) %-> (term · (A⁺ true)) %. +%term focL %pi (stable Suc) %-> (hyp A⁻) %-> (spine A⁻ Suc) %-> (term · Suc) %. +%term η⁺ %pi (%pi (value (c Q⁺)) %-> (term _Ω Suc)) %-> (term (c Q⁺ , _Ω) Suc) %. +%term ↓L %pi (%pi (hyp A⁻) %-> (term _Ω Suc)) %-> (term (↓ A⁻ , _Ω) Suc) %. +%term ⊥L term (⊥ , _Ω) Suc %. +%term ∨L %pi (term (A⁺ , _Ω) Suc) %-> (term (B⁺ , _Ω) Suc) %-> (term (A⁺ ∨ B⁺ , _Ω) Suc) %. +%term ⊤⁺L %pi (term _Ω Suc) %-> (term (⊤⁺ , _Ω) Suc) %. +%term ∧⁺L %pi (term (A⁺ , B⁺ , _Ω) Suc) %-> (term (A⁺ ∧⁺ B⁺ , _Ω) Suc) %. +%term η⁻ %pi (term · (c Q⁻ susp)) %-> (term · (c Q⁻ inv)) %. +%term ↑R %pi (term · (A⁺ true)) %-> (term · (↑ A⁺ inv)) %. +%term ⊃R %pi (term (A⁺ , ·) (B⁻ inv)) %-> (term · (A⁺ ⊃ B⁻ inv)) %. +%term ⊤⁻R term · (⊤⁻ inv) %. +%term ∧⁻R %pi (term · (A⁻ inv)) %-> (term · (B⁻ inv)) %-> (term · (A⁻ ∧⁻ B⁻ inv)) %. +%{! ### Spines !}% +%{! +``Sp ::= nil | pm N | V;Sp | π₁;Sp | π₂;Sp`` +!}% +%term id⁻ spine A⁻ (A⁻ susp) %. +%term ↑L %pi (stable Suc) %-> (term (A⁺ , ·) Suc) %-> (spine (↑ A⁺) Suc) %. +%term ⊃L %pi (value A⁺) %-> (spine B⁻ Suc) %-> (spine (A⁺ ⊃ B⁻) Suc) %. +%term ∧⁻L₁ %pi (spine A⁻ Suc) %-> (spine (A⁻ ∧⁻ B⁻) Suc) %. +%term ∧⁻L₂ %pi (spine B⁻ Suc) %-> (spine (A⁻ ∧⁻ B⁻) Suc) %. +%{! ### Negative focal substitution !}% +%{! Our encoding gives us positive focal substitution for free - it's +very natural to describe suspended propositions `⟨A⁺⟩` in the +hypothetical context as variables of type `rfoc A⁺`, which is +adequate and gives us the `id⁺` rule for free. By choosing a more +traditional way of describing negative suspended propositions (instead +of the complicated encoding that gives us `id⁻` for free), we have +forced ourselves to prove a theorem, negative focal substitution, that +we could have avoided. !}% +%sort subst⁻ %. +%mode subst⁻ %in %in %in %out %. +%term _ %pi (subst⁻ Pf (focL _ X Sp0) Sp (focL Pf X Sp0')) %<- (subst⁻ Pf Sp0 Sp Sp0') %. +%term _ + %pi (subst⁻ Pf (η⁺ ([z] N z)) Sp (η⁺ ([z] N' z))) + %<- ({z} subst⁻ Pf (N z) Sp (N' z)) %. +%term _ + %pi (subst⁻ Pf (↓L ([x] N x)) Sp (↓L ([x] N' x))) + %<- ({x} subst⁻ Pf (N x) Sp (N' x)) %. +%term _ subst⁻ Pf ⊥L Sp ⊥L %. +%term _ + %pi (subst⁻ Pf (∨L N1 N2) Sp (∨L N1' N2')) + %<- (subst⁻ Pf N1 Sp N1') + %<- (subst⁻ Pf N2 Sp N2') %. +%term _ %pi (subst⁻ Pf (⊤⁺L N) Sp (⊤⁺L N')) %<- (subst⁻ Pf N Sp N') %. +%term _ %pi (subst⁻ Pf (∧⁺L N) Sp (∧⁺L N')) %<- (subst⁻ Pf N Sp N') %. +%term _ subst⁻ Pf id⁻ Sp Sp %. +%term _ %pi (subst⁻ Pf (↑L _ N) Sp (↑L Pf N')) %<- (subst⁻ Pf N Sp N') %. +%term _ %pi (subst⁻ Pf (⊃L V Sp0) Sp (⊃L V Sp0')) %<- (subst⁻ Pf Sp0 Sp Sp0') %. +%term _ %pi (subst⁻ Pf (∧⁻L₁ Sp0) Sp (∧⁻L₁ Sp0')) %<- (subst⁻ Pf Sp0 Sp Sp0') %. +%term _ %pi (subst⁻ Pf (∧⁻L₂ Sp0) Sp (∧⁻L₂ Sp0')) %<- (subst⁻ Pf Sp0 Sp Sp0') %. +%worlds (gamma) (subst⁻ _ _ _ _) %. +%total E (subst⁻ _ E _ _) %. +%{! ## Cut admissibility !}% +%{! Cut admissibility has a couple of mutually inductive theorems, but +no more than is absolutely necessary given the syntactic classes we're +dealing with. Furthermore, the different theorems we use neatly sort +out the informal division of cases that we are used to dealing with +when proving cut admissibility theorems. The "principal" substitutions +capture the principal cuts: + +* ``V • Ni`` - positive cut formula, and +* ``N • Sp`` - negative cut formula. !}% +%sort cut⁺ {A⁺} %. +%sort cut⁻ {A⁻} %. +%mode cut⁺ %in %in %in %in %in %out %. +%mode cut⁻ %in %in %in %in %in %out %. +%{! The "rightist" substitution ``[[N/x]]E`` captures all right +commutative cuts. Each inductive call within this group decreases the +size of the expression E that we are substituting into. !}% +%sort rsubst {A⁻} %. +%mode rsubst %in %in %in %in %in %out %. +%{! The "leftist" substitution ``<>N`` captures all left commutative +cuts. Each inductive call within this group decreases the size of the +expression E that we are substituting in, hence "leftist." !}% +%sort lsubst {A⁺} %. +%mode lsubst %in %in %in %in %in %out %. +%{! ### Principal substitution/cuts !}% +%{! ``(V • N) = N'`` !}% +%term _ cut⁺ (c Q⁺) S _ X (η⁺ ([z] N z)) (N X) %. +%term _ + %pi (cut⁺ (↓ A⁻) s Pf (↓R M) (↓L N) N') + %<- (rsubst A⁻ (b s) (snl Pf) M N (%the (term _Ω U) N')) %. +%term _ + %pi (cut⁺ (A⁺ ∨ B⁺) S Pf (∨R₁ V) (∨L N₁ N₂) N') + %<- (cut⁺ A⁺ S Pf V N₁ (%the (term _Ω U) N')) %. +%term _ + %pi (cut⁺ (A⁺ ∨ B⁺) S Pf (∨R₂ V) (∨L N₁ N₂) N') + %<- (cut⁺ B⁺ S Pf V N₂ (%the (term _Ω U) N')) %. +%term _ cut⁺ ⊤⁺ S Pf ⊤⁺R (⊤⁺L N) N %. +%term _ + %pi (cut⁺ (A⁺ ∧⁺ B⁺) S Pf (∧⁺R V₁ V₂) (∧⁺L N) N') + %<- (cut⁺ A⁺ S Pf V₁ N (%the (term (B⁺ , _Ω) U) NB)) + %<- (cut⁺ B⁺ S Pf V₂ NB (%the (term _Ω U) N')) %. +%{! ``(M • Sp) = N'`` !}% +%term _ cut⁻ (c Q⁻) S Pf (η⁻ N) id⁻ N %. +%term _ %pi (cut⁻ (↑ A⁺) s Pf (↑R N) (↑L _ M) N') %<- (lsubst A⁺ (b s) Pf N M N') %. +%term _ + %pi (cut⁻ (A⁺ ⊃ B⁻) S (ss Pf1 Pf2) (⊃R N) (⊃L V Sp) N') + %<- (cut⁺ A⁺ S snn V N (%the (term · (B⁻ inv)) NB)) + %<- (cut⁻ B⁻ S (ss Pf1 Pf2) NB Sp (%the (term · U) N')) %. +%term _ + %pi (cut⁻ (A⁻ ∧⁻ B⁻) S Pf (∧⁻R N₁ N₂) (∧⁻L₁ Sp) N') + %<- (cut⁻ A⁻ S Pf N₁ Sp (%the (term · U) N')) %. +%term _ + %pi (cut⁻ (A⁻ ∧⁻ B⁻) S Pf (∧⁻R N₁ N₂) (∧⁻L₂ Sp) N') + %<- (cut⁻ B⁻ S Pf N₂ Sp (%the (term · U) N')) %. +%{! ### Rightist substitution (right commutative cuts) !}% +%{! ``[[M/x]]V = V' `` !}% +%term _ rsubst A⁻ S snr M ([x] V) (%the (value (c Q⁺)) V) %. +%term _ + %pi (rsubst A⁻ S snr M ([x] ↓R (N x)) (↓R N')) + %<- (rsubst A⁻ S (snl snn) M ([x] N x) (%the (term · (C⁻ inv)) N')) %. +%term _ + %pi (rsubst A⁻ S snr M ([x] ∨R₁ (V x)) (∨R₁ V')) + %<- (rsubst A⁻ S snr M ([x] V x) (%the (value C₁⁺) V')) %. +%term _ + %pi (rsubst A⁻ S snr M ([x] ∨R₂ (V x)) (∨R₂ V')) + %<- (rsubst A⁻ S snr M ([x] V x) (%the (value C₂⁺) V')) %. +%term _ rsubst A⁻ S snr M ([x] ⊤⁺R) ⊤⁺R %. +%term _ + %pi (rsubst A⁻ S snr M ([x] ∧⁺R (V₁ x) (V₂ x)) (∧⁺R V₁' V₂')) + %<- (rsubst A⁻ S snr M ([x] V₁ x) (%the (value C₁⁺) V₁')) + %<- (rsubst A⁻ S snr M ([x] V₂ x) (%the (value C₂⁺) V₂')) %. +%{! ``[[M/x]]N = N' `` !}% +% Entering principal substitution (cut⁻) +%term _ + %pi (rsubst A⁻ (b s) (snl Pf) M ([x] focR (V x)) (focR V')) + %<- (rsubst A⁻ (b s) snr M ([x] V x) V') %. +%term _ + %pi (rsubst A⁻ (b s) (snl Pf) M ([x] focL Pf' x (Sp x)) N') + %<- (rsubst A⁻ (b s) (snl Pf) M ([x] Sp x) (%the (spine A⁻ U) Sp')) + %<- (cut⁻ A⁻ s (ss Pf' Pf) M Sp' (%the (term · U) N')) %. +%term _ + %pi (rsubst A⁻ S (snl Pf) M ([x] focL Pf' X' (Sp x)) (focL Pf' X' Sp')) + %<- (rsubst A⁻ S (snl Pf) M ([x] Sp x) (%the (spine B⁻ U) Sp')) %. +%term _ + %pi (rsubst A⁻ S (snl Pf) M ([x] η⁺ ([z] N x z)) (η⁺ ([z] N' z))) + %<- ({z value (c Q⁺)} rsubst A⁻ S (snl Pf) M ([x] N x z) (%the (term _Ω U) (N' z))) %. +%term _ + %pi (rsubst A⁻ S (snl Pf) M ([x] ↓L ([x'] N x x')) (↓L ([x'] N' x'))) + %<- ({x' hyp B⁻} rsubst A⁻ S (snl Pf) M ([x] N x x') (%the (term _Ω U) (N' x'))) %. +%term _ rsubst A⁻ S (snl Pf) M ([x] ⊥L) ⊥L %. +%term _ + %pi (rsubst A⁻ S (snl Pf) M ([x] ∨L (N₁ x) (N₂ x)) (∨L N₁' N₂')) + %<- (rsubst A⁻ S (snl Pf) M ([x] N₁ x) (%the (term (B₁ , _Ω) U) N₁')) + %<- (rsubst A⁻ S (snl Pf) M ([x] N₂ x) (%the (term (B₂ , _Ω) U) N₂')) %. +%term _ + %pi (rsubst A⁻ S (snl Pf) M ([x] ⊤⁺L (N x)) (⊤⁺L N')) + %<- (rsubst A⁻ S (snl Pf) M ([x] N x) (%the (term _Ω U) N')) %. +%term _ + %pi (rsubst A⁻ S (snl Pf) M ([x] ∧⁺L (N x)) (∧⁺L N')) + %<- (rsubst A⁻ S (snl Pf) M ([x] N x) (%the (term (A⁺ , B⁺ , _Ω) U) N')) %. +%term _ + %pi (rsubst A⁻ S (snl snn) M ([x] η⁻ (N x)) (η⁻ N')) + %<- (rsubst A⁻ S (snl sna) M ([x] N x) (%the (term · (c Q⁻ susp)) N')) %. +%term _ + %pi (rsubst A⁻ S (snl snn) M ([x] ↑R (N x)) (↑R N')) + %<- (rsubst A⁻ S (snl snp) M ([x] N x) (%the (term · (C⁺ true)) N')) %. +%term _ + %pi (rsubst A⁻ S (snl snn) M ([x] ⊃R (N x)) (⊃R N')) + %<- (rsubst A⁻ S (snl snn) M ([x] N x) (%the (term (C₁⁺ , ·) (C₂⁻ inv)) N')) %. +%term _ rsubst A⁻ S (snl snn) M ([x] ⊤⁻R) ⊤⁻R %. +%term _ + %pi (rsubst A⁻ S (snl snn) M ([x] ∧⁻R (N₁ x) (N₂ x)) (∧⁻R N₁' N₂')) + %<- (rsubst A⁻ S (snl snn) M ([x] N₁ x) (%the (term · (C₁⁻ inv)) N₁')) + %<- (rsubst A⁻ S (snl snn) M ([x] N₂ x) (%the (term · (C₂⁻ inv)) N₂')) %. +%{! ``[[M/x]]Sp = Sp' `` !}% +%term _ rsubst A⁻ S (snl sna) M ([x] id⁻) id⁻ %. +%term _ + %pi (rsubst A⁻ S (snl Pf) M ([x] ↑L Pf' (N x)) (↑L Pf' N')) + %<- (rsubst A⁻ S (snl Pf) M ([x] N x) (%the (term (B⁺ , ·) U) N')) %. +%term _ + %pi (rsubst A⁻ S (snl Pf) M ([x] ⊃L (V x) (Sp x)) (⊃L V' Sp')) + %<- (rsubst A⁻ S snr M ([x] V x) (%the (value B₁⁺) V')) + %<- (rsubst A⁻ S (snl Pf) M ([x] Sp x) (%the (spine B₂⁻ U) Sp')) %. +%term _ + %pi (rsubst A⁻ S (snl Pf) M ([x] ∧⁻L₁ (Sp x)) (∧⁻L₁ Sp')) + %<- (rsubst A⁻ S (snl Pf) M ([x] Sp x) (%the (spine B₁⁻ U) Sp')) %. +%term _ + %pi (rsubst A⁻ S (snl Pf) M ([x] ∧⁻L₂ (Sp x)) (∧⁻L₂ Sp')) + %<- (rsubst A⁻ S (snl Pf) M ([x] Sp x) (%the (spine B₂⁻ U) Sp')) %. +%{! ### Leftist substitution (left commutative cuts) !}% +%{! ``<>N = M' `` !}% +% Entering principal substitution (cut⁺) +%term _ %pi (lsubst A⁺ (b s) (ss Pf Pf') (focR V) N N') %<- (cut⁺ A⁺ s Pf' V N N') %. +%term _ + %pi (lsubst A⁺ S (ss Pf Pf') (focL _ X Sp) N (focL Pf X Sp')) + %<- (lsubst A⁺ S (ss Pf Pf') Sp N (%the (spine B⁻ U) Sp')) %. +%term _ + %pi (lsubst A⁺ S Pf (η⁺ ([z] M z)) N (η⁺ ([z] M' z))) + %<- ({z value (c Q⁺)} lsubst A⁺ S Pf (M z) N (%the (term _Ω U) (M' z))) %. +%term _ + %pi (lsubst A⁺ S (ss Pf Pf') (↓L ([x'] M x')) N (↓L ([x'] M' x'))) + %<- ({x' hyp B⁻} lsubst A⁺ S (ss Pf Pf') (M x') N (%the (term _Ω U) (M' x'))) %. +%term _ lsubst A⁺ S Pf ⊥L N ⊥L %. +%term _ + %pi (lsubst A⁺ S Pf (∨L M₁ M₂) N (∨L M₁' M₂')) + %<- (lsubst A⁺ S Pf M₁ N (%the (term (B₁⁺ , _Ω) U) M₁')) + %<- (lsubst A⁺ S Pf M₂ N (%the (term (B₂⁺ , _Ω) U) M₂')) %. +%term _ + %pi (lsubst A⁺ S Pf (⊤⁺L M) N (⊤⁺L M')) + %<- (lsubst A⁺ S Pf M N (%the (term _Ω U) M')) %. +%term _ + %pi (lsubst A⁺ S Pf (∧⁺L M) N (∧⁺L M')) + %<- (lsubst A⁺ S Pf M N (%the (term (B₁⁺ , B₂⁺ , _Ω) U) M')) %. +%{! ``<>N = Sp' `` !}% +%term _ + %pi (lsubst A⁺ S (ss Pf Pf') (↑L _ M) N (↑L Pf M')) + %<- (lsubst A⁺ S (ss Pf Pf') M N (%the (term (B⁺ , ·) U) M')) %. +%term _ + %pi (lsubst A⁺ S Pf (⊃L V Sp) N (⊃L V Sp')) + %<- (lsubst A⁺ S Pf Sp N (%the (spine B₂⁻ U) Sp')) %. +%term _ + %pi (lsubst A⁺ S Pf (∧⁻L₁ Sp) N (∧⁻L₁ Sp')) + %<- (lsubst A⁺ S Pf Sp N (%the (spine B₁⁻ U) Sp')) %. +%term _ + %pi (lsubst A⁺ S Pf (∧⁻L₂ Sp) N (∧⁻L₂ Sp')) + %<- (lsubst A⁺ S Pf Sp N (%the (spine B₂⁻ U) Sp')) %. +%{! ### Wrap-up !}% +%worlds (gamma_suspnormal) (cut⁺ _ _ _ _ _ _) (cut⁻ _ _ _ _ _ _) (rsubst _ _ _ _ _ _) (lsubst _ _ _ _ _ _) %. +%{! With the exception of the "big/small" metric that allows the +leftist and rightist substitutions to call principal substitutions at +the same type, this is the usual induction metric for structural cut +elimination arguments. !}% +%total {(A1 A2 A3 A4) {(S1 S2 S3 S4) [(V N⁻ MR EL) (N⁺ S ER NL)]}} (cut⁺ A1 S1 _ V N⁺ _) (cut⁻ A2 S2 _ N⁻ S _) (rsubst A3 S3 _ MR ER _) (lsubst A4 S4 _ EL NL _) %. +%{! The following simpler induction metric also works, emphasizing that +the "derivation" metric matters not at all in the principal cases, +that the rightist substitutions are structurally inductive over the +second given derivation (the "right" derivation) and that the leftist +substitutions are structurally inductive over the first given +derivation (the "left" derivation). !}% +%total {(A1 A2 A3 A4) {(S1 S2 S3 S4) (S1 S2 ER EL)}} (cut⁻ A1 S1 _ _ _ _) (cut⁺ A2 S2 _ _ _ _) (rsubst A3 S3 _ _ ER _) (lsubst A4 S4 _ EL _ _) %. +%{! ## Expansion !}% +%sort expand⁺ {A⁺} %. +%sort expand⁻ {A⁻} %. +%mode expand⁺ %in %in %out %. +%mode expand⁻ %in %in %out %. +%{! ``η(z.N) = N' ``!}% +%term _ expand⁺ (c Q⁺) ([z value (c Q⁺)] N z) (η⁺ ([z value (c Q⁺)] N z)) %. +%term _ + %pi (expand⁺ (↓ A⁻) ([z value (↓ A⁻)] N z) (↓L ([x hyp A⁻] N (↓R (N' x))))) + %<- ({x hyp A⁻} expand⁻ A⁻ (focL sn x id⁻) (%the (term · (A⁻ inv)) (N' x))) %. +%term _ expand⁺ ⊥ ([z value ⊥] N z) ⊥L %. +%term _ + %pi (expand⁺ (A⁺ ∨ B⁺) ([z value (A⁺ ∨ B⁺)] N z) (∨L N₁ N₂)) + %<- (expand⁺ A⁺ ([z₁ value A⁺] N (∨R₁ z₁)) (%the (term (A⁺ , _Ω) U) N₁)) + %<- (expand⁺ B⁺ ([z₂ value B⁺] N (∨R₂ z₂)) (%the (term (B⁺ , _Ω) U) N₂)) %. +%term _ expand⁺ ⊤⁺ ([z value ⊤⁺] N z) (⊤⁺L (N ⊤⁺R)) %. +%term _ + %pi (expand⁺ (A⁺ ∧⁺ B⁺) ([z value (A⁺ ∧⁺ B⁺)] N z) (∧⁺L N₂)) + %<- ({z₁ value A⁺} expand⁺ B⁺ ([z₂ value B⁺] N (∧⁺R z₁ z₂)) (%the (term (B⁺ , _Ω) U) (N₁ z₁))) + %<- (expand⁺ A⁺ ([z₁ value A⁺] N₁ z₁) (%the (term (A⁺ , B⁺ , _Ω) U) N₂)) %. +%{! ``η(s.N) = N' `` !}% +%term _ expand⁻ (c Q⁻) N (η⁻ N) %. +%term _ + %pi (expand⁻ (↑ A⁺) N (↑R N'')) + %<- (expand⁺ A⁺ ([z value A⁺] focR z) (%the (term (A⁺ , ·) (A⁺ true)) N')) + %<- (subst⁻ sp N (↑L sp N') (%the (term · (A⁺ true)) N'')) %. +%term _ + %pi (expand⁻ (A⁺ ⊃ B⁻) N (⊃R N₃)) + %<- ({z value A⁺} subst⁻ sn N (⊃L z id⁻) (%the (term · (B⁻ susp)) (N₁ z))) + %<- ({z value A⁺} expand⁻ B⁻ (N₁ z) (%the (term · (B⁻ inv)) (N₂ z))) + %<- (expand⁺ A⁺ N₂ (%the (term (A⁺ , ·) (B⁻ inv)) N₃)) %. +%term _ expand⁻ ⊤⁻ N ⊤⁻R %. +%term _ + %pi (expand⁻ (A⁻ ∧⁻ B⁻) N (∧⁻R N₁' N₂')) + %<- (subst⁻ sn N (∧⁻L₁ id⁻) (%the (term · (A⁻ susp)) N₁)) + %<- (expand⁻ A⁻ N₁ (%the (term · (A⁻ inv)) N₁')) + %<- (subst⁻ sn N (∧⁻L₂ id⁻) (%the (term · (B⁻ susp)) N₂)) + %<- (expand⁻ B⁻ N₂ (%the (term · (B⁻ inv)) N₂')) %. +%worlds (gamma) (expand⁺ _ _ _) (expand⁻ _ _ _) %. +%total (A⁺ A⁻) (expand⁺ A⁺ _ _) (expand⁻ A⁻ _ _) %. +%{! Identity is a corollary, though we'll only use it at the very end +when we prove the identity principle for the unfocused sequent +calculus: !}% +%sort identity⁻ %. +%mode {%in A⁻} {%out N %pi (hyp A⁻) %-> (term · (A⁻ inv))} identity⁻ N %. +%term _ + %pi (identity⁻ N) + %<- ({x hyp A⁻} expand⁻ A⁻ (focL sn x id⁻) (%the (term · (A⁻ inv)) (N x))) %. +%worlds (gamma) (identity⁻ _) %. +%total [] (identity⁻ _) %. +%sort identity⁺ %. +%mode {%in A⁺} {%out N term (A⁺ , ·) (A⁺ true)} identity⁺ N %. +%term _ + %pi (identity⁺ N) + %<- (expand⁺ A⁺ ([z value A⁺] focR z) (%the (term (A⁺ , ·) (A⁺ true)) N)) %. +%worlds (gamma) (identity⁺ _) %. +%total [] (identity⁺ _) %. +%{! ## Unfocused admissibility !}% +%{! The key lemmas for establishing the completeness of the focused +sequent calculus are the "unfocused admissibility" lemmas, which +establish that the normal rules of the sequent calculus are usable in +the context of the focused calculus. + +These lemmas are all provable by use of the cut and identity +principles; while the proofs are hardly straightforward, they share +a certain high-level structure. !}% +%{! ### Initial rules !}% +%sort adm-initsusp⁻ %. +%mode {%in Q} {%out N %pi (hyp (c Q)) %-> (term · (c Q susp))} adm-initsusp⁻ N %. +%term _ adm-initsusp⁻ ([x hyp (c Q)] focL sn x id⁻) %. +%worlds (gamma_suspnormal) (adm-initsusp⁻ _) %. +%total [] (adm-initsusp⁻ _) %. +%sort adm-init⁻ %. +%mode {%in Q} {%out N %pi (hyp (c Q)) %-> (term · (↓ (c Q) true))} adm-init⁻ N %. +%term _ adm-init⁻ ([x hyp (c Q)] focR (↓R (η⁻ (focL sn x id⁻)))) %. +%worlds (gamma_suspnormal) (adm-init⁻ _) %. +%total [] (adm-init⁻ _) %. +%sort adm-initsusp⁺ %. +%mode {%in Q} {%out N %pi (value (c Q)) %-> (term · (c Q true))} adm-initsusp⁺ N %. +%term _ adm-initsusp⁺ ([z value (c Q)] focR z) %. +%worlds (gamma_suspnormal) (adm-initsusp⁺ _) %. +%total [] (adm-initsusp⁺ _) %. +%sort adm-init⁺ %. +%mode {%in Q} {%out N %pi (hyp (↑ (c Q))) %-> (term · (c Q true))} adm-init⁺ N %. +%term _ adm-init⁺ ([x hyp (↑ (c Q))] focL sp x (↑L sp (η⁺ ([z] focR z)))) %. +%worlds (gamma_suspnormal) (adm-init⁺ _) %. +%total [] (adm-init⁺ _) %. +%{! ### Disjunction !}% +%sort adm-⊥L %. +%mode adm-⊥L %in %out %. +%term _ adm-⊥L Pf ([x hyp (↑ ⊥)] focL Pf x (↑L Pf ⊥L)) %. +%worlds (gamma_suspnormal) (adm-⊥L _ _) %. +%total [] (adm-⊥L _ _) %. +%sort adm-∨R₁ %. +%mode {%in A⁺} {%in B⁺} {%in N₁} {%out N' term · (A⁺ ∨ B⁺ true)} adm-∨R₁ N₁ N' %. +%term _ + %pi (adm-∨R₁ (%the (term · (A⁺ true)) N₁) N') + %<- (expand⁺ A⁺ ([z value A⁺] focR (∨R₁ z)) (%the (term (A⁺ , ·) (A⁺ ∨ B⁺ true)) NId₁)) + %<- (lsubst A⁺ (b s) (ss sp snp) N₁ NId₁ (%the (term · (A⁺ ∨ B⁺ true)) N')) %. +%worlds (gamma_suspnormal) (adm-∨R₁ _ _) %. +%total [] (adm-∨R₁ _ _) %. +%sort adm-∨R₂ %. +%mode {%in A⁺} {%in B⁺} {%in N₂} {%out N' term · (A⁺ ∨ B⁺ true)} adm-∨R₂ N₂ N' %. +%term _ + %pi (adm-∨R₂ (%the (term · (B⁺ true)) N₂) N') + %<- (expand⁺ B⁺ ([z value B⁺] focR (∨R₂ z)) (%the (term (B⁺ , ·) (A⁺ ∨ B⁺ true)) NId₂)) + %<- (lsubst B⁺ (b s) (ss sp snp) N₂ NId₂ (%the (term · (A⁺ ∨ B⁺ true)) N')) %. +%worlds (gamma_suspnormal) (adm-∨R₂ _ _) %. +%total [] (adm-∨R₂ _ _) %. +%sort adm-∨L %. +%mode adm-∨L %in %in %in %out %. +%term _ + %pi (adm-∨L (ss Pf Pf') (%the (%pi (hyp (↑ A⁺)) %-> (term · U)) N₁) (%the (%pi (hyp (↑ B⁺)) %-> (term · U)) N₂) ([x hyp (↑ (A⁺ ∨ B⁺))] focL Pf x (↑L Pf N'))) + %<- (expand⁺ A⁺ ([z₁ value A⁺] focR (∨R₁ (↓R (↑R (focR z₁))))) (%the (term (A⁺ , ·) (↓ (↑ A⁺) ∨ ↓ (↑ B⁺) true)) NId₁)) + %<- (expand⁺ B⁺ ([z₂ value B⁺] focR (∨R₂ (↓R (↑R (focR z₂))))) (%the (term (B⁺ , ·) (↓ (↑ A⁺) ∨ ↓ (↑ B⁺) true)) NId₂)) + %<- (lsubst (↓ (↑ A⁺) ∨ ↓ (↑ B⁺)) (b s) (ss Pf Pf') (∨L NId₁ NId₂) (∨L (↓L N₁) (↓L N₂)) (%the (term (A⁺ ∨ B⁺ , ·) U) N')) %. +%worlds (gamma_suspnormal) (adm-∨L _ _ _ _) %. +%total [] (adm-∨L _ _ _ _) %. +%{! ### Positive conjunction !}% +%sort adm-⊤⁺R %. +%mode adm-⊤⁺R %out %. +%term _ adm-⊤⁺R (focR ⊤⁺R) %. +%worlds (gamma_suspnormal) (adm-⊤⁺R _) %. +%total [] (adm-⊤⁺R _) %. +%sort adm-⊤⁺L %. +%mode adm-⊤⁺L %in %in %out %. +%term _ adm-⊤⁺L Pf N₁ ([x hyp (↑ ⊤⁺)] focL Pf x (↑L Pf (⊤⁺L N₁))) %. +%worlds (gamma_suspnormal) (adm-⊤⁺L _ _ _) %. +%total [] (adm-⊤⁺L _ _ _) %. +%sort adm-∧⁺R %. +%mode adm-∧⁺R %in %in %out %. +%term _ + %pi (adm-∧⁺R (%the (term · (A⁺ true)) N₁) (%the (term · (B⁺ true)) N₂) N') + %<- ({v₂ value B⁺} expand⁺ A⁺ ([v₁ value A⁺] focR (∧⁺R v₁ v₂)) (%the (term (A⁺ , ·) (A⁺ ∧⁺ B⁺ true)) (NIdB v₂))) + %<- ({x hyp (↑ A⁺)} expand⁺ B⁺ ([z₂ value B⁺] focL sp x (↑L sp (NIdB z₂))) (%the (term (B⁺ , ·) (A⁺ ∧⁺ B⁺ true)) (NId x))) + %<- ({x hyp (↑ A⁺)} lsubst B⁺ (b s) (ss sp snp) N₂ (NId x) (%the (term · (A⁺ ∧⁺ B⁺ true)) (NA x))) + %<- (rsubst (↑ A⁺) (b s) (snl snp) (↑R N₁) ([x hyp (↑ A⁺)] NA x) N') %. +%worlds (gamma_suspnormal) (adm-∧⁺R _ _ _) %. +%total [] (adm-∧⁺R _ _ _) %. +%sort adm-∧⁺L %. +%mode adm-∧⁺L %in %in %out %. +%term _ + %pi (adm-∧⁺L (ss Pf Pf') (%the (%pi (hyp (↑ A⁺)) %-> (hyp (↑ B⁺)) %-> (term · U)) N₁) ([x hyp (↑ (A⁺ ∧⁺ B⁺))] focL Pf x (↑L Pf N'))) + %<- ({z₁ value A⁺} expand⁺ B⁺ ([z₂ value B⁺] focR (∧⁺R (↓R (↑R (focR z₁))) (↓R (↑R (focR z₂))))) (%the (term (B⁺ , ·) (↓ (↑ A⁺) ∧⁺ ↓ (↑ B⁺) true)) (NIdA z₁))) + %<- (expand⁺ A⁺ ([z₁ value A⁺] NIdA z₁) (%the (term (A⁺ , B⁺ , ·) (↓ (↑ A⁺) ∧⁺ ↓ (↑ B⁺) true)) NId)) + %<- (lsubst (↓ (↑ A⁺) ∧⁺ ↓ (↑ B⁺)) (b s) (ss Pf Pf') (∧⁺L NId) (∧⁺L (↓L ([x₁ hyp (↑ A⁺)] ↓L ([x₂ hyp (↑ B⁺)] N₁ x₁ x₂)))) (%the (term (A⁺ ∧⁺ B⁺ , ·) U) N')) %. +%worlds (gamma_suspnormal) (adm-∧⁺L _ _ _) %. +%total [] (adm-∧⁺L _ _ _) %. +%{! ### Implication !}% +%sort adm-⊃R %. +%mode adm-⊃R %in %out %. +%term _ + %pi (adm-⊃R (%the (%pi (hyp (↑ A⁺)) %-> (term · (↓ B⁻ true))) N₁) (focR (↓R N'))) + %<- ({x hyp (↓ (↑ A⁺) ⊃ ↑ (↓ B⁻))} {z value A⁺} expand⁻ B⁻ (focL sn x (⊃L (↓R (↑R (focR z))) (↑L sn (↓L ([x' hyp B⁻] focL sn x' id⁻))))) (%the (term · (B⁻ inv)) (NIdA x z))) + %<- ({x hyp (↓ (↑ A⁺) ⊃ ↑ (↓ B⁻))} expand⁺ A⁺ ([z value A⁺] NIdA x z) (%the (term (A⁺ , ·) (B⁻ inv)) (NId x))) + %<- (rsubst (↓ (↑ A⁺) ⊃ ↑ (↓ B⁻)) (b s) (snl snn) (⊃R (↓L ([x₁ hyp (↑ A⁺)] ↑R (N₁ x₁)))) ([x hyp (↓ (↑ A⁺) ⊃ ↑ (↓ B⁻))] ⊃R (NId x)) (%the (term · (A⁺ ⊃ B⁻ inv)) N')) %. +%worlds (gamma_suspnormal) (adm-⊃R _ _) %. +%total [] (adm-⊃R _ _) %. +%sort adm-⊃L %. +%mode adm-⊃L %in %in %in %out %. +%term _ + %pi (adm-⊃L (ss Pf Pf') (%the (term · (A⁺ true)) N₁) (%the (%pi (hyp B⁻) %-> (term · U)) N₂) N') + %<- ({x hyp (A⁺ ⊃ B⁻)} {z value A⁺} expand⁻ B⁻ (focL sn x (⊃L z id⁻)) (%the (term · (B⁻ inv)) (NIdB x z))) + %<- ({x hyp (A⁺ ⊃ B⁻)} expand⁺ A⁺ ([z value A⁺] focR (↓R (NIdB x z))) (%the (term (A⁺ , ·) (↓ B⁻ true)) (NId x))) + %<- ({x hyp (A⁺ ⊃ B⁻)} lsubst A⁺ (b s) (ss sp snp) N₁ (NId x) (%the (term · (↓ B⁻ true)) (NB x))) + %<- ({x hyp (A⁺ ⊃ B⁻)} lsubst (↓ B⁻) (b s) (ss Pf Pf') (NB x) (↓L ([x₂ hyp B⁻] N₂ x₂)) (%the (term · U) (N' x))) %. +%worlds (gamma_suspnormal) (adm-⊃L _ _ _ _) %. +%total [] (adm-⊃L _ _ _ _) %. +%{! ### Negative conjunction !}% +%sort adm-⊤⁻R %. +%mode adm-⊤⁻R %out %. +%term _ adm-⊤⁻R (focR (↓R ⊤⁻R)) %. +%worlds (gamma_suspnormal) (adm-⊤⁻R _) %. +%total [] (adm-⊤⁻R _) %. +%sort adm-∧⁻R %. +%mode adm-∧⁻R %in %in %out %. +%term _ + %pi (adm-∧⁻R N₁ N₂ (focR (↓R N'))) + %<- ({x hyp (↑ (↓ A⁻) ∧⁻ ↑ (↓ B⁻))} expand⁻ A⁻ (focL sn x (∧⁻L₁ (↑L sn (↓L ([y] focL sn y id⁻))))) (%the (term · (A⁻ inv)) (NId₁ x))) + %<- ({x hyp (↑ (↓ A⁻) ∧⁻ ↑ (↓ B⁻))} expand⁻ B⁻ (focL sn x (∧⁻L₂ (↑L sn (↓L ([y] focL sn y id⁻))))) (%the (term · (B⁻ inv)) (NId₂ x))) + %<- (rsubst (↑ (↓ A⁻) ∧⁻ ↑ (↓ B⁻)) (b s) (snl snn) (∧⁻R (↑R N₁) (↑R N₂)) ([x hyp (↑ (↓ A⁻) ∧⁻ ↑ (↓ B⁻))] ∧⁻R (NId₁ x) (NId₂ x)) (%the (term · (A⁻ ∧⁻ B⁻ inv)) N')) %. +%worlds (gamma_suspnormal) (adm-∧⁻R _ _ _) %. +%total [] (adm-∧⁻R _ _ _) %. +%sort adm-∧⁻L₁ %. +%mode {%in A⁻} {%in B⁻} {%in U} {%in Pf} {%in N₁} {%out N' %pi (hyp (A⁻ ∧⁻ B⁻)) %-> (term · U)} adm-∧⁻L₁ Pf N₁ N' %. +%term _ + %pi (adm-∧⁻L₁ Pf (%the (%pi (hyp A⁻) %-> (term · U)) N₁) N') + %<- ({x hyp (A⁻ ∧⁻ B⁻)} expand⁻ A⁻ (focL sn x (∧⁻L₁ id⁻)) (%the (term · (A⁻ inv)) (NId x))) + %<- ({x hyp (A⁻ ∧⁻ B⁻)} rsubst A⁻ (b s) (snl Pf) (NId x) N₁ (%the (term · U) (N' x))) %. +%worlds (gamma_suspnormal) (adm-∧⁻L₁ _ _ _) %. +%total [] (adm-∧⁻L₁ _ _ _) %. +%sort adm-∧⁻L₂ %. +%mode {%in A⁻} {%in B⁻} {%in U} {%in Pf} {%in N₂} {%out N' %pi (hyp (A⁻ ∧⁻ B⁻)) %-> (term · U)} adm-∧⁻L₂ Pf N₂ N' %. +%term _ + %pi (adm-∧⁻L₂ Pf (%the (%pi (hyp B⁻) %-> (term · U)) N₂) N') + %<- ({x hyp (A⁻ ∧⁻ B⁻)} expand⁻ B⁻ (focL sn x (∧⁻L₂ id⁻)) (%the (term · (B⁻ inv)) (NId x))) + %<- ({x hyp (A⁻ ∧⁻ B⁻)} rsubst B⁻ (b s) (snl Pf) (NId x) N₂ (%the (term · U) (N' x))) %. +%worlds (gamma_suspnormal) (adm-∧⁻L₂ _ _ _) %. +%total [] (adm-∧⁻L₂ _ _ _) %. +%{! ## Unfocused system !}% +%{! ### Syntax !}% +%sort prop %. +%name prop %. +%term a %pi (atom P) %-> prop %. +%term ff prop %. +%term or %pi prop %-> prop %-> prop %. +%term tt prop %. +%term and %pi prop %-> prop %-> prop %. +%term imp %pi prop %-> prop %-> prop %. +%sort props %. +%name props %. +%term ○ props %. +%term ; %pi prop %-> props %-> props %. +%prec %right 3 ; %. +%{! ### Erasure !}% +%{! The judgment ``t P A`` relates unpolarized propositions ``P`` and +polarized propositions ``A``. The judgment can be effectively run as +an erasure function from polarized to unpolarized propositions. !}% +%sort t %. +%name t %. +%term tQ t (c Q) (a Q) %. +%term t↑ %pi (t A⁺ P) %-> (t (↑ A⁺) P) %. +%term t⊥ t ⊥ ff %. +%term t∨ %pi (t A⁺ P₁) %-> (t B⁺ P₂) %-> (t (A⁺ ∨ B⁺) (or P₁ P₂)) %. +%term t⊤⁺ t ⊤⁺ tt %. +%term t∧⁺ %pi (t A⁺ P₁) %-> (t B⁺ P₂) %-> (t (A⁺ ∧⁺ B⁺) (and P₁ P₂)) %. +%term t↓ %pi (t A⁻ P) %-> (t (↓ A⁻) P) %. +%term t⊃ %pi (t A⁺ P₁) %-> (t B⁻ P₂) %-> (t (A⁺ ⊃ B⁻) (imp P₁ P₂)) %. +%term t⊤⁻ t ⊤⁻ tt %. +%term t∧⁻ %pi (t A⁻ P₁) %-> (t B⁻ P₂) %-> (t (A⁻ ∧⁻ B⁻) (and P₁ P₂)) %. +%mode t %in %out %. +%worlds (atom⁺ atom⁻) (t _ _) %. +%total A (t A _) %. +%sort tΩ %. +%name tΩ %. +%term t· tΩ · ○ %. +%term t, %pi (t A⁺ P) %-> (tΩ _Ω _Ψ) %-> (tΩ (A⁺ , _Ω) (P ; _Ψ)) %. +%mode tΩ %in %out %. +%worlds (gamma) (tΩ _ _) %. +%total _Ω (tΩ _Ω _) %. +%sort tU %. +%name tU %. +%term tp %pi (tU (A⁺ true) P) %<- (t A⁺ P) %. +%term tn %pi (tU (A⁻ inv) P) %<- (t A⁻ P) %. +%term ta tU (c Q⁻ susp) (a Q⁻) %. +%sort tseq %. +%term tV %pi (tseq (rfoc A⁺) ○ P) %<- (t A⁺ P) %. +%term tN %pi (tseq (hasleft (ininv _Ω) U) _Ψ Q) %<- (tU U Q) %<- (tΩ _Ω _Ψ) %. +%term tS %pi (tseq (hasleft (infoc A⁻) U) (P ; ○) Q) %<- (tU U Q) %<- (t A⁻ P) %. +%{! ### Shift removal !}% +%{! There may be a more elegant way to do this, but the last missing +piece of our unfocused admissibility lemma - the last thing we need to +know about the polarized calculus - is that if we need to prove (or +use) a deeply-shifted proposition, then it suffices to prove (or use) +a non-deeply-shifted proposition that erases to the same thing. !}% +%sort not-doubleshifted %. +%term ndQ not-doubleshifted (c Q) %. +%term ndQ⁺ not-doubleshifted (↓ (c Q)) %. +%term ndQ⁻ not-doubleshifted (↑ (c Q)) %. +%term nd⊥⁺ not-doubleshifted ⊥ %. +%term nd⊥⁻ not-doubleshifted (↑ ⊥) %. +%term nd∨⁺ not-doubleshifted (A⁺ ∨ B⁺) %. +%term nd∨⁻ not-doubleshifted (↑ (A⁺ ∨ B⁺)) %. +%term nd⊤⁺⁺ not-doubleshifted ⊤⁺ %. +%term nd⊤⁺⁻ not-doubleshifted (↑ ⊤⁺) %. +%term nd∧⁺⁺ not-doubleshifted (A⁺ ∧⁺ B⁺) %. +%term nd∧⁺⁻ not-doubleshifted (↑ (A⁺ ∧⁺ B⁺)) %. +%term nd⊃⁺ not-doubleshifted (↓ (A⁺ ⊃ B⁻)) %. +%term nd⊃⁻ not-doubleshifted (A⁺ ⊃ B⁻) %. +%term nd⊤⁻⁺ not-doubleshifted (↓ ⊤⁻) %. +%term nd⊤⁻⁻ not-doubleshifted ⊤⁻ %. +%term nd∧⁻⁺ not-doubleshifted (↓ (A⁻ ∧⁻ B⁻)) %. +%term nd∧⁻⁻ not-doubleshifted (A⁻ ∧⁻ B⁻) %. +%sort rshifty %. +%mode rshifty %in %out %out %out %. +%term _ + %pi (rshifty (t↓ (t↑ T)) T' ND ([n] focR (↓R (↑R (N n))))) + %<- (rshifty T T' ND N) %. +%term _ rshifty tQ tQ ndQ ([n] n) %. +%term _ rshifty t⊥ t⊥ nd⊥⁺ ([n] n) %. +%term _ rshifty (t∨ T₁ T₂) (t∨ T₁ T₂) nd∨⁺ ([n] n) %. +%term _ rshifty t⊤⁺ t⊤⁺ nd⊤⁺⁺ ([n] n) %. +%term _ rshifty (t∧⁺ T₁ T₂) (t∧⁺ T₁ T₂) nd∧⁺⁺ ([n] n) %. +%term _ rshifty (t↓ tQ) (t↓ tQ) ndQ⁺ ([n] n) %. +%term _ rshifty (t↓ (t⊃ T₁ T₂)) (t↓ (t⊃ T₁ T₂)) nd⊃⁺ ([n] n) %. +%term _ rshifty (t↓ t⊤⁻) (t↓ t⊤⁻) nd⊤⁻⁺ ([n] n) %. +%term _ rshifty (t↓ (t∧⁻ T₁ T₂)) (t↓ (t∧⁻ T₁ T₂)) nd∧⁻⁺ ([n] n) %. +%worlds (gamma_suspnormal) (rshifty _ _ _ _) %. +%total T (rshifty T _ _ _) %. +%sort lshifty %. +%mode lshifty %in %out %out %out %. +%term _ + %pi (lshifty (t↑ (t↓ T)) T' ND ([u] [pf] [n] [x] focL pf x (↑L pf (↓L (N u pf n))))) + %<- (lshifty T T' ND N) %. +%term _ lshifty (t↑ tQ) (t↑ tQ) ndQ⁻ ([u] [pf] [n] n) %. +%term _ lshifty (t↑ t⊥) (t↑ t⊥) nd⊥⁻ ([u] [pf] [n] n) %. +%term _ lshifty (t↑ (t∨ T₁ T₂)) (t↑ (t∨ T₁ T₂)) nd∨⁻ ([u] [pf] [n] n) %. +%term _ lshifty (t↑ t⊤⁺) (t↑ t⊤⁺) nd⊤⁺⁻ ([u] [pf] [n] n) %. +%term _ lshifty (t↑ (t∧⁺ T₁ T₂)) (t↑ (t∧⁺ T₁ T₂)) nd∧⁺⁻ ([u] [pf] [n] n) %. +%term _ lshifty tQ tQ ndQ ([u] [pf] [n] n) %. +%term _ lshifty (t⊃ T₁ T₂) (t⊃ T₁ T₂) nd⊃⁻ ([u] [pf] [n] n) %. +%term _ lshifty t⊤⁻ t⊤⁻ nd⊤⁻⁻ ([u] [pf] [n] n) %. +%term _ lshifty (t∧⁻ T₁ T₂) (t∧⁻ T₁ T₂) nd∧⁻⁻ ([u] [pf] [n] n) %. +%worlds (gamma_suspnormal) (lshifty _ _ _ _) %. +%total T (lshifty T _ _ _) %. +%{! ### Sequent rules !}% +%sort left %. +%name left %. +%sort right %. +%name right %. +%sort right' %. +%name right' %. +%block h {P} [h left P]%. +%term init %pi (left (a Q)) %-> (right (a Q)) %. +%term ffL %pi (left ff) %-> (right P) %. +%term orR₁ %pi (right P₁) %-> (right (or P₁ P₂)) %. +%term orR₂ %pi (right P₂) %-> (right (or P₁ P₂)) %. +%term orL + %pi (%pi (left P₁) %-> (right Q)) + %-> (%pi (left P₂) %-> (right Q)) + %-> (left (or P₁ P₂)) + %-> (right Q) %. +%term ttR right tt %. +%term andR %pi (right P₁) %-> (right P₂) %-> (right (and P₁ P₂)) %. +%term andL₁ %pi (%pi (left P₁) %-> (right Q)) %-> (left (and P₁ P₂)) %-> (right Q) %. +%term andL₂ %pi (%pi (left P₂) %-> (right Q)) %-> (left (and P₁ P₂)) %-> (right Q) %. +%term impR %pi (%pi (left P₁) %-> (right P₂)) %-> (right (imp P₁ P₂)) %. +%term impL + %pi (right P₁) + %-> (%pi (left P₂) %-> (right Q)) + %-> (left (imp P₁ P₂)) + %-> (right Q) %. +%term nil %pi (right P) %-> (right' ○ P) %. +%term cons %pi (%pi (left P) %-> (right' _Ψ Q)) %-> (right' (P ; _Ψ) Q) %. +%{! Syntactic identity on derivations is used for completeness -- it +keeps the derivation size the same when we do case analysis. !}% +%sort id %. +%term refl id D D %. +%{! ## De-focalization (soundness) !}% +%{! Soundness is established in a context where each polarized +hypothesis is mapped to its erasure. !}% +%sort soundhyp %. +%block soundhyps {A⁻} {P} {T t A⁻ P} [x hyp A⁻] [h left P] [_ soundhyp x T h]%. +%sort soundsusp %. +%block soundsusps {Q⁺} [z value (c Q⁺)] [h left (a Q⁺)] [_ soundsusp z h]%. +%block soundctx%. +%mode {%in A⁻} {%out P} {%in X hyp A⁻} {%out T t A⁻ P} {%out H left P} soundhyp X T H %. +%worlds (soundctx) (soundhyp _ _ _) %. +%total [] (soundhyp _ _ _) %. +%mode {%in Q⁺} {%in Z value (c Q⁺)} {%out H left (a Q⁺)} soundsusp Z H %. +%worlds (soundctx) (soundsusp _ _) %. +%total [] (soundsusp _ _) %. +%sort sound %. +%mode sound %in %in %out %. +%term _ %pi (sound (%the (value (c Q⁺)) Z) (tV tQ) (nil (init H))) %<- (soundsusp Z H) %. +%term _ %pi (sound (↓R N) (tV (t↓ T)) (nil D)) %<- (sound N (tN t· (tn T)) (nil D)) %. +%term _ %pi (sound (∨R₁ V) (tV (t∨ T₁ _)) (nil (orR₁ D))) %<- (sound V (tV T₁) (nil D)) %. +%term _ %pi (sound (∨R₂ V) (tV (t∨ _ T₂)) (nil (orR₂ D))) %<- (sound V (tV T₂) (nil D)) %. +%term _ sound ⊤⁺R (tV t⊤⁺) (nil ttR) %. +%term _ + %pi (sound (∧⁺R V₁ V₂) (tV (t∧⁺ T₁ T₂)) (nil (andR D₁ D₂))) + %<- (sound V₁ (tV T₁) (nil D₁)) + %<- (sound V₂ (tV T₂) (nil D₂)) %. +%term _ %pi (sound (focR V) (tN t· (tp T)) D) %<- (sound V (tV T) D) %. +%term _ + %pi (sound (focL _ X Sp) (tN t· (%the (tU U Q) TCQ)) (nil (D H))) + %<- (soundhyp X (%the (t A⁻ P) TAP) (%the (left P) H)) + %<- (sound Sp (tS TAP TCQ) (cons ([x left P] nil (%the (right Q) (D x))))) %. +%term _ + %pi (sound (η⁺ ([z value (c Q⁺)] N z)) (tN (t, tQ TΩ) TCQ) (cons D)) + %<- ({z value (c Q⁺)} {h left (a Q⁺)} %pi (soundsusp z h) %-> (sound (N z) (tN TΩ TCQ) (%the (right' _Ψ Q) (D h)))) %. +%term _ + %pi (sound (↓L ([x hyp A⁻] N x)) (tN (t, (t↓ T) TΩ) TCQ) (cons D)) + %<- ({x hyp A⁻} {h left P} %pi (soundhyp x T h) %-> (sound (N x) (tN TΩ TCQ) (%the (right' _Ψ Q) (D h)))) %. +%sort sound-ffL {_Ψ} %. +%term _ sound-ffL _ H (nil (ffL H)) %. +%term _ %pi (sound-ffL _ H (cons D)) %<- ({h} sound-ffL _ H (D h)) %. +%mode {%in _Ψ} {%in Q} {%in H left ff} {%out D right' _Ψ Q} sound-ffL _Ψ H D %. +%worlds (atom⁺ atom⁻ h) (sound-ffL _ _ _) %. +%total _Ψ (sound-ffL _Ψ _ _) %. +%term _ %pi (sound ⊥L (tN TΩ TCQ) (cons D)) %<- ({h left ff} sound-ffL _ h (D h)) %. +%sort sound-orL {_Ψ} %. +%term _ sound-orL _ H ([h₁] nil (D₁ h₁)) ([h₂] nil (D₂ h₂)) (nil (orL D₁ D₂ H)) %. +%term _ + %pi (sound-orL _ H ([h₁] cons (D₁ h₁)) ([h₂] cons (D₂ h₂)) (cons D)) + %<- ({h} sound-orL _ H ([h₁] D₁ h₁ h) ([h₂] D₂ h₂ h) (D h)) %. +%mode sound-orL %in %in %in %in %out %. +%worlds (atom⁺ atom⁻ h) (sound-orL _ _ _ _ _) %. +%total _Ψ (sound-orL _Ψ _ _ _ _) %. +%term _ + %pi (sound (∨L N₁ N₂) (tN (t, (t∨ T₁ T₂) TΩ) TCQ) (cons D)) + %<- (sound N₁ (tN (t, T₁ TΩ) TCQ) (cons (%the (%pi (left P₁) %-> (right' _Ψ Q)) D₁))) + %<- (sound N₂ (tN (t, T₂ TΩ) TCQ) (cons (%the (%pi (left P₂) %-> (right' _Ψ Q)) D₂))) + %<- ({h left (or P₁ P₂)} sound-orL _ h D₁ D₂ (D h)) %. +%term _ + %pi (sound (⊤⁺L N) (tN (t, t⊤⁺ TΩ) TCQ) (cons ([h left tt] D))) + %<- (sound N (tN TΩ TCQ) (%the (right' _Ψ Q) D)) %. +%sort sound-andL {_Ψ} %. +%term _ sound-andL _ H ([h₁] [h₂] nil (D₁ h₁ h₂)) (nil (andL₁ ([h₁] andL₂ ([h₂] D₁ h₁ h₂) H) H)) %. +%term _ + %pi (sound-andL _ H ([h₁] [h₂] cons (D₁ h₁ h₂)) (cons D)) + %<- ({h} sound-andL _ H ([h₁] [h₂] D₁ h₁ h₂ h) (D h)) %. +%mode sound-andL %in %in %in %out %. +%worlds (atom⁺ atom⁻ h) (sound-andL _ _ _ _) %. +%total _Ψ (sound-andL _Ψ _ _ _) %. +%term _ + %pi (sound (∧⁺L N) (tN (t, (t∧⁺ T₁ T₂) TΩ) TCQ) (cons D)) + %<- (sound N (tN (t, T₁ (t, T₂ TΩ)) TCQ) (cons ([h₁] cons ([h₂] D₁ h₁ h₂)))) + %<- ({h left (and P₁ P₂)} sound-andL _ h D₁ (%the (right' _Ψ Q) (D h))) %. +%term _ %pi (sound (η⁻ N) (tN t· (tn tQ)) (nil D)) %<- (sound N (tN t· ta) (nil D)) %. +%term _ + %pi (sound (↑R N) (tN t· (tn (t↑ T))) (nil D)) + %<- (sound N (tN t· (tp T)) (nil D)) %. +%term _ + %pi (sound (⊃R N) (tN t· (tn (t⊃ T₁ T₂))) (nil (impR D))) + %<- (sound N (tN (t, T₁ t·) (tn T₂)) (cons ([h left P₁] nil (%the (right P₂) (D h))))) %. +%term _ sound ⊤⁻R (tN t· (tn t⊤⁻)) (nil ttR) %. +%term _ + %pi (sound (∧⁻R N₁ N₂) (tN t· (tn (t∧⁻ T₁ T₂))) (nil (andR D₁ D₂))) + %<- (sound N₁ (tN t· (tn T₁)) (nil D₁)) + %<- (sound N₂ (tN t· (tn T₂)) (nil D₂)) %. +%term _ sound id⁻ (tS tQ ta) (cons ([h] nil (init h))) %. +%term _ + %pi (sound (↑L _ N) (tS (t↑ T₁) TCQ) D) + %<- (sound N (tN (t, T₁ t·) TCQ) (%the (right' (P ; ○) Q) D)) %. +%term _ + %pi (sound (⊃L V Sp) (tS (t⊃ T₁ T₂) TCQ) (cons ([h] nil (impL D₁ D₂ h)))) + %<- (sound V (tV T₁) (nil (%the (right P₁) D₁))) + %<- (sound Sp (tS T₂ TCQ) (cons ([h left P₂] nil (%the (right Q) (D₂ h))))) %. +%term _ + %pi (sound (∧⁻L₁ Sp) (tS (t∧⁻ T₁ T₂) TCQ) (cons ([h left (and P₁ P₂)] nil (andL₁ D h)))) + %<- (sound Sp (tS T₁ TCQ) (cons ([h₁ left P₁] nil (%the (right Q) (D h₁))))) %. +%term _ + %pi (sound (∧⁻L₂ Sp) (tS (t∧⁻ T₁ T₂) TCQ) (cons ([h left (and P₁ P₂)] nil (andL₂ D h)))) + %<- (sound Sp (tS T₂ TCQ) (cons ([h₂ left P₂] nil (%the (right Q) (D h₂))))) %. +%worlds (soundctx) (sound _ _ _) %. +%total (E) (sound E _ _) %. +%{! ## Focalization (completeness) !}% +%{! Completeness is established in a context where each erased +hypothesis is mapped to some polarization. This is the almost same, +type-wise, as the context for soundness, but the computational +interpretation runs the opposite way 'round. !}% +%sort completehyp_res %. +%term cn %pi (hyp A⁻) %-> (completehyp_res A⁻) %. +%term ca %pi (value (c Q⁺)) %-> (completehyp_res (c Q⁺)) %. +%sort completehyp %. +%block completehypn {A⁻ typ ⁻} {P} {T t A⁻ P} [x hyp A⁻] [h left P] [_ completehyp h T (cn x)]%. +%block completehypa {Q⁺ atom ⁺} {P} {T t (c Q⁺) P} [z value (c Q⁺)] [h left P] [_ completehyp h T (ca z)]%. +%block completectx%. +%sort transnormal %. +%mode transnormal %in %out %. +%term _ transnormal (tp _) snp %. +%term _ transnormal (tn _) snn %. +%term _ transnormal ta sna %. +%worlds (atom⁺ atom⁻) (transnormal _ _) %. +%total [] (transnormal _ _) %. +%mode completehyp %in %out %out %. +%worlds (completectx) (completehyp _ _ _) %. +%total [] (completehyp _ _ _) %. +%sort complete %. +%mode complete %in %in %in %out %. +%{! ### Initial rules !}% +%sort comp-init-susp %. +%term _ %pi (comp-init-susp _ tQ M) %<- (adm-initsusp⁻ M) %. +%mode comp-init-susp %in %in %out %. +%worlds (completectx) (comp-init-susp _ _ _) %. +%total [] (comp-init-susp _ _ _) %. +%term _ + %pi (complete Pf (init H) ta (NS _ sn M X)) + %<- (completehyp H (%the (t A (a Q)) TH) (cn (%the (hyp A) X))) + %<- (lshifty TH (%the (t A' (a Q)) TH') (%the (not-doubleshifted A') ND) (%the ({U} %pi (stable U) %-> (%pi (hyp A') %-> (term · U)) %-> (hyp A) %-> (term · U)) NS)) + %<- (comp-init-susp ND TH' M) %. +%sort comp-init-case %. +%mode comp-init-case %in %in %in %in %out %. +%term _ %pi (comp-init-case ndQ tQ ndQ⁺ (t↓ tQ) M) %<- (adm-init⁻ M) %. +%term _ %pi (comp-init-case ndQ⁻ (t↑ tQ) ndQ tQ M) %<- (adm-init⁺ M) %. +%worlds (completectx) (comp-init-case _ _ _ _ _) %. +%total [] (comp-init-case _ _ _ _ _) %. +%sort comp-init %. +%term _ %pi (comp-init tQ (ca Z) ndQ tQ (M Z)) %<- (adm-initsusp⁺ M) %. +%term _ + %pi (comp-init TH (cn X) ND T' (NS _ sp M X)) + %<- (lshifty TH (%the (t Aleft' (a Q)) TH') (%the (not-doubleshifted Aleft') NDH) (%the ({U} %pi (stable U) %-> (%pi (hyp Aleft') %-> (term · U)) %-> (hyp Aleft) %-> (term · U)) NS)) + %<- (comp-init-case NDH TH' ND T' M) %. +%mode comp-init %in %in %in %in %out %. +%worlds (completectx) (comp-init _ _ _ _ _) %. +%total [] (comp-init _ _ _ _ _) %. +%term _ + %pi (complete Pf (init H) (tp (%the (t Aright (a Q)) T)) (NSright M)) + %<- (rshifty T (%the (t Aright' (a Q)) T') (%the (not-doubleshifted Aright') NDright) (%the (%pi (term · (Aright' true)) %-> (term · (Aright true))) NSright)) + %<- (completehyp H (%the (t Aleft (a Q)) TH) (%the (completehyp_res Aleft) HypRes)) + %<- (comp-init TH HypRes NDright T' M) %. +%{! ### Disjunction !}% +%sort comp-ffL %. +%term _ %pi (comp-ffL Pf nd⊥⁻ (t↑ t⊥) M) %<- (adm-⊥L Pf M) %. +%mode comp-ffL %in %in %in %out %. +%worlds (completectx) (comp-ffL _ _ _ _) %. +%total [] (comp-ffL _ _ _ _) %. +%term _ + %pi (complete Pf (ffL H) T (NS _ Pf M X)) + %<- (completehyp H (%the (t A ff) TH) (cn (%the (hyp A) X))) + %<- (lshifty TH (%the (t A' ff) TH') (%the (not-doubleshifted A') ND) (%the ({U} %pi (stable U) %-> (%pi (hyp A') %-> (term · U)) %-> (hyp A) %-> (term · U)) NS)) + %<- (comp-ffL Pf ND TH' M) %. +%sort comp-orR₁ {D right (or P₁ P₂)} %. +%mode comp-orR₁ %in %in %in %in %out %. +%term _ + %pi (comp-orR₁ (orR₁ D₁) refl nd∨⁺ (t∨ (%the (t A⁺ P₁) T₁) (%the (t B⁺ P₂) T₂)) M) + %<- (complete sp D₁ (tp T₁) (%the (term · (A⁺ true)) N₁)) + %<- (adm-∨R₁ N₁ (%the (term · (A⁺ ∨ B⁺ true)) M)) %. +%term _ + %pi (complete Pf (orR₁ D₁) (tp T) (NS M)) + %<- (rshifty T (%the (t A' (or P₁ P₂)) T') (%the (not-doubleshifted A') ND) (%the (%pi (term · (A' true)) %-> (term · (A true))) NS)) + %<- (comp-orR₁ (orR₁ D₁) refl ND T' M) %. +%sort comp-orR₂ {D right (or P₁ P₂)} %. +%mode comp-orR₂ %in %in %in %in %out %. +%term _ + %pi (comp-orR₂ (orR₂ D₂) refl nd∨⁺ (t∨ (%the (t A⁺ P₁) T₁) (%the (t B⁺ P₂) T₂)) M) + %<- (complete sp D₂ (tp T₂) (%the (term · (B⁺ true)) N₂)) + %<- (adm-∨R₂ N₂ (%the (term · (A⁺ ∨ B⁺ true)) M)) %. +%term _ + %pi (complete Pf (orR₂ D₂) (tp T) (NS M)) + %<- (rshifty T (%the (t A' (or P₁ P₂)) T') (%the (not-doubleshifted A') ND) (%the (%pi (term · (A' true)) %-> (term · (A true))) NS)) + %<- (comp-orR₂ (orR₂ D₂) refl ND T' M) %. +%sort comp-orL %. +%mode comp-orL %in %in %in %in %in %in %out %. +%term _ + %pi (comp-orL Pf (orL D₁ D₂ H) refl nd∨⁻ (t↑ (t∨ T₁ T₂)) T M) + %<- ({x₁ hyp (↑ B₁⁺)} {h₁ left P₁} + %pi (completehyp h₁ (t↑ T₁) (cn x₁)) + %-> (complete Pf (D₁ h₁) T (%the (term · U) (N₁ x₁)))) + %<- ({x₂ hyp (↑ B₂⁺)} {h₂ left P₂} + %pi (completehyp h₂ (t↑ T₂) (cn x₂)) + %-> (complete Pf (D₂ h₂) T (%the (term · U) (N₂ x₂)))) + %<- (transnormal T Pf') + %<- (adm-∨L (ss Pf Pf') N₁ N₂ (%the (%pi (hyp (↑ (B₁⁺ ∨ B₂⁺))) %-> (term · U)) M)) %. +%term _ + %pi (complete Pf (orL D₁ D₂ H) (%the (tU U Q) T) (NS _ Pf M X)) + %<- (completehyp H (%the (t A (or P₁ P₂)) TH) (cn (%the (hyp A) X))) + %<- (lshifty TH TH' ND NS) + %<- (comp-orL Pf (orL D₁ D₂ H) refl ND TH' T M) %. +%{! ### Conjunction !}% +%sort comp-ttR {D right tt} %. +%term _ %pi (comp-ttR ttR refl nd⊤⁺⁺ t⊤⁺ M) %<- (adm-⊤⁺R M) %. +%term _ %pi (comp-ttR ttR refl nd⊤⁻⁺ (t↓ t⊤⁻) M) %<- (adm-⊤⁻R M) %. +%mode comp-ttR %in %in %in %in %out %. +%worlds (completectx) (comp-ttR _ _ _ _ _) %. +%total [] (comp-ttR _ _ _ _ _) %. +%term _ + %pi (complete Pf ttR (tp T) (NS M)) + %<- (rshifty T (%the (t A' tt) T') (%the (not-doubleshifted A') ND) (%the (%pi (term · (A' true)) %-> (term · (A true))) NS)) + %<- (comp-ttR ttR refl ND T' M) %. +%sort comp-andR {D right (and P₁ P₂)} %. +%mode comp-andR %in %in %in %in %out %. +%term _ + %pi (comp-andR (andR D₁ D₂) refl nd∧⁺⁺ (t∧⁺ (%the (t A⁺ P₁) T₁) (%the (t B⁺ P₂) T₂)) M) + %<- (complete sp D₁ (tp T₁) (%the (term · (A⁺ true)) N₁)) + %<- (complete sp D₂ (tp T₂) (%the (term · (B⁺ true)) N₂)) + %<- (adm-∧⁺R N₁ N₂ M) %. +%term _ + %pi (comp-andR (andR D₁ D₂) refl nd∧⁻⁺ (t↓ (t∧⁻ (%the (t A⁻ P₁) T₁) (%the (t B⁻ P₂) T₂))) M) + %<- (complete sp D₁ (tp (t↓ T₁)) (%the (term · (↓ A⁻ true)) N₁)) + %<- (complete sp D₂ (tp (t↓ T₂)) (%the (term · (↓ B⁻ true)) N₂)) + %<- (adm-∧⁻R N₁ N₂ M) %. +%term _ + %pi (complete Pf (andR D₁ D₂) (tp T) (NS M)) + %<- (rshifty T (%the (t A' (and P₁ P₂)) T') (%the (not-doubleshifted A') ND) (%the (%pi (term · (A' true)) %-> (term · (A true))) NS)) + %<- (comp-andR (andR D₁ D₂) refl ND T' M) %. +%sort comp-andL₁ %. +%mode comp-andL₁ %in %in %in %in %in %in %out %. +%term _ + %pi (comp-andL₁ Pf (andL₁ D₁ H) refl nd∧⁺⁻ (t↑ (t∧⁺ (%the (t A⁺ P₁) T₁) (%the (t B⁺ P₂) T₂))) T M) + %<- ({x hyp (↑ A⁺)} {h left P₁} + %pi (completehyp h (t↑ T₁) (cn x)) + %-> (complete Pf (D₁ h) T (%the (term · U) (N₁ x)))) + %<- (transnormal T Pf') + %<- (adm-∧⁺L (ss Pf Pf') ([x] [y] N₁ x) M) %. +%term _ + %pi (comp-andL₁ Pf (andL₁ D₁ H) refl nd∧⁻⁻ (t∧⁻ (%the (t A⁻ P₁) T₁) (%the (t B⁻ P₂) T₂)) T M) + %<- ({x hyp A⁻} {h left P₁} %pi (completehyp h T₁ (cn x)) %-> (complete Pf (D₁ h) T (%the (term · U) (N₁ x)))) + %<- (transnormal T Pf') + %<- (adm-∧⁻L₁ Pf' N₁ M) %. +%term _ + %pi (complete Pf (andL₁ D₁ H) (%the (tU U Q) T) (NS _ Pf M X)) + %<- (completehyp H (%the (t A (and P₁ P₂)) TH) (cn (%the (hyp A) X))) + %<- (lshifty TH TH' ND NS) + %<- (comp-andL₁ Pf (andL₁ D₁ H) refl ND TH' T M) %. +%sort comp-andL₂ %. +%mode comp-andL₂ %in %in %in %in %in %in %out %. +%term _ + %pi (comp-andL₂ Pf (andL₂ D₂ H) refl nd∧⁺⁻ (t↑ (t∧⁺ (%the (t A⁺ P₁) T₁) (%the (t B⁺ P₂) T₂))) T M) + %<- ({x hyp (↑ B⁺)} {h left P₂} + %pi (completehyp h (t↑ T₂) (cn x)) + %-> (complete Pf (D₂ h) T (%the (term · U) (N₂ x)))) + %<- (transnormal T Pf') + %<- (adm-∧⁺L (ss Pf Pf') ([x] [y] N₂ y) M) %. +%term _ + %pi (comp-andL₂ Pf (andL₂ D₂ H) refl nd∧⁻⁻ (t∧⁻ (%the (t A⁻ P₁) T₁) (%the (t B⁻ P₂) T₂)) T M) + %<- ({x hyp B⁻} {h left P₂} %pi (completehyp h T₂ (cn x)) %-> (complete Pf (D₂ h) T (%the (term · U) (N₂ x)))) + %<- (transnormal T Pf') + %<- (adm-∧⁻L₂ Pf' N₂ M) %. +%term _ + %pi (complete Pf (andL₂ D₂ H) (%the (tU U Q) T) (NS _ Pf M X)) + %<- (completehyp H (%the (t A (and P₁ P₂)) TH) (cn (%the (hyp A) X))) + %<- (lshifty TH TH' ND NS) + %<- (comp-andL₂ Pf (andL₂ D₂ H) refl ND TH' T M) %. +%{! ### Implication !}% +%sort comp-impR {D right (imp P₁ P₂)} %. +%mode comp-impR %in %in %in %in %out %. +%term _ + %pi (comp-impR (impR D₁) refl nd⊃⁺ (t↓ (t⊃ (%the (t A⁺ P₁) T₁) (%the (t B⁻ P₂) T₂))) M) + %<- ({x hyp (↑ A⁺)} {h left P₁} %pi (completehyp h (t↑ T₁) (cn x)) %-> (complete sp (D₁ h) (tp (t↓ T₂)) (N₁ x))) + %<- (adm-⊃R N₁ M) %. +%term _ + %pi (complete Pf (impR D₁) (tp T) (NS M)) + %<- (rshifty T (%the (t A' (imp P₁ P₂)) T') (%the (not-doubleshifted A') ND) (%the (%pi (term · (A' true)) %-> (term · (A true))) NS)) + %<- (comp-impR (impR D₁) refl ND T' M) %. +%sort comp-impL %. +%mode comp-impL %in %in %in %in %in %in %out %. +%term _ + %pi (comp-impL Pf (impL D₁ D₂ _) refl nd⊃⁻ (t⊃ (%the (t A⁺ P₁) T₁) (%the (t B⁻ P₂) T₂)) T M) + %<- (complete sp D₁ (tp T₁) (%the (term · (A⁺ true)) N₁)) + %<- ({x hyp B⁻} {h left P₂} %pi (completehyp h T₂ (cn x)) %-> (complete Pf (D₂ h) T (N₂ x))) + %<- (transnormal T Pf') + %<- (adm-⊃L (ss Pf Pf') N₁ N₂ M) %. +%term _ + %pi (complete Pf (impL D₁ D₂ H) T (NS _ Pf M X)) + %<- (completehyp H (%the (t A (imp P₁ P₂)) TH) (cn (%the (hyp A) X))) + %<- (lshifty TH TH' ND NS) + %<- (comp-impL Pf (impL D₁ D₂ H) refl ND TH' T M) %. +%worlds (completectx) (comp-orR₁ _ _ _ _ _) (comp-orR₂ _ _ _ _ _) (comp-orL _ _ _ _ _ _ _) (comp-andR _ _ _ _ _) (comp-andL₁ _ _ _ _ _ _ _) (comp-andL₂ _ _ _ _ _ _ _) (comp-impR _ _ _ _ _) (comp-impL _ _ _ _ _ _ _) (complete _ _ _ _) %. +%total (D D1 D2 D3 D4 D5 D6 D7 D8) (comp-orR₁ D1 _ _ _ _) (comp-orR₂ D2 _ _ _ _) (comp-orL _ D3 _ _ _ _ _) (comp-andR D4 _ _ _ _) (comp-andL₁ _ D5 _ _ _ _ _) (comp-andL₂ _ D6 _ _ _ _ _) (comp-impR D7 _ _ _ _) (comp-impL _ D8 _ _ _ _ _) (complete _ D _ _) %. +%{! ## Inheriting the focused calculus's metatheory !}% +%{! We need the existance of some polarization strategy; we intentionally pick +a middling, undistinghished sort of translation that translates everything as +a negative proposition. It will work great on hereditary Harrop formulas and +not so good if you have lots of disjunction and positive propositions; the only +interesting thing about it is that its results are reminiscent of Howe's +semi-focused lax logic. !}% +%sort polarize {P} %. +%mode polarize %in %out %. +%term _ polarize (a Q) tQ %. +%term _ polarize (a Q) (t↑ tQ) %. +%term _ polarize ff (t↑ t⊥) %. +%term _ + %pi (polarize (or P₁ P₂) (t↑ (t∨ (t↓ T₁) (t↓ T₂)))) + %<- (polarize P₁ T₁) + %<- (polarize P₂ T₂) %. +%term _ polarize tt t⊤⁻ %. +%term _ %pi (polarize (and P₁ P₂) (t∧⁻ T₁ T₂)) %<- (polarize P₁ T₁) %<- (polarize P₂ T₂) %. +%term _ + %pi (polarize (imp P₁ P₂) (t⊃ (t↓ T₁) T₂)) + %<- (polarize P₁ T₁) + %<- (polarize P₂ T₂) %. +%worlds (atom⁺ atom⁻) (polarize _ _) %. +%total P (polarize P _) %. +%block translate {A} {P} {TH t A P} [x hyp A] [h left P] [_ soundhyp x TH h] [_ completehyp h TH (cn x)]%. +%sort unfocused-cut %. +%mode unfocused-cut %in %in %out %. +%term _ + %pi (unfocused-cut (%the (right P) D) (%the (%pi (left P) %-> (right Q)) E) F) + %<- (polarize P (%the (t A⁻ P) TP)) + %<- (complete sp D (tp (t↓ TP)) (%the (term · (↓ A⁻ true)) M)) + %<- (polarize Q (%the (t C⁻ Q) TQ)) + %<- ({x hyp A⁻} {h left P} + %pi (completehyp h TP (cn x)) + %-> (complete sp (E h) (tp (t↓ TQ)) (%the (term · (↓ C⁻ true)) (N x)))) + %<- (lsubst _ (b s) (ss sp snp) M (↓L N) N') + %<- (sound N' (tN t· (tp (t↓ TQ))) (nil (%the (right Q) F))) %. +%worlds (translate) (unfocused-cut _ _ _) %. +%total [] (unfocused-cut _ _ _) %. +%sort unfocused-identity %. +%mode {%in P} {%out D %pi (left P) %-> (right P)} unfocused-identity D %. +%term _ + %pi (unfocused-identity D) + %<- (polarize P (%the (t A⁻ P) TP)) + %<- (identity⁻ (%the (%pi (hyp A⁻) %-> (term · (A⁻ inv))) N)) + %<- ({x hyp A⁻} {h left P} + %pi (soundhyp x TP h) + %-> (sound (N x) (tN t· (tn TP)) (nil (%the (right P) (D h))))) %. +%worlds (translate) (unfocused-identity _) %. +%total [] (unfocused-identity _) %. +%{! ## Running the theorems !}% +%{! Here are two unfocused derivations of +`(p ∧ q) ⊃ (r ∧ s) ⊃ (p ∧ t)`. !}% +%define d₁ ({P} {p atom P} {q atom P} {r atom P} {s atom P} right (imp (and (a p) (a q)) (imp (and (a r) (a s)) (and (a p) (a r))))) [P] [p] [q] [r] [s] impR ([h₁ left (and (a p) (a q))] impR ([h₂ left (and (a r) (a s))] andR (andL₁ ([h₁' left (a p)] init h₁') h₁) (andL₁ ([h₂' left (a r)] init h₂') h₂))) %. +%define d₂ ({P} {p atom P} {q atom P} {r atom P} {s atom P} right (imp (and (a p) (a q)) (imp (and (a r) (a s)) (and (a p) (a r))))) [P] [p] [q] [r] [s] impR ([h₁ left (and (a p) (a q))] impR ([h₂ left (and (a r) (a s))] andL₁ ([h₁' left (a p)] andR (andL₂ ([_] andL₂ ([_] init h₁') h₁) h₁) (andL₁ ([h₂' left (a r)] init h₂') h₂)) h₁)) %. +%{! Here are three different polarizations. !}% +%solve t⁻ : {p} {q} {r} {s} t (↓ (c p ∧⁻ c q) ⊃ (↓ (c r ∧⁻ c s) ⊃ (c p ∧⁻ c r))) _ %. +%solve t⁺ : {p} {q} {r} {s} t (c p ∧⁺ c q ⊃ (c r ∧⁺ c s ⊃ ↑ (c p ∧⁺ c r))) _ %. +%solve t⁼ : {p} {q} {r} {s} t (↓ (↑ (c p) ∧⁻ ↑ (c q)) ⊃ ↑ (↓ (↓ (↑ (c r) ∧⁻ ↑ (c s)) ⊃ ↑ (↓ (↑ (c p) ∧⁻ ↑ (c r)))))) _ %. +%{! Under the same "good" polarization, different derivations translate to a +single unique proof. !}% +%{!! begin checked !!}% +%query 1 _ _ {p} {q} {r} {s} complete _ (d₁ ⁻ p q r s) (tp (t↓ (t⁻ p q r s))) (N p q r s) %. +%query 1 _ _ {p} {q} {r} {s} complete _ (d₂ ⁻ p q r s) (tp (t↓ (t⁻ p q r s))) (N p q r s) %. +%{!! end checked !!}% +%{! Different polarizations may lead to very differently shaped derivations. +!}% +%{!! begin checked !!}% +%query 1 _ _ {p} {q} {r} {s} complete _ (d₂ ⁺ p q r s) (tp (t↓ (t⁺ p q r s))) (N p q r s) %. +%query 1 _ _ {p} {q} {r} {s} complete _ (d₂ ⁺ p q r s) (tp (t↓ (t⁼ p q r s))) (N p q r s) %. +%{!! end checked !!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/function.lf b/new-tests/stelf-output/pages/function.lf new file mode 100644 index 0000000..ad2e7ab --- /dev/null +++ b/new-tests/stelf-output/pages/function.lf @@ -0,0 +1,10 @@ +% +%{! A [relation](/wiki/relation/) behaves like a **function** if a [uniqueness lemma](/wiki/uniqueness-lemma/) and an [effectiveness lemma](/wiki/effectiveness-lemma/) can be proven assuming a particular set of positions in the relation are inputs. + +\{\{stub\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Function). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/general-description-of-twelf.lf b/new-tests/stelf-output/pages/general-description-of-twelf.lf new file mode 100644 index 0000000..912a82f --- /dev/null +++ b/new-tests/stelf-output/pages/general-description-of-twelf.lf @@ -0,0 +1,20 @@ +% +% +%{! _This quick introduction to Twelf is aimed at people without any specific technical background. If you want more information, you can find it on the [documentation](/wiki/documentation/) page._ + + +**Twelf** is a piece of computer software, and it is also a computer language understood by the Twelf software. This is a common ambiguity in computer systems - you can "install Java on your computer" (Java, a piece of computer software), or you can "write programs in Java for work" (Java, the computer language that is understood and interpreted by the software). + +
+![A Twelf file open in Emacs, showing Twelf's output and the "Server OK" message that means Twelf loaded the file with no errors.](/src/assets/serverok.png) + +_Using Twelf with the Emacs text editor. The red, blue, and black text at the top is Twelf code, and the black text at the bottom is the output from the Twelf program reading the code._ +
+ +C code and Java code describe programs, HTML code describes graphical web pages, and Twelf code describes _logical systems._ Logical systems are sometimes strange to think about, because the only logical system that most people have used is basic arithmetic (addition, subtraction, division, etc.), or maybe set theory if they're particularly ambitious. Most people that use Twelf aren’t interested in using it to do basic arithmetic, because there are a great deal more interesting logical systems than just the ones that we learn about in high school. + +The reason someone might want to use Twelf code to describe a logical system is that once they’ve described it, they can write more Twelf code that uses that logical system. You could use Twelf to write out a statement about basic arithmetic (for instance, “if a + b = c, then b + a = c”), and then use Twelf to write out a justification of why that statement is true (i.e. a _proof_). When you do so, Twelf will check your proof, making sure that what you said actually is true! + +Twelf is by no means the only program you can use to do this sort of thing. ACL2, Agda, AUTOMATH, HOL, HOL Light, Lean, LEGO, Isabelle, MetaPRL, NuPRL PVS, Rocq, and TPS are just a few (!) of the systems that will let you define logical systems and prove things with them. A lot of really amazing work is done using these different systems: one project at the University of Pittsburgh used HOL Light to [check a proof of the Kepler conjecture](https://arxiv.org/abs/1501.02155). [The Economist wrote an article about it](http://www.economist.com/science/displayStory.cfm?story_id=3809661), though that article is unfortunately no longer freely available. + +But here’s where, for people that use Twelf, it gets interesting: it turns out that while basic arithmetic, set theory, and interesting logics are logical systems, _programming languages_ are also logical systems - and Twelf has a couple of unique features that make it a great tool to use when the logical systems you are working with are programming languages... !}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/glossary.lf b/new-tests/stelf-output/pages/glossary.lf new file mode 100644 index 0000000..f5b6341 --- /dev/null +++ b/new-tests/stelf-output/pages/glossary.lf @@ -0,0 +1,59 @@ +% +%{! This **glossary** should include most of the unfamiliar concepts that +someone with an undergraduate-level understanding of type theory would +encounter in the process of learning to use Twelf. Some of these links point to +extensive tutorials, but in those cases the first paragraph or two should +include a definition of the concept. + +* [Adequacy](/wiki/adequacy/) +* [Alpha-equivalence](/wiki/alpha-equivalence/) (i.e. "α-equivalence") +* [Beta-equivalence](/wiki/beta-equivalence/) (i.e. "β-equivalence") +* [Canonical form](/wiki/canonical-form/) +* [Canonical forms lemma](/wiki/canonical-forms-lemma/) +* [Compatibility lemma](/wiki/respects-lemma/) +* [Congruence relation](/wiki/congruence-relation/) +* [Congruence lemma](/wiki/respects-lemma/) +* [Constraint domain](/wiki/constraint-domain/) +* [Coverage checking](/wiki/coverage-checking/) +* [Effectiveness lemma](/wiki/effectiveness-lemma/) +* [Equality](/wiki/equality/) +* [Equivalence relation](/wiki/equivalence-relation/) +* [Eta-equivalence](/wiki/eta-equivalence/) (i.e. "η-equivalence") +* [Eta-long form](/wiki/eta-equivalence/) (i.e. "η-long form") +* [Exchange lemma](/wiki/exchange-lemma/) +* [Explicit parameter](/wiki/implicit-and-explicit-parameters/) +* [Explicit context](/wiki/explicit-context/) +* [Extrinsic encoding](/wiki/intrinsic-and-extrinsic-encodings/) +* [Function](/wiki/function/) +* [Ground](/wiki/ground/) +* [Higher-order abstract syntax](/wiki/higher-order-abstract-syntax/) +* [Higher-order judgement](/wiki/higher-order-judgements/) +* [Hereditary substitution](/wiki/hereditary-substitution/) +* [Hypothetical judgment](/wiki/hypothetical-judgment/) +* [Implicit parameter](/wiki/implicit-and-explicit-parameters/) +* [Intrinsic encoding](/wiki/intrinsic-and-extrinsic-encodings/) +* [Judgment](/wiki/judgment/) +* [Judgments as types](/wiki/judgment/) +* [LF](/wiki/lf/) +* [Logic programming](/wiki/logic-programming/) +* [Meta-logic](/wiki/meta-logic/) +* [Metatheorem](/wiki/metatheorem/) +* [Mode checking](/wiki/percent-mode/) +* [Object logic](/wiki/object-logic/) +* [Relation](/wiki/relation/) +* [Regular worlds](/wiki/percent-worlds/) +* [Respects lemma](/wiki/respects-lemma/) +* [Substitution lemma](/wiki/substitution-lemma/) +* [Subordination](/wiki/subordination/) +* [Tabled logic programming](/wiki/tabled-logic-programming/) +* [Tactical theorem proving](/wiki/tactical-theorem-proving/) +* [Theorem prover](/wiki/theorem-prover/) +* [Totality assertion](/wiki/totality-assertion/) +* [Totality checking](/wiki/percent-total/) +* [Type family](/wiki/type-family/) +* [Twelf signature](/wiki/twelf-signature/) +* [Unification](/wiki/unification/) +* [Uniqueness lemma](/wiki/uniqueness-lemma/) +* [Weakening lemma](/wiki/weakening-lemma/) +* [World checking](/wiki/percent-worlds/) +* [World subsumption](/wiki/world-subsumption/) !}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/ground.lf b/new-tests/stelf-output/pages/ground.lf new file mode 100644 index 0000000..c52716d --- /dev/null +++ b/new-tests/stelf-output/pages/ground.lf @@ -0,0 +1,21 @@ +% +%{! A term is **ground** if it does not contain any existential ([unification](/wiki/unification/)) variables. For example, we can make the following [queries](/wiki/percent-solve/) on [``plus``](/wiki/natural-numbers/): !}% +%{! (options removed from twelftag: hidden="true") !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort plus %. +%term plus-z plus z N N %. +%term plus-s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%{! !}% +%{! (options removed from twelftag: discard="true" check="decl") !}% +%mode plus %in %in %out %. +%solve D1 : plus (s z) (s (s z)) X %. +%solve D2 : plus (s z) (s Y) (s (s z)) %. +%{! In the query for ``D1``, the first two arguments to ``plus`` are ground. The third is not; it is an existential variable. This is the typical situation: that the input arguments are ground and the output is an unconstrained existential variable. A [%total](/wiki/percent-total/) declaration for ``plus`` guarantees that such queries will always succeed. However, we can still issue queries like the second, where the middle argument is not ground. In this case, the query does succeed, but the totality declaration does not guarantee that it will. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Ground). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/hereditary-substitution-for-the-stlc-part-2.lf b/new-tests/stelf-output/pages/hereditary-substitution-for-the-stlc-part-2.lf new file mode 100644 index 0000000..6034283 --- /dev/null +++ b/new-tests/stelf-output/pages/hereditary-substitution-for-the-stlc-part-2.lf @@ -0,0 +1,1136 @@ +% +%{! This is part 2 of the case study [Hereditary substitution for the STLC](/wiki/hereditary-substitution-for-the-stlc/). !}% +%{! (options removed from twelftag: hidden="true") !}% +%sort tp %. +%term b tp %. +%term arrow %pi tp %-> tp %-> tp %. +%term prod %pi tp %-> tp %-> tp %. +%sort rtm %. +%sort mtm %. +%term app %pi rtm %-> mtm %-> rtm %. +%term c rtm %. +%term fst %pi rtm %-> rtm %. +%term snd %pi rtm %-> rtm %. +%term asm %pi rtm %-> mtm %. +%term lam %pi (%pi rtm %-> mtm) %-> mtm %. +%term pair %pi mtm %-> mtm %-> mtm %. +%block rtm_block [x rtm]%. +%worlds (rtm_block) (tp) (rtm) (mtm) %. +%sort atom %. +%mode atom %in %out %. +%sort canon %. +%mode canon %in %in %. +% atomic +%term atom_c atom c b %. +%term atom_app %pi (atom (app R1 M2) A) %<- (atom R1 (arrow A2 A)) %<- (canon M2 A2) %. +%term atom_fst %pi (atom (fst R) A1) %<- (atom R (prod A1 A2)) %. +%term atom_snd %pi (atom (snd R) A2) %<- (atom R (prod A1 A2)) %. +%% canonical +%term canon_asm %pi (canon (asm R) b) %<- (atom R b) %. +%term canon_lam + %pi (canon (lam M) (arrow A2 A)) + %<- ({x rtm} %pi (atom x A2) %-> (canon (M x) A)) %. +%term canon_pair %pi (canon (pair M1 M2) (prod A1 A2)) %<- (canon M1 A1) %<- (canon M2 A2) %. +%block atom_block {A tp} [x rtm] [dx atom x A]%. +%worlds (atom_block) (atom _ _) (canon _ _) %. +%terminates (R M) (atom R _) (canon M _) %. +%sort hsubst_m %. +%mode hsubst_m %in %in %in %out %. +%sort hsubst_r %. +%mode hsubst_r %in %in %in %out %. +%sort hsubst_rr %. +%mode hsubst_rr %in %in %in %out %out %. +%% Ms +%term hsubst_m_r %pi (hsubst_m M0 A0 ([x] asm (R x)) (asm R')) %<- (hsubst_r M0 A0 R R') %. +%term hsubst_m_rr %pi (hsubst_m M0 A0 ([x] asm (R x)) M') %<- (hsubst_rr M0 A0 R M' _) %. +%term hsubst_m_lam + %pi (hsubst_m M0 A0 ([x] lam ([y] M x y)) (lam M')) + %<- ({y rtm} hsubst_m M0 A0 ([x] M x y) (M' y)) %. +%term hsubst_m_pair + %pi (hsubst_m M0 A0 ([x] pair (M1 x) (M2 x)) (pair M1' M2')) + %<- (hsubst_m M0 A0 M1 M1') + %<- (hsubst_m M0 A0 M2 M2') %. +%% R not head +%term hsubst_r_closed hsubst_r M0 A0 ([x] E) E %. +%term hsubst_r_app + %pi (hsubst_r M0 A0 ([x] app (R x) (M x)) (app R' M')) + %<- (hsubst_r M0 A0 R R') + %<- (hsubst_m M0 A0 M M') %. +%term hsubst_r_fst %pi (hsubst_r M0 A0 ([x] fst (R x)) (fst R')) %<- (hsubst_r M0 A0 R R') %. +%term hsubst_r_snd %pi (hsubst_r M0 A0 ([x] snd (R x)) (snd R')) %<- (hsubst_r M0 A0 R R') %. +%% r head +%term hsubst_rr_var hsubst_rr M0 A0 ([x] x) M0 A0 %. +%term hsubst_rr_app + %pi (hsubst_rr M0 A0 ([x] app (R1 x) (M2 x)) M'' A) + %<- (hsubst_rr M0 A0 R1 (lam M') (arrow A2 A)) + %<- (hsubst_m M0 A0 M2 M2') + %<- (hsubst_m M2' A2 M' M'') %. +%term hsubst_rr_fst + %pi (hsubst_rr M0 A0 ([x] fst (R x)) M1' A1') + %<- (hsubst_rr M0 A0 R (pair M1' M2') (prod A1' A2')) %. +%term hsubst_rr_snd + %pi (hsubst_rr M0 A0 ([x] snd (R x)) M2' A2') + %<- (hsubst_rr M0 A0 R (pair M1' M2') (prod A1' A2')) %. +%worlds (rtm_block) (hsubst_m _ _ _ _) (hsubst_r _ _ _ _) (hsubst_rr _ _ _ _ _) %. +%reduces <= A' A0 (hsubst_rr _ A0 _ _ A') %. +%terminates {(A0 A0' A0'') (M R R')} (hsubst_m _ A0 M _) (hsubst_r _ A0' R _) (hsubst_rr _ A0'' R' _ _) %. +%sort expand %. +%mode expand %in %in %out %. +%term expand_b expand b R (asm R) %. +%term expand_arrow + %pi (expand (arrow A2 A) R (lam M)) + %<- ({x rtm} expand A2 x (M2 x)) + %<- ({x rtm} expand A (app R (M2 x)) (M x)) %. +%term expand_prod + %pi (expand (prod A1 A2) R (pair M1 M2)) + %<- (expand A1 (fst R) M1) + %<- (expand A2 (snd R) M2) %. +%worlds (rtm_block) (expand _ _ _) %. +%total A (expand A _ _) %. +%sort hsubst_rr_size {A2} {A'} %. +%mode hsubst_rr_size %in %in %in %. +%term _ hsubst_rr_size A2 A2 hsubst_rr_var %. +%term _ + %pi (hsubst_rr_size A2 A4 (hsubst_rr_app _ _ Drr)) + %<- (hsubst_rr_size A2 (arrow A3 A4) Drr) %. +%term _ + %pi (hsubst_rr_size A2 Al (hsubst_rr_fst Drr)) + %<- (hsubst_rr_size A2 (prod Al Ar) Drr) %. +%term _ + %pi (hsubst_rr_size A2 Ar (hsubst_rr_snd Drr)) + %<- (hsubst_rr_size A2 (prod Al Ar) Drr) %. +%worlds (rtm_block) (hsubst_rr_size _ _ _) %. +%total D (hsubst_rr_size _ _ D) %. +%reduces <= A' A2 (hsubst_rr_size A2 A' _) %. +%sort hsubst_r_exists_sum %. +%term hsubst_r_exists_sum_r %pi (hsubst_r_exists_sum M2 A2 R A) %<- (hsubst_r M2 A2 R R') %<- (atom R' A) %. +%term hsubst_r_exists_sum_rr + %pi (hsubst_r_exists_sum M2 A2 R A') + %<- (hsubst_rr M2 A2 R M' A') + %<- (canon M' A') %. +%sort hsubst_m_exists_asm %. +%mode hsubst_m_exists_asm %in %out %out %. +%term _ hsubst_m_exists_asm (hsubst_r_exists_sum_r DaR' DsR) (hsubst_m_r DsR) (canon_asm DaR') %. +%term _ hsubst_m_exists_asm (hsubst_r_exists_sum_rr DcM' DsR) (hsubst_m_rr DsR) DcM' %. +%worlds (atom_block) (hsubst_m_exists_asm _ _ _) %. +%total {} (hsubst_m_exists_asm _ _ _) %. +%sort hsubst_r_exists_fst %. +%mode hsubst_r_exists_fst %in %out %. +%term _ hsubst_r_exists_fst (hsubst_r_exists_sum_r DaR' DsR) (hsubst_r_exists_sum_r (atom_fst DaR') (hsubst_r_fst DsR)) %. +%term _ hsubst_r_exists_fst (hsubst_r_exists_sum_rr (canon_pair _ DcMl') DsR) (hsubst_r_exists_sum_rr DcMl' (hsubst_rr_fst DsR)) %. +%worlds (atom_block) (hsubst_r_exists_fst _ _) %. +%total {} (hsubst_r_exists_fst _ _) %. +%sort hsubst_r_exists_snd %. +%mode hsubst_r_exists_snd %in %out %. +%term _ hsubst_r_exists_snd (hsubst_r_exists_sum_r DaR' DsR) (hsubst_r_exists_sum_r (atom_snd DaR') (hsubst_r_snd DsR)) %. +%term _ hsubst_r_exists_snd (hsubst_r_exists_sum_rr (canon_pair DcMr' _) DsR) (hsubst_r_exists_sum_rr DcMr' (hsubst_rr_snd DsR)) %. +%worlds (atom_block) (hsubst_r_exists_snd _ _) %. +%total {} (hsubst_r_exists_snd _ _) %. +%sort hsubst_m_exists {A2} {y rtm} {dy atom y A2} %. +%mode hsubst_m_exists %in %in %in %out %out %. +%sort hsubst_r_exists {A2} {y rtm} {dx atom y A2} %. +%mode hsubst_r_exists %in %in %in %out %. +%sort hsubst_m_exists_app {A2} {y rtm} {dx atom y A2} %. +%mode hsubst_m_exists_app %in %in %in %in %in %out %. +%% canonical terms +%term _ + %pi (hsubst_m_exists A2 ([x] [dx] canon_asm (DaR x dx)) DcM2 DsR DcM') + %<- (hsubst_r_exists A2 DaR DcM2 Dsum) + %<- (hsubst_m_exists_asm Dsum DsR DcM') %. +%term _ + %pi (hsubst_m_exists A2 ([x] [dx] %the (canon (lam (M x)) (arrow Af At)) (canon_lam ([y] [dy] DcM x dx y dy))) (%the (canon M2 A2) DcM2) (hsubst_m_lam DsM) (canon_lam DcM')) + %<- ({y} {dy atom y Af} hsubst_m_exists A2 ([x] [dx] DcM x dx y dy) DcM2 (DsM y) (DcM' y dy)) %. +%term _ + %pi (hsubst_m_exists A2 ([x] [dx] canon_pair (DcMr x dx) (DcMl x dx)) DcM2 (hsubst_m_pair DsMr DsMl) (canon_pair DcMr' DcMl')) + %<- (hsubst_m_exists A2 DcMl DcM2 DsMl DcMl') + %<- (hsubst_m_exists A2 DcMr DcM2 DsMr DcMr') %. +%% atomic terms +%term _ hsubst_r_exists A2 ([x] [dx] D) _ (hsubst_r_exists_sum_r D hsubst_r_closed) %. +%term _ hsubst_r_exists A2 ([x] [dx] dx) DcM2 (hsubst_r_exists_sum_rr DcM2 hsubst_rr_var) %. +%term _ + %pi (hsubst_r_exists A2 ([x] [dx] atom_app (DcM x dx) (DaR x dx)) DcM2 DsumApp) + %<- (hsubst_m_exists A2 DcM DcM2 DsM' DcM') + %<- (hsubst_r_exists A2 DaR DcM2 DsumR') + %<- (hsubst_m_exists_app A2 DaR DsumR' DsM' DcM' DsumApp) %. +%term _ + %pi (hsubst_r_exists A2 ([x] [dx] atom_fst (DcR x dx)) DcM2 DsumFst) + %<- (hsubst_r_exists A2 DcR DcM2 DsumR) + %<- (hsubst_r_exists_fst DsumR DsumFst) %. +%term _ + %pi (hsubst_r_exists A2 ([x] [dx] atom_snd (DcR x dx)) DcM2 DsumSnd) + %<- (hsubst_r_exists A2 DcR DcM2 DsumR) + %<- (hsubst_r_exists_snd DsumR DsumSnd) %. +%% app factoring lemma +%term _ hsubst_m_exists_app A2 _ (hsubst_r_exists_sum_r DaR1 DsR1) DsM3 DcM3 (hsubst_r_exists_sum_r (atom_app DcM3 DaR1) (hsubst_r_app DsM3 DsR1)) %. +%term _ + %pi (hsubst_m_exists_app A2 _ (hsubst_r_exists_sum_rr (canon_lam DcM4') DsR1) DsM3 DcM3 (hsubst_r_exists_sum_rr DcM4'' (hsubst_rr_app DsM4' DsM3 DsR1))) + %<- (hsubst_rr_size A2 (arrow A3 A4) DsR1) + %<- (hsubst_m_exists A3 DcM4' DcM3 DsM4' DcM4'') %. +%worlds (atom_block) (hsubst_m_exists_app _ _ _ _ _ _) (hsubst_r_exists _ _ _ _) (hsubst_m_exists _ _ _ _ _) %. +%total {(A2 A2' A2'') (D D' D'')} (hsubst_m_exists A2' D' _ _ _) (hsubst_m_exists_app A2'' D'' _ _ _ _) (hsubst_r_exists A2 D _ _) %. +%sort id/mtm %. +%term id/mtm_refl id/mtm M M %. +%sort id/rtm %. +%term id/rtm_refl id/rtm R R %. +%sort id/tp %. +%term id/tp_refl id/tp A A %. +%sort id/rtm_sym %. +%mode id/rtm_sym %in %out %. +%term _ id/rtm_sym id/rtm_refl id/rtm_refl %. +%worlds (rtm_block) (id/rtm_sym _ _) %. +%total {} (id/rtm_sym _ _) %. +%sort id/tp_arrow_inv %. +%mode id/tp_arrow_inv %in %out %out %. +%term _ id/tp_arrow_inv _ id/tp_refl id/tp_refl %. +%worlds (rtm_block) (id/tp_arrow_inv _ _ _) %. +%total {} (id/tp_arrow_inv _ _ _) %. +%sort id/tp_prod_inv %. +%mode id/tp_prod_inv %in %out %out %. +%term _ id/tp_prod_inv _ id/tp_refl id/tp_refl %. +%worlds (rtm_block) (id/tp_prod_inv _ _ _) %. +%total {} (id/tp_prod_inv _ _ _) %. +%sort id/rtm_app_cong %. +%mode id/rtm_app_cong %in %in %out %. +%term _ id/rtm_app_cong _ _ id/rtm_refl %. +%worlds (rtm_block) (id/rtm_app_cong _ _ _) %. +%total {} (id/rtm_app_cong _ _ _) %. +%sort id/rtm_fst_cong %. +%mode id/rtm_fst_cong %in %out %. +%term _ id/rtm_fst_cong _ id/rtm_refl %. +%worlds (rtm_block) (id/rtm_fst_cong _ _) %. +%total {} (id/rtm_fst_cong _ _) %. +%sort id/rtm_snd_cong %. +%mode id/rtm_snd_cong %in %out %. +%term _ id/rtm_snd_cong _ id/rtm_refl %. +%worlds (rtm_block) (id/rtm_snd_cong _ _) %. +%total {} (id/rtm_snd_cong _ _) %. +%sort id/mtm_lam_cong {x rtm} %. +%mode id/mtm_lam_cong %in %out %. +%term _ id/mtm_lam_cong _ id/mtm_refl %. +%worlds (rtm_block) (id/mtm_lam_cong _ _) %. +%total {} (id/mtm_lam_cong _ _) %. +%sort id/mtm_lam_inv %. +%mode id/mtm_lam_inv %in %out %. +%term _ id/mtm_lam_inv _ ([_] id/mtm_refl) %. +%worlds (rtm_block) (id/mtm_lam_inv _ _) %. +%total {} (id/mtm_lam_inv _ _) %. +%sort id/mtm_pair_cong %. +%mode id/mtm_pair_cong %in %in %out %. +%term _ id/mtm_pair_cong _ _ id/mtm_refl %. +%worlds (rtm_block) (id/mtm_pair_cong _ _ _) %. +%total {} (id/mtm_pair_cong _ _ _) %. +%sort id/mtm_pair_inv %. +%mode id/mtm_pair_inv %in %out %out %. +%term _ id/mtm_pair_inv _ id/mtm_refl id/mtm_refl %. +%worlds (rtm_block) (id/mtm_pair_inv _ _ _) %. +%total {} (id/mtm_pair_inv _ _ _) %. +%sort expand_respects_id %. +%mode expand_respects_id %in %in %in %out %. +%term _ expand_respects_id _ _ D D %. +%worlds (rtm_block) (expand_respects_id _ _ _ _) %. +%total {} (expand_respects_id _ _ _ _) %. +%reduces = D1 D2 (expand_respects_id _ _ D2 D1) %. +%sort canon_respects_id %. +%mode canon_respects_id %in %in %in %out %. +%term _ canon_respects_id _ _ D D %. +%worlds (atom_block) (canon_respects_id _ _ _ _) %. +%total {} (canon_respects_id _ _ _ _) %. +%reduces = D1 D2 (canon_respects_id _ _ D2 D1) %. +%sort hsubst_m_respects_id %. +%mode hsubst_m_respects_id %in %in %in %in %out %. +%term _ hsubst_m_respects_id _ _ _ D D %. +%worlds (rtm_block) (hsubst_m_respects_id _ _ _ _ _) %. +%total {} (hsubst_m_respects_id _ _ _ _ _) %. +%reduces = D1 D2 (hsubst_m_respects_id _ _ _ D2 D1) %. +%sort id/mtm_asm_cong %. +%mode id/mtm_asm_cong %in %out %. +%term _ id/mtm_asm_cong _ id/mtm_refl %. +%worlds (rtm_block) (id/mtm_asm_cong _ _) %. +%total {} (id/mtm_asm_cong _ _) %. +%sort false %. +%freeze false %. +%sort hsubst_rr_closed_contra %. +%mode hsubst_rr_closed_contra %in %out %. +%term _ + %pi (hsubst_rr_closed_contra (hsubst_rr_app _ _ Dr) X) + %<- (hsubst_rr_closed_contra Dr X) %. +%term _ + %pi (hsubst_rr_closed_contra (hsubst_rr_fst D) X) + %<- (hsubst_rr_closed_contra D X) %. +%term _ + %pi (hsubst_rr_closed_contra (hsubst_rr_snd D) X) + %<- (hsubst_rr_closed_contra D X) %. +%worlds (rtm_block) (hsubst_rr_closed_contra _ _) %. +%total D (hsubst_rr_closed_contra D _) %. +%% contradiction of root and non-root +%sort hsubst_r_rr_contra %. +%mode hsubst_r_rr_contra %in %in %out %. +%term _ + %pi (hsubst_r_rr_contra hsubst_r_closed Drr X) + %<- (hsubst_rr_closed_contra Drr X) %. +%term _ + %pi (hsubst_r_rr_contra (hsubst_r_app _ DsrR) (hsubst_rr_app _ _ DsrrR) X) + %<- (hsubst_r_rr_contra DsrR DsrrR X) %. +%term _ + %pi (hsubst_r_rr_contra (hsubst_r_fst DsrR) (hsubst_rr_fst DsrrR) X) + %<- (hsubst_r_rr_contra DsrR DsrrR X) %. +%term _ + %pi (hsubst_r_rr_contra (hsubst_r_snd DsrR) (hsubst_rr_snd DsrrR) X) + %<- (hsubst_r_rr_contra DsrR DsrrR X) %. +%worlds (rtm_block) (hsubst_r_rr_contra _ _ _) %. +%% bar in extra block for lemma below's future use +%total (D) (hsubst_r_rr_contra D _ _) %. +%% false implies id +%sort false_implies_id/mtm {M} {M'} %. +%mode false_implies_id/mtm %in %in %in %out %. +%worlds (rtm_block) (false_implies_id/mtm _ _ _ _) %. +%% bar in extra block for lemma below's future use +%total {} (false_implies_id/mtm _ _ _ _) %. +%sort hsubst_r_vacuous_id %. +%mode hsubst_r_vacuous_id %in %out %. +%sort hsubst_m_vacuous_id %. +%mode hsubst_m_vacuous_id %in %out %. +%% R +%term _ hsubst_r_vacuous_id hsubst_r_closed id/rtm_refl %. +%term _ + %pi (hsubst_r_vacuous_id (hsubst_r_app Dm Dr) Did') + %<- (hsubst_r_vacuous_id Dr DidR) + %<- (hsubst_m_vacuous_id Dm DidM) + %<- (id/rtm_app_cong DidR DidM Did') %. +%term _ + %pi (hsubst_r_vacuous_id (hsubst_r_fst D) Did') + %<- (hsubst_r_vacuous_id D Did) + %<- (id/rtm_fst_cong Did Did') %. +%term _ + %pi (hsubst_r_vacuous_id (hsubst_r_snd D) Did') + %<- (hsubst_r_vacuous_id D Did) + %<- (id/rtm_snd_cong Did Did') %. +%% M +%term _ + %pi (hsubst_m_vacuous_id (hsubst_m_r Dr) Did') + %<- (hsubst_r_vacuous_id Dr Did) + %<- (id/mtm_asm_cong Did Did') %. +%term _ + %pi (hsubst_m_vacuous_id (hsubst_m_rr D) Did) + %<- (hsubst_rr_closed_contra D X) + %<- (false_implies_id/mtm _ _ X Did) %. +%term _ + %pi (hsubst_m_vacuous_id (hsubst_m_lam D) Did') + %<- ({x} hsubst_m_vacuous_id (D x) (Did x)) + %<- (id/mtm_lam_cong Did Did') %. +%term _ + %pi (hsubst_m_vacuous_id (hsubst_m_pair D2 D1) Did') + %<- (hsubst_m_vacuous_id D1 Did1) + %<- (hsubst_m_vacuous_id D2 Did2) + %<- (id/mtm_pair_cong Did1 Did2 Did') %. +%worlds (rtm_block) (hsubst_r_vacuous_id _ _) (hsubst_m_vacuous_id _ _) %. +%total (D1 D2) (hsubst_r_vacuous_id D1 _) (hsubst_m_vacuous_id D2 _) %. +%sort hsubst_m_unique %. +%mode hsubst_m_unique %in %in %out %. +%sort hsubst_r_unique %. +%mode hsubst_r_unique %in %in %out %. +%sort hsubst_rr_unique %. +%mode hsubst_rr_unique %in %in %out %out %. +%% M +%term _ + %pi (hsubst_m_unique (hsubst_m_r Dsr) (hsubst_m_r Dsr') DidAsm) + %<- (hsubst_r_unique Dsr Dsr' DidR) + %<- (id/mtm_asm_cong DidR DidAsm) %. +%term _ + %pi (hsubst_m_unique (hsubst_m_rr Dsr) (hsubst_m_rr Dsr') Did) + %<- (hsubst_rr_unique Dsr Dsr' Did _) %. +%% contradict mismatch +%term _ + %pi (hsubst_m_unique (hsubst_m_r Dsr) (hsubst_m_rr Dsrr) Did) + %<- (hsubst_r_rr_contra Dsr Dsrr X) + %<- (false_implies_id/mtm _ _ X Did) %. +%% contradict mismatch +%term _ + %pi (hsubst_m_unique (hsubst_m_rr Dsrr) (hsubst_m_r Dsr) Did) + %<- (hsubst_r_rr_contra Dsr Dsrr X) + %<- (false_implies_id/mtm _ _ X Did) %. +%term _ + %pi (hsubst_m_unique (hsubst_m_pair DsRight DsLeft) (hsubst_m_pair DsRight' DsLeft') DidPair) + %<- (hsubst_m_unique DsLeft DsLeft' DidLeft) + %<- (hsubst_m_unique DsRight DsRight' DidRight) + %<- (id/mtm_pair_cong DidLeft DidRight DidPair) %. +%term _ + %pi (hsubst_m_unique (hsubst_m_lam Ds) (hsubst_m_lam Ds') DidLam) + %<- ({y rtm} hsubst_m_unique (Ds y) (Ds' y) (Did y)) + %<- (id/mtm_lam_cong Did DidLam) %. +%% R non-root +%term _ hsubst_r_unique D D id/rtm_refl %. +%term _ + %pi (hsubst_r_unique (%the (hsubst_r M2 A2 ([_] R) R) hsubst_r_closed) (%the (hsubst_r M2 A2 ([_] R) R') D) Did') + %<- (hsubst_r_vacuous_id D Did) + %<- (id/rtm_sym Did Did') %. +%term _ + %pi (hsubst_r_unique (%the (hsubst_r M2 A2 ([_] R) R') D) (%the (hsubst_r M2 A2 ([_] R) R) hsubst_r_closed) Did) + %<- (hsubst_r_vacuous_id D Did) %. +%term _ + %pi (hsubst_r_unique (hsubst_r_app DsM' DsR') (hsubst_r_app DsM'' DsR'') DidApp) + %<- (hsubst_r_unique DsR' DsR'' DidR) + %<- (hsubst_m_unique DsM' DsM'' DidM) + %<- (id/rtm_app_cong DidR DidM DidApp) %. +%term _ + %pi (hsubst_r_unique (hsubst_r_fst DsR') (hsubst_r_fst DsR'') DidFst) + %<- (hsubst_r_unique DsR' DsR'' DidR) + %<- (id/rtm_fst_cong DidR DidFst) %. +%term _ + %pi (hsubst_r_unique (hsubst_r_snd DsR') (hsubst_r_snd DsR'') DidSnd) + %<- (hsubst_r_unique DsR' DsR'' DidR) + %<- (id/rtm_snd_cong DidR DidSnd) %. +%% R root +%term _ hsubst_rr_unique hsubst_rr_var hsubst_rr_var id/mtm_refl id/tp_refl %. +%term _ + %pi (hsubst_rr_unique (hsubst_rr_app DsM4' DsM3 DsR1) (hsubst_rr_app DsM4'-2 DsM3-2 DsR1-2) DidM4'' DidA4) + %<- (hsubst_rr_unique DsR1 DsR1-2 DidLam DidArrow) + %<- (hsubst_m_unique DsM3 DsM3-2 DidM3') + %<- (id/tp_arrow_inv DidArrow DidA3 DidA4) + %<- (id/mtm_lam_inv DidLam DidM4') + %<- (hsubst_m_respects_id DidM3' DidA3 DidM4' DsM4' DsM4'-1) + %<- (hsubst_m_unique DsM4'-1 DsM4'-2 DidM4'') %. +%term _ + %pi (hsubst_rr_unique (hsubst_rr_fst DsR') (hsubst_rr_fst DsR'') DidMl DidAl) + %<- (hsubst_rr_unique DsR' DsR'' DidPair DidProd) + %<- (id/tp_prod_inv DidProd DidAl _) + %<- (id/mtm_pair_inv DidPair DidMl _) %. +%term _ + %pi (hsubst_rr_unique (hsubst_rr_snd DsR') (hsubst_rr_snd DsR'') DidMr DidAr) + %<- (hsubst_rr_unique DsR' DsR'' DidPair DidProd) + %<- (id/tp_prod_inv DidProd _ DidAr) + %<- (id/mtm_pair_inv DidPair _ DidMr) %. +%worlds (rtm_block) (hsubst_m_unique _ _ _) (hsubst_r_unique _ _ _) (hsubst_rr_unique _ _ _ _) %. +%total (D1 D2 D3) (hsubst_r_unique D2 _ _) (hsubst_rr_unique D3 _ _ _) (hsubst_m_unique D1 _ _) %. +%sort expand_exists {A} %. +%mode expand_exists %in %in %out %out %. +%term _ expand_exists b D expand_b (canon_asm D) %. +%term _ + %pi (expand_exists (arrow A2 A) DaR (expand_arrow DeApp DeX) (canon_lam DcApp)) + %<- ({x rtm} {dx atom x A2} expand_exists A2 dx (DeX x) (DcM x dx)) + %<- ({x rtm} {dx atom x A2} expand_exists A (atom_app (DcM x dx) DaR) (DeApp x) (DcApp x dx)) %. +%term _ + %pi (expand_exists (prod A1 A2) DaR (expand_prod DeS DeF) (canon_pair DcS DcF)) + %<- (expand_exists A1 (atom_fst DaR) DeF DcF) + %<- (expand_exists A2 (atom_snd DaR) DeS DcS) %. +%worlds (atom_block) (expand_exists _ _ _ _) %. +%total D (expand_exists D _ _ _) %. +%sort expand_unique %. +%mode expand_unique %in %in %out %. +%term _ expand_unique expand_b expand_b id/mtm_refl %. +%term _ + %pi (expand_unique (expand_arrow (%the ({x rtm} expand A (app R (Mx x)) (MApp x)) DeApp) (%the ({x rtm} expand A2 x (Mx x)) DeX)) (expand_arrow (%the ({x rtm} expand A (app R (Mx' x)) (MApp' x)) DeApp') (%the ({x rtm} expand A2 x (Mx' x)) DeX')) DidLam) + %<- ({x rtm} expand_unique (DeX x) (DeX' x) (DidX x)) + %<- ({x rtm} id/rtm_app_cong (%the (id/rtm R R) id/rtm_refl) (DidX x) (DidApp x)) + %<- ({x rtm} expand_respects_id (DidApp x) id/mtm_refl (DeApp x) (DeApp-2 x)) + %<- ({x rtm} expand_unique (DeApp-2 x) (DeApp' x) (DidAppExp x)) + %<- (id/mtm_lam_cong DidAppExp DidLam) %. +%term _ + %pi (expand_unique (expand_prod DeS DeF) (expand_prod DeS' DeF') DidPair) + %<- (expand_unique DeF DeF' DidF) + %<- (expand_unique DeS DeS' DidS) + %<- (id/mtm_pair_cong DidF DidS DidPair) %. +%worlds (rtm_block) (expand_unique _ _ _) %. +%total D (expand_unique _ D _) %. +%sort tp %. +%term b tp %. +%term arrow %pi tp %-> tp %-> tp %. +%term prod %pi tp %-> tp %-> tp %. +%sort rtm %. +%sort mtm %. +%term app %pi rtm %-> mtm %-> rtm %. +%term c rtm %. +%term fst %pi rtm %-> rtm %. +%term snd %pi rtm %-> rtm %. +%term asm %pi rtm %-> mtm %. +%term lam %pi (%pi rtm %-> mtm) %-> mtm %. +%term pair %pi mtm %-> mtm %-> mtm %. +%block rtm_block [x rtm]%. +%worlds (rtm_block) (tp) (rtm) (mtm) %. +%sort atom %. +%mode atom %in %out %. +%sort canon %. +%mode canon %in %in %. +% atomic +%term atom_c atom c b %. +%term atom_app %pi (atom (app R1 M2) A) %<- (atom R1 (arrow A2 A)) %<- (canon M2 A2) %. +%term atom_fst %pi (atom (fst R) A1) %<- (atom R (prod A1 A2)) %. +%term atom_snd %pi (atom (snd R) A2) %<- (atom R (prod A1 A2)) %. +%% canonical +%term canon_asm %pi (canon (asm R) b) %<- (atom R b) %. +%term canon_lam + %pi (canon (lam M) (arrow A2 A)) + %<- ({x rtm} %pi (atom x A2) %-> (canon (M x) A)) %. +%term canon_pair %pi (canon (pair M1 M2) (prod A1 A2)) %<- (canon M1 A1) %<- (canon M2 A2) %. +%block atom_block {A tp} [x rtm] [dx atom x A]%. +%worlds (atom_block) (atom _ _) (canon _ _) %. +%terminates (R M) (atom R _) (canon M _) %. +%sort hsubst_m %. +%mode hsubst_m %in %in %in %out %. +%sort hsubst_r %. +%mode hsubst_r %in %in %in %out %. +%sort hsubst_rr %. +%mode hsubst_rr %in %in %in %out %out %. +%% Ms +%term hsubst_m_r %pi (hsubst_m M0 A0 ([x] asm (R x)) (asm R')) %<- (hsubst_r M0 A0 R R') %. +%term hsubst_m_rr %pi (hsubst_m M0 A0 ([x] asm (R x)) M') %<- (hsubst_rr M0 A0 R M' _) %. +%term hsubst_m_lam + %pi (hsubst_m M0 A0 ([x] lam ([y] M x y)) (lam M')) + %<- ({y rtm} hsubst_m M0 A0 ([x] M x y) (M' y)) %. +%term hsubst_m_pair + %pi (hsubst_m M0 A0 ([x] pair (M1 x) (M2 x)) (pair M1' M2')) + %<- (hsubst_m M0 A0 M1 M1') + %<- (hsubst_m M0 A0 M2 M2') %. +%% R not head +%term hsubst_r_closed hsubst_r M0 A0 ([x] E) E %. +%term hsubst_r_app + %pi (hsubst_r M0 A0 ([x] app (R x) (M x)) (app R' M')) + %<- (hsubst_r M0 A0 R R') + %<- (hsubst_m M0 A0 M M') %. +%term hsubst_r_fst %pi (hsubst_r M0 A0 ([x] fst (R x)) (fst R')) %<- (hsubst_r M0 A0 R R') %. +%term hsubst_r_snd %pi (hsubst_r M0 A0 ([x] snd (R x)) (snd R')) %<- (hsubst_r M0 A0 R R') %. +%% r head +%term hsubst_rr_var hsubst_rr M0 A0 ([x] x) M0 A0 %. +%term hsubst_rr_app + %pi (hsubst_rr M0 A0 ([x] app (R1 x) (M2 x)) M'' A) + %<- (hsubst_rr M0 A0 R1 (lam M') (arrow A2 A)) + %<- (hsubst_m M0 A0 M2 M2') + %<- (hsubst_m M2' A2 M' M'') %. +%term hsubst_rr_fst + %pi (hsubst_rr M0 A0 ([x] fst (R x)) M1' A1') + %<- (hsubst_rr M0 A0 R (pair M1' M2') (prod A1' A2')) %. +%term hsubst_rr_snd + %pi (hsubst_rr M0 A0 ([x] snd (R x)) M2' A2') + %<- (hsubst_rr M0 A0 R (pair M1' M2') (prod A1' A2')) %. +%worlds (rtm_block) (hsubst_m _ _ _ _) (hsubst_r _ _ _ _) (hsubst_rr _ _ _ _ _) %. +%reduces <= A' A0 (hsubst_rr _ A0 _ _ A') %. +%terminates {(A0 A0' A0'') (M R R')} (hsubst_m _ A0 M _) (hsubst_r _ A0' R _) (hsubst_rr _ A0'' R' _ _) %. +%sort expand %. +%mode expand %in %in %out %. +%term expand_b expand b R (asm R) %. +%term expand_arrow + %pi (expand (arrow A2 A) R (lam M)) + %<- ({x rtm} expand A2 x (M2 x)) + %<- ({x rtm} expand A (app R (M2 x)) (M x)) %. +%term expand_prod + %pi (expand (prod A1 A2) R (pair M1 M2)) + %<- (expand A1 (fst R) M1) + %<- (expand A2 (snd R) M2) %. +%worlds (rtm_block) (expand _ _ _) %. +%total A (expand A _ _) %. +%sort hsubst_rr_size {A2} {A'} %. +%mode hsubst_rr_size %in %in %in %. +%term _ hsubst_rr_size A2 A2 hsubst_rr_var %. +%term _ + %pi (hsubst_rr_size A2 A4 (hsubst_rr_app _ _ Drr)) + %<- (hsubst_rr_size A2 (arrow A3 A4) Drr) %. +%term _ + %pi (hsubst_rr_size A2 Al (hsubst_rr_fst Drr)) + %<- (hsubst_rr_size A2 (prod Al Ar) Drr) %. +%term _ + %pi (hsubst_rr_size A2 Ar (hsubst_rr_snd Drr)) + %<- (hsubst_rr_size A2 (prod Al Ar) Drr) %. +%worlds (rtm_block) (hsubst_rr_size _ _ _) %. +%total D (hsubst_rr_size _ _ D) %. +%reduces <= A' A2 (hsubst_rr_size A2 A' _) %. +%sort hsubst_r_exists_sum %. +%term hsubst_r_exists_sum_r %pi (hsubst_r_exists_sum M2 A2 R A) %<- (hsubst_r M2 A2 R R') %<- (atom R' A) %. +%term hsubst_r_exists_sum_rr + %pi (hsubst_r_exists_sum M2 A2 R A') + %<- (hsubst_rr M2 A2 R M' A') + %<- (canon M' A') %. +%sort hsubst_m_exists_asm %. +%mode hsubst_m_exists_asm %in %out %out %. +%term _ hsubst_m_exists_asm (hsubst_r_exists_sum_r DaR' DsR) (hsubst_m_r DsR) (canon_asm DaR') %. +%term _ hsubst_m_exists_asm (hsubst_r_exists_sum_rr DcM' DsR) (hsubst_m_rr DsR) DcM' %. +%worlds (atom_block) (hsubst_m_exists_asm _ _ _) %. +%total {} (hsubst_m_exists_asm _ _ _) %. +%sort hsubst_r_exists_fst %. +%mode hsubst_r_exists_fst %in %out %. +%term _ hsubst_r_exists_fst (hsubst_r_exists_sum_r DaR' DsR) (hsubst_r_exists_sum_r (atom_fst DaR') (hsubst_r_fst DsR)) %. +%term _ hsubst_r_exists_fst (hsubst_r_exists_sum_rr (canon_pair _ DcMl') DsR) (hsubst_r_exists_sum_rr DcMl' (hsubst_rr_fst DsR)) %. +%worlds (atom_block) (hsubst_r_exists_fst _ _) %. +%total {} (hsubst_r_exists_fst _ _) %. +%sort hsubst_r_exists_snd %. +%mode hsubst_r_exists_snd %in %out %. +%term _ hsubst_r_exists_snd (hsubst_r_exists_sum_r DaR' DsR) (hsubst_r_exists_sum_r (atom_snd DaR') (hsubst_r_snd DsR)) %. +%term _ hsubst_r_exists_snd (hsubst_r_exists_sum_rr (canon_pair DcMr' _) DsR) (hsubst_r_exists_sum_rr DcMr' (hsubst_rr_snd DsR)) %. +%worlds (atom_block) (hsubst_r_exists_snd _ _) %. +%total {} (hsubst_r_exists_snd _ _) %. +%sort hsubst_m_exists {A2} {y rtm} {dy atom y A2} %. +%mode hsubst_m_exists %in %in %in %out %out %. +%sort hsubst_r_exists {A2} {y rtm} {dx atom y A2} %. +%mode hsubst_r_exists %in %in %in %out %. +%sort hsubst_m_exists_app {A2} {y rtm} {dx atom y A2} %. +%mode hsubst_m_exists_app %in %in %in %in %in %out %. +%% canonical terms +%term _ + %pi (hsubst_m_exists A2 ([x] [dx] canon_asm (DaR x dx)) DcM2 DsR DcM') + %<- (hsubst_r_exists A2 DaR DcM2 Dsum) + %<- (hsubst_m_exists_asm Dsum DsR DcM') %. +%term _ + %pi (hsubst_m_exists A2 ([x] [dx] %the (canon (lam (M x)) (arrow Af At)) (canon_lam ([y] [dy] DcM x dx y dy))) (%the (canon M2 A2) DcM2) (hsubst_m_lam DsM) (canon_lam DcM')) + %<- ({y} {dy atom y Af} hsubst_m_exists A2 ([x] [dx] DcM x dx y dy) DcM2 (DsM y) (DcM' y dy)) %. +%term _ + %pi (hsubst_m_exists A2 ([x] [dx] canon_pair (DcMr x dx) (DcMl x dx)) DcM2 (hsubst_m_pair DsMr DsMl) (canon_pair DcMr' DcMl')) + %<- (hsubst_m_exists A2 DcMl DcM2 DsMl DcMl') + %<- (hsubst_m_exists A2 DcMr DcM2 DsMr DcMr') %. +%% atomic terms +%term _ hsubst_r_exists A2 ([x] [dx] D) _ (hsubst_r_exists_sum_r D hsubst_r_closed) %. +%term _ hsubst_r_exists A2 ([x] [dx] dx) DcM2 (hsubst_r_exists_sum_rr DcM2 hsubst_rr_var) %. +%term _ + %pi (hsubst_r_exists A2 ([x] [dx] atom_app (DcM x dx) (DaR x dx)) DcM2 DsumApp) + %<- (hsubst_m_exists A2 DcM DcM2 DsM' DcM') + %<- (hsubst_r_exists A2 DaR DcM2 DsumR') + %<- (hsubst_m_exists_app A2 DaR DsumR' DsM' DcM' DsumApp) %. +%term _ + %pi (hsubst_r_exists A2 ([x] [dx] atom_fst (DcR x dx)) DcM2 DsumFst) + %<- (hsubst_r_exists A2 DcR DcM2 DsumR) + %<- (hsubst_r_exists_fst DsumR DsumFst) %. +%term _ + %pi (hsubst_r_exists A2 ([x] [dx] atom_snd (DcR x dx)) DcM2 DsumSnd) + %<- (hsubst_r_exists A2 DcR DcM2 DsumR) + %<- (hsubst_r_exists_snd DsumR DsumSnd) %. +%% app factoring lemma +%term _ hsubst_m_exists_app A2 _ (hsubst_r_exists_sum_r DaR1 DsR1) DsM3 DcM3 (hsubst_r_exists_sum_r (atom_app DcM3 DaR1) (hsubst_r_app DsM3 DsR1)) %. +%term _ + %pi (hsubst_m_exists_app A2 _ (hsubst_r_exists_sum_rr (canon_lam DcM4') DsR1) DsM3 DcM3 (hsubst_r_exists_sum_rr DcM4'' (hsubst_rr_app DsM4' DsM3 DsR1))) + %<- (hsubst_rr_size A2 (arrow A3 A4) DsR1) + %<- (hsubst_m_exists A3 DcM4' DcM3 DsM4' DcM4'') %. +%worlds (atom_block) (hsubst_m_exists_app _ _ _ _ _ _) (hsubst_r_exists _ _ _ _) (hsubst_m_exists _ _ _ _ _) %. +%total {(A2 A2' A2'') (D D' D'')} (hsubst_m_exists A2' D' _ _ _) (hsubst_m_exists_app A2'' D'' _ _ _ _) (hsubst_r_exists A2 D _ _) %. +%sort id/mtm %. +%term id/mtm_refl id/mtm M M %. +%sort id/rtm %. +%term id/rtm_refl id/rtm R R %. +%sort id/tp %. +%term id/tp_refl id/tp A A %. +%sort id/rtm_sym %. +%mode id/rtm_sym %in %out %. +%term _ id/rtm_sym id/rtm_refl id/rtm_refl %. +%worlds (rtm_block) (id/rtm_sym _ _) %. +%total {} (id/rtm_sym _ _) %. +%sort id/tp_arrow_inv %. +%mode id/tp_arrow_inv %in %out %out %. +%term _ id/tp_arrow_inv _ id/tp_refl id/tp_refl %. +%worlds (rtm_block) (id/tp_arrow_inv _ _ _) %. +%total {} (id/tp_arrow_inv _ _ _) %. +%sort id/tp_prod_inv %. +%mode id/tp_prod_inv %in %out %out %. +%term _ id/tp_prod_inv _ id/tp_refl id/tp_refl %. +%worlds (rtm_block) (id/tp_prod_inv _ _ _) %. +%total {} (id/tp_prod_inv _ _ _) %. +%sort id/rtm_app_cong %. +%mode id/rtm_app_cong %in %in %out %. +%term _ id/rtm_app_cong _ _ id/rtm_refl %. +%worlds (rtm_block) (id/rtm_app_cong _ _ _) %. +%total {} (id/rtm_app_cong _ _ _) %. +%sort id/rtm_fst_cong %. +%mode id/rtm_fst_cong %in %out %. +%term _ id/rtm_fst_cong _ id/rtm_refl %. +%worlds (rtm_block) (id/rtm_fst_cong _ _) %. +%total {} (id/rtm_fst_cong _ _) %. +%sort id/rtm_snd_cong %. +%mode id/rtm_snd_cong %in %out %. +%term _ id/rtm_snd_cong _ id/rtm_refl %. +%worlds (rtm_block) (id/rtm_snd_cong _ _) %. +%total {} (id/rtm_snd_cong _ _) %. +%sort id/mtm_lam_cong {x rtm} %. +%mode id/mtm_lam_cong %in %out %. +%term _ id/mtm_lam_cong _ id/mtm_refl %. +%worlds (rtm_block) (id/mtm_lam_cong _ _) %. +%total {} (id/mtm_lam_cong _ _) %. +%sort id/mtm_lam_inv %. +%mode id/mtm_lam_inv %in %out %. +%term _ id/mtm_lam_inv _ ([_] id/mtm_refl) %. +%worlds (rtm_block) (id/mtm_lam_inv _ _) %. +%total {} (id/mtm_lam_inv _ _) %. +%sort id/mtm_pair_cong %. +%mode id/mtm_pair_cong %in %in %out %. +%term _ id/mtm_pair_cong _ _ id/mtm_refl %. +%worlds (rtm_block) (id/mtm_pair_cong _ _ _) %. +%total {} (id/mtm_pair_cong _ _ _) %. +%sort id/mtm_pair_inv %. +%mode id/mtm_pair_inv %in %out %out %. +%term _ id/mtm_pair_inv _ id/mtm_refl id/mtm_refl %. +%worlds (rtm_block) (id/mtm_pair_inv _ _ _) %. +%total {} (id/mtm_pair_inv _ _ _) %. +%sort expand_respects_id %. +%mode expand_respects_id %in %in %in %out %. +%term _ expand_respects_id _ _ D D %. +%worlds (rtm_block) (expand_respects_id _ _ _ _) %. +%total {} (expand_respects_id _ _ _ _) %. +%reduces = D1 D2 (expand_respects_id _ _ D2 D1) %. +%sort canon_respects_id %. +%mode canon_respects_id %in %in %in %out %. +%term _ canon_respects_id _ _ D D %. +%worlds (atom_block) (canon_respects_id _ _ _ _) %. +%total {} (canon_respects_id _ _ _ _) %. +%reduces = D1 D2 (canon_respects_id _ _ D2 D1) %. +%sort hsubst_m_respects_id %. +%mode hsubst_m_respects_id %in %in %in %in %out %. +%term _ hsubst_m_respects_id _ _ _ D D %. +%worlds (rtm_block) (hsubst_m_respects_id _ _ _ _ _) %. +%total {} (hsubst_m_respects_id _ _ _ _ _) %. +%reduces = D1 D2 (hsubst_m_respects_id _ _ _ D2 D1) %. +%sort id/mtm_asm_cong %. +%mode id/mtm_asm_cong %in %out %. +%term _ id/mtm_asm_cong _ id/mtm_refl %. +%worlds (rtm_block) (id/mtm_asm_cong _ _) %. +%total {} (id/mtm_asm_cong _ _) %. +%sort false %. +%freeze false %. +%sort hsubst_rr_closed_contra %. +%mode hsubst_rr_closed_contra %in %out %. +%term _ + %pi (hsubst_rr_closed_contra (hsubst_rr_app _ _ Dr) X) + %<- (hsubst_rr_closed_contra Dr X) %. +%term _ + %pi (hsubst_rr_closed_contra (hsubst_rr_fst D) X) + %<- (hsubst_rr_closed_contra D X) %. +%term _ + %pi (hsubst_rr_closed_contra (hsubst_rr_snd D) X) + %<- (hsubst_rr_closed_contra D X) %. +%worlds (rtm_block) (hsubst_rr_closed_contra _ _) %. +%total D (hsubst_rr_closed_contra D _) %. +%% contradiction of root and non-root +%sort hsubst_r_rr_contra %. +%mode hsubst_r_rr_contra %in %in %out %. +%term _ + %pi (hsubst_r_rr_contra hsubst_r_closed Drr X) + %<- (hsubst_rr_closed_contra Drr X) %. +%term _ + %pi (hsubst_r_rr_contra (hsubst_r_app _ DsrR) (hsubst_rr_app _ _ DsrrR) X) + %<- (hsubst_r_rr_contra DsrR DsrrR X) %. +%term _ + %pi (hsubst_r_rr_contra (hsubst_r_fst DsrR) (hsubst_rr_fst DsrrR) X) + %<- (hsubst_r_rr_contra DsrR DsrrR X) %. +%term _ + %pi (hsubst_r_rr_contra (hsubst_r_snd DsrR) (hsubst_rr_snd DsrrR) X) + %<- (hsubst_r_rr_contra DsrR DsrrR X) %. +%worlds (rtm_block) (hsubst_r_rr_contra _ _ _) %. +%% bar in extra block for lemma below's future use +%total (D) (hsubst_r_rr_contra D _ _) %. +%% false implies id +%sort false_implies_id/mtm {M} {M'} %. +%mode false_implies_id/mtm %in %in %in %out %. +%worlds (rtm_block) (false_implies_id/mtm _ _ _ _) %. +%% bar in extra block for lemma below's future use +%total {} (false_implies_id/mtm _ _ _ _) %. +%sort hsubst_r_vacuous_id %. +%mode hsubst_r_vacuous_id %in %out %. +%sort hsubst_m_vacuous_id %. +%mode hsubst_m_vacuous_id %in %out %. +%% R +%term _ hsubst_r_vacuous_id hsubst_r_closed id/rtm_refl %. +%term _ + %pi (hsubst_r_vacuous_id (hsubst_r_app Dm Dr) Did') + %<- (hsubst_r_vacuous_id Dr DidR) + %<- (hsubst_m_vacuous_id Dm DidM) + %<- (id/rtm_app_cong DidR DidM Did') %. +%term _ + %pi (hsubst_r_vacuous_id (hsubst_r_fst D) Did') + %<- (hsubst_r_vacuous_id D Did) + %<- (id/rtm_fst_cong Did Did') %. +%term _ + %pi (hsubst_r_vacuous_id (hsubst_r_snd D) Did') + %<- (hsubst_r_vacuous_id D Did) + %<- (id/rtm_snd_cong Did Did') %. +%% M +%term _ + %pi (hsubst_m_vacuous_id (hsubst_m_r Dr) Did') + %<- (hsubst_r_vacuous_id Dr Did) + %<- (id/mtm_asm_cong Did Did') %. +%term _ + %pi (hsubst_m_vacuous_id (hsubst_m_rr D) Did) + %<- (hsubst_rr_closed_contra D X) + %<- (false_implies_id/mtm _ _ X Did) %. +%term _ + %pi (hsubst_m_vacuous_id (hsubst_m_lam D) Did') + %<- ({x} hsubst_m_vacuous_id (D x) (Did x)) + %<- (id/mtm_lam_cong Did Did') %. +%term _ + %pi (hsubst_m_vacuous_id (hsubst_m_pair D2 D1) Did') + %<- (hsubst_m_vacuous_id D1 Did1) + %<- (hsubst_m_vacuous_id D2 Did2) + %<- (id/mtm_pair_cong Did1 Did2 Did') %. +%worlds (rtm_block) (hsubst_r_vacuous_id _ _) (hsubst_m_vacuous_id _ _) %. +%total (D1 D2) (hsubst_r_vacuous_id D1 _) (hsubst_m_vacuous_id D2 _) %. +%sort hsubst_m_unique %. +%mode hsubst_m_unique %in %in %out %. +%sort hsubst_r_unique %. +%mode hsubst_r_unique %in %in %out %. +%sort hsubst_rr_unique %. +%mode hsubst_rr_unique %in %in %out %out %. +%% M +%term _ + %pi (hsubst_m_unique (hsubst_m_r Dsr) (hsubst_m_r Dsr') DidAsm) + %<- (hsubst_r_unique Dsr Dsr' DidR) + %<- (id/mtm_asm_cong DidR DidAsm) %. +%term _ + %pi (hsubst_m_unique (hsubst_m_rr Dsr) (hsubst_m_rr Dsr') Did) + %<- (hsubst_rr_unique Dsr Dsr' Did _) %. +%% contradict mismatch +%term _ + %pi (hsubst_m_unique (hsubst_m_r Dsr) (hsubst_m_rr Dsrr) Did) + %<- (hsubst_r_rr_contra Dsr Dsrr X) + %<- (false_implies_id/mtm _ _ X Did) %. +%% contradict mismatch +%term _ + %pi (hsubst_m_unique (hsubst_m_rr Dsrr) (hsubst_m_r Dsr) Did) + %<- (hsubst_r_rr_contra Dsr Dsrr X) + %<- (false_implies_id/mtm _ _ X Did) %. +%term _ + %pi (hsubst_m_unique (hsubst_m_pair DsRight DsLeft) (hsubst_m_pair DsRight' DsLeft') DidPair) + %<- (hsubst_m_unique DsLeft DsLeft' DidLeft) + %<- (hsubst_m_unique DsRight DsRight' DidRight) + %<- (id/mtm_pair_cong DidLeft DidRight DidPair) %. +%term _ + %pi (hsubst_m_unique (hsubst_m_lam Ds) (hsubst_m_lam Ds') DidLam) + %<- ({y rtm} hsubst_m_unique (Ds y) (Ds' y) (Did y)) + %<- (id/mtm_lam_cong Did DidLam) %. +%% R non-root +%term _ hsubst_r_unique D D id/rtm_refl %. +%term _ + %pi (hsubst_r_unique (%the (hsubst_r M2 A2 ([_] R) R) hsubst_r_closed) (%the (hsubst_r M2 A2 ([_] R) R') D) Did') + %<- (hsubst_r_vacuous_id D Did) + %<- (id/rtm_sym Did Did') %. +%term _ + %pi (hsubst_r_unique (%the (hsubst_r M2 A2 ([_] R) R') D) (%the (hsubst_r M2 A2 ([_] R) R) hsubst_r_closed) Did) + %<- (hsubst_r_vacuous_id D Did) %. +%term _ + %pi (hsubst_r_unique (hsubst_r_app DsM' DsR') (hsubst_r_app DsM'' DsR'') DidApp) + %<- (hsubst_r_unique DsR' DsR'' DidR) + %<- (hsubst_m_unique DsM' DsM'' DidM) + %<- (id/rtm_app_cong DidR DidM DidApp) %. +%term _ + %pi (hsubst_r_unique (hsubst_r_fst DsR') (hsubst_r_fst DsR'') DidFst) + %<- (hsubst_r_unique DsR' DsR'' DidR) + %<- (id/rtm_fst_cong DidR DidFst) %. +%term _ + %pi (hsubst_r_unique (hsubst_r_snd DsR') (hsubst_r_snd DsR'') DidSnd) + %<- (hsubst_r_unique DsR' DsR'' DidR) + %<- (id/rtm_snd_cong DidR DidSnd) %. +%% R root +%term _ hsubst_rr_unique hsubst_rr_var hsubst_rr_var id/mtm_refl id/tp_refl %. +%term _ + %pi (hsubst_rr_unique (hsubst_rr_app DsM4' DsM3 DsR1) (hsubst_rr_app DsM4'-2 DsM3-2 DsR1-2) DidM4'' DidA4) + %<- (hsubst_rr_unique DsR1 DsR1-2 DidLam DidArrow) + %<- (hsubst_m_unique DsM3 DsM3-2 DidM3') + %<- (id/tp_arrow_inv DidArrow DidA3 DidA4) + %<- (id/mtm_lam_inv DidLam DidM4') + %<- (hsubst_m_respects_id DidM3' DidA3 DidM4' DsM4' DsM4'-1) + %<- (hsubst_m_unique DsM4'-1 DsM4'-2 DidM4'') %. +%term _ + %pi (hsubst_rr_unique (hsubst_rr_fst DsR') (hsubst_rr_fst DsR'') DidMl DidAl) + %<- (hsubst_rr_unique DsR' DsR'' DidPair DidProd) + %<- (id/tp_prod_inv DidProd DidAl _) + %<- (id/mtm_pair_inv DidPair DidMl _) %. +%term _ + %pi (hsubst_rr_unique (hsubst_rr_snd DsR') (hsubst_rr_snd DsR'') DidMr DidAr) + %<- (hsubst_rr_unique DsR' DsR'' DidPair DidProd) + %<- (id/tp_prod_inv DidProd _ DidAr) + %<- (id/mtm_pair_inv DidPair _ DidMr) %. +%worlds (rtm_block) (hsubst_m_unique _ _ _) (hsubst_r_unique _ _ _) (hsubst_rr_unique _ _ _ _) %. +%total (D1 D2 D3) (hsubst_r_unique D2 _ _) (hsubst_rr_unique D3 _ _ _) (hsubst_m_unique D1 _ _) %. +%sort expand_exists {A} %. +%mode expand_exists %in %in %out %out %. +%term _ expand_exists b D expand_b (canon_asm D) %. +%term _ + %pi (expand_exists (arrow A2 A) DaR (expand_arrow DeApp DeX) (canon_lam DcApp)) + %<- ({x rtm} {dx atom x A2} expand_exists A2 dx (DeX x) (DcM x dx)) + %<- ({x rtm} {dx atom x A2} expand_exists A (atom_app (DcM x dx) DaR) (DeApp x) (DcApp x dx)) %. +%term _ + %pi (expand_exists (prod A1 A2) DaR (expand_prod DeS DeF) (canon_pair DcS DcF)) + %<- (expand_exists A1 (atom_fst DaR) DeF DcF) + %<- (expand_exists A2 (atom_snd DaR) DeS DcS) %. +%worlds (atom_block) (expand_exists _ _ _ _) %. +%total D (expand_exists D _ _ _) %. +%sort expand_unique %. +%mode expand_unique %in %in %out %. +%term _ expand_unique expand_b expand_b id/mtm_refl %. +%term _ + %pi (expand_unique (expand_arrow (%the ({x rtm} expand A (app R (Mx x)) (MApp x)) DeApp) (%the ({x rtm} expand A2 x (Mx x)) DeX)) (expand_arrow (%the ({x rtm} expand A (app R (Mx' x)) (MApp' x)) DeApp') (%the ({x rtm} expand A2 x (Mx' x)) DeX')) DidLam) + %<- ({x rtm} expand_unique (DeX x) (DeX' x) (DidX x)) + %<- ({x rtm} id/rtm_app_cong (%the (id/rtm R R) id/rtm_refl) (DidX x) (DidApp x)) + %<- ({x rtm} expand_respects_id (DidApp x) id/mtm_refl (DeApp x) (DeApp-2 x)) + %<- ({x rtm} expand_unique (DeApp-2 x) (DeApp' x) (DidAppExp x)) + %<- (id/mtm_lam_cong DidAppExp DidLam) %. +%term _ + %pi (expand_unique (expand_prod DeS DeF) (expand_prod DeS' DeF') DidPair) + %<- (expand_unique DeF DeF' DidF) + %<- (expand_unique DeS DeS' DidS) + %<- (id/mtm_pair_cong DidF DidS DidPair) %. +%worlds (rtm_block) (expand_unique _ _ _) %. +%total D (expand_unique _ D _) %. +%{! ## External language: syntax and judgements + +### Syntax + +The external language syntax is a copy of the internal language, where we additionally include canonincal terms ``M`` into atomic terms ``R``. To preserve the modes of the typing judgements, these canonical terms are annotated with the type that they should be checked against. !}% +%sort el_rtm %. +%sort el_mtm %. +%term el_app %pi el_rtm %-> el_mtm %-> el_rtm %. +%term el_c el_rtm %. +%term el_fst %pi el_rtm %-> el_rtm %. +%term el_snd %pi el_rtm %-> el_rtm %. +%term el_annot %pi el_mtm %-> tp %-> el_rtm %. +%term el_asm %pi el_rtm %-> el_mtm %. +%term el_lam %pi (%pi el_rtm %-> el_mtm) %-> el_mtm %. +%term el_pair %pi el_mtm %-> el_mtm %-> el_mtm %. +%block el_rtm_block [x el_rtm]%. +%{! Modulo the placement of type annotations, this grammar lets you write any term that you can write without the syntactic stratification. + +### Typing judgements + +The typing judgements for the external language extand the judgements for canonical forms with a rule ``synth_annot`` for synthesizing the type of an annotated term. Additionally, the rules do not force terms to be η-long. !}% +%sort synth %. +%mode synth %in %out %. +%sort check %. +%mode check %in %in %. +% synthesize +%term synth_c synth el_c b %. +%term synth_app %pi (synth (el_app R1 M2) A) %<- (synth R1 (arrow A2 A)) %<- (check M2 A2) %. +%term synth_fst %pi (synth (el_fst R) A1) %<- (synth R (prod A1 A2)) %. +%term synth_snd %pi (synth (el_snd R) A2) %<- (synth R (prod A1 A2)) %. +%% admit non-canonical forms with an annotation +%term synth_annot %pi (synth (el_annot M A) A) %<- (check M A) %. +%% check +%% in the EL anything that synths also checks, no matter the type +%term check_asm %pi (check (el_asm R) A) %<- (synth R A) %. +%term check_lam + %pi (check (el_lam M) (arrow A2 A)) + %<- ({x el_rtm} %pi (synth x A2) %-> (check (M x) A)) %. +%term check_pair %pi (check (el_pair M1 M2) (prod A1 A2)) %<- (check M1 A1) %<- (check M2 A2) %. +%block synth_block {A tp} [x el_rtm] [dx synth x A]%. +%worlds (synth_block) (synth _ _) (check _ _) %. +%{! It is possible to show that any well-typed term in a standard presentation of the λ-calculus can be translated into a term in this syntax that is well-typed by these rules. However, we have not formulated this step here, choosing instead to take these rules as the definition of the external language. + +### Elaboration + +We now define elaboration. We define three mutually recursive elaboration judgements, one for checked terms, and two for synthesizing terms. The two judgements for synthesizing terms differ in whether the elaboration of the synthesizing term is an atomic term or a canonical term. !}% +%sort elab_m %. +%mode elab_m %in %out %in %. +%sort elab_r_to_r %. +%mode elab_r_to_r %in %out %out %. +%sort elab_r_to_m %. +%mode elab_r_to_m %in %out %out %. +% M +%term elab_m_asm_r2r %pi (elab_m (el_asm ER) M A) %<- (elab_r_to_r ER R A) %<- (expand A R M) %. +%term elab_m_asm_r2m %pi (elab_m (el_asm ER) M A) %<- (elab_r_to_m ER M A) %. +%term elab_m_lam + %pi (elab_m (el_lam EM) (lam M) (arrow A2 A)) + %<- ({el_x el_rtm} {x rtm} %pi (elab_r_to_r el_x x A2) %-> (elab_m (EM el_x) (M x) A)) %. +%term elab_m_pair + %pi (elab_m (el_pair EM1 EM2) (pair M1 M2) (prod A1 A2)) + %<- (elab_m EM1 M1 A1) + %<- (elab_m EM2 M2 A2) %. +% R to R +%term elab_r_to_r_c elab_r_to_r el_c c b %. +%term elab_r_to_r_app + %pi (elab_r_to_r (el_app ER1 EM2) (app R1 M2) A) + %<- (elab_r_to_r ER1 R1 (arrow A2 A)) + %<- (elab_m EM2 M2 A2) %. +%term elab_r_to_r_fst %pi (elab_r_to_r (el_fst ER) (fst R) A1) %<- (elab_r_to_r ER R (prod A1 A2)) %. +%term elab_r_to_r_snd %pi (elab_r_to_r (el_snd ER) (snd R) A2) %<- (elab_r_to_r ER R (prod A1 A2)) %. +% R root +%term elab_r_to_m_annot %pi (elab_r_to_m (el_annot EM A) M A) %<- (elab_m EM M A) %. +%term elab_r_to_m_app + %pi (elab_r_to_m (el_app ER1 EM2) M' A) + %<- (elab_r_to_m ER1 (lam M) (arrow A2 A)) + %<- (elab_m EM2 M2 A2) + %<- (hsubst_m M2 A2 M M') %. +%term elab_r_to_m_fst + %pi (elab_r_to_m (el_fst ER) M1 A1) + %<- (elab_r_to_m ER (pair M1 M2) (prod A1 A2)) %. +%term elab_r_to_m_snd + %pi (elab_r_to_m (el_snd ER) M2 A2) + %<- (elab_r_to_m ER (pair M1 M2) (prod A1 A2)) %. +%block elab_block {A tp} [el_x el_rtm] [x rtm] [dex elab_r_to_r el_x x A]%. +%{! The inductive structure of the elaboration judgements is similar to the structure of hereditary substitution. Elaboration uses hereditary substitution in the key rule ``elab_r_to_m_app`` to compute the canonical result of a function application. Elaboration also uses expansion to expand atomic terms into canonical ones. + +## External Language: Metatheory + +### Static Correctness of Elaboration + +First, we prove that the elaboration of a well-typed term is well-typed. !}% +%sort elab_m_reg %. +%mode elab_m_reg %in %out %out %. +%sort elab_r2r_reg %. +%mode elab_r2r_reg %in %out %out %. +%sort elab_r2m_reg %. +%mode elab_r2m_reg %in %out %out %. +%% M to M +%term _ + %pi (elab_m_reg (elab_m_asm_r2r Dex Del) (check_asm DsynER) DcanM) + %<- (elab_r2r_reg Del DsynER DaR) + %<- (expand_exists _ DaR Dex-2 DcanM-2) + %<- (expand_unique Dex-2 Dex Did) + %<- (canon_respects_id Did id/tp_refl DcanM-2 DcanM) %. +%term _ + %pi (elab_m_reg (elab_m_asm_r2m Del) (check_asm DsynER) DcanM) + %<- (elab_r2m_reg Del DsynER DcanM) %. +%term _ + %pi (elab_m_reg (elab_m_lam Del) (check_lam Dcheck) (canon_lam Dcanon)) + %<- ({el_x el_rtm} {delx synth el_x A} {x rtm} {dx atom x A} {delabx elab_r_to_r el_x x A} {_ elab_r2r_reg delabx delx dx} elab_m_reg (Del el_x x delabx) (Dcheck el_x delx) (Dcanon x dx)) %. +%term _ + %pi (elab_m_reg (elab_m_pair De2 De1) (check_pair Dcheck2 Dcheck1) (canon_pair Dcanon2 Dcanon1)) + %<- (elab_m_reg De2 Dcheck2 Dcanon2) + %<- (elab_m_reg De1 Dcheck1 Dcanon1) %. +%% R to R +%term _ elab_r2r_reg elab_r_to_r_c synth_c atom_c %. +%term _ + %pi (elab_r2r_reg (elab_r_to_r_app De2 De1) (synth_app Dcheck2 Dsynth1) (atom_app Dcanon2 Datom1)) + %<- (elab_m_reg De2 Dcheck2 Dcanon2) + %<- (elab_r2r_reg De1 Dsynth1 Datom1) %. +%term _ + %pi (elab_r2r_reg (elab_r_to_r_fst De1) (synth_fst Dsynth1) (atom_fst Datom1)) + %<- (elab_r2r_reg De1 Dsynth1 Datom1) %. +%term _ + %pi (elab_r2r_reg (elab_r_to_r_snd De1) (synth_snd Dsynth1) (atom_snd Datom1)) + %<- (elab_r2r_reg De1 Dsynth1 Datom1) %. +%% R to M +%term _ + %pi (elab_r2m_reg (elab_r_to_m_annot De) (synth_annot DcheckM) DcanonM) + %<- (elab_m_reg De DcheckM DcanonM) %. +%term _ + %pi (elab_r2m_reg (elab_r_to_m_app DsubstM2 DeM2 DeR1) (synth_app DcheckM2 DsynthR1) DcanonM') + %<- (elab_r2m_reg DeR1 DsynthR1 (canon_lam DcanonM)) + %<- (elab_m_reg DeM2 DcheckM2 DcanonM2) + %<- (hsubst_m_exists _ DcanonM DcanonM2 DsubstM2-2 DcanonM'-2) + %<- (hsubst_m_unique DsubstM2-2 DsubstM2 Did) + %<- (canon_respects_id Did id/tp_refl DcanonM'-2 DcanonM') %. +%term _ + %pi (elab_r2m_reg (elab_r_to_m_fst DeR) (synth_fst DsynthR) DcM1) + %<- (elab_r2m_reg DeR DsynthR (canon_pair DcM2 DcM1)) %. +%term _ + %pi (elab_r2m_reg (elab_r_to_m_snd DeR) (synth_snd DsynthR) DcM2) + %<- (elab_r2m_reg DeR DsynthR (canon_pair DcM2 DcM1)) %. +%% because world subsumption doesn't do exchange, +%% you can't get this block to match both +%% atom_block/synth_block and elab_block. +%% +%% this order matches atom_block/synth_block but not elab_block. +%block elab_reg_block {A tp} [el_x el_rtm] [delx synth el_x A] [x rtm] [dx atom x A] [delabx elab_r_to_r el_x x A] [_ elab_r2r_reg delabx delx dx]%. +%worlds (elab_reg_block) (elab_m_reg _ _ _) (elab_r2r_reg _ _ _) (elab_r2m_reg _ _ _) %. +%total (D1 D2 D3) (elab_m_reg D1 _ _) (elab_r2r_reg D2 _ _) (elab_r2m_reg D3 _ _) %. +%{! The proof is a straightforward induction, using ``hsubst_m_exists`` and ``expand_exists``. Because these two lemmas _output_ a hereditary substitution or expansion, along with the typing derivation for the result of the operation, we use uniqueness here to show well-typedness of the outputs of the hereditary substitution and expansion derivations given by elaboration. In retrospect, we should have factored this reasoning into separate lemmas that are like ``hsubst_m_exists`` and ``expand_exists`` but take the the hereditary substitution or expansion as an input. + +### Existence of elaboration + +We require one addtional [respects lemma](/wiki/respects-lemma/). !}% +%sort elab_r2m_respects_id %. +%mode elab_r2m_respects_id %in %in %in %out %. +%term _ elab_r2m_respects_id _ _ D D %. +%worlds (elab_block elab_reg_block) (elab_r2m_respects_id _ _ _ _) %. +%% bar in other block +%total {} (elab_r2m_respects_id _ _ _ _) %. +%reduces = D1 D2 (elab_r2m_respects_id _ _ D2 D1) %. +%{! The worlds of this lemma require a little explanation. ``elab_reg_block`` should equal ``elab_block`` for this type family. However, [world subsumption](/wiki/world-subsumption/) does not notice this relationship because doing so would require permutation. Consequently, we prove the theorem for both blocks directly. + +Next, we prove that all well-typed terms elaborate. Analogously to hereditary substitution, the theorem for synthesizing terms computes a sum of the two ways it could elaborate: !}% +%sort elab_r_exists_sum %. +%term elab_r_exists_sum_r2r %pi (elab_r_exists_sum ER A) %<- (elab_r_to_r ER R A) %. +%term elab_r_exists_sum_r2m %pi (elab_r_exists_sum ER A) %<- (elab_r_to_m ER M A) %. +%{! We begin with some [output factoring](/wiki/output-factoring/) and [canonical forms](/wiki/canonical-form/) lemmas: !}% +%sort elab_m_exists_asm %. +%mode elab_m_exists_asm %in %out %. +%term _ elab_m_exists_asm (elab_r_exists_sum_r2m DeR) (elab_m_asm_r2m DeR) %. +%term _ + %pi (elab_m_exists_asm (elab_r_exists_sum_r2r DeR) (elab_m_asm_r2r Dex DeR)) + %<- (elab_r2r_reg DeR _ DaR) + %<- (expand_exists _ DaR Dex _) %. +%worlds (elab_reg_block) (elab_m_exists_asm _ _) %. +%total {} (elab_m_exists_asm _ _) %. +%sort canon_prod_is_pair %. +%mode canon_prod_is_pair %in %out %. +%term _ canon_prod_is_pair _ id/mtm_refl %. +%worlds (elab_reg_block) (canon_prod_is_pair _ _) %. +%total {} (canon_prod_is_pair _ _) %. +%sort elab_r_exists_fst %. +%mode elab_r_exists_fst %in %out %. +%term _ elab_r_exists_fst (elab_r_exists_sum_r2r DeR) (elab_r_exists_sum_r2r (elab_r_to_r_fst DeR)) %. +%term _ + %pi (elab_r_exists_fst (elab_r_exists_sum_r2m DeR) (elab_r_exists_sum_r2m (elab_r_to_m_fst DeR-2))) + %<- (elab_r2m_reg DeR _ DcM) + %<- (canon_prod_is_pair DcM Did) + %<- (elab_r2m_respects_id Did id/tp_refl DeR DeR-2) %. +%worlds (elab_reg_block) (elab_r_exists_fst _ _) %. +%total {} (elab_r_exists_fst _ _) %. +%sort elab_r_exists_snd %. +%mode elab_r_exists_snd %in %out %. +%term _ elab_r_exists_snd (elab_r_exists_sum_r2r DeR) (elab_r_exists_sum_r2r (elab_r_to_r_snd DeR)) %. +%term _ + %pi (elab_r_exists_snd (elab_r_exists_sum_r2m DeR) (elab_r_exists_sum_r2m (elab_r_to_m_snd DeR-2))) + %<- (elab_r2m_reg DeR _ DcM) + %<- (canon_prod_is_pair DcM Did) + %<- (elab_r2m_respects_id Did id/tp_refl DeR DeR-2) %. +%worlds (elab_reg_block) (elab_r_exists_snd _ _) %. +%total {} (elab_r_exists_snd _ _) %. +%sort canon_arrow_is_lam %. +%mode canon_arrow_is_lam %in %out %. +%term _ canon_arrow_is_lam _ id/mtm_refl %. +%worlds (elab_reg_block) (canon_arrow_is_lam _ _) %. +%total {} (canon_arrow_is_lam _ _) %. +%sort elab_r_exists_app %. +%mode elab_r_exists_app %in %in %out %. +%term _ elab_r_exists_app (elab_r_exists_sum_r2r DeR1) DeM2 (elab_r_exists_sum_r2r (elab_r_to_r_app DeM2 DeR1)) %. +%term _ + %pi (elab_r_exists_app (elab_r_exists_sum_r2m DeR1) DeM2 (elab_r_exists_sum_r2m (elab_r_to_m_app Ds DeM2 DeR1-2))) + %<- (elab_r2m_reg DeR1 _ DcM1) + %<- (elab_m_reg DeM2 _ DcM2) + %<- (canon_arrow_is_lam DcM1 Did) + %<- (canon_respects_id Did id/tp_refl DcM1 (canon_lam DcBody)) + %<- (elab_r2m_respects_id Did id/tp_refl DeR1 DeR1-2) + %<- (hsubst_m_exists _ DcBody DcM2 Ds _) %. +%worlds (elab_reg_block) (elab_r_exists_app _ _ _) %. +%total {} (elab_r_exists_app _ _ _) %. +%{! Finally, we prove the main results: !}% +%sort elab_m_exists %. +%mode elab_m_exists %in %out %. +%sort elab_r_exists %. +%mode elab_r_exists %in %out %. +%% M +%term _ + %pi (elab_m_exists (check_asm Dsyn) DeRes) + %<- (elab_r_exists Dsyn Dsum) + %<- (elab_m_exists_asm Dsum DeRes) %. +%term _ + %pi (elab_m_exists (check_lam DcM) (elab_m_lam DelabM)) + %<- ({el_x el_rtm} {delx synth el_x A} {x rtm} {dx atom x A} {delabx elab_r_to_r el_x x A} {_ elab_r2r_reg delabx delx dx} {_ elab_r_exists delx (elab_r_exists_sum_r2r delabx)} elab_m_exists (DcM el_x delx) (DelabM el_x x delabx)) %. +%term _ + %pi (elab_m_exists (check_pair Dc2 Dc1) (elab_m_pair De2 De1)) + %<- (elab_m_exists Dc1 De1) + %<- (elab_m_exists Dc2 De2) %. +%% R +%term _ elab_r_exists synth_c (elab_r_exists_sum_r2r elab_r_to_r_c) %. +%term _ + %pi (elab_r_exists (synth_app DchM DsynR) DsumApp) + %<- (elab_r_exists DsynR DsumR) + %<- (elab_m_exists DchM DeM) + %<- (elab_r_exists_app DsumR DeM DsumApp) %. +%term _ + %pi (elab_r_exists (synth_fst Dsyn) Dsum') + %<- (elab_r_exists Dsyn Dsum) + %<- (elab_r_exists_fst Dsum Dsum') %. +%term _ + %pi (elab_r_exists (synth_snd Dsyn) Dsum') + %<- (elab_r_exists Dsyn Dsum) + %<- (elab_r_exists_snd Dsum Dsum') %. +%term _ + %pi (elab_r_exists (synth_annot DcM) (elab_r_exists_sum_r2m (elab_r_to_m_annot DeM))) + %<- (elab_m_exists DcM DeM) %. +% maintains all the invariants from elab_reg_block because the above lemmas use call regularity +%block elab_exists_block {A tp} [el_x el_rtm] [delx synth el_x A] [x rtm] [dx atom x A] [delabx elab_r_to_r el_x x A] [_ elab_r2r_reg delabx delx dx] [_ elab_r_exists delx (elab_r_exists_sum_r2r delabx)]%. +%worlds (elab_exists_block) (elab_m_exists _ _) (elab_r_exists _ _) %. +%total (D1 D2) (elab_m_exists D1 _) (elab_r_exists D2 _) %. +%{! ``>All Twelf code for this tutorial.`` +`` check="true">See Twelf's output.`` + +\{\{case study\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Hereditary_substitution_for_the_STLC_(part_2)). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/hereditary-substitution-for-the-stlc.lf b/new-tests/stelf-output/pages/hereditary-substitution-for-the-stlc.lf new file mode 100644 index 0000000..e7f844f --- /dev/null +++ b/new-tests/stelf-output/pages/hereditary-substitution-for-the-stlc.lf @@ -0,0 +1,558 @@ +% +%{! _You may wish to read the tutorial on [admissibility of cut](/wiki/admissibility-of-cut/) and/or the article on [verifications and uses](/wiki/verifications-and-uses/) first._ + +In this tutorial, we recast the proof of cut admissibility as an algorithm for normalizing terms in the simply-typed λ-calculus. This algorithm is called _hereditary substitution_; it is used in the definition of LF itself, as well as in many other type theories. To apply hereditary substitution, it is necessary to: +- Define a language of [canonical form](/wiki/canonical-form/)s. In programming language terms, canonical forms correspond to terms that are not β-reducible ([beta-normal](/wiki/beta-equivalence/)) and then are η-expanded as much as possible ([eta-long](/wiki/eta-equivalence/)); logically, canonical forms correspond to the cut-free sequent calculus proofs. +- Define hereditary substitution, which computes the canonical result of substituting one canonical form into another. In programming language terms, hereditary substitution is part of a normalization algorithm; logically, it is the computational content of the proof of cut admissibility. +- Define an eta-expansion judgement. In programming language terms, eta-expansion is part of a normalization algorithm; logically, it is the computational content of the identity theorem . +- Define an external language that admits non-canonical forms by elaboration into the canonical forms. In programming language terms, this elaboration relation corresponds to a normalization algorithm; logically, it is the computational content of the proof of cut elimination. + +In this article, we formalize hereditary substitution and elaboration in Twelf. We prove several results: +- It is decidable whether or not a hereditary substitution exists. (This property is proved automatically by Twelf.) +- Under the appropriate typing conditions, hereditary substitutions exist and preserve types. Moreover, hereditary substitutions compute a unique result. +- Eta-expansions exist, preserve types, and are unique. +- All well-typed non-canonical forms elaborate to a canonical form of the same type. + +This example brings together a number of Twelf proof techniques: +* using [%reduces](/wiki/percent-reduces/) for termination +* [mutual induction](/wiki/mutual-induction/) and lexicographic induction +* reasoning with [equality](/wiki/equality/) and [respects lemma](/wiki/respects-lemma/)s +* proving [uniqueness lemma](/wiki/uniqueness-lemma/)s +* [output factoring](/wiki/output-factoring/) +* [reasoning from false](/wiki/reasoning-from-false/) +* [catch-all case](/wiki/catch-all-case/)s. +You may wish to read the individual tutorials on these techniques before reading this example. + +The canonical forms language is described on this page; the external language and elaboration are described on the [next page](/wiki/hereditary-substitution-for-the-stlc-part-2/). + +**Experience report ([DRL](/wiki/user-drl/)):** _This tutorial might seem like it takes a lot of Twelf code. At the time I wrote this code, I was an experienced Twelf user and familiar with the basic ideas of the canonical forms approach. However, I hadn't gone through the details of these proofs before (that's why I wrote the code). That said, it took no more than a work-day to work out and formalize the metatheory the canonical forms approach (everything on this page), and no more than another day to do the elaboration on the next page._ + +## Canonical forms language: syntax and judgements + +### Syntax + +We consider a calculus with function and pair types, as well as a single base type: !}% +%sort tp %. +%term b tp %. +%term arrow %pi tp %-> tp %-> tp %. +%term prod %pi tp %-> tp %-> tp %. +%sort rtm %. +%sort mtm %. +%term app %pi rtm %-> mtm %-> rtm %. +%term c rtm %. +%term fst %pi rtm %-> rtm %. +%term snd %pi rtm %-> rtm %. +%term asm %pi rtm %-> mtm %. +%term lam %pi (%pi rtm %-> mtm) %-> mtm %. +%term pair %pi mtm %-> mtm %-> mtm %. +%block rtm_block [x rtm]%. +%worlds (rtm_block) (tp) (rtm) (mtm) %. +%{! The syntax of types is standard. There are a few subtleties in the syntax of terms: +* The syntax of terms is stratified into atomic terms ``rtm`` and canonical terms ``mtm`` in order to syntactically prevent β-redices. For example, pairs are canonical but the argument of a projection (``fst`` and ``snd``) is atomic, so we can never project from a term that is syntactically a pair. +* We name the syntactic classes ``rtm`` and ``mtm`` after the metavariables ``R`` and ``M`` that we will use to refer to them. +* Because variables are considered to be atomic terms, the body of a ``lam`` binds a variable of type ``rtm``. This means that the substitution operation that we get "for free" from the encoding is only the substitution of an ``rtm`` for a variable; hereditary substitution defines the substitution of an ``mtm`` for a variable. +* The constant ``asm`` represents an injection from ``rtm`` into ``mtm``. + +### Typing judgements !}% +%sort atom %. +%mode atom %in %out %. +%sort canon %. +%mode canon %in %in %. +% atomic +%term atom_c atom c b %. +%term atom_app %pi (atom (app R1 M2) A) %<- (atom R1 (arrow A2 A)) %<- (canon M2 A2) %. +%term atom_fst %pi (atom (fst R) A1) %<- (atom R (prod A1 A2)) %. +%term atom_snd %pi (atom (snd R) A2) %<- (atom R (prod A1 A2)) %. +%% canonical +%term canon_asm %pi (canon (asm R) b) %<- (atom R b) %. +%term canon_lam + %pi (canon (lam M) (arrow A2 A)) + %<- ({x rtm} %pi (atom x A2) %-> (canon (M x) A)) %. +%term canon_pair %pi (canon (pair M1 M2) (prod A1 A2)) %<- (canon M1 A1) %<- (canon M2 A2) %. +%block atom_block {A tp} [x rtm] [dx atom x A]%. +%worlds (atom_block) (atom _ _) (canon _ _) %. +%terminates (R M) (atom R _) (canon M _) %. +%{! We define two mutually-recursive judgements, ``atom R A`` and ``canon M A``. The typing judgements for the two syntactic categories have a bidirectional operational interpretation: an ``mtm`` is checked against a type, whereas an ``rtm`` synthesizes a type. The rules are the standard typing rules for the λ-calculus, modified to follow the restrictions of the syntax. Additionally, the rule ``canon_asm`` applies only at base type, which forces an ``mtm`` to be η-long. + +These judgements are manifestly terminating, which we verify with the ``%terminates``. + +### Hereditary substitution + +We define three hereditary substitution judgements: +* ``hsubst_m M0 A0 ([x0] M) M'``: compute the canonical form ``M'`` of substituting ``M0`` for ``x0`` in ``M``. ``A0`` should be the type of ``M0``; it is used to show that hereditary substitution is decidable. +* ``hsubst_r M0 A0 ([x0] R) R'``: compute the atomic term ``R'`` resulting from substituting ``M0`` for ``x0`` in ``R``. This judgement applies when ``x0`` is not the head variable of the atomic term ``R``. +* ``hsubst_rr M0 A0 ([x0] R) M' A'``: compute a new canonical form term ``M'`` resulting from substituting ``M0`` for ``x0`` in ``R``. This judgement applies when ``x0`` is the head variable of ``R``. + +The head variable of an atomic term ``R`` is the variable at the root of a series of eliminations. E.g., the head of ``fst (app x c)`` is ``x``. When a canonical form is substituted for a head variable, hereditary substitution continues reducing. E.g., the following judgement is derivable: + +``` +hsubst_rr + (lam ([x] (pair c c))) + (arrow b (prod b b)) + ([x] fst (app x c)) + c + b +``` + +Intuitively, we substitute ``(lam ([x] (pair c c)))`` to get ``fst (app (lam ([x] (pair c c))) c)`` and then reduce this to ``c``. + +``hsubst_rr`` also computes the type of the canonical form it produces, which is needed for the termination metric for hereditary substitution. + +Here are the judgements: !}% +%sort hsubst_m %. +%mode hsubst_m %in %in %in %out %. +%sort hsubst_r %. +%mode hsubst_r %in %in %in %out %. +%sort hsubst_rr %. +%mode hsubst_rr %in %in %in %out %out %. +%% Ms +%term hsubst_m_r %pi (hsubst_m M0 A0 ([x] asm (R x)) (asm R')) %<- (hsubst_r M0 A0 R R') %. +%term hsubst_m_rr %pi (hsubst_m M0 A0 ([x] asm (R x)) M') %<- (hsubst_rr M0 A0 R M' _) %. +%term hsubst_m_lam + %pi (hsubst_m M0 A0 ([x] lam ([y] M x y)) (lam M')) + %<- ({y rtm} hsubst_m M0 A0 ([x] M x y) (M' y)) %. +%term hsubst_m_pair + %pi (hsubst_m M0 A0 ([x] pair (M1 x) (M2 x)) (pair M1' M2')) + %<- (hsubst_m M0 A0 M1 M1') + %<- (hsubst_m M0 A0 M2 M2') %. +%% R not head +%term hsubst_r_closed hsubst_r M0 A0 ([x] E) E %. +%term hsubst_r_app + %pi (hsubst_r M0 A0 ([x] app (R x) (M x)) (app R' M')) + %<- (hsubst_r M0 A0 R R') + %<- (hsubst_m M0 A0 M M') %. +%term hsubst_r_fst %pi (hsubst_r M0 A0 ([x] fst (R x)) (fst R')) %<- (hsubst_r M0 A0 R R') %. +%term hsubst_r_snd %pi (hsubst_r M0 A0 ([x] snd (R x)) (snd R')) %<- (hsubst_r M0 A0 R R') %. +%% r head +%term hsubst_rr_var hsubst_rr M0 A0 ([x] x) M0 A0 %. +%term hsubst_rr_app + %pi (hsubst_rr M0 A0 ([x] app (R1 x) (M2 x)) M'' A) + %<- (hsubst_rr M0 A0 R1 (lam M') (arrow A2 A)) + %<- (hsubst_m M0 A0 M2 M2') + %<- (hsubst_m M2' A2 M' M'') %. +%term hsubst_rr_fst + %pi (hsubst_rr M0 A0 ([x] fst (R x)) M1' A1') + %<- (hsubst_rr M0 A0 R (pair M1' M2') (prod A1' A2')) %. +%term hsubst_rr_snd + %pi (hsubst_rr M0 A0 ([x] snd (R x)) M2' A2') + %<- (hsubst_rr M0 A0 R (pair M1' M2') (prod A1' A2')) %. +%worlds (rtm_block) (hsubst_m _ _ _ _) (hsubst_r _ _ _ _) (hsubst_rr _ _ _ _ _) %. +%reduces <= A' A0 (hsubst_rr _ A0 _ _ A') %. +%terminates {(A0 A0' A0'') (M R R')} (hsubst_m _ A0 M _) (hsubst_r _ A0' R _) (hsubst_rr _ A0'' R' _ _) %. +%{! The rule ``hsubst_r_closed`` is similar to a [catch-all case](/wiki/catch-all-case/): rather than putting cases of hereditary substitution into the LF context, we give a single rule that applies whenever the variable is not free in the term we are substituting into. This covers both the variables ``y:rtm`` in the context and the constant ``c``, as well as any series of applications or projections made up from these terms—so the downside is that we have made hereditary substitution non-deterministic. However, keeping hereditary substitution out of the context simplifies some proofs below. + +Other than that, the judgements ``hsubst_m`` and ``hsubst_r`` are defined in the straightforward compositional manner. The interesting judgement is ``hsubst_rr``, which in each case performs some reduction on the output of the recursive call in order to compute a new canonical form. The most interesting rule is ``hsubst_rr_app``, which continues the hereditary substitution into the body of the function. + +When processing the ``%terminates`` declaration, Twelf proves that it is decidable whether or not a hereditary substitution exists: the hereditary substitution judgements define a terminating logic program. The auxiliary ``%reduces`` proves that the result type of ``hsubst_rr`` is always a subterm of the type of the term being substituted for. This is true because ``hsubst_rr`` only applies when the cut variable is the head of a sequence of eliminations, and applying one of these elimination forms always decreases the type of the term. With this reduces information, we check that the hereditary substitution judgements terminate by mutual lexicographic induction—mutual because we show termination for all three judgements simultaneously, and lexicographic because, in each case, the termination metric is first the type of the substituted term, and then the term being substituted into. This metric is expressed in Twelf by writing ``{(A0 A0' A0_) (M R R')}``. The curly-braces mean lexicographic induction; the parentheses mean mutual induction. + +### Expansion + +Expansion shows that we can turn an atomic term of any type into a canonical term of that type. It is a total judgement defined by induction on the structure of the type: !}% +%sort expand %. +%mode expand %in %in %out %. +%term expand_b expand b R (asm R) %. +%term expand_arrow + %pi (expand (arrow A2 A) R (lam M)) + %<- ({x rtm} expand A2 x (M2 x)) + %<- ({x rtm} expand A (app R (M2 x)) (M x)) %. +%term expand_prod + %pi (expand (prod A1 A2) R (pair M1 M2)) + %<- (expand A1 (fst R) M1) + %<- (expand A2 (snd R) M2) %. +%worlds (rtm_block) (expand _ _ _) %. +%total A (expand A _ _) %. +%{! In the ``expand_arrow`` case, we first expand a variable at the argument type, and then we expand the application at the result type. + +## Canonical forms language: Metatheory + +### Hereditary substitution + +#### Existence of hereditary substitution + +Even though we annotated ``hsubst_rr`` with a [%reduces](/wiki/percent-reduces/) above, we will need an [effectiveness lemma](/wiki/effectiveness-lemma/) version of this [%reduces](/wiki/percent-reduces/) to use that fact in the proof below. !}% +%sort hsubst_rr_size {A2} {A'} %. +%mode hsubst_rr_size %in %in %in %. +%term _ hsubst_rr_size A2 A2 hsubst_rr_var %. +%term _ + %pi (hsubst_rr_size A2 A4 (hsubst_rr_app _ _ Drr)) + %<- (hsubst_rr_size A2 (arrow A3 A4) Drr) %. +%term _ + %pi (hsubst_rr_size A2 Al (hsubst_rr_fst Drr)) + %<- (hsubst_rr_size A2 (prod Al Ar) Drr) %. +%term _ + %pi (hsubst_rr_size A2 Ar (hsubst_rr_snd Drr)) + %<- (hsubst_rr_size A2 (prod Al Ar) Drr) %. +%worlds (rtm_block) (hsubst_rr_size _ _ _) %. +%total D (hsubst_rr_size _ _ D) %. +%reduces <= A' A2 (hsubst_rr_size A2 A' _) %. +%{! All the terms are inputs; the only output is the ``%reduces``. + +The main theorem proves that under the appropriate typing conditions, hereditary substitutions exist and preserve types. For atomic terms, this means that one of the two kinds of hereditary substitutions exists. Thus, we define the following sum type: !}% +%sort hsubst_r_exists_sum %. +%term hsubst_r_exists_sum_r %pi (hsubst_r_exists_sum M2 A2 R A) %<- (hsubst_r M2 A2 R R') %<- (atom R' A) %. +%term hsubst_r_exists_sum_rr + %pi (hsubst_r_exists_sum M2 A2 R A') + %<- (hsubst_rr M2 A2 R M' A') + %<- (canon M' A') %. +%{! Because we are proving a theorem that concludes a disjunction, it should not be surprising that we need some [output factoring](/wiki/output-factoring/) lemmas. The following lemmas may be proved independently: !}% +%sort hsubst_m_exists_asm %. +%mode hsubst_m_exists_asm %in %out %out %. +%term _ hsubst_m_exists_asm (hsubst_r_exists_sum_r DaR' DsR) (hsubst_m_r DsR) (canon_asm DaR') %. +%term _ hsubst_m_exists_asm (hsubst_r_exists_sum_rr DcM' DsR) (hsubst_m_rr DsR) DcM' %. +%worlds (atom_block) (hsubst_m_exists_asm _ _ _) %. +%total {} (hsubst_m_exists_asm _ _ _) %. +%sort hsubst_r_exists_fst %. +%mode hsubst_r_exists_fst %in %out %. +%term _ hsubst_r_exists_fst (hsubst_r_exists_sum_r DaR' DsR) (hsubst_r_exists_sum_r (atom_fst DaR') (hsubst_r_fst DsR)) %. +%term _ hsubst_r_exists_fst (hsubst_r_exists_sum_rr (canon_pair _ DcMl') DsR) (hsubst_r_exists_sum_rr DcMl' (hsubst_rr_fst DsR)) %. +%worlds (atom_block) (hsubst_r_exists_fst _ _) %. +%total {} (hsubst_r_exists_fst _ _) %. +%sort hsubst_r_exists_snd %. +%mode hsubst_r_exists_snd %in %out %. +%term _ hsubst_r_exists_snd (hsubst_r_exists_sum_r DaR' DsR) (hsubst_r_exists_sum_r (atom_snd DaR') (hsubst_r_snd DsR)) %. +%term _ hsubst_r_exists_snd (hsubst_r_exists_sum_rr (canon_pair DcMr' _) DsR) (hsubst_r_exists_sum_rr DcMr' (hsubst_rr_snd DsR)) %. +%worlds (atom_block) (hsubst_r_exists_snd _ _) %. +%total {} (hsubst_r_exists_snd _ _) %. +%{! Next, we prove the overall theorems, as well as a factoring lemma for application. This last factoring lemma +must be in the mutually recursive loop with the main theorems because of the final premise of ``hsubst_rr_app``, which continues the hereditary substitution. !}% +%sort hsubst_m_exists {A2} {y rtm} {dy atom y A2} %. +%mode hsubst_m_exists %in %in %in %out %out %. +%sort hsubst_r_exists {A2} {y rtm} {dx atom y A2} %. +%mode hsubst_r_exists %in %in %in %out %. +%sort hsubst_m_exists_app {A2} {y rtm} {dx atom y A2} %. +%mode hsubst_m_exists_app %in %in %in %in %in %out %. +%% canonical terms +%term _ + %pi (hsubst_m_exists A2 ([x] [dx] canon_asm (DaR x dx)) DcM2 DsR DcM') + %<- (hsubst_r_exists A2 DaR DcM2 Dsum) + %<- (hsubst_m_exists_asm Dsum DsR DcM') %. +%term _ + %pi (hsubst_m_exists A2 ([x] [dx] %the (canon (lam (M x)) (arrow Af At)) (canon_lam ([y] [dy] DcM x dx y dy))) (%the (canon M2 A2) DcM2) (hsubst_m_lam DsM) (canon_lam DcM')) + %<- ({y} {dy atom y Af} hsubst_m_exists A2 ([x] [dx] DcM x dx y dy) DcM2 (DsM y) (DcM' y dy)) %. +%term _ + %pi (hsubst_m_exists A2 ([x] [dx] canon_pair (DcMr x dx) (DcMl x dx)) DcM2 (hsubst_m_pair DsMr DsMl) (canon_pair DcMr' DcMl')) + %<- (hsubst_m_exists A2 DcMl DcM2 DsMl DcMl') + %<- (hsubst_m_exists A2 DcMr DcM2 DsMr DcMr') %. +%% atomic terms +%term _ hsubst_r_exists A2 ([x] [dx] D) _ (hsubst_r_exists_sum_r D hsubst_r_closed) %. +%term _ hsubst_r_exists A2 ([x] [dx] dx) DcM2 (hsubst_r_exists_sum_rr DcM2 hsubst_rr_var) %. +%term _ + %pi (hsubst_r_exists A2 ([x] [dx] atom_app (DcM x dx) (DaR x dx)) DcM2 DsumApp) + %<- (hsubst_m_exists A2 DcM DcM2 DsM' DcM') + %<- (hsubst_r_exists A2 DaR DcM2 DsumR') + %<- (hsubst_m_exists_app A2 DaR DsumR' DsM' DcM' DsumApp) %. +%term _ + %pi (hsubst_r_exists A2 ([x] [dx] atom_fst (DcR x dx)) DcM2 DsumFst) + %<- (hsubst_r_exists A2 DcR DcM2 DsumR) + %<- (hsubst_r_exists_fst DsumR DsumFst) %. +%term _ + %pi (hsubst_r_exists A2 ([x] [dx] atom_snd (DcR x dx)) DcM2 DsumSnd) + %<- (hsubst_r_exists A2 DcR DcM2 DsumR) + %<- (hsubst_r_exists_snd DsumR DsumSnd) %. +%% app factoring lemma +%term _ hsubst_m_exists_app A2 _ (hsubst_r_exists_sum_r DaR1 DsR1) DsM3 DcM3 (hsubst_r_exists_sum_r (atom_app DcM3 DaR1) (hsubst_r_app DsM3 DsR1)) %. +%term _ + %pi (hsubst_m_exists_app A2 _ (hsubst_r_exists_sum_rr (canon_lam DcM4') DsR1) DsM3 DcM3 (hsubst_r_exists_sum_rr DcM4'' (hsubst_rr_app DsM4' DsM3 DsR1))) + %<- (hsubst_rr_size A2 (arrow A3 A4) DsR1) + %<- (hsubst_m_exists A3 DcM4' DcM3 DsM4' DcM4'') %. +%worlds (atom_block) (hsubst_m_exists_app _ _ _ _ _ _) (hsubst_r_exists _ _ _ _) (hsubst_m_exists _ _ _ _ _) %. +%total {(A2 A2' A2'') (D D' D'')} (hsubst_m_exists A2' D' _ _ _) (hsubst_m_exists_app A2'' D'' _ _ _ _) (hsubst_r_exists A2 D _ _) %. +%{! As you can see, the proof is a straightforward induction, where in each case we use the appropriate factoring lemma. In the final case of the application factoring lemma, we use the ``hsubst_rr_size`` lemma for its "side effect" of justifying the recursive call back to ``hsubst_m_exists``. The mutual lexicographic termination metric is the same as before. + +#### Uniqueness of hereditary substitution + +As usual for a [uniqueness lemma](/wiki/uniqueness-lemma/), we must begin by defining the appropriate identity types: !}% +%sort id/mtm %. +%term id/mtm_refl id/mtm M M %. +%sort id/rtm %. +%term id/rtm_refl id/rtm R R %. +%sort id/tp %. +%term id/tp_refl id/tp A A %. +%{! We elide the trivial proofs of the equivalence relation, congruence, inversion, and respects lemmas; see the complete Twelf code at the end for their statements. !}% +%{!! begin hidden !!}% +%sort id/rtm_sym %. +%mode id/rtm_sym %in %out %. +%term _ id/rtm_sym id/rtm_refl id/rtm_refl %. +%worlds (rtm_block) (id/rtm_sym _ _) %. +%total {} (id/rtm_sym _ _) %. +%sort id/tp_arrow_inv %. +%mode id/tp_arrow_inv %in %out %out %. +%term _ id/tp_arrow_inv _ id/tp_refl id/tp_refl %. +%worlds (rtm_block) (id/tp_arrow_inv _ _ _) %. +%total {} (id/tp_arrow_inv _ _ _) %. +%sort id/tp_prod_inv %. +%mode id/tp_prod_inv %in %out %out %. +%term _ id/tp_prod_inv _ id/tp_refl id/tp_refl %. +%worlds (rtm_block) (id/tp_prod_inv _ _ _) %. +%total {} (id/tp_prod_inv _ _ _) %. +%sort id/rtm_app_cong %. +%mode id/rtm_app_cong %in %in %out %. +%term _ id/rtm_app_cong _ _ id/rtm_refl %. +%worlds (rtm_block) (id/rtm_app_cong _ _ _) %. +%total {} (id/rtm_app_cong _ _ _) %. +%sort id/rtm_fst_cong %. +%mode id/rtm_fst_cong %in %out %. +%term _ id/rtm_fst_cong _ id/rtm_refl %. +%worlds (rtm_block) (id/rtm_fst_cong _ _) %. +%total {} (id/rtm_fst_cong _ _) %. +%sort id/rtm_snd_cong %. +%mode id/rtm_snd_cong %in %out %. +%term _ id/rtm_snd_cong _ id/rtm_refl %. +%worlds (rtm_block) (id/rtm_snd_cong _ _) %. +%total {} (id/rtm_snd_cong _ _) %. +%sort id/mtm_lam_cong {x rtm} %. +%mode id/mtm_lam_cong %in %out %. +%term _ id/mtm_lam_cong _ id/mtm_refl %. +%worlds (rtm_block) (id/mtm_lam_cong _ _) %. +%total {} (id/mtm_lam_cong _ _) %. +%sort id/mtm_lam_inv %. +%mode id/mtm_lam_inv %in %out %. +%term _ id/mtm_lam_inv _ ([_] id/mtm_refl) %. +%worlds (rtm_block) (id/mtm_lam_inv _ _) %. +%total {} (id/mtm_lam_inv _ _) %. +%sort id/mtm_pair_cong %. +%mode id/mtm_pair_cong %in %in %out %. +%term _ id/mtm_pair_cong _ _ id/mtm_refl %. +%worlds (rtm_block) (id/mtm_pair_cong _ _ _) %. +%total {} (id/mtm_pair_cong _ _ _) %. +%sort id/mtm_pair_inv %. +%mode id/mtm_pair_inv %in %out %out %. +%term _ id/mtm_pair_inv _ id/mtm_refl id/mtm_refl %. +%worlds (rtm_block) (id/mtm_pair_inv _ _ _) %. +%total {} (id/mtm_pair_inv _ _ _) %. +%sort expand_respects_id %. +%mode expand_respects_id %in %in %in %out %. +%term _ expand_respects_id _ _ D D %. +%worlds (rtm_block) (expand_respects_id _ _ _ _) %. +%total {} (expand_respects_id _ _ _ _) %. +%reduces = D1 D2 (expand_respects_id _ _ D2 D1) %. +%sort canon_respects_id %. +%mode canon_respects_id %in %in %in %out %. +%term _ canon_respects_id _ _ D D %. +%worlds (atom_block) (canon_respects_id _ _ _ _) %. +%total {} (canon_respects_id _ _ _ _) %. +%reduces = D1 D2 (canon_respects_id _ _ D2 D1) %. +%sort hsubst_m_respects_id %. +%mode hsubst_m_respects_id %in %in %in %in %out %. +%term _ hsubst_m_respects_id _ _ _ D D %. +%worlds (rtm_block) (hsubst_m_respects_id _ _ _ _ _) %. +%total {} (hsubst_m_respects_id _ _ _ _ _) %. +%reduces = D1 D2 (hsubst_m_respects_id _ _ _ D2 D1) %. +%sort id/mtm_asm_cong %. +%mode id/mtm_asm_cong %in %out %. +%term _ id/mtm_asm_cong _ id/mtm_refl %. +%worlds (rtm_block) (id/mtm_asm_cong _ _) %. +%total {} (id/mtm_asm_cong _ _) %. +%{!! end hidden !!}% +%{! The hardest part of proving uniqueness of hereditary substitution is proving that the rules ``hsubst_m_r`` and ``hsubst_m_rr`` are mutually exclusive. We do by using [reasoning with contradiction](/wiki/reasoning-from-false/). !}% +%sort false %. +%freeze false %. +%{! First, we prove that ``hsubst_rr`` is never inhabited when the variable is not free in the term being substituted into: !}% +%sort hsubst_rr_closed_contra %. +%mode hsubst_rr_closed_contra %in %out %. +%term _ + %pi (hsubst_rr_closed_contra (hsubst_rr_app _ _ Dr) X) + %<- (hsubst_rr_closed_contra Dr X) %. +%term _ + %pi (hsubst_rr_closed_contra (hsubst_rr_fst D) X) + %<- (hsubst_rr_closed_contra D X) %. +%term _ + %pi (hsubst_rr_closed_contra (hsubst_rr_snd D) X) + %<- (hsubst_rr_closed_contra D X) %. +%worlds (rtm_block) (hsubst_rr_closed_contra _ _) %. +%total D (hsubst_rr_closed_contra D _) %. +%{! This is an example of deriving a contradiction by induction—the coverage checker rules out all the base cases. + +Next, we show that the two atomic term judgements are exclusive. In the only base case that is not ruled out by coverage checking, we use the previous lemma. !}% +%% contradiction of root and non-root +%sort hsubst_r_rr_contra %. +%mode hsubst_r_rr_contra %in %in %out %. +%term _ + %pi (hsubst_r_rr_contra hsubst_r_closed Drr X) + %<- (hsubst_rr_closed_contra Drr X) %. +%term _ + %pi (hsubst_r_rr_contra (hsubst_r_app _ DsrR) (hsubst_rr_app _ _ DsrrR) X) + %<- (hsubst_r_rr_contra DsrR DsrrR X) %. +%term _ + %pi (hsubst_r_rr_contra (hsubst_r_fst DsrR) (hsubst_rr_fst DsrrR) X) + %<- (hsubst_r_rr_contra DsrR DsrrR X) %. +%term _ + %pi (hsubst_r_rr_contra (hsubst_r_snd DsrR) (hsubst_rr_snd DsrrR) X) + %<- (hsubst_r_rr_contra DsrR DsrrR X) %. +%worlds (rtm_block) (hsubst_r_rr_contra _ _ _) %. +%% bar in extra block for lemma below's future use +%total (D) (hsubst_r_rr_contra D _ _) %. +%{! Next, we write a little lemma showing that ``false`` implies any identity that we need. !}% +%% false implies id +%sort false_implies_id/mtm {M} {M'} %. +%mode false_implies_id/mtm %in %in %in %out %. +%worlds (rtm_block) (false_implies_id/mtm _ _ _ _) %. +%% bar in extra block for lemma below's future use +%total {} (false_implies_id/mtm _ _ _ _) %. +%{! The cost of defining hereditary substitution in a non-deterministic manner, which we did to avoid putting cases of hereditary substitution in the context, is that we now need to prove the following lemmas. These lemmas establish that any way of deriving a hereditary substitution for a closed term produces the same result as ``hsubst_r_closed``. !}% +%sort hsubst_r_vacuous_id %. +%mode hsubst_r_vacuous_id %in %out %. +%sort hsubst_m_vacuous_id %. +%mode hsubst_m_vacuous_id %in %out %. +%% R +%term _ hsubst_r_vacuous_id hsubst_r_closed id/rtm_refl %. +%term _ + %pi (hsubst_r_vacuous_id (hsubst_r_app Dm Dr) Did') + %<- (hsubst_r_vacuous_id Dr DidR) + %<- (hsubst_m_vacuous_id Dm DidM) + %<- (id/rtm_app_cong DidR DidM Did') %. +%term _ + %pi (hsubst_r_vacuous_id (hsubst_r_fst D) Did') + %<- (hsubst_r_vacuous_id D Did) + %<- (id/rtm_fst_cong Did Did') %. +%term _ + %pi (hsubst_r_vacuous_id (hsubst_r_snd D) Did') + %<- (hsubst_r_vacuous_id D Did) + %<- (id/rtm_snd_cong Did Did') %. +%% M +%term _ + %pi (hsubst_m_vacuous_id (hsubst_m_r Dr) Did') + %<- (hsubst_r_vacuous_id Dr Did) + %<- (id/mtm_asm_cong Did Did') %. +%term _ + %pi (hsubst_m_vacuous_id (hsubst_m_rr D) Did) + %<- (hsubst_rr_closed_contra D X) + %<- (false_implies_id/mtm _ _ X Did) %. +%term _ + %pi (hsubst_m_vacuous_id (hsubst_m_lam D) Did') + %<- ({x} hsubst_m_vacuous_id (D x) (Did x)) + %<- (id/mtm_lam_cong Did Did') %. +%term _ + %pi (hsubst_m_vacuous_id (hsubst_m_pair D2 D1) Did') + %<- (hsubst_m_vacuous_id D1 Did1) + %<- (hsubst_m_vacuous_id D2 Did2) + %<- (id/mtm_pair_cong Did1 Did2 Did') %. +%worlds (rtm_block) (hsubst_r_vacuous_id _ _) (hsubst_m_vacuous_id _ _) %. +%total (D1 D2) (hsubst_r_vacuous_id D1 _) (hsubst_m_vacuous_id D2 _) %. +%{! Now, we prove the top-level results: !}% +%sort hsubst_m_unique %. +%mode hsubst_m_unique %in %in %out %. +%sort hsubst_r_unique %. +%mode hsubst_r_unique %in %in %out %. +%sort hsubst_rr_unique %. +%mode hsubst_rr_unique %in %in %out %out %. +%% M +%term _ + %pi (hsubst_m_unique (hsubst_m_r Dsr) (hsubst_m_r Dsr') DidAsm) + %<- (hsubst_r_unique Dsr Dsr' DidR) + %<- (id/mtm_asm_cong DidR DidAsm) %. +%term _ + %pi (hsubst_m_unique (hsubst_m_rr Dsr) (hsubst_m_rr Dsr') Did) + %<- (hsubst_rr_unique Dsr Dsr' Did _) %. +%% contradict mismatch +%term _ + %pi (hsubst_m_unique (hsubst_m_r Dsr) (hsubst_m_rr Dsrr) Did) + %<- (hsubst_r_rr_contra Dsr Dsrr X) + %<- (false_implies_id/mtm _ _ X Did) %. +%% contradict mismatch +%term _ + %pi (hsubst_m_unique (hsubst_m_rr Dsrr) (hsubst_m_r Dsr) Did) + %<- (hsubst_r_rr_contra Dsr Dsrr X) + %<- (false_implies_id/mtm _ _ X Did) %. +%term _ + %pi (hsubst_m_unique (hsubst_m_pair DsRight DsLeft) (hsubst_m_pair DsRight' DsLeft') DidPair) + %<- (hsubst_m_unique DsLeft DsLeft' DidLeft) + %<- (hsubst_m_unique DsRight DsRight' DidRight) + %<- (id/mtm_pair_cong DidLeft DidRight DidPair) %. +%term _ + %pi (hsubst_m_unique (hsubst_m_lam Ds) (hsubst_m_lam Ds') DidLam) + %<- ({y rtm} hsubst_m_unique (Ds y) (Ds' y) (Did y)) + %<- (id/mtm_lam_cong Did DidLam) %. +%% R non-root +%term _ hsubst_r_unique D D id/rtm_refl %. +%term _ + %pi (hsubst_r_unique (%the (hsubst_r M2 A2 ([_] R) R) hsubst_r_closed) (%the (hsubst_r M2 A2 ([_] R) R') D) Did') + %<- (hsubst_r_vacuous_id D Did) + %<- (id/rtm_sym Did Did') %. +%term _ + %pi (hsubst_r_unique (%the (hsubst_r M2 A2 ([_] R) R') D) (%the (hsubst_r M2 A2 ([_] R) R) hsubst_r_closed) Did) + %<- (hsubst_r_vacuous_id D Did) %. +%term _ + %pi (hsubst_r_unique (hsubst_r_app DsM' DsR') (hsubst_r_app DsM'' DsR'') DidApp) + %<- (hsubst_r_unique DsR' DsR'' DidR) + %<- (hsubst_m_unique DsM' DsM'' DidM) + %<- (id/rtm_app_cong DidR DidM DidApp) %. +%term _ + %pi (hsubst_r_unique (hsubst_r_fst DsR') (hsubst_r_fst DsR'') DidFst) + %<- (hsubst_r_unique DsR' DsR'' DidR) + %<- (id/rtm_fst_cong DidR DidFst) %. +%term _ + %pi (hsubst_r_unique (hsubst_r_snd DsR') (hsubst_r_snd DsR'') DidSnd) + %<- (hsubst_r_unique DsR' DsR'' DidR) + %<- (id/rtm_snd_cong DidR DidSnd) %. +%% R root +%term _ hsubst_rr_unique hsubst_rr_var hsubst_rr_var id/mtm_refl id/tp_refl %. +%term _ + %pi (hsubst_rr_unique (hsubst_rr_app DsM4' DsM3 DsR1) (hsubst_rr_app DsM4'-2 DsM3-2 DsR1-2) DidM4'' DidA4) + %<- (hsubst_rr_unique DsR1 DsR1-2 DidLam DidArrow) + %<- (hsubst_m_unique DsM3 DsM3-2 DidM3') + %<- (id/tp_arrow_inv DidArrow DidA3 DidA4) + %<- (id/mtm_lam_inv DidLam DidM4') + %<- (hsubst_m_respects_id DidM3' DidA3 DidM4' DsM4' DsM4'-1) + %<- (hsubst_m_unique DsM4'-1 DsM4'-2 DidM4'') %. +%term _ + %pi (hsubst_rr_unique (hsubst_rr_fst DsR') (hsubst_rr_fst DsR'') DidMl DidAl) + %<- (hsubst_rr_unique DsR' DsR'' DidPair DidProd) + %<- (id/tp_prod_inv DidProd DidAl _) + %<- (id/mtm_pair_inv DidPair DidMl _) %. +%term _ + %pi (hsubst_rr_unique (hsubst_rr_snd DsR') (hsubst_rr_snd DsR'') DidMr DidAr) + %<- (hsubst_rr_unique DsR' DsR'' DidPair DidProd) + %<- (id/tp_prod_inv DidProd _ DidAr) + %<- (id/mtm_pair_inv DidPair _ DidMr) %. +%worlds (rtm_block) (hsubst_m_unique _ _ _) (hsubst_r_unique _ _ _) (hsubst_rr_unique _ _ _ _) %. +%total (D1 D2 D3) (hsubst_r_unique D2 _ _) (hsubst_rr_unique D3 _ _ _) (hsubst_m_unique D1 _ _) %. +%{! ### Expansion + +#### Existence of expansion + +This lemma is in part an [effectiveness lemma](/wiki/effectiveness-lemma/) for expansion, but it also shows that expansion preserves types. !}% +%sort expand_exists {A} %. +%mode expand_exists %in %in %out %out %. +%term _ expand_exists b D expand_b (canon_asm D) %. +%term _ + %pi (expand_exists (arrow A2 A) DaR (expand_arrow DeApp DeX) (canon_lam DcApp)) + %<- ({x rtm} {dx atom x A2} expand_exists A2 dx (DeX x) (DcM x dx)) + %<- ({x rtm} {dx atom x A2} expand_exists A (atom_app (DcM x dx) DaR) (DeApp x) (DcApp x dx)) %. +%term _ + %pi (expand_exists (prod A1 A2) DaR (expand_prod DeS DeF) (canon_pair DcS DcF)) + %<- (expand_exists A1 (atom_fst DaR) DeF DcF) + %<- (expand_exists A2 (atom_snd DaR) DeS DcS) %. +%worlds (atom_block) (expand_exists _ _ _ _) %. +%total D (expand_exists D _ _ _) %. +%{! #### Uniqueness of expansion + +Uniqueness is proved by a straightforward induction using some identity lemmas: !}% +%sort expand_unique %. +%mode expand_unique %in %in %out %. +%term _ expand_unique expand_b expand_b id/mtm_refl %. +%term _ + %pi (expand_unique (expand_arrow (%the ({x rtm} expand A (app R (Mx x)) (MApp x)) DeApp) (%the ({x rtm} expand A2 x (Mx x)) DeX)) (expand_arrow (%the ({x rtm} expand A (app R (Mx' x)) (MApp' x)) DeApp') (%the ({x rtm} expand A2 x (Mx' x)) DeX')) DidLam) + %<- ({x rtm} expand_unique (DeX x) (DeX' x) (DidX x)) + %<- ({x rtm} id/rtm_app_cong (%the (id/rtm R R) id/rtm_refl) (DidX x) (DidApp x)) + %<- ({x rtm} expand_respects_id (DidApp x) id/mtm_refl (DeApp x) (DeApp-2 x)) + %<- ({x rtm} expand_unique (DeApp-2 x) (DeApp' x) (DidAppExp x)) + %<- (id/mtm_lam_cong DidAppExp DidLam) %. +%term _ + %pi (expand_unique (expand_prod DeS DeF) (expand_prod DeS' DeF') DidPair) + %<- (expand_unique DeF DeF' DidF) + %<- (expand_unique DeS DeS' DidS) + %<- (id/mtm_pair_cong DidF DidS DidPair) %. +%worlds (rtm_block) (expand_unique _ _ _) %. +%total D (expand_unique _ D _) %. +%{! ## Elaboration + +In [part 2](/wiki/hereditary-substitution-for-the-stlc-part-2/), we discuss elaboration. + +The complete Twelf code for this tutorial contains the identity lemmas that we elided above. + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Hereditary_substitution_for_the_STLC). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/hereditary-substitution-with-a-zipper.lf b/new-tests/stelf-output/pages/hereditary-substitution-with-a-zipper.lf new file mode 100644 index 0000000..556dbba --- /dev/null +++ b/new-tests/stelf-output/pages/hereditary-substitution-with-a-zipper.lf @@ -0,0 +1,151 @@ +% +%{! This article represents a (partially unsuccessful) attempt to remove the "ugly" portion from the global soundness proof in the [verifications and uses](/wiki/verifications-and-uses/) article. +The language of propositions and rules (and, as a result, the argument for global completeness) is unchanged from the [verifications and uses](/wiki/verifications-and-uses/), and so we omit +it here. + +This article is an intermediate point between the [verifications and uses](/wiki/verifications-and-uses/) article and the [verifications and uses with zippers](/wiki/verifications-and-uses-with-zippers/) article. | hidden = true !}% +%sort prop %. +%sort atom %. +%block bl_atom [qp atom]%. +%term a %pi atom %-> prop %. +%term ⊃ %pi prop %-> prop %-> prop %. +%prec %right 9 ⊃ %. +%term ∧ %pi prop %-> prop %-> prop %. +%prec %right 8 ∧ %. +%sort hyp %. +%sort verif %. +%sort use %. +%block bl_hyp {A prop} [x hyp A]%. +%term var %pi (hyp A) %-> (use A) %. +%term atm %pi (use (a Q)) %-> (verif (a Q)) %. +%term ⊃I %pi (%pi (hyp A) %-> (verif B)) %-> (verif (A ⊃ B)) %. +%term ⊃E %pi (use (A ⊃ B)) %-> (verif A) %-> (use B) %. +%term ∧I %pi (verif A) %-> (verif B) %-> (verif (A ∧ B)) %. +%term ∧E₁ %pi (use (A ∧ B)) %-> (use A) %. +%term ∧E₂ %pi (use (A ∧ B)) %-> (use B) %. +%{! Rather than the (somewhat ugly) process used to find the main variable +in the [verifications and uses](/wiki/verifications-and-uses/) example, in this example we will create +a sort-of zipper data structure that allows us to "pull out" the +head variable from the inside of a term. !}% +%{! ## Defining pseudo-zippers !}% +%{! A [[w:zipper|zipper data structure]] is a way of describing paths into +complex structures. A "real" zipper over an atomic term has the structure of a +_spine_ in a [[spine form]] presentation of logic; what we present +here isn't "really" a spine (or a zipper). !}% +%sort zip %. +%term end zip A A %. +%term ⊃Z %pi (zip A (B₁ ⊃ B₂)) %-> (verif B₁) %-> (zip A B₂) %. +%term ∧Z₁ %pi (zip A (B₁ ∧ B₂)) %-> (zip A B₁) %. +%term ∧Z₂ %pi (zip A (B₁ ∧ B₂)) %-> (zip A B₂) %. +%sort use' %. +%term · %pi (hyp A) %-> (zip A B) %-> (use' B) %. +%prec %none 10 · %. +%{! For instance, the ``use'`` +corresponding to ``(⊃E (⊃E (⊃E (var x) N<sub>1</sub>) N<sub>2</sub>) +N<sub>3</sub>)`` is +``x · (⊃Z (⊃Z (⊃Z end N<sub>1</sub>) N<sub>2</sub>) N<sub>3</sub>)`` --- +the head variable ``x`` has been brought out to the top of the term, but +the subterm ``N<sub>1</sub>`` is still nested more deeply +than the subterms ``N<sub>2</sub>`` and +``N<sub>3</sub>``. In a conversion to spine form, +we would not only expose the head variable ``x`` but would make +``N<sub>1</sub>`` the "least deeply nested" subterm and make +``N<sub>3</sub>`` the "most deeply nested" subterm. !}% +%{! ## Zipping and unzipping !}% +%{! We need to both show that we can zip and unzip a ``use`` into +a ``use'``, and vice versa. These two proofs are essentially the +same logic program run in opposite directions, but Twelf only allows +us to assign a single mode to a metatheorem, so rather than just +copying and pasted we have "cleaned up" both the ``unzip`` +and ``rezip`` functions a bit. !}% +%sort unzip %. +%term _ %pi (unzip (⊃E R N) (X · ⊃Z Z N)) %<- (unzip R (X · Z)) %. +%term _ %pi (unzip (∧E₁ R) (X · ∧Z₁ Z)) %<- (unzip R (X · Z)) %. +%term _ %pi (unzip (∧E₂ R) (X · ∧Z₂ Z)) %<- (unzip R (X · Z)) %. +%term _ unzip (var X) (X · end) %. +%mode unzip %in %out %. +%worlds (bl_atom bl_hyp) (unzip _ _) %. +%total R (unzip R _) %. +%sort rezip %. +%term _ %pi (rezip X (⊃Z Z N) (⊃E R N)) %<- (rezip X Z R) %. +%term _ %pi (rezip X (∧Z₁ Z) (∧E₁ R)) %<- (rezip X Z R) %. +%term _ %pi (rezip X (∧Z₂ Z) (∧E₂ R)) %<- (rezip X Z R) %. +%term _ rezip X end (var X) %. +%mode rezip %in %in %out %. +%worlds (bl_atom bl_hyp) (rezip _ _ _) %. +%total Z (rezip _ Z _) %. +%{! ## Global soundness !}% +%sort hsubst_n {A} %. +%sort hsubst_r {A} %. +%sort hsubst_rr {A} %. +%sort hsubst_rn {A} {B} %. +%mode hsubst_n %in %in %in %out %. +%mode hsubst_r %in %in %in %out %. +%mode hsubst_rr %in %in %in %out %. +%mode hsubst_rn %in %in %in %in %out %. +%term _ + %pi (hsubst_n A M₀ ([x] ⊃I ([y] M x y)) (⊃I ([y] N y))) + %<- ({y hyp B₁} hsubst_n A M₀ ([x] M x y) (%the (verif B₂) (N y))) %. +%term _ + %pi (hsubst_n A M₀ ([x] ∧I (M₁ x) (M₂ x)) (∧I N₁ N₂)) + %<- (hsubst_n A M₀ ([x] M₁ x) (%the (verif B₁) N₁)) + %<- (hsubst_n A M₀ ([x] M₂ x) (%the (verif B₂) N₂)) %. +%term _ + %pi (hsubst_n A M₀ ([x] atm (R x)) N) + %<- ({x hyp A} unzip (R x) (R' x)) + %<- (hsubst_r A M₀ ([x] R' x) N) %. +%term _ %pi (hsubst_r A M₀ ([x] x · Z x) N) %<- (hsubst_rn A _ M₀ ([x] Z x) N) %. +%term _ + %pi (hsubst_r A M₀ ([x] Y · Z x) (atm R)) + %<- (hsubst_rr A M₀ ([x] Z x) Z') + %<- (rezip Y Z' R) %. +%term _ + %pi (hsubst_rr A M₀ ([x] ⊃Z (Z x) (M x)) (⊃Z Z' N)) + %<- (hsubst_rr A M₀ ([x] Z x) Z') + %<- (hsubst_n A M₀ ([x] M x) N) %. +%term _ %pi (hsubst_rr A M₀ ([x] ∧Z₁ (Z x)) (∧Z₁ Z')) %<- (hsubst_rr A M₀ ([x] Z x) Z') %. +%term _ %pi (hsubst_rr A M₀ ([x] ∧Z₂ (Z x)) (∧Z₂ Z')) %<- (hsubst_rr A M₀ ([x] Z x) Z') %. +%term _ hsubst_rr A M₀ ([x] end) end %. +%term _ + %pi (hsubst_rn A _ M₀ ([x] ⊃Z (Z x) (M x)) N') + %<- (hsubst_rn A _ M₀ ([x] Z x) (%the (verif (B₁ ⊃ B₂)) (⊃I ([y] N y)))) + %<- (hsubst_n A M₀ ([x] M x) (%the (verif B₁) M')) + %<- (hsubst_n B₁ M' ([y] N y) (%the (verif B₂) N')) %. +%term _ + %pi (hsubst_rn A _ M₀ ([x] ∧Z₁ (Z x)) N₁) + %<- (hsubst_rn A _ M₀ ([x] Z x) (%the (verif (B₁ ∧ B₂)) (∧I N₁ N₂))) %. +%term _ + %pi (hsubst_rn A _ M₀ ([x] ∧Z₂ (Z x)) N₂) + %<- (hsubst_rn A _ M₀ ([x] Z x) (%the (verif (B₁ ∧ B₂)) (∧I N₁ N₂))) %. +%term _ hsubst_rn A _ M₀ ([x] end) M₀ %. +%worlds (bl_atom bl_hyp) (hsubst_n _ _ _ _) (hsubst_rr _ _ _ _) (hsubst_rn _ _ _ _ _) (hsubst_r _ _ _ _) %. +%reduces <= B A (hsubst_rn A B _ _ _) %. +%{! ## Failure of termination checking !}% +%{! The fact that our representation uses both proofs ``use A`` and +proofs ``use' A`` means that we will run afowl of Twelf's termination +checker --- an unzipped term has a different size than the corresponding +zipped term. We could certainly convince Twelf that zipping and unzipping +preserved size by using the same tree-like [structural metric](/wiki/structural-metrics/) used in the +[concrete representation](/wiki/concrete-representation/) case study, but that would be notationally +heavy and unenlightening. + +Another option is to ask Twelf to trust us: !}% +%reduces = R' R (unzip R R') %. +%{! However, for some reason +(possibly because the types of the two terms is different) this does not +work. and we still get an error message from the final `%total` +declaration. + +```checkedtwelf +%total {(A B C D) (M R S T)} +(hsubst_n A _ M _) +(hsubst_rr C _ S _) +(hsubst_rn D _ _ T _) +(hsubst_r B _ R _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Hereditary_substitution_with_a_zipper). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/hereditary-substitution.lf b/new-tests/stelf-output/pages/hereditary-substitution.lf new file mode 100644 index 0000000..5c8664b --- /dev/null +++ b/new-tests/stelf-output/pages/hereditary-substitution.lf @@ -0,0 +1,31 @@ +% +%{! [LF](/wiki/lf/) is defined so that only [canonical form](/wiki/canonical-form/)s exist. However, canonical forms are not closed under substitution. **Hereditary substitution** is an algorithm that directly computes the canonical result of an ordinary substiution of one canonical form into another. This algorithm has been applied in several additional type theories as well. + +\{\{needs|citations of said type theories.\}\} + +## Example + +Substitution of one canonical form into another does not necessarily produce a canonical result. For example: + + + +Even though both terms are canonical forms, the result is [beta-reducible](/wiki/beta-equivalence/). + +However, _hereditary substitution_ directly computes the canonical result of an ordinary substitution. When ordinary substitution would return a non-canonical form, hereditary substitution continues to reduce by substituting the argument into the body of the function. In the above example, the hereditary substitution + + + +Whenever we use the notation for LF, we mean hereditary substitution. + +\{\{needs|an explanation of the hereditary substitution algorithm and its metatheory.\}\} + +## See also +* The tutorial on [hereditary substitution for the STLC](/wiki/hereditary-substitution-for-the-stlc/) formalizes the hereditary substitution algorithm for a simply typed lambda-calculus in Twelf. + +\{\{stub\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Hereditary_substitution). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/higher-order-abstract-syntax.lf b/new-tests/stelf-output/pages/higher-order-abstract-syntax.lf new file mode 100644 index 0000000..d847827 --- /dev/null +++ b/new-tests/stelf-output/pages/higher-order-abstract-syntax.lf @@ -0,0 +1,42 @@ +% +%{! **Higher-order abstract syntax** is a technique for the representation of [abstract syntax](/wiki/abstract-syntax/) trees for languages with bound variables. It is often abbreviated as **HOAS**. In the context of mechanizing metatheory, HOAS refers to a specific technique where binding constructs in the [object language](/wiki/object-logic/) are represented by the binding construct of the [meta-language](/wiki/meta-logic/). When the meta language is Twelf, this is the λ construct. + +An abstract syntax tree is _abstract_ because it is a mathematical object that has certain structure by its very nature. For instance, in first-order abstract syntax (FOAS) trees, as commonly used in compilers, the tree structure implies the subexpression relation, meaning that no parentheses are required to disambiguate programs (as they are in the concrete syntax). HOAS exposes additional structure: the relationship between variables and their binding sites. In FOAS representations, a variable is typically represented with an identifier and the relation between binding site and use is indicated by the use of the _same_ identifier. FOAS representations thus need to formalize the rules of identifier lookup, and often need to formalize other conventions of language metatheory, such as [alpha-equivalence](/wiki/alpha-equivalence/), [weakening](/wiki/weakening-lemma/), [exchange](/wiki/exchange-lemma/), and type-preserving [substitution](/wiki/substitution-lemma/). When it can be used, HOAS is powerful because the metalanguage can provide these properties for free, reducing the amount of boilerplate code in an encoding. + +## Example +For example, LF has a λ construct, which has arrow (→) type. A first-order encoding of an object language construct ``let`` might be (using Twelf syntax): !}% +%sort exp %. +%sort var %. +%term v %pi var %-> exp %. +%term let %pi exp %-> var %-> exp %-> exp %. +%{! Here, ``exp`` is the family of object language expressions. The family ``var`` is the representation of variables (implemented perhaps as natural numbers, which is not shown); the constant ``v`` witnesses the fact that variables are expressions. The constant ``let`` is an expression that takes three arguments: an expression (that is being bound), a variable (that it is bound to) and another expression (that the variable is bound within). + +The natural HOAS representation of the same object language would be: !}% +%sort exp %. +%term let %pi exp %-> (%pi exp %-> exp) %-> exp %. +%{! In this representation, object level variables do not appear explicitly. The constant ``let`` takes an expression (that is being bound) and a meta-level function ``exp`` → ``exp`` (the body of the let). This function is the _higher-order_ part: an expression with a free variable is represented as an expression with _holes_ that are filled in by the meta-level function when applied. As a concrete example, we would construct the object level expression + +``` +let x = 1 + 2 +in x + 3 +``` + +(assuming the natural constructors for numbers and addition) using the HOAS signature above as + +``` +let (plus 1 2) ([y] plus y 3) +``` + +where ``[y] e`` is Twelf's syntax for the function _λy.e_. + +Because this technique reuses the mechanism of the meta-language to encode a concept in the object language, it is generally only applicable when the meta-language and object-language notions of binding coincide. This is often the case, but not always: for instance, it is unlikely that a HOAS encoding of dynamic scope such as in Lisp would be possible in a statically-scoped language like LF. + +## See also +* [Higher-order judgment](/wiki/higher-order-judgements/)s: a related LF representation technique, in which object-language [hypothetical judgment](/wiki/hypothetical-judgment/)s are represented using LF binding. +* [Equality](/wiki/equality/): when an object language is represented using HOAS, equality of LF terms corresponds with [α-equivalence](/wiki/alpha-equivalence/) of object language terms. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Higher-order_abstract_syntax). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/higher-order-judgements.lf b/new-tests/stelf-output/pages/higher-order-judgements.lf new file mode 100644 index 0000000..4717c36 --- /dev/null +++ b/new-tests/stelf-output/pages/higher-order-judgements.lf @@ -0,0 +1,99 @@ +% +% +%{! When representing [judgment](/wiki/judgment/)s in [LF](/wiki/lf/), it is often possible to represent [hypothetical judgment](/wiki/hypothetical-judgment/)s using LF binding. We call this representation technique _higher-order judgments_ because judgments are represented using higher-order types in LF. Higher-order representations are advantageous because hypothetical judgment properties such as [weakening](/wiki/weakening-lemma/), [exchange](/wiki/exchange-lemma/), and [substitution](/wiki/substitution-lemma/) are inherited "for free" from the corresponding properties of LF. + +## Example + +### Hypothetical judgment in standard notation + +As an example, we use the typing judgment for the [simply-typed lambda calculus](/wiki/simply-typed-lambda-calculus/). This calculus has the following syntax: + +```math +\tau ::= \texttt{unit} \,|\, \tau_1 \rightarrow \tau_2 +``` +```math +\texttt{e} ::= x \,|\, \langle\rangle \,|\, \lambda x :\tau . e\ |\, e_1\ e_2 +``` + +The terms are the variable , the empty pair (which has type ``unit``), lambda abstraction (with a type annotation), and application. + +The typing rules for the simply typed lambda calculus use a typing context ; containing assumptions of the form . Such a context is well-formed when all variables in it are distinct. + +```math +{ + \over + \gamma, x : \tau, \gamma' \vdash x : \tau +}\textit{of-var} +\qquad +{ + \gamma, x: \tau_2 \vdash e : \tau + \over + \gamma \vdash \lambda x:\tau_2.e : (\tau_2 \rightarrow \tau) +}\textit{of-lam} +``` +```math +{ + \over + \gamma \vdash \langle\rangle : \texttt{unit} +}\textit{of-empty} +\qquad +{ + \gamma \vdash e_1 : (\tau_2 \rightarrow \tau) + \qquad + \gamma \vdash e_2 : \tau_2 + \over + \gamma \vdash \lambda x:\tau_2.e : (\tau_2 \rightarrow \tau) +}\textit{of-app} +``` + +This is a _hypothetical judgement_, which means that the following structural properties are true: +* Hypothesis: is derivable. +* Weakening: if and is fresh then . +* Exchange: if then . +* Substitution: if and then . + +Hypothesis is derivable by the rule . Weakening, exchange, and substitution are admissible. + +### LF representation + +We represent the syntax of this calculus with the following LF signature: !}% +%sort tp %. +%term arrow %pi tp %-> tp %-> tp %. +%term unit tp %. +%sort tm %. +%term empty tm %. +%term app %pi tm %-> tm %-> tm %. +%term lam %pi tp %-> (%pi tm %-> tm) %-> tm %. +%{! Terms are represented using [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/). + +As an example of higher-order representations of judgments, we use LF binding to represent the object-language typing judgement. The following LF signature represents the above judgement with the LF type family ``of``. !}% +%sort of %. +%term of-empty of empty unit %. +%term of-lam + %pi (of (lam T2 ([x] E x)) (arrow T2 T)) + %<- ({x tm} %pi (of x T2) %-> (of (E x) T)) %. +%term of-app %pi (of (app E1 E2) T) %<- (of E1 (arrow T2 T)) %<- (of E2 T2) %. +%{! The first thing to note is that the type family is indexed by a ``tm`` and a ``tp`` but **not** a representation of the context ;. The reason for this is that we identify the object-language context with the LF context. Specifically, an object-language assumption does two things: +* It binds the variable . +* It declares a typing assumption . + +Thus, an object-language assumption is represented by the following two LF assumptions: + +``x : tm, dx : of x T `` (where ``T`` is the encoding of ). + +The first LF variable represents an object-language term ``x``, as per the encoding of syntax in the previous section. The second variable represents a derivation that ``of x T``. Consequently, there is no LF constant corresponding to the rule ; uses of this rule are represented by uses of the corresponding LF variable ``dx``. + +This representation of hypotheses gives rise to the higher-order premise of the constant ``of-lam``, which has type + +``{x: tm} of x T2 -> of (E x) T`` + +An LF term of this type has the form ``([x] [dx: of x T2] M)``, where ``M : of (E x) T`` _in an LF context extended with ``x : tm, dx : of x T2``_. Thus, ``M`` is the representation of an object-language derivation under the additional assumption . + +The constants ``of-empty`` and ``of-app`` correspond to the informal inference rules of the same name. + +## See also +* the [introductions to Twelf](/wiki/introductions-to-twelf/) for more discussion of higher-order representations of hypothetical judgments. +* [Reformulating languages to use hypothetical judgements](/wiki/reformulating-languages-to-use-hypothetical-judgements/) +* [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/) + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/hoas-nat-bijection.lf b/new-tests/stelf-output/pages/hoas-nat-bijection.lf new file mode 100644 index 0000000..14e113d --- /dev/null +++ b/new-tests/stelf-output/pages/hoas-nat-bijection.lf @@ -0,0 +1,4391 @@ +% +%{! # Bijection from HOAS to natural numbers !}% +%{! ## Preliminaries !}% +%{! +This file shows a simple HOAS term type and a mapping from these +terms to the natural numbers. It depends on John Boyland's +library signatures, in particular void, bool, nat and natpair. + +Such mappings are useful so that the library signatures +for set and map can be used to form sets or maps using the +HOAS terms as keys. + +The mapping is proved total, deterministic, onto and one2one in +four separate metatheorems. (The actual relation itself is not +proved total or deterministic directly.). Thus the mapping is +a bijection. + +This file is a proof of concept that this sort of mapping can be done +and more importantly, proved correct, in Twelf. The HOAS terms themselves +are uninteresting. The purpose in doing these proofs is to help me +(John Boyland) write a general package to generate such mappings and +their proofs. + +In writing the proofs, I am greatly indebted to Rob Simmons who took +up my challenge to write bijections involving HOAS terms. However, +I go against his explicit advice in using the "%theorem" syntax +extensively which I find much more clear that the "preferred" Twelf +way of purity. +!}% +%{! | hidden=true !}% +%%%% Imports +%%%%%% Proposed Twelf Standard Library +%%%%%% John Boyland +%%%%%% You may freely use, modify and distribute this file without restrictions. +% This file includes some simple types that should +% be standardized. Here is my proposal. +%%%% Structure +%%% The uninhabited type +%sort void %. +%freeze void %. +%%% The uninteresting type +%sort unit %. +%term unit/ unit %. +%freeze unit %. +%%% The three-way comparison type +%sort comp %. +%term less comp %. +%term greater comp %. +%term equal comp %. +%freeze comp %. +%%%%% bool.elf +%%%%% Boolean literals +%%%%% John Boyland +%%%% Definitions +%sort bool %. +%term true bool %. +%term false bool %. +%freeze bool %. +%sort eq %. +%term eq/ eq B B %. +%sort ne %. +%term ne/TF ne true false %. +%term ne/FT ne false true %. +%sort eq? %. +%term eq?/yes eq? B B true %. +%term eq?/no %pi (eq? B1 B2 false) %<- (ne B1 B2) %. +%%%% Theorems +%%% theorems about eq +%theorem false-implies-eq : forall* {X1} {X2} forall {F void} exists {E eq X1 X2} true %. +%worlds () (false-implies-eq _ _) %. +%total {} (false-implies-eq _ _) %. +%theorem meta-eq : forall {X1} {X2} {E eq X1 X2} true %. +%term _ meta-eq _ _ eq/ %. +%worlds () (meta-eq _ _ _) %. +%total {} (meta-eq _ _ _) %. +%reduces = X Y (meta-eq X Y _) %. +%theorem eq-reflexive : forall {X} exists {E eq X X} true %. +%term _ eq-reflexive _ eq/ %. +%worlds () (eq-reflexive _ _) %. +%total {} (eq-reflexive _ _) %. +%theorem eq-symmetric : forall* {X} {Y} forall {E eq X Y} exists {F eq Y X} true %. +%term _ eq-symmetric eq/ eq/ %. +%worlds () (eq-symmetric _ _) %. +%total {} (eq-symmetric _ _) %. +%theorem eq-transitive : forall* {X} {Y} {Z} forall {E1 eq X Y} {E2 eq Y Z} exists {F eq X Z} true %. +%term _ eq-transitive eq/ eq/ eq/ %. +%worlds () (eq-transitive _ _ _) %. +%total {} (eq-transitive _ _ _) %. +%%% theorems about ne +%theorem false-implies-ne : forall* {X1} {X2} forall {F void} exists {G ne X1 X2} true %. +%worlds () (false-implies-ne _ _) %. +%total {} (false-implies-ne _ _) %. +%theorem ne-respects-eq : forall* {X1} {X2} {Y1} {Y2} forall {D1 ne X1 X2} {E1 eq X1 Y1} {E2 eq X2 Y2} exists {D2 ne Y1 Y2} true %. +%term _ ne-respects-eq X1<>X2 eq/ eq/ X1<>X2 %. +%worlds () (ne-respects-eq _ _ _ _) %. +%total {} (ne-respects-eq _ _ _ _) %. +%theorem ne-anti-reflexive : forall* {B} forall {R ne B B} exists {F void} true %. +%worlds () (ne-anti-reflexive _ _) %. +%total {} (ne-anti-reflexive _ _) %. +%theorem ne-symmetric : forall* {B1} {B2} forall {R1 ne B1 B2} exists {R2 ne B2 B1} true %. +%term _ ne-symmetric ne/TF ne/FT %. +%term _ ne-symmetric ne/FT ne/TF %. +%worlds () (ne-symmetric _ _) %. +%total {} (ne-symmetric _ _) %. +%theorem eq-ne-implies-false : forall* {B1} {B2} forall {D1 eq B1 B2} {D2 ne B1 B2} exists {F void} true %. +%worlds () (eq-ne-implies-false _ _ _) %. +%total {} (eq-ne-implies-false _ _ _) %. +%%% theorems about eq? +%theorem eq?-total* : forall {B1} {B2} exists {B} {EQ? eq? B1 B2 B} true %. +%term _ eq?-total* true true _ eq?/yes %. +%term _ eq?-total* false false _ eq?/yes %. +%term _ eq?-total* true false _ (eq?/no ne/TF) %. +%term _ eq?-total* false true _ (eq?/no ne/FT) %. +%worlds () (eq?-total* _ _ _ _) %. +%total {} (eq?-total* _ _ _ _) %. +%inline eq?-total eq?-total* _ _ _ %. +%inline bool`bool bool %. +%inline bool`true true %. +%inline bool`false false %. +%inline bool`eq eq %. +%inline bool`eq/ eq/ %. +%inline bool`ne ne %. +%inline bool`ne/TF ne/TF %. +%inline bool`ne/FT ne/FT %. +%inline bool`eq? eq? %. +%inline bool`eq?/yes eq?/yes %. +%inline bool`eq?/no eq?/no %. +%inline bool`false-implies-eq false-implies-eq %. +%inline bool`meta-eq meta-eq %. +%inline bool`eq-reflexive eq-reflexive %. +%inline bool`eq-symmetric eq-symmetric %. +%inline bool`eq-transitive eq-transitive %. +%inline bool`false-implies-ne false-implies-ne %. +%inline bool`ne-respects-eq ne-respects-eq %. +%inline bool`ne-anti-reflexive ne-anti-reflexive %. +%inline bool`ne-symmetric ne-symmetric %. +%inline bool`eq-ne-implies-false eq-ne-implies-false %. +%inline bool`eq?-total* eq?-total* %. +%inline bool`eq?-total eq?-total %. +%%%%% Natural numbers +%%%%% John Boyland +%%%%% Anyone may use, copy or modify this software without restriction +%%%%% This file requires std.elf +% The natural numbers signature comes in several pieces, +% all of which are concatenated in nat.elf: +% - nat-base.elf (basic definitions, relations and operations) +% - nat-comp.elf (composed orders: ge, ne) +% - nat-inv.elf (inverse operation: minus) +% - nat-less.elf (inverse orders: lt, le) +% - nat-inv-comp.elf (theorems about minus and composed relations) +% - nat-inv-less.elf (theorems about minus and inverse orders) +% - nat-divrem.elf (quotient/remainder operation) +% - nat-minmax-elf (min/max operations) +% With the exception of the nat-inv-XXX.elf files, +% all later files depend on (require) only the nat-base.elf file. +% The nat-inv-XXX.elf files depend also on nat-inv.elf and nat-XXX.elf. +% The theorems in this signature mostly fall into the following groups: +% false-implies-XXX: one can derive XXX after a contradiction +% XXX-respects-eq: one can substitute equal terms in a relation XXX +% XXX-total: effectiveness lemma for XXX +% XXX-deterministic: uniqueness lemma for XXX +% XXX-reflexive, XXX-symmetric, XXX-transitive: properties of an equivalence +% XXX-anti-reflexive, XXX-anti-symmetric: properties of a partial order +% XXX-commutative, XXX-associative: properties of a binary operation +% XXX-left/right-distributes-over-YYY: distribution theorem +% XXX-left/right-factors-over-YYY: converse of distribution theorem +% XXX-left/right-preserves-ORD: If X ORD Y then we can apply Z to both sides +% XXX-left/right-cancels: cancellation property of binary operator XXX +% XXX-left/right-cancels-ORD: cancellation property w.r.t. order ORD +% XXX-contradiction: case where XXX can never happen +% XXX-implies-YYY: if XXX is true, we show YYY is true +% TTT-converse: converse of theorem TTT +% Additionally there are varieties of theorems with star appended to the name. +% These versions of the theorems typically require more inputs. +%%%%% nat-base.elf +%%%%% Basic definitions, operations and theorems +%%%%% This file is part of the nat.elf signature +%%%% Definitions +%%% Natural numbers: +%sort nat %. +%name nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%freeze nat %. +%%% Operations on natural numbers +%sort plus %. +%term plus/z plus z Y Y %. +%term plus/s %pi (plus (s X) Y (s Z)) %<- (plus X Y Z) %. +%sort times %. +%term times/z times z X z %. +%term times/s %pi (times (s X) Y Z) %<- (plus T Y Z) %<- (times X Y T) %. +%sort eq %. +%term eq/ eq N N %. +%sort gt %. +%term gt/1 gt (s M) M %. +%term gt/> %pi (gt (s M) N) %<- (gt M N) %. +%%% Using the conditional for natural numbers +%sort compare %. +%term compare/= compare N N equal %. +%term compare/< %pi (compare M N less) %<- (gt N M) %. +%term compare/> %pi (compare M N greater) %<- (gt M N) %. +%%%% Theorems +%%% Theorems about eq +%reduces = X Y (eq X Y) %. +%theorem meta-eq : forall {M} {N} {E eq M N} true %. +%term _ meta-eq N N eq/ %. +%worlds () (meta-eq _ _ _) %. +%total {} (meta-eq _ _ _) %. +%reduces = X Y (meta-eq X Y _) %. +%theorem false-implies-eq : forall* {M} {N} forall {P void} exists {Q eq M N} true %. +%worlds () (false-implies-eq _ M=N) %. +%total {} (false-implies-eq _ _) %. +%theorem eq-symmetric : forall* {M nat} {N nat} forall {E eq M N} exists {F eq N M} true %. +%term _ eq-symmetric eq/ eq/ %. +%worlds () (eq-symmetric M>N N>M) %. +%total {} (eq-symmetric _ _) %. +%theorem eq-transitive : forall* {M nat} {N nat} {P nat} forall {E1 eq M N} {E2 eq N P} exists {F eq M P} true %. +%term _ eq-transitive eq/ eq/ eq/ %. +%worlds () (eq-transitive M>N N>P M>P) %. +%total {} (eq-transitive _ _ _) %. +%theorem succ-deterministic : forall* {N1 nat} {N2 nat} forall {E eq N1 N2} exists {F eq (s N1) (s N2)} true %. +%term _ succ-deterministic eq/ eq/ %. +%worlds () (succ-deterministic N1=N2 N1+1=N2+1) %. +%total {} (succ-deterministic E _) %. +%theorem succ-cancels : forall* {N1 nat} {N2 nat} forall {E eq (s N1) (s N2)} exists {F eq N1 N2} true %. +%term _ succ-cancels eq/ eq/ %. +%worlds () (succ-cancels N1+1=N2+1 N1=N2) %. +%total {} (succ-cancels E _) %. +%theorem eq-contradiction : forall* {N} forall {E eq z (s N)} exists {F void} true %. +%worlds () (eq-contradiction ZERO=N+1 _) %. +%total {} (eq-contradiction _ _) %. +%%% Theorems about gt +%reduces < M N (gt N M) %. +%% If we want to prove the termination of a theorem using gt, +%% we need the gt relation lifted to the meta level: +%theorem meta-gt : forall {M} {N} {G gt M N} true %. +%term _ meta-gt (s M) M gt/1 %. +%term _ %pi (meta-gt (s M) N (gt/> G)) %<- (meta-gt M N G) %. +%worlds () (meta-gt _ _ _) %. +%total M (meta-gt M _ _) %. +%reduces < M N (meta-gt N M _) %. +%theorem false-implies-gt : forall* {M} {N} forall {P void} exists {Q gt M N} true %. +%worlds () (false-implies-gt _ M>N) %. +%total {} (false-implies-gt _ _) %. +%theorem gt-respects-eq : forall* {M1 nat} {M2 nat} {N1 nat} {N2 nat} forall {P gt M1 N1} {E1 eq M1 M2} {E2 eq N1 N2} exists {Q gt M2 N2} true %. +%term _ gt-respects-eq M1>N1 eq/ eq/ M1>N1 %. +%worlds () (gt-respects-eq M1>N1 M1=M2 N1=N2 M2>N2) %. +%total {} (gt-respects-eq _ _ _ _) %. +%theorem succ-implies-gt : forall* {X} {X'} forall {E eq X (s X')} exists {G gt X X'} true %. +%term _ succ-implies-gt eq/ gt/1 %. +%worlds () (succ-implies-gt X=sX' X>X') %. +%total {} (succ-implies-gt _ _) %. +%theorem succ-implies-gt-zero : forall {M} exists {G gt (s M) z} true %. +%term _ succ-implies-gt-zero z gt/1 %. +%term _ %pi (succ-implies-gt-zero (s M) (gt/> SM>0)) %<- (succ-implies-gt-zero M SM>0) %. +%worlds () (succ-implies-gt-zero M SM>0) %. +%total M (succ-implies-gt-zero M _) %. +%theorem succ-preserves-gt : forall* {M} {N} forall {G1 gt M N} exists {G2 gt (s M) (s N)} true %. +%term _ succ-preserves-gt gt/1 gt/1 %. +%term _ %pi (succ-preserves-gt (gt/> M>N) (gt/> SM>SN)) %<- (succ-preserves-gt M>N SM>SN) %. +%worlds () (succ-preserves-gt M>N SM>SN) %. +%total G1 (succ-preserves-gt G1 _) %. +%theorem succ-preserves-gt-converse : forall* {M} {N} forall {G1 gt (s M) (s N)} exists {G2 gt M N} true %. +%term _ succ-preserves-gt-converse gt/1 gt/1 %. +%term _ + %pi (succ-preserves-gt-converse (gt/> SM>SN) (gt/> M>N)) + %<- (succ-preserves-gt-converse SM>SN M>N) %. +%worlds () (succ-preserves-gt-converse SM>SN M>N) %. +%total G1 (succ-preserves-gt-converse G1 _) %. +%theorem gt-implies-positive : forall* {M} {N} forall {G gt M N} exists {M'} {E eq M (s M')} true %. +%term _ gt-implies-positive gt/1 M eq/ %. +%term _ gt-implies-positive (gt/> (%the (gt M N) G)) M eq/ %. +%worlds () (gt-implies-positive M>N M' M=sM') %. +%total {} (gt-implies-positive _ _ _) %. +%theorem gt-anti-reflexive* : forall {M} {G gt M M} exists {F void} true %. +%term _ + %pi (gt-anti-reflexive* (s M) (%the (gt (s M) (s M)) G) F) + %<- (succ-preserves-gt-converse G G') + %<- (gt-anti-reflexive* M G' F) %. +%worlds () (gt-anti-reflexive* M M>M _) %. +%total M (gt-anti-reflexive* M _ _) %. +%inline gt-anti-reflexive gt-anti-reflexive* _ %. +%theorem gt-transitive : forall* {M} {N} {P} forall {G1 gt M N} {G2 gt N P} exists {G3 gt M P} true %. +%term _ gt-transitive gt/1 G (gt/> G) %. +%term _ %pi (gt-transitive (gt/> M>N) N>P (gt/> M>P)) %<- (gt-transitive M>N N>P M>P) %. +%worlds () (gt-transitive M>N N>P M>P) %. +%total (G1) (gt-transitive G1 _ _) %. +%theorem gt-anti-symmetric : forall* {M} {N} forall {G1 gt M N} {G2 gt N M} exists {F void} true %. +%term _ + %pi (gt-anti-symmetric M>N N>M F) + %<- (gt-transitive M>N N>M M>M) + %<- (gt-anti-reflexive M>M F) %. +%worlds () (gt-anti-symmetric M>N N>M _) %. +%total {} (gt-anti-symmetric _ _ _) %. +%theorem gt-implies-plus : forall* {M} {N} forall {G gt M N} exists {D} {P plus (s D) N M} true %. +%term _ gt-implies-plus gt/1 z (plus/s plus/z) %. +%term _ + %pi (gt-implies-plus (gt/> M>N) (s D) (plus/s SD+N=M)) + %<- (gt-implies-plus M>N D SD+N=M) %. +%worlds () (gt-implies-plus M>N D SD+N=M) %. +%total G (gt-implies-plus G _ _) %. +%theorem gt-contradiction : forall* {M} forall {P gt z M} exists {Q void} true %. +%worlds () (gt-contradiction ZERO>N _) %. +%total {} (gt-contradiction _ _) %. +%%% Theorems about compare +%theorem false-implies-compare : forall* {M} {N} {C} forall {P void} exists {Q compare M N C} true %. +%worlds () (false-implies-compare _ _) %. +%total {} (false-implies-compare _ _) %. +%theorem succ-preserves-compare : forall* {M} {N} {C} forall {CMP compare M N C} exists {CMP' compare (s M) (s N) C} true %. +%term _ succ-preserves-compare compare/= compare/= %. +%term _ + %pi (succ-preserves-compare (compare/< M>N) (compare/< M+1>N+1)) + %<- (succ-preserves-gt M>N M+1>N+1) %. +%term _ + %pi (succ-preserves-compare (compare/> M>N) (compare/> M+1>N+1)) + %<- (succ-preserves-gt M>N M+1>N+1) %. +%worlds () (succ-preserves-compare _ _) %. +%total {} (succ-preserves-compare _ _) %. +%theorem compare-total* : forall {M} {N} exists {CMP} {P compare M N CMP} true %. +%term _ compare-total* z z equal compare/= %. +%term _ + %pi (compare-total* z (s M) less (compare/< M+1>0)) + %<- (succ-implies-gt-zero M M+1>0) %. +%term _ + %pi (compare-total* (s M) z greater (compare/> M+1>0)) + %<- (succ-implies-gt-zero M M+1>0) %. +%term _ + %pi (compare-total* (s M) (s N) R M+1-R-N+1) + %<- (compare-total* M N R M-R-N) + %<- (succ-preserves-compare M-R-N M+1-R-N+1) %. +%worlds () (compare-total* _ _ _ _) %. +%total (M) (compare-total* M _ _ _) %. +%inline compare-total compare-total* _ _ _ %. +%theorem greater-implies-gt : forall* {M} {N} forall {C compare M N greater} exists {G gt M N} true %. +%term _ greater-implies-gt (compare/> G) G %. +%worlds () (greater-implies-gt M>N M-gt-N) %. +%total C (greater-implies-gt C _) %. +%theorem less-is-reverse-greater : forall* {M} {N} forall {C1 compare M N less} exists {C2 compare N M greater} true %. +%term _ less-is-reverse-greater (compare/< G) (compare/> G) %. +%worlds () (less-is-reverse-greater MM) %. +%total C (less-is-reverse-greater C _) %. +%theorem less-implies-lt : forall* {M} {N} forall {C compare M N less} exists {G gt N M} true %. +%term _ less-implies-lt (compare/< G) G %. +%worlds () (less-implies-lt MN2 plus/z plus/z N1>N2 %. +%term _ + %pi (plus-left-preserves-gt* N1>N2 (plus/s M+N1=O1) (plus/s M+N2=O2) SO1>SO2) + %<- (plus-left-preserves-gt* N1>N2 M+N1=O1 M+N2=O2 O1>O2) + %<- (succ-preserves-gt O1>O2 SO1>SO2) %. +%worlds () (plus-left-preserves-gt* N1>N2 M+N1=O1 M+N2=O2 O1>O2) %. +%total P1 (plus-left-preserves-gt* _ P1 _ _) %. +%theorem plus-left-cancels-gt : forall* {X1 nat} {X2 nat} {Y nat} {Z nat} {S1 nat} {S2 nat} forall {P1 plus X1 Y S1} {P2 plus X2 Z S2} {EX eq X1 X2} {G1 gt S1 S2} exists {G2 gt Y Z} true %. +%term _ plus-left-cancels-gt plus/z plus/z eq/ G G %. +%term _ + %pi (plus-left-cancels-gt (plus/s X+Y1=Z1) (plus/s X+Y2=Z2) eq/ SZ1>SZ2 Y1>Y2) + %<- (succ-preserves-gt-converse SZ1>SZ2 Z1>Z2) + %<- (plus-left-cancels-gt X+Y1=Z1 X+Y2=Z2 eq/ Z1>Z2 Y1>Y2) %. +%worlds () (plus-left-cancels-gt X1+Y1=Z1 X2+Y2=Z2 X1=X2 Z1>Z2 Y1>Y2) %. +%total P1 (plus-left-cancels-gt P1 _ _ _ _) %. +%theorem plus-left-preserves-gt : forall* {X1} {X2} {X4} forall {G gt X2 X4} exists {X3} {X5} {O1 plus X1 X2 X3} {O2 plus X1 X4 X5} {G2 gt X3 X5} true %. +%term _ + %pi (plus-left-preserves-gt X2>X4 X3 X5 X1+X2=A3 X1+X4=X5 X3>X5) + %<- (plus-total X1+X2=A3) + %<- (plus-total X1+X4=X5) + %<- (plus-left-preserves-gt* X2>X4 X1+X2=A3 X1+X4=X5 X3>X5) %. +%worlds () (plus-left-preserves-gt X2>X4 X3 X5 X1+X2=A3 X1+X4=X5 X3>X5) %. +%total {} (plus-left-preserves-gt _ _ _ _ _ _) %. +%theorem plus-right-preserves-gt* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 gt X1 X2} {O1 plus X1 X3 X4} {O2 plus X2 X3 X5} exists {G2 gt X4 X5} true %. +%term _ + %pi (plus-right-preserves-gt* X1>X2 X1+X3=X4 X2+X3=X5 X4>X5) + %<- (plus-commutative X1+X3=X4 X3+X1=X4) + %<- (plus-commutative X2+X3=X5 X3+X2=X5) + %<- (plus-left-preserves-gt* X1>X2 X3+X1=X4 X3+X2=X5 X4>X5) %. +%worlds () (plus-right-preserves-gt* X1>X2 X1+X3=X4 X2+X3=X5 X4>X5) %. +%total {} (plus-right-preserves-gt* _ _ _ _) %. +%theorem plus-right-preserves-gt : forall* {X1} {X2} {X3} forall {G1 gt X1 X2} exists {X4} {X5} {O1 plus X1 X3 X4} {O2 plus X2 X3 X5} {G2 gt X4 X5} true %. +%term _ + %pi (plus-right-preserves-gt X1>X2 X4 X5 X1+X3=X4 X2+X3=X5 X4>X5) + %<- (plus-total X1+X3=X4) + %<- (plus-total X2+X3=X5) + %<- (plus-right-preserves-gt* X1>X2 X1+X3=X4 X2+X3=X5 X4>X5) %. +%worlds () (plus-right-preserves-gt X1>X2 X4 X5 X1+X3=X4 X2+X3=X5 X4>X5) %. +%total {} (plus-right-preserves-gt _ _ _ _ _ _) %. +%theorem plus-preserves-gt* : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {G1 gt X1 Y1} {G2 gt X2 Y2} {MX plus X1 X2 X3} {MY plus Y1 Y2 Y3} exists {G3 gt X3 Y3} true %. +%term _ + %pi (plus-preserves-gt* X1>Y1 X2>Y2 X1+X2=X3 Y1+Y2=Y3 X3>Y3) + %<- (plus-total Y1+X2=X) + %<- (plus-right-preserves-gt* X1>Y1 X1+X2=X3 Y1+X2=X X3>X) + %<- (plus-left-preserves-gt* X2>Y2 Y1+X2=X Y1+Y2=Y3 X>Y3) + %<- (gt-transitive X3>X X>Y3 X3>Y3) %. +%worlds () (plus-preserves-gt* X1>Y1 X2>Y2 X1+X2=X3 Y1+Y2=Y3 X3>Y3) %. +%total {} (plus-preserves-gt* _ _ _ _ _) %. +%theorem plus-preserves-gt : forall* {X1} {X2} {Y1} {Y2} forall {G1 gt X1 Y1} {G2 gt X2 Y2} exists {X3} {Y3} {MX plus X1 X2 X3} {MY plus Y1 Y2 Y3} {G3 gt X3 Y3} true %. +%term _ + %pi (plus-preserves-gt X1>Y1 X2>Y2 X3 Y3 X1+X2=X3 Y1+Y2=Y3 X3>Y3) + %<- (plus-total X1+X2=X3) + %<- (plus-total Y1+Y2=Y3) + %<- (plus-preserves-gt* X1>Y1 X2>Y2 X1+X2=X3 Y1+Y2=Y3 X3>Y3) %. +%worlds () (plus-preserves-gt X1>Y1 X2>Y2 X3 Y3 X1+X2=X3 Y1+Y2=Y3 X3>Y3) %. +%total {} (plus-preserves-gt _ _ _ _ _ _ _) %. +%theorem plus-right-cancels-gt : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {OP1 plus X1 X2 X3} {OP2 plus Y1 Y2 Y3} {E2 eq X2 Y2} {G3 gt X3 Y3} exists {G1 gt X1 Y1} true %. +%term _ + %pi (plus-right-cancels-gt X1+X2=X3 Y1+Y2=Y3 X2=Y2 X3>Y3 X1>Y1) + %<- (plus-commutative X1+X2=X3 X2+X1=X3) + %<- (plus-commutative Y1+Y2=Y3 Y2+Y1=Y3) + %<- (plus-left-cancels-gt X2+X1=X3 Y2+Y1=Y3 X2=Y2 X3>Y3 X1>Y1) %. +%worlds () (plus-right-cancels-gt X1+X2=X3 Y1+Y2=Y3 X2=Y2 X3>Y3 X1>Y1) %. +%total {} (plus-right-cancels-gt _ _ _ _ _) %. +%theorem plus-implies-gt : forall* {M} {N} {O} {M'} forall {P plus M N O} {E eq M (s M')} exists {G gt O N} true %. +%term _ + %pi (plus-implies-gt X+Y=Z eq/ Z>Y) + %<- (succ-implies-gt-zero _ X>0) + %<- (plus-right-preserves-gt* X>0 X+Y=Z plus/z Z>Y) %. +%worlds () (plus-implies-gt X+Y=Z X=sX' Z>Y) %. +%total {} (plus-implies-gt _ _ _) %. +%theorem plus-gt-contradiction : forall* {M} {N} {O} forall {P plus M N O} {G gt M O} exists {F void} true %. +%term _ + %pi (plus-gt-contradiction M+0=O M>O F) + %<- (plus-right-identity _ M+0=M) + %<- (plus-deterministic M+0=O M+0=M eq/ eq/ O=M) + %<- (gt-respects-eq M>O eq/ O=M M>M) + %<- (gt-anti-reflexive M>M F) %. +%term _ + %pi (plus-gt-contradiction M+N=O M>O F) + %<- (plus-commutative M+N=O N+M=O) + %<- (plus-implies-gt N+M=O eq/ O>M) + %<- (gt-anti-symmetric M>O O>M F) %. +%worlds () (plus-gt-contradiction M+N=O M>O _) %. +%total {} (plus-gt-contradiction _ _ _) %. +%%% Theorems about times +%theorem false-implies-times : forall* {M} {N} {O} forall {P void} exists {Q times M N O} true %. +%worlds () (false-implies-times _ _) %. +%total {} (false-implies-times _ _) %. +%theorem times-respects-eq : forall* {M1 nat} {M2 nat} {N1 nat} {N2 nat} {P1 nat} {P2 nat} forall {P times M1 N1 P1} {E1 eq M1 M2} {E2 eq N1 N2} {E3 eq P1 P2} exists {Q times M2 N2 P2} true %. +%term _ times-respects-eq M1*N1=P1 eq/ eq/ eq/ M1*N1=P1 %. +%worlds () (times-respects-eq M1*N1=P1 M1=M2 N1=N2 P1=P2 M2*N2=P2) %. +%total {} (times-respects-eq _ _ _ _ _) %. +%theorem times-total* : forall {N1 nat} {N2 nat} exists {N3 nat} {T times N1 N2 N3} true %. +%term _ times-total* z N2 z times/z %. +%term _ + %pi (times-total* (s X) Y Z (times/s X*Y=Z' Z'+Y=Z)) + %<- (times-total* X Y Z' X*Y=Z') + %<- (plus-total Z'+Y=Z) %. +%worlds () (times-total* N1 N2 N3 N1*N2=N3) %. +%total (N1) (times-total* N1 _ _ _) %. +%inline times-total times-total* _ _ _ %. +%theorem times-deterministic : forall* {N1 nat} {N1' nat} {N2 nat} {N2' nat} {N3 nat} {N3' nat} forall {P times N1 N2 N3} {P' times N1' N2' N3'} {E1 eq N1 N1'} {E2 eq N2 N2'} exists {E3 eq N3 N3'} true %. +%term _ times-deterministic times/z times/z eq/ eq/ eq/ %. +%term _ + %pi (times-deterministic (times/s X*Y=Z1 Z1+Y=Z1') (times/s X*Y=Z2 Z2+Y=Z2') eq/ eq/ Z1'=Z2') + %<- (times-deterministic X*Y=Z1 X*Y=Z2 eq/ eq/ Z1=Z2) + %<- (plus-deterministic Z1+Y=Z1' Z2+Y=Z2' Z1=Z2 eq/ Z1'=Z2') %. +%worlds () (times-deterministic X1*Y1=Z1 X2*Y2=Z2 X1=X2 Y1=Y2 Z1=Z2) %. +%total P (times-deterministic P _ _ _ _) %. +%theorem times-left-identity : forall {N nat} exists {T times (s z) N N} true %. +%term _ times-left-identity N (times/s times/z plus/z) %. +%worlds () (times-left-identity N ONE*N=N) %. +%total {} (times-left-identity _ _) %. +%theorem times-right-identity : forall {N nat} exists {T times N (s z) N} true %. +%term _ times-right-identity z times/z %. +%term _ + %pi (times-right-identity (s M) (times/s M*1=M M+1=sM)) + %<- (times-right-identity M M*1=M) + %<- (plus-right-identity M M+0=M) + %<- (plus-right-increase M+0=M M+1=sM) %. +%worlds () (times-right-identity N N*1=N) %. +%total M (times-right-identity M _) %. +%theorem times-right-zero : forall {N nat} exists {T times N z z} true %. +%term _ times-right-zero z times/z %. +%term _ + %pi (times-right-zero (s M) (times/s M*0=0 plus/z)) + %<- (times-right-zero M M*0=0) %. +%worlds () (times-right-zero N N*0=0) %. +%total M (times-right-zero M _) %. +%theorem times-preserves-positive : forall {M} {N} exists {P} {T times (s M) (s N) (s P)} true %. +%term _ {T1 times M (s N) O} {P1 plus O (s N) (s P)} {P2 plus O N P} + %pi (times-preserves-positive M N P (times/s T1 P1)) + %<- (times-total T1) + %<- (plus-total P2) + %<- (plus-right-increase P2 P1) %. +%worlds () (times-preserves-positive M N P SM*SN=SP) %. +%total {} (times-preserves-positive _ _ _ _) %. +%theorem times-preserves-positive* : forall* {M} {N} {P} {M'} {N'} forall {T times M N P} {M+ eq M (s M')} {N+ eq N (s N')} exists {P'} {P+ eq P (s P')} true %. +%term _ + %pi (times-preserves-positive* M*N=P M=sM' N=sN' P' P=sP') + %<- (times-respects-eq M*N=P M=sM' N=sN' eq/ (times/s M'*sN'=O' O'+sN'=P)) + %<- (plus-right-decrease O'+sN'=P P' P=sP' _) %. +%worlds () (times-preserves-positive* M*N=P M=sM' N=sN' P' P=sP') %. +%total {} (times-preserves-positive* _ _ _ _ _) %. +%theorem times-positive-implies-positive : forall* {M} {N} {P} {P'} forall {T times M N P} {P+ eq P (s P')} exists {M'} {M+ eq M (s M')} {N'} {N+ eq N (s N')} true %. +%term _ times-positive-implies-positive (times/s M'*N=T plus/z) eq/ M' eq/ P' eq/ %. +%term _ + %pi (times-positive-implies-positive (times/s M'*N=sT' (plus/s T'+N=P')) eq/ M' eq/ N' N=sN') + %<- (times-positive-implies-positive M'*N=sT' eq/ _ _ N' N=sN') %. +%worlds () (times-positive-implies-positive M*N=P P=sP' M' M=sM' N' N=sN') %. +%total T (times-positive-implies-positive T _ _ _ _ _) %. +%theorem times-left-increase : forall* {M} {N} {O} {X} forall {T times M N O} {P plus O N X} exists {U times (s M) N X} true %. +%term _ times-left-increase T P (times/s T P) %. +%worlds () (times-left-increase M*N=O O+N=X SM*N=X) %. +%total {} (times-left-increase _ _ _) %. +%theorem times-right-increase : forall* {M nat} {N nat} {O nat} {X nat} forall {T times M N O} {P plus M O X} exists {U times M (s N) X} true %. +%term _ times-right-increase times/z plus/z times/z %. +%term _ {M*N=O times M N O} {M+O=Z plus M O Z} + %pi (times-right-increase (times/s M*N=O O+N=O1) (plus/s M+O1=Y) (times/s M*sN=Z Z+sN=sY)) + %<- (plus-associative-converse O+N=O1 M+O1=Y Z M+O=Z Z+N=Y) + %<- (times-right-increase M*N=O M+O=Z M*sN=Z) + %<- (plus-right-increase Z+N=Y Z+sN=sY) %. +%worlds () (times-right-increase M*N=O M+O=X M*sN=X) %. +%total T (times-right-increase T _ _) %. +%theorem times-left-decrease : forall* {X} {Y} {Z} forall {T1 times (s X) Y Z} exists {Z1} {T2 times X Y Z1} {P plus Z1 Y Z} true %. +%term _ times-left-decrease (times/s T P) _ T P %. +%worlds () (times-left-decrease SX*Y=Z Z1 X*Y=Z1 Z1+Y=Z) %. +%total {} (times-left-decrease _ _ _ _) %. +%theorem times-right-decrease : forall* {M} {N} {X} forall {T times M (s N) X} exists {O} {U times M N O} {P plus M O X} true %. +%term _ times-right-decrease times/z z times/z plus/z %. +%term _ + %pi (times-right-decrease (times/s M*sN=Y Y+sN=X) O (times/s M*N=P P+N=O) SM+O=X) + %<- (times-right-decrease M*sN=Y P M*N=P M+P=Y) + %<- (plus-total P+N=O) + %<- (plus-right-increase P+N=O P+sN=sO) + %<- (plus-associative* M+P=Y Y+sN=X P+sN=sO M+sO=X) + %<- (plus-swap-succ-converse M+sO=X SM+O=X) %. +%worlds () (times-right-decrease M*sN=X O M*N=O M+O=X) %. +%total (T) (times-right-decrease T _ _ _) %. +%theorem times-commutative : forall* {N1} {N2} {N3} forall {T times N1 N2 N3} exists {U times N2 N1 N3} true %. +%term _ %pi (times-commutative times/z T) %<- (times-right-zero N2 T) %. +%term _ {T1 times N1' N2 N3'} {P2 plus N3' N2 N3} {T1c times N2 N1' N3'} {P2c plus N2 N3' N3} {Tc times N2 (s N1') N3} + %pi (times-commutative (times/s T1 P2) Tc) + %<- (plus-commutative P2 P2c) + %<- (times-commutative T1 T1c) + %<- (times-right-increase T1c P2c Tc) %. +%worlds () (times-commutative N1*N2=N3 N2*N1=N3) %. +%total T (times-commutative T _) %. +%theorem times-right-distributes-over-plus : forall* {N1} {N2} {N3} {N12} {N123} forall {P1 plus N1 N2 N12} {T1 times N12 N3 N123} exists {N13} {N23} {T13 times N1 N3 N13} {T23 times N2 N3 N23} {P123 plus N13 N23 N123} true %. +%term _ times-right-distributes-over-plus plus/z Y*Z=YZ z YZ times/z Y*Z=YZ plus/z %. +%term _ + %pi (times-right-distributes-over-plus (plus/s X+Y=XY) (times/s XY*Z=XYZ XYZ+Z=SXYZ) SXZ YZ (times/s X*Z=XZ XZ+Z=SXZ) Y*Z=YZ SXZ+YZ=SXYZ) + %<- (times-right-distributes-over-plus X+Y=XY XY*Z=XYZ XZ YZ X*Z=XZ Y*Z=YZ XZ+YZ=XYZ) + %<- (plus-commutative XZ+YZ=XYZ YZ+XZ=XYZ) + %<- (plus-associative YZ+XZ=XYZ XYZ+Z=SXYZ SXZ XZ+Z=SXZ YZ+SXZ=SXYZ) + %<- (plus-commutative YZ+SXZ=SXYZ SXZ+YZ=SXYZ) %. +%worlds () (times-right-distributes-over-plus X+Y=XY XY*Z=XYZ XZ YZ X*Z=XZ Y*Z=YZ XZ+YZ=XYZ) %. +%total (P) (times-right-distributes-over-plus P _ _ _ _ _ _) %. +%theorem times-right-distributes-over-plus* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {A12 plus X1 X2 X3} {M34 times X3 X4 X7} {M14 times X1 X4 X5} {M24 times X2 X4 X6} exists {A56 plus X5 X6 X7} true %. +%term _ + %pi (times-right-distributes-over-plus* X1+X2=X3 X3*X4=X7 X1*X4=X5 X2*X4=X6 X5+X6=X7) + %<- (times-right-distributes-over-plus X1+X2=X3 X3*X4=X7 Y5 Y6 X1*X4=Y5 X2*X4=Y6 Y5+Y6=X7) + %<- (times-deterministic X1*X4=Y5 X1*X4=X5 eq/ eq/ Y5=X5) + %<- (times-deterministic X2*X4=Y6 X2*X4=X6 eq/ eq/ Y6=X6) + %<- (plus-respects-eq Y5+Y6=X7 Y5=X5 Y6=X6 eq/ X5+X6=X7) %. +%worlds () (times-right-distributes-over-plus* X1+X2=X3 X3*X4=X7 X1*X4=X5 X2*X4=X6 X5+X6=X7) %. +%total {} (times-right-distributes-over-plus* _ _ _ _ _) %. +%theorem times-left-distributes-over-plus* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {A12 plus X2 X4 X6} {M34 times X1 X6 X7} {M14 times X1 X2 X3} {M24 times X1 X4 X5} exists {A56 plus X3 X5 X7} true %. +%term _ + %pi (times-left-distributes-over-plus* X2+X4=X6 X1*X6=X7 X1*X2=X3 X1*X4=X5 X3+X5=X7) + %<- (times-commutative X1*X6=X7 X6*X1=X7) + %<- (times-commutative X1*X2=X3 X2*X1=X3) + %<- (times-commutative X1*X4=X5 X4*X1=X5) + %<- (times-right-distributes-over-plus* X2+X4=X6 X6*X1=X7 X2*X1=X3 X4*X1=X5 X3+X5=X7) %. +%worlds () (times-left-distributes-over-plus* X2+X4=X6 X1*X6=X7 X1*X2=X3 X1*X4=X5 X3+X5=X7) %. +%total {} (times-left-distributes-over-plus* _ _ _ _ _) %. +%theorem times-left-distributes-over-plus : forall* {X1} {X2} {X4} {X6} {X7} forall {A12 plus X2 X4 X6} {M34 times X1 X6 X7} exists {X3} {X5} {M14 times X1 X2 X3} {M24 times X1 X4 X5} {A56 plus X3 X5 X7} true %. +%term _ + %pi (times-left-distributes-over-plus X2+X4=X6 X1*X6=X7 X3 X5 X1*X2=X3 X1*X4=X5 X3+X5=X7) + %<- (times-total X1*X2=X3) + %<- (times-total X1*X4=X5) + %<- (times-left-distributes-over-plus* X2+X4=X6 X1*X6=X7 X1*X2=X3 X1*X4=X5 X3+X5=X7) %. +%worlds () (times-left-distributes-over-plus X2+X4=X6 X1*X6=X7 X3 X5 X1*X2=X3 X1*X4=X5 X3+X5=X7) %. +%total {} (times-left-distributes-over-plus _ _ _ _ _ _ _) %. +%theorem times-right-factors-over-plus : forall* {X1} {X2} {X4} {X5} {X6} {X7} forall {M14 times X1 X4 X5} {M24 times X2 X4 X6} {A56 plus X5 X6 X7} exists {X3} {A12 plus X1 X2 X3} {M34 times X3 X4 X7} true %. +%term _ + %pi (times-right-factors-over-plus X1*X4=X5 X2*X4=X6 X5+X6=X7 X3 X1+X2=X3 X3*X4=X7) + %<- (plus-total X1+X2=X3) + %<- (times-total X3*X4=Y7) + %<- (times-right-distributes-over-plus* X1+X2=X3 X3*X4=Y7 X1*X4=X5 X2*X4=X6 X5+X6=Y7) + %<- (plus-deterministic X5+X6=Y7 X5+X6=X7 eq/ eq/ Y7=X7) + %<- (times-respects-eq X3*X4=Y7 eq/ eq/ Y7=X7 X3*X4=X7) %. +%worlds () (times-right-factors-over-plus X1*X4=X5 X2*X4=X6 X5+X6=X7 X3 X1+X2=X3 X3*X4=X7) %. +%total {} (times-right-factors-over-plus _ _ _ _ _ _) %. +%theorem times-right-factors-over-plus* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {M14 times X1 X4 X5} {M24 times X2 X4 X6} {A56 plus X5 X6 X7} {A12 plus X1 X2 X3} exists {M34 times X3 X4 X7} true %. +%term _ + %pi (times-right-factors-over-plus* X1*X4=X5 X2*X4=X6 X5+X6=X7 X1+X2=X3 X3*X4=X7) + %<- (times-total X3*X4=Y7) + %<- (times-right-distributes-over-plus* X1+X2=X3 X3*X4=Y7 X1*X4=X5 X2*X4=X6 X5+X6=Y7) + %<- (plus-deterministic X5+X6=Y7 X5+X6=X7 eq/ eq/ Y7=X7) + %<- (times-respects-eq X3*X4=Y7 eq/ eq/ Y7=X7 X3*X4=X7) %. +%worlds () (times-right-factors-over-plus* X1*X4=X5 X2*X4=X6 X5+X6=X7 X1+X2=X3 X3*X4=X7) %. +%total {} (times-right-factors-over-plus* _ _ _ _ _) %. +%theorem times-left-factors-over-plus : forall* {X1} {X2} {X3} {X4} {X5} {X7} forall {M12 times X1 X2 X3} {M14 times X1 X4 X5} {A35 plus X3 X5 X7} exists {X6} {A24 plus X2 X4 X6} {M16 times X1 X6 X7} true %. +%term _ + %pi (times-left-factors-over-plus X1*X2=X3 X1*X4=X5 X3+X5=X7 X6 X2+X4=X6 X1*X6=X7) + %<- (times-commutative X1*X2=X3 X2*X1=X3) + %<- (times-commutative X1*X4=X5 X4*X1=X5) + %<- (times-right-factors-over-plus X2*X1=X3 X4*X1=X5 X3+X5=X7 X6 X2+X4=X6 X6*X1=X7) + %<- (times-commutative X6*X1=X7 X1*X6=X7) %. +%worlds () (times-left-factors-over-plus X1*X2=X3 X1*X4=X5 X3+X5=X7 X6 X2+X4=X6 X1*X6=X7) %. +%total {} (times-left-factors-over-plus _ _ _ _ _ _) %. +%theorem times-left-factors-over-plus* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {M12 times X1 X2 X3} {M14 times X1 X4 X5} {A35 plus X3 X5 X7} {A24 plus X2 X4 X6} exists {M16 times X1 X6 X7} true %. +%term _ + %pi (times-left-factors-over-plus* X1*X2=X3 X1*X4=X5 X3+X5=X7 X2+X4=X6 X1*X6=X7) + %<- (times-total X1*X6=Y7) + %<- (times-left-distributes-over-plus* X2+X4=X6 X1*X6=Y7 X1*X2=X3 X1*X4=X5 X3+X5=Y7) + %<- (plus-deterministic X3+X5=Y7 X3+X5=X7 eq/ eq/ Y7=X7) + %<- (times-respects-eq X1*X6=Y7 eq/ eq/ Y7=X7 X1*X6=X7) %. +%worlds () (times-left-factors-over-plus* X1*X2=X3 X1*X4=X5 X3+X5=X7 X2+X4=X6 X1*X6=X7) %. +%total {} (times-left-factors-over-plus* _ _ _ _ _) %. +%theorem times-associative : forall* {N1} {N2} {N3} {N12} {N123} forall {T1 times N1 N2 N12} {T12 times N12 N3 N123} exists {N23} {T2 times N2 N3 N23} {T123 times N1 N23 N123} true %. +%term _ {T2 times N2 N3 N23} %pi (times-associative times/z times/z N23 T2 times/z) %<- (times-total T2) %. +%term _ {T1 times N1' N2 N1'2} {P2 plus N1'2 N2 N12} {T3 times N12 N3 N123} {T4 times N2 N3 N23} {T5 times N1' N23 N1'23} {P6 plus N1'23 N23 N123} {T7 times N1'2 N3 N1'23} + %pi (times-associative (times/s T1 P2) T3 N23 T4 (times/s T5 P6)) + %<- (times-right-distributes-over-plus P2 T3 N1'23 N23 T7 T4 P6) + %<- (times-associative T1 T7 N23' T4' T5') + %<- (times-deterministic T4' T4 eq/ eq/ N23'=N23) + %<- (times-respects-eq T5' eq/ N23'=N23 eq/ T5) %. +%worlds () (times-associative _ _ _ _ _) %. +%total T1 (times-associative T1 _ _ _ _) %. +%theorem times-associative* : forall* {X1} {X2} {X12} {X3} {X23} {X123} forall {OP12 times X1 X2 X12} {OP12-3 times X12 X3 X123} {OP23 times X2 X3 X23} exists {OP1-23 times X1 X23 X123} true %. +%term _ + %pi (times-associative* X1*X2=X3 X3*X4=X7 X2*X4=X6 X1*X6=X7) + %<- (times-associative X1*X2=X3 X3*X4=X7 Y6 X2*X4=Y6 X1*Y6=X7) + %<- (times-deterministic X2*X4=Y6 X2*X4=X6 eq/ eq/ Y6=X6) + %<- (times-respects-eq X1*Y6=X7 eq/ Y6=X6 eq/ X1*X6=X7) %. +%worlds () (times-associative* _ _ _ _) %. +%total {} (times-associative* _ _ _ _) %. +%theorem times-associative-converse : forall* {X1} {X2} {X4} {X6} {X7} forall {OP24 times X2 X4 X6} {OP16 times X1 X6 X7} exists {X3} {OP12 times X1 X2 X3} {OP34 times X3 X4 X7} true %. +%term _ + %pi (times-associative-converse X2*X4=X6 X1*X6=X7 _ X1*X2=X3 X3*X4=X7) + %<- (times-commutative X2*X4=X6 X4*X2=X6) + %<- (times-commutative X1*X6=X7 X6*X1=X7) + %<- (times-associative X4*X2=X6 X6*X1=X7 _ X2*X1=X3 X4*X3=X7) + %<- (times-commutative X2*X1=X3 X1*X2=X3) + %<- (times-commutative X4*X3=X7 X3*X4=X7) %. +%worlds () (times-associative-converse X2*X4=X6 X1*X6=X7 X3 X1*X2=X3 X3*X4=X7) %. +%total {} (times-associative-converse _ _ _ _ _) %. +%theorem times-associative-converse* : forall* {X1} {X2} {X3} {X4} {X6} {X7} forall {OP24 times X2 X4 X6} {OP16 times X1 X6 X7} {OP12 times X1 X2 X3} exists {OP34 times X3 X4 X7} true %. +%term _ + %pi (times-associative-converse* X2*X4=X6 X1*X6=X7 X1*X2=X3 X3*X4=X7) + %<- (times-associative-converse X2*X4=X6 X1*X6=X7 X3P X1*X2=X3P X3P*X4=X7) + %<- (times-deterministic X1*X2=X3P X1*X2=X3 eq/ eq/ X3P=X3) + %<- (times-respects-eq X3P*X4=X7 X3P=X3 eq/ eq/ X3*X4=X7) %. +%worlds () (times-associative-converse* X2*X4=X6 X1*X6=X7 X1*X2=X3 X3*X4=X7) %. +%total {} (times-associative-converse* _ _ _ _) %. +%theorem times-assoc-commutative* : forall* {X1} {X2} {X3} {X4} {X5} {X7} forall {OP1 times X1 X2 X3} {OP2 times X3 X4 X7} {OP3 times X1 X4 X5} exists {OP4 times X5 X2 X7} true %. +%term _ + %pi (times-assoc-commutative* X1*X2=X3 X3*X4=X7 X1*X4=X5 X5*X2=X7) + %<- (times-associative X1*X2=X3 X3*X4=X7 X6 X2*X4=X6 X1*X6=X7) + %<- (times-commutative X2*X4=X6 X4*X2=X6) + %<- (times-associative-converse* X4*X2=X6 X1*X6=X7 X1*X4=X5 X5*X2=X7) %. +%worlds () (times-assoc-commutative* X1*X2=X3 X3*X4=X7 X1*X4=X5 X5*X2=X7) %. +%total {} (times-assoc-commutative* _ _ _ _) %. +%theorem times-assoc-commutative : forall* {X1} {X2} {X3} {X4} {X7} forall {OP1 times X1 X2 X3} {OP2 times X3 X4 X7} exists {X5} {OP3 times X1 X4 X5} {OP4 times X5 X2 X7} true %. +%term _ + %pi (times-assoc-commutative X1*X2=X3 X3*X4=X7 X5 X1*X4=X5 X5*X2=X7) + %<- (times-associative X1*X2=X3 X3*X4=X7 X6 X2*X4=X6 X1*X6=X7) + %<- (times-commutative X2*X4=X6 X4*X2=X6) + %<- (times-associative-converse X4*X2=X6 X1*X6=X7 X5 X1*X4=X5 X5*X2=X7) %. +%worlds () (times-assoc-commutative X1*X2=X3 X3*X4=X7 X5 X1*X4=X5 X5*X2=X7) %. +%total {} (times-assoc-commutative _ _ _ _ _) %. +%theorem times-double-associative* : forall* {A} {B} {C} {D} {A+B} {C+D} {A+C} {B+D} {X} forall {AB times A B A+B} {CD times C D C+D} {ABCD times A+B C+D X} {AC times A C A+C} {BD times B D B+D} exists {ACBD times A+C B+D X} true %. +%term _ + %pi (times-double-associative* X1*X2=X3 X4*X8=XC X3*XC=XF X1*X4=X5 X2*X8=XA X5*XA=XF) + %<- (times-associative X1*X2=X3 X3*XC=XF XE X2*XC=XE X1*XE=XF) + %<- (times-commutative X4*X8=XC X8*X4=XC) + %<- (times-associative-converse* X8*X4=XC X2*XC=XE X2*X8=XA XA*X4=XE) + %<- (times-commutative XA*X4=XE X4*XA=XE) + %<- (times-associative-converse* X4*XA=XE X1*XE=XF X1*X4=X5 X5*XA=XF) %. +%worlds () (times-double-associative* X1*X2=X3 X4*X8=XC X3*XC=XF X1*X4=X5 X2*X8=XA X5*XA=XF) %. +%total {} (times-double-associative* _ _ _ _ _ _) %. +%theorem times-double-associative : forall* {A} {B} {C} {D} {A+B} {C+D} {X} forall {AB times A B A+B} {CD times C D C+D} {ABCD times A+B C+D X} exists {A+C} {B+D} {AC times A C A+C} {BD times B D B+D} {ACBD times A+C B+D X} true %. +%term _ + %pi (times-double-associative X1*X2=X3 X4*X8=XC X3*XC=XF X5 XA X1*X4=X5 X2*X8=XA X5*XA=XF) + %<- (times-associative X1*X2=X3 X3*XC=XF XE X2*XC=XE X1*XE=XF) + %<- (times-commutative X4*X8=XC X8*X4=XC) + %<- (times-associative-converse X8*X4=XC X2*XC=XE XA X2*X8=XA XA*X4=XE) + %<- (times-commutative XA*X4=XE X4*XA=XE) + %<- (times-associative-converse X4*XA=XE X1*XE=XF X5 X1*X4=X5 X5*XA=XF) %. +%worlds () (times-double-associative _ _ _ _ _ _ _ _) %. +%total {} (times-double-associative _ _ _ _ _ _ _ _) %. +%theorem times-right-cancels : forall* {X1} {Y1} {Z1} {X2} {Y2} {Z2} forall {T1 times X1 (s Y1) Z1} {T2 times X2 (s Y2) Z2} {EY eq Y1 Y2} {EZ eq Z1 Z2} exists {EX eq X1 X2} true %. +%term _ times-right-cancels times/z times/z EY eq/ eq/ %. +%term _ {T1 times X1 (s Y1) Z1'} {P1 plus Z1' (s Y1) Z1} {T2 times X2 (s Y2) Z2'} {P2 plus Z2' (s Y2) Z2} {EY eq Y1 Y2} {EZ eq Z1 Z2} {EX eq X1 X2} + %pi (times-right-cancels (times/s T1 P1) (times/s T2 P2) EY EZ EX') + %<- (succ-deterministic EY EY') + %<- (plus-right-cancels P1 P2 EY' EZ EZ') + %<- (times-right-cancels T1 T2 EY EZ' EX) + %<- (succ-deterministic EX EX') %. +%worlds () (times-right-cancels X1*sY1=Z1 X2*sY2=Z2 Y1=Y2 Z1=Z2 X1=X2) %. +%total T1 (times-right-cancels T1 _ _ _ _) %. +%theorem times-right-cancels* : forall* {X1} {Y} {Y-1} {Z1} {X2} {Z2} forall {T1 times X1 Y Z1} {T2 times X2 Y Z2} {EY eq Y (s Y-1)} {EZ eq Z1 Z2} exists {EX eq X1 X2} true %. +%term _ + %pi (times-right-cancels* X1*Y=Z1 X2*Y=Z2 Y+ Z1=Z2 X1=X2) + %<- (times-respects-eq X1*Y=Z1 eq/ Y+ eq/ X1*Y+=Z1) + %<- (times-respects-eq X2*Y=Z2 eq/ Y+ eq/ X2*Y+=Z2) + %<- (times-right-cancels X1*Y+=Z1 X2*Y+=Z2 eq/ Z1=Z2 X1=X2) %. +%worlds () (times-right-cancels* X1*Y=Z1 X2*Y=Z2 Y+ Z1=Z2 X1=X2) %. +%total {} (times-right-cancels* _ _ _ _ _) %. +%theorem times-right-cancels** : forall* {X1} {Y1} {Z} {X2} {Y2} {Z-} forall {T1 times X1 Y1 Z} {T2 times X2 Y2 Z} {EY eq Y1 Y2} {EZ eq Z (s Z-)} exists {EX eq X1 X2} true %. +%term _ + %pi (times-right-cancels** X1*0=sZ X2*0=sZ eq/ eq/ X1=X2) + %<- (times-right-zero _ X1*0=0) + %<- (times-deterministic X1*0=0 X1*0=sZ eq/ eq/ ZERO=sZ) + %<- (succ-implies-gt ZERO=sZ ZERO>sZ) + %<- (gt-contradiction ZERO>sZ F) + %<- (false-implies-eq F X1=X2) %. +%term _ + %pi (times-right-cancels** X1*Y1-=sZ X2*Y1-=sZ eq/ eq/ X1=X2) + %<- (times-right-cancels X1*Y1-=sZ X2*Y1-=sZ eq/ eq/ X1=X2) %. +%worlds () (times-right-cancels** X1*Y1=Z X2*Y2=Z Y1=Y2 Z+ X1=X2) %. +%total {} (times-right-cancels** _ _ _ _ _) %. +%theorem times-left-cancels : forall* {X1} {Y1} {Z1} {X2} {Y2} {Z2} forall {T1 times (s X1) Y1 Z1} {T2 times (s X2) Y2 Z2} {E1 eq X1 X2} {E2 eq Z1 Z2} exists {F eq Y1 Y2} true %. +%term _ + %pi (times-left-cancels SX1*Y1=Z1 SX2*Y2=Z2 X1=X2 Z1=Z2 Y1=Y2) + %<- (times-commutative SX1*Y1=Z1 Y1*sX1=Z1) + %<- (times-commutative SX2*Y2=Z2 Y2*sX2=Z2) + %<- (times-right-cancels Y1*sX1=Z1 Y2*sX2=Z2 X1=X2 Z1=Z2 Y1=Y2) %. +%worlds () (times-left-cancels SX1*Y1=Z1 SX2*Y2=Z2 X1=X2 Z1=Z2 Y1=Y2) %. +%total {} (times-left-cancels _ _ _ _ _) %. +%theorem times-left-cancels* : forall* {X} {Y1} {Z1} {X-} {Y2} {Z2} forall {T1 times X Y1 Z1} {T2 times X Y2 Z2} {E1 eq X (s X-)} {E2 eq Z1 Z2} exists {F eq Y1 Y2} true %. +%term _ + %pi (times-left-cancels* X*Y1=Z1 X*Y2=Z2 X+ Z1=Z2 Y1=Y2) + %<- (times-commutative X*Y1=Z1 Y1*X=Z1) + %<- (times-commutative X*Y2=Z2 Y2*X=Z2) + %<- (times-right-cancels* Y1*X=Z1 Y2*X=Z2 X+ Z1=Z2 Y1=Y2) %. +%worlds () (times-left-cancels* X*Y1=Z1 X*Y2=Z2 X+ Z1=Z2 Y1=Y2) %. +%total {} (times-left-cancels* _ _ _ _ _) %. +%theorem times-left-preserves-gt : forall* {M} {N1} {N2} {P1} {P2} forall {GN gt N1 N2} {T1 times (s M) N1 P1} {T2 times (s M) N2 P2} exists {GP gt P1 P2} true %. +%term _ times-left-preserves-gt N1>N2 (times/s times/z plus/z) (times/s times/z plus/z) N1>N2 %. +%term _ + %pi (times-left-preserves-gt N1>N2 (times/s (%the (times (s M) N1 X1) T1) X1+N1=O1) (times/s (%the (times (s M) N2 X2) T2) X2+N2=O2) O1>O2) + %<- (times-left-preserves-gt N1>N2 T1 T2 X1>X2) + %<- (plus-preserves-gt* X1>X2 N1>N2 X1+N1=O1 X2+N2=O2 O1>O2) %. +%worlds () (times-left-preserves-gt N1>N2 SM*N1=P1 SM*N2=P2 P1>P2) %. +%total T1 (times-left-preserves-gt _ T1 _ _) %. +%theorem times-left-preserves-gt* : forall* {M} {M-} {N1} {N2} {P1} {P2} forall {GN gt N1 N2} {T1 times M N1 P1} {T2 times M N2 P2} {M+ eq M (s M-)} exists {GP gt P1 P2} true %. +%term _ + %pi (times-left-preserves-gt* N1>N2 M*N1=P1 M*N2=P2 M+ P1>P2) + %<- (times-respects-eq M*N1=P1 M+ eq/ eq/ SM-*N1=P1) + %<- (times-respects-eq M*N2=P2 M+ eq/ eq/ SM-*N2=P2) + %<- (times-left-preserves-gt N1>N2 SM-*N1=P1 SM-*N2=P2 P1>P2) %. +%worlds () (times-left-preserves-gt* N1>N2 M*N1=P1 M*N2=P2 M+ P1>P2) %. +%total {} (times-left-preserves-gt* _ _ _ _ _) %. +%theorem times-right-preserves-gt : forall* {M1} {M2} {N} {P1} {P2} forall {G1 gt M1 M2} {T1 times M1 (s N) P1} {T2 times M2 (s N) P2} exists {G2 gt P1 P2} true %. +%term _ + %pi (times-right-preserves-gt M1>M2 M1*sN=P1 M2*sN=P2 P1>P2) + %<- (times-commutative M1*sN=P1 SN*M1=P1) + %<- (times-commutative M2*sN=P2 SN*M2=P2) + %<- (times-left-preserves-gt M1>M2 SN*M1=P1 SN*M2=P2 P1>P2) %. +%worlds () (times-right-preserves-gt M1>M2 M1*sN=P1 M2*sN=P2 P1>P2) %. +%total {} (times-right-preserves-gt _ _ _ _) %. +%theorem times-right-preserves-gt* : forall* {M1} {M2} {N} {N-1} {P1} {P2} forall {G1 gt M1 M2} {T1 times M1 N P1} {T2 times M2 N P2} {N+ eq N (s N-1)} exists {G2 gt P1 P2} true %. +%term _ + %pi (times-right-preserves-gt* M1>M2 M1*N=P1 M2*N=P2 N=sN-1 P1>P2) + %<- (times-respects-eq M1*N=P1 eq/ N=sN-1 eq/ M1*N+=P1) + %<- (times-respects-eq M2*N=P2 eq/ N=sN-1 eq/ M2*N+=P2) + %<- (times-right-preserves-gt M1>M2 M1*N+=P1 M2*N+=P2 P1>P2) %. +%worlds () (times-right-preserves-gt* M1>M2 M1*N=P1 M2*N=P2 N=sN-1 P1>P2) %. +%total {} (times-right-preserves-gt* _ _ _ _ _) %. +%theorem times-preserves-gt : forall* {M1} {N1} {P1} {M2} {N2} {P2} forall {GM gt M1 M2} {GN gt N1 N2} {T1 times M1 N1 P1} {T2 times M2 N2 P2} exists {GP gt P1 P2} true %. +%term _ {0=0' eq z _} + %pi (times-preserves-gt (%the (gt M1 M2) M1>M2) (%the (gt N1 z) N1>0) (%the (times M1 N1 P1) M1*N1=P1) M2*0=0' P1>0') + %<- (gt-implies-positive M1>M2 M1' M1=sM1') + %<- (times-respects-eq M1*N1=P1 M1=sM1' eq/ eq/ SM1'*N1=P1) + %<- (times-right-zero M2 M2*0=0) + %<- (times-deterministic M2*0=0 M2*0=0' eq/ eq/ 0=0') + %<- (times-right-zero (s M1') SM1'*0=0) + %<- (times-left-preserves-gt N1>0 SM1'*N1=P1 SM1'*0=0 P1>0) + %<- (gt-respects-eq P1>0 eq/ 0=0' P1>0') %. +%term _ + %pi (times-preserves-gt M1>M2 (%the (gt N1 (s N2')) N1>sN2') M1*N1=P1 M2*sN2'=P2 P1>P2) + %<- (gt-implies-positive M1>M2 M1' M1=sM1') + %<- (times-respects-eq M1*N1=P1 M1=sM1' eq/ eq/ SM1'*N1=P1) + %<- (times-total (%the (times (s M1') (s N2') _) SM1'*sN2'=PX)) + %<- (times-left-preserves-gt N1>sN2' SM1'*N1=P1 SM1'*sN2'=PX P1>PX) + %<- (eq-symmetric M1=sM1' SM1'=M1) + %<- (times-respects-eq SM1'*sN2'=PX SM1'=M1 eq/ eq/ M1*sN2'=PX) + %<- (times-right-preserves-gt M1>M2 M1*sN2'=PX M2*sN2'=P2 PX>P2) + %<- (gt-transitive P1>PX PX>P2 P1>P2) %. +%worlds () (times-preserves-gt M1>M2 N1>N2 M1*N1=P1 M2*N2=P2 P1>P2) %. +%total {} (times-preserves-gt _ _ _ _ _) %. +%theorem times-right-cancels-gt : forall* {X1 nat} {X2 nat} {Y1 nat} {Y2 nat} {Z1 nat} {Z2 nat} forall {P1 times X1 Y1 Z1} {P2 times X2 Y2 Z2} {EY eq Y1 Y2} {G1 gt Z1 Z2} exists {G2 gt X1 X2} true %. +%term _ + %pi (times-right-cancels-gt (times/s X1*Y=N1 N1+Y=Z1) times/z eq/ Z1>0 SX1>0) + %<- (succ-implies-gt-zero _ SX1>0) %. +%term _ + %pi (times-right-cancels-gt (times/s X1*Y=N1 N1+Y=Z1) (times/s X2*Y=N2 N2+Y=Z2) eq/ Z1>Z2 SX1>SX2) + %<- (plus-right-cancels-gt N1+Y=Z1 N2+Y=Z2 eq/ Z1>Z2 N1>N2) + %<- (times-right-cancels-gt X1*Y=N1 X2*Y=N2 eq/ N1>N2 X1>X2) + %<- (succ-preserves-gt X1>X2 SX1>SX2) %. +%worlds () (times-right-cancels-gt X1*Y1=Z1 X2*Y2=Z2 Y1=Y2 Z1>Z2 X1>X2) %. +%total [P1 P2] (times-right-cancels-gt P1 P2 _ _ _) %. +%theorem times-left-cancels-gt : forall* {X1 nat} {X2 nat} {Y1 nat} {Y2 nat} {Z1 nat} {Z2 nat} forall {P1 times X1 Y1 Z1} {P2 times X2 Y2 Z2} {EX eq X1 X2} {G1 gt Z1 Z2} exists {G2 gt Y1 Y2} true %. +%term _ + %pi (times-left-cancels-gt X1*Y1=Z1 X2*Y2=Z2 X1=X2 Z1>Z2 Y1>Y2) + %<- (times-commutative X1*Y1=Z1 Y1*X1=Z1) + %<- (times-commutative X2*Y2=Z2 Y2*X2=Z2) + %<- (times-right-cancels-gt Y1*X1=Z1 Y2*X2=Z2 X1=X2 Z1>Z2 Y1>Y2) %. +%worlds () (times-left-cancels-gt X1*Y1=Z1 X2*Y2=Z2 X1=X2 Z1>Z2 Y1>Y2) %. +%total P1 (times-left-cancels-gt P1 _ _ _ _) %. +%%%%% nat-inv.elf +%%%%% Minus for natural numbers +%%%%% This file is part of the nat.elf signature +%%%% Definitions +%inline minus [X1] [X2] [X3] plus X3 X2 X1 %. +%%%% Theorems +%%% Theorems about minus +%inline false-implies-minus false-implies-plus %. +%theorem minus-respects-eq : forall* {X1} {X2} {X3} {X4} {X5} {X6} forall {D minus X1 X2 X3} {E1 eq X1 X4} {E2 eq X2 X5} {E3 eq X3 X6} exists {DP minus X4 X5 X6} true %. +%term _ minus-respects-eq S eq/ eq/ eq/ S %. +%worlds () (minus-respects-eq _ _ _ _ _) %. +%total {} (minus-respects-eq _ _ _ _ _) %. +%theorem minus-deterministic : forall* {X1} {X2} {X3} {X4} {X5} {X6} forall {S minus X1 X2 X3} {SP minus X4 X5 X6} {E1 eq X1 X4} {E2 eq X2 X5} exists {E3 eq X3 X6} true %. +%term _ + %pi (minus-deterministic X3+X2=X1 X6+X5=X4 X1=X4 X2=X5 X3=X6) + %<- (plus-right-cancels X3+X2=X1 X6+X5=X4 X2=X5 X1=X4 X3=X6) %. +%worlds () (minus-deterministic X1-X2=X3 X4-X5=X6 X1=X4 X2=X5 X3=X6) %. +%total {} (minus-deterministic _ _ _ _ _) %. +%theorem plus-associates-with-minus* : forall* {X1} {X2} {X3} {X4} {X6} {X7} forall {OP1 plus X1 X2 X3} {IOP1 minus X3 X4 X7} {IOP2 minus X2 X4 X6} exists {OP2 plus X1 X6 X7} true %. +%term _ + %pi (plus-associates-with-minus* X1+X2=X3 X7+X4=X3 X6+X4=X2 X1+X6=X7) + %<- (plus-associative-converse X6+X4=X2 X1+X2=X3 X7P X1+X6=X7P X7P+X4=X3) + %<- (plus-right-cancels X7P+X4=X3 X7+X4=X3 eq/ eq/ X7P=X7) + %<- (plus-respects-eq X1+X6=X7P eq/ eq/ X7P=X7 X1+X6=X7) %. +%worlds () (plus-associates-with-minus* X1+X2=X3 X3-X4=X7 X2-X4=X6 X1+X6=X7) %. +%total {} (plus-associates-with-minus* _ _ _ _) %. +%theorem plus-associates-with-minus-converse* : forall* {X1} {X2} {X3} {X4} {X6} {X7} forall {IOP2 minus X2 X4 X6} {OP2 plus X1 X6 X7} {OP1 plus X1 X2 X3} exists {IOP1 minus X3 X4 X7} true %. +%term _ + %pi (plus-associates-with-minus-converse* X6+X4=X2 X1+X6=X7 X1+X2=X3 X7+X4=X3) + %<- (plus-associative-converse* X6+X4=X2 X1+X2=X3 X1+X6=X7 X7+X4=X3) %. +%worlds () (plus-associates-with-minus-converse* X2-X4=X6 X1+X6=X7 X1+X2=X3 X3-X4=X7) %. +%total {} (plus-associates-with-minus-converse* _ _ _ _) %. +%theorem plus-associates-with-minus-converse : forall* {X1} {X2} {X4} {X6} {X7} forall {IOP2 minus X2 X4 X6} {OP2 plus X1 X6 X7} exists {X3} {OP1 plus X1 X2 X3} {IOP1 minus X3 X4 X7} true %. +%term _ + %pi (plus-associates-with-minus-converse X6+X4=X2 X1+X6=X7 X3 X1+X2=X3 X7+X4=X3) + %<- (plus-total X1+X2=X3) + %<- (plus-associates-with-minus-converse* X6+X4=X2 X1+X6=X7 X1+X2=X3 X7+X4=X3) %. +%worlds () (plus-associates-with-minus-converse X2-X4=X6 X1+X6=X7 X3 X1+X2=X3 X3-X4=X7) %. +%total {} (plus-associates-with-minus-converse _ _ _ _ _) %. +%theorem minus-associates-from-plus* : forall* {X1} {X2} {X3} {X4} {X6} {X7} forall {IOP1 minus X1 X2 X3} {OP1 plus X3 X4 X7} {IOP2 minus X2 X4 X6} exists {IOP3 minus X1 X6 X7} true %. +%term _ + %pi (minus-associates-from-plus* X3+X2=X1 X3+X4=X7 X6+X4=X2 X7+X6=X1) + %<- (plus-commutative X6+X4=X2 X4+X6=X2) + %<- (plus-associative-converse* X4+X6=X2 X3+X2=X1 X3+X4=X7 X7+X6=X1) %. +%worlds () (minus-associates-from-plus* X1-X2=X3 X3+X4=X7 X2-X4=X6 X1-X6=X7) %. +%total {} (minus-associates-from-plus* _ _ _ _) %. +%theorem minus-associates-from-plus-converse* : forall* {X1} {X2} {X3} {X4} {X6} {X7} forall {IOP2 minus X2 X4 X6} {IOP3 minus X1 X6 X7} {IOP1 minus X1 X2 X3} exists {OP1 plus X3 X4 X7} true %. +%term _ + %pi (minus-associates-from-plus-converse* X6+X4=X2 X7+X6=X1 X3+X2=X1 X3+X4=X7) + %<- (plus-commutative X6+X4=X2 X4+X6=X2) + %<- (plus-associative-converse X4+X6=X2 X3+X2=X1 X7P X3+X4=X7P X7P+X6=X1) + %<- (plus-right-cancels X7P+X6=X1 X7+X6=X1 eq/ eq/ X7P=X7) + %<- (plus-respects-eq X3+X4=X7P eq/ eq/ X7P=X7 X3+X4=X7) %. +%worlds () (minus-associates-from-plus-converse* X2-X4=X6 X1-X6=X7 X1-X2=X3 X3+X4=X7) %. +%total {} (minus-associates-from-plus-converse* _ _ _ _) %. +%theorem minus-associates-to-plus* : forall* {X1} {X2} {X3} {X4} {X6} {X7} forall {IOP1 minus X1 X2 X3} {IOP2 minus X3 X4 X7} {OP1 plus X2 X4 X6} exists {IOP3 minus X1 X6 X7} true %. +%term _ + %pi (minus-associates-to-plus* X3+X2=X1 X7+X4=X3 X2+X4=X6 X7+X6=X1) + %<- (plus-commutative X2+X4=X6 X4+X2=X6) + %<- (plus-associative* X7+X4=X3 X3+X2=X1 X4+X2=X6 X7+X6=X1) %. +%worlds () (minus-associates-to-plus* X1-X2=X3 X3-X4=X7 X2+X4=X6 X1-X6=X7) %. +%total {} (minus-associates-to-plus* _ _ _ _) %. +%theorem minus-associates-to-plus : forall* {X1} {X2} {X3} {X4} {X7} forall {IOP1 minus X1 X2 X3} {IOP2 minus X3 X4 X7} exists {X6} {OP1 plus X2 X4 X6} {IOP3 minus X1 X6 X7} true %. +%term _ + %pi (minus-associates-to-plus X3+X2=X1 X7+X4=X3 X6 X2+X4=X6 X7+X6=X1) + %<- (plus-associative X7+X4=X3 X3+X2=X1 X6 X4+X2=X6 X7+X6=X1) + %<- (plus-commutative X4+X2=X6 X2+X4=X6) %. +%worlds () (minus-associates-to-plus X1-X2=X3 X3-X4=X7 X6 X2+X4=X6 X1-X6=X7) %. +%total {} (minus-associates-to-plus _ _ _ _ _) %. +%theorem minus-associates-to-plus-converse* : forall* {X1} {X2} {X3} {X4} {X6} {X7} forall {OP1 plus X2 X4 X6} {IOP3 minus X1 X6 X7} {IOP1 minus X1 X2 X3} exists {IOP2 minus X3 X4 X7} true %. +%term _ + %pi (minus-associates-to-plus-converse* X2+X4=X6 X7+X6=X1 X3+X2=X1 X7+X4=X3) + %<- (plus-commutative X2+X4=X6 X4+X2=X6) + %<- (plus-associative-converse X4+X2=X6 X7+X6=X1 X3P X7+X4=X3P X3P+X2=X1) + %<- (plus-right-cancels X3P+X2=X1 X3+X2=X1 eq/ eq/ X3P=X3) + %<- (plus-respects-eq X7+X4=X3P eq/ eq/ X3P=X3 X7+X4=X3) %. +%worlds () (minus-associates-to-plus-converse* X2+X4=X6 X1-X6=X7 X1-X2=X3 X3-X4=X7) %. +%total {} (minus-associates-to-plus-converse* _ _ _ _) %. +%theorem minus-associates-to-plus-converse : forall* {X1} {X2} {X4} {X6} {X7} forall {OP1 plus X2 X4 X6} {IOP3 minus X1 X6 X7} exists {X3} {IOP1 minus X1 X2 X3} {IOP2 minus X3 X4 X7} true %. +%term _ + %pi (minus-associates-to-plus-converse X2+X4=X6 X7+X6=X1 X3 X3+X2=X1 X7+X4=X3) + %<- (plus-commutative X2+X4=X6 X4+X2=X6) + %<- (plus-associative-converse X4+X2=X6 X7+X6=X1 X3 X7+X4=X3 X3+X2=X1) %. +%worlds () (minus-associates-to-plus-converse X2+X4=X6 X1-X6=X7 X3 X1-X2=X3 X3-X4=X7) %. +%total {} (minus-associates-to-plus-converse _ _ _ _ _) %. +%theorem minus-is-zero-implies-eq : forall* {N1} {N2} {N3} forall {P minus N1 N2 N3} {E3 eq N3 z} exists {E1 eq N1 N2} true %. +%term _ minus-is-zero-implies-eq plus/z eq/ eq/ %. +%worlds () (minus-is-zero-implies-eq X-Y=Z Z=0 X=Y) %. +%total {} (minus-is-zero-implies-eq _ _ _) %. +%inline minus-implies-gt plus-implies-gt %. +%theorem minus-left-cancels : forall* {X1} {X2} {X3} {X4} {X5} {X6} forall {IOP1 minus X1 X2 X3} {IOP2 minus X4 X5 X6} {E1 eq X1 X4} {E3 eq X3 X6} exists {E2 eq X2 X5} true %. +%term _ + %pi (minus-left-cancels X3+X2=X1 X6+X5=X4 X1=X4 X3=X6 X2=X5) + %<- (plus-left-cancels X3+X2=X1 X6+X5=X4 X3=X6 X1=X4 X2=X5) %. +%worlds () (minus-left-cancels X1-X2=X3 X4-X5=X6 X1=X4 X3=X6 X2=X5) %. +%total {} (minus-left-cancels _ _ _ _ _) %. +%theorem minus-right-cancels : forall* {X1} {X2} {X3} {X4} {X5} {X6} forall {IOP1 minus X1 X2 X3} {IOP2 minus X4 X5 X6} {E2 eq X2 X5} {E3 eq X3 X6} exists {E1 eq X1 X4} true %. +%term _ + %pi (minus-right-cancels X3+X2=X1 X6+X5=X4 X2=X5 X3=X6 X1=X4) + %<- (plus-deterministic X3+X2=X1 X6+X5=X4 X3=X6 X2=X5 X1=X4) %. +%worlds () (minus-right-cancels X1-X2=X3 X4-X5=X6 X2=X5 X3=X6 X1=X4) %. +%total {} (minus-right-cancels _ _ _ _ _) %. +%theorem minus-left-inverts-gt* : forall* {X1} {X2} {X3} {X4} {X5} forall {G gt X2 X4} {IOP1 minus X1 X2 X3} {IOP2 minus X1 X4 X5} exists {GP gt X5 X3} true %. +%term _ + %pi (minus-left-inverts-gt* X2>X4 X3+X2=X1 X5+X4=X1 X5>X3) + %<- (plus-total X3+X4=X7) + %<- (plus-left-preserves-gt* X2>X4 X3+X2=X1 X3+X4=X7 X1>X7) + %<- (plus-right-cancels-gt X5+X4=X1 X3+X4=X7 eq/ X1>X7 X5>X3) %. +%worlds () (minus-left-inverts-gt* X2>X4 X1-X2=X3 X1-X4=X5 X5>X3) %. +%total {} (minus-left-inverts-gt* _ _ _ _) %. +%theorem minus-right-preserves-gt* : forall* {X1} {X2} {X3} {X4} {X5} forall {G gt X1 X2} {IOP1 minus X1 X3 X4} {IOP2 minus X2 X3 X5} exists {GP gt X4 X5} true %. +%term _ + %pi (minus-right-preserves-gt* X1>X2 X4+X3=X1 X5+X3=X2 X4>X5) + %<- (plus-right-cancels-gt X4+X3=X1 X5+X3=X2 eq/ X1>X2 X4>X5) %. +%worlds () (minus-right-preserves-gt* X1>X2 X1-X3=X4 X2-X3=X5 X4>X5) %. +%total {} (minus-right-preserves-gt* _ _ _ _) %. +%theorem minus-left-cancels-inverts-gt : forall* {X1} {X2} {X3} {X4} {X5} {X6} forall {IOP1 minus X1 X2 X3} {IOP2 minus X4 X5 X6} {E eq X1 X4} {G gt X3 X6} exists {GP gt X5 X2} true %. +%term _ + %pi (minus-left-cancels-inverts-gt X3+X2=X1 X6+X5=X4 X1=X4 X3>X6 X5>X2) + %<- (plus-total X6+X2=X7) + %<- (plus-right-preserves-gt* X3>X6 X3+X2=X1 X6+X2=X7 X1>X7) + %<- (eq-symmetric X1=X4 X4=X1) + %<- (plus-respects-eq X6+X5=X4 eq/ eq/ X4=X1 X6+X5=X1) + %<- (plus-left-cancels-gt X6+X5=X1 X6+X2=X7 eq/ X1>X7 X5>X2) %. +%worlds () (minus-left-cancels-inverts-gt X1-X2=X3 X4-X5=X6 X1=X4 X3>X6 X5>X2) %. +%total {} (minus-left-cancels-inverts-gt _ _ _ _ _) %. +%theorem minus-right-cancels-gt : forall* {X1} {X2} {X3} {X4} {X5} {X6} forall {IOP1 minus X1 X2 X3} {IOP2 minus X4 X5 X6} {E2 eq X2 X5} {G3 gt X3 X6} exists {G1 gt X1 X4} true %. +%term _ + %pi (minus-right-cancels-gt X3+X2=X1 X6+X5=X4 X2=X5 X3>X6 X1>X4) + %<- (plus-respects-eq X3+X2=X1 eq/ X2=X5 eq/ X3+X5=X1) + %<- (plus-right-preserves-gt* X3>X6 X3+X5=X1 X6+X5=X4 X1>X4) %. +%worlds () (minus-right-cancels-gt X1-X2=X3 X4-X5=X6 X2=X5 X3>X6 X1>X4) %. +%total {} (minus-right-cancels-gt _ _ _ _ _) %. +%theorem times-right-distributes-over-minus : forall* {X1} {X2} {X3} {X4} {X7} forall {S12 minus X1 X2 X3} {M34 times X3 X4 X7} exists {X5} {X6} {M14 times X1 X4 X5} {M24 times X2 X4 X6} {S56 minus X5 X6 X7} true %. +%term _ + %pi (times-right-distributes-over-minus X3+X2=X1 X3*X4=X7 _ _ X1*X4=X5 X2*X4=X6 X7+X6=X5) + %<- (times-total X1*X4=X5) + %<- (times-right-distributes-over-plus X3+X2=X1 X1*X4=X5 _ _ X3*X4=Y7 X2*X4=X6 Y7+X6=X5) + %<- (times-deterministic X3*X4=Y7 X3*X4=X7 eq/ eq/ Y7=X7) + %<- (plus-respects-eq Y7+X6=X5 Y7=X7 eq/ eq/ X7+X6=X5) %. +%worlds () (times-right-distributes-over-minus X1-X2=X3 X3*X4=X7 X5 X6 X1*X4=X5 X2*X4=X6 X5-X6=X7) %. +%total {} (times-right-distributes-over-minus _ _ _ _ _ _ _) %. +%theorem times-right-distributes-over-minus* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {A12 minus X1 X2 X3} {M34 times X3 X4 X7} {M14 times X1 X4 X5} {M24 times X2 X4 X6} exists {A56 minus X5 X6 X7} true %. +%term _ + %pi (times-right-distributes-over-minus* X1-X2=X3 X3*X4=X7 X1*X4=X5 X2*X4=X6 X5-X6=X7) + %<- (times-right-distributes-over-minus X1-X2=X3 X3*X4=X7 Y5 Y6 X1*X4=Y5 X2*X4=Y6 Y5-Y6=X7) + %<- (times-deterministic X1*X4=Y5 X1*X4=X5 eq/ eq/ Y5=X5) + %<- (times-deterministic X2*X4=Y6 X2*X4=X6 eq/ eq/ Y6=X6) + %<- (minus-respects-eq Y5-Y6=X7 Y5=X5 Y6=X6 eq/ X5-X6=X7) %. +%worlds () (times-right-distributes-over-minus* X1-X2=X3 X3*X4=X7 X1*X4=X5 X2*X4=X6 X5-X6=X7) %. +%total {} (times-right-distributes-over-minus* _ _ _ _ _) %. +%theorem times-left-distributes-over-minus* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {A12 minus X2 X4 X6} {M34 times X1 X6 X7} {M14 times X1 X2 X3} {M24 times X1 X4 X5} exists {A56 minus X3 X5 X7} true %. +%term _ + %pi (times-left-distributes-over-minus* X2-X4=X6 X1*X6=X7 X1*X2=X3 X1*X4=X5 X3-X5=X7) + %<- (times-commutative X1*X6=X7 X6*X1=X7) + %<- (times-commutative X1*X2=X3 X2*X1=X3) + %<- (times-commutative X1*X4=X5 X4*X1=X5) + %<- (times-right-distributes-over-minus* X2-X4=X6 X6*X1=X7 X2*X1=X3 X4*X1=X5 X3-X5=X7) %. +%worlds () (times-left-distributes-over-minus* X2-X4=X6 X1*X6=X7 X1*X2=X3 X1*X4=X5 X3-X5=X7) %. +%total {} (times-left-distributes-over-minus* _ _ _ _ _) %. +%theorem times-left-distributes-over-minus : forall* {X1} {X2} {X4} {X6} {X7} forall {A12 minus X2 X4 X6} {M34 times X1 X6 X7} exists {X3} {X5} {M14 times X1 X2 X3} {M24 times X1 X4 X5} {A56 minus X3 X5 X7} true %. +%term _ + %pi (times-left-distributes-over-minus X2-X4=X6 X1*X6=X7 X3 X5 X1*X2=X3 X1*X4=X5 X3-X5=X7) + %<- (times-total X1*X2=X3) + %<- (times-total X1*X4=X5) + %<- (times-left-distributes-over-minus* X2-X4=X6 X1*X6=X7 X1*X2=X3 X1*X4=X5 X3-X5=X7) %. +%worlds () (times-left-distributes-over-minus X2-X4=X6 X1*X6=X7 X3 X5 X1*X2=X3 X1*X4=X5 X3-X5=X7) %. +%total {} (times-left-distributes-over-minus _ _ _ _ _ _ _) %. +%theorem times-right-factors-over-minus* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {M14 times X1 X4 X5} {M24 times X2 X4 X6} {A56 minus X5 X6 X7} {A12 minus X1 X2 X3} exists {M34 times X3 X4 X7} true %. +%term _ + %pi (times-right-factors-over-minus* X1*X4=X5 X2*X4=X6 X5-X6=X7 X1-X2=X3 X3*X4=X7) + %<- (times-total X3*X4=Y7) + %<- (times-right-distributes-over-minus* X1-X2=X3 X3*X4=Y7 X1*X4=X5 X2*X4=X6 X5-X6=Y7) + %<- (minus-deterministic X5-X6=Y7 X5-X6=X7 eq/ eq/ Y7=X7) + %<- (times-respects-eq X3*X4=Y7 eq/ eq/ Y7=X7 X3*X4=X7) %. +%worlds () (times-right-factors-over-minus* X1*X4=X5 X2*X4=X6 X5-X6=X7 X1-X2=X3 X3*X4=X7) %. +%total {} (times-right-factors-over-minus* _ _ _ _ _) %. +%theorem times-left-factors-over-minus* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {M12 times X1 X2 X3} {M14 times X1 X4 X5} {A35 minus X3 X5 X7} {A24 minus X2 X4 X6} exists {M16 times X1 X6 X7} true %. +%term _ + %pi (times-left-factors-over-minus* X1*X2=X3 X1*X4=X5 X3-X5=X7 X2-X4=X6 X1*X6=X7) + %<- (times-total X1*X6=Y7) + %<- (times-left-distributes-over-minus* X2-X4=X6 X1*X6=Y7 X1*X2=X3 X1*X4=X5 X3-X5=Y7) + %<- (minus-deterministic X3-X5=Y7 X3-X5=X7 eq/ eq/ Y7=X7) + %<- (times-respects-eq X1*X6=Y7 eq/ eq/ Y7=X7 X1*X6=X7) %. +%worlds () (times-left-factors-over-minus* X1*X2=X3 X1*X4=X5 X3-X5=X7 X2-X4=X6 X1*X6=X7) %. +%total {} (times-left-factors-over-minus* _ _ _ _ _) %. +%theorem times-right-factors-over-minus : forall* {Y} {Z} {XY} {XZ} {YZ} {XYZ} {Z-} forall {TXY times XY Z XYZ} {TY times Y Z YZ} {M minus XYZ YZ XZ} {EZ eq Z (s Z-)} exists {X} {M' minus XY Y X} {TX times X Z XZ} true %. +% minus isn't total, so this is harder. +%term _ + %pi (times-right-factors-over-minus XY*Z=YZ Y*Z=YZ plus/z eq/ z ZERO+Y=XY times/z) + %<- (times-right-cancels* XY*Z=YZ Y*Z=YZ eq/ eq/ XY=Y) + %<- (plus-respects-eq plus/z eq/ XY=Y eq/ ZERO+Y=XY) %. +%term _ + %pi (times-right-factors-over-minus XY*Z=XYZ Y*Z=YZ XZ+YZ=XYZ _ (s X-) X+Y=XY X*Z=XZ) + %<- (plus-implies-gt XZ+YZ=XYZ eq/ XYZ>YZ) + %<- (times-right-cancels-gt XY*Z=XYZ Y*Z=YZ eq/ XYZ>YZ XY>Y) + %<- (gt-implies-plus XY>Y X- X+Y=XY) + %<- (times-right-factors-over-minus* XY*Z=XYZ Y*Z=YZ XZ+YZ=XYZ X+Y=XY X*Z=XZ) %. +%worlds () (times-right-factors-over-minus XY*Z=XYZ Y*Z=YZ XYZ-YZ=XZ Z+ X XY-Y=X X*Z=XZ) %. +%total {} (times-right-factors-over-minus _ _ _ _ _ _ _) %. +%theorem times-left-factors-over-minus : forall* {X} {Y} {Z} {XY} {XZ} {XYZ} {X-} forall {TXY times X Y XY} {TXZ times X Z XZ} {M minus XY XZ XYZ} {EX eq X (s X-)} exists {YZ} {MYZ minus Y Z YZ} {TXYZ times X YZ XYZ} true %. +%term _ + %pi (times-left-factors-over-minus X*Y=XY X*Z=XZ XY-XZ=XYZ X=sX- YZ Y-Z=YZ X*YZ=XYZ) + %<- (times-commutative X*Y=XY Y*X=XY) + %<- (times-commutative X*Z=XZ Z*X=XZ) + %<- (times-right-factors-over-minus Y*X=XY Z*X=XZ XY-XZ=XYZ X=sX- YZ Y-Z=YZ YZ*X=XYZ) + %<- (times-commutative YZ*X=XYZ X*YZ=XYZ) %. +%worlds () (times-left-factors-over-minus X*Y=XY X*Z=XZ XY-XZ=XYZ X=sX- YZ Y-Z=YZ X*YZ=XYZ) %. +%total {} (times-left-factors-over-minus _ _ _ _ _ _ _) %. +%%%%% nat-comp.elf +%%%%% Composed relations for natural numbers +%%%%% This file is part of the nat.elf signature +%%%% Definitions +%sort ge %. +%term ge/= %pi (ge X Y) %<- (eq X Y) %. +%term ge/> %pi (ge X Y) %<- (gt X Y) %. +%%%% Theorems +%%% Theorems about ge +%theorem false-implies-ge : forall* {X1} {X2} forall {F void} exists {G ge X1 X2} true %. +%worlds () (false-implies-ge _ _) %. +%total {} (false-implies-ge _ _) %. +%theorem ge-respects-eq : forall* {X1} {X2} {Y1} {Y2} forall {D1 ge X1 X2} {E1 eq X1 Y1} {E2 eq X2 Y2} exists {D2 ge Y1 Y2} true %. +%term _ ge-respects-eq X1>=X2 eq/ eq/ X1>=X2 %. +%worlds () (ge-respects-eq _ _ _ _) %. +%total {} (ge-respects-eq _ _ _ _) %. +%theorem ge-reflexive : forall {X} exists {G ge X X} true %. +%term _ ge-reflexive _ (ge/= eq/) %. +%worlds () (ge-reflexive X X>=X) %. +%total {} (ge-reflexive _ _) %. +%theorem ge-transitive : forall* {X1} {X2} {X3} forall {G1 ge X1 X2} {G2 ge X2 X3} exists {G3 ge X1 X3} true %. +%term _ ge-transitive (ge/= eq/) (ge/= eq/) (ge/= eq/) %. +%term _ ge-transitive (ge/= eq/) (ge/> X>X3) (ge/> X>X3) %. +%term _ ge-transitive (ge/> X1>X) (ge/= eq/) (ge/> X1>X) %. +%term _ + %pi (ge-transitive (ge/> X1>X2) (ge/> X2>X3) (ge/> X1>X3)) + %<- (gt-transitive X1>X2 X2>X3 X1>X3) %. +%worlds () (ge-transitive X1>=X2 X2>=X3 X1>=X3) %. +%total {} (ge-transitive _ _ _) %. +%theorem ge-anti-symmetric : forall* {X1} {X2} forall {G1 ge X1 X2} {G2 ge X2 X1} exists {E eq X1 X2} true %. +%term _ ge-anti-symmetric (ge/= eq/) _ eq/ %. +%term _ ge-anti-symmetric _ (ge/= eq/) eq/ %. +%term _ + %pi (ge-anti-symmetric (ge/> X1>X2) (ge/> X2>X1) X1=X2) + %<- (gt-anti-symmetric X1>X2 X2>X1 F) + %<- (false-implies-eq F X1=X2) %. +%worlds () (ge-anti-symmetric X1>=X2 X2>=X1 X1=X2) %. +%total {} (ge-anti-symmetric _ _ _) %. +%theorem ge-transitive-gt : forall* {X1} {X2} {X3} forall {G1 ge X1 X2} {G2 gt X2 X3} exists {G3 gt X1 X3} true %. +%term _ ge-transitive-gt (ge/= eq/) X>X3 X>X3 %. +%term _ + %pi (ge-transitive-gt (ge/> X1>X2) X2>X3 X1>X3) + %<- (gt-transitive X1>X2 X2>X3 X1>X3) %. +%worlds () (ge-transitive-gt X1>=X2 X2>X3 X1>X3) %. +%total {} (ge-transitive-gt _ _ _) %. +%theorem gt-transitive-ge : forall* {X1} {X2} {X3} forall {G1 gt X1 X2} {G2 ge X2 X3} exists {G3 gt X1 X3} true %. +%term _ gt-transitive-ge X1>X2 (ge/= eq/) X1>X2 %. +%term _ + %pi (gt-transitive-ge X1>X2 (ge/> X2>X3) X1>X3) + %<- (gt-transitive X1>X2 X2>X3 X1>X3) %. +%worlds () (gt-transitive-ge X1>X2 X2>=X3 X1>X3) %. +%total {} (gt-transitive-ge _ _ _) %. +%theorem meta-ge : forall {M} {N} {G ge M N} true %. +%term _ meta-ge _ _ (ge/= eq/) %. +%term _ %pi (meta-ge _ _ (ge/> M>N)) %<- (meta-gt _ _ M>N) %. +%worlds () (meta-ge _ _ _) %. +%total {} (meta-ge _ _ _) %. +%reduces <= N M (meta-ge M N _) %. +%theorem succ-preserves-ge : forall* {M} {N} forall {G ge M N} exists {G' ge (s M) (s N)} true %. +%term _ succ-preserves-ge (ge/= eq/) (ge/= eq/) %. +%term _ + %pi (succ-preserves-ge (ge/> N>M) (ge/> N+1>M+1)) + %<- (succ-preserves-gt N>M N+1>M+1) %. +%worlds () (succ-preserves-ge M>=N M+1>=N+1) %. +%total {} (succ-preserves-ge _ _) %. +%theorem succ-preserves-ge-converse : forall* {M} {N} forall {G' ge (s M) (s N)} exists {G ge M N} true %. +%term _ succ-preserves-ge-converse (ge/= eq/) (ge/= eq/) %. +%term _ + %pi (succ-preserves-ge-converse (ge/> N+1>M+1) (ge/> N>M)) + %<- (succ-preserves-gt-converse N+1>M+1 N>M) %. +%worlds () (succ-preserves-ge-converse M+1>=N+1 M>=N) %. +%total {} (succ-preserves-ge-converse _ _) %. +%theorem ge-succ-implies-gt : forall* {N1} {N2} forall {G ge N1 (s N2)} exists {G' gt N1 N2} true %. +%term _ ge-succ-implies-gt (ge/= eq/) gt/1 %. +%term _ + %pi (ge-succ-implies-gt (ge/> N1>sN2) N1>N2) + %<- (gt-transitive N1>sN2 gt/1 N1>N2) %. +%worlds () (ge-succ-implies-gt _ _) %. +%total {} (ge-succ-implies-gt _ _) %. +%theorem ge-implies-succ-gt : forall* {N1} {N2} forall {G ge N1 N2} exists {G' gt (s N1) N2} true %. +%term _ + %pi (ge-implies-succ-gt N1>=N2 N1+1>N2) + %<- (succ-preserves-ge N1>=N2 N1+1>=N2+1) + %<- (ge-succ-implies-gt N1+1>=N2+1 N1+1>N2) %. +%worlds () (ge-implies-succ-gt _ _) %. +%total {} (ge-implies-succ-gt _ _) %. +%theorem succ-gt-implies-ge : forall* {N1} {N2} forall {G gt (s N1) N2} exists {G' ge N1 N2} true %. +%term _ succ-gt-implies-ge gt/1 (ge/= eq/) %. +%term _ succ-gt-implies-ge (gt/> N1>N2) (ge/> N1>N2) %. +%worlds () (succ-gt-implies-ge _ _) %. +%total {} (succ-gt-implies-ge _ _) %. +%theorem gt-implies-ge-succ : forall* {N1} {N2} forall {G' gt N1 N2} exists {G ge N1 (s N2)} true %. +%term _ + %pi (gt-implies-ge-succ N1>N2 N1>=N2+1) + %<- (succ-preserves-gt N1>N2 N1+1>N2+1) + %<- (succ-gt-implies-ge N1+1>N2+1 N1>=N2+1) %. +%worlds () (gt-implies-ge-succ _ _) %. +%total {} (gt-implies-ge-succ _ _) %. +%theorem ge-implies-plus : forall* {N1} {N2} forall {G ge N2 N1} exists {N0} {P plus N0 N1 N2} true %. +%term _ ge-implies-plus (ge/= eq/) z plus/z %. +%term _ %pi (ge-implies-plus (ge/> N2>N1) (s N0) P) %<- (gt-implies-plus N2>N1 N0 P) %. +%worlds () (ge-implies-plus N2>=N1 N0 N0+N1=N2) %. +%total {} (ge-implies-plus _ _ _) %. +%theorem plus-implies-ge : forall* {N0} {N1} {N2} forall {P plus N0 N1 N2} exists {G ge N2 N1} true %. +%term _ plus-implies-ge plus/z (ge/= eq/) %. +%term _ %pi (plus-implies-ge P (ge/> N2>N1)) %<- (plus-implies-gt P eq/ N2>N1) %. +%worlds () (plus-implies-ge N0+N1=N2 N2>=N1) %. +%total {} (plus-implies-ge _ _) %. +%theorem ge-zero-always : forall {N} exists {G ge N z} true %. +%term _ + %pi (ge-zero-always _ N>=0) + %<- (plus-right-identity _ N+0=N) + %<- (plus-implies-ge N+0=N N>=0) %. +%worlds () (ge-zero-always _ _) %. +%total {} (ge-zero-always _ _) %. +%theorem nonzero-times-implies-ge : forall* {N0} {N1} {N2} forall {P times (s N0) N1 N2} exists {G ge N2 N1} true %. +%term _ + %pi (nonzero-times-implies-ge (times/s _ X+N1=N2) N2>=N1) + %<- (plus-implies-ge X+N1=N2 N2>=N1) %. +%worlds () (nonzero-times-implies-ge N0*N1=N2 N2>=N1) %. +%total {} (nonzero-times-implies-ge _ _) %. +%theorem times-nonzero-implies-ge : forall* {N0} {N1} {N2} forall {P times N0 (s N1) N2} exists {G ge N2 N0} true %. +%term _ + %pi (times-nonzero-implies-ge A*B=C C>=A) + %<- (times-commutative A*B=C B*A=C) + %<- (nonzero-times-implies-ge B*A=C C>=A) %. +%worlds () (times-nonzero-implies-ge _ _) %. +%total {} (times-nonzero-implies-ge _ _) %. +%theorem non-trivial-times-implies-much-gt* : forall* {N1} {N2} {N3} forall {D times (s (s N1)) (s (s N2)) N3} exists {G gt N3 (s (s (s N1)))} true %. +%term _ + %pi (non-trivial-times-implies-much-gt* (times/s (times/s N1*ssN2=P1 P1+ssN2=P2) P2+ssN2=N3) N3>sssN1) + %<- (times-nonzero-implies-ge N1*ssN2=P1 P1>=N1) + %<- (succ-preserves-ge P1>=N1 SP1>=sN1) + %<- (plus-swap-succ-converse P1+ssN2=P2 SP1+sN2=P2) + %<- (plus-commutative SP1+sN2=P2 SN2+sP1=P2) + %<- (plus-implies-gt SN2+sP1=P2 eq/ P2>sP1) + %<- (gt-transitive-ge P2>sP1 SP1>=sN1 P2>sN1) + %<- (succ-preserves-gt P2>sN1 SP2>ssN1) + %<- (gt-implies-ge-succ SP2>ssN1 SP2>=sssN1) + %<- (plus-commutative P2+ssN2=N3 SSN2+P2=N3) + %<- (plus-swap-succ SSN2+P2=N3 SN2+sP2=N3) + %<- (plus-implies-gt SN2+sP2=N3 eq/ N3>sP2) + %<- (gt-transitive-ge N3>sP2 SP2>=sssN1 N3>sssN1) %. +%worlds () (non-trivial-times-implies-much-gt* _ _) %. +%total {} (non-trivial-times-implies-much-gt* _ _) %. +%theorem non-trivial-times-implies-much-gt : forall* {N1} {N2} {N3} forall {D times (s (s N1)) (s (s N2)) N3} exists {G1 gt N3 (s (s (s N1)))} {G2 gt N3 (s (s (s N2)))} true %. +%term _ + %pi (non-trivial-times-implies-much-gt T G1 G2) + %<- (non-trivial-times-implies-much-gt* T G1) + %<- (times-commutative T Tc) + %<- (non-trivial-times-implies-much-gt* Tc G2) %. +%worlds () (non-trivial-times-implies-much-gt _ _ _) %. +%total {} (non-trivial-times-implies-much-gt _ _ _) %. +%theorem plus-left-preserves-ge* : forall* {X1} {X2} {X3} {X4} {X5} forall {G ge X2 X4} {OP1 plus X1 X2 X3} {OP2 plus X1 X4 X5} exists {G2 ge X3 X5} true %. +%term _ + %pi (plus-left-preserves-ge* (ge/= eq/) X1+X2=X3 X1+X2=X5 (ge/= X3=X5)) + %<- (plus-deterministic X1+X2=X3 X1+X2=X5 eq/ eq/ X3=X5) %. +%term _ + %pi (plus-left-preserves-ge* (ge/> X2>X4) X1+X2=X3 X1+X4=X5 (ge/> X3>X5)) + %<- (plus-left-preserves-gt* X2>X4 X1+X2=X3 X1+X4=X5 X3>X5) %. +%worlds () (plus-left-preserves-ge* X2>=X4 X1+X2=X3 X1+X4=X5 X3>=X5) %. +%total {} (plus-left-preserves-ge* _ _ _ _) %. +%theorem plus-left-cancels-ge : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {OP1 plus X1 X2 X3} {OP2 plus Y1 Y2 Y3} {E1 eq X1 Y1} {G3 ge X3 Y3} exists {G2 ge X2 Y2} true %. +%term _ + %pi (plus-left-cancels-ge X1+X2=X3 X1+Y2=X3 eq/ (ge/= eq/) (ge/= X2=Y2)) + %<- (plus-left-cancels X1+X2=X3 X1+Y2=X3 eq/ eq/ X2=Y2) %. +%term _ + %pi (plus-left-cancels-ge X1+X2=X3 X1+Y2=Y3 eq/ (ge/> X3>Y3) (ge/> X2>Y2)) + %<- (plus-left-cancels-gt X1+X2=X3 X1+Y2=Y3 eq/ X3>Y3 X2>Y2) %. +%worlds () (plus-left-cancels-ge X1+X2=X3 Y1+Y2=Y3 X1=Y1 X3>=Y3 X2>=Y2) %. +%total {} (plus-left-cancels-ge _ _ _ _ _) %. +%theorem plus-left-preserves-ge : forall* {X1} {X2} {X4} forall {G ge X2 X4} exists {X3} {X5} {O1 plus X1 X2 X3} {O2 plus X1 X4 X5} {G2 ge X3 X5} true %. +%term _ + %pi (plus-left-preserves-ge X2>=X4 X3 X5 X1+X2=A3 X1+X4=X5 X3>=X5) + %<- (plus-total X1+X2=A3) + %<- (plus-total X1+X4=X5) + %<- (plus-left-preserves-ge* X2>=X4 X1+X2=A3 X1+X4=X5 X3>=X5) %. +%worlds () (plus-left-preserves-ge X2>=X4 X3 X5 X1+X2=A3 X1+X4=X5 X3>=X5) %. +%total {} (plus-left-preserves-ge _ _ _ _ _ _) %. +%theorem plus-right-preserves-ge* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 ge X1 X2} {O1 plus X1 X3 X4} {O2 plus X2 X3 X5} exists {G2 ge X4 X5} true %. +%term _ + %pi (plus-right-preserves-ge* X1>=X2 X1+X3=X4 X2+X3=X5 X4>=X5) + %<- (plus-commutative X1+X3=X4 X3+X1=X4) + %<- (plus-commutative X2+X3=X5 X3+X2=X5) + %<- (plus-left-preserves-ge* X1>=X2 X3+X1=X4 X3+X2=X5 X4>=X5) %. +%worlds () (plus-right-preserves-ge* X1>=X2 X1+X3=X4 X2+X3=X5 X4>=X5) %. +%total {} (plus-right-preserves-ge* _ _ _ _) %. +%theorem plus-right-preserves-ge : forall* {X1} {X2} {X3} forall {G1 ge X1 X2} exists {X4} {X5} {O1 plus X1 X3 X4} {O2 plus X2 X3 X5} {G2 ge X4 X5} true %. +%term _ + %pi (plus-right-preserves-ge X1>=X2 X4 X5 X1+X3=X4 X2+X3=X5 X4>=X5) + %<- (plus-total X1+X3=X4) + %<- (plus-total X2+X3=X5) + %<- (plus-right-preserves-ge* X1>=X2 X1+X3=X4 X2+X3=X5 X4>=X5) %. +%worlds () (plus-right-preserves-ge X1>=X2 X4 X5 X1+X3=X4 X2+X3=X5 X4>=X5) %. +%total {} (plus-right-preserves-ge _ _ _ _ _ _) %. +%theorem plus-preserves-ge* : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {G1 ge X1 Y1} {G2 ge X2 Y2} {MX plus X1 X2 X3} {MY plus Y1 Y2 Y3} exists {G3 ge X3 Y3} true %. +%term _ + %pi (plus-preserves-ge* X1>=Y1 X2>=Y2 X1+X2=X3 Y1+Y2=Y3 X3>=Y3) + %<- (plus-total Y1+X2=X) + %<- (plus-right-preserves-ge* X1>=Y1 X1+X2=X3 Y1+X2=X X3>=X) + %<- (plus-left-preserves-ge* X2>=Y2 Y1+X2=X Y1+Y2=Y3 X>=Y3) + %<- (ge-transitive X3>=X X>=Y3 X3>=Y3) %. +%worlds () (plus-preserves-ge* X1>=Y1 X2>=Y2 X1+X2=X3 Y1+Y2=Y3 X3>=Y3) %. +%total {} (plus-preserves-ge* _ _ _ _ _) %. +%theorem plus-preserves-ge : forall* {X1} {X2} {Y1} {Y2} forall {G1 ge X1 Y1} {G2 ge X2 Y2} exists {X3} {Y3} {MX plus X1 X2 X3} {MY plus Y1 Y2 Y3} {G3 ge X3 Y3} true %. +%term _ + %pi (plus-preserves-ge X1>=Y1 X2>=Y2 X3 Y3 X1+X2=X3 Y1+Y2=Y3 X3>=Y3) + %<- (plus-total X1+X2=X3) + %<- (plus-total Y1+Y2=Y3) + %<- (plus-preserves-ge* X1>=Y1 X2>=Y2 X1+X2=X3 Y1+Y2=Y3 X3>=Y3) %. +%worlds () (plus-preserves-ge X1>=Y1 X2>=Y2 X3 Y3 X1+X2=X3 Y1+Y2=Y3 X3>=Y3) %. +%total {} (plus-preserves-ge _ _ _ _ _ _ _) %. +%theorem plus-right-cancels-ge : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {OP1 plus X1 X2 X3} {OP2 plus Y1 Y2 Y3} {E2 eq X2 Y2} {G3 ge X3 Y3} exists {G1 ge X1 Y1} true %. +%term _ + %pi (plus-right-cancels-ge X1+X2=X3 Y1+Y2=Y3 X2=Y2 X3>=Y3 X1>=Y1) + %<- (plus-commutative X1+X2=X3 X2+X1=X3) + %<- (plus-commutative Y1+Y2=Y3 Y2+Y1=Y3) + %<- (plus-left-cancels-ge X2+X1=X3 Y2+Y1=Y3 X2=Y2 X3>=Y3 X1>=Y1) %. +%worlds () (plus-right-cancels-ge X1+X2=X3 Y1+Y2=Y3 X2=Y2 X3>=Y3 X1>=Y1) %. +%total {} (plus-right-cancels-ge _ _ _ _ _) %. +% Times preserves ge only because multiplying with zero yields equality. +%theorem times-left-preserves-ge* : forall* {X1} {X2} {X3} {X4} {X5} forall {G ge X2 X4} {OP1 times X1 X2 X3} {OP2 times X1 X4 X5} exists {G2 ge X3 X5} true %. +%term _ times-left-preserves-ge* _ times/z times/z (ge/= eq/) %. +%term _ + %pi (times-left-preserves-ge* (ge/= eq/) X1*X2=X3 X1*X2=X5 (ge/= X3=X5)) + %<- (times-deterministic X1*X2=X3 X1*X2=X5 eq/ eq/ X3=X5) %. +%term _ + %pi (times-left-preserves-ge* (ge/> X2>X4) X1*X2=X3 X1*X4=X5 (ge/> X3>X5)) + %<- (times-left-preserves-gt X2>X4 X1*X2=X3 X1*X4=X5 X3>X5) %. +%worlds () (times-left-preserves-ge* X2>=X4 X1*X2=X3 X1*X4=X5 X3>=X5) %. +%total {} (times-left-preserves-ge* _ _ _ _) %. +%theorem times-left-preserves-ge : forall* {X1} {X2} {X4} forall {G ge X2 X4} exists {X3} {X5} {O1 times X1 X2 X3} {O2 times X1 X4 X5} {G2 ge X3 X5} true %. +%term _ + %pi (times-left-preserves-ge X2>=X4 X3 X5 X1*X2=A3 X1*X4=X5 X3>=X5) + %<- (times-total X1*X2=A3) + %<- (times-total X1*X4=X5) + %<- (times-left-preserves-ge* X2>=X4 X1*X2=A3 X1*X4=X5 X3>=X5) %. +%worlds () (times-left-preserves-ge X2>=X4 X3 X5 X1*X2=A3 X1*X4=X5 X3>=X5) %. +%total {} (times-left-preserves-ge _ _ _ _ _ _) %. +%theorem times-right-preserves-ge* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 ge X1 X2} {O1 times X1 X3 X4} {O2 times X2 X3 X5} exists {G2 ge X4 X5} true %. +%term _ + %pi (times-right-preserves-ge* X1>=X2 X1*X3=X4 X2*X3=X5 X4>=X5) + %<- (times-commutative X1*X3=X4 X3*X1=X4) + %<- (times-commutative X2*X3=X5 X3*X2=X5) + %<- (times-left-preserves-ge* X1>=X2 X3*X1=X4 X3*X2=X5 X4>=X5) %. +%worlds () (times-right-preserves-ge* X1>=X2 X1*X3=X4 X2*X3=X5 X4>=X5) %. +%total {} (times-right-preserves-ge* _ _ _ _) %. +%theorem times-right-preserves-ge : forall* {X1} {X2} {X3} forall {G1 ge X1 X2} exists {X4} {X5} {O1 times X1 X3 X4} {O2 times X2 X3 X5} {G2 ge X4 X5} true %. +%term _ + %pi (times-right-preserves-ge X1>=X2 X4 X5 X1*X3=X4 X2*X3=X5 X4>=X5) + %<- (times-total X1*X3=X4) + %<- (times-total X2*X3=X5) + %<- (times-right-preserves-ge* X1>=X2 X1*X3=X4 X2*X3=X5 X4>=X5) %. +%worlds () (times-right-preserves-ge X1>=X2 X4 X5 X1*X3=X4 X2*X3=X5 X4>=X5) %. +%total {} (times-right-preserves-ge _ _ _ _ _ _) %. +%%%% Definitions +%sort ne %. +%term ne/< %pi (ne X Y) %<- (gt Y X) %. +%term ne/> %pi (ne X Y) %<- (gt X Y) %. +%sort eq? %. +%term eq?/yes eq? X X true %. +%term eq?/no %pi (eq? X Y false) %<- (ne X Y) %. +%%%% Theorems +%%% Theorems about ne +%theorem false-implies-ne : forall* {X1} {X2} forall {F void} exists {G ne X1 X2} true %. +%worlds () (false-implies-ne _ _) %. +%total {} (false-implies-ne _ _) %. +%theorem ne-respects-eq : forall* {X1} {X2} {Y1} {Y2} forall {D1 ne X1 X2} {E1 eq X1 Y1} {E2 eq X2 Y2} exists {D2 ne Y1 Y2} true %. +%term _ ne-respects-eq X1<>X2 eq/ eq/ X1<>X2 %. +%worlds () (ne-respects-eq _ _ _ _) %. +%total {} (ne-respects-eq _ _ _ _) %. +%theorem ne-anti-reflexive : forall* {X} forall {R ne X X} exists {F void} true %. +%term _ %pi (ne-anti-reflexive (ne/< X X>X) F) %<- (gt-anti-reflexive X>X F) %. +%worlds () (ne-anti-reflexive X<>X _) %. +%total {} (ne-anti-reflexive _ _) %. +%theorem ne-symmetric : forall* {X} {Y} forall {R1 ne X Y} exists {R2 ne Y X} true %. +%term _ ne-symmetric (ne/< X X X>Y) (ne/< X>Y) %. +%worlds () (ne-symmetric X<>Y Y<>X) %. +%total {} (ne-symmetric _ _) %. +%theorem eq-ne-implies-false : forall* {X} {Y} forall {D1 eq X Y} {D2 ne X Y} exists {F void} true %. +%term _ %pi (eq-ne-implies-false eq/ X<>X F) %<- (ne-anti-reflexive X<>X F) %. +%worlds () (eq-ne-implies-false X=Y X<>Y _) %. +%total {} (eq-ne-implies-false _ _ _) %. +%theorem ge-ne-implies-gt : forall* {X} {Y} forall {D1 ge X Y} {D2 ne X Y} exists {D3 gt X Y} true %. +%term _ ge-ne-implies-gt (ge/> X>Y) _ X>Y %. +%term _ + %pi (ge-ne-implies-gt (ge/= eq/) X<>X X>X) + %<- (ne-anti-reflexive X<>X F) + %<- (false-implies-gt F X>X) %. +%worlds () (ge-ne-implies-gt X>=Y X<>Y X>Y) %. +%total {} (ge-ne-implies-gt _ _ _) %. +%theorem eq?-total* : forall {M} {N} exists {B} {T eq? M N B} true %. +%theorem eq?-total*/L : forall* {M} {N} {C} forall {CMP compare M N C} exists {B} {T eq? M N B} true %. +%term _ eq?-total*/L compare/= true eq?/yes %. +%term _ eq?-total*/L (compare/< X X>Y) false (eq?/no (ne/> X>Y)) %. +%worlds () (eq?-total*/L _ _ _) %. +%total {} (eq?-total*/L _ _ _) %. +%term _ %pi (eq?-total* M N B T) %<- (compare-total CMP) %<- (eq?-total*/L CMP B T) %. +%worlds () (eq?-total* _ _ _ _) %. +%total {} (eq?-total* _ _ _ _) %. +%inline eq?-total eq?-total* _ _ _ %. +%theorem succ-preserves-ne : forall* {M} {N} forall {D ne M N} exists {D' ne (s M) (s N)} true %. +%term _ + %pi (succ-preserves-ne (ne/< N>M) (ne/< N+1>M+1)) + %<- (succ-preserves-gt N>M N+1>M+1) %. +%term _ + %pi (succ-preserves-ne (ne/> N>M) (ne/> N+1>M+1)) + %<- (succ-preserves-gt N>M N+1>M+1) %. +%worlds () (succ-preserves-ne M<>N M+1<>N+1) %. +%total {} (succ-preserves-ne _ _) %. +%theorem succ-preserves-ne-converse : forall* {M} {N} forall {D' ne (s M) (s N)} exists {D ne M N} true %. +%term _ + %pi (succ-preserves-ne-converse (ne/< N+1>M+1) (ne/< N>M)) + %<- (succ-preserves-gt-converse N+1>M+1 N>M) %. +%term _ + %pi (succ-preserves-ne-converse (ne/> N+1>M+1) (ne/> N>M)) + %<- (succ-preserves-gt-converse N+1>M+1 N>M) %. +%worlds () (succ-preserves-ne-converse M+1<>N+1 M<>N) %. +%total {} (succ-preserves-ne-converse _ _) %. +%theorem plus-left-preserves-ne* : forall* {X1} {X2} {X3} {X4} {X5} forall {G ne X2 X4} {OP1 plus X1 X2 X3} {OP2 plus X1 X4 X5} exists {G2 ne X3 X5} true %. +%term _ + %pi (plus-left-preserves-ne* (ne/< X4>X2) X1+X2=X3 X1+X4=X5 (ne/< X5>X3)) + %<- (plus-left-preserves-gt* X4>X2 X1+X4=X5 X1+X2=X3 X5>X3) %. +%term _ + %pi (plus-left-preserves-ne* (ne/> X2>X4) X1+X2=X3 X1+X4=X5 (ne/> X3>X5)) + %<- (plus-left-preserves-gt* X2>X4 X1+X2=X3 X1+X4=X5 X3>X5) %. +%worlds () (plus-left-preserves-ne* X2<>X4 X1+X2=X3 X1+X4=X5 X3<>X5) %. +%total {} (plus-left-preserves-ne* _ _ _ _) %. +%theorem plus-left-cancels-ne : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {OP1 plus X1 X2 X3} {OP2 plus Y1 Y2 Y3} {E1 eq X1 Y1} {G3 ne X3 Y3} exists {G2 ne X2 Y2} true %. +%term _ + %pi (plus-left-cancels-ne X1+X2=X3 X1+Y2=Y3 eq/ (ne/< Y3>X3) (ne/< Y2>X2)) + %<- (plus-left-cancels-gt X1+Y2=Y3 X1+X2=X3 eq/ Y3>X3 Y2>X2) %. +%term _ + %pi (plus-left-cancels-ne X1+X2=X3 X1+Y2=Y3 eq/ (ne/> X3>Y3) (ne/> X2>Y2)) + %<- (plus-left-cancels-gt X1+X2=X3 X1+Y2=Y3 eq/ X3>Y3 X2>Y2) %. +%worlds () (plus-left-cancels-ne X1+X2=X3 Y1+Y2=Y3 X1=Y1 X3<>Y3 X2<>Y2) %. +%total {} (plus-left-cancels-ne _ _ _ _ _) %. +%theorem plus-left-preserves-ne : forall* {X1} {X2} {X4} forall {G ne X2 X4} exists {X3} {X5} {O1 plus X1 X2 X3} {O2 plus X1 X4 X5} {G2 ne X3 X5} true %. +%term _ + %pi (plus-left-preserves-ne X2<>X4 X3 X5 X1+X2=A3 X1+X4=X5 X3<>X5) + %<- (plus-total X1+X2=A3) + %<- (plus-total X1+X4=X5) + %<- (plus-left-preserves-ne* X2<>X4 X1+X2=A3 X1+X4=X5 X3<>X5) %. +%worlds () (plus-left-preserves-ne X2<>X4 X3 X5 X1+X2=A3 X1+X4=X5 X3<>X5) %. +%total {} (plus-left-preserves-ne _ _ _ _ _ _) %. +%theorem plus-right-preserves-ne* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 ne X1 X2} {O1 plus X1 X3 X4} {O2 plus X2 X3 X5} exists {G2 ne X4 X5} true %. +%term _ + %pi (plus-right-preserves-ne* X1<>X2 X1+X3=X4 X2+X3=X5 X4<>X5) + %<- (plus-commutative X1+X3=X4 X3+X1=X4) + %<- (plus-commutative X2+X3=X5 X3+X2=X5) + %<- (plus-left-preserves-ne* X1<>X2 X3+X1=X4 X3+X2=X5 X4<>X5) %. +%worlds () (plus-right-preserves-ne* X1<>X2 X1+X3=X4 X2+X3=X5 X4<>X5) %. +%total {} (plus-right-preserves-ne* _ _ _ _) %. +%theorem plus-right-preserves-ne : forall* {X1} {X2} {X3} forall {G1 ne X1 X2} exists {X4} {X5} {O1 plus X1 X3 X4} {O2 plus X2 X3 X5} {G2 ne X4 X5} true %. +%term _ + %pi (plus-right-preserves-ne X1<>X2 X4 X5 X1+X3=X4 X2+X3=X5 X4<>X5) + %<- (plus-total X1+X3=X4) + %<- (plus-total X2+X3=X5) + %<- (plus-right-preserves-ne* X1<>X2 X1+X3=X4 X2+X3=X5 X4<>X5) %. +%worlds () (plus-right-preserves-ne X1<>X2 X4 X5 X1+X3=X4 X2+X3=X5 X4<>X5) %. +%total {} (plus-right-preserves-ne _ _ _ _ _ _) %. +%theorem plus-right-cancels-ne : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {OP1 plus X1 X2 X3} {OP2 plus Y1 Y2 Y3} {E2 eq X2 Y2} {G3 ne X3 Y3} exists {G1 ne X1 Y1} true %. +%term _ + %pi (plus-right-cancels-ne X1+X2=X3 Y1+Y2=Y3 X2=Y2 X3<>Y3 X1<>Y1) + %<- (plus-commutative X1+X2=X3 X2+X1=X3) + %<- (plus-commutative Y1+Y2=Y3 Y2+Y1=Y3) + %<- (plus-left-cancels-ne X2+X1=X3 Y2+Y1=Y3 X2=Y2 X3<>Y3 X1<>Y1) %. +%worlds () (plus-right-cancels-ne X1+X2=X3 Y1+Y2=Y3 X2=Y2 X3<>Y3 X1<>Y1) %. +%total {} (plus-right-cancels-ne _ _ _ _ _) %. +%%%%% nat-less.elf +%%%%% Inverse relations for natural numbers +%%%%% This file is part of the nat.elf signature +%%%% Definitions +%inline lt [X] [Y] gt Y X %. +%%%% Theorems about lt +%theorem false-implies-lt : forall* {X1} {X2} forall {F void} exists {G lt X1 X2} true %. +%worlds () (false-implies-lt _ _) %. +%total {} (false-implies-lt _ _) %. +%theorem lt-respects-eq : forall* {X1} {X2} {Y1} {Y2} forall {D1 lt X1 X2} {E1 eq X1 Y1} {E2 eq X2 Y2} exists {D2 lt Y1 Y2} true %. +%term _ lt-respects-eq X1X1 X1>X2 R) %<- (gt-anti-symmetric X1>X2 X2>X1 R) %. +%worlds () (lt-anti-symmetric _ _ _) %. +%total {} (lt-anti-symmetric _ _ _) %. +%theorem lt-transitive : forall* {X1} {X2} {X3} forall {G1 lt X1 X2} {G2 lt X2 X3} exists {G3 lt X1 X3} true %. +%term _ %pi (lt-transitive X1X2 X1+X2=X3 X1+X4=X5 X5>X3) + %<- (plus-left-preserves-gt* X4>X2 X1+X4=X5 X1+X2=X3 X5>X3) %. +%worlds () (plus-left-preserves-lt* X2X3 Y2>X2) + %<- (plus-left-cancels-gt X1+Y2=X3 X1+X2=X3 eq/ Y3>X3 Y2>X2) %. +%worlds () (plus-left-cancels-lt X1+X2=X3 Y1+Y2=Y3 X1=Y1 X3=X1 X1>=X2 R) %<- (ge-anti-symmetric X1>=X2 X2>=X1 R) %. +%worlds () (le-anti-symmetric _ _ _) %. +%total {} (le-anti-symmetric _ _ _) %. +%theorem le-transitive : forall* {X1} {X2} {X3} forall {G1 le X1 X2} {G2 le X2 X3} exists {G3 le X1 X3} true %. +%term _ %pi (le-transitive X1<=X2 X2<=X3 X1<=X3) %<- (ge-transitive X2<=X3 X1<=X2 X1<=X3) %. +%worlds () (le-transitive X1<=X2 X2<=X3 X1<=X3) %. +%total {} (le-transitive _ _ _) %. +%inline le-reflexive ge-reflexive %. +%theorem le-transitive-lt : forall* {X1} {X2} {X3} forall {L1 le X1 X2} {L2 lt X2 X3} exists {L3 lt X1 X3} true %. +%term _ + %pi (le-transitive-lt X2>=X1 X3>X2 X3>X1) + %<- (gt-transitive-ge X3>X2 X2>=X1 X3>X1) %. +%worlds () (le-transitive-lt X1<=X2 X2X1 X3>=X2 X3>X1) + %<- (ge-transitive-gt X3>=X2 X2>X1 X3>X1) %. +%worlds () (lt-transitive-le X1=X2 X1+X2=X3 X1+X4=X5 X5>=X3) + %<- (plus-left-preserves-ge* X4>=X2 X1+X4=X5 X1+X2=X3 X5>=X3) %. +%worlds () (plus-left-preserves-le* X2<=X4 X1+X2=X3 X1+X4=X5 X3<=X5) %. +%total {} (plus-left-preserves-le* _ _ _ _) %. +%theorem plus-left-cancels-le : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {OP1 plus X1 X2 X3} {OP2 plus Y1 Y2 Y3} {E1 eq X1 Y1} {R3 le X3 Y3} exists {R2 le X2 Y2} true %. +%term _ + %pi (plus-left-cancels-le X1+X2=X3 X1+Y2=X3 eq/ Y3>=X3 Y2>=X2) + %<- (plus-left-cancels-ge X1+Y2=X3 X1+X2=X3 eq/ Y3>=X3 Y2>=X2) %. +%worlds () (plus-left-cancels-le X1+X2=X3 Y1+Y2=Y3 X1=Y1 X3<=Y3 X2<=Y2) %. +%total {} (plus-left-cancels-le _ _ _ _ _) %. +%theorem plus-left-preserves-le : forall* {X1} {X2} {X4} forall {G le X2 X4} exists {X3} {X5} {O1 plus X1 X2 X3} {O2 plus X1 X4 X5} {G2 le X3 X5} true %. +%term _ + %pi (plus-left-preserves-le X2<=X4 X3 X5 X1+X2=A3 X1+X4=X5 X3<=X5) + %<- (plus-total X1+X2=A3) + %<- (plus-total X1+X4=X5) + %<- (plus-left-preserves-le* X2<=X4 X1+X2=A3 X1+X4=X5 X3<=X5) %. +%worlds () (plus-left-preserves-le X2<=X4 X3 X5 X1+X2=A3 X1+X4=X5 X3<=X5) %. +%total {} (plus-left-preserves-le _ _ _ _ _ _) %. +%theorem plus-right-preserves-le* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 le X1 X2} {O1 plus X1 X3 X4} {O2 plus X2 X3 X5} exists {G2 le X4 X5} true %. +%term _ + %pi (plus-right-preserves-le* X1<=X2 X1+X3=X4 X2+X3=X5 X4<=X5) + %<- (plus-commutative X1+X3=X4 X3+X1=X4) + %<- (plus-commutative X2+X3=X5 X3+X2=X5) + %<- (plus-left-preserves-le* X1<=X2 X3+X1=X4 X3+X2=X5 X4<=X5) %. +%worlds () (plus-right-preserves-le* X1<=X2 X1+X3=X4 X2+X3=X5 X4<=X5) %. +%total {} (plus-right-preserves-le* _ _ _ _) %. +%theorem plus-right-preserves-le : forall* {X1} {X2} {X3} forall {G1 le X1 X2} exists {X4} {X5} {O1 plus X1 X3 X4} {O2 plus X2 X3 X5} {G2 le X4 X5} true %. +%term _ + %pi (plus-right-preserves-le X1<=X2 X4 X5 X1+X3=X4 X2+X3=X5 X4<=X5) + %<- (plus-total X1+X3=X4) + %<- (plus-total X2+X3=X5) + %<- (plus-right-preserves-le* X1<=X2 X1+X3=X4 X2+X3=X5 X4<=X5) %. +%worlds () (plus-right-preserves-le X1<=X2 X4 X5 X1+X3=X4 X2+X3=X5 X4<=X5) %. +%total {} (plus-right-preserves-le _ _ _ _ _ _) %. +%theorem plus-preserves-le* : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {G1 le X1 Y1} {G2 le X2 Y2} {MX plus X1 X2 X3} {MY plus Y1 Y2 Y3} exists {G3 le X3 Y3} true %. +%term _ + %pi (plus-preserves-le* X1<=Y1 X2<=Y2 X1+X2=X3 Y1+Y2=Y3 X3<=Y3) + %<- (plus-total Y1+X2=X) + %<- (plus-right-preserves-le* X1<=Y1 X1+X2=X3 Y1+X2=X X3<=X) + %<- (plus-left-preserves-le* X2<=Y2 Y1+X2=X Y1+Y2=Y3 X<=Y3) + %<- (le-transitive X3<=X X<=Y3 X3<=Y3) %. +%worlds () (plus-preserves-le* X1<=Y1 X2<=Y2 X1+X2=X3 Y1+Y2=Y3 X3<=Y3) %. +%total {} (plus-preserves-le* _ _ _ _ _) %. +%theorem plus-preserves-le : forall* {X1} {X2} {Y1} {Y2} forall {G1 le X1 Y1} {G2 le X2 Y2} exists {X3} {Y3} {MX plus X1 X2 X3} {MY plus Y1 Y2 Y3} {G3 le X3 Y3} true %. +%term _ + %pi (plus-preserves-le X1<=Y1 X2<=Y2 X3 Y3 X1+X2=X3 Y1+Y2=Y3 X3<=Y3) + %<- (plus-total X1+X2=X3) + %<- (plus-total Y1+Y2=Y3) + %<- (plus-preserves-le* X1<=Y1 X2<=Y2 X1+X2=X3 Y1+Y2=Y3 X3<=Y3) %. +%worlds () (plus-preserves-le X1<=Y1 X2<=Y2 X3 Y3 X1+X2=X3 Y1+Y2=Y3 X3<=Y3) %. +%total {} (plus-preserves-le _ _ _ _ _ _ _) %. +%theorem plus-right-cancels-le : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {OP1 plus X1 X2 X3} {OP2 plus Y1 Y2 Y3} {E2 eq X2 Y2} {G3 le X3 Y3} exists {G1 le X1 Y1} true %. +%term _ + %pi (plus-right-cancels-le X1+X2=X3 Y1+Y2=Y3 X2=Y2 X3<=Y3 X1<=Y1) + %<- (plus-commutative X1+X2=X3 X2+X1=X3) + %<- (plus-commutative Y1+Y2=Y3 Y2+Y1=Y3) + %<- (plus-left-cancels-le X2+X1=X3 Y2+Y1=Y3 X2=Y2 X3<=Y3 X1<=Y1) %. +%worlds () (plus-right-cancels-le X1+X2=X3 Y1+Y2=Y3 X2=Y2 X3<=Y3 X1<=Y1) %. +%total {} (plus-right-cancels-le _ _ _ _ _) %. +%theorem times-left-preserves-le* : forall* {X1} {X2} {X3} {X4} {X5} forall {R1 le X2 X4} {OP1 times X1 X2 X3} {OP2 times X1 X4 X5} exists {R2 le X3 X5} true %. +%term _ + %pi (times-left-preserves-le* X4>=X2 X1*X2=X3 X1*X4=X5 X5>=X3) + %<- (times-left-preserves-ge* X4>=X2 X1*X4=X5 X1*X2=X3 X5>=X3) %. +%worlds () (times-left-preserves-le* X2<=X4 X1*X2=X3 X1*X4=X5 X3<=X5) %. +%total {} (times-left-preserves-le* _ _ _ _) %. +%theorem times-left-preserves-le : forall* {X1} {X2} {X4} forall {G le X2 X4} exists {X3} {X5} {O1 times X1 X2 X3} {O2 times X1 X4 X5} {G2 le X3 X5} true %. +%term _ + %pi (times-left-preserves-le X2<=X4 X3 X5 X1*X2=A3 X1*X4=X5 X3<=X5) + %<- (times-total X1*X2=A3) + %<- (times-total X1*X4=X5) + %<- (times-left-preserves-le* X2<=X4 X1*X2=A3 X1*X4=X5 X3<=X5) %. +%worlds () (times-left-preserves-le X2<=X4 X3 X5 X1*X2=A3 X1*X4=X5 X3<=X5) %. +%total {} (times-left-preserves-le _ _ _ _ _ _) %. +%theorem times-right-preserves-le* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 le X1 X2} {O1 times X1 X3 X4} {O2 times X2 X3 X5} exists {G2 le X4 X5} true %. +%term _ + %pi (times-right-preserves-le* X1<=X2 X1*X3=X4 X2*X3=X5 X4<=X5) + %<- (times-commutative X1*X3=X4 X3*X1=X4) + %<- (times-commutative X2*X3=X5 X3*X2=X5) + %<- (times-left-preserves-le* X1<=X2 X3*X1=X4 X3*X2=X5 X4<=X5) %. +%worlds () (times-right-preserves-le* X1<=X2 X1*X3=X4 X2*X3=X5 X4<=X5) %. +%total {} (times-right-preserves-le* _ _ _ _) %. +%theorem times-right-preserves-le : forall* {X1} {X2} {X3} forall {G1 le X1 X2} exists {X4} {X5} {O1 times X1 X3 X4} {O2 times X2 X3 X5} {G2 le X4 X5} true %. +%term _ + %pi (times-right-preserves-le X1<=X2 X4 X5 X1*X3=X4 X2*X3=X5 X4<=X5) + %<- (times-total X1*X3=X4) + %<- (times-total X2*X3=X5) + %<- (times-right-preserves-le* X1<=X2 X1*X3=X4 X2*X3=X5 X4<=X5) %. +%worlds () (times-right-preserves-le X1<=X2 X4 X5 X1*X3=X4 X2*X3=X5 X4<=X5) %. +%total {} (times-right-preserves-le _ _ _ _ _ _) %. +%theorem times-preserves-le* : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {G1 le X1 Y1} {G2 le X2 Y2} {MX times X1 X2 X3} {MY times Y1 Y2 Y3} exists {G3 le X3 Y3} true %. +%term _ + %pi (times-preserves-le* X1<=Y1 X2<=Y2 X1*X2=X3 Y1*Y2=Y3 X3<=Y3) + %<- (times-total Y1*X2=X) + %<- (times-right-preserves-le* X1<=Y1 X1*X2=X3 Y1*X2=X X3<=X) + %<- (times-left-preserves-le* X2<=Y2 Y1*X2=X Y1*Y2=Y3 X<=Y3) + %<- (le-transitive X3<=X X<=Y3 X3<=Y3) %. +%worlds () (times-preserves-le* X1<=Y1 X2<=Y2 X1*X2=X3 Y1*Y2=Y3 X3<=Y3) %. +%total {} (times-preserves-le* _ _ _ _ _) %. +%theorem times-preserves-le : forall* {X1} {X2} {Y1} {Y2} forall {G1 le X1 Y1} {G2 le X2 Y2} exists {X3} {Y3} {MX times X1 X2 X3} {MY times Y1 Y2 Y3} {G3 le X3 Y3} true %. +%term _ + %pi (times-preserves-le X1<=Y1 X2<=Y2 X3 Y3 X1*X2=X3 Y1*Y2=Y3 X3<=Y3) + %<- (times-total X1*X2=X3) + %<- (times-total Y1*Y2=Y3) + %<- (times-preserves-le* X1<=Y1 X2<=Y2 X1*X2=X3 Y1*Y2=Y3 X3<=Y3) %. +%worlds () (times-preserves-le X1<=Y1 X2<=Y2 X3 Y3 X1*X2=X3 Y1*Y2=Y3 X3<=Y3) %. +%total {} (times-preserves-le _ _ _ _ _ _ _) %. +%%%%% nat-inv-comp.elf +%%%%% Theorems about minus and composed relations +%%%%% This file is part of the nat.elf signature +%%%% Theorems +%%% Theorems about minus +%theorem minus-left-inverts-ge* : forall* {X1} {X2} {X3} {X4} {X5} forall {G ge X2 X4} {IOP1 minus X1 X2 X3} {IOP2 minus X1 X4 X5} exists {GP ge X5 X3} true %. +%term _ + %pi (minus-left-inverts-ge* X2>=X4 X3+X2=X1 X5+X4=X1 X5>=X3) + %<- (plus-total X3+X4=X7) + %<- (plus-left-preserves-ge* X2>=X4 X3+X2=X1 X3+X4=X7 X1>=X7) + %<- (plus-right-cancels-ge X5+X4=X1 X3+X4=X7 eq/ X1>=X7 X5>=X3) %. +%worlds () (minus-left-inverts-ge* X2>=X4 X1-X2=X3 X1-X4=X5 X5>=X3) %. +%total {} (minus-left-inverts-ge* _ _ _ _) %. +%theorem minus-right-preserves-ge* : forall* {X1} {X2} {X3} {X4} {X5} forall {G ge X1 X2} {IOP1 minus X1 X3 X4} {IOP2 minus X2 X3 X5} exists {GP ge X4 X5} true %. +%term _ + %pi (minus-right-preserves-ge* X1>=X2 X4+X3=X1 X5+X3=X2 X4>=X5) + %<- (plus-right-cancels-ge X4+X3=X1 X5+X3=X2 eq/ X1>=X2 X4>=X5) %. +%worlds () (minus-right-preserves-ge* X1>=X2 X1-X3=X4 X2-X3=X5 X4>=X5) %. +%total {} (minus-right-preserves-ge* _ _ _ _) %. +%theorem minus-left-cancels-inverts-ge : forall* {X1} {X2} {X3} {X4} {X5} {X6} forall {IOP1 minus X1 X2 X3} {IOP2 minus X4 X5 X6} {E eq X1 X4} {G ge X3 X6} exists {GP ge X5 X2} true %. +%term _ + %pi (minus-left-cancels-inverts-ge X3+X2=X1 X6+X5=X4 X1=X4 X3>=X6 X5>=X2) + %<- (plus-total X6+X2=X7) + %<- (plus-right-preserves-ge* X3>=X6 X3+X2=X1 X6+X2=X7 X1>=X7) + %<- (eq-symmetric X1=X4 X4=X1) + %<- (plus-respects-eq X6+X5=X4 eq/ eq/ X4=X1 X6+X5=X1) + %<- (plus-left-cancels-ge X6+X5=X1 X6+X2=X7 eq/ X1>=X7 X5>=X2) %. +%worlds () (minus-left-cancels-inverts-ge X1-X2=X3 X4-X5=X6 X1=X4 X3>=X6 X5>=X2) %. +%total {} (minus-left-cancels-inverts-ge _ _ _ _ _) %. +%theorem minus-right-cancels-ge : forall* {X1} {X2} {X3} {X4} {X5} {X6} forall {IOP1 minus X1 X2 X3} {IOP2 minus X4 X5 X6} {E2 eq X2 X5} {G3 ge X3 X6} exists {G1 ge X1 X4} true %. +%term _ + %pi (minus-right-cancels-ge X3+X2=X1 X6+X5=X4 X2=X5 X3>=X6 X1>=X4) + %<- (plus-respects-eq X3+X2=X1 eq/ X2=X5 eq/ X3+X5=X1) + %<- (plus-right-preserves-ge* X3>=X6 X3+X5=X1 X6+X5=X4 X1>=X4) %. +%worlds () (minus-right-cancels-ge X1-X2=X3 X4-X5=X6 X2=X5 X3>=X6 X1>=X4) %. +%total {} (minus-right-cancels-ge _ _ _ _ _) %. +%theorem minus-left-preserves-ne* : forall* {X1} {X2} {X3} {X4} {X5} forall {G ne X2 X4} {IOP1 minus X1 X2 X3} {IOP2 minus X1 X4 X5} exists {GP ne X3 X5} true %. +%term _ + %pi (minus-left-preserves-ne* X2<>X4 X3+X2=X1 X5+X4=X1 X3<>X5) + %<- (plus-total X3+X4=X7) + %<- (plus-left-preserves-ne* X2<>X4 X3+X2=X1 X3+X4=X7 X1<>X7) + %<- (plus-right-cancels-ne X5+X4=X1 X3+X4=X7 eq/ X1<>X7 X5<>X3) + %<- (ne-symmetric X5<>X3 X3<>X5) %. +%worlds () (minus-left-preserves-ne* X2<>X4 X1-X2=X3 X1-X4=X5 X3<>X5) %. +%total {} (minus-left-preserves-ne* _ _ _ _) %. +%theorem minus-right-preserves-ne* : forall* {X1} {X2} {X3} {X4} {X5} forall {G ne X1 X2} {IOP1 minus X1 X3 X4} {IOP2 minus X2 X3 X5} exists {GP ne X4 X5} true %. +%term _ + %pi (minus-right-preserves-ne* X1<>X2 X4+X3=X1 X5+X3=X2 X4<>X5) + %<- (plus-right-cancels-ne X4+X3=X1 X5+X3=X2 eq/ X1<>X2 X4<>X5) %. +%worlds () (minus-right-preserves-ne* X1<>X2 X1-X3=X4 X2-X3=X5 X4<>X5) %. +%total {} (minus-right-preserves-ne* _ _ _ _) %. +%theorem minus-left-cancels-ne : forall* {X1} {X2} {X3} {X4} {X5} {X6} forall {IOP1 minus X1 X2 X3} {IOP2 minus X4 X5 X6} {E eq X1 X4} {G ne X3 X6} exists {GP ne X2 X5} true %. +%term _ + %pi (minus-left-cancels-ne X3+X2=X1 X6+X5=X4 X1=X4 X3<>X6 X2<>X5) + %<- (plus-total X6+X2=X7) + %<- (plus-right-preserves-ne* X3<>X6 X3+X2=X1 X6+X2=X7 X1<>X7) + %<- (eq-symmetric X1=X4 X4=X1) + %<- (plus-respects-eq X6+X5=X4 eq/ eq/ X4=X1 X6+X5=X1) + %<- (plus-left-cancels-ne X6+X5=X1 X6+X2=X7 eq/ X1<>X7 X5<>X2) + %<- (ne-symmetric X5<>X2 X2<>X5) %. +%worlds () (minus-left-cancels-ne X1-X2=X3 X4-X5=X6 X1=X4 X3<>X6 X2<>X5) %. +%total {} (minus-left-cancels-ne _ _ _ _ _) %. +%theorem minus-right-cancels-ne : forall* {X1} {X2} {X3} {X4} {X5} {X6} forall {IOP1 minus X1 X2 X3} {IOP2 minus X4 X5 X6} {E2 eq X2 X5} {G3 ne X3 X6} exists {G1 ne X1 X4} true %. +%term _ + %pi (minus-right-cancels-ne X3+X2=X1 X6+X5=X4 X2=X5 X3<>X6 X1<>X4) + %<- (plus-respects-eq X3+X2=X1 eq/ X2=X5 eq/ X3+X5=X1) + %<- (plus-right-preserves-ne* X3<>X6 X3+X5=X1 X6+X5=X4 X1<>X4) %. +%worlds () (minus-right-cancels-ne X1-X2=X3 X4-X5=X6 X2=X5 X3<>X6 X1<>X4) %. +%total {} (minus-right-cancels-ne _ _ _ _ _) %. +%%%%% nat-inv-less.elf +%%%%% Theorems about minus and inverted relations +%%%%% This file is part of the nat.elf signature +%%%% Theorems +%%% Theorems about minus +%theorem minus-left-inverts-lt* : forall* {X1} {X2} {X3} {X4} {X5} forall {G lt X2 X4} {IOP1 minus X1 X2 X3} {IOP2 minus X1 X4 X5} exists {GP lt X5 X3} true %. +%term _ + %pi (minus-left-inverts-lt* X2M)) + %<- (equal-implies-eq CMP M=N-1) + %<- (succ-implies-gt eq/ N>N-1) + %<- (eq-symmetric M=N-1 N-1=M) + %<- (gt-respects-eq N>N-1 eq/ N-1=M N>M) %. +%term _ + %pi (divrem-total** M N-1 less CMP z M (divrem/z N>M)) + %<- (less-implies-lt CMP N-1>M) + %<- (succ-implies-gt eq/ N>N-1) + %<- (gt-transitive N>N-1 N-1>M N>M) %. +%term _ + %pi (divrem-total** M N-1 greater CMP (s Q) R (divrem/s D/N=Q,R D+N=M)) + %<- (greater-implies-gt CMP M>N-1) + %<- (gt-implies-plus M>N-1 D SD+N-1=M) + %<- (plus-swap-succ SD+N-1=M D+N=M) + %<- (plus-commutative D+N=M N+D=M) + %<- (plus-implies-gt N+D=M eq/ M>D) + %<- (meta-gt M D M>D) + %<- (compare-total* D N-1 C' CMP') + %<- (divrem-total** D N-1 C' CMP' Q R D/N=Q,R) %. +%worlds () (divrem-total** M N-1 C MCN Q R M/N=Q,R) %. +%total (M) (divrem-total** M _ _ _ _ _ _) %. +%theorem divrem-total* : forall {M} {N-} exists {Q} {R} {D divrem M (s N-) Q R} true %. +%term _ + %pi (divrem-total* M N-1 Q R M/N=Q,R) + %<- (compare-total* M N-1 C CMP) + %<- (divrem-total** M N-1 C CMP Q R M/N=Q,R) %. +%worlds () (divrem-total* M N-1 Q R M/N=Q,R) %. +%total {} (divrem-total* _ _ _ _ _) %. +%inline divrem-total divrem-total* _ _ _ _ %. +%theorem divrem-deterministic : forall* {M} {N} {Q} {R} {M'} {N'} {Q'} {R'} forall {D divrem M N Q R} {D' divrem M' N' Q' R'} {E1 eq M M'} {E2 eq N N'} exists {E3 eq Q Q'} {E4 eq R R'} true %. +%term _ divrem-deterministic (divrem/z _) (divrem/z _) eq/ eq/ eq/ eq/ %. +%term _ + %pi (divrem-deterministic (divrem/s D/N=Q,R D+N=M) (divrem/s D'/N=Q',R' D'+N=M) eq/ eq/ SQ=SQ' R=R') + %<- (plus-right-cancels D+N=M D'+N=M eq/ eq/ D=D') + %<- (divrem-deterministic D/N=Q,R D'/N=Q',R' D=D' eq/ Q=Q' R=R') + %<- (succ-deterministic Q=Q' SQ=SQ') %. +%% contradiction cases: +%term _ + %pi (divrem-deterministic (divrem/z N>M) (divrem/s _ D+N=M) eq/ eq/ Q=Q' R=R') + %<- (plus-commutative D+N=M N+D=M) + %<- (plus-gt-contradiction N+D=M N>M F) + %<- (false-implies-eq F Q=Q') + %<- (false-implies-eq F R=R') %. +%term _ + %pi (divrem-deterministic (divrem/s _ D+N=M) (divrem/z N>M) eq/ eq/ Q=Q' R=R') + %<- (plus-commutative D+N=M N+D=M) + %<- (plus-gt-contradiction N+D=M N>M F) + %<- (false-implies-eq F Q=Q') + %<- (false-implies-eq F R=R') %. +%worlds () (divrem-deterministic M/N=Q,R M'/N'=Q'/R' M=M' N=N' Q=Q' R=R') %. +%total (D) (divrem-deterministic D _ _ _ _ _) %. +%theorem divrem-implies-positive : forall* {M} {N} {Q} {R} forall {D divrem M N Q R} exists {N-1} {E eq N (s N-1)} true %. +%term _ + %pi (divrem-implies-positive (divrem/z N>M) N-1 N=sN-1) + %<- (gt-implies-positive N>M N-1 N=sN-1) %. +%term _ divrem-implies-positive (divrem/s _ _) _ eq/ %. +%worlds () (divrem-implies-positive M/N=Q,R N-1 N=sN-1) %. +%total {} (divrem-implies-positive _ _ _) %. +%theorem divrem-implies-gt : forall* {M} {N} {Q} {R} forall {D divrem M N Q R} exists {G gt N R} true %. +%term _ divrem-implies-gt (divrem/z N>M) N>M %. +%term _ + %pi (divrem-implies-gt (divrem/s D/N=Q,R _) N>R) + %<- (divrem-implies-gt D/N=Q,R N>R) %. +%worlds () (divrem-implies-gt M/N=Q,R N>R) %. +%total D (divrem-implies-gt D _) %. +%theorem divrem-contradiction : forall* {M} {N} {Q} {R} {X} forall {D divrem M N Q R} {P plus N X R} exists {F void} true %. +%term _ + %pi (divrem-contradiction D P F) + %<- (divrem-implies-gt D N>R) + %<- (plus-commutative P Pc) + %<- (plus-implies-ge Pc R>=N) + %<- (gt-transitive-ge N>R R>=N N>N) + %<- (gt-anti-reflexive N>N F) %. +%worlds () (divrem-contradiction _ _ _) %. +%total {} (divrem-contradiction _ _ _) %. +%theorem divrem-can-be-inverted : forall* {M} {N} {Q} {R} forall {D divrem M N Q R} exists {X} {T times Q N X} {P plus X R M} true %. +%term _ divrem-can-be-inverted (divrem/z _) z times/z plus/z %. +%term _ + %pi (divrem-can-be-inverted (divrem/s D/N=Q,R D+N=M) X (times/s Q*N=Y Y+N=X) X+R=M) + %<- (divrem-can-be-inverted D/N=Q,R Y Q*N=Y Y+R=D) + %<- (plus-commutative Y+R=D R+Y=D) + %<- (plus-associative R+Y=D D+N=M X Y+N=X R+X=M) + %<- (plus-commutative R+X=M X+R=M) %. +%worlds () (divrem-can-be-inverted M/N=Q,R X Q*N=X X+R=M) %. +%total (D) (divrem-can-be-inverted D _ _ _) %. +%theorem div-can-be-inverted : forall* {M} {N} {Q} forall {D divrem M N Q z} exists {T times Q N M} true %. +%term _ div-can-be-inverted (divrem/z _) times/z %. +%term _ + %pi (div-can-be-inverted (divrem/s D/N=Q,z D+N=M) (times/s Q*N=D D+N=M)) + %<- (div-can-be-inverted D/N=Q,z Q*N=D) %. +%worlds () (div-can-be-inverted _ _) %. +%total (D) (div-can-be-inverted D _) %. +%theorem divrem-can-be-constructed : forall* {M} {N} {Q} {R} {X} forall {T times Q N X} {P plus X R M} {G gt N R} exists {D divrem M N Q R} true %. +%term _ divrem-can-be-constructed times/z plus/z N>R (divrem/z N>R) %. +%term _ + %pi (divrem-can-be-constructed (times/s Q*N=Y Y+N=X) X+R=M N>R (divrem/s Z/N=Q,R Z+N=M)) + %<- (plus-commutative Y+N=X N+Y=X) + %<- (plus-associative N+Y=X X+R=M Z Y+R=Z N+Z=M) + %<- (plus-commutative N+Z=M Z+N=M) + %<- (divrem-can-be-constructed Q*N=Y Y+R=Z N>R Z/N=Q,R) %. +%term _ + %pi (divrem-can-be-constructed _ _ ZERO>R D) + %<- (gt-contradiction ZERO>R F) + %<- (false-implies-divrem F D) %. +%worlds () (divrem-can-be-constructed Q*N=X X+R=M N>R M/N=Q,R) %. +%total (T) (divrem-can-be-constructed T _ _ _) %. +%theorem div-can-be-constructed : forall* {M} {N} {Q} forall {T times Q (s N) M} exists {D divrem M (s N) Q z} true %. +%term _ + %pi (div-can-be-constructed times/z (divrem/z N+1>0)) + %<- (succ-implies-gt-zero _ N+1>0) %. +%term _ + %pi (div-can-be-constructed (times/s Q*sN=D D+sN=M) (divrem/s D/sN=Q,z D+sN=M)) + %<- (div-can-be-constructed Q*sN=D D/sN=Q,z) %. +%worlds () (div-can-be-constructed _ _) %. +%total (T) (div-can-be-constructed T _) %. +%theorem remainder-implies-gt-quotient : forall* {M} {N} {Q} {R} forall {D divrem M N Q (s R)} exists {G gt M Q} true %. +%term _ + %pi (remainder-implies-gt-quotient (divrem/z _) R+1>0) + %<- (succ-implies-gt-zero _ R+1>0) %. +%term _ + %pi (remainder-implies-gt-quotient (divrem/s D/N=Q,sR D+N=M) M>sQ) + %<- (remainder-implies-gt-quotient D/N=Q,sR D>Q) + %<- (gt-implies-ge-succ D>Q D>=sQ) + %<- (plus-commutative D+N=M N+D=M) + %<- (plus-implies-gt N+D=M eq/ M>D) + %<- (gt-transitive-ge M>D D>=sQ M>sQ) %. +%worlds () (remainder-implies-gt-quotient _ _) %. +%total (D) (remainder-implies-gt-quotient D _) %. +%theorem quotient-of-nonzero-is-smaller : forall* {M} {N} {Q} {R} {M-} forall {DR divrem M (s (s N)) Q R} {EN eq M (s M-)} exists {G gt M Q} true %. +%term _ %pi (quotient-of-nonzero-is-smaller _ eq/ M>0) %<- (succ-implies-gt-zero _ M>0) %. +%term _ + %pi (quotient-of-nonzero-is-smaller M/N=Q,R eq/ M>Q) + %<- (divrem-can-be-inverted M/N=Q,R X Q*N=X X+R=M) + %<- (times-right-identity _ Q*1=Q) + %<- (succ-implies-gt-zero _ N->0) + %<- (succ-preserves-gt N->0 N>1) + %<- (times-left-preserves-gt N>1 Q*N=X Q*1=Q X>Q) + %<- (plus-commutative X+R=M R+X=M) + %<- (plus-implies-ge R+X=M M>=X) + %<- (ge-transitive-gt M>=X X>Q M>Q) %. +%worlds () (quotient-of-nonzero-is-smaller _ _ _) %. +%total {} (quotient-of-nonzero-is-smaller _ _ _) %. +%theorem quotient-is-no-greater : forall* {M} {N} {Q} {R} forall {DR divrem M N Q R} exists {ge ge M Q} true %. +%term _ + %pi (quotient-is-no-greater M/N=Q,R M>=Q) + %<- (divrem-can-be-inverted M/N=Q,R X Q*N=X X+R=M) + %<- (divrem-implies-positive M/N=Q,R NN N=NN+1) + %<- (eq-symmetric N=NN+1 NN+1=N) + %<- (succ-implies-gt-zero NN NN+1>0) + %<- (gt-respects-eq NN+1>0 NN+1=N eq/ N>0) + %<- (gt-implies-ge-succ N>0 N>=1) + %<- (times-right-identity _ Q*1=Q) + %<- (times-left-preserves-ge* N>=1 Q*N=X Q*1=Q X>=Q) + %<- (plus-commutative X+R=M R+X=M) + %<- (plus-implies-ge R+X=M M>=X) + %<- (ge-transitive M>=X X>=Q M>=Q) %. +%worlds () (quotient-is-no-greater _ _) %. +%total {} (quotient-is-no-greater _ _) %. +%%%%% minmax.elf +%%%%% Minimum and Maximum functor +%%%%% John Boyland +% Minimum and maximum defined given anti-reflexive gt total order. +% We assume compare three-way comparison. +% We assume that ge is defined from gt and eq. +%%%% Definitions +%sort min %. +%term min/= min X X X %. +%term min/> %pi (gt X1 X2) %-> (min X1 X2 X2) %. +%term min/< %pi (gt X2 X1) %-> (min X1 X2 X1) %. +%sort max %. +%term max/= max X X X %. +%term max/> %pi (gt X1 X2) %-> (max X1 X2 X1) %. +%term max/< %pi (gt X2 X1) %-> (max X1 X2 X2) %. +%%%% Theorems +%%% Theorems about min +%theorem false-implies-min : forall* {X1} {X2} {X3} forall {F void} exists {M min X1 X2 X3} true %. +%worlds () (false-implies-min _ _) %. +%total {} (false-implies-min _ _) %. +%theorem min-respects-eq : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {MX min X1 X2 X3} {E1 eq X1 Y1} {E2 eq X2 Y2} {E3 eq X3 Y3} exists {MY min Y1 Y2 Y3} true %. +%term _ min-respects-eq M eq/ eq/ eq/ M %. +%worlds () (min-respects-eq _ _ _ _ _) %. +%total {} (min-respects-eq _ _ _ _ _) %. +%theorem min-total** : forall* {X1} {X2} {C} forall {D compare X1 X2 C} exists {X3} {M min X1 X2 X3} true %. +%term _ min-total** compare/= _ min/= %. +%term _ min-total** (compare/> X1>X2) _ (min/> X1>X2) %. +%term _ min-total** (compare/< X2>X1) _ (min/< X2>X1) %. +%worlds () (min-total** _ _ _) %. +%total {} (min-total** _ _ _) %. +%theorem min-total* : forall {X1} {X2} exists {X3} {M min X1 X2 X3} true %. +%term _ %pi (min-total* X1 X2 X3 M) %<- (compare-total D) %<- (min-total** D X3 M) %. +%worlds () (min-total* _ _ _ _) %. +%total {} (min-total* _ _ _ _) %. +%inline min-total min-total* _ _ _ %. +%theorem min-deterministic : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {MX min X1 X2 X3} {MY min Y1 Y2 Y3} {E1 eq X1 Y1} {E2 eq X2 Y2} exists {E3 eq X3 Y3} true %. +%term _ min-deterministic min/= min/= eq/ eq/ eq/ %. +%term _ min-deterministic min/= (min/> X>X) eq/ eq/ eq/ %. +%term _ min-deterministic min/= (min/< X>X) eq/ eq/ eq/ %. +%term _ min-deterministic (min/> X>X) min/= eq/ eq/ eq/ %. +%term _ min-deterministic (min/> X>Y) (min/> X>YP) eq/ eq/ eq/ %. +%term _ + %pi (min-deterministic (min/> X>Y) (min/< Y>X) eq/ eq/ E) + %<- (gt-anti-symmetric X>Y Y>X F) + %<- (false-implies-eq F E) %. +%term _ min-deterministic (min/< X>X) min/= eq/ eq/ eq/ %. +%term _ + %pi (min-deterministic (min/< X>Y) (min/> Y>X) eq/ eq/ E) + %<- (gt-anti-symmetric X>Y Y>X F) + %<- (false-implies-eq F E) %. +%term _ min-deterministic (min/< X>Y) (min/< X>YP) eq/ eq/ eq/ %. +%worlds () (min-deterministic _ _ _ _ _) %. +%total {} (min-deterministic _ _ _ _ _) %. +%theorem min-commutative : forall* {X1} {X2} {X3} forall {M min X1 X2 X3} exists {Mc min X2 X1 X3} true %. +%term _ min-commutative min/= min/= %. +%term _ min-commutative (min/> X>Y) (min/< X>Y) %. +%term _ min-commutative (min/< X>Y) (min/> X>Y) %. +%worlds () (min-commutative _ _) %. +%total {} (min-commutative _ _) %. +%theorem ge-implies-min : forall* {X1} {X2} forall {G ge X1 X2} exists {M min X1 X2 X2} true %. +%term _ ge-implies-min (ge/> X1>X2) (min/> X1>X2) %. +%term _ ge-implies-min (ge/= eq/) min/= %. +%worlds () (ge-implies-min _ _) %. +%total {} (ge-implies-min _ _) %. +%theorem le-implies-min : forall* {X1} {X2} forall {G le X1 X2} exists {M min X1 X2 X1} true %. +%term _ + %pi (le-implies-min X2>=X1 M) + %<- (ge-implies-min X2>=X1 Mc) + %<- (min-commutative Mc M) %. +%worlds () (le-implies-min _ _) %. +%total {} (le-implies-min _ _) %. +%theorem min-implies-ge : forall* {X1} {X2} {X3} forall {M min X1 X2 X3} exists {G1 ge X1 X3} {G2 ge X2 X3} true %. +%term _ min-implies-ge min/= (ge/= eq/) (ge/= eq/) %. +%term _ min-implies-ge (min/> X1>X2) (ge/> X1>X2) (ge/= eq/) %. +%term _ min-implies-ge (min/< X2>X1) (ge/= eq/) (ge/> X2>X1) %. +%worlds () (min-implies-ge _ _ _) %. +%total {} (min-implies-ge _ _ _) %. +%theorem min-left-preserves-ge* : forall* {X1} {X2} {X3} {X4} {X5} forall {G ge X2 X4} {M12 min X1 X2 X3} {M14 min X1 X4 X5} exists {G ge X3 X5} true %. +%term _ %pi (min-left-preserves-ge* _ min/= M G) %<- (min-implies-ge M G _) %. +%term _ + %pi (min-left-preserves-ge* X2>=X4 (min/> X1>X2) X1&X4=X5 X2>=X5) + %<- (min-implies-ge X1&X4=X5 _ X4>=X5) + %<- (ge-transitive X2>=X4 X4>=X5 X2>=X5) %. +%term _ + %pi (min-left-preserves-ge* _ (min/< _) X1&X4=X5 X1>=X5) + %<- (min-implies-ge X1&X4=X5 X1>=X5 _) %. +%worlds () (min-left-preserves-ge* _ _ _ _) %. +%total {} (min-left-preserves-ge* _ _ _ _) %. +%theorem min-left-preserves-le* : forall* {X1} {X2} {X3} {X4} {X5} forall {G le X2 X4} {M12 min X1 X2 X3} {M14 min X1 X4 X5} exists {G le X3 X5} true %. +%term _ + %pi (min-left-preserves-le* G1 M12 M14 G2) + %<- (min-left-preserves-ge* G1 M14 M12 G2) %. +%worlds () (min-left-preserves-le* _ _ _ _) %. +%total {} (min-left-preserves-le* _ _ _ _) %. +%theorem min-left-preserves-ge : forall* {X1} {X2} {X4} forall {G ge X2 X4} exists {X3} {X5} {O1 min X1 X2 X3} {O2 min X1 X4 X5} {G2 ge X3 X5} true %. +%term _ + %pi (min-left-preserves-ge X2>=X4 X3 X5 X1&X2=A3 X1&X4=X5 X3>=X5) + %<- (min-total X1&X2=A3) + %<- (min-total X1&X4=X5) + %<- (min-left-preserves-ge* X2>=X4 X1&X2=A3 X1&X4=X5 X3>=X5) %. +%worlds () (min-left-preserves-ge X2>=X4 X3 X5 X1&X2=A3 X1&X4=X5 X3>=X5) %. +%total {} (min-left-preserves-ge _ _ _ _ _ _) %. +%theorem min-right-preserves-ge* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 ge X1 X2} {O1 min X1 X3 X4} {O2 min X2 X3 X5} exists {G2 ge X4 X5} true %. +%term _ + %pi (min-right-preserves-ge* X1>=X2 X1&X3=X4 X2&X3=X5 X4>=X5) + %<- (min-commutative X1&X3=X4 X3&X1=X4) + %<- (min-commutative X2&X3=X5 X3&X2=X5) + %<- (min-left-preserves-ge* X1>=X2 X3&X1=X4 X3&X2=X5 X4>=X5) %. +%worlds () (min-right-preserves-ge* X1>=X2 X1&X3=X4 X2&X3=X5 X4>=X5) %. +%total {} (min-right-preserves-ge* _ _ _ _) %. +%theorem min-right-preserves-ge : forall* {X1} {X2} {X3} forall {G1 ge X1 X2} exists {X4} {X5} {O1 min X1 X3 X4} {O2 min X2 X3 X5} {G2 ge X4 X5} true %. +%term _ + %pi (min-right-preserves-ge X1>=X2 X4 X5 X1&X3=X4 X2&X3=X5 X4>=X5) + %<- (min-total X1&X3=X4) + %<- (min-total X2&X3=X5) + %<- (min-right-preserves-ge* X1>=X2 X1&X3=X4 X2&X3=X5 X4>=X5) %. +%worlds () (min-right-preserves-ge X1>=X2 X4 X5 X1&X3=X4 X2&X3=X5 X4>=X5) %. +%total {} (min-right-preserves-ge _ _ _ _ _ _) %. +%theorem min-preserves-ge* : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {G1 ge X1 Y1} {G2 ge X2 Y2} {MX min X1 X2 X3} {MY min Y1 Y2 Y3} exists {G3 ge X3 Y3} true %. +%term _ + %pi (min-preserves-ge* X1>=Y1 X2>=Y2 X1&X2=X3 Y1&Y2=Y3 X3>=Y3) + %<- (min-total Y1&X2=X) + %<- (min-right-preserves-ge* X1>=Y1 X1&X2=X3 Y1&X2=X X3>=X) + %<- (min-left-preserves-ge* X2>=Y2 Y1&X2=X Y1&Y2=Y3 X>=Y3) + %<- (ge-transitive X3>=X X>=Y3 X3>=Y3) %. +%worlds () (min-preserves-ge* X1>=Y1 X2>=Y2 X1&X2=X3 Y1&Y2=Y3 X3>=Y3) %. +%total {} (min-preserves-ge* _ _ _ _ _) %. +%theorem min-preserves-ge : forall* {X1} {X2} {Y1} {Y2} forall {G1 ge X1 Y1} {G2 ge X2 Y2} exists {X3} {Y3} {MX min X1 X2 X3} {MY min Y1 Y2 Y3} {G3 ge X3 Y3} true %. +%term _ + %pi (min-preserves-ge X1>=Y1 X2>=Y2 X3 Y3 X1&X2=X3 Y1&Y2=Y3 X3>=Y3) + %<- (min-total X1&X2=X3) + %<- (min-total Y1&Y2=Y3) + %<- (min-preserves-ge* X1>=Y1 X2>=Y2 X1&X2=X3 Y1&Y2=Y3 X3>=Y3) %. +%worlds () (min-preserves-ge X1>=Y1 X2>=Y2 X3 Y3 X1&X2=X3 Y1&Y2=Y3 X3>=Y3) %. +%total {} (min-preserves-ge _ _ _ _ _ _ _) %. +%theorem min-left-preserves-le : forall* {X1} {X2} {X4} forall {G le X2 X4} exists {X3} {X5} {O1 min X1 X2 X3} {O2 min X1 X4 X5} {G2 le X3 X5} true %. +%term _ + %pi (min-left-preserves-le X2<=X4 X3 X5 X1&X2=A3 X1&X4=X5 X3<=X5) + %<- (min-total X1&X2=A3) + %<- (min-total X1&X4=X5) + %<- (min-left-preserves-le* X2<=X4 X1&X2=A3 X1&X4=X5 X3<=X5) %. +%worlds () (min-left-preserves-le X2<=X4 X3 X5 X1&X2=A3 X1&X4=X5 X3<=X5) %. +%total {} (min-left-preserves-le _ _ _ _ _ _) %. +%theorem min-right-preserves-le* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 le X1 X2} {O1 min X1 X3 X4} {O2 min X2 X3 X5} exists {G2 le X4 X5} true %. +%term _ + %pi (min-right-preserves-le* X1<=X2 X1&X3=X4 X2&X3=X5 X4<=X5) + %<- (min-commutative X1&X3=X4 X3&X1=X4) + %<- (min-commutative X2&X3=X5 X3&X2=X5) + %<- (min-left-preserves-le* X1<=X2 X3&X1=X4 X3&X2=X5 X4<=X5) %. +%worlds () (min-right-preserves-le* X1<=X2 X1&X3=X4 X2&X3=X5 X4<=X5) %. +%total {} (min-right-preserves-le* _ _ _ _) %. +%theorem min-right-preserves-le : forall* {X1} {X2} {X3} forall {G1 le X1 X2} exists {X4} {X5} {O1 min X1 X3 X4} {O2 min X2 X3 X5} {G2 le X4 X5} true %. +%term _ + %pi (min-right-preserves-le X1<=X2 X4 X5 X1&X3=X4 X2&X3=X5 X4<=X5) + %<- (min-total X1&X3=X4) + %<- (min-total X2&X3=X5) + %<- (min-right-preserves-le* X1<=X2 X1&X3=X4 X2&X3=X5 X4<=X5) %. +%worlds () (min-right-preserves-le X1<=X2 X4 X5 X1&X3=X4 X2&X3=X5 X4<=X5) %. +%total {} (min-right-preserves-le _ _ _ _ _ _) %. +%theorem min-preserves-le* : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {G1 le X1 Y1} {G2 le X2 Y2} {MX min X1 X2 X3} {MY min Y1 Y2 Y3} exists {G3 le X3 Y3} true %. +%term _ + %pi (min-preserves-le* X1<=Y1 X2<=Y2 X1&X2=X3 Y1&Y2=Y3 X3<=Y3) + %<- (min-total Y1&X2=X) + %<- (min-right-preserves-le* X1<=Y1 X1&X2=X3 Y1&X2=X X3<=X) + %<- (min-left-preserves-le* X2<=Y2 Y1&X2=X Y1&Y2=Y3 X<=Y3) + %<- (le-transitive X3<=X X<=Y3 X3<=Y3) %. +%worlds () (min-preserves-le* X1<=Y1 X2<=Y2 X1&X2=X3 Y1&Y2=Y3 X3<=Y3) %. +%total {} (min-preserves-le* _ _ _ _ _) %. +%theorem min-preserves-le : forall* {X1} {X2} {Y1} {Y2} forall {G1 le X1 Y1} {G2 le X2 Y2} exists {X3} {Y3} {MX min X1 X2 X3} {MY min Y1 Y2 Y3} {G3 le X3 Y3} true %. +%term _ + %pi (min-preserves-le X1<=Y1 X2<=Y2 X3 Y3 X1&X2=X3 Y1&Y2=Y3 X3<=Y3) + %<- (min-total X1&X2=X3) + %<- (min-total Y1&Y2=Y3) + %<- (min-preserves-le* X1<=Y1 X2<=Y2 X1&X2=X3 Y1&Y2=Y3 X3<=Y3) %. +%worlds () (min-preserves-le X1<=Y1 X2<=Y2 X3 Y3 X1&X2=X3 Y1&Y2=Y3 X3<=Y3) %. +%total {} (min-preserves-le _ _ _ _ _ _ _) %. +%theorem min-is-glb : forall* {X1} {X2} {X3} {X4} forall {M min X1 X2 X3} {G1 ge X1 X4} {G2 ge X2 X4} exists {G3 ge X3 X4} true %. +%term _ min-is-glb min/= G _ G %. +%term _ min-is-glb (min/> X1>X2) _ G G %. +%term _ min-is-glb (min/< X2>X1) G _ G %. +%worlds () (min-is-glb _ _ _ _) %. +%total {} (min-is-glb _ _ _ _) %. +%theorem min-associative : forall* {X1} {X2} {X3} {X4} {X7} forall {M12 min X1 X2 X3} {M34 min X3 X4 X7} exists {X6} {M24 min X2 X4 X6} {M16 min X1 X6 X7} true %. +%term _ min-associative min/= min/= _ min/= min/= %. +%term _ min-associative min/= (min/> X3>X4) _ (min/> X3>X4) (min/> X3>X4) %. +%term _ min-associative min/= (min/< X4>X3) _ (min/< X4>X3) min/= %. +%term _ min-associative (min/> X1>X2) min/= _ min/= (min/> X1>X2) %. +%term _ + %pi (min-associative (min/> X1>X2) (min/> X2>X4) _ (min/> X2>X4) (min/> X1>X4)) + %<- (gt-transitive X1>X2 X2>X4 X1>X4) %. +%term _ min-associative (min/> X1>X2) (min/< X4>X2) _ (min/< X4>X2) (min/> X1>X2) %. +%term _ min-associative (min/< X2>X1) min/= _ (min/> X2>X1) min/= %. +%term _ + %pi (min-associative (min/< X2>X1) (min/> X1>X4) _ (min/> X2>X4) (min/> X1>X4)) + %<- (gt-transitive X2>X1 X1>X4 X2>X4) %. +%term _ + %pi (min-associative (min/< X2>X1) (min/< X4>X1) _ M24 M16) + %<- (min-total M24) + %<- (min-is-glb M24 (ge/> X2>X1) (ge/> X4>X1) (%the (ge X6 X1) X6>=X1)) + %<- (ge-implies-min X6>=X1 M16c) + %<- (min-commutative M16c M16) %. +%worlds () (min-associative _ _ _ _ _) %. +%total {} (min-associative _ _ _ _ _) %. +%theorem min-associative* : forall* {X1} {X2} {X12} {X3} {X23} {X123} forall {OP12 min X1 X2 X12} {OP12-3 min X12 X3 X123} {OP23 min X2 X3 X23} exists {OP1-23 min X1 X23 X123} true %. +%term _ + %pi (min-associative* X1&X2=X3 X3&X4=X7 X2&X4=X6 X1&X6=X7) + %<- (min-associative X1&X2=X3 X3&X4=X7 Y6 X2&X4=Y6 X1&Y6=X7) + %<- (min-deterministic X2&X4=Y6 X2&X4=X6 eq/ eq/ Y6=X6) + %<- (min-respects-eq X1&Y6=X7 eq/ Y6=X6 eq/ X1&X6=X7) %. +%worlds () (min-associative* _ _ _ _) %. +%total {} (min-associative* _ _ _ _) %. +%theorem min-associative-converse : forall* {X1} {X2} {X4} {X6} {X7} forall {OP24 min X2 X4 X6} {OP16 min X1 X6 X7} exists {X3} {OP12 min X1 X2 X3} {OP34 min X3 X4 X7} true %. +%term _ + %pi (min-associative-converse X2&X4=X6 X1&X6=X7 _ X1&X2=X3 X3&X4=X7) + %<- (min-commutative X2&X4=X6 X4&X2=X6) + %<- (min-commutative X1&X6=X7 X6&X1=X7) + %<- (min-associative X4&X2=X6 X6&X1=X7 _ X2&X1=X3 X4&X3=X7) + %<- (min-commutative X2&X1=X3 X1&X2=X3) + %<- (min-commutative X4&X3=X7 X3&X4=X7) %. +%worlds () (min-associative-converse X2&X4=X6 X1&X6=X7 X3 X1&X2=X3 X3&X4=X7) %. +%total {} (min-associative-converse _ _ _ _ _) %. +%theorem min-associative-converse* : forall* {X1} {X2} {X3} {X4} {X6} {X7} forall {OP24 min X2 X4 X6} {OP16 min X1 X6 X7} {OP12 min X1 X2 X3} exists {OP34 min X3 X4 X7} true %. +%term _ + %pi (min-associative-converse* X2&X4=X6 X1&X6=X7 X1&X2=X3 X3&X4=X7) + %<- (min-associative-converse X2&X4=X6 X1&X6=X7 X3P X1&X2=X3P X3P&X4=X7) + %<- (min-deterministic X1&X2=X3P X1&X2=X3 eq/ eq/ X3P=X3) + %<- (min-respects-eq X3P&X4=X7 X3P=X3 eq/ eq/ X3&X4=X7) %. +%worlds () (min-associative-converse* X2&X4=X6 X1&X6=X7 X1&X2=X3 X3&X4=X7) %. +%total {} (min-associative-converse* _ _ _ _) %. +%theorem min-assoc-commutative* : forall* {X1} {X2} {X3} {X4} {X5} {X7} forall {OP1 min X1 X2 X3} {OP2 min X3 X4 X7} {OP3 min X1 X4 X5} exists {OP4 min X5 X2 X7} true %. +%term _ + %pi (min-assoc-commutative* X1&X2=X3 X3&X4=X7 X1&X4=X5 X5&X2=X7) + %<- (min-associative X1&X2=X3 X3&X4=X7 X6 X2&X4=X6 X1&X6=X7) + %<- (min-commutative X2&X4=X6 X4&X2=X6) + %<- (min-associative-converse* X4&X2=X6 X1&X6=X7 X1&X4=X5 X5&X2=X7) %. +%worlds () (min-assoc-commutative* X1&X2=X3 X3&X4=X7 X1&X4=X5 X5&X2=X7) %. +%total {} (min-assoc-commutative* _ _ _ _) %. +%theorem min-assoc-commutative : forall* {X1} {X2} {X3} {X4} {X7} forall {OP1 min X1 X2 X3} {OP2 min X3 X4 X7} exists {X5} {OP3 min X1 X4 X5} {OP4 min X5 X2 X7} true %. +%term _ + %pi (min-assoc-commutative X1&X2=X3 X3&X4=X7 X5 X1&X4=X5 X5&X2=X7) + %<- (min-associative X1&X2=X3 X3&X4=X7 X6 X2&X4=X6 X1&X6=X7) + %<- (min-commutative X2&X4=X6 X4&X2=X6) + %<- (min-associative-converse X4&X2=X6 X1&X6=X7 X5 X1&X4=X5 X5&X2=X7) %. +%worlds () (min-assoc-commutative X1&X2=X3 X3&X4=X7 X5 X1&X4=X5 X5&X2=X7) %. +%total {} (min-assoc-commutative _ _ _ _ _) %. +%theorem min-double-associative* : forall* {A} {B} {C} {D} {A+B} {C+D} {A+C} {B+D} {X} forall {AB min A B A+B} {CD min C D C+D} {ABCD min A+B C+D X} {AC min A C A+C} {BD min B D B+D} exists {ACBD min A+C B+D X} true %. +%term _ + %pi (min-double-associative* X1&X2=X3 X4&X8=XC X3&XC=XF X1&X4=X5 X2&X8=XA X5&XA=XF) + %<- (min-associative X1&X2=X3 X3&XC=XF XE X2&XC=XE X1&XE=XF) + %<- (min-commutative X4&X8=XC X8&X4=XC) + %<- (min-associative-converse* X8&X4=XC X2&XC=XE X2&X8=XA XA&X4=XE) + %<- (min-commutative XA&X4=XE X4&XA=XE) + %<- (min-associative-converse* X4&XA=XE X1&XE=XF X1&X4=X5 X5&XA=XF) %. +%worlds () (min-double-associative* X1&X2=X3 X4&X8=XC X3&XC=XF X1&X4=X5 X2&X8=XA X5&XA=XF) %. +%total {} (min-double-associative* _ _ _ _ _ _) %. +%theorem min-double-associative : forall* {A} {B} {C} {D} {A+B} {C+D} {X} forall {AB min A B A+B} {CD min C D C+D} {ABCD min A+B C+D X} exists {A+C} {B+D} {AC min A C A+C} {BD min B D B+D} {ACBD min A+C B+D X} true %. +%term _ + %pi (min-double-associative X1&X2=X3 X4&X8=XC X3&XC=XF X5 XA X1&X4=X5 X2&X8=XA X5&XA=XF) + %<- (min-associative X1&X2=X3 X3&XC=XF XE X2&XC=XE X1&XE=XF) + %<- (min-commutative X4&X8=XC X8&X4=XC) + %<- (min-associative-converse X8&X4=XC X2&XC=XE XA X2&X8=XA XA&X4=XE) + %<- (min-commutative XA&X4=XE X4&XA=XE) + %<- (min-associative-converse X4&XA=XE X1&XE=XF X5 X1&X4=X5 X5&XA=XF) %. +%worlds () (min-double-associative _ _ _ _ _ _ _ _) %. +%total {} (min-double-associative _ _ _ _ _ _ _ _) %. +%%% Theorems about max +%theorem false-implies-max : forall* {X1} {X2} {X3} forall {F void} exists {M max X1 X2 X3} true %. +%worlds () (false-implies-max _ _) %. +%total {} (false-implies-max _ _) %. +%theorem max-respects-eq : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {MX max X1 X2 X3} {E1 eq X1 Y1} {E2 eq X2 Y2} {E3 eq X3 Y3} exists {MY max Y1 Y2 Y3} true %. +%term _ max-respects-eq M eq/ eq/ eq/ M %. +%worlds () (max-respects-eq _ _ _ _ _) %. +%total {} (max-respects-eq _ _ _ _ _) %. +%theorem max-total** : forall* {X1} {X2} {C} forall {D compare X1 X2 C} exists {X3} {M max X1 X2 X3} true %. +%term _ max-total** compare/= _ max/= %. +%term _ max-total** (compare/> X1>X2) _ (max/> X1>X2) %. +%term _ max-total** (compare/< X2>X1) _ (max/< X2>X1) %. +%worlds () (max-total** _ _ _) %. +%total {} (max-total** _ _ _) %. +%theorem max-total* : forall {X1} {X2} exists {X3} {M max X1 X2 X3} true %. +%term _ %pi (max-total* X1 X2 X3 M) %<- (compare-total D) %<- (max-total** D X3 M) %. +%worlds () (max-total* _ _ _ _) %. +%total {} (max-total* _ _ _ _) %. +%inline max-total max-total* _ _ _ %. +%theorem max-deterministic : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {MX max X1 X2 X3} {MY max Y1 Y2 Y3} {E1 eq X1 Y1} {E2 eq X2 Y2} exists {E3 eq X3 Y3} true %. +%term _ max-deterministic max/= max/= eq/ eq/ eq/ %. +%term _ max-deterministic max/= (max/> X>X) eq/ eq/ eq/ %. +%term _ max-deterministic max/= (max/< X>X) eq/ eq/ eq/ %. +%term _ max-deterministic (max/> X>X) max/= eq/ eq/ eq/ %. +%term _ max-deterministic (max/> X>Y) (max/> X>YP) eq/ eq/ eq/ %. +%term _ + %pi (max-deterministic (max/> X>Y) (max/< Y>X) eq/ eq/ E) + %<- (gt-anti-symmetric X>Y Y>X F) + %<- (false-implies-eq F E) %. +%term _ max-deterministic (max/< X>X) max/= eq/ eq/ eq/ %. +%term _ + %pi (max-deterministic (max/< X>Y) (max/> Y>X) eq/ eq/ E) + %<- (gt-anti-symmetric X>Y Y>X F) + %<- (false-implies-eq F E) %. +%term _ max-deterministic (max/< X>Y) (max/< X>YP) eq/ eq/ eq/ %. +%worlds () (max-deterministic _ _ _ _ _) %. +%total {} (max-deterministic _ _ _ _ _) %. +%theorem max-commutative : forall* {X1} {X2} {X3} forall {M max X1 X2 X3} exists {Mc max X2 X1 X3} true %. +%term _ max-commutative max/= max/= %. +%term _ max-commutative (max/> X>Y) (max/< X>Y) %. +%term _ max-commutative (max/< X>Y) (max/> X>Y) %. +%worlds () (max-commutative _ _) %. +%total {} (max-commutative _ _) %. +%theorem ge-implies-max : forall* {X1} {X2} forall {G ge X1 X2} exists {M max X1 X2 X1} true %. +%term _ ge-implies-max (ge/> X1>X2) (max/> X1>X2) %. +%term _ ge-implies-max (ge/= eq/) max/= %. +%worlds () (ge-implies-max _ _) %. +%total {} (ge-implies-max _ _) %. +%theorem le-implies-max : forall* {X1} {X2} forall {G le X1 X2} exists {M max X1 X2 X2} true %. +%term _ + %pi (le-implies-max X2>=X1 M) + %<- (ge-implies-max X2>=X1 Mc) + %<- (max-commutative Mc M) %. +%worlds () (le-implies-max _ _) %. +%total {} (le-implies-max _ _) %. +%theorem max-implies-ge : forall* {X1} {X2} {X3} forall {M max X1 X2 X3} exists {G1 ge X3 X1} {G2 ge X3 X2} true %. +%term _ max-implies-ge max/= (ge/= eq/) (ge/= eq/) %. +%term _ max-implies-ge (max/> X1>X2) (ge/= eq/) (ge/> X1>X2) %. +%term _ max-implies-ge (max/< X2>X1) (ge/> X2>X1) (ge/= eq/) %. +%worlds () (max-implies-ge _ _ _) %. +%total {} (max-implies-ge _ _ _) %. +%theorem max-is-lub : forall* {X0} {X1} {X2} {X3} forall {M max X1 X2 X3} {G1 ge X0 X1} {G2 ge X0 X2} exists {G3 ge X0 X3} true %. +%term _ max-is-lub max/= G _ G %. +%term _ max-is-lub (max/> X1>X2) G _ G %. +%term _ max-is-lub (max/< X2>X1) _ G G %. +%worlds () (max-is-lub _ _ _ _) %. +%total {} (max-is-lub _ _ _ _) %. +%theorem max-left-preserves-ge* : forall* {X1} {X2} {X3} {X4} {X5} forall {G ge X2 X4} {M12 max X1 X2 X3} {M14 max X1 X4 X5} exists {G ge X3 X5} true %. +%term _ + %pi (max-left-preserves-ge* X>=X4 max/= X|X4=X5 X>=X5) + %<- (ge-reflexive _ X>=X) + %<- (max-is-lub X|X4=X5 X>=X X>=X4 X>=X5) %. +%term _ + %pi (max-left-preserves-ge* X2>=X4 (max/> X1>X2) X1|X4=X5 X1>=X5) + %<- (ge-transitive (ge/> X1>X2) X2>=X4 X1>=X4) + %<- (ge-reflexive _ X1>=X1) + %<- (max-is-lub X1|X4=X5 X1>=X1 X1>=X4 X1>=X5) %. +%term _ + %pi (max-left-preserves-ge* X2>=X4 (max/< X2>X1) X1|X4=X5 X2>=X5) + %<- (max-is-lub X1|X4=X5 (ge/> X2>X1) X2>=X4 X2>=X5) %. +%worlds () (max-left-preserves-ge* _ _ _ _) %. +%total {} (max-left-preserves-ge* _ _ _ _) %. +%theorem max-left-preserves-le* : forall* {X1} {X2} {X3} {X4} {X5} forall {G le X2 X4} {M12 max X1 X2 X3} {M14 max X1 X4 X5} exists {G le X3 X5} true %. +%term _ + %pi (max-left-preserves-le* G1 M12 M14 G2) + %<- (max-left-preserves-ge* G1 M14 M12 G2) %. +%worlds () (max-left-preserves-le* _ _ _ _) %. +%total {} (max-left-preserves-le* _ _ _ _) %. +%theorem max-left-preserves-ge : forall* {X1} {X2} {X4} forall {G ge X2 X4} exists {X3} {X5} {O1 max X1 X2 X3} {O2 max X1 X4 X5} {G2 ge X3 X5} true %. +%term _ + %pi (max-left-preserves-ge X2>=X4 X3 X5 X1&X2=A3 X1&X4=X5 X3>=X5) + %<- (max-total X1&X2=A3) + %<- (max-total X1&X4=X5) + %<- (max-left-preserves-ge* X2>=X4 X1&X2=A3 X1&X4=X5 X3>=X5) %. +%worlds () (max-left-preserves-ge X2>=X4 X3 X5 X1&X2=A3 X1&X4=X5 X3>=X5) %. +%total {} (max-left-preserves-ge _ _ _ _ _ _) %. +%theorem max-right-preserves-ge* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 ge X1 X2} {O1 max X1 X3 X4} {O2 max X2 X3 X5} exists {G2 ge X4 X5} true %. +%term _ + %pi (max-right-preserves-ge* X1>=X2 X1&X3=X4 X2&X3=X5 X4>=X5) + %<- (max-commutative X1&X3=X4 X3&X1=X4) + %<- (max-commutative X2&X3=X5 X3&X2=X5) + %<- (max-left-preserves-ge* X1>=X2 X3&X1=X4 X3&X2=X5 X4>=X5) %. +%worlds () (max-right-preserves-ge* X1>=X2 X1&X3=X4 X2&X3=X5 X4>=X5) %. +%total {} (max-right-preserves-ge* _ _ _ _) %. +%theorem max-right-preserves-ge : forall* {X1} {X2} {X3} forall {G1 ge X1 X2} exists {X4} {X5} {O1 max X1 X3 X4} {O2 max X2 X3 X5} {G2 ge X4 X5} true %. +%term _ + %pi (max-right-preserves-ge X1>=X2 X4 X5 X1&X3=X4 X2&X3=X5 X4>=X5) + %<- (max-total X1&X3=X4) + %<- (max-total X2&X3=X5) + %<- (max-right-preserves-ge* X1>=X2 X1&X3=X4 X2&X3=X5 X4>=X5) %. +%worlds () (max-right-preserves-ge X1>=X2 X4 X5 X1&X3=X4 X2&X3=X5 X4>=X5) %. +%total {} (max-right-preserves-ge _ _ _ _ _ _) %. +%theorem max-preserves-ge* : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {G1 ge X1 Y1} {G2 ge X2 Y2} {MX max X1 X2 X3} {MY max Y1 Y2 Y3} exists {G3 ge X3 Y3} true %. +%term _ + %pi (max-preserves-ge* X1>=Y1 X2>=Y2 X1&X2=X3 Y1&Y2=Y3 X3>=Y3) + %<- (max-total Y1&X2=X) + %<- (max-right-preserves-ge* X1>=Y1 X1&X2=X3 Y1&X2=X X3>=X) + %<- (max-left-preserves-ge* X2>=Y2 Y1&X2=X Y1&Y2=Y3 X>=Y3) + %<- (ge-transitive X3>=X X>=Y3 X3>=Y3) %. +%worlds () (max-preserves-ge* X1>=Y1 X2>=Y2 X1&X2=X3 Y1&Y2=Y3 X3>=Y3) %. +%total {} (max-preserves-ge* _ _ _ _ _) %. +%theorem max-preserves-ge : forall* {X1} {X2} {Y1} {Y2} forall {G1 ge X1 Y1} {G2 ge X2 Y2} exists {X3} {Y3} {MX max X1 X2 X3} {MY max Y1 Y2 Y3} {G3 ge X3 Y3} true %. +%term _ + %pi (max-preserves-ge X1>=Y1 X2>=Y2 X3 Y3 X1&X2=X3 Y1&Y2=Y3 X3>=Y3) + %<- (max-total X1&X2=X3) + %<- (max-total Y1&Y2=Y3) + %<- (max-preserves-ge* X1>=Y1 X2>=Y2 X1&X2=X3 Y1&Y2=Y3 X3>=Y3) %. +%worlds () (max-preserves-ge X1>=Y1 X2>=Y2 X3 Y3 X1&X2=X3 Y1&Y2=Y3 X3>=Y3) %. +%total {} (max-preserves-ge _ _ _ _ _ _ _) %. +%theorem max-left-preserves-le : forall* {X1} {X2} {X4} forall {G le X2 X4} exists {X3} {X5} {O1 max X1 X2 X3} {O2 max X1 X4 X5} {G2 le X3 X5} true %. +%term _ + %pi (max-left-preserves-le X2<=X4 X3 X5 X1&X2=A3 X1&X4=X5 X3<=X5) + %<- (max-total X1&X2=A3) + %<- (max-total X1&X4=X5) + %<- (max-left-preserves-le* X2<=X4 X1&X2=A3 X1&X4=X5 X3<=X5) %. +%worlds () (max-left-preserves-le X2<=X4 X3 X5 X1&X2=A3 X1&X4=X5 X3<=X5) %. +%total {} (max-left-preserves-le _ _ _ _ _ _) %. +%theorem max-right-preserves-le* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 le X1 X2} {O1 max X1 X3 X4} {O2 max X2 X3 X5} exists {G2 le X4 X5} true %. +%term _ + %pi (max-right-preserves-le* X1<=X2 X1&X3=X4 X2&X3=X5 X4<=X5) + %<- (max-commutative X1&X3=X4 X3&X1=X4) + %<- (max-commutative X2&X3=X5 X3&X2=X5) + %<- (max-left-preserves-le* X1<=X2 X3&X1=X4 X3&X2=X5 X4<=X5) %. +%worlds () (max-right-preserves-le* X1<=X2 X1&X3=X4 X2&X3=X5 X4<=X5) %. +%total {} (max-right-preserves-le* _ _ _ _) %. +%theorem max-right-preserves-le : forall* {X1} {X2} {X3} forall {G1 le X1 X2} exists {X4} {X5} {O1 max X1 X3 X4} {O2 max X2 X3 X5} {G2 le X4 X5} true %. +%term _ + %pi (max-right-preserves-le X1<=X2 X4 X5 X1&X3=X4 X2&X3=X5 X4<=X5) + %<- (max-total X1&X3=X4) + %<- (max-total X2&X3=X5) + %<- (max-right-preserves-le* X1<=X2 X1&X3=X4 X2&X3=X5 X4<=X5) %. +%worlds () (max-right-preserves-le X1<=X2 X4 X5 X1&X3=X4 X2&X3=X5 X4<=X5) %. +%total {} (max-right-preserves-le _ _ _ _ _ _) %. +%theorem max-preserves-le* : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {G1 le X1 Y1} {G2 le X2 Y2} {MX max X1 X2 X3} {MY max Y1 Y2 Y3} exists {G3 le X3 Y3} true %. +%term _ + %pi (max-preserves-le* X1<=Y1 X2<=Y2 X1&X2=X3 Y1&Y2=Y3 X3<=Y3) + %<- (max-total Y1&X2=X) + %<- (max-right-preserves-le* X1<=Y1 X1&X2=X3 Y1&X2=X X3<=X) + %<- (max-left-preserves-le* X2<=Y2 Y1&X2=X Y1&Y2=Y3 X<=Y3) + %<- (le-transitive X3<=X X<=Y3 X3<=Y3) %. +%worlds () (max-preserves-le* X1<=Y1 X2<=Y2 X1&X2=X3 Y1&Y2=Y3 X3<=Y3) %. +%total {} (max-preserves-le* _ _ _ _ _) %. +%theorem max-preserves-le : forall* {X1} {X2} {Y1} {Y2} forall {G1 le X1 Y1} {G2 le X2 Y2} exists {X3} {Y3} {MX max X1 X2 X3} {MY max Y1 Y2 Y3} {G3 le X3 Y3} true %. +%term _ + %pi (max-preserves-le X1<=Y1 X2<=Y2 X3 Y3 X1&X2=X3 Y1&Y2=Y3 X3<=Y3) + %<- (max-total X1&X2=X3) + %<- (max-total Y1&Y2=Y3) + %<- (max-preserves-le* X1<=Y1 X2<=Y2 X1&X2=X3 Y1&Y2=Y3 X3<=Y3) %. +%worlds () (max-preserves-le X1<=Y1 X2<=Y2 X3 Y3 X1&X2=X3 Y1&Y2=Y3 X3<=Y3) %. +%total {} (max-preserves-le _ _ _ _ _ _ _) %. +%theorem max-associative : forall* {X1} {X2} {X3} {X4} {X7} forall {M12 max X1 X2 X3} {M34 max X3 X4 X7} exists {X6} {M24 max X2 X4 X6} {M16 max X1 X6 X7} true %. +%term _ max-associative max/= max/= _ max/= max/= %. +%term _ max-associative max/= (max/> X3>X4) _ (max/> X3>X4) max/= %. +%term _ max-associative max/= (max/< X4>X3) _ (max/< X4>X3) (max/< X4>X3) %. +%term _ max-associative (max/> X1>X2) max/= _ (max/< X1>X2) max/= %. +%term _ + %pi (max-associative (max/> X1>X2) (max/> X1>X4) _ M24 M16) + %<- (max-total M24) + %<- (max-is-lub M24 (ge/> X1>X2) (ge/> X1>X4) X1>=X6) + %<- (ge-implies-max X1>=X6 M16) %. +%term _ + %pi (max-associative (max/> X1>X2) (max/< X4>X1) _ (max/< X4>X2) (max/< X4>X1)) + %<- (gt-transitive X4>X1 X1>X2 X4>X2) %. +%term _ max-associative (max/< X2>X1) max/= _ max/= (max/< X2>X1) %. +%term _ max-associative (max/< X2>X1) (max/> X2>X4) _ (max/> X2>X4) (max/< X2>X1) %. +%term _ + %pi (max-associative (max/< X2>X1) (max/< X4>X2) _ (max/< X4>X2) (max/< X4>X1)) + %<- (gt-transitive X4>X2 X2>X1 X4>X1) %. +%worlds () (max-associative _ _ _ _ _) %. +%total {} (max-associative _ _ _ _ _) %. +%theorem max-associative* : forall* {X1} {X2} {X12} {X3} {X23} {X123} forall {OP12 max X1 X2 X12} {OP12-3 max X12 X3 X123} {OP23 max X2 X3 X23} exists {OP1-23 max X1 X23 X123} true %. +%term _ + %pi (max-associative* X1|X2=X3 X3|X4=X7 X2|X4=X6 X1|X6=X7) + %<- (max-associative X1|X2=X3 X3|X4=X7 Y6 X2|X4=Y6 X1|Y6=X7) + %<- (max-deterministic X2|X4=Y6 X2|X4=X6 eq/ eq/ Y6=X6) + %<- (max-respects-eq X1|Y6=X7 eq/ Y6=X6 eq/ X1|X6=X7) %. +%worlds () (max-associative* _ _ _ _) %. +%total {} (max-associative* _ _ _ _) %. +%theorem max-associative-converse : forall* {X1} {X2} {X4} {X6} {X7} forall {OP24 max X2 X4 X6} {OP16 max X1 X6 X7} exists {X3} {OP12 max X1 X2 X3} {OP34 max X3 X4 X7} true %. +%term _ + %pi (max-associative-converse X2|X4=X6 X1|X6=X7 _ X1|X2=X3 X3|X4=X7) + %<- (max-commutative X2|X4=X6 X4|X2=X6) + %<- (max-commutative X1|X6=X7 X6|X1=X7) + %<- (max-associative X4|X2=X6 X6|X1=X7 _ X2|X1=X3 X4|X3=X7) + %<- (max-commutative X2|X1=X3 X1|X2=X3) + %<- (max-commutative X4|X3=X7 X3|X4=X7) %. +%worlds () (max-associative-converse X2|X4=X6 X1|X6=X7 X3 X1|X2=X3 X3|X4=X7) %. +%total {} (max-associative-converse _ _ _ _ _) %. +%theorem max-associative-converse* : forall* {X1} {X2} {X3} {X4} {X6} {X7} forall {OP24 max X2 X4 X6} {OP16 max X1 X6 X7} {OP12 max X1 X2 X3} exists {OP34 max X3 X4 X7} true %. +%term _ + %pi (max-associative-converse* X2|X4=X6 X1|X6=X7 X1|X2=X3 X3|X4=X7) + %<- (max-associative-converse X2|X4=X6 X1|X6=X7 X3P X1|X2=X3P X3P|X4=X7) + %<- (max-deterministic X1|X2=X3P X1|X2=X3 eq/ eq/ X3P=X3) + %<- (max-respects-eq X3P|X4=X7 X3P=X3 eq/ eq/ X3|X4=X7) %. +%worlds () (max-associative-converse* X2|X4=X6 X1|X6=X7 X1|X2=X3 X3|X4=X7) %. +%total {} (max-associative-converse* _ _ _ _) %. +%theorem max-assoc-commutative* : forall* {X1} {X2} {X3} {X4} {X5} {X7} forall {OP1 max X1 X2 X3} {OP2 max X3 X4 X7} {OP3 max X1 X4 X5} exists {OP4 max X5 X2 X7} true %. +%term _ + %pi (max-assoc-commutative* X1|X2=X3 X3|X4=X7 X1|X4=X5 X5|X2=X7) + %<- (max-associative X1|X2=X3 X3|X4=X7 X6 X2|X4=X6 X1|X6=X7) + %<- (max-commutative X2|X4=X6 X4|X2=X6) + %<- (max-associative-converse* X4|X2=X6 X1|X6=X7 X1|X4=X5 X5|X2=X7) %. +%worlds () (max-assoc-commutative* X1|X2=X3 X3|X4=X7 X1|X4=X5 X5|X2=X7) %. +%total {} (max-assoc-commutative* _ _ _ _) %. +%theorem max-assoc-commutative : forall* {X1} {X2} {X3} {X4} {X7} forall {OP1 max X1 X2 X3} {OP2 max X3 X4 X7} exists {X5} {OP3 max X1 X4 X5} {OP4 max X5 X2 X7} true %. +%term _ + %pi (max-assoc-commutative X1|X2=X3 X3|X4=X7 X5 X1|X4=X5 X5|X2=X7) + %<- (max-associative X1|X2=X3 X3|X4=X7 X6 X2|X4=X6 X1|X6=X7) + %<- (max-commutative X2|X4=X6 X4|X2=X6) + %<- (max-associative-converse X4|X2=X6 X1|X6=X7 X5 X1|X4=X5 X5|X2=X7) %. +%worlds () (max-assoc-commutative X1|X2=X3 X3|X4=X7 X5 X1|X4=X5 X5|X2=X7) %. +%total {} (max-assoc-commutative _ _ _ _ _) %. +%theorem max-double-associative* : forall* {A} {B} {C} {D} {A+B} {C+D} {A+C} {B+D} {X} forall {AB max A B A+B} {CD max C D C+D} {ABCD max A+B C+D X} {AC max A C A+C} {BD max B D B+D} exists {ACBD max A+C B+D X} true %. +%term _ + %pi (max-double-associative* X1|X2=X3 X4|X8=XC X3|XC=XF X1|X4=X5 X2|X8=XA X5|XA=XF) + %<- (max-associative X1|X2=X3 X3|XC=XF XE X2|XC=XE X1|XE=XF) + %<- (max-commutative X4|X8=XC X8|X4=XC) + %<- (max-associative-converse* X8|X4=XC X2|XC=XE X2|X8=XA XA|X4=XE) + %<- (max-commutative XA|X4=XE X4|XA=XE) + %<- (max-associative-converse* X4|XA=XE X1|XE=XF X1|X4=X5 X5|XA=XF) %. +%worlds () (max-double-associative* X1|X2=X3 X4|X8=XC X3|XC=XF X1|X4=X5 X2|X8=XA X5|XA=XF) %. +%total {} (max-double-associative* _ _ _ _ _ _) %. +%theorem max-double-associative : forall* {A} {B} {C} {D} {A+B} {C+D} {X} forall {AB max A B A+B} {CD max C D C+D} {ABCD max A+B C+D X} exists {A+C} {B+D} {AC max A C A+C} {BD max B D B+D} {ACBD max A+C B+D X} true %. +%term _ + %pi (max-double-associative X1|X2=X3 X4|X8=XC X3|XC=XF X5 XA X1|X4=X5 X2|X8=XA X5|XA=XF) + %<- (max-associative X1|X2=X3 X3|XC=XF XE X2|XC=XE X1|XE=XF) + %<- (max-commutative X4|X8=XC X8|X4=XC) + %<- (max-associative-converse X8|X4=XC X2|XC=XE XA X2|X8=XA XA|X4=XE) + %<- (max-commutative XA|X4=XE X4|XA=XE) + %<- (max-associative-converse X4|XA=XE X1|XE=XF X5 X1|X4=X5 X5|XA=XF) %. +%worlds () (max-double-associative _ _ _ _ _ _ _ _) %. +%total {} (max-double-associative _ _ _ _ _ _ _ _) %. +%%% Distributivity theorems +%theorem min-right-distributes-over-max : forall* {X1} {X2} {X3} {X4} {X7} forall {A12 max X1 X2 X3} {M34 min X3 X4 X7} exists {X5} {X6} {M14 min X1 X4 X5} {M24 min X2 X4 X6} {A56 max X5 X6 X7} true %. +%term _ min-right-distributes-over-max max/= M _ _ M M max/= %. +%term _ + %pi (min-right-distributes-over-max (max/> X1>X2) X1&X4=X7 _ _ X1&X4=X7 X2&X4=X6 X7|X6=X7) + %<- (min-total X2&X4=X6) + %<- (min-right-preserves-ge* (ge/> X1>X2) X1&X4=X7 X2&X4=X6 X7>=X6) + %<- (ge-implies-max X7>=X6 X7|X6=X7) %. +%term _ + %pi (min-right-distributes-over-max (max/< X2>X1) X2&X4=X7 _ _ X1&X4=X5 X2&X4=X7 X5|X7=X7) + %<- (min-total X1&X4=X5) + %<- (min-right-preserves-ge* (ge/> X2>X1) X2&X4=X7 X1&X4=X5 X7>=X5) + %<- (ge-implies-max X7>=X5 X7|X5=X7) + %<- (max-commutative X7|X5=X7 X5|X7=X7) %. +%worlds () (min-right-distributes-over-max _ _ _ _ _ _ _) %. +%total {} (min-right-distributes-over-max _ _ _ _ _ _ _) %. +%theorem max-right-distributes-over-min : forall* {X1} {X2} {X3} {X4} {X7} forall {A12 min X1 X2 X3} {M34 max X3 X4 X7} exists {X5} {X6} {M14 max X1 X4 X5} {M24 max X2 X4 X6} {A56 min X5 X6 X7} true %. +%term _ max-right-distributes-over-min min/= M _ _ M M min/= %. +%term _ + %pi (max-right-distributes-over-min (min/> X1>X2) X2|X4=X7 _ _ X1|X4=X5 X2|X4=X7 X5&X7=X7) + %<- (max-total X1|X4=X5) + %<- (max-right-preserves-ge* (ge/> X1>X2) X1|X4=X5 X2|X4=X7 X5>=X7) + %<- (ge-implies-min X5>=X7 X5&X7=X7) %. +%term _ + %pi (max-right-distributes-over-min (min/< X2>X1) X1|X4=X7 _ _ X1|X4=X7 X2|X4=X6 X7&X6=X7) + %<- (max-total X2|X4=X6) + %<- (max-right-preserves-ge* (ge/> X2>X1) X2|X4=X6 X1|X4=X7 X6>=X7) + %<- (le-implies-min X6>=X7 X7&X6=X7) %. +%worlds () (max-right-distributes-over-min _ _ _ _ _ _ _) %. +%total {} (max-right-distributes-over-min _ _ _ _ _ _ _) %. +%theorem min-right-distributes-over-max* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {A12 max X1 X2 X3} {M34 min X3 X4 X7} {M14 min X1 X4 X5} {M24 min X2 X4 X6} exists {A56 max X5 X6 X7} true %. +%term _ + %pi (min-right-distributes-over-max* X1|X2=X3 X3&X4=X7 X1&X4=X5 X2&X4=X6 X5|X6=X7) + %<- (min-right-distributes-over-max X1|X2=X3 X3&X4=X7 Y5 Y6 X1&X4=Y5 X2&X4=Y6 Y5|Y6=X7) + %<- (min-deterministic X1&X4=Y5 X1&X4=X5 eq/ eq/ Y5=X5) + %<- (min-deterministic X2&X4=Y6 X2&X4=X6 eq/ eq/ Y6=X6) + %<- (max-respects-eq Y5|Y6=X7 Y5=X5 Y6=X6 eq/ X5|X6=X7) %. +%worlds () (min-right-distributes-over-max* X1|X2=X3 X3&X4=X7 X1&X4=X5 X2&X4=X6 X5|X6=X7) %. +%total {} (min-right-distributes-over-max* _ _ _ _ _) %. +%theorem min-left-distributes-over-max* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {A12 max X2 X4 X6} {M34 min X1 X6 X7} {M14 min X1 X2 X3} {M24 min X1 X4 X5} exists {A56 max X3 X5 X7} true %. +%term _ + %pi (min-left-distributes-over-max* X2|X4=X6 X1&X6=X7 X1&X2=X3 X1&X4=X5 X3|X5=X7) + %<- (min-commutative X1&X6=X7 X6&X1=X7) + %<- (min-commutative X1&X2=X3 X2&X1=X3) + %<- (min-commutative X1&X4=X5 X4&X1=X5) + %<- (min-right-distributes-over-max* X2|X4=X6 X6&X1=X7 X2&X1=X3 X4&X1=X5 X3|X5=X7) %. +%worlds () (min-left-distributes-over-max* X2|X4=X6 X1&X6=X7 X1&X2=X3 X1&X4=X5 X3|X5=X7) %. +%total {} (min-left-distributes-over-max* _ _ _ _ _) %. +%theorem min-left-distributes-over-max : forall* {X1} {X2} {X4} {X6} {X7} forall {A12 max X2 X4 X6} {M34 min X1 X6 X7} exists {X3} {X5} {M14 min X1 X2 X3} {M24 min X1 X4 X5} {A56 max X3 X5 X7} true %. +%term _ + %pi (min-left-distributes-over-max X2|X4=X6 X1&X6=X7 X3 X5 X1&X2=X3 X1&X4=X5 X3|X5=X7) + %<- (min-total X1&X2=X3) + %<- (min-total X1&X4=X5) + %<- (min-left-distributes-over-max* X2|X4=X6 X1&X6=X7 X1&X2=X3 X1&X4=X5 X3|X5=X7) %. +%worlds () (min-left-distributes-over-max X2|X4=X6 X1&X6=X7 X3 X5 X1&X2=X3 X1&X4=X5 X3|X5=X7) %. +%total {} (min-left-distributes-over-max _ _ _ _ _ _ _) %. +%theorem min-right-factors-over-max : forall* {X1} {X2} {X4} {X5} {X6} {X7} forall {M14 min X1 X4 X5} {M24 min X2 X4 X6} {A56 max X5 X6 X7} exists {X3} {A12 max X1 X2 X3} {M34 min X3 X4 X7} true %. +%term _ + %pi (min-right-factors-over-max X1&X4=X5 X2&X4=X6 X5|X6=X7 X3 X1|X2=X3 X3&X4=X7) + %<- (max-total X1|X2=X3) + %<- (min-total X3&X4=Y7) + %<- (min-right-distributes-over-max* X1|X2=X3 X3&X4=Y7 X1&X4=X5 X2&X4=X6 X5|X6=Y7) + %<- (max-deterministic X5|X6=Y7 X5|X6=X7 eq/ eq/ Y7=X7) + %<- (min-respects-eq X3&X4=Y7 eq/ eq/ Y7=X7 X3&X4=X7) %. +%worlds () (min-right-factors-over-max X1&X4=X5 X2&X4=X6 X5|X6=X7 X3 X1|X2=X3 X3&X4=X7) %. +%total {} (min-right-factors-over-max _ _ _ _ _ _) %. +%theorem min-right-factors-over-max* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {M14 min X1 X4 X5} {M24 min X2 X4 X6} {A56 max X5 X6 X7} {A12 max X1 X2 X3} exists {M34 min X3 X4 X7} true %. +%term _ + %pi (min-right-factors-over-max* X1&X4=X5 X2&X4=X6 X5|X6=X7 X1|X2=X3 X3&X4=X7) + %<- (min-total X3&X4=Y7) + %<- (min-right-distributes-over-max* X1|X2=X3 X3&X4=Y7 X1&X4=X5 X2&X4=X6 X5|X6=Y7) + %<- (max-deterministic X5|X6=Y7 X5|X6=X7 eq/ eq/ Y7=X7) + %<- (min-respects-eq X3&X4=Y7 eq/ eq/ Y7=X7 X3&X4=X7) %. +%worlds () (min-right-factors-over-max* X1&X4=X5 X2&X4=X6 X5|X6=X7 X1|X2=X3 X3&X4=X7) %. +%total {} (min-right-factors-over-max* _ _ _ _ _) %. +%theorem min-left-factors-over-max : forall* {X1} {X2} {X3} {X4} {X5} {X7} forall {M12 min X1 X2 X3} {M14 min X1 X4 X5} {A35 max X3 X5 X7} exists {X6} {A24 max X2 X4 X6} {M16 min X1 X6 X7} true %. +%term _ + %pi (min-left-factors-over-max X1&X2=X3 X1&X4=X5 X3|X5=X7 X6 X2|X4=X6 X1&X6=X7) + %<- (min-commutative X1&X2=X3 X2&X1=X3) + %<- (min-commutative X1&X4=X5 X4&X1=X5) + %<- (min-right-factors-over-max X2&X1=X3 X4&X1=X5 X3|X5=X7 X6 X2|X4=X6 X6&X1=X7) + %<- (min-commutative X6&X1=X7 X1&X6=X7) %. +%worlds () (min-left-factors-over-max X1&X2=X3 X1&X4=X5 X3|X5=X7 X6 X2|X4=X6 X1&X6=X7) %. +%total {} (min-left-factors-over-max _ _ _ _ _ _) %. +%theorem min-left-factors-over-max* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {M12 min X1 X2 X3} {M14 min X1 X4 X5} {A35 max X3 X5 X7} {A24 max X2 X4 X6} exists {M16 min X1 X6 X7} true %. +%term _ + %pi (min-left-factors-over-max* X1&X2=X3 X1&X4=X5 X3|X5=X7 X2|X4=X6 X1&X6=X7) + %<- (min-total X1&X6=Y7) + %<- (min-left-distributes-over-max* X2|X4=X6 X1&X6=Y7 X1&X2=X3 X1&X4=X5 X3|X5=Y7) + %<- (max-deterministic X3|X5=Y7 X3|X5=X7 eq/ eq/ Y7=X7) + %<- (min-respects-eq X1&X6=Y7 eq/ eq/ Y7=X7 X1&X6=X7) %. +%worlds () (min-left-factors-over-max* X1&X2=X3 X1&X4=X5 X3|X5=X7 X2|X4=X6 X1&X6=X7) %. +%total {} (min-left-factors-over-max* _ _ _ _ _) %. +%theorem max-right-distributes-over-min* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {A12 min X1 X2 X3} {M34 max X3 X4 X7} {M14 max X1 X4 X5} {M24 max X2 X4 X6} exists {A56 min X5 X6 X7} true %. +%term _ + %pi (max-right-distributes-over-min* X1&X2=X3 X3|X4=X7 X1|X4=X5 X2|X4=X6 X5&X6=X7) + %<- (max-right-distributes-over-min X1&X2=X3 X3|X4=X7 Y5 Y6 X1|X4=Y5 X2|X4=Y6 Y5&Y6=X7) + %<- (max-deterministic X1|X4=Y5 X1|X4=X5 eq/ eq/ Y5=X5) + %<- (max-deterministic X2|X4=Y6 X2|X4=X6 eq/ eq/ Y6=X6) + %<- (min-respects-eq Y5&Y6=X7 Y5=X5 Y6=X6 eq/ X5&X6=X7) %. +%worlds () (max-right-distributes-over-min* X1&X2=X3 X3|X4=X7 X1|X4=X5 X2|X4=X6 X5&X6=X7) %. +%total {} (max-right-distributes-over-min* _ _ _ _ _) %. +%theorem max-left-distributes-over-min* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {A12 min X2 X4 X6} {M34 max X1 X6 X7} {M14 max X1 X2 X3} {M24 max X1 X4 X5} exists {A56 min X3 X5 X7} true %. +%term _ + %pi (max-left-distributes-over-min* X2&X4=X6 X1|X6=X7 X1|X2=X3 X1|X4=X5 X3&X5=X7) + %<- (max-commutative X1|X6=X7 X6|X1=X7) + %<- (max-commutative X1|X2=X3 X2|X1=X3) + %<- (max-commutative X1|X4=X5 X4|X1=X5) + %<- (max-right-distributes-over-min* X2&X4=X6 X6|X1=X7 X2|X1=X3 X4|X1=X5 X3&X5=X7) %. +%worlds () (max-left-distributes-over-min* X2&X4=X6 X1|X6=X7 X1|X2=X3 X1|X4=X5 X3&X5=X7) %. +%total {} (max-left-distributes-over-min* _ _ _ _ _) %. +%theorem max-left-distributes-over-min : forall* {X1} {X2} {X4} {X6} {X7} forall {A12 min X2 X4 X6} {M34 max X1 X6 X7} exists {X3} {X5} {M14 max X1 X2 X3} {M24 max X1 X4 X5} {A56 min X3 X5 X7} true %. +%term _ + %pi (max-left-distributes-over-min X2&X4=X6 X1|X6=X7 X3 X5 X1|X2=X3 X1|X4=X5 X3&X5=X7) + %<- (max-total X1|X2=X3) + %<- (max-total X1|X4=X5) + %<- (max-left-distributes-over-min* X2&X4=X6 X1|X6=X7 X1|X2=X3 X1|X4=X5 X3&X5=X7) %. +%worlds () (max-left-distributes-over-min X2&X4=X6 X1|X6=X7 X3 X5 X1|X2=X3 X1|X4=X5 X3&X5=X7) %. +%total {} (max-left-distributes-over-min _ _ _ _ _ _ _) %. +%theorem max-right-factors-over-min : forall* {X1} {X2} {X4} {X5} {X6} {X7} forall {M14 max X1 X4 X5} {M24 max X2 X4 X6} {A56 min X5 X6 X7} exists {X3} {A12 min X1 X2 X3} {M34 max X3 X4 X7} true %. +%term _ + %pi (max-right-factors-over-min X1|X4=X5 X2|X4=X6 X5&X6=X7 X3 X1&X2=X3 X3|X4=X7) + %<- (min-total X1&X2=X3) + %<- (max-total X3|X4=Y7) + %<- (max-right-distributes-over-min* X1&X2=X3 X3|X4=Y7 X1|X4=X5 X2|X4=X6 X5&X6=Y7) + %<- (min-deterministic X5&X6=Y7 X5&X6=X7 eq/ eq/ Y7=X7) + %<- (max-respects-eq X3|X4=Y7 eq/ eq/ Y7=X7 X3|X4=X7) %. +%worlds () (max-right-factors-over-min X1|X4=X5 X2|X4=X6 X5&X6=X7 X3 X1&X2=X3 X3|X4=X7) %. +%total {} (max-right-factors-over-min _ _ _ _ _ _) %. +%theorem max-right-factors-over-min* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {M14 max X1 X4 X5} {M24 max X2 X4 X6} {A56 min X5 X6 X7} {A12 min X1 X2 X3} exists {M34 max X3 X4 X7} true %. +%term _ + %pi (max-right-factors-over-min* X1|X4=X5 X2|X4=X6 X5&X6=X7 X1&X2=X3 X3|X4=X7) + %<- (max-total X3|X4=Y7) + %<- (max-right-distributes-over-min* X1&X2=X3 X3|X4=Y7 X1|X4=X5 X2|X4=X6 X5&X6=Y7) + %<- (min-deterministic X5&X6=Y7 X5&X6=X7 eq/ eq/ Y7=X7) + %<- (max-respects-eq X3|X4=Y7 eq/ eq/ Y7=X7 X3|X4=X7) %. +%worlds () (max-right-factors-over-min* X1|X4=X5 X2|X4=X6 X5&X6=X7 X1&X2=X3 X3|X4=X7) %. +%total {} (max-right-factors-over-min* _ _ _ _ _) %. +%theorem max-left-factors-over-min : forall* {X1} {X2} {X3} {X4} {X5} {X7} forall {M12 max X1 X2 X3} {M14 max X1 X4 X5} {A35 min X3 X5 X7} exists {X6} {A24 min X2 X4 X6} {M16 max X1 X6 X7} true %. +%term _ + %pi (max-left-factors-over-min X1|X2=X3 X1|X4=X5 X3&X5=X7 X6 X2&X4=X6 X1|X6=X7) + %<- (max-commutative X1|X2=X3 X2|X1=X3) + %<- (max-commutative X1|X4=X5 X4|X1=X5) + %<- (max-right-factors-over-min X2|X1=X3 X4|X1=X5 X3&X5=X7 X6 X2&X4=X6 X6|X1=X7) + %<- (max-commutative X6|X1=X7 X1|X6=X7) %. +%worlds () (max-left-factors-over-min X1|X2=X3 X1|X4=X5 X3&X5=X7 X6 X2&X4=X6 X1|X6=X7) %. +%total {} (max-left-factors-over-min _ _ _ _ _ _) %. +%theorem max-left-factors-over-min* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {M12 max X1 X2 X3} {M14 max X1 X4 X5} {A35 min X3 X5 X7} {A24 min X2 X4 X6} exists {M16 max X1 X6 X7} true %. +%term _ + %pi (max-left-factors-over-min* X1|X2=X3 X1|X4=X5 X3&X5=X7 X2&X4=X6 X1|X6=X7) + %<- (max-total X1|X6=Y7) + %<- (max-left-distributes-over-min* X2&X4=X6 X1|X6=Y7 X1|X2=X3 X1|X4=X5 X3&X5=Y7) + %<- (min-deterministic X3&X5=Y7 X3&X5=X7 eq/ eq/ Y7=X7) + %<- (max-respects-eq X1|X6=Y7 eq/ eq/ Y7=X7 X1|X6=X7) %. +%worlds () (max-left-factors-over-min* X1|X2=X3 X1|X4=X5 X3&X5=X7 X2&X4=X6 X1|X6=X7) %. +%total {} (max-left-factors-over-min* _ _ _ _ _) %. +%inline nat`nat nat %. +%inline nat`z z %. +%inline nat`s s %. +%inline nat`plus plus %. +%inline nat`plus/z plus/z %. +%inline nat`plus/s plus/s %. +%inline nat`times times %. +%inline nat`times/z times/z %. +%inline nat`times/s times/s %. +%inline nat`eq eq %. +%inline nat`eq/ eq/ %. +%inline nat`gt gt %. +%inline nat`gt/1 gt/1 %. +%inline nat`gt/> gt/> %. +%inline nat`compare compare %. +%inline nat`compare/= compare/= %. +%inline nat`compare/< compare/< %. +%inline nat`compare/> compare/> %. +%inline nat`meta-eq meta-eq %. +%inline nat`false-implies-eq false-implies-eq %. +%inline nat`eq-symmetric eq-symmetric %. +%inline nat`eq-transitive eq-transitive %. +%inline nat`succ-deterministic succ-deterministic %. +%inline nat`succ-cancels succ-cancels %. +%inline nat`eq-contradiction eq-contradiction %. +%inline nat`meta-gt meta-gt %. +%inline nat`false-implies-gt false-implies-gt %. +%inline nat`gt-respects-eq gt-respects-eq %. +%inline nat`succ-implies-gt succ-implies-gt %. +%inline nat`succ-implies-gt-zero succ-implies-gt-zero %. +%inline nat`succ-preserves-gt succ-preserves-gt %. +%inline nat`succ-preserves-gt-converse succ-preserves-gt-converse %. +%inline nat`gt-implies-positive gt-implies-positive %. +%inline nat`gt-anti-reflexive* gt-anti-reflexive* %. +%inline nat`gt-anti-reflexive gt-anti-reflexive %. +%inline nat`gt-transitive gt-transitive %. +%inline nat`gt-anti-symmetric gt-anti-symmetric %. +%inline nat`gt-implies-plus gt-implies-plus %. +%inline nat`gt-contradiction gt-contradiction %. +%inline nat`false-implies-compare false-implies-compare %. +%inline nat`succ-preserves-compare succ-preserves-compare %. +%inline nat`compare-total* compare-total* %. +%inline nat`compare-total compare-total %. +%inline nat`greater-implies-gt greater-implies-gt %. +%inline nat`less-is-reverse-greater less-is-reverse-greater %. +%inline nat`less-implies-lt less-implies-lt %. +%inline nat`equal-implies-eq equal-implies-eq %. +%inline nat`false-implies-plus false-implies-plus %. +%inline nat`plus-respects-eq plus-respects-eq %. +%inline nat`plus-total* plus-total* %. +%inline nat`plus-total plus-total %. +%inline nat`plus-deterministic plus-deterministic %. +%inline nat`plus-left-identity plus-left-identity %. +%inline nat`plus-left-increase plus-left-increase %. +%inline nat`plus-right-identity plus-right-identity %. +%inline nat`plus-right-increase plus-right-increase %. +%inline nat`plus-left-decrease plus-left-decrease %. +%inline nat`plus-right-decrease plus-right-decrease %. +%inline nat`plus-swap-succ plus-swap-succ %. +%inline nat`plus-swap-succ-converse plus-swap-succ-converse %. +%inline nat`plus-left-preserves-positive plus-left-preserves-positive %. +%inline nat`plus-right-preserves-positive plus-right-preserves-positive %. +%inline nat`plus-is-zero-implies-zero plus-is-zero-implies-zero %. +%inline nat`plus-commutative plus-commutative %. +%inline nat`plus-associative plus-associative %. +%inline nat`plus-associative* plus-associative* %. +%inline nat`plus-associative-converse plus-associative-converse %. +%inline nat`plus-associative-converse* plus-associative-converse* %. +%inline nat`plus-assoc-commutative* plus-assoc-commutative* %. +%inline nat`plus-assoc-commutative plus-assoc-commutative %. +%inline nat`plus-double-associative* plus-double-associative* %. +%inline nat`plus-double-associative plus-double-associative %. +%inline nat`plus-left-cancels plus-left-cancels %. +%inline nat`plus-right-cancels* plus-right-cancels* %. +%inline nat`plus-right-cancels plus-right-cancels %. +%inline nat`plus-left-preserves-gt* plus-left-preserves-gt* %. +%inline nat`plus-left-cancels-gt plus-left-cancels-gt %. +%inline nat`plus-left-preserves-gt plus-left-preserves-gt %. +%inline nat`plus-right-preserves-gt* plus-right-preserves-gt* %. +%inline nat`plus-right-preserves-gt plus-right-preserves-gt %. +%inline nat`plus-preserves-gt* plus-preserves-gt* %. +%inline nat`plus-preserves-gt plus-preserves-gt %. +%inline nat`plus-right-cancels-gt plus-right-cancels-gt %. +%inline nat`plus-implies-gt plus-implies-gt %. +%inline nat`plus-gt-contradiction plus-gt-contradiction %. +%inline nat`false-implies-times false-implies-times %. +%inline nat`times-respects-eq times-respects-eq %. +%inline nat`times-total* times-total* %. +%inline nat`times-total times-total %. +%inline nat`times-deterministic times-deterministic %. +%inline nat`times-left-identity times-left-identity %. +%inline nat`times-right-identity times-right-identity %. +%inline nat`times-right-zero times-right-zero %. +%inline nat`times-preserves-positive times-preserves-positive %. +%inline nat`times-preserves-positive* times-preserves-positive* %. +%inline nat`times-positive-implies-positive times-positive-implies-positive %. +%inline nat`times-left-increase times-left-increase %. +%inline nat`times-right-increase times-right-increase %. +%inline nat`times-left-decrease times-left-decrease %. +%inline nat`times-right-decrease times-right-decrease %. +%inline nat`times-commutative times-commutative %. +%inline nat`times-right-distributes-over-plus times-right-distributes-over-plus %. +%inline nat`times-right-distributes-over-plus* times-right-distributes-over-plus* %. +%inline nat`times-left-distributes-over-plus* times-left-distributes-over-plus* %. +%inline nat`times-left-distributes-over-plus times-left-distributes-over-plus %. +%inline nat`times-right-factors-over-plus times-right-factors-over-plus %. +%inline nat`times-right-factors-over-plus* times-right-factors-over-plus* %. +%inline nat`times-left-factors-over-plus times-left-factors-over-plus %. +%inline nat`times-left-factors-over-plus* times-left-factors-over-plus* %. +%inline nat`times-associative times-associative %. +%inline nat`times-associative* times-associative* %. +%inline nat`times-associative-converse times-associative-converse %. +%inline nat`times-associative-converse* times-associative-converse* %. +%inline nat`times-assoc-commutative* times-assoc-commutative* %. +%inline nat`times-assoc-commutative times-assoc-commutative %. +%inline nat`times-double-associative* times-double-associative* %. +%inline nat`times-double-associative times-double-associative %. +%inline nat`times-right-cancels times-right-cancels %. +%inline nat`times-right-cancels* times-right-cancels* %. +%inline nat`times-right-cancels** times-right-cancels** %. +%inline nat`times-left-cancels times-left-cancels %. +%inline nat`times-left-cancels* times-left-cancels* %. +%inline nat`times-left-preserves-gt times-left-preserves-gt %. +%inline nat`times-left-preserves-gt* times-left-preserves-gt* %. +%inline nat`times-right-preserves-gt times-right-preserves-gt %. +%inline nat`times-right-preserves-gt* times-right-preserves-gt* %. +%inline nat`times-preserves-gt times-preserves-gt %. +%inline nat`times-right-cancels-gt times-right-cancels-gt %. +%inline nat`times-left-cancels-gt times-left-cancels-gt %. +%inline nat`minus minus %. +%inline nat`false-implies-minus false-implies-minus %. +%inline nat`minus-respects-eq minus-respects-eq %. +%inline nat`minus-deterministic minus-deterministic %. +%inline nat`plus-associates-with-minus* plus-associates-with-minus* %. +%inline nat`plus-associates-with-minus-converse* plus-associates-with-minus-converse* %. +%inline nat`plus-associates-with-minus-converse plus-associates-with-minus-converse %. +%inline nat`minus-associates-from-plus* minus-associates-from-plus* %. +%inline nat`minus-associates-from-plus-converse* minus-associates-from-plus-converse* %. +%inline nat`minus-associates-to-plus* minus-associates-to-plus* %. +%inline nat`minus-associates-to-plus minus-associates-to-plus %. +%inline nat`minus-associates-to-plus-converse* minus-associates-to-plus-converse* %. +%inline nat`minus-associates-to-plus-converse minus-associates-to-plus-converse %. +%inline nat`minus-is-zero-implies-eq minus-is-zero-implies-eq %. +%inline nat`minus-implies-gt minus-implies-gt %. +%inline nat`minus-left-cancels minus-left-cancels %. +%inline nat`minus-right-cancels minus-right-cancels %. +%inline nat`minus-left-inverts-gt* minus-left-inverts-gt* %. +%inline nat`minus-right-preserves-gt* minus-right-preserves-gt* %. +%inline nat`minus-left-cancels-inverts-gt minus-left-cancels-inverts-gt %. +%inline nat`minus-right-cancels-gt minus-right-cancels-gt %. +%inline nat`times-right-distributes-over-minus times-right-distributes-over-minus %. +%inline nat`times-right-distributes-over-minus* times-right-distributes-over-minus* %. +%inline nat`times-left-distributes-over-minus* times-left-distributes-over-minus* %. +%inline nat`times-left-distributes-over-minus times-left-distributes-over-minus %. +%inline nat`times-right-factors-over-minus* times-right-factors-over-minus* %. +%inline nat`times-left-factors-over-minus* times-left-factors-over-minus* %. +%inline nat`times-right-factors-over-minus times-right-factors-over-minus %. +%inline nat`times-left-factors-over-minus times-left-factors-over-minus %. +%inline nat`ge ge %. +%inline nat`ge/= ge/= %. +%inline nat`ge/> ge/> %. +%inline nat`false-implies-ge false-implies-ge %. +%inline nat`ge-respects-eq ge-respects-eq %. +%inline nat`ge-reflexive ge-reflexive %. +%inline nat`ge-transitive ge-transitive %. +%inline nat`ge-anti-symmetric ge-anti-symmetric %. +%inline nat`ge-transitive-gt ge-transitive-gt %. +%inline nat`gt-transitive-ge gt-transitive-ge %. +%inline nat`meta-ge meta-ge %. +%inline nat`succ-preserves-ge succ-preserves-ge %. +%inline nat`succ-preserves-ge-converse succ-preserves-ge-converse %. +%inline nat`ge-succ-implies-gt ge-succ-implies-gt %. +%inline nat`ge-implies-succ-gt ge-implies-succ-gt %. +%inline nat`succ-gt-implies-ge succ-gt-implies-ge %. +%inline nat`gt-implies-ge-succ gt-implies-ge-succ %. +%inline nat`ge-implies-plus ge-implies-plus %. +%inline nat`plus-implies-ge plus-implies-ge %. +%inline nat`ge-zero-always ge-zero-always %. +%inline nat`nonzero-times-implies-ge nonzero-times-implies-ge %. +%inline nat`times-nonzero-implies-ge times-nonzero-implies-ge %. +%inline nat`non-trivial-times-implies-much-gt* non-trivial-times-implies-much-gt* %. +%inline nat`non-trivial-times-implies-much-gt non-trivial-times-implies-much-gt %. +%inline nat`plus-left-preserves-ge* plus-left-preserves-ge* %. +%inline nat`plus-left-cancels-ge plus-left-cancels-ge %. +%inline nat`plus-left-preserves-ge plus-left-preserves-ge %. +%inline nat`plus-right-preserves-ge* plus-right-preserves-ge* %. +%inline nat`plus-right-preserves-ge plus-right-preserves-ge %. +%inline nat`plus-preserves-ge* plus-preserves-ge* %. +%inline nat`plus-preserves-ge plus-preserves-ge %. +%inline nat`plus-right-cancels-ge plus-right-cancels-ge %. +%inline nat`times-left-preserves-ge* times-left-preserves-ge* %. +%inline nat`times-left-preserves-ge times-left-preserves-ge %. +%inline nat`times-right-preserves-ge* times-right-preserves-ge* %. +%inline nat`times-right-preserves-ge times-right-preserves-ge %. +%inline nat`ne ne %. +%inline nat`ne/< ne/< %. +%inline nat`ne/> ne/> %. +%inline nat`eq? eq? %. +%inline nat`eq?/yes eq?/yes %. +%inline nat`eq?/no eq?/no %. +%inline nat`false-implies-ne false-implies-ne %. +%inline nat`ne-respects-eq ne-respects-eq %. +%inline nat`ne-anti-reflexive ne-anti-reflexive %. +%inline nat`ne-symmetric ne-symmetric %. +%inline nat`eq-ne-implies-false eq-ne-implies-false %. +%inline nat`ge-ne-implies-gt ge-ne-implies-gt %. +%inline nat`eq?-total* eq?-total* %. +%inline nat`eq?-total*/L eq?-total*/L %. +%inline nat`eq?-total eq?-total %. +%inline nat`succ-preserves-ne succ-preserves-ne %. +%inline nat`succ-preserves-ne-converse succ-preserves-ne-converse %. +%inline nat`plus-left-preserves-ne* plus-left-preserves-ne* %. +%inline nat`plus-left-cancels-ne plus-left-cancels-ne %. +%inline nat`plus-left-preserves-ne plus-left-preserves-ne %. +%inline nat`plus-right-preserves-ne* plus-right-preserves-ne* %. +%inline nat`plus-right-preserves-ne plus-right-preserves-ne %. +%inline nat`plus-right-cancels-ne plus-right-cancels-ne %. +%inline nat`lt lt %. +%inline nat`false-implies-lt false-implies-lt %. +%inline nat`lt-respects-eq lt-respects-eq %. +%inline nat`lt-anti-symmetric lt-anti-symmetric %. +%inline nat`lt-transitive lt-transitive %. +%inline nat`lt-anti-reflexive lt-anti-reflexive %. +%inline nat`plus-left-preserves-lt* plus-left-preserves-lt* %. +%inline nat`plus-left-cancels-lt plus-left-cancels-lt %. +%inline nat`plus-left-preserves-lt plus-left-preserves-lt %. +%inline nat`plus-right-preserves-lt* plus-right-preserves-lt* %. +%inline nat`plus-right-preserves-lt plus-right-preserves-lt %. +%inline nat`plus-preserves-lt* plus-preserves-lt* %. +%inline nat`plus-preserves-lt plus-preserves-lt %. +%inline nat`plus-right-cancels-lt plus-right-cancels-lt %. +%inline nat`le le %. +%inline nat`false-implies-le false-implies-le %. +%inline nat`le-respects-eq le-respects-eq %. +%inline nat`le-anti-symmetric le-anti-symmetric %. +%inline nat`le-transitive le-transitive %. +%inline nat`le-reflexive le-reflexive %. +%inline nat`le-transitive-lt le-transitive-lt %. +%inline nat`lt-transitive-le lt-transitive-le %. +%inline nat`plus-left-preserves-le* plus-left-preserves-le* %. +%inline nat`plus-left-cancels-le plus-left-cancels-le %. +%inline nat`plus-left-preserves-le plus-left-preserves-le %. +%inline nat`plus-right-preserves-le* plus-right-preserves-le* %. +%inline nat`plus-right-preserves-le plus-right-preserves-le %. +%inline nat`plus-preserves-le* plus-preserves-le* %. +%inline nat`plus-preserves-le plus-preserves-le %. +%inline nat`plus-right-cancels-le plus-right-cancels-le %. +%inline nat`times-left-preserves-le* times-left-preserves-le* %. +%inline nat`times-left-preserves-le times-left-preserves-le %. +%inline nat`times-right-preserves-le* times-right-preserves-le* %. +%inline nat`times-right-preserves-le times-right-preserves-le %. +%inline nat`times-preserves-le* times-preserves-le* %. +%inline nat`times-preserves-le times-preserves-le %. +%inline nat`minus-left-inverts-ge* minus-left-inverts-ge* %. +%inline nat`minus-right-preserves-ge* minus-right-preserves-ge* %. +%inline nat`minus-left-cancels-inverts-ge minus-left-cancels-inverts-ge %. +%inline nat`minus-right-cancels-ge minus-right-cancels-ge %. +%inline nat`minus-left-preserves-ne* minus-left-preserves-ne* %. +%inline nat`minus-right-preserves-ne* minus-right-preserves-ne* %. +%inline nat`minus-left-cancels-ne minus-left-cancels-ne %. +%inline nat`minus-right-cancels-ne minus-right-cancels-ne %. +%inline nat`minus-left-inverts-lt* minus-left-inverts-lt* %. +%inline nat`minus-right-preserves-lt* minus-right-preserves-lt* %. +%inline nat`minus-left-cancels-inverts-lt minus-left-cancels-inverts-lt %. +%inline nat`minus-right-cancels-lt minus-right-cancels-lt %. +%inline nat`minus-left-inverts-le* minus-left-inverts-le* %. +%inline nat`minus-right-preserves-le* minus-right-preserves-le* %. +%inline nat`minus-left-cancels-inverts-le minus-left-cancels-inverts-le %. +%inline nat`minus-right-cancels-le minus-right-cancels-le %. +%inline nat`divrem divrem %. +%inline nat`divrem/z divrem/z %. +%inline nat`divrem/s divrem/s %. +%inline nat`false-implies-divrem false-implies-divrem %. +%inline nat`divrem-respects-eq divrem-respects-eq %. +%inline nat`divrem-total** divrem-total** %. +%inline nat`divrem-total* divrem-total* %. +%inline nat`divrem-total divrem-total %. +%inline nat`divrem-deterministic divrem-deterministic %. +%inline nat`divrem-implies-positive divrem-implies-positive %. +%inline nat`divrem-implies-gt divrem-implies-gt %. +%inline nat`divrem-contradiction divrem-contradiction %. +%inline nat`divrem-can-be-inverted divrem-can-be-inverted %. +%inline nat`div-can-be-inverted div-can-be-inverted %. +%inline nat`divrem-can-be-constructed divrem-can-be-constructed %. +%inline nat`div-can-be-constructed div-can-be-constructed %. +%inline nat`remainder-implies-gt-quotient remainder-implies-gt-quotient %. +%inline nat`quotient-of-nonzero-is-smaller quotient-of-nonzero-is-smaller %. +%inline nat`quotient-is-no-greater quotient-is-no-greater %. +%inline nat`min min %. +%inline nat`min/= min/= %. +%inline nat`min/> min/> %. +%inline nat`min/< min/< %. +%inline nat`max max %. +%inline nat`max/= max/= %. +%inline nat`max/> max/> %. +%inline nat`max/< max/< %. +%inline nat`false-implies-min false-implies-min %. +%inline nat`min-respects-eq min-respects-eq %. +%inline nat`min-total** min-total** %. +%inline nat`min-total* min-total* %. +%inline nat`min-total min-total %. +%inline nat`min-deterministic min-deterministic %. +%inline nat`min-commutative min-commutative %. +%inline nat`ge-implies-min ge-implies-min %. +%inline nat`le-implies-min le-implies-min %. +%inline nat`min-implies-ge min-implies-ge %. +%inline nat`min-left-preserves-ge* min-left-preserves-ge* %. +%inline nat`min-left-preserves-le* min-left-preserves-le* %. +%inline nat`min-left-preserves-ge min-left-preserves-ge %. +%inline nat`min-right-preserves-ge* min-right-preserves-ge* %. +%inline nat`min-right-preserves-ge min-right-preserves-ge %. +%inline nat`min-preserves-ge* min-preserves-ge* %. +%inline nat`min-preserves-ge min-preserves-ge %. +%inline nat`min-left-preserves-le min-left-preserves-le %. +%inline nat`min-right-preserves-le* min-right-preserves-le* %. +%inline nat`min-right-preserves-le min-right-preserves-le %. +%inline nat`min-preserves-le* min-preserves-le* %. +%inline nat`min-preserves-le min-preserves-le %. +%inline nat`min-is-glb min-is-glb %. +%inline nat`min-associative min-associative %. +%inline nat`min-associative* min-associative* %. +%inline nat`min-associative-converse min-associative-converse %. +%inline nat`min-associative-converse* min-associative-converse* %. +%inline nat`min-assoc-commutative* min-assoc-commutative* %. +%inline nat`min-assoc-commutative min-assoc-commutative %. +%inline nat`min-double-associative* min-double-associative* %. +%inline nat`min-double-associative min-double-associative %. +%inline nat`false-implies-max false-implies-max %. +%inline nat`max-respects-eq max-respects-eq %. +%inline nat`max-total** max-total** %. +%inline nat`max-total* max-total* %. +%inline nat`max-total max-total %. +%inline nat`max-deterministic max-deterministic %. +%inline nat`max-commutative max-commutative %. +%inline nat`ge-implies-max ge-implies-max %. +%inline nat`le-implies-max le-implies-max %. +%inline nat`max-implies-ge max-implies-ge %. +%inline nat`max-is-lub max-is-lub %. +%inline nat`max-left-preserves-ge* max-left-preserves-ge* %. +%inline nat`max-left-preserves-le* max-left-preserves-le* %. +%inline nat`max-left-preserves-ge max-left-preserves-ge %. +%inline nat`max-right-preserves-ge* max-right-preserves-ge* %. +%inline nat`max-right-preserves-ge max-right-preserves-ge %. +%inline nat`max-preserves-ge* max-preserves-ge* %. +%inline nat`max-preserves-ge max-preserves-ge %. +%inline nat`max-left-preserves-le max-left-preserves-le %. +%inline nat`max-right-preserves-le* max-right-preserves-le* %. +%inline nat`max-right-preserves-le max-right-preserves-le %. +%inline nat`max-preserves-le* max-preserves-le* %. +%inline nat`max-preserves-le max-preserves-le %. +%inline nat`max-associative max-associative %. +%inline nat`max-associative* max-associative* %. +%inline nat`max-associative-converse max-associative-converse %. +%inline nat`max-associative-converse* max-associative-converse* %. +%inline nat`max-assoc-commutative* max-assoc-commutative* %. +%inline nat`max-assoc-commutative max-assoc-commutative %. +%inline nat`max-double-associative* max-double-associative* %. +%inline nat`max-double-associative max-double-associative %. +%inline nat`min-right-distributes-over-max min-right-distributes-over-max %. +%inline nat`max-right-distributes-over-min max-right-distributes-over-min %. +%inline nat`min-right-distributes-over-max* min-right-distributes-over-max* %. +%inline nat`min-left-distributes-over-max* min-left-distributes-over-max* %. +%inline nat`min-left-distributes-over-max min-left-distributes-over-max %. +%inline nat`min-right-factors-over-max min-right-factors-over-max %. +%inline nat`min-right-factors-over-max* min-right-factors-over-max* %. +%inline nat`min-left-factors-over-max min-left-factors-over-max %. +%inline nat`min-left-factors-over-max* min-left-factors-over-max* %. +%inline nat`max-right-distributes-over-min* max-right-distributes-over-min* %. +%inline nat`max-left-distributes-over-min* max-left-distributes-over-min* %. +%inline nat`max-left-distributes-over-min max-left-distributes-over-min %. +%inline nat`max-right-factors-over-min max-right-factors-over-min %. +%inline nat`max-right-factors-over-min* max-right-factors-over-min* %. +%inline nat`max-left-factors-over-min max-left-factors-over-min %. +%inline nat`max-left-factors-over-min* max-left-factors-over-min* %. +%%%%% natpair.elf +%%%%% Pairs of natural numbers +%%%%% John Boyland +% Pairs of natural numbers are mapped one-to-one to the natural numbers. +% We use the binary merging technique, e.g.: +% (x3x2x1x0,y3y2y1y0) <-> x3y3x2y2x1y1x0y0 +%%%% Functor use +%%%%% pair.elf +%%%%% a pseudo-functor +%%%%% John Boyland +% We require the following definitions: +% nat : equality type. +% nat : equality type. +% The result is an equality type too. +%%%% Definitions +%sort pair %. +%term pair/ %pi nat %-> nat %-> pair %. +%sort eq %. +%term eq/ eq P P %. +%sort ne %. +%term ne/1 %pi (ne (pair/ X1 Y1) (pair/ X2 Y2)) %<- (nat`ne X1 X2) %. +%term ne/2 %pi (ne (pair/ X1 Y1) (pair/ X2 Y2)) %<- (nat`ne Y1 Y2) %. +%sort eq? %. +%term eq?/yes eq? P P true %. +%term eq?/no %pi (eq? P1 P2 false) %<- (ne P1 P2) %. +%%%% Theorems +%%% theorems about eq +%theorem false-implies-eq : forall* {X1} {X2} forall {F void} exists {E eq X1 X2} true %. +%worlds () (false-implies-eq _ _) %. +%total {} (false-implies-eq _ _) %. +%theorem meta-eq : forall {X1} {X2} {E eq X1 X2} true %. +%term _ meta-eq _ _ eq/ %. +%worlds () (meta-eq _ _ _) %. +%total {} (meta-eq _ _ _) %. +%reduces = X Y (meta-eq X Y _) %. +%theorem eq-reflexive : forall {X} exists {E eq X X} true %. +%term _ eq-reflexive _ eq/ %. +%worlds () (eq-reflexive _ _) %. +%total {} (eq-reflexive _ _) %. +%theorem eq-symmetric : forall* {X} {Y} forall {E eq X Y} exists {F eq Y X} true %. +%term _ eq-symmetric eq/ eq/ %. +%worlds () (eq-symmetric _ _) %. +%total {} (eq-symmetric _ _) %. +%theorem eq-transitive : forall* {X} {Y} {Z} forall {E1 eq X Y} {E2 eq Y Z} exists {F eq X Z} true %. +%term _ eq-transitive eq/ eq/ eq/ %. +%worlds () (eq-transitive _ _ _) %. +%total {} (eq-transitive _ _ _) %. +%theorem pair-eq-implies-eq : forall* {D1a} {D1b} {D2a} {D2b} forall {E eq (pair/ D1a D2a) (pair/ D1b D2b)} exists {E1 nat`eq D1a D1b} {E2 nat`eq D2a D2b} true %. +%term _ pair-eq-implies-eq eq/ nat`eq/ nat`eq/ %. +%worlds () (pair-eq-implies-eq _ _ _) %. +%total {} (pair-eq-implies-eq _ _ _) %. +%theorem pair-preserves-eq : forall* {D1a} {D1b} {D2a} {D2b} forall {E1 nat`eq D1a D1b} {E2 nat`eq D2a D2b} exists {E eq (pair/ D1a D2a) (pair/ D1b D2b)} true %. +%term _ pair-preserves-eq nat`eq/ nat`eq/ eq/ %. +%worlds () (pair-preserves-eq _ _ _) %. +%total {} (pair-preserves-eq _ _ _) %. +%%% theorems about ne +%theorem false-implies-ne : forall* {X1} {X2} forall {F void} exists {G ne X1 X2} true %. +%worlds () (false-implies-ne _ _) %. +%total {} (false-implies-ne _ _) %. +%theorem ne-respects-eq : forall* {X1} {X2} {Y1} {Y2} forall {D1 ne X1 X2} {E1 eq X1 Y1} {E2 eq X2 Y2} exists {D2 ne Y1 Y2} true %. +%term _ ne-respects-eq X1<>X2 eq/ eq/ X1<>X2 %. +%worlds () (ne-respects-eq _ _ _ _) %. +%total {} (ne-respects-eq _ _ _ _) %. +%theorem ne-anti-reflexive : forall* {P} forall {R ne P P} exists {F void} true %. +%term _ %pi (ne-anti-reflexive (ne/1 X<>X) F) %<- (nat`ne-anti-reflexive X<>X F) %. +%term _ %pi (ne-anti-reflexive (ne/2 Y<>Y) F) %<- (nat`ne-anti-reflexive Y<>Y F) %. +%worlds () (ne-anti-reflexive _ _) %. +%total {} (ne-anti-reflexive _ _) %. +%theorem ne-symmetric : forall* {P1} {P2} forall {R1 ne P1 P2} exists {R2 ne P2 P1} true %. +%term _ + %pi (ne-symmetric (ne/1 X1<>X2) (ne/1 X2<>X1)) + %<- (nat`ne-symmetric X1<>X2 X2<>X1) %. +%term _ + %pi (ne-symmetric (ne/2 Y1<>Y2) (ne/2 Y2<>Y1)) + %<- (nat`ne-symmetric Y1<>Y2 Y2<>Y1) %. +%worlds () (ne-symmetric _ _) %. +%total {} (ne-symmetric _ _) %. +%theorem eq-ne-implies-false : forall* {P1} {P2} forall {D1 eq P1 P2} {D2 ne P1 P2} exists {F void} true %. +%term _ %pi (eq-ne-implies-false eq/ X<>X F) %<- (ne-anti-reflexive X<>X F) %. +%worlds () (eq-ne-implies-false _ _ _) %. +%total {} (eq-ne-implies-false _ _ _) %. +%%% theorems about eq? +%theorem eq?-total* : forall {P1} {P2} exists {B} {T eq? P1 P2 B} true %. +%theorem eq?-total*/L : forall* {X1} {Y1} {X2} {Y2} {B1} {B2} forall {T1 nat`eq? X1 Y1 B1} {T2 nat`eq? X2 Y2 B2} exists {B} {T eq? (pair/ X1 X2) (pair/ Y1 Y2) B} true %. +%term _ eq?-total*/L nat`eq?/yes nat`eq?/yes _ eq?/yes %. +%term _ eq?-total*/L (nat`eq?/no X1<>Y1) _ _ (eq?/no (ne/1 X1<>Y1)) %. +%term _ eq?-total*/L _ (nat`eq?/no X2<>Y2) _ (eq?/no (ne/2 X2<>Y2)) %. +%worlds () (eq?-total*/L _ _ _ _) %. +%total {} (eq?-total*/L _ _ _ _) %. +%term _ + %pi (eq?-total* _ _ _ T) + %<- (nat`eq?-total E?1) + %<- (nat`eq?-total E?2) + %<- (eq?-total*/L E?1 E?2 _ T) %. +%worlds () (eq?-total* _ _ _ _) %. +%total {} (eq?-total* _ _ _ _) %. +%inline eq?-total eq?-total* _ _ _ %. +%%%% Definitions +%%% local abbreviations: +%inline ssN>N gt/> gt/1 %. +%inline 2>0 (gt (s (s z)) z) ssN>N %. +%inline 1*N=N times/s times/z plus/z %. +%inline 1*2=2 (times (s z) (s (s z)) (s (s z))) 1*N=N %. +%inline 2*2=4 times/s 1*N=N (plus/s (plus/s plus/z)) %. +%inline 2+1=3 (plus (s (s z)) (s z) (s (s (s z)))) plus/s (plus/s plus/z) %. +%%% mapping from a pair to a nat +%sort pair2nat %. +%term pair2nat/00 pair2nat (pair/ z z) z %. +%term pair2nat/XX + %pi (plus Z3 Y0 Z) + %-> (plus Z2 X2 Z3) + %-> (times X0 (s (s z)) X2) + %-> (times Z1 (s (s (s (s z)))) Z2) + %-> (pair2nat (pair/ X1 Y1) Z1) + %-> (divrem Y (s (s z)) Y1 Y0) + %-> (divrem X (s (s z)) X1 X0) + %-> (pair2nat (pair/ X Y) Z) %. +%%% mapping from a nat to pair +%inline nat2pair (%pi nat %-> pair %-> %type) [N] [P] pair2nat P N %. +%%%% Theorems +%theorem false-implies-pair2nat : forall* {P} {N} forall {F void} exists {P2N pair2nat P N} true %. +%worlds () (false-implies-pair2nat _ _) %. +%total {} (false-implies-pair2nat _ _) %. +%inline false-implies-not2pair false-implies-pair2nat %. +%theorem pair2nat-respects-eq : forall* {P1} {N1} {P2} {N2} forall {D1 pair2nat P1 N1} {EP eq P1 P2} {EN nat`eq N1 N2} exists {D2 pair2nat P2 N2} true %. +%term _ pair2nat-respects-eq P2N eq/ nat`eq/ P2N %. +%worlds () (pair2nat-respects-eq _ _ _ _) %. +%total {} (pair2nat-respects-eq _ _ _ _) %. +%reduces = D1 D2 (pair2nat-respects-eq D1 _ _ D2) %. +%inline nat2pair-respects-eq ( + %pi (nat2pair N1 P1) + %-> (nat`eq N1 N2) + %-> (eq P1 P2) + %-> (nat2pair N2 P2) + %-> %type) [D1] [EN] [EP] [D2] pair2nat-respects-eq D1 EP EN D2 %. +%theorem pair2nat-total** : forall {X nat} {Y nat} exists {Z} {P2N pair2nat (pair/ X Y) Z} true %. +%term _ pair2nat-total** z z z pair2nat/00 %. +%term _ + %pi (pair2nat-total** (s X-) Y Z (pair2nat/XX Z3+Y0=Z Z2+X2=Z3 X0*2=X2 Z1*4=Z2 (%the (pair2nat (pair/ X1 Y1) Z1) P2N) Y/2=Y1,Y0 X/2=X1,X0)) + %<- (divrem-total X/2=X1,X0) + %<- (divrem-total Y/2=Y1,Y0) + %<- (quotient-of-nonzero-is-smaller X/2=X1,X0 nat`eq/ X>X1) + %<- (quotient-is-no-greater Y/2=Y1,Y0 Y>=Y1) + %<- (meta-gt _ _ X>X1) + %<- (meta-ge _ _ Y>=Y1) + %<- (pair2nat-total** _ _ _ P2N) + %<- (times-total Z1*4=Z2) + %<- (times-total X0*2=X2) + %<- (plus-total Z2+X2=Z3) + %<- (plus-total Z3+Y0=Z) %. +%term _ + %pi (pair2nat-total** X (s Y-) Z (pair2nat/XX Z3+Y0=Z Z2+X2=Z3 X0*2=X2 Z1*4=Z2 (%the (pair2nat (pair/ X1 Y1) Z1) P2N) Y/2=Y1,Y0 X/2=X1,X0)) + %<- (divrem-total X/2=X1,X0) + %<- (divrem-total Y/2=Y1,Y0) + %<- (quotient-is-no-greater X/2=X1,X0 X>=X1) + %<- (quotient-of-nonzero-is-smaller Y/2=Y1,Y0 nat`eq/ Y>Y1) + %<- (meta-ge _ _ X>=X1) + %<- (meta-gt _ _ Y>Y1) + %<- (pair2nat-total** _ _ _ P2N) + %<- (times-total Z1*4=Z2) + %<- (times-total X0*2=X2) + %<- (plus-total Z2+X2=Z3) + %<- (plus-total Z3+Y0=Z) %. +%worlds () (pair2nat-total** _ _ _ _) %. +%total [X Y] (pair2nat-total** X Y _ _) %. +%theorem pair2nat-total* : forall {P pair} exists {N nat} {P2N pair2nat P N} true %. +%term _ %pi (pair2nat-total* (pair/ X Y) Z P2N) %<- (pair2nat-total** X Y Z P2N) %. +%worlds () (pair2nat-total* _ _ _) %. +%total {} (pair2nat-total* _ _ _) %. +%inline pair2nat-total pair2nat-total* _ _ %. +%theorem nat2pair-total* : forall {N nat} exists {P pair} {N2P nat2pair N P} true %. +%term _ nat2pair-total* z (pair/ z z) pair2nat/00 %. +%term _ + %pi (nat2pair-total* (s Z-) (pair/ X Y) (pair2nat/XX Z3+Y0=Z Z2+X2=Z3 X0*2=X2 Z1*4=Z2 (%the (pair2nat (pair/ X1 Y1) Z1) P2N) Y/2=Y1,Y0 X/2=X1,X0)) + %<- (divrem-total Z/4=Z1,R) + %<- (divrem-implies-gt Z/4=Z1,R FOUR>R) + %<- (divrem-can-be-inverted Z/4=Z1,R Z2 Z1*4=Z2 Z2+R=Z) + %<- (divrem-total R/2=X0,Y0) + %<- (divrem-implies-gt R/2=X0,Y0 TWO>Y0) + %<- (divrem-can-be-inverted R/2=X0,Y0 X2 X0*2=X2 X2+Y0=R) + %<- (plus-associative-converse X2+Y0=R Z2+R=Z Z3 Z2+X2=Z3 Z3+Y0=Z) + %<- (plus-commutative X2+Y0=R Y0+X2=R) + %<- (plus-implies-ge Y0+X2=R R>=X2) + %<- (gt-transitive-ge FOUR>R R>=X2 FOUR>X2) + %<- (times-right-cancels-gt 2*2=4 X0*2=X2 nat`eq/ FOUR>X2 TWO>X0) + %<- (quotient-of-nonzero-is-smaller Z/4=Z1,R nat`eq/ Z>Z1) + %<- (meta-gt _ _ Z>Z1) + %<- (nat2pair-total* Z1 (pair/ X1 Y1) P2N) + %<- (times-total X1*2=XE) + %<- (times-total Y1*2=YE) + %<- (plus-total XE+X0=X) + %<- (plus-total YE+Y0=Y) + %<- (divrem-can-be-constructed X1*2=XE XE+X0=X TWO>X0 X/2=X1,X0) + %<- (divrem-can-be-constructed Y1*2=YE YE+Y0=Y TWO>Y0 Y/2=Y1,Y0) %. +%worlds () (nat2pair-total* _ _ _) %. +%total (Z) (nat2pair-total* Z _ _) %. +% %reduces X <= N (nat2pair-total* N (pair/ X _) _). +% %reduces Y <= N (pair2nat-total* N (pair/ _ Y) _). +%inline nat2pair-total nat2pair-total* _ _ %. +%theorem pair2nat-deterministic : forall* {P1} {P2} {N1} {N2} forall {D1 pair2nat P1 N1} {D2 pair2nat P2 N2} {EP eq P1 P2} exists {EN nat`eq N1 N2} true %. +%term _ pair2nat-deterministic pair2nat/00 pair2nat/00 eq/ nat`eq/ %. +% lemma +%theorem pair2nat-deterministic/00 : forall* {Z} forall {P2N pair2nat (pair/ z z) Z} exists {E nat`eq Z z} true %. +%term _ pair2nat-deterministic/00 pair2nat/00 nat`eq/ %. +%term _ + %pi (pair2nat-deterministic/00 (pair2nat/XX Z3+Y0=Z Z2+X2=Z3 X0*2=X2 Z1*4=Z2 (%the (pair2nat (pair/ X1 Y1) Z1) P2N) ZERO/2=Y1,Y0 ZERO/2=X1,X0) Z=0) + %<- (divrem-deterministic ZERO/2=X1,X0 (divrem/z 2>0) nat`eq/ nat`eq/ X1=0 X0=0) + %<- (divrem-deterministic ZERO/2=Y1,Y0 (divrem/z 2>0) nat`eq/ nat`eq/ Y1=0 Y0=0) + %<- (pair-preserves-eq X1=0 Y1=0 X1,Y1=0,0) + %<- (pair2nat-respects-eq P2N X1,Y1=0,0 nat`eq/ ZERO,ZERO->Z1) + %<- (pair2nat-deterministic/00 ZERO,ZERO->Z1 Z1=0) + %<- (times-deterministic Z1*4=Z2 times/z Z1=0 nat`eq/ Z2=0) + %<- (times-deterministic X0*2=X2 times/z X0=0 nat`eq/ X2=0) + %<- (plus-deterministic Z2+X2=Z3 plus/z Z2=0 X2=0 Z3=0) + %<- (plus-deterministic Z3+Y0=Z plus/z Z3=0 Y0=0 Z=0) %. +%worlds () (pair2nat-deterministic/00 _ _) %. +%total (D) (pair2nat-deterministic/00 D _) %. +%term _ + %pi (pair2nat-deterministic pair2nat/00 P2N eq/ ZERO=Z) + %<- (pair2nat-deterministic/00 P2N Z=0) + %<- (nat`eq-symmetric Z=0 ZERO=Z) %. +%term _ + %pi (pair2nat-deterministic P2N pair2nat/00 eq/ Z=0) + %<- (pair2nat-deterministic/00 P2N Z=0) %. +%term _ + %pi (pair2nat-deterministic (pair2nat/XX Z3+Y0=Z Z2+X2=Z3 X0*2=X2 Z1*4=Z2 (%the (pair2nat (pair/ X1 Y1) Z1) P2N) Y/2=Y1,Y0 X/2=X1,X0) (pair2nat/XX Z3'+Y0'=Z' Z2'+X2'=Z3' X0'*2=X2' Z1'*4=Z2' (%the (pair2nat (pair/ X1' Y1') Z1') P2N') Y/2=Y1',Y0' X/2=X1',X0') eq/ Z=Z') + %<- (divrem-deterministic X/2=X1,X0 X/2=X1',X0' nat`eq/ nat`eq/ X1=X1' X0=X0') + %<- (divrem-deterministic Y/2=Y1,Y0 Y/2=Y1',Y0' nat`eq/ nat`eq/ Y1=Y1' Y0=Y0') + %<- (pair-preserves-eq X1=X1' Y1=Y1' X1,Y1=X1',Y1') + %<- (pair2nat-deterministic P2N P2N' X1,Y1=X1',Y1' Z1=Z1') + %<- (times-deterministic Z1*4=Z2 Z1'*4=Z2' Z1=Z1' nat`eq/ Z2=Z2') + %<- (times-deterministic X0*2=X2 X0'*2=X2' X0=X0' nat`eq/ X2=X2') + %<- (plus-deterministic Z2+X2=Z3 Z2'+X2'=Z3' Z2=Z2' X2=X2' Z3=Z3') + %<- (plus-deterministic Z3+Y0=Z Z3'+Y0'=Z' Z3=Z3' Y0=Y0' Z=Z') %. +%worlds () (pair2nat-deterministic _ _ _ _) %. +%total (D) (pair2nat-deterministic D _ _ _) %. +%theorem nat2pair-deterministic : forall* {P1} {P2} {N1} {N2} forall {D1 nat2pair N1 P1} {D2 nat2pair N2 P2} {EN nat`eq N1 N2} exists {EP eq P1 P2} true %. +% lemma +%theorem nat2pair-deterministic/0 : forall* {P} forall {D nat2pair z P} exists {EP eq P (pair/ z z)} true %. +%term _ nat2pair-deterministic/0 pair2nat/00 eq/ %. +%term _ + %pi (nat2pair-deterministic/0 (pair2nat/XX Z3+Y0=0 Z2+X2=Z3 X0*2=X2 Z1*4=Z2 (%the (pair2nat (pair/ X1 Y1) Z1) P2N) Y/2=Y1,Y0 X/2=X1,X0) X,Y=0,0) + %<- (plus-is-zero-implies-zero Z3+Y0=0 nat`eq/ Z3=0 Y0=0) + %<- (plus-is-zero-implies-zero Z2+X2=Z3 Z3=0 Z2=0 X2=0) + %<- (times-right-cancels X0*2=X2 times/z nat`eq/ X2=0 X0=0) + %<- (times-right-cancels Z1*4=Z2 times/z nat`eq/ Z2=0 Z1=0) + %<- (nat2pair-respects-eq P2N Z1=0 eq/ N2P) + %<- (nat2pair-deterministic/0 N2P X1,Y1=0,0) + %<- (pair-eq-implies-eq X1,Y1=0,0 X1=0 Y1=0) + %<- (divrem-can-be-inverted X/2=X1,X0 XM X1*2=XM XM+X0=X) + %<- (divrem-can-be-inverted Y/2=Y1,Y0 YM Y1*2=YM YM+Y0=Y) + %<- (times-deterministic X1*2=XM times/z X1=0 nat`eq/ XM=0) + %<- (times-deterministic Y1*2=YM times/z Y1=0 nat`eq/ YM=0) + %<- (plus-deterministic XM+X0=X plus/z XM=0 X0=0 X=0) + %<- (plus-deterministic YM+Y0=Y plus/z YM=0 Y0=0 Y=0) + %<- (pair-preserves-eq X=0 Y=0 X,Y=0,0) %. +%worlds () (nat2pair-deterministic/0 _ _) %. +%total (D) (nat2pair-deterministic/0 D _) %. +%term _ + %pi (nat2pair-deterministic N2P N2P' nat`eq/ X,Y=X',Y') + %<- (nat2pair-deterministic/0 N2P X,Y=0,0) + %<- (nat2pair-deterministic/0 N2P' X',Y'=0,0) + %<- (eq-symmetric X',Y'=0,0 ZERO,ZERO=X',Y') + %<- (eq-transitive X,Y=0,0 ZERO,ZERO=X',Y' X,Y=X',Y') %. +%term _ + %pi (nat2pair-deterministic (pair2nat/XX Z3+Y0=Z Z2+X2=Z3 X0*2=X2 Z1*4=Z2 (%the (pair2nat (pair/ X1 Y1) Z1) P2N) Y/2=Y1,Y0 X/2=X1,X0) (pair2nat/XX Z3'+Y0'=Z Z2'+X2'=Z3' X0'*2=X2' Z1'*4=Z2' (%the (pair2nat (pair/ X1' Y1') Z1') P2N') Y'/2=Y1',Y0' X'/2=X1',X0') nat`eq/ X,Y=X',Y') + %<- (plus-associative Z2+X2=Z3 Z3+Y0=Z R X2+Y0=R Z2+R=Z) + %<- (divrem-implies-gt X/2=X1,X0 TWO>X0) + %<- (divrem-implies-gt Y/2=Y1,Y0 TWO>Y0) + %<- (succ-gt-implies-ge TWO>X0 ONE>=X0) + %<- (succ-gt-implies-ge TWO>Y0 ONE>=Y0) + %<- (times-right-preserves-ge* ONE>=X0 1*2=2 X0*2=X2 TWO>=X2) + %<- (plus-preserves-ge* TWO>=X2 ONE>=Y0 2+1=3 X2+Y0=R THREE>=R) + %<- (ge-implies-succ-gt THREE>=R FOUR>R) + %<- (divrem-can-be-constructed Z1*4=Z2 Z2+R=Z FOUR>R Z/4=Z1,R) + %<- (plus-associative Z2'+X2'=Z3' Z3'+Y0'=Z R' X2'+Y0'=R' Z2'+R'=Z) + %<- (divrem-implies-gt X'/2=X1',X0' TWO>X0') + %<- (divrem-implies-gt Y'/2=Y1',Y0' TWO>Y0') + %<- (succ-gt-implies-ge TWO>X0' ONE>=X0') + %<- (succ-gt-implies-ge TWO>Y0' ONE>=Y0') + %<- (times-right-preserves-ge* ONE>=X0' 1*2=2 X0'*2=X2' TWO>=X2') + %<- (plus-preserves-ge* TWO>=X2' ONE>=Y0' 2+1=3 X2'+Y0'=R' THREE>=R') + %<- (ge-implies-succ-gt THREE>=R' FOUR>R') + %<- (divrem-can-be-constructed Z1'*4=Z2' Z2'+R'=Z FOUR>R' Z/4=Z1',R') + %<- (divrem-deterministic Z/4=Z1,R Z/4=Z1',R' nat`eq/ nat`eq/ Z1=Z1' R=R') + %<- (divrem-can-be-constructed X0*2=X2 X2+Y0=R TWO>Y0 R/2=X0,Y0) + %<- (divrem-can-be-constructed X0'*2=X2' X2'+Y0'=R' TWO>Y0' R'/2=X0',Y0') + %<- (divrem-deterministic R/2=X0,Y0 R'/2=X0',Y0' R=R' nat`eq/ X0=X0' Y0=Y0') + %<- (nat2pair-deterministic P2N P2N' Z1=Z1' X1,Y1=X1',Y1') + %<- (pair-eq-implies-eq X1,Y1=X1',Y1' X1=X1' Y1=Y1') + %<- (divrem-can-be-inverted X/2=X1,X0 XM X1*2=XM XM+X0=X) + %<- (divrem-can-be-inverted Y/2=Y1,Y0 YM Y1*2=YM YM+Y0=Y) + %<- (divrem-can-be-inverted X'/2=X1',X0' XM' X1'*2=XM' XM'+X0'=X') + %<- (divrem-can-be-inverted Y'/2=Y1',Y0' YM' Y1'*2=YM' YM'+Y0'=Y') + %<- (times-deterministic X1*2=XM X1'*2=XM' X1=X1' nat`eq/ XM=XM') + %<- (plus-deterministic XM+X0=X XM'+X0'=X' XM=XM' X0=X0' X=X') + %<- (times-deterministic Y1*2=YM Y1'*2=YM' Y1=Y1' nat`eq/ YM=YM') + %<- (plus-deterministic YM+Y0=Y YM'+Y0'=Y' YM=YM' Y0=Y0' Y=Y') + %<- (pair-preserves-eq X=X' Y=Y' X,Y=X',Y') %. +%worlds () (nat2pair-deterministic _ _ _ _) %. +%total (D) (nat2pair-deterministic D _ _ _) %. +%theorem pair2nat-preserves-ne* : forall* {P1} {P2} {N1} {N2} forall {PNE ne P1 P2} {T1 pair2nat P1 N1} {T2 pair2nat P2 N2} exists {NE nat`ne N1 N2} true %. +%theorem pair2nat-preserves-ne*/L : forall* {P1} {P2} {N1} {N2} {B} forall {PNE ne P1 P2} {T1 pair2nat P1 N1} {T2 pair2nat P2 N2} {NT nat`eq? N1 N2 B} exists {NE nat`ne N1 N2} true %. +%term _ + %pi (pair2nat-preserves-ne* P1<>P2 P1->N1 P2->N2 N1<>N2) + %<- (nat`eq?-total EQ?) + %<- (pair2nat-preserves-ne*/L P1<>P2 P1->N1 P2->N2 EQ? N1<>N2) %. +%term _ pair2nat-preserves-ne*/L _ _ _ (nat`eq?/no N1<>N2) N1<>N2 %. +%term _ + %pi (pair2nat-preserves-ne*/L P1<>P2 P1->N P2->N nat`eq?/yes N<>N) + %<- (nat2pair-deterministic P1->N P2->N nat`eq/ P1=P2) + %<- (eq-ne-implies-false P1=P2 P1<>P2 F) + %<- (nat`false-implies-ne F N<>N) %. +%worlds () (pair2nat-preserves-ne*/L _ _ _ _ _) %. +%total {} (pair2nat-preserves-ne*/L _ _ _ _ _) %. +%worlds () (pair2nat-preserves-ne* _ _ _ _) %. +%total {} (pair2nat-preserves-ne* _ _ _ _) %. +%theorem pair2nat-preserves-ne : forall* {P1} {P2} forall {PNE ne P1 P2} exists {N1} {N2} {T1 pair2nat P1 N1} {T2 pair2nat P2 N2} {NE nat`ne N1 N2} true %. +%term _ + %pi (pair2nat-preserves-ne P1<>P2 N1 N2 T1 T2 N1<>N2) + %<- (pair2nat-total T1) + %<- (pair2nat-total T2) + %<- (pair2nat-preserves-ne* P1<>P2 T1 T2 N1<>N2) %. +%worlds () (pair2nat-preserves-ne _ _ _ _ _ _) %. +%total {} (pair2nat-preserves-ne _ _ _ _ _ _) %. +%theorem nat2pair-preserves-ne* : forall* {P1} {P2} {N1} {N2} forall {NE nat`ne N1 N2} {T1 nat2pair N1 P1} {T2 nat2pair N2 P2} exists {PNE ne P1 P2} true %. +%theorem nat2pair-preserves-ne*/L : forall* {P1} {P2} {N1} {N2} {B} forall {NE nat`ne N1 N2} {T1 nat2pair N1 P1} {T2 nat2pair N2 P2} {PT eq? P1 P2 B} exists {PNE ne P1 P2} true %. +%term _ + %pi (nat2pair-preserves-ne* N1<>N2 N1->P1 N2->P2 P1<>P2) + %<- (eq?-total EP?) + %<- (nat2pair-preserves-ne*/L N1<>N2 N1->P1 N2->P2 EP? P1<>P2) %. +%term _ nat2pair-preserves-ne*/L _ _ _ (eq?/no P1<>P2) P1<>P2 %. +%term _ + %pi (nat2pair-preserves-ne*/L N1<>N2 N1->P N2->P eq?/yes P<>P) + %<- (pair2nat-deterministic N1->P N2->P eq/ N1=N2) + %<- (nat`eq-ne-implies-false N1=N2 N1<>N2 F) + %<- (false-implies-ne F P<>P) %. +%worlds () (nat2pair-preserves-ne*/L _ _ _ _ _) %. +%total {} (nat2pair-preserves-ne*/L _ _ _ _ _) %. +%worlds () (nat2pair-preserves-ne* _ _ _ _) %. +%total {} (nat2pair-preserves-ne* _ _ _ _) %. +%theorem nat2pair-preserves-ne : forall* {N1} {N2} forall {NNE nat`ne N1 N2} exists {P1} {P2} {T1 nat2pair N1 P1} {T2 nat2pair N2 P2} {PE ne P1 P2} true %. +%term _ + %pi (nat2pair-preserves-ne N1<>N2 P1 P2 T1 T2 P1<>P2) + %<- (nat2pair-total T1) + %<- (nat2pair-total T2) + %<- (nat2pair-preserves-ne* N1<>N2 T1 T2 P1<>P2) %. +%worlds () (nat2pair-preserves-ne _ _ _ _ _ _) %. +%total {} (nat2pair-preserves-ne _ _ _ _ _ _) %. +%theorem nonzero-nat2pair-implies-gt-ge : forall* {N} {X} {Y} forall {D nat2pair (s N) (pair/ X Y)} exists {G1 gt (s N) X} {G2 ge (s N) Y} true %. +%term _ + %pi (nonzero-nat2pair-implies-gt-ge (pair2nat/XX plus/z plus/z times/z times/z _ (divrem/z _) (divrem/z _)) N+1>0 (nat`ge/= nat`eq/)) + %<- (succ-implies-gt-zero _ N+1>0) %. +%term _ + %pi (nonzero-nat2pair-implies-gt-ge (pair2nat/XX X2+Y=sN plus/z (times/s X-1*2=X2-2 X2-2+2=X2) times/z _ (divrem/z _) (divrem/z TWO>X)) N+1>X N+1>=Y) + %<- (plus-implies-ge X2+Y=sN (%the (ge (s N) Y) N+1>=Y)) + %<- (succ-gt-implies-ge TWO>X ONE>=X) + %<- (ge-succ-implies-gt ONE>=X ONE>X-1) + %<- (succ-gt-implies-ge ONE>X-1 ZERO>=X-1) + %<- (ge-zero-always _ X-1>=0) + %<- (ge-anti-symmetric ZERO>=X-1 X-1>=0 ZERO=X-1) + %<- (succ-deterministic ZERO=X-1 ONE=X) + %<- (times-deterministic times/z X-1*2=X2-2 ZERO=X-1 nat`eq/ ZERO=X2-2) + %<- (plus-deterministic plus/z X2-2+2=X2 ZERO=X2-2 nat`eq/ TWO=X2) + %<- (gt-respects-eq gt/1 TWO=X2 ONE=X X2>X) + %<- (plus-commutative X2+Y=sN Y+X2=sN) + %<- (plus-implies-ge Y+X2=sN N+1>=X2) + %<- (ge-transitive-gt N+1>=X2 X2>X N+1>X) %. +%term _ + %pi (nonzero-nat2pair-implies-gt-ge (pair2nat/XX _ _ _ _ (%the (pair2nat (pair/ (s _) _) z) P2z) _ _) GT GE) + %<- (nat2pair-deterministic pair2nat/00 P2z nat`eq/ ZERO,0=sN,_) + %<- (pair-eq-implies-eq ZERO,0=sN,_ ZERO=sN _) + %<- (nat`eq-contradiction ZERO=sN F) + %<- (nat`false-implies-gt F GT) + %<- (nat`false-implies-ge F GE) %. +%term _ + %pi (nonzero-nat2pair-implies-gt-ge (pair2nat/XX _ _ _ _ (%the (pair2nat (pair/ _ (s _)) z) P2z) _ _) GT GE) + %<- (nat2pair-deterministic pair2nat/00 P2z nat`eq/ ZERO,0=_,sN) + %<- (pair-eq-implies-eq ZERO,0=_,sN _ ZERO=sN) + %<- (nat`eq-contradiction ZERO=sN F) + %<- (nat`false-implies-gt F GT) + %<- (nat`false-implies-ge F GE) %. +%term _ + %pi (nonzero-nat2pair-implies-gt-ge (pair2nat/XX Z3+Y0=sN Z2+X2=Z3 X0*2=X2 Z1*4=Z2 Z1->X1,Y1 Y/2=Y1,Y0 X/2=X1,X0) N+1>X (ge/> N+1>Y)) + %<- (plus-commutative Z3+Y0=sN Y0+Z3=sN) + %<- (plus-implies-ge Y0+Z3=sN N+1>=Z3) + %<- (plus-commutative Z2+X2=Z3 X2+Z2=Z3) + %<- (plus-implies-ge X2+Z2=Z3 Z3>=Z2) + %<- (ge-transitive N+1>=Z3 Z3>=Z2 N+1>=Z2) + %<- (nonzero-nat2pair-implies-gt-ge Z1->X1,Y1 Z1>X1 Z1>=Y1) + %<- (times-associative-converse 2*2=4 Z1*4=Z2 Z2/2 Z1*2=Z2/2 Z2/2*2=Z2) + %<- (divrem-can-be-inverted Y/2=Y1,Y0 Y12 Y1*2=Y12 Y12+Y0=Y) + %<- (divrem-can-be-inverted X/2=X1,X0 X12 X1*2=X12 X12+X0=X) + %<- (times-right-preserves-gt* Z1>X1 Z1*2=Z2/2 X1*2=X12 nat`eq/ Z2/2>X12) + %<- (succ-implies-gt-zero _ Z1>0) + %<- (gt-implies-ge-succ Z1>0 Z1>=1) + %<- (times-right-preserves-ge* Z1>=1 Z1*2=Z2/2 1*2=2 Z2/2>=2) + %<- (ge-implies-plus Z2/2>=2 ZZ ZZ+2=Z2/2) + %<- (plus-commutative ZZ+2=Z2/2 TWO+ZZ=Z2/2) + %<- (plus-deterministic TWO+ZZ=Z2/2 (plus/s (plus/s plus/z)) nat`eq/ nat`eq/ Z2/2=ssZZ) + %<- (times-respects-eq Z2/2*2=Z2 Z2/2=ssZZ nat`eq/ nat`eq/ SSZZ*2=Z2) + %<- (non-trivial-times-implies-much-gt* SSZZ*2=Z2 Z2>sssZZ) + %<- (divrem-implies-gt X/2=X1,X0 TWO>X0) + %<- (succ-gt-implies-ge TWO>X0 ONE>=X0) + %<- (plus-right-identity _ X12+0=X12) + %<- (plus-right-increase X12+0=X12 X12+1=sX12) + %<- (plus-left-preserves-ge* ONE>=X0 X12+1=sX12 X12+X0=X SX12>=X) + %<- (gt-implies-ge-succ Z2/2>X12 Z2/2>=sX12) + %<- (ge-transitive Z2/2>=sX12 SX12>=X Z2/2>=X) + %<- (ge-respects-eq Z2/2>=X Z2/2=ssZZ nat`eq/ SSZZ>=X) + %<- (ge-implies-succ-gt SSZZ>=X SSSZZ>X) + %<- (gt-transitive Z2>sssZZ SSSZZ>X Z2>X) + %<- (ge-transitive-gt N+1>=Z2 Z2>X N+1>X) + %<- (times-right-preserves-ge* Z1>=Y1 Z1*2=Z2/2 Y1*2=Y12 Z2/2>=Y12) + %<- (ge-respects-eq Z2/2>=Y12 Z2/2=ssZZ nat`eq/ SSZZ>=Y12) + %<- (divrem-implies-gt Y/2=Y1,Y0 TWO>Y0) + %<- (succ-gt-implies-ge TWO>Y0 ONE>=Y0) + %<- (plus-right-identity _ Y12+0=Y12) + %<- (plus-right-increase Y12+0=Y12 Y12+1=sY12) + %<- (plus-left-preserves-ge* ONE>=Y0 Y12+1=sY12 Y12+Y0=Y SY12>=Y) + %<- (succ-preserves-ge SSZZ>=Y12 SSSZZ>=SY12) + %<- (ge-transitive SSSZZ>=SY12 SY12>=Y SSSZZ>=Y) + %<- (gt-transitive-ge Z2>sssZZ SSSZZ>=Y Z2>Y) + %<- (ge-transitive-gt N+1>=Z2 Z2>Y N+1>Y) %. +%worlds () (nonzero-nat2pair-implies-gt-ge _ _ _) %. +%total (N) (nonzero-nat2pair-implies-gt-ge N _ _) %. +%theorem nat2pair-implies-ge : forall* {N} {X} {Y} forall {D nat2pair N (pair/ X Y)} exists {G1 ge N X} {G2 ge N Y} true %. +%term _ + %pi (nat2pair-implies-ge N2P (ge/> N>X) N>=Y) + %<- (nonzero-nat2pair-implies-gt-ge N2P N>X N>=Y) %. +%term _ + %pi (nat2pair-implies-ge Z2P (ge/= ZERO=X) (ge/= ZERO=Y)) + %<- (nat2pair-deterministic pair2nat/00 Z2P nat`eq/ ZERO,ZERO=X,Y) + %<- (pair-eq-implies-eq ZERO,ZERO=X,Y ZERO=X ZERO=Y) %. +%worlds () (nat2pair-implies-ge _ _ _) %. +%total {} (nat2pair-implies-ge _ _ _) %. +%theorem constrained1-pair2nat-unbounded : forall {N1} {B} exists {N2} {N} {D pair2nat (pair/ N1 N2) N} {G gt N B} true %. +%term _ + %pi (constrained1-pair2nat-unbounded N1 B (s B) N N1,N2->N N>B) + %<- (pair2nat-total N1,N2->N) + %<- (nat2pair-implies-ge N1,N2->N N>=N1 N>=N2) + %<- (ge-succ-implies-gt N>=N2 N>B) %. +%worlds () (constrained1-pair2nat-unbounded _ _ _ _ _ _) %. +%total {} (constrained1-pair2nat-unbounded _ _ _ _ _ _) %. +%theorem constrained2-pair2nat-unbounded : forall {N2} {B} exists {N1} {N} {D pair2nat (pair/ N1 N2) N} {G gt N B} true %. +%term _ + %pi (constrained2-pair2nat-unbounded N2 B (s B) N N1,N2->N N>B) + %<- (pair2nat-total N1,N2->N) + %<- (nat2pair-implies-ge N1,N2->N N>=N1 N>=N2) + %<- (ge-succ-implies-gt N>=N1 N>B) %. +%worlds () (constrained2-pair2nat-unbounded _ _ _ _ _ _) %. +%total {} (constrained2-pair2nat-unbounded _ _ _ _ _ _) %. +%%%% Renamings +%inline natpair pair %. +%inline natpair/ pair/ %. +%%%% Exports +%inline natpair`pair pair %. +%inline natpair`pair/ pair/ %. +%inline natpair`eq eq %. +%inline natpair`eq/ eq/ %. +%inline natpair`ne ne %. +%inline natpair`ne/1 ne/1 %. +%inline natpair`ne/2 ne/2 %. +%inline natpair`eq? eq? %. +%inline natpair`eq?/yes eq?/yes %. +%inline natpair`eq?/no eq?/no %. +%inline natpair`false-implies-eq false-implies-eq %. +%inline natpair`meta-eq meta-eq %. +%inline natpair`eq-reflexive eq-reflexive %. +%inline natpair`eq-symmetric eq-symmetric %. +%inline natpair`eq-transitive eq-transitive %. +%inline natpair`pair-eq-implies-eq pair-eq-implies-eq %. +%inline natpair`pair-preserves-eq pair-preserves-eq %. +%inline natpair`false-implies-ne false-implies-ne %. +%inline natpair`ne-respects-eq ne-respects-eq %. +%inline natpair`ne-anti-reflexive ne-anti-reflexive %. +%inline natpair`ne-symmetric ne-symmetric %. +%inline natpair`eq-ne-implies-false eq-ne-implies-false %. +%inline natpair`eq?-total* eq?-total* %. +%inline natpair`eq?-total*/L eq?-total*/L %. +%inline natpair`eq?-total eq?-total %. +%inline natpair`pair2nat pair2nat %. +%inline natpair`pair2nat/00 pair2nat/00 %. +%inline natpair`pair2nat/XX pair2nat/XX %. +%inline natpair`nat2pair nat2pair %. +%inline natpair`false-implies-pair2nat false-implies-pair2nat %. +%inline natpair`false-implies-not2pair false-implies-not2pair %. +%inline natpair`pair2nat-respects-eq pair2nat-respects-eq %. +%inline natpair`nat2pair-respects-eq nat2pair-respects-eq %. +%inline natpair`pair2nat-total** pair2nat-total** %. +%inline natpair`pair2nat-total* pair2nat-total* %. +%inline natpair`pair2nat-total pair2nat-total %. +%inline natpair`nat2pair-total* nat2pair-total* %. +%inline natpair`nat2pair-total nat2pair-total %. +%inline natpair`pair2nat-deterministic pair2nat-deterministic %. +%inline natpair`pair2nat-deterministic/00 pair2nat-deterministic/00 %. +%inline natpair`nat2pair-deterministic nat2pair-deterministic %. +%inline natpair`nat2pair-deterministic/0 nat2pair-deterministic/0 %. +%inline natpair`pair2nat-preserves-ne* pair2nat-preserves-ne* %. +%inline natpair`pair2nat-preserves-ne*/L pair2nat-preserves-ne*/L %. +%inline natpair`pair2nat-preserves-ne pair2nat-preserves-ne %. +%inline natpair`nat2pair-preserves-ne* nat2pair-preserves-ne* %. +%inline natpair`nat2pair-preserves-ne*/L nat2pair-preserves-ne*/L %. +%inline natpair`nat2pair-preserves-ne nat2pair-preserves-ne %. +%inline natpair`nonzero-nat2pair-implies-gt-ge nonzero-nat2pair-implies-gt-ge %. +%inline natpair`nat2pair-implies-ge nat2pair-implies-ge %. +%inline natpair`constrained1-pair2nat-unbounded constrained1-pair2nat-unbounded %. +%inline natpair`constrained2-pair2nat-unbounded constrained2-pair2nat-unbounded %. +%inline natpair`natpair natpair %. +%inline natpair`natpair/ natpair/ %. +%{! ## Definitions !}% +%{! ### The syntax !}% +%{! +The HOAS defined here is uninteresting. There isn't even any way to use +more than one variable (although the proofs use techniques that can +handle any number of variables). +!}% +%sort t %. +%term a t %. +%term b %pi t %-> t %. +%term f %pi (%pi t %-> t) %-> t %. +%block blocksimple [v t]%. +%{! ### Equality !}% +%sort eq %. +%term eq/ eq T T %. +%{! ### Variable levels !}% +%{! +A variable level is the (nonzero) natural number for a variable. +This value is used to determine the mapping for a variable. +!}% +%sort varlevel %. +%block blockvar {l} [v] [vl varlevel v (s l)]%. +%{! ### Mapping !}% +%{! +The bijection from t to nat is called "tonat". In its more general +form the relation takes a natural number indicating how deep we are +inside functions. +!}% +%sort tonat* %. +%inline tonat tonat* z %. +%term tonat/v %pi (varlevel V L) %-> (plus M L N) %-> (tonat* N V M) %. +%term tonat/a tonat* N a N %. +%term tonat/b + %pi (tonat* N T M) + %-> (times (s (s z)) M TM) + %-> (plus (s N) TM M') + %-> (tonat* N (b T) M') %. +%term tonat/f + %pi ({v} %pi (varlevel v (s N)) %-> (tonat* (s N) (F v) M)) + %-> (times (s (s z)) M TM) + %-> (plus (s (s N)) TM M') + %-> (tonat* N (f F) M') %. +%{! ### Utility lemmas !}% +%{! +The following theorems prove obvious simple things about the +basic relations. They following the conventions established in +John Boyland's library signatures. +!}% +%theorem false-implies-varlevel : forall* {V} {L} forall {F void} exists {VL varlevel V L} true %. +%worlds (blockvar) (false-implies-varlevel _ _) %. +%total {} (false-implies-varlevel _ _) %. +%theorem varlevel-respects-eq : forall* {V} {L1} {L2} forall {VL1 varlevel V L1} {E nat`eq L1 L2} exists {VL2 varlevel V L2} true %. +%term _ varlevel-respects-eq VL nat`eq/ VL %. +%worlds (blocksimple blockvar) (varlevel-respects-eq _ _ _) %. +%total {} (varlevel-respects-eq _ _ _) %. +%theorem false-implies-tonat : forall* {N} {T} {M} forall {F void} exists {TN tonat* N T M} true %. +%worlds (blockvar) (false-implies-tonat _ _) %. +%total {} (false-implies-tonat _ _) %. +%{! ## Proof of totality of tonat !}% +%{! +The difficulty here is that we need to prove that when +we get to a variable (and exactly how we tell this +in Twelf is tricky because variables can't be captured in +case analysis), we need to ensure that (1) the variable +has a level associated with it and (2) the level is +in the range 1..N where N is the block nesting we are in. +Blocks are useful for (1) but not for (2) because there's +no way to connect the context with the current nesting level. + +Instead we use a technique (I learn from Rob Simmons) to handle +one level of variable a time in a separate lemma. This works since +for each particular HOAS function we know that the variable +is bound legally. + +We package this approach up into an auxiliary relation that incidentally +makes it easy to capture variables in case analysis. This makes +for a wordy series of proofs. Perhaps we can get rid of 'case' +in general. +!}% +%{! ### Auxiliary definitions !}% +%{! #### raw variables !}% +%{! +A variable is raw if we haven't verified that it has a level in range. +Non variables are not raw. +!}% +%sort israw %. +%inline rawvar [T] israw T true %. +%term israw/a israw a false %. +%term israw/b israw (b _) false %. +%term israw/f israw (f _) false %. +%{! #### case analysis !}% +%{! +We case analysis terms with two cases for variables. +The raw case is used only internally and can be ignored in +"clients" that don't use israw. +!}% +%sort case %. +%term case/a case _ a %. +%term case/b %pi (case N T) %-> (case N (b T)) %. +%term case/f %pi ({v} %pi (varlevel v (s N)) %-> (case (s N) (F v))) %-> (case N (f F)) %. +%term case/var %pi (varlevel V L) %-> (nat`ge N L) %-> (case N V) %. +%term case/raw %pi (rawvar V) %-> (case N V) %. +%{! ### Theorems about auxiliary definitions !}% +%{! +What follows first is a theorem that says that in a context where +all variables are raw, we can campute the rawness of all terms. +In a Twelf idiom that will be seen several times in this file +(and which I learned from Rob Simmons), we have to put the theorem +for the the variable case in the same context that defines the variable +as raw. (This is rather annoying: Twelf should be smart enough to +see that the context that defines the variable has the +necessary relation.) This somewhat contorted idiom falls afoul of +autofreezing in Twelf 1.5r3. I consider this a bug in Twelf, +but fortunately it can be worked around by defining a fake +circular dependency. (Again, the idea from Rob Simmons.) +For some reason, unlike Rob's examples and the later instances in +this file, I need to add israw to the fake dependencies as well. +The definition "fake" is never used again. It has no "meaning." +!}% +%theorem israw-total* : forall {T} exists {B} {I israw T B} true %. +%inline israw-total israw-total* _ _ %. +%term _ israw-total israw/a %. +%term _ israw-total israw/b %. +%term _ israw-total israw/f %. +%sort fake %. +%term _ %pi fake %<- ({i israw-total* T B I} israw-total* T' B' I') %. +%term _ %pi fake %<- ({i israw-total* T B I} israw T' B') %. +%block blockraw [v] [rv rawvar v] [irt israw-total rv]%. +%worlds (blockraw) (israw-total* _ _ _) %. +%total {} (israw-total* _ _ _) %. +%{! +The following theorem handles one variable converting it from raw to +handle a level that is in the required range. This is an important +technique for handle variables in Twelf: one at a time. +!}% +%theorem var-gets-level : forall* {N} {T} {L} forall {F {v} {rv rawvar v} {i israw-total rv} case N (T v)} {GE nat`ge N L} exists {F' {v} %pi (varlevel v L) %-> (case N (T v))} true %. +%term _ var-gets-level ([v] [r] [i] case/raw r) N>=L ([v] [vl] case/var vl N>=L) %. +%term _ var-gets-level ([v] [r] [i] case/raw R) _ ([v] [vl] case/raw R) %. +%term _ var-gets-level ([v] [c] [i] case/var VL N>=L) _ ([v] [vl] case/var VL N>=L) %. +%term _ var-gets-level ([v] [c] [i] case/a) _ ([v] [vl] case/a) %. +%term _ + %pi (var-gets-level ([v] [c] [i] case/b (C v c i)) N>=L ([v] [vl] case/b (C' v vl))) + %<- (var-gets-level ([v] [c] [i] C v c i) N>=L ([v] [vl] C' v vl)) %. +%term _ + %pi (var-gets-level ([v] [c] [i] case/f ([v'] [vl'] C v' vl' v c i)) N>=L ([v] [vl] case/f ([v'] [vl'] C' v' vl' v vl))) + %<- (ge-implies-succ-gt N>=L N+1>L) + %<- ({v'} {vl' varlevel v' _} var-gets-level ([v] [c] [i] C v' vl' v c i) (nat`ge/> N+1>L) ([v] [vl] C' v' vl' v vl)) %. +%worlds (blockvar blockraw) (var-gets-level _ _ _) %. +%total F (var-gets-level F _ _) %. +%{! +We are now ready to prove that we can always "case" a term. +This code follows John Boyland's library convention of defining a +"-total" metatheorem as having implicit arguments that are explicit +in a "-total*" version. In the main lemma, the interesting case +is when we have a "f" term: after ensuring that the subterm can be +be tested for rawness, we recurse while the variable is assumed raw. +Once this is done, we convert the variable into one with a level +using "var-gets-level". Note that the var levels don't go into +the context for this theorem. +!}% +%theorem case-total* : forall {T} exists {C case z T} true %. +%inline case-total case-total* _ %. +%theorem case-total/L : forall* {B} forall {N} {T} {I israw T B} exists {C case N T} true %. +%term _ case-total/L _ _ _ case/a %. +%term _ %pi (case-total/L _ _ _ (case/b C)) %<- (israw-total I) %<- (case-total/L _ _ I C) %. +%term _ + %pi (case-total/L _ (f ([v] F v)) israw/f (case/f ([v] [vl] C' v vl))) + %<- ({v} {r rawvar v} {i israw-total r} israw-total (I v r i)) + %<- ({v} {r rawvar v} {i israw-total r} case-total/L _ (F v) (%the (israw (F v) B) (I v r i)) (C v r i)) + %<- (var-gets-level C (nat`ge/= nat`eq/) C') %. +%term _ case-total/L _ V R (case/raw R) %. +%worlds (blockraw) (case-total/L _ _ _ _) %. +%total T (case-total/L _ T _ _) %. +%term _ %pi (case-total* T C) %<- (israw-total I) %<- (case-total/L z T I C) %. +%worlds () (case-total* _ _) %. +%total {} (case-total* _ _) %. +%{! ### Main theorem !}% +%{! +We are ready now to prove totality of the relation. +We case the term first and then have everything we need +to push through totality. +!}% +%theorem tonat-total* : forall {T t} exists {M nat} {D tonat T M} true %. +%inline tonat-total tonat-total* _ _ %. +%theorem tonat-total/L : forall {N nat} {T t} {C case N T} exists {M nat} {D tonat* N T M} true %. +%term _ + %pi (tonat-total/L _ _ (case/var VL GE) _ (tonat/v VL P)) + %<- (ge-implies-plus GE _ P) %. +%term _ tonat-total/L _ _ case/a _ tonat/a %. +%term _ + %pi (tonat-total/L _ _ (case/b C) _ (tonat/b TN T P)) + %<- (tonat-total/L _ _ C _ TN) + %<- (times-total T) + %<- (plus-total P) %. +%term _ + %pi (tonat-total/L _ _ (case/f ([v] [vl] C v vl)) _ (tonat/f ([v] [vl] TN v vl) T P)) + %<- ({v} {vl varlevel v (s N)} tonat-total/L _ _ (C v vl) _ (TN v vl)) + %<- (times-total T) + %<- (plus-total P) %. +%worlds (blockvar) (tonat-total/L _ _ _ _ _) %. +%total (C) (tonat-total/L _ _ C _ _) %. +%term _ %pi (tonat-total TN) %<- (case-total C) %<- (tonat-total/L _ _ C _ TN) %. +%worlds () (tonat-total* _ _ _) %. +%total {} (tonat-total* _ _ _) %. +%{! ## Proof of the determinicity of the mapping !}% +%{! +In this section, we prove that tonat gives only one value +(hence it is a function). This sort of theorem is called +a "uniqueness" theorem. Here the name I use for it comes from +Twelf's "%deterministic" declaration. (I find the term "unique" +might refer to the "one2one" aspect, proved later.) + +This aspect is much easier to prove that any of the others. +That probably reflects the fact that the relation was written +in a functional style. +!}% +%{! ### Auxiliary theorems !}% +%{! +We prove that variable levels are "unique" and that they are never zero. +The proofs are trivial: Twelf can accept them from the context alone. +!}% +%theorem varlevel-deterministic : forall* {V} {L1} {L2} forall {VL1 varlevel V L1} {VL2 varlevel V L2} exists {E nat`eq L1 L2} true %. +%term _ varlevel-deterministic _ _ nat`eq/ %. +%worlds (blockvar) (varlevel-deterministic _ _ _) %. +%total {} (varlevel-deterministic _ _ _) %. +%theorem varlevel-contradiction : forall* {V} {L} forall {VL varlevel V L} {E nat`eq L z} exists {F void} true %. +%worlds (blockvar) (varlevel-contradiction _ _ _) %. +%total {} (varlevel-contradiction _ _ _) %. +%{! ### Main Theorem !}% +%theorem tonat-deterministic : forall* {T1} {T2} {N1} {N2} forall {TN1 tonat T1 N1} {TN2 tonat T2 N2} {E eq T1 T2} exists {E nat`eq N1 N2} true %. +%theorem tonat-deterministic/L : forall* {T} {N} {N1} {N2} forall {TN1 tonat* N T N1} {TN2 tonat* N T N2} exists {E nat`eq N1 N2} true %. +%term _ + %pi (tonat-deterministic/L (tonat/v VL1 P1) (tonat/v VL2 P2) N1=N2) + %<- (varlevel-deterministic VL1 VL2 L1=L2) + %<- (plus-right-cancels P1 P2 L1=L2 nat`eq/ N1=N2) %. +%term _ tonat-deterministic/L tonat/a tonat/a nat`eq/ %. +%term _ + %pi (tonat-deterministic/L (tonat/b TN1 T1 P1) (tonat/b TN2 T2 P2) M1'=M2') + %<- (tonat-deterministic/L TN1 TN2 M1=M2) + %<- (times-deterministic T1 T2 nat`eq/ M1=M2 TM1=TM2) + %<- (plus-deterministic P1 P2 nat`eq/ TM1=TM2 M1'=M2') %. +%term _ + %pi (tonat-deterministic/L (tonat/f ([v] [vl] TN1 v vl) T1 P1) (tonat/f ([v] [vl] TN2 v vl) T2 P2) M1'=M2') + %<- ({v} {vl varlevel v (s N)} tonat-deterministic/L (TN1 v vl) (TN2 v vl) M1=M2) + %<- (times-deterministic T1 T2 nat`eq/ M1=M2 TM1=TM2) + %<- (plus-deterministic P1 P2 nat`eq/ TM1=TM2 M1'=M2') %. +%worlds (blockvar) (tonat-deterministic/L _ _ _) %. +%total (T) (tonat-deterministic/L T _ _) %. +%term _ + %pi (tonat-deterministic TN1 TN2 eq/ N1=N2) + %<- (tonat-deterministic/L TN1 TN2 N1=N2) %. +%worlds () (tonat-deterministic _ _ _ _) %. +%total {} (tonat-deterministic _ _ _ _) %. +%{! ## Proving that the mapping is onto. !}% +%{! +Here we use the mathematical term "onto": a function is "onto" if +its range is equal to its co-domain: that is if every value in the +co-domain has a value in the domain that maps to it. + +The tricky aspect here is that we need to show that every level that the +reverse mapping has a variable associated with it. We do this using a +helper relation, as opposed to putting something in the context, since +as explained earlier, the context is useless to connect variables with +the nesting level. +!}% +%{! ### Auxiliary definitions !}% +%{! +We define a relation that builds on the context relation. +I find this rather interesting because it uses the context relation but +is not itself in the context. This is rather rare in my limited +experience. + +upto N says that we have a variable for all levels 1..N. +!}% +%sort upto %. +%term upto/z upto z %. +%term upto/s %pi (upto N) %-> (varlevel V (s N)) %-> (upto (s N)) %. +%{! ### Theorems about auxiliary definitions !}% +%{! #### The obvious lemma that makes use of the main purpose of the relation: !}% +%theorem upto-implies-varlevel : forall* {N} {L} forall {U upto N} {LT nat`gt N L} exists {V} {VL varlevel V (s L)} true %. +%term _ + %pi (upto-implies-varlevel upto/z ZERO>L a VL) + %<- (nat`gt-contradiction ZERO>L F) + %<- (false-implies-varlevel F VL) %. +%term _ upto-implies-varlevel (upto/s _ VL) gt/1 _ VL %. +%term _ + %pi (upto-implies-varlevel (upto/s U _) (gt/> G) _ VL) + %<- (upto-implies-varlevel U G _ VL) %. +%worlds (blockvar) (upto-implies-varlevel _ _ _ _) %. +%total (U) (upto-implies-varlevel U _ _ _) %. +%{! ### Main theorem !}% +%{! +We prove the main result using two lemmas that do the case +analysis on the number against the nesting level and the +parity. (In general, one would use a divisor counting all +cases that are recursive.) The proofs are long but simply +arithmetic manipulation. Proving termination uses meta-gt +for strong induction over the natural numbers. +!}% +%theorem tonat-onto* : forall {N nat} exists {T} {TN tonat T N} true %. +%inline tonat-onto tonat-onto* _ _ %. +%theorem tonat-onto/L1 : forall* {C} forall {N nat} {U upto N} {M nat} {CMP nat`compare N M C} exists {T} {TN tonat* N T M} true %. +%theorem tonat-onto/L2 : forall* {TM} forall {N nat} {U upto N} {M nat} {P plus N (s TM) M} {Q} {R} {DR divrem TM (s (s z)) Q R} exists {T} {TN tonat* N T M} true %. +%term _ + %pi (tonat-onto/L1 N U M (compare/> N>M) _ (tonat/v VL M+L+1=N)) + %<- (nat`gt-implies-plus N>M L L+1+M=N) + %<- (plus-commutative L+1+M=N M+L+1=N) + %<- (plus-implies-ge M+L+1=N N>=L+1) + %<- (ge-succ-implies-gt N>=L+1 N>L) + %<- (upto-implies-varlevel U N>L _ VL) %. +%term _ tonat-onto/L1 N U _ compare/= a tonat/a %. +%term _ + %pi (tonat-onto/L1 N U M (compare/< M>N) _ TN) + %<- (gt-implies-plus M>N TM TM+1+N=M) + %<- (plus-commutative TM+1+N=M N+TM+1=M) + %<- (divrem-total DR) + %<- (tonat-onto/L2 N U M N+TM+1=M _ _ DR _ TN) %. +%term _ + %pi (tonat-onto/L2 _ U M' N+TM+1=M' M z TM/2=M _ (tonat/b TN TWO*M=TM N+1+TM=M')) + %<- (div-can-be-inverted TM/2=M M*2=TM) + %<- (times-commutative M*2=TM TWO*M=TM) + %<- (plus-swap-succ-converse N+TM+1=M' N+1+TM=M') + %<- (plus-implies-gt N+1+TM=M' nat`eq/ M'>TM) + %<- (times-nonzero-implies-ge M*2=TM TM>=M) + %<- (nat`gt-transitive-ge M'>TM TM>=M M'>M) + %<- (meta-gt _ _ M'>M) + %<- (compare-total CMP) + %<- (tonat-onto/L1 _ U M CMP _ TN) %. +%term _ + %pi (tonat-onto/L2 N U M' (%the (plus N (s TM') M') N+TM'+1=M') M (s z) TM'/2=M,1 (f ([v] F v)) (tonat/f ([v] [vl] TN v vl) TWO*M=TM N+2+TM=M')) + %<- (divrem-can-be-inverted TM'/2=M,1 TM M*2=TM TM+ONE=TM') + %<- (times-commutative M*2=TM TWO*M=TM) + %<- (plus-commutative (plus/s plus/z) TM+ONE=TM+1) + %<- (plus-deterministic TM+ONE=TM' TM+ONE=TM+1 nat`eq/ nat`eq/ TM'=TM+1) + %<- (succ-deterministic TM'=TM+1 (%the (nat`eq (s TM') (s (s TM))) TM'+1=TM+2)) + %<- (plus-respects-eq N+TM'+1=M' nat`eq/ TM'+1=TM+2 nat`eq/ N+TM+2=M') + %<- (plus-swap-succ-converse N+TM+2=M' N+1+TM+1=M') + %<- (plus-swap-succ-converse N+1+TM+1=M' N+2+TM=M') + %<- (plus-implies-ge N+TM'+1=M' M'>=TM'+1) + %<- (ge-succ-implies-gt M'>=TM'+1 M'>TM') + %<- (quotient-of-nonzero-is-smaller TM'/2=M,1 TM'=TM+1 TM'>M) + %<- (nat`gt-transitive M'>TM' TM'>M M'>M) + %<- (meta-gt _ _ M'>M) + %<- (nat`compare-total CMP) + %<- ({v} {vl varlevel v (s N)} tonat-onto/L1 (s N) (upto/s U vl) M CMP (F v) (TN v vl)) %. +%term _ + %pi (tonat-onto/L2 _ _ _ _ _ (s (s _)) DR a TN) + %<- (divrem-implies-gt DR TWO>R+2) + %<- (succ-preserves-gt-converse TWO>R+2 ONE>R+1) + %<- (succ-preserves-gt-converse ONE>R+1 ZERO>R) + %<- (gt-contradiction ZERO>R F) + %<- (false-implies-tonat F TN) %. +%worlds (blockvar) (tonat-onto/L1 _ _ _ _ _ _) (tonat-onto/L2 _ _ _ _ _ _ _ _ _) %. +%total (M1 M2) (tonat-onto/L2 _ _ M2 _ _ _ _ _ _) (tonat-onto/L1 _ _ M1 _ _ _) %. +%{! ## Proof that mapping is "one to one" !}% +%{! +This is the most involved proof. The sketch is that we first prove +that two terms that reduce to the same natural number are "eql" in a way +that only requires that the variables have the same level, but not that +they are the same. Of course, there is only one variable for each level, +which means that the terms are truly identical. But this is impossible +to express in the context. Instead, we chip away at the variables from +the "outside", each time reducing the level of the remaining variables. +!}% +%{! ### Auxiliary definitions !}% +%{! #### Equality (permitting variables with the same level). !}% +%sort eql* %. +%inline eql eql* z %. +%term eql/eq %pi (eq T1 T2) %-> (eql* N T1 T2) %. +%term eql/b %pi (eql* N T1 T2) %-> (eql* N (b T1) (b T2)) %. +%term eql/f1 %pi ({v t} eql* N (F1 v) (F2 v)) %-> (eql* N (f F1) (f F2)) %. +%term eql/f2 + %pi ({v t} {vl varlevel v (s N)} eql* (s N) (F1 v) (F2 v)) + %-> (eql* N (f F1) (f F2)) %. +%term eql/v %pi (varlevel V1 L) %-> (varlevel V2 L) %-> (eql* N V1 V2) %. +%{! #### Measure of eql sizes. !}% +%{! +We use this measure to be able to prove termination. +We need eqlsize/v = eqlsize/eq, eqlsize/f1 = eqlsize/f2. +(Less than is ok in each case but would require that we +rephrase the lemmas.) +!}% +%sort eqlsize %. +%term eqlsize/eq eqlsize (eql/eq _) z %. +%term eqlsize/b %pi (eqlsize E N) %-> (eqlsize (eql/b E) (s N)) %. +%term eqlsize/f1 %pi ({v} eqlsize (E v) N) %-> (eqlsize (eql/f1 ([v] E v)) (s N)) %. +%term eqlsize/f2 %pi ({v} {vl} eqlsize (E v vl) N) %-> (eqlsize (eql/f2 ([v] [vl] E v vl)) (s N)) %. +%term eqlsize/v eqlsize (eql/v _ _) z %. +%{! #### Copied definitions !}% +%{! +It turns out that switching variables to an earlier level +will cause mixup (because we need to have both the old +and new levels in the context at the same time) unless +we use a different definition. For that reason, we define +alternatives for varlevel, eql and eqlsize. +!}% +%sort varlevel' %. +%sort eql*' %. +%term eql'/eq %pi (eq T1 T2) %-> (eql*' N T1 T2) %. +%term eql'/b %pi (eql*' N T1 T2) %-> (eql*' N (b T1) (b T2)) %. +%term eql'/f1 %pi ({v t} eql*' N (F1 v) (F2 v)) %-> (eql*' N (f F1) (f F2)) %. +%term eql'/f2 + %pi ({v t} {vl varlevel' v (s N)} eql*' (s N) (F1 v) (F2 v)) + %-> (eql*' N (f F1) (f F2)) %. +%term eql'/v %pi (varlevel' V1 L) %-> (varlevel' V2 L) %-> (eql*' N V1 V2) %. +%sort eqlsize' %. +%term eqlsize'/eq eqlsize' (eql'/eq _) z %. +%term eqlsize'/b %pi (eqlsize' E N) %-> (eqlsize' (eql'/b E) (s N)) %. +%term eqlsize'/f1 %pi ({v} eqlsize' (E v) N) %-> (eqlsize' (eql'/f1 ([v] E v)) (s N)) %. +%term eqlsize'/f2 + %pi ({v} {vl} eqlsize' (E v vl) N) + %-> (eqlsize' (eql'/f2 ([v] [vl] E v vl)) (s N)) %. +%term eqlsize'/v eqlsize' (eql'/v _ _) z %. +%{! ### Theorems about auxiliary definitions !}% +%theorem false-implies-eql : forall* {T1} {T2} {N} forall {F void} exists {E eql* N T1 T2} true %. +%worlds (blockvar blocksimple) (false-implies-eql _ _) %. +%total {} (false-implies-eql _ _) %. +%theorem eqlsize-total* : forall* {N} {T1} {T2} forall {E eql* N T1 T2} exists {S} {ES eqlsize E S} true %. +%inline eqlsize-total eqlsize-total* _ _ %. +%term _ eqlsize-total eqlsize/eq %. +%term _ %pi (eqlsize-total (eqlsize/b ES)) %<- (eqlsize-total ES) %. +%term _ %pi (eqlsize-total (eqlsize/f1 ([v] ES v))) %<- ({v} eqlsize-total (ES v)) %. +%term _ + %pi (eqlsize-total (eqlsize/f2 ([v] [vl] ES v vl))) + %<- ({v} {vl} eqlsize-total (ES v vl)) %. +%term _ eqlsize-total eqlsize/v %. +%worlds (blocksimple blockvar) (eqlsize-total* _ _ _) %. +%total (E) (eqlsize-total* E _ _) %. +%{! +The following block is used when we remove the outmost variable: +all other variables are at least level 2. "blockvar2" +makes this context explicit. +!}% +%block blockvar2 {l} [v] [vl varlevel v (s (s l))]%. +%theorem remove-one-var : forall* {F1} {F2} {N} {S} forall {E {v} {vl varlevel v (s z)} eql* (s N) (F1 v) (F2 v)} {ES {v} {vl} eqlsize (E v vl) S} exists {E' {v} eql* (s N) (F1 v) (F2 v)} {ES' {v} eqlsize (E' v) S} true %. +%term _ remove-one-var ([v] [vl] eql/eq eq/) ([v] [vl] eqlsize/eq) ([v] eql/eq eq/) ([v] eqlsize/eq) %. +%term _ + %pi (remove-one-var ([v] [vl] eql/b (F v vl)) ([v] [vl] eqlsize/b (FS v vl)) ([v] eql/b (F' v)) ([v] eqlsize/b (FS' v))) + %<- (remove-one-var F FS F' FS') %. +%term _ + %pi (remove-one-var ([v] [vl] eql/f1 ([v'] F v' v vl)) ([v] [vl] eqlsize/f1 ([v'] FS v' v vl)) ([v] eql/f1 ([v'] F' v' v)) ([v] eqlsize/f1 ([v'] FS' v' v))) + %<- ({v'} remove-one-var (F v') (FS v') (F' v') (FS' v')) %. +%term _ + %pi (remove-one-var ([v] [vl] eql/f2 ([v'] [vl'] F v' vl' v vl)) ([v] [vl] eqlsize/f2 ([v'] [vl'] FS v' vl' v vl)) ([v] eql/f2 ([v'] [vl'] F' v' vl' v)) ([v] eqlsize/f2 ([v'] [vl'] FS' v' vl' v))) + %<- ({v'} {vl'} remove-one-var (F v' vl') (FS v' vl') (F' v' vl') (FS' v' vl')) %. +%term _ remove-one-var ([v] [vl] eql/v vl vl) ([v] [vl] eqlsize/v) ([v] eql/eq eq/) ([v] eqlsize/eq) %. +%term _ remove-one-var ([v] [vl] eql/v VL1 VL2) ([v] [vl] eqlsize/v) ([v] eql/v VL1 VL2) ([v] eqlsize/v) %. +%worlds (blocksimple blockvar2) (remove-one-var _ _ _ _) %. +%total (E) (remove-one-var E _ _ _) %. +%{! +Next follows the tortuous shift down of levels: +we shift down and change to use the alternative definitions, +and then we go to back the normal definitions (with no shift). +The context we use will have both the old and new varlevels. +It order to shift from one to the other, we need theorems +that let us go from one to the other. Unfortunately, Twelf +is not smart enough to infer that the context has what the +theorem needs, so (as explained above) we need to put the theorem +in context itself and add fake dependencies. + +Adding to the complexity is the fact that we need to track the +size of the equality rules for termination proof (later). +!}% +%theorem varlevel-shifts-down : forall* {V} {L} forall {VL varlevel V (s (s L))} exists {VL' varlevel' V (s L)} true %. +%block shiftdown {l} [v] [vl varlevel v (s (s l))] [vl' varlevel' v (s l)] [vsd varlevel-shifts-down vl vl']%. +%sort fake %. +%term _ %pi fake %<- ({x varlevel-shifts-down X Y} varlevel-shifts-down X' Y') %. +%worlds (blocksimple shiftdown) (varlevel-shifts-down _ _) %. +%total {} (varlevel-shifts-down _ _) %. +%theorem shift-varlevel/L1 : forall* {N} {T1} {T2} {S} forall {E eql* (s N) T1 T2} {ES eqlsize E S} exists {E' eql*' N T1 T2} {ES' eqlsize' E' S} true %. +%term _ shift-varlevel/L1 (eql/eq eq/) eqlsize/eq (eql'/eq eq/) eqlsize'/eq %. +%term _ + %pi (shift-varlevel/L1 (eql/b E) (eqlsize/b ES) (eql'/b E') (eqlsize'/b ES')) + %<- (shift-varlevel/L1 E ES E' ES') %. +%term _ + %pi (shift-varlevel/L1 (eql/f1 ([v] F v)) (eqlsize/f1 FS) (eql'/f1 ([v] F' v)) (eqlsize'/f1 FS')) + %<- ({v} shift-varlevel/L1 (F v) (FS v) (F' v) (FS' v)) %. +%term _ + %pi (shift-varlevel/L1 (eql/f2 ([v] [vl varlevel v (s (s N))] F v vl)) (eqlsize/f2 FS) (eql'/f2 ([v] [vl varlevel' v (s N)] F' v vl)) (eqlsize'/f2 FS')) + %<- ({v} {vl} {vl' varlevel' v (s N)} {vsd varlevel-shifts-down vl vl'} shift-varlevel/L1 (F v vl) (FS v vl) (F' v vl') (FS' v vl')) %. +%term _ + %pi (shift-varlevel/L1 (eql/v VL1 VL2) eqlsize/v (eql'/v VL1' VL2') eqlsize'/v) + %<- (varlevel-shifts-down VL1 VL1') + %<- (varlevel-shifts-down VL2 VL2') %. +%worlds (blocksimple shiftdown) (shift-varlevel/L1 _ _ _ _) %. +%total (E) (shift-varlevel/L1 E _ _ _) %. +%theorem varlevel-shifts-back : forall* {V} {L} forall {VL' varlevel' V (s L)} exists {VL varlevel V (s L)} true %. +%block shiftback {l} [v] [vl' varlevel' v (s l)] [vl varlevel v (s l)] [vsb varlevel-shifts-back vl' vl]%. +%sort fake %. +%term _ %pi fake %<- ({x varlevel-shifts-back X Y} varlevel-shifts-back X' Y') %. +%worlds (blocksimple shiftback) (varlevel-shifts-back _ _) %. +%total {} (varlevel-shifts-back _ _) %. +%theorem shift-varlevel/L2 : forall* {N} {T1} {T2} {S} forall {E' eql*' N T1 T2} {ES' eqlsize' E' S} exists {E eql* N T1 T2} {ES eqlsize E S} true %. +%term _ shift-varlevel/L2 (eql'/eq eq/) eqlsize'/eq (eql/eq eq/) eqlsize/eq %. +%term _ + %pi (shift-varlevel/L2 (eql'/b E) (eqlsize'/b ES) (eql/b E') (eqlsize/b ES')) + %<- (shift-varlevel/L2 E ES E' ES') %. +%term _ + %pi (shift-varlevel/L2 (eql'/f1 ([v] F v)) (eqlsize'/f1 FS) (eql/f1 ([v] F' v)) (eqlsize/f1 FS')) + %<- ({v} shift-varlevel/L2 (F v) (FS v) (F' v) (FS' v)) %. +%term _ + %pi (shift-varlevel/L2 (eql'/f2 ([v] [vl varlevel' v (s N)] F v vl)) (eqlsize'/f2 FS) (eql/f2 ([v] [vl varlevel v (s N)] F' v vl)) (eqlsize/f2 FS')) + %<- ({v} {vl'} {vl} {vsb varlevel-shifts-back vl' vl} shift-varlevel/L2 (F v vl') (FS v vl') (F' v vl) (FS' v vl)) %. +%term _ + %pi (shift-varlevel/L2 (eql'/v VL1 VL2) eqlsize'/v (eql/v VL1' VL2') eqlsize/v) + %<- (varlevel-shifts-back VL1 VL1') + %<- (varlevel-shifts-back VL2 VL2') %. +%worlds (blocksimple shiftback) (shift-varlevel/L2 _ _ _ _) %. +%total (E) (shift-varlevel/L2 E _ _ _) %. +%{! +Now we put these two parts in one that hides the alternative definitions. +!}% +%theorem shift-varlevel : forall* {N} {T1} {T2} {S} forall {E eql* (s N) T1 T2} {ES eqlsize E S} exists {E' eql* N T1 T2} {ES' eqlsize E' S} true %. +%term _ + %pi (shift-varlevel E1 ES1 E3 ES3) + %<- (shift-varlevel/L1 E1 ES1 E2 ES2) + %<- (shift-varlevel/L2 E2 ES2 E3 ES3) %. +%worlds (blocksimple) (shift-varlevel _ _ _ _) %. +%total {} (shift-varlevel _ _ _ _) %. +%{! +The following two theorems relate structural equality with +identity based equality. One of them is called "Leibnitz" equality, +I have heard, but apparently haven't remembered which one. +!}% +%theorem b-preserves-eq : forall* {T1} {T2} forall {E eq T1 T2} exists {BE eq (b T1) (b T2)} true %. +%term _ b-preserves-eq eq/ eq/ %. +%worlds (blocksimple) (b-preserves-eq _ _) %. +%total {} (b-preserves-eq _ _) %. +%theorem f-preserves-eq : forall* {F1} {F2} forall {E {v} eq (F1 v) (F2 v)} exists {E eq (f F1) (f F2)} true %. +%term _ f-preserves-eq ([v] eq/) eq/ %. +%worlds (blocksimple) (f-preserves-eq _ _) %. +%total {} (f-preserves-eq _ _) %. +%{! +Next the main lemma that says we can avoid looking at varlevels in checking +equality. We remove the outside variable, shift remaining variables, +and then recurse (hence the need for tracking eqlsize). Note that we +never put var levels in the context. +!}% +%theorem eql-implies-eq : forall* {T1} {T2} forall {E eql T1 T2} exists {E' eq T1 T2} true %. +%theorem eql-implies-eq/L : forall* {T1} {T2} forall {E eql T1 T2} {S} {ES eqlsize E S} exists {E' eq T1 T2} true %. +%term _ eql-implies-eq/L (eql/eq E) _ _ E %. +%term _ + %pi (eql-implies-eq/L (eql/b E1) _ (eqlsize/b ES) E') + %<- (eql-implies-eq/L E1 _ ES E1') + %<- (b-preserves-eq E1' E') %. +%term _ + %pi (eql-implies-eq/L (eql/f1 ([v] E1 v)) _ (eqlsize/f1 FS) E') + %<- ({v} eql-implies-eq/L (E1 v) _ (FS v) (E1' v)) + %<- (f-preserves-eq E1' E') %. +%term _ + %pi (eql-implies-eq/L (eql/f2 ([v] [vl] F1 v vl)) _ (eqlsize/f2 FS1) E') + %<- (remove-one-var F1 FS1 ([v] F2 v) FS2) + %<- ({v} shift-varlevel (F2 v) (FS2 v) (F3 v) (FS3 v)) + %<- ({v} eql-implies-eq/L (F3 v) _ (FS3 v) (F4 v)) + %<- (f-preserves-eq F4 E') %. +%worlds (blocksimple) (eql-implies-eq/L _ _ _ _) %. +%total (S) (eql-implies-eq/L _ S _ _) %. +%term _ %pi (eql-implies-eq E E') %<- (eqlsize-total ES) %<- (eql-implies-eq/L E _ ES E') %. +%worlds (blocksimple) (eql-implies-eq _ _) %. +%total {} (eql-implies-eq _ _) %. +%{! ### Main Theorem !}% +%{! +Finally the statement of the main theorem of this section. +It is proved by using eql as a between station. In this case, +we do have var levels in the context. This theorem (or rather +its main lemma) uses reasoning-from-false extensively because the +cases cannot be distinguished by Twelf's case analysis. +It also uses the "divrem" part of the nat signature extensively +as well as theorems about plus and times. The proofs of the cases are +uninteresting arithmetic fiddling. +!}% +%theorem tonat-one2one : forall* {T1} {N1} {T2} {N2} forall {TN1 tonat T1 N1} {TN2 tonat T2 N2} {E nat`eq N1 N2} exists {ET eq T1 T2} true %. +%theorem tonat-one2one/L : forall* {N} {T1} {M1} {T2} {M2} forall {TN1 tonat* N T1 M1} {TN2 tonat* N T2 M2} {E nat`eq M1 M2} exists {ET eql* N T1 T2} true %. +%term _ tonat-one2one/L tonat/a tonat/a _ (eql/eq eq/) %. +%term _ + %pi (tonat-one2one/L tonat/a (tonat/b _ _ N+1+TM=N) nat`eq/ E) + %<- (plus-swap-succ N+1+TM=N N+TM+1=N) + %<- (plus-commutative N+TM+1=N TM+1+N=N) + %<- (plus-implies-gt TM+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/b _ _ N+1+TM=N) tonat/a nat`eq/ E) + %<- (plus-swap-succ N+1+TM=N N+TM+1=N) + %<- (plus-commutative N+TM+1=N TM+1+N=N) + %<- (plus-implies-gt TM+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L tonat/a (tonat/f _ _ N+2+TM=N) nat`eq/ E) + %<- (plus-swap-succ N+2+TM=N N+1+TM+1=N) + %<- (plus-swap-succ N+1+TM+1=N N+TM+2=N) + %<- (plus-commutative N+TM+2=N TM+2+N=N) + %<- (plus-implies-gt TM+2+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/f _ _ N+2+TM=N) tonat/a nat`eq/ E) + %<- (plus-swap-succ N+2+TM=N N+1+TM+1=N) + %<- (plus-swap-succ N+1+TM+1=N N+TM+2=N) + %<- (plus-commutative N+TM+2=N TM+2+N=N) + %<- (plus-implies-gt TM+2+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L tonat/a (tonat/v VL N+L=N) nat`eq/ E) + %<- (plus-right-identity N N+0=N) + %<- (plus-left-cancels N+L=N N+0=N nat`eq/ nat`eq/ L=0) + %<- (varlevel-contradiction VL L=0 F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/v VL N+L=N) tonat/a nat`eq/ E) + %<- (plus-right-identity N N+0=N) + %<- (plus-left-cancels N+L=N N+0=N nat`eq/ nat`eq/ L=0) + %<- (varlevel-contradiction VL L=0 F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/b TN1 TWO*M1=TM1 N+1+TM1=M) (tonat/b TN2 TWO*M2=TM2 N+1+TM2=M) nat`eq/ (eql/b E)) + %<- (plus-left-cancels N+1+TM1=M N+1+TM2=M nat`eq/ nat`eq/ TM1=TM2) + %<- (times-left-cancels TWO*M1=TM1 TWO*M2=TM2 nat`eq/ TM1=TM2 M1=M2) + %<- (tonat-one2one/L TN1 TN2 M1=M2 E) %. +%term _ + %pi (tonat-one2one/L (tonat/b _ TWO*M1=TM1 N+1+TM1=M) (tonat/f _ TWO*M2=TM2 N+2+TM2=M) nat`eq/ E) + %<- (plus-swap-succ N+2+TM2=M N+1+TM2+1=M) + %<- (plus-left-cancels N+1+TM1=M N+1+TM2+1=M nat`eq/ nat`eq/ TM1=TM2+1) + %<- (times-commutative TWO*M1=TM1 M1*2=TM1) + %<- (plus-right-identity _ TM1+0=TM1) + %<- (divrem-can-be-constructed M1*2=TM1 TM1+0=TM1 (gt/> gt/1) TM1/2=M1,0) + %<- (times-commutative TWO*M2=TM2 M2*2=TM2) + %<- (nat`eq-symmetric TM1=TM2+1 TM2+1=TM1) + %<- (plus-respects-eq (plus/s plus/z) nat`eq/ nat`eq/ TM2+1=TM1 ONE+TM2=TM1) + %<- (plus-commutative ONE+TM2=TM1 TM2+ONE=TM1) + %<- (divrem-can-be-constructed M2*2=TM2 TM2+ONE=TM1 gt/1 TM1/2=M2,1) + %<- (divrem-deterministic TM1/2=M1,0 TM1/2=M2,1 nat`eq/ nat`eq/ _ ZERO=ONE) + %<- (nat`eq-contradiction ZERO=ONE F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/f _ TWO*M2=TM2 N+2+TM2=M) (tonat/b _ TWO*M1=TM1 N+1+TM1=M) nat`eq/ E) + %<- (plus-swap-succ N+2+TM2=M N+1+TM2+1=M) + %<- (plus-left-cancels N+1+TM1=M N+1+TM2+1=M nat`eq/ nat`eq/ TM1=TM2+1) + %<- (times-commutative TWO*M1=TM1 M1*2=TM1) + %<- (plus-right-identity _ TM1+0=TM1) + %<- (divrem-can-be-constructed M1*2=TM1 TM1+0=TM1 (gt/> gt/1) TM1/2=M1,0) + %<- (times-commutative TWO*M2=TM2 M2*2=TM2) + %<- (nat`eq-symmetric TM1=TM2+1 TM2+1=TM1) + %<- (plus-respects-eq (plus/s plus/z) nat`eq/ nat`eq/ TM2+1=TM1 ONE+TM2=TM1) + %<- (plus-commutative ONE+TM2=TM1 TM2+ONE=TM1) + %<- (divrem-can-be-constructed M2*2=TM2 TM2+ONE=TM1 gt/1 TM1/2=M2,1) + %<- (divrem-deterministic TM1/2=M1,0 TM1/2=M2,1 nat`eq/ nat`eq/ _ ZERO=ONE) + %<- (nat`eq-contradiction ZERO=ONE F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/b _ _ N+1+TM=M) (tonat/v _ M+L=N) nat`eq/ E) + %<- (plus-swap-succ N+1+TM=M N+TM+1=M) + %<- (plus-commutative N+TM+1=M TM+1+N=M) + %<- (plus-implies-gt TM+1+N=M nat`eq/ M>N) + %<- (plus-commutative M+L=N L+M=N) + %<- (plus-implies-ge L+M=N N>=M) + %<- (nat`gt-transitive-ge M>N N>=M M>M) + %<- (nat`gt-anti-reflexive M>M F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/v _ M+L=N) (tonat/b _ _ N+1+TM=M) nat`eq/ E) + %<- (plus-swap-succ N+1+TM=M N+TM+1=M) + %<- (plus-commutative N+TM+1=M TM+1+N=M) + %<- (plus-implies-gt TM+1+N=M nat`eq/ M>N) + %<- (plus-commutative M+L=N L+M=N) + %<- (plus-implies-ge L+M=N N>=M) + %<- (nat`gt-transitive-ge M>N N>=M M>M) + %<- (nat`gt-anti-reflexive M>M F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/f ([v] [vl] F1 v vl) TWO*M1=TM1 N+2+TM1=M) (tonat/f ([v] [vl] F2 v vl) TWO*M2=TM2 N+2+TM2=M) nat`eq/ (eql/f2 FE)) + %<- (plus-left-cancels N+2+TM1=M N+2+TM2=M nat`eq/ nat`eq/ TM1=TM2) + %<- (times-left-cancels TWO*M1=TM1 TWO*M2=TM2 nat`eq/ TM1=TM2 M1=M2) + %<- ({v} {vl varlevel v (s N)} tonat-one2one/L (F1 v vl) (F2 v vl) M1=M2 (FE v vl)) %. +%term _ + %pi (tonat-one2one/L (tonat/f _ _ N+2+TM=M) (tonat/v _ M+L=N) nat`eq/ E) + %<- (plus-swap-succ N+2+TM=M N+1+TM+1=M) + %<- (plus-swap-succ N+1+TM+1=M N+TM+2=M) + %<- (plus-commutative N+TM+2=M TM+2+N=M) + %<- (plus-implies-gt TM+2+N=M nat`eq/ M>N) + %<- (plus-commutative M+L=N L+M=N) + %<- (plus-implies-ge L+M=N N>=M) + %<- (nat`gt-transitive-ge M>N N>=M M>M) + %<- (nat`gt-anti-reflexive M>M F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/v _ M+L=N) (tonat/f _ _ N+2+TM=M) nat`eq/ E) + %<- (plus-swap-succ N+2+TM=M N+1+TM+1=M) + %<- (plus-swap-succ N+1+TM+1=M N+TM+2=M) + %<- (plus-commutative N+TM+2=M TM+2+N=M) + %<- (plus-implies-gt TM+2+N=M nat`eq/ M>N) + %<- (plus-commutative M+L=N L+M=N) + %<- (plus-implies-ge L+M=N N>=M) + %<- (nat`gt-transitive-ge M>N N>=M M>M) + %<- (nat`gt-anti-reflexive M>M F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/v VL1 M+L1=N) (tonat/v VL2 M+L2=N) nat`eq/ (eql/v VL1' VL2)) + %<- (plus-left-cancels M+L1=N M+L2=N nat`eq/ nat`eq/ L1=L2) + %<- (varlevel-respects-eq VL1 L1=L2 VL1') %. +%worlds (blockvar) (tonat-one2one/L _ _ _ _) %. +%total (T) (tonat-one2one/L T _ _ _) %. +%term _ + %pi (tonat-one2one TN1 TN2 EQ TEQ) + %<- (tonat-one2one/L TN1 TN2 EQ EQL) + %<- (eql-implies-eq EQL TEQ) %. +%worlds () (tonat-one2one _ _ _ _) %. +%total {} (tonat-one2one _ _ _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/HOAS_nat_bijection). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/hypothetical-judgment.lf b/new-tests/stelf-output/pages/hypothetical-judgment.lf new file mode 100644 index 0000000..eb62931 --- /dev/null +++ b/new-tests/stelf-output/pages/hypothetical-judgment.lf @@ -0,0 +1,31 @@ +% +% +%{! A [judgment](/wiki/judgment/) is a statement derivable in a particular object logic. A **hypothetical judgment** is a judgment that makes use of hypothetical assumptions. On paper, we often represent a set of hypothetical assumptions using a context: + +```math +{ + \Gamma, A \textrm{ true} \vdash B \textrm{ true} + \over + \Gamma \vdash A \Rightarrow B \textrm{ true} +} +``` + +We name the judgment that proves the truth of a proposition , which can be read as "Under the assumptions , is true." The context is the sequence of assumptions . + +With an ordinary hypothetical context, we may freely coalesce repeated assumptions, add extra unused assumptions, and to reorder the assumptions in the context. (Formally, these are the properties of contraction, weakening, and exchange.) + +Additionally, a context in a hypothetical judgment should satisfy the identity and substitution properties. Identity simply means that we should be able to use our hypotheses -- . Substitution means that if we have a proof tree showing , and another proof tree , then we should be able to substitute the second tree into the first, replacing hypothesis and producing a new tree proving . + +This sample rule is the implication introduction rule, which says that is true if we can show is true under the hypothetical assumption . + +In the LF methodology, we represent the hypothetical context of our object language's judgment using the LF context itself. (See [higher-order judgment](/wiki/higher-order-judgements/).) + +For example, we can represent the introduction rule as follows: !}% +%sort prop %. +%term ==> %pi prop %-> prop %-> prop %. +%prec %none 10 ==> %. +%sort true %. +%term imp/intro %pi (true (A ==> B)) %<- (%pi (true A) %-> (true B)) %. +%{! Here, we represent the hypothetical assumption that using an LF hypothesis of type ``A true``. We can see this clearly if we write down the proof term for the proof of the tautology ``P ==> P``: !}% +%define taut (true (P ==> P)) imp/intro ([ptrue true P] ptrue) %. +%{! Within the scope of the function argument to ``imp/intro``, we assume that we have a proof of ``P true`` (which we creatively named ``ptrue``). This assumption lets us show that ``P true`` holds, and that lets us show that ``(P ==> P) true``. !}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/implicit-and-explicit-parameters.lf b/new-tests/stelf-output/pages/implicit-and-explicit-parameters.lf new file mode 100644 index 0000000..702e8f2 --- /dev/null +++ b/new-tests/stelf-output/pages/implicit-and-explicit-parameters.lf @@ -0,0 +1,41 @@ +% +%{! When declaring a type family or constant, the Twelf programmer often needs to use **universally quantified** parameters, and has the choice of making the parameter **implicit** or **explicit**. The article on [converting between implicit and explicit parameters](/wiki/converting-between-implicit-and-explicit-parameters/) discusses how these two ways of doing things can be derived from each other. + +Using fully **explicit parameters** means that all variables are _bound_ by being placed within curly braces ``\{\}``. The type family or constant generally has more arguments when using this form. + +Using **implicit parameters** means that variables are not put within curly braces. Twelf assumes an identifier is meant to be a universally quantified parameter if the identifier starts with an uppercase letter _and_ if that identifier has not been bound or defined anywhere else. This is essentially an interface aspect allowing for simpler code—internally, Twelf applies [[type reconstruction]] to identify the type of the universally quantified variables and converts the implicit parameters into explicit parameters. + +The implicit parameters style is more concise and is often cleaner; however, in some cases it is necessary to use explicit parameters (for instance, a [``%terminates``](/wiki/percent-terminates/) or [``%total``](/wiki/percent-total/) declaration can refer to explicit parameters, not implicit parameters), and it may make stylistic sense in other situations as well. + +Examples of the two different styles follow, using the language from the article on [converting between implicit and explicit parameters](/wiki/converting-between-implicit-and-explicit-parameters/) which is omitted for brevity. The Twelf output is included - note that the Twelf output _always_ writes out explicit parameters,\{\{discuss|Always prints out the parameters\}\} even if the definition is using implicit parameters; therefore the two examples have almost exactly the same output from Twelf. + +### Implicit parameters !}% +%{! (options removed from twelftag: hidden=true) !}% +%sort exp %. +%sort typ %. +%sort of %. +%term 0 exp %. +%term 1 exp %. +%term bit typ %. +%term void typ %. +%term of1 of 1 bit %. +%term of0 of 0 bit %. +%{! !}% +%{! (options removed from twelftag: discard=true check=decl) !}% +%sort translate-i %. +%mode translate-i %in %out %. +%{! ### Explicit parameters + +Note that in this example we use slightly bad [style](/wiki/twelf-style-guide/), capitalizing our bound variables ``M``, ``A``, and ``M'``. !}% +%{! (options removed from twelftag: discard=true check=decl) !}% +%sort translate-e {M exp} {A typ} {M' exp} %. +%mode translate-e %in %in %out %in %out %. +%{! ## See also + +* \{\{proving|title=Full LF\}\} discusses implicit and explicit parameters in the section "Twelf conveniences." !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Implicit_and_explicit_parameters). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/incremental-metatheorem-development.lf b/new-tests/stelf-output/pages/incremental-metatheorem-development.lf new file mode 100644 index 0000000..edb9c33 --- /dev/null +++ b/new-tests/stelf-output/pages/incremental-metatheorem-development.lf @@ -0,0 +1,61 @@ +% +%{! Typically, metatheorems must be verified as total before they can be used to prove other metatheorems. However, one often wishes to develop a proof without first proving some intermediate lemmas. For example, suppose a programmer believes that if metatheorem A is true, he will be able to prove B. However, A may be difficult to prove, and the effort to prove it may be wasted if it does not lead to a proof of B. Instead, the programmer wants to prove B first (assuming A) and then, if successful, proceed to prove A. Therefore, he would like to create a proof of B with a "hole" for A, but still be able to check B. + +The newest versions of Twelf provide a direct way to create such holes. Users of earlier versions may take advantage of a curious feature. + +## Twelf CVS: %trustme + +The [``%trustme``](/wiki/percent-trustme/) declaration instructs Twelf to run the immediately following declaration but suppress any errors that occur. It is commonly used on [``%total``](/wiki/percent-total/) declarations to introduce lemmas that have not yet been proved, in order to develop a proof without first proving some necessary lemmas. + +If Twelf is in [unsafe mode](/wiki/unsafe-mode/), it will accept ``%trustme`` before the ``%total`` directive of a metatheorem with an incomplete proof. Twelf will consider such metatheorems total for the purpose of the totality checks of subsequent metatheorems. The following is an example of ``%trustme`` in action. !}% +%sort nat %. +%term nat/z nat %. +%term nat/s %pi nat %-> nat %. +%sort nat-less %. +%term nat-less/z nat-less nat/z (nat/s N) %. +%term nat-less/s %pi (nat-less (nat/s N1) (nat/s N2)) %<- (nat-less N1 N2) %. +%sort nat-less-immsucc {N nat} %. +%mode nat-less-immsucc %in %out %. +%term _ nat-less-immsucc nat/z nat-less/z %. +%worlds () (nat-less-immsucc _ _) %. +%total (D1) (nat-less-immsucc D1 _) %. +%{! When ``%trustme`` directives are no longer needed, unsafe mode can be disabled by inputting the following line into the Twelf server. + +<code>set unsafe false</code> + +The %trustme declaration makes Twelf's deductions unsound, of course, so should only be thought of as a development and debugging tool. + +## Twelf 1.5R1 and earlier + +In older versions of Twelf, the following technique can be used. In the following code, "foo" is obviously not a total relation—it has no cases at all defined. However, after Twelf tries to check the line ``%total I (foo I _).`` and fails, it will actually allow ``foo`` to be used to check ``bar``'s totality. + +```twelf +thing : type. + +a : thing. +b : thing. + +foo : thing -> thing -> type. +%mode foo +I -O. +%worlds () (foo _ _). +%total I (foo I _). + +bar : thing -> thing -> type. + +- : bar I O + <- foo I O. + +%mode bar +I -O. +%worlds () (bar _ _). +%total I (bar I _). +``` + +To reiterate, in order to use this "feature" you get Twelf to reject the ``%total`` declaration for foo. Then, it will believe that foo is total in checking _subsequent_ theorems. It won't believe that foo is total if you try to rerun ``%total`` for foo itself—if you re-check the ``%total``, it reruns the totality check, and finds that foo still isn't total. + +\{\{tutorial\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Incremental_metatheorem_development). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/indexed-hoas-nat-bijection.lf b/new-tests/stelf-output/pages/indexed-hoas-nat-bijection.lf new file mode 100644 index 0000000..7efbc82 --- /dev/null +++ b/new-tests/stelf-output/pages/indexed-hoas-nat-bijection.lf @@ -0,0 +1,9576 @@ +% +%{! # Bijection from indexed HOAS to natural numbers !}% +%{! +This file shows an *indexed* HOAS term type and a mapping from these +terms to the natural numbers. It depends on John Boyland's +library signatures, in particular void, bool, nat, natpair and multiset. + +This file shows some of the extensions needed to handle indexed HOAS +terms above and beyond those of "regular" HOAS terms (which is the topic of +the [HOAS_nat_bijection] page). In the comments here, I emphasize those +different aspects rather than repeat the same discussion. +!}% +%{! | hidden = true !}% +% contents included from ../library/std.elf +%%%%%% Proposed Twelf Standard Library +%%%%%% John Boyland +%%%%%% You may freely use, modify and distribute this file without restrictions. +%%%% Structure +%%% The uninhabited type +%sort void %. +%freeze void %. +%%% The uninteresting type +%sort unit %. +%term unit/ unit %. +%freeze unit %. +%%% The three-way comparison type +%sort comp %. +%term less comp %. +%term greater comp %. +%term equal comp %. +%freeze comp %. +%{! | hidden = true !}% +% contents included from ../library/bool.elf +%%%%% bool.elf +%%%%% Boolean literals +%%%%% John Boyland +%%%% Definitions +%sort bool %. +%term true bool %. +%term false bool %. +%freeze bool %. +%sort eq? %. +%inline eq [B1] [B2] eq? B1 B2 true %. +%inline ne [B1] [B2] eq? B1 B2 false %. +%term eq/ eq B B %. +%term ne/TF ne true false %. +%term ne/FT ne false true %. +%inline eq?/yes eq/ %. +%inline eq?/no [NE ne B1 B2] NE %. +%sort not %. +%term not/T not true false %. +%term not/F not false true %. +%sort and %. +%term and/FX and false B false %. +%term and/XF and B false false %. +%term and/TT and true true true %. +%sort or %. +%term or/TX or true B true %. +%term or/XT or B true true %. +%term or/FF or false false false %. +%%%% Theorems +%%% theorems about eq +%theorem false-implies-eq : forall* {X1} {X2} forall {F void} exists {E eq X1 X2} true %. +%worlds () (false-implies-eq _ _) %. +%total {} (false-implies-eq _ _) %. +%theorem meta-eq : forall {X1} {X2} {E eq X1 X2} true %. +%term _ meta-eq _ _ eq/ %. +%worlds () (meta-eq _ _ _) %. +%total {} (meta-eq _ _ _) %. +%reduces = X Y (meta-eq X Y _) %. +%theorem eq-reflexive : forall {X} exists {E eq X X} true %. +%term _ eq-reflexive _ eq/ %. +%worlds () (eq-reflexive _ _) %. +%total {} (eq-reflexive _ _) %. +%theorem eq-symmetric : forall* {X} {Y} forall {E eq X Y} exists {F eq Y X} true %. +%term _ eq-symmetric eq/ eq/ %. +%worlds () (eq-symmetric _ _) %. +%total {} (eq-symmetric _ _) %. +%theorem eq-transitive : forall* {X} {Y} {Z} forall {E1 eq X Y} {E2 eq Y Z} exists {F eq X Z} true %. +%term _ eq-transitive eq/ eq/ eq/ %. +%worlds () (eq-transitive _ _ _) %. +%total {} (eq-transitive _ _ _) %. +%%% theorems about ne +%theorem false-implies-ne : forall* {X1} {X2} forall {F void} exists {G ne X1 X2} true %. +%worlds () (false-implies-ne _ _) %. +%total {} (false-implies-ne _ _) %. +%theorem ne-respects-eq : forall* {X1} {X2} {Y1} {Y2} forall {D1 ne X1 X2} {E1 eq X1 Y1} {E2 eq X2 Y2} exists {D2 ne Y1 Y2} true %. +%term _ ne-respects-eq X1<>X2 eq/ eq/ X1<>X2 %. +%worlds () (ne-respects-eq _ _ _ _) %. +%total {} (ne-respects-eq _ _ _ _) %. +%theorem ne-anti-reflexive : forall* {B} forall {R ne B B} exists {F void} true %. +%worlds () (ne-anti-reflexive _ _) %. +%total {} (ne-anti-reflexive _ _) %. +%theorem ne-symmetric : forall* {B1} {B2} forall {R1 ne B1 B2} exists {R2 ne B2 B1} true %. +%term _ ne-symmetric ne/TF ne/FT %. +%term _ ne-symmetric ne/FT ne/TF %. +%worlds () (ne-symmetric _ _) %. +%total {} (ne-symmetric _ _) %. +%theorem eq-ne-implies-false : forall* {B1} {B2} forall {D1 eq B1 B2} {D2 ne B1 B2} exists {F void} true %. +%worlds () (eq-ne-implies-false _ _ _) %. +%total {} (eq-ne-implies-false _ _ _) %. +%%% theorems about eq? +%theorem eq?-total* : forall {B1} {B2} exists {B} {EQ? eq? B1 B2 B} true %. +%term _ eq?-total* true true _ eq?/yes %. +%term _ eq?-total* false false _ eq?/yes %. +%term _ eq?-total* true false _ (eq?/no ne/TF) %. +%term _ eq?-total* false true _ (eq?/no ne/FT) %. +%worlds () (eq?-total* _ _ _ _) %. +%total {} (eq?-total* _ _ _ _) %. +%inline eq?-total eq?-total* _ _ _ %. +%%% Theorems about not, and, or +%theorem not-total* : forall {B1} exists {B2} {N not B1 B2} true %. +%inline not-total not-total* _ _ %. +%term _ not-total not/T %. +%term _ not-total not/F %. +%worlds () (not-total* _ _ _) %. +%theorem and-total* : forall {B1} {B2} exists {B3} {A and B1 B2 B3} true %. +%inline and-total and-total* _ _ _ %. +%term _ and-total and/FX %. +%term _ and-total and/XF %. +%term _ and-total and/TT %. +%worlds () (and-total* _ _ _ _) %. +%total {} (and-total* _ _ _ _) %. +%theorem or-total* : forall {B1} {B2} exists {B3} {A or B1 B2 B3} true %. +%inline or-total or-total* _ _ _ %. +%term _ or-total or/TX %. +%term _ or-total or/XT %. +%term _ or-total or/FF %. +%worlds () (or-total* _ _ _ _) %. +%total {} (or-total* _ _ _ _) %. +%inline bool`bool bool %. +%inline bool`true true %. +%inline bool`false false %. +%inline bool`eq? eq? %. +%inline bool`eq eq %. +%inline bool`ne ne %. +%inline bool`eq/ eq/ %. +%inline bool`ne/TF ne/TF %. +%inline bool`ne/FT ne/FT %. +%inline bool`eq?/yes eq?/yes %. +%inline bool`eq?/no eq?/no %. +%inline bool`not not %. +%inline bool`not/T not/T %. +%inline bool`not/F not/F %. +%inline bool`and and %. +%inline bool`and/FX and/FX %. +%inline bool`and/XF and/XF %. +%inline bool`and/TT and/TT %. +%inline bool`or or %. +%inline bool`or/TX or/TX %. +%inline bool`or/XT or/XT %. +%inline bool`or/FF or/FF %. +%inline bool`false-implies-eq false-implies-eq %. +%inline bool`meta-eq meta-eq %. +%inline bool`eq-reflexive eq-reflexive %. +%inline bool`eq-symmetric eq-symmetric %. +%inline bool`eq-transitive eq-transitive %. +%inline bool`false-implies-ne false-implies-ne %. +%inline bool`ne-respects-eq ne-respects-eq %. +%inline bool`ne-anti-reflexive ne-anti-reflexive %. +%inline bool`ne-symmetric ne-symmetric %. +%inline bool`eq-ne-implies-false eq-ne-implies-false %. +%inline bool`eq?-total* eq?-total* %. +%inline bool`eq?-total eq?-total %. +%inline bool`not-total* not-total* %. +%inline bool`not-total not-total %. +%inline bool`and-total* and-total* %. +%inline bool`and-total and-total %. +%inline bool`or-total* or-total* %. +%inline bool`or-total or-total %. +%{! | hidden = true !}% +% contents included from ../library/nat.elf +%%%%% Natural numbers +%%%%% John Boyland +%%%%% Anyone may use, copy or modify this software without restriction +%%%%% This file requires std.elf +%%%%% nat-base.elf +%%%%% Basic definitions, operations and theorems +%%%%% This file is part of the nat.elf signature +%%%% Definitions +%%% Natural numbers: +%sort nat %. +%name nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%freeze nat %. +%%% Operations on natural numbers +%sort plus %. +%term plus/z plus z Y Y %. +%term plus/s %pi (plus (s X) Y (s Z)) %<- (plus X Y Z) %. +%sort times %. +%term times/z times z X z %. +%term times/s %pi (times (s X) Y Z) %<- (plus T Y Z) %<- (times X Y T) %. +%sort eq %. +%term eq/ eq N N %. +%sort gt %. +%term gt/1 gt (s M) M %. +%term gt/> %pi (gt (s M) N) %<- (gt M N) %. +%%% Using the conditional for natural numbers +%sort compare %. +%term compare/= compare N N equal %. +%term compare/< %pi (compare M N less) %<- (gt N M) %. +%term compare/> %pi (compare M N greater) %<- (gt M N) %. +%%%% Theorems +%%% Theorems about eq +%reduces = X Y (eq X Y) %. +%theorem meta-eq : forall {M} {N} {E eq M N} true %. +%term _ meta-eq N N eq/ %. +%worlds () (meta-eq _ _ _) %. +%total {} (meta-eq _ _ _) %. +%reduces = X Y (meta-eq X Y _) %. +%theorem false-implies-eq : forall* {M} {N} forall {P void} exists {Q eq M N} true %. +%worlds () (false-implies-eq _ M=N) %. +%total {} (false-implies-eq _ _) %. +%theorem eq-symmetric : forall* {M nat} {N nat} forall {E eq M N} exists {F eq N M} true %. +%term _ eq-symmetric eq/ eq/ %. +%worlds () (eq-symmetric M>N N>M) %. +%total {} (eq-symmetric _ _) %. +%theorem eq-transitive : forall* {M nat} {N nat} {P nat} forall {E1 eq M N} {E2 eq N P} exists {F eq M P} true %. +%term _ eq-transitive eq/ eq/ eq/ %. +%worlds () (eq-transitive M>N N>P M>P) %. +%total {} (eq-transitive _ _ _) %. +%theorem succ-deterministic : forall* {N1 nat} {N2 nat} forall {E eq N1 N2} exists {F eq (s N1) (s N2)} true %. +%term _ succ-deterministic eq/ eq/ %. +%worlds () (succ-deterministic N1=N2 N1+1=N2+1) %. +%total {} (succ-deterministic E _) %. +%theorem succ-cancels : forall* {N1 nat} {N2 nat} forall {E eq (s N1) (s N2)} exists {F eq N1 N2} true %. +%term _ succ-cancels eq/ eq/ %. +%worlds () (succ-cancels N1+1=N2+1 N1=N2) %. +%total {} (succ-cancels E _) %. +%theorem succ-contradiction : forall* {N} forall {E eq N (s N)} exists {F void} true %. +%worlds () (succ-contradiction _ _) %. +%total {} (succ-contradiction _ _) %. +%theorem eq-contradiction : forall* {N} forall {E eq z (s N)} exists {F void} true %. +%worlds () (eq-contradiction ZERO=N+1 _) %. +%total {} (eq-contradiction _ _) %. +%%% Theorems about gt +%reduces < M N (gt N M) %. +%% If we want to prove the termination of a theorem using gt, +%% we need the gt relation lifted to the meta level: +%theorem meta-gt : forall {M} {N} {G gt M N} true %. +%term _ meta-gt (s M) M gt/1 %. +%term _ %pi (meta-gt (s M) N (gt/> G)) %<- (meta-gt M N G) %. +%worlds () (meta-gt _ _ _) %. +%total M (meta-gt M _ _) %. +%reduces < M N (meta-gt N M _) %. +%theorem false-implies-gt : forall* {M} {N} forall {P void} exists {Q gt M N} true %. +%worlds () (false-implies-gt _ M>N) %. +%total {} (false-implies-gt _ _) %. +%theorem gt-respects-eq : forall* {M1 nat} {M2 nat} {N1 nat} {N2 nat} forall {P gt M1 N1} {E1 eq M1 M2} {E2 eq N1 N2} exists {Q gt M2 N2} true %. +%term _ gt-respects-eq M1>N1 eq/ eq/ M1>N1 %. +%worlds () (gt-respects-eq M1>N1 M1=M2 N1=N2 M2>N2) %. +%total {} (gt-respects-eq _ _ _ _) %. +%theorem succ-implies-gt : forall* {X} {X'} forall {E eq X (s X')} exists {G gt X X'} true %. +%term _ succ-implies-gt eq/ gt/1 %. +%worlds () (succ-implies-gt X=sX' X>X') %. +%total {} (succ-implies-gt _ _) %. +%theorem succ-implies-gt-zero : forall {M} exists {G gt (s M) z} true %. +%term _ succ-implies-gt-zero z gt/1 %. +%term _ %pi (succ-implies-gt-zero (s M) (gt/> SM>0)) %<- (succ-implies-gt-zero M SM>0) %. +%worlds () (succ-implies-gt-zero M SM>0) %. +%total M (succ-implies-gt-zero M _) %. +%theorem succ-preserves-gt : forall* {M} {N} forall {G1 gt M N} exists {G2 gt (s M) (s N)} true %. +%term _ succ-preserves-gt gt/1 gt/1 %. +%term _ %pi (succ-preserves-gt (gt/> M>N) (gt/> SM>SN)) %<- (succ-preserves-gt M>N SM>SN) %. +%worlds () (succ-preserves-gt M>N SM>SN) %. +%total G1 (succ-preserves-gt G1 _) %. +%theorem succ-preserves-gt-converse : forall* {M} {N} forall {G1 gt (s M) (s N)} exists {G2 gt M N} true %. +%term _ succ-preserves-gt-converse gt/1 gt/1 %. +%term _ + %pi (succ-preserves-gt-converse (gt/> SM>SN) (gt/> M>N)) + %<- (succ-preserves-gt-converse SM>SN M>N) %. +%worlds () (succ-preserves-gt-converse SM>SN M>N) %. +%total G1 (succ-preserves-gt-converse G1 _) %. +%theorem gt-implies-positive : forall* {M} {N} forall {G gt M N} exists {M'} {E eq M (s M')} true %. +%term _ gt-implies-positive gt/1 M eq/ %. +%term _ gt-implies-positive (gt/> (%the (gt M N) G)) M eq/ %. +%worlds () (gt-implies-positive M>N M' M=sM') %. +%total {} (gt-implies-positive _ _ _) %. +%theorem gt-anti-reflexive* : forall {M} {G gt M M} exists {F void} true %. +%term _ + %pi (gt-anti-reflexive* (s M) (%the (gt (s M) (s M)) G) F) + %<- (succ-preserves-gt-converse G G') + %<- (gt-anti-reflexive* M G' F) %. +%worlds () (gt-anti-reflexive* M M>M _) %. +%total M (gt-anti-reflexive* M _ _) %. +%inline gt-anti-reflexive gt-anti-reflexive* _ %. +%theorem gt-transitive : forall* {M} {N} {P} forall {G1 gt M N} {G2 gt N P} exists {G3 gt M P} true %. +%term _ gt-transitive gt/1 G (gt/> G) %. +%term _ %pi (gt-transitive (gt/> M>N) N>P (gt/> M>P)) %<- (gt-transitive M>N N>P M>P) %. +%worlds () (gt-transitive M>N N>P M>P) %. +%total (G1) (gt-transitive G1 _ _) %. +%theorem gt-anti-symmetric : forall* {M} {N} forall {G1 gt M N} {G2 gt N M} exists {F void} true %. +%term _ + %pi (gt-anti-symmetric M>N N>M F) + %<- (gt-transitive M>N N>M M>M) + %<- (gt-anti-reflexive M>M F) %. +%worlds () (gt-anti-symmetric M>N N>M _) %. +%total {} (gt-anti-symmetric _ _ _) %. +%theorem gt-implies-plus : forall* {M} {N} forall {G gt M N} exists {D} {P plus (s D) N M} true %. +%term _ gt-implies-plus gt/1 z (plus/s plus/z) %. +%term _ + %pi (gt-implies-plus (gt/> M>N) (s D) (plus/s SD+N=M)) + %<- (gt-implies-plus M>N D SD+N=M) %. +%worlds () (gt-implies-plus M>N D SD+N=M) %. +%total G (gt-implies-plus G _ _) %. +%theorem gt-contradiction : forall* {M} forall {P gt z M} exists {Q void} true %. +%worlds () (gt-contradiction ZERO>N _) %. +%total {} (gt-contradiction _ _) %. +%%% Theorems about compare +%theorem false-implies-compare : forall* {M} {N} {C} forall {P void} exists {Q compare M N C} true %. +%worlds () (false-implies-compare _ _) %. +%total {} (false-implies-compare _ _) %. +%theorem succ-preserves-compare : forall* {M} {N} {C} forall {CMP compare M N C} exists {CMP' compare (s M) (s N) C} true %. +%term _ succ-preserves-compare compare/= compare/= %. +%term _ + %pi (succ-preserves-compare (compare/< M>N) (compare/< M+1>N+1)) + %<- (succ-preserves-gt M>N M+1>N+1) %. +%term _ + %pi (succ-preserves-compare (compare/> M>N) (compare/> M+1>N+1)) + %<- (succ-preserves-gt M>N M+1>N+1) %. +%worlds () (succ-preserves-compare _ _) %. +%total {} (succ-preserves-compare _ _) %. +%theorem compare-total* : forall {M} {N} exists {CMP} {P compare M N CMP} true %. +%term _ compare-total* z z equal compare/= %. +%term _ + %pi (compare-total* z (s M) less (compare/< M+1>0)) + %<- (succ-implies-gt-zero M M+1>0) %. +%term _ + %pi (compare-total* (s M) z greater (compare/> M+1>0)) + %<- (succ-implies-gt-zero M M+1>0) %. +%term _ + %pi (compare-total* (s M) (s N) R M+1-R-N+1) + %<- (compare-total* M N R M-R-N) + %<- (succ-preserves-compare M-R-N M+1-R-N+1) %. +%worlds () (compare-total* _ _ _ _) %. +%total (M) (compare-total* M _ _ _) %. +%inline compare-total compare-total* _ _ _ %. +%theorem greater-implies-gt : forall* {M} {N} forall {C compare M N greater} exists {G gt M N} true %. +%term _ greater-implies-gt (compare/> G) G %. +%worlds () (greater-implies-gt M>N M-gt-N) %. +%total C (greater-implies-gt C _) %. +%theorem less-is-reverse-greater : forall* {M} {N} forall {C1 compare M N less} exists {C2 compare N M greater} true %. +%term _ less-is-reverse-greater (compare/< G) (compare/> G) %. +%worlds () (less-is-reverse-greater MM) %. +%total C (less-is-reverse-greater C _) %. +%theorem less-implies-lt : forall* {M} {N} forall {C compare M N less} exists {G gt N M} true %. +%term _ less-implies-lt (compare/< G) G %. +%worlds () (less-implies-lt MN2 plus/z plus/z N1>N2 %. +%term _ + %pi (plus-left-preserves-gt* N1>N2 (plus/s M+N1=O1) (plus/s M+N2=O2) SO1>SO2) + %<- (plus-left-preserves-gt* N1>N2 M+N1=O1 M+N2=O2 O1>O2) + %<- (succ-preserves-gt O1>O2 SO1>SO2) %. +%worlds () (plus-left-preserves-gt* N1>N2 M+N1=O1 M+N2=O2 O1>O2) %. +%total P1 (plus-left-preserves-gt* _ P1 _ _) %. +%theorem plus-left-cancels-gt : forall* {X1 nat} {X2 nat} {Y nat} {Z nat} {S1 nat} {S2 nat} forall {P1 plus X1 Y S1} {P2 plus X2 Z S2} {EX eq X1 X2} {G1 gt S1 S2} exists {G2 gt Y Z} true %. +%term _ plus-left-cancels-gt plus/z plus/z eq/ G G %. +%term _ + %pi (plus-left-cancels-gt (plus/s X+Y1=Z1) (plus/s X+Y2=Z2) eq/ SZ1>SZ2 Y1>Y2) + %<- (succ-preserves-gt-converse SZ1>SZ2 Z1>Z2) + %<- (plus-left-cancels-gt X+Y1=Z1 X+Y2=Z2 eq/ Z1>Z2 Y1>Y2) %. +%worlds () (plus-left-cancels-gt X1+Y1=Z1 X2+Y2=Z2 X1=X2 Z1>Z2 Y1>Y2) %. +%total P1 (plus-left-cancels-gt P1 _ _ _ _) %. +%theorem plus-left-preserves-gt : forall* {X1} {X2} {X4} forall {G gt X2 X4} exists {X3} {X5} {O1 plus X1 X2 X3} {O2 plus X1 X4 X5} {G2 gt X3 X5} true %. +%term _ + %pi (plus-left-preserves-gt X2>X4 X3 X5 X1+X2=A3 X1+X4=X5 X3>X5) + %<- (plus-total X1+X2=A3) + %<- (plus-total X1+X4=X5) + %<- (plus-left-preserves-gt* X2>X4 X1+X2=A3 X1+X4=X5 X3>X5) %. +%worlds () (plus-left-preserves-gt X2>X4 X3 X5 X1+X2=A3 X1+X4=X5 X3>X5) %. +%total {} (plus-left-preserves-gt _ _ _ _ _ _) %. +%theorem plus-right-preserves-gt* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 gt X1 X2} {O1 plus X1 X3 X4} {O2 plus X2 X3 X5} exists {G2 gt X4 X5} true %. +%term _ + %pi (plus-right-preserves-gt* X1>X2 X1+X3=X4 X2+X3=X5 X4>X5) + %<- (plus-commutative X1+X3=X4 X3+X1=X4) + %<- (plus-commutative X2+X3=X5 X3+X2=X5) + %<- (plus-left-preserves-gt* X1>X2 X3+X1=X4 X3+X2=X5 X4>X5) %. +%worlds () (plus-right-preserves-gt* X1>X2 X1+X3=X4 X2+X3=X5 X4>X5) %. +%total {} (plus-right-preserves-gt* _ _ _ _) %. +%theorem plus-right-preserves-gt : forall* {X1} {X2} {X3} forall {G1 gt X1 X2} exists {X4} {X5} {O1 plus X1 X3 X4} {O2 plus X2 X3 X5} {G2 gt X4 X5} true %. +%term _ + %pi (plus-right-preserves-gt X1>X2 X4 X5 X1+X3=X4 X2+X3=X5 X4>X5) + %<- (plus-total X1+X3=X4) + %<- (plus-total X2+X3=X5) + %<- (plus-right-preserves-gt* X1>X2 X1+X3=X4 X2+X3=X5 X4>X5) %. +%worlds () (plus-right-preserves-gt X1>X2 X4 X5 X1+X3=X4 X2+X3=X5 X4>X5) %. +%total {} (plus-right-preserves-gt _ _ _ _ _ _) %. +%theorem plus-preserves-gt* : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {G1 gt X1 Y1} {G2 gt X2 Y2} {MX plus X1 X2 X3} {MY plus Y1 Y2 Y3} exists {G3 gt X3 Y3} true %. +%term _ + %pi (plus-preserves-gt* X1>Y1 X2>Y2 X1+X2=X3 Y1+Y2=Y3 X3>Y3) + %<- (plus-total Y1+X2=X) + %<- (plus-right-preserves-gt* X1>Y1 X1+X2=X3 Y1+X2=X X3>X) + %<- (plus-left-preserves-gt* X2>Y2 Y1+X2=X Y1+Y2=Y3 X>Y3) + %<- (gt-transitive X3>X X>Y3 X3>Y3) %. +%worlds () (plus-preserves-gt* X1>Y1 X2>Y2 X1+X2=X3 Y1+Y2=Y3 X3>Y3) %. +%total {} (plus-preserves-gt* _ _ _ _ _) %. +%theorem plus-preserves-gt : forall* {X1} {X2} {Y1} {Y2} forall {G1 gt X1 Y1} {G2 gt X2 Y2} exists {X3} {Y3} {MX plus X1 X2 X3} {MY plus Y1 Y2 Y3} {G3 gt X3 Y3} true %. +%term _ + %pi (plus-preserves-gt X1>Y1 X2>Y2 X3 Y3 X1+X2=X3 Y1+Y2=Y3 X3>Y3) + %<- (plus-total X1+X2=X3) + %<- (plus-total Y1+Y2=Y3) + %<- (plus-preserves-gt* X1>Y1 X2>Y2 X1+X2=X3 Y1+Y2=Y3 X3>Y3) %. +%worlds () (plus-preserves-gt X1>Y1 X2>Y2 X3 Y3 X1+X2=X3 Y1+Y2=Y3 X3>Y3) %. +%total {} (plus-preserves-gt _ _ _ _ _ _ _) %. +%theorem plus-right-cancels-gt : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {OP1 plus X1 X2 X3} {OP2 plus Y1 Y2 Y3} {E2 eq X2 Y2} {G3 gt X3 Y3} exists {G1 gt X1 Y1} true %. +%term _ + %pi (plus-right-cancels-gt X1+X2=X3 Y1+Y2=Y3 X2=Y2 X3>Y3 X1>Y1) + %<- (plus-commutative X1+X2=X3 X2+X1=X3) + %<- (plus-commutative Y1+Y2=Y3 Y2+Y1=Y3) + %<- (plus-left-cancels-gt X2+X1=X3 Y2+Y1=Y3 X2=Y2 X3>Y3 X1>Y1) %. +%worlds () (plus-right-cancels-gt X1+X2=X3 Y1+Y2=Y3 X2=Y2 X3>Y3 X1>Y1) %. +%total {} (plus-right-cancels-gt _ _ _ _ _) %. +%theorem plus-implies-gt : forall* {M} {N} {O} {M'} forall {P plus M N O} {E eq M (s M')} exists {G gt O N} true %. +%term _ + %pi (plus-implies-gt X+Y=Z eq/ Z>Y) + %<- (succ-implies-gt-zero _ X>0) + %<- (plus-right-preserves-gt* X>0 X+Y=Z plus/z Z>Y) %. +%worlds () (plus-implies-gt X+Y=Z X=sX' Z>Y) %. +%total {} (plus-implies-gt _ _ _) %. +%theorem plus-gt-contradiction : forall* {M} {N} {O} forall {P plus M N O} {G gt M O} exists {F void} true %. +%term _ + %pi (plus-gt-contradiction M+0=O M>O F) + %<- (plus-right-identity _ M+0=M) + %<- (plus-deterministic M+0=O M+0=M eq/ eq/ O=M) + %<- (gt-respects-eq M>O eq/ O=M M>M) + %<- (gt-anti-reflexive M>M F) %. +%term _ + %pi (plus-gt-contradiction M+N=O M>O F) + %<- (plus-commutative M+N=O N+M=O) + %<- (plus-implies-gt N+M=O eq/ O>M) + %<- (gt-anti-symmetric M>O O>M F) %. +%worlds () (plus-gt-contradiction M+N=O M>O _) %. +%total {} (plus-gt-contradiction _ _ _) %. +%%% Theorems about times +%theorem false-implies-times : forall* {M} {N} {O} forall {P void} exists {Q times M N O} true %. +%worlds () (false-implies-times _ _) %. +%total {} (false-implies-times _ _) %. +%theorem times-respects-eq : forall* {M1 nat} {M2 nat} {N1 nat} {N2 nat} {P1 nat} {P2 nat} forall {P times M1 N1 P1} {E1 eq M1 M2} {E2 eq N1 N2} {E3 eq P1 P2} exists {Q times M2 N2 P2} true %. +%term _ times-respects-eq M1*N1=P1 eq/ eq/ eq/ M1*N1=P1 %. +%worlds () (times-respects-eq M1*N1=P1 M1=M2 N1=N2 P1=P2 M2*N2=P2) %. +%total {} (times-respects-eq _ _ _ _ _) %. +%theorem times-total* : forall {N1 nat} {N2 nat} exists {N3 nat} {T times N1 N2 N3} true %. +%term _ times-total* z N2 z times/z %. +%term _ + %pi (times-total* (s X) Y Z (times/s X*Y=Z' Z'+Y=Z)) + %<- (times-total* X Y Z' X*Y=Z') + %<- (plus-total Z'+Y=Z) %. +%worlds () (times-total* N1 N2 N3 N1*N2=N3) %. +%total (N1) (times-total* N1 _ _ _) %. +%inline times-total times-total* _ _ _ %. +%theorem times-deterministic : forall* {N1 nat} {N1' nat} {N2 nat} {N2' nat} {N3 nat} {N3' nat} forall {P times N1 N2 N3} {P' times N1' N2' N3'} {E1 eq N1 N1'} {E2 eq N2 N2'} exists {E3 eq N3 N3'} true %. +%term _ times-deterministic times/z times/z eq/ eq/ eq/ %. +%term _ + %pi (times-deterministic (times/s X*Y=Z1 Z1+Y=Z1') (times/s X*Y=Z2 Z2+Y=Z2') eq/ eq/ Z1'=Z2') + %<- (times-deterministic X*Y=Z1 X*Y=Z2 eq/ eq/ Z1=Z2) + %<- (plus-deterministic Z1+Y=Z1' Z2+Y=Z2' Z1=Z2 eq/ Z1'=Z2') %. +%worlds () (times-deterministic X1*Y1=Z1 X2*Y2=Z2 X1=X2 Y1=Y2 Z1=Z2) %. +%total P (times-deterministic P _ _ _ _) %. +%theorem times-left-identity : forall {N nat} exists {T times (s z) N N} true %. +%term _ times-left-identity N (times/s times/z plus/z) %. +%worlds () (times-left-identity N ONE*N=N) %. +%total {} (times-left-identity _ _) %. +%theorem times-right-identity : forall {N nat} exists {T times N (s z) N} true %. +%term _ times-right-identity z times/z %. +%term _ + %pi (times-right-identity (s M) (times/s M*1=M M+1=sM)) + %<- (times-right-identity M M*1=M) + %<- (plus-right-identity M M+0=M) + %<- (plus-right-increase M+0=M M+1=sM) %. +%worlds () (times-right-identity N N*1=N) %. +%total M (times-right-identity M _) %. +%theorem times-right-zero : forall {N nat} exists {T times N z z} true %. +%term _ times-right-zero z times/z %. +%term _ + %pi (times-right-zero (s M) (times/s M*0=0 plus/z)) + %<- (times-right-zero M M*0=0) %. +%worlds () (times-right-zero N N*0=0) %. +%total M (times-right-zero M _) %. +%theorem times-preserves-positive : forall {M} {N} exists {P} {T times (s M) (s N) (s P)} true %. +%term _ {T1 times M (s N) O} {P1 plus O (s N) (s P)} {P2 plus O N P} + %pi (times-preserves-positive M N P (times/s T1 P1)) + %<- (times-total T1) + %<- (plus-total P2) + %<- (plus-right-increase P2 P1) %. +%worlds () (times-preserves-positive M N P SM*SN=SP) %. +%total {} (times-preserves-positive _ _ _ _) %. +%theorem times-preserves-positive* : forall* {M} {N} {P} {M'} {N'} forall {T times M N P} {M+ eq M (s M')} {N+ eq N (s N')} exists {P'} {P+ eq P (s P')} true %. +%term _ + %pi (times-preserves-positive* M*N=P M=sM' N=sN' P' P=sP') + %<- (times-respects-eq M*N=P M=sM' N=sN' eq/ (times/s M'*sN'=O' O'+sN'=P)) + %<- (plus-right-decrease O'+sN'=P P' P=sP' _) %. +%worlds () (times-preserves-positive* M*N=P M=sM' N=sN' P' P=sP') %. +%total {} (times-preserves-positive* _ _ _ _ _) %. +%theorem times-positive-implies-positive : forall* {M} {N} {P} {P'} forall {T times M N P} {P+ eq P (s P')} exists {M'} {M+ eq M (s M')} {N'} {N+ eq N (s N')} true %. +%term _ times-positive-implies-positive (times/s M'*N=T plus/z) eq/ M' eq/ P' eq/ %. +%term _ + %pi (times-positive-implies-positive (times/s M'*N=sT' (plus/s T'+N=P')) eq/ M' eq/ N' N=sN') + %<- (times-positive-implies-positive M'*N=sT' eq/ _ _ N' N=sN') %. +%worlds () (times-positive-implies-positive M*N=P P=sP' M' M=sM' N' N=sN') %. +%total T (times-positive-implies-positive T _ _ _ _ _) %. +%theorem times-left-increase : forall* {M} {N} {O} {X} forall {T times M N O} {P plus O N X} exists {U times (s M) N X} true %. +%term _ times-left-increase T P (times/s T P) %. +%worlds () (times-left-increase M*N=O O+N=X SM*N=X) %. +%total {} (times-left-increase _ _ _) %. +%theorem times-right-increase : forall* {M nat} {N nat} {O nat} {X nat} forall {T times M N O} {P plus M O X} exists {U times M (s N) X} true %. +%term _ times-right-increase times/z plus/z times/z %. +%term _ {M*N=O times M N O} {M+O=Z plus M O Z} + %pi (times-right-increase (times/s M*N=O O+N=O1) (plus/s M+O1=Y) (times/s M*sN=Z Z+sN=sY)) + %<- (plus-associative-converse O+N=O1 M+O1=Y Z M+O=Z Z+N=Y) + %<- (times-right-increase M*N=O M+O=Z M*sN=Z) + %<- (plus-right-increase Z+N=Y Z+sN=sY) %. +%worlds () (times-right-increase M*N=O M+O=X M*sN=X) %. +%total T (times-right-increase T _ _) %. +%theorem times-left-decrease : forall* {X} {Y} {Z} forall {T1 times (s X) Y Z} exists {Z1} {T2 times X Y Z1} {P plus Z1 Y Z} true %. +%term _ times-left-decrease (times/s T P) _ T P %. +%worlds () (times-left-decrease SX*Y=Z Z1 X*Y=Z1 Z1+Y=Z) %. +%total {} (times-left-decrease _ _ _ _) %. +%theorem times-right-decrease : forall* {M} {N} {X} forall {T times M (s N) X} exists {O} {U times M N O} {P plus M O X} true %. +%term _ times-right-decrease times/z z times/z plus/z %. +%term _ + %pi (times-right-decrease (times/s M*sN=Y Y+sN=X) O (times/s M*N=P P+N=O) SM+O=X) + %<- (times-right-decrease M*sN=Y P M*N=P M+P=Y) + %<- (plus-total P+N=O) + %<- (plus-right-increase P+N=O P+sN=sO) + %<- (plus-associative* M+P=Y Y+sN=X P+sN=sO M+sO=X) + %<- (plus-swap-succ-converse M+sO=X SM+O=X) %. +%worlds () (times-right-decrease M*sN=X O M*N=O M+O=X) %. +%total (T) (times-right-decrease T _ _ _) %. +%theorem times-commutative : forall* {N1} {N2} {N3} forall {T times N1 N2 N3} exists {U times N2 N1 N3} true %. +%term _ %pi (times-commutative times/z T) %<- (times-right-zero N2 T) %. +%term _ {T1 times N1' N2 N3'} {P2 plus N3' N2 N3} {T1c times N2 N1' N3'} {P2c plus N2 N3' N3} {Tc times N2 (s N1') N3} + %pi (times-commutative (times/s T1 P2) Tc) + %<- (plus-commutative P2 P2c) + %<- (times-commutative T1 T1c) + %<- (times-right-increase T1c P2c Tc) %. +%worlds () (times-commutative N1*N2=N3 N2*N1=N3) %. +%total T (times-commutative T _) %. +%theorem times-right-distributes-over-plus : forall* {N1} {N2} {N3} {N12} {N123} forall {P1 plus N1 N2 N12} {T1 times N12 N3 N123} exists {N13} {N23} {T13 times N1 N3 N13} {T23 times N2 N3 N23} {P123 plus N13 N23 N123} true %. +%term _ times-right-distributes-over-plus plus/z Y*Z=YZ z YZ times/z Y*Z=YZ plus/z %. +%term _ + %pi (times-right-distributes-over-plus (plus/s X+Y=XY) (times/s XY*Z=XYZ XYZ+Z=SXYZ) SXZ YZ (times/s X*Z=XZ XZ+Z=SXZ) Y*Z=YZ SXZ+YZ=SXYZ) + %<- (times-right-distributes-over-plus X+Y=XY XY*Z=XYZ XZ YZ X*Z=XZ Y*Z=YZ XZ+YZ=XYZ) + %<- (plus-commutative XZ+YZ=XYZ YZ+XZ=XYZ) + %<- (plus-associative YZ+XZ=XYZ XYZ+Z=SXYZ SXZ XZ+Z=SXZ YZ+SXZ=SXYZ) + %<- (plus-commutative YZ+SXZ=SXYZ SXZ+YZ=SXYZ) %. +%worlds () (times-right-distributes-over-plus X+Y=XY XY*Z=XYZ XZ YZ X*Z=XZ Y*Z=YZ XZ+YZ=XYZ) %. +%total (P) (times-right-distributes-over-plus P _ _ _ _ _ _) %. +%theorem times-right-distributes-over-plus* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {A12 plus X1 X2 X3} {M34 times X3 X4 X7} {M14 times X1 X4 X5} {M24 times X2 X4 X6} exists {A56 plus X5 X6 X7} true %. +%term _ + %pi (times-right-distributes-over-plus* X1+X2=X3 X3*X4=X7 X1*X4=X5 X2*X4=X6 X5+X6=X7) + %<- (times-right-distributes-over-plus X1+X2=X3 X3*X4=X7 Y5 Y6 X1*X4=Y5 X2*X4=Y6 Y5+Y6=X7) + %<- (times-deterministic X1*X4=Y5 X1*X4=X5 eq/ eq/ Y5=X5) + %<- (times-deterministic X2*X4=Y6 X2*X4=X6 eq/ eq/ Y6=X6) + %<- (plus-respects-eq Y5+Y6=X7 Y5=X5 Y6=X6 eq/ X5+X6=X7) %. +%worlds () (times-right-distributes-over-plus* X1+X2=X3 X3*X4=X7 X1*X4=X5 X2*X4=X6 X5+X6=X7) %. +%total {} (times-right-distributes-over-plus* _ _ _ _ _) %. +%theorem times-left-distributes-over-plus* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {A12 plus X2 X4 X6} {M34 times X1 X6 X7} {M14 times X1 X2 X3} {M24 times X1 X4 X5} exists {A56 plus X3 X5 X7} true %. +%term _ + %pi (times-left-distributes-over-plus* X2+X4=X6 X1*X6=X7 X1*X2=X3 X1*X4=X5 X3+X5=X7) + %<- (times-commutative X1*X6=X7 X6*X1=X7) + %<- (times-commutative X1*X2=X3 X2*X1=X3) + %<- (times-commutative X1*X4=X5 X4*X1=X5) + %<- (times-right-distributes-over-plus* X2+X4=X6 X6*X1=X7 X2*X1=X3 X4*X1=X5 X3+X5=X7) %. +%worlds () (times-left-distributes-over-plus* X2+X4=X6 X1*X6=X7 X1*X2=X3 X1*X4=X5 X3+X5=X7) %. +%total {} (times-left-distributes-over-plus* _ _ _ _ _) %. +%theorem times-left-distributes-over-plus : forall* {X1} {X2} {X4} {X6} {X7} forall {A12 plus X2 X4 X6} {M34 times X1 X6 X7} exists {X3} {X5} {M14 times X1 X2 X3} {M24 times X1 X4 X5} {A56 plus X3 X5 X7} true %. +%term _ + %pi (times-left-distributes-over-plus X2+X4=X6 X1*X6=X7 X3 X5 X1*X2=X3 X1*X4=X5 X3+X5=X7) + %<- (times-total X1*X2=X3) + %<- (times-total X1*X4=X5) + %<- (times-left-distributes-over-plus* X2+X4=X6 X1*X6=X7 X1*X2=X3 X1*X4=X5 X3+X5=X7) %. +%worlds () (times-left-distributes-over-plus X2+X4=X6 X1*X6=X7 X3 X5 X1*X2=X3 X1*X4=X5 X3+X5=X7) %. +%total {} (times-left-distributes-over-plus _ _ _ _ _ _ _) %. +%theorem times-right-factors-over-plus : forall* {X1} {X2} {X4} {X5} {X6} {X7} forall {M14 times X1 X4 X5} {M24 times X2 X4 X6} {A56 plus X5 X6 X7} exists {X3} {A12 plus X1 X2 X3} {M34 times X3 X4 X7} true %. +%term _ + %pi (times-right-factors-over-plus X1*X4=X5 X2*X4=X6 X5+X6=X7 X3 X1+X2=X3 X3*X4=X7) + %<- (plus-total X1+X2=X3) + %<- (times-total X3*X4=Y7) + %<- (times-right-distributes-over-plus* X1+X2=X3 X3*X4=Y7 X1*X4=X5 X2*X4=X6 X5+X6=Y7) + %<- (plus-deterministic X5+X6=Y7 X5+X6=X7 eq/ eq/ Y7=X7) + %<- (times-respects-eq X3*X4=Y7 eq/ eq/ Y7=X7 X3*X4=X7) %. +%worlds () (times-right-factors-over-plus X1*X4=X5 X2*X4=X6 X5+X6=X7 X3 X1+X2=X3 X3*X4=X7) %. +%total {} (times-right-factors-over-plus _ _ _ _ _ _) %. +%theorem times-right-factors-over-plus* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {M14 times X1 X4 X5} {M24 times X2 X4 X6} {A56 plus X5 X6 X7} {A12 plus X1 X2 X3} exists {M34 times X3 X4 X7} true %. +%term _ + %pi (times-right-factors-over-plus* X1*X4=X5 X2*X4=X6 X5+X6=X7 X1+X2=X3 X3*X4=X7) + %<- (times-total X3*X4=Y7) + %<- (times-right-distributes-over-plus* X1+X2=X3 X3*X4=Y7 X1*X4=X5 X2*X4=X6 X5+X6=Y7) + %<- (plus-deterministic X5+X6=Y7 X5+X6=X7 eq/ eq/ Y7=X7) + %<- (times-respects-eq X3*X4=Y7 eq/ eq/ Y7=X7 X3*X4=X7) %. +%worlds () (times-right-factors-over-plus* X1*X4=X5 X2*X4=X6 X5+X6=X7 X1+X2=X3 X3*X4=X7) %. +%total {} (times-right-factors-over-plus* _ _ _ _ _) %. +%theorem times-left-factors-over-plus : forall* {X1} {X2} {X3} {X4} {X5} {X7} forall {M12 times X1 X2 X3} {M14 times X1 X4 X5} {A35 plus X3 X5 X7} exists {X6} {A24 plus X2 X4 X6} {M16 times X1 X6 X7} true %. +%term _ + %pi (times-left-factors-over-plus X1*X2=X3 X1*X4=X5 X3+X5=X7 X6 X2+X4=X6 X1*X6=X7) + %<- (times-commutative X1*X2=X3 X2*X1=X3) + %<- (times-commutative X1*X4=X5 X4*X1=X5) + %<- (times-right-factors-over-plus X2*X1=X3 X4*X1=X5 X3+X5=X7 X6 X2+X4=X6 X6*X1=X7) + %<- (times-commutative X6*X1=X7 X1*X6=X7) %. +%worlds () (times-left-factors-over-plus X1*X2=X3 X1*X4=X5 X3+X5=X7 X6 X2+X4=X6 X1*X6=X7) %. +%total {} (times-left-factors-over-plus _ _ _ _ _ _) %. +%theorem times-left-factors-over-plus* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {M12 times X1 X2 X3} {M14 times X1 X4 X5} {A35 plus X3 X5 X7} {A24 plus X2 X4 X6} exists {M16 times X1 X6 X7} true %. +%term _ + %pi (times-left-factors-over-plus* X1*X2=X3 X1*X4=X5 X3+X5=X7 X2+X4=X6 X1*X6=X7) + %<- (times-total X1*X6=Y7) + %<- (times-left-distributes-over-plus* X2+X4=X6 X1*X6=Y7 X1*X2=X3 X1*X4=X5 X3+X5=Y7) + %<- (plus-deterministic X3+X5=Y7 X3+X5=X7 eq/ eq/ Y7=X7) + %<- (times-respects-eq X1*X6=Y7 eq/ eq/ Y7=X7 X1*X6=X7) %. +%worlds () (times-left-factors-over-plus* X1*X2=X3 X1*X4=X5 X3+X5=X7 X2+X4=X6 X1*X6=X7) %. +%total {} (times-left-factors-over-plus* _ _ _ _ _) %. +%theorem times-associative : forall* {N1} {N2} {N3} {N12} {N123} forall {T1 times N1 N2 N12} {T12 times N12 N3 N123} exists {N23} {T2 times N2 N3 N23} {T123 times N1 N23 N123} true %. +%term _ {T2 times N2 N3 N23} %pi (times-associative times/z times/z N23 T2 times/z) %<- (times-total T2) %. +%term _ {T1 times N1' N2 N1'2} {P2 plus N1'2 N2 N12} {T3 times N12 N3 N123} {T4 times N2 N3 N23} {T5 times N1' N23 N1'23} {P6 plus N1'23 N23 N123} {T7 times N1'2 N3 N1'23} + %pi (times-associative (times/s T1 P2) T3 N23 T4 (times/s T5 P6)) + %<- (times-right-distributes-over-plus P2 T3 N1'23 N23 T7 T4 P6) + %<- (times-associative T1 T7 N23' T4' T5') + %<- (times-deterministic T4' T4 eq/ eq/ N23'=N23) + %<- (times-respects-eq T5' eq/ N23'=N23 eq/ T5) %. +%worlds () (times-associative _ _ _ _ _) %. +%total T1 (times-associative T1 _ _ _ _) %. +%theorem times-associative* : forall* {X1} {X2} {X12} {X3} {X23} {X123} forall {OP12 times X1 X2 X12} {OP12-3 times X12 X3 X123} {OP23 times X2 X3 X23} exists {OP1-23 times X1 X23 X123} true %. +%term _ + %pi (times-associative* X1*X2=X3 X3*X4=X7 X2*X4=X6 X1*X6=X7) + %<- (times-associative X1*X2=X3 X3*X4=X7 Y6 X2*X4=Y6 X1*Y6=X7) + %<- (times-deterministic X2*X4=Y6 X2*X4=X6 eq/ eq/ Y6=X6) + %<- (times-respects-eq X1*Y6=X7 eq/ Y6=X6 eq/ X1*X6=X7) %. +%worlds () (times-associative* _ _ _ _) %. +%total {} (times-associative* _ _ _ _) %. +%theorem times-associative-converse : forall* {X1} {X2} {X4} {X6} {X7} forall {OP24 times X2 X4 X6} {OP16 times X1 X6 X7} exists {X3} {OP12 times X1 X2 X3} {OP34 times X3 X4 X7} true %. +%term _ + %pi (times-associative-converse X2*X4=X6 X1*X6=X7 _ X1*X2=X3 X3*X4=X7) + %<- (times-commutative X2*X4=X6 X4*X2=X6) + %<- (times-commutative X1*X6=X7 X6*X1=X7) + %<- (times-associative X4*X2=X6 X6*X1=X7 _ X2*X1=X3 X4*X3=X7) + %<- (times-commutative X2*X1=X3 X1*X2=X3) + %<- (times-commutative X4*X3=X7 X3*X4=X7) %. +%worlds () (times-associative-converse X2*X4=X6 X1*X6=X7 X3 X1*X2=X3 X3*X4=X7) %. +%total {} (times-associative-converse _ _ _ _ _) %. +%theorem times-associative-converse* : forall* {X1} {X2} {X3} {X4} {X6} {X7} forall {OP24 times X2 X4 X6} {OP16 times X1 X6 X7} {OP12 times X1 X2 X3} exists {OP34 times X3 X4 X7} true %. +%term _ + %pi (times-associative-converse* X2*X4=X6 X1*X6=X7 X1*X2=X3 X3*X4=X7) + %<- (times-associative-converse X2*X4=X6 X1*X6=X7 X3P X1*X2=X3P X3P*X4=X7) + %<- (times-deterministic X1*X2=X3P X1*X2=X3 eq/ eq/ X3P=X3) + %<- (times-respects-eq X3P*X4=X7 X3P=X3 eq/ eq/ X3*X4=X7) %. +%worlds () (times-associative-converse* X2*X4=X6 X1*X6=X7 X1*X2=X3 X3*X4=X7) %. +%total {} (times-associative-converse* _ _ _ _) %. +%theorem times-assoc-commutative* : forall* {X1} {X2} {X3} {X4} {X5} {X7} forall {OP1 times X1 X2 X3} {OP2 times X3 X4 X7} {OP3 times X1 X4 X5} exists {OP4 times X5 X2 X7} true %. +%term _ + %pi (times-assoc-commutative* X1*X2=X3 X3*X4=X7 X1*X4=X5 X5*X2=X7) + %<- (times-associative X1*X2=X3 X3*X4=X7 X6 X2*X4=X6 X1*X6=X7) + %<- (times-commutative X2*X4=X6 X4*X2=X6) + %<- (times-associative-converse* X4*X2=X6 X1*X6=X7 X1*X4=X5 X5*X2=X7) %. +%worlds () (times-assoc-commutative* X1*X2=X3 X3*X4=X7 X1*X4=X5 X5*X2=X7) %. +%total {} (times-assoc-commutative* _ _ _ _) %. +%theorem times-assoc-commutative : forall* {X1} {X2} {X3} {X4} {X7} forall {OP1 times X1 X2 X3} {OP2 times X3 X4 X7} exists {X5} {OP3 times X1 X4 X5} {OP4 times X5 X2 X7} true %. +%term _ + %pi (times-assoc-commutative X1*X2=X3 X3*X4=X7 X5 X1*X4=X5 X5*X2=X7) + %<- (times-associative X1*X2=X3 X3*X4=X7 X6 X2*X4=X6 X1*X6=X7) + %<- (times-commutative X2*X4=X6 X4*X2=X6) + %<- (times-associative-converse X4*X2=X6 X1*X6=X7 X5 X1*X4=X5 X5*X2=X7) %. +%worlds () (times-assoc-commutative X1*X2=X3 X3*X4=X7 X5 X1*X4=X5 X5*X2=X7) %. +%total {} (times-assoc-commutative _ _ _ _ _) %. +%theorem times-double-associative* : forall* {A} {B} {C} {D} {A+B} {C+D} {A+C} {B+D} {X} forall {AB times A B A+B} {CD times C D C+D} {ABCD times A+B C+D X} {AC times A C A+C} {BD times B D B+D} exists {ACBD times A+C B+D X} true %. +%term _ + %pi (times-double-associative* X1*X2=X3 X4*X8=XC X3*XC=XF X1*X4=X5 X2*X8=XA X5*XA=XF) + %<- (times-associative X1*X2=X3 X3*XC=XF XE X2*XC=XE X1*XE=XF) + %<- (times-commutative X4*X8=XC X8*X4=XC) + %<- (times-associative-converse* X8*X4=XC X2*XC=XE X2*X8=XA XA*X4=XE) + %<- (times-commutative XA*X4=XE X4*XA=XE) + %<- (times-associative-converse* X4*XA=XE X1*XE=XF X1*X4=X5 X5*XA=XF) %. +%worlds () (times-double-associative* X1*X2=X3 X4*X8=XC X3*XC=XF X1*X4=X5 X2*X8=XA X5*XA=XF) %. +%total {} (times-double-associative* _ _ _ _ _ _) %. +%theorem times-double-associative : forall* {A} {B} {C} {D} {A+B} {C+D} {X} forall {AB times A B A+B} {CD times C D C+D} {ABCD times A+B C+D X} exists {A+C} {B+D} {AC times A C A+C} {BD times B D B+D} {ACBD times A+C B+D X} true %. +%term _ + %pi (times-double-associative X1*X2=X3 X4*X8=XC X3*XC=XF X5 XA X1*X4=X5 X2*X8=XA X5*XA=XF) + %<- (times-associative X1*X2=X3 X3*XC=XF XE X2*XC=XE X1*XE=XF) + %<- (times-commutative X4*X8=XC X8*X4=XC) + %<- (times-associative-converse X8*X4=XC X2*XC=XE XA X2*X8=XA XA*X4=XE) + %<- (times-commutative XA*X4=XE X4*XA=XE) + %<- (times-associative-converse X4*XA=XE X1*XE=XF X5 X1*X4=X5 X5*XA=XF) %. +%worlds () (times-double-associative _ _ _ _ _ _ _ _) %. +%total {} (times-double-associative _ _ _ _ _ _ _ _) %. +%theorem times-right-cancels : forall* {X1} {Y1} {Z1} {X2} {Y2} {Z2} forall {T1 times X1 (s Y1) Z1} {T2 times X2 (s Y2) Z2} {EY eq Y1 Y2} {EZ eq Z1 Z2} exists {EX eq X1 X2} true %. +%term _ times-right-cancels times/z times/z EY eq/ eq/ %. +%term _ {T1 times X1 (s Y1) Z1'} {P1 plus Z1' (s Y1) Z1} {T2 times X2 (s Y2) Z2'} {P2 plus Z2' (s Y2) Z2} {EY eq Y1 Y2} {EZ eq Z1 Z2} {EX eq X1 X2} + %pi (times-right-cancels (times/s T1 P1) (times/s T2 P2) EY EZ EX') + %<- (succ-deterministic EY EY') + %<- (plus-right-cancels P1 P2 EY' EZ EZ') + %<- (times-right-cancels T1 T2 EY EZ' EX) + %<- (succ-deterministic EX EX') %. +%worlds () (times-right-cancels X1*sY1=Z1 X2*sY2=Z2 Y1=Y2 Z1=Z2 X1=X2) %. +%total T1 (times-right-cancels T1 _ _ _ _) %. +%theorem times-right-cancels* : forall* {X1} {Y} {Y-1} {Z1} {X2} {Z2} forall {T1 times X1 Y Z1} {T2 times X2 Y Z2} {EY eq Y (s Y-1)} {EZ eq Z1 Z2} exists {EX eq X1 X2} true %. +%term _ + %pi (times-right-cancels* X1*Y=Z1 X2*Y=Z2 Y+ Z1=Z2 X1=X2) + %<- (times-respects-eq X1*Y=Z1 eq/ Y+ eq/ X1*Y+=Z1) + %<- (times-respects-eq X2*Y=Z2 eq/ Y+ eq/ X2*Y+=Z2) + %<- (times-right-cancels X1*Y+=Z1 X2*Y+=Z2 eq/ Z1=Z2 X1=X2) %. +%worlds () (times-right-cancels* X1*Y=Z1 X2*Y=Z2 Y+ Z1=Z2 X1=X2) %. +%total {} (times-right-cancels* _ _ _ _ _) %. +%theorem times-right-cancels** : forall* {X1} {Y1} {Z} {X2} {Y2} {Z-} forall {T1 times X1 Y1 Z} {T2 times X2 Y2 Z} {EY eq Y1 Y2} {EZ eq Z (s Z-)} exists {EX eq X1 X2} true %. +%term _ + %pi (times-right-cancels** X1*0=sZ X2*0=sZ eq/ eq/ X1=X2) + %<- (times-right-zero _ X1*0=0) + %<- (times-deterministic X1*0=0 X1*0=sZ eq/ eq/ ZERO=sZ) + %<- (succ-implies-gt ZERO=sZ ZERO>sZ) + %<- (gt-contradiction ZERO>sZ F) + %<- (false-implies-eq F X1=X2) %. +%term _ + %pi (times-right-cancels** X1*Y1-=sZ X2*Y1-=sZ eq/ eq/ X1=X2) + %<- (times-right-cancels X1*Y1-=sZ X2*Y1-=sZ eq/ eq/ X1=X2) %. +%worlds () (times-right-cancels** X1*Y1=Z X2*Y2=Z Y1=Y2 Z+ X1=X2) %. +%total {} (times-right-cancels** _ _ _ _ _) %. +%theorem times-left-cancels : forall* {X1} {Y1} {Z1} {X2} {Y2} {Z2} forall {T1 times (s X1) Y1 Z1} {T2 times (s X2) Y2 Z2} {E1 eq X1 X2} {E2 eq Z1 Z2} exists {F eq Y1 Y2} true %. +%term _ + %pi (times-left-cancels SX1*Y1=Z1 SX2*Y2=Z2 X1=X2 Z1=Z2 Y1=Y2) + %<- (times-commutative SX1*Y1=Z1 Y1*sX1=Z1) + %<- (times-commutative SX2*Y2=Z2 Y2*sX2=Z2) + %<- (times-right-cancels Y1*sX1=Z1 Y2*sX2=Z2 X1=X2 Z1=Z2 Y1=Y2) %. +%worlds () (times-left-cancels SX1*Y1=Z1 SX2*Y2=Z2 X1=X2 Z1=Z2 Y1=Y2) %. +%total {} (times-left-cancels _ _ _ _ _) %. +%theorem times-left-cancels* : forall* {X} {Y1} {Z1} {X-} {Y2} {Z2} forall {T1 times X Y1 Z1} {T2 times X Y2 Z2} {E1 eq X (s X-)} {E2 eq Z1 Z2} exists {F eq Y1 Y2} true %. +%term _ + %pi (times-left-cancels* X*Y1=Z1 X*Y2=Z2 X+ Z1=Z2 Y1=Y2) + %<- (times-commutative X*Y1=Z1 Y1*X=Z1) + %<- (times-commutative X*Y2=Z2 Y2*X=Z2) + %<- (times-right-cancels* Y1*X=Z1 Y2*X=Z2 X+ Z1=Z2 Y1=Y2) %. +%worlds () (times-left-cancels* X*Y1=Z1 X*Y2=Z2 X+ Z1=Z2 Y1=Y2) %. +%total {} (times-left-cancels* _ _ _ _ _) %. +%theorem times-left-preserves-gt : forall* {M} {N1} {N2} {P1} {P2} forall {GN gt N1 N2} {T1 times (s M) N1 P1} {T2 times (s M) N2 P2} exists {GP gt P1 P2} true %. +%term _ times-left-preserves-gt N1>N2 (times/s times/z plus/z) (times/s times/z plus/z) N1>N2 %. +%term _ + %pi (times-left-preserves-gt N1>N2 (times/s (%the (times (s M) N1 X1) T1) X1+N1=O1) (times/s (%the (times (s M) N2 X2) T2) X2+N2=O2) O1>O2) + %<- (times-left-preserves-gt N1>N2 T1 T2 X1>X2) + %<- (plus-preserves-gt* X1>X2 N1>N2 X1+N1=O1 X2+N2=O2 O1>O2) %. +%worlds () (times-left-preserves-gt N1>N2 SM*N1=P1 SM*N2=P2 P1>P2) %. +%total T1 (times-left-preserves-gt _ T1 _ _) %. +%theorem times-left-preserves-gt* : forall* {M} {M-} {N1} {N2} {P1} {P2} forall {GN gt N1 N2} {T1 times M N1 P1} {T2 times M N2 P2} {M+ eq M (s M-)} exists {GP gt P1 P2} true %. +%term _ + %pi (times-left-preserves-gt* N1>N2 M*N1=P1 M*N2=P2 M+ P1>P2) + %<- (times-respects-eq M*N1=P1 M+ eq/ eq/ SM-*N1=P1) + %<- (times-respects-eq M*N2=P2 M+ eq/ eq/ SM-*N2=P2) + %<- (times-left-preserves-gt N1>N2 SM-*N1=P1 SM-*N2=P2 P1>P2) %. +%worlds () (times-left-preserves-gt* N1>N2 M*N1=P1 M*N2=P2 M+ P1>P2) %. +%total {} (times-left-preserves-gt* _ _ _ _ _) %. +%theorem times-right-preserves-gt : forall* {M1} {M2} {N} {P1} {P2} forall {G1 gt M1 M2} {T1 times M1 (s N) P1} {T2 times M2 (s N) P2} exists {G2 gt P1 P2} true %. +%term _ + %pi (times-right-preserves-gt M1>M2 M1*sN=P1 M2*sN=P2 P1>P2) + %<- (times-commutative M1*sN=P1 SN*M1=P1) + %<- (times-commutative M2*sN=P2 SN*M2=P2) + %<- (times-left-preserves-gt M1>M2 SN*M1=P1 SN*M2=P2 P1>P2) %. +%worlds () (times-right-preserves-gt M1>M2 M1*sN=P1 M2*sN=P2 P1>P2) %. +%total {} (times-right-preserves-gt _ _ _ _) %. +%theorem times-right-preserves-gt* : forall* {M1} {M2} {N} {N-1} {P1} {P2} forall {G1 gt M1 M2} {T1 times M1 N P1} {T2 times M2 N P2} {N+ eq N (s N-1)} exists {G2 gt P1 P2} true %. +%term _ + %pi (times-right-preserves-gt* M1>M2 M1*N=P1 M2*N=P2 N=sN-1 P1>P2) + %<- (times-respects-eq M1*N=P1 eq/ N=sN-1 eq/ M1*N+=P1) + %<- (times-respects-eq M2*N=P2 eq/ N=sN-1 eq/ M2*N+=P2) + %<- (times-right-preserves-gt M1>M2 M1*N+=P1 M2*N+=P2 P1>P2) %. +%worlds () (times-right-preserves-gt* M1>M2 M1*N=P1 M2*N=P2 N=sN-1 P1>P2) %. +%total {} (times-right-preserves-gt* _ _ _ _ _) %. +%theorem times-preserves-gt : forall* {M1} {N1} {P1} {M2} {N2} {P2} forall {GM gt M1 M2} {GN gt N1 N2} {T1 times M1 N1 P1} {T2 times M2 N2 P2} exists {GP gt P1 P2} true %. +%term _ {0=0' eq z _} + %pi (times-preserves-gt (%the (gt M1 M2) M1>M2) (%the (gt N1 z) N1>0) (%the (times M1 N1 P1) M1*N1=P1) M2*0=0' P1>0') + %<- (gt-implies-positive M1>M2 M1' M1=sM1') + %<- (times-respects-eq M1*N1=P1 M1=sM1' eq/ eq/ SM1'*N1=P1) + %<- (times-right-zero M2 M2*0=0) + %<- (times-deterministic M2*0=0 M2*0=0' eq/ eq/ 0=0') + %<- (times-right-zero (s M1') SM1'*0=0) + %<- (times-left-preserves-gt N1>0 SM1'*N1=P1 SM1'*0=0 P1>0) + %<- (gt-respects-eq P1>0 eq/ 0=0' P1>0') %. +%term _ + %pi (times-preserves-gt M1>M2 (%the (gt N1 (s N2')) N1>sN2') M1*N1=P1 M2*sN2'=P2 P1>P2) + %<- (gt-implies-positive M1>M2 M1' M1=sM1') + %<- (times-respects-eq M1*N1=P1 M1=sM1' eq/ eq/ SM1'*N1=P1) + %<- (times-total (%the (times (s M1') (s N2') _) SM1'*sN2'=PX)) + %<- (times-left-preserves-gt N1>sN2' SM1'*N1=P1 SM1'*sN2'=PX P1>PX) + %<- (eq-symmetric M1=sM1' SM1'=M1) + %<- (times-respects-eq SM1'*sN2'=PX SM1'=M1 eq/ eq/ M1*sN2'=PX) + %<- (times-right-preserves-gt M1>M2 M1*sN2'=PX M2*sN2'=P2 PX>P2) + %<- (gt-transitive P1>PX PX>P2 P1>P2) %. +%worlds () (times-preserves-gt M1>M2 N1>N2 M1*N1=P1 M2*N2=P2 P1>P2) %. +%total {} (times-preserves-gt _ _ _ _ _) %. +%theorem times-right-cancels-gt : forall* {X1 nat} {X2 nat} {Y1 nat} {Y2 nat} {Z1 nat} {Z2 nat} forall {P1 times X1 Y1 Z1} {P2 times X2 Y2 Z2} {EY eq Y1 Y2} {G1 gt Z1 Z2} exists {G2 gt X1 X2} true %. +%term _ + %pi (times-right-cancels-gt (times/s X1*Y=N1 N1+Y=Z1) times/z eq/ Z1>0 SX1>0) + %<- (succ-implies-gt-zero _ SX1>0) %. +%term _ + %pi (times-right-cancels-gt (times/s X1*Y=N1 N1+Y=Z1) (times/s X2*Y=N2 N2+Y=Z2) eq/ Z1>Z2 SX1>SX2) + %<- (plus-right-cancels-gt N1+Y=Z1 N2+Y=Z2 eq/ Z1>Z2 N1>N2) + %<- (times-right-cancels-gt X1*Y=N1 X2*Y=N2 eq/ N1>N2 X1>X2) + %<- (succ-preserves-gt X1>X2 SX1>SX2) %. +%worlds () (times-right-cancels-gt X1*Y1=Z1 X2*Y2=Z2 Y1=Y2 Z1>Z2 X1>X2) %. +%total [P1 P2] (times-right-cancels-gt P1 P2 _ _ _) %. +%theorem times-left-cancels-gt : forall* {X1 nat} {X2 nat} {Y1 nat} {Y2 nat} {Z1 nat} {Z2 nat} forall {P1 times X1 Y1 Z1} {P2 times X2 Y2 Z2} {EX eq X1 X2} {G1 gt Z1 Z2} exists {G2 gt Y1 Y2} true %. +%term _ + %pi (times-left-cancels-gt X1*Y1=Z1 X2*Y2=Z2 X1=X2 Z1>Z2 Y1>Y2) + %<- (times-commutative X1*Y1=Z1 Y1*X1=Z1) + %<- (times-commutative X2*Y2=Z2 Y2*X2=Z2) + %<- (times-right-cancels-gt Y1*X1=Z1 Y2*X2=Z2 X1=X2 Z1>Z2 Y1>Y2) %. +%worlds () (times-left-cancels-gt X1*Y1=Z1 X2*Y2=Z2 X1=X2 Z1>Z2 Y1>Y2) %. +%total P1 (times-left-cancels-gt P1 _ _ _ _) %. +%%%%% nat-inv.elf +%%%%% Minus for natural numbers +%%%%% This file is part of the nat.elf signature +%%%% Definitions +%inline minus [X1] [X2] [X3] plus X3 X2 X1 %. +%%%% Theorems +%%% Theorems about minus +%inline false-implies-minus false-implies-plus %. +%theorem minus-respects-eq : forall* {X1} {X2} {X3} {X4} {X5} {X6} forall {D minus X1 X2 X3} {E1 eq X1 X4} {E2 eq X2 X5} {E3 eq X3 X6} exists {DP minus X4 X5 X6} true %. +%term _ minus-respects-eq S eq/ eq/ eq/ S %. +%worlds () (minus-respects-eq _ _ _ _ _) %. +%total {} (minus-respects-eq _ _ _ _ _) %. +%theorem minus-deterministic : forall* {X1} {X2} {X3} {X4} {X5} {X6} forall {S minus X1 X2 X3} {SP minus X4 X5 X6} {E1 eq X1 X4} {E2 eq X2 X5} exists {E3 eq X3 X6} true %. +%term _ + %pi (minus-deterministic X3+X2=X1 X6+X5=X4 X1=X4 X2=X5 X3=X6) + %<- (plus-right-cancels X3+X2=X1 X6+X5=X4 X2=X5 X1=X4 X3=X6) %. +%worlds () (minus-deterministic X1-X2=X3 X4-X5=X6 X1=X4 X2=X5 X3=X6) %. +%total {} (minus-deterministic _ _ _ _ _) %. +%theorem plus-associates-with-minus* : forall* {X1} {X2} {X3} {X4} {X6} {X7} forall {OP1 plus X1 X2 X3} {IOP1 minus X3 X4 X7} {IOP2 minus X2 X4 X6} exists {OP2 plus X1 X6 X7} true %. +%term _ + %pi (plus-associates-with-minus* X1+X2=X3 X7+X4=X3 X6+X4=X2 X1+X6=X7) + %<- (plus-associative-converse X6+X4=X2 X1+X2=X3 X7P X1+X6=X7P X7P+X4=X3) + %<- (plus-right-cancels X7P+X4=X3 X7+X4=X3 eq/ eq/ X7P=X7) + %<- (plus-respects-eq X1+X6=X7P eq/ eq/ X7P=X7 X1+X6=X7) %. +%worlds () (plus-associates-with-minus* X1+X2=X3 X3-X4=X7 X2-X4=X6 X1+X6=X7) %. +%total {} (plus-associates-with-minus* _ _ _ _) %. +%theorem plus-associates-with-minus-converse* : forall* {X1} {X2} {X3} {X4} {X6} {X7} forall {IOP2 minus X2 X4 X6} {OP2 plus X1 X6 X7} {OP1 plus X1 X2 X3} exists {IOP1 minus X3 X4 X7} true %. +%term _ + %pi (plus-associates-with-minus-converse* X6+X4=X2 X1+X6=X7 X1+X2=X3 X7+X4=X3) + %<- (plus-associative-converse* X6+X4=X2 X1+X2=X3 X1+X6=X7 X7+X4=X3) %. +%worlds () (plus-associates-with-minus-converse* X2-X4=X6 X1+X6=X7 X1+X2=X3 X3-X4=X7) %. +%total {} (plus-associates-with-minus-converse* _ _ _ _) %. +%theorem plus-associates-with-minus-converse : forall* {X1} {X2} {X4} {X6} {X7} forall {IOP2 minus X2 X4 X6} {OP2 plus X1 X6 X7} exists {X3} {OP1 plus X1 X2 X3} {IOP1 minus X3 X4 X7} true %. +%term _ + %pi (plus-associates-with-minus-converse X6+X4=X2 X1+X6=X7 X3 X1+X2=X3 X7+X4=X3) + %<- (plus-total X1+X2=X3) + %<- (plus-associates-with-minus-converse* X6+X4=X2 X1+X6=X7 X1+X2=X3 X7+X4=X3) %. +%worlds () (plus-associates-with-minus-converse X2-X4=X6 X1+X6=X7 X3 X1+X2=X3 X3-X4=X7) %. +%total {} (plus-associates-with-minus-converse _ _ _ _ _) %. +%theorem minus-associates-from-plus* : forall* {X1} {X2} {X3} {X4} {X6} {X7} forall {IOP1 minus X1 X2 X3} {OP1 plus X3 X4 X7} {IOP2 minus X2 X4 X6} exists {IOP3 minus X1 X6 X7} true %. +%term _ + %pi (minus-associates-from-plus* X3+X2=X1 X3+X4=X7 X6+X4=X2 X7+X6=X1) + %<- (plus-commutative X6+X4=X2 X4+X6=X2) + %<- (plus-associative-converse* X4+X6=X2 X3+X2=X1 X3+X4=X7 X7+X6=X1) %. +%worlds () (minus-associates-from-plus* X1-X2=X3 X3+X4=X7 X2-X4=X6 X1-X6=X7) %. +%total {} (minus-associates-from-plus* _ _ _ _) %. +%theorem minus-associates-from-plus-converse* : forall* {X1} {X2} {X3} {X4} {X6} {X7} forall {IOP2 minus X2 X4 X6} {IOP3 minus X1 X6 X7} {IOP1 minus X1 X2 X3} exists {OP1 plus X3 X4 X7} true %. +%term _ + %pi (minus-associates-from-plus-converse* X6+X4=X2 X7+X6=X1 X3+X2=X1 X3+X4=X7) + %<- (plus-commutative X6+X4=X2 X4+X6=X2) + %<- (plus-associative-converse X4+X6=X2 X3+X2=X1 X7P X3+X4=X7P X7P+X6=X1) + %<- (plus-right-cancels X7P+X6=X1 X7+X6=X1 eq/ eq/ X7P=X7) + %<- (plus-respects-eq X3+X4=X7P eq/ eq/ X7P=X7 X3+X4=X7) %. +%worlds () (minus-associates-from-plus-converse* X2-X4=X6 X1-X6=X7 X1-X2=X3 X3+X4=X7) %. +%total {} (minus-associates-from-plus-converse* _ _ _ _) %. +%theorem minus-associates-to-plus* : forall* {X1} {X2} {X3} {X4} {X6} {X7} forall {IOP1 minus X1 X2 X3} {IOP2 minus X3 X4 X7} {OP1 plus X2 X4 X6} exists {IOP3 minus X1 X6 X7} true %. +%term _ + %pi (minus-associates-to-plus* X3+X2=X1 X7+X4=X3 X2+X4=X6 X7+X6=X1) + %<- (plus-commutative X2+X4=X6 X4+X2=X6) + %<- (plus-associative* X7+X4=X3 X3+X2=X1 X4+X2=X6 X7+X6=X1) %. +%worlds () (minus-associates-to-plus* X1-X2=X3 X3-X4=X7 X2+X4=X6 X1-X6=X7) %. +%total {} (minus-associates-to-plus* _ _ _ _) %. +%theorem minus-associates-to-plus : forall* {X1} {X2} {X3} {X4} {X7} forall {IOP1 minus X1 X2 X3} {IOP2 minus X3 X4 X7} exists {X6} {OP1 plus X2 X4 X6} {IOP3 minus X1 X6 X7} true %. +%term _ + %pi (minus-associates-to-plus X3+X2=X1 X7+X4=X3 X6 X2+X4=X6 X7+X6=X1) + %<- (plus-associative X7+X4=X3 X3+X2=X1 X6 X4+X2=X6 X7+X6=X1) + %<- (plus-commutative X4+X2=X6 X2+X4=X6) %. +%worlds () (minus-associates-to-plus X1-X2=X3 X3-X4=X7 X6 X2+X4=X6 X1-X6=X7) %. +%total {} (minus-associates-to-plus _ _ _ _ _) %. +%theorem minus-associates-to-plus-converse* : forall* {X1} {X2} {X3} {X4} {X6} {X7} forall {OP1 plus X2 X4 X6} {IOP3 minus X1 X6 X7} {IOP1 minus X1 X2 X3} exists {IOP2 minus X3 X4 X7} true %. +%term _ + %pi (minus-associates-to-plus-converse* X2+X4=X6 X7+X6=X1 X3+X2=X1 X7+X4=X3) + %<- (plus-commutative X2+X4=X6 X4+X2=X6) + %<- (plus-associative-converse X4+X2=X6 X7+X6=X1 X3P X7+X4=X3P X3P+X2=X1) + %<- (plus-right-cancels X3P+X2=X1 X3+X2=X1 eq/ eq/ X3P=X3) + %<- (plus-respects-eq X7+X4=X3P eq/ eq/ X3P=X3 X7+X4=X3) %. +%worlds () (minus-associates-to-plus-converse* X2+X4=X6 X1-X6=X7 X1-X2=X3 X3-X4=X7) %. +%total {} (minus-associates-to-plus-converse* _ _ _ _) %. +%theorem minus-associates-to-plus-converse : forall* {X1} {X2} {X4} {X6} {X7} forall {OP1 plus X2 X4 X6} {IOP3 minus X1 X6 X7} exists {X3} {IOP1 minus X1 X2 X3} {IOP2 minus X3 X4 X7} true %. +%term _ + %pi (minus-associates-to-plus-converse X2+X4=X6 X7+X6=X1 X3 X3+X2=X1 X7+X4=X3) + %<- (plus-commutative X2+X4=X6 X4+X2=X6) + %<- (plus-associative-converse X4+X2=X6 X7+X6=X1 X3 X7+X4=X3 X3+X2=X1) %. +%worlds () (minus-associates-to-plus-converse X2+X4=X6 X1-X6=X7 X3 X1-X2=X3 X3-X4=X7) %. +%total {} (minus-associates-to-plus-converse _ _ _ _ _) %. +%theorem minus-is-zero-implies-eq : forall* {N1} {N2} {N3} forall {P minus N1 N2 N3} {E3 eq N3 z} exists {E1 eq N1 N2} true %. +%term _ minus-is-zero-implies-eq plus/z eq/ eq/ %. +%worlds () (minus-is-zero-implies-eq X-Y=Z Z=0 X=Y) %. +%total {} (minus-is-zero-implies-eq _ _ _) %. +%inline minus-implies-gt plus-implies-gt %. +%theorem minus-left-cancels : forall* {X1} {X2} {X3} {X4} {X5} {X6} forall {IOP1 minus X1 X2 X3} {IOP2 minus X4 X5 X6} {E1 eq X1 X4} {E3 eq X3 X6} exists {E2 eq X2 X5} true %. +%term _ + %pi (minus-left-cancels X3+X2=X1 X6+X5=X4 X1=X4 X3=X6 X2=X5) + %<- (plus-left-cancels X3+X2=X1 X6+X5=X4 X3=X6 X1=X4 X2=X5) %. +%worlds () (minus-left-cancels X1-X2=X3 X4-X5=X6 X1=X4 X3=X6 X2=X5) %. +%total {} (minus-left-cancels _ _ _ _ _) %. +%theorem minus-right-cancels : forall* {X1} {X2} {X3} {X4} {X5} {X6} forall {IOP1 minus X1 X2 X3} {IOP2 minus X4 X5 X6} {E2 eq X2 X5} {E3 eq X3 X6} exists {E1 eq X1 X4} true %. +%term _ + %pi (minus-right-cancels X3+X2=X1 X6+X5=X4 X2=X5 X3=X6 X1=X4) + %<- (plus-deterministic X3+X2=X1 X6+X5=X4 X3=X6 X2=X5 X1=X4) %. +%worlds () (minus-right-cancels X1-X2=X3 X4-X5=X6 X2=X5 X3=X6 X1=X4) %. +%total {} (minus-right-cancels _ _ _ _ _) %. +%theorem minus-left-inverts-gt* : forall* {X1} {X2} {X3} {X4} {X5} forall {G gt X2 X4} {IOP1 minus X1 X2 X3} {IOP2 minus X1 X4 X5} exists {GP gt X5 X3} true %. +%term _ + %pi (minus-left-inverts-gt* X2>X4 X3+X2=X1 X5+X4=X1 X5>X3) + %<- (plus-total X3+X4=X7) + %<- (plus-left-preserves-gt* X2>X4 X3+X2=X1 X3+X4=X7 X1>X7) + %<- (plus-right-cancels-gt X5+X4=X1 X3+X4=X7 eq/ X1>X7 X5>X3) %. +%worlds () (minus-left-inverts-gt* X2>X4 X1-X2=X3 X1-X4=X5 X5>X3) %. +%total {} (minus-left-inverts-gt* _ _ _ _) %. +%theorem minus-right-preserves-gt* : forall* {X1} {X2} {X3} {X4} {X5} forall {G gt X1 X2} {IOP1 minus X1 X3 X4} {IOP2 minus X2 X3 X5} exists {GP gt X4 X5} true %. +%term _ + %pi (minus-right-preserves-gt* X1>X2 X4+X3=X1 X5+X3=X2 X4>X5) + %<- (plus-right-cancels-gt X4+X3=X1 X5+X3=X2 eq/ X1>X2 X4>X5) %. +%worlds () (minus-right-preserves-gt* X1>X2 X1-X3=X4 X2-X3=X5 X4>X5) %. +%total {} (minus-right-preserves-gt* _ _ _ _) %. +%theorem minus-left-cancels-inverts-gt : forall* {X1} {X2} {X3} {X4} {X5} {X6} forall {IOP1 minus X1 X2 X3} {IOP2 minus X4 X5 X6} {E eq X1 X4} {G gt X3 X6} exists {GP gt X5 X2} true %. +%term _ + %pi (minus-left-cancels-inverts-gt X3+X2=X1 X6+X5=X4 X1=X4 X3>X6 X5>X2) + %<- (plus-total X6+X2=X7) + %<- (plus-right-preserves-gt* X3>X6 X3+X2=X1 X6+X2=X7 X1>X7) + %<- (eq-symmetric X1=X4 X4=X1) + %<- (plus-respects-eq X6+X5=X4 eq/ eq/ X4=X1 X6+X5=X1) + %<- (plus-left-cancels-gt X6+X5=X1 X6+X2=X7 eq/ X1>X7 X5>X2) %. +%worlds () (minus-left-cancels-inverts-gt X1-X2=X3 X4-X5=X6 X1=X4 X3>X6 X5>X2) %. +%total {} (minus-left-cancels-inverts-gt _ _ _ _ _) %. +%theorem minus-right-cancels-gt : forall* {X1} {X2} {X3} {X4} {X5} {X6} forall {IOP1 minus X1 X2 X3} {IOP2 minus X4 X5 X6} {E2 eq X2 X5} {G3 gt X3 X6} exists {G1 gt X1 X4} true %. +%term _ + %pi (minus-right-cancels-gt X3+X2=X1 X6+X5=X4 X2=X5 X3>X6 X1>X4) + %<- (plus-respects-eq X3+X2=X1 eq/ X2=X5 eq/ X3+X5=X1) + %<- (plus-right-preserves-gt* X3>X6 X3+X5=X1 X6+X5=X4 X1>X4) %. +%worlds () (minus-right-cancels-gt X1-X2=X3 X4-X5=X6 X2=X5 X3>X6 X1>X4) %. +%total {} (minus-right-cancels-gt _ _ _ _ _) %. +%theorem times-right-distributes-over-minus : forall* {X1} {X2} {X3} {X4} {X7} forall {S12 minus X1 X2 X3} {M34 times X3 X4 X7} exists {X5} {X6} {M14 times X1 X4 X5} {M24 times X2 X4 X6} {S56 minus X5 X6 X7} true %. +%term _ + %pi (times-right-distributes-over-minus X3+X2=X1 X3*X4=X7 _ _ X1*X4=X5 X2*X4=X6 X7+X6=X5) + %<- (times-total X1*X4=X5) + %<- (times-right-distributes-over-plus X3+X2=X1 X1*X4=X5 _ _ X3*X4=Y7 X2*X4=X6 Y7+X6=X5) + %<- (times-deterministic X3*X4=Y7 X3*X4=X7 eq/ eq/ Y7=X7) + %<- (plus-respects-eq Y7+X6=X5 Y7=X7 eq/ eq/ X7+X6=X5) %. +%worlds () (times-right-distributes-over-minus X1-X2=X3 X3*X4=X7 X5 X6 X1*X4=X5 X2*X4=X6 X5-X6=X7) %. +%total {} (times-right-distributes-over-minus _ _ _ _ _ _ _) %. +%theorem times-right-distributes-over-minus* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {A12 minus X1 X2 X3} {M34 times X3 X4 X7} {M14 times X1 X4 X5} {M24 times X2 X4 X6} exists {A56 minus X5 X6 X7} true %. +%term _ + %pi (times-right-distributes-over-minus* X1-X2=X3 X3*X4=X7 X1*X4=X5 X2*X4=X6 X5-X6=X7) + %<- (times-right-distributes-over-minus X1-X2=X3 X3*X4=X7 Y5 Y6 X1*X4=Y5 X2*X4=Y6 Y5-Y6=X7) + %<- (times-deterministic X1*X4=Y5 X1*X4=X5 eq/ eq/ Y5=X5) + %<- (times-deterministic X2*X4=Y6 X2*X4=X6 eq/ eq/ Y6=X6) + %<- (minus-respects-eq Y5-Y6=X7 Y5=X5 Y6=X6 eq/ X5-X6=X7) %. +%worlds () (times-right-distributes-over-minus* X1-X2=X3 X3*X4=X7 X1*X4=X5 X2*X4=X6 X5-X6=X7) %. +%total {} (times-right-distributes-over-minus* _ _ _ _ _) %. +%theorem times-left-distributes-over-minus* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {A12 minus X2 X4 X6} {M34 times X1 X6 X7} {M14 times X1 X2 X3} {M24 times X1 X4 X5} exists {A56 minus X3 X5 X7} true %. +%term _ + %pi (times-left-distributes-over-minus* X2-X4=X6 X1*X6=X7 X1*X2=X3 X1*X4=X5 X3-X5=X7) + %<- (times-commutative X1*X6=X7 X6*X1=X7) + %<- (times-commutative X1*X2=X3 X2*X1=X3) + %<- (times-commutative X1*X4=X5 X4*X1=X5) + %<- (times-right-distributes-over-minus* X2-X4=X6 X6*X1=X7 X2*X1=X3 X4*X1=X5 X3-X5=X7) %. +%worlds () (times-left-distributes-over-minus* X2-X4=X6 X1*X6=X7 X1*X2=X3 X1*X4=X5 X3-X5=X7) %. +%total {} (times-left-distributes-over-minus* _ _ _ _ _) %. +%theorem times-left-distributes-over-minus : forall* {X1} {X2} {X4} {X6} {X7} forall {A12 minus X2 X4 X6} {M34 times X1 X6 X7} exists {X3} {X5} {M14 times X1 X2 X3} {M24 times X1 X4 X5} {A56 minus X3 X5 X7} true %. +%term _ + %pi (times-left-distributes-over-minus X2-X4=X6 X1*X6=X7 X3 X5 X1*X2=X3 X1*X4=X5 X3-X5=X7) + %<- (times-total X1*X2=X3) + %<- (times-total X1*X4=X5) + %<- (times-left-distributes-over-minus* X2-X4=X6 X1*X6=X7 X1*X2=X3 X1*X4=X5 X3-X5=X7) %. +%worlds () (times-left-distributes-over-minus X2-X4=X6 X1*X6=X7 X3 X5 X1*X2=X3 X1*X4=X5 X3-X5=X7) %. +%total {} (times-left-distributes-over-minus _ _ _ _ _ _ _) %. +%theorem times-right-factors-over-minus* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {M14 times X1 X4 X5} {M24 times X2 X4 X6} {A56 minus X5 X6 X7} {A12 minus X1 X2 X3} exists {M34 times X3 X4 X7} true %. +%term _ + %pi (times-right-factors-over-minus* X1*X4=X5 X2*X4=X6 X5-X6=X7 X1-X2=X3 X3*X4=X7) + %<- (times-total X3*X4=Y7) + %<- (times-right-distributes-over-minus* X1-X2=X3 X3*X4=Y7 X1*X4=X5 X2*X4=X6 X5-X6=Y7) + %<- (minus-deterministic X5-X6=Y7 X5-X6=X7 eq/ eq/ Y7=X7) + %<- (times-respects-eq X3*X4=Y7 eq/ eq/ Y7=X7 X3*X4=X7) %. +%worlds () (times-right-factors-over-minus* X1*X4=X5 X2*X4=X6 X5-X6=X7 X1-X2=X3 X3*X4=X7) %. +%total {} (times-right-factors-over-minus* _ _ _ _ _) %. +%theorem times-left-factors-over-minus* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {M12 times X1 X2 X3} {M14 times X1 X4 X5} {A35 minus X3 X5 X7} {A24 minus X2 X4 X6} exists {M16 times X1 X6 X7} true %. +%term _ + %pi (times-left-factors-over-minus* X1*X2=X3 X1*X4=X5 X3-X5=X7 X2-X4=X6 X1*X6=X7) + %<- (times-total X1*X6=Y7) + %<- (times-left-distributes-over-minus* X2-X4=X6 X1*X6=Y7 X1*X2=X3 X1*X4=X5 X3-X5=Y7) + %<- (minus-deterministic X3-X5=Y7 X3-X5=X7 eq/ eq/ Y7=X7) + %<- (times-respects-eq X1*X6=Y7 eq/ eq/ Y7=X7 X1*X6=X7) %. +%worlds () (times-left-factors-over-minus* X1*X2=X3 X1*X4=X5 X3-X5=X7 X2-X4=X6 X1*X6=X7) %. +%total {} (times-left-factors-over-minus* _ _ _ _ _) %. +%theorem times-right-factors-over-minus : forall* {Y} {Z} {XY} {XZ} {YZ} {XYZ} {Z-} forall {TXY times XY Z XYZ} {TY times Y Z YZ} {M minus XYZ YZ XZ} {EZ eq Z (s Z-)} exists {X} {M' minus XY Y X} {TX times X Z XZ} true %. +% minus isn't total, so this is harder. +%term _ + %pi (times-right-factors-over-minus XY*Z=YZ Y*Z=YZ plus/z eq/ z ZERO+Y=XY times/z) + %<- (times-right-cancels* XY*Z=YZ Y*Z=YZ eq/ eq/ XY=Y) + %<- (plus-respects-eq plus/z eq/ XY=Y eq/ ZERO+Y=XY) %. +%term _ + %pi (times-right-factors-over-minus XY*Z=XYZ Y*Z=YZ XZ+YZ=XYZ _ (s X-) X+Y=XY X*Z=XZ) + %<- (plus-implies-gt XZ+YZ=XYZ eq/ XYZ>YZ) + %<- (times-right-cancels-gt XY*Z=XYZ Y*Z=YZ eq/ XYZ>YZ XY>Y) + %<- (gt-implies-plus XY>Y X- X+Y=XY) + %<- (times-right-factors-over-minus* XY*Z=XYZ Y*Z=YZ XZ+YZ=XYZ X+Y=XY X*Z=XZ) %. +%worlds () (times-right-factors-over-minus XY*Z=XYZ Y*Z=YZ XYZ-YZ=XZ Z+ X XY-Y=X X*Z=XZ) %. +%total {} (times-right-factors-over-minus _ _ _ _ _ _ _) %. +%theorem times-left-factors-over-minus : forall* {X} {Y} {Z} {XY} {XZ} {XYZ} {X-} forall {TXY times X Y XY} {TXZ times X Z XZ} {M minus XY XZ XYZ} {EX eq X (s X-)} exists {YZ} {MYZ minus Y Z YZ} {TXYZ times X YZ XYZ} true %. +%term _ + %pi (times-left-factors-over-minus X*Y=XY X*Z=XZ XY-XZ=XYZ X=sX- YZ Y-Z=YZ X*YZ=XYZ) + %<- (times-commutative X*Y=XY Y*X=XY) + %<- (times-commutative X*Z=XZ Z*X=XZ) + %<- (times-right-factors-over-minus Y*X=XY Z*X=XZ XY-XZ=XYZ X=sX- YZ Y-Z=YZ YZ*X=XYZ) + %<- (times-commutative YZ*X=XYZ X*YZ=XYZ) %. +%worlds () (times-left-factors-over-minus X*Y=XY X*Z=XZ XY-XZ=XYZ X=sX- YZ Y-Z=YZ X*YZ=XYZ) %. +%total {} (times-left-factors-over-minus _ _ _ _ _ _ _) %. +%%%%% nat-comp.elf +%%%%% Composed relations for natural numbers +%%%%% This file is part of the nat.elf signature +%%%% Definitions +%sort ge %. +%term ge/= %pi (ge X Y) %<- (eq X Y) %. +%term ge/> %pi (ge X Y) %<- (gt X Y) %. +%%%% Theorems +%%% Theorems about ge +%theorem false-implies-ge : forall* {X1} {X2} forall {F void} exists {G ge X1 X2} true %. +%worlds () (false-implies-ge _ _) %. +%total {} (false-implies-ge _ _) %. +%theorem ge-respects-eq : forall* {X1} {X2} {Y1} {Y2} forall {D1 ge X1 X2} {E1 eq X1 Y1} {E2 eq X2 Y2} exists {D2 ge Y1 Y2} true %. +%term _ ge-respects-eq X1>=X2 eq/ eq/ X1>=X2 %. +%worlds () (ge-respects-eq _ _ _ _) %. +%total {} (ge-respects-eq _ _ _ _) %. +%theorem ge-reflexive : forall {X} exists {G ge X X} true %. +%term _ ge-reflexive _ (ge/= eq/) %. +%worlds () (ge-reflexive X X>=X) %. +%total {} (ge-reflexive _ _) %. +%theorem ge-transitive : forall* {X1} {X2} {X3} forall {G1 ge X1 X2} {G2 ge X2 X3} exists {G3 ge X1 X3} true %. +%term _ ge-transitive (ge/= eq/) (ge/= eq/) (ge/= eq/) %. +%term _ ge-transitive (ge/= eq/) (ge/> X>X3) (ge/> X>X3) %. +%term _ ge-transitive (ge/> X1>X) (ge/= eq/) (ge/> X1>X) %. +%term _ + %pi (ge-transitive (ge/> X1>X2) (ge/> X2>X3) (ge/> X1>X3)) + %<- (gt-transitive X1>X2 X2>X3 X1>X3) %. +%worlds () (ge-transitive X1>=X2 X2>=X3 X1>=X3) %. +%total {} (ge-transitive _ _ _) %. +%theorem ge-anti-symmetric : forall* {X1} {X2} forall {G1 ge X1 X2} {G2 ge X2 X1} exists {E eq X1 X2} true %. +%term _ ge-anti-symmetric (ge/= eq/) _ eq/ %. +%term _ ge-anti-symmetric _ (ge/= eq/) eq/ %. +%term _ + %pi (ge-anti-symmetric (ge/> X1>X2) (ge/> X2>X1) X1=X2) + %<- (gt-anti-symmetric X1>X2 X2>X1 F) + %<- (false-implies-eq F X1=X2) %. +%worlds () (ge-anti-symmetric X1>=X2 X2>=X1 X1=X2) %. +%total {} (ge-anti-symmetric _ _ _) %. +%theorem ge-transitive-gt : forall* {X1} {X2} {X3} forall {G1 ge X1 X2} {G2 gt X2 X3} exists {G3 gt X1 X3} true %. +%term _ ge-transitive-gt (ge/= eq/) X>X3 X>X3 %. +%term _ + %pi (ge-transitive-gt (ge/> X1>X2) X2>X3 X1>X3) + %<- (gt-transitive X1>X2 X2>X3 X1>X3) %. +%worlds () (ge-transitive-gt X1>=X2 X2>X3 X1>X3) %. +%total {} (ge-transitive-gt _ _ _) %. +%theorem gt-transitive-ge : forall* {X1} {X2} {X3} forall {G1 gt X1 X2} {G2 ge X2 X3} exists {G3 gt X1 X3} true %. +%term _ gt-transitive-ge X1>X2 (ge/= eq/) X1>X2 %. +%term _ + %pi (gt-transitive-ge X1>X2 (ge/> X2>X3) X1>X3) + %<- (gt-transitive X1>X2 X2>X3 X1>X3) %. +%worlds () (gt-transitive-ge X1>X2 X2>=X3 X1>X3) %. +%total {} (gt-transitive-ge _ _ _) %. +%theorem meta-ge : forall {M} {N} {G ge M N} true %. +%term _ meta-ge _ _ (ge/= eq/) %. +%term _ %pi (meta-ge _ _ (ge/> M>N)) %<- (meta-gt _ _ M>N) %. +%worlds () (meta-ge _ _ _) %. +%total {} (meta-ge _ _ _) %. +%reduces <= N M (meta-ge M N _) %. +%theorem succ-preserves-ge : forall* {M} {N} forall {G ge M N} exists {G' ge (s M) (s N)} true %. +%term _ succ-preserves-ge (ge/= eq/) (ge/= eq/) %. +%term _ + %pi (succ-preserves-ge (ge/> N>M) (ge/> N+1>M+1)) + %<- (succ-preserves-gt N>M N+1>M+1) %. +%worlds () (succ-preserves-ge M>=N M+1>=N+1) %. +%total {} (succ-preserves-ge _ _) %. +%theorem succ-preserves-ge-converse : forall* {M} {N} forall {G' ge (s M) (s N)} exists {G ge M N} true %. +%term _ succ-preserves-ge-converse (ge/= eq/) (ge/= eq/) %. +%term _ + %pi (succ-preserves-ge-converse (ge/> N+1>M+1) (ge/> N>M)) + %<- (succ-preserves-gt-converse N+1>M+1 N>M) %. +%worlds () (succ-preserves-ge-converse M+1>=N+1 M>=N) %. +%total {} (succ-preserves-ge-converse _ _) %. +%theorem ge-succ-implies-gt : forall* {N1} {N2} forall {G ge N1 (s N2)} exists {G' gt N1 N2} true %. +%term _ ge-succ-implies-gt (ge/= eq/) gt/1 %. +%term _ + %pi (ge-succ-implies-gt (ge/> N1>sN2) N1>N2) + %<- (gt-transitive N1>sN2 gt/1 N1>N2) %. +%worlds () (ge-succ-implies-gt _ _) %. +%total {} (ge-succ-implies-gt _ _) %. +%theorem ge-implies-succ-gt : forall* {N1} {N2} forall {G ge N1 N2} exists {G' gt (s N1) N2} true %. +%term _ + %pi (ge-implies-succ-gt N1>=N2 N1+1>N2) + %<- (succ-preserves-ge N1>=N2 N1+1>=N2+1) + %<- (ge-succ-implies-gt N1+1>=N2+1 N1+1>N2) %. +%worlds () (ge-implies-succ-gt _ _) %. +%total {} (ge-implies-succ-gt _ _) %. +%theorem succ-gt-implies-ge : forall* {N1} {N2} forall {G gt (s N1) N2} exists {G' ge N1 N2} true %. +%term _ succ-gt-implies-ge gt/1 (ge/= eq/) %. +%term _ succ-gt-implies-ge (gt/> N1>N2) (ge/> N1>N2) %. +%worlds () (succ-gt-implies-ge _ _) %. +%total {} (succ-gt-implies-ge _ _) %. +%theorem gt-implies-ge-succ : forall* {N1} {N2} forall {G' gt N1 N2} exists {G ge N1 (s N2)} true %. +%term _ + %pi (gt-implies-ge-succ N1>N2 N1>=N2+1) + %<- (succ-preserves-gt N1>N2 N1+1>N2+1) + %<- (succ-gt-implies-ge N1+1>N2+1 N1>=N2+1) %. +%worlds () (gt-implies-ge-succ _ _) %. +%total {} (gt-implies-ge-succ _ _) %. +%theorem ge-implies-plus : forall* {N1} {N2} forall {G ge N2 N1} exists {N0} {P plus N0 N1 N2} true %. +%term _ ge-implies-plus (ge/= eq/) z plus/z %. +%term _ %pi (ge-implies-plus (ge/> N2>N1) (s N0) P) %<- (gt-implies-plus N2>N1 N0 P) %. +%worlds () (ge-implies-plus N2>=N1 N0 N0+N1=N2) %. +%total {} (ge-implies-plus _ _ _) %. +%theorem plus-implies-ge : forall* {N0} {N1} {N2} forall {P plus N0 N1 N2} exists {G ge N2 N1} true %. +%term _ plus-implies-ge plus/z (ge/= eq/) %. +%term _ %pi (plus-implies-ge P (ge/> N2>N1)) %<- (plus-implies-gt P eq/ N2>N1) %. +%worlds () (plus-implies-ge N0+N1=N2 N2>=N1) %. +%total {} (plus-implies-ge _ _) %. +%theorem ge-zero-always : forall {N} exists {G ge N z} true %. +%term _ + %pi (ge-zero-always _ N>=0) + %<- (plus-right-identity _ N+0=N) + %<- (plus-implies-ge N+0=N N>=0) %. +%worlds () (ge-zero-always _ _) %. +%total {} (ge-zero-always _ _) %. +%theorem nonzero-times-implies-ge : forall* {N0} {N1} {N2} forall {P times (s N0) N1 N2} exists {G ge N2 N1} true %. +%term _ + %pi (nonzero-times-implies-ge (times/s _ X+N1=N2) N2>=N1) + %<- (plus-implies-ge X+N1=N2 N2>=N1) %. +%worlds () (nonzero-times-implies-ge N0*N1=N2 N2>=N1) %. +%total {} (nonzero-times-implies-ge _ _) %. +%theorem times-nonzero-implies-ge : forall* {N0} {N1} {N2} forall {P times N0 (s N1) N2} exists {G ge N2 N0} true %. +%term _ + %pi (times-nonzero-implies-ge A*B=C C>=A) + %<- (times-commutative A*B=C B*A=C) + %<- (nonzero-times-implies-ge B*A=C C>=A) %. +%worlds () (times-nonzero-implies-ge _ _) %. +%total {} (times-nonzero-implies-ge _ _) %. +%theorem non-trivial-times-implies-much-gt* : forall* {N1} {N2} {N3} forall {D times (s (s N1)) (s (s N2)) N3} exists {G gt N3 (s (s (s N1)))} true %. +%term _ + %pi (non-trivial-times-implies-much-gt* (times/s (times/s N1*ssN2=P1 P1+ssN2=P2) P2+ssN2=N3) N3>sssN1) + %<- (times-nonzero-implies-ge N1*ssN2=P1 P1>=N1) + %<- (succ-preserves-ge P1>=N1 SP1>=sN1) + %<- (plus-swap-succ-converse P1+ssN2=P2 SP1+sN2=P2) + %<- (plus-commutative SP1+sN2=P2 SN2+sP1=P2) + %<- (plus-implies-gt SN2+sP1=P2 eq/ P2>sP1) + %<- (gt-transitive-ge P2>sP1 SP1>=sN1 P2>sN1) + %<- (succ-preserves-gt P2>sN1 SP2>ssN1) + %<- (gt-implies-ge-succ SP2>ssN1 SP2>=sssN1) + %<- (plus-commutative P2+ssN2=N3 SSN2+P2=N3) + %<- (plus-swap-succ SSN2+P2=N3 SN2+sP2=N3) + %<- (plus-implies-gt SN2+sP2=N3 eq/ N3>sP2) + %<- (gt-transitive-ge N3>sP2 SP2>=sssN1 N3>sssN1) %. +%worlds () (non-trivial-times-implies-much-gt* _ _) %. +%total {} (non-trivial-times-implies-much-gt* _ _) %. +%theorem non-trivial-times-implies-much-gt : forall* {N1} {N2} {N3} forall {D times (s (s N1)) (s (s N2)) N3} exists {G1 gt N3 (s (s (s N1)))} {G2 gt N3 (s (s (s N2)))} true %. +%term _ + %pi (non-trivial-times-implies-much-gt T G1 G2) + %<- (non-trivial-times-implies-much-gt* T G1) + %<- (times-commutative T Tc) + %<- (non-trivial-times-implies-much-gt* Tc G2) %. +%worlds () (non-trivial-times-implies-much-gt _ _ _) %. +%total {} (non-trivial-times-implies-much-gt _ _ _) %. +%theorem plus-left-preserves-ge* : forall* {X1} {X2} {X3} {X4} {X5} forall {G ge X2 X4} {OP1 plus X1 X2 X3} {OP2 plus X1 X4 X5} exists {G2 ge X3 X5} true %. +%term _ + %pi (plus-left-preserves-ge* (ge/= eq/) X1+X2=X3 X1+X2=X5 (ge/= X3=X5)) + %<- (plus-deterministic X1+X2=X3 X1+X2=X5 eq/ eq/ X3=X5) %. +%term _ + %pi (plus-left-preserves-ge* (ge/> X2>X4) X1+X2=X3 X1+X4=X5 (ge/> X3>X5)) + %<- (plus-left-preserves-gt* X2>X4 X1+X2=X3 X1+X4=X5 X3>X5) %. +%worlds () (plus-left-preserves-ge* X2>=X4 X1+X2=X3 X1+X4=X5 X3>=X5) %. +%total {} (plus-left-preserves-ge* _ _ _ _) %. +%theorem plus-left-cancels-ge : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {OP1 plus X1 X2 X3} {OP2 plus Y1 Y2 Y3} {E1 eq X1 Y1} {G3 ge X3 Y3} exists {G2 ge X2 Y2} true %. +%term _ + %pi (plus-left-cancels-ge X1+X2=X3 X1+Y2=X3 eq/ (ge/= eq/) (ge/= X2=Y2)) + %<- (plus-left-cancels X1+X2=X3 X1+Y2=X3 eq/ eq/ X2=Y2) %. +%term _ + %pi (plus-left-cancels-ge X1+X2=X3 X1+Y2=Y3 eq/ (ge/> X3>Y3) (ge/> X2>Y2)) + %<- (plus-left-cancels-gt X1+X2=X3 X1+Y2=Y3 eq/ X3>Y3 X2>Y2) %. +%worlds () (plus-left-cancels-ge X1+X2=X3 Y1+Y2=Y3 X1=Y1 X3>=Y3 X2>=Y2) %. +%total {} (plus-left-cancels-ge _ _ _ _ _) %. +%theorem plus-left-preserves-ge : forall* {X1} {X2} {X4} forall {G ge X2 X4} exists {X3} {X5} {O1 plus X1 X2 X3} {O2 plus X1 X4 X5} {G2 ge X3 X5} true %. +%term _ + %pi (plus-left-preserves-ge X2>=X4 X3 X5 X1+X2=A3 X1+X4=X5 X3>=X5) + %<- (plus-total X1+X2=A3) + %<- (plus-total X1+X4=X5) + %<- (plus-left-preserves-ge* X2>=X4 X1+X2=A3 X1+X4=X5 X3>=X5) %. +%worlds () (plus-left-preserves-ge X2>=X4 X3 X5 X1+X2=A3 X1+X4=X5 X3>=X5) %. +%total {} (plus-left-preserves-ge _ _ _ _ _ _) %. +%theorem plus-right-preserves-ge* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 ge X1 X2} {O1 plus X1 X3 X4} {O2 plus X2 X3 X5} exists {G2 ge X4 X5} true %. +%term _ + %pi (plus-right-preserves-ge* X1>=X2 X1+X3=X4 X2+X3=X5 X4>=X5) + %<- (plus-commutative X1+X3=X4 X3+X1=X4) + %<- (plus-commutative X2+X3=X5 X3+X2=X5) + %<- (plus-left-preserves-ge* X1>=X2 X3+X1=X4 X3+X2=X5 X4>=X5) %. +%worlds () (plus-right-preserves-ge* X1>=X2 X1+X3=X4 X2+X3=X5 X4>=X5) %. +%total {} (plus-right-preserves-ge* _ _ _ _) %. +%theorem plus-right-preserves-ge : forall* {X1} {X2} {X3} forall {G1 ge X1 X2} exists {X4} {X5} {O1 plus X1 X3 X4} {O2 plus X2 X3 X5} {G2 ge X4 X5} true %. +%term _ + %pi (plus-right-preserves-ge X1>=X2 X4 X5 X1+X3=X4 X2+X3=X5 X4>=X5) + %<- (plus-total X1+X3=X4) + %<- (plus-total X2+X3=X5) + %<- (plus-right-preserves-ge* X1>=X2 X1+X3=X4 X2+X3=X5 X4>=X5) %. +%worlds () (plus-right-preserves-ge X1>=X2 X4 X5 X1+X3=X4 X2+X3=X5 X4>=X5) %. +%total {} (plus-right-preserves-ge _ _ _ _ _ _) %. +%theorem plus-preserves-ge* : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {G1 ge X1 Y1} {G2 ge X2 Y2} {MX plus X1 X2 X3} {MY plus Y1 Y2 Y3} exists {G3 ge X3 Y3} true %. +%term _ + %pi (plus-preserves-ge* X1>=Y1 X2>=Y2 X1+X2=X3 Y1+Y2=Y3 X3>=Y3) + %<- (plus-total Y1+X2=X) + %<- (plus-right-preserves-ge* X1>=Y1 X1+X2=X3 Y1+X2=X X3>=X) + %<- (plus-left-preserves-ge* X2>=Y2 Y1+X2=X Y1+Y2=Y3 X>=Y3) + %<- (ge-transitive X3>=X X>=Y3 X3>=Y3) %. +%worlds () (plus-preserves-ge* X1>=Y1 X2>=Y2 X1+X2=X3 Y1+Y2=Y3 X3>=Y3) %. +%total {} (plus-preserves-ge* _ _ _ _ _) %. +%theorem plus-preserves-ge : forall* {X1} {X2} {Y1} {Y2} forall {G1 ge X1 Y1} {G2 ge X2 Y2} exists {X3} {Y3} {MX plus X1 X2 X3} {MY plus Y1 Y2 Y3} {G3 ge X3 Y3} true %. +%term _ + %pi (plus-preserves-ge X1>=Y1 X2>=Y2 X3 Y3 X1+X2=X3 Y1+Y2=Y3 X3>=Y3) + %<- (plus-total X1+X2=X3) + %<- (plus-total Y1+Y2=Y3) + %<- (plus-preserves-ge* X1>=Y1 X2>=Y2 X1+X2=X3 Y1+Y2=Y3 X3>=Y3) %. +%worlds () (plus-preserves-ge X1>=Y1 X2>=Y2 X3 Y3 X1+X2=X3 Y1+Y2=Y3 X3>=Y3) %. +%total {} (plus-preserves-ge _ _ _ _ _ _ _) %. +%theorem plus-right-cancels-ge : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {OP1 plus X1 X2 X3} {OP2 plus Y1 Y2 Y3} {E2 eq X2 Y2} {G3 ge X3 Y3} exists {G1 ge X1 Y1} true %. +%term _ + %pi (plus-right-cancels-ge X1+X2=X3 Y1+Y2=Y3 X2=Y2 X3>=Y3 X1>=Y1) + %<- (plus-commutative X1+X2=X3 X2+X1=X3) + %<- (plus-commutative Y1+Y2=Y3 Y2+Y1=Y3) + %<- (plus-left-cancels-ge X2+X1=X3 Y2+Y1=Y3 X2=Y2 X3>=Y3 X1>=Y1) %. +%worlds () (plus-right-cancels-ge X1+X2=X3 Y1+Y2=Y3 X2=Y2 X3>=Y3 X1>=Y1) %. +%total {} (plus-right-cancels-ge _ _ _ _ _) %. +% Times preserves ge only because multiplying with zero yields equality. +%theorem times-left-preserves-ge* : forall* {X1} {X2} {X3} {X4} {X5} forall {G ge X2 X4} {OP1 times X1 X2 X3} {OP2 times X1 X4 X5} exists {G2 ge X3 X5} true %. +%term _ times-left-preserves-ge* _ times/z times/z (ge/= eq/) %. +%term _ + %pi (times-left-preserves-ge* (ge/= eq/) X1*X2=X3 X1*X2=X5 (ge/= X3=X5)) + %<- (times-deterministic X1*X2=X3 X1*X2=X5 eq/ eq/ X3=X5) %. +%term _ + %pi (times-left-preserves-ge* (ge/> X2>X4) X1*X2=X3 X1*X4=X5 (ge/> X3>X5)) + %<- (times-left-preserves-gt X2>X4 X1*X2=X3 X1*X4=X5 X3>X5) %. +%worlds () (times-left-preserves-ge* X2>=X4 X1*X2=X3 X1*X4=X5 X3>=X5) %. +%total {} (times-left-preserves-ge* _ _ _ _) %. +%theorem times-left-preserves-ge : forall* {X1} {X2} {X4} forall {G ge X2 X4} exists {X3} {X5} {O1 times X1 X2 X3} {O2 times X1 X4 X5} {G2 ge X3 X5} true %. +%term _ + %pi (times-left-preserves-ge X2>=X4 X3 X5 X1*X2=A3 X1*X4=X5 X3>=X5) + %<- (times-total X1*X2=A3) + %<- (times-total X1*X4=X5) + %<- (times-left-preserves-ge* X2>=X4 X1*X2=A3 X1*X4=X5 X3>=X5) %. +%worlds () (times-left-preserves-ge X2>=X4 X3 X5 X1*X2=A3 X1*X4=X5 X3>=X5) %. +%total {} (times-left-preserves-ge _ _ _ _ _ _) %. +%theorem times-right-preserves-ge* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 ge X1 X2} {O1 times X1 X3 X4} {O2 times X2 X3 X5} exists {G2 ge X4 X5} true %. +%term _ + %pi (times-right-preserves-ge* X1>=X2 X1*X3=X4 X2*X3=X5 X4>=X5) + %<- (times-commutative X1*X3=X4 X3*X1=X4) + %<- (times-commutative X2*X3=X5 X3*X2=X5) + %<- (times-left-preserves-ge* X1>=X2 X3*X1=X4 X3*X2=X5 X4>=X5) %. +%worlds () (times-right-preserves-ge* X1>=X2 X1*X3=X4 X2*X3=X5 X4>=X5) %. +%total {} (times-right-preserves-ge* _ _ _ _) %. +%theorem times-right-preserves-ge : forall* {X1} {X2} {X3} forall {G1 ge X1 X2} exists {X4} {X5} {O1 times X1 X3 X4} {O2 times X2 X3 X5} {G2 ge X4 X5} true %. +%term _ + %pi (times-right-preserves-ge X1>=X2 X4 X5 X1*X3=X4 X2*X3=X5 X4>=X5) + %<- (times-total X1*X3=X4) + %<- (times-total X2*X3=X5) + %<- (times-right-preserves-ge* X1>=X2 X1*X3=X4 X2*X3=X5 X4>=X5) %. +%worlds () (times-right-preserves-ge X1>=X2 X4 X5 X1*X3=X4 X2*X3=X5 X4>=X5) %. +%total {} (times-right-preserves-ge _ _ _ _ _ _) %. +%%%% Definitions +%sort ne %. +%term ne/< %pi (ne X Y) %<- (gt Y X) %. +%term ne/> %pi (ne X Y) %<- (gt X Y) %. +%sort eq? %. +%term eq?/yes eq? X X true %. +%term eq?/no %pi (eq? X Y false) %<- (ne X Y) %. +%%%% Theorems +%%% Theorems about ne +%theorem false-implies-ne : forall* {X1} {X2} forall {F void} exists {G ne X1 X2} true %. +%worlds () (false-implies-ne _ _) %. +%total {} (false-implies-ne _ _) %. +%theorem ne-respects-eq : forall* {X1} {X2} {Y1} {Y2} forall {D1 ne X1 X2} {E1 eq X1 Y1} {E2 eq X2 Y2} exists {D2 ne Y1 Y2} true %. +%term _ ne-respects-eq X1<>X2 eq/ eq/ X1<>X2 %. +%worlds () (ne-respects-eq _ _ _ _) %. +%total {} (ne-respects-eq _ _ _ _) %. +%theorem ne-anti-reflexive : forall* {X} forall {R ne X X} exists {F void} true %. +%term _ %pi (ne-anti-reflexive (ne/< X X>X) F) %<- (gt-anti-reflexive X>X F) %. +%worlds () (ne-anti-reflexive X<>X _) %. +%total {} (ne-anti-reflexive _ _) %. +%theorem ne-symmetric : forall* {X} {Y} forall {R1 ne X Y} exists {R2 ne Y X} true %. +%term _ ne-symmetric (ne/< X X X>Y) (ne/< X>Y) %. +%worlds () (ne-symmetric X<>Y Y<>X) %. +%total {} (ne-symmetric _ _) %. +%theorem eq-ne-implies-false : forall* {X} {Y} forall {D1 eq X Y} {D2 ne X Y} exists {F void} true %. +%term _ %pi (eq-ne-implies-false eq/ X<>X F) %<- (ne-anti-reflexive X<>X F) %. +%worlds () (eq-ne-implies-false X=Y X<>Y _) %. +%total {} (eq-ne-implies-false _ _ _) %. +%theorem ge-ne-implies-gt : forall* {X} {Y} forall {D1 ge X Y} {D2 ne X Y} exists {D3 gt X Y} true %. +%term _ ge-ne-implies-gt (ge/> X>Y) _ X>Y %. +%term _ + %pi (ge-ne-implies-gt (ge/= eq/) X<>X X>X) + %<- (ne-anti-reflexive X<>X F) + %<- (false-implies-gt F X>X) %. +%worlds () (ge-ne-implies-gt X>=Y X<>Y X>Y) %. +%total {} (ge-ne-implies-gt _ _ _) %. +%theorem eq?-total* : forall {M} {N} exists {B} {T eq? M N B} true %. +%theorem eq?-total*/L : forall* {M} {N} {C} forall {CMP compare M N C} exists {B} {T eq? M N B} true %. +%term _ eq?-total*/L compare/= true eq?/yes %. +%term _ eq?-total*/L (compare/< X X>Y) false (eq?/no (ne/> X>Y)) %. +%worlds () (eq?-total*/L _ _ _) %. +%total {} (eq?-total*/L _ _ _) %. +%term _ %pi (eq?-total* M N B T) %<- (compare-total CMP) %<- (eq?-total*/L CMP B T) %. +%worlds () (eq?-total* _ _ _ _) %. +%total {} (eq?-total* _ _ _ _) %. +%inline eq?-total eq?-total* _ _ _ %. +%theorem succ-preserves-ne : forall* {M} {N} forall {D ne M N} exists {D' ne (s M) (s N)} true %. +%term _ + %pi (succ-preserves-ne (ne/< N>M) (ne/< N+1>M+1)) + %<- (succ-preserves-gt N>M N+1>M+1) %. +%term _ + %pi (succ-preserves-ne (ne/> N>M) (ne/> N+1>M+1)) + %<- (succ-preserves-gt N>M N+1>M+1) %. +%worlds () (succ-preserves-ne M<>N M+1<>N+1) %. +%total {} (succ-preserves-ne _ _) %. +%theorem succ-preserves-ne-converse : forall* {M} {N} forall {D' ne (s M) (s N)} exists {D ne M N} true %. +%term _ + %pi (succ-preserves-ne-converse (ne/< N+1>M+1) (ne/< N>M)) + %<- (succ-preserves-gt-converse N+1>M+1 N>M) %. +%term _ + %pi (succ-preserves-ne-converse (ne/> N+1>M+1) (ne/> N>M)) + %<- (succ-preserves-gt-converse N+1>M+1 N>M) %. +%worlds () (succ-preserves-ne-converse M+1<>N+1 M<>N) %. +%total {} (succ-preserves-ne-converse _ _) %. +%theorem plus-left-preserves-ne* : forall* {X1} {X2} {X3} {X4} {X5} forall {G ne X2 X4} {OP1 plus X1 X2 X3} {OP2 plus X1 X4 X5} exists {G2 ne X3 X5} true %. +%term _ + %pi (plus-left-preserves-ne* (ne/< X4>X2) X1+X2=X3 X1+X4=X5 (ne/< X5>X3)) + %<- (plus-left-preserves-gt* X4>X2 X1+X4=X5 X1+X2=X3 X5>X3) %. +%term _ + %pi (plus-left-preserves-ne* (ne/> X2>X4) X1+X2=X3 X1+X4=X5 (ne/> X3>X5)) + %<- (plus-left-preserves-gt* X2>X4 X1+X2=X3 X1+X4=X5 X3>X5) %. +%worlds () (plus-left-preserves-ne* X2<>X4 X1+X2=X3 X1+X4=X5 X3<>X5) %. +%total {} (plus-left-preserves-ne* _ _ _ _) %. +%theorem plus-left-cancels-ne : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {OP1 plus X1 X2 X3} {OP2 plus Y1 Y2 Y3} {E1 eq X1 Y1} {G3 ne X3 Y3} exists {G2 ne X2 Y2} true %. +%term _ + %pi (plus-left-cancels-ne X1+X2=X3 X1+Y2=Y3 eq/ (ne/< Y3>X3) (ne/< Y2>X2)) + %<- (plus-left-cancels-gt X1+Y2=Y3 X1+X2=X3 eq/ Y3>X3 Y2>X2) %. +%term _ + %pi (plus-left-cancels-ne X1+X2=X3 X1+Y2=Y3 eq/ (ne/> X3>Y3) (ne/> X2>Y2)) + %<- (plus-left-cancels-gt X1+X2=X3 X1+Y2=Y3 eq/ X3>Y3 X2>Y2) %. +%worlds () (plus-left-cancels-ne X1+X2=X3 Y1+Y2=Y3 X1=Y1 X3<>Y3 X2<>Y2) %. +%total {} (plus-left-cancels-ne _ _ _ _ _) %. +%theorem plus-left-preserves-ne : forall* {X1} {X2} {X4} forall {G ne X2 X4} exists {X3} {X5} {O1 plus X1 X2 X3} {O2 plus X1 X4 X5} {G2 ne X3 X5} true %. +%term _ + %pi (plus-left-preserves-ne X2<>X4 X3 X5 X1+X2=A3 X1+X4=X5 X3<>X5) + %<- (plus-total X1+X2=A3) + %<- (plus-total X1+X4=X5) + %<- (plus-left-preserves-ne* X2<>X4 X1+X2=A3 X1+X4=X5 X3<>X5) %. +%worlds () (plus-left-preserves-ne X2<>X4 X3 X5 X1+X2=A3 X1+X4=X5 X3<>X5) %. +%total {} (plus-left-preserves-ne _ _ _ _ _ _) %. +%theorem plus-right-preserves-ne* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 ne X1 X2} {O1 plus X1 X3 X4} {O2 plus X2 X3 X5} exists {G2 ne X4 X5} true %. +%term _ + %pi (plus-right-preserves-ne* X1<>X2 X1+X3=X4 X2+X3=X5 X4<>X5) + %<- (plus-commutative X1+X3=X4 X3+X1=X4) + %<- (plus-commutative X2+X3=X5 X3+X2=X5) + %<- (plus-left-preserves-ne* X1<>X2 X3+X1=X4 X3+X2=X5 X4<>X5) %. +%worlds () (plus-right-preserves-ne* X1<>X2 X1+X3=X4 X2+X3=X5 X4<>X5) %. +%total {} (plus-right-preserves-ne* _ _ _ _) %. +%theorem plus-right-preserves-ne : forall* {X1} {X2} {X3} forall {G1 ne X1 X2} exists {X4} {X5} {O1 plus X1 X3 X4} {O2 plus X2 X3 X5} {G2 ne X4 X5} true %. +%term _ + %pi (plus-right-preserves-ne X1<>X2 X4 X5 X1+X3=X4 X2+X3=X5 X4<>X5) + %<- (plus-total X1+X3=X4) + %<- (plus-total X2+X3=X5) + %<- (plus-right-preserves-ne* X1<>X2 X1+X3=X4 X2+X3=X5 X4<>X5) %. +%worlds () (plus-right-preserves-ne X1<>X2 X4 X5 X1+X3=X4 X2+X3=X5 X4<>X5) %. +%total {} (plus-right-preserves-ne _ _ _ _ _ _) %. +%theorem plus-right-cancels-ne : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {OP1 plus X1 X2 X3} {OP2 plus Y1 Y2 Y3} {E2 eq X2 Y2} {G3 ne X3 Y3} exists {G1 ne X1 Y1} true %. +%term _ + %pi (plus-right-cancels-ne X1+X2=X3 Y1+Y2=Y3 X2=Y2 X3<>Y3 X1<>Y1) + %<- (plus-commutative X1+X2=X3 X2+X1=X3) + %<- (plus-commutative Y1+Y2=Y3 Y2+Y1=Y3) + %<- (plus-left-cancels-ne X2+X1=X3 Y2+Y1=Y3 X2=Y2 X3<>Y3 X1<>Y1) %. +%worlds () (plus-right-cancels-ne X1+X2=X3 Y1+Y2=Y3 X2=Y2 X3<>Y3 X1<>Y1) %. +%total {} (plus-right-cancels-ne _ _ _ _ _) %. +%%%%% nat-less.elf +%%%%% Inverse relations for natural numbers +%%%%% This file is part of the nat.elf signature +%%%% Definitions +%inline lt [X] [Y] gt Y X %. +%%%% Theorems about lt +%theorem false-implies-lt : forall* {X1} {X2} forall {F void} exists {G lt X1 X2} true %. +%worlds () (false-implies-lt _ _) %. +%total {} (false-implies-lt _ _) %. +%theorem lt-respects-eq : forall* {X1} {X2} {Y1} {Y2} forall {D1 lt X1 X2} {E1 eq X1 Y1} {E2 eq X2 Y2} exists {D2 lt Y1 Y2} true %. +%term _ lt-respects-eq X1X1 X1>X2 R) %<- (gt-anti-symmetric X1>X2 X2>X1 R) %. +%worlds () (lt-anti-symmetric _ _ _) %. +%total {} (lt-anti-symmetric _ _ _) %. +%theorem lt-transitive : forall* {X1} {X2} {X3} forall {G1 lt X1 X2} {G2 lt X2 X3} exists {G3 lt X1 X3} true %. +%term _ %pi (lt-transitive X1X2 X1+X2=X3 X1+X4=X5 X5>X3) + %<- (plus-left-preserves-gt* X4>X2 X1+X4=X5 X1+X2=X3 X5>X3) %. +%worlds () (plus-left-preserves-lt* X2X3 Y2>X2) + %<- (plus-left-cancels-gt X1+Y2=X3 X1+X2=X3 eq/ Y3>X3 Y2>X2) %. +%worlds () (plus-left-cancels-lt X1+X2=X3 Y1+Y2=Y3 X1=Y1 X3=X1 X1>=X2 R) %<- (ge-anti-symmetric X1>=X2 X2>=X1 R) %. +%worlds () (le-anti-symmetric _ _ _) %. +%total {} (le-anti-symmetric _ _ _) %. +%theorem le-transitive : forall* {X1} {X2} {X3} forall {G1 le X1 X2} {G2 le X2 X3} exists {G3 le X1 X3} true %. +%term _ %pi (le-transitive X1<=X2 X2<=X3 X1<=X3) %<- (ge-transitive X2<=X3 X1<=X2 X1<=X3) %. +%worlds () (le-transitive X1<=X2 X2<=X3 X1<=X3) %. +%total {} (le-transitive _ _ _) %. +%inline le-reflexive ge-reflexive %. +%theorem le-transitive-lt : forall* {X1} {X2} {X3} forall {L1 le X1 X2} {L2 lt X2 X3} exists {L3 lt X1 X3} true %. +%term _ + %pi (le-transitive-lt X2>=X1 X3>X2 X3>X1) + %<- (gt-transitive-ge X3>X2 X2>=X1 X3>X1) %. +%worlds () (le-transitive-lt X1<=X2 X2X1 X3>=X2 X3>X1) + %<- (ge-transitive-gt X3>=X2 X2>X1 X3>X1) %. +%worlds () (lt-transitive-le X1=X2 X1+X2=X3 X1+X4=X5 X5>=X3) + %<- (plus-left-preserves-ge* X4>=X2 X1+X4=X5 X1+X2=X3 X5>=X3) %. +%worlds () (plus-left-preserves-le* X2<=X4 X1+X2=X3 X1+X4=X5 X3<=X5) %. +%total {} (plus-left-preserves-le* _ _ _ _) %. +%theorem plus-left-cancels-le : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {OP1 plus X1 X2 X3} {OP2 plus Y1 Y2 Y3} {E1 eq X1 Y1} {R3 le X3 Y3} exists {R2 le X2 Y2} true %. +%term _ + %pi (plus-left-cancels-le X1+X2=X3 X1+Y2=X3 eq/ Y3>=X3 Y2>=X2) + %<- (plus-left-cancels-ge X1+Y2=X3 X1+X2=X3 eq/ Y3>=X3 Y2>=X2) %. +%worlds () (plus-left-cancels-le X1+X2=X3 Y1+Y2=Y3 X1=Y1 X3<=Y3 X2<=Y2) %. +%total {} (plus-left-cancels-le _ _ _ _ _) %. +%theorem plus-left-preserves-le : forall* {X1} {X2} {X4} forall {G le X2 X4} exists {X3} {X5} {O1 plus X1 X2 X3} {O2 plus X1 X4 X5} {G2 le X3 X5} true %. +%term _ + %pi (plus-left-preserves-le X2<=X4 X3 X5 X1+X2=A3 X1+X4=X5 X3<=X5) + %<- (plus-total X1+X2=A3) + %<- (plus-total X1+X4=X5) + %<- (plus-left-preserves-le* X2<=X4 X1+X2=A3 X1+X4=X5 X3<=X5) %. +%worlds () (plus-left-preserves-le X2<=X4 X3 X5 X1+X2=A3 X1+X4=X5 X3<=X5) %. +%total {} (plus-left-preserves-le _ _ _ _ _ _) %. +%theorem plus-right-preserves-le* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 le X1 X2} {O1 plus X1 X3 X4} {O2 plus X2 X3 X5} exists {G2 le X4 X5} true %. +%term _ + %pi (plus-right-preserves-le* X1<=X2 X1+X3=X4 X2+X3=X5 X4<=X5) + %<- (plus-commutative X1+X3=X4 X3+X1=X4) + %<- (plus-commutative X2+X3=X5 X3+X2=X5) + %<- (plus-left-preserves-le* X1<=X2 X3+X1=X4 X3+X2=X5 X4<=X5) %. +%worlds () (plus-right-preserves-le* X1<=X2 X1+X3=X4 X2+X3=X5 X4<=X5) %. +%total {} (plus-right-preserves-le* _ _ _ _) %. +%theorem plus-right-preserves-le : forall* {X1} {X2} {X3} forall {G1 le X1 X2} exists {X4} {X5} {O1 plus X1 X3 X4} {O2 plus X2 X3 X5} {G2 le X4 X5} true %. +%term _ + %pi (plus-right-preserves-le X1<=X2 X4 X5 X1+X3=X4 X2+X3=X5 X4<=X5) + %<- (plus-total X1+X3=X4) + %<- (plus-total X2+X3=X5) + %<- (plus-right-preserves-le* X1<=X2 X1+X3=X4 X2+X3=X5 X4<=X5) %. +%worlds () (plus-right-preserves-le X1<=X2 X4 X5 X1+X3=X4 X2+X3=X5 X4<=X5) %. +%total {} (plus-right-preserves-le _ _ _ _ _ _) %. +%theorem plus-preserves-le* : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {G1 le X1 Y1} {G2 le X2 Y2} {MX plus X1 X2 X3} {MY plus Y1 Y2 Y3} exists {G3 le X3 Y3} true %. +%term _ + %pi (plus-preserves-le* X1<=Y1 X2<=Y2 X1+X2=X3 Y1+Y2=Y3 X3<=Y3) + %<- (plus-total Y1+X2=X) + %<- (plus-right-preserves-le* X1<=Y1 X1+X2=X3 Y1+X2=X X3<=X) + %<- (plus-left-preserves-le* X2<=Y2 Y1+X2=X Y1+Y2=Y3 X<=Y3) + %<- (le-transitive X3<=X X<=Y3 X3<=Y3) %. +%worlds () (plus-preserves-le* X1<=Y1 X2<=Y2 X1+X2=X3 Y1+Y2=Y3 X3<=Y3) %. +%total {} (plus-preserves-le* _ _ _ _ _) %. +%theorem plus-preserves-le : forall* {X1} {X2} {Y1} {Y2} forall {G1 le X1 Y1} {G2 le X2 Y2} exists {X3} {Y3} {MX plus X1 X2 X3} {MY plus Y1 Y2 Y3} {G3 le X3 Y3} true %. +%term _ + %pi (plus-preserves-le X1<=Y1 X2<=Y2 X3 Y3 X1+X2=X3 Y1+Y2=Y3 X3<=Y3) + %<- (plus-total X1+X2=X3) + %<- (plus-total Y1+Y2=Y3) + %<- (plus-preserves-le* X1<=Y1 X2<=Y2 X1+X2=X3 Y1+Y2=Y3 X3<=Y3) %. +%worlds () (plus-preserves-le X1<=Y1 X2<=Y2 X3 Y3 X1+X2=X3 Y1+Y2=Y3 X3<=Y3) %. +%total {} (plus-preserves-le _ _ _ _ _ _ _) %. +%theorem plus-right-cancels-le : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {OP1 plus X1 X2 X3} {OP2 plus Y1 Y2 Y3} {E2 eq X2 Y2} {G3 le X3 Y3} exists {G1 le X1 Y1} true %. +%term _ + %pi (plus-right-cancels-le X1+X2=X3 Y1+Y2=Y3 X2=Y2 X3<=Y3 X1<=Y1) + %<- (plus-commutative X1+X2=X3 X2+X1=X3) + %<- (plus-commutative Y1+Y2=Y3 Y2+Y1=Y3) + %<- (plus-left-cancels-le X2+X1=X3 Y2+Y1=Y3 X2=Y2 X3<=Y3 X1<=Y1) %. +%worlds () (plus-right-cancels-le X1+X2=X3 Y1+Y2=Y3 X2=Y2 X3<=Y3 X1<=Y1) %. +%total {} (plus-right-cancels-le _ _ _ _ _) %. +%theorem times-left-preserves-le* : forall* {X1} {X2} {X3} {X4} {X5} forall {R1 le X2 X4} {OP1 times X1 X2 X3} {OP2 times X1 X4 X5} exists {R2 le X3 X5} true %. +%term _ + %pi (times-left-preserves-le* X4>=X2 X1*X2=X3 X1*X4=X5 X5>=X3) + %<- (times-left-preserves-ge* X4>=X2 X1*X4=X5 X1*X2=X3 X5>=X3) %. +%worlds () (times-left-preserves-le* X2<=X4 X1*X2=X3 X1*X4=X5 X3<=X5) %. +%total {} (times-left-preserves-le* _ _ _ _) %. +%theorem times-left-preserves-le : forall* {X1} {X2} {X4} forall {G le X2 X4} exists {X3} {X5} {O1 times X1 X2 X3} {O2 times X1 X4 X5} {G2 le X3 X5} true %. +%term _ + %pi (times-left-preserves-le X2<=X4 X3 X5 X1*X2=A3 X1*X4=X5 X3<=X5) + %<- (times-total X1*X2=A3) + %<- (times-total X1*X4=X5) + %<- (times-left-preserves-le* X2<=X4 X1*X2=A3 X1*X4=X5 X3<=X5) %. +%worlds () (times-left-preserves-le X2<=X4 X3 X5 X1*X2=A3 X1*X4=X5 X3<=X5) %. +%total {} (times-left-preserves-le _ _ _ _ _ _) %. +%theorem times-right-preserves-le* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 le X1 X2} {O1 times X1 X3 X4} {O2 times X2 X3 X5} exists {G2 le X4 X5} true %. +%term _ + %pi (times-right-preserves-le* X1<=X2 X1*X3=X4 X2*X3=X5 X4<=X5) + %<- (times-commutative X1*X3=X4 X3*X1=X4) + %<- (times-commutative X2*X3=X5 X3*X2=X5) + %<- (times-left-preserves-le* X1<=X2 X3*X1=X4 X3*X2=X5 X4<=X5) %. +%worlds () (times-right-preserves-le* X1<=X2 X1*X3=X4 X2*X3=X5 X4<=X5) %. +%total {} (times-right-preserves-le* _ _ _ _) %. +%theorem times-right-preserves-le : forall* {X1} {X2} {X3} forall {G1 le X1 X2} exists {X4} {X5} {O1 times X1 X3 X4} {O2 times X2 X3 X5} {G2 le X4 X5} true %. +%term _ + %pi (times-right-preserves-le X1<=X2 X4 X5 X1*X3=X4 X2*X3=X5 X4<=X5) + %<- (times-total X1*X3=X4) + %<- (times-total X2*X3=X5) + %<- (times-right-preserves-le* X1<=X2 X1*X3=X4 X2*X3=X5 X4<=X5) %. +%worlds () (times-right-preserves-le X1<=X2 X4 X5 X1*X3=X4 X2*X3=X5 X4<=X5) %. +%total {} (times-right-preserves-le _ _ _ _ _ _) %. +%theorem times-preserves-le* : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {G1 le X1 Y1} {G2 le X2 Y2} {MX times X1 X2 X3} {MY times Y1 Y2 Y3} exists {G3 le X3 Y3} true %. +%term _ + %pi (times-preserves-le* X1<=Y1 X2<=Y2 X1*X2=X3 Y1*Y2=Y3 X3<=Y3) + %<- (times-total Y1*X2=X) + %<- (times-right-preserves-le* X1<=Y1 X1*X2=X3 Y1*X2=X X3<=X) + %<- (times-left-preserves-le* X2<=Y2 Y1*X2=X Y1*Y2=Y3 X<=Y3) + %<- (le-transitive X3<=X X<=Y3 X3<=Y3) %. +%worlds () (times-preserves-le* X1<=Y1 X2<=Y2 X1*X2=X3 Y1*Y2=Y3 X3<=Y3) %. +%total {} (times-preserves-le* _ _ _ _ _) %. +%theorem times-preserves-le : forall* {X1} {X2} {Y1} {Y2} forall {G1 le X1 Y1} {G2 le X2 Y2} exists {X3} {Y3} {MX times X1 X2 X3} {MY times Y1 Y2 Y3} {G3 le X3 Y3} true %. +%term _ + %pi (times-preserves-le X1<=Y1 X2<=Y2 X3 Y3 X1*X2=X3 Y1*Y2=Y3 X3<=Y3) + %<- (times-total X1*X2=X3) + %<- (times-total Y1*Y2=Y3) + %<- (times-preserves-le* X1<=Y1 X2<=Y2 X1*X2=X3 Y1*Y2=Y3 X3<=Y3) %. +%worlds () (times-preserves-le X1<=Y1 X2<=Y2 X3 Y3 X1*X2=X3 Y1*Y2=Y3 X3<=Y3) %. +%total {} (times-preserves-le _ _ _ _ _ _ _) %. +%%%%% nat-inv-comp.elf +%%%%% Theorems about minus and composed relations +%%%%% This file is part of the nat.elf signature +%%%% Theorems +%%% Theorems about minus +%theorem minus-left-inverts-ge* : forall* {X1} {X2} {X3} {X4} {X5} forall {G ge X2 X4} {IOP1 minus X1 X2 X3} {IOP2 minus X1 X4 X5} exists {GP ge X5 X3} true %. +%term _ + %pi (minus-left-inverts-ge* X2>=X4 X3+X2=X1 X5+X4=X1 X5>=X3) + %<- (plus-total X3+X4=X7) + %<- (plus-left-preserves-ge* X2>=X4 X3+X2=X1 X3+X4=X7 X1>=X7) + %<- (plus-right-cancels-ge X5+X4=X1 X3+X4=X7 eq/ X1>=X7 X5>=X3) %. +%worlds () (minus-left-inverts-ge* X2>=X4 X1-X2=X3 X1-X4=X5 X5>=X3) %. +%total {} (minus-left-inverts-ge* _ _ _ _) %. +%theorem minus-right-preserves-ge* : forall* {X1} {X2} {X3} {X4} {X5} forall {G ge X1 X2} {IOP1 minus X1 X3 X4} {IOP2 minus X2 X3 X5} exists {GP ge X4 X5} true %. +%term _ + %pi (minus-right-preserves-ge* X1>=X2 X4+X3=X1 X5+X3=X2 X4>=X5) + %<- (plus-right-cancels-ge X4+X3=X1 X5+X3=X2 eq/ X1>=X2 X4>=X5) %. +%worlds () (minus-right-preserves-ge* X1>=X2 X1-X3=X4 X2-X3=X5 X4>=X5) %. +%total {} (minus-right-preserves-ge* _ _ _ _) %. +%theorem minus-left-cancels-inverts-ge : forall* {X1} {X2} {X3} {X4} {X5} {X6} forall {IOP1 minus X1 X2 X3} {IOP2 minus X4 X5 X6} {E eq X1 X4} {G ge X3 X6} exists {GP ge X5 X2} true %. +%term _ + %pi (minus-left-cancels-inverts-ge X3+X2=X1 X6+X5=X4 X1=X4 X3>=X6 X5>=X2) + %<- (plus-total X6+X2=X7) + %<- (plus-right-preserves-ge* X3>=X6 X3+X2=X1 X6+X2=X7 X1>=X7) + %<- (eq-symmetric X1=X4 X4=X1) + %<- (plus-respects-eq X6+X5=X4 eq/ eq/ X4=X1 X6+X5=X1) + %<- (plus-left-cancels-ge X6+X5=X1 X6+X2=X7 eq/ X1>=X7 X5>=X2) %. +%worlds () (minus-left-cancels-inverts-ge X1-X2=X3 X4-X5=X6 X1=X4 X3>=X6 X5>=X2) %. +%total {} (minus-left-cancels-inverts-ge _ _ _ _ _) %. +%theorem minus-right-cancels-ge : forall* {X1} {X2} {X3} {X4} {X5} {X6} forall {IOP1 minus X1 X2 X3} {IOP2 minus X4 X5 X6} {E2 eq X2 X5} {G3 ge X3 X6} exists {G1 ge X1 X4} true %. +%term _ + %pi (minus-right-cancels-ge X3+X2=X1 X6+X5=X4 X2=X5 X3>=X6 X1>=X4) + %<- (plus-respects-eq X3+X2=X1 eq/ X2=X5 eq/ X3+X5=X1) + %<- (plus-right-preserves-ge* X3>=X6 X3+X5=X1 X6+X5=X4 X1>=X4) %. +%worlds () (minus-right-cancels-ge X1-X2=X3 X4-X5=X6 X2=X5 X3>=X6 X1>=X4) %. +%total {} (minus-right-cancels-ge _ _ _ _ _) %. +%theorem minus-left-preserves-ne* : forall* {X1} {X2} {X3} {X4} {X5} forall {G ne X2 X4} {IOP1 minus X1 X2 X3} {IOP2 minus X1 X4 X5} exists {GP ne X3 X5} true %. +%term _ + %pi (minus-left-preserves-ne* X2<>X4 X3+X2=X1 X5+X4=X1 X3<>X5) + %<- (plus-total X3+X4=X7) + %<- (plus-left-preserves-ne* X2<>X4 X3+X2=X1 X3+X4=X7 X1<>X7) + %<- (plus-right-cancels-ne X5+X4=X1 X3+X4=X7 eq/ X1<>X7 X5<>X3) + %<- (ne-symmetric X5<>X3 X3<>X5) %. +%worlds () (minus-left-preserves-ne* X2<>X4 X1-X2=X3 X1-X4=X5 X3<>X5) %. +%total {} (minus-left-preserves-ne* _ _ _ _) %. +%theorem minus-right-preserves-ne* : forall* {X1} {X2} {X3} {X4} {X5} forall {G ne X1 X2} {IOP1 minus X1 X3 X4} {IOP2 minus X2 X3 X5} exists {GP ne X4 X5} true %. +%term _ + %pi (minus-right-preserves-ne* X1<>X2 X4+X3=X1 X5+X3=X2 X4<>X5) + %<- (plus-right-cancels-ne X4+X3=X1 X5+X3=X2 eq/ X1<>X2 X4<>X5) %. +%worlds () (minus-right-preserves-ne* X1<>X2 X1-X3=X4 X2-X3=X5 X4<>X5) %. +%total {} (minus-right-preserves-ne* _ _ _ _) %. +%theorem minus-left-cancels-ne : forall* {X1} {X2} {X3} {X4} {X5} {X6} forall {IOP1 minus X1 X2 X3} {IOP2 minus X4 X5 X6} {E eq X1 X4} {G ne X3 X6} exists {GP ne X2 X5} true %. +%term _ + %pi (minus-left-cancels-ne X3+X2=X1 X6+X5=X4 X1=X4 X3<>X6 X2<>X5) + %<- (plus-total X6+X2=X7) + %<- (plus-right-preserves-ne* X3<>X6 X3+X2=X1 X6+X2=X7 X1<>X7) + %<- (eq-symmetric X1=X4 X4=X1) + %<- (plus-respects-eq X6+X5=X4 eq/ eq/ X4=X1 X6+X5=X1) + %<- (plus-left-cancels-ne X6+X5=X1 X6+X2=X7 eq/ X1<>X7 X5<>X2) + %<- (ne-symmetric X5<>X2 X2<>X5) %. +%worlds () (minus-left-cancels-ne X1-X2=X3 X4-X5=X6 X1=X4 X3<>X6 X2<>X5) %. +%total {} (minus-left-cancels-ne _ _ _ _ _) %. +%theorem minus-right-cancels-ne : forall* {X1} {X2} {X3} {X4} {X5} {X6} forall {IOP1 minus X1 X2 X3} {IOP2 minus X4 X5 X6} {E2 eq X2 X5} {G3 ne X3 X6} exists {G1 ne X1 X4} true %. +%term _ + %pi (minus-right-cancels-ne X3+X2=X1 X6+X5=X4 X2=X5 X3<>X6 X1<>X4) + %<- (plus-respects-eq X3+X2=X1 eq/ X2=X5 eq/ X3+X5=X1) + %<- (plus-right-preserves-ne* X3<>X6 X3+X5=X1 X6+X5=X4 X1<>X4) %. +%worlds () (minus-right-cancels-ne X1-X2=X3 X4-X5=X6 X2=X5 X3<>X6 X1<>X4) %. +%total {} (minus-right-cancels-ne _ _ _ _ _) %. +%%%%% nat-inv-less.elf +%%%%% Theorems about minus and inverted relations +%%%%% This file is part of the nat.elf signature +%%%% Theorems +%%% Theorems about minus +%theorem minus-left-inverts-lt* : forall* {X1} {X2} {X3} {X4} {X5} forall {G lt X2 X4} {IOP1 minus X1 X2 X3} {IOP2 minus X1 X4 X5} exists {GP lt X5 X3} true %. +%term _ + %pi (minus-left-inverts-lt* X2M)) + %<- (equal-implies-eq CMP M=N-1) + %<- (succ-implies-gt eq/ N>N-1) + %<- (eq-symmetric M=N-1 N-1=M) + %<- (gt-respects-eq N>N-1 eq/ N-1=M N>M) %. +%term _ + %pi (divrem-total** M N-1 less CMP z M (divrem/z N>M)) + %<- (less-implies-lt CMP N-1>M) + %<- (succ-implies-gt eq/ N>N-1) + %<- (gt-transitive N>N-1 N-1>M N>M) %. +%term _ + %pi (divrem-total** M N-1 greater CMP (s Q) R (divrem/s D/N=Q,R D+N=M)) + %<- (greater-implies-gt CMP M>N-1) + %<- (gt-implies-plus M>N-1 D SD+N-1=M) + %<- (plus-swap-succ SD+N-1=M D+N=M) + %<- (plus-commutative D+N=M N+D=M) + %<- (plus-implies-gt N+D=M eq/ M>D) + %<- (meta-gt M D M>D) + %<- (compare-total* D N-1 C' CMP') + %<- (divrem-total** D N-1 C' CMP' Q R D/N=Q,R) %. +%worlds () (divrem-total** M N-1 C MCN Q R M/N=Q,R) %. +%total (M) (divrem-total** M _ _ _ _ _ _) %. +%theorem divrem-total* : forall {M} {N-} exists {Q} {R} {D divrem M (s N-) Q R} true %. +%term _ + %pi (divrem-total* M N-1 Q R M/N=Q,R) + %<- (compare-total* M N-1 C CMP) + %<- (divrem-total** M N-1 C CMP Q R M/N=Q,R) %. +%worlds () (divrem-total* M N-1 Q R M/N=Q,R) %. +%total {} (divrem-total* _ _ _ _ _) %. +%inline divrem-total divrem-total* _ _ _ _ %. +%theorem divrem-deterministic : forall* {M} {N} {Q} {R} {M'} {N'} {Q'} {R'} forall {D divrem M N Q R} {D' divrem M' N' Q' R'} {E1 eq M M'} {E2 eq N N'} exists {E3 eq Q Q'} {E4 eq R R'} true %. +%term _ divrem-deterministic (divrem/z _) (divrem/z _) eq/ eq/ eq/ eq/ %. +%term _ + %pi (divrem-deterministic (divrem/s D/N=Q,R D+N=M) (divrem/s D'/N=Q',R' D'+N=M) eq/ eq/ SQ=SQ' R=R') + %<- (plus-right-cancels D+N=M D'+N=M eq/ eq/ D=D') + %<- (divrem-deterministic D/N=Q,R D'/N=Q',R' D=D' eq/ Q=Q' R=R') + %<- (succ-deterministic Q=Q' SQ=SQ') %. +%% contradiction cases: +%term _ + %pi (divrem-deterministic (divrem/z N>M) (divrem/s _ D+N=M) eq/ eq/ Q=Q' R=R') + %<- (plus-commutative D+N=M N+D=M) + %<- (plus-gt-contradiction N+D=M N>M F) + %<- (false-implies-eq F Q=Q') + %<- (false-implies-eq F R=R') %. +%term _ + %pi (divrem-deterministic (divrem/s _ D+N=M) (divrem/z N>M) eq/ eq/ Q=Q' R=R') + %<- (plus-commutative D+N=M N+D=M) + %<- (plus-gt-contradiction N+D=M N>M F) + %<- (false-implies-eq F Q=Q') + %<- (false-implies-eq F R=R') %. +%worlds () (divrem-deterministic M/N=Q,R M'/N'=Q'/R' M=M' N=N' Q=Q' R=R') %. +%total (D) (divrem-deterministic D _ _ _ _ _) %. +%theorem divrem-implies-positive : forall* {M} {N} {Q} {R} forall {D divrem M N Q R} exists {N-1} {E eq N (s N-1)} true %. +%term _ + %pi (divrem-implies-positive (divrem/z N>M) N-1 N=sN-1) + %<- (gt-implies-positive N>M N-1 N=sN-1) %. +%term _ divrem-implies-positive (divrem/s _ _) _ eq/ %. +%worlds () (divrem-implies-positive M/N=Q,R N-1 N=sN-1) %. +%total {} (divrem-implies-positive _ _ _) %. +%theorem divrem-implies-gt : forall* {M} {N} {Q} {R} forall {D divrem M N Q R} exists {G gt N R} true %. +%term _ divrem-implies-gt (divrem/z N>M) N>M %. +%term _ + %pi (divrem-implies-gt (divrem/s D/N=Q,R _) N>R) + %<- (divrem-implies-gt D/N=Q,R N>R) %. +%worlds () (divrem-implies-gt M/N=Q,R N>R) %. +%total D (divrem-implies-gt D _) %. +%theorem divrem-contradiction : forall* {M} {N} {Q} {R} {X} forall {D divrem M N Q R} {P plus N X R} exists {F void} true %. +%term _ + %pi (divrem-contradiction D P F) + %<- (divrem-implies-gt D N>R) + %<- (plus-commutative P Pc) + %<- (plus-implies-ge Pc R>=N) + %<- (gt-transitive-ge N>R R>=N N>N) + %<- (gt-anti-reflexive N>N F) %. +%worlds () (divrem-contradiction _ _ _) %. +%total {} (divrem-contradiction _ _ _) %. +%theorem divrem-can-be-inverted : forall* {M} {N} {Q} {R} forall {D divrem M N Q R} exists {X} {T times Q N X} {P plus X R M} true %. +%term _ divrem-can-be-inverted (divrem/z _) z times/z plus/z %. +%term _ + %pi (divrem-can-be-inverted (divrem/s D/N=Q,R D+N=M) X (times/s Q*N=Y Y+N=X) X+R=M) + %<- (divrem-can-be-inverted D/N=Q,R Y Q*N=Y Y+R=D) + %<- (plus-commutative Y+R=D R+Y=D) + %<- (plus-associative R+Y=D D+N=M X Y+N=X R+X=M) + %<- (plus-commutative R+X=M X+R=M) %. +%worlds () (divrem-can-be-inverted M/N=Q,R X Q*N=X X+R=M) %. +%total (D) (divrem-can-be-inverted D _ _ _) %. +%theorem div-can-be-inverted : forall* {M} {N} {Q} forall {D divrem M N Q z} exists {T times Q N M} true %. +%term _ div-can-be-inverted (divrem/z _) times/z %. +%term _ + %pi (div-can-be-inverted (divrem/s D/N=Q,z D+N=M) (times/s Q*N=D D+N=M)) + %<- (div-can-be-inverted D/N=Q,z Q*N=D) %. +%worlds () (div-can-be-inverted _ _) %. +%total (D) (div-can-be-inverted D _) %. +%theorem divrem-can-be-constructed : forall* {M} {N} {Q} {R} {X} forall {T times Q N X} {P plus X R M} {G gt N R} exists {D divrem M N Q R} true %. +%term _ divrem-can-be-constructed times/z plus/z N>R (divrem/z N>R) %. +%term _ + %pi (divrem-can-be-constructed (times/s Q*N=Y Y+N=X) X+R=M N>R (divrem/s Z/N=Q,R Z+N=M)) + %<- (plus-commutative Y+N=X N+Y=X) + %<- (plus-associative N+Y=X X+R=M Z Y+R=Z N+Z=M) + %<- (plus-commutative N+Z=M Z+N=M) + %<- (divrem-can-be-constructed Q*N=Y Y+R=Z N>R Z/N=Q,R) %. +%term _ + %pi (divrem-can-be-constructed _ _ ZERO>R D) + %<- (gt-contradiction ZERO>R F) + %<- (false-implies-divrem F D) %. +%worlds () (divrem-can-be-constructed Q*N=X X+R=M N>R M/N=Q,R) %. +%total (T) (divrem-can-be-constructed T _ _ _) %. +%theorem div-can-be-constructed : forall* {M} {N} {Q} forall {T times Q (s N) M} exists {D divrem M (s N) Q z} true %. +%term _ + %pi (div-can-be-constructed times/z (divrem/z N+1>0)) + %<- (succ-implies-gt-zero _ N+1>0) %. +%term _ + %pi (div-can-be-constructed (times/s Q*sN=D D+sN=M) (divrem/s D/sN=Q,z D+sN=M)) + %<- (div-can-be-constructed Q*sN=D D/sN=Q,z) %. +%worlds () (div-can-be-constructed _ _) %. +%total (T) (div-can-be-constructed T _) %. +%theorem remainder-implies-gt-quotient : forall* {M} {N} {Q} {R} forall {D divrem M N Q (s R)} exists {G gt M Q} true %. +%term _ + %pi (remainder-implies-gt-quotient (divrem/z _) R+1>0) + %<- (succ-implies-gt-zero _ R+1>0) %. +%term _ + %pi (remainder-implies-gt-quotient (divrem/s D/N=Q,sR D+N=M) M>sQ) + %<- (remainder-implies-gt-quotient D/N=Q,sR D>Q) + %<- (gt-implies-ge-succ D>Q D>=sQ) + %<- (plus-commutative D+N=M N+D=M) + %<- (plus-implies-gt N+D=M eq/ M>D) + %<- (gt-transitive-ge M>D D>=sQ M>sQ) %. +%worlds () (remainder-implies-gt-quotient _ _) %. +%total (D) (remainder-implies-gt-quotient D _) %. +%theorem quotient-of-nonzero-is-smaller : forall* {M} {N} {Q} {R} {M-} forall {DR divrem M (s (s N)) Q R} {EN eq M (s M-)} exists {G gt M Q} true %. +%term _ %pi (quotient-of-nonzero-is-smaller _ eq/ M>0) %<- (succ-implies-gt-zero _ M>0) %. +%term _ + %pi (quotient-of-nonzero-is-smaller M/N=Q,R eq/ M>Q) + %<- (divrem-can-be-inverted M/N=Q,R X Q*N=X X+R=M) + %<- (times-right-identity _ Q*1=Q) + %<- (succ-implies-gt-zero _ N->0) + %<- (succ-preserves-gt N->0 N>1) + %<- (times-left-preserves-gt N>1 Q*N=X Q*1=Q X>Q) + %<- (plus-commutative X+R=M R+X=M) + %<- (plus-implies-ge R+X=M M>=X) + %<- (ge-transitive-gt M>=X X>Q M>Q) %. +%worlds () (quotient-of-nonzero-is-smaller _ _ _) %. +%total {} (quotient-of-nonzero-is-smaller _ _ _) %. +%theorem quotient-is-no-greater : forall* {M} {N} {Q} {R} forall {DR divrem M N Q R} exists {ge ge M Q} true %. +%term _ + %pi (quotient-is-no-greater M/N=Q,R M>=Q) + %<- (divrem-can-be-inverted M/N=Q,R X Q*N=X X+R=M) + %<- (divrem-implies-positive M/N=Q,R NN N=NN+1) + %<- (eq-symmetric N=NN+1 NN+1=N) + %<- (succ-implies-gt-zero NN NN+1>0) + %<- (gt-respects-eq NN+1>0 NN+1=N eq/ N>0) + %<- (gt-implies-ge-succ N>0 N>=1) + %<- (times-right-identity _ Q*1=Q) + %<- (times-left-preserves-ge* N>=1 Q*N=X Q*1=Q X>=Q) + %<- (plus-commutative X+R=M R+X=M) + %<- (plus-implies-ge R+X=M M>=X) + %<- (ge-transitive M>=X X>=Q M>=Q) %. +%worlds () (quotient-is-no-greater _ _) %. +%total {} (quotient-is-no-greater _ _) %. +%%%%% minmax.elf +%%%%% Minimum and Maximum functor +%%%%% John Boyland +%%%% Definitions +%sort min %. +%term min/= min X X X %. +%term min/> %pi (gt X1 X2) %-> (min X1 X2 X2) %. +%term min/< %pi (gt X2 X1) %-> (min X1 X2 X1) %. +%sort max %. +%term max/= max X X X %. +%term max/> %pi (gt X1 X2) %-> (max X1 X2 X1) %. +%term max/< %pi (gt X2 X1) %-> (max X1 X2 X2) %. +%%%% Theorems +%%% Theorems about min +%theorem false-implies-min : forall* {X1} {X2} {X3} forall {F void} exists {M min X1 X2 X3} true %. +%worlds () (false-implies-min _ _) %. +%total {} (false-implies-min _ _) %. +%theorem min-respects-eq : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {MX min X1 X2 X3} {E1 eq X1 Y1} {E2 eq X2 Y2} {E3 eq X3 Y3} exists {MY min Y1 Y2 Y3} true %. +%term _ min-respects-eq M eq/ eq/ eq/ M %. +%worlds () (min-respects-eq _ _ _ _ _) %. +%total {} (min-respects-eq _ _ _ _ _) %. +%theorem min-total** : forall* {X1} {X2} {C} forall {D compare X1 X2 C} exists {X3} {M min X1 X2 X3} true %. +%term _ min-total** compare/= _ min/= %. +%term _ min-total** (compare/> X1>X2) _ (min/> X1>X2) %. +%term _ min-total** (compare/< X2>X1) _ (min/< X2>X1) %. +%worlds () (min-total** _ _ _) %. +%total {} (min-total** _ _ _) %. +%theorem min-total* : forall {X1} {X2} exists {X3} {M min X1 X2 X3} true %. +%term _ %pi (min-total* X1 X2 X3 M) %<- (compare-total D) %<- (min-total** D X3 M) %. +%worlds () (min-total* _ _ _ _) %. +%total {} (min-total* _ _ _ _) %. +%inline min-total min-total* _ _ _ %. +%theorem min-deterministic : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {MX min X1 X2 X3} {MY min Y1 Y2 Y3} {E1 eq X1 Y1} {E2 eq X2 Y2} exists {E3 eq X3 Y3} true %. +%term _ min-deterministic min/= min/= eq/ eq/ eq/ %. +%term _ min-deterministic min/= (min/> X>X) eq/ eq/ eq/ %. +%term _ min-deterministic min/= (min/< X>X) eq/ eq/ eq/ %. +%term _ min-deterministic (min/> X>X) min/= eq/ eq/ eq/ %. +%term _ min-deterministic (min/> X>Y) (min/> X>YP) eq/ eq/ eq/ %. +%term _ + %pi (min-deterministic (min/> X>Y) (min/< Y>X) eq/ eq/ E) + %<- (gt-anti-symmetric X>Y Y>X F) + %<- (false-implies-eq F E) %. +%term _ min-deterministic (min/< X>X) min/= eq/ eq/ eq/ %. +%term _ + %pi (min-deterministic (min/< X>Y) (min/> Y>X) eq/ eq/ E) + %<- (gt-anti-symmetric X>Y Y>X F) + %<- (false-implies-eq F E) %. +%term _ min-deterministic (min/< X>Y) (min/< X>YP) eq/ eq/ eq/ %. +%worlds () (min-deterministic _ _ _ _ _) %. +%total {} (min-deterministic _ _ _ _ _) %. +%theorem min-commutative : forall* {X1} {X2} {X3} forall {M min X1 X2 X3} exists {Mc min X2 X1 X3} true %. +%term _ min-commutative min/= min/= %. +%term _ min-commutative (min/> X>Y) (min/< X>Y) %. +%term _ min-commutative (min/< X>Y) (min/> X>Y) %. +%worlds () (min-commutative _ _) %. +%total {} (min-commutative _ _) %. +%theorem ge-implies-min : forall* {X1} {X2} forall {G ge X1 X2} exists {M min X1 X2 X2} true %. +%term _ ge-implies-min (ge/> X1>X2) (min/> X1>X2) %. +%term _ ge-implies-min (ge/= eq/) min/= %. +%worlds () (ge-implies-min _ _) %. +%total {} (ge-implies-min _ _) %. +%theorem le-implies-min : forall* {X1} {X2} forall {G le X1 X2} exists {M min X1 X2 X1} true %. +%term _ + %pi (le-implies-min X2>=X1 M) + %<- (ge-implies-min X2>=X1 Mc) + %<- (min-commutative Mc M) %. +%worlds () (le-implies-min _ _) %. +%total {} (le-implies-min _ _) %. +%theorem min-implies-ge : forall* {X1} {X2} {X3} forall {M min X1 X2 X3} exists {G1 ge X1 X3} {G2 ge X2 X3} true %. +%term _ min-implies-ge min/= (ge/= eq/) (ge/= eq/) %. +%term _ min-implies-ge (min/> X1>X2) (ge/> X1>X2) (ge/= eq/) %. +%term _ min-implies-ge (min/< X2>X1) (ge/= eq/) (ge/> X2>X1) %. +%worlds () (min-implies-ge _ _ _) %. +%total {} (min-implies-ge _ _ _) %. +%theorem min-left-preserves-ge* : forall* {X1} {X2} {X3} {X4} {X5} forall {G ge X2 X4} {M12 min X1 X2 X3} {M14 min X1 X4 X5} exists {G ge X3 X5} true %. +%term _ %pi (min-left-preserves-ge* _ min/= M G) %<- (min-implies-ge M G _) %. +%term _ + %pi (min-left-preserves-ge* X2>=X4 (min/> X1>X2) X1&X4=X5 X2>=X5) + %<- (min-implies-ge X1&X4=X5 _ X4>=X5) + %<- (ge-transitive X2>=X4 X4>=X5 X2>=X5) %. +%term _ + %pi (min-left-preserves-ge* _ (min/< _) X1&X4=X5 X1>=X5) + %<- (min-implies-ge X1&X4=X5 X1>=X5 _) %. +%worlds () (min-left-preserves-ge* _ _ _ _) %. +%total {} (min-left-preserves-ge* _ _ _ _) %. +%theorem min-left-preserves-le* : forall* {X1} {X2} {X3} {X4} {X5} forall {G le X2 X4} {M12 min X1 X2 X3} {M14 min X1 X4 X5} exists {G le X3 X5} true %. +%term _ + %pi (min-left-preserves-le* G1 M12 M14 G2) + %<- (min-left-preserves-ge* G1 M14 M12 G2) %. +%worlds () (min-left-preserves-le* _ _ _ _) %. +%total {} (min-left-preserves-le* _ _ _ _) %. +%theorem min-left-preserves-ge : forall* {X1} {X2} {X4} forall {G ge X2 X4} exists {X3} {X5} {O1 min X1 X2 X3} {O2 min X1 X4 X5} {G2 ge X3 X5} true %. +%term _ + %pi (min-left-preserves-ge X2>=X4 X3 X5 X1&X2=A3 X1&X4=X5 X3>=X5) + %<- (min-total X1&X2=A3) + %<- (min-total X1&X4=X5) + %<- (min-left-preserves-ge* X2>=X4 X1&X2=A3 X1&X4=X5 X3>=X5) %. +%worlds () (min-left-preserves-ge X2>=X4 X3 X5 X1&X2=A3 X1&X4=X5 X3>=X5) %. +%total {} (min-left-preserves-ge _ _ _ _ _ _) %. +%theorem min-right-preserves-ge* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 ge X1 X2} {O1 min X1 X3 X4} {O2 min X2 X3 X5} exists {G2 ge X4 X5} true %. +%term _ + %pi (min-right-preserves-ge* X1>=X2 X1&X3=X4 X2&X3=X5 X4>=X5) + %<- (min-commutative X1&X3=X4 X3&X1=X4) + %<- (min-commutative X2&X3=X5 X3&X2=X5) + %<- (min-left-preserves-ge* X1>=X2 X3&X1=X4 X3&X2=X5 X4>=X5) %. +%worlds () (min-right-preserves-ge* X1>=X2 X1&X3=X4 X2&X3=X5 X4>=X5) %. +%total {} (min-right-preserves-ge* _ _ _ _) %. +%theorem min-right-preserves-ge : forall* {X1} {X2} {X3} forall {G1 ge X1 X2} exists {X4} {X5} {O1 min X1 X3 X4} {O2 min X2 X3 X5} {G2 ge X4 X5} true %. +%term _ + %pi (min-right-preserves-ge X1>=X2 X4 X5 X1&X3=X4 X2&X3=X5 X4>=X5) + %<- (min-total X1&X3=X4) + %<- (min-total X2&X3=X5) + %<- (min-right-preserves-ge* X1>=X2 X1&X3=X4 X2&X3=X5 X4>=X5) %. +%worlds () (min-right-preserves-ge X1>=X2 X4 X5 X1&X3=X4 X2&X3=X5 X4>=X5) %. +%total {} (min-right-preserves-ge _ _ _ _ _ _) %. +%theorem min-preserves-ge* : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {G1 ge X1 Y1} {G2 ge X2 Y2} {MX min X1 X2 X3} {MY min Y1 Y2 Y3} exists {G3 ge X3 Y3} true %. +%term _ + %pi (min-preserves-ge* X1>=Y1 X2>=Y2 X1&X2=X3 Y1&Y2=Y3 X3>=Y3) + %<- (min-total Y1&X2=X) + %<- (min-right-preserves-ge* X1>=Y1 X1&X2=X3 Y1&X2=X X3>=X) + %<- (min-left-preserves-ge* X2>=Y2 Y1&X2=X Y1&Y2=Y3 X>=Y3) + %<- (ge-transitive X3>=X X>=Y3 X3>=Y3) %. +%worlds () (min-preserves-ge* X1>=Y1 X2>=Y2 X1&X2=X3 Y1&Y2=Y3 X3>=Y3) %. +%total {} (min-preserves-ge* _ _ _ _ _) %. +%theorem min-preserves-ge : forall* {X1} {X2} {Y1} {Y2} forall {G1 ge X1 Y1} {G2 ge X2 Y2} exists {X3} {Y3} {MX min X1 X2 X3} {MY min Y1 Y2 Y3} {G3 ge X3 Y3} true %. +%term _ + %pi (min-preserves-ge X1>=Y1 X2>=Y2 X3 Y3 X1&X2=X3 Y1&Y2=Y3 X3>=Y3) + %<- (min-total X1&X2=X3) + %<- (min-total Y1&Y2=Y3) + %<- (min-preserves-ge* X1>=Y1 X2>=Y2 X1&X2=X3 Y1&Y2=Y3 X3>=Y3) %. +%worlds () (min-preserves-ge X1>=Y1 X2>=Y2 X3 Y3 X1&X2=X3 Y1&Y2=Y3 X3>=Y3) %. +%total {} (min-preserves-ge _ _ _ _ _ _ _) %. +%theorem min-left-preserves-le : forall* {X1} {X2} {X4} forall {G le X2 X4} exists {X3} {X5} {O1 min X1 X2 X3} {O2 min X1 X4 X5} {G2 le X3 X5} true %. +%term _ + %pi (min-left-preserves-le X2<=X4 X3 X5 X1&X2=A3 X1&X4=X5 X3<=X5) + %<- (min-total X1&X2=A3) + %<- (min-total X1&X4=X5) + %<- (min-left-preserves-le* X2<=X4 X1&X2=A3 X1&X4=X5 X3<=X5) %. +%worlds () (min-left-preserves-le X2<=X4 X3 X5 X1&X2=A3 X1&X4=X5 X3<=X5) %. +%total {} (min-left-preserves-le _ _ _ _ _ _) %. +%theorem min-right-preserves-le* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 le X1 X2} {O1 min X1 X3 X4} {O2 min X2 X3 X5} exists {G2 le X4 X5} true %. +%term _ + %pi (min-right-preserves-le* X1<=X2 X1&X3=X4 X2&X3=X5 X4<=X5) + %<- (min-commutative X1&X3=X4 X3&X1=X4) + %<- (min-commutative X2&X3=X5 X3&X2=X5) + %<- (min-left-preserves-le* X1<=X2 X3&X1=X4 X3&X2=X5 X4<=X5) %. +%worlds () (min-right-preserves-le* X1<=X2 X1&X3=X4 X2&X3=X5 X4<=X5) %. +%total {} (min-right-preserves-le* _ _ _ _) %. +%theorem min-right-preserves-le : forall* {X1} {X2} {X3} forall {G1 le X1 X2} exists {X4} {X5} {O1 min X1 X3 X4} {O2 min X2 X3 X5} {G2 le X4 X5} true %. +%term _ + %pi (min-right-preserves-le X1<=X2 X4 X5 X1&X3=X4 X2&X3=X5 X4<=X5) + %<- (min-total X1&X3=X4) + %<- (min-total X2&X3=X5) + %<- (min-right-preserves-le* X1<=X2 X1&X3=X4 X2&X3=X5 X4<=X5) %. +%worlds () (min-right-preserves-le X1<=X2 X4 X5 X1&X3=X4 X2&X3=X5 X4<=X5) %. +%total {} (min-right-preserves-le _ _ _ _ _ _) %. +%theorem min-preserves-le* : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {G1 le X1 Y1} {G2 le X2 Y2} {MX min X1 X2 X3} {MY min Y1 Y2 Y3} exists {G3 le X3 Y3} true %. +%term _ + %pi (min-preserves-le* X1<=Y1 X2<=Y2 X1&X2=X3 Y1&Y2=Y3 X3<=Y3) + %<- (min-total Y1&X2=X) + %<- (min-right-preserves-le* X1<=Y1 X1&X2=X3 Y1&X2=X X3<=X) + %<- (min-left-preserves-le* X2<=Y2 Y1&X2=X Y1&Y2=Y3 X<=Y3) + %<- (le-transitive X3<=X X<=Y3 X3<=Y3) %. +%worlds () (min-preserves-le* X1<=Y1 X2<=Y2 X1&X2=X3 Y1&Y2=Y3 X3<=Y3) %. +%total {} (min-preserves-le* _ _ _ _ _) %. +%theorem min-preserves-le : forall* {X1} {X2} {Y1} {Y2} forall {G1 le X1 Y1} {G2 le X2 Y2} exists {X3} {Y3} {MX min X1 X2 X3} {MY min Y1 Y2 Y3} {G3 le X3 Y3} true %. +%term _ + %pi (min-preserves-le X1<=Y1 X2<=Y2 X3 Y3 X1&X2=X3 Y1&Y2=Y3 X3<=Y3) + %<- (min-total X1&X2=X3) + %<- (min-total Y1&Y2=Y3) + %<- (min-preserves-le* X1<=Y1 X2<=Y2 X1&X2=X3 Y1&Y2=Y3 X3<=Y3) %. +%worlds () (min-preserves-le X1<=Y1 X2<=Y2 X3 Y3 X1&X2=X3 Y1&Y2=Y3 X3<=Y3) %. +%total {} (min-preserves-le _ _ _ _ _ _ _) %. +%theorem min-is-glb : forall* {X1} {X2} {X3} {X4} forall {M min X1 X2 X3} {G1 ge X1 X4} {G2 ge X2 X4} exists {G3 ge X3 X4} true %. +%term _ min-is-glb min/= G _ G %. +%term _ min-is-glb (min/> X1>X2) _ G G %. +%term _ min-is-glb (min/< X2>X1) G _ G %. +%worlds () (min-is-glb _ _ _ _) %. +%total {} (min-is-glb _ _ _ _) %. +%theorem min-associative : forall* {X1} {X2} {X3} {X4} {X7} forall {M12 min X1 X2 X3} {M34 min X3 X4 X7} exists {X6} {M24 min X2 X4 X6} {M16 min X1 X6 X7} true %. +%term _ min-associative min/= min/= _ min/= min/= %. +%term _ min-associative min/= (min/> X3>X4) _ (min/> X3>X4) (min/> X3>X4) %. +%term _ min-associative min/= (min/< X4>X3) _ (min/< X4>X3) min/= %. +%term _ min-associative (min/> X1>X2) min/= _ min/= (min/> X1>X2) %. +%term _ + %pi (min-associative (min/> X1>X2) (min/> X2>X4) _ (min/> X2>X4) (min/> X1>X4)) + %<- (gt-transitive X1>X2 X2>X4 X1>X4) %. +%term _ min-associative (min/> X1>X2) (min/< X4>X2) _ (min/< X4>X2) (min/> X1>X2) %. +%term _ min-associative (min/< X2>X1) min/= _ (min/> X2>X1) min/= %. +%term _ + %pi (min-associative (min/< X2>X1) (min/> X1>X4) _ (min/> X2>X4) (min/> X1>X4)) + %<- (gt-transitive X2>X1 X1>X4 X2>X4) %. +%term _ + %pi (min-associative (min/< X2>X1) (min/< X4>X1) _ M24 M16) + %<- (min-total M24) + %<- (min-is-glb M24 (ge/> X2>X1) (ge/> X4>X1) (%the (ge X6 X1) X6>=X1)) + %<- (ge-implies-min X6>=X1 M16c) + %<- (min-commutative M16c M16) %. +%worlds () (min-associative _ _ _ _ _) %. +%total {} (min-associative _ _ _ _ _) %. +%theorem min-associative* : forall* {X1} {X2} {X12} {X3} {X23} {X123} forall {OP12 min X1 X2 X12} {OP12-3 min X12 X3 X123} {OP23 min X2 X3 X23} exists {OP1-23 min X1 X23 X123} true %. +%term _ + %pi (min-associative* X1&X2=X3 X3&X4=X7 X2&X4=X6 X1&X6=X7) + %<- (min-associative X1&X2=X3 X3&X4=X7 Y6 X2&X4=Y6 X1&Y6=X7) + %<- (min-deterministic X2&X4=Y6 X2&X4=X6 eq/ eq/ Y6=X6) + %<- (min-respects-eq X1&Y6=X7 eq/ Y6=X6 eq/ X1&X6=X7) %. +%worlds () (min-associative* _ _ _ _) %. +%total {} (min-associative* _ _ _ _) %. +%theorem min-associative-converse : forall* {X1} {X2} {X4} {X6} {X7} forall {OP24 min X2 X4 X6} {OP16 min X1 X6 X7} exists {X3} {OP12 min X1 X2 X3} {OP34 min X3 X4 X7} true %. +%term _ + %pi (min-associative-converse X2&X4=X6 X1&X6=X7 _ X1&X2=X3 X3&X4=X7) + %<- (min-commutative X2&X4=X6 X4&X2=X6) + %<- (min-commutative X1&X6=X7 X6&X1=X7) + %<- (min-associative X4&X2=X6 X6&X1=X7 _ X2&X1=X3 X4&X3=X7) + %<- (min-commutative X2&X1=X3 X1&X2=X3) + %<- (min-commutative X4&X3=X7 X3&X4=X7) %. +%worlds () (min-associative-converse X2&X4=X6 X1&X6=X7 X3 X1&X2=X3 X3&X4=X7) %. +%total {} (min-associative-converse _ _ _ _ _) %. +%theorem min-associative-converse* : forall* {X1} {X2} {X3} {X4} {X6} {X7} forall {OP24 min X2 X4 X6} {OP16 min X1 X6 X7} {OP12 min X1 X2 X3} exists {OP34 min X3 X4 X7} true %. +%term _ + %pi (min-associative-converse* X2&X4=X6 X1&X6=X7 X1&X2=X3 X3&X4=X7) + %<- (min-associative-converse X2&X4=X6 X1&X6=X7 X3P X1&X2=X3P X3P&X4=X7) + %<- (min-deterministic X1&X2=X3P X1&X2=X3 eq/ eq/ X3P=X3) + %<- (min-respects-eq X3P&X4=X7 X3P=X3 eq/ eq/ X3&X4=X7) %. +%worlds () (min-associative-converse* X2&X4=X6 X1&X6=X7 X1&X2=X3 X3&X4=X7) %. +%total {} (min-associative-converse* _ _ _ _) %. +%theorem min-assoc-commutative* : forall* {X1} {X2} {X3} {X4} {X5} {X7} forall {OP1 min X1 X2 X3} {OP2 min X3 X4 X7} {OP3 min X1 X4 X5} exists {OP4 min X5 X2 X7} true %. +%term _ + %pi (min-assoc-commutative* X1&X2=X3 X3&X4=X7 X1&X4=X5 X5&X2=X7) + %<- (min-associative X1&X2=X3 X3&X4=X7 X6 X2&X4=X6 X1&X6=X7) + %<- (min-commutative X2&X4=X6 X4&X2=X6) + %<- (min-associative-converse* X4&X2=X6 X1&X6=X7 X1&X4=X5 X5&X2=X7) %. +%worlds () (min-assoc-commutative* X1&X2=X3 X3&X4=X7 X1&X4=X5 X5&X2=X7) %. +%total {} (min-assoc-commutative* _ _ _ _) %. +%theorem min-assoc-commutative : forall* {X1} {X2} {X3} {X4} {X7} forall {OP1 min X1 X2 X3} {OP2 min X3 X4 X7} exists {X5} {OP3 min X1 X4 X5} {OP4 min X5 X2 X7} true %. +%term _ + %pi (min-assoc-commutative X1&X2=X3 X3&X4=X7 X5 X1&X4=X5 X5&X2=X7) + %<- (min-associative X1&X2=X3 X3&X4=X7 X6 X2&X4=X6 X1&X6=X7) + %<- (min-commutative X2&X4=X6 X4&X2=X6) + %<- (min-associative-converse X4&X2=X6 X1&X6=X7 X5 X1&X4=X5 X5&X2=X7) %. +%worlds () (min-assoc-commutative X1&X2=X3 X3&X4=X7 X5 X1&X4=X5 X5&X2=X7) %. +%total {} (min-assoc-commutative _ _ _ _ _) %. +%theorem min-double-associative* : forall* {A} {B} {C} {D} {A+B} {C+D} {A+C} {B+D} {X} forall {AB min A B A+B} {CD min C D C+D} {ABCD min A+B C+D X} {AC min A C A+C} {BD min B D B+D} exists {ACBD min A+C B+D X} true %. +%term _ + %pi (min-double-associative* X1&X2=X3 X4&X8=XC X3&XC=XF X1&X4=X5 X2&X8=XA X5&XA=XF) + %<- (min-associative X1&X2=X3 X3&XC=XF XE X2&XC=XE X1&XE=XF) + %<- (min-commutative X4&X8=XC X8&X4=XC) + %<- (min-associative-converse* X8&X4=XC X2&XC=XE X2&X8=XA XA&X4=XE) + %<- (min-commutative XA&X4=XE X4&XA=XE) + %<- (min-associative-converse* X4&XA=XE X1&XE=XF X1&X4=X5 X5&XA=XF) %. +%worlds () (min-double-associative* X1&X2=X3 X4&X8=XC X3&XC=XF X1&X4=X5 X2&X8=XA X5&XA=XF) %. +%total {} (min-double-associative* _ _ _ _ _ _) %. +%theorem min-double-associative : forall* {A} {B} {C} {D} {A+B} {C+D} {X} forall {AB min A B A+B} {CD min C D C+D} {ABCD min A+B C+D X} exists {A+C} {B+D} {AC min A C A+C} {BD min B D B+D} {ACBD min A+C B+D X} true %. +%term _ + %pi (min-double-associative X1&X2=X3 X4&X8=XC X3&XC=XF X5 XA X1&X4=X5 X2&X8=XA X5&XA=XF) + %<- (min-associative X1&X2=X3 X3&XC=XF XE X2&XC=XE X1&XE=XF) + %<- (min-commutative X4&X8=XC X8&X4=XC) + %<- (min-associative-converse X8&X4=XC X2&XC=XE XA X2&X8=XA XA&X4=XE) + %<- (min-commutative XA&X4=XE X4&XA=XE) + %<- (min-associative-converse X4&XA=XE X1&XE=XF X5 X1&X4=X5 X5&XA=XF) %. +%worlds () (min-double-associative _ _ _ _ _ _ _ _) %. +%total {} (min-double-associative _ _ _ _ _ _ _ _) %. +%%% Theorems about max +%theorem false-implies-max : forall* {X1} {X2} {X3} forall {F void} exists {M max X1 X2 X3} true %. +%worlds () (false-implies-max _ _) %. +%total {} (false-implies-max _ _) %. +%theorem max-respects-eq : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {MX max X1 X2 X3} {E1 eq X1 Y1} {E2 eq X2 Y2} {E3 eq X3 Y3} exists {MY max Y1 Y2 Y3} true %. +%term _ max-respects-eq M eq/ eq/ eq/ M %. +%worlds () (max-respects-eq _ _ _ _ _) %. +%total {} (max-respects-eq _ _ _ _ _) %. +%theorem max-total** : forall* {X1} {X2} {C} forall {D compare X1 X2 C} exists {X3} {M max X1 X2 X3} true %. +%term _ max-total** compare/= _ max/= %. +%term _ max-total** (compare/> X1>X2) _ (max/> X1>X2) %. +%term _ max-total** (compare/< X2>X1) _ (max/< X2>X1) %. +%worlds () (max-total** _ _ _) %. +%total {} (max-total** _ _ _) %. +%theorem max-total* : forall {X1} {X2} exists {X3} {M max X1 X2 X3} true %. +%term _ %pi (max-total* X1 X2 X3 M) %<- (compare-total D) %<- (max-total** D X3 M) %. +%worlds () (max-total* _ _ _ _) %. +%total {} (max-total* _ _ _ _) %. +%inline max-total max-total* _ _ _ %. +%theorem max-deterministic : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {MX max X1 X2 X3} {MY max Y1 Y2 Y3} {E1 eq X1 Y1} {E2 eq X2 Y2} exists {E3 eq X3 Y3} true %. +%term _ max-deterministic max/= max/= eq/ eq/ eq/ %. +%term _ max-deterministic max/= (max/> X>X) eq/ eq/ eq/ %. +%term _ max-deterministic max/= (max/< X>X) eq/ eq/ eq/ %. +%term _ max-deterministic (max/> X>X) max/= eq/ eq/ eq/ %. +%term _ max-deterministic (max/> X>Y) (max/> X>YP) eq/ eq/ eq/ %. +%term _ + %pi (max-deterministic (max/> X>Y) (max/< Y>X) eq/ eq/ E) + %<- (gt-anti-symmetric X>Y Y>X F) + %<- (false-implies-eq F E) %. +%term _ max-deterministic (max/< X>X) max/= eq/ eq/ eq/ %. +%term _ + %pi (max-deterministic (max/< X>Y) (max/> Y>X) eq/ eq/ E) + %<- (gt-anti-symmetric X>Y Y>X F) + %<- (false-implies-eq F E) %. +%term _ max-deterministic (max/< X>Y) (max/< X>YP) eq/ eq/ eq/ %. +%worlds () (max-deterministic _ _ _ _ _) %. +%total {} (max-deterministic _ _ _ _ _) %. +%theorem max-commutative : forall* {X1} {X2} {X3} forall {M max X1 X2 X3} exists {Mc max X2 X1 X3} true %. +%term _ max-commutative max/= max/= %. +%term _ max-commutative (max/> X>Y) (max/< X>Y) %. +%term _ max-commutative (max/< X>Y) (max/> X>Y) %. +%worlds () (max-commutative _ _) %. +%total {} (max-commutative _ _) %. +%theorem ge-implies-max : forall* {X1} {X2} forall {G ge X1 X2} exists {M max X1 X2 X1} true %. +%term _ ge-implies-max (ge/> X1>X2) (max/> X1>X2) %. +%term _ ge-implies-max (ge/= eq/) max/= %. +%worlds () (ge-implies-max _ _) %. +%total {} (ge-implies-max _ _) %. +%theorem le-implies-max : forall* {X1} {X2} forall {G le X1 X2} exists {M max X1 X2 X2} true %. +%term _ + %pi (le-implies-max X2>=X1 M) + %<- (ge-implies-max X2>=X1 Mc) + %<- (max-commutative Mc M) %. +%worlds () (le-implies-max _ _) %. +%total {} (le-implies-max _ _) %. +%theorem max-implies-ge : forall* {X1} {X2} {X3} forall {M max X1 X2 X3} exists {G1 ge X3 X1} {G2 ge X3 X2} true %. +%term _ max-implies-ge max/= (ge/= eq/) (ge/= eq/) %. +%term _ max-implies-ge (max/> X1>X2) (ge/= eq/) (ge/> X1>X2) %. +%term _ max-implies-ge (max/< X2>X1) (ge/> X2>X1) (ge/= eq/) %. +%worlds () (max-implies-ge _ _ _) %. +%total {} (max-implies-ge _ _ _) %. +%theorem max-is-lub : forall* {X0} {X1} {X2} {X3} forall {M max X1 X2 X3} {G1 ge X0 X1} {G2 ge X0 X2} exists {G3 ge X0 X3} true %. +%term _ max-is-lub max/= G _ G %. +%term _ max-is-lub (max/> X1>X2) G _ G %. +%term _ max-is-lub (max/< X2>X1) _ G G %. +%worlds () (max-is-lub _ _ _ _) %. +%total {} (max-is-lub _ _ _ _) %. +%theorem max-left-preserves-ge* : forall* {X1} {X2} {X3} {X4} {X5} forall {G ge X2 X4} {M12 max X1 X2 X3} {M14 max X1 X4 X5} exists {G ge X3 X5} true %. +%term _ + %pi (max-left-preserves-ge* X>=X4 max/= X|X4=X5 X>=X5) + %<- (ge-reflexive _ X>=X) + %<- (max-is-lub X|X4=X5 X>=X X>=X4 X>=X5) %. +%term _ + %pi (max-left-preserves-ge* X2>=X4 (max/> X1>X2) X1|X4=X5 X1>=X5) + %<- (ge-transitive (ge/> X1>X2) X2>=X4 X1>=X4) + %<- (ge-reflexive _ X1>=X1) + %<- (max-is-lub X1|X4=X5 X1>=X1 X1>=X4 X1>=X5) %. +%term _ + %pi (max-left-preserves-ge* X2>=X4 (max/< X2>X1) X1|X4=X5 X2>=X5) + %<- (max-is-lub X1|X4=X5 (ge/> X2>X1) X2>=X4 X2>=X5) %. +%worlds () (max-left-preserves-ge* _ _ _ _) %. +%total {} (max-left-preserves-ge* _ _ _ _) %. +%theorem max-left-preserves-le* : forall* {X1} {X2} {X3} {X4} {X5} forall {G le X2 X4} {M12 max X1 X2 X3} {M14 max X1 X4 X5} exists {G le X3 X5} true %. +%term _ + %pi (max-left-preserves-le* G1 M12 M14 G2) + %<- (max-left-preserves-ge* G1 M14 M12 G2) %. +%worlds () (max-left-preserves-le* _ _ _ _) %. +%total {} (max-left-preserves-le* _ _ _ _) %. +%theorem max-left-preserves-ge : forall* {X1} {X2} {X4} forall {G ge X2 X4} exists {X3} {X5} {O1 max X1 X2 X3} {O2 max X1 X4 X5} {G2 ge X3 X5} true %. +%term _ + %pi (max-left-preserves-ge X2>=X4 X3 X5 X1&X2=A3 X1&X4=X5 X3>=X5) + %<- (max-total X1&X2=A3) + %<- (max-total X1&X4=X5) + %<- (max-left-preserves-ge* X2>=X4 X1&X2=A3 X1&X4=X5 X3>=X5) %. +%worlds () (max-left-preserves-ge X2>=X4 X3 X5 X1&X2=A3 X1&X4=X5 X3>=X5) %. +%total {} (max-left-preserves-ge _ _ _ _ _ _) %. +%theorem max-right-preserves-ge* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 ge X1 X2} {O1 max X1 X3 X4} {O2 max X2 X3 X5} exists {G2 ge X4 X5} true %. +%term _ + %pi (max-right-preserves-ge* X1>=X2 X1&X3=X4 X2&X3=X5 X4>=X5) + %<- (max-commutative X1&X3=X4 X3&X1=X4) + %<- (max-commutative X2&X3=X5 X3&X2=X5) + %<- (max-left-preserves-ge* X1>=X2 X3&X1=X4 X3&X2=X5 X4>=X5) %. +%worlds () (max-right-preserves-ge* X1>=X2 X1&X3=X4 X2&X3=X5 X4>=X5) %. +%total {} (max-right-preserves-ge* _ _ _ _) %. +%theorem max-right-preserves-ge : forall* {X1} {X2} {X3} forall {G1 ge X1 X2} exists {X4} {X5} {O1 max X1 X3 X4} {O2 max X2 X3 X5} {G2 ge X4 X5} true %. +%term _ + %pi (max-right-preserves-ge X1>=X2 X4 X5 X1&X3=X4 X2&X3=X5 X4>=X5) + %<- (max-total X1&X3=X4) + %<- (max-total X2&X3=X5) + %<- (max-right-preserves-ge* X1>=X2 X1&X3=X4 X2&X3=X5 X4>=X5) %. +%worlds () (max-right-preserves-ge X1>=X2 X4 X5 X1&X3=X4 X2&X3=X5 X4>=X5) %. +%total {} (max-right-preserves-ge _ _ _ _ _ _) %. +%theorem max-preserves-ge* : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {G1 ge X1 Y1} {G2 ge X2 Y2} {MX max X1 X2 X3} {MY max Y1 Y2 Y3} exists {G3 ge X3 Y3} true %. +%term _ + %pi (max-preserves-ge* X1>=Y1 X2>=Y2 X1&X2=X3 Y1&Y2=Y3 X3>=Y3) + %<- (max-total Y1&X2=X) + %<- (max-right-preserves-ge* X1>=Y1 X1&X2=X3 Y1&X2=X X3>=X) + %<- (max-left-preserves-ge* X2>=Y2 Y1&X2=X Y1&Y2=Y3 X>=Y3) + %<- (ge-transitive X3>=X X>=Y3 X3>=Y3) %. +%worlds () (max-preserves-ge* X1>=Y1 X2>=Y2 X1&X2=X3 Y1&Y2=Y3 X3>=Y3) %. +%total {} (max-preserves-ge* _ _ _ _ _) %. +%theorem max-preserves-ge : forall* {X1} {X2} {Y1} {Y2} forall {G1 ge X1 Y1} {G2 ge X2 Y2} exists {X3} {Y3} {MX max X1 X2 X3} {MY max Y1 Y2 Y3} {G3 ge X3 Y3} true %. +%term _ + %pi (max-preserves-ge X1>=Y1 X2>=Y2 X3 Y3 X1&X2=X3 Y1&Y2=Y3 X3>=Y3) + %<- (max-total X1&X2=X3) + %<- (max-total Y1&Y2=Y3) + %<- (max-preserves-ge* X1>=Y1 X2>=Y2 X1&X2=X3 Y1&Y2=Y3 X3>=Y3) %. +%worlds () (max-preserves-ge X1>=Y1 X2>=Y2 X3 Y3 X1&X2=X3 Y1&Y2=Y3 X3>=Y3) %. +%total {} (max-preserves-ge _ _ _ _ _ _ _) %. +%theorem max-left-preserves-le : forall* {X1} {X2} {X4} forall {G le X2 X4} exists {X3} {X5} {O1 max X1 X2 X3} {O2 max X1 X4 X5} {G2 le X3 X5} true %. +%term _ + %pi (max-left-preserves-le X2<=X4 X3 X5 X1&X2=A3 X1&X4=X5 X3<=X5) + %<- (max-total X1&X2=A3) + %<- (max-total X1&X4=X5) + %<- (max-left-preserves-le* X2<=X4 X1&X2=A3 X1&X4=X5 X3<=X5) %. +%worlds () (max-left-preserves-le X2<=X4 X3 X5 X1&X2=A3 X1&X4=X5 X3<=X5) %. +%total {} (max-left-preserves-le _ _ _ _ _ _) %. +%theorem max-right-preserves-le* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 le X1 X2} {O1 max X1 X3 X4} {O2 max X2 X3 X5} exists {G2 le X4 X5} true %. +%term _ + %pi (max-right-preserves-le* X1<=X2 X1&X3=X4 X2&X3=X5 X4<=X5) + %<- (max-commutative X1&X3=X4 X3&X1=X4) + %<- (max-commutative X2&X3=X5 X3&X2=X5) + %<- (max-left-preserves-le* X1<=X2 X3&X1=X4 X3&X2=X5 X4<=X5) %. +%worlds () (max-right-preserves-le* X1<=X2 X1&X3=X4 X2&X3=X5 X4<=X5) %. +%total {} (max-right-preserves-le* _ _ _ _) %. +%theorem max-right-preserves-le : forall* {X1} {X2} {X3} forall {G1 le X1 X2} exists {X4} {X5} {O1 max X1 X3 X4} {O2 max X2 X3 X5} {G2 le X4 X5} true %. +%term _ + %pi (max-right-preserves-le X1<=X2 X4 X5 X1&X3=X4 X2&X3=X5 X4<=X5) + %<- (max-total X1&X3=X4) + %<- (max-total X2&X3=X5) + %<- (max-right-preserves-le* X1<=X2 X1&X3=X4 X2&X3=X5 X4<=X5) %. +%worlds () (max-right-preserves-le X1<=X2 X4 X5 X1&X3=X4 X2&X3=X5 X4<=X5) %. +%total {} (max-right-preserves-le _ _ _ _ _ _) %. +%theorem max-preserves-le* : forall* {X1} {X2} {X3} {Y1} {Y2} {Y3} forall {G1 le X1 Y1} {G2 le X2 Y2} {MX max X1 X2 X3} {MY max Y1 Y2 Y3} exists {G3 le X3 Y3} true %. +%term _ + %pi (max-preserves-le* X1<=Y1 X2<=Y2 X1&X2=X3 Y1&Y2=Y3 X3<=Y3) + %<- (max-total Y1&X2=X) + %<- (max-right-preserves-le* X1<=Y1 X1&X2=X3 Y1&X2=X X3<=X) + %<- (max-left-preserves-le* X2<=Y2 Y1&X2=X Y1&Y2=Y3 X<=Y3) + %<- (le-transitive X3<=X X<=Y3 X3<=Y3) %. +%worlds () (max-preserves-le* X1<=Y1 X2<=Y2 X1&X2=X3 Y1&Y2=Y3 X3<=Y3) %. +%total {} (max-preserves-le* _ _ _ _ _) %. +%theorem max-preserves-le : forall* {X1} {X2} {Y1} {Y2} forall {G1 le X1 Y1} {G2 le X2 Y2} exists {X3} {Y3} {MX max X1 X2 X3} {MY max Y1 Y2 Y3} {G3 le X3 Y3} true %. +%term _ + %pi (max-preserves-le X1<=Y1 X2<=Y2 X3 Y3 X1&X2=X3 Y1&Y2=Y3 X3<=Y3) + %<- (max-total X1&X2=X3) + %<- (max-total Y1&Y2=Y3) + %<- (max-preserves-le* X1<=Y1 X2<=Y2 X1&X2=X3 Y1&Y2=Y3 X3<=Y3) %. +%worlds () (max-preserves-le X1<=Y1 X2<=Y2 X3 Y3 X1&X2=X3 Y1&Y2=Y3 X3<=Y3) %. +%total {} (max-preserves-le _ _ _ _ _ _ _) %. +%theorem max-associative : forall* {X1} {X2} {X3} {X4} {X7} forall {M12 max X1 X2 X3} {M34 max X3 X4 X7} exists {X6} {M24 max X2 X4 X6} {M16 max X1 X6 X7} true %. +%term _ max-associative max/= max/= _ max/= max/= %. +%term _ max-associative max/= (max/> X3>X4) _ (max/> X3>X4) max/= %. +%term _ max-associative max/= (max/< X4>X3) _ (max/< X4>X3) (max/< X4>X3) %. +%term _ max-associative (max/> X1>X2) max/= _ (max/< X1>X2) max/= %. +%term _ + %pi (max-associative (max/> X1>X2) (max/> X1>X4) _ M24 M16) + %<- (max-total M24) + %<- (max-is-lub M24 (ge/> X1>X2) (ge/> X1>X4) X1>=X6) + %<- (ge-implies-max X1>=X6 M16) %. +%term _ + %pi (max-associative (max/> X1>X2) (max/< X4>X1) _ (max/< X4>X2) (max/< X4>X1)) + %<- (gt-transitive X4>X1 X1>X2 X4>X2) %. +%term _ max-associative (max/< X2>X1) max/= _ max/= (max/< X2>X1) %. +%term _ max-associative (max/< X2>X1) (max/> X2>X4) _ (max/> X2>X4) (max/< X2>X1) %. +%term _ + %pi (max-associative (max/< X2>X1) (max/< X4>X2) _ (max/< X4>X2) (max/< X4>X1)) + %<- (gt-transitive X4>X2 X2>X1 X4>X1) %. +%worlds () (max-associative _ _ _ _ _) %. +%total {} (max-associative _ _ _ _ _) %. +%theorem max-associative* : forall* {X1} {X2} {X12} {X3} {X23} {X123} forall {OP12 max X1 X2 X12} {OP12-3 max X12 X3 X123} {OP23 max X2 X3 X23} exists {OP1-23 max X1 X23 X123} true %. +%term _ + %pi (max-associative* X1|X2=X3 X3|X4=X7 X2|X4=X6 X1|X6=X7) + %<- (max-associative X1|X2=X3 X3|X4=X7 Y6 X2|X4=Y6 X1|Y6=X7) + %<- (max-deterministic X2|X4=Y6 X2|X4=X6 eq/ eq/ Y6=X6) + %<- (max-respects-eq X1|Y6=X7 eq/ Y6=X6 eq/ X1|X6=X7) %. +%worlds () (max-associative* _ _ _ _) %. +%total {} (max-associative* _ _ _ _) %. +%theorem max-associative-converse : forall* {X1} {X2} {X4} {X6} {X7} forall {OP24 max X2 X4 X6} {OP16 max X1 X6 X7} exists {X3} {OP12 max X1 X2 X3} {OP34 max X3 X4 X7} true %. +%term _ + %pi (max-associative-converse X2|X4=X6 X1|X6=X7 _ X1|X2=X3 X3|X4=X7) + %<- (max-commutative X2|X4=X6 X4|X2=X6) + %<- (max-commutative X1|X6=X7 X6|X1=X7) + %<- (max-associative X4|X2=X6 X6|X1=X7 _ X2|X1=X3 X4|X3=X7) + %<- (max-commutative X2|X1=X3 X1|X2=X3) + %<- (max-commutative X4|X3=X7 X3|X4=X7) %. +%worlds () (max-associative-converse X2|X4=X6 X1|X6=X7 X3 X1|X2=X3 X3|X4=X7) %. +%total {} (max-associative-converse _ _ _ _ _) %. +%theorem max-associative-converse* : forall* {X1} {X2} {X3} {X4} {X6} {X7} forall {OP24 max X2 X4 X6} {OP16 max X1 X6 X7} {OP12 max X1 X2 X3} exists {OP34 max X3 X4 X7} true %. +%term _ + %pi (max-associative-converse* X2|X4=X6 X1|X6=X7 X1|X2=X3 X3|X4=X7) + %<- (max-associative-converse X2|X4=X6 X1|X6=X7 X3P X1|X2=X3P X3P|X4=X7) + %<- (max-deterministic X1|X2=X3P X1|X2=X3 eq/ eq/ X3P=X3) + %<- (max-respects-eq X3P|X4=X7 X3P=X3 eq/ eq/ X3|X4=X7) %. +%worlds () (max-associative-converse* X2|X4=X6 X1|X6=X7 X1|X2=X3 X3|X4=X7) %. +%total {} (max-associative-converse* _ _ _ _) %. +%theorem max-assoc-commutative* : forall* {X1} {X2} {X3} {X4} {X5} {X7} forall {OP1 max X1 X2 X3} {OP2 max X3 X4 X7} {OP3 max X1 X4 X5} exists {OP4 max X5 X2 X7} true %. +%term _ + %pi (max-assoc-commutative* X1|X2=X3 X3|X4=X7 X1|X4=X5 X5|X2=X7) + %<- (max-associative X1|X2=X3 X3|X4=X7 X6 X2|X4=X6 X1|X6=X7) + %<- (max-commutative X2|X4=X6 X4|X2=X6) + %<- (max-associative-converse* X4|X2=X6 X1|X6=X7 X1|X4=X5 X5|X2=X7) %. +%worlds () (max-assoc-commutative* X1|X2=X3 X3|X4=X7 X1|X4=X5 X5|X2=X7) %. +%total {} (max-assoc-commutative* _ _ _ _) %. +%theorem max-assoc-commutative : forall* {X1} {X2} {X3} {X4} {X7} forall {OP1 max X1 X2 X3} {OP2 max X3 X4 X7} exists {X5} {OP3 max X1 X4 X5} {OP4 max X5 X2 X7} true %. +%term _ + %pi (max-assoc-commutative X1|X2=X3 X3|X4=X7 X5 X1|X4=X5 X5|X2=X7) + %<- (max-associative X1|X2=X3 X3|X4=X7 X6 X2|X4=X6 X1|X6=X7) + %<- (max-commutative X2|X4=X6 X4|X2=X6) + %<- (max-associative-converse X4|X2=X6 X1|X6=X7 X5 X1|X4=X5 X5|X2=X7) %. +%worlds () (max-assoc-commutative X1|X2=X3 X3|X4=X7 X5 X1|X4=X5 X5|X2=X7) %. +%total {} (max-assoc-commutative _ _ _ _ _) %. +%theorem max-double-associative* : forall* {A} {B} {C} {D} {A+B} {C+D} {A+C} {B+D} {X} forall {AB max A B A+B} {CD max C D C+D} {ABCD max A+B C+D X} {AC max A C A+C} {BD max B D B+D} exists {ACBD max A+C B+D X} true %. +%term _ + %pi (max-double-associative* X1|X2=X3 X4|X8=XC X3|XC=XF X1|X4=X5 X2|X8=XA X5|XA=XF) + %<- (max-associative X1|X2=X3 X3|XC=XF XE X2|XC=XE X1|XE=XF) + %<- (max-commutative X4|X8=XC X8|X4=XC) + %<- (max-associative-converse* X8|X4=XC X2|XC=XE X2|X8=XA XA|X4=XE) + %<- (max-commutative XA|X4=XE X4|XA=XE) + %<- (max-associative-converse* X4|XA=XE X1|XE=XF X1|X4=X5 X5|XA=XF) %. +%worlds () (max-double-associative* X1|X2=X3 X4|X8=XC X3|XC=XF X1|X4=X5 X2|X8=XA X5|XA=XF) %. +%total {} (max-double-associative* _ _ _ _ _ _) %. +%theorem max-double-associative : forall* {A} {B} {C} {D} {A+B} {C+D} {X} forall {AB max A B A+B} {CD max C D C+D} {ABCD max A+B C+D X} exists {A+C} {B+D} {AC max A C A+C} {BD max B D B+D} {ACBD max A+C B+D X} true %. +%term _ + %pi (max-double-associative X1|X2=X3 X4|X8=XC X3|XC=XF X5 XA X1|X4=X5 X2|X8=XA X5|XA=XF) + %<- (max-associative X1|X2=X3 X3|XC=XF XE X2|XC=XE X1|XE=XF) + %<- (max-commutative X4|X8=XC X8|X4=XC) + %<- (max-associative-converse X8|X4=XC X2|XC=XE XA X2|X8=XA XA|X4=XE) + %<- (max-commutative XA|X4=XE X4|XA=XE) + %<- (max-associative-converse X4|XA=XE X1|XE=XF X5 X1|X4=X5 X5|XA=XF) %. +%worlds () (max-double-associative _ _ _ _ _ _ _ _) %. +%total {} (max-double-associative _ _ _ _ _ _ _ _) %. +%%% Distributivity theorems +%theorem min-right-distributes-over-max : forall* {X1} {X2} {X3} {X4} {X7} forall {A12 max X1 X2 X3} {M34 min X3 X4 X7} exists {X5} {X6} {M14 min X1 X4 X5} {M24 min X2 X4 X6} {A56 max X5 X6 X7} true %. +%term _ min-right-distributes-over-max max/= M _ _ M M max/= %. +%term _ + %pi (min-right-distributes-over-max (max/> X1>X2) X1&X4=X7 _ _ X1&X4=X7 X2&X4=X6 X7|X6=X7) + %<- (min-total X2&X4=X6) + %<- (min-right-preserves-ge* (ge/> X1>X2) X1&X4=X7 X2&X4=X6 X7>=X6) + %<- (ge-implies-max X7>=X6 X7|X6=X7) %. +%term _ + %pi (min-right-distributes-over-max (max/< X2>X1) X2&X4=X7 _ _ X1&X4=X5 X2&X4=X7 X5|X7=X7) + %<- (min-total X1&X4=X5) + %<- (min-right-preserves-ge* (ge/> X2>X1) X2&X4=X7 X1&X4=X5 X7>=X5) + %<- (ge-implies-max X7>=X5 X7|X5=X7) + %<- (max-commutative X7|X5=X7 X5|X7=X7) %. +%worlds () (min-right-distributes-over-max _ _ _ _ _ _ _) %. +%total {} (min-right-distributes-over-max _ _ _ _ _ _ _) %. +%theorem max-right-distributes-over-min : forall* {X1} {X2} {X3} {X4} {X7} forall {A12 min X1 X2 X3} {M34 max X3 X4 X7} exists {X5} {X6} {M14 max X1 X4 X5} {M24 max X2 X4 X6} {A56 min X5 X6 X7} true %. +%term _ max-right-distributes-over-min min/= M _ _ M M min/= %. +%term _ + %pi (max-right-distributes-over-min (min/> X1>X2) X2|X4=X7 _ _ X1|X4=X5 X2|X4=X7 X5&X7=X7) + %<- (max-total X1|X4=X5) + %<- (max-right-preserves-ge* (ge/> X1>X2) X1|X4=X5 X2|X4=X7 X5>=X7) + %<- (ge-implies-min X5>=X7 X5&X7=X7) %. +%term _ + %pi (max-right-distributes-over-min (min/< X2>X1) X1|X4=X7 _ _ X1|X4=X7 X2|X4=X6 X7&X6=X7) + %<- (max-total X2|X4=X6) + %<- (max-right-preserves-ge* (ge/> X2>X1) X2|X4=X6 X1|X4=X7 X6>=X7) + %<- (le-implies-min X6>=X7 X7&X6=X7) %. +%worlds () (max-right-distributes-over-min _ _ _ _ _ _ _) %. +%total {} (max-right-distributes-over-min _ _ _ _ _ _ _) %. +%theorem min-right-distributes-over-max* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {A12 max X1 X2 X3} {M34 min X3 X4 X7} {M14 min X1 X4 X5} {M24 min X2 X4 X6} exists {A56 max X5 X6 X7} true %. +%term _ + %pi (min-right-distributes-over-max* X1|X2=X3 X3&X4=X7 X1&X4=X5 X2&X4=X6 X5|X6=X7) + %<- (min-right-distributes-over-max X1|X2=X3 X3&X4=X7 Y5 Y6 X1&X4=Y5 X2&X4=Y6 Y5|Y6=X7) + %<- (min-deterministic X1&X4=Y5 X1&X4=X5 eq/ eq/ Y5=X5) + %<- (min-deterministic X2&X4=Y6 X2&X4=X6 eq/ eq/ Y6=X6) + %<- (max-respects-eq Y5|Y6=X7 Y5=X5 Y6=X6 eq/ X5|X6=X7) %. +%worlds () (min-right-distributes-over-max* X1|X2=X3 X3&X4=X7 X1&X4=X5 X2&X4=X6 X5|X6=X7) %. +%total {} (min-right-distributes-over-max* _ _ _ _ _) %. +%theorem min-left-distributes-over-max* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {A12 max X2 X4 X6} {M34 min X1 X6 X7} {M14 min X1 X2 X3} {M24 min X1 X4 X5} exists {A56 max X3 X5 X7} true %. +%term _ + %pi (min-left-distributes-over-max* X2|X4=X6 X1&X6=X7 X1&X2=X3 X1&X4=X5 X3|X5=X7) + %<- (min-commutative X1&X6=X7 X6&X1=X7) + %<- (min-commutative X1&X2=X3 X2&X1=X3) + %<- (min-commutative X1&X4=X5 X4&X1=X5) + %<- (min-right-distributes-over-max* X2|X4=X6 X6&X1=X7 X2&X1=X3 X4&X1=X5 X3|X5=X7) %. +%worlds () (min-left-distributes-over-max* X2|X4=X6 X1&X6=X7 X1&X2=X3 X1&X4=X5 X3|X5=X7) %. +%total {} (min-left-distributes-over-max* _ _ _ _ _) %. +%theorem min-left-distributes-over-max : forall* {X1} {X2} {X4} {X6} {X7} forall {A12 max X2 X4 X6} {M34 min X1 X6 X7} exists {X3} {X5} {M14 min X1 X2 X3} {M24 min X1 X4 X5} {A56 max X3 X5 X7} true %. +%term _ + %pi (min-left-distributes-over-max X2|X4=X6 X1&X6=X7 X3 X5 X1&X2=X3 X1&X4=X5 X3|X5=X7) + %<- (min-total X1&X2=X3) + %<- (min-total X1&X4=X5) + %<- (min-left-distributes-over-max* X2|X4=X6 X1&X6=X7 X1&X2=X3 X1&X4=X5 X3|X5=X7) %. +%worlds () (min-left-distributes-over-max X2|X4=X6 X1&X6=X7 X3 X5 X1&X2=X3 X1&X4=X5 X3|X5=X7) %. +%total {} (min-left-distributes-over-max _ _ _ _ _ _ _) %. +%theorem min-right-factors-over-max : forall* {X1} {X2} {X4} {X5} {X6} {X7} forall {M14 min X1 X4 X5} {M24 min X2 X4 X6} {A56 max X5 X6 X7} exists {X3} {A12 max X1 X2 X3} {M34 min X3 X4 X7} true %. +%term _ + %pi (min-right-factors-over-max X1&X4=X5 X2&X4=X6 X5|X6=X7 X3 X1|X2=X3 X3&X4=X7) + %<- (max-total X1|X2=X3) + %<- (min-total X3&X4=Y7) + %<- (min-right-distributes-over-max* X1|X2=X3 X3&X4=Y7 X1&X4=X5 X2&X4=X6 X5|X6=Y7) + %<- (max-deterministic X5|X6=Y7 X5|X6=X7 eq/ eq/ Y7=X7) + %<- (min-respects-eq X3&X4=Y7 eq/ eq/ Y7=X7 X3&X4=X7) %. +%worlds () (min-right-factors-over-max X1&X4=X5 X2&X4=X6 X5|X6=X7 X3 X1|X2=X3 X3&X4=X7) %. +%total {} (min-right-factors-over-max _ _ _ _ _ _) %. +%theorem min-right-factors-over-max* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {M14 min X1 X4 X5} {M24 min X2 X4 X6} {A56 max X5 X6 X7} {A12 max X1 X2 X3} exists {M34 min X3 X4 X7} true %. +%term _ + %pi (min-right-factors-over-max* X1&X4=X5 X2&X4=X6 X5|X6=X7 X1|X2=X3 X3&X4=X7) + %<- (min-total X3&X4=Y7) + %<- (min-right-distributes-over-max* X1|X2=X3 X3&X4=Y7 X1&X4=X5 X2&X4=X6 X5|X6=Y7) + %<- (max-deterministic X5|X6=Y7 X5|X6=X7 eq/ eq/ Y7=X7) + %<- (min-respects-eq X3&X4=Y7 eq/ eq/ Y7=X7 X3&X4=X7) %. +%worlds () (min-right-factors-over-max* X1&X4=X5 X2&X4=X6 X5|X6=X7 X1|X2=X3 X3&X4=X7) %. +%total {} (min-right-factors-over-max* _ _ _ _ _) %. +%theorem min-left-factors-over-max : forall* {X1} {X2} {X3} {X4} {X5} {X7} forall {M12 min X1 X2 X3} {M14 min X1 X4 X5} {A35 max X3 X5 X7} exists {X6} {A24 max X2 X4 X6} {M16 min X1 X6 X7} true %. +%term _ + %pi (min-left-factors-over-max X1&X2=X3 X1&X4=X5 X3|X5=X7 X6 X2|X4=X6 X1&X6=X7) + %<- (min-commutative X1&X2=X3 X2&X1=X3) + %<- (min-commutative X1&X4=X5 X4&X1=X5) + %<- (min-right-factors-over-max X2&X1=X3 X4&X1=X5 X3|X5=X7 X6 X2|X4=X6 X6&X1=X7) + %<- (min-commutative X6&X1=X7 X1&X6=X7) %. +%worlds () (min-left-factors-over-max X1&X2=X3 X1&X4=X5 X3|X5=X7 X6 X2|X4=X6 X1&X6=X7) %. +%total {} (min-left-factors-over-max _ _ _ _ _ _) %. +%theorem min-left-factors-over-max* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {M12 min X1 X2 X3} {M14 min X1 X4 X5} {A35 max X3 X5 X7} {A24 max X2 X4 X6} exists {M16 min X1 X6 X7} true %. +%term _ + %pi (min-left-factors-over-max* X1&X2=X3 X1&X4=X5 X3|X5=X7 X2|X4=X6 X1&X6=X7) + %<- (min-total X1&X6=Y7) + %<- (min-left-distributes-over-max* X2|X4=X6 X1&X6=Y7 X1&X2=X3 X1&X4=X5 X3|X5=Y7) + %<- (max-deterministic X3|X5=Y7 X3|X5=X7 eq/ eq/ Y7=X7) + %<- (min-respects-eq X1&X6=Y7 eq/ eq/ Y7=X7 X1&X6=X7) %. +%worlds () (min-left-factors-over-max* X1&X2=X3 X1&X4=X5 X3|X5=X7 X2|X4=X6 X1&X6=X7) %. +%total {} (min-left-factors-over-max* _ _ _ _ _) %. +%theorem max-right-distributes-over-min* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {A12 min X1 X2 X3} {M34 max X3 X4 X7} {M14 max X1 X4 X5} {M24 max X2 X4 X6} exists {A56 min X5 X6 X7} true %. +%term _ + %pi (max-right-distributes-over-min* X1&X2=X3 X3|X4=X7 X1|X4=X5 X2|X4=X6 X5&X6=X7) + %<- (max-right-distributes-over-min X1&X2=X3 X3|X4=X7 Y5 Y6 X1|X4=Y5 X2|X4=Y6 Y5&Y6=X7) + %<- (max-deterministic X1|X4=Y5 X1|X4=X5 eq/ eq/ Y5=X5) + %<- (max-deterministic X2|X4=Y6 X2|X4=X6 eq/ eq/ Y6=X6) + %<- (min-respects-eq Y5&Y6=X7 Y5=X5 Y6=X6 eq/ X5&X6=X7) %. +%worlds () (max-right-distributes-over-min* X1&X2=X3 X3|X4=X7 X1|X4=X5 X2|X4=X6 X5&X6=X7) %. +%total {} (max-right-distributes-over-min* _ _ _ _ _) %. +%theorem max-left-distributes-over-min* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {A12 min X2 X4 X6} {M34 max X1 X6 X7} {M14 max X1 X2 X3} {M24 max X1 X4 X5} exists {A56 min X3 X5 X7} true %. +%term _ + %pi (max-left-distributes-over-min* X2&X4=X6 X1|X6=X7 X1|X2=X3 X1|X4=X5 X3&X5=X7) + %<- (max-commutative X1|X6=X7 X6|X1=X7) + %<- (max-commutative X1|X2=X3 X2|X1=X3) + %<- (max-commutative X1|X4=X5 X4|X1=X5) + %<- (max-right-distributes-over-min* X2&X4=X6 X6|X1=X7 X2|X1=X3 X4|X1=X5 X3&X5=X7) %. +%worlds () (max-left-distributes-over-min* X2&X4=X6 X1|X6=X7 X1|X2=X3 X1|X4=X5 X3&X5=X7) %. +%total {} (max-left-distributes-over-min* _ _ _ _ _) %. +%theorem max-left-distributes-over-min : forall* {X1} {X2} {X4} {X6} {X7} forall {A12 min X2 X4 X6} {M34 max X1 X6 X7} exists {X3} {X5} {M14 max X1 X2 X3} {M24 max X1 X4 X5} {A56 min X3 X5 X7} true %. +%term _ + %pi (max-left-distributes-over-min X2&X4=X6 X1|X6=X7 X3 X5 X1|X2=X3 X1|X4=X5 X3&X5=X7) + %<- (max-total X1|X2=X3) + %<- (max-total X1|X4=X5) + %<- (max-left-distributes-over-min* X2&X4=X6 X1|X6=X7 X1|X2=X3 X1|X4=X5 X3&X5=X7) %. +%worlds () (max-left-distributes-over-min X2&X4=X6 X1|X6=X7 X3 X5 X1|X2=X3 X1|X4=X5 X3&X5=X7) %. +%total {} (max-left-distributes-over-min _ _ _ _ _ _ _) %. +%theorem max-right-factors-over-min : forall* {X1} {X2} {X4} {X5} {X6} {X7} forall {M14 max X1 X4 X5} {M24 max X2 X4 X6} {A56 min X5 X6 X7} exists {X3} {A12 min X1 X2 X3} {M34 max X3 X4 X7} true %. +%term _ + %pi (max-right-factors-over-min X1|X4=X5 X2|X4=X6 X5&X6=X7 X3 X1&X2=X3 X3|X4=X7) + %<- (min-total X1&X2=X3) + %<- (max-total X3|X4=Y7) + %<- (max-right-distributes-over-min* X1&X2=X3 X3|X4=Y7 X1|X4=X5 X2|X4=X6 X5&X6=Y7) + %<- (min-deterministic X5&X6=Y7 X5&X6=X7 eq/ eq/ Y7=X7) + %<- (max-respects-eq X3|X4=Y7 eq/ eq/ Y7=X7 X3|X4=X7) %. +%worlds () (max-right-factors-over-min X1|X4=X5 X2|X4=X6 X5&X6=X7 X3 X1&X2=X3 X3|X4=X7) %. +%total {} (max-right-factors-over-min _ _ _ _ _ _) %. +%theorem max-right-factors-over-min* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {M14 max X1 X4 X5} {M24 max X2 X4 X6} {A56 min X5 X6 X7} {A12 min X1 X2 X3} exists {M34 max X3 X4 X7} true %. +%term _ + %pi (max-right-factors-over-min* X1|X4=X5 X2|X4=X6 X5&X6=X7 X1&X2=X3 X3|X4=X7) + %<- (max-total X3|X4=Y7) + %<- (max-right-distributes-over-min* X1&X2=X3 X3|X4=Y7 X1|X4=X5 X2|X4=X6 X5&X6=Y7) + %<- (min-deterministic X5&X6=Y7 X5&X6=X7 eq/ eq/ Y7=X7) + %<- (max-respects-eq X3|X4=Y7 eq/ eq/ Y7=X7 X3|X4=X7) %. +%worlds () (max-right-factors-over-min* X1|X4=X5 X2|X4=X6 X5&X6=X7 X1&X2=X3 X3|X4=X7) %. +%total {} (max-right-factors-over-min* _ _ _ _ _) %. +%theorem max-left-factors-over-min : forall* {X1} {X2} {X3} {X4} {X5} {X7} forall {M12 max X1 X2 X3} {M14 max X1 X4 X5} {A35 min X3 X5 X7} exists {X6} {A24 min X2 X4 X6} {M16 max X1 X6 X7} true %. +%term _ + %pi (max-left-factors-over-min X1|X2=X3 X1|X4=X5 X3&X5=X7 X6 X2&X4=X6 X1|X6=X7) + %<- (max-commutative X1|X2=X3 X2|X1=X3) + %<- (max-commutative X1|X4=X5 X4|X1=X5) + %<- (max-right-factors-over-min X2|X1=X3 X4|X1=X5 X3&X5=X7 X6 X2&X4=X6 X6|X1=X7) + %<- (max-commutative X6|X1=X7 X1|X6=X7) %. +%worlds () (max-left-factors-over-min X1|X2=X3 X1|X4=X5 X3&X5=X7 X6 X2&X4=X6 X1|X6=X7) %. +%total {} (max-left-factors-over-min _ _ _ _ _ _) %. +%theorem max-left-factors-over-min* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {M12 max X1 X2 X3} {M14 max X1 X4 X5} {A35 min X3 X5 X7} {A24 min X2 X4 X6} exists {M16 max X1 X6 X7} true %. +%term _ + %pi (max-left-factors-over-min* X1|X2=X3 X1|X4=X5 X3&X5=X7 X2&X4=X6 X1|X6=X7) + %<- (max-total X1|X6=Y7) + %<- (max-left-distributes-over-min* X2&X4=X6 X1|X6=Y7 X1|X2=X3 X1|X4=X5 X3&X5=Y7) + %<- (min-deterministic X3&X5=Y7 X3&X5=X7 eq/ eq/ Y7=X7) + %<- (max-respects-eq X1|X6=Y7 eq/ eq/ Y7=X7 X1|X6=X7) %. +%worlds () (max-left-factors-over-min* X1|X2=X3 X1|X4=X5 X3&X5=X7 X2&X4=X6 X1|X6=X7) %. +%total {} (max-left-factors-over-min* _ _ _ _ _) %. +%theorem succ-preserves-min : forall* {N1} {N2} {N3} forall {M min N1 N2 N3} exists {MS min (s N1) (s N2) (s N3)} true %. +%term _ succ-preserves-min min/= min/= %. +%term _ + %pi (succ-preserves-min (min/< N2>N1) (min/< N2+1>N1+1)) + %<- (succ-preserves-gt N2>N1 N2+1>N1+1) %. +%term _ + %pi (succ-preserves-min (min/> N1>N2) (min/> N1+1>N2+1)) + %<- (succ-preserves-gt N1>N2 N1+1>N2+1) %. +%worlds () (succ-preserves-min _ _) %. +%total {} (succ-preserves-min _ _) %. +%theorem succ-preserves-min-converse : forall* {N1} {N2} {N3} forall {MS min (s N1) (s N2) (s N3)} exists {M min N1 N2 N3} true %. +%term _ succ-preserves-min-converse min/= min/= %. +%term _ + %pi (succ-preserves-min-converse (min/< N2+1>N1+1) (min/< N2>N1)) + %<- (succ-preserves-gt-converse N2+1>N1+1 N2>N1) %. +%term _ + %pi (succ-preserves-min-converse (min/> N1+1>N2+1) (min/> N1>N2)) + %<- (succ-preserves-gt-converse N1+1>N2+1 N1>N2) %. +%worlds () (succ-preserves-min-converse _ _) %. +%total {} (succ-preserves-min-converse _ _) %. +%theorem succ-preserves-max : forall* {N1} {N2} {N3} forall {M max N1 N2 N3} exists {MS max (s N1) (s N2) (s N3)} true %. +%term _ succ-preserves-max max/= max/= %. +%term _ + %pi (succ-preserves-max (max/< N2>N1) (max/< N2+1>N1+1)) + %<- (succ-preserves-gt N2>N1 N2+1>N1+1) %. +%term _ + %pi (succ-preserves-max (max/> N1>N2) (max/> N1+1>N2+1)) + %<- (succ-preserves-gt N1>N2 N1+1>N2+1) %. +%worlds () (succ-preserves-max _ _) %. +%total {} (succ-preserves-max _ _) %. +%theorem succ-preserves-max-converse : forall* {N1} {N2} {N3} forall {MS max (s N1) (s N2) (s N3)} exists {M max N1 N2 N3} true %. +%term _ succ-preserves-max-converse max/= max/= %. +%term _ + %pi (succ-preserves-max-converse (max/< N2+1>N1+1) (max/< N2>N1)) + %<- (succ-preserves-gt-converse N2+1>N1+1 N2>N1) %. +%term _ + %pi (succ-preserves-max-converse (max/> N1+1>N2+1) (max/> N1>N2)) + %<- (succ-preserves-gt-converse N1+1>N2+1 N1>N2) %. +%worlds () (succ-preserves-max-converse _ _) %. +%total {} (succ-preserves-max-converse _ _) %. +%inline nat`nat nat %. +%inline nat`z z %. +%inline nat`s s %. +%inline nat`plus plus %. +%inline nat`plus/z plus/z %. +%inline nat`plus/s plus/s %. +%inline nat`times times %. +%inline nat`times/z times/z %. +%inline nat`times/s times/s %. +%inline nat`eq eq %. +%inline nat`eq/ eq/ %. +%inline nat`gt gt %. +%inline nat`gt/1 gt/1 %. +%inline nat`gt/> gt/> %. +%inline nat`compare compare %. +%inline nat`compare/= compare/= %. +%inline nat`compare/< compare/< %. +%inline nat`compare/> compare/> %. +%inline nat`meta-eq meta-eq %. +%inline nat`false-implies-eq false-implies-eq %. +%inline nat`eq-symmetric eq-symmetric %. +%inline nat`eq-transitive eq-transitive %. +%inline nat`succ-deterministic succ-deterministic %. +%inline nat`succ-cancels succ-cancels %. +%inline nat`succ-contradiction succ-contradiction %. +%inline nat`eq-contradiction eq-contradiction %. +%inline nat`meta-gt meta-gt %. +%inline nat`false-implies-gt false-implies-gt %. +%inline nat`gt-respects-eq gt-respects-eq %. +%inline nat`succ-implies-gt succ-implies-gt %. +%inline nat`succ-implies-gt-zero succ-implies-gt-zero %. +%inline nat`succ-preserves-gt succ-preserves-gt %. +%inline nat`succ-preserves-gt-converse succ-preserves-gt-converse %. +%inline nat`gt-implies-positive gt-implies-positive %. +%inline nat`gt-anti-reflexive* gt-anti-reflexive* %. +%inline nat`gt-anti-reflexive gt-anti-reflexive %. +%inline nat`gt-transitive gt-transitive %. +%inline nat`gt-anti-symmetric gt-anti-symmetric %. +%inline nat`gt-implies-plus gt-implies-plus %. +%inline nat`gt-contradiction gt-contradiction %. +%inline nat`false-implies-compare false-implies-compare %. +%inline nat`succ-preserves-compare succ-preserves-compare %. +%inline nat`compare-total* compare-total* %. +%inline nat`compare-total compare-total %. +%inline nat`greater-implies-gt greater-implies-gt %. +%inline nat`less-is-reverse-greater less-is-reverse-greater %. +%inline nat`less-implies-lt less-implies-lt %. +%inline nat`equal-implies-eq equal-implies-eq %. +%inline nat`false-implies-plus false-implies-plus %. +%inline nat`plus-respects-eq plus-respects-eq %. +%inline nat`plus-total* plus-total* %. +%inline nat`plus-total plus-total %. +%inline nat`plus-deterministic plus-deterministic %. +%inline nat`plus-left-identity plus-left-identity %. +%inline nat`plus-left-increase plus-left-increase %. +%inline nat`plus-right-identity plus-right-identity %. +%inline nat`plus-right-increase plus-right-increase %. +%inline nat`plus-left-decrease plus-left-decrease %. +%inline nat`plus-right-decrease plus-right-decrease %. +%inline nat`plus-swap-succ plus-swap-succ %. +%inline nat`plus-swap-succ-converse plus-swap-succ-converse %. +%inline nat`plus-left-preserves-positive plus-left-preserves-positive %. +%inline nat`plus-right-preserves-positive plus-right-preserves-positive %. +%inline nat`plus-is-zero-implies-zero plus-is-zero-implies-zero %. +%inline nat`plus-commutative plus-commutative %. +%inline nat`plus-associative plus-associative %. +%inline nat`plus-associative* plus-associative* %. +%inline nat`plus-associative-converse plus-associative-converse %. +%inline nat`plus-associative-converse* plus-associative-converse* %. +%inline nat`plus-assoc-commutative* plus-assoc-commutative* %. +%inline nat`plus-assoc-commutative plus-assoc-commutative %. +%inline nat`plus-double-associative* plus-double-associative* %. +%inline nat`plus-double-associative plus-double-associative %. +%inline nat`plus-left-cancels plus-left-cancels %. +%inline nat`plus-right-cancels* plus-right-cancels* %. +%inline nat`plus-right-cancels plus-right-cancels %. +%inline nat`plus-left-preserves-gt* plus-left-preserves-gt* %. +%inline nat`plus-left-cancels-gt plus-left-cancels-gt %. +%inline nat`plus-left-preserves-gt plus-left-preserves-gt %. +%inline nat`plus-right-preserves-gt* plus-right-preserves-gt* %. +%inline nat`plus-right-preserves-gt plus-right-preserves-gt %. +%inline nat`plus-preserves-gt* plus-preserves-gt* %. +%inline nat`plus-preserves-gt plus-preserves-gt %. +%inline nat`plus-right-cancels-gt plus-right-cancels-gt %. +%inline nat`plus-implies-gt plus-implies-gt %. +%inline nat`plus-gt-contradiction plus-gt-contradiction %. +%inline nat`false-implies-times false-implies-times %. +%inline nat`times-respects-eq times-respects-eq %. +%inline nat`times-total* times-total* %. +%inline nat`times-total times-total %. +%inline nat`times-deterministic times-deterministic %. +%inline nat`times-left-identity times-left-identity %. +%inline nat`times-right-identity times-right-identity %. +%inline nat`times-right-zero times-right-zero %. +%inline nat`times-preserves-positive times-preserves-positive %. +%inline nat`times-preserves-positive* times-preserves-positive* %. +%inline nat`times-positive-implies-positive times-positive-implies-positive %. +%inline nat`times-left-increase times-left-increase %. +%inline nat`times-right-increase times-right-increase %. +%inline nat`times-left-decrease times-left-decrease %. +%inline nat`times-right-decrease times-right-decrease %. +%inline nat`times-commutative times-commutative %. +%inline nat`times-right-distributes-over-plus times-right-distributes-over-plus %. +%inline nat`times-right-distributes-over-plus* times-right-distributes-over-plus* %. +%inline nat`times-left-distributes-over-plus* times-left-distributes-over-plus* %. +%inline nat`times-left-distributes-over-plus times-left-distributes-over-plus %. +%inline nat`times-right-factors-over-plus times-right-factors-over-plus %. +%inline nat`times-right-factors-over-plus* times-right-factors-over-plus* %. +%inline nat`times-left-factors-over-plus times-left-factors-over-plus %. +%inline nat`times-left-factors-over-plus* times-left-factors-over-plus* %. +%inline nat`times-associative times-associative %. +%inline nat`times-associative* times-associative* %. +%inline nat`times-associative-converse times-associative-converse %. +%inline nat`times-associative-converse* times-associative-converse* %. +%inline nat`times-assoc-commutative* times-assoc-commutative* %. +%inline nat`times-assoc-commutative times-assoc-commutative %. +%inline nat`times-double-associative* times-double-associative* %. +%inline nat`times-double-associative times-double-associative %. +%inline nat`times-right-cancels times-right-cancels %. +%inline nat`times-right-cancels* times-right-cancels* %. +%inline nat`times-right-cancels** times-right-cancels** %. +%inline nat`times-left-cancels times-left-cancels %. +%inline nat`times-left-cancels* times-left-cancels* %. +%inline nat`times-left-preserves-gt times-left-preserves-gt %. +%inline nat`times-left-preserves-gt* times-left-preserves-gt* %. +%inline nat`times-right-preserves-gt times-right-preserves-gt %. +%inline nat`times-right-preserves-gt* times-right-preserves-gt* %. +%inline nat`times-preserves-gt times-preserves-gt %. +%inline nat`times-right-cancels-gt times-right-cancels-gt %. +%inline nat`times-left-cancels-gt times-left-cancels-gt %. +%inline nat`minus minus %. +%inline nat`false-implies-minus false-implies-minus %. +%inline nat`minus-respects-eq minus-respects-eq %. +%inline nat`minus-deterministic minus-deterministic %. +%inline nat`plus-associates-with-minus* plus-associates-with-minus* %. +%inline nat`plus-associates-with-minus-converse* plus-associates-with-minus-converse* %. +%inline nat`plus-associates-with-minus-converse plus-associates-with-minus-converse %. +%inline nat`minus-associates-from-plus* minus-associates-from-plus* %. +%inline nat`minus-associates-from-plus-converse* minus-associates-from-plus-converse* %. +%inline nat`minus-associates-to-plus* minus-associates-to-plus* %. +%inline nat`minus-associates-to-plus minus-associates-to-plus %. +%inline nat`minus-associates-to-plus-converse* minus-associates-to-plus-converse* %. +%inline nat`minus-associates-to-plus-converse minus-associates-to-plus-converse %. +%inline nat`minus-is-zero-implies-eq minus-is-zero-implies-eq %. +%inline nat`minus-implies-gt minus-implies-gt %. +%inline nat`minus-left-cancels minus-left-cancels %. +%inline nat`minus-right-cancels minus-right-cancels %. +%inline nat`minus-left-inverts-gt* minus-left-inverts-gt* %. +%inline nat`minus-right-preserves-gt* minus-right-preserves-gt* %. +%inline nat`minus-left-cancels-inverts-gt minus-left-cancels-inverts-gt %. +%inline nat`minus-right-cancels-gt minus-right-cancels-gt %. +%inline nat`times-right-distributes-over-minus times-right-distributes-over-minus %. +%inline nat`times-right-distributes-over-minus* times-right-distributes-over-minus* %. +%inline nat`times-left-distributes-over-minus* times-left-distributes-over-minus* %. +%inline nat`times-left-distributes-over-minus times-left-distributes-over-minus %. +%inline nat`times-right-factors-over-minus* times-right-factors-over-minus* %. +%inline nat`times-left-factors-over-minus* times-left-factors-over-minus* %. +%inline nat`times-right-factors-over-minus times-right-factors-over-minus %. +%inline nat`times-left-factors-over-minus times-left-factors-over-minus %. +%inline nat`ge ge %. +%inline nat`ge/= ge/= %. +%inline nat`ge/> ge/> %. +%inline nat`false-implies-ge false-implies-ge %. +%inline nat`ge-respects-eq ge-respects-eq %. +%inline nat`ge-reflexive ge-reflexive %. +%inline nat`ge-transitive ge-transitive %. +%inline nat`ge-anti-symmetric ge-anti-symmetric %. +%inline nat`ge-transitive-gt ge-transitive-gt %. +%inline nat`gt-transitive-ge gt-transitive-ge %. +%inline nat`meta-ge meta-ge %. +%inline nat`succ-preserves-ge succ-preserves-ge %. +%inline nat`succ-preserves-ge-converse succ-preserves-ge-converse %. +%inline nat`ge-succ-implies-gt ge-succ-implies-gt %. +%inline nat`ge-implies-succ-gt ge-implies-succ-gt %. +%inline nat`succ-gt-implies-ge succ-gt-implies-ge %. +%inline nat`gt-implies-ge-succ gt-implies-ge-succ %. +%inline nat`ge-implies-plus ge-implies-plus %. +%inline nat`plus-implies-ge plus-implies-ge %. +%inline nat`ge-zero-always ge-zero-always %. +%inline nat`nonzero-times-implies-ge nonzero-times-implies-ge %. +%inline nat`times-nonzero-implies-ge times-nonzero-implies-ge %. +%inline nat`non-trivial-times-implies-much-gt* non-trivial-times-implies-much-gt* %. +%inline nat`non-trivial-times-implies-much-gt non-trivial-times-implies-much-gt %. +%inline nat`plus-left-preserves-ge* plus-left-preserves-ge* %. +%inline nat`plus-left-cancels-ge plus-left-cancels-ge %. +%inline nat`plus-left-preserves-ge plus-left-preserves-ge %. +%inline nat`plus-right-preserves-ge* plus-right-preserves-ge* %. +%inline nat`plus-right-preserves-ge plus-right-preserves-ge %. +%inline nat`plus-preserves-ge* plus-preserves-ge* %. +%inline nat`plus-preserves-ge plus-preserves-ge %. +%inline nat`plus-right-cancels-ge plus-right-cancels-ge %. +%inline nat`times-left-preserves-ge* times-left-preserves-ge* %. +%inline nat`times-left-preserves-ge times-left-preserves-ge %. +%inline nat`times-right-preserves-ge* times-right-preserves-ge* %. +%inline nat`times-right-preserves-ge times-right-preserves-ge %. +%inline nat`ne ne %. +%inline nat`ne/< ne/< %. +%inline nat`ne/> ne/> %. +%inline nat`eq? eq? %. +%inline nat`eq?/yes eq?/yes %. +%inline nat`eq?/no eq?/no %. +%inline nat`false-implies-ne false-implies-ne %. +%inline nat`ne-respects-eq ne-respects-eq %. +%inline nat`ne-anti-reflexive ne-anti-reflexive %. +%inline nat`ne-symmetric ne-symmetric %. +%inline nat`eq-ne-implies-false eq-ne-implies-false %. +%inline nat`ge-ne-implies-gt ge-ne-implies-gt %. +%inline nat`eq?-total* eq?-total* %. +%inline nat`eq?-total*/L eq?-total*/L %. +%inline nat`eq?-total eq?-total %. +%inline nat`succ-preserves-ne succ-preserves-ne %. +%inline nat`succ-preserves-ne-converse succ-preserves-ne-converse %. +%inline nat`plus-left-preserves-ne* plus-left-preserves-ne* %. +%inline nat`plus-left-cancels-ne plus-left-cancels-ne %. +%inline nat`plus-left-preserves-ne plus-left-preserves-ne %. +%inline nat`plus-right-preserves-ne* plus-right-preserves-ne* %. +%inline nat`plus-right-preserves-ne plus-right-preserves-ne %. +%inline nat`plus-right-cancels-ne plus-right-cancels-ne %. +%inline nat`lt lt %. +%inline nat`false-implies-lt false-implies-lt %. +%inline nat`lt-respects-eq lt-respects-eq %. +%inline nat`lt-anti-symmetric lt-anti-symmetric %. +%inline nat`lt-transitive lt-transitive %. +%inline nat`lt-anti-reflexive lt-anti-reflexive %. +%inline nat`plus-left-preserves-lt* plus-left-preserves-lt* %. +%inline nat`plus-left-cancels-lt plus-left-cancels-lt %. +%inline nat`plus-left-preserves-lt plus-left-preserves-lt %. +%inline nat`plus-right-preserves-lt* plus-right-preserves-lt* %. +%inline nat`plus-right-preserves-lt plus-right-preserves-lt %. +%inline nat`plus-preserves-lt* plus-preserves-lt* %. +%inline nat`plus-preserves-lt plus-preserves-lt %. +%inline nat`plus-right-cancels-lt plus-right-cancels-lt %. +%inline nat`le le %. +%inline nat`false-implies-le false-implies-le %. +%inline nat`le-respects-eq le-respects-eq %. +%inline nat`le-anti-symmetric le-anti-symmetric %. +%inline nat`le-transitive le-transitive %. +%inline nat`le-reflexive le-reflexive %. +%inline nat`le-transitive-lt le-transitive-lt %. +%inline nat`lt-transitive-le lt-transitive-le %. +%inline nat`plus-left-preserves-le* plus-left-preserves-le* %. +%inline nat`plus-left-cancels-le plus-left-cancels-le %. +%inline nat`plus-left-preserves-le plus-left-preserves-le %. +%inline nat`plus-right-preserves-le* plus-right-preserves-le* %. +%inline nat`plus-right-preserves-le plus-right-preserves-le %. +%inline nat`plus-preserves-le* plus-preserves-le* %. +%inline nat`plus-preserves-le plus-preserves-le %. +%inline nat`plus-right-cancels-le plus-right-cancels-le %. +%inline nat`times-left-preserves-le* times-left-preserves-le* %. +%inline nat`times-left-preserves-le times-left-preserves-le %. +%inline nat`times-right-preserves-le* times-right-preserves-le* %. +%inline nat`times-right-preserves-le times-right-preserves-le %. +%inline nat`times-preserves-le* times-preserves-le* %. +%inline nat`times-preserves-le times-preserves-le %. +%inline nat`minus-left-inverts-ge* minus-left-inverts-ge* %. +%inline nat`minus-right-preserves-ge* minus-right-preserves-ge* %. +%inline nat`minus-left-cancels-inverts-ge minus-left-cancels-inverts-ge %. +%inline nat`minus-right-cancels-ge minus-right-cancels-ge %. +%inline nat`minus-left-preserves-ne* minus-left-preserves-ne* %. +%inline nat`minus-right-preserves-ne* minus-right-preserves-ne* %. +%inline nat`minus-left-cancels-ne minus-left-cancels-ne %. +%inline nat`minus-right-cancels-ne minus-right-cancels-ne %. +%inline nat`minus-left-inverts-lt* minus-left-inverts-lt* %. +%inline nat`minus-right-preserves-lt* minus-right-preserves-lt* %. +%inline nat`minus-left-cancels-inverts-lt minus-left-cancels-inverts-lt %. +%inline nat`minus-right-cancels-lt minus-right-cancels-lt %. +%inline nat`minus-left-inverts-le* minus-left-inverts-le* %. +%inline nat`minus-right-preserves-le* minus-right-preserves-le* %. +%inline nat`minus-left-cancels-inverts-le minus-left-cancels-inverts-le %. +%inline nat`minus-right-cancels-le minus-right-cancels-le %. +%inline nat`divrem divrem %. +%inline nat`divrem/z divrem/z %. +%inline nat`divrem/s divrem/s %. +%inline nat`false-implies-divrem false-implies-divrem %. +%inline nat`divrem-respects-eq divrem-respects-eq %. +%inline nat`divrem-total** divrem-total** %. +%inline nat`divrem-total* divrem-total* %. +%inline nat`divrem-total divrem-total %. +%inline nat`divrem-deterministic divrem-deterministic %. +%inline nat`divrem-implies-positive divrem-implies-positive %. +%inline nat`divrem-implies-gt divrem-implies-gt %. +%inline nat`divrem-contradiction divrem-contradiction %. +%inline nat`divrem-can-be-inverted divrem-can-be-inverted %. +%inline nat`div-can-be-inverted div-can-be-inverted %. +%inline nat`divrem-can-be-constructed divrem-can-be-constructed %. +%inline nat`div-can-be-constructed div-can-be-constructed %. +%inline nat`remainder-implies-gt-quotient remainder-implies-gt-quotient %. +%inline nat`quotient-of-nonzero-is-smaller quotient-of-nonzero-is-smaller %. +%inline nat`quotient-is-no-greater quotient-is-no-greater %. +%inline nat`min min %. +%inline nat`min/= min/= %. +%inline nat`min/> min/> %. +%inline nat`min/< min/< %. +%inline nat`max max %. +%inline nat`max/= max/= %. +%inline nat`max/> max/> %. +%inline nat`max/< max/< %. +%inline nat`false-implies-min false-implies-min %. +%inline nat`min-respects-eq min-respects-eq %. +%inline nat`min-total** min-total** %. +%inline nat`min-total* min-total* %. +%inline nat`min-total min-total %. +%inline nat`min-deterministic min-deterministic %. +%inline nat`min-commutative min-commutative %. +%inline nat`ge-implies-min ge-implies-min %. +%inline nat`le-implies-min le-implies-min %. +%inline nat`min-implies-ge min-implies-ge %. +%inline nat`min-left-preserves-ge* min-left-preserves-ge* %. +%inline nat`min-left-preserves-le* min-left-preserves-le* %. +%inline nat`min-left-preserves-ge min-left-preserves-ge %. +%inline nat`min-right-preserves-ge* min-right-preserves-ge* %. +%inline nat`min-right-preserves-ge min-right-preserves-ge %. +%inline nat`min-preserves-ge* min-preserves-ge* %. +%inline nat`min-preserves-ge min-preserves-ge %. +%inline nat`min-left-preserves-le min-left-preserves-le %. +%inline nat`min-right-preserves-le* min-right-preserves-le* %. +%inline nat`min-right-preserves-le min-right-preserves-le %. +%inline nat`min-preserves-le* min-preserves-le* %. +%inline nat`min-preserves-le min-preserves-le %. +%inline nat`min-is-glb min-is-glb %. +%inline nat`min-associative min-associative %. +%inline nat`min-associative* min-associative* %. +%inline nat`min-associative-converse min-associative-converse %. +%inline nat`min-associative-converse* min-associative-converse* %. +%inline nat`min-assoc-commutative* min-assoc-commutative* %. +%inline nat`min-assoc-commutative min-assoc-commutative %. +%inline nat`min-double-associative* min-double-associative* %. +%inline nat`min-double-associative min-double-associative %. +%inline nat`false-implies-max false-implies-max %. +%inline nat`max-respects-eq max-respects-eq %. +%inline nat`max-total** max-total** %. +%inline nat`max-total* max-total* %. +%inline nat`max-total max-total %. +%inline nat`max-deterministic max-deterministic %. +%inline nat`max-commutative max-commutative %. +%inline nat`ge-implies-max ge-implies-max %. +%inline nat`le-implies-max le-implies-max %. +%inline nat`max-implies-ge max-implies-ge %. +%inline nat`max-is-lub max-is-lub %. +%inline nat`max-left-preserves-ge* max-left-preserves-ge* %. +%inline nat`max-left-preserves-le* max-left-preserves-le* %. +%inline nat`max-left-preserves-ge max-left-preserves-ge %. +%inline nat`max-right-preserves-ge* max-right-preserves-ge* %. +%inline nat`max-right-preserves-ge max-right-preserves-ge %. +%inline nat`max-preserves-ge* max-preserves-ge* %. +%inline nat`max-preserves-ge max-preserves-ge %. +%inline nat`max-left-preserves-le max-left-preserves-le %. +%inline nat`max-right-preserves-le* max-right-preserves-le* %. +%inline nat`max-right-preserves-le max-right-preserves-le %. +%inline nat`max-preserves-le* max-preserves-le* %. +%inline nat`max-preserves-le max-preserves-le %. +%inline nat`max-associative max-associative %. +%inline nat`max-associative* max-associative* %. +%inline nat`max-associative-converse max-associative-converse %. +%inline nat`max-associative-converse* max-associative-converse* %. +%inline nat`max-assoc-commutative* max-assoc-commutative* %. +%inline nat`max-assoc-commutative max-assoc-commutative %. +%inline nat`max-double-associative* max-double-associative* %. +%inline nat`max-double-associative max-double-associative %. +%inline nat`min-right-distributes-over-max min-right-distributes-over-max %. +%inline nat`max-right-distributes-over-min max-right-distributes-over-min %. +%inline nat`min-right-distributes-over-max* min-right-distributes-over-max* %. +%inline nat`min-left-distributes-over-max* min-left-distributes-over-max* %. +%inline nat`min-left-distributes-over-max min-left-distributes-over-max %. +%inline nat`min-right-factors-over-max min-right-factors-over-max %. +%inline nat`min-right-factors-over-max* min-right-factors-over-max* %. +%inline nat`min-left-factors-over-max min-left-factors-over-max %. +%inline nat`min-left-factors-over-max* min-left-factors-over-max* %. +%inline nat`max-right-distributes-over-min* max-right-distributes-over-min* %. +%inline nat`max-left-distributes-over-min* max-left-distributes-over-min* %. +%inline nat`max-left-distributes-over-min max-left-distributes-over-min %. +%inline nat`max-right-factors-over-min max-right-factors-over-min %. +%inline nat`max-right-factors-over-min* max-right-factors-over-min* %. +%inline nat`max-left-factors-over-min max-left-factors-over-min %. +%inline nat`max-left-factors-over-min* max-left-factors-over-min* %. +%inline nat`succ-preserves-min succ-preserves-min %. +%inline nat`succ-preserves-min-converse succ-preserves-min-converse %. +%inline nat`succ-preserves-max succ-preserves-max %. +%inline nat`succ-preserves-max-converse succ-preserves-max-converse %. +%{! | hidden = true !}% +% contents included from ../library/natpair.elf +%%%%% natpair.elf +%%%%% Pairs of natural numbers +%%%%% John Boyland +%%%% Functor use +%%%%% pair.elf +%%%%% a pseudo-functor +%%%%% John Boyland +%%%% Definitions +%sort pair %. +%term pair/ %pi nat %-> nat %-> pair %. +%sort eq %. +%term eq/ eq P P %. +%sort ne %. +%term ne/1 %pi (ne (pair/ X1 Y1) (pair/ X2 Y2)) %<- (nat`ne X1 X2) %. +%term ne/2 %pi (ne (pair/ X1 Y1) (pair/ X2 Y2)) %<- (nat`ne Y1 Y2) %. +%sort eq? %. +%term eq?/yes eq? P P true %. +%term eq?/no %pi (eq? P1 P2 false) %<- (ne P1 P2) %. +%%%% Theorems +%%% theorems about eq +%theorem false-implies-eq : forall* {X1} {X2} forall {F void} exists {E eq X1 X2} true %. +%worlds () (false-implies-eq _ _) %. +%total {} (false-implies-eq _ _) %. +%theorem meta-eq : forall {X1} {X2} {E eq X1 X2} true %. +%term _ meta-eq _ _ eq/ %. +%worlds () (meta-eq _ _ _) %. +%total {} (meta-eq _ _ _) %. +%reduces = X Y (meta-eq X Y _) %. +%theorem eq-reflexive : forall {X} exists {E eq X X} true %. +%term _ eq-reflexive _ eq/ %. +%worlds () (eq-reflexive _ _) %. +%total {} (eq-reflexive _ _) %. +%theorem eq-symmetric : forall* {X} {Y} forall {E eq X Y} exists {F eq Y X} true %. +%term _ eq-symmetric eq/ eq/ %. +%worlds () (eq-symmetric _ _) %. +%total {} (eq-symmetric _ _) %. +%theorem eq-transitive : forall* {X} {Y} {Z} forall {E1 eq X Y} {E2 eq Y Z} exists {F eq X Z} true %. +%term _ eq-transitive eq/ eq/ eq/ %. +%worlds () (eq-transitive _ _ _) %. +%total {} (eq-transitive _ _ _) %. +%theorem pair-eq-implies-eq : forall* {D1a} {D1b} {D2a} {D2b} forall {E eq (pair/ D1a D2a) (pair/ D1b D2b)} exists {E1 nat`eq D1a D1b} {E2 nat`eq D2a D2b} true %. +%term _ pair-eq-implies-eq eq/ nat`eq/ nat`eq/ %. +%worlds () (pair-eq-implies-eq _ _ _) %. +%total {} (pair-eq-implies-eq _ _ _) %. +%theorem pair-preserves-eq : forall* {D1a} {D1b} {D2a} {D2b} forall {E1 nat`eq D1a D1b} {E2 nat`eq D2a D2b} exists {E eq (pair/ D1a D2a) (pair/ D1b D2b)} true %. +%term _ pair-preserves-eq nat`eq/ nat`eq/ eq/ %. +%worlds () (pair-preserves-eq _ _ _) %. +%total {} (pair-preserves-eq _ _ _) %. +%%% theorems about ne +%theorem false-implies-ne : forall* {X1} {X2} forall {F void} exists {G ne X1 X2} true %. +%worlds () (false-implies-ne _ _) %. +%total {} (false-implies-ne _ _) %. +%theorem ne-respects-eq : forall* {X1} {X2} {Y1} {Y2} forall {D1 ne X1 X2} {E1 eq X1 Y1} {E2 eq X2 Y2} exists {D2 ne Y1 Y2} true %. +%term _ ne-respects-eq X1<>X2 eq/ eq/ X1<>X2 %. +%worlds () (ne-respects-eq _ _ _ _) %. +%total {} (ne-respects-eq _ _ _ _) %. +%theorem ne-anti-reflexive : forall* {P} forall {R ne P P} exists {F void} true %. +%term _ %pi (ne-anti-reflexive (ne/1 X<>X) F) %<- (nat`ne-anti-reflexive X<>X F) %. +%term _ %pi (ne-anti-reflexive (ne/2 Y<>Y) F) %<- (nat`ne-anti-reflexive Y<>Y F) %. +%worlds () (ne-anti-reflexive _ _) %. +%total {} (ne-anti-reflexive _ _) %. +%theorem ne-symmetric : forall* {P1} {P2} forall {R1 ne P1 P2} exists {R2 ne P2 P1} true %. +%term _ + %pi (ne-symmetric (ne/1 X1<>X2) (ne/1 X2<>X1)) + %<- (nat`ne-symmetric X1<>X2 X2<>X1) %. +%term _ + %pi (ne-symmetric (ne/2 Y1<>Y2) (ne/2 Y2<>Y1)) + %<- (nat`ne-symmetric Y1<>Y2 Y2<>Y1) %. +%worlds () (ne-symmetric _ _) %. +%total {} (ne-symmetric _ _) %. +%theorem eq-ne-implies-false : forall* {P1} {P2} forall {D1 eq P1 P2} {D2 ne P1 P2} exists {F void} true %. +%term _ %pi (eq-ne-implies-false eq/ X<>X F) %<- (ne-anti-reflexive X<>X F) %. +%worlds () (eq-ne-implies-false _ _ _) %. +%total {} (eq-ne-implies-false _ _ _) %. +%%% theorems about eq? +%theorem eq?-total* : forall {P1} {P2} exists {B} {T eq? P1 P2 B} true %. +%theorem eq?-total*/L : forall* {X1} {Y1} {X2} {Y2} {B1} {B2} forall {T1 nat`eq? X1 Y1 B1} {T2 nat`eq? X2 Y2 B2} exists {B} {T eq? (pair/ X1 X2) (pair/ Y1 Y2) B} true %. +%term _ eq?-total*/L nat`eq?/yes nat`eq?/yes _ eq?/yes %. +%term _ eq?-total*/L (nat`eq?/no X1<>Y1) _ _ (eq?/no (ne/1 X1<>Y1)) %. +%term _ eq?-total*/L _ (nat`eq?/no X2<>Y2) _ (eq?/no (ne/2 X2<>Y2)) %. +%worlds () (eq?-total*/L _ _ _ _) %. +%total {} (eq?-total*/L _ _ _ _) %. +%term _ + %pi (eq?-total* _ _ _ T) + %<- (nat`eq?-total E?1) + %<- (nat`eq?-total E?2) + %<- (eq?-total*/L E?1 E?2 _ T) %. +%worlds () (eq?-total* _ _ _ _) %. +%total {} (eq?-total* _ _ _ _) %. +%inline eq?-total eq?-total* _ _ _ %. +%%%% Definitions +%%% local abbreviations: +%inline ssN>N gt/> gt/1 %. +%inline 2>0 (gt (s (s z)) z) ssN>N %. +%inline 1*N=N times/s times/z plus/z %. +%inline 1*2=2 (times (s z) (s (s z)) (s (s z))) 1*N=N %. +%inline 2*2=4 times/s 1*N=N (plus/s (plus/s plus/z)) %. +%inline 2+1=3 (plus (s (s z)) (s z) (s (s (s z)))) plus/s (plus/s plus/z) %. +%%% mapping from a pair to a nat +%sort pair2nat %. +%term pair2nat/00 pair2nat (pair/ z z) z %. +%term pair2nat/XX + %pi (plus Z3 Y0 Z) + %-> (plus Z2 X2 Z3) + %-> (times X0 (s (s z)) X2) + %-> (times Z1 (s (s (s (s z)))) Z2) + %-> (pair2nat (pair/ X1 Y1) Z1) + %-> (divrem Y (s (s z)) Y1 Y0) + %-> (divrem X (s (s z)) X1 X0) + %-> (pair2nat (pair/ X Y) Z) %. +%%% mapping from a nat to pair +%inline nat2pair (%pi nat %-> pair %-> %type) [N] [P] pair2nat P N %. +%%%% Theorems +%theorem false-implies-pair2nat : forall* {P} {N} forall {F void} exists {P2N pair2nat P N} true %. +%worlds () (false-implies-pair2nat _ _) %. +%total {} (false-implies-pair2nat _ _) %. +%inline false-implies-not2pair false-implies-pair2nat %. +%theorem pair2nat-respects-eq : forall* {P1} {N1} {P2} {N2} forall {D1 pair2nat P1 N1} {EP eq P1 P2} {EN nat`eq N1 N2} exists {D2 pair2nat P2 N2} true %. +%term _ pair2nat-respects-eq P2N eq/ nat`eq/ P2N %. +%worlds () (pair2nat-respects-eq _ _ _ _) %. +%total {} (pair2nat-respects-eq _ _ _ _) %. +%reduces = D1 D2 (pair2nat-respects-eq D1 _ _ D2) %. +%inline nat2pair-respects-eq ( + %pi (nat2pair N1 P1) + %-> (nat`eq N1 N2) + %-> (eq P1 P2) + %-> (nat2pair N2 P2) + %-> %type) [D1] [EN] [EP] [D2] pair2nat-respects-eq D1 EP EN D2 %. +%theorem pair2nat-total** : forall {X nat} {Y nat} exists {Z} {P2N pair2nat (pair/ X Y) Z} true %. +%term _ pair2nat-total** z z z pair2nat/00 %. +%term _ + %pi (pair2nat-total** (s X-) Y Z (pair2nat/XX Z3+Y0=Z Z2+X2=Z3 X0*2=X2 Z1*4=Z2 (%the (pair2nat (pair/ X1 Y1) Z1) P2N) Y/2=Y1,Y0 X/2=X1,X0)) + %<- (divrem-total X/2=X1,X0) + %<- (divrem-total Y/2=Y1,Y0) + %<- (quotient-of-nonzero-is-smaller X/2=X1,X0 nat`eq/ X>X1) + %<- (quotient-is-no-greater Y/2=Y1,Y0 Y>=Y1) + %<- (meta-gt _ _ X>X1) + %<- (meta-ge _ _ Y>=Y1) + %<- (pair2nat-total** _ _ _ P2N) + %<- (times-total Z1*4=Z2) + %<- (times-total X0*2=X2) + %<- (plus-total Z2+X2=Z3) + %<- (plus-total Z3+Y0=Z) %. +%term _ + %pi (pair2nat-total** X (s Y-) Z (pair2nat/XX Z3+Y0=Z Z2+X2=Z3 X0*2=X2 Z1*4=Z2 (%the (pair2nat (pair/ X1 Y1) Z1) P2N) Y/2=Y1,Y0 X/2=X1,X0)) + %<- (divrem-total X/2=X1,X0) + %<- (divrem-total Y/2=Y1,Y0) + %<- (quotient-is-no-greater X/2=X1,X0 X>=X1) + %<- (quotient-of-nonzero-is-smaller Y/2=Y1,Y0 nat`eq/ Y>Y1) + %<- (meta-ge _ _ X>=X1) + %<- (meta-gt _ _ Y>Y1) + %<- (pair2nat-total** _ _ _ P2N) + %<- (times-total Z1*4=Z2) + %<- (times-total X0*2=X2) + %<- (plus-total Z2+X2=Z3) + %<- (plus-total Z3+Y0=Z) %. +%worlds () (pair2nat-total** _ _ _ _) %. +%total [X Y] (pair2nat-total** X Y _ _) %. +%theorem pair2nat-total* : forall {P pair} exists {N nat} {P2N pair2nat P N} true %. +%term _ %pi (pair2nat-total* (pair/ X Y) Z P2N) %<- (pair2nat-total** X Y Z P2N) %. +%worlds () (pair2nat-total* _ _ _) %. +%total {} (pair2nat-total* _ _ _) %. +%inline pair2nat-total pair2nat-total* _ _ %. +%theorem nat2pair-total* : forall {N nat} exists {P pair} {N2P nat2pair N P} true %. +%term _ nat2pair-total* z (pair/ z z) pair2nat/00 %. +%term _ + %pi (nat2pair-total* (s Z-) (pair/ X Y) (pair2nat/XX Z3+Y0=Z Z2+X2=Z3 X0*2=X2 Z1*4=Z2 (%the (pair2nat (pair/ X1 Y1) Z1) P2N) Y/2=Y1,Y0 X/2=X1,X0)) + %<- (divrem-total Z/4=Z1,R) + %<- (divrem-implies-gt Z/4=Z1,R FOUR>R) + %<- (divrem-can-be-inverted Z/4=Z1,R Z2 Z1*4=Z2 Z2+R=Z) + %<- (divrem-total R/2=X0,Y0) + %<- (divrem-implies-gt R/2=X0,Y0 TWO>Y0) + %<- (divrem-can-be-inverted R/2=X0,Y0 X2 X0*2=X2 X2+Y0=R) + %<- (plus-associative-converse X2+Y0=R Z2+R=Z Z3 Z2+X2=Z3 Z3+Y0=Z) + %<- (plus-commutative X2+Y0=R Y0+X2=R) + %<- (plus-implies-ge Y0+X2=R R>=X2) + %<- (gt-transitive-ge FOUR>R R>=X2 FOUR>X2) + %<- (times-right-cancels-gt 2*2=4 X0*2=X2 nat`eq/ FOUR>X2 TWO>X0) + %<- (quotient-of-nonzero-is-smaller Z/4=Z1,R nat`eq/ Z>Z1) + %<- (meta-gt _ _ Z>Z1) + %<- (nat2pair-total* Z1 (pair/ X1 Y1) P2N) + %<- (times-total X1*2=XE) + %<- (times-total Y1*2=YE) + %<- (plus-total XE+X0=X) + %<- (plus-total YE+Y0=Y) + %<- (divrem-can-be-constructed X1*2=XE XE+X0=X TWO>X0 X/2=X1,X0) + %<- (divrem-can-be-constructed Y1*2=YE YE+Y0=Y TWO>Y0 Y/2=Y1,Y0) %. +%worlds () (nat2pair-total* _ _ _) %. +%total (Z) (nat2pair-total* Z _ _) %. +% %reduces X <= N (nat2pair-total* N (pair/ X _) _). +% %reduces Y <= N (pair2nat-total* N (pair/ _ Y) _). +%inline nat2pair-total nat2pair-total* _ _ %. +%theorem pair2nat-deterministic : forall* {P1} {P2} {N1} {N2} forall {D1 pair2nat P1 N1} {D2 pair2nat P2 N2} {EP eq P1 P2} exists {EN nat`eq N1 N2} true %. +%term _ pair2nat-deterministic pair2nat/00 pair2nat/00 eq/ nat`eq/ %. +% lemma +%theorem pair2nat-deterministic/00 : forall* {Z} forall {P2N pair2nat (pair/ z z) Z} exists {E nat`eq Z z} true %. +%term _ pair2nat-deterministic/00 pair2nat/00 nat`eq/ %. +%term _ + %pi (pair2nat-deterministic/00 (pair2nat/XX Z3+Y0=Z Z2+X2=Z3 X0*2=X2 Z1*4=Z2 (%the (pair2nat (pair/ X1 Y1) Z1) P2N) ZERO/2=Y1,Y0 ZERO/2=X1,X0) Z=0) + %<- (divrem-deterministic ZERO/2=X1,X0 (divrem/z 2>0) nat`eq/ nat`eq/ X1=0 X0=0) + %<- (divrem-deterministic ZERO/2=Y1,Y0 (divrem/z 2>0) nat`eq/ nat`eq/ Y1=0 Y0=0) + %<- (pair-preserves-eq X1=0 Y1=0 X1,Y1=0,0) + %<- (pair2nat-respects-eq P2N X1,Y1=0,0 nat`eq/ ZERO,ZERO->Z1) + %<- (pair2nat-deterministic/00 ZERO,ZERO->Z1 Z1=0) + %<- (times-deterministic Z1*4=Z2 times/z Z1=0 nat`eq/ Z2=0) + %<- (times-deterministic X0*2=X2 times/z X0=0 nat`eq/ X2=0) + %<- (plus-deterministic Z2+X2=Z3 plus/z Z2=0 X2=0 Z3=0) + %<- (plus-deterministic Z3+Y0=Z plus/z Z3=0 Y0=0 Z=0) %. +%worlds () (pair2nat-deterministic/00 _ _) %. +%total (D) (pair2nat-deterministic/00 D _) %. +%term _ + %pi (pair2nat-deterministic pair2nat/00 P2N eq/ ZERO=Z) + %<- (pair2nat-deterministic/00 P2N Z=0) + %<- (nat`eq-symmetric Z=0 ZERO=Z) %. +%term _ + %pi (pair2nat-deterministic P2N pair2nat/00 eq/ Z=0) + %<- (pair2nat-deterministic/00 P2N Z=0) %. +%term _ + %pi (pair2nat-deterministic (pair2nat/XX Z3+Y0=Z Z2+X2=Z3 X0*2=X2 Z1*4=Z2 (%the (pair2nat (pair/ X1 Y1) Z1) P2N) Y/2=Y1,Y0 X/2=X1,X0) (pair2nat/XX Z3'+Y0'=Z' Z2'+X2'=Z3' X0'*2=X2' Z1'*4=Z2' (%the (pair2nat (pair/ X1' Y1') Z1') P2N') Y/2=Y1',Y0' X/2=X1',X0') eq/ Z=Z') + %<- (divrem-deterministic X/2=X1,X0 X/2=X1',X0' nat`eq/ nat`eq/ X1=X1' X0=X0') + %<- (divrem-deterministic Y/2=Y1,Y0 Y/2=Y1',Y0' nat`eq/ nat`eq/ Y1=Y1' Y0=Y0') + %<- (pair-preserves-eq X1=X1' Y1=Y1' X1,Y1=X1',Y1') + %<- (pair2nat-deterministic P2N P2N' X1,Y1=X1',Y1' Z1=Z1') + %<- (times-deterministic Z1*4=Z2 Z1'*4=Z2' Z1=Z1' nat`eq/ Z2=Z2') + %<- (times-deterministic X0*2=X2 X0'*2=X2' X0=X0' nat`eq/ X2=X2') + %<- (plus-deterministic Z2+X2=Z3 Z2'+X2'=Z3' Z2=Z2' X2=X2' Z3=Z3') + %<- (plus-deterministic Z3+Y0=Z Z3'+Y0'=Z' Z3=Z3' Y0=Y0' Z=Z') %. +%worlds () (pair2nat-deterministic _ _ _ _) %. +%total (D) (pair2nat-deterministic D _ _ _) %. +%theorem nat2pair-deterministic : forall* {P1} {P2} {N1} {N2} forall {D1 nat2pair N1 P1} {D2 nat2pair N2 P2} {EN nat`eq N1 N2} exists {EP eq P1 P2} true %. +% lemma +%theorem nat2pair-deterministic/0 : forall* {P} forall {D nat2pair z P} exists {EP eq P (pair/ z z)} true %. +%term _ nat2pair-deterministic/0 pair2nat/00 eq/ %. +%term _ + %pi (nat2pair-deterministic/0 (pair2nat/XX Z3+Y0=0 Z2+X2=Z3 X0*2=X2 Z1*4=Z2 (%the (pair2nat (pair/ X1 Y1) Z1) P2N) Y/2=Y1,Y0 X/2=X1,X0) X,Y=0,0) + %<- (plus-is-zero-implies-zero Z3+Y0=0 nat`eq/ Z3=0 Y0=0) + %<- (plus-is-zero-implies-zero Z2+X2=Z3 Z3=0 Z2=0 X2=0) + %<- (times-right-cancels X0*2=X2 times/z nat`eq/ X2=0 X0=0) + %<- (times-right-cancels Z1*4=Z2 times/z nat`eq/ Z2=0 Z1=0) + %<- (nat2pair-respects-eq P2N Z1=0 eq/ N2P) + %<- (nat2pair-deterministic/0 N2P X1,Y1=0,0) + %<- (pair-eq-implies-eq X1,Y1=0,0 X1=0 Y1=0) + %<- (divrem-can-be-inverted X/2=X1,X0 XM X1*2=XM XM+X0=X) + %<- (divrem-can-be-inverted Y/2=Y1,Y0 YM Y1*2=YM YM+Y0=Y) + %<- (times-deterministic X1*2=XM times/z X1=0 nat`eq/ XM=0) + %<- (times-deterministic Y1*2=YM times/z Y1=0 nat`eq/ YM=0) + %<- (plus-deterministic XM+X0=X plus/z XM=0 X0=0 X=0) + %<- (plus-deterministic YM+Y0=Y plus/z YM=0 Y0=0 Y=0) + %<- (pair-preserves-eq X=0 Y=0 X,Y=0,0) %. +%worlds () (nat2pair-deterministic/0 _ _) %. +%total (D) (nat2pair-deterministic/0 D _) %. +%term _ + %pi (nat2pair-deterministic N2P N2P' nat`eq/ X,Y=X',Y') + %<- (nat2pair-deterministic/0 N2P X,Y=0,0) + %<- (nat2pair-deterministic/0 N2P' X',Y'=0,0) + %<- (eq-symmetric X',Y'=0,0 ZERO,ZERO=X',Y') + %<- (eq-transitive X,Y=0,0 ZERO,ZERO=X',Y' X,Y=X',Y') %. +%term _ + %pi (nat2pair-deterministic (pair2nat/XX Z3+Y0=Z Z2+X2=Z3 X0*2=X2 Z1*4=Z2 (%the (pair2nat (pair/ X1 Y1) Z1) P2N) Y/2=Y1,Y0 X/2=X1,X0) (pair2nat/XX Z3'+Y0'=Z Z2'+X2'=Z3' X0'*2=X2' Z1'*4=Z2' (%the (pair2nat (pair/ X1' Y1') Z1') P2N') Y'/2=Y1',Y0' X'/2=X1',X0') nat`eq/ X,Y=X',Y') + %<- (plus-associative Z2+X2=Z3 Z3+Y0=Z R X2+Y0=R Z2+R=Z) + %<- (divrem-implies-gt X/2=X1,X0 TWO>X0) + %<- (divrem-implies-gt Y/2=Y1,Y0 TWO>Y0) + %<- (succ-gt-implies-ge TWO>X0 ONE>=X0) + %<- (succ-gt-implies-ge TWO>Y0 ONE>=Y0) + %<- (times-right-preserves-ge* ONE>=X0 1*2=2 X0*2=X2 TWO>=X2) + %<- (plus-preserves-ge* TWO>=X2 ONE>=Y0 2+1=3 X2+Y0=R THREE>=R) + %<- (ge-implies-succ-gt THREE>=R FOUR>R) + %<- (divrem-can-be-constructed Z1*4=Z2 Z2+R=Z FOUR>R Z/4=Z1,R) + %<- (plus-associative Z2'+X2'=Z3' Z3'+Y0'=Z R' X2'+Y0'=R' Z2'+R'=Z) + %<- (divrem-implies-gt X'/2=X1',X0' TWO>X0') + %<- (divrem-implies-gt Y'/2=Y1',Y0' TWO>Y0') + %<- (succ-gt-implies-ge TWO>X0' ONE>=X0') + %<- (succ-gt-implies-ge TWO>Y0' ONE>=Y0') + %<- (times-right-preserves-ge* ONE>=X0' 1*2=2 X0'*2=X2' TWO>=X2') + %<- (plus-preserves-ge* TWO>=X2' ONE>=Y0' 2+1=3 X2'+Y0'=R' THREE>=R') + %<- (ge-implies-succ-gt THREE>=R' FOUR>R') + %<- (divrem-can-be-constructed Z1'*4=Z2' Z2'+R'=Z FOUR>R' Z/4=Z1',R') + %<- (divrem-deterministic Z/4=Z1,R Z/4=Z1',R' nat`eq/ nat`eq/ Z1=Z1' R=R') + %<- (divrem-can-be-constructed X0*2=X2 X2+Y0=R TWO>Y0 R/2=X0,Y0) + %<- (divrem-can-be-constructed X0'*2=X2' X2'+Y0'=R' TWO>Y0' R'/2=X0',Y0') + %<- (divrem-deterministic R/2=X0,Y0 R'/2=X0',Y0' R=R' nat`eq/ X0=X0' Y0=Y0') + %<- (nat2pair-deterministic P2N P2N' Z1=Z1' X1,Y1=X1',Y1') + %<- (pair-eq-implies-eq X1,Y1=X1',Y1' X1=X1' Y1=Y1') + %<- (divrem-can-be-inverted X/2=X1,X0 XM X1*2=XM XM+X0=X) + %<- (divrem-can-be-inverted Y/2=Y1,Y0 YM Y1*2=YM YM+Y0=Y) + %<- (divrem-can-be-inverted X'/2=X1',X0' XM' X1'*2=XM' XM'+X0'=X') + %<- (divrem-can-be-inverted Y'/2=Y1',Y0' YM' Y1'*2=YM' YM'+Y0'=Y') + %<- (times-deterministic X1*2=XM X1'*2=XM' X1=X1' nat`eq/ XM=XM') + %<- (plus-deterministic XM+X0=X XM'+X0'=X' XM=XM' X0=X0' X=X') + %<- (times-deterministic Y1*2=YM Y1'*2=YM' Y1=Y1' nat`eq/ YM=YM') + %<- (plus-deterministic YM+Y0=Y YM'+Y0'=Y' YM=YM' Y0=Y0' Y=Y') + %<- (pair-preserves-eq X=X' Y=Y' X,Y=X',Y') %. +%worlds () (nat2pair-deterministic _ _ _ _) %. +%total (D) (nat2pair-deterministic D _ _ _) %. +%theorem pair2nat-preserves-ne* : forall* {P1} {P2} {N1} {N2} forall {PNE ne P1 P2} {T1 pair2nat P1 N1} {T2 pair2nat P2 N2} exists {NE nat`ne N1 N2} true %. +%theorem pair2nat-preserves-ne*/L : forall* {P1} {P2} {N1} {N2} {B} forall {PNE ne P1 P2} {T1 pair2nat P1 N1} {T2 pair2nat P2 N2} {NT nat`eq? N1 N2 B} exists {NE nat`ne N1 N2} true %. +%term _ + %pi (pair2nat-preserves-ne* P1<>P2 P1->N1 P2->N2 N1<>N2) + %<- (nat`eq?-total EQ?) + %<- (pair2nat-preserves-ne*/L P1<>P2 P1->N1 P2->N2 EQ? N1<>N2) %. +%term _ pair2nat-preserves-ne*/L _ _ _ (nat`eq?/no N1<>N2) N1<>N2 %. +%term _ + %pi (pair2nat-preserves-ne*/L P1<>P2 P1->N P2->N nat`eq?/yes N<>N) + %<- (nat2pair-deterministic P1->N P2->N nat`eq/ P1=P2) + %<- (eq-ne-implies-false P1=P2 P1<>P2 F) + %<- (nat`false-implies-ne F N<>N) %. +%worlds () (pair2nat-preserves-ne*/L _ _ _ _ _) %. +%total {} (pair2nat-preserves-ne*/L _ _ _ _ _) %. +%worlds () (pair2nat-preserves-ne* _ _ _ _) %. +%total {} (pair2nat-preserves-ne* _ _ _ _) %. +%theorem pair2nat-preserves-ne : forall* {P1} {P2} forall {PNE ne P1 P2} exists {N1} {N2} {T1 pair2nat P1 N1} {T2 pair2nat P2 N2} {NE nat`ne N1 N2} true %. +%term _ + %pi (pair2nat-preserves-ne P1<>P2 N1 N2 T1 T2 N1<>N2) + %<- (pair2nat-total T1) + %<- (pair2nat-total T2) + %<- (pair2nat-preserves-ne* P1<>P2 T1 T2 N1<>N2) %. +%worlds () (pair2nat-preserves-ne _ _ _ _ _ _) %. +%total {} (pair2nat-preserves-ne _ _ _ _ _ _) %. +%theorem nat2pair-preserves-ne* : forall* {P1} {P2} {N1} {N2} forall {NE nat`ne N1 N2} {T1 nat2pair N1 P1} {T2 nat2pair N2 P2} exists {PNE ne P1 P2} true %. +%theorem nat2pair-preserves-ne*/L : forall* {P1} {P2} {N1} {N2} {B} forall {NE nat`ne N1 N2} {T1 nat2pair N1 P1} {T2 nat2pair N2 P2} {PT eq? P1 P2 B} exists {PNE ne P1 P2} true %. +%term _ + %pi (nat2pair-preserves-ne* N1<>N2 N1->P1 N2->P2 P1<>P2) + %<- (eq?-total EP?) + %<- (nat2pair-preserves-ne*/L N1<>N2 N1->P1 N2->P2 EP? P1<>P2) %. +%term _ nat2pair-preserves-ne*/L _ _ _ (eq?/no P1<>P2) P1<>P2 %. +%term _ + %pi (nat2pair-preserves-ne*/L N1<>N2 N1->P N2->P eq?/yes P<>P) + %<- (pair2nat-deterministic N1->P N2->P eq/ N1=N2) + %<- (nat`eq-ne-implies-false N1=N2 N1<>N2 F) + %<- (false-implies-ne F P<>P) %. +%worlds () (nat2pair-preserves-ne*/L _ _ _ _ _) %. +%total {} (nat2pair-preserves-ne*/L _ _ _ _ _) %. +%worlds () (nat2pair-preserves-ne* _ _ _ _) %. +%total {} (nat2pair-preserves-ne* _ _ _ _) %. +%theorem nat2pair-preserves-ne : forall* {N1} {N2} forall {NNE nat`ne N1 N2} exists {P1} {P2} {T1 nat2pair N1 P1} {T2 nat2pair N2 P2} {PE ne P1 P2} true %. +%term _ + %pi (nat2pair-preserves-ne N1<>N2 P1 P2 T1 T2 P1<>P2) + %<- (nat2pair-total T1) + %<- (nat2pair-total T2) + %<- (nat2pair-preserves-ne* N1<>N2 T1 T2 P1<>P2) %. +%worlds () (nat2pair-preserves-ne _ _ _ _ _ _) %. +%total {} (nat2pair-preserves-ne _ _ _ _ _ _) %. +%theorem nonzero-nat2pair-implies-gt-ge : forall* {N} {X} {Y} forall {D nat2pair (s N) (pair/ X Y)} exists {G1 gt (s N) X} {G2 ge (s N) Y} true %. +%term _ + %pi (nonzero-nat2pair-implies-gt-ge (pair2nat/XX plus/z plus/z times/z times/z _ (divrem/z _) (divrem/z _)) N+1>0 (nat`ge/= nat`eq/)) + %<- (succ-implies-gt-zero _ N+1>0) %. +%term _ + %pi (nonzero-nat2pair-implies-gt-ge (pair2nat/XX X2+Y=sN plus/z (times/s X-1*2=X2-2 X2-2+2=X2) times/z _ (divrem/z _) (divrem/z TWO>X)) N+1>X N+1>=Y) + %<- (plus-implies-ge X2+Y=sN (%the (ge (s N) Y) N+1>=Y)) + %<- (succ-gt-implies-ge TWO>X ONE>=X) + %<- (ge-succ-implies-gt ONE>=X ONE>X-1) + %<- (succ-gt-implies-ge ONE>X-1 ZERO>=X-1) + %<- (ge-zero-always _ X-1>=0) + %<- (ge-anti-symmetric ZERO>=X-1 X-1>=0 ZERO=X-1) + %<- (succ-deterministic ZERO=X-1 ONE=X) + %<- (times-deterministic times/z X-1*2=X2-2 ZERO=X-1 nat`eq/ ZERO=X2-2) + %<- (plus-deterministic plus/z X2-2+2=X2 ZERO=X2-2 nat`eq/ TWO=X2) + %<- (gt-respects-eq gt/1 TWO=X2 ONE=X X2>X) + %<- (plus-commutative X2+Y=sN Y+X2=sN) + %<- (plus-implies-ge Y+X2=sN N+1>=X2) + %<- (ge-transitive-gt N+1>=X2 X2>X N+1>X) %. +%term _ + %pi (nonzero-nat2pair-implies-gt-ge (pair2nat/XX _ _ _ _ (%the (pair2nat (pair/ (s _) _) z) P2z) _ _) GT GE) + %<- (nat2pair-deterministic pair2nat/00 P2z nat`eq/ ZERO,0=sN,_) + %<- (pair-eq-implies-eq ZERO,0=sN,_ ZERO=sN _) + %<- (nat`eq-contradiction ZERO=sN F) + %<- (nat`false-implies-gt F GT) + %<- (nat`false-implies-ge F GE) %. +%term _ + %pi (nonzero-nat2pair-implies-gt-ge (pair2nat/XX _ _ _ _ (%the (pair2nat (pair/ _ (s _)) z) P2z) _ _) GT GE) + %<- (nat2pair-deterministic pair2nat/00 P2z nat`eq/ ZERO,0=_,sN) + %<- (pair-eq-implies-eq ZERO,0=_,sN _ ZERO=sN) + %<- (nat`eq-contradiction ZERO=sN F) + %<- (nat`false-implies-gt F GT) + %<- (nat`false-implies-ge F GE) %. +%term _ + %pi (nonzero-nat2pair-implies-gt-ge (pair2nat/XX Z3+Y0=sN Z2+X2=Z3 X0*2=X2 Z1*4=Z2 Z1->X1,Y1 Y/2=Y1,Y0 X/2=X1,X0) N+1>X (ge/> N+1>Y)) + %<- (plus-commutative Z3+Y0=sN Y0+Z3=sN) + %<- (plus-implies-ge Y0+Z3=sN N+1>=Z3) + %<- (plus-commutative Z2+X2=Z3 X2+Z2=Z3) + %<- (plus-implies-ge X2+Z2=Z3 Z3>=Z2) + %<- (ge-transitive N+1>=Z3 Z3>=Z2 N+1>=Z2) + %<- (nonzero-nat2pair-implies-gt-ge Z1->X1,Y1 Z1>X1 Z1>=Y1) + %<- (times-associative-converse 2*2=4 Z1*4=Z2 Z2/2 Z1*2=Z2/2 Z2/2*2=Z2) + %<- (divrem-can-be-inverted Y/2=Y1,Y0 Y12 Y1*2=Y12 Y12+Y0=Y) + %<- (divrem-can-be-inverted X/2=X1,X0 X12 X1*2=X12 X12+X0=X) + %<- (times-right-preserves-gt* Z1>X1 Z1*2=Z2/2 X1*2=X12 nat`eq/ Z2/2>X12) + %<- (succ-implies-gt-zero _ Z1>0) + %<- (gt-implies-ge-succ Z1>0 Z1>=1) + %<- (times-right-preserves-ge* Z1>=1 Z1*2=Z2/2 1*2=2 Z2/2>=2) + %<- (ge-implies-plus Z2/2>=2 ZZ ZZ+2=Z2/2) + %<- (plus-commutative ZZ+2=Z2/2 TWO+ZZ=Z2/2) + %<- (plus-deterministic TWO+ZZ=Z2/2 (plus/s (plus/s plus/z)) nat`eq/ nat`eq/ Z2/2=ssZZ) + %<- (times-respects-eq Z2/2*2=Z2 Z2/2=ssZZ nat`eq/ nat`eq/ SSZZ*2=Z2) + %<- (non-trivial-times-implies-much-gt* SSZZ*2=Z2 Z2>sssZZ) + %<- (divrem-implies-gt X/2=X1,X0 TWO>X0) + %<- (succ-gt-implies-ge TWO>X0 ONE>=X0) + %<- (plus-right-identity _ X12+0=X12) + %<- (plus-right-increase X12+0=X12 X12+1=sX12) + %<- (plus-left-preserves-ge* ONE>=X0 X12+1=sX12 X12+X0=X SX12>=X) + %<- (gt-implies-ge-succ Z2/2>X12 Z2/2>=sX12) + %<- (ge-transitive Z2/2>=sX12 SX12>=X Z2/2>=X) + %<- (ge-respects-eq Z2/2>=X Z2/2=ssZZ nat`eq/ SSZZ>=X) + %<- (ge-implies-succ-gt SSZZ>=X SSSZZ>X) + %<- (gt-transitive Z2>sssZZ SSSZZ>X Z2>X) + %<- (ge-transitive-gt N+1>=Z2 Z2>X N+1>X) + %<- (times-right-preserves-ge* Z1>=Y1 Z1*2=Z2/2 Y1*2=Y12 Z2/2>=Y12) + %<- (ge-respects-eq Z2/2>=Y12 Z2/2=ssZZ nat`eq/ SSZZ>=Y12) + %<- (divrem-implies-gt Y/2=Y1,Y0 TWO>Y0) + %<- (succ-gt-implies-ge TWO>Y0 ONE>=Y0) + %<- (plus-right-identity _ Y12+0=Y12) + %<- (plus-right-increase Y12+0=Y12 Y12+1=sY12) + %<- (plus-left-preserves-ge* ONE>=Y0 Y12+1=sY12 Y12+Y0=Y SY12>=Y) + %<- (succ-preserves-ge SSZZ>=Y12 SSSZZ>=SY12) + %<- (ge-transitive SSSZZ>=SY12 SY12>=Y SSSZZ>=Y) + %<- (gt-transitive-ge Z2>sssZZ SSSZZ>=Y Z2>Y) + %<- (ge-transitive-gt N+1>=Z2 Z2>Y N+1>Y) %. +%worlds () (nonzero-nat2pair-implies-gt-ge _ _ _) %. +%total (N) (nonzero-nat2pair-implies-gt-ge N _ _) %. +%theorem nat2pair-implies-ge : forall* {N} {X} {Y} forall {D nat2pair N (pair/ X Y)} exists {G1 ge N X} {G2 ge N Y} true %. +%term _ + %pi (nat2pair-implies-ge N2P (ge/> N>X) N>=Y) + %<- (nonzero-nat2pair-implies-gt-ge N2P N>X N>=Y) %. +%term _ + %pi (nat2pair-implies-ge Z2P (ge/= ZERO=X) (ge/= ZERO=Y)) + %<- (nat2pair-deterministic pair2nat/00 Z2P nat`eq/ ZERO,ZERO=X,Y) + %<- (pair-eq-implies-eq ZERO,ZERO=X,Y ZERO=X ZERO=Y) %. +%worlds () (nat2pair-implies-ge _ _ _) %. +%total {} (nat2pair-implies-ge _ _ _) %. +%theorem constrained1-pair2nat-unbounded : forall {N1} {B} exists {N2} {N} {D pair2nat (pair/ N1 N2) N} {G gt N B} true %. +%term _ + %pi (constrained1-pair2nat-unbounded N1 B (s B) N N1,N2->N N>B) + %<- (pair2nat-total N1,N2->N) + %<- (nat2pair-implies-ge N1,N2->N N>=N1 N>=N2) + %<- (ge-succ-implies-gt N>=N2 N>B) %. +%worlds () (constrained1-pair2nat-unbounded _ _ _ _ _ _) %. +%total {} (constrained1-pair2nat-unbounded _ _ _ _ _ _) %. +%theorem constrained2-pair2nat-unbounded : forall {N2} {B} exists {N1} {N} {D pair2nat (pair/ N1 N2) N} {G gt N B} true %. +%term _ + %pi (constrained2-pair2nat-unbounded N2 B (s B) N N1,N2->N N>B) + %<- (pair2nat-total N1,N2->N) + %<- (nat2pair-implies-ge N1,N2->N N>=N1 N>=N2) + %<- (ge-succ-implies-gt N>=N1 N>B) %. +%worlds () (constrained2-pair2nat-unbounded _ _ _ _ _ _) %. +%total {} (constrained2-pair2nat-unbounded _ _ _ _ _ _) %. +%%%% Renamings +%inline natpair pair %. +%inline natpair/ pair/ %. +%%%% Exports +%inline natpair`pair pair %. +%inline natpair`pair/ pair/ %. +%inline natpair`eq eq %. +%inline natpair`eq/ eq/ %. +%inline natpair`ne ne %. +%inline natpair`ne/1 ne/1 %. +%inline natpair`ne/2 ne/2 %. +%inline natpair`eq? eq? %. +%inline natpair`eq?/yes eq?/yes %. +%inline natpair`eq?/no eq?/no %. +%inline natpair`false-implies-eq false-implies-eq %. +%inline natpair`meta-eq meta-eq %. +%inline natpair`eq-reflexive eq-reflexive %. +%inline natpair`eq-symmetric eq-symmetric %. +%inline natpair`eq-transitive eq-transitive %. +%inline natpair`pair-eq-implies-eq pair-eq-implies-eq %. +%inline natpair`pair-preserves-eq pair-preserves-eq %. +%inline natpair`false-implies-ne false-implies-ne %. +%inline natpair`ne-respects-eq ne-respects-eq %. +%inline natpair`ne-anti-reflexive ne-anti-reflexive %. +%inline natpair`ne-symmetric ne-symmetric %. +%inline natpair`eq-ne-implies-false eq-ne-implies-false %. +%inline natpair`eq?-total* eq?-total* %. +%inline natpair`eq?-total*/L eq?-total*/L %. +%inline natpair`eq?-total eq?-total %. +%inline natpair`pair2nat pair2nat %. +%inline natpair`pair2nat/00 pair2nat/00 %. +%inline natpair`pair2nat/XX pair2nat/XX %. +%inline natpair`nat2pair nat2pair %. +%inline natpair`false-implies-pair2nat false-implies-pair2nat %. +%inline natpair`false-implies-not2pair false-implies-not2pair %. +%inline natpair`pair2nat-respects-eq pair2nat-respects-eq %. +%inline natpair`nat2pair-respects-eq nat2pair-respects-eq %. +%inline natpair`pair2nat-total** pair2nat-total** %. +%inline natpair`pair2nat-total* pair2nat-total* %. +%inline natpair`pair2nat-total pair2nat-total %. +%inline natpair`nat2pair-total* nat2pair-total* %. +%inline natpair`nat2pair-total nat2pair-total %. +%inline natpair`pair2nat-deterministic pair2nat-deterministic %. +%inline natpair`pair2nat-deterministic/00 pair2nat-deterministic/00 %. +%inline natpair`nat2pair-deterministic nat2pair-deterministic %. +%inline natpair`nat2pair-deterministic/0 nat2pair-deterministic/0 %. +%inline natpair`pair2nat-preserves-ne* pair2nat-preserves-ne* %. +%inline natpair`pair2nat-preserves-ne*/L pair2nat-preserves-ne*/L %. +%inline natpair`pair2nat-preserves-ne pair2nat-preserves-ne %. +%inline natpair`nat2pair-preserves-ne* nat2pair-preserves-ne* %. +%inline natpair`nat2pair-preserves-ne*/L nat2pair-preserves-ne*/L %. +%inline natpair`nat2pair-preserves-ne nat2pair-preserves-ne %. +%inline natpair`nonzero-nat2pair-implies-gt-ge nonzero-nat2pair-implies-gt-ge %. +%inline natpair`nat2pair-implies-ge nat2pair-implies-ge %. +%inline natpair`constrained1-pair2nat-unbounded constrained1-pair2nat-unbounded %. +%inline natpair`constrained2-pair2nat-unbounded constrained2-pair2nat-unbounded %. +%inline natpair`natpair natpair %. +%inline natpair`natpair/ natpair/ %. +%{! | hidden = true !}% +% contents included from ../library/multiset.elf +%%%%%% Multisets of natural numbers +%%%%%% John Boyland +%%%%%% You may freely use, modify and distribute this file without restrictions. +%%%%% This file requires the "nat.elf" signature +%%%%% multiset-help.elf +%%%%% Theorems needed to help 'multiset' use 'map.' +%%%%% This file is part of the multiset.elf signature +%%%% Renamings +%inline nat`leq nat`le %. +%inline nat`false-implies-leq nat`false-implies-le %. +%inline nat`leq-reflexive nat`le-reflexive %. +%inline nat`leq-transitive nat`le-transitive %. +%inline nat`leq-anti-symmetric nat`le-anti-symmetric %. +%inline nat`union nat`max %. +%inline nat`false-implies-union nat`false-implies-max %. +%inline nat`union-deterministic nat`max-deterministic %. +%inline nat`union-total* nat`max-total* %. +%inline nat`union-commutative nat`max-commutative %. +%inline nat`union-associative nat`max-associative %. +%inline nat`union-associative* nat`max-associative* %. +%inline nat`union-left-preserves-leq* nat`max-left-preserves-le* %. +%inline nat`union-preserves-leq nat`max-preserves-le %. +%inline nat`union-implies-leq* [G] [L] nat`max-implies-ge G L IG %. +%inline nat`union-implies-leq nat`max-implies-ge %. +%inline nat`union-is-lub nat`max-is-lub %. +%inline nat`intersection nat`min %. +%inline nat`false-implies-intersection nat`false-implies-min %. +%inline nat`intersection-deterministic nat`min-deterministic %. +%inline nat`intersection-total* nat`min-total* %. +%inline nat`intersection-commutative nat`min-commutative %. +%inline nat`intersection-associative nat`min-associative %. +%inline nat`intersection-associative* nat`min-associative* %. +%inline nat`intersection-implies-leq* [M] [L] nat`min-implies-ge M L IG %. +%inline nat`intersection-left-preserves-leq* nat`min-left-preserves-le* %. +%inline nat`intersection-is-glb nat`min-is-glb %. +%inline nat`intersection-right-distributes-over-union nat`min-right-distributes-over-max %. +%inline nat`union-right-distributes-over-intersection nat`max-right-distributes-over-min %. +%%%% Functor Use +%%%% Definitions of Maps +%sort map %. +%term map/0 map %. +%term map/+ %pi nat %-> nat %-> map %-> map %. +%%%% Relations on maps +%sort eq %. +%term eq/ eq M M %. +%sort ne %. +%term ne/L ne map/0 (map/+ _ _ _) %. +%term ne/R ne (map/+ _ _ _) map/0 %. +%term ne/N %pi (nat`ne N1 N2) %-> (ne (map/+ N1 _ _) (map/+ N2 _ _)) %. +%term ne/D %pi (nat`ne D1 D2) %-> (ne (map/+ _ D1 _) (map/+ _ D2 _)) %. +%term ne/+ %pi (ne M1 M2) %-> (ne (map/+ _ _ M1) (map/+ _ _ M2)) %. +%sort eq? %. +%term eq?/yes eq? X X true %. +%term eq?/no %pi (eq? X Y false) %<- (ne X Y) %. +%sort lookup %. +%term lookup/= %pi (lookup (map/+ N1 D _) N2 D) %<- (nat`eq N1 N2) %. +%term lookup/> %pi (lookup (map/+ N1 _ F) N2 D) %<- (plus (s N0) N1 N2) %<- (lookup F N0 D) %. +%sort not-member %. +%term not-member/0 not-member map/0 M %. +%term not-member/< %pi (not-member (map/+ N _ F) M) %<- (gt N M) %. +%term not-member/> %pi (not-member (map/+ N _ F) M) %<- (plus (s M1) N M) %<- (not-member F M1) %. +%sort member? %. +%term member?/in %pi (member? M N true) %<- (lookup M N _) %. +%term member?/out %pi (member? M N false) %<- (not-member M N) %. +%sort disjoint %. +%term disjoint/L disjoint map/0 M %. +%term disjoint/R disjoint M map/0 %. +%term disjoint/< + %pi (disjoint (map/+ N1 D1 M1) (map/+ N2 D2 M2)) + %<- (nat`plus (s N0) N1 N2) + %<- (disjoint M1 (map/+ N0 D2 M2)) %. +%term disjoint/> + %pi (disjoint (map/+ N1 D1 M1) (map/+ N2 D2 M2)) + %<- (nat`plus (s N3) N2 N1) + %<- (disjoint (map/+ N3 D1 M1) M2) %. +%sort disjoint? %. +%term disjoint?/yes %pi (disjoint M1 M2) %-> (disjoint? M1 M2 true) %. +%term disjoint?/no %pi (lookup M1 N D1) %-> (lookup M2 N D2) %-> (disjoint? M1 M2 false) %. +%sort size %. +%term size/0 size map/0 z %. +%term size/+ %pi (size (map/+ _ _ M) (s N)) %<- (size M N) %. +%% useful for proving termination on map operations: +%sort bound %. +%term bound/0 bound map/0 z %. +%term bound/+ %pi (bound (map/+ N1 D M) N3) %<- (bound M N2) %<- (plus (s N1) N2 N3) %. +%sort shift %. +%term shift/0 shift _ map/0 map/0 %. +%term shift/+ %pi (shift N1 (map/+ N2 D M) (map/+ N3 D M)) %<- (plus (s N1) N2 N3) %. +%sort update %. +%term update/0 update map/0 N D (map/+ N D map/0) %. +%term update/= %pi (update (map/+ N1 _ F) N2 D (map/+ N2 D F)) %<- (nat`eq N1 N2) %. +%term update/< + %pi (update (map/+ N1 D1 F) N2 D2 (map/+ N2 D2 (map/+ N3 D1 F))) + %<- (plus (s N3) N2 N1) %. +%term update/> + %pi (update (map/+ N1 D1 F1) N2 D2 (map/+ N1 D1 F2)) + %<- (plus (s N0) N1 N2) + %<- (update F1 N0 D2 F2) %. +%%%% Theorems +%%% Theorems about eq +%theorem meta-eq : forall {M} {N} {E eq M N} true %. +%term _ meta-eq M M eq/ %. +%worlds () (meta-eq _ _ _) %. +%total {} (meta-eq _ _ _) %. +%reduces = M N (meta-eq M N _) %. +%theorem false-implies-eq : forall* {M} {M'} forall {F void} exists {E eq M M'} true %. +%worlds () (false-implies-eq _ M=M') %. +%total {} (false-implies-eq _ _) %. +%theorem eq-reflexive : forall {M} exists {E eq M M} true %. +%term _ eq-reflexive _ eq/ %. +%worlds () (eq-reflexive M M=M) %. +%total {} (eq-reflexive _ _) %. +%theorem eq-symmetric : forall* {M} {M'} forall {E1 eq M M'} exists {E2 eq M' M} true %. +%term _ eq-symmetric eq/ eq/ %. +%worlds () (eq-symmetric M=M' M'=M) %. +%total {} (eq-symmetric _ _) %. +%theorem eq-transitive : forall* {M1} {M2} {M3} forall {E12 eq M1 M2} {E23 eq M2 M3} exists {E13 eq M1 M3} true %. +%term _ eq-transitive eq/ eq/ eq/ %. +%worlds () (eq-transitive M1=M2 M2=M3 M1=M3) %. +%total {} (eq-transitive _ _ _) %. +%theorem map/+-preserves-eq : forall* {N} {NP} {D} {DP} {F} {FP} forall {EN nat`eq N NP} {ED nat`eq D DP} {EF eq F FP} exists {E eq (map/+ N D F) (map/+ NP DP FP)} true %. +%term _ map/+-preserves-eq nat`eq/ nat`eq/ eq/ eq/ %. +%worlds () (map/+-preserves-eq N=N' D=D' F=F' NDF=N'D'F') %. +%total {} (map/+-preserves-eq _ _ _ _) %. +%theorem map/+-preserves-eq-converse : forall* {N} {NP} {D} {DP} {F} {FP} forall {E eq (map/+ N D F) (map/+ NP DP FP)} exists {EN nat`eq N NP} {ED nat`eq D DP} {EF eq F FP} true %. +%term _ map/+-preserves-eq-converse eq/ nat`eq/ nat`eq/ eq/ %. +%worlds () (map/+-preserves-eq-converse _ _ _ _) %. +%total {} (map/+-preserves-eq-converse _ _ _ _) %. +%theorem eq-no-occur : forall* {M} {N} {D} forall {E eq M (map/+ N D M)} exists {F void} true %. +%worlds () (eq-no-occur _ _) %. +%total {} (eq-no-occur _ _) %. +%theorem eq-contradiction : forall* {N} {D} {M} forall {E eq map/0 (map/+ N D M)} exists {F void} true %. +%worlds () (eq-contradiction _ _) %. +%total {} (eq-contradiction _ _) %. +%%% Theorems about ne +%theorem false-implies-ne : forall* {M1} {M2} forall {F void} exists {N ne M1 M2} true %. +%worlds () (false-implies-ne _ _) %. +%total {} (false-implies-ne _ _) %. +%theorem ne-respects-eq : forall* {M11} {M12} {M21} {M22} forall {N1 ne M11 M12} {E1 eq M11 M21} {E2 eq M12 M22} exists {N2 ne M21 M22} true %. +%term _ ne-respects-eq N eq/ eq/ N %. +%worlds () (ne-respects-eq _ _ _ _) %. +%total {} (ne-respects-eq _ _ _ _) %. +%theorem ne-anti-reflexive : forall* {M} forall {N ne M M} exists {F void} true %. +%term _ %pi (ne-anti-reflexive (ne/N N) F) %<- (nat`ne-anti-reflexive N F) %. +%term _ %pi (ne-anti-reflexive (ne/D N) F) %<- (nat`ne-anti-reflexive N F) %. +%term _ %pi (ne-anti-reflexive (ne/+ N) F) %<- (ne-anti-reflexive N F) %. +%worlds () (ne-anti-reflexive _ _) %. +%total (N) (ne-anti-reflexive N _) %. +%theorem ne-symmetric : forall* {M1} {M2} forall {N1 ne M1 M2} exists {N2 ne M2 M1} true %. +%term _ ne-symmetric ne/L ne/R %. +%term _ ne-symmetric ne/R ne/L %. +%term _ %pi (ne-symmetric (ne/N N1) (ne/N N2)) %<- (nat`ne-symmetric N1 N2) %. +%term _ %pi (ne-symmetric (ne/D N1) (ne/D N2)) %<- (nat`ne-symmetric N1 N2) %. +%term _ %pi (ne-symmetric (ne/+ N1) (ne/+ N2)) %<- (ne-symmetric N1 N2) %. +%worlds () (ne-symmetric _ _) %. +%total (N) (ne-symmetric N _) %. +%theorem eq-ne-implies-false : forall* {X} {Y} forall {D1 eq X Y} {D2 ne X Y} exists {F void} true %. +%term _ %pi (eq-ne-implies-false eq/ X<>X F) %<- (ne-anti-reflexive X<>X F) %. +%worlds () (eq-ne-implies-false _ _ _) %. +%total {} (eq-ne-implies-false _ _ _) %. +%theorem eq?-total* : forall {M} {N} exists {B} {T eq? M N B} true %. +%inline eq?-total eq?-total* _ _ _ %. +%theorem eq?-total/+ : forall* {N1} {D1} {N2} {D2} {M2} {EN} {ED} {EM} forall {M1} {EN? nat`eq? N1 N2 EN} {ED? nat`eq? D1 D2 ED} {EM? eq? M1 M2 EM} exists {B} {E? eq? (map/+ N1 D1 M1) (map/+ N2 D2 M2) B} true %. +%term _ eq?-total eq?/yes %. +%term _ eq?-total (eq?/no ne/L) %. +%term _ eq?-total (eq?/no ne/R) %. +%term _ + %pi (eq?-total E?) + %<- (nat`eq?-total EN?) + %<- (nat`eq?-total ED?) + %<- (eq?-total EM?) + %<- (eq?-total/+ _ EN? ED? EM? _ E?) %. +%term _ eq?-total/+ _ nat`eq?/yes nat`eq?/yes eq?/yes _ eq?/yes %. +%term _ eq?-total/+ _ (nat`eq?/no N) _ _ _ (eq?/no (ne/N N)) %. +%term _ eq?-total/+ _ _ (nat`eq?/no N) _ _ (eq?/no (ne/D N)) %. +%term _ eq?-total/+ _ _ _ (eq?/no N) _ (eq?/no (ne/+ N)) %. +%worlds () (eq?-total* _ _ _ _) (eq?-total/+ _ _ _ _ _ _) %. +%total (M W) (eq?-total* M _ _ _) (eq?-total/+ W _ _ _ _ _) %. +%%% Theorems about lookup +%theorem false-implies-lookup : forall* {M} {N} {D} forall {F void} exists {L lookup M N D} true %. +%worlds () (false-implies-lookup _ F^N=D) %. +%total {} (false-implies-lookup _ _) %. +%theorem lookup-respects-eq : forall* {M} {N} {D} {MP} {NP} {DP} forall {L lookup M N D} {EM eq M MP} {EN nat`eq N NP} {ED nat`eq D DP} exists {LP lookup MP NP DP} true %. +%term _ lookup-respects-eq L eq/ nat`eq/ nat`eq/ L %. +%worlds () (lookup-respects-eq M^N=D M=M' N=N' D=D' M'^N'=D') %. +%total {} (lookup-respects-eq _ _ _ _ _) %. +%theorem lookup-deterministic : forall* {M} {N} {D} {MP} {NP} {DP} forall {L lookup M N D} {LP lookup MP NP DP} {EM eq M MP} {EN nat`eq N NP} exists {ED nat`eq D DP} true %. +%term _ lookup-deterministic (lookup/= nat`eq/) (lookup/= nat`eq/) eq/ nat`eq/ nat`eq/ %. +%term _ + %pi (lookup-deterministic (lookup/> F^N0=D N0+1+N1=N2) (lookup/> F^N0'=D' N0'+1+N1=N2) eq/ nat`eq/ D=D') + %<- (plus-right-cancels N0+1+N1=N2 N0'+1+N1=N2 nat`eq/ nat`eq/ N0+1=N0'+1) + %<- (succ-cancels N0+1=N0'+1 N0=N0') + %<- (lookup-deterministic F^N0=D F^N0'=D' eq/ N0=N0' D=D') %. +%% contradiction cases +%term _ + %pi (lookup-deterministic (lookup/= nat`eq/) (lookup/> _ N0+1+N=N) eq/ nat`eq/ D=D') + %<- (plus-implies-gt N0+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N FALSE) + %<- (nat`false-implies-eq FALSE D=D') %. +%term _ + %pi (lookup-deterministic (lookup/> _ N0+1+N=N) (lookup/= nat`eq/) eq/ nat`eq/ D=D') + %<- (plus-implies-gt N0+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N FALSE) + %<- (nat`false-implies-eq FALSE D=D') %. +%worlds () (lookup-deterministic M^N=D M'^N'=D' M=M' N=N' D=D') %. +%total (L) (lookup-deterministic L _ _ _ _) %. +%% lookup is NOT total +%theorem lookup-contradiction : forall* {N} {D} forall {L lookup map/0 N D} exists {F void} true %. +%worlds () (lookup-contradiction _ _) %. +%total {} (lookup-contradiction _ _) %. +%theorem lookup-one-choice : forall* {N1} {D1} {N2} {D2} forall {L lookup (map/+ N1 D1 map/0) N2 D2} exists {NE nat`eq N1 N2} {DE nat`eq D1 D2} true %. +%term _ lookup-one-choice (lookup/= nat`eq/) nat`eq/ nat`eq/ %. +%worlds () (lookup-one-choice _ _ _) %. +%total {} (lookup-one-choice _ _ _) %. +%theorem lookup-ne-implies-ne : forall* {M1} {N1} {D1} {M2} {N2} {D2} forall {L1 lookup M1 N1 D1} {L2 lookup M2 N2 D2} {EN nat`eq N1 N2} {ND nat`ne D1 D2} exists {NM ne M1 M2} true %. +%theorem lookup-ne-implies-ne/L : forall* {M1} {N1} {D1} {M2} {N2} {D2} {B} forall {L1 lookup M1 N1 D1} {L2 lookup M2 N2 D2} {EN nat`eq N1 N2} {ND nat`ne D1 D2} {EM? eq? M1 M2 B} exists {NM ne M1 M2} true %. +%term _ + %pi (lookup-ne-implies-ne L1 L2 EN ND NM) + %<- (eq?-total EM?) + %<- (lookup-ne-implies-ne/L L1 L2 EN ND EM? NM) %. +%term _ lookup-ne-implies-ne/L L1 L2 _ _ (eq?/no NM) NM %. +%term _ + %pi (lookup-ne-implies-ne/L L1 L2 nat`eq/ D1<>D2 eq?/yes NM) + %<- (lookup-deterministic L1 L2 eq/ nat`eq/ D1=D2) + %<- (nat`eq-ne-implies-false D1=D2 D1<>D2 F) + %<- (false-implies-ne F NM) %. +%worlds () (lookup-ne-implies-ne/L _ _ _ _ _ _) %. +%total {} (lookup-ne-implies-ne/L _ _ _ _ _ _) %. +%worlds () (lookup-ne-implies-ne _ _ _ _ _) %. +%total {} (lookup-ne-implies-ne _ _ _ _ _) %. +%%% Theorems about not-member +%theorem false-implies-not-member : forall* {M} {N} forall {F void} exists {D not-member M N} true %. +%worlds () (false-implies-not-member _ N-not-in-member-M) %. +%total {} (false-implies-not-member _ _) %. +%theorem not-member-respects-eq : forall* {M} {N} {MP} {NP} forall {D not-member M N} {EM eq M MP} {EN nat`eq N NP} exists {DP not-member MP NP} true %. +%term _ not-member-respects-eq D eq/ nat`eq/ D %. +%worlds () (not-member-respects-eq _ _ _ _) %. +%total {} (not-member-respects-eq _ _ _ _) %. +%% not-member is NOT deterministic +%theorem not-member-total* : forall {M} exists {N} {F not-member M N} true %. +%term _ not-member-total* map/0 z not-member/0 %. +%term _ + %pi (not-member-total* (map/+ N1 _ M) N3 (not-member/> F N+1+N1=N3)) + %<- (not-member-total* M N F) + %<- (plus-total* (s N) N1 N3 N+1+N1=N3) %. +%worlds () (not-member-total* M N N-not-in-member-of-M) %. +%total (M) (not-member-total* M _ _) %. +%inline not-member-total not-member-total* _ _ %. +%theorem not-member-lookup-not-equal : forall* {M} {N1} {N2} {D2} forall {F not-member M N1} {L lookup M N2 D2} exists {NE nat`ne N1 N2} true %. +%term _ not-member-lookup-not-equal (not-member/< N2>N1) (lookup/= nat`eq/) (nat`ne/< N2>N1) %. +%term _ + %pi (not-member-lookup-not-equal (not-member/< N1>N3) (lookup/> _ N0+1+N1=N2) (nat`ne/< N2>N3)) + %<- (plus-implies-gt N0+1+N1=N2 nat`eq/ N2>N1) + %<- (gt-transitive N2>N1 N1>N3 N2>N3) %. +%term _ + %pi (not-member-lookup-not-equal (not-member/> _ X+1+N2=N1) (lookup/= nat`eq/) (nat`ne/> N1>N2)) + %<- (plus-implies-gt X+1+N2=N1 nat`eq/ N1>N2) %. +%term _ + %pi (not-member-lookup-not-equal (not-member/> F N4+1+N1=N3) (lookup/> L N0+1+N1=N2) N3<>N2) + %<- (not-member-lookup-not-equal F L N4<>N0) + %<- (succ-preserves-ne N4<>N0 N4+1<>N0+1) + %<- (plus-right-preserves-ne* N4+1<>N0+1 N4+1+N1=N3 N0+1+N1=N2 N3<>N2) %. +%worlds () (not-member-lookup-not-equal N1-not-in-member-of-M M^N2=D N1<>N2) %. +%total (F) (not-member-lookup-not-equal F _ _) %. +%theorem not-member-contradiction : forall* {M} {N} {D} forall {F not-member (map/+ N D M) N} exists {V void} true %. +%term _ + %pi (not-member-contradiction (not-member/< N>N) V) + %<- (nat`gt-anti-reflexive N>N V) %. +%term _ + %pi (not-member-contradiction (not-member/> _ N0+1+N=N) V) + %<- (nat`plus-implies-gt N0+1+N=N nat`eq/ N>N) + %<- (nat`gt-anti-reflexive N>N V) %. +%worlds () (not-member-contradiction _ _) %. +%total {} (not-member-contradiction _ _) %. +%theorem ne-implies-unit-map-not-member : forall* {N1} {D} {N2} forall {NE nat`ne N1 N2} exists {F not-member (map/+ N1 D map/0) N2} true %. +%term _ + %pi (ne-implies-unit-map-not-member (nat`ne/< N1 not-member/0 N0+1+N1=N2)) + %<- (nat`gt-implies-plus N1 N1>N2) (not-member/< N1>N2) %. +%worlds () (ne-implies-unit-map-not-member _ _) %. +%total {} (ne-implies-unit-map-not-member _ _) %. +%theorem plus-right-preserves-not-member* : forall* {M} {N1} {D} {N2} {N} {N3} {N4} forall {F not-member (map/+ N1 D M) N2} {P1 plus N1 N N3} {P2 plus N2 N N4} exists {FP not-member (map/+ N3 D M) N4} true %. +%term _ + %pi (plus-right-preserves-not-member* (not-member/< N2>N1) N1+N=N3 N2+N=N4 (not-member/< N4>N3)) + %<- (nat`plus-right-preserves-gt* N2>N1 N1+N=N3 N2+N=N4 N4>N3) %. +%term _ + %pi (plus-right-preserves-not-member* (not-member/> F10 N0+1+N1=N2) N1+N=N3 N2+N=N4 (not-member/> F10 N0+1+N3=N4)) + %<- (nat`plus-associative* N0+1+N1=N2 N2+N=N4 N1+N=N3 N0+1+N3=N4) %. +%worlds () (plus-right-preserves-not-member* _ _ _ _) %. +%total {} (plus-right-preserves-not-member* _ _ _ _) %. +%theorem not-member-lookup-implies-ne : forall* {M1} {N1} {M2} {N2} {D2} forall {L1 not-member M1 N1} {L2 lookup M2 N2 D2} {EN nat`eq N1 N2} exists {NM ne M1 M2} true %. +%theorem not-member-lookup-implies-ne/L : forall* {M1} {N1} {M2} {N2} {D2} {B} forall {L1 not-member M1 N1} {L2 lookup M2 N2 D2} {EN nat`eq N1 N2} {EM? eq? M1 M2 B} exists {NM ne M1 M2} true %. +%term _ + %pi (not-member-lookup-implies-ne F1 L2 EN NM) + %<- (eq?-total EM?) + %<- (not-member-lookup-implies-ne/L F1 L2 EN EM? NM) %. +%term _ not-member-lookup-implies-ne/L _ _ _ (eq?/no NM) NM %. +%term _ + %pi (not-member-lookup-implies-ne/L F1 L2 nat`eq/ eq?/yes NM) + %<- (not-member-lookup-not-equal F1 L2 N<>N) + %<- (nat`ne-anti-reflexive N<>N F) + %<- (false-implies-ne F NM) %. +%worlds () (not-member-lookup-implies-ne/L _ _ _ _ _) %. +%total {} (not-member-lookup-implies-ne/L _ _ _ _ _) %. +%worlds () (not-member-lookup-implies-ne _ _ _ _) %. +%total {} (not-member-lookup-implies-ne _ _ _ _) %. +%%% Theorems about map/member +%theorem false-implies-member? : forall* {M} {N} {D} forall {F void} exists {MD member? M N D} true %. +%worlds () (false-implies-member? _ _) %. +%total {} (false-implies-member? _ _) %. +%theorem member?-respects-eq : forall* {M1} {N1} {B1} {M2} {N2} {B2} forall {MD1 member? M1 N1 B1} {EM eq M1 M2} {EN nat`eq N1 N2} {BE bool`eq B1 B2} exists {MD2 member? M2 N2 B2} true %. +%term _ member?-respects-eq MD eq/ nat`eq/ bool`eq/ MD %. +%worlds () (member?-respects-eq _ _ _ _ _) %. +%total {} (member?-respects-eq _ _ _ _ _) %. +%theorem member?-deterministic : forall* {M1} {N1} {B1} {M2} {N2} {B2} forall {MD1 member? M1 N1 B1} {MD2 member? M2 N2 B2} {EM eq M1 M2} {EN nat`eq N1 N2} exists {BE bool`eq B1 B2} true %. +%term _ member?-deterministic _ _ _ _ bool`eq/ %. +%term _ + %pi (member?-deterministic (member?/in L) (member?/out F) eq/ nat`eq/ BE) + %<- (not-member-lookup-not-equal F L NE) + %<- (nat`ne-anti-reflexive NE V) + %<- (bool`false-implies-eq V BE) %. +%term _ + %pi (member?-deterministic (member?/out F) (member?/in L) eq/ nat`eq/ BE) + %<- (not-member-lookup-not-equal F L NE) + %<- (nat`ne-anti-reflexive NE V) + %<- (bool`false-implies-eq V BE) %. +%worlds () (member?-deterministic _ _ _ _ _) %. +%total {} (member?-deterministic _ _ _ _ _) %. +%theorem member?-total* : forall {M} {N} exists {B} {MD member? M N B} true %. +%% we need a lemma +%theorem member?-map/+-total : forall {N1} {D1} {M1} {N2} {C} {CMP nat`compare N1 N2 C} exists {B} {MD member? (map/+ N1 D1 M1) N2 B} true %. +%% and this lemma needs a lemma +%theorem member?-map/+-complete : forall {N1} {D1} {M1} {N2} {N0} {P plus (s N0) N1 N2} {B} {MD1 member? M1 N0 B} exists {MD member? (map/+ N1 D1 M1) N2 B} true %. +%term _ member?-total* map/0 N false (member?/out not-member/0) %. +%term _ + %pi (member?-total* (map/+ N1 D1 M1) N2 B MD) + %<- (nat`compare-total* N1 N2 C CMP) + %<- (member?-map/+-total N1 D1 M1 N2 C CMP B MD) %. +%term _ + %pi (member?-map/+-total N1 D1 M1 N2 equal CMP true (member?/in (lookup/= N1=N2))) + %<- (equal-implies-eq CMP N1=N2) %. +%term _ + %pi (member?-map/+-total N1 D1 M1 N2 greater CMP false (member?/out (not-member/< N1>N2))) + %<- (greater-implies-gt CMP N1>N2) %. +%term _ + %pi (member?-map/+-total N1 D1 M1 N2 less CMP B MD) + %<- (less-implies-lt CMP N2>N1) + %<- (gt-implies-plus N2>N1 N0 N0+1+N1=N2) + %<- (member?-total* M1 N0 B MD1) + %<- (member?-map/+-complete N1 D1 M1 N2 N0 N0+1+N1=N2 B MD1 MD) %. +%term _ member?-map/+-complete N1 D1 M1 N2 N0 N0+1+N1=N2 true (member?/in L1) (member?/in (lookup/> L1 N0+1+N1=N2)) %. +%term _ member?-map/+-complete N1 D1 M1 N2 N0 N0+1+N1=N2 false (member?/out F1) (member?/out (not-member/> F1 N0+1+N1=N2)) %. +%worlds () (member?-map/+-complete _ _ _ _ _ _ _ _ _) %. +%total {} (member?-map/+-complete _ _ _ _ _ _ _ _ _) %. +%worlds () (member?-total* _ _ _ _) (member?-map/+-total _ _ _ _ _ _ _ _) %. +%total (M M1) (member?-total* M _ _ _) (member?-map/+-total _ _ M1 _ _ _ _ _) %. +%inline member?-total member?-total* _ _ _ %. +%theorem in-implies-lookup : forall* {M} {N} forall {MD member? M N true} exists {D} {L lookup M N D} true %. +%term _ in-implies-lookup (member?/in L) _ L %. +%worlds () (in-implies-lookup _ _ _) %. +%total {} (in-implies-lookup _ _ _) %. +%theorem out-implies-not-member : forall* {M} {N} forall {MD member? M N false} exists {F not-member M N} true %. +%term _ out-implies-not-member (member?/out F) F %. +%worlds () (out-implies-not-member _ _) %. +%total {} (out-implies-not-member _ _) %. +%%% Theorems about disjoint +%theorem false-implies-disjoint : forall* {M1} {M2} forall {F void} exists {D disjoint M1 M2} true %. +%worlds () (false-implies-disjoint _ _) %. +%total {} (false-implies-disjoint _ _) %. +%theorem disjoint-respects-eq : forall* {M1} {M2} {M1P} {M2P} forall {A disjoint M1 M2} {E1 eq M1 M1P} {E2 eq M2 M2P} exists {AP disjoint M1P M2P} true %. +%term _ disjoint-respects-eq A eq/ eq/ A %. +%worlds () (disjoint-respects-eq _ _ _ _) %. +%total {} (disjoint-respects-eq _ _ _ _) %. +%reduces = A AP (disjoint-respects-eq A _ _ AP) %. +%theorem disjoint/=-contradiction : forall* {N1} {D1} {M1} {N2} {D2} {M2} forall {A disjoint (map/+ N1 D1 M1) (map/+ N2 D2 M2)} {G nat`eq N1 N2} exists {F void} true %. +%term _ + %pi (disjoint/=-contradiction (disjoint/< _ N0+1+N=N) nat`eq/ F) + %<- (nat`plus-implies-gt N0+1+N=N nat`eq/ N>N) + %<- (nat`gt-anti-reflexive N>N F) %. +%term _ + %pi (disjoint/=-contradiction (disjoint/> _ N3+1+N=N) nat`eq/ F) + %<- (nat`plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (nat`gt-anti-reflexive N>N F) %. +%worlds () (disjoint/=-contradiction _ _ _) %. +%total {} (disjoint/=-contradiction _ _ _) %. +%theorem disjoint/<-inversion : forall* {N1} {D1} {M1} {N2} {D2} {M2} {N0} forall {A disjoint (map/+ N1 D1 M1) (map/+ N2 D2 M2)} {P plus (s N0) N1 N2} exists {AP disjoint M1 (map/+ N0 D2 M2)} true %. +%term _ + %pi (disjoint/<-inversion (disjoint/< A P) P' A') + %<- (nat`plus-right-cancels P P' nat`eq/ nat`eq/ N0+1=N0'+1) + %<- (succ-cancels N0+1=N0'+1 N0=N0P) + %<- (map/+-preserves-eq N0=N0P nat`eq/ eq/ M022=M022') + %<- (disjoint-respects-eq A eq/ M022=M022' A') %. +%term _ + %pi (disjoint/<-inversion (disjoint/> A' N3+1+N2=N1) N0+1+N1=N2 A) + %<- (nat`plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) + %<- (nat`plus-implies-gt N0+1+N1=N2 nat`eq/ N2>N1) + %<- (nat`gt-anti-symmetric N1>N2 N2>N1 F) + %<- (false-implies-eq F M311=M1) + %<- (false-implies-eq F M2=M022) + %<- (disjoint-respects-eq A' M311=M1 M2=M022 A) %. +%worlds () (disjoint/<-inversion _ _ _) %. +%total {} (disjoint/<-inversion _ _ _) %. +%reduces < AP A (disjoint/<-inversion A _ AP) %. +%theorem disjoint/>-inversion : forall* {N1} {D1} {M1} {N2} {D2} {M2} {N3} forall {A disjoint (map/+ N1 D1 M1) (map/+ N2 D2 M2)} {P plus (s N3) N2 N1} exists {AP disjoint (map/+ N3 D1 M1) M2} true %. +%term _ + %pi (disjoint/>-inversion (disjoint/> A P) P' A') + %<- (nat`plus-right-cancels P P' nat`eq/ nat`eq/ N3+1=N3'+1) + %<- (succ-cancels N3+1=N3'+1 N3=N3P) + %<- (map/+-preserves-eq N3=N3P nat`eq/ eq/ M311=M311') + %<- (disjoint-respects-eq A M311=M311' eq/ A') %. +%term _ + %pi (disjoint/>-inversion (disjoint/< A' N0+1+N1=N2) N3+1+N2=N1 A) + %<- (nat`plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) + %<- (nat`plus-implies-gt N0+1+N1=N2 nat`eq/ N2>N1) + %<- (nat`gt-anti-symmetric N1>N2 N2>N1 F) + %<- (false-implies-eq F M1=M311) + %<- (false-implies-eq F M022=M2) + %<- (disjoint-respects-eq A' M1=M311 M022=M2 A) %. +%worlds () (disjoint/>-inversion _ _ _) %. +%total {} (disjoint/>-inversion _ _ _) %. +%reduces < AP A (disjoint/>-inversion A _ AP) %. +%theorem disjoint-anti-reflexive : forall* {M} forall {D disjoint M M} exists {E eq map/0 M} true %. +%term _ disjoint-anti-reflexive disjoint/L eq/ %. +%term _ disjoint-anti-reflexive disjoint/R eq/ %. +%term _ + %pi (disjoint-anti-reflexive (%the (disjoint (map/+ N D M) (map/+ N D M)) A) E) + %<- (disjoint/=-contradiction A nat`eq/ F) + %<- (false-implies-eq F E) %. +%worlds () (disjoint-anti-reflexive _ _) %. +%total {} (disjoint-anti-reflexive _ _) %. +%theorem disjoint-symmetric : forall* {M1} {M2} forall {D disjoint M1 M2} exists {D disjoint M2 M1} true %. +%term _ disjoint-symmetric disjoint/L disjoint/R %. +%term _ disjoint-symmetric disjoint/R disjoint/L %. +%term _ + %pi (disjoint-symmetric (disjoint/< D P) (disjoint/> D' P)) + %<- (disjoint-symmetric D D') %. +%term _ + %pi (disjoint-symmetric (disjoint/> D P) (disjoint/< D' P)) + %<- (disjoint-symmetric D D') %. +%worlds () (disjoint-symmetric _ _) %. +%total (D) (disjoint-symmetric D _) %. +%theorem disjoint-lookup-contradiction : forall* {M1} {M2} {N} {D1} {D2} forall {A disjoint M1 M2} {L1 lookup M1 N D1} {L2 lookup M2 N D2} exists {F void} true %. +%term _ + %pi (disjoint-lookup-contradiction disjoint/L L _ F) + %<- (lookup-contradiction L F) %. +%term _ + %pi (disjoint-lookup-contradiction disjoint/R _ L F) + %<- (lookup-contradiction L F) %. +%term _ + %pi (disjoint-lookup-contradiction (disjoint/< _ N0+1+N=N) (lookup/= nat`eq/) (lookup/= nat`eq/) F) + %<- (plus-implies-gt N0+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) %. +%term _ + %pi (disjoint-lookup-contradiction (disjoint/< _ N0+1+N1=N2) (lookup/= nat`eq/) (lookup/> _ N3+1+N2=N1) F) + %<- (plus-implies-gt N0+1+N1=N2 nat`eq/ N2>N1) + %<- (plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) + %<- (gt-anti-symmetric N2>N1 N1>N2 F) %. +%term _ + %pi (disjoint-lookup-contradiction (disjoint/< D N0+1+N1=N2) (lookup/> L1P N0P+1+N1=N2) (lookup/= nat`eq/) F) + %<- (plus-right-cancels N0P+1+N1=N2 N0+1+N1=N2 nat`eq/ nat`eq/ N0P+1=N0+1) + %<- (succ-cancels N0P+1=N0+1 N0P=N0) + %<- (lookup-respects-eq L1P eq/ N0P=N0 nat`eq/ L1) + %<- (disjoint-lookup-contradiction D L1 (lookup/= nat`eq/) F) %. +%term _ + %pi (disjoint-lookup-contradiction (disjoint/< D N0+1+N1=N2) (lookup/> L1 N1P+1+N1=N) (lookup/> L2 N2P+1+N2=N) F) + %<- (plus-swap-succ N0+1+N1=N2 N0+N1+1=N2) + %<- (plus-associative-converse N0+N1+1=N2 N2P+1+N2=N NX N2P+1+N0=NX NX+N1+1=N) + %<- (plus-swap-succ N1P+1+N1=N N1P+N1+1=N) + %<- (plus-right-cancels NX+N1+1=N N1P+N1+1=N nat`eq/ nat`eq/ NX=N1P) + %<- (plus-respects-eq N2P+1+N0=NX nat`eq/ nat`eq/ NX=N1P N2P+1+N0=N1P) + %<- (disjoint-lookup-contradiction D L1 (lookup/> L2 N2P+1+N0=N1P) F) %. +%term _ + %pi (disjoint-lookup-contradiction (disjoint/> _ N3+1+N=N) (lookup/= nat`eq/) (lookup/= nat`eq/) F) + %<- (plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) %. +%term _ + %pi (disjoint-lookup-contradiction (disjoint/> _ N3+1+N2=N1) (lookup/> _ N3+1+N1=N2) (lookup/= nat`eq/) F) + %<- (plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) + %<- (plus-implies-gt N3+1+N1=N2 nat`eq/ N2>N1) + %<- (gt-anti-symmetric N1>N2 N2>N1 F) %. +%term _ + %pi (disjoint-lookup-contradiction (disjoint/> D N3+1+N2=N1) (lookup/= nat`eq/) (lookup/> L2P N3P+1+N2=N1) F) + %<- (plus-right-cancels N3P+1+N2=N1 N3+1+N2=N1 nat`eq/ nat`eq/ N3P+1=N3+1) + %<- (succ-cancels N3P+1=N3+1 N3P=N3) + %<- (lookup-respects-eq L2P eq/ N3P=N3 nat`eq/ L2) + %<- (disjoint-lookup-contradiction D (lookup/= nat`eq/) L2 F) %. +%term _ + %pi (disjoint-lookup-contradiction (disjoint/> D N3+1+N2=N1) (lookup/> L1 N1P+1+N1=N) (lookup/> L2 N2P+1+N2=N) F) + %<- (plus-swap-succ N3+1+N2=N1 N3+N2+1=N1) + %<- (plus-associative-converse N3+N2+1=N1 N1P+1+N1=N NX N1P+1+N3=NX NX+N2+1=N) + %<- (plus-swap-succ N2P+1+N2=N N2P+N2+1=N) + %<- (plus-right-cancels NX+N2+1=N N2P+N2+1=N nat`eq/ nat`eq/ NX=N2P) + %<- (plus-respects-eq N1P+1+N3=NX nat`eq/ nat`eq/ NX=N2P N1P+1+N3=N2P) + %<- (disjoint-lookup-contradiction D (lookup/> L1 N1P+1+N3=N2P) L2 F) %. +%worlds () (disjoint-lookup-contradiction _ _ _ _) %. +%total (D) (disjoint-lookup-contradiction D _ _ _) %. +%theorem shift-left-preserves-disjoint : forall* {N} {D} {M1} {M2} {SM1} forall {A disjoint M1 M2} {S1 shift N M1 SM1} exists {SA disjoint SM1 (map/+ N D M2)} true %. +%term _ shift-left-preserves-disjoint _ shift/0 disjoint/L %. +%term _ + %pi (shift-left-preserves-disjoint M111*M2 (shift/+ N+1+N1=N1P) (disjoint/> M111*M2 N1+1+N=N1P)) + %<- (plus-swap-succ N+1+N1=N1P N+N1+1=N1P) + %<- (plus-commutative N+N1+1=N1P N1+1+N=N1P) %. +%worlds () (shift-left-preserves-disjoint _ _ _) %. +%total {} (shift-left-preserves-disjoint _ _ _) %. +%theorem shift-left-preserves-disjoint-converse : forall* {N} {D} {M1} {M2} {SM1} forall {SA disjoint SM1 (map/+ N D M2)} {S1 shift N M1 SM1} exists {A disjoint M1 M2} true %. +%term _ shift-left-preserves-disjoint-converse _ shift/0 disjoint/L %. +%term _ + %pi (shift-left-preserves-disjoint-converse M111*M222 (shift/+ N2+1+N3=N1) M311*M2) + %<- (plus-swap-succ N2+1+N3=N1 N2+N3+1=N1) + %<- (plus-commutative N2+N3+1=N1 N3+1+N2=N1) + %<- (disjoint/>-inversion M111*M222 N3+1+N2=N1 M311*M2) %. +%worlds () (shift-left-preserves-disjoint-converse _ _ _) %. +%total {} (shift-left-preserves-disjoint-converse _ _ _) %. +%theorem shift-right-preserves-disjoint : forall* {N} {D} {M1} {M2} {SM2} forall {A disjoint M1 M2} {S2 shift N M2 SM2} exists {SA disjoint (map/+ N D M1) SM2} true %. +%term _ shift-right-preserves-disjoint _ shift/0 disjoint/R %. +%term _ + %pi (shift-right-preserves-disjoint M1*M222 (shift/+ N+1+N2=N2P) (disjoint/< M1*M222 N2+1+N=N2P)) + %<- (plus-swap-succ N+1+N2=N2P N+N2+1=N2P) + %<- (plus-commutative N+N2+1=N2P N2+1+N=N2P) %. +%worlds () (shift-right-preserves-disjoint _ _ _) %. +%total {} (shift-right-preserves-disjoint _ _ _) %. +%theorem shift-right-preserves-disjoint-converse : forall* {N} {D} {M1} {M2} {SM2} forall {SA disjoint (map/+ N D M1) SM2} {S2 shift N M2 SM2} exists {A disjoint M1 M2} true %. +%term _ shift-right-preserves-disjoint-converse _ shift/0 disjoint/R %. +%term _ + %pi (shift-right-preserves-disjoint-converse M111*M322 (shift/+ N1+1+N2=N3) M1*M222) + %<- (plus-swap-succ N1+1+N2=N3 N1+N2+1=N3) + %<- (plus-commutative N1+N2+1=N3 N2+1+N1=N3) + %<- (disjoint/<-inversion M111*M322 N2+1+N1=N3 M1*M222) %. +%worlds () (shift-right-preserves-disjoint-converse _ _ _) %. +%total {} (shift-right-preserves-disjoint-converse _ _ _) %. +%theorem shift-preserves-disjoint : forall* {N} {M1} {M2} {SM1} {SM2} forall {A disjoint M1 M2} {S1 shift N M1 SM1} {S2 shift N M2 SM2} exists {SA disjoint SM1 SM2} true %. +%term _ shift-preserves-disjoint _ shift/0 _ disjoint/L %. +%term _ shift-preserves-disjoint _ _ shift/0 disjoint/R %. +%term _ + %pi (shift-preserves-disjoint (disjoint/< M1*M022 N0+1+N1=N2) (shift/+ N+1+N1=N4) (shift/+ N+1+N2=N5) (disjoint/< M1*M022 N0+1+N4=N5)) + %<- (plus-swap-succ N+1+N1=N4 N+N1+1=N4) + %<- (plus-commutative N+N1+1=N4 N1+1+N=N4) + %<- (plus-commutative N0+1+N1=N2 N1+N0+1=N2) + %<- (plus-associative-converse* N1+N0+1=N2 N+1+N2=N5 N+1+N1=N4 N4+N0+1=N5) + %<- (plus-commutative N4+N0+1=N5 N0+1+N4=N5) %. +%term _ + %pi (shift-preserves-disjoint (disjoint/> M311*M2 N3+1+N2=N1) (shift/+ N+1+N1=N4) (shift/+ N+1+N2=N5) (disjoint/> M311*M2 N3+1+N5=N4)) + %<- (plus-swap-succ N+1+N2=N5 N+N2+1=N5) + %<- (plus-commutative N+N2+1=N5 N2+1+N=N5) + %<- (plus-commutative N3+1+N2=N1 N2+N3+1=N1) + %<- (plus-associative-converse* N2+N3+1=N1 N+1+N1=N4 N+1+N2=N5 N5+N3+1=N4) + %<- (plus-commutative N5+N3+1=N4 N3+1+N5=N4) %. +%worlds () (shift-preserves-disjoint _ _ _ _) %. +%total {} (shift-preserves-disjoint _ _ _ _) %. +%theorem shift-preserves-disjoint-converse : forall* {N} {M1} {M2} {SM1} {SM2} forall {SA disjoint SM1 SM2} {S1 shift N M1 SM1} {S2 shift N M2 SM2} exists {A disjoint M1 M2} true %. +%term _ shift-preserves-disjoint-converse _ shift/0 _ disjoint/L %. +%term _ shift-preserves-disjoint-converse _ _ shift/0 disjoint/R %. +%term _ + %pi (shift-preserves-disjoint-converse (disjoint/< M1*M055 N0+1+N4=N5) (shift/+ N+1+N1=N4) (shift/+ N+1+N2=N5) (disjoint/< M1*M055 N0+1+N1=N2)) + %<- (plus-commutative N+1+N1=N4 N1+N+1=N4) + %<- (plus-swap-succ-converse N1+N+1=N4 N1+1+N=N4) + %<- (plus-associative-converse N1+N+1=N4 N0+1+N4=N5 N2P N0+1+N1=N2P N2P+N+1=N5) + %<- (plus-commutative N+1+N2=N5 N2+N+1=N5) + %<- (plus-right-cancels N2P+N+1=N5 N2+N+1=N5 nat`eq/ nat`eq/ N2P=N2) + %<- (plus-respects-eq N0+1+N1=N2P nat`eq/ nat`eq/ N2P=N2 N0+1+N1=N2) %. +%term _ + %pi (shift-preserves-disjoint-converse (disjoint/> M611*M2 N6+1+N5=N4) (shift/+ N+1+N1=N4) (shift/+ N+1+N2=N5) (disjoint/> M611*M2 N6+1+N2=N1)) + %<- (plus-commutative N+1+N2=N5 N2+N+1=N5) + %<- (plus-swap-succ-converse N2+N+1=N5 N2+1+N=N5) + %<- (plus-associative-converse N2+N+1=N5 N6+1+N5=N4 N1P N6+1+N2=N1P N1P+N+1=N4) + %<- (plus-commutative N+1+N1=N4 N1+N+1=N4) + %<- (plus-right-cancels N1P+N+1=N4 N1+N+1=N4 nat`eq/ nat`eq/ N1P=N1) + %<- (plus-respects-eq N6+1+N2=N1P nat`eq/ nat`eq/ N1P=N1 N6+1+N2=N1) %. +%worlds () (shift-preserves-disjoint-converse _ _ _ _) %. +%total {} (shift-preserves-disjoint-converse _ _ _ _) %. +%theorem ne-implies-disjoint : forall* {N1} {D1} {N2} {D2} forall {NE nat`ne N1 N2} exists {D disjoint (map/+ N1 D1 map/0) (map/+ N2 D2 map/0)} true %. +%term _ + %pi (ne-implies-disjoint (nat`ne/< N1 N1>N2) (disjoint/> disjoint/R N3+1+N2=N1)) + %<- (gt-implies-plus N1>N2 _ N3+1+N2=N1) %. +%worlds () (ne-implies-disjoint _ _) %. +%total {} (ne-implies-disjoint _ _) %. +%%% Theorems about size +%theorem false-implies-size : forall* {M} {N} forall {F void} exists {SZ size M N} true %. +%worlds () (false-implies-size _ _) %. +%total {} (false-implies-size _ _) %. +%theorem size-total* : forall {M} exists {N} {MX size M N} true %. +%term _ size-total* map/0 _ size/0 %. +%term _ %pi (size-total* _ _ (size/+ SZ)) %<- (size-total* _ _ SZ) %. +%worlds () (size-total* _ _ _) %. +%total (M) (size-total* M _ _) %. +%inline size-total size-total* _ _ %. +%theorem size-deterministic : forall* {M1} {M2} {N1} {N2} forall {SZ1 size M1 N1} {SZ2 size M2 N2} {EM eq M1 M2} exists {EN nat`eq N1 N2} true %. +%term _ size-deterministic size/0 size/0 eq/ nat`eq/ %. +%term _ + %pi (size-deterministic (size/+ N1=|M1|) (size/+ N2=|M2|) eq/ N1+1=N2+1) + %<- (size-deterministic N1=|M1| N2=|M2| eq/ N1=N2) + %<- (succ-deterministic N1=N2 N1+1=N2+1) %. +%worlds () (size-deterministic _ _ _ _) %. +%total (S) (size-deterministic S _ _ _) %. +%%% Theorems about bound +%theorem false-implies-bound : forall* {M} {N} forall {F void} exists {MX bound M N} true %. +%worlds () (false-implies-bound _ _) %. +%total {} (false-implies-bound _ _) %. +%theorem bound-total* : forall {M} exists {N} {MX bound M N} true %. +%term _ bound-total* map/0 _ bound/0 %. +%term _ %pi (bound-total* _ _ (bound/+ P MX)) %<- (bound-total* _ _ MX) %<- (plus-total P) %. +%worlds () (bound-total* _ _ _) %. +%total (M) (bound-total* M _ _) %. +%inline bound-total bound-total* _ _ %. +%theorem ge-bound-implies-not-member : forall* {M} {X} {N} forall {B bound M X} {G nat`ge N X} exists {F not-member M N} true %. +%term _ ge-bound-implies-not-member bound/0 _ not-member/0 %. +%term _ + %pi (ge-bound-implies-not-member (bound/+ M1+1+X1=X B) N>=X (not-member/> F1 N1+1+M1=N)) + %<- (nat`ge-implies-plus N>=X Y1 Y1+X=N) + %<- (nat`plus-commutative M1+1+X1=X X1+M1+1=X) + %<- (nat`plus-associative-converse X1+M1+1=X Y1+X=N N1 Y1+X1=N1 N1+M1+1=N) + %<- (plus-swap-succ-converse N1+M1+1=N N1+1+M1=N) + %<- (plus-implies-ge Y1+X1=N1 N1>=X1) + %<- (ge-bound-implies-not-member B N1>=X1 F1) %. +%worlds () (ge-bound-implies-not-member _ _ _) %. +%total (B) (ge-bound-implies-not-member B _ _) %. +%%% Theorems about shift +%theorem false-implies-shift : forall* {M} {N} {M'} forall {F void} exists {S shift N M M'} true %. +%worlds () (false-implies-shift _ _) %. +%total {} (false-implies-shift _ _) %. +%theorem shift-respects-eq : forall* {N} {M1} {M2} {N'} {M1'} {M2'} forall {S shift N M1 M2} {EN nat`eq N N'} {E1 eq M1 M1'} {E2 eq M2 M2'} exists {S' shift N' M1' M2'} true %. +%term _ shift-respects-eq S nat`eq/ eq/ eq/ S %. +%worlds () (shift-respects-eq _ _ _ _ _) %. +%total {} (shift-respects-eq _ _ _ _ _) %. +%theorem shift-total* : forall {N} {M1} exists {M2} {S shift N M1 M2} true %. +%term _ shift-total* N map/0 map/0 shift/0 %. +%term _ + %pi (shift-total* N1 (map/+ N2 D M) (map/+ N3 D M) (shift/+ N1+1+N2=N3)) + %<- (plus-total N1+1+N2=N3) %. +%worlds () (shift-total* _ _ _ _) %. +%total {} (shift-total* _ _ _ _) %. +%inline shift-total shift-total* _ _ _ %. +%theorem shift-deterministic : forall* {N} {M1} {M2} {N'} {M1'} {M2'} forall {S shift N M1 M2} {S' shift N' M1' M2'} {EN nat`eq N N'} {EM1 eq M1 M1'} exists {EM2 eq M2 M2'} true %. +%term _ shift-deterministic shift/0 shift/0 nat`eq/ eq/ eq/ %. +%term _ + %pi (shift-deterministic (shift/+ N1+1+N2=N3) (shift/+ N1+1+N2=N3') nat`eq/ eq/ E) + %<- (plus-deterministic N1+1+N2=N3 N1+1+N2=N3' nat`eq/ nat`eq/ N3=N3P) + %<- (map/+-preserves-eq N3=N3P nat`eq/ eq/ E) %. +%worlds () (shift-deterministic _ _ _ _ _) %. +%total {} (shift-deterministic _ _ _ _ _) %. +%theorem shifts-add : forall* {N1} {N2} {N3} {M0} {M1} {M3} forall {S1 shift N1 M0 M1} {S2 shift N2 M1 M3} {P plus (s N1) N2 N3} exists {S3 shift N3 M0 M3} true %. +%term _ shifts-add shift/0 shift/0 _ shift/0 %. +%term _ + %pi (shifts-add (shift/+ N1+1+N4=N5) (shift/+ N2+1+N5=N7) N1+1+N2=N3 (shift/+ N3+1+N4=N7)) + %<- (plus-total N3+1+N4=N7') + %<- (plus-swap-succ N3+1+N4=N7' N3+N4+1=N7') + %<- (plus-swap-succ N1+1+N2=N3 N1+N2+1=N3) + %<- (plus-swap-succ N1+1+N4=N5 N1+N4+1=N5) + %<- (plus-commutative N1+N2+1=N3 N2+1+N1=N3) + %<- (plus-associative* N2+1+N1=N3 N3+N4+1=N7' N1+N4+1=N5 N2+1+N5=N7') + %<- (plus-deterministic N2+1+N5=N7' N2+1+N5=N7 nat`eq/ nat`eq/ N7'=N7) + %<- (plus-respects-eq N3+1+N4=N7' nat`eq/ nat`eq/ N7'=N7 N3+1+N4=N7) %. +%worlds () (shifts-add _ _ _ _) %. +%total {} (shifts-add _ _ _ _) %. +%theorem shifts-add-converse : forall* {N1} {N2} {N3} {M0} {M3} forall {S3 shift N3 M0 M3} {P plus (s N1) N2 N3} exists {M1} {S1 shift N1 M0 M1} {S2 shift N2 M1 M3} true %. +%term _ + %pi (shifts-add-converse S3 P M1 S1 S2) + %<- (shift-total S1) + %<- (shift-total S2') + %<- (shifts-add S1 S2' P S3') + %<- (shift-deterministic S3' S3 nat`eq/ eq/ M3'=M3) + %<- (shift-respects-eq S2' nat`eq/ eq/ M3'=M3 S2) %. +%worlds () (shifts-add-converse _ _ _ _ _) %. +%total {} (shifts-add-converse _ _ _ _ _) %. +%theorem shift-preserves-not-member-converse* : forall* {M1} {N1} {N2} {M2} {N0} forall {L2 not-member M2 N2} {S shift N0 M1 M2} {P plus (s N0) N1 N2} exists {L1 not-member M1 N1} true %. +%term _ shift-preserves-not-member-converse* not-member/0 shift/0 _ not-member/0 %. +%term _ + %pi (shift-preserves-not-member-converse* (not-member/< N1>N) (shift/+ S+1+N2=N1) S+1+N'=N (not-member/< N2>N')) + %<- (plus-left-cancels-gt S+1+N2=N1 S+1+N'=N nat`eq/ N1>N N2>N') %. +%term _ + %pi (shift-preserves-not-member-converse* (not-member/> F NX+1+N2=N) (shift/+ S+1+N1=N2) S+1+N'=N (not-member/> F NX+1+N1=N')) + %<- (plus-commutative S+1+N1=N2 N1+S+1=N2) + %<- (plus-associative-converse N1+S+1=N2 NX+1+N2=N N'' NX+1+N1=N'' N''+S+1=N) + %<- (plus-commutative N''+S+1=N S+1+N''=N) + %<- (plus-left-cancels S+1+N''=N S+1+N'=N nat`eq/ nat`eq/ N''=N') + %<- (plus-respects-eq NX+1+N1=N'' nat`eq/ nat`eq/ N''=N' NX+1+N1=N') %. +%worlds () (shift-preserves-not-member-converse* _ _ _ _) %. +%total {} (shift-preserves-not-member-converse* _ _ _ _) %. +%theorem shift-preserves-lookup : forall* {M1} {N1} {D} {N0} {M2} forall {L1 lookup M1 N1 D} {S shift N0 M1 M2} exists {N2} {P plus (s N0) N1 N2} {L2 lookup M2 N2 D} true %. +%term _ shift-preserves-lookup (lookup/= nat`eq/) (shift/+ N0+1+N1=N2) _ N0+1+N1=N2 (lookup/= nat`eq/) %. +%term _ + %pi (shift-preserves-lookup (lookup/> L N3+1+N1=N4) (shift/+ N0+1+N1=N2) _ N0+1+N4=N5 (lookup/> L N3+1+N2=N5)) + %<- (plus-total N0+1+N4=N5) + %<- (plus-commutative N3+1+N1=N4 N1+N3+1=N4) + %<- (plus-associative-converse* N1+N3+1=N4 N0+1+N4=N5 N0+1+N1=N2 N2+N3+1=N5) + %<- (plus-commutative N2+N3+1=N5 N3+1+N2=N5) %. +%worlds () (shift-preserves-lookup _ _ _ _ _) %. +%total {} (shift-preserves-lookup _ _ _ _ _) %. +%theorem shift-preserves-lookup* : forall* {M1} {N1} {D} {N0} {M2} {N2} forall {L1 lookup M1 N1 D} {S shift N0 M1 M2} {P plus (s N0) N1 N2} exists {L2 lookup M2 N2 D} true %. +%term _ + %pi (shift-preserves-lookup* L1 S P L2) + %<- (shift-preserves-lookup L1 S _ P' L2') + %<- (plus-deterministic P' P nat`eq/ nat`eq/ N2'=N2) + %<- (lookup-respects-eq L2' eq/ N2'=N2 nat`eq/ L2) %. +%worlds () (shift-preserves-lookup* _ _ _ _) %. +%total {} (shift-preserves-lookup* _ _ _ _) %. +%theorem shift-preserves-lookup-converse : forall* {M1} {N0} {D} {N2} {M2} forall {L2 lookup M2 N2 D} {S shift N0 M1 M2} exists {N1} {P plus (s N0) N1 N2} {L1 lookup M1 N1 D} true %. +%term _ shift-preserves-lookup-converse (lookup/= nat`eq/) (shift/+ N0+1+N1=N2) _ N0+1+N1=N2 (lookup/= nat`eq/) %. +%term _ + %pi (shift-preserves-lookup-converse (lookup/> L N3+1+N2=N5) (shift/+ N0+1+N1=N2) _ N0+1+N4=N5 (lookup/> L N3+1+N1=N4)) + %<- (plus-commutative N0+1+N1=N2 N1+N0+1=N2) + %<- (plus-associative-converse N1+N0+1=N2 N3+1+N2=N5 N4 N3+1+N1=N4 N4+N0+1=N5) + %<- (plus-commutative N4+N0+1=N5 N0+1+N4=N5) %. +%worlds () (shift-preserves-lookup-converse _ _ _ _ _) %. +%total {} (shift-preserves-lookup-converse _ _ _ _ _) %. +%theorem shift-preserves-lookup-converse* : forall* {M1} {N1} {D} {N2} {M2} {N0} forall {L2 lookup M2 N2 D} {S shift N0 M1 M2} {P plus (s N0) N1 N2} exists {L1 lookup M1 N1 D} true %. +%term _ + %pi (shift-preserves-lookup-converse* L2 S P L1) + %<- (shift-preserves-lookup-converse L2 S _ P' L1') + %<- (plus-left-cancels P' P nat`eq/ nat`eq/ N1'=N1) + %<- (lookup-respects-eq L1' eq/ N1'=N1 nat`eq/ L1) %. +%worlds () (shift-preserves-lookup-converse* _ _ _ _) %. +%total {} (shift-preserves-lookup-converse* _ _ _ _) %. +%theorem shift-preserves-size : forall* {M} {N1} {N2} {S2M} forall {SZ size M N1} {SH shift N2 M S2M} exists {SHSZ size S2M N1} true %. +%term _ shift-preserves-size size/0 shift/0 size/0 %. +%term _ shift-preserves-size (size/+ SZ) (shift/+ _) (size/+ SZ) %. +%worlds () (shift-preserves-size _ _ _) %. +%total {} (shift-preserves-size _ _ _) %. +%%% Theorems about disjoint? +%theorem disjoint?-total* : forall {M1} {M2} exists {B} {D disjoint? M1 M2 B} true %. +%term _ disjoint?-total* _ _ _ (disjoint?/yes disjoint/L) %. +%term _ disjoint?-total* _ _ _ (disjoint?/yes disjoint/R) %. +%theorem disjoint?-total*/+ : forall* {N1} {D1} {M1} {N2} {D2} {M2} {C} forall {S1} {S2} {SZ1 size M1 S1} {SZ2 size M2 S2} {CMP nat`compare N1 N2 C} exists {B} {D disjoint? (map/+ N1 D1 M1) (map/+ N2 D2 M2) B} true %. +%theorem disjoint?-total*/< : forall* {N1} {D1} {M1} {N2} {D2} {M2} {N0} {B1} forall {P plus (s N0) N1 N2} {D?1 disjoint? M1 (map/+ N0 D2 M2) B1} exists {B} {D disjoint? (map/+ N1 D1 M1) (map/+ N2 D2 M2) B} true %. +%theorem disjoint?-total*/> : forall* {N1} {D1} {M1} {N2} {D2} {M2} {N3} {B1} forall {P plus (s N3) N2 N1} {D?1 disjoint? (map/+ N3 D1 M1) M2 B1} exists {B} {D disjoint? (map/+ N1 D1 M1) (map/+ N2 D2 M2) B} true %. +%term _ + %pi (disjoint?-total* _ _ _ D?) + %<- (size-total SZ1) + %<- (size-total SZ2) + %<- (nat`compare-total CMP) + %<- (disjoint?-total*/+ _ _ SZ1 SZ2 CMP _ D?) %. +%term _ disjoint?-total*/+ _ _ _ _ nat`compare/= _ (disjoint?/no (lookup/= nat`eq/) (lookup/= nat`eq/)) %. +%term _ + %pi (disjoint?-total*/+ _ _ _ _ (nat`compare/< N2>N1) _ D?) + %<- (gt-implies-plus N2>N1 _ N0+1+N1=N2) + %<- (disjoint?-total*/< N0+1+N1=N2 (disjoint?/yes disjoint/L) _ D?) %. +%term _ + %pi (disjoint?-total*/+ _ _ (size/+ SZ1) SZ2 (nat`compare/< N2>N1) _ D?) + %<- (gt-implies-plus N2>N1 _ N0+1+N1=N2) + %<- (nat`compare-total CMP) + %<- (disjoint?-total*/+ _ _ SZ1 SZ2 CMP _ D?1) + %<- (disjoint?-total*/< N0+1+N1=N2 D?1 _ D?) %. +%term _ disjoint?-total*/< N0+1+N1=N2 (disjoint?/yes M1*M022) _ (disjoint?/yes (disjoint/< M1*M022 N0+1+N1=N2)) %. +%term _ + %pi (disjoint?-total*/< N0+1+N1=N2 (disjoint?/no M1^N3=D1 M022^N3=D2) _ (disjoint?/no (lookup/> M1^N3=D1 N3+1+N1=N4) M222^N4=D2)) + %<- (plus-total N3+1+N1=N4) + %<- (plus-swap-succ N3+1+N1=N4 N3+N1+1=N4) + %<- (plus-commutative N3+N1+1=N4 N1+1+N3=N4) + %<- (plus-swap-succ N0+1+N1=N2 N0+N1+1=N2) + %<- (plus-commutative N0+N1+1=N2 N1+1+N0=N2) + %<- (shift-preserves-lookup* M022^N3=D2 (shift/+ N1+1+N0=N2) N1+1+N3=N4 M222^N4=D2) %. +%worlds () (disjoint?-total*/< _ _ _ _) %. +%total {} (disjoint?-total*/< _ _ _ _) %. +%term _ + %pi (disjoint?-total*/+ _ _ _ _ (nat`compare/> N1>N2) _ D?) + %<- (gt-implies-plus N1>N2 _ N3+1+N2=N1) + %<- (disjoint?-total*/> N3+1+N2=N1 (disjoint?/yes disjoint/R) _ D?) %. +%term _ + %pi (disjoint?-total*/+ _ _ SZ1 (size/+ SZ2) (nat`compare/> N1>N2) _ D?) + %<- (gt-implies-plus N1>N2 _ N3+1+N2=N1) + %<- (nat`compare-total CMP) + %<- (disjoint?-total*/+ _ _ SZ1 SZ2 CMP _ D?1) + %<- (disjoint?-total*/> N3+1+N2=N1 D?1 _ D?) %. +%term _ disjoint?-total*/> P (disjoint?/yes D) _ (disjoint?/yes (disjoint/> D P)) %. +%term _ + %pi (disjoint?-total*/> N3+1+N2=N1 (disjoint?/no M311^N4=D1 M2^N4=D2) _ (disjoint?/no M111^N5=D1 (lookup/> M2^N4=D2 N4+1+N2=N5))) + %<- (plus-total N4+1+N2=N5) + %<- (plus-swap-succ N4+1+N2=N5 N4+N2+1=N5) + %<- (plus-commutative N4+N2+1=N5 N2+1+N4=N5) + %<- (plus-swap-succ N3+1+N2=N1 N3+N2+1=N1) + %<- (plus-commutative N3+N2+1=N1 N2+1+N3=N1) + %<- (shift-preserves-lookup* M311^N4=D1 (shift/+ N2+1+N3=N1) N2+1+N4=N5 M111^N5=D1) %. +%worlds () (disjoint?-total*/> _ _ _ _) %. +%total {} (disjoint?-total*/> _ _ _ _) %. +%worlds () (disjoint?-total*/+ _ _ _ _ _ _ _) %. +%total [S1 S2] (disjoint?-total*/+ S1 S2 _ _ _ _ _) %. +%worlds () (disjoint?-total* _ _ _ _) %. +%total {} (disjoint?-total* _ _ _ _) %. +%inline disjoint?-total disjoint?-total* _ _ _ %. +%%% Theorems about update +%theorem false-implies-update : forall* {M} {N} {D} {M'} forall {F void} exists {U update M N D M'} true %. +%worlds () (false-implies-update _ M^N=D->M') %. +%total {} (false-implies-update _ _) %. +%theorem update-respects-eq : forall* {M1} {N} {D} {M2} {M1P} {NP} {DP} {M2P} forall {U update M1 N D M2} {EM1 eq M1 M1P} {EN nat`eq N NP} {ED nat`eq D DP} {EM2 eq M2 M2P} exists {UP update M1P NP DP M2P} true %. +%term _ update-respects-eq U eq/ nat`eq/ nat`eq/ eq/ U %. +%worlds () (update-respects-eq M1^N=D->M2 M1=M1' N=N' D=D' M2=M2' M1'^N'=D'->M2') %. +%total {} (update-respects-eq _ _ _ _ _ _) %. +%reduces = U U' (update-respects-eq U _ _ _ _ U') %. +%%% technical lemmas to help prove reduction arguments +%sort update-eq {M1} {N1} {D1} {M1'} {M2} {N2} {D2} {M2'} %. +%term update-eq/ update-eq M1 N1 D1 M1' M1 N1 D1 M1' U U %. +%theorem false-implies-update-eq : forall* {M1} {N1} {D1} {M1'} {M2} {N2} {D2} {M2'} {U} {U'} forall {F void} exists {UE update-eq M1 N1 D1 M1' M2 N2 D2 M2' U U'} true %. +%worlds () (false-implies-update-eq _ _) %. +%total {} (false-implies-update-eq _ _) %. +%theorem meta-update-eq : forall* {M1} {N1} {D1} {M1'} {M2} {N2} {D2} {M2'} forall {U} {U'} {UE update-eq M1 N1 D1 M1' M2 N2 D2 M2' U U'} true %. +%term _ meta-update-eq U U update-eq/ %. +%worlds () (meta-update-eq _ _ _) %. +%total {} (meta-update-eq _ _ _) %. +%reduces = U U' (meta-update-eq U U' _) %. +%%% inversion lemmas +%theorem update/=-inversion : forall* {N1} {D1} {M1} {N2} {D2} {M2} forall {U update (map/+ N1 D1 M1) N2 D2 M2} {E nat`eq N1 N2} exists {EM eq (map/+ N2 D2 M1) M2} true %. +%term _ update/=-inversion (update/= nat`eq/) nat`eq/ eq/ %. +%term _ + %pi (update/=-inversion (update/< N3+1+N=N) nat`eq/ E) + %<- (nat`plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (nat`gt-anti-reflexive N>N F) + %<- (false-implies-eq F E) %. +%term _ + %pi (update/=-inversion (update/> U1022 N3+1+N=N) nat`eq/ E) + %<- (nat`plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (nat`gt-anti-reflexive N>N F) + %<- (false-implies-eq F E) %. +%worlds () (update/=-inversion _ _ _) %. +%total {} (update/=-inversion _ _ _) %. +%theorem update/<-inversion : forall* {N1} {D1} {M1} {N2} {D2} {M2} {N3} forall {U update (map/+ N1 D1 M1) N2 D2 M2} {P plus (s N3) N2 N1} exists {E eq (map/+ N2 D2 (map/+ N3 D1 M1)) M2} true %. +%term _ + %pi (update/<-inversion (update/= nat`eq/) N3+1+N=N E) + %<- (nat`plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (nat`gt-anti-reflexive N>N F) + %<- (false-implies-eq F E) %. +%term _ + %pi (update/<-inversion (update/< N3+1+N2=N1) N3P+1+N2=N1 E) + %<- (nat`plus-right-cancels N3P+1+N2=N1 N3+1+N2=N1 nat`eq/ nat`eq/ N3P+1=N3+1) + %<- (nat`succ-cancels N3P+1=N3+1 N3P=N3) + %<- (map/+-preserves-eq N3P=N3 nat`eq/ eq/ M311P=M311) + %<- (map/+-preserves-eq nat`eq/ nat`eq/ M311P=M311 E) %. +%term _ + %pi (update/<-inversion (update/> _ N0+1+N1=N2) N3+1+N2=N1 E) + %<- (nat`plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) + %<- (nat`plus-implies-gt N0+1+N1=N2 nat`eq/ N2>N1) + %<- (nat`gt-anti-symmetric N1>N2 N2>N1 F) + %<- (false-implies-eq F E) %. +%worlds () (update/<-inversion _ _ _) %. +%total {} (update/<-inversion _ _ _) %. +%theorem update/>-inversion : forall* {N1} {D1} {M1} {N2} {D2} {M} {N0} forall {U update (map/+ N1 D1 M1) N2 D2 M} {P plus (s N0) N1 N2} exists {M2} {UP update M1 N0 D2 M2} {E eq (map/+ N1 D1 M2) M} true %. +% a little more complex than might be expected +% because we want to prove reduction +%term _ + %pi (update/>-inversion (%the (update (map/+ N D1 M1) N D2 (map/+ N D2 M1)) (update/= nat`eq/)) N0+1+N=N M1 U' E) + %<- (plus-implies-gt N0+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-eq F E) + %<- (false-implies-update F U') + %<- (false-implies-update-eq F (%the (update-eq (map/+ N D1 M1) N D2 (map/+ N D2 M1) (map/+ N D1 M1) N D2 (map/+ N D1 M1) _ _) UE)) + %<- (meta-update-eq (update/= nat`eq/) (update/> U' N0+1+N=N) UE) %. +%term _ + %pi (update/>-inversion (%the (update (map/+ N1 D1 M1) _ _ _) (update/< N3+1+N2=N1)) N0+1+N1=N2 M1 U' E) + %<- (nat`plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) + %<- (nat`plus-implies-gt N0+1+N1=N2 nat`eq/ N2>N1) + %<- (nat`gt-anti-symmetric N1>N2 N2>N1 F) + %<- (false-implies-update F U') + %<- (false-implies-eq F E) + %<- (false-implies-update-eq F (%the (update-eq (map/+ N1 D1 M1) N2 D2 (map/+ N2 D2 (map/+ N3 D1 M1)) (map/+ N1 D1 M1) N2 D2 (map/+ N1 D1 M1) _ _) UE)) + %<- (meta-update-eq (update/< N3+1+N2=N1) (update/> U' N0+1+N1=N2) UE) %. +%term _ + %pi (update/>-inversion (update/> U N0+1+N1=N2) N0P+1+N1=N2 _ UP eq/) + %<- (nat`plus-right-cancels N0+1+N1=N2 N0P+1+N1=N2 nat`eq/ nat`eq/ N0+1=N0P+1) + %<- (nat`succ-cancels N0+1=N0P+1 N0=N0P) + %<- (update-respects-eq U eq/ N0=N0P nat`eq/ eq/ UP) %. +%worlds () (update/>-inversion _ _ _ _ _) %. +%total {} (update/>-inversion _ _ _ _ _) %. +%reduces < U' U (update/>-inversion U _ _ U' _) %. +%theorem update-deterministic : forall* {M1} {N1} {D1} {M1'} {M2} {N2} {D2} {M2'} forall {U1 update M1 N1 D1 M1'} {U2 update M2 N2 D2 M2'} {EM eq M1 M2} {EN nat`eq N1 N2} {ED nat`eq D1 D2} exists {EM' eq M1' M2'} true %. +%term _ update-deterministic update/0 update/0 eq/ nat`eq/ nat`eq/ eq/ %. +%term _ update-deterministic (update/= nat`eq/) (update/= nat`eq/) eq/ nat`eq/ nat`eq/ eq/ %. +%term _ + %pi (update-deterministic (update/< N3+1+N2=N1) (update/< N3'+1+N2=N1) eq/ nat`eq/ nat`eq/ M1'=M2') + %<- (plus-right-cancels N3+1+N2=N1 N3'+1+N2=N1 nat`eq/ nat`eq/ SN3=SN3') + %<- (succ-cancels SN3=SN3' N3E) + %<- (map/+-preserves-eq N3E nat`eq/ eq/ MM1=MM2) + %<- (map/+-preserves-eq nat`eq/ nat`eq/ MM1=MM2 M1'=M2') %. +%term _ + %pi (update-deterministic (update/> F1^N0=D2->F2 N0+1+N1=N2) (update/> F1^N0'=D2->F2' N0'+1+N1=N2) eq/ nat`eq/ nat`eq/ M1'=M2') + %<- (plus-right-cancels N0+1+N1=N2 N0'+1+N1=N2 nat`eq/ nat`eq/ N0+1=N0'+1) + %<- (succ-cancels N0+1=N0'+1 N0=N0') + %<- (update-deterministic F1^N0=D2->F2 F1^N0'=D2->F2' eq/ N0=N0' nat`eq/ F2=F2') + %<- (map/+-preserves-eq nat`eq/ nat`eq/ F2=F2' M1'=M2') %. +%% contradiction cases: +%term _ + %pi (update-deterministic (update/= nat`eq/) (update/< N3+1+N=N) eq/ nat`eq/ nat`eq/ E) + %<- (plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-eq F E) %. +%term _ + %pi (update-deterministic (update/= nat`eq/) (update/> _ N0+1+N=N) eq/ nat`eq/ nat`eq/ E) + %<- (plus-implies-gt N0+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-eq F E) %. +%term _ + %pi (update-deterministic (update/< N3+1+N=N) (update/= nat`eq/) eq/ nat`eq/ nat`eq/ E) + %<- (plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-eq F E) %. +%term _ + %pi (update-deterministic (update/< N3+1+N2=N1) (update/> _ N0+1+N1=N2) eq/ nat`eq/ nat`eq/ E) + %<- (nat`plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) + %<- (nat`plus-implies-gt N0+1+N1=N2 nat`eq/ N2>N1) + %<- (nat`gt-anti-symmetric N1>N2 N2>N1 F) + %<- (false-implies-eq F E) %. +%term _ + %pi (update-deterministic (update/> _ N0+1+N=N) (update/= nat`eq/) eq/ nat`eq/ nat`eq/ E) + %<- (plus-implies-gt N0+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-eq F E) %. +%term _ + %pi (update-deterministic (update/> _ N0+1+N1=N2) (update/< N3+1+N2=N1) eq/ nat`eq/ nat`eq/ E) + %<- (nat`plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) + %<- (nat`plus-implies-gt N0+1+N1=N2 nat`eq/ N2>N1) + %<- (nat`gt-anti-symmetric N1>N2 N2>N1 F) + %<- (false-implies-eq F E) %. +%worlds () (update-deterministic M1^N1=D1->M1' M2^N2=D2->M2' M1=M2 N1=N2 D1=D2 M1'=M2') %. +%total (U) (update-deterministic U _ _ _ _ _) %. +%theorem update-total* : forall {M} {N} {D} exists {M'} {U update M N D M'} true %. +%% we need a mutually recursive lemma +%theorem update-map/+-total : forall {N1} {D1} {M1} {N2} {D2} {C} {CMP nat`compare N1 N2 C} exists {M2} {U update (map/+ N1 D1 M1) N2 D2 M2} true %. +%term _ update-total* map/0 N D (map/+ N D map/0) update/0 %. +%term _ + %pi (update-total* (map/+ N1 D1 M1) N2 D2 M2 U) + %<- (nat`compare-total* N1 N2 C CMP) + %<- (update-map/+-total N1 D1 M1 N2 D2 C CMP M2 U) %. +%term _ + %pi (update-map/+-total N1 D1 M1 N2 D2 equal CMP (map/+ N2 D2 M1) (update/= N1=N2)) + %<- (equal-implies-eq CMP N1=N2) %. +%term _ + %pi (update-map/+-total N1 D1 M1 N2 D2 less CMP (map/+ N1 D1 M1') (update/> U1 N0+1+N1=N2)) + %<- (less-implies-lt CMP N2>N1) + %<- (gt-implies-plus N2>N1 N0 N0+1+N1=N2) + %<- (update-total* M1 N0 D2 M1' U1) %. +%term _ + %pi (update-map/+-total N1 D1 M1 N2 D2 greater CMP (map/+ N2 D2 (map/+ N3 D1 M1)) (update/< N3+1+N2=N1)) + %<- (greater-implies-gt CMP N1>N2) + %<- (gt-implies-plus N1>N2 N3 N3+1+N2=N1) %. +%worlds () (update-total* M N D M' M^N=D->M') (update-map/+-total _ _ _ _ _ _ _ _ _) %. +%total (M1 M2) (update-total* M1 _ _ _ _) (update-map/+-total _ _ M2 _ _ _ _ _ _) %. +%inline update-total update-total* _ _ _ _ %. +%theorem lookup-implies-update : forall* {F} {N} {D} forall {L lookup F N D} exists {U update F N D F} true %. +%term _ lookup-implies-update (lookup/= nat`eq/) (update/= nat`eq/) %. +%term _ + %pi (lookup-implies-update (lookup/> L P) (update/> U P)) + %<- (lookup-implies-update L U) %. +%worlds () (lookup-implies-update _ _) %. +%total (L) (lookup-implies-update L _) %. +%theorem update-implies-lookup : forall* {F} {N} {D} {F'} forall {U update F N D F'} exists {L lookup F' N D} true %. +%term _ update-implies-lookup update/0 (lookup/= nat`eq/) %. +%term _ update-implies-lookup (update/= nat`eq/) (lookup/= nat`eq/) %. +%term _ update-implies-lookup (update/< _) (lookup/= nat`eq/) %. +%term _ + %pi (update-implies-lookup (update/> F^N0=D2->F' N0+1+N1=N2) (lookup/> F'^N0=D2 N0+1+N1=N2)) + %<- (update-implies-lookup F^N0=D2->F' F'^N0=D2) %. +%worlds () (update-implies-lookup F^N=D->F' F'^N=D) %. +%total (U) (update-implies-lookup U _) %. +%theorem update-preserves-lookup : forall* {F} {N1} {D1} {F'} {N2} {D2} forall {L lookup F N2 D2} {U update F N1 D1 F'} {X nat`ne N2 N1} exists {L' lookup F' N2 D2} true %. +%% update/0 is impossible +%term _ + %pi (update-preserves-lookup (lookup/= nat`eq/) (update/= nat`eq/) N<>N L') + %<- (nat`ne-anti-reflexive N<>N FALSE) + %<- (false-implies-lookup FALSE L') %. +%term _ update-preserves-lookup (lookup/> L1 P1) (update/= nat`eq/) _ (lookup/> L1 P1) %. +%term _ update-preserves-lookup (lookup/= nat`eq/) (update/< N3+1+N2=N1) _ (lookup/> (lookup/= nat`eq/) N3+1+N2=N1) %. +%term _ + %pi (update-preserves-lookup (lookup/> L N0+1+N1=N2') (update/< N3+1+N2=N1) _ (lookup/> (lookup/> L N0+1+N3=N4) N4+1+N2=N2')) + %<- (plus-left-decrease N3+1+N2=N1 N1-1 N1=N1-1+1 N3+N2=N1-1) + %<- (plus-right-increase N3+N2=N1-1 N3+N2+1=N1-1+1) + %<- (nat`eq-symmetric N1=N1-1+1 N1-1+1=N1) + %<- (plus-respects-eq N3+N2+1=N1-1+1 nat`eq/ nat`eq/ N1-1+1=N1 N3+N2+1=N1) + %<- (plus-associative-converse N3+N2+1=N1 N0+1+N1=N2' N4 N0+1+N3=N4 N4+N2+1=N2') + %<- (plus-swap-succ-converse N4+N2+1=N2' N4+1+N2=N2') %. +%term _ update-preserves-lookup (lookup/= nat`eq/) (update/> _ _) _ (lookup/= nat`eq/) %. +%term _ + %pi (update-preserves-lookup (lookup/> L N0+1+N1=N2) (%the (update (map/+ N1 D1 M1) N2' D' (map/+ N1 D1 M1')) (update/> U N0'+1+N1=N2')) N2<>N2' (%the (lookup (map/+ N1 D1 M1') N2 D) (lookup/> L' N0+1+N1=N2))) + %<- (plus-right-cancels-ne N0+1+N1=N2 N0'+1+N1=N2' nat`eq/ N2<>N2' N0+1<>N0'+1) + %<- (succ-preserves-ne-converse N0+1<>N0'+1 N0<>N0') + %<- (update-preserves-lookup L U N0<>N0' L') %. +%worlds () (update-preserves-lookup F^N2=D2 F^N1=D1->F' N1<>N2 F'^N2=D2) %. +%total (L) (update-preserves-lookup L _ _ _) %. +%theorem update-preserves-lookup-converse : forall* {F1} {N1} {D1} {F2} {N2} {D2} forall {L2 lookup F2 N2 D2} {U update F1 N1 D1 F2} {X nat`ne N2 N1} exists {L1 lookup F1 N2 D2} true %. +%term _ + %pi (update-preserves-lookup-converse (lookup/= nat`eq/) update/0 N<>N L1) + %<- (nat`ne-anti-reflexive N<>N F) + %<- (false-implies-lookup F L1) %. +%term _ + %pi (update-preserves-lookup-converse (lookup/= nat`eq/) (update/= nat`eq/) N<>N L1) + %<- (nat`ne-anti-reflexive N<>N F) + %<- (false-implies-lookup F L1) %. +%term _ + %pi (update-preserves-lookup-converse (lookup/= nat`eq/) (update/< N3+1+N2=N1) N<>N L1) + %<- (nat`ne-anti-reflexive N<>N F) + %<- (false-implies-lookup F L1) %. +%term _ update-preserves-lookup-converse (lookup/= nat`eq/) (update/> _ _) _ (lookup/= nat`eq/) %. +%term _ update-preserves-lookup-converse (lookup/> L1 P) (update/= nat`eq/) _ (lookup/> L1 P) %. +%term _ + %pi (update-preserves-lookup-converse (lookup/> (lookup/= nat`eq/) N3+1+N2=N4) (update/< N3+1+N2=N1) _ (lookup/= N1=N4)) + %<- (plus-deterministic N3+1+N2=N1 N3+1+N2=N4 nat`eq/ nat`eq/ N1=N4) %. +%term _ + %pi (update-preserves-lookup-converse (lookup/> (lookup/> L1 N6+1+N3=N5) N5+1+N2=N4) (update/< N3+1+N2=N1) _ (lookup/> L1 N6+1+N1=N4)) + %<- (plus-swap-succ N3+1+N2=N1 N3+N2+1=N1) + %<- (plus-swap-succ N5+1+N2=N4 N5+N2+1=N4) + %<- (plus-associative* N6+1+N3=N5 N5+N2+1=N4 N3+N2+1=N1 N6+1+N1=N4) %. +%term _ + %pi (update-preserves-lookup-converse (lookup/> L2 N5+1+N1=N4) (update/> U1 N0+1+N1=N2) N4<>N2 (lookup/> L1 N5+1+N1=N4)) + %<- (plus-right-cancels-ne N5+1+N1=N4 N0+1+N1=N2 nat`eq/ N4<>N2 N5+1<>N0+1) + %<- (succ-preserves-ne-converse N5+1<>N0+1 N5<>N0) + %<- (update-preserves-lookup-converse L2 U1 N5<>N0 L1) %. +%worlds () (update-preserves-lookup-converse _ _ _ _) %. +%total (L) (update-preserves-lookup-converse L _ _ _) %. +%theorem update-preserves-not-member : forall* {M1} {N1} {N2} {D} {M2} forall {F1 not-member M1 N1} {U update M1 N2 D M2} {N nat`ne N1 N2} exists {F2 not-member M2 N1} true %. +%term _ update-preserves-not-member not-member/0 update/0 (nat`ne/< N>M) (not-member/< N>M) %. +%term _ + %pi (update-preserves-not-member not-member/0 update/0 (nat`ne/> M>N) (not-member/> not-member/0 M1+1+N=M)) + %<- (gt-implies-plus M>N M1 M1+1+N=M) %. +%term _ update-preserves-not-member (not-member/< N>M) (update/= nat`eq/) _ (not-member/< N>M) %. +%term _ update-preserves-not-member (not-member/< N1>M) (update/< N3+1+N2=N1) (nat`ne/< N2>M) (not-member/< N2>M) %. +%term _ + %pi (update-preserves-not-member (not-member/< N1>M) (update/< N3+1+N2=N1) (nat`ne/> M>N2) (not-member/> (not-member/< N3>M1) M1+1+N2=M)) + %<- (gt-implies-plus M>N2 M1 M1+1+N2=M) + %<- (plus-swap-succ N3+1+N2=N1 N3+N2+1=N1) + %<- (plus-swap-succ M1+1+N2=M M1+N2+1=M) + %<- (plus-right-cancels-gt N3+N2+1=N1 M1+N2+1=M nat`eq/ N1>M N3>M1) %. +%term _ update-preserves-not-member (not-member/< N1>M) (update/> _ _) _ (not-member/< N1>M) %. +%term _ update-preserves-not-member (not-member/> F P) (update/= nat`eq/) _ (not-member/> F P) %. +%term _ + %pi (update-preserves-not-member (not-member/> F M1+1+N1=M) (update/< N3+1+N2=N1) _ (not-member/> (not-member/> F M1+1+N3=MM) MM+1+N2=M)) + %<- (plus-swap-succ N3+1+N2=N1 N3+N2+1=N1) + %<- (plus-associative-converse N3+N2+1=N1 M1+1+N1=M MM M1+1+N3=MM MM+N2+1=M) + %<- (plus-swap-succ-converse MM+N2+1=M MM+1+N2=M) %. +%term _ + %pi (update-preserves-not-member (not-member/> F M1+1+N1=M) (update/> U N0+1+N1=N2) M<>N2 (not-member/> F' M1+1+N1=M)) + %<- (plus-right-cancels-ne M1+1+N1=M N0+1+N1=N2 nat`eq/ M<>N2 M1+1<>N0+1) + %<- (succ-preserves-ne-converse M1+1<>N0+1 M1<>N0) + %<- (update-preserves-not-member F U M1<>N0 F') %. +%worlds () (update-preserves-not-member N1-not-member-M1 M1^N2=D->M2 N1<>N2 N1-not-member-M2) %. +%total (F) (update-preserves-not-member F _ _ _) %. +%theorem update-preserves-not-member-converse : forall* {M1} {N1} {N2} {D} {M2} forall {F1 not-member M2 N1} {U update M1 N2 D M2} exists {F2 not-member M1 N1} true %. +%theorem update-preserves-not-member-converse-helper : forall* {M1} {N1} {N2} {D} {M2} {B} {B2} forall {F1 not-member M2 N1} {U update M1 N2 D M2} {D member? M1 N1 B} {E nat`eq? N1 N2 B2} exists {F2 not-member M1 N1} true %. +%term _ update-preserves-not-member-converse-helper _ _ (member?/out F) _ F %. +%term _ + %pi (update-preserves-not-member-converse-helper F2 U (member?/in L1) (nat`eq?/no N) F1) + %<- (update-preserves-lookup L1 U N L2) + %<- (not-member-lookup-not-equal F2 L2 N<>N) + %<- (nat`ne-anti-reflexive N<>N F) + %<- (false-implies-not-member F F1) %. +%term _ + %pi (update-preserves-not-member-converse-helper F2 U _ nat`eq?/yes F1) + %<- (update-implies-lookup U L2) + %<- (not-member-lookup-not-equal F2 L2 N<>N) + %<- (nat`ne-anti-reflexive N<>N F) + %<- (false-implies-not-member F F1) %. +%worlds () (update-preserves-not-member-converse-helper _ _ _ _ _) %. +%total {} (update-preserves-not-member-converse-helper _ _ _ _ _) %. +%term _ + %pi (update-preserves-not-member-converse F2 U F1) + %<- (member?-total D) + %<- (nat`eq?-total E) + %<- (update-preserves-not-member-converse-helper F2 U D E F1) %. +%worlds () (update-preserves-not-member-converse _ _ _) %. +%total {} (update-preserves-not-member-converse _ _ _) %. +%theorem update-is-cause-of-change : forall* {M1} {N1} {N2} {M2} {D1} {D2} forall {F not-member M1 N1} {U update M1 N2 D2 M2} {L lookup M2 N1 D1} exists {EN nat`eq N1 N2} {ED nat`eq D1 D2} true %. +%theorem update-is-cause-of-change/L : forall* {M1} {N1} {N2} {M2} {D1} {D2} {B} forall {F not-member M1 N1} {U update M1 N2 D2 M2} {L lookup M2 N1 D1} {E nat`eq? N1 N2 B} exists {EN nat`eq N1 N2} {ED nat`eq D1 D2} true %. +%term _ + %pi (update-is-cause-of-change F U L EN ED) + %<- (nat`eq?-total E?) + %<- (update-is-cause-of-change/L F U L E? EN ED) %. +%term _ + %pi (update-is-cause-of-change/L F U L nat`eq?/yes nat`eq/ ED) + %<- (update-implies-lookup U L') + %<- (lookup-deterministic L L' eq/ nat`eq/ ED) %. +%term _ + %pi (update-is-cause-of-change/L F U L (nat`eq?/no N1<>N2) EN ED) + %<- (update-preserves-not-member F U N1<>N2 F') + %<- (not-member-lookup-not-equal F' L N1<>N1) + %<- (nat`ne-anti-reflexive N1<>N1 V) + %<- (nat`false-implies-eq V EN) + %<- (nat`false-implies-eq V ED) %. +%worlds () (update-is-cause-of-change/L _ _ _ _ _ _) %. +%total {} (update-is-cause-of-change/L _ _ _ _ _ _) %. +%worlds () (update-is-cause-of-change _ _ _ _ _) %. +%total {} (update-is-cause-of-change _ _ _ _ _) %. +%theorem update-preserves-membership : forall* {M1} {N1} {B} {N2} {D} {M2} forall {MD1 member? M1 N1 B} {U update M1 N2 D M2} {N nat`ne N1 N2} exists {MD2 member? M2 N1 B} true %. +%term _ + %pi (update-preserves-membership (member?/in L) U NE (member?/in L')) + %<- (update-preserves-lookup L U NE L') %. +%term _ + %pi (update-preserves-membership (member?/out F) U NE (member?/out F')) + %<- (update-preserves-not-member F U NE F') %. +%worlds () (update-preserves-membership _ _ _ _) %. +%total {} (update-preserves-membership _ _ _ _) %. +%theorem update-preserves-membership-converse : forall* {M1} {N1} {B} {N2} {D} {M2} forall {MD2 member? M2 N1 B} {U update M1 N2 D M2} {N nat`ne N1 N2} exists {MD1 member? M1 N1 B} true %. +%term _ + %pi (update-preserves-membership-converse (member?/in L2) U NE (member?/in L1)) + %<- (update-preserves-lookup-converse L2 U NE L1) %. +%term _ + %pi (update-preserves-membership-converse (member?/out F2) U NE (member?/out F1)) + %<- (update-preserves-not-member-converse F2 U F1) %. +%worlds () (update-preserves-membership-converse _ _ _ _) %. +%total {} (update-preserves-membership-converse _ _ _ _) %. +%theorem lookup-update-preserves-membership : forall* {M1} {N1} {B} {N2} {D1} {D2} {M2} forall {MD1 member? M1 N1 B} {L lookup M1 N2 D1} {U update M1 N2 D2 M2} exists {MD2 member? M2 N1 B} true %. +%theorem lookup-update-preserves-membership/L : forall* {M1} {N1} {B} {N2} {D1} {D2} {M2} {B2} forall {MD1 member? M1 N1 B} {L lookup M1 N2 D1} {U update M1 N2 D2 M2} {EQ? nat`eq? N1 N2 B2} exists {MD2 member? M2 N1 B} true %. +%term _ + %pi (lookup-update-preserves-membership/L MD1 _ Ux1 (nat`eq?/no N1<>N2) MD2) + %<- (update-preserves-membership MD1 Ux1 N1<>N2 MD2) %. +%term _ + %pi (lookup-update-preserves-membership/L (member?/in _) _ U nat`eq?/yes (member?/in L2)) + %<- (update-implies-lookup U L2) %. +%term _ + %pi (lookup-update-preserves-membership/L (member?/out F1) L1 _ nat`eq?/yes (member?/out F2)) + %<- (not-member-lookup-not-equal F1 L1 NE) + %<- (nat`ne-anti-reflexive NE F) + %<- (false-implies-not-member F F2) %. +%worlds () (lookup-update-preserves-membership/L _ _ _ _ _) %. +%total {} (lookup-update-preserves-membership/L _ _ _ _ _) %. +%term _ + %pi (lookup-update-preserves-membership MD1 L1 U MD2) + %<- (nat`eq?-total EQ?) + %<- (lookup-update-preserves-membership/L MD1 L1 U EQ? MD2) %. +%worlds () (lookup-update-preserves-membership _ _ _ _) %. +%total {} (lookup-update-preserves-membership _ _ _ _) %. +%theorem lookup-update-preserves-membership-converse : forall* {M1} {N1} {B} {N2} {D1} {D2} {M2} forall {MD1 member? M2 N1 B} {L lookup M1 N2 D1} {U update M1 N2 D2 M2} exists {MD2 member? M1 N1 B} true %. +%term _ + %pi (lookup-update-preserves-membership-converse MD2 ML MU MD1) + %<- (member?-total MD1') + %<- (lookup-update-preserves-membership MD1' ML MU MD2') + %<- (member?-deterministic MD2' MD2 eq/ nat`eq/ B'=B) + %<- (member?-respects-eq MD1' eq/ nat`eq/ B'=B MD1) %. +%worlds () (lookup-update-preserves-membership-converse _ _ _ _) %. +%total {} (lookup-update-preserves-membership-converse _ _ _ _) %. +%theorem update-preserves-in-member : forall* {M1} {N1} {N2} {D} {M2} forall {MD1 member? M1 N1 true} {U update M1 N2 D M2} exists {MD2 member? M2 N1 true} true %. +%theorem update-preserves-in-member/L : forall* {M1} {N1} {N2} {D} {M2} {B} forall {MD1 member? M1 N1 true} {U update M1 N2 D M2} {E nat`eq? N1 N2 B} exists {MD2 member? M2 N1 true} true %. +%term _ + %pi (update-preserves-in-member/L (member?/in ML1) U (nat`eq?/no N1<>N2) (member?/in ML2)) + %<- (update-preserves-lookup ML1 U N1<>N2 ML2) %. +%term _ + %pi (update-preserves-in-member/L _ U nat`eq?/yes (member?/in ML)) + %<- (update-implies-lookup U ML) %. +%worlds () (update-preserves-in-member/L _ _ _ _) %. +%total {} (update-preserves-in-member/L _ _ _ _) %. +%term _ + %pi (update-preserves-in-member MD1 U MD2) + %<- (nat`eq?-total E) + %<- (update-preserves-in-member/L MD1 U E MD2) %. +%worlds () (update-preserves-in-member _ _ _) %. +%total {} (update-preserves-in-member _ _ _) %. +%theorem shift-preserves-update : forall* {M1} {N1} {D} {M1'} {N0} {M2} forall {U1 update M1 N1 D M1'} {S shift N0 M1 M2} exists {N2} {M2'} {P plus (s N0) N1 N2} {SS shift N0 M1' M2'} {U2 update M2 N2 D M2'} true %. +%term _ + %pi (shift-preserves-update update/0 shift/0 _ _ P (shift/+ P) update/0) + %<- (plus-total P) %. +%term _ shift-preserves-update (update/= nat`eq/) (shift/+ P) _ _ P (shift/+ P) (update/= nat`eq/) %. +%term _ + %pi (shift-preserves-update (update/< N4+1+N1=N3) (shift/+ N0+1+N3=N5) _ _ N0+1+N1=N2 (shift/+ N0+1+N1=N2) (update/< N4+1+N2=N5)) + %<- (plus-commutative N4+1+N1=N3 N1+N4+1=N3) + %<- (plus-associative-converse N1+N4+1=N3 N0+1+N3=N5 _ N0+1+N1=N2 N2+N4+1=N5) + %<- (plus-commutative N2+N4+1=N5 N4+1+N2=N5) %. +%term _ + %pi (shift-preserves-update (update/> U N4+1+N3=N1) (shift/+ N0+1+N3=N5) _ _ N0+1+N1=N2 (shift/+ N0+1+N3=N5) (update/> U N4+1+N5=N2)) + %<- (plus-total N0+1+N1=N2) + %<- (plus-commutative N4+1+N3=N1 N3+N4+1=N1) + %<- (plus-associative-converse* N3+N4+1=N1 N0+1+N1=N2 N0+1+N3=N5 N5+N4+1=N2) + %<- (plus-commutative N5+N4+1=N2 N4+1+N5=N2) %. +%worlds () (shift-preserves-update _ _ _ _ _ _ _) %. +%total {} (shift-preserves-update _ _ _ _ _ _ _) %. +%theorem shift-preserves-update* : forall* {M1} {N1} {D} {M1'} {N0} {M2} {N2} {M2'} forall {U1 update M1 N1 D M1'} {S shift N0 M1 M2} {P plus (s N0) N1 N2} {SS shift N0 M1' M2'} exists {U2 update M2 N2 D M2'} true %. +%term _ + %pi (shift-preserves-update* U1 S P SS U2) + %<- (shift-preserves-update U1 S _ _ P' SS' U2') + %<- (plus-deterministic P' P nat`eq/ nat`eq/ N2'=N2) + %<- (shift-deterministic SS' SS nat`eq/ eq/ M2'=M2) + %<- (update-respects-eq U2' eq/ N2'=N2 nat`eq/ M2'=M2 U2) %. +%worlds () (shift-preserves-update* _ _ _ _ _) %. +%total {} (shift-preserves-update* _ _ _ _ _) %. +%theorem shift-preserves-update-converse : forall* {M1} {N1} {D} {N0} {M2} {N2} {M2'} forall {U2 update M2 N2 D M2'} {S shift N0 M1 M2} {P plus (s N0) N1 N2} exists {M1'} {SS shift N0 M1' M2'} {U1 update M1 N1 D M1'} true %. +%term _ + %pi (shift-preserves-update-converse U2 S P _ SS U1) + %<- (update-total U1) + %<- (shift-preserves-update U1 S _ _ P' SS' U2') + %<- (plus-deterministic P' P nat`eq/ nat`eq/ N2'=N2) + %<- (update-deterministic U2' U2 eq/ N2'=N2 nat`eq/ M2'=M2) + %<- (shift-respects-eq SS' nat`eq/ eq/ M2'=M2 SS) %. +%worlds () (shift-preserves-update-converse _ _ _ _ _ _) %. +%total {} (shift-preserves-update-converse _ _ _ _ _ _) %. +%theorem update-overwrites : forall* {M1} {N1} {D1} {M2} {N2} {D2} {M3} forall {U1 update M1 N1 D1 M2} {U2 update M2 N2 D2 M3} {E nat`eq N1 N2} exists {U12 update M1 N1 D2 M3} true %. +%term _ update-overwrites update/0 (update/= nat`eq/) nat`eq/ update/0 %. +%term _ update-overwrites (update/= nat`eq/) (update/= nat`eq/) nat`eq/ (update/= nat`eq/) %. +%term _ update-overwrites (update/< P) (update/= nat`eq/) nat`eq/ (update/< P) %. +%term _ + %pi (update-overwrites (update/> U1 P) (update/> U2 P') nat`eq/ (update/> U3 P)) + %<- (plus-right-cancels P P' nat`eq/ nat`eq/ N0+1=N0'+1) + %<- (succ-cancels N0+1=N0'+1 N0=N0') + %<- (update-overwrites U1 U2 N0=N0' U3) %. +%% contradiction cases +%term _ + %pi (update-overwrites update/0 (update/< N3+1+N=N) nat`eq/ U) + %<- (plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-update F U) %. +%term _ + %pi (update-overwrites update/0 (update/> _ N0+1+N=N) nat`eq/ U) + %<- (plus-implies-gt N0+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-update F U) %. +%term _ + %pi (update-overwrites (update/= nat`eq/) (update/< N3+1+N=N) nat`eq/ U) + %<- (plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-update F U) %. +%term _ + %pi (update-overwrites (update/= nat`eq/) (update/> _ N0+1+N=N) nat`eq/ U) + %<- (plus-implies-gt N0+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-update F U) %. +%term _ + %pi (update-overwrites (update/< _) (update/< N3+1+N=N) nat`eq/ U) + %<- (plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-update F U) %. +%term _ + %pi (update-overwrites (update/< _) (update/> _ N0+1+N=N) nat`eq/ U) + %<- (plus-implies-gt N0+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-update F U) %. +%term _ + %pi (update-overwrites (update/> _ N0+1+N=N) (update/= nat`eq/) nat`eq/ U) + %<- (plus-implies-gt N0+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-update F U) %. +%term _ + %pi (update-overwrites (update/> _ N0+1+N1=N2) (update/< N3+1+N2=N1) nat`eq/ U) + %<- (nat`plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) + %<- (nat`plus-implies-gt N0+1+N1=N2 nat`eq/ N2>N1) + %<- (nat`gt-anti-symmetric N1>N2 N2>N1 F) + %<- (false-implies-update F U) %. +%worlds () (update-overwrites M1^N1=D1->M2 M2^N2=D2->M3 N1=N2 M1^N1=D2->M3) %. +%total (U) (update-overwrites U _ _ _) %. +%theorem update-overwrites-converse : forall* {M1} {N1} {D1} {M2} {D2} {M3} forall {U12 update M1 N1 D1 M3} {U1 update M1 N1 D2 M2} exists {U2 update M2 N1 D1 M3} true %. +%term _ + %pi (update-overwrites-converse U12 U1 U2) + %<- (update-total U2P) + %<- (update-overwrites U1 U2P nat`eq/ U12P) + %<- (update-deterministic U12P U12 eq/ nat`eq/ nat`eq/ M2P=M2) + %<- (update-respects-eq U2P eq/ nat`eq/ nat`eq/ M2P=M2 U2) %. +%worlds () (update-overwrites-converse _ _ _) %. +%total {} (update-overwrites-converse _ _ _) %. +%theorem update-may-have-no-effect : forall* {M1} {N} {D} {M2} forall {L lookup M1 N D} {U update M1 N D M2} exists {E eq M1 M2} true %. +%term _ + %pi (update-may-have-no-effect (lookup/= nat`eq/) U E) + %<- (update/=-inversion U nat`eq/ E) %. +%term _ + %pi (update-may-have-no-effect (lookup/> L1 N0+1+N1=N2) U E) + %<- (update/>-inversion U N0+1+N1=N2 _ U1 M112=M2) + %<- (update-may-have-no-effect L1 U1 M1=M2) + %<- (map/+-preserves-eq nat`eq/ nat`eq/ M1=M2 M111=M112) + %<- (eq-transitive M111=M112 M112=M2 E) %. +%worlds () (update-may-have-no-effect _ _ _) %. +%total (L) (update-may-have-no-effect L _ _) %. +%theorem update-idempotent : forall* {M1} {N1} {D1} {M2} {N2} {D2} {M3} forall {U1 update M1 N1 D1 M2} {U2 update M2 N2 D2 M3} {EN nat`eq N1 N2} {ED nat`eq D1 D2} exists {EM eq M2 M3} true %. +%term _ + %pi (update-idempotent U1 U2 nat`eq/ nat`eq/ M2=M3) + %<- (update-overwrites U1 U2 nat`eq/ M1^N=D->M3) + %<- (update-deterministic U1 M1^N=D->M3 eq/ nat`eq/ nat`eq/ M2=M3) %. +%worlds () (update-idempotent M1^N1=D1->M2 M2^N2=D2->M3 N1=N2 D1=D2 M2=M3) %. +%total {} (update-idempotent _ _ _ _ _) %. +%theorem update-commutes : forall* {M} {N1} {D1} {M1} {N2} {D2} {M12} forall {U1 update M N1 D1 M1} {U12 update M1 N2 D2 M12} {NE nat`ne N1 N2} exists {M2} {U2 update M N2 D2 M2} {U21 update M2 N1 D1 M12} true %. +%term _ + %pi (update-commutes update/0 (update/= nat`eq/) N<>N map/0 U2 U21) + %<- (nat`ne-anti-reflexive N<>N F) + %<- (false-implies-update F U2) + %<- (false-implies-update F U21) %. +%term _ update-commutes update/0 (update/< N'+1+N2=N1) _ _ update/0 (update/> update/0 N'+1+N2=N1) %. +%term _ update-commutes update/0 (update/> update/0 N'+1+N1=N2) _ _ update/0 (update/< N'+1+N1=N2) %. +%term _ + %pi (update-commutes (update/= nat`eq/) (update/= nat`eq/) N<>N map/0 U2 U21) + %<- (nat`ne-anti-reflexive N<>N F) + %<- (false-implies-update F U2) + %<- (false-implies-update F U21) %. +%term _ update-commutes (update/= nat`eq/) (update/< N'+1+N2=N1) _ _ (update/< N'+1+N2=N1) (update/> (update/= nat`eq/) N'+1+N2=N1) %. +%term _ update-commutes (update/= nat`eq/) (update/> U N'+1+N1=N2) _ _ (update/> U N'+1+N1=N2) (update/= nat`eq/) %. +%term _ + %pi (update-commutes (update/< _) (update/= nat`eq/) N<>N map/0 U2 U21) + %<- (nat`ne-anti-reflexive N<>N F) + %<- (false-implies-update F U2) + %<- (false-implies-update F U21) %. +%term _ + %pi (update-commutes (update/< N1'+1+N1=N) (update/< N2'+1+N2=N1) _ _ (update/< N2''+1+N2=N) (update/> (update/< N1'+1+N2'=N2'') N2'+1+N2=N1)) + %<- (plus-swap-succ N2'+1+N2=N1 N2'+N2+1=N1) + %<- (plus-associative-converse N2'+N2+1=N1 N1'+1+N1=N N2'' N1'+1+N2'=N2'' N2''+N2+1=N) + %<- (plus-swap-succ-converse N2''+N2+1=N N2''+1+N2=N) %. +%term _ + %pi (update-commutes (%the (update (map/+ N D M) _ _ _) (update/< N11+1+N1=N)) (update/> (update/= nat`eq/) N11+1+N1=N2) _ (map/+ N2 D2 M) (update/= N=N2) (%the (update _ N1 D1 _) (update/< N11+1+N1=N2))) + %<- (plus-deterministic N11+1+N1=N N11+1+N1=N2 nat`eq/ nat`eq/ N=N2) %. +%term _ + %pi (update-commutes (update/< N11+1+N1=N) (update/> (update/< N2''+1+N2'=N11) N2'+1+N1=N2) _ _ (update/< N2''+1+N2=N) (update/< N2'+1+N1=N2)) + %<- (plus-swap-succ N11+1+N1=N N11+N1+1=N) + %<- (plus-swap-succ N2'+1+N1=N2 N2'+N1+1=N2) + %<- (plus-associative* N2''+1+N2'=N11 N11+N1+1=N N2'+N1+1=N2 N2''+1+N2=N) %. +%term _ + %pi (update-commutes (update/< N11+1+N1=N) (update/> (update/> U N2''+1+N11=N2') N2'+1+N1=N2) _ _ (update/> U N2''+1+N=N2) (update/< N11+1+N1=N)) + %<- (plus-swap-succ N11+1+N1=N N11+N1+1=N) + %<- (plus-swap-succ N2'+1+N1=N2 N2'+N1+1=N2) + %<- (plus-associative* N2''+1+N11=N2' N2'+N1+1=N2 N11+N1+1=N N2''+1+N=N2) %. +%term _ update-commutes (update/> U N11+1+N=N1) (update/= nat`eq/) _ _ (update/= nat`eq/) (update/> U N11+1+N=N1) %. +%term _ + %pi (update-commutes (update/> U N11+1+N=N1) (update/< N2'+1+N2=N) _ _ (update/< N2'+1+N2=N) (update/> (update/> U N11+1+N2'=N11') N11'+1+N2=N1)) + %<- (plus-swap-succ N2'+1+N2=N N2'+N2+1=N) + %<- (plus-associative-converse N2'+N2+1=N N11+1+N=N1 N11' N11+1+N2'=N11' N11'+N2+1=N1) + %<- (plus-swap-succ-converse N11'+N2+1=N1 N11'+1+N2=N1) %. +%term _ + %pi (update-commutes (update/> U1 N11+1+N=N1) (update/> U12 N2'+1+N=N2) N1<>N2 (map/+ N D M2) (update/> U2 N2'+1+N=N2) (update/> U21 N11+1+N=N1)) + %<- (plus-right-cancels-ne N11+1+N=N1 N2'+1+N=N2 nat`eq/ N1<>N2 N11+1<>N2'+1) + %<- (succ-preserves-ne-converse N11+1<>N2'+1 N11<>N2') + %<- (update-commutes U1 U12 N11<>N2' M2 U2 U21) %. +%worlds () (update-commutes M^N1=D1->M1 M1^D2=N2->M12 N1<>N2 M2 M^N2=D2->M2 M2^N1=D1->M12) %. +%total (U1) (update-commutes U1 _ _ _ _ _) %. +%theorem update-commutes* : forall* {M} {N1} {D1} {M1} {N2} {D2} {M12} {M2} forall {U1 update M N1 D1 M1} {U12 update M1 N2 D2 M12} {NE nat`ne N1 N2} {U2 update M N2 D2 M2} exists {U21 update M2 N1 D1 M12} true %. +%term _ + %pi (update-commutes* M^N1=D1->M1 M1^D2=N2->M12 N1<>N2 M^N2=D2->M2 M2^N1=D1->M12) + %<- (update-commutes M^N1=D1->M1 M1^D2=N2->M12 N1<>N2 M2' M^N2=D2->M2' M2'^N1=D1->M12) + %<- (update-deterministic M^N2=D2->M2' M^N2=D2->M2 eq/ nat`eq/ nat`eq/ M2'=M2) + %<- (update-respects-eq M2'^N1=D1->M12 M2'=M2 nat`eq/ nat`eq/ eq/ M2^N1=D1->M12) %. +%worlds () (update-commutes* M^N1=D1->M1 M1^D2=N2->M12 N1<>N2 M^N2=D2->M2 M2^N1=D1->M12) %. +%total {} (update-commutes* _ _ _ _ _) %. +%% The following theorem is needed if you want to iteratively +%% do something with a set. It says that you can take out an element +%% and (using the update-preserves-X-converse theorems) get a smaller set +%% that differs only for this element: +%theorem can-remove : forall* {M} {S} {N} {D} forall {SZ size M S} {L lookup M N D} exists {M-} {S-} {SZ- size M- S-} {E nat`eq (s S-) S} {U update M- N D M} {F not-member M- N} true %. +%term _ can-remove (size/+ _) (lookup/= _) _ _ size/0 nat`eq/ update/0 not-member/0 %. +%term _ + %pi (can-remove (size/+ (size/+ SZ)) (lookup/= nat`eq/) _ _ (size/+ SZ) nat`eq/ (update/< N3+1+N2=N1) (not-member/< N1>N2)) + %<- (plus-total N3+1+N2=N1) + %<- (plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) %. +%term _ + %pi (can-remove (size/+ SZ) (lookup/> LK N0+1+N1=N2) _ _ (size/+ SZ2) E (update/> U2 N0+1+N1=N2) (not-member/> F2 N0+1+N1=N2)) + %<- (can-remove SZ LK M- S- SZ2 E2 U2 F2) + %<- (succ-deterministic E2 E) %. +%worlds () (can-remove _ _ _ _ _ _ _ _) %. +%total (L) (can-remove _ L _ _ _ _ _ _) %. +%%%% Map comparison +%%% Definition of leq +%sort leq %. +%term leq/0 leq map/0 M %. +%term leq/= + %pi (leq (map/+ N1 D1 M1) (map/+ N2 D2 M2)) + %<- (nat`eq N1 N2) + %<- (nat`leq D1 D2) + %<- (leq M1 M2) %. +%term leq/> + %pi (leq (map/+ N1 D1 M1) (map/+ N2 D2 M2)) + %<- (nat`plus (s N3) N2 N1) + %<- (leq (map/+ N3 D1 M1) M2) %. +%%% Theorems about leq +%theorem false-implies-leq : forall* {M1} {M2} forall {F void} exists {L leq M1 M2} true %. +%worlds () (false-implies-leq _ _) %. +%total {} (false-implies-leq _ _) %. +%theorem leq-respects-eq : forall* {M1} {M2} {M1'} {M2'} forall {L leq M1 M2} {E1 eq M1 M1'} {E2 eq M2 M2'} exists {LP leq M1' M2'} true %. +%term _ leq-respects-eq L eq/ eq/ L %. +%worlds () (leq-respects-eq _ _ _ _) %. +%total {} (leq-respects-eq _ _ _ _) %. +%reduces = L1 L2 (leq-respects-eq L1 _ _ L2) %. +%theorem leq/0-inversion : forall* {M1} {M2} forall {L leq M1 M2} {E2 eq M2 map/0} exists {E1 eq M1 map/0} true %. +%term _ leq/0-inversion leq/0 eq/ eq/ %. +%worlds () (leq/0-inversion _ _ _) %. +%total {} (leq/0-inversion _ _ _) %. +%theorem leq/=-inversion : forall* {N1} {D1} {M1} {N2} {D2} {M2} forall {L leq (map/+ N1 D1 M1) (map/+ N2 D2 M2)} {EN nat`eq N1 N2} exists {ED nat`leq D1 D2} {EM leq M1 M2} true %. +%term _ leq/=-inversion (leq/= M1<=M2 D1<=D2 nat`eq/) nat`eq/ D1<=D2 M1<=M2 %. +%term _ + %pi (leq/=-inversion (leq/> M311<=M2 N3+1+N=N) nat`eq/ ED M1<=M2) + %<- (plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (nat`false-implies-leq F ED) + %<- (false-implies-eq F M311=M1) + %<- (leq-respects-eq M311<=M2 M311=M1 eq/ M1<=M2) %. +%worlds () (leq/=-inversion _ _ _ _) %. +%total {} (leq/=-inversion _ _ _ _) %. +%reduces < L1 L (leq/=-inversion L _ _ L1) %. +%theorem leq/>-inversion : forall* {N1} {D1} {M1} {N2} {D2} {M2} {N3} forall {L leq (map/+ N1 D1 M1) (map/+ N2 D2 M2)} {P plus (s N3) N2 N1} exists {EM leq (map/+ N3 D1 M1) M2} true %. +%term _ + %pi (leq/>-inversion (leq/= M1<=M2 D1<=D2 nat`eq/) N3+1+N=N M311<=M2) + %<- (plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-eq F M1=M311) + %<- (leq-respects-eq M1<=M2 M1=M311 eq/ M311<=M2) %. +%term _ + %pi (leq/>-inversion (leq/> M311'<=M2 N3P+1+N2=N1) N3+1+N2=N1 M311<=M2) + %<- (plus-right-cancels N3P+1+N2=N1 N3+1+N2=N1 nat`eq/ nat`eq/ N3P+1=N3+1) + %<- (succ-cancels N3P+1=N3+1 N3P=N3) + %<- (map/+-preserves-eq N3P=N3 nat`eq/ eq/ M311'=M311) + %<- (leq-respects-eq M311'<=M2 M311'=M311 eq/ M311<=M2) %. +%worlds () (leq/>-inversion _ _ _) %. +%total {} (leq/>-inversion _ _ _) %. +%reduces < LP L (leq/>-inversion L _ LP) %. +%theorem leq-contradiction : forall* {N1} {D1} {M1} {N2} {D2} {M2} {N0} forall {L leq (map/+ N1 D1 M1) (map/+ N2 D2 M2)} {P plus (s N0) N1 N2} exists {F void} true %. +%term _ + %pi (leq-contradiction (leq/= _ _ nat`eq/) N0+1+N=N F) + %<- (plus-implies-gt N0+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) %. +%term _ + %pi (leq-contradiction (leq/> _ N3+1+N2=N1) N0+1+N1=N2 F) + %<- (nat`plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) + %<- (nat`plus-implies-gt N0+1+N1=N2 nat`eq/ N2>N1) + %<- (nat`gt-anti-symmetric N1>N2 N2>N1 F) %. +%worlds () (leq-contradiction _ _ _) %. +%total {} (leq-contradiction _ _ _) %. +%theorem leq-reflexive : forall {M} exists {L leq M M} true %. +%term _ leq-reflexive map/0 leq/0 %. +%term _ + %pi (leq-reflexive (map/+ N D M) (leq/= M<=M D<=D nat`eq/)) + %<- (nat`leq-reflexive D D<=D) + %<- (leq-reflexive M M<=M) %. +%worlds () (leq-reflexive _ _) %. +%total (M) (leq-reflexive M _) %. +%theorem leq-anti-symmetric : forall* {M1} {M2} forall {L1 leq M1 M2} {L2 leq M2 M1} exists {E eq M1 M2} true %. +%term _ leq-anti-symmetric leq/0 leq/0 eq/ %. +%term _ + %pi (leq-anti-symmetric (leq/= M1<=M2 D1<=D2 nat`eq/) (leq/= M2<=M1 D2<=D1 nat`eq/) E) + %<- (nat`leq-anti-symmetric D1<=D2 D2<=D1 D1=D2) + %<- (leq-anti-symmetric M1<=M2 M2<=M1 M1=M2) + %<- (map/+-preserves-eq nat`eq/ D1=D2 M1=M2 E) %. +%term _ + %pi (leq-anti-symmetric (leq/= _ _ nat`eq/) (leq/> _ N3+1+N=N) E) + %<- (nat`plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (nat`gt-anti-reflexive N>N F) + %<- (false-implies-eq F E) %. +%term _ + %pi (leq-anti-symmetric (leq/> _ N3+1+N=N) (leq/= _ _ nat`eq/) E) + %<- (nat`plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (nat`gt-anti-reflexive N>N F) + %<- (false-implies-eq F E) %. +%term _ + %pi (leq-anti-symmetric (leq/> _ N3+1+N2=N1) (leq/> _ N0+1+N1=N2) E) + %<- (nat`plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) + %<- (nat`plus-implies-gt N0+1+N1=N2 nat`eq/ N2>N1) + %<- (nat`gt-anti-symmetric N1>N2 N2>N1 F) + %<- (false-implies-eq F E) %. +%worlds () (leq-anti-symmetric _ _ _) %. +%total (L) (leq-anti-symmetric L _ _) %. +%theorem leq-transitive : forall* {M1} {M2} {M3} forall {L1 leq M1 M2} {L2 leq M2 M3} exists {L3 leq M1 M3} true %. +%term _ leq-transitive leq/0 _ leq/0 %. +%term _ + %pi (leq-transitive (leq/= M1<=M2 D1<=D2 nat`eq/) (leq/= M2<=M3 D2<=D3 nat`eq/) (leq/= M1<=M3 D1<=D3 nat`eq/)) + %<- (nat`leq-transitive D1<=D2 D2<=D3 D1<=D3) + %<- (leq-transitive M1<=M2 M2<=M3 M1<=M3) %. +%term _ + %pi (leq-transitive (leq/= M1<=M2 D1<=D2 nat`eq/) (leq/> M522<=M3 N5+1+N3=N2) (leq/> M511<=M3 N5+1+N3=N2)) + %<- (leq-transitive (leq/= M1<=M2 D1<=D2 nat`eq/) M522<=M3 M511<=M3) %. +%term _ + %pi (leq-transitive (leq/> M011<=M2 N0+1+N=N1) (leq/= M2<=M3 D2<=D3 nat`eq/) (leq/> M011<=M3 N0+1+N=N1)) + %<- (leq-transitive M011<=M2 M2<=M3 M011<=M3) %. +%term _ + %pi (leq-transitive (leq/> M011<=M2 N0+1+N2=N1) (leq/> M522<=M3 N5+1+N3=N2) (leq/> M411<=M3 N4+1+N3=N1)) + %<- (nat`plus-swap-succ N5+1+N3=N2 N5+N3+1=N2) + %<- (nat`plus-associative-converse N5+N3+1=N2 N0+1+N2=N1 N4 N0+1+N5=N4 N4+N3+1=N1) + %<- (nat`plus-swap-succ-converse N4+N3+1=N1 N4+1+N3=N1) + %<- (leq-transitive (leq/> M011<=M2 N0+1+N5=N4) M522<=M3 M411<=M3) %. +%worlds () (leq-transitive _ _ _) %. +%total (L) (leq-transitive _ L _) %. +%theorem map/+-preserves-leq : forall* {N1} {N2} {D1} {D2} {M1} {M2} forall {E nat`eq N1 N2} {LD nat`leq D1 D2} {LM leq M1 M2} exists {L leq (map/+ N1 D1 M1) (map/+ N2 D2 M2)} true %. +%term _ map/+-preserves-leq N1=N2 D1<=D2 M1<=M2 (leq/= M1<=M2 D1<=D2 N1=N2) %. +%worlds () (map/+-preserves-leq _ _ _ _) %. +%total {} (map/+-preserves-leq _ _ _ _) %. +%theorem lookup-respects-leq : forall* {M} {N} {D} {MP} forall {L lookup M N D} {ME leq M MP} exists {DP} {L lookup MP N DP} {DE nat`leq D DP} true %. +%term _ lookup-respects-leq (lookup/= nat`eq/) (leq/= _ D1<=D2 nat`eq/) _ (lookup/= nat`eq/) D1<=D2 %. +%term _ + %pi (lookup-respects-leq (lookup/= nat`eq/) (leq/> M311<=M2 N3+1+N2=N1) D2 (lookup/> L13 N3+1+N2=N1) D1<=D2) + %<- (lookup-respects-leq (lookup/= nat`eq/) M311<=M2 D2 L13 D1<=D2) %. +%term _ + %pi (lookup-respects-leq (lookup/> L13 N3+1+N2=N1) (leq/= M1<=M2 _ nat`eq/) D2 (lookup/> L13' N3+1+N2=N1) D1<=D2) + %<- (lookup-respects-leq L13 M1<=M2 D2 L13' D1<=D2) %. +%term _ + %pi (lookup-respects-leq (lookup/> L10 N0+1+N1=N) (leq/> M311<=M2 N3+1+N2=N1) D2 (lookup/> L14' N4+1+N2=N) D1<=D2) + %<- (nat`plus-swap-succ N3+1+N2=N1 N3+N2+1=N1) + %<- (nat`plus-associative-converse N3+N2+1=N1 N0+1+N1=N N4 N0+1+N3=N4 N4+N2+1=N) + %<- (nat`plus-swap-succ-converse N4+N2+1=N N4+1+N2=N) + %<- (lookup-respects-leq (lookup/> L10 N0+1+N3=N4) M311<=M2 D2 L14' D1<=D2) %. +%worlds () (lookup-respects-leq _ _ _ _ _) %. +%total (L) (lookup-respects-leq _ L _ _ _) %. +%theorem not-member-respects-geq : forall* {M} {MP} {N} forall {FP not-member MP N} {L leq M MP} exists {F not-member M N} true %. +%term _ not-member-respects-geq F leq/0 not-member/0 %. +%term _ not-member-respects-geq (not-member/< R) (leq/= _ _ nat`eq/) (not-member/< R) %. +%term _ + %pi (not-member-respects-geq (not-member/< N2>N) (leq/> _ N3+1+N2=N1) (not-member/< N1>N)) + %<- (nat`plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) + %<- (gt-transitive N1>N2 N2>N N1>N) %. +%term _ + %pi (not-member-respects-geq (not-member/> F20 N0+1+N2=N) (leq/= M1<=M2 _ nat`eq/) (not-member/> F10 N0+1+N2=N)) + %<- (not-member-respects-geq F20 M1<=M2 F10) %. +%term _ {F3110 not-member (map/+ N3 D1 M1) N0} {N0+1+N2=N plus (s N0) N2 N} {N3+1+N2=N1 plus (s N3) N2 N1} + %pi (not-member-respects-geq (%the (not-member (map/+ N2 D2 M2) N) (not-member/> F20 N0+1+N2=N)) (leq/> M311<=M2 N3+1+N2=N1) F') + %<- (not-member-respects-geq F20 M311<=M2 F3110) + %<- (nat`plus-swap-succ N0+1+N2=N N0+N2+1=N) + %<- (nat`plus-swap-succ N3+1+N2=N1 N3+N2+1=N1) + %<- (plus-right-preserves-not-member* F3110 N3+N2+1=N1 N0+N2+1=N F') %. +%worlds () (not-member-respects-geq _ _ _) %. +%total (F) (not-member-respects-geq F _ _) %. +%theorem leq-implies-size-le : forall* {M1} {M2} {N1} {N2} forall {L leq M1 M2} {SZ1 size M1 N1} {SZ2 size M2 N2} exists {G ge N2 N1} true %. +%term _ + %pi (leq-implies-size-le leq/0 size/0 N2=|M2| N2>=0) + %<- (plus-commutative (%the (plus z N2 N2) plus/z) N2+0=N2) + %<- (plus-implies-ge N2+0=N2 N2>=0) %. +%term _ + %pi (leq-implies-size-le (leq/= M1<=M2 _ _) (size/+ N1=|M1|) (size/+ N2=|M2|) N2+1>=N1+1) + %<- (leq-implies-size-le M1<=M2 N1=|M1| N2=|M2| N2>=N1) + %<- (succ-preserves-ge N2>=N1 N2+1>=N1+1) %. +%term _ + %pi (leq-implies-size-le (leq/> M311<=M2 _) (size/+ N1=|M1|) (size/+ N2=|M2|) (ge/> N2+1>N1+1)) + %<- (leq-implies-size-le M311<=M2 (size/+ N1=|M1|) N2=|M2| N2>=N1+1) + %<- (succ-implies-gt nat`eq/ N2+1>N2) + %<- (gt-transitive-ge N2+1>N2 N2>=N1+1 N2+1>N1+1) %. +%worlds () (leq-implies-size-le _ _ _ _) %. +%total (L) (leq-implies-size-le L _ _ _) %. +%theorem leq-implies-bound-le : forall* {M1} {M2} {N1} {N2} forall {L leq M1 M2} {BD1 bound M1 N1} {BD2 bound M2 N2} exists {G ge N2 N1} true %. +%term _ + %pi (leq-implies-bound-le leq/0 bound/0 _ N2>=0) + %<- (plus-commutative plus/z N2+0=N2) + %<- (plus-implies-ge N2+0=N2 N2>=0) %. +%term _ + %pi (leq-implies-bound-le (leq/= M1<=M2 _ nat`eq/) (bound/+ N+1+N1=N3 DM1=N3) + %<- (leq-implies-bound-le M1<=M2 DM1=N1) + %<- (plus-left-preserves-ge* N2>=N1 N+1+N2=N4 N+1+N1=N3 N4>=N3) %. +%term _ + %pi (leq-implies-bound-le (leq/> M311<=M2 N3+1+N2=N1) (bound/+ N1+1+N10=N11 DM1=N11) + %<- (plus-total N3+1+N10=N13) + %<- (leq-implies-bound-le M311<=M2 (bound/+ N3+1+N10=N13 DM1=N13) + %<- (plus-swap-succ N1+1+N10=N11 N1+N10+1=N11) + %<- (plus-swap-succ N3+1+N2=N1 N3+N2+1=N1) + %<- (plus-commutative N3+N2+1=N1 N2+1+N3=N1) + %<- (plus-swap-succ N3+1+N10=N13 N3+N10+1=N13) + %<- (plus-associative* N2+1+N3=N1 N1+N10+1=N11 N3+N10+1=N13 N2+1+N13=N11) + %<- (plus-left-preserves-ge* N20>=N13 N2+1+N20=N22 N2+1+N13=N11 N22>=N11) %. +%worlds () (leq-implies-bound-le _ _ _ _) %. +%total (L) (leq-implies-bound-le L _ _ _) %. +%theorem shift-left-preserves-leq* : forall* {M1} {M2} {N} {SM1} {D} forall {L leq M1 M2} {S1 shift N M1 SM1} exists {SL leq SM1 (map/+ N D M2)} true %. +%term _ shift-left-preserves-leq* _ shift/0 leq/0 %. +%term _ + %pi (shift-left-preserves-leq* M111<=M2 (shift/+ N+1+N1=N1') (leq/> M111<=M2 N1+1+N=N1')) + %<- (plus-swap-succ N+1+N1=N1' N+N1+1=N1') + %<- (plus-commutative N+N1+1=N1' N1+1+N=N1') %. +%worlds () (shift-left-preserves-leq* _ _ _) %. +%total {} (shift-left-preserves-leq* _ _ _) %. +%theorem shift-preserves-leq* : forall* {M1} {M2} {N} {SM1} {SM2} forall {L leq M1 M2} {S1 shift N M1 SM1} {S1 shift N M2 SM2} exists {SL leq SM1 SM2} true %. +%term _ shift-preserves-leq* _ shift/0 _ leq/0 %. +%term _ + %pi (shift-preserves-leq* (leq/= M1<=M2 D1<=D2 nat`eq/) (shift/+ N+1+N1=N1') (shift/+ N+1+N1=N2') (leq/= M1<=M2 D1<=D2 N1'=N2')) + %<- (plus-deterministic N+1+N1=N1' N+1+N1=N2' nat`eq/ nat`eq/ N1'=N2') %. +%term _ + %pi (shift-preserves-leq* (leq/> M311<=M2 N3+1+N2=N1) (shift/+ N+1+N1=N1') (shift/+ N+1+N2=N2') (leq/> M311<=M2 N3+1+N2'=N1')) + %<- (plus-commutative N3+1+N2=N1 N2+N3+1=N1) + %<- (plus-associative-converse* N2+N3+1=N1 N+1+N1=N1' N+1+N2=N2' N2'+N3+1=N1') + %<- (plus-commutative N2'+N3+1=N1' N3+1+N2'=N1') %. +%worlds () (shift-preserves-leq* _ _ _ _) %. +%total {} (shift-preserves-leq* _ _ _ _) %. +%theorem update-left-preserves-leq* : forall* {M1} {M2} {N} {D} {M1'} {M2'} forall {L leq M1 M2} {U1 update M1 N D M1'} {U2 update M2 N D M2'} exists {LP leq M1' M2'} true %. +%term _ + %pi (update-left-preserves-leq* leq/0 update/0 update/0 (leq/= leq/0 D<=D nat`eq/)) + %<- (nat`leq-reflexive _ D<=D) %. +%term _ + %pi (update-left-preserves-leq* leq/0 update/0 (update/= nat`eq/) (leq/= leq/0 D<=D nat`eq/)) + %<- (nat`leq-reflexive _ D<=D) %. +%term _ + %pi (update-left-preserves-leq* leq/0 update/0 (update/< _) (leq/= leq/0 D<=D nat`eq/)) + %<- (nat`leq-reflexive _ D<=D) %. +%term _ + %pi (update-left-preserves-leq* leq/0 update/0 (update/> U2505 N5+1+N2=N) (leq/> M500<=M5 N5+1+N2=N)) + %<- (update-left-preserves-leq* leq/0 update/0 U2505 M500<=M5) %. +%term _ + %pi (update-left-preserves-leq* (leq/= M1<=M2 D1<=D2 nat`eq/) (update/= nat`eq/) (%the (update (map/+ N D2 M2) N D M) U2) M001<=M) + %<- (update/=-inversion U2 nat`eq/ M002=M) + %<- (nat`leq-reflexive _ D<=D) + %<- (leq-respects-eq (leq/= M1<=M2 D<=D nat`eq/) eq/ M002=M M001<=M) %. +%term _ + %pi (update-left-preserves-leq* (leq/= M1<=M2 D1<=D2 nat`eq/) (update/< N3+1+N=N1) (%the (update (map/+ N1 D2 M2) N D M) U2) M00M311<=M) + %<- (update/<-inversion U2 N3+1+N=N1 M00M322=M) + %<- (nat`leq-reflexive _ D<=D) + %<- (leq-respects-eq (leq/= (leq/= M1<=M2 D1<=D2 nat`eq/) D<=D nat`eq/) eq/ M00M322=M M00M311<=M) %. +%term _ + %pi (update-left-preserves-leq* (leq/= M1<=M2 D1<=D2 nat`eq/) (update/> U1303 N3+1+N1=N) (%the (update (map/+ N1 D2 M2) N D M) U2) M113<=M) + %<- (update/>-inversion U2 N3+1+N1=N M4 U2304 M124=M) + %<- (update-left-preserves-leq* M1<=M2 U1303 U2304 M3<=M4) + %<- (leq-respects-eq (leq/= M3<=M4 D1<=D2 nat`eq/) eq/ M124=M M113<=M) %. +%term _ + %pi (update-left-preserves-leq* (leq/> M311<=M2 N3+1+N2=N1) (update/= nat`eq/) (%the (update (map/+ N2 D2 M2) N1 D M) U2) M101<=M) + %<- (update/>-inversion U2 N3+1+N2=N1 M4 U2304 M224=M) + %<- (update-left-preserves-leq* M311<=M2 (update/= nat`eq/) U2304 M301<=M4) + %<- (leq-respects-eq (leq/> M301<=M4 N3+1+N2=N1) eq/ M224=M M101<=M) %. +%term _ + %pi (update-left-preserves-leq* (leq/> M311<=M2 N3+1+N=N1) (update/< N4+1+N=N1) (update/= nat`eq/) (leq/= M411<=M2 D<=D nat`eq/)) + %<- (nat`plus-right-cancels N3+1+N=N1 N4+1+N=N1 nat`eq/ nat`eq/ N3+1=N4+1) + %<- (nat`succ-cancels N3+1=N4+1 N3=N4) + %<- (map/+-preserves-eq N3=N4 nat`eq/ eq/ M311=M411) + %<- (leq-respects-eq M311<=M2 M311=M411 eq/ M411<=M2) + %<- (nat`leq-reflexive _ D<=D) %. +%term _ + %pi (update-left-preserves-leq* (leq/> M311<=M2 N3+1+N2=N1) (update/< N4+1+N=N1) (update/< N5+1+N=N2) (leq/= (leq/> M311<=M2 N3+1+N5=N4) D<=D nat`eq/)) + %<- (nat`plus-swap-succ N5+1+N=N2 N5+N+1=N2) + %<- (nat`plus-associative-converse N5+N+1=N2 N3+1+N2=N1 N4' N3+1+N5=N4' N4'+N+1=N1) + %<- (nat`plus-swap-succ N4+1+N=N1 N4+N+1=N1) + %<- (nat`plus-right-cancels N4'+N+1=N1 N4+N+1=N1 nat`eq/ nat`eq/ N4'=N4) + %<- (nat`plus-respects-eq N3+1+N5=N4' nat`eq/ nat`eq/ N4'=N4 N3+1+N5=N4) + %<- (nat`leq-reflexive _ D<=D) %. +%term _ + %pi (update-left-preserves-leq* (leq/> M311<=M2 N3+1+N2=N1) (update/< N4+1+N=N1) (update/> U2505 N5+1+N2=N) (leq/> M50M411<=M5 N5+1+N2=N)) + %<- (nat`plus-swap-succ N5+1+N2=N N5+N2+1=N) + %<- (nat`plus-associative-converse N5+N2+1=N N4+1+N=N1 N3P N4+1+N5=N3P N3P+N2+1=N1) + %<- (nat`plus-swap-succ N3+1+N2=N1 N3+N2+1=N1) + %<- (nat`plus-right-cancels N3P+N2+1=N1 N3+N2+1=N1 nat`eq/ nat`eq/ N3P=N3) + %<- (nat`plus-respects-eq N4+1+N5=N3P nat`eq/ nat`eq/ N3P=N3 N4+1+N5=N3) + %<- (update-left-preserves-leq* M311<=M2 (update/< N4+1+N5=N3) U2505 M50M411<=M5) %. +%term _ + %pi (update-left-preserves-leq* (leq/> M311<=M2 N3+1+N2=N1) (update/> U1404 N4+1+N1=N) (%the (update (map/+ N2 D2 M2) N D M) U2) M114<=M) + %<- (nat`plus-swap-succ N3+1+N2=N1 N3+N2+1=N1) + %<- (nat`plus-associative-converse N3+N2+1=N1 N4+1+N1=N N5 N4+1+N3=N5 N5+N2+1=N) + %<- (nat`plus-swap-succ-converse N5+N2+1=N N5+1+N2=N) + %<- (update/>-inversion U2 N5+1+N2=N M5 U2505 M225=M) + %<- (update-left-preserves-leq* M311<=M2 (update/> U1404 N4+1+N3=N5) U2505 M314<=M5) + %<- (leq-respects-eq (leq/> M314<=M5 N3+1+N2=N1) eq/ M225=M M114<=M) %. +%worlds () (update-left-preserves-leq* _ _ _ _) %. +%total (U) (update-left-preserves-leq* _ _ U _) %. +%theorem update-right-preserves-leq* : forall* {M} {N} {D1} {D2} {M1'} {M2'} forall {L nat`leq D1 D2} {U1 update M N D1 M1'} {U2 update M N D2 M2'} exists {LP leq M1' M2'} true %. +%term _ update-right-preserves-leq* D1<=D2 update/0 update/0 (leq/= leq/0 D1<=D2 nat`eq/) %. +%term _ + %pi (update-right-preserves-leq* D1<=D2 (update/= nat`eq/) U2 L) + %<- (update/=-inversion U2 nat`eq/ M221=M2') + %<- (leq-reflexive _ M1<=M1) + %<- (leq-respects-eq (leq/= M1<=M1 D1<=D2 nat`eq/) eq/ M221=M2' L) %. +%term _ + %pi (update-right-preserves-leq* D1<=D2 (update/< P) U2 L) + %<- (update/<-inversion U2 P M22311=M2') + %<- (leq-reflexive _ M311<=M311) + %<- (leq-respects-eq (leq/= M311<=M311 D1<=D2 nat`eq/) eq/ M22311=M2' L) %. +%term _ + %pi (update-right-preserves-leq* D1<=D2 (update/> U1 P) U22 L) + %<- (update/>-inversion U22 P M2' U2 M112=M) + %<- (update-right-preserves-leq* D1<=D2 U1 U2 M1'<=M2') + %<- (nat`leq-reflexive _ D<=D) + %<- (leq-respects-eq (leq/= M1'<=M2' D<=D nat`eq/) eq/ M112=M L) %. +%worlds () (update-right-preserves-leq* _ _ _ _) %. +%total (U) (update-right-preserves-leq* _ U _ _) %. +%theorem update-preserves-leq* : forall* {M1} {M2} {N} {D1} {D2} {M1'} {M2'} forall {L leq M1 M2} {L nat`leq D1 D2} {U1 update M1 N D1 M1'} {U2 update M2 N D2 M2'} exists {LP leq M1' M2'} true %. +%term _ + %pi (update-preserves-leq* M1<=M2 D1<=D2 M1^N=D1->M1' M2^N=D2->M2' M1'<=M2') + %<- (update-total M2^N=D1->MM) + %<- (update-left-preserves-leq* M1<=M2 M1^N=D1->M1' M2^N=D1->MM M1'<=MM) + %<- (update-right-preserves-leq* D1<=D2 M2^N=D1->MM M2^N=D2->M2' MM<=M2') + %<- (leq-transitive M1'<=MM MM<=M2' M1'<=M2') %. +%worlds () (update-preserves-leq* _ _ _ _ _) %. +%total {} (update-preserves-leq* _ _ _ _ _) %. +%theorem not-member-update-implies-leq : forall* {M1} {N} {D} {M2} forall {F not-member M1 N} {U update M1 N D M2} exists {L leq M1 M2} true %. +%term _ not-member-update-implies-leq not-member/0 U leq/0 %. +%term _ + %pi (not-member-update-implies-leq (not-member/< N2 M311<=M311 N3+1+N2=N1) eq/ M22311=M2 L) %. +%term _ + %pi (not-member-update-implies-leq (not-member/> F1 N0+1+N1=N2) U L) + %<- (update/>-inversion U N0+1+N1=N2 _ U1 M112=M) + %<- (not-member-update-implies-leq F1 U1 L1) + %<- (nat`leq-reflexive _ DL) + %<- (leq-respects-eq (leq/= L1 DL nat`eq/) eq/ M112=M L) %. +%worlds () (not-member-update-implies-leq _ _ _) %. +%total (F) (not-member-update-implies-leq F _ _) %. +%theorem lookup-update-preserves-leq : forall* {M1} {N} {D1} {D2} {M2} forall {L lookup M1 N D1} {U update M1 N D2 M2} {L nat`leq D1 D2} exists {L leq M1 M2} true %. +%term _ + %pi (lookup-update-preserves-leq (lookup/= nat`eq/) U D1<=D2 L) + %<- (update/=-inversion U nat`eq/ M221=M2) + %<- (leq-reflexive _ M1<=M1) + %<- (leq-respects-eq (leq/= M1<=M1 D1<=D2 nat`eq/) eq/ M221=M2 L) %. +%term _ + %pi (lookup-update-preserves-leq (lookup/> L1 N0+1+N1=N2) U D1<=D2 L) + %<- (update/>-inversion U N0+1+N1=N2 M2 U1 M112=M) + %<- (lookup-update-preserves-leq L1 U1 D1<=D2 L1') + %<- (nat`leq-reflexive _ D1<=D1) + %<- (leq-respects-eq (leq/= L1' D1<=D1 nat`eq/) eq/ M112=M L) %. +%worlds () (lookup-update-preserves-leq _ _ _ _) %. +%total (L) (lookup-update-preserves-leq L _ _ _) %. +%%%% Map addition +%%% Definition of union +%sort union %. +%term union/L union map/0 M M %. +%term union/R union M map/0 M %. +%term union/= + %pi (union (map/+ N1 D1 M1) (map/+ N2 D2 M2) (map/+ N1 D3 M3)) + %<- (nat`eq N1 N2) + %<- (nat`union D1 D2 D3) + %<- (union M1 M2 M3) %. +%term union/< + %pi (union (map/+ N1 D1 M1) (map/+ N2 D2 M2) (map/+ N1 D1 M3)) + %<- (nat`plus (s N0) N1 N2) + %<- (union M1 (map/+ N0 D2 M2) M3) %. +%term union/> + %pi (union (map/+ N1 D1 M1) (map/+ N2 D2 M2) (map/+ N2 D2 M3)) + %<- (nat`plus (s N3) N2 N1) + %<- (union (map/+ N3 D1 M1) M2 M3) %. +%%% Theorems about union +%theorem false-implies-union : forall* {M1} {M2} {M3} forall {F void} exists {D union M1 M2 M3} true %. +%worlds () (false-implies-union _ _) %. +%total {} (false-implies-union _ _) %. +%theorem union-respects-eq : forall* {M1} {M2} {M3} {M1P} {M2P} {M3P} forall {A union M1 M2 M3} {E1 eq M1 M1P} {E2 eq M2 M2P} {E3 eq M3 M3P} exists {AP union M1P M2P M3P} true %. +%term _ union-respects-eq A eq/ eq/ eq/ A %. +%worlds () (union-respects-eq _ _ _ _ _) %. +%total {} (union-respects-eq _ _ _ _ _) %. +%reduces = A AP (union-respects-eq A _ _ _ AP) %. +%% Inversion lemmas for union +%theorem union/=-inversion : forall* {N1} {D1} {M1} {N2} {D2} {M2} {M} forall {A union (map/+ N1 D1 M1) (map/+ N2 D2 M2) M} {G nat`eq N1 N2} exists {D3} {M3} {D nat`union D1 D2 D3} {AP union M1 M2 M3} {E eq M (map/+ N1 D3 M3)} true %. +%term _ union/=-inversion (union/= MM DD nat`eq/) _ _ _ DD MM eq/ %. +%term _ + %pi (union/=-inversion (union/< (%the (union _ (map/+ N0 D2 M2) M3) JP) N0+1+N=N) nat`eq/ D2 M3 DJ MJ ME) + %<- (plus-implies-gt N0+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (nat`false-implies-union F DJ) + %<- (false-implies-eq F (%the (eq (map/+ N0 D2 M2) M2) M022=M2)) + %<- (union-respects-eq JP eq/ M022=M2 eq/ MJ) + %<- (false-implies-eq F ME) %. +%term _ + %pi (union/=-inversion (union/> (%the (union (map/+ _ D1 M1) M2 M3) JP) N3+1+N=N) nat`eq/ D1 M3 DJ MJ ME) + %<- (nat`plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (nat`gt-anti-reflexive N>N F) + %<- (nat`false-implies-union F DJ) + %<- (false-implies-eq F (%the (eq (map/+ N3 D1 M1) M1) M311=M1)) + %<- (union-respects-eq JP M311=M1 eq/ eq/ MJ) + %<- (false-implies-eq F ME) %. +%worlds () (union/=-inversion _ _ _ _ _ _ _) %. +%total {} (union/=-inversion _ _ _ _ _ _ _) %. +%reduces < JP J (union/=-inversion J _ _ _ _ JP _) %. +%theorem union/<-inversion : forall* {N1} {D1} {M1} {N2} {D2} {M2} {M} {N0} forall {A union (map/+ N1 D1 M1) (map/+ N2 D2 M2) M} {P plus (s N0) N1 N2} exists {M3} {AP union M1 (map/+ N0 D2 M2) M3} {E eq M (map/+ N1 D1 M3)} true %. +%term _ + %pi (union/<-inversion (union/< J P) P' _ J' eq/) + %<- (nat`plus-right-cancels P P' nat`eq/ nat`eq/ N0+1=N0'+1) + %<- (succ-cancels N0+1=N0'+1 N0=N0P) + %<- (map/+-preserves-eq N0=N0P nat`eq/ eq/ M022=M022') + %<- (union-respects-eq J eq/ M022=M022' eq/ J') %. +%term _ + %pi (union/<-inversion (union/= J' _ nat`eq/) N0+1+N=N M3 J E) + %<- (plus-implies-gt N0+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-eq F M2=M022) + %<- (union-respects-eq J' eq/ M2=M022 eq/ J) + %<- (false-implies-eq F E) %. +%term _ + %pi (union/<-inversion (union/> J' N3+1+N2=N1) N0+1+N1=N2 M3 J E) + %<- (nat`plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) + %<- (nat`plus-implies-gt N0+1+N1=N2 nat`eq/ N2>N1) + %<- (nat`gt-anti-symmetric N1>N2 N2>N1 F) + %<- (false-implies-eq F M311=M1) + %<- (false-implies-eq F M2=M022) + %<- (union-respects-eq J' M311=M1 M2=M022 eq/ J) + %<- (false-implies-eq F E) %. +%worlds () (union/<-inversion _ _ _ _ _) %. +%total {} (union/<-inversion _ _ _ _ _) %. +%reduces < JP J (union/<-inversion J _ _ JP _) %. +%theorem union/>-inversion : forall* {N1} {D1} {M1} {N2} {D2} {M2} {N3} {M} forall {A union (map/+ N1 D1 M1) (map/+ N2 D2 M2) M} {P plus (s N3) N2 N1} exists {M3} {AP union (map/+ N3 D1 M1) M2 M3} {E eq M (map/+ N2 D2 M3)} true %. +%term _ + %pi (union/>-inversion (union/> J P) P' _ J' eq/) + %<- (nat`plus-right-cancels P P' nat`eq/ nat`eq/ N3+1=N3'+1) + %<- (succ-cancels N3+1=N3'+1 N3=N3P) + %<- (map/+-preserves-eq N3=N3P nat`eq/ eq/ M311=M311') + %<- (union-respects-eq J M311=M311' eq/ eq/ J') %. +%term _ + %pi (union/>-inversion (union/= J' _ nat`eq/) N3+1+N=N M3 J E) + %<- (nat`plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-eq F M1=M311) + %<- (union-respects-eq J' M1=M311 eq/ eq/ J) + %<- (false-implies-eq F E) %. +%term _ + %pi (union/>-inversion (union/< J' N0+1+N1=N2) N3+1+N2=N1 M3 J E) + %<- (nat`plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) + %<- (nat`plus-implies-gt N0+1+N1=N2 nat`eq/ N2>N1) + %<- (nat`gt-anti-symmetric N1>N2 N2>N1 F) + %<- (false-implies-eq F M1=M311) + %<- (false-implies-eq F M022=M2) + %<- (union-respects-eq J' M1=M311 M022=M2 eq/ J) + %<- (false-implies-eq F E) %. +%worlds () (union/>-inversion _ _ _ _ _) %. +%total {} (union/>-inversion _ _ _ _ _) %. +%reduces < JP J (union/>-inversion J _ _ JP _) %. +%theorem union-deterministic : forall* {M1} {M2} {M3} {M1P} {M2P} {M3P} forall {A union M1 M2 M3} {AP union M1P M2P M3P} {E1 eq M1 M1P} {E2 eq M2 M2P} exists {E3 eq M3 M3P} true %. +%term _ union-deterministic union/L union/L eq/ eq/ eq/ %. +%term _ union-deterministic union/L union/R eq/ eq/ eq/ %. +%term _ union-deterministic union/R union/L eq/ eq/ eq/ %. +%term _ union-deterministic union/R union/R eq/ eq/ eq/ %. +%term _ + %pi (union-deterministic (union/= M1+M2=M3 D1+D2=D3 nat`eq/) (union/= M1+M2=M3' D1+D2=D3P nat`eq/) eq/ eq/ M=M') + %<- (nat`union-deterministic D1+D2=D3 D1+D2=D3P nat`eq/ nat`eq/ D3=D3') + %<- (union-deterministic M1+M2=M3 M1+M2=M3' eq/ eq/ M3=M3') + %<- (map/+-preserves-eq nat`eq/ D3=D3' M3=M3' M=M') %. +%term _ + %pi (union-deterministic (union/< M1+MT=M3 N0+1+N1=N2) (union/< M1+MT'=M3' N0'+1+N1=N2) eq/ eq/ M=M') + %<- (plus-right-cancels N0+1+N1=N2 N0'+1+N1=N2 nat`eq/ nat`eq/ N0+1=N0'+1) + %<- (succ-cancels N0+1=N0'+1 N0=N0P) + %<- (map/+-preserves-eq N0=N0P nat`eq/ eq/ MT=MT') + %<- (union-deterministic M1+MT=M3 M1+MT'=M3' eq/ MT=MT' M3=M3') + %<- (map/+-preserves-eq nat`eq/ nat`eq/ M3=M3' M=M') %. +%term _ + %pi (union-deterministic (union/> MT+M2=M3 N3+1+N2=N1) (union/> MT'+M2=M3' N3'+1+N2=N1) eq/ eq/ E) + %<- (plus-right-cancels N3+1+N2=N1 N3'+1+N2=N1 nat`eq/ nat`eq/ N3+1=N3'+1) + %<- (succ-cancels N3+1=N3'+1 N3=N3P) + %<- (map/+-preserves-eq N3=N3P nat`eq/ eq/ MT=MT') + %<- (union-deterministic MT+M2=M3 MT'+M2=M3' MT=MT' eq/ M3=M3') + %<- (map/+-preserves-eq nat`eq/ nat`eq/ M3=M3' E) %. +%% contradiction cases: +%term _ + %pi (union-deterministic (union/= _ _ nat`eq/) (union/< _ N'+1+N=N) eq/ eq/ E) + %<- (plus-implies-gt N'+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-eq F E) %. +%term _ + %pi (union-deterministic (union/= _ _ nat`eq/) (union/> _ N'+1+N=N) eq/ eq/ E) + %<- (plus-implies-gt N'+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-eq F E) %. +%term _ + %pi (union-deterministic (union/< _ NP+1+N=N) (union/= _ _ nat`eq/) eq/ eq/ E) + %<- (plus-implies-gt NP+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-eq F E) %. +%term _ + %pi (union-deterministic (union/< _ N0+1+N1=N2) (union/> _ N3+1+N2=N1) eq/ eq/ E) + %<- (nat`plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) + %<- (nat`plus-implies-gt N0+1+N1=N2 nat`eq/ N2>N1) + %<- (nat`gt-anti-symmetric N1>N2 N2>N1 F) + %<- (false-implies-eq F E) %. +%term _ + %pi (union-deterministic (union/> _ NP+1+N=N) (union/= _ _ nat`eq/) eq/ eq/ E) + %<- (plus-implies-gt NP+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-eq F E) %. +%term _ + %pi (union-deterministic (union/> _ N3+1+N2=N1) (union/< _ N0+1+N1=N2) eq/ eq/ E) + %<- (nat`plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) + %<- (nat`plus-implies-gt N0+1+N1=N2 nat`eq/ N2>N1) + %<- (nat`gt-anti-symmetric N1>N2 N2>N1 F) + %<- (false-implies-eq F E) %. +%worlds () (union-deterministic _ _ _ _ _) %. +%total (A) (union-deterministic A _ _ _ _) %. +%theorem union-total* : forall {M1} {M2} exists {M3} {A union M1 M2 M3} true %. +%% we need some lemmas +%% We need them to ensure termination because +%% union substitutes new maps on recursive calls which +%% makes it hard to prove the arguments get smaller. +%theorem union-map/+-M-total* : forall {N1} {D1} {M1} {M2} exists {M3} {A union (map/+ N1 D1 M1) M2 M3} true %. +%theorem union-M-map/+-total* : forall {M1} {N2} {D2} {M2} exists {M3} {A union M1 (map/+ N2 D2 M2) M3} true %. +%theorem union-map/+-map/+-total* : forall {N1} {D1} {M1} {N2} {D2} {M2} {C} {CMP nat`compare N1 N2 C} exists {M3} {A union (map/+ N1 D1 M1) (map/+ N2 D2 M2) M3} true %. +%term _ union-total* map/0 M M union/L %. +%term _ union-total* M map/0 M union/R %. +%term _ + %pi (union-total* (map/+ N1 D1 M1) (map/+ N2 D2 M2) M3 A) + %<- (nat`compare-total* N1 N2 C CMP) + %<- (union-map/+-map/+-total* N1 D1 M1 N2 D2 M2 C CMP M3 A) %. +%term _ + %pi (union-map/+-map/+-total* N1 D1 M1 N2 D2 M2 equal CMP (map/+ N1 D3 M3) (union/= M1+M2=M3 D1+D2=D3 N1=N2)) + %<- (equal-implies-eq CMP N1=N2) + %<- (nat`union-total* D1 D2 D3 D1+D2=D3) + %<- (union-total* M1 M2 M3 M1+M2=M3) %. +%term _ + %pi (union-map/+-map/+-total* N1 D1 M1 N2 D2 M2 less CMP (map/+ N1 D1 M3) (union/< M1+T=M3 N0+1+N1=N2)) + %<- (less-implies-lt CMP N2>N1) + %<- (gt-implies-plus N2>N1 _ N0+1+N1=N2) + %<- (union-M-map/+-total* M1 N0 D2 M2 M3 M1+T=M3) %. +%term _ + %pi (union-map/+-map/+-total* N1 D1 M1 N2 D2 M2 greater CMP (map/+ N2 D2 M3) (union/> T+M2=M3 N3+1+N2=N1)) + %<- (greater-implies-gt CMP N1>N2) + %<- (gt-implies-plus N1>N2 _ N3+1+N2=N1) + %<- (union-map/+-M-total* N3 D1 M1 M2 M3 T+M2=M3) %. +%term _ union-M-map/+-total* map/0 N2 D2 M2 (map/+ N2 D2 M2) union/L %. +%term _ + %pi (union-M-map/+-total* (map/+ N1 D1 M1) N2 D2 M2 M3 A) + %<- (nat`compare-total* N1 N2 C CMP) + %<- (union-map/+-map/+-total* N1 D1 M1 N2 D2 M2 C CMP M3 A) %. +%term _ union-map/+-M-total* N1 D1 M1 map/0 (map/+ N1 D1 M1) union/R %. +%term _ + %pi (union-map/+-M-total* N1 D1 M1 (map/+ N2 D2 M2) M3 A) + %<- (nat`compare-total* N1 N2 C CMP) + %<- (union-map/+-map/+-total* N1 D1 M1 N2 D2 M2 C CMP M3 A) %. +%worlds () (union-total* _ _ _ _) (union-M-map/+-total* _ _ _ _ _ _) (union-map/+-M-total* _ _ _ _ _ _) (union-map/+-map/+-total* _ _ _ _ _ _ _ _ _ _) %. +%total [(M1a M1b M1c M1d) (M2a M2b M2c M2d)] (union-total* M1d M2d _ _) (union-M-map/+-total* M1c _ _ M2c _ _) (union-map/+-M-total* _ _ M1b M2b _ _) (union-map/+-map/+-total* _ _ M1a _ _ M2a _ _ _ _) %. +%inline union-total union-total* _ _ _ %. +%theorem disjoint-union-total : forall* {M1} {M2} forall {D disjoint M1 M2} exists {M3} {A union M1 M2 M3} true %. +%term _ disjoint-union-total disjoint/L _ union/L %. +%term _ disjoint-union-total disjoint/R _ union/R %. +%term _ + %pi (disjoint-union-total (disjoint/< D P) _ (union/< J P)) + %<- (disjoint-union-total D _ J) %. +%term _ + %pi (disjoint-union-total (disjoint/> D P) _ (union/> J P)) + %<- (disjoint-union-total D _ J) %. +%worlds () (disjoint-union-total _ _ _) %. +%total (D) (disjoint-union-total D _ _) %. +%theorem union-empty-implies-empty : forall* {M1} {M2} forall {A union M1 M2 map/0} exists {E1 eq M1 map/0} {E2 eq M2 map/0} true %. +%term _ union-empty-implies-empty union/L eq/ eq/ %. +%term _ union-empty-implies-empty union/R eq/ eq/ %. +%worlds () (union-empty-implies-empty _ _ _) %. +%total {} (union-empty-implies-empty _ _ _) %. +%theorem union-preserves-disjoint* : forall* {M1} {M2} {M3} {M4} forall {D1 disjoint M1 M4} {D2 disjoint M2 M4} {A union M1 M2 M3} exists {D3 disjoint M3 M4} true %. +% a lemma that counts the size of maps to help prove termination +%theorem union-preserves-disjoint*/L : forall* {M1} {M2} {M3} {M4} forall {S1} {S2} {SZ1 size M1 S1} {SZ2 size M2 S2} {D1 disjoint M1 M4} {D2 disjoint M2 M4} {A union M1 M2 M3} exists {D3 disjoint M3 M4} true %. +%term _ + %pi (union-preserves-disjoint* D1 D2 J D3) + %<- (size-total SZ1) + %<- (size-total SZ2) + %<- (union-preserves-disjoint*/L _ _ SZ1 SZ2 D1 D2 J D3) %. +%term _ union-preserves-disjoint*/L _ _ _ _ disjoint/R _ _ disjoint/R %. +%term _ union-preserves-disjoint*/L _ _ _ _ _ disjoint/R _ disjoint/R %. +%term _ union-preserves-disjoint*/L _ _ _ _ _ D union/L D %. +%term _ union-preserves-disjoint*/L _ _ _ _ D _ union/R D %. +%term _ + %pi (union-preserves-disjoint*/L (s S1) (s S2) (size/+ SZ1) (size/+ SZ2) (disjoint/< D1 N5+1+N1=N4) D2X (union/= J _ nat`eq/) (disjoint/< D3 N5+1+N1=N4)) + %<- (disjoint/<-inversion D2X N5+1+N1=N4 D2) + %<- (union-preserves-disjoint*/L S1 S2 SZ1 SZ2 D1 D2 J D3) %. +%term _ + %pi (union-preserves-disjoint*/L (s S1) (s S2) (size/+ SZ1) (size/+ SZ2) (disjoint/< D1 N5+1+N1=N4) D2X (union/> J N3+1+N2=N1) (disjoint/< D3 N6+1+N2=N4)) + %<- (plus-swap-succ N3+1+N2=N1 N3+N2+1=N1) + %<- (plus-associative-converse N3+N2+1=N1 N5+1+N1=N4 N6 N5+1+N3=N6 N6+N2+1=N4) + %<- (plus-swap-succ-converse N6+N2+1=N4 N6+1+N2=N4) + %<- (disjoint/<-inversion D2X N6+1+N2=N4 D2) + %<- (plus-swap-succ N5+1+N3=N6 N5+N3+1=N6) + %<- (plus-commutative N5+N3+1=N6 N3+1+N5=N6) + %<- (shift-right-preserves-disjoint D1 (shift/+ N3+1+N5=N6) D1< D1 N5+1+N4=N1) D2X (union/= J DJ nat`eq/) (disjoint/> D3 N5+1+N4=N1)) + %<- (disjoint/>-inversion D2X N5+1+N4=N1 D2) + %<- (union-preserves-disjoint*/L _ _ (size/+ SZ1) (size/+ SZ2) D1 D2 (union/= J DJ nat`eq/) D3) %. +%term _ + %pi (union-preserves-disjoint*/L (s S1) (s S2) (size/+ SZ1) (size/+ SZ2) (disjoint/> D1 N5+1+N4=N1) D2X (union/< J N0+1+N1=N2) (disjoint/> D3 N5+1+N4=N1)) + %<- (plus-swap-succ N5+1+N4=N1 N5+N4+1=N1) + %<- (plus-associative-converse N5+N4+1=N1 N0+1+N1=N2 N6 N0+1+N5=N6 N6+N4+1=N2) + %<- (plus-swap-succ-converse N6+N4+1=N2 N6+1+N4=N2) + %<- (disjoint/>-inversion D2X N6+1+N4=N2 D2) + %<- (union-preserves-disjoint*/L _ _ (size/+ SZ1) (size/+ SZ2) D1 D2 (union/< J N0+1+N5=N6) D3) %. +%term _ + %pi (union-preserves-disjoint*/L (s S1) (s S2) (size/+ SZ1) (size/+ SZ2) D1X (disjoint/< D2 N6+1+N2=N4) (union/< J N0+1+N1=N2) (disjoint/< D3 N5+1+N1=N4)) + %<- (plus-swap-succ N0+1+N1=N2 N0+N1+1=N2) + %<- (plus-associative-converse N0+N1+1=N2 N6+1+N2=N4 N5 N6+1+N0=N5 N5+N1+1=N4) + %<- (plus-swap-succ-converse N5+N1+1=N4 N5+1+N1=N4) + %<- (disjoint/<-inversion D1X N5+1+N1=N4 D1) + %<- (plus-swap-succ N6+1+N0=N5 N6+N0+1=N5) + %<- (plus-commutative N6+N0+1=N5 N0+1+N6=N5) + %<- (shift-right-preserves-disjoint D2 (shift/+ N0+1+N6=N5) D2< D2 N6+1+N4=N2) (union/> J N3+1+N2=N1) (disjoint/> D3 N6+1+N4=N2)) + %<- (plus-swap-succ N6+1+N4=N2 N6+N4+1=N2) + %<- (plus-associative-converse N6+N4+1=N2 N3+1+N2=N1 N5 N3+1+N6=N5 N5+N4+1=N1) + %<- (plus-swap-succ-converse N5+N4+1=N1 N5+1+N4=N1) + %<- (disjoint/>-inversion D1X N5+1+N4=N1 D1) + %<- (union-preserves-disjoint*/L _ _ (size/+ SZ1) (size/+ SZ2) D1 D2 (union/> J N3+1+N6=N5) D3) %. +%term _ + %pi (union-preserves-disjoint*/L (s S1) (s S2) (size/+ SZ1) (size/+ SZ2) (disjoint/< D1 N5+1+N1=N4) (disjoint/> D2 N6+1+N4=N2) JX D3X) + %<- (plus-swap-succ N5+1+N1=N4 N5+N1+1=N4) + %<- (plus-associative-converse N5+N1+1=N4 N6+1+N4=N2 N0 N6+1+N5=N0 N0+N1+1=N2) + %<- (plus-swap-succ-converse N0+N1+1=N2 N0+1+N1=N2) + %<- (union/<-inversion JX N0+1+N1=N2 _ J M=M113) + %<- (eq-symmetric M=M113 M113=M) + %<- (union-preserves-disjoint*/L S1 (s S2) SZ1 (size/+ SZ2) D1 (disjoint/> D2 N6+1+N5=N0) J D3) + %<- (disjoint-respects-eq (disjoint/< D3 N5+1+N1=N4) M113=M eq/ D3X) %. +%term _ + %pi (union-preserves-disjoint*/L (s S1) (s S2) (size/+ SZ1) (size/+ SZ2) (disjoint/> D1 N5+1+N4=N1) (disjoint/< D2 N6+1+N2=N4) JX D3X) + %<- (plus-swap-succ N6+1+N2=N4 N6+N2+1=N4) + %<- (plus-associative-converse N6+N2+1=N4 N5+1+N4=N1 N3 N5+1+N6=N3 N3+N2+1=N1) + %<- (plus-swap-succ-converse N3+N2+1=N1 N3+1+N2=N1) + %<- (union/>-inversion JX N3+1+N2=N1 _ J M=M223) + %<- (eq-symmetric M=M223 M223=M) + %<- (union-preserves-disjoint*/L _ _ (size/+ SZ1) SZ2 (disjoint/> D1 N5+1+N6=N3) D2 J D3) + %<- (disjoint-respects-eq (disjoint/< D3 N6+1+N2=N4) M223=M eq/ D3X) %. +%worlds () (union-preserves-disjoint*/L _ _ _ _ _ _ _ _) %. +%total {S1 S2 D1} (union-preserves-disjoint*/L S1 S2 _ _ D1 _ _ _) %. +%worlds () (union-preserves-disjoint* _ _ _ _) %. +%total {} (union-preserves-disjoint* _ _ _ _) %. +%theorem shift-left-preserves-union : forall* {N} {D} {M1} {M2} {M3} {SM1} forall {A union M1 M2 M3} {S1 shift N M1 SM1} exists {SA union SM1 (map/+ N D M2) (map/+ N D M3)} true %. +%term _ shift-left-preserves-union union/L shift/0 union/L %. +%term _ shift-left-preserves-union union/R shift/0 union/L %. +%term _ + %pi (shift-left-preserves-union M111+M2=M3 (shift/+ N+1+N1=N1') (union/> M111+M2=M3 N1+1+N=N1')) + %<- (plus-swap-succ N+1+N1=N1' N+N1+1=N1') + %<- (plus-commutative N+N1+1=N1' N1+1+N=N1') %. +%worlds () (shift-left-preserves-union _ _ _) %. +%total {} (shift-left-preserves-union _ _ _) %. +%theorem shift-left-preserves-union-converse : forall* {N} {D} {M1} {M2} {SM1} {SM3} forall {SA union SM1 (map/+ N D M2) SM3} {S1 shift N M1 SM1} exists {M3} {A union M1 M2 M3} {E eq (map/+ N D M3) SM3} true %. +%term _ shift-left-preserves-union-converse union/L shift/0 _ union/L eq/ %. +%term _ + %pi (shift-left-preserves-union-converse M111+M222=SM3 (shift/+ N2+1+N3=N1) M3 M311+M2=M3 M223=SM3) + %<- (plus-swap-succ N2+1+N3=N1 N2+N3+1=N1) + %<- (plus-commutative N2+N3+1=N1 N3+1+N2=N1) + %<- (union/>-inversion M111+M222=SM3 N3+1+N2=N1 M3 M311+M2=M3 SM3=M223) + %<- (eq-symmetric SM3=M223 M223=SM3) %. +%worlds () (shift-left-preserves-union-converse _ _ _ _ _) %. +%total {} (shift-left-preserves-union-converse _ _ _ _ _) %. +%theorem shift-right-preserves-union : forall* {N} {D} {M1} {M2} {M3} {SM2} forall {A union M1 M2 M3} {S2 shift N M2 SM2} exists {SA union (map/+ N D M1) SM2 (map/+ N D M3)} true %. +%term _ shift-right-preserves-union union/L shift/0 union/R %. +%term _ shift-right-preserves-union union/R shift/0 union/R %. +%term _ + %pi (shift-right-preserves-union M1+M222=M3 (shift/+ N+1+N2=N2') (union/< M1+M222=M3 N2+1+N=N2')) + %<- (plus-swap-succ N+1+N2=N2' N+N2+1=N2') + %<- (plus-commutative N+N2+1=N2' N2+1+N=N2') %. +%worlds () (shift-right-preserves-union _ _ _) %. +%total {} (shift-right-preserves-union _ _ _) %. +%theorem shift-right-preserves-union-converse : forall* {N} {D} {M1} {M2} {SM2} {SM3} forall {SA union (map/+ N D M1) SM2 SM3} {S2 shift N M2 SM2} exists {M3} {A union M1 M2 M3} {E eq (map/+ N D M3) SM3} true %. +%term _ shift-right-preserves-union-converse union/R shift/0 _ union/R eq/ %. +%term _ + %pi (shift-right-preserves-union-converse M111+M322=SM3 (shift/+ N1+1+N2=N3) M3 M1+M222=M3 M133=SM3) + %<- (plus-swap-succ N1+1+N2=N3 N1+N2+1=N3) + %<- (plus-commutative N1+N2+1=N3 N2+1+N1=N3) + %<- (union/<-inversion M111+M322=SM3 N2+1+N1=N3 M3 M1+M222=M3 SM3=M133) + %<- (eq-symmetric SM3=M133 M133=SM3) %. +%worlds () (shift-right-preserves-union-converse _ _ _ _ _) %. +%total {} (shift-right-preserves-union-converse _ _ _ _ _) %. +%theorem shift-preserves-union : forall* {N} {M1} {M2} {M3} {SM1} {SM2} {SM3} forall {A union M1 M2 M3} {S1 shift N M1 SM1} {S2 shift N M2 SM2} {S3 shift N M3 SM3} exists {SA union SM1 SM2 SM3} true %. +%term _ + %pi (shift-preserves-union union/L shift/0 M2< M311+M2=M3 N3+1+N2=N1) (shift/+ N+1+N1=N4) (shift/+ N+1+N2=N5) (shift/+ N+1+N2=N6) M411+M522=M623) + %<- (plus-deterministic N+1+N2=N5 N+1+N2=N6 nat`eq/ nat`eq/ N5=N6) + %<- (plus-swap-succ N+1+N2=N5 N+N2+1=N5) + %<- (plus-commutative N+N2+1=N5 N2+1+N=N5) + %<- (plus-commutative N3+1+N2=N1 N2+N3+1=N1) + %<- (plus-associative-converse* N2+N3+1=N1 N+1+N1=N4 N+1+N2=N5 N5+N3+1=N4) + %<- (plus-commutative N5+N3+1=N4 N3+1+N5=N4) + %<- (map/+-preserves-eq N5=N6 nat`eq/ eq/ M523=M623) + %<- (union-respects-eq (union/> M311+M2=M3 N3+1+N5=N4) eq/ eq/ M523=M623 M411+M522=M623) %. +%worlds () (shift-preserves-union _ _ _ _ _) %. +%total {} (shift-preserves-union _ _ _ _ _) %. +%theorem shift-preserves-union-converse : forall* {N} {M1} {M2} {SM1} {SM2} {SM3} forall {SA union SM1 SM2 SM3} {S1 shift N M1 SM1} {S2 shift N M2 SM2} exists {M3} {A union M1 M2 M3} {S3 shift N M3 SM3} true %. +%term _ shift-preserves-union-converse union/L shift/0 M2< M611+M2=M3 N6+1+N5=N4) (shift/+ N+1+N1=N4) (shift/+ N+1+N2=N5) _ (union/> M611+M2=M3 N6+1+N2=N1) (shift/+ N+1+N2=N5)) + %<- (plus-commutative N+1+N2=N5 N2+N+1=N5) + %<- (plus-swap-succ-converse N2+N+1=N5 N2+1+N=N5) + %<- (plus-associative-converse N2+N+1=N5 N6+1+N5=N4 N1' N6+1+N2=N1' N1'+N+1=N4) + %<- (plus-commutative N+1+N1=N4 N1+N+1=N4) + %<- (plus-right-cancels N1'+N+1=N4 N1+N+1=N4 nat`eq/ nat`eq/ N1'=N1) + %<- (plus-respects-eq N6+1+N2=N1' nat`eq/ nat`eq/ N1'=N1 N6+1+N2=N1) %. +%worlds () (shift-preserves-union-converse _ _ _ _ _ _) %. +%total {} (shift-preserves-union-converse _ _ _ _ _ _) %. +%theorem union-commutative : forall* {M1} {M2} {M3} forall {A union M1 M2 M3} exists {AP union M2 M1 M3} true %. +%term _ union-commutative union/L union/R %. +%term _ union-commutative union/R union/L %. +%term _ + %pi (union-commutative (union/= M1+M2=M3 D1+D2=D3 nat`eq/) (union/= M2+M1=M3 D2+D1=D3 nat`eq/)) + %<- (nat`union-commutative D1+D2=D3 D2+D1=D3) + %<- (union-commutative M1+M2=M3 M2+M1=M3) %. +%term _ + %pi (union-commutative (union/< M1+MT=M3 N0+1+N1=N2) (union/> MT+M1=M3 N0+1+N1=N2)) + %<- (union-commutative M1+MT=M3 MT+M1=M3) %. +%term _ + %pi (union-commutative (union/> MT+M2=M3 N3+1+N2=N1) (union/< M2+MT=M3 N3+1+N2=N1)) + %<- (union-commutative MT+M2=M3 M2+MT=M3) %. +%worlds () (union-commutative _ _) %. +%total (A) (union-commutative A _) %. +%theorem union-associative : forall* {M1} {M2} {M3} {M4} {M7} forall {A12 union M1 M2 M3} {A34 union M3 M4 M7} exists {M6} {A24 union M2 M4 M6} {A16 union M1 M6 M7} true %. +%% a lemma +%theorem union-associative-union/<-union/< : forall* {N1} {D1} {M1} {N2} {D2} {M2} {N0} {N5} {N4} {D4} {M4} {M6} {M7} forall {PLUS012 nat`plus (s N0) N1 N2} {PLUS514 nat`plus (s N5) N1 N4} {JOIN246 union (map/+ N0 D2 M2) (map/+ N5 D4 M4) M6} {JOIN167 union M1 M6 M7} exists {M} {JOIN24 union (map/+ N2 D2 M2) (map/+ N4 D4 M4) M} {JOIN union (map/+ N1 D1 M1) M (map/+ N1 D1 M7)} true %. +%term _ union-associative union/L A _ A union/L %. +%term _ union-associative A union/R _ union/R A %. +%term _ union-associative union/R A _ union/L A %. +%term _ + %pi (union-associative (union/= M1+M2=M3 D1+D2=D3 nat`eq/) (union/= M3+M4=M7 D3+D4=D7 nat`eq/) (map/+ _ D6 M6) (union/= M2+M4=M6 D2+D4=D6 nat`eq/) (union/= M1+M6=M7 D1+D6=D7 nat`eq/)) + %<- (nat`union-associative D1+D2=D3 D3+D4=D7 D6 D2+D4=D6 D1+D6=D7) + %<- (union-associative M1+M2=M3 M3+M4=M7 M6 M2+M4=M6 M1+M6=M7) %. +%term _ + %pi (union-associative (union/= M1+M2=M3 D1+D2=D3 nat`eq/) (union/< M3+M044=M7 N0+1+N3=N4) (map/+ _ _ M6) (union/< M2+M044=M6 N0+1+N3=N4) (union/= M1+M6=M7 D1+D2=D3 nat`eq/)) + %<- (union-associative M1+M2=M3 M3+M044=M7 M6 M2+M044=M6 M1+M6=M7) %. +%term _ + %pi (union-associative (union/= M1+M2=M3 D1+D2=D3 nat`eq/) (union/> M533+M4=M7 N5+1+N4=N3) (map/+ _ _ M6) (union/> M522+M4=M6 N5+1+N4=N3) (union/> M511+M6=M7 N5+1+N4=N3)) + %<- (union-associative (union/= M1+M2=M3 D1+D2=D3 nat`eq/) M533+M4=M7 M6 M522+M4=M6 M511+M6=M7) %. +%term _ + %pi (union-associative (union/< M1+M022=M3 N0+1+N1=N2) (union/= M3+M4=M7 D1+D4=D7 nat`eq/) (map/+ _ _ M6) (union/> M022+M3=M6 N0+1+N1=N2) (union/= M1+M6=M7 D1+D4=D7 nat`eq/)) + %<- (union-associative M1+M022=M3 M3+M4=M7 M6 M022+M3=M6 M1+M6=M7) %. +%% the hardest of all 11 cases! +%term _ {M111+M=M117 union (map/+ N1 D1 M1) M (map/+ N1 D1 M7)} {M3+M544=M7 union M3 (map/+ N5 D4 M4) M7} {M1+M022=M3 union M1 (map/+ N0 D2 M2) M3} {M1+M6=M7 union M1 M6 M7} + %pi (union-associative (union/< M1+M022=M3 N0+1+N1=N2) (union/< M3+M544=M7 N5+1+N1=N4) M M222+M444=M M111+M=M117) + %<- (union-associative M1+M022=M3 M3+M544=M7 M6 M022+M544=M6 M1+M6=M7) + %<- (union-associative-union/<-union/< N0+1+N1=N2 N5+1+N1=N4 M022+M544=M6 M1+M6=M7 M M222+M444=M M111+M=M117) %. +%term _ + %pi (union-associative-union/<-union/< N+1+N1=N2 N+1+N1=N4 (union/= M2+M4=M6 D2+D4=D6 nat`eq/) M1+M066=M7 (map/+ N2 D6 M6) (union/= M2+M4=M6 D2+D4=D6 N2=N4) (union/< M1+M066=M7 N+1+N1=N2)) + %<- (nat`plus-deterministic N+1+N1=N2 N+1+N1=N4 nat`eq/ nat`eq/ N2=N4) %. +%term _ + %pi (union-associative-union/<-union/< N0+1+N1=N2 N5+1+N1=N4 (union/< M2+M744=M6 N7+1+N0=N5) M1+M026=M7 (map/+ N2 D2 M6) (union/< M2+M744=M6 N7+1+N2=N4) (union/< M1+M026=M7 N0+1+N1=N2)) + %<- (nat`plus-swap-succ N5+1+N1=N4 N5+N1+1=N4) + %<- (nat`plus-swap-succ N0+1+N1=N2 N0+N1+1=N2) + %<- (nat`plus-associative* N7+1+N0=N5 N5+N1+1=N4 N0+N1+1=N2 N7+1+N2=N4) %. +%term _ + %pi (union-associative-union/<-union/< N0+1+N1=N2 N5+1+N1=N4 (union/> M722+M4=M6 N7+1+N5=N0) M1+M546=M7 (map/+ N4 D4 M6) (union/> M722+M4=M6 N7+1+N4=N2) (union/< M1+M546=M7 N5+1+N1=N4)) + %<- (nat`plus-swap-succ N0+1+N1=N2 N0+N1+1=N2) + %<- (nat`plus-swap-succ N5+1+N1=N4 N5+N1+1=N4) + %<- (nat`plus-associative* N7+1+N5=N0 N0+N1+1=N2 N5+N1+1=N4 N7+1+N4=N2) %. +%% and now we return to the main theorem +%term _ + %pi (union-associative (union/< M1+M022=M3 N0+1+N1=N2) (union/> M513+M4=M7 N5+1+N3=N1) (map/+ _ _ M6) (union/> M622+M4=M6 N6+1+N3=N2) (union/> M511+M6=M7 N5+1+N3=N1)) + %<- (nat`plus-swap-succ N5+1+N3=N1 N5+N3+1=N1) + %<- (nat`plus-associative-converse N5+N3+1=N1 N0+1+N1=N2 N6 N0+1+N5=N6 N6+N3+1=N2) + %<- (nat`plus-swap-succ-converse N6+N3+1=N2 N6+1+N3=N2) + %<- (union-associative (union/< M1+M022=M3 N0+1+N5=N6) M513+M4=M7 M6 M622+M4=M6 M511+M6=M7) %. +%term _ + %pi (union-associative (union/> M311+M2=M3 N3+1+N2=N1) (union/= M3+M4=M7 D2+D4=D7 nat`eq/) (map/+ _ _ M6) (union/= M2+M4=M6 D2+D4=D7 nat`eq/) (union/> M311+M6=M7 N3+1+N2=N1)) + %<- (union-associative M311+M2=M3 M3+M4=M7 M6 M2+M4=M6 M311+M6=M7) %. +%term _ + %pi (union-associative (union/> M311+M2=M3 N3+1+N2=N1) (union/< M3+M044=M7 N0+1+N2=N4) (map/+ _ _ M6) (union/< M2+M044=M6 N0+1+N2=N4) (union/> M311+M6=M7 N3+1+N2=N1)) + %<- (union-associative M311+M2=M3 M3+M044=M7 M6 M2+M044=M6 M311+M6=M7) %. +%term _ + %pi (union-associative (union/> M311+M2=M3 N3+1+N2=N1) (union/> M523+M4=M7 N5+1+N4=N2) (map/+ _ _ M6) (union/> M522+M4=M6 N5+1+N4=N2) (union/> M711+M6=M7 N7+1+N4=N1)) + %<- (nat`plus-swap-succ N5+1+N4=N2 N5+N4+1=N2) + %<- (nat`plus-associative-converse N5+N4+1=N2 N3+1+N2=N1 N7 N3+1+N5=N7 N7+N4+1=N1) + %<- (nat`plus-swap-succ-converse N7+N4+1=N1 N7+1+N4=N1) + %<- (union-associative (union/> M311+M2=M3 N3+1+N5=N7) M523+M4=M7 M6 M522+M4=M6 M711+M6=M7) %. +%worlds () (union-associative-union/<-union/< _ _ _ _ _ _ _) %. +%total {} (union-associative-union/<-union/< _ _ _ _ _ _ _) %. +%worlds () (union-associative _ _ _ _ _) %. +%total (J) (union-associative _ J _ _ _) %. +%theorem union-associative* : forall* {X1} {X2} {X12} {X3} {X23} {X123} forall {OP12 union X1 X2 X12} {OP12-3 union X12 X3 X123} {OP23 union X2 X3 X23} exists {OP1-23 union X1 X23 X123} true %. +%term _ + %pi (union-associative* X1+X2=X3 X3+X4=X7 X2+X4=X6 X1+X6=X7) + %<- (union-associative X1+X2=X3 X3+X4=X7 Y6 X2+X4=Y6 X1+Y6=X7) + %<- (union-deterministic X2+X4=Y6 X2+X4=X6 eq/ eq/ Y6=X6) + %<- (union-respects-eq X1+Y6=X7 eq/ Y6=X6 eq/ X1+X6=X7) %. +%worlds () (union-associative* _ _ _ _) %. +%total {} (union-associative* _ _ _ _) %. +%theorem union-associative-converse : forall* {X1} {X2} {X4} {X6} {X7} forall {OP24 union X2 X4 X6} {OP16 union X1 X6 X7} exists {X3} {OP12 union X1 X2 X3} {OP34 union X3 X4 X7} true %. +%term _ + %pi (union-associative-converse X2+X4=X6 X1+X6=X7 _ X1+X2=X3 X3+X4=X7) + %<- (union-commutative X2+X4=X6 X4+X2=X6) + %<- (union-commutative X1+X6=X7 X6+X1=X7) + %<- (union-associative X4+X2=X6 X6+X1=X7 _ X2+X1=X3 X4+X3=X7) + %<- (union-commutative X2+X1=X3 X1+X2=X3) + %<- (union-commutative X4+X3=X7 X3+X4=X7) %. +%worlds () (union-associative-converse X2+X4=X6 X1+X6=X7 X3 X1+X2=X3 X3+X4=X7) %. +%total {} (union-associative-converse _ _ _ _ _) %. +%theorem union-associative-converse* : forall* {X1} {X2} {X3} {X4} {X6} {X7} forall {OP24 union X2 X4 X6} {OP16 union X1 X6 X7} {OP12 union X1 X2 X3} exists {OP34 union X3 X4 X7} true %. +%term _ + %pi (union-associative-converse* X2+X4=X6 X1+X6=X7 X1+X2=X3 X3+X4=X7) + %<- (union-associative-converse X2+X4=X6 X1+X6=X7 X3P X1+X2=X3P X3P+X4=X7) + %<- (union-deterministic X1+X2=X3P X1+X2=X3 eq/ eq/ X3P=X3) + %<- (union-respects-eq X3P+X4=X7 X3P=X3 eq/ eq/ X3+X4=X7) %. +%worlds () (union-associative-converse* X2+X4=X6 X1+X6=X7 X1+X2=X3 X3+X4=X7) %. +%total {} (union-associative-converse* _ _ _ _) %. +%% The following two theorems are useful for reordering elements +%% is a left-associative sequence of operations. +%theorem union-assoc-commutative* : forall* {X1} {X2} {X3} {X4} {X5} {X7} forall {OP1 union X1 X2 X3} {OP2 union X3 X4 X7} {OP3 union X1 X4 X5} exists {OP4 union X5 X2 X7} true %. +%term _ + %pi (union-assoc-commutative* X1+X2=X3 X3+X4=X7 X1+X4=X5 X5+X2=X7) + %<- (union-associative X1+X2=X3 X3+X4=X7 X6 X2+X4=X6 X1+X6=X7) + %<- (union-commutative X2+X4=X6 X4+X2=X6) + %<- (union-associative-converse* X4+X2=X6 X1+X6=X7 X1+X4=X5 X5+X2=X7) %. +%worlds () (union-assoc-commutative* X1+X2=X3 X3+X4=X7 X1+X4=X5 X5+X2=X7) %. +%total {} (union-assoc-commutative* _ _ _ _) %. +%theorem union-assoc-commutative : forall* {X1} {X2} {X3} {X4} {X7} forall {OP1 union X1 X2 X3} {OP2 union X3 X4 X7} exists {X5} {OP3 union X1 X4 X5} {OP4 union X5 X2 X7} true %. +%term _ + %pi (union-assoc-commutative X1+X2=X3 X3+X4=X7 X5 X1+X4=X5 X5+X2=X7) + %<- (union-associative X1+X2=X3 X3+X4=X7 X6 X2+X4=X6 X1+X6=X7) + %<- (union-commutative X2+X4=X6 X4+X2=X6) + %<- (union-associative-converse X4+X2=X6 X1+X6=X7 X5 X1+X4=X5 X5+X2=X7) %. +%worlds () (union-assoc-commutative X1+X2=X3 X3+X4=X7 X5 X1+X4=X5 X5+X2=X7) %. +%total {} (union-assoc-commutative _ _ _ _ _) %. +%% The following theorem is a useful shortcut to +%% re-associate (AB)(CD) to (AC)(BD): +%theorem union-double-associative* : forall* {A} {B} {C} {D} {A+B} {C+D} {A+C} {B+D} {X} forall {AB union A B A+B} {CD union C D C+D} {ABCD union A+B C+D X} {AC union A C A+C} {BD union B D B+D} exists {ACBD union A+C B+D X} true %. +%term _ + %pi (union-double-associative* X1+X2=X3 X4+X8=XC X3+XC=XF X1+X4=X5 X2+X8=XA X5+XA=XF) + %<- (union-associative X1+X2=X3 X3+XC=XF XE X2+XC=XE X1+XE=XF) + %<- (union-commutative X4+X8=XC X8+X4=XC) + %<- (union-associative-converse* X8+X4=XC X2+XC=XE X2+X8=XA XA+X4=XE) + %<- (union-commutative XA+X4=XE X4+XA=XE) + %<- (union-associative-converse* X4+XA=XE X1+XE=XF X1+X4=X5 X5+XA=XF) %. +%worlds () (union-double-associative* X1+X2=X3 X4+X8=XC X3+XC=XF X1+X4=X5 X2+X8=XA X5+XA=XF) %. +%total {} (union-double-associative* _ _ _ _ _ _) %. +%theorem union-double-associative : forall* {A} {B} {C} {D} {A+B} {C+D} {X} forall {AB union A B A+B} {CD union C D C+D} {ABCD union A+B C+D X} exists {A+C} {B+D} {AC union A C A+C} {BD union B D B+D} {ACBD union A+C B+D X} true %. +%term _ + %pi (union-double-associative X1+X2=X3 X4+X8=XC X3+XC=XF X5 XA X1+X4=X5 X2+X8=XA X5+XA=XF) + %<- (union-associative X1+X2=X3 X3+XC=XF XE X2+XC=XE X1+XE=XF) + %<- (union-commutative X4+X8=XC X8+X4=XC) + %<- (union-associative-converse X8+X4=XC X2+XC=XE XA X2+X8=XA XA+X4=XE) + %<- (union-commutative XA+X4=XE X4+XA=XE) + %<- (union-associative-converse X4+XA=XE X1+XE=XF X5 X1+X4=X5 X5+XA=XF) %. +%worlds () (union-double-associative _ _ _ _ _ _ _ _) %. +%total {} (union-double-associative _ _ _ _ _ _ _ _) %. +%theorem lookup-implies-union : forall* {M} {N} {D} forall {L lookup M N D} exists {M-} {F not-member M- N} {A union (map/+ N D map/0) M- M} true %. +%term _ lookup-implies-union (lookup/= nat`eq/) _ not-member/0 union/R %. +%term _ + %pi (lookup-implies-union (lookup/= nat`eq/) _ (not-member/< N2>N1) (union/< union/L N0+1+N1=N2)) + %<- (nat`plus-total N0+1+N1=N2) + %<- (nat`plus-implies-gt N0+1+N1=N2 nat`eq/ N2>N1) %. +%term _ + %pi (lookup-implies-union (lookup/> L N0+1+N1=N2) _ (not-member/> F N0+1+N1=N2) (union/> A N0+1+N1=N2)) + %<- (lookup-implies-union L _ F A) %. +%worlds () (lookup-implies-union _ _ _ _) %. +%total (L) (lookup-implies-union L _ _ _) %. +%theorem union-joins-lookup : forall* {M1} {M2} {M3} {N} {D1} {D2} forall {L1 lookup M1 N D1} {L2 lookup M2 N D2} {A union M1 M2 M3} exists {D3} {J nat`union D1 D2 D3} {L3 lookup M3 N D3} true %. +%term _ + %pi (union-joins-lookup (lookup/= nat`eq/) (lookup/= nat`eq/) A _ AD L3) + %<- (union/=-inversion A nat`eq/ _ _ AD _ M=M133) + %<- (eq-symmetric M=M133 M133=M) + %<- (lookup-respects-eq (lookup/= nat`eq/) M133=M nat`eq/ nat`eq/ L3) %. +%term _ + %pi (union-joins-lookup (lookup/> L1 N0+1+N1=N2) (lookup/= nat`eq/) A _ AD L3P) + %<- (union/<-inversion A N0+1+N1=N2 M3 M1+M022=M3 M=M113) + %<- (union-joins-lookup L1 (lookup/= nat`eq/) M1+M022=M3 _ AD L3) + %<- (eq-symmetric M=M113 M113=M) + %<- (lookup-respects-eq (lookup/> L3 N0+1+N1=N2) M113=M nat`eq/ nat`eq/ L3P) %. +%term _ + %pi (union-joins-lookup (lookup/= nat`eq/) (lookup/> L2 N3+1+N2=N1) A _ AD L3P) + %<- (union/>-inversion A N3+1+N2=N1 M3 M311+M2=M3 M=M223) + %<- (union-joins-lookup (lookup/= nat`eq/) L2 M311+M2=M3 _ AD L3) + %<- (eq-symmetric M=M223 M223=M) + %<- (lookup-respects-eq (lookup/> L3 N3+1+N2=N1) M223=M nat`eq/ nat`eq/ L3P) %. +%term _ + %pi (union-joins-lookup (lookup/> L1P N4+1+N1=N) (lookup/> L2 N5+1+N2=N) (union/= M1+M2=M3 _ nat`eq/) _ AD (lookup/> L N5+1+N2=N)) + %<- (plus-right-cancels N4+1+N1=N N5+1+N2=N nat`eq/ nat`eq/ N4+1=N5+1) + %<- (succ-cancels N4+1=N5+1 N4=N5) + %<- (lookup-respects-eq L1P eq/ N4=N5 nat`eq/ L1) + %<- (union-joins-lookup L1 L2 M1+M2=M3 _ AD L) %. +%term _ + %pi (union-joins-lookup (lookup/> L1 N4+1+N1=N) (lookup/> L2 N5+1+N2=N) (union/< M1+M022=M3 N0+1+N1=N2) _ AD (lookup/> L3 N4+1+N1=N)) + %<- (plus-swap-succ N0+1+N1=N2 N0+N1+1=N2) + %<- (plus-associative-converse N0+N1+1=N2 N5+1+N2=N N6 N5+1+N0=N6 N6+N1+1=N) + %<- (plus-swap-succ N4+1+N1=N N4+N1+1=N) + %<- (plus-right-cancels N6+N1+1=N N4+N1+1=N nat`eq/ nat`eq/ N6=N4) + %<- (plus-respects-eq N5+1+N0=N6 nat`eq/ nat`eq/ N6=N4 N5+1+N0=N4) + %<- (union-joins-lookup L1 (lookup/> L2 N5+1+N0=N4) M1+M022=M3 _ AD L3) %. +%term _ + %pi (union-joins-lookup (lookup/> L1 N4+1+N1=N) (lookup/> L2 N5+1+N2=N) (union/> M311+M2=M3 N3+1+N2=N1) _ AD (lookup/> L3 N5+1+N2=N)) + %<- (plus-swap-succ N3+1+N2=N1 N3+N2+1=N1) + %<- (plus-associative-converse N3+N2+1=N1 N4+1+N1=N N6 N4+1+N3=N6 N6+N2+1=N) + %<- (plus-swap-succ N5+1+N2=N N5+N2+1=N) + %<- (plus-right-cancels N6+N2+1=N N5+N2+1=N nat`eq/ nat`eq/ N6=N5) + %<- (plus-respects-eq N4+1+N3=N6 nat`eq/ nat`eq/ N6=N5 N4+1+N3=N5) + %<- (union-joins-lookup (lookup/> L1 N4+1+N3=N5) L2 M311+M2=M3 _ AD L3) %. +%worlds () (union-joins-lookup _ _ _ _ _ _) %. +%total (A) (union-joins-lookup _ _ A _ _ _) %. +%theorem union-preserves-not-member* : forall* {M1} {M2} {M3} {N} forall {F1 not-member M1 N} {F2 not-member M2 N} {A union M1 M2 M3} exists {F3 not-member M3 N} true %. +%term _ union-preserves-not-member* _ F union/L F %. +%term _ union-preserves-not-member* F _ union/R F %. +%term _ union-preserves-not-member* (not-member/< N _ _) (not-member/< N F2 N4+1+N2=N) AX F3X) + %<- (gt-implies-plus N-inversion AX N3+1+N2=N1 M3 A M=M223) + %<- (plus-implies-gt N0+1+N4=N3 nat`eq/ N4 F3 N4+1+N2=N) M223=M nat`eq/ F3X) %. +%term _ + %pi (union-preserves-not-member* (not-member/> F1 N3+1+N1=N) (not-member/< N F3 N3+1+N1=N) M113=M nat`eq/ F3X) %. +%term _ + %pi (union-preserves-not-member* (not-member/> F1 N3+1+N1=N) (not-member/> F2P N4+1+N2=N) (union/= A _ N1=N2) (not-member/> F3 N3+1+N1=N)) + %<- (plus-right-cancels N3+1+N1=N N4+1+N2=N N1=N2 nat`eq/ N3+1=N4+1) + %<- (succ-cancels N3+1=N4+1 N3=N4) + %<- (nat`eq-symmetric N3=N4 N4=N3) + %<- (not-member-respects-eq F2P eq/ N4=N3 F2) + %<- (union-preserves-not-member* F1 F2 A F3) %. +%term _ + %pi (union-preserves-not-member* (not-member/> F1 N3+1+N1=N) (not-member/> F2 N4+1+N2=N) (union/< A N0+1+N1=N2) (not-member/> F3 N3+1+N1=N)) + %<- (plus-swap-succ N0+1+N1=N2 N0+N1+1=N2) + %<- (plus-associative-converse N0+N1+1=N2 N4+1+N2=N N6 N4+1+N0=N6 N6+N1+1=N) + %<- (plus-swap-succ N3+1+N1=N N3+N1+1=N) + %<- (plus-right-cancels N6+N1+1=N N3+N1+1=N nat`eq/ nat`eq/ N6=N3) + %<- (plus-respects-eq N4+1+N0=N6 nat`eq/ nat`eq/ N6=N3 N4+1+N0=N3) + %<- (union-preserves-not-member* F1 (not-member/> F2 N4+1+N0=N3) A F3) %. +%term _ + %pi (union-preserves-not-member* (not-member/> F1 N3+1+N1=N) (not-member/> F2 N4+1+N2=N) (union/> A N0+1+N2=N1) (not-member/> F3 N4+1+N2=N)) + %<- (plus-swap-succ N0+1+N2=N1 N0+N2+1=N1) + %<- (plus-associative-converse N0+N2+1=N1 N3+1+N1=N N6 N3+1+N0=N6 N6+N2+1=N) + %<- (plus-swap-succ N4+1+N2=N N4+N2+1=N) + %<- (plus-right-cancels N6+N2+1=N N4+N2+1=N nat`eq/ nat`eq/ N6=N4) + %<- (plus-respects-eq N3+1+N0=N6 nat`eq/ nat`eq/ N6=N4 N3+1+N0=N4) + %<- (union-preserves-not-member* (not-member/> F1 N3+1+N0=N4) F2 A F3) %. +%worlds () (union-preserves-not-member* _ _ _ _) %. +%total (A) (union-preserves-not-member* _ _ A _) %. +%theorem not-member-union-left-preserves-lookup* : forall* {M1} {M2} {M3} {N} {D} forall {F1 not-member M1 N} {L2 lookup M2 N D} {A union M1 M2 M3} exists {L3 lookup M3 N D} true %. +%term _ not-member-union-left-preserves-lookup* _ L union/L L %. +%term _ + %pi (not-member-union-left-preserves-lookup* (not-member/< N2-inversion AX N3+1+N2=N1 M3 A M=M223) + %<- (eq-symmetric M=M223 M223=M) + %<- (lookup-respects-eq (lookup/= nat`eq/) M223=M nat`eq/ nat`eq/ L3X) %. +%term _ + %pi (not-member-union-left-preserves-lookup* (not-member/< N L2 N4+1+N2=N) AX L3X) + %<- (gt-implies-plus N-inversion AX N3+1+N2=N1 M3 A M=M223) + %<- (eq-symmetric M=M223 M223=M) + %<- (plus-implies-gt N0+1+N4=N3 nat`eq/ N3>N4) + %<- (not-member-union-left-preserves-lookup* (not-member/< N3>N4) L2 A L3) + %<- (lookup-respects-eq (lookup/> L3 N4+1+N2=N) M223=M nat`eq/ nat`eq/ L3X) %. +%term _ + %pi (not-member-union-left-preserves-lookup* (not-member/> F1 N0+1+N1=N2) (lookup/= nat`eq/) AX L3X) + %<- (union/<-inversion AX N0+1+N1=N2 M3 A M=M113) + %<- (eq-symmetric M=M113 M113=M) + %<- (not-member-union-left-preserves-lookup* F1 (lookup/= nat`eq/) A L3) + %<- (lookup-respects-eq (lookup/> L3 N0+1+N1=N2) M113=M nat`eq/ nat`eq/ L3X) %. +%term _ + %pi (not-member-union-left-preserves-lookup* (not-member/> F1P N4+1+N1=N) (lookup/> L2 N5+1+N1=N) (union/= A _ nat`eq/) (lookup/> L3 N5+1+N1=N)) + %<- (plus-right-cancels N4+1+N1=N N5+1+N1=N nat`eq/ nat`eq/ N4+1=N5+1) + %<- (succ-cancels N4+1=N5+1 N4=N5) + %<- (not-member-respects-eq F1P eq/ N4=N5 F1) + %<- (not-member-union-left-preserves-lookup* F1 L2 A L3) %. +%term _ + %pi (not-member-union-left-preserves-lookup* (not-member/> F1 N4+1+N1=N) (lookup/> L2 N5+1+N2=N) (union/< A N0+1+N1=N2) (lookup/> L3 N4+1+N1=N)) + %<- (plus-swap-succ N0+1+N1=N2 N0+N1+1=N2) + %<- (plus-associative-converse N0+N1+1=N2 N5+1+N2=N N3 N5+1+N0=N3 N3+N1+1=N) + %<- (plus-swap-succ N4+1+N1=N N4+N1+1=N) + %<- (plus-right-cancels N3+N1+1=N N4+N1+1=N nat`eq/ nat`eq/ N3=N4) + %<- (plus-respects-eq N5+1+N0=N3 nat`eq/ nat`eq/ N3=N4 N5+1+N0=N4) + %<- (not-member-union-left-preserves-lookup* F1 (lookup/> L2 N5+1+N0=N4) A L3) %. +%term _ + %pi (not-member-union-left-preserves-lookup* (not-member/> F1 N4+1+N1=N) (lookup/> L2 N5+1+N2=N) (union/> A N3+1+N2=N1) (lookup/> L3 N5+1+N2=N)) + %<- (plus-swap-succ N3+1+N2=N1 N3+N2+1=N1) + %<- (plus-associative-converse N3+N2+1=N1 N4+1+N1=N N6 N4+1+N3=N6 N6+N2+1=N) + %<- (plus-swap-succ N5+1+N2=N N5+N2+1=N) + %<- (plus-right-cancels N6+N2+1=N N5+N2+1=N nat`eq/ nat`eq/ N6=N5) + %<- (plus-respects-eq N4+1+N3=N6 nat`eq/ nat`eq/ N6=N5 N4+1+N3=N5) + %<- (not-member-union-left-preserves-lookup* (not-member/> F1 N4+1+N3=N5) L2 A L3) %. +%worlds () (not-member-union-left-preserves-lookup* _ _ _ _) %. +%total (A) (not-member-union-left-preserves-lookup* _ _ A _) %. +%theorem not-member-union-left-preserves-lookup-converse : forall* {M1} {M2} {M3} {N} {D} forall {F1 not-member M1 N} {L3 lookup M3 N D} {A union M1 M2 M3} exists {L2 lookup M2 N D} true %. +%theorem not-member-union-left-preserves-lookup-converse/L : forall* {M1} {M2} {M3} {N} {D} {B} forall {F1 not-member M1 N} {L3 lookup M3 N D} {A union M1 M2 M3} {D? member? M2 N B} exists {L2 lookup M2 N D} true %. +%term _ + %pi (not-member-union-left-preserves-lookup-converse F1 L3 A L2) + %<- (member?-total D?) + %<- (not-member-union-left-preserves-lookup-converse/L F1 L3 A D? L2) %. +%term _ + %pi (not-member-union-left-preserves-lookup-converse/L F1 L3 A (member?/in L2') L2) + %<- (not-member-union-left-preserves-lookup* F1 L2' A L3') + %<- (lookup-deterministic L3' L3 eq/ nat`eq/ D'=D) + %<- (lookup-respects-eq L2' eq/ nat`eq/ D'=D L2) %. +%term _ + %pi (not-member-union-left-preserves-lookup-converse/L F1 L3 A (member?/out F2) L2) + %<- (union-preserves-not-member* F1 F2 A F3) + %<- (not-member-lookup-not-equal F3 L3 N<>N) + %<- (nat`ne-anti-reflexive N<>N F) + %<- (false-implies-lookup F L2) %. +%worlds () (not-member-union-left-preserves-lookup-converse/L _ _ _ _ _) %. +%total {} (not-member-union-left-preserves-lookup-converse/L _ _ _ _ _) %. +%worlds () (not-member-union-left-preserves-lookup-converse _ _ _ _) %. +%total {} (not-member-union-left-preserves-lookup-converse _ _ _ _) %. +%theorem union-left-affects-lookup : forall* {M1} {N} {D2} {M2} {M3} forall {L lookup M2 N D2} {A union M1 M2 M3} exists {D3} {N lookup M3 N D3} true %. +%theorem union-left-affects-lookup/L : forall* {M1} {N} {D2} {M2} {M3} {B} forall {L lookup M2 N D2} {A union M1 M2 M3} {D member? M1 N B} exists {D3} {N lookup M3 N D3} true %. +%term _ + %pi (union-left-affects-lookup/L L2 A (member?/in L1) _ L3) + %<- (union-joins-lookup L1 L2 A _ _ L3) %. +%term _ + %pi (union-left-affects-lookup/L L2 A (member?/out F1) _ L3) + %<- (not-member-union-left-preserves-lookup* F1 L2 A L3) %. +%worlds () (union-left-affects-lookup/L _ _ _ _ _) %. +%total {} (union-left-affects-lookup/L _ _ _ _ _) %. +%term _ + %pi (union-left-affects-lookup L2 A _ L3) + %<- (member?-total D) + %<- (union-left-affects-lookup/L L2 A D _ L3) %. +%worlds () (union-left-affects-lookup _ _ _ _) %. +%total {} (union-left-affects-lookup _ _ _ _) %. +%theorem not-member-union-right-preserves-lookup* : forall* {M1} {M2} {M3} {N} {D} forall {L1 lookup M1 N D} {F2 not-member M2 N} {A union M1 M2 M3} exists {L3 lookup M3 N D} true %. +%term _ + %pi (not-member-union-right-preserves-lookup* L1 F2 A L3) + %<- (union-commutative A Ac) + %<- (not-member-union-left-preserves-lookup* F2 L1 Ac L3) %. +%worlds () (not-member-union-right-preserves-lookup* _ _ _ _) %. +%total {} (not-member-union-right-preserves-lookup* _ _ _ _) %. +%theorem not-member-union-right-preserves-lookup-converse : forall* {M1} {M2} {M3} {N} {D} forall {L3 lookup M3 N D} {F2 not-member M2 N} {A union M1 M2 M3} exists {L1 lookup M1 N D} true %. +%theorem not-member-union-right-preserves-lookup-converse/L : forall* {M1} {M2} {M3} {N} {D} {B} forall {L3 lookup M3 N D} {F2 not-member M2 N} {A union M1 M2 M3} {D? member? M1 N B} exists {L1 lookup M1 N D} true %. +%term _ + %pi (not-member-union-right-preserves-lookup-converse L3 F2 A L1) + %<- (member?-total D?) + %<- (not-member-union-right-preserves-lookup-converse/L L3 F2 A D? L1) %. +%term _ + %pi (not-member-union-right-preserves-lookup-converse/L L3 F2 A (member?/in L1') L1) + %<- (not-member-union-right-preserves-lookup* L1' F2 A L3') + %<- (lookup-deterministic L3' L3 eq/ nat`eq/ D'=D) + %<- (lookup-respects-eq L1' eq/ nat`eq/ D'=D L1) %. +%term _ + %pi (not-member-union-right-preserves-lookup-converse/L L3 F2 A (member?/out F1) L1) + %<- (union-preserves-not-member* F1 F2 A F3) + %<- (not-member-lookup-not-equal F3 L3 N<>N) + %<- (nat`ne-anti-reflexive N<>N F) + %<- (false-implies-lookup F L1) %. +%worlds () (not-member-union-right-preserves-lookup-converse/L _ _ _ _ _) %. +%total {} (not-member-union-right-preserves-lookup-converse/L _ _ _ _ _) %. +%worlds () (not-member-union-right-preserves-lookup-converse _ _ _ _) %. +%total {} (not-member-union-right-preserves-lookup-converse _ _ _ _) %. +%theorem union-right-affects-lookup : forall* {M1} {N} {D1} {M2} {M3} forall {L lookup M1 N D1} {A union M1 M2 M3} exists {D3} {N lookup M3 N D3} true %. +%theorem union-right-affects-lookup/L : forall* {M1} {N} {D1} {M2} {M3} {B} forall {L lookup M1 N D1} {A union M1 M2 M3} {D member? M2 N B} exists {D3} {N lookup M3 N D3} true %. +%term _ + %pi (union-right-affects-lookup/L L1 A (member?/in L2) _ L3) + %<- (union-joins-lookup L1 L2 A _ _ L3) %. +%term _ + %pi (union-right-affects-lookup/L L1 A (member?/out F2) _ L3) + %<- (not-member-union-right-preserves-lookup* L1 F2 A L3) %. +%worlds () (union-right-affects-lookup/L _ _ _ _ _) %. +%total {} (union-right-affects-lookup/L _ _ _ _ _) %. +%term _ + %pi (union-right-affects-lookup L1 A _ L3) + %<- (member?-total D) + %<- (union-right-affects-lookup/L L1 A D _ L3) %. +%worlds () (union-right-affects-lookup _ _ _ _) %. +%total {} (union-right-affects-lookup _ _ _ _) %. +%theorem union-preserves-not-member-converse* : forall* {M1} {M2} {M3} {N} forall {F3 not-member M3 N} {A union M1 M2 M3} exists {F1 not-member M1 N} {F2 not-member M2 N} true %. +%theorem union-preserves-not-member-converse/L : forall* {M1} {M2} {M3} {N} {B1} {B2} forall {F3 not-member M3 N} {A union M1 M2 M3} {D1 member? M1 N B1} {D2 member? M2 N B2} exists {F1 not-member M1 N} {F2 not-member M2 N} true %. +%term _ + %pi (union-preserves-not-member-converse* F3 A F1 F2) + %<- (member?-total D1) + %<- (member?-total D2) + %<- (union-preserves-not-member-converse/L F3 A D1 D2 F1 F2) %. +%term _ union-preserves-not-member-converse/L _ _ (member?/out F1) (member?/out F2) F1 F2 %. +%term _ + %pi (union-preserves-not-member-converse/L F3 A (member?/out F1) (member?/in L2) F1 F2) + %<- (not-member-union-left-preserves-lookup* F1 L2 A L3) + %<- (not-member-lookup-not-equal F3 L3 N<>N) + %<- (nat`ne-anti-reflexive N<>N F) + %<- (false-implies-not-member F F2) %. +%term _ + %pi (union-preserves-not-member-converse/L F3 A (member?/in L1) (member?/out F2) F1 F2) + %<- (not-member-union-right-preserves-lookup* L1 F2 A L3) + %<- (not-member-lookup-not-equal F3 L3 N<>N) + %<- (nat`ne-anti-reflexive N<>N F) + %<- (false-implies-not-member F F1) %. +%term _ + %pi (union-preserves-not-member-converse/L F3 A (member?/in L1) (member?/in L2) F1 F2) + %<- (union-joins-lookup L1 L2 A _ _ L3) + %<- (not-member-lookup-not-equal F3 L3 N<>N) + %<- (nat`ne-anti-reflexive N<>N F) + %<- (false-implies-not-member F F1) + %<- (false-implies-not-member F F2) %. +%worlds () (union-preserves-not-member-converse/L _ _ _ _ _ _) %. +%total {} (union-preserves-not-member-converse/L _ _ _ _ _ _) %. +%worlds () (union-preserves-not-member-converse* _ _ _ _) %. +%total {} (union-preserves-not-member-converse* _ _ _ _) %. +%theorem disjoint-union-left-preserves-lookup* : forall* {M1} {M2} {M3} {N} {D} forall {L2 lookup M2 N D} {X disjoint M1 M2} {A union M1 M2 M3} exists {L3 lookup M3 N D} true %. +%theorem disjoint-union-left-preserves-lookup*/L : forall* {M1} {M2} {M3} {N} {D} {B} forall {L2 lookup M2 N D} {X disjoint M1 M2} {A union M1 M2 M3} {D? member? M1 N B} exists {L3 lookup M3 N D} true %. +%term _ + %pi (disjoint-union-left-preserves-lookup* L2 X A L3) + %<- (member?-total D?) + %<- (disjoint-union-left-preserves-lookup*/L L2 X A D? L3) %. +%term _ + %pi (disjoint-union-left-preserves-lookup*/L L2 X A (member?/in L1) L3) + %<- (disjoint-lookup-contradiction X L1 L2 F) + %<- (false-implies-lookup F L3) %. +%term _ + %pi (disjoint-union-left-preserves-lookup*/L L2 X A (member?/out F1) L3) + %<- (not-member-union-left-preserves-lookup* F1 L2 A L3) %. +%worlds () (disjoint-union-left-preserves-lookup*/L _ _ _ _ _) %. +%total {} (disjoint-union-left-preserves-lookup*/L _ _ _ _ _) %. +%worlds () (disjoint-union-left-preserves-lookup* _ _ _ _) %. +%total {} (disjoint-union-left-preserves-lookup* _ _ _ _) %. +%theorem disjoint-union-right-preserves-lookup* : forall* {M1} {M2} {M3} {N} {D} forall {L1 lookup M1 N D} {X disjoint M1 M2} {A union M1 M2 M3} exists {L3 lookup M3 N D} true %. +%theorem disjoint-union-right-preserves-lookup*/L : forall* {M1} {M2} {M3} {N} {D} {B} forall {L1 lookup M1 N D} {X disjoint M1 M2} {A union M1 M2 M3} {D? member? M2 N B} exists {L3 lookup M3 N D} true %. +%term _ + %pi (disjoint-union-right-preserves-lookup* L1 X A L3) + %<- (member?-total D?) + %<- (disjoint-union-right-preserves-lookup*/L L1 X A D? L3) %. +%term _ + %pi (disjoint-union-right-preserves-lookup*/L L1 X A (member?/in L2) L3) + %<- (disjoint-lookup-contradiction X L1 L2 F) + %<- (false-implies-lookup F L3) %. +%term _ + %pi (disjoint-union-right-preserves-lookup*/L L1 X A (member?/out F2) L3) + %<- (not-member-union-right-preserves-lookup* L1 F2 A L3) %. +%worlds () (disjoint-union-right-preserves-lookup*/L _ _ _ _ _) %. +%total {} (disjoint-union-right-preserves-lookup*/L _ _ _ _ _) %. +%worlds () (disjoint-union-right-preserves-lookup* _ _ _ _) %. +%total {} (disjoint-union-right-preserves-lookup* _ _ _ _) %. +%theorem union-implies-leq* : forall* {M1} {M2} {M3} forall {J union M1 M2 M3} exists {L leq M1 M3} true %. +%term _ union-implies-leq* union/L leq/0 %. +%term _ %pi (union-implies-leq* union/R M2=M2) %<- (leq-reflexive _ M2=M2) %. +%term _ + %pi (union-implies-leq* (union/= M1+M2=M3 D1+D2=D3 nat`eq/) (leq/= M1<=M3 D1<=D3 nat`eq/)) + %<- (nat`union-implies-leq* D1+D2=D3 D1<=D3) + %<- (union-implies-leq* M1+M2=M3 M1<=M3) %. +%term _ + %pi (union-implies-leq* (union/< M1+_=M3 _) (leq/= M1<=M3 D1<=D1 nat`eq/)) + %<- (nat`leq-reflexive _ D1<=D1) + %<- (union-implies-leq* M1+_=M3 M1<=M3) %. +%term _ + %pi (union-implies-leq* (union/> M311+M2=M3 P) (leq/> M311<=M3 P)) + %<- (union-implies-leq* M311+M2=M3 M311<=M3) %. +%worlds () (union-implies-leq* _ _) %. +%total (J) (union-implies-leq* J _) %. +%theorem union-implies-leq : forall* {M1} {M2} {M3} forall {A union M1 M2 M3} exists {L1 leq M1 M3} {L2 leq M2 M3} true %. +%term _ + %pi (union-implies-leq M1*M2=M3 M1<=M3 M2<=M3) + %<- (union-implies-leq* M1*M2=M3 M1<=M3) + %<- (union-commutative M1*M2=M3 M2*M1=M3) + %<- (union-implies-leq* M2*M1=M3 M2<=M3) %. +%worlds () (union-implies-leq _ _ _) %. +%total {} (union-implies-leq _ _ _) %. +%theorem union-is-lub : forall* {M1} {M2} {M3} {M4} forall {J union M1 M2 M3} {L1 leq M1 M4} {L2 leq M2 M4} exists {L3 leq M3 M4} true %. +%term _ union-is-lub union/L _ L L %. +%term _ union-is-lub union/R L _ L %. +%term _ + %pi (union-is-lub (union/= M1+M2=M3 D1+D2=D3 nat`eq/) (leq/= M1<=M4 D1<=D4 nat`eq/) (leq/= M2<=M4 D2<=D4 nat`eq/) (leq/= M3<=M4 D3<=D4 nat`eq/)) + %<- (nat`union-is-lub D1+D2=D3 D1<=D4 D2<=D4 D3<=D4) + %<- (union-is-lub M1+M2=M3 M1<=M4 M2<=M4 M3<=M4) %. +%term _ + %pi (union-is-lub (union/= _ _ nat`eq/) (leq/= _ _ nat`eq/) (leq/> _ N3+1+N=N) L) + %<- (nat`plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-leq F L) %. +%term _ + %pi (union-is-lub (union/= _ _ nat`eq/) (leq/> _ N3+1+N=N) (leq/= _ _ nat`eq/) L) + %<- (nat`plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-leq F L) %. +%term _ + %pi (union-is-lub (union/= M1+M2=M3 D1+D2=D3 nat`eq/) (leq/> M511<=M4 N5+1+N4=N) (leq/> M622<=M4 N6+1+N4=N) (leq/> M633<=M4 N6+1+N4=N)) + %<- (nat`plus-right-cancels N5+1+N4=N N6+1+N4=N nat`eq/ nat`eq/ N5+1=N6+1) + %<- (nat`succ-cancels N5+1=N6+1 N5=N6) + %<- (map/+-preserves-eq N5=N6 nat`eq/ eq/ M511=M611) + %<- (leq-respects-eq M511<=M4 M511=M611 eq/ M611<=M4) + %<- (union-is-lub (union/= M1+M2=M3 D1+D2=D3 nat`eq/) M611<=M4 M622<=M4 M633<=M4) %. +%term _ + %pi (union-is-lub (union/< _ N0+1+N=N) (leq/= _ _ nat`eq/) (leq/= _ _ nat`eq/) L) + %<- (nat`plus-implies-gt N0+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-leq F L) %. +%term _ + %pi (union-is-lub (union/< M1+M022=M3 N0+1+N1=N2) (leq/= M1<=M4 D1<=D4 nat`eq/) (leq/> M622<=M4 N6+1+N1=N2) (leq/= M3<=M4 D1<=D4 nat`eq/)) + %<- (nat`plus-right-cancels N6+1+N1=N2 N0+1+N1=N2 nat`eq/ nat`eq/ N6+1=N0+1) + %<- (succ-cancels N6+1=N0+1 N6=N0) + %<- (map/+-preserves-eq N6=N0 nat`eq/ eq/ M622=M022) + %<- (leq-respects-eq M622<=M4 M622=M022 eq/ M022<=M4) + %<- (union-is-lub M1+M022=M3 M1<=M4 M022<=M4 M3<=M4) %. +%term _ + %pi (union-is-lub (union/< _ N0+1+N1=N2) (leq/> _ N3+1+N2=N1) (leq/= _ _ nat`eq/) L) + %<- (nat`plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) + %<- (nat`plus-implies-gt N0+1+N1=N2 nat`eq/ N2>N1) + %<- (nat`gt-anti-symmetric N1>N2 N2>N1 F) + %<- (false-implies-leq F L) %. +%term _ + %pi (union-is-lub (union/< M1+M022=M3 N0+1+N1=N2) (leq/> M511<=M4 N5+1+N4=N1) (leq/> M622<=M4 N6+1+N4=N2) (leq/> M513<=M4 N5+1+N4=N1)) + %<- (nat`plus-swap-succ N5+1+N4=N1 N5+N4+1=N1) + %<- (nat`plus-associative-converse N5+N4+1=N1 N0+1+N1=N2 N6' N0+1+N5=N6' N6'+N4+1=N2) + %<- (nat`plus-swap-succ-converse N6'+N4+1=N2 N6'+1+N4=N2) + %<- (nat`plus-right-cancels N6'+1+N4=N2 N6+1+N4=N2 nat`eq/ nat`eq/ N6'+1=N6+1) + %<- (nat`succ-cancels N6'+1=N6+1 N6'=N6) + %<- (nat`plus-respects-eq N0+1+N5=N6' nat`eq/ nat`eq/ N6'=N6 N0+1+N5=N6) + %<- (union-is-lub (union/< M1+M022=M3 N0+1+N5=N6) M511<=M4 M622<=M4 M513<=M4) %. +%term _ + %pi (union-is-lub (union/> _ N3+1+N=N) (leq/= _ _ nat`eq/) (leq/= _ _ nat`eq/) L) + %<- (nat`plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-leq F L) %. +%term _ + %pi (union-is-lub (union/> _ N3+1+N2=N1) (leq/= _ _ nat`eq/) (leq/> _ N0+1+N1=N2) L) + %<- (nat`plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) + %<- (nat`plus-implies-gt N0+1+N1=N2 nat`eq/ N2>N1) + %<- (nat`gt-anti-symmetric N1>N2 N2>N1 F) + %<- (false-implies-leq F L) %. +%term _ + %pi (union-is-lub (union/> M311+M2=M3 N3+1+N2=N1) (leq/> M511<=M4 N5+1+N2=N1) (leq/= M2<=M4 D2<=D4 nat`eq/) (leq/= M3<=M4 D2<=D4 nat`eq/)) + %<- (nat`plus-right-cancels N3+1+N2=N1 N5+1+N2=N1 nat`eq/ nat`eq/ N3+1=N5+1) + %<- (nat`succ-cancels N3+1=N5+1 N3=N5) + %<- (map/+-preserves-eq N3=N5 nat`eq/ eq/ M311=M511) + %<- (union-respects-eq M311+M2=M3 M311=M511 eq/ eq/ M511+M2=M3) + %<- (union-is-lub M511+M2=M3 M511<=M4 M2<=M4 M3<=M4) %. +%term _ + %pi (union-is-lub (union/> M311+M2=M3 N3+1+N2=N1) (leq/> M511<=M4 N5+1+N4=N1) (leq/> M622<=M4 N6+1+N4=N2) (leq/> M623<=M4 N6+1+N4=N2)) + %<- (nat`plus-swap-succ N6+1+N4=N2 N6+N4+1=N2) + %<- (nat`plus-associative-converse N6+N4+1=N2 N3+1+N2=N1 N5' N3+1+N6=N5' N5'+N4+1=N1) + %<- (nat`plus-swap-succ N5+1+N4=N1 N5+N4+1=N1) + %<- (nat`plus-right-cancels N5'+N4+1=N1 N5+N4+1=N1 nat`eq/ nat`eq/ N5'=N5) + %<- (nat`plus-respects-eq N3+1+N6=N5' nat`eq/ nat`eq/ N5'=N5 N3+1+N6=N5) + %<- (union-is-lub (union/> M311+M2=M3 N3+1+N6=N5) M511<=M4 M622<=M4 M623<=M4) %. +%worlds () (union-is-lub _ _ _ _) %. +%total (L) (union-is-lub _ L _ _) %. +%theorem union-idempotent : forall* {S} exists {J union S S S} true %. +%term _ + %pi (union-idempotent S+S=S) + %<- (union-total S+S=S') + %<- (union-implies-leq* S+S=S' S<=S') + %<- (leq-reflexive _ S<=S) + %<- (union-is-lub S+S=S' S<=S S<=S S'<=S) + %<- (leq-anti-symmetric S'<=S S<=S' S'=S) + %<- (union-respects-eq S+S=S' eq/ eq/ S'=S S+S=S) %. +%worlds () (union-idempotent _) %. +%total {} (union-idempotent _) %. +%theorem leq-implies-union : forall* {M1} {M2} forall {L leq M1 M2} exists {J union M1 M2 M2} true %. +%term _ + %pi (leq-implies-union M1<=M2 M1+M2=M2) + %<- (union-total M1+M2=M3) + %<- (leq-reflexive _ M2<=M2) + %<- (union-is-lub M1+M2=M3 M1<=M2 M2<=M2 M3<=M2) + %<- (union-implies-leq M1+M2=M3 _ M2<=M3) + %<- (leq-anti-symmetric M3<=M2 M2<=M3 M3=M2) + %<- (union-respects-eq M1+M2=M3 eq/ eq/ M3=M2 M1+M2=M2) %. +%worlds () (leq-implies-union _ _) %. +%total {} (leq-implies-union _ _) %. +%theorem disjoint-leq-implies-union-leq* : forall* {C1} {C2} {C} {C3} forall {D disjoint C1 C2} {L1 leq C1 C} {L2 leq C2 C} {J union C1 C2 C3} exists {L3 leq C3 C} true %. +%term _ + %pi (disjoint-leq-implies-union-leq* disjoint/L _ C2<=C Z+C2=C3 C3<=C) + %<- (union-deterministic union/L Z+C2=C3 eq/ eq/ C2=C3) + %<- (leq-respects-eq C2<=C C2=C3 eq/ C3<=C) %. +%term _ + %pi (disjoint-leq-implies-union-leq* disjoint/R C1<=C _ C1+0=C3 C3<=C) + %<- (union-deterministic union/R C1+0=C3 eq/ eq/ C1=C3) + %<- (leq-respects-eq C1<=C C1=C3 eq/ C3<=C) %. +%term _ + %pi (disjoint-leq-implies-union-leq* (disjoint/< C1^C022 P) (leq/= C1<=C4 D1<=D4 nat`eq/) C222<=C444 C111+C222=C333 C333<=C444) + %<- (union/<-inversion C111+C222=C333 P _ C1+C022=C5 C333=C115) + %<- (leq/>-inversion C222<=C444 P C022<=C4) + %<- (disjoint-leq-implies-union-leq* C1^C022 C1<=C4 C022<=C4 C1+C022=C5 C5<=C4) + %<- (eq-symmetric C333=C115 C115=C333) + %<- (leq-respects-eq (leq/= C5<=C4 D1<=D4 nat`eq/) C115=C333 eq/ C333<=C444) %. +%term _ + %pi (disjoint-leq-implies-union-leq* (disjoint/< C1^C022 N0+1+N1=N2) (leq/> C611<=C4 N6+1+N4=N1) C222<=C444 C111+C222=C333 C333<=C444) + %<- (union/<-inversion C111+C222=C333 N0+1+N1=N2 _ C1+C022=C5 C333=C115) + %<- (plus-swap-succ N6+1+N4=N1 N6+N4+1=N1) + %<- (plus-associative-converse N6+N4+1=N1 N0+1+N1=N2 N7 N0+1+N6=N7 N7+N4+1=N2) + %<- (plus-swap-succ-converse N7+N4+1=N2 N7+1+N4=N2) + %<- (leq/>-inversion C222<=C444 N7+1+N4=N2 C722<=C4) + %<- (disjoint-leq-implies-union-leq* (disjoint/< C1^C022 N0+1+N6=N7) C611<=C4 C722<=C4 (union/< C1+C022=C5 N0+1+N6=N7) C615<=C4) + %<- (eq-symmetric C333=C115 C115=C333) + %<- (leq-respects-eq (leq/> C615<=C4 N6+1+N4=N1) C115=C333 eq/ C333<=C444) %. +%term _ + %pi (disjoint-leq-implies-union-leq* (disjoint/> C011^C2 P) C111<=C444 (leq/= C2<=C4 D2<=D4 nat`eq/) C111+C222=C333 C333<=C444) + %<- (union/>-inversion C111+C222=C333 P _ C011+C2=C5 C333=C225) + %<- (leq/>-inversion C111<=C444 P C011<=C4) + %<- (disjoint-leq-implies-union-leq* C011^C2 C011<=C4 C2<=C4 C011+C2=C5 C5<=C4) + %<- (eq-symmetric C333=C225 C225=C333) + %<- (leq-respects-eq (leq/= C5<=C4 D2<=D4 nat`eq/) C225=C333 eq/ C333<=C444) %. +%term _ + %pi (disjoint-leq-implies-union-leq* (disjoint/> C011^C2 N0+1+N2=N1) C111<=C444 (leq/> C622<=C4 N6+1+N4=N2) C111+C222=C333 C333<=C444) + %<- (union/>-inversion C111+C222=C333 N0+1+N2=N1 _ C011+C2=C5 C333=C225) + %<- (plus-swap-succ N6+1+N4=N2 N6+N4+1=N2) + %<- (plus-associative-converse N6+N4+1=N2 N0+1+N2=N1 N7 N0+1+N6=N7 N7+N4+1=N1) + %<- (plus-swap-succ-converse N7+N4+1=N1 N7+1+N4=N1) + %<- (leq/>-inversion C111<=C444 N7+1+N4=N1 C711<=C4) + %<- (disjoint-leq-implies-union-leq* (disjoint/> C011^C2 N0+1+N6=N7) C711<=C4 C622<=C4 (union/> C011+C2=C5 N0+1+N6=N7) C625<=C4) + %<- (eq-symmetric C333=C225 C225=C333) + %<- (leq-respects-eq (leq/> C625<=C4 N6+1+N4=N2) C225=C333 eq/ C333<=C444) %. +%worlds () (disjoint-leq-implies-union-leq* _ _ _ _ _) %. +%total (L) (disjoint-leq-implies-union-leq* _ L _ _ _) %. +%theorem union-left-preserves-leq* : forall* {M1} {M2} {M3} {M4} {M5} forall {L1 leq M2 M4} {J union M1 M2 M3} {JP union M1 M4 M5} exists {L3 leq M3 M5} true %. +%term _ union-left-preserves-leq* L union/L union/L L %. +%term _ union-left-preserves-leq* _ union/L union/R leq/0 %. +%term _ + %pi (union-left-preserves-leq* leq/0 union/R M1+M4=M5 M1<=M5) + %<- (union-implies-leq* M1+M4=M5 M1<=M5) %. +%term _ + %pi (union-left-preserves-leq* (leq/= M2<=M4 D2<=D4 nat`eq/) (union/= M1+M2=M3 D1+D2=D3 nat`eq/) M111+M144=M M133<=M) + %<- (union/=-inversion M111+M144=M nat`eq/ D5 M5 D1+D4=D5 M1+M4=M5 M=M155) + %<- (eq-symmetric M=M155 M155=M) + %<- (meta-eq (map/+ N1 D5 M5) M M155=M) + %<- (nat`union-left-preserves-leq* D2<=D4 D1+D2=D3 D1+D4=D5 D3<=D5) + %<- (union-left-preserves-leq* M2<=M4 M1+M2=M3 M1+M4=M5 M3<=M5) + %<- (leq-respects-eq (leq/= M3<=M5 D3<=D5 nat`eq/) eq/ M155=M M133<=M) %. +%term _ {M1+M044=M5 union M1 (map/+ N0 D4 M4) M5} {M115=M eq (map/+ N1 D1 M5) M} + %pi (union-left-preserves-leq* (leq/= M2<=M4 D2<=D4 nat`eq/) (union/< M1+M022=M3 N0+1+N1=N2) M111+M244=M M113<=M) + %<- (union/<-inversion M111+M244=M N0+1+N1=N2 M5 M1+M044=M5 M=M115) + %<- (eq-symmetric M=M115 M115=M) + %<- (meta-eq (map/+ N1 D1 M5) M M115=M) + %<- (union-left-preserves-leq* (leq/= M2<=M4 D2<=D4 nat`eq/) M1+M022=M3 M1+M044=M5 M3<=M5) + %<- (nat`leq-reflexive _ D1<=D1) + %<- (leq-respects-eq (leq/= M3<=M5 D1<=D1 nat`eq/) eq/ M115=M M113<=M) %. +%term _ + %pi (union-left-preserves-leq* (leq/= M2<=M4 D2<=D4 nat`eq/) (union/> M311+M2=M3 N3+1+N2=N1) M111+M244=M M223<=M) + %<- (union/>-inversion M111+M244=M N3+1+N2=N1 M5 M311+M4=M5 M=M245) + %<- (eq-symmetric M=M245 M245=M) + %<- (meta-eq (map/+ N2 D4 M5) M M245=M) + %<- (union-left-preserves-leq* M2<=M4 M311+M2=M3 M311+M4=M5 M3<=M5) + %<- (leq-respects-eq (leq/= M3<=M5 D2<=D4 nat`eq/) eq/ M245=M M223<=M) %. +%term _ + %pi (union-left-preserves-leq* (leq/> M622<=M4 N6+1+N4=N2) (union/= M1+M2=M3 D1+D2=D3 nat`eq/) M211+M444=M M233<=M) + %<- (union/>-inversion M211+M444=M N6+1+N4=N2 M5 M611+M4=M5 M=M445) + %<- (eq-symmetric M=M445 M445=M) + %<- (meta-eq (map/+ N4 D4 M5) M M445=M) + %<- (union-left-preserves-leq* M622<=M4 (union/= M1+M2=M3 D1+D2=D3 nat`eq/) M611+M4=M5 M633<=M5) + %<- (leq-respects-eq (leq/> M633<=M5 N6+1+N4=N2) eq/ M445=M M233<=M) %. +%term _ + %pi (union-left-preserves-leq* (leq/> M622<=M4 N6+1+N4=N2) (union/< M1+M022=M3 N0+1+N4=N2) (union/= M1+M4=M5 D1+D4=D5 nat`eq/) (leq/= M3<=M5 D1<=D5 nat`eq/)) + %<- (nat`plus-right-cancels N6+1+N4=N2 N0+1+N4=N2 nat`eq/ nat`eq/ N6+1=N0+1) + %<- (nat`succ-cancels N6+1=N0+1 N6=N0) + %<- (map/+-preserves-eq N6=N0 nat`eq/ eq/ M622=M022) + %<- (leq-respects-eq M622<=M4 M622=M022 eq/ M022<=M4) + %<- (union-left-preserves-leq* M022<=M4 M1+M022=M3 M1+M4=M5 M3<=M5) + %<- (nat`union-implies-leq* D1+D4=D5 D1<=D5) %. +%term _ + %pi (union-left-preserves-leq* (leq/> M622<=M4 N6+1+N4=N2) (union/< M1+M022=M3 N0+1+N1=N2) (union/< M1+M544=M5 N5+1+N1=N4) (leq/= M3<=M5 D1<=D1 nat`eq/)) + %<- (nat`plus-swap-succ N5+1+N1=N4 N5+N1+1=N4) + %<- (nat`plus-associative-converse N5+N1+1=N4 N6+1+N4=N2 N0' N6+1+N5=N0' N0'+N1+1=N2) + %<- (nat`plus-swap-succ N0+1+N1=N2 N0+N1+1=N2) + %<- (nat`plus-right-cancels N0'+N1+1=N2 N0+N1+1=N2 nat`eq/ nat`eq/ N0'=N0) + %<- (nat`plus-respects-eq N6+1+N5=N0' nat`eq/ nat`eq/ N0'=N0 N6+1+N5=N0) + %<- (union-left-preserves-leq* (leq/> M622<=M4 N6+1+N5=N0) M1+M022=M3 M1+M544=M5 M3<=M5) + %<- (nat`leq-reflexive _ D1<=D1) %. +% for some reason, twelf needs a lot of help inferring types here: +%term _ + %pi (union-left-preserves-leq* (%the (leq (map/+ N2 D2 M2) (map/+ N4 D4 M4)) (leq/> M622<=M4 N6+1+N4=N2)) (union/< M1+M022=M3 N0+1+N1=N2) (union/> M511+M4=M5 N5+1+N4=N1) (leq/> M513<=M5 N5+1+N4=N1)) + %<- (nat`plus-swap-succ N5+1+N4=N1 N5+N4+1=N1) + %<- (nat`plus-associative-converse N5+N4+1=N1 N0+1+N1=N2 N6' N0+1+N5=N6' N6'+N4+1=N2) + %<- (nat`plus-swap-succ N6+1+N4=N2 N6+N4+1=N2) + %<- (nat`plus-right-cancels N6'+N4+1=N2 N6+N4+1=N2 nat`eq/ nat`eq/ N6'=N6) + %<- (nat`plus-respects-eq N0+1+N5=N6' nat`eq/ nat`eq/ N6'=N6 N0+1+N5=N6) + %<- (union-left-preserves-leq* M622<=M4 (%the (union (map/+ N5 D1 M1) (map/+ N6 D2 M2) (map/+ N5 D1 M3)) (union/< M1+M022=M3 N0+1+N5=N6)) M511+M4=M5 M513<=M5) %. +%term _ + %pi (union-left-preserves-leq* (leq/> M622<=M4 N6+1+N4=N2) (union/> M311+M2=M3 N3+1+N2=N1) M111+M444=M M223<=M) + %<- (nat`plus-swap-succ N6+1+N4=N2 N6+N4+1=N2) + %<- (nat`plus-associative-converse N6+N4+1=N2 N3+1+N2=N1 N5 N3+1+N6=N5 N5+N4+1=N1) + %<- (nat`plus-swap-succ-converse N5+N4+1=N1 N5+1+N4=N1) + %<- (union/>-inversion M111+M444=M N5+1+N4=N1 M5 M511+M4=M5 M=M445) + %<- (eq-symmetric M=M445 M445=M) + %<- (meta-eq _ _ M445=M) + %<- (union-left-preserves-leq* M622<=M4 (union/> M311+M2=M3 N3+1+N6=N5) M511+M4=M5 M623<=M5) + %<- (leq-respects-eq (leq/> M623<=M5 N6+1+N4=N2) eq/ M445=M M223<=M) %. +%worlds () (union-left-preserves-leq* _ _ _ _) %. +%total (J) (union-left-preserves-leq* _ _ J _) %. +%theorem union-right-preserves-leq* : forall* {X1} {X2} {X3} {X4} {X5} forall {G1 leq X1 X2} {O1 union X1 X3 X4} {O2 union X2 X3 X5} exists {G2 leq X4 X5} true %. +%term _ + %pi (union-right-preserves-leq* X1<=X2 X1+X3=X4 X2+X3=X5 X4<=X5) + %<- (union-commutative X1+X3=X4 X3+X1=X4) + %<- (union-commutative X2+X3=X5 X3+X2=X5) + %<- (union-left-preserves-leq* X1<=X2 X3+X1=X4 X3+X2=X5 X4<=X5) %. +%worlds () (union-right-preserves-leq* X1<=X2 X1+X3=X4 X2+X3=X5 X4<=X5) %. +%total {} (union-right-preserves-leq* _ _ _ _) %. +%%%% Map ``multiplication'' +%%% Definition of intersection +%sort intersection %. +%term intersection/L intersection map/0 M map/0 %. +%term intersection/R intersection M map/0 map/0 %. +%term intersection/= + %pi (intersection (map/+ N1 D1 M1) (map/+ N2 D2 M2) (map/+ N1 D3 M3)) + %<- (nat`eq N1 N2) + %<- (nat`intersection D1 D2 D3) + %<- (intersection M1 M2 M3) %. +%term intersection/< + %pi (intersection (map/+ N1 D1 M1) (map/+ N2 D2 M2) S1M3) + %<- (nat`plus (s N0) N1 N2) + %<- (intersection M1 (map/+ N0 D2 M2) M3) + %<- (shift N1 M3 S1M3) %. +%term intersection/> + %pi (intersection (map/+ N1 D1 M1) (map/+ N2 D2 M2) S2M3) + %<- (nat`plus (s N3) N2 N1) + %<- (intersection (map/+ N3 D1 M1) M2 M3) + %<- (shift N2 M3 S2M3) %. +%%% Theorems about intersection +%theorem false-implies-intersection : forall* {M1} {M2} {M3} forall {F void} exists {D intersection M1 M2 M3} true %. +%worlds () (false-implies-intersection _ _) %. +%total {} (false-implies-intersection _ _) %. +%theorem intersection-respects-eq : forall* {M1} {M2} {M3} {M1P} {M2P} {M3P} forall {A intersection M1 M2 M3} {E1 eq M1 M1P} {E2 eq M2 M2P} {E3 eq M3 M3P} exists {AP intersection M1P M2P M3P} true %. +%term _ intersection-respects-eq A eq/ eq/ eq/ A %. +%worlds () (intersection-respects-eq _ _ _ _ _) %. +%total {} (intersection-respects-eq _ _ _ _ _) %. +%reduces = A AP (intersection-respects-eq A _ _ _ AP) %. +%% Inversion lemmas for intersection +%theorem intersection/L-inversion : forall* {M1} {M2} {M3} forall {A intersection M1 M2 M3} {E1 eq map/0 M1} exists {E3 eq map/0 M3} true %. +%term _ intersection/L-inversion intersection/L eq/ eq/ %. +%term _ intersection/L-inversion intersection/R eq/ eq/ %. +%worlds () (intersection/L-inversion _ _ _) %. +%total {} (intersection/L-inversion _ _ _) %. +%theorem intersection/R-inversion : forall* {M1} {M2} {M3} forall {A intersection M1 M2 M3} {E1 eq map/0 M2} exists {E3 eq map/0 M3} true %. +%term _ intersection/R-inversion intersection/L eq/ eq/ %. +%term _ intersection/R-inversion intersection/R eq/ eq/ %. +%worlds () (intersection/R-inversion _ _ _) %. +%total {} (intersection/R-inversion _ _ _) %. +%theorem intersection/=-inversion : forall* {N1} {D1} {M1} {N2} {D2} {M2} {M} forall {A intersection (map/+ N1 D1 M1) (map/+ N2 D2 M2) M} {G nat`eq N1 N2} exists {D3} {M3} {D nat`intersection D1 D2 D3} {AP intersection M1 M2 M3} {E eq (map/+ N1 D3 M3) M} true %. +%term _ intersection/=-inversion (intersection/= MM DD nat`eq/) _ _ _ DD MM eq/ %. +%term _ + %pi (intersection/=-inversion (intersection/< S A' N0+1+N=N) nat`eq/ D2 M3 DA MA ME) + %<- (nat`plus-implies-gt N0+1+N=N nat`eq/ N>N) + %<- (nat`gt-anti-reflexive N>N F) + %<- (nat`false-implies-intersection F DA) + %<- (false-implies-eq F (%the (eq (map/+ N0 D2 M2) M2) M022=M2)) + %<- (intersection-respects-eq A' eq/ M022=M2 eq/ MA) + %<- (false-implies-eq F ME) %. +%term _ + %pi (intersection/=-inversion (intersection/> S (%the (intersection (map/+ _ D1 M1) M2 _) AP) N3+1+N=N) nat`eq/ D1 M3 DA MA ME) + %<- (nat`plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (nat`gt-anti-reflexive N>N F) + %<- (nat`false-implies-intersection F DA) + %<- (false-implies-eq F (%the (eq (map/+ N3 D1 M1) M1) M311=M1)) + %<- (intersection-respects-eq AP M311=M1 eq/ eq/ MA) + %<- (false-implies-eq F ME) %. +%worlds () (intersection/=-inversion _ _ _ _ _ _ _) %. +%total {} (intersection/=-inversion _ _ _ _ _ _ _) %. +%reduces < AP A (intersection/=-inversion A _ _ _ _ AP _) %. +%theorem intersection/<-inversion : forall* {N1} {D1} {M1} {N2} {D2} {M2} {S1M3} {N0} forall {A intersection (map/+ N1 D1 M1) (map/+ N2 D2 M2) S1M3} {P plus (s N0) N1 N2} exists {M3} {AP intersection M1 (map/+ N0 D2 M2) M3} {S shift N1 M3 S1M3} true %. +%term _ + %pi (intersection/<-inversion (intersection/< S A P) P' _ A' S) + %<- (nat`plus-right-cancels P P' nat`eq/ nat`eq/ N0+1=N0'+1) + %<- (succ-cancels N0+1=N0'+1 N0=N0P) + %<- (map/+-preserves-eq N0=N0P nat`eq/ eq/ M022=M022') + %<- (intersection-respects-eq A eq/ M022=M022' eq/ A') %. +%term _ + %pi (intersection/<-inversion (intersection/= A' _ nat`eq/) N0+1+N=N map/0 A S) + %<- (nat`plus-implies-gt N0+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-eq F M2=M022) + %<- (false-implies-eq F M3=M333) + %<- (intersection-respects-eq A' eq/ M2=M022 M3=M333 A) + %<- (false-implies-shift F S) %. +%term _ + %pi (intersection/<-inversion (intersection/> S A' N3+1+N2=N1) N0+1+N1=N2 _ A S') + %<- (nat`plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) + %<- (nat`plus-implies-gt N0+1+N1=N2 nat`eq/ N2>N1) + %<- (nat`gt-anti-symmetric N1>N2 N2>N1 F) + %<- (false-implies-eq F M311=M1) + %<- (false-implies-eq F M2=M022) + %<- (intersection-respects-eq A' M311=M1 M2=M022 eq/ A) + %<- (false-implies-shift F S') %. +%worlds () (intersection/<-inversion _ _ _ _ _) %. +%total {} (intersection/<-inversion _ _ _ _ _) %. +%reduces < AP A (intersection/<-inversion A _ _ AP _) %. +%theorem intersection/>-inversion : forall* {N1} {D1} {M1} {N2} {D2} {M2} {N3} {S2M3} forall {A intersection (map/+ N1 D1 M1) (map/+ N2 D2 M2) S2M3} {P plus (s N3) N2 N1} exists {M3} {AP intersection (map/+ N3 D1 M1) M2 M3} {S shift N2 M3 S2M3} true %. +%term _ + %pi (intersection/>-inversion (intersection/> S A P) P' _ A' S) + %<- (nat`plus-right-cancels P P' nat`eq/ nat`eq/ N3+1=N3'+1) + %<- (succ-cancels N3+1=N3'+1 N3=N3P) + %<- (map/+-preserves-eq N3=N3P nat`eq/ eq/ M311=M311') + %<- (intersection-respects-eq A M311=M311' eq/ eq/ A') %. +%term _ + %pi (intersection/>-inversion (intersection/= A' _ nat`eq/) N3+1+N=N map/0 A S) + %<- (nat`plus-implies-gt N3+1+N=N nat`eq/ N>N) + %<- (gt-anti-reflexive N>N F) + %<- (false-implies-eq F M1=M311) + %<- (false-implies-eq F M3=M333) + %<- (intersection-respects-eq A' M1=M311 eq/ M3=M333 A) + %<- (false-implies-shift F S) %. +%term _ + %pi (intersection/>-inversion (intersection/< S A' N0+1+N1=N2) N3+1+N2=N1 _ A S') + %<- (nat`plus-implies-gt N3+1+N2=N1 nat`eq/ N1>N2) + %<- (nat`plus-implies-gt N0+1+N1=N2 nat`eq/ N2>N1) + %<- (nat`gt-anti-symmetric N1>N2 N2>N1 F) + %<- (false-implies-eq F M1=M311) + %<- (false-implies-eq F M022=M2) + %<- (intersection-respects-eq A' M1=M311 M022=M2 eq/ A) + %<- (false-implies-shift F S') %. +%worlds () (intersection/>-inversion _ _ _ _ _) %. +%total {} (intersection/>-inversion _ _ _ _ _) %. +%reduces < AP A (intersection/>-inversion A _ _ AP _) %. +%theorem intersection-implies-ge : forall* {N1} {D1} {M1} {N2} {D2} {M2} {N3} {D3} {M3} forall {A intersection (map/+ N1 D1 M1) (map/+ N2 D2 M2) (map/+ N3 D3 M3)} exists {G1 ge N3 N1} {G2 ge N3 N2} true %. +%term _ intersection-implies-ge (intersection/= _ _ nat`eq/) (ge/= nat`eq/) (ge/= nat`eq/) %. +%term _ + %pi (intersection-implies-ge (intersection/< (shift/+ N1+1+N3=N4) M1*M022=M333 N0+1+N1=N2) (ge/> N4>N1) N4>=N2) + %<- (plus-swap-succ N1+1+N3=N4 N1+N3+1=N4) + %<- (plus-commutative N1+N3+1=N4 N3+1+N1=N4) + %<- (plus-implies-gt N3+1+N1=N4 nat`eq/ N4>N1) + %<- (intersection-implies-ge M1*M022=M333 _ N3>=N0) + %<- (succ-preserves-ge N3>=N0 N3+1>=N0+1) + %<- (plus-right-preserves-ge* N3+1>=N0+1 N3+1+N1=N4 N0+1+N1=N2 N4>=N2) %. +%term _ + %pi (intersection-implies-ge (intersection/> (shift/+ N2+1+N3=N5) M011*M2=M333 N0+1+N2=N1) N5>=N1 (ge/> N5>N2)) + %<- (plus-swap-succ N2+1+N3=N5 N2+N3+1=N5) + %<- (plus-commutative N2+N3+1=N5 N3+1+N2=N5) + %<- (plus-implies-gt N3+1+N2=N5 nat`eq/ N5>N2) + %<- (intersection-implies-ge M011*M2=M333 N3>=N0 _) + %<- (succ-preserves-ge N3>=N0 N3+1>=N0+1) + %<- (plus-right-preserves-ge* N3+1>=N0+1 N3+1+N2=N5 N0+1+N2=N1 N5>=N1) %. +%worlds () (intersection-implies-ge _ _ _) %. +%total (A) (intersection-implies-ge A _ _) %. +%theorem intersection-deterministic : forall* {M1} {M2} {M3} {M1P} {M2P} {M3P} forall {A intersection M1 M2 M3} {AP intersection M1P M2P M3P} {E1 eq M1 M1P} {E2 eq M2 M2P} exists {E3 eq M3 M3P} true %. +%term _ intersection-deterministic intersection/L intersection/L eq/ eq/ eq/ %. +%term _ intersection-deterministic intersection/L intersection/R eq/ eq/ eq/ %. +%term _ intersection-deterministic intersection/R intersection/L eq/ eq/ eq/ %. +%term _ intersection-deterministic intersection/R intersection/R eq/ eq/ eq/ %. +%term _ + %pi (intersection-deterministic (intersection/= M1*M2=M3 D1*D2=D3 nat`eq/) (%the (intersection _ _ M') AP) eq/ eq/ M133=M') + %<- (intersection/=-inversion AP nat`eq/ D3' M3' D1*D2=D3P M1*M2=M3' M133'=M') + %<- (nat`intersection-deterministic D1*D2=D3 D1*D2=D3P nat`eq/ nat`eq/ D3=D3') + %<- (intersection-deterministic M1*M2=M3 M1*M2=M3' eq/ eq/ M3=M3') + %<- (map/+-preserves-eq nat`eq/ D3=D3' M3=M3' M133=M133') + %<- (eq-transitive M133=M133' M133'=M' M133=M') %. +%term _ + %pi (intersection-deterministic (intersection/< M3< M3<-inversion A' N3+1+N2=N1 M3' M311*M2=M3' M3'<N1) + %<- (gt-implies-plus N2>N1 _ N0+1+N1=N2) + %<- (intersection-M-map/+-total* M1 N0 D2 M2 M3 M1*M022=M3) + %<- (shift-total* N1 M3 S1M3 M3< M3<N2) + %<- (gt-implies-plus N1>N2 _ N3+1+N2=N1) + %<- (intersection-map/+-M-total* N3 D1 M1 M2 M3 M311*M2=M3) + %<- (shift-total* N2 M3 S2M3 M3< M3< M3< M3<-inversion M111*M222=SM3 N3+1+N2=N1 M3 M311*M2=M3 M3< M3< M3< M3< M3<B1) + %<- (meta-gt _ _ B>B1) + %<- (intersection-associativeM _ BD1 M1*M2=M3 M3*M4=M7 M2*M4=M6 M1*M6=M7) + %<- (intersection-respects-eq (intersection/= M1*M6=M7 D1*D6=D7 nat`eq/) eq/ M166=M24 M177=M34 A16) %. +%term _ + %pi (intersection-associativeM* B (bound/+ N1+1+B1=B BD1) nat`compare/= (nat`compare/< N4>N1) _ A12 A34 A24 A16) + %<- (intersection/=-inversion A12 nat`eq/ D3 M3 D1*D2=D3 M1*M2=M3 M133=M12) + %<- (eq-symmetric M133=M12 M12=M133) + %<- (intersection-respects-eq A34 M12=M133 eq/ eq/ A34') + %<- (gt-implies-plus N4>N1 N5 N5+1+N1=N4) + %<- (intersection/<-inversion A34' N5+1+N1=N4 M7 M3*M544=M7 M7<B1) + %<- (meta-gt _ _ B>B1) + %<- (intersection-associativeM _ BD1 M1*M2=M3 M3*M544=M7 M2*M544=M6 M1*M6=M7) + %<- (shift-right-preserves-intersection M1*M6=M7 M6< N1>N4) _ A12 A34 A24 A16) + %<- (intersection/=-inversion A12 nat`eq/ D3 M3 D1*D2=D3 M1*M2=M3 M133=M12) + %<- (eq-symmetric M133=M12 M12=M133) + %<- (intersection-respects-eq A34 M12=M133 eq/ eq/ A34') + %<- (gt-implies-plus N1>N4 N6 N6+1+N4=N1) + %<- (intersection/>-inversion A34' N6+1+N4=N1 M7 M633*M4=M7 M7<-inversion A24 N6+1+N4=N1 M6 M622*M4=M6 M6<N6) + %<- (succ-preserves-gt N1>N6 N1+1>N6+1) + %<- (plus-total N6+1+B1=B6) + %<- (plus-right-preserves-gt* N1+1>N6+1 N1+1+B1=B N6+1+B1=B6 B>B6) + %<- (meta-gt _ _ B>B6) + %<- (intersection-associativeM _ (bound/+ N6+1+B1=B6 BD1) (intersection/= M1*M2=M3 D1*D2=D3 nat`eq/) M633*M4=M7 M622*M4=M6 M611*M6=M7) + %<- (shift-preserves-intersection M611*M6=M7 (shift/+ N4+1+N6=N1) M6<N1) nat`compare/= _ A12 A34 A24 A16) + %<- (gt-implies-plus N2>N1 N0 N0+1+N1=N2) + %<- (intersection/<-inversion A12 N0+1+N1=N2 M3 M1*M022=M3 M3<B1) + %<- (meta-gt _ _ B>B1) + %<- (intersection-associativeM _ BD1 M1*M022=M3 M3*M044=M7 (intersection/= M2*M4=M6 D2*D4=D6 nat`eq/) M1*M066=M7) + %<- (shift-right-preserves-intersection M1*M066=M7 (shift/+ N1+1+N0=N2) M7< N1>N2) nat`compare/= _ A12 A34 A24 A16) + %<- (gt-implies-plus N1>N2 N3 N3+1+N2=N1) + %<- (intersection/>-inversion A12 N3+1+N2=N1 M3 M311*M2=M3 M3<N3) + %<- (succ-preserves-gt N1>N3 N1+1>N3+1) + %<- (plus-total N3+1+B1=B3) + %<- (plus-right-preserves-gt* N1+1>N3+1 N1+1+B1=B N3+1+B1=B3 B>B3) + %<- (meta-gt _ _ B>B3) + %<- (intersection-associativeM _ (bound/+ N3+1+B1=B3 BD1) M311*M2=M3 M3*M4=M7 M2*M4=M6 M311*M6=M7) + %<- (intersection-respects-eq (intersection/> M7<N1) _ nat`compare/= A12 A34 A24 A16) + %<- (gt-implies-plus N2>N1 N0 N0+1+N1=N2) + %<- (intersection/<-inversion A12 N0+1+N1=N2 M3 M1*M022=M3 M3<-inversion A24 N0+1+N1=N2 M6 M022*M4=M6 M6<B1) + %<- (meta-gt _ _ B>B1) + %<- (intersection-associativeM _ BD1 M1*M022=M3 M3*M4=M7 M022*M4=M6 M1*M6=M7) + %<- (shift-right-preserves-intersection M1*M6=M7 M6< N1>N2) _ nat`compare/= A12 A34 A24 A16) + %<- (gt-implies-plus N1>N2 N3 N3+1+N2=N1) + %<- (plus-swap-succ N3+1+N2=N1 N3+N2+1=N1) + %<- (plus-commutative N3+N2+1=N1 N2+1+N3=N1) + %<- (intersection/>-inversion A12 N3+1+N2=N1 M3 M311*M2=M3 M3<N3) + %<- (succ-preserves-gt N1>N3 N1+1>N3+1) + %<- (plus-total N3+1+B1=B3) + %<- (plus-right-preserves-gt* N1+1>N3+1 N1+1+B1=B N3+1+B1=B3 B>B3) + %<- (meta-gt _ _ B>B3) + %<- (intersection-associativeM _ (bound/+ N3+1+B1=B3 BD1) M311*M2=M3 M3*M344=M7 M2*M344=M6 M311*M6=M7) + %<- (shift-preserves-intersection M311*M6=M7 (shift/+ N2+1+N3=N1) M6<N1) (nat`compare/< N4>N2) _ A12 A34 A24 A16) + %<- (gt-implies-plus N2>N1 N0 N0+1+N1=N2) + %<- (intersection/<-inversion A12 N0+1+N1=N2 M3 M1*M022=M3 M3<N2 N6 N6+1+N2=N4) + %<- (intersection/<-inversion A24 N6+1+N2=N4 M6 M2*M644=M6 M6<B1) + %<- (meta-gt _ _ B>B1) + %<- (intersection-associativeM _ BD1 M1*M022=M3 M3*M744=M7 (intersection/< M6< N1>N2) (nat`compare/> N2>N4) _ A12 A34 A24 A16) + %<- (gt-implies-plus N1>N2 N3 N3+1+N2=N1) + %<- (intersection/>-inversion A12 N3+1+N2=N1 M3 M311*M2=M3 M3<N4 N6 N6+1+N4=N2) + %<- (intersection/>-inversion A24 N6+1+N4=N2 M6 M622*M4=M6 M6<N9) + %<- (succ-preserves-gt N1>N9 N1+1>N9+1) + %<- (plus-total N9+1+B1=B9) + %<- (plus-right-preserves-gt* N1+1>N9+1 N1+1+B1=B N9+1+B1=B9 B>B9) + %<- (meta-gt _ _ B>B9) + %<- (intersection-associativeM _ (bound/+ N9+1+B1=B9 BD1) M911*M622=S6M3 S6M3*M4=M7 M622*M4=M6 M911*M6=M7) + %<- (shift-preserves-intersection M911*M6=M7 (shift/+ N4+1+N9=N1) M6< N2>N4) (nat`compare/< N4>N1) A12 A34 A24 A16) + %<- (gt-implies-plus N2>N4 N6 N6+1+N4=N2) + %<- (gt-implies-plus N4>N1 N5 N5+1+N1=N4) + %<- (plus-swap-succ N5+1+N1=N4 N5+N1+1=N4) + %<- (plus-associative-converse N5+N1+1=N4 N6+1+N4=N2 N0 N6+1+N5=N0 N0+N1+1=N2) + %<- (plus-swap-succ-converse N0+N1+1=N2 N0+1+N1=N2) + %<- (intersection/<-inversion A12 N0+1+N1=N2 M3 M1*M022=M3 M3<-inversion A24 N6+1+N4=N2 M6 M622*M4=M6 M6<B1) + %<- (meta-gt _ _ B>B1) + %<- (intersection-associativeM _ BD1 M1*M022=M3 M3*M544=M7 M022*M544=S5M6 M1*S5M6=M7) + %<- (shift-right-preserves-intersection M1*S5M6=M7 S5M6<N2) (nat`compare/> N1>N4) A12 A34 A24 A16) + %<- (gt-implies-plus N4>N2 N6 N6+1+N2=N4) + %<- (gt-implies-plus N1>N4 N5 N5+1+N4=N1) + %<- (plus-swap-succ N6+1+N2=N4 N6+N2+1=N4) + %<- (plus-associative-converse N6+N2+1=N4 N5+1+N4=N1 N3 N5+1+N6=N3 N3+N2+1=N1) + %<- (plus-swap-succ-converse N3+N2+1=N1 N3+1+N2=N1) + %<- (intersection/>-inversion A12 N3+1+N2=N1 M3 M311*M2=M3 M3<N3) + %<- (succ-preserves-gt N1>N3 N1+1>N3+1) + %<- (plus-total N3+1+B1=B3) + %<- (plus-right-preserves-gt* N1+1>N3+1 N1+1+B1=B N3+1+B1=B3 B>B3) + %<- (meta-gt _ _ B>B3) + %<- (intersection-associativeM _ (bound/+ N3+1+B1=B3 BD1) M311*M2=M3 M3*M644=M7 M2*M644=M6 M311*M6=M7) + %<- (shift-preserves-intersection M311*M6=M7 (shift/+ N2+1+N3=N1) M6< N1>N2) _ (nat`compare/< N4>N1) A12 A34 A24 A16) + %<- (gt-implies-plus N1>N2 N3 N3+1+N2=N1) + %<- (gt-implies-plus N4>N1 N5 N5+1+N1=N4) + %<- (plus-swap-succ N3+1+N2=N1 N3+N2+1=N1) + %<- (plus-associative-converse N3+N2+1=N1 N5+1+N1=N4 N6 N5+1+N3=N6 N6+N2+1=N4) + %<- (plus-swap-succ-converse N6+N2+1=N4 N6+1+N2=N4) + %<- (intersection/>-inversion A12 N3+1+N2=N1 M3 M311*M2=M3 M3<N3) + %<- (succ-preserves-gt N1>N3 N1+1>N3+1) + %<- (plus-total N3+1+B1=B3) + %<- (plus-right-preserves-gt* N1+1>N3+1 N1+1+B1=B N3+1+B1=B3 B>B3) + %<- (meta-gt _ _ B>B3) + %<- (intersection-associativeM _ (bound/+ N3+1+B1=B3 BD1) M311*M2=M3 M3*M644=M7 M2*M644=M6 M311*M6=M7) + %<- (shift-preserves-intersection M311*M6=M7 (shift/+ N2+1+N3=N1) M6<N1) _ (nat`compare/> N1>N4) A12 A34 A24 A16) + %<- (gt-implies-plus N2>N1 N0 N0+1+N1=N2) + %<- (gt-implies-plus N1>N4 N5 N5+1+N4=N1) + %<- (plus-swap-succ N5+1+N4=N1 N5+N4+1=N1) + %<- (plus-associative-converse N5+N4+1=N1 N0+1+N1=N2 N6 N0+1+N5=N6 N6+N4+1=N2) + %<- (plus-swap-succ-converse N6+N4+1=N2 N6+1+N4=N2) + %<- (intersection/<-inversion A12 N0+1+N1=N2 M3 M1*M022=M3 M3<-inversion A24 N6+1+N4=N2 M6 M622*M4=M6 M6<N5) + %<- (succ-preserves-gt N1>N5 N1+1>N5+1) + %<- (plus-total N5+1+B1=B5) + %<- (plus-right-preserves-gt* N1+1>N5+1 N1+1+B1=B N5+1+B1=B5 B>B5) + %<- (meta-gt _ _ B>B5) + %<- (intersection-associativeM _ (bound/+ N5+1+B1=B5 BD1) M511*M622=S5M3 S5M3*M4=M7 M622*M4=M6 M511*M6=M7) + %<- (shift-preserves-intersection M511*M6=M7 (shift/+ N4+1+N5=N1) M6< M3< M400<=M1 N4+1+N1=N0) L2 (leq/> M400<=M3 N4+1+N1=N0)) + %<- (leq/>-inversion L2 N4+1+N1=N0 M400<=M2) + %<- (intersection-is-glb M1*M2=M3 M400<=M1 M400<=M2 M400<=M3) %. +%term _ + %pi (intersection-is-glb (intersection/< M3< M500<=M1 N5+1+N1=N2) (leq/= M0<=M2 D0<=D2 nat`eq/) M200<=S1M3) + %<- (plus-right-cancels N5+1+N1=N2 N3+1+N1=N2 nat`eq/ nat`eq/ N5+1=N3+1) + %<- (succ-cancels N5+1=N3+1 N5=N3) + %<- (map/+-preserves-eq N5=N3 nat`eq/ eq/ M500=M300) + %<- (leq-respects-eq M500<=M1 M500=M300 eq/ M300<=M1) + %<- (intersection-is-glb M1*M322=M3 M300<=M1 (leq/= M0<=M2 D0<=D2 nat`eq/) M300<=M3) + %<- (plus-swap-succ N3+1+N1=N2 N3+N1+1=N2) + %<- (plus-commutative N3+N1+1=N2 N1+1+N3=N2) + %<- (shift-preserves-leq* M300<=M3 (shift/+ N1+1+N3=N2) M3< M500<=M1 N5+1+N1=N0) (leq/> M600<=M2 N6+1+N2=N0) M000<=S1M3) + %<- (plus-swap-succ N3+1+N1=N2 N3+N1+1=N2) + %<- (plus-associative-converse N3+N1+1=N2 N6+1+N2=N0 N5' N6+1+N3=N5' N5'+N1+1=N0) + %<- (plus-swap-succ N5+1+N1=N0 N5+N1+1=N0) + %<- (plus-right-cancels N5'+N1+1=N0 N5+N1+1=N0 nat`eq/ nat`eq/ N5'=N5) + %<- (plus-respects-eq N6+1+N3=N5' nat`eq/ nat`eq/ N5'=N5 N6+1+N3=N5) + %<- (plus-swap-succ N6+1+N3=N5 N6+N3+1=N5) + %<- (plus-commutative N6+N3+1=N5 N3+1+N6=N5) + %<- (shift-left-preserves-leq* M600<=M2 (shift/+ N3+1+N6=N5) M500<=M322) + %<- (intersection-is-glb M1*M322=M3 M500<=M1 M500<=M322 M500<=M3) + %<- (plus-commutative N5+N1+1=N0 N1+1+N5=N0) + %<- (shift-preserves-leq* M500<=M3 (shift/+ N1+1+N5=N0) M3< M3<-inversion L2 N3+1+N2=N1 M300<=M2) + %<- (intersection-is-glb M311*M2=M3 (leq/= M0<=M1 D0<=D1 nat`eq/) M300<=M2 M300<=M3) + %<- (plus-swap-succ N3+1+N2=N1 N3+N2+1=N1) + %<- (plus-commutative N3+N2+1=N1 N2+1+N3=N1) + %<- (shift-preserves-leq* M300<=M3 (shift/+ N2+1+N3=N1) M3< M3< M500<=M1 N5+1+N1=N0) L2 M000<=S2M3) + %<- (plus-swap-succ N3+1+N2=N1 N3+N2+1=N1) + %<- (plus-associative-converse N3+N2+1=N1 N5+1+N1=N0 N6 N5+1+N3=N6 N6+N2+1=N0) + %<- (plus-swap-succ-converse N6+N2+1=N0 N6+1+N2=N0) + %<- (leq/>-inversion L2 N6+1+N2=N0 M600<=M2) + %<- (plus-swap-succ N5+1+N3=N6 N5+N3+1=N6) + %<- (plus-commutative N5+N3+1=N6 N3+1+N5=N6) + %<- (shift-left-preserves-leq* M500<=M1 (shift/+ N3+1+N5=N6) M600<=M311) + %<- (intersection-is-glb M311*M2=M3 M600<=M311 M600<=M2 M600<=M3) + %<- (plus-commutative N6+N2+1=N0 N2+1+N6=N0) + %<- (shift-preserves-leq* M600<=M3 (shift/+ N2+1+N6=N0) M3< M3<-inversion M111*M244=M N3+1+N2=N1 M5 M311*M4=M5 M5< M622<=M4 N6+1+N4=N2) (intersection/= M1*M2=M3 D1*D2=D3 nat`eq/) M211*M444=M M233<=M) + %<- (intersection/>-inversion M211*M444=M N6+1+N4=N2 M5 M611*M4=M5 M5< M622<=M4 N6+1+N4=N2) (intersection/> M3<-inversion M111*M444=M N5+1+N4=N1 M5 M511*M4=M5 M5< M3< M622<=M4 N6+1+N4=N2) A1 (intersection/= M1*M4=M5 D1*D4=D5 nat`eq/) S1M3<=M455) + %<- (intersection/<-inversion A1 N6+1+N4=N2 M3 M1*M622=M3 M3< M622<=M4 N6+1+N4=N2) A1 (intersection/< M5< M5<-inversion L N6+1+N4=N2 M622<=M4) + %<- (plus-swap-succ N0+1+N5=N6 N0+N5+1=N6) + %<- (plus-commutative N0+N5+1=N6 N5+1+N0=N6) + %<- (shift-total M3< M7< M5< M6< M6< M7< M5< M6< M311+M2=M3 N3+1+N2=N1) (intersection/= M3*M4=M7 D2*D4=D7 nat`eq/) _ _ (intersection/> M5< M311+M2=M3 N3+1+N2=N1) (intersection/< M7< M311+M2=M3 N3+1+N2=N1) (intersection/> M7< M5< M6< (count M N (s C)) %. +%term count/z %pi (not-member M N) %-> (count M N z) %. +%sort add %. +%term add/0 %pi (not-member M N) %-> (update M N z M') %-> (add M N M') %. +%term add/+ %pi (lookup M N C) %-> (update M N (s C) M') %-> (add M N M') %. +%%%% Theorems +%%% Theorems about count +%theorem false-implies-count : forall* {M} {N} {C} forall {F void} exists {MC count M N C} true %. +%worlds () (false-implies-count _ _) %. +%total {} (false-implies-count _ _) %. +%theorem count-respects-eq : forall* {M1} {N1} {C1} {M2} {N2} {C2} forall {MC1 count M1 N1 C1} {EM eq M1 M2} {EN nat`eq N1 N2} {EC nat`eq C1 C2} exists {MC2 count M2 N2 C2} true %. +%term _ count-respects-eq C eq/ nat`eq/ nat`eq/ C %. +%worlds () (count-respects-eq _ _ _ _ _) %. +%total {} (count-respects-eq _ _ _ _ _) %. +%theorem count-total** : forall* {M} {N} {B} forall {D? member? M N B} exists {C} {MC count M N C} true %. +%term _ count-total** (member?/in L) _ (count/s L) %. +%term _ count-total** (member?/out F) _ (count/z F) %. +%worlds () (count-total** _ _ _) %. +%total {} (count-total** _ _ _) %. +%theorem count-total* : forall {M} {N} exists {C} {MC count M N C} true %. +%term _ %pi (count-total* M N _ MC) %<- (member?-total M?) %<- (count-total** M? _ MC) %. +%worlds () (count-total* _ _ _ _) %. +%total {} (count-total* _ _ _ _) %. +%inline count-total count-total* _ _ _ %. +%theorem count-deterministic : forall* {M1} {N1} {C1} {M2} {N2} {C2} forall {MC1 count M1 N1 C1} {MC2 count M2 N2 C2} {EM eq M1 M2} {EN nat`eq N1 N2} exists {EC nat`eq C1 C2} true %. +%term _ count-deterministic (count/z _) (count/z _) _ _ nat`eq/ %. +%term _ + %pi (count-deterministic (count/s L) (count/s L') eq/ nat`eq/ SC=sC') + %<- (lookup-deterministic L L' eq/ nat`eq/ C=C') + %<- (succ-deterministic C=C' SC=sC') %. +%term _ + %pi (count-deterministic (count/z F) (count/s L) eq/ nat`eq/ E) + %<- (not-member-lookup-not-equal F L N<>N) + %<- (nat`ne-anti-reflexive N<>N V) + %<- (nat`false-implies-eq V E) %. +%term _ + %pi (count-deterministic (count/s L) (count/z F) eq/ nat`eq/ E) + %<- (not-member-lookup-not-equal F L N<>N) + %<- (nat`ne-anti-reflexive N<>N V) + %<- (nat`false-implies-eq V E) %. +%worlds () (count-deterministic _ _ _ _ _) %. +%total {} (count-deterministic _ _ _ _ _) %. +%theorem count-empty-is-zero : forall* {N} {M} forall {K count map/0 N M} exists {E nat`eq M z} true %. +%term _ count-empty-is-zero (count/z _) nat`eq/ %. +%worlds () (count-empty-is-zero _ _) %. +%total {} (count-empty-is-zero _ _) %. +%%% Theorems about add +%theorem false-implies-add : forall* {M} {N} {M'} forall {F void} exists {A add M N M'} true %. +%worlds () (false-implies-add _ _) %. +%total {} (false-implies-add _ _) %. +%theorem add-respects-eq : forall* {M1} {N1} {M1'} {M2} {N2} {M2'} forall {A1 add M1 N1 M1'} {EM eq M1 M2} {EN nat`eq N1 N2} {EM' eq M1' M2'} exists {A2 add M2 N2 M2'} true %. +%term _ add-respects-eq A eq/ nat`eq/ eq/ A %. +%worlds () (add-respects-eq _ _ _ _ _) %. +%total {} (add-respects-eq _ _ _ _ _) %. +%theorem add-total* : forall {M} {N} exists {M'} {A add M N M'} true %. +%inline add-total add-total* _ _ _ %. +%theorem add-total/L : forall* {M} {N} {B} forall {M? member? M N B} exists {M'} {A add M N M'} true %. +%term _ %pi (add-total/L (member?/out MF) _ (add/0 MF MU)) %<- (update-total MU) %. +%term _ %pi (add-total/L (member?/in ML) _ (add/+ ML MU)) %<- (update-total MU) %. +%worlds () (add-total/L _ _ _) %. +%total {} (add-total/L _ _ _) %. +%term _ %pi (add-total A) %<- (member?-total M) %<- (add-total/L M _ A) %. +%worlds () (add-total* _ _ _ _) %. +%total {} (add-total* _ _ _ _) %. +%theorem add-deterministic : forall* {M1} {N1} {M1'} {M2} {N2} {M2'} forall {A1 add M1 N1 M1'} {A2 add M2 N2 M2'} {EM eq M1 M2} {EN nat`eq N1 N2} exists {EM' eq M1' M2'} true %. +%term _ + %pi (add-deterministic (add/0 _ U) (add/0 _ U') eq/ nat`eq/ EM') + %<- (update-deterministic U U' eq/ nat`eq/ nat`eq/ EM') %. +%term _ + %pi (add-deterministic (add/0 NM _) (add/+ M _) eq/ nat`eq/ EM) + %<- (not-member-lookup-not-equal NM M NE) + %<- (nat`ne-anti-reflexive NE F) + %<- (false-implies-eq F EM) %. +%term _ + %pi (add-deterministic (add/+ M _) (add/0 NM _) eq/ nat`eq/ EM) + %<- (not-member-lookup-not-equal NM M NE) + %<- (nat`ne-anti-reflexive NE F) + %<- (false-implies-eq F EM) %. +%term _ + %pi (add-deterministic (add/+ ML1 U1) (add/+ ML2 U2) EM EN EM') + %<- (lookup-deterministic ML1 ML2 EM EN EC) + %<- (succ-deterministic EC SEC) + %<- (update-deterministic U1 U2 EM EN SEC EM') %. +%worlds () (add-deterministic _ _ _ _ _) %. +%total {} (add-deterministic _ _ _ _ _) %. +%theorem add-commutes : forall* {M0} {N1} {M1} {N2} {M2} forall {A01 add M0 N1 M1} {A12 add M1 N2 M2} exists {M3} {A02 add M0 N2 M3} {A32 add M3 N1 M2} true %. +%theorem add-commutes/L : forall* {M0} {N1} {M1} {N2} {M2} {B} forall {A01 add M0 N1 M1} {A12 add M1 N2 M2} {EQ? nat`eq? N1 N2 B} exists {M3} {A02 add M0 N2 M3} {A32 add M3 N1 M2} true %. +%term _ add-commutes/L A0 A1 nat`eq?/yes _ A0 A1 %. +%term _ + %pi (add-commutes/L (add/0 NM1 U01) (add/0 NM2 U12) (nat`eq?/no N1<>N2) _ (add/0 NM2' U02) (add/0 NM1' U32)) + %<- (update-preserves-not-member-converse NM2 U01 NM2') + %<- (update-total U02) + %<- (update-preserves-not-member NM1 U02 N1<>N2 NM1') + %<- (update-commutes* U01 U12 N1<>N2 U02 U32) %. +%term _ + %pi (add-commutes/L (add/0 NM1 U01) (add/+ L2 U12) (nat`eq?/no N1<>N2) _ (add/+ L2' U02) (add/0 NM1' U32)) + %<- (nat`ne-symmetric N1<>N2 N2<>N1) + %<- (update-preserves-lookup-converse L2 U01 N2<>N1 L2') + %<- (update-total U02) + %<- (update-preserves-not-member NM1 U02 N1<>N2 NM1') + %<- (update-commutes* U01 U12 N1<>N2 U02 U32) %. +%term _ + %pi (add-commutes/L (add/+ L1 U01) (add/0 NM2 U12) (nat`eq?/no N1<>N2) _ (add/0 NM2' U02) (add/+ L1' U32)) + %<- (update-preserves-not-member-converse NM2 U01 NM2') + %<- (update-total U02) + %<- (update-preserves-lookup L1 U02 N1<>N2 L1') + %<- (update-commutes* U01 U12 N1<>N2 U02 U32) %. +%term _ + %pi (add-commutes/L (add/+ L1 U01) (add/+ L2 U12) (nat`eq?/no N1<>N2) _ (add/+ L2' U02) (add/+ L1' U32)) + %<- (nat`ne-symmetric N1<>N2 N2<>N1) + %<- (update-preserves-lookup-converse L2 U01 N2<>N1 L2') + %<- (update-total U02) + %<- (update-preserves-lookup L1 U02 N1<>N2 L1') + %<- (update-commutes* U01 U12 N1<>N2 U02 U32) %. +%worlds () (add-commutes/L _ _ _ _ _ _) %. +%total {} (add-commutes/L _ _ _ _ _ _) %. +%term _ + %pi (add-commutes A01 A12 _ A02 A23) + %<- (nat`eq?-total EQ?) + %<- (add-commutes/L A01 A12 EQ? _ A02 A23) %. +%worlds () (add-commutes _ _ _ _ _) %. +%total {} (add-commutes _ _ _ _ _) %. +%theorem count-add-implies-count : forall* {M} {N} {C} {M'} forall {K count M N C} {A add M N M'} exists {K' count M' N (s C)} true %. +%term _ + %pi (count-add-implies-count (count/z NM) A K') + %<- (update-total U) + %<- (add-deterministic (add/0 NM U) A eq/ nat`eq/ EM') + %<- (update-implies-lookup U L) + %<- (count-respects-eq (count/s L) EM' nat`eq/ nat`eq/ K') %. +%term _ + %pi (count-add-implies-count (count/s L) A K') + %<- (update-total U) + %<- (add-deterministic (add/+ L U) A eq/ nat`eq/ EM') + %<- (update-implies-lookup U L') + %<- (count-respects-eq (count/s L') EM' nat`eq/ nat`eq/ K') %. +%worlds () (count-add-implies-count _ _ _) %. +%total {} (count-add-implies-count _ _ _) %. +%theorem update-preserves-count : forall* {M} {N} {C} {M'} {N'} {C'} forall {K count M N C} {A update M N' C' M'} {NE nat`ne N N'} exists {K' count M' N C} true %. +%term _ + %pi (update-preserves-count (count/z NM) U NE (count/z NM')) + %<- (update-preserves-not-member NM U NE NM') %. +%term _ + %pi (update-preserves-count (count/s L) U NE (count/s L')) + %<- (update-preserves-lookup L U NE L') %. +%worlds () (update-preserves-count _ _ _ _) %. +%total {} (update-preserves-count _ _ _ _) %. +%theorem add-preserves-count : forall* {M} {N} {C} {M'} {N'} forall {K count M N C} {A add M N' M'} {NE nat`ne N N'} exists {K' count M' N C} true %. +%term _ + %pi (add-preserves-count K (add/0 _ U) NE K') + %<- (update-preserves-count K U NE K') %. +%term _ + %pi (add-preserves-count K (add/+ _ U) NE K') + %<- (update-preserves-count K U NE K') %. +%worlds () (add-preserves-count _ _ _ _) %. +%total {} (add-preserves-count _ _ _ _) %. +%theorem add-preserves-count-converse : forall* {M} {N} {C} {M'} {N'} forall {K' count M' N C} {A add M N' M'} {NE nat`ne N N'} exists {K count M N C} true %. +%term _ + %pi (add-preserves-count-converse K2 A NE K1) + %<- (count-total K1') + %<- (add-preserves-count K1' A NE K2') + %<- (count-deterministic K2' K2 eq/ nat`eq/ EQ) + %<- (count-respects-eq K1' eq/ nat`eq/ EQ K1) %. +%worlds () (add-preserves-count-converse _ _ _ _) %. +%total {} (add-preserves-count-converse _ _ _ _) %. +%theorem shift-preserves-count-converse : forall* {N} {M1} {N1} {M2} {N2} {C} forall {C2 count M2 N2 C} {MS shift N M1 M2} {P plus (s N) N1 N2} exists {C1 count M1 N1 C} true %. +%term _ + %pi (shift-preserves-count-converse (count/z MF) MS P (count/z MF')) + %<- (shift-preserves-not-member-converse* MF MS P MF') %. +%term _ + %pi (shift-preserves-count-converse (count/s ML) MS P (count/s ML')) + %<- (shift-preserves-lookup-converse* ML MS P ML') %. +%worlds () (shift-preserves-count-converse _ _ _ _) %. +%total {} (shift-preserves-count-converse _ _ _ _) %. +%theorem shift-preserves-add-converse : forall* {N} {M1} {N1} {M2} {N2} {M2'} forall {A2 add M2 N2 M2'} {MS shift N M1 M2} {P plus (s N) N1 N2} exists {M1'} {MS' shift N M1' M2'} {A1 add M1 N1 M1'} true %. +%term _ + %pi (shift-preserves-add-converse (add/+ ML MU) MS P _ MS' (add/+ ML' MU')) + %<- (shift-preserves-update-converse MU MS P _ MS' MU') + %<- (shift-preserves-lookup-converse* ML MS P ML') %. +%term _ + %pi (shift-preserves-add-converse (add/0 MF MU) MS P _ MS' (add/0 MF' MU')) + %<- (shift-preserves-update-converse MU MS P _ MS' MU') + %<- (shift-preserves-not-member-converse* MF MS P MF') %. +%worlds () (shift-preserves-add-converse _ _ _ _ _ _) %. +%total {} (shift-preserves-add-converse _ _ _ _ _ _) %. +%%%%% multiset-extra.elf +%%%%% Extra theorems about multisets not derived from map. +%%%%% This file is part of the multiset.elf signature +%%%% Theorems +%theorem union-right-distributes-over-intersection : forall* {S1} {S2} {S3} {S4} {S7} forall {I12 intersection S1 S2 S3} {U34 union S3 S4 S7} exists {S5} {S6} {U14 union S1 S4 S5} {U24 union S2 S4 S6} {I56 intersection S5 S6 S7} true %. +%term _ + %pi (union-right-distributes-over-intersection S1*S2=S3 S3+S4=S7 S5 S6 S1+S4=S5 S2+S4=S6 S5*S6=S7) + %<- (union-total S1+S4=S5) + %<- (union-total S2+S4=S6) + %<- (intersection-total S5*S6=S7') + %<- (union-implies-leq S2+S4=S6 _ S4<=S6) + %<- (leq-implies-intersection S4<=S6 S4*S6=S4) + %<- (intersection-total S1*S6=S8) + %<- (intersection-total S1*S4=S9) + %<- (intersection-right-distributes-over-union* S1+S4=S5 S5*S6=S7' S1*S6=S8 S4*S6=S4 S8+S4=S7') + %<- (intersection-left-distributes-over-union* S2+S4=S6 S1*S6=S8 S1*S2=S3 S1*S4=S9 S3+S9=S8) + %<- (intersection-implies-leq S1*S4=S9 _ S9<=S4) + %<- (leq-implies-union S9<=S4 S9+S4=S4) + %<- (union-associative* S3+S9=S8 S8+S4=S7' S9+S4=S4 S3+S4=S7') + %<- (union-deterministic S3+S4=S7' S3+S4=S7 eq/ eq/ S7'=S7) + %<- (intersection-respects-eq S5*S6=S7' eq/ eq/ S7'=S7 S5*S6=S7) %. +%worlds () (union-right-distributes-over-intersection _ _ _ _ _ _ _) %. +%total {} (union-right-distributes-over-intersection _ _ _ _ _ _ _) %. +%theorem union-right-distributes-over-intersection* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {A12 intersection X1 X2 X3} {M34 union X3 X4 X7} {M14 union X1 X4 X5} {M24 union X2 X4 X6} exists {A56 intersection X5 X6 X7} true %. +%term _ + %pi (union-right-distributes-over-intersection* X1*X2=X3 X3+X4=X7 X1+X4=X5 X2+X4=X6 X5*X6=X7) + %<- (union-right-distributes-over-intersection X1*X2=X3 X3+X4=X7 Y5 Y6 X1+X4=Y5 X2+X4=Y6 Y5*Y6=X7) + %<- (union-deterministic X1+X4=Y5 X1+X4=X5 eq/ eq/ Y5=X5) + %<- (union-deterministic X2+X4=Y6 X2+X4=X6 eq/ eq/ Y6=X6) + %<- (intersection-respects-eq Y5*Y6=X7 Y5=X5 Y6=X6 eq/ X5*X6=X7) %. +%worlds () (union-right-distributes-over-intersection* X1*X2=X3 X3+X4=X7 X1+X4=X5 X2+X4=X6 X5*X6=X7) %. +%total {} (union-right-distributes-over-intersection* _ _ _ _ _) %. +%theorem union-left-distributes-over-intersection* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {A12 intersection X2 X4 X6} {M34 union X1 X6 X7} {M14 union X1 X2 X3} {M24 union X1 X4 X5} exists {A56 intersection X3 X5 X7} true %. +%term _ + %pi (union-left-distributes-over-intersection* X2*X4=X6 X1+X6=X7 X1+X2=X3 X1+X4=X5 X3*X5=X7) + %<- (union-commutative X1+X6=X7 X6+X1=X7) + %<- (union-commutative X1+X2=X3 X2+X1=X3) + %<- (union-commutative X1+X4=X5 X4+X1=X5) + %<- (union-right-distributes-over-intersection* X2*X4=X6 X6+X1=X7 X2+X1=X3 X4+X1=X5 X3*X5=X7) %. +%worlds () (union-left-distributes-over-intersection* X2*X4=X6 X1+X6=X7 X1+X2=X3 X1+X4=X5 X3*X5=X7) %. +%total {} (union-left-distributes-over-intersection* _ _ _ _ _) %. +%theorem union-left-distributes-over-intersection : forall* {X1} {X2} {X4} {X6} {X7} forall {A12 intersection X2 X4 X6} {M34 union X1 X6 X7} exists {X3} {X5} {M14 union X1 X2 X3} {M24 union X1 X4 X5} {A56 intersection X3 X5 X7} true %. +%term _ + %pi (union-left-distributes-over-intersection X2*X4=X6 X1+X6=X7 X3 X5 X1+X2=X3 X1+X4=X5 X3*X5=X7) + %<- (union-total X1+X2=X3) + %<- (union-total X1+X4=X5) + %<- (union-left-distributes-over-intersection* X2*X4=X6 X1+X6=X7 X1+X2=X3 X1+X4=X5 X3*X5=X7) %. +%worlds () (union-left-distributes-over-intersection X2*X4=X6 X1+X6=X7 X3 X5 X1+X2=X3 X1+X4=X5 X3*X5=X7) %. +%total {} (union-left-distributes-over-intersection _ _ _ _ _ _ _) %. +%theorem union-right-factors-over-intersection : forall* {X1} {X2} {X4} {X5} {X6} {X7} forall {M14 union X1 X4 X5} {M24 union X2 X4 X6} {A56 intersection X5 X6 X7} exists {X3} {A12 intersection X1 X2 X3} {M34 union X3 X4 X7} true %. +%term _ + %pi (union-right-factors-over-intersection X1+X4=X5 X2+X4=X6 X5*X6=X7 X3 X1*X2=X3 X3+X4=X7) + %<- (intersection-total X1*X2=X3) + %<- (union-total X3+X4=Y7) + %<- (union-right-distributes-over-intersection* X1*X2=X3 X3+X4=Y7 X1+X4=X5 X2+X4=X6 X5*X6=Y7) + %<- (intersection-deterministic X5*X6=Y7 X5*X6=X7 eq/ eq/ Y7=X7) + %<- (union-respects-eq X3+X4=Y7 eq/ eq/ Y7=X7 X3+X4=X7) %. +%worlds () (union-right-factors-over-intersection X1+X4=X5 X2+X4=X6 X5*X6=X7 X3 X1*X2=X3 X3+X4=X7) %. +%total {} (union-right-factors-over-intersection _ _ _ _ _ _) %. +%theorem union-right-factors-over-intersection* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {M14 union X1 X4 X5} {M24 union X2 X4 X6} {A56 intersection X5 X6 X7} {A12 intersection X1 X2 X3} exists {M34 union X3 X4 X7} true %. +%term _ + %pi (union-right-factors-over-intersection* X1+X4=X5 X2+X4=X6 X5*X6=X7 X1*X2=X3 X3+X4=X7) + %<- (union-total X3+X4=Y7) + %<- (union-right-distributes-over-intersection* X1*X2=X3 X3+X4=Y7 X1+X4=X5 X2+X4=X6 X5*X6=Y7) + %<- (intersection-deterministic X5*X6=Y7 X5*X6=X7 eq/ eq/ Y7=X7) + %<- (union-respects-eq X3+X4=Y7 eq/ eq/ Y7=X7 X3+X4=X7) %. +%worlds () (union-right-factors-over-intersection* X1+X4=X5 X2+X4=X6 X5*X6=X7 X1*X2=X3 X3+X4=X7) %. +%total {} (union-right-factors-over-intersection* _ _ _ _ _) %. +%theorem union-left-factors-over-intersection : forall* {X1} {X2} {X3} {X4} {X5} {X7} forall {M12 union X1 X2 X3} {M14 union X1 X4 X5} {A35 intersection X3 X5 X7} exists {X6} {A24 intersection X2 X4 X6} {M16 union X1 X6 X7} true %. +%term _ + %pi (union-left-factors-over-intersection X1+X2=X3 X1+X4=X5 X3*X5=X7 X6 X2*X4=X6 X1+X6=X7) + %<- (union-commutative X1+X2=X3 X2+X1=X3) + %<- (union-commutative X1+X4=X5 X4+X1=X5) + %<- (union-right-factors-over-intersection X2+X1=X3 X4+X1=X5 X3*X5=X7 X6 X2*X4=X6 X6+X1=X7) + %<- (union-commutative X6+X1=X7 X1+X6=X7) %. +%worlds () (union-left-factors-over-intersection X1+X2=X3 X1+X4=X5 X3*X5=X7 X6 X2*X4=X6 X1+X6=X7) %. +%total {} (union-left-factors-over-intersection _ _ _ _ _ _) %. +%theorem union-left-factors-over-intersection* : forall* {X1} {X2} {X3} {X4} {X5} {X6} {X7} forall {M12 union X1 X2 X3} {M14 union X1 X4 X5} {A35 intersection X3 X5 X7} {A24 intersection X2 X4 X6} exists {M16 union X1 X6 X7} true %. +%term _ + %pi (union-left-factors-over-intersection* X1+X2=X3 X1+X4=X5 X3*X5=X7 X2*X4=X6 X1+X6=X7) + %<- (union-total X1+X6=Y7) + %<- (union-left-distributes-over-intersection* X2*X4=X6 X1+X6=Y7 X1+X2=X3 X1+X4=X5 X3*X5=Y7) + %<- (intersection-deterministic X3*X5=Y7 X3*X5=X7 eq/ eq/ Y7=X7) + %<- (union-respects-eq X1+X6=Y7 eq/ eq/ Y7=X7 X1+X6=X7) %. +%worlds () (union-left-factors-over-intersection* X1+X2=X3 X1+X4=X5 X3*X5=X7 X2*X4=X6 X1+X6=X7) %. +%total {} (union-left-factors-over-intersection* _ _ _ _ _) %. +%%%% Redefinitions +%inline multiset map %. +%inline multiset/0 map/0 %. +%inline multiset/+ map/+ %. +%inline member [S] [N] lookup S N C %. +%%% Theorem renamings. +%% using "member" +%inline false-implies-member false-implies-lookup %. +%inline not-member-member-implies-ne not-member-lookup-not-equal %. +%inline in-implies-member in-implies-lookup %. +%inline member-respects-eq [L member M N] [EM eq M MP] [EN nat`eq N NP] [LP member MP NP] lookup-respects-eq L EM EN nat`eq/ LP %. +%inline multiset`map map %. +%inline multiset`map/0 map/0 %. +%inline multiset`map/+ map/+ %. +%inline multiset`eq eq %. +%inline multiset`eq/ eq/ %. +%inline multiset`ne ne %. +%inline multiset`ne/L ne/L %. +%inline multiset`ne/R ne/R %. +%inline multiset`ne/N ne/N %. +%inline multiset`ne/D ne/D %. +%inline multiset`ne/+ ne/+ %. +%inline multiset`eq? eq? %. +%inline multiset`eq?/yes eq?/yes %. +%inline multiset`eq?/no eq?/no %. +%inline multiset`lookup lookup %. +%inline multiset`lookup/= lookup/= %. +%inline multiset`lookup/> lookup/> %. +%inline multiset`not-member not-member %. +%inline multiset`not-member/0 not-member/0 %. +%inline multiset`not-member/< not-member/< %. +%inline multiset`not-member/> not-member/> %. +%inline multiset`member? member? %. +%inline multiset`member?/in member?/in %. +%inline multiset`member?/out member?/out %. +%inline multiset`disjoint disjoint %. +%inline multiset`disjoint/L disjoint/L %. +%inline multiset`disjoint/R disjoint/R %. +%inline multiset`disjoint/< disjoint/< %. +%inline multiset`disjoint/> disjoint/> %. +%inline multiset`disjoint? disjoint? %. +%inline multiset`disjoint?/yes disjoint?/yes %. +%inline multiset`disjoint?/no disjoint?/no %. +%inline multiset`size size %. +%inline multiset`size/0 size/0 %. +%inline multiset`size/+ size/+ %. +%inline multiset`bound bound %. +%inline multiset`bound/0 bound/0 %. +%inline multiset`bound/+ bound/+ %. +%inline multiset`shift shift %. +%inline multiset`shift/0 shift/0 %. +%inline multiset`shift/+ shift/+ %. +%inline multiset`update update %. +%inline multiset`update/0 update/0 %. +%inline multiset`update/= update/= %. +%inline multiset`update/< update/< %. +%inline multiset`update/> update/> %. +%inline multiset`meta-eq meta-eq %. +%inline multiset`false-implies-eq false-implies-eq %. +%inline multiset`eq-reflexive eq-reflexive %. +%inline multiset`eq-symmetric eq-symmetric %. +%inline multiset`eq-transitive eq-transitive %. +%inline multiset`map/+-preserves-eq map/+-preserves-eq %. +%inline multiset`map/+-preserves-eq-converse map/+-preserves-eq-converse %. +%inline multiset`eq-no-occur eq-no-occur %. +%inline multiset`eq-contradiction eq-contradiction %. +%inline multiset`false-implies-ne false-implies-ne %. +%inline multiset`ne-respects-eq ne-respects-eq %. +%inline multiset`ne-anti-reflexive ne-anti-reflexive %. +%inline multiset`ne-symmetric ne-symmetric %. +%inline multiset`eq-ne-implies-false eq-ne-implies-false %. +%inline multiset`eq?-total* eq?-total* %. +%inline multiset`eq?-total eq?-total %. +%inline multiset`eq?-total/+ eq?-total/+ %. +%inline multiset`false-implies-lookup false-implies-lookup %. +%inline multiset`lookup-respects-eq lookup-respects-eq %. +%inline multiset`lookup-deterministic lookup-deterministic %. +%inline multiset`lookup-contradiction lookup-contradiction %. +%inline multiset`lookup-one-choice lookup-one-choice %. +%inline multiset`lookup-ne-implies-ne lookup-ne-implies-ne %. +%inline multiset`lookup-ne-implies-ne/L lookup-ne-implies-ne/L %. +%inline multiset`false-implies-not-member false-implies-not-member %. +%inline multiset`not-member-respects-eq not-member-respects-eq %. +%inline multiset`not-member-total* not-member-total* %. +%inline multiset`not-member-total not-member-total %. +%inline multiset`not-member-lookup-not-equal not-member-lookup-not-equal %. +%inline multiset`not-member-contradiction not-member-contradiction %. +%inline multiset`ne-implies-unit-map-not-member ne-implies-unit-map-not-member %. +%inline multiset`plus-right-preserves-not-member* plus-right-preserves-not-member* %. +%inline multiset`not-member-lookup-implies-ne not-member-lookup-implies-ne %. +%inline multiset`not-member-lookup-implies-ne/L not-member-lookup-implies-ne/L %. +%inline multiset`false-implies-member? false-implies-member? %. +%inline multiset`member?-respects-eq member?-respects-eq %. +%inline multiset`member?-deterministic member?-deterministic %. +%inline multiset`member?-total* member?-total* %. +%inline multiset`member?-map/+-total member?-map/+-total %. +%inline multiset`member?-map/+-complete member?-map/+-complete %. +%inline multiset`member?-total member?-total %. +%inline multiset`in-implies-lookup in-implies-lookup %. +%inline multiset`out-implies-not-member out-implies-not-member %. +%inline multiset`false-implies-disjoint false-implies-disjoint %. +%inline multiset`disjoint-respects-eq disjoint-respects-eq %. +%inline multiset`disjoint/=-contradiction disjoint/=-contradiction %. +%inline multiset`disjoint/<-inversion disjoint/<-inversion %. +%inline multiset`disjoint/>-inversion disjoint/>-inversion %. +%inline multiset`disjoint-anti-reflexive disjoint-anti-reflexive %. +%inline multiset`disjoint-symmetric disjoint-symmetric %. +%inline multiset`disjoint-lookup-contradiction disjoint-lookup-contradiction %. +%inline multiset`shift-left-preserves-disjoint shift-left-preserves-disjoint %. +%inline multiset`shift-left-preserves-disjoint-converse shift-left-preserves-disjoint-converse %. +%inline multiset`shift-right-preserves-disjoint shift-right-preserves-disjoint %. +%inline multiset`shift-right-preserves-disjoint-converse shift-right-preserves-disjoint-converse %. +%inline multiset`shift-preserves-disjoint shift-preserves-disjoint %. +%inline multiset`shift-preserves-disjoint-converse shift-preserves-disjoint-converse %. +%inline multiset`ne-implies-disjoint ne-implies-disjoint %. +%inline multiset`false-implies-size false-implies-size %. +%inline multiset`size-total* size-total* %. +%inline multiset`size-total size-total %. +%inline multiset`size-deterministic size-deterministic %. +%inline multiset`false-implies-bound false-implies-bound %. +%inline multiset`bound-total* bound-total* %. +%inline multiset`bound-total bound-total %. +%inline multiset`ge-bound-implies-not-member ge-bound-implies-not-member %. +%inline multiset`false-implies-shift false-implies-shift %. +%inline multiset`shift-respects-eq shift-respects-eq %. +%inline multiset`shift-total* shift-total* %. +%inline multiset`shift-total shift-total %. +%inline multiset`shift-deterministic shift-deterministic %. +%inline multiset`shifts-add shifts-add %. +%inline multiset`shifts-add-converse shifts-add-converse %. +%inline multiset`shift-preserves-not-member-converse* shift-preserves-not-member-converse* %. +%inline multiset`shift-preserves-lookup shift-preserves-lookup %. +%inline multiset`shift-preserves-lookup* shift-preserves-lookup* %. +%inline multiset`shift-preserves-lookup-converse shift-preserves-lookup-converse %. +%inline multiset`shift-preserves-lookup-converse* shift-preserves-lookup-converse* %. +%inline multiset`shift-preserves-size shift-preserves-size %. +%inline multiset`disjoint?-total* disjoint?-total* %. +%inline multiset`disjoint?-total*/+ disjoint?-total*/+ %. +%inline multiset`disjoint?-total*/< disjoint?-total*/< %. +%inline multiset`disjoint?-total*/> disjoint?-total*/> %. +%inline multiset`disjoint?-total disjoint?-total %. +%inline multiset`false-implies-update false-implies-update %. +%inline multiset`update-respects-eq update-respects-eq %. +%inline multiset`update-eq update-eq %. +%inline multiset`update-eq/ update-eq/ %. +%inline multiset`false-implies-update-eq false-implies-update-eq %. +%inline multiset`meta-update-eq meta-update-eq %. +%inline multiset`update/=-inversion update/=-inversion %. +%inline multiset`update/<-inversion update/<-inversion %. +%inline multiset`update/>-inversion update/>-inversion %. +%inline multiset`update-deterministic update-deterministic %. +%inline multiset`update-total* update-total* %. +%inline multiset`update-map/+-total update-map/+-total %. +%inline multiset`update-total update-total %. +%inline multiset`lookup-implies-update lookup-implies-update %. +%inline multiset`update-implies-lookup update-implies-lookup %. +%inline multiset`update-preserves-lookup update-preserves-lookup %. +%inline multiset`update-preserves-lookup-converse update-preserves-lookup-converse %. +%inline multiset`update-preserves-not-member update-preserves-not-member %. +%inline multiset`update-preserves-not-member-converse update-preserves-not-member-converse %. +%inline multiset`update-preserves-not-member-converse-helper update-preserves-not-member-converse-helper %. +%inline multiset`update-is-cause-of-change update-is-cause-of-change %. +%inline multiset`update-is-cause-of-change/L update-is-cause-of-change/L %. +%inline multiset`update-preserves-membership update-preserves-membership %. +%inline multiset`update-preserves-membership-converse update-preserves-membership-converse %. +%inline multiset`lookup-update-preserves-membership lookup-update-preserves-membership %. +%inline multiset`lookup-update-preserves-membership/L lookup-update-preserves-membership/L %. +%inline multiset`lookup-update-preserves-membership-converse lookup-update-preserves-membership-converse %. +%inline multiset`update-preserves-in-member update-preserves-in-member %. +%inline multiset`update-preserves-in-member/L update-preserves-in-member/L %. +%inline multiset`shift-preserves-update shift-preserves-update %. +%inline multiset`shift-preserves-update* shift-preserves-update* %. +%inline multiset`shift-preserves-update-converse shift-preserves-update-converse %. +%inline multiset`update-overwrites update-overwrites %. +%inline multiset`update-overwrites-converse update-overwrites-converse %. +%inline multiset`update-may-have-no-effect update-may-have-no-effect %. +%inline multiset`update-idempotent update-idempotent %. +%inline multiset`update-commutes update-commutes %. +%inline multiset`update-commutes* update-commutes* %. +%inline multiset`can-remove can-remove %. +%inline multiset`leq leq %. +%inline multiset`leq/0 leq/0 %. +%inline multiset`leq/= leq/= %. +%inline multiset`leq/> leq/> %. +%inline multiset`false-implies-leq false-implies-leq %. +%inline multiset`leq-respects-eq leq-respects-eq %. +%inline multiset`leq/0-inversion leq/0-inversion %. +%inline multiset`leq/=-inversion leq/=-inversion %. +%inline multiset`leq/>-inversion leq/>-inversion %. +%inline multiset`leq-contradiction leq-contradiction %. +%inline multiset`leq-reflexive leq-reflexive %. +%inline multiset`leq-anti-symmetric leq-anti-symmetric %. +%inline multiset`leq-transitive leq-transitive %. +%inline multiset`map/+-preserves-leq map/+-preserves-leq %. +%inline multiset`lookup-respects-leq lookup-respects-leq %. +%inline multiset`not-member-respects-geq not-member-respects-geq %. +%inline multiset`leq-implies-size-le leq-implies-size-le %. +%inline multiset`leq-implies-bound-le leq-implies-bound-le %. +%inline multiset`shift-left-preserves-leq* shift-left-preserves-leq* %. +%inline multiset`shift-preserves-leq* shift-preserves-leq* %. +%inline multiset`update-left-preserves-leq* update-left-preserves-leq* %. +%inline multiset`update-right-preserves-leq* update-right-preserves-leq* %. +%inline multiset`update-preserves-leq* update-preserves-leq* %. +%inline multiset`not-member-update-implies-leq not-member-update-implies-leq %. +%inline multiset`lookup-update-preserves-leq lookup-update-preserves-leq %. +%inline multiset`union union %. +%inline multiset`union/L union/L %. +%inline multiset`union/R union/R %. +%inline multiset`union/= union/= %. +%inline multiset`union/< union/< %. +%inline multiset`union/> union/> %. +%inline multiset`false-implies-union false-implies-union %. +%inline multiset`union-respects-eq union-respects-eq %. +%inline multiset`union/=-inversion union/=-inversion %. +%inline multiset`union/<-inversion union/<-inversion %. +%inline multiset`union/>-inversion union/>-inversion %. +%inline multiset`union-deterministic union-deterministic %. +%inline multiset`union-total* union-total* %. +%inline multiset`union-map/+-M-total* union-map/+-M-total* %. +%inline multiset`union-M-map/+-total* union-M-map/+-total* %. +%inline multiset`union-map/+-map/+-total* union-map/+-map/+-total* %. +%inline multiset`union-total union-total %. +%inline multiset`disjoint-union-total disjoint-union-total %. +%inline multiset`union-empty-implies-empty union-empty-implies-empty %. +%inline multiset`union-preserves-disjoint* union-preserves-disjoint* %. +%inline multiset`union-preserves-disjoint*/L union-preserves-disjoint*/L %. +%inline multiset`shift-left-preserves-union shift-left-preserves-union %. +%inline multiset`shift-left-preserves-union-converse shift-left-preserves-union-converse %. +%inline multiset`shift-right-preserves-union shift-right-preserves-union %. +%inline multiset`shift-right-preserves-union-converse shift-right-preserves-union-converse %. +%inline multiset`shift-preserves-union shift-preserves-union %. +%inline multiset`shift-preserves-union-converse shift-preserves-union-converse %. +%inline multiset`union-commutative union-commutative %. +%inline multiset`union-associative union-associative %. +%inline multiset`union-associative-union/<-union/< union-associative-union/<-union/< %. +%inline multiset`union-associative* union-associative* %. +%inline multiset`union-associative-converse union-associative-converse %. +%inline multiset`union-associative-converse* union-associative-converse* %. +%inline multiset`union-assoc-commutative* union-assoc-commutative* %. +%inline multiset`union-assoc-commutative union-assoc-commutative %. +%inline multiset`union-double-associative* union-double-associative* %. +%inline multiset`union-double-associative union-double-associative %. +%inline multiset`lookup-implies-union lookup-implies-union %. +%inline multiset`union-joins-lookup union-joins-lookup %. +%inline multiset`union-preserves-not-member* union-preserves-not-member* %. +%inline multiset`not-member-union-left-preserves-lookup* not-member-union-left-preserves-lookup* %. +%inline multiset`not-member-union-left-preserves-lookup-converse not-member-union-left-preserves-lookup-converse %. +%inline multiset`not-member-union-left-preserves-lookup-converse/L not-member-union-left-preserves-lookup-converse/L %. +%inline multiset`union-left-affects-lookup union-left-affects-lookup %. +%inline multiset`union-left-affects-lookup/L union-left-affects-lookup/L %. +%inline multiset`not-member-union-right-preserves-lookup* not-member-union-right-preserves-lookup* %. +%inline multiset`not-member-union-right-preserves-lookup-converse not-member-union-right-preserves-lookup-converse %. +%inline multiset`not-member-union-right-preserves-lookup-converse/L not-member-union-right-preserves-lookup-converse/L %. +%inline multiset`union-right-affects-lookup union-right-affects-lookup %. +%inline multiset`union-right-affects-lookup/L union-right-affects-lookup/L %. +%inline multiset`union-preserves-not-member-converse* union-preserves-not-member-converse* %. +%inline multiset`union-preserves-not-member-converse/L union-preserves-not-member-converse/L %. +%inline multiset`disjoint-union-left-preserves-lookup* disjoint-union-left-preserves-lookup* %. +%inline multiset`disjoint-union-left-preserves-lookup*/L disjoint-union-left-preserves-lookup*/L %. +%inline multiset`disjoint-union-right-preserves-lookup* disjoint-union-right-preserves-lookup* %. +%inline multiset`disjoint-union-right-preserves-lookup*/L disjoint-union-right-preserves-lookup*/L %. +%inline multiset`union-implies-leq* union-implies-leq* %. +%inline multiset`union-implies-leq union-implies-leq %. +%inline multiset`union-is-lub union-is-lub %. +%inline multiset`union-idempotent union-idempotent %. +%inline multiset`leq-implies-union leq-implies-union %. +%inline multiset`disjoint-leq-implies-union-leq* disjoint-leq-implies-union-leq* %. +%inline multiset`union-left-preserves-leq* union-left-preserves-leq* %. +%inline multiset`union-right-preserves-leq* union-right-preserves-leq* %. +%inline multiset`intersection intersection %. +%inline multiset`intersection/L intersection/L %. +%inline multiset`intersection/R intersection/R %. +%inline multiset`intersection/= intersection/= %. +%inline multiset`intersection/< intersection/< %. +%inline multiset`intersection/> intersection/> %. +%inline multiset`false-implies-intersection false-implies-intersection %. +%inline multiset`intersection-respects-eq intersection-respects-eq %. +%inline multiset`intersection/L-inversion intersection/L-inversion %. +%inline multiset`intersection/R-inversion intersection/R-inversion %. +%inline multiset`intersection/=-inversion intersection/=-inversion %. +%inline multiset`intersection/<-inversion intersection/<-inversion %. +%inline multiset`intersection/>-inversion intersection/>-inversion %. +%inline multiset`intersection-implies-ge intersection-implies-ge %. +%inline multiset`intersection-deterministic intersection-deterministic %. +%inline multiset`intersection-total* intersection-total* %. +%inline multiset`intersection-map/+-M-total* intersection-map/+-M-total* %. +%inline multiset`intersection-M-map/+-total* intersection-M-map/+-total* %. +%inline multiset`intersection-map/+-map/+-total* intersection-map/+-map/+-total* %. +%inline multiset`intersection-total intersection-total %. +%inline multiset`intersection-commutative intersection-commutative %. +%inline multiset`shift-left-preserves-intersection shift-left-preserves-intersection %. +%inline multiset`shift-left-preserves-intersection-converse shift-left-preserves-intersection-converse %. +%inline multiset`shift-right-preserves-intersection shift-right-preserves-intersection %. +%inline multiset`shift-right-preserves-intersection-converse shift-right-preserves-intersection-converse %. +%inline multiset`shift-preserves-intersection shift-preserves-intersection %. +%inline multiset`shift-preserves-intersection-converse shift-preserves-intersection-converse %. +%inline multiset`intersection-associativeM intersection-associativeM %. +%inline multiset`intersection-associativeM* intersection-associativeM* %. +%inline multiset`intersection-associative intersection-associative %. +%inline multiset`intersection-associative* intersection-associative* %. +%inline multiset`intersection-associative-converse intersection-associative-converse %. +%inline multiset`intersection-associative-converse* intersection-associative-converse* %. +%inline multiset`intersection-assoc-commutative* intersection-assoc-commutative* %. +%inline multiset`intersection-assoc-commutative intersection-assoc-commutative %. +%inline multiset`intersection-double-associative* intersection-double-associative* %. +%inline multiset`intersection-double-associative intersection-double-associative %. +%inline multiset`intersection-implies-leq* intersection-implies-leq* %. +%inline multiset`intersection-implies-leq intersection-implies-leq %. +%inline multiset`intersection-is-glb intersection-is-glb %. +%inline multiset`intersection-idempotent intersection-idempotent %. +%inline multiset`leq-implies-intersection leq-implies-intersection %. +%inline multiset`intersection-left-preserves-leq* intersection-left-preserves-leq* %. +%inline multiset`intersection-right-preserves-leq* intersection-right-preserves-leq* %. +%inline multiset`intersection-right-distributes-over-union intersection-right-distributes-over-union %. +%inline multiset`intersection-right-distributes-over-union* intersection-right-distributes-over-union* %. +%inline multiset`intersection-left-distributes-over-union* intersection-left-distributes-over-union* %. +%inline multiset`intersection-left-distributes-over-union intersection-left-distributes-over-union %. +%inline multiset`intersection-right-factors-over-union* intersection-right-factors-over-union* %. +%inline multiset`intersection-left-factors-over-union* intersection-left-factors-over-union* %. +%inline multiset`count count %. +%inline multiset`count/s count/s %. +%inline multiset`count/z count/z %. +%inline multiset`add add %. +%inline multiset`add/0 add/0 %. +%inline multiset`add/+ add/+ %. +%inline multiset`false-implies-count false-implies-count %. +%inline multiset`count-respects-eq count-respects-eq %. +%inline multiset`count-total** count-total** %. +%inline multiset`count-total* count-total* %. +%inline multiset`count-total count-total %. +%inline multiset`count-deterministic count-deterministic %. +%inline multiset`count-empty-is-zero count-empty-is-zero %. +%inline multiset`false-implies-add false-implies-add %. +%inline multiset`add-respects-eq add-respects-eq %. +%inline multiset`add-total* add-total* %. +%inline multiset`add-total add-total %. +%inline multiset`add-total/L add-total/L %. +%inline multiset`add-deterministic add-deterministic %. +%inline multiset`add-commutes add-commutes %. +%inline multiset`add-commutes/L add-commutes/L %. +%inline multiset`count-add-implies-count count-add-implies-count %. +%inline multiset`add-preserves-count add-preserves-count %. +%inline multiset`add-preserves-count-converse add-preserves-count-converse %. +%inline multiset`shift-preserves-count-converse shift-preserves-count-converse %. +%inline multiset`shift-preserves-add-converse shift-preserves-add-converse %. +%inline multiset`union-right-distributes-over-intersection union-right-distributes-over-intersection %. +%inline multiset`union-right-distributes-over-intersection* union-right-distributes-over-intersection* %. +%inline multiset`union-left-distributes-over-intersection* union-left-distributes-over-intersection* %. +%inline multiset`union-left-distributes-over-intersection union-left-distributes-over-intersection %. +%inline multiset`union-right-factors-over-intersection union-right-factors-over-intersection %. +%inline multiset`union-right-factors-over-intersection* union-right-factors-over-intersection* %. +%inline multiset`union-left-factors-over-intersection union-left-factors-over-intersection %. +%inline multiset`union-left-factors-over-intersection* union-left-factors-over-intersection* %. +%inline multiset`multiset multiset %. +%inline multiset`multiset/0 multiset/0 %. +%inline multiset`multiset/+ multiset/+ %. +%inline multiset`member member %. +%inline multiset`false-implies-member false-implies-member %. +%inline multiset`not-member-member-implies-ne not-member-member-implies-ne %. +%inline multiset`in-implies-member in-implies-member %. +%inline multiset`member-respects-eq member-respects-eq %. +%{! ## Definitions !}% +%{! ### The syntax !}% +%{! +Here we define multi-arity functions that are typed to avoid +type errors (hence, no higher-order functions in the syntax -- +but see "rec" which is the typed Y combinator). This syntax is +rather impoverished, but it would be fairly easily to add "succ", +"ifzero" terms. By "easy", I mean the addition would cause no new +technical issues. It would still change al he arithmetic parts of +the proofs dramatically. For this reason, I would like to +implement a tool that would generate such proofs for any HOAS family +of terms. +!}% +%sort term %. +%inline t term z %. +%term lit %pi nat %-> t %. +%term app %pi (term (s N)) %-> t %-> (term N) %. +%term lam %pi (%pi t %-> (term N)) %-> (term (s N)) %. +%term rec %pi (%pi (term (s N)) %-> (term (s N))) %-> (term (s N)) %. +%block blocksimple {n} [v term n]%. +%{! ### Equality !}% +%sort eq %. +%term eq/ eq T T %. +%{! ### Variable levels !}% +%{! +As with the non-indexed case, +a variable level is the (nonzero) natural number for a variable. +But unlike before 'varlevel' is now an abbreviation for a more +complex relation that permits adjustment of the index level. +!}% +%sort varadjlevel %. +%inline varlevel (%pi (term N) %-> nat %-> %type) [T] [L] varadjlevel T N L %. +%block blockvar {n} {l} [v term n] [vl varlevel v (s l)]%. +%{! ### Mapping !}% +%{! +The bijection from terms to nat is called "tonat". There +is a separate mapping for each index, which makes sense +becaue each index represents a different type. +Unlike the non-indexed case, we need to handle an unbounded +number of different types of variables, so we can't just pass +one extra int, or even a fixed number of ints. Instead we need +to pass a structure that represents an unbounded number of ints. +We use `multiset' to represent this structure. The elements of the +multiset represent the levels that came "before". + +Another interesting aspect is that not all constructors for the +indexed type are available for all indices. For instane `lit' is +only for the 0-index and rec only for non-zero indices. On the other +hand, `app' can occur for any index. Since the mapping is separate +for every index, and the mapping must be one-to-one as well as onto, +we must work carefully that every natural number is mapped onto exactly +once. Variables are handled (as before) by reserving the first VN spots +for variables, where (unlike before) VN depends on the index. +Ignoring variables, `lit' takes the veen numbers and `app' takes the odd +numbers for the 0-indexed terms. For the nonzero-indexed terms, app +takes the odd numbers (of course) and lam and rec share the even numbers. +!}% +%sort tonat* {N} %. +%inline tonat tonat* _ multiset/0 %. +%term tonat/var %pi (count MS N VN) %-> (varlevel V L) %-> (plus M L VN) %-> (tonat* N MS V M) %. +%term tonat/lit + %pi (count MS z VN) + %-> (times (s (s z)) M TM) + %-> (plus VN TM M') + %-> (tonat* z MS (lit M) M') %. +%term tonat/app + %pi (count MS N VN) + %-> (tonat* _ MS T1 M1) + %-> (tonat* _ MS T2 M2) + %-> (natpair`pair2nat (natpair/ M1 M2) M) + %-> (times (s (s z)) M TM) + %-> (plus VN (s TM) M') + %-> (tonat* N MS (app T1 T2) M') %. +%term tonat/lam + %pi (count MS (s N) VN) + %-> (count MS z ZN) + %-> (multiset`add MS z MS') + %-> ({v term z} %pi (varlevel v (s ZN)) %-> (tonat* N MS' (F v) M)) + %-> (times (s (s (s (s z)))) M TM) + %-> (plus VN TM M') + %-> (tonat* (s N) MS (lam F) M') %. +%term tonat/rec + %pi (count MS (s N) VN) + %-> (multiset`add MS (s N) MS') + %-> ({f} %pi (varlevel f (s VN)) %-> (tonat* (s N) MS' (F f) M)) + %-> (times (s (s (s (s z)))) M TM) + %-> (plus VN (s (s TM)) M') + %-> (tonat* (s N) MS (rec F) M') %. +%{! ### Utility lemmas !}% +%{! +The following theorems prove obvious simple things about the +basic relations. They following the conventions established in +John Boyland's library signatures. +!}% +%theorem false-implies-eq : forall* {N1} {N2} {T1 term N1} {T2 term N2} forall {F void} exists {EQ eq T1 T2} true %. +%worlds (blocksimple) (false-implies-eq _ _) %. +%total {} (false-implies-eq _ _) %. +%theorem false-implies-varlevel : forall* {N} {V term N} {L} forall {F void} exists {VL varlevel V L} true %. +%worlds (blocksimple blockvar) (false-implies-varlevel _ _) %. +%total {} (false-implies-varlevel _ _) %. +%theorem varlevel-respects-eq : forall* {N} {V term N} {L1} {L2} forall {VL1 varlevel V L1} {E nat`eq L1 L2} exists {VL2 varlevel V L2} true %. +%term _ varlevel-respects-eq VL nat`eq/ VL %. +%worlds (blocksimple blockvar) (varlevel-respects-eq _ _ _) %. +%total {} (varlevel-respects-eq _ _ _) %. +%theorem false-implies-tonat : forall* {N} {MS} {T} {M} forall {F void} exists {TN tonat* N MS T M} true %. +%worlds (blocksimple blockvar) (false-implies-tonat _ _) %. +%total {} (false-implies-tonat _ _) %. +%theorem tonat-respects-eq : forall* {N1} {MS1} {T1} {M1} {N2} {MS2} {T2} {M2} forall {TN1 tonat* N1 MS1 T1 M1} {EM multiset`eq MS1 MS2} {ET eq T1 T2} {EN nat`eq M1 M2} exists {TN2 tonat* N2 MS2 T2 M2} true %. +%term _ tonat-respects-eq TN multiset`eq/ eq/ nat`eq/ TN %. +%worlds (blocksimple blockvar) (tonat-respects-eq _ _ _ _ _) %. +%total {} (tonat-respects-eq _ _ _ _ _) %. +%{! ## Proof of totality of tonat !}% +%{! +As with the non-indexed case, proving totality is +surprisingly tricky. The problem is that when we get to +a variable, we need to make sure that (1) the variable +has a level associated with it and (2) the level is +in the range 1..VN where VN is the number of variables +of this index that have been seen in the context. +Twelf's blocks are useful for (1) but not for (2) because there's +no way to connect the context with the current nesting level. + +For this proof, I generalized/extended the "case" technique +to handle indexed variables. The proof is almost the same. +The multisets add extra paremeters but the basic structure is the same. +!}% +%{! ### Auxiliary definitions !}% +%{! #### raw variables !}% +%{! +A variable is raw if we haven't verified that it has a level in range. +Non variables are not raw. +!}% +%sort israw* {N} %. +%inline israw israw* _ %. +%inline rawvar [T] israw T true %. +%term israw/lit israw (lit _) false %. +%term israw/app israw (app _ _) false %. +%term israw/lam israw (lam _) false %. +%term israw/rec israw (rec _) false %. +%{! #### case analysis !}% +%{! +We case analysis terms with two cases for variables. +The raw case is used only internally and can be ignored in +"clients" that don't use israw. +!}% +%sort case* {N} %. +%inline case case* _ %. +%term case/lit case _ (lit _) %. +%term case/app %pi (case MS T1) %-> (case MS T2) %-> (case MS (app T1 T2)) %. +%term case/lam + %pi (count MS z VN) + %-> (add MS z MS') + %-> ({v} %pi (varlevel v (s VN)) %-> (case MS' (F v))) + %-> (case MS (lam F)) %. +%term case/rec + %pi (count MS (s N) VN) + %-> (multiset`add MS (s N) MS') + %-> ({f} %pi (varlevel f (s VN)) %-> (case MS' (F f))) + %-> (case* (s N) MS (rec F)) %. +%term case/var %pi (count MS N VN) %-> (varlevel V FL) %-> (nat`ge VN FL) %-> (case* N MS V) %. +%term case/raw %pi (rawvar V) %-> (case _ V) %. +%{! ### Theorems about auxiliary definitions !}% +%theorem israw-total* : forall* {N} forall {T term N} exists {B} {I israw T B} true %. +%inline israw-total israw-total* _ _ %. +%term _ israw-total israw/lit %. +%term _ israw-total israw/app %. +%term _ israw-total israw/lam %. +%term _ israw-total israw/rec %. +%sort fake %. +%term _ %pi fake %<- ({i israw-total* T B I} israw-total* T' B' I') %. +%term _ %pi fake %<- ({i israw-total* T B I} israw T' B') %. +%block blockraw {n} [v term n] [rv rawvar v] [irt israw-total rv]%. +%worlds (blockraw) (israw-total* _ _ _) %. +%total {} (israw-total* _ _ _) %. +%{! +The following theorem handles one variable converting it from raw to +handle a level that is in the required range. This is an important +technique for handle variables in Twelf: one at a time. Unlike the +one2one theorem (see later), we don't have to do extraordinary things +to handle having multiple types. (Although we do need some extra lemmas +to ahndle facts about multisets.) +!}% +%theorem var-gets-level : forall* {M} {N} {VN} {T %pi (term M) %-> (term N)} {L} {MS} forall {F {v term M} {rv rawvar v} {i israw-total rv} case MS (T v)} {MC count MS M VN} {GE nat`ge VN L} exists {F' {v term M} %pi (varlevel v L) %-> (case MS (T v))} true %. +%term _ var-gets-level ([f] [r] [i] case/raw r) MC GE ([f] [fl] case/var MC fl GE) %. +%term _ var-gets-level ([f] [r] [i] case/raw R) _ _ ([f] [fl] case/raw R) %. +%term _ var-gets-level ([f] [r] [i] case/var MC VL GE) _ _ ([f] [fl] case/var MC VL GE) %. +%term _ var-gets-level ([f] [r] [i] %the (case MS (lit O)) case/lit) _ _ ([f] [fl] case/lit) %. +%term _ + %pi (var-gets-level ([f] [r] [i] case/app (C1 f r i) (C2 f r i)) MC GE ([f] [fl] case/app (C1' f fl) (C2' f fl))) + %<- (var-gets-level C1 MC GE C1') + %<- (var-gets-level C2 MC GE C2') %. +%theorem var-gets-level/L : forall* {MS} {N1} {FN1} {N2} {FN2} {MS'} {L1} {B} forall {MC1 multiset`count MS N1 FN1} {MC2 multiset`count MS N2 FN2} {MA2 multiset`add MS N2 MS'} {GE nat`ge FN1 L1} {EQ? nat`eq? N1 N2 B} exists {FN1'} {MC1' multiset`count MS' N1 FN1'} {GE' nat`ge FN1' L1} true %. +%term _ + %pi (var-gets-level/L MC _ MA FN>=L nat`eq?/yes _ MC' (ge/> FN+1>L)) + %<- (ge-implies-succ-gt FN>=L FN+1>L) + %<- (multiset`count-add-implies-count MC MA MC') %. +%term _ + %pi (var-gets-level/L MC1 _ MA2 GE (nat`eq?/no N1<>N2) _ MC1' GE) + %<- (multiset`add-preserves-count MC1 MA2 N1<>N2 MC1') %. +%worlds () (var-gets-level/L _ _ _ _ _ _ _ _) %. +%total {} (var-gets-level/L _ _ _ _ _ _ _ _) %. +%term _ + %pi (var-gets-level ([f] [r] [i] case/lam MS^0=FN' MSU ([v] [vl] C v vl f r i)) MC GE ([f] [fl] case/lam MS^0=FN' MSU ([v] [vl] C' v vl f fl))) + %<- (nat`eq?-total EQ?) + %<- (var-gets-level/L MC MS^0=FN' MSU GE EQ? _ MC' GE') + %<- ({v} {vl varlevel v _} var-gets-level (C v vl) MC' GE' (C' v vl)) %. +%term _ + %pi (var-gets-level ([f] [r] [i] case/rec MS^N'=FN' MSU ([f'] [fl'] C f' fl' f r i)) MS^N=FN FN>=L ([f] [fl] case/rec MS^N'=FN' MSU ([f'] [fl'] C' f' fl' f fl))) + %<- (nat`eq?-total EQ?) + %<- (var-gets-level/L MS^N=FN MS^N'=FN' MSU FN>=L EQ? _ MS'^N=FN'' FN''>=L) + %<- ({f'} {fl'} var-gets-level (C f' fl') MS'^N=FN'' FN''>=L (C' f' fl')) %. +%worlds (blockvar blockraw) (var-gets-level _ _ _ _) %. +%total F (var-gets-level F _ _ _) %. +%{! +We are now ready to prove that we can always "case" a term. +This works almost the same as for non-indexed terms. +!}% +%theorem case-total* : forall* {N} forall {T term N} exists {C case multiset/0 T} true %. +%inline case-total case-total* _ %. +%theorem case-total/L : forall* {B} {N} {MS} forall {T term N} {I israw T B} exists {C case MS T} true %. +%term _ case-total/L _ _ case/lit %. +%term _ + %pi (case-total/L _ _ (case/app C1 C2)) + %<- (israw-total I1) + %<- (case-total/L _ I1 C1) + %<- (israw-total I2) + %<- (case-total/L _ I2 C2) %. +%term _ + %pi (case-total/L _ _ (case/lam MC MA ([f] [fl] C' f fl))) + %<- (multiset`count-total MC) + %<- (multiset`add-total MA) + %<- ({v} {r rawvar v} {i israw-total r} israw-total (%the (israw _ B) (I v r i))) + %<- ({v} {r rawvar v} {i israw-total r} case-total/L _ (I v r i) (C v r i)) + %<- (multiset`count-add-implies-count MC MA MC') + %<- (var-gets-level C MC' (nat`ge/= nat`eq/) C') %. +%term _ + %pi (case-total/L _ _ (case/rec MC MA ([f] [fl] C' f fl))) + %<- (multiset`count-total MC) + %<- (multiset`add-total MA) + %<- ({v} {r rawvar v} {i israw-total r} israw-total (%the (israw _ B) (I v r i))) + %<- ({v} {r rawvar v} {i israw-total r} case-total/L _ (I v r i) (C v r i)) + %<- (multiset`count-add-implies-count MC MA MC') + %<- (var-gets-level C MC' (nat`ge/= nat`eq/) C') %. +%term _ case-total/L V R (case/raw R) %. +%worlds (blockraw) (case-total/L _ _ _) %. +%total T (case-total/L T _ _) %. +%term _ %pi (case-total* T C) %<- (israw-total I) %<- (case-total/L T I C) %. +%worlds () (case-total* _ _) %. +%total {} (case-total* _ _) %. +%{! ### Main theorem !}% +%{! +We are ready now to prove totality of the relation. +We case the term first and then have everything we need +to push through totality. +!}% +%theorem tonat-total* : forall* {N} forall {T term N} exists {M nat} {D tonat T M} true %. +%inline tonat-total tonat-total* _ _ %. +%theorem tonat-total/L : forall* {N nat} {T term N} {MS} forall {C case MS T} exists {M nat} {D tonat* N MS T M} true %. +%term _ + %pi (tonat-total/L (case/var MC VL GE) _ (tonat/var MC VL P)) + %<- (ge-implies-plus GE _ P) %. +%term _ + %pi (tonat-total/L case/lit _ (tonat/lit MC T P)) + %<- (count-total MC) + %<- (times-total T) + %<- (plus-total P) %. +%term _ + %pi (tonat-total/L (case/app C1 C2) _ (tonat/app MC TN1 TN2 P2N T P)) + %<- (count-total MC) + %<- (tonat-total/L C1 _ TN1) + %<- (tonat-total/L C2 _ TN2) + %<- (pair2nat-total P2N) + %<- (times-total T) + %<- (plus-total P) %. +%term _ + %pi (tonat-total/L (case/lam MCz MSU ([v] [vl] C v vl)) _ (tonat/lam MC MCz MSU ([v] [vl] TN v vl) T P)) + %<- ({v} {vl varlevel v (s N)} tonat-total/L (C v vl) _ (TN v vl)) + %<- (count-total MC) + %<- (times-total T) + %<- (plus-total P) %. +%term _ + %pi (tonat-total/L (case/rec MC MA ([v] [vl] C v vl)) _ (tonat/rec MC MA ([v] [vl] TN v vl) T P)) + %<- ({v} {vl} tonat-total/L (C v vl) _ (TN v vl)) + %<- (times-total T) + %<- (plus-total P) %. +%worlds (blockvar) (tonat-total/L _ _ _) %. +%total (C) (tonat-total/L C _ _) %. +%term _ %pi (tonat-total TN) %<- (case-total C) %<- (tonat-total/L C _ TN) %. +%worlds () (tonat-total* _ _ _) %. +%total {} (tonat-total* _ _ _) %. +%{! ## Proof of the determinicity of the mapping !}% +%{! +As with the non-indexed case, this aspect is easy to prove. +!}% +%{! ### Auxiliary theorems !}% +%{! +We prove that variable levels are "unique" and that they are never zero. +The proofs are trivial: Twelf can accept them from the context alone. +!}% +%theorem varlevel-deterministic : forall* {N} {V term N} {L1} {L2} forall {VL1 varlevel V L1} {VL2 varlevel V L2} exists {E nat`eq L1 L2} true %. +%term _ varlevel-deterministic _ _ nat`eq/ %. +%worlds (blockvar) (varlevel-deterministic _ _ _) %. +%total {} (varlevel-deterministic _ _ _) %. +%theorem varlevel-contradiction : forall* {N} {V term N} {L} forall {VL varlevel V L} {E nat`eq L z} exists {F void} true %. +%worlds (blockvar) (varlevel-contradiction _ _ _) %. +%total {} (varlevel-contradiction _ _ _) %. +%{! ### Main Theorem !}% +%theorem tonat-deterministic : forall* {N1} {N2} {T1 term N1} {T2 term N2} {M1} {M2} forall {TN1 tonat T1 M1} {TN2 tonat T2 M2} {E eq T1 T2} exists {E nat`eq M1 M2} true %. +%theorem tonat-deterministic/L : forall* {N} {T term N} {MS1} {MS2} {M1} {M2} forall {TN1 tonat* N MS1 T M1} {TN2 tonat* N MS2 T M2} {E multiset`eq MS1 MS2} exists {E nat`eq M1 M2} true %. +%term _ + %pi (tonat-deterministic/L (tonat/var MC1 VL1 P1) (tonat/var MC2 VL2 P2) _ N1=N2) + %<- (count-deterministic MC1 MC2 multiset`eq/ nat`eq/ C1=C2) + %<- (varlevel-deterministic VL1 VL2 L1=L2) + %<- (plus-right-cancels P1 P2 L1=L2 C1=C2 N1=N2) %. +%term _ + %pi (tonat-deterministic/L (tonat/lit MC1 T1 P1) (tonat/lit MC2 T2 P2) _ M1'=M2') + %<- (count-deterministic MC1 MC2 multiset`eq/ nat`eq/ C1=C2) + %<- (times-deterministic T1 T2 nat`eq/ nat`eq/ TM1=TM2) + %<- (plus-deterministic P1 P2 C1=C2 TM1=TM2 M1'=M2') %. +%term _ + %pi (tonat-deterministic/L (tonat/app MC1 TN1a TN1b P2N1 T1 P1) (tonat/app MC2 TN2a TN2b P2N2 T2 P2) MSE M1'=M2') + %<- (count-deterministic MC1 MC2 multiset`eq/ nat`eq/ C1=C2) + %<- (tonat-deterministic/L TN1a TN2a MSE M1a=M2a) + %<- (tonat-deterministic/L TN1b TN2b MSE M1b=M2b) + %<- (natpair`pair-preserves-eq M1a=M2a M1b=M2b P1=P2) + %<- (pair2nat-deterministic P2N1 P2N2 P1=P2 M1=M2) + %<- (times-deterministic T1 T2 nat`eq/ M1=M2 TM1=TM2) + %<- (succ-deterministic TM1=TM2 TM1+1=TM2+1) + %<- (plus-deterministic P1 P2 C1=C2 TM1+1=TM2+1 M1'=M2') %. +%theorem tonat-deterministic/L2 : forall* {N1} {N2} {M1} {M2} {MS} {F %pi (term N1) %-> (term N2)} {M} forall {F1 {f term N1} {fl varlevel f (s M1)} tonat* N2 MS (F f) M} {ME nat`eq M1 M2} exists {F2 {f term N1} {fl varlevel f (s M2)} tonat* N2 MS (F f) M} true %. +%term _ tonat-deterministic/L2 F nat`eq/ F %. +%worlds (blockvar) (tonat-deterministic/L2 _ _ _) %. +%total {} (tonat-deterministic/L2 _ _ _) %. +%term _ + %pi (tonat-deterministic/L (tonat/lam MC1 MCz1 MA1 ([v] [vl] TN1 v vl) T1 P1) (tonat/lam MC2 MCz2 MA2 ([v] [vl] TN2 v vl) T2 P2) MS1=MS2 M1'=M2') + %<- (multiset`count-deterministic MC1 MC2 MS1=MS2 nat`eq/ VN1=VN2) + %<- (multiset`count-deterministic MCz1 MCz2 MS1=MS2 nat`eq/ ZN1=ZN2) + %<- (multiset`add-deterministic MA1 MA2 MS1=MS2 nat`eq/ MS1'=MS2') + %<- (tonat-deterministic/L2 TN1 ZN1=ZN2 TN1') + %<- ({v} {vl varlevel v (s ZN2)} tonat-deterministic/L (TN1' v vl) (TN2 v vl) MS1'=MS2' M1=M2) + %<- (times-deterministic T1 T2 nat`eq/ M1=M2 TM1=TM2) + %<- (plus-deterministic P1 P2 VN1=VN2 TM1=TM2 M1'=M2') %. +%term _ + %pi (tonat-deterministic/L (tonat/rec MC1 MA1 ([f] [fl] TN1 f fl) T1 P1) (tonat/rec MC2 MA2 ([f] [fl] TN2 f fl) T2 P2) MS1=MS2 M1'=M2') + %<- (multiset`count-deterministic MC1 MC2 MS1=MS2 nat`eq/ FN1=FN2) + %<- (multiset`add-deterministic MA1 MA2 MS1=MS2 nat`eq/ MS1'=MS2') + %<- (tonat-deterministic/L2 TN1 FN1=FN2 TN1') + %<- ({f} {fl varlevel f (s N)} tonat-deterministic/L (TN1' f fl) (TN2 f fl) MS1'=MS2' M1=M2) + %<- (times-deterministic T1 T2 nat`eq/ M1=M2 TM1=TM2) + %<- (succ-deterministic TM1=TM2 TM1'=TM2') + %<- (succ-deterministic TM1'=TM2' TM1''=TM2'') + %<- (plus-deterministic P1 P2 FN1=FN2 TM1''=TM2'' M1'=M2') %. +%worlds (blockvar) (tonat-deterministic/L _ _ _ _) %. +%total (T) (tonat-deterministic/L _ T _ _) %. +%term _ + %pi (tonat-deterministic TN1 TN2 eq/ N1=N2) + %<- (tonat-deterministic/L TN1 TN2 multiset`eq/ N1=N2) %. +%worlds () (tonat-deterministic _ _ _ _) %. +%total {} (tonat-deterministic _ _ _ _) %. +%{! ## Proving that the mapping is onto. !}% +%{! +Again, this aspect has the same structure as with non-indexed terms. +!}% +%{! ### Auxiliary definitions !}% +%{! +We define a relation that builds on the context relation. +In the non-indexed case, it took a nat, here it takes the +multiset representing the levels of variables in the context. +!}% +%sort upto %. +%term upto/0 upto multiset/0 %. +%term upto/+ + %pi (upto MS) + %-> (multiset`count MS N FN) + %-> (varlevel (%the (term N) V) (s FN)) + %-> (multiset`add MS N MS') + %-> (upto MS') %. +%{! ### Lemmas about regular definitions !}% +%theorem term-inhabited : forall {N} exists {T term N} true %. +%term _ term-inhabited _ (lit z) %. +%term _ %pi (term-inhabited _ (lam ([x] T))) %<- (term-inhabited _ T) %. +%worlds (blockvar) (term-inhabited _ _) %. +%total (N) (term-inhabited N _) %. +%{! ### Theorems about auxiliary definitions !}% +%{! +The obvious lemma that makes use of the main purpose of the relation: +that a variable is always available. +!}% +%theorem upto-implies-varlevel : forall* {MS} {N} {L} {FN} forall {U upto MS} {MC multiset`count MS N FN} {LT nat`gt FN L} exists {V term N} {VL varlevel V (s L)} true %. +%term _ + %pi (upto-implies-varlevel upto/0 MC FN>L T FL) + %<- (count-empty-is-zero MC FN=0) + %<- (nat`gt-respects-eq FN>L FN=0 nat`eq/ ZERO>L) + %<- (term-inhabited _ T) + %<- (nat`gt-contradiction ZERO>L F) + %<- (false-implies-varlevel F FL) %. +%theorem upto-implies-varlevel/L : forall* {MS1} {N1} {L} {FN1} {MS2} {N2} {FN2} {V1 term N1} {B} forall {U upto MS1} {MC1 multiset`count MS1 N1 FN1} {FL1 varlevel V1 (s FN1)} {MA multiset`add MS1 N1 MS2} {MC2 multiset`count MS2 N2 FN2} {LT nat`gt FN2 L} {EQ? nat`eq? N1 N2 B} exists {V term N2} {VL varlevel V (s L)} true %. +%term _ + %pi (upto-implies-varlevel/L _ MC1 FL1 MA MC2 gt/1 nat`eq?/yes _ FL2) + %<- (count-add-implies-count MC1 MA MC2') + %<- (count-deterministic MC2' MC2 multiset`eq/ nat`eq/ FN1+1=FN2+1) + %<- (varlevel-respects-eq FL1 FN1+1=FN2+1 FL2) %. +%term _ + %pi (upto-implies-varlevel/L U MC1 FL1 MA MC2 (gt/> FN2>L) nat`eq?/yes _ FL2) + %<- (count-add-implies-count MC1 MA MC2') + %<- (count-deterministic MC2 MC2' multiset`eq/ nat`eq/ FN2+1=FN1+1) + %<- (succ-cancels FN2+1=FN1+1 FN2=FN1) + %<- (gt-respects-eq FN2>L FN2=FN1 nat`eq/ FN1>L) + %<- (upto-implies-varlevel U MC1 FN1>L _ FL2) %. +%term _ + %pi (upto-implies-varlevel/L U _ _ MA MC2 GT (nat`eq?/no N1<>N2) _ FL) + %<- (nat`ne-symmetric N1<>N2 N2<>N1) + %<- (add-preserves-count-converse MC2 MA N2<>N1 MC1) + %<- (upto-implies-varlevel U MC1 GT _ FL) %. +%term _ + %pi (upto-implies-varlevel (upto/+ U MC1 FL1 MA) MC2 GT _ FL) + %<- (nat`eq?-total EQ?) + %<- (upto-implies-varlevel/L U MC1 FL1 MA MC2 GT EQ? _ FL) %. +%worlds (blockvar) (upto-implies-varlevel/L _ _ _ _ _ _ _ _ _) (upto-implies-varlevel _ _ _ _ _) %. +%total (U V) (upto-implies-varlevel U _ _ _ _) (upto-implies-varlevel/L V _ _ _ _ _ _ _ _) %. +%{! ### Main theorem !}% +%{! +We prove the main result using two lemmas that do the case +analysis on the number against the nesting level and the +parity. (In general, one would use a divisor counting all +cases that are recursive.) The proofs are long but simply +arithmetic manipulation. Proving termination uses meta-gt +for strong induction over the natural numbers. +!}% +%theorem tonat-onto* : forall* {N} forall {M nat} exists {T term N} {TN tonat T M} true %. +%inline tonat-onto tonat-onto* _ _ %. +%theorem tonat-onto/L : forall* {N} {MS multiset} forall {U upto MS} {M nat} exists {T term N} {TN tonat* _ MS T M} true %. +%theorem tonat-onto/L0 : forall* {N} {MS multiset} {VN} {C} forall {U upto MS} {M nat} {MC count MS N VN} {CMP nat`compare VN M C} exists {T} {TN tonat* N MS T M} true %. +%term _ + %pi (tonat-onto/L U M _ TN) + %<- (multiset`count-total MC) + %<- (nat`compare-total CMP) + %<- (tonat-onto/L0 U M MC CMP _ TN) %. +%term _ + %pi (tonat-onto/L0 U M MC (compare/> VN>M) _ (tonat/var MC VL M+L+1=VN)) + %<- (nat`gt-implies-plus VN>M L L+1+M=VN) + %<- (plus-commutative L+1+M=VN M+L+1=VN) + %<- (plus-implies-ge M+L+1=VN VN>=L+1) + %<- (ge-succ-implies-gt VN>=L+1 VN>L) + %<- (upto-implies-varlevel U MC VN>L _ VL) %. +%theorem tonat-onto/L1 : forall* {TM} {MS} {N} {VN} forall {UF upto MS} {M nat} {MC count MS N VN} {P plus VN TM M} {Q} {R} {DR divrem TM (s (s z)) Q R} exists {T} {TN tonat* N MS T M} true %. +%term _ + %pi (tonat-onto/L0 U M MC compare/= _ TN) + %<- (plus-right-identity _ P) + %<- (divrem-total M/2=Q,R) + %<- (tonat-onto/L1 U M MC P Q R M/2=Q,R _ TN) %. +%term _ + %pi (tonat-onto/L0 U M MC (compare/< M>VN) _ TN) + %<- (gt-implies-plus M>VN _ Pc) + %<- (plus-commutative Pc P) + %<- (divrem-total M/2=Q,R) + %<- (tonat-onto/L1 U M MC P Q R M/2=Q,R _ TN) %. +%term _ + %pi (tonat-onto/L1 _ _ _ _ _ (s (s _)) DR T TN) + %<- (divrem-contradiction DR (plus/s (plus/s plus/z)) F) + %<- (term-inhabited _ T) + %<- (false-implies-tonat F TN) %. +%term _ + %pi (tonat-onto/L1 U M' MC VN+TM'=M' M (s z) TM'/2=M,1 _ (tonat/app MC TN1 TN2 P2N TWO*M=TM VN+TM+1=M')) + %<- (plus-implies-ge VN+TM'=M' M'>=TM') + %<- (divrem-can-be-inverted TM'/2=M,1 TM M*2=TM TM+ONE=TM') + %<- (times-commutative M*2=TM TWO*M=TM) + %<- (plus-commutative (plus/s plus/z) TM+ONE=TM+1) + %<- (plus-deterministic TM+ONE=TM' TM+ONE=TM+1 nat`eq/ nat`eq/ TM'=TM+1) + %<- (plus-respects-eq VN+TM'=M' nat`eq/ TM'=TM+1 nat`eq/ VN+TM+1=M') + %<- (nat2pair-total P2N) + %<- (quotient-of-nonzero-is-smaller TM'/2=M,1 TM'=TM+1 TM'>M) + %<- (nat`ge-transitive-gt M'>=TM' TM'>M M'>M) + %<- (nat2pair-implies-ge P2N M>=M1 M>=M2) + %<- (nat`gt-transitive-ge M'>M M>=M1 M'>M1) + %<- (nat`gt-transitive-ge M'>M M>=M2 M'>M2) + %<- (meta-gt _ _ M'>M1) + %<- (meta-gt _ _ M'>M2) + %<- (tonat-onto/L U M1 _ TN1) + %<- (tonat-onto/L U M2 _ TN2) %. +%term _ + %pi (tonat-onto/L1 _ M' MC VN+TM=M' M z TM/2=M _ (tonat/lit MC TWO*M=TM VN+TM=M')) + %<- (div-can-be-inverted TM/2=M M*2=TM) + %<- (times-commutative M*2=TM TWO*M=TM) %. +%theorem tonat-onto/L2 : forall* {TM} {MS} {N} {VN} {Q2} forall {U upto MS} {M nat} {MC count MS (s N) VN} {P plus VN TM M} {DR divrem TM (s (s z)) Q2 z} {Q4} {R} {DR divrem Q2 (s (s z)) Q4 R} exists {T} {TN tonat* (s N) MS T M} true %. +%term _ + %pi (tonat-onto/L1 U M' MC VN+FM=M' TM z FM/2=TM _ TN) + %<- (divrem-total TM/2=Q,R) + %<- (tonat-onto/L2 U M' MC VN+FM=M' FM/2=TM Q R TM/2=Q,R _ TN) %. +%inline 2*2=4 times/s (times/s times/z plus/z) (plus/s (plus/s plus/z)) %. +%term _ + %pi (tonat-onto/L2 _ _ _ _ _ _ (s (s _)) DR T TN) + %<- (divrem-contradiction DR (plus/s (plus/s plus/z)) F) + %<- (term-inhabited _ T) + %<- (false-implies-tonat F TN) %. +%term _ + %pi (tonat-onto/L2 U M' MC VN+FM'=M' FM'/2=TM' M (s z) TM'/2=M,1 _ (tonat/rec MC MA ([f] [fl] TN f fl) FOUR*M=FM VN+FM+2=M')) + %<- (divrem-can-be-inverted TM'/2=M,1 TM M*2=TM TM+ONE=TM') + %<- (div-can-be-inverted FM'/2=TM' TM'*2=FM') + %<- (times-total* TM (s (s z)) FM TM*2=FM) + %<- (times-right-distributes-over-plus* TM+ONE=TM' TM'*2=FM' TM*2=FM (times/s times/z plus/z) FM+TWO=FM') + %<- (times-associative* M*2=TM TM*2=FM 2*2=4 M*4=FM) + %<- (times-commutative M*4=FM FOUR*M=FM) + %<- (plus-commutative (plus/s (plus/s plus/z)) FM+TWO=FM+2) + %<- (plus-deterministic FM+TWO=FM' FM+TWO=FM+2 nat`eq/ nat`eq/ FM'=FM+2) + %<- (plus-respects-eq VN+FM'=M' nat`eq/ FM'=FM+2 nat`eq/ VN+FM+2=M') + %<- (plus-swap-succ-converse VN+FM+2=M' VN+1+FM+1=M') + %<- (plus-swap-succ-converse VN+1+FM+1=M' VN+2+FM=M') + %<- (multiset`add-total MA) + %<- (plus-implies-gt VN+2+FM=M' nat`eq/ M'>FM) + %<- (times-nonzero-implies-ge M*4=FM FM>=M) + %<- (nat`gt-transitive-ge M'>FM FM>=M M'>M) + %<- (meta-gt _ _ M'>M) + %<- ({f} {fl} tonat-onto/L (upto/+ U MC fl MA) M _ (TN f fl)) %. +% 0 needs a special case for termination: +%theorem tonat-onto/0 : forall* {MS} forall {N} {U upto MS} {VN} {MC count MS N VN} exists {T} {TN tonat* N MS T z} true %. +%inline 2*0=0 times/s (times/s times/z plus/z) plus/z %. +%inline 4*0=0 times/s (times/s 2*0=0 plus/z) plus/z %. +%term _ tonat-onto/0 _ U z MC _ (tonat/lit MC 2*0=0 plus/z) %. +%term _ + %pi (tonat-onto/0 _ U z MC _ (tonat/lam MC MCz MA TN 4*0=0 plus/z)) + %<- (multiset`count-total MCz) + %<- (multiset`add-total MA) + %<- (multiset`count-total MC') + %<- ({v} {vl} tonat-onto/0 _ (upto/+ U MCz vl MA) _ MC' _ (TN v vl)) %. +%term _ + %pi (tonat-onto/0 N U (s VN-1) MC _ (tonat/var MC VL plus/z)) + %<- (upto-implies-varlevel U MC gt/1 _ VL) %. +%worlds (blockvar) (tonat-onto/0 _ _ _ _ _ _) %. +%total (N) (tonat-onto/0 N _ _ _ _ _) %. +%term _ %pi (tonat-onto/L2 U z MC _ _ _ _ _ _ TN) %<- (tonat-onto/0 _ U _ MC _ TN) %. +%theorem tonat-onto/L2/L : forall* {M} {X} {Y} {Z} {XM} forall {T times M (s (s X)) XM} {P plus Y XM (s Z)} exists {G gt (s Z) M} true %. +%term _ + %pi (tonat-onto/L2/L M*2=XM plus/z XM>M) + %<- (div-can-be-constructed M*2=XM XM/2=M) + %<- (quotient-of-nonzero-is-smaller XM/2=M nat`eq/ XM>M) %. +%term _ + %pi (tonat-onto/L2/L M*2=XM Y+XM=X' X'>M) + %<- (plus-implies-gt Y+XM=X' nat`eq/ X'>MX) + %<- (times-nonzero-implies-ge M*2=XM XM>=M) + %<- (nat`gt-transitive-ge X'>MX XM>=M X'>M) %. +%worlds () (tonat-onto/L2/L _ _ _) %. +%total {} (tonat-onto/L2/L _ _ _) %. +%term _ + %pi (tonat-onto/L2 U (s X) MC VN+FM=M' FM/2=TM M z TM/2=M _ (tonat/lam MC MCz MA TN FOUR*M=FM VN+FM=M')) + %<- (div-can-be-inverted TM/2=M M*2=TM) + %<- (div-can-be-inverted FM/2=TM TM*2=FM) + %<- (times-associative* M*2=TM TM*2=FM 2*2=4 M*4=FM) + %<- (times-commutative M*4=FM FOUR*M=FM) + %<- (multiset`count-total MCz) + %<- (multiset`add-total MA) + %<- (tonat-onto/L2/L M*4=FM VN+FM=M' M'>M) + %<- (meta-gt (s X) M M'>M) + %<- ({v} {vl} tonat-onto/L (upto/+ U MCz vl MA) _ _ (TN v vl)) %. +%worlds (blockvar) (tonat-onto/L2 _ _ _ _ _ _ _ _ _ _) (tonat-onto/L1 _ _ _ _ _ _ _ _ _) (tonat-onto/L0 _ _ _ _ _ _) (tonat-onto/L _ _ _ _) %. +%total (M2 M1 M0 M) (tonat-onto/L2 _ M2 _ _ _ _ _ _ _ _) (tonat-onto/L1 _ M1 _ _ _ _ _ _ _) (tonat-onto/L0 _ M0 _ _ _ _) (tonat-onto/L _ M _ _) %. +%{! ## Proof that mapping is "one to one" !}% +%{! +As with the simple HOAS syntax, this is the hardest of the four theorems +to prove. As before, the basic structure is the same: we first show +that the terms that result in the same number must be identical +except that two variables may be equal just by having the same level. + +As before we then 'chip' away at variables with low levels and shift +the others down. We need to work at low levels because contexts cannot +be parameterized by an 'N' and must be absolute. The additional problem +with indexed terms is that the first variable we encounter may be +of a higher-index than zero. In any case, there can be level +1 variables inside that have a different index than the one we are removing. +So the context cannot simply assert that all variables will have level 2 +or higher, as with did in the non-indexed case. + +The basic idea is that we handle one index level at a time. But again +because Twelf contexts cannot have be parameterized, we need to +shift index levels down too. Of course, we cannot actually change +index values. Instead, we keep a index-adjustment. Then a variable +of index type (term N) has two parts to its level M L, where M+A=N +and L is the original level. The multiset that keeps track of the +next level is similarly shifted. We then do a shift such as we had +for non-indexed terms while removing all variables of adjusted index +zero (M = z). Then when these are all removed, we can adjust the indices +one more step until there are no more variables of any index needing +levels. +!}% +%{! ### Auxiliary definitions !}% +%{! +In this section, we use the full generality of varadjlevel +because we need to recurse in two different directions: +the original var level and the term level. +!}% +%block blockvaradj {m} {n} {l} [v term n] [vl varadjlevel v m (s l)]%. +%{! #### Equality (permitting variables with the same level). !}% +%{! +Some variables are are bound with levels, others are not given levels. +We do this in order to gradually squeeze out those that need levels. +The lam1 and rec1 cases use variables that don't need levels, whereas +the lam2, rec2 cases handle variables that still need level. +We also take a natural number A that indicates the minimum term level +that will be used. Notice that lam2 is only legal while A is still zero. +!}% +%sort eql* {N} {A nat} %. +%inline eql eql* _ %. +%term eql/eq %pi (eq T1 T2) %-> (eql A MS T1 T2) %. +%term eql/app %pi (eql A MS F1 F2) %-> (eql A MS A1 A2) %-> (eql A MS (app F1 A1) (app F2 A2)) %. +%term eql/lam1 %pi ({v} eql A MS (F1 v) (F2 v)) %-> (eql A MS (lam F1) (lam F2)) %. +%term eql/lam2 + %pi (count MS z VN) + %-> (add MS z MS') + %-> ({v} {vl varadjlevel v z (s VN)} eql z MS' (F1 v) (F2 v)) + %-> (eql z MS (lam F1) (lam F2)) %. +%term eql/rec1 %pi ({v} eql A MS (F1 v) (F2 v)) %-> (eql A MS (rec F1) (rec F2)) %. +%term eql/rec2 + %pi (plus A M (s N)) + %-> (count MS M VN) + %-> (add MS M MS') + %-> ({v} {vl varadjlevel v M (s VN)} eql A MS' (F1 v) (F2 v)) + %-> (eql A MS (rec F1) (rec F2)) %. +%term eql/var + %pi (plus A M N) + %-> (varadjlevel V1 M L) + %-> (varadjlevel V2 M L) + %-> (eql* N A MS V1 V2) %. +%{! #### Measure of eql sizes. !}% +%{! +We use this measure to be able to prove termination. +We need eqlsize/var = eqlsize/eq, eqlsize/lam1 = eqlsize/lam2. +(Less than is ok in each case but would require that we +rephrase the lemmas.) +We have three measures: N B M. + +- N is the height of the term + +- B is the minimum of any M for a varlevel (or (s z)) + +- M is greater than the M for any varlevel used. + +The first measure is used to ensure that regular recursion through the +tree terminates. We don't use normal structure recursion because +everytime we change variable levels, the tree changes. + +The second measure is not used for termination, but when non-zero +means there are no variables with that need adjusted index zero. +Once this is the case, we can adjust all levels down one notch. + +The third measure is used to ensure that chipping away +at indices eventually terminates: M is the maximum adjusted +index used. Once M drops to zero, it means no variables +need levels for equality and we can convert to an 'eq' proof +easily. +!}% +%sort eqlsize %. +%term eqlsize/eq eqlsize (eql/eq _) z (s z) z %. +%term eqlsize/app + %pi (eqlsize E1 N1 B1 M1) + %-> (eqlsize E2 N2 B2 M2) + %-> (nat`max N1 N2 N) + %-> (nat`min B1 B2 B) + %-> (nat`max M1 M2 M) + %-> (eqlsize (eql/app E1 E2) (s N) B M) %. +%term eqlsize/lam1 %pi ({v} eqlsize (E v) N B M) %-> (eqlsize (eql/lam1 E) (s N) B M) %. +%term eqlsize/lam2 + %pi ({v} {vl} eqlsize (E v vl) N B M) + %-> (nat`max (s z) M M') + %-> (eqlsize (eql/lam2 _ _ E) (s N) z M') %. +%term eqlsize/rec1 %pi ({v} eqlsize (E v) N B M) %-> (eqlsize (eql/rec1 E) (s N) B M) %. +%term eqlsize/rec2 + %pi ({v} {vl varadjlevel v M _} eqlsize (E v vl) N B1 M1) + %-> (nat`min M B1 B2) + %-> (nat`max (s M) M1 M2) + %-> (eqlsize (eql/rec2 _ _ _ E) (s N) B2 M2) %. +%term eqlsize/var eqlsize (eql/var _ _ _) z (s z) z %. +%{! #### Copied definitions !}% +%{! +As with the non-indexed case, we need to use an alternate form +for levels as we shift them down or adjust them down. +(Remember there are two dimensions to a level!) +!}% +%sort varadjlevel' %. +%sort eql*' {N} {A nat} %. +%inline eql' eql*' _ %. +%term eql'/eq %pi (eq T1 T2) %-> (eql' A MS T1 T2) %. +%term eql'/app + %pi (eql' A MS F1 F2) + %-> (eql' A MS A1 A2) + %-> (eql' A MS (app F1 A1) (app F2 A2)) %. +%term eql'/lam1 %pi ({v} eql' A MS (F1 v) (F2 v)) %-> (eql' A MS (lam F1) (lam F2)) %. +%term eql'/lam2 + %pi (count MS z VN) + %-> (add MS z MS') + %-> ({v} {vl varadjlevel' v z (s VN)} eql' z MS' (F1 v) (F2 v)) + %-> (eql' z MS (lam F1) (lam F2)) %. +%term eql'/rec1 %pi ({v} eql' A MS (F1 v) (F2 v)) %-> (eql' A MS (rec F1) (rec F2)) %. +%term eql'/rec2 + %pi (plus A M (s N)) + %-> (count MS M VN) + %-> (add MS M MS') + %-> ({v} {vl varadjlevel' v M (s VN)} eql' A MS' (F1 v) (F2 v)) + %-> (eql' A MS (rec F1) (rec F2)) %. +%term eql'/var + %pi (plus A M N) + %-> (varadjlevel' V1 M L) + %-> (varadjlevel' V2 M L) + %-> (eql*' N A MS V1 V2) %. +%sort eqlsize' %. +%term eqlsize'/eq eqlsize' (eql'/eq _) z (s z) z %. +%term eqlsize'/app + %pi (eqlsize' E1 N1 B1 M1) + %-> (eqlsize' E2 N2 B2 M2) + %-> (nat`max N1 N2 N) + %-> (nat`min B1 B2 B) + %-> (nat`max M1 M2 M) + %-> (eqlsize' (eql'/app E1 E2) (s N) B M) %. +%term eqlsize'/lam1 %pi ({v} eqlsize' (E v) N B M) %-> (eqlsize' (eql'/lam1 E) (s N) B M) %. +%term eqlsize'/lam2 + %pi ({v} {vl} eqlsize' (E v vl) N B M1) + %-> (nat`max (s z) M1 M2) + %-> (eqlsize' (eql'/lam2 _ _ E) (s N) z M2) %. +%term eqlsize'/rec1 %pi ({v} eqlsize' (E v) N B M) %-> (eqlsize' (eql'/rec1 E) (s N) B M) %. +%term eqlsize'/rec2 + %pi ({v} {vl varadjlevel' v M _} eqlsize' (E v vl) N B1 M1) + %-> (nat`min M B1 B2) + %-> (nat`max (s M) M1 M2) + %-> (eqlsize' (eql'/rec2 _ _ _ E) (s N) B2 M2) %. +%term eqlsize'/var eqlsize' (eql'/var _ _ _) z (s z) z %. +%{! ### Theorems about auxiliary definitions !}% +%{! +Some of the lemmas that would normally belong here are +proved later when we have more blocks available. +!}% +%theorem false-implies-eql : forall* {MS} {N} {A} {T1} {T2} forall {F void} exists {E eql* N A MS T1 T2} true %. +%worlds (blocksimple blockvaradj) (false-implies-eql _ _) %. +%total {} (false-implies-eql _ _) %. +%theorem false-implies-eqlsize : forall* {N} {A} {MS} {T1} {T2} {S} {B} {M} {E eql* N A MS T1 T2} forall {F void} exists {ES1 eqlsize E S B M} true %. +%worlds (blocksimple blockvaradj) (false-implies-eqlsize _ _) %. +%total {} (false-implies-eqlsize _ _) %. +%theorem eqlsize-total* : forall* {N} {A} {MS} {T1} {T2} forall {E eql* N A MS T1 T2} exists {S} {M} {B} {ES eqlsize E S B M} true %. +%inline eqlsize-total eqlsize-total* _ _ _ _ %. +%term _ eqlsize-total eqlsize/eq %. +%term _ + %pi (eqlsize-total (eqlsize/app S1 S2 M B M2)) + %<- (eqlsize-total S1) + %<- (eqlsize-total S2) + %<- (nat`max-total M) + %<- (nat`min-total B) + %<- (nat`max-total M2) %. +%term _ %pi (eqlsize-total (eqlsize/lam1 F)) %<- ({v} eqlsize-total (F v)) %. +%term _ + %pi (eqlsize-total (eqlsize/lam2 F M)) + %<- ({v} {vl} eqlsize-total (F v vl)) + %<- (nat`max-total M) %. +%term _ %pi (eqlsize-total (eqlsize/rec1 F)) %<- ({v} eqlsize-total (F v)) %. +%term _ + %pi (eqlsize-total (eqlsize/rec2 F MN M)) + %<- ({v} {vl} eqlsize-total (F v vl)) + %<- (nat`min-total MN) + %<- (nat`max-total M) %. +%term _ eqlsize-total eqlsize/var %. +%worlds (blocksimple blockvaradj) (eqlsize-total* _ _ _ _ _) %. +%total (E) (eqlsize-total* E _ _ _ _) %. +%{! ### Shifting varlevels down !}% +%{! +Now as with the non-indexed case, we shift variable levels down. +As before, we need to use the alternate definitions. + +The astute reader will notice that we don't have a block that +just handles nonzero (adjusted) index variables being shifted +over. Instead we handle a double context "noshift" that handles +both directions. That's because when these lemmas are used +to shift levels down, the outer context may have non-zero (adjusted) +index that will need to handle being shifted in both directions. +!}% +%theorem varlevel-zero-shifts-down : forall* {N} {V term N} {L} forall {VL varadjlevel V z (s (s L))} exists {VL' varadjlevel' V z (s L)} true %. +%theorem varlevel-nonzero-shifts-over : forall* {N} {V term N} {M} {L} forall {VL varadjlevel V (s M) (s L)} exists {VL' varadjlevel' V (s M) (s L)} true %. +%theorem varlevel-shifts-back : forall* {N} {V term N} {M} {L} forall {VL' varadjlevel' V M (s L)} exists {VL varadjlevel V M (s L)} true %. +%block shiftdown {l} {n} [v term n] [vl varadjlevel v z (s (s l))] [vl' varadjlevel' v z (s l)] [vsd varlevel-zero-shifts-down vl vl']%. +%block noshift {l} {m} {n} [v term n] [vl varadjlevel v (s m) (s l)] [vl' varadjlevel' v (s m) (s l)] [vsd varlevel-nonzero-shifts-over vl vl'] [vsd varlevel-shifts-back vl' vl]%. +%block shiftback {n} {m} {l} [v term n] [vl' varadjlevel' v m (s l)] [vl varadjlevel v m (s l)] [vsb varlevel-shifts-back vl' vl]%. +%sort fake %. +%term _ %pi fake %<- ({x varlevel-zero-shifts-down X Y} varlevel-zero-shifts-down X' Y') %. +%term _ + %pi fake + %<- ({x varlevel-nonzero-shifts-over X Y} varlevel-nonzero-shifts-over X' Y') %. +%term _ %pi fake %<- ({x varlevel-shifts-back X Y} varlevel-shifts-back X' Y') %. +%worlds (blocksimple shiftdown noshift) (varlevel-zero-shifts-down _ _) %. +%worlds (blocksimple shiftdown noshift) (varlevel-nonzero-shifts-over _ _) %. +%worlds (blocksimple shiftback noshift) (varlevel-shifts-back _ _) %. +%total {} (varlevel-zero-shifts-down _ _) %. +%total {} (varlevel-nonzero-shifts-over _ _) %. +%total {} (varlevel-shifts-back _ _) %. +%theorem shift-varlevel/L1 : forall* {N} {A} {M0} {M1} {T1} {T2} {S} {MN} {MM} forall {E eql* N A M1 T1 T2} {ES eqlsize E S MN MM} {MA multiset`add M0 z M1} exists {E' eql*' N A M0 T1 T2} {ES' eqlsize' E' S MN MM} true %. +%term _ shift-varlevel/L1 (eql/eq eq/) eqlsize/eq _ (eql'/eq eq/) eqlsize'/eq %. +%term _ + %pi (shift-varlevel/L1 (eql/app E1 E2) (eqlsize/app ES1 ES2 MX MN MX2) MA (eql'/app E1' E2') (eqlsize'/app ES1' ES2' MX MN MX2)) + %<- (shift-varlevel/L1 E1 ES1 MA E1' ES1') + %<- (shift-varlevel/L1 E2 ES2 MA E2' ES2') %. +%term _ + %pi (shift-varlevel/L1 (eql/lam1 ([v] F v)) (eqlsize/lam1 FS) MA (eql'/lam1 ([v] F' v)) (eqlsize'/lam1 FS')) + %<- ({v} shift-varlevel/L1 (F v) (FS v) MA (F' v) (FS' v)) %. +%theorem shift-varlevel/L1/rec : forall* {N0} {N} {N'} {N1} {F1} {F2} {A} {S} {M1} {M2} {MN} {MM} forall {F {v term N0} {vl varadjlevel v z (s N)} eql* N1 A M2 (F1 v) (F2 v)} {FS {v term N0} {vl varadjlevel v z (s N)} eqlsize (F v vl) S MN MM} {MA1 multiset`add M1 z M2} {EQ nat`eq N (s N')} exists {F' {v term N0} {vl' varadjlevel' v z (s N')} eql*' N1 A M1 (F1 v) (F2 v)} {FS' {v term N0} {vl' varadjlevel' v z (s N')} eqlsize' (F' v vl') S MN MM} true %. +%term _ + %pi (shift-varlevel/L1/rec F FS MA1 nat`eq/ F' FS') + %<- ({v} {vl} {vl' varadjlevel' v z (s N)} {vsd varlevel-zero-shifts-down vl vl'} shift-varlevel/L1 (F v vl) (FS v vl) MA1 (F' v vl') (FS' v vl')) %. +%term _ + %pi (shift-varlevel/L1 (eql/lam2 MC1 MA1 ([v] [vl varadjlevel v z (s N)] F v vl)) (eqlsize/lam2 FS MM) MA0 (eql'/lam2 MC0 MA0 ([v] [vl varadjlevel' v z (s N')] F' v vl)) (eqlsize'/lam2 FS' MM)) + %<- (multiset`count-total MC0) + %<- (count-add-implies-count MC0 MA0 MC1') + %<- (count-deterministic MC1 MC1' multiset`eq/ nat`eq/ EQ) + %<- (shift-varlevel/L1/rec F FS MA1 EQ F' FS') %. +%term _ + %pi (shift-varlevel/L1 (eql/rec1 ([v] F v)) (eqlsize/rec1 FS) MA (eql'/rec1 ([v] F' v)) (eqlsize'/rec1 FS')) + %<- ({v} shift-varlevel/L1 (F v) (FS v) MA (F' v) (FS' v)) %. +%term _ + %pi (shift-varlevel/L1 (eql/rec2 P MC1 MA1 ([v] [vl varadjlevel v z (s N)] F v vl)) (eqlsize/rec2 FS MN MM) MA0 (eql'/rec2 P MC0 MA0 ([v] [vl varadjlevel' v z (s N')] F' v vl)) (eqlsize'/rec2 FS' MN MM)) + %<- (multiset`count-total MC0) + %<- (count-add-implies-count MC0 MA0 MC1') + %<- (count-deterministic MC1 MC1' multiset`eq/ nat`eq/ EQ) + %<- (shift-varlevel/L1/rec F FS MA1 EQ F' FS') %. +%term _ + %pi (shift-varlevel/L1 (eql/rec2 P MC1 MA1 ([v] [vl varadjlevel v (s M) (s N)] F v vl)) (eqlsize/rec2 FS MN MM) MA (eql'/rec2 P MC1' MA1' ([v] [vl varadjlevel' v (s M) (s N)] F' v vl)) (eqlsize'/rec2 FS' MN MM)) + %<- (add-commutes MA MA1 _ MA1' MA') + %<- (succ-implies-gt-zero _ GT) + %<- (add-preserves-count-converse MC1 MA (nat`ne/> GT) MC1') + %<- ({v} {vl} {vl' varadjlevel' v (s M) (s N)} {vsd varlevel-nonzero-shifts-over vl vl'} {vsb varlevel-shifts-back vl' vl} shift-varlevel/L1 (F v vl) (FS v vl) MA' (F' v vl') (FS' v vl')) %. +%term _ + %pi (shift-varlevel/L1 (eql/var P VL1 VL2) eqlsize/var MA (eql'/var P VL1' VL2') eqlsize'/var) + %<- (varlevel-zero-shifts-down VL1 VL1') + %<- (varlevel-zero-shifts-down VL2 VL2') %. +%term _ + %pi (shift-varlevel/L1 (eql/var P VL1 VL2) eqlsize/var MA (eql'/var P VL1' VL2') eqlsize'/var) + %<- (varlevel-nonzero-shifts-over VL1 VL1') + %<- (varlevel-nonzero-shifts-over VL2 VL2') %. +%worlds (blocksimple shiftdown noshift) (shift-varlevel/L1 _ _ _ _ _) (shift-varlevel/L1/rec _ _ _ _ _ _) %. +%total (E El) (shift-varlevel/L1 E _ _ _ _) (shift-varlevel/L1/rec El _ _ _ _ _) %. +%theorem shift-varlevel/L2 : forall* {N} {M} {A} {T1} {T2} {S} {MN} {MM} forall {E' eql*' N M A T1 T2} {ES' eqlsize' E' S MN MM} exists {E eql* N M A T1 T2} {ES eqlsize E S MN MM} true %. +%term _ shift-varlevel/L2 (eql'/eq eq/) eqlsize'/eq (eql/eq eq/) eqlsize/eq %. +%term _ + %pi (shift-varlevel/L2 (eql'/app E1' E2') (eqlsize'/app ES1' ES2' MAX MIN MAX2) (eql/app E1 E2) (eqlsize/app ES1 ES2 MAX MIN MAX2)) + %<- (shift-varlevel/L2 E1' ES1' E1 ES1) + %<- (shift-varlevel/L2 E2' ES2' E2 ES2) %. +%term _ + %pi (shift-varlevel/L2 (eql'/lam1 ([v] F v)) (eqlsize'/lam1 FS) (eql/lam1 ([v] F' v)) (eqlsize/lam1 FS')) + %<- ({v} shift-varlevel/L2 (F v) (FS v) (F' v) (FS' v)) %. +%term _ + %pi (shift-varlevel/L2 (eql'/lam2 MC MA ([v] [vl'] F v vl')) (eqlsize'/lam2 FS MM) (eql/lam2 MC MA ([v] [vl] F' v vl)) (eqlsize/lam2 FS' MM)) + %<- ({v} {vl'} {vl} {vsb varlevel-shifts-back vl' vl} shift-varlevel/L2 (F v vl') (FS v vl') (F' v vl) (FS' v vl)) %. +%term _ + %pi (shift-varlevel/L2 (eql'/rec1 ([v] F v)) (eqlsize'/rec1 FS) (eql/rec1 ([v] F' v)) (eqlsize/rec1 FS')) + %<- ({v} shift-varlevel/L2 (F v) (FS v) (F' v) (FS' v)) %. +%term _ + %pi (shift-varlevel/L2 (eql'/rec2 P MC MA ([v] [vl'] F v vl')) (eqlsize'/rec2 FS MN MM) (eql/rec2 P MC MA ([v] [vl] F' v vl)) (eqlsize/rec2 FS' MN MM)) + %<- ({v} {vl'} {vl} {vsb varlevel-shifts-back vl' vl} shift-varlevel/L2 (F v vl') (FS v vl') (F' v vl) (FS' v vl)) %. +%term _ + %pi (shift-varlevel/L2 (eql'/var P VL1 VL2) eqlsize'/var (eql/var P VL1' VL2') eqlsize/var) + %<- (varlevel-shifts-back VL1 VL1') + %<- (varlevel-shifts-back VL2 VL2') %. +%worlds (blocksimple shiftback noshift) (shift-varlevel/L2 _ _ _ _) %. +%total (E) (shift-varlevel/L2 E _ _ _) %. +%{! +Now we put these two parts into one that hides the alternative definitions. +!}% +%theorem shift-varlevel : forall* {N} {A} {M0} {M1} {T1} {T2} {S} {MN} {MM} forall {E eql* N A M1 T1 T2} {ES eqlsize E S MN MM} {MA multiset`add M0 z M1} exists {E' eql* N A M0 T1 T2} {ES' eqlsize E' S MN MM} true %. +%term _ + %pi (shift-varlevel E1 ES1 MA E3 ES3) + %<- (shift-varlevel/L1 E1 ES1 MA E2 ES2) + %<- (shift-varlevel/L2 E2 ES2 E3 ES3) %. +%worlds (blocksimple noshift) (shift-varlevel _ _ _ _ _) %. +%total {} (shift-varlevel _ _ _ _ _) %. +%{! +Now the whole reason for the two stage var-levels: +we have a lemma that says we can adjust the M of +all variables down if the minimum varlevel used is greater than zero. +First we need some helper lemmas and the ubiquitous blocks. +!}% +%theorem varlevel-adjusts-down : forall* {N} {V term N} {M} {L} forall {VL varadjlevel V (s M) (s L)} exists {VL' varadjlevel' V M (s L)} true %. +%block adjustdown {n} {m} {l} [v term n] [vl varadjlevel v (s m) (s l)] [vl' varadjlevel' v m (s l)] [vsd varlevel-adjusts-down vl vl']%. +%sort fake %. +%term _ %pi fake %<- ({x varlevel-adjusts-down X Y} varlevel-adjusts-down X' Y') %. +%worlds (blocksimple adjustdown) (varlevel-adjusts-down _ _) %. +%total {} (varlevel-adjusts-down _ _) %. +%theorem false-implies-eql' : forall* {MS} {N} {A} {T1} {T2} forall {F void} exists {E eql*' N A MS T1 T2} true %. +%worlds (blocksimple adjustdown) (false-implies-eql' _ _) %. +%total {} (false-implies-eql' _ _) %. +%theorem false-implies-eqlsize' : forall* {N} {A} {MS} {T1} {T2} {S} {B} {M} {E eql*' N A MS T1 T2} forall {F void} exists {ES1 eqlsize' E S B M} true %. +%worlds (blocksimple adjustdown) (false-implies-eqlsize' _ _) %. +%total {} (false-implies-eqlsize' _ _) %. +%theorem eqlsize-respects-eq : forall* {N} {A} {MS} {T1} {T2} {S1} {B1} {M1} {S2} {B2} {M2} {E eql* N A MS T1 T2} forall {ES1 eqlsize E S1 B1 M1} {SE nat`eq S1 S2} {BE nat`eq B1 B2} {ME nat`eq M1 M2} exists {ES2 eqlsize E S2 B2 M2} true %. +%term _ eqlsize-respects-eq ES nat`eq/ nat`eq/ nat`eq/ ES %. +%worlds (blocksimple blockvaradj adjustdown) (eqlsize-respects-eq _ _ _ _ _) %. +%total {} (eqlsize-respects-eq _ _ _ _ _) %. +%theorem adjust-varlevel/L0 : forall* {N} {A} {M0} {M1} {T1} {T2} {S} {MN} forall {E eql* N A M1 T1 T2} {ES eqlsize E S (s MN) z} exists {E' eql*' N (s A) M0 T1 T2} {MN'} {ES' eqlsize' E' S MN' z} true %. +%term _ adjust-varlevel/L0 (eql/eq eq/) eqlsize/eq (eql'/eq eq/) _ eqlsize'/eq %. +%term _ + %pi (adjust-varlevel/L0 (eql/app E1 E2) (eqlsize/app ES1 ES2 MAX MIN MAX2) (eql'/app E1' E2') _ (eqlsize'/app ES1'' ES2'' MAX MIN' nat`max/=)) + %<- (max-implies-ge MAX2 ZERO>=M1 ZERO>=M2) + %<- (ge-zero-always M1 M1>=0) + %<- (ge-zero-always M2 M2>=0) + %<- (nat`ge-anti-symmetric M1>=0 ZERO>=M1 M1=0) + %<- (nat`ge-anti-symmetric M2>=0 ZERO>=M2 M2=0) + %<- (min-implies-ge MIN B1>=B+1 B2>=B+1) + %<- (ge-succ-implies-gt B1>=B+1 B1>B) + %<- (ge-succ-implies-gt B2>=B+1 B2>B) + %<- (gt-implies-positive B1>B B1' B1=B1'+1) + %<- (gt-implies-positive B2>B B2' B2=B2'+1) + %<- (eqlsize-respects-eq ES1 nat`eq/ B1=B1'+1 M1=0 ES1') + %<- (eqlsize-respects-eq ES2 nat`eq/ B2=B2'+1 M2=0 ES2') + %<- (adjust-varlevel/L0 E1 ES1' E1' _ ES1'') + %<- (adjust-varlevel/L0 E2 ES2' E2' _ ES2'') + %<- (nat`min-total MIN') %. +%term _ + %pi (adjust-varlevel/L0 (eql/lam1 ([v] F v)) (eqlsize/lam1 FS) (eql'/lam1 ([v] F' v)) MN' (eqlsize'/lam1 FS')) + %<- ({v} adjust-varlevel/L0 (F v) (FS v) (F' v) MN' (FS' v)) %. +%term _ + %pi (adjust-varlevel/L0 (eql/rec1 ([v] F v)) (eqlsize/rec1 FS) (eql'/rec1 ([v] F' v)) MN' (eqlsize'/rec1 FS')) + %<- ({v} adjust-varlevel/L0 (F v) (FS v) (F' v) MN' (FS' v)) %. +%term _ + %pi (adjust-varlevel/L0 (eql/var P VL1 VL2) eqlsize/var (eql'/var P' VL1' VL2') _ eqlsize'/var) + %<- (plus-swap-succ-converse P P') + %<- (varlevel-adjusts-down VL1 VL1') + %<- (varlevel-adjusts-down VL2 VL2') %. +%worlds (blocksimple adjustdown) (adjust-varlevel/L0 _ _ _ _ _) %. +%total (E) (adjust-varlevel/L0 E _ _ _ _) %. +%theorem adjust-varlevel/L1 : forall* {N} {A} {M0} {M1} {T1} {T2} {S} {MN} {MM} forall {E eql* N A M1 T1 T2} {ES eqlsize E S (s MN) (s MM)} {MS multiset`shift z M0 M1} exists {E' eql*' N (s A) M0 T1 T2} {MN'} {ES' eqlsize' E' S MN' MM} true %. +% MAX2 = max z z case (contradiction) +%term _ + %pi (adjust-varlevel/L1 (eql/app E1 E2) (eqlsize/app ES1 ES2 MAX MIN (%the (nat`max z z (s MM)) MAX2)) _ E' z ES') + %<- (nat`max-deterministic nat`max/= MAX2 nat`eq/ nat`eq/ ZERO=MM+1) + %<- (nat`eq-contradiction ZERO=MM+1 F) + %<- (false-implies-eql' F E') + %<- (false-implies-eqlsize' F ES') %. +% MAX2 = max z (s MM2) (s MM) +%term _ + %pi (adjust-varlevel/L1 (eql/app E1 E2) (eqlsize/app ES1 ES2 MAX MIN MAX2) MS (eql'/app E1' E2') _ (eqlsize'/app ES1'' ES2'' MAX MIN' MAX2')) + %<- (min-implies-ge MIN B1>=B+1 B2>=B+1) + %<- (ge-succ-implies-gt B1>=B+1 B1>B) + %<- (ge-succ-implies-gt B2>=B+1 B2>B) + %<- (gt-implies-positive B1>B B1' B1=B1'+1) + %<- (gt-implies-positive B2>B B2' B2=B2'+1) + %<- (eqlsize-respects-eq ES1 nat`eq/ B1=B1'+1 nat`eq/ ES1') + %<- (eqlsize-respects-eq ES2 nat`eq/ B2=B2'+1 nat`eq/ ES2') + %<- (adjust-varlevel/L0 E1 ES1' E1' _ ES1'') + %<- (adjust-varlevel/L1 E2 ES2' MS E2' _ ES2'') + %<- (succ-implies-gt-zero _ MM2+1>0) + %<- (nat`max-deterministic (max/< MM2+1>0) MAX2 nat`eq/ nat`eq/ MM2+1=MM+1) + %<- (succ-cancels MM2+1=MM+1 MM2=MM) + %<- (nat`ge-zero-always _ MM2>=0) + %<- (nat`le-implies-max MM2>=0 MAX2'') + %<- (nat`max-respects-eq MAX2'' nat`eq/ nat`eq/ MM2=MM MAX2') + %<- (nat`min-total MIN') %. +% MAX2 = max (s MM1) z (s MM) +%term _ + %pi (adjust-varlevel/L1 (eql/app E1 E2) (eqlsize/app ES1 ES2 MAX MIN MAX2) MS (eql'/app E1' E2') _ (eqlsize'/app ES1'' ES2'' MAX MIN' MAX2')) + %<- (min-implies-ge MIN B1>=B+1 B2>=B+1) + %<- (ge-succ-implies-gt B1>=B+1 B1>B) + %<- (ge-succ-implies-gt B2>=B+1 B2>B) + %<- (gt-implies-positive B1>B B1' B1=B1'+1) + %<- (gt-implies-positive B2>B B2' B2=B2'+1) + %<- (eqlsize-respects-eq ES1 nat`eq/ B1=B1'+1 nat`eq/ ES1') + %<- (eqlsize-respects-eq ES2 nat`eq/ B2=B2'+1 nat`eq/ ES2') + %<- (adjust-varlevel/L1 E1 ES1' MS E1' _ ES1'') + %<- (adjust-varlevel/L0 E2 ES2' E2' _ ES2'') + %<- (succ-implies-gt-zero _ MM1+1>0) + %<- (nat`max-deterministic (max/> MM1+1>0) MAX2 nat`eq/ nat`eq/ MM1+1=MM+1) + %<- (succ-cancels MM1+1=MM+1 MM1=MM) + %<- (nat`ge-zero-always _ MM1>=0) + %<- (nat`ge-implies-max MM1>=0 MAX2'') + %<- (nat`max-respects-eq MAX2'' nat`eq/ nat`eq/ MM1=MM MAX2') + %<- (nat`min-total MIN') %. +%term _ + %pi (adjust-varlevel/L1 (eql/app E1 E2) (eqlsize/app ES1 ES2 MAX MIN MAX2) MS (eql'/app E1' E2') _ (eqlsize'/app ES1'' ES2'' MAX MIN' MAX2')) + %<- (min-implies-ge MIN B1>=B+1 B2>=B+1) + %<- (ge-succ-implies-gt B1>=B+1 B1>B) + %<- (ge-succ-implies-gt B2>=B+1 B2>B) + %<- (gt-implies-positive B1>B B1' B1=B1'+1) + %<- (gt-implies-positive B2>B B2' B2=B2'+1) + %<- (eqlsize-respects-eq ES1 nat`eq/ B1=B1'+1 nat`eq/ ES1') + %<- (eqlsize-respects-eq ES2 nat`eq/ B2=B2'+1 nat`eq/ ES2') + %<- (adjust-varlevel/L1 E1 ES1' MS E1' _ ES1'') + %<- (adjust-varlevel/L1 E2 ES2' MS E2' _ ES2'') + %<- (succ-preserves-max-converse MAX2 MAX2') + %<- (nat`min-total MIN') %. +%term _ + %pi (adjust-varlevel/L1 (eql/lam1 ([v] F v)) (eqlsize/lam1 FS) MS (eql'/lam1 ([v] F' v)) MN' (eqlsize'/lam1 FS')) + %<- ({v} adjust-varlevel/L1 (F v) (FS v) MS (F' v) MN' (FS' v)) %. +%term _ + %pi (adjust-varlevel/L1 (eql/rec1 ([v] F v)) (eqlsize/rec1 FS) MS (eql'/rec1 ([v] F' v)) MN' (eqlsize'/rec1 FS')) + %<- ({v} adjust-varlevel/L1 (F v) (FS v) MS (F' v) MN' (FS' v)) %. +%term _ + %pi (adjust-varlevel/L1 (eql/rec2 P MC1 MA1 ([v] [vl varadjlevel v z (s N)] F v vl)) (eqlsize/rec2 FS MN MM) _ EQ z ES) + %<- (min-implies-ge MN ZERO>=B+1 _) + %<- (ge-succ-implies-gt ZERO>=B+1 ZERO>B) + %<- (gt-contradiction ZERO>B V) + %<- (false-implies-eql' V EQ) + %<- (false-implies-eqlsize' V ES) %. +%term _ + %pi (adjust-varlevel/L1 (eql/rec2 P MC1 MA1 ([v] [vl varadjlevel v (s M) (s N)] F v vl)) (eqlsize/rec2 FS MN MM) MS (eql'/rec2 P' MC1' MA1' ([v] [vl varadjlevel' v M (s N)] F' v vl)) _ (eqlsize'/rec2 FS'' MN' MM')) + %<- (plus-swap-succ-converse P P') + %<- (nat`min-implies-ge MN _ B2>=B+1) + %<- (ge-succ-implies-gt B2>=B+1 B2>B) + %<- (gt-implies-positive B2>B B2' B2=B2'+1) + %<- ({v} {vl} eqlsize-respects-eq (FS v vl) nat`eq/ B2=B2'+1 nat`eq/ (FS' v vl)) + %<- (shift-preserves-add-converse MA1 MS (plus/s plus/z) _ MS' MA1') + %<- ({v} {vl} {vl' varadjlevel' v M (s N)} {vsd varlevel-adjusts-down vl vl'} adjust-varlevel/L0 (F v vl) (FS' v vl) (F' v vl') _ (FS'' v vl')) + %<- (shift-preserves-count-converse MC1 MS (plus/s plus/z) MC1') + %<- (succ-implies-gt-zero _ MM1+1>0) + %<- (nat`max-deterministic (max/> MM1+1>0) MM nat`eq/ nat`eq/ MM1+1=MM+1) + %<- (succ-cancels MM1+1=MM+1 MM1=MM) + %<- (nat`ge-zero-always _ MM1>=0) + %<- (nat`ge-implies-max MM1>=0 MAX2'') + %<- (nat`max-respects-eq MAX2'' nat`eq/ nat`eq/ MM1=MM MM') + %<- (nat`min-total MN') %. +%term _ + %pi (adjust-varlevel/L1 (eql/rec2 P MC1 MA1 ([v] [vl varadjlevel v (s M) (s N)] F v vl)) (eqlsize/rec2 FS MN MM) MS (eql'/rec2 P' MC1' MA1' ([v] [vl varadjlevel' v M (s N)] F' v vl)) _ (eqlsize'/rec2 FS'' MN' MM')) + %<- (plus-swap-succ-converse P P') + %<- (nat`min-implies-ge MN _ B2>=B+1) + %<- (ge-succ-implies-gt B2>=B+1 B2>B) + %<- (gt-implies-positive B2>B B2' B2=B2'+1) + %<- ({v} {vl} eqlsize-respects-eq (FS v vl) nat`eq/ B2=B2'+1 nat`eq/ (FS' v vl)) + %<- (shift-preserves-add-converse MA1 MS (plus/s plus/z) _ MS' MA1') + %<- ({v} {vl} {vl' varadjlevel' v M (s N)} {vsd varlevel-adjusts-down vl vl'} adjust-varlevel/L1 (F v vl) (FS' v vl) MS' (F' v vl') _ (FS'' v vl')) + %<- (shift-preserves-count-converse MC1 MS (plus/s plus/z) MC1') + %<- (nat`min-total MN') + %<- (succ-preserves-max-converse MM MM') %. +%worlds (blocksimple adjustdown) (adjust-varlevel/L1 _ _ _ _ _ _) %. +%total (E) (adjust-varlevel/L1 E _ _ _ _ _) %. +%theorem adjust-varlevel : forall* {N} {A} {M0} {M1} {T1} {T2} {S} {MN} {MM} forall {E eql* N A M1 T1 T2} {ES eqlsize E S (s MN) (s MM)} {MS multiset`shift z M0 M1} exists {E' eql* N (s A) M0 T1 T2} {MN'} {ES' eqlsize E' S MN' MM} true %. +%term _ + %pi (adjust-varlevel E ES MS E'' _ ES'') + %<- (adjust-varlevel/L1 E ES MS E' _ ES') + %<- (shift-varlevel/L2 E' ES' E'' ES'') %. +%worlds (blocksimple) (adjust-varlevel _ _ _ _ _ _) %. +%total {} (adjust-varlevel _ _ _ _ _ _) %. +%{! ### Removing variable levels altogether !}% +%{! +The following blocks are used when we remove the level for +the outermost (level 1) variable of adjusted index zero. +All other variables are at least level 2 or non-zero adjusted index. +"blockvar2" and "blockvaradj" makes this context explicit. +(I probably could substitute noshift for blockvaradj +!}% +%block blockvar2 {l} {n} [v term n] [vl varadjlevel v z (s (s l))]%. +%block blockvaradj {m} {l} {n} [v term n] [vl varadjlevel v (s m) (s l)]%. +%theorem remove-one-var : forall* {N} {N'} {F1} {F2} {S} {A} {M} {NZ} {MN} {MM} forall {E {v term N} {vl varadjlevel v z (s z)} eql* N' A M (F1 v) (F2 v)} {ES {v} {vl} eqlsize (E v vl) S MN MM} {MC multiset`count M z (s NZ)} exists {E' {v term N} eql A M (F1 v) (F2 v)} {ES' {v} eqlsize (E' v) S MN MM} true %. +%term _ remove-one-var ([v] [vl] eql/eq eq/) ([v] [vl] eqlsize/eq) _ ([v] eql/eq eq/) ([v] eqlsize/eq) %. +%term _ + %pi (remove-one-var ([v] [vl] eql/app (FE v vl) (AE v vl)) ([v] [vl] eqlsize/app (ESF v vl) (ESA v vl) MX MN MX2) MC ([v] eql/app (FE' v) (AE' v)) ([v] eqlsize/app (ESF' v) (ESA' v) MX MN MX2)) + %<- (remove-one-var FE ESF MC FE' ESF') + %<- (remove-one-var AE ESA MC AE' ESA') %. +%term _ + %pi (remove-one-var ([v] [vl] eql/lam1 ([v'] F v' v vl)) ([v] [vl] eqlsize/lam1 ([v'] FS v' v vl)) MC ([v] eql/lam1 ([v'] F' v' v)) ([v] eqlsize/lam1 ([v'] FS' v' v))) + %<- ({v'} remove-one-var (F v') (FS v') MC (F' v') (FS' v')) %. +%theorem remove-one-var/lam2 : forall* {N0} {N1} {N2} {N3} {N1'} {S} {A} {M0} {M1} {F1} {F2} {MN} {MM} forall {E {v' term N0} {vl' varadjlevel v' z (s N1)} {v term N2} {vl varadjlevel v z (s z)} eql* N3 A M1 (F1 v' v) (F2 v' v)} {ES {v' term N0} {vl' varadjlevel v' z (s N1)} {v term N2} {vl varadjlevel v z (s z)} eqlsize (E v' vl' v vl) S MN MM} {MC multiset`count M0 z N1} {MA multiset`add M0 z M1} {EQ nat`eq N1 (s N1')} exists {E' {v' term N0} {vl' varadjlevel v' z (s N1)} {v term N2} eql* N3 A M1 (F1 v' v) (F2 v' v)} {ES' {v' term N0} {vl' varadjlevel v' z (s N1)} {v term N2} eqlsize (E' v' vl' v) S MN MM} true %. +%term _ + %pi (remove-one-var/lam2 E ES MC0 MA nat`eq/ E' ES') + %<- (count-add-implies-count MC0 MA MC1) + %<- ({v'} {vl'} remove-one-var (E v' vl') (ES v' vl') MC1 (E' v' vl') (ES' v' vl')) %. +%term _ + %pi (remove-one-var ([v] [vl] eql/lam2 MC MA ([v'] [vl'] F v' vl' v vl)) ([v] [vl] eqlsize/lam2 ([v'] [vl'] FS v' vl' v vl) MM) MC0 ([v] eql/lam2 MC MA ([v'] [vl'] F' v' vl' v)) ([v] eqlsize/lam2 ([v'] [vl'] FS' v' vl' v) MM)) + %<- (count-deterministic MC MC0 multiset`eq/ nat`eq/ EQ) + %<- (remove-one-var/lam2 F FS MC (%the (add M0 z M1) MA) EQ F' FS') %. +%term _ + %pi (remove-one-var ([v] [vl] eql/rec1 ([v'] F v' v vl)) ([v] [vl] eqlsize/rec1 ([v'] FS v' v vl)) MC ([v] eql/rec1 ([v'] F' v' v)) ([v] eqlsize/rec1 ([v'] FS' v' v))) + %<- ({v'} remove-one-var (F v') (FS v') MC (F' v') (FS' v')) %. +%term _ + %pi (remove-one-var ([v] [vl] eql/rec2 P MC MA ([v'] [vl'] F v' vl' v vl)) ([v] [vl] eqlsize/rec2 ([v'] [vl'] FS v' vl' v vl) MN MM) MC0 ([v] eql/rec2 P MC MA ([v'] [vl'] F' v' vl' v)) ([v] eqlsize/rec2 ([v'] [vl'] FS' v' vl' v) MN MM)) + %<- (count-deterministic MC MC0 multiset`eq/ nat`eq/ EQ) + %<- (remove-one-var/lam2 F FS MC MA EQ F' FS') %. +%term _ + %pi (remove-one-var ([v] [vl] eql/rec2 P MC MA ([v'] [vl'] F v' vl' v vl)) ([v] [vl] eqlsize/rec2 ([v'] [vl'] FS v' vl' v vl) MN MM) MC0 ([v] eql/rec2 P MC MA ([v'] [vl'] F' v' vl' v)) ([v] eqlsize/rec2 ([v'] [vl'] FS' v' vl' v) MN MM)) + %<- (succ-implies-gt-zero _ GT) + %<- (add-preserves-count MC0 MA (nat`ne/< GT) MC1) + %<- ({v'} {vl'} remove-one-var (F v' vl') (FS v' vl') MC1 (F' v' vl') (FS' v' vl')) %. +%term _ remove-one-var ([v] [vl] eql/var P vl vl) ([v] [vl] eqlsize/var) _ ([v] eql/eq eq/) ([v] eqlsize/eq) %. +%term _ remove-one-var ([v] [vl] eql/var P VL1 VL2) ([v] [vl] eqlsize/var) _ ([v] eql/var P VL1 VL2) ([v] eqlsize/var) %. +%worlds (blocksimple blockvar2 blockvaradj noshift) (remove-one-var _ _ _ _ _) (remove-one-var/lam2 _ _ _ _ _ _ _) %. +%total (E El) (remove-one-var E _ _ _ _) (remove-one-var/lam2 El _ _ _ _ _ _) %. +%{! +We now have the machinery needed to get rid of all level 0 variables +in one recursive sweep. +!}% +%theorem remove-zero-index-vars : forall* {N} {A} {MS} {T1} {T2} {MN} {MM} forall {E eql* N A MS T1 T2} {S} {ES eqlsize E S MN MM} {MC multiset`count MS z z} exists {E' eql* N A MS T1 T2} {MN'} {MM'} {ES' eqlsize E' S (s MN') MM'} {G nat`ge MM MM'} true %. +%term _ remove-zero-index-vars (eql/eq E) _ eqlsize/eq _ (eql/eq E) _ _ eqlsize/eq (nat`ge/= nat`eq/) %. +%term _ + %pi (remove-zero-index-vars (eql/app E1 E2) (s S) (eqlsize/app ES1 ES2 MAX1 _ MAX2) MC (eql/app E1' E2') _ _ (eqlsize/app ES1' ES2' MAX1 MIN' MAX2') GE) + %<- (nat`max-implies-ge MAX1 S>=S1 S>=S2) + %<- (ge-implies-succ-gt S>=S1 S+1>S1) + %<- (ge-implies-succ-gt S>=S2 S+1>S2) + %<- (nat`meta-gt _ _ S+1>S1) + %<- (nat`meta-gt _ _ S+1>S2) + %<- (remove-zero-index-vars E1 S1 ES1 MC E1' _ _ ES1' GE1) + %<- (remove-zero-index-vars E2 S2 ES2 MC E2' _ _ ES2' GE2) + %<- (nat`min-total MIN) + %<- (succ-preserves-min MIN MIN') + %<- (nat`max-total MAX2') + %<- (max-preserves-ge* GE1 GE2 MAX2 MAX2' GE) %. +%term _ + %pi (remove-zero-index-vars (eql/lam1 FE) _ (eqlsize/lam1 FES) MC (eql/lam1 FE') _ _ (eqlsize/lam1 FES') GE) + %<- ({v} remove-zero-index-vars (FE v) _ (FES v) MC (FE' v) _ _ (FES' v) GE) %. +%term _ + %pi (remove-zero-index-vars (eql/lam2 MC0 (%the (add MS0 z MS1) MA0) ([v] [vl] FE0 v vl)) _ (eqlsize/lam2 ([v] [vl] FES0 v vl) MAX) MC (eql/lam1 ([v] FE3 v)) _ _ (eqlsize/lam1 ([v] FES3 v)) GE') + %<- (count-add-implies-count MC0 MA0 MC1) + %<- (remove-one-var FE0 FES0 MC1 FE1 FES1) + %<- ({v} shift-varlevel (FE1 v) (FES1 v) MA0 (FE2 v) (FES2 v)) + %<- ({v} remove-zero-index-vars (FE2 v) S' (FES2 v) MC (FE3 v) _ _ (FES3 v) GE) + %<- (nat`max-implies-ge MAX _ GE1) + %<- (nat`ge-transitive GE1 GE GE') %. +%term _ + %pi (remove-zero-index-vars (eql/lam2 (%the (count MS0 z (s C)) MC0) MA0 ([v] [vl] FE0 v vl)) _ (eqlsize/lam2 ([v] [vl] FES0 v vl) MAX) MC EQ z z EQS GE) + %<- (multiset`count-deterministic MC MC0 multiset`eq/ nat`eq/ ZERO=C+1) + %<- (nat`eq-contradiction ZERO=C+1 F) + %<- (false-implies-eql F EQ) + %<- (false-implies-eqlsize F EQS) + %<- (nat`false-implies-ge F GE) %. +%term _ + %pi (remove-zero-index-vars (eql/rec1 FE) _ (eqlsize/rec1 FES) MC (eql/rec1 FE') _ _ (eqlsize/rec1 FES') GE) + %<- ({v} remove-zero-index-vars (FE v) _ (FES v) MC (FE' v) _ _ (FES' v) GE) %. +%term _ + %pi (remove-zero-index-vars (eql/rec2 P MC0 (%the (add MS0 z MS1) MA0) ([v] [vl] FE0 v vl)) _ (eqlsize/rec2 ([v] [vl] FES0 v vl) MIN MAX) MC (eql/rec1 ([v] FE3 v)) _ _ (eqlsize/rec1 ([v] FES3 v)) GE') + %<- (count-add-implies-count MC0 MA0 MC1) + %<- (remove-one-var FE0 FES0 MC1 FE1 FES1) + %<- ({v} shift-varlevel (FE1 v) (FES1 v) MA0 (FE2 v) (FES2 v)) + %<- ({v} remove-zero-index-vars (FE2 v) S' (FES2 v) MC (FE3 v) _ _ (FES3 v) GE) + %<- (nat`max-implies-ge MAX _ GE1) + %<- (nat`ge-transitive GE1 GE GE') %. +%term _ + %pi (remove-zero-index-vars (eql/rec2 P MC0 (%the (add MS0 z MS1) MA0) ([v] [vl] FE0 v vl)) _ (eqlsize/rec2 ([v] [vl] FES0 v vl) MIN MAX) MC EQ z z EQS GE) + %<- (multiset`count-deterministic MC MC0 multiset`eq/ nat`eq/ ZERO=C+1) + %<- (nat`eq-contradiction ZERO=C+1 F) + %<- (false-implies-eql F EQ) + %<- (false-implies-eqlsize F EQS) + %<- (nat`false-implies-ge F GE) %. +%term _ + %pi (remove-zero-index-vars (eql/rec2 P MC0 (%the (add MS0 (s M) MS1) MA0) ([v] [vl] FE0 v vl)) _ (eqlsize/rec2 ([v] [vl] FES0 v vl) _ (%the (max (s (s M)) MM2 MM) MAX)) MC (eql/rec2 P MC0 MA0 ([v] [vl] FE1 v vl)) _ _ (eqlsize/rec2 ([v] [vl] FES1 v vl) MIN' MAX') GE') + %<- (ge-zero-always M M>=0) + %<- (ge-implies-succ-gt M>=0 M+1>0) + %<- (add-preserves-count MC MA0 (nat`ne/< M+1>0) MC') + %<- ({v} {vl} {vl'} + %pi (varlevel-nonzero-shifts-over vl vl') + %-> (varlevel-shifts-back vl' vl) + %-> (remove-zero-index-vars (FE0 v vl) S' (FES0 v vl) MC' (FE1 v vl) _ _ (FES1 v vl) GE)) + %<- (nat`min-total MIN) + %<- (succ-preserves-min MIN MIN') + %<- (nat`max-total MAX') + %<- (max-preserves-ge* (nat`ge/= nat`eq/) GE MAX MAX' GE') %. +%term _ remove-zero-index-vars (eql/var P VL1 VL2) _ eqlsize/var _ (eql/var P VL1 VL2) _ _ eqlsize/var (nat`ge/= nat`eq/) %. +%worlds (blocksimple noshift) (remove-zero-index-vars _ _ _ _ _ _ _ _ _) %. +%total (S) (remove-zero-index-vars _ S _ _ _ _ _ _ _) %. +%theorem lit-preserves-eq : forall* {M1} {M2} forall {EQ nat`eq M1 M2} exists {EQ' eq (lit M1) (lit M2)} true %. +%term _ lit-preserves-eq nat`eq/ eq/ %. +%worlds (blocksimple blockvar) (lit-preserves-eq _ _) %. +%total {} (lit-preserves-eq _ _) %. +%theorem app-preserves-eq : forall* {N1} {N2} {T1 term (s N1)} {T2 term (s N2)} {T3} {T4} forall {E eq T1 T2} {E eq T3 T4} exists {BE eq (app T1 T3) (app T2 T4)} true %. +%term _ app-preserves-eq eq/ eq/ eq/ %. +%worlds (blocksimple) (app-preserves-eq _ _ _) %. +%total {} (app-preserves-eq _ _ _) %. +%theorem lam-preserves-eq : forall* {N1} {N2} {F1 %pi t %-> (term N1)} {F2 %pi t %-> (term N2)} forall {E {v} eq (F1 v) (F2 v)} exists {E eq (lam F1) (lam F2)} true %. +%term _ lam-preserves-eq ([v] eq/) eq/ %. +%worlds (blocksimple) (lam-preserves-eq _ _) %. +%total {} (lam-preserves-eq _ _) %. +%theorem rec-preserves-eq : forall* {N} {F1 %pi (term (s N)) %-> (term (s N))} {F2 %pi (term (s N)) %-> (term (s N))} forall {E {v} eq (F1 v) (F2 v)} exists {E eq (rec F1) (rec F2)} true %. +%term _ rec-preserves-eq ([v] eq/) eq/ %. +%worlds (blocksimple) (rec-preserves-eq _ _) %. +%total {} (rec-preserves-eq _ _) %. +%{! +Next the main lemma that says we can avoid looking at varlevels in checking +equality. We remove the outside variable, shift remaining variables, +and then recurse (hence the need for tracking eqlsize). Note that we +never put var levels in the context. +!}% +%theorem eql-implies-eq : forall* {N} {T1 term N} {T2 term N} forall {E eql z multiset`map/0 T1 T2} exists {E' eq T1 T2} true %. +%theorem eql-implies-eq/L0 : forall* {N} {A} {MS} {T1} {T2} {S} {MN} forall {E eql* N A MS T1 T2} {ES eqlsize E S MN z} exists {E eq T1 T2} true %. +%term _ eql-implies-eq/L0 (eql/eq E) _ E %. +%term _ + %pi (eql-implies-eq/L0 (eql/app E1 E2) (eqlsize/app ES1 ES2 MAX1 MIN MAX2) E') + %<- (max-implies-ge MAX2 ZERO>=M1 ZERO>=M2) + %<- (ge-zero-always _ M1>=0) + %<- (ge-zero-always _ M2>=0) + %<- (ge-anti-symmetric M1>=0 ZERO>=M1 M1=0) + %<- (ge-anti-symmetric M2>=0 ZERO>=M2 M2=0) + %<- (eqlsize-respects-eq ES1 nat`eq/ nat`eq/ M1=0 ES1') + %<- (eqlsize-respects-eq ES2 nat`eq/ nat`eq/ M2=0 ES2') + %<- (eql-implies-eq/L0 E1 ES1' E1') + %<- (eql-implies-eq/L0 E2 ES2' E2') + %<- (app-preserves-eq E1' E2' E') %. +%term _ + %pi (eql-implies-eq/L0 (eql/lam1 ([v] E v)) (eqlsize/lam1 ([v] ES v)) E'') + %<- ({v} eql-implies-eq/L0 (E v) (ES v) (E' v)) + %<- (lam-preserves-eq E' E'') %. +%term _ + %pi (eql-implies-eq/L0 (eql/lam2 _ _ _) (eqlsize/lam2 _ MAX) E) + %<- (nat`max-implies-ge MAX ZERO>=ONE _) + %<- (ge-succ-implies-gt ZERO>=ONE ZERO>ZERO) + %<- (nat`gt-contradiction ZERO>ZERO F) + %<- (false-implies-eq F E) %. +%term _ + %pi (eql-implies-eq/L0 (eql/rec1 ([v] E v)) (eqlsize/rec1 ([v] ES v)) E'') + %<- ({v} eql-implies-eq/L0 (E v) (ES v) (E' v)) + %<- (rec-preserves-eq E' E'') %. +%term _ + %pi (eql-implies-eq/L0 (eql/rec2 _ _ _ _) (eqlsize/rec2 _ _ MAX) E) + %<- (nat`max-implies-ge MAX ZERO>=ONE _) + %<- (ge-succ-implies-gt ZERO>=ONE ZERO>ZERO) + %<- (nat`gt-contradiction ZERO>ZERO F) + %<- (false-implies-eq F E) %. +%worlds (blocksimple) (eql-implies-eq/L0 _ _ _) %. +%total (E) (eql-implies-eq/L0 E _ _) %. +%theorem eql-implies-eq/L1 : forall* {N} {A} {T1} {T2} {S} {MN} forall {E eql* N A multiset`map/0 T1 T2} {MM} {ES eqlsize E S MN MM} exists {E eq T1 T2} true %. +%term _ %pi (eql-implies-eq/L1 EQL z ES E) %<- (eql-implies-eq/L0 EQL ES E) %. +%theorem eql-implies-eq/L2 : forall* {N} {A} {T1} {T2} {S} {MN} forall {E eql* N A multiset`map/0 T1 T2} {MM} {ES eqlsize E S (s MN) MM} exists {E eq T1 T2} true %. +%term _ %pi (eql-implies-eq/L2 EQL z ES E) %<- (eql-implies-eq/L0 EQL ES E) %. +%term _ + %pi (eql-implies-eq/L2 EQL (s M) ES E) + %<- (adjust-varlevel EQL ES multiset`shift/0 E' _ ES') + %<- (eql-implies-eq/L1 E' _ ES' E) %. +%term _ + %pi (eql-implies-eq/L1 EQL (s M) ES E) + %<- (remove-zero-index-vars EQL _ ES (multiset`count/z multiset`not-member/0) EQL' _ _ ES' GE) + %<- (meta-ge _ _ GE) + %<- (eql-implies-eq/L2 EQL' _ ES' E) %. +%worlds () (eql-implies-eq/L1 _ _ _ _) (eql-implies-eq/L2 _ _ _ _) %. +%total (M1 M2) (eql-implies-eq/L2 _ M2 _ _) (eql-implies-eq/L1 _ M1 _ _) %. +%term _ %pi (eql-implies-eq E E') %<- (eqlsize-total ES) %<- (eql-implies-eq/L1 E _ ES E') %. +%worlds () (eql-implies-eq _ _) %. +%total {} (eql-implies-eq _ _) %. +%{! ### Main Theorem !}% +%{! +Finally the statement of the main theorem of this section. +As with the non-indexed case, it is proved by using eql as a between station. +Again, as with the non-indexed code, this theorem (or rather +its main lemma) uses reasoning-from-false extensively because the +cases cannot be distinguished by Twelf's case analysis. +It also uses the "divrem" part of the nat signature extensively +as well as theorems about plus and times. The proofs of the cases are +uninteresting arithmetic fiddling. +!}% +%theorem tonat-one2one : forall* {N} {T1 term N} {N1} {T2 term N} {N2} forall {TN1 tonat T1 N1} {TN2 tonat T2 N2} {E nat`eq N1 N2} exists {ET eq T1 T2} true %. +%theorem tonat-one2one/L : forall* {N} {MS} {T1} {M1} {T2} {M2} forall {TN1 tonat* N MS T1 M1} {TN2 tonat* N MS T2 M2} {E nat`eq M1 M2} exists {ET eql* N z MS T1 T2} true %. +%term _ + %pi (tonat-one2one/L (tonat/var MC1 VL1 P1) (tonat/var MC2 VL2 P2) nat`eq/ (eql/var plus/z VL1' VL2)) + %<- (count-deterministic MC1 MC2 multiset`eq/ nat`eq/ VN1=VN2) + %<- (plus-left-cancels P1 P2 nat`eq/ VN1=VN2 L1=L2) + %<- (varlevel-respects-eq VL1 L1=L2 VL1') %. +%term _ + %pi (tonat-one2one/L (tonat/var MC VL M+L=VN) (tonat/lit MC' _ VN'+X=M) nat`eq/ E) + %<- (count-deterministic MC' MC multiset`eq/ nat`eq/ VN'=VN) + %<- (plus-respects-eq VN'+X=M VN'=VN nat`eq/ nat`eq/ VN+X=M) + %<- (plus-commutative VN+X=M X+VN=M) + %<- (plus-commutative M+L=VN L+M=VN) + %<- (plus-implies-ge X+VN=M M>=VN) + %<- (plus-implies-ge L+M=VN VN>=M) + %<- (nat`ge-anti-symmetric M>=VN VN>=M M=VN) + %<- (plus-right-cancels L+M=VN plus/z M=VN nat`eq/ L=0) + %<- (varlevel-contradiction VL L=0 F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/lit MC' _ VN'+X=M) (tonat/var MC VL M+L=VN) nat`eq/ E) + %<- (count-deterministic MC' MC multiset`eq/ nat`eq/ VN'=VN) + %<- (plus-respects-eq VN'+X=M VN'=VN nat`eq/ nat`eq/ VN+X=M) + %<- (plus-commutative VN+X=M X+VN=M) + %<- (plus-commutative M+L=VN L+M=VN) + %<- (plus-implies-ge X+VN=M M>=VN) + %<- (plus-implies-ge L+M=VN VN>=M) + %<- (nat`ge-anti-symmetric M>=VN VN>=M M=VN) + %<- (plus-right-cancels L+M=VN plus/z M=VN nat`eq/ L=0) + %<- (varlevel-contradiction VL L=0 F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/var MC VL M+L=VN) (tonat/app MC' _ _ _ _ VN'+X=M) nat`eq/ E) + %<- (count-deterministic MC' MC multiset`eq/ nat`eq/ VN'=VN) + %<- (plus-respects-eq VN'+X=M VN'=VN nat`eq/ nat`eq/ VN+X=M) + %<- (plus-commutative VN+X=M X+VN=M) + %<- (plus-commutative M+L=VN L+M=VN) + %<- (plus-implies-ge X+VN=M M>=VN) + %<- (plus-implies-ge L+M=VN VN>=M) + %<- (nat`ge-anti-symmetric M>=VN VN>=M M=VN) + %<- (plus-right-cancels L+M=VN plus/z M=VN nat`eq/ L=0) + %<- (varlevel-contradiction VL L=0 F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/app MC' _ _ _ _ VN'+X=M) (tonat/var MC VL M+L=VN) nat`eq/ E) + %<- (count-deterministic MC' MC multiset`eq/ nat`eq/ VN'=VN) + %<- (plus-respects-eq VN'+X=M VN'=VN nat`eq/ nat`eq/ VN+X=M) + %<- (plus-commutative VN+X=M X+VN=M) + %<- (plus-commutative M+L=VN L+M=VN) + %<- (plus-implies-ge X+VN=M M>=VN) + %<- (plus-implies-ge L+M=VN VN>=M) + %<- (nat`ge-anti-symmetric M>=VN VN>=M M=VN) + %<- (plus-right-cancels L+M=VN plus/z M=VN nat`eq/ L=0) + %<- (varlevel-contradiction VL L=0 F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/var MC VL M+L=VN) (tonat/lam MC' _ _ _ _ VN'+X=M) nat`eq/ E) + %<- (count-deterministic MC' MC multiset`eq/ nat`eq/ VN'=VN) + %<- (plus-respects-eq VN'+X=M VN'=VN nat`eq/ nat`eq/ VN+X=M) + %<- (plus-commutative VN+X=M X+VN=M) + %<- (plus-commutative M+L=VN L+M=VN) + %<- (plus-implies-ge X+VN=M M>=VN) + %<- (plus-implies-ge L+M=VN VN>=M) + %<- (nat`ge-anti-symmetric M>=VN VN>=M M=VN) + %<- (plus-right-cancels L+M=VN plus/z M=VN nat`eq/ L=0) + %<- (varlevel-contradiction VL L=0 F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/lam MC' _ _ _ _ VN'+X=M) (tonat/var MC VL M+L=VN) nat`eq/ E) + %<- (count-deterministic MC' MC multiset`eq/ nat`eq/ VN'=VN) + %<- (plus-respects-eq VN'+X=M VN'=VN nat`eq/ nat`eq/ VN+X=M) + %<- (plus-commutative VN+X=M X+VN=M) + %<- (plus-commutative M+L=VN L+M=VN) + %<- (plus-implies-ge X+VN=M M>=VN) + %<- (plus-implies-ge L+M=VN VN>=M) + %<- (nat`ge-anti-symmetric M>=VN VN>=M M=VN) + %<- (plus-right-cancels L+M=VN plus/z M=VN nat`eq/ L=0) + %<- (varlevel-contradiction VL L=0 F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/var MC VL M+L=VN) (tonat/rec MC' _ _ _ VN'+X=M) nat`eq/ E) + %<- (count-deterministic MC' MC multiset`eq/ nat`eq/ VN'=VN) + %<- (plus-respects-eq VN'+X=M VN'=VN nat`eq/ nat`eq/ VN+X=M) + %<- (plus-commutative VN+X=M X+VN=M) + %<- (plus-commutative M+L=VN L+M=VN) + %<- (plus-implies-ge X+VN=M M>=VN) + %<- (plus-implies-ge L+M=VN VN>=M) + %<- (nat`ge-anti-symmetric M>=VN VN>=M M=VN) + %<- (plus-right-cancels L+M=VN plus/z M=VN nat`eq/ L=0) + %<- (varlevel-contradiction VL L=0 F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/rec MC' _ _ _ VN'+X=M) (tonat/var MC VL M+L=VN) nat`eq/ E) + %<- (count-deterministic MC' MC multiset`eq/ nat`eq/ VN'=VN) + %<- (plus-respects-eq VN'+X=M VN'=VN nat`eq/ nat`eq/ VN+X=M) + %<- (plus-commutative VN+X=M X+VN=M) + %<- (plus-commutative M+L=VN L+M=VN) + %<- (plus-implies-ge X+VN=M M>=VN) + %<- (plus-implies-ge L+M=VN VN>=M) + %<- (nat`ge-anti-symmetric M>=VN VN>=M M=VN) + %<- (plus-right-cancels L+M=VN plus/z M=VN nat`eq/ L=0) + %<- (varlevel-contradiction VL L=0 F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/lit MC1 T1 P1) (tonat/lit MC2 T2 P2) nat`eq/ (eql/eq T1=T2)) + %<- (count-deterministic MC1 MC2 multiset`eq/ nat`eq/ VN1=VN2) + %<- (plus-left-cancels P1 P2 VN1=VN2 nat`eq/ TM1=TM2) + %<- (times-left-cancels T1 T2 nat`eq/ TM1=TM2 M1=M2) + %<- (lit-preserves-eq M1=M2 T1=T2) %. +%inline 2>1 gt/1 %. +%term _ + %pi (tonat-one2one/L (tonat/lit MC T P) (tonat/app MC' _ _ _ T' P') nat`eq/ E) + %<- (count-deterministic MC' MC multiset`eq/ nat`eq/ VN'=VN) + %<- (plus-left-cancels P' P VN'=VN nat`eq/ TM'+1=TM) + %<- (times-commutative T Tc) + %<- (div-can-be-constructed Tc DR) + %<- (plus-commutative (plus/s plus/z) TM'+ONE=TM'+1) + %<- (plus-respects-eq TM'+ONE=TM'+1 nat`eq/ nat`eq/ TM'+1=TM TM'+ONE=TM) + %<- (times-commutative T' Tc') + %<- (divrem-can-be-constructed Tc' TM'+ONE=TM 2>1 DR') + %<- (divrem-deterministic DR DR' nat`eq/ nat`eq/ QE ZERO=ONE) + %<- (nat`eq-contradiction ZERO=ONE F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/app MC' _ _ _ T' P') (tonat/lit MC T P) nat`eq/ E) + %<- (count-deterministic MC' MC multiset`eq/ nat`eq/ VN'=VN) + %<- (plus-left-cancels P' P VN'=VN nat`eq/ TM'+1=TM) + %<- (times-commutative T Tc) + %<- (div-can-be-constructed Tc DR) + %<- (plus-commutative (plus/s plus/z) TM'+ONE=TM'+1) + %<- (plus-respects-eq TM'+ONE=TM'+1 nat`eq/ nat`eq/ TM'+1=TM TM'+ONE=TM) + %<- (times-commutative T' Tc') + %<- (divrem-can-be-constructed Tc' TM'+ONE=TM 2>1 DR') + %<- (divrem-deterministic DR DR' nat`eq/ nat`eq/ QE ZERO=ONE) + %<- (nat`eq-contradiction ZERO=ONE F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/app MC1 TN1a TN1b P2N1 T1 P1) (tonat/app MC2 TN2a TN2b P2N2 T2 P2) nat`eq/ (eql/app E1 E2)) + %<- (count-deterministic MC1 MC2 multiset`eq/ nat`eq/ VN1=VN2) + %<- (plus-left-cancels P1 P2 VN1=VN2 nat`eq/ TM1+1=TM2+1) + %<- (succ-cancels TM1+1=TM2+1 TM1=TM2) + %<- (times-left-cancels T1 T2 nat`eq/ TM1=TM2 (%the (nat`eq M1 M2) M1=M2)) + %<- (nat2pair-deterministic P2N1 P2N2 M1=M2 PR1=PR2) + %<- (natpair`pair-eq-implies-eq PR1=PR2 EQa EQb) + %<- (tonat-one2one/L TN1a TN2a EQa E1) + %<- (tonat-one2one/L TN1b TN2b EQb E2) %. +%inline 2+2=4 (plus (s (s z)) (s (s z)) (s (s (s (s z))))) plus/s (plus/s plus/z) %. +%inline 2x2=4 (times (s (s z)) (s (s z)) (s (s (s (s z))))) times/s (times/s times/z plus/z) 2+2=4 %. +%term _ + %pi (tonat-one2one/L (tonat/app MC1 TN1a TN1b P2N1 T1 P1) (tonat/lam MC2 MC0 MA0 _ T2 P2) nat`eq/ E) + %<- (count-deterministic MC1 MC2 multiset`eq/ nat`eq/ VN1=VN2) + %<- (plus-left-cancels P1 P2 VN1=VN2 nat`eq/ TM1+1=TM2) + %<- (times-associative 2x2=4 T2 _ T2' T2'') + %<- (times-commutative T2'' T2''c) + %<- (div-can-be-constructed T2''c DR2) + %<- (plus-commutative (plus/s plus/z) TM1+ONE=TM1+1) + %<- (plus-respects-eq TM1+ONE=TM1+1 nat`eq/ nat`eq/ TM1+1=TM2 TM1+ONE=TM2) + %<- (times-commutative T1 T1c) + %<- (divrem-can-be-constructed T1c TM1+ONE=TM2 2>1 DR1) + %<- (divrem-deterministic DR2 DR1 nat`eq/ nat`eq/ QE ZERO=ONE) + %<- (nat`eq-contradiction ZERO=ONE F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/lam MC2 MC0 MA0 _ T2 P2) (tonat/app MC1 TN1a TN1b P2N1 T1 P1) nat`eq/ E) + %<- (count-deterministic MC1 MC2 multiset`eq/ nat`eq/ VN1=VN2) + %<- (plus-left-cancels P1 P2 VN1=VN2 nat`eq/ TM1+1=TM2) + %<- (times-associative 2x2=4 T2 _ T2' T2'') + %<- (times-commutative T2'' T2''c) + %<- (div-can-be-constructed T2''c DR2) + %<- (plus-commutative (plus/s plus/z) TM1+ONE=TM1+1) + %<- (plus-respects-eq TM1+ONE=TM1+1 nat`eq/ nat`eq/ TM1+1=TM2 TM1+ONE=TM2) + %<- (times-commutative T1 T1c) + %<- (divrem-can-be-constructed T1c TM1+ONE=TM2 2>1 DR1) + %<- (divrem-deterministic DR2 DR1 nat`eq/ nat`eq/ QE ZERO=ONE) + %<- (nat`eq-contradiction ZERO=ONE F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/app MC1 _ _ _ T1 P1) (tonat/rec MC2 _ _ T2 P2) nat`eq/ E) + %<- (count-deterministic MC1 MC2 multiset`eq/ nat`eq/ VN1=VN2) + %<- (plus-left-cancels P1 P2 VN1=VN2 nat`eq/ TM1+1=TM2+2) + %<- (times-associative 2x2=4 T2 _ _ T2') + %<- (times-commutative T2' T2'c) + %<- (plus-commutative (plus/s (plus/s plus/z)) P2'c) + %<- (div-can-be-constructed (times/s T2'c P2'c) DR2) + %<- (plus-commutative (plus/s plus/z) TM1+ONE=TM1+1) + %<- (plus-respects-eq TM1+ONE=TM1+1 nat`eq/ nat`eq/ TM1+1=TM2+2 TM1+ONE=TM2+2) + %<- (times-commutative T1 T1c) + %<- (divrem-can-be-constructed T1c TM1+ONE=TM2+2 2>1 DR1) + %<- (divrem-deterministic DR2 DR1 nat`eq/ nat`eq/ QE ZERO=ONE) + %<- (nat`eq-contradiction ZERO=ONE F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/rec MC2 _ _ T2 P2) (tonat/app MC1 _ _ _ T1 P1) nat`eq/ E) + %<- (count-deterministic MC1 MC2 multiset`eq/ nat`eq/ VN1=VN2) + %<- (plus-left-cancels P1 P2 VN1=VN2 nat`eq/ TM1+1=TM2+2) + %<- (times-associative 2x2=4 T2 _ _ T2') + %<- (times-commutative T2' T2'c) + %<- (plus-commutative (plus/s (plus/s plus/z)) P2'c) + %<- (div-can-be-constructed (times/s T2'c P2'c) DR2) + %<- (plus-commutative (plus/s plus/z) TM1+ONE=TM1+1) + %<- (plus-respects-eq TM1+ONE=TM1+1 nat`eq/ nat`eq/ TM1+1=TM2+2 TM1+ONE=TM2+2) + %<- (times-commutative T1 T1c) + %<- (divrem-can-be-constructed T1c TM1+ONE=TM2+2 2>1 DR1) + %<- (divrem-deterministic DR2 DR1 nat`eq/ nat`eq/ QE ZERO=ONE) + %<- (nat`eq-contradiction ZERO=ONE F) + %<- (false-implies-eql F E) %. +%theorem tonat-one2one/L/eq : forall* {N} {VN1} {VN2} {N1} {MS1} {MS2} {M} {F} forall {FTN {v term N} %pi (varlevel v (s VN1)) %-> (tonat* N1 MS1 (F v) M)} {EQ nat`eq VN1 VN2} {ME multiset`eq MS1 MS2} exists {FTN' {v term N} %pi (varlevel v (s VN2)) %-> (tonat* N1 MS2 (F v) M)} true %. +%term _ tonat-one2one/L/eq FTN nat`eq/ multiset`eq/ FTN %. +%worlds (blocksimple blockvar) (tonat-one2one/L/eq _ _ _ _) %. +%total {} (tonat-one2one/L/eq _ _ _ _) %. +%reduces = F1 F2 (tonat-one2one/L/eq F1 _ _ F2) %. +%term _ + %pi (tonat-one2one/L (tonat/lam MC1 MC01 MA1 ([v] [vl] F1 v vl) T1 P1) (tonat/lam MC2 MC02 MA2 ([v] [vl] F2 v vl) T2 P2) nat`eq/ (eql/lam2 MC02 MA2 FE)) + %<- (count-deterministic MC1 MC2 multiset`eq/ nat`eq/ VN1=VN2) + %<- (plus-left-cancels P1 P2 VN1=VN2 nat`eq/ TM1=TM2) + %<- (times-left-cancels T1 T2 nat`eq/ TM1=TM2 M1=M2) + %<- (count-deterministic MC01 MC02 multiset`eq/ nat`eq/ ZN1=ZN2) + %<- (add-deterministic MA1 MA2 multiset`eq/ nat`eq/ MS1=MS2) + %<- (tonat-one2one/L/eq F1 ZN1=ZN2 MS1=MS2 F1') + %<- ({v} {vl varlevel v (s ZN2)} tonat-one2one/L (F1' v vl) (F2 v vl) M1=M2 (FE v vl)) %. +%inline 4>2 (nat`gt (s (s (s (s z)))) (s (s z))) gt/> gt/1 %. +%term _ + %pi (tonat-one2one/L (tonat/lam MC1 _ _ _ T1 P1) (tonat/rec MC2 _ _ T2 P2) nat`eq/ E) + %<- (count-deterministic MC2 MC1 multiset`eq/ nat`eq/ VN2=VN1) + %<- (plus-left-cancels P2 P1 VN2=VN1 nat`eq/ TM2+2=TM1) + %<- (times-commutative T1 T1c) + %<- (div-can-be-constructed T1c DR1) + %<- (plus-commutative (plus/s (plus/s plus/z)) TM2+TWO=TM2+2) + %<- (plus-respects-eq TM2+TWO=TM2+2 nat`eq/ nat`eq/ TM2+2=TM1 TM2+TWO=TM1) + %<- (times-commutative T2 T2c) + %<- (divrem-can-be-constructed T2c TM2+TWO=TM1 4>2 DR2) + %<- (divrem-deterministic DR1 DR2 nat`eq/ nat`eq/ QE ZERO=TWO) + %<- (nat`eq-contradiction ZERO=TWO F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/rec MC2 _ _ T2 P2) (tonat/lam MC1 _ _ _ T1 P1) nat`eq/ E) + %<- (count-deterministic MC2 MC1 multiset`eq/ nat`eq/ VN2=VN1) + %<- (plus-left-cancels P2 P1 VN2=VN1 nat`eq/ TM2+2=TM1) + %<- (times-commutative T1 T1c) + %<- (div-can-be-constructed T1c DR1) + %<- (plus-commutative (plus/s (plus/s plus/z)) TM2+TWO=TM2+2) + %<- (plus-respects-eq TM2+TWO=TM2+2 nat`eq/ nat`eq/ TM2+2=TM1 TM2+TWO=TM1) + %<- (times-commutative T2 T2c) + %<- (divrem-can-be-constructed T2c TM2+TWO=TM1 4>2 DR2) + %<- (divrem-deterministic DR1 DR2 nat`eq/ nat`eq/ QE ZERO=TWO) + %<- (nat`eq-contradiction ZERO=TWO F) + %<- (false-implies-eql F E) %. +%term _ + %pi (tonat-one2one/L (tonat/rec MC1 MA1 ([v] [vl] F1 v vl) T1 P1) (tonat/rec MC2 MA2 ([v] [vl] F2 v vl) T2 P2) nat`eq/ (eql/rec2 plus/z MC2 MA2 FE)) + %<- (count-deterministic MC1 MC2 multiset`eq/ nat`eq/ VN1=VN2) + %<- (plus-left-cancels P1 P2 VN1=VN2 nat`eq/ TM1+2=TM2+2) + %<- (succ-cancels TM1+2=TM2+2 TM1+1=TM2+1) + %<- (succ-cancels TM1+1=TM2+1 TM1=TM2) + %<- (times-left-cancels T1 T2 nat`eq/ TM1=TM2 M1=M2) + %<- (add-deterministic MA1 MA2 multiset`eq/ nat`eq/ MS1=MS2) + %<- (tonat-one2one/L/eq F1 VN1=VN2 MS1=MS2 F1') + %<- ({v} {vl varlevel v (s VN2)} tonat-one2one/L (F1' v vl) (F2 v vl) M1=M2 (FE v vl)) %. +%worlds (blockvar) (tonat-one2one/L _ _ _ _) %. +%total (T) (tonat-one2one/L T _ _ _) %. +%term _ + %pi (tonat-one2one TN1 TN2 EQ TEQ) + %<- (tonat-one2one/L TN1 TN2 EQ EQL) + %<- (eql-implies-eq EQL TEQ) %. +%worlds () (tonat-one2one _ _ _ _) %. +%total {} (tonat-one2one _ _ _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Indexed_HOAS_nat_bijection). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/indexed-lists.lf b/new-tests/stelf-output/pages/indexed-lists.lf new file mode 100644 index 0000000..73d2e48 --- /dev/null +++ b/new-tests/stelf-output/pages/indexed-lists.lf @@ -0,0 +1,88 @@ +% +%{! !}% +%{! (options removed from twelftag: check=true) !}% +% Lists +% By Carsten Varming 2006 +%sort tp %. +%name tp %. +%sort list %. +%name list %. +%term stuff tp %. +%freeze tp %. +%term cons {Tp} %pi (list Tp) %-> (list Tp) %. +%term nil {Tp} list Tp %. +%freeze list %. +%sort eq %. +%mode eq %in %out %. +%term eq_ref eq Ls Ls %. +%worlds () (eq _ _) %. +%freeze eq %. +%sort eq_symm %. +%mode eq_symm %in %out %. +%term eq_symm_rule eq_symm eq_ref eq_ref %. +%worlds () (eq_symm _ _) %. +%freeze eq_symm %. +%total {} (eq_symm _ _) %. +%sort eq_trans %. +%mode eq_trans %in %in %out %. +%term eq_trans_rule eq_trans eq_ref eq_ref eq_ref %. +%worlds () (eq_trans _ _ _) %. +%freeze eq_trans %. +%total {} (eq_trans _ _ _) %. +%sort rev %. +%mode rev %in %in %out %. +%term rev_nil rev (nil _) Ls' Ls' %. +%term rev_cons %pi (rev (cons E Ls) Ls'' Ls') %<- (rev Ls (cons E Ls'') Ls') %. +%worlds () (rev _ _ _) %. +%freeze rev %. +%total D (rev D _ _) %. +%sort rev_exists {Ls} {Ls'} %. +%mode rev_exists %in %in %out %. +%term rev_exists_nil rev_exists (nil _) _ rev_nil %. +%term rev_exists_cons + %pi (rev_exists (cons E Ls) Ls' (rev_cons Ls'')) + %<- (rev_exists Ls (cons E Ls') Ls'') %. +%worlds () (rev_exists _ _ _) %. +%freeze rev_exists %. +%total D (rev_exists D _ _) %. +%sort revDet %. +%mode revDet %in %in %out %. +%term revDet_nil revDet rev_nil _ eq_ref %. +%term revDet_cons %pi (revDet (rev_cons R) (rev_cons R') Q) %<- (revDet R R' Q) %. +%worlds () (revDet _ _ _) %. +%freeze revDet %. +%total D (revDet D _ _) %. +%sort revrev_id_lem %. +%mode revrev_id_lem %in %in %in %out %. +%term revrev_id_lem_nil %pi (revrev_id_lem rev_nil F F' Q) %<- (revDet F' F Q) %. +%term revrev_id_lem_cons + %pi (revrev_id_lem (rev_cons R) R' R'' Q) + %<- (revrev_id_lem R R' (rev_cons R'') Q) %. +%worlds () (revrev_id_lem _ _ _ _) %. +%freeze revrev_id_lem %. +%total D (revrev_id_lem D _ _ _) %. +%sort revrev_id %. +%mode revrev_id %in %in %out %. +%term revrev_id_rule %pi (revrev_id R R' Q) %<- (revrev_id_lem R R' rev_nil Q) %. +%worlds () (revrev_id _ _ _) %. +%freeze revrev_id %. +%total {} (revrev_id _ _ _) %. +%sort rev_injective %. +%mode rev_injective %in %in %out %. +%term rev_injective_rule + %pi (rev_injective (%the (rev Ls (nil Tp) Ls') R) R' Q) + %<- (rev_exists Ls' (nil Tp) Rev) + %<- (revrev_id R Rev Q') + %<- (eq_symm Q' Q''') + %<- (revrev_id R' Rev Q'') + %<- (eq_trans Q'' Q''' Q1) + %<- (eq_symm Q1 Q) %. +%worlds () (rev_injective _ _ _) %. +%freeze rev_injective %. +%total D (rev_injective D _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Indexed_lists). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/intrinsic-and-extrinsic-encodings.lf b/new-tests/stelf-output/pages/intrinsic-and-extrinsic-encodings.lf new file mode 100644 index 0000000..251bc58 --- /dev/null +++ b/new-tests/stelf-output/pages/intrinsic-and-extrinsic-encodings.lf @@ -0,0 +1,78 @@ +% +%{! It often comes to pass that you have a judgement and a second judgement which picks out some of the subjects of the first. For instance, terms and values, or terms and well-typed terms, or finite maps and well-formed finite maps (e.g. maps without duplicate keys). Depending on what you're doing, you may then have a choice as to whether to encode the judgements _intrinsically_ or _extrinsically_. + +In an extrinsic encoding, the second judgement is represented as a refinement of the first: a type family which is indexed by the type family representing the first judgement. For example, the simply-typed lambda calculus, extrinsically: !}% +%sort tm %. +%term lam %pi (%pi tm %-> tm) %-> tm %. +%term app %pi tm %-> tm %-> tm %. +%sort tp %. +%term arr %pi tp %-> tp %-> tp %. +%sort of %. +%term of-lam %pi ({x tm} {d of x T} of (B x) T') %-> (of (lam B) (arr T T')) %. +%term of-app %pi (of M' (arr T T')) %-> (of M T) %-> (of (app M' M) T') %. +%{! In an intrinsic encoding, the second judgement is rolled into the definition of the first; there is no representation of subjects of the first judgement which are not subjects of the second. For example, the simply-typed lambda calculus, intrinsically: !}% +%sort tp %. +%term arr %pi tp %-> tp %-> tp %. +%sort tm %. +%term lam %pi (%pi (tm T) %-> (tm T')) %-> (tm (arr T T')) %. +%term app %pi (tm (arr T T')) %-> (tm T) %-> (tm T') %. +%{! Strictly speaking this is not just a different encoding; it adequately encodes only the typing judgement, since there are ill-typed terms which can be written down in the previous signature but not in this one. + + + +You may wish to work with both judgements (for instance, terms and values). In that case you have the choice of representing values extrinsically, as a judgement indexed by terms, or intrinsically, as a separate judgement. Extrinsically: !}% +%sort tm %. +%term lam %pi (%pi tm %-> tm) %-> tm %. +%term app %pi tm %-> tm %-> tm %. +%sort val %. +%term val-lam val (lam B) %. +%{! Intrinsically: !}% +%sort tm %. +%sort val %. +%term lam %pi (%pi val %-> tm) %-> val %. +%term app %pi tm %-> tm %-> tm %. +%term inj %pi val %-> tm %. +%{! In this representation values must be explicitly injected into terms with ``inj``. (Note that taking ``val`` as argument to the body of ``lam`` makes the language call-by-value.) An advantage of the intrinsic encoding is that you need only pass around a single ``val`` derivation to operate on values, while in the extrinsic encoding you must pass both a ``tm`` and a ``value``. A disadvantage is that you have to deal with the term/value distinction everywhere you work with terms. + +Another possibility is to have a completely separate secondary judgement and inject its subjects into the primary judgement via a metatheorem rather than a constructor: !}% +%sort tm %. +%term lam %pi (%pi tm %-> tm) %-> tm %. +%term app %pi tm %-> tm %-> tm %. +%sort val %. +%term val-lam %pi (%pi tm %-> tm) %-> val %. +%{! As a final example, consider finite maps. On paper you typically work with finite maps which are implicitly well-defined (they have at most one binding for a key). To encode them as LF terms in Twelf you have the choice of an extrinsic encoding, in which there is a separate well-formedness judgement, or an intrinsic one, in which only well-formed maps can be written down. Extrinsically: !}% +%sort key %. +% pretend that key and value have some inhabitants +%sort value %. +%sort map %. +%term empty map %. +%term bind %pi map %-> key %-> value %-> map %. +%sort key-neq %. +% holds if keys are not equal +%sort key-not-in-map %. +%term _ key-not-in-map K empty %. +%term _ %pi (key-not-in-map K (bind M K' V)) %<- (key-neq K K') %<- (key-not-in-map K M) %. +%sort wf-map %. +%term _ wf-map empty %. +%term _ %pi (wf-map (bind M K V)) %<- (key-not-in-map K M) %<- (wf-map M) %. +%{! Intrinsically: !}% +%sort key %. +% pretend that key and value have some inhabitants +%sort value %. +%sort map %. +%sort key-neq %. +% holds if keys are not equal +%term empty map %. +%sort key-not-in-map %. +%term bind {M map} {K key} %pi value %-> (key-not-in-map K M) %-> map %. +%term _ key-not-in-map K empty %. +%term _ %pi (key-not-in-map K (bind M K' V _)) %<- (key-neq K K') %<- (key-not-in-map K M) %. +%{! Here the extrinsic encoding has the advantage that you need not prove that well-formedness is maintained as you go (or even maintain it), which may save a lot of work. The intrinsic encoding has the advantage that since only well-formed maps can be written down, you never need to prove that a type-family maintains well-formedness as an invariant, which may save a lot of work. + +For an example of an intrinsic encoding, see [CPS conversion](/wiki/cps-conversion/), where the distinction between values and expressions is encoded intrinsically, and where all terms are manifestly well-typed. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Intrinsic_and_extrinsic_encodings). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/introductions-to-twelf.lf b/new-tests/stelf-output/pages/introductions-to-twelf.lf new file mode 100644 index 0000000..716ee44 --- /dev/null +++ b/new-tests/stelf-output/pages/introductions-to-twelf.lf @@ -0,0 +1,60 @@ +% +% +%{! + +### Recommended + +- Our recommended introduction to Twelf is [Proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/). If you have some background in programming languages but no prior experience with LF and Twelf, then _Proving metatheorems_ is probably right for you! + +### Other tutorials + +- Andrew Appel's [Hints on Proving Theorems in + Twelf](http://www.cs.princeton.edu/~appel/twelf-tutorial/) describes a + particular methodology for using Twelf that is rather different than the + strategy of proving [metatheorems](/wiki/metatheorem/) that predominates on + this wiki (also known as "the particular strange way they do it at + Princeton"). +- John Boyland's [Using Twelf to Prove Type + Theorems](https://cs.nju.edu.cn/_upload/tpl/00/aa/170/template170/proof/using-twelf.pdf) + is a tutorial and experience report. +- Dan Licata and Bob Harper have written a survey article called [Mechanizing + Metatheory in a Logical Framework](http://www.cs.cmu.edu/~drl/pubs/hl06mechanizing/hl06mechanizing.pdf). + This paper provides a more formal introduction to the modern way of thinking + about LF and Twelf than this site does. +- Alberto Momigliano's [A Practical Approach to Co-induction in + Twelf](http://www.cs.nott.ac.uk/types06/slides/am.pdf) describes a technique + for encoding Twelf-unfriendly co-inductive proofs as Twelf-friendly induction + proofs (slides from a talk at TYPES 2006). +- John Altidor's + [tutorial](http://jgaltidor.github.io/typetheory_paper.pdf) and + [slides](http://jgaltidor.github.io/twelf_slides.pdf) are accessible to folks + without a strong background in programming language foundations. + +### Experience reports and commentary + +- [The POPLmark Challenge](https://www.seas.upenn.edu/~plclub/poplmark/) has + commentary on the [POPLmark submission from Carnegie + Mellon](https://www.seas.upenn.edu/~plclub/poplmark/cmu.html) that uses Twelf. +- Andrew Appel and Xavier Leroy's [A list-machine benchmark for mechanized + metatheory](https://www.cs.princeton.edu/~appel/listmachine/) serves as both + a tutorial and an experience report on using Twelf to prove theroems about + compilers. + +### Curriculum from past Twelf tutorials + +- [POPL Tutorial](/wiki/popl-tutorial/) 2009: course materials from a Twelf + tutorial at POPL 2009. This path through the material is the best + introduction to Twelf, but it may be harder to follow along with online than + [Proving metatheorems](/wiki/proving-metatheorems-with-twelf/). +- [Summer school 2008](/wiki/summer-school-2008/): notes from a Twelf course + at [University of Oregon Summer School on Logic and Theorem Proving in + Programming Languages, July + 2008](http://www.cs.uoregon.edu/research/summerschool/summer08/). + +### Twelf user's guide + +The Twelf [User's Guide](/wiki/users-guide/) was the basic reference manual +for Twelf prior to the Twelf Wiki, and is still the authoritative source for +some topics. + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/iterated-inductive-definitions-and-defunctionalization.lf b/new-tests/stelf-output/pages/iterated-inductive-definitions-and-defunctionalization.lf new file mode 100644 index 0000000..f6206ad --- /dev/null +++ b/new-tests/stelf-output/pages/iterated-inductive-definitions-and-defunctionalization.lf @@ -0,0 +1,65 @@ +% +%%%%%%%%%%%% +% RUN-TIME % +%%%%%%%%%%%% +%sort i %. +%term z i %. +%term s %pi i %-> i %. +%prec %prefix 10 s %. +%sort add %. +%term add/z add z N N %. +%term add/s %pi (add (s M) N (s P)) %<- (add M N P) %. +%mode add %in %in %out %. +%worlds () (add M _ _) %. +%total (M) (add M _ _) %. +%unique add %in %in %out %. +%sort mult %. +%term mult/z mult z N z %. +%term mult/s %pi (mult (s M) N P') %<- (mult M N P) %<- (add P N P') %. +%mode mult %in %in %out %. +%worlds () (mult M _ _) %. +%total (M) (mult M _ _) %. +%unique mult %in %in %out %. +%%%%%%%%%%%% +% SYNTAX % +%%%%%%%%%%%% +%sort tm %. +%sort binop %. +%term n %pi i %-> tm %. +%term let %pi tm %-> (%pi tm %-> tm) %-> tm %. +%term @ %pi binop %-> tm %-> tm %-> tm %. +%sort apply %. +%mode apply %in %in %in %out %. +%worlds () (apply F M1 M2 _) %. +%total {F M1 M2} (apply F M1 M2 _) %. +%unique apply %in %in %in %out %. +%%%%%%%%%%%%% +% SEMANTICS % +%%%%%%%%%%%%% +%sort eval %. +%mode eval %in %out %. +%term eval/n eval (n N) N %. +%term eval/let %pi (eval (let E E*) N) %<- (eval (E* E) N) %. +%term eval/@ %pi (eval (@ F E1 E2) N) %<- (eval E1 M1) %<- (eval E2 M2) %<- (apply F M1 M2 N) %. +%worlds () (eval E _) %. +%covers eval %in %out %. +%unique eval %in %out %. +%%%%%%%%%%%% +% EXAMPLES % +%%%%%%%%%%%% +%term plus binop %. +%term plus/_ %pi (apply plus M N P) %<- (add M N P) %. +%term times binop %. +%term times/_ %pi (apply times M N P) %<- (mult M N P) %. +%total (M1) (apply F M1 M2 _) %. +%unique apply %in %in %in %out %. +%worlds () (eval E _) %. +%covers eval %in %out %. +%unique eval %in %out %. +%query 1 _ _ eval (let (n (s s z)) ([two] let (n (s s s z)) ([three] @ plus two (@ times two three)))) N %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Iterated_inductive_definitions_and_defunctionalization). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/iterated-let-bindings.lf b/new-tests/stelf-output/pages/iterated-let-bindings.lf new file mode 100644 index 0000000..9f45c48 --- /dev/null +++ b/new-tests/stelf-output/pages/iterated-let-bindings.lf @@ -0,0 +1,78 @@ +% +%{! + +Let with iterated bindings. + +## Preliminaries +!}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort add %. +%mode add %in %in %out %. +%term add/z add z N N %. +%term add/s %pi (add (s N) M (s P)) %<- (add N M P) %. +%worlds () (add _ _ _) %. +%total M (add M _ _) %. +%{! +## Syntax +!}% +%sort exp %. +%% oexp N is morally exp^N -> exp, but curried +%sort oexp %. +%% binds N M is a list of length M - N with types +%% (oexp N , oexp (s N) , ... , oexp (M - 1)) +%sort binds %. +%term oexp/done %pi exp %-> (oexp z) %. +%term oexp/bind %pi (%pi exp %-> (oexp N)) %-> (oexp (s N)) %. +%term binds/done binds N N %. +%term binds/cons %pi (oexp F) %-> (binds (s F) L) %-> (binds F L) %. +%term let* %pi (binds z N) %-> (oexp N) %-> exp %. +%term num %pi nat %-> exp %. +%term plus %pi exp %-> exp %-> exp %. +%{! +## Evaluation +!}% +%% substitute the expression for the first var in each of the binds +%sort map-subst %. +%mode map-subst %in %in %out %. +%term _ map-subst E binds/done binds/done %. +%term _ + %pi (map-subst E (binds/cons (oexp/bind OE) B) (binds/cons (OE E) B')) + %<- (map-subst E B B') %. +%worlds () (map-subst _ _ _) %. +%total D (map-subst _ D _) %. +%% contradict the existence of binds (s N) z; +%% return a nat +%sort binds-imposs %. +%mode binds-imposs %in %out %. +%term _ %pi (binds-imposs (binds/cons _ B) X) %<- (binds-imposs B X) %. +%worlds () (binds-imposs _ _) %. +%total D (binds-imposs D _) %. +%sort eval %. +%mode eval %in %out %. +%term _ %pi (eval (let* binds/done (oexp/done E)) N) %<- (eval E N) %. +%term _ + %pi (eval (let* (binds/cons (oexp/done E) Bs) (oexp/bind E')) N) + %<- (map-subst E Bs Bs') + %<- (eval (let* Bs' (E' E)) N) %. +%term _ %pi (eval (plus E1 E2) N) %<- (eval E1 N1) %<- (eval E2 N2) %<- (add N1 N2 N) %. +%term _ eval (num N) N %. +%% silly contradictory case so it coverage checks +%term _ %pi (eval (let* (binds/cons _ B) (oexp/done E)) X) %<- (binds-imposs B X) %. +%worlds () (eval _ _) %. +%covers eval %in %out %. +%{! +## Example +!}% +%% let* x = 3 +%% y = x + 2 +%% in x + y +%% N should be 8 +%solve D : eval (let* (binds/cons (oexp/done (num (s (s (s z))))) (binds/cons (oexp/bind ([x] oexp/done (plus x (num (s (s z)))))) binds/done)) (oexp/bind ([x] oexp/bind ([y] oexp/done (plus x y))))) N %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Iterated_Let_Bindings). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/judgment.lf b/new-tests/stelf-output/pages/judgment.lf new file mode 100644 index 0000000..ac81ec3 --- /dev/null +++ b/new-tests/stelf-output/pages/judgment.lf @@ -0,0 +1,64 @@ +% +% +%{! In the context of this wiki, we use the word **judgment** (or **judgement**) to refer to a relation that is defined inductively by a collection of inference rules. The **judgments as types principle** is a name for the methodology by which judgments are represented in LF. + +## A judgment in standard notation + +For example, we can define a judgment that a [natural number](/wiki/natural-numbers/) is even. The judgement holds when is even. It is inductively defined by the following inference rules: + +```math +{ + \over + \mathsf{even}(\mathsf{zero}) +} +\qquad +{ + \mathsf{even}(n) + \over + \mathsf{even}(\mathsf{succ}(\mathsf{succ}(n))) +} +``` + +## Judgments as types + +A judgment is represented in LF using the **judgments as types** methodology: we represent a judgment with an LF type, where the inhabitants of this type correspond exactly to derivations of the judgement. + +For example, we represent the judgment using the following signature: !}% +%{!! begin hidden !!}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%{!! end hidden !!}% +%sort even %. +%term even-z even z %. +%term even-s {N nat} %pi (even N) %-> (even (s (s N))) %. +%{! The first declaration says that ``even`` is a family of types indexed by a ``nat``. This means that for every term ``N : nat``, there is a type ``even N``. Note that the syntax ``->`` is overloaded: it is used to classify both type-level families and term-level functions. We then use this type family to define the types of two term constants. + +The first term constant, ``even-z``, has type ``even z``. This constant represents the derivation that consists of the first inference rule above, which concludes . + +The second term constant ``even-s``, corresponds to the second inference rule above, which, for any , constructs a derivation of from a derivation of . To encode this inference rule, the constant ``even-s`` is given a [dependent function type](/wiki/dependent-types/). + +For example, the LF term +{/* syntax highlighting looks weird when it's not actually Twelf code */} +``` +even-s z even-z +``` + +represents the derivation + +```math +{ + \overline{\mathsf{even}(\mathsf{zero})} + \over + \mathsf{even}(\mathsf{succ}(\mathsf{succ}(\mathsf{zero}))) +} +``` + +The term ``even-s (s (s z)) (even-s z even-z)`` represents a derivation that 4 is even, and so on. + +## See also + +* [Hypothetical judgement](/wiki/hypothetical-judgment/)s can be represented in LF in a [higher-order](/wiki/higher-order-judgements/) manner, using LF binding to represent hypotheses. +* The [introductions to Twelf](/wiki/introductions-to-twelf/) discuss how judgments are represented in LF in more detail. + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/lax-logic.lf b/new-tests/stelf-output/pages/lax-logic.lf new file mode 100644 index 0000000..7be0e54 --- /dev/null +++ b/new-tests/stelf-output/pages/lax-logic.lf @@ -0,0 +1,1244 @@ +% +%{! In this article, we will take a presentation of a logic very close to +that of Fairtlough and Mendler's lax logic<ref >\{\{bibtex:fm97lax\}\}</ref> and +show a translation into a polarized version of Pfenning and Davies' judgmental +reconstruction of lax logic.<ref>\{\{bibtex:pd01modal\}\}</ref> We will refer to +the non-judgmental, non-polarized system as the _source logic_ and to the +judgmentally reconstructed, polarized system as the _target logic._ +While we tend to see _judgmentally reconstructed_ logics as Good +Things<sup>TM</sup>, there's nothing inherently interesting about +_polarization_ of the the target +logic on its own -- the target logic is polarized because it will make the +proof of completeness for [[focused lax logic]] clearer. + +Therefore, we have three separate things that are happening simultaneously. +* First, we are formally establishing the connection between Fairtlough and Mendler's system and Pfenning and Davies' system. +* Second, because the source logic has a cut rule and the target logic does not, we prove the [admissibility of cut](/wiki/admissibility-of-cut/) in the target logic and use it to eliminate instances of the cut rule during translation. Alternatively, we could have (presumably) proven cut elimination within Fairtlough and Mendler's system, or we could have left a cut rule in the target logic and then proven cut elimination there. +* Third, we are showing the connection between a non-polarized system and a polarized one. +All three of these things could be done separately, but we did not find that +it added to the overall complexity to do them all together. + +The third point above, translation into a non-polarized system, +is where most of the +technical difficulty of the completeness proof happens, because a general +translation from a non-polarized logic to a polarized one may add an +arbitrary number of shift operations. + +<references/> + +<br/> !}% +%{! ## Preliminaries !}% +%{! We will be [using a metric](/wiki/structural-metrics/) to make certain arguments about translation +simpler - we can think of this metric as a tree ``T`` that is either +a leaf (``o``), a parent with one child tree (``x T``), or a parent +with two child trees (``T1 | T2``). +!}% +%sort metric %. +%name metric %. +%term o metric %. +%term x %pi metric %-> metric %. +%term | %pi metric %-> metric %-> metric %. +%prec %right 10 | %. +%sort id %. +%name id %. +%term id/refl id M M %. +%{! We will also be, throughout, working in a universe of atomic propositions. +Each atomic proposition has a defined polarity (``pos`` or ``neg``), +though this is mostly irrelevant in the target logic and totally irrelevant in +the source logic. Every theorem we prove will assume a universe of arbitrary +atomic propositions of both polarities, so every [``%worlds``](/wiki/percent-worlds/) +declaration will include at least the blocks ``bl_atmpos`` (positive +atomic propositions) and ``bl_atmneg`` (negative atomic propositions). !}% +%sort polarity %. +%name polarity %. +%term pos polarity %. +%term neg polarity %. +%sort atm %. +%name atm %. +%block bl_atmpos [qp atm pos]%. +%block bl_atmneg [qn atm neg]%. +%{! ## Fairtlough and Mendler's lax logic !}% +%{! Fairtlough and Mendler's presentation of lax logic has a slight +inconsistancy: the logic's syntax is defined as + + + +However, rules are given for neither nor +. A reasonable approach seemed to be to define + as . !}% +%sort prop' %. +%name prop' %. +%term atom' %pi (atm S) %-> prop' %. +%term tt' prop' %. +%term /\ %pi prop' %-> prop' %-> prop' %. +%prec %right 9 /\ %. +%term \/ %pi prop' %-> prop' %-> prop' %. +%prec %right 8 \/ %. +%term => %pi prop' %-> prop' %-> prop' %. +%prec %right 7 => %. +%term not' %pi prop' %-> prop' %. +%term circ' %pi prop' %-> prop' %. +%define ff' prop' not' tt' %. +%{! ### Sequent calculus !}% +%{! Fairtlough and Mendler present a Genzen-style intuitionstic sequent +calculus for propositional lax logic in terms of sequents +, where is a finite +list of _hypotheses_ and is either 0 or 1 +_assertions_. To prove is to say that the +hypotheses in are contradictory, to prove + is to say that the hypotheses in + establish the proposition . + +We represent the hypotheses using the LF context; each + becomes an LF variable of type ``hyp' M``, and +we use the world ``bl_hyp'`` to describe all such contexts. !}% +%sort hyp' %. +%block bl_hyp' {M prop'} [h hyp' M]%. +%{! We represent the conclusion as ``conc' T D``, where +* ``T`` is a metric capturing the shape of the derivation +* ``D`` = ``conc-#'`` represents proving a contradiction ; we abbreviate ``conc' T conc-#'`` as ``# T``. +* ``D`` = ``conc-true' M`` represents proving ; we abbreviate ``conc' T (conc-true' M)`` as ``true' T M``. !}% +%sort conseq' %. +%name conseq' %. +%term conc-# conseq' %. +% Contradiction, length 0 +%term conc-true' %pi prop' %-> conseq' %. +% Conclusion, length 1 +%sort conc' %. +%inline # [m] conc' m conc-# %. +%inline true' [m] [x] conc' m (conc-true' x) %. +%{! While our use of a [structural metric](/wiki/structural-metrics/) embedded into the conclusion of +a sequent needs to be justified by [adequacy](/wiki/adequacy/), it is less of an issue than +other changes we have made to the system. + +As previously mentioned, +Fairtlough and Mendler's system had no rule for , +and we need no rule for as we defined it in terms +of . We do not write rules for +weakening on the left or exchange on the left, as those are provided "for free" +by the encoding into LF. In addition, we embed contraction into every +left rule: + +<center>F&M left conjunction rule: +            +Our left conjunction rule: </center> + +With those caveats, we will continue to consider our source logic, presented +here, to have the essential character of Fairtlough and Mendler's. + +#### Rules on paper + +<center>                      </center> + +<br/> + +<center>                      </center> + +<br/> + +<center>                                 </center> + +<br/> + +<center>           </center> + +<br/> + +<center>                      </center> !}% +%{! #### Rules in Twelf !}% +%term trueR' true' o tt' %. +% no trueL' +%term /\R' %pi (true' Tm M) %-> (true' Tn N) %-> (true' (Tm | Tn) (M /\ N)) %. +%term /\L' + %pi (%pi (hyp' M) %-> (hyp' N) %-> (conc' T D)) + %-> (hyp' (M /\ N)) + %-> (conc' (x (x T)) D) %. +%term \/R1' %pi (true' T M) %-> (true' (x T) (M \/ N)) %. +%term \/R2' %pi (true' T N) %-> (true' (x T) (M \/ N)) %. +%term \/L' + %pi (%pi (hyp' M) %-> (conc' Tm D)) + %-> (%pi (hyp' N) %-> (conc' Tn D)) + %-> (hyp' (M \/ N)) + %-> (conc' (x (Tm | Tn)) D) %. +%term =>R' %pi (%pi (hyp' M) %-> (true' T N)) %-> (true' (x T) (M => N)) %. +%term =>L' + %pi (true' Tm M) + %-> (%pi (hyp' N) %-> (conc' Tn D)) + %-> (hyp' (M => N)) + %-> (conc' (x (Tm | Tn)) D) %. +%term notR' %pi (%pi (hyp' M) %-> (# T)) %-> (true' (x T) (not' M)) %. +%term notL' %pi (true' T M) %-> (hyp' (not' M)) %-> (# (x (x T))) %. +%term circR' %pi (true' T M) %-> (true' (x T) (circ' M)) %. +%term circL' + %pi (%pi (hyp' M) %-> (true' T (circ' N))) + %-> (hyp' (circ' M)) + %-> (true' (x (x T)) (circ' N)) %. +%term id' %pi (hyp' M) %-> (true' o M) %. +%term cut' %pi (true' Ta M) %-> (%pi (hyp' M) %-> (conc' T D)) %-> (conc' (Ta | T) D) %. +%term weakR' %pi (# T) %-> (true' (x T) M) %. +%worlds (bl_atmpos bl_atmneg bl_hyp') (hyp' M) (conc' T M) %. +% = (# T) (true' T A) +%{! ## Polarized judgmental lax logic !}% +%{! Polarized lax logic loses the but distinguishes +between two different kinds of conjunction, and +, and two variants of +which are the units of the two kinds of conjunction. + +_Every_ proposition has a polarity, not just atomic propositions, and +this limits what we can write down. For instance, we cannot +write , because expectes two +positive things and is negative. We could either write +, or else we could write +. + +The full language of propositions is: + + + + + + + +The Twelf versions of the the connectives resemble ones from linear logic, +but we mean unrestricted implication +when we write ``P -o N``, not linear implication. !}% +%sort prop %. +%name prop %. +%term atom %pi (atm S) %-> (prop S) %. +%term up %pi (prop pos) %-> (prop neg) %. +%term down %pi (prop neg) %-> (prop pos) %. +%term top prop neg %. +%term 1 prop pos %. +%term 0 prop pos %. +%term * %pi (prop pos) %-> (prop pos) %-> (prop pos) %. +%prec %right 9 * %. +%term & %pi (prop neg) %-> (prop neg) %-> (prop neg) %. +%prec %right 8 & %. +%term + %pi (prop pos) %-> (prop pos) %-> (prop pos) %. +%prec %right 7 + %. +%scope + %term o %pi (prop pos) %-> (prop neg) %-> (prop neg) %. +%prec %right 6 -o %. +%term circ %pi (prop pos) %-> (prop neg) %. +%{! ### Sequent calculus !}% +%{! Our assumptions are, as before, represented by LF variables, this time +of type ``hyp A``. !}% +%sort hyp %. +%block bl_hyp {S polarity} {A prop S} [h hyp A]%. +%{! We are always trying to prove one of two _judgments,_ and so our +sequents either take the form ( +is true) or ( is true +under a constraint). !}% +%sort conseq %. +%name conseq %. +%term conc-lax %pi (prop S) %-> conseq %. +% A lax +%term conc-true %pi (prop S) %-> conseq %. +% A true +%sort conc %. +%inline lax [x] conc (conc-lax x) %. +%inline true [x] conc (conc-true x) %. +%{! Later on in the proof of soundness we will use identity of _derivations_ +in order to make a termination argument work; with this one trick, we avoid +any need for a [structural metric](/wiki/structural-metrics/) like we used before. !}% +%sort idconc %. +%term idconc/refl idconc D D %. +%{! #### Rules on paper + +<center>                      </center> + +<br/> + +<center>                                 </center> + +<br/> + +<center>                      </center> + +<br/> + +<center>           </center> + +<br/> + +<center>           </center> + +<br/> + +<center>                      </center> + +<br/> + +<center>           </center> + +<br/> + +<center>           </center> !}% +%{! #### Rules in Twelf !}% +%term init+ %pi (hyp (atom Qp)) %-> (true (atom (%the (atm pos) Qp))) %. +%term init- %pi (hyp (atom Qn)) %-> (true (atom (%the (atm neg) Qn))) %. +%term laxR %pi (true A) %-> (lax A) %. +%term upR %pi (true P) %-> (true (up P)) %. +%term upL %pi (%pi (hyp P) %-> (conc J)) %-> (hyp (up P)) %-> (conc J) %. +%term downR %pi (true N) %-> (true (down N)) %. +%term downL %pi (%pi (hyp N) %-> (conc J)) %-> (hyp (down N)) %-> (conc J) %. +%term 1R true 1 %. +% no 1L +%term *R %pi (true P1) %-> (true P2) %-> (true (P1 * P2)) %. +%term *L %pi (%pi (hyp P1) %-> (hyp P2) %-> (conc J)) %-> (hyp (P1 * P2)) %-> (conc J) %. +%term topR true top %. +% no topL +%term &R %pi (true N1) %-> (true N2) %-> (true (N1 & N2)) %. +%term &L1 %pi (%pi (hyp N1) %-> (conc J)) %-> (hyp (N1 & N2)) %-> (conc J) %. +%term &L2 %pi (%pi (hyp N2) %-> (conc J)) %-> (hyp (N1 & N2)) %-> (conc J) %. +% no 0R +%term 0L %pi (hyp 0) %-> (conc J) %. +%term +R1 %pi (true P1) %-> (true (P1 + P2)) %. +%term +R2 %pi (true P2) %-> (true (P1 + P2)) %. +%term +L + %pi (%pi (hyp P1) %-> (conc J)) + %-> (%pi (hyp P2) %-> (conc J)) + %-> (hyp (P1 + P2)) + %-> (conc J) %. +%scope +L %term oR %pi (%pi (hyp P) %-> (true N)) %-> (true (P -o N)) %. +%term oL %pi (true P) %-> (%pi (hyp N) %-> (conc J)) %-> (hyp (P -o N)) %-> (conc J) %. +%term circR %pi (lax P) %-> (true (circ P)) %. +%term circL %pi (%pi (hyp P) %-> (lax A)) %-> (hyp (circ P)) %-> (lax A) %. +%worlds (bl_atmpos bl_atmneg bl_hyp) (hyp A) (conc J) %. +%{! ## Translation !}% +%{! Now we can define a nondeterminstic translation between our variant of +Fairtlough and Mendler's propositional lax logic and our target logic, +polarized, judgmental lax logic. We will use a metric that essentially +captures the structure of the term in the target logic (in fact, it would +be worth seeing if we could use the term in the target logic in lieu of +the metric.) !}% +%{! ### Propositions !}% +%{! The intent of the first relation we define is to be maximally general, which +means that the ``t+-`` and ``t-+`` rules can add an arbitrary +number of shifts. Read in the reverse direction, the translation is not +very nondeterminstic; however "``P -o up 0``" can still be translated +to either "``not M``" or "``M => (not' tt')``," which are different +source logic expressions. !}% +%sort trans %. +%name trans %. +%term t+- %pi (trans T M pos P) %-> (trans (x T) M neg (up P)) %. +%term t-+ %pi (trans T M neg N) %-> (trans (x T) M pos (down N)) %. +%term tatom trans o (atom' Q) S (atom Q) %. +%term t+true trans o tt' pos 1 %. +%term t-true trans o tt' neg top %. +%term t+false trans o ff' pos 0 %. +%term t+and + %pi (trans T1 M1 pos P1) + %-> (trans T2 M2 pos P2) + %-> (trans (T1 | T2) (M1 /\ M2) pos (P1 * P2)) %. +%term t-and + %pi (trans T1 M1 neg N1) + %-> (trans T2 M2 neg N2) + %-> (trans (T1 | T2) (M1 /\ M2) neg (N1 & N2)) %. +%term t+or + %pi (trans T1 M1 pos P1) + %-> (trans T2 M2 pos P2) + %-> (trans (T1 | T2) (M1 \/ M2) pos (P1 + P2)) %. +%term t-imp + %pi (trans T1 M1 pos P) + %-> (trans T2 M2 neg N) + %-> (trans (T1 | T2) (M1 => M2) neg (P -o N)) %. +%term t+not %pi (trans T M pos P) %-> (trans (x T) (not' M) neg (P -o up 0)) %. +%term t+circ %pi (trans T M pos P) %-> (trans (x T) (circ' M) neg (circ P)) %. +%worlds (bl_atmpos bl_atmneg) (trans _ _ _ _) %. +%{! ### Conclusions !}% +%{! We additionally define translation between right-hand-sides +``D`` in the source logic and judgments ``J`` in the target +logic. !}% +%sort trans-conc %. +%name trans-conc %. +%term tc %pi (trans-conc T (conc-true' M) (conc-true A)) %<- (trans T M S A) %. +%term tc# trans-conc o conc-# (conc-true (up 0)) %. +%term tclax %pi (trans-conc T (conc-true' (circ' M)) (conc-lax A)) %<- (trans T M S A) %. +%worlds (bl_atmpos bl_atmneg) (trans-conc _ _ _) %. +%{! ### Hypotheses !}% +%{! When we are showing soundness and completeness, we will need +to know that for every hypothesis in the \{source, target\} logic we have +a hypothesis in the \{target, source\} logic such that one is the translation +of the other. + +What we would like to do is define a single ``trans-hyp`` that allows us +to translate from source logic hypotheses to target logic hypotheses, and +vice versa. However, because we cannot assign multiple modes to the same +relation, we'll define two different relations, and use them to define two +different worlds. + +* ``trans-soundhyp`` takes a target logic hypotheis ``H`` and obtains a translation ``TR`` and a source logic hypothesis ``H'``. +* ``trans-complhyp`` takes a source logic hypothesis ``H'`` and obtains a translation ``TR`` and a target logic hypothesis ``H``. !}% +%sort trans-soundhyp %. +%mode trans-soundhyp %in %out %out %. +%block bl_trans-soundhyp {M metric} {S polarity} {A prop S} {A' prop'} {T trans M A' S A} [h hyp A] [h' hyp' A'] [t trans-soundhyp h T h']%. +%worlds (bl_atmpos bl_atmneg bl_trans-soundhyp) (trans-soundhyp _ _ _) %. +%total [] (trans-soundhyp _ _ _) %. +%sort trans-complhyp %. +%mode trans-complhyp %in %out %out %. +%block bl_trans-complhyp {M metric} {A prop'} {S polarity} {C prop S} {T trans M A S C} [h' hyp' A] [h hyp C] [t trans-complhyp h' T h]%. +%worlds (bl_atmpos bl_atmneg bl_trans-complhyp) (trans-complhyp _ _ _) %. +%total [] (trans-complhyp _ _ _) %. +%{! ### Correspondence !}% +%{! A corollary to the theorem we will state below will say "If + and , then +." However, if there is no such that +, then the theorem will be vacuously true, +which is not our intent. Therefore, we need to establish that our translations +of propositions and conclusions are _correspondences_ +(left- and right-total). + +Twelf can verify that translation is _right-total_, that there is an +``M`` for every ``A`` and a ``D`` for every ``J``, without +any further work on our part: we just assign modes to the relation +and state a [totality assertion](/wiki/totality-assertion/). !}% +%mode trans %out %out %in %in %. +%mode trans-conc %out %out %in %. +%total (A C) (trans _ _ _ A) (trans-conc _ _ C) %. +%{! However, in the forward direction, to show that the relation is +_left-total_ we will need to write an +[effectiveness lemma](/wiki/effectiveness-lemma/), which we will call ``can-trans``. + +The two lemmas ``can-switch`` are really just case analysis to +establish that if you can translate something as a positive formula, you can +apply an up shift to make it a negative formula, and vice versa. !}% +%sort can-switch+ %. +%term _ can-switch+ T T %. +%term _ can-switch+ T (t-+ T) %. +%mode can-switch+ %in %out %. +%worlds (bl_atmpos bl_atmneg) (can-switch+ _ _) %. +%total [] (can-switch+ _ _) %. +%sort can-switch- %. +%term _ can-switch- T T %. +%term _ can-switch- T (t+- T) %. +%mode can-switch- %in %out %. +%worlds (bl_atmpos bl_atmneg) (can-switch- _ _) %. +%total [] (can-switch- _ _) %. +%sort can-trans {A} %. +%mode can-trans %in %out %. +%term _ can-trans (atom' Qp) tatom %. +%term _ can-trans tt' t-true %. +%term _ + %pi (can-trans (A /\ B) (t-and T1' T2')) + %<- (can-trans A T1) + %<- (can-switch- T1 T1') + %<- (can-trans B T2) + %<- (can-switch- T2 T2') %. +%term _ + %pi (can-trans (A \/ B) (t+or T1' T2')) + %<- (can-trans A T1) + %<- (can-switch+ T1 T1') + %<- (can-trans B T2) + %<- (can-switch+ T2 T2') %. +%term _ + %pi (can-trans (A => B) (t-imp T1' T2')) + %<- (can-trans A T1) + %<- (can-switch+ T1 T1') + %<- (can-trans B T2) + %<- (can-switch- T2 T2') %. +%term _ %pi (can-trans (not' A) (t+not T')) %<- (can-trans A T) %<- (can-switch+ T T') %. +%term _ %pi (can-trans (circ' A) (t+circ T')) %<- (can-trans A T) %<- (can-switch+ T T') %. +%worlds (bl_atmpos bl_atmneg) (can-trans _ _) %. +%total A (can-trans A _) %. +%sort can-trans-conc {D} %. +%mode can-trans-conc %in %out %. +%term _ can-trans-conc conc-# tc# %. +%term _ %pi (can-trans-conc (conc-true' M) (tc TR)) %<- (can-trans M TR) %. +%worlds (bl_atmpos bl_atmneg) (can-trans-conc _ _) %. +%total A (can-trans-conc A _) %. +%{! ## Metatheory of the target logic !}% +%{! The correctness of translation will rest on the standard metatheoretic +results: the _identity principle_ and the _admissibility of cut_. Note +that we could have added a ``cut`` and ``id`` rule to the target +logic, but we still would have eventually needed to do the work in this +section in order to have confidence that our target logic was reasonable. !}% +%{! ### Identity !}% +%{! For all propositions , there exists a derivation of +. + +The proof is by induction on the formula . !}% +%sort identity {A} %. +%mode identity %in %out %. +%term _ identity (atom Q) init+ %. +%term _ identity (atom Q) init- %. +%term _ + %pi (identity (up P) ([h] upL ([h'] upR (D h')) h)) + %<- (identity P (%the (%pi (hyp P) %-> (true P)) D)) %. +%term _ + %pi (identity (down N) ([h] downL ([h'] downR (D h')) h)) + %<- (identity N (%the (%pi (hyp N) %-> (true N)) D)) %. +%term _ identity top ([_] topR) %. +%term _ identity 1 ([_] 1R) %. +%term _ identity 0 ([h] 0L h) %. +%term _ + %pi (identity (P1 * P2) ([h] *L ([h1] [h2] *R (D1 h1) (D2 h2)) h)) + %<- (identity P1 (%the (%pi (hyp P1) %-> (true P1)) D1)) + %<- (identity P2 (%the (%pi (hyp P2) %-> (true P2)) D2)) %. +%term _ + %pi (identity (N1 & N2) ([h] &L1 ([h1] &L2 ([h2] &R (D1 h1) (D2 h2)) h) h)) + %<- (identity N1 (%the (%pi (hyp N1) %-> (true N1)) D1)) + %<- (identity N2 (%the (%pi (hyp N2) %-> (true N2)) D2)) %. +%term _ + %pi (identity (P1 + P2) ([h] +L ([h1] +R1 (D1 h1)) ([h2] +R2 (D2 h2)) h)) + %<- (identity P1 (%the (%pi (hyp P1) %-> (true P1)) D1)) + %<- (identity P2 (%the (%pi (hyp P2) %-> (true P2)) D2)) %. +%term _ + %pi (identity (P -o N) ([h] -oR ([h1] -oL (D1 h1) ([h2] D2 h2) h))) + %<- (identity P (%the (%pi (hyp P) %-> (true P)) D1)) + %<- (identity N (%the (%pi (hyp N) %-> (true N)) D2)) %. +%term _ + %pi (identity (circ P) ([h] circR (circL ([h'] laxR (D1 h')) h))) + %<- (identity P (%the (%pi (hyp P) %-> (true P)) D1)) %. +%worlds (bl_atmpos bl_atmneg bl_hyp) (identity _ _) %. +%total T (identity T _) %. +%{! ### Cut admissibility !}% +%{! There are two cut principles which must be proven simultaneously; +``cut`` is the main theorem, and ``lcut`` is the cut principle +for lax truth. + +* If and , then (``cut``) +* If and , then (``lcut``) + +The proofs are by mutual lexographic induction; either the principal cut +formula gets smaller, or else the principal cut formula +stays the same and one or both of the input derivations get smaller. The proof +is entirely standard other than the need for extra commutative cuts for +``tlax``; more explanation is available at the article on +the [admissibility of cut](/wiki/admissibility-of-cut/). !}% +%sort cut {A} %. +%sort lcut {A} %. +%mode cut %in %in %in %out %. +%mode lcut %in %in %in %out %. +%{! #### Principal cuts !}% +%term _ cut (atom Q) (init+ H) ([h] init+ h) (init+ H) %. +%term _ cut (atom Q) (init- H) ([h] init- h) (init- H) %. +%term _ + %pi (cut (up P) (upR (%the (true P) D)) ([h] upL (%the (%pi (hyp P) %-> (conc J)) (E h)) h) G) + %<- ({h' hyp P} cut (up P) (upR D) ([h] E h h') (F h')) + %<- (cut P D ([h'] F h') (%the (conc J) G)) %. +%term _ + %pi (cut (down N) (downR (%the (true N) D)) ([h] downL (%the (%pi (hyp N) %-> (conc J)) (E h)) h) G) + %<- ({h' hyp N} cut (down N) (downR D) ([h] E h h') (F h')) + %<- (cut N D ([h'] F h') (%the (conc J) G)) %. +%term _ + %pi (cut (P1 * P2) (*R (%the (true P1) D1) (%the (true P2) D2)) ([h] *L (%the (%pi (hyp P1) %-> (hyp P2) %-> (conc J)) (E h)) h) G) + %<- ({h1 hyp P1} {h2 hyp P2} cut (P1 * P2) (*R D1 D2) ([h] E h h1 h2) (F h1 h2)) + %<- ({h2 hyp P2} cut P1 D1 ([h1] F h1 h2) (F1 h2)) + %<- (cut P2 D2 ([h2] F1 h2) (%the (conc J) G)) %. +%term _ + %pi (cut (N1 & N2) (&R (%the (true N1) D1) (%the (true N2) D2)) ([h] &L1 (%the (%pi (hyp N1) %-> (conc J)) (E h)) h) G) + %<- ({h1 hyp N1} cut (N1 & N2) (&R D1 D2) ([h] E h h1) (F h1)) + %<- (cut N1 D1 ([h1] F h1) (%the (conc J) G)) %. +%term _ + %pi (cut (N1 & N2) (&R (%the (true N1) D1) (%the (true N2) D2)) ([h] &L2 (%the (%pi (hyp N2) %-> (conc J)) (E h)) h) G) + %<- ({h2 hyp N2} cut (N1 & N2) (&R D1 D2) ([h] E h h2) (F h2)) + %<- (cut N2 D2 ([h2] F h2) (%the (conc J) G)) %. +%term _ + %pi (cut (P1 + P2) (+R1 (%the (true P1) D1)) ([h] +L (%the (%pi (hyp P1) %-> (conc J)) (E1 h)) _ h) G) + %<- ({h1 hyp P1} cut (P1 + P2) (+R1 D1) ([h] E1 h h1) (F h1)) + %<- (cut P1 D1 ([h1] F h1) G) %. +%term _ + %pi (cut (P1 + P2) (+R2 (%the (true P2) D2)) ([h] +L _ (%the (%pi (hyp P2) %-> (conc J)) (E2 h)) h) G) + %<- ({h2 hyp P2} cut (P1 + P2) (+R2 D2) ([h] E2 h h2) (F h2)) + %<- (cut P2 D2 ([h2] F h2) G) %. +%term _ + %pi (cut (P -o N) (-oR (%the (%pi (hyp P) %-> (true N)) D)) ([h] -oL (%the (true P) (E1 h)) (%the (%pi (hyp N) %-> (conc J)) (E2 h)) h) G) + %<- (cut (P -o N) (-oR D) ([h] E1 h) (%the (true P) F1)) + %<- ({h' hyp N} cut (P -o N) (-oR D) ([h] E2 h h') (%the (conc J) (F2 h'))) + %<- (cut P F1 ([h'] D h') (%the (true N) G1)) + %<- (cut N G1 ([h'] F2 h') (%the (conc J) G)) %. +%term _ + %pi (cut (circ P) (circR (%the (lax P) D)) ([h] circL (%the (%pi (hyp P) %-> (lax C)) (E h)) h) G) + %<- ({h' hyp P} cut (circ P) (circR D) ([h] E h h') (%the (lax C) (F h'))) + %<- (lcut P D ([h'] F h') G) %. +%{! #### Left commutative cuts !}% +%term _ %pi (lcut P (laxR (%the (true P) D)) E F) %<- (cut P D E F) %. +%term _ %pi (cut A (upL D H) E (upL F H)) %<- ({h1} cut A (D h1) E (F h1)) %. +%term _ %pi (cut A (downL D H) E (downL F H)) %<- ({h1} cut A (D h1) E (F h1)) %. +%term _ cut A (0L H) E (0L H) %. +%term _ %pi (cut A (*L D H) E (*L F H)) %<- ({h1} {h2} cut A (D h1 h2) E (F h1 h2)) %. +%term _ %pi (cut A (&L1 D H) E (&L1 F H)) %<- ({h1} cut A (D h1) E (F h1)) %. +%term _ %pi (cut A (&L2 D H) E (&L2 F H)) %<- ({h2} cut A (D h2) E (F h2)) %. +%term _ + %pi (cut A (+L D1 D2 H) E (+L F1 F2 H)) + %<- ({h1} cut A (D1 h1) E (F1 h1)) + %<- ({h2} cut A (D2 h2) E (F2 h2)) %. +%term _ %pi (cut A (-oL D1 D2 H) E (-oL D1 F2 H)) %<- ({h'} cut A (D2 h') E (F2 h')) %. +%{! #### Monadic left commutative cuts !}% +%term _ + %pi (lcut A (circL ([h] D h) H) E (circL ([h] F h) H)) + %<- ({h'} lcut A (D h') E (F h')) %. +%term _ %pi (lcut A (upL D H) E (upL F H)) %<- ({h1} lcut A (D h1) E (F h1)) %. +%term _ %pi (lcut A (downL D H) E (downL F H)) %<- ({h1} lcut A (D h1) E (F h1)) %. +%term _ lcut A (0L H) E (0L H) %. +%term _ %pi (lcut A (*L D H) E (*L F H)) %<- ({h1} {h2} lcut A (D h1 h2) E (F h1 h2)) %. +%term _ %pi (lcut A (&L1 D H) E (&L1 F H)) %<- ({h1} lcut A (D h1) E (F h1)) %. +%term _ %pi (lcut A (&L2 D H) E (&L2 F H)) %<- ({h2} lcut A (D h2) E (F h2)) %. +%term _ + %pi (lcut A (+L D1 D2 H) E (+L F1 F2 H)) + %<- ({h1} lcut A (D1 h1) E (F1 h1)) + %<- ({h2} lcut A (D2 h2) E (F2 h2)) %. +%term _ %pi (lcut A (-oL D1 D2 H) E (-oL D1 F2 H)) %<- ({h'} lcut A (D2 h') E (F2 h')) %. +%{! #### Right commutative cuts !}% +%term _ cut A D ([h] init+ H) (init+ H) %. +%term _ cut A D ([h] init- H) (init- H) %. +%term _ %pi (cut A D ([h] laxR (E h)) (laxR F)) %<- (cut A D ([h] E h) F) %. +%term _ %pi (cut A D ([h] upR (E h)) (upR F)) %<- (cut A D ([h] E h) F) %. +%term _ + %pi (cut A D ([h] upL ([h'] E h h') H) (upL F H)) + %<- ({h'} cut A D ([h] E h h') (F h')) %. +%term _ %pi (cut A D ([h] downR (E h)) (downR F)) %<- (cut A D ([h] E h) F) %. +%term _ + %pi (cut A D ([h] downL ([h'] E h h') H) (downL F H)) + %<- ({h'} cut A D ([h] E h h') (F h')) %. +%term _ cut A D ([h] topR) topR %. +%term _ cut A D ([h] 1R) 1R %. +%term _ cut A D ([h] 0L H) (0L H) %. +%term _ + %pi (cut A D ([h] *R (E1 h) (E2 h)) (*R F1 F2)) + %<- (cut A D ([h] E1 h) F1) + %<- (cut A D ([h] E2 h) F2) %. +%term _ + %pi (cut A D ([h] *L ([h1] [h2] E h h1 h2) H) (*L ([h1] [h2] F h1 h2) H)) + %<- ({h1} {h2} cut A D ([h] E h h1 h2) (F h1 h2)) %. +%term _ + %pi (cut A D ([h] &R (E1 h) (E2 h)) (&R F1 F2)) + %<- (cut A D ([h] E1 h) F1) + %<- (cut A D ([h] E2 h) F2) %. +%term _ + %pi (cut A D ([h] &L1 ([h1] E1 h h1) H) (&L1 ([h1] F1 h1) H)) + %<- ({h1} cut A D ([h] E1 h h1) (F1 h1)) %. +%term _ + %pi (cut A D ([h] &L2 ([h2] E2 h h2) H) (&L2 ([h2] F2 h2) H)) + %<- ({h2} cut A D ([h] E2 h h2) (F2 h2)) %. +%term _ %pi (cut A D ([h] +R1 (E1 h)) (+R1 F1)) %<- (cut A D ([h] E1 h) F1) %. +%term _ %pi (cut A D ([h] +R2 (E2 h)) (+R2 F2)) %<- (cut A D ([h] E2 h) F2) %. +%term _ + %pi (cut A D ([h] +L ([h1] E1 h h1) ([h2] E2 h h2) H) (+L F1 F2 H)) + %<- ({h1} cut A D ([h] E1 h h1) (F1 h1)) + %<- ({h2} cut A D ([h] E2 h h2) (F2 h2)) %. +%term _ + %pi (cut A D ([h] -oR ([h'] E h h')) (-oR ([h'] F h'))) + %<- ({h'} cut A D ([h] E h h') (F h')) %. +%term _ + %pi (cut A D ([h] -oL (E1 h) ([h'] E2 h h') H) (-oL F1 ([h'] F2 h') H)) + %<- (cut A D ([h] E1 h) F1) + %<- ({h'} cut A D ([h] E2 h h') (F2 h')) %. +%term _ %pi (cut A D ([h] circR (E h)) (circR F)) %<- (cut A D ([h] E h) F) %. +%term _ + %pi (cut A D ([h] circL ([h'] E h h') H) (circL ([h'] F h') H)) + %<- ({h'} cut A D ([h] E h h') (F h')) %. +%{! #### Monadic right commutive cuts !}% +%term _ + %pi (lcut A D ([h] upL ([h'] E h h') H) (upL F H)) + %<- ({h'} lcut A D ([h] E h h') (F h')) %. +%term _ + %pi (lcut A D ([h] downL ([h'] E h h') H) (downL F H)) + %<- ({h'} lcut A D ([h] E h h') (F h')) %. +%term _ + %pi (lcut A D ([h] *L ([h1] [h2] E h h1 h2) H) (*L ([h1] [h2] F h1 h2) H)) + %<- ({h1} {h2} lcut A D ([h] E h h1 h2) (F h1 h2)) %. +%term _ + %pi (lcut A D ([h] &L1 ([h1] E1 h h1) H) (&L1 ([h1] F1 h1) H)) + %<- ({h1} lcut A D ([h] E1 h h1) (F1 h1)) %. +%term _ + %pi (lcut A D ([h] &L2 ([h2] E2 h h2) H) (&L2 ([h2] F2 h2) H)) + %<- ({h2} lcut A D ([h] E2 h h2) (F2 h2)) %. +%term _ + %pi (lcut A D ([h] +L ([h1] E1 h h1) ([h2] E2 h h2) H) (+L F1 F2 H)) + %<- ({h1} lcut A D ([h] E1 h h1) (F1 h1)) + %<- ({h2} lcut A D ([h] E2 h h2) (F2 h2)) %. +%worlds (bl_atmpos bl_atmneg bl_hyp) (cut _ _ _ _) (lcut _ _ _ _) %. +%total {(A1 A2) [(D1 D2) (E1 E2)]} (cut A1 D1 E1 F1) (lcut A2 D2 E2 F2) %. +%{! ### Inversion lemmas !}% +%{! Proving invertability is an application of both cut and identity; it +establishes that that for some rules, whenver the conclusion is true then +the premises are always true as well. + +In general, negative propositions are invertible on the right and positive +propositions are invertible on the left, though in unfocused intutionstic +logic ``up`` and ``down`` are invertible in both directions and +both variants of conjunction are invertible on the right. This is not the full +set of inversion lemmas, just the ones we needed. !}% +%sort invupR %. +%term _ + %pi (invupR D F) + %<- (identity P (%the (%pi (hyp P) %-> (true P)) EP)) + %<- (cut (up P) D (upL ([h] EP h)) F) %. +%mode invupR %in %out %. +%worlds (bl_atmpos bl_atmneg bl_hyp) (invupR _ _) %. +%total [] (invupR _ _) %. +%sort invupL %. +%term _ + %pi (invupL ([h] D h) ([h] F h)) + %<- (identity P (%the (%pi (hyp P) %-> (true P)) EP)) + %<- ({h hyp P} cut (up P) (upR (EP h)) D (F h)) %. +%mode invupL %in %out %. +%worlds (bl_atmpos bl_atmneg bl_hyp) (invupL _ _) %. +%total [] (invupL _ _) %. +%sort invdownR %. +%term _ + %pi (invdownR D F) + %<- (identity P (%the (%pi (hyp P) %-> (true P)) EP)) + %<- (cut (down P) D (downL ([h] EP h)) F) %. +%mode invdownR %in %out %. +%worlds (bl_atmpos bl_atmneg bl_hyp) (invdownR _ _) %. +%total [] (invdownR _ _) %. +%sort invdownL %. +%term _ + %pi (invdownL ([h] D h) ([h] F h)) + %<- (identity P (%the (%pi (hyp P) %-> (true P)) EP)) + %<- ({h hyp P} cut (down P) (downR (EP h)) D (F h)) %. +%mode invdownL %in %out %. +%worlds (bl_atmpos bl_atmneg bl_hyp) (invdownL _ _) %. +%total [] (invdownL _ _) %. +%sort inv*R %. +%term _ + %pi (inv*R D F1 F2) + %<- (identity (P1 * P2) (%the (%pi (hyp (P1 * P2)) %-> (true (P1 * P2))) E)) + %<- (identity P1 (%the (%pi (hyp P1) %-> (true P1)) E1)) + %<- (identity P2 (%the (%pi (hyp P2) %-> (true P2)) E2)) + %<- (cut (P1 * P2) D (*L ([h1] [h2] E1 h1)) F1) + %<- (cut (P1 * P2) D (*L ([h1] [h2] E2 h2)) F2) %. +%mode inv*R %in %out %out %. +%worlds (bl_atmpos bl_atmneg bl_hyp) (inv*R _ _ _) %. +%total [] (inv*R _ _ _) %. +%sort inv&R %. +%term _ + %pi (inv&R D F1 F2) + %<- (identity (N1 & N2) (%the (%pi (hyp (N1 & N2)) %-> (true (N1 & N2))) E)) + %<- (identity N1 (%the (%pi (hyp N1) %-> (true N1)) E1)) + %<- (identity N2 (%the (%pi (hyp N2) %-> (true N2)) E2)) + %<- (cut (N1 & N2) D (&L1 ([h1] E1 h1)) F1) + %<- (cut (N1 & N2) D (&L2 ([h2] E2 h2)) F2) %. +%mode inv&R %in %out %out %. +%worlds (bl_atmpos bl_atmneg bl_hyp) (inv&R _ _ _) %. +%total [] (inv&R _ _ _) %. +%sort inv-oR %. +%term _ + %pi (inv-oR D ([h] F h)) + %<- (identity P (%the (%pi (hyp P) %-> (true P)) EP)) + %<- (identity N (%the (%pi (hyp N) %-> (true N)) EN)) + %<- ({h' hyp P} cut (P -o N) D (-oL (EP h') ([h] EN h)) (F h')) %. +%mode inv-oR %in %out %. +%worlds (bl_atmpos bl_atmneg bl_hyp) (inv-oR _ _) %. +%total [] (inv-oR _ _) %. +%sort invcircR %. +%term _ + %pi (invcircR D F) + %<- (identity P (%the (%pi (hyp P) %-> (true P)) E)) + %<- (lcut (circ P) (laxR D) (circL ([h'] laxR (E h'))) F) %. +%mode invcircR %in %out %. +%worlds (bl_atmpos bl_atmneg bl_hyp) (invcircR _ _) %. +%total [] (invcircR _ _) %. +%sort invcircL %. +%term _ + %pi (invcircL D F) + %<- (identity P (%the (%pi (hyp P) %-> (true P)) E)) + %<- ({h' hyp P} cut (circ P) (circR (laxR (E h'))) ([h] D h) (F h')) %. +%{! ## Correctness of translation !}% +%{! ### Equivalence !}% +%{! This is an interesting theorem, because it captures the idea that the +translation, while nondeterminstic (a single source logic proposition +corresponds to many target logic propositions), is not significant +(all target logic propositions in the image of a single source logic +proposition are equivalent). Equivalence will be necessary for the completeness +theorem in order to translate the identity rule in the source logic into the +target logic. + +* If and and , then . + +The theorem is proved by simultaneous induction on the unordered pair of the +two translations, and therefore suffers from an annoying lack of expressivity +in the Twelf termination checker. There is no way to express the argument +"the unordered pair of X and Y gets smaller," and so when we need to switch +the position of the arguments in the lambda case, we cannot convince Twelf +that something is getting smaller. The only way to do this seems to be to +cut and paste the lemma into two copies which then call each other at the +critical case; Twelf can handle the unordered pair as mutual induction. + +First we need a minor strenghtening lemma first, that expresses that if a +hypothesis translated from true, then it isn't needed. !}% +%sort trans-equiv-str %. +%term _ + %pi (trans-equiv-str D (t-+ (t+- TR)) E) + %<- (invdownL D D') + %<- (invupL D' D'') + %<- (trans-equiv-str D'' TR E) %. +%term _ %pi (trans-equiv-str D (t-+ t-true) E) %<- (cut (down top) (downR topR) D E) %. +%term _ %pi (trans-equiv-str D t+true E) %<- (cut 1 1R D E) %. +%mode trans-equiv-str %in %in %out %. +%worlds (bl_atmpos bl_atmneg bl_hyp) (trans-equiv-str _ _ _) %. +%total TR (trans-equiv-str _ TR _) %. +%sort trans-equiv %. +%sort trans-equivX %. +%mode trans-equiv %in %in %in %out %. +%mode trans-equivX %in %in %in %out %. +%term _ + %pi (trans-equiv (t+- T1) D1 T2 D2) + %<- (invupR D1 D1') + %<- (trans-equiv T1 D1' T2 D2) %. +%term _ + %pi (trans-equiv (t-+ T1) D1 T2 D2) + %<- (invdownR D1 D1') + %<- (trans-equiv T1 D1' T2 D2) %. +%term _ %pi (trans-equiv T1 D1 (t+- T2) (upR D2)) %<- (trans-equiv T1 D1 T2 D2) %. +%term _ %pi (trans-equiv T1 D1 (t-+ T2) (downR D2)) %<- (trans-equiv T1 D1 T2 D2) %. +%term _ trans-equiv tatom D tatom D %. +%term _ trans-equiv t+true D t+true D %. +%term _ trans-equiv t+true D t-true topR %. +%term _ trans-equiv t-true D t+true 1R %. +%term _ trans-equiv t-true D t-true D %. +%term _ %pi (trans-equiv t+false D _ E) %<- (cut 0 D ([h] 0L h) E) %. +%term _ + %pi (trans-equiv (t+not T) D t+false E) + %<- (inv-oR D (%the (%pi (hyp P1) %-> (true (up 0))) Da)) + %<- ({h} invupR (Da h) (Db h)) + %<- (trans-equiv-str Db T E) %. +%term _ + %pi (trans-equiv (t+and Ta Tb) D (t+and Sa Sb) (*R Ea Eb)) + %<- (inv*R D Da Db) + %<- (trans-equiv Ta Da Sa Ea) + %<- (trans-equiv Tb Db Sb Eb) %. +%term _ + %pi (trans-equiv (t-and Ta Tb) D (t+and Sa Sb) (*R Ea Eb)) + %<- (inv&R D Da Db) + %<- (trans-equiv Ta Da Sa Ea) + %<- (trans-equiv Tb Db Sb Eb) %. +%term _ + %pi (trans-equiv (t+and Ta Tb) D (t-and Sa Sb) (&R Ea Eb)) + %<- (inv*R D Da Db) + %<- (trans-equiv Ta Da Sa Ea) + %<- (trans-equiv Tb Db Sb Eb) %. +%term _ + %pi (trans-equiv (t-and Ta Tb) D (t-and Sa Sb) (&R Ea Eb)) + %<- (inv&R D Da Db) + %<- (trans-equiv Ta Da Sa Ea) + %<- (trans-equiv Tb Db Sb Eb) %. +%term _ + %pi (trans-equiv (t+or Ta Tb) D (t+or Sa Sb) E) + %<- (identity P1 (%the (%pi (hyp P1) %-> (true P1)) D1)) + %<- ({h1} trans-equiv Ta (D1 h1) Sa (E1 h1)) + %<- (identity P2 (%the (%pi (hyp P2) %-> (true P2)) D2)) + %<- ({h2} trans-equiv Tb (D2 h2) Sb (E2 h2)) + %<- (cut (P1 + P2) D (+L ([h1] +R1 (E1 h1)) ([h2] +R2 (E2 h2))) E) %. +%term _ + %pi (trans-equiv (t-imp Ta Tb) D (t-imp Sa Sb) (-oR E)) + %<- (identity P2 (%the (%pi (hyp P2) %-> (true P2)) D1)) + %<- ({h1} trans-equivX Sa (D1 h1) Ta (E1 h1)) + %<- (inv-oR D (%the (%pi (hyp P1) %-> (true N1)) D2)) + %<- ({h1} trans-equiv Tb (D2 h1) Sb (E2 h1)) + %<- ({h2} cut P1 (E1 h2) ([h1] E2 h1) (%the (true N2) (E h2))) %. +%term _ + %pi (trans-equiv (t+not T) D (t+not S) (-oR E)) + %<- (identity P2 (%the (%pi (hyp P2) %-> (true P2)) D1)) + %<- ({h1} trans-equivX S (D1 h1) T (E1 h1)) + %<- (inv-oR D (%the (%pi (hyp P1) %-> (true (up 0))) D2)) + %<- ({h2} cut P1 (E1 h2) ([h1] D2 h1) (%the (true (up 0)) (E h2))) %. +%term _ + %pi (trans-equiv (t+circ T) (%the (true (circ P)) D) (t+circ S) (circR F)) + %<- (invcircR D (%the (lax P) D1)) + %<- (identity P (%the (%pi (hyp P) %-> (true P)) D2)) + %<- ({h1} trans-equiv T (D2 h1) S (%the (true B) (E h1))) + %<- (lcut P D1 ([h] laxR (E h)) F) %. +%% Cut and paste: +%term _ + %pi (trans-equivX (t+- T1) D1 T2 D2) + %<- (invupR D1 D1') + %<- (trans-equivX T1 D1' T2 D2) %. +%term _ + %pi (trans-equivX (t-+ T1) D1 T2 D2) + %<- (invdownR D1 D1') + %<- (trans-equivX T1 D1' T2 D2) %. +%term _ %pi (trans-equivX T1 D1 (t+- T2) (upR D2)) %<- (trans-equivX T1 D1 T2 D2) %. +%term _ %pi (trans-equivX T1 D1 (t-+ T2) (downR D2)) %<- (trans-equivX T1 D1 T2 D2) %. +%term _ trans-equivX tatom D tatom D %. +%term _ trans-equivX t+true D t+true D %. +%term _ trans-equivX t+true D t-true topR %. +%term _ trans-equivX t-true D t+true 1R %. +%term _ trans-equivX t-true D t-true D %. +%term _ %pi (trans-equivX t+false D _ E) %<- (cut 0 D ([h] 0L h) E) %. +%term _ + %pi (trans-equivX (t+not T) D t+false E) + %<- (inv-oR D (%the (%pi (hyp P1) %-> (true (up 0))) Da)) + %<- ({h} invupR (Da h) (Db h)) + %<- (trans-equiv-str Db T E) %. +%term _ + %pi (trans-equivX (t+and Ta Tb) D (t+and Sa Sb) (*R Ea Eb)) + %<- (inv*R D Da Db) + %<- (trans-equivX Ta Da Sa Ea) + %<- (trans-equivX Tb Db Sb Eb) %. +%term _ + %pi (trans-equivX (t-and Ta Tb) D (t+and Sa Sb) (*R Ea Eb)) + %<- (inv&R D Da Db) + %<- (trans-equivX Ta Da Sa Ea) + %<- (trans-equivX Tb Db Sb Eb) %. +%term _ + %pi (trans-equivX (t+and Ta Tb) D (t-and Sa Sb) (&R Ea Eb)) + %<- (inv*R D Da Db) + %<- (trans-equivX Ta Da Sa Ea) + %<- (trans-equivX Tb Db Sb Eb) %. +%term _ + %pi (trans-equivX (t-and Ta Tb) D (t-and Sa Sb) (&R Ea Eb)) + %<- (inv&R D Da Db) + %<- (trans-equivX Ta Da Sa Ea) + %<- (trans-equivX Tb Db Sb Eb) %. +%term _ + %pi (trans-equivX (t+or Ta Tb) D (t+or Sa Sb) E) + %<- (identity P1 (%the (%pi (hyp P1) %-> (true P1)) D1)) + %<- ({h1} trans-equivX Ta (D1 h1) Sa (E1 h1)) + %<- (identity P2 (%the (%pi (hyp P2) %-> (true P2)) D2)) + %<- ({h2} trans-equivX Tb (D2 h2) Sb (E2 h2)) + %<- (cut (P1 + P2) D (+L ([h1] +R1 (E1 h1)) ([h2] +R2 (E2 h2))) E) %. +%term _ + %pi (trans-equivX (t-imp Ta Tb) D (t-imp Sa Sb) (-oR E)) + %<- (identity P2 (%the (%pi (hyp P2) %-> (true P2)) D1)) + %<- ({h1} trans-equiv Sa (D1 h1) Ta (E1 h1)) + %<- (inv-oR D (%the (%pi (hyp P1) %-> (true N1)) D2)) + %<- ({h1} trans-equivX Tb (D2 h1) Sb (E2 h1)) + %<- ({h2} cut P1 (E1 h2) ([h1] E2 h1) (%the (true N2) (E h2))) %. +%term _ + %pi (trans-equivX (t+not T) D (t+not S) (-oR E)) + %<- (identity P2 (%the (%pi (hyp P2) %-> (true P2)) D1)) + %<- ({h1} trans-equiv S (D1 h1) T (E1 h1)) + %<- (inv-oR D (%the (%pi (hyp P1) %-> (true (up 0))) D2)) + %<- ({h2} cut P1 (E1 h2) ([h1] D2 h1) (%the (true (up 0)) (E h2))) %. +%term _ + %pi (trans-equivX (t+circ T) (%the (true (circ P)) D) (t+circ S) (circR F)) + %<- (invcircR D (%the (lax P) D1)) + %<- (identity P (%the (%pi (hyp P) %-> (true P)) D2)) + %<- ({h1} trans-equivX T (D2 h1) S (%the (true B) (E h1))) + %<- (lcut P D1 ([h] laxR (E h)) F) %. +%worlds (bl_atmpos bl_atmneg bl_hyp) (trans-equiv _ _ _ _) (trans-equivX _ _ _ _) %. +%total [(T1 T2) (S1 S2)] (trans-equiv T1 _ S1 _) (trans-equivX S2 _ T2 _) %. +%{! ### Soundness !}% +%{! * If and and , then (``trans-sound``) + +Proved by induction on the target logic derivation . !}% +%sort trans-sound %. +%mode trans-sound %in %in %out %. +%{! Every case of translating a left rule needs to translate a hypothesis, +and we can usefully predict the form of the output, except in the cases where +the reverse translation is not determinstic. The first lemma, +``trans-init``, is essentally just an [output factoring](/wiki/output-factoring/) +lemma because the case of translating +atomic formula really consists of two cases, the positive atoms and the +negative atoms, and this triggers Twelf's [output freeness](/wiki/output-freeness/) check. + +The second nondetermistic case is where we are translating an implication +in the target logic and may encounter either an implication or a +negation in the source logic. This case (``trans-imp``) +becomes mutually inductive +with the main theorem, and establishes a pattern that will continue in the +completeness theorem. + +One case in particular is worth pointing out. We have + in the target logic using the +rule ``upR``, which gives us a subderivation +, and we are +translating into a contradiction in the +target logic (let be the translated +). + +By the induction hypothesis we have +, +which we have defined to be . +We need to prove, however, . Let +[me](/wiki/user-rsimmons/) know if I'm wrong, but the only way I could figure out +how to do this was by using the cut rule in the source logic, which means +that we do _not_ get cut elimination "for free" in the source logic by virtue +of cut admissibility in the target logic. + + + +This awkwardness is a result of the slight mismatch between falsehood in +the target logic and contradiction in the source logic, and is part of +the reason we didn't include falsehood as a primitive in the source logic. +!}% +%sort trans-init %. +%term _ trans-init tatom H H %. +%mode trans-init %in %in %out %. +%worlds (bl_atmpos bl_atmneg bl_trans-soundhyp) (trans-init _ _ _) %. +%total [] (trans-init _ _ _) %. +%term _ + %pi (trans-sound (init+ H) (tc tatom) (id' H'')) + %<- (trans-soundhyp H T H') + %<- (trans-init T H' H'') %. +%term _ + %pi (trans-sound (init- H) (tc tatom) (id' H'')) + %<- (trans-soundhyp H T H') + %<- (trans-init T H' H'') %. +%term _ %pi (trans-sound (laxR D) (tclax T) (circR' E)) %<- (trans-sound D (tc T) E) %. +%term _ %pi (trans-sound (upR D) (tc (t+- T)) E) %<- (trans-sound D (tc T) E) %. +%term _ + %pi (trans-sound (upR D) tc# (cut' E ([h] notL' trueR' h))) + %<- (trans-sound D (tc t+false) E) %. +%term _ + %pi (trans-sound (upL D H) T (E H')) + %<- (trans-soundhyp H (t+- Th) H') + %<- ({h} {h'} %pi (trans-soundhyp h Th h') %-> (trans-sound (D h) T (%the (conc' M C) (E h')))) %. +%term _ %pi (trans-sound (downR D) (tc (t-+ T)) E) %<- (trans-sound D (tc T) E) %. +%term _ + %pi (trans-sound (downL D H) T (E H')) + %<- (trans-soundhyp H (t-+ Th) H') + %<- ({h} {h'} %pi (trans-soundhyp h Th h') %-> (trans-sound (D h) T (%the (conc' M C) (E h')))) %. +%term _ trans-sound topR (tc t-true) trueR' %. +%term _ trans-sound 1R (tc t+true) trueR' %. +%term _ %pi (trans-sound (0L H) _ (notL' trueR' H')) %<- (trans-soundhyp H t+false H') %. +%term _ + %pi (trans-sound (0L H) _ (weakR' (notL' trueR' H'))) + %<- (trans-soundhyp H t+false H') %. +%term _ + %pi (trans-sound (*R D1 D2) (tc (t+and T1 T2)) (/\R' E1 E2)) + %<- (trans-sound D1 (tc T1) E1) + %<- (trans-sound D2 (tc T2) E2) %. +%term _ + %pi (trans-sound (*L D H) T (/\L' ([h1] [h2] E h1 h2) H')) + %<- (trans-soundhyp H (t+and Th1 Th2) H') + %<- ({h1} {h1'} + %pi (trans-soundhyp h1 Th1 h1') + %-> ({h2} {h2'} %pi (trans-soundhyp h2 Th2 h2') %-> (trans-sound (D h1 h2) T (E h1' h2')))) %. +%term _ + %pi (trans-sound (&R D1 D2) (tc (t-and T1 T2)) (/\R' E1 E2)) + %<- (trans-sound D1 (tc T1) E1) + %<- (trans-sound D2 (tc T2) E2) %. +%term _ + %pi (trans-sound (&L1 D H) T (/\L' ([h1] [h2] E h1) H')) + %<- (trans-soundhyp H (t-and Th _) H') + %<- ({h} {h'} %pi (trans-soundhyp h Th h') %-> (trans-sound (D h) T (E h'))) %. +%term _ + %pi (trans-sound (&L2 D H) T (/\L' ([h1] [h2] E h2) H')) + %<- (trans-soundhyp H (t-and _ Th) H') + %<- ({h} {h'} %pi (trans-soundhyp h Th h') %-> (trans-sound (D h) T (E h'))) %. +%term _ %pi (trans-sound (+R1 D) (tc (t+or T _)) (\/R1' E)) %<- (trans-sound D (tc T) E) %. +%term _ %pi (trans-sound (+R2 D) (tc (t+or _ T)) (\/R2' E)) %<- (trans-sound D (tc T) E) %. +%term _ + %pi (trans-sound (+L D1 D2 H) T (\/L' E1 E2 H')) + %<- (trans-soundhyp H (t+or Th1 Th2) H') + %<- ({h1} {h1'} %pi (trans-soundhyp h1 Th1 h1') %-> (trans-sound (D1 h1) T (E1 h1'))) + %<- ({h2} {h2'} %pi (trans-soundhyp h2 Th2 h2') %-> (trans-sound (D2 h2) T (E2 h2'))) %. +%term _ + %pi (trans-sound (-oR D) (tc (t-imp T1 T2)) (=>R' E)) + %<- ({h} {h'} %pi (trans-soundhyp h T1 h') %-> (trans-sound (D h) (tc T2) (E h'))) %. +%term _ + %pi (trans-sound (-oR D) (tc (t+not T)) (notR' E)) + %<- ({h hyp A} {h' hyp' A'} %pi (trans-soundhyp h T h') %-> (trans-sound (D h) tc# (E h'))) %. +%sort trans-imp {D conc C} %. +%mode trans-imp %in %in %in %in %in %out %. +%term _ + %pi (trans-imp (-oL D1 D2 H) idconc/refl T (t-imp Th1 Th2) H' (=>L' E1 E2 H')) + %<- (trans-sound D1 (tc Th1) E1) + %<- ({h} {h'} %pi (trans-soundhyp h Th2 h') %-> (trans-sound (D2 h) T (E2 h'))) %. +%term _ + %pi (trans-imp (-oL D1 D2 H) idconc/refl T (t+not Th) (%the (hyp' (not' A')) H') (weakR' (notL' E1 H'))) + %<- (trans-sound D1 (tc Th) (%the (true' Md A') E1)) %. +%term _ + %pi (trans-imp (-oL D1 D2 H) idconc/refl tc# (t+not Th) (%the (hyp' (not' A')) H') (notL' E1 H')) + %<- (trans-sound D1 (tc Th) (%the (true' Md A') E1)) %. +%term _ + %pi (trans-sound (-oL D1 D2 H) T F) + %<- (trans-soundhyp H Th H') + %<- (trans-imp (-oL D1 D2 H) idconc/refl T Th H' F) %. +%term _ %pi (trans-sound (circR D) (tc (t+circ T)) E) %<- (trans-sound D (tclax T) E) %. +%term _ + %pi (trans-sound (circL D H) (tclax T) (circL' E H')) + %<- (trans-soundhyp H (t+circ Th) H') + %<- ({h} {h'} %pi (trans-soundhyp h Th h') %-> (trans-sound (D h) (tclax T) (E h'))) %. +%worlds (bl_atmpos bl_atmneg bl_trans-soundhyp) (trans-sound _ _ _) (trans-imp _ _ _ _ _ _) %. +%total (D E) (trans-imp E _ _ _ _ _) (trans-sound D _ _) %. +%{! ### Completeness !}% +%{! * If and and , then . + +By lexographic induction on the first derivation's metric and then +the second derivation's metric. For every left rule, we need a mutually +inductive lemma that strips off any unnecessary shifts from the translation +of the hypothesis; within these lemmas, the first derivation will always +stay the same, but the derivation of the translation of the hypothesis +will get smaller. !}% +%sort trans-compl {Td} {Tt} %. +%mode trans-compl %in %in %in %in %out %. +%inline trans-compl' [d conc' Td D] [t trans-conc Tt D J] [e conc J] trans-compl Td Tt d t e %. +%{! #### Shifts !}% +%term _ %pi (trans-compl' D (tc (t+- T)) (upR E)) %<- (trans-compl' D (tc T) E) %. +%term _ %pi (trans-compl' D (tc (t-+ T)) (downR E)) %<- (trans-compl' D (tc T) E) %. +%{! #### Truth !}% +%term _ trans-compl' trueR' (tc t+true) 1R %. +%term _ trans-compl' trueR' (tc t-true) topR %. +%{! #### Conjunction !}% +%term _ + %pi (trans-compl' (/\R' D1 D2) (tc (t+and T1 T2)) (*R E1 E2)) + %<- (trans-compl' D1 (tc T1) E1) + %<- (trans-compl' D2 (tc T2) E2) %. +%term _ + %pi (trans-compl' (/\R' D1 D2) (tc (t-and T1 T2)) (&R E1 E2)) + %<- (trans-compl' D1 (tc T1) E1) + %<- (trans-compl' D2 (tc T2) E2) %. +%sort trans-andL {TD} {Th} %. +%mode trans-andL %in %in %in %in %in %in %out %. +%term _ + %pi (trans-andL _ _ id/refl (%the (%pi (hyp' M1) %-> (hyp' M2) %-> (conc' Td D')) D) (%the (trans-conc Tt D' J) T) (t-and T1 T2) ([h] &L1 ([h1] &L2 ([h2] E h1 h2) h) h)) + %<- ({h1'} {h1} + %pi (trans-complhyp h1' T1 h1) + %-> ({h2'} {h2} %pi (trans-complhyp h2' T2 h2) %-> (trans-compl' (D h1' h2') T (E h1 h2)))) %. +%term _ + %pi (trans-andL _ _ id/refl (%the (%pi (hyp' M1) %-> (hyp' M2) %-> (conc' Td D')) D) (%the (trans-conc Tt D' J) T) (t+and T1 T2) ([h] *L ([h1] [h2] E h1 h2) h)) + %<- ({h1'} {h1} + %pi (trans-complhyp h1' T1 h1) + %-> ({h2'} {h2} %pi (trans-complhyp h2' T2 h2) %-> (trans-compl' (D h1' h2') T (E h1 h2)))) %. +%term _ %pi (trans-andL _ _ Id D T (t+- Th) (upL E)) %<- (trans-andL _ _ Id D T Th E) %. +%term _ %pi (trans-andL _ _ Id D T (t-+ Th) (downL E)) %<- (trans-andL _ _ Id D T Th E) %. +%term _ + %pi (trans-compl' (/\L' D H') T (E H)) + %<- (trans-complhyp H' Th (%the (hyp A) H)) + %<- (trans-andL _ _ id/refl D T Th (%the (%pi (hyp A) %-> (conc D')) E)) %. +%{! #### Disjunction !}% +%term _ + %pi (trans-compl' (\/R1' (%the (true' Tt M1) D)) (tc (t+or T1 _)) (+R1 E)) + %<- (trans-compl' D (tc T1) E) %. +%term _ + %pi (trans-compl' (\/R2' (%the (true' Tt M2) D)) (tc (t+or _ T2)) (+R2 E)) + %<- (trans-compl' D (tc T2) E) %. +%sort trans-orL {TD} {Th} %. +%mode trans-orL %in %in %in %in %in %in %in %out %. +%term _ + %pi (trans-orL _ _ id/refl D1 D2 T (t+or Th1 Th2) (+L ([h1] E1 h1) ([h2] E2 h2))) + %<- ({h1' hyp' M1} {h1 hyp P1} + %pi (trans-complhyp h1' Th1 h1) + %-> (trans-compl' (D1 h1') T (%the (conc J) (E1 h1)))) + %<- ({h2' hyp' M2} {h2 hyp P2} + %pi (trans-complhyp h2' Th2 h2) + %-> (trans-compl' (D2 h2') T (%the (conc J) (E2 h2)))) %. +%term _ + %pi (trans-orL _ _ id/refl D1 D2 T (t+- Th) (upL E)) + %<- (trans-orL _ _ id/refl D1 D2 T Th E) %. +%term _ + %pi (trans-orL _ _ id/refl D1 D2 T (t-+ Th) (downL E)) + %<- (trans-orL _ _ id/refl D1 D2 T Th E) %. +%term _ + %pi (trans-compl' (\/L' D1 D2 H') T (E H)) + %<- (trans-complhyp H' Th H) + %<- (trans-orL _ _ id/refl D1 D2 T Th E) %. +%{! #### Implication !}% +%term _ + %pi (trans-compl' (=>R' (%the (%pi (hyp' M1) %-> (true' Tt M2)) D)) (tc (t-imp T1 T2)) (-oR E)) + %<- ({h'} {h} %pi (trans-complhyp h' T1 h) %-> (trans-compl' (D h') (tc T2) (E h))) %. +%sort trans-impL {TD} {Th} %. +%mode trans-impL %in %in %in %in %in %in %in %out %. +%term _ + %pi (trans-impL _ _ id/refl (%the (true' Td1 M1) D1) (%the (%pi (hyp' M2) %-> (conc' Td2 D')) D2) (%the (trans-conc Mt D' J) T) (t-imp T1 T2) (-oL E1 ([h] E2 h))) + %<- (trans-compl' D1 (tc T1) (%the (true P) E1)) + %<- ({h' hyp' M2} {h hyp N} {t trans-complhyp h' T2 h} trans-compl' (D2 h') T (%the (conc J) (E2 h))) %. +%term _ + %pi (trans-impL _ _ id/refl D1 D2 T (t+- Th) (upL E)) + %<- (trans-impL _ _ id/refl D1 D2 T Th E) %. +%term _ + %pi (trans-impL _ _ id/refl D1 D2 T (t-+ Th) (downL E)) + %<- (trans-impL _ _ id/refl D1 D2 T Th E) %. +%term _ + %pi (trans-compl' (=>L' D1 D2 (%the (hyp' (M1 => M2)) H')) T (E H)) + %<- (trans-complhyp H' Th (%the (hyp A) H)) + %<- (trans-impL _ _ id/refl D1 D2 T Th E) %. +%{! #### Negation !}% +%term _ + %pi (trans-compl' (notR' (%the (%pi (hyp' tt') %-> (# _)) D)) (tc t+false) G) + %<- ({h'} {h} %pi (trans-complhyp h' t+true h) %-> (trans-compl' (D h') tc# (E h))) + %<- (cut 1 1R ([h] E h) F) + %<- (invupR F G) %. +%term _ + %pi (trans-compl' (notR' (%the (%pi (hyp' M) %-> (# _)) D)) (tc (t+not T)) (-oR ([h] E h))) + %<- ({h'} {h} %pi (trans-complhyp h' T h) %-> (trans-compl' (D h') tc# (E h))) %. +%sort trans-notL {TD} {Tt} %. +%mode trans-notL %in %in %in %in %in %out %. +%term _ + %pi (trans-notL _ _ id/refl (%the (true' Td M) D) (t+not T) (-oL E (upL 0L))) + %<- (trans-compl' D (tc T) (%the (true P) E)) %. +%term _ trans-notL _ _ id/refl D t+false 0L %. +%term _ %pi (trans-notL _ _ Id D (t+- T) (upL E)) %<- (trans-notL _ _ Id D T E) %. +%term _ %pi (trans-notL _ _ Id D (t-+ T) (downL E)) %<- (trans-notL _ _ Id D T E) %. +%term _ + %pi (trans-compl' (notL' (%the (true' Td M) D) (%the (hyp' (not' M)) H')) tc# (E H)) + %<- (trans-complhyp H' Th (%the (hyp A) H)) + %<- (trans-notL _ _ id/refl D Th E) %. +%{! #### Lax modality !}% +%term _ + %pi (trans-compl' (circR' D) (tc (t+circ T)) (circR (laxR E))) + %<- (trans-compl' D (tc T) E) %. +%term _ %pi (trans-compl' (circR' D) (tclax T) (laxR E)) %<- (trans-compl' D (tc T) E) %. +%sort trans-circL {TD} {Th} %. +%mode trans-circL %in %in %in %in %in %in %out %. +%term _ + %pi (trans-circL _ _ id/refl D T (t+circ Th) (circL E)) + %<- ({h' hyp' M} {h hyp P} {t trans-complhyp h' Th h} trans-compl' (D h') (tclax T) (%the (lax C) (E h))) %. +%term _ %pi (trans-circL _ _ Id D T (t+- Th) (upL F)) %<- (trans-circL _ _ Id D T Th F) %. +%term _ %pi (trans-circL _ _ Id D T (t-+ Th) (downL F)) %<- (trans-circL _ _ Id D T Th F) %. +%term _ + %pi (trans-compl' (circL' (%the (%pi (hyp' M) %-> (true' Td (circ' D'))) D) H') (tc (t+circ (%the (trans Tt D' pos J) T))) (circR (E H))) + %<- (trans-complhyp H' Th (%the (hyp A) H)) + %<- (trans-circL _ _ id/refl D T Th E) %. +%term _ + %pi (trans-compl' (circL' (%the (%pi (hyp' M) %-> (true' Td (circ' D'))) D) H') (tclax (%the (trans Tt D' S J) T)) (E H)) + %<- (trans-complhyp H' Th (%the (hyp A) H)) + %<- (trans-circL _ _ id/refl D T Th E) %. +%{! #### Structural Rules !}% +%{! For translating a cut, we need to come up with a translation of a source +logc expression to a target logic expression; we have already established +this was possible when we proved ``can-trans`` to show that translation +was left-total. !}% +%term _ + %pi (trans-compl' (cut' (%the (true' _ M) Da) Dc) T F) + %<- (can-trans M (%the (trans _ M S A) TRa)) + %<- (trans-compl' Da (tc TRa) (%the (true A) Ea)) + %<- ({h'} {h} %pi (trans-complhyp h' TRa h) %-> (trans-compl' (Dc h') T (Ec h))) + %<- (cut A Ea Ec F) %. +%{! Due to the mismatch between +the lax judgment and the circle, we need three cases for translating identity. +The first case is the most important one and is straightforward, however: we +are translating and need to show +, +where . + +Identity is a terminal rule, so we will not use induction here. +We know because is a hypothesis that there is +a hypothesis such that +, and so by the identity principle we can prove +; then, ``trans-equiv`` allows +us to prove , which is what we +actually need. !}% +%term _ + %pi (trans-compl' (id' (%the (hyp' M) H')) (tc (%the (trans _ M S Am) T)) E) + %<- (trans-complhyp H' Th (%the (hyp A) H)) + %<- (identity A (%the (%pi (hyp A) %-> (true A)) D2)) + %<- (trans-equiv Th (D2 H) T (%the (true Am) E)) %. +%{! When we are translating to a lax judgment in the target language, what we +get out of the translation isn't exactly what we want, we need + but the best we can get is + or +. However, we can +use inversion lemmas (and lax cut, in the second case) to get what we need. +!}% +%term _ + %pi (trans-compl' (id' (%the (hyp' (circ' M)) H')) (tclax (%the (trans _ M pos Pm) T)) F) + %<- (trans-complhyp H' Th (%the (hyp P) H)) + %<- (identity P (%the (%pi (hyp P) %-> (true P)) D2)) + %<- (trans-equiv Th (D2 H) (t+circ T) (%the (true (circ Pm)) E)) + %<- (invcircR E (%the (lax Pm) F)) %. +%term _ + %pi (trans-compl' (id' (%the (hyp' (circ' M)) H')) (tclax (%the (trans _ M neg Nm) T)) G) + %<- (trans-complhyp H' Th (%the (hyp N) H)) + %<- (identity N (%the (%pi (hyp N) %-> (true N)) D2)) + %<- (trans-equiv Th (D2 H) (t+circ (t-+ T)) (%the (true (circ (down Nm))) E)) + %<- (invcircR E (%the (lax (down Nm)) F)) + %<- (identity _ (%the (%pi (hyp Nm) %-> (true Nm)) Id)) + %<- (lcut _ F ([h] laxR (downL ([h'] Id h') h)) (%the (lax Nm) G)) %. +%term _ + %pi (trans-compl' (weakR' D) _ F) + %<- (trans-compl' D tc# E) + %<- (cut (up 0) E ([h] upL 0L h) F) %. +%worlds (bl_atmpos bl_atmneg bl_trans-complhyp) (trans-compl _ _ _ _ _) (trans-andL _ _ _ _ _ _ _) (trans-impL _ _ _ _ _ _ _ _) (trans-orL _ _ _ _ _ _ _ _) (trans-notL _ _ _ _ _ _) (trans-circL _ _ _ _ _ _ _) %. +%total {(D1 D2 D3 D4 D5 D6) (T1 T2 T3 T4 T5 T6)} (trans-compl D1 T1 _ _ _) (trans-andL D2 T2 _ _ _ _ _) (trans-orL D3 T3 _ _ _ _ _ _) (trans-impL D4 T4 _ _ _ _ _ _) (trans-notL D5 T5 _ _ _ _) (trans-circL D6 T6 _ _ _ _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Lax_logic). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/letrec.lf b/new-tests/stelf-output/pages/letrec.lf new file mode 100644 index 0000000..6c75eac --- /dev/null +++ b/new-tests/stelf-output/pages/letrec.lf @@ -0,0 +1,295 @@ +% +%{! An example of encoding ``letrec``, i.e. let-binding a bundle of mutually recursive expressions !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort tp %. +%term natt tp %. +%term arrow %pi tp %-> tp %-> tp %. +%sort tplist %. +%term tplist/z tplist z %. +%term tplist/s %pi tp %-> (tplist N) %-> (tplist (s N)) %. +%sort exp %. +%sort oexp %. +%sort explist %. +%{! +oexplist N M, is a list of M expressions with N bound variables +!}% +%sort oexplist %. +%term zero exp %. +%term succ %pi exp %-> exp %. +%term case %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%term let %pi exp %-> (%pi exp %-> exp) %-> exp %. +%term letrec %pi (oexplist N N) %-> (oexp N) %-> exp %. +%term bproj %pi (oexplist N N) %-> nat %-> exp %. +% projecting from a bundle +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term oexp/z %pi exp %-> (oexp z) %. +%term oexp/s %pi (%pi exp %-> (oexp N)) %-> (oexp (s N)) %. +%term explist/z explist z %. +%term explist/s %pi exp %-> (explist N) %-> (explist (s N)) %. +%term oexplist/z %pi (explist N) %-> (oexplist z N) %. +%term oexplist/s %pi (%pi exp %-> (oexplist N M)) %-> (oexplist (s N) M) %. +%sort tplist-get %. +%term tplist-get/hit tplist-get z (tplist/s T TL) T %. +%term tplist-get/miss %pi (tplist-get (s N) (tplist/s T TL) T') %<- (tplist-get N TL T') %. +%sort of-exp %. +%sort of-oexp %. +%sort of-explist %. +%sort of-oexplist %. +%term of-exp/zero of-exp zero natt %. +%term of-exp/succ %pi (of-exp (succ E) natt) %<- (of-exp E natt) %. +%term of-exp/case + %pi (of-exp (case E1 E2 E3) T) + %<- (of-exp E1 natt) + %<- (of-exp E2 T) + %<- ({x} %pi (of-exp x natt) %-> (of-exp (E3 x) T)) %. +%term of-exp/let + %pi (of-exp (let E1 E2) T') + %<- (of-exp E1 T) + %<- ({x} %pi (of-exp x T) %-> (of-exp (E2 x) T')) %. +%term of-exp/letrec %pi (of-exp (letrec OEL OE) T) %<- (of-oexplist TL OEL TL) %<- (of-oexp TL OE T) %. +%term of-exp/bproj %pi (of-exp (bproj OE N) T) %<- (of-oexplist TL OE TL) %<- (tplist-get N TL T) %. +%term of-exp/lam + %pi (of-exp (lam E) (arrow T1 T2)) + %<- ({x} %pi (of-exp x T1) %-> (of-exp (E x) T2)) %. +%term of-exp/app %pi (of-exp (app E1 E2) T2) %<- (of-exp E1 (arrow T1 T2)) %<- (of-exp E2 T1) %. +%term of-oexp/z %pi (of-oexp tplist/z (oexp/z E) T) %<- (of-exp E T) %. +%term of-oexp/s + %pi (of-oexp (tplist/s T TL) (oexp/s ([x] EL x)) T') + %<- ({x} %pi (of-exp x T) %-> (of-oexp TL (EL x) T')) %. +%sort of-explist %. +%term of-explist/z of-explist explist/z tplist/z %. +%term of-explist/s + %pi (of-explist (explist/s E EL) (tplist/s T TL)) + %<- (of-exp E T) + %<- (of-explist EL TL) %. +%term of-oexplist/z %pi (of-oexplist tplist/z (oexplist/z EL) TL) %<- (of-explist EL TL) %. +%term of-oexplist/s + %pi (of-oexplist (tplist/s T TL) (oexplist/s OEL) TL') + %<- ({x} %pi (of-exp x T) %-> (of-oexplist TL (OEL x) TL')) %. +%sort value %. +%term value/zero value zero %. +%term value/succ %pi (value (succ E)) %<- (value E) %. +%term value/lam value (lam E) %. +%sort subst-oexp %. +%term subst-oexp/z subst-oexp explist/z (oexp/z E) E %. +%term subst-oexp/s + %pi (subst-oexp (explist/s E EL) (oexp/s ([x] OE x)) E') + %<- (subst-oexp EL (OE E) E') %. +%sort expand-oexplist %. +%term expand-oexplist/z expand-oexplist _ _ (oexplist/z EL) EL %. +%term expand-oexplist/s + %pi (expand-oexplist N OEL (oexplist/s OEL') EL) + %<- (expand-oexplist (s N) OEL (OEL' (bproj OEL N)) EL) %. +%sort explist-get %. +%term explist-get/hit explist-get z (explist/s E EL) E %. +%term explist-get/miss %pi (explist-get (s N) (explist/s E EL) E') %<- (explist-get N EL E') %. +%sort step %. +%term step/succ %pi (step (succ E) (succ E')) %<- (step E E') %. +%term step/case %pi (step (case E1 E2 E3) (case E1' E2 E3)) %<- (step E1 E1') %. +%term step/case-beta-1 step (case zero E2 E3) E2 %. +%term step/case-beta-2 %pi (step (case (succ E1) E2 E3) (E3 E1)) %<- (value E1) %. +%term step/let %pi (step (let E1 E2) (let E1' E2)) %<- (step E1 E1') %. +%term step/let-beta %pi (step (let E1 E2) (E2 E1)) %<- (value E1) %. +%term step/letrec + %pi (step (letrec OEL OE) E) + %<- (expand-oexplist z OEL OEL EL) + %<- (subst-oexp EL OE E) %. +%term step/bproj + %pi (step (bproj OEL N) E) + %<- (expand-oexplist z OEL OEL EL) + %<- (explist-get N EL E) %. +%term step/app-1 %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app-2 %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app-beta %pi (step (app (lam E) E2) (E E2)) %<- (value E2) %. +%sort nat-plus %. +%term nat-plus/z nat-plus z N N %. +%term nat-plus/s %pi (nat-plus (s N1) N2 (s N3)) %<- (nat-plus N1 N2 N3) %. +%sort nat-plus-move-s %. +%mode nat-plus-move-s %in %out %. +%term _ nat-plus-move-s (nat-plus/s nat-plus/z) nat-plus/z %. +%term _ %pi (nat-plus-move-s (nat-plus/s D1) (nat-plus/s D2)) %<- (nat-plus-move-s D1 D2) %. +%worlds () (nat-plus-move-s _ _) %. +%total (D1) (nat-plus-move-s D1 _) %. +%sort can-expand-oexplist {N} {OEL oexplist NN NN} {OEL' oexplist N NN} %. +%mode can-expand-oexplist %in %in %in %in %out %. +%term _ can-expand-oexplist z OEL (oexplist/z EL) nat-plus/z expand-oexplist/z %. +%term _ + %pi (can-expand-oexplist (s N) _ _ NP (expand-oexplist/s DEO)) + %<- (nat-plus-move-s NP NP') + %<- (can-expand-oexplist N _ _ NP' DEO) %. +%worlds () (can-expand-oexplist _ _ _ _ _) %. +%total (D1) (can-expand-oexplist D1 _ _ _ _) %. +%sort nat-plus-z {N} %. +%mode nat-plus-z %in %out %. +%term _ nat-plus-z _ nat-plus/z %. +%term _ %pi (nat-plus-z _ (nat-plus/s D1)) %<- (nat-plus-z _ D1) %. +%worlds () (nat-plus-z _ _) %. +%total (D1) (nat-plus-z D1 _) %. +%sort notstuck %. +%term notstuck/value %pi (notstuck E) %<- (value E) %. +%term notstuck/step %pi (notstuck E) %<- (step E E') %. +%sort can-subst-oexp {EL explist N} {OE oexp N} %. +%mode can-subst-oexp %in %in %out %. +%term _ can-subst-oexp _ _ subst-oexp/z %. +%term _ + %pi (can-subst-oexp (explist/s E EL) _ (subst-oexp/s D1)) + %<- (can-subst-oexp EL _ D1) %. +%worlds () (can-subst-oexp _ _ _) %. +%total (D1) (can-subst-oexp D1 _ _) %. +%sort can-explist-get {EL explist N'} %. +%mode can-explist-get %in %in %out %. +%term _ can-explist-get _ tplist-get/hit explist-get/hit %. +%term _ + %pi (can-explist-get _ (tplist-get/miss D2) (explist-get/miss D')) + %<- (can-explist-get _ D2 D') %. +%worlds () (can-explist-get _ _ _) %. +%total (D2) (can-explist-get _ D2 _) %. +%sort progress-case {E1} {E2} %. +%mode progress-case %in %in %in %in %out %. +%term _ progress-case _ _ _ (notstuck/step S) (notstuck/step (step/case S)) %. +%term _ progress-case _ _ of-exp/zero (notstuck/value _) (notstuck/step step/case-beta-1) %. +%term _ progress-case _ _ (of-exp/succ _) (notstuck/value (value/succ DV)) (notstuck/step (step/case-beta-2 DV)) %. +%worlds () (progress-case _ _ _ _ _) %. +%total {} (progress-case _ _ _ _ _) %. +%sort progress-succ %. +%mode progress-succ %in %out %. +%term _ progress-succ (notstuck/step DS) (notstuck/step (step/succ DS)) %. +%term _ progress-succ (notstuck/value DV1) (notstuck/value (value/succ DV1)) %. +%worlds () (progress-succ _ _) %. +%total {} (progress-succ _ _) %. +%sort progress-app %. +%mode progress-app %in %in %in %in %out %. +%term _ progress-app _ _ (notstuck/step DS) _ (notstuck/step (step/app-1 DS)) %. +%term _ progress-app _ _ (notstuck/value V) (notstuck/step DS) (notstuck/step (step/app-2 DS V)) %. +%term _ progress-app (of-exp/lam DM) D2 (notstuck/value value/lam) (notstuck/value DV) (notstuck/step (step/app-beta DV)) %. +%worlds () (progress-app _ _ _ _ _) %. +%total {} (progress-app _ _ _ _ _) %. +%sort progress-let {E2} %. +%mode progress-let %in %in %out %. +%term _ progress-let _ (notstuck/step S) (notstuck/step (step/let S)) %. +%term _ progress-let _ (notstuck/value V) (notstuck/step (step/let-beta V)) %. +%worlds () (progress-let _ _ _) %. +%total {} (progress-let _ _ _) %. +%sort progress %. +%mode progress %in %out %. +%term _ progress (of-exp/lam _) (notstuck/value value/lam) %. +%term _ + %pi (progress (of-exp/app D2 D1) NS3) + %<- (progress D1 NS1) + %<- (progress D2 NS2) + %<- (progress-app D1 D2 NS1 NS2 NS3) %. +%term _ progress of-exp/zero (notstuck/value value/zero) %. +%term _ + %pi (progress (of-exp/succ D1) NS) + %<- (progress D1 NS1) + %<- (progress-succ NS1 NS) %. +%term _ + %pi (progress (of-exp/case _ _ D1) NS) + %<- (progress D1 NS1) + %<- (progress-case _ _ D1 NS1 NS) %. +%term _ + %pi (progress (of-exp/let _ D1) NS) + %<- (progress D1 NS1) + %<- (progress-let _ NS1 NS) %. +%term _ + %pi (progress (of-exp/letrec D2 D1) (notstuck/step (step/letrec DS DX))) + %<- (nat-plus-z _ DNP) + %<- (can-expand-oexplist _ _ _ DNP DX) + %<- (can-subst-oexp _ _ DS) %. +%term _ + %pi (progress (of-exp/bproj D2 D1) (notstuck/step (step/bproj DS DX))) + %<- (nat-plus-z _ DNP) + %<- (can-expand-oexplist _ _ _ DNP DX) + %<- (can-explist-get _ D2 DS) %. +%worlds () (progress _ _) %. +%total (D1) (progress D1 _) %. +%sort tplist-prefix %. +%term tplist-prefix/z tplist-prefix z TL TL %. +%term tplist-prefix/s %pi (tplist-prefix (s N) (tplist/s T TL) TL') %<- (tplist-prefix N TL TL') %. +%sort tplist-prefix-get %. +%mode tplist-prefix-get %in %out %. +%term _ tplist-prefix-get tplist-prefix/z tplist-get/hit %. +%term _ + %pi (tplist-prefix-get (tplist-prefix/s D1) (tplist-get/miss D2)) + %<- (tplist-prefix-get D1 D2) %. +%worlds () (tplist-prefix-get _ _) %. +%total (D1) (tplist-prefix-get D1 _) %. +%sort tplist-prefix-s %. +%mode tplist-prefix-s %in %out %. +%term _ tplist-prefix-s tplist-prefix/z (tplist-prefix/s tplist-prefix/z) %. +%term _ + %pi (tplist-prefix-s (tplist-prefix/s D1) (tplist-prefix/s D2)) + %<- (tplist-prefix-s D1 D2) %. +%worlds () (tplist-prefix-s _ _) %. +%total (D1) (tplist-prefix-s D1 _) %. +%sort preservation-expand-oexplist %. +%mode preservation-expand-oexplist %in %in %in %in %out %. +%term _ preservation-expand-oexplist _ DOE (of-oexplist/z DEL) expand-oexplist/z DEL %. +%term _ + %pi (preservation-expand-oexplist DO DOE (of-oexplist/s DOE') (expand-oexplist/s DOX) DEL) + %<- (tplist-prefix-s DO DO') + %<- (tplist-prefix-get DO DG) + %<- (preservation-expand-oexplist DO' DOE (DOE' _ (of-exp/bproj DG DOE)) DOX DEL) %. +%worlds () (preservation-expand-oexplist _ _ _ _ _) %. +%total (D1) (preservation-expand-oexplist _ _ _ D1 _) %. +%sort preservation-subst-oexp %. +%mode preservation-subst-oexp %in %in %in %out %. +%term _ preservation-subst-oexp _ (of-oexp/z D1) subst-oexp/z D1 %. +%term _ + %pi (preservation-subst-oexp (of-explist/s D1 D) (of-oexp/s D2) (subst-oexp/s D3) D4) + %<- (preservation-subst-oexp D1 (D2 _ D) D3 D4) %. +%worlds () (preservation-subst-oexp _ _ _ _) %. +%total (D1) (preservation-subst-oexp _ _ D1 _) %. +%sort preservation-get %. +%mode preservation-get %in %in %in %out %. +%term _ preservation-get (of-explist/s _ D) explist-get/hit tplist-get/hit D %. +%term _ + %pi (preservation-get (of-explist/s DL _) (explist-get/miss D') (tplist-get/miss D'') D) + %<- (preservation-get DL D' D'' D) %. +%worlds () (preservation-get _ _ _ _) %. +%total (D1) (preservation-get _ _ D1 _) %. +%sort preservation %. +%mode preservation %in %in %out %. +%term _ + %pi (preservation (of-exp/app D2 D1) (step/app-1 DS) (of-exp/app D2 D1')) + %<- (preservation D1 DS D1') %. +%term _ + %pi (preservation (of-exp/app D2 D1) (step/app-2 DS V) (of-exp/app D2' D1)) + %<- (preservation D2 DS D2') %. +%term _ preservation (of-exp/app D2 (of-exp/lam D1)) (step/app-beta V) (D1 _ D2) %. +%term _ + %pi (preservation (of-exp/succ D1) (step/succ DS) (of-exp/succ D1')) + %<- (preservation D1 DS D1') %. +%term _ + %pi (preservation (of-exp/case D3 D2 D1) (step/case DS) (of-exp/case D3 D2 D1')) + %<- (preservation D1 DS D1') %. +%term _ preservation (of-exp/case D3 D2 _) step/case-beta-1 D2 %. +%term _ preservation (of-exp/case D3 _ (of-exp/succ D1)) (step/case-beta-2 _) (D3 _ D1) %. +%term _ + %pi (preservation (of-exp/let D2 D1) (step/let DS) (of-exp/let D2 D1')) + %<- (preservation D1 DS D1') %. +%term _ preservation (of-exp/let D2 D1) (step/let-beta _) (D2 _ D1) %. +%term _ + %pi (preservation (of-exp/letrec D2 D1) (step/letrec DOS DOX) D) + %<- (preservation-expand-oexplist tplist-prefix/z D1 D1 DOX D1') + %<- (preservation-subst-oexp D1' D2 DOS D) %. +%term _ + %pi (preservation (of-exp/bproj D2 D1) (step/bproj DOS DOX) D) + %<- (preservation-expand-oexplist tplist-prefix/z D1 D1 DOX D1') + %<- (preservation-get D1' DOS D2 D) %. +%worlds () (preservation _ _ _) %. +%total (D1) (preservation _ D1 _) %. +%{! +TODO: commentary. + +\{\{stub\}\} +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Letrec). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/lexicographical-orderings-with-density.lf b/new-tests/stelf-output/pages/lexicographical-orderings-with-density.lf new file mode 100644 index 0000000..c156c53 --- /dev/null +++ b/new-tests/stelf-output/pages/lexicographical-orderings-with-density.lf @@ -0,0 +1,97 @@ +% +%{! This is a fairly advanced example, but it illustrates a number of useful things. First, it adheres very closely to the [Twelf style guide](/wiki/twelf-style-guide/). Second, it has proofs of ``nat-less-trans`` and ``nat-less-immsucc`` which you usually need to get any mileage out of the natural numbers. Third, it is an example of how to prove ``uninhabited/void/false/absurd`` in order to user [reasoning from false](/wiki/reasoning-from-false/). + +This is a lexicographical ordering with the property that it has a less-than relation that is irreflexive and transitive, like the natural numbers. In addition, it has the additional property that the ordering is "dense" in that there exists an index between any two indices. This ordering should be isomorphic to the non-negative rationals. Density is admissible in this ordering without having to deal with multiplication, as would be necessary in a more obvious encoding of the non-negative rationals. !}% +%{! (options removed from twelftag: check="true") !}% +% natural numbers +%sort nat %. +%term nat/z nat %. +%term nat/s %pi nat %-> nat %. +% a list of natural numbers. we will use it as a lexicographical ordering +%sort lex %. +%term lex/z lex nat/z %. +%term lex/s %pi nat %-> (lex N) %-> (lex (nat/s N)) %. +% the less-than relation on natural numbers +%sort nat-less %. +%term nat-less/z nat-less nat/z (nat/s N) %. +%term nat-less/s %pi (nat-less (nat/s N1) (nat/s N2)) %<- (nat-less N1 N2) %. +% the less-than relation on our lexicographical ordering. +%sort lex-less %. +%term lex-less/z lex-less lex/z (lex/s _ _) %. +%term lex-less/eq %pi (lex-less (lex/s N1 NL1) (lex/s N1 NL2)) %<- (lex-less NL1 NL2) %. +%term lex-less/gt %pi (lex-less (lex/s N1 _) (lex/s N2 _)) %<- (nat-less N2 N1) %. +%sort nat-less-trans %. +%mode nat-less-trans %in %in %out %. +%term _ nat-less-trans nat-less/z _ nat-less/z %. +%term _ + %pi (nat-less-trans (nat-less/s N1) (nat-less/s N2) (nat-less/s N3)) + %<- (nat-less-trans N1 N2 N3) %. +%worlds () (nat-less-trans _ _ _) %. +%total {D1} (nat-less-trans D1 _ _) %. +%sort nat-less-immsucc {N} %. +%mode nat-less-immsucc %in %out %. +%term _ nat-less-immsucc nat/z nat-less/z %. +%term _ %pi (nat-less-immsucc (nat/s N1) (nat-less/s NL)) %<- (nat-less-immsucc N1 NL) %. +%worlds () (nat-less-immsucc _ _) %. +%total {D1} (nat-less-immsucc D1 _) %. +%sort lex-less-trans %. +%mode lex-less-trans %in %in %out %. +%term _ lex-less-trans lex-less/z _ lex-less/z %. +%term _ + %pi (lex-less-trans (lex-less/eq NL1) (lex-less/eq NL2) (lex-less/eq NL3)) + %<- (lex-less-trans NL1 NL2 NL3) %. +%term _ + %pi (lex-less-trans (lex-less/gt NL1) (lex-less/gt NL2) (lex-less/gt NL3)) + %<- (nat-less-trans NL2 NL1 NL3) %. +%term _ lex-less-trans (lex-less/gt NL) (lex-less/eq _) (lex-less/gt NL) %. +%term _ lex-less-trans (lex-less/eq _) (lex-less/gt NL) (lex-less/gt NL) %. +%worlds () (lex-less-trans _ _ _) %. +%total {D1} (lex-less-trans D1 _ _) %. +% for any lexicographical index LL, there exists an index LL' that is greater +%sort lex-less-succ {LL} %. +%mode lex-less-succ %in %out %. +%term _ lex-less-succ lex/z (%the (lex-less _ (lex/s nat/z lex/z)) lex-less/z) %. +%term _ %pi (lex-less-succ (lex/s N NL) (lex-less/eq NLL)) %<- (lex-less-succ NL NLL) %. +%worlds () (lex-less-succ _ _) %. +%total {N} (lex-less-succ N _) %. +%sort lex-less-dense %. +%mode lex-less-dense %in %out %out %. +%term _ + %pi (lex-less-dense lex-less/z lex-less/z (%the (lex-less (lex/s (nat/s N) lex/z) _) (lex-less/gt NL))) + %<- (nat-less-immsucc N NL) %. +%term _ + %pi (lex-less-dense (lex-less/eq LL) (lex-less/eq LL1) (lex-less/eq LL2)) + %<- (lex-less-dense LL LL1 LL2) %. +%term _ + %pi (lex-less-dense (lex-less/gt NL) (lex-less/eq LL) (lex-less/gt NL)) + %<- (lex-less-succ _ LL) %. +%worlds () (lex-less-dense _ _ _) %. +%total {D1} (lex-less-dense D1 _ _) %. +%sort uninhabited %. +%freeze uninhabited %. +%sort nat-less-refl-uninhabited %. +%mode nat-less-refl-uninhabited %in %out %. +%term _ + %pi (nat-less-refl-uninhabited (nat-less/s NL) DU) + %<- (nat-less-refl-uninhabited NL DU) %. +%worlds () (nat-less-refl-uninhabited _ _) %. +%total {D1} (nat-less-refl-uninhabited D1 _) %. +%sort lex-less-refl-uninhabited %. +%mode lex-less-refl-uninhabited %in %out %. +%term _ + %pi (lex-less-refl-uninhabited (lex-less/eq LL) DU) + %<- (lex-less-refl-uninhabited LL DU) %. +%term _ + %pi (lex-less-refl-uninhabited (lex-less/gt DL) DU) + %<- (nat-less-refl-uninhabited DL DU) %. +%worlds () (lex-less-refl-uninhabited _ _) %. +%total {DU} (lex-less-refl-uninhabited DU _) %. +%{! [Category:Twelf code](/wiki/category-twelf-code/) + +\{\{case study\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Lexicographical_orderings_with_density). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/lf.lf b/new-tests/stelf-output/pages/lf.lf new file mode 100644 index 0000000..929b840 --- /dev/null +++ b/new-tests/stelf-output/pages/lf.lf @@ -0,0 +1,32 @@ +% +%{! **LF** is the abbreviation commonly used for the **Edinburgh Logical Framework** presented by Harper, Honsell, and Plotkin in _A Framework for Defining Logics_<ref>\{\{bibtex:hhp93lf\}\}</ref>. Harper, Honsell, and Plotkin introduce both LF, which is a [dependently typed](/wiki/dependent-types/) lambda-calculus, and a systematic methodology for representing [deductive system](/wiki/object-logic/)s, such as programming languages and logics, in this lambda-calculus. This methodology is often called the [judgments as types](/wiki/judgment/) principle, because a judgment in a deductive system is represented as an LF type family classifiying (the representations of) derivations of the judgement. Derivations in a deductive system can be checked by type checking their LF representations. An LF representation is [adequate](/wiki/adequacy/) iff it is isomorphic to the original description of the deductive system. + +Twelf includes: +* an implementation of the LF logical framework, which can be used to type check LF representations +* a [logic programming](/wiki/logic-programming/) language based on LF +* a [metatheorem](/wiki/metatheorem/) checker, which can be used to verify proofs of theorems about LF representations + + +## See also + +* Read the [Introductions to Twelf](/wiki/introductions-to-twelf/) to learn more about LF and Twelf. +* [Canonical form](/wiki/canonical-form/)s: describes the modern way of thinking about the LF type theory. +* [Higher-order abstract syntax](/wiki/higher-order-abstract-syntax/) +* [Judgments as types](/wiki/judgment/) +* [Higher-order judgements](/wiki/higher-order-judgements/) +* [Bibliography of LF](/bibliography/) +* [[w:LF (logical framework)|LF (logical framework)]] on Wikipedia + + +## References + +<references /> + + +\{\{stub\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/LF). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/lily.lf b/new-tests/stelf-output/pages/lily.lf new file mode 100644 index 0000000..93c5a7f --- /dev/null +++ b/new-tests/stelf-output/pages/lily.lf @@ -0,0 +1,2599 @@ +% +%{! This [case study](/wiki/case-studies/) concerns the metatheory of the [linear](/wiki/linear-logic/) Lily language. + +It is incomplete; ask [User:varming](/wiki/user-varming/) for updates. + + +In this example I will show examples of the following: +- Encoding of a linear type system. +- Definition of an abstract machine using evaluation frames. +- Correspondence between the big step semantics and the machine. +- Complete induction on the height of a derivation. + +## Lily - the language +### Grammar +I define the grammar for the types !}% +% Lily meta theory +% By Carsten Varming 2006 +%sort tp %. +%name tp %. +%term func %pi tp %-> tp %-> tp %. +%term all %pi (%pi tp %-> tp) %-> tp %. +%term bang %pi tp %-> tp %. +%term i tp %. +%term tensor %pi tp %-> tp %-> tp %. +%freeze tp %. +%{! and the grammar for the terms and values !}% +%sort term %. +%name term %. +%term lam %pi tp %-> (%pi term %-> term) %-> term %. +%term app %pi term %-> term %-> term %. +%term tlam %pi (%pi tp %-> term) %-> term %. +%term tapp %pi term %-> tp %-> term %. +%term thunk %pi tp %-> (%pi term %-> term) %-> term %. +%term letb %pi term %-> (%pi term %-> term) %-> term %. +%term unit term %. +%term letu %pi term %-> term %-> term %. +%term tens %pi term %-> term %-> term %. +%term lett %pi term %-> (%pi term %-> term %-> term) %-> term %. +%freeze term %. +%sort value %. +%mode value %in %. +%term val_lam value (lam _ _) %. +%term val_tlam value (tlam _) %. +%term val_thunk value (thunk _ _) %. +%term val_unit value unit %. +%term val_tens value (tens _ _) %. +%worlds () (value _) %. +%terminates {} (value _) %. +%freeze value %. +%{! Next I define the call-by-name big step evaluation relation !}% +%sort \n/ %. +%mode \n/ %in %out %. +%name \n/ %. +%prec %none 500 \n/ %. +%term ev_lam lam T M \n/ lam T M %. +%term ev_tlam tlam M \n/ tlam M %. +%term ev_thunk thunk T M \n/ thunk T M %. +%term ev_unit unit \n/ unit %. +%term ev_tens tens M N \n/ tens M N %. +%term ev_app %pi (app M N \n/ V) %<- (M \n/ lam _ M') %<- (M' N \n/ V) %. +%term ev_tapp %pi (tapp M T \n/ V) %<- (M \n/ tlam M') %<- (M' T \n/ V) %. +%term ev_letb %pi (letb M N \n/ V) %<- (M \n/ thunk T M') %<- (N (letb (thunk T M') M') \n/ V) %. +%term ev_letu %pi (letu M N \n/ V) %<- (M \n/ unit) %<- (N \n/ V) %. +%term ev_lett %pi (lett M N \n/ V) %<- (M \n/ tens M' M'') %<- (N M' M'' \n/ V) %. +%worlds () (\n/ _ _) %. +%covers \n/ %in %out %. +%freeze \n/ %. +%sort \/ %. +%prec %postfix 500 \/ %. +%mode \/ %in %. +%term terminate %pi (M \/) %<- (M \n/ V) %. +%freeze \/ %. +%{! I also have an strict (call-by-value) evaluation relation for Lily. !}% +%sort \s/ %. +%mode \s/ %in %out %. +%name \s/ %. +%prec %none 500 \s/ %. +%term evs_lam lam T M \s/ lam T M %. +%term evs_tlam tlam M \s/ tlam M %. +%term evs_thunk thunk T M \s/ thunk T M %. +%term evs_unit unit \s/ unit %. +%term evs_tens tens M N \s/ tens M N %. +%term evs_app %pi (app M N \s/ V) %<- (M \s/ lam _ M') %<- (N \s/ V') %<- (M' V' \s/ V) %. +%term evs_tapp %pi (tapp M T \s/ V) %<- (M \s/ tlam M') %<- (M' T \s/ V) %. +%term evs_letb %pi (letb M N \s/ V) %<- (M \s/ thunk T M') %<- (N (letb (thunk T M') M') \s/ V) %. +%term evs_letu %pi (letu M N \s/ V) %<- (M \s/ unit) %<- (N \s/ V) %. +%term evs_lett %pi (lett M N \s/ V) %<- (M \s/ tens M' M'') %<- (N M' M'' \s/ V) %. +%worlds () (\s/ _ _) %. +%covers \s/ %in %out %. +%freeze \s/ %. +%{! ### Value soundness +By now I can prove value soundness for the two evaluation relations. !}% +%sort value_soundness %. +%mode value_soundness %in %out %. +%term vs_lam value_soundness ev_lam val_lam %. +%term vs_tlam value_soundness ev_tlam val_tlam %. +%term vs_thunk value_soundness ev_thunk val_thunk %. +%term vs_unit value_soundness ev_unit val_unit %. +%term vs_tens value_soundness ev_tens val_tens %. +%term vs_app %pi (value_soundness (ev_app D' _) V) %<- (value_soundness D' V) %. +%term vs_tapp %pi (value_soundness (ev_tapp D' _) V) %<- (value_soundness D' V) %. +%term vs_letb %pi (value_soundness (ev_letb D' _) V) %<- (value_soundness D' V) %. +%term vs_letu %pi (value_soundness (ev_letu D' _) V) %<- (value_soundness D' V) %. +%term vs_lett %pi (value_soundness (ev_lett D' _) V) %<- (value_soundness D' V) %. +%worlds () (value_soundness _ _) %. +%freeze value_soundness %. +%total D (value_soundness D _) %. +%sort value_soundness_s %. +%mode value_soundness_s %in %out %. +%term vs_lam value_soundness_s evs_lam val_lam %. +%term vs_tlam value_soundness_s evs_tlam val_tlam %. +%term vs_thunk value_soundness_s evs_thunk val_thunk %. +%term vs_unit value_soundness_s evs_unit val_unit %. +%term vs_tens value_soundness_s evs_tens val_tens %. +%term vs_app %pi (value_soundness_s (evs_app D' _ _) V) %<- (value_soundness_s D' V) %. +%term vs_tapp %pi (value_soundness_s (evs_tapp D' _) V) %<- (value_soundness_s D' V) %. +%term vs_letb %pi (value_soundness_s (evs_letb D' _) V) %<- (value_soundness_s D' V) %. +%term vs_letu %pi (value_soundness_s (evs_letu D' _) V) %<- (value_soundness_s D' V) %. +%term vs_lett %pi (value_soundness_s (evs_lett D' _) V) %<- (value_soundness_s D' V) %. +%worlds () (value_soundness_s _ _) %. +%freeze value_soundness_s %. +%total D (value_soundness_s D _) %. +%{! It is also neat to know that values evaluate to themselves. !}% +%sort selfeval %. +%mode selfeval %in %out %. +%term selfeval_lam selfeval val_lam ev_lam %. +%term selfeval_tlam selfeval val_tlam ev_tlam %. +%term selfeval_thunk selfeval val_thunk ev_thunk %. +%term selfeval_tens selfeval val_tens ev_tens %. +%term selfeval_unit selfeval val_unit ev_unit %. +%worlds () (selfeval _ _) %. +%freeze selfeval %. +%total V (selfeval V _) %. +%sort selfevals %. +%mode selfevals %in %out %. +%term selfevals_lam selfevals val_lam evs_lam %. +%term selfevals_tlam selfevals val_tlam evs_tlam %. +%term selfevals_thunk selfevals val_thunk evs_thunk %. +%term selfevals_tens selfevals val_tens evs_tens %. +%term selfevals_unit selfevals val_unit evs_unit %. +%worlds () (selfevals _ _) %. +%freeze selfevals %. +%total V (selfevals V _) %. +%{! ### Equalities +I need some equalities on types and terms: !}% +%sort eqt %. +%mode eqt %in %out %. +%term eqt_ref eqt T T %. +%worlds () (eqt _ _) %. +%freeze eqt %. +%total D (eqt D _) %. +%sort eqt_symm %. +%mode eqt_symm %in %out %. +%term eqt_symm_rule eqt_symm eqt_ref eqt_ref %. +%worlds () (eqt_symm _ _) %. +%freeze eqt_symm %. +%total {} (eqt_symm _ _) %. +%sort eqt_ctx %. +%mode eqt_ctx %in %in %out %. +%term eqt_ctx_ref eqt_ctx eqt_ref _ eqt_ref %. +%worlds () (eqt_ctx _ _ _) %. +%freeze eqt_ctx %. +%total D (eqt_ctx D _ _) %. +%sort eqt_ctx2 %. +%mode eqt_ctx2 %in %in %in %out %. +%term eqt_ctx2_ref eqt_ctx2 eqt_ref eqt_ref _ eqt_ref %. +%worlds () (eqt_ctx2 _ _ _ _) %. +%freeze eqt_ctx2 %. +%total {} (eqt_ctx2 _ _ _ _) %. +%sort eq %. +%mode eq %in %out %. +%term eq_ref eq M M %. +%worlds () (eq _ _) %. +%freeze eq %. +%total D (eq D _) %. +%sort eq_ctx %. +%mode eq_ctx %in %in %out %. +%term eq_ctx_ref eq_ctx eq_ref _ eq_ref %. +%worlds () (eq_ctx _ _ _) %. +%freeze eq_ctx %. +%total D (eq_ctx D _ _) %. +%sort eq_ctx2 %. +%mode eq_ctx2 %in %in %in %out %. +%term eq_ctx2_ref eq_ctx2 eq_ref eq_ref _ eq_ref %. +%worlds () (eq_ctx2 _ _ _ _) %. +%freeze eq_ctx2 %. +%total D (eq_ctx2 D _ _ _) %. +%sort eq_eval %. +%mode eq_eval %in %in %out %. +%term eq_eval_rule eq_eval eq_ref E E %. +%worlds () (eq_eval _ _ _) %. +%freeze eq_eval %. +%total D (eq_eval D _ _) %. +%sort eq_eval2 %. +%mode eq_eval2 %in %in %out %. +%term eq_eval2_rule eq_eval2 eq_ref E E %. +%worlds () (eq_eval2 _ _ _) %. +%freeze eq_eval2 %. +%total D (eq_eval2 D _ _) %. +%sort eq_res_s %. +%mode eq_res_s %in %in %out %. +%term eq_res_s_rile eq_res_s eq_ref E E %. +%worlds () (eq_res_s _ _ _) %. +%freeze eq_res_s %. +%total {} (eq_res_s _ _ _) %. +%sort eq_evaluations %. +%mode eq_evaluations %in %out %. +%term eq_evaluations_ref eq_evaluations E E %. +%worlds () (eq_evaluations _ _) %. +%freeze eq_evaluations %. +%total D (eq_evaluations D _) %. +%sort eq_tens %. +%mode eq_tens %in %out %out %. +%term eq_tens_rule eq_tens eq_ref eq_ref eq_ref %. +%worlds () (eq_tens _ _ _) %. +%freeze eq_tens %. +%total D (eq_tens D _ _) %. +%sort eq_thunk %. +%mode eq_thunk %in %out %. +%term eq_thunk_rule eq_thunk eq_ref eq_ref %. +%worlds () (eq_thunk _ _) %. +%freeze eq_thunk %. +%total D (eq_thunk D _) %. +%sort eq_lam %. +%mode eq_lam %in %in %out %. +%term eq_lam_rule eq_lam eq_ref eq_ref eq_ref %. +%worlds () (eq_lam _ _ _) %. +%freeze eq_lam %. +%total D (eq_lam D _ _) %. +%sort eq_tlam %. +%mode eq_tlam %in %in %out %. +%term eq_tlam_rule eq_tlam eq_ref eqt_ref eq_ref %. +%worlds () (eq_tlam _ _ _) %. +%freeze eq_tlam %. +%total D (eq_tlam D _ _) %. +%sort eq_sym %. +%mode eq_sym %in %out %. +%term eq_sym_rule eq_sym eq_ref eq_ref %. +%worlds () (eq_sym _ _) %. +%freeze eq_sym %. +%total D (eq_sym D _) %. +%{! ### Determinism +Evaluation is deterministic. !}% +%sort eval_determ %. +%mode eval_determ %in %in %out %. +%term eval_determ_unit eval_determ ev_unit ev_unit eq_ref %. +%term eval_determ_tens eval_determ ev_tens ev_tens eq_ref %. +%term eval_determ_tlam eval_determ ev_tlam ev_tlam eq_ref %. +%term eval_determ_lam eval_determ ev_lam ev_lam eq_ref %. +%term eval_determ_thunk eval_determ ev_thunk ev_thunk eq_ref %. +%term eval_determ_app + %pi (eval_determ (%the (app M1 M2 \n/ V) (ev_app EV2 EV1)) (%the (app M1 M2 \n/ V') (ev_app EV2' EV1')) Q) + %<- (eval_determ EV1 EV1' Q1) + %<- (eq_lam Q1 (%the (eq M2 M2) eq_ref) Q3) + %<- (eq_sym Q3 Q4) + %<- (eq_eval Q4 EV2' EV2s) + %<- (eval_determ EV2 EV2s Q) %. +%term eval_determ_tapp + %pi (eval_determ (%the (tapp M1 T2 \n/ V) (ev_tapp EV2 EV1)) (%the (tapp M1 T2 \n/ V') (ev_tapp EV2' EV1')) Q) + %<- (eval_determ EV1 EV1' Q1) + %<- (eq_tlam Q1 (%the (eqt T2 T2) eqt_ref) Q3) + %<- (eq_sym Q3 Q4) + %<- (eq_eval Q4 EV2' EV2s) + %<- (eval_determ EV2 EV2s Q) %. +%term eval_determ_letb + %pi (eval_determ (%the (letb M1 M2 \n/ V) (ev_letb EV2 EV1)) (%the (letb M1 M2 \n/ V') (ev_letb EV2' EV1')) Q) + %<- (eval_determ EV1 EV1' Q1) + %<- (eq_thunk Q1 Q2) + %<- (eq_ctx Q2 M2 Q3) + %<- (eq_sym Q3 Q4) + %<- (eq_eval Q4 EV2' EV2s) + %<- (eval_determ EV2 EV2s Q) %. +%term eval_determ_letu + %pi (eval_determ (%the (letu M1 M2 \n/ V) (ev_letu EV2 EV1)) (%the (letu M1 M2 \n/ V') (ev_letu EV2' EV1')) Q) + %<- (eval_determ EV2 EV2' Q) %. +%term eval_determ_lett + %pi (eval_determ (%the (lett M1 M2 \n/ V) (ev_lett EV2 EV1)) (%the (lett M1 M2 \n/ V') (ev_lett EV2' EV1')) Q) + %<- (eval_determ EV1 EV1' Q1) + %<- (eq_tens Q1 Q2 Q3) + %<- (eq_ctx2 Q2 Q3 M2 Q4) + %<- (eq_sym Q4 Q4') + %<- (eq_eval Q4' EV2' EV2s) + %<- (eval_determ EV2 EV2s Q) %. +%worlds () (eval_determ _ _ _) %. +%freeze eval_determ %. +%total D (eval_determ D _ _) %. +%{! And transitive: !}% +%sort eval_trans %. +%mode eval_trans %in %in %out %. +%term eval_trans_rule + %pi (eval_trans EV EV' EV'') + %<- (value_soundness EV Vv) + %<- (selfeval Vv EVv') + %<- (eval_determ EVv' EV' Q) + %<- (eq_eval2 Q EV EV'') %. +%worlds () (eval_trans _ _ _) %. +%total {} (eval_trans _ _ _) %. +%sort eq_val %. +%mode eq_val %in %in %out %. +%term eq_val_rule eq_val eq_ref V V %. +%worlds () (eq_val _ _ _) %. +%freeze eval_trans %. +%total {} (eq_val _ _ _) %. +%{! ## Frame stack semantics +The abstract machine is defined as a relation on tuples consisting of a frame stack and a term to evaluate in that stack. + +### Evaluation frames +The frame stack is a stack of evaluation frames that provides a context on which we will continue evaluation after we have computed a value for the current term. !}% +%sort frame %. +%name frame %. +%term fletb {M} frame ([a] letb a M) %. +%term fapp {M} frame ([a] app a M) %. +%term ftapp {T} frame ([a] tapp a T) %. +%term fletu {M} frame ([a] letu a M) %. +%term flett {M} frame ([a] lett a M) %. +%freeze frame %. +%{! Evaluation frames are functions from terms to terms and sometimes I need to apply such a function to a term. !}% +%sort frameapp %. +%mode frameapp %in %in %out %. +%term frameapp_app frameapp (%the (frame F') F) M (F' M) %. +%worlds () (frameapp _ _ _) %. +%freeze frameapp %. +%total D (frameapp D _ _) %. +%sort frameapp_exists {F} {M} %. +%mode frameapp_exists %in %in %out %. +%term frameapp_exists_rule frameapp_exists F M frameapp_app %. +%worlds () (frameapp_exists _ _ _) %. +%freeze frameapp_exists %. +%total D (frameapp_exists D _ _) %. +%{! The definition of the frame stack and application of frame stacks to terms. !}% +%sort framestack %. +%name framestack %. +%term cons %pi (frame F) %-> framestack %-> framestack %. +%term nil framestack %. +%freeze framestack %. +%sort frameapply %. +%mode frameapply %in %in %out %. +%term frameapply_nil frameapply nil M M %. +%term frameapply_cons + %pi (frameapply (cons F Fs) M M') + %<- (frameapp F M M'') + %<- (frameapply Fs M'' M') %. +%worlds () (frameapply _ _ _) %. +%freeze frameapply %. +%total D (frameapply D _ _) %. +%{! I should move you: !}% +%block blam [y term]%. +%{! ### Equalities on frame stacks +I also need some equational reasoning on frame stacks: !}% +%sort eqf %. +%mode eqf %in %out %. +%term eqf_ref eqf Fs Fs %. +%freeze eqf %. +%sort eqf_symm %. +%mode eqf_symm %in %out %. +%term eqf_symm_rule eqf_symm eqf_ref eqf_ref %. +%worlds (blam) (eqf_symm _ _) %. +%freeze eqf_symm %. +%total {} (eqf_symm _ _) %. +%sort eqf_trans %. +%mode eqf_trans %in %in %out %. +%term eqf_trans_rule eqf_trans eqf_ref eqf_ref eqf_ref %. +%worlds (blam) (eqf_trans _ _ _) %. +%freeze eqf_trans %. +%total {} (eqf_trans _ _ _) %. +%sort eqf_extend {F} %. +%mode eqf_extend %in %in %out %. +%term eqf_extend_rule eqf_extend _ eqf_ref eqf_ref %. +%worlds (blam) (eqf_extend _ _ _) %. +%freeze eqf_extend %. +%total {} (eqf_extend _ _ _) %. +%sort frameapply_eq %. +%mode frameapply_eq %in %in %out %. +%term frameapply_eq_rule frameapply_eq eqf_ref FA FA %. +%worlds (blam) (frameapply_eq _ _ _) %. +%freeze frameapply_eq %. +%total {} (frameapply_eq _ _ _) %. +%sort frameapply_nil_eq %. +%mode frameapply_nil_eq %in %out %. +%term frameapply_nil_eq_rule frameapply_nil_eq frameapply_nil eq_ref %. +%worlds (blam) (frameapply_nil_eq _ _) %. +%freeze frameapply_nil_eq %. +%total {} (frameapply_nil_eq _ _) %. +%{! #### More metatheorems about frame application !}% +%sort frameapply_exists {Fs} {M} %. +%mode frameapply_exists %in %in %out %. +%term frameapply_exists_nil frameapply_exists nil M frameapply_nil %. +%term frameapply_exists_cons_letu + %pi (frameapply_exists (cons (fletu N) Fs) M (frameapply_cons FsA frameapp_app)) + %<- (frameapply_exists Fs (letu M N) FsA) %. +%term frameapply_exists_cons_letb + %pi (frameapply_exists (cons (fletb N) Fs) M (frameapply_cons FsA frameapp_app)) + %<- (frameapply_exists Fs (letb M N) FsA) %. +%term frameapply_exists_cons_lett + %pi (frameapply_exists (cons (flett N) Fs) M (frameapply_cons FsA frameapp_app)) + %<- (frameapply_exists Fs (lett M N) FsA) %. +%term frameapply_exists_cons_app + %pi (frameapply_exists (cons (fapp N) Fs) M (frameapply_cons FsA frameapp_app)) + %<- (frameapply_exists Fs (app M N) FsA) %. +%term frameapply_exists_cons_tapp + %pi (frameapply_exists (cons (ftapp N) Fs) M (frameapply_cons FsA frameapp_app)) + %<- (frameapply_exists Fs (tapp M N) FsA) %. +%worlds () (frameapply_exists _ _ _) %. +%freeze frameapply_exists %. +%total D (frameapply_exists D _ _) %. +%{! ### Frame stack evaluation +Here I define the evaluation relation for the abstract machine. !}% +%scope frameapply_exists_cons_tapp %sort -> %. +%mode --> %in %in %out %out %. +%term evfs_letu --> Fs (letu M N) (cons (fletu N) Fs) M %. +%term evfs_letb --> Fs (letb M N) (cons (fletb N) Fs) M %. +%term evfs_lett --> Fs (lett M N) (cons (flett N) Fs) M %. +%term evfs_app --> Fs (app M N) (cons (fapp N) Fs) M %. +%term evfs_tapp --> Fs (tapp M T) (cons (ftapp T) Fs) M %. +%term evfs_lam --> (cons (fapp N) Fs) (lam _ M') Fs (M' N) %. +%term evfs_tlam --> (cons (ftapp T) Fs) (tlam M') Fs (M' T) %. +%term evfs_unit --> (cons (fletu N) Fs) unit Fs N %. +%term evfs_tens --> (cons (flett N) Fs) (tens M1 M2) Fs (N M1 M2) %. +%term evfs_thunk --> (cons (fletb N) Fs) (thunk T M) Fs (N (letb (thunk T M) M)) %. +%worlds () (--> _ _ _ _) %. +%covers --> %out %in %out %out %. +%freeze --> %. +%sort eq_step %. +%mode eq_step %in %in %out %. +%term eq_step_rule eq_step eq_ref S S %. +%worlds () (eq_step _ _ _) %. +%total {} (eq_step _ _ _) %. +%scope eq_step_rule %sort ->* %. +%term ->*_ref -->* Fs M Fs M %. +%scope eq_step_rule %term ->*_step %pi (-->* Fs M Fs' M') %<- (--> Fs'' M'' Fs' M') %<- (-->* Fs M Fs'' M'') %. +%freeze -->* %. +%{! ### Meta theory of the abstract machine +Here are some basic properties of the abstract machine: !}% +%sort concat-->* %. +%mode concat-->* %in %in %out %. +%term concatref concat-->* S -->*_ref S %. +%term concatstep %pi (concat-->* S' (-->*_step Ss S) (-->*_step Sc S)) %<- (concat-->* S' Ss Sc) %. +%worlds () (concat-->* _ _ _) %. +%freeze concat-->* %. +%total D (concat-->* _ D _) %. +%scope concatstep %sort ->*_impossible %. +%mode {%in V term} {%in Fs framestack} {%in M term} {%in Fs' framestack} {%in M' term} {%in Fs'' framestack} {%in M'' term} {%in S --> nil V Fs M} {%in V1 value V} {%out R -->* Fs' M' Fs'' M''} -->*_impossible S V1 R %. +%worlds () (-->*_impossible _ _ _) %. +%freeze -->*_impossible %. +%total {} (-->*_impossible _ _ _) %. +%sort lemma44 {Fs framestack} {M} %. +%mode lemma44 %in %in %in %out %. +%term lemma44_nil lemma44 nil M frameapply_nil -->*_ref %. +%term lemma44_cons_letu + %pi (lemma44 (cons (fletu N) Fs) M (frameapply_cons FA frameapp_app) (-->*_step MS evfs_letu)) + %<- (lemma44 Fs (letu M N) FA MS) %. +%term lemma44_cons_letb + %pi (lemma44 (cons (fletb N) Fs) M (frameapply_cons FA frameapp_app) (-->*_step MS evfs_letb)) + %<- (lemma44 Fs (letb M N) FA MS) %. +%term lemma44_cons_lett + %pi (lemma44 (cons (flett N) Fs) M (frameapply_cons FA frameapp_app) (-->*_step MS evfs_lett)) + %<- (lemma44 Fs (lett M N) FA MS) %. +%term lemma44_cons_app + %pi (lemma44 (cons (fapp N) Fs) M (frameapply_cons FA frameapp_app) (-->*_step MS evfs_app)) + %<- (lemma44 Fs (app M N) FA MS) %. +%term lemma44_cons_tapp + %pi (lemma44 (cons (ftapp N) Fs) M (frameapply_cons FA frameapp_app) (-->*_step MS evfs_tapp)) + %<- (lemma44 Fs (tapp M N) FA MS) %. +%worlds () (lemma44 _ _ _ _) %. +%freeze lemma44 %. +%total D (lemma44 D _ _ _) %. +%sort frameApplyUnique %. +%mode frameApplyUnique %in %in %out %. +%term frameApplyUnique_nil frameApplyUnique frameapply_nil frameapply_nil eq_ref %. +%term frameApplyUnique_cons + %pi (frameApplyUnique (frameapply_cons FsA frameapp_app) (frameapply_cons FsA' frameapp_app) Q) + %<- (frameApplyUnique FsA FsA' Q) %. +%worlds () (frameApplyUnique _ _ _) %. +%freeze frameApplyUnique %. +%total D (frameApplyUnique D _ _) %. +%{! ## Correspondence between the big step semantics and the abstract machine !}% +%sort lemma48 {Fs} {M} %. +%mode lemma48 %in %in %in %out %. +%term lemma48_lam lemma48 _ _ ev_lam -->*_ref %. +%term lemma48_tens lemma48 _ _ ev_tens -->*_ref %. +%term lemma48_thunk lemma48 _ _ ev_thunk -->*_ref %. +%term lemma48_tlam lemma48 _ _ ev_tlam -->*_ref %. +%term lemma48_unit lemma48 _ _ ev_unit -->*_ref %. +%term lemma48_app + %pi (lemma48 Fs (app M1 M2) (ev_app EM' EM1) Sr) + %<- (lemma48 (cons (fapp M2) Fs) M1 (%the (M1 \n/ (lam _ M')) EM1) SM1) + %<- (lemma48 Fs (M' M2) (%the (M' M2 \n/ V) EM') SM') + %<- (concat-->* (-->*_step SM1 evfs_lam) SM' S) + %<- (concat-->* (-->*_step -->*_ref evfs_app) S Sr) %. +%term lemma48_tapp + %pi (lemma48 Fs (tapp M1 T) (ev_tapp EM' EM1) Sr) + %<- (lemma48 (cons (ftapp T) Fs) M1 (%the (M1 \n/ (tlam M')) EM1) SM1) + %<- (lemma48 Fs (M' T) (%the (M' T \n/ V) EM') SM') + %<- (concat-->* (-->*_step SM1 evfs_tlam) SM' S) + %<- (concat-->* (-->*_step -->*_ref evfs_tapp) S Sr) %. +%term lemma48_letu + %pi (lemma48 Fs (letu M1 M2) (ev_letu EM' EM1) Sr) + %<- (lemma48 (cons (fletu M2) Fs) M1 (%the (M1 \n/ unit) EM1) SM1) + %<- (lemma48 Fs M2 (%the (M2 \n/ V) EM') SM') + %<- (concat-->* (-->*_step SM1 evfs_unit) SM' S) + %<- (concat-->* (-->*_step -->*_ref evfs_letu) S Sr) %. +%term lemma48_letb + %pi (lemma48 Fs (letb M1 M2) (ev_letb EM' EM1) Sr) + %<- (lemma48 (cons (fletb M2) Fs) M1 (%the (M1 \n/ (thunk T M')) EM1) SM1) + %<- (lemma48 Fs (M2 (letb (thunk T M') M')) (%the (M2 (letb (thunk T M') M') \n/ V) EM') SM') + %<- (concat-->* (-->*_step SM1 evfs_thunk) SM' S) + %<- (concat-->* (-->*_step -->*_ref evfs_letb) S Sr) %. +%term lemma48_lett + %pi (lemma48 Fs (lett M1 M2) (ev_lett EM' EM1) Sr) + %<- (lemma48 (cons (flett M2) Fs) M1 (%the (M1 \n/ (tens M' M'')) EM1) SM1) + %<- (lemma48 Fs (M2 M' M'') (%the (M2 M' M'' \n/ V) EM') SM') + %<- (concat-->* (-->*_step SM1 evfs_tens) SM' S) + %<- (concat-->* (-->*_step -->*_ref evfs_lett) S Sr) %. +%worlds () (lemma48 _ _ _ _) %. +%freeze lemma48 %. +%total D (lemma48 _ _ D _) %. +%sort lemma46a {Fs framestack} %. +%sort lemma46b {Fs framestack} %. +%mode lemma46a %in %in %in %out %out %out %. +%mode lemma46b %in %in %in %in %in %out %. +%term lemma46a_nil + %pi (lemma46a nil E frameapply_nil frameapply_nil E EV) + %<- (value_soundness E Vv) + %<- (selfeval Vv EV) %. +%term lemma46a_letu + %pi (lemma46a (cons (fletu N) Fs) (%the (M' \n/ V) E) (frameapply_cons FA frameapp_app) (frameapply_cons FAu frameapp_app) RM E') + %<- (lemma46a Fs E FA FA' (ev_letu (%the (N \n/ V') RN) RM) (%the (_ \n/ V) R2)) + %<- (frameapply_exists Fs (letu unit N) FAu) + %<- (lemma46b Fs (%the (letu unit N \n/ V') (ev_letu RN ev_unit)) R2 FAu FA' E') %. +%term lemma46a_app + %pi (lemma46a (cons (fapp N) Fs) (%the (M' \n/ V) E) (frameapply_cons FA frameapp_app) (frameapply_cons FAu frameapp_app) RM E') + %<- (lemma46a Fs E FA FA' (ev_app RN (%the (M \n/ (lam T M1)) RM)) R2) + %<- (frameapply_exists Fs (app (lam T M1) N) FAu) + %<- (lemma46b Fs (ev_app RN ev_lam) R2 FAu FA' E') %. +%term lemma46a_tapp + %pi (lemma46a (cons (ftapp N) Fs) (%the (M' \n/ V) E) (frameapply_cons FA frameapp_app) (frameapply_cons FAu frameapp_app) RM E') + %<- (lemma46a Fs E FA FA' (ev_tapp RN (%the (M \n/ (tlam T)) RM)) R2) + %<- (frameapply_exists Fs (tapp (tlam T) N) FAu) + %<- (lemma46b Fs (ev_tapp RN ev_tlam) R2 FAu FA' E') %. +%term lemma46a_letb + %pi (lemma46a (cons (fletb N) Fs) (%the (M' \n/ V) E) (frameapply_cons FA frameapp_app) (frameapply_cons FAu frameapp_app) RM E') + %<- (lemma46a Fs E FA FA' (ev_letb RN (%the (M \n/ (thunk T M1)) RM)) R2) + %<- (frameapply_exists Fs (letb (thunk T M1) N) FAu) + %<- (lemma46b Fs (ev_letb RN ev_thunk) R2 FAu FA' E') %. +%term lemma46a_lett + %pi (lemma46a (cons (flett N) Fs) (%the (M' \n/ V) E) (frameapply_cons FA frameapp_app) (frameapply_cons FAu frameapp_app) RM E') + %<- (lemma46a Fs E FA FA' (ev_lett RN (%the (M \n/ (tens M1 M2)) RM)) R2) + %<- (frameapply_exists Fs (lett (tens M1 M2) N) FAu) + %<- (lemma46b Fs (ev_lett RN ev_tens) R2 FAu FA' E') %. +%term lemma46b_nil + %pi (lemma46b nil (%the (M1 \n/ M2) E) E' frameapply_nil frameapply_nil E'') + %<- (value_soundness E V) + %<- (selfeval V EV) + %<- (eval_determ EV E' Q) + %<- (eq_eval2 Q E E'') %. +%term lemma46b_letu + %pi (lemma46b (cons (fletu M6) Fs1) E1 E2 (frameapply_cons X1 frameapp_app) (frameapply_cons X2 frameapp_app) E3) + %<- (lemma46a Fs1 E2 X2 X3 (ev_letu EM6 EM2) EN) + %<- (eval_trans E1 EM2 EM1) + %<- (lemma46b Fs1 (ev_letu EM6 EM1) EN X1 X3 E3) %. +%term lemma46b_lett + %pi (lemma46b (cons (flett M6) Fs1) E1 E2 (frameapply_cons X1 frameapp_app) (frameapply_cons X2 frameapp_app) E3) + %<- (lemma46a Fs1 E2 X2 X3 (ev_lett EM6 EM2) EN) + %<- (eval_trans E1 EM2 EM1) + %<- (lemma46b Fs1 (ev_lett EM6 EM1) EN X1 X3 E3) %. +%term lemma46b_app + %pi (lemma46b (cons (fapp M6) Fs1) E1 E2 (frameapply_cons X1 frameapp_app) (frameapply_cons X2 frameapp_app) E3) + %<- (lemma46a Fs1 E2 X2 X3 (ev_app EM6 EM2) EN) + %<- (eval_trans E1 EM2 EM1) + %<- (lemma46b Fs1 (ev_app EM6 EM1) EN X1 X3 E3) %. +%term lemma46b_tapp + %pi (lemma46b (cons (ftapp M6) Fs1) E1 E2 (frameapply_cons X1 frameapp_app) (frameapply_cons X2 frameapp_app) E3) + %<- (lemma46a Fs1 E2 X2 X3 (ev_tapp EM6 EM2) EN) + %<- (eval_trans E1 EM2 EM1) + %<- (lemma46b Fs1 (ev_tapp EM6 EM1) EN X1 X3 E3) %. +%term lemma46b_letb + %pi (lemma46b (cons (fletb M6) Fs1) E1 E2 (frameapply_cons X1 frameapp_app) (frameapply_cons X2 frameapp_app) E3) + %<- (lemma46a Fs1 E2 X2 X3 (ev_letb EM6 EM2) EN) + %<- (eval_trans E1 EM2 EM1) + %<- (lemma46b Fs1 (ev_letb EM6 EM1) EN X1 X3 E3) %. +%worlds () (lemma46a _ _ _ _ _ _) (lemma46b _ _ _ _ _ _) %. +%freeze lemma46b lemma46a %. +%total (D E) (lemma46a D _ _ _ _ _) (lemma46b E _ _ _ _ _) %. +%sort lemma47 %. +%mode lemma47 %in %in %out %in %out %. +%term lemma47_letu lemma47 evfs_letu (frameapply_cons X1 frameapp_app) X1 EV' EV' %. +%term lemma47_letb lemma47 evfs_letb (frameapply_cons X1 frameapp_app) X1 EV' EV' %. +%term lemma47_lett lemma47 evfs_lett (frameapply_cons X1 frameapp_app) X1 EV' EV' %. +%term lemma47_app lemma47 evfs_app (frameapply_cons X1 frameapp_app) X1 EV' EV' %. +%term lemma47_tapp lemma47 evfs_tapp (frameapply_cons X1 frameapp_app) X1 EV' EV' %. +%term lemma47_unit + %pi (lemma47 evfs_unit X1 (frameapply_cons X3 frameapp_app) EV' EV) + %<- (lemma46a Fs EV' X1 X2 (%the (M2 \n/ V') EM2) EN) + %<- (frameapply_exists Fs (letu unit M2) X3) + %<- (lemma46b Fs (ev_letu EM2 ev_unit) EN X3 X2 EV) %. +%term lemma47_tens + %pi (lemma47 evfs_tens X1 (frameapply_cons X3 frameapp_app) EV' EV) + %<- (lemma46a Fs EV' X1 X2 EM2 EN) + %<- (frameapply_exists Fs (lett (tens M3 M4) M2) X3) + %<- (lemma46b Fs (ev_lett EM2 ev_tens) EN X3 X2 EV) %. +%term lemma47_thunk + %pi (lemma47 evfs_thunk X1 (frameapply_cons X3 frameapp_app) EV' EV) + %<- (lemma46a Fs EV' X1 X2 EM2 EN) + %<- (frameapply_exists Fs (letb (thunk T M4) M2) X3) + %<- (lemma46b Fs (ev_letb EM2 ev_thunk) EN X3 X2 EV) %. +%term lemma47_lam + %pi (lemma47 evfs_lam X1 (frameapply_cons X3 frameapp_app) EV' EV) + %<- (lemma46a Fs EV' X1 X2 EM2 EN) + %<- (frameapply_exists Fs (app (lam T M4) M2) X3) + %<- (lemma46b Fs (ev_app EM2 ev_lam) EN X3 X2 EV) %. +%term lemma47_tlam + %pi (lemma47 evfs_tlam X1 (frameapply_cons X3 frameapp_app) EV' EV) + %<- (lemma46a Fs EV' X1 X2 EM2 EN) + %<- (frameapply_exists Fs (tapp (tlam M4) M2) X3) + %<- (lemma46b Fs (ev_tapp EM2 ev_tlam) EN X3 X2 EV) %. +%worlds () (lemma47 _ _ _ _ _) %. +%freeze lemma47 %. +%total S (lemma47 S _ _ _ _) %. +%scope lemma47_tlam %sort ->r* %. +%term ->r*_ref -->r* Fs M Fs M %. +%scope lemma47_tlam %term ->r*_step %pi (-->r* Fs M Fs' M') %<- (--> Fs M Fs'' M'') %<- (-->r* Fs'' M'' Fs' M') %. +%freeze -->r* %. +%sort concat-->r* %. +%mode concat-->r* %in %in %out %. +%term concat-->r*_ref concat-->r* -->r*_ref D D %. +%term concat-->r*_step %pi (concat-->r* (-->r*_step R S) D (-->r*_step D' S)) %<- (concat-->r* R D D') %. +%worlds () (concat-->r* _ _ _) %. +%freeze concat-->r* %. +%total D (concat-->r* D _ _) %. +%scope concat-->r*_step %sort ->r*_impossible %. +%mode {%in V term} {%in Fs framestack} {%in M term} {%in Fs' framestack} {%in M' term} {%in Fs'' framestack} {%in M'' term} {%in S --> nil V Fs M} {%in V1 value V} {%out R -->r* Fs' M' Fs'' M''} -->r*_impossible S V1 R %. +%worlds () (-->r*_impossible _ _ _) %. +%freeze -->r*_impossible %. +%total {} (-->r*_impossible _ _ _) %. +%sort concat-->r*_exists {R -->r* F1 M1 Fs M} {S -->r* (%the framestack Fs) (%the term M) (%the framestack Fs2) (%the term M2)} %. +%mode concat-->r*_exists %in %in %out %. +%term concat-->r*_exists_ref concat-->r*_exists -->r*_ref _ concat-->r*_ref %. +%term concat-->r*_exists_step + %pi (concat-->r*_exists (-->r*_step R S) B (concat-->r*_step E)) + %<- (concat-->r*_exists R B E) %. +%worlds () (concat-->r*_exists _ _ _) %. +%freeze concat-->r*_exists %. +%total D (concat-->r*_exists D _ _) %. +%scope concat-->r*_exists_step %sort ->*_to_-->r*_lem %. +%mode -->*_to_-->r*_lem %in %out %. +%scope concat-->r*_exists_step %term ->*_to_-->r*_lem_ref -->*_to_-->r*_lem -->*_ref -->r*_ref %. +%term ->*_to_-->r*_lem_step + %pi (-->*_to_-->r*_lem (-->*_step R S) R'') + %<- (-->*_to_-->r*_lem R R') + %<- (concat-->r* R' (-->r*_step -->r*_ref S) R'') %. +%worlds () (-->*_to_-->r*_lem _ _) %. +%freeze -->*_to_-->r*_lem %. +%total D (-->*_to_-->r*_lem D _) %. +%scope concat-->r*_exists_step %sort ->r*_to_-->*_lem %. +%mode -->r*_to_-->*_lem %in %out %. +%scope concat-->r*_exists_step %term ->r*_to_-->*_lem_ref -->r*_to_-->*_lem -->r*_ref -->*_ref %. +%term ->r*_to_-->*_lem_step + %pi (-->r*_to_-->*_lem (-->r*_step R S) R'') + %<- (-->r*_to_-->*_lem R R') + %<- (concat-->* (-->*_step -->*_ref S) R' R'') %. +%worlds () (-->r*_to_-->*_lem _ _) %. +%freeze -->r*_to_-->*_lem %. +%total D (-->r*_to_-->*_lem D _) %. +%scope concat-->r*_exists_step %sort ->r*_add_step %. +%mode -->r*_add_step %in %in %in %out %. +%scope concat-->r*_exists_step %term ->r*_add_step_ref %pi (-->r*_add_step S -->r*_ref V R) %<- (-->r*_impossible S V R) %. +%term ->r*_add_step_step -->r*_add_step S (-->r*_step Rr S) V Rr %. +%worlds () (-->r*_add_step _ _ _ _) %. +%freeze -->r*_add_step %. +%total {} (-->r*_add_step _ _ _ _) %. +%scope concat-->r*_exists_step %sort ->*_add_step %. +%mode -->*_add_step %in %in %in %out %. +%scope concat-->r*_exists_step %term ->*_add_step_rule + %pi (-->*_add_step S R V R') + %<- (-->*_to_-->r*_lem R Rr) + %<- (-->r*_add_step S Rr V R'r) + %<- (-->r*_to_-->*_lem R'r R') %. +%worlds () (-->*_add_step _ _ _ _) %. +%freeze -->*_add_step %. +%total {} (-->*_add_step _ _ _ _) %. +%scope concat-->r*_exists_step %sort ->*_to_-->r*_lem_exists {S -->* _ _ _ _} %. +%mode -->*_to_-->r*_lem_exists %in %out %. +%scope concat-->r*_exists_step %term ->*_to_-->r*_lem_exists_ref -->*_to_-->r*_lem_exists -->*_ref -->*_to_-->r*_lem_ref %. +%term ->*_to_-->r*_lem_exists_step + %pi (-->*_to_-->r*_lem_exists (-->*_step R S) (-->*_to_-->r*_lem_step R'' R')) + %<- (-->*_to_-->r*_lem_exists R (%the (-->*_to_-->r*_lem R Rr) R')) + %<- (concat-->r*_exists Rr (-->r*_step -->r*_ref S) R'') %. +%worlds () (-->*_to_-->r*_lem_exists _ _) %. +%freeze -->*_to_-->r*_lem_exists %. +%total D (-->*_to_-->r*_lem_exists D _) %. +%sort eq_frame_eval_lemma %. +%mode eq_frame_eval_lemma %in %in %out %out %. +%term eq_frame_eval_lemma_z %pi (eq_frame_eval_lemma -->r*_ref V frameapply_nil EV) %<- (selfeval V EV) %. +%term eq_frame_eval_lemma_s + %pi (eq_frame_eval_lemma (-->r*_step R S) V FA EV) + %<- (eq_frame_eval_lemma R V FA' EV') + %<- (lemma47 S FA' FA EV' EV) %. +%worlds () (eq_frame_eval_lemma _ _ _ _) %. +%freeze eq_frame_eval_lemma %. +%total D (eq_frame_eval_lemma D _ _ _) %. +%sort eq_frame_eval %. +%mode eq_frame_eval %in %in %out %out %. +%term eq_frame_eval_rule + %pi (eq_frame_eval FE V FA E) + %<- (-->*_to_-->r*_lem FE R) + %<- (eq_frame_eval_lemma R V FA E) %. +%worlds () (eq_frame_eval _ _ _ _) %. +%freeze eq_frame_eval %. +%total D (eq_frame_eval D _ _ _) %. +%scope eq_frame_eval_rule %sort ->*_skew %. +%mode -->*_skew %in %in %in %out %. +%scope eq_frame_eval_rule %term ->*_skew_ref -->*_skew R V -->*_ref R %. +%term ->*_skew_step + %pi (-->*_skew R V (-->*_step Rest S) RI) + %<- (-->*_skew R V Rest RI') + %<- (-->*_add_step S RI' V RI) %. +%worlds () (-->*_skew _ _ _ _) %. +%freeze -->*_skew %. +%total D (-->*_skew _ _ D _) %. +%sort eq_eval_frame %. +%mode eq_eval_frame %in %in %out %. +%term eq_eval_frame_rule + %pi (eq_eval_frame (%the (frameapply Fs M FsM) FA) EV R) + %<- (lemma44 Fs M FA R'') + %<- (lemma48 nil FsM EV R') + %<- (value_soundness EV V) + %<- (-->*_skew R' V R'' R) %. +%worlds () (eq_eval_frame _ _ _) %. +%freeze eq_eval_frame %. +%total {} (eq_eval_frame _ _ _) %. +%{! #### Termination relation on the abstract machine !}% +%sort frameterm %. +%term frameterm_val %pi (frameterm nil V) %<- (value V) %. +%term frameterm_app %pi (frameterm Fs (app M1 M2)) %<- (frameterm (cons (fapp M2) Fs) M1) %. +%term frameterm_tapp %pi (frameterm Fs (tapp M T)) %<- (frameterm (cons (ftapp T) Fs) M) %. +%term frameterm_letu %pi (frameterm Fs (letu M1 M2)) %<- (frameterm (cons (fletu M2) Fs) M1) %. +%term frameterm_letb %pi (frameterm Fs (letb M1 M2)) %<- (frameterm (cons (fletb M2) Fs) M1) %. +%term frameterm_lett %pi (frameterm Fs (lett M1 M2)) %<- (frameterm (cons (flett M2) Fs) M1) %. +%term frameterm_lam %pi (frameterm (cons (fapp M2) Fs) (lam T M1)) %<- (frameterm Fs (M1 M2)) %. +%term frameterm_tlam %pi (frameterm (cons (ftapp T) Fs) (tlam M1)) %<- (frameterm Fs (M1 T)) %. +%term frameterm_tens %pi (frameterm (cons (flett M3) Fs) (tens M1 M2)) %<- (frameterm Fs (M3 M1 M2)) %. +%term frameterm_thunk + %pi (frameterm (cons (fletb M2) Fs) (thunk T M1)) + %<- (frameterm Fs (M2 (letb (thunk T M1) M1))) %. +%term frameterm_unit %pi (frameterm (cons (fletu M3) Fs) unit) %<- (frameterm Fs M3) %. +%freeze frameterm %. +%sort frameterm_eq_frames %. +%mode frameterm_eq_frames %in %in %in %out %. +%term frameterm_eq_frames_rule frameterm_eq_frames eqf_ref eq_ref Ft Ft %. +%worlds () (frameterm_eq_frames _ _ _ _) %. +%freeze frameterm_eq_frames %. +%total {} (frameterm_eq_frames _ _ _ _) %. +%sort eq_frameterms %. +%mode eq_frameterms %in %out %. +%term eq_frameterms_ref eq_frameterms F F %. +%worlds () (eq_frameterms _ _) %. +%freeze eq_frameterms %. +%total {} (eq_frameterms _ _) %. +%sort inverse_ft_app %. +%mode inverse_ft_app %in %out %. +%term inverse_ft_app_rule inverse_ft_app (frameterm_app Ft) Ft %. +%worlds () (inverse_ft_app _ _) %. +%reduces < D E (inverse_ft_app E D) %. +%total {} (inverse_ft_app _ _) %. +%sort inverse_ft_tapp %. +%mode inverse_ft_tapp %in %out %. +%term inverse_ft_tapp_rule inverse_ft_tapp (frameterm_tapp Ft) Ft %. +%worlds () (inverse_ft_tapp _ _) %. +%reduces < D E (inverse_ft_tapp E D) %. +%total {} (inverse_ft_tapp _ _) %. +%sort inverse_ft_letu %. +%mode inverse_ft_letu %in %out %. +%term inverse_ft_letu_rule inverse_ft_letu (frameterm_letu Ft) Ft %. +%worlds () (inverse_ft_letu _ _) %. +%reduces < D E (inverse_ft_letu E D) %. +%total {} (inverse_ft_letu _ _) %. +%sort inverse_ft_lett %. +%mode inverse_ft_lett %in %out %. +%term inverse_ft_lett_rule inverse_ft_lett (frameterm_lett Ft) Ft %. +%worlds () (inverse_ft_lett _ _) %. +%reduces < D E (inverse_ft_lett E D) %. +%total {} (inverse_ft_lett _ _) %. +%sort inverse_ft_letb %. +%mode inverse_ft_letb %in %out %. +%term inverse_ft_letb_rule inverse_ft_letb (frameterm_letb Ft) Ft %. +%worlds () (inverse_ft_letb _ _) %. +%reduces < D E (inverse_ft_letb E D) %. +%total {} (inverse_ft_letb _ _) %. +%{! The termination relation is really about the machine. !}% +%sort frameterm_eq_f %. +%mode frameterm_eq_f %in %out %out %. +%term frameterm_eq_f_val frameterm_eq_f (frameterm_val V) -->r*_ref V %. +%term frameterm_eq_f_app + %pi (frameterm_eq_f (frameterm_app FT) (-->r*_step R evfs_app) V) + %<- (frameterm_eq_f FT R V) %. +%term frameterm_eq_f_tapp + %pi (frameterm_eq_f (frameterm_tapp FT) (-->r*_step R evfs_tapp) V) + %<- (frameterm_eq_f FT R V) %. +%term frameterm_eq_f_letu + %pi (frameterm_eq_f (frameterm_letu FT) (-->r*_step R evfs_letu) V) + %<- (frameterm_eq_f FT R V) %. +%term frameterm_eq_f_letb + %pi (frameterm_eq_f (frameterm_letb FT) (-->r*_step R evfs_letb) V) + %<- (frameterm_eq_f FT R V) %. +%term frameterm_eq_f_lett + %pi (frameterm_eq_f (frameterm_lett FT) (-->r*_step R evfs_lett) V) + %<- (frameterm_eq_f FT R V) %. +%term frameterm_eq_f_lam + %pi (frameterm_eq_f (frameterm_lam FT) (-->r*_step R evfs_lam) V) + %<- (frameterm_eq_f FT R V) %. +%term frameterm_eq_f_tlam + %pi (frameterm_eq_f (frameterm_tlam FT) (-->r*_step R evfs_tlam) V) + %<- (frameterm_eq_f FT R V) %. +%term frameterm_eq_f_tens + %pi (frameterm_eq_f (frameterm_tens FT) (-->r*_step R evfs_tens) V) + %<- (frameterm_eq_f FT R V) %. +%term frameterm_eq_f_unit + %pi (frameterm_eq_f (frameterm_unit FT) (-->r*_step R evfs_unit) V) + %<- (frameterm_eq_f FT R V) %. +%term frameterm_eq_f_thunk + %pi (frameterm_eq_f (frameterm_thunk FT) (-->r*_step R evfs_thunk) V) + %<- (frameterm_eq_f FT R V) %. +%worlds () (frameterm_eq_f _ _ _) %. +%freeze frameterm_eq_f %. +%total D (frameterm_eq_f D _ _) %. +%sort frameterm_eq1 %. +%mode frameterm_eq1 %in %out %out %. +%term frameterm_eq1_rule + %pi (frameterm_eq1 FT R V) + %<- (frameterm_eq_f FT R' V) + %<- (-->r*_to_-->*_lem R' R) %. +%worlds () (frameterm_eq1 _ _ _) %. +%freeze frameterm_eq1 %. +%total {} (frameterm_eq1 _ _ _) %. +%sort frameterm_eq2_r %. +%mode frameterm_eq2_r %in %in %out %. +%term frameterm_eq2_r_ref frameterm_eq2_r -->r*_ref V (frameterm_val V) %. +%term frameterm_eq2_r_app + %pi (frameterm_eq2_r (-->r*_step R evfs_app) V (frameterm_app F)) + %<- (frameterm_eq2_r R V F) %. +%term frameterm_eq2_r_tapp + %pi (frameterm_eq2_r (-->r*_step R evfs_tapp) V (frameterm_tapp F)) + %<- (frameterm_eq2_r R V F) %. +%term frameterm_eq2_r_letu + %pi (frameterm_eq2_r (-->r*_step R evfs_letu) V (frameterm_letu F)) + %<- (frameterm_eq2_r R V F) %. +%term frameterm_eq2_r_letb + %pi (frameterm_eq2_r (-->r*_step R evfs_letb) V (frameterm_letb F)) + %<- (frameterm_eq2_r R V F) %. +%term frameterm_eq2_r_lett + %pi (frameterm_eq2_r (-->r*_step R evfs_lett) V (frameterm_lett F)) + %<- (frameterm_eq2_r R V F) %. +%term frameterm_eq2_r_lam + %pi (frameterm_eq2_r (-->r*_step R evfs_lam) V (frameterm_lam F)) + %<- (frameterm_eq2_r R V F) %. +%term frameterm_eq2_r_tlam + %pi (frameterm_eq2_r (-->r*_step R evfs_tlam) V (frameterm_tlam F)) + %<- (frameterm_eq2_r R V F) %. +%term frameterm_eq2_r_tens + %pi (frameterm_eq2_r (-->r*_step R evfs_tens) V (frameterm_tens F)) + %<- (frameterm_eq2_r R V F) %. +%term frameterm_eq2_r_unit + %pi (frameterm_eq2_r (-->r*_step R evfs_unit) V (frameterm_unit F)) + %<- (frameterm_eq2_r R V F) %. +%term frameterm_eq2_r_thunk + %pi (frameterm_eq2_r (-->r*_step R evfs_thunk) V (frameterm_thunk F)) + %<- (frameterm_eq2_r R V F) %. +%worlds () (frameterm_eq2_r _ _ _) %. +%freeze frameterm_eq2_r %. +%total D (frameterm_eq2_r D _ _) %. +%sort frameterm_eq2 %. +%mode frameterm_eq2 %in %in %out %. +%term frameterm_eq2_rule + %pi (frameterm_eq2 R V F) + %<- (-->*_to_-->r*_lem R R') + %<- (frameterm_eq2_r R' V F) %. +%worlds () (frameterm_eq2 _ _ _) %. +%freeze frameterm_eq2 %. +%total {} (frameterm_eq2 _ _ _) %. +%sort eval_frameterm %. +%mode eval_frameterm %in %out %. +%term eval_frameterm_rule + %pi (eval_frameterm EV FT) + %<- (value_soundness EV Vv) + %<- (eq_eval_frame frameapply_nil EV R) + %<- (frameterm_eq2 R Vv FT) %. +%worlds () (eval_frameterm _ _) %. +%freeze eval_frameterm %. +%total {} (eval_frameterm _ _) %. +%sort frameterm_eval %. +%mode frameterm_eval %in %out %. +%term frameterm_eval_rule + %pi (frameterm_eval FT EV) + %<- (frameterm_eq1 FT R V) + %<- (eq_frame_eval R V FA EV') + %<- (frameapply_nil_eq FA Q) + %<- (eq_sym Q Q') + %<- (eq_eval Q' EV' EV) %. +%worlds () (frameterm_eval _ _) %. +%freeze frameterm_eval %. +%total {} (frameterm_eval _ _) %. +%{! Excellent. We now know that termination in the frame stack semantics is the same as evaluation in the big step semantics. The nice thing about the termination relation is that it supplies us with an inductive description of termination. + +#### Termination is invariant under evaluation !}% +%sort lemma413a %. +%mode lemma413a %in %in %out %. +%term lemma413a_ref lemma413a -->*_ref FT FT %. +%term lemma413a_thunk + %pi (lemma413a (-->*_step R evfs_thunk) FT FT') + %<- (lemma413a R FT (frameterm_thunk FT')) %. +%term lemma413a_app + %pi (lemma413a (-->*_step R evfs_app) FT FT') + %<- (lemma413a R FT (frameterm_app FT')) %. +%term lemma413a_tapp + %pi (lemma413a (-->*_step R evfs_tapp) FT FT') + %<- (lemma413a R FT (frameterm_tapp FT')) %. +%term lemma413a_letu + %pi (lemma413a (-->*_step R evfs_letu) FT FT') + %<- (lemma413a R FT (frameterm_letu FT')) %. +%term lemma413a_letb + %pi (lemma413a (-->*_step R evfs_letb) FT FT') + %<- (lemma413a R FT (frameterm_letb FT')) %. +%term lemma413a_lett + %pi (lemma413a (-->*_step R evfs_lett) FT FT') + %<- (lemma413a R FT (frameterm_lett FT')) %. +%term lemma413a_lam + %pi (lemma413a (-->*_step R evfs_lam) FT FT') + %<- (lemma413a R FT (frameterm_lam FT')) %. +%term lemma413a_tlam + %pi (lemma413a (-->*_step R evfs_tlam) FT FT') + %<- (lemma413a R FT (frameterm_tlam FT')) %. +%term lemma413a_tens + %pi (lemma413a (-->*_step R evfs_tens) FT FT') + %<- (lemma413a R FT (frameterm_tens FT')) %. +%term lemma413a_unit + %pi (lemma413a (-->*_step R evfs_unit) FT FT') + %<- (lemma413a R FT (frameterm_unit FT')) %. +%worlds () (lemma413a _ _ _) %. +%freeze lemma413a %. +%total D (lemma413a D _ _) %. +%sort lemma413b %. +%mode lemma413b %in %in %out %. +%term lemma413b_ref lemma413b -->r*_ref FT FT %. +%term lemma413b_thunk + %pi (lemma413b (-->r*_step R evfs_thunk) FT (frameterm_thunk FT')) + %<- (lemma413b R FT FT') %. +%term lemma413b_app + %pi (lemma413b (-->r*_step R evfs_app) FT (frameterm_app FT')) + %<- (lemma413b R FT FT') %. +%term lemma413b_tapp + %pi (lemma413b (-->r*_step R evfs_tapp) FT (frameterm_tapp FT')) + %<- (lemma413b R FT FT') %. +%term lemma413b_letu + %pi (lemma413b (-->r*_step R evfs_letu) FT (frameterm_letu FT')) + %<- (lemma413b R FT FT') %. +%term lemma413b_letb + %pi (lemma413b (-->r*_step R evfs_letb) FT (frameterm_letb FT')) + %<- (lemma413b R FT FT') %. +%term lemma413b_lett + %pi (lemma413b (-->r*_step R evfs_lett) FT (frameterm_lett FT')) + %<- (lemma413b R FT FT') %. +%term lemma413b_lam + %pi (lemma413b (-->r*_step R evfs_lam) FT (frameterm_lam FT')) + %<- (lemma413b R FT FT') %. +%term lemma413b_tlam + %pi (lemma413b (-->r*_step R evfs_tlam) FT (frameterm_tlam FT')) + %<- (lemma413b R FT FT') %. +%term lemma413b_tens + %pi (lemma413b (-->r*_step R evfs_tens) FT (frameterm_tens FT')) + %<- (lemma413b R FT FT') %. +%term lemma413b_unit + %pi (lemma413b (-->r*_step R evfs_unit) FT (frameterm_unit FT')) + %<- (lemma413b R FT FT') %. +%worlds () (lemma413b _ _ _) %. +%freeze lemma413b %. +%total D (lemma413b D _ _) %. +%sort lemma413c %. +%mode lemma413c %in %in %out %. +%term lemma413b_rule %pi (lemma413c R FT FT') %<- (-->*_to_-->r*_lem R R') %<- (lemma413b R' FT FT') %. +%worlds () (lemma413c _ _ _) %. +%freeze lemma413c %. +%total {} (lemma413c _ _ _) %. +%{! ## The Lily type system +#### Linearity +I encode linearity as a predicate on LF functions. Thus if linear([x].F) then we know that F is linear in x. !}% +%sort linear %. +%name linear %. +% mode linear +E. +%term linear_id linear ([x] x) %. +%term linear_app1 %pi (linear ([x] app (E1 x) E2)) %<- (linear E1) %. +%term linear_app2 %pi (linear ([x] app E1 (E2 x))) %<- (linear E2) %. +%term linear_lam %pi (linear ([x] lam T ([y] E x y))) %<- ({y} linear ([x] E x y)) %. +%term linear_tlam %pi (linear ([x] tlam ([y] F x y))) %<- ({t tp} linear ([z] F z t)) %. +%term linear_tapp %pi (linear ([x] tapp (E x) T)) %<- (linear ([y] E y)) %. +% linear_thunk : No Rule as thunks have no free linear variables. +%term linear_letb1 %pi (linear ([x] letb (E1 x) E2)) %<- (linear E1) %. +%term linear_letb2 %pi (linear ([x] letb E1 ([y] E2 x y))) %<- ({y} linear ([x] E2 x y)) %. +% linear_unit : No Rule as units has no free linear variables. +%term linear_letu1 %pi (linear ([x] letu (E1 x) E2)) %<- (linear ([x] E1 x)) %. +%term linear_letu2 %pi (linear ([x] letu E1 (E2 x))) %<- (linear ([x] E2 x)) %. +%term linear_tens1 %pi (linear ([x] tens (E1 x) E2)) %<- (linear ([x] E1 x)) %. +%term linear_tens2 %pi (linear ([x] tens E1 (E2 x))) %<- (linear ([x] E2 x)) %. +%term linear_lett1 %pi (linear ([x] lett (E x) F)) %<- (linear ([x] E x)) %. +%term linear_lett2 %pi (linear ([x] lett E ([y] [z] F x y z))) %<- ({y} {z} linear ([x] F x y z)) %. +%block btlam [t tp]%. +%worlds (blam btlam) (linear _) %. +%freeze linear %. +%sort sub_linear %. +%mode sub_linear %in %in %out %. +%term sub_linear_id sub_linear linear_id L2 L2 %. +%term sub_linear_app1 %pi (sub_linear (linear_app1 L1) L2 (linear_app1 L3)) %<- (sub_linear L1 L2 L3) %. +%term sub_linear_app2 %pi (sub_linear (linear_app2 L1) L2 (linear_app2 L3)) %<- (sub_linear L1 L2 L3) %. +%term sub_linear_lam + %pi (sub_linear (linear_lam L1) L2 (linear_lam ([y] L3 y))) + %<- ({y} sub_linear (L1 y) L2 (L3 y)) %. +%term sub_linear_tlam + %pi (sub_linear (linear_tlam L1) L2 (linear_tlam ([t] L3 t))) + %<- ({t} sub_linear (L1 t) L2 (L3 t)) %. +%term sub_linear_tapp %pi (sub_linear (linear_tapp L1) L2 (linear_tapp L3)) %<- (sub_linear L1 L2 L3) %. +%term sub_linear_letb1 %pi (sub_linear (linear_letb1 L1) L2 (linear_letb1 L3)) %<- (sub_linear L1 L2 L3) %. +%term sub_linear_letb2 + %pi (sub_linear (linear_letb2 L1) L2 (linear_letb2 L3)) + %<- ({y} sub_linear (L1 y) L2 (L3 y)) %. +%term sub_linear_letu1 %pi (sub_linear (linear_letu1 L1) L2 (linear_letu1 L3)) %<- (sub_linear L1 L2 L3) %. +%term sub_linear_letu2 %pi (sub_linear (linear_letu2 L1) L2 (linear_letu2 L3)) %<- (sub_linear L1 L2 L3) %. +%term sub_linear_lett1 %pi (sub_linear (linear_lett1 L1) L2 (linear_lett1 L3)) %<- (sub_linear L1 L2 L3) %. +%term sub_linear_lett2 + %pi (sub_linear (linear_lett2 L1) L2 (linear_lett2 L3)) + %<- ({y} {z} sub_linear (L1 y z) L2 (L3 y z)) %. +%term sub_linear_tens1 %pi (sub_linear (linear_tens1 L1) L2 (linear_tens1 L3)) %<- (sub_linear L1 L2 L3) %. +%term sub_linear_tens2 %pi (sub_linear (linear_tens2 L1) L2 (linear_tens2 L3)) %<- (sub_linear L1 L2 L3) %. +%block bsub_linear_lam1 [y term]%. +%block bsub_linear_lam2 [y tp]%. +%worlds (bsub_linear_lam1 bsub_linear_lam2) (sub_linear _ _ _) %. +%freeze sub_linear %. +%total D (sub_linear D _ _) %. +%{! Some proofs are easier with relevance than with linearity. !}% +%sort relavant %. +%name relavant %. +%mode relavant %in %. +%term relavant_id relavant ([x] x) %. +%term relavant_app1 %pi (relavant ([x] app (E1 x) (E2 x))) %<- (relavant E1) %. +%term relavant_app2 %pi (relavant ([x] app (E1 x) (E2 x))) %<- (relavant E2) %. +%term relavant_lam %pi (relavant ([x] lam T ([y] F x y))) %<- ({y} relavant ([x] F x y)) %. +%term relavant_tlam %pi (relavant ([x] tlam ([y] F x y))) %<- ({t tp} relavant ([z] F z t)) %. +%term relavant_tapp %pi (relavant ([x] tapp (E x) T)) %<- (relavant E) %. +%term relavant_letb1 %pi (relavant ([x] letb (E1 x) ([y] E2 x y))) %<- (relavant E1) %. +%term relavant_letb2 %pi (relavant ([x] letb (E1 x) ([y] E2 x y))) %<- ({y} relavant ([x] E2 x y)) %. +% linear_unit : No Rule as units has no free variables. +%term relavant_thunk %pi (relavant ([x] thunk T ([y] E x y))) %<- ({y} relavant ([x] E x y)) %. +%term relavant_letu1 %pi (relavant ([x] letu (E1 x) (E2 x))) %<- (relavant E1) %. +%term relavant_letu2 %pi (relavant ([x] letu (E1 x) (E2 x))) %<- (relavant ([x] E2 x)) %. +%term relavant_tens1 %pi (relavant ([x] tens (E1 x) (E2 x))) %<- (relavant E1) %. +%term relavant_tens2 %pi (relavant ([x] tens (E1 x) (E2 x))) %<- (relavant E2) %. +%term relavant_lett1 %pi (relavant ([x] lett (E1 x) ([y] [z] E2 x y z))) %<- (relavant E1) %. +%term relavant_lett2 + %pi (relavant ([x] lett (E1 x) ([y] [z] E2 x y z))) + %<- ({y} {z} relavant ([x] E2 x y z)) %. +%worlds (blam btlam) (relavant _) %. +%freeze relavant %. +%sort linear_relavant %. +%mode linear_relavant %in %out %. +%term linear_relavant_id linear_relavant linear_id relavant_id %. +%term linear_relavant_app1 %pi (linear_relavant (linear_app1 L) (relavant_app1 R)) %<- (linear_relavant L R) %. +%term linear_relavant_app2 %pi (linear_relavant (linear_app2 L) (relavant_app2 R)) %<- (linear_relavant L R) %. +%term linear_relavant_lam + %pi (linear_relavant (linear_lam L) (relavant_lam R)) + %<- ({y} linear_relavant (L y) (R y)) %. +%term linear_relavant_tlam + %pi (linear_relavant (linear_tlam L) (relavant_tlam R)) + %<- ({y} linear_relavant (L y) (R y)) %. +%term linear_relavant_tapp %pi (linear_relavant (linear_tapp L) (relavant_tapp R)) %<- (linear_relavant L R) %. +%term linear_relavant_letu1 + %pi (linear_relavant (linear_letu1 L) (relavant_letu1 R)) + %<- (linear_relavant L R) %. +%term linear_relavant_letu2 + %pi (linear_relavant (linear_letu2 L) (relavant_letu2 R)) + %<- (linear_relavant L R) %. +%term linear_relavant_letb1 + %pi (linear_relavant (linear_letb1 L) (relavant_letb1 R)) + %<- (linear_relavant L R) %. +%term linear_relavant_letb2 + %pi (linear_relavant (linear_letb2 L) (relavant_letb2 R)) + %<- ({y} linear_relavant (L y) (R y)) %. +%term linear_relavant_tens1 + %pi (linear_relavant (linear_tens1 L) (relavant_tens1 R)) + %<- (linear_relavant L R) %. +%term linear_relavant_tens2 + %pi (linear_relavant (linear_tens2 L) (relavant_tens2 R)) + %<- (linear_relavant L R) %. +%term linear_relavant_lett1 + %pi (linear_relavant (linear_lett1 L) (relavant_lett1 R)) + %<- (linear_relavant L R) %. +%term linear_relavant_lett2 + %pi (linear_relavant (linear_lett2 L) (relavant_lett2 R)) + %<- ({y} {z} linear_relavant (L y z) (R y z)) %. +%worlds (blam btlam) (linear_relavant _ _) %. +%freeze linear_relavant %. +%total D (linear_relavant D _) %. +%sort eq_relavant {m} %. +%mode eq_relavant %in %in %out %. +%term eq_relavant_rule eq_relavant ([m] eq_ref) R R %. +%worlds (blam) (eq_relavant _ _ _) %. +%freeze eq_relavant %. +%total {} (eq_relavant _ _ _) %. +%sort eq_linear {m} %. +%mode eq_linear %in %in %out %. +%term eq_linear_rule eq_linear ([m] eq_ref) R R %. +%worlds (blam) (eq_linear _ _ _) %. +%freeze eq_linear %. +%total {} (eq_linear _ _ _) %. +%{! #### Typing rules !}% +%sort ? %. +% mode ? +M -T. +%name ? %. +%prec %none 500 ? %. +%term of_lam + %pi (lam T ([x] M x) ? func T T') + %<- ({x term} %pi (x ? T) %-> (M x ? T')) + %<- (linear M) %. +%term of_app %pi (app M1 M2 ? T) %<- (M1 ? func T' T) %<- (M2 ? T') %. +%term of_tlam %pi (tlam M ? all T) %<- ({t tp} M t ? T t) %. +%term of_tapp {t tp} %pi (tapp M t ? T t) %<- (M ? all T) %. +%term of_thunk %pi (thunk T M ? bang T) %<- ({x} %pi (x ? T) %-> (M x ? T)) %. +%term of_letb %pi (letb M N ? T') %<- (M ? bang T) %<- ({x} %pi (x ? T) %-> (N x ? T')) %. +%term of_unit unit ? i %. +%term of_letu %pi (letu M N ? T) %<- (M ? i) %<- (N ? T) %. +%term of_tens %pi (tens M N ? tensor T T') %<- (M ? T) %<- (N ? T') %. +%term of_lett + %pi (lett M N ? T'') + %<- (M ? tensor T T') + %<- ({x} %pi (x ? T) %-> ({y} %pi (y ? T') %-> (N x y ? T''))) + %<- ({y} linear ([x] N x y)) + %<- ({x} linear ([y] N x y)) %. +%block blam_t {T tp} [x term] [u x ? T]%. +%block btlam_t [t tp]%. +%worlds (blam_t btlam_t) (? _ _) %. +%covers ? %in %out %. +%freeze ? %. +%{! Take note of how I require functions and products to be linear. + +### Type preservation +As we have a type system we should prove soundness of it. !}% +%sort tpres %. +%mode tpres %in %in %out %. +%term tpres_lam tpres (of_lam L D) ev_lam (of_lam L D) %. +%term tpres_tlam tpres (of_tlam D) ev_tlam (of_tlam D) %. +%term tpres_thunk tpres (of_thunk D) ev_thunk (of_thunk D) %. +%term tpres_unit tpres of_unit ev_unit of_unit %. +%term tpres_tens tpres (of_tens D' D) ev_tens (of_tens D' D) %. +%sort tpres_app_aux %. +%mode tpres_app_aux %in %out %. +%term tpres_app_aux_rule tpres_app_aux (of_lam _ F) F %. +%worlds () (tpres_app_aux _ _) %. +%freeze tpres_app_aux %. +%total D (tpres_app_aux D _) %. +%term tpres_app + %pi (tpres (of_app D2 D1) (ev_app E' E) D) + %<- (tpres D1 E D') + %<- (tpres_app_aux D' F) + %<- (tpres (F _ D2) E' D) %. +%term tpres_tapp {D'' {t tp} M t ? T t} + %pi (tpres (of_tapp _ D') (ev_tapp E' E) D) + %<- (tpres D' E (of_tlam D'')) + %<- (tpres (D'' _) E' D) %. +%sort tpres_letb_aux %. +%mode tpres_letb_aux %in %out %. +%term tpres_letb_aux_rule tpres_letb_aux (of_thunk F) F %. +%worlds () (tpres_letb_aux _ _) %. +%freeze tpres_letb_aux %. +%total D (tpres_letb_aux D _) %. +%term tpres_letb + %pi (tpres (of_letb D' D) (ev_letb E' E) D'') + %<- (tpres D E DT) + %<- (tpres_letb_aux DT Dt) + %<- (tpres (D' _ (of_letb Dt DT)) E' D'') %. +%term tpres_letu %pi (tpres (of_letu D' _) (ev_letu E' _) D'') %<- (tpres D' E' D'') %. +%term tpres_lett + %pi (tpres (of_lett _ _ D' D) (ev_lett E' E) D'') + %<- (tpres D E (of_tens Dn Dm)) + %<- (tpres (D' _ Dm _ Dn) E' D'') %. +%worlds () (tpres _ _ _) %. +%freeze tpres %. +%total D (tpres _ D _) %. +%sort tpres_s %. +%mode tpres_s %in %in %out %. +%term tpres_s_lam tpres_s (of_lam L D) evs_lam (of_lam L D) %. +%term tpres_s_tlam tpres_s (of_tlam D) evs_tlam (of_tlam D) %. +%term tpres_s_thunk tpres_s (of_thunk D) evs_thunk (of_thunk D) %. +%term tpres_s_unit tpres_s of_unit evs_unit of_unit %. +%term tpres_s_tens tpres_s (of_tens D' D) evs_tens (of_tens D' D) %. +%sort tpres_s_app_aux %. +%mode tpres_s_app_aux %in %out %. +%term tpres_s_app_aux_rule tpres_s_app_aux (of_lam _ F) F %. +%worlds () (tpres_s_app_aux _ _) %. +%freeze tpres_s_app_aux %. +%total D (tpres_s_app_aux D _) %. +%term tpres_s_app + %pi (tpres_s (of_app D2 D1) (evs_app E3 E2 E1) D) + %<- (tpres_s D1 E1 D') + %<- (tpres_s_app_aux D' F) + %<- (tpres_s D2 E2 D4) + %<- (tpres_s (F _ D4) E3 D) %. +%term tpres_s_tapp {D'' {t tp} M t ? T t} + %pi (tpres_s (of_tapp _ D') (evs_tapp E' E) D) + %<- (tpres_s D' E (of_tlam D'')) + %<- (tpres_s (D'' _) E' D) %. +%sort tpres_s_letb_aux %. +%mode tpres_s_letb_aux %in %out %. +%term tpres_s_letb_aux_rule tpres_s_letb_aux (of_thunk F) F %. +%worlds () (tpres_s_letb_aux _ _) %. +%freeze tpres_s_letb_aux %. +%total D (tpres_s_letb_aux D _) %. +%term tpres_s_letb + %pi (tpres_s (of_letb D' D) (evs_letb E' E) D'') + %<- (tpres_s D E DT) + %<- (tpres_s_letb_aux DT Dt) + %<- (tpres_s (D' _ (of_letb Dt DT)) E' D'') %. +%term tpres_s_letu %pi (tpres_s (of_letu D' _) (evs_letu E' _) D'') %<- (tpres_s D' E' D'') %. +%term tpres_s_lett + %pi (tpres_s (of_lett _ _ D' D) (evs_lett E' E) D'') + %<- (tpres_s D E (of_tens Dn Dm)) + %<- (tpres_s (D' _ Dm _ Dn) E' D'') %. +%worlds () (tpres_s _ _ _) %. +%freeze tpres_s %. +%total D (tpres_s _ D _) %. +%{! ### Equalities and inversions !}% +%block bts_tlam [t tp]%. +%sort eq_typings %. +%mode eq_typings %in %out %. +%term eq_typings_ref eq_typings D D %. +%worlds (blam bts_tlam) (eq_typings _ _) %. +%freeze eq_typings %. +%total D (eq_typings D _) %. +%sort eq_type %. +%mode eq_type %in %in %out %. +%term eq_typing_rule eq_type eq_ref D D %. +%worlds () (eq_type _ _ _) %. +%freeze eq_type %. +%total {} (eq_type _ _ _) %. +%sort eq_type1 %. +%mode eq_type1 %in %in %out %. +%term eq_type1_rule eq_type1 eqt_ref D D %. +%worlds () (eq_type1 _ _ _) %. +%freeze eq_type1 %. +%total {} (eq_type1 _ _ _) %. +%sort inverse_lam %. +%mode inverse_lam %in %out %out %. +%term inverse_lam_rule inverse_lam (of_lam L D) D L %. +%worlds () (inverse_lam _ _ _) %. +%total {} (inverse_lam _ _ _) %. +%sort inverse_tapp %. +%mode inverse_tapp %in %out %. +%term inverse_tapp_rule inverse_tapp (of_tapp T D) D %. +%worlds () (inverse_tapp _ _) %. +%total {} (inverse_tapp _ _) %. +%sort eq_thunk_type %. +%mode eq_thunk_type %in %out %. +%term eq_thunk_type_rule eq_thunk_type (of_thunk _) eqt_ref %. +%worlds () (eq_thunk_type _ _) %. +%freeze eq_thunk_type %. +%total {} (eq_thunk_type _ _) %. +%sort eq_val_funs %. +%mode eq_val_funs %in %in %out %. +%term eq_val_funs_rule eq_val_funs val_lam (of_lam _ _) eq_ref %. +%worlds () (eq_val_funs _ _ _) %. +%freeze eq_val_funs %. +%total {} (eq_val_funs _ _ _) %. +%sort eq_val_all %. +%mode eq_val_all %in %in %out %. +%term eq_val_all_rule eq_val_all val_tlam (of_tlam _) eq_ref %. +%worlds () (eq_val_all _ _ _) %. +%freeze eq_val_all %. +%total {} (eq_val_all _ _ _) %. +%sort eq_val_unit %. +%mode eq_val_unit %in %in %out %. +%term eq_val_unit_rule eq_val_unit val_unit of_unit eq_ref %. +%worlds () (eq_val_unit _ _ _) %. +%freeze eq_val_unit %. +%total {} (eq_val_unit _ _ _) %. +%sort eq_val_tensor %. +%mode eq_val_tensor %in %in %out %. +%term eq_val_tensor_rule eq_val_tensor val_tens (of_tens _ _) eq_ref %. +%worlds () (eq_val_tensor _ _ _) %. +%freeze eq_val_tensor %. +%total {} (eq_val_tensor _ _ _) %. +%sort eq_val_bang %. +%mode eq_val_bang %in %in %out %. +%term eq_val_bang_rule eq_val_bang val_thunk (of_thunk _) eq_ref %. +%worlds () (eq_val_bang _ _ _) %. +%freeze eq_val_bang %. +%total {} (eq_val_bang _ _ _) %. +%{! ## Abstract machine types !}% +%sort fstp %. +% mode fstp +FST +T -T'. +%name fstp %. +%term ftp_nil fstp nil T T %. +%term ftp_cons + %pi (fstp (cons (%the (frame F') F) Fs) T T') + %<- ({a} %pi (a ? T) %-> (F' a ? T'')) + %<- (fstp Fs T'' T') %. +%worlds () (fstp _ _ _) %. +%covers fstp %in %in %out %. +%freeze fstp %. +%sort eqf_fstp %. +%mode eqf_fstp %in %in %out %. +%term eqf_fstp_rule eqf_fstp eqf_ref F F %. +%worlds (blam) (eqf_fstp _ _ _) %. +%freeze eqf_fstp %. +%total {} (eqf_fstp _ _ _) %. +%sort applysound {Fs framestack} %. +%mode applysound %in %in %in %in %out %. +%term as_ftp_nil applysound nil ftp_nil D frameapply_nil D %. +%term as_ftp_cons_letb + %pi (applysound (cons (fletb N) Fs) (ftp_cons FsT (%the ({m term} %pi (m ? T) %-> (letb m N ? T'')) Ft)) D1 (frameapply_cons Fap frameapp_app) D2) + %<- (applysound Fs FsT (Ft M D1) Fap D2) %. +%term as_ftp_cons_app + %pi (applysound (cons (fapp N) Fs) (ftp_cons FsT (%the ({m term} %pi (m ? T) %-> (app m N ? T'')) Ft)) D1 (frameapply_cons Fap frameapp_app) D2) + %<- (applysound Fs FsT (Ft M D1) Fap D2) %. +%term as_ftp_cons_fletu + %pi (applysound (cons (fletu N) Fs) (ftp_cons FsT (%the ({m term} %pi (m ? T) %-> (letu m N ? T'')) Ft)) D1 (frameapply_cons Fap frameapp_app) D2) + %<- (applysound Fs FsT (Ft M D1) Fap D2) %. +%term as_ftp_cons_flett + %pi (applysound (cons (flett N) Fs) (ftp_cons FsT (%the ({m term} %pi (m ? T) %-> (lett m N ? T'')) Ft)) D1 (frameapply_cons Fap frameapp_app) D2) + %<- (applysound Fs FsT (Ft M D1) Fap D2) %. +%term as_ftp_cons_ftapp + %pi (applysound (cons (ftapp T3) Fs) (ftp_cons FsT (%the ({m term} %pi (m ? T1) %-> (tapp m T3 ? T4)) Ft)) D1 (frameapply_cons Fap frameapp_app) D2) + %<- (applysound Fs FsT (Ft M D1) Fap D2) %. +%worlds () (applysound _ _ _ _ _) %. +%freeze applysound %. +%total D (applysound D _ _ _ _) %. +%sort eqt_tapp %. +%mode eqt_tapp %in %out %out %. +%term eqt_tapp_rule eqt_tapp (of_tapp T D) D eqt_ref %. +%worlds () (eqt_tapp _ _ _) %. +%freeze eqt_tapp %. +%total {} (eqt_tapp _ _ _) %. +%{! ### Preservation +Now I can prove soundness of the type system for the frame stack semantics: !}% +%sort typed_step %. +%mode typed_step %in %in %in %out %out %. +%term typed_step_letu %type d_step FsT (of_letu DN Di) evfs_letu (ftp_cons FsT ([a] [da] of_letu DN da)) Di %. +%term typed_step_letb %type d_step FsT (of_letb DN DM) evfs_letb (ftp_cons FsT ([a] [da] of_letb DN da)) DM %. +%term typed_step_lett %type d_step FsT (of_lett L1 L2 DN DM) evfs_lett (ftp_cons FsT ([a] [da] of_lett L1 L2 DN da)) DM %. +%term typed_step_fapp %type d_step FsT (of_app DN DM) evfs_app (ftp_cons FsT ([a] [da] of_app DN da)) DM %. +%term typed_step_ftapp %type d_step FsT (of_tapp T DM) evfs_tapp (ftp_cons FsT ([a] [da] of_tapp T da)) DM %. +%term typed_step_unit + %pi (%type d_step (ftp_cons FsT Ft) of_unit evfs_unit FsT D) + %<- (eq_typings (Ft unit of_unit) (of_letu D _)) %. +%term typed_step_lam + %pi (%type d_step (ftp_cons FsT Ft) D1 evfs_lam FsT (D2 _ D)) + %<- (eq_typings (Ft _ D1) (of_app D (of_lam _ D2))) %. +%term typed_step_thunk + %pi (%type d_step (ftp_cons FsT Ft) (of_thunk D1) evfs_thunk FsT (DN _ (of_letb DT (of_thunk DT)))) + %<- (eq_typings (Ft _ (of_thunk D1)) (of_letb DN (of_thunk DT))) %. +%term typed_step_tlam + %pi (%type d_step (ftp_cons FsT Ft) D1 (%the (--> (cons (ftapp T1) Fs) _ _ _) evfs_tlam) FsT D5) + %<- (eqt_tapp (Ft (tlam M1) D1) (of_tlam F) Q) + %<- (eq_type1 Q (F T1) D5) %. +%term typed_step_tens + %pi (%type d_step (ftp_cons FsT Ft) (of_tens D2 D1) evfs_tens FsT (D' M1 D1ss M2 D2ss)) + %<- (eq_typings (Ft (tens M1 M2) (of_tens D2 D1)) (of_lett L1 L2 D' (of_tens D2ss D1ss))) %. +%worlds () (typed_step _ _ _ _ _) %. +%freeze typed_step %. +%total {} (typed_step _ _ _ _ _) %. +%sort frame_pres_lem %. +%mode frame_pres_lem %in %in %in %out %out %. +%term frame_pres_lem_ref frame_pres_lem -->r*_ref FsTp D FsTp D %. +%term frame_pres_lem_step + %pi (frame_pres_lem (-->r*_step Ss S) FsTp D FsTp' D') + %<- (%type d_step FsTp D S FsTp'' D'') + %<- (frame_pres_lem Ss FsTp'' D'' FsTp' D') %. +%worlds () (frame_pres_lem _ _ _ _ _) %. +%total Ss (frame_pres_lem Ss _ _ _ _) %. +%sort frame_pres %. +%mode frame_pres %in %in %in %out %out %. +%term frame_pres_rule + %pi (frame_pres Ss FsTp D FsTp' D') + %<- (-->*_to_-->r*_lem Ss Ss') + %<- (frame_pres_lem Ss' FsTp D FsTp' D') %. +%worlds () (frame_pres _ _ _ _ _) %. +%total Ss (frame_pres Ss _ _ _ _) %. +%{! #### Progress +Progress and preservation seem to be in fashion. +I will not rely on progress elsewhere. !}% +%sort progress_good %. +%mode progress_good %in %in %. +%term progress_good_v %pi (progress_good nil V) %<- (value V) %. +%term progress_good_s %pi (progress_good Fs M) %<- (--> Fs M Fs' M') %. +%freeze progress_good %. +%sort frame_progress {F' frame F} {Fs} %. +%mode frame_progress %in %in %in %in %out %. +%term frame_progress_app + %pi (frame_progress (fapp M2) Fs (of_app D2 D1) V S) + %<- (eq_val_funs V D1 Q) + %<- (eq_sym Q Q') + %<- (eq_step Q' evfs_lam S) %. +%term frame_progress_tapp + %pi (frame_progress (ftapp T) Fs (of_tapp T D1) V S) + %<- (eq_val_all V D1 Q) + %<- (eq_sym Q Q') + %<- (eq_step Q' evfs_tlam S) %. +%term frame_progress_lett + %pi (frame_progress (flett M2) Fs (of_lett _ _ _ D1) V S) + %<- (eq_val_tensor V D1 Q) + %<- (eq_sym Q Q') + %<- (eq_step Q' evfs_tens S) %. +%term frame_progress_letb + %pi (frame_progress (fletb M2) Fs (of_letb _ D1) V S) + %<- (eq_val_bang V D1 Q) + %<- (eq_sym Q Q') + %<- (eq_step Q' evfs_thunk S) %. +%term frame_progress_letu + %pi (frame_progress (fletu M2) Fs (of_letu _ D1) V S) + %<- (eq_val_unit V D1 Q) + %<- (eq_sym Q Q') + %<- (eq_step Q' evfs_unit S) %. +%worlds () (frame_progress _ _ _ _ _) %. +%total {} (frame_progress _ _ _ _ _) %. +%sort progress_val %. +%mode progress_val %in %in %in %out %. +%term progress_nil_val progress_val ftp_nil D V (progress_good_v V) %. +%term progress_cons_rule + %pi (progress_val (%the (fstp (cons F Fs) _ _) (ftp_cons FsTp Ftp)) D V (progress_good_s S)) + %<- (frame_progress F Fs (Ftp _ D) V S) %. +%worlds () (progress_val _ _ _ _) %. +%total {} (progress_val _ _ _ _) %. +%sort progress %. +%mode progress %in %in %out %. +%term progress_lam %pi (progress FsTp D G) %<- (progress_val FsTp D val_lam G) %. +%term progress_tlam %pi (progress FsTp D G) %<- (progress_val FsTp D val_tlam G) %. +%term progress_thunk %pi (progress FsTp D G) %<- (progress_val FsTp D val_thunk G) %. +%term progress_tens %pi (progress FsTp D G) %<- (progress_val FsTp D val_tens G) %. +%term progress_unit %pi (progress FsTp D G) %<- (progress_val FsTp D val_unit G) %. +%term progress_app progress _ _ (progress_good_s evfs_app) %. +%term progress_tapp progress _ _ (progress_good_s evfs_tapp) %. +%term progress_letu progress _ _ (progress_good_s evfs_letu) %. +%term progress_letb progress _ _ (progress_good_s evfs_letb) %. +%term progress_lett progress _ _ (progress_good_s evfs_lett) %. +%worlds () (progress _ _ _) %. +%total {} (progress _ _ _) %. +%{! ## Strictness lemma +### Stack manipulations +There are two natural ways to define application of a frame stack to a term. I need them both and I need to show how they relate. As stacks are represented as lists this code is a nice example of how to prove classic theorems about lists. !}% +%sort frameapply' %. +%mode frameapply' %in %in %out %. +%term frameapply'_nil frameapply' nil M M %. +%term frameapply'_cons + %pi (frameapply' (cons F Fs') M M') + %<- (frameapply' Fs' M M'') + %<- (frameapp F M'' M') %. +%worlds () (frameapply' _ _ _) %. +%freeze frameapply' %. +%total D (frameapply' D _ _) %. +%sort frameapply'_exists {Fs} {M} %. +%mode frameapply'_exists %in %in %out %. +%term frameapply'_exists_nil frameapply'_exists nil M frameapply'_nil %. +%term frameapply'_exists_cons_letu + %pi (frameapply'_exists (cons (fletu N) Fs) M (frameapply'_cons frameapp_app FsA)) + %<- (frameapply'_exists Fs M FsA) %. +%term frameapply'_exists_cons_letb + %pi (frameapply'_exists (cons (fletb N) Fs) M (frameapply'_cons frameapp_app FsA)) + %<- (frameapply'_exists Fs M FsA) %. +%term frameapply'_exists_cons_lett + %pi (frameapply'_exists (cons (flett N) Fs) M (frameapply'_cons frameapp_app FsA)) + %<- (frameapply'_exists Fs M FsA) %. +%term frameapply'_exists_cons_app + %pi (frameapply'_exists (cons (fapp N) Fs) M (frameapply'_cons frameapp_app FsA)) + %<- (frameapply'_exists Fs M FsA) %. +%term frameapply'_exists_cons_tapp + %pi (frameapply'_exists (cons (ftapp N) Fs) M (frameapply'_cons frameapp_app FsA)) + %<- (frameapply'_exists Fs M FsA) %. +%worlds (blam) (frameapply'_exists _ _ _) %. +%freeze frameapply'_exists %. +%total D (frameapply'_exists D _ _) %. +%sort frameapply'_eq %. +%mode frameapply'_eq %in %in %out %. +%term frameapply'_eq_rule frameapply'_eq eqf_ref FA FA %. +%worlds (blam) (frameapply'_eq _ _ _) %. +%freeze frameapply'_eq %. +%total {} (frameapply'_eq _ _ _) %. +%sort frameapply'_nil_eq %. +%mode frameapply'_nil_eq %in %out %. +%term frameapply'_nil_eq_rule frameapply'_nil_eq frameapply'_nil eq_ref %. +%worlds (blam) (frameapply'_nil_eq _ _) %. +%freeze frameapply'_nil_eq %. +%total {} (frameapply'_nil_eq _ _) %. +%sort revFs %. +%mode revFs %in %in %out %. +%term revFs_nil revFs nil Fs' Fs' %. +%term revFs_cons %pi (revFs (cons F Fs) Fs'' Fs') %<- (revFs Fs (cons F Fs'') Fs') %. +%worlds () (revFs _ _ _) %. +%freeze revFs %. +%total D (revFs D _ _) %. +%sort revFs_exists {Fs} {Fs'} %. +%mode revFs_exists %in %in %out %. +%term revFs_exists_nil revFs_exists nil _ revFs_nil %. +%term revFs_exists_cons + %pi (revFs_exists (cons F Fs) Fs' (revFs_cons Fs'')) + %<- (revFs_exists Fs (cons F Fs') Fs'') %. +%worlds (blam) (revFs_exists _ _ _) %. +%freeze revFs_exists %. +%total D (revFs_exists D _ _) %. +%sort revDet %. +%mode revDet %in %in %out %. +%term revDet_nil revDet revFs_nil _ eqf_ref %. +%term revDet_cons %pi (revDet (revFs_cons R) (revFs_cons R') Q) %<- (revDet R R' Q) %. +%worlds (blam) (revDet _ _ _) %. +%freeze revDet %. +%total D (revDet D _ _) %. +%sort revrev_id_lem %. +%mode revrev_id_lem %in %in %in %out %. +%term revrev_id_lem_nil %pi (revrev_id_lem revFs_nil F F' Q) %<- (revDet F' F Q) %. +%term revrev_id_lem_cons + %pi (revrev_id_lem (revFs_cons R) R' R'' Q) + %<- (revrev_id_lem R R' (revFs_cons R'') Q) %. +%worlds (blam) (revrev_id_lem _ _ _ _) %. +%freeze revrev_id_lem %. +%total D (revrev_id_lem D _ _ _) %. +%sort revrev_id %. +%mode revrev_id %in %in %out %. +%term revrev_id_rule %pi (revrev_id R R' Q) %<- (revrev_id_lem R R' revFs_nil Q) %. +%worlds (blam) (revrev_id _ _ _) %. +%freeze revrev_id %. +%total {} (revrev_id _ _ _) %. +%sort rev_injective %. +%mode rev_injective %in %in %out %. +%term rev_injective_rule + %pi (rev_injective (%the (revFs Fs nil Fs') R) R' Q) + %<- (revFs_exists Fs' nil Rev) + %<- (revrev_id R Rev Q') + %<- (eqf_symm Q' Q''') + %<- (revrev_id R' Rev Q'') + %<- (eqf_trans Q'' Q''' Q1) + %<- (eqf_symm Q1 Q) %. +%worlds (blam) (rev_injective _ _ _) %. +%freeze rev_injective %. +%total D (rev_injective D _ _) %. +%sort rev_eq1 %. +%mode rev_eq1 %in %in %out %. +%term rev_eq1_rule rev_eq1 eqf_ref R R %. +%worlds (blam) (rev_eq1 _ _ _) %. +%freeze rev_eq1 %. +%total {} (rev_eq1 _ _ _) %. +%sort rev_eq2 %. +%mode rev_eq2 %in %in %out %. +%term rev_eq2_rule rev_eq2 eqf_ref R R %. +%worlds (blam) (rev_eq2 _ _ _) %. +%freeze rev_eq2 %. +%total {} (rev_eq2 _ _ _) %. +%sort frameapply_e %. +%mode frameapply_e %in %in %in %out %. +%term frameapply_e_nil frameapply_e revFs_nil FA frameapply_nil FA %. +%term frameapply_e_cons + %pi (frameapply_e (revFs_cons R) FA' (frameapply_cons FA frameapp_app) FA'') + %<- (frameapply_e R (frameapply'_cons frameapp_app FA') FA FA'') %. +%worlds (blam) (frameapply_e _ _ _ _) %. +%freeze frameapply_e %. +%total D (frameapply_e _ _ D _) %. +%sort frameapply_e2 %. +%mode frameapply_e2 %in %in %out %. +%term frameapply_e2_rule %pi (frameapply_e2 R FA' FA) %<- (frameapply_e R frameapply'_nil FA' FA) %. +%worlds (blam) (frameapply_e2 _ _ _) %. +%freeze frameapply_e2 %. +%total {} (frameapply_e2 _ _ _) %. +%sort frameapply'_e %. +%mode frameapply'_e %in %in %in %in %out %. +%term frameapply'_e_nil + %pi (frameapply'_e R R' frameapply'_nil FA FA') + %<- (revrev_id R R' Q) + %<- (frameapply_eq Q FA FA') %. +%term frameapply'_e_cons + %pi (frameapply'_e R R' (frameapply'_cons frameapp_app FA') FA FA'') + %<- (frameapply'_e (revFs_cons R) R' FA' (frameapply_cons FA frameapp_app) FA'') %. +%worlds (blam) (frameapply'_e _ _ _ _ _) %. +%freeze frameapply'_e %. +%total D (frameapply'_e _ _ D _ _) %. +%sort frameapply_e1 %. +%mode frameapply_e1 %in %in %out %. +%term frameapply_e1_rule + %pi (frameapply_e1 R FA' FA) + %<- (frameapply'_e revFs_nil R FA' frameapply_nil FA) %. +%worlds (blam) (frameapply_e1 _ _ _) %. +%freeze frameapply_e1 %. +%total {} (frameapply_e1 _ _ _) %. +%{! ### Some degenerate lemmas +The next lemma is degenerate and needed to take care of other degenerate cases. !}% +%sort lem45 {M %pi term %-> term} {m term} %. +%mode lem45 %in %in %out %. +%term lem45_nil lem45 _ ([m] frameapply'_nil) ([m] frameapply'_nil) %. +%term lem45_app + %pi (lem45 M ([m] frameapply'_cons (%the (frameapp (fapp _) _ _) frameapp_app) (FA m)) ([m] frameapply'_cons frameapp_app (FA' m))) + %<- (lem45 M FA FA') %. +%term lem45_tapp + %pi (lem45 M ([m] frameapply'_cons (%the (frameapp (ftapp _) _ _) frameapp_app) (FA m)) ([m] frameapply'_cons frameapp_app (FA' m))) + %<- (lem45 M FA FA') %. +%term lem45_letb + %pi (lem45 M ([m] frameapply'_cons (%the (frameapp (fletb _) _ _) frameapp_app) (FA m)) ([m] frameapply'_cons frameapp_app (FA' m))) + %<- (lem45 M FA FA') %. +%term lem45_lett + %pi (lem45 M ([m] frameapply'_cons (%the (frameapp (flett _) _ _) frameapp_app) (FA m)) ([m] frameapply'_cons frameapp_app (FA' m))) + %<- (lem45 M FA FA') %. +%term lem45_letu + %pi (lem45 M ([m] frameapply'_cons (%the (frameapp (fletu _) _ _) frameapp_app) (FA m)) ([m] frameapply'_cons frameapp_app (FA' m))) + %<- (lem45 M FA FA') %. +%worlds (blam) (lem45 _ _ _) %. +%freeze lem45 %. +%total D (lem45 _ D _) %. +%{! ### Frames are linear !}% +%sort redex %. +%name redex %. +% mode redex +M +M'. +%term redex_app redex (app (lam T M1) M2) (M1 M2) %. +%term redex_tapp redex (tapp (tlam M) T) (M T) %. +%term redex_lett redex (lett (tens M1 M2) M3) (M3 M1 M2) %. +%term redex_letu redex (letu unit M) M %. +%term redex_letb redex (letb (thunk T M1) M2) (M2 (letb (thunk T M1) M1)) %. +%freeze redex %. +%sort ctp %. +%term ctp_0 ctp %. +%term ctp_1 %pi (linear M) %-> ctp %. +%term ctp_2 + %pi ({m term} linear ([n term] M n m)) + %-> ({m term} linear ([n term] M m n)) + %-> ctp %. +%freeze ctp %. +%sort redex_tp {m term} %. +%term redex_tp_app {L linear (M N)} redex_tp ([m term] %the (redex (app (lam T (M m)) (M2 m)) _) redex_app) (ctp_1 L) %. +%term redex_tp_tapp redex_tp ([m] %the (redex (tapp (tlam (M m)) T) (M m T)) redex_tapp) ctp_0 %. +%term redex_tp_lett {L1 {m term} linear ([n term] M3 N n m)} {L2 {m term} linear ([n term] M3 N m n)} redex_tp ([m term] %the (redex (lett (tens (M1 m) (M2 m)) (M3 m)) _) redex_lett) (ctp_2 L1 L2) %. +%term redex_tp_letu redex_tp ([m] redex_letu) ctp_0 %. +%term redex_tp_letb redex_tp ([m] %the (redex (letb (thunk T (M1 m)) (M2 m)) (M2 m (letb (thunk T (M1 m)) (M1 m)))) redex_letb) ctp_0 %. +%freeze redex_tp %. +%sort lem33 {m term} %. +%mode lem33 %in %in %out %. +%term lem33_nil lem33 C ([m] frameapply'_nil) ([m] frameapply'_nil) %. +%term lem33_app + %pi (lem33 C ([m] frameapply'_cons (%the (frameapp (fapp _) _ _) frameapp_app) (FA m)) ([m] frameapply'_cons frameapp_app (FA' m))) + %<- (lem33 C FA FA') %. +%term lem33_tapp + %pi (lem33 C ([m] frameapply'_cons (%the (frameapp (ftapp _) _ _) frameapp_app) (FA m)) ([m] frameapply'_cons frameapp_app (FA' m))) + %<- (lem33 C FA FA') %. +%term lem33_letb + %pi (lem33 C ([m] frameapply'_cons (%the (frameapp (fletb _) _ _) frameapp_app) (FA m)) ([m] frameapply'_cons frameapp_app (FA' m))) + %<- (lem33 C FA FA') %. +%term lem33_lett + %pi (lem33 C ([m] frameapply'_cons (%the (frameapp (flett _) _ _) frameapp_app) (FA m)) ([m] frameapply'_cons frameapp_app (FA' m))) + %<- (lem33 C FA FA') %. +%term lem33_letu + %pi (lem33 C ([m] frameapply'_cons (%the (frameapp (fletu _) _ _) frameapp_app) (FA m)) ([m] frameapply'_cons frameapp_app (FA' m))) + %<- (lem33 C FA FA') %. +%worlds (blam) (lem33 _ _ _) %. +%total D (lem33 _ D _) %. +%sort lem31 {m term} %. +%mode lem31 %in %in %out %. +%term lem31_rule lem31 _ ([m] frameapply'_nil) ([m] eq_ref) %. +%worlds (blam) (lem31 _ _ _) %. +%total {} (lem31 _ _ _) %. +%sort lem32 {Fs} {M} {m} %. +%mode lem32 %in %in %in %in %in %out %. +%worlds (blam) (lem32 _ _ _ _ _ _) %. +%total {} (lem32 _ _ _ _ _ _) %. +%sort lem34 {a term} %. +%mode lem34 %in %out %. +%term lem34_rule lem34 ([m] frameapply'_nil) ([m] eqf_ref) %. +%worlds (blam) (lem34 _ _) %. +%total {} (lem34 _ _) %. +%sort lem35 {Fs} {M} {m term} %. +%mode lem35 %in %in %in %in %out %. +%worlds (blam) (lem35 _ _ _ _ _) %. +%total {} (lem35 _ _ _ _ _) %. +%{! Here I am building up rather technical lemmas that are needed in the strictness lemma. +Take note of how easy it is to deal with linearity. !}% +%sort frame_linear_lem2 %. +%mode frame_linear_lem2 %in %in %in %out %out %. +%term frame_linear_lem2_nil1 frame_linear_lem2 _ ([m] %the (frameapply' _ (app _ M2) _) frameapply'_nil) (linear_app1 (linear_lam L)) ([m] frameapply'_nil) (L M2) %. +%term frame_linear_lem2_nil2 + %pi (frame_linear_lem2 (redex_tp_app L2) ([m] %the (frameapply' _ (app _ (M' m)) _) frameapply'_nil) (linear_app2 L) ([m] frameapply'_nil) L') + %<- (sub_linear L2 L L') %. +%term frame_linear_tens_lem2_nil1 + %pi (frame_linear_lem2 (redex_tp_lett L1 L2) ([m] %the (frameapply' _ (lett (tens _ M2) M3) _) frameapply'_nil) (linear_lett1 (linear_tens1 L)) ([m] frameapply'_nil) L') + %<- (sub_linear (L1 M2) L L') %. +%term frame_linear_tens_lem2_nil2 + %pi (frame_linear_lem2 (redex_tp_lett L1 L2) ([m] %the (frameapply' _ (lett (tens M1 _) M3) _) frameapply'_nil) (linear_lett1 (linear_tens2 L)) ([m] frameapply'_nil) L') + %<- (sub_linear (L2 M1) L L') %. +%term frame_linear_tens_lem2_nil3 frame_linear_lem2 C ([m] %the (frameapply' _ (lett (tens M1 M2) (M3 m)) _) frameapply'_nil) (linear_lett2 L) ([m] frameapply'_nil) (L M1 M2) %. +%term frame_linear_unit_lem_nil2 frame_linear_lem2 C ([m] %the (frameapply' _ (letu unit (M3 m)) _) frameapply'_nil) (linear_letu2 L) ([m] frameapply'_nil) L %. +%term frame_linear_thunk_lem_nil2 frame_linear_lem2 C ([m] %the (frameapply' _ (letb (thunk T M1) (M2 m)) _) frameapply'_nil) (linear_letb2 L) ([m] frameapply'_nil) (L (letb (thunk T M1) M1)) %. +%term frame_linear_tlam_lem_nil1 frame_linear_lem2 C ([m] %the (frameapply' _ (tapp _ T) _) frameapply'_nil) (linear_tapp (linear_tlam L)) ([m] frameapply'_nil) (L T) %. +%term frame_linear_cons_app1 + %pi (frame_linear_lem2 C ([m] frameapply'_cons (%the (frameapp (fapp M2) (M1 m) _) frameapp_app) (FA m)) (linear_app1 L) ([m] frameapply'_cons frameapp_app (FA' m)) (linear_app1 L')) + %<- (frame_linear_lem2 C FA L FA' L') %. +%term frame_linear_cons_tapp1 + %pi (frame_linear_lem2 C ([m] frameapply'_cons (%the (frameapp (ftapp T) (M1 m) _) frameapp_app) (FA m)) (linear_tapp L) ([m] frameapply'_cons frameapp_app (FA' m)) (linear_tapp L')) + %<- (frame_linear_lem2 C FA L FA' L') %. +%term frame_linear_cons_letu1 + %pi (frame_linear_lem2 C ([m] frameapply'_cons (%the (frameapp (fletu T) (M1 m) _) frameapp_app) (FA m)) (linear_letu1 L) ([m] frameapply'_cons frameapp_app (FA' m)) (linear_letu1 L')) + %<- (frame_linear_lem2 C FA L FA' L') %. +%term frame_linear_cons_letb1 + %pi (frame_linear_lem2 C ([m] frameapply'_cons (%the (frameapp (fletb T) (M1 m) _) frameapp_app) (FA m)) (linear_letb1 L) ([m] frameapply'_cons frameapp_app (FA' m)) (linear_letb1 L')) + %<- (frame_linear_lem2 C FA L FA' L') %. +%term frame_linear_cons_lett1 + %pi (frame_linear_lem2 C ([m] frameapply'_cons (%the (frameapp (flett T) (M1 m) _) frameapp_app) (FA m)) (linear_lett1 L) ([m] frameapply'_cons frameapp_app (FA' m)) (linear_lett1 L')) + %<- (frame_linear_lem2 C FA L FA' L') %. +%term frame_linear_cons_letu2 + %pi (frame_linear_lem2 (%the (redex_tp C' _) C) ([m] frameapply'_cons (%the (frameapp (fletu (M2 m)) M1 _) frameapp_app) (FA m)) (linear_letu2 L) ([m] frameapply'_cons frameapp_app (FA' m)) (linear_letu2 L)) + %<- (lem33 C' FA FA') %. +%term frame_linear_cons_letb2 + %pi (frame_linear_lem2 (%the (redex_tp C' _) C) ([m] frameapply'_cons (%the (frameapp (fletb (M2 m)) M1 _) frameapp_app) (FA m)) (linear_letb2 L) ([m] frameapply'_cons frameapp_app (FA' m)) (linear_letb2 L)) + %<- (lem33 C' FA FA') %. +%term frame_linear_cons_lett2 + %pi (frame_linear_lem2 (%the (redex_tp C' _) C) ([m] frameapply'_cons (%the (frameapp (flett (M2 m)) M1 _) frameapp_app) (FA m)) (linear_lett2 L) ([m] frameapply'_cons frameapp_app (FA' m)) (linear_lett2 L)) + %<- (lem33 C' FA FA') %. +%term frame_linear_cons_app2 + %pi (frame_linear_lem2 (%the (redex_tp C' _) C) ([m] frameapply'_cons (%the (frameapp (fapp (M2 m)) M1 _) frameapp_app) (FA m)) (linear_app2 L) ([m] frameapply'_cons frameapp_app (FA' m)) (linear_app2 L)) + %<- (lem33 C' FA FA') %. +%term frame_linear_cons_tens1 + %pi (frame_linear_lem2 (%the (redex_tp C' _) C) (%the ({m} frameapply' (Fs m) (M1 m) (tens (M3 m) M4)) FA) (linear_tens1 L) FA' (linear_tens1 L)) + %<- (lem31 ([m] val_tens) FA Q) + %<- (lem32 Fs ([m] tens (M3 m) M4) ([m] val_tens) C' Q FA') %. +%term frame_linear_cons_tens2 + %pi (frame_linear_lem2 (%the (redex_tp C' _) C) (%the ({m} frameapply' (Fs m) (M1 m) (tens M3 (M4 m))) FA) (linear_tens2 L) FA' (linear_tens2 L)) + %<- (lem31 ([m] val_tens) FA Q) + %<- (lem32 Fs ([m] tens M3 (M4 m)) ([m] val_tens) C' Q FA') %. +%term frame_linear_cons_tlam + %pi (frame_linear_lem2 (%the (redex_tp C' _) C) (%the ({m} frameapply' (Fs m) (M1 m) (tlam (M4 m))) FA) (linear_tlam L) FA' (linear_tlam L)) + %<- (lem31 ([m] val_tlam) FA Q) + %<- (lem32 Fs ([m] tlam (M4 m)) ([m] val_tlam) C' Q FA') %. +%term frame_linear_cons_lam_lem2 + %pi (frame_linear_lem2 (%the (redex_tp C' _) C) (%the ({m term} frameapply' (Fs m) (M1 m) (lam T1 (M4 m))) FA) (linear_lam L) FA' (linear_lam L)) + %<- (lem31 ([m] val_lam) FA Q) + %<- (lem32 Fs ([m] lam T1 (M4 m)) ([m] val_lam) C' Q FA') %. +%term frame_linear_lam_lem2_id + %pi (frame_linear_lem2 (%the (redex_tp C' _) C) (%the ({m} frameapply' (Fs m) (M1 m) m) FA) linear_id FA' linear_id) + %<- (lem34 FA Q) + %<- ({m} frameapply'_eq (Q m) (FA m) (FA'' m)) + %<- ({m} frameapply'_nil_eq (FA'' m) (Q' m)) + %<- (lem35 Fs ([m] m) C' Q' FA') %. +%worlds (blam) (frame_linear_lem2 _ _ _ _ _) %. +%total FA (frame_linear_lem2 _ FA _ _ _) %. +%sort frame_linear_lem1 %. +%mode frame_linear_lem1 %in %in %in %out %out %. +%term frame_linear_lem1_rule + %pi (frame_linear_lem1 C (%the ({m} frameapply (Fs m) _ _) FA) LR FA' LR') + %<- ({m} revFs_exists (Fs m) nil (%the (revFs (Fs m) nil (Fs' m)) (Rev m))) + %<- ({m} frameapply_e2 (Rev m) (FA m) (FA'' m)) + %<- (frame_linear_lem2 C FA'' LR FA3 LR') + %<- ({m} revFs_exists (Fs' m) nil (Rev' m)) + %<- ({m} revrev_id (Rev m) (Rev' m) (Q m)) + %<- ({m} frameapply_e1 (Rev' m) (FA3 m) (FA4 m)) + %<- ({m} eqf_symm (Q m) (Q' m)) + %<- ({m} frameapply_eq (Q' m) (FA4 m) (FA' m)) %. +%worlds (blam) (frame_linear_lem1 _ _ _ _ _) %. +%total {} (frame_linear_lem1 _ _ _ _ _) %. +%sort frame_linear {m} %. +%mode frame_linear %in %in %in %in %out %out %. +%term frame_linear_app + %pi (frame_linear ([m] redex_app) (of_app D2 (of_lam LM _)) FA LR FA' LR') + %<- (frame_linear_lem1 (%the (redex_tp ([m term] %the (redex (app (lam T (M1 m)) _) _) redex_app) _) (redex_tp_app LM)) FA LR FA' LR') %. +%term frame_linear_lett + %pi (frame_linear ([m] redex_lett) (of_lett L1 L2 _ _) FA LR FA' LR') + %<- (frame_linear_lem1 (%the (redex_tp ([m term] %the (redex (lett (tens (M1 m) (M2 m)) (M3 m)) _) redex_lett) _) (redex_tp_lett L2 L1)) FA LR FA' LR') %. +%term frame_linear_letu + %pi (frame_linear ([m] redex_letu) _ FA LR FA' LR') + %<- (frame_linear_lem1 redex_tp_letu FA LR FA' LR') %. +%term frame_linear_letb + %pi (frame_linear ([m] redex_letb) _ FA LR FA' LR') + %<- (frame_linear_lem1 redex_tp_letb FA LR FA' LR') %. +%term frame_linear_tapp + %pi (frame_linear ([m] redex_tapp) _ FA LR FA' LR') + %<- (frame_linear_lem1 redex_tp_tapp FA LR FA' LR') %. +%worlds () (frame_linear _ _ _ _ _ _) %. +%total {} (frame_linear _ _ _ _ _ _) %. +%{! ### Termination is invariant to redexes !}% +%sort add_redex_steps %. +%mode add_redex_steps %in %in %out %. +%term add_redex_steps_app add_redex_steps redex_app Ft (frameterm_app (frameterm_lam Ft)) %. +%term add_redex_steps_tapp add_redex_steps redex_tapp Ft (frameterm_tapp (frameterm_tlam Ft)) %. +%term add_redex_steps_lett add_redex_steps redex_lett Ft (frameterm_lett (frameterm_tens Ft)) %. +%term add_redex_steps_letu add_redex_steps redex_letu Ft (frameterm_letu (frameterm_unit Ft)) %. +%term add_redex_steps_letb add_redex_steps redex_letb Ft (frameterm_letb (frameterm_thunk Ft)) %. +%worlds () (add_redex_steps _ _ _) %. +%total {} (add_redex_steps _ _ _) %. +%sort lem42 {Fs} {N} %. +%mode lem42 %in %in %in %in %out %. +%term lem42_val + %pi (lem42 ([a] nil) ([a] a) R (frameterm_val V) Ft) + %<- (add_redex_steps R (frameterm_val V) Ft) %. +%term lem42_base1 lem42 ([a term] nil) ([a] lam T _) _ (frameterm_val _) (frameterm_val val_lam) %. +%term lem42_base2 lem42 ([a term] nil) ([a] tlam _) _ (frameterm_val _) (frameterm_val val_tlam) %. +%term lem42_base3 lem42 ([a term] nil) ([a] tens _ _) _ (frameterm_val _) (frameterm_val val_tens) %. +%term lem42_base4 lem42 ([a term] nil) ([a] thunk T _) _ (frameterm_val _) (frameterm_val val_thunk) %. +%term lem42_base5 lem42 ([a term] nil) ([a] unit) _ (frameterm_val _) (frameterm_val val_unit) %. +%term lem42_app + %pi (lem42 Fs ([a] a) R (frameterm_app Ft) Ft') + %<- (lem42 ([a] cons (fapp M2) (Fs a)) ([a] M1) R Ft Ft'') + %<- (add_redex_steps R (frameterm_app Ft'') Ft') %. +%term lem42_tapp + %pi (lem42 Fs ([a] a) R (frameterm_tapp Ft) Ft') + %<- (lem42 ([a] cons (ftapp T) (Fs a)) ([a] M1) R Ft Ft'') + %<- (add_redex_steps R (frameterm_tapp Ft'') Ft') %. +%term lem42_lett + %pi (lem42 Fs ([a] a) R (frameterm_lett Ft) Ft') + %<- (lem42 ([a] cons (flett T) (Fs a)) ([a] M1) R Ft Ft'') + %<- (add_redex_steps R (frameterm_lett Ft'') Ft') %. +%term lem42_letu + %pi (lem42 Fs ([a] a) R (frameterm_letu Ft) Ft') + %<- (lem42 ([a] cons (fletu T) (Fs a)) ([a] M1) R Ft Ft'') + %<- (add_redex_steps R (frameterm_letu Ft'') Ft') %. +%term lem42_letb + %pi (lem42 Fs ([a] a) R (frameterm_letb Ft) Ft') + %<- (lem42 ([a] cons (fletb T) (Fs a)) ([a] M1) R Ft Ft'') + %<- (add_redex_steps R (frameterm_letb Ft'') Ft') %. +%term lem42_unit + %pi (lem42 ([a] cons (fletu (M2 a)) (Fs a)) ([a] a) R (frameterm_unit Ft) Ft') + %<- (lem42 Fs ([a] M2 a) R Ft Ft'') + %<- (add_redex_steps R (frameterm_unit Ft'') Ft') %. +%term lem42_tlam + %pi (lem42 ([a] cons (ftapp T) (Fs a)) ([a] a) R (%the (frameterm _ (tlam M)) (frameterm_tlam Ft)) Ft') + %<- (lem42 Fs ([a] M T) R Ft Ft'') + %<- (add_redex_steps R (frameterm_tlam Ft'') Ft') %. +%term lem42_lam + %pi (lem42 ([a] cons (fapp (M2 a)) (Fs a)) ([a] a) R (%the (frameterm _ (lam T M1')) (frameterm_lam Ft)) Ft') + %<- (lem42 Fs ([a] M1' (M2 a)) R Ft Ft'') + %<- (add_redex_steps R (frameterm_lam Ft'') Ft') %. +%term lem42_tens + %pi (lem42 ([a] cons (flett (M3 a)) (Fs a)) ([a] a) R (%the (frameterm _ (tens M1 M2)) (frameterm_tens Ft)) Ft') + %<- (lem42 Fs ([a] M3 a M1 M2) R Ft Ft'') + %<- (add_redex_steps R (frameterm_tens Ft'') Ft') %. +%term lem42_thunk + %pi (lem42 ([a] cons (fletb (M2 a)) (Fs a)) ([a] a) R (%the (frameterm _ (thunk T M1)) (frameterm_thunk Ft)) Ft') + %<- (lem42 Fs ([a] M2 a (letb (thunk T M1) M1)) R Ft Ft'') + %<- (add_redex_steps R (frameterm_thunk Ft'') Ft') %. +%term lem42_lam2 + %pi (lem42 ([a] cons (fapp (M2 a)) (Fs a)) ([a] lam T (M1' a)) R (frameterm_lam Ft) (frameterm_lam Ft')) + %<- (lem42 Fs ([a] M1' a (M2 a)) R Ft Ft') %. +%term lem42_tens2 + %pi (lem42 ([a] cons (flett (M3 a)) (Fs a)) ([a] tens (M1 a) (M2 a)) R (frameterm_tens Ft) (frameterm_tens Ft')) + %<- (lem42 Fs ([a] M3 a (M1 a) (M2 a)) R Ft Ft') %. +%term lem42_tlam2 + %pi (lem42 ([a] cons (ftapp T) (Fs a)) ([a] tlam (M1' a)) R (frameterm_tlam Ft) (frameterm_tlam Ft')) + %<- (lem42 Fs ([a] M1' a T) R Ft Ft') %. +%term lem42_unit2 + %pi (lem42 ([a] cons (fletu (M2 a)) (Fs a)) ([a] unit) R (frameterm_unit Ft) (frameterm_unit Ft')) + %<- (lem42 Fs ([a] M2 a) R Ft Ft') %. +%term lem42_thunk2 + %pi (lem42 ([a] cons (fletb (M2 a)) (Fs a)) ([a] thunk T (M1 a)) R (frameterm_thunk Ft) (frameterm_thunk Ft')) + %<- (lem42 Fs ([a] M2 a (letb (thunk T (M1 a)) (M1 a))) R Ft Ft') %. +%term lem42_app2 + %pi (lem42 Fs ([a] app (M1 a) (M2 a)) R Ft (frameterm_app Ft')) + %<- (inverse_ft_app Ft Ft2) + %<- (lem42 ([a] cons (fapp (M2 a)) (Fs a)) M1 R Ft2 Ft') %. +%term lem42_tapp2 + %pi (lem42 Fs ([a] tapp (M1 a) T) R Ft (frameterm_tapp Ft')) + %<- (inverse_ft_tapp Ft Ft2) + %<- (lem42 ([a] cons (ftapp T) (Fs a)) M1 R Ft2 Ft') %. +%term lem42_letu2 + %pi (lem42 Fs ([a] letu (M1 a) (M2 a)) R Ft (frameterm_letu Ft')) + %<- (inverse_ft_letu Ft Ft2) + %<- (lem42 ([a] cons (fletu (M2 a)) (Fs a)) M1 R Ft2 Ft') %. +%term lem42_letb2 + %pi (lem42 Fs ([a] letb (M1 a) (M2 a)) R Ft (frameterm_letb Ft')) + %<- (inverse_ft_letb Ft Ft2) + %<- (lem42 ([a] cons (fletb (M2 a)) (Fs a)) M1 R Ft2 Ft') %. +%term lem42_lett2 + %pi (lem42 Fs ([a] lett (M1 a) (M2 a)) R Ft (frameterm_lett Ft')) + %<- (inverse_ft_lett Ft Ft2) + %<- (lem42 ([a] cons (flett (M2 a)) (Fs a)) M1 R Ft2 Ft') %. +%worlds () (lem42 _ _ _ _ _) %. +%total D (lem42 _ _ _ D _) %. +%sort lem20 {M term} {M' term} {a term} %. +%mode lem20 %in %in %in %in %out %. +%term lem20_rule lem20 M M' _ ([a] frameapply'_nil) eqf_ref %. +%worlds (blam) (lem20 _ _ _ _ _) %. +%freeze lem20 %. +%total {} (lem20 _ _ _ _ _) %. +%sort lem26 {F1} {F2} {m} %. +%mode lem26 %in %in %in %out %. +%worlds (blam) (lem26 _ _ _ _) %. +%freeze lem26 %. +%total {} (lem26 _ _ _ _) %. +%{! ### More on linear frames !}% +%sort lem27 %. +%mode lem27 %in %in %in %out %. +%term lem27_app1 %pi (lem27 (relavant_app1 R) M M' Q) %<- (lem27 R M M' Q) %. +%term lem27_app2 %pi (lem27 (relavant_app2 R) M M' Q) %<- (lem27 R M M' Q) %. +%term lem27_lam %pi (lem27 (relavant_lam R) M M' Q) %<- (lem27 (R unit) M M' Q) %. +%term lem27_tlam %pi (lem27 (relavant_tlam R) M M' Q) %<- (lem27 (R i) M M' Q) %. +%term lem27_tapp %pi (lem27 (relavant_tapp R) M M' Q) %<- (lem27 R M M' Q) %. +%term lem27_letb1 %pi (lem27 (relavant_letb1 R) M M' Q) %<- (lem27 R M M' Q) %. +%term lem27_leb2 %pi (lem27 (relavant_letb2 R) M M' Q) %<- (lem27 (R unit) M M' Q) %. +%term lem27_thunk %pi (lem27 (relavant_thunk R) M M' Q) %<- (lem27 (R unit) M M' Q) %. +%term lem27_letu1 %pi (lem27 (relavant_letu1 R) M M' Q) %<- (lem27 R M M' Q) %. +%term lem27_letu2 %pi (lem27 (relavant_letu2 R) M M' Q) %<- (lem27 R M M' Q) %. +%term lem27_tens1 %pi (lem27 (relavant_tens1 R) M M' Q) %<- (lem27 R M M' Q) %. +%term lem27_tens2 %pi (lem27 (relavant_tens2 R) M M' Q) %<- (lem27 R M M' Q) %. +%term lem27_lett1 %pi (lem27 (relavant_lett1 R) M M' Q) %<- (lem27 R M M' Q) %. +%term lem27_lett2 %pi (lem27 (relavant_lett2 R) M M' Q) %<- (lem27 (R unit unit) M M' Q) %. +%worlds (blam) (lem27 _ _ _ _) %. +%freeze lem27 %. +%total D (lem27 D _ _ _) %. +%sort lem28 {Fs'} {Fs''} {m term} %. +%mode lem28 %in %in %in %in %out %. +%term lem28_id %pi (lem28 Fs' Fs'' ([m] frameapply_cons _ (F m)) _ Q) %<- (lem26 _ _ F Q) %. +%term lem28_app1 + %pi (lem28 Fs' Fs'' ([m] frameapply_cons (FA m) (%the (frameapp (fapp (N m)) (M3 m) (app (M3 m) (N m))) frameapp_app)) R Q) + %<- (lem28 Fs' Fs'' FA (relavant_app1 R) Q) %. +%term lem28_tapp + %pi (lem28 Fs' Fs'' ([m] frameapply_cons (FA m) (%the (frameapp (ftapp T) (M3 m) (tapp (M3 m) T)) frameapp_app)) R Q) + %<- (lem28 Fs' Fs'' FA (relavant_tapp R) Q) %. +%term lem28_letb + %pi (lem28 Fs' Fs'' ([m] frameapply_cons (FA m) (%the (frameapp (fletb (N m)) (M3 m) (letb (M3 m) (N m))) frameapp_app)) R Q) + %<- (lem28 Fs' Fs'' FA (relavant_letb1 R) Q) %. +%term lem28_lett + %pi (lem28 Fs' Fs'' ([m] frameapply_cons (FA m) (%the (frameapp (flett (N m)) (M3 m) (lett (M3 m) (N m))) frameapp_app)) R Q) + %<- (lem28 Fs' Fs'' FA (relavant_lett1 R) Q) %. +%term lem28_letu + %pi (lem28 Fs' Fs'' ([m] frameapply_cons (FA m) (%the (frameapp (fletu (N m)) (M3 m) (letu (M3 m) (N m))) frameapp_app)) R Q) + %<- (lem28 Fs' Fs'' FA (relavant_letu1 R) Q) %. +%term lem28_nil + %pi (lem28 Fs' Fs'' ([m term] %the (frameapply nil (N2 m) N1) (FA m)) R Q) + %<- ({m} frameapply_nil_eq (FA m) (Q' m)) + %<- (eq_relavant Q' R R') + %<- (lem27 R' Fs' Fs'' Q) %. +%worlds (blam) (lem28 _ _ _ _ _) %. +%freeze lem28 %. +%total D (lem28 _ _ D _ _) %. +%sort lem19 {M' term} {M term} {a} %. +%mode lem19 %in %in %in %in %out %. +%term lem19_nil lem19 M M' ([a] frameapply'_nil) L eqf_ref %. +%term lem19_app1 + %pi (lem19 M M' ([a] frameapply'_cons (%the (frameapp (fapp N2) _ _) frameapp_app) (FA a)) (linear_app1 L) Q) + %<- (lem19 M M' FA L Q') + %<- (eqf_extend (fapp N2) Q' Q) %. +%term lem19_letu1 + %pi (lem19 M M' ([a] frameapply'_cons (%the (frameapp (fletu N2) _ _) frameapp_app) (FA a)) (linear_letu1 L) Q) + %<- (lem19 M M' FA L Q') + %<- (eqf_extend (fletu N2) Q' Q) %. +%term lem19_lett1 + %pi (lem19 M M' ([a] frameapply'_cons (%the (frameapp (flett N2) _ _) frameapp_app) (FA a)) (linear_lett1 L) Q) + %<- (lem19 M M' FA L Q') + %<- (eqf_extend (flett N2) Q' Q) %. +%term lem19_letb1 + %pi (lem19 M M' ([a] frameapply'_cons (%the (frameapp (fletb N2) _ _) frameapp_app) (FA a)) (linear_letb1 L) Q) + %<- (lem19 M M' FA L Q') + %<- (eqf_extend (fletb N2) Q' Q) %. +%term lem19_tapp + %pi (lem19 M M' ([a] frameapply'_cons (%the (frameapp (ftapp N2) _ _) frameapp_app) (FA a)) (linear_tapp L) Q) + %<- (lem19 M M' FA L Q') + %<- (eqf_extend (ftapp N2) Q' Q) %. +%term lem19_tens1 %pi (lem19 M M' FA (linear_tens1 L) Q) %<- (lem20 M M' ([a] val_tens) FA Q) %. +%term lem19_tens2 %pi (lem19 M M' FA (linear_tens2 L) Q) %<- (lem20 M M' ([a] val_tens) FA Q) %. +%term lem19_lam %pi (lem19 M M' FA (linear_lam L) Q) %<- (lem20 M M' ([a] val_lam) FA Q) %. +%term lem19_tlam %pi (lem19 M M' FA (linear_tlam L) Q) %<- (lem20 M M' ([a] val_tlam) FA Q) %. +%term lem19_app2 + %pi (lem19 M M' ([a] frameapply'_cons (%the (frameapp (fapp (N2 a)) _ _) frameapp_app) (%the (frameapply' (Fs a) a M1) (FA a))) (linear_app2 L) Q) + %<- ({y term} revFs_exists (Fs y) nil (Rev y)) + %<- ({y term} frameapply_e1 (Rev y) (FA y) (FA' y)) + %<- (lem28 _ _ FA' relavant_id Q) %. +%term lem19_letb2 + %pi (lem19 M M' ([a] frameapply'_cons (%the (frameapp (fletb (N2 a)) _ _) frameapp_app) (%the (frameapply' (Fs a) a M1) (FA a))) (linear_letb2 L) Q) + %<- ({y term} revFs_exists (Fs y) nil (Rev y)) + %<- ({y term} frameapply_e1 (Rev y) (FA y) (FA' y)) + %<- (lem28 _ _ FA' relavant_id Q) %. +%term lem19_letu2 + %pi (lem19 M M' ([a] frameapply'_cons (%the (frameapp (fletu (N2 a)) _ _) frameapp_app) (%the (frameapply' (Fs a) a M1) (FA a))) (linear_letu2 L) Q) + %<- ({y term} revFs_exists (Fs y) nil (Rev y)) + %<- ({y term} frameapply_e1 (Rev y) (FA y) (FA' y)) + %<- (lem28 _ _ FA' relavant_id Q) %. +%term lem19_lett2 + %pi (lem19 M M' ([a] frameapply'_cons (%the (frameapp (flett (N2 a)) _ _) frameapp_app) (%the (frameapply' (Fs a) a M1) (FA a))) (linear_lett2 L) Q) + %<- ({y term} revFs_exists (Fs y) nil (Rev y)) + %<- ({y term} frameapply_e1 (Rev y) (FA y) (FA' y)) + %<- (lem28 _ _ FA' relavant_id Q) %. +%worlds (blam) (lem19 _ _ _ _ _) %. +%total D (lem19 _ _ _ D _) %. +%sort lem18 {M' term} {M term} {a} %. +%mode lem18 %in %in %in %in %out %. +%term lem18_rule + %pi (lem18 M M' (%the ({a} frameapply (Fs a) _ _) FA) L Q) + %<- ({a} revFs_exists (Fs a) nil (Rev a)) + %<- ({a} frameapply_e2 (Rev a) (FA a) (FA' a)) + %<- (lem19 M M' FA' L Q') + %<- (rev_eq2 Q' (Rev M) R) + %<- (rev_injective R (Rev M') Q) %. +%worlds (blam) (lem18 _ _ _ _ _) %. +%total {} (lem18 _ _ _ _ _) %. +%sort apply_linear %. +%mode apply_linear %in %in %in %in %out %out %. +%term apply_linear_nil + %pi (apply_linear LM LN LR ([a term] frameapply'_nil) ([a term] frameapply'_nil) LR') + %<- (sub_linear LM LN LR') %. +%term apply_linear_app1 + %pi (apply_linear LM LN (linear_app1 L) ([a term] frameapply'_cons (%the (frameapp (fapp M2) _ _) frameapp_app) (FA a)) ([a term] frameapply'_cons frameapp_app (FA' a)) (linear_app1 LR')) + %<- (apply_linear LM LN L FA FA' LR') %. +%term apply_linear_tapp1 + %pi (apply_linear LM LN (linear_tapp L) ([a term] frameapply'_cons (%the (frameapp (ftapp M2) _ _) frameapp_app) (FA a)) ([a term] frameapply'_cons frameapp_app (FA' a)) (linear_tapp LR')) + %<- (apply_linear LM LN L FA FA' LR') %. +%term apply_linear_letu1 + %pi (apply_linear LM LN (linear_letu1 L) ([a term] frameapply'_cons (%the (frameapp (fletu M2) _ _) frameapp_app) (FA a)) ([a term] frameapply'_cons frameapp_app (FA' a)) (linear_letu1 LR')) + %<- (apply_linear LM LN L FA FA' LR') %. +%term apply_linear_letb1 + %pi (apply_linear LM LN (linear_letb1 L) ([a term] frameapply'_cons (%the (frameapp (fletb M2) _ _) frameapp_app) (FA a)) ([a term] frameapply'_cons frameapp_app (FA' a)) (linear_letb1 LR')) + %<- (apply_linear LM LN L FA FA' LR') %. +%term apply_linear_lett1 + %pi (apply_linear LM LN (linear_lett1 L) ([a term] frameapply'_cons (%the (frameapp (flett M2) _ _) frameapp_app) (FA a)) ([a term] frameapply'_cons frameapp_app (FA' a)) (linear_lett1 LR')) + %<- (apply_linear LM LN L FA FA' LR') %. +%term apply_linear_app2 + %pi (apply_linear (%the (linear M) LM) LN (linear_app2 L) ([a term] frameapply'_cons (%the (frameapp (fapp _) _ _) frameapp_app) (FA a)) ([a term] frameapply'_cons frameapp_app (FA' a)) (linear_app2 L)) + %<- (lem45 M FA FA') %. +%term apply_linear_letu2 + %pi (apply_linear (%the (linear M) LM) LN (linear_letu2 L) ([a term] frameapply'_cons (%the (frameapp (fletu _) _ _) frameapp_app) (FA a)) ([a term] frameapply'_cons frameapp_app (FA' a)) (linear_letu2 L)) + %<- (lem45 M FA FA') %. +%term apply_linear_letb2 + %pi (apply_linear (%the (linear M) LM) LN (linear_letb2 L) ([a term] frameapply'_cons (%the (frameapp (fletb _) _ _) frameapp_app) (FA a)) ([a term] frameapply'_cons frameapp_app (FA' a)) (linear_letb2 L)) + %<- (lem45 M FA FA') %. +%term apply_linear_lett2 + %pi (apply_linear (%the (linear M) LM) LN (linear_lett2 L) ([a term] frameapply'_cons (%the (frameapp (flett _) _ _) frameapp_app) (FA a)) ([a term] frameapply'_cons frameapp_app (FA' a)) (linear_lett2 L)) + %<- (lem45 M FA FA') %. +%worlds () (apply_linear _ _ _ _ _ _) %. +%freeze apply_linear %. +%total D (apply_linear _ _ _ D _ _) %. +%sort apply_lin %. +%mode apply_lin %in %in %in %in %out %out %. +%term apply_lin_rule + %pi (apply_lin LM LN LR (%the ({m} frameapply (Fs m) (N m) _) FA) FA4 L2) + %<- ({m} revFs_exists (Fs m) nil (%the (revFs (Fs m) nil (Fs' m)) (Rev m))) + %<- ({m} frameapply_e2 (Rev m) (FA m) (FA' m)) + %<- (apply_linear LM LN LR FA' FA2 L2) + %<- ({m} revFs_exists (Fs' m) nil (Rev' m)) + %<- ({m} frameapply_e1 (Rev' m) (FA2 m) (FA3 m)) + %<- ({m} revrev_id (Rev m) (Rev' m) (Q m)) + %<- ({m} eqf_symm (Q m) (Q' m)) + %<- ({m} frameapply_eq (Q' m) (FA3 m) (FA4 m)) %. +%worlds () (apply_lin _ _ _ _ _ _) %. +%freeze apply_lin %. +%total D (apply_lin _ _ _ D _ _) %. +%sort add_frame {F} {m term} %. +%mode add_frame %in %in %in %out %out %. +%term add_frame_app + %pi (add_frame (fapp M) FA LR ([m] frameapply_cons (FA' m) frameapp_app) L') + %<- (apply_lin (%the (linear ([a] app a M)) (linear_app1 linear_id)) linear_id LR FA FA' L') %. +%term add_frame_tapp + %pi (add_frame (ftapp T) FA LR ([m] frameapply_cons (FA' m) frameapp_app) L') + %<- (apply_lin (%the (linear ([a] tapp a T)) (linear_tapp linear_id)) linear_id LR FA FA' L') %. +%term add_frame_lett + %pi (add_frame (flett M) FA LR ([m] frameapply_cons (FA' m) frameapp_app) L') + %<- (apply_lin (%the (linear ([a] lett a M)) (linear_lett1 linear_id)) linear_id LR FA FA' L') %. +%term add_frame_letb + %pi (add_frame (fletb M) FA LR ([m] frameapply_cons (FA' m) frameapp_app) L') + %<- (apply_lin (%the (linear ([a] letb a M)) (linear_letb1 linear_id)) linear_id LR FA FA' L') %. +%term add_frame_letu + %pi (add_frame (fletu M) FA LR ([m] frameapply_cons (FA' m) frameapp_app) L') + %<- (apply_lin (%the (linear ([a] letu a M)) (linear_letu1 linear_id)) linear_id LR FA FA' L') %. +%worlds () (add_frame _ _ _ _ _) %. +%freeze add_frame %. +%total {} (add_frame _ _ _ _ _) %. +%{! ### Strictness lemma preparations +Now we are ready for the strictness lemma. + +### Height preservation +The proof relies on complete induction on the height of a derivation which is a little troublesome in Twelf. !}% +%sort nat %. +%term zero nat %. +%term succ %pi nat %-> nat %. +%sort height_ok %. +%term height_ok_bot height_ok (frameterm_val _) N %. +%term height_ok_app %pi (height_ok (frameterm_app F) (succ N)) %<- (height_ok F N) %. +%term height_ok_tapp %pi (height_ok (frameterm_tapp F) (succ N)) %<- (height_ok F N) %. +%term height_ok_letu %pi (height_ok (frameterm_letu F) (succ N)) %<- (height_ok F N) %. +%term height_ok_lett %pi (height_ok (frameterm_lett F) (succ N)) %<- (height_ok F N) %. +%term height_ok_letb %pi (height_ok (frameterm_letb F) (succ N)) %<- (height_ok F N) %. +%term height_ok_lam %pi (height_ok (frameterm_lam F) (succ N)) %<- (height_ok F N) %. +%term height_ok_tlam %pi (height_ok (frameterm_tlam F) (succ N)) %<- (height_ok F N) %. +%term height_ok_unit %pi (height_ok (frameterm_unit F) (succ N)) %<- (height_ok F N) %. +%term height_ok_tens %pi (height_ok (frameterm_tens F) (succ N)) %<- (height_ok F N) %. +%term height_ok_thunk %pi (height_ok (frameterm_thunk F) (succ N)) %<- (height_ok F N) %. +%sort height_ok_exists {ft frameterm Fs M} %. +%mode height_ok_exists %in %out %. +%term height_ok_exists_bot height_ok_exists (frameterm_val _) (%the (height_ok _ zero) height_ok_bot) %. +%term height_ok_exists_app + %pi (height_ok_exists (frameterm_app Ft) (height_ok_app H)) + %<- (height_ok_exists Ft H) %. +%term height_ok_exists_tapp + %pi (height_ok_exists (frameterm_tapp Ft) (height_ok_tapp H)) + %<- (height_ok_exists Ft H) %. +%term height_ok_exists_letu + %pi (height_ok_exists (frameterm_letu Ft) (height_ok_letu H)) + %<- (height_ok_exists Ft H) %. +%term height_ok_exists_lett + %pi (height_ok_exists (frameterm_lett Ft) (height_ok_lett H)) + %<- (height_ok_exists Ft H) %. +%term height_ok_exists_letb + %pi (height_ok_exists (frameterm_letb Ft) (height_ok_letb H)) + %<- (height_ok_exists Ft H) %. +%term height_ok_exists_lam + %pi (height_ok_exists (frameterm_lam Ft) (height_ok_lam H)) + %<- (height_ok_exists Ft H) %. +%term height_ok_exists_tlam + %pi (height_ok_exists (frameterm_tlam Ft) (height_ok_tlam H)) + %<- (height_ok_exists Ft H) %. +%term height_ok_exists_unit + %pi (height_ok_exists (frameterm_unit Ft) (height_ok_unit H)) + %<- (height_ok_exists Ft H) %. +%term height_ok_exists_tens + %pi (height_ok_exists (frameterm_tens Ft) (height_ok_tens H)) + %<- (height_ok_exists Ft H) %. +%term height_ok_exists_thunk + %pi (height_ok_exists (frameterm_thunk Ft) (height_ok_thunk H)) + %<- (height_ok_exists Ft H) %. +%worlds () (height_ok_exists _ _) %. +%total D (height_ok_exists D _) %. +%sort succ_height_ok %. +%mode succ_height_ok %in %out %. +%term succ_height_ok_bot succ_height_ok height_ok_bot height_ok_bot %. +%term succ_height_ok_app + %pi (succ_height_ok (height_ok_app H) (height_ok_app H')) + %<- (succ_height_ok H H') %. +%term succ_height_ok_tapp + %pi (succ_height_ok (height_ok_tapp H) (height_ok_tapp H')) + %<- (succ_height_ok H H') %. +%term succ_height_ok_letu + %pi (succ_height_ok (height_ok_letu H) (height_ok_letu H')) + %<- (succ_height_ok H H') %. +%term succ_height_ok_lett + %pi (succ_height_ok (height_ok_lett H) (height_ok_lett H')) + %<- (succ_height_ok H H') %. +%term succ_height_ok_letb + %pi (succ_height_ok (height_ok_letb H) (height_ok_letb H')) + %<- (succ_height_ok H H') %. +%term succ_height_ok_lam + %pi (succ_height_ok (height_ok_lam H) (height_ok_lam H')) + %<- (succ_height_ok H H') %. +%term succ_height_ok_tlam + %pi (succ_height_ok (height_ok_tlam H) (height_ok_tlam H')) + %<- (succ_height_ok H H') %. +%term succ_height_ok_unit + %pi (succ_height_ok (height_ok_unit H) (height_ok_unit H')) + %<- (succ_height_ok H H') %. +%term succ_height_ok_tens + %pi (succ_height_ok (height_ok_tens H) (height_ok_tens H')) + %<- (succ_height_ok H H') %. +%term succ_height_ok_thunk + %pi (succ_height_ok (height_ok_thunk H) (height_ok_thunk H')) + %<- (succ_height_ok H H') %. +%worlds () (succ_height_ok _ _) %. +%total D (succ_height_ok D _) %. +%sort frameterm_eq_framesNum %. +%mode frameterm_eq_framesNum %in %in %in %in %out %out %. +%term frameterm_eq_framesNum_rule frameterm_eq_framesNum eqf_ref eq_ref Ft H Ft H %. +%worlds () (frameterm_eq_framesNum _ _ _ _ _ _) %. +%total {} (frameterm_eq_framesNum _ _ _ _ _ _) %. +%{! #### Height reducing inversions !}% +%sort up_app {ft frameterm Fs (app M1 M2)} %. +%mode up_app %in %in %out %out %. +%term up_app_rule up_app (frameterm_app F) (height_ok_app H) F H %. +%worlds () (up_app _ _ _ _) %. +%total D (up_app D _ _ _) %. +%sort up_tapp {ft frameterm Fs (tapp M T)} %. +%mode up_tapp %in %in %out %out %. +%term up_tapp_rule up_tapp (frameterm_tapp F) (height_ok_tapp H) F H %. +%worlds () (up_tapp _ _ _ _) %. +%total D (up_tapp D _ _ _) %. +%sort up_lett {ft frameterm Fs (lett M1 M2)} %. +%mode up_lett %in %in %out %out %. +%term up_lett_rule up_lett (frameterm_lett F) (height_ok_lett H) F H %. +%worlds () (up_lett _ _ _ _) %. +%total D (up_lett D _ _ _) %. +%sort up_letu {ft frameterm Fs (letu M1 M2)} %. +%mode up_letu %in %in %out %out %. +%term up_letu_rule up_letu (frameterm_letu F) (height_ok_letu H) F H %. +%worlds () (up_letu _ _ _ _) %. +%total D (up_letu D _ _ _) %. +%sort up_letb {ft frameterm Fs (letb M1 M2)} %. +%mode up_letb %in %in %out %out %. +%term up_letb_rule up_letb (frameterm_letb F) (height_ok_letb H) F H %. +%worlds () (up_letb _ _ _ _) %. +%total D (up_letb D _ _ _) %. +%sort up_lam {ft frameterm (cons (fapp M2) Fs) (lam T M1')} %. +%mode up_lam %in %in %out %out %. +%term up_lam_rule up_lam (frameterm_lam F) (height_ok_lam H) F H %. +%worlds () (up_lam _ _ _ _) %. +%total D (up_lam D _ _ _) %. +%sort up_tlam {ft frameterm (cons (ftapp T) Fs) (tlam M)} %. +%mode up_tlam %in %in %out %out %. +%term up_tlam_rule up_tlam (frameterm_tlam F) (height_ok_tlam H) F H %. +%worlds () (up_tlam _ _ _ _) %. +%total D (up_tlam D _ _ _) %. +%sort up_tens {ft frameterm (cons (flett M3) Fs) (tens M1 M2)} %. +%mode up_tens %in %in %out %out %. +%term up_tens_rule up_tens (frameterm_tens F) (height_ok_tens H) F H %. +%worlds () (up_tens _ _ _ _) %. +%total D (up_tens D _ _ _) %. +%sort up_unit {ft frameterm (cons (fletu M) Fs) unit} %. +%mode up_unit %in %in %out %out %. +%term up_unit_rule up_unit (frameterm_unit F) (height_ok_unit H) F H %. +%worlds () (up_unit _ _ _ _) %. +%total D (up_unit D _ _ _) %. +%sort up_thunk {ft frameterm (cons (fletb M2) Fs) (thunk T M1)} %. +%mode up_thunk %in %in %out %out %. +%term up_thunk_rule up_thunk (frameterm_thunk F) (height_ok_thunk H) F H %. +%worlds () (up_thunk _ _ _ _) %. +%total D (up_thunk D _ _ _) %. +%{! ### Special cases of type preservation !}% +%sort types_pres_app1_s %. +%mode types_pres_app1_s %in %in %out %. +%term types_pres_app1_s_rule %pi (%type s_pres_app1_s E (of_app D2 D1) (of_app D2 D1')) %<- (tpres_s D1 E D1') %. +%worlds () (types_pres_app1_s _ _ _) %. +%total {} (types_pres_app1_s _ _ _) %. +%sort types_pres_app2_s %. +%mode types_pres_app2_s %in %in %out %. +%term types_pres_app2_s_rule %pi (%type s_pres_app2_s E (of_app D2 D1) (of_app D2' D1)) %<- (tpres_s D2 E D2') %. +%worlds () (types_pres_app2_s _ _ _) %. +%total {} (types_pres_app2_s _ _ _) %. +%sort types_pres_tapp_s %. +%mode types_pres_tapp_s %in %in %out %. +%term types_pres_tapp_s_rule %pi (%type s_pres_tapp_s E (of_tapp T D1) (of_tapp T D1')) %<- (tpres_s D1 E D1') %. +%worlds () (types_pres_tapp_s _ _ _) %. +%total {} (types_pres_tapp_s _ _ _) %. +%sort types_pres_letu_s %. +%mode types_pres_letu_s %in %in %out %. +%term types_pres_letu_s_rule + %pi (%type s_pres_letu_s E (of_letu D2 D1) (of_letu D2 D1')) + %<- (tpres_s D1 E D1') %. +%worlds () (types_pres_letu_s _ _ _) %. +%total {} (types_pres_letu_s _ _ _) %. +%sort types_pres_lett_s %. +%mode types_pres_lett_s %in %in %out %. +%term types_pres_lett_s_rule + %pi (%type s_pres_lett_s E (of_lett L1 L2 D2 D1) (of_lett L1 L2 D2 D1')) + %<- (tpres_s D1 E D1') %. +%worlds () (types_pres_lett_s _ _ _) %. +%total {} (types_pres_lett_s _ _ _) %. +%sort types_pres_letb_s %. +%mode types_pres_letb_s %in %in %out %. +%term types_pres_letb_s_rule + %pi (%type s_pres_letb_s E (of_letb D2 D1) (of_letb D2 D1')) + %<- (tpres_s D1 E D1') %. +%worlds () (types_pres_letb_s _ _ _) %. +%freeze types_pres_letb_s %. +%total {} (types_pres_letb_s _ _ _) %. +%sort redex_rev_type %. +%mode redex_rev_type %in %in %out %. +%term redex_rev_type_app + %pi (redex_rev_type redex_app (of_app D2 D1) (D1' _ D2)) + %<- (inverse_lam D1 D1' _) %. +%term redex_rev_type_tapp redex_rev_type redex_tapp (of_tapp T (of_tlam D1)) (D1 T) %. +%term redex_rev_type_letu redex_rev_type redex_letu (of_letu D2 D1) D2 %. +%term redex_rev_type_lett redex_rev_type redex_lett (of_lett _ _ D3 (of_tens D2 D1)) (D3 _ D1 _ D2) %. +%term redex_rev_type_letb redex_rev_type redex_letb (of_letb D2 (of_thunk D1)) (D2 _ (of_letb D1 (of_thunk D1))) %. +%worlds () (redex_rev_type _ _ _) %. +%total {} (redex_rev_type _ _ _) %. +%{! ### The actual lemma !}% +%sort strictness_base {N} {M} %. +%mode strictness_base %in %in %in %in %in %out %. +%term strictness_base_rule strictness_base _ _ _ linear_id V V %. +%worlds () (strictness_base _ _ _ _ _ _) %. +%total {} (strictness_base _ _ _ _ _ _) %. +%sort strictness_lem1 {N} {Fs} {Num nat} {Ft frameterm (Fs M) (N M)} %. +%mode strictness_lem1 %in %in %in %in %in %in %in %in %in %out %out %out %. +%term strictness_lem1_base + %pi (strictness_lem1 N ([a] nil) Num (frameterm_val V) H FA L FsTp DNM (frameterm_val V) Evs H) + %<- (frameapply_nil_eq (FA M) Q) + %<- (eq_val Q V V') + %<- (applysound nil FsTp DNM (FA M) DNMB) + %<- (strictness_base _ M DNMB L V' VM) + %<- (selfevals VM Evs) %. +%term strictness_lem1_app2 + %pi (strictness_lem1 ([a] a) Fs _ (frameterm_app Ft) (height_ok_app H) FA L FsTp D3 Ft6 (evs_app EvM12 EvM2 EvM1') H7) + %<- (lem18 (app M1 M2) M1 FA L Q) + %<- (frameterm_eq_framesNum Q eq_ref (frameterm_app Ft) (height_ok_app H) Ft1 H1) + %<- (up_app Ft1 H1 Ft2 H2) + %<- (eqf_fstp Q FsTp FsTp1) + %<- (%type d_step FsTp1 D3 evfs_app FsTp' D3') + %<- (add_frame (fapp M2) FA L FA' L') + %<- (eq_typings D3 (of_app _ D1)) + %<- (strictness_lem1 ([a] a) ([a] cons (fapp M2) (Fs a)) _ Ft2 H2 FA' L' FsTp' D3' Ft2' (%the (M1 \s/ V1) EvM1) H2') + %<- (tpres_s D1 EvM1 D1') + %<- (value_soundness_s EvM1 Val1) + %<- (eq_val_funs Val1 D1' Q1) + %<- (eq_type Q1 D1' D1'') + %<- (eq_typings D1'' (of_lam (%the (linear M1') L1) D1''')) + %<- (succ_height_ok H2' H2'') + %<- (frameterm_eq_framesNum eqf_ref Q1 Ft2' H2'' Ft3 H3) + %<- (up_lam Ft3 H3 Ft3' H3') + %<- (lem18 V1 M2 FA L Q2) + %<- (frameterm_eq_framesNum Q2 eq_ref Ft3' H3' Ft4 H4) + %<- (apply_lin L1 linear_id L FA FA2 L2) + %<- (%type s_pres_app1_s EvM1 D3 D3'') + %<- (eq_ctx Q1 ([a] app a M2) Q1') + %<- (eq_type Q1' D3'' D4) + %<- (redex_rev_type redex_app D4 D4') + %<- (lem18 (app M1 M2) M2 FA L Q3) + %<- (eqf_fstp Q3 FsTp FsTp2) + %<- (strictness_lem1 M1' Fs _ Ft4 H4 FA2 L2 FsTp2 D4' Ft4' (%the (M2 \s/ V2) EvM2) H4') + %<- (lem18 V2 (M1' V2) FA L Q5) + %<- (frameterm_eq_framesNum Q5 eq_ref Ft4' H4' Ft5 H5) + %<- (%type s_pres_app2_s EvM2 D4 D5) + %<- (redex_rev_type redex_app D5 D5') + %<- (lem18 M2 (M1' V2) FA L Q6) + %<- (eqf_fstp Q6 FsTp2 FsTp3) + %<- (strictness_lem1 ([a] a) Fs _ Ft5 H5 FA L FsTp3 D5' Ft6 EvM12 H6) + %<- (eq_res_s Q1 EvM1 EvM1') + %<- (succ_height_ok H6 H7) %. +%term strictness_lem1_tapp2 + %pi (strictness_lem1 ([a] a) Fs _ Ft H FA L FsTp D2 Ft4' (evs_tapp EvM1'T EvM1') H5) + %<- (lem18 (tapp M1 T1) M1 FA L Q) + %<- (frameterm_eq_framesNum Q eq_ref Ft H Ft1 H1) + %<- (up_tapp Ft1 H1 Ft2 H2) + %<- (eqf_fstp Q FsTp FsTp1) + %<- (inverse_tapp D2 D1) + %<- (%type d_step FsTp1 D2 evfs_tapp FsTp' D2') + %<- (add_frame (ftapp T1) FA L FA' L') + %<- (strictness_lem1 ([a] a) ([a] cons (ftapp T1) (Fs a)) _ Ft2 H2 FA' L' FsTp' D2' Ft2' (%the (M1 \s/ V1) EvM1) H2') + %<- (tpres_s D1 EvM1 D1') + %<- (value_soundness_s EvM1 Val1) + %<- (eq_val_all Val1 D1' Q1) + %<- (eq_type Q1 D1' D1'') + %<- (eq_typings D1'' (of_tlam (%the ({t tp} M1' t ? _) D1'''))) + %<- (succ_height_ok H2' H2'') + %<- (frameterm_eq_framesNum eqf_ref Q1 Ft2' H2'' Ft2'' H2''') + %<- (up_tlam Ft2'' H2''' Ft3 H3) + %<- (%type s_pres_tapp_s EvM1 D2 D3) + %<- (eq_ctx Q1 ([a] tapp a T1) Q1') + %<- (eq_type Q1' D3 D4) + %<- (redex_rev_type redex_tapp D4 D4') + %<- (lem18 M1 (M1' T1) FA L Q2) + %<- (eqf_fstp Q2 FsTp1 FsTp1') + %<- (lem18 V1 (M1' T1) FA L Q3) + %<- (frameterm_eq_framesNum Q3 eq_ref Ft3 H3 Ft4 H4) + %<- (strictness_lem1 ([a] a) Fs _ Ft4 H4 FA L FsTp1' D4' Ft4' EvM1'T H4') + %<- (eq_res_s Q1 EvM1 EvM1') + %<- (succ_height_ok H4' H5) %. +%term strictness_lem1_lett2 + %pi (strictness_lem1 ([a] a) Fs _ Ft H FA L FsTp D3 Ft4' (evs_lett EvM2 EvM1') H5) + %<- (lem18 (lett M1 M2) M1 FA L Q) + %<- (frameterm_eq_framesNum Q eq_ref Ft H Ft1 H1) + %<- (up_lett Ft1 H1 Ft2 H2) + %<- (eqf_fstp Q FsTp FsTp1) + %<- (%type d_step FsTp1 D3 evfs_lett FsTp' D3') + %<- (add_frame (flett M2) FA L FA' L') + %<- (strictness_lem1 ([a] a) ([a] cons (flett M2) (Fs a)) _ Ft2 H2 FA' L' FsTp' D3' Ft2' (%the (M1 \s/ V1) EvM1) H2') + %<- (eq_typings D3 (of_lett _ _ _ D1)) + %<- (tpres_s D1 EvM1 D1') + %<- (value_soundness_s EvM1 Val1) + %<- (eq_val_tensor Val1 D1' Q1) + %<- (eq_type Q1 D1' D1'') + %<- (eq_typings D1'' (of_tens (%the (Mt2 ? _) Dt2) (%the (Mt1 ? _) Dt1))) + %<- (succ_height_ok H2' H2'') + %<- (frameterm_eq_framesNum eqf_ref Q1 Ft2' H2'' Ft3 H3) + %<- (up_tens Ft3 H3 Ft3' H3') + %<- (%type s_pres_lett_s EvM1 D3 D4) + %<- (eq_ctx Q1 ([a] lett a M2) Q1') + %<- (eq_type Q1' D4 D4'') + %<- (redex_rev_type redex_lett D4'' D4') + %<- (lem18 M1 (M2 Mt1 Mt2) FA L Q2) + %<- (eqf_fstp Q2 FsTp1 FsTp1') + %<- (lem18 V1 (M2 Mt1 Mt2) FA L Q3) + %<- (frameterm_eq_framesNum Q3 eq_ref Ft3' H3' Ft4 H4) + %<- (strictness_lem1 ([a] a) Fs _ Ft4 H4 FA L FsTp1' D4' Ft4' EvM2 H4') + %<- (eq_res_s Q1 EvM1 EvM1') + %<- (succ_height_ok H4' H5) %. +%term strictness_lem1_letu2 + %pi (strictness_lem1 ([a] a) Fs _ Ft H FA L FsTp D3 Ft4' (evs_letu EvM2 EvM1') H5) + %<- (lem18 (letu M1 M2) M1 FA L Q) + %<- (frameterm_eq_framesNum Q eq_ref Ft H Ft1 H1) + %<- (up_letu Ft1 H1 Ft2 H2) + %<- (eqf_fstp Q FsTp FsTp1) + %<- (%type d_step FsTp1 D3 evfs_letu FsTp' D3') + %<- (add_frame (fletu M2) FA L FA' L') + %<- (strictness_lem1 ([a] a) ([a] cons (fletu M2) (Fs a)) _ Ft2 H2 FA' L' FsTp' D3' Ft2' (%the (M1 \s/ V1) EvM1) H2') + %<- (eq_typings D3 (of_letu _ D1)) + %<- (tpres_s D1 EvM1 D1') + %<- (value_soundness_s EvM1 Val1) + %<- (eq_val_unit Val1 D1' Q1) + %<- (succ_height_ok H2' H2'') + %<- (frameterm_eq_framesNum eqf_ref Q1 Ft2' H2'' Ft3 H3) + %<- (up_unit Ft3 H3 Ft3' H3') + %<- (lem18 V1 M2 FA L Q2) + %<- (lem18 M1 M2 FA L Q2') + %<- (frameterm_eq_framesNum Q2 eq_ref Ft3' H3' Ft4 H4) + %<- (eqf_fstp Q2' FsTp1 FsTp1') + %<- (%type s_pres_letu_s EvM1 D3 D3'') + %<- (eq_ctx Q1 ([a] letu a M2) Q3) + %<- (eq_type Q3 D3'' D4) + %<- (redex_rev_type redex_letu D4 D4') + %<- (strictness_lem1 ([a] a) Fs _ Ft4 H4 FA L FsTp1' D4' Ft4' EvM2 H4') + %<- (succ_height_ok H4' H5) + %<- (eq_res_s Q1 EvM1 EvM1') %. +%term strictness_lem1_letb2 + %pi (strictness_lem1 ([a] a) Fs _ Ft H FA L FsTp D3 Ft4' (evs_letb EvM2 EvM1') H5) + %<- (lem18 (letb M1 M2) M1 FA L Q) + %<- (frameterm_eq_framesNum Q eq_ref Ft H Ft1 H1) + %<- (up_letb Ft1 H1 Ft2 H2) + %<- (eqf_fstp Q FsTp FsTp1) + %<- (%type d_step FsTp1 D3 evfs_letb FsTp' D3') + %<- (add_frame (fletb M2) FA L FA' L') + %<- (strictness_lem1 ([a] a) ([a] cons (fletb M2) (Fs a)) _ Ft2 H2 FA' L' FsTp' D3' Ft2' (%the (M1 \s/ V1) EvM1) H2') + %<- (eq_typings D3 (of_letb _ D1)) + %<- (tpres_s D1 EvM1 D1') + %<- (value_soundness_s EvM1 Val1) + %<- (eq_val_bang Val1 D1' Q1) + %<- (eq_type Q1 D1' D1'') + %<- (eq_thunk_type D1'' Q') + %<- (eqt_symm Q' Q'') + %<- (eqt_ctx Q'' ([t] bang t) Q''') + %<- (eq_type1 Q''' D1'' D1''') + %<- (eq_typings D1''' (%the (thunk Ty _ ? _) (of_thunk (%the ({m term} {d m ? _} Mt1 m ? _) Dt1)))) + %<- (succ_height_ok H2' H2'') + %<- (frameterm_eq_framesNum eqf_ref Q1 Ft2' H2'' Ft3 H3) + %<- (up_thunk Ft3 H3 Ft3' H3') + %<- (%type s_pres_letb_s EvM1 D3 D4) + %<- (eq_ctx Q1 ([a] letb a M2) Q1') + %<- (eq_type Q1' D4 D4'') + %<- (redex_rev_type redex_letb D4'' D4') + %<- (lem18 M1 (M2 (letb (thunk Ty Mt1) Mt1)) FA L Q2) + %<- (eqf_fstp Q2 FsTp1 FsTp1') + %<- (lem18 V1 (M2 (letb (thunk Ty Mt1) Mt1)) FA L Q3) + %<- (frameterm_eq_framesNum Q3 eq_ref Ft3' H3' Ft4 H4) + %<- (strictness_lem1 ([a] a) Fs _ Ft4 H4 FA L FsTp1' D4' Ft4' EvM2 H4') + %<- (succ_height_ok H4' H5) + %<- (eq_res_s Q1 EvM1 EvM1') %. +%term strictness_lem1_app1 + %pi (strictness_lem1 ([a] app (N1 a) (N2 a)) Fs _ (frameterm_app Ft) (height_ok_app H) FA L FsTp (of_app DN2 DN1) (frameterm_app Ft') Evs (height_ok_app H')) + %<- (strictness_lem1 N1 ([a] cons (fapp (N2 a)) (Fs a)) _ Ft H ([a] frameapply_cons (FA a) (%the (frameapp _ _ _) frameapp_app)) L (ftp_cons FsTp ([a] [da] of_app DN2 da)) DN1 Ft' Evs H') %. +%term strictness_lem1_tapp1 + %pi (strictness_lem1 ([a] tapp (N1 a) T) Fs _ (frameterm_tapp Ft) (height_ok_tapp H) FA L FsTp (of_tapp _ DN1) (frameterm_tapp Ft') Evs (height_ok_tapp H')) + %<- (strictness_lem1 N1 ([a] cons (ftapp T) (Fs a)) _ Ft H ([a] frameapply_cons (FA a) (%the (frameapp _ _ _) frameapp_app)) L (ftp_cons FsTp ([a] [da] of_tapp _ da)) DN1 Ft' Evs H') %. +%term strictness_lem1_letu1 + %pi (strictness_lem1 ([a] letu (N1 a) (N2 a)) Fs _ (frameterm_letu Ft) (height_ok_letu H) FA L FsTp (of_letu DN2 DN1) (frameterm_letu Ft') Evs (height_ok_letu H')) + %<- (strictness_lem1 N1 ([a] cons (fletu (N2 a)) (Fs a)) _ Ft H ([a] frameapply_cons (FA a) (%the (frameapp _ _ _) frameapp_app)) L (ftp_cons FsTp ([a] [da] of_letu DN2 da)) DN1 Ft' Evs H') %. +%term strictness_lem1_letb1 + %pi (strictness_lem1 ([a] letb (N1 a) (N2 a)) Fs _ (frameterm_letb Ft) (height_ok_letb H) FA L FsTp (of_letb DN2 DN1) (frameterm_letb Ft') Evs (height_ok_letb H')) + %<- (strictness_lem1 N1 ([a] cons (fletb (N2 a)) (Fs a)) _ Ft H ([a] frameapply_cons (FA a) (%the (frameapp _ _ _) frameapp_app)) L (ftp_cons FsTp ([a] [da] of_letb DN2 da)) DN1 Ft' Evs H') %. +%term strictness_lem1_lett1 + %pi (strictness_lem1 ([a] lett (N1 a) (N2 a)) Fs _ (frameterm_lett Ft) (height_ok_lett H) FA L FsTp (of_lett L1 L2 DN2 DN1) (frameterm_lett Ft') Evs (height_ok_lett H')) + %<- (strictness_lem1 N1 ([a] cons (flett (N2 a)) (Fs a)) _ Ft H ([a] frameapply_cons (FA a) (%the (frameapp _ _ _) frameapp_app)) L (ftp_cons FsTp ([a] [da] of_lett L1 L2 DN2 da)) DN1 Ft' Evs H') %. +%term strictness_lem1_lam + %pi (strictness_lem1 ([a] lam T (M4 a)) ([a] cons (fapp (M5 a)) (Fs a)) _ (frameterm_lam Ft) (height_ok_lam H) ([a] frameapply_cons (FA a) (%the (frameapp (fapp (M5 a)) _ _) frameapp_app)) L (ftp_cons FsTp FTp) D2 (frameterm_lam Ft') Evs (height_ok_lam H')) + %<- (eq_typings ((%the ({m} %pi (m ? func _ _) %-> (app m _ ? T2)) FTp) _ D2) D1) + %<- (redex_rev_type redex_app D1 D') + %<- (frame_linear ([m] redex_app) D1 FA L FA1 L1) + %<- (strictness_lem1 ([a] M4 a (M5 a)) Fs _ Ft H FA1 L1 FsTp D' Ft' Evs H') %. +%term strictness_lem1_tens + %pi (strictness_lem1 ([a] tens (M3 a) (M4 a)) ([a] cons (flett (M5 a)) (Fs a)) _ (frameterm_tens Ft) (height_ok_tens H) ([a] frameapply_cons (FA a) (%the (frameapp (flett (M5 a)) _ _) frameapp_app)) L (ftp_cons FsTp FTp) D2 (frameterm_tens Ft') Evs (height_ok_tens H')) + %<- (eq_typings ((%the ({m} %pi (m ? tensor _ _) %-> (lett m _ ? T2)) FTp) _ D2) Dt) + %<- (redex_rev_type redex_lett Dt D') + %<- (frame_linear ([m] redex_lett) Dt FA L FA1 L') + %<- (strictness_lem1 ([a] M5 a (M3 a) (M4 a)) Fs _ Ft H FA1 L' FsTp D' Ft' Evs H') %. +%term strictness_lem1_unit + %pi (strictness_lem1 ([a] unit) ([a] cons (fletu (M5 a)) (Fs a)) _ (frameterm_unit Ft) (height_ok_unit H) ([a] frameapply_cons (FA a) (%the (frameapp (fletu (M5 a)) _ _) frameapp_app)) L (ftp_cons FsTp FTp) D2 (frameterm_unit Ft') Evs (height_ok_unit H')) + %<- (eq_typings ((%the ({m} %pi (m ? i) %-> (letu m _ ? T2)) FTp) _ D2) Dt) + %<- (redex_rev_type redex_letu Dt D') + %<- (frame_linear ([m] redex_letu) Dt FA L FA1 L') + %<- (strictness_lem1 ([a] M5 a) Fs _ Ft H FA1 L' FsTp D' Ft' Evs H') %. +%term strictness_lem1_thunk + %pi (strictness_lem1 ([a] thunk T (M4 a)) ([a] cons (fletb (M5 a)) (Fs a)) _ (frameterm_thunk Ft) (height_ok_thunk H) ([a] frameapply_cons (FA a) (%the (frameapp (fletb (M5 a)) _ _) frameapp_app)) L (ftp_cons FsTp FTp) D2 (frameterm_thunk Ft') Evs (height_ok_thunk H')) + %<- (eq_typings ((%the ({m} %pi (m ? bang _) %-> (letb m _ ? T2)) FTp) _ D2) Dt) + %<- (redex_rev_type redex_letb Dt D') + %<- (frame_linear ([m] redex_letb) Dt FA L FA1 L') + %<- (strictness_lem1 ([a] M5 a (letb (thunk T (M4 a)) (M4 a))) Fs _ Ft H FA1 L' FsTp D' Ft' Evs H') %. +%term strictness_lem1_tlam + %pi (strictness_lem1 ([a] tlam (M4 a)) ([a] cons (ftapp T) (Fs a)) _ (frameterm_tlam Ft) (height_ok_tlam H) ([a] frameapply_cons (FA a) (%the (frameapp (ftapp T) _ _) frameapp_app)) L (ftp_cons FsTp FTp) D2 (frameterm_tlam Ft') Evs (height_ok_tlam H')) + %<- (eq_typings ((%the ({m} %pi (m ? all _) %-> (tapp m _ ? T2)) FTp) _ D2) D1) + %<- (redex_rev_type redex_tapp D1 D') + %<- (frame_linear ([m] redex_tapp) D1 FA L FA1 L1) + %<- (strictness_lem1 ([a] M4 a T) Fs _ Ft H FA1 L1 FsTp D' Ft' Evs H') %. +%term strictness_lem1_unit2 + %pi (strictness_lem1 ([a] a) Fs _ Ft H FA L FsTp D2 Ft Evs H) + %<- (selfevals val_unit Evs) %. +%term strictness_lem1_lam2 + %pi (strictness_lem1 ([a] a) Fs _ Ft H FA L FsTp D2 Ft Evs H) + %<- (selfevals val_lam Evs) %. +%term strictness_lem1_tlam2 + %pi (strictness_lem1 ([a] a) Fs _ Ft H FA L FsTp D2 Ft Evs H) + %<- (selfevals val_tlam Evs) %. +%term strictness_lem1_tens2 + %pi (strictness_lem1 ([a] a) Fs _ Ft H FA L FsTp D2 Ft Evs H) + %<- (selfevals val_tens Evs) %. +%term strictness_lem1_thunk2 + %pi (strictness_lem1 ([a] a) Fs _ Ft H FA L FsTp D2 Ft Evs H) + %<- (selfevals val_thunk Evs) %. +%worlds () (strictness_lem1 _ _ _ _ _ _ _ _ _ _ _ _) %. +% covers strictness_lem1 +N +Fs +Num +FtM +H +FA +L +FD +FsT -FtV -EVs -H'. +% terminates D (strictness_lem1 _ _ D _ _ _ _ _ _ _ _ _). +%total D (strictness_lem1 _ _ D _ _ _ _ _ _ _ _ _) %. +%sort strictness1 %. +%mode strictness1 %in %in %in %out %out %. +%term strictness1_rule + %pi (strictness1 D1 L (terminate Ev) Evs (terminate Ev')) + %<- (eval_frameterm Ev Ft) + %<- (height_ok_exists Ft (%the (height_ok _ Num) H)) + %<- (strictness_lem1 N ([a term] nil) Num Ft H ([a] frameapply_nil) L ftp_nil D1 Ft' Evs H') + %<- (frameterm_eval Ft' Ev') %. +%worlds () (strictness1 _ _ _ _ _) %. +% covers strictness1 +D +L +Ev -S -Term. +%total {} (strictness1 _ _ _ _ _) %. +%sort frameterm_eq_frames %. +%mode frameterm_eq_frames %in %in %in %out %. +%term frameterm_eq_frames_rule frameterm_eq_frames eqf_ref eq_ref Ft Ft %. +%worlds () (frameterm_eq_frames _ _ _ _) %. +%total {} (frameterm_eq_frames _ _ _ _) %. +%sort strictness2_lem1 {N} {Fs} %. +%mode strictness2_lem1 %in %in %in %in %out %. +%term strictness2_lem1_tens_val strictness2_lem1 ([a] tens _ _) ([a] nil) (frameterm_val _) _ (frameterm_val val_tens) %. +%term strictness2_lem1_unit_val strictness2_lem1 ([a] unit) ([a] nil) (frameterm_val _) _ (frameterm_val val_unit) %. +%term strictness2_lem1_lam_val strictness2_lem1 ([a] lam T _) ([a] nil) (frameterm_val _) _ (frameterm_val val_lam) %. +%term strictness2_lem1_tlam_val strictness2_lem1 ([a] tlam _) ([a] nil) (frameterm_val _) _ (frameterm_val val_tlam) %. +%term strictness2_lem1_thunk_val strictness2_lem1 ([a] thunk T _) ([a] nil) (frameterm_val _) _ (frameterm_val val_thunk) %. +%term strictness2_lem1_unit2 strictness2_lem1 ([a] a) Fs Ft evs_unit Ft %. +%term strictness2_lem1_lam2 strictness2_lem1 ([a] a) Fs Ft evs_lam Ft %. +%term strictness2_lem1_tlam2 strictness2_lem1 ([a] a) Fs Ft evs_tlam Ft %. +%term strictness2_lem1_tens2 strictness2_lem1 ([a] a) Fs Ft evs_tens Ft %. +%term strictness2_lem1_thunk2 strictness2_lem1 ([a] a) Fs Ft evs_thunk Ft %. +%term strictness2_lem1_app1 + %pi (strictness2_lem1 ([a] app (N1 a) (N2 a)) Fs (frameterm_app Ft) Evs (frameterm_app Ft')) + %<- (strictness2_lem1 N1 ([a] cons (fapp (N2 a)) (Fs a)) Ft Evs Ft') %. +%term strictness2_lem1_tapp1 + %pi (strictness2_lem1 ([a] tapp (N1 a) T) Fs (frameterm_tapp Ft) Evs (frameterm_tapp Ft')) + %<- (strictness2_lem1 N1 ([a] cons (ftapp T) (Fs a)) Ft Evs Ft') %. +%term strictness2_lem1_letu1 + %pi (strictness2_lem1 ([a] letu (N1 a) (N2 a)) Fs (frameterm_letu Ft) Evs (frameterm_letu Ft')) + %<- (strictness2_lem1 N1 ([a] cons (fletu (N2 a)) (Fs a)) Ft Evs Ft') %. +%term strictness2_lem1_letb1 + %pi (strictness2_lem1 ([a] letb (N1 a) (N2 a)) Fs (frameterm_letb Ft) Evs (frameterm_letb Ft')) + %<- (strictness2_lem1 N1 ([a] cons (fletb (N2 a)) (Fs a)) Ft Evs Ft') %. +%term strictness2_lem1_lett1 + %pi (strictness2_lem1 ([a] lett (N1 a) (N2 a)) Fs (frameterm_lett Ft) Evs (frameterm_lett Ft')) + %<- (strictness2_lem1 N1 ([a] cons (flett (N2 a)) (Fs a)) Ft Evs Ft') %. +%term strictness2_lem1_lam + %pi (strictness2_lem1 ([a] lam T (M4 a)) ([a] cons (fapp (M5 a)) (Fs a)) (frameterm_lam Ft) Evs (frameterm_lam Ft')) + %<- (strictness2_lem1 ([a] M4 a (M5 a)) Fs Ft Evs Ft') %. +%term strictness2_lem1_tens + %pi (strictness2_lem1 ([a] tens (M3 a) (M4 a)) ([a] cons (flett (M5 a)) (Fs a)) (frameterm_tens Ft) Evs (frameterm_tens Ft')) + %<- (strictness2_lem1 ([a] M5 a (M3 a) (M4 a)) Fs Ft Evs Ft') %. +%term strictness2_lem1_unit + %pi (strictness2_lem1 ([a] unit) ([a] cons (fletu (M5 a)) (Fs a)) (frameterm_unit Ft) Evs (frameterm_unit Ft')) + %<- (strictness2_lem1 ([a] M5 a) Fs Ft Evs Ft') %. +%term strictness2_lem1_thunk + %pi (strictness2_lem1 ([a] thunk T (M4 a)) ([a] cons (fletb (M5 a)) (Fs a)) (frameterm_thunk Ft) Evs (frameterm_thunk Ft')) + %<- (strictness2_lem1 ([a] M5 a (letb (thunk T (M4 a)) (M4 a))) Fs Ft Evs Ft') %. +%term strictness2_lem1_tlam + %pi (strictness2_lem1 ([a] tlam (M4 a)) ([a] cons (ftapp T) (Fs a)) (frameterm_tlam Ft) Evs (frameterm_tlam Ft')) + %<- (strictness2_lem1 ([a] M4 a T) Fs Ft Evs Ft') %. +%term strictness2_lem1_app2 + %pi (strictness2_lem1 ([a] a) Fs Ft1 (evs_app EvM12 EvM2 EvM1) (frameterm_app Ft7)) + %<- (strictness2_lem1 ([a] a) Fs Ft1 EvM12 Ft2) + %<- (strictness2_lem1 ([a] M1' a) ([a] Fs (M1' a)) Ft2 EvM2 Ft4) + %<- (lem42 ([a] Fs a) ([a] a) (%the (redex (app (lam T M1') M2) _) redex_app) Ft4 Ft5) + %<- (inverse_ft_app Ft5 Ft6) + %<- (strictness2_lem1 ([a] a) ([a] cons (fapp M2) (Fs (app a M2))) Ft6 EvM1 Ft7) %. +%term strictness2_lem1_tapp2 + %pi (strictness2_lem1 ([a] a) Fs Ft1 (evs_tapp EvM12 EvM1) (frameterm_tapp Ft5)) + %<- (strictness2_lem1 ([a] a) Fs Ft1 EvM12 Ft2) + %<- (lem42 ([a] Fs a) ([a] a) (%the (redex (tapp (tlam M1') T) _) redex_tapp) Ft2 Ft3) + %<- (inverse_ft_tapp Ft3 Ft4) + %<- (strictness2_lem1 ([a] a) ([a] cons (ftapp T) (Fs (tapp a T))) Ft4 EvM1 Ft5) %. +%term strictness2_lem1_letu2 + %pi (strictness2_lem1 ([a] a) Fs Ft1 (evs_letu EvM2 EvM1) (frameterm_letu Ft5)) + %<- (strictness2_lem1 ([a] a) Fs Ft1 EvM2 Ft2) + %<- (lem42 ([a] Fs a) ([a] a) (%the (redex (letu unit M2) _) redex_letu) Ft2 Ft3) + %<- (inverse_ft_letu Ft3 Ft4) + %<- (strictness2_lem1 ([a] a) ([a] cons (fletu M2) (Fs (letu a M2))) Ft4 EvM1 Ft5) %. +%term strictness2_lem1_lett2 + %pi (strictness2_lem1 ([a] a) Fs Ft1 (evs_lett EvM2 EvM1) (frameterm_lett Ft5)) + %<- (strictness2_lem1 ([a] a) Fs Ft1 EvM2 Ft2) + %<- (lem42 ([a] Fs a) ([a] a) (%the (redex (lett (tens M1 M2) M3) _) redex_lett) Ft2 Ft3) + %<- (inverse_ft_lett Ft3 Ft4) + %<- (strictness2_lem1 ([a] a) ([a] cons (flett M3) (Fs (lett a M3))) Ft4 EvM1 Ft5) %. +%term strictness2_lem1_letb2 + %pi (strictness2_lem1 ([a] a) Fs Ft1 (evs_letb EvM2 EvM1) (frameterm_letb Ft5)) + %<- (strictness2_lem1 ([a] a) Fs Ft1 EvM2 Ft2) + %<- (lem42 ([a] Fs a) ([a] a) (%the (redex (letb (thunk T M1) M2) _) redex_letb) Ft2 Ft3) + %<- (inverse_ft_letb Ft3 Ft4) + %<- (strictness2_lem1 ([a] a) ([a] cons (fletb M2) (Fs (letb a M2))) Ft4 EvM1 Ft5) %. +%worlds () (strictness2_lem1 _ _ _ _ _) %. +% covers strictness2_lem1 +N +Fs +FtV +EVs -FtM. +% terminates {E D} (strictness2_lem1 _ _ D E _). +%total {E D} (strictness2_lem1 _ _ D E _) %. +%sort strictness21 {N} %. +%mode strictness21 %in %in %in %out %. +%term strictness21_rule + %pi (strictness21 N (terminate Ev) Evs (terminate Ev')) + %<- (eval_frameterm Ev Ft) + %<- (strictness2_lem1 N ([a term] nil) Ft Evs Ft') + %<- (frameterm_eval Ft' Ev') %. +%worlds () (strictness21 _ _ _ _) %. +% covers strictness21 +N +Ev -S -Term. +%total {} (strictness21 _ _ _ _) %. +%{! ``>See all code for this case study``. `` check="true">See Twelf's output``. + +[Category:Twelf code](/wiki/category-twelf-code/) +\{\{case study\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Lily). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/linear-logic.lf b/new-tests/stelf-output/pages/linear-logic.lf new file mode 100644 index 0000000..6cdc7fb --- /dev/null +++ b/new-tests/stelf-output/pages/linear-logic.lf @@ -0,0 +1,759 @@ +% +%{! The fact that the LF type theory uses only unrestricted assumptions has led some to assume that it cannot encode substructural logics such as linear logic. In fact, LF admits a very elegant encoding of linear logic, given below. The key idea is a judgement used to enforce the linear use of each linear assumption. + +### The Encoding + +#### Syntax !}% +%{! (options removed from twelftag: hidden="true") !}% +%%%%% Syntax %%%%% +%{! !}% +%%% Types %%% +%sort tp %. +%name tp %. +%scope tp %term o %pi tp %-> tp %-> tp %. +%prec %right 7 -o %. +%term * %pi tp %-> tp %-> tp %. +%prec %right 10 * %. +%term & %pi tp %-> tp %-> tp %. +%prec %right 9 & %. +%term + %pi tp %-> tp %-> tp %. +%prec %right 8 + %. +%term ! %pi tp %-> tp %. +%term zero tp %. +%term top tp %. +%% 1 need not be primitive. +%define one tp ! top %. +%%% Terms %%% +%sort term %. +%name term %. +%term lam %pi (%pi term %-> term) %-> term %. +%term app %pi term %-> term %-> term %. +%term tensor %pi term %-> term %-> term %. +%term lett %pi term %-> (%pi term %-> term %-> term) %-> term %. +%term pair %pi term %-> term %-> term %. +%term pi1 %pi term %-> term %. +%term pi2 %pi term %-> term %. +%term in1 %pi term %-> term %. +%term in2 %pi term %-> term %. +%term case %pi term %-> (%pi term %-> term) %-> (%pi term %-> term) %-> term %. +%term bang %pi term %-> term %. +%term letb %pi term %-> (%pi term %-> term) %-> term %. +%term any %pi term %-> term %. +%term unit term %. +%% derived syntax for 1 +%define star term bang unit %. +%define leto (%pi term %-> term %-> term) [t1] [t2] letb t1 ([x] t2) %. +%{! #### Linearity + +Linearity is enforced by employing a linearity judgement for each linear assumption. The linearity judgement ensures that a variable is used linearly (roughly speaking, exactly once) within its scope. (For an example of its use, see the rule ``of/lam`` for linear lambda abstraction below, which ensures that its argument is used linearly within its body.) !}% +%{! (options removed from twelftag: hidden="true") !}% +%%%%% Linearity %%%%% +%{! !}% +%sort linear %. +%{! For example, a variable is used linearly in a single occurrence of that variable: !}% +%term linear/var linear ([x] x) %. +%{! A variable is used linearly in a lambda abstraction if it is used linearly in the body. (The abstraction's argument must also be used linearly in the body, but that is handled separately, by the abstraction's typing rule.) !}% +%term linear/lam %pi (linear ([x] lam (M x))) %<- ({y} linear ([x] M x y)) %. +%{! A variable is used linearly in application, if it is used linearly in one of the two subterms and does not appear at all in the other: !}% +%term linear/app1 %pi (linear ([x] app (M1 x) M2)) %<- (linear M1) %. +%term linear/app2 %pi (linear ([x] app M1 (M2 x))) %<- (linear M2) %. +%{! The remaining constructs use the same technique. Note that there is no rule for ``bang``; no linear variable may appear within a ``bang``. Also note that a linear variable need not appear within an ``any`` (as with ``unit``); however, if it does appear, it must be used linearly. !}% +%term linear/tensor1 %pi (linear ([x] tensor (M1 x) M2)) %<- (linear M1) %. +%term linear/tensor2 %pi (linear ([x] tensor M1 (M2 x))) %<- (linear M2) %. +%term linear/lett1 %pi (linear ([x] lett (M1 x) M2)) %<- (linear M1) %. +%term linear/lett2 %pi (linear ([x] lett M1 (M2 x))) %<- ({y} {z} linear ([x] M2 x y z)) %. +%term linear/pair %pi (linear ([x] pair (M1 x) (M2 x))) %<- (linear M1) %<- (linear M2) %. +%term linear/pi1 %pi (linear ([x] pi1 (M x))) %<- (linear M) %. +%term linear/pi2 %pi (linear ([x] pi2 (M x))) %<- (linear M) %. +%term linear/in1 %pi (linear ([x] in1 (M x))) %<- (linear M) %. +%term linear/in2 %pi (linear ([x] in2 (M x))) %<- (linear M) %. +%term linear/case1 %pi (linear ([x] case (M x) M1 M2)) %<- (linear M) %. +%term linear/case2 + %pi (linear ([x] case M (M1 x) (M2 x))) + %<- ({y} linear ([x] M1 x y)) + %<- ({y} linear ([x] M2 x y)) %. +%term linear/letb1 %pi (linear ([x] letb (M1 x) M2)) %<- (linear M1) %. +%term linear/letb2 %pi (linear ([x] letb M1 (M2 x))) %<- ({y} linear ([x] M2 x y)) %. +%term linear/any1 %pi (linear ([x] any (M x))) %<- (linear M) %. +%term linear/any2 linear ([x] any M) %. +%term linear/unit linear ([x] unit) %. +%% derived rules for 1 +%define linear/leto1 (%pi (linear ([x] leto (M1 x) M2)) %<- (linear M1)) [d] linear/letb1 d %. +%define linear/leto2 (%pi (linear ([x] leto M1 (M2 x))) %<- (linear M2)) [d] linear/letb2 ([y] d) %. +%{! #### Typing rules + +The typing rules are standard for linear logic, except that they must check that the linearity invariants are maintained. Thus, the typing rule for any construct that binds a linear variable (``lam``, for example) has a premise (or premises) ensuring that such variable(s) are used linearly in their scope. However, ``letb``, which binds an unrestricted variable, does not check for linear usage of that variable. !}% +%{! (options removed from twelftag: hidden="true") !}% +%%%%% Typing Rules %%%%% +%{! !}% +%sort of %. +%term of/lam + %pi (of (lam M) (T1 -o T2)) + %<- ({x} %pi (of x T1) %-> (of (M x) T2)) + %<- (linear M) %. +%term of/app %pi (of (app M1 M2) T2) %<- (of M1 (T1 -o T2)) %<- (of M2 T1) %. +%term of/tensor %pi (of (tensor M1 M2) (T1 * T2)) %<- (of M1 T1) %<- (of M2 T2) %. +%term of/lett + %pi (of (lett M1 M2) T) + %<- (of M1 (T1 * T2)) + %<- ({x} %pi (of x T1) %-> ({y} %pi (of y T2) %-> (of (M2 x y) T))) + %<- ({y} linear ([x] M2 x y)) + %<- ({x} linear ([y] M2 x y)) %. +%term of/pair %pi (of (pair M1 M2) (T1 & T2)) %<- (of M1 T1) %<- (of M2 T2) %. +%term of/pi1 %pi (of (pi1 M) T1) %<- (of M (T1 & T2)) %. +%term of/pi2 %pi (of (pi2 M) T2) %<- (of M (T1 & T2)) %. +%term of/in1 %pi (of (in1 M) (T1 + T2)) %<- (of M T1) %. +%term of/in2 %pi (of (in2 M) (T1 + T2)) %<- (of M T2) %. +%term of/case + %pi (of (case M M1 M2) T) + %<- (of M (T1 + T2)) + %<- ({x} %pi (of x T1) %-> (of (M1 x) T)) + %<- ({x} %pi (of x T2) %-> (of (M2 x) T)) + %<- (linear M1) + %<- (linear M2) %. +%term of/bang %pi (of (bang M) (! T)) %<- (of M T) %. +%term of/letb + %pi (of (letb M1 M2) T2) + %<- (of M1 (! T1)) + %<- ({x} %pi (of x T1) %-> (of (M2 x) T2)) %. +%term of/any %pi (of (any M) T) %<- (of M zero) %. +%term of/unit of unit top %. +%% derived rules for 1 +%define of/star (of star one) of/bang of/unit %. +%define of/leto (%pi (of (leto M1 M2) T) %<- (of M1 one) %<- (of M2 T)) [d2] [d1] of/letb ([x] [d] d2) d1 %. +%{! ### Subject Reduction + +As a metatheoretic example, we may prove the subject reduction property using this encoding. First we define the relevant worlds: !}% +%{! (options removed from twelftag: hidden="true") !}% +%%%%% Worlds %%%%% +%{! !}% +%block block [x term]%. +%block bind {t tp} [x term] [d of x t]%. +%{! Next we define reduction: !}% +%{! (options removed from twelftag: hidden="true") !}% +%%%%% Reduction %%%%% +%{! !}% +%sort reduce %. +%term red/refl reduce M M %. +%term red/trans %pi (reduce M1 M3) %<- (reduce M1 M2) %<- (reduce M2 M3) %. +%% beta rules +%term red/beta reduce (app (lam M1) M2) (M1 M2) %. +%term red/beta* reduce (lett (tensor M1 M2) M) (M M1 M2) %. +%term red/beta&1 reduce (pi1 (pair M1 M2)) M1 %. +%term red/beta&2 reduce (pi2 (pair M1 M2)) M2 %. +%term red/beta+1 reduce (case (in1 M) M1 M2) (M1 M) %. +%term red/beta+2 reduce (case (in2 M) M1 M2) (M2 M) %. +%term red/beta! reduce (letb (bang M1) M2) (M2 M1) %. +%% compatibility +%term red/lam %pi (reduce (lam M) (lam M')) %<- ({x} reduce (M x) (M' x)) %. +%term red/app %pi (reduce (app M1 M2) (app M1' M2')) %<- (reduce M1 M1') %<- (reduce M2 M2') %. +%term red/tensor + %pi (reduce (tensor M1 M2) (tensor M1' M2')) + %<- (reduce M1 M1') + %<- (reduce M2 M2') %. +%term red/lett + %pi (reduce (lett M1 M2) (lett M1' M2')) + %<- (reduce M1 M1') + %<- ({x} {y} reduce (M2 x y) (M2' x y)) %. +%term red/pair %pi (reduce (pair M1 M2) (pair M1' M2')) %<- (reduce M1 M1') %<- (reduce M2 M2') %. +%term red/pi1 %pi (reduce (pi1 M) (pi1 M')) %<- (reduce M M') %. +%term red/pi2 %pi (reduce (pi2 M) (pi2 M')) %<- (reduce M M') %. +%term red/in1 %pi (reduce (in1 M) (in1 M')) %<- (reduce M M') %. +%term red/in2 %pi (reduce (in2 M) (in2 M')) %<- (reduce M M') %. +%term red/case + %pi (reduce (case M M1 M2) (case M' M1' M2')) + %<- (reduce M M') + %<- ({x} reduce (M1 x) (M1' x)) + %<- ({x} reduce (M2 x) (M2' x)) %. +%term red/bang %pi (reduce (bang M) (bang M')) %<- (reduce M M') %. +%term red/letb + %pi (reduce (letb M1 M2) (letb M1' M2')) + %<- (reduce M1 M1') + %<- ({x} reduce (M2 x) (M2' x)) %. +%term red/any %pi (reduce (any M) (any M')) %<- (reduce M M') %. +%% commuting conversions +%term red/app/lett reduce (app (lett M1 M2) M3) (lett M1 ([x] [y] app (M2 x y) M3)) %. +%term red/app/case reduce (app (case M1 M2a M2b) M3) (case M1 ([x] app (M2a x) M3) ([x] app (M2b x) M3)) %. +%term red/app/letb reduce (app (letb M1 M2) M3) (letb M1 ([x] app (M2 x) M3)) %. +%term red/app/any reduce (app (any M1) M2) (any M1) %. +%term red/pi1/lett reduce (pi1 (lett M1 M2)) (lett M1 ([x] [y] pi1 (M2 x y))) %. +%term red/pi1/case reduce (pi1 (case M1 M2a M2b)) (case M1 ([x] pi1 (M2a x)) ([x] pi1 (M2b x))) %. +%term red/pi1/letb reduce (pi1 (letb M1 M2)) (letb M1 ([x] pi1 (M2 x))) %. +%term red/pi1/any reduce (pi1 (any M)) (any M) %. +%term red/pi2/lett reduce (pi2 (lett M1 M2)) (lett M1 ([x] [y] pi2 (M2 x y))) %. +%term red/pi2/case reduce (pi2 (case M1 M2a M2b)) (case M1 ([x] pi2 (M2a x)) ([x] pi2 (M2b x))) %. +%term red/pi2/letb reduce (pi2 (letb M1 M2)) (letb M1 ([x] pi2 (M2 x))) %. +%term red/pi2/any reduce (pi2 (any M)) (any M) %. +%term red/lett/lett reduce (lett (lett M1 M2) M3) (lett M1 ([x] [y] lett (M2 x y) M3)) %. +%term red/lett/case reduce (lett (case M1 M2a M2b) M3) (case M1 ([x] lett (M2a x) M3) ([x] lett (M2b x) M3)) %. +%term red/lett/letb reduce (lett (letb M1 M2) M3) (letb M1 ([x] lett (M2 x) M3)) %. +%term red/lett/any reduce (lett (any M1) M2) (any M1) %. +%term red/case/lett reduce (case (lett M1 M2) M3a M3b) (lett M1 ([x] [y] case (M2 x y) M3a M3b)) %. +%term red/case/case reduce (case (case M1 M2a M2b) M3a M3b) (case M1 ([x] case (M2a x) M3a M3b) ([x] case (M2b x) M3a M3b)) %. +%term red/case/letb reduce (case (letb M1 M2) M3a M3b) (letb M1 ([x] case (M2 x) M3a M3b)) %. +%term red/case/any reduce (case (any M1) M2a M2b) (any M1) %. +%term red/letb/lett reduce (letb (lett M1 M2) M3) (lett M1 ([x] [y] letb (M2 x y) M3)) %. +%term red/letb/case reduce (letb (case M1 M2a M2b) M3) (case M1 ([x] letb (M2a x) M3) ([x] letb (M2b x) M3)) %. +%term red/letb/letb reduce (letb (letb M1 M2) M3) (letb M1 ([x] letb (M2 x) M3)) %. +%term red/letb/any reduce (letb (any M1) M2) (any M1) %. +%term red/any/lett reduce (any (lett M1 M2)) (lett M1 ([x] [y] any (M2 x y))) %. +%term red/any/case reduce (any (case M1 M2a M2b)) (case M1 ([x] any (M2a x)) ([x] any (M2b x))) %. +%term red/any/letb reduce (any (letb M1 M2)) (letb M1 ([x] any (M2 x))) %. +%term red/any/any reduce (any (any M)) (any M) %. +%{! Next we define syntactic equality and establish some uninteresting properties about it. !}% +%{! (options removed from twelftag: hidden="true") !}% +%%%%% Equality %%%%% +%{! !}% +%sort tp-eq %. +%term tp-eq/i tp-eq T T %. +%sort term-eq %. +%term term-eq/i term-eq M M %. +%{! !}% +%{! (options removed from twelftag: hidden="true") !}% +%%%%% Cons Lemmas %%%%% +%sort any-resp %. +%mode any-resp %in %out %. +%term _ any-resp term-eq/i term-eq/i %. +%worlds (block bind) (any-resp _ _) %. +%total {} (any-resp _ _) %. +%sort app-resp %. +%mode app-resp %in %in %out %. +%term _ app-resp term-eq/i term-eq/i term-eq/i %. +%worlds (block bind) (app-resp _ _ _) %. +%total {} (app-resp _ _ _) %. +%sort bang-resp %. +%mode bang-resp %in %out %. +%term _ bang-resp term-eq/i term-eq/i %. +%worlds (block bind) (bang-resp _ _) %. +%total {} (bang-resp _ _) %. +%sort case-resp %. +%mode case-resp %in %in %in %out %. +%term _ case-resp term-eq/i ([_] term-eq/i) ([_] term-eq/i) term-eq/i %. +%worlds (block bind) (case-resp _ _ _ _) %. +%total {} (case-resp _ _ _ _) %. +%sort in1-resp %. +%mode in1-resp %in %out %. +%term _ in1-resp term-eq/i term-eq/i %. +%worlds (block bind) (in1-resp _ _) %. +%total {} (in1-resp _ _) %. +%sort in2-resp %. +%mode in2-resp %in %out %. +%term _ in2-resp term-eq/i term-eq/i %. +%worlds (block bind) (in2-resp _ _) %. +%total {} (in2-resp _ _) %. +%sort lam-resp {x} %. +%mode lam-resp %in %out %. +%term _ lam-resp ([x] term-eq/i) term-eq/i %. +%worlds (block bind) (lam-resp _ _) %. +%total {} (lam-resp _ _) %. +%sort letb-resp %. +%mode letb-resp %in %in %out %. +%term _ letb-resp term-eq/i ([x] term-eq/i) term-eq/i %. +%worlds (block bind) (letb-resp _ _ _) %. +%total {} (letb-resp _ _ _) %. +%sort lett-resp %. +%mode lett-resp %in %in %out %. +%term _ lett-resp term-eq/i ([x] [y] term-eq/i) term-eq/i %. +%worlds (block bind) (lett-resp _ _ _) %. +%total {} (lett-resp _ _ _) %. +%sort pair-resp %. +%mode pair-resp %in %in %out %. +%term _ pair-resp term-eq/i term-eq/i term-eq/i %. +%worlds (block bind) (pair-resp _ _ _) %. +%total {} (pair-resp _ _ _) %. +%sort pi1-resp %. +%mode pi1-resp %in %out %. +%term _ pi1-resp term-eq/i term-eq/i %. +%worlds (block bind) (pi1-resp _ _) %. +%total {} (pi1-resp _ _) %. +%sort pi2-resp %. +%mode pi2-resp %in %out %. +%term _ pi2-resp term-eq/i term-eq/i %. +%worlds (block bind) (pi2-resp _ _) %. +%total {} (pi2-resp _ _) %. +%sort tensor-resp %. +%mode tensor-resp %in %in %out %. +%term _ tensor-resp term-eq/i term-eq/i term-eq/i %. +%worlds (block bind) (tensor-resp _ _ _) %. +%total {} (tensor-resp _ _ _) %. +%%%%% Respects Lemmas %%%%% +%sort linear-resp {x} %. +%mode linear-resp %in %in %out %. +%term _ linear-resp ([x] term-eq/i) D D %. +%worlds (block bind) (linear-resp _ _ _) %. +%total {} (linear-resp _ _ _) %. +%sort reduce-resp %. +%mode reduce-resp %in %in %in %out %. +%term _ reduce-resp term-eq/i term-eq/i D D %. +%worlds (block bind) (reduce-resp _ _ _ _) %. +%total {} (reduce-resp _ _ _ _) %. +%reduces <= D1 D2 (reduce-resp _ _ D2 D1) %. +%%%%% Down to Business %%%%% +%{! Next we prove an important lemma: when a variable x is used linearly in a term M1, and M1 is substituted for a linear variable in a term M2 that does not mention x, than x is used linearly in the result. !}% +%sort compose-linear %. +%mode compose-linear %in %in %out %. +%{! !}% +%{! (options removed from twelftag: hidden="true") !}% +%term _ compose-linear linear/var D D %. +%term _ + %pi (compose-linear (linear/lam D1) D2 (linear/lam D)) + %<- ({y} compose-linear (D1 y) D2 (D y)) %. +%term _ + %pi (compose-linear (linear/app1 D1) D2 (linear/app1 D)) + %<- (compose-linear D1 D2 D) %. +%term _ + %pi (compose-linear (linear/app2 D1) D2 (linear/app2 D)) + %<- (compose-linear D1 D2 D) %. +%term _ + %pi (compose-linear (linear/tensor1 D1) D2 (linear/tensor1 D)) + %<- (compose-linear D1 D2 D) %. +%term _ + %pi (compose-linear (linear/tensor2 D1) D2 (linear/tensor2 D)) + %<- (compose-linear D1 D2 D) %. +%term _ + %pi (compose-linear (linear/lett1 D1) D2 (linear/lett1 D)) + %<- (compose-linear D1 D2 D) %. +%term _ + %pi (compose-linear (linear/lett2 D1) D2 (linear/lett2 D)) + %<- ({y} {z} compose-linear (D1 y z) D2 (D y z)) %. +%term _ + %pi (compose-linear (linear/pair D1b D1a) D2 (linear/pair Db Da)) + %<- (compose-linear D1a D2 Da) + %<- (compose-linear D1b D2 Db) %. +%term _ + %pi (compose-linear (linear/pi1 D1) D2 (linear/pi1 D)) + %<- (compose-linear D1 D2 D) %. +%term _ + %pi (compose-linear (linear/pi2 D1) D2 (linear/pi2 D)) + %<- (compose-linear D1 D2 D) %. +%term _ + %pi (compose-linear (linear/in1 D1) D2 (linear/in1 D)) + %<- (compose-linear D1 D2 D) %. +%term _ + %pi (compose-linear (linear/in2 D1) D2 (linear/in2 D)) + %<- (compose-linear D1 D2 D) %. +%term _ + %pi (compose-linear (linear/case1 D1) D2 (linear/case1 D)) + %<- (compose-linear D1 D2 D) %. +%term _ + %pi (compose-linear (linear/case2 D1b D1a) D2 (linear/case2 Db Da)) + %<- ({y} compose-linear (D1a y) D2 (Da y)) + %<- ({y} compose-linear (D1b y) D2 (Db y)) %. +%term _ + %pi (compose-linear (linear/letb1 D1) D2 (linear/letb1 D)) + %<- (compose-linear D1 D2 D) %. +%term _ + %pi (compose-linear (linear/letb2 D1) D2 (linear/letb2 D)) + %<- ({y} compose-linear (D1 y) D2 (D y)) %. +%term _ + %pi (compose-linear (linear/any1 D1) D2 (linear/any1 D)) + %<- (compose-linear D1 D2 D) %. +%term _ compose-linear linear/any2 _ linear/any2 %. +%term _ compose-linear linear/unit _ linear/unit %. +%worlds (block bind) (compose-linear _ _ _) %. +%total D (compose-linear D _ _) %. +%{! Next we prove a technical lemma. It states that if x does not appear in M1, and M1 reduces to M2, then x does not appear in M2. !}% +%sort reduce-closed {x term} %. +%mode reduce-closed %in %out %. +%{! !}% +%{! (options removed from twelftag: hidden="true") !}% +%term _ reduce-closed ([x] red/refl) ([x] term-eq/i) %. +%sort reduce-closed!trans {x term} %. +%mode reduce-closed!trans %in %in %out %. +%term _ + %pi (reduce-closed ([x] red/trans (D2 x) (D1 x)) Deq2) + %<- (reduce-closed D1 Deq1) + %<- (reduce-closed!trans Deq1 D2 Deq2) %. +%term _ %pi (reduce-closed!trans ([x] term-eq/i) D Deq) %<- (reduce-closed D Deq) %. +%term _ reduce-closed ([x] red/beta) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/beta*) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/beta&1) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/beta&2) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/beta+1) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/beta+2) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/beta!) ([x] term-eq/i) %. +%term _ + %pi (reduce-closed ([x] red/lam (Dred x)) Deq') + %<- ({y} reduce-closed ([x] Dred x y) ([x] Deq x y)) + %<- ({x} lam-resp (Deq x) (Deq' x)) %. +%term _ + %pi (reduce-closed ([x] red/app (Dred2 x) (Dred1 x)) Deq) + %<- (reduce-closed Dred1 Deq1) + %<- (reduce-closed Dred2 Deq2) + %<- ({x} app-resp (Deq1 x) (Deq2 x) (Deq x)) %. +%term _ + %pi (reduce-closed ([x] red/tensor (Dred2 x) (Dred1 x)) Deq) + %<- (reduce-closed Dred1 Deq1) + %<- (reduce-closed Dred2 Deq2) + %<- ({x} tensor-resp (Deq1 x) (Deq2 x) (Deq x)) %. +%term _ + %pi (reduce-closed ([x] red/lett (Dred2 x) (Dred1 x)) Deq) + %<- (reduce-closed Dred1 Deq1) + %<- ({y} {z} reduce-closed ([x] Dred2 x y z) ([x] Deq2 x y z)) + %<- ({x} lett-resp (Deq1 x) (Deq2 x) (Deq x)) %. +%term _ + %pi (reduce-closed ([x] red/pair (Dred2 x) (Dred1 x)) Deq) + %<- (reduce-closed Dred1 Deq1) + %<- (reduce-closed Dred2 Deq2) + %<- ({x} pair-resp (Deq1 x) (Deq2 x) (Deq x)) %. +%term _ + %pi (reduce-closed ([x] red/pi1 (Dred x)) Deq') + %<- (reduce-closed Dred Deq) + %<- ({x} pi1-resp (Deq x) (Deq' x)) %. +%term _ + %pi (reduce-closed ([x] red/pi2 (Dred x)) Deq') + %<- (reduce-closed Dred Deq) + %<- ({x} pi2-resp (Deq x) (Deq' x)) %. +%term _ + %pi (reduce-closed ([x] red/in1 (Dred x)) Deq') + %<- (reduce-closed Dred Deq) + %<- ({x} in1-resp (Deq x) (Deq' x)) %. +%term _ + %pi (reduce-closed ([x] red/in2 (Dred x)) Deq') + %<- (reduce-closed Dred Deq) + %<- ({x} in2-resp (Deq x) (Deq' x)) %. +%term _ + %pi (reduce-closed ([x] red/case (Dred3 x) (Dred2 x) (Dred1 x)) Deq') + %<- (reduce-closed Dred1 Deq1) + %<- ({y} reduce-closed ([x] Dred2 x y) ([x] Deq2 x y)) + %<- ({y} reduce-closed ([x] Dred3 x y) ([x] Deq3 x y)) + %<- ({x} case-resp (Deq1 x) (Deq2 x) (Deq3 x) (Deq' x)) %. +%term _ + %pi (reduce-closed ([x] red/bang (Dred x)) Deq') + %<- (reduce-closed Dred Deq) + %<- ({x} bang-resp (Deq x) (Deq' x)) %. +%term _ + %pi (reduce-closed ([x] red/letb (Dred2 x) (Dred1 x)) Deq) + %<- (reduce-closed Dred1 Deq1) + %<- ({y} reduce-closed ([x] Dred2 x y) ([x] Deq2 x y)) + %<- ({x} letb-resp (Deq1 x) (Deq2 x) (Deq x)) %. +%term _ + %pi (reduce-closed ([x] red/any (Dred x)) Deq') + %<- (reduce-closed Dred Deq) + %<- ({x} any-resp (Deq x) (Deq' x)) %. +%term _ reduce-closed ([x] red/app/lett) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/app/case) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/app/letb) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/app/any) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/pi1/lett) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/pi1/case) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/pi1/letb) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/pi1/any) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/pi2/lett) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/pi2/case) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/pi2/letb) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/pi2/any) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/lett/lett) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/lett/case) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/lett/letb) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/lett/any) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/case/lett) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/case/case) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/case/letb) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/case/any) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/letb/lett) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/letb/case) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/letb/letb) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/letb/any) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/any/lett) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/any/case) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/any/letb) ([x] term-eq/i) %. +%term _ reduce-closed ([x] red/any/any) ([x] term-eq/i) %. +%worlds (block bind) (reduce-closed _ _) (reduce-closed!trans _ _ _) %. +%total (D1 D2) (reduce-closed D1 _) (reduce-closed!trans _ D2 _) %. +%{! With these lemmas in hand, we can prove the subject reduction theorem. In fact, there are two theorems to be proved simultaneously. One (``sr``) states that types are preserved by reduction, and the other (``srl``) states that linearity is preserved by reduction. !}% +%sort srl {x} %. +%sort sr %. +%mode srl %in %in %in %out %. +%mode sr %in %in %out %. +%{! !}% +%{! (options removed from twelftag: hidden="true") !}% +%scope sr %term refl srl ([x] red/refl) _ D D %. +%term trans + %pi (srl ([x] red/trans (Dred23 x) (Dred12 x)) Dof1 Dlin1 Dlin3) + %<- (srl Dred12 Dof1 Dlin1 Dlin2) + %<- ({x} {d} sr (Dred12 x) (Dof1 x d) (Dof2 x d)) + %<- (srl Dred23 Dof2 Dlin2 Dlin3) %. +%scope sr %term beta1 srl ([x] %the (reduce (app (lam (M1 x)) M2) _) red/beta) ([x] [d] of/app (%the (of _ T2) (Dof2 x d)) (of/lam (Dlin1 x) (Dof1 x d))) (linear/app1 (linear/lam Dlin)) (Dlin M2) %. +%term beta2 + %pi (srl ([x] %the (reduce (app (lam M1) (M2 x)) _) red/beta) ([x] [d] of/app (%the (of _ T2) (Dof2 x d)) (of/lam (Dlin1 x) (Dof1 x d))) (linear/app2 Dlin) Dlin') + %<- (compose-linear (Dlin1 unit) Dlin Dlin') %. +%scope sr %term bet*11 + %pi (srl ([x] %the (reduce (lett (tensor (M1 x) M2) M) _) red/beta*) ([x] [d] of/lett (Dlin2 x) (Dlin1 x) (Dof2 x d) (%the (of _ (T1 * T2)) (Dof1 x d))) (linear/lett1 (linear/tensor1 Dlin)) Dlin') + %<- (compose-linear (Dlin1 unit M2) Dlin Dlin') %. +%term bet*12 + %pi (srl ([x] %the (reduce (lett (tensor M1 (M2 x)) M) _) red/beta*) ([x] [d] of/lett (Dlin2 x) (Dlin1 x) (Dof2 x d) (%the (of _ (T1 * T2)) (Dof1 x d))) (linear/lett1 (linear/tensor2 Dlin)) Dlin') + %<- (compose-linear (Dlin2 unit M1) Dlin Dlin') %. +%scope sr %term bet*2 srl ([x] %the (reduce (lett (tensor M1 M2) (M x)) _) red/beta*) ([x] [d] of/lett (Dlin2 x) (Dlin1 x) (Dof2 x d) (%the (of _ (T1 * T2)) (Dof1 x d))) (linear/lett2 Dlin) (Dlin M1 M2) %. +%term beta&1 srl ([x] %the (reduce (pi1 (pair (M1 x) (M2 x))) _) red/beta&1) ([x] [d] of/pi1 (of/pair (%the (of _ T1) (Dof2 x d)) (%the (of _ T2) (Dof1 x d)))) (linear/pi1 (linear/pair Dlin2 Dlin1)) Dlin1 %. +%scope sr %term beta&2 srl ([x] %the (reduce (pi2 (pair (M1 x) (M2 x))) _) red/beta&2) ([x] [d] of/pi2 (of/pair (%the (of _ T1) (Dof2 x d)) (%the (of _ T2) (Dof1 x d)))) (linear/pi2 (linear/pair Dlin2 Dlin1)) Dlin2 %. +%term bt+1-1 + %pi (srl ([x] %the (reduce (case (in1 (M x)) M1 M2) _) red/beta+1) ([x] [d] of/case (Dlin2 x) (Dlin1 x) _ _ (%the (of _ (T1 + T2)) _)) (linear/case1 (linear/in1 Dlin)) Dlin') + %<- (compose-linear (Dlin1 unit) Dlin Dlin') %. +%scope sr %term bt+1-2 srl ([x] %the (reduce (case (in1 M) (M1 x) (M2 x)) _) red/beta+1) ([x] [d] of/case (Dlin2 x) (Dlin1 x) _ _ (%the (of _ (T1 + T2)) _)) (linear/case2 Dlin2' Dlin1') (Dlin1' M) %. +%term bt+2-1 + %pi (srl ([x] %the (reduce (case (in2 (M x)) M1 M2) _) red/beta+2) ([x] [d] of/case (Dlin2 x) (Dlin1 x) _ _ (%the (of _ (T1 + T2)) _)) (linear/case1 (linear/in2 Dlin)) Dlin') + %<- (compose-linear (Dlin2 unit) Dlin Dlin') %. +%scope sr %term bt+2-2 srl ([x] %the (reduce (case (in2 M) (M1 x) (M2 x)) _) red/beta+2) ([x] [d] of/case (Dlin2 x) (Dlin1 x) _ _ (%the (of _ (T1 + T2)) _)) (linear/case2 Dlin2' Dlin1') (Dlin2' M) %. +%term beta! srl ([x] %the (reduce (letb (bang M1) (M2 x)) _) red/beta!) ([x] [d] of/letb (Dof2 x d) (of/bang (%the (of _ T) (Dof1 x d)))) (linear/letb2 Dlin) (Dlin M1) %. +%scope sr %term lam + %pi (srl ([x] red/lam (Dred x)) ([x] [d] of/lam (Dlin1 x) (Dof x d)) (linear/lam Dlin) (linear/lam Dlin')) + %<- ({y} {e} srl ([x] Dred x y) ([x] [d] Dof x d y e) (Dlin y) (Dlin' y)) %. +%term app1 + %pi (srl ([x] red/app (Dred2 x) (Dred1 x)) ([x] [d] of/app (Dof2 x d) (Dof1 x d)) (linear/app1 Dlin) Dlin'') + %<- (srl Dred1 Dof1 Dlin Dlin') + %<- (reduce-closed Dred2 Deq) + %<- ({x} app-resp term-eq/i (Deq x) (Deq' x)) + %<- (linear-resp Deq' (linear/app1 Dlin') Dlin'') %. +%scope sr %term app2 + %pi (srl ([x] red/app (Dred2 x) (Dred1 x)) ([x] [d] of/app (Dof2 x d) (Dof1 x d)) (linear/app2 Dlin) Dlin'') + %<- (srl Dred2 Dof2 Dlin Dlin') + %<- (reduce-closed Dred1 Deq) + %<- ({x} app-resp (Deq x) term-eq/i (Deq' x)) + %<- (linear-resp Deq' (linear/app2 Dlin') Dlin'') %. +%term tens1 + %pi (srl ([x] red/tensor (Dred2 x) (Dred1 x)) ([x] [d] of/tensor (Dof2 x d) (Dof1 x d)) (linear/tensor1 Dlin) Dlin'') + %<- (srl Dred1 Dof1 Dlin Dlin') + %<- (reduce-closed Dred2 Deq) + %<- ({x} tensor-resp term-eq/i (Deq x) (Deq' x)) + %<- (linear-resp Deq' (linear/tensor1 Dlin') Dlin'') %. +%scope sr %term tens2 + %pi (srl ([x] red/tensor (Dred2 x) (Dred1 x)) ([x] [d] of/tensor (Dof2 x d) (Dof1 x d)) (linear/tensor2 Dlin) Dlin'') + %<- (srl Dred2 Dof2 Dlin Dlin') + %<- (reduce-closed Dred1 Deq) + %<- ({x} tensor-resp (Deq x) term-eq/i (Deq' x)) + %<- (linear-resp Deq' (linear/tensor2 Dlin') Dlin'') %. +%term lett1 + %pi (srl ([x] red/lett (Dred2 x) (Dred1 x)) ([x] [d] of/lett (Dlin2 x) (Dlin1 x) (Dof2 x d) (Dof1 x d)) (linear/lett1 Dlin) Dlin'') + %<- (srl Dred1 Dof1 Dlin Dlin') + %<- ({y} {z} reduce-closed ([x] Dred2 x y z) ([x] Deq x y z)) + %<- ({x} lett-resp term-eq/i (Deq x) (Deq' x)) + %<- (linear-resp Deq' (linear/lett1 Dlin') Dlin'') %. +%scope sr %term lett2 + %pi (srl ([x] red/lett (Dred2 x) (Dred1 x)) ([x] [d] of/lett (Dlin2 x) (Dlin1 x) (Dof2 x d) (Dof1 x d)) (linear/lett2 Dlin) Dlin'') + %<- ({y} {e of y T1} {z} {f of z T2} srl ([x] Dred2 x y z) ([x] [d] Dof2 x d y e z f) (Dlin y z) (Dlin' y z)) + %<- (reduce-closed Dred1 Deq) + %<- ({x} lett-resp (Deq x) ([_] [_] term-eq/i) (Deq' x)) + %<- (linear-resp Deq' (linear/lett2 Dlin') Dlin'') %. +%term pair + %pi (srl ([x] red/pair (Dred2 x) (Dred1 x)) ([x] [d] of/pair (Dof2 x d) (Dof1 x d)) (linear/pair Dlin2 Dlin1) (linear/pair Dlin2' Dlin1')) + %<- (srl Dred1 Dof1 Dlin1 Dlin1') + %<- (srl Dred2 Dof2 Dlin2 Dlin2') %. +%scope sr %term pi1 + %pi (srl ([x] red/pi1 (Dred x)) ([x] [d] of/pi1 (Dof x d)) (linear/pi1 Dlin) (linear/pi1 Dlin')) + %<- (srl Dred Dof Dlin Dlin') %. +%term pi2 + %pi (srl ([x] red/pi2 (Dred x)) ([x] [d] of/pi2 (Dof x d)) (linear/pi2 Dlin) (linear/pi2 Dlin')) + %<- (srl Dred Dof Dlin Dlin') %. +%scope sr %term in1 + %pi (srl ([x] red/in1 (Dred x)) ([x] [d] of/in1 (Dof x d)) (linear/in1 Dlin) (linear/in1 Dlin')) + %<- (srl Dred Dof Dlin Dlin') %. +%term in2 + %pi (srl ([x] red/in2 (Dred x)) ([x] [d] of/in2 (Dof x d)) (linear/in2 Dlin) (linear/in2 Dlin')) + %<- (srl Dred Dof Dlin Dlin') %. +%scope sr %term case1 + %pi (srl ([x] red/case (Dred3 x) (Dred2 x) (Dred1 x)) ([x] [d] of/case (Dlin3 x) (Dlin2 x) (Dof3 x d) (Dof2 x d) (Dof1 x d)) (linear/case1 Dlin) Dlin'') + %<- (srl Dred1 Dof1 Dlin Dlin') + %<- ({y} reduce-closed ([x] Dred2 x y) ([x] Deq2 x y)) + %<- ({y} reduce-closed ([x] Dred3 x y) ([x] Deq3 x y)) + %<- ({x} case-resp term-eq/i (Deq2 x) (Deq3 x) (Deq' x)) + %<- (linear-resp Deq' (linear/case1 Dlin') Dlin'') %. +%term case2 + %pi (srl ([x] red/case (Dred3 x) (Dred2 x) (Dred1 x)) ([x] [d] of/case (Dlin3 x) (Dlin2 x) (Dof3 x d) (Dof2 x d) (Dof1 x d)) (linear/case2 Dlin3' Dlin2') Dlin'') + %<- ({x'} {d'} srl ([x] Dred2 x x') ([x] [d] Dof2 x d x' d') (Dlin2' x') (Dlin2'' x')) + %<- ({x'} {d'} srl ([x] Dred3 x x') ([x] [d] Dof3 x d x' d') (Dlin3' x') (Dlin3'' x')) + %<- (reduce-closed Dred1 Deq) + %<- ({x} case-resp (Deq x) ([y] term-eq/i) ([y] term-eq/i) (Deq' x)) + %<- (linear-resp Deq' (linear/case2 Dlin3'' Dlin2'') Dlin'') %. +%scope sr %term letb1 + %pi (srl ([x] red/letb (Dred2 x) (Dred1 x)) ([x] [d] of/letb (Dof2 x d) (Dof1 x d)) (linear/letb1 Dlin) Dlin'') + %<- (srl Dred1 Dof1 Dlin Dlin') + %<- ({y} reduce-closed ([x] Dred2 x y) ([x] Deq x y)) + %<- ({x} letb-resp term-eq/i (Deq x) (Deq' x)) + %<- (linear-resp Deq' (linear/letb1 Dlin') Dlin'') %. +%term letb1 + %pi (srl ([x] red/letb (Dred2 x) (Dred1 x)) ([x] [d] of/letb (Dof2 x d) (Dof1 x d)) (linear/letb2 Dlin) Dlin'') + %<- ({x'} {d'} srl ([x] Dred2 x x') ([x] [d] Dof2 x d x' d') (Dlin x') (Dlin' x')) + %<- (reduce-closed Dred1 Deq) + %<- ({x} letb-resp (Deq x) ([y] term-eq/i) (Deq' x)) + %<- (linear-resp Deq' (linear/letb2 Dlin') Dlin'') %. +%scope sr %term any1 + %pi (srl ([x] red/any (Dred x)) ([x] [d] of/any (Dof x d)) (linear/any1 Dlin) (linear/any1 Dlin')) + %<- (srl Dred Dof Dlin Dlin') %. +%term any2 + %pi (srl ([x] red/any (Dred x)) ([x] [d] of/any (Dof x d)) linear/any2 Dlin) + %<- (reduce-closed Dred Deq) + %<- ({x} any-resp (Deq x) (Deq' x)) + %<- (linear-resp Deq' linear/any2 Dlin) %. +%scope sr %term Cl*11 srl ([x] %the (reduce (app (lett (M1 x) M2) M3) _) red/app/lett) _ (linear/app1 (linear/lett1 Dlin)) (linear/lett1 Dlin) %. +%term Cl*12 srl ([x] %the (reduce (app (lett M1 (M2 x)) M3) _) red/app/lett) _ (linear/app1 (linear/lett2 Dlin)) (linear/lett2 ([x] [y] linear/app1 (Dlin x y))) %. +%scope sr %term Cl*2 srl ([x] %the (reduce (app (lett M1 M2) (M3 x)) _) red/app/lett) _ (linear/app2 Dlin) (linear/lett2 ([x] [y] linear/app2 Dlin)) %. +%term Cl+11 srl ([x] %the (reduce (app (case (M1 x) M2a M2b) M3) _) red/app/case) _ (linear/app1 (linear/case1 Dlin)) (linear/case1 Dlin) %. +%scope sr %term Cl+12 srl ([x] %the (reduce (app (case M1 (M2a x) (M2b x)) M3) _) red/app/case) _ (linear/app1 (linear/case2 DlinB DlinA)) (linear/case2 ([y] linear/app1 (DlinB y)) ([y] linear/app1 (DlinA y))) %. +%term Cl+2 srl ([x] %the (reduce (app (case M1 M2a M2b) (M3 x)) _) red/app/case) _ (linear/app2 Dlin) (linear/case2 ([y] linear/app2 Dlin) ([y] linear/app2 Dlin)) %. +%scope sr %term Cl!11 srl ([x] %the (reduce (app (letb (M1 x) M2) M3) _) red/app/letb) _ (linear/app1 (linear/letb1 Dlin)) (linear/letb1 Dlin) %. +%term Cl!12 srl ([x] %the (reduce (app (letb M1 (M2 x)) M3) _) red/app/letb) _ (linear/app1 (linear/letb2 Dlin)) (linear/letb2 ([y] linear/app1 (Dlin y))) %. +%scope sr %term Cl!2 srl ([x] %the (reduce (app (letb M1 M2) (M3 x)) _) red/app/letb) _ (linear/app2 Dlin) (linear/letb2 ([y] linear/app2 Dlin)) %. +%term Cl011 srl ([x] %the (reduce (app (any (M1 x)) M2) _) red/app/any) _ (linear/app1 (linear/any1 Dlin)) (linear/any1 Dlin) %. +%scope sr %term Cl012 srl ([x] %the (reduce (app (any M1) M2) _) red/app/any) _ (linear/app1 linear/any2) linear/any2 %. +%term Cl02 srl ([x] %the (reduce (app (any M1) (M2 x)) _) red/app/any) _ (linear/app2 _) linear/any2 %. +%scope sr %term C&1*1 srl ([x] %the (reduce (pi1 (lett (M1 x) M2)) _) red/pi1/lett) _ (linear/pi1 (linear/lett1 Dlin)) (linear/lett1 Dlin) %. +%term C&1*2 srl ([x] %the (reduce (pi1 (lett M1 (M2 x))) _) red/pi1/lett) _ (linear/pi1 (linear/lett2 Dlin)) (linear/lett2 ([x] [y] linear/pi1 (Dlin x y))) %. +%scope sr %term C&1+1 srl ([x] %the (reduce (pi1 (case (M1 x) M2a M2b)) _) red/pi1/case) _ (linear/pi1 (linear/case1 Dlin)) (linear/case1 Dlin) %. +%term C&1+2 srl ([x] %the (reduce (pi1 (case M1 (M2a x) (M2b x))) _) red/pi1/case) _ (linear/pi1 (linear/case2 DlinB DlinA)) (linear/case2 ([x] linear/pi1 (DlinB x)) ([x] linear/pi1 (DlinA x))) %. +%scope sr %term C&1!1 srl ([x] %the (reduce (pi1 (letb (M1 x) M2)) _) red/pi1/letb) _ (linear/pi1 (linear/letb1 Dlin)) (linear/letb1 Dlin) %. +%term C&1!2 srl ([x] %the (reduce (pi1 (letb M1 (M2 x))) _) red/pi1/letb) _ (linear/pi1 (linear/letb2 Dlin)) (linear/letb2 ([x] linear/pi1 (Dlin x))) %. +%scope sr %term C&101 srl ([x] %the (reduce (pi1 (any (M x))) _) red/pi1/any) _ (linear/pi1 (linear/any1 Dlin)) (linear/any1 Dlin) %. +%term C&102 srl ([x] %the (reduce (pi1 (any M)) _) red/pi1/any) _ (linear/pi1 linear/any2) linear/any2 %. +%scope sr %term C&2*1 srl ([x] %the (reduce (pi2 (lett (M1 x) M2)) _) red/pi2/lett) _ (linear/pi2 (linear/lett1 Dlin)) (linear/lett1 Dlin) %. +%term C&2*2 srl ([x] %the (reduce (pi2 (lett M1 (M2 x))) _) red/pi2/lett) _ (linear/pi2 (linear/lett2 Dlin)) (linear/lett2 ([x] [y] linear/pi2 (Dlin x y))) %. +%scope sr %term C&2+1 srl ([x] %the (reduce (pi2 (case (M1 x) M2a M2b)) _) red/pi2/case) _ (linear/pi2 (linear/case1 Dlin)) (linear/case1 Dlin) %. +%term C&2+2 srl ([x] %the (reduce (pi2 (case M1 (M2a x) (M2b x))) _) red/pi2/case) _ (linear/pi2 (linear/case2 DlinB DlinA)) (linear/case2 ([x] linear/pi2 (DlinB x)) ([x] linear/pi2 (DlinA x))) %. +%scope sr %term C&2!1 srl ([x] %the (reduce (pi2 (letb (M1 x) M2)) _) red/pi2/letb) _ (linear/pi2 (linear/letb1 Dlin)) (linear/letb1 Dlin) %. +%term C&2!2 srl ([x] %the (reduce (pi2 (letb M1 (M2 x))) _) red/pi2/letb) _ (linear/pi2 (linear/letb2 Dlin)) (linear/letb2 ([x] linear/pi2 (Dlin x))) %. +%scope sr %term C&201 srl ([x] %the (reduce (pi2 (any (M x))) _) red/pi2/any) _ (linear/pi2 (linear/any1 Dlin)) (linear/any1 Dlin) %. +%term C&202 srl ([x] %the (reduce (pi2 (any M)) _) red/pi2/any) _ (linear/pi2 linear/any2) linear/any2 %. +%scope sr %term C**11 srl ([x] %the (reduce (lett (lett (M1 x) M2) M3) _) red/lett/lett) _ (linear/lett1 (linear/lett1 Dlin)) (linear/lett1 Dlin) %. +%term C**12 srl ([x] %the (reduce (lett (lett M1 (M2 x)) M3) _) red/lett/lett) _ (linear/lett1 (linear/lett2 Dlin)) (linear/lett2 ([x] [y] linear/lett1 (Dlin x y))) %. +%scope sr %term C**2 srl ([x] %the (reduce (lett (lett M1 M2) (M3 x)) _) red/lett/lett) _ (linear/lett2 Dlin) (linear/lett2 ([x] [y] linear/lett2 Dlin)) %. +%term C*+11 srl ([x] %the (reduce (lett (case (M1 x) M2a M2b) M3) _) red/lett/case) _ (linear/lett1 (linear/case1 Dlin)) (linear/case1 Dlin) %. +%scope sr %term C*+12 srl ([x] %the (reduce (lett (case M1 (M2a x) (M2b x)) M3) _) red/lett/case) _ (linear/lett1 (linear/case2 DlinB DlinA)) (linear/case2 ([y] linear/lett1 (DlinB y)) ([y] linear/lett1 (DlinA y))) %. +%term C*+2 srl ([x] %the (reduce (lett (case M1 M2a M2b) (M3 x)) _) red/lett/case) _ (linear/lett2 Dlin) (linear/case2 ([y] linear/lett2 Dlin) ([y] linear/lett2 Dlin)) %. +%scope sr %term C*!11 srl ([x] %the (reduce (lett (letb (M1 x) M2) M3) _) red/lett/letb) _ (linear/lett1 (linear/letb1 Dlin)) (linear/letb1 Dlin) %. +%term C*!12 srl ([x] %the (reduce (lett (letb M1 (M2 x)) M3) _) red/lett/letb) _ (linear/lett1 (linear/letb2 Dlin)) (linear/letb2 ([x] linear/lett1 (Dlin x))) %. +%scope sr %term C*!2 srl ([x] %the (reduce (lett (letb M1 M2) (M3 x)) _) red/lett/letb) _ (linear/lett2 Dlin) (linear/letb2 ([x] linear/lett2 Dlin)) %. +%term C*011 srl ([x] %the (reduce (lett (any (M1 x)) M2) _) red/lett/any) _ (linear/lett1 (linear/any1 Dlin)) (linear/any1 Dlin) %. +%scope sr %term C*012 srl ([x] %the (reduce (lett (any M1) M2) _) red/lett/any) _ (linear/lett1 linear/any2) linear/any2 %. +%term C*02 srl ([x] %the (reduce (lett (any M1) (M2 x)) _) red/lett/any) _ (linear/lett2 Dlin) linear/any2 %. +%scope sr %term C+*11 srl ([x] %the (reduce (case (lett (M1 x) M2) M3a M3b) _) red/case/lett) _ (linear/case1 (linear/lett1 Dlin)) (linear/lett1 Dlin) %. +%term C+*12 srl ([x] %the (reduce (case (lett M1 (M2 x)) M3a M3b) _) red/case/lett) _ (linear/case1 (linear/lett2 Dlin)) (linear/lett2 ([y] [z] linear/case1 (Dlin y z))) %. +%scope sr %term C+*2 srl ([x] %the (reduce (case (lett M1 M2) (M3a x) (M3b x)) _) red/case/lett) _ (linear/case2 DlinB DlinA) (linear/lett2 ([y] [z] linear/case2 DlinB DlinA)) %. +%term C++11 srl ([x] %the (reduce (case (case (M1 x) M2a M2b) M3a M3b) _) red/case/case) _ (linear/case1 (linear/case1 Dlin)) (linear/case1 Dlin) %. +%scope sr %term C++12 srl ([x] %the (reduce (case (case M1 (M2a x) (M2b x)) M3a M3b) _) red/case/case) _ (linear/case1 (linear/case2 DlinB DlinA)) (linear/case2 ([y] linear/case1 (DlinB y)) ([y] linear/case1 (DlinA y))) %. +%term C++2 srl ([x] %the (reduce (case (case M1 M2a M2b) (M3a x) (M3b x)) _) red/case/case) _ (linear/case2 DlinB DlinA) (linear/case2 ([y] linear/case2 DlinB DlinA) ([y] linear/case2 DlinB DlinA)) %. +%scope sr %term C+!11 srl ([x] %the (reduce (case (letb (M1 x) M2) M3a M3b) _) red/case/letb) _ (linear/case1 (linear/letb1 Dlin)) (linear/letb1 Dlin) %. +%term C+!12 srl ([x] %the (reduce (case (letb M1 (M2 x)) M3a M3b) _) red/case/letb) _ (linear/case1 (linear/letb2 Dlin)) (linear/letb2 ([x] linear/case1 (Dlin x))) %. +%scope sr %term C+!2 srl ([x] %the (reduce (case (letb M1 M2) (M3a x) (M3b x)) _) red/case/letb) _ (linear/case2 DlinB DlinA) (linear/letb2 ([x] linear/case2 DlinB DlinA)) %. +%term C+011 srl ([x] %the (reduce (case (any (M1 x)) M2a M2b) _) red/case/any) _ (linear/case1 (linear/any1 Dlin)) (linear/any1 Dlin) %. +%scope sr %term C+012 srl ([x] %the (reduce (case (any M1) M2a M2b) _) red/case/any) _ (linear/case1 linear/any2) linear/any2 %. +%term C+02 srl ([x] %the (reduce (case (any M1) (M2a x) (M2b x)) _) red/case/any) _ (linear/case2 DlinB DlinA) linear/any2 %. +%scope sr %term C!*11 srl ([x] %the (reduce (letb (lett (M1 x) M2) M3) _) red/letb/lett) _ (linear/letb1 (linear/lett1 Dlin)) (linear/lett1 Dlin) %. +%term C!*12 srl ([x] %the (reduce (letb (lett M1 (M2 x)) M3) _) red/letb/lett) _ (linear/letb1 (linear/lett2 Dlin)) (linear/lett2 ([x] [y] linear/letb1 (Dlin x y))) %. +%scope sr %term C!*2 srl ([x] %the (reduce (letb (lett M1 M2) (M3 x)) _) red/letb/lett) _ (linear/letb2 Dlin) (linear/lett2 ([x] [y] linear/letb2 Dlin)) %. +%term C!+11 srl ([x] %the (reduce (letb (case (M1 x) M2a M2b) M3) _) red/letb/case) _ (linear/letb1 (linear/case1 Dlin)) (linear/case1 Dlin) %. +%scope sr %term C!+12 srl ([x] %the (reduce (letb (case M1 (M2a x) (M2b x)) M3) _) red/letb/case) _ (linear/letb1 (linear/case2 DlinB DlinA)) (linear/case2 ([x] linear/letb1 (DlinB x)) ([x] linear/letb1 (DlinA x))) %. +%term C!+2 srl ([x] %the (reduce (letb (case M1 M2a M2b) (M3 x)) _) red/letb/case) _ (linear/letb2 Dlin) (linear/case2 ([x] linear/letb2 Dlin) ([x] linear/letb2 Dlin)) %. +%scope sr %term C!!11 srl ([x] %the (reduce (letb (letb (M1 x) M2) M3) _) red/letb/letb) _ (linear/letb1 (linear/letb1 Dlin)) (linear/letb1 Dlin) %. +%term C!!12 srl ([x] %the (reduce (letb (letb M1 (M2 x)) M3) _) red/letb/letb) _ (linear/letb1 (linear/letb2 Dlin)) (linear/letb2 ([x] linear/letb1 (Dlin x))) %. +%scope sr %term C!!2 srl ([x] %the (reduce (letb (letb M1 M2) (M3 x)) _) red/letb/letb) _ (linear/letb2 Dlin) (linear/letb2 ([x] linear/letb2 Dlin)) %. +%term C!011 srl ([x] %the (reduce (letb (any (M1 x)) M2) _) red/letb/any) _ (linear/letb1 (linear/any1 Dlin)) (linear/any1 Dlin) %. +%scope sr %term C!012 srl ([x] %the (reduce (letb (any M1) M2) _) red/letb/any) _ (linear/letb1 linear/any2) linear/any2 %. +%term C!02 srl ([x] %the (reduce (letb (any M1) (M2 x)) _) red/letb/any) _ (linear/letb2 Dlin) linear/any2 %. +%scope sr %term C0*11 srl ([x] %the (reduce (any (lett (M1 x) M2)) _) red/any/lett) _ (linear/any1 (linear/lett1 Dlin)) (linear/lett1 Dlin) %. +%term C0*12 srl ([x] %the (reduce (any (lett M1 (M2 x))) _) red/any/lett) _ (linear/any1 (linear/lett2 Dlin)) (linear/lett2 ([x] [y] linear/any1 (Dlin x y))) %. +%scope sr %term C0*2 srl ([x] %the (reduce (any (lett M1 M2)) _) red/any/lett) _ linear/any2 (linear/lett2 ([x] [y] linear/any2)) %. +%term C0+11 srl ([x] %the (reduce (any (case (M1 x) M2a M2b)) _) red/any/case) _ (linear/any1 (linear/case1 Dlin)) (linear/case1 Dlin) %. +%scope sr %term C0+12 srl ([x] %the (reduce (any (case M1 (M2a x) (M2b x))) _) red/any/case) _ (linear/any1 (linear/case2 DlinB DlinA)) (linear/case2 ([x] linear/any1 (DlinB x)) ([x] linear/any1 (DlinA x))) %. +%term C0+2 srl ([x] %the (reduce (any (case M1 M2a M2b)) _) red/any/case) _ linear/any2 (linear/case2 ([x] linear/any2) ([x] linear/any2)) %. +%scope sr %term C0!11 srl ([x] %the (reduce (any (letb (M1 x) M2)) _) red/any/letb) _ (linear/any1 (linear/letb1 Dlin)) (linear/letb1 Dlin) %. +%term C0!12 srl ([x] %the (reduce (any (letb M1 (M2 x))) _) red/any/letb) _ (linear/any1 (linear/letb2 Dlin)) (linear/letb2 ([x] linear/any1 (Dlin x))) %. +%scope sr %term C0!2 srl ([x] %the (reduce (any (letb M1 M2)) _) red/any/letb) _ linear/any2 (linear/letb2 ([x] linear/any2)) %. +%term C0011 srl ([x] %the (reduce (any (any (M x))) _) red/any/any) _ (linear/any1 (linear/any1 Dlin)) (linear/any1 Dlin) %. +%scope sr %term C0012 srl ([x] %the (reduce (any (any M)) _) red/any/any) _ (linear/any1 linear/any2) linear/any2 %. +%term C002 srl ([x] %the (reduce (any (any M)) _) red/any/any) _ linear/any2 linear/any2 %. +%scope sr %term refl sr red/refl D D %. +%term trans + %pi (sr (red/trans Dred23 Dred12) D1 D3) + %<- (sr Dred12 D1 D2) + %<- (sr Dred23 D2 D3) %. +%scope sr %term beta sr red/beta (of/app Dof2 (of/lam _ Dof1)) (Dof1 _ Dof2) %. +%term beta* sr red/beta* (of/lett Dlin2 Dlin1 Dof (of/tensor Dof2 Dof1)) (Dof _ Dof1 _ Dof2) %. +%scope sr %term beta&1 sr red/beta&1 (of/pi1 (of/pair Dof2 Dof1)) Dof1 %. +%term beta&2 sr red/beta&2 (of/pi2 (of/pair Dof2 Dof1)) Dof2 %. +%scope sr %term beta+1 sr red/beta+1 (of/case _ _ Dof2 Dof1 (of/in1 Dof)) (Dof1 _ Dof) %. +%term beta+2 sr red/beta+2 (of/case _ _ Dof2 Dof1 (of/in2 Dof)) (Dof2 _ Dof) %. +%scope sr %term beta! sr red/beta! (of/letb Dof2 (of/bang Dof1)) (Dof2 _ Dof1) %. +%term lam + %pi (sr (red/lam Dred) (of/lam Dlin Dof) (of/lam Dlin' Dof')) + %<- ({x} {d} sr (Dred x) (Dof x d) (Dof' x d)) + %<- (srl Dred Dof Dlin Dlin') %. +%scope sr %term app + %pi (sr (red/app Dred2 Dred1) (of/app Dof2 Dof1) (of/app Dof2' Dof1')) + %<- (sr Dred1 Dof1 Dof1') + %<- (sr Dred2 Dof2 Dof2') %. +%term tensor + %pi (sr (red/tensor Dred2 Dred1) (of/tensor Dof2 Dof1) (of/tensor Dof2' Dof1')) + %<- (sr Dred1 Dof1 Dof1') + %<- (sr Dred2 Dof2 Dof2') %. +%scope sr %term lett + %pi (sr (red/lett Dred2 Dred1) (of/lett Dlin2 Dlin1 Dof2 Dof1) (of/lett Dlin2' Dlin1' Dof2' Dof1')) + %<- (sr Dred1 Dof1 Dof1') + %<- ({x} {d of x T1} {y} {e of y T2} sr (Dred2 x y) (Dof2 x d y e) (Dof2' x d y e)) + %<- ({y} {e of y T2} srl ([x] Dred2 x y) ([x] [d] Dof2 x d y e) (Dlin1 y) (Dlin1' y)) + %<- ({x} {d of x T1} srl ([y] Dred2 x y) ([y] [e] Dof2 x d y e) (Dlin2 x) (Dlin2' x)) %. +%term pair + %pi (sr (red/pair Dred2 Dred1) (of/pair Dof2 Dof1) (of/pair Dof2' Dof1')) + %<- (sr Dred1 Dof1 Dof1') + %<- (sr Dred2 Dof2 Dof2') %. +%scope sr %term pi1 %pi (sr (red/pi1 Dred) (of/pi1 Dof) (of/pi1 Dof')) %<- (sr Dred Dof Dof') %. +%term pi2 %pi (sr (red/pi2 Dred) (of/pi2 Dof) (of/pi2 Dof')) %<- (sr Dred Dof Dof') %. +%scope sr %term in1 %pi (sr (red/in1 Dred) (of/in1 Dof) (of/in1 Dof')) %<- (sr Dred Dof Dof') %. +%term in2 %pi (sr (red/in2 Dred) (of/in2 Dof) (of/in2 Dof')) %<- (sr Dred Dof Dof') %. +%scope sr %term case + %pi (sr (red/case Dred3 Dred2 Dred1) (of/case Dlin3 Dlin2 Dof3 Dof2 Dof1) (of/case Dlin3' Dlin2' Dof3' Dof2' Dof1')) + %<- (sr Dred1 Dof1 Dof1') + %<- ({x} {d} sr (Dred2 x) (Dof2 x d) (Dof2' x d)) + %<- ({x} {d} sr (Dred3 x) (Dof3 x d) (Dof3' x d)) + %<- (srl Dred2 Dof2 Dlin2 Dlin2') + %<- (srl Dred3 Dof3 Dlin3 Dlin3') %. +%term letb + %pi (sr (red/letb Dred2 Dred1) (of/letb Dof2 Dof1) (of/letb Dof2' Dof1')) + %<- (sr Dred1 Dof1 Dof1') + %<- ({x} {d} sr (Dred2 x) (Dof2 x d) (Dof2' x d)) %. +%scope sr %term bang %pi (sr (red/bang Dred) (of/bang Dof) (of/bang Dof')) %<- (sr Dred Dof Dof') %. +%term any %pi (sr (red/any Dred) (of/any Dof) (of/any Dof')) %<- (sr Dred Dof Dof') %. +%scope sr %term Cl* sr red/app/lett (of/app Dof3 (of/lett Dlin2 Dlin1 Dof2 Dof1)) (of/lett ([x] linear/app1 (Dlin2 x)) ([x] linear/app1 (Dlin1 x)) ([x1] [d1] [x2] [d2] of/app Dof3 (Dof2 x1 d1 x2 d2)) Dof1) %. +%term Cl+ sr red/app/case (of/app Dof3 (of/case DlinB DlinA Dof2b Dof2a Dof1)) (of/case (linear/app1 DlinB) (linear/app1 DlinA) ([x] [d] of/app Dof3 (Dof2b x d)) ([x] [d] of/app Dof3 (Dof2a x d)) Dof1) %. +%scope sr %term Cl! sr red/app/letb (of/app Dof3 (of/letb Dof2 Dof1)) (of/letb ([x] [d] of/app Dof3 (Dof2 x d)) Dof1) %. +%term Cl0 sr red/app/any (of/app _ (of/any Dof)) (of/any Dof) %. +%scope sr %term C&1* sr red/pi1/lett (of/pi1 (of/lett Dlin2' Dlin2 Dof2 Dof1)) (of/lett ([x] linear/pi1 (Dlin2' x)) ([x] linear/pi1 (Dlin2 x)) ([x1] [d1] [x2] [d2] of/pi1 (Dof2 x1 d1 x2 d2)) Dof1) %. +%term C&1+ sr red/pi1/case (of/pi1 (of/case Dlin2b Dlin2a Dof2b Dof2a Dof1)) (of/case (linear/pi1 Dlin2b) (linear/pi1 Dlin2a) ([x] [d] of/pi1 (Dof2b x d)) ([x] [d] of/pi1 (Dof2a x d)) Dof1) %. +%scope sr %term C&1! sr red/pi1/letb (of/pi1 (of/letb Dof2 Dof1)) (of/letb ([x] [d] of/pi1 (Dof2 x d)) Dof1) %. +%term C&10 sr red/pi1/any (of/pi1 (of/any Dof)) (of/any Dof) %. +%scope sr %term C&2* sr red/pi2/lett (of/pi2 (of/lett Dlin2' Dlin2 Dof2 Dof1)) (of/lett ([x] linear/pi2 (Dlin2' x)) ([x] linear/pi2 (Dlin2 x)) ([x1] [d1] [x2] [d2] of/pi2 (Dof2 x1 d1 x2 d2)) Dof1) %. +%term C&2+ sr red/pi2/case (of/pi2 (of/case Dlin2b Dlin2a Dof2b Dof2a Dof1)) (of/case (linear/pi2 Dlin2b) (linear/pi2 Dlin2a) ([x] [d] of/pi2 (Dof2b x d)) ([x] [d] of/pi2 (Dof2a x d)) Dof1) %. +%scope sr %term C&2! sr red/pi2/letb (of/pi2 (of/letb Dof2 Dof1)) (of/letb ([x] [d] of/pi2 (Dof2 x d)) Dof1) %. +%term C&20 sr red/pi2/any (of/pi2 (of/any Dof)) (of/any Dof) %. +%scope sr %term C** sr red/lett/lett (of/lett Dlin3' Dlin3 Dof3 (of/lett Dlin2' Dlin2 Dof2 Dof1)) (of/lett ([x] linear/lett1 (Dlin2' x)) ([x] linear/lett1 (Dlin2 x)) ([x1] [d1] [x2] [d2] of/lett Dlin3' Dlin3 Dof3 (Dof2 x1 d1 x2 d2)) Dof1) %. +%term C*+ sr red/lett/case (of/lett Dlin3' Dlin3 Dof3 (of/case Dlin2b Dlin2a Dof2b Dof2a Dof1)) (of/case (linear/lett1 Dlin2b) (linear/lett1 Dlin2a) ([x] [d] of/lett Dlin3' Dlin3 Dof3 (Dof2b x d)) ([x] [d] of/lett Dlin3' Dlin3 Dof3 (Dof2a x d)) Dof1) %. +%scope sr %term C*! sr red/lett/letb (of/lett Dlin3' Dlin3 Dof3 (of/letb Dof2 Dof1)) (of/letb ([x] [d] of/lett Dlin3' Dlin3 Dof3 (Dof2 x d)) Dof1) %. +%term C*0 sr red/lett/any (of/lett _ _ _ (of/any Dof)) (of/any Dof) %. +%scope sr %term C+* sr red/case/lett (of/case Dlin3b Dlin3a Dof3b Dof3a (of/lett Dlin2' Dlin2 Dof2 Dof1)) (of/lett ([x] linear/case1 (Dlin2' x)) ([y] linear/case1 (Dlin2 y)) ([x1] [d1] [x2] [d2] of/case Dlin3b Dlin3a Dof3b Dof3a (Dof2 x1 d1 x2 d2)) Dof1) %. +%term C++ sr red/case/case (of/case Dlin3b Dlin3a Dof3b Dof3a (of/case Dlin2b Dlin2a Dof2b Dof2a Dof1)) (of/case (linear/case1 Dlin2b) (linear/case1 Dlin2a) ([x] [d] of/case Dlin3b Dlin3a Dof3b Dof3a (Dof2b x d)) ([x] [d] of/case Dlin3b Dlin3a Dof3b Dof3a (Dof2a x d)) Dof1) %. +%scope sr %term C+! sr red/case/letb (of/case Dlin3b Dlin3a Dof3b Dof3a (of/letb Dof2 Dof1)) (of/letb ([x] [d] of/case Dlin3b Dlin3a Dof3b Dof3a (Dof2 x d)) Dof1) %. +%term C+0 sr red/case/any (of/case _ _ _ _ (of/any Dof)) (of/any Dof) %. +%scope sr %term C!* sr red/letb/lett (of/letb Dof3 (of/lett Dlin2' Dlin2 Dof2 Dof1)) (of/lett ([x] linear/letb1 (Dlin2' x)) ([x] linear/letb1 (Dlin2 x)) ([x1] [d1] [x2] [d2] of/letb Dof3 (Dof2 x1 d1 x2 d2)) Dof1) %. +%term C!+ sr red/letb/case (of/letb Dof3 (of/case Dlin2b Dlin2a Dof2b Dof2a Dof1)) (of/case (linear/letb1 Dlin2b) (linear/letb1 Dlin2a) ([x] [d] of/letb Dof3 (Dof2b x d)) ([x] [d] of/letb Dof3 (Dof2a x d)) Dof1) %. +%scope sr %term C!! sr red/letb/letb (of/letb Dof3 (of/letb Dof2 Dof1)) (of/letb ([x] [d] of/letb Dof3 (Dof2 x d)) Dof1) %. +%term C!0 sr red/letb/any (of/letb _ (of/any Dof)) (of/any Dof) %. +%scope sr %term C0* sr red/any/lett (of/any (of/lett Dlin2' Dlin2 Dof2 Dof1)) (of/lett ([x] linear/any1 (Dlin2' x)) ([x] linear/any1 (Dlin2 x)) ([x1] [d1] [x2] [d2] of/any (Dof2 x1 d1 x2 d2)) Dof1) %. +%term C0+ sr red/any/case (of/any (of/case Dlin2b Dlin2a Dof2b Dof2a Dof1)) (of/case (linear/any1 Dlin2b) (linear/any1 Dlin2a) ([x] [d] of/any (Dof2b x d)) ([x] [d] of/any (Dof2a x d)) Dof1) %. +%scope sr %term C0! sr red/any/letb (of/any (of/letb Dof2 Dof1)) (of/letb ([x] [d] of/any (Dof2 x d)) Dof1) %. +%term C00 sr red/any/any (of/any (of/any Dof)) (of/any Dof) %. +%worlds (bind) (srl _ _ _ _) (sr _ _ _) %. +%total (D1 D2) (srl D1 _ _ _) (sr D2 _ _) %. +%{! ``> +Show complete code. +`` + +`` check="true"> +Show Twelf output. +`` + +\{\{case study\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Linear_logic). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/lists.lf b/new-tests/stelf-output/pages/lists.lf new file mode 100644 index 0000000..d8d5ea4 --- /dev/null +++ b/new-tests/stelf-output/pages/lists.lf @@ -0,0 +1,235 @@ +% +%{! This page describes the data structure of lists as well as a large number of operations and theorems about them. + +## Definition + +This standard definition of lists in Twelf . We first define what the list elements are: in this example they are just an arbitrary type ``elem``. This example is parametric in the type of the elements, so you can change this definition to anything you like. + + !}% +%{! (options removed from twelftag: hidden="true") !}% +% Lists +% By Carsten Varming 2006 +%{! !}% +%sort elem %. +%term a elem %. +%term b elem %. +%term c %pi elem %-> elem %. +%{! The important defintion is the one of lists. There are two ways of constructing a Twelf object of type ``list``. The first is the constructor for the empty list ``nil``. The second is the constructor ``cons`` that given an object of type ``elem`` and an object of type ``list`` creates a list with elem as head and list as tail. + + !}% +%sort list %. +%name list %. +%term nil list %. +%term cons %pi elem %-> list %-> list %. +%freeze list %. +%{! ## Equalities + +Equality is defined as [identity](/wiki/equality/), which makes the symmetry and transitivity proofs entirely trivial. One [congruence lemma](/wiki/respects-lemma/) needs to be defined ``eq-cong-cons``, expressing that if the tails of two lists are equal, and the heads of the lists are identical, the lists are equal. !}% +%sort eq %. +%mode eq %in %out %. +%term eq/ref eq Ls Ls %. +%worlds () (eq _ _) %. +%freeze eq %. +%sort eq-symm %. +%mode eq-symm %in %out %. +%term _ eq-symm eq/ref eq/ref %. +%worlds () (eq-symm _ _) %. +%freeze eq-symm %. +%total {} (eq-symm _ _) %. +%sort eq-trans %. +%mode eq-trans %in %in %out %. +%term _ eq-trans eq/ref eq/ref eq/ref %. +%worlds () (eq-trans _ _ _) %. +%freeze eq-trans %. +%total {} (eq-trans _ _ _) %. +%sort eq-cong-cons {E} %. +%mode eq-cong-cons %in %in %out %. +%term _ eq-cong-cons _ eq/ref eq/ref %. +%worlds () (eq-cong-cons _ _ _) %. +%total {} (eq-cong-cons _ _ _) %. +%{! ## Reversal +There are at least two very common operations on lists: Reverse and Append. Lists form a semigroup with the opeation append and the empty list as right and left unit. If we define an operation then reverse is an isomorphism between the semigroups (lists,,nil) and (lists,f,nil). +### Definition +I will use the standard logic programming way of encoding reversal. In the following ``rev`` will be a ternary relation where the first two elements are input and the third is output. The invariant is , where is concatenation, thus is obtained by ``rev a nil b``. We will later see that can be thought of as . !}% +%sort rev %. +%mode rev %in %in %out %. +%term rev/nil rev nil Ls' Ls' %. +%term rev/cons %pi (rev (cons E Ls) Ls'' Ls') %<- (rev Ls (cons E Ls'') Ls') %. +%worlds () (rev _ _ _) %. +%freeze rev %. +%total D (rev D _ _) %. +%sort can-rev {Ls} {Ls'} %. +%mode can-rev %in %in %out %. +%term _ can-rev nil _ rev/nil %. +%term _ %pi (can-rev (cons E Ls) Ls' (rev/cons Ls'')) %<- (can-rev Ls (cons E Ls') Ls'') %. +%worlds () (can-rev _ _ _) %. +%freeze can-rev %. +%total D (can-rev D _ _) %. +%{! ### Equality !}% +%sort rev-eq %. +%mode rev-eq %in %in %in %in %out %. +%term _ rev-eq eq/ref eq/ref eq/ref R R %. +%worlds () (rev-eq _ _ _ _ _) %. +%total {} (rev-eq _ _ _ _ _) %. +%{! ### Determinism +As the Twelf definition only states that rev is a relation we need to prove that it indeed is a function. Namely . !}% +%sort rev-determ %. +%mode rev-determ %in %in %out %. +%term _ rev-determ rev/nil _ eq/ref %. +%term _ %pi (rev-determ (rev/cons R) (rev/cons R') Q) %<- (rev-determ R R' Q) %. +%worlds () (rev-determ _ _ _) %. +%freeze rev-determ %. +%total D (rev-determ D _ _) %. +%{! ### Double reversal +Reverse is an isomorphism that turn out to be its own inverse. +. !}% +%sort revrev-id-lem %. +%mode revrev-id-lem %in %in %in %out %. +%term _ %pi (revrev-id-lem rev/nil F F' Q) %<- (rev-determ F' F Q) %. +%term _ + %pi (revrev-id-lem (rev/cons R) R' R'' Q) + %<- (revrev-id-lem R R' (rev/cons R'') Q) %. +%worlds () (revrev-id-lem _ _ _ _) %. +%freeze revrev-id-lem %. +%total D (revrev-id-lem D _ _ _) %. +%sort revrev-id %. +%mode revrev-id %in %out %. +%term _ + %pi (revrev-id R R') + %<- (can-rev _ nil R1) + %<- (revrev-id-lem R R1 rev/nil E) + %<- (eq-symm E E') + %<- (rev-eq eq/ref eq/ref E' R1 R') %. +%worlds () (revrev-id _ _) %. +%freeze revrev-id %. +%total {} (revrev-id _ _) %. +%{! ### Injectivity +As reverse has an inverse it is injective . !}% +%sort rev-injective %. +%mode rev-injective %in %in %out %. +%term _ + %pi (rev-injective (%the (rev Ls nil Ls') R) R' Q) + %<- (revrev-id R R1) + %<- (revrev-id R' R1') + %<- (rev-determ R1 R1' Q) %. +%worlds () (rev-injective _ _ _) %. +%freeze rev-injective %. +%total D (rev-injective D _ _) %. +%{! ## Append + +### Definition !}% +%sort append %. +%name append %. +%term append/nil append nil L L %. +%term append/cons %pi (append (cons E L1) L2 (cons E L3)) %<- (append L1 L2 L3) %. +%freeze append %. +%sort can-append {L1} {L2} %. +%mode can-append %in %in %out %. +%term _ can-append nil L append/nil %. +%term _ %pi (can-append (cons E L1) L2 (append/cons A)) %<- (can-append L1 L2 A) %. +%worlds () (can-append _ _ _) %. +%total D (can-append D _ _) %. +%{! ### Equality !}% +%sort append-eq %. +%mode append-eq %in %in %in %in %out %. +%term _ append-eq eq/ref eq/ref eq/ref A A %. +%worlds () (append-eq _ _ _ _ _) %. +%total {} (append-eq _ _ _ _ _) %. +%{! ### Units +We need to prove that ``nil`` is the right and left unit of the semigroup . The later is by axiom and the former is by this lemma. !}% +%sort append-eq-nil {L1} %. +%mode append-eq-nil %in %out %. +%term _ append-eq-nil nil append/nil %. +%term _ %pi (append-eq-nil (cons E L) (append/cons A)) %<- (append-eq-nil L A) %. +%worlds () (append-eq-nil _ _) %. +%total D (append-eq-nil D _) %. +%{! Sometimes it is helpful to reason with equalities like . !}% +%sort append-eq-nil %. +%mode append-eq-nil %in %out %. +%term _ append-eq-nil append/nil eq/ref %. +%term _ + %pi (append-eq-nil (append/cons A) E) + %<- (append-eq-nil A E1) + %<- (eq-cong-cons _ E1 E) %. +%worlds () (append-eq-nil _ _) %. +%total D (append-eq-nil D _) %. +%{! ### Determinism +As with reverse we need to show that append is indeed a function . !}% +%sort append-determ %. +%mode append-determ %in %in %out %. +%term _ append-determ append/nil append/nil eq/ref %. +%term _ + %pi (append-determ (append/cons A1) (append/cons A2) E) + %<- (append-determ A1 A2 E2) + %<- (eq-cong-cons _ E2 E) %. +%worlds () (append-determ _ _ _) %. +%total D (append-determ D _ _) %. +%{! ### Associativity +We need to prove that append is associative . !}% +%sort append-assoc1 %. +%mode append-assoc1 %in %in %out %out %. +%term _ append-assoc1 append/nil A1 A1 append/nil %. +%term _ + %pi (append-assoc1 (append/cons A1) (append/cons A2) A3 (append/cons A4)) + %<- (append-assoc1 A1 A2 A3 A4) %. +%worlds () (append-assoc1 _ _ _ _) %. +%total D (append-assoc1 D _ _ _) %. +%sort append-assoc2 %. +%mode append-assoc2 %in %in %in %in %out %. +%term _ + %pi (append-assoc2 A1 A2 A3 A4 E) + %<- (append-assoc1 A1 A2 A3' A4') + %<- (append-determ A3 A3' E3) + %<- (append-eq eq/ref E3 eq/ref A4 A4'') + %<- (append-determ A4' A4'' E) %. +%worlds () (append-assoc2 _ _ _ _ _) %. +%total {} (append-assoc2 _ _ _ _ _) %. +%{! ## Append and reversal +I promissed to prove that ``rev`` is an isomorphism. Lets start by a few lemmas. +First . !}% +%sort append-rev1 %. +%mode append-rev1 %in %in %in %out %. +%term _ append-rev1 append/nil rev/nil R R %. +%term _ + %pi (append-rev1 (append/cons A) (rev/cons R1) R3 (rev/cons R2)) + %<- (append-rev1 A R1 R3 R2) %. +%worlds () (append-rev1 _ _ _ _) %. +%total D (append-rev1 D _ _ _) %. +%{! And second . If we let then we get , which shows how we can think of as . !}% +%sort append-rev2 %. +%mode append-rev2 %in %in %in %out %. +%term _ + %pi (append-rev2 rev/nil A1 A2 R) + %<- (append-determ A1 A2 E) + %<- (rev-eq eq/ref eq/ref E rev/nil R) %. +%term _ + %pi (append-rev2 (rev/cons R) A1 A2 (rev/cons R2)) + %<- (append-rev2 R (append/cons A1) A2 R2) %. +%worlds () (append-rev2 _ _ _ _) %. +%total D (append-rev2 D _ _ _) %. +%{! And here we go. which boils down to +, whence reverse is a homeomorphism. This combined with the fact that reverse is its own inverse we get that reverse is an isomorphism. !}% +%sort append-rev %. +%mode append-rev %in %in %in %in %in %in %out %. +%term _ + %pi (append-rev R1 (rev/cons R2) A1 R3 R4 A2 R5) + %<- (append-rev R1 R2 A1 R3 (rev/cons R4) A2 R5) %. +%term _ + %pi (append-rev (rev/cons R1) rev/nil A1 R2 R3 A2 R5) + %<- (append-rev R1 rev/nil A1 (rev/cons R2) R3 A2 R5) %. +%term _ + %pi (append-rev rev/nil rev/nil A1 R3 R4 A2 R5) + %<- (append-rev2 R3 append/nil A2 RC) + %<- (append-rev1 A1 R4 RC RB) + %<- (revrev-id RB R5) %. +%worlds () (append-rev _ _ _ _ _ _ _) %. +%total [D E] (append-rev E D _ _ _ _ _) %. +%{! ``>See the Twelf code for this example`` - `` check=true>See Twelf's output`` + +\{\{case study\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Lists). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/logic-programming.lf b/new-tests/stelf-output/pages/logic-programming.lf new file mode 100644 index 0000000..48775ce --- /dev/null +++ b/new-tests/stelf-output/pages/logic-programming.lf @@ -0,0 +1,26 @@ +% +%{! Most of the articles and examples on this wiki are focused towards using Twelf as a specification language, useful for encoding an [object logic](/wiki/object-logic/) and then stating and proving [metatheorems](/wiki/metatheorem/) about that object logic. However, Twelf originated from [[Elf]], and like Elf it is a dependently-typed, higher-order **logic programming language** based on the logic [LF](/wiki/lf/). + +A Twelf [signature](/wiki/twelf-signature/) may contain, among other things, some **type family** declarations and some **constants** inhabiting those type families. A logic programming language takes a _type_ as an input, and it then uses a simple search strategy to try and discover a term that has that type using the constants that have been defined. By the [judgments as types](/wiki/judgment/) principle, this means that Twelf is searching for a derivation - a **proof witness** - of a particular judgment. + +Twelf's logic programming engine is activated by [``%solve``](/wiki/percent-solve/) and [``%query``](/wiki/percent-query/) directives. It can also be activated directly inside ML; see the section on \{\{guide|title=interactive queries|chapter=5|section=22\}\} for more information. Twelf's [tabled logic programming](/wiki/tabled-logic-programming/) engine is activated by the [``%querytabled``](/wiki/percent-querytabled/) directive. + +## Uses of logic programming in Twelf + +* **Implementing a reference typechecker/evaluator** - If the static and dynamic semantics of a programming language are written correctly, then the specification of the static semantics can be . This was a component to the [[poplmark:The POPLmark Challenge|The POPLmark Challenge]] challenge; the case study of a [language with references](/wiki/mutable-state/), among other examples, demonstarte Twelf in this capacity. +* Writing programs and then proving things about them: The TALT project <ref>\{\{bibtex:crary03talt\}\}</ref> includes a code checker written as a Twelf logic program, along with a proof in Twelf that code that passes the checker is safe according to a safety policy. +* [Tactical theorem proving](/wiki/tactical-theorem-proving/) - in the style described by Appel and Felty in <ref>\{\{bibtex:af04dependent\}\}</ref>. + +## See also +* \{\{guide|title=Logic Programming|chapter=5|section=22\}\} +* [[w:Logic programming|Logic programming]] at Wikipedia + +## References + +<references /> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Logic_programming). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/mailing-lists.lf b/new-tests/stelf-output/pages/mailing-lists.lf new file mode 100644 index 0000000..48768d2 --- /dev/null +++ b/new-tests/stelf-output/pages/mailing-lists.lf @@ -0,0 +1,12 @@ +% +%{! We maintain two mailing lists for the Twelf project through mailman. The moderated mailing list which is used for announcments related to Twelf is called **twelf-list**. The other is called **twelf-developer**. It serves as discussion forum for people who are working on the Twelf implementation. + +If you want to join twelf mailing list visit [http://www.itu.dk/mailman/listinfo/twelf-list http://www.itu.dk/mailman/listinfo/twelf-list]. + +If you want to join twelf-developer mailing list visit [http://www.itu.dk/mailman/listinfo/twelf-developer http://www.itu.dk/mailman/listinfo/twelf-developer]. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Mailing_lists). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/manipulating-proof-witnesses-as-inputs.lf b/new-tests/stelf-output/pages/manipulating-proof-witnesses-as-inputs.lf new file mode 100644 index 0000000..0b05c27 --- /dev/null +++ b/new-tests/stelf-output/pages/manipulating-proof-witnesses-as-inputs.lf @@ -0,0 +1,141 @@ +% +%{! Christopher asks... + +
+There are things which I can express as a sequence of two "%solve" +commands that I can't express in one: +e.g. + +```twelf +%solve A : foo. +%solve B : bar A. +``` + +I'd like to be able to write a type quux which combines the two +searches, like: + +```twelf +quux : { A : foo } bar A -> type. +%mode quux -A -B. +quux_con : { A : foo } { B : foo A } quux A B +% or maybe if there was syntax like +% quux_con2 : quux A B <- A : foo <- B : foo A +``` +
+ +## Response + +I believe what you want to do is doable - you were on the right track with introducing A as an [explicit parameter](/wiki/implicit-and-explicit-parameters/) here. However, I would caution you against what you're suggesting, because - as far as I know, what you're asking requires essentially duplicating your entire program as [effectiveness lemmas](/wiki/effectiveness-lemma/). + +I had trouble working out what the meaning of the example was based on abstract examples, so included the natural numbers with plus. We also encode the metatheorem for the commutivity of plus, but we're not interested in the fact that it is a metatheorem here - ``plus`` here is our analog for ``foo`` above, and so ``plus-comm`` is our ``bar``. !}% +%{!! begin hidden !!}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort plus %. +%mode plus %in %in %out %. +%term plus/z plus z N N %. +%term plus/s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%sort plus-zero {N} %. +%mode plus-zero %in %out %. +%term plus-zero/z plus-zero z plus/z %. +%term plus-zero/s %pi (plus-zero (s N) (plus/s D)) %<- (plus-zero N D) %. +%sort plus-succ %. +%mode plus-succ %in %out %. +%term plus-succ/z plus-succ plus/z plus/z %. +%term plus-succ/s %pi (plus-succ (plus/s D1) (plus/s D2)) %<- (plus-succ D1 D2) %. +%{!! end hidden !!}% +%sort plus-comm %. +%mode plus-comm %in %out %. +%term plus-comm/z %pi (plus-comm plus/z (%the (plus N z N) D)) %<- (plus-zero N D) %. +%term plus-comm/s %pi (plus-comm (plus/s D1) D2) %<- (plus-comm D1 D3) %<- (plus-succ D3 D2) %. +%{! +Now, in our example, we have a new version of the first code example: + +```checkedtwelf +%solve derivA : plus (s (s z)) (s z) _. +%solve derivB : plus-comm derivA C. +``` + +As far as I can tell, the equivalent of your relation above in my example is one that takes two natural numbers ``A`` and ``B`` and produces + +* The natural number ``C`` that is the sum of ``A`` and ``B`` +* A derivation ``D`` of ``plus A B C`` +* A derivation of the commuted sum ``D'``, i.e. ``plus B A C`` +* A derivation of ``plus-comm D D'`` of that + +This is realized in the following relation: + +```twelf +twoplus : {A}{B}{C}{D: plus A B C}{D': plus B A C} plus-comm D D' -> type. +%mode twoplus +A +B -C -D -D' -D''. +``` + +### Lots of effectiveness lemmas + +In order to get what we want, we are going to have to encode a bunch of [effectiveness lemmas](/wiki/effectiveness-lemma/)—not having written them those may have been what was giving you trouble before. In your example you seem to indicate that the relations you were working with were not necessarily total relations from inputs to outputs - if that is the case, then you will define what could more accurately be called _effectiveness relations_. What effectiveness lemmas/relations do is express _how to build a proof witness given certain inputs_ - in some sense making the proof search process explicit rather than letting ``%solve`` handle it. + +Our effectivness lemma for ``plus`` looks like this. Pay careful attention to what is an input and what is an output. !}% +%sort can-plus {A} {B} {C} %. +%mode can-plus %in %in %out %out %. +%term can-plus/z can-plus z B B plus/z %. +%term can-plus/s %pi (can-plus (s A) B (s C) (plus/s D)) %<- (can-plus A B C D) %. +%worlds () (can-plus _ _ _ _) %. +%total T (can-plus T _ _ _) %. +%{! The totality checks on these and the other effectiveness lemmas are just a sanity check here - this technique would work even if you were working with relations which were not total. As before, I will skip over the effectivenes lemmas for ``plus-zero`` and ``plus-succ`` and skip to the proof for !}% +%{!! begin hidden !!}% +%sort can-plus-zero {N} {D plus N z N} %. +%mode can-plus-zero %in %out %out %. +%term can-plus-zero/z can-plus-zero z plus/z plus-zero/z %. +%term can-plus-zero/s %pi (can-plus-zero (s N) (plus/s D) (plus-zero/s DZ)) %<- (can-plus-zero N D DZ) %. +%worlds () (can-plus-zero _ _ _) %. +%total T (can-plus-zero T _ _) %. +%sort can-plus-succ {D plus A B C} {D' plus A (s B) (s C)} %. +%mode can-plus-succ %in %out %out %. +%term can-plus-succ/z can-plus-succ plus/z plus/z plus-succ/z %. +%term can-plus-succ/s + %pi (can-plus-succ (plus/s D) (plus/s D') (plus-succ/s DS)) + %<- (can-plus-succ D D' DS) %. +%worlds () (can-plus-succ _ _ _) %. +%total T (can-plus-succ T _ _) %. +%{!! end hidden !!}% +%sort can-plus-comm {D plus A B C} {D' plus B A C} %. +%mode can-plus-comm %in %out %out %. +%term can-plus-comm/z + %pi (can-plus-comm plus/z (%the (plus N z N) D) (plus-comm/z DZ)) + %<- (can-plus-zero N D (%the (plus-zero N D) DZ)) %. +%term can-plus-succ/s + %pi (can-plus-comm (plus/s D) D'' (plus-comm/s DS DC)) + %<- (can-plus-comm D D' DC) + %<- (can-plus-succ D' D'' DS) %. +%worlds () (can-plus-comm _ _ _) %. +%total T (can-plus-comm T _ _) %. +%{! + +### Putting together the ``twoplus`` relation + +Now that we have all the effectiveness lemmas we could ever need or want, it is relatively simple to define twoplus. Furthermore, because we established totality of ``plus`` and ``plus-comm``, it is simple to establish totality of ``twoplus``. !}% +%sort twoplus {A} {B} {C} {D plus A B C} {D' plus B A C} %. +%mode twoplus %in %in %out %out %out %out %. +%term twoplus/i %pi (twoplus A B C D D' DC) %<- (can-plus A B C D) %<- (can-plus-comm D D' DC) %. +%worlds () (twoplus _ _ _ _ _ _) %. +%total {} (twoplus _ _ _ _ _ _) %. +%{! + +## Side note about ``%define`` + +As a side note, look at Twelf's output when we solved ``%solve B : plus-comm A C.`` It didn't allow us to see the type of ``C``, and it didn't allow us to use ``C`` later in the program. We can bind outputs of functions and use them later in the signature by using [%define](/wiki/define-declaration/), as the following example shows: + +```checkedtwelf +%solve deriv1 : plus (s (s z)) (s z) _. +%define deriv2 = C +%solve deriv3 : plus-comm deriv1 C. + +test = plus/s deriv2. +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Manipulating_proof_witnesses_as_inputs). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/mediawiki-copyrightwarning.lf b/new-tests/stelf-output/pages/mediawiki-copyrightwarning.lf new file mode 100644 index 0000000..3c90ddb --- /dev/null +++ b/new-tests/stelf-output/pages/mediawiki-copyrightwarning.lf @@ -0,0 +1,9 @@ +% +%{! By saving, you assert that you have the right and intention to license your submission to us under the $2 (see $1 for details). +<strong>Do not submit copyrighted work without permission!</strong> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/MediaWiki:Copyrightwarning). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/mediawiki-copyrightwarning2.lf b/new-tests/stelf-output/pages/mediawiki-copyrightwarning2.lf new file mode 100644 index 0000000..01d4f82 --- /dev/null +++ b/new-tests/stelf-output/pages/mediawiki-copyrightwarning2.lf @@ -0,0 +1,11 @@ +% +%{! By saving, you assert that you have the right and intention to license your submission to us under the $2 (see $1 for details). +<strong>Do not submit copyrighted work without permission!</strong> + +<ajalf>unused</ajalf><br/> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/MediaWiki:Copyrightwarning2). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/mediawiki-edittools.lf b/new-tests/stelf-output/pages/mediawiki-edittools.lf new file mode 100644 index 0000000..aecc5a8 --- /dev/null +++ b/new-tests/stelf-output/pages/mediawiki-edittools.lf @@ -0,0 +1,22 @@ +% +%{! <!-- Text here will be shown below edit and upload forms. --> +<!-- nb. the charinsert extension doesn't work, so these have to be copy and pasted manually. we should probably install charinsert. --> +<div id="editpage-specialchars" class="plainlinks" style="margin-top: 15px; border-width: 1px; border-style: solid; border-color: #aaaaaa; padding: 2px;"> +**Logic:** <charinsert>← → λ Π Σ α β γ Γ Δ η θ σ Ψ Ω ≡ ¬ ∧ ∨ ∩ ∪ ⊃ ⊂</charinsert> + +**Workarounds:** \{<nowiki>\{vdash\}</nowiki>\} \{<nowiki>\{dia\}</nowiki>\} \{<nowiki>\{box\}</nowiki>\} \{<nowiki>\{lolli\}</nowiki>\} \{<nowiki>\{otimes\}</nowiki>\} \{<nowiki>\{top\}</nowiki>\} \{<nowiki>\{darrow\}</nowiki>\} \{<nowiki>\{forall\}</nowiki>\} \{<nowiki>\{exists\}</nowiki>\} \{<nowiki>\{oplus\}</nowiki>\} \{<nowiki>\{bottom\}</nowiki>\} + +**Symbols:** <charinsert>~ | ¡ ¿ † ‡ ↔ ↑ ↓ •</charinsert>   <charinsert>– — … ° ≈ ± − × ÷ ← → · §</charinsert>   <charinsert># ¹ ² ³ ½ ⅓ ⅔ ¼ ¾ ⅛ ⅜ ⅝ ⅞</charinsert>   <charinsert>¢ $ € £ ¥</charinsert> + +**Letters:** <span class="latinx"> Á á Ć ć É é Í í Ĺ ĺ Ń ń Ó ó Ŕ ŕ Ś ś Ú ú Ý ý Ź ź   À à È è Ì ì Ò ò Ù ù    â Ĉ ĉ Ê ê Ĝ ĝ Ĥ ĥ Î î Ĵ ĵ Ô ô Ŝ ŝ Û û Ŵ ŵ Ŷ ŷ   Ä ä Ë ë Ï ï Ö ö Ü ü Ÿ ÿ   ß   à ã Ẽ ẽ Ĩ ĩ Ñ ñ Õ õ Ũ ũ Ỹ ỹ   Ç ç Ģ ģ Ķ ķ Ļ ļ Ņ ņ Ŗ ŗ Ş ş Ţ ţ   Đ đ   Ů ů   Ǎ ǎ Č č Ď ď Ě ě Ǐ ǐ Ľ ľ Ň ň Ǒ ǒ Ř ř Š š Ť ť Ǔ ǔ Ž ž   Ā ā Ē ē Ī ī Ō ō Ū ū   ǖ ǘ ǚ ǜ   Ă ă Ĕ ĕ Ğ ğ Ĭ ĭ Ŏ ŏ Ŭ ŭ   Ċ ċ Ė ė Ġ ġ İ ı Ż ż   Ł ł   Ő ő Ű ű   Ŀ ŀ   Ħ ħ   Ð ð Þ þ   Œ œ   Æ æ Ø ø Å å   Ə ə </span> + +**Greek:** <charinsert>Α α Β β Γ γ Δ δ</charinsert>   <charinsert>Ε ε Ζ ζ Η η Θ θ</charinsert>   <charinsert>Ι ι Κ κ Λ λ Μ μ</charinsert>   <charinsert>Ν ν Ξ ξ Ο ο Π π</charinsert>   <charinsert>Ρ ρ Σ σ ς Τ τ Υ υ</charinsert>   <charinsert>Φ φ Χ χ Ψ ψ Ω ω</charinsert> +</div> + +On **talk pages**, please sign your comment by typing four tildes (<nowiki>~~~~</nowiki>). !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/MediaWiki:Edittools). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/mediawiki-fileinfo.lf b/new-tests/stelf-output/pages/mediawiki-fileinfo.lf new file mode 100644 index 0000000..baee6cb --- /dev/null +++ b/new-tests/stelf-output/pages/mediawiki-fileinfo.lf @@ -0,0 +1,8 @@ +% +%{! $1KB, MIME type: ``$2`` !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/MediaWiki:Fileinfo). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/mediawiki-formatting-description.lf b/new-tests/stelf-output/pages/mediawiki-formatting-description.lf new file mode 100644 index 0000000..be66d64 --- /dev/null +++ b/new-tests/stelf-output/pages/mediawiki-formatting-description.lf @@ -0,0 +1,10 @@ +% +%{! Guide for writing in mediawiki syntax will be posted here + +[http://www.mediawiki.com Official Mediawiki Site] !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/MediaWiki_formatting_description). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/mediawiki-loginreqpagetext.lf b/new-tests/stelf-output/pages/mediawiki-loginreqpagetext.lf new file mode 100644 index 0000000..40926c5 --- /dev/null +++ b/new-tests/stelf-output/pages/mediawiki-loginreqpagetext.lf @@ -0,0 +1,12 @@ +% +%{! You are most likely seeing this message because you are trying to access a page starting with "Elves:". These pages are restricted to users that have the permission "party-elf." + +If you are not logged you may need to $1. + +If you are logged in, let an [[Project:Administrators|administrator]], preferably one with "Bureaucrat" status, know that you need access to pages in this namespace. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/MediaWiki:Loginreqpagetext). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/mediawiki-monobook-css.lf b/new-tests/stelf-output/pages/mediawiki-monobook-css.lf new file mode 100644 index 0000000..d76ce3d --- /dev/null +++ b/new-tests/stelf-output/pages/mediawiki-monobook-css.lf @@ -0,0 +1,16 @@ +% +%{! /* edit this file to customize the monobook skin for the entire site */ + + #p-logo a \{ background: url(/images/mediumelf.png) center center no-repeat !important; \} + #p-logo a:hover \{ background: url(/images/mediumelf.png) center center no-repeat !important; \} + body \{background: #f9f9f9 url(/images/raster.gif) 0 0 repeat-x; \} + + #n-Documentation \{ + margin: 0em 0em .5em 0em; + \} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/MediaWiki:Monobook.css). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/mediawiki-newwindow.lf b/new-tests/stelf-output/pages/mediawiki-newwindow.lf new file mode 100644 index 0000000..eb1c7d0 --- /dev/null +++ b/new-tests/stelf-output/pages/mediawiki-newwindow.lf @@ -0,0 +1,8 @@ +% +%{! (opens in new window) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/MediaWiki:Newwindow). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/mediawiki-previewnote.lf b/new-tests/stelf-output/pages/mediawiki-previewnote.lf new file mode 100644 index 0000000..46743dd --- /dev/null +++ b/new-tests/stelf-output/pages/mediawiki-previewnote.lf @@ -0,0 +1,8 @@ +% +%{! <strong>This is only a preview; changes have not yet been saved!</strong> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/MediaWiki:Previewnote). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/mediawiki-privacy.lf b/new-tests/stelf-output/pages/mediawiki-privacy.lf new file mode 100644 index 0000000..d98228f --- /dev/null +++ b/new-tests/stelf-output/pages/mediawiki-privacy.lf @@ -0,0 +1,8 @@ +% +%{! Download Twelf !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/MediaWiki:Privacy). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/mediawiki-privacypage.lf b/new-tests/stelf-output/pages/mediawiki-privacypage.lf new file mode 100644 index 0000000..9f53990 --- /dev/null +++ b/new-tests/stelf-output/pages/mediawiki-privacypage.lf @@ -0,0 +1,8 @@ +% +%{! Download !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/MediaWiki:Privacypage). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/mediawiki-sidebar.lf b/new-tests/stelf-output/pages/mediawiki-sidebar.lf new file mode 100644 index 0000000..370f793 --- /dev/null +++ b/new-tests/stelf-output/pages/mediawiki-sidebar.lf @@ -0,0 +1,26 @@ +% +%{! * the twelf wiki +** mainpage|mainpage +** About The Twelf Project|The Twelf Project +** Download|Download Twelf +** Documentation|Documentation +<!-- The space here is controlled by Mediawiki:Monobook.css --> +** recentchanges-url|recentchanges +** Project:Contributing|Contributing + +* learn twelf +** Introductions to Twelf|Introductions +** Tutorials|Tutorials +** Case studies|Case studies +** Glossary|Twelf glossary +** Ask Twelf Elf|Ask Twelf Elf + +* reference +** Bibliography of LF|LF bibliography +** Research projects using Twelf|Research with Twelf !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/MediaWiki:Sidebar). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/mediawiki-talk-sidebar.lf b/new-tests/stelf-output/pages/mediawiki-talk-sidebar.lf new file mode 100644 index 0000000..4e03c80 --- /dev/null +++ b/new-tests/stelf-output/pages/mediawiki-talk-sidebar.lf @@ -0,0 +1,11 @@ +% +%{! Here's my attempt at rationalizing the organization of the sidebar: +* Everything on the wiki should be reachable from the links in the top navigation section. Right now, this amounts to saying that [documentation](/wiki/documentation/) indexes mostly everything. +* the remaining space above the search box should be used for quick links to things we think people will go to often (right now, the main documentation sections). +[Drl](/wiki/user-drl/) 11:43, 15 March 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/MediaWiki_talk:Sidebar). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/meta-logic.lf b/new-tests/stelf-output/pages/meta-logic.lf new file mode 100644 index 0000000..8df88b7 --- /dev/null +++ b/new-tests/stelf-output/pages/meta-logic.lf @@ -0,0 +1,14 @@ +% +%{! A **meta-logic** is a logic for reasoning about another logic. When working with Twelf, there are two common usages for the term "meta-logic." + +The first meaning of this term is "a logic used to encode a [object logic](/wiki/object-logic/)." When using Twelf, [LF](/wiki/lf/) is the meta-logic used to create an [adequate](/wiki/adequacy/) encoding of an object logic. When used this way, a meta-logic may also be called a **meta-language.** + +The second meaning of the term is that a meta-logic is a logic in which we state and prove [metatheorem](/wiki/metatheorem/)s. In this case Twelf's facility used to state and verify [totality assertion](/wiki/totality-assertion/)s is the meta-logic used to reason about derivations in object logics. + +\{\{stub\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Meta-logic). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/metatheorem.lf b/new-tests/stelf-output/pages/metatheorem.lf new file mode 100644 index 0000000..f4973f7 --- /dev/null +++ b/new-tests/stelf-output/pages/metatheorem.lf @@ -0,0 +1,81 @@ +% +%{! A **metatheorem** is a theorem about an [object language](/wiki/object-logic/). This is a very general statement, but **metatheorem** is a very general term. Many interesting metatheorems can be posed as _∀∃-statements,_ and these are the kind of metatheorems that can be verified in Twelf. Another kind of metatheorem is a _[totality assertion](/wiki/totality-assertion/)_; these are more limited because any totality assertion can also be posed as a ∀∃-statement. + +Twelf can prove ∀∃-metatheorems in one of two ways. The first method, the [theorem prover](/wiki/theorem-prover/), is incomplete and not currently recommended for use. It allows the user to directly specify a ∀∃-statement about LF terms, and then ask Twelf to verify that statement. The other method is to write out a proof of the ∀∃ statement in Twelf, and then use Twelf's ability to state and verify [totality assertions](/wiki/totality-assertion/) to show that the proof is correct. + +## Object language + +The object language is "the object of study": a logic, programming language, or other deductive system that the user wishes to reason about with Twelf. Simple examples of object languages are [natural numbers](/wiki/natural-numbers/) and the [simply-typed lambda calculus](/wiki/simply-typed-lambda-calculus/), more complex examples can be found in the [case studies](/wiki/case-studies/) or [research projects using Twelf](/wiki/research-projects-using-twelf/). Object languages like the simply-typed lambda calculus and first-order logic demonstrate the power of Twelf's [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/), but the example of natural numbers with addition is used here for its simplicity. The syntax and judgments over the natural numbers can be simply presented in [[w: Backus-Naur form| Backus-Naur form]]: + + + +We then declare to be a judgment that relates three natural numbers. We can define what kind of judgments we are allowed to relate by using inference rules: + +           + +## Theorems + +The usual interpretation of inference rules like the ones above is that we can consider some fact _A_ to be true if we can create a complete derivation that has _A_ a result. Truths that can be verified by writing a derivation are called _theorems._ If the object language is the [natural numbers](/wiki/natural-numbers/), then the following is a theorem theorem proving that 2 + 1 = 3. + + + +Similarly, if the object language is the [simply-typed lambda calculus](/wiki/simply-typed-lambda-calculus/), then a theorem might state that the expression steps to . This can be shown by using the rules ``step_app_beta`` and ``value_empty``. + +## Totality assertions + +The term "metatheorem," as previously stated, is a very general term. We first want to think about a specific kind of metatheorem, a [totality assertion](/wiki/totality-assertion/). + +A totality assertion for the informal deductive system presented above might be that, for any two natural numbers and , there is a natural number for which is derivable. This means that the judgment forms a total relation, mapping any inputs in the first and second positions of the relation to some output in the third position. Note that the output need not be unique; the totality assertion holds even if there are many such numbers. + +The totality assertions in Twelf establish a stronger property. The Twelf [[encoding]] of the judgment can be run as a logic program, and totality assertions that are verified by [``%total``](/wiki/percent-total/) directives verify that, if an encoding is run as a higher-order logic program in Twelf, it will act as a function that, given appropriate inputs, will find an output in a finite amount of time. Twelf's totality analysis is incomplete in that some relations that are total cannot be verified by a ``%total`` declaration. + +Twelf, when used in this way, can be thought of as a "theorem prover" in the limited sense that it _proves_ that a relation is total by doing a program analysis that shows that, given appropriate inputs, it produces outputs when run as a logic program in Twelf. However, it is more common to describe Twelf as _verifying_ the totality assertions, both because Twelf does not produce a proof witness and because the analysis is not as sophisticated as most theorem proving procedures. Usually, the relations about which programmers intend to prove totality assertions are specifically constructed to be analyzable by Twelf's ``%total`` directive. These relations often correspond closely to an informal proof by structural induction of the same fact, making Twelf's process of verifying totality similar to the human process of verifying that a proof is valid. + +## ∀∃-statements + +Totality assertions seem very limited in scope. We have merely shown that, given a judgment like , we can interpret as the judgment as a relation from some inputs to some outputs and prove that the relation, given inputs, have rules that will always allow us to find outputs. But we can't, for instance, write a totality assertion on that will allow us to prove that the relation is commutative or associative. + +The theorem "addition is commutative" can be specified more precisely like this: for all natural numbers , , and derivation of the judgment , there exists a derivation of the judgment . The previous statement, at a high level, said "for all (some things) there exist (some other things." Statement with this forms are called ∀∃-statement. + +The page about Twelf's [theorem prover](/wiki/theorem-prover/) shows how the theorem prover could be used to state, and prove, this statement, but the currently recommended way of doing this is by using a totality assertion. The series of tutorials on [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/) explain in detail how to do this; the remainder of this article will only give a very general view based on the notes from [http://www.cs.cmu.edu/~fp/courses/lp/lectures/18-proofs.html Lecture 18] of Frank Pfenning's course on Logic Programming. + +### Simple types + +First, think back at our presentation of the structure of natural numbers: + + + +We will now think of this BNF grammar as defining the members of the _type_ . In this view is an object of type , and is a constructor that, given an object of type , produces an object of type . + +### Proof terms + +Now, look back at this derivation + + + +and notice that applying the rule twice to the rule , as we do here, gives us a way to derive for _any_ natural number ``N``; in the example above, ``N`` happens to be . + +Natural numbers are objects, and we can think of derivations as objects as well: we can represent this process of applying the rule twice to the rule using standard notation for application: . These objects are called generally called _proof terms_, and the _type_ of a proof term is the judgment it can produce. This idea that judgments can be types is one of the important observations of the [[w:Curry-Howard isomorphism|Curry-Howard isomorphism]]. + +The proof term can be thought of as having the type , but it can also be thought of as having the type +. Remembering our analysis above, we can see that the most general type we can give the proof object is . + +### Proving ∀∃-statements using totality assertions + +Recall that when we first defined the judgment , we mentioned the type of objects that it related: + + + +We will now write a judgment that, instead of relating objects with type , relates derivations of and . Call this derivation . + + + +Describing the rules that define this judgment is beyond the scope of this article; see Frank Pfenning's notes [http://www.cs.cmu.edu/~fp/courses/lp/lectures/18-proofs.html notes] for a continuation of this approach, or [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/) for a description that is closer to how Twelf is used in practice. However, without describing these rules, if they were written and written correctly, we could verify a totality assertion that stated that ``plus-comm`` is a total relation with derivations of as an input and derivations of as an output. + +This means that, for any three natural numbers , , and , and any proof object with the type , there is a proof object with type . This is equivalent to the ∀∃-statement for the commutativity of addition that we started out with; the only difference is that we are now speaking in terms of proof objects and types instead of derivations and judgments. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Metatheorem). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/minmltominhaskell.lf b/new-tests/stelf-output/pages/minmltominhaskell.lf new file mode 100644 index 0000000..52aec1b --- /dev/null +++ b/new-tests/stelf-output/pages/minmltominhaskell.lf @@ -0,0 +1,138 @@ +% +%{! + +Translation from MinML (unencapsulated effects) to MiniHaskell (monadic effects). +Uses third-order coverage checking. + +!}% +%{! + +## MiniHaskell + +!}% +%sort tp %. +%term unit tp %. +%term arr %pi tp %-> tp %-> tp %. +%term plus %pi tp %-> tp %-> tp %. +%term circ %pi tp %-> tp %. +%sort conc %. +%term true %pi tp %-> conc %. +%term lax %pi tp %-> conc %. +%sort | %. +%prec %prefix 0 | %. +%term <> | true unit %. +%term lam %pi (%pi (| true A) %-> (| true B)) %-> (| true (arr A B)) %. +%term app %pi (| true (arr A B)) %-> (| true A) %-> (| true B) %. +%term inl %pi (| true A) %-> (| true (plus A B)) %. +%term inr %pi (| true B) %-> (| true (plus A B)) %. +%term case + %pi (| true (plus A B)) + %-> (%pi (| true A) %-> (| J)) + %-> (%pi (| true B) %-> (| J)) + %-> (| J) %. +%term comp %pi (| lax A) %-> (| true (circ A)) %. +%term return %pi (| true A) %-> (| lax A) %. +%term let %pi (| true (circ A)) %-> (%pi (| true A) %-> (| lax C)) %-> (| lax C) %. +%term print %pi (| lax C) %-> (| lax C) %. +%block trueb {A _} [x | true A]%. +%worlds (trueb) (| _) %. +%{! + +## MinML + +!}% +%sort mtp %. +%term munit mtp %. +%term marr %pi mtp %-> mtp %-> mtp %. +%term mplus %pi mtp %-> mtp %-> mtp %. +%sort mtrue %. +%term m<> mtrue munit %. +%term mlam %pi (%pi (mtrue A) %-> (mtrue B)) %-> (mtrue (marr A B)) %. +%term mapp %pi (mtrue (marr A B)) %-> (mtrue A) %-> (mtrue B) %. +%term minl %pi (mtrue A) %-> (mtrue (mplus A B)) %. +%term minr %pi (mtrue B) %-> (mtrue (mplus A B)) %. +%term mcase + %pi (mtrue (mplus A B)) + %-> (%pi (mtrue A) %-> (mtrue C)) + %-> (%pi (mtrue B) %-> (mtrue C)) + %-> (mtrue C) %. +%term mprint %pi (mtrue C) %-> (mtrue C) %. +%{! + +## Translation + +!}% +%sort tptrans %. +%mode tptrans %in %out %. +%term tptrans/unit tptrans munit unit %. +%term tptrans/arr %pi (tptrans (marr A B) (arr A' (circ B'))) %<- (tptrans A A') %<- (tptrans B B') %. +%term tptrans/plus %pi (tptrans (mplus A B) (plus A' B')) %<- (tptrans A A') %<- (tptrans B B') %. +%worlds () (tptrans _ _) %. +%total A (tptrans A _) %. +%unique tptrans %in %out %. +%sort id-tp %. +%term id-tp/refl id-tp A A %. +%sort trueresp %. +%mode trueresp %in %in %out %. +%term _ trueresp E id-tp/refl E %. +%worlds (trueb) (trueresp _ _ _) %. +%total {} (trueresp _ _ _) %. +%sort laxresp %. +%mode laxresp %in %in %out %. +%term _ laxresp E id-tp/refl E %. +%worlds (trueb) (laxresp _ _ _) %. +%total {} (laxresp _ _ _) %. +%sort can-tptrans {A} %. +%mode can-tptrans %in %out %. +%worlds () (can-tptrans _ _) %. +%total A (can-tptrans A _) %. +%sort tptrans-unique %. +%mode tptrans-unique %in %in %out %. +%worlds () (tptrans-unique _ _ _) %. +%total D (tptrans-unique D _ _) %. +%sort trans %. +%mode trans %in %in %out %. +%term _ trans m<> tptrans/unit (return <>) %. +%term _ + %pi (trans (mlam E) (tptrans/arr (%the (tptrans B B') DB) DA) (return (lam ([x] comp (E' x))))) + %<- ({x mtrue A} {x' | true A'} {_ {A'' _} {DA'' tptrans A A''} {Did id-tp A' A''} {E'' | lax A''} + %pi (trans x DA'' E'') + %<- (tptrans-unique DA DA'' Did) + %<- (laxresp (return x') Did E'')} trans (E x) DB (E' x')) %. +%term _ + %pi (trans (mapp (%the (mtrue (marr A B)) E1) E2) DB (let (comp E1') ([x1] let (comp E2') ([x2] let (app x1 x2) ([r] return r))))) + %<- (can-tptrans A DA) + %<- (trans E1 (tptrans/arr DB DA) E1') + %<- (trans E2 DA E2') %. +%term _ + %pi (trans (minl E) (tptrans/plus DB DA) (let (comp E') ([x] return (inl x)))) + %<- (trans E DA E') %. +%term _ + %pi (trans (minr E) (tptrans/plus DB DA) (let (comp E') ([x] return (inr x)))) + %<- (trans E DB E') %. +%term _ + %pi (trans (mcase (%the (mtrue (mplus A B)) E) E1 E2) DC (let (comp E') ([x] case x E1' E2'))) + %<- (can-tptrans A DA) + %<- (can-tptrans B DB) + %<- (trans E (tptrans/plus DB DA) E') + %<- ({x mtrue A} {x' | true A'} {_ {A'' _} {DA'' tptrans A A''} {Did id-tp A' A''} {E'' | lax A''} + %pi (trans x DA'' E'') + %<- (tptrans-unique DA DA'' Did) + %<- (laxresp (return x') Did E'')} trans (E1 x) DC (E1' x')) + %<- ({x mtrue B} {x' | true B'} {_ {B'' _} {DB'' tptrans B B''} {Did id-tp B' B''} {E'' | lax B''} + %pi (trans x DB'' E'') + %<- (tptrans-unique DB DB'' Did) + %<- (laxresp (return x') Did E'')} trans (E2 x) DC (E2' x')) %. +%term _ %pi (trans (mprint E) DC (print E')) %<- (trans E DC E') %. +%block transb {A _} {A' _} {DA tptrans A A'} [x mtrue A] [x' | true A'] [_ {A'' _} {DA'' tptrans A A''} {Did id-tp A' A''} {E'' | lax A''} + %pi (trans x DA'' E'') + %<- (tptrans-unique DA DA'' Did) + %<- (laxresp (return x') Did E'')]%. +%worlds (transb) (trans _ _ _) %. +%total E (trans E _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/MinMLToMinHaskell). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/modal-logic.lf b/new-tests/stelf-output/pages/modal-logic.lf new file mode 100644 index 0000000..d131853 --- /dev/null +++ b/new-tests/stelf-output/pages/modal-logic.lf @@ -0,0 +1,95 @@ +% +%{! In this page we show how to encode modal logic in LF. Similarly to [linear logic](/wiki/linear-logic/), the main idea here is to define a judgment that explicitly restricts the use of non-necessity assumptions (any assumption of a proof for a judgment that is not necessary). + +## The Encoding +### Syntax !}% +% Terms +%sort tm %. +%term unit tm %. +%term app %pi tm %-> tm %-> tm %. +%term lam %pi (%pi tm %-> tm) %-> tm %. +%term bx %pi tm %-> tm %. +%term letbox %pi tm %-> (%pi tm %-> tm) %-> tm %. +% Types +%sort tp %. +%term o tp %. +%term arrow %pi tp %-> tp %-> tp %. +%term box %pi tp %-> tp %. +%term dia %pi tp %-> tp %. +%{! ### Local Judgment +The correct use of an assumption inside a proof term can be enforced by a ``local`` judgment. This jugment ensures that a variable is never used inside any term `` bx M `` that exists within its scope (we say that the variable is local). !}% +%sort local %. +%{! The simplest cases are the one for variables. !}% +%term local/var local ([x] x) %. +%{! !}% +%term local/closed local ([x] M) %. +%{! A variable is local in an abstraction if it is local inside the abstraction's body. Note that the abstraction's argument (which must be local as well) is handled separately by the abstraction's typing rule. !}% +%term local/lam %pi (local ([x] lam ([y] N x y))) %<- ({y} local ([x] N x y)) %. +%{! A variable is local in an application if it is local inside its subterms. !}% +%term local/app %pi (local ([x] app (M x) (N x))) %<- (local ([x] M x)) %<- (local ([x] N x)) %. +%{! The same idea applies to letbox. !}% +%term local/letbox + %pi (local ([x] letbox (M x) ([y] N x y))) + %<- (local ([x] M x)) + %<- ({y} local ([x] N x y)) %. +%{! Significantly, there is no rule for bx. Local variables are not permitted to appear within box terms. + +### Typing rules +The key idea introduced by our enconding of modal logic is that typing rules need to check (using the local judgment described above) that non-necessary assumptions are never used inside a term ``bx M ``. In our case, abstractions are the only terms that bind restricted (non-necessary) variables, and therefore the rule for ``letbox`` does not need to check whether its variable is local or not. !}% +%sort of %. +%term of/unit of unit o %. +%term of/bx %pi (of (bx M) (box T)) %<- (of M T) %. +%term of/lam + %pi (of (lam M) (arrow A B)) + %<- ({n tm} %pi (of n A) %-> (of (M n) B)) + %<- (local M) %. +%term of/letbox + %pi (of (letbox N M) B) + %<- (of N (box A)) + %<- ({n tm} %pi (of n A) %-> (of (M n) B)) %. +%term of/app %pi (of M (arrow T1 T2)) %-> (of N T1) %-> (of (app M N) T2) %. +%{! ``>Source code for the encoding`` + +# Type Preservation +As an example, we can prove preservation using our encoding. + +## Evaluation Rules +First, we define the ``value`` judgment and evaluation rules. !}% +%sort value %. +%term value/unit value unit %. +%term value/lam value (lam M) %. +%term value/bx %pi (value (bx M)) %<- (value M) %. +%%%%%% Evaluation rules %%%%%% +%sort step %. +%term step/beta %pi (step (app (lam N) M) (N M)) %<- (value M) %. +%term step/app1 %pi (step (app M N) (app M1 N)) %<- (step M M1) %<- (value N) %. +%term step/app2 %pi (step (app M N) (app M N1)) %<- (step N N1) %. +%term step/bx %pi (step (bx M) (bx N)) %<- (step M N) %. +%term step/letbox %pi (step (letbox M N) (letbox M' N)) %<- (step M M') %. +%term step/letbox/beta %pi (step (letbox (bx M) N) (N M)) %<- (value M) %. +%{! ## Type Preservation Proof !}% +%sort step-type %. +%mode step-type %in %in %out %. +%term step-type/beta step-type (of/app (of/lam LN P) OM) (%the (step (app (lam N) M) (N M)) (step/beta _)) (P M OM) %. +%term step-type/app1 + %pi (step-type (of/app OM ON) (step/app1 _ R) (of/app OM1 ON)) + %<- (step-type OM R OM1) %. +%term step-type/app2 + %pi (step-type (of/app ON OM) (step/app2 R) (of/app ON OM1)) + %<- (step-type OM R OM1) %. +%term step-type/bx %pi (step-type (of/bx PM) (step/bx S) (of/bx PN)) %<- (step-type PM S PN) %. +%term step-type/letbox + %pi (step-type (of/letbox ON OM) (step/letbox R) (of/letbox ON OM1)) + %<- (step-type OM R OM1) %. +%term step-type/letbox/beta step-type (of/letbox P (of/bx OM)) (step/letbox/beta _) (P M OM) %. +%worlds () (step-type _ _ _) %. +%total (R) (step-type R _ _) %. +%{! ``>Complete Source code`` + +`` check=true>Twelf Output`` !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Modal_logic). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/modally-propositional-logic.lf b/new-tests/stelf-output/pages/modally-propositional-logic.lf new file mode 100644 index 0000000..5a841b7 --- /dev/null +++ b/new-tests/stelf-output/pages/modally-propositional-logic.lf @@ -0,0 +1,121 @@ +% +%{! + +## Modally-Propositional Logic + +!}% +%sort world %. +%name world %. +%term world1 world %. +%term succ %pi world %-> world %. +%% so it doesn't split worlds, which makes the coverage checking output annoying +%block worldb [w world]%. +%worlds (worldb) (world) %. +%sort acc %. +%term refl acc W W %. +%term trans %pi (acc W1 W2) %-> (acc W2 W3) %-> (acc W1 W3) %. +%block accb {W1 _} {W2 _} [a acc W1 W2]%. +%worlds (worldb accb) (acc _ _) %. +%sort prop %. +%name prop %. +%inline boxprop (%pi world %-> %type) [w] {w'} %pi (acc w w') %-> (prop w') %. +%term box %pi (boxprop W) %-> (prop W) %. +%term imp %pi (prop W) %-> (prop W) %-> (prop W) %. +%term at %pi (prop W') %-> (prop W) %. +%term down %pi (boxprop W) %-> (prop W) %. +%block propb {W world} [a prop W]%. +%worlds (worldb accb propb) (prop _) %. +%sort hyp %. +%sort conc %. +%term impL %pi (conc C) %<- (hyp (imp A B)) %<- (conc A) %<- (%pi (hyp B) %-> (conc C)) %. +%term impR %pi (conc (imp A B)) %<- (%pi (hyp A) %-> (conc B)) %. +%term atL %pi (conc C) %<- (hyp (at A)) %<- (%pi (hyp A) %-> (conc C)) %. +%term atR %pi (conc (at A)) %<- (conc A) %. +%term boxR %pi (conc (box (%the (boxprop W) A))) %<- ({w'} {a acc W w'} conc (A w' a)) %. +%term boxL {a acc W W'} + %pi (conc C) + %<- (hyp (box (%the (boxprop W) A))) + %<- (%pi (hyp (A W' a)) %-> (conc C)) %. +%term downR %pi (conc (%the (prop W) (down A))) %<- (conc (A W refl)) %. +%term downL + %pi (conc C) + %<- (hyp (%the (prop W) (down A))) + %<- (%pi (hyp (A W refl)) %-> (conc C)) %. +%block hypb {W _} {A prop W} [x hyp A]%. +%block prophypb {W _} [a prop W] [h %pi (hyp a) %-> (conc a)]%. +%worlds (worldb accb prophypb hypb) (hyp _) (conc _) %. +%sort id {A prop W} %. +%mode id %in %out %. +%term _ %pi (id (imp A B) ([f] impR ([x] impL E' (E x) f))) %<- (id A E) %<- (id B E') %. +%term _ + %pi (id (box (%the (boxprop W) A)) ([b] boxR ([w'] [a] boxL a (E w' a) b))) + %<- ({w} {a acc W w} id (A w a) (E w a)) %. +%term _ + %pi (id (down A) ([d] downR (downL (E _ refl) d))) + %<- ({w} {a acc W w} id (A w a) (E w a)) %. +%% ambipolar: +% - : id (down A) ([d] downL ([x] downR (E _ refl x)) d) +% <- {w} {a : acc W w} id (A w a) (E w a). +%term _ %pi (id (at A) ([a] atR (atL E a))) %<- (id A E) %. +%% ambipolar: +% - : id (at A) ([a] (atL ([x1] atR (E x1)) a)) +% <- id A E. +%block idcase {W _} [a prop W] [h %pi (hyp a) %-> (conc a)] [_ id a h]%. +%worlds (worldb accb idcase hypb) (id _ _) %. +%total A (id A _) %. +%sort ca {A} %. +%mode ca %in %in %in %out %. +%term _ + %pi (ca _ (atR D) ([x] atL ([y] E x y) x) E'') + %<- ({y} ca _ (atR D) ([x] E x y) (E' y)) + %<- (ca _ D E' E'') %. +%term _ + %pi (ca _ (boxR D) ([x] boxL A ([y] E x y) x) E'') + %<- ({y} ca _ (boxR D) ([x] E x y) (E' y)) + %<- (ca _ (D _ A) E' E'') %. +%term _ + %pi (ca _ (downR D) ([x] downL ([y] E x y) x) E'') + %<- ({y} ca _ (downR D) ([x] E x y) (E' y)) + %<- (ca _ D E' E'') %. +%term _ + %pi (ca (%the (prop WAB) (imp A B)) (impR D) ([x] impL ([y] E x y) (Arg x) x) (%the (conc (%the (prop WC) C)) E'')) + %<- ({y} ca (imp A B) (impR D) ([x] E x y) (E' y)) + %<- (ca (imp A B) (impR D) ([x] Arg x) Arg') + %<- (ca A Arg' D D') + %<- (ca B D' E' E'') %. +%% left commutative +%term _ %pi (ca _ (atL D D') E (atL D1 D')) %<- ({y} ca _ (D y) E (D1 y)) %. +%term _ %pi (ca _ (boxL A D D') E (boxL A D1 D')) %<- ({y} ca _ (D y) E (D1 y)) %. +%term _ %pi (ca _ (impL D A D') E (impL D1 A D')) %<- ({y} ca _ (D y) E (D1 y)) %. +%term _ %pi (ca _ (downL D D') E (downL D1 D')) %<- ({y} ca _ (D y) E (D1 y)) %. +%% right commutative +%term _ %pi (ca _ D ([x] impR ([y] E x y)) (impR F)) %<- ({y} ca _ D ([x] E x y) (F y)) %. +%term _ %pi (ca _ D ([x] atR (E x)) (atR F)) %<- (ca _ D ([x] E x) F) %. +%term _ %pi (ca _ D ([x] downR (E x)) (downR F)) %<- (ca _ D ([x] E x) F) %. +%term _ + %pi (ca _ D ([x] boxR ([w'] [a] E w' a x)) (boxR F)) + %<- ({w'} {a} ca _ D ([x] E w' a x) (F w' a)) %. +%term _ + %pi (ca _ D ([x] boxL A ([y] E x y) Y) (boxL A F Y)) + %<- ({y} ca _ D ([x] E x y) (F y)) %. +%term _ + %pi (ca _ D ([x] downL ([y] E x y) Y) (downL F Y)) + %<- ({y} ca _ D ([x] E x y) (F y)) %. +%term _ %pi (ca _ D ([x] atL ([y] E x y) Y) (atL F Y)) %<- ({y} ca _ D ([x] E x y) (F y)) %. +%term _ + %pi (ca _ D ([x] impL ([y] E x y) (Arg x) Y) (impL F Arg' Y)) + %<- ({y} ca _ D ([x] E x y) (F y)) + %<- (ca _ D ([x] Arg x) Arg') %. +%block capropb {W _} [a prop W] [init %pi (hyp a) %-> (conc a)] [_ {y hyp a} ca a (init y) init (init y)] [_ {W' world} {A prop W'} {D conc A} {y hyp a} ca A D ([_] init y) (init y)]%. +%% FIXME: this shouldn't pass: +%% propb is not equivalent to prophypb for hyp and conc. +%% does twelf only check world subsumption on subgoals? +%% %worlds (worldb | accb | propb | hypb) (ca _ _ _ _). +%worlds (worldb accb capropb hypb) (ca _ _ _ _) %. +%total {A D E} (ca A D E _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Modally_Propositional_Logic). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/modes-of-use.lf b/new-tests/stelf-output/pages/modes-of-use.lf new file mode 100644 index 0000000..2e61361 --- /dev/null +++ b/new-tests/stelf-output/pages/modes-of-use.lf @@ -0,0 +1,89 @@ +% +%{! + +Twelf is a very flexible system that lends itself to a wide variety of [deductive systems](/wiki/object-logic/). This flexibility can be confusing, as different uses of Twelf look very different from one another. Here, we attempt to describe a number of the major modes of use of Twelf. + +* **[[#A framework for defining logics|A framework for defining logics]]** - The judgments of a logic are encoded in Twelf so that Twelf's type checker can check the correctness of proofs. The Foundational Proof Carrying Code project at Princeton used this method extensively to write proofs about programs that could be verified by a small, trusted checker. The primary tutorial for this style is Andrew Appel's [http://www.cs.princeton.edu/~appel/twelf-tutorial/ Hints on Proving Theorems in Twelf] +* **[[#Proving properties about deductive systems|Proving properties about deductive systems]] - This is a relatively recent use of Twelf, but it is also perhaps the most widespread. The judgments of a deductive system, typically a programming language, are encoded in Twelf, and then Twelf is used to verify [metatheorems](/wiki/metatheorem/) about that deductive system. The primary tutorial for this style is [Dan Licata's](/wiki/user-drl/) [Proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/), though several others can be found on the [Documentation](/wiki/documentation/) page. +* Typed logic programming - Twelf can also be used as an advanced typed logic programming language, a variant of Lambda Prolog with dependent types. + +Most large projects integrate several of these modes of use. There are some limitations to this integration, particularly in that the [constraint domains](/wiki/constraint-domain/) and [[type level definitions]] cannot be used in projects that want to prove [metatheorems](/wiki/metatheorem/). + + +## A framework for defining logics + +The "theorem style" use of Twelf described in Andrew Appel's notes defines a deductive system, typically a **logic**: first a set of propositions (propositions often have type ``o`` for historical reasons), and then axioms about how propositions can be proven. Appel's FPCC project uses higher-order logic, but here we define a much simpler propositional logic. + +!}% +%sort o %. +% Propositions +%term true o %. +%term imp %pi o %-> o %-> o %. +%prec %right 10 imp %. +%term and %pi o %-> o %-> o %. +%prec %right 11 and %. +% Judgments +%sort pf %. +%term true-i pf true %. +%term imp-i %pi (%pi (pf A) %-> (pf B)) %-> (pf (A imp B)) %. +%term imp-e %pi (pf (A imp B)) %-> (pf A) %-> (pf B) %. +%term and-i %pi (pf A) %-> (pf B) %-> (pf (A and B)) %. +%term and-e1 %pi (pf (A and B)) %-> (pf A) %. +%term and-e2 %pi (pf (A and B)) %-> (pf B) %. +%{! Having defined this a logic, the point of the exercise is to write out proofs by hand (perhaps with the assistance of [tactical theorem proving](/wiki/tactical-theorem-proving/)). These proofs can then be automatically verified by Twelf, or by an extremely small independent checker. + +Here are some examples from the first lecture of Hints on Proving Theorems in Twelf: !}% +%define symm-and (%pi (pf (A and B)) %-> (pf (B and A))) [p1 pf (A and B)] and-i (and-e2 p1) (and-e1 p1) %. +%define and-l (%pi (pf (A and B)) %-> (%pi (pf A) %-> (pf B) %-> (pf C)) %-> (pf C)) [p1 pf (A and B)] [p2 %pi (pf A) %-> (pf B) %-> (pf C)] imp-e (imp-e (imp-i ([p3] imp-i (p2 p3))) (and-e1 p1)) (and-e2 p1) %. +%define example-abc (%pi (pf (A and B)) %-> (pf C) %-> (pf (B and C and (A and C)))) [p1 pf (A and B)] [p2 pf C] and-l p1 ([p3 pf A] [p4 pf B] and-i (and-i p4 p2) (and-i p3 p2)) %. +%{! Even this logic allows us to prove a number of interesting theorems; the definition of a more complex logic (without much commentary) can be found at [Zermelo Frankel](/wiki/zermelo-frankel/). + + +## Proving properties about deductive systems + +The "metatheorem style" use of Twelf defines a deductive system, typically a **programming language**: first the abstract syntax of the language, and then the static and dynamic semantics. Here is an example of a simple lambda calculus with product types (we don't define the dynamic semantics here, see \{\{proving|title=Representing the judgements of the STLC\}\} for a similar example). +!}% +%sort exp %. +%sort tp %. +% Expressions +%term exp/unit exp %. +%term exp/lam %pi tp %-> (%pi exp %-> exp) %-> exp %. +%term exp/app %pi exp %-> exp %-> exp %. +%term exp/pair %pi exp %-> exp %-> exp %. +%term exp/fst %pi exp %-> exp %. +%term exp/snd %pi exp %-> exp %. +% Types +%term unit tp %. +%term arrow %pi tp %-> tp %-> tp %. +%prec %right 10 arrow %. +%term pair %pi tp %-> tp %-> tp %. +%prec %right 11 pair %. +% Static semantics +%sort of %. +%term of/unit of exp/unit unit %. +%term of/lam + %pi (of (exp/lam T1 ([x] E x)) (T1 arrow T2)) + %<- ({x exp} %pi (of x T1) %-> (of (E x) T2)) %. +%term of/app %pi (of (exp/app E E') T) %<- (of E' T') %<- (of E (T' arrow T)) %. +%term of/pair %pi (of (exp/pair E1 E2) (T1 pair T2)) %<- (of E2 T2) %<- (of E1 T1) %. +%term of/fst %pi (of (exp/fst E) T1) %<- (of E (T1 pair T2)) %. +%term of/snd %pi (of (exp/snd E) T2) %<- (of E (T1 pair T2)) %. +% Dynamic semantics +%sort step %. +% ... and so on +%{! + +Having defined this programming language, we are generally interested in proving [metatheorems](/wiki/metatheorem/) relating that programming language to a particular dynamic semantics - showing, for instance, that a certain evaluation strategy will preserve the type of the terms it evaluates, and that if a closed can be given a type, it can either be reduced or it is already a value. We also may be interested in using the [logic programming](/wiki/logic-programming/) engine of Twelf to _run_ our typing judgments as a logic program, giving our language a reference type checker. + + +### Discussion + + + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Modes_of_use). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/mutable-state.lf b/new-tests/stelf-output/pages/mutable-state.lf new file mode 100644 index 0000000..eb10da3 --- /dev/null +++ b/new-tests/stelf-output/pages/mutable-state.lf @@ -0,0 +1,505 @@ +% +%{! +This article presents a simple language with mutable state—it has +``let``-statements but, unlike many other examples such as the +[simply-typed lambda calculus](/wiki/simply-typed-lambda-calculus/), it does not have functions, though this is for simplicity and is not an inherent limitation. It is based on +Chapter 2 of [Rob Simmons's](/wiki/user-rsimmons/) undergraduate thesis,<ref><bibtex>@TechReport\{simmons05thesis, + author = \{Robert J. Simmons\}, + title = \{Twelf as a Unified Framework for Language Formalization and Implementation\}, + Institution = \{Princeton University\}, + year = \{2005\}, + note = \{Undergraduate Senior Thesis 18679\}, +\}</bibtex></ref> which contains more commentary—this account includes a simple extension, the ability to update state. That account is in turn based on Pierce's description in TAPL.<ref>\{\{bibtex:pierce02tapl\}\}</ref> This example can be seen as an extension of the one from the tutorial on [strengthening](/wiki/strengthening/); that example defines a language with references but no way to use them. + +## Language definition + +### Natural numbers + +First we need [natural numbers](/wiki/natural-numbers/), which we will use in the object language +to represent both locations and actual numbers (this is why we define +``sum`` as well.) +!}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort sum %. +%mode sum %in %in %out %. +%term sum-z sum z N N %. +%term sum-s %pi (sum (s N1) N2 (s N3)) %<- (sum N1 N2 N3) %. +%worlds () (sum A B C) %. +%total A (sum A B C) %. +%{! +### Syntax + +Using this we can define the syntax of the language; natural numbers will be +used both for locations (which the programmer cannot access) and as numbers, +which can be added—the typing rules will ensure that natural numbers +representing heap locations cannot be confused with natural numbers that may +be added. + +* ``n N`` - integrates natural numbers into the language as , where is a natural number 0, 1, 2... +* ``E1 + E2`` - addition, +* ``ref E`` - creating a reference cell, +* ``! E`` - dereferencing a reference cell, +* ``gets E1 E2`` - updating a reference cell, +* ``loc L`` - locations (not available to the programmer), , where is an abstract location. +* ``let E1 ([x] E2 x)`` - let statements +!}% +%inline location nat %. +%sort exp %. +%term n %pi nat %-> exp %. +%term + %pi exp %-> exp %-> exp %. +%prec %left 10 + %. +%term ref %pi exp %-> exp %. +%term ! %pi exp %-> exp %. +%term gets %pi exp %-> exp %-> exp %. +%term loc %pi location %-> exp %. +%term let %pi exp %-> (%pi exp %-> exp) %-> exp %. +%{! +### Types + +Because we do not have functions, our language has a very simple language of types, only integers and references to other types. +!}% +%sort tp %. +%term int-tp tp %. +%term ref-tp %pi tp %-> tp %. +%{! +### Lists + +We will represent the store as a list of expressions, and define several +operations on lists—projecting the _n_th element from a list, appending a new element to the end of a list, updating the _n_th element of the list, and having a list that is the subset of another list. We also define a similar list of types that is omitted. +!}% +%sort explist %. +%term $exp %pi exp %-> explist %-> explist %. +%prec %right 5 $exp %. +%term nil-exp explist %. +%sort proj-exp %. +%mode proj-exp %in %in %out %. +%term proj-exp-z proj-exp (E $exp EL) z E %. +%term proj-exp-s %pi (proj-exp (E $exp EL) (s N) E’) %<- (proj-exp EL N E’) %. +%sort append-exp %. +%mode append-exp %in %in %out %out %. +%term append-exp-z append-exp nil-exp E (E $exp nil-exp) z %. +%term append-exp-s %pi (append-exp (E’ $exp EL) E (E’ $exp EL’) (s N)) %<- (append-exp EL E EL’ N) %. +%sort subset-exp %. +%mode subset-exp %in %in %. +%term subset-exp-z subset-exp nil-exp E %. +%term subset-exp-s %pi (subset-exp (E $exp EL) (E $exp EL’)) %<- (subset-exp EL EL’) %. +%sort update-exp %. +%mode update-exp %in %in %in %out %. +%term update-exp-z update-exp (E $exp EL) z E' (E' $exp EL) %. +%term update-exp-s %pi (update-exp (E $exp EL) (s N) E' (E $exp EL')) %<- (update-exp EL N E' EL') %. +%{! |hidden=true !}% +%sort tplist %. +%term $tp %pi tp %-> tplist %-> tplist %. +%prec %right 5 $tp %. +%term nil-tp tplist %. +%sort proj-tp %. +%mode proj-tp %in %in %out %. +%term proj-tp-z proj-tp (T $tp TL) z T %. +%term proj-tp-s %pi (proj-tp (T $tp TL) (s N) T’) %<- (proj-tp TL N T’) %. +%sort append-tp %. +%mode append-tp %in %in %out %out %. +%term append-tp-z append-tp nil-tp T (T $tp nil-tp) z %. +%term append-tp-s %pi (append-tp (T’ $tp TL) T (T’ $tp TL’) (s N)) %<- (append-tp TL T TL’ N) %. +%sort subset-tp %. +%term subset-tp-z subset-tp nil-tp T %. +%term subset-tp-s %pi (subset-tp (T $tp TL) (T $tp TL’)) %<- (subset-tp TL TL’) %. +%sort update-tp %. +%mode update-tp %in %in %in %out %. +%term update-tp-z update-tp (T $tp TL) z T' (T' $tp TL) %. +%term update-tp-s %pi (update-tp (T $tp TL) (s N) T' (T $tp TL')) %<- (update-tp TL N T' TL') %. +%{! +We create abbreviations to indicate that we are using expression +lists to represent stores and type lists to represent store typings. +!}% +%inline store explist %. +%inline storetp tplist %. +%{! +### Values and evaluation + +The predicate ``isval`` +!}% +%sort isval %. +%mode isval %in %. +%term v-int isval (n N) %. +%term v-loc isval (loc L) %. +%sort isval-list %. +%mode isval-list %in %. +%term vl-z isval-list nil-exp %. +%term vl-s %pi (isval-list (E $exp EL)) %<- (isval E) %<- (isval-list EL) %. +%sort eval %. +%mode eval %in %in %out %out %. +%term s1-add %pi (eval S (E1 + E2) S’ (E1’ + E2)) %<- (eval S E1 S’ E1’) %. +%term s2-add %pi (eval S (E1 + E2) S’ (E1 + E2’)) %<- (isval E1) %<- (eval S E2 S’ E2’) %. +%term e-add %pi (eval S (n N1 + n N2) S (n N3)) %<- (sum N1 N2 N3) %. +%term s-ref %pi (eval S (ref E) S’ (ref E’)) %<- (eval S E S’ E’) %. +%term e-ref %pi (eval S (ref E) S’ (loc L)) %<- (isval E) %<- (append-exp S E S’ L) %. +%term s-bang %pi (eval S (! E) S’ (! E’)) %<- (eval S E S’ E’) %. +%term e-bang %pi (eval S (! (loc L)) S E) %<- (proj-exp S L E) %. +%term s1-gets %pi (eval S (gets E1 E2) S' (gets E1' E2)) %<- (eval S E1 S' E1') %. +%term s2-gets %pi (eval S (gets E1 E2) S' (gets E1 E2')) %<- (isval E1) %<- (eval S E2 S' E2') %. +%term e-gets %pi (eval S (gets (loc L1) E) S' E) %<- (isval E) %<- (update-exp S L1 E S') %. +%term s-let %pi (eval S (let E EF) S’ (let E’ EF)) %<- (eval S E S’ E’) %. +%term e-let %pi (eval S (let E EF) S (EF E)) %<- (isval E) %. +%{! +### Typing judgment + +As we do in the article on [strengthening](/wiki/strengthening/), we need to define a separate +judgment ``var-of`` that we will use to represent hypothetical typing +judgments—the hypothetical judgment simply associates an expression with +a type, whereas the typing judgment ``of`` associates an expression with +a type in a specific store typing. +!}% +%sort var-of %. +%mode var-of %in %out %. +%sort of %. +%mode of %in %in %out %. +%sort of-list %. +%mode of-list %in %in %out %. +%term tl-z of-list ST nil-exp nil-tp %. +%term tl-s %pi (of-list ST (E $exp EL) (T $tp TL)) %<- (of ST E T) %<- (of-list ST EL TL) %. +%term t-int of ST (n N) int-tp %. +%term t-add %pi (of ST (E1 + E2) int-tp) %<- (of ST E1 int-tp) %<- (of ST E2 int-tp) %. +%term t-ref %pi (of ST (ref E) (ref-tp T)) %<- (of ST E T) %. +%term t-bang %pi (of ST (! E) T) %<- (of ST E (ref-tp T)) %. +%term t-gets %pi (of ST (gets E1 E2) T) %<- (of ST E1 (ref-tp T)) %<- (of ST E2 T) %. +%term t-loc %pi (of ST (loc L) (ref-tp T)) %<- (proj-tp ST L T) %. +%term t-let + %pi (of ST (let E EF) T) + %<- (of ST E T’) + %<- ({x} %pi (var-of x T’) %-> (of ST (EF x) T)) %. +%term t-var %pi (of ST E T) %<- (var-of E T) %. +%{! The typing judgment exists in an LF context where new expression +variables can be introduced if they are related with ``var-of`` +to a well-formed type. This requirement is expressed by the [block](/wiki/percent-block/) +``var-tp``. !}% +%block var-tp {T tp} [x exp] [v var-of x T]%. +%{! +We also define a judgment ``of-store``, that expresses a store being +well-typed and containing only values. +!}% +%sort of-store %. +%term &of-store %pi (of-store TL EL) %<- (isval-list EL) %<- (of-list TL EL TL) %. +%{! ### Multi-step evaluation !}% +%sort run %. +%mode run %in %in %out %out %. +%term run-step %pi (run S E S’’ E’’) %<- (eval S E S’ E’) %<- (run S’ E’ S’’ E’’) %. +%term run-end run S E S E %. +%{! ## Language theory !}% +%{! +### Effectiveness lemmas + +We need a few [effectiveness lemmas](/wiki/effectiveness-lemma/) that show that the ``sum`` and ``append`` judgments are always derivable. + +The ``can-update`` theorem is slightly different, establishing that if a particular store has a particular type, then any index that has a corresponding type can be updated. +!}% +%sort can-sum {A nat} {B nat} %. +%mode can-sum %in %in %out %. +%term & can-sum z N sum-z %. +%term & %pi (can-sum (s N1) N2 (sum-s SUM)) %<- (can-sum N1 N2 SUM) %. +%worlds () (can-sum _ _ _) %. +%total T (can-sum T _ _) %. +%sort can-append-exp {EL explist} {E exp} %. +%mode can-append-exp %in %in %out %. +%term & can-append-exp nil-exp E append-exp-z %. +%term & + %pi (can-append-exp (E $exp EL) E' (append-exp-s APPEND)) + %<- (can-append-exp EL E' APPEND) %. +%worlds () (can-append-exp _ _ _) %. +%total T (can-append-exp T _ _) %. +%{! ### Other auxillary lemmas !}% +%sort of-projection %. +%mode of-projection %in %in %out %out %. +%term & of-projection (tl-s TL T) proj-tp-z proj-exp-z T %. +%term & + %pi (of-projection (tl-s TL T’) (proj-tp-s PT) (proj-exp-s PE) T) + %<- (of-projection TL PT PE T) %. +%worlds () (of-projection _ _ _ _) %. +%total PT (of-projection _ PT _ _) %. +%sort of-projection' %. +%mode of-projection' %in %in %in %out %. +%term & of-projection' (tl-s TL T) proj-tp-z proj-exp-z T %. +%term & + %pi (of-projection' (tl-s TL T’) (proj-tp-s PT) (proj-exp-s PE) T) + %<- (of-projection' TL PT PE T) %. +%worlds () (of-projection' _ _ _ _) %. +%total PT (of-projection' _ PT _ _) %. +%sort subset-projectable %. +%mode subset-projectable %in %in %out %. +%term & subset-projectable proj-tp-z (subset-tp-s SUB) proj-tp-z %. +%term & + %pi (subset-projectable (proj-tp-s P) (subset-tp-s SUB) (proj-tp-s P’)) + %<- (subset-projectable P SUB P’) %. +%worlds () (subset-projectable _ _ _) %. +%total P (subset-projectable P S P’) %. +%sort subset-refl {ST} %. +%mode subset-refl %in %out %. +%term & subset-refl nil-tp subset-tp-z %. +%term & %pi (subset-refl (T $tp ST) (subset-tp-s SUB)) %<- (subset-refl ST SUB) %. +%worlds () (subset-refl _ _) %. +%total T (subset-refl T _) %. +%sort of-update %. +%mode of-update %in %in %in %out %. +%term & of-update (tl-s _ _) proj-tp-z _ update-exp-z %. +%term & + %pi (of-update (tl-s T _) (proj-tp-s PT) E (update-exp-s PE)) + %<- (of-update T PT E PE) %. +%worlds () (of-update _ _ _ _) %. +%total T (of-update T _ _ _) %. +%sort append-lemma %. +%mode append-lemma %in %in %in %in %in %out %out %out %out %. +%term & append-lemma V T vl-z tl-z append-exp-z subset-tp-z (vl-s vl-z V) (tl-s tl-z T) proj-tp-z %. +%term & + %pi (append-lemma V T (vl-s VL V*) (tl-s TL T*) (append-exp-s AE) (subset-tp-s S) (vl-s VL’ V*) (tl-s TL’ T*) (proj-tp-s P)) + %<- (append-lemma V T VL TL AE S VL’ TL’ P) %. +%worlds () (append-lemma _ _ _ _ _ _ _ _ _) %. +%total [VL TL AE] (append-lemma V T VL TL AE AT VL’ TL’ P) %. +%sort update-lemma %. +%mode update-lemma %in %in %in %in %in %in %out %out %. +%term & update-lemma V T (vl-s VL V*) (tl-s TL T*) proj-tp-z update-exp-z (vl-s VL V) (tl-s TL T) %. +%term & + %pi (update-lemma V T (vl-s VL V*) (tl-s TL T*) (proj-tp-s P) (update-exp-s UE) (vl-s VL' V*) (tl-s TL' T*)) + %<- (update-lemma V T VL TL P UE VL' TL') %. +%worlds () (update-lemma _ _ _ _ _ _ _ _) %. +%total T (update-lemma _ _ T _ _ _ _ _) %. +%{! +### Progress + +Progress, as usual, relies on an auxillary notion of what it means for an +expression to not be stuck that is captured by the judgement ``notstuck``, +which states that in a certain store ``S`` an expression ``E`` is +either a value or can take a step. +!}% +%sort notstuck %. +%term ns-steps %pi (notstuck S E) %<- (eval S E S’ E’) %. +%term ns-isval %pi (notstuck S E) %<- (isval E) %. +%{! The statement of the progress theorem then utilizes the progress +theorem in a straightforward manner. !}% +%sort progress %. +%mode progress %in %in %out %. +%{! A natural number is already a value !}% +%term prog-int progress t-int _ (ns-isval v-int) %. +%{! To show progress for references, we show that a new reference cell can +always be created on the end of the list we use to represent the heap. !}% +%sort lemma %. +%mode lemma %in %out %. +%term & lemma (ns-steps E) (ns-steps (s-ref E)) %. +%term & + %pi (lemma (ns-isval V) (ns-steps (e-ref APPEND V))) + %<- (can-append-exp EL E APPEND) %. +%worlds () (lemma _ _) %. +%total NS (lemma NS _) %. +%term prog-ref %pi (progress (t-ref T) TS NS) %<- (progress T TS NS1) %<- (lemma NS1 NS) %. +%{! To prove progress for addition, we will need to use the fact that the subexpressions both have type ``int-tp``—that way, once they are reduced to values, they must be numbers, not locations. !}% +%sort lemma %. +%mode lemma %in %in %in %in %out %. +%term & lemma _ (ns-steps E) _ _ (ns-steps (s1-add E)) %. +%term & lemma _ (ns-isval V) _ (ns-steps E) (ns-steps (s2-add E V)) %. +%term & + %pi (lemma t-int (ns-isval v-int) t-int (ns-isval v-int) (ns-steps (e-add SUM))) + %<- (can-sum N1 N2 SUM) %. +%worlds () (lemma _ _ _ _ _) %. +%total [NS1 NS2] (lemma T1 NS1 T2 NS2 NS) %. +%term prog-add + %pi (progress (t-add T2 T1) TS NS) + %<- (progress T1 TS NS1) + %<- (progress T2 TS NS2) + %<- (lemma T1 NS1 T2 NS2 NS) %. +%{! We use both the notstuckness of the subexpressions and the ability to always project from a well-typed store (``of-projection``) to prove the ``gets`` case. !}% +%sort lemma %. +%mode lemma %in %in %in %out %. +%term & lemma _ _ (ns-steps E) (ns-steps (s-bang E)) %. +%term & + %pi (lemma (&of-store TL VL) (t-loc PROJ-T) (ns-isval v-loc) (ns-steps (e-bang PROJ-E))) + %<- (of-projection TL PROJ-T PROJ-E _) %. +%worlds () (lemma _ _ _ _) %. +%total NS1 (lemma ST T NS1 NS) %. +%term prog-bang %pi (progress (t-bang T) TS NS) %<- (progress T TS NS1) %<- (lemma TS T NS1 NS) %. +%{! A location is already a value. !}% +%term prog-loc progress (t-loc PROJ-T) TS (ns-isval v-loc) %. +%{! We use both the notstuckness of the subexpressions and the ability to always update from a well-typed store (``of-update``) to prove the ``gets`` case. !}% +%sort lemma %. +%mode lemma %in %in %in %in %out %. +%term & lemma _ _ (ns-steps E) _ (ns-steps (s1-gets E)) %. +%term & lemma _ _ (ns-isval V) (ns-steps E) (ns-steps (s2-gets E V)) %. +%term & + %pi (lemma (&of-store TL _) (t-loc PROJ) (ns-isval v-loc) (ns-isval (%the (isval E) V)) (ns-steps (e-gets UPD V))) + %<- (of-update TL PROJ E UPD) %. +%worlds () (lemma _ _ _ _ _) %. +%total {} (lemma _ _ _ _ _) %. +%term proj-gets + %pi (progress (t-gets T2 T1) TS NS) + %<- (progress T1 TS NS1) + %<- (progress T2 TS NS2) + %<- (lemma TS T1 NS1 NS2 NS) %. +%{! We use a lemma about the nonstuckness of the subexpression to prove the ``let`` case. !}% +%sort lemma {EF %pi exp %-> exp} %. +%mode lemma %in %in %out %. +%term & lemma _ (ns-steps E) (ns-steps (s-let E)) %. +%term & lemma _ (ns-isval V) (ns-steps (e-let V)) %. +%worlds () (lemma _ _ _) %. +%total NS1 (lemma T NS1 NS) %. +%term prog-let %pi (progress (t-let _ T) TS NS) %<- (progress T TS NS1) %<- (lemma EF NS1 NS) %. +%worlds () (progress _ _ _) %. +%total T (progress T TS NS) %. +%{! +### Substitution + +As in the article on [strengthening](/wiki/strengthening/), progress will need to appeal to a simple substitution lemma. +!}% +%sort substitute {x} %. +%mode substitute %in %in %out %. +%term sub-refl substitute ([x] [v var-of x Tp] T) T* T %. +%term sub-sum + %pi (substitute ([x] [v var-of x Tp] t-add (T2 x v) (T1 x v)) T* (t-add T2’ T1’)) + %<- (substitute ([x] [v var-of x Tp] T2 x v) T* T2’) + %<- (substitute ([x] [v var-of x Tp] T1 x v) T* T1’) %. +%term sub-ref + %pi (substitute ([x] [v var-of x Tp] t-ref (T x v)) T* (t-ref T’)) + %<- (substitute ([x] [v var-of x Tp] T x v) T* T’) %. +%term sub-bang + %pi (substitute ([x] [v var-of x Tp] t-bang (T x v)) T* (t-bang T’)) + %<- (substitute ([x] [v var-of x Tp] T x v) T* T’) %. +%term sub-gets + %pi (substitute ([x] [v var-of x Tp] t-gets (T2 x v) (T1 x v)) T* (t-gets T2' T1')) + %<- (substitute ([x] [v var-of x Tp] T2 x v) T* T2') + %<- (substitute ([x] [v var-of x Tp] T1 x v) T* T1') %. +%term sub-let + %pi (substitute ([x] [v var-of x Tp] t-let (F x v) (T x v)) T* (t-let F’ T’)) + %<- (substitute ([x] [v var-of x Tp] T x v) T* T’) + %<- ({x’} {v’ var-of x’ Tp2} substitute ([x] [v var-of x Tp] F x v x’ v’) T* (F’ x’ v’)) %. +%term sub-var substitute ([x] [v var-of x Tp] t-var v) T* T* %. +%worlds (var-tp) (substitute _ _ _) %. +%total F (substitute F T T’) %. +%{! +### Weakening + +In order to be able to evaluate reference cells, we need to be able to show +that adding new things to the store typing will leave all current programs +well-typed. The only interesting case is ``weak-loc``. +!}% +%sort weakening %. +%mode weakening %in %in %out %. +%term weak-var weakening (t-var V) _ (t-var V) %. +%term weak-int weakening t-int _ t-int %. +%term weak-ref %pi (weakening (t-ref T) S (t-ref T')) %<- (weakening T S T') %. +%term weak-sum + %pi (weakening (t-add T2 T1) S (t-add T2’ T1’)) + %<- (weakening T1 S T1’) + %<- (weakening T2 S T2’) %. +%term weak-bang %pi (weakening (t-bang T) S (t-bang T')) %<- (weakening T S T') %. +%term weak-loc + %pi (weakening (t-loc PROJ) S (t-loc PROJ’)) + %<- (subset-projectable PROJ S PROJ’) %. +%term weak-gets + %pi (weakening (t-gets T2 T1) S (t-gets T2' T1')) + %<- (weakening T2 S T2') + %<- (weakening T1 S T1') %. +%term weak-let + %pi (weakening (t-let F T) S (t-let F’ T’)) + %<- (weakening T S T’) + %<- ({x} {v var-of x Tp} weakening (F x v) S (F’ x v)) %. +%worlds (var-tp) (weakening _ _ _) %. +%total T (weakening T S T’) %. +%{! A comparable notion of weakening must be defined for lists: !}% +%sort weakening-list %. +%mode weakening-list %in %in %out %. +%term weak-z weakening-list tl-z S tl-z %. +%term weak-s + %pi (weakening-list (tl-s TL T) S (tl-s TL’ T’)) + %<- (weakening T S T’) + %<- (weakening-list TL S TL’) %. +%worlds () (weakening-list _ _ _) %. +%total T (weakening-list T S T’) %. +%{! +### Preservation + +!}% +%sort preservation %. +%mode preservation %in %in %in %out %out %out %. +%{! In the cases where evaluation is passed on to a later step, the cases are mostly a straightforward call to the induction hypothesis and the weakening lemma. !}% +%term pres-s1-add + %pi (preservation (t-add T2 T1) ST (s1-add E) S (t-add T2’ T1’) ST’) + %<- (preservation T1 ST E S T1’ ST’) + %<- (weakening T2 S T2’) %. +%term pres-s2-add + %pi (preservation (t-add T2 T1) ST (s2-add E V) S (t-add T2’ T1’) ST’) + %<- (preservation T2 ST E S T2’ ST’) + %<- (weakening T1 S T1’) %. +%term pres-s-ref + %pi (preservation (t-ref T) ST (s-ref E) S (t-ref T’) ST’) + %<- (preservation T ST E S T’ ST’) %. +%term pres-s-bang + %pi (preservation (t-bang T) ST (s-bang E) S (t-bang T’) ST’) + %<- (preservation T ST E S T’ ST’) %. +%term pres-s1-gets + %pi (preservation (t-gets T2 T1) ST (s1-gets E) S (t-gets T2' T1') ST') + %<- (preservation T1 ST E S T1' ST') + %<- (weakening T2 S T2') %. +%term pres-s2-gets + %pi (preservation (t-gets T2 T1) ST (s2-gets E V) S (t-gets T2' T1') ST') + %<- (preservation T2 ST E S T2' ST') + %<- (weakening T1 S T1') %. +%term pres-s-let + %pi (preservation (t-let F T) ST (s-let E) S (t-let F’ T’) ST’) + %<- (preservation T ST E S T’ ST’) + %<- ({x} {v} weakening (F x v) S (F’ x v)) %. +%{! The cases where evaluation happens are more involved—for instance, the ``ref`` case relies on the very involved ``append-lemma`` from the auxillary lemmas. !}% +%term pres-e-add + %pi (preservation (t-add t-int t-int) ST (e-add SUM) S t-int ST) + %<- (subset-refl _ S) %. +%term pres-e-ref + %pi (preservation (t-ref T) (&of-store TL VL) (e-ref AE V) S (t-loc P) (&of-store TL’ VL’)) + %<- (append-lemma V T VL TL AE S VL’ TL’’ P) + %<- (weakening-list TL’’ S TL’) %. +%term pres-e-bang + %pi (preservation (t-bang (t-loc PROJ-T)) (&of-store TL VL) (e-bang PROJ-E) S T (&of-store TL VL)) + %<- (subset-refl _ S) + %<- (of-projection' TL PROJ-T PROJ-E T) %. +%term pres-e-gets + %pi (preservation (t-gets T (t-loc PROJ-T)) (&of-store TL VL) (e-gets UL V) S T (&of-store TL' VL')) + %<- (subset-refl _ S) + %<- (update-lemma V T VL TL PROJ-T UL VL' TL') %. +%term pres-e-let + %pi (preservation (t-let F T’) ST (e-let V) S T ST) + %<- (substitute F T’ T) + %<- (subset-refl _ S) %. +%worlds () (preservation _ _ _ _ _ _) %. +%total T (preservation T ST E S T’ ST’) %. +%{! ### Safety !}% +%sort safety %. +%mode safety %in %in %in %out %. +%term safe-end %pi (safety T ST run-end NS) %<- (progress T ST NS) %. +%term safe-step + %pi (safety T ST (run-step R E) NS) + %<- (preservation T ST E S T’ ST’) + %<- (safety T’ ST’ R NS) %. +%worlds () (safety _ _ _ _) %. +%total R (safety T TS R NS) %. +%{! +## Example + +We will define the following simple program as ``prog1``, and use Twelf's [logic programming](/wiki/logic-programming/) abilities to actually run the typechecker ``of E T`` and multi-step evaluation ``run S E S' E'``. Because ``gets`` returns the result of the assignment in the same way C does, should have the same value as at the end of the program, namely 4. + + +!}% +%define prog1 let (ref (n (s (s z)) + n (s z))) ([x] let (gets x (! x + n (s z))) ([y] ! x + n (s z))) %. +%{! We can typecheck the program (in the empty store typing ``nil-tp``). The syntax we use for [``%query``](/wiki/percent-query/) indicates that we only expect there to be one solution (because type checking should be syntax directed): |check=decl !}% +%query 1 _ _ of nil-tp prog1 T %. +%{! We can then run the program to completion (starting in the empty store ``nil-exp``). We expect the final state ``S`` to include the number 4, and we expect the program to evaluate to value ``V`` 5. The multi-step evaluation judgment ``run`` is defined to run arbitrarily until stopping, so we only want the first solution—it will be the solution resulting in running ``run-step`` as many times as possible: |check=decl !}% +%query 1 1 _ run nil-exp prog1 S V %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Mutable_state). +!}% +%. + +== References == + + + +{{case study}} + diff --git a/new-tests/stelf-output/pages/mutual-induction.lf b/new-tests/stelf-output/pages/mutual-induction.lf new file mode 100644 index 0000000..7c51e1f --- /dev/null +++ b/new-tests/stelf-output/pages/mutual-induction.lf @@ -0,0 +1,80 @@ +% +% +%{! It is common to write two or more mutually recursive type families. For example, an object language may have mutually recursive judgements, or a proof may require mutually recursive theorems. To check the totality of mutually recursive type families, you need to usual a mutual termination metric. This tutorial presents an example of doing so. + +## Motivating example + +As a motivating example, we define a simple subtyping relation on the types of a programming language. The language includes integers, floating point numbers, and functions. We consider ``int`` to be a subtype of ``float``, and we give the usual contravariant rule for functions: !}% +%sort tp %. +%term int tp %. +%term float tp %. +%term arrow %pi tp %-> tp %-> tp %. +%sort sub %. +%term sub-ii sub int int %. +%term sub-ff sub float float %. +%term sub-if sub int float %. +%term sub-arrow %pi (sub (arrow T S) (arrow T' S')) %<- (sub T' T) %<- (sub S S') %. +%{! Let's prove that this subtyping relation is transitive. Our first proof attempt goes as follows: + +```checkedtwelf +sub-trans : sub T1 T2 -> sub T2 T3 -> sub T1 T3 -> type. +%mode sub-trans +X1 +X2 -X3. + +- : sub-trans (D : sub T T) (D' : sub T T) D. +- : sub-trans sub-ii sub-if sub-if. +- : sub-trans sub-if sub-ff sub-if. + +- : sub-trans + (sub-arrow + (DS12 : sub S1 S2) + (DT21 : sub T2 T1)) + (sub-arrow + (DS23 : sub S2 S3) + (DT32 : sub T3 T2)) + (sub-arrow DS13 DT31) + <- sub-trans DT32 DT21 DT31 + <- sub-trans DS12 DS23 DS13. + +%total D (sub-trans D _ _). +``` + +The base cases do the appropriate thing. The case for ``sub-arrow`` against ``sub-arrow`` makes two inductive calls, corresponding to the two components of the arrow type. + +However, the totality check fails! + +To deal with the contravariance, we swapped the two argument derivations in the first inductive call of ``sub-arrow``. However, this means that we have not given a proof by induction on the derivation of ``sub T1 T2``, since in this premise that derivation is not a subterm of the input. This relation cannot be regarded as a proof by induction on the second derivation for the same reason. + +One solution would be to induct over the sum of the sizes of the two derivations, which certainly gets smaller in this case. + +## Mutual induction + +However, a simpler solution, which happens to work in this case, is to prove the transitivity theorem _twice_, once by induction on the first input, and again by induction on the second input, in a mutually inductive loop. These two proofs will call each other in the contravariant premise, so the induction will go through. Here's what the revised proof looks like: !}% +%sort sub-trans %. +%mode sub-trans %in %in %out %. +%sort sub-trans' %. +%mode sub-trans' %in %in %out %. +%% normal +%term _ sub-trans (%the (sub T T) D) (%the (sub T T) D') D %. +%term _ sub-trans sub-ii sub-if sub-if %. +%term _ sub-trans sub-if sub-ff sub-if %. +%term _ + %pi (sub-trans (sub-arrow (%the (sub S1 S2) DS12) (%the (sub T2 T1) DT21)) (sub-arrow (%the (sub S2 S3) DS23) (%the (sub T3 T2) DT32)) (sub-arrow DS13 DT31)) + %<- (sub-trans' DT32 DT21 DT31) + %<- (sub-trans DS12 DS23 DS13) %. +%% prime +%term _ sub-trans' D D D %. +%term _ sub-trans' sub-ii sub-if sub-if %. +%term _ sub-trans' sub-if sub-ff sub-if %. +%term _ + %pi (sub-trans' (sub-arrow (%the (sub S1 S2) DS12) (%the (sub T2 T1) DT21)) (sub-arrow (%the (sub S2 S3) DS23) (%the (sub T3 T2) DT32)) (sub-arrow DS13 DT31)) + %<- (sub-trans DT32 DT21 DT31) + %<- (sub-trans' DS12 DS23 DS13) %. +%worlds () (sub-trans _ _ _) (sub-trans' _ _ _) %. +%total (D D') (sub-trans D _ _) (sub-trans' _ D' _) %. +%{! Because each relation refers to the other, we must use a mutual termination metric. A mutual termination metric has the form ``(D1 D2 ... Dn)``, where there must be as many variables in the termination metric as there are mutual relations. Then, the remainder of the ``%total`` lists each mutual relation, and identifies which argument to each relation is considered to be the induction position. In this case, we say that ``sum-trans`` is by induction on the first argument, whereas ``sum-trans'`` is by induction on the second (though the symmetric declaration would also work). + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Mutual_induction). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/naming-conventions.lf b/new-tests/stelf-output/pages/naming-conventions.lf new file mode 100644 index 0000000..7764536 --- /dev/null +++ b/new-tests/stelf-output/pages/naming-conventions.lf @@ -0,0 +1,102 @@ +% +%{! \{\{needswork|Incomplete commentary; furthermore, a consensus needs to exist around this document before we present it generally\}\} + +These are **naming conventions** for Twelf code. This article should be considered an appendix to/chapter of the [style guide](/wiki/twelf-style-guide/). + +## Syntax + +### The uninhabited type + +Defining an uninhabited type is important for [reasoning from false](/wiki/reasoning-from-false/) - however, the name of that type varies from user to user, with ``void``, ``false``, and ``absurd`` perhaps being the most common. + +Standard usage is to use ``void``, and to freeze the type immediately after its introduction, like this: !}% +%sort void %. +%freeze void %. +%{! ## Judgments + +### Factored judgments + +[Factoring](/wiki/output-factoring/) is a technique used to reduce backtracking. If a rule does some computation, it might then sends the result to another rule that continues based on the output of its computation. The style introduced here works well with factoring - if you are creating "helper judgment," just replace the forward slash in the primary [judgment](/wiki/judgment/) with a hyphen to come up with the name. + +```twelf +eval : state -> state -> type. + +eval-branch-if-nil : list -> state -> state -> type. + +... + +eval-branch-if-nil/nil : eval-branch-if-nil list/nil ... +eval-branch-if-nil/cons : eval-branch-if-nil (list/cons _ _) ... + +eval/branch-if-nil : eval S1 S2 + <- do-some-computation-on-state S1 L + <- eval-branch-if-nil L S1 S2. +``` + +### Introduction rules + +If a type family is inhabited by exactly one rule, it is recommended to use ``i`` (short for introduction) as the case name for the lone inference rule. For example, the [identity](/wiki/equality/) (syntactic equality) [relation](/wiki/relation/) for natural numbers would be defined in the following way. + +```twelf +seq-nat : nat -> nat -> type. + +seq-nat/i : seq-nat N N. +``` + +This also applies to pieces of syntax in the [object language](/wiki/object-logic/), but it appears less frequently there. + +### A note on word order + +Here as in the previous section, there is a legitimate case for reversing the word order, calling a list of natural numbers ``nat-list`` instead of ``list-nat`` on the other, and for calling the three-place relation implementing addition of natural numbers ``nat-plus`` instead of ``plus-nat`` + +The construction that is not used here looks more like declarations would look were a module system is in place (i.e. ``nat::list``, but the first is more readable after combination - for instance ``list-list-nat`` can be read out loud as "a list of lists of nats," whereas the equivalent formulation does not compose as well - ``nat-list-list`` can best be read as "a natlist list." + +Internal consistancy is most important here, but the construction ``(structure)-(object)`` for syntax (i.e. ``list-nat``) and ``(adjective/verb/operation)-(object)`` (i.e. ``wellformed-env``, ``sort-list``, or ``plus-nat``) is preferred. + +### Different types of equality + + +## Metatheorems + +### Metatheorem names + +One way to think about a naming strategy for many metatheorems is that many simple theorems act like admissibility arguments for other rules. For instance, imagine an alternate form of ``plus-nat/z`` that has the zero in the _second_ place instead of the first. It might be written like this if it were a real definition: + +```twelf +plus-nat/z-alt : plus N z N. +``` + +Following that logic, a lemma establishing the admissibility argument might be titled + +```twelf +plus-nat-z-alt : {N} plus-nat N z N -> type. +``` + +Beyond that suggestion, the properties that we wish to prove about our inference rules vary to the point where some measure of consistency is the only reasonable guideline. + +### ``can-`` and ``-unique`` metatheorems + +A judgment behaves like a function (in the meta-logical sense) on particular "input" derivations if it satisfies two properties. The first is that given the "input" derivations, the judgment can be derived, i.e. an [effectiveness lemma](/wiki/effectiveness-lemma/). The second is that given two derivations of the judgments with the same "input" derivations, the results are actually the same, i.e. a [uniqueness lemma](/wiki/uniqueness-lemma/). It is common practice to name the metatheorems proving the first property ``can-judgmentname`` and the second property ``judgmentname-unique``. + +```twelf +can-plus-nat : {N1:nat} {N2:nat} + plus-nat N1 N2 N3 + -> type. +%mode can-plus-nat +D1 +D2 -D3. + +... + +plus-nat-unique : plus-nat N1 N2 N3 + -> plus-nat N1 N2 N3' + -> seq-nat N3 N3 + -> type. +%mode plus-nat-unique +D1 +D2 -D3. + +... +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Naming_conventions). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/natural-numbers-with-inequality.lf b/new-tests/stelf-output/pages/natural-numbers-with-inequality.lf new file mode 100644 index 0000000..ddbc7ec --- /dev/null +++ b/new-tests/stelf-output/pages/natural-numbers-with-inequality.lf @@ -0,0 +1,202 @@ +% +%{! +A signature for natural numbers, adapted from a number of sources, +in particular the TALT project. +There is no arithmetic, just the theory of inequality. +This code uses an uninhabited type, "void", in order to express +[proofs by reductio ad absurdum](/wiki/reasoning-from-false/). +!}% +%{! ## Syntax !}% +%{! ### Natural numbers !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%{! ### The uninhabited type !}% +%sort void %. +%freeze void %. +%{! ### Constants !}% +%define 0 z %. +%define 1 s 0 %. +%{! ... and continuing on to 25 !}% +%{!! begin hidden !!}% +%define 2 s 1 %. +%define 3 s 2 %. +%define 4 s 3 %. +%define 5 s 4 %. +%define 6 s 5 %. +%define 7 s 6 %. +%define 8 s 7 %. +%define 9 s 8 %. +%define 10 s 9 %. +%define 11 s 10 %. +%define 12 s 11 %. +%define 13 s 12 %. +%define 14 s 13 %. +%define 15 s 14 %. +%define 16 s 15 %. +%define 17 s 16 %. +%define 18 s 17 %. +%define 19 s 18 %. +%define 20 s 19 %. +%define 21 s 20 %. +%define 22 s 21 %. +%define 23 s 22 %. +%define 24 s 23 %. +%define 25 s 24 %. +%{!! end hidden !!}% +%{! ## Equality and inequality !}% +%{! ### Definitions !}% +%% equality +%sort id-nat %. +%term id-nat/refl id-nat N N %. +%% less than +%sort lt-nat %. +%term lt-nat/z lt-nat z (s N) %. +%term lt-nat/s %pi (lt-nat (s N1) (s N2)) %<- (lt-nat N1 N2) %. +%% less than or equal to +%sort leq-nat %. +%term leq-nat/eq %pi (leq-nat N1 N2) %<- (id-nat N1 N2) %. +%term leq-nat/lt %pi (leq-nat N1 N2) %<- (lt-nat N1 N2) %. +%% not equal +%sort neq-nat %. +%term neq-nat/gt %pi (neq-nat N1 N2) %<- (lt-nat N2 N1) %. +%term neq-nat/lt %pi (neq-nat N1 N2) %<- (lt-nat N1 N2) %. +%{! ### Theorems !}% +%{! #### False implies anything !}% +%sort false-imp-id-nat %. +%mode false-imp-id-nat %in %in %in %out %. +%worlds () (false-imp-id-nat _ _ _ _) %. +%total {} (false-imp-id-nat _ _ _ _) %. +%sort false-imp-neq-nat %. +%mode false-imp-neq-nat %in %in %in %out %. +%worlds () (false-imp-neq-nat _ _ _ _) %. +%total {} (false-imp-neq-nat _ _ _ _) %. +%sort false-imp-lt-nat %. +%mode false-imp-lt-nat %in %in %in %out %. +%worlds () (false-imp-lt-nat _ _ _ _) %. +%total {} (false-imp-lt-nat _ _ _ _) %. +%sort false-imp-leq-nat %. +%mode false-imp-leq-nat %in %in %in %out %. +%worlds () (false-imp-leq-nat _ _ _ _) %. +%total {} (false-imp-leq-nat _ _ _ _) %. +%{! #### Basic properties !}% +%sort lt-nat-succ {N} %. +%mode lt-nat-succ %in %out %. +%term _ lt-nat-succ z lt-nat/z %. +%term _ %pi (lt-nat-succ (s N) (lt-nat/s D)) %<- (lt-nat-succ N D) %. +%worlds () (lt-nat-succ _ _) %. +%total T (lt-nat-succ T _) %. +%{! #### Reflexivity and symmetry !}% +%sort id-nat/symm %. +%mode id-nat/symm %in %out %. +%term _ id-nat/symm id-nat/refl id-nat/refl %. +%worlds () (id-nat/symm _ _) %. +%total {} (id-nat/symm _ _) %. +%sort id-nat/trans %. +%mode id-nat/trans %in %in %out %. +%term _ id-nat/trans id-nat/refl id-nat/refl id-nat/refl %. +%worlds () (id-nat/trans _ _ _) %. +%total {} (id-nat/trans _ _ _) %. +%sort neq-nat/symm %. +%mode neq-nat/symm %in %out %. +%term _ neq-nat/symm (neq-nat/lt D) (neq-nat/gt D) %. +%term _ neq-nat/symm (neq-nat/gt D) (neq-nat/lt D) %. +%worlds () (neq-nat/symm _ _) %. +%total {} (neq-nat/symm _ _) %. +%sort lt-nat/trans %. +%mode lt-nat/trans %in %in %out %. +%term _ lt-nat/trans lt-nat/z _ lt-nat/z %. +%term _ + %pi (lt-nat/trans (lt-nat/s D1) (lt-nat/s D2) (lt-nat/s D3)) + %<- (lt-nat/trans D1 D2 D3) %. +%worlds () (lt-nat/trans _ _ _) %. +%total T (lt-nat/trans T _ _) %. +%sort leq-nat/trans %. +%mode leq-nat/trans %in %in %out %. +%term _ leq-nat/trans (leq-nat/eq _) D D %. +%term _ leq-nat/trans D (leq-nat/eq _) D %. +%term _ + %pi (leq-nat/trans (leq-nat/lt D1) (leq-nat/lt D2) (leq-nat/lt D3)) + %<- (lt-nat/trans D1 D2 D3) %. +%worlds () (leq-nat/trans _ _ _) %. +%total {} (leq-nat/trans _ _ _) %. +%{! #### Respects lemmas !}% +%{! This is an instance of the generalization technique +described in the page on [respects lemma](/wiki/respects-lemma/)s. !}% +%sort id-nat/compat {F %pi nat %-> nat} %. +%mode id-nat/compat %in %in %out %. +%term _ id-nat/compat _ id-nat/refl id-nat/refl %. +%worlds () (id-nat/compat _ _ _) %. +%total {} (id-nat/compat _ _ _) %. +%inline id-nat/inc (%pi (id-nat N1 N2) %-> (id-nat (s N1) (s N2)) %-> %type) id-nat/compat s %. +%sort id-nat/dec %. +%mode id-nat/dec %in %out %. +%term _ id-nat/dec id-nat/refl id-nat/refl %. +%worlds () (id-nat/dec _ _) %. +%total {} (id-nat/dec _ _) %. +%sort leq-nat-resp %. +%mode leq-nat-resp %in %in %in %out %. +%term _ leq-nat-resp id-nat/refl id-nat/refl D D %. +%worlds () (leq-nat-resp _ _ _ _) %. +%total {} (leq-nat-resp _ _ _ _) %. +%sort lt-nat-resp %. +%mode lt-nat-resp %in %in %in %out %. +%term _ lt-nat-resp id-nat/refl id-nat/refl D D %. +%worlds () (lt-nat-resp _ _ _ _) %. +%total {} (lt-nat-resp _ _ _ _) %. +%sort neq-nat-resp %. +%mode neq-nat-resp %in %in %in %out %. +%term _ neq-nat-resp id-nat/refl id-nat/refl D D %. +%worlds () (neq-nat-resp _ _ _ _) %. +%total {} (neq-nat-resp _ _ _ _) %. +%{! #### Contradictions !}% +%sort lt-nat-contr %. +%mode lt-nat-contr %in %out %. +%term _ %pi (lt-nat-contr (lt-nat/s D) F) %<- (lt-nat-contr D F) %. +%worlds () (lt-nat-contr _ _) %. +%total T (lt-nat-contr T _) %. +%sort neq-nat-contr %. +%mode neq-nat-contr %in %out %. +%term _ %pi (neq-nat-contr (neq-nat/lt D) F) %<- (lt-nat-contr D F) %. +%term _ %pi (neq-nat-contr (neq-nat/gt D) F) %<- (lt-nat-contr D F) %. +%worlds () (neq-nat-contr _ _) %. +%total T (neq-nat-contr T _) %. +%sort lt-gt-nat-contr %. +%mode lt-gt-nat-contr %in %in %out %. +%term _ %pi (lt-gt-nat-contr (lt-nat/s D1) (lt-nat/s D2) F) %<- (lt-gt-nat-contr D1 D2 F) %. +%worlds () (lt-gt-nat-contr _ _ _) %. +%total T (lt-gt-nat-contr T _ _) %. +%sort lt-leq-nat-contr %. +%mode lt-leq-nat-contr %in %in %out %. +%term _ %pi (lt-leq-nat-contr D1 (leq-nat/lt D2) F) %<- (lt-gt-nat-contr D1 D2 F) %. +%term _ %pi (lt-leq-nat-contr D1 (leq-nat/eq D2) F) %<- (lt-nat-contr D1 F) %. +%worlds () (lt-leq-nat-contr _ _ _) %. +%total T (lt-leq-nat-contr T _ _) %. +%{! #### Dichotomy !}% +%{! The property that any two numbers are comparable by lt, eq, or lt +the other way around. Because leq and neq are defined in terms of lt and eq, +this should allow relatively simple assessment of these cases as well. !}% +%sort dichotomy-nat %. +%term dichotomy-nat/lt %pi (lt-nat N1 N2) %-> (dichotomy-nat N1 N2) %. +%term dichotomy-nat/gt %pi (lt-nat N1 N2) %-> (dichotomy-nat N2 N1) %. +%term dichotomy-nat/id %pi (id-nat N1 N2) %-> (dichotomy-nat N1 N2) %. +%sort can-dichotomy-nat/s %. +%mode can-dichotomy-nat/s %in %out %. +%term _ can-dichotomy-nat/s (dichotomy-nat/gt D) (dichotomy-nat/gt (lt-nat/s D)) %. +%term _ can-dichotomy-nat/s (dichotomy-nat/lt D) (dichotomy-nat/lt (lt-nat/s D)) %. +%term _ + %pi (can-dichotomy-nat/s (dichotomy-nat/id D) (dichotomy-nat/id D')) + %<- (id-nat/inc D D') %. +%worlds () (can-dichotomy-nat/s _ _) %. +%total {} (can-dichotomy-nat/s _ _) %. +%sort can-dichotomy-nat {N1} {N2} %. +%mode can-dichotomy-nat %in %in %out %. +%term _ can-dichotomy-nat z z (dichotomy-nat/id id-nat/refl) %. +%term _ can-dichotomy-nat (s _) z (dichotomy-nat/gt lt-nat/z) %. +%term _ can-dichotomy-nat z (s _) (dichotomy-nat/lt lt-nat/z) %. +%term _ + %pi (can-dichotomy-nat (s N1) (s N2) D') + %<- (can-dichotomy-nat N1 N2 D) + %<- (can-dichotomy-nat/s D D') %. +%worlds () (can-dichotomy-nat _ _ _) %. +%total T (can-dichotomy-nat T _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/pages/natural-numbers.lf b/new-tests/stelf-output/pages/natural-numbers.lf new file mode 100644 index 0000000..5f2c199 --- /dev/null +++ b/new-tests/stelf-output/pages/natural-numbers.lf @@ -0,0 +1,86 @@ +% +%{! The **natural numbers** are the numbers 0, 1, 2, etc. The term is generally used to indicate a specific technique of representing natural numbers as either zero or the successor of some other natural number - 0, s(0), s(s(0)), etc - as in Peano arithmetic, a technique also sometimes referred to as **unary numbers.** + +## Natural numbers in Twelf + +Natural numbers in Twelf are usually defined in a similar way. Mathematically, natural numbers can be defined as zero or the successor of some other natural number: + +```math +n ::= 0 \,|\, \texttt{s}(n) +``` + +This representation translates easily into Twelf: !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%{! The first line declares that ``nat`` is a type. The second line declares ``z`` (zero) to be an object of type ``nat``, and the third line declars ``s`` (successor) to be a type constructor that takes an object ``N`` of type ``nat`` and produces another object ``(s N)`` of type ``nat``. + +## Addition of natural numbers in Twelf + +The addition of these natural numbers is defined by the judgment , where , , and are natural numbers. In the definition below, capital letters stand for metavariables that can range over all natural numbers. + +```math +{ + \over + \texttt{plus}(0,N,N) +}\mathit{p\textit{-}z} +``` + +```math +{ + \texttt{plus}(N_1,N_2,N_3) + \over + \texttt{plus}(\texttt{s}(N_1),N_2,\texttt{s}(N_3)) +}{\mathit{p\textit{-}s}} +``` + +These judgments also translate cleanly into Twelf: !}% +%{!! begin checked !!}% +%sort plus %. +%term p-z plus z N N %. +%term p-s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%{!! end checked !!}% +%{! The first line defines the judgment, declaring ``plus`` to be a type family indexed by three terms of type ``nat``. + +The second line declares that for any natural number ``N``, ``p-z`` is an object of type ``plus z N N``, which corresponds to the axiom p-z above. The ``N`` is an [implicit parameter](/wiki/implicit-and-explicit-parameters/) - it is treated as a bound variable by Twelf, which you can see by looking at Twelf's output after checking the above code. + +The third line says that ``p-s`` is a type constructor that, given an object ``D`` of type ``plus N1 N2 N3`` (where ``N1``, ``N2``, and ``N3`` are all implicit parameters that can +be treated as metavariables), produces an object, ``p-s D``, with type ``plus (s N1) N2 (s N3)``. This corresponds to the rule p-s, which given a derivation of allows us to conclude . + +Consider this derivation which encodes the fact that : + + +```math +{ + { + \over + { + \texttt{plus}(0,\;\;\texttt{s}(0),\;\;\texttt{s}(0)) + } + }{\mathit{p\textit{-}z}} + \over + { + { + \texttt{plus}(\texttt{s}(0),\;\;\texttt{s}(0),\;\;\texttt{s}(\texttt{s}(0))) + } + \over + { + \texttt{plus}(\texttt{s}(\texttt{s}(0)),\;\;\texttt{s}(0),\;\;\texttt{s}(\texttt{s}(\texttt{s}(0)))) + } + }{\mathit{p\textit{-}s}} +}{\mathit{p\textit{-}s}} +``` + +This can be represented in Twelf by applying the type constructor ``p-s`` to the object ``p-z`` twice: !}% +%{!! begin checked !!}% +%define 2+1=3 (plus (s (s z)) (s z) (s (s (s z)))) p-s (p-s p-z) %. +%{!! end checked !!}% +%{! ## See also + +* [Natural numbers with inequality](/wiki/natural-numbers-with-inequality/) +* [Division over the natural numbers](/wiki/division-over-the-natural-numbers/) +* [Proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/), which uses natural numbers as an example, and also discusses the [adequacy](/wiki/adequacy/) of the encoding. + +## External link + +* [Natural numbers at Wikipedia](http://en.wikipedia.org/wiki/Natural_numbers) !}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/negation-as-failure.lf b/new-tests/stelf-output/pages/negation-as-failure.lf new file mode 100644 index 0000000..02f1a3c --- /dev/null +++ b/new-tests/stelf-output/pages/negation-as-failure.lf @@ -0,0 +1,71 @@ +% +%{! +Negation as failure. It is possible to do negation-as-failure using Twelf's [logic programming](/wiki/logic-programming/) engine, with the use of [%deterministic](/wiki/deterministic-declaration/). As an example, we will define logic programs to compute the less-than and greater-than-or-equal-to functions. The less-than function will be defined in a standard way. The greater-than-or-equal-to function will be defined using a negation-as-failure interpretation of the less-function. +!}% +%{! +### Definitions + +Natural numbers and booleans. +!}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort bool %. +%term true bool %. +%term false bool %. +%{! +``less`` is defined inductively in the standard way. +!}% +%sort less %. +%mode less %in %in %. +%term less/z less z (s _) %. +%term less/s %pi (less (s N1) (s N2)) %<- (less N1 N2) %. +%{! +We will need a judgment that tests whether a boolean is ``false`` in order to use negation-as-failure. +!}% +%sort isfalse %. +%mode isfalse %in %. +%term isfalse/i isfalse false %. +%{! +### Coding negation-as-failure + +We define a logic program that when given two numbers returns ``true`` if the first is less than the second, and ``false`` otherwise. We use ``%deterministic`` to make the logic programming engine commit to the first solution it finds. +!}% +%sort less-bool %. +%mode less-bool %in %in %out %. +%deterministic less-bool %. +%{! +Because the ``less-bool/true`` case is first, it will attempt to find a proof that the first number is less than the second. +!}% +%term less-bool/true %pi (less-bool N1 N2 true) %<- (less N1 N2) %. +%{! +Because of the ``%deterministic`` declaration, when searching for a proof of ``less N1 N2 B``, only executes when ``less-bool/true`` fails. However, because of pattern matching a search for a proof of ``less N1 N2 false`` always succeeds. +!}% +%term less-bool/false less-bool N1 N2 false %. +%{! +We will now define ``gte`` using ``less-bool``. It has only one rule, which makes a call to ``less-bool N1 N2 B``. It is important to make sure that the result ``B`` is not directly identified as ``false`` so that it executes ``less-bool`` in the appropriate order. We use the call to ``isfalse B`` to verify that the output really is ``false``. +!}% +%sort gte %. +%mode gte %in %in %. +%term gte/i %pi (gte N1 N2) %<- (less-bool N1 N2 B) %<- (isfalse B) %. +%{! +We can use a number of ``%solve`` declarations to test our ``less`` and ``gte`` judgments. +!}% +%solve deriv : less (s z) (s (s z)) %. +%solve deriv1 : gte (s (s z)) (s z) %. +%solve deriv2 : gte z z %. +% solve deriv3 : gte (s z) (s (s z)). % should fail +%{! +It is important to note that while these definitions work as intended as logic programs, proving appropriate [[meta-theorems]] about judgments that use negation as failure is problematic or impossible. + +**Note from dklee: I had to run home to take care of some stuff. I will finish documenting this later in the evening.** + +\{\{stub\}\} + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Negation_as_failure). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/numeric-termination-metrics.lf b/new-tests/stelf-output/pages/numeric-termination-metrics.lf new file mode 100644 index 0000000..d9fd73d --- /dev/null +++ b/new-tests/stelf-output/pages/numeric-termination-metrics.lf @@ -0,0 +1,158 @@ +% +% +%{! Sometimes, a proof proceeds not by a direct induction on some assumption, but by induction on some size function computed from an assumption. To mechanize such a proof in Twelf, you must make the size function explicit in the statement of the theorem. + +This tutorial presents an example of such a proof. We show a fragment of a proof of confluence for a λ-calculus with typed η-expansion. The proof inducts on the size of a reduction derivation. Moreover, the proof uses [%reduces](/wiki/percent-reduces/) to tell the termination checker that addends are subterms of their sum. In general, a [%reduces](/wiki/percent-reduces/) declaration is necessary whenever the computation of a numeric termination metric uses an auxiliary relation like addition or maximum. See the tutorial on [structural termination metrics](/wiki/structural-metrics/) for another approach to termination metrics. + +## Language Definition + +The syntax, typing judgement, and reduction relation for the language are straightforward: !}% +%% Syntax +%sort tp %. +%name tp %. +%term o tp %. +%term arrow %pi tp %-> tp %-> tp %. +%sort exp %. +%name exp %. +%term lam %pi tp %-> (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%% Static Semantics +%sort of %. +%term of_lam %pi (of (lam T1 E) (arrow T1 T2)) %<- ({x exp} %pi (of x T1) %-> (of (E x) T2)) %. +%term of_app %pi (of (app E1 E2) T2) %<- (of E1 (arrow T1 T2)) %<- (of E2 T1) %. +%% Dynamic Semantics +%sort reduce %. +%term reduce_id reduce E E %. +%term reduce_lam + %pi (reduce (lam T E) (lam T E')) + %<- ({x exp} %pi (of x T) %-> (reduce (E x) (E' x))) %. +%term reduce_app %pi (reduce (app E1 E2) (app E1' E2')) %<- (reduce E1 E1') %<- (reduce E2 E2') %. +%term reduce_beta + %pi (reduce (app (lam T E1) E2) (E1' E2')) + %<- ({x exp} %pi (of x T) %-> (reduce (E1 x) (E1' x))) + %<- (reduce E2 E2') %. +%term reduce_eta %pi (reduce E (lam T1 ([x] app E' x))) %<- (of E (arrow T1 T2)) %<- (reduce E E') %. +%{! The judgement ``reduce`` defines a parallel, reflexive reduction relation with typed η-expansion. + +## Facts about numbers + +In the proof below, we induct on the size of a reduction derivation. To get this induction to go through, we require some facts about addition on natural numbers. + +First, we define addition: !}% +%sort nat %. +%name nat %. +%term 0 nat %. +%term s %pi nat %-> nat %. +%define 1 nat s 0 %. +%sort sum %. +%term sum_0 sum 0 N N %. +%term sum_s %pi (sum (s N1) N2 (s N3)) %<- (sum N1 N2 N3) %. +%{! For the proof below, we need a way to tell Twelf's termination checker that summands are subterms of their sum. We do that by proving a lemma with a [%reduces](/wiki/percent-reduces/) declaration. + +We prove the lemma for the second summand first. Note that all arguments of this lemma are inputs; the only "output" is the fact that the ``%reduces`` holds: !}% +%sort sum_reduces2 {N1 nat} {N2 nat} {N3 nat} %. +%mode sum_reduces2 %in %in %in %in %. +%term _ sum_reduces2 _ _ _ sum_0 %. +%term _ %pi (sum_reduces2 (s N1) N2 (s N3) (sum_s D)) %<- (sum_reduces2 N1 N2 N3 D) %. +%worlds () (sum_reduces2 _ _ _ _) %. +%total D (sum_reduces2 _ _ _ D) %. +%reduces <= N2 N3 (sum_reduces2 N1 N2 N3 _) %. +%{! The easiest way to prove the lemma for the first summand is to commute the addition and appeal to the previous lemma. We state commutativity as + +```twelf +sum_commute : sum N1 N2 N3 -> sum N2 N1 N3 -> type. +%mode sum_commute +D1 -D2. +%worlds () (sum_commute _ _). +``` + +but elide its proof. !}% +%{!! begin hidden !!}% +%sort sum_ident {n nat} %. +%mode sum_ident %in %out %. +%scope sum_ident %term 0 sum_ident 0 sum_0 %. +%term s %pi (sum_ident (s N) (sum_s D)) %<- (sum_ident N D) %. +%worlds () (sum_ident _ _) %. +%total N (sum_ident N _) %. +%sort sum_inc %. +%mode sum_inc %in %out %. +%scope sum_inc %term 0 sum_inc sum_0 sum_0 %. +%term s %pi (sum_inc (sum_s D) (sum_s D')) %<- (sum_inc D D') %. +%worlds () (sum_inc _ _) %. +%total D (sum_inc D _) %. +%sort sum_commute %. +%mode sum_commute %in %out %. +%scope sum_commute %term 0 %pi (sum_commute (%the (sum 0 N N) sum_0) D) %<- (sum_ident N D) %. +%term s + %pi (sum_commute (%the (sum (s N1) N2 (s N3)) (sum_s D)) D'') + %<- (sum_commute D (%the (sum N2 N1 N3) D')) + %<- (sum_inc D' D'') %. +%worlds () (sum_commute _ _) %. +%total D (sum_commute D _) %. +%{!! end hidden !!}% +%sort sum_reduces1 {N1 nat} {N2 nat} {N3 nat} %. +%mode sum_reduces1 %in %in %in %in %. +%term _ + %pi (sum_reduces1 N1 N2 N3 Dsum) + %<- (sum_commute Dsum Dsum') + %<- (sum_reduces2 N2 N1 N3 Dsum') %. +%worlds () (sum_reduces1 _ _ _ _) %. +%total {} (sum_reduces1 _ _ _ _) %. +%reduces <= N1 N3 (sum_reduces1 N1 N2 N3 _) %. +%{! ## Proof of confluence + +We now show part of the proof of the diamond property for this notion of reduction. The proof requires a metric computing the size of a reduction derivation. + +### Definition of the metric !}% +%sort reduce_metric %. +%term reduce_metric_id reduce_metric reduce_id 1 %. +%term reduce_metric_lam + %pi (reduce_metric (reduce_lam D) (s N)) + %<- ({x exp} {d of x T} reduce_metric (D x d) N) %. +%term reduce_metric_app + %pi (reduce_metric (reduce_app D2 D1) (s N)) + %<- (reduce_metric D1 N1) + %<- (reduce_metric D2 N2) + %<- (sum N1 N2 N) %. +%term reduce_metric_beta + %pi (reduce_metric (reduce_beta D2 D1) (s N)) + %<- ({x exp} {d of x T} reduce_metric (D1 x d) N1) + %<- (reduce_metric D2 N2) + %<- (sum N1 N2 N) %. +%term reduce_metric_eta %pi (reduce_metric (reduce_eta D _) (s N)) %<- (reduce_metric D N) %. +%{! ### Excerpt of the proof !}% +%sort diamond {N1 nat} {N2 nat} {D1 reduce E E1} {D2 reduce E E2} %. +%mode diamond %in %in %in %in %in %in %out %out %. +%term _ diamond _ _ reduce_id D _ _ D reduce_id %. +%term _ diamond _ _ D reduce_id _ _ reduce_id D %. +%term _ + %pi (diamond (s N1) (s N2) (reduce_lam D1) (reduce_lam D2) (reduce_metric_lam DM1) (reduce_metric_lam DM2) (reduce_lam D1') (reduce_lam D2')) + %<- ({x exp} {d of x T} diamond N1 N2 (D1 x d) (D2 x d) (DM1 x d) (DM2 x d) (D1' x d) (D2' x d)) %. +%term _ + %pi (diamond (s N1) (s N2) (reduce_app (%the (reduce E2 E21) D21) (%the (reduce E1 E11) D11)) (reduce_app (%the (reduce E2 E22) D22) (%the (reduce E1 E12) D12)) (reduce_metric_app (%the (sum N11 N21 N1) Dsum1) (%the (reduce_metric D21 N21) DM21) (%the (reduce_metric D11 N11) DM11)) (reduce_metric_app (%the (sum N12 N22 N2) Dsum2) (%the (reduce_metric D22 N22) DM22) (%the (reduce_metric D12 N12) DM12)) (reduce_app D21' D11') (reduce_app D22' D12')) + %<- (sum_reduces1 N11 N21 N1 Dsum1) + %<- (sum_reduces2 N11 N21 N1 Dsum1) + %<- (sum_reduces1 N12 N22 N2 Dsum2) + %<- (sum_reduces2 N12 N22 N2 Dsum2) + %<- (diamond N11 N12 D11 D12 DM11 DM12 D11' D12') + %<- (diamond N21 N22 D21 D22 DM21 DM22 D21' D22') %. +%% fill in remaining cases +%block bind {t tp} [x exp] [d of x t]%. +%worlds (bind) (diamond _ _ _ _ _ _ _ _) %. +%terminates [N1 N2] (diamond N1 N2 _ _ _ _ _ _) %. +%{! The ``reduce_app`` against ``reduce_app`` case illustrates why we need to know that summands are subterms of their sum: the inductive calls are on the summands that add up to the size of the overall derivation. If we elided the calls to ``sum_reduces*``, the case would not termination-check, because Twelf would not be able to tell that, for example, ``N11 < (s N1)``. + +In other cases, which we have elided, the termination metric gets smaller but the reduction derivations themselves do not. + +### Cleanup + +We would like an overall theorem: !}% +%sort diamond/clean %. +%mode diamond/clean %in %in %out %out %. +%worlds (bind) (diamond/clean _ _ _ _) %. +%{! It is simple to prove this theorem using the above if we prove an [effectiveness lemma](/wiki/effectiveness-lemma/) for ``reduce_metric``. + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Numeric_termination_metrics). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/object-logic.lf b/new-tests/stelf-output/pages/object-logic.lf new file mode 100644 index 0000000..e689916 --- /dev/null +++ b/new-tests/stelf-output/pages/object-logic.lf @@ -0,0 +1,17 @@ +% +%{! An **object logic** is a logic encoded as an [LF](/wiki/lf/) signature, with the intent of studying it with Twelf. The term "logic" is used very broadly here, as it can refer to any **deductive system** with an [abstract syntax](/wiki/abstract-syntax/) and a set of [relation](/wiki/relation/)s that can be encoded in LF. We use the term **object language** as a synonym. Examples of object logics include arithmetic over [natural numbers](/wiki/natural-numbers/), a theory of [lists](/wiki/lists/), propositional logic, [linear logic](/wiki/linear-logic/), or programming languages like the [simply-typed lambda calculus](/wiki/simply-typed-lambda-calculus/). + +In general, anything defined using an LF signature is an object logic of some form. The term object logic is used to contrast the logic from the [meta-logic](/wiki/meta-logic/) used to encode it. + +## See also + +* [Proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/) +* [Meta-logic](/wiki/meta-logic/) + +\{\{stub\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Object_logic). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/output-factoring.lf b/new-tests/stelf-output/pages/output-factoring.lf new file mode 100644 index 0000000..8338b51 --- /dev/null +++ b/new-tests/stelf-output/pages/output-factoring.lf @@ -0,0 +1,134 @@ +% +% +%{! When checking coverage of [metatheorem](/wiki/metatheorem/)s, a common problem arises because the [output coverage](/wiki/coverage-checking/) checker only considers each rule in isolation. The proof transformation technique for addressing this problem is known as **output factoring**. + +The primary symptom of the problem is if Twelf fails when checking an %total declaration and gives this sort of error: + +``` +Totality: Output of subgoal not covered +Output coverage error ... +``` + +## Even and odd + +In this section, we prove that every natural number is even or odd. First, we define the judgements: !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort odd %. +%sort even %. +%term z-e even z %. +%term s-o %pi (odd (s X)) %<- (even X) %. +%term s-e %pi (even (s X)) %<- (odd X) %. +%sort even_or_odd %. +%term eoo-e %pi (even_or_odd X) %<- (even X) %. +%term eoo-o %pi (even_or_odd X) %<- (odd X) %. +%{! + +### Incorrect proof + +Next, we attempt the following theorem: + +```twelf +always_even_or_odd : {N:nat} even_or_odd N -> type. +%mode always_even_or_odd +D -P. + +aeo_zero : always_even_or_odd z (eoo-e z-e). + +aeo_even : always_even_or_odd (s X) (eoo-e (s-e Y)) + <- always_even_or_odd X (eoo-o Y). + +aeo_odd : always_even_or_odd (s X) (eoo-o (s-o Y)) + <- always_even_or_odd X (eoo-e Y). + +%worlds () (always_even_or_odd D P). + +%total D (always_even_or_odd D _). +``` + +Unfortunately, the totality check fails. + +The line number in Twelf's error message points to the premise of ``aeo_even``. The output of ``always_even_or_odd`` can be either ``eoo-o _`` or ``eoo-e _``, but this premise pattern-matches as if the output is always ``eoo-o _``. The constant ``aeo_odd`` covers the alternative case when the output is ``eoo-e _``. + +Unfortunately, Twelf's output-coverage checker does not notice this sort of multiple-constant output coverage: the output-coverage checker only accepts a relation if each constant covers all outputs of its premises. + +### Correct proof + +Nonetheless, we need to case-analyze this output in order to complete the proof. How can we do so? Well, Twelf does not allow output coverage to be split across constants, but it certainly allows **input coverage** to be split across constants. Thus, we can solve the problem by turning an output-coverage checking problem into an input-coverage checking problem. We do this by writing an additional helper lemma. + +Looking at what the two constants do with the output of the inductive call, we can see that in each case they turn a derivation of ``even_or_odd X`` into a derivation of ``even_or_odd (s X)``—it's just that they do it differently depending on what the output is. Thus, we factor this reasoning into a lemma: !}% +%sort lemma %. +%mode lemma %in %out %. +%term _ lemma (eoo-o Y) (eoo-e (s-e Y)) %. +%term _ lemma (eoo-e Y) (eoo-o (s-o Y)) %. +%worlds () (lemma _ _) %. +%total X (lemma X _) %. +%{! Using this lemma, we complete the proof as follows: !}% +%sort always_even_or_odd {N nat} %. +%mode always_even_or_odd %in %out %. +%term aeo_zero always_even_or_odd z (eoo-e z-e) %. +%term aeo_succ %pi (always_even_or_odd (s X) D') %<- (always_even_or_odd X D) %<- (lemma D D') %. +%worlds () (always_even_or_odd D P) %. +%total D (always_even_or_odd D _) %. +%{! ## Progress + +Another example where output factoring comes up is the progress theorem for a programming language: + +: If then or . + +A typical case of progress makes an inductive call on a subderivation and then case-analyzes whether the result is a value or takes a step. This reasoning must be factored off into lemmas to avoid output coverage problems. + +Using the simply typed λ-calculus defined in \{\{proving|title=Representing the judgements of the STLC\}\}, we now show how output factoring is used to prove progress. For review, here is the LF signature for the STLC: !}% +%% Syntax +%sort tp %. +%term unit tp %. +%term arrow %pi tp %-> tp %-> tp %. +%sort tm %. +%term empty tm %. +%term lam %pi tp %-> (%pi tm %-> tm) %-> tm %. +%term app %pi tm %-> tm %-> tm %. +%% Static Semantics +%sort of %. +%term of_empty of empty unit %. +%term of_lam %pi (of (lam T2 ([x] E x)) (arrow T2 T)) %<- ({x tm} {dx of x T2} of (E x) T) %. +%term of_app %pi (of (app E1 E2) T) %<- (of E2 T2) %<- (of E1 (arrow T2 T)) %. +%% Dynamic Semantics +%sort value %. +%term value_empty value empty %. +%term value_lam value (lam T2 ([x] E x)) %. +%sort step %. +%term step_app_1 %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step_app_2 %pi (step (app E1 E2) (app E1 E2')) %<- (step E2 E2') %<- (value E1) %. +%term step_app_beta %pi (step (app (lam T2 ([x] E x)) E2) (E E2)) %<- (value E2) %. +%{! The result of progress is represented by the following sum type: !}% +%sort val_or_step %. +%term vos_val %pi (val_or_step E) %<- (value E) %. +%term vos_step %pi (val_or_step E) %<- (step E E') %. +%{! As discussed above, we first prove a factoring lemma that does all the work after the inductive call of the application case: !}% +%sort progress/app %. +%mode progress/app %in %in %in %out %. +%term pa_step_1 progress/app _ (vos_step DstepE1) _ (vos_step (step_app_1 DstepE1)) %. +%term pa_val_1_step_2 progress/app _ (vos_val DvalE1) (vos_step DstepE2) (vos_step (step_app_2 DvalE1 DstepE2)) %. +%term pa_val_val progress/app (%the (of (lam T2' ([x] E x)) (arrow T2 T)) DofE1) (vos_val (%the (value (lam T2' ([x] E x))) DvalE1)) (vos_val DvalE2) (vos_step (step_app_beta DvalE2)) %. +%worlds () (progress/app _ _ _ _) %. +%total {} (progress/app _ _ _ _) %. +%{! We give this lemma the typing derivation for ``E1`` so that we can learn in the case ``pa_val_val`` that the ``E1`` is a ``lam``, which is necessary to apply ``step_app_beta``. Note that there is no need for an explicit value inversion/canonical forms lemma: in the case ``pa_val_val``, we simply assume that ``E1`` is a ``lam``, and the coverage checker justifies this assumption because it is both a ``value`` and has type ``arrow T2 T``. + +Using this lemma, the proof of progress is quite simple: !}% +%sort progress %. +%mode progress %in %out %. +%term prog_empty progress of_empty (vos_val value_empty) %. +%term prog_lam progress (of_lam ([x] [dx] DofE x dx)) (vos_val value_lam) %. +%term prog_app + %pi (progress (of_app (%the (of E1 (arrow T2 T)) DofE1) (%the (of E2 T2) DofE2)) DvosApp) + %<- (progress DofE1 (%the (val_or_step E1) DvosE1)) + %<- (progress DofE2 (%the (val_or_step E2) DvosE2)) + %<- (progress/app DofE1 DvosE1 DvosE2 DvosApp) %. +%worlds () (progress _ _) %. +%total D (progress D _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Output_factoring). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/output-freeness.lf b/new-tests/stelf-output/pages/output-freeness.lf new file mode 100644 index 0000000..563f237 --- /dev/null +++ b/new-tests/stelf-output/pages/output-freeness.lf @@ -0,0 +1,80 @@ +% +% +%{! +When Twelf [proves that an LF type family defines a total relation](/wiki/percent-total/), it checks that the output of each premise of each constant can never fail to unify with an output that is actually produced. This is called [output coverage](/wiki/coverage-checking/) checking. One way in which a premise can fail output coverage checking is if its output is a metavariable that is constrained by appearing elsewhere in the constant. + +**Output freeness checking** ensures that no outputs are constrained in such a manner. More precisely, the output freeness check ensures that a metavariable in an output position does not occur in any previous input or output (where "previous" is the same order used in [%mode](/wiki/percent-mode/) checking—i.e., the search order of the [logic programming](/wiki/logic-programming/) operational semantics). + +Prior to Twelf 1.5, output freeness checking was not implemented, allowing some false metatheorems to check. + +## Simple example + +As a first output freeness violation, consider the natural numbers + +!}% +%sort nat %. +%term s %pi nat %-> nat %. +%term z nat %. +%{! + +and following buggy definition of the addition relation: + +```checkedtwelf +add : nat -> nat -> nat -> type. +%mode add +M +N -O. + +add/z : add M z M. + +add/s-incorrect : add M (s N) (s N) + <- add M N N. + +%worlds () (add _ _ _). +%total N (add _ N _). +``` + +This type family ``add`` clearly does not define a total relation: the constant ``add/s-incorrect`` only applies when ``add M N N``, but in general it is possible to derive ``add M N O`` for ``O`` not equal to ``N`` (and the constant ``add-s-incorrect`` is the only constant that covers the case when the second number is a successor, so the relation is clearly not total without this constant). + +Thus, Twelf reports an output coverage error on this constant when we attempt to check totality. +Specifically, an output freeness error: the output metavariable ``N`` occurs previously because ``N`` is an input to the relation. + +## Uniqueness example + +It is common to encounter output freeness errors when working with relations that have unique outputs. For example, consider a correct definition of ``add``: !}% +%sort add %. +%mode add %in %in %out %. +%term add/z add M z M %. +%term add/s-incorrect %pi (add M (s N) (s O)) %<- (add M N O) %. +%worlds () (add _ _ _) %. +%total N (add _ N _) %. +%{! This relation satisfies a [uniqueness lemma](/wiki/uniqueness-lemma/) stating that the first two indices (the summands) uniquely determine the third (the sum). + +Now, suppose we use ``add`` to define another relation: + +```checkedtwelf +add2 : nat -> nat -> type. +%mode add2 +N -O. + +- : add2 N O + <- add (s (s z)) N O + <- add (s (s z)) N O. + +%worlds () (add2 _ _). +%total {} (add2 _ _). +``` + +Here, we have unnecessarily copied the ``add`` premise twice. However, because ``add`` satisfies the aforementioned uniqueness lemma, it is, in fact, correct to insist that we get the same output ``O`` in each case. Unfortunately, Twelf is not aware of this uniqueness lemma, so it flags an error. +The second output occurrence of ``O`` occurs previously in the first one. + +This example, where we explicitly repeat a premise, is clearly avoidable. However, analogous situations do come up in practice, and to work around them, you must prove and use the [uniqueness lemma](/wiki/uniqueness-lemma/) explicitly. + +## Non-free implicit outputs + +Another common source of output freeness errors is constraining implicit outputs that appear in the types of other outputs. + +This is where output freeness violations usually come up in connection with unique relations: if we have a derivation ``D : add M N O`` and a premise that returns another derivation of ``D' : add M N O'``, a common error is to insist that ``O'`` be ``O`` (which in fact it must be, but Twelf doesn't know this). Even though we match the output derivation ``D'`` with a fresh metavariable, we get an output freeness error because we constrain the type of ``D'``. + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Output_freeness). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/pattern-matching.lf b/new-tests/stelf-output/pages/pattern-matching.lf new file mode 100644 index 0000000..418d4af --- /dev/null +++ b/new-tests/stelf-output/pages/pattern-matching.lf @@ -0,0 +1,533 @@ +% +%{! +This is a case study on pattern matching using Twelf. +!}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +% +%sort map %. +%term map/z map z %. +%term map/s %pi nat %-> (map N) %-> (map (s N)) %. +%sort tp %. +%term o tp %. +%term arrow %pi tp %-> tp %-> tp %. +%term prod %pi tp %-> tp %-> tp %. +%term sum %pi tp %-> tp %-> tp %. +%sort tplist %. +%term tplist/z tplist ([n] n) %. +%term tplist/s %pi tp %-> (tplist N) %-> (tplist ([n] s (N n))) %. +%{! A ``pat N`` is a pattern that binds (N z) variables. !}% +%sort pat %. +%term pat/underscore pat ([n] n) %. +%term pat/pair %pi (pat N1) %-> (pat N2) %-> (pat ([n] N1 (N2 n))) %. +%term pat/inl %pi (pat N) %-> (pat N) %. +%term pat/inr %pi (pat N) %-> (pat N) %. +%term pat/var pat s %. +%term pat/as %pi (pat N1) %-> (pat N2) %-> (pat ([n] N1 (N2 n))) %. +%term pat/or %pi (pat N) %-> (map (N z)) %-> (pat N) %-> (pat N) %. +%sort exp %. +%sort oexp %. +% oexp N is an expression with N bound vars +%sort match %. +%term exp/unit exp %. +%term exp/lam %pi match %-> exp %. +%term exp/app %pi exp %-> exp %-> exp %. +%term exp/pair %pi exp %-> exp %-> exp %. +%term exp/inl %pi exp %-> exp %. +%term exp/inr %pi exp %-> exp %. +%term exp/handle %pi exp %-> exp %-> exp %. +%term oexp/z %pi exp %-> (oexp z) %. +%term oexp/s %pi (%pi exp %-> (oexp N)) %-> (oexp (s N)) %. +%term match/nil match %. +%term match/cons %pi (pat N) %-> (oexp (N z)) %-> match %-> match %. +%{! static semantics !}% +%sort tplist-append %. +%term tplist-append/z tplist-append tplist/z TL TL %. +%term tplist-append/s + %pi (tplist-append (tplist/s T TL) TL' (tplist/s T TL'')) + %<- (tplist-append TL TL' TL'') %. +%sort tplist-get %. +%term tplist-get/hit tplist-get z (tplist/s T TL) T %. +%term tplist-get/miss %pi (tplist-get (s N) (tplist/s T TL) T') %<- (tplist-get N TL T') %. +%sort tplist-map %. +%term tplist-map/z tplist-map TL map/z tplist/z %. +%term tplist-map/s + %pi (tplist-map TL (map/s N M) (tplist/s T TL')) + %<- (tplist-get N TL T) + %<- (tplist-map TL M TL') %. +%sort of-pat %. +%term of-pat/underscore of-pat pat/underscore T tplist/z %. +%term of-pat/pair + %pi (of-pat (pat/pair P1 P2) (prod T1 T2) TL') + %<- (of-pat P1 T1 TL1) + %<- (of-pat P2 T2 TL2) + %<- (tplist-append TL1 TL2 TL') %. +%term of-pat/inl %pi (of-pat (pat/inl P) (sum T1 T2) TL) %<- (of-pat P T1 TL) %. +%term of-pat/inr %pi (of-pat (pat/inr P) (sum T1 T2) TL) %<- (of-pat P T2 TL) %. +%term of-pat/var of-pat pat/var T (tplist/s T tplist/z) %. +%term of-pat/as + %pi (of-pat (pat/as P1 P2) T TL3) + %<- (of-pat P1 T TL1) + %<- (of-pat P2 T TL2) + %<- (tplist-append TL1 TL2 TL3) %. +%term of-pat/or + %pi (of-pat (pat/or P1 NM P2) T TL) + %<- (of-pat P1 T (%the (tplist N) TL)) + %<- (of-pat P2 T (%the (tplist N) TL')) + %<- (tplist-map TL' NM TL) %. +% or patterns are limited +%sort of-exp %. +%sort of-match %. +%sort of-oexp %. +%term of-oexp/z %pi (of-oexp tplist/z (oexp/z E) T) %<- (of-exp E T) %. +%term of-oexp/s + %pi (of-oexp (tplist/s T TL) (oexp/s ([x] EL x)) T') + %<- ({x} %pi (of-exp x T) %-> (of-oexp TL (EL x) T')) %. +%term of-match/nil of-match match/nil T T' %. +%term of-match/cons + %pi (of-match (match/cons P OE M) T T') + %<- (of-pat P T TL) + %<- (of-oexp TL OE T') + %<- (of-match M T T') %. +%term of-exp/unit of-exp exp/unit o %. +%term of-exp/lam %pi (of-exp (exp/lam M) (arrow T1 T2)) %<- (of-match M T1 T2) %. +%term of-exp/app %pi (of-exp (exp/app E1 E2) T2) %<- (of-exp E1 (arrow T1 T2)) %<- (of-exp E2 T1) %. +%term of-exp/pair %pi (of-exp (exp/pair E1 E2) (prod T1 T2)) %<- (of-exp E1 T1) %<- (of-exp E2 T2) %. +%term of-exp/inl %pi (of-exp (exp/inl E) (sum T1 T2)) %<- (of-exp E T1) %. +%term of-exp/inr %pi (of-exp (exp/inr E) (sum T1 T2)) %<- (of-exp E T2) %. +%term of-exp/handle %pi (of-exp (exp/handle E1 E2) T) %<- (of-exp E1 T) %<- (of-exp E2 T) %. +% syntax only needed for dynamic semantics of the language +%sort explist %. +%term explist/z explist ([n] n) %. +%term explist/s %pi exp %-> (explist N) %-> (explist ([n] s (N n))) %. +% static semantics of explists +%sort of-explist %. +%term of-explist/z of-explist explist/z tplist/z %. +%term of-explist/s + %pi (of-explist (explist/s E EL) (tplist/s T TL)) + %<- (of-exp E T) + %<- (of-explist EL TL) %. +%sort subst-oexp %. +%term subst-oexp/z subst-oexp explist/z (oexp/z E) E %. +%term subst-oexp/s + %pi (subst-oexp (explist/s E EL) (oexp/s ([x] OE x)) E') + %<- (subst-oexp EL (OE E) E') %. +%sort explist-get %. +%term explist-get/hit explist-get z (explist/s T TL) T %. +%term explist-get/miss %pi (explist-get (s N) (explist/s T TL) T') %<- (explist-get N TL T') %. +%sort explist-map %. +%term explist-map/z explist-map TL map/z explist/z %. +%term explist-map/s + %pi (explist-map TL (map/s N M) (explist/s T TL')) + %<- (explist-get N TL T) + %<- (explist-map TL M TL') %. +%sort explist-append %. +%term explist-append/z explist-append explist/z EL EL %. +%term explist-append/s + %pi (explist-append (explist/s E EL) EL' (explist/s E EL'')) + %<- (explist-append EL EL' EL'') %. +%sort apply-pat %. +% only have to define failure over well-typed pattern/expression pairs +% failures arise from inl/inr mismatches +%sort fail-pat %. +%term apply-pat/underscore apply-pat pat/underscore E explist/z %. +%term apply-pat/pair + %pi (apply-pat (pat/pair P1 P2) (exp/pair E1 E2) EL') + %<- (apply-pat P1 E1 EL1) + %<- (apply-pat P2 E2 EL2) + %<- (explist-append EL1 EL2 EL') %. +%term apply-pat/inl %pi (apply-pat (pat/inl P) (exp/inl E) EL) %<- (apply-pat P E EL) %. +%term apply-pat/inr %pi (apply-pat (pat/inr P) (exp/inr E) EL) %<- (apply-pat P E EL) %. +%term apply-pat/var apply-pat pat/var E (explist/s E explist/z) %. +%term apply-pat/as + %pi (apply-pat (pat/as P1 P2) E EL') + %<- (apply-pat P1 E EL1) + %<- (apply-pat P2 E EL2) + %<- (explist-append EL1 EL2 EL') %. +%term apply-pat/or-1 %pi (apply-pat (pat/or P1 M P2) E EL) %<- (apply-pat P1 E EL) %. +%term apply-pat/or-2 + %pi (apply-pat (pat/or P1 M P2) E EL') + %<- (fail-pat P1 E) + %<- (apply-pat P2 E EL) + %<- (explist-map EL M EL') %. +%term fail-pat/pair-1 %pi (fail-pat (pat/pair P1 P2) (exp/pair E1 E2)) %<- (fail-pat P1 E1) %. +%term fail-pat/pair-2 + %pi (fail-pat (pat/pair P1 P2) (exp/pair E1 E2)) + %<- (apply-pat P1 E1 EL) + %<- (fail-pat P2 E2) %. +%term fail-pat/inl-t %pi (fail-pat (pat/inl P) (exp/inl E)) %<- (fail-pat P E) %. +%term fail-pat/inl-f fail-pat (pat/inl P) (exp/inr E) %. +%term fail-pat/inr-t %pi (fail-pat (pat/inr P) (exp/inr E)) %<- (fail-pat P E) %. +%term fail-pat/inr-f fail-pat (pat/inr P) (exp/inl E) %. +%term fail-pat/as-1 %pi (fail-pat (pat/as P1 P2) E) %<- (fail-pat P1 E) %. +%term fail-pat/as-2 %pi (fail-pat (pat/as P1 P2) E) %<- (apply-pat P1 E EL) %<- (fail-pat P2 E) %. +%term fail-pat/or %pi (fail-pat (pat/or P1 M P2) E) %<- (fail-pat P1 E) %<- (fail-pat P2 E) %. +%sort apply-or-fail-pat %. +%term apply-or-fail-pat/apply %pi (apply-or-fail-pat P E) %<- (apply-pat P E EL) %. +%term apply-or-fail-pat/fail %pi (apply-or-fail-pat P E) %<- (fail-pat P E) %. +%sort apply-match %. +%term apply-match/cons-1 + %pi (apply-match (match/cons P OE M) E E') + %<- (apply-pat P E EL) + %<- (subst-oexp EL OE E') %. +%term apply-match/cons-2 + %pi (apply-match (match/cons P OE M) E E') + %<- (fail-pat P E) + %<- (apply-match M E E') %. +%sort fail-match %. +%term fail-match/nil fail-match match/nil E %. +%term fail-match/cons %pi (fail-match (match/cons P OE M) E) %<- (fail-pat P E) %<- (fail-match M E) %. +%sort apply-or-fail-match %. +%term apply-or-fail-match/apply %pi (apply-or-fail-match M E) %<- (apply-match M E E') %. +%term apply-or-fail-match/fail %pi (apply-or-fail-match M E) %<- (fail-match M E) %. +%sort exception %. +%term exception/match exception %. +%sort value %. +%term value/unit value exp/unit %. +%term value/lam value (exp/lam M) %. +%term value/pair %pi (value (exp/pair E1 E2)) %<- (value E1) %<- (value E2) %. +%term value/inl %pi (value (exp/inl E)) %<- (value E) %. +%term value/inr %pi (value (exp/inr E)) %<- (value E) %. +%sort raises %. +%term raises/app-1 %pi (raises (exp/app E1 E2) X) %<- (raises E1 X) %. +%term raises/app-2 %pi (raises (exp/app E1 E2) X) %<- (value E1) %<- (raises E2 X) %. +%term raises/app-fail + %pi (raises (exp/app (exp/lam M) E2) exception/match) + %<- (value E2) + %<- (fail-match M E2) %. +%term raises/pair-1 %pi (raises (exp/pair E1 E2) X) %<- (raises E1 X) %. +%term raises/pair-2 %pi (raises (exp/pair E1 E2) X) %<- (value E1) %<- (raises E2 X) %. +%term raises/inl %pi (raises (exp/inl E) X) %<- (raises E X) %. +%term raises/inr %pi (raises (exp/inr E) X) %<- (raises E X) %. +%sort step %. +%term step/app-1 %pi (step (exp/app E1 E2) (exp/app E1' E2)) %<- (step E1 E1') %. +%term step/app-2 %pi (step (exp/app E1 E2) (exp/app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app-beta %pi (step (exp/app (exp/lam M) E) E') %<- (value E2) %<- (apply-match M E E') %. +%term step/pair-1 %pi (step (exp/pair E1 E2) (exp/pair E1' E2)) %<- (step E1 E1') %. +%term step/pair-2 %pi (step (exp/pair E1 E2) (exp/pair E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/inl %pi (step (exp/inl E) (exp/inl E')) %<- (step E E') %. +%term step/inr %pi (step (exp/inr E) (exp/inr E')) %<- (step E E') %. +%term step/handle %pi (step (exp/handle E1 E2) (exp/handle E1' E2)) %<- (step E1 E1') %. +%term step/handle-beta %pi (step (exp/handle E1 E2) E1) %<- (value E1) %. +%term step/handle-fail %pi (step (exp/handle E1 E2) E2) %<- (raises E1 X) %. +%sort notstuck %. +%term notstuck/value %pi (notstuck E) %<- (value E) %. +%term notstuck/raises %pi (notstuck E) %<- (raises E X) %. +%term notstuck/step %pi (notstuck E) %<- (step E E') %. +%sort can-explist-append {EL1 explist N1} {EL2 explist N2} %. +%mode can-explist-append %in %in %out %. +%term _ can-explist-append _ _ explist-append/z %. +%term _ + %pi (can-explist-append (explist/s E EL) EL' (explist-append/s D1)) + %<- (can-explist-append EL EL' D1) %. +%worlds () (can-explist-append _ _ _) %. +%total (D1) (can-explist-append D1 _ _) %. +%sort can-explist-get {EL explist N'} %. +%mode can-explist-get %in %in %out %. +%term _ can-explist-get _ tplist-get/hit explist-get/hit %. +%term _ + %pi (can-explist-get _ (tplist-get/miss D2) (explist-get/miss D')) + %<- (can-explist-get _ D2 D') %. +%worlds () (can-explist-get _ _ _) %. +%total (D2) (can-explist-get _ D2 _) %. +%sort can-explist-map {EL explist N'} %. +%mode can-explist-map %in %in %out %. +%term _ can-explist-map _ tplist-map/z explist-map/z %. +%term _ + %pi (can-explist-map D1 (tplist-map/s D2 D) (explist-map/s D3 D')) + %<- (can-explist-get D1 D D') + %<- (can-explist-map D1 D2 D3) %. +%worlds () (can-explist-map _ _ _) %. +%total (D1) (can-explist-map _ D1 _) %. +%sort progress-pat-pair %. +%mode progress-pat-pair %in %in %out %. +%term _ progress-pat-pair (apply-or-fail-pat/fail DF) _ (apply-or-fail-pat/fail (fail-pat/pair-1 DF)) %. +%term _ progress-pat-pair (apply-or-fail-pat/apply DA) (apply-or-fail-pat/fail DF) (apply-or-fail-pat/fail (fail-pat/pair-2 DF DA)) %. +%term _ + %pi (progress-pat-pair (apply-or-fail-pat/apply DA) (apply-or-fail-pat/apply DA') (apply-or-fail-pat/apply (apply-pat/pair Dap DA' DA))) + %<- (can-explist-append _ _ Dap) %. +%worlds () (progress-pat-pair _ _ _) %. +%total {} (progress-pat-pair _ _ _) %. +%sort progress-pat-as %. +%mode progress-pat-as %in %in %out %. +%term _ progress-pat-as (apply-or-fail-pat/fail DF) _ (apply-or-fail-pat/fail (fail-pat/as-1 DF)) %. +%term _ progress-pat-as (apply-or-fail-pat/apply DA) (apply-or-fail-pat/fail DF) (apply-or-fail-pat/fail (fail-pat/as-2 DF DA)) %. +%term _ + %pi (progress-pat-as (apply-or-fail-pat/apply DA) (apply-or-fail-pat/apply DA') (apply-or-fail-pat/apply (apply-pat/as Dap DA' DA))) + %<- (can-explist-append _ _ Dap) %. +%worlds () (progress-pat-as _ _ _) %. +%total {} (progress-pat-as _ _ _) %. +%sort progress-pat-inl %. +%mode progress-pat-inl %in %out %. +%term _ progress-pat-inl (apply-or-fail-pat/fail DF) (apply-or-fail-pat/fail (fail-pat/inl-t DF)) %. +%term _ progress-pat-inl (apply-or-fail-pat/apply DA) (apply-or-fail-pat/apply (apply-pat/inl DA)) %. +%worlds () (progress-pat-inl _ _) %. +%total {} (progress-pat-inl _ _) %. +%sort progress-pat-inr %. +%mode progress-pat-inr %in %out %. +%term _ progress-pat-inr (apply-or-fail-pat/fail DF) (apply-or-fail-pat/fail (fail-pat/inr-t DF)) %. +%term _ progress-pat-inr (apply-or-fail-pat/apply DA) (apply-or-fail-pat/apply (apply-pat/inr DA)) %. +%worlds () (progress-pat-inr _ _) %. +%total {} (progress-pat-inr _ _) %. +%sort progress-pat-or %. +%mode progress-pat-or %in %in %in %out %. +%term _ progress-pat-or _ (apply-or-fail-pat/fail DF1) (apply-or-fail-pat/fail DF2) (apply-or-fail-pat/fail (fail-pat/or DF2 DF1)) %. +%term _ progress-pat-or _ (apply-or-fail-pat/apply DA1) _ (apply-or-fail-pat/apply (apply-pat/or-1 DA1)) %. +%term _ + %pi (progress-pat-or DM (apply-or-fail-pat/fail DF1) (apply-or-fail-pat/apply DA2) (apply-or-fail-pat/apply (apply-pat/or-2 DMap DA2 DF1))) + %<- (can-explist-map _ DM DMap) %. +%worlds () (progress-pat-or _ _ _ _) %. +%total {} (progress-pat-or _ _ _ _) %. +%sort progress-pat %. +%mode progress-pat %in %in %in %out %. +%term _ progress-pat _ D1 of-pat/underscore (apply-or-fail-pat/apply apply-pat/underscore) %. +%term _ progress-pat _ D1 of-pat/var (apply-or-fail-pat/apply apply-pat/var) %. +%term _ + %pi (progress-pat (value/pair DV2 DV1) (of-exp/pair D2 D1) (of-pat/pair _ D2' D1') DAF3) + %<- (progress-pat DV1 D1 D1' DAF1) + %<- (progress-pat DV2 D2 D2' DAF2) + %<- (progress-pat-pair DAF1 DAF2 DAF3) %. +%term _ + %pi (progress-pat DV D1 (of-pat/as _ D2' D1') DAF3) + %<- (progress-pat DV D1 D1' DAF1) + %<- (progress-pat DV D1 D2' DAF2) + %<- (progress-pat-as DAF1 DAF2 DAF3) %. +%term _ + %pi (progress-pat (value/inl DV1) (of-exp/inl D1) (of-pat/inl D1') DAF2) + %<- (progress-pat DV1 D1 D1' DAF1) + %<- (progress-pat-inl DAF1 DAF2) %. +%term _ + %pi (progress-pat (value/inr DV1) (of-exp/inr D1) (of-pat/inr D1') DAF2) + %<- (progress-pat DV1 D1 D1' DAF1) + %<- (progress-pat-inr DAF1 DAF2) %. +%term _ progress-pat (value/inr DV1) (of-exp/inr D1) (of-pat/inl D1') (apply-or-fail-pat/fail fail-pat/inl-f) %. +%term _ progress-pat (value/inl DV1) (of-exp/inl D1) (of-pat/inr D1') (apply-or-fail-pat/fail fail-pat/inr-f) %. +%term _ + %pi (progress-pat DV D1 (of-pat/or DM D2' D1') DAF3) + %<- (progress-pat DV D1 D1' (%the (apply-or-fail-pat X4 X1) DAF1)) + %<- (progress-pat DV D1 D2' (%the (apply-or-fail-pat X6 X1) DAF2)) + %<- (progress-pat-or DM DAF1 DAF2 DAF3) %. +%worlds () (progress-pat _ _ _ _) %. +%total (D1) (progress-pat _ _ D1 _) %. +%sort can-subst-oexp {EL explist N} {OE oexp (N z)} %. +%mode can-subst-oexp %in %in %out %. +%term _ can-subst-oexp _ _ subst-oexp/z %. +%term _ + %pi (can-subst-oexp (explist/s E EL) _ (subst-oexp/s D1)) + %<- (can-subst-oexp EL _ D1) %. +%worlds () (can-subst-oexp _ _ _) %. +%total (D1) (can-subst-oexp D1 _ _) %. +%sort progress-match-cons {OE} %. +%mode progress-match-cons %in %in %in %out %. +%term _ progress-match-cons _ (apply-or-fail-pat/fail DF) (apply-or-fail-match/fail DF') (apply-or-fail-match/fail (fail-match/cons DF' DF)) %. +%term _ progress-match-cons _ (apply-or-fail-pat/fail DF) (apply-or-fail-match/apply DA) (apply-or-fail-match/apply (apply-match/cons-2 DA DF)) %. +%term _ + %pi (progress-match-cons _ (apply-or-fail-pat/apply DA) _ (apply-or-fail-match/apply (apply-match/cons-1 DS DA))) + %<- (can-subst-oexp _ _ DS) %. +%worlds () (progress-match-cons _ _ _ _) %. +%total {} (progress-match-cons _ _ _ _) %. +%sort progress-match %. +%mode progress-match %in %in %in %out %. +%term _ progress-match _ _ of-match/nil (apply-or-fail-match/fail fail-match/nil) %. +%term _ + %pi (progress-match DV DE (of-match/cons D2 _ D1) DAF3) + %<- (progress-pat DV DE D1 DAF1) + %<- (progress-match DV DE D2 DAF2) + %<- (progress-match-cons _ DAF1 DAF2 DAF3) %. +%worlds () (progress-match _ _ _ _) %. +%total (D1) (progress-match _ _ D1 _) %. +%sort progress-app-beta %. +%mode progress-app-beta %in %in %out %. +%term _ progress-app-beta DV (apply-or-fail-match/fail DF) (notstuck/raises (raises/app-fail DF DV)) %. +%term _ progress-app-beta DV (apply-or-fail-match/apply DA) (notstuck/step (step/app-beta DA DV)) %. +%worlds () (progress-app-beta _ _ _) %. +%total {} (progress-app-beta _ _ _) %. +%sort progress-app %. +%mode progress-app %in %in %in %in %out %. +%term _ progress-app _ _ (notstuck/step DS) _ (notstuck/step (step/app-1 DS)) %. +%term _ progress-app _ _ (notstuck/value V) (notstuck/step DS) (notstuck/step (step/app-2 DS V)) %. +%term _ progress-app _ _ (notstuck/raises DS) _ (notstuck/raises (raises/app-1 DS)) %. +%term _ progress-app _ _ (notstuck/value V) (notstuck/raises DS) (notstuck/raises (raises/app-2 DS V)) %. +%term _ + %pi (progress-app (of-exp/lam DM) D2 (notstuck/value value/lam) (notstuck/value DV) NS) + %<- (progress-match DV D2 DM DAF) + %<- (progress-app-beta DV DAF NS) %. +%worlds () (progress-app _ _ _ _ _) %. +%total {} (progress-app _ _ _ _ _) %. +%sort progress-pair %. +%mode progress-pair %in %in %out %. +%term _ progress-pair (notstuck/step DS) _ (notstuck/step (step/pair-1 DS)) %. +%term _ progress-pair (notstuck/value V) (notstuck/step DS) (notstuck/step (step/pair-2 DS V)) %. +%term _ progress-pair (notstuck/raises DS) _ (notstuck/raises (raises/pair-1 DS)) %. +%term _ progress-pair (notstuck/value V) (notstuck/raises DS) (notstuck/raises (raises/pair-2 DS V)) %. +%term _ progress-pair (notstuck/value DV1) (notstuck/value DV2) (notstuck/value (value/pair DV2 DV1)) %. +%worlds () (progress-pair _ _ _) %. +%total {} (progress-pair _ _ _) %. +%sort progress-inl %. +%mode progress-inl %in %out %. +%term _ progress-inl (notstuck/step DS) (notstuck/step (step/inl DS)) %. +%term _ progress-inl (notstuck/raises DS) (notstuck/raises (raises/inl DS)) %. +%term _ progress-inl (notstuck/value DV1) (notstuck/value (value/inl DV1)) %. +%worlds () (progress-inl _ _) %. +%total {} (progress-inl _ _) %. +%sort progress-inr %. +%mode progress-inr %in %out %. +%term _ progress-inr (notstuck/step DS) (notstuck/step (step/inr DS)) %. +%term _ progress-inr (notstuck/raises DS) (notstuck/raises (raises/inr DS)) %. +%term _ progress-inr (notstuck/value DV1) (notstuck/value (value/inr DV1)) %. +%worlds () (progress-inr _ _) %. +%total {} (progress-inr _ _) %. +%sort progress-handle {E2} %. +%mode progress-handle %in %in %out %. +%term _ progress-handle _ (notstuck/step DS) (notstuck/step (step/handle DS)) %. +%term _ progress-handle _ (notstuck/raises DR) (notstuck/step (step/handle-fail DR)) %. +%term _ progress-handle _ (notstuck/value DV1) (notstuck/step (step/handle-beta DV1)) %. +%worlds () (progress-handle _ _ _) %. +%total {} (progress-handle _ _ _) %. +%sort progress %. +%mode progress %in %out %. +%term _ progress of-exp/unit (notstuck/value value/unit) %. +%term _ progress (of-exp/lam _) (notstuck/value value/lam) %. +%term _ + %pi (progress (of-exp/app D2 D1) NS3) + %<- (progress D1 NS1) + %<- (progress D2 NS2) + %<- (progress-app D1 D2 NS1 NS2 NS3) %. +%term _ + %pi (progress (of-exp/pair D2 D1) NS3) + %<- (progress D1 NS1) + %<- (progress D2 NS2) + %<- (progress-pair NS1 NS2 NS3) %. +%term _ + %pi (progress (of-exp/inl D1) NS2) + %<- (progress D1 NS1) + %<- (progress-inl NS1 NS2) %. +%term _ + %pi (progress (of-exp/inr D1) NS2) + %<- (progress D1 NS1) + %<- (progress-inr NS1 NS2) %. +%term _ + %pi (progress (of-exp/handle D2 D1) NS2) + %<- (progress D1 NS1) + %<- (progress-handle _ NS1 NS2) %. +%worlds () (progress _ _) %. +%total (D1) (progress D1 _) %. +%sort preservation-subst-oexp %. +%mode preservation-subst-oexp %in %in %in %out %. +%term _ preservation-subst-oexp _ (of-oexp/z D1) subst-oexp/z D1 %. +%term _ + %pi (preservation-subst-oexp (of-explist/s D1 D) (of-oexp/s D2) (subst-oexp/s D3) D4) + %<- (preservation-subst-oexp D1 (D2 _ D) D3 D4) %. +%worlds () (preservation-subst-oexp _ _ _ _) %. +%total (D1) (preservation-subst-oexp _ _ D1 _) %. +%sort preservation-append %. +%mode preservation-append %in %in %in %in %out %. +%term _ preservation-append _ D explist-append/z tplist-append/z D %. +%term _ + %pi (preservation-append (of-explist/s D1 D) D2 (explist-append/s D3) (tplist-append/s D4) (of-explist/s D5 D)) + %<- (preservation-append D1 D2 D3 D4 D5) %. +%worlds () (preservation-append _ _ _ _ _) %. +%total (D1) (preservation-append _ _ _ D1 _) %. +%sort preservation-get %. +%mode preservation-get %in %in %in %out %. +%term _ preservation-get (of-explist/s _ D) explist-get/hit tplist-get/hit D %. +%term _ + %pi (preservation-get (of-explist/s DL _) (explist-get/miss D') (tplist-get/miss D'') D) + %<- (preservation-get DL D' D'' D) %. +%worlds () (preservation-get _ _ _ _) %. +%total (D1) (preservation-get _ _ D1 _) %. +%sort preservation-map %. +%mode preservation-map %in %in %in %out %. +%term _ preservation-map DL explist-map/z tplist-map/z of-explist/z %. +%term _ + %pi (preservation-map DL (explist-map/s DEM DEG) (tplist-map/s DTM DTG) (of-explist/s DL' D1')) + %<- (preservation-get DL DEG DTG D1') + %<- (preservation-map DL DEM DTM DL') %. +%worlds () (preservation-map _ _ _ _) %. +%total (D1) (preservation-map _ _ D1 _) %. +%sort preservation-apply-pat %. +%mode preservation-apply-pat %in %in %in %out %. +%term _ preservation-apply-pat D1 of-pat/underscore apply-pat/underscore of-explist/z %. +%term _ + %pi (preservation-apply-pat (of-exp/pair DE2 DE1) (of-pat/pair DTA DP2 DP1) (apply-pat/pair DEA DA2 DA1) D3) + %<- (preservation-apply-pat DE1 DP1 DA1 D1) + %<- (preservation-apply-pat DE2 DP2 DA2 D2) + %<- (preservation-append D1 D2 DEA DTA D3) %. +%term _ + %pi (preservation-apply-pat DE (of-pat/as DTA DP2 DP1) (apply-pat/as DEA DA2 DA1) D3) + %<- (preservation-apply-pat DE DP1 DA1 D1) + %<- (preservation-apply-pat DE DP2 DA2 D2) + %<- (preservation-append D1 D2 DEA DTA D3) %. +%term _ + %pi (preservation-apply-pat (of-exp/inl DE) (of-pat/inl DP) (apply-pat/inl DA) D) + %<- (preservation-apply-pat DE DP DA D) %. +%term _ + %pi (preservation-apply-pat (of-exp/inr DE) (of-pat/inr DP) (apply-pat/inr DA) D) + %<- (preservation-apply-pat DE DP DA D) %. +%term _ preservation-apply-pat D1 of-pat/var apply-pat/var (of-explist/s of-explist/z D1) %. +%term _ + %pi (preservation-apply-pat DE (of-pat/or _ _ DP) (apply-pat/or-1 DA) D) + %<- (preservation-apply-pat DE DP DA D) %. +%term _ + %pi (preservation-apply-pat DE (of-pat/or DTM DP _) (apply-pat/or-2 DEM DA _) D') + %<- (preservation-apply-pat DE DP DA D) + %<- (preservation-map D DEM DTM D') %. +%worlds () (preservation-apply-pat _ _ _ _) %. +%total (D1) (preservation-apply-pat _ _ D1 _) %. +%sort preservation-apply-match %. +%mode preservation-apply-match %in %in %in %out %. +%term _ + %pi (preservation-apply-match D1 (of-match/cons _ DOE D2) (apply-match/cons-1 DS DA) D') + %<- (preservation-apply-pat D1 D2 DA D) + %<- (preservation-subst-oexp D DOE DS D') %. +%term _ + %pi (preservation-apply-match D1 (of-match/cons D2 _ _) (apply-match/cons-2 DA _) D') + %<- (preservation-apply-match D1 D2 DA D') %. +%worlds () (preservation-apply-match _ _ _ _) %. +%total (D1) (preservation-apply-match _ _ D1 _) %. +%sort preservation %. +%mode preservation %in %in %out %. +%term _ + %pi (preservation (of-exp/app D2 D1) (step/app-1 DS) (of-exp/app D2 D1')) + %<- (preservation D1 DS D1') %. +%term _ + %pi (preservation (of-exp/app D2 D1) (step/app-2 DS V) (of-exp/app D2' D1)) + %<- (preservation D2 DS D2') %. +%term _ + %pi (preservation (of-exp/app D2 (of-exp/lam D1)) (step/app-beta DA _) D) + %<- (preservation-apply-match D2 D1 DA D) %. +%term _ + %pi (preservation (of-exp/pair D2 D1) (step/pair-1 DS) (of-exp/pair D2 D1')) + %<- (preservation D1 DS D1') %. +%term _ + %pi (preservation (of-exp/pair D2 D1) (step/pair-2 DS V) (of-exp/pair D2' D1)) + %<- (preservation D2 DS D2') %. +%term _ + %pi (preservation (of-exp/handle D2 D1) (step/handle DS) (of-exp/handle D2 D1')) + %<- (preservation D1 DS D1') %. +%term _ + %pi (preservation (of-exp/inl D1) (step/inl DS) (of-exp/inl D1')) + %<- (preservation D1 DS D1') %. +%term _ + %pi (preservation (of-exp/inr D1) (step/inr DS) (of-exp/inr D1')) + %<- (preservation D1 DS D1') %. +%term _ + %pi (preservation (of-exp/handle D2 D1) (step/handle DS) (of-exp/handle D2 D1')) + %<- (preservation D1 DS D1') %. +%term _ preservation (of-exp/handle D2 D1) (step/handle-beta _) D1 %. +%term _ preservation (of-exp/handle D2 D1) (step/handle-fail _) D2 %. +%worlds () (preservation _ _ _) %. +%total (D1) (preservation _ D1 _) %. +%{! +--[DanielKLee](/wiki/user-danielklee/) 01:46, 11 October 2007 (EDT) + +TODO: Finish commentary. +\{\{stub\}\} +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Pattern_matching). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-assert.lf b/new-tests/stelf-output/pages/percent-assert.lf new file mode 100644 index 0000000..c2043d6 --- /dev/null +++ b/new-tests/stelf-output/pages/percent-assert.lf @@ -0,0 +1,30 @@ +% +% +%{! The **``%assert``** directive instructs the [theorem prover](/wiki/theorem-prover/) to accept a previously stated theorem as true without trying to prove that theorem. This directive is thus analogous to a [`%trustme`](/wiki/percent-trustme/)[`%total`](/wiki/percent-total/) directive. Because it instructs Twelf to treat something as true without a proof, Twelf must be in [unsafe mode](/wiki/unsafe-mode/) to use this directive. + +## Example + +The example from the [theorem prover](/wiki/theorem-prover/) article can be modified to use ``%assert`` instead of ``%prove`` for the two lemmas that are needed to establish the commutativity of addition. !}% +%sort nat %. +%term s %pi nat %-> nat %. +%term z nat %. +%sort plus %. +%term plus/z plus z N N %. +%term plus/s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%{! This website checks Twelf in safe mode, so `%assert` directives cause Twelf to display an error. !}% +%{!! begin checked !!}% +%theorem plus-z : forall {N nat} exists {D plus N z N} true %. +%assert (plus-z N D) %. +%theorem plus-s : forall* {N1 nat} {N2 nat} {N3 nat} forall {D1 plus N1 N2 N3} exists {D2 plus N1 (s N2) (s N3)} true %. +%assert (plus-s D1 D2) %. +%theorem plus-comm : forall* {N1 nat} {N2 nat} {N3 nat} forall {D1 plus N1 N2 N3} exists {D2 plus N2 N1 N3} true %. +%prove 5 D1 (plus-comm D1 D2) %. +%{!! end checked !!}% +%{! ## See also + +* [Theorem prover](/wiki/theorem-prover/) +* Theorem Prover + + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-block.lf b/new-tests/stelf-output/pages/percent-block.lf new file mode 100644 index 0000000..01e4606 --- /dev/null +++ b/new-tests/stelf-output/pages/percent-block.lf @@ -0,0 +1,46 @@ +% +% +%{! A **``%block``** declaration names a **block**, which is a partial description of an LF context. Blocks are composed into [regular worlds](/wiki/percent-worlds/) with [``%worlds``](/wiki/percent-worlds/) declarations, which describe the complete set of possible contexts for a type family. This description of the context is an important part of the [adequacy](/wiki/adequacy/) of the [metatheorem](/wiki/metatheorem/) established by a [``%total``](/wiki/percent-total/) declaration. + +A ``%block`` declaration acts more like a definition than other keywords, which generally cause Twelf to _do_ something rather than _define_ something. The obvious exception to this is [%define](/wiki/define-declaration/). + + +## Sample ``%block`` declarations + +These sample ``%block`` declarations are in the context of the definitions of ``exp`` and ``height`` in the [``%worlds``](/wiki/percent-worlds/) article. + +Using this block in a ``%worlds`` declaration means that arbitrary variables representing expressions can appear in the context: +```twelf +%block var-rand : block {x : exp}. +``` +Using this block in a ``%worlds`` declaration means that variables representing expressions can appear in the context, but only if they are accompanied by a judgment that defines the height of that variable to be one. +```twelf +%block var-height : block {x : exp}{_ : height x (s z)}. +``` +Using this block in a ``%worlds`` declaration means that variables representing expressions can appear in the context, but only if they are accompanied by a judgment that defines the height of that variable to be some natural number. +```twelf +%block var-heightN : some {N: nat} block {x : exp}{_ : height x N}. +``` +## Block definitions + +Block definitions allow multiple blocks to be combined: +!}% +%sort typ %. +%sort exp %. +%block typ-var [a typ]%. +%block exp-var {T typ} [x exp T]%. +%block vars%. +%{! Using ``vars`` in a [``%worlds``](/wiki/percent-worlds/) declaration is the same as using ``typ-var | exp-var``, and in fact when Twelf prints out the ``%worlds`` declaration, block definitions will be automatically expanded. !}% +%{!! begin checked !!}% +%worlds (vars) (exp _) %. +%{!! end checked !!}% +%{! ## See also + +* [``%worlds``](/wiki/percent-worlds/) +* [Proving totality assertions in non-empty contexts](/wiki/proving-metatheorems-proving-totality-assertions-in-non-empty-contexts/) in the Proving Metatheorems tutorial +* [Totality assertion](/wiki/totality-assertion/) +* Regular Worlds + + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-clause.lf b/new-tests/stelf-output/pages/percent-clause.lf new file mode 100644 index 0000000..afa5828 --- /dev/null +++ b/new-tests/stelf-output/pages/percent-clause.lf @@ -0,0 +1,14 @@ +% +% +%{! The **``%clause``** keyword allows a constant that is defined to be used by Twelf during proof search. If **``%clause``** is not used, then Twelf will only perform proof search on constants that are declared in the [signature](/wiki/twelf-signature/). + +This function of Twelf is incompatible with Twelf's ability to prove [metatheorems](/wiki/metatheorem/), and so type families that use **``%clause``** or depend on other type families that use **``%clause``** cannot be a part of directives that specify [totality assertions](/wiki/totality-assertion/) such as [``%total``](/wiki/percent-total/). It has mostly been used to do [tactical theorem proving](/wiki/tactical-theorem-proving/) in Twelf, and the article on that subject has an example of ``%clause`` being used. + +## See also + +- Clause Definitions +- [Tactical theorem proving](/wiki/tactical-theorem-proving/) + + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-covers.lf b/new-tests/stelf-output/pages/percent-covers.lf new file mode 100644 index 0000000..e4380c9 --- /dev/null +++ b/new-tests/stelf-output/pages/percent-covers.lf @@ -0,0 +1,75 @@ +% +% +%{! A **``%covers``** declaration checks [input coverage](/wiki/coverage-checking/) for a type family, which is one of the two kinds of [coverage checking](/wiki/coverage-checking/) that Twelf performs. Running a [``%total``](/wiki/percent-total/) declaration checks input coverage as well as [output coverage](/wiki/coverage-checking/). + +The ``%covers`` declaration essentially checks that your type family does an ML-style exhaustive match on its inputs; this does not establish that running your program will succeed, or even terminate! + +## Example + +For this example, we define a type ``elem`` with three elements, ``a``, ``b``, ``c``, and a relation ``rel`` with three positions. We will then use ``%covers`` to check coverage on this relation given a number of different ways of assigning positions as either inputs or outputs. !}% +%sort elem %. +%term a elem %. +%term b elem %. +%term c elem %. +%sort rel %. +%sort fail %. +%term rel1 rel a X c %. +%term rel2 %pi (rel b Y b) %<- fail %. +%term rel3 %pi (rel c c c) %<- (rel c a c) %. +%worlds () (rel _ _ _) (fail) %. +%{! The first ``%covers`` declaration below checks that ``rel`` covers the first position - it does, as there is one case for each of the three possibilities. This is true even though ``rel b X Y`` will fail and the query ``rel c X Y`` will loop forever. The second declaration checks that ``rel`` covers the second position, which happens trivially because the first case, labeled ``rel1``, and the second case ``rel2``, have a [[metavariable]] in the second position that matches any possible input. + +```checkedtwelf +%covers rel +X1 -X2 -X3. +%covers rel -X1 +X2 -X3. +``` + +If we try to check coverage on the third position, however, we get an error; the third position does not have a case that handles ``a``. + +```checkedtwelf +%covers rel -X1 -X2 +X3. +``` + +Furthermore, even though we cover the first and second positions individually, if we consider them together we leave out two cases, which is what Twelf's response below explains: + +```checkedtwelf +%covers rel +X1 +X2 -X3. +``` + +## Limitations + +Coverage checking is a weak test; it only checks that at least one constant will immediately apply during proof search. For example, the following program coverage checks: !}% +%sort bit %. +%term 0 bit %. +%term 1 bit %. +%sort nope %. +%mode nope %in %. +%sort not %. +%mode not %in %out %. +%term n0 not 0 1 %. +%term n1 not 1 0 %. +%sort zero %. +%mode zero %in %out %. +%term cn %pi (zero X 0) %<- (nope X) %. +%term cz %pi (zero X 0) %<- (zero X 1) %. +%term cf %pi (zero X 0) %<- (not X X) %. +%worlds () (nope _) (not _ _) (zero _ _) %. +%{! The ``zero`` predicate passes the coverage check. !}% +%{!! begin checked !!}% +%covers zero %in %out %. +%{!! end checked !!}% +%{! Despite passing the input coverage check, calling `zero B _` for any `B` will always fail. ``cn`` matches all inputs, but then appeals to a subgoal (``nope``) that is empty and immediately fails. ``cz`` appeals recursively to ``zero``, but insists that its output be ``1``, so this also fails. ``cf`` appeals to the total relation ``not``, but in a way that constrains the output so that it also always fails (see [output freeness](/wiki/output-freeness/)). To get all of these tests, you must use the [%total](/wiki/percent-total/) declaration. Here, totality fails as expected: `nope` is not defined as total (because it's not!) + +```checkedtwelf +%total D (zero D _). +``` + +## See also + +* Input Coverage +* [Input coverage](/wiki/coverage-checking/) +* [Coverage checking](/wiki/coverage-checking/) + + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-eof.lf b/new-tests/stelf-output/pages/percent-eof.lf new file mode 100644 index 0000000..71037a9 --- /dev/null +++ b/new-tests/stelf-output/pages/percent-eof.lf @@ -0,0 +1,24 @@ +% +% +%{! The **``%.``** keyword instructs Twelf to stop reading input in the current file. It is most useful in [emacs mode](/wiki/twelf-with-emacs/) to check some prefix of a file that you are currently working on. + +## Example + +The following example shows a simple Twelf declaration at top, and the text below it [in green](/wiki/twelftag/) shows the response from Twelf. Notice that the last three lines are not considered by Twelf, because it stops reading input when it reaches the ``%.`` keyword. + +```checkedtwelf +nat : type. + +%. + +This line will not be checked. +Neither will anything else that gets put after the %. +%total T (pretendlemma T _ _). +``` + +## See also +* Lexical Conventions + + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-establish.lf b/new-tests/stelf-output/pages/percent-establish.lf new file mode 100644 index 0000000..b589fb9 --- /dev/null +++ b/new-tests/stelf-output/pages/percent-establish.lf @@ -0,0 +1,12 @@ +% +% +%{! The **``%establish``** directive attempts to prove a theorem specified by a [``%theorem``](/wiki/percent-theorem/) directive using the [theorem prover](/wiki/theorem-prover/). It is therefore similar to the [``%prove``](/wiki/percent-prove/) directive. However, unlike the ``%prove`` directive, theorems shown to be true by the ``%establish`` directive are not used to prove other theorems. + +## See also + +* [Theorem prover](/wiki/theorem-prover/) +* Theorem Prover + + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-freeze.lf b/new-tests/stelf-output/pages/percent-freeze.lf new file mode 100644 index 0000000..faf98e8 --- /dev/null +++ b/new-tests/stelf-output/pages/percent-freeze.lf @@ -0,0 +1,61 @@ +% +% +%{! The **``%freeze``** declaration _freezes_ a set of type families. A frozen family cannot be extended: new constants at that type cannot be added, nor can the [subordination](/wiki/subordination/) relation be extended such that the family could depend on other types. The [``%thaw``](/wiki/percent-thaw/) declaration can be used to reenable the extension of a type family. + +## Syntax + +The syntax is as follows: + +``%freeze t1 t2 ... tn.`` + +The type families ``t1``–``tn`` are frozen. + +## Example + +Suppose we define addition in the natural way: !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%prec %prefix 9999 s %. +%sort plus %. +%mode plus %in %in %out %. +%term plus/z plus z N N %. +%term plus/s %pi (plus (s N) M (s P)) %<- (plus N M P) %. +%{! At this point, we may still extend the definition of addition: + +```checkedtwelf +plus/zz : plus M z M. +``` + +However, if we freeze ``plus`` then this will not be allowed: + +```checkedtwelf +%freeze plus. +plus/zzz : plus z z z. +``` + +More subtly, we will not be able to extend the [subordination](/wiki/subordination/) relation for ``plus``: + +```checkedtwelf +%freeze plus. +thing : type. + +oops : (thing -> plus _ _ _) -> type. +``` + +The subordination relation can be extended such that other non-frozen types depend on a frozen type: we can make a new `thing` depend on `plus`, just not the other way around. + +```checkedtwelf +%freeze plus. +thing : type. + +okay : (plus _ _ _ -> thing) -> type. +``` + +## Autofreeze + +Because types are automatically frozen for any family for which there has been a [``%worlds``](/wiki/percent-worlds/) declaration, or for any type family that depends on that family. This prevents mistakes where a metatheorem is proved for a type family but then that type family is extended, invalidating the theorem. + + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-mode.lf b/new-tests/stelf-output/pages/percent-mode.lf new file mode 100644 index 0000000..e6690e5 --- /dev/null +++ b/new-tests/stelf-output/pages/percent-mode.lf @@ -0,0 +1,52 @@ +% +% +%{! The **``%mode``** declaration specifies the way in which a relation is intended to be used, by specifying those arguments that are _inputs_ or _outputs_. Twelf then checks that all constants defining the relation respect the mode specified (are "well-moded"). + +The ``%mode`` declaration is an important part of the specification of a [totality assertion](/wiki/totality-assertion/). In the "for all/exists" statement about a relation, the "for all" terms are the input arguments to the relation, and the "exists" terms are its outputs. + +## Syntax + +A mode declaration for the family _id_ with _n_ arguments usually takes the following ("short") form: + +``` +%mode id a1 ... an. +``` + +Each argument ``ai`` is ``+ID``, ``-ID``, or ``*ID`` for some distinct identifier `ID`. + + - ``+`` indicates that the argument is an input, + - ``-`` indicates that the argument is an output, and + - ``*`` indicates that the argument is unrestricted. (The ``*`` mode is rarely used.) + + For example, a natural mode for the ``plus`` relation on [natural numbers](/wiki/natural-numbers/) is: + +```twelf +plus : nat -> nat -> nat -> type. +%mode plus +N1 +N2 -N3. +``` + +The names ``N1``..``N3`` are arbitrary, but each must be distinct. + +### Full form + +The short mode declaration above only specifies the mode for explicit parameters; the modes for implicit parameters are then assigned automatically. Occasionally, one needs to use a "full" form to specify the modes of implicit parameters: + +``` +%mode p1 ... pm term. +``` + +Each parameter ``pi`` is ``+\{``_ID_`` : term\}``, ``-\{``_ID_`` : term\}``, or ``*\{``_ID_`` : term\}`` for distinct identifiers _ID_. Terms may refer to the variables bound in the previous parameters. For example, the full mode declaration corresponding to the short one above would be: + +```twelf +plus : nat -> nat -> nat -> type. +%mode +{N1:nat} +{N2:nat} -{N3:nat} plus N1 N2 N3. +``` + +## See also +* Modes +* [Troubleshooting mode checking errors](/wiki/debugging-mode-checking-errors/) +* The section on [mode](/wiki/proving-metatheorems-proving-totality-assertions-about-the-natural-numbers/) in the _Proving metatheorems_ introduction to Twelf + + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-name.lf b/new-tests/stelf-output/pages/percent-name.lf new file mode 100644 index 0000000..a676fe3 --- /dev/null +++ b/new-tests/stelf-output/pages/percent-name.lf @@ -0,0 +1,49 @@ +% +% +%{! A **``%name``** declaration allows us to direct Twelf to name unnamed variables in a certain way, which can make it much easier to understand Twelf's output. Using a ``%name`` declaration is never required, but it often makes the task of proving [metatheorem](/wiki/metatheorem/)s significantly easier. More information can be found in the section on name preferences. + +Two examples show the use of ``%name`` - the first shows its use to give a default name for universally quantified variables, and the second example shows its use to give a default name for both universally quantified variables and bound variables. + +## Example 1: Natural numbers + +We start with a standard presentation of unary numbers and addition: !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort plus %. +%term plus/z plus z N N %. +%term plus/s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%{! If we give Twelf a derivation, but do not give it information about the type of that derivation, it will infer the type. In order to return the most general possible type, Twelf considers the second type to be an "anonymous" (or [universally quantified](/wiki/implicit-and-explicit-parameters/)) variable. If Twelf is given no other information, it will automatically name all anonymous variables ``X1``, ``X2``, etc... !}% +%{!! begin checked !!}% +%define _ plus/s (plus/s plus/z) %. +%{!! end checked !!}% +%{! These type reconstructions can become rather complicated, particularly when we are dealing with metatheorems that have multiple types. A good way to deal with this complexity is by giving Twelf a different prefix for anonymous variables of different types using the ``%name`` declaration. !}% +%name nat %. +%{! This identifier _must_ start with an uppercase letter, and often only a single uppercase letter suffices; however, any identifier starting with an uppercase letter works. Given this information, Twelf will change the prefix of anonymous variables from ``X`` to whatever was defined in the ``%name`` declaration. !}% +%{!! begin checked !!}% +%define _ plus/s (plus/s plus/z) %. +%{!! end checked !!}% +%{! ## Example 2: Call-by-name lambda calculus + +We can also use the ``%name`` declaration to define the default name for bound variables. !}% +%sort exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%sort step %. +%term step/app %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/appabs step (app (lam E1) E2) (E1 E2) %. +%{! The default prefix for universally quantified variables is again ``X``. Also, observe that the default prefix for bound variables is ``x``. !}% +%{!! begin checked !!}% +%define _ step/app (step/app step/appabs) %. +%{!! end checked !!}% +%{! This ``%name`` declaration causes no output from Twelf, but it changes the prefixes of universally quantified and bound variables to ``E`` and ``e``, respectively. !}% +%{!! begin checked !!}% +%name exp %. +%define _ step/app (step/app step/appabs) %. +%{!! end checked !!}% +%{! ## See also +* Name preferences + + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-prove.lf b/new-tests/stelf-output/pages/percent-prove.lf new file mode 100644 index 0000000..f9178b3 --- /dev/null +++ b/new-tests/stelf-output/pages/percent-prove.lf @@ -0,0 +1,12 @@ +% +% +%{! The **``%prove``** directive attempts to prove a theorem specified by a [``%theorem``](/wiki/percent-theorem/) directive using the [theorem prover](/wiki/theorem-prover/). It is therefore similar to the [``%establish``](/wiki/percent-establish/) directive. However, unlike the ``%establish`` directive, theorems shown to be true by the ``%prove`` directive are considered as potential lemmas when the theorem prover attempts to prove future theorems. + +## See also + +* [Theorem prover](/wiki/theorem-prover/) +* Theorem Prover + + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-query.lf b/new-tests/stelf-output/pages/percent-query.lf new file mode 100644 index 0000000..20a36e2 --- /dev/null +++ b/new-tests/stelf-output/pages/percent-query.lf @@ -0,0 +1,82 @@ +% +% +%{! The **``%query``** declarations specify a type (that corresponds to a judgment by the [judgments as types](/wiki/judgment/) principle), and tells Twelf's [logic programming](/wiki/logic-programming/) engine to search for a proof of that judgment. + +A query declaration also takes two other inputs, which can either be a number or a star "``*``". The meaning of this is a bit awkward and counterintuitive: + +* The first input is the _expected number of solutions_ - a star here means "as many solutions as you let Twelf look for" +* The second input is the _number of solutions to find_ - a star here means "keep looking until you have found all solutions" + +This means that some possible queries will always fail: + +* ``%query 5 4 ...`` will always fail because there is no way to come up with 5 solutions in 4 tries +* ``%query 6 6 ...`` means the same thing as ``%query * 6 ...``, because both mean "try six times, and find a solution every time" +* ``%query * * ...`` will either keep coming up with more solutions forever (and will therefore not terminatie) or it will eventually fail to come up with a solution and so will fail. + +It also means there are often multiple ways to write the same query, as the examples below demonstrate. + +## Examples + +We will use the example of list membership to demonstrate ``%query`` because there may be multiple ways to find an element in a list (if that element occurs multiple times). !}% +%sort elem %. +%term a elem %. +%term b elem %. +%term c elem %. +%term d elem %. +%sort list %. +%term nil list %. +%term , %pi elem %-> list %-> list %. +%prec %right 10 , %. +%define sample-list list a , c , a , b , a , c , a , a , b , c , c , d , nil %. +%sort member %. +%term member/hit member A (A , L) %. +%term member/skip %pi (member A (B , L)) %<- (member A L) %. +%{! + +### Confirming there is one "d" in the list +```checkedtwelf +%query 1 * member d sample-list. +``` + +### Confirming there are at least three "a's" in the list +```checkedtwelf +%query 3 3 member a sample-list. +``` + +### Confirming there are at least two "b's" in the list +```checkedtwelf +%query * 2 member b sample-list. +``` + +### Confirming there are at least zero "c's" in the list (the query isn't even run) +```checkedtwelf +%query * 0 member c sample-list. % +``` + +### Confirming there are at least zero "c's" in the list (the query isn't even run) +```checkedtwelf +%query 8 0 member c sample-list. +``` + +### Confirming there are three "a's" in the list (fails, there are more) +```checkedtwelf +%query 3 * member a sample-list. +``` + +### Getting output from ``%query`` + +Unlike [``%solve``](/wiki/percent-solve/), ``%query`` declarations do not allow you to insert the output of a query into the [signature](/wiki/twelf-signature/). However, you can inspect the output in the Twelf buffer. For example, if we leave the first argument of ``%query`` as a metavariable, the following query will output the first two elements it finds in the list (the first two elements in the list). !}% +%{!! begin checked !!}% +%query _ 2 _ member E sample-list %. +%{!! end checked !!}% +%{! Furthermore, if you change the format of a query from ``%query 1 * ...`` to ``%query 1 * D : ...``, where ``D`` is some uppercase identifier, it will print the proof term that caused the query to succeed. The proof terms below indicate that the first two instances of ``b`` is located in the fourth position in the list (the search skips three times and then hits) and in the ninth position in the list (the search skips eight times and then hits). !}% +%{!! begin checked !!}% +%query _ 2 _ %the (member b sample-list) D %. +%{!! end checked !!}% +%{! ## See also + +* Query Declaration + + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-querytabled.lf b/new-tests/stelf-output/pages/percent-querytabled.lf new file mode 100644 index 0000000..13a97bd --- /dev/null +++ b/new-tests/stelf-output/pages/percent-querytabled.lf @@ -0,0 +1,11 @@ +% +% +%{! The **``%querytabled``** directive starts a query using Twelf's [tabled logic programming](/wiki/tabled-logic-programming/) engine, so that instances of a type family marked with [``%tabled``](/wiki/percent-tabled/) directive will be derived at most once during proof search. It otherwise looks the same as [``%query``](/wiki/percent-query/). + +## See also + +* [Tabled logic programming](/wiki/tabled-logic-programming/) +* Tabled Logic Programming + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-reduces.lf b/new-tests/stelf-output/pages/percent-reduces.lf new file mode 100644 index 0000000..5b87f2f --- /dev/null +++ b/new-tests/stelf-output/pages/percent-reduces.lf @@ -0,0 +1,13 @@ +% +% +%{! + +The **``%reduces``** declaration checks a specified subderivation relationship between two derivations in a judgment. Once a ``%reduces`` relationship has been established for a judgment, the termination checker can use that information to verify that inductive calls are always on smaller derivations. Its primary use is for termination checking in inductive proofs that induct on the output of some other judgment, rather than directly on a subderivation. + +## See also +* The case study on [division over the natural numbers](/wiki/division-over-the-natural-numbers/) uses `%reduces` +* %reduces declaration + + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-solve.lf b/new-tests/stelf-output/pages/percent-solve.lf new file mode 100644 index 0000000..214fffb --- /dev/null +++ b/new-tests/stelf-output/pages/percent-solve.lf @@ -0,0 +1,29 @@ +% +% +%{! A [``%solve``](/wiki/percent-solve/) declaration specifies a type and then uses Twelf's [logic programming](/wiki/logic-programming/) engine to search for an term with that type. It is different from [``%query``](/wiki/percent-query/), both because it can only cause Twelf to search for the _first_ proof it can find and because it can add the result of the search to the [Twelf signature](/wiki/twelf-signature/). + +## Example + +We can define [natural numbers](/wiki/natural-numbers/) with addition in the standard manner: !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort plus %. +%term pz plus z N N %. +%term ps %pi (plus N1 N2 N3) %-> (plus (s N1) N2 (s N3)) %. +%{! Then we can use ``%solve`` and ``%define`` to insert the result of adding two and two to the signature as ``four``, and the derivation itself to the signature as ``deriv``. !}% +%{!! begin checked !!}% +%define four = N %solve deriv : plus (s (s z)) (s (s z)) N %. +%{!! end checked !!}% +%{! We can then use ``five`` and ``deriv`` as a defined constants for the rest of the program: !}% +%{!! begin checked !!}% +%define six nat s (s four) %. +%define deriv2 (plus four (s (s z)) six) ps (ps deriv) %. +%{!! end checked !!}% +%{! ## See also + +* Solve Declaration + + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-subord.lf b/new-tests/stelf-output/pages/percent-subord.lf new file mode 100644 index 0000000..cde9cb1 --- /dev/null +++ b/new-tests/stelf-output/pages/percent-subord.lf @@ -0,0 +1,19 @@ +% +% +%{! +The **``%subord``** declaration adds edges to the [subordination](/wiki/subordination/) relation. + +## Syntax + +The syntax is as follows: + +``` +%subord (t1 t1') ... (tn tn'). +``` + +The type families ``ti'`` will then be treated as being dependent on the type family ``ti``. + + + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-tabled.lf b/new-tests/stelf-output/pages/percent-tabled.lf new file mode 100644 index 0000000..cc08156 --- /dev/null +++ b/new-tests/stelf-output/pages/percent-tabled.lf @@ -0,0 +1,11 @@ +% +% +%{! The ``%tabled`` directive informs Twelf's [tabled logic programming](/wiki/tabled-logic-programming/) engine to record derivations for a certain type family _at most once_ during a query started with a [``%querytabled``](/wiki/percent-querytabled/) directive. It is therefore a bit like [``%deterministic``](/wiki/deterministic-declaration/) in that it changes Twelf's search behavior for a single type family. + +## See also + +* [Tabled logic programming](/wiki/tabled-logic-programming/) +* Tabled Logic Programming + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-terminates.lf b/new-tests/stelf-output/pages/percent-terminates.lf new file mode 100644 index 0000000..4c4b700 --- /dev/null +++ b/new-tests/stelf-output/pages/percent-terminates.lf @@ -0,0 +1,154 @@ +% +% +%{! A `%terminates` declaration checks that a program will either succeed or fail in a finite amount of time when given [ground](/wiki/ground/) inputs. + +A [``%total``](/wiki/percent-total/) declaration uses the same syntax as a ``%terminates`` declaration. + +## Termination checking in Twelf + +Termination is in general an undecidable problem, and so Twelf uses a simple strategy of requiring the user to define some **termination ordering**, and then checking that every recursive subgoal makes that ordering smaller. + +Twelf considers a term to be smaller than another term if it can inspect the two terms to see that the first is a strict subterm or if it follows from a [``%reduces``](/wiki/percent-reduces/) declaration that the first is smaller. Twelf also uses [mode](/wiki/percent-mode/) information to ensure that it is reasoning about [ground](/wiki/ground/) terms. + +A term A is only considered to be smaller than B if A is a strict subterm of B. So, for instance, ``(s (s z))``, ``(s z)``, and ``z`` are all subterms of ``(f (s (s z)) (s z))``, but ``(f z z)`` is not. + +As usual, we will use the [natural numbers](/wiki/natural-numbers/) as the basis for our example. !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%{! ### Non-ordering + +The simplest termination ordering says that no termination argument is needed, because there are no recursive calls! We can write such an ordering for ``triv``, which takes a number and can either return a number one greater or one less. !}% +%sort triv %. +%mode triv %in %out %. +%term triv/ triv (s N) N %. +%term triv/ triv N (s N) %. +%{! +```checkedtwelf +%terminates {} (triv _ _). +``` +!}% +%{! ### Simple ordering + +A simple ordering says that one term always gets smaller in a subgoal, even if the other ones get larger. The first argument to ``a`` always gets smaller in this example (even though the second may get bigger). We only need to be able to reason about the first argument, but we define both to be inputs. !}% +%sort a %. +%mode a %in %in %. +%term a0 a z z %. +%term a1 %pi (a (s N1) N) %<- (a N1 (s (s (s N)))) %. +%{! +```checkedtwelf +%terminates N (a N _). +``` +!}% +%{! Alternatively, the second argument to ``b`` always gets smaller (even though the first may stay the same), and the termination declarations capture this information. The ``*A`` in the [``%mode``](/wiki/percent-mode/) declaration means that we do not care whether the first argument to ``b`` is an input or an output. + +Notice that because we already showed ``a`` to be terminating, we can use it as a subgoal to ``b`` as long as it is only called with ground terms. !}% +%sort b %. +%mode b %star %in %. +%term b0 b (s z) (s z) %. +%term b1 %pi (b N (s (s M))) %<- (b N M) %<- (a (s (s (s M))) M) %. +%{! +```checkedtwelf +%terminates N (b _ N). +``` +!}% +%{! ### Simultaneous ordering + +We can define a relation ``c`` that non-deterministically counts three numbers down to zero: !}% +%sort c %. +%mode c %in %in %in %. +%term c0 c z z z %. +%term c1 %pi (c (s N1) N2 N3) %<- (c N1 N2 N3) %. +%term c2 %pi (c N1 (s N2) N3) %<- (c N1 N2 N3) %. +%term c3 %pi (c N1 N2 (s N3)) %<- (c N1 N2 N3) %. +%{! No single term gets smaller at any one step, but _some_ term gets smaller at every step. We can express this using a simultaneous ordering: + +```checkedtwelf +%terminates [N1 N2 N3] (c N1 N2 N3). +``` +!}% +%{! ### Lexicographic ordering + +Lexicographic orders generalize simultaneous orders - using a lexicographic ordering we can define ``d`` in a manner similar to ``c``, but we can allow the second and third numbers to count _up_ whenever the first number counts down, and allow the third number to count up whenever the second number counts down. However, the first number must stay constant when the second counts down, and the first and second both must stay constant when the third counts down. !}% +%sort d %. +%mode d %in %in %in %. +%term d0 d z z z %. +%term d1 %pi (d (s N1) N2 N3) %<- (d N1 (s (s (s (s N2)))) (s (s N3))) %. +%term d2 %pi (d N1 (s N2) N3) %<- (d N1 N2 (s N3)) %. +%term d3 %pi (d N1 N2 (s N3)) %<- (d N1 N2 N3) %. +%{! +```checkedtwelf +%terminates {N1 N2 N3} (d N1 N2 N3). +``` +!}% +%{! ### Lexicographic ordering II + +Another example that is (slightly) less contrived arises with lists of natural numbers. If we want to directly represent the sum of a natural number ``N`` and a list ``L``, then we can define it as follows: + +* The sum of ``z`` and ``nil`` is ``z``. +* The sum of ``z`` and ``cons N L`` is the sum of ``N`` and ``L``. +* The sum of ``(s N)`` and ``L`` is ``s M``, if ``M`` is the sume of ``N`` and ``L``. + +This is a lexicographic induction - either the list gets smaller, or the natural number gets smaller and the list stays the same size. !}% +%sort list %. +%term nil list %. +%term cons %pi nat %-> list %-> list %. +%sort listsum %. +%mode listsum %in %in %out %. +%term lsz listsum z nil z %. +%term lsl %pi (listsum z (cons N L) M) %<- (listsum N L M) %. +%term lss %pi (listsum (s N) L (s M)) %<- (listsum N L M) %. +%{!! begin checked !!}% +%terminates {L N} (listsum N L M) %. +%{!! end checked !!}% +%{! ### Mutual, lexicographic ordering + +In this case, we have a program that is non-deterministic to illustrate more possibilities. We will describe a "big" lemma and a "small" lemma, each with two arguments. Twelf decides what is "big" or "small" based on the way we order things - from small to large - in the ``%terminates`` declaration below. !}% +%sort big %. +%sort small %. +%mode big %in %in %out %. +%mode small %in %in %out %. +%{! We can split up the possibilities into four cases: + +* We "bottom out" in a base case or a call to another theorem: !}% +%term & small z N N %. +%term & %pi (big N M P) %<- (listsum N (cons M (cons M nil)) P) %. +%{! * The second argument gets smaller (nothing else matters) !}% +%term & %pi (small N1 (s N2) N3) %<- (big (s (s N1)) N2 N3) %. +%term & %pi (small N1 (s N2) N3) %<- (big N2 N2 N3) %. +%term & %pi (big N1 (s (s N2)) N3) %<- (big N1 N2 N3) %. +%{! * The first argument gets smaller, and the second argument gets no bigger. !}% +%term & %pi (small (s N1) (s N2) (s N3)) %<- (big N1 N2 N3) %. +%term & %pi (big (s N1) N2 N3) %<- (big N1 N2 N3) %. +%{! * The "lemma" gets smaller and everything else gets no bigger. !}% +%term & %pi (big N1 N2 N3) %<- (small N1 N2 N3) %. +%term & %pi (big (s N1) (s N2) N3) %<- (small N1 N2 N3) %. +%{! These requirements are encoded in the declaration below. The ordering of the patterns declares ``small`` to be smaller than ``big``, and either the second argument (``B1`` and ``B2``) gets smaller, or else it gets no bigger and the first +argument (``A1`` and ``A2``) gets smaller. !}% +%{!! begin checked !!}% +%terminates {(B1 B2) (A1 A2)} (small A2 B2 _) (big A1 B1 _) %. +%{!! end checked !!}% +%{! ## Troubleshooting + +The simplest example of running afoul of the Twelf termination checker is if the recursive call uses an argument unrelated to the original argument. + +```checkedtwelf +e: nat -> nat -> type. +- : e N M <- e M N. +%mode e +N +M. +%terminates N (e N M). +``` + +As the error message indicates, Twelf requires that ``M`` be less than ``N`` in the recursive call in order for termination analysis to hold, but it has +no way of establishing this. If, for some reason, the un-fufilled termination requirement is actually true, it can often be established with a [``%reduces``](/wiki/percent-reduces/) +declaration. + +## See also + +* Termination +* [``%total``](/wiki/percent-total/) +* Lexicographic induction +* [Mutual induction](/wiki/mutual-induction/) + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-thaw.lf b/new-tests/stelf-output/pages/percent-thaw.lf new file mode 100644 index 0000000..218c171 --- /dev/null +++ b/new-tests/stelf-output/pages/percent-thaw.lf @@ -0,0 +1,28 @@ +% +%{! +The **``%thaw``** directive allows previously [frozen](/wiki/percent-freeze/) type families to be extended with new [canonical forms](/wiki/canonical-form/). Because this can easily be used to invalidate [metatheorem](/wiki/metatheorem/)s, it is a directive that can only be used in [unsafe mode](/wiki/unsafe-mode/). + +If Twelf is in unsafe mode, the following code can be run to demonstrate **``%thaw``**: !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%freeze nat %. +%{! With `nat` frozen, it's impossible to extend `nat` without error. + +```checkedtwelf +veryS: nat -> nat -> nat. +``` + +Because the Twelf Wiki runs in safe mode, this example won't work, but if you click the elf's hat to run this code in the live Twelf editor, you can toggle unsafe mode and see that `%thaw` allows `q` to be defined as a new `nat`. + +```checkedtwelf +%thaw nat. +q : nat. +``` + +## See also + +* [`%freeze`](/wiki/percent-freeze/) + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-theorem.lf b/new-tests/stelf-output/pages/percent-theorem.lf new file mode 100644 index 0000000..489171d --- /dev/null +++ b/new-tests/stelf-output/pages/percent-theorem.lf @@ -0,0 +1,16 @@ +% +% +%{! The **``%theorem``** declaration is an alternative way of specifying metatheorems in Twelf. It is not fully supported or documented, but +some prefer the syntax for its clarity when proving [metatheorems](/wiki/metatheorem/). A ``%theorem`` declaration used in this style handles the type family declation and the [``%mode``](/wiki/percent-mode/) declaration parts of a [totality assertion](/wiki/totality-assertion/). + +## Use with the theorem prover + +When using the Twelf [theorem prover](/wiki/theorem-prover/), a ``%theorem`` declaration establishes a proposition that the theorem prover may later attempt to establish. A **``%theorem``** declaration is thus somewhat analogous to the three-part specification of a [totality assertions](/wiki/totality-assertion/), the type family definition, the [`%mode`](/wiki/percent-mode/) declaration, and the [`%worlds`](/wiki/percent-worlds/) declaration. + +## See also + +* [Theorem prover](/wiki/theorem-prover/) +* Theorem Prover + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-total.lf b/new-tests/stelf-output/pages/percent-total.lf new file mode 100644 index 0000000..e211262 --- /dev/null +++ b/new-tests/stelf-output/pages/percent-total.lf @@ -0,0 +1,22 @@ +% +% +%{! In Twelf, a [totality assertion](/wiki/totality-assertion/) for an LF [type family](/wiki/type-family/) is specified by a [``%mode``](/wiki/percent-mode/) declaration and a [``%worlds``](/wiki/percent-worlds/) declarations. A **``%total``** declaration for a type family causes Twelf to attempt to verify the specified totality assertion. Twelf does so by attempting to prove that the type family defines a total logic program. The ``%total`` declaration is used both to verify properties of object-language [judgment](/wiki/judgment/)s and to check proofs of general [metatheorem](/wiki/metatheorem/)s. + +A ``%total`` declaration requires that [`%mode`](/wiki/percent-mode/) checking and [`%worlds`](/wiki/percent-worlds/) checking have already succeeded. +Then a **``%total``** declaration causes Twelf to run a number of analyses: +* **[Termination analysis](/wiki/percent-terminates/)** - The syntax of a ``%total`` declaration is the same as a [``%terminates``](/wiki/percent-terminates/) declaration, and Twelf runs its termination analysis with that information. This verifies that the logic program always terminates when given [ground](/wiki/ground/) inputs. +* **[Input coverage checking](/wiki/coverage-checking/)** - Twelf uses the modes specified by the [``%mode``](/wiki/percent-mode/) declaration ([``%mode``](/wiki/percent-mode/) and [``%covers``](/wiki/percent-covers/) declarations also have the same syntax) to check input coverage. This verifies that the logic program will match all possible ground inputs in the specified set of contexts. +* **[Output coverage checking](/wiki/coverage-checking/)** - Twelf checks that the output of a [[subgoal]] can never fail to unify. First, it checks for incorrect constant pattern-matching; next, it checks for [output freeness](/wiki/output-freeness/) violations. + +Taken together, these analyses verify that if the type family is run as a [logic program](/wiki/logic-programming/) in Twelf with ground derivations in the input positions, then it the execution will terminate successfully and will derive ground derivations in the output positions. This proves the [totality assertion](/wiki/totality-assertion/) for the type family: in any context conforming to the ``%worlds`` declaration, for any ground derivations in the input positions (as specified by the ``%mode`` declaration), there exist ground derivations for the output positions such that the type family is inhabited. + +## See also +* Read the [introductions to Twelf](/wiki/introductions-to-twelf/) to learn more about ``%total``. The following sections of [Proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/) are particularly relevant: + - [Proving totality assertions about the natural numbers](/wiki/proving-metatheorems-proving-totality-assertions-about-the-natural-numbers/) + - [Proving metatheorems about the natural numbers](/wiki/proving-metatheorems-proving-metatheorems-about-the-natural-numbers/) + - [Proving totality assertions in non-empty contexts](/wiki/proving-metatheorems-proving-totality-assertions-in-non-empty-contexts/). +* Totality + + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-trustme.lf b/new-tests/stelf-output/pages/percent-trustme.lf new file mode 100644 index 0000000..d3d309b --- /dev/null +++ b/new-tests/stelf-output/pages/percent-trustme.lf @@ -0,0 +1,10 @@ +% +% +%{! The **``%trustme``** keyword, when it precedes another directive such as a **``%total``** directive, causes Twelf to believe that the directive succeeded even if it did not. This feature is only available in [unsafe mode](/wiki/unsafe-mode/), and can be used to easily create [holes in metatheorems](/wiki/incremental-metatheorem-development/). + +## See also + +* [Holes in metatheorems](/wiki/incremental-metatheorem-development/) + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-unique.lf b/new-tests/stelf-output/pages/percent-unique.lf new file mode 100644 index 0000000..a3c9bdc --- /dev/null +++ b/new-tests/stelf-output/pages/percent-unique.lf @@ -0,0 +1,98 @@ +% +% +%{! + +A **``%unique``** declaration attempts to automatically check whether some positions of a relation (its outputs) are uniquely determined by some other positions (its inputs). Its syntax is similar to that of [``%mode``](/wiki/percent-mode/), except that in addition to being able to specify an argument to be an input (``+``), an output (``-``), or unmoded (``*``), you may also specify an argument to be a unique output (``-1``). + +Successful ``%unique`` declarations are used to simplify [coverage checking](/wiki/coverage-checking/), and they can be transformed into first-class [uniqueness lemmas](/wiki/uniqueness-lemma/). Unfortunately, there is no automatic way of doing this transformation! + +## Example + +We define the oft-used example of addition of [natural numbers](/wiki/natural-numbers/): !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort plus %. +%term pz plus z N N %. +%term ps %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%{! We can then check for uniqueness using ``%unique`` (a [``%worlds``](/wiki/percent-worlds/) declaration is also required). + +```checkedtwelf +%worlds () (plus _ _ _). +%unique plus +N1 +N2 -1N3. +``` + +If we had created a non-unique definition of ``plus``, for instance by adding an additional, broken version of ``ps2``, Twelf would have indicated an error upon checking for uniqueness: + +```checkedtwelf +ps2 : plus (s N1) N2 N3 + <- plus N1 N2 N3. +%worlds () (plus _ _ _). +%unique plus +N1 +N2 -1N3. +``` + +## Mutual recursion + +Checking the uniqueness of mutually recursive predicates +creates a problem, because uniqueness (unlike [``%mode``](/wiki/percent-mode/)) cannot +be checked incrementally. We therefore introduce a simultaneous +form of uniqueness declarations, in analogy with other +simultaneous declarations. !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort div2 %. +%sort div2' %. +%term d2s %pi (div2 (s N) (s M)) %<- (div2' N M) %. +%term d2z div2 z z %. +%term d2's %pi (div2' (s N) M) %<- (div2 N M) %. +%worlds () (div2 _ _) (div2' _ _) %. +%unique div2 %in %out %. +%unique div2' %in %out %. +%{! ## Coverage checking + +Successful ``%unique`` declarations are taken into account to simplify certain kinds of coverage goals in [coverage checking](/wiki/coverage-checking/). Suppose you have a goal with two hypotheses of the form + ... \{x : a N1 ... Nk M \} ... \{y : a N1 ... Nk M' \} ... +where the inputs ``N1 ... Nk`` are all equal, and suppose further that you have declared (and Twelf checked) + %unique a +X1 ... +Xk -1Y. +Then, Twelf's coverage checker will unify the unique outputs ``M`` and ``M'`` in the coverage goal, which may cause some otherwise non-exhaustive pattern match to be recognized as exhaustive. (If ``M`` and ``M'`` do not unify, the coverage goal is impossible, and no case will be needed to cover it.) + +As a simple example, consider proving a first-class [uniqueness lemma](/wiki/uniqueness-lemma/) for the original ``plus`` relation defined above. !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort plus %. +%term pz plus z N N %. +%term ps %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%{! To state such a lemma , we first define [identity](/wiki/equality/) on natural numbers with just one constructor, reflexivity. !}% +%sort id %. +%term refl id N N %. +%{! Then we can say that for any and , if both and , then in fact . !}% +%sort plus-unique %. +%mode plus-unique %in %in %out %. +%{! We might naively hope to be able to prove this theorem with a single case: !}% +%term _ plus-unique D D' refl %. +%{! However, before taking any uniqueness information into account, this pattern match isn't sufficiently general to cover all cases, since matching the third argument as ``refl`` requires the types of ``D`` and ``D'`` to be equal. + +```checkedtwelf +%worlds () (plus-unique _ _ _). +%total {} (plus-unique _ _ _). +``` + +If we can show that ``plus``'s output is unique, though, the coverage checker can determine that the coverage goal need not be so generic: the types of ``D`` and ``D'`` _can_ be considered equal, since the uniqueness declaration says that they _will_ be. !}% +%worlds () (plus _ _ _) %. +%unique plus %in %in %out %. +%{! + +```checkedtwelf +%worlds () (plus-unique _ _ _). +%total {} (plus-unique _ _ _). +``` + +## See also + +- [_Verifying Uniqueness in a Logical Framework_](https://www.cs.cmu.edu/~fp/papers/tphols04.pdf), by Penny Anderson and Frank Pfenning + + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-use.lf b/new-tests/stelf-output/pages/percent-use.lf new file mode 100644 index 0000000..52fa3de --- /dev/null +++ b/new-tests/stelf-output/pages/percent-use.lf @@ -0,0 +1,13 @@ +% +% +%{! A **``%use``** directive adds a [constraint domain](/wiki/constraint-domain/) to the current Twelf [signature](/wiki/twelf-signature/). + +Like some other directives, for instance [``%clause``](/wiki/percent-clause/), it is incompatible with Twelf's ability to verify [totality assertions](/wiki/totality-assertion/) and [metatheorems](/wiki/metatheorem/). + +## See also + +* [Constraint domains](/wiki/constraint-domain/) +* Constraint domains + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/percent-worlds.lf b/new-tests/stelf-output/pages/percent-worlds.lf new file mode 100644 index 0000000..2b898cf --- /dev/null +++ b/new-tests/stelf-output/pages/percent-worlds.lf @@ -0,0 +1,125 @@ +% +% +%{! A **world** (or **worlds**) is a set of [LF](/wiki/lf/) contexts. + +The representation of an [object language](/wiki/object-logic/) in LF is specified not just by an LF [signature](/wiki/twelf-signature/), but also by the world in which that signature is considered. This is because a representation that is [adequate](/wiki/adequacy/) in one set of contexts may not be adequate in another. Similarly, a [totality assertion](/wiki/totality-assertion/) about an LF type family may be true in one set of contexts but false in another. + +Twelf includes a **``%worlds``** declaration that serves two purposes. First, it declares the world for which a [totality assertion](/wiki/totality-assertion/) about a type family is proved. Second, it verifies that the constants inhabiting a type family stay within contexts in the specified world. + +In Twelf, a world is specified by first defining a set of _blocks_ using a [%block](/wiki/percent-block/) declaration. A block is a pattern that describes a fragment of an LF context. Then a world is specified by a regular expression ``(b1 | b2 | ... | bn)*`` for blocks ``b1`` through ``bn``. This regular expression matches any LF context consisting of any number of these blocks in any order. Hence, Twelf supports the specification of **regular worlds**. + +## Specifying the world of a totality assertion + +We can define natural numbers and a relation ``max`` that defines the larger of two natural numbers. !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort max %. +%mode max %in %in %out %. +%term mzz max z z z %. +%term mzs max z (s N) (s N) %. +%term msz max (s N) z (s N) %. +%term mss %pi (max (s N1) (s N2) (s N)) %<- (max N1 N2 N) %. +%{! As discussed in the introduction article on [Representing the judgments of the natural numbers](/wiki/proving-metatheorems-representing-the-judgements-of-the-natural-numbers/), the type ``nat`` in this signature [adequately](/wiki/adequacy/) represents the natural numbers only if the LF context does not contain variables of type ``nat``. + +### Totality of ``max`` in the empty context + +The judgement ``max`` defined in this signature is a [total](/wiki/percent-total/) relation on natural numbers, or LF terms of type ``nat`` in the empty LF context. We specify this totality assertion in Twelf with the following declarations: + +```checkedtwelf +%worlds () (max _ _ _). +%total T (max T _ _). +``` + +The ``%worlds`` declaration does not mention any blocks; the world ``()*`` describes only the empty LF context. + +### Non-totality of ``max`` in an extended context + +What happens if we consider ``max`` in an LF context that includes variables of type ``nat``? (Note that in this world ``nat`` no longer represents the natural numbers.) + +The following [``%block``](/wiki/percent-block/) declaration defines the block ``random_nat``, which allows such variables: + +```checkedtwelf +%block random_nat : block {x: nat}. + +%worlds (random_nat) (max _ _ _). +%total T (max T _ _). +``` + +If we attempt to verify the totality assertion for ``max`` in these worlds, Twelf reports an error: ``max`` is not defined in the "natural numbers" arising from LF variables in the context. + +## World checking + +Up until this point, we have focused on the role of a ``%worlds`` declaration has in specifying a totality assertion. However, just as ``%mode`` both specifies the mode of a totality assertion and mode-checks a type family, ``%worlds`` both specifies the world of a totality assertion and _world checks_ a type family. + +What property does a world declaration specify? Intuitively, a world declaration for a type family circumscribes the contexts in which we consider inhabitants of that type family. For example, the world declaration for ``max`` says that we should only ever consider terms of type ``max`` in the empty LF context. +Thus, it is a world error if some piece of Twelf code introduces a variable relevant to ``max``. World checking ensures that we only consider a term in the contexts declared for its type. + +When Twelf processes a ``%worlds`` declaration, it world checks each constant in the type family. A constant _world checks_ under the following condition: whenever the ambient LF context is of the form specified by the world declaration, each premise of the constant occurs in an LF context that is within the world specified for the premise's type family. + +### Example world violation + +For example, consider the following signature for the lambda-calculus: !}% +%sort exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%{! We define a height relation on open lambda-terms as a [hypothetical judgement](/wiki/hypothetical-judgment/); note that the height rule for variables is represented as an LF assumption: !}% +%sort height %. +%term height/lam + %pi (height (lam ([x] E x)) (s N)) + %<- ({x exp} %pi (height x (s z)) %-> (height (E x) N)) %. +%term height/app + %pi (height (app E1 E2) (s N)) + %<- (height E1 N1) + %<- (height E2 N2) + %<- (max N1 N2 N) %. +%{! If we try to declare ``height`` to be defined in the closed world, as we did for ``max``, Twelf will fail with a ``World violation`` [error message](/wiki/error-messages/): + +```checkedtwelf +%worlds () (height _ _). +``` + +This is because ``height/lam`` adds variables to the LF context, so saying that the context should always be empty is wrong! Additionally, this world declaration is incorrect for another reason: the adequacy argument for ``exp`` says LF terms in contexts including variables of type ``tm`` adequately represent open object language terms; so if we wish ``height`` to represent a judgement on open terms, it must be declared in a world including such variables. Declaring ``height`` as above, even if it succeeded, would not capture the idea we have in mind that ``height`` is a relation on open terms. + +### Example world success + +The correct world for this type family consists of blocks of the form defined in the following [``%block``](/wiki/percent-block/) declaration called ``var-height``. These blocks specify that whenever a new variable with type ``exp`` is added to the context, it is added along with a derivation of ``height x (s z)``. The type family ``height`` adequately represents the notion of height that we have in mind in this world. Moreover, it also world-checks in this world: !}% +%{!! begin hidden !!}% +%worlds () (max _ _ _) %. +%total T (max T _ _) %. +%{!! end hidden !!}% +%{!! begin checked !!}% +%block var-height [x exp] [_ height x (s z)]%. +%mode height %in %out %. +%worlds (var-height) (height _ _) %. +%{!! end checked !!}% +%{! Indeed, the type family is total in this world as well: !}% +%{!! begin checked !!}% +%total T (height T _) %. +%{!! end checked !!}% +%{! ### Subordination and world subsumption + +It should be interesting that ``max`` can be called by ``height``, even though ``height`` is defined for non-empty LF contexts. Technically, these contexts are not in the world for ``max``, so the call to ``max`` from ``height`` would seem to be a world violation. However, Twelf knows, by keeping track of the [subordination](/wiki/subordination/) relation, that any ``var-height`` blocks can never change the [canonical forms](/wiki/canonical-form/) of ``nat`` or ``max``. The Twelf server will print out the current subordination relation if you type in ``Print.subord``. For the example above, this is the subordination relation: + +``` +%% OK %% +Print.subord +nat #> nat +max #> max nat +exp #> exp +height #> height exp max nat +%% OK %% +``` + +The line ``max #> max nat`` means that the canonical forms (i.e. the possible derivations) of ``max`` will not change unless the definitions of ``max`` or ``nat`` are changed. Because the block ``var-height`` does not add any declarations that are subordinate to either of these types, it is permissible to call ``max`` from within an LF context that includes blocks of the form ``var-height``. +This is an instance of what is called [world subsumption](/wiki/world-subsumption/), which is the criterion under which a type family in one world can be called from a type family in another. + +## See also + +* [``%block``](/wiki/percent-block/) +* [Totality assertion](/wiki/totality-assertion/) +* Regular Worlds +* The case study [Church-Rosser via complete development](/wiki/church-rosser-via-complete-development/) uses regular worlds heavily. + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/pltheory-introduction-to-twelf.lf b/new-tests/stelf-output/pages/pltheory-introduction-to-twelf.lf new file mode 100644 index 0000000..c1acaea --- /dev/null +++ b/new-tests/stelf-output/pages/pltheory-introduction-to-twelf.lf @@ -0,0 +1,16 @@ +% +%{! This article is an introduction to Twelf and how it can be used to represent basic data structures and logical systems—in particular the judgment-based syntax and semantics of programming languages—as well as represent and check proofs of their properties. Additional [introductions to Twelf](/wiki/introductions-to-twelf/) and many [tutorials](/wiki/tutorials/) that explain representation and proof techniques can be found elsewhere. These notes assume that you have already installed Twelf and are familiar with the process of starting and interacting with a Twelf server. Additional information on getting started with Twelf can be found in the [Documentation](/wiki/documentation/) section, and in particular the [User's Guide](/wiki/users-guide/) (which also comes with the Twelf distribution). + +## Table of Contents for Part 1 + +- [First-order encodings](/wiki/first-order-encodings/) +- [Signatures as logic programs](/wiki/signatures-as-logic-programs/) +- [Signature checking](/wiki/signature-checking/): modes, termination, coverage, totality +- [[Higher-order encodings]] +- [[Common patterns and pitfalls]] !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/PLTheory:Introduction_to_Twelf). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/polarized-pcf.lf b/new-tests/stelf-output/pages/polarized-pcf.lf new file mode 100644 index 0000000..65cac68 --- /dev/null +++ b/new-tests/stelf-output/pages/polarized-pcf.lf @@ -0,0 +1,229 @@ +% +%{! + +Polarized PCF with a stack machine semantics + +- Interalizes the composition/leftist substitution principles (left-commutative cuts), so there are no meta-ops in evaluation other than LF substitution +- force+ binds a positive value variable, rather than requiring an immediate left inversion +- Proves completeness of focusing, so you can see what direct-style terms mean +- Does purely negative stream using (non-canonical) negative continuation variables; completeness of focusing does some sort of CPS conversion + +!}% +%{! ## Syntax !}% +%sort pos %. +%sort neg %. +%term down %pi neg %-> pos %. +%term nat pos %. +%term up %pi pos %-> neg %. +%term arr %pi pos %-> neg %-> neg %. +%term stream %pi neg %-> neg %. +%sort val+ %. +%sort cont+ %. +%sort val- %. +%sort cont- %. +%sort exp %. +%% positive values +%term zero val+ nat %. +%term succ %pi (val+ nat) %-> (val+ nat) %. +%term delay- %pi (val- A-) %-> (val+ (down A-)) %. +%% positive conts +%term ifz %pi (exp A+) %-> (%pi (val+ nat) %-> (exp A+)) %-> (cont+ nat A+) %. +%term force- %pi (%pi (val- A-) %-> (exp C+)) %-> (cont+ (down A-) C+) %. +%% internalize composition +%term comp+ %pi (cont+ A+ B+) %-> (%pi (val+ B+) %-> (exp C+)) %-> (cont+ A+ C+) %. +%% negative values +%term lam %pi (%pi (val+ A+) %-> (val- B-)) %-> (val- (arr A+ B-)) %. +%term delay+ %pi (exp A+) %-> (val- (up A+)) %. +%term fix %pi (%pi (val- A-) %-> (val- A-)) %-> (val- A-) %. +%term cons + %pi ({C+} %pi (cont- A- C+) %-> (exp C+)) + %-> ({C+} %pi (cont- (stream A-) C+) %-> (exp C+)) + %-> (val- (stream A-)) %. +%% negative conts +%term app %pi (val+ A+) %-> (cont- B- C+) %-> (cont- (arr A+ B-) C+) %. +%term force+ %pi (%pi (val+ A+) %-> (exp C+)) %-> (cont- (up A+) C+) %. +%% internalize composition +%term comp- %pi (cont- A- B+) %-> (%pi (val+ B+) %-> (exp C+)) %-> (cont- A- C+) %. +%term head %pi (cont- A- C+) %-> (cont- (stream A-) C+) %. +%term tail %pi (cont- (stream A-) A+) %-> (cont- (stream A-) A+) %. +%% expressions +%term ret %pi (val+ A+) %-> (exp A+) %. +%% internalize arbitrary cuts. +%% these two are canonical when the value is a variable, +%% but if we'ere only running closed terms, then that +%% distinction is moot. +%term cut+ %pi (val+ A+) %-> (cont+ A+ C+) %-> (exp C+) %. +%term cut- %pi (val- A-) %-> (cont- A- C+) %-> (exp C+) %. +%% internalize composition +%term let %pi (exp A+) %-> (%pi (val+ A+) %-> (exp C+)) %-> (exp C+) %. +%% these terms are not eta-long, because we use the same +%% val judgements on the left as on the right +%block posb {A+ pos} [x val+ A+]%. +%block negb {A- neg} [x val- A-]%. +%block negcb {A- neg} [A+ pos] [k cont- A- A+]%. +%worlds (posb negb negcb) (val+ _) (cont+ _ _) (val- _) (cont- _ _) (exp _) %. +%% FIXME: weirldly asymetric: we need both pos and neg assumptions, +%% but not both kinds of conclusion?? +%{! + +### Step + +Note that all rules are axioms. + +!}% +%sort step %. +%mode step %in %out %. +%term step/ifz-0 step (cut+ zero (ifz E0 E1)) E0 %. +%term step/ifz-1 step (cut+ (succ V) (ifz E0 E1)) (E1 V) %. +%term step/force- step (cut+ (delay- V-) (force- E)) (E V-) %. +%term step/fix step (cut- (fix E) K-) (cut- (E (fix E)) K-) %. +%term step/app step (cut- (lam V-) (app V+ K-)) (cut- (V- V+) K-) %. +%term step/force+ step (cut- (delay+ E1) (force+ E)) (let E1 E) %. +%term step/head step (cut- (cons E1 E2) (head K-)) (E1 _ K-) %. +%term step/tail step (cut- (cons E1 E2) (tail K-)) (E2 _ K-) %. +%% left commutative cuts: +%term step/letret step (let (ret V+) E2) (E2 V+) %. +%term step/letcut+ step (let (cut+ V+ K+) E2) (cut+ V+ (comp+ K+ E2)) %. +%term step/letcut- step (let (cut- V- K-) E2) (cut- V- (comp- K- E2)) %. +%term step/letlet step (let (let E1 E2) E3) (let E1 ([x] let (E2 x) E3)) %. +%term step/cconv-ifz step (cut+ V+ (comp+ (ifz E0 E1) E)) (cut+ V+ (ifz (let E0 E) ([y] let (E1 y) E))) %. +%term step/cconv-force- step (cut+ V+ (comp+ (force- E1) E)) (cut+ V+ (force- ([y] let (E1 y) E))) %. +%term step/cconv-comp+ step (cut+ V+ (comp+ (comp+ E1 E2) E)) (cut+ V+ (comp+ E1 ([y] let (E2 y) E))) %. +%term step/cconv-app step (cut- V- (comp- (app V+ K-) E)) (cut- V- (app V+ (comp- K- E))) %. +%term step/cconv-force+ step (cut- V- (comp- (force+ E1) E)) (cut- V- (force+ ([y] let (E1 y) E))) %. +%term step/cconv-comp- step (cut- V- (comp- (comp- K- E1) E2)) (cut- V- (comp- K- ([y] let (E1 y) E2))) %. +%term step/cconv-head step (cut- V- (comp- (head K-) E)) (cut- V- (head (comp- K- E))) %. +%term step/cconv-tail step (cut- V- (comp- (tail K-) E)) (cut- V- (tail (comp- K- E))) %. +%worlds () (step _ _) %. +%{! + +## Progress + +!}% +%sort ok %. +%term ok/final ok (ret V+) %. +%term ok/step %pi (ok E) %<- (step E E') %. +%sort progress {E exp A+} %. +%mode progress %in %out %. +%term _ progress _ ok/final %. +%term _ progress _ (ok/step step/ifz-0) %. +%term _ progress _ (ok/step step/ifz-1) %. +%term _ progress _ (ok/step step/force-) %. +%term _ progress _ (ok/step step/fix) %. +%term _ progress _ (ok/step step/app) %. +%term _ progress _ (ok/step step/force+) %. +%term _ progress _ (ok/step step/head) %. +%term _ progress _ (ok/step step/tail) %. +%term _ progress _ (ok/step step/letret) %. +%term _ progress _ (ok/step step/letcut+) %. +%term _ progress _ (ok/step step/letcut-) %. +%term _ progress _ (ok/step step/letlet) %. +%term _ progress _ (ok/step step/cconv-ifz) %. +%term _ progress _ (ok/step step/cconv-force-) %. +%term _ progress _ (ok/step step/cconv-comp+) %. +%term _ progress _ (ok/step step/cconv-app) %. +%term _ progress _ (ok/step step/cconv-force+) %. +%term _ progress _ (ok/step step/cconv-head) %. +%term _ progress _ (ok/step step/cconv-tail) %. +%term _ progress _ (ok/step step/cconv-comp-) %. +%worlds () (progress _ _) %. +%total {} (progress _ _) %. +%{! ## EL !}% +%{! ### Syntax !}% +%sort etp %. +%name etp %. +%term enat etp %. +%term earr %pi etp %-> etp %-> etp %. +%term estream %pi etp %-> etp %. +%sort eexp %. +%name eexp %. +%prec %postfix 1 eexp %. +%term ez enat eexp %. +%term es %pi (enat eexp) %-> (enat eexp) %. +%term eifz %pi (enat eexp) %-> (T eexp) %-> (%pi (enat eexp) %-> (T eexp)) %-> (T eexp) %. +%term efun {T1 etp} {T2 etp} %pi (%pi (earr T1 T2 eexp) %-> (T1 eexp) %-> (T2 eexp)) %-> (earr T1 T2 eexp) %. +%term eapp %pi (earr T1 T2 eexp) %-> (T1 eexp) %-> (T2 eexp) %. +%term econs %pi (T eexp) %-> (estream T eexp) %-> (estream T eexp) %. +%term estreamfix %pi (%pi (estream T eexp) %-> (estream T eexp)) %-> (estream T eexp) %. +%term ehead %pi (estream T eexp) %-> (T eexp) %. +%term etail %pi (estream T eexp) %-> (estream T eexp) %. +%{! ### Completeness of focusing !}% +%sort poltp %. +%mode poltp %in %out %. +%term poltp/nat poltp enat nat %. +%term poltp/arr + %pi (poltp (earr T1 T2) (down (arr A1 (up A2)))) + %<- (poltp T1 A1) + %<- (poltp T2 A2) %. +%term poltp/stream %pi (poltp (estream T) (down (stream (up A)))) %<- (poltp T A) %. +%worlds () (poltp _ _) %. +%total T (poltp T _) %. +%unique poltp %in %out %. +%sort can-poltp {T} %. +%mode can-poltp %in %out %. +%worlds () (can-poltp _ _) %. +% not really a trustme: Twelf proved this totality above! +%total {} (can-poltp _ _) %. +%sort id %. +%term refl id A A %. +%sort unique-poltp %. +%mode unique-poltp %in %in %out %. +%worlds () (unique-poltp _ _ _) %. +% not really a trustme: Twelf proved this totality above! +%total {} (unique-poltp _ _ _) %. +%sort retype %. +%mode retype %in %in %out %. +%term _ retype D refl D %. +%worlds (posb negb) (retype _ _ _) %. +%total {} (retype _ _ _) %. +%sort complete %. +%mode complete %in %in %out %. +%term complete/z complete poltp/nat ez (ret zero) %. +%term complete/s + %pi (complete poltp/nat (es E) (let E' ([x] ret (succ x)))) + %<- (complete poltp/nat E E') %. +%term complete/ifz + %pi (complete (%the (poltp T A) Dpol) (eifz E E0 E1) (let E' ([x] cut+ x (ifz E0' E1')))) + %<- (complete poltp/nat E E') + %<- (complete Dpol E0 E0') + %<- ({x enat eexp} {x' val+ nat} {d {A1' pos} {D' poltp enat A1'} {Did id nat A1'} {E exp A1'} + %pi (complete D' x E) + %<- (unique-poltp D' poltp/nat Did) + %<- (retype (ret x') Did E)} complete Dpol (E1 x) (E1' x')) %. +%term complete/fun + %pi (complete (poltp/arr (%the (poltp T2 A2) Dp2) (%the (poltp T1 A1) Dp1)) (efun T1 T2 E) (ret (delay- (fix ([f'] lam ([x'] delay+ (E' (delay- f') x'))))))) + %<- ({f earr T1 T2 eexp} {f' val+ (down (arr A1 (up A2)))} {d {A1' pos} {D' poltp (earr T1 T2) A1'} {Did id (down (arr A1 (up A2))) A1'} {E exp A1'} + %pi (complete D' f E) + %<- (unique-poltp D' (poltp/arr Dp2 Dp1) Did) + %<- (retype (ret f') Did E)} {x T1 eexp} {x' val+ A1} {d {A1' pos} {D' poltp T1 A1'} {Did id A1 A1'} {E exp A1'} %pi (complete D' x E) %<- (unique-poltp D' Dp1 Did) %<- (retype (ret x') Did E)} complete Dp2 (E f x) (E' f' x')) %. +%term complete/app + %pi (complete (%the (poltp T A) D) (eapp (%the (earr T2 T eexp) E1) E2) (let E1' ([f] let E2' ([x] cut+ f (force- ([y] cut- y (app x (force+ ([z] ret z))))))))) + %<- (can-poltp T2 (%the (poltp T2 A2) Dp2)) + %<- (complete (poltp/arr D Dp2) E1 E1') + %<- (complete Dp2 E2 E2') %. +%% cons is lazy in both components +%term complete/cons + %pi (complete (poltp/stream Dp) (econs E1 E2) (ret (delay- (cons ([C+] [k] let E1' ([y] cut- (delay+ (ret y)) k)) ([C+] [k] let E2' ([y] cut+ y (force- ([w] cut- w k)))))))) + %<- (complete Dp E1 E1') + %<- (complete (poltp/stream Dp) E2 E2') %. +%term complete/head + %pi (complete D (ehead E) (let E' ([x] cut+ x (force- ([y] cut- y (head (force+ ([z] ret z)))))))) + %<- (complete (poltp/stream D) E E') %. +%term complete/tail + %pi (complete (poltp/stream D) (etail E) (let E' ([x] cut+ x (force- ([y] ret (delay- (cons ([C] [hobs] cut- y (tail (head hobs))) ([C] [tobs] cut- y (tail (tail tobs)))))))))) + %<- (complete (poltp/stream D) E E') %. +%term complete/streamfix + %pi (complete (poltp/stream D) (estreamfix E1) (ret (delay- (fix ([f] cons ([C] [hobs] let (E1' (delay- f)) ([r] cut+ r (force- ([r'] cut- r' (head hobs))))) ([C] [tobs] let (E1' (delay- f)) ([r] cut+ r (force- ([r'] cut- r' (tail tobs)))))))))) + %<- ({x _ eexp} {x' val+ _} {d {A1' pos} {D' poltp _ A1'} {Did id _ A1'} {E exp A1'} + %pi (complete D' x E) + %<- (unique-poltp D' (poltp/stream D) Did) + %<- (retype (ret x') Did E)} complete (poltp/stream D) (E1 x) (E1' x')) %. +%block completeb {T1 etp} {A1 pos} {D poltp T1 A1} [x T1 eexp] [x' val+ A1] [d {A1' pos} {D' poltp T1 A1'} {Did id A1 A1'} {E exp A1'} %pi (complete D' x E) %<- (unique-poltp D' D Did) %<- (retype (ret x') Did E)]%. +%worlds (completeb) (complete _ E _) %. +%total (E) (complete _ E _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Polarized_PCF). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-basic-error-messages.lf b/new-tests/stelf-output/pages/popl-tutorial-basic-error-messages.lf new file mode 100644 index 0000000..badc993 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-basic-error-messages.lf @@ -0,0 +1,227 @@ +% +%{! ## How Twelf checks assertions + +Twelf proves a totality assertion for a type family such as +``mult`` by checking several properties. These properties, taken +together, constitute a proof by induction on canonical forms that the +type family defines a total relation. + +### Running Example + +We'll start from the following code: !}% +%sort nat %. +%term zero nat %. +%term succ %pi nat %-> nat %. +%sort add %. +%mode add %in %in %out %. +%term add/z add zero N N %. +%term add/s %pi (add (succ M) N (succ P)) %<- (add M N P) %. +%worlds () (add _ _ _) %. +%total M (add M _ _) %. +%sort mult %. +%mode mult %in %in %out %. +%term mult/z mult zero N zero %. +%term mult/s %pi (mult (succ M) N P') %<- (mult M N P) %<- (add N P P') %. +%{! ### Mode + +Twelf checks that each constant inhabiting the type family is +_well-moded_. Roughly, this means that the inputs to the conclusion +of a constant determine the inputs of the first premise, and that these +together with the outputs of the first premise determine the inputs of +the second premise, and so on, until the outputs of all the premises +determine the outputs of the conclusion. +For example, the constant + + +```checkedtwelf +mult/s : mult (succ M) N P' + <- mult M N P + <- add N P P'. +``` + +has mode ``+M +N -P`` because the input ``M`` and ``N`` +in the conclusion determine the inputs of the premise, and the +``P`` output by the premise determines the first input to the second +premise (add), and the output of that determines the conclusion. +On the other hand, a constant + +```checkedtwelf +mult/bad-mode-output : mult zero N P. +``` + +is not well-moded---the output ``P`` is not determined by the +inputs. Similarly, + +```checkedtwelf +mult/bad-mode-input : mult (succ M) N zero + <- mult M Q P. +``` + +is not well-moded---the second input to the premise is not determined +by the inputs of the conclusion. + +Note that subgoal order matters: + +```checkedtwelf +mult/s : mult (succ M) N Q + <- add P N Q + <- mult M N P. +``` + +The premises are mode-checked in order; the outputs from one subgoal are +consider ground in subsequent premises. + +### Worlds + +Twelf checks that each constant inhabiting the type family obeys the +worlds declaration. Because we are only proving theorems about closed +terms right now, we will not run across any problems with world +checks. + +### Termination + +Twelf checks that each constant inhabiting the type family obeys the +induction order specified in the ``%total`` declaration. In each +inductive premise of a constant, the specified induction position must +be a strict subterm of the corresponding argument in the conclusion. +For example, the constant + +```checkedtwelf +mult/s : mult (s M) N Q + <- mult M N P + <- add P N Q. +``` + +obeys the induction order ``M`` specified in the above totality +assertion because the term ``M`` is a strict subterm of the term +``(s M)``. + +Recuring on exactly the same term is obviously a termination error: + +```checkedtwelf +mult/bad-termination-1 : mult M N P + <- mult M N P. +%total M (mult M _ _). +``` + +As is recuring on an entirely unrelated argumenet: + +```checkedtwelf +mult/bad-termination-2 : mult M N P + <- mult N N P. +%total M (mult M _ _). +``` + +For example Twelf would not accept the totality of ``mult`` +if ``N`` were used as the induction order—the same term +``N`` in the conclusion of this constant appears in the premise: + +```checkedtwelf +%total N (mult _ N _). +``` + +%In addition to the subterm ordering on a single argument, Twelf supports +[mutual induction](/wiki/mutual-induction/) and lexicographic induction. + +### Output coverage + +In the definition of a type family, you may pattern-match the outputs of +a premise. For example, we might write + +```checkedtwelf +mult/bad-output-cov : mult (succ M) N zero + <- mult M N (succ P). +%worlds () (mult _ _ _). +%total N1 (mult N1 N2 N3). +``` + +Here we have insisted that the output of the premise has the form +``succ P`` for some ``P``. Twelf correctly reports an output +coverage error because this condition can fail. + +Pattern-matching the output of a premise is like an inversion step +in a proof: you're insisting that the premise derivation must conclude a +particular fact that is more specific than the judgement form itself. +For Twelf to accept a relation as total, Twelf must notice that all of +these inversions are permissible. Twelf permits such inversions when it +is readily apparent that they are justified, and those inversions that +Twelf does not accept can be proved explicitly. + +In this example, we got an output coverage error because we constrained +the output of the premise by insisting it be formed by a particular +constant. The other way to get output coverage errors is to insist that +the output of a premise be a variable that occurs elsewhere in the type. +For example: + +```checkedtwelf +mult/bad-output-free : mult (succ M) N zero + <- mult M N N. +%worlds () (mult _ _ _). +%total N1 (mult N1 N2 N3). +``` + +Here, we insisted that the output of the premise be the number +``N`` that we put in. Twelf is very conservative in checking +[output freeness](/wiki/output-freeness/): a type family will not be judged total if you +constrain the outputs of any premise at all in this manner. + +### Input coverage + +Mode, worlds, termination, and output coverage ensure that each constant +really does cover the part of the relation indicated by its conclusion. +For example, if ``mult`` passes these four checks, we know that +``mult/z`` and ``mult/s`` cover ``(zero, N, _)`` and ``(succ M, +N, _)``, respectively. What else is necessary to know that +``mult`` defines a total relation? We need to know that all the +constants inhabiting ``mult``, taken together, cover all of the +inputs. Input coverage checks exactly this. + +For example, if we forgot ``mult/z``, input coverage for +``mult`` would fail. For example: + +```checkedtwelf +mult' : nat -> nat -> nat -> type. +%mode mult' +N1 +N2 -X3. + +mult'/s : mult' (succ M) N Q + <- mult' M N P + <- add P N Q. + +%worlds () (mult' _ _ _). +%total M (mult' M _ _). +``` + +Here's an analogy that might be helpful: You can think of each constant +of a type as being a clause in an ML pattern matching declaration. Then +input coverage is like the exhaustiveness checker for pattern matching. + +Twelf checks input coverage by splitting the input types to +case-analyze the various constants that could have been used to inhabit +them. For ``add``, Twelf splits the first ``nat`` argument +``M``, and then checks that the cases ``plus z N N`` and +``add (succ M) N P`` are covered. Fortunately, these are exactly +the cases we wrote down. If we had case-analyzed further in the +definition of the judgement (e.g., if the definition of ``plus`` +case-analyzed the second argument as well), Twelf would continue +splitting the input space. Because Twelf separates termination checking +and coverage checking, the constants defining a type family do not need +to follow any particular primitive recursion schema-the constants may +pattern-match the inputs in a general manner. + +## When Twelf checks what + +To a first approximation, you can think of the ``%mode`` and ``%worlds`` declarations as specifying a totality assertion and the ``%total`` declaration as checking it. This isn't exactly how Twelf works, though: + +- When a ``%mode`` declaration is entered, Twelf checks that all previous constants inhabiting the specified type family are well-moded; further, it then mode-checks any subsequent constants inhabiting that family. +- When a ``%worlds`` declaration is entered, Twelf world-checks the type family; further, it then reports an error if any new constants contributing to the family at all are added. +- When a ``%total`` declaration is entered, Twelf checks termination, then input coverage, then output coverage. When checking output coverage, Twelf checks for unjustified constant pattern-matching in a first pass and then output freeness problems in a second pass. + +This separation allows you to, for example, check that each constant in a family is well-moded (i.e., takes specified inputs to specified outputs) without checking that the entire type family is total. You can also use the declarations [%terminates](/wiki/percent-terminates/) and [%covers](/wiki/percent-covers/) to check termination and input coverage independently. + +If any constant in a type family fails mode, worlds, or output coverage, then mode, worlds, or totality checking fails for the whole type family. One could imagine that Twelf instead would just disregard the offending constant: it is possible that the type family as a whole satisfies a totality assertion without that constant, and, in a mathematical sense, adding additional constants never invalidates the fact a totality assertion is true of a family. The reason Twelf does not work this way is that ``%total`` actually has a more specific meaning, as we discuss in the next section. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +You can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Basic_error_messages). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-basics-answer.lf b/new-tests/stelf-output/pages/popl-tutorial-basics-answer.lf new file mode 100644 index 0000000..f827b00 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-basics-answer.lf @@ -0,0 +1,116 @@ +% +%{! + +## Natural numbers + +!}% +%sort nat %. +%term zero nat %. +%term succ %pi nat %-> nat %. +%{! + +## Addition + +!}% +%sort add %. +%term add/z add zero N N %. +%term add/s %pi (add (succ M) N (succ P)) %<- (add M N P) %. +%{! + +### Example derivations + +!}% +%define 1 nat succ zero %. +%define 2 nat succ 1 %. +%define 1+1is2 (add 1 1 2) add/s add/z %. +%% explicit version of add/z +%% add/z-explicit : {n:nat} add zero n n. +%% 1+1is2-explicit : add 1 1 2 = add/s (add/z-explicit 1). +%{! + +## Exercise: Multiplication + +!}% +%sort mult %. +%term mult/z mult zero N zero %. +%term mult/s %pi (mult (succ M) N P') %<- (mult M N P) %<- (add N P P') %. +%% note that the arguments are "backwards" +%define 1*2is2 (mult 1 2 2) mult/s (add/s (add/s add/z)) mult/z %. +%{! + +## Mode, worlds total + +!}% +%mode add %in %in %out %. +%worlds () (add _ _ _) %. +%total M (add M _ _) %. +%solve 1+1is2' : add 1 1 N %. +%% Examples of errors: +%% +%% mult/bad-mode-output : mult zero N Q. +%% +%% mult/bad-mode-input : mult (succ M) N zero +%% <- mult M Q P. +%% +%% %% do input coverage by removing cases +%% +%% mult/bad-termination-1 : mult M N P +%% <- mult M N P. +%% mult/bad-termination-2 : mult M N P +%% <- mult N N P. +%% +%% mult/bad-output-free : mult (succ M) N zero +%% <- mult M N N. +%% mult/bad-output-cov : mult (succ M) N zero +%% <- mult M N (succ P). +%mode mult %in %in %out %. +%worlds () (mult _ _ _) %. +%total M (mult M _ _) %. +%{! + +## Right-hand zero + +!}% +%sort rhzero {M nat} %. +%mode rhzero %in %out %. +%term _ rhzero zero add/z %. +%term _ %pi (rhzero (succ M) (add/s D)) %<- (rhzero M (%the (add M zero M) D)) %. +%worlds () (rhzero _ _) %. +%total M (rhzero M _) %. +%{! + +## Right-hand succ + +!}% +%sort rhsucc %. +%mode rhsucc %in %out %. +%term _ rhsucc (%the (add zero M M) add/z) (%the (add zero (succ M) (succ M)) add/z) %. +%term _ + %pi (rhsucc (add/s (%the (add M N P) D1)) (add/s D2)) + %<- (rhsucc D1 (%the (add M (succ N) (succ P)) D2)) %. +%% remark that type annotations are optional: +%% - : rhsucc add/z add/z. +%% - : rhsucc (add/s D1) (add/s D2) +%% <- rhsucc D1 D2. +%worlds () (rhsucc _ _) %. +%total M (rhsucc M _) %. +%{! + +## Exercise: addition is commutative + +!}% +%sort commute %. +%mode commute %in %out %. +%term _ %pi (commute (%the (add zero M M) add/z) D) %<- (rhzero M D) %. +%term _ + %pi (commute (add/s (%the (add M N P) D)) D'') + %<- (commute D (%the (add N M P) D')) + %<- (rhsucc D' (%the (add N (succ M) (succ P)) D'')) %. +%worlds () (commute _ _) %. +%total D (commute D _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Basics_Answer). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-basics-starter.lf b/new-tests/stelf-output/pages/popl-tutorial-basics-starter.lf new file mode 100644 index 0000000..bc3c83b --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-basics-starter.lf @@ -0,0 +1,104 @@ +% +%{! + +## Natural numbers + +!}% +%sort nat %. +%term zero nat %. +%term succ %pi nat %-> nat %. +%{! + +## Addition + +!}% +%sort add %. +%term add/z add zero N N %. +%term add/s %pi (add (succ M) N (succ P)) %<- (add M N P) %. +%{! + +### Example derivations + +!}% +%define 1 nat succ zero %. +%define 2 nat succ 1 %. +%define 1+1is2 (add 1 1 2) add/s add/z %. +%{! + +## Exercise: Multiplication + +!}% +%sort mult %. +%% The syntax '% .' (without the space) +%% causes Twelf to stop processing the file at this point +%% remove once you have completed the exercise +%% note that the arguments are "backwards" +%{! + +## Mode, worlds total + +!}% +%{! + +## Right-hand zero + +!}% +%{! + +## Right-hand succ + +!}% +%{! + +## Exercise: Prove that addition is commutative + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Basics_Starter). +!}% +%. + +%% note that the arguments are "backwards" +1*2is2 : mult 1 2 2 = mult/s (add/s (add/s add/z)) mult/z. + + + +%mode add +M +N -P. +%worlds () (add _ _ _). +%total M (add M _ _). + +%solve 1+1is2' : add 1 1 N. + +%mode mult +M +N -P. +%worlds () (mult _ _ _). +%total M (mult M _ _). + + + +rhzero : {M : nat} add M zero M -> type. +%mode rhzero +M -D. + +- : rhzero zero add/z. +- : rhzero (succ M) (add/s D) + <- rhzero M (D : add M zero M). + +%worlds () (rhzero _ _). +%total M (rhzero M _). + + + +rhsucc : add M N P -> add M (succ N) (succ P) -> type. +%mode rhsucc +D1 -D2. + +- : rhsucc (add/z : add zero M M) (add/z : add zero (succ M) (succ M)). +- : rhsucc (add/s (D1 : add M N P)) (add/s D2) + <- rhsucc D1 (D2 : add M (succ N) (succ P)). + +%worlds () (rhsucc _ _). +%total M (rhsucc M _). + + + diff --git a/new-tests/stelf-output/pages/popl-tutorial-basics.lf b/new-tests/stelf-output/pages/popl-tutorial-basics.lf new file mode 100644 index 0000000..5d0afe0 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-basics.lf @@ -0,0 +1,10 @@ +% +%{! * [Examples of basic error messages](/wiki/popl-tutorial-basic-error-messages/) +* [http://twelf.org/w/index.php?title=POPL_Tutorial/Basics_Starter&action=raw&ctype=text/css Starter code]: start from here if you don't want to type along +* [Solutions to exercises](/wiki/popl-tutorial-basics-answer/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Basics). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-big-step-small-step-solution.lf b/new-tests/stelf-output/pages/popl-tutorial-big-step-small-step-solution.lf new file mode 100644 index 0000000..4ccbfb8 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-big-step-small-step-solution.lf @@ -0,0 +1,241 @@ +% +%{! This is the solution to [this exercise](/wiki/popl-tutorial-big-step-small-step/). +|hidden = true !}% +%% Syntax %% +%sort tp %. +%name tp %. +%term nat tp %. +%term arr %pi tp %-> tp %-> tp %. +%sort exp %. +%name exp %. +%term fn %pi tp %-> (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term z exp %. +%term s %pi exp %-> exp %. +%term ifz %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%% Small-step dynamic semantics %% +%sort value %. +%name value %. +%mode value %in %. +%term value/z value z %. +%term value/s %pi (value (s E)) %<- (value E) %. +%term value/fn value (fn _ _) %. +%sort step %. +%name step %. +%mode step %in %out %. +%term step/app/fn %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app/arg %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app/beta %pi (step (app (fn _ ([x] E x)) E2) (E E2)) %<- (value E2) %. +%term step/s %pi (step (s E) (s E')) %<- (step E E') %. +%term step/ifz/arg %pi (step (ifz E E1 ([x] E2 x)) (ifz E' E1 ([x] E2 x))) %<- (step E E') %. +%term step/ifz/z step (ifz z E1 ([x] E2 x)) E1 %. +%term step/ifz/s %pi (step (ifz (s E) E1 ([x] E2 x)) (E2 E)) %<- (value E) %. +%%% Multi-step %%% +%sort step* %. +%name step* %. +%term step*/step %pi (step* E E') %<- (step E E') %. +%term step*/refl step* E E %. +%term step*/trans %pi (step* E E'') %<- (step* E' E'') %<- (step* E E') %. +%inline @ (%pi (step* E E') %-> (step* E' E'') %-> (step* E E'')) [d1] [d2] step*/trans d1 d2 %. +%prec %right 10 @ %. +%% Big-step dynamic semantics %% +%sort eval %. +%name eval %. +%term eval/z eval z z %. +%term eval/s %pi (eval (s E) (s V)) %<- (eval E V) %. +%term eval/ifz/z %pi (eval (ifz E E0 ([x] E1 x)) V0) %<- (eval E z) %<- (eval E0 V0) %. +%term eval/ifz/s %pi (eval (ifz E E0 ([x] E1 x)) V1) %<- (eval E (s V)) %<- (eval (E1 V) V1) %. +%term eval/fn eval (fn T1 ([x] E x)) (fn T1 ([x] E x)) %. +%term eval/app + %pi (eval (app E1 E2) V) + %<- (eval E1 (fn T1 ([x] E x))) + %<- (eval E2 V2) + %<- (eval (E V2) V) %. +%%% Lemmas about big-step dynamic semantics and values %%% +%sort eval-val %. +%mode eval-val %in %out %. +%term _ eval-val eval/z value/z %. +%term _ %pi (eval-val (eval/s D) (value/s Dval)) %<- (eval-val D Dval) %. +%term _ + %pi (eval-val (eval/ifz/z (%the (eval E0 V0) D0) (%the (eval E z) Dz)) Dval0) + %<- (eval-val D0 (%the (value V0) Dval0)) %. +%term _ + %pi (eval-val (%the (eval (ifz E E0 ([x] E1 x)) V1) (eval/ifz/s (%the (eval (E1 V) V1) D1) (%the (eval E (s V)) Ds))) Dval1) + %<- (eval-val D1 (%the (value V1) Dval1)) %. +%term _ eval-val eval/fn value/fn %. +%term _ + %pi (eval-val (eval/app (%the (eval (E V2) V) Dsub) (%the (eval E2 V2) D2) (%the (eval E1 (fn T1 ([x] E x))) D1)) DvalV) + %<- (eval-val Dsub (%the (value V) DvalV)) %. +%worlds () (eval-val _ _) %. +%total D (eval-val D _) %. +%sort val-eval %. +%mode val-eval %in %out %. +%term _ val-eval value/z eval/z %. +%term _ %pi (val-eval (value/s Dval) (eval/s Dev)) %<- (val-eval Dval Dev) %. +%term _ val-eval value/fn eval/fn %. +%worlds () (val-eval _ _) %. +%total D (val-eval D _) %. +%{! ## Big-step implies multi-step !}% +%{! ### Compatibility lemmas === |hidden=true !}% +%sort step*/s %. +%mode step*/s %in %out %. +%term _ step*/s (step*/step (%the (step E E') D)) (step*/step (step/s D)) %. +%term _ step*/s (%the (step* E E) step*/refl) (%the (step* (s E) (s E)) step*/refl) %. +%term _ + %pi (step*/s (step*/trans (%the (step* E E') D*) (%the (step* E' E'') D*')) (step*/trans D*s D*'s)) + %<- (step*/s D* (%the (step* (s E) (s E')) D*s)) + %<- (step*/s D*' (%the (step* (s E') (s E'')) D*'s)) %. +%worlds () (step*/s _ _) %. +%total D (step*/s D _) %. +%sort step*/ifz/arg %. +%mode {%in E exp} {%in E' exp} {%in E0 exp} {%in E1 %pi exp %-> exp} {%in D step* E E'} {%out Difz step* (ifz E E0 ([x exp] E1 x)) (ifz E' E0 ([x exp] E1 x))} step*/ifz/arg D Difz %. +%term _ step*/ifz/arg (step*/step (%the (step E E') D)) (step*/step (step/ifz/arg D)) %. +%term _ step*/ifz/arg step*/refl step*/refl %. +%term _ + %pi (step*/ifz/arg (step*/trans D* D*') (step*/trans D*ifz D*'ifz)) + %<- (step*/ifz/arg D* D*ifz) + %<- (step*/ifz/arg D*' D*'ifz) %. +%worlds () (step*/ifz/arg _ _) %. +%total D (step*/ifz/arg D _) %. +%sort step*/app/fn %. +%mode {%in E1 exp} {%in E1' exp} {%in E2 exp} {%in D1 step* E1 E1'} {%out Dapp step* (app E1 E2) (app E1' E2)} step*/app/fn D1 Dapp %. +%term _ step*/app/fn (step*/step (%the (step E1 E1') D1)) (step*/step (step/app/fn D1)) %. +%term _ step*/app/fn step*/refl step*/refl %. +%term _ + %pi (step*/app/fn (step*/trans D* D*') (step*/trans D*app D*'app)) + %<- (step*/app/fn D* D*app) + %<- (step*/app/fn D*' D*'app) %. +%worlds () (step*/app/fn _ _) %. +%total D (step*/app/fn D _) %. +%{! #### TASK 1: Fill in the cases of the unfinished compatibility proof !}% +%sort step*/app/arg %. +%mode step*/app/arg %in %in %out %. +%term _ step*/app/arg (step*/step (%the (step E2 E2') D2)) (%the (value V1) DvalV1) (step*/step (step/app/arg D2 DvalV1)) %. +%term _ step*/app/arg step*/refl _ step*/refl %. +%term _ + %pi (step*/app/arg (step*/trans D* D*') (%the (value V1) DvalV1) (step*/trans D*app D*'app)) + %<- (step*/app/arg D* DvalV1 D*app) + %<- (step*/app/arg D*' DvalV1 D*'app) %. +%worlds () (step*/app/arg _ _ _) %. +%total D (step*/app/arg D _ _) %. +%{! |hidden=true !}% +%{! ### Main theorem !}% +%{! #### TASK 2: Fill in the missing cases of the theorem !}% +%sort eval-multi %. +%mode eval-multi %in %out %. +%{! |hidden=true !}% +%term _ eval-multi (%the (eval z z) eval/z) step*/refl %. +%term _ + %pi (eval-multi (%the (eval (s E) (s V)) (eval/s D)) D*s) + %<- (eval-multi D (%the (step* E V) D*)) + %<- (step*/s D* (%the (step* (s E) (s V)) D*s)) %. +%term _ + %pi (eval-multi (eval/ifz/z (%the (eval E0 V0) D0) (%the (eval E z) Dz)) (Difz* @ step*/step step/ifz/z @ D0*)) + %<- (eval-multi Dz (%the (step* E z) Dz*)) + %<- (eval-multi D0 (%the (step* E0 V0) D0*)) + %<- (step*/ifz/arg Dz* (%the (step* (ifz E E0 ([x] E1 x)) (ifz z E0 ([x] E1 x))) Difz*)) %. +%term _ + %pi (eval-multi (eval/ifz/s (%the (eval (E1 V) V1) D1) (%the (eval E (s V)) Ds)) (Difz* @ step*/step (step/ifz/s DvalV) @ D1*)) + %<- (eval-multi Ds (%the (step* E (s V)) Ds*)) + %<- (eval-multi D1 (%the (step* (E1 V) V1) D1*)) + %<- (step*/ifz/arg Ds* (%the (step* (ifz E E0 ([x] E1 x)) (ifz (s V) E0 ([x] E1 x))) Difz*)) + %<- (eval-val Ds (value/s (%the (value V) DvalV))) %. +%{! ...snip... !}% +%term _ eval-multi eval/fn step*/refl %. +%term _ + %pi (eval-multi (eval/app (%the (eval (E V2) V) Dsub) (%the (eval E2 V2) D2) (%the (eval E1 (fn T1 ([x] E x))) D1)) (Dfun* @ Darg* @ step*/step (step/app/beta DvalV2) @ Dsub*)) + %<- (eval-multi D1 (%the (step* E1 (fn T1 ([x] E x))) D1*)) + %<- (eval-multi D2 (%the (step* E2 V2) D2*)) + %<- (eval-multi Dsub (%the (step* (E V2) V) Dsub*)) + %<- (step*/app/fn D1* (%the (step* (app E1 E2) (app (fn T1 ([x] E x)) E2)) Dfun*)) + %<- (step*/app/arg D2* value/fn (%the (step* (app (fn T1 ([x] E x)) E2) (app (fn T1 ([x] E x)) V2)) Darg*)) + %<- (eval-val D2 DvalV2) %. +%worlds () (eval-multi _ _) %. +%total D (eval-multi D _) %. +%{! ## Multi-step implies big step == |hidden=true !}% +%sort step** %. +%name step** %. +%term step**/refl step** E E %. +%term step**/cons %pi (step** E E'') %<- (step** E' E'') %<- (step E E') %. +%sort step**/trans %. +%mode step**/trans %in %in %out %. +%term _ step**/trans step**/refl (%the (step** E E'') Dstep**) Dstep** %. +%term _ + %pi (step**/trans (step**/cons (%the (step E E1) Dstep1) (%the (step** E1 E') Dstep**1)) (%the (step** E' E'') Dstep**2) (step**/cons Dstep1 Dstep**)) + %<- (step**/trans Dstep**1 Dstep**2 (%the (step** E1 E'') Dstep**)) %. +%worlds () (step**/trans _ _ _) %. +%total D (step**/trans D _ _) %. +%sort multi-step** %. +%mode multi-step** %in %out %. +%term _ multi-step** (step*/step Dstep) (step**/cons Dstep step**/refl) %. +%term _ multi-step** step*/refl step**/refl %. +%term _ + %pi (multi-step** (step*/trans (%the (step* E E') D1) (%the (step* E' E'') D2)) D**) + %<- (multi-step** D1 (%the (step** E E') D1**)) + %<- (multi-step** D2 (%the (step** E' E'') D2**)) + %<- (step**/trans D1** D2** (%the (step** E E'') D**)) %. +%worlds () (multi-step** _ _) %. +%total D (multi-step** D _) %. +%{! ### Expansion lemma !}% +%{! #### TASK 3: Prove the three missing cases of the expansion lemma !}% +%sort expansion %. +%mode expansion %in %in %out %. +%{! |hidden=true !}% +%term _ + %pi (expansion (%the (eval (s E') (s V)) (eval/s (%the (eval E' V) Deval'))) (%the (step (s E) (s E')) (step/s (%the (step E E') Dstep))) (%the (eval (s E) (s V)) (eval/s (%the (eval E V) Deval)))) + %<- (expansion Deval' Dstep (%the (eval E V) Deval)) %. +%term _ + %pi (expansion (%the (eval (ifz E' E0 ([x] E1 x)) V0) (eval/ifz/z (%the (eval E0 V0) D0) (%the (eval E' z) Dz'))) (%the (step (ifz E E0 ([x] E1 x)) (ifz E' E0 ([x] E1 x))) (step/ifz/arg (%the (step E E') Dstep))) (%the (eval (ifz E E0 ([x] E1 x)) V0) (eval/ifz/z D0 Dz))) + %<- (expansion Dz' Dstep (%the (eval E z) Dz)) %. +%{! ...snip... !}% +%term _ + %pi (expansion (%the (eval (ifz E' E0 ([x] E1 x)) V1) (eval/ifz/s (%the (eval (E1 V) V1) D1) (%the (eval E' (s V)) Ds'))) (%the (step (ifz E E0 ([x] E1 x)) (ifz E' E0 ([x] E1 x))) (step/ifz/arg (%the (step E E') Dstep))) (%the (eval (ifz E E0 ([x] E1 x)) V1) (eval/ifz/s D1 Ds))) + %<- (expansion Ds' Dstep (%the (eval E (s V)) Ds)) %. +%{! |hidden=true !}% +%term _ expansion (%the (eval E0 V0) Deval') (%the (step (ifz z E0 ([x] E1 x)) E0) step/ifz/z) (eval/ifz/z Deval' eval/z) %. +%term _ + %pi (expansion (%the (eval (E1 V) V1) Deval') (%the (step (ifz (s V) E0 ([x] E1 x)) (E1 V)) (step/ifz/s (%the (value V) Dval))) (eval/ifz/s Deval' (eval/s DevalV))) + %<- (val-eval Dval (%the (eval V V) DevalV)) %. +%term _ + %pi (expansion (%the (eval (E1 V) V1) Deval') (%the (step (ifz (s V) E0 ([x] E1 x)) (E1 V)) (step/ifz/s (%the (value V) Dval))) (eval/ifz/s Deval' (eval/s DevalV))) + %<- (val-eval Dval (%the (eval V V) DevalV)) %. +%term _ + %pi (expansion (%the (eval (app E1' E2) V) (eval/app (%the (eval (E V2) V) Dsub) (%the (eval E2 V2) D2) (%the (eval E1' (fn T1 ([x] E x))) D1'))) (%the (step (app E1 E2) (app E1' E2)) (step/app/fn (%the (step E1 E1') Dstep1))) (eval/app Dsub D2 D1)) + %<- (expansion D1' Dstep1 (%the (eval E1 (fn T1 ([x] E x))) D1)) %. +%{! ...snip... !}% +%term _ + %pi (expansion (%the (eval (app V1 E2') V) (eval/app (%the (eval (E V2) V) Dsub) (%the (eval E2' V2) D2') (%the (eval V1 (fn T1 ([x] E x))) D1))) (%the (step (app V1 E2) (app V1 E2')) (step/app/arg (%the (step E2 E2') Dstep2) (%the (value V1) Dval1))) (eval/app Dsub D2 D1)) + %<- (expansion D2' Dstep2 (%the (eval E2 V2) D2)) %. +%term _ + %pi (expansion (%the (eval (E V2) V) Deval') (%the (step (app (fn T1 ([x] E x)) V2) (E V2)) (step/app/beta (%the (value V2) Dval2))) (eval/app Deval' DevalV2 eval/fn)) + %<- (val-eval Dval2 (%the (eval V2 V2) DevalV2)) %. +%worlds () (expansion _ _ _) %. +%total D (expansion _ D _) %. +%{! ### Main theorem !}% +%{! #### TASK 4: Prove main theorem in the forward direction !}% +%sort multi**-eval %. +%mode multi**-eval %in %in %out %. +%term _ + %pi (multi**-eval step**/refl (%the (value V) Dval) Deval) + %<- (val-eval Dval (%the (eval V V) Deval)) %. +%term _ + %pi (multi**-eval (step**/cons (%the (step E E') Dstep) (%the (step** E' V) Dstep**)) (%the (value V) Dval) Deval) + %<- (multi**-eval Dstep** Dval (%the (eval E' V) Deval')) + %<- (expansion Deval' Dstep (%the (eval E V) Deval)) %. +%worlds () (multi**-eval _ _ _) %. +%total D (multi**-eval D _ _) %. +%{! #### TASK 5: Complete proof by composing previous lemmas !}% +%sort multi-eval %. +%mode multi-eval %in %in %out %. +%term _ + %pi (multi-eval (%the (step* E V) D*) (%the (value V) Dval) Dev) + %<- (multi-step** D* (%the (step** E V) D**)) + %<- (multi**-eval D** Dval (%the (eval E V) Dev)) %. +%worlds () (multi-eval _ _ _) %. +%total {} (multi-eval _ _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Big_step,_small_step:_Solution). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-big-step-small-step.lf b/new-tests/stelf-output/pages/popl-tutorial-big-step-small-step.lf new file mode 100644 index 0000000..9e897bd --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-big-step-small-step.lf @@ -0,0 +1,353 @@ +% +%{! In this exercise, we explore the relationship between big-step evaluation and +small-step transition semantics for the MinML language we worked with before. +The theroem we want to establish is that an expression e evaluates +to a value v in many little steps if and only if it evaluates to v in one +big step. + +There are five tasks: +* Complete a compatibility lemma that if , then +* Prove the backward direction (big-step evaluation implies small-step evaluation) +* Prove three cases of the expansion lemma. +* Prove the forward direction (small-step evaluation implies big-step evaluation) for a "convienent" definition of multi-step. +* Prove the forward direction for the original definition of multi-step. + +The solution is [here](/wiki/popl-tutorial-big-step-small-step-solution/). + +## Syntax + +Types: + +!}% +%sort tp %. +%name tp %. +%term nat tp %. +%term arr %pi tp %-> tp %-> tp %. +%{! + +Expressions + +!}% +%sort exp %. +%name exp %. +%term fn %pi tp %-> (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term z exp %. +%term s %pi exp %-> exp %. +%term ifz %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%{! + +## Small-step dynamic semantics + +!}% +%sort value %. +%name value %. +%mode value %in %. +%term value/z value z %. +%term value/s %pi (value (s E)) %<- (value E) %. +%term value/fn value (fn _ _) %. +%sort step %. +%name step %. +%mode step %in %out %. +%term step/app/fn %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app/arg %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app/beta %pi (step (app (fn _ ([x] E x)) E2) (E E2)) %<- (value E2) %. +%term step/s %pi (step (s E) (s E')) %<- (step E E') %. +%term step/ifz/arg %pi (step (ifz E E1 ([x] E2 x)) (ifz E' E1 ([x] E2 x))) %<- (step E E') %. +%term step/ifz/z step (ifz z E1 ([x] E2 x)) E1 %. +%term step/ifz/s %pi (step (ifz (s E) E1 ([x] E2 x)) (E2 E)) %<- (value E) %. +%{! + +### Multi-step + +!}% +%sort step* %. +%name step* %. +%term step*/step %pi (step* E E') %<- (step E E') %. +%term step*/refl step* E E %. +%term step*/trans %pi (step* E E'') %<- (step* E' E'') %<- (step* E E') %. +%{! + +We can define ``D1 @ D2`` to be an abbreviation for ``step*/trans +D1 D2``. This is convenient for chaining together several derivations +using transitivity. + +!}% +%inline @ (%pi (step* E E') %-> (step* E' E'') %-> (step* E E'')) [d1] [d2] step*/trans d1 d2 %. +%prec %right 10 @ %. +%{! + +## Big-step dynamic semantics + +!}% +%sort eval %. +%name eval %. +%term eval/z eval z z %. +%term eval/s %pi (eval (s E) (s V)) %<- (eval E V) %. +%term eval/ifz/z %pi (eval (ifz E E0 ([x] E1 x)) V0) %<- (eval E z) %<- (eval E0 V0) %. +%term eval/ifz/s %pi (eval (ifz E E0 ([x] E1 x)) V1) %<- (eval E (s V)) %<- (eval (E1 V) V1) %. +%term eval/fn eval (fn T1 ([x] E x)) (fn T1 ([x] E x)) %. +%term eval/app + %pi (eval (app E1 E2) V) + %<- (eval E1 (fn T1 ([x] E x))) + %<- (eval E2 V2) + %<- (eval (E V2) V) %. +%{! + +## Lemmas + +There are two easy lemmas we'll need later that relate evaluation to +the value judgement from above. + +### Evaluation returns a value + +!}% +%sort eval-val %. +%mode eval-val %in %out %. +%term _ eval-val eval/z value/z %. +%term _ %pi (eval-val (eval/s D) (value/s Dval)) %<- (eval-val D Dval) %. +%term _ + %pi (eval-val (eval/ifz/z (%the (eval E0 V0) D0) (%the (eval E z) Dz)) Dval0) + %<- (eval-val D0 (%the (value V0) Dval0)) %. +%term _ + %pi (eval-val (%the (eval (ifz E E0 ([x] E1 x)) V1) (eval/ifz/s (%the (eval (E1 V) V1) D1) (%the (eval E (s V)) Ds))) Dval1) + %<- (eval-val D1 (%the (value V1) Dval1)) %. +%term _ eval-val eval/fn value/fn %. +%term _ + %pi (eval-val (eval/app (%the (eval (E V2) V) Dsub) (%the (eval E2 V2) D2) (%the (eval E1 (fn T1 ([x] E x))) D1)) DvalV) + %<- (eval-val Dsub (%the (value V) DvalV)) %. +%worlds () (eval-val _ _) %. +%total D (eval-val D _) %. +%{! + +### Values evaluate to themselves + +!}% +%sort val-eval %. +%mode val-eval %in %out %. +%term _ val-eval value/z eval/z %. +%term _ %pi (val-eval (value/s Dval) (eval/s Dev)) %<- (val-eval Dval Dev) %. +%term _ val-eval value/fn eval/fn %. +%worlds () (val-eval _ _) %. +%total D (val-eval D _) %. +%{! + +## Big-step implies multi-step + +Our first main theorem is to show that big-step evaluation implies +multi-step evaluation, i.e. that if ``eval E V`` then ``step* +E V``. + +### Compatibility lemmas + +To prove this, we'll need some compatibility lemmas for multi-step +transition which say that multi-step transitions in a sub-term of a +term can be bubbled up to the term itself. + +Each argument proceeds by induction on the derivation that +``step* E E'``. + +!}% +%sort step*/s %. +%mode step*/s %in %out %. +%term _ step*/s (step*/step (%the (step E E') D)) (step*/step (step/s D)) %. +%term _ step*/s (%the (step* E E) step*/refl) (%the (step* (s E) (s E)) step*/refl) %. +%term _ + %pi (step*/s (step*/trans (%the (step* E E') D*) (%the (step* E' E'') D*')) (step*/trans D*s D*'s)) + %<- (step*/s D* (%the (step* (s E) (s E')) D*s)) + %<- (step*/s D*' (%the (step* (s E') (s E'')) D*'s)) %. +%worlds () (step*/s _ _) %. +%total D (step*/s D _) %. +%sort step*/ifz/arg %. +%mode {%in E exp} {%in E' exp} {%in E0 exp} {%in E1 %pi exp %-> exp} {%in D step* E E'} {%out Difz step* (ifz E E0 ([x exp] E1 x)) (ifz E' E0 ([x exp] E1 x))} step*/ifz/arg D Difz %. +%term _ step*/ifz/arg (step*/step (%the (step E E') D)) (step*/step (step/ifz/arg D)) %. +%term _ step*/ifz/arg step*/refl step*/refl %. +%term _ + %pi (step*/ifz/arg (step*/trans D* D*') (step*/trans D*ifz D*'ifz)) + %<- (step*/ifz/arg D* D*ifz) + %<- (step*/ifz/arg D*' D*'ifz) %. +%worlds () (step*/ifz/arg _ _) %. +%total D (step*/ifz/arg D _) %. +%sort step*/app/fn %. +%mode {%in E1 exp} {%in E1' exp} {%in E2 exp} {%in D1 step* E1 E1'} {%out Dapp step* (app E1 E2) (app E1' E2)} step*/app/fn D1 Dapp %. +%term _ step*/app/fn (step*/step (%the (step E1 E1') D1)) (step*/step (step/app/fn D1)) %. +%term _ step*/app/fn step*/refl step*/refl %. +%term _ + %pi (step*/app/fn (step*/trans D* D*') (step*/trans D*app D*'app)) + %<- (step*/app/fn D* D*app) + %<- (step*/app/fn D*' D*'app) %. +%worlds () (step*/app/fn _ _) %. +%total D (step*/app/fn D _) %. +%{! #### TASK 1: Fill in the cases of the unfinished compatibility proof !}% +%sort step*/app/arg %. +%mode step*/app/arg %in %in %out %. +%%% fill in here. +%worlds () (step*/app/arg _ _ _) %. +%total D (step*/app/arg D _ _) %. +%{! +### Main theorem + +Now we can prove our first main theorem by induction on the derivation +of ``eval E V`` using the multi-step compatibility lemmas above. + +#### TASK 2: Fill in the missing cases of the theorem + +!}% +%sort eval-multi %. +%mode eval-multi %in %out %. +%term _ eval-multi (%the (eval z z) eval/z) step*/refl %. +%term _ + %pi (eval-multi (%the (eval (s E) (s V)) (eval/s D)) D*s) + %<- (eval-multi D (%the (step* E V) D*)) + %<- (step*/s D* (%the (step* (s E) (s V)) D*s)) %. +%term _ + %pi (eval-multi (eval/ifz/z (%the (eval E0 V0) D0) (%the (eval E z) Dz)) (Difz* @ step*/step step/ifz/z @ D0*)) + %<- (eval-multi Dz (%the (step* E z) Dz*)) + %<- (eval-multi D0 (%the (step* E0 V0) D0*)) + %<- (step*/ifz/arg Dz* (%the (step* (ifz E E0 ([x] E1 x)) (ifz z E0 ([x] E1 x))) Difz*)) %. +%term _ + %pi (eval-multi (eval/ifz/s (%the (eval (E1 V) V1) D1) (%the (eval E (s V)) Ds)) (Difz* @ step*/step (step/ifz/s DvalV) @ D1*)) + %<- (eval-multi Ds (%the (step* E (s V)) Ds*)) + %<- (eval-multi D1 (%the (step* (E1 V) V1) D1*)) + %<- (step*/ifz/arg Ds* (%the (step* (ifz E E0 ([x] E1 x)) (ifz (s V) E0 ([x] E1 x))) Difz*)) + %<- (eval-val Ds (value/s (%the (value V) DvalV))) %. +%term _ eval-multi eval/fn XXX %. +%%% fill in here. +%term _ eval-multi (eval/app (%the (eval (E (fn T1 T2 ([f] [x] E f x)) V2) V) Dsub) (%the (eval E2 V2) D2) (%the (eval E1 (fn T1 T2 ([f] [x] E f x))) D1)) XXX %. +%%% fill in here. +%worlds () (eval-multi _ _) %. +%total D (eval-multi D _) %. +%{! + +## Multi-step implies big step + +### Left-linearized multi-step + +For the reverse direction, it turns out we'd rather have the inductive +definition of the multi-step relation be a little bit different. We call +the more convienent direction "left-linearlized multi-step" and +prove it equivlaent to the previous definition of multi-step. + +!}% +%sort step** %. +%name step** %. +%term step**/refl step** E E %. +%term step**/cons %pi (step** E E'') %<- (step** E' E'') %<- (step E E') %. +%{! + +Lemma: Multi-step implies linearized multi-step. This is tantamount +to showing that transitivity is admissible for linearized derivations. + +!}% +%sort step**/trans %. +%mode step**/trans %in %in %out %. +%term _ step**/trans step**/refl (%the (step** E E'') Dstep**) Dstep** %. +%term _ + %pi (step**/trans (step**/cons (%the (step E E1) Dstep1) (%the (step** E1 E') Dstep**1)) (%the (step** E' E'') Dstep**2) (step**/cons Dstep1 Dstep**)) + %<- (step**/trans Dstep**1 Dstep**2 (%the (step** E1 E'') Dstep**)) %. +%worlds () (step**/trans _ _ _) %. +%total D (step**/trans D _ _) %. +%{! + +Lemma: Multi-step implies linearized multi-step. This is tantamount +to showing that transitivity is admissible for linearized derivations. + +!}% +%sort multi-step** %. +%mode multi-step** %in %out %. +%term _ multi-step** (step*/step Dstep) (step**/cons Dstep step**/refl) %. +%term _ multi-step** step*/refl step**/refl %. +%term _ + %pi (multi-step** (step*/trans (%the (step* E E') D1) (%the (step* E' E'') D2)) D**) + %<- (multi-step** D1 (%the (step** E E') D1**)) + %<- (multi-step** D2 (%the (step** E' E'') D2**)) + %<- (step**/trans D1** D2** (%the (step** E E'') D**)) %. +%worlds () (multi-step** _ _) %. +%total D (multi-step** D _) %. +%{! + +### Expansion lemma + +The main lemma is that evaluation is closed under single-step expansion. +This is a key lemma in the proof that evaluation in the small-step +semantics implies evaluation in the big-step evaluation. + +We proceed by induction on the second derivation, the derivation +that ``step E E'``. + +#### TASK 3: Prove the three missing cases of the expansion lemma + +!}% +%sort expansion %. +%mode expansion %in %in %out %. +%{! Case ``step/s``: !}% +%term _ + %pi (expansion (%the (eval (s E') (s V)) (eval/s (%the (eval E' V) Deval'))) (%the (step (s E) (s E')) (step/s (%the (step E E') Dstep))) (%the (eval (s E) (s V)) (eval/s (%the (eval E V) Deval)))) + %<- (expansion Deval' Dstep (%the (eval E V) Deval)) %. +%{! Case ``step/ifz/arg``: !}% +%% By inversion, either eval E' V by eval/ifz/z or by eval/ifz/s. +%term _ + %pi (expansion (%the (eval (ifz E' E0 ([x] E1 x)) V0) (eval/ifz/z (%the (eval E0 V0) D0) (%the (eval E' z) Dz'))) (%the (step (ifz E E0 ([x] E1 x)) (ifz E' E0 ([x] E1 x))) (step/ifz/arg (%the (step E E') Dstep))) (%the (eval (ifz E E0 ([x] E1 x)) V0) (eval/ifz/z D0 Dz))) + %<- (expansion Dz' Dstep (%the (eval E z) Dz)) %. +%term _ expansion XXX (%the (step (ifz E E0 ([x] E1 x)) (ifz E' E0 ([x] E1 x))) (step/ifz/arg (%the (step E E') Dstep))) YYY %. +%%% fill in here. +%{! Case ``step/ifz/z``: !}% +%term _ expansion (%the (eval E0 V0) Deval') (%the (step (ifz z E0 ([x] E1 x)) E0) step/ifz/z) (eval/ifz/z Deval' eval/z) %. +%{! Case ``step/ifz/s``: !}% +%term _ + %pi (expansion (%the (eval (E1 V) V1) Deval') (%the (step (ifz (s V) E0 ([x] E1 x)) (E1 V)) (step/ifz/s (%the (value V) Dval))) (eval/ifz/s Deval' (eval/s DevalV))) + %<- (val-eval Dval (%the (eval V V) DevalV)) %. +%{! Case ``step/app/fn``: !}% +%term _ + %pi (expansion (%the (eval (app E1' E2) V) (eval/app (%the (eval (E V2) V) Dsub) (%the (eval E2 V2) D2) (%the (eval E1' (fn T1 ([x] E x))) D1'))) (%the (step (app E1 E2) (app E1' E2)) (step/app/fn (%the (step E1 E1') Dstep1))) (eval/app Dsub D2 D1)) + %<- (expansion D1' Dstep1 (%the (eval E1 (fn T1 ([x] E x))) D1)) %. +%{! Case ``step/app/arg``: !}% +%term _ expansion XXX (%the (step (app V1 E2) (app V1 E2')) (step/app/arg (%the (step E2 E2') Dstep2) (%the (value V1) Dval1))) YYY %. +%%% fill in here. +%{! Case ``step/app/beta-v``: !}% +%term _ expansion XXX (%the (step (app (fn T1 T2 ([f] [x] E f x)) V2) (E (fn T1 T2 ([f] [x] E f x)) V2)) (step/app/beta-v (%the (value V2) Dval2))) YYY %. +%%% fill in here. +%worlds () (expansion _ _ _) %. +%total D (expansion _ D _) %. +%{! + +### Main theorem + +Now we will use our "more convienent" left-linearized multi-step definition +and induct over it in order to prove the forward direction of our primary +theorem, that small step evaluation implies big step evaluation. + +The proof is by induction on the derivation that ``step** E V``, +using closure under single-step expansion. There are only two cases +due to the simple definition of ``step**``. + +#### TASK 4: Prove main theorem in the forward direction + +!}% +%sort multi**-eval %. +%mode multi**-eval %in %in %out %. +%%% fill in here. +%worlds () (multi**-eval _ _ _) %. +%total D (multi**-eval D _ _) %. +%{! + +### Corollary + +Using one of the theorems we proved about left-linearized multi-step, +we can easly show the final result; our proof needs to have only one case +and does not use induction. + +#### TASK 5: Complete proof by composing previous lemmas + +!}% +%sort multi-eval %. +%mode multi-eval %in %in %out %. +%%% fill in here. +%worlds () (multi-eval _ _ _) %. +%total {} (multi-eval _ _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Big_step,_small_step). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-church-rosser-problem.lf b/new-tests/stelf-output/pages/popl-tutorial-church-rosser-problem.lf new file mode 100644 index 0000000..9452b93 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-church-rosser-problem.lf @@ -0,0 +1,283 @@ +% +%{! + +In this exercise, we will prove the [[w:Church-Rosser theorem|Church-Rosser theorem]]. You are +provided with a syntax and two substitution lemmas, and your task will be +to implement two cases of the diamond lemma - the +**Application-Application** case, and the **Beta-Application** case. + +You will be able to uncomment the ``%total`` at the end when you are +done. + +The solution is [here](/wiki/church-rosser-w-identity-reduction/). + +!}% +%{! ## Syntax !}% +%sort exp %. +%name exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%{! + +When we use this ``%block``, it expresses that we can be working in a +context with arbitrary expression variables. This will be the only world we +will need for this exercise. + +!}% +%block exps [x exp]%. +%{! ## Reduction !}% +%{! + +We can always reduce an expression to itself (which means that we don't +explicitly need a case to handle variables). We can reduce under binders and +reduce both sides of an application "in parallel." If we have a β-redex +``(λx.ea) eb``, then after reducing ``ea`` (with ``x`` free) to +``ea'`` (with ``x`` free) and reducing ``eb`` to ``eb'``, +we can return ``[eb'/x]ea'``, the substitution of ``eb'`` into +``ea'``. + +The ``%worlds`` explicitly states that we will be reducing in a setting +with free variables. + +!}% +%sort reduce %. +%term reduce/id reduce E E %. +%term reduce/lam %pi (reduce (lam E) (lam E')) %<- ({x exp} reduce (E x) (E' x)) %. +%term reduce/app %pi (reduce (app E1 E2) (app E1' E2')) %<- (reduce E1 E1') %<- (reduce E2 E2') %. +%term reduce/beta + %pi (reduce (app (lam E1) E2) (E1' E2')) + %<- ({x exp} reduce (E1 x) (E1' x)) + %<- (reduce E2 E2') %. +%worlds (exps) (reduce _ _) %. +%{! ## Substitution !}% +%{! + +The first substitution theorem says that if we have a term ``e1`` with +``x`` free and a term ``e2`` that reduces to term ``e2'``, +then ``[e2/x]e1`` reduces to ``[e2'/x]e1``. + +The interesting cases are really the first two - if the term is just the free +variable being substituted for, the result follows from the hypothesis. In the +case that the free variable is _not_ free (i.e. the term has no hole), then +the result follows from the reflexivity of reduction. This is more than +sufficient to handle the case of a variable that is not the distinguished free +variable, and all other cases are covered. + +The proof is by induction on the structure of the term with the free variable. + +!}% +%sort substitute1 {E1 %pi exp %-> exp} %. +%mode substitute1 %in %in %out %. +%term _ substitute1 ([x] x) D D %. +%term _ substitute1 ([x] E) _ reduce/id %. +%term _ + %pi (substitute1 ([x] lam ([y] E x y)) D (reduce/lam D')) + %<- ({y} substitute1 ([x] E x y) D (D' y)) %. +%term _ + %pi (substitute1 ([x] app (E1 x) (E2 x)) D (reduce/app D2 D1)) + %<- (substitute1 E1 D D1) + %<- (substitute1 E2 D D2) %. +%worlds (exps) (substitute1 _ _ _) %. +%total E (substitute1 E _ _) %. +%{! + +The second substitution theorem is similar, except that we are reducing +``e1`` with ``x`` free to ``e1'`` as well as by reducing +``e2`` to ``e2'``, showing that ``[e2/x]e1`` reduces to +``[e2'/x]e1'``. + +Proof is by induction on the structure of the reduction of ``e1`` to +``e1'``. We call to the ``substitute1`` lemma +when we "bottom out" at the reflexive case ``reduce/id``. + +!}% +%sort substitute2 {x exp} %. +%mode substitute2 %in %in %out %. +%term _ + %pi (substitute2 ([x] %the (reduce (E x) (E x)) reduce/id) D D') + %<- (substitute1 E D D') %. +%term _ + %pi (substitute2 ([x] reduce/lam ([y] D1 x y)) D2 (reduce/lam D)) + %<- ({y} substitute2 ([x] D1 x y) D2 (D y)) %. +%term _ + %pi (substitute2 ([x] reduce/app (D2 x) (D1 x)) D (reduce/app D2' D1')) + %<- (substitute2 D1 D D1') + %<- (substitute2 D2 D D2') %. +%term _ + %pi (substitute2 ([x] reduce/beta (D2 x) ([y] D1 x y)) D (reduce/beta D2' D1')) + %<- ({y} substitute2 ([x] D1 x y) D (D1' y)) + %<- (substitute2 D2 D D2') %. +%worlds (exps) (substitute2 _ _ _) %. +%total D (substitute2 D _ _) %. +%{! ## The Diamond Property !}% +%{! + +Now we come to the interesting part: the diamond property. + + E + / \ + / \ + E1 E2 + \ / + \ / + E' + +If ``E`` reduces to both ``E1``, and ``E2``, then there is a +common E' such that ``E1`` and ``E2`` both reduce to it. + +!}% +%sort diamond %. +%mode diamond %in %in %out %out %. +%{! ### Identity !}% +%{! + +If either case is the identity, then we are done. + + id: E D: D: E id: + e=>e / \ e=>e2 e=>e1 / \ e=>e + / \ / \ + E E2 E1 E + D: \ /id: id: \ /D: + e=>e2 \ / e2=>e2 e1=>e1\ / e=>e1 + e2 E1 + +!}% +%term _ diamond reduce/id D D reduce/id %. +%term _ diamond D reduce/id reduce/id D %. +%{! ### Lambda-Lambda !}% +%{! + +If both cases are reductions under a binder, we pull the result straight +from the induction hypothesis. + + λx.e by induction: + reduce/lam / \ reduce/lam D1, D2 ---> D1': e1'=>e' + (D1: e=>e1) / \ (D2: e=>e2) D2': e2'=>e' + / \ + λx.e1 λx.e2 + \ / + reduce/lam \ / reduce/lam + D1' \ / D2' + λx.e' + +Note the oversimplification being made in the graphical presentation, in that +the subterms and sub-derivations are not clearly shown to have a free variable. +Twelf will, of course, not allow this sloppiness. + +!}% +%term _ + %pi (diamond (%the (reduce (lam E) (lam E1)) (reduce/lam (%the ({x exp} reduce (E x) (E1 x)) D1))) (%the (reduce (lam E) (lam E2)) (reduce/lam (%the ({x exp} reduce (E x) (E2 x)) D2))) (reduce/lam D1') (reduce/lam D2')) + %<- ({x exp} diamond (D1 x) (D2 x) (%the (reduce (E1 x) (E' x)) (D1' x)) (%the (reduce (E2 x) (E' x)) (D2' x))) %. +%{! ### Application-Application !}% +%{! + +If both cases are applications, we pull the result straight from the +induction hypothesis. + + ea eb by induction + reduce/app / \ reduce/app D1a, D2a ---> D1a': e1a=>ea' + (D1b: eb=>e1b) / \ (D2b: eb=>e2b) D2a': e2a=>ea' + (D1a: ea=>e1a) / \ (D2a: ea=>e2a) D1b, D2b ---> D1b': e1b=>eb' + e1a e1b e2a e2b D2b': e2b=>eb' + \ / + reduce/app \ / reduce/app + D1b' D1a' \ / D2b' D2a' + ea' eb' +!}% +%% WRITE THIS CASE +%{! ### Beta-Beta !}% +%{! + +If both cases are beta reductions, we get the result from performing two +substitutions. + + (λx.ea) eb by induction + reduce/beta / \ reduce/beta D1a, D2a ---> D1a': e1a=>ea' + (D1b: eb=>e1b) / \ (D2b: eb=>e2b) D2a': e2a=>ea' + (D1a: ea=>e1a) / \ (D2a: ea=>e1a) D1b, D2b ---> D1b': e1b=>eb' + [e1b/x]e1a [e2b/x]e2a D2b': e2b=>eb' + \ / + substitute \ / substitute + D1b' into D1a' \ / D2b' into D2a' + [eb'/x]ea +!}% +%term _ + %pi (diamond (%the (reduce (app (lam Ea) Eb) (E1a E1b)) (reduce/beta (%the (reduce Eb E1b) D1b) (%the ({x} reduce (Ea x) (E1a x)) D1a))) (%the (reduce (app (lam Ea) Eb) (E2a E2b)) (reduce/beta (%the (reduce Eb E2b) D2b) (%the ({x} reduce (Ea x) (E2a x)) D2a))) D1 D2) + %<- ({x} diamond (D1a x) (D2a x) (%the (reduce (E1a x) (Ea' x)) (D1a' x)) (%the (reduce (E2a x) (Ea' x)) (D2a' x))) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) + %<- (substitute2 D1a' D1b' (%the (reduce (E1a E1b) (Ea' Eb')) D1)) + %<- (substitute2 D2a' D2b' (%the (reduce (E2a E2b) (Ea' Eb')) D2)) %. +%{! ### Beta-Application !}% +%{! + +If the left-hand side is a β-reduction +``(λx.ea) eb => [e1b/x] e1a`` but the right-hand side is not, then we +know that the right-hand side reduction must be +``(λx.ea) eb => (λx.e2a) e2b``. + +Unfortunately, then we have two options - either ``λx.ea => λx.e2a`` by +``reduce/refl`` and ``reduce/lam``. The ``reduce/refl`` case +is essentially just an inconvenience caused by allowing identity reductions +anywhere (not just at variables). + +The first subcase: + + (λx.ea) eb by induction + reduce/beta / \ reduce/app D1a, D2a ---> D1a': e1a=>ea' + (D1b: eb=>e1b) / \ (D2b: eb=>e2b) D2a': e2a=>ea' + (D1a: ea=>e1a) / \ (reduce/lam D1b, D2b ---> D1b': e1b=>eb' + / \ (D2a: ea=>e2a)) D2b': e1b=>eb' + [e1b/x]e1a (λx.e2a) e2b + \ / + substitute \ / reduce/beta + D1b' into D2a' \ / D2b' D2a' + \ / + [eb'/x]ea' +!}% +%% WRITE THIS CASE +%{! + +The second subcase: + + (λx.ea) eb by induction + reduce/beta / \ reduce/app D1b, D2b ---> D1b': e1b=>eb' + (D1b: eb=>e1b) / \ (D2b: eb=>e2b) D2b': e1b=>eb' + (D1a: ea=>e1a) / \ (reduce/id + / \ ea=>ea)) + [e1b/x]e1a (λx.ea) e2b + \ / + substitute \ / reduce/beta + D1b' into e1a \ / D2b' D1a + \ / + [eb'/x]e1a +!}% +%term _ + %pi (diamond (%the (reduce (app (lam Ea) Eb) (E1a E1b)) (reduce/beta (%the (reduce Eb E1b) D1b) (%the ({x} reduce (Ea x) (E1a x)) D1a))) (%the (reduce (app (lam Ea) Eb) (app (lam Ea) E2b)) (reduce/app (%the (reduce Eb E2b) D2b) (%the (reduce (lam Ea) (lam Ea)) reduce/id))) D1 (reduce/beta D2b' D1a)) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) + %<- (substitute1 E1a D1b' (%the (reduce (E1a E1b) (E1a Eb')) D1)) %. +%{! ### Application-Beta !}% +%{! + +If the right-hand hand side is a β-reduction but the left-hand side is not, we +have to do the same two cases in reverse; we omit the graphics for those +two cases as they are symmetric. + +!}% +%term _ + %pi (diamond (%the (reduce (app (lam Ea) Eb) (app (lam Ea) E1b)) (reduce/app (%the (reduce Eb E1b) D1b) (%the (reduce (lam Ea) (lam Ea)) reduce/id))) (%the (reduce (app (lam Ea) Eb) (E2a E2b)) (reduce/beta (%the (reduce Eb E2b) D2b) (%the ({x} reduce (Ea x) (E2a x)) D2a))) (reduce/beta D1b' D2a) D2) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) + %<- (substitute1 E2a D2b' (%the (reduce (E2a E2b) (E2a Eb')) D2)) %. +%term _ + %pi (diamond (%the (reduce (app (lam Ea) Eb) (app (lam E1a) E1b)) (reduce/app (%the (reduce Eb E1b) D1b) (reduce/lam (%the ({x} reduce (Ea x) (E1a x)) D1a)))) (%the (reduce (app (lam Ea) Eb) (E2a E2b)) (reduce/beta (%the (reduce Eb E2b) D2b) (%the ({x} reduce (Ea x) (E2a x)) D2a))) (reduce/beta D1b' D1a') D2) + %<- ({x} diamond (D1a x) (D2a x) (%the (reduce (E1a x) (Ea' x)) (D1a' x)) (%the (reduce (E2a x) (Ea' x)) (D2a' x))) + %<- (diamond D1b D2b (%the (reduce E1b Eb') D1b') (%the (reduce E2b Eb') D2b')) + %<- (substitute2 D2a' D2b' (%the (reduce (E2a E2b) (Ea' Eb')) D2)) %. +%{! Now we are done! We check in the ``exps`` world with free variables. !}% +%worlds (exps) (diamond _ _ _ _) %. +%% %total D1 (diamond D1 D2 _ _). +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Church_Rosser_(Problem)). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-combinators-answer.lf b/new-tests/stelf-output/pages/popl-tutorial-combinators-answer.lf new file mode 100644 index 0000000..5020fca --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-combinators-answer.lf @@ -0,0 +1,112 @@ +% +%{! +Adapted from the case study on [Typed combinators soundness and completeness](/wiki/typed-combinators-soundness-and-completeness/). + +## Syntax + +!}% +% lambda calculus +%sort term %. +%name term %. +%term app %pi term %-> term %-> term %. +%term lam %pi (%pi term %-> term) %-> term %. +%block term-block [x term]%. +%worlds (term-block) (term) %. +% combinator calculus +%sort comb %. +%name comb %. +%term s comb %. +%term k comb %. +%term i comb %. +%term capp %pi comb %-> comb %-> comb %. +%block comb-block [y comb]%. +%worlds (comb-block) (comb) %. +%{! + +## Translation + +!}% +% bracket abstraction +%sort bracket %. +%mode bracket %in %out %. +%term bracket/var bracket ([y] y) i %. +%term bracket/i bracket ([y] i) (capp k i) %. +%term bracket/k bracket ([y] k) (capp k k) %. +%term bracket/s bracket ([y] s) (capp k s) %. +%term bracket/app + %pi (bracket ([y] capp (A y) (B y)) (capp (capp s A') B')) + %<- (bracket ([y] A y) A') + %<- (bracket ([y] B y) B') %. +%block bracket-block [y comb] [bracket/y bracket ([z] y) (capp k y)]%. +%worlds (bracket-block) (bracket _ _) %. +%total A (bracket A _) %. +% translation +%sort translate %. +%mode translate %in %out %. +%term translate/app %pi (translate (app M N) (capp A B)) %<- (translate M A) %<- (translate N B) %. +%term translate/lam + %pi (translate (lam ([x] M x)) A') + %<- ({x} {y} %pi (bracket ([z] y) (capp k y)) %-> (translate x y) %-> (translate (M x) (A y))) + %<- (bracket ([y] A y) A') %. +%block translate-block [x term] [y comb] [bracket/y bracket ([z] y) (capp k y)] [translate/x translate x y]%. +%worlds (translate-block) (translate _ _) %. +%total M (translate M _) %. +%{! + +## Equational theory + +!}% +% lambda term equality +%sort teq %. +% beta +%term teq/beta teq (app (lam ([x] M x)) N) (M N) %. +% extensionality (eta) +%term teq/ext %pi (teq M M') %<- ({x term} teq (app M x) (app M' x)) %. +% compatibilities +%term teq/app %pi (teq (app M N) (app M' N')) %<- (teq M M') %<- (teq N N') %. +%term teq/lam %pi (teq (lam ([x] M x)) (lam ([x] M' x))) %<- ({x term} teq (M x) (M' x)) %. +% equivalence +%term teq/refl teq M M %. +%term teq/symm %pi (teq M M') %<- (teq M' M) %. +%term teq/trans %pi (teq M M') %<- (teq M N) %<- (teq N M') %. +%worlds (term-block) (teq _ _) %. +% combinator equality +%sort ceq %. +% betas +%term ceq/i ceq (capp i A) A %. +%term ceq/k ceq (capp (capp k A) B) A %. +%term ceq/s ceq (capp (capp (capp s A) B) C) (capp (capp A C) (capp B C)) %. +% extensionality +%term ceq/ext %pi (ceq A A') %<- ({y comb} ceq (capp A y) (capp A' y)) %. +% compatibility +%term ceq/app %pi (ceq (capp A B) (capp A' B')) %<- (ceq A A') %<- (ceq B B') %. +% equivalence +%term ceq/refl ceq A A %. +%term ceq/symm %pi (ceq A A') %<- (ceq A' A) %. +%term ceq/trans %pi (ceq A A') %<- (ceq A B) %<- (ceq B A') %. +%worlds (comb-block) (ceq _ _) %. +%{! + +## Correctness of the translation + +!}% +% substitution lemma +%sort subst %. +%mode subst %in %in %out %. +%term _ subst (%the (bracket ([y] y) i) bracket/var) C (%the (ceq (capp i C) C) ceq/i) %. +%term _ subst (%the (bracket ([y] i) (capp k i)) bracket/i) C (%the (ceq (capp (capp k i) C) i) ceq/k) %. +%term _ subst (%the (bracket ([y] k) (capp k k)) bracket/k) C (%the (ceq (capp (capp k k) C) k) ceq/k) %. +%term _ subst (%the (bracket ([y] s) (capp k s)) bracket/s) C (%the (ceq (capp (capp k s) C) s) ceq/k) %. +%term _ + %pi (subst (bracket/app (%the (bracket ([y] B y) B') Dbrack2) (%the (bracket ([y] A y) A') Dbrack1)) C (ceq/trans (ceq/app Dceq2 Dceq1) ceq/s)) + %<- (subst Dbrack1 C (%the (ceq (capp A' C) (A C)) Dceq1)) + %<- (subst Dbrack2 C (%the (ceq (capp B' C) (B C)) Dceq2)) %. +%block subst-block [y comb] [dbrack bracket ([z] y) (capp k y)] [thm-subst {C comb} subst dbrack C ceq/k]%. +%worlds (subst-block) (subst _ _ _) %. +%total D (subst D _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Combinators_Answer). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-combinators-karl.lf b/new-tests/stelf-output/pages/popl-tutorial-combinators-karl.lf new file mode 100644 index 0000000..c6dcfac --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-combinators-karl.lf @@ -0,0 +1,131 @@ +% +%{! ## Lambda Terms !}% +%sort exp %. +%name exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%{! ## Reduction !}% +%sort reduce %. +%term reduce/refl reduce E E %. +%term reduce/trans %pi (reduce E1 E3) %<- (reduce E1 E2) %<- (reduce E2 E3) %. +%term reduce/lam %pi (reduce (lam ([x] E x)) (lam ([x] E' x))) %<- ({x} reduce (E x) (E' x)) %. +%term reduce/app %pi (reduce (app E1 E2) (app E1' E2')) %<- (reduce E1 E1') %<- (reduce E2 E2') %. +%term reduce/beta reduce (app (lam ([x] E1 x)) E2) (E1 E2) %. +%{! ## Combinators !}% +%define s exp lam ([a] lam ([b] lam ([x] app (app a x) (app b x)))) %. +%define k exp lam ([x] lam ([y] x)) %. +%define i exp lam ([x] x) %. +%{! ## Bracket Abstraction !}% +%sort abstract %. +%mode abstract %in %out %. +%term abstract/var abstract ([x] x) i %. +%term abstract/closed abstract ([x] E) (app k E) %. +%term abstract/app + %pi (abstract ([x] app (E1 x) (E2 x)) (app (app s E1') E2')) + %<- (abstract E1 E1') + %<- (abstract E2 E2') %. +%sort translate %. +%mode translate %in %out %. +%term translate/lam + %pi (translate (lam ([x] E x)) E'') + %<- ({x exp} %pi (translate x x) %-> (translate (E x) (E' x))) + %<- (abstract ([x] E' x) E'') %. +%term translate/app + %pi (translate (app E1 E2) (app E1' E2')) + %<- (translate E1 E1') + %<- (translate E2 E2') %. +%{! ## Soundness !}% +%sort abstract-sound %. +%mode abstract-sound %in %out %. +%block abstract-sound-block [x exp]%. +%term _ abstract-sound (%the (abstract ([x] x) i) abstract/var) (%the (reduce i (lam ([x] x))) reduce/refl) %. +%term _ abstract-sound (%the (abstract ([x] E) (app k E)) abstract/closed) (%the (reduce (app k E) (lam ([x] E))) reduce/beta) %. +%term _ + %pi (abstract-sound (abstract/app (%the (abstract ([x] E2 x) E2') Dabstract2) (%the (abstract ([x] E1 x) E1') Dabstract1)) (reduce/trans (reduce/lam ([x] reduce/app reduce/beta reduce/beta)) (reduce/trans reduce/beta (reduce/app Dreduce2 (reduce/trans reduce/beta (reduce/app Dreduce1 reduce/refl)))))) + %<- (abstract-sound Dabstract1 (%the (reduce E1' (lam ([x] E1 x))) Dreduce1)) + %<- (abstract-sound Dabstract2 (%the (reduce E2' (lam ([x] E2 x))) Dreduce2)) %. +%worlds (abstract-sound-block) (abstract-sound _ _) %. +%total D (abstract-sound D _) %. +%sort translate-sound %. +%mode translate-sound %in %out %. +%block translate-sound-block [x exp] [d_translate translate x x] [_ translate-sound d_translate reduce/refl]%. +%term _ + %pi (translate-sound (%the (translate (lam ([x] E x)) E'') (translate/lam (%the (abstract ([x] E' x) E'') Dabstract) (%the ({x exp} %pi (translate x x) %-> (translate (E x) (E' x))) Dtranslate))) (reduce/trans (reduce/lam ([x] Dreduce1 x)) Dreduce2)) + %<- ({x} {d_translate translate x x} + %pi (translate-sound d_translate reduce/refl) + %-> (translate-sound (Dtranslate x d_translate) (%the (reduce (E' x) (E x)) (Dreduce1 x)))) + %<- (abstract-sound Dabstract (%the (reduce E'' (lam ([x] E' x))) Dreduce2)) %. +%term _ + %pi (translate-sound (%the (translate (app E1 E2) (app E1' E2')) (translate/app (%the (translate E2 E2') Dtranslate2) (%the (translate E1 E1') Dtranslate1))) (reduce/app Dreduce2 Dreduce1)) + %<- (translate-sound Dtranslate1 (%the (reduce E1' E1) Dreduce1)) + %<- (translate-sound Dtranslate2 (%the (reduce E2' E2) Dreduce2)) %. +%worlds (translate-sound-block) (translate-sound _ _) %. +%total D (translate-sound D _) %. +%{! ## Completeness !}% +%sort clean %. +%term clean/s clean s %. +%term clean/k clean k %. +%term clean/i clean i %. +%term clean/app %pi (clean (app E1 E2)) %<- (clean E1) %<- (clean E2) %. +%sort abstract-clean {x} %. +%mode abstract-clean %in %in %out %. +%block abstract-clean-block [x exp] [d_clean clean x]%. +%term _ abstract-clean _ (%the (abstract ([x] x) i) abstract/var) clean/i %. +%term _ abstract-clean (%the ({x} %pi (clean x) %-> (clean E)) Dclean) abstract/closed (clean/app (%the (clean E) (Dclean i clean/i)) clean/k) %. +%term _ + %pi (abstract-clean ([x] [d_clean clean x] clean/app (%the (clean (E2 x)) (Dclean2 x d_clean)) (%the (clean (E1 x)) (Dclean1 x d_clean))) (abstract/app (%the (abstract ([x] E2 x) E2') Dabstract2) (%the (abstract ([x] E1 x) E1') Dabstract1)) (%the (clean (app (app s E1') E2')) (clean/app Dclean2' (clean/app Dclean1' clean/s)))) + %<- (abstract-clean Dclean1 Dabstract1 (%the (clean E1') Dclean1')) + %<- (abstract-clean Dclean2 Dabstract2 (%the (clean E2') Dclean2')) %. +%worlds (abstract-clean-block) (abstract-clean _ _ _) %. +%total D (abstract-clean _ D _) %. +%sort translate-clean %. +%mode translate-clean %in %out %. +%block translate-clean-block [x exp] [d_translate translate x x] [d_clean clean x] [_ translate-clean d_translate d_clean]%. +%term _ + %pi (translate-clean (%the (translate (lam ([x] E x)) E'') (translate/lam (%the (abstract ([x] E' x) E'') Dabstract) (%the ({x exp} %pi (translate x x) %-> (translate (E x) (E' x))) Dtranslate))) Dclean') + %<- ({x} {d_translate translate x x} {d_clean clean x} + %pi (translate-clean d_translate d_clean) + %-> (translate-clean (Dtranslate x d_translate) (%the (clean (E' x)) (Dclean x d_clean)))) + %<- (abstract-clean Dclean Dabstract (%the (clean E'') Dclean')) %. +%term _ + %pi (translate-clean (%the (translate (app E1 E2) (app E1' E2')) (translate/app (%the (translate E2 E2') Dtranslate2) (%the (translate E1 E1') Dtranslate1))) (%the (clean (app E1' E2')) (clean/app Dclean2 Dclean1))) + %<- (translate-clean Dtranslate1 (%the (clean E1') Dclean1)) + %<- (translate-clean Dtranslate2 (%the (clean E2') Dclean2)) %. +%worlds (translate-clean-block) (translate-clean _ _) %. +%total D (translate-clean D _) %. +%sort can-abstract {x} %. +%mode can-abstract %in %out %. +%block can-abstract-block [x exp] [d_clean clean x]%. +%term _ can-abstract ([x] [d clean x] d) abstract/var %. +%term _ can-abstract (%the ({x} %pi (clean x) %-> (clean E)) _) abstract/closed %. +%term _ + %pi (can-abstract ([x] [d_clean clean x] clean/app (%the (clean (E2 x)) (Dclean2 x d_clean)) (%the (clean (E1 x)) (Dclean1 x d_clean))) (abstract/app Dabstract2 Dabstract1)) + %<- (can-abstract Dclean1 (%the (abstract ([x] E1 x) E1') Dabstract1)) + %<- (can-abstract Dclean2 (%the (abstract ([x] E2 x) E2') Dabstract2)) %. +%worlds (can-abstract-block) (can-abstract _ _) %. +%total D (can-abstract D _) %. +%sort can-translate {E exp} %. +%mode can-translate %in %out %. +%block can-translate-block [x exp] [d_translate translate x x] [d_clean clean x] [_ translate-clean d_translate d_clean] [_ can-translate x d_translate]%. +%term _ + %pi (can-translate (lam ([x] E x)) (%the (translate (lam ([x] E x)) E'') (translate/lam Dabstract Dtranslate))) + %<- ({x} {d_translate translate x x} {d_clean clean x} + %pi (translate-clean d_translate d_clean) + %-> (can-translate x d_translate) + %-> (can-translate (E x) (%the (translate (E x) (E' x)) (Dtranslate x d_translate)))) + %<- ({x} {d_translate translate x x} {d_clean clean x} + %pi (translate-clean d_translate d_clean) + %-> (translate-clean (Dtranslate x d_translate) (%the (clean (E' x)) (Dclean x d_clean)))) + %<- (can-abstract Dclean (%the (abstract ([x] E' x) E'') Dabstract)) %. +%term _ + %pi (can-translate (app E1 E2) (%the (translate (app E1 E2) (app E1' E2')) (translate/app Dtranslate2 Dtranslate1))) + %<- (can-translate E1 (%the (translate E1 E1') Dtranslate1)) + %<- (can-translate E2 (%the (translate E2 E2') Dtranslate2)) %. +%worlds (can-translate-block) (can-translate _ _) %. +%total E (can-translate E _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Combinators_(karl)). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-combinators-session-answers.lf b/new-tests/stelf-output/pages/popl-tutorial-combinators-session-answers.lf new file mode 100644 index 0000000..6d0ecc8 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-combinators-session-answers.lf @@ -0,0 +1,110 @@ +% +%{! +Adapted from the case study on [Typed combinators soundness and completeness](/wiki/typed-combinators-soundness-and-completeness/). + +## Syntax + +!}% +% lambda calculus +%sort term %. +%name term %. +%term app %pi term %-> term %-> term %. +%term lam %pi (%pi term %-> term) %-> term %. +% combinator calculus +%sort comb %. +%name comb %. +%term s comb %. +%term k comb %. +%term i comb %. +%term capp %pi comb %-> comb %-> comb %. +%{! + +## Translation + +!}% +% bracket abstraction +%sort bracket %. +%mode bracket %in %out %. +%term bracket/var bracket ([y] y) i %. +%term bracket/i bracket ([y] i) (capp k i) %. +%term bracket/k bracket ([y] k) (capp k k) %. +%term bracket/s bracket ([y] s) (capp k s) %. +%term bracket/app + %pi (bracket ([y] capp (A y) (B y)) (capp (capp s A') B')) + %<- (bracket ([y] A y) A') + %<- (bracket ([y] B y) B') %. +%block bracket-block [y comb] [bracket/y bracket ([z] y) (capp k y)]%. +%worlds (bracket-block) (bracket _ _) %. +%total A (bracket A _) %. +% translation +%sort translate %. +%mode translate %in %out %. +%term translate/app %pi (translate (app M N) (capp A B)) %<- (translate M A) %<- (translate N B) %. +%term translate/lam + %pi (translate (lam ([x] M x)) A') + %<- ({x} {y} %pi (bracket ([z] y) (capp k y)) %-> (translate x y) %-> (translate (M x) (A y))) + %<- (bracket ([y] A y) A') %. +%block translate-block [x term] [y comb] [bracket/y bracket ([z] y) (capp k y)] [translate/x translate x y]%. +%worlds (translate-block) (translate _ _) %. +%total M (translate M _) %. +%{! + +## Equational theory + +!}% +% lambda term equality +%sort teq %. +% beta +%term teq/beta teq (app (lam ([x] M x)) N) (M N) %. +% extensionality (eta) +%term teq/ext %pi (teq M M') %<- ({x term} teq (app M x) (app M' x)) %. +% compatibilities +%term teq/app %pi (teq (app M N) (app M' N')) %<- (teq M M') %<- (teq N N') %. +%term teq/lam %pi (teq (lam ([x] M x)) (lam ([x] M' x))) %<- ({x term} teq (M x) (M' x)) %. +% equivalence +%term teq/refl teq M M %. +%term teq/symm %pi (teq M M') %<- (teq M' M) %. +%term teq/trans %pi (teq M M') %<- (teq M N) %<- (teq N M') %. +%block term-block [x term]%. +%worlds (term-block) (teq _ _) %. +% combinator equality +%sort ceq %. +% betas +%term ceq/i ceq (capp i A) A %. +%term ceq/k ceq (capp (capp k A) _) A %. +%term ceq/s ceq (capp (capp (capp s A) B) C) (capp (capp A C) (capp B C)) %. +% extensionality +%term ceq/ext %pi (ceq A A') %<- ({y comb} ceq (capp A y) (capp A' y)) %. +% compatibility +%term ceq/app %pi (ceq (capp A B) (capp A' B')) %<- (ceq A A') %<- (ceq B B') %. +% equivalence +%term ceq/refl ceq A A %. +%term ceq/symm %pi (ceq A A') %<- (ceq A' A) %. +%term ceq/trans %pi (ceq A A') %<- (ceq A B) %<- (ceq B A') %. +%block comb-block [y comb]%. +%worlds (comb-block) (ceq _ _) %. +%{! + +## Correctness of the translation + +!}% +% substitution lemma +%sort subst %. +%mode subst %in %in %out %. +%term _ subst (%the (bracket ([y] y) i) bracket/var) C (%the (ceq (capp i C) C) ceq/i) %. +%term _ subst (%the (bracket ([y] i) (capp k i)) bracket/i) C (%the (ceq (capp (capp k i) C) i) ceq/k) %. +%term _ subst (%the (bracket ([y] k) (capp k k)) bracket/k) C (%the (ceq (capp (capp k k) C) k) ceq/k) %. +%term _ subst (%the (bracket ([y] s) (capp k s)) bracket/s) C (%the (ceq (capp (capp k s) C) s) ceq/k) %. +%term _ + %pi (subst (bracket/app (%the (bracket ([y] B y) B') Dbrack2) (%the (bracket ([y] A y) A') Dbrack1)) C (ceq/trans (ceq/app Dceq2 Dceq1) ceq/s)) + %<- (subst Dbrack1 C (%the (ceq (capp A' C) (A C)) Dceq1)) + %<- (subst Dbrack2 C (%the (ceq (capp B' C) (B C)) Dceq2)) %. +%block subst-block [y comb] [dbrack bracket ([z] y) (capp k y)] [thm-subst {C comb} subst dbrack C ceq/k]%. +%worlds (subst-block) (subst _ _ _) %. +%total D (subst D _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Combinators_session_(answers)). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-combinators-session.lf b/new-tests/stelf-output/pages/popl-tutorial-combinators-session.lf new file mode 100644 index 0000000..341278f --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-combinators-session.lf @@ -0,0 +1,70 @@ +% +%{! +Adapted from the case study on [Typed combinators soundness and completeness](/wiki/typed-combinators-soundness-and-completeness/). + +## Syntax + +!}% +% lambda calculus +%sort term %. +%name term %. +%term app %pi term %-> term %-> term %. +%term lam %pi (%pi term %-> term) %-> term %. +% combinator calculus +%sort comb %. +%name comb %. +%term s comb %. +%term k comb %. +%term i comb %. +%term capp %pi comb %-> comb %-> comb %. +%{! + +## Translation + +!}% +% bracket abstraction +% XXX fill in +% translation +% XXX fill in +%{! + +## Equational theory + +!}% +% lambda term equality +% (elided) +% combinator equality +%sort ceq %. +% betas +%term ceq/i ceq (capp i A) A %. +%term ceq/k ceq (capp (capp k A) _) A %. +%term ceq/s ceq (capp (capp (capp s A) B) C) (capp (capp A C) (capp B C)) %. +% extensionality +%term ceq/ext %pi (ceq A A') %<- ({y comb} ceq (capp A y) (capp A' y)) %. +% compatibility +%term ceq/app %pi (ceq (capp A B) (capp A' B')) %<- (ceq A A') %<- (ceq B B') %. +% equivalence +%term ceq/refl ceq A A %. +%term ceq/symm %pi (ceq A A') %<- (ceq A' A) %. +%term ceq/trans %pi (ceq A A') %<- (ceq A B) %<- (ceq B A') %. +%block comb-block [y comb]%. +%worlds (comb-block) (ceq _ _) %. +%{! + +## Correctness of the translation + +!}% +% substitution lemma +% XXX fill in +%{! + + +[Solution](/wiki/popl-tutorial-combinators-session-answers/) + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Combinators_session). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-combinators-support.lf b/new-tests/stelf-output/pages/popl-tutorial-combinators-support.lf new file mode 100644 index 0000000..003bc20 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-combinators-support.lf @@ -0,0 +1,64 @@ +% +%{! +Adapted from the case study on [Typed combinators soundness and completeness](/wiki/typed-combinators-soundness-and-completeness/). + +## Syntax + +!}% +% lambda calculus +%sort term %. +%name term %. +%term app %pi term %-> term %-> term %. +%term lam %pi (%pi term %-> term) %-> term %. +% combinator calculus +%sort comb %. +%name comb %. +%term s comb %. +%term k comb %. +%term i comb %. +%term capp %pi comb %-> comb %-> comb %. +%{! + +## Translation + +!}% +% bracket abstraction +% XXX fill in +% translation +% XXX fill in +%{! + +## Equational theory + +!}% +% lambda term equality +% (elided) +% combinator equality +%sort ceq %. +% betas +%term ceq/i ceq (capp i A) A %. +%term ceq/k ceq (capp (capp k A) B) A %. +%term ceq/s ceq (capp (capp (capp s A) B) C) (capp (capp A C) (capp B C)) %. +% extensionality +%term ceq/ext %pi (ceq A A') %<- ({y comb} ceq (capp A y) (capp A' y)) %. +% compatibility +%term ceq/app %pi (ceq (capp A B) (capp A' B')) %<- (ceq A A') %<- (ceq B B') %. +% equivalence +%term ceq/refl ceq A A %. +%term ceq/symm %pi (ceq A A') %<- (ceq A' A) %. +%term ceq/trans %pi (ceq A A') %<- (ceq A B) %<- (ceq B A') %. +%block comb-block [y comb]%. +%worlds (comb-block) (ceq _ _) %. +%{! + +## Correctness of the translation + +!}% +% substitution lemma +% XXX fill in +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Combinators_Support). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-combinators.lf b/new-tests/stelf-output/pages/popl-tutorial-combinators.lf new file mode 100644 index 0000000..c4dd49b --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-combinators.lf @@ -0,0 +1,9 @@ +% +%{! * [Support code](/wiki/popl-tutorial-combinators-support/) +* [Solutions to interactive exercises](/wiki/popl-tutorial-combinators-answer/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Combinators). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-control-machine.lf b/new-tests/stelf-output/pages/popl-tutorial-control-machine.lf new file mode 100644 index 0000000..cc572f8 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-control-machine.lf @@ -0,0 +1,82 @@ +% +%{! !}% +%sort exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%sort ev %. +%term ev_lam ev (lam ([x] E x)) (lam ([x] E x)) %. +%term ev_app %pi (ev (app E1 E2) V) %<- (ev E1 (lam ([x] E x))) %<- (ev E2 V2) %<- (ev (E V2) V) %. +%sort stack %. +%term # stack %. +%term app1 %pi exp %-> stack %-> stack %. +%term app2 %pi exp %-> stack %-> stack %. +%sort state %. +%term eval %pi stack %-> exp %-> state %. +%term return %pi stack %-> exp %-> state %. +%sort step %. +%term step_lam step (eval K (lam E)) (return K (lam E)) %. +%term step_app1 step (eval K (app E1 E2)) (eval (app1 E2 K) E1) %. +%term step_app2 step (return (app1 E2 K) V) (eval (app2 V K) E2) %. +%term step_red step (return (app2 (lam E) K) V) (eval K (E V)) %. +%sort steps %. +%term steps_refl steps S S %. +%term steps_trans %pi (step S S') %-> (steps S' S'') %-> (steps S S'') %. +%sort steps_steps %. +%term _ steps_steps steps_refl W W %. +%term _ + %pi (steps_steps (steps_trans W Ws) Ws' (steps_trans W Ws'')) + %<- (steps_steps Ws Ws' Ws'') %. +%mode steps_steps %in %in %out %. +%worlds () (steps_steps _ _ _) %. +%total T (steps_steps T _ _) %. +%sort sound %. +%mode {%in E exp} {%in V exp} {%in K stack} {%in E1 ev E V} {%out V1 steps (eval K E) (return K V)} sound E1 V1 %. +%term _ sound ev_lam (steps_trans step_lam steps_refl) %. +%term _ + %pi (sound (ev_app D D2 D1) W123) + %<- (sound D1 (%the (steps (eval (app1 E2 K) E1) (return (app1 E2 K) (lam E))) W1)) + %<- (sound D2 (%the (steps (eval (app2 (lam E) K) E2) (return (app2 (lam E) K) V2)) W2)) + %<- (sound D (%the (steps (eval K (E V2)) (return K V)) W3)) + %<- (steps_steps (steps_trans step_app1 W1) (steps_trans step_app2 W2) (%the (steps (eval K (app E1 E2)) (return (app2 (lam E) K) V2)) W12)) + %<- (steps_steps W12 (steps_trans step_red W3) (%the (steps (eval K (app E1 E2)) (return K V)) W123)) %. +%worlds () (sound _ _) %. +%total T (sound T _) %. +%sort invert_app2 %. +%term _ invert_app2 (steps_trans step_app2 W) W %. +%mode invert_app2 %in %out %. +%worlds () (invert_app2 _ _) %. +%total {} (invert_app2 _ _) %. +%reduces < C A (invert_app2 A C) %. +%sort id %. +%term refl id E E %. +%sort invert_red %. +%term _ invert_red (steps_trans step_red W) refl W %. +%mode invert_red %in %out %out %. +%worlds () (invert_red _ _ _) %. +%total {} (invert_red _ _ _) %. +%reduces < C A (invert_red A _ C) %. +%sort use_id %. +%term _ use_id refl E1 E2 E (ev_app E E2 E1) %. +%mode use_id %in %in %in %in %out %. +%worlds () (use_id _ _ _ _ _) %. +%total {} (use_id _ _ _ _ _) %. +%sort complete %. +%mode complete %in %out %out %. +%term _ complete (%the (steps (eval K (lam E)) (return # V0)) (steps_trans step_lam W)) ev_lam W %. +%term _ + %pi (complete (%the (steps (eval K (app E1 E2)) (return # V0)) (steps_trans step_app1 W)) D' W5) + %<- (complete (%the (steps (eval (app1 E2 K) E1) (return # V0)) W) (%the (ev E1 V1) D1) (%the (steps (return (app1 E2 K) V1) (return # V0)) W1)) + %<- (invert_app2 W1 (%the (steps (eval (app2 V1 K) E2) (return # V0)) W2)) + %<- (complete W2 (%the (ev E2 V2) D2) (%the (steps (return (app2 V1 K) V2) (return # V0)) W3)) + %<- (invert_red W3 (%the (id V1 (lam E)) ID) (%the (steps (eval K (E V2)) (return # V0)) W4)) + %<- (complete W4 (%the (ev (E V2) V) D) (%the (steps (return K V) (return # V0)) W5)) + %<- (use_id ID D1 D2 D D') %. +%worlds () (complete _ _ _) %. +%reduces < C A (complete A _ C) %. +%total T (complete T _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Control_machine). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-cost-semantics.lf b/new-tests/stelf-output/pages/popl-tutorial-cost-semantics.lf new file mode 100644 index 0000000..a29625f --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-cost-semantics.lf @@ -0,0 +1,208 @@ +% +%{! ## Syntax !}% +%sort tp %. +%name tp %. +%term tunit tp %. +%term arr %pi tp %-> tp %-> tp %. +%sort exp %. +%name exp %. +%term unit exp %. +%term fun %pi tp %-> tp %-> (%pi exp %-> exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%{! ## Series-parallel graphs !}% +%sort graph %. +%term 0 graph %. +%term 1 graph %. +%term + %pi graph %-> graph %-> graph %. +%prec %right 5 + %. +%term * %pi graph %-> graph %-> graph %. +%prec %right 5 * %. +%{! !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%{! ### Size of graphs !}% +%sort plus %. +%term plus/z plus z N N %. +%term plus/s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%sort sizeof %. +%term sizeof/0 sizeof 0 z %. +%term sizeof/1 sizeof 1 (s z) %. +%term sizeof/+ %pi (sizeof (G1 + G2) C3) %<- (sizeof G1 C1) %<- (sizeof G2 C2) %<- (plus C2 C1 C3) %. +%term sizeof/* %pi (sizeof (G1 * G2) C3) %<- (sizeof G1 C1) %<- (sizeof G2 C2) %<- (plus C2 C1 C3) %. +%{! ### Depth of graphs !}% +%sort max %. +%term max/zz max z z z %. +%term max/sz %pi (max (s N) z (s M)) %<- (max N z M) %. +%term max/zs %pi (max z (s N) (s M)) %<- (max z N M) %. +%term max/ss %pi (max (s N) (s M) (s P)) %<- (max N M P) %. +%sort depthof %. +%term depthof/0 depthof 0 z %. +%term depthof/1 depthof 1 (s z) %. +%term depthof/+ + %pi (depthof (G1 + G2) C) + %<- (depthof G1 C1) + %<- (depthof G2 C2) + %<- (plus C1 C2 C) %. +%term depthof/* %pi (depthof (G1 * G2) C) %<- (depthof G1 C1) %<- (depthof G2 C2) %<- (max C1 C2 C) %. +%{! ## Cost Semantics !}% +%sort evcost %. +%term evcost/unit evcost unit unit 0 %. +%term evcost/fun evcost (fun T1 T2 E) (fun T1 T2 E) 0 %. +%term evcost/app + %pi (evcost (app E1 E2) V (G1 * G2 + 1 + G)) + %<- (evcost E1 (fun T1 T2 ([f] [x] E f x)) G1) + %<- (evcost E2 V2 G2) + %<- (evcost (E (fun T1 T2 ([f] [x] E f x)) V2) V G) %. +%{! ## Sequential Evaluation !}% +%sort value %. +%name value %. +%mode value %in %. +%term value/unit value unit %. +%term value/fun value (fun _ _ _) %. +%sort step %. +%name step %. +%mode step %in %out %. +%term step/app/fun %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app/arg %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app/beta-v + %pi (step (app (fun T1 T2 ([f] [x] E f x)) E2) (E (fun T1 T2 ([f] [x] E f x)) E2)) + %<- (value E2) %. +%{! !}% +%sort steps %. +%mode steps %in %out %out %. +%term steps/refl steps E E z %. +%term steps/trans %pi (steps E E'' (s C)) %<- (step E E') %<- (steps E' E'' C) %. +%{! ## Parallel Evaluation !}% +%sort pstep %. +%term pstep/app/ %pi (pstep (app E1 E2) (app E1' E2')) %<- (pstep E1 E1') %<- (pstep E2 E2') %. +%term pstep/app/fun %pi (pstep (app E1 E2) (app E1' E2)) %<- (pstep E1 E1') %<- (value E2) %. +%term pstep/app/arg %pi (pstep (app E1 E2) (app E1 E2')) %<- (pstep E2 E2') %<- (value E1) %. +%term pstep/app/beta-v + %pi (pstep (app (fun T1 T2 ([f] [x] E f x)) E2) (E (fun T1 T2 ([f] [x] E f x)) E2)) + %<- (value E2) %. +%{! !}% +%sort psteps %. +%term psteps/refl psteps E E z %. +%term psteps/trans %pi (psteps E E'' (s C)) %<- (pstep E E') %<- (psteps E' E'' C) %. +%{! ### Cost of parallel evaluation !}% +%sort par_app %. +%mode par_app %in %in %in %in %in %out %. +%term _ par_app max/zz psteps/refl Val1 psteps/refl Val2 psteps/refl %. +%term _ + %pi (par_app (max/sz (%the (max C1 z Cmax) Max)) (psteps/trans (%the (psteps E1' V1 C1) P1s) (%the (pstep E1 E1') Step1)) (%the (value V1) Val1) (%the (psteps V2 V2 z) psteps/refl) (%the (value V2) Val2) (psteps/trans Psteps (pstep/app/fun Val2 Step1))) + %<- (par_app Max P1s Val1 psteps/refl Val2 (%the (psteps (app E1' V2) (app V1 V2) Cmax) Psteps)) %. +%term _ + %pi (par_app (max/zs (%the (max z C2 Cmax) Max)) (%the (psteps V1 V1 z) psteps/refl) (%the (value V1) Val1) (psteps/trans (%the (psteps E2' V2 C2) P2s) (%the (pstep E2 E2') Step2)) (%the (value V2) Val2) (psteps/trans Psteps (pstep/app/arg Val1 Step2))) + %<- (par_app Max psteps/refl Val1 P2s Val2 (%the (psteps (app V1 E2') (app V1 V2) Cmax) Psteps)) %. +%term _ + %pi (par_app (max/ss (%the (max C1 C2 Cmax) Max)) (psteps/trans (%the (psteps E1' V1 C1) P1s) (%the (pstep E1 E1') Step1)) (%the (value V1) Val1) (psteps/trans (%the (psteps E2' V2 C2) P2s) (%the (pstep E2 E2') Step2)) (%the (value V2) Val2) (psteps/trans Psteps (pstep/app/ Step2 Step1))) + %<- (par_app Max P1s Val1 P2s Val2 (%the (psteps (app E1' E2') (app V1 V2) Cmax) Psteps)) %. +%worlds () (par_app _ _ _ _ _ _) %. +%total T (par_app T _ _ _ _ _) %. +%sort compose %. +%mode compose %in %in %in %out %. +%term _ compose plus/z psteps/refl Psteps Psteps %. +%term _ + %pi (compose (plus/s D) (psteps/trans Ps S) Ps' (psteps/trans Ps'' S)) + %<- (compose D Ps Ps' Ps'') %. +%worlds () (compose _ _ _ _) %. +%total T (compose T _ _ _) %. +%sort schedule_parallel %. +%mode schedule_parallel %in %in %out %out %. +%term _ schedule_parallel evcost/unit depthof/0 psteps/refl value/unit %. +%term _ schedule_parallel evcost/fun depthof/0 psteps/refl value/fun %. +%term _ + %pi (schedule_parallel (evcost/app (%the (evcost (E (fun T1 T2 E) V2) V G) Dcost) (%the (evcost E2 V2 G2) Dcost2) (%the (evcost E1 (fun T1 T2 E) G1) Dcost1)) (depthof/+ (%the (plus DepthMax (s DepthG) Depth) Dplus) (depthof/+ (plus/s plus/z) (%the (depthof G DepthG) DdepthG) depthof/1) (depthof/* (%the (max Depth1 Depth2 DepthMax) Dmax) (%the (depthof G2 Depth2) DdepthG2) (%the (depthof G1 Depth1) DdepthG1))) Dstep'' Val) + %<- (schedule_parallel Dcost1 DdepthG1 (%the (psteps E1 (fun T1 T2 E) Depth1) Dstep1) (%the (value (fun T1 T2 E)) Val1)) + %<- (schedule_parallel Dcost2 DdepthG2 (%the (psteps E2 V2 Depth2) Dstep2) (%the (value V2) Val2)) + %<- (schedule_parallel Dcost DdepthG (%the (psteps (E (fun T1 T2 E) V2) V DepthG) Dstep) (%the (value V) Val)) + %<- (par_app Dmax Dstep1 Val1 Dstep2 Val2 (%the (psteps (app E1 E2) (app (fun T1 T2 E) V2) DepthMax) Dstep')) + %<- (compose Dplus Dstep' (psteps/trans Dstep (pstep/app/beta-v Val2)) (%the (psteps (app E1 E2) V Depth) Dstep'')) %. +%worlds () (schedule_parallel _ _ _ _) %. +%total T (schedule_parallel T _ _ _) %. +%{! ### Random Plus Lemmas !}% +%sort plus_lemma %. +%mode plus_lemma %out %in %. +%term _ %pi (plus_lemma (plus/s Dplus) (plus/s Dplus')) %<- (plus_lemma Dplus Dplus') %. +%term _ plus_lemma (plus/s plus/z) plus/z %. +%worlds () (plus_lemma _ _) %. +%total [D] (plus_lemma _ D) %. +%sort plus_lemma' %. +%mode plus_lemma' %in %out %. +%term _ %pi (plus_lemma' (plus/s Dplus) (plus/s Dplus')) %<- (plus_lemma' Dplus Dplus') %. +%term _ plus_lemma' (plus/s plus/z) plus/z %. +%worlds () (plus_lemma' _ _) %. +%total [D] (plus_lemma' D _) %. +%sort plus_zero {N} %. +%mode plus_zero %in %out %. +%term _ plus_zero z plus/z %. +%term _ %pi (plus_zero (s N) (plus/s Dplus)) %<- (plus_zero N Dplus) %. +%worlds () (plus_zero _ _) %. +%total [N] (plus_zero N _) %. +%sort plus_commute %. +%mode plus_commute %in %out %. +%term _ + %pi (plus_commute (plus/s Dplus) Dplus'') + %<- (plus_commute Dplus Dplus') + %<- (plus_lemma' (plus/s Dplus') Dplus'') %. +%term _ %pi (plus_commute plus/z Dplus) %<- (plus_zero _ Dplus) %. +%worlds () (plus_commute _ _) %. +%total [D] (plus_commute D _) %. +%{! ### Cost of sequential evaluation !}% +%sort steps_lemma %. +%mode steps_lemma %in %in %out %. +%term _ steps_lemma steps/refl E'step (steps/trans steps/refl E'step) %. +%term _ + %pi (steps_lemma (steps/trans E'steps Estep) E''step (steps/trans E'steps' Estep)) + %<- (steps_lemma E'steps E''step E'steps') %. +%worlds () (steps_lemma _ _ _) %. +%total [D] (steps_lemma D _ _) %. +%sort steps_lemma2 %. +%mode steps_lemma2 %in %in %out %out %. +%term _ steps_lemma2 Esteps steps/refl plus/z Esteps %. +%term _ + %pi (steps_lemma2 Esteps (steps/trans E''steps E'step) Dplus' Esteps'') + %<- (steps_lemma Esteps E'step Esteps') + %<- (steps_lemma2 Esteps' E''steps Dplus Esteps'') + %<- (plus_lemma Dplus' Dplus) %. +%worlds () (steps_lemma2 _ _ _ _) %. +%total [D] (steps_lemma2 _ D _ _) %. +%sort app_lemma1 %. +%mode {%in E1 exp} {%in E1' exp} {%in E2 exp} {%in C nat} {%in D steps E1 E1' C} {%out D' steps (app E1 E2) (app E1' E2) C} app_lemma1 D D' %. +%term _ app_lemma1 steps/refl steps/refl %. +%term _ + %pi (app_lemma1 (steps/trans E'steps Estep) (steps/trans E'steps' (step/app/fun Estep))) + %<- (app_lemma1 E'steps E'steps') %. +%worlds () (app_lemma1 _ _) %. +%total [D] (app_lemma1 D _) %. +%sort app_lemma2 %. +%mode {%in V1 exp} {%in E2 exp} {%in E2' exp} {%in C nat} {%in D value V1} {%in D' steps E2 E2' C} {%out D'' steps (app V1 E2) (app V1 E2') C} app_lemma2 D D' D'' %. +%term _ app_lemma2 _ steps/refl steps/refl %. +%term _ + %pi (app_lemma2 Vvalue (steps/trans E'steps Estep) (steps/trans E'steps' (step/app/arg Estep Vvalue))) + %<- (app_lemma2 Vvalue E'steps E'steps') %. +%worlds () (app_lemma2 _ _ _) %. +%total [D] (app_lemma2 _ D _) %. +%sort schedule_serial %. +%mode schedule_serial %in %out %out %out %. +%term _ schedule_serial evcost/unit sizeof/0 steps/refl value/unit %. +%term _ schedule_serial evcost/fun sizeof/0 steps/refl value/fun %. +%term _ + %pi (schedule_serial (evcost/app Ecost E2cost E1cost) (sizeof/+ Dplus' (sizeof/+ Dplus'' Esize sizeof/1) (sizeof/* Dplus E2size E1size)) E1steps''' E'value) + %<- (schedule_serial E1cost E1size E1steps _) + %<- (schedule_serial E2cost E2size E2steps E2'value) + %<- (schedule_serial Ecost Esize Esteps E'value) + %<- (app_lemma1 E1steps E1steps') + %<- (app_lemma2 value/fun E2steps E2steps') + %<- (steps_lemma2 E1steps' E2steps' Dplus E1steps'') + %<- (steps_lemma2 E1steps'' (steps/trans Esteps (step/app/beta-v E2'value)) Dplus' E1steps''') + %<- (plus_commute (plus/s plus/z) Dplus'') %. +%worlds () (schedule_serial _ _ _ _) %. +%total [D] (schedule_serial D _ _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Cost_semantics). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-cps-ctp.lf b/new-tests/stelf-output/pages/popl-tutorial-cps-ctp.lf new file mode 100644 index 0000000..d57d9e8 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-cps-ctp.lf @@ -0,0 +1,54 @@ +% +%{! ## CPS conversion with target types !}% +%% Source types +%sort tp %. +%term o tp %. +%term => %pi tp %-> tp %-> tp %. +%prec %right 3 => %. +%% Destination types +%sort ctp %. +%term co ctp %. +%term cnot %pi ctp %-> ctp %. +%term cand %pi ctp %-> ctp %-> ctp %. +%prec %right 3 cand %. +%% Source expressions and values annotated by the translation of their types. +%sort e %. +%sort v %. +%term app %pi (e (cnot (A cand (cnot B)))) %-> (e A) %-> (e B) %. +%term lam %pi (%pi (v A) %-> (e B)) %-> (v (cnot (A cand (cnot B)))) %. +%term inj %pi (v A) %-> (e A) %. +%block sourceb {A ctp} [x v A]%. +%worlds (sourceb) (e _) (v _) %. +%% Destination expression and values annotated by their types. +%sort ce %. +%sort cv %. +% This is the only kind of conjunction we need but it seems pretty odd. +%term capp %pi (cv A) %-> (cv (cnot B)) %-> (cv (cnot (cnot (A cand (cnot B))))) %. +%term clam %pi (%pi (cv A) %-> (cv (cnot B)) %-> ce) %-> (cv (cnot (A cand (cnot B)))) %. +%term cnoti %pi (%pi (cv A) %-> ce) %-> (cv (cnot A)) %. +%term throw %pi (cv A) %-> (cv (cnot A)) %-> ce %. +%block targetb {A ctp} [x cv A]%. +%worlds (targetb) (ce) (cv _) %. +%sort cps %. +%mode cps %in %out %. +%sort cpse %. +%mode cpse %in %out %. +%term cps/lam + %pi (cps (lam (%the (%pi (v A) %-> (e B)) E)) (clam (%the (%pi (cv A) %-> (cv (cnot B)) %-> ce) E'))) + %<- ({x v A} {x' cv A} %pi (cps x x') %-> (cpse (E x) (E' x'))) %. +%term cpse/app + %pi (cpse (app (%the (e (cnot (B cand (cnot A)))) E1) (%the (e B) E2)) ([c cv (cnot A)] E2' (cnoti ([arg cv B] E1' (capp arg c))))) + %<- (cpse E1 (%the (%pi (cv (cnot (cnot (B cand (cnot A))))) %-> ce) E1')) + %<- (cpse E2 (%the (%pi (cv (cnot B)) %-> ce) E2')) %. +%term cpse/inj + %pi (cpse (inj (%the (v A) V)) ([c cv (cnot A)] throw V' c)) + %<- (cps V (%the (cv A) V')) %. +%block cpsb {A ctp} [x v A] [x' cv A] [d cps x x']%. +%worlds (cpsb) (cps _ _) (cpse _ _) %. +%total (E V) (cps E _) (cpse V _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/cps-ctp). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-cps-problem.lf b/new-tests/stelf-output/pages/popl-tutorial-cps-problem.lf new file mode 100644 index 0000000..4360e75 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-cps-problem.lf @@ -0,0 +1,61 @@ +% +%{! + +## Problem 1: CPS Conversion with Administrative Redices + +In this problem we define a translation from the following simply-typed +λ-calculus to a language in continuation passing style (CPS). + +!}% +%sort tp %. +%term o tp %. +%term => %pi tp %-> tp %-> tp %. +%prec %right 3 => %. +%sort exp %. +%sort value %. +%term app %pi (exp (A => B)) %-> (exp A) %-> (exp B) %. +%term lam %pi (%pi (value A) %-> (exp B)) %-> (value (A => B)) %. +%term ret %pi (value A) %-> (exp A) %. +%block sourceb {A tp} [x value A]%. +%worlds (sourceb) (exp _) (value _) %. +%{! + +The terms are divided into expressions and values as in the previous exercise, but there is also an injection of values into expressions. We define the language so that an expression or value of type A will have the type exp A or value A, respectively. We will initially define a translation into the following CPS language. + +!}% +%sort contra %. +%sort cvalue %. +%sort ccont %. +%term capp %pi (cvalue (A => B)) %-> (cvalue A) %-> (ccont B) %-> contra %. +%term clam %pi (%pi (cvalue A) %-> (ccont B) %-> contra) %-> (cvalue (A => B)) %. +%term cconti %pi (%pi (cvalue A) %-> contra) %-> (ccont A) %. +%term cthrow %pi (ccont A) %-> (cvalue A) %-> contra %. +%block targetb1 {A tp} [x cvalue A]%. +%block targetb2 {A tp} [x ccont A]%. +%worlds (targetb1 targetb2) (contra) (cvalue _) (ccont _) %. +%{! + +Here we have divided the terms into values, continuations, and contradictions. Values have a similar interpretation as in the source language. The continuations represent the remainder of the computation given a value. From a logical perspective, a continuation of type A may be viewed as a proof of not A. Notice that the introduction and elimination rules for continuations are inverses. Intuitively, a contradiction is a computation. + +A value of type A => B can be interpreted as proofs of (not (A and (not B))). Consequently, the function in the target language is defined by deriving a contradiction from A value of type A and a continuation of type B, and the application in the target language is the inverse of the function introduction taking a value of type A => B and giving back a proof that a value of type A and a continuation of type B yield a contradiction. + +It remains to define a translation from the source to the target language: + +!}% +%sort cps %. +%mode cps %in %out %. +%sort cpse %. +%mode cpse %in %out %. +%{! + +We translate values in the source language to values in the target language of the same type. We translate expressions in the source language to computations in the target language which depend on a continuation of the given type. For this problem, fill in the three cases of the translation and check that it is total. The translation defined on the bottom of page 147 in [http://homepages.inf.ed.ac.uk/gdp/publications/cbn_cbv_lambda.pdf Plotkin's Call-by-name, Call-by-value and the λ-Calculus] may be helpful. Note that the translation of functions given there is the composition of the translations of ret and lam in our setting. + +When you complete this problem you can learn another technique in the second part: [POPL Tutorial/cps-problem2](/wiki/popl-tutorial-cps-problem2/) + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/cps-problem). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-cps-problem2.lf b/new-tests/stelf-output/pages/popl-tutorial-cps-problem2.lf new file mode 100644 index 0000000..37e2c45 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-cps-problem2.lf @@ -0,0 +1,45 @@ +% +%{! +## Problem 2: Elimination of Administrative Redices + +Rather than introducing the administrative redices of explicitly introducing and eliminating continuations (see [POPL Tutorial/cps-problem](/wiki/popl-tutorial-cps-problem/)), we can simply use the function from values to computations itself wherever we previously used a continuation. Therefore, the new definition of the CPS language is obtained by replacing all instances of ccont A with (cvalue A -> contra). +| hidden="true" !}% +%sort tp %. +%term o tp %. +%term => %pi tp %-> tp %-> tp %. +%prec %right 3 => %. +%sort exp %. +%sort value %. +%term app %pi (exp (A => B)) %-> (exp A) %-> (exp B) %. +%term lam %pi (%pi (value A) %-> (exp B)) %-> (value (A => B)) %. +%term ret %pi (value A) %-> (exp A) %. +%block sourceb {A tp} [x value A]%. +%worlds (sourceb) (exp _) (value _) %. +%{! !}% +%sort contra %. +%sort cvalue %. +%term capp %pi (cvalue (A => B)) %-> (cvalue A) %-> (%pi (cvalue B) %-> contra) %-> contra %. +%term clam + %pi (%pi (cvalue A) %-> (%pi (cvalue B) %-> contra) %-> contra) + %-> (cvalue (A => B)) %. +%block targetb1 {A tp} [x cvalue A]%. +%block targetb2 {A tp} [y %pi (cvalue A) %-> contra]%. +%worlds (targetb1 targetb2) (contra) (cvalue _) %. +%{! +Notice that we no longer need terms for continuations as they are implicitly represented by the functions from values to contradictions. + +For this problem, write the translation to this version of the CPS language: +!}% +%sort cps %. +%mode cps %in %out %. +%sort cpse %. +%mode cpse %in %out %. +%{! +The code for this problem should be similar to the code from the previous problem except that the introduction of continuations with cfalsei and their elimination with throw is replaced by Twelf function introduction and application. +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/cps-problem2). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-cps-rp.lf b/new-tests/stelf-output/pages/popl-tutorial-cps-rp.lf new file mode 100644 index 0000000..85dde41 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-cps-rp.lf @@ -0,0 +1,86 @@ +% +%{! !}% +%sort tp %. +%term o tp %. +%term => %pi tp %-> tp %-> tp %. +%prec %right 3 => %. +%sort e %. +%sort v %. +%term app %pi (e (A => B)) %-> (e A) %-> (e B) %. +%term lam %pi (%pi (v A) %-> (e B)) %-> (v (A => B)) %. +%term inj %pi (v A) %-> (e A) %. +%block sourceb {A tp} [x v A]%. +%worlds (sourceb) (e _) (v _) %. +%sort ctp %. +%term co ctp %. +%term carr %pi ctp %-> ctp %-> ctp %. +%sort ce %. +%sort cv %. +%term capp %pi (cv (carr A B)) %-> (cv A) %-> (%pi (cv B) %-> ce) %-> ce %. +%term clam %pi (%pi (cv A) %-> (%pi (cv B) %-> ce) %-> ce) %-> (cv (carr A B)) %. +%block targetb1 {A ctp} [x cv A]%. +%block targetb2 {A ctp} [y %pi (cv A) %-> ce]%. +%worlds (targetb1 targetb2) (ce) (cv _) %. +%sort trans-tp %. +%term trans-tp/o trans-tp o co %. +%term trans-tp/arr %pi (trans-tp (T1 => T2) (carr P1 P2)) %<- (trans-tp T1 P1) %<- (trans-tp T2 P2) %. +%sort can-trans-tp {A tp} %. +%mode can-trans-tp %in %out %. +%term _ can-trans-tp _ trans-tp/o %. +%term _ + %pi (can-trans-tp (A => B) (trans-tp/arr D2 D1)) + %<- (can-trans-tp A D1) + %<- (can-trans-tp B D2) %. +%worlds () (can-trans-tp _ _) %. +%total (D1) (can-trans-tp D1 _) %. +%sort ctp-eq %. +%term ctp-eq/i ctp-eq A' A' %. +%sort ctp-eq-resp {C %pi ctp %-> ctp %-> ctp} %. +%mode ctp-eq-resp %in %in %in %out %. +%term _ ctp-eq-resp C (%the (ctp-eq A A) ctp-eq/i) (%the (ctp-eq B B) ctp-eq/i) (%the (ctp-eq (C A B) (C A B)) ctp-eq/i) %. +%worlds () (ctp-eq-resp _ _ _ _) %. +%total {} (ctp-eq-resp _ _ _ _) %. +%sort trans-tp-unique %. +%mode trans-tp-unique %in %in %out %. +%term _ trans-tp-unique trans-tp/o trans-tp/o ctp-eq/i %. +%term _ + %pi (trans-tp-unique (trans-tp/arr D2 D1) (trans-tp/arr D2' D1') DQ3) + %<- (trans-tp-unique D1 D1' DQ1) + %<- (trans-tp-unique D2 D2' DQ2) + %<- (ctp-eq-resp carr DQ1 DQ2 DQ3) %. +%worlds () (trans-tp-unique _ _ _) %. +%total (D1) (trans-tp-unique D1 _ _) %. +%sort ceo-resp-ctp-eq %. +%mode ceo-resp-ctp-eq %in %in %out %. +%term _ ceo-resp-ctp-eq _ D1 D1 %. +%worlds (targetb1 targetb2) (ceo-resp-ctp-eq _ _ _) %. +%total {} (ceo-resp-ctp-eq _ _ _) %. +%sort cps %. +%mode cps %in %out %out %. +%sort cpse %. +%mode cpse %in %out %out %. +%sort cpse+ %. +%mode cpse+ %in %in %out %. +%term cps/lam + %pi (cps (lam E) (trans-tp/arr D2 D1) (clam E')) + %<- (can-trans-tp _ D1) + %<- ({x v A} {x' cv A'} %pi (cps x D1 x') %-> (cpse (E x) D2 (E' x'))) %. +%term cpse/app + %pi (cpse (app E1 E2) D2 ([c %pi (cv A') %-> ce] E1' ([w1] E2' ([w2] capp w1 w2 c)))) + %<- (cpse E1 (trans-tp/arr D2 D1) E1') + %<- (cpse+ E2 D1 E2') %. +%term cpse/inj %pi (cpse (inj E) D1 ([c %pi (cv A) %-> ce] c E')) %<- (cps E D1 E') %. +%term cpse+/i + %pi (cpse+ E D1 E'') + %<- (cpse E D1' E') + %<- (trans-tp-unique D1' D1 DQ) + %<- (ceo-resp-ctp-eq DQ E' E'') %. +%block cpsb {A tp} {A' ctp} {D1 trans-tp A A'} [x v A] [x' cv A'] [d cps x D1 x']%. +%worlds (cpsb) (cps _ _ _) (cpse _ _ _) (cpse+ _ _ _) %. +%total (E V E') (cps E _ _) (cpse V _ _) (cpse+ E' _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/cps-rp). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-cps-solution2.lf b/new-tests/stelf-output/pages/popl-tutorial-cps-solution2.lf new file mode 100644 index 0000000..ecd32d1 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-cps-solution2.lf @@ -0,0 +1,50 @@ +% +%{! +## Problem 2: Elimination of Administrative Redices + +!}% +%sort tp %. +%term o tp %. +%term => %pi tp %-> tp %-> tp %. +%prec %right 3 => %. +%sort exp %. +%sort value %. +%term app %pi (exp (A => B)) %-> (exp A) %-> (exp B) %. +%term lam %pi (%pi (value A) %-> (exp B)) %-> (value (A => B)) %. +%term ret %pi (value A) %-> (exp A) %. +%block sourceb {A tp} [x value A]%. +%worlds (sourceb) (exp _) (value _) %. +%sort contra %. +%sort cvalue %. +%term capp %pi (cvalue (A => B)) %-> (cvalue A) %-> (%pi (cvalue B) %-> contra) %-> contra %. +%term clam + %pi (%pi (cvalue A) %-> (%pi (cvalue B) %-> contra) %-> contra) + %-> (cvalue (A => B)) %. +%block targetb1 {A tp} [x cvalue A]%. +%block targetb2 {A tp} [y %pi (cvalue A) %-> contra]%. +%worlds (targetb1 targetb2) (contra) (cvalue _) %. +%sort cps %. +%mode cps %in %out %. +%sort cpse %. +%mode cpse %in %out %. +%term cps/lam + %pi (cps (lam (%the (%pi (value A) %-> (exp B)) E)) (clam (%the (%pi (cvalue A) %-> (%pi (cvalue B) %-> contra) %-> contra) E'))) + %<- ({x value A} {x' cvalue A} %pi (cps x x') %-> (cpse (E x) (E' x'))) %. +%term cpse/app + %pi (cpse (app (%the (exp (B => A)) E1) (%the (exp B) E2)) ([c %pi (cvalue A) %-> contra] E1' ([f cvalue (B => A)] E2' ([x cvalue B] capp f x c)))) + %<- (cpse E1 (%the (%pi (%pi (cvalue (B => A)) %-> contra) %-> contra) E1')) + %<- (cpse E2 (%the (%pi (%pi (cvalue B) %-> contra) %-> contra) E2')) %. +%term cpse/ret + %pi (cpse (ret (%the (value A) V)) ([c %pi (cvalue A) %-> contra] c V')) + %<- (cps V (%the (cvalue A) V')) %. +%block cpsb {A tp} [x value A] [x' cvalue A] [d cps x x']%. +%worlds (cpsb) (cps _ _) (cpse _ _) %. +%total (E V) (cps E _) (cpse V _) %. +%{! +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/CPS_Solution2). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-cps-solutions.lf b/new-tests/stelf-output/pages/popl-tutorial-cps-solutions.lf new file mode 100644 index 0000000..d6cd368 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-cps-solutions.lf @@ -0,0 +1,53 @@ +% +%{! + +## Problem 1: CPS Conversion with Administrative Redices + +!}% +%sort tp %. +%term o tp %. +%term => %pi tp %-> tp %-> tp %. +%prec %right 3 => %. +%sort exp %. +%sort value %. +%term app %pi (exp (A => B)) %-> (exp A) %-> (exp B) %. +%term lam %pi (%pi (value A) %-> (exp B)) %-> (value (A => B)) %. +%term ret %pi (value A) %-> (exp A) %. +%block sourceb {A tp} [x value A]%. +%worlds (sourceb) (exp _) (value _) %. +%sort contra %. +%sort cvalue %. +%sort ccont %. +%term capp %pi (cvalue (A => B)) %-> (cvalue A) %-> (ccont B) %-> contra %. +%term clam %pi (%pi (cvalue A) %-> (ccont B) %-> contra) %-> (cvalue (A => B)) %. +%term cconti %pi (%pi (cvalue A) %-> contra) %-> (ccont A) %. +%term cthrow %pi (ccont A) %-> (cvalue A) %-> contra %. +%block targetb1 {A tp} [x cvalue A]%. +%block targetb2 {A tp} [x ccont A]%. +%worlds (targetb1 targetb2) (contra) (cvalue _) (ccont _) %. +%sort cps %. +%mode cps %in %out %. +%sort cpse %. +%mode cpse %in %out %. +%term cps/lam + %pi (cps (lam (%the (%pi (value A) %-> (exp B)) E)) (clam (%the (%pi (cvalue A) %-> (ccont B) %-> contra) E'))) + %<- ({x value A} {x' cvalue A} %pi (cps x x') %-> (cpse (E x) (E' x'))) %. +%term cpse/app + %pi (cpse (app (%the (exp (B => A)) E1) (%the (exp B) E2)) ([c ccont A] E1' (cconti ([f cvalue (B => A)] E2' (cconti ([x cvalue B] capp f x c)))))) + %<- (cpse E1 (%the (%pi (ccont (B => A)) %-> contra) E1')) + %<- (cpse E2 (%the (%pi (ccont B) %-> contra) E2')) %. +%term cpse/ret + %pi (cpse (ret (%the (value A) V)) ([c ccont A] cthrow c V')) + %<- (cps V (%the (cvalue A) V')) %. +%block cpsb {A tp} [x value A] [x' cvalue A] [d cps x x']%. +%worlds (cpsb) (cps _ _) (cpse _ _) %. +%total (E V) (cps E _) (cpse V _) %. +%{! +See [POPL Tutorial/CPS_Solution2](/wiki/popl-tutorial-cps-solution2/) for the solution to the second CPS problem. +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/CPS_Solutions). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-cps-truefalse.lf b/new-tests/stelf-output/pages/popl-tutorial-cps-truefalse.lf new file mode 100644 index 0000000..32dbed3 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-cps-truefalse.lf @@ -0,0 +1,46 @@ +% +%{! ## CPS Conversion !}% +%sort tp %. +%term o tp %. +%term => %pi tp %-> tp %-> tp %. +%prec %right 3 => %. +%sort exp %. +%sort value %. +%term app %pi (exp (A => B)) %-> (exp A) %-> (exp B) %. +%term lam %pi (%pi (value A) %-> (exp B)) %-> (value (A => B)) %. +%term ret %pi (value A) %-> (exp A) %. +%block sourceb {A tp} [x value A]%. +%worlds (sourceb) (exp _) (value _) %. +%sort contra %. +%sort cvalue %. +%sort ccont %. +%term capp %pi (cvalue (A => B)) %-> (cvalue A) %-> (ccont B) %-> contra %. +%term clam %pi (%pi (cvalue A) %-> (ccont B) %-> contra) %-> (cvalue (A => B)) %. +%term cfalsei %pi (%pi (cvalue A) %-> contra) %-> (ccont A) %. +%term cthrow %pi (ccont A) %-> (cvalue A) %-> contra %. +%block targetb1 {A tp} [x cvalue A]%. +%block targetb2 {A tp} [x ccont A]%. +%worlds (targetb1 targetb2) (contra) (cvalue _) (ccont _) %. +%sort cps %. +%mode cps %in %out %. +%sort cpse %. +%mode cpse %in %out %. +%term cps/lam + %pi (cps (lam (%the (%pi (value A) %-> (exp B)) E)) (clam (%the (%pi (cvalue A) %-> (ccont B) %-> contra) E'))) + %<- ({x value A} {x' cvalue A} %pi (cps x x') %-> (cpse (E x) (E' x'))) %. +%term cpse/app + %pi (cpse (app (%the (exp (B => A)) E1) (%the (exp B) E2)) ([c ccont A] E1' (cfalsei ([f cvalue (B => A)] E2' (cfalsei ([x cvalue B] capp f x c)))))) + %<- (cpse E1 (%the (%pi (ccont (B => A)) %-> contra) E1')) + %<- (cpse E2 (%the (%pi (ccont B) %-> contra) E2')) %. +%term cpse/ret + %pi (cpse (ret (%the (value A) V)) ([c ccont A] cthrow c V')) + %<- (cps V (%the (cvalue A) V')) %. +%block cpsb {A tp} [x value A] [x' cvalue A] [d cps x x']%. +%worlds (cpsb) (cps _ _) (cpse _ _) %. +%total (E V) (cps E _) (cpse V _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/cps-truefalse). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-cps.lf b/new-tests/stelf-output/pages/popl-tutorial-cps.lf new file mode 100644 index 0000000..961257d --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-cps.lf @@ -0,0 +1,45 @@ +% +%{! + +## CPS Conversion + +!}% +%sort tp %. +%term o tp %. +%term => %pi tp %-> tp %-> tp %. +%prec %right 3 => %. +%sort e %. +%sort v %. +%term app %pi (e (A => B)) %-> (e A) %-> (e B) %. +%term lam %pi (%pi (v A) %-> (e B)) %-> (v (A => B)) %. +%term inj %pi (v A) %-> (e A) %. +%block sourceb {A tp} [x v A]%. +%worlds (sourceb) (e _) (v _) %. +%sort ce %. +%sort cv %. +%term capp %pi (cv (A => B)) %-> (cv A) %-> (%pi (cv B) %-> ce) %-> ce %. +%term clam %pi (%pi (cv A) %-> (%pi (cv B) %-> ce) %-> ce) %-> (cv (A => B)) %. +%block targetb1 {A tp} [x cv A]%. +%block targetb2 {A tp} [y %pi (cv A) %-> ce]%. +%worlds (targetb1 targetb2) (ce) (cv _) %. +%sort cps %. +%mode cps %in %out %. +%sort cpse %. +%mode cpse %in %out %. +%term cps/lam + %pi (cps (lam E) (clam E')) + %<- ({x v A} {x' cv A} %pi (cps x x') %-> (cpse (E x) (E' x'))) %. +%term cpse/app + %pi (cpse (app E1 E2) ([c %pi (cv A) %-> ce] E1' ([w1] E2' ([w2] capp w1 w2 c)))) + %<- (cpse E1 E1') + %<- (cpse E2 E2') %. +%term cpse/inj %pi (cpse (inj E) ([c %pi (cv A) %-> ce] c E')) %<- (cps E E') %. +%block cpsb {A tp} [x v A] [x' cv A] [d cps x x']%. +%worlds (cpsb) (cps _ _) (cpse _ _) %. +%total (E V) (cps E _) (cpse V _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/cps). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-evaluation-contexts-intrinsic.lf b/new-tests/stelf-output/pages/popl-tutorial-evaluation-contexts-intrinsic.lf new file mode 100644 index 0000000..a8469cd --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-evaluation-contexts-intrinsic.lf @@ -0,0 +1,73 @@ +% +%{! + +## Evaluation Contexts, Intrinsically Typed + +!}% +%sort tp %. +%term b tp %. +%term arr %pi tp %-> tp %-> tp %. +%sort tm %. +%sort val %. +%term app %pi (tm (arr A B)) %-> (tm A) %-> (tm B) %. +%term ret %pi (val A) %-> (tm A) %. +%term lam %pi (%pi (val A) %-> (tm B)) %-> (val (arr A B)) %. +%block tmb {A tp} [x val A]%. +%worlds (tmb) (tm _) (val _) %. +%sort ec %. +%term ec/var ec ([x] x) %. +%term ec/app1 %pi (ec ([x] app (E x) M)) %<- (ec E) %. +%term ec/app2 %pi (ec ([x] app (ret V) (E x))) %<- (ec E) %. +%sort redex %. +%mode redex %in %. +%term redex/beta redex (app (ret V1) (ret V2)) %. +%sort factored %. +%term f/val factored (ret V) %. +%term f/ec %pi (factored (E M0)) %<- (ec E) %<- (redex M0) %. +%sort factorapp %. +%mode factorapp %in %in %out %. +%term _ factorapp f/val f/val (f/ec redex/beta ec/var) %. +%term _ factorapp (f/ec (%the (redex M0) Dred0) (%the (ec E) Dec)) (%the (factored M2) _) (f/ec Dred0 (ec/app1 Dec)) %. +%term _ factorapp f/val (f/ec (%the (redex M0) Dred0) (%the (ec E) Dec)) (f/ec Dred0 (ec/app2 Dec)) %. +%worlds () (factorapp _ _ _) %. +%total {} (factorapp _ _ _) %. +%unique factorapp %in %in %out %. +%sort factor {M tm A} %. +%mode factor %in %out %. +%term factor/val factor (ret V) f/val %. +%term factor/app + %pi (factor (app E1 E2) F) + %<- (factor E1 F1) + %<- (factor E2 F2) + %<- (factorapp F1 F2 F) %. +%worlds () (factor _ _) %. +%total M (factor M _) %. +%unique factor %in %out %. +%sort result %. +%term done result (ret V) %. +%term stepped %pi (tm A) %-> (result M) %. +%sort contract %. +%mode contract %in %out %. +%term _ contract (%the (redex (app (ret (lam M)) (ret V))) redex/beta) (M V) %. +%worlds () (contract _ _) %. +%total {} (contract _ _) %. +%unique contract %in %out %. +%sort stepf %. +%mode stepf %in %out %. +%term _ stepf f/val done %. +%term _ %pi (stepf (f/ec Dr (%the (ec E) Dec)) (stepped (E Mred))) %<- (contract Dr Mred) %. +%worlds () (stepf _ _) %. +%total {} (stepf _ _) %. +%unique stepf %in %out %. +%sort step {M tm A} %. +%mode step %in %out %. +%term step/i %pi (step M M') %<- (factor M F) %<- (stepf F M') %. +%worlds () (step _ _) %. +%total {} (step _ _) %. +%unique step %in %out %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Evaluation_Contexts_Intrinsic). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-exceptions-problem.lf b/new-tests/stelf-output/pages/popl-tutorial-exceptions-problem.lf new file mode 100644 index 0000000..d742d97 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-exceptions-problem.lf @@ -0,0 +1,231 @@ +% +%{! + +Type safety for MinML: call-by-value, with recursive functions, in extrinsic form, with exceptions. + +In this example, we will take the MinML language from earlier and extend it with constructs for raising and handling exceptions. + +The static semantics will be extended with rules to handle these new expressions. The structured operational semantics from before will be extended with a new judgment that denotes when an expression raises an uncaught exception. + +The preservation and progress proofs must then be updated to +accommodate the extensions to the language. + +## Syntax + +Types: + +!}% +%sort tp %. +%name tp %. +%term nat tp %. +%term arr %pi tp %-> tp %-> tp %. +%{! + +Raw expressions, which admit ill-typed terms + +!}% +%sort exp %. +%name exp %. +%term z exp %. +%term s %pi exp %-> exp %. +%term ifz %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%term fun %pi tp %-> tp %-> (%pi exp %-> exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%{! + +We will extend MinML with two constructs. The first, (raise T), is a term of type T that will raise an exception. + +The second, (handle E1 E2), is a term that executes E1. If E1 +evaluates to a value V, then (handle E1 E2) evaluates to V. If E1 +raises an exception, then the handler E2 is executed. + +!}% +%term raise %pi tp %-> exp %. +%term handle %pi exp %-> exp %-> exp %. +%{! + +## Static semantics + +A judgement picking out the well-typed terms: + +!}% +%sort of %. +%name of %. +%mode of %in %out %. +%term of/z of z nat %. +%term of/s %pi (of (s E) nat) %<- (of E nat) %. +%term of/ifz + %pi (of (ifz E E1 ([x] E2 x)) T) + %<- (of E nat) + %<- (of E1 T) + %<- ({x exp} %pi (of x nat) %-> (of (E2 x) T)) %. +%term of/fun + %pi (of (fun T1 T2 ([f] [x] E f x)) (arr T1 T2)) + %<- ({f exp} %pi (of f (arr T1 T2)) %-> ({x exp} %pi (of x T1) %-> (of (E f x) T2))) %. +%term of/app %pi (of (app E1 E2) T) %<- (of E1 (arr T2 T)) %<- (of E2 T2) %. +%{! + +The following are the new typing rules for (raise T) and (handle E1 E2). + +!}% +%term of/raise of (raise T) T %. +%term of/handle %pi (of (handle E1 E2) T) %<- (of E1 T) %<- (of E2 T) %. +%{! + +## Dynamic semantics + +!}% +%sort value %. +%name value %. +%mode value %in %. +%term value/z value z %. +%term value/s %pi (value (s E)) %<- (value E) %. +%term value/fun value (fun _ _ _) %. +%sort raises %. +%mode raises %in %. +%term raises/raise raises (raise T) %. +%term raises/app/fun %pi (raises (app E1 E2)) %<- (raises E1) %. +%term raises/app/arg %pi (raises (app E1 E2)) %<- (value E1) %<- (raises E2) %. +%{! +Exercise: Give two rules, raises/s and raises/ifz, that propogate raises through the (s E) and (ifz E1 E2 E3) expressions. +!}% +%sort step %. +%name step %. +%mode step %in %out %. +%term step/s %pi (step (s E) (s E')) %<- (step E E') %. +%term step/ifz/arg %pi (step (ifz E E1 ([x] E2 x)) (ifz E' E1 ([x] E2 x))) %<- (step E E') %. +%term step/ifz/z step (ifz z E1 ([x] E2 x)) E1 %. +%term step/ifz/s %pi (step (ifz (s E) E1 ([x] E2 x)) (E2 E)) %<- (value E) %. +%term step/app/fun %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app/arg %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app/beta-v + %pi (step (app (fun T1 T2 ([f] [x] E f x)) E2) (E (fun T1 T2 ([f] [x] E f x)) E2)) + %<- (value E2) %. +%{! + +The following are new step rules for the (handle E1 E2) +expression. There is one rule for stepping the body, another rule for +handling a raised exception, and a third rule for when the body is a +value. + +!}% +%term step/handle/body %pi (step (handle E1 E2) (handle E1' E2)) %<- (step E1 E1') %. +%term step/handle/raise %pi (step (handle E1 E2) E2) %<- (raises E1) %. +%term step/handle/body-v %pi (step (handle E1 E2) E1) %<- (value E1) %. +%{! + +## Preservation + +With this encoding, we have to prove preservation explicitly, as the +type of ``step`` doesn't guarantee it. + +!}% +%sort pres %. +%name pres %. +%mode pres %in %in %out %. +%term _ %pi (pres (step/s Dstep) (of/s Dof) (of/s Dof')) %<- (pres Dstep Dof Dof') %. +%term _ + %pi (pres (step/ifz/arg Dstep) (of/ifz ([x] [dx] Dof2 x dx) Dof1 Dof) (of/ifz ([x] [dx] Dof2 x dx) Dof1 Dof')) + %<- (pres Dstep Dof Dof') %. +%term _ pres step/ifz/z (of/ifz _ Dof1 _) Dof1 %. +%term _ pres (step/ifz/s (%the (value E) _)) (of/ifz ([x] [dx] Dof2 x dx) _ (of/s Dof)) (Dof2 E Dof) %. +%term _ + %pi (pres (step/app/fun Dstep1) (of/app Dof2 Dof1) (of/app Dof2 Dof1')) + %<- (pres Dstep1 Dof1 Dof1') %. +%term _ + %pi (pres (step/app/arg Dstep2 _) (of/app Dof2 Dof1) (of/app Dof2' Dof1)) + %<- (pres Dstep2 Dof2 Dof2') %. +%term _ pres (step/app/beta-v _) (of/app Dof2 (of/fun ([f] [df] [x] [dx] Dof1 f df x dx))) (Dof1 _ (of/fun ([f] [df] [x] [dx] Dof1 f df x dx)) _ Dof2) %. +%{! + +Exercise: Give proof cases for step/handle/body, step/handle/body-v, and step/handle/raise. + +!}% +%worlds () (pres Dstep Dof Dof') %. +%total Dstep (pres Dstep _ _) %. +%{! + +## Progress + +!}% +%sort val-or-raises-or-step %. +%name val-or-raises-or-step %. +%term vrs/val %pi (val-or-raises-or-step E) %<- (value E) %. +%term vrs/step %pi (val-or-raises-or-step E) %<- (step E _) %. +%{! + +val-or-raises-or-step must have a case for when an expression raises an exception. + +!}% +%term vrs/raises %pi (val-or-raises-or-step E) %<- (raises E) %. +%{! + +### [Factoring lemmas](/wiki/output-factoring/) + +!}% +%sort prog/s %. +%mode prog/s %in %out %. +%term _ prog/s (vrs/step Dstep) (vrs/step (step/s Dstep)) %. +%term _ prog/s (vrs/val Dval) (vrs/val (value/s Dval)) %. +%{! + +A new case for when the input is an unhandled exception must be provided. + +!}% +%term _ prog/s (vrs/raises Draises) (vrs/raises (raises/s Draises)) %. +%worlds () (prog/s _ _) %. +%total {} (prog/s _ _) %. +%sort prog/ifz %. +%mode prog/ifz %in %in %in %in %out %. +%term _ prog/ifz _ (vrs/step Dstep) _ _ (vrs/step (step/ifz/arg Dstep)) %. +%term _ prog/ifz _ (vrs/val value/z) _ _ (vrs/step step/ifz/z) %. +%term _ prog/ifz _ (vrs/val (value/s Dval)) _ _ (vrs/step (step/ifz/s Dval)) %. +%{! A new case for when the input is an unhandled exception must be provided. !}% +%term _ prog/ifz _ (vrs/raises Draises) _ _ (vrs/raises (raises/ifz Draises)) %. +%worlds () (prog/ifz _ _ _ _ _) %. +%total {} (prog/ifz _ _ _ _ _) %. +%sort prog/app %. +%mode prog/app %in %in %in %out %. +%term _ prog/app _ (vrs/step Dstep1) _ (vrs/step (step/app/fun Dstep1)) %. +%term _ prog/app _ (vrs/val Dval1) (vrs/step Dstep2) (vrs/step (step/app/arg Dstep2 Dval1)) %. +%term _ prog/app _ (vrs/val Dval1) (vrs/val Dval2) (vrs/step (step/app/beta-v Dval2)) %. +%{! + +Exercise: Give the missing cases for when the inputs are unhandled exceptions. + +!}% +%worlds () (prog/app _ _ _ _) %. +%total {} (prog/app _ _ _ _) %. +%{! + +### Main theorem + +!}% +%sort prog %. +%name prog %. +%mode prog %in %out %. +%term _ prog of/z (vrs/val value/z) %. +%term _ %pi (prog (of/s Dof) Dvrs') %<- (prog Dof Dvrs) %<- (prog/s Dvrs Dvrs') %. +%term _ + %pi (prog (of/ifz ([x] [dx] Dof2 x dx) Dof1 Dof) Dvrs') + %<- (prog Dof Dvrs) + %<- (prog/ifz Dof Dvrs _ _ Dvrs') %. +%term _ prog (of/fun _) (vrs/val value/fun) %. +%term _ + %pi (prog (of/app Dof2 Dof1) Dvrs3) + %<- (prog Dof1 Dvrs1) + %<- (prog Dof2 Dvrs2) + %<- (prog/app Dof1 Dvrs1 Dvrs2 Dvrs3) %. +%{! + +Exercise: Give the missing cases for of/handle and of/raise. It may be necessary to use a factoring lemma in the solution for the case corresponding to of/handle. + +!}% +%worlds () (prog _ _) %. +%total Dof (prog Dof _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Exceptions-problem). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-exceptions.lf b/new-tests/stelf-output/pages/popl-tutorial-exceptions.lf new file mode 100644 index 0000000..6f06c5e --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-exceptions.lf @@ -0,0 +1,199 @@ +% +%{! + +Type safety for MinML: call-by-value, with recursive functions, in extrinsic form, with exceptions + +## Syntax + +Types: + +!}% +%sort tp %. +%name tp %. +%term nat tp %. +%term arr %pi tp %-> tp %-> tp %. +%{! + +Raw expressions, which admit ill-typed terms + +!}% +%sort exp %. +%name exp %. +%term z exp %. +%term s %pi exp %-> exp %. +%term ifz %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%term fun %pi tp %-> tp %-> (%pi exp %-> exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term raise %pi tp %-> exp %. +%term handle %pi exp %-> exp %-> exp %. +%{! + +## Static semantics + +A judgement picking out the well-typed terms: + +!}% +%sort of %. +%name of %. +%mode of %in %out %. +%term of/z of z nat %. +%term of/s %pi (of (s E) nat) %<- (of E nat) %. +%term of/ifz + %pi (of (ifz E E1 ([x] E2 x)) T) + %<- (of E nat) + %<- (of E1 T) + %<- ({x exp} %pi (of x nat) %-> (of (E2 x) T)) %. +%term of/fun + %pi (of (fun T1 T2 ([f] [x] E f x)) (arr T1 T2)) + %<- ({f exp} %pi (of f (arr T1 T2)) %-> ({x exp} %pi (of x T1) %-> (of (E f x) T2))) %. +%term of/app %pi (of (app E1 E2) T) %<- (of E1 (arr T2 T)) %<- (of E2 T2) %. +%term of/raise of (raise T) T %. +%term of/handle %pi (of (handle E1 E2) T) %<- (of E1 T) %<- (of E2 T) %. +%{! + +## Dynamic semantics + +!}% +%sort value %. +%name value %. +%mode value %in %. +%term value/z value z %. +%term value/s %pi (value (s E)) %<- (value E) %. +%term value/fun value (fun _ _ _) %. +%sort raises %. +%mode raises %in %. +%term raises/raise raises (raise T) %. +%term raises/app/fun %pi (raises (app E1 E2)) %<- (raises E1) %. +%term raises/app/arg %pi (raises (app E1 E2)) %<- (value E1) %<- (raises E2) %. +%term raises/s %pi (raises (s E)) %<- (raises E) %. +%term raises/ifz %pi (raises (ifz E1 E2 E3)) %<- (raises E1) %. +%sort step %. +%name step %. +%mode step %in %out %. +%term step/s %pi (step (s E) (s E')) %<- (step E E') %. +%term step/ifz/arg %pi (step (ifz E E1 ([x] E2 x)) (ifz E' E1 ([x] E2 x))) %<- (step E E') %. +%term step/ifz/z step (ifz z E1 ([x] E2 x)) E1 %. +%term step/ifz/s %pi (step (ifz (s E) E1 ([x] E2 x)) (E2 E)) %<- (value E) %. +%term step/app/fun %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app/arg %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app/beta-v + %pi (step (app (fun T1 T2 ([f] [x] E f x)) E2) (E (fun T1 T2 ([f] [x] E f x)) E2)) + %<- (value E2) %. +%term step/handle/body %pi (step (handle E1 E2) (handle E1' E2)) %<- (step E1 E1') %. +%term step/handle/raise %pi (step (handle E1 E2) E2) %<- (raises E1) %. +%term step/handle/body-v %pi (step (handle E1 E2) E1) %<- (value E1) %. +%{! + +## Preservation + +With this encoding, we have to prove preservation explicitly, as the +type of ``step`` doesn't guarantee it. + +!}% +%sort pres %. +%name pres %. +%mode pres %in %in %out %. +%term _ %pi (pres (step/s Dstep) (of/s Dof) (of/s Dof')) %<- (pres Dstep Dof Dof') %. +%term _ + %pi (pres (step/ifz/arg Dstep) (of/ifz ([x] [dx] Dof2 x dx) Dof1 Dof) (of/ifz ([x] [dx] Dof2 x dx) Dof1 Dof')) + %<- (pres Dstep Dof Dof') %. +%term _ pres step/ifz/z (of/ifz _ Dof1 _) Dof1 %. +%term _ pres (step/ifz/s (%the (value E) _)) (of/ifz ([x] [dx] Dof2 x dx) _ (of/s Dof)) (Dof2 E Dof) %. +%term _ + %pi (pres (step/app/fun Dstep1) (of/app Dof2 Dof1) (of/app Dof2 Dof1')) + %<- (pres Dstep1 Dof1 Dof1') %. +%term _ + %pi (pres (step/app/arg Dstep2 _) (of/app Dof2 Dof1) (of/app Dof2' Dof1)) + %<- (pres Dstep2 Dof2 Dof2') %. +%term _ pres (step/app/beta-v _) (of/app Dof2 (of/fun ([f] [df] [x] [dx] Dof1 f df x dx))) (Dof1 _ (of/fun ([f] [df] [x] [dx] Dof1 f df x dx)) _ Dof2) %. +%term _ + %pi (pres (step/handle/body Dstep) (of/handle Dof2 Dof1) (of/handle Dof2 Dof1')) + %<- (pres Dstep Dof1 Dof1') %. +%term _ pres (step/handle/raise Draise) (of/handle Dof2 Dof1) Dof2 %. +%term _ pres (step/handle/body-v Dval) (of/handle Dof2 Dof1) Dof1 %. +%worlds () (pres Dstep Dof Dof') %. +%total Dstep (pres Dstep _ _) %. +%{! + +## Progress + +!}% +%sort val-or-raises-or-step %. +%name val-or-raises-or-step %. +%term vrs/val %pi (val-or-raises-or-step E) %<- (value E) %. +%term vrs/step %pi (val-or-raises-or-step E) %<- (step E _) %. +%term vrs/raises %pi (val-or-raises-or-step E) %<- (raises E) %. +%{! + +### [Factoring lemmas](/wiki/output-factoring/) + +!}% +%sort prog/s %. +%mode prog/s %in %out %. +%term _ prog/s (vrs/step Dstep) (vrs/step (step/s Dstep)) %. +%term _ prog/s (vrs/val Dval) (vrs/val (value/s Dval)) %. +%term _ prog/s (vrs/raises Draises) (vrs/raises (raises/s Draises)) %. +%worlds () (prog/s _ _) %. +%total {} (prog/s _ _) %. +%sort prog/ifz %. +%mode prog/ifz %in %in %in %in %out %. +%term _ prog/ifz _ (vrs/step Dstep) _ _ (vrs/step (step/ifz/arg Dstep)) %. +%term _ prog/ifz _ (vrs/val value/z) _ _ (vrs/step step/ifz/z) %. +%term _ prog/ifz _ (vrs/val (value/s Dval)) _ _ (vrs/step (step/ifz/s Dval)) %. +%term _ prog/ifz _ (vrs/raises Draises) _ _ (vrs/raises (raises/ifz Draises)) %. +%worlds () (prog/ifz _ _ _ _ _) %. +%total {} (prog/ifz _ _ _ _ _) %. +%sort prog/app %. +%mode prog/app %in %in %in %out %. +%term _ prog/app _ (vrs/step Dstep1) _ (vrs/step (step/app/fun Dstep1)) %. +%term _ prog/app _ (vrs/val Dval1) (vrs/step Dstep2) (vrs/step (step/app/arg Dstep2 Dval1)) %. +%term _ prog/app _ (vrs/val Dval1) (vrs/val Dval2) (vrs/step (step/app/beta-v Dval2)) %. +%term _ prog/app _ (vrs/raises Draise1) _ (vrs/raises (raises/app/fun Draise1)) %. +%term _ prog/app _ (vrs/val Dval1) (vrs/raises Draise2) (vrs/raises (raises/app/arg Draise2 Dval1)) %. +%worlds () (prog/app _ _ _ _) %. +%total {} (prog/app _ _ _ _) %. +%sort prog/handle %. +%mode prog/handle %in %in %out %. +%term _ prog/handle (vrs/val Dval) _ (vrs/step (step/handle/body-v Dval)) %. +%term _ prog/handle (vrs/raises Draise) _ (vrs/step (step/handle/raise Draise)) %. +%term _ prog/handle (vrs/step Dstep) _ (vrs/step (step/handle/body Dstep)) %. +%worlds () (prog/handle _ _ _) %. +%total {} (prog/handle _ _ _) %. +%{! + +### Main theorem + +!}% +%sort prog %. +%name prog %. +%mode prog %in %out %. +%term _ prog of/z (vrs/val value/z) %. +%term _ %pi (prog (of/s Dof) Dvrs') %<- (prog Dof Dvrs) %<- (prog/s Dvrs Dvrs') %. +%term _ + %pi (prog (of/ifz ([x] [dx] Dof2 x dx) Dof1 Dof) Dvrs') + %<- (prog Dof Dvrs) + %<- (prog/ifz Dof Dvrs _ _ Dvrs') %. +%term _ prog (of/fun _) (vrs/val value/fun) %. +%term _ + %pi (prog (of/app Dof2 Dof1) Dvrs3) + %<- (prog Dof1 Dvrs1) + %<- (prog Dof2 Dvrs2) + %<- (prog/app Dof1 Dvrs1 Dvrs2 Dvrs3) %. +%term _ + %pi (prog (of/handle Dof2 Dof1) Dvrs2) + %<- (prog Dof1 Dvrs1) + %<- (prog/handle Dvrs1 _ Dvrs2) %. +%term _ prog of/raise (vrs/raises raises/raise) %. +%worlds () (prog _ _) %. +%total Dof (prog Dof _) %. +%{! + +And thus we have proved type safety for minml with exceptions! + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Exceptions). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-minml-answer.lf b/new-tests/stelf-output/pages/popl-tutorial-minml-answer.lf new file mode 100644 index 0000000..adfe49e --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-minml-answer.lf @@ -0,0 +1,163 @@ +% +%{! + +## Syntax + +!}% +%sort tp %. +%name tp %. +%term num tp %. +%term arr %pi tp %-> tp %-> tp %. +%% you can world-check syntax if you want +%worlds () (tp) %. +%sort exp %. +%name exp %. +%term fn %pi tp %-> (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term z exp %. +%term s %pi exp %-> exp %. +%{! +#### Exercise: constant for ifz +!}% +%term ifz %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%% you can world-check syntax if you want +%block exp_block [x exp]%. +%worlds (exp_block) (exp) %. +%{! + +## Static semantics + +!}% +%sort of %. +%name of %. +%term of/z of z num %. +%term of/app %pi (of (app E1 E2) T) %<- (of E1 (arr T' T)) %<- (of E2 T') %. +%term of/fn + %pi (of (fn T1 ([x] E x)) (arr T1 T2)) + %<- ({x exp} %pi (of x T1) %-> (of (E x) T2)) %. +%{! +#### Exercise: typing rules for s and ifz +!}% +%term of/s %pi (of (s E) num) %<- (of E num) %. +%term of/ifz + %pi (of (ifz E E0 ([x] E1 x)) T) + %<- (of E num) + %<- (of E0 T) + %<- ({x exp} %pi (of x num) %-> (of (E1 x) T)) %. +%block of_block {T tp} [x exp] [dx of x T]%. +%worlds (of_block) (of _ _) %. +%{! + +## Dynamic semantics + +### value judgement + + +!}% +%sort value %. +%name value %. +%term value/fn value (fn T ([x] E x)) %. +%{! +#### Exercise: value rules for z and s +!}% +%term value/z value z %. +%term value/s %pi (value (s E)) %<- (value E) %. +%{! + +### step judgement + +!}% +%sort step %. +%name step %. +%term step/app/fn %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app/arg %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app/beta %pi (step (app (fn T ([x] E x)) E2) (E E2)) %<- (value E2) %. +%{! +#### Exercise: step rules for s and ifz +!}% +%term step/s %pi (step (s E) (s E')) %<- (step E E') %. +%term step/ifz/arg %pi (step (ifz E E0 ([x] E1 x)) (ifz E' E0 ([x] E1 x))) %<- (step E E') %. +%term step/ifz/z step (ifz z E0 ([x] E1 x)) E0 %. +%term step/ifz/s %pi (step (ifz (s E) E0 ([x] E1 x)) (E1 E)) %<- (value E) %. +%{! + +## Progress theorem + +### Sum type for the result + +!}% +%sort val-or-step %. +%name val-or-step %. +%term vos/value %pi (val-or-step E) %<- (value E) %. +%term vos/step %pi (val-or-step E) %<- (step E E') %. +%{! + +### Lemmas + +!}% +%sort prog/app %. +%mode prog/app %in %in %in %out %. +%term _ prog/app _ (vos/step (%the (step E1 E1') Dstep1)) _ (vos/step (step/app/fn Dstep1)) %. +%term _ prog/app _ (vos/value (%the (value E1) Dval1)) (vos/step (%the (step E2 E2') Dstep2)) (vos/step (step/app/arg Dstep2 Dval1)) %. +%term _ prog/app (%the (of (fn T ([x] E' x)) (arr T T')) (of/fn _)) (vos/value (%the (value (fn T ([x] E' x))) Dval1)) (vos/value (%the (value E2) Dval2)) (vos/step (step/app/beta Dval2)) %. +%worlds () (prog/app _ _ _ _) %. +%total {} (prog/app _ _ _ _) %. +%{! + +#### Exercise: lemma for s + +!}% +%sort prog/s %. +%mode prog/s %in %out %. +%term _ prog/s (vos/step (%the (step E E') Dstep)) (vos/step (step/s Dstep)) %. +%term _ prog/s (vos/value (%the (value E) Dval)) (vos/value (value/s Dval)) %. +%worlds () (prog/s _ _) %. +%total {} (prog/s _ _) %. +%{! + +#### Exercise: lemma for ifz + +!}% +%sort prog/ifz %. +%mode prog/ifz %in %in %in %in %out %. +%mode prog/ifz %in %in %in %in %out %. +%term _ prog/ifz _ _ _ (vos/step Dstep) (step/ifz/arg Dstep) %. +%term _ prog/ifz _ _ _ (vos/value value/z) step/ifz/z %. +%term _ prog/ifz _ _ _ (vos/value (value/s Dval)) (step/ifz/s Dval) %. +%worlds () (prog/ifz _ _ _ _ _) %. +%total {} (prog/ifz _ _ _ _ _) %. +%{! + +### Main theorem + +!}% +%sort prog %. +%mode prog %in %out %. +%term _ prog (%the (of z num) of/z) (vos/value (%the (value z) value/z)) %. +%term _ prog (of/fn _) (vos/value value/fn) %. +%term _ + %pi (prog (of/app (%the (of E2 T') D2) (%the (of E1 (arr T' T)) D1)) DvosApp) + %<- (prog D1 (%the (val-or-step E1) Dvos1)) + %<- (prog D2 (%the (val-or-step E2) Dvos2)) + %<- (prog/app D1 Dvos1 Dvos2 DvosApp) %. +%{! + +#### Exercise: cases for s and ifz + +!}% +%term _ + %pi (prog (of/s (%the (of E num) D)) DvosS) + %<- (prog D (%the (val-or-step E) Dvos)) + %<- (prog/s Dvos DvosS) %. +%term _ + %pi (prog (of/ifz ([x] [dx of x num] %the (of (E1 x) T) (D1 x dx)) (%the (of E0 T) D0) (%the (of E num) D)) (vos/step DstepIfz)) + %<- (prog D (%the (val-or-step E) Dvos)) + %<- (prog/ifz D E0 ([x] E1 x) Dvos DstepIfz) %. +%worlds () (prog _ _) %. +%total Dof (prog Dof _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/MinML_Answer). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-minml-encoding.lf b/new-tests/stelf-output/pages/popl-tutorial-minml-encoding.lf new file mode 100644 index 0000000..5266fd3 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-minml-encoding.lf @@ -0,0 +1,141 @@ +% +%{! + +## Syntax + +!}% +%sort tp %. +%name tp %. +%term num tp %. +%term arr %pi tp %-> tp %-> tp %. +%% you can world-check syntax if you want +%worlds () (tp) %. +%sort exp %. +%name exp %. +%term fn %pi tp %-> (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term z exp %. +%term s %pi exp %-> exp %. +%{! +#### Exercise: constant for ifz +!}% +%term ifz %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%% you can world-check syntax if you want +%block exp_block [x exp]%. +%worlds (exp_block) (exp) %. +%{! + +## Static semantics + +!}% +%sort of %. +%name of %. +%term of/z of z num %. +%term of/app %pi (of (app E1 E2) T) %<- (of E1 (arr T' T)) %<- (of E2 T') %. +%term of/fn + %pi (of (fn T1 ([x] E x)) (arr T1 T2)) + %<- ({x exp} %pi (of x T1) %-> (of (E x) T2)) %. +%{! +#### Exercise: typing rules for s and ifz +!}% +%term of/s %pi (of (s E) num) %<- (of E num) %. +%term of/ifz + %pi (of (ifz E E0 ([x] E1 x)) T) + %<- (of E num) + %<- (of E0 T) + %<- ({x exp} %pi (of x num) %-> (of (E1 x) T)) %. +%block of_block {T tp} [x exp] [dx of x T]%. +%worlds (of_block) (of _ _) %. +%{! + +## Dynamic semantics + +### value judgement + + +!}% +%sort value %. +%name value %. +%term value/fn value (fn T ([x] E x)) %. +%{! +#### Exercise: value rules for z and s +!}% +%term value/z value z %. +%term value/s %pi (value (s E)) %<- (value E) %. +%{! + +### step judgement + +!}% +%sort step %. +%name step %. +%term step/app/fn %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app/arg %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app/beta %pi (step (app (fn T ([x] E x)) E2) (E E2)) %<- (value E2) %. +%{! +#### Exercise: step rules for s and ifz +!}% +%term step/s %pi (step (s E) (s E')) %<- (step E E') %. +%term step/ifz/arg %pi (step (ifz E E0 ([x] E1 x)) (ifz E' E0 ([x] E1 x))) %<- (step E E') %. +%term step/ifz/z step (ifz z E0 ([x] E1 x)) E0 %. +%term step/ifz/s %pi (step (ifz (s E) E0 ([x] E1 x)) (E1 E)) %<- (value E) %. +%{! + +## Progress theorem + +### Sum type for the result + +!}% +%sort val-or-step %. +%name val-or-step %. +%term vos/value %pi (val-or-step E) %<- (value E) %. +%term vos/step %pi (val-or-step E) %<- (step E E') %. +%{! + +### Lemmas + +!}% +%sort prog/app %. +%mode prog/app %in %in %in %out %. +%term _ prog/app _ (vos/step (%the (step E1 E1') Dstep1)) _ (vos/step (step/app/fn Dstep1)) %. +%term _ prog/app _ (vos/value (%the (value E1) Dval1)) (vos/step (%the (step E2 E2') Dstep2)) (vos/step (step/app/arg Dstep2 Dval1)) %. +%term _ prog/app (%the (of (fn T ([x] E' x)) (arr T T')) (of/fn _)) (vos/value (%the (value (fn T ([x] E' x))) Dval1)) (vos/value (%the (value E2) Dval2)) (vos/step (step/app/beta Dval2)) %. +%worlds () (prog/app _ _ _ _) %. +%total {} (prog/app _ _ _ _) %. +%{! + +#### Exercise: lemma for s + +!}% +%{! + +#### Exercise: lemma for ifz + +!}% +%{! + +### Main theorem + +!}% +%sort prog %. +%mode prog %in %out %. +%term _ prog (%the (of z num) of/z) (vos/value (%the (value z) value/z)) %. +%term _ prog (of/fn _) (vos/value value/fn) %. +%term _ + %pi (prog (of/app (%the (of E2 T') D2) (%the (of E1 (arr T' T)) D1)) DvosApp) + %<- (prog D1 (%the (val-or-step E1) Dvos1)) + %<- (prog D2 (%the (val-or-step E2) Dvos2)) + %<- (prog/app D1 Dvos1 Dvos2 DvosApp) %. +%{! + +#### Exercise: cases for s and ifz + +!}% +%worlds () (prog _ _) %. +%total Dof (prog Dof _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/MinML_encoding). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-minml-preservation-theorem-solution.lf b/new-tests/stelf-output/pages/popl-tutorial-minml-preservation-theorem-solution.lf new file mode 100644 index 0000000..053b3f6 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-minml-preservation-theorem-solution.lf @@ -0,0 +1,80 @@ +% +%{! This is the solution to [this exercise](/wiki/popl-tutorial-minml-preservation-theorem/). + +Type safety for MinML + +|hidden = true !}% +%% Syntax %% +%sort tp %. +%name tp %. +%term nat tp %. +%term arr %pi tp %-> tp %-> tp %. +%% Expressions %% +%sort exp %. +%name exp %. +%term fn %pi tp %-> (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term z exp %. +%term s %pi exp %-> exp %. +%term ifz %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%% Static semantics %% +%sort of %. +%name of %. +%mode of %in %out %. +%term of/z of z nat %. +%term of/fn + %pi (of (fn T1 ([x] E x)) (arr T1 T2)) + %<- ({x exp} %pi (of x T1) %-> (of (E x) T2)) %. +%term of/app %pi (of (app E1 E2) T) %<- (of E1 (arr T2 T)) %<- (of E2 T2) %. +%term of/s %pi (of (s E) nat) %<- (of E nat) %. +%term of/ifz + %pi (of (ifz E E1 ([x] E2 x)) T) + %<- (of E nat) + %<- (of E1 T) + %<- ({x exp} %pi (of x nat) %-> (of (E2 x) T)) %. +%% Dynamic semantics %% +%sort value %. +%name value %. +%mode value %in %. +%term value/z value z %. +%term value/s %pi (value (s E)) %<- (value E) %. +%term value/fn value (fn _ _) %. +%sort step %. +%name step %. +%mode step %in %out %. +%term step/app/fn %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app/arg %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app/beta %pi (step (app (fn _ ([x] E x)) E2) (E E2)) %<- (value E2) %. +%term step/s %pi (step (s E) (s E')) %<- (step E E') %. +%term step/ifz/arg %pi (step (ifz E E1 ([x] E2 x)) (ifz E' E1 ([x] E2 x))) %<- (step E E') %. +%term step/ifz/z step (ifz z E1 ([x] E2 x)) E1 %. +%term step/ifz/s %pi (step (ifz (s E) E1 ([x] E2 x)) (E2 E)) %<- (value E) %. +%{! + +## Preservation + +!}% +%sort pres %. +%name pres %. +%mode pres %in %in %out %. +%term _ %pi (pres (step/s Dstep) (of/s Dof) (of/s Dof')) %<- (pres Dstep Dof Dof') %. +%term _ + %pi (pres (step/ifz/arg Dstep) (of/ifz ([x] [dx] Dof2 x dx) Dof1 Dof) (of/ifz ([x] [dx] Dof2 x dx) Dof1 Dof')) + %<- (pres Dstep Dof Dof') %. +%term _ pres step/ifz/z (of/ifz _ Dof1 _) Dof1 %. +%term _ pres (step/ifz/s (%the (value E) _)) (of/ifz ([x] [dx] Dof2 x dx) _ (of/s Dof)) (Dof2 E Dof) %. +%term _ + %pi (pres (step/app/fn Dstep1) (of/app Dof2 Dof1) (of/app Dof2 Dof1')) + %<- (pres Dstep1 Dof1 Dof1') %. +%term _ + %pi (pres (step/app/arg Dstep2 _) (of/app Dof2 Dof1) (of/app Dof2' Dof1)) + %<- (pres Dstep2 Dof2 Dof2') %. +%term _ pres (step/app/beta _) (of/app Dof2 (of/fn ([x] [dx] Dof1 x dx))) (Dof1 _ Dof2) %. +%worlds () (pres Dstep Dof Dof') %. +%total Dstep (pres Dstep _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/MinML_Preservation_Theorem:_Solution). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-minml-preservation-theorem.lf b/new-tests/stelf-output/pages/popl-tutorial-minml-preservation-theorem.lf new file mode 100644 index 0000000..a0a2c9a --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-minml-preservation-theorem.lf @@ -0,0 +1,90 @@ +% +%{! + +In this exercise, we prove the preservation theorem for the MinML langauge we worked with before. + +There is one task: +* Complete the cases of the preservation theorem + +The solution is [here](/wiki/popl-tutorial-minml-preservation-theorem-solution/). + +## Syntax + +Types: + +!}% +%sort tp %. +%name tp %. +%term nat tp %. +%term arr %pi tp %-> tp %-> tp %. +%{! + +Expressions + +!}% +%sort exp %. +%name exp %. +%term fn %pi tp %-> (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term z exp %. +%term s %pi exp %-> exp %. +%term ifz %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%{! + +## Static semantics + +!}% +%sort of %. +%name of %. +%mode of %in %out %. +%term of/z of z nat %. +%term of/fn + %pi (of (fn T1 ([x] E x)) (arr T1 T2)) + %<- ({x exp} %pi (of x T1) %-> (of (E x) T2)) %. +%term of/app %pi (of (app E1 E2) T) %<- (of E1 (arr T2 T)) %<- (of E2 T2) %. +%term of/s %pi (of (s E) nat) %<- (of E nat) %. +%term of/ifz + %pi (of (ifz E E1 ([x] E2 x)) T) + %<- (of E nat) + %<- (of E1 T) + %<- ({x exp} %pi (of x nat) %-> (of (E2 x) T)) %. +%{! + +## Dynamic semantics + +!}% +%sort value %. +%name value %. +%mode value %in %. +%term value/z value z %. +%term value/s %pi (value (s E)) %<- (value E) %. +%term value/fn value (fn _ _) %. +%sort step %. +%name step %. +%mode step %in %out %. +%term step/app/fn %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app/arg %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app/beta %pi (step (app (fn _ ([x] E x)) E2) (E E2)) %<- (value E2) %. +%term step/s %pi (step (s E) (s E')) %<- (step E E') %. +%term step/ifz/arg %pi (step (ifz E E1 ([x] E2 x)) (ifz E' E1 ([x] E2 x))) %<- (step E E') %. +%term step/ifz/z step (ifz z E1 ([x] E2 x)) E1 %. +%term step/ifz/s %pi (step (ifz (s E) E1 ([x] E2 x)) (E2 E)) %<- (value E) %. +%{! + +## Preservation + +#### TASK 1: Fill in missing cases + +!}% +%sort pres %. +%name pres %. +%mode pres %in %in %out %. +% fill in here. +%worlds () (pres Dstep Dof Dof') %. +%total Dstep (pres Dstep _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/MinML_Preservation_Theorem). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-minml-starter.lf b/new-tests/stelf-output/pages/popl-tutorial-minml-starter.lf new file mode 100644 index 0000000..0ca9c3f --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-minml-starter.lf @@ -0,0 +1,236 @@ +% +%{! + +## Syntax + +!}% +%sort tp %. +%name tp %. +%term num tp %. +%term arr %pi tp %-> tp %-> tp %. +%sort exp %. +%name exp %. +%term fn %pi tp %-> (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term z exp %. +%term s %pi exp %-> exp %. +%{! +#### Exercise: constant for ifz +!}% +%% The syntax '% .' (without the space) +%% causes Twelf to stop processing the file at this point +%% remove once you have completed the exercise +%{! + +## Static semantics + +!}% +%{! +#### Exercise: typing rules for s and ifz +!}% +%{! + +## Dynamic semantics + +### value judgement + + +!}% +%{! +#### Exercise: value rules for z and s +!}% +%{! + +### step judgement + +!}% +%{! +#### Exercise: step rules for s and ifz +!}% +%{! + +## Progress theorem + +### Sum type for the result + +!}% +%{! + +### Lemmas + +!}% +%{! + +#### Exercise: lemma for s + +!}% +%{! + +#### Exercise: lemma for ifz + +!}% +%{! + +### Main theorem + +!}% +%{! + +#### Exercise: cases for s and ifz + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/MinML_Starter). +!}% +%. + + + +of : exp -> tp -> type. %name of Dof. + +of/z : of z num. + +of/app : of (app E1 E2) T + <- of E1 (arr T' T) + <- of E2 T'. + +of/fn : of (fn T1 ([x] E x)) (arr T1 T2) + <- ({x:exp} of x T1 -> of (E x) T2). + + + + + + + + + + + + +%block of_block : some {T:tp} block {x:exp}{dx: of x T}. +%worlds (of_block) (of _ _). + + + +value : exp -> type. %name value Dval. + +value/fn : value (fn T ([x] E x)). + + + + + + + + + + +step : exp -> exp -> type. %name step Dstep. + +step/app/fn : step (app E1 E2) (app E1' E2) + <- step E1 E1'. + +step/app/arg : step (app E1 E2) (app E1 E2') + <- value E1 + <- step E2 E2'. + +step/app/beta : step (app (fn T ([x] E x)) E2) (E E2) + <- value E2. + + + + + + + + + + +val-or-step : exp -> type. %name val-or-step Dvos. + +vos/value : val-or-step E + <- value E. +vos/step : val-or-step E + <- step E E'. + + + +prog/app : of E1 (arr T' T) + -> val-or-step E1 + -> val-or-step E2 + -> val-or-step (app E1 E2) + -> type. +%mode prog/app +Dof +Dvos1 +Dvos2 -Dvos. + +- : prog/app + _ + (vos/step (Dstep1 : step E1 E1')) + _ + (vos/step (step/app/fn Dstep1)). + +- : prog/app + _ + (vos/value (Dval1 : value E1)) + (vos/step (Dstep2 : step E2 E2')) + (vos/step (step/app/arg Dstep2 Dval1)). + +- : prog/app + (of/fn _ : of (fn T ([x] E' x)) (arr T T')) + (vos/value (Dval1 : value (fn T ([x] E' x)))) + (vos/value (Dval2 : value E2)) + (vos/step (step/app/beta Dval2)). + +%worlds () (prog/app _ _ _ _). +%total {} (prog/app _ _ _ _). + + + + + + + + + + + + + + + + + + + + + +prog : of E T -> val-or-step E -> type. +%mode prog +Dof -Dvos. + +- : prog (of/z : of z num) (vos/value (value/z : value z)). + +- : prog (of/fn _) (vos/value value/fn). + +- : prog (of/app + (D2 : of E2 T') + (D1 : of E1 (arr T' T))) + DvosApp + <- prog D1 (Dvos1 : val-or-step E1) + <- prog D2 (Dvos2 : val-or-step E2) + <- prog/app D1 Dvos1 Dvos2 DvosApp. + + + + + + + + + + +%worlds () (prog _ _). +%total Dof (prog Dof _). + diff --git a/new-tests/stelf-output/pages/popl-tutorial-minml.lf b/new-tests/stelf-output/pages/popl-tutorial-minml.lf new file mode 100644 index 0000000..74ff446 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-minml.lf @@ -0,0 +1,11 @@ +% +%{! * [MinML Syntax and judgments---the code we just wrote](/wiki/popl-tutorial-minml-encoding/) + +* [http://twelf.org/w/index.php?title=POPL_Tutorial/MinML_Starter&action=raw&ctype=text/css Starter code]: start here if you don't want to type along +* [Solutions to exercises](/wiki/popl-tutorial-minml-answer/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/MinML). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-nat.lf b/new-tests/stelf-output/pages/popl-tutorial-nat.lf new file mode 100644 index 0000000..ff7f718 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-nat.lf @@ -0,0 +1,75 @@ +% +%{! + +## Nats + +!}% +%sort nat %. +%term zero nat %. +%term succ %pi nat %-> nat %. +%{! + +## Add + +!}% +%sort add %. +%term add/z add zero N N %. +%term add/s %pi (add (succ M) N (succ O)) %<- (add M N O) %. +%mode add %in %in %out %. +%worlds () (add _ _ _) %. +%total M (add M _ _) %. +%{! + +## Exercise: Mult + +!}% +%sort mult %. +%term mult/z mult zero N zero %. +%term mult/s %pi (mult (succ M) N O') %<- (mult M N O) %<- (add N O O') %. +%mode mult %in %in %out %. +%worlds () (mult _ _ _) %. +%total M (mult M _ _) %. +%{! + +## Right-hand Identity + +!}% +%sort rhzero {M nat} %. +%mode rhzero %in %out %. +%term _ rhzero zero add/z %. +%term _ %pi (rhzero (succ M) (add/s D)) %<- (rhzero M (%the (add M zero M) D)) %. +%worlds () (rhzero _ _) %. +%total M (rhzero M _) %. +%{! + +## Right-hand succ + +!}% +%sort rhsucc %. +%mode rhsucc %in %out %. +%term _ rhsucc (%the (add zero M M) add/z) (%the (add zero (succ M) (succ M)) add/z) %. +%term _ + %pi (rhsucc (%the (add (succ M) N (succ O)) (add/s (%the (add M N O) D1))) (add/s D2)) + %<- (rhsucc D1 (%the (add M (succ N) (succ O)) D2)) %. +%worlds () (rhsucc _ _) %. +%total M (rhsucc M _) %. +%{! + +## Exercise: put it all together + +!}% +%sort commute %. +%mode commute %in %out %. +%term _ %pi (commute (%the (add zero M M) add/z) D) %<- (rhzero M D) %. +%term _ + %pi (commute (add/s (%the (add M N O) D)) D'') + %<- (commute D (%the (add N M O) D')) + %<- (rhsucc D' (%the (add N (succ M) (succ O)) D'')) %. +%worlds () (commute _ _) %. +%total D (commute D _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Nat). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-new-language.lf b/new-tests/stelf-output/pages/popl-tutorial-new-language.lf new file mode 100644 index 0000000..20c8dc4 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-new-language.lf @@ -0,0 +1,224 @@ +% +%{! + +## Arithmetic Primitives + +!}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort add %. +%mode add %in %in %out %. +%term add/z add z N N %. +%term add/s %pi (add (s M) N (s P)) %<- (add M N P) %. +%worlds () (add _ _ _) %. +%total M (add M _ _) %. +%sort mult %. +%mode mult %in %in %out %. +%term mult/z mult z N z %. +%term mult/s %pi (mult (s M) N Q) %<- (mult M N P) %<- (add P N Q) %. +%worlds () (mult _ _ _) %. +%total M (mult M _ _) %. +%{! + +## Language Syntax + +!}% +%sort tp %. +%term num tp %. +%term arrow %pi tp %-> tp %-> tp %. +%term prod %pi tp %-> tp %-> tp %. +%sort exp %. +%term numeral %pi nat %-> exp %. +%term plus %pi exp %-> exp %-> exp %. +%term times %pi exp %-> exp %-> exp %. +%term lam %pi tp %-> (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%term pair %pi exp %-> exp %-> exp %. +%term split %pi exp %-> (%pi exp %-> exp %-> exp) %-> exp %. +%{! + +## Dynamic Semantics + +!}% +%sort value %. +%sort step %. +%%% argh, too many rules! +%term value/numeral value (numeral N) %. +%term value/lam value (lam T ([x] E x)) %. +%term value/pair %pi (value (pair E1 E2)) %<- (value E1) %<- (value E2) %. +%term step/plus/1 %pi (step (plus E1 E2) (plus E1' E2)) %<- (step E1 E1') %. +%term step/plus/2 %pi (step (plus V1 E2) (plus V1 E2')) %<- (value V1) %<- (step E2 E2') %. +%term step/plus/add %pi (step (plus (numeral N1) (numeral N2)) (numeral N3)) %<- (add N1 N2 N3) %. +%term step/times/1 %pi (step (times E1 E2) (times E1' E2)) %<- (step E1 E1') %. +%term step/times/2 %pi (step (times V1 E2) (times V1 E2')) %<- (value V1) %<- (step E2 E2') %. +%term step/times/mult %pi (step (times (numeral N1) (numeral N2)) (numeral N3)) %<- (mult N1 N2 N3) %. +%term step/app/fun %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app/arg %pi (step (app V1 E2) (app V1 E2')) %<- (value V1) %<- (step E2 E2') %. +%term step/app/beta %pi (step (app (lam T ([x] E x)) V) (E V)) %<- (value V) %. +%term step/pair/1 %pi (step (pair E1 E2) (pair E1' E2)) %<- (step E1 E1') %. +%term step/pair/2 %pi (step (pair V1 E2) (pair V1 E2')) %<- (value V1) %<- (step E2 E2') %. +%term step/split/arg + %pi (step (split E1 ([x] [y] E2 x y)) (split E1' ([x] [y] E2 x y))) + %<- (step E1 E1') %. +%term step/split/pair + %pi (step (split (pair V1 V2) ([x] [y] E x y)) (E V1 V2)) + %<- (value V1) + %<- (value V2) %. +%worlds () (value _) %. +%worlds () (step _ _) %. +%{! + +## Static Semantics + +!}% +%sort of %. +%term of/numeral of (numeral N) num %. +%term of/plus %pi (of (plus E1 E2) num) %<- (of E1 num) %<- (of E2 num) %. +%term of/times %pi (of (times E1 E2) num) %<- (of E1 num) %<- (of E2 num) %. +%term of/lam + %pi (of (lam T1 ([x] E x)) (arrow T1 T2)) + %<- ({x} %pi (of x T1) %-> (of (E x) T2)) %. +%term of/app %pi (of (app E1 E2) T) %<- (of E1 (arrow T2 T)) %<- (of E2 T2) %. +%term of/pair %pi (of (pair E1 E2) (prod T1 T2)) %<- (of E1 T1) %<- (of E2 T2) %. +%term of/split + %pi (of (split E ([x] [y] Ebody x y)) T) + %<- (of E (prod T1 T2)) + %<- ({x} %pi (of x T1) %-> ({y} %pi (of y T2) %-> (of (Ebody x y) T))) %. +%block tbind {T tp} [x exp] [dx of x T]%. +%worlds (tbind) (of _ _) %. +%{! + +## Progress + +!}% +%sort notstuck %. +%term notstuck/value %pi (notstuck E) %<- (value E) %. +%term notstuck/step %pi (notstuck E) %<- (step E E') %. +%sort progress-app %. +%mode progress-app %in %in %in %out %. +%term _ progress-app Dof (notstuck/step Dstep) Dns2 (notstuck/step (step/app/fun Dstep)) %. +%term _ progress-app Dof (notstuck/value Dval) (notstuck/step Dstep) (notstuck/step (step/app/arg Dstep Dval)) %. +%term _ progress-app (of/lam Dof') (notstuck/value value/lam) (notstuck/value Dval2) (notstuck/step (step/app/beta Dval2)) %. +%worlds () (progress-app _ _ _ _) %. +%total {} (progress-app _ _ _ _) %. +%sort progress-pair %. +%mode progress-pair %in %in %out %. +%term _ progress-pair (notstuck/step Dstep) Dns2 (notstuck/step (step/pair/1 Dstep)) %. +%term _ progress-pair (notstuck/value Dval1) (notstuck/step Dstep2) (notstuck/step (step/pair/2 Dstep2 Dval1)) %. +%term _ progress-pair (notstuck/value Dval1) (notstuck/value Dval2) (notstuck/value (value/pair Dval2 Dval1)) %. +%worlds () (progress-pair _ _ _) %. +%total {} (progress-pair _ _ _) %. +%sort progress-split %. +%mode progress-split %in %in %in %out %. +%term _ progress-split Dof (notstuck/step Dstep) Ebody (notstuck/step (step/split/arg Dstep)) %. +%term _ progress-split (of/pair Dof2 Dof1) (notstuck/value (value/pair Dval2 Dval1)) Ebody (notstuck/step (step/split/pair Dval2 Dval1)) %. +%worlds () (progress-split _ _ _ _) %. +%total {} (progress-split _ _ _ _) %. +%sort can-add {N1} {N2} %. +%mode can-add %in %in %out %. +%term _ can-add z N add/z %. +%term _ %pi (can-add (s N1) N2 (add/s Dadd)) %<- (can-add N1 N2 Dadd) %. +%worlds () (can-add _ _ _) %. +%total (D1) (can-add D1 _ _) %. +%sort progress-plus %. +%mode progress-plus %in %in %in %in %out %. +%term _ progress-plus Dof1 Dof2 (notstuck/step Dstep) Dns2 (notstuck/step (step/plus/1 Dstep)) %. +%term _ progress-plus Dof1 Dof2 (notstuck/value Dval1) (notstuck/step Dstep2) (notstuck/step (step/plus/2 Dstep2 Dval1)) %. +%term _ + %pi (progress-plus of/numeral of/numeral (notstuck/value value/numeral) (notstuck/value value/numeral) (notstuck/step (step/plus/add Dadd))) + %<- (can-add N1 N2 (%the (add N1 N2 N3) Dadd)) %. +%worlds () (progress-plus _ _ _ _ _) %. +%total {} (progress-plus _ _ _ _ _) %. +%sort can-mult {N1} {N2} %. +%mode can-mult %in %in %out %. +%term _ can-mult z N mult/z %. +%term _ + %pi (can-mult (s N1) N2 (mult/s Dadd Dmult)) + %<- (can-mult N1 N2 (%the (mult N1 N2 N3) Dmult)) + %<- (can-add N3 N2 Dadd) %. +%worlds () (can-mult _ _ _) %. +%total (D1) (can-mult D1 _ _) %. +%sort progress-times %. +%mode progress-times %in %in %in %in %out %. +%term _ progress-times Dof1 Dof2 (notstuck/step Dstep) Dns2 (notstuck/step (step/times/1 Dstep)) %. +%term _ progress-times Dof1 Dof2 (notstuck/value Dval1) (notstuck/step Dstep2) (notstuck/step (step/times/2 Dstep2 Dval1)) %. +%term _ + %pi (progress-times of/numeral of/numeral (notstuck/value value/numeral) (notstuck/value value/numeral) (notstuck/step (step/times/mult Dmult))) + %<- (can-mult N1 N2 (%the (mult N1 N2 N3) Dmult)) %. +%worlds () (progress-times _ _ _ _ _) %. +%total {} (progress-times _ _ _ _ _) %. +%sort progress %. +%mode progress %in %out %. +%term _ progress of/numeral (notstuck/value value/numeral) %. +%term _ + %pi (progress (of/plus Dof2 Dof1) Dns3) + %<- (progress Dof1 Dns1) + %<- (progress Dof2 Dns2) + %<- (progress-plus Dof1 Dof2 Dns1 Dns2 Dns3) %. +%term _ + %pi (progress (of/times Dof2 Dof1) Dns3) + %<- (progress Dof1 Dns1) + %<- (progress Dof2 Dns2) + %<- (progress-times Dof1 Dof2 Dns1 Dns2 Dns3) %. +%term _ progress (of/lam Dof1) (notstuck/value value/lam) %. +%term _ + %pi (progress (of/app Dof2 Dof1) Dns3) + %<- (progress Dof1 Dns1) + %<- (progress Dof2 Dns2) + %<- (progress-app Dof1 Dns1 Dns2 Dns3) %. +%term _ + %pi (progress (of/pair Dof2 Dof1) Dns3) + %<- (progress Dof1 Dns1) + %<- (progress Dof2 Dns2) + %<- (progress-pair Dns1 Dns2 Dns3) %. +%term _ + %pi (progress (of/split (%the ({x} %pi (of x T1) %-> ({y} %pi (of y T2) %-> (of (Ebody x y) T))) Dbody) Dof1) Dns2) + %<- (progress Dof1 Dns1) + %<- (progress-split Dof1 Dns1 Ebody Dns2) %. +%worlds () (progress _ _) %. +%total (D1) (progress D1 _) %. +%{! +## Preservation +!}% +%sort preservation %. +%mode preservation %in %in %out %. +%term _ + %pi (preservation (of/plus Dof2 Dof1) (step/plus/1 Dstep1) (of/plus Dof2 Dof1')) + %<- (preservation Dof1 Dstep1 Dof1') %. +%term _ + %pi (preservation (of/plus Dof2 Dof1) (step/plus/2 Dstep2 Dval1) (of/plus Dof2' Dof1)) + %<- (preservation Dof2 Dstep2 Dof2') %. +%term _ preservation (of/plus Dof2 Dof1) (step/plus/add Dadd) of/numeral %. +%term _ + %pi (preservation (of/times Dof2 Dof1) (step/times/1 Dstep1) (of/times Dof2 Dof1')) + %<- (preservation Dof1 Dstep1 Dof1') %. +%term _ + %pi (preservation (of/times Dof2 Dof1) (step/times/2 Dstep2 Dval1) (of/times Dof2' Dof1)) + %<- (preservation Dof2 Dstep2 Dof2') %. +%term _ preservation (of/times Dof2 Dof1) (step/times/mult Dmult) of/numeral %. +%term _ + %pi (preservation (of/pair Dof2 Dof1) (step/pair/1 Dstep1) (of/pair Dof2 Dof1')) + %<- (preservation Dof1 Dstep1 Dof1') %. +%term _ + %pi (preservation (of/pair Dof2 Dof1) (step/pair/2 Dstep2 Dval) (of/pair Dof2' Dof1)) + %<- (preservation Dof2 Dstep2 Dof2') %. +%term _ + %pi (preservation (of/split Dbody Dof1) (step/split/arg Dstep1) (of/split Dbody Dof1')) + %<- (preservation Dof1 Dstep1 Dof1') %. +%term _ preservation (of/split Dbody (of/pair Dof2 Dof1)) (step/split/pair Dval2 Dval1) (Dbody E1 Dof1 E2 Dof2) %. +%term _ + %pi (preservation (of/app Dof2 Dof1) (step/app/fun Dstep1) (of/app Dof2 Dof1')) + %<- (preservation Dof1 Dstep1 Dof1') %. +%term _ + %pi (preservation (of/app Dof2 Dof1) (step/app/arg Dstep2 Dval) (of/app Dof2' Dof1)) + %<- (preservation Dof2 Dstep2 Dof2') %. +%term _ preservation (of/app Dof2 (of/lam Dbody)) (step/app/beta Dval1) (Dbody E2 Dof2) %. +%worlds () (preservation _ _ _) %. +%total (D1) (preservation _ D1 _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/New_language). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-next.lf b/new-tests/stelf-output/pages/popl-tutorial-next.lf new file mode 100644 index 0000000..9df1fea --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-next.lf @@ -0,0 +1,22 @@ +% +%{! If you want to read more about Twelf, you can check out the [introductions](/wiki/introductions-to-twelf/), [tutorials](/wiki/tutorials/), and [case studies](/wiki/case-studies/) on the wiki. + +We strongly recommend that you work on some of the following exercises: you'll get a flavor for the experience of defining systems and proving theorems in Twelf. If we have time at the end of the tutorial, we'll break into small groups and you can pick an interesting problem or two to work on. If you get stuck, ask a TA! Refer to the solutions to check your work (though Server OK from Twelf should be assurance enough!) Feel free to ask us for help during the rest of the conference. + +## Beginner problems (Recommended) +* [Preservation for MinML](/wiki/popl-tutorial-minml-preservation-theorem/) -- Prove type preservation for MinML. ([Solution](/wiki/popl-tutorial-minml-preservation-theorem-solution/)) +* [Sequent calculus vs. natural deduction](/wiki/popl-tutorial-sequent-vs-natural-deduction/) -- Prove the logical equivalence of Gentzen's intuitionistic sequent calculus and intuitionistic natural deduction. ([Solution](/wiki/popl-tutorial-sequent-vs-natural-deduction-solution/)) +* [Big-step evaluation vs. small-step transition semantics](/wiki/popl-tutorial-big-step-small-step/) -- Prove the operational equivalence of the big-step and small-step views of operational semantics. ([Solution](/wiki/popl-tutorial-big-step-small-step-solution/)) + +## Intermediate problems +* [Call-by-value CPS transform, with administrative redexes](/wiki/popl-tutorial-cps-problem/) -- Define an algorithm for converting lambda terms to continuation-passing style ([Solution](/wiki/popl-tutorial-cps-solutions/)) +* [Higher-order call-by-value CPS transform, no administrative redexes](/wiki/popl-tutorial-cps-problem2/) -- Define a higher-order variant of the CPS conversion that never creates administrative redexes ([Solution](/wiki/popl-tutorial-cps-solution2/)) +* [Church-Rosser](/wiki/popl-tutorial-church-rosser-problem/) -- Prove Church and Rosser's seminal result via a Diamond Lemma for the untyped lambda calculus. ([Solution](/wiki/church-rosser-w-identity-reduction/)) +* [MinML with exceptions](/wiki/popl-tutorial-exceptions-problem/) -- Prove type safety for an extension of MinML with exceptions defined using structural operational semantics with a "raises" judgement. ([Solution](/wiki/popl-tutorial-exceptions/)) +* [Lambda calculus and combinators](/wiki/popl-tutorial-typed-bracket-abstraction/) -- Explore the relation between the simply-typed lambda calculus and combinatory logic using Curry's classic bracket abstraction algorithm. ([Solution](/wiki/popl-tutorial-typed-bracket-abstraction-solution/)) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Next). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-pattern-matching.lf b/new-tests/stelf-output/pages/popl-tutorial-pattern-matching.lf new file mode 100644 index 0000000..54a0db7 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-pattern-matching.lf @@ -0,0 +1,285 @@ +% +%{! +This is a case study on pattern matching using Twelf. +!}% +% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort tp %. +%term o tp %. +%term arrow %pi tp %-> tp %-> tp %. +%term prod %pi tp %-> tp %-> tp %. +%term sum %pi tp %-> tp %-> tp %. +%sort tplist %. +%term tplist/nil tplist %. +%term tplist/cons %pi tp %-> tplist %-> tplist %. +%sort pat %. +%term pat/underscore pat %. +%term pat/pair %pi pat %-> pat %-> pat %. +%term pat/inl %pi pat %-> pat %. +%term pat/inr %pi pat %-> pat %. +%term pat/var pat %. +%term pat/as %pi pat %-> pat %-> pat %. +%sort exp %. +%sort oexp %. +% oexp N is an expression with N bound vars +%sort match %. +%term exp/unit exp %. +%term exp/lam %pi match %-> exp %. +%term exp/app %pi exp %-> exp %-> exp %. +%term exp/pair %pi exp %-> exp %-> exp %. +%term exp/inl %pi exp %-> exp %. +%term exp/inr %pi exp %-> exp %. +%term exp/handle %pi exp %-> exp %-> exp %. +%term oexp/c %pi exp %-> oexp %. +%term oexp/o %pi (%pi exp %-> oexp) %-> oexp %. +%term match/nil match %. +%term match/cons %pi pat %-> oexp %-> match %-> match %. +%{! static semantics !}% +%sort tplist-append %. +%term tplist-append/nil tplist-append tplist/nil TL TL %. +%term tplist-append/cons + %pi (tplist-append (tplist/cons T TL) TL' (tplist/cons T TL'')) + %<- (tplist-append TL TL' TL'') %. +%sort tplist-get %. +%term tplist-get/hit tplist-get z (tplist/cons T TL) T %. +%term tplist-get/miss %pi (tplist-get (s N) (tplist/cons T TL) T') %<- (tplist-get N TL T') %. +%sort of-pat %. +%term of-pat/underscore of-pat pat/underscore T tplist/nil %. +%term of-pat/pair + %pi (of-pat (pat/pair P1 P2) (prod T1 T2) TL') + %<- (of-pat P1 T1 TL1) + %<- (of-pat P2 T2 TL2) + %<- (tplist-append TL1 TL2 TL') %. +%term of-pat/inl %pi (of-pat (pat/inl P) (sum T1 T2) TL) %<- (of-pat P T1 TL) %. +%term of-pat/inr %pi (of-pat (pat/inr P) (sum T1 T2) TL) %<- (of-pat P T2 TL) %. +%term of-pat/var of-pat pat/var T (tplist/cons T tplist/nil) %. +%term of-pat/as + %pi (of-pat (pat/as P1 P2) T TL3) + %<- (of-pat P1 T TL1) + %<- (of-pat P2 T TL2) + %<- (tplist-append TL1 TL2 TL3) %. +%sort of-exp %. +%sort of-match %. +%sort of-oexp %. +%term of-oexp/c %pi (of-oexp tplist/nil (oexp/c E) T) %<- (of-exp E T) %. +%term of-oexp/o + %pi (of-oexp (tplist/cons T TL) (oexp/o ([x] EL x)) T') + %<- ({x} %pi (of-exp x T) %-> (of-oexp TL (EL x) T')) %. +%term of-match/nil of-match match/nil T T' %. +%term of-match/cons + %pi (of-match (match/cons P OE M) T T') + %<- (of-pat P T TL) + %<- (of-oexp TL OE T') + %<- (of-match M T T') %. +%term of-exp/unit of-exp exp/unit o %. +%term of-exp/lam %pi (of-exp (exp/lam M) (arrow T1 T2)) %<- (of-match M T1 T2) %. +%term of-exp/app %pi (of-exp (exp/app E1 E2) T2) %<- (of-exp E1 (arrow T1 T2)) %<- (of-exp E2 T1) %. +%term of-exp/pair %pi (of-exp (exp/pair E1 E2) (prod T1 T2)) %<- (of-exp E1 T1) %<- (of-exp E2 T2) %. +%term of-exp/inl %pi (of-exp (exp/inl E) (sum T1 T2)) %<- (of-exp E T1) %. +%term of-exp/inr %pi (of-exp (exp/inr E) (sum T1 T2)) %<- (of-exp E T2) %. +%term of-exp/handle %pi (of-exp (exp/handle E1 E2) T) %<- (of-exp E1 T) %<- (of-exp E2 T) %. +% syntax only needed for dynamic semantics of the language +%sort explist %. +%term explist/nil explist %. +%term explist/cons %pi exp %-> explist %-> explist %. +% static semantics of explists +%sort of-explist %. +%term of-explist/nil of-explist explist/nil tplist/nil %. +%term of-explist/cons + %pi (of-explist (explist/cons E EL) (tplist/cons T TL)) + %<- (of-exp E T) + %<- (of-explist EL TL) %. +%sort subst-oexp %. +%term subst-oexp/c subst-oexp explist/nil (oexp/c E) E %. +%term subst-oexp/o + %pi (subst-oexp (explist/cons E EL) (oexp/o ([x] OE x)) E') + %<- (subst-oexp EL (OE E) E') %. +%sort explist-get %. +%term explist-get/hit explist-get z (explist/cons T TL) T %. +%term explist-get/miss %pi (explist-get (s N) (explist/cons T TL) T') %<- (explist-get N TL T') %. +%sort explist-append %. +%term explist-append/nil explist-append explist/nil EL EL %. +%term explist-append/cons + %pi (explist-append (explist/cons E EL) EL' (explist/cons E EL'')) + %<- (explist-append EL EL' EL'') %. +%sort apply-pat %. +% only have to define failure over well-typed pattern/expression pairs +% failures arise from inl/inr mismatches +%sort fail-pat %. +%term apply-pat/underscore apply-pat pat/underscore E explist/nil %. +%term apply-pat/pair + %pi (apply-pat (pat/pair P1 P2) (exp/pair E1 E2) EL') + %<- (apply-pat P1 E1 EL1) + %<- (apply-pat P2 E2 EL2) + %<- (explist-append EL1 EL2 EL') %. +%term apply-pat/inl %pi (apply-pat (pat/inl P) (exp/inl E) EL) %<- (apply-pat P E EL) %. +%term apply-pat/inr %pi (apply-pat (pat/inr P) (exp/inr E) EL) %<- (apply-pat P E EL) %. +%term apply-pat/var apply-pat pat/var E (explist/cons E explist/nil) %. +%term apply-pat/as + %pi (apply-pat (pat/as P1 P2) E EL') + %<- (apply-pat P1 E EL1) + %<- (apply-pat P2 E EL2) + %<- (explist-append EL1 EL2 EL') %. +%term fail-pat/pair-1 %pi (fail-pat (pat/pair P1 P2) (exp/pair E1 E2)) %<- (fail-pat P1 E1) %. +%term fail-pat/pair-2 + %pi (fail-pat (pat/pair P1 P2) (exp/pair E1 E2)) + %<- (apply-pat P1 E1 EL) + %<- (fail-pat P2 E2) %. +%term fail-pat/inl-t %pi (fail-pat (pat/inl P) (exp/inl E)) %<- (fail-pat P E) %. +%term fail-pat/inl-f fail-pat (pat/inl P) (exp/inr E) %. +%term fail-pat/inr-t %pi (fail-pat (pat/inr P) (exp/inr E)) %<- (fail-pat P E) %. +%term fail-pat/inr-f fail-pat (pat/inr P) (exp/inl E) %. +%term fail-pat/as-1 %pi (fail-pat (pat/as P1 P2) E) %<- (fail-pat P1 E) %. +%term fail-pat/as-2 %pi (fail-pat (pat/as P1 P2) E) %<- (apply-pat P1 E EL) %<- (fail-pat P2 E) %. +%sort apply-or-fail-pat %. +%term apply-or-fail-pat/apply %pi (apply-or-fail-pat P E) %<- (apply-pat P E EL) %. +%term apply-or-fail-pat/fail %pi (apply-or-fail-pat P E) %<- (fail-pat P E) %. +%sort apply-match %. +%term apply-match/cons-1 + %pi (apply-match (match/cons P OE M) E E') + %<- (apply-pat P E EL) + %<- (subst-oexp EL OE E') %. +%term apply-match/cons-2 + %pi (apply-match (match/cons P OE M) E E') + %<- (fail-pat P E) + %<- (apply-match M E E') %. +%sort fail-match %. +%term fail-match/nil fail-match match/nil E %. +%term fail-match/cons %pi (fail-match (match/cons P OE M) E) %<- (fail-pat P E) %<- (fail-match M E) %. +%sort apply-or-fail-match %. +%term apply-or-fail-match/apply %pi (apply-or-fail-match M E) %<- (apply-match M E E') %. +%term apply-or-fail-match/fail %pi (apply-or-fail-match M E) %<- (fail-match M E) %. +%sort exception %. +%term exception/match exception %. +%sort value %. +%term value/unit value exp/unit %. +%term value/lam value (exp/lam M) %. +%term value/pair %pi (value (exp/pair E1 E2)) %<- (value E1) %<- (value E2) %. +%term value/inl %pi (value (exp/inl E)) %<- (value E) %. +%term value/inr %pi (value (exp/inr E)) %<- (value E) %. +%sort raises %. +%term raises/app-1 %pi (raises (exp/app E1 E2) X) %<- (raises E1 X) %. +%term raises/app-2 %pi (raises (exp/app E1 E2) X) %<- (value E1) %<- (raises E2 X) %. +%term raises/app-fail + %pi (raises (exp/app (exp/lam M) E2) exception/match) + %<- (value E2) + %<- (fail-match M E2) %. +%term raises/pair-1 %pi (raises (exp/pair E1 E2) X) %<- (raises E1 X) %. +%term raises/pair-2 %pi (raises (exp/pair E1 E2) X) %<- (value E1) %<- (raises E2 X) %. +%term raises/inl %pi (raises (exp/inl E) X) %<- (raises E X) %. +%term raises/inr %pi (raises (exp/inr E) X) %<- (raises E X) %. +%sort step %. +%term step/app-1 %pi (step (exp/app E1 E2) (exp/app E1' E2)) %<- (step E1 E1') %. +%term step/app-2 %pi (step (exp/app E1 E2) (exp/app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app-beta %pi (step (exp/app (exp/lam M) E) E') %<- (value E2) %<- (apply-match M E E') %. +%term step/pair-1 %pi (step (exp/pair E1 E2) (exp/pair E1' E2)) %<- (step E1 E1') %. +%term step/pair-2 %pi (step (exp/pair E1 E2) (exp/pair E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/inl %pi (step (exp/inl E) (exp/inl E')) %<- (step E E') %. +%term step/inr %pi (step (exp/inr E) (exp/inr E')) %<- (step E E') %. +%term step/handle %pi (step (exp/handle E1 E2) (exp/handle E1' E2)) %<- (step E1 E1') %. +%term step/handle-beta %pi (step (exp/handle E1 E2) E1) %<- (value E1) %. +%term step/handle-fail %pi (step (exp/handle E1 E2) E2) %<- (raises E1 X) %. +%sort preservation-subst-oexp %. +%mode preservation-subst-oexp %in %in %in %out %. +%term _ preservation-subst-oexp _ (of-oexp/c D1) subst-oexp/c D1 %. +%term _ + %pi (preservation-subst-oexp (of-explist/cons D1 D) (of-oexp/o D2) (subst-oexp/o D3) D4) + %<- (preservation-subst-oexp D1 (D2 _ D) D3 D4) %. +%worlds () (preservation-subst-oexp _ _ _ _) %. +%total (D1) (preservation-subst-oexp _ _ D1 _) %. +%sort preservation-append %. +%mode preservation-append %in %in %in %in %out %. +%term _ preservation-append _ D explist-append/nil tplist-append/nil D %. +%term _ + %pi (preservation-append (of-explist/cons D1 D) D2 (explist-append/cons D3) (tplist-append/cons D4) (of-explist/cons D5 D)) + %<- (preservation-append D1 D2 D3 D4 D5) %. +%worlds () (preservation-append _ _ _ _ _) %. +%total (D1) (preservation-append _ _ _ D1 _) %. +%sort preservation-get %. +%mode preservation-get %in %in %in %out %. +%term _ preservation-get (of-explist/cons _ D) explist-get/hit tplist-get/hit D %. +%term _ + %pi (preservation-get (of-explist/cons DL _) (explist-get/miss D') (tplist-get/miss D'') D) + %<- (preservation-get DL D' D'' D) %. +%worlds () (preservation-get _ _ _ _) %. +%total (D1) (preservation-get _ _ D1 _) %. +%sort preservation-apply-pat %. +%mode preservation-apply-pat %in %in %in %out %. +%term _ preservation-apply-pat D1 of-pat/underscore apply-pat/underscore of-explist/nil %. +%term _ + %pi (preservation-apply-pat (of-exp/pair DE2 DE1) (of-pat/pair DTA DP2 DP1) (apply-pat/pair DEA DA2 DA1) D3) + %<- (preservation-apply-pat DE1 DP1 DA1 D1) + %<- (preservation-apply-pat DE2 DP2 DA2 D2) + %<- (preservation-append D1 D2 DEA DTA D3) %. +%term _ + %pi (preservation-apply-pat DE (of-pat/as DTA DP2 DP1) (apply-pat/as DEA DA2 DA1) D3) + %<- (preservation-apply-pat DE DP1 DA1 D1) + %<- (preservation-apply-pat DE DP2 DA2 D2) + %<- (preservation-append D1 D2 DEA DTA D3) %. +%term _ + %pi (preservation-apply-pat (of-exp/inl DE) (of-pat/inl DP) (apply-pat/inl DA) D) + %<- (preservation-apply-pat DE DP DA D) %. +%term _ + %pi (preservation-apply-pat (of-exp/inr DE) (of-pat/inr DP) (apply-pat/inr DA) D) + %<- (preservation-apply-pat DE DP DA D) %. +%term _ preservation-apply-pat D1 of-pat/var apply-pat/var (of-explist/cons of-explist/nil D1) %. +%worlds () (preservation-apply-pat _ _ _ _) %. +%total (D1) (preservation-apply-pat _ _ D1 _) %. +%sort preservation-apply-match %. +%mode preservation-apply-match %in %in %in %out %. +%term _ + %pi (preservation-apply-match D1 (of-match/cons _ DOE D2) (apply-match/cons-1 DS DA) D') + %<- (preservation-apply-pat D1 D2 DA D) + %<- (preservation-subst-oexp D DOE DS D') %. +%term _ + %pi (preservation-apply-match D1 (of-match/cons D2 _ _) (apply-match/cons-2 DA _) D') + %<- (preservation-apply-match D1 D2 DA D') %. +%worlds () (preservation-apply-match _ _ _ _) %. +%total (D1) (preservation-apply-match _ _ D1 _) %. +%sort preservation %. +%mode preservation %in %in %out %. +%term _ + %pi (preservation (of-exp/app D2 D1) (step/app-1 DS) (of-exp/app D2 D1')) + %<- (preservation D1 DS D1') %. +%term _ + %pi (preservation (of-exp/app D2 D1) (step/app-2 DS V) (of-exp/app D2' D1)) + %<- (preservation D2 DS D2') %. +%term _ + %pi (preservation (of-exp/app D2 (of-exp/lam D1)) (step/app-beta DA _) D) + %<- (preservation-apply-match D2 D1 DA D) %. +%term _ + %pi (preservation (of-exp/pair D2 D1) (step/pair-1 DS) (of-exp/pair D2 D1')) + %<- (preservation D1 DS D1') %. +%term _ + %pi (preservation (of-exp/pair D2 D1) (step/pair-2 DS V) (of-exp/pair D2' D1)) + %<- (preservation D2 DS D2') %. +%term _ + %pi (preservation (of-exp/handle D2 D1) (step/handle DS) (of-exp/handle D2 D1')) + %<- (preservation D1 DS D1') %. +%term _ + %pi (preservation (of-exp/inl D1) (step/inl DS) (of-exp/inl D1')) + %<- (preservation D1 DS D1') %. +%term _ + %pi (preservation (of-exp/inr D1) (step/inr DS) (of-exp/inr D1')) + %<- (preservation D1 DS D1') %. +%term _ + %pi (preservation (of-exp/handle D2 D1) (step/handle DS) (of-exp/handle D2 D1')) + %<- (preservation D1 DS D1') %. +%term _ preservation (of-exp/handle D2 D1) (step/handle-beta _) D1 %. +%term _ preservation (of-exp/handle D2 D1) (step/handle-fail _) D2 %. +%worlds () (preservation _ _ _) %. +%total (D1) (preservation _ D1 _) %. +%{! +--[DanielKLee](/wiki/user-danielklee/) 01:46, 11 October 2007 (EDT) + +TODO: Finish commentary. +\{\{stub\}\} +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Pattern_matching). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-problems.lf b/new-tests/stelf-output/pages/popl-tutorial-problems.lf new file mode 100644 index 0000000..b027778 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-problems.lf @@ -0,0 +1,35 @@ +% +%{! * [POPL Tutorial/cps](/wiki/popl-tutorial-cps/) by jcreed +* [POPL Tutorial/cps-rp](/wiki/popl-tutorial-cps-rp/) by dklee, based on code by jcreed +* [POPL Tutorial/cps-problem](/wiki/popl-tutorial-cps-problem/) by arbob, based on code by jcreed +** [POPL Tutorial/CPS_Solutions](/wiki/popl-tutorial-cps-solutions/) by arbob, based on code by jcreed +* [POPL Tutorial/cps-problem2](/wiki/popl-tutorial-cps-problem2/) by arbob, based on code by jcreed +** [POPL Tutorial/CPS_Solution2](/wiki/popl-tutorial-cps-solution2/) by arbob, based on code by jcreed +* [POPL_Tutorial/Sequent_vs_Natural_Deduction](/wiki/popl-tutorial-sequent-vs-natural-deduction/) by cmartens/wlovas +** [POPL Tutorial/Sequent vs Natural Deduction: Solution](/wiki/popl-tutorial-sequent-vs-natural-deduction-solution/) +* [POPL_Tutorial/Pattern_matching](/wiki/popl-tutorial-pattern-matching/) by dklee +* [POPL_Tutorial/Control machine](/wiki/popl-tutorial-control-machine/) by [Rob](/wiki/user-rsimmons/) +* [POPL Tutorial/Church Rosser (Problem)](/wiki/popl-tutorial-church-rosser-problem/) by [Rob](/wiki/user-rsimmons/), based on code by Karl +** [POPL Tutorial/Church Rosser](/wiki/church-rosser-w-identity-reduction/) by [Rob](/wiki/user-rsimmons/) solution to the above +* [POPL_Tutorial/Properties_of_Typing_and_Reduction](/wiki/popl-tutorial-properties-of-typing-and-reduction/) by [Todd](/wiki/user-twilson/). +* [POPL_Tutorial/Cost_semantics](/wiki/popl-tutorial-cost-semantics/) by Rob and mpa +* [POPL_Tutorial/Combinators](/wiki/popl-tutorial-combinators/) by Karl +* [Bracket abstraction](/wiki/bracket-abstraction/) by wlovas (can this be made into a problem?) +* [POPL Tutorial/Typed bracket abstraction](/wiki/popl-tutorial-typed-bracket-abstraction/) by wlovas (types make it combinatory logic) +* [POPL Tutorial/Typed bracket abstraction with equivalence](/wiki/popl-tutorial-typed-bracket-abstraction-with-equivalence/) by wlovas (full beta/eta equivalence) +* [POPL Tutorial/Big step, small step](/wiki/popl-tutorial-big-step-small-step/) by wlovas +** [POPL Tutorial/Big step, small step: Solution](/wiki/popl-tutorial-big-step-small-step-solution/) +* [POPL Tutorial/Exceptions-problem](/wiki/popl-tutorial-exceptions-problem/) by dklee +** [POPL Tutorial/Exceptions](/wiki/popl-tutorial-exceptions/) by dklee, a solution to the above problem. +* [POPL Tutorial/Evaluation_Contexts_Intrinsic](/wiki/popl-tutorial-evaluation-contexts-intrinsic/) by drl + +* CPS variations (Dklee and arbob are working on this for wednesday) +* Stack machine vs. SOS vs. evaluation (Will and Chris are working on this for Weds) +* Authorization logic +* ML5 !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Problems). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-properties-of-typing-and-reduction.lf b/new-tests/stelf-output/pages/popl-tutorial-properties-of-typing-and-reduction.lf new file mode 100644 index 0000000..678fbd3 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-properties-of-typing-and-reduction.lf @@ -0,0 +1,392 @@ +% +%{! This problem (or problems) involves proving three properties of typing and reduction for the language L\{num str\} +from Bob Harper's book, [http://www.cs.cmu.edu/~rwh/plbook/book.pdf Practical Foundations for Programming Languages]. + +## The System L\{num str\} !}% +% Natural numbers +%sort nat %. +%term 0 nat %. +%term s %pi nat %-> nat %. +% Addition for natural numbers +%sort plus_op %. +%term plus_op/0 plus_op 0 N N %. +%term plus_op/s %pi (plus_op N M L) %-> (plus_op (s N) M (s L)) %. +% Multiplication for natural numbers +%sort times_op %. +%term times_op/0 times_op 0 N 0 %. +%term times_op/s %pi (times_op (s N) M P) %<- (times_op N M L) %<- (plus_op L M P) %. +%%%%%%%%%%%%%%%% L{num str}: Syntax %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Types +%sort typ %. +%term num typ %. +%term str typ %. +% Expressions. For simplicity, strings are just natural numbers, +% concatenation is addition, and len is the identity. +%sort exp %. +%term nume %pi nat %-> exp %. +%term stre %pi nat %-> exp %. +%term plus %pi exp %-> exp %-> exp %. +%term times %pi exp %-> exp %-> exp %. +%term cat %pi exp %-> exp %-> exp %. +%term len %pi exp %-> exp %. +%term let %pi exp %-> (%pi exp %-> exp) %-> exp %. +%%%%%%%%%%%%%%%% L{num str}: Static semantics (typing judgment) %%%%%%%%%%%%%%% +%sort of %. +%term of/stre of (stre S) str %. +%term of/nume of (nume N) num %. +%term of/plus %pi (of E1 num) %-> (of E2 num) %-> (of (plus E1 E2) num) %. +%term of/times %pi (of E1 num) %-> (of E2 num) %-> (of (times E1 E2) num) %. +%term of/cat %pi (of E1 str) %-> (of E2 str) %-> (of (cat E1 E2) str) %. +%term of/len %pi (of E1 str) %-> (of (len E1) num) %. +%term of/let + %pi (of E1 T1) + %-> ({x} %pi (of x T1) %-> (of (E2 x) T2)) + %-> (of (let E1 ([x] E2 x)) T2) %. +% Assumption block for typing judgment (coming from of/let) +%block of_bind {T typ} [x exp] [ofx of x T]%. +%%%%%%%%%%%%%%%% L{num str}: Dynamic Semantics %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Values +%sort val %. +%term val/stre val (stre N) %. +%term val/nume val (nume N) %. +% Reduction +%sort red %. +%term red/plus/num %pi (plus_op N1 N2 N) %-> (red (plus (nume N1) (nume N2)) (nume N)) %. +%term red/plus1 %pi (red E1 E1') %-> (red (plus E1 E2) (plus E1' E2)) %. +%term red/plus2 %pi (val E1) %-> (red E2 E2') %-> (red (plus E1 E2) (plus E1 E2')) %. +%term red/times/num %pi (times_op N1 N2 N) %-> (red (times (nume N1) (nume N2)) (nume N)) %. +%term red/times1 %pi (red E1 E1') %-> (red (times E1 E2) (times E1' E2)) %. +%term red/times2 %pi (val E1) %-> (red E2 E2') %-> (red (times E1 E2) (times E1 E2')) %. +%term red/cat/str %pi (plus_op S1 S2 S) %-> (red (cat (stre S1) (stre S2)) (stre S)) %. +%term red/cat1 %pi (red E1 E1') %-> (red (cat E1 E2) (cat E1' E2)) %. +%term red/cat2 %pi (val E1) %-> (red E2 E2') %-> (red (cat E1 E2) (cat E1 E2')) %. +%term red/len/str red (len (stre S)) (nume S) %. +%term red/len %pi (red E1 E1') %-> (red (len E1) (len E1')) %. +%term red/let/val %pi (val E1) %-> (red (let E1 ([x] E2 x)) (E2 E1)) %. +%term red/let %pi (red E1 E1') %-> (red (let E1 ([x] E2 x)) (let E1' ([x] E2 x))) %. +%{! ## Problem 1: Unicity of Typing !}% +%%%%%%%%%%%%%%%% Lemma 9.1 (Unicity of typing) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% + +% If G |- E : T1 and G |- E : T2, then T1 = T2. +% Equality of types +%sort eqtyp %. +%term refl_t eqtyp T T %. +% Congruence for typing assumptions. Although the proof doesn't +% introduce any parameters or hypotheses, it is used by a theorem that +% does, and so we need to check it in the extended worlds. +%sort cong_of %. +%mode cong_of %in %in %out %. +%term _ cong_of refl_t OE1 OE1 %. +%worlds (of_bind) (cong_of _ _ _) %. +%total OE1 (cong_of _ OE1 _) %. +% Statement and proof of Lemma. Notice that the worlds in this proof +% are an extension (technically, they subsume) the worlds in the +% congruence lemma above. +%sort lemma9-1 %. +%mode lemma9-1 %in %in %out %. +%term _ lemma9-1 of/stre of/stre refl_t %. +%term _ lemma9-1 of/nume of/nume refl_t %. +%term _ lemma9-1 (of/plus OjE11 OjE12) (of/plus OjE21 OjE22) refl_t %. +%term _ lemma9-1 (of/times OjE11 OjE12) (of/times OjE21 OjE22) refl_t %. +%term _ lemma9-1 (of/cat OjE11 OjE12) (of/cat OjE21 OjE22) refl_t %. +%term _ lemma9-1 (of/len OjE1) (of/len OjE2) refl_t %. +%term _ + %pi (lemma9-1 (of/let OjE11 ([x] [ofx1] OjE12 x ofx1)) (of/let OjE21 ([x] [ofx2] OjE22 x ofx2)) Q) + %<- (lemma9-1 OjE11 OjE21 E1) + %<- (cong_of E1 OjE12 OjE12') + %<- ({x} {ofx2} %pi (lemma9-1 ofx2 ofx2 refl_t) %-> (lemma9-1 (OjE12' x ofx2) (OjE22 x ofx2) Q)) %. +%block lemma_block {T typ} [x exp] [ofx of x T] [u lemma9-1 ofx ofx refl_t]%. +%worlds (lemma_block) (lemma9-1 _ _ _) %. +%total OjT2 (lemma9-1 _ OjT2 _) %. +%{! ## Problem 2: Determinacy of Reduction !}% +%%%%%%%%%%%%%%%% Lemma 10.1 (Determinacy of Reduction) %%%%%%%%%%%%%%%%%%%%%%% +% + +% If E |-> E' and E |-> E'', then E' = E''. +% Equality and congruence for nats +%sort eqnat %. +%term refl_n eqnat N N %. +%sort cong_s %. +%mode cong_s %in %out %. +%term _ cong_s refl_n refl_n %. +%worlds () (cong_s _ _) %. +%total {} (cong_s _ _) %. +% Equality for expressions +%sort eqexp %. +%term refl_e eqexp E E %. +% Congruences for expressions +%sort cong_nume %. +%mode cong_nume %in %out %. +%term _ cong_nume refl_n refl_e %. +%worlds () (cong_nume _ _) %. +%total E (cong_nume E _) %. +%sort cong_stre %. +%mode cong_stre %in %out %. +%term _ cong_stre refl_n refl_e %. +%worlds () (cong_stre _ _) %. +%total E (cong_stre E _) %. +%sort cong_plus1 %. +%mode {%in E1 exp} {%in E2 exp} {%in F exp} {%in E eqexp E1 E2} {%out E' eqexp (plus E1 F) (plus E2 F)} cong_plus1 E E' %. +%term _ cong_plus1 refl_e refl_e %. +%worlds () (cong_plus1 _ _) %. +%total E (cong_plus1 E _) %. +%sort cong_plus2 %. +%mode {%in F1 exp} {%in F2 exp} {%in E exp} {%in E1 eqexp F1 F2} {%out E' eqexp (plus E F1) (plus E F2)} cong_plus2 E1 E' %. +%term _ cong_plus2 refl_e refl_e %. +%worlds () (cong_plus2 _ _) %. +%total E (cong_plus2 E _) %. +%sort cong_times1 %. +%mode {%in E1 exp} {%in E2 exp} {%in F exp} {%in E eqexp E1 E2} {%out E' eqexp (times E1 F) (times E2 F)} cong_times1 E E' %. +%term _ cong_times1 refl_e refl_e %. +%worlds () (cong_times1 _ _) %. +%total E (cong_times1 E _) %. +%sort cong_times2 %. +%mode {%in F1 exp} {%in F2 exp} {%in E exp} {%in E2 eqexp F1 F2} {%out E' eqexp (times E F1) (times E F2)} cong_times2 E2 E' %. +%term _ cong_times2 refl_e refl_e %. +%worlds () (cong_times2 _ _) %. +%total E (cong_times2 E _) %. +%sort cong_cat1 %. +%mode {%in E1 exp} {%in E2 exp} {%in F exp} {%in E eqexp E1 E2} {%out E' eqexp (cat E1 F) (cat E2 F)} cong_cat1 E E' %. +%term _ cong_cat1 refl_e refl_e %. +%worlds () (cong_cat1 _ _) %. +%total E (cong_cat1 E _) %. +%sort cong_cat2 %. +%mode {%in F1 exp} {%in F2 exp} {%in E exp} {%in E3 eqexp F1 F2} {%out E' eqexp (cat E F1) (cat E F2)} cong_cat2 E3 E' %. +%term _ cong_cat2 refl_e refl_e %. +%worlds () (cong_cat2 _ _) %. +%total E (cong_cat2 E _) %. +%sort cong_len %. +%mode cong_len %in %out %. +%term _ cong_len refl_e refl_e %. +%worlds () (cong_len _ _) %. +%total E (cong_len E _) %. +%sort cong_let %. +%mode {%in E1 exp} {%in E2 exp} {%in F %pi exp %-> exp} {%in E eqexp E1 E2} {%out E' eqexp (let E1 ([x exp] F x)) (let E2 ([x exp] F x))} cong_let E E' %. +%term _ cong_let refl_e refl_e %. +%worlds () (cong_let _ _) %. +%total E (cong_let E _) %. +% Uniqueness of plus +%sort plus! %. +%mode plus! %in %in %out %. +%term _ plus! plus_op/0 plus_op/0 refl_n %. +%term _ + %pi (plus! (plus_op/s Pj1) (plus_op/s Pj2) ES) + %<- (plus! Pj1 Pj2 EL) + %<- (cong_s EL ES) %. +%worlds () (plus! _ _ _) %. +%total Pj1 (plus! Pj1 _ _) %. +% Congruence of plus +%sort plus_cong %. +%mode plus_cong %in %in %in %out %. +%term _ plus_cong refl_n plus_op/0 plus_op/0 refl_n %. +%term _ + %pi (plus_cong refl_n (plus_op/s Pj1) (plus_op/s Pj2) ES) + %<- (plus_cong refl_n Pj1 Pj2 EL) + %<- (cong_s EL ES) %. +%worlds () (plus_cong _ _ _ _) %. +%total PjL (plus_cong _ PjL _ _) %. +% Uniqueness of times +%sort times! %. +%mode times! %in %in %out %. +%term _ times! times_op/0 times_op/0 refl_n %. +%term _ + %pi (times! (times_op/s Pj1 Tj1) (times_op/s Pj2 Tj2) ES) + %<- (times! Tj1 Tj2 EL) + %<- (plus_cong EL Pj1 Pj2 ES) %. +%worlds () (times! _ _ _) %. +%total Tj1 (times! Tj1 _ _) %. +% Proof of Lemma 10.1 +%sort lemma10-1 %. +%mode lemma10-1 %in %in %out %. +%term _ + %pi (lemma10-1 (red/plus/num PjN1) (red/plus/num PjN2) EE') + %<- (plus! PjN1 PjN2 EL) + %<- (cong_nume EL EE') %. +%term _ + %pi (lemma10-1 (red/plus1 RjE1) (red/plus1 RjE2) EE') + %<- (lemma10-1 RjE1 RjE2 EE) + %<- (cong_plus1 EE EE') %. +%term _ + %pi (lemma10-1 (red/plus2 _ RjF1) (red/plus2 _ RjF2) EE') + %<- (lemma10-1 RjF1 RjF2 EF) + %<- (cong_plus2 EF EE') %. +%term _ + %pi (lemma10-1 (red/times/num TjN1) (red/times/num TjN2) EE') + %<- (times! TjN1 TjN2 EE) + %<- (cong_nume EE EE') %. +%term _ + %pi (lemma10-1 (red/times1 RjE1) (red/times1 RjE2) EE') + %<- (lemma10-1 RjE1 RjE2 EE) + %<- (cong_times1 EE EE') %. +%term _ + %pi (lemma10-1 (red/times2 _ RjF1) (red/times2 _ RjF2) EE') + %<- (lemma10-1 RjF1 RjF2 EF) + %<- (cong_times2 EF EE') %. +%term _ + %pi (lemma10-1 (red/cat/str PjN1) (red/cat/str PjN2) EE') + %<- (plus! PjN1 PjN2 EE) + %<- (cong_stre EE EE') %. +%term _ + %pi (lemma10-1 (red/cat1 RjE1) (red/cat1 RjE2) EE') + %<- (lemma10-1 RjE1 RjE2 EE) + %<- (cong_cat1 EE EE') %. +%term _ + %pi (lemma10-1 (red/cat2 _ RjF1) (red/cat2 _ RjF2) EE') + %<- (lemma10-1 RjF1 RjF2 EF) + %<- (cong_cat2 EF EE') %. +%term _ + %pi (lemma10-1 (%the (red (len (stre S)) (nume S)) red/len/str) red/len/str EE') + %<- (cong_nume refl_n EE') %. +%term _ + %pi (lemma10-1 (red/len RjE1) (red/len RjE2) EE') + %<- (lemma10-1 RjE1 RjE2 EE) + %<- (cong_len EE EE') %. +%term _ lemma10-1 (red/let/val VjE11) (red/let/val VjE12) refl_e %. +%term _ + %pi (lemma10-1 (red/let RjE1) (red/let RjE2) EE') + %<- (lemma10-1 RjE1 RjE2 EE) + %<- (cong_let EE EE') %. +%worlds () (lemma10-1 _ _ _) %. +%total RjE (lemma10-1 RjE _ _) %. +%{! ## Problem 3: Contextual Semantics + +In this problem, we define the contextual semantics for L\{num str\} and proof that it is equivalent to (defines the same relation as) the given dynamic semantics (reduction). !}% +%%%%%%%%%%%%%%%% L{num str}: Contextual Semantics %%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Instructions: E0 ~> E0' +%sort instr %. +%term instr/plus %pi (plus_op N1 N2 N) %-> (instr (plus (nume N1) (nume N2)) (nume N)) %. +%term instr/times %pi (times_op N1 N2 N) %-> (instr (times (nume N1) (nume N2)) (nume N)) %. +%term instr/cat %pi (plus_op S1 S2 S) %-> (instr (cat (stre S1) (stre S2)) (stre S)) %. +%term instr/len instr (len (stre S)) (nume S) %. +%term instr/let %pi (val E1) %-> (instr (let E1 ([x] E2 x)) (E2 E1)) %. +% Expression contexts, represented as "expressions with holes", i.e., +% functions on expressions. +%sort ectxt %. +%term ectxt/hole ectxt ([x] x) %. +%term ectxt/plus1 %pi (ectxt ([x] C1 x)) %-> (ectxt ([x] plus (C1 x) E2)) %. +%term ectxt/plus2 %pi (val E1) %-> (ectxt ([x] C2 x)) %-> (ectxt ([x] plus E1 (C2 x))) %. +%term ectxt/times1 %pi (ectxt ([x] C1 x)) %-> (ectxt ([x] times (C1 x) E2)) %. +%term ectxt/times2 %pi (val E1) %-> (ectxt ([x] C2 x)) %-> (ectxt ([x] times E1 (C2 x))) %. +%term ectxt/cat1 %pi (ectxt ([x] C1 x)) %-> (ectxt ([x] cat (C1 x) E2)) %. +%term ectxt/cat2 %pi (val E1) %-> (ectxt ([x] C2 x)) %-> (ectxt ([x] cat E1 (C2 x))) %. +%term ectxt/len %pi (ectxt ([x] C x)) %-> (ectxt ([x] len (C x))) %. +%term ectxt/let %pi (ectxt ([x] C1 x)) %-> (ectxt ([x] let (C1 x) ([y] E2 y))) %. +% Filling the hole of an expression context with an expression +%sort fill %. +%term fill/hole fill ectxt/hole E E %. +%term fill/plus1 + %pi (fill ECC1 E C1@E) + %-> (fill (%the (ectxt ([x] plus (C1 x) E2)) (ectxt/plus1 ECC1)) E (plus C1@E E2)) %. +%term fill/plus2 {VjE1 val E1} %pi (fill ECC2 E C2@E) %-> (fill (ectxt/plus2 VjE1 ECC2) E (plus E1 C2@E)) %. +%term fill/times1 + %pi (fill ECC1 E C1@E) + %-> (fill (%the (ectxt ([x] times (C1 x) E2)) (ectxt/times1 ECC1)) E (times C1@E E2)) %. +%term fill/times2 {VjE1 val E1} %pi (fill ECC2 E C2@E) %-> (fill (ectxt/times2 VjE1 ECC2) E (times E1 C2@E)) %. +%term fill/cat1 + %pi (fill ECC1 E C1@E) + %-> (fill (%the (ectxt ([x] cat (C1 x) E2)) (ectxt/cat1 ECC1)) E (cat C1@E E2)) %. +%term fill/cat2 {VjE1 val E1} %pi (fill ECC2 E C2@E) %-> (fill (ectxt/cat2 VjE1 ECC2) E (cat E1 C2@E)) %. +%term fill/len %pi (fill ECC1 E C1@E) %-> (fill (ectxt/len ECC1) E (len C1@E)) %. +%term fill/let + %pi (fill ECC1 E C1@E) + %-> (fill (%the (ectxt ([x] let (C1 x) ([y] E2 y))) (ectxt/let ECC1)) E (let C1@E ([y] E2 y))) %. +% Contextual reduction rule +%sort cred %. +%term cred/step + %pi (fill EC E0 C@E0) + %-> (instr E0 E0') + %-> (fill EC E0' C@E0') + %-> (cred C@E0 C@E0') %. +%%%%%%%%%%%%%%%% Theorem 10.2.1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% + +% If E |-> E', then E |->c E'. +%sort thm10-2-1 %. +%mode thm10-2-1 %in %out %. +%term _ thm10-2-1 (red/plus/num PjN) (cred/step fill/hole (instr/plus PjN) fill/hole) %. +%term _ + %pi (thm10-2-1 (red/plus1 RjE1) (cred/step (fill/plus1 FjC1@E0) I (fill/plus1 FjC1@E0'))) + %<- (thm10-2-1 RjE1 (cred/step FjC1@E0 I FjC1@E0')) %. +%term _ + %pi (thm10-2-1 (red/plus2 VjE1 RjE2) (cred/step (fill/plus2 VjE1 FjC2@E0) I (fill/plus2 VjE1 FjC2@E0'))) + %<- (thm10-2-1 RjE2 (cred/step FjC2@E0 I FjC2@E0')) %. +%term _ thm10-2-1 (red/times/num TjN) (cred/step fill/hole (instr/times TjN) fill/hole) %. +%term _ + %pi (thm10-2-1 (red/times1 RjE1) (cred/step (fill/times1 FjC1@E0) I (fill/times1 FjC1@E0'))) + %<- (thm10-2-1 RjE1 (cred/step FjC1@E0 I FjC1@E0')) %. +%term _ + %pi (thm10-2-1 (red/times2 VjE1 RjE2) (cred/step (fill/times2 VjE1 FjC2@E0) I (fill/times2 VjE1 FjC2@E0'))) + %<- (thm10-2-1 RjE2 (cred/step FjC2@E0 I FjC2@E0')) %. +%term _ thm10-2-1 (red/cat/str PjN) (cred/step fill/hole (instr/cat PjN) fill/hole) %. +%term _ + %pi (thm10-2-1 (red/cat1 RjE1) (cred/step (fill/cat1 FjC1@E0) I (fill/cat1 FjC1@E0'))) + %<- (thm10-2-1 RjE1 (cred/step FjC1@E0 I FjC1@E0')) %. +%term _ + %pi (thm10-2-1 (red/cat2 VjE1 RjE2) (cred/step (fill/cat2 VjE1 FjC2@E0) I (fill/cat2 VjE1 FjC2@E0'))) + %<- (thm10-2-1 RjE2 (cred/step FjC2@E0 I FjC2@E0')) %. +%term _ thm10-2-1 red/len/str (cred/step fill/hole instr/len fill/hole) %. +%term _ + %pi (thm10-2-1 (red/len RjE1) (cred/step (fill/len FjC1@E0) I (fill/len FjC1@E0'))) + %<- (thm10-2-1 RjE1 (cred/step FjC1@E0 I FjC1@E0')) %. +%term _ thm10-2-1 (red/let/val VjE1) (cred/step fill/hole (instr/let VjE1) fill/hole) %. +%term _ + %pi (thm10-2-1 (red/let RjE1) (cred/step (fill/let FjC1@E0) I (fill/let FjC1@E0'))) + %<- (thm10-2-1 RjE1 (cred/step FjC1@E0 I FjC1@E0')) %. +%worlds () (thm10-2-1 _ _) %. +%total RjE (thm10-2-1 RjE _) %. +%%%%%%%%%%%%%%%% Theorem 10.2.2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% + +% If E |->c E', then E |-> E'. +%sort thm10-2-2 %. +%mode thm10-2-2 %in %out %. +% Lemma. If C{E0} = E, E0 ~> E0', and C{E0'} = E', then E |-> E'. +%sort lem10-2-2 %. +%mode lem10-2-2 %in %in %in %out %. +%term _ lem10-2-2 fill/hole (instr/plus PjN) fill/hole (red/plus/num PjN) %. +%term _ lem10-2-2 fill/hole (instr/times TjN) fill/hole (red/times/num TjN) %. +%term _ lem10-2-2 fill/hole (instr/cat PjN) fill/hole (red/cat/str PjN) %. +%term _ lem10-2-2 fill/hole instr/len fill/hole red/len/str %. +%term _ lem10-2-2 fill/hole (instr/let VjE1) fill/hole (red/let/val VjE1) %. +%term _ + %pi (lem10-2-2 (fill/plus1 FjC1@E0) I (fill/plus1 FjC1@E0') (red/plus1 RjE)) + %<- (lem10-2-2 FjC1@E0 I FjC1@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/plus2 VjE1 FjC2@E0) I (fill/plus2 VjE1 FjC2@E0') (red/plus2 VjE1 RjE)) + %<- (lem10-2-2 FjC2@E0 I FjC2@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/times1 FjC1@E0) I (fill/times1 FjC1@E0') (red/times1 RjE)) + %<- (lem10-2-2 FjC1@E0 I FjC1@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/times2 VjE1 FjC2@E0) I (fill/times2 VjE1 FjC2@E0') (red/times2 VjE1 RjE)) + %<- (lem10-2-2 FjC2@E0 I FjC2@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/cat1 FjC1@E0) I (fill/cat1 FjC1@E0') (red/cat1 RjE)) + %<- (lem10-2-2 FjC1@E0 I FjC1@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/cat2 VjE1 FjC2@E0) I (fill/cat2 VjE1 FjC2@E0') (red/cat2 VjE1 RjE)) + %<- (lem10-2-2 FjC2@E0 I FjC2@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/len FjC1@E0) I (fill/len FjC1@E0') (red/len RjE)) + %<- (lem10-2-2 FjC1@E0 I FjC1@E0' RjE) %. +%term _ + %pi (lem10-2-2 (fill/let FjC2@E0) I (fill/let FjC2@E0') (red/let RjE)) + %<- (lem10-2-2 FjC2@E0 I FjC2@E0' RjE) %. +%worlds () (lem10-2-2 _ _ _ _) %. +%total FjE (lem10-2-2 FjE _ _ _) %. +% Proof of main theorem +%term _ + %pi (thm10-2-2 (cred/step FjC@E0 I FjC@E0') RjE) + %<- (lem10-2-2 FjC@E0 I FjC@E0' RjE) %. +%worlds () (thm10-2-2 _ _) %. +%total {} (thm10-2-2 _ _) %. +%{! ## Twelf Output + +The `` check="true">output`` for the above. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Properties_of_Typing_and_Reduction). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-saturday.lf b/new-tests/stelf-output/pages/popl-tutorial-saturday.lf new file mode 100644 index 0000000..33b1ba7 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-saturday.lf @@ -0,0 +1,63 @@ +% +%{! ## Notes from Saturday + +Please fill out our <big>**_[https://spreadsheets.google.com/viewform?key=p0-89EvFXQhUXF827jjk_Dg feedback form]**_</big>. + +If you have questions, you can email a TA at ``twelfelf@gmail.com``. + +Check out [Session 5](/wiki/popl-tutorial-session-5/)'s material: do some practice problems, email us feedback. + +## Mechanizing Metatheory with LF and Twelf + +Do you want to learn how to use [Twelf](/) to specify, implement, and prove +properties about programming languages? + +Come to the Twelf tutorial on **January 19, 2009**, co-located with POPL 2009, in +**Savannah, Georgia**. + +Learn to: +* Represent [languages and logics](/wiki/object-logic/) in [LF](/wiki/lf/) +* Prove [metatheorem](/wiki/metatheorem/)s with [Twelf](/) +under the helpful guidance of Twelf experts. + +The tutorial will be a highly interactive introduction to LF and Twelf +aimed at programming languages researchers. No prior experience with LF +and Twelf is presumed. Participants will leave the workshop with +experience in reading and writing LF representations of programming +languages, and experience reading, writing, and debugging Twelf proofs. + +Register at the [http://www.regmaster.com/conf/popl2009.html POPL 2009 registration site]! +The early registration deadline is **December 19**. + +The tutorial is organized and presented by the CMU Principles of +Programming group. + +## Schedule + +* Session 1 (11AM-noon): [http://www.cs.cmu.edu/~drl/tmp/twelflectures.pdf Overview of LF and Twelf] +* Session 2 (12:15-1:15PM): [Working with Twelf](/wiki/popl-tutorial-session-2/) +* Session 3 (1:30PM-2:30PM): [http://www.cs.cmu.edu/~drl/tmp/twelflectures.pdf Adequacy] +* Pizza break +* Session 4 (3:00PM-4:30PM): [Type safety for MinML](/wiki/popl-tutorial-session-4/) +* Session 5 (4:45PM-6:15PM): [Work on exercies](/wiki/popl-tutorial-session-5/) + +## Get Twelf before the tutorial! + +The tutorial will be interactive, with participants writing Twelf code, so you should come with Twelf installed on your laptop. + +Pre-built binaries of Twelf are available for most operating systems through the [http://twelf.plparty.org/builds Twelf Night(ly)]. +* [http://twelf.plparty.org/builds/twelf-linux.tar.gz Linux (tgz)] +* [http://twelf.plparty.org/builds/twelf.exe Windows (exe)] +* [http://twelf.plparty.org/builds/twelf-osx-intel.dmg Mac OS 10.4 or 10.5 (Intel only) (dmg)] + +Otherwise: +* you can build Twelf from the [http://twelf.plparty.org/builds/twelf-src.tar.gz source tarball]. You will need [http://www.mlton.org MLton] or [http://www.smlnj.org sml/nj]. +* you can make yourself an account on the wiki, and do the exercises on your User:<login> page (linked at the top after you log in). + +Then see [Twelf with Emacs](/wiki/twelf-with-emacs/) for the basics of interacting with Twelf. (You can also use [Twelf without Emacs](/wiki/twelf-without-emacs/), by interacting with the Twelf server directly.) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Saturday). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-sequent-vs-natural-deduction-solution.lf b/new-tests/stelf-output/pages/popl-tutorial-sequent-vs-natural-deduction-solution.lf new file mode 100644 index 0000000..808d3db --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-sequent-vs-natural-deduction-solution.lf @@ -0,0 +1,101 @@ +% +%{! This is the solution to [this exercise](/wiki/popl-tutorial-sequent-vs-natural-deduction/). +|hidden = true !}% +%% Syntax for Propositions %% +%sort prop %. +%name prop %. +%term top prop %. +%term and %pi prop %-> prop %-> prop %. +%term imp %pi prop %-> prop %-> prop %. +%% Natural Deduction Inference Rules %% +%sort true %. +%term topI true top %. +%term andI %pi (true (and A B)) %<- (true B) %<- (true A) %. +%term andE1 %pi (true A) %<- (true (and A B)) %. +%term andE2 %pi (true B) %<- (true (and A B)) %. +%term impI %pi (true (imp A B)) %<- (%pi (true A) %-> (true B)) %. +%term impE %pi (true B) %<- (true (imp A B)) %<- (true A) %. +%% Sequent Calculus %% +%sort hyp %. +%sort conc %. +%term init %pi (conc A) %<- (hyp A) %. +%term topR conc top %. +%term andL %pi (%pi (hyp (and A B)) %-> (conc C)) %<- (%pi (hyp A) %-> (hyp B) %-> (conc C)) %. +%term andR %pi (conc (and A B)) %<- (conc B) %<- (conc A) %. +%term impL %pi (%pi (hyp (imp A B)) %-> (conc C)) %<- (conc A) %<- (%pi (hyp B) %-> (conc C)) %. +%term impR %pi (conc (imp A B)) %<- (%pi (hyp A) %-> (conc B)) %. +%term cut %pi (conc B) %<- (conc A) %<- (%pi (hyp A) %-> (conc B)) %. +%% blocks %% +%block bhyp {A prop} [H hyp A]%. +%{! + +## Translation: Natural Deduction to Sequent Calculus + +!}% +%sort nd-to-seq %. +%mode nd-to-seq %in %out %. +%scope nd-to-seq %term top nd-to-seq topI topR %. +%term andI + %pi (nd-to-seq (andI (%the (true A) DtrueA) (%the (true B) DtrueB)) (andR DconcA DconcB)) + %<- (nd-to-seq DtrueA (%the (conc A) DconcA)) + %<- (nd-to-seq DtrueB (%the (conc B) DconcB)) %. +%scope nd-to-seq %term andE1 + %pi (nd-to-seq (andE1 (%the (true (and A B)) DtrueAB)) (cut (andL ([dA hyp A] [_] init dA)) DconcAB)) + %<- (nd-to-seq DtrueAB (%the (conc (and A B)) DconcAB)) %. +%term andE2 + %pi (nd-to-seq (andE2 (%the (true (and A B)) DtrueAB)) (cut (andL ([_] [dB hyp B] init dB)) DconcAB)) + %<- (nd-to-seq DtrueAB (%the (conc (and A B)) DconcAB)) %. +%scope nd-to-seq %term impI + %pi (nd-to-seq (impI ([dA true A] %the (true B) (Dimp dA))) (impR ([hA hyp A] DconcB hA))) + %<- ({dA true A} {hA hyp A} {dtrans nd-to-seq dA (init hA)} nd-to-seq (Dimp dA) (%the (conc B) (DconcB hA))) %. +%term impE + %pi (nd-to-seq (impE (%the (true A) DtrueA) (%the (true (imp A B)) DtrueAB)) (cut (impL ([hB hyp B] init hB) DconcA) DconcAB)) + %<- (nd-to-seq DtrueA (%the (conc A) DconcA)) + %<- (nd-to-seq DtrueAB (%the (conc (imp A B)) DconcAB)) %. +%block truetohyp {A prop} [DA true A] [HA hyp A] [Dtrans nd-to-seq DA (init HA)]%. +%worlds (truetohyp) (nd-to-seq _ _) %. +%total D (nd-to-seq D _) %. +%{! + +## Translation: Sequent Calculus to Natural Deduction + +!}% +%sort hyp-to-true %. +%mode hyp-to-true %in %out %. +%block hyptotrue {A prop} [HA hyp A] [DA true A] [Dtrans hyp-to-true HA DA]%. +%worlds (hyptotrue) (hyp-to-true _ _) %. +%total {} (hyp-to-true _ _) %. +%sort seq-to-nd %. +%mode seq-to-nd %in %out %. +%scope seq-to-nd %term init + %pi (seq-to-nd (init (%the (hyp A) DhypA)) DtrueA) + %<- (hyp-to-true DhypA (%the (true A) DtrueA)) %. +%term top seq-to-nd (%the (conc top) topR) topI %. +%scope seq-to-nd %term andR + %pi (seq-to-nd (%the (conc (and A B)) (andR (%the (conc A) DconcA) (%the (conc B) DconcB))) (andI DtrueA DtrueB)) + %<- (seq-to-nd DconcA (%the (true A) DtrueA)) + %<- (seq-to-nd DconcB (%the (true B) DtrueB)) %. +%term andL + %pi (seq-to-nd (%the (conc C) (andL ([hA hyp A] [hB hyp B] %the (conc C) (DconcC hA hB)) (%the (hyp (and A B)) Hab))) (DtrueC (andE1 DtrueAB) (andE2 DtrueAB))) + %<- ({hA} {dA true A} {dtransA hyp-to-true hA dA} {hB} {dB true B} {dtransB hyp-to-true hB dB} seq-to-nd (DconcC hA hB) (%the (true C) (DtrueC dA dB))) + %<- (hyp-to-true Hab (%the (true (and A B)) DtrueAB)) %. +%scope seq-to-nd %term impR + %pi (seq-to-nd (%the (conc (imp A B)) (impR ([hA] %the (conc B) (DconcB hA)))) (impI ([dA] DtrueB dA))) + %<- ({hA} {dA true A} {dtrans hyp-to-true hA dA} seq-to-nd (DconcB hA) (%the (true B) (DtrueB dA))) %. +%term impL + %pi (seq-to-nd (impL ([hB hyp B] %the (conc C) (DconcC hB)) (%the (conc A) DconcA) (%the (hyp (imp A B)) Hab)) (DtrueC (impE DtrueA DtrueAB))) + %<- ({hB} {dB true B} {dtrans hyp-to-true hB dB} seq-to-nd (DconcC hB) (%the (true C) (DtrueC dB))) + %<- (seq-to-nd DconcA (%the (true A) DtrueA)) + %<- (hyp-to-true Hab (%the (true (imp A B)) DtrueAB)) %. +%scope seq-to-nd %term cut + %pi (seq-to-nd (cut ([hA hyp A] %the (conc B) (DconcB hA)) (%the (conc A) DconcA)) (DtrueB DtrueA)) + %<- ({hA} {dA true A} {dtrans hyp-to-true hA dA} seq-to-nd (DconcB hA) (%the (true B) (DtrueB dA))) + %<- (seq-to-nd DconcA (%the (true A) DtrueA)) %. +%worlds (hyptotrue) (seq-to-nd _ _) %. +%total D (seq-to-nd D _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Sequent_vs_Natural_Deduction:_Solution). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-sequent-vs-natural-deduction.lf b/new-tests/stelf-output/pages/popl-tutorial-sequent-vs-natural-deduction.lf new file mode 100644 index 0000000..5a05ae1 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-sequent-vs-natural-deduction.lf @@ -0,0 +1,149 @@ +% +%{! + +In this exercise, we will present rules for natural deduction and for a sequent calculus presentation of intuitionistic logic, and give an two incomplete proofs. The first establishes that derivability in natural deduction implies derivability in the sequent calculus, the second establishes the converse. + +There are two tasks: +* Complete the three cases of the natural deduction -> sequent calculus proof relating to conjunction. +* Complete the three cases of the sequent calculus -> natural deduction proof relating to implication and cut. + +The solution is [here](/wiki/popl-tutorial-sequent-vs-natural-deduction-solution/). + +## Syntax for Propositions + +!}% +%sort prop %. +%name prop %. +%term top prop %. +%term and %pi prop %-> prop %-> prop %. +%term imp %pi prop %-> prop %-> prop %. +%{! + +## Natural Deduction Inference Rules + +Natural deduction deals with a single judgment, . + +!}% +%sort true %. +%{! + +Six rules describe a natural deduction system for the language considered here. + +!}% +%term topI true top %. +%term andI %pi (true (and A B)) %<- (true B) %<- (true A) %. +%term andE1 %pi (true A) %<- (true (and A B)) %. +%term andE2 %pi (true B) %<- (true (and A B)) %. +%term impI %pi (true (imp A B)) %<- (%pi (true A) %-> (true B)) %. +%term impE %pi (true B) %<- (true (imp A B)) %<- (true A) %. +%{! + +## Sequent Calculus + +With the same grammar of propositions, we can define a sequent +calculus with different judgments for the left and the right of +the sequent ( for the left and for the right). + +!}% +%sort hyp %. +%sort conc %. +%{! + +This particular sequent calculus includes the cut rule. We can show +separately that it is possible to eliminate all uses of cut by proving +the [admissibility of cut](/wiki/admissibility-of-cut/). + +!}% +%term init %pi (conc A) %<- (hyp A) %. +%term topR conc top %. +%term andL %pi (%pi (hyp (and A B)) %-> (conc C)) %<- (%pi (hyp A) %-> (hyp B) %-> (conc C)) %. +%term andR %pi (conc (and A B)) %<- (conc B) %<- (conc A) %. +%term impL %pi (%pi (hyp (imp A B)) %-> (conc C)) %<- (conc A) %<- (%pi (hyp B) %-> (conc C)) %. +%term impR %pi (conc (imp A B)) %<- (%pi (hyp A) %-> (conc B)) %. +%term cut %pi (conc B) %<- (conc A) %<- (%pi (hyp A) %-> (conc B)) %. +%{! + +The natural deduction system exists in a world with arbitrary assumptions . + +!}% +%block bhyp {A prop} [H hyp A]%. +%{! + +## Translation: Natural Deduction to Sequent Calculus + +The following (incomplete) proof establishes that whenever a proposition is true in the natural deduction system, it is derivable as a conclusion in the sequent calculus. + +The cases for top and implication are filled in for you; all that remains is +the cases for conjunction. + +The -impI case provides and example of using a theorem case. You will need +something similar in the translation for the other direction. + +#### TASK 1: Fill in the three missing cases for conjunction + +!}% +%sort nd-to-seq %. +%mode nd-to-seq %in %out %. +%scope nd-to-seq %term top nd-to-seq topI topR %. +%term andI nd-to-seq (andI (%the (true A) DtrueA) (%the (true B) DtrueB)) XXX %. +% fill in here. +% fill in here. +%scope nd-to-seq %term andE1 nd-to-seq (andE1 (%the (true (and A B)) DtrueAB)) XXX %. +%term andE2 nd-to-seq (andE2 (%the (true (and A B)) DtrueAB)) XXX %. +% fill in here. +%scope nd-to-seq %term impI + %pi (nd-to-seq (impI ([dA true A] %the (true B) (Dimp dA))) (impR ([hA hyp A] DconcB hA))) + %<- ({dA true A} {hA hyp A} {dtrans nd-to-seq dA (init hA)} nd-to-seq (Dimp dA) (%the (conc B) (DconcB hA))) %. +%term impE + %pi (nd-to-seq (impE (%the (true A) DtrueA) (%the (true (imp A B)) DtrueAB)) (cut (impL ([hB hyp B] init hB) DconcA) DconcAB)) + %<- (nd-to-seq DtrueA (%the (conc A) DconcA)) + %<- (nd-to-seq DtrueAB (%the (conc (imp A B)) DconcAB)) %. +%block truetohyp {A prop} [DA true A] [HA hyp A] [Dtrans nd-to-seq DA (init HA)]%. +%worlds (truetohyp) (nd-to-seq _ _) %. +%total D (nd-to-seq D _) %. +%{! + +## Translation: Sequent Calculus to Natural Deduction + + +The following (incomplete) proof establishes that whenever a proposition is derivable as a conclusion in the sequent calculus, it is true in the natural deduction system. + +#### TASK 2: Complete the theorem with cases for implication and cut + +!}% +%sort hyp-to-true %. +%mode hyp-to-true %in %out %. +%block hyptotrue {A prop} [HA hyp A] [DA true A] [Dtrans hyp-to-true HA DA]%. +%worlds (hyptotrue) (hyp-to-true _ _) %. +%total {} (hyp-to-true _ _) %. +%sort seq-to-nd %. +%mode seq-to-nd %in %out %. +%scope seq-to-nd %term init + %pi (seq-to-nd (init (%the (hyp A) DhypA)) DtrueA) + %<- (hyp-to-true DhypA (%the (true A) DtrueA)) %. +%term top seq-to-nd (%the (conc top) topR) topI %. +%scope seq-to-nd %term andR + %pi (seq-to-nd (%the (conc (and A B)) (andR (%the (conc A) DconcA) (%the (conc B) DconcB))) (andI DtrueA DtrueB)) + %<- (seq-to-nd DconcA (%the (true A) DtrueA)) + %<- (seq-to-nd DconcB (%the (true B) DtrueB)) %. +%term andL + %pi (seq-to-nd (%the (conc C) (andL ([hA hyp A] [hB hyp B] %the (conc C) (DconcC hA hB)) (%the (hyp (and A B)) Hab))) (DtrueC (andE1 DtrueAB) (andE2 DtrueAB))) + %<- ({hA} {dA true A} {dtransA hyp-to-true hA dA} {hB} {dB true B} {dtransB hyp-to-true hB dB} seq-to-nd (DconcC hA hB) (%the (true C) (DtrueC dA dB))) + %<- (hyp-to-true Hab (%the (true (and A B)) DtrueAB)) %. +%% fill in here +%scope seq-to-nd %term impR seq-to-nd (%the (conc (imp A B)) (impR ([hA] %the (conc B) (DconcB hA)))) XXX %. +%term impL seq-to-nd (impL ([hB hyp B] %the (conc C) (DconcC hB)) (%the (conc A) DconcA) (%the (hyp (imp A B)) Hab)) XXX %. +%% fill in here +%% fill in here +%scope seq-to-nd %term cut seq-to-nd (cut ([hA hyp A] %the (conc B) (DconcB hA)) (%the (conc A) DconcA)) XXX %. +%worlds (hyptotrue) (seq-to-nd _ _) %. +%total D (seq-to-nd D _) %. +%{! +[Solution](/wiki/popl-tutorial-sequent-vs-natural-deduction-solution/) +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Sequent_vs_Natural_Deduction). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-session-2-answer.lf b/new-tests/stelf-output/pages/popl-tutorial-session-2-answer.lf new file mode 100644 index 0000000..f9a3069 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-session-2-answer.lf @@ -0,0 +1,482 @@ +% +%{! + +## Arithmetic primitives + +!}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort add %. +%mode add %in %in %out %. +%term add/z add z N N %. +%term add/s %pi (add (s M) N (s P)) %<- (add M N P) %. +%worlds () (add _ _ _) %. +%total M (add M _ _) %. +%{! + +## Multiplication + +(Interactively) First, let's write multiplication: + +Answer: + +!}% +%sort mult %. +%term mult/z mult z N z %. +%term mult/s %pi (mult (s M) N Q) %<- (mult M N P) %<- (add P N Q) %. +%{! + +Now let's check that multiplication is total. + +First, we specify the mode: which arguments are inputs, and which +arguments are outputs? + +Answer: + +!}% +%mode mult %in %in %out %. +%{! + ++ means input (univeral); - means output (existential). + +Next the world: + +!}% +%worlds () (mult _ _ _) %. +%{! + +This means we consider terms of type mult in the empty LF context only. + +Finally, we ask Twelf to check that it is total by induction over the +first argument: + +!}% +%total M (mult M _ _) %. +%{! + +What does this really mean? "For any two LF terms M and N in the empty +context, there exists a term P and a deriation of ``mult M N P`` in +the empty context." + +## How Twelf checks assertions + +Twelf proves a totality assertion for a type family such as +``mult`` by checking several properties. These properties, taken +together, constitute a proof by induction on canonical forms that the +type family defines a total relation. + +### Mode + +Twelf checks that each constant inhabiting the type family is +_well-moded_. Roughly, this means that the inputs to the conclusion +of a constant determine the inputs of the first premise, and that these +together with the outputs of the first premise determine the inputs of +the second premise, and so on, until the outputs of all the premises +determine the outputs of the conclusion. + +<twelf name="base" hidden="true"> +%% so we don't get freezing errors +nat : type. +z : nat. +s : nat -> nat. + +add : nat -> nat -> nat -> type. +%mode add +M +N -P. + +add/z : add z N N. +add/s : add (s M) N (s P) <- add M N P. + +%worlds () (add _ _ _). +%total M (add M _ _). +</twelf> + +<twelf name="m" hidden="true"> +mult : nat -> nat -> nat -> type. +%mode mult +M +N -P. + +mult/z : mult z N z. +mult/s : mult (s M) N Q + <- mult M N P + <- add P N Q. +</twelf> + +For example, the constant +<twelf name="examplesucc" ignore="true"> +mult/s : mult (s M) N Q + <- mult M N P + <- add P N Q. +</twelf> + +has mode ``+M +N -P`` because the input ``M`` and ``N`` +in the conclusion determine the inputs of the premise, and the +``P`` output by the premise determines the first input to the second +premise (add), and the output of that determines the conclusion. +On the other hand, a constant + +<twelf include="m" name="badmode" check="decl"> +mult-bad-mode : mult M N P. +</twelf> +is not well-moded---the output ``P`` is not determined by the +inputs. Similarly, +<twelf include="m" name="badmode2" check="decl"> +mult-badmode2 : mult N1 N2 N3 + <- mult N4 N2 N3. +</twelf> +is not well-moded---first input to the premise is not determined +by the inputs of the conclusion. + +Subgoal order matters: + +<twelf include="m" name="subgoal" check="decl"> +mult/s : mult (s M) N Q + <- add P N Q + <- mult M N P. +</twelf> + +### Worlds + +Twelf checks that each constant inhabiting the type family obeys the +worlds declaration. Because we are only proving theorems about closed +terms right now, we will not run across any problems with world +checks. + +### Termination + +Twelf checks that each constant inhabiting the type family obeys the +induction order specified in the ``%total`` declaration. In each +inductive premise of a constant, the specified induction position must +be a strict subterm of the corresponding argument in the conclusion. +For example, the constant + +<twelf name="termination" ignore="true"> +mult/s : mult (s M) N Q + <- mult M N P + <- add P N Q. +</twelf> +obeys the induction order ``M`` specified in the above totality +assertion because the term ``M`` is a strict subterm of the term +``(s M)``. + +On the other hand, Twelf would not accept the totality of ``mult`` +if ``N`` were used as the induction order—the same term +``N`` in the conclusion of this constant appears in the premise: + +<twelf include="m" name="badtotal" check="decl"> +%total N (mult _ N _). +</twelf> + +In addition to the subterm ordering on a single argument, Twelf supports +[mutual induction](/wiki/mutual-induction/) and [[lexicographic induction]]. + +### Output coverage + +In the definition of a type family, you may pattern-match the outputs of +a premise. For example, we might write +<twelf include="m" name="badoutput" check="decl"> +mult-bad-output : mult (s N1) N2 (s (s N3)) + <- mult N1 N2 (s N3). +%worlds () (mult _ _ _). +%total N1 (mult N1 N2 N3). +</twelf> + +Here we have insisted that the output of the premise has the form ``s +N3`` for some ``N3``. Twelf correctly reports an output +coverage error because this condition can fail.. + +Pattern-matching the output of a premise is like an [[inversion]] step +in a proof: you're insisting that the premise derivation must conclude a +particular fact that is more specific than the judgement form itself. +For Twelf to accept a relation as total, Twelf must notice that all of +these inversions are permissible. Twelf permits such inversions when it +is readily apparent that they are justified, and those inversions that +Twelf does not accept can be proved explicitly. + +In this example, we got an output coverage error because we constrained +the output of the premise by insisting it be formed by a particular +constant. The other way to get output coverage errors is to insist that +the output of a premise be a variable that occurs elsewhere in the type. +For example: + +<twelf include="m" name="badoutputfreeness" check="decl"> +mult-bad-output-freeness : mult (s N1) N2 (s N2) + <- mult N1 N2 N2. +%worlds () (mult _ _ _). +%total N1 (mult N1 N2 N3). +</twelf> + +Here, we insisted that the output of the premise be the number +``N2`` that we put in. Twelf is very conservative in checking +[output freeness](/wiki/output-freeness/): a type family will not be judged total if you +constrain the outputs of any premise at all in this manner. + +### Input coverage + +Mode, worlds, termination, and output coverage ensure that each constant +really does cover the part of the relation indicated by its conclusion. +For example, if ``mult`` passes these four checks, we know that +``mult/z`` and ``mult/s`` cover ``(z, N, _)`` and ``(s N1, +N2, _)``, respectively. What else is necessary to know that +``mult`` defines a total relation? We need to know that all the +constants inhabiting ``mult``, taken together, cover all of the +inputs. Input coverage checks exactly this. + +For example, if we forgot ``mult/z``, input coverage for +``mult`` would fail. For example: + +<twelf include="base" name="inputcov" check="decl"> +mult' : nat -> nat -> nat -> type. +%mode mult' +N1 +N2 -X3. + +mult'/s : mult' (s M) N Q + <- mult' M N P + <- add P N Q. + +%worlds () (mult' _ _ _). +%total M (mult' M _ _). +</twelf> + +Here's an analogy that might be helpful: You can think of each constant +of a type as being a clause in an ML pattern matching declaration. Then +input coverage is like the exhaustiveness checker for pattern matching. + +Twelf checks input coverage by [[splitting]] the input types to +case-analyze the various constants that could have been used to inhabit +them. For ``plus``, Twelf splits the first ``nat`` argument +``N1``, and then checks that the cases ``plus z N2 N2`` and +``plus (s N1) N2 N3`` are covered. Fortunately, these are exactly +the cases we wrote down. If we had case-analyzed further in the +definition of the judgement (e.g., if the definition of ``plus`` +case-analyzed the second argument as well), Twelf would continue +splitting the input space. Because Twelf separates termination checking +and coverage checking, the constants defining a type family do not need +to follow any particular primitive recursion schema-the constants may +pattern-match the inputs in a general manner. + +## When Twelf checks what + +To a first approximation, you can think of the ``%mode`` and ``%worlds`` declarations as specifying a totality assertion and the ``%total`` declaration as checking it. This isn't exactly how Twelf works, though: + +- When a ``%mode`` declaration is entered, Twelf checks that all previous constants inhabiting the specified type family are well-moded; further, it then mode-checks any subsequent constants inhabiting that family. +- When a ``%worlds`` declaration is entered, Twelf world-checks the type family; further, it then reports an error if any new constants contributing to the family at all are added. +- When a ``%total`` declaration is entered, Twelf checks termination, then input coverage, then output coverage. When checking output coverage, Twelf checks for unjustified constant pattern-matching in a first pass and then output freeness problems in a second pass. + +This separation allows you to, for example, check that each constant in a family is well-moded (i.e., takes specified inputs to specified outputs) without checking that the entire type family is total. You can also use the declarations [%terminates](/wiki/percent-terminates/) and [%covers](/wiki/percent-covers/) to check termination and input coverage independently. + +If any constant in a type family fails mode, worlds, or output coverage, then mode, worlds, or totality checking fails for the whole type family. One could imagine that Twelf instead would just disregard the offending constant: it is possible that the type family as a whole satisfies a totality assertion without that constant, and, in a mathematical sense, adding additional constants never invalidates the fact a totality assertion is true of a family. The reason Twelf does not work this way is that ``%total`` actually has a more specific meaning, as we discuss in the next section. + +!}% +%{! + +## Expressions from before + +!}% +%sort val %. +%term num %pi nat %-> val %. +%sort exp %. +%term ret %pi val %-> exp %. +%term plus %pi exp %-> exp %-> exp %. +%term let %pi exp %-> (%pi val %-> exp) %-> exp %. +%sort eval %. +%mode eval %in %out %. +%term eval/val eval (ret V) V %. +%term eval/plus + %pi (eval (plus E1 E2) (num N)) + %<- (eval E1 (num N1)) + %<- (eval E2 (num N2)) + %<- (add N1 N2 N) %. +%term eval/let %pi (eval (let E1 ([x] E2 x)) A) %<- (eval E1 V) %<- (eval (E2 V) A) %. +%worlds () (eval _ _) %. +%total E (eval E _) %. +%{! + +## Expressions with times + +We'll add syntax for 'times' and an evaluation rule + +!}% +%sort val %. +%term num %pi nat %-> val %. +%sort exp %. +%term ret %pi val %-> exp %. +%term plus %pi exp %-> exp %-> exp %. +%term let %pi exp %-> (%pi val %-> exp) %-> exp %. +%% NEW +%term times %pi exp %-> exp %-> exp %. +%sort eval %. +%mode eval %in %out %. +%term eval/val eval (ret V) V %. +%term eval/plus + %pi (eval (plus E1 E2) (num N)) + %<- (eval E1 (num N1)) + %<- (eval E2 (num N2)) + %<- (add N1 N2 N) %. +%term eval/let %pi (eval (let E1 ([x] E2 x)) A) %<- (eval E1 V) %<- (eval (E2 V) A) %. +%% NEW +%term eval/times + %pi (eval (times E1 E2) (num N)) + %<- (eval E1 (num N1)) + %<- (eval E2 (num N2)) + %<- (mult N1 N2 N) %. +%worlds () (eval _ _) %. +%total E (eval E _) %. +%{! + +## Expressions with pairs + +!}% +%sort val %. +%term num %pi nat %-> val %. +%% NEW +%term pair %pi val %-> val %-> val %. +%sort exp %. +%term ret %pi val %-> exp %. +%term plus %pi exp %-> exp %-> exp %. +%term let %pi exp %-> (%pi val %-> exp) %-> exp %. +%term times %pi exp %-> exp %-> exp %. +%% NEW +%term fst %pi exp %-> exp %. +%term snd %pi exp %-> exp %. +%sort eval %. +%mode eval %in %out %. +%term eval/val eval (ret V) V %. +%term eval/plus + %pi (eval (plus E1 E2) (num N)) + %<- (eval E1 (num N1)) + %<- (eval E2 (num N2)) + %<- (add N1 N2 N) %. +%term eval/let %pi (eval (let E1 ([x] E2 x)) A) %<- (eval E1 V) %<- (eval (E2 V) A) %. +%term eval/times + %pi (eval (times E1 E2) (num N)) + %<- (eval E1 (num N1)) + %<- (eval E2 (num N2)) + %<- (mult N1 N2 N) %. +%% NEW +%term eval/fst %pi (eval (fst E) V1) %<- (eval E (pair V1 V2)) %. +%% NEW +%term eval/snd %pi (eval (snd E) V2) %<- (eval E (pair V1 V2)) %. +%worlds () (eval _ _) %. +%{! + +<twelf name="evalnottotal" check="decl"> +%total E (eval E _). +</twelf> + +We get an output coverage error for ``eval-plus``. + +Why? Not every expression evaluates to a ``num``, e.g. ``(plus (ret (pair 6 7)) 8)``. + +## Typing 1 + +We'll introduce a typing judgement. + +!}% +%sort tp %. +%term natural tp %. +%term prod %pi tp %-> tp %-> tp %. +%% relates a value to a type. +%sort of-val %. +%term of-val/num of-val (num N) natural %. +%term of-val/pair %pi (of-val (pair V1 V2) (prod T1 T2)) %<- (of-val V1 T1) %<- (of-val V2 T2) %. +%% synthesis +%mode of-val %in %out %. +%worlds () (of-val _ _) %. +%sort of %. +%mode of %in %out %. +%term of/ret %pi (of (ret V) T) %<- (of-val V T) %. +%term of/plus %pi (of (plus E1 E2) natural) %<- (of E1 natural) %<- (of E2 natural) %. +%term of/times %pi (of (times E1 E2) natural) %<- (of E1 natural) %<- (of E2 natural) %. +%term of/fst %pi (of (fst E) T1) %<- (of E (prod T1 T2)) %. +%term of/snd %pi (of (snd E) T2) %<- (of E (prod T1 T2)) %. +%term of/let %pi (of (let E1 ([x] E2 x)) T) %<- (of E1 T1) %<- ({x val} of (E2 x) T) %. +%{! + +<twelf check="decl" name="badworlds"> +%worlds () (of _ _). +</twelf> + +We promised we'd stay in the empty LF context, but we don't. + +## Typing 2a + +!}% +%% relates a value to a type. +%sort of-val %. +%term of-val/num of-val (num N) natural %. +%term of-val/pair %pi (of-val (pair V1 V2) (prod T1 T2)) %<- (of-val V1 T1) %<- (of-val V2 T2) %. +%% synthesis +%mode of-val %in %out %. +%worlds () (of-val _ _) %. +%sort of %. +%term of/ret %pi (of (ret V) T) %<- (of-val V T) %. +%term of/plus %pi (of (plus E1 E2) natural) %<- (of E1 natural) %<- (of E2 natural) %. +%term of/times %pi (of (times E1 E2) natural) %<- (of E1 natural) %<- (of E2 natural) %. +%term of/fst %pi (of (fst E) T1) %<- (of E (prod T1 T2)) %. +%term of/snd %pi (of (snd E) T2) %<- (of E (prod T1 T2)) %. +%term of/let %pi (of (let E1 ([x] E2 x)) T) %<- (of E1 T1) %<- ({x val} of (E2 x) T) %. +%mode of %in %out %. +%block valb [x val]%. +%{! + +<twelf name="worldsubserror" check="decl"> +%worlds (valb) (of _ _). +</twelf> + +## Typing 2 + +!}% +%% relates a value to a type. +%sort of-val %. +%term of-val/num of-val (num N) natural %. +%term of-val/pair %pi (of-val (pair V1 V2) (prod T1 T2)) %<- (of-val V1 T1) %<- (of-val V2 T2) %. +%% synthesis +%mode of-val %in %out %. +%block valb [x val]%. +%worlds (valb) (of-val _ _) %. +%sort of %. +%term of/ret %pi (of (ret V) T) %<- (of-val V T) %. +%term of/plus %pi (of (plus E1 E2) natural) %<- (of E1 natural) %<- (of E2 natural) %. +%term of/times %pi (of (times E1 E2) natural) %<- (of E1 natural) %<- (of E2 natural) %. +%term of/fst %pi (of (fst E) T1) %<- (of E (prod T1 T2)) %. +%term of/snd %pi (of (snd E) T2) %<- (of E (prod T1 T2)) %. +%term of/let %pi (of (let E1 ([x] E2 x)) T) %<- (of E1 T1) %<- ({x val} of (E2 x) T) %. +%mode of %in %out %. +%worlds (valb) (of _ _) %. +%{! + +## Typing 3 + +<twelf check="decl" name="trytotypelet"> +example : of (let (ret (num z)) ([y] (ret y))) natural + = of/let ([x] (of/ret (XXX x))) (of/ret of-val/num). +</twelf> + +We need a derivation of ``of-val x natural``! + +!}% +%sort of-val %. +%term of-val/num of-val (num N) natural %. +%term of-val/pair %pi (of-val (pair V1 V2) (prod T1 T2)) %<- (of-val V1 T1) %<- (of-val V2 T2) %. +%mode of-val %in %out %. +%block valb {T tp} [x val] [dx of-val x T]%. +%worlds (valb) (of-val _ _) %. +%sort of %. +%term of/ret %pi (of (ret V) T) %<- (of-val V T) %. +%term of/plus %pi (of (plus E1 E2) natural) %<- (of E1 natural) %<- (of E2 natural) %. +%term of/times %pi (of (times E1 E2) natural) %<- (of E1 natural) %<- (of E2 natural) %. +%term of/fst %pi (of (fst E) T1) %<- (of E (prod T1 T2)) %. +%term of/snd %pi (of (snd E) T2) %<- (of E (prod T1 T2)) %. +%term of/let + %pi (of (let E1 ([x] E2 x)) T) + %<- (of E1 T1) + %<- ({x val} %pi (of-val x T1) %-> (of (E2 x) T)) %. +%mode of %in %out %. +%worlds (valb) (of _ _) %. +%% now we can finish it: +%define example (of (let (ret (num z)) ([y] ret y)) natural) of/let ([x] [dx of-val x natural] of/ret dx) (of/ret of-val/num) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Session_2_Answer). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-session-2-script.lf b/new-tests/stelf-output/pages/popl-tutorial-session-2-script.lf new file mode 100644 index 0000000..f4acbac --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-session-2-script.lf @@ -0,0 +1,169 @@ +% +%{! + +```twelf +% Arithmetic primitives + +nat : type. +z : nat. +s : nat -> nat. + +add : nat -> nat -> nat -> type. +%mode add +M +N -P. + +add/z : add z N N. +add/s : add (s M) N (s P) <- add M N P. + +%worlds () (add _ _ _). +%total M (add M _ _). + +% Multiplication + +mult : nat -> nat -> nat -> type. + +mult/z : mult z N z. +mult/s : mult (s M) N Q + <- mult M N P + <- add P N Q. + +%mode mult +M +N -P. +%worlds () (mult _ _ _). +%total M (mult M _ _). + +%% Mode failures: +mult-bad-mode : mult M N P. +mult-badmode2 : mult N1 N2 N3 + <- mult N4 N2 N3. +mult/s : mult (s M) N Q + <- add P N Q + <- mult M N P. + +%% Termination failure +%total N (mult _ N _). + +%% Output cov: +mult-bad-output : mult (s N1) N2 (s (s N3)) + <- mult N1 N2 (s N3). +mult-bad-output-freeness : mult (s N1) N2 (s N2) + <- mult N1 N2 N2. + +%% Input coverage: +mult' : nat -> nat -> nat -> type. +%mode mult' +N1 +N2 -X3. + +mult'/s : mult' (s M) N Q + <- mult' M N P + <- add P N Q. + +%worlds () (mult' _ _ _). +%total M (mult'' M _ _). + + +% Expressions with times, pairs + +% - Add syntax for times; eval is still total +% - Add syntax for pairs; eval is no longer total + +val : type. +num : nat -> val. +%% NEW +pair : val -> val -> val. + +exp : type. +ret : val -> exp. +plus : exp -> exp -> exp. +let : exp -> (val -> exp) -> exp. +%% NEW +times : exp -> exp -> exp. +fst : exp -> exp. +snd : exp -> exp. + +eval : exp -> val -> type. +%mode eval +E1 -E2. + +eval/val : eval (ret V) V. + +eval/plus : eval (plus E1 E2) (num N) + <- eval E1 (num N1) %% OUTPUT COV ERROR HERE + <- eval E2 (num N2) + <- add N1 N2 N. + +eval/let : eval (let E1 ([x] E2 x)) A + <- eval E1 V + <- eval (E2 V) A. + +%% NEW +eval/times : eval (times E1 E2) (num N) + <- eval E1 (num N1) + <- eval E2 (num N2) + <- mult N1 N2 N. + +eval/fst : eval (fst E) V1 + <- eval E (pair V1 V2). + +eval/snd : eval (snd E) V2 + <- eval E (pair V1 V2). + +%worlds () (eval _ _). +%% %total E (eval E _). + +% Typing judgement +% +% - Write the rules, don't add typing assumption in of/let, get a world error +% - Define block, get a world subsumption error for of-val +% - Put of-val in the right world; now of world-checks +% - Try example; get stuck +% - Add typing assumption, change of/let, finish example + +% Finished product: + +tp : type. +natural : tp. +prod : tp -> tp -> tp. + +%% relates a value to a type. +of/val : val -> tp -> type. + +of/num : of/val (num N) natural. +of/pair : of/val (pair V1 V2) (prod T1 T2) + <- of/val V1 T1 + <- of/val V2 T2. + +%% synthesis +%mode of/val +E -T. +%block ofb : some {T : tp} block {x : val} {dx : of/val x T}. +%worlds (ofb) (of/val _ _). + +of : exp -> tp -> type. + +of/ret : of (ret V) T + <- of/val V T. +of/plus : of (plus E1 E2) natural + <- of E1 natural + <- of E2 natural. +of/times : of (times E1 E2) natural + <- of E1 natural + <- of E2 natural. +of/fst : of (fst E) T1 + <- of E (prod T1 T2). +of/snd : of (snd E) T2 + <- of E (prod T1 T2). +of/let : of (let E1 ([x] E2 x)) T + <- of E1 T1 + <- ({x : val} of/val x T1 -> of (E2 x) T). + +%mode of +E -T. +%worlds (ofb) (of _ _). + + +% example : of (let (ret (num z)) ([y] (ret y))) natural +% = of/let ([x] (of/ret (XXX x))) (of/ret of/num). +example : of (let (ret (num z)) ([y] (ret y))) natural + = of/let ([x] [dx : of/val x natural] (of/ret dx)) (of/ret of/num). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Session_2_Script). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-session-2-starter.lf b/new-tests/stelf-output/pages/popl-tutorial-session-2-starter.lf new file mode 100644 index 0000000..043ec1a --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-session-2-starter.lf @@ -0,0 +1,45 @@ +% +%{! + +## Starter code for Session 2 + +## Arithmetic primitives + +!}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort add %. +%mode add %in %in %out %. +%term add/z add z N N %. +%term add/s %pi (add (s M) N (s P)) %<- (add M N P) %. +%worlds () (add _ _ _) %. +%total M (add M _ _) %. +%{! + +## Expressions + +!}% +%sort val %. +%term num %pi nat %-> val %. +%sort exp %. +%term ret %pi val %-> exp %. +%term plus %pi exp %-> exp %-> exp %. +%term let %pi exp %-> (%pi val %-> exp) %-> exp %. +%sort eval %. +%mode eval %in %out %. +%term eval/val eval (ret V) V %. +%term eval/plus + %pi (eval (plus E1 E2) (num N)) + %<- (eval E1 (num N1)) + %<- (eval E2 (num N2)) + %<- (add N1 N2 N) %. +%term eval/let %pi (eval (let E1 ([x] E2 x)) A) %<- (eval E1 V) %<- (eval (E2 V) A) %. +%worlds () (eval _ _) %. +%total E (eval E _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Session_2_Starter). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-session-2.lf b/new-tests/stelf-output/pages/popl-tutorial-session-2.lf new file mode 100644 index 0000000..23afc41 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-session-2.lf @@ -0,0 +1,9 @@ +% +%{! - Start from the [(starter code)](/wiki/popl-tutorial-session-2-starter/) +- [Notes and solutions](/wiki/popl-tutorial-session-2-answer/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Session_2). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-session-4-answer.lf b/new-tests/stelf-output/pages/popl-tutorial-session-4-answer.lf new file mode 100644 index 0000000..4e3dd39 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-session-4-answer.lf @@ -0,0 +1,100 @@ +% +%{! + +## Session 4: Type Preservation Answer + +### Syntax + +!}% +%sort tp %. +%name tp %. +%term nat tp %. +%term arr %pi tp %-> tp %-> tp %. +%sort exp %. +%name exp %. +%term z exp %. +%term fun %pi tp %-> tp %-> (%pi exp %-> exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%{! +TASK 1: +!}% +%term s %pi exp %-> exp %. +%term ifz %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%{! + +### Static Semantics + +!}% +%sort of %. +%name of %. +%term of/z of z nat %. +%term of/fun + %pi (of (fun T1 T2 ([f] [x] E f x)) (arr T1 T2)) + %<- ({f} %pi (of f (arr T1 T2)) %-> ({x} %pi (of x T1) %-> (of (E f x) T2))) %. +%term of/app %pi (of (app E1 E2) T') %<- (of E1 (arr T T')) %<- (of E2 T) %. +%{! +TASK 2: +!}% +%term of/s %pi (of (s E) nat) %<- (of E nat) %. +%term of/ifz + %pi (of (ifz E E1 ([x] E2 x)) T) + %<- (of E nat) + %<- (of E1 T) + %<- ({x} %pi (of x nat) %-> (of (E2 x) T)) %. +%{! + +### Dynamic Semantics + +!}% +%sort value %. +%name value %. +%term value/z value z %. +%term value/s %pi (value (s E)) %<- (value E) %. +%term value/fun value (fun _ _ _) %. +%sort step %. +%name step %. +%term step/app/fun %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app/arg %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app/beta-v + %pi (step (app (fun T1 T2 ([f] [x] E1 f x)) E2) (E1 (fun T1 T2 ([f] [x] E1 f x)) E2)) + %<- (value E2) %. +%{! +TASK 3: +!}% +%term step/s %pi (step (s E) (s E')) %<- (step E E') %. +%term step/ifz/arg %pi (step (ifz E E1 ([x] E2 x)) (ifz E' E1 ([x] E2 x))) %<- (step E E') %. +%term step/ifz/z step (ifz z E1 ([x] E2 x)) E1 %. +%term step/ifz/s %pi (step (ifz (s E) E1 ([x] E2 x)) (E2 E)) %<- (value E) %. +%{! + +### Preservation + +!}% +%sort preservation %. +%mode preservation %in %in %out %. +%term _ + %pi (preservation (of/app (%the (of E2 T) DofE2) (%the (of E1 (arr T T')) DofE1)) (%the (step (app E1 E2) (app E1' E2)) (step/app/fun (%the (step E1 E1') Dstep))) (of/app DofE2 DofE1')) + %<- (preservation DofE1 Dstep (%the (of E1' (arr T T')) DofE1')) %. +%term _ + %pi (preservation (of/app (%the (of E2 T) DofE2) (%the (of E1 (arr T T')) DofE1)) (%the (step (app E1 E2) (app E1 E2')) (step/app/arg (%the (step E2 E2') Dstep) _)) (of/app DofE2' DofE1)) + %<- (preservation DofE2 Dstep (%the (of E2' T) DofE2')) %. +%term _ preservation (of/app (%the (of E2 T1) DofE2) (%the (of (fun T1 T2 ([f] [x] E1 f x)) (arr T1 T2)) (of/fun (%the ({f} %pi (of f (arr T1 T2)) %-> ({x} %pi (of x T1) %-> (of (E1 f x) T2))) DofE1)))) (%the (step (app (fun T1 T2 ([f] [x] E1 f x)) E2) (E1 (fun T1 T2 ([f] [x] E1 f x)) E2)) (step/app/beta-v _)) (DofE1 (fun T1 T2 ([f] [x] E1 f x)) (of/fun DofE1) E2 DofE2) %. +%{! +TASK 4: +!}% +%term _ + %pi (preservation (of/s (%the (of E nat) Dof)) (%the (step (s E) (s E')) (step/s (%the (step E E') Dstep))) (of/s Dof')) + %<- (preservation Dof Dstep (%the (of E' nat) Dof')) %. +%term _ + %pi (preservation (of/ifz (%the ({x} %pi (of x nat) %-> (of (E2 x) T)) DofE2) (%the (of E1 T) DofE1) (%the (of E nat) DofE)) (%the (step (ifz E E1 ([x] E2 x)) (ifz E' E1 ([x] E2 x))) (step/ifz/arg (%the (step E E') Dstep))) (of/ifz DofE2 DofE1 DofE')) + %<- (preservation DofE Dstep (%the (of E' nat) DofE')) %. +%term _ preservation (of/ifz (%the ({x} %pi (of x nat) %-> (of (E2 x) T)) DofE2) (%the (of E1 T) DofE1) of/z) (%the (step (ifz z E1 ([x] E2 x)) E1) step/ifz/z) DofE1 %. +%term _ preservation (of/ifz (%the ({x} %pi (of x nat) %-> (of (E2 x) T)) DofE2) (%the (of E1 T) DofE1) (of/s (%the (of E nat) DofE))) (%the (step (ifz (s E) E1 ([x] E2 x)) (E2 E)) (step/ifz/s _)) (DofE2 E DofE) %. +%worlds () (preservation _ _ _) %. +%total D (preservation _ D _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Session_4_Answer). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-session-4-live.lf b/new-tests/stelf-output/pages/popl-tutorial-session-4-live.lf new file mode 100644 index 0000000..35c6dde --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-session-4-live.lf @@ -0,0 +1,235 @@ +% +%{! + +**_Please fill [https://spreadsheets.google.com/viewform?key=p0-89EvFXQhUXF827jjk_Dg feedback form]**_ + +```twelf +## Syntax + +%% Types: + +tp : type. %name tp T. +nat : tp. +arr : tp -> tp -> tp. + +%% Terms: + +exp : type. %name exp E. + +%% variables represented by LF variables + +z : exp. + +fun : tp -> tp -> (exp -> exp -> exp) -> exp. +app : exp -> exp -> exp. + +%% TASK 1 Answer: + +s : exp -> exp. +ifz : exp -> exp -> (exp -> exp) -> exp. +%% not: ifz : exp -> exp -> exp -> exp. + +%{! + +## Typing + +!}% + +%% Note: not total! Not all terms are well-typed +%% Don't need mode, worlds right now. +of : exp -> tp -> type. + +%% represent variables as LF variables + +of/z : of z nat. + +of/app : of (app E1 E2) T' + <- of E1 (arr T T') + <- of E2 T. + +%% really: +%% of E2 T -> of E1 (arr T T') -> of/app : of (app E1 E2) T' + +%% not eta-long: +%% of/fun' : of (fun T1 T2 E) _. + +%% eta-long: +of/fun : of (fun T1 T2 ([f] [x] E f x)) (arr T1 T2) + <- ({f : exp} of f (arr T1 T2) + -> {x : exp} of x T1 + -> of (E f x) T2). + +% of/fun : of (fun T1 T2 ([f] [x] E f x)) (arr T1 T2) +% %% make assumptions in a different order +% <- ({f : exp} {x : exp} +% -> of f (arr T1 T2) -> of x T1 +% -> of (E f x) T2). + +%% Task: give typing rules for +%% s, ifz + +%{! + e : nat +--------- +s e : nat + +e : nat +e0 : T +x exp , x : nat |- e1 : T +-------------------------- +ifz(e,e0,x.e1) : T +!}% + +of/s : of (s E) nat + <- of E nat. + +of/ifz : of (ifz E E0 ([x] Es x)) T + <- of E nat + <- of E0 T + <- ({x : exp} of x nat + -> of (Es x) T). + +%{! + +## Operational semantics + +!}% + +%% Which expressions are values? +%% Use a predicate + +value : exp -> type. +%mode value +E. + +value/z : value z. +value/s : value (s E) + <- value E. +%% value/fun : value (fun T1 T2 ([f] [x] E f x)). +value/fun : value (fun _ _ _). + +%% Small-step evaluation + +step : exp -> exp -> type. +%mode step +E -E'. + +step/app/fun : step (app E1 E2) (app E1' E2) + <- step E1 E1'. + +step/app/arg : step (app E1 E2) (app E1 E2') + <- value E1 + <- step E2 E2'. + +step/app/beta : step + (app + (fun T1 T2 ([f] [x] E1 f x)) + E2) + %% substitution = LF function application + (E1 (fun T1 T2 ([f] [x] E1 f x)) E2) + <- value E2. + + +%% Task 3: +%% (s E) takes a step if E takes a step + +step/s : step (s E) (s E') + <- step E E'. + + +%% when E takes a step then +%% ifz E _ _ takes a step + +step/ifz/arg : step + (ifz E E0 ([x] Es x)) + (ifz E' E0 ([x] Es x)) + <- step E E'. + +%% (ifz z E0 Es) steps to E0 + +step/ifz/z : step + (ifz z E0 ([x] Es x)) + E0. + +%% (ifz (s E) _ x.Es) steps to [E/x]Es +%% when E is a value + +step/ifz/s : step + (ifz (s E) E0 ([x] Es x)) + (Es E) %% substitution + <- value E. + +%{! + +## Proof of type preservation + +If e : T +and step e e' +then e' : T. + +(eval relates expressions to answers) + +Type preservation relates derivations of judgements +to other derivations + +Relate: e : T and step e e' to e' : T + +Key idea: define total relations + +!}% + +%% Type family: + +pres : of E T + -> step E E' + -> of E' T + -> type. +%mode pres +Dof +Dstep -Dof'. + +%% gives cases + +- : pres + ((of/app + (DofE2 : of E2 T2) + (DofE1 : of E1 (arr T2 T1))) + : of (app E1 E2) T1) + (step/app/fun + (Dstep : step E1 E1') + : step (app E1 E2) (app E1' E2)) + (of/app DofE2 DofE1') + <- pres DofE1 Dstep (DofE1' : of E1' (arr T2 T1)). + +- : pres + ((of/app + (DofE2 : of E2 T2) + (DofE1 : of E1 (arr T2 T1))) + : of (app E1 E2) T1) + (step/app/arg + (Dstep : step E2 E2') + (Dval : value E1) + : step (app E1 E2) (app E1 E2')) + (of/app DofE2' DofE1) + <- pres DofE2 Dstep (DofE2' : of E2' T2). + +- : pres + ((of/app + (DofE2 : of E2 T2) + ((of/fun + (DofE1 : {f} of f (arr T2 T1) + -> {x} of x T2 + -> of (E1 f x) T1)) + : of (fun T2 T1 ([f] [x] E1 f x)) (arr T2 T1))) + : of (app (fun T2 T1 ([f] [x] E1 f x)) E2) T1) + (step/app/beta + (Dval : value E2) + : step (app (fun T2 T1 ([f] [x] E1 f x)) E2) + (E1 (fun T2 T1 ([f] [x] E1 f x)) E2)) + (DofE1 (fun T2 T1 ([f] [x] E1 f x)) (of/fun DofE1) E2 DofE2). + +%worlds () (pres _ _ _). +%total D (pres _ D _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Session_4_Live). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-session-4-starter.lf b/new-tests/stelf-output/pages/popl-tutorial-session-4-starter.lf new file mode 100644 index 0000000..2470d6c --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-session-4-starter.lf @@ -0,0 +1,91 @@ +% +%{! + +## Session 4: Type Preservation Starter + +### Syntax + +!}% +%sort tp %. +%name tp %. +%term nat tp %. +%term arr %pi tp %-> tp %-> tp %. +%sort exp %. +%name exp %. +%term z exp %. +%term fun %pi tp %-> tp %-> (%pi exp %-> exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%{! +TASK 1: + +Add constants for successor and ifz: + +!}% +%{! + +### Static Semantics + +!}% +%sort of %. +%name of %. +%term of/z of z nat %. +%term of/fun + %pi (of (fun T1 T2 ([f] [x] E f x)) (arr T1 T2)) + %<- ({f} %pi (of f (arr T1 T2)) %-> ({x} %pi (of x T1) %-> (of (E f x) T2))) %. +%term of/app %pi (of (app E1 E2) T') %<- (of E1 (arr T T')) %<- (of E2 T) %. +%{! +TASK 2: + +Add two new typing rules for successor and ifz + +!}% +%{! + +### Dynamic Semantics + +!}% +%sort value %. +%name value %. +%term value/z value z %. +%{! +Task 3a: Add a value rule for successor +!}% +%term value/fun value (fun _ _ _) %. +%sort step %. +%name step %. +%term step/app/fun %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app/arg %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app/beta-v + %pi (step (app (fun T1 T2 ([f] [x] E1 f x)) E2) (E1 (fun T1 T2 ([f] [x] E1 f x)) E2)) + %<- (value E2) %. +%{! +TASK 3: + +Add four more operational semantics rules for successor and ifz. + +!}% +%{! + +### Preservation + +!}% +%sort preservation %. +%mode preservation %in %in %out %. +%term _ + %pi (preservation (of/app (%the (of E2 T) DofE2) (%the (of E1 (arr T T')) DofE1)) (%the (step (app E1 E2) (app E1' E2)) (step/app/fun (%the (step E1 E1') Dstep))) (of/app DofE2 DofE1')) + %<- (preservation DofE1 Dstep (%the (of E1' (arr T T')) DofE1')) %. +%term _ + %pi (preservation (of/app (%the (of E2 T) DofE2) (%the (of E1 (arr T T')) DofE1)) (%the (step (app E1 E2) (app E1 E2')) (step/app/arg (%the (step E2 E2') Dstep) _)) (of/app DofE2' DofE1)) + %<- (preservation DofE2 Dstep (%the (of E2' T) DofE2')) %. +%term _ preservation (of/app (%the (of E2 T1) DofE2) (%the (of (fun T1 T2 ([f] [x] E1 f x)) (arr T1 T2)) (of/fun (%the ({f} %pi (of f (arr T1 T2)) %-> ({x} %pi (of x T1) %-> (of (E1 f x) T2))) DofE1)))) (%the (step (app (fun T1 T2 ([f] [x] E1 f x)) E2) (E1 (fun T1 T2 ([f] [x] E1 f x)) E2)) (step/app/beta-v _)) (DofE1 (fun T1 T2 ([f] [x] E1 f x)) (of/fun DofE1) E2 DofE2) %. +%{! +TASK 4: Finish the proof! +!}% +%worlds () (preservation _ _ _) %. +%total D (preservation _ D _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Session_4_Starter). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-session-4.lf b/new-tests/stelf-output/pages/popl-tutorial-session-4.lf new file mode 100644 index 0000000..7a44653 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-session-4.lf @@ -0,0 +1,13 @@ +% +%{! ## Type preservation for MinML + +In this session, we will prove type preservation for MinML. + +- We recommend that you type in what we type, so your fingers get used to Twelf. But if you want to save your self some typing, start from the [(starter code)](/wiki/popl-tutorial-session-4-starter/) +- Answers to the exercises: [(answer)](/wiki/popl-tutorial-session-4-answer/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Session_4). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-session-5.lf b/new-tests/stelf-output/pages/popl-tutorial-session-5.lf new file mode 100644 index 0000000..8914c25 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-session-5.lf @@ -0,0 +1,23 @@ +% +%{! In this session, you'll get a flavor for the experience of defining systems and proving theorems in Twelf. Break into small groups and pick an interesting problem or two to work on. If you get stuck, ask a TA! Refer to the solutions to check your work (though Server OK from Twelf should be assurance enough!) + +<big>**_Please fill [https://spreadsheets.google.com/viewform?key=p0-89EvFXQhUXF827jjk_Dg feedback form]**_</big> + +## Beginner problems (Recommended) +* [Sequent calculus vs. natural deduction](/wiki/popl-tutorial-sequent-vs-natural-deduction/) -- Prove the logical equivalence of Gentzen's intuitionistic sequent calculus and intuitionistic natural deduction. ([Solution](/wiki/popl-tutorial-sequent-vs-natural-deduction-solution/)) +* [Big-step evaluation vs. small-step transition semantics](/wiki/popl-tutorial-big-step-small-step/) -- Prove the operational equivalence of the big-step and small-step views of operational semantics. ([Solution](/wiki/popl-tutorial-big-step-small-step-solution/)) + +## Intermediate problems +* [Call-by-value CPS transform, with administrative redexes](/wiki/popl-tutorial-cps-problem/) -- Define an algorithm for converting lambda terms to continuation-passing style ([Solution](/wiki/popl-tutorial-cps-solutions/)) +* [Higher-order call-by-value CPS transform, no administrative redexes](/wiki/popl-tutorial-cps-problem2/) -- Define a higher-order variant of the CPS conversion that never creates administrative redexes ([Solution](/wiki/popl-tutorial-cps-solution2/)) +* [Lambda calculus and combinators](/wiki/popl-tutorial-typed-bracket-abstraction/) -- Explore the relation between the simply-typed lambda calculus and combinatory logic using Curry's classic bracket abstraction algorithm. ([Solution](/wiki/popl-tutorial-typed-bracket-abstraction-solution/)) +* [Church-Rosser](/wiki/popl-tutorial-church-rosser-problem/) -- Prove Church and Rosser's seminal result via a Diamond Lemma for the untyped lambda calculus. ([Solution](/wiki/church-rosser-w-identity-reduction/)) + +## Advanced problems +* [MinML with exceptions](/wiki/popl-tutorial-exceptions-problem/) -- Prove type safety for an extension of MinML with exceptions defined using structural operational semantics with a "raises" judgement. ([Solution](/wiki/popl-tutorial-exceptions/)) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Session_5). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-thursday.lf b/new-tests/stelf-output/pages/popl-tutorial-thursday.lf new file mode 100644 index 0000000..5a23cdf --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-thursday.lf @@ -0,0 +1,61 @@ +% +%{! ## Mechanizing Metatheory with LF and Twelf + +Do you want to learn how to use [Twelf](/) to specify, implement, and prove +properties about programming languages? + +Come to the Twelf tutorial on **January 19, 2009**, co-located with POPL 2009, in +**Savannah, Georgia**. + +Learn to: +* Represent [languages and logics](/wiki/object-logic/) in [LF](/wiki/lf/) +* Prove [metatheorem](/wiki/metatheorem/)s with [Twelf](/) +under the helpful guidance of Twelf experts. + +The tutorial will be a highly interactive introduction to LF and Twelf +aimed at programming languages researchers. No prior experience with LF +and Twelf is presumed. Participants will leave the workshop with +experience in reading and writing LF representations of programming +languages, and experience reading, writing, and debugging Twelf proofs. + +Register at the [http://www.regmaster.com/conf/popl2009.html POPL 2009 registration site]! +The early registration deadline is **December 19**. + +The tutorial is organized and presented by the CMU Principles of +Programming group. + +## Schedule ([http://www.cs.cmu.edu/~drl/tmp/lectures.pdf all slides]) + +* Overview ([Answers](/wiki/popl-tutorial-nat/)) +** Session A1: Representation +** Session A2: Totality ([notes](/wiki/popl-tutorial-session-2-answer/)) +** Session A3: Metatheory + +* Languages ([Answers](/wiki/popl-tutorial-minml/)) +** Session B1: Representation and adequacy +** Session B2: Metatheory + +* Advanced examples +** Session C1: [Combinators](/wiki/popl-tutorial-combinators-session/) +** Session C2: [Work on exercies](/wiki/popl-tutorial-session-5/) + +## Get Twelf before the tutorial! + +The tutorial will be interactive, with participants writing Twelf code, so you should come with Twelf installed on your laptop. + +Pre-built binaries of Twelf are available for most operating systems through the [http://twelf.plparty.org/builds Twelf Night(ly)]. +* [http://twelf.plparty.org/builds/twelf-linux.tar.gz Linux (tgz)] +* [http://twelf.plparty.org/builds/twelf.exe Windows (exe)] +* [http://twelf.plparty.org/builds/twelf-osx-intel.dmg Mac OS 10.4 or 10.5 (Intel only) (dmg)] + +Otherwise: +* you can build Twelf from the [http://twelf.plparty.org/builds/twelf-src.tar.gz source tarball]. You will need [http://www.mlton.org MLton] or [http://www.smlnj.org sml/nj]. +* you can make yourself an account on the wiki, and do the exercises on your User:<login> page (linked at the top after you log in). + +Then see [Twelf with Emacs](/wiki/twelf-with-emacs/) for the basics of interacting with Twelf. (You can also use [Twelf without Emacs](/wiki/twelf-without-emacs/), by interacting with the Twelf server directly.) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Thursday). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-typed-bracket-abstraction-solution.lf b/new-tests/stelf-output/pages/popl-tutorial-typed-bracket-abstraction-solution.lf new file mode 100644 index 0000000..e76520b --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-typed-bracket-abstraction-solution.lf @@ -0,0 +1,243 @@ +% +%{! + +This is a case study translating the simply lambda calculus into S, K, and I +combinatory logic. The correctness of the translation is proven in the +following sense: if a term steps to a reduct, its translation multi-steps to +its reduct's translation. + +[William Lovas](/wiki/user-william-lovas/) + +!}% +%{! + +## Syntax and static semantics + +!}% +%{! + +First we define the syntax of the two languages. ``term A`` is the +type of simply-typed lambda-calculus terms of type A; ``cterm A`` is +the type of simply-typed combinator terms of type A. Combinators are of +interest primarily because they have no binding structure; despite this +apparent limitation, we can translate any lambda-calculus term to an +operationally and logically related combinator term. + +Note that since we're using an +[intrinsic encoding](/wiki/intrinsic-and-extrinsic-encodings/), these syntax +definitions double as definitions of the languages' static semantics. + +!}% +%sort tp %. +%term i tp %. +%term => %pi tp %-> tp %-> tp %. +%prec %right 10 => %. +%sort term %. +%name term %. +%term app %pi (term (A => B)) %-> (term A) %-> (term B) %. +%term lam %pi (%pi (term A) %-> (term B)) %-> (term (A => B)) %. +%sort cterm %. +%name cterm %. +%term s cterm (A => B => C => (A => B) => A => C) %. +%term k cterm (A => B => A) %. +%term i cterm (A => A) %. +%term capp %pi (cterm (A => B)) %-> (cterm A) %-> (cterm B) %. +%{! + +We can use Twelf's [abbreviation mechanism](/wiki/abbrev-declaration/) and +[fixity declarations](/wiki/fixity-declaration/) to obtain some cute syntax. + +!}% +%inline @ app %. +%prec %left 10 @ %. +%inline @@ capp %. +%prec %left 10 @@ %. +%{! + +## Dynamic semantics + +!}% +%{! + +Then we define reduction relations on both languages. ``step`` is the +single-step reduction relation on lambda terms; ``cstep`` is the +single-step reduction relation on combinator terms. For lambda-calculus +terms, we do not reduce under binders. + +!}% +%sort step %. +%term s-beta step (app (lam ([x] M1 x)) M2) (M1 M2) %. +%term s-1 %pi (step (app M1 M2) (app M1' M2)) %<- (step M1 M1') %. +%term s-2 %pi (step (app M1 M2) (app M1 M2')) %<- (step M2 M2') %. +%sort cstep %. +%term cs-i cstep (capp i X) X %. +%term cs-k cstep (capp (capp k X) Y) X %. +%term cs-s cstep (capp (capp (capp s X) Y) Z) (capp (capp X Z) (capp Y Z)) %. +%term cs-1 %pi (cstep (capp X Y) (capp X' Y)) %<- (cstep X X') %. +%term cs-2 %pi (cstep (capp X Y) (capp X Y')) %<- (cstep Y Y') %. +%{! + +We also define multi-step reduction on combinator terms. + +Our simulation will relate single-step derivations in the lambda-calculus +to multi-step derivations on the translated terms. + +!}% +%sort cstep* %. +%term cs-cons %pi (cstep* N N'') %<- (cstep N N') %<- (cstep* N' N'') %. +%term cs-nil cstep* N N %. +%{! + +We can pre-emptively prove some compatibility lemmas about multi-step +reduction. + +!}% +%sort cs-1* %. +%mode {%in A tp} {%in B tp} {%in N1 cterm (A => B)} {%in N1' cterm (A => B)} {%in N2 cterm A} {%in CS1 cstep* N1 N1'} {%out CS2 cstep* (capp N1 N2) (capp N1' N2)} cs-1* CS1 CS2 %. +%term _ cs-1* cs-nil cs-nil %. +%term _ %pi (cs-1* (cs-cons CS C) (cs-cons CS' (cs-1 C))) %<- (cs-1* CS CS') %. +%worlds () (cs-1* _ _) %. +%total {CS} (cs-1* CS _) %. +%sort cs-2* %. +%mode {%in A tp} {%in B tp} {%in N1 cterm (A => B)} {%in N2 cterm A} {%in N2' cterm A} {%in CS1 cstep* N2 N2'} {%out CS2 cstep* (capp N1 N2) (capp N1 N2')} cs-2* CS1 CS2 %. +%term _ cs-2* cs-nil cs-nil %. +%term _ %pi (cs-2* (cs-cons CS C) (cs-cons CS' (cs-2 C))) %<- (cs-2* CS CS') %. +%worlds () (cs-2* _ _) %. +%total {CS} (cs-2* CS _) %. +%sort cs-trans %. +%mode cs-trans %in %in %out %. +%term _ cs-trans cs-nil Cs2 Cs2 %. +%term _ %pi (cs-trans (cs-cons Cs1 C) Cs2 (cs-cons Cs12 C)) %<- (cs-trans Cs1 Cs2 Cs12) %. +%worlds () (cs-trans _ _ _) %. +%total {Cs} (cs-trans Cs _ _) %. +%{! + +## Translation + +!}% +%{! + +We can now define a compositional translation from lambda terms to combinator +terms in the standard way. The translation appeals to a function called +_bracket abstraction_ which simulates binding in the combinator calculus. + +Bracket abstraction is usually written where +is a combinator term, and is a variable potentially free in +. (Be careful not to confuse these brackets with Twelf's syntax +for lambda abstraction.) It is defined inductively over the term +: + +<blockquote> + <br> + + (where not free in )<br> + + <br> +</blockquote> + +Using it, we can define a translation on lambda-terms, , where + is a lambda-term, as follows: + +<blockquote> +<br> + = + <br> + +</blockquote> + +Note that in the definition of bracket abstraction, we need not consider a +case for lambda-abstraction since bracket abstraction works over combinator +terms, in which lambda-abstraction has already been eliminated. + +The main translation is represented in LF by a judgement (an LF type family) +``trans M N``. Bracket abstraction is represented by a judgement +relating LF-level abstractions in the combinator language to closed combinator +terms, ``bracket ([x] N x) N``. This definition is essentially an +instance of [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/). + +!}% +%sort trans %. +%sort bracket %. +%term t-app %pi (trans (app M1 M2) (capp N1 N2)) %<- (trans M1 N1) %<- (trans M2 N2) %. +%{! + +The translation on lambda-abstractions has to work under an extended context +with a lambda-term variable, a combinator-term variable, and an assumption +that the one translates to the other. See the ``%worlds`` declaration +below. + +!}% +%term t-lam + %pi (trans (lam ([x] M x)) N') + %<- ({x term A} {y cterm A} %pi (trans x y) %-> (trans (M x) (N y))) + %<- (bracket ([y] N y) N') %. +%term b-i bracket ([x] x) i %. +%term b-k bracket ([x] Y) (capp k Y) %. +%term b-s + %pi (bracket ([x] capp (N1 x) (N2 x)) (capp (capp s N1') N2')) + %<- (bracket ([x] N1 x) N1') + %<- (bracket ([x] N2 x) N2') %. +%block tbind {A tp} [x term A] [y cterm A] [dtrans trans x y]%. +%worlds (tbind) (bracket _ _) %. +%worlds (tbind) (trans _ _) %. +%{! + +## Correctness of the translation + +!}% +%{! + +First, we prove the correctness of bracket abstraction itself: +the application of a bracket abstraction reduces to a substitution. + +!}% +%sort subst %. +%mode subst %in %in %out %. +%term _ subst b-i N0 (cs-cons cs-nil cs-i) %. +%term _ subst b-k N0 (cs-cons cs-nil cs-k) %. +% developing incrementally, it's useful to write down the type of each output +%term _ + %pi (subst (b-s (%the (bracket ([x] N2 x) N2') B2) (%the (bracket ([x] N1 x) N1') B1)) N0 (cs-cons CS12' cs-s)) + %<- (subst B1 N0 (%the (cstep* (N1' @@ N0) (N1 N0)) CS1)) + %<- (subst B2 N0 (%the (cstep* (N2' @@ N0) (N2 N0)) CS2)) + %<- (cs-1* CS1 (%the (cstep* (N1' @@ N0 @@ (N2' @@ N0)) (N1 N0 @@ (N2' @@ N0))) CS1')) + %<- (cs-2* CS2 (%the (cstep* (N1 N0 @@ (N2' @@ N0)) (N1 N0 @@ N2 N0)) CS2')) + %<- (cs-trans CS1' CS2' CS12') %. +%worlds () (subst _ _ _) %. +%total {B} (subst B _ _) %. +%{! + +Then, we can prove simulation, the correctness of translation, by a +straightforward induction on single-step derivations in the lambda-calculus, +using the correctness of bracket abstraction as a lemma in the case of a +beta-reduction. + +!}% +%sort simulate %. +%mode simulate %in %in %out %out %. +%{! + +NB: in this step, on paper, you have to prove two compositionality lemmas. +in Twelf, using higher-order abstract syntax, we get them for free -- see +(T1 _ _ T2) output. + +!}% +%term _ + %pi (simulate s-beta (t-app T2 (t-lam B1 T1)) (T1 _ _ T2) CS) + %<- (subst B1 N2 CS) %. +%term _ + %pi (simulate (s-1 S1) (t-app T2 T1) (t-app T2 T1') CSapp) + %<- (simulate S1 T1 T1' CS1) + %<- (cs-1* CS1 CSapp) %. +%term _ + %pi (simulate (s-2 S2) (t-app T2 T1) (t-app T2' T1) CSapp) + %<- (simulate S2 T2 T2' CS2) + %<- (cs-2* CS2 CSapp) %. +%worlds () (simulate _ _ _ _) %. +%total D (simulate D _ _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Typed_bracket_abstraction_(solution)). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-typed-bracket-abstraction-with-equivalence.lf b/new-tests/stelf-output/pages/popl-tutorial-typed-bracket-abstraction-with-equivalence.lf new file mode 100644 index 0000000..fc40129 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-typed-bracket-abstraction-with-equivalence.lf @@ -0,0 +1,312 @@ +% +%{! + +This is a case study translating the simply lambda calculus into S, K, and I +combinatory logic. The correctness of the translation is proven in the +following sense: if a two terms are beta-eta equal, then their translations +are beta-eta equal. + +[William Lovas](/wiki/user-william-lovas/) + +!}% +%{! + +## Syntax and static semantics + +!}% +%{! + +First we define the syntax of the two languages. ``term A`` is the +type of simply-typed lambda-calculus terms of type A; ``comb A`` is +the type of simply-typed combinator terms of type A. Combinators are of +interest primarily because they have no binding structure; despite this +apparent limitation, we can translate any lambda-calculus term to an +operationally and logically related combinator term. + +Note that since we're using an +[intrinsic encoding](/wiki/intrinsic-and-extrinsic-encodings/), these syntax +definitions double as definitions of the languages' static semantics. + +!}% +%sort tp %. +%name tp %. +%term o tp %. +%term => %pi tp %-> tp %-> tp %. +%prec %right 10 => %. +%sort term %. +%name term %. +%% tm : type. %name tm M x. +%% %abbrev term : tp -> type = [x] tm. +%term app %pi (term (A => B)) %-> (term A) %-> (term B) %. +%term lam %pi (%pi (term A) %-> (term B)) %-> (term (A => B)) %. +%sort comb %. +%name comb %. +%% cm : type. %name cm N y. +%% %abbrev comb : tp -> type = [x] cm. +%term s comb (A => B => C => (A => B) => A => C) %. +%term k comb (A => B => A) %. +%term i comb (A => A) %. +%term capp %pi (comb (A => B)) %-> (comb A) %-> (comb B) %. +%{! + +We can use Twelf's [abbreviation mechanism](/wiki/abbrev-declaration/) and +[fixity declarations](/wiki/fixity-declaration/) to obtain some cute syntax. + +!}% +%inline @ app %. +%prec %left 10 @ %. +%inline @@ capp %. +%prec %left 10 @@ %. +%{! + +## Equational theory + +!}% +%{! + +Then we define equality relations on both languages. ``teq`` is +definitional equality for lambda terms; ``ceq`` is definitional +equality for combinator terms. In both cases, the equality relation +amounts to beta-eta equivalence. (We use extensionality instead of a +rule based on eta-expansion.) + +!}% +% definitional equality on terms (better name?) +%sort teq %. +%mode teq %star %star %. +% spurious mode declaration necessary? see unique-block.. +% beta +%term eq/beta teq (app (lam ([x] M1 x)) M2) (M1 M2) %. +% eta +%term eq/eta %pi (teq M1 M2) %<- ({x} teq (app M1 x) (app M2 x)) %. +% XXX interesting note: beta + ext ⊦ lam. lam + beta + eta ⊦ ext. +% others? cut out some cases? +% compatibilities +%term eq/lam %pi (teq (lam ([x] M x)) (lam ([x] M' x))) %<- ({x} teq (M x) (M' x)) %. +%term eq/app %pi (teq (app M1 M2) (app M1' M2')) %<- (teq M1 M1') %<- (teq M2 M2') %. +% equivalence +%term eq/refl teq M M %. +%term eq/symm %pi (teq M M') %<- (teq M' M) %. +%term eq/trans %pi (teq M M'') %<- (teq M' M'') %<- (teq M M') %. +%block teq-block {A tp} [x term A]%. +%worlds (teq-block) (teq _ _) %. +% definitional equality on combs +%sort ceq %. +% betas +%term ceq/i ceq (capp i N) N %. +%term ceq/k ceq (capp (capp k N1) N2) N1 %. +%term ceq/s ceq (capp (capp (capp s N1) N2) N3) (capp (capp N1 N3) (capp N2 N3)) %. +% eta +%term ceq/eta %pi (ceq N1 N2) %<- ({y} ceq (capp N1 y) (capp N2 y)) %. +% compatibility +%term ceq/app %pi (ceq (capp N1 N2) (capp N1' N2')) %<- (ceq N1 N1') %<- (ceq N2 N2') %. +% equivalence +%term ceq/refl ceq N N %. +%term ceq/symm %pi (ceq N N') %<- (ceq N' N) %. +%term ceq/trans %pi (ceq N N'') %<- (ceq N' N'') %<- (ceq N N') %. +%inline ; (%pi (ceq N N') %-> (ceq N' N'') %-> (ceq N N'')) [d1] [d2] ceq/trans d1 d2 %. +%prec %right 5 ; %. +%block ceq-block {A tp} [y comb A]%. +%worlds (ceq-block) (ceq _ _) %. +%{! + +## Translation + +!}% +%{! + +We can now define a compositional translation from lambda terms to combinator +terms in the standard way. The translation appeals to a function called +_bracket abstraction_ which simulates binding in the combinator calculus. + +Bracket abstraction is usually written where +is a combinator term, and is a variable potentially free in +. (Be careful not to confuse these brackets with Twelf's syntax +for lambda abstraction.) It is defined inductively over the term +: + +<blockquote> + <br> + + (where not free in )<br> + + <br> +</blockquote> + +Using it, we can define a translation on lambda-terms, , where + is a lambda-term, as follows: + +<blockquote> +<br> + = + <br> + +</blockquote> + +Note that in the definition of bracket abstraction, we need not consider a +case for lambda-abstraction since bracket abstraction works over combinator +terms, in which lambda-abstraction has already been eliminated. + +The main translation is represented in LF by a judgement (an LF type family) +``translate M N``. Bracket abstraction is represented by a judgement +relating LF-level abstractions in the combinator language to closed combinator +terms, ``bracket ([x] N x) N``. This definition is essentially an +instance of [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/). + +!}% +%sort translate %. +%mode translate %in %out %. +%name translate %. +%sort bracket %. +%mode bracket %in %out %. +%name bracket %. +%term t/app + %pi (translate (app M1 M2) (capp N1 N2)) + %<- (translate M1 N1) + %<- (translate M2 N2) %. +%{! + +The translation on lambda-abstractions has to work under an extended context +with a lambda-term variable, a combinator-term variable, and an assumption +that the one translates to the other. See the ``%worlds`` declaration +below. + +!}% +%term t/lam + %pi (translate (lam ([x] M x)) N*) + %<- ({x term A} {y comb A} %pi (translate x y) %-> (translate (M x) (N y))) + %<- (bracket ([y] N y) N*) %. +%term b/i bracket ([x] x) i %. +%term b/k bracket ([x] Y) (capp k Y) %. +%term b/s + %pi (bracket ([x] capp (N1 x) (N2 x)) (capp (capp s N1') N2')) + %<- (bracket ([x] N1 x) N1') + %<- (bracket ([x] N2 x) N2') %. +%block trans-block {A tp} [x term A] [y comb A] [dtrans translate x y]%. +%worlds (trans-block) (bracket _ _) %. +%worlds (trans-block) (translate _ _) %. +%{! + +Both relations are effective (though we cannot use this fact in proofs). + +!}% +%total N (bracket N _) %. +%total M (translate M _) %. +%{! + +Instead, we need effectiveness lemmas. XXX explain this better... + +!}% +%sort can-bracket {N %pi (comb A) %-> (comb B)} %. +%mode can-bracket %in %out %. +%sort can-translate {M term A} %. +%mode can-translate %in %out %. +%term _ can-bracket ([x] x) b/i %. +%term _ can-bracket ([x] N) b/k %. +%term _ + %pi (can-bracket ([x] capp (N1 x) (N2 x)) (b/s Dbrack2 Dbrack1)) + %<- (can-bracket ([x] N1 x) Dbrack1) + %<- (can-bracket ([x] N2 x) Dbrack2) %. +%term _ + %pi (can-translate (app M1 M2) (t/app Dtrans2 Dtrans1)) + %<- (can-translate M1 (%the (translate M1 N1) Dtrans1)) + %<- (can-translate M2 (%the (translate M2 N2) Dtrans2)) %. +%term _ + %pi (can-translate (lam ([x] M x)) (t/lam Dbrack Dtrans)) + %<- ({x} {y} {dtrans translate x y} {thm can-translate x dtrans} can-translate (M x) (%the (translate (M x) (N y)) (Dtrans x y dtrans))) + %<- (can-bracket ([y] N y) (%the (bracket ([y] N y) N*) Dbrack)) %. +%block can-block {A tp} [x term A] [y comb A] [dtrans translate x y] [thm can-translate x dtrans]%. +%worlds (can-block) (can-bracket _ _) %. +%total N (can-bracket N _) %. +%worlds (can-block) (can-translate _ _) %. +%total M (can-translate M _) %. +%{! + +## Correctness of the translation + +!}% +%{! + +First, we prove the correctness of bracket abstraction itself: +the application of a bracket abstraction reduces to a substitution. + +!}% +%sort subst %. +%mode subst %in %in %out %. +%term _ subst (%the (bracket ([x] x) i) b/i) N0 (%the (ceq (i @@ N0) N0) ceq/i) %. +%term _ subst (%the (bracket ([x] Y) (k @@ Y)) b/k) N0 (%the (ceq (k @@ Y @@ N0) Y) ceq/k) %. +% developing incrementally, it's useful to write down the type of each output +%term _ + %pi (subst (%the (bracket ([x] N1 x @@ N2 x) (s @@ N1' @@ N2')) (b/s (%the (bracket ([x] N2 x) N2') B2) (%the (bracket ([x] N1 x) N1') B1))) N0 (ceq/trans (%the (ceq (s @@ N1' @@ N2' @@ N0) (N1' @@ N0 @@ (N2' @@ N0))) ceq/s) (%the (ceq (N1' @@ N0 @@ (N2' @@ N0)) (N1 N0 @@ N2 N0)) (ceq/app Dceq2 Dceq1)))) + %<- (subst B1 N0 (%the (ceq (N1' @@ N0) (N1 N0)) Dceq1)) + %<- (subst B2 N0 (%the (ceq (N2' @@ N0) (N2 N0)) Dceq2)) %. +%worlds (ceq-block) (subst _ _ _) %. +%total {B} (subst B _ _) %. +%{! + +Next, we need to know that the translation of a term is unique up to +equivalence. + +!}% +%sort translate-unique %. +%mode translate-unique %in %in %out %. +%term _ + %pi (translate-unique (%the (translate (M1 @ M2) (N1 @@ N2)) (t/app (%the (translate M2 N2) Dtrans2) (%the (translate M1 N1) Dtrans1))) (%the (translate (M1 @ M2) (N1' @@ N2')) (t/app (%the (translate M2 N2') Dtrans2') (%the (translate M1 N1') Dtrans1'))) (ceq/app Dceq2 Dceq1)) + %<- (translate-unique Dtrans1 Dtrans1' (%the (ceq N1 N1') Dceq1)) + %<- (translate-unique Dtrans2 Dtrans2' (%the (ceq N2 N2') Dceq2)) %. +%term _ + %pi (translate-unique (%the (translate (lam ([x] M x)) N*) (t/lam (%the (bracket ([y] N y) N*) Dbrack) ([x] [y] [dt] %the (translate (M x) (N y)) (Dtrans x y dt)))) (%the (translate (lam ([x] M x)) N'*) (t/lam (%the (bracket ([y] N' y) N'*) Dbrack') ([x] [y] [dt] %the (translate (M x) (N' y)) (Dtrans' x y dt)))) (%the (ceq N* N'*) (ceq/eta ([y] %the (ceq (N* @@ y) (N'* @@ y)) ((%the (ceq (N* @@ y) (N y)) (Dceq y)) ; (%the (ceq (N y) (N' y)) (Dceqtrans y)) ; (%the (ceq (N' y) (N'* @@ y)) (ceq/symm (Dceq' y)))))))) + %<- ({x} {y} {dtrans translate x y} {thm-unique translate-unique dtrans dtrans ceq/refl} translate-unique (Dtrans x y dtrans) (Dtrans' x y dtrans) (%the (ceq (N y) (N' y)) (Dceqtrans y))) + %<- ({z} subst Dbrack z (%the (ceq (N* @@ z) (N z)) (Dceq z))) + %<- ({z} subst Dbrack' z (%the (ceq (N'* @@ z) (N' z)) (Dceq' z))) %. +%block unique-block {A tp} [x term A] [y comb A] [dtrans translate x y] [thm translate-unique dtrans dtrans ceq/refl]%. +%worlds (unique-block) (translate-unique _ _ _) %. +%total D (translate-unique D _ _) %. +%{! + +Then, we can prove simulation, the correctness of translation, by a +straightforward induction on equality derivations in the lambda-calculus, +using the correctness of bracket abstraction as a lemma in the case of a +beta-reduction. + +!}% +%sort simulate' %. +%mode simulate' %in %in %in %out %. +%inline simulate (%pi (teq M M') %-> (translate M N) %-> (translate M' N') %-> (ceq N N') %-> %type) [deq] [dt] [dt'] [ceq] simulate' dt dt' deq ceq %. +%%mode simulate +Deq +Dt +Dt' -Dceq. +%term _ + %pi (simulate (%the (teq (app (lam ([x] M1 x)) M2) (M1 M2)) eq/beta) (%the (translate (app (lam ([x] M1 x)) M2) (capp N1* N2)) (t/app (%the (translate M2 N2) Dtrans2) (t/lam (%the (bracket ([y] N1 y) N1*) Dbrack) ([x term A2] [y comb A2] [dtrans translate x y] %the (translate (M1 x) (N1 y)) (Dtrans1 x y dtrans))))) (%the (translate (M1 M2) N3) Dtrans3) (%the (ceq (capp N1* N2) N3) (Dceq ; Dceq3))) + %<- (subst Dbrack N2 (%the (ceq (capp N1* N2) (N1 N2)) Dceq)) + %<- (translate-unique (Dtrans1 M2 N2 Dtrans2) Dtrans3 (%the (ceq (N1 N2) N3) Dceq3)) %. +%term _ + %pi (simulate (%the (teq M1 M2) (eq/eta ([x] %the (teq (app M1 x) (app M2 x)) (Deq x)))) (%the (translate M1 N1) Dtrans1) (%the (translate M2 N2) Dtrans2) (%the (ceq N1 N2) (ceq/eta ([y] Dceq y)))) + %<- ({x} {y} {dtrans translate x y} {thm-can can-translate x dtrans} {thm-unique translate-unique dtrans dtrans ceq/refl} {thm-simulate simulate eq/refl dtrans dtrans ceq/refl} simulate (%the (teq (app M1 x) (app M2 x)) (Deq x)) (%the (translate (app M1 x) (capp N1 y)) (t/app dtrans Dtrans1)) (%the (translate (app M2 x) (capp N2 y)) (t/app dtrans Dtrans2)) (%the (ceq (capp N1 y) (capp N2 y)) (Dceq y))) %. +%term _ + %pi (simulate (%the (teq (lam ([x] M x)) (lam ([x] M' x))) (eq/lam ([x] %the (teq (M x) (M' x)) (Deq x)))) (%the (translate (lam ([x] M x)) N*) (t/lam (%the (bracket ([y] N y) N*) Dbrack) ([x] [y] [dtrans translate x y] %the (translate (M x) (N y)) (Dtrans x y dtrans)))) (%the (translate (lam ([x] M' x)) N*') (t/lam (%the (bracket ([y] N' y) N*') Dbrack') ([x] [y] [dtrans translate x y] %the (translate (M' x) (N' y)) (Dtrans' x y dtrans)))) (%the (ceq N* N*') (ceq/eta ([y] (%the (ceq (N* @@ y) (N y)) (Dceq* y)) ; (%the (ceq (N y) (N' y)) (Dceq y)) ; (%the (ceq (N' y) (N*' @@ y)) (ceq/symm (Dceq*' y))))))) + %<- ({x} {y} {dtrans translate x y} {thm-can can-translate x dtrans} {thm-unique translate-unique dtrans dtrans ceq/refl} {thm-simulate simulate eq/refl dtrans dtrans ceq/refl} simulate (%the (teq (M x) (M' x)) (Deq x)) (%the (translate (M x) (N y)) (Dtrans x y dtrans)) (%the (translate (M' x) (N' y)) (Dtrans' x y dtrans)) (%the (ceq (N y) (N' y)) (Dceq y))) + %<- ({y} subst Dbrack y (%the (ceq (N* @@ y) (N y)) (Dceq* y))) + %<- ({y} subst Dbrack' y (%the (ceq (N*' @@ y) (N' y)) (Dceq*' y))) %. +%term _ + %pi (simulate (%the (teq (M1 @ M2) (M1' @ M2')) (eq/app (%the (teq M2 M2') Deq2) (%the (teq M1 M1') Deq1))) (%the (translate (M1 @ M2) (N1 @@ N2)) (t/app (%the (translate M2 N2) Dtrans2) (%the (translate M1 N1) Dtrans1))) (%the (translate (M1' @ M2') (N1' @@ N2')) (t/app (%the (translate M2' N2') Dtrans2') (%the (translate M1' N1') Dtrans1'))) (%the (ceq (N1 @@ N2) (N1' @@ N2')) (ceq/app Dceq2 Dceq1))) + %<- (simulate Deq1 Dtrans1 Dtrans1' (%the (ceq N1 N1') Dceq1)) + %<- (simulate Deq2 Dtrans2 Dtrans2' (%the (ceq N2 N2') Dceq2)) %. +%term _ + %pi (simulate (%the (teq M M) eq/refl) (%the (translate M N) Dtrans) (%the (translate M N') Dtrans') Dceq) + %<- (translate-unique Dtrans Dtrans' (%the (ceq N N') Dceq)) %. +%term _ + %pi (simulate (%the (teq M M') (eq/symm (%the (teq M' M) Deq))) (%the (translate M N) Dtrans) (%the (translate M' N') Dtrans') (%the (ceq N N') (ceq/symm Dceq))) + %<- (simulate Deq Dtrans' Dtrans (%the (ceq N' N) Dceq)) %. +%term _ + %pi (simulate (%the (teq M M'') (eq/trans (%the (teq M M') Deq1) (%the (teq M' M'') Deq2))) (%the (translate M N) Dtrans) (%the (translate M'' N'') Dtrans'') (%the (ceq N N'') (ceq/trans Dceq1 Dceq2))) + %<- (can-translate M' (%the (translate M' N') Dtrans')) + %<- (simulate Deq1 Dtrans Dtrans' (%the (ceq N N') Dceq1)) + %<- (simulate Deq2 Dtrans' Dtrans'' (%the (ceq N' N'') Dceq2)) %. +%block simulate-block {A tp} [x term A] [y comb A] [dtrans translate x y] [thm-can can-translate x dtrans] [thm-unique translate-unique dtrans dtrans ceq/refl] [thm-simulate simulate' dtrans dtrans eq/refl ceq/refl]%. +%worlds (simulate-block) (simulate' _ _ _ _) %. +%total D (simulate' _ _ D _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Typed_bracket_abstraction_with_equivalence). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial-typed-bracket-abstraction.lf b/new-tests/stelf-output/pages/popl-tutorial-typed-bracket-abstraction.lf new file mode 100644 index 0000000..46acc43 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial-typed-bracket-abstraction.lf @@ -0,0 +1,236 @@ +% +%{! + +This is a case study translating the simply lambda calculus into S, K, and I +combinatory logic. The correctness of the translation is proven in the +following sense: if a term steps to a reduct, its translation multi-steps to +its reduct's translation. + +[William Lovas](/wiki/user-william-lovas/) + +!}% +%{! + +## Syntax and static semantics + +!}% +%{! + +First we define the syntax of the two languages. ``term A`` is the +type of simply-typed lambda-calculus terms of type A; ``cterm A`` is +the type of simply-typed combinator terms of type A. Combinators are of +interest primarily because they have no binding structure; despite this +apparent limitation, we can translate any lambda-calculus term to an +operationally and logically related combinator term. + +Note that since we're using an +[intrinsic encoding](/wiki/intrinsic-and-extrinsic-encodings/), these syntax +definitions double as definitions of the languages' static semantics. + +!}% +%sort tp %. +%term i tp %. +%term => %pi tp %-> tp %-> tp %. +%prec %right 10 => %. +%sort term %. +%name term %. +%term app %pi (term (A => B)) %-> (term A) %-> (term B) %. +%term lam %pi (%pi (term A) %-> (term B)) %-> (term (A => B)) %. +%sort cterm %. +%name cterm %. +%term s cterm (A => B => C => (A => B) => A => C) %. +%term k cterm (A => B => A) %. +%term i cterm (A => A) %. +%term capp %pi (cterm (A => B)) %-> (cterm A) %-> (cterm B) %. +%{! + +We can use Twelf's [abbreviation mechanism](/wiki/abbrev-declaration/) and +[fixity declarations](/wiki/fixity-declaration/) to obtain some cute syntax. + +!}% +%inline @ app %. +%prec %left 10 @ %. +%inline @@ capp %. +%prec %left 10 @@ %. +%{! + +## Dynamic semantics + +!}% +%{! + +Then we define reduction relations on both languages. ``step`` is the +single-step reduction relation on lambda terms; ``cstep`` is the +single-step reduction relation on combinator terms. For lambda-calculus +terms, we do not reduce under binders. + +!}% +%sort step %. +%term s-beta step (app (lam ([x] M1 x)) M2) (M1 M2) %. +%term s-1 %pi (step (app M1 M2) (app M1' M2)) %<- (step M1 M1') %. +%term s-2 %pi (step (app M1 M2) (app M1 M2')) %<- (step M2 M2') %. +%sort cstep %. +%term cs-i cstep (capp i X) X %. +%term cs-k cstep (capp (capp k X) Y) X %. +%term cs-s cstep (capp (capp (capp s X) Y) Z) (capp (capp X Z) (capp Y Z)) %. +%term cs-1 %pi (cstep (capp X Y) (capp X' Y)) %<- (cstep X X') %. +%term cs-2 %pi (cstep (capp X Y) (capp X Y')) %<- (cstep Y Y') %. +%{! + +We also define multi-step reduction on combinator terms. + +Our simulation will relate single-step derivations in the lambda-calculus +to multi-step derivations on the translated terms. + +!}% +%sort cstep* %. +%term cs-cons %pi (cstep* N N'') %<- (cstep N N') %<- (cstep* N' N'') %. +%term cs-nil cstep* N N %. +%{! + +We can pre-emptively prove some compatibility lemmas about multi-step +reduction. + +!}% +%sort cs-1* %. +%mode {%in A tp} {%in B tp} {%in N1 cterm (A => B)} {%in N1' cterm (A => B)} {%in N2 cterm A} {%in CS1 cstep* N1 N1'} {%out CS2 cstep* (capp N1 N2) (capp N1' N2)} cs-1* CS1 CS2 %. +%term _ cs-1* cs-nil cs-nil %. +%term _ %pi (cs-1* (cs-cons CS C) (cs-cons CS' (cs-1 C))) %<- (cs-1* CS CS') %. +%worlds () (cs-1* _ _) %. +%total {CS} (cs-1* CS _) %. +%sort cs-2* %. +%mode {%in A tp} {%in B tp} {%in N1 cterm (A => B)} {%in N2 cterm A} {%in N2' cterm A} {%in CS1 cstep* N2 N2'} {%out CS2 cstep* (capp N1 N2) (capp N1 N2')} cs-2* CS1 CS2 %. +%term _ cs-2* cs-nil cs-nil %. +%term _ %pi (cs-2* (cs-cons CS C) (cs-cons CS' (cs-2 C))) %<- (cs-2* CS CS') %. +%worlds () (cs-2* _ _) %. +%total {CS} (cs-2* CS _) %. +%sort cs-trans %. +%mode cs-trans %in %in %out %. +%term _ cs-trans cs-nil Cs2 Cs2 %. +%term _ %pi (cs-trans (cs-cons Cs1 C) Cs2 (cs-cons Cs12 C)) %<- (cs-trans Cs1 Cs2 Cs12) %. +%worlds () (cs-trans _ _ _) %. +%total {Cs} (cs-trans Cs _ _) %. +%{! + +## Translation + +!}% +%{! + +We can now define a compositional translation from lambda terms to combinator +terms in the standard way. The translation appeals to a function called +_bracket abstraction_ which simulates binding in the combinator calculus. + +Bracket abstraction is usually written where +is a combinator term, and is a variable potentially free in +. (Be careful not to confuse these brackets with Twelf's syntax +for lambda abstraction.) It is defined inductively over the term +: + +<blockquote> + <br> + + (where not free in )<br> + + <br> +</blockquote> + +Using it, we can define a translation on lambda-terms, , where + is a lambda-term, as follows: + +<blockquote> +<br> + = + <br> + +</blockquote> + +Note that in the definition of bracket abstraction, we need not consider a +case for lambda-abstraction since bracket abstraction works over combinator +terms, in which lambda-abstraction has already been eliminated. + +The main translation is represented in LF by a judgement (an LF type family) +``trans M N``. Bracket abstraction is represented by a judgement +relating LF-level abstractions in the combinator language to closed combinator +terms, ``bracket ([x] N x) N``. This definition is essentially an +instance of [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/). + +!}% +%sort trans %. +%sort bracket %. +%term t-app %pi (trans (app M1 M2) (capp N1 N2)) %<- (trans M1 N1) %<- (trans M2 N2) %. +%{! + +The translation on lambda-abstractions has to work under an extended context +with a lambda-term variable, a combinator-term variable, and an assumption +that the one translates to the other. See the ``%worlds`` declaration +below. + +!}% +%%% +%%% TODO: complete the translation case for lambda-abstraction +%%% +%term t-lam trans (lam ([x] M x)) XXX-FILL_IN-XXX %. +%term b-i bracket ([x] x) i %. +%term b-k bracket ([x] Y) (capp k Y) %. +%%% +%%% TODO: complete the application case for bracket abstraction +%%% +%term b-s bracket XXX-FILL_IN-XXX YYY-FILL_IN-YYY %. +%block tbind {A tp} [x term A] [y cterm A] [dtrans trans x y]%. +%worlds (tbind) (bracket _ _) %. +%worlds (tbind) (trans _ _) %. +%{! + +## Correctness of the translation + +!}% +%{! + +First, we prove the correctness of bracket abstraction itself: +the application of a bracket abstraction reduces to a substitution. + +!}% +%sort subst %. +%mode subst %in %in %out %. +%term _ subst b-i N0 (cs-cons cs-nil cs-i) %. +%term _ subst b-k N0 (cs-cons cs-nil cs-k) %. +%%% +%%% TODO: complete the case for b-s +%% +%term _ subst (b-s (%the (bracket ([x] N2 x) N2') B2) (%the (bracket ([x] N1 x) N1') B1)) N0 XXX-FILL_IN-XXX %. +%worlds () (subst _ _ _) %. +%total {B} (subst B _ _) %. +%{! + +Then, we can prove simulation, the correctness of translation, by a +straightforward induction on single-step derivations in the lambda-calculus, +using the correctness of bracket abstraction as a lemma in the case of a +beta-reduction. + +!}% +%sort simulate %. +%mode simulate %in %in %out %out %. +%{! + +NB: in the ``s-beta`` case on paper, you have to prove two +compositionality lemmas. In Twelf, using higher-order abstract +syntax, we get them for free. + +!}% +%%% +%%% TODO: prove simulation; there are three cases. +%%% +%worlds () (simulate _ _ _ _) %. +%total D (simulate D _ _ _) %. +%{! + +[POPL Tutorial/Typed bracket abstraction (solution)](/wiki/popl-tutorial-typed-bracket-abstraction-solution/) + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial/Typed_bracket_abstraction). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/popl-tutorial.lf b/new-tests/stelf-output/pages/popl-tutorial.lf new file mode 100644 index 0000000..20625a0 --- /dev/null +++ b/new-tests/stelf-output/pages/popl-tutorial.lf @@ -0,0 +1,80 @@ +% +%{! <!-- +<div style="font-size:162%;border:none;margin: 0;padding:.1em;color:#F00">**_We want your input!**_</div> + +Help us out by filling out our handy dandy [http://spreadsheets.google.com/viewform?key=pff4VF1v6KVNcwC_c2qFAGA feedback form]! +--> + +## Mechanizing Metatheory with LF and Twelf + +Do you want to learn how to use [Twelf](/) to specify, implement, and prove +properties about programming languages? + +Come to the Twelf tutorial on **January 19, 2009**, co-located with POPL 2009, in +**Savannah, Georgia**. + +Learn to: +* Represent [languages and logics](/wiki/object-logic/) in [LF](/wiki/lf/) +* Prove [metatheorem](/wiki/metatheorem/)s with [Twelf](/) +under the helpful guidance of Twelf experts. + +The tutorial will be a highly interactive introduction to LF and Twelf +aimed at programming languages researchers. No prior experience with LF +and Twelf is presumed. Participants will leave the workshop with +experience in reading and writing LF representations of programming +languages, and experience reading, writing, and debugging Twelf proofs. + +Register at the [http://www.regmaster.com/conf/popl2009.html POPL 2009 registration site]! + +The tutorial is organized and presented by the CMU Principles of Programming group. The presenters and TAs at POPL will be Dan Licata, William Lovas, Chris Martens, Rob Simmons, Bob Harper, and Karl Crary. + +## Schedule + +The tutorial will begin at **9:00AM**. Get [http://twelf.plparty.org/tutorialslides/lectures.pdf the slides]! + +* Part 1: [Basic Twelf Skills](/wiki/popl-tutorial-basics/) +* Part 2: [Mechanizing MinML](/wiki/popl-tutorial-minml/) +* Part 3: [Combinators: Worlds and Adequacy](/wiki/popl-tutorial-combinators/) +* Coda: [What's next?](/wiki/popl-tutorial-next/) + +There will be a morning coffee break (10AM), lunch (12:30PM-1:30PM), and an afternoon coffee break (3PM). + +## Get Twelf before the tutorial! + +The tutorial will be interactive, with participants writing Twelf code, so you should come with Twelf installed on your laptop. + +Pre-built binaries of Twelf are available for most operating systems: see the [download page](/download/). + +Otherwise: +* you can build Twelf from the [http://twelf.plparty.org/builds/twelf-src.tar.gz source tarball]. You will need [http://www.mlton.org MLton] or [http://www.smlnj.org sml/nj]. +* you can make yourself an account on the wiki, and do the exercises on your User:<login> page (linked at the top after you log in). + +Then see [Twelf with Emacs](/wiki/twelf-with-emacs/) for the basics of interacting with Twelf. (You can also use [Twelf without Emacs](/wiki/twelf-without-emacs/), by interacting with the Twelf server directly.) + +## Sponsors + +Thanks to our sponsors: +[http://www.cs.cmu.edu Carnegie Mellon School of Computer Science], +[http://www.research.ibm.com IBM Research], +[http://research.microsoft.com/ Microsoft Research], +[http://www.intel.com/ Intel], +[http://www.docomolabs-usa.com/ DOCOMO USA Labs], +[http://www.mozilla.org/ Mozilla], +[http://www.google.com/ Google]. + +[[Image:POPL_Tutorial-cmu.gif|250px]] [[Image:POPL_Tutorial-scslogo.gif]] + +[[Image:POPL_Tutorial-IBM.jpg|600px]] + +[[Image:POPL Tutorial-msr.jpg]] [[Image:POPL Tutorial-intel.jpg|300px]] + + +[[Image:POPL_Tutorial-docomo-logo.gif|250px]] [[Image:POPL Tutorial-mozilla.jpg|300px]] + +[[Image:POPL Tutorial-google-logo.gif]] !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/POPL_Tutorial). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/programming-language-theory-with-twelf.lf b/new-tests/stelf-output/pages/programming-language-theory-with-twelf.lf new file mode 100644 index 0000000..c2d19df --- /dev/null +++ b/new-tests/stelf-output/pages/programming-language-theory-with-twelf.lf @@ -0,0 +1,77 @@ +% +%{! ## Preface + +The study of programming languages generally proceeds in two parallel directions. In one direction, +students are introduced to several specific programming languages representing different +_programming paradigms_ (functional, object-oriented, logic programming, scripting, etc.) and are +shown many examples of how these languages are used. They learn about the strengths and weaknesses +of the languages and about techniques that can be used to program effectively with them. In the +other direction, students are introduced to _programming language theory_, a body of knowledge +that attempts to organize and analyze the wide variety of programming languages and programming +language features. Such knowledge can be used to explain why certain languages _adhere_ better +than others, or why certain kinds of run-time errors in a particular language are impossible, and it +can serve as a reliable guide to designing new programming languages. + +Although several approaches to programming language theory exist, the most effective is the approach +that represents both the abstract syntax and the static and dynamic semantics of a programming +language using inductively-defined hypothetical judgments given by sets of rules—essentially +treating the language as a formal system. Coupled with a systematic classification of program +fragments using types, even when types are not explicitly part of the language under consideration, +a powerfully uniform treatment of programming language theory emerges. This approach is +developed in the two textbooks + +* PFPL: [http://www.cs.cmu.edu/~rwh/plbook/book.pdf Practical Foundations for Programming Languages] (PDF), by Robert Harper (draft, 2008), and +* TAPL: [http://www.cis.upenn.edu/~bcpierce/tapl/ Types and Programming Languages], by Benjamin Pierce (MIT Press, 2002). + +Despite the success of this approach to programming language theory, one of its disadvantages, +especially for students seeing it for the first time, is that it can be quite abstract. This is +where Twelf comes in. The rules for the hypothetical judgments associated with a particular +programming language, and much of the resulting metatheory (i.e., the theorems about these +representations and their relations), are very naturally represented using Twelf. Further, the +correctness of the proofs can often be checked by Twelf, with feedback given when the proof is +incomplete or incorrect. This gives students concrete artifacts they can manipulate, interact and +experiment with, and test ideas on. + +The aim of these notes is to act as a kind of "laboratory handbook" for the above two books, +introducing students of programming language theory to the Twelf system so that they can use it as a +supplement to their study, as well as, eventually, a tool for programming language research. Our +goal here is not to _teach_ programming language theory—that is done very well already by +the two books above—but to _illustrate_ it and make it more concrete. To that end, after a +tutorial introduction to Twelf, we will give complete and carefully explained formalizations of the +various systems, examples, proofs, and exercises in each of the above two books, pausing frequently +to discuss the techniques used in the formalizations and the obstacles faced producing them. In the +case of TAPL, we will be making use of some notes of Benjamin Pierce for his course at Penn, +[http://www.seas.upenn.edu/~cis500/cis500-f07/schedule.html CIS 500], which uses the +[http://coq.inria.fr Coq system] where we have used Twelf. + +## Table of Contents + +Part 1: [[Introduction to Twelf]] +- [First-order encodings](/wiki/first-order-encodings/) +- [Signatures as logic programs](/wiki/signatures-as-logic-programs/) +- [Signature checking](/wiki/signature-checking/): modes, termination, coverage, totality +- [[Higher-order encodings]] +- [[Common patterns and pitfalls]] + +Part 2: [[Logical Foundations of Programming Languages]] +- [[LFPL:Chapter 1|Chapter 1]]: Inductive Definitions +- [[LFPL:Chapter 4|Chapter 4]]: Transition Systems +- [[LFPL:Chapter 5|Chapter 5]]: Basic Syntactic Objects +- [[LFPL:Chapter 6|Chapter 6]]: Binding and Scope +- [[LFPL:Chapter 7|Chapter 7]]: Concrete Syntax +- [[LFPL:Chapter 8|Chapter 8]]: Abstract Syntax +- [[LFPL:Chapter 9|Chapter 9]]: Static Semantics +- [[LFPL:Chapter 10|Chapter 10]]: Dynamic Semantics +- [[LFPL:Chapter 11|Chapter 11]]: Type Safety +- [[LFPL:Chapter 12|Chapter 12]]: Evaluation Semantics +- [[LFPL:Chapter 14|Chapter 14]]: Functions +- [[LFPL:Chapter 15|Chapter 15]]: Gödel's System T +- [[LFPL:Chapter 16|Chapter 16]]: Plotkin's PCF + +Part 3: [[Types and Programming Languages]] !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Programming_language_theory_with_Twelf). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-full-lf.lf b/new-tests/stelf-output/pages/proving-metatheorems-full-lf.lf new file mode 100644 index 0000000..4f54866 --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-full-lf.lf @@ -0,0 +1,164 @@ +% +% +% +% +% +% +% +% +%{! + +_This page is part of the introduction to [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)._ + +In the [previous section](/wiki/proving-metatheorems-representing-the-judgements-of-the-natural-numbers/), we saw how dependently-typed LF can be used to represent object-language judgements. We now discuss dependently-typed LF in more detail. + +## Definition of LF + +Full LF has the following syntax: + +```math +\begin{align*} +K & ::= \mathtt{type} \mid \Pi x{:}A. K +\\ +A & ::= b \mid A \, M \mid \Pi x{:}A_1. A_2 +\\ +R & ::= c \mid x \mid R \, M +\\ +M & ::= R \mid \lambda x. M +\end{align*} +``` + +In Twelf's concrete syntax, the type is written ``{x:A1} A2``, the kind is written ``{x:A} K``, and the term is written ``[x] M``. The terms are unchanged from the previous grammar. The type families now include both dependent function types and applications of type families to terms. The kind level classifies type families. Type families of kind classify terms. Type families that are not yet fully instantiated have kinds ``{x:A} K``. In both types and kinds, we use ``->`` as an abbreviation when the argument is not free in the result. + +For example, in the previous section, we declared the type family ``plus`` to have kind ``nat -> nat -> nat -> type``. This means that when ``plus`` is applied to three terms of type ``nat``, it forms a type. This kind is syntactic sugar for the kind ``{_:nat} {_:nat} {_:nat} type``. The partial application ``plus z`` has kind ``nat -> nat -> type``. Below, we will see examples where the type of one index of a type family depends on a previous argument, which motivates permitting general ``{x:A} K`` kinds in the general case. + +We do not present the typing rules for LF in detail here. However, we do call attention to the typing rule for application. In a dependently typed language, the application typing rules must substitute the argument into the body of the dependent function type: + +```math +R : \Pi x {:} A. B \qquad M : A \over R \; M : [M/x]B +``` + +In LF, this substitution is in fact a [hereditary substitution](/wiki/hereditary-substitution/), ensuring that the terms embedded in the result of the substitution are in canonical form. + +This typing rule is how constants representing inference rules are specialized to particular instances. For example, constant ``even-s`` from the previous section has type ``{N:nat} even N ->; even (s (s N))``, so the application ``even-s z`` has type ``even z -> even (s (s z))``. + +The typing rule for family applications has a similar substitution into the result kind. + +```math +A : \Pi x {:} B. K \qquad M : B \over A \; M : [M/x]K +``` + + +## Twelf conveniences + +Twelf permits several syntactic conveniences for writing LF signatures. The purpose of this list is to introduce you to these conveniences; we will see examples of them later in this introduction or elsewhere on the wiki. + +### Definitions + +In Twelf, you can define an LF constant for later use. For example assuming the [last section's definition of natural numbers](/wiki/proving-metatheorems-representing-the-judgements-of-the-natural-numbers/), we can write this: !}% +%{!! begin hidden !!}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort even %. +%term even-z even z %. +%term even-s {N nat} %pi (even N) %-> (even (s (s N))) %. +%sort plus %. +%{!! end hidden !!}% +%{!! begin checked !!}% +%define 2 nat s (s z) %. +%define even-2 (even 2) even-s z even-z %. +%{!! end checked !!}% +%{! The first constant names a natural number ``2``; the second names a derivation that ``2`` is even. (Twelf treats numeric identifiers the same as any other constants as long as no [constraint domains](/wiki/constraint-domain/) are active, so naming a constant `2` is no different, from Twelf's perspective, than naming it `two`.) + +You can name terms of higher type as well: !}% +%{!! begin checked !!}% +%define ss (%pi nat %-> nat) [x nat] s (s x) %. +%{!! end checked !!}% +%{! This constant might be used as follows: !}% +%{!! begin checked !!}% +%define even-4 (even (ss 2)) even-s 2 even-2 %. +%{!! end checked !!}% +%{! Definitions like this do not create any new numbers (i.e. any new canonical forms of type `nat`), they just name what can be formed with existing constants. + +### Non-canonical forms + +Twelf permits non-canonical forms, which are treated as syntactic sugar for the associated canonical form. Here are some examples: + +#### [Beta-equivalence](/wiki/beta-equivalence/) + +The term ``ss 2``, which, if we expand the definitions, reads ``([x] (s (s x))) (s (s z))``, has the canonical form ``s (s (s (s z)))``. So we might expect Twelf to automatically expand that definition in its output: !}% +%{!! begin checked !!}% +%define four ss 2 %. +%{!! end checked !!}% +%{! +The expected expansion doesn't actually happen! Twelf works very hard _not_ to expand definitions, while still always returning the same results as if those definitions had been expanded. For all purposes `ss 2` should be thought of as another way of writing the `s (s (s (s z)))`. + +#### [Eta-equivalence](/wiki/eta-equivalence/) + +The term ``s``, which has type ``nat ->; nat`` can be used instead of its eta-expansion ``[x] s x``. Eta expansions will always be performed in Twelf's output: !}% +%{!! begin checked !!}% +%define other-s s %. +%{!! end checked !!}% +%{! +We will discuss eta-equivalence more in the context of [higher-order representations.](/wiki/proving-metatheorems-representing-the-judgements-of-the-stlc/) + +### Implicit arguments + +Writing out all of the parameters to a constant becomes tedious, so Twelf permits these parameters to be left implicit. Specifically, if a variable starting with a lower-case letter is left unbound in a constant's type, Twelf reports an error. If a variable beginning with an upper-case letter is left unbound, Twelf implicitly binds it in a ``\{N\}`` at the outside of the type. For example, the following two ways of declaring the constant ``plus-s`` give it the same type: !}% +%{!! begin checked !!}% +%% explicit +%term plus-s-explicit {N1 nat} {N2 nat} {N3 nat} %pi (plus N1 N2 N3) %-> (plus (s N1) N2 (s N3)) %. +%% implicit +%term plus-s-implicit %pi (plus N1 N2 N3) %-> (plus (s N1) N2 (s N3)) %. +%{!! end checked !!}% +%{! +Similarly, we can make the parameter to ``plus-z`` implicit: +!}% +%{!! begin checked !!}% +% +% +%{!! end checked !!}% +%{! +In most circumstances, Twelf can infer the types of the implicit parameters from their use in the constant's type. If it cannot, it reports an error---the type of a constant must be fully determined when the constant is processed. + +The application of a constant to its implicit parameters is then inferred from context. For example, using the fully explicit definition of ``plus``, the derivation that 2 + 1 = 3 is written as follows: !}% +%{!! begin checked !!}% +%define d2+1e plus-s-explicit (s z) (s z) (s (s z)) (plus-s-explicit z (s z) (s z) (plus-z-explicit (s z))) %. +%{!! end checked !!}% +%{! Using the implicit version, it is much shorter: !}% +%{!! begin checked !!}% +%define d2+1i plus-s-implicit (plus-s-implicit plus-z-implicit) %. +%{!! end checked !!}% +%{! The type of this term is ambiguous, but Twelf will attempt to use [unification](/wiki/unification/) to infer the most general possible type. The fully general result has an additional parameter, `X1:nat`, and establishes not that but the more general fact + +### Backwards arrow + +Twelf permits a type ``A -> B`` to be written ``B <- A``. This makes it easier to see the result of applying a constant to all of its arguments. For example: + +```twelf +plus-s : plus (s N1) N2 (s N3) + <- plus N1 N2 N3. +``` + +When we write ``C <- Pn <- ... <- P1``, we will refer to ``C`` as the _conclusion_ of the constant and each ``Pi`` as a _premise_, by analogy with the terminology for the inference rule that this constant represents. + +The backwards-arrow also has implications on the logic programming operational semantics for LF, as we discuss below; the conclusion/premise terminology is consistent with the use of these words in logic programming as well. + +### Type and term inference + +Twelf permits type annotations, which are written ``M:A``, on all terms. Type annotations can be used for documentation. Additionally, in some cases they are necessary to help guide type inference. We saw before that +the implicitly typed version of plus resulted in an ambiguous type: !}% +%{!! begin checked !!}% +%define d2+1i plus-s-implicit (plus-s-implicit plus-z-implicit) %. +%{!! end checked !!}% +%{! If for some reason this was undesirable, and we really wanted this to be proof that , one way to achieve this would be to use a type annotation to say that `plus-z` should be used specifically as a proof that : !}% +%{!! begin checked !!}% +%define d2+1i' plus-s-implicit (plus-s-implicit (%the (plus z (s z) (s z)) plus-z-implicit)) %. +%{!! end checked !!}% +%{! Type annotations on the arguments of lambda-terms ``[x:A] M`` are sometimes useful, even though they aren't present in the formal definition of LF. Type annotations on Pi-types ``\{x:A2\} A`` can sometimes be elided when the type of the variable is determined by its use, even though they _are_ required in the formal definition of LF. + +When they are determined from context, terms can be elided by writing an underscore. For example, if the constants defining ``plus`` were declared with explicit ``nat`` parameters, we could still write the derivation of 2+1=3 as follows: !}% +%{!! begin checked !!}% +%define d2+1e' plus-s-explicit _ _ _ (plus-s-explicit _ _ _ (plus-z-explicit (s z))) %. +%{!! end checked !!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-natural-numbers-answers-to-exercises.lf b/new-tests/stelf-output/pages/proving-metatheorems-natural-numbers-answers-to-exercises.lf new file mode 100644 index 0000000..3b80592 --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-natural-numbers-answers-to-exercises.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you should can find the exercises and their solutions [here](/wiki/proving-metatheorems-summary-the-natural-numbers/). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-proving-metatheorems-about-the-natural-numbers.lf b/new-tests/stelf-output/pages/proving-metatheorems-proving-metatheorems-about-the-natural-numbers.lf new file mode 100644 index 0000000..8e4d352 --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-proving-metatheorems-about-the-natural-numbers.lf @@ -0,0 +1,116 @@ +% +% +% +% +% +% +% +% +%{! + +_This page is part of the introduction to [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)._ + +In the previous section, we saw how to ask Twelf to verify totality assertions, which are a particular form of statement one can make about a deductive system. However, not all that many metatheorems are totality assertions about the judgements of an object language. Moreover, even if a judgement does in fact satisfy a total assertion, the judgement often will not be written in the particular form that allows Twelf to automatically check mode, worlds, termination, and coverage. So why are totality assertions so important? !}% +%{!! begin hidden !!}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort even %. +%term even-z even z %. +%term even-s %pi (even (s (s N))) %<- (even N) %. +%sort plus %. +%term plus-z plus z N2 N2 %. +%term plus-s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%{!! end hidden !!}% +%{! ## How to prove metatheorems + +The reason is that we can use Twelf's ability to _prove_ totality assertions to _check_ proofs of more general metatheorems. Let's illustrate this with an example. Say we want to prove that the sum of two even numbers is even. More precisely, let's prove the following statement: + +> If and and then . + +We can prove this statement by rule induction on the first premise, the derivation that . The computational content of this proof will be a function that transforms the derivations of the premises into a derivation of the conclusion. + +Now, let's recast this informal theorem statement as a statement about LF terms. By the adequacy of our LF representations as [discussed in a previous section](/wiki/proving-metatheorems-representing-the-judgements-of-the-natural-numbers/), this informal statement is equivalent to a statement about the canonical forms of the corresponding types: + +> For all ``DevenN1 : even N1`` and ``DevenN2 : even N2`` and ``Dplus : plus N1 N2 N3`` there exists a ``DevenN3 : even N3``. + +We can recast the informal proof by rule induction on the derivation of as a proof by induction on the canonical forms of type ``even N1``. The computational content of this proof will be a transformation from LF terms of type ``even N1``, ``even N2``, and ``plus N1 N2 N3`` to an LF term of type ``even N3``. + +The key to the way we use Twelf to prove general metatheorems is that _we can represent this transformation within LF itself_. Specifically, we represent the theorem statement as a type family, which defines a relation on the subjects of the theorem. Then, we represent the proof as LF constants inhabiting that type family. A proof is correct iff it defines a total relation (because the original metatheorem is implied by the totality assertion for the relation). So we deploy Twelf's ability to verify totality assertions in order to check the proof of the metatheorem. Thus, you already have all the tools you need to prove metatheorems with Twelf; you just need to use them in the right way. + +## Our first metatheorem + +This will all be clearer once we do an example. + +### Representing the theorem statement + +We represent the above theorem statement in Twelf as a type family relating the appropriate derivations: !}% +%sort sum-evens %. +%mode sum-evens %in %in %in %out %. +%{! The kind of ``sum-evens`` is _dependent_, unlike all the kinds we have seen so far. The dependency ensures that the ``sum-evens`` relation relates derivations of the appropriate facts. + +The mode declaration for this type family makes the "for all" theorem subjects into inputs, and the "there exists" theorem subject into an output. Intuitively, this is the right thing to do because then the relation defined by ``sum-evens`` will transform these three "for all" derivations into the "there exists" derivation. More formally, observe that the mode and worlds declarations declare the following totality assertion: + +> For all ``D1 : even N1`` and ``D2 : even N2`` and ``Dplus : plus N1 N2 N3`` there exist ``D3 : even N3`` and ``D : sum-evens D1 D2 Dplus D3``. + +The above metatheorem statement is clearly a corollary of this totality assertion. + +### Informally describing the proof + +Before we transcribe the informal proof of the metatheorem as LF constants of the appropriate types, let's review the informal proof. + +The proof is by induction over the derivation of . + +Case for : + +In this case, is , so by inversion the derivation of must derive . Then the assumption of gives the result. + +Case for : + +In this case, is , so we have a derivation of . By inversion on this derivation, is and we have a derivation of . Then the inductive hypothesis applied to this derivation and the derivation of yields a derivation of . So we can derive + +```math +\mathsf{even}(N_3’) \over \mathsf{even}(\mathsf{succ}(\mathsf{succ}(N_3’)) +``` + +to finish the case. + +### LF Representation of the proof + +We represent this informal proof as LF constants inhabiting the type family ``sum-evens``. We use the implicit-arguments versions of the definitions from [Proving_metatheorems:Full_LF](/wiki/proving-metatheorems-full-lf/). There will be one constant for each case of the informal proof. The constant corresponding to the first case is: !}% +%term sez sum-evens even-z (%the (even N2) DevenN2) (%the (plus z N2 N2) plus-z) DevenN2 %. +%{! We can read the type of this constant as the above case: when the term of type ``even N1`` is ``even-z``, then ``N1`` is ``z`` and the term of type ``plus N1 N2 N3`` must be ``plus-z``. Then the assumed term of type ``even N2`` has the appropriate type for the result. + +The constant corresponding to the second case is: !}% +%term ses + %pi (sum-evens (%the (even (s (s N1'))) (even-s DevenN1')) (%the (even N2) DevenN2) (%the (plus (s (s N1')) N2 (s (s N3'))) (plus-s (plus-s Dplus))) (even-s DevenN3')) + %<- (sum-evens DevenN1' DevenN2 Dplus DevenN3') %. +%{! We invert the derivation of ``plus`` by pattern-matching it as two applications of the constant ``plus-s``. The inductive call to the theorem is represented as a premise of type ``sum-evens`` on a term ``DevenN1'`` that is a subterm of the input term ``(even-s DevenN1')``. The result of the case is created by applying the constant ``even-s``, which corresponds to the rule used in the informal case, to the result of the inductive call. + +You should try to get used to the three equivalent ways of thinking about these constants: +- **Informal proof:** They are a direct transcription of the cases of the informal proof. +- **Type inhabitation:** They define the relation ``sum-evens`` by type inhabitation, in such a way that Twelf can prove that the family is inhabited for all inputs. +- **Logic programming:** They define a total logic program from the inputs to the outputs. + +Now that we have represented the proof, all that's left is to ask Twelf to verify this proof by induction on the first argument: !}% +%{!! begin checked !!}% +%worlds () (sum-evens _ _ _ _) %. +%total D (sum-evens D _ _ _) %. +%{!! end checked !!}% +%{! We call attention to a few of Twelf's checks: +* **Termination:** In the premise of the constant ``even-s``, the term ``DevenN1'`` is a subterm of the input derivation, so the termination check succeeds. In this case, it happens that the term ``Dplus`` is also smaller, so we could have checked termination on this argument instead. +* **Input Coverage:** Because Twelf separates termination checking from coverage checking, the constants defining a type family may case-analyze more arguments than just the induction one. This subsidiary case-analyses correspond to inversion steps in an informal proof. If an inversion is not justified (e.g., if in ``sez`` there were another way to derive ``plus z N2 N3`` besides ``plus-z``), Twelf reports an input coverage error—not all possible input terms are covered. + +## What metatheorems can we prove this way? + +The metatheorems that can be proved in Twelf have the form + +> For all LF terms of some input types, there exist LF terms of some output types. + +These metatheorems are called -statements, and they are the theorems whose proofs consist of total relations between the input types and the output types. + +This means that there are many theorems that cannot be mechanized in Twelf (e.g., theorems requiring higher quantifier complexity). However, in practice, -statements have been shown to be a useful class of metatheorems about programming languages (for example, they are enough to [prove type safety for all of Standard ML](https://dl.acm.org/doi/10.1145/1190215.1190245)). + +It is important to recognize that the restriction to -statements is a consequence of representing proofs of metatheorems as total relations among LF types. It is _not_ a consequence of representing object languages in LF. Any on-paper proof about an informal description of an object language can be translated into an on-paper proof about the associated adequate LF representation. But only some of these proofs can be mechanically verified in Twelf using the techniques presented above. + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-proving-metatheorems-about-the-stlc.lf b/new-tests/stelf-output/pages/proving-metatheorems-proving-metatheorems-about-the-stlc.lf new file mode 100644 index 0000000..5ad7fc5 --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-proving-metatheorems-about-the-stlc.lf @@ -0,0 +1,83 @@ +% +% +% +% +% +% +% +% +%{! + +_This page is part of the introduction to [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)._ + +In this section, we show how to prove type preservation for the STLC. As you will see, this theorem is really no harder than the metatheorems about the natural numbers that we proved above. For reference, we recap the entire LF signature representing the STLC, both the [syntax](/wiki/proving-metatheorems-representing-the-syntax-of-the-stlc/) and [judgements](/wiki/proving-metatheorems-representing-the-judgements-of-the-stlc/): !}% +%% Syntax +%sort tp %. +%term unit tp %. +%term arrow %pi tp %-> tp %-> tp %. +%sort tm %. +%term empty tm %. +%term app %pi tm %-> tm %-> tm %. +%term lam %pi tp %-> (%pi tm %-> tm) %-> tm %. +%% Static Semantics +%sort of %. +%term of-empty of empty unit %. +%term of-lam + %pi (of (lam T2 ([x] E x)) (arrow T2 T)) + %<- ({x tm} %pi (of x T2) %-> (of (E x) T)) %. +%term of-app %pi (of (app E1 E2) T) %<- (of E1 (arrow T2 T)) %<- (of E2 T2) %. +%% Dynamic Semantics +%sort value %. +%term value-empty value empty %. +%term value-lam value (lam T ([x] E x)) %. +%sort step %. +%term step-app-1 %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step-app-2 %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step-app-beta %pi (step (app (lam T2 ([x] E x)) E2) (E E2)) %<- (value E2) %. +%{! ## Preservation + +Here is a complete Twelf proof of preservation for the STLC: !}% +%sort preserv %. +%mode preserv %in %in %out %. +%term preserv-app-1 + %pi (preserv (step-app-1 (%the (step E1 E1') DstepE1)) (of-app (%the (of E2 T2) DofE2) (%the (of E1 (arrow T2 T)) DofE1)) (of-app DofE2 DofE1')) + %<- (preserv DstepE1 DofE1 (%the (of E1' (arrow T2 T)) DofE1')) %. +%term preserv-app-2 + %pi (preserv (step-app-2 (%the (step E2 E2') DstepE2) (%the (value E1) DvalE1)) (of-app (%the (of E2 T2) DofE2) (%the (of E1 (arrow T2 T)) DofE1)) (of-app DofE2' DofE1)) + %<- (preserv DstepE2 DofE2 (%the (of E2' T2) DofE2')) %. +%term preserv-app-beta preserv (step-app-beta (%the (value E2) Dval)) (of-app (%the (of E2 T2) DofE2) (of-lam (%the ({x tm} {dx of x T2} of (E x) T) ([x] [dx] DofE x dx)))) (DofE E2 DofE2) %. +%{!! begin checked !!}% +%worlds () (preserv _ _ _) %. +%total D (preserv D _ _) %. +%{!! end checked !!}% +%{! Now, let's examine the pieces of the proof in more detail. + +### Correctness of the theorem statement. + +We should check that this Twelf theorem statement actually corresponds to the notion of type preservation that we have in mind, which is stated as follows: + +> If and , then . + +This corresponds to the following statement about LF terms: + +> If ``Dstep : step E E'`` and ``Dof : of E T`` then ``Dof' : of E' T``. + +By adequacy, these two theorem statements are equivalent; note that closed (i.e. no free variables) LF terms correspond to closed object-language terms and typing derivations in the empty object-language context. + +The above totality assertion for the type family ``preserv`` clearly implies this statement. Thus, the Twelf proof proves the informal statement of preservation. + +### Reading the cases + +As we have discussed, the LF constants implementing a proof correspond to the cases of an informal proof. For preservation, the three LF constants correspond to the three cases of an informal proof by induction on the dynamic semantics derivation, where in each case we invert the static semantics derivation. For example, in the case ``preserv-app-1`` for ``step-app-1``, we invert the typing derivation for the application, appeal to the inductive hypothesis (the premise of type ``preserv``), and then use ``of-app`` to derive the result. The other two cases use similar inversions. Because of the higher-order representation of the hypothetical typing judgement, there is no need for a substitution lemma in the case ``preserv-app-beta``: the LF term ``DofE`` representing the hypothetical derivation can simply be applied to the derivation ``DofE2``. + +The ``%total`` directive asks Twelf to verify that ``preserv`` defines a total relation—i.e., that it is a correct proof. We call attention to a few of Twelf's checks: +* **Termination:** In all premises, the dynamic semantics derivation is a subterm of the input dynamic semantics derivation. In this proof, it happens that the typing derivation is always smaller as well, so we could equivalently have verified totality by induction on the second argument. +* **Input coverage:** The inversion steps in the above reading of the proof create input-coverage-checking obligations. For example, Twelf's coverage checker justifies the inversion of the typing derivation in ``preserv-app-1`` by using index information (i.e., that the subject of the derivation is ``app E1 E2``) to show that ``of-app`` is the only way to construct an LF term of the appropriate type. If there were another way to derive the well-typedness of an application, the inversion of the typing derivation in ``step-app-1`` would not be justified. This would manifest itself as an input coverage error, as not all possible inputs would be derived by ``of-app``. + +## Other theorems + +Other properties of the STLC are proved elsewhere on this wiki. Several [tutorials](/wiki/tutorials/) use theorems about the STLC to illustrate various Twelf devices. For example, proving progress for the STLC uses [output factoring](/wiki/output-factoring/); proving determinacy of the operational semantics is an example of a [uniqueness lemma](/wiki/uniqueness-lemma/). You may wish to read these tutorials at this point. Neither requires any more machinery than what we have already introduced; the tutorials simply show how to use Twelf in a fashion that may be somewhat subtle until you've seen it once. + +The next section of this introduction describes one new piece of Twelf machinery: the ability to prove totality assertions for non-empty contexts. + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-proving-metatheorems-in-non-empty-contexts.lf b/new-tests/stelf-output/pages/proving-metatheorems-proving-metatheorems-in-non-empty-contexts.lf new file mode 100644 index 0000000..50bb0f1 --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-proving-metatheorems-in-non-empty-contexts.lf @@ -0,0 +1,275 @@ +% +% +% +% +% +% +% +%{! + +_This page is part of the introduction to [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)._ + +By using Twelf's ability to prove totality assertions for type families in general worlds, we can mechanize metatheorems about open LF terms. We prove a simple metatheorem about the judgement ``size E N`` as an example. !}% +%{!! begin hidden !!}% +%sort tp %. +%term unit tp %. +%term arrow %pi tp %-> tp %-> tp %. +%sort tm %. +%term empty tm %. +%term lam %pi tp %-> (%pi tm %-> tm) %-> tm %. +%term app %pi tm %-> tm %-> tm %. +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort plus %. +%mode plus %in %in %out %. +%term plus-z plus z N2 N2 %. +%term plus-s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%worlds () (plus _ _ _) %. +%total N (plus N _ _) %. +%{!! end hidden !!}% +%{!! begin hidden !!}% +%sort plus-exists {N1} {N2} %. +%mode plus-exists %in %in %out %. +%term _ plus-exists z _ plus-z %. +%term _ %pi (plus-exists (s N1) N2 (plus-s D)) %<- (plus-exists N1 N2 D) %. +%worlds () (plus-exists _ _ _) %. +%total N1 (plus-exists N1 _ _) %. +%sort plus-z-rh {n nat} %. +%mode plus-z-rh %in %out %. +%term _ plus-z-rh z plus-z %. +%term _ %pi (plus-z-rh (s N) (plus-s D)) %<- (plus-z-rh N (%the (plus N z N) D)) %. +%worlds () (plus-z-rh _ _) %. +%total N (plus-z-rh N _) %. +%sort plus-s-rh %. +%mode plus-s-rh %in %out %. +%term _ plus-s-rh plus-z plus-z %. +%term _ + %pi (plus-s-rh (plus-s (%the (plus N1' N2 N3') D)) (plus-s D')) + %<- (plus-s-rh D (%the (plus N1' (s N2) (s N3')) D')) %. +%worlds () (plus-s-rh _ _) %. +%total D (plus-s-rh D _) %. +%sort plus-commute %. +%mode plus-commute %in %out %. +%term _ %pi (plus-commute (%the (plus z N N) plus-z) D) %<- (plus-z-rh N D) %. +%term _ + %pi (plus-commute (%the (plus (s N1) N2 (s N3)) (plus-s D)) D'') + %<- (plus-commute D (%the (plus N2 N1 N3) D')) + %<- (plus-s-rh D' (%the (plus N2 (s N1) (s N3)) D'')) %. +%worlds () (plus-commute _ _) %. +%total D (plus-commute D _) %. +%sort id/nat %. +%term id/nat-refl id/nat N N %. +%sort id/nat-s-cong %. +%mode id/nat-s-cong %in %out %. +%term _ id/nat-s-cong id/nat-refl id/nat-refl %. +%worlds () (id/nat-s-cong _ _) %. +%total {} (id/nat-s-cong _ _) %. +%sort plus-unique %. +%mode plus-unique %in %in %out %. +%term _ plus-unique D D id/nat-refl %. +%term _ + %pi (plus-unique (plus-s D) (plus-s D') DidS) + %<- (plus-unique D D' Did) + %<- (id/nat-s-cong Did DidS) %. +%worlds () (plus-unique _ _ _) %. +%total D (plus-unique D _ _) %. +%sort plus-respects-id %. +%mode plus-respects-id %in %in %in %in %out %. +%term _ plus-respects-id D id/nat-refl id/nat-refl id/nat-refl D %. +%worlds () (plus-respects-id _ _ _ _ _) %. +%total {} (plus-respects-id _ _ _ _ _) %. +%sort id/nat-sym %. +%mode id/nat-sym %in %out %. +%term _ id/nat-sym id/nat-refl id/nat-refl %. +%worlds () (id/nat-sym _ _) %. +%total {} (id/nat-sym _ _) %. +%sort plus-assoc %. +%mode plus-assoc %in %in %in %out %. +%term _ + %pi (plus-assoc plus-z DplusB-C DplusB-C' Dplus) + %<- (plus-unique DplusB-C DplusB-C' Did) + %<- (plus-respects-id plus-z id/nat-refl id/nat-refl Did Dplus) %. +%term _ + %pi (plus-assoc (plus-s DplusA'-B) DplusB-C (plus-s DplusA'B-C) (plus-s DplusA'-BC)) + %<- (plus-assoc DplusA'-B DplusB-C DplusA'B-C DplusA'-BC) %. +%worlds () (plus-assoc _ _ _ _) %. +%total D (plus-assoc D _ _ _) %. +%sort plus-assoc2 %. +%mode plus-assoc2 %in %in %out %in %. +%term _ + %pi (plus-assoc2 D1 D2 D3' D4) + %<- (plus-exists _ _ D3) + %<- (plus-assoc D1 D2 D3 D4') + %<- (plus-unique D4 D4' Did) + %<- (id/nat-sym Did Did') + %<- (plus-respects-id D3 id/nat-refl id/nat-refl Did' D3') %. +%worlds () (plus-assoc2 _ _ _ _) %. +%total {} (plus-assoc2 _ _ _ _) %. +%sort lemma %. +%mode lemma %in %in %in %in %out %. +%term _ + %pi (lemma (%the (plus N1 N2 Nsum) D) (%the (plus Ndiff1 N1 N1') D1pre) (%the (plus Ndiff2 N2 N2') D2pre) (%the (plus N1' N2' Nsum') D') Dres') + %<- (plus-commute D1pre D1) + %<- (plus-commute D2pre D2) + %<- (plus-exists Ndiff1 N2' (%the (plus Ndiff1 N2' Ndiff1+2') Ddiff1+2')) + %<- (plus-assoc D1 Ddiff1+2' D' (%the (plus N1 Ndiff1+2' Nsum') Dassoc)) + %<- (plus-commute D2 (%the (plus Ndiff2 N2 N2') D2')) + %<- (plus-exists Ndiff1 Ndiff2 (%the (plus Ndiff1 Ndiff2 Ndiff1+2) Ddiff1+2)) + %<- (plus-assoc2 Ddiff1+2 D2' (%the (plus Ndiff1+2 N2 Ndiff1+2') Dassoc') Ddiff1+2') + %<- (plus-commute Dassoc' (%the (plus N2 Ndiff1+2 Ndiff1+2') Dassoc'')) + %<- (plus-assoc2 D Dassoc'' Dres Dassoc) + %<- (plus-commute Dres Dres') %. +%worlds () (lemma _ _ _ _ _) %. +%total {} (lemma _ _ _ _ _) %. +%sort size %. +%mode size %in %out %. +%term size-empty size empty (s z) %. +%term size-lam %pi (size (lam _ E) (s N)) %<- ({x} {dx size x (s z)} size (E x) N) %. +%term size-app %pi (size (app E1 E2) (s N)) %<- (size E1 N1) %<- (size E2 N2) %<- (plus N1 N2 N) %. +%block size-block [x tm] [dx size x (s z)]%. +%worlds (size-block) (size _ _) %. +%total E (size E _) %. +%{!! end hidden !!}% +%{! ## Lower bound on size of substitution: Theorem statement + +We prove that substitution never decreases the size of a term. The informal theorem statement is as follows: + +> For all terms and , . + +We can restate this theorem relationally as follows: + +> For all terms and , if and then there exists an such that . + +For pedagogical parsimony, we have defined greater-than in terms of addition, so that we do not need to introduce any additional judgements on numbers here. Additionally, because ``size`` defines a function, we are free to assume or conclude the derivations of ``size`` as is convenient; in this case, the proof is slightly simpler if we assume these derivations. + +By adequacy, this theorem can be recast as the following statement about LF terms: + +> For all Γ in ``(size-block)*``, if ``Γ ⊦ E' : tm`` and ``Γ ⊦ D : {x:tm} size x (s z) -> size (E x) N`` and ``Γ ⊦ D' : size (E E') N'`` then there exist ``Ndiff`` and ``Dplus`` such that ``Γ ⊦ Dplus : plus Ndiff N N'``. + +Considering arbitrary contexts in the world ``(size-block)*`` is necessary to capture the informal theorem statement, which ranges over all possibly-open terms and . The ``size`` derivation for the term with a distinguished free variable is represented by an LF term of higher type ``{x:tm} size x (s z) -> size (E x) N``. + +We can prove this theorem in Twelf by giving a relation that satisfies the following totality assertion: !}% +%sort subst-size {E' tm} {x tm} %. +%mode subst-size %in %in %in %out %. +%worlds (size-block) (subst-size _ _ _ _) %. +%{! ## Proof attempt + +For pedagogical purposes, we first present an incorrect proof attempt; the change required to fix this attempt is small. The proof uses some straightforward lemmas about arithmetic: + +```twelf +plus-s-rh : plus N1 N2 N3 -> plus N1 (s N2) (s N3) -> type. +%mode plus-s-rh +D1 -D2. +%worlds () (plus-s-rh _ _). + +plus-commute : plus N1 N2 N3 -> plus N2 N1 N3 -> type. +%mode plus-commute +D1 -D2. +%worlds () (plus-commute _ _). + +lemma : plus N1 N2 Nsum + -> plus Ndiff1 N1 N1' + -> plus Ndiff2 N2 N2' + -> plus N1' N2' Nsum' + -> plus Ndiff Nsum Nsum' + -> type. +%mode lemma +X1 +X2 +X3 +X4 -X5. +%worlds () (lemma _ _ _ _ _). +``` + +The first two would be part of an arithmetic library; the third is a simple lemma that is proved directly using commutativity and associativity of addition. + +Additionally, the proof of the theorem uses the following easy lemma: !}% +%{!! begin checked !!}% +%sort size-at-least-one %. +%mode size-at-least-one %in %out %. +%term _ size-at-least-one _ (plus-s plus-z) %. +%worlds (size-block) (size-at-least-one _ _) %. +%total {} (size-at-least-one _ _) %. +%{!! end checked !!}% +%{! This relation is total because every way to create a term of type ``size E N`` syntactically produces a term of type ``size E (s N')`` for some ``N'``. The input coverage checker notices this fact by splitting and therefore validates this simple proof. + +Next, we attempt a proof of the theorem: !}% +%sort subst-size {E' tm} {x tm} %. +%mode subst-size %in %in %in %out %. +%term _ + %pi (subst-size E' ([x] [dx] dx) D Dplus') + %<- (size-at-least-one D Dplus) + %<- (plus-commute Dplus Dplus') %. +%term _ subst-size E' ([x] [dx] size-empty) size-empty plus-z %. +%term _ + %pi (subst-size E' ([x] [dx] %the (size (lam T _) _) (size-lam ([y] [dy] D x dx y dy))) (size-lam D') Dplus') + %<- ({y tm} {dy size y (s z)} subst-size E' ([x] [dx] D x dx y dy) (D' y dy) Dplus) + %<- (plus-s-rh Dplus Dplus') %. +%term _ + %pi (subst-size E' ([x] [dx] size-app (%the (plus N1 N2 Nsum) Dplus) (%the (size (E2 x) N2) (D2 x dx)) (%the (size (E1 x) N1) (D1 x dx))) (size-app (%the (plus N1' N2' Nsum') Dplus') (%the (size (E2 E') N2') D2') (%the (size (E1 E') N1') D1')) DplusRes') + %<- (subst-size E' D1 (%the (size (E1 E') N1') D1') (%the (plus Ndiff1 N1 N1') Dplus1)) + %<- (subst-size E' D2 (%the (size (E2 E') N2') D2') (%the (plus Ndiff2 N2 N2') Dplus2)) + %<- (lemma Dplus Dplus1 Dplus2 Dplus' DplusRes) + %<- (plus-s-rh DplusRes DplusRes') %. +%worlds (size-block) (subst-size _ _ _ _) %. +%{! By now, you should be getting good at reading relational Twelf proofs of metatheorems as the informal proofs they represent. This proof is slightly more involved than those that we have seen before, but it involves no new machinery. We call out some of the tricky parts of the proof: +* The proof inducts over and case-analyzes the LF term of type ``{x:tm} size x (s z) -> size (E x) N``. By inversion, such a term has the form ``[x] [dx] M`` where ``M : size (E x) N``. Thus, the proof includes one case for each possible ``M``. This includes each constructor of type ``size``, along with the distinguished variable ``x``. +* Make sure you understand why each constant world checks: The ``subst-size`` premise of the case for ``size-lam`` is in an extended context, but that this context stays in the appropriate world. Additionally, by world subsumption, ``(size-block)*`` subsumes the world containing only the empty context for the arithmetic lemmas. + +However, the totality check fails: + +```checkedtwelf +%total D (subst-size _ D _ _). +``` + +This error message tells us that this proof does not cover the case for variables in the context itself. That is, we forgot to cover the case when the size derivation is a variable (representing the use of a hypothesis) but is not the distinguished variable ``x``. + +## Corrected proof + +To fix the proof, we need to cover the case for other variables in the LF context. But where can we put such a theorem case, since it needs to talk about the variable in the context? The answer is that we can use LF hypotheses: we assume a case of the theorem ``subst-size`` in the LF context along with every assumption of ``x:tm,dx:size x (s z)``. This technique works because proofs of metatheorems are just LF constants of particular types. + +To correct the proof, we work over contexts of a different form. Only the ``size-lam`` case needs to change, so we elide the others: !}% +%sort subst-size {E' tm} {x tm} %. +%mode subst-size %in %in %in %out %. +%term _ + %pi (subst-size E' ([x] [dx] %the (size (lam T _) _) (size-lam ([y] [dy] D x dx y dy))) (size-lam D') Dplus') + %<- ({y tm} {dy size y (s z)} {_ {E' tm} subst-size E' ([x tm] [dx size x (s z)] dy) dy plus-z} subst-size E' ([x] [dx] D x dx y dy) (D' y dy) Dplus) + %<- (plus-s-rh Dplus Dplus') %. +%% ... +%{! !}% +%{!! begin hidden !!}% +%term _ + %pi (subst-size E' ([x] [dx] dx) D Dplus') + %<- (size-at-least-one D Dplus) + %<- (plus-commute Dplus Dplus') %. +%term _ subst-size E' ([x] [dx] size-empty) size-empty plus-z %. +%term _ + %pi (subst-size E' ([x] [dx] size-app (%the (plus N1 N2 Nsum) Dplus) (%the (size (E2 x) N2) (D2 x dx)) (%the (size (E1 x) N1) (D1 x dx))) (size-app (%the (plus N1' N2' Nsum') Dplus') (%the (size (E2 E') N2') D2') (%the (size (E1 E') N1') D1')) DplusRes') + %<- (subst-size E' D1 (%the (size (E1 E') N1') D1') (%the (plus Ndiff1 N1 N1') Dplus1)) + %<- (subst-size E' D2 (%the (size (E2 E') N2') D2') (%the (plus Ndiff2 N2 N2') Dplus2)) + %<- (lemma Dplus Dplus1 Dplus2 Dplus' DplusRes) + %<- (plus-s-rh DplusRes DplusRes') %. +%{!! end hidden !!}% +%{!! begin checked !!}% +%block ssblock [y tm] [dy size y (s z)] [_ {E' tm} subst-size E' ([x] [dx] dy) dy plus-z]%. +%worlds (ssblock) (subst-size _ _ _ _) %. +%total D (subst-size _ D _ _) %. +%{!! end checked !!}% +%{! The block ``ssblock`` extends ``size-block`` with a case of the theorem: when the size derivation is ``dy`` from the context, the term ``E`` must be ``[_] y``, so by inversion the derivation of ``size (([_] y) E') N'`` must be ``dy`` as well, in which case ``plus-z`` derives ``plus z (s z) (s z)``. The only other change is that the premise of the case for ``size-lam`` adds this extra assumption to the context. In this world, the relation is indeed total. + +## Correctness of the revised theorem statement + +We set out to prove + +> For all Γ in ``(size-block)*``, if ``Γ ⊦ E' : tm`` and ``Γ ⊦ D : {x:tm} size x (s z) -> size (E x) N`` and ``Γ ⊦ D' : size (E E') N'`` then there exist ``Ndiff`` and ``Dplus`` such that ``Γ ⊦ Dplus : plus Ndiff N N'``. + +But the corrected Twelf proof actually proves + +> For all Γ in ``(ssblock)*``, if ``Γ ⊦ E' : tm`` and ``Γ ⊦ D : {x:tm} size x (s z) -> size (E x) N`` and ``Γ ⊦ D' : size (E E') N'`` then there exist ``Ndiff`` and ``Dplus`` such that ``Γ ⊦ Dplus : plus Ndiff N N'``. + +Did we prove the right theorem? Not necessarily: it is possible that the contexts in ``(ssblock)*`` do not cover all open terms, or that they include inadequate derivations of ``plus Ndiff N N'``. Thus, there is a danger here: we need to check that the second statement implies the first to see that we proved the right theorem. + +In this case, the implication holds. The reason is that every context in ``(ssblock)*`` corresponds to a context in ``(size-block)*`` when restricted to those declarations subordinate to the types in the theorem statement. That is, the context extension in ``(ssblock)*`` does not change the inhabitants of the types in the theorem statement—all it does is add a case of the proof type family ``subst-size``, which is not subordinate to any of the subjects of the proof. This is another argument that can be made formal using [subordination](/wiki/subordination/). + +When you recast an informal theorem statement as a Twelf theorem statement, you need to check that the world of the theorem actually yields the theorem you had in mind. However, in a large Twelf project, this obligation is usually not very large: Usually, a Twelf proof consists of a few top-level theorems of interest and many supporting lemmas. In this case, you only need to check that these theorems correspond to what you had in mind; the various lemmas can mean whatever they mean, as long as they are strong enough to prove the overall theorem. + +## An even better proof + +Putting a case of a theorem in the context is a general technique that is useful in many circumstances. However, in this example, we can avoid putting the theorem case in the context by using a [catch-all case](/wiki/catch-all-case/), as the tutorial on that Twelf proof device explains. + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-proving-totality-assertions-about-the-natural-numbers.lf b/new-tests/stelf-output/pages/proving-metatheorems-proving-totality-assertions-about-the-natural-numbers.lf new file mode 100644 index 0000000..eac285c --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-proving-totality-assertions-about-the-natural-numbers.lf @@ -0,0 +1,208 @@ +% +% +% +% +% +% +% +% +%{! + +_This page is part of the introduction to [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)._ + +In this section, we will use Twelf to mechanize proofs of [metatheorems](/wiki/metatheorem/), which are statements about an object language. First, we recap the signature we have defined to this point, using the conveniences afforded by Twelf: !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort even %. +%term even-z even z %. +%term even-s %pi (even (s (s N))) %<- (even N) %. +%sort plus %. +%term plus-z plus z N2 N2 %. +%term plus-s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%{! ## Totality assertions + +A type family such as ``plus`` defines a relation among its indices, where indices are related iff their instance of the family is inhabited by an LF term. For example, ``plus`` defines a relation between three terms of type ``nat``. The terms (``s (s z)``, ``s z``, ``s (s (s z))``) are related by this relation (because ), but the terms (``z``, ``s z``, ``s (s (s z))``) are not. + +[Totality assertions](/wiki/totality-assertion/) are one class of statements about LF type families. A totality assertion for a type family is specified by designating some index positions as inputs and others positions as outputs. We call this specification the [mode](/wiki/percent-mode/) of the totality assertion. Given a mode specification, the totality assertion asserts that for all inputs, there exist outputs that stand in the relation. Using ``plus`` as an example, if we designate the first two positions as inputs and the third position as an output, this specifies the following totality assertion: + +> For all ``N1:nat`` and ``N2:nat``, there exist ``N3:nat`` where ``D:plus N1 N2 N3``. + +That is, the sum of any two natural numbers exists. Designating instead the first position as output and second and third positions as inputs defines the following totality assertion: + +> For all ``N2:nat`` and ``N3:nat``, there exist ``N1:nat`` where ``D:plus N1 N2 N3``. + +This second totality assertion is false, as ``N3 - N2`` might not be a natural number. Specifically, there is no ``N1`` such that ``plus N1 (s z) z`` is inhabited. + +A totality assertion for an LF type family corresponds with the standard notion of totality for the relation defined by the family. Proving a relation total is different from showing that the relation defines a function, as the outputs of a total relation are not necessarily unique—the relation may relate particular inputs to more than one collection of outputs. + +We may prove a totality assertion by induction on canonical forms. For example, we can prove the first above totality assertion by induction on ``N1:nat``. + +Totality assertions for LF type families are actually more general than we have discussed so far. The above examples consider only terms that are well-typed without mentioning LF variables. In general, it is useful to state totality assertions about all terms that are well-typed in any LF context in a particular [world](/wiki/percent-worlds/), which is a set of LF contexts. Thus, a totality assertion is specified by both a mode declaration and a world declaration. For the time being, we will consider totality assertions that are stated for the world containing only the empty LF context (i.e., totality assertions about closed terms), so we defer further discussion of worlds. + +## Totality assertions in Twelf + +You can use Twelf to verify totality assertions about LF type families. For example, to verify the totality assertion + +> For all ``N1:nat`` and ``N2:nat``, there exist ``N3:nat`` and ``D:plus N1 N2 N3``. + +you enter the following Twelf declarations: + +```checkedtwelf +%% declare totality assertion +%mode plus +N1 +N2 -N3. +%worlds () (plus _ _ _). + +%% check totality assertion +%total N1 (plus N1 _ _). +``` + +Let's break down what each of these declarations means: +* The [``%mode``](/wiki/percent-mode/) declaration specifies the mode of a type family—in this case, that ``N1`` and ``N2`` are universally quantified (``+``) and that ``N3`` is existentially quantified (``-``). We will sometimes refer to the universally-quantified types as _inputs_ and the existentially-quantified types as _outputs_. +* The [``%worlds``](/wiki/percent-worlds/) declaration states the totality assertion for LF terms in the empty context, which adequately represent natural numbers and the addition judgement. +* The [``%total``](/wiki/percent-total/) declaration asks Twelf to prove the totality assertion by induction on the canonical forms ``N1`` of type ``nat``. Note that it is the position of the variable named in the ``%total`` that determines the induction index. + +In this case, Twelf succeeds in proving this assertion. + +## How Twelf checks assertions + +Twelf proves a totality assertion for a type family such as ``plus`` by checking several properties. These properties, taken together, constitute a proof by induction on canonical forms that the type family defines a total relation. + +### Mode + +Twelf checks that each constant inhabiting the type family is _well-moded_. Roughly, this means that the inputs to the conclusion of a constant determine the inputs of the first premise, and that these together with the outputs of the first premise determine the inputs of the second premise, and so on, until the outputs of all the premises determine the outputs of the conclusion. + +For example, the constant + +!}% +%{!! begin hidden !!}% +%mode plus %in %in %out %. +%{!! end hidden !!}% +%{! + +```twelf +plus-s : plus (s N1) N2 (s N3) + <- plus N1 N2 N3. +``` + +has mode ``+N1 +N2 -N3`` because the input ``N1`` and ``N2`` in the conclusion determine the inputs of the premise, and the ``N3`` output by the premise determines the output of the conclusion. On the other hand, a constant + +```checkedtwelf +plus-bad-mode : plus N1 N2 N3. +``` + +is not well-moded—the output ``N3`` is not determined by the inputs. Similarly, + +```checkedtwelf +plus-bad-mode2 : plus N1 N2 N3 + <- plus N4 N2 N3. +``` + +is not well-moded—the first input to the premise is not determined by the inputs of the conclusion. + +### Worlds + +Twelf checks that each constant inhabiting the type family obeys the worlds declaration. Because we are only proving theorems about closed terms in this section, we will not run across any problems with world checks. Consequently, we defer discussion of world checking until [later](/wiki/proving-metatheorems-proving-totality-assertions-in-non-empty-contexts/). + +### Termination + +Twelf checks that each constant inhabiting the type family obeys the induction order specified in the ``%total`` declaration. In each inductive premise of a constant, the specified induction position must be a strict subterm of the corresponding argument in the conclusion. For example, the constant + +```twelf +plus-s : plus (s N1) N2 (s N3) + <- plus N1 N2 N3. +``` + +obeys the induction order ``N1`` specified in the above totality assertion because the term ``N1`` is a strict subterm of the term ``(s N1)``. On the other hand, Twelf would not accept the totality of ``plus`` if ``N2`` were used as the induction order—the same term ``N2`` in the conclusion of this constant appears in the premise: + +```checkedtwelf +%total N2 (plus _ N2 _). +``` + +In addition to the subterm ordering on a single argument, Twelf supports [mutual induction](/wiki/mutual-induction/) and [lexicographic induction](/wiki/percent-terminates/). + +### Output coverage !}% +%{! In the definition of a type family, you may pattern-match the outputs of a premise. For example, we might write + +```checkedtwelf +%mode plus +N1 +N2 -N3. +plus-bad-output : plus (s N1) N2 (s (s N3)) + <- plus N1 N2 (s N3). +%worlds () (plus _ _ _). +%total N1 (plus N1 N2 N3). +``` + +Here we have insisted that the output of the premise has the form ``s N3`` for some ``N3``. Twelf correctly reports an output coverage error because this condition can fail (for example, if the premise was ``plus-z : plus z z z``). + +Pattern-matching the output of a premise is like an inversion step in a proof: you're insisting that the premise derivation must conclude a particular fact that is more specific than the judgement form itself. For Twelf to accept a relation as total, Twelf must notice that all of these inversions are permissible. Twelf permits such inversions when it is readily apparent that they are justified, and those inversions that Twelf does not accept can be proved explicitly. + +In this example, we got an output coverage error because we constrained the output of the premise by insisting it be formed by a particular constant. The other way to get output coverage errors is to insist that the output of a premise be a variable that occurs elsewhere in the type. For example: + +```checkedtwelf +%mode plus +N1 +N2 -N3. +plus-bad-output-freeness : plus (s N1) N2 (s N2) + <- plus N1 N2 N2. +%worlds () (plus _ _ _). +%total N1 (plus N1 N2 N3). +``` + +Here, we insisted that the output of the premise be the number ``N2`` that we put in. Twelf is very conservative in checking [output freeness](/wiki/output-freeness/): a type family will not be judged total if you constrain the outputs of any premise at all in this manner. + +### Input coverage + +Mode, worlds, termination, and output coverage ensure that each constant really does cover the part of the relation indicated by its conclusion. For example, if ``plus`` passes these four checks, we know that ``plus-z`` and ``plus-s`` cover ``(z, N2, N2)`` and ``(s N1, N2, s N3)``, respectively. What else is necessary to know that ``plus`` defines a total relation? We need to know that all the constants inhabiting ``plus``, taken together, cover all of the inputs. Input coverage checks exactly this. + +For example, if we forgot ``plus-z``, input coverage for ``plus`` would fail. That would look like this: + +```checkedtwelf +plus : nat -> nat -> nat -> type. +%mode plus +N1 +N2 -X3. + +plus-s : plus (s N1) N2 (s N3) + <- plus N1 N2 N3. + +%worlds () (plus _ _ _). +%total N1 (plus N1 _ _). +``` + +Here's an analogy that might be helpful: You can think of each constant of a type as being a clause in an ML pattern matching declaration. Then input coverage is like the exhaustiveness checker for pattern matching. +The type family ``plus`` is exhaustive, but this ``plus'`` is not. + +Twelf checks input coverage by splitting the input types to case-analyze the various constants that could have been used to inhabit them. For ``plus``, Twelf splits the first ``nat`` argument ``N1``, and then checks that the cases ``plus z N2 N2`` and ``plus (s N1) N2 N3`` are covered. Fortunately, these are exactly the cases we wrote down. If we had case-analyzed further in the definition of the judgement (e.g., if the definition of ``plus`` case-analyzed the second argument as well), Twelf would continue splitting the input space. Because Twelf separates termination checking and coverage checking, the constants defining a type family do not need to follow any particular primitive recursion schema—the constants may pattern-match the inputs in a general manner. + +## When Twelf checks what + +To a first approximation, you can think of the ``%mode`` and ``%worlds`` declarations as specifying a totality assertion and the ``%total`` declaration as checking it. This isn't exactly how Twelf works, though: + +- When a ``%mode`` declaration is entered, Twelf checks that all previous constants inhabiting the specified type family are well-moded; further, it then mode-checks any subsequent constants inhabiting that family. +- When a ``%worlds`` declaration is entered, Twelf world-checks the type family; further, it then reports an error if any new constants contributing to the family at all are added. +- When a ``%total`` declaration is entered, Twelf checks termination, then input coverage, then output coverage. When checking output coverage, Twelf checks for unjustified constant pattern-matching in a first pass and then output freeness problems in a second pass. + +This separation allows you to, for example, check that each constant in a family is well-moded (i.e., takes specified inputs to specified outputs) without checking that the entire type family is total. You can also use the declarations [%terminates](/wiki/percent-terminates/) and [%covers](/wiki/percent-covers/) to check termination and input coverage independently. + +If any constant in a type family fails mode, worlds, or output coverage, then mode, worlds, or totality checking fails for the whole type family. One could imagine that Twelf instead would just disregard the offending constant: it is possible that the type family as a whole satisfies a totality assertion without that constant, and, in a mathematical sense, adding additional constants never invalidates the fact a totality assertion is true of a family. The reason Twelf does not work this way is that ``%total`` actually has a more specific meaning, as we discuss in the next section. + +## Logic programming + +An LF signature can be interpreted as a logic program, where the constants defining a type family are viewed as rules for searching for derivations of that type family. For example, we can use Twelf to add numbers as follows: !}% +%{!! begin checked !!}% +%solve D : plus (s z) (s z) N %. +%{!! end checked !!}% +%{! This [%solve](/wiki/percent-solve/) declaration asks Twelf to find an ``N`` and a derivation ``D`` that ``plus (s z) (s z) N``. Of course, Twelf responds that 1 + 1 = 2. + +This logic programming interpretation of an LF signature is very useful: you can directly run the judgements defining your programming language. However, because the focus of this introduction is proving metatheorems, an interested reader should refer to the tutorial on [logic programming](/wiki/logic-programming/). + +The reason for bringing up logic programming here is that, when Twelf proves a totality assertion via ``%total``, it actually proves a stronger statement than the totality assertion itself: Twelf proves that the type family defines a total logic program. The above checks have an operational interpretation: +* Mode: when called on [ground](/wiki/ground/) inputs (i.e., concrete terms without [unification](/wiki/unification/) variables), the logic program returns ground outputs. +* Worlds: all premises of every constant are in a context in the world specified by the premise's world declaration. +* Termination: search for a derivation always terminates. +* Output coverage: the output of a recursive call never fails to unify. +* Input coverage: the logic program covers all possible inputs. + +To start proving metatheorems, it is not necessary to have a thorough understanding of the logic programming interpretation of the LF signature. However, the logic programming view does explain some of the specifics of what Twelf will and will not deem total. For example, mode checking considers premises in the order that it does (rather than, say, the reverse order) because that is the order in which the logic programming engine searches for derivations. + +## Summary + +In this section, we have seen how to ask Twelf to verify totality assertions for LF type families. To see why we have spent so much time on totality assertions, proceed to the next section. + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-proving-totality-assertions-in-non-empty-contexts.lf b/new-tests/stelf-output/pages/proving-metatheorems-proving-totality-assertions-in-non-empty-contexts.lf new file mode 100644 index 0000000..bf1cd1f --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-proving-totality-assertions-in-non-empty-contexts.lf @@ -0,0 +1,211 @@ +% +% +% +% +% +% +% +%{! + +_This page is part of the introduction to [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)._ + +So far, we have only proved relations total in the empty LF context. In this section, we discuss proving totality assertions for relations in more general contexts. As a motivating example, we define a judgement that computes the size of a STLC term. + +## Definition of Size + +The definition of size uses the definitions of [natural numbers](/wiki/proving-metatheorems-representing-the-syntax-of-the-natural-numbers/), [addition judgement](/wiki/proving-metatheorems-representing-the-judgements-of-the-natural-numbers/), and [STLC term syntax](/wiki/proving-metatheorems-representing-the-syntax-of-the-stlc/) from earlier in this introduction: !}% +%{!! begin hidden !!}% +%sort tp %. +%term unit tp %. +%term arrow %pi tp %-> tp %-> tp %. +%sort tm %. +%term empty tm %. +%term lam %pi tp %-> (%pi tm %-> tm) %-> tm %. +%term app %pi tm %-> tm %-> tm %. +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort plus %. +%mode plus %in %in %out %. +%term plus-z plus z N2 N2 %. +%term plus-s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%worlds () (plus _ _ _) %. +%total N (plus N _ _) %. +%{!! end hidden !!}% +%{! Informally, we define the size of a term as follows: +* +* +* +* + +We define this function in LF using the following judgement: !}% +%{!! begin checked !!}% +%sort size %. +%mode size %in %out %. +%term size-empty size empty (s z) %. +%term size-lam %pi (size (lam _ E) (s N)) %<- ({x tm} {_ size x (s z)} size (E x) N) %. +%term size-app %pi (size (app E1 E2) (s N)) %<- (size E1 N1) %<- (size E2 N2) %<- (plus N1 N2 N) %. +%{!! end checked !!}% +%{! The only subtlety in this judgement is that we have recast size as a hypothetical judgement: each variable ``x:tm`` in the context is assumed to have size 1. Correspondingly, the premise of the ``lam`` case extends the context with such an assumption for the bound variable. + +## World Declarations + +We would like to assert that every term has a size. However, because the definition of the size judgement extends the LF context, our usual worlds declaration fails: + +```checkedtwelf +%worlds () (size _ _). +``` + +The worlds declaration fails because ``size`` does not stay in the empty context. + +To give an appropriate worlds declaration for this type family, we must first specify the form of the contexts in which we consider it. We do so with a block declaration: !}% +%block size-block [x tm] [_ size x (s z)]%. +%worlds (size-block) (size _ _) %. +%{! The block declaration names a context block comprised of the assumptions ``x:tm, _:size x (s z)``. The world declaration declares ``size`` for any context composed of any number of ``size-blocks``. That +is, it specifies that we consider ``size`` in the [regular world](/wiki/percent-worlds/) ``(size-block)*``. + +In general, Twelf permits regular world declarations of the form ``(b1 | b2 | ... | bn)*`` for blocks ``b1`` through ``bn`` defined by ``%block`` declarations. + +## World Checking + +Up until this point, we have focused on the role a ``%worlds`` declaration has in specifying a totality assertion. However, just as ``%mode`` both specifies the mode of a totality assertion and mode-checks a type family, ``%worlds`` both specifies the world of a totality assertion and _world checks_ a type family. + +What property does a world declaration specify? Intuitively, a world declaration for a type family circumscribes the contexts in which we consider inhabitants of that type family. For example, the world declaration for ``size`` says that we should only ever consider terms of type ``size`` in contexts in ``(sizeblock)*``. Thus, it is a world error if some piece of Twelf code introduces a variable ``x:tm`` but not a derivation ``_:size x (s z)``, or if it introduces a variable ``_:size x z``, or if it introduces a variable ``_:size empty N``, and so on. World checking ensures that we only consider a term in the contexts declared for its type. + +When Twelf processes a ``%worlds`` declaration, it world checks each constant in the type family. A constant _world checks_ under the following condition: whenever the ambient LF context is of the form specified by the world declaration, each premise of the constant occurs in an LF context that is within the world specified for the premise's type family. For example: +* ``size-empty`` has no premises, so it vacuously world checks. +* the ``size`` premise of ``size-lam`` is in an extended context, but adding ``x:tm,_:size x (s z)`` to a context in ``(size-block)*`` creates another context in ``(size-block)*``. +* the ``size`` premises of ``size-app`` do not extend the LF context, so they are in ``(size-block)*`` whenever the ambient context is. +* the ``plus`` premise of ``size-app`` is **not** directly in a context in ``plus``'s world (the world containing only the empty context) because the ambient LF context will contain variables of type ``tm`` and ``size``. However, the call to ``plus`` is permissible because of something called [world subsumption](/wiki/world-subsumption/). + +### Subordination and world subsumption + +A context in ``(size-block)*`` has the form + +``x:tm, _:size x (s z), ...`` + +These contexts are not in the world for ``plus``, which contains only the empty LF context. Thus, it would seem that the call to ``plus`` from ``size`` should be deemed a world violation. + +However, considering derivations of ``plus`` in contexts of this form is in fact no different from considering these derivations in the empty context, because variables of type ``tm`` and ``size`` do not influence the set of terms of type ``plus``. That is, we are considering contexts outside of the world of ``plus``, but the extra assumptions in the context are irrelevant to ``plus``. By improving our definition of world checking to account for circumstances like these, we can allow more world correct signatures, while maintaining the property that the world declaration circumcribes the **relevant** parts of the contexts in which we consider the terms of that type. + +Formalizing this reasoning requires knowing when terms of one type can appear in terms of another. This information is tracked by a [subordination](/wiki/subordination/) relation, which is a pre-order (reflexive and transitive binary relation) on type family constants. Terms of a type ``a`` are only permitted to appear in terms of type ``b`` if ``a`` is subordinate to ``b``. In practice, Twelf infers a subordination relation from the Twelf signature that permits all relationships implied by the signature. + +Using subordination, we define world subsumption as follows: + +> **World subsumption:** A world ``W'`` subsumes a world ``W`` for a type family ``a`` iff for all contexts Γ' in ``W'``, there exists a context Γ in ``W`` such that Γ and Γ' are equal when restricted to only the declarations at types subordinate to ``a``. + +For example, ``(size-block)*`` subsumes the world containing only the empty context for ``plus``. On the other hand, the world ``({x:nat})*`` does not subsume the world containing only the empty context for ``plus``, because adding variables representing more natural numbers is relevant to ``plus``. + +Using world subsumption, we refine our notion of world checking as follows. A constant world checks iff +- The world of a type family subsumes the world of each of its premises. +- Each premise's local assumptions stay in the world for that type family. + +These conditions ensure that whenever the ambient LF context is of the form specified by the world declaration, for each premise of the constant, the world for the premise contains the restriction of the ambient context to the premise appended with the local assumptions made by the premise. + +Under this more permissive definition, the ``plus`` premise of ``size`` is deemed world-correct. + +## Totality assertions in non-empty worlds + +We now return to the role a world declaration has in specifying totality assertions. The above mode and world declarations for ``size`` declare the following totality assertion: + +> For all LF contexts Γ in ``(size-block)*``, if ``Γ ⊢ E : tm`` then there exist ``N`` such that ``Γ ⊢ N : nat`` and ``D`` such that ``Γ ⊢ D : size E N``. + +There are two things to note about this totality assertion: +- It generalizes those that we have seen before, where the world always contained only the empty context and consequently the assertion was about closed terms. +- The context is quantified at the outside and is consistent across all the indices to the type family. That is, given inputs in a particular context, the relation returns outputs in that same context. + +Twelf verifies totality assertions in non-empty contexts in the same manner as before. If a type family world checks, then the inductive hypothesis of a totality relation for a recursive premise (or the previously established totality assertion for a prior type family) will never fail to apply because of the form of the context. Thus, the only differences in checking totality assertions for general contexts are that world checking may fail, and that coverage checking must account for variables from the context. + +In this case, Twelf succeeds in proving the totality of ``size``. !}% +%{!! begin checked !!}% +%total E (size E _) %. +%{!! end checked !!}% +%{! ## Example world errors + +In this section, we show some example world errors. For comparison, we collect the complete correct definition of ``size`` here: + +```checkedtwelf +size : tm -> nat -> type. +%mode size +E -N. + +size-empty : size empty (s z). +size-lam : size (lam _ E) (s N) + <- ({x : tm} + {_ : size x (s z)} + size (E x) N). +size-app : size (app E1 E2) (s N) + <- size E1 N1 + <- size E2 N2 + <- plus N1 N2 N. + +%block size-block : block {x : tm} {_ : size x (s z)}. +%worlds (size-block) (size _ _). +%total E (size E _). +``` + +### Premise in an incorrect context + +One possible world error is to write a premise whose local context extensions do not stay in the appropriate world. For example, if we forgot the ``size`` assumption in the ``lam`` case, Twelf would report an error: + +!}% +%{!! begin hidden !!}% +%sort size %. +%mode size %in %out %. +%term size-empty size empty (s z) %. +%term size-app %pi (size (app E1 E2) (s N)) %<- (size E1 N1) %<- (size E2 N2) %<- (plus N1 N2 N) %. +%{!! end hidden !!}% +%{! + +```checkedtwelf +size-lam : size (lam _ E) (s N) + <- ({x : tm} + size (E x) N). +%block size-block : block {x : tm} {_ : size x (s z)}. +%worlds (size-block) (size _ _). +``` + +### World subsumption failure + +Another possible world error is that world subsumption could fail. For example, if ``size`` were stated for a world containing natural number assumptions (which of course ruins the adequacy of ``nat`` for the natural numbers), it would not be permissible to call ``plus``: + +!}% +%{!! begin hidden !!}% +%sort size %. +%mode size %in %out %. +%term size-empty size empty (s z) %. +%term size-lam %pi (size (lam _ E) (s N)) %<- ({x tm} {_ size x (s z)} size (E x) N) %. +%term size-app %pi (size (app E1 E2) (s N)) %<- (size E1 N1) %<- (size E2 N2) %<- (plus N1 N2 N) %. +%{!! end hidden !!}% +%{! + +```checkedtwelf +%block size-block : block {x : tm} {_ : size x (s z)}. +%block natblock : block {x:nat}. +%worlds (natblock | size-block) (size _ _). +``` + +### Totality failure because of a forgotten assumption + +Finally, a type family can be world correct but fail to define a total relation if we consider the theorem in an improper world. For example, say we had defined ``size`` as follows: + +```checkedtwelf +size : tm -> nat -> type. +%mode size +E -N. + +size-empty : size empty (s z). +size-lam : size (lam _ E) (s N) + <- ({x : tm} size (E x) N). +size-app : size (app E1 E2) (s N) + <- size E1 N1 + <- size E2 N2 + <- plus N1 N2 N. + +%block size-block : block {x : tm}. +%worlds (size-block) (size _ _). +%total E (size E _). +``` + +This definition world checks because the context in ``size-lam`` is of the form specified by the world declaration. However, the type family ``size`` does not define a total relation because it does not cover +variables ``x:tm`` in the LF context. A total relation must cover both the LF constants in the signature and the LF variables in all contexts in the world. + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-representing-the-judgements-of-the-natural-numbers.lf b/new-tests/stelf-output/pages/proving-metatheorems-representing-the-judgements-of-the-natural-numbers.lf new file mode 100644 index 0000000..cfff615 --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-representing-the-judgements-of-the-natural-numbers.lf @@ -0,0 +1,130 @@ +% +% +% +% +% +% +% +% +%{! + +_This page is part of the introduction to [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)._ + +Now that we've seen how to represent the syntax of a deductive system, we turn to representing judgements. Our first example is the judgement that a number is even. + +## Even numbers + +### Informal definition + +The judgement holds when is even. It is inductively defined by the following inference rules: + +```math +{ + \over + {\mathsf{even}(\mathsf{zero})} +} +\qquad +{ + {\mathsf{even}(n)} + \over + {\mathsf{even}(\mathsf{succ}(\mathsf{succ}(n)))} +} +``` + + +### LF representation + +In the previous sections, we saw how to represent object-language syntax as the inhabitants of LF types. At a high level, the LF methodology for representing judgements is exactly analogous: we represent an object-language judgement with an LF type, where the inhabitants of this type correspond exactly to derivations of the judgement. + +However, to adequately represent judgements, we must generalize from simply-typed LF to dependently-typed LF. A moment's thought reveals why a dependently-typed language is appropriate: Object-language judgements are parametrized by their subjects—for example is parametrized by the number being judged to be even. Consequently, to represent judgements themselves as LF types, we should consider LF types that are parametrized by the subjects of object-language judgements. But the judgement subjects—the syntax of the language—are represented as LF terms. Thus, to represent judgements themselves as LF types, it is natural to consider families of LF types parameterized by LF terms. + +We represent the judgement using the following signature: !}% +%{!! begin hidden !!}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%{!! end hidden !!}% +%sort even %. +%term even-z even z %. +%term even-s {N nat} %pi (even N) %-> (even (s (s N))) %. +%{! The first declaration says that ``even`` is a family of types indexed by a ``nat``. This means that for every term ``N : nat``, there is a type ``even N``. Note that the syntax ``->`` is overloaded: it is used to classify both type-level families and and term-level functions. We then use this type +family to define the types of two term constants. + +The first term constant, ``even-z``, has type ``even z``. This constant represents the derivation that consists of the first inference rule above, which concludes . + +The second term constant ``even-s``, corresponds to the second inference rule above, which, for any , constructs a derivation of from a derivation of . To encode this inference rule, the constant ``even-s`` is given a [dependent function type](/wiki/dependent-types/). + +The syntax ``{x:A1} A2`` is used to give a type to `even-s` (replacing the term variable `x` with `N`, the type `A1` with `nat`, and the type `A2` with `even N -> even (s (s N))`). This syntax represents a _dependent function type_, which is a generalization of the ordinary function type ``A1 -> A2`` that allows the argument term ``x`` to appear in ``A2``. We write the ordinary function type ``->`` as a synonym when the argument is not free in the result type (i.e., ``A1 -> A2`` means ``{_:A1} A2``). Just as with the ordinary function type, LF terms of dependent function type can be applied to a term of the argument type to form a new term. + +As an example, the constant ``even-s`` can be applied to a term ``N:nat`` and then a term of type +``even N`` _for that ``N``_ to create a term of type ``even (s (s N))`` (again, for that ``N``). It looks like this: !}% +%{!! begin checked !!}% +%define test even-s N D %. +%{!! end checked !!}% +%{! The dependent function type is used to bind the variable ``N`` in the type ``(even N -> even (s (s N)))``, expressing that the inference rule is schematic in . + +When a term of dependent function type is applied to an argument, the argument is substituted into the result type. For example, the term ``even-s z`` has type ``even z -> even (s (s z))``—i.e., it is the specialization of the inference rule to . That looks like this: !}% +%{!! begin checked !!}% +%define two-is-even even-s z even-z %. +%{!! end checked !!}% +%{! + +This represents the derivation + +```math +{\overline{\mathsf{even}(\mathsf{zero})}} +\over +{\mathsf{even}(\mathsf{succ}(\mathsf{succ}(\mathsf{zero})))} +``` + +The term ``even-s (s (s z)) (even-s z even-z)`` represents a derivation that 4 is even, and so on. + +To summarize, dependently-typed LF refines the type level of simply-typed LF with dependent function types and families of types parametrized by LF terms. The terms of LF remain unchanged from the grammar presented in the previous section. These changes are sufficient for representing judgements as LF type families and derivations as LF terms. + +### Adequacy + +To check that this LF representation is correct ([adequate](/wiki/adequacy/)), we must verify that there is a bijection between the derivations of and the LF terms of type ``even N`` where . In particular, this LF signature adequately represents natural numbers and derivations of when considered in the empty LF context. It would not be adequate in contexts containing assumptions of the form ``x : nat`` (which don't correspond to any natural number) or ``d : even N`` (which don't correspond to any of the rules for deriving ). + +Part of establishing this bijection requires reasoning by induction on the canonical forms of type ``even N``, which corresponds to rule induction on the informal judgement. The definition of the encoding and the proof of adequacy are simple; interested readers are referred to the literature for examples. + +## Addition + +### Informal definition + +Addition can be represented as a judgement relating two natural numbers to their sum: + +```math +{ + \over + {\mathsf{plus}(\mathsf{zero},n_2,n_2)} +} +\qquad +{ + {\mathsf{plus}(n_1,n_2,n_3)} + \over + {\mathsf{plus}(\mathsf{succ}(n_1), n_2, \mathsf{succ}(n_3))} +} +``` + +This judgement defines addition by induction on the first argument. + +### LF representation + +This judgement is represented by the following LF signature: !}% +%sort plus %. +%term plus-z {N2 nat} plus z N2 N2 %. +%term plus-s {N1 nat} {N2 nat} {N3 nat} %pi (plus N1 N2 N3) %-> (plus (s N1) N2 (s N3)) %. +%{! The type family ``plus`` is indexed by three terms of type ``nat`` because the informal judgement has three parameters. The constants correspond to the two inference rules, inhabiting the type family with terms representing derivations. For example, this term represents a derivation that . !}% +%{!! begin checked !!}% +%define test2 plus-s (s z) (s z) (s (s z)) (plus-s z (s z) (s z) (plus-z (s z))) %. +%{!! end checked !!}% +%{! + +### Adequacy + +To check correctness of this representation, we must verify that there is a bijection between derivations of and LF terms of type +``plus N1 N2 N3`` where . Once again, the definition of the encoding judgement and the adequacy proof are simple. + +These adequacy theorems will be more precise after we review the full language of dependently typed LF in the next section. + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-representing-the-judgements-of-the-stlc.lf b/new-tests/stelf-output/pages/proving-metatheorems-representing-the-judgements-of-the-stlc.lf new file mode 100644 index 0000000..f459045 --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-representing-the-judgements-of-the-stlc.lf @@ -0,0 +1,185 @@ +% +% +% +% +% +% +% +% +%{! + +_This page is part of the introduction to [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)._ + +## Static semantics + +### Informal definition + +The typing rules for the simply typed lambda calculus use a typing context containing assumptions of the form . Such a context is well-formed when all variables in it are distinct. + +```math +{ + \over + \gamma \vdash \langle\rangle : \mathsf{unit} +}\mathit{of\textit{-}empty} +\qquad +{ + x : \tau \in \gamma + \over + \gamma \vdash x : \tau +}\mathit{of\textit{-}var} +``` +```math +{ + \gamma, x :\tau_2 \vdash e : \tau + \over + \gamma \vdash \lambda x{:}\tau_2.e : (\tau_2 \rightarrow \tau) +}\mathit{of\textit{-}lam} +\qquad +{ + \gamma \vdash e_1 : (\tau_2 \rightarrow \tau) + \qquad + \gamma \vdash e_2 : \tau_2 + \over + \gamma \vdash e_1\,e_2 : \tau +}\mathit{of\textit{-}app} +``` + +This is a _hypothetical judgement_, which means that the following structural properties are true: +* Hypothesis: if is in , then . +* Weakening: if and is fresh then . +* Exchange: if then . +* Substitution: if and then + +Hypothesis is derivable by the rule . Weakening, exchange, and substitution are admissible. + +### LF representation + +In the previous section, we used LF binding to represent object-language variables. In this section, we use LF binding to represent object-language hypothetical judgements. The following LF signature represents the above judgement with the LF type family ``of``. !}% +%{!! begin hidden !!}% +%sort tp %. +%term unit tp %. +%term arrow %pi tp %-> tp %-> tp %. +%sort tm %. +%term empty tm %. +%term app %pi tm %-> tm %-> tm %. +%term lam %pi tp %-> (%pi tm %-> tm) %-> tm %. +%{!! end hidden !!}% +%sort of %. +%term of-empty of empty unit %. +%term of-lam + %pi (of (lam T2 ([x] E x)) (arrow T2 T)) + %<- ({x tm} %pi (of x T2) %-> (of (E x) T)) %. +%term of-app %pi (of (app E1 E2) T) %<- (of E1 (arrow T2 T)) %<- (of E2 T2) %. +%{! The first thing to note is that the type family is indexed by a ``tm`` and a ``tp`` but **not** a representation of the context . The reason for this is that we identify the object-language context with the LF context. Specifically, an object-language assumption does two things: +* It binds the variable . +* It declares a typing assumption that has type . + +Thus, an object-language assumption is represented by the following two LF assumptions: ``x : tm`` and ``dx : of x T``, where ``T`` is the encoding of . + +The first LF variable represents an object-language term ``x``, as per the encoding of syntax in the previous section. The second variable represents a derivation that ``of x T``. Consequently, there is no LF constant corresponding to the rule because uses of this rule are represented by uses of the corresponding LF variable ``dx``. + +This representation of hypotheses gives rise to the higher-order premise of the constant ``of-lam``, which has type ``{x: tm} of x T2 -> of (E x) T`` + +An LF term of this type has the form ``([x] [dx: of x T2] M)``, where ``M : of (E x) T`` _in an LF context extended with ``x : tm, dx : of x T2``_. Thus, ``M`` is the representation of an object-language derivation under the additional assumption >. + +The constants ``of-empty`` and ``of-app`` correspond to the informal inference rules of the same name; they are no more complicated than the rules for the judgements on natural numbers from the previous sections. + +#### Eta-expansion + +As we mentioned [before in the discussion of full LF](/wiki/proving-metatheorems-full-lf/), Twelf permits programmers to write non-eta-expanded terms, which it treats as syntactic sugar for their eta-expansions. For example, we could equivalently have written: + +```twelf +of-lam : of (lam T2 E) (arrow T2 T) + <- ... +``` + +This version differs from ``of-lam`` above in that the term ``E`` is not eta-expanded. Some consider it good Twelf style to write terms in a fully eta-expanded ([eta-long](/wiki/eta-equivalence/)) form, because (1) the eta-expanded terms are the canonical forms of LF, and (2) the eta-expansion makes the binding structure apparent in the syntax, without knowing the types of the involved constants. On the other hand, readers who know the types of the constants may find the eta-short version easier to read. + +### Theorems for free + +There are several advantages to using this higher-order representation of a hypothetical judgement, as opposed to a first-order representation where the context is treated explicitly. + +First, the representation is quite succinct: there is no need define and work with auxiliary data structures like lists of distinct assumptions. + +More importantly, this representation gives the structural properties of the hypothetical judgement for free: the structural properties for the object language are inherited from the corresponding properties of LF. For instance, there is no need to prove a substitution lemma about the object language. As an example, we can derive + +``` +T : tp, +T2 : tp, +E : tm -> tm, +E2 : tm, +D1 : {x : tm} of x T2 -> of (E x) T, +D2 : of E2 T2 + |- D1 E2 D2 : of (E E2) T +``` + +The term ``D1 E2 D2`` represents a derivation that the substitution ``(E E2)`` is well-typed. We call the LF application ``(E E2)`` a substitution because when ``E`` is substituted by an actual term ``[x] M``, the application will reduce to ``[E2/x]M``. + +### Adequacy + +The adequacy theorem for typing derivations is as follows: + +> There is a compositional bijection between informal derivations of and LF terms ``D`` such that ``x1 : tm, dx1 : of x1 T1, ... |- D : of E T``, where , , and , ... + +To prove this adequacy theorem, it is necessary to show that STLC terms and types remain adequately represented when considered in LF contexts of this form. Intuitively, adding canonical forms of type ``of`` does not change the canonical forms of types ``tp`` and ``tm`` because STLC typing derivations cannot appear in STLC types or terms. This kind of argument can be made precise using [subordination](/wiki/subordination/), and Harper and Licata's [_Mechanizing Metatheory in a Logical Framework_](https://www.cambridge.org/core/journals/journal-of-functional-programming/article/mechanizing-metatheory-in-a-logical-framework/E4EC2F77CEF3DC56EABCF6F6658B179E) discusses all these details. + +## Dynamic semantics + +### Informal definition + +We define the dynamic semantics of the STLC by a call-by-value, left-to-right structural operational semantics on closed terms. + +The judgement identifies the values: + +```math +{ + \over + \langle\rangle\,\mathtt{value} +}\mathit{value\textit{-}empty} +\qquad +{ + \over + \gamma x{:}\tau.e\,\mathtt{value} +}\mathit{value\textit{-}lam} +``` + +Next, we define the operational semantics with a judgement : + +```math +{ + e_1 \mapsto e_1' + \over + e_1\,e_2 \mapsto e_1'\,e_2 +}\mathit{step\textit{-}app_1} +\qquad +{ + e_1\,\mathtt{value} + \qquad + e_2 \mapsto e_2' + \over + e_1\,e_2 \mapsto e_1\,e_2' +}\mathit{step\textit{-}app_2} +``` +```math +{ + e_2\,\mathtt{value} + \over + (\lambda x{:}\tau.e)\,e_2 \mapsto e[e_2/x] +}\mathit{step\textit{-}app\textit{-}beta} +``` + +### LF representation + +These judgements are represented in LF in the usual fashion. The closed LF terms of these types adequately represent the above judgements. !}% +%sort value %. +%term value-empty value empty %. +%term value-lam value (lam T ([x] E x)) %. +%sort step %. +%term step-app-1 %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step-app-2 %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step-app-beta %pi (step (app (lam T2 ([x] E x)) E2) (E E2)) %<- (value E2) %. +%{! In ``step-app-beta``, the right-hand term is the application ``E E2``, which represents the substitution on the right-hand side of the informal rule. + +And that completes our representation of the STLC in LF. Now, we can start proving metatheorems about the language using Twelf. + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-representing-the-syntax-of-the-natural-numbers.lf b/new-tests/stelf-output/pages/proving-metatheorems-representing-the-syntax-of-the-natural-numbers.lf new file mode 100644 index 0000000..5bbb3ca --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-representing-the-syntax-of-the-natural-numbers.lf @@ -0,0 +1,84 @@ +% +% +% +% +% +% +% +% +%{! + +_This page is part of the introduction to [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)._ + +To use Twelf, you need to know how to represent [deductive systems](/wiki/object-logic/) using the LF logical framework. LF is a very convenient language for representing deductive systems that involve variable binding and [hypothetical judgements](/wiki/hypothetical-judgment/), such as programming languages and logics. However, we won't see any examples of such uses of LF until [later](/wiki/proving-metatheorems-representing-the-syntax-of-the-stlc/). The goal of the first part of this guide is to take a broad sweep through LF and Twelf; you will see how to represent and prove properties of a very simple language, which will prepare you for more sophisticated (and more impressive) content later on. + +Our first task is to introduce LF and see how to represent the syntax of a deductive system in it. We refer to a language that we are formalizing in LF as an [object language](/wiki/object-logic/) (the language that it is the _object_ of our study). In contrast, we sometimes refer to LF as the [meta-language](/wiki/meta-logic/). To keep things simple, we use the natural numbers as first example object language. + +## Natural numbers + +### Informal definition + +In informal mathematical notation ("informal" is used here in the sense of "not formalized"), the syntax of the natural numbers is defined by the following grammar: + +```math +n ::= \mathsf{zero} \mid \mathsf{succ}(n) +``` + +That is, is a natural number, and if is a natural number, then is as well. + +### LF representation + +LF is a typed lambda-calculus. We represent an object language in LF by writing an LF signature. A signature declares type and term constants. For example, this LF signature defines a type representing the natural numbers: + +!}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%{! + +Intuitively, the LF type `nat` classifies the LF representations of natural numbers. The LF constant `z` corresponds to and the LF constant `s` corresponds to . + +- The signature declares that `z` has type `nat`, which makes sense because is a natural number. + +- The signature declares that `s` has function type `nat -> nat`. An LF term of function type can be applied to another LF term of the appropriate type to form a new term. For example, the constant `s` can be applied to the constant `z` to form the term `s z` representing the number . Then `s` can be applied to this term to form `s (s z)`, and so on. An informal natural number is represented by the LF term `s N` where is represented by `N`. + +We can state the relationship between the informal presentation of the object language and its LF representation by giving an encoding judgement relating an informal object with its representation: + +```math +{ + {\,} + \over + {\mathsf{zero} \gg \mathtt{z}} +} + +\qquad + +{ + {n \gg N } + \over + {\mathsf{succ}(n) \gg \mathtt{s}(N)} +} + +``` + +## Adequacy + +Is this a _correct_ encoding of the natural numbers? In the LF methodology, the correctness criterion for an encoding is called [adequacy](/wiki/adequacy/): + +> **Adequacy**: An encoding is adequate iff it is an isomorphism between the informal object-language entities and the LF terms of the appropriate type. + +Adequacy permits any reasoning in or about an object language to be "ported" between the informal and LF descriptions. For example, if we prove a property about the LF representation, by adequacy, that property holds for the informal description as well. Adequacy ensures that the work we do using the formalization has bearing on the actual language that we had in mind to begin with. + +In this example, the informal object-language entity is the syntax of the natural numbers, and the appropriate LF type is `nat`. Thus, to satisfy adequacy, the above encoding judgement must be an isomorphism between informal natural numbers and LF terms of type `nat`. Because both the informal and LF representations are first order (there is no binding structure), adequacy requires only a simple bijection. + +To establish a bijection, we must show three things: + +1. If then is a well-formed number and is an LF term of type `nat`. +2. For all , there exists a unique LF term such that . +3. For all LF terms `N : nat`, there exists a unique such that . + +The first part establishes that the judgement relates informal numbers to LF terms of the appropriate type. The second and third show that this relation defines a function in both directions. Because we show that the same relation is functional in both directions, these functions are mutually inverse, establishing a bijection. + +The first part can be proved by a simple induction on the derivation of the encoding. The second can be proved by structural induction on . The third part states that all LF terms of type `nat` represent a unique informal number. It can be proved by induction on the structure of LF terms. To do so, we will need to give a more precise account of LF. + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-representing-the-syntax-of-the-stlc.lf b/new-tests/stelf-output/pages/proving-metatheorems-representing-the-syntax-of-the-stlc.lf new file mode 100644 index 0000000..f9d250b --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-representing-the-syntax-of-the-stlc.lf @@ -0,0 +1,85 @@ +% +% +% +% +% +% +% +% +%{! + +_This page is part of the introduction to [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)._ + +In this section, we discuss how to represent the syntax of a simply typed λ-calculus in LF. + +## Representing STLC types + +We consider a λ-calculus with as a base type: + +```math +\tau ::= \mathsf{unit} \mid \tau_1 \rightarrow \tau_2 +``` +The representation of types is no more complicated than the representation of natural numbers in the previous section: !}% +%sort tp %. +%term unit tp %. +%term arrow %pi tp %-> tp %-> tp %. +%{! We leave it to the reader to formulate an appropriate encoding judgement between STLC types and closed LF terms of type ``tp`` and to prove that this encoding is adequate in the usual sense. + +## Representing STLC terms + +### Informal definition + +The terms of the STLC are variables , the 0-element tuple , lambda abstraction (with a type annotation), and application: + +```math +e ::= x \mid \langle\rangle \mid \lambda x{:}\tau.e \mid e_1\,e_2 +``` + +The variable is considered to be bound by the expression . Following standard practice, we implicitly consider terms modulo renaming of bound variables. + +### LF representation + +We represent these terms in LF with the following signature: !}% +%sort tm %. +%term empty tm %. +%term app %pi tm %-> tm %-> tm %. +%term lam %pi tp %-> (%pi tm %-> tm) %-> tm %. +%{! The representation of as ``empty`` and application as ``app`` is no different than what we did when representing natural numbers. Two questions you are may be asking are: + +1. How does this LF signature represent variables , since these don't appear in the LF signature, and +2. What's going on with the type of ``lam``? + +The answer is that we are using a representation technique called _higher-order abstract syntax_ (HOAS): we use LF variables to represent object-language variables. Here are some example representations: + + - The term + is represented in LF as `lam unit ([x] x)`. + - The term is represented in LF as `lam (arrow unit unit) ([x] app x empty)` + + +As you can see, we represent an object-language variable by an LF variable with the same name. To represent an object-language term , we first represent the body as an LF term ``M`` in which the LF variable ``x`` occurs free. Then, we bind this variable using an LF abstraction ``[x] M``. This abstraction, which represents the body of the object-language lambda, has LF type ``tm -> tm``. That's where the higher-order type of the constant ``lam`` comes from. + +Using higher-order abstract syntax (HOAS) in this fashion makes two operations on syntax primitive: +* **α-equivalence**: Because LF terms are considered modulo α-renaming of bound variables, and LF bound variables are used to model object-language bound variables, the type ``tm`` represents α-equivalence classes of object-language terms. For example, because there is no way to distinguish ``([x] app x empty)`` and ``([y] app y empty)`` in LF, so there is no way to distinguish ``lam (arrow unit unit) ([x] app x empty)`` and ``lam (arrow unit unit) ([y] app y empty)``. +* **capture-avoiding substitution**: an object-language substitution, written , is modeled by an LF substitution ``[E2/x] E``. + +## Adequacy + +The adequacy theorem for terms shows that the LF operations correctly implement the object-language operations: + +### Adequacy for terms + +There is a bijection between (α-equivalence classes of) STLC terms with free variables in and (α-equivalence classes of) LF terms ``E : tm`` in the LF context ``x1:tm, ..., xn:tm``. + +Moreover, this bijection is compositional in the following sense: if and then . + +If you're interested in the details of how this adequacy theorem is stated and proved, you can read more about it Harper and Licata's paper [_Mechanizing Metatheory in a Logical Framework_](https://www.cambridge.org/core/journals/journal-of-functional-programming/article/mechanizing-metatheory-in-a-logical-framework/E4EC2F77CEF3DC56EABCF6F6658B179E). However, there are a few high-level ideas in this proof that we should call out here: + +**Adequacy for terms requires adequacy for types.** Because the syntax of terms mentions the syntax of types, the proof of adequacy for terms requires knowing that ``tp`` adequately represents types . Above, we (well, we left it as an exercise for you) proved that types are adequately represented by _closed_ LF terms of type ``tp``. Thus, there is a danger here: the syntax of STLC types might not be adequately represented in the LF contexts we consider here, which have the form ``x1:tm, ..., xn:tm``. + +In fact, the representation of STLC types remains adequate in contexts of this form because adding variables of type ``tm`` to the context does not change the canonical forms of type ``tp``. This kind of argument can be made precise using [subordination](/wiki/subordination/), which tracks when terms of one type may appear in terms of another. Informally, the idea is that the canonical forms of a type only depend on the canonical forms of subordinate types. Here, ``tm`` is not subordinate to ``tp`` (because STLC terms do not appear in STLC types), so adding canonical forms of type ``tm`` does not change the canonical forms of type ``tp``. We will discuss another application of subordination [later in this introduction when we discuss totality assertions in non-empty contexts](/wiki/proving-metatheorems-proving-totality-assertions-in-non-empty-contexts/). + +**The power of weak frameworks.** When we consider closed LF terms of type ``nat`` or ``tp``, the only possible terms of those types are applications of the appropriate constants. The reason our higher-order representation of STLC terms works in LF is that, when we consider open LF terms in contexts containing assumptions ``x:tm``, the only LF terms of type ``tm`` are either those variables or applications of the appropriate constants. + +Specifically, it is important that LF does not provide any elimination forms for constants. For example, suppose LF provided a ``case`` construct for case-analyzing the terms of a type, as one would have if ``tm`` were considered to be an inductive definition within the type theory. Then the term ``([x:tm] case x of empty => empty | ...)`` would need to be considered a canonical form of type ``tm -> tm`` (without knowing what the variable stands for, there is no way to reduce the case). However, this term does not correspond to any object-language term-with-a-free-variable, so having such a case construct would ruin the adequacy of the representation. + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-simply-typed-lf.lf b/new-tests/stelf-output/pages/proving-metatheorems-simply-typed-lf.lf new file mode 100644 index 0000000..2a0652f --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-simply-typed-lf.lf @@ -0,0 +1,72 @@ +% +% +% +% +% +% +% +% +%{! + +_This page is part of the introduction to [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)._ + +In the [previous section](/wiki/proving-metatheorems-representing-the-syntax-of-the-natural-numbers/), we discussed representing object-language syntax in LF. As we saw, to prove that an LF representation is correct, you need to be able to reason by induction on the [canonical forms](/wiki/canonical-form/) of LF. To do so, you will need a basic understanding of the LF type theory. This understanding will pay off later on, as we will also use induction on canonical forms to [prove metatheorems](/wiki/proving-metatheorems-proving-totality-assertions-about-the-natural-numbers/) about encoded languages. + +In this section, we present simply-typed LF in more detail. + +## Syntax + +The syntax of simply-typed LF is as follows: + +```math +\begin{align*} +A & ::= a \mid A_1 \rightarrow A_2 +\\ +R & ::= c \mid x \mid R \, M +\\ +M & ::= R \, | \, \lambda x.M +\end{align*} +``` + + +The types consist of base types , which are declared in the signature (e.g., ``nat``), and function types . The terms of LF consist of variables , constants declared in the signature (e.g., `z` and `s`), application , and lambda-abstractions , which in Twelf's concrete syntax are written ``[x] M``. This much should be familiar to you if you've seen the simply-typed lambda-calculus before. Likely more unfamiliar is that the grammar is stratified into and . We discuss the reason for this stratification now. + +## Canonical forms + +The above syntax describes what we call _[canonical forms](/wiki/canonical-form/)_ (as long as they are well-typed, more on this below). Note what is _not_ a canonical form: there is no syntactic way to apply a lambda-abstraction to an argument. Based on your past experience with programming languages, it may seem strange to define LF so that only canonical forms exist: we are not allowed to write down any programs that have any computation left to do. However, this restriction makes sense if you think about our methodology for representing object languages in LF. In the previous section, we represented natural numbers with the following LF signature: !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%{! For this representation to be adequate, the only LF terms of type ``nat`` must be ``z``, ``s z``, ``s (s z)``, and so on. Non-canonical LF terms interfere with this encoding. For example, the LF term would have type ``nat`` were it allowed, and that is not the representation of any informal natural number. This would contradict the third part of the adequacy argument in the previous section. Restricting LF to canonical forms avoids these counterexamples. + +We won't present all the typing rules for canonical forms here, but you can see Harper and Licata's paper [_Mechanizing Metatheory in a Logical Framework_](https://www.cs.cmu.edu/~rwh/papers/mech/jfp07.pdf) for all the details. The only detail we need to note here is that terms are only canonical at base type , not at function type. For example, the constant ``s`` is _not_ a canonical form of type ``nat -> nat``. The term , however, is a canonical form. + +If you have encountered [beta-reduction](/wiki/beta-equivalence/) and [eta-expansion](/wiki/eta-equivalence/) before in your study of programming languages, it may help your intuition to know that the canonical forms of LF coincide with the beta-normal, eta-long terms of the lambda calculus. What we were saying above is that the syntax of canonical forms forces them to be beta-normal, and that the typing rules for canonical forms ensure that they are eta-long. In logic, canonical forms correspond to normal and neutral natural deduction proofs and cut-free sequent calculus proofs. + +### Hereditary Substitution + +Substitution of one canonical form into another does not necessarily produce a canonical result. For example: + +```math +[\lambda y.y/x](x \; z) = (\lambda y.y) \; z +``` + +Even though both and are canonical forms, the result of the substitution is beta-reducible. + +However, it is possible to define a notion of _hereditary substitution_, which directly computes the canonical result of an ordinary substitution. When ordinary substitution would return a non-canonical form, hereditary substitution continues to reduce by substituting the argument into the body of the function. In the above example, the hereditary substitution + +```math +[\lambda y.y/x](x \; z) = [z/y]y = z +``` + +Whenever we use the notation for LF in this guide, we mean hereditary substitution. + +## Induction on canonical forms + +The above syntax constitutes an inductive definition of the canonical forms of LF. Consequently, we can reason by induction on canonical forms using structural induction over this syntax. When a type adequately represents some informal object-language syntax, induction on the canonical forms of the type corresponds to the structural induction principle for the informal syntax. For example, the induction principle for the type ``nat`` defined in the previous section corresponds to the standard induction principle for the natural numbers. As we discussed in the previous section, induction on canonical forms is used to prove adequacy. Additionally, as we discuss below, induction on canonical forms is used to prove metatheorems about deductive systems. + +## Why a lambda calculus? + +At this point, you may be wondering what we gain by using a lambda-calculus to represent other deductive systems—to represent the natural numbers, all we needed was constants and application; we never even used a term of the form . At a high level, the answer is that using a meta-language with variable binding permits clean representations of object languages with variable binding. However, we defer a real answer to this question until we talk about representing the [syntax](/wiki/proving-metatheorems-representing-the-syntax-of-the-stlc/) and [judgements](/wiki/proving-metatheorems-representing-the-judgements-of-the-stlc/) of a language with binding. + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-solution-plus-comm.lf b/new-tests/stelf-output/pages/proving-metatheorems-solution-plus-comm.lf new file mode 100644 index 0000000..bbd6858 --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-solution-plus-comm.lf @@ -0,0 +1,51 @@ +% +% +%{! + +_This page is part of the introduction to [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)._ + +Exercise 1 from [the part 1 exercises](/wiki/proving-metatheorems-summary-the-natural-numbers/): State and prove a metatheorem showing that ``plus`` is commutative. + +Starting with the preliminaries: + +!}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort plus %. +%term plus-z plus z N2 N2 %. +%term plus-s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%{! The theorem may be stated: !}% +%sort plus-commutes %. +%mode plus-commutes %in %out %. +%{! However, before we prove this theorem, we first prove two lemmas. The first says that for any natural number , . This is similar to the constant ``plus-z``, but the order of the arguments to ``plus`` has been changed: !}% +%sort plus-zero-id {N1 nat} %. +%mode plus-zero-id %in %out %. +%term pzidz plus-zero-id z plus-z %. +%term pzids + %pi (plus-zero-id (s N) (%the (plus (s N) z (s N)) (plus-s D))) + %<- (plus-zero-id N D) %. +%worlds () (plus-zero-id _ _) %. +%total N (plus-zero-id N _) %. +%{! Our second lemma states that if then . This lemma is similar to the constant ``plus-s``, but the order of the arguments to ``plus`` has been changed: !}% +%sort plus-flip %. +%mode plus-flip %in %out %. +%term pfz plus-flip _ plus-z %. +%term pfs + %pi (plus-flip (%the (plus (s N1) N2 (s N3)) (plus-s Dplus)) (%the (plus (s N1) (s N2) (s (s N3))) (plus-s DIH))) + %<- (plus-flip Dplus DIH) %. +%worlds () (plus-flip _ _) %. +%total D (plus-flip D _) %. +%{! Finally, using these two lemmas, we may prove the theorem itself: !}% +%sort plus-commutes %. +%mode plus-commutes %in %out %. +%term pcz %pi (plus-commutes _ D) %<- (plus-zero-id N1 D) %. +%term pcs + %pi (plus-commutes (%the (plus (s N1') N2 (s N3')) (plus-s Dplus)) D) + %<- (plus-commutes Dplus DIH) + %<- (plus-flip DIH D) %. +%{! The theorem checks: given a derivation of `plus N1 N2 N3`, there exists a derivation of `plus N2 N1 N3`. !}% +%{!! begin checked !!}% +%worlds () (plus-commutes _ _) %. +%total D (plus-commutes D _) %. +%{!! end checked !!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-solutions-define-odd.lf b/new-tests/stelf-output/pages/proving-metatheorems-solutions-define-odd.lf new file mode 100644 index 0000000..288a555 --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-solutions-define-odd.lf @@ -0,0 +1,24 @@ +% +% +%{! + +_This page is part of the introduction to [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)._ + +Exercise 2 from [the part 1 exercises](/wiki/proving-metatheorems-summary-the-natural-numbers/): Define the odd numbers with a judgement analogous to the ``even`` judgement defined above. + +Starting with the preliminaries: + +!}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort plus %. +%term plus-z plus z N2 N2 %. +%term plus-s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%sort even %. +%term even-z even z %. +%term even-s %pi (even (s (s N))) %<- (even N) %. +%{! The definition of `odd` looks much like the definition of `even`. !}% +%sort odd %. +%term odd-1 odd (s z) %. +%term odd-s %pi (odd N) %-> (odd (s (s N))) %. \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-solutions-odd-even-plus.lf b/new-tests/stelf-output/pages/proving-metatheorems-solutions-odd-even-plus.lf new file mode 100644 index 0000000..20293bf --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-solutions-odd-even-plus.lf @@ -0,0 +1,129 @@ +% +% +%{! +_This page is part of the introduction to [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)._ + +Exercise 4 from [the part 1 exercises](/wiki/proving-metatheorems-summary-the-natural-numbers/): Prove remaining properties of how evenness and oddness interacts with addition: +- State and prove the theorem `sum-even-odd` that shows that the sum of an even and an odd number results in an odd number +- State and prove a theorem `sum-odd-even` that shows that the sum of an odd plus an even number produces an odd number +- Finally, state and prove a theorem `sum-odds` that shows that the sum of two odd numbers produces an even number + +Starting with the preliminaries, including the [definition of `odd` from exercise 2](/wiki/proving-metatheorems-solutions-define-odd/): + +!}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort plus %. +%term plus-z plus z N2 N2 %. +%term plus-s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%sort even %. +%term even-z even z %. +%term even-s %pi (even (s (s N))) %<- (even N) %. +%sort odd %. +%term odd-1 odd (s z) %. +%term odd-s %pi (odd N) %-> (odd (s (s N))) %. +%{! ## The sum of an even and odd number is odd !}% +%{!! begin checked !!}% +%sort sum-even-odd %. +%mode sum-even-odd %in %in %in %out %. +%term seoz sum-even-odd even-z OddN2 plus-z OddN2 %. +%term seos + %pi (sum-even-odd (even-s EvenN1) OddN2 (plus-s (plus-s PlusN1N2N3)) (odd-s OddN3)) + %<- (sum-even-odd EvenN1 OddN2 PlusN1N2N3 OddN3) %. +%worlds () (sum-even-odd _ _ _ _) %. +%total D (sum-even-odd D _ _ _) %. +%{!! end checked !!}% +%{! ## The sum of an odd and an even number is odd + +Unlike the previous proof that the sum of an even number and on odd, the base case of this proof is easiest to prove by reusing one of the lemmas a previous exercise. + +Neither answer is better than the other! It's valuable to think through the answer that you didn't select, as it's important to be able to think through multiple ways of proving the same theorem. + +### Likely proof, reusing `succ-even` + +If you've just written `sum-even-odd`, and try to adapt the structure of that proof, the easiest approach is to reuse [`succ-even` from Exercise 3](/wiki/proving-metatheorems-solutions-odd-even-succ/), the proof that the successor of an even number is odd. + +We'll first repeat that proof: +!}% +%sort succ-even %. +%mode succ-even %in %out %. +%term sez succ-even even-z odd-1 %. +%term ses %pi (succ-even (even-s EvenA) (odd-s OddA)) %<- (succ-even EvenA OddA) %. +%worlds () (succ-even _ _) %. +%total D (succ-even D _) %. +%{! With `succ-even` in hand, the proof of `sum-odd-even` has the same structure as the proof of `sum-even-odd`: !}% +%{!! begin checked !!}% +%sort sum-odd-even %. +%mode sum-odd-even %in %in %in %out %. +%term soe1 %pi (sum-odd-even odd-1 EvenN2 _ OddN3) %<- (succ-even EvenN2 OddN3) %. +%term soes + %pi (sum-odd-even (odd-s OddN1) EvenN2 (plus-s (plus-s PlusN1N2N3)) (odd-s OddN3)) + %<- (sum-odd-even OddN1 EvenN2 PlusN1N2N3 OddN3) %. +%worlds () (sum-odd-even _ _ _ _) %. +%total D (sum-odd-even D _ _ _) %. +%{!! end checked !!}% +%{! ### Alternative solution, reusing `succ-even-odd` and `plus-commutes` + +Another proof of `sum-odd-even` uses the previously proved `sum-even-odd` along with the proof of the [proof of the commutativity of plus from Exercise 1](/wiki/proving-metatheorems-solution-plus-comm/). The `sum-even-odd` proof is a bit involved: + +!}% +%sort plus-zero-id {N1 nat} %. +%mode plus-zero-id %in %out %. +%term pzidz plus-zero-id z plus-z %. +%term pzids + %pi (plus-zero-id (s N) (%the (plus (s N) z (s N)) (plus-s D))) + %<- (plus-zero-id N D) %. +%worlds () (plus-zero-id _ _) %. +%total N (plus-zero-id N _) %. +%sort plus-flip %. +%mode plus-flip %in %out %. +%term pfz plus-flip _ plus-z %. +%term pfs + %pi (plus-flip (%the (plus (s N1) N2 (s N3)) (plus-s Dplus)) (%the (plus (s N1) (s N2) (s (s N3))) (plus-s DIH))) + %<- (plus-flip Dplus DIH) %. +%worlds () (plus-flip _ _) %. +%total D (plus-flip D _) %. +%sort plus-commutes %. +%mode plus-commutes %in %out %. +%term pcz %pi (plus-commutes _ D) %<- (plus-zero-id N1 D) %. +%term pcs + %pi (plus-commutes (%the (plus (s N1') N2 (s N3')) (plus-s Dplus)) D) + %<- (plus-commutes Dplus DIH) + %<- (plus-flip DIH D) %. +%worlds () (plus-commutes _ _) %. +%total D (plus-commutes D _) %. +%{! Once that proof is available, the proof of `sum-odd-even` requires no additional induction: !}% +%{!! begin checked !!}% +%sort sum-odd-even %. +%mode sum-odd-even %in %in %in %out %. +%term soe + %pi (sum-odd-even (%the (odd M) O) (%the (even N) E) (%the (plus M N P) A) (%the (odd P) O0)) + %<- (plus-commutes A A0) + %<- (sum-even-odd E O A0 O0) %. +%worlds () (sum-odd-even _ _ _ _) %. +%total [] (sum-odd-even D _ _ _) %. +%{!! end checked !!}% +%{! ## The sum of two odd numbers is even + +The most straightforward approach here reuses [`succ-odd` from Exercise 3](/wiki/proving-metatheorems-solutions-odd-even-succ/) in exactly the same way as the "expected" solution to the previous theorem reused `succ-even`. + +First, we'll repeat the proof of `succ-odd` from Exercise 3: +!}% +%sort succ-odd %. +%mode succ-odd %in %out %. +%term so1 succ-odd odd-1 (even-s even-z) %. +%term sos %pi (succ-odd (odd-s OddA) (even-s EvenA)) %<- (succ-odd OddA EvenA) %. +%worlds () (succ-odd _ _) %. +%total D (succ-odd D _) %. +%{! With that out of the way, the proof of `sum-odds` has the same structure as the previous solutions: !}% +%{!! begin checked !!}% +%sort sum-odds %. +%mode sum-odds %in %in %in %out %. +%term soz %pi (sum-odds odd-1 OddN2 _ EvenN3) %<- (succ-odd OddN2 EvenN3) %. +%term sos + %pi (sum-odds (odd-s OddN1) OddN2 (plus-s (plus-s PlusN1N2N3)) (even-s EvenN3)) + %<- (sum-odds OddN1 OddN2 PlusN1N2N3 EvenN3) %. +%worlds () (sum-odds _ _ _ _) %. +%total D (sum-odds D _ _ _) %. +%{!! end checked !!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-solutions-odd-even-succ.lf b/new-tests/stelf-output/pages/proving-metatheorems-solutions-odd-even-succ.lf new file mode 100644 index 0000000..7c66ccf --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-solutions-odd-even-succ.lf @@ -0,0 +1,42 @@ +% +% +%{! + +_This page is part of the introduction to [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)._ + +Exercise 3 from [the part 1 exercises](/wiki/proving-metatheorems-summary-the-natural-numbers/): State and prove a theorem ``succ-even`` that shows that the successor of an even number is an odd number. After that, state and prove a theorem ``succ-odd`` that shows that the successor of an odd number is an even number. + +Starting with the preliminaries, including the [definition of `odd` from exercise 2](/wiki/proving-metatheorems-solutions-define-odd/): + +!}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort plus %. +%term plus-z plus z N2 N2 %. +%term plus-s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%sort even %. +%term even-z even z %. +%term even-s %pi (even (s (s N))) %<- (even N) %. +%sort odd %. +%term odd-1 odd (s z) %. +%term odd-s %pi (odd N) %-> (odd (s (s N))) %. +%{! Because both the `even` and `odd` judgements are defined independently, it's necessary to perform induction on the `even` judgement to prove that the first theorem, and induction on the `odd` judgement to prove the second theorem: !}% +%{! ## The successor of an even number is odd !}% +%{!! begin checked !!}% +%sort succ-even %. +%mode succ-even %in %out %. +%term sez succ-even even-z odd-1 %. +%term ses %pi (succ-even (even-s EvenA) (odd-s OddA)) %<- (succ-even EvenA OddA) %. +%worlds () (succ-even _ _) %. +%total D (succ-even D _) %. +%{!! end checked !!}% +%{! ## The successor of an odd number is even !}% +%{!! begin checked !!}% +%sort succ-odd %. +%mode succ-odd %in %out %. +%term so1 succ-odd odd-1 (even-s even-z) %. +%term sos %pi (succ-odd (odd-s OddA) (even-s EvenA)) %<- (succ-odd OddA EvenA) %. +%worlds () (succ-odd _ _) %. +%total D (succ-odd D _) %. +%{!! end checked !!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-stlc-answers-to-exercises.lf b/new-tests/stelf-output/pages/proving-metatheorems-stlc-answers-to-exercises.lf new file mode 100644 index 0000000..47c6e82 --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-stlc-answers-to-exercises.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-summary-the-natural-numbers.lf b/new-tests/stelf-output/pages/proving-metatheorems-summary-the-natural-numbers.lf new file mode 100644 index 0000000..84696b3 --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-summary-the-natural-numbers.lf @@ -0,0 +1,43 @@ +% +% +% +% +% +% +% +% +%{! + +_This page is part of the introduction to [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)._ + + +## Take-aways + +The three things to remember from this section are: + +- Object-language syntax and judgements are represented as LF terms of corresponding types. These representations are adequate iff they are isomorphic to the original description of the object language. +- An LF type family defines a relation on its indices. Twelf has the ability to verify that a type family represents a total relation. +- Proofs of -statements can be mechanized and verified in Twelf by representing them as LF type families. + +At a high level, that's all there is to know about proving metatheorems with Twelf. + +## What's next? + +So far, we have used only first-order LF representations (we haven't used LF variables or lambdas for anything yet). That's reasonable, since the natural numbers and the judgements we defined about them are first-order things. + +However, one of the main benefits of LF is that the above methodology scales gracefully to programming languages with binding structure. This is the subject of the next section. + +## Test yourself + +Before going on to the next section, you may wish to test yourself with the following exercises. + +1. State and prove a metatheorem showing that ``plus`` is commutative. ([Solution](/wiki/proving-metatheorems-solution-plus-comm/)) +2. Define the odd numbers with a judgement analogous to the ``even`` judgement defined above. ([Solution](/wiki/proving-metatheorems-solutions-define-odd/)) +3. State and prove a theorem ``succ-even`` that shows that the successor of an even number is an odd number. After that, state and prove a theorem ``succ-odd`` that shows that the successor of an odd number is an even number. ([Solution](/wiki/proving-metatheorems-solutions-odd-even-succ/)) +4. Prove remaining properties of how evenness and oddness interacts with addition. _(Hint: some of the most straightforward solutions to these exercises reuse one or more results from previous exercises.)_ ([Solution](/wiki/proving-metatheorems-solutions-odd-even-plus/)): + - State and prove the theorem `sum-even-odd` that shows that the sum of an even and an odd number results in an odd number + - State and prove a theorem `sum-odd-even` that shows that the sum of an odd plus an even number produces an odd number + - Finally, state and prove a theorem `sum-odds` that shows that the sum of two odd numbers produces an even number + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-summary-the-stlc.lf b/new-tests/stelf-output/pages/proving-metatheorems-summary-the-stlc.lf new file mode 100644 index 0000000..635f241 --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-summary-the-stlc.lf @@ -0,0 +1,119 @@ +% +% +% +% +% +% +% +% +%{! + +_This page is part of the introduction to [proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)._ + +## Take-aways + +The lessons you should take from this section are: + +- LF's binding structure enables elegant representations of object-language binding and hypothetical judgements. +- World declarations permit totality assertions about type families in non-empty contexts. +- Such totality assertions can be used to prove general metatheorems about object-language entities that are represented using higher-order LF encodings. + +## What's next? + +Now that you've made it through this introduction, you have the basic tools you need to start proving metatheorems with Twelf. You've seen all of the basic machinery that Twelf provides, and you are in a good position to understand the other documentation on this wiki. Your next step should be to read some of the [tutorials](/wiki/tutorials/), which describe clever modes of use of the techniques we have talked about here, or to read some of the [case studies](/wiki/case-studies/), which describe interesting applications of Twelf. + +## Test yourself + +Here are a few suggestions for additional explorations: + +### Adding features to the simply-typed lambda calculus + +A standard extension of the simply-typed lambda calculus is _pairs_: + +```math +{ + \Gamma \vdash e_1 : \tau_1 + \qquad + \Gamma \vdash e_2 : \tau_2 + \over + \Gamma \vdash \langle e_1, e_2 \rangle : \tau_1 \times \tau_2 +}\mathit{of\textit{-}pair} +``` +```math +{ + \Gamma \vdash e : \tau_1 \times \tau_2 + \over + \Gamma \vdash \pi_1\,e : \tau_1 +}\mathit{of\textit{-}proj_1} +\qquad +{ + \Gamma \vdash e : \tau_1 \times \tau_2 + \over + \Gamma \vdash \pi_2\,e : \tau_1 +}\mathit{of\textit{-}proj_2} +``` + +The dynamic semantics of pairs have a number of rules, but they are all relatively straightforward: + +```math +{ + e_1 \, \mathtt{value} + \qquad + e_2 \, \mathtt{value} + \over + \langle e_1, e_2 \rangle \, \mathtt{value} +}\mathit{value\textit{-}pair} +``` +```math +{ + e_1 \mapsto e_1' + \over + \langle e_1 , e_2 \rangle \mapsto \langle e_1', e_2 \rangle +}\mathit{step\textit{-}pair_1} +\qquad +{ + e_1 \, \mathtt{value} + \qquad + e_2 \mapsto e_2' + \over + \langle e_1 , e_2 \rangle \mapsto \langle e_1, e_2' \rangle +}\mathit{step\textit{-}pair_2} +``` +```math +{ + e \mapsto e' + \over + \pi_1 \, e \mapsto \pi_1 \, e' +}\mathit{step\textit{-}proj_1} +\qquad +{ + e \mapsto e' + \over + \pi_2\, e \mapsto \pi_2 \, e' +}\mathit{step\textit{-}proj_2} +``` +```math +{ + e_1 \, \mathtt{value} + \qquad + e_2 \, \mathtt{value} + \over + \pi_1 \, \langle e_1 , e_2 \rangle \mapsto e_1 +}\mathit{step\textit{-}proj\textit{-}beta_1} +\qquad +{ + e_1 \, \mathtt{value} + \qquad + e_2 \, \mathtt{value} + \over + \pi_2 \, \langle e_1 , e_2 \rangle \mapsto e_2 +}\mathit{step\textit{-}proj\textit{-}beta_2} +``` + +Add these to your formalization of the simply-typed lambda calculus and extend the proof of preservation to cover this new type. + +## Next steps + +You now have everything you need to start proving metatheorems with Twelf! The [proof techniques covered in the tutorials](/wiki/tutorials/) are likely to be valuable as you continue. If you're interested in proving progress for the simply typed lambda calculus formulation, you'll do well to start by learning about [output factoring](/wiki/output-factoring/). + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-talk-representing-the-judgements-of-the-natural-numbers.lf b/new-tests/stelf-output/pages/proving-metatheorems-talk-representing-the-judgements-of-the-natural-numbers.lf new file mode 100644 index 0000000..c8f7f2e --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-talk-representing-the-judgements-of-the-natural-numbers.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-talk-representing-the-judgements-of-the-stlc.lf b/new-tests/stelf-output/pages/proving-metatheorems-talk-representing-the-judgements-of-the-stlc.lf new file mode 100644 index 0000000..2d46d9d --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-talk-representing-the-judgements-of-the-stlc.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-talk-representing-the-syntax-of-the-stlc.lf b/new-tests/stelf-output/pages/proving-metatheorems-talk-representing-the-syntax-of-the-stlc.lf new file mode 100644 index 0000000..47a98ab --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-talk-representing-the-syntax-of-the-stlc.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-talk-simply-typed-lf.lf b/new-tests/stelf-output/pages/proving-metatheorems-talk-simply-typed-lf.lf new file mode 100644 index 0000000..6075aa1 --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-talk-simply-typed-lf.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/proving-metatheorems-with-twelf.lf b/new-tests/stelf-output/pages/proving-metatheorems-with-twelf.lf new file mode 100644 index 0000000..ac85a9e --- /dev/null +++ b/new-tests/stelf-output/pages/proving-metatheorems-with-twelf.lf @@ -0,0 +1,54 @@ +% +% +% +% +% +%{! + +The story of a typical [deductive system](/wiki/object-logic/) goes something like this: + - One day, a programming language designer has an idea for a new programming language. + - She writes down the language's [abstract syntax](/wiki/abstract-syntax/). + - Then, she gives the syntax meaning by defining some [judgement](/wiki/judgment/)s for it, such as a type system or an operational semantics. + - Next, she investigates the language's properties by proving some theorems. Maybe she fiddles with the language some to make the desired theorems true; maybe the idea doesn't work at all and she goes on to something else. + - Maybe she implements the language to try out some examples. + - Eventually, if the idea works out, she writes up a paper about the language. If it's a good idea, then she, or maybe someone who read the paper, will someday want to extend the language, or incorporate the idea into another language. + +This introduction assumes this story is somewhat familiar to you. If not, you should read a textbook such as [TAPL](http://www.cis.upenn.edu/~bcpierce/tapl/) or [PFPL](https://www.cs.cmu.edu/~rwh/pfpl/). The Twelf version of the story goes something like this: + - One day, a programming language designer has an idea. + - She formalizes the syntax and judgements of the language in the [LF](/wiki/lf/) logical framework. + - She uses Twelf to check her proofs of the theorems. + - She uses Twelf to run the language definition itself to try out some examples. + - Her paper includes a Twelf appendix, which makes her and her readers much more confident that the theorems in the paper are actually true. + - When someone goes to extend the language, he has a formal, machine-checkable artifact that he can study or perhaps even reuse directly. + +This tutorial pays a lot of attention to the question "how do I know that what I wrote down and proved on paper, and what I wrote down and proved in Twelf, are exactly the same thing?" Checking that these two things are the same is called [adequacy](/wiki/adequacy/), and it's an important problem _whenever_ you're using a theorem proving like Twelf, Agda, or Rocq. But it definitely has the flavor of "checking lots of fiddly details," and so it's reasonable to skip discussions of adequacy in this tutorial if you're exploring Twelf for the first time. + +### First order representations: The natural numbers + +The first layer of this introduction uses a _very_ simple deductive system, the [natural numbers](/wiki/natural-numbers/) with addition, to introduce the Twelf methodology. + + 1. [Representing syntax](/wiki/proving-metatheorems-representing-the-syntax-of-the-natural-numbers/) + 2. [Simply typed LF](/wiki/proving-metatheorems-simply-typed-lf/) + 3. [Representing judgements](/wiki/proving-metatheorems-representing-the-judgements-of-the-natural-numbers/) + 4. [Full LF](/wiki/proving-metatheorems-full-lf/) + 5. [Proving totality assertions](/wiki/proving-metatheorems-proving-totality-assertions-about-the-natural-numbers/) + 6. [Proving metatheorems](/wiki/proving-metatheorems-proving-metatheorems-about-the-natural-numbers/) + 7. [Summary and exercises](/wiki/proving-metatheorems-summary-the-natural-numbers/) + +### Higher-order representations: The simply-typed lambda calculus + +The second layer tells the same story for a programming language with variable binding (the [simply-typed lambda calculus](/wiki/simply-typed-lambda-calculus/)), which is where Twelf really shines. + + 8. [Representing syntax](/wiki/proving-metatheorems-representing-the-syntax-of-the-stlc/) + 9. [Representing judgements](/wiki/proving-metatheorems-representing-the-judgements-of-the-stlc/) + 10. [Proving metatheorems](/wiki/proving-metatheorems-proving-metatheorems-about-the-stlc/) + 11. [Summary and exercises](/wiki/proving-metatheorems-summary-the-stlc/) + +### Beyond the simply-typed lambda calculus + +The third layer presents some more-interesting proofs and introduces one additional feature of Twelf, the ability to do proofs about open terms. + + 12. [Proving totality assertions in non-empty contexts](/wiki/proving-metatheorems-proving-totality-assertions-in-non-empty-contexts/) + 13. [Proving metatheorems in non-empty contexts](/wiki/proving-metatheorems-proving-metatheorems-in-non-empty-contexts/) + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/reasoning-from-false.lf b/new-tests/stelf-output/pages/reasoning-from-false.lf new file mode 100644 index 0000000..b0eb182 --- /dev/null +++ b/new-tests/stelf-output/pages/reasoning-from-false.lf @@ -0,0 +1,78 @@ +% +%{! Twelf's coverage checker often rules out contradictory cases for you (see, for example, the [proof of determinacy for the simply typed λ-calculus](/wiki/uniqueness-lemma/)). However, you will sometimes need to reason from contradictory assumptions yourself. You can do this by declaring an _uninhabited type_, proving that certain assumptions are contradictory, and then writing lemmas that conclude anything from a term of the uninhabited type. + +## The uninhabited type !}% +%sort uninhabited %. +%freeze uninhabited %. +%{! Freezing the intentionally uninhabited type (which is also sometimes called ``void``, ``false``, or ``absurd``) is not strictly necessary, but it clarifies the programmer's intent—Twelf will signal an error if any later part of the signature introduces a term of this type. + +## Proving contradictions + +We use some simple judgements on natural numbers to illustrate proving contradictions. !}% +%sort nat %. +%term nat/z nat %. +%term nat/s %pi nat %-> nat %. +%sort nat-less %. +%term nat-less/z nat-less nat/z (nat/s N) %. +%term nat-less/s %pi (nat-less (nat/s N1) (nat/s N2)) %<- (nat-less N1 N2) %. +%sort nat-plus %. +%term nat-plus/z nat-plus nat/z N N %. +%term nat-plus/s %pi (nat-plus (nat/s N1) N2 (nat/s N3)) %<- (nat-plus N1 N2 N3) %. +%{! ### Contradiction by coverage checking + +First, we prove that no number is less than zero: !}% +%{! (options removed from twelftag: check="true") !}% +%sort nat-less-z-uninhabited %. +%mode nat-less-z-uninhabited %in %out %. +%worlds () (nat-less-z-uninhabited _ _) %. +%total {} (nat-less-z-uninhabited _ _) %. +%{! The coverage checker accepts this proof without any cases because there are no rules for deriving ``nat-less _ nat/z``. This is the simplest way to create a contradiction. + +### Inductive contradictions + +Some contradictions require an inductive argument. For example, we can prove that if ``N1 < N2`` then there is no ``N3`` such that ``N2 + N3 = N1``. !}% +%{! (options removed from twelftag: check="true") !}% +%sort nat-less-plus-uninhabited %. +%mode nat-less-plus-uninhabited %in %in %out %. +%term _ + %pi (nat-less-plus-uninhabited (nat-less/s NL) (nat-plus/s NP) DU) + %<- (nat-less-plus-uninhabited NL NP DU) %. +%worlds () (nat-less-plus-uninhabited _ _ _) %. +%total (D1) (nat-less-plus-uninhabited D1 _ _) %. +%{! In this theorem, the coverage checker rules out all cases but the one where the first input is of the form +<code> +nat-less N1 N2 +------------------------------nat-less/s +nat-less (nat/s N1) (nat/s N2) +</code> +and the second input is of the form +<code> +nat-plus N2 N3 N1 +----------------------------------nat-plus/s +nat-plus (nat/s N2) N3 (nat/s N1) +</code> +To show that this remaining case is contradictory. we apply the inductive hypothesis on the subderivations to derive ``uninhabited``, and then return the resulting derivation of ``uninhabited`` to meet our proof obligations. + +With this theorem, the coverage checker rules out all the base cases, so we are able to return a term of type ``uninhabited`` in the one remaining case by writing a loop. + +## Reasoning from contradiction + +Once you have a contradiction, it is easy to write a lemma that reasons from it. For example, from ``uninhabited`` we can conclude that any two numbers are ``nat-less``: !}% +%{! (options removed from twelftag: check="true") !}% +%sort uninhabited-nat-less {N1 nat} {N2 nat} %. +%mode uninhabited-nat-less %in %in %in %out %. +%worlds () (uninhabited-nat-less _ _ _ _) %. +%total {} (uninhabited-nat-less _ _ _ _) %. +%{! This proof requires no cases because any case would involve a canonical form of type ``uninhabited``. + +This lemma could be used in a proof that requires coming up with a derivation of some ``less-than`` fact in contradictory circumstances. + +For an example where reasoning from contradiction is necessary, see the tutorial on [hereditary substitution for the STLC](/wiki/hereditary-substitution-for-the-stlc/). + +\{\{tutorial\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Reasoning_from_false). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/reformulating-languages-to-use-hypothetical-judgements.lf b/new-tests/stelf-output/pages/reformulating-languages-to-use-hypothetical-judgements.lf new file mode 100644 index 0000000..9c4c306 --- /dev/null +++ b/new-tests/stelf-output/pages/reformulating-languages-to-use-hypothetical-judgements.lf @@ -0,0 +1,558 @@ +% +% +%{! It's easy to represent [hypothetical judgements](/wiki/hypothetical-judgment/) in LF, exploiting [higher-order](/wiki/higher-order-judgements/) representation techniques. This tutorial presents some object-language judgements which are **not** typically phrased as hypothetical judgements, but can easily be reformulated as such, making the correspondence with their LF representation quite clear. In particular, we discuss parallel reduction and complete development for the lambda-calculus; thanks to Randy Pollack for suggesting this example. + +Before reading this tutorial, you should learn about hypothetical judgements and their representation in LF: + + - [Representing the judgements of the STLC](/wiki/proving-metatheorems-representing-the-judgements-of-the-stlc/) shows how object-language hypothetical judgements can be represented using LF binding. + - [Proving totality assertions in non-empty contexts](/wiki/proving-metatheorems-proving-totality-assertions-in-non-empty-contexts/) shows an additional example, and discusses proving totality assertions about higher-order judgements. + - [Proving metatheorems in non-empty contexts](/wiki/proving-metatheorems-proving-metatheorems-in-non-empty-contexts/) shows how to use totality assertions to prove metatheorems about higher-order judgements. + +## Syntax is a hypothetical judgement + +First, we need to define the syntax of the untyped lambda-calculus: + +```math +M ::= x \mid \mathsf{lam}(x.M) \mid \mathsf{app}(M_1, M_2) +``` + +A traditional story about this definition is that variables are some piece of first-order data such as strings or de Bruijn indices, is a binder (which means it can be -converted and can be substituted for) and so on. + +However, suppose we were given a single untyped datatype of tree-structured data with binding as a primitive notion, where such trees consist of variables , binders , and applications of constants such as and . Then we can save ourselves the trouble of recapitulating the construction of binding for each object language by simply carving out those trees that represent the language in question. We can do so with a hypothetical judgement of the form + +```math +x_1\,\mathtt{trm} \; \ldots \; x_n\,\mathtt{trm} \vdash M\,\mathtt{trm} +``` + +where the subjects of the judgement "" are untyped binding trees. This judgement is defined as follows: + +```math +{ + {x\,\mathtt{trm} \in \Gamma} + \over + {\Gamma \vdash x\,\mathtt{trm}} +} +\qquad +{ + {\Gamma \vdash M_1\,\mathsf{trm} \qquad \Gamma \vdash M_2\,\mathsf{trm}} + \over + {\Gamma \vdash \mathsf{app}(M_1, M_2)\,\mathtt{trm}} +} +``` + +```math +{ + {\Gamma, x\,\mathtt{trm} \vdash M\,\mathtt{trm}} + \over + {\Gamma \vdash \mathsf{lam}(x.M)\,\mathtt{trm}} +} +``` + +A variable is a term if it was assumed to be a term; at a binder, we extend the context by assuming a new term. The important point about this style of definition is that variables are inherently _scoped_ and given meaning only by assumption: is only meaningful if we are in a context where we have some assumptions about it. Consequently, + +```math +x_1\,\mathtt{trm} \; \ldots \; x_n\,\mathtt{trm} \vdash M\,\mathtt{trm} +``` + +captures exactly the terms with free variables .... + +When you're working with an inherently scoped type of binding trees, you can't give an unconditional definition of what it means to be a term with rules like + +```math +\color{red} +{ + \over + {x\,\mathtt{trm}} +} +\qquad +{ + M\,\mathtt{trm} + \over + \mathsf{lam}(x.M)\,\mathtt{trm} +} +``` + +where the first rule means "all those trees that happen to be variables are terms": it would break the abstraction of variables-as-scoped-data to state a rule about all those trees that happen to be variables. + +The moral of the story is that syntax with binding can be thought of as a hypothetical judgement. +The LF encoding of this syntax can be thought of as an [intrinsic encoding](/wiki/intrinsic-and-extrinsic-encodings/) of the above judgement . LF provides _typed_ binding trees, so we can define terms by specifying typed operators, rather than a predicate over untyped trees: !}% +%sort trm %. +%term lam %pi (%pi trm %-> trm) %-> trm %. +%term app %pi trm %-> trm %-> trm %. +%{! Then this judgement: + +```math +x_1\,\mathtt{trm} \; \ldots \; x_n\,\mathtt{trm} \vdash M\,\mathtt{trm} +``` + +becomes the following LF judgment: + +```math +x_1 : \mathtt{trm} \; \ldots \; x_n : \mathtt{trm} \vdash M : \mathtt{trm} +``` + +where the and are LF variables and terms. + +## Parallel reduction + +Parallel reduction is traditionally defined as follows. For conciseness, we write instead of and instead of + +```math +{ + \over + x \Rightarrow x +}\mathit{var} +\qquad +{ + M \Rightarrow M' \qquad N \Rightarrow N' + \over + (\lambda x.M)\,N \Rightarrow M'[N'/x] +}\mathit{beta} +``` + +```math +{ + M \Rightarrow M' \qquad N \Rightarrow N' + \over + M\,N \Rightarrow M'\,N' +}\mathit{app} +\qquad +{ + M \Rightarrow M' + \over + \lambda x.M \Rightarrow \lambda x.M' +}\mathit{lam} +``` + +But there's a problem with this presentation: it's not explicit about scoping! The first rule has the exact same problem as the "bad" rule above that had as a premise and no conclusion. + +If we want to treat variables as scoped data, we must be explicit about scoping. This suggests the following presentation: + +```math +\Gamma ::= \cdot \mid \Gamma, x\,\mathtt{trm} +``` +```math +{ + x\,\mathtt{trm} \in \Gamma + \over + \Gamma \vdash x \Rightarrow x +}\mathit{var} +\qquad +{ + \Gamma, x\,\mathtt{trm} \vdash M \Rightarrow M' + \qquad + \Gamma \vdash N \Rightarrow N' + \over + \Gamma \vdash (\lambda x.M)\,N \Rightarrow M'[N'/x] +}\mathit{beta} +``` +```math +{ + \Gamma \vdash M \Rightarrow M' + \qquad + \Gamma \vdash N \Rightarrow N' + \over + \Gamma \vdash M\,N \Rightarrow M'\,N' +}\mathit{app} +\qquad +{ + \Gamma, x\,\mathtt{trm} \vdash M \Rightarrow M' + \over + \Gamma \vdash \lambda x.M \Rightarrow \lambda x.M' +}\mathit{lam} +``` + +Now at least the judgement only talks about well-scoped data. However, the rule that concludes can be a stumbling block for encoding in LF. It has the form "derive if " where and are _different judgements_. That's not one of the structural principles of a [hypothetical judgement](/wiki/hypothetical-judgment/), and allowing this strange sort of access to the context could invalidate the substitution principle (if I substitute for , I can no longer derive !). So what are we to do? + +We'll give three possibilities here. The third is in some sense the "most faithful" to the informal specification above. However, when one is not committed to a specific presentation, it's often best to let the formalization guide the informal specification! + +### Reformulation 1: Hypothetical reductions + +While a rule that says "derive if " is suspicious, a rule that says +"derive if " is just the usual hypothesis/identity axiom that we expect from all hypothetical judgements. So, one solution is to change the notion of context we consider so that is just an instance of hypothesis. Whenever we assume a variable , we also _assume_ a derivation : + +```math +\Gamma ::= \cdot \mid \Gamma, x\,\mathtt{trm}, x \Rightarrow x +``` +```math +{ + x \Rightarrow x \in \Gamma + \over + \Gamma \vdash x \Rightarrow x +}\mathit{var}_1 +\qquad +{ + \Gamma, x\,\mathtt{trm}, x \Rightarrow x \vdash M \Rightarrow M' + \qquad + \Gamma \vdash N \Rightarrow N' + \over + \Gamma \vdash (\lambda x.M)\,N \Rightarrow M'[N'/x] +}\mathit{beta}_1 +``` +```math +{ + \Gamma \vdash M \Rightarrow M' + \qquad + \Gamma \vdash N \Rightarrow N' + \over + \Gamma \vdash M\,N \Rightarrow M'\,N' +}\mathit{app}_1 +\qquad +{ + \Gamma, x\,\mathtt{trm}, x \Rightarrow x \vdash M \Rightarrow M' + \over + \Gamma \vdash \lambda x.M \Rightarrow \lambda x.M' +}\mathit{lam}_1 +``` + +In this presentation, the rule is totally unobjectionable. In the premies of the rules and , which deal with binding forms, the context is extended with the assumption that for the bound variable . The derivations of this version are isomorphic to the first definition of , but the rule here works as a use of hypothetical judgements. + +The LF representation of this formulation is quite direct: !}% +%sort => %. +%prec %none 10 => %. +%term =>/beta + %pi (app (lam M) N => M' N') + %<- ({x trm} %pi (x => x) %-> (M x => M' x)) + %<- (N => N') %. +%term =>/app %pi (app M N => (app M' N')) %<- (N => N') %<- (M => M') %. +%term =>/lam %pi (lam M => lam M') %<- ({x trm} %pi (x => x) %-> (M x => M' x)) %. +%block =>b [x trm] [=>/x x => x]%. +%worlds (=>b) (=> _ _) %. +%{! Derivations using are represented by LF variables representing the reduction assumptions in . The Twelf ``%worlds`` declaration documents the form of in our informal definition. + +#### Adequacy + +This is an [adequate encoding](/wiki/adequacy/) of the informal judgment we started with: there's a 1-to-1 correspondence between derivations like this with the rules subscripted with : + +```math +x_1\,\mathtt{trm}, x_1 \Rightarrow x_1, \ldots, x_k\,\mathtt{trm}, x_k \Rightarrow x_k \vdash M \Rightarrow M' +``` + +and with LF derivations $D$ such that the following is derivable according to the rules of LF: + +```math +x_1 : \mathtt{trm}, d_1 : x_1 \Rightarrow x_1, \ldots x_k : \mathtt{trm}, d_k : x_k \Rightarrow x_k \vdash D : M \Rightarrow M' +``` + + +#### Substitution + +This reformulation elucidates a substitution principle for parallel reduction, as an instance of the general substitution principle for hypothetical judgements: + +> If \ +> and \ +> and ,\ +> then . + +In the LF representation, this substitution principle comes "for free" from the general substitution principle for LF terms. + +### Reformulation 2: Change the relation + +Another option is to change the definition of the judgement so that it doesn't have a variable-specific rule. In this case, we can generalize the variable rule to a general reflexivity rule: + +```math +\Gamma ::= \cdot \mid \Gamma, x\,\mathtt{trm} +``` +```math +{ + \over + \Gamma \vdash M \Rightarrow M +}\mathit{refl}_2 +\qquad +{ + \Gamma, x\,\mathtt{trm} \vdash M \Rightarrow M' + \qquad + \Gamma \vdash N \Rightarrow N' + \over + \Gamma \vdash (\lambda x.M)\,N \Rightarrow M'[N'/x] +}\mathit{beta}_2 +``` +```math +{ + \Gamma \vdash M \Rightarrow M' + \qquad + \Gamma \vdash N \Rightarrow N' + \over + \Gamma \vdash M\,N \Rightarrow M'\,N' +}\mathit{app}_2 +\qquad +{ + \Gamma, x\,\mathtt{trm} \vdash M \Rightarrow M' + \over + \Gamma \vdash \lambda x.M \Rightarrow \lambda x.M' +}\mathit{lam}_2 +``` + +This is a _different definition_ than the informal one we started with: derivations +with the rules subscripted are _not_ isomorphic to those subscripted . However, the while this change changes the possible derivations, it does not change the relation , because reflexivity was admissible before. Maybe that's an acceptable change, that actually simplifies the definition. + +The LF representation looks like this: !}% +%sort => %. +%prec %none 10 => %. +%term =>/refl M => M %. +%term =>/beta %pi (app (lam M) N => M' N') %<- ({x trm} M x => M' x) %<- (N => N') %. +%term =>/app %pi (app M N => (app M' N')) %<- (N => N') %<- (M => M') %. +%term =>/lam %pi (lam M => lam M') %<- ({x trm} M x => M' x) %. +%block trmb [x trm]%. +%worlds (trmb) (=> _ _) %. +%{! + +#### Substitution + +The substitution principle implied by this LF development looks a bit simpler than the previous one, though this is only because it lacks the premise , which as we've previously mentioned is always derivable even in the first reformulation. + +> If \ +> and \ +> then . + +### Reformulation 3 : Tagged variables + +If we're really committed to the original informal specification, with the rule that said we could immediately derive the reflexive conclusion when was specifically a variable in the context, then it's possible to capture that by having an new judgment , which is only ever derivable from a hypothetical judgment. + +```math +\Gamma ::= \cdot \mid \Gamma, x\,\mathtt{trm}, x\,\mathtt{isvar} +``` + +Then, we can use the judgment as a regular, uncontroversial hypothetical judgment, where a premise allows you to prove the conclusion + +```math +{ + x\,\mathtt{isvar} \in \Gamma + \over + \Gamma \vdash x\,\mathtt{isvar} +} +\qquad +{ + \Gamma \vdash M\,\mathtt{isvar} + \over + \Gamma \vdash M \Rightarrow M +} +``` + +This presentation is pedantically appropriate, but it's entirely more common to take these two rules and compress them into the single on-paper rule shown below: + +```math +{ + x\,\mathtt{isvar} \in \Gamma + \over + \Gamma \vdash x \Rightarrow x +}\mathit{var}_3 +\qquad +{ + \Gamma, x\,\mathtt{trm}, x\,\mathtt{isvar} \vdash M \Rightarrow M' + \qquad + \Gamma \vdash N \Rightarrow N' + \over + \Gamma \vdash (\lambda x.M)\,N \Rightarrow M'[N'/x] +}\mathit{beta}_3 +``` +```math +{ + \Gamma \vdash M \Rightarrow M' + \qquad + \Gamma \vdash N \Rightarrow N' + \over + \Gamma \vdash M\,N \Rightarrow M'\,N' +}\mathit{app}_3 +\qquad +{ + \Gamma, x\,\mathtt{trm}, x\,\mathtt{isvar} \vdash M \Rightarrow M' + \over + \Gamma \vdash \lambda x.M \Rightarrow \lambda x.M' +}\mathit{lam}_3 +``` + +When all is said and done, this actually looks a lot like the first reformulation, but we're using a one place relation instead of the two-place relation where hypothetical occurrences are, by invariant, always reflexive. + +Unlike the first reformulation, where the rule came "for free" from LF, this formulation has four rules. The rule looks a little different in LF than in the on-paper derivation, because we use the uppercase `M` in LF instead of the lowercase in , but the meaning is the same, since we will only ever derive a term of type `isvar M` when `M` is a variable in the context. + +!}% +%sort => %. +%prec %none 10 => %. +%sort isvar %. +%term =>/var %pi (M => M) %<- (isvar M) %. +%term =>/beta + %pi (app (lam M) N => M' N') + %<- ({x trm} %pi (isvar x) %-> (M x => M' x)) + %<- (N => N') %. +%term =>/app %pi (app M N => (app M' N')) %<- (N => N') %<- (M => M') %. +%term =>/lam %pi (lam M => lam M') %<- ({x trm} %pi (isvar x) %-> (M x => M' x)) %. +%block isvarb [x trm] [isvar/x isvar x]%. +%worlds (isvarb) (isvar _) (=> _ _) %. +%{! + +#### Substitution + +As always when we are defining judgments in non-closed worlds, LF provides us a substitution principle for free. But, this time, the substitution principle is basically useless: + +> If \ +> and \ +> and ,\ +> then . + +We only ever get to prove when is a variable! The for-free substitution properties in the last two formulations actually told us something, but here we've got. It _is_ possible to prove the substitution theorem that we expect --- the same substitution theorem that LF gave for free in the first reformulation --- but we have to prove a metatheorem to do so. It's quite a bit of development for what is, at the end of the day, a straightforward proof by induction: + +!}% +%sort subst {x trm} {d isvar x} %. +%mode subst %in %in %out %. +%term _ subst ([x] [dx isvar x] =>/var dx) (%the (M' => M') D2) D2 %. +%term _ subst ([x] [dx isvar x] =>/var D) (%the (M' => M') D2) (=>/var D) %. +%term _ + %pi (subst ([x] [dx isvar x] =>/beta (%the (M2 x => N2 x) (D1 x dx)) ([y] [dy isvar y] %the (M1 x y => N1 x y) (D1' x dx y dy))) (%the (M' => M') D2) (%the (app (lam ([y] M1 M' y)) (M2 M') => N1 M' (N2 M')) (=>/beta D3 D3'))) + %<- (subst D1 D2 (%the (M2 M' => N2 M') D3)) + %<- ({y} {dy isvar y} subst ([x] [dx isvar x] D1' x dx y dy) D2 (%the (M1 M' y => N1 M' y) (D3' y dy))) %. +%term _ + %pi (subst ([x] [dx isvar x] =>/app (%the (M1 x => N1 x) (D1 x dx)) (%the (M2 x => N2 x) (D1' x dx))) (%the (M' => M') D2) (%the (app (M1 M') (M2 M') => app (N1 M') (N2 M')) (=>/app D3 D3'))) + %<- (subst D1 D2 (%the (M1 M' => N1 M') D3)) + %<- (subst D1' D2 (%the (M2 M' => N2 M') D3')) %. +%term _ + %pi (subst ([x] [dx isvar x] =>/lam ([y] [dy isvar y] %the (M x y => N x y) (D1 x dx y dy))) (%the (M' => M') D2) (%the (lam ([y] M M' y) => (lam ([y] N M' y))) (=>/lam D3))) + %<- ({y} {dy isvar y} subst ([x] [dx isvar x] D1 x dx y dy) D2 (%the (M M' y => N M' y) (D3 y dy))) %. +%{! The `%worlds` for the metatheorem is the same `isvarb` block used in the third reformulation. !}% +%{!! begin checked !!}% +%worlds (isvarb) (subst _ _ _) %. +%total T (subst T _ _) %. +%{!! end checked !!}% +%{! + +### Conclusion: the cost of losing substitution + +The third reformulation above was almost without a doubt the one most akin to the initial presentation. But the cost was that the substitution principle needed to be established separately, which requires a bit of work. + +This is sometimes the correct call: in the development of [structural focalization](/wiki/focusing/) on the wiki, there were two "focal substitution principles." It was ultimately the most convenient to balance clarity and convenience prove one of the two principles by hand (as in the third reformulation above) while letting the other be inhereted from LF's substitution principles (as in the first two reformulations above). + +## Complete development + +Parallel reduction is non-deterministic: any left-hand term that can be reduced by the ``beta`` rule can also be reduced by the ``app`` rule, which is why the relation can be reflexive. Complete development is a restriction of parallel reduction where ``beta`` takes precedence over ``app``. In each step of complete development, all of the beta-redices in the left-hand term are reduced. + +Informally, we add a side condition to the ``app`` rule: + +```math +{ + \Gamma \vdash M \Rightarrow M' + \qquad + \Gamma \vdash N \Rightarrow N' + \qquad + (M \textit{ is not a lambda}) + \over + \Gamma \vdash M\,N \Rightarrow M'\,N' +}\mathit{app} +``` + + +How can we state this side condition more precisely? We need a judgement which holds whenever does not have the form . It's easy to define this as a hypothetical judgement if we choose our contexts correctly: + +```math +\Gamma ::= \cdot \mid \Gamma, x\,\mathtt{trm}, x\,\mathtt{notlam} +``` +```math +{ + x\,\mathtt{notlam} \in \Gamma + \over + \Gamma \vdash x\,\mathtt{notlam} +} +\qquad +{ + \over + \Gamma \vdash (M_1\,M_2)\,\mathtt{notlam} +} +``` + +That is, with each variable, we make the additional assumption that it is not a lambda. + +Then complete development is easy to define: + +```math +\Gamma ::= \cdot \mid \Gamma, x\,\mathtt{trm}, x\,\mathtt{notlam}, x \Rightarrow x +``` +```math +{ + x \Rightarrow x \in \Gamma + \over + \Gamma \vdash x \Rightarrow x +}\mathit{var}_c +``` +```math +{ + \Gamma, x\,\mathtt{trm}, x\,\mathtt{notlam}, x \Rightarrow x \vdash M \Rightarrow M' + \qquad + \Gamma \vdash N \Rightarrow N' + \over + \Gamma \vdash (\lambda x.M)\,N \Rightarrow M'[N'/x] +}\mathit{beta}_c +``` +```math +{ + \Gamma \vdash M \Rightarrow M' + \qquad + \Gamma \vdash N \Rightarrow N' + \qquad + M\,\mathtt{notlam} + \over + \Gamma \vdash M\,N \Rightarrow M'\,N' +}\mathit{app}_c +``` +```math +{ + \Gamma, x\,\mathtt{trm}, x\,\mathtt{notlam}, x \Rightarrow x \vdash M \Rightarrow M' + \over + \Gamma \vdash \lambda x.M \Rightarrow \lambda x.M' +}\mathit{lam}_c +``` + +The LF representation is direct: !}% +%sort notlam %. +%term notlam/app notlam (app _ _) %. +%block nlb [x trm] [nlx notlam x]%. +%worlds (nlb) (notlam _) %. +%{! Whenever we add a ``trm`` assumption to the LF context, we assume that it is not a lambda. The Twelf ``%worlds`` declaration documents this fact, and causes Twelf to complain if we ever violate this convention. + +Then complete development is a simple twist on parallel reduction: !}% +%sort ==> %. +%prec %none 10 ==> %. +%term ==>/beta + %pi (app (lam M) N ==> M' N') + %<- ({x trm} %pi (notlam x) %-> (x ==> x) %-> (M x ==> M' x)) + %<- (N ==> N') %. +%term ==>/app %pi (app M N ==> (app M' N')) %<- (N ==> N') %<- (M ==> M') %<- (notlam M) %. +%term ==>/lam + %pi (lam M ==> lam M') + %<- ({x trm} %pi (notlam x) %-> (x ==> x) %-> (M x ==> M' x)) %. +%block ==>b [x trm] [nlx notlam x] [==>/x x ==> x]%. +%worlds (==>b) (==> _ _) %. +%{! Every time we extend the context, we add an assumption ``notlam x`` for that variable. We also add a reflexivity assumption for each variable because the informal definition of complete development still has a reflexivity rule for variables (even though the relation is not reflexive in general). + +#### Substitution + +As above, this formulation gives a "free" substitution principle for complete development: + +> If \ +> and \ +> and \ +> and ,\ +> then . + +The form of the context ensures that we need a derivation of to make this substitution. This restriction is similar to the restriction that made the third reformulation's "free" substitution principle useless, but it's more reasonable in this case: it's not obvious that you can substitute lambdas for variables while preserving complete development, since you'd have to replace instances of ``app`` with something else. + +#### Examples + +We can give a few examples of parallel reduction in action using the [`%query`](/wiki/percent-query/) mechanism. + +!}% +%{!! begin checked !!}% +%query 1 _ _ app (lam ([x] app x x)) (lam ([x] app x x)) ==> N %. +%{!! end checked !!}% +%{!! begin checked !!}% +%define z lam ([f] lam ([x] x)) %. +%define s lam ([n] lam ([f] lam ([x] app f (app (app n f) x)))) %. +%query 1 _ _ app (app s z) (lam ([f] lam ([x] app f (app f x)))) ==> N %. +%{!! end checked !!}% +%{! + +## See also + +* [Hypothetical judgements](/wiki/hypothetical-judgment/) +* [Higher-order judgements](/wiki/higher-order-judgements/) +* The case study on [Church-Rosser via complete development](/wiki/church-rosser-via-complete-development/) for some proofs about the judgements defined here, illustrating the use of [regular worlds](/wiki/percent-worlds/). + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/relation.lf b/new-tests/stelf-output/pages/relation.lf new file mode 100644 index 0000000..59b2a00 --- /dev/null +++ b/new-tests/stelf-output/pages/relation.lf @@ -0,0 +1,14 @@ +% +%{! It is common to refer to a an LF [type family](/wiki/type-family/) as a **relation**, because a type family defines a relation on its indices where indices are related iff their instance of the type family is inhabited. A htype family is also commonly referred to as a [judgment](/wiki/judgment/). + +## See also + +* [Judgment](/wiki/judgment/) +* [Judgments as types](/wiki/judgment/) +\{\{stub\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Relation). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/release-history.lf b/new-tests/stelf-output/pages/release-history.lf new file mode 100644 index 0000000..a572049 --- /dev/null +++ b/new-tests/stelf-output/pages/release-history.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you should go to the [new version](/release-history/) +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/research-projects-using-twelf.lf b/new-tests/stelf-output/pages/research-projects-using-twelf.lf new file mode 100644 index 0000000..97f96fa --- /dev/null +++ b/new-tests/stelf-output/pages/research-projects-using-twelf.lf @@ -0,0 +1,36 @@ +% +%{! + +Please add your research using Twelf to the links below. We hope this +collection of links will encourage collaboration among Twelf users and record +how Twelf is being used in practice. + +The links on this page are grouped into two rough categories: larger projects +(e.g., a multi-year endeavor) and smaller projects (e.g., some Twelf code that +accompanies a research paper). The line between these two is fuzzy, so add +your work wherever you think it fits best. + +## Research using Twelf + +### Larger projects + +- [Proof Carrying Code](http://www.cs.princeton.edu/sip/projects/pcc/) - scaling up the idea of Proof Carrying Code to real programming languages, using a logic encoded in Twelf's dependent type system. +* [The SeLF Project](http://www.cs.cmu.edu/~self/) - encodings of security policies and proofs of security properties for the [Grey project](http://www.ece.cmu.edu/~grey/). +* [Mechanizing the Metatheory of Standard ML](http://www.cs.cmu.edu/~dklee/tslf/) - a formalization of Standard ML and verification of its metatheory in Twelf. +* [Modal Types for Mobile Code](http://tom7.org/proposal/) - [Tom](/wiki/user-tom7/)'s Ph.D. thesis project +* [LATIN](https://web.archive.org/web/20150316072122/https://trac.omdoc.org/LATIN/) - a project aimed at the creation of a large library of logics and logic translations supported by a web-scalable infrastructure + +### Smaller projects + +* [Dependent Types for ML](http://www.cs.cmu.edu/~drl/pubs/lh05dmlproofs-tr/lh05dmlproofs-tr.pdf) - A paper by [Dan Licata](/wiki/user-drl/) and Bob Harper about enriching an ML-like type system with a form of dependent types. Much of the metatheory of the language is mechanized in Twelf. +* [Substructural Languages](http://www.cs.cornell.edu/people/fluet/twelf/index.html) - Twelf code from some of Matthew Fluet's papers on substructural languages. +* [Self-Adjusting Computation](http://www.cs.cmu.edu/~jdonham/aml-proof/) - [Jake Donham's](/wiki/user-jaked/) formalization of a consistency proof of non-deterministic semantics. +* [Elaborating Intersection and Union Types](https://arxiv.org/abs/1206.5386) — Jana Dunfield's proofs about an elaboration semantics. + +## Related projects + +* [The POPLmark Challenge](https://www.seas.upenn.edu/~plclub/poplmark/cmu.html) - A project aimed at increasing the overall use of theorem provers like Twelf in programming language design. A solution to the full challenge using Twelf submitted by a team at CMU [http://fling-l.seas.upenn.edu/~plclub/cgi-bin/poplmark/index.php?title=Submission_from_Carnegie_Mellon]. +* [Delphin](http://delphin.poswolsky.com/) - An integration of concepts from Twelf and functional programming languages. +* [Elf](https://web.archive.org/web/20071012232336/http://www.cs.cmu.edu/~fp/elf.html) - The predecessor to Twelf. + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/respects-lemma.lf b/new-tests/stelf-output/pages/respects-lemma.lf new file mode 100644 index 0000000..530c5c4 --- /dev/null +++ b/new-tests/stelf-output/pages/respects-lemma.lf @@ -0,0 +1,118 @@ +% +%{! _You should read the tutorial on [representing equality of LF terms as a type family](/wiki/equality/) before reading this one._ + +At a high level, a **respects lemma** shows that a type family or term constructor "respects" some relations on its arguments. Specifically, we use the term _respects lemma_ to refer to: +- a lemma proving that the indices to a type family "respect" some relations on those indices, in the sense that inhabitation of the type family is preserved by replacing related indices. +- a **compatibility lemma** for a term constructor ``f``, which states that if some relations hold for some arguments, then a relation holds for ``f`` applied to those arguments. + +Often, the relations in question are [equality of LF terms](/wiki/equality/). + +## Example: Proving that a type family respects equality + +It is often necessary to prove a respects lemma showing that a type family respects equality of its indices. +For example, we can define the standard equality relation ``bin-eq`` for binary numbers, the standard ``and`` relation on binary numbers, and then prove a _respects lemma_ that ``and`` respects equality: !}% +%sort bin %. +%term e bin %. +%term 1 %pi bin %-> bin %. +%term 0 %pi bin %-> bin %. +%sort bin-eq %. +%term bin-eq/ bin-eq N N %. +% note: not defined unless the numbers are the same length +%sort and %. +%mode and %in %in %out %. +%term and/e and e e e %. +%term and/00 %pi (and (0 M) (0 N) (0 P)) %<- (and M N P) %. +%term and/01 %pi (and (0 M) (1 N) (0 P)) %<- (and M N P) %. +%term and/10 %pi (and (1 M) (0 N) (0 P)) %<- (and M N P) %. +%term and/11 %pi (and (1 M) (1 N) (1 P)) %<- (and M N P) %. +%{ +The respects lemma: the and relation respects equality +}% +%sort and-resp %. +%mode and-resp %in %in %in %in %out %. +%term _ and-resp bin-eq/ bin-eq/ bin-eq/ A A %. +%worlds () (and-resp _ _ _ _ _) %. +%total {} (and-resp _ _ _ _ _) %. +%{! The theorem ``and-resp`` states that if M and M' are equal, and N and N' are equal, and O and O' are equal, and M & N = O, then M' & N' = O'. The proof is trivial: since the only way to conclude ``bin-eq`` is using ``bin-eq/`` where both binary numbers must be identical, the same derivation of ``and`` that we got as input works for the output. + +If ``bin-eq`` were some other equivalence relation (_e.g._, ignoring leading zeros), the respects lemma would require a more sophisticated proof (e.g., an inductive argument over the derivation of the equality). + +Although we proved that ``and`` respects equality in all three arguments, we can easily use this lemma even when we only have an equality derivation for one pair of arguments, as long as the other two pairs of arguments are already syntactically equal. For example, we can prove: !}% +%sort and-first-resp %. +%mode and-first-resp %in %in %out %. +%term _ %pi (and-first-resp EQ A A') %<- (and-resp EQ bin-eq/ bin-eq/ A A') %. +%worlds () (and-first-resp _ _ _) %. +%total D (and-first-resp D _ _) %. +%{! (This theorem is trivial and could have been proven directly, but it illustrates the point.) When we invoke the ``and-resp`` lemma as a subgoal, we pass along the equality derivation for the first two arguments, and pass along the constant ``bin-eq/`` for the others, which are known to be identical. Therefore, it suffices to define the more general respects lemmas and then invoke them in specialized ways, if necessary. + +\{\{needs|an example where a respects lemma is needed\}\} + +## Example: proving compatibility lemmas + +Another kind of respects lemma is a **compatibility lemma**, which states that a relation is respected by the application of term constructors. (This is also called a **congruence lemma** when the relation in question is an equivalence relation.) For example, we can prove compatibility lemmas about the constructors ``1`` and ``0``: !}% +%sort 1-compat %. +%mode 1-compat %in %out %. +%term _ 1-compat bin-eq/ bin-eq/ %. +%sort 0-compat %. +%mode 0-compat %in %out %. +%term _ 0-compat bin-eq/ bin-eq/ %. +%worlds () (1-compat _ _) (0-compat _ _) %. +%total (D E) (1-compat D _) (0-compat E _) %. +%{! We might use these compatibility lemmas in proving some theorems about binary numbers. For instance, we might want to prove that ``not`` is its own inverse: !}% +%sort not %. +%mode not %in %out %. +%term not/e not e e %. +%term not/1 %pi (not (1 B) (0 B')) %<- (not B B') %. +%term not/0 %pi (not (0 B) (1 B')) %<- (not B B') %. +%sort notnot %. +%mode notnot %in %in %out %. +%term notnot/e notnot not/e not/e bin-eq/ %. +%term notnot/1 + %pi (notnot (not/1 N1) (not/0 N2) EQ') + %<- (notnot N1 N2 (%the (bin-eq A' C') EQ)) + %<- (1-compat EQ (%the (bin-eq (1 A') (1 C')) EQ')) %. +%term notnot/0 + %pi (notnot (not/0 N1) (not/1 N2) EQ') + %<- (notnot N1 N2 (%the (bin-eq A' C') EQ)) + %<- (0-compat EQ (%the (bin-eq (0 A') (0 C')) EQ')) %. +%worlds () (notnot _ _ _) %. +%total D (notnot D _ _) %. +%{! In the inductive cases of ``notnot``, we have a derivation ``EQ`` that two smaller numbers are equal, and need to use this derivation to prove that if we apply the ``1`` constructor (for each) to each of the terms, they are still equal. + +### Generalized compatibility lemmas + +Compatibility lemmas can be generalized somewhat so that we needn't prove so many. Instead of defining the ``0-compat`` and ``1-compat`` cases separately, we can abstract over the constructor as an LF function: !}% +%sort f-compat {EQ bin-eq M M'} {F %pi bin %-> bin} {EQ' bin-eq (F M) (F M')} %. +%mode f-compat %in %in %out %. +%term _ f-compat bin-eq/ _ bin-eq/ %. +%worlds () (f-compat _ _ _) %. +%total D (f-compat D _ _) %. +%{! Here, we instead prove that if ``M`` and ``M'`` are equal, then ``F M`` and ``F M'`` are equal for any ``F``. Now we can prove the ``notnot`` theorem again using only this generalized compatibility lemma: !}% +%sort notnot %. +%mode notnot %in %in %out %. +%term notnot/e notnot not/e not/e bin-eq/ %. +%term notnot/1 + %pi (notnot (not/1 N1) (not/0 N2) EQ') + %<- (notnot N1 N2 EQ) + %<- (f-compat EQ 1 EQ') %. +%term notnot/0 + %pi (notnot (not/0 N1) (not/1 N2) EQ') + %<- (notnot N1 N2 EQ) + %<- (f-compat EQ 0 EQ') %. +%worlds () (notnot _ _ _) %. +%total D (notnot D _ _) %. +%{! Here we pass in the constructors ``1 : bin -> bin`` and ``0 : bin -> bin`` as ``F``. The functions could arbitrary complex, like ``[b] 0 (1 (1 b))``. + +This generalization technique usually only works when equality is defined as Twelf identity, because of the arbitrariness of the function passed in. When it can be used, it can substantially reduce the number of compatibility lemmas that one needs to state. + +\{\{needs|a description of some more advanced/tasteless generalization tricks\}\} + +``>All code from this tutorial``. `` check="true">Twelf's output from this tutorial``. + +\{\{tutorial\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Respects_lemma). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/sets-and-supersets.lf b/new-tests/stelf-output/pages/sets-and-supersets.lf new file mode 100644 index 0000000..e9c1fb5 --- /dev/null +++ b/new-tests/stelf-output/pages/sets-and-supersets.lf @@ -0,0 +1,122 @@ +% +%{! \{\{needs|this is really about multisets encoded as lists . s/set/multiset/g. BTW didn't I author the part to do with reflexivity? [Varming](/wiki/user-varming/) 03:52, 13 October 2006 (EDT)\}\} +\{\{needs|editing for context and tone\}\} + +This is a simplification of your concept further to take away the need for subtyping. I think this is essentially as simple as you can get, and from here you can do one of two things: + +* Make the entries in the sets not strictly equal, but subject to a subtype relation (this is what you did, essentially) +* Make the sets ordered + +Once you've done both of these things, which seem like they are relatively orthogonal to each other, *then* we've got the environments that Appel is looking for. I think what would be interesting is to see the progression of all of them, so what I'm probably going to do next (unless you want to crack at it) is to redo this file will well-ordered sets. !}% +%% Map subtyping - concept progression +%% 1) Notion of sets and supersets of natural numbers +%% Daniel K. Lee and Robert J. Simmons +%% Carnegie Mellon University. August 2006. +% Natural numbers +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +% Sets of natural numbers +%sort set %. +%term set/nil set %. +%term set/cons %pi nat %-> set %-> set %. +% The containment judgment +% + +% Succeeds if a natural number is one of the items in the set +%sort in-set %. +%term in-set/hit in-set N (set/cons N SET) %. +%term in-set/miss %pi (in-set N (set/cons N' SET)) %<- (in-set N SET) %. +%{ +The superset relation + +This definition is "extensional" in that S1 is a superset of S2 if for +every natural number n in S2, that natural number also exists in S1. + +This relation says nothing about the frequency of these entries, so +(1,2,1) is a superset of (2,2,2) (a strict superset, though we do not define the +notion of "strict") +}% +%sort superset %. +%term superset/nil superset SET set/nil %. +%term superset/cons %pi (superset SET1 (set/cons N SET2)) %<- (in-set N SET1) %<- (superset SET1 SET2) %. +% BEGIN META-THEOREMS +%{ +Weakening + +Weakening is the crucial lemma for the reflexivity - it states that if SET1 is +a superset of SET2, you can add something to SET1 and the result is still a +superset of SET2. + +The second case goes as follows (using A >= B as shorthand for superset A B). +In the case that we have: + + IN:(in-set N' SET1) SUP:(SET1 >= SET2) +---------------------------------------- superset/cons + SET1 >= (set/cons N' SET2) + +we can get from induction that, for some arbitrary N, +SUP':((set/cons N SET1) >= SET2) + +We can then build a derivation of the result that we want like so: + + IN:(in-set N' SET1) +----------------------------- in-set/miss + in-set N' (set/cons N SET1) SUP':((set/cons N SET1) >= SET2) +------------------------------------------------------------------ superset/cons + (set/cons N SET1) >= (set/cons N' SET2) +}% +%sort superset-wkn {N} %. +%mode superset-wkn %in %in %out %. +%term _ superset-wkn _ superset/nil superset/nil %. +%term _ + %pi (superset-wkn N (superset/cons SUP IN) (superset/cons SUP' (in-set/miss IN))) + %<- (superset-wkn N SUP SUP') %. +%worlds () (superset-wkn _ _ _) %. +%total T (superset-wkn _ T _) %. +% Reflexivity of the superset relation +%sort superset-refl {SET} %. +%mode superset-refl %in %out %. +%term _ superset-refl set/nil superset/nil %. +%term _ + %pi (superset-refl (set/cons N SET) (superset/cons SUP' in-set/hit)) + %<- (superset-refl SET (%the (superset SET SET) SUP)) + %<- (superset-wkn N SUP (%the (superset (set/cons N SET) SET) SUP')) %. +%worlds () (superset-refl _ _) %. +%total T (superset-refl T _) %. +%{ +The "Entry-in-superset" lemma + +Just as weakening was the crucial lemma for reflexivity, this is the crucial +lemma for the transitivity lemma. It's statmement is simple - if N is in SET2, +and SET1 is a supertype of SET2, then N is in SET1. + +The proof is pretty straightforward induction - if the derivation of "in-set" +is the hit, you're done, if it's not, keep looking. +}% +%sort in-superset %. +%mode in-superset %in %in %out %. +%term _ in-superset in-set/hit (superset/cons SUP IN) IN %. +%term _ + %pi (in-superset (in-set/miss IN) (superset/cons SUP _) IN') + %<- (in-superset IN SUP IN') %. +%worlds () (in-superset _ _ _) %. +%total T (in-superset T _ _) %. +% Transitivity of the superset relation +%sort superset-trans %. +%mode superset-trans %in %in %out %. +%term _ superset-trans _ superset/nil superset/nil %. +%term _ + %pi (superset-trans SUP1 (superset/cons SUP2 IN) (superset/cons SUP3 IN')) + %<- (superset-trans SUP1 SUP2 SUP3) + %<- (in-superset IN SUP1 IN') %. +%worlds () (superset-trans _ _ _) %. +%total T (superset-trans _ T _) %. +%{! [Category:Twelf code](/wiki/category-twelf-code/) +\{\{stub tutorial\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Sets_and_supersets). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/signature-checking.lf b/new-tests/stelf-output/pages/signature-checking.lf new file mode 100644 index 0000000..0a096f0 --- /dev/null +++ b/new-tests/stelf-output/pages/signature-checking.lf @@ -0,0 +1,14 @@ +% +%{! \{\{needswork|To be completed\}\} +If all Twelf had were the features we've introduced so far, it would still be a useful tool: we can represent a large variety of complex types and do searches and logic programming over them. But where Twelf really shines is in its signature checks. Having created a Twelf signature—which, as we've seen, we can interpret as a labeled logic program—we can then ask Twelf to check certain properties of that signature. In particular, as we will see shortly, we can designate the arguments of a type family as inputs and outputs, and then ask +* whether this designation is consistent with the reading of the signature as a logic program, that is, whether the outputs of the constructor will be ground (involve no logic variables) if the inputs are all ground (a **mode** check), +* whether the constructor will produce at most one combination of outputs for any given combination of ground inputs (a **uniqueness** check), +* whether the logic program will always terminate if it is given ground inputs (a **termination** check), or +* whether the constructor will always produce at least one combination of ground outputs for any possible combination of ground inputs (a **totality** check). +The totality check subsumes termination, but it has a couple of additional elements: **input coverage** and **output coverage**. Input coverage involves checking whether there are enough clauses for the constructor to match any possible combination of inputs; we then say that all possible inputs are _covered_. Output coverage involves checking that certain assumptions made about the form of outputs from "subroutine" calls are justified. Let's now take a look at these checks in more detail and see some examples. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Signature_checking). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/signatures-as-logic-programs.lf b/new-tests/stelf-output/pages/signatures-as-logic-programs.lf new file mode 100644 index 0000000..d7c137d --- /dev/null +++ b/new-tests/stelf-output/pages/signatures-as-logic-programs.lf @@ -0,0 +1,127 @@ +% +%{! Recall from the previous section the signature for representing the days of the week, the "next day" function, and the "day after tomorrow" function: !}% +% Days of the week +%sort day %. +%term sunday day %. +%term monday day %. +%term tuesday day %. +%term wednesday day %. +%term thursday day %. +%term friday day %. +%term saturday day %. +% Next day function +%sort next_day %. +%term next_day_sun next_day sunday monday %. +%term next_day_mon next_day monday tuesday %. +%term next_day_tue next_day tuesday wednesday %. +%term next_day_wed next_day wednesday thursday %. +%term next_day_thu next_day thursday friday %. +%term next_day_fri next_day friday saturday %. +%term next_day_sat next_day saturday sunday %. +% Day after tomorrow function +%sort dat %. +%term dat_def %pi (next_day D1 D2) %-> (next_day D2 D3) %-> (dat D1 D3) %. +%{! Readers familiar with logic programming may have noticed a strong similarity between this signature and how the relations "next day" and "day after tomorrow" might be defined in Prolog. This is no coincidence. Indeed, consider the following Prolog definitions for these two relations: + +```prolog +% Prolog definitions of next_day and dat +next_day(sunday, monday). +next_day(monday, tuesday). +next_day(tuesday, wednesday). +next_day(wednesday, thursday). +next_day(thursday, friday). +next_day(friday, saturday). +next_day(saturday, sunday). + +dat(D1, D3) :- next_day(D1, D2), next_day(D2, D3). +``` + +No types are involved here; the logical constants are introduced when they are needed, rather than declared in advance; and ``D1``, ``D2``, and ``D3`` are logic variables. The Twelf declarations for these relations have essentially the same form, except that they are curried and associate an identifier with each clause (``next_day_sun``, ``next_day_mon``, etc., and ``dat_def``), making a Twelf signature a kind of "labeled" logic program. It may thus be said that, in the analogy between Prolog and Twelf, Prolog is concerned only with which instances of relations are "true", whereas Twelf is concerned with _how_ they are true—i.e., what constitutes evidence for them. Each successful Prolog search for a satisfying substitution to a goal query corresponds to a Twelf term of the type associated with the instantiated goal. Prolog searches can only succeed or fail, but Twelf terms are concrete certificates of successful searches that can be further manipulated. + +Another way of making this point is to say that every logic program has both a _declarative_ reading, in which we are asserting some facts about certain relations, and an _operational_ reading, which specifies how the clause "database" is searched when a query is given. Twelf signatures have similar readings: on the one hand, we can view a Twelf signature as specifying certain families of types and elements, representing data structures and relations with evidence; on the other, we can view a Twelf signature as a labeled logic program that specifies how to search for terms of a given type. + +The Twelf server, besides being able to check Twelf source files for type-correctness, also has a logic-programming engine that can be used in three different ways: +* It can be run stand-alone or as an inferior process as part of the Emacs interface and be used interactively, as a top-level loop, like most Prolog interpreters. In this mode, you type in queries—which are just types, possibly with free variables (upper-case identifiers or, as we might call them, "logic variables")—and the Twelf server _solves_ them, i.e., looks for elements of the given type. If it finds any elements, it will print out a satisfying substitution for the free variables and give you the option of looking for more solutions. +* As part of a Twelf source file, you can use a ``%query`` directive to search, in "batch mode", for a given number of solutions to a query, printing out satisfying substitutions if they exist. +* Also as part of a Twelf source file, you can use a ``%solve`` directive to search for an element of a given type and define an identifier to be the first such element found. You can also, with an accompanying ``%define`` directive, define identifiers to be the elements that get substituted for particular logic variables in the satisfying solution. +See the Twelf [User's Guide](/wiki/users-guide/) for more information on interacting with the Twelf server using the top-level loop. Here we will give some examples of using ``%query``, ``%solve``, and ``%define``. + +### The ``%query`` directive + +The format of a ``%query`` declaration is ``%query`` _solns_ _tries_ _type_, where _solns_ is the expected number of solutions, _tries_ is a limit to the number of solutions that will be listed (or "``*``" if there is no limit), and _type_ is the type, possibly containing logic variables, of which Twelf will search for elements. Of course, it only makes sense to speak of elements of real types, without free variables—we call them _ground_ types—but the point is that, searching for elements of the given type using the declarations in the signature will cause these logic variables to be instantiated to particular terms, and it is discovering these substitutions that are the main purpose of the ``%query`` declaration. Here are some examples: + +Check that `dat sunday tuesday` has exactly one solution +```checkedtwelf +%query 1 1 dat sunday tuesday. +``` + +Check that `dat monday X` has exactly one solution, and report the value of X +```checkedtwelf +%query 1 1 dat monday X. +``` + +Same for the inverse relation +```checkedtwelf +%query 1 1 dat X monday. +``` + +Check that `dat X Y` has exactly seven solutions +```checkedtwelf +%query 7 7 dat X Y. +``` + +Look for at least one solution to "dat X X", FAILS! +```checkedtwelf +%query 1 * dat X X. +``` + +### The ``%solve`` and ``%define`` directives + +The format of a ``%solve`` declaration is ``%solve`` _ident_ ``:`` _type_. This directive will search for a solution to the query (i.e., an element of the type) _type_, and define _ident_ to be this element. Any logic variables in _type_ will be instantiated, and you can seen them in the type when Twelf echoes the definition in its output, but they are not reported separately, as they are with the ``%query`` directive. If you want to capture these instantiations of free variables as definitions, you can prefix the ``%solve`` directive with one or more ``%define`` directives, the format of which is ``%define`` _ident_ ``=`` _var_ ``:`` _type_, where _ident_ is the identifier to be defined, _var_ is the free variable in the subsequent ``%solve`` declaration whose instantiation you want to capture, and _type_ is the type of that variable. Please note that the ``%define`` directives do not end in a period ("``.``") but are the beginning part of a single ``%solve`` directive. Here are some examples, the first of which repeats the definition of ``dat_sunday`` above, except that Twelf figures out the element (or "evidence") itself: !}% +%{!! begin checked !!}% +%solve dat_sunday : dat sunday tuesday %. +%{!! end checked !!}% +%{!! begin checked !!}% +%solve dat_monday : dat monday X %. +%{!! end checked !!}% +%{!! begin checked !!}% +%define d = X : day %solve dat_tuesday : dat tuesday X %. +%{!! end checked !!}% +%{!! begin checked !!}% +%define first_day = X : day %define second_day = Y : day %solve dat_first_second : dat X Y %. +%{!! end checked !!}% +%{! In the first ``%solve``, the identifier ``dat_sunday`` gets bound to ``dat_def next_day_sun next_day_mon``. In the second, ``X`` gets instantiated to ``wednesday`` (which we can see in the type ``dat monday wednesday``) and ``dat_monday`` gets bound to ``dat_def next_day_mon next_day_tue``. In the next query, ``d`` gets bound to ``thursday``, since this is what ``X`` gets instantiated to in the ``%solve``, and ``dat_tuesday`` gets bound to ``dat_def next_day_tue next_day_wed``. Finally, in the last query, ``X`` and ``Y`` get instantiated to ``saturday`` and ``monday`` respectively, and ``first_day`` and ``second_day`` get bound to these, while ``dat_first_second`` gets bound to ``dat_def next_day_sat next_day_sun``. + +### Operational Semantics + +Let's look more closely at what Twelf does to solve a query, for this will clarify the operational meaning of a Twelf signature (see Section 5.5 of the [User's Guide](/wiki/users-guide/) for more information). First, some terminology. A _goal_ is a type, possibly involving free variables, for which we are interested in finding an element. There are three kinds of goal: +* an _atomic_ goal, which is a fully-applied type family, i.e., a type constructor with a declaration like ``a : B1 -> ... -> Bm -> type`` applied to ``m`` terms, such as ``a M1 ... Mm``, resulting in a type; +* a _hypothetical_ goal, which is a type of the form ``A -> B``, with ``A`` and ``B`` types; and +* a _parametric_ goal, which is a type of the form ``{x:A} B``, where ``x`` occurs in ``B`` (if it does not, then ``{x:A} B`` is just ``A -> B``, and it is treated as a hypothetical goal). +The type families used in such goals are called _predicates_. A value declaration in a signature that is available during search is called a _clause_. A clause will always have the form + `c : An -> ... -> A1 -> a M1 ... Mm.` % or, equivalently, `c : a M1 ... Mm <- A1 <- ... <- An.` +where the atomic type ``a M1 ... Mm`` is the _head_ of the clause and the atomic, hypothetical, or parametric types ``A1`` through ``An`` are the _subgoals_. Note, as mentioned in the comment, that Twelf allows us to write the arrows in these types in either orientation, with ``->`` associating to the right and `` <-`` associating to the left. The latter is especially useful from a logic-programming perspective, because it puts the head of the clause at the front and the subgoals linearly afterward, as is the case in Prolog (read the first back-arrow as ``:-``, and read the rest of the back-arrows as commas). + +How does Twelf go about solving a goal? That is, given a type, possibly involving free variables, how does Twelf find a substitution instance for these variables and an element of the resulting type? Here is a simplified description of the algorithm. Throughout the search process, Twelf maintains several pieces of information: +* a _current substitution_ and _element trace_, which are, respectively, a partial instantiation of the free variables in the query and a partial term of the resulting type; +* a list of _subgoals_, whose joint solution entails a solution to the original query; and +* the _context_, a list of both _local assumptions_, which are goals that are temporarily assumed during the search, and _local parameters_, which are new constants that are introduced during the search. +Twelf also maintains a list of equational _constraints_ that arise during unification and a stack of backtrack points that it will use to find alternate solutions, but we will not go into detail about these here. Initially, the current substitution, element trace, and context are all empty, and the list of subgoals consists of a single element, namely the query itself. The first element of the list of subgoals is the _current subgoal_. The search process consists of repeating the following steps: +1. if the list of subgoals is emtpy, then the current substitution and element trace are complete and provide a solution to the query; +2. otherwise, consider the current subgoal: + + - if it is a hypothetical subgoal, of the form ``A -> B``, then add ``A`` to the context as a local assumption and take ``B`` as the new current subgoal; + - if it is a parametric subgoal, of the form `{x:A} B`, then choose a new constant, say ``c``, add ``c:A`` to the context as a local parameter, and take as the new current subgoal the result of replacing every instance of ``x`` in ``B`` with ``c``; + - if it is an atomic subgoal, of the form ``a M1 ... Mm``, then perform _backchaining_: look for a clause with a head that unifies with the subgoal; the resulting substitution is then "composed" with the current substitution, the element trace is updated using the constructor for the clause, and the current subgoal is replaced by the list of subgoals of the clause, in order from "inside out" (for example, in the clause for ``c`` displayed above, the order of subgoals is ``A1``, ..., ``An``, i.e., the order they appear when the clause is written with back-arrows, Prolog-style). Clauses in the local assumptions are tried first, starting with the most recent assumption and proceeding to the earlier assumptions, and then the clauses from the signature are tried, in the order they appear in the input file. As a result, + + 1. If a unifying clause is found, it is pushed onto the stack of backtrack points, so that if backtracking later reaches this point, a later unifying clause can be sought. + 2. If no more clauses can be found to match the current subgoal, then backtrack: pop the stack of backtrack points and resume the previous search for a clause from that point + 3. If the stack of backtracking points is empty, then fail. + +Note, again, that this method is very similar to the method Prolog uses to solve a query. The main difference is that each clause that gets used in a search results in its constructor becoming part of the partial term that is being constructed, so that when the search succeeds, we not only have an instantiation of the free variables in the query but also a term of the given type. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Signatures_as_logic_programs). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/simplifying-dynamic-clauses.lf b/new-tests/stelf-output/pages/simplifying-dynamic-clauses.lf new file mode 100644 index 0000000..8563d87 --- /dev/null +++ b/new-tests/stelf-output/pages/simplifying-dynamic-clauses.lf @@ -0,0 +1,91 @@ +% +% +%{! ### Summary + +When writing proofs about relations that introduce hypotheses, it is sometimes necessary to introduce dynamic clauses for the proof along with the hypothetical variable, to encode informal reasoning "at the variable case". In some cases, this reasoning is somewhat complicated, and can be streamlined by a judicious choice of lemma, so that the dynamic clause, while still present, is of a much simpler form, and the complicated reasoning is pushed to the top-level, where it can be more easily encoded. + +### Example + +Suppose that we have lists of items, some of which may be colored red. !}% +%sort elt %. +%term cherry elt %. +%term strawberry elt %. +%term blueberry elt %. +%sort list %. +%term nil list %. +%term cons %pi elt %-> list %-> list %. +%sort is-red %. +%term is-red/cherry is-red cherry %. +%term is-red/strawberry is-red strawberry %. +%{! A property that a list may satisfy is that all of its elements are red, which is encoded as follows: !}% +%sort all-red %. +%term all-red/nil all-red nil %. +%term all-red/cons %pi (all-red (cons H L)) %<- (all-red L) %<- (is-red H) %. +%{! We now define a nondeterministic function that take in a list, and outputs a list derived from the input by some combination of permutation, duplication, and dropping of its elements. !}% +%sort mix %. +%sort seed %. +%term mix/sow %pi (%pi (seed E) %-> (mix L L')) %-> (mix (cons E L) L') %. +%term mix/reap %pi (mix nil L) %-> (seed E) %-> (mix nil (cons E L)) %. +%term mix/nil mix nil nil %. +%{! The behavior of ``mix`` is as follows: it decomposes the input list, turning every element ``E`` in it into a hypothesis of type ``seed E``. Once the input list is empty, it builds up a new list by consing on some number of elements to the empty list, but each element must have come from some ``seed E``. Since there is no restriction on how many times a hypothesis is used, the resulting list may have many (or one, or no) copies of each element in the original list, and in any order. However, since the function ``mix`` cannot introduce elements <I>ex nihilo</I>, we can state and prove some properties of it. For instance, when applied to an all-red list, it must yield an all-red list. !}% +%sort mix-pres %. +%mode mix-pres %in %in %out %. +%{! #### First proof + +The proof of this theorem can be completed in the following way: + +```checkedtwelf +- : mix-pres + (mix/sow ([s] MIX s)) + (all-red/cons RH AR) + AR' + <- ({sd:seed E} + {dynclause: {L: list} {Mix : mix nil L} {Ared:all-red nil} {Ared':all-red L} + mix-pres (mix/reap Mix sd) Ared (all-red/cons RH Ared') + <- mix-pres Mix Ared Ared'} + mix-pres (MIX sd) AR AR'). + +- : mix-pres + mix/nil + all-red/nil + all-red/nil. + +%block complicated_block : some {E:elt} {RH:is-red E} block {sd:seed E} + {dynclause: {L: list} {Mix : mix nil L} {Ared:all-red nil} {Ared':all-red L} + mix-pres (mix/reap Mix sd) Ared (all-red/cons RH Ared') + <- mix-pres Mix Ared Ared'}. +%worlds (complicated_block) (mix-pres _ _ _). +%total X (mix-pres X _ _). +``` + +In this proof of the theorem, the block of variables introduced includes a _dynamic clause_; for the relation ``mix-pres`` that covers the variable case. It expresses the reasoning, "if the last rule used to derive ``mix`` was ``mix-pres`` applied to the seed ``sd`` just introduced, then appeal to the induction hypothesis on the smaller derivation of ``mix``". Without this dynamic clause, coverage would fail, and the theorem would not go through. !}% +%{! Writing dynamic clauses like this can be annoying, because Twelf's type reconstruction cannot be used to elide implicit s. If we left a variable such as ``Mix`` implicitly quantified inside the dynamic clause, it would be quantified at the very outside, which would be incorrect. + +#### Second proof + +We can instead prove the same theorem in a different way, which has the advantage of working in older verisons of Twelf, and allowing more leverage of type reconstruction in any event, simplifying the presentation. +The technique is to introduce another lemma that makes explicit the invariant represented by the worlds declaration for the main theorem. In this case, we know that for every ``seed E`` in the context, there is a derivation ``RH`` of type ``is-red E``. !}% +%sort has-rh %. +%mode has-rh %in %out %. +%{! We can prove the `has-rh` theorem in a `%world` that is much simpler than the complicated world used before: !}% +%block simpler_block {E elt} {RH is-red E} [sd seed E] [clause has-rh sd RH]%. +%worlds (simpler_block) (has-rh _ _) %. +%total X (has-rh X _) %. +%{! The proof of the main theorem can be stated more simply in this less complicated world: !}% +%term _ + %pi (mix-pres (mix/reap Mix SEED) Ared (all-red/cons RH Ared')) + %<- (has-rh SEED RH) + %<- (mix-pres Mix Ared Ared') %. +%term _ + %pi (mix-pres (mix/sow ([s] MIX s)) (all-red/cons RH AR) AR') + %<- ({sd seed E} {dynclause has-rh sd RH} mix-pres (MIX sd) AR AR') %. +%term _ mix-pres mix/nil all-red/nil all-red/nil %. +%worlds (simpler_block) (mix-pres _ _ _) %. +%total X (mix-pres X _ _) %. +%{! Note how there is now an "explicit variable case", which covers any possible derivation ``SEED`` of ``seed E``, and appeals to the above lemma ``has-rh`` in order to get the derivation of ``is-red H``. The second case, which introduces a parametric ``sd : seed E``, needs only supply the comparatively simple dynamic clause ``{dynclause: has-rh sd RH}`` to ensure the proof of ``has-rh`` goes through. + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +You can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Simplifying_dynamic_clauses). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/simply-typed-lambda-calculus.lf b/new-tests/stelf-output/pages/simply-typed-lambda-calculus.lf new file mode 100644 index 0000000..53ca022 --- /dev/null +++ b/new-tests/stelf-output/pages/simply-typed-lambda-calculus.lf @@ -0,0 +1,144 @@ +% +% +%{! The **simply-typed lambda calculus** (or STLC) is a common example of a simple typed programming language. This article discusses its encoding in Twelf. + +If you're trying to learn Twelf from this example, you may wish to read the discussion starting in ["Representing the syntax of the STLC" from the Introduction to Proving Metatheorems in Twelf](/wiki/proving-metatheorems-representing-the-syntax-of-the-stlc/). That introductory guide discusses this representation of the STLC and why it works in more detail. This page summarizes the judgements of the STLC and the corresponding LF code for reference, but does not explain them in detail. + +## What is illustrated by this example? + +There are simpler examples of LF representations (see, e.g., the [natural numbers](/wiki/natural-numbers/)). However, the STLC is a good first example of a representation that uses [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/) and [higher-order judgments](/wiki/higher-order-judgements/). These two representation techniques drastically simplify the process of representing and proving theorems about many programming languages and logics. The idea is that the binding structure of LF is used to represent the binding structure of the object language. At the level of syntax, this gives [alpha-equivalence](/wiki/alpha-equivalence/) and capture-avoiding substitution "for free" from the representation. At the level of judgements, this gives the properties of a [hypothetical judgement](/wiki/hypothetical-judgment/), such as weakening and substitution, for free. + +This encoding of the STLC is adapted from [_Mechanizing Metatheory in a Logical Framework_ by Harper and Licata](https://www.cambridge.org/core/journals/journal-of-functional-programming/article/mechanizing-metatheory-in-a-logical-framework/E4EC2F77CEF3DC56EABCF6F6658B179E). The Twelf in this article closely tracks Figure 7 and Figure 13 from that paper, and Section 3 of that paper exhaustively discusses the [adequacy](/wiki/adequacy/) of this encoding. + +## Encoding of syntax + +The types of the simply typed lambda calculus are simply the unit type and the arrow or function type. + +```math +\tau ::= \mathsf{unit} \mid \tau_1 \rightarrow \tau_2 +``` +!}% +%sort tp %. +%term arrow %pi tp %-> tp %-> tp %. +%term unit tp %. +%worlds () (tp) %. +%{! The terms are the variable , the empty pair (which has type ), lambda abstraction (with a type annotation), and application. + +```math +e ::= x \mid \langle\rangle \mid \lambda x{:}\tau.e \mid e_1\,e_2 +``` + !}% +%sort tm %. +%term empty tm %. +%term app %pi tm %-> tm %-> tm %. +%term lam %pi tp %-> (%pi tm %-> tm) %-> tm %. +%block tmvar [x tm]%. +%worlds (tmvar) (tm) %. +%{! ## Encoding of judgments + +### Static semantics + +The typing rules for the simply typed lambda calculus use a typing context to record the type annotations that have been encountered at lambda-bindings. + +```math +{ + \over + \gamma \vdash \langle\rangle : \mathsf{unit} +}\mathit{of\textit{-}empty} +\qquad +{ + x : \tau \in \gamma + \over + \gamma \vdash x : \tau +}\mathit{of\textit{-}var} +``` +```math +{ + \gamma, x :\tau_2 \vdash e : \tau + \over + \gamma \vdash \lambda x{:}\tau_2.e : (\tau_2 \rightarrow \tau) +}\mathit{of\textit{-}lam} +\qquad +{ + \gamma \vdash e_1 : (\tau_2 \rightarrow \tau) + \qquad + \gamma \vdash e_2 : \tau_2 + \over + \gamma \vdash e_1\,e_2 : \tau +}\mathit{of\textit{-}app} +``` + +This judgement is represented by the following LF signature: !}% +%sort of %. +%term of-empty of empty unit %. +%term of-lam + %pi (of (lam T2 ([x] E x)) (arrow T2 T)) + %<- ({x tm} %pi (of x T2) %-> (of (E x) T)) %. +%term of-app %pi (of (app E1 E2) T) %<- (of E1 (arrow T2 T)) %<- (of E2 T2) %. +%block tmof {T tp} [x tm] [d of x T]%. +%worlds (tmof) (of _ _) %. +%{! Assumptions in the typing context are represented as a pair of objects `x:tm` and `d: of x T` in the LF context (where `T` is the LF representation of some type ). This can be seen in the encoding of the judgment ``of-lam``. There is no need to write out the equivalent of the `of-var` rule, because the necessary judgment is directly assumed by the ``of-lam`` rule. This method avoids the need for a [substitution lemma](/wiki/substitution-lemma/): the article on that subject discusses the matter further. + +### Dynamic semantics + +We define the dynamic semantics of the STLC by a call-by-value, left-to-right structural operational semantics on closed terms. + +The judgement identifies the values: + +```math +{ + \over + \langle\rangle\,\mathtt{value} +}\mathit{value\textit{-}empty} +\qquad +{ + \over + \gamma x{:}\tau.e\,\mathtt{value} +}\mathit{value\textit{-}lam} +``` + +Next, we define the operational semantics with a judgement : + +```math +{ + e_1 \mapsto e_1' + \over + e_1\,e_2 \mapsto e_1'\,e_2 +}\mathit{step\textit{-}app_1} +\qquad +{ + e_1\,\mathtt{value} + \qquad + e_2 \mapsto e_2' + \over + e_1\,e_2 \mapsto e_1\,e_2' +}\mathit{step\textit{-}app_2} +``` +```math +{ + e_2\,\mathtt{value} + \over + (\lambda x{:}\tau.e)\,e_2 \mapsto e[e_2/x] +}\mathit{step\textit{-}app\textit{-}beta} +``` + +These judgments are represented by the following Twelf signature: !}% +%sort value %. +%term value-empty value empty %. +%term value-lam value (lam T ([x] E x)) %. +%sort step %. +%term step-app-1 %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step-app-2 %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step-app-beta %pi (step (app (lam T2 ([x] E x)) E2) (E E2)) %<- (value E2) %. +%worlds () (value _) (step _ _) %. +%{! +## Metatheorems + +Several metatheorems about this formulation of the simply typed lambda-calculus are proved on this wiki. See, for example: +* [Proving metatheorems about the STLC](/wiki/proving-metatheorems-proving-metatheorems-about-the-stlc/) for a proof of type preservation +* [Output factoring](/wiki/output-factoring/) for a proof of progress +* [Uniqueness lemma](/wiki/uniqueness-lemma/)s for a proof of determinacy of the operational semantics +* [Proving totality assertions in non-empty contexts](/wiki/proving-metatheorems-proving-totality-assertions-in-non-empty-contexts/) for a height judgement +* [Proving metatheorems in non-empty contexts](/wiki/proving-metatheorems-proving-metatheorems-in-non-empty-contexts/) for a proof that the height of a term can only be increased by substitution. + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/strengthening.lf b/new-tests/stelf-output/pages/strengthening.lf new file mode 100644 index 0000000..e975890 --- /dev/null +++ b/new-tests/stelf-output/pages/strengthening.lf @@ -0,0 +1,205 @@ +% +% +%{! **Strengthening** lemmas prove that something which holds under assumptions also holds without them. + +When proving type families total using [%total](/wiki/percent-total/), strengthening lemmas are sometimes necessary because Twelf estimates dependencies conservatively using the [subordination](/wiki/subordination/) relation. Specifically, for Twelf's [output coverage](/wiki/coverage-checking/) check to succeed, the output variables of higher-order premises must mention all assumptions of subordinate types. If an assumption's type is subordinate to an output variable's type, but additional reasoning shows that the assumption cannot occur in the output variable in a particular case, you must prove a strengthening lemma expressing this reasoning. + +In this tutorial, we'll show a proof where the need for strengthening arises. We'll define a language with locations, stores which map locations to terms in the language, and a notion of well-formedness with respect to a store (no dangling references and no loops). Then we'll define evaluation for the language, and prove that well-formed terms evaluate to well-formed terms; we will need strengthening for this proof. + +## A little language with locations + +Locations and terms: !}% +%sort loc %. +%name loc %. +%term loc/z loc %. +%term loc/s %pi loc %-> loc %. +%sort tm %. +%name tm %. +%term tm/loc %pi loc %-> tm %. +%term tm/lam %pi (%pi tm %-> tm) %-> tm %. +%term tm/app %pi tm %-> tm %-> tm %. +%{! This is a silly little language; it has locations but no way to use them. + +(The [``%name``](/wiki/percent-name/) declarations give a prefix which Twelf uses to name unnamed variables for display; we prefix the names with ``_`` so that it's clear which variables Twelf named, and also because Twelf will rename your variables if they collide with the naming prefix, which can be very confusing.) + +Stores and store lookup: !}% +%sort st %. +%name st %. +%term st/nil st %. +%term st/cons %pi loc %-> tm %-> st %-> st %. +%sort st-lookup %. +%name st-lookup %. +%term st-lookup/1 st-lookup (st/cons L T _) L T %. +%term st-lookup/2 %pi (st-lookup (st/cons L T S) L' T') %<- (st-lookup S L' T') %. +%{! Stores are just lists of location/term pairs. The lookup relation might look a little strange, since both ``st-lookup/1`` and ``st-lookup/2`` apply if ``L = L'``, but for our purpose it is OK and this avoids having to define disequality of locations. + +Well-formedness of terms with respect to a store: !}% +%sort tm-wf %. +%name tm-wf %. +%sort tm-var %. +%term tm-wf/var %pi (tm-wf V S) %<- (tm-var V) %. +%term tm-wf/loc %pi (tm-wf (tm/loc L) S) %<- (st-lookup S L T) %<- (tm-wf T S) %. +%term tm-wf/lam %pi (tm-wf (tm/lam ([x tm] T x)) S) %<- ({v tm} {d tm-var v} tm-wf (T v) S) %. +%term tm-wf/app %pi (tm-wf (tm/app T1 T2) S) %<- (tm-wf T1 S) %<- (tm-wf T2 S) %. +%{! A term is well-formed with respect to a store if its locations are all defined in the store, and if the terms bound to its locations are all well-formed with respect to the store. + +How to define well-formedness of the open term that appears in the ``lam`` case? We assume some term, and mark it is a variable using ``tm-var``, then check the well-formedness of the lambda body applied to this new term. Then we give a case saying that any term marked as a variable is well-formed. + +An alternative would be to directly assume ``tm-wf v S``. This example was extracted from the [[AML proof]], where we need to prove that if we have a store ``S'`` which extends ``S``, and a term ``T`` is well-formed in ``S``, then ``T`` is also well-formed in ``S'``. That proof is difficult if we assume ``tm-wf v S``, because we know nothing about ``v`` in ``S'``. With the ``tm-var`` approach we can just apply the ``tm-wf/var`` rule to get ``tm-wf v S'``. + +However, as we will see, this approach forces us to prove a substitution lemma, which requires the strengthening lemma. + +### Evaluation semantics + +Now we give a definition for how terms evaluate. +Evaluation is just beta-reduction; we do nothing with locations. + +!}% +%sort eval %. +%name eval %. +%term eval/loc eval (tm/loc L) (tm/loc L) %. +%term eval/lam eval (tm/lam ([x tm] T x)) (tm/lam ([x tm] T x)) %. +%term eval/app + %pi (eval (tm/app T1 T2) T') + %<- (eval T1 (tm/lam ([x tm] T1' x))) + %<- (eval (T1' T2) T') %. +%{! + +### Substitution Lemma + +If a term is well-formed, then what it evaluates to should be, too. +We can try to state the preservation theorm, and start working on the application case. + +```twelf +eval-pres-wf : eval T T' -> tm-wf T S -> tm-wf T' S -> type. +%mode eval-pres-wf +X1 +X2 -X3. + +-/app : eval-pres-wf (eval/app Deval2 Deval1) (tm-wf/app Dwf2 Dwf1) Dwf'' + <- eval-pres-wf Deval1 Dwf1 (tm-wf/lam Dwf1') + <- % ??? +``` + +We have ``Dwf1' : {v:tm}{d:tm-var v} tm-wf (T1 v) S`` and ``Dwf2 : +tm-wf T2 S``; we need ``Dwf' : tm-wf (T1 T2) S``. Now, had we defined +``tm-wf/lam`` to assume ``d:tm-wf v S`` instead of ``d:tm-var v``, +then we could just write ``(Dwf1' T2 Dwf2)`` in place of ``Dwf'``. +But, as previously noted, that gets us into a different kind of +trouble. So we'll need to prove an explicit substitution lemma first. + + !}% +%sort tm-wf-subst {v tm} {d tm-var v} %. +%mode tm-wf-subst %in %in %out %. +%scope tm-wf-subst %term /app + %pi (tm-wf-subst ([v] [d] tm-wf/app (Dwf2 v d) (Dwf1 v d)) Dwf (tm-wf/app Dwf2' Dwf1')) + %<- (tm-wf-subst Dwf1 Dwf Dwf1') + %<- (tm-wf-subst Dwf2 Dwf Dwf2') %. +%term /lam + %pi (tm-wf-subst ([v] [d] tm-wf/lam (Dwf1 v d)) Dwf2 (tm-wf/lam Dwf1')) + %<- ({v' tm} {d' tm-var v'} tm-wf-subst ([v] [d] Dwf1 v d v' d') Dwf2 (Dwf1' v' d')) %. +%scope tm-wf-subst %term /var tm-wf-subst ([v] [d] tm-wf/var d) Dwf Dwf %. +%term /bind tm-wf-subst ([v] [d] Dwf) _ Dwf %. +%{! This lemma gives us exactly what we'll need in the application +%case of preservation. + +The ``lam`` case is complicated only by the fact that we need to go +under a binder; here ``v`` is the variable we're substituting for and +``v'`` is the bound variable. Since we call the lemma recursively +under assumptions, we need to prove it in worlds where those +assumptions are present. + +In the ``var`` case we have found the variable we're substituting for. +The ``bind`` case handles assumptions; there is no way to explicitly +refer to them, but they are closed, so we can catch them with this +case for closed terms. + +The ``loc`` case is why we're here: Twelf determines that ``Dlook`` +can depend on ``v`` and ``Dwf1`` on ``v`` and ``d``, even though they +can't. So we need two strengthening lemmas to get the terms we need. +The proof case will look like: + +```twelf +-/loc : tm-wf-subst ([v][d] tm-wf/loc (Dwf1 v d) (Dlook v)) Dwf2 (tm-wf/loc Dwf1' Dlook') + <- st-lookup-strengthen Dlook Dlook' Deq + <- tm-wf-strengthen Dwf1 Deq Dwf1'. +``` + +Why can't ``Dlook`` and ``Dwf1`` depend on ``v`` and ``d``? We are +about to give proofs, but here is the intuition: The store ``S`` has +no dependencies in the declaration of the lemma, and ``st-lookup`` +just walks over the store, so ``Dlook`` has no dependencies; neither +does the looked-up term. Now ``tm-wf`` just walks over the term +looking up locations in the store, so it also has no dependencies. + +### Preservation theorem + +!}% +%sort eval-pres-wf %. +%mode eval-pres-wf %in %in %out %. +%scope eval-pres-wf %term /loc eval-pres-wf eval/loc D D %. +%term /lam eval-pres-wf eval/lam D D %. +%scope eval-pres-wf %term /app + %pi (eval-pres-wf (eval/app Deval2 Deval1) (tm-wf/app Dwf2 Dwf1) Dwf'') + %<- (eval-pres-wf Deval1 Dwf1 (tm-wf/lam Dwf1')) + %<- (tm-wf-subst Dwf1' Dwf2 Dwf') + %<- (eval-pres-wf Deval2 Dwf' Dwf'') %. +%{! + +### Strengthening lemmas + +Here is the first: !}% +%sort tm-eq %. +%term tm-eq_ tm-eq T T %. +%sort st-lookup-strengthen {v tm} %. +%mode st-lookup-strengthen %in %out %out %. +%scope st-lookup-strengthen %term /1 st-lookup-strengthen ([v] st-lookup/1) st-lookup/1 ([v] tm-eq_) %. +%term /2 + %pi (st-lookup-strengthen ([v] st-lookup/2 (Dlook v)) (st-lookup/2 Dlook') Deq) + %<- (st-lookup-strengthen Dlook Dlook' Deq) %. +%block tm-bind [v tm] [d tm-var v]%. +%worlds (tm-bind) (st-lookup-strengthen _ _ _) %. +%total D (st-lookup-strengthen D _ _) %. +%{! It would be nice if we could just get rid of the ``v`` assumption on the ``st-lookup`` derivation, but we have in addition that the returned term ``T`` can depend on ``v``, so we must invent a new non-dependent ``T'`` and return an equality. Otherwise, this is a straightforward induction that demonstrates that we never use the assumption. + +The second strengthening lemma is a little different: !}% +%sort tm-wf-strengthen {v tm} {d tm-var v} %. +%mode tm-wf-strengthen %in %in %out %. +%scope tm-wf-strengthen %term /app + %pi (tm-wf-strengthen ([v] [d] tm-wf/app (Dwf2 v d) (Dwf1 v d)) ([v] tm-eq_) (tm-wf/app Dwf2' Dwf1')) + %<- (tm-wf-strengthen Dwf2 ([v] tm-eq_) Dwf2') + %<- (tm-wf-strengthen Dwf1 ([v] tm-eq_) Dwf1') %. +%term /lam + %pi (tm-wf-strengthen ([v] [d] tm-wf/lam (Dwf v d)) ([v tm] tm-eq_) (tm-wf/lam Dwf')) + %<- ({v' tm} {d' tm-var v'} tm-wf-strengthen ([v] [d] Dwf v d v' d') ([v] tm-eq_) (Dwf' v' d')) %. +%scope tm-wf-strengthen %term /loc + %pi (tm-wf-strengthen ([v] [d] tm-wf/loc (Dwf v d) (Dlook v)) ([v] tm-eq_) (tm-wf/loc Dwf' Dlook')) + %<- (st-lookup-strengthen Dlook Dlook' Deq) + %<- (tm-wf-strengthen Dwf Deq Dwf') %. +%term /bind tm-wf-strengthen ([v] [d] Dwf) _ Dwf %. +%worlds (tm-bind) (tm-wf-strengthen _ _ _) %. +%total D (tm-wf-strengthen D _ _) %. +%{! Here it is a priori possible that something of type ``tm-wf (T v) S`` could depend on ``v`` and ``d`` (e.g. it could be ``(tm-wf/var d)``), but in conjunction with the equality it cannot. In the ``loc`` case we're doing essentially the same thing that we do in the substitution lemma, and must appeal to ``st-lookup-strengthen`` and ``tm-wf-strengthen`` (but we're not inducting over the well-formed term here). The coverage checker is able to rule out the ``var`` case since ``T`` does not depend on ``v``. + +### Finishing the proof + +Now that we have the strengthening lemmas, we can implement the final clause of our substitution lemma: + +!}% +%scope tm-wf-strengthen %term /loc + %pi (tm-wf-subst ([v] [d] tm-wf/loc (Dwf1 v d) (Dlook v)) Dwf2 (tm-wf/loc Dwf1' Dlook')) + %<- (st-lookup-strengthen Dlook Dlook' Deq) + %<- (tm-wf-strengthen Dwf1 Deq Dwf1') %. +%{! + +and check that the substitution lemma and the preservation theorem are actually correct: + +!}% +%worlds (tm-bind) (tm-wf-subst _ _ _) %. +%total D (tm-wf-subst D _ _) %. +%worlds () (eval-pres-wf _ _ _) %. +%total D (eval-pres-wf D _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Strengthening). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/structural-metrics.lf b/new-tests/stelf-output/pages/structural-metrics.lf new file mode 100644 index 0000000..b8c4ff2 --- /dev/null +++ b/new-tests/stelf-output/pages/structural-metrics.lf @@ -0,0 +1,136 @@ +% +%{! Twelf is able to prove metatheorems using induction over a sub-term ordering on a particular input. However, there are some theorems you would like to prove where you want to induct on an object that is not necessarily a subterm of your induction position, but in some sense this object is "smaller". What you can do is define a metric describing the size of the input that you would like to induct on, and use a termination argument that uses the metric in the induction position. The first choice for a metric that comes to mind is the [natural numbers](/wiki/natural-numbers/); see the tutorial on [numeric termination metrics](/wiki/numeric-termination-metrics/) for an example. Using the natural numbers as a metric has the advantage that you can use nat-less with %reduces in order to do strong induction. + +However, the natural numbers are not the only thing you can use as a metric. Oftentimes it is more convenient to use a metric that captures the structure of your derivations, instead of defining a numerical "size" or "height" with the natural numbers. Structural termination metrics correspond more closely to the original derivation, and they save you from having to prove [%reduces](/wiki/percent-reduces/) lemmas for the auxiliary operations used to define a numerical metric. Structural metrics for most [type families](/wiki/type-family/) are tree-like structures (but the structural metric for type families that look like lists is the natural numbers). Typing derivations for lambda calculi are an obvious example of a type family whose structural metric resembles a tree. + +In the following example, we will define a type family for colors and for trees with colors at the leaves. We will then mutually recursively define two recoloring operations over the colored trees. In order to show the [effectiveness lemma](/wiki/effectiveness-lemma/) for these recoloring operations, we will need to use a structural metric describing the shape of the colored trees as part of our induction. This technique is particularly important when proving properties about an operation that may change a derivation, but preserves "shape". + +We will first define some colors and trees over colors. !}% +%{! (options removed from twelftag: export="trees") !}% +%sort color %. +%term color/red color %. +%term color/blue color %. +%term color/white color %. +%sort colortree %. +%term colortree/node %pi colortree %-> colortree %-> colortree %. +%term colortree/leaf %pi color %-> colortree %. +%{! We will also define two operations to recolor colors. We will use these operations to define two operations to recolor trees. Notice how they are defined mutually recursively. This mutual recursion in the definition is what will ultimately force us to require a metric. !}% +%{! (options removed from twelftag: import="trees") !}% +%sort recolor1 %. +%term recolor1/rb recolor1 color/red color/blue %. +%term recolor1/bw recolor1 color/blue color/white %. +%term recolor1/wr recolor1 color/white color/red %. +%sort recolor2 %. +%term recolor2/rr recolor2 color/red color/red %. +%term recolor2/br recolor2 color/blue color/red %. +%term recolor2/wb recolor2 color/white color/blue %. +%sort recolortree1 %. +%sort recolortree2 %. +%term recolortree1/node + %pi (recolortree2 TA TA') + %-> (recolortree1 TA' TA'') + %-> (recolortree1 (colortree/node TA TB) (colortree/node TA'' TB)) %. +%term recolortree1/leaf %pi (recolor1 C1 C2) %-> (recolortree1 (colortree/leaf C1) (colortree/leaf C2)) %. +%term recolortree2/node + %pi (recolortree1 TB TB') + %-> (recolortree2 TB' TB'') + %-> (recolortree2 (colortree/node TA TB) (colortree/node TA TB'')) %. +%term recolortree2/leaf %pi (recolor2 C1 C2) %-> (recolortree2 (colortree/leaf C1) (colortree/leaf C2)) %. +%{! At a high level, we can see that the recoloring operations to not change the height or "shape" of the trees, just the coloring. So this operation should be total. We can easily show an [effectiveness lemma](/wiki/effectiveness-lemma/) for each of the recoloring operations for colors. !}% +%{! (options removed from twelftag: import="trees") !}% +%sort can-recolor1 {C} %. +%mode can-recolor1 %in %out %. +%term _ can-recolor1 color/red recolor1/rb %. +%term _ can-recolor1 color/blue recolor1/bw %. +%term _ can-recolor1 color/white recolor1/wr %. +%worlds () (can-recolor1 _ _) %. +%total {} (can-recolor1 _ _) %. +%sort can-recolor2 {C} %. +%mode can-recolor2 %in %out %. +%term _ can-recolor2 color/red recolor2/rr %. +%term _ can-recolor2 color/blue recolor2/br %. +%term _ can-recolor2 color/white recolor2/wb %. +%worlds () (can-recolor2 _ _) %. +%total {} (can-recolor2 _ _) %. +%{! Can we prove an effectiveness lemma about the recoloring operations for trees? We will pre-emptively avoid an early pitfull and try to prove both lemmas simultaneously (which is necessary because one must call the other). !}% +%{! (options removed from twelftag: import="trees") !}% +%sort can-recolortree1 {T} {T'} %. +%mode can-recolortree1 %in %out %out %. +%sort can-recolortree2 {T} {T'} %. +%mode can-recolortree2 %in %out %out %. +%term _ + %pi (can-recolortree1 (colortree/node T1 T2) (colortree/node T1'' T2) (recolortree1/node D1 D2)) + %<- (can-recolortree2 T1 T1' D1) + %<- (can-recolortree1 T1' T1'' D2) %. +% we can't even get the above case to pass termination checking +%worlds () (can-recolortree1 _ _ _) (can-recolortree2 _ _ _) %. +% %total (T) (can-recolortree1 T _ _) (can-recolortree2 T _ _). +%{! What went wrong when in the above case? The first inductive call to ``can-recolortree2`` was valid. But in the second inductive call to ``can-recolortree2`` we had the _output_ of the first inductive call in the induction position of the second call. This is not valid, because the tree that was outputted was not a sub-term of the input tree. So we will need a metric. It is possible to use a natural number representing the height or size of the tree as a metric, but for this particular proof such a metric would be a bit cumbersome, because we would need to know things about arithmetic to make the induction go through. Instead, we will exploit the fact that the "shape" of the tree is unchanged by the operations and use a tree metric describing the shape of the tree. !}% +%{! (options removed from twelftag: import="trees") !}% +%sort tree %. +%term tree/node %pi tree %-> tree %-> tree %. +%term tree/leaf tree %. +%{! The type ``tree`` we've defined above describes trees with no interesting information at the nodes or leaves. They are only "shapes". We can relate ``colortrees`` to ``trees`` by shape. We can also prove an [effectiveness lemma](/wiki/effectiveness-lemma/) showing this operation is always possible. !}% +%{! (options removed from twelftag: import="trees") !}% +%sort colortree-shape %. +%term colortree-shape/node + %pi (colortree-shape T1 T1') + %-> (colortree-shape T2 T2') + %-> (colortree-shape (colortree/node T1 T2) (tree/node T1' T2')) %. +%term colortree-shape/leaf colortree-shape (colortree/leaf _) tree/leaf %. +%sort can-colortree-shape {C} {T} %. +%mode can-colortree-shape %in %out %out %. +%term _ + %pi (can-colortree-shape (colortree/node C1 C2) (tree/node T1 T2) (colortree-shape/node D1 D2)) + %<- (can-colortree-shape C1 T1 D1) + %<- (can-colortree-shape C2 T2 D2) %. +%term _ can-colortree-shape (colortree/leaf _) tree/leaf colortree-shape/leaf %. +%worlds () (can-colortree-shape _ _ _) %. +%total (C) (can-colortree-shape C _ _) %. +%{! We will now use ``colortree`` to strengthen our induction hypothesis for the effectiveness of recoloring trees. We will use them to state that not only is it always possible to recolor a tree, but the resulting tree is always the same shape. We will induct over the structure of the ``tree``. !}% +%{! (options removed from twelftag: import="trees") !}% +%sort can-recolortree1* {T tree} %. +%mode can-recolortree1* %in %in %out %out %. +%sort can-recolortree2* {T tree} %. +%mode can-recolortree2* %in %in %out %out %. +%term _ + %pi (can-recolortree1* (tree/node T1 T2) (colortree-shape/node DCS1 DCS2) (colortree-shape/node DCS1'' DCS2) (recolortree1/node DR1 DR2)) + %<- (can-recolortree2* T1 DCS1 DCS1' DR1) + %<- (can-recolortree1* T1 DCS1' DCS1'' DR2) %. +%term _ + %pi (can-recolortree1* tree/leaf colortree-shape/leaf colortree-shape/leaf (recolortree1/leaf DR)) + %<- (can-recolor1 _ DR) %. +%term _ + %pi (can-recolortree2* (tree/node T1 T2) (colortree-shape/node DCS1 DCS2) (colortree-shape/node DCS1 DCS2'') (recolortree2/node DR1 DR2)) + %<- (can-recolortree1* T2 DCS2 DCS2' DR1) + %<- (can-recolortree2* T2 DCS2' DCS2'' DR2) %. +%term _ + %pi (can-recolortree2* tree/leaf colortree-shape/leaf colortree-shape/leaf (recolortree2/leaf DR)) + %<- (can-recolor2 _ DR) %. +%worlds () (can-recolortree1* _ _ _ _) (can-recolortree2* _ _ _ _) %. +%total (T1 T2) (can-recolortree1* T1 _ _ _) (can-recolortree2* T2 _ _ _) %. +%{! Now we have proven effectiveness lemmas for the recoloring tree operations conditional on the fact that we can compute the size metric. Since we proved this is possible, we can write nicer versions of the lemmas that do not mention the metric. !}% +%{! (options removed from twelftag: import="trees" check="true") !}% +%sort can-recolortree1 {T} {T'} %. +%mode can-recolortree1 %in %out %out %. +%term _ + %pi (can-recolortree1 CT _ DR) + %<- (can-colortree-shape CT T DCS) + %<- (can-recolortree1* T DCS _ DR) %. +%worlds () (can-recolortree1 _ _ _) %. +%total {} (can-recolortree1 _ _ _) %. +%sort can-recolortree2 {T} {T'} %. +%mode can-recolortree2 %in %out %out %. +%term _ + %pi (can-recolortree2 CT _ DR) + %<- (can-colortree-shape CT T DCS) + %<- (can-recolortree2* T DCS _ DR) %. +%worlds () (can-recolortree2 _ _ _) %. +%total {} (can-recolortree2 _ _ _) %. +%{! \{\{tutorial\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Structural_metrics). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/subordination.lf b/new-tests/stelf-output/pages/subordination.lf new file mode 100644 index 0000000..0d50df5 --- /dev/null +++ b/new-tests/stelf-output/pages/subordination.lf @@ -0,0 +1,41 @@ +% +%{! A **subordination relation** determines when terms of one LF type may appear in terms of another. Subordination is used in Twelf to determine how the inhabitants of a type change when considered in different LF contexts. + +Twelf infers a subordination relation from a signature. For example, consider the following signature for the [simply-typed lambda calculus](/wiki/simply-typed-lambda-calculus/): !}% +%sort tp %. +%term unit tp %. +%term arrow %pi tp %-> tp %-> tp %. +%sort tm %. +%term empty tm %. +%term lam %pi tp %-> (%pi tm %-> tm) %-> tm %. +%term app %pi tm %-> tm %-> tm %. +%{! \{\{needs|syntax consistent with Print.subord\}\} + +Twelf infers the following subordination relation: +<pre> +tm <| tm +tp <| tp +tp <| tm +</pre> +This subordination relation says that ``tm`` is subordinate to ``tm``, and ``tp`` is subordinate to ``tp`` and ``tm``. That is, terms can appear in terms, types can appear in types, and types can appear in terms, but terms cannot appear in types. This makes sense for the language that we have given. + +However, if we add dependent types to this object language, Twelf extends the subordination relation with +<pre> +tm <| tp +</pre> +because now terms may appear in types. + +\{\{needs|a formal account of subordination\}\} + +Subordination has several uses, including: +* It enables modular [adequacy](/wiki/adequacy/) proofs. +* It enables a Twelf [metatheorem](/wiki/metatheorem/) that is proved for one set of LF contexts to be reused in other sets of contexts. This is called [world subsumption](/wiki/world-subsumption/). +The reason is that subordination gives rise to a general condition under which the terms of a type remain invariant when the type is considered in different LF contexts. Roughly, the terms of a type are invariant when the context is changed by adding or deleting declarations that are **not** subordination to that type. + +\{\{stub\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Subordination). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/substitution-lemma.lf b/new-tests/stelf-output/pages/substitution-lemma.lf new file mode 100644 index 0000000..3006644 --- /dev/null +++ b/new-tests/stelf-output/pages/substitution-lemma.lf @@ -0,0 +1,200 @@ +% +% +%{! Type-preserving **substitution** is the property of a hypothetical judgment that if and , then . In other words, if we have a well-typed term that depends on a hypothesis of type , and we have a well-typed term , then we can substitute for the variable within to produce a well-typed term that does not depend on . + +Depending on how the type system of an object language is encoded, there are a number of different techniques for proving a **substitution lemma** for the language. +## Substitution via application in LF + +When an object language is encoded using [higher-order representations of hypothetical judgements](/wiki/higher-order-judgements/), substitution comes for free: object-language variables and typing assumptions are represented by LF lambdas, so substitution follows from LF application. + +Consider the following encoding of a simply typed λ-calculus: !}% +%% Syntax +%sort tp %. +%term tp/unit tp %. +%term tp/arrow %pi tp %-> tp %-> tp %. +%sort exp %. +%term exp/unit exp %. +%term exp/lam %pi tp %-> (%pi exp %-> exp) %-> exp %. +%term exp/app %pi exp %-> exp %-> exp %. +%% Typing judgement +%sort of %. +%term of/unit of exp/unit tp/unit %. +%term of/lam + %pi (of (exp/lam T E) (tp/arrow T T')) + %<- ({x exp} %pi (of x T) %-> (of (E x) T')) %. +%term of/app %pi (of (exp/app E1 E2) T') %<- (of E2 T) %<- (of E1 (tp/arrow T T')) %. +%{! We can give a direct proof of substitution: !}% +%sort subst %. +%mode subst %in %in %out %. +%term _ subst D1 D2 (D2 E1 D1) %. +%block of-block {T tp} [x exp] [dx of x T]%. +%worlds (of-block) (subst _ _ _) %. +%total {} (subst _ _ _) %. +%{! The typing derivation for the substitution is created by applying the LF function ``D2``, which represents the hypothetical typing derivation, to the LF term ``D1``, which represents the typing derivation for the substituted term. + +Although we proved substitution as a metatheorem ``subst`` here, in practice it is unnecessary to state substitution as a lemma. The substitution can be performed via application whenever it is necessary. + +## Substitution lemmas with a "var" rule + +In the above language, there was no explicit typing rule in the system for typing variables. Instead, typing derivations for variables were put directly into the LF context. However, for some languages it is necessary to use separate judgments for assumptions about variables and for typing derivations. This is common for languages with references and stores which require a lemma that shows it is admissible to weaken typing derivations with respect to the store. + +In the following example, the ``assm`` judgment is used in conjunction with the ``oftp/var`` rule to give ``of`` typing derivations for variables. A derivation of ``{x:exp} assm x T -> of (E2 x) T'`` and a derivation of ``of E1 T`` _cannot_ be used to show ``of (E2 E1) T'`` via application in LF. This is because a hypothetical judgment expecting an ``assm`` derivation cannot be applied to an ``of`` derivation. That is, the LF representation does not directly tell us that substitution holds, as the assumption is of a different type. + +However, in this case, the desired substitution principle can be proved via an induction over the structure of the hypothetical judgment. The key to making the following proof work is the fact that [exchange](/wiki/exchange-lemma/) is admissible for this language via its encoding in LF. In general, exchange is admissible whenever assumptions about variables cannot depend on assumptions earlier in the context. + +Here is an encoding that uses a different judgement for assumptions: !}% +%% Syntax +%sort tp %. +%term tp/unit tp %. +%term tp/arrow %pi tp %-> tp %-> tp %. +%sort exp %. +%term exp/unit exp %. +%term exp/lam %pi tp %-> (%pi exp %-> exp) %-> exp %. +%term exp/app %pi exp %-> exp %-> exp %. +%% Typing judgements +%sort assm %. +%sort of %. +%term of/var %pi (of E T) %<- (assm E T) %. +%term of/unit of exp/unit tp/unit %. +%term of/lam + %pi (of (exp/lam T E) (tp/arrow T T')) + %<- ({x exp} %pi (assm x T) %-> (of (E x) T')) %. +%term of/app %pi (of (exp/app E1 E2) T') %<- (of E2 T) %<- (of E1 (tp/arrow T T')) %. +%{! Now, we prove the substitution theorem: !}% +%sort subst %. +%mode subst %in %in %out %. +% case for substituting out the assumption +%term _ subst D1 ([x] [dx] of/var dx) D1 %. +% catch-all case for of/unit or instances of of/var for a different assumption +%term _ subst D1 ([x] [dx] D2) D2 %. +%{! The following case is the key for the proof of this metatheorem. It works because the language admits the property of exchange. + +In the following case, D2 has the type + +``` +{x:exp}{dx:of x T}{y:exp}{dy:of y T'} of (E' x y) T'' +``` + +The principle of exchange can be applied to produce a derivation of + +``` +{y:exp}{dy:of y T'}{x:exp}{dx:of x T} of (E' x y) T'' +``` + +On the inductive call to subst, the assumptions + +``` +{y:exp}{dy:of y T'} +``` + +are pushed into the LF context, so that a derivation of type + +``` +{x:exp}{dx:of x T} of (E' x y) T'' +``` + +can be given as the second argument to the inductive call. !}% +%term _ + %pi (subst D1 ([x] [dx] of/lam ([y] [dy] D2 x dx y dy)) (of/lam D2')) + %<- ({y} {dy} subst D1 ([x] [dx] D2 x dx y dy) (D2' y dy)) %. +%term _ + %pi (subst D1 ([x] [dx] of/app (D2 x dx) (D3 x dx)) (of/app D2' D3')) + %<- (subst D1 D2 D2') + %<- (subst D1 D3 D3') %. +%block assm-block {T tp} [x exp] [dx assm x T]%. +%worlds (assm-block) (subst _ _ _) %. +%total (D1) (subst _ D1 _) %. +%{! ## Substitution lemmas with dependent types [Advanced topic] + +The preceding technique for showing substitution lemmas works fine for systems without dependent types. However, it would not work for a system with dependent types, because its typing assumptions could not admit exchange, in general. A recent discovery is that a similarly general technique is available for languages with dependent types and a var rule, despite the apparent absence of an exchange property. Of course, if the system has the same judgment for assumptions and typing derivations, then substitution can be performed via application in LF. + +The following example studies a very simple singleton calculus with the dependent Π type for functions. The presence of dependent types means typing assumptions about expressions can depend on expression variables introduced in earlier assumptions. + +Explicit contexts could be used to do substition in this case, but this technique is much simpler in practice. !}% +%sort tp %. +%sort exp %. +%term tp/sing %pi exp %-> tp %. +%term tp/unit tp %. +%term tp/pi %pi tp %-> (%pi exp %-> tp) %-> tp %. +%term exp/unit exp %. +%term exp/lam %pi tp %-> (%pi exp %-> exp) %-> exp %. +%term exp/app %pi exp %-> exp %-> exp %. +%sort assm %. +%sort of %. +%term of/var %pi (of E T) %<- (assm E T) %. +%term of/unit of exp/unit tp/unit %. +%term of/lam + %pi (of (exp/lam T E) (tp/pi T T')) + %<- ({x exp} %pi (assm x T) %-> (of (E x) (T' x))) %. +%term of/app %pi (of (exp/app E1 E2) (T' E2)) %<- (of E2 T) %<- (of E1 (tp/pi T T')) %. +%term of/sing %pi (of E (tp/sing E)) %<- (of E tp/unit) %. +%block assm-block {T tp} [x exp] [dx assm x T]%. +%{! The following metatheorem is actually true, but the naive attempt at proving it will fail. !}% +%sort subst-wontwork %. +%mode subst-wontwork %in %in %out %. +% case for substituting out the assumption +%term _ subst-wontwork D1 ([x] [dx] of/var dx) D1 %. +% catch-all case for of/unit or instances of of/var for a different assumption +%term _ subst-wontwork D1 ([x] [dx] D2) D2 %. +%term _ + %pi (subst-wontwork D1 ([x] [dx] of/app (D2 x dx) (D3 x dx)) (of/app D2' D3')) + %<- (subst-wontwork D1 D2 D2') + %<- (subst-wontwork D1 D3 D3') %. +%term _ + %pi (subst-wontwork D1 ([x] [dx] of/sing (D2 x dx)) (of/sing D2')) + %<- (subst-wontwork D1 D2 D2') %. +%{! The proof fails because this case is not general enough to cover all lambdas. Specifically, this case only applies if the type assigned to the variable y does not depend on the variable x. + +```checkedtwelf +- : subst-wontwork D1 ([x][dx] of/lam ([y][dy:assm y T'] D2 x dx y dy)) (of/lam D2') + <- ({y}{dy} subst-wontwork D1 ([x][dx] D2 x dx y dy) (D2' y dy)). + +%worlds (assm-block) (subst-wontwork _ _ _). +%total (D1) (subst-wontwork _ D1 _). +``` + +In the blocker case for the above proof attempt, the problem was that +new assumptions could depend on the x. However, E1 is going to be +substituted in for x anyway. If it is substituted in first, then these +dependencies disappear. What remains is to show that we can swap in a +derivation for of E1 T into a derivation of (assm E1 T -> of (E2 E1) (T' E1)). + +In the statement of the following metatheorem, (E2 E1) and (T' E1) +are generalized by the schematic variables E3 and T_, respectively. !}% +%sort subst* %. +%mode subst* %in %in %out %. +% case for substituting out the assumption +%term _ subst* D1 ([dx] of/var dx) D1 %. +% catch-all case for of/unit or instances of of/var for a different assumption +%term _ subst* D1 ([dx] D2) D2 %. +%term _ + %pi (subst* D1 ([dx] of/app (D2 dx) (D3 dx)) (of/app D2' D3')) + %<- (subst* D1 D2 D2') + %<- (subst* D1 D3 D3') %. +%term _ %pi (subst* D1 ([dx] of/sing (D2 dx)) (of/sing D2')) %<- (subst* D1 D2 D2') %. +%term _ + %pi (subst* D1 ([dx] of/lam ([y] [dy] D2 dx y dy)) (of/lam D2')) + %<- ({y} {dy} subst* D1 ([dx] D2 dx y dy) (D2' y dy)) %. +%worlds (assm-block) (subst* _ _ _) %. +%total (D1) (subst* _ D1 _) %. +%{! Having proved the preceding lemma, the general substitution principle can be proven by first substituting E1 in for x, and then applying subst* to eliminate the dependency on assm E1 T. !}% +%sort subst %. +%mode subst %in %in %out %. +%term _ + %pi (subst (%the (of E1 T) D1) (%the ({x} {dx assm x T} of (E2 x) (T' x)) D2) D2') + %<- (subst* D1 (D2 E1) D2') %. +%worlds (assm-block) (subst _ _ _) %. +%total {} (subst _ _ _) %. +%{! ### Remarks about adequacy + +The reader may have concerns as to whether this proof technique affects [adequacy](/wiki/adequacy/) for the object language. Although the second input to the ``subst*`` is not an adequate encoding of a typing derivation that makes a proper extension to the LF context (because it is missing an ``\{x:exp\}`` argument), the result is a valid typing derivation in the object logic. ``subst*`` is a correct lemma about LF terms of a particular form. Doing a proof of ``subst`` using ``subst*`` does not "break" the object language in any way, because 1) the deviations from adequate encodings occured only in intermediate stages of the proof and 2) the proof produces results that are adequate encodings of the object language. + +### Comparison to the POPLMark 1a narrowing proof + +This technique is similar to how the narrowing lemma was proven for POPLMark challenge 1a. However, because we are proving a substitution lemma instead of narrowing the type of a variable, we have an actual term to substitute in for the variable. Because the variable can be substituted away and does not have to be put into the LF context, this proof does not need to deal with any special worlds. + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +You can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Substitution_lemma). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/sudoku.lf b/new-tests/stelf-output/pages/sudoku.lf new file mode 100644 index 0000000..fc734bc --- /dev/null +++ b/new-tests/stelf-output/pages/sudoku.lf @@ -0,0 +1,106 @@ +% +%{! [[Image:sudokuexample.svg|thumb|right|300px|Example Sudoku puzzle]] +In the **Sudoku** puzzle, a 9x9 board must be filled with the numbers such that no row, column, or box contains the same digit twice. + +\{\{needs|to be completed\}\} !}% +% Sudoku solver, brute force +%sort cell %. +%term 1 cell %. +%term 2 cell %. +%term 3 cell %. +%term 4 cell %. +%term 5 cell %. +%term 6 cell %. +%term 7 cell %. +%term 8 cell %. +%term 9 cell %. +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%prec %prefix 5 s %. +%define nine s s s s s s s s s z %. +%sort cells %. +%term // %pi cell %-> (cells N) %-> (cells (s N)) %. +%prec %right 5 // %. +%term nil cells z %. +%sort cellss %. +%term /// %pi (cells nine) %-> (cellss N) %-> (cellss (s N)) %. +%prec %right 4 /// %. +%term nill cellss z %. +%sort remove %. +%mode remove %in %in %out %. +%term rem-hit remove (X // C) X C %. +% okay that we can also skip this one +%term rem-miss %pi (remove (X // C) Y (X // C')) %<- (remove C Y C') %. +%worlds () (remove _ _ _) %. +%terminates D (remove D _ _) %. +%sort nodups %. +%mode nodups %in %in %. +%term nd-nil nodups _ nil %. +%term nd-hit %pi (nodups OK (X // C)) %<- (remove OK X OK') %<- (nodups OK' C) %. +%worlds () (nodups _ _) %. +%terminates D (nodups _ D) %. +% valid if there are no duplicates +%define allcells 1 // 2 // 3 // 4 // 5 // 6 // 7 // 8 // 9 // nil %. +%sort unitok %. +%term uo %pi (unitok C) %<- (nodups allcells C) %. +% the nine rows +%inline board cellss nine %. +% extract three boxes from three rows +%sort boxes %. +%mode boxes %in %in %in %out %out %out %. +%term b boxes (A1 // B1 // C1 // D1 // E1 // F1 // G1 // H1 // I1 // nil) (A2 // B2 // C2 // D2 // E2 // F2 // G2 // H2 // I2 // nil) (A3 // B3 // C3 // D3 // E3 // F3 // G3 // H3 // I3 // nil) (A1 // B1 // C1 // A2 // B2 // C2 // A3 // B3 // C3 // nil) (D1 // E1 // F1 // D2 // E2 // F2 // D3 // E3 // F3 // nil) (G1 // H1 // I1 // G2 // H2 // I2 // G3 // H3 // I3 // nil) %. +%worlds () (boxes _ _ _ _ _ _) %. +%total {} (boxes _ _ _ _ _ _) %. +% transpose a whole board +%sort transpose %. +%mode transpose %in %out %. +%term tr transpose (A1 // B1 // C1 // D1 // E1 // F1 // G1 // H1 // I1 // nil /// (A2 // B2 // C2 // D2 // E2 // F2 // G2 // H2 // I2 // nil) /// (A3 // B3 // C3 // D3 // E3 // F3 // G3 // H3 // I3 // nil) /// (A4 // B4 // C4 // D4 // E4 // F4 // G4 // H4 // I4 // nil) /// (A5 // B5 // C5 // D5 // E5 // F5 // G5 // H5 // I5 // nil) /// (A6 // B6 // C6 // D6 // E6 // F6 // G6 // H6 // I6 // nil) /// (A7 // B7 // C7 // D7 // E7 // F7 // G7 // H7 // I7 // nil) /// (A8 // B8 // C8 // D8 // E8 // F8 // G8 // H8 // I8 // nil) /// (A9 // B9 // C9 // D9 // E9 // F9 // G9 // H9 // I9 // nil) /// nill) (A1 // A2 // A3 // A4 // A5 // A6 // A7 // A8 // A9 // nil /// (B1 // B2 // B3 // B4 // B5 // B6 // B7 // B8 // B9 // nil) /// (C1 // C2 // C3 // C4 // C5 // C6 // C7 // C8 // C9 // nil) /// (D1 // D2 // D3 // D4 // D5 // D6 // D7 // D8 // D9 // nil) /// (E1 // E2 // E3 // E4 // E5 // E6 // E7 // E8 // E9 // nil) /// (F1 // F2 // F3 // F4 // F5 // F6 // F7 // F8 // F9 // nil) /// (G1 // G2 // G3 // G4 // G5 // G6 // G7 // G8 // G9 // nil) /// (H1 // H2 // H3 // H4 // H5 // H6 // H7 // H8 // H9 // nil) /// (I1 // I2 // I3 // I4 // I5 // I6 // I7 // I8 // I9 // nil) /// nill) %. +%worlds () (transpose _ _) %. +%total {} (transpose C _) %. +%sort boardok %. +%term bo + %pi (boardok (A /// B /// C /// D /// E /// F /// G /// H /// I /// nill)) + %<- (unitok A) + %<- (unitok B) + %<- (unitok C) + %<- (unitok D) + %<- (unitok E) + %<- (unitok F) + %<- (unitok G) + %<- (unitok H) + %<- (unitok I) + %<- (boxes A B C Ab Bb Cb) + %<- (unitok Ab) + %<- (unitok Bb) + %<- (unitok Cb) + %<- (boxes D E F Db Eb Fb) + %<- (unitok Da) + %<- (unitok Ea) + %<- (unitok Fa) + %<- (boxes G H I Gb Hb Ib) + %<- (unitok Gb) + %<- (unitok Hb) + %<- (unitok Ib) + %<- (transpose (A /// B /// C /// D /// E /// F /// G /// H /// I /// nill) (At /// Bt /// Ct /// Dt /// Et /// Ft /// Gt /// Ht /// It /// nill)) + %<- (unitok At) + %<- (unitok Bt) + %<- (unitok Ct) + %<- (unitok Dt) + %<- (unitok Et) + %<- (unitok Ft) + %<- (unitok Gt) + %<- (unitok Ht) + %<- (unitok It) %. +%solve _ : boardok (8 // 2 // 6 // 7 // 1 // 4 // 9 // 5 // 3 // nil /// (3 // 4 // 5 // 8 // 6 // 9 // 2 // 1 // 7 // nil) /// (9 // 7 // 1 // 2 // 5 // 3 // 4 // 8 // 6 // nil) /// (1 // 6 // 3 // 9 // 2 // 5 // 7 // 4 // 8 // nil) /// (5 // 9 // 4 // 6 // 7 // 8 // 1 // 3 // 2 // nil) /// (2 // 8 // 7 // 3 // 4 // 1 // 5 // 6 // 9 // nil) /// (7 // 5 // 9 // 4 // 3 // 6 // 8 // 2 // 1 // nil) /// (4 // 3 // 2 // 1 // 8 // 7 // 6 // 9 // 5 // nil) /// (6 // 1 // 8 // 5 // 9 // 2 // 3 // 7 // 4 // nil) /// nill) %. +%solve _ : boardok (8 // 2 // 6 // 7 // 1 // 4 // 9 // 5 // 3 // nil /// (3 // 4 // 5 // 8 // 6 // 9 // 2 // 1 // 7 // nil) /// (9 // 7 // 1 // 2 // 5 // 3 // 4 // 8 // 6 // nil) /// (1 // 6 // 3 // _ // 2 // 5 // 7 // 4 // 8 // nil) /// (5 // 9 // 4 // _ // 7 // 8 // 1 // 3 // 2 // nil) /// (2 // 8 // 7 // _ // _ // 1 // 5 // 6 // 9 // nil) /// (7 // 5 // 9 // 4 // 3 // 6 // 8 // 2 // 1 // nil) /// (4 // 3 // 2 // 1 // 8 // 7 // 6 // 9 // 5 // nil) /// (6 // 1 // 8 // 5 // 9 // 2 // 3 // 7 // 4 // nil) /// nill) %. +%{! ``>All code from this tutorial``. `` check="true">Twelf's output from this tutorial``. + + +\{\{case study\}\}[Category:Twelf code](/wiki/category-twelf-code/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Sudoku). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008-alternate-typed-arithmetic-expressions-with-sums.lf b/new-tests/stelf-output/pages/summer-school-2008-alternate-typed-arithmetic-expressions-with-sums.lf new file mode 100644 index 0000000..17e00a4 --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008-alternate-typed-arithmetic-expressions-with-sums.lf @@ -0,0 +1,129 @@ +% +%{! + +\{\{Summer school 2008 + |prev=Typed arithmetic expressions + |prevname=Typed arithmetic expressions + |nonext= +\}\} + +Arithmetic expressions with pairs and sums. (There are two alternate solutions, [one by William](/wiki/summer-school-2008-typed-arithmetic-expressions-with-sums-2/) (involving output factoring) and [one by Dan](/wiki/summer-school-2008-typed-arithmetic-expressions-with-sums/) (involving output factoring and identity types).) + +Numbers and strings are unchanged. + +|hidden="true" !}% +%sort nat %. +%name nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort add %. +%mode add %in %in %out %. +%term add/z add z N N %. +%term add/s %pi (add (s M) N (s P)) %<- (add M N P) %. +%worlds () (add _ _ _) %. +%total M (add M _ _) %. +%sort char %. +%name char %. +%term a char %. +%term b char %. +%sort str %. +%name str %. +%term emp str %. +%term cons %pi char %-> str %-> str %. +%sort cat %. +%mode cat %in %in %out %. +%term cat/e cat emp S S %. +%term cat/c %pi (cat (cons X S1) S2 (cons X S3)) %<- (cat S1 S2 S3) %. +%worlds () (cat _ _ _) %. +%total S (cat S _ _) %. +%{! + +## Typed expressions + +Add a type for disjoint sums: + +!}% +%sort tp %. +%name tp %. +%term number tp %. +%term string tp %. +%term prod %pi tp %-> tp %-> tp %. +%term sum %pi tp %-> tp %-> tp %. +%{! + +Add injections as values, and case as an expression. Note that you can only case on a value. This makes evaluation slightly easier; +for versions with expression-case as a primitive; see the above alternate solutions. + +!}% +%sort val %. +%name val %. +%prec %postfix 1 val %. +%term num %pi nat %-> (number val) %. +%term lit %pi str %-> (string val) %. +%term pair %pi (T val) %-> (U val) %-> (prod T U val) %. +%term inl %pi (T val) %-> (sum T U val) %. +%term inr %pi (U val) %-> (sum T U val) %. +%sort exp %. +%name exp %. +%prec %postfix 1 exp %. +%term ret %pi (T val) %-> (T exp) %. +%term plus %pi (number exp) %-> (number exp) %-> (number exp) %. +%term append %pi (string exp) %-> (string exp) %-> (string exp) %. +%term let %pi (T exp) %-> (%pi (T val) %-> (U exp)) %-> (U exp) %. +%term fst %pi (prod T U exp) %-> (T exp) %. +%term snd %pi (prod T U exp) %-> (U exp) %. +%term case + %pi (sum T U val) + %-> (%pi (T val) %-> (V exp)) + %-> (%pi (U val) %-> (V exp)) + %-> (V exp) %. +%{! + +Note that if we want to case on an expression, we can let-bind it: + +!}% +%define ecase ( + %pi (sum T U exp) + %-> (%pi (T val) %-> (V exp)) + %-> (%pi (U val) %-> (V exp)) + %-> (V exp)) [esum] [e1] [e2] let esum ([x] case x e1 e2) %. +%{! + +Add eval cases for case. + +!}% +%sort eval %. +%mode eval %in %out %. +%term eval/val eval (ret V) V %. +%term eval/plus + %pi (eval (plus E1 E2) (num N)) + %<- (eval E1 (num N1)) + %<- (eval E2 (num N2)) + %<- (add N1 N2 N) %. +%term eval/append + %pi (eval (append E1 E2) (lit S)) + %<- (eval E1 (lit S1)) + %<- (eval E2 (lit S2)) + %<- (cat S1 S2 S) %. +%term eval/let %pi (eval (let E1 ([x] E2 x)) A) %<- (eval E1 V) %<- (eval (E2 V) A) %. +%term eval/fst %pi (eval (fst E) A) %<- (eval E (pair A _)) %. +%term eval/snd %pi (eval (snd E) B) %<- (eval E (pair _ B)) %. +%term eval/case1 %pi (eval (case (inl A) ([x] E1 x) _) A1) %<- (eval (E1 A) A1) %. +%term eval/case2 %pi (eval (case (inr A) _ ([x] E2 x)) A2) %<- (eval (E2 A) A2) %. +%worlds () (eval _ _) %. +%total E (eval E _) %. +%{! + +\{\{Summer school 2008 + |prev=Exercises 1 + |prevname=Exercises 1 + |nonext= +\}\} + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008:Alternate_typed_arithmetic_expressions_with_sums). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008-arithmetic-expressions-with-call-by-value-let-binding.lf b/new-tests/stelf-output/pages/summer-school-2008-arithmetic-expressions-with-call-by-value-let-binding.lf new file mode 100644 index 0000000..e085689 --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008-arithmetic-expressions-with-call-by-value-let-binding.lf @@ -0,0 +1,101 @@ +% +%{! + +\{\{Summer school 2008 + |prev=Arithmetic expressions with let-binding + |prevname=Arithmetic expressions with let-binding + |next=Arithmetic expressions with let-binding (hypothetical evaluation) + |nextname=Defining evaluation with a hypothetical judgement\}\} + +We define a simple expression language with let-bindings, making explicit the +invariant that variables stand for values into the syntax. This allows +Twelf to prove termination of evaluation. + +Natural numbers and addition are defined as before. + +|hidden="true" !}% +%sort nat %. +%name nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort add %. +%mode add %in %in %out %. +%term add/z add z N N %. +%term add/s %pi (add (s M) N (s P)) %<- (add M N P) %. +%worlds () (add _ _ _) %. +%total M (add M _ _) %. +%{! + +## Call-by-value arithmetic expressions + +First, we define a type of ``open values``, which are either a +numeral or a variable. Numerals are represented by a constant +``num``; variables are represented by LF variables. + +!}% +%sort val %. +%name val %. +%term num %pi nat %-> val %. +%{! +Next, we define expressions, with a constant ``ret`` including +values into expressions. +!}% +%sort exp %. +%name exp %. +%term ret %pi val %-> exp %. +%term plus %pi exp %-> exp %-> exp %. +%term let %pi exp %-> (%pi val %-> exp) %-> exp %. +%{! + +Note that the ``let``-bound variable stands for a value. + +## Evaluation, using substitution + +As before, the type of answers includes only numerals. The code for +evaluation is exactly the same as [before](/wiki/summer-school-2008-arithmetic-expressions-with-let-binding/). + +!}% +%sort ans %. +%name ans %. +%term anum %pi nat %-> ans %. +%sort eval %. +%mode eval %in %out %. +%term eval/val eval (ret (num N)) (anum N) %. +%term eval/plus + %pi (eval (plus E1 E2) (anum N)) + %<- (eval E1 (anum N1)) + %<- (eval E2 (anum N2)) + %<- (add N1 N2 N) %. +%term eval/let %pi (eval (let E1 ([x] E2 x)) A) %<- (eval E1 (anum N)) %<- (eval (E2 (num N)) A) %. +%{! + +But now, Twelf can prove evaluation total: + +!}% +%worlds () (eval _ _) %. +%total E (eval E _) %. +%{! + +Why does this work? When termination-checking ``eval/let``, Twelf +observes that no expressions can appear in variables (this observation +is based on [subordination](/wiki/subordination/)), so it's possible to view all +substitution instances of ``E2`` as having the same size as +``E2``. This justifies the recursive call. If Twelf didn't build +in this reasoning, one could justify it explcitly by using a [numeric termination metric](/wiki/numeric-termination-metrics/) where the size of any value = the size of a +variable = 1. + + + +\{\{Summer school 2008 + |prev=Arithmetic expressions with let-binding + |prevname=Arithmetic expressions with let-binding + |next=Arithmetic expressions with let-binding (hypothetical evaluation) + |nextname=Defining evaluation with a hypothetical judgement\}\} + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008:Arithmetic_expressions_with_call-by-value_let-binding). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008-arithmetic-expressions-with-let-binding-hypothetical-evaluation.lf b/new-tests/stelf-output/pages/summer-school-2008-arithmetic-expressions-with-let-binding-hypothetical-evaluation.lf new file mode 100644 index 0000000..eb86b54 --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008-arithmetic-expressions-with-let-binding-hypothetical-evaluation.lf @@ -0,0 +1,104 @@ +% +%{! + +\{\{Summer school 2008 + |prev=Arithmetic expressions with call-by-value let-binding + |prevname=Call-by-value let-binding syntax + |next=Typed arithmetic expressions + |nextname=Typed arithmetic expressions\}\} + +Numbers and addition are as before. + +|hidden="true" !}% +%sort nat %. +%name nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort add %. +%mode add %in %in %out %. +%term add/z add z N N %. +%term add/s %pi (add (s M) N (s P)) %<- (add M N P) %. +%% addition is a total function on closed terms of type nat +%worlds () (add _ _ _) %. +%total M (add M _ _) %. +%{! + +## Evaluation using a hypothetical judgement + +We use the call-by-value syntax for expressions here. + +Values, expressions, answers, and the first two cases of evaluation are as before: + +!}% +%sort val %. +%name val %. +%term num %pi nat %-> val %. +%sort exp %. +%name exp %. +%term ret %pi val %-> exp %. +%term plus %pi exp %-> exp %-> exp %. +%term let %pi exp %-> (%pi val %-> exp) %-> exp %. +%%% evaluation, using hypothetical judgements +%sort ans %. +%name ans %. +%term anum %pi nat %-> ans %. +%sort eval %. +%mode eval %in %out %. +%term eval/val eval (ret (num N)) (anum N) %. +%term eval/plus + %pi (eval (plus E1 E2) (anum N)) + %<- (eval E1 (anum N1)) + %<- (eval E2 (anum N2)) + %<- (add N1 N2 N) %. +%term eval/let + %pi (eval (let E1 ([x] E2 x)) A2) + %<- (eval E1 A1) + %<- ({x val} %pi (eval (ret x) A1) %-> (eval (E2 x) A2)) %. +%{! + +``eval/let`` deserves some explanation: the second recursive call says that we evaluate ``(E2 x)`` in an extended LF context. In particular, we extend the context with ``x:val``, a variable ranging over values, and a derivation of ``eval (ret x) A1``. In the scope of these assumptions, the expression ``ret x`` therefore evaluates to ``A1``. In the terminology of [http://www.cs.cmu.edu/~rwh/plbook/book.pdf Practical Foundations for Programming Languages], ``eval`` is a [hypothetical](/wiki/hypothetical-judgment/) (because we add ``eval`` assumptions to the context) and general (because we add variables to the context) judgement. The context of ``eval`` is represented by the LF context, a technique called [higher-order judgements](/wiki/higher-order-judgements/). + +### Totality in non-empty worlds + +Because evaluation recurs in an extend context, we must prove it total not just for the empty context, as we have done above, but for a [world](/wiki/percent-worlds/) of a particular form. + +If we tried to prove it total in the empty context, Twelf would complain: + +<twelf check="decl"> +%worlds () (eval _ _). +</twelf> + +This error means "you said ``eval`` stays in the empty context, but it doesn't!". + +In what contexts in ``eval`` total? Not in every context: if we ever assumed a variable ``x:val`` without also assuming ``eval (ret x) A`` for some ``A``, then ``ret x`` would be an expression without a value. So we want to describe the invariant that the context looks like + +``x1:val, d1:eval (ret x) A1, ...... , xn:val, dn:eval (ret x) An`` + +for some ``A1, ... , An``. + +We do this by +- defining a block ``eval_block`` describing that pattern +- stating ``eval`` for a world containing contexts made up of ``eval_block``s +!}% +%block eval_block {A ans} [x val] [_ eval (ret x) A]%. +%worlds (eval_block) (eval _ _) %. +%{! + +Now Twelf can verify the totality of ``eval``: + +!}% +%total E (eval E _) %. +%{! + +\{\{Summer school 2008 + |prev=Arithmetic expressions with call-by-value let-binding + |prevname=Call-by-value let-binding syntax + |next=Typed arithmetic expressions + |nextname=Typed arithmetic expressions\}\} +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008:Arithmetic_expressions_with_let-binding_(hypothetical_evaluation)). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008-arithmetic-expressions-with-let-binding.lf b/new-tests/stelf-output/pages/summer-school-2008-arithmetic-expressions-with-let-binding.lf new file mode 100644 index 0000000..e9612ed --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008-arithmetic-expressions-with-let-binding.lf @@ -0,0 +1,108 @@ +% +%{! + +\{\{Summer school 2008 + |prev=Arithmetic expressions + |prevname=Arithmetic expressions + |next=Arithmetic expressions with call-by-value let-binding + |nextname=Arithmetic expressions with call-by-value let-binding +\}\} + +Next, we add let-binding to our expression language. Natural numbers +and addition are the same as before. + +|hidden="true" !}% +%sort nat %. +%name nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort add %. +%mode add %in %in %out %. +%term add/z add z N N %. +%term add/s %pi (add (s M) N (s P)) %<- (add M N P) %. +%% addition is a total function on closed terms of type nat +%worlds () (add _ _ _) %. +%total M (add M _ _) %. +%{! + +## Arithmetic expressions with let-binding + +### Syntax + +First, the syntax: + +!}% +%sort exp %. +%name exp %. +%term num %pi nat %-> exp %. +%term plus %pi exp %-> exp %-> exp %. +%term let %pi exp %-> (%pi exp %-> exp) %-> exp %. +%{! + +Let-binding is represented using [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/): + is +represented by ``let e_1 ([x] e_2)``; an LF variable is used to +represent the bound-variable. So the body of the ``let`` has LF +type ``(exp -> exp)``. + +!}% +%{! + +### Evaluation, using substitution + +!}% +%sort ans %. +%name ans %. +%term anum %pi nat %-> ans %. +%sort eval %. +%mode eval %in %out %. +%term eval/num eval (num N) (anum N) %. +%term eval/plus + %pi (eval (plus E1 E2) (anum N)) + %<- (eval E1 (anum N1)) + %<- (eval E2 (anum N2)) + %<- (add N1 N2 N) %. +%term eval/let %pi (eval (let E1 ([x] E2 x)) A) %<- (eval E1 (anum N)) %<- (eval (E2 (num N)) A) %. +%{! +That is, to evaluate a ``let``, we +* evaluate the let-bound term ``E1`` to an answer ``anum N`` +* substitute its value into the body. Substitution is represented by the LF application of ``E2`` to ``(num N)``. +* evaluate the result + +Twelf cannot prove this total without some help, because it's not +obvious that the substitution instance ``(E2 (num N))`` is smaller +than the input expression. + +!}% +%worlds () (eval _ _) %. +%{! + +<twelf ignore="true" check="decl"> +%total E (eval E _). +</twelf> + +However, evaluation does terminate. There are two different ways to see +this: + +- Observe that we only substitute _values_ for variables. Consequently, it is possible to give a size metric on terms where all the substitution instances of ``E2`` are the same size as ``E2``, by taking the size of a variable = the size of a value = one. We can formalize this reasoning in Twelf in two ways: +-# We can prove termination ourselves as a [metatheorem](/wiki/metatheorem/). We'll learn about this in class 3. +-# We can make the invariant that variables stand for values explicit in the syntax of the language, in which case Twelf can prove termination itself. See [Variation: Call-by-value let-binding syntax](/wiki/summer-school-2008-arithmetic-expressions-with-call-by-value-let-binding/) +- Rather than recursively evaluating the substitution instance, we can give an environment semantics where the values of variables are tracked off to the side. To evaluate a ``let``, we recursively evaluate body (so evaluation is structurally inductive on the expression) in an extended environment. See [Variation: Defining evaluation with a hypothetical judgement](/wiki/summer-school-2008-arithmetic-expressions-with-let-binding-hypothetical-evaluation/) + +At this point, you should explore one or both of these variations, and then proceed to see how we represent typed arithmetic expressions. + + +\{\{Summer school 2008 + |prev=Arithmetic expressions + |prevname=Arithmetic expressions + |next=Arithmetic expressions with call-by-value let-binding + |nextname=Arithmetic expressions with call-by-value let-binding +\}\} + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008:Arithmetic_expressions_with_let-binding). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008-arithmetic-expressions.lf b/new-tests/stelf-output/pages/summer-school-2008-arithmetic-expressions.lf new file mode 100644 index 0000000..19427b7 --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008-arithmetic-expressions.lf @@ -0,0 +1,131 @@ +% +%{! + +\{\{Summer school 2008 + |noprevious = + |next = Arithmetic expressions with let-binding + |nextname = Arithmetic expressions with let-binding +\}\} + +## Rudimentary arithmetic + +In on-paper notation, the syntax of the natural numbers is given as follows: + + + +That is, is a natural number, and if is a natural number, then is as well. + +We represent natural numbers as an LF type ``nat`` with two constants generating LF terms of that type: + +!}% +%sort nat %. +%name nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%{! + +The constant ``z`` constructs a ``nat``; the constant ``s`` constructs a ``nat`` from a ``nat``. + +### Addition + +Next, we define addition as a judgement relating two natural numbers to their sum: + +<table style="margin-left:auto;margin-right:auto" cellpadding="24"> +<tr><td> + +</td><td> + +</td></tr> +</table> + +This judgement is represented in LF as follows: + +!}% +%sort add %. +%mode add %in %in %out %. +%term add/z add z N N %. +%term add/s %pi (add (s M) N (s P)) %<- (add M N P) %. +%{! + +The first line says that the type family ``add`` relates three natural numbers. The second line says that the first two (the addends) determine the third (the sum). The constants ``add/z`` and ``add/s`` correspond to the inference rules above. + +#### Totality + +Twelf verifies that addition is a total function on closed terms of type nat: + +!}% +%worlds () (add _ _ _) %. +%total M (add M _ _) %. +%{! +The declarations should be read as follows: +* [%mode](/wiki/percent-mode/): The two addends are inputs; the sum is an output. +* [%worlds](/wiki/percent-worlds/): ``add`` is defined on closed LF terms +* [%total](/wiki/percent-total/): For all ``M`` and ``N``, there exists a ``P`` such that ``add M N P`` is derivable. + +This proves a [totality assertion](/wiki/totality-assertion/) for ``add``. +!}% +%{! + +## Simple arithmetic expressions + +!}% +%sort exp %. +%name exp %. +%term num %pi nat %-> exp %. +%term plus %pi exp %-> exp %-> exp %. +%{! +### Evaluation + +First, we define a syntactic category of answers, which in this case is +just natural numbers: + +!}% +%sort ans %. +%name ans %. +%term anum %pi nat %-> ans %. +%{! +Next, we define the evaluation judgement relating an expression to an answer. +!}% +%sort eval %. +%name eval %. +%mode eval %in %out %. +%term eval/num eval (num N) (anum N) %. +%term eval/plus + %pi (eval (plus E1 E2) (anum N)) + %<- (eval E1 (anum N1)) + %<- (eval E2 (anum N2)) + %<- (add N1 N2 N) %. +%worlds () (eval _ _) %. +%total E (eval E _) %. +%{! +Twelf verifies that evaluation is total: +* the mode declaration says that the expression is an input, and the answer is an output. +* the worlds declaration says that we're only considering closed expressions +* the total declaration asks Twelf to verify that ``eval`` defines a total relation from closed expressions to closed answers. +!}% +%{! +### Solving for derivations + +We can use logic programming to [solve](/wiki/percent-solve/) for derivations. + +!}% +%define two nat s (s z) %. +%solve _ : add two two N %. +%define two_exp exp num two %. +%solve _ : eval (plus two_exp two_exp) E %. +%{! + + +\{\{Summer school 2008 + |noprevious = + |next = Arithmetic expressions with let-binding + |nextname = Arithmetic expressions with let-binding +\}\} + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008:Arithmetic_expressions). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008-encoding-of-system-f.lf b/new-tests/stelf-output/pages/summer-school-2008-encoding-of-system-f.lf new file mode 100644 index 0000000..ab80e52 --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008-encoding-of-system-f.lf @@ -0,0 +1,68 @@ +% +%{! + +\{\{Summer school 2008 + |prev=Exercises 2 + |prevname=Exercises 2 + |nonext= +\}\} + +## Syntax + +Types: + +!}% +%sort tp %. +%term arrow %pi tp %-> tp %-> tp %. +%term forall %pi (%pi tp %-> tp) %-> tp %. +%{! + +Terms: + +!}% +%sort tm %. +%term fn %pi tp %-> (%pi tm %-> tm) %-> tm %. +%term app %pi tm %-> tm %-> tm %. +%term tfn %pi (%pi tp %-> tm) %-> tm %. +%term tapp %pi tm %-> tp %-> tm %. +%{! + +## Static semantics + +!}% +%sort of %. +%term of-fn %pi (of (fn T2 ([x tm] E x)) (arrow T2 T)) %<- ({x tm} {dx of x T2} of (E x) T) %. +%term of-app %pi (of (app E1 E2) T) %<- (of E1 (arrow T2 T)) %<- (of E2 T2) %. +%term of-tfn %pi (of (tfn ([u tp] E u)) (forall ([u tp] T u))) %<- ({u tp} of (E u) (T u)) %. +%term of-tapp %pi (of (tapp E T2) (T T2)) %<- (of E (forall ([u] T u))) %. +%{! + +## Dynamic semantics + +!}% +%sort value %. +%term value-fn value (fn A E) %. +%term value-tfn value (tfn E) %. +%sort step %. +%term step-app-1 %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step-app-2 %pi (step (app V1 E2) (app V1 E2')) %<- (value V1) %<- (step E2 E2') %. +%term step-app-beta %pi (step (app (fn T2 ([x tm] E x)) E2) (E E2)) %<- (value E2) %. +%term step-tapp-1 %pi (step (tapp E1 T) (tapp E1' T)) %<- (step E1 E1') %. +%term step-tapp-beta step (tapp (tfn ([u tp] E u)) T2) (E T2) %. +%{! + + +\{\{Summer school 2008 + |prev=Exercises 2 + |prevname=Exercises 2 + |nonext= +\}\} + + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008:Encoding_of_System_F). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008-exercises-1.lf b/new-tests/stelf-output/pages/summer-school-2008-exercises-1.lf new file mode 100644 index 0000000..897b9bb --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008-exercises-1.lf @@ -0,0 +1,34 @@ +% +%{! \{\{Summer school 2008 + |noprevious = + |nonext = \}\} + +## Add pairs [(solution)](/wiki/summer-school-2008-typed-arithmetic-expressions-with-pairs/) + +Extend the [Typed arithmetic expressions (value)](/wiki/summer-school-2008-typed-arithmetic-expressions-value/) language with binary pairs (see [http://www.cs.cmu.edu/~rwh/plbook/book.pdf PFPL Chapter 16] if you need a refresher on the syntax and typing rules for pairs). + +This will be easiest if you start from [Typed arithmetic expressions (value)](/wiki/summer-school-2008-typed-arithmetic-expressions-value/). Use the "Code: here" link in the top-left corner to download the code. + +You will need to add: + +- a type ``prod T U`` +- expression constructors for pairing ``(pair E1 E2)``, and first and second projection ``(fst E)`` and ``(snd E)``. +- new cases for evaluation + +Get Twelf to verify the totality of your extended evaluation judgement. + +If you're feeling ambitious, add disjoint sums ([http://www.cs.cmu.edu/~rwh/plbook/book.pdf PFPL Chapter 17]) as well! [(solution)](/wiki/summer-school-2008-alternate-typed-arithmetic-expressions-with-sums/) + +## Getting familiar with Twelf + +Read the discussion of Twelf's totality checker on [this page](/wiki/proving-metatheorems-proving-totality-assertions-about-the-natural-numbers/) so that you understand Twelf's error messages better. + +## Call-by-name + +Do a call-by-name version of evaluation for [Arithmetic expressions with let-binding](/wiki/summer-school-2008-arithmetic-expressions-with-let-binding/). How would you prove evaluation terminates? !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008:Exercises_1). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008-exercises-2.lf b/new-tests/stelf-output/pages/summer-school-2008-exercises-2.lf new file mode 100644 index 0000000..a9584eb --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008-exercises-2.lf @@ -0,0 +1,31 @@ +% +%{! \{\{Summer school 2008 + |noprevious= + |nonext= +\}\} + +## System F [(solution)](/wiki/summer-school-2008-encoding-of-system-f/) + +- Give an LF encoding of System F (the polymorphic lambda-calculus). Use [http://www.cs.cmu.edu/~rwh/plbook/book.pdf PFPL Chapter 25 as the on-paper definition]. +- Prove adequacy. You may want to follow the adequacy proof in [http://www.cs.cmu.edu/~drl/pubs/hl07mechanizing/hl07mechanizing.pdf MMLF], section 3.2, page 21. + +## System F, Intrinsic Encoding + +- Use an [intrinsic encoding](/wiki/intrinsic-and-extrinsic-encodings/) to represent only the well-typed System F terms. +- What can you say about adequacy for this encoding? + +## Hereditary substitution + +Hereditary substitution is used in the definition of LF to compute the canonical result of substituting one canonical term into another. Read and understand the rules for hereditary substitution in [http://www.cs.cmu.edu/~drl/pubs/hl07mechanizing/hl07mechanizing.pdf MMLF]. + + +\{\{Summer school 2008 + |noprevious= + |nonext= +\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008:Exercises_2). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008-exercises-3.lf b/new-tests/stelf-output/pages/summer-school-2008-exercises-3.lf new file mode 100644 index 0000000..c4085d7 --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008-exercises-3.lf @@ -0,0 +1,32 @@ +% +%{! ## Polymorphic MinML [(solution)](/wiki/summer-school-2008-type-safety-for-polymorphic-minml-intrinsic-encoding/) + +Add explicit polymorphism to MinML and prove type safety: + +- Add a type +- Add terms and , with the appropriate typing rules and dynamic semantics +- Extend the type safety proof to cover the new cases. + +You can start from either encoding: +* [The intrinsic encoding](/wiki/summer-school-2008-type-safety-for-minml-intrinsic-encoding/) will be easier, because it takes full advantage of dependent types. +* [The extrinsic encoding](/wiki/summer-school-2008-type-safety-for-minml-extrinsic-encoding/) is closer to what people usually write on paper. + +## Monadic Effects [(solution)](/wiki/summer-school-2008-type-safety-for-minml-with-monadic-effects/) + +Add monadically encapsulated effects (i.e., an ``IO`` monad) to MinML, following [http://www.cs.cmu.edu/~rwh/plbook/book.pdf PFPL Chapter 39]. + +If you're feeling ambitious, add the following specific effects [(solution)](/wiki/summer-school-2008-type-safety-for-minml-with-monadic-effects-putngetn/): +* ``getn ; x.m`` reads a natural number from an input buffer (i.e., stdin). ``x`` stands for the number in ``m``. +* ``putn(e) ; m`` writes a natural number, the value of ``e`` to an output buffer (i.e., stdout). + +These primitives are simpler than full mutable references because their dynamic semantics can be given as a labeled transition system with labels ``read n`` and ``write n``---you do not need a store as part of the judgement form. + +## Determinacy of Evaluation + +Prove that the dynamic semantics of MinML are deterministic (every expression steps to at most one other expression). Hint: you will need to use [equality](/wiki/equality/) to state the theorem. See the discussion of [uniqueness lemma](/wiki/uniqueness-lemma/)s for a proof of determinacy for the simply-typed λ-calculus. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008:Exercises_3). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008-type-safety-for-minml-extrinsic-encoding.lf b/new-tests/stelf-output/pages/summer-school-2008-type-safety-for-minml-extrinsic-encoding.lf new file mode 100644 index 0000000..10181dc --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008-type-safety-for-minml-extrinsic-encoding.lf @@ -0,0 +1,200 @@ +% +%{! + +\{\{Summer school 2008 + |prev=Type safety for MinML (intrinsic encoding) + |prevname=Type safety for MinML (intrinsic encoding) + |next=Exercises 3 + |nextname=Exercises 3 +\}\} + +Type safety for MinML: call-by-value, with recursive functions, in extrinsic form + +## Syntax + +Types: + +!}% +%sort tp %. +%name tp %. +%term nat tp %. +%term arr %pi tp %-> tp %-> tp %. +%{! + +Raw expressions, which admit ill-typed terms + +!}% +%sort exp %. +%name exp %. +%term z exp %. +%term s %pi exp %-> exp %. +%term ifz %pi exp %-> exp %-> (%pi exp %-> exp) %-> exp %. +%term fun %pi tp %-> tp %-> (%pi exp %-> exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%{! + +## Static semantics + +A judgement picking out the well-typed terms: + +!}% +%sort of %. +%name of %. +%mode of %in %out %. +%term of/z of z nat %. +%term of/s %pi (of (s E) nat) %<- (of E nat) %. +%term of/ifz + %pi (of (ifz E E1 ([x] E2 x)) T) + %<- (of E nat) + %<- (of E1 T) + %<- ({x exp} %pi (of x nat) %-> (of (E2 x) T)) %. +%term of/fun + %pi (of (fun T1 T2 ([f] [x] E f x)) (arr T1 T2)) + %<- ({f exp} %pi (of f (arr T1 T2)) %-> ({x exp} %pi (of x T1) %-> (of (E f x) T2))) %. +%term of/app %pi (of (app E1 E2) T) %<- (of E1 (arr T2 T)) %<- (of E2 T2) %. +%{! + +## Dynamic semantics + +!}% +%sort value %. +%name value %. +%mode value %in %. +%term value/z value z %. +%term value/s %pi (value (s E)) %<- (value E) %. +%term value/fun value (fun _ _ _) %. +%sort step %. +%name step %. +%mode step %in %out %. +%term step/s %pi (step (s E) (s E')) %<- (step E E') %. +%term step/ifz/arg %pi (step (ifz E E1 ([x] E2 x)) (ifz E' E1 ([x] E2 x))) %<- (step E E') %. +%term step/ifz/z step (ifz z E1 ([x] E2 x)) E1 %. +%term step/ifz/s %pi (step (ifz (s E) E1 ([x] E2 x)) (E2 E)) %<- (value E) %. +%term step/app/fun %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app/arg %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app/beta-v + %pi (step (app (fun T1 T2 ([f] [x] E f x)) E2) (E (fun T1 T2 ([f] [x] E f x)) E2)) + %<- (value E2) %. +%{! + +## Preservation + +With this encoding, we have to prove preservation explicitly, as the +type of ``step`` doesn't guarantee it. + +!}% +%sort pres %. +%name pres %. +%mode pres %in %in %out %. +%term _ %pi (pres (step/s Dstep) (of/s Dof) (of/s Dof')) %<- (pres Dstep Dof Dof') %. +%term _ + %pi (pres (step/ifz/arg Dstep) (of/ifz ([x] [dx] Dof2 x dx) Dof1 Dof) (of/ifz ([x] [dx] Dof2 x dx) Dof1 Dof')) + %<- (pres Dstep Dof Dof') %. +%term _ pres step/ifz/z (of/ifz _ Dof1 _) Dof1 %. +%term _ pres (step/ifz/s (%the (value E) _)) (of/ifz ([x] [dx] Dof2 x dx) _ (of/s Dof)) (Dof2 E Dof) %. +%term _ + %pi (pres (step/app/fun Dstep1) (of/app Dof2 Dof1) (of/app Dof2 Dof1')) + %<- (pres Dstep1 Dof1 Dof1') %. +%term _ + %pi (pres (step/app/arg Dstep2 _) (of/app Dof2 Dof1) (of/app Dof2' Dof1)) + %<- (pres Dstep2 Dof2 Dof2') %. +%term _ pres (step/app/beta-v _) (of/app Dof2 (of/fun ([f] [df] [x] [dx] Dof1 f df x dx))) (Dof1 _ (of/fun ([f] [df] [x] [dx] Dof1 f df x dx)) _ Dof2) %. +%worlds () (pres Dstep Dof Dof') %. +%total Dstep (pres Dstep _ _) %. +%{! + +## Progress + +!}% +%sort val-or-step %. +%name val-or-step %. +%term vos/val %pi (val-or-step E) %<- (value E) %. +%term vos/step %pi (val-or-step E) %<- (step E _) %. +%{! + +### [Factoring lemmas](/wiki/output-factoring/) + +!}% +%sort prog/s %. +%mode prog/s %in %out %. +%term _ prog/s (vos/step Dstep) (vos/step (step/s Dstep)) %. +%term _ prog/s (vos/val Dval) (vos/val (value/s Dval)) %. +%worlds () (prog/s _ _) %. +%total {} (prog/s _ _) %. +%sort prog/ifz %. +%mode prog/ifz %in %in %in %in %out %. +%term _ prog/ifz _ (vos/step Dstep) _ _ (step/ifz/arg Dstep) %. +%term _ prog/ifz _ (vos/val value/z) _ _ step/ifz/z %. +%term _ prog/ifz _ (vos/val (value/s Dval)) _ _ (step/ifz/s Dval) %. +%worlds () (prog/ifz _ _ _ _ _) %. +%total {} (prog/ifz _ _ _ _ _) %. +%sort prog/app %. +%mode prog/app %in %in %in %out %. +%term _ prog/app _ (vos/step Dstep1) _ (step/app/fun Dstep1) %. +%term _ prog/app _ (vos/val Dval1) (vos/step Dstep2) (step/app/arg Dstep2 Dval1) %. +%term _ prog/app _ (vos/val Dval1) (vos/val Dval2) (step/app/beta-v Dval2) %. +%worlds () (prog/app _ _ _ _) %. +%total {} (prog/app _ _ _ _) %. +%{! + +### Main theorem + +!}% +%sort prog %. +%name prog %. +%mode prog %in %out %. +%term _ prog of/z (vos/val value/z) %. +%term _ %pi (prog (of/s Dof) Dvos') %<- (prog Dof Dvos) %<- (prog/s Dvos Dvos') %. +%term _ + %pi (prog (of/ifz ([x] [dx] Dof2 x dx) Dof1 Dof) (vos/step Dstep)) + %<- (prog Dof Dvos) + %<- (prog/ifz Dof Dvos _ _ Dstep) %. +%term _ prog (of/fun _) (vos/val value/fun) %. +%term _ + %pi (prog (of/app Dof2 Dof1) (vos/step Dstep)) + %<- (prog Dof1 Dvos1) + %<- (prog Dof2 Dvos2) + %<- (prog/app Dof1 Dvos1 Dvos2 Dstep) %. +%worlds () (prog _ _) %. +%total Dof (prog Dof _) %. +%{! + +And thus we have proved type safety for minml! + +## Examples + +!}% +%inline plus (exp) fun nat (arr nat nat) ([plus] [x] ifz x (fun nat nat ([_] [y] y)) ([predx] fun nat nat ([_] [y] s (app (app plus predx) y)))) %. +%solve D : of plus T %. +%inline mult (exp) fun nat (arr nat nat) ([mult] [x] fun nat nat ([_] [y] ifz y z ([predy] app (app plus x) (app (app mult x) predy)))) %. +%solve Dmult : of mult T %. +%inline fact (exp) fun nat nat ([fact] [x] ifz x (s z) ([predx] app (app mult x) (app fact predx))) %. +%{! +|check="decl" !}% +%solve Dfact : of fact T %. +%{! !}% +%sort stepv %. +%term stepv/v %pi (stepv E E) %<- (value E) %. +%term stepv/s %pi (stepv E E'') %<- (step E E') %<- (stepv E' E'') %. +%solve D : stepv (app (app plus (s (s z))) (s (s z))) E %. +%solve D : stepv (app (app mult (s (s (s z)))) (s (s z))) E %. +%{! +|check="decl" !}% +%solve D : stepv (app fact (s (s (s z)))) E %. +%{! !}% +%{! + +\{\{Summer school 2008 + |prev=Type safety for MinML (intrinsic encoding) + |prevname=Type safety for MinML (intrinsic encoding) + |next=Exercises 3 + |nextname=Exercises 3 +\}\} + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008:Type_safety_for_MinML_(extrinsic_encoding)). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008-type-safety-for-minml-intrinsic-encoding.lf b/new-tests/stelf-output/pages/summer-school-2008-type-safety-for-minml-intrinsic-encoding.lf new file mode 100644 index 0000000..f00e5b4 --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008-type-safety-for-minml-intrinsic-encoding.lf @@ -0,0 +1,141 @@ +% +%{! + +\{\{Summer school 2008 + |noprevious= + |next=Type safety for MinML (extrinsic encoding) + |nextname=Type safety for MinML (extrinsic encoding) +\}\} + +Type safety for MinML: call-by-value, with recursive functions, in intrinsic form + +!}% +%{! + +## Syntax / static semantics + +Types: + +!}% +%sort tp %. +%name tp %. +%term nat tp %. +%term arr %pi tp %-> tp %-> tp %. +%{! + +Typed expressions (i.e., typing derivations): + +!}% +%sort exp %. +%name exp %. +%prec %postfix 1 exp %. +%term z nat exp %. +%term s %pi (nat exp) %-> (nat exp) %. +%term ifz %pi (nat exp) %-> (T exp) %-> (%pi (nat exp) %-> (T exp)) %-> (T exp) %. +%term fun {T1 tp} {T2 tp} %pi (%pi (arr T1 T2 exp) %-> (T1 exp) %-> (T2 exp)) %-> (arr T1 T2 exp) %. +%term app %pi (arr T1 T2 exp) %-> (T1 exp) %-> (T2 exp) %. +%{! + +## Dynamic semantics / preservation + +Values: + +!}% +%sort value %. +%name value %. +%mode value %in %. +%term value/z value z %. +%term value/s %pi (value (s E)) %<- (value E) %. +%term value/fun value (fun _ _ _) %. +%{! + +Transition relation: + +!}% +%sort step %. +%name step %. +%mode step %in %out %. +%term step/s %pi (step (s E) (s E')) %<- (step E E') %. +%term step/ifz/arg %pi (step (ifz E E1 ([x] E2 x)) (ifz E' E1 ([x] E2 x))) %<- (step E E') %. +%term step/ifz/z step (ifz z E1 ([x] E2 x)) E1 %. +%term step/ifz/s %pi (step (ifz (s E) E1 ([x] E2 x)) (E2 E)) %<- (value E) %. +%term step/app/fun %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app/arg %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app/beta-v + %pi (step (app (fun T1 T2 ([f] [x] E f x)) E2) (E (fun T1 T2 ([f] [x] E f x)) E2)) + %<- (value E2) %. +%{! + +## Progress + +The sum type we return: + +!}% +%sort val-or-step %. +%name val-or-step %. +%term vos/val %pi (val-or-step E) %<- (value E) %. +%term vos/step %pi (val-or-step E) %<- (step E _) %. +%{! + +### [Factoring](/wiki/output-factoring/) lemmas + +!}% +%sort prog/s %. +%mode prog/s %in %out %. +%term _ prog/s (vos/step Dstep) (vos/step (step/s Dstep)) %. +%term _ prog/s (vos/val Dval) (vos/val (value/s Dval)) %. +%worlds () (prog/s _ _) %. +%total {} (prog/s _ _) %. +%sort prog/ifz %. +%mode prog/ifz %in %in %in %out %. +%term _ prog/ifz (vos/step Dstep) _ _ (step/ifz/arg Dstep) %. +%term _ prog/ifz (vos/val value/z) _ _ step/ifz/z %. +%term _ prog/ifz (vos/val (value/s Dval)) _ _ (step/ifz/s Dval) %. +%worlds () (prog/ifz _ _ _ _) %. +%total {} (prog/ifz _ _ _ _) %. +%sort prog/app %. +%mode prog/app %in %in %out %. +%term _ prog/app (vos/step Dstep1) _ (step/app/fun Dstep1) %. +%term _ prog/app (vos/val Dval1) (vos/step Dstep2) (step/app/arg Dstep2 Dval1) %. +%term _ prog/app (vos/val Dval1) (vos/val Dval2) (step/app/beta-v Dval2) %. +%worlds () (prog/app _ _ _) %. +%total {} (prog/app _ _ _) %. +%{! + +### Main theorem + +!}% +%sort prog {E T exp} %. +%name prog %. +%mode prog %in %out %. +%term _ prog z (vos/val value/z) %. +%term _ %pi (prog (s E) Dvos') %<- (prog E Dvos) %<- (prog/s Dvos Dvos') %. +%term _ + %pi (prog (ifz E E1 ([x] E2 x)) (vos/step Dstep)) + %<- (prog E Dvos) + %<- (prog/ifz Dvos _ _ Dstep) %. +%term _ prog (fun _ _ _) (vos/val value/fun) %. +%term _ + %pi (prog (app E1 E2) (vos/step Dstep)) + %<- (prog E1 Dvos1) + %<- (prog E2 Dvos2) + %<- (prog/app Dvos1 Dvos2 Dstep) %. +%worlds () (prog _ _) %. +%total Dof (prog Dof _) %. +%{! +And thus we have proved type safety for MinML! + + +\{\{Summer school 2008 + |noprevious= + |next=Type safety for MinML (extrinsic encoding) + |nextname=Type safety for MinML (extrinsic encoding) +\}\} + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008:Type_safety_for_MinML_(intrinsic_encoding)). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008-type-safety-for-minml-with-monadic-effects-putngetn.lf b/new-tests/stelf-output/pages/summer-school-2008-type-safety-for-minml-with-monadic-effects-putngetn.lf new file mode 100644 index 0000000..e057ec2 --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008-type-safety-for-minml-with-monadic-effects-putngetn.lf @@ -0,0 +1,239 @@ +% +%{! + +\{\{Summer school 2008 + |prev=Exercises 3 + |prevname=Exercies 3 + |nonext= +\}\} + +Type safety for MinML with monadic effects, with putn and getn primitives. + +!}% +%{! + +## Syntax / static semantics + +Types: + +!}% +%sort tp %. +%name tp %. +%term nat tp %. +%term arr %pi tp %-> tp %-> tp %. +%term circle %pi tp %-> tp %. +%{! + +Typed expressions (i.e., typing derivations): + +!}% +%sort exp %. +%name exp %. +%prec %postfix 1 exp %. +%sort command %. +%name command %. +%prec %postfix 1 command %. +%term z nat exp %. +%term s %pi (nat exp) %-> (nat exp) %. +%term ifz %pi (nat exp) %-> (T exp) %-> (%pi (nat exp) %-> (T exp)) %-> (T exp) %. +%term fun {T1 tp} {T2 tp} %pi (%pi (arr T1 T2 exp) %-> (T1 exp) %-> (T2 exp)) %-> (arr T1 T2 exp) %. +%term app %pi (arr T1 T2 exp) %-> (T1 exp) %-> (T2 exp) %. +%term comp %pi (T command) %-> (circle T exp) %. +%term return %pi (T exp) %-> (T command) %. +%term let %pi (circle T exp) %-> (%pi (T exp) %-> (U command)) %-> (U command) %. +%term getn %pi (%pi (nat exp) %-> (U command)) %-> (U command) %. +%term putn %pi (nat exp) %-> (U command) %-> (U command) %. +%{! + +## Dynamic semantics / preservation + +### Values and final states + +!}% +%sort value %. +%name value %. +%mode value %in %. +%term value/z value z %. +%term value/s %pi (value (s E)) %<- (value E) %. +%term value/fun value (fun _ _ _) %. +%term value/comp value (comp _) %. +%sort final %. +%name final %. +%mode final %in %. +%term final/ret %pi (final (return E)) %<- (value E) %. +%{! + +### Transition relation for expressions + +!}% +%sort step %. +%name step %. +%mode step %in %out %. +%term step/s %pi (step (s E) (s E')) %<- (step E E') %. +%term step/ifz/arg %pi (step (ifz E E1 ([x] E2 x)) (ifz E' E1 ([x] E2 x))) %<- (step E E') %. +%term step/ifz/z step (ifz z E1 ([x] E2 x)) E1 %. +%term step/ifz/s %pi (step (ifz (s E) E1 ([x] E2 x)) (E2 E)) %<- (value E) %. +%term step/app/fun %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app/arg %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app/beta-v + %pi (step (app (fun T1 T2 ([f] [x] E f x)) E2) (E (fun T1 T2 ([f] [x] E f x)) E2)) + %<- (value E2) %. +%{! + +### Labelled transition relation for commands + +!}% +%sort label %. +%name label %. +%term label/read %pi (value (%the (nat exp) N)) %-> label %. +%term label/write %pi (value (%the (nat exp) N)) %-> label %. +%term label/sil label %. +%% silent transition +%sort mstep %. +%name mstep %. +%mode mstep %in %star %star %. +%term mstep/ret %pi (mstep (return E) (return E') label/sil) %<- (step E E') %. +%term mstep/let-e %pi (mstep (let E ([x] M x)) (let E' ([x] M x)) label/sil) %<- (step E E') %. +%term mstep/let-m + %pi (mstep (let (comp M1) ([x] M2 x)) (let (comp M1') ([x] M2 x)) L) + %<- (mstep M1 M1' L) %. +%term mstep/let-beta + %pi (mstep (let (comp (return E)) ([x] M x)) (M E) label/sil) + %<- (final (return E)) %. +%term mstep/getn mstep (getn ([x] M x)) (M E) (label/read (%the (value E) Dval)) %. +%term mstep/putn %pi (mstep (putn E M) (putn E' M) label/sil) %<- (step E E') %. +%term mstep/putn-put mstep (putn E M) M (label/write (%the (value E) Dval)) %. +%{! + +## Progress + +### Return sums + +!}% +%sort val-or-step %. +%name val-or-step %. +%term vos/val %pi (val-or-step E) %<- (value E) %. +%term vos/step %pi (val-or-step E) %<- (step E _) %. +%sort fin-or-mstep %. +%name fin-or-mstep %. +%term fos/fin %pi (fin-or-mstep M) %<- (final M) %. +%term fos/step %pi (fin-or-mstep M) %<- (mstep M _ _) %. +%{! + +### [Factoring](/wiki/output-factoring/) lemmas + +!}% +%sort prog/s %. +%mode prog/s %in %out %. +%term _ prog/s (vos/step Dstep) (vos/step (step/s Dstep)) %. +%term _ prog/s (vos/val Dval) (vos/val (value/s Dval)) %. +%worlds () (prog/s _ _) %. +%total {} (prog/s _ _) %. +%sort prog/ifz %. +%mode prog/ifz %in %in %in %out %. +%term _ prog/ifz (vos/step Dstep) _ _ (step/ifz/arg Dstep) %. +%term _ prog/ifz (vos/val value/z) _ _ step/ifz/z %. +%term _ prog/ifz (vos/val (value/s Dval)) _ _ (step/ifz/s Dval) %. +%worlds () (prog/ifz _ _ _ _) %. +%total {} (prog/ifz _ _ _ _) %. +%sort prog/app %. +%mode prog/app %in %in %out %. +%term _ prog/app (vos/step Dstep1) _ (step/app/fun Dstep1) %. +%term _ prog/app (vos/val Dval1) (vos/step Dstep2) (step/app/arg Dstep2 Dval1) %. +%term _ prog/app (vos/val Dval1) (vos/val Dval2) (step/app/beta-v Dval2) %. +%worlds () (prog/app _ _ _) %. +%total {} (prog/app _ _ _) %. +%{! + +### Progress for expressions + +!}% +%sort prog {E T exp} %. +%name prog %. +%mode prog %in %out %. +%term _ prog z (vos/val value/z) %. +%term _ %pi (prog (s E) Dvos') %<- (prog E Dvos) %<- (prog/s Dvos Dvos') %. +%term _ + %pi (prog (ifz E E1 ([x] E2 x)) (vos/step Dstep)) + %<- (prog E Dvos) + %<- (prog/ifz Dvos _ _ Dstep) %. +%term _ prog (fun _ _ _) (vos/val value/fun) %. +%term _ + %pi (prog (app E1 E2) (vos/step Dstep)) + %<- (prog E1 Dvos1) + %<- (prog E2 Dvos2) + %<- (prog/app Dvos1 Dvos2 Dstep) %. +%term _ prog (comp M) (vos/val value/comp) %. +%worlds () (prog _ _) %. +%total Dof (prog Dof _) %. +%{! + +### Factoring lemmas for commands + +!}% +%sort mprog/ret %. +%mode mprog/ret %in %out %. +%term _ mprog/ret (vos/val Dval) (fos/fin (final/ret Dval)) %. +%term _ mprog/ret (vos/step Dstep) (fos/step (mstep/ret Dstep)) %. +%worlds () (mprog/ret _ _) %. +%total {} (mprog/ret _ _) %. +%sort mprog/let-comp %. +%mode mprog/let-comp %in %in %out %. +%term _ mprog/let-comp (fos/fin (%the (final (return E)) (final/ret Dval))) M2 (%the (mstep (let (comp (return E)) M2) (M2 E) label/sil) (mstep/let-beta (final/ret Dval))) %. +%term _ mprog/let-comp (fos/step (%the (mstep M1 M1' L) Dmstep)) M2 (%the (mstep (let (comp M1) M2) (let (comp M1') M2) L) (mstep/let-m Dmstep)) %. +%worlds () (mprog/let-comp _ _ _) %. +%total {} (mprog/let-comp _ _ _) %. +%sort mprog/putn %. +%mode mprog/putn %in %in %out %. +%term _ mprog/putn (vos/val Dval) M (%the (mstep (putn E M) M (label/write Dval)) mstep/putn-put) %. +%term _ mprog/putn (vos/step Dstep) _ (mstep/putn Dstep) %. +%worlds () (mprog/putn _ _ _) %. +%total {} (mprog/putn _ _ _) %. +%{! + +### Progress for commands + +One of the factoring lemmas is mutually recursive with the main theorem: + +!}% +%sort mprog/let {E circle T exp} %. +%mode mprog/let %in %in %in %out %. +%sort mprog {M T command} %. +%mode mprog %in %out %. +%% mprog/let +%term _ + %pi (mprog/let (comp M) (vos/val value/comp) _ Dmstep) + %<- (mprog M Dfos) + %<- (mprog/let-comp Dfos _ Dmstep) %. +%term _ mprog/let _ (vos/step Dstep) _ (mstep/let-e Dstep) %. +%% mprog +%term _ %pi (mprog (return E) Dfos) %<- (prog E Dvos) %<- (mprog/ret Dvos Dfos) %. +%term _ + %pi (mprog (let (%the (circle T exp) E) (%the (%pi (T exp) %-> (U command)) M)) (fos/step Dmstep)) + %<- (prog E (%the (val-or-step E) Dvos)) + %<- (mprog/let E Dvos M (%the (mstep (let E M) M' L) Dmstep)) %. +% +%% maybe this isn't the right theorem +%term _ mprog (getn M) (fos/step (%the (mstep (getn M) (M z) (label/read value/z)) mstep/getn)) %. +%term _ + %pi (mprog (putn E M) (fos/step Dmstep)) + %<- (prog E Dvos) + %<- (mprog/putn Dvos M Dmstep) %. +%worlds () (mprog _ _) (mprog/let _ _ _ _) %. +%total (M E) (mprog M _) (mprog/let E _ _ _) %. +%{! +And thus we have proved type safety for MinML with monadic effects! + +\{\{Summer school 2008 + |prev=Exercises 3 + |prevname=Exercies 3 + |nonext= +\}\} + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008:Type_safety_for_MinML_with_monadic_effects_(putngetn)). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008-type-safety-for-minml-with-monadic-effects.lf b/new-tests/stelf-output/pages/summer-school-2008-type-safety-for-minml-with-monadic-effects.lf new file mode 100644 index 0000000..995e1bf --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008-type-safety-for-minml-with-monadic-effects.lf @@ -0,0 +1,229 @@ +% +%{! + +\{\{Summer school 2008 + |prev=Exercises 3 + |prevname=Exercies 3 + |nonext= +\}\} + +Type safety for MinML with monadic effects + +!}% +%{! + +## Syntax / static semantics + +Types: + +!}% +%sort tp %. +%name tp %. +%term nat tp %. +%term arr %pi tp %-> tp %-> tp %. +%term circle %pi tp %-> tp %. +%% called 'comp' in PFPL; +%% the lax modality is often rendered as a circle +%{! + +Next, we define typed expressions (i.e., typing derivations). There are +two syntactic classes: pure expressions, and possibly-effectful +commands. + +!}% +%sort exp %. +%name exp %. +%prec %postfix 1 exp %. +%sort command %. +%name command %. +%prec %postfix 1 command %. +%term z nat exp %. +%term s %pi (nat exp) %-> (nat exp) %. +%term ifz %pi (nat exp) %-> (T exp) %-> (%pi (nat exp) %-> (T exp)) %-> (T exp) %. +%term fun {T1 tp} {T2 tp} %pi (%pi (arr T1 T2 exp) %-> (T1 exp) %-> (T2 exp)) %-> (arr T1 T2 exp) %. +%term app %pi (arr T1 T2 exp) %-> (T1 exp) %-> (T2 exp) %. +%term comp %pi (T command) %-> (circle T exp) %. +%term return %pi (T exp) %-> (T command) %. +%term let %pi (circle T exp) %-> (%pi (T exp) %-> (U command)) %-> (U command) %. +%{! + +## Dynamic semantics / preservation + +### Values and final states + +!}% +%sort value %. +%name value %. +%mode value %in %. +%term value/z value z %. +%term value/s %pi (value (s E)) %<- (value E) %. +%term value/fun value (fun _ _ _) %. +%term value/comp value (comp _) %. +%sort final %. +%name final %. +%mode final %in %. +%term final/ret %pi (final (return E)) %<- (value E) %. +%{! + +### Transition relation for expressions + +!}% +%sort step %. +%name step %. +%mode step %in %out %. +%term step/s %pi (step (s E) (s E')) %<- (step E E') %. +%term step/ifz/arg %pi (step (ifz E E1 ([x] E2 x)) (ifz E' E1 ([x] E2 x))) %<- (step E E') %. +%term step/ifz/z step (ifz z E1 ([x] E2 x)) E1 %. +%term step/ifz/s %pi (step (ifz (s E) E1 ([x] E2 x)) (E2 E)) %<- (value E) %. +%term step/app/fun %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app/arg %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app/beta-v + %pi (step (app (fun T1 T2 ([f] [x] E f x)) E2) (E (fun T1 T2 ([f] [x] E f x)) E2)) + %<- (value E2) %. +%{! + +### Transition relation for commands + +!}% +%sort mstep %. +%name mstep %. +%mode mstep %in %out %. +%term mstep/ret %pi (mstep (return E) (return E')) %<- (step E E') %. +%term mstep/let-e %pi (mstep (let E ([x] M x)) (let E' ([x] M x))) %<- (step E E') %. +%term mstep/let-m + %pi (mstep (let (comp M1) ([x] M2 x)) (let (comp M1') ([x] M2 x))) + %<- (mstep M1 M1') %. +%term mstep/let-beta %pi (mstep (let (comp (return E)) ([x] M x)) (M E)) %<- (final (return E)) %. +%{! + +## Progress + +The progress theorem for expressions does not require a corresponding +theorem for commands, because commands are suspended inside expressions +(``comp M`` is always a value). We prove progress for expressions +first, and for commands below. + +### Return sums + +!}% +%sort val-or-step %. +%name val-or-step %. +%term vos/val %pi (val-or-step E) %<- (value E) %. +%term vos/step %pi (val-or-step E) %<- (step E _) %. +%sort fin-or-mstep %. +%name fin-or-mstep %. +%term fos/fin %pi (fin-or-mstep M) %<- (final M) %. +%term fos/step %pi (fin-or-mstep M) %<- (mstep M _) %. +%{! + +### [Factoring](/wiki/output-factoring/) lemmas + +!}% +%sort prog/s %. +%mode prog/s %in %out %. +%term _ prog/s (vos/step Dstep) (vos/step (step/s Dstep)) %. +%term _ prog/s (vos/val Dval) (vos/val (value/s Dval)) %. +%worlds () (prog/s _ _) %. +%total {} (prog/s _ _) %. +%sort prog/ifz %. +%mode prog/ifz %in %in %in %out %. +%term _ prog/ifz (vos/step Dstep) _ _ (step/ifz/arg Dstep) %. +%term _ prog/ifz (vos/val value/z) _ _ step/ifz/z %. +%term _ prog/ifz (vos/val (value/s Dval)) _ _ (step/ifz/s Dval) %. +%worlds () (prog/ifz _ _ _ _) %. +%total {} (prog/ifz _ _ _ _) %. +%sort prog/app %. +%mode prog/app %in %in %out %. +%term _ prog/app (vos/step Dstep1) _ (step/app/fun Dstep1) %. +%term _ prog/app (vos/val Dval1) (vos/step Dstep2) (step/app/arg Dstep2 Dval1) %. +%term _ prog/app (vos/val Dval1) (vos/val Dval2) (step/app/beta-v Dval2) %. +%worlds () (prog/app _ _ _) %. +%total {} (prog/app _ _ _) %. +%{! + +### Progress for expressions + +!}% +%sort prog {E T exp} %. +%name prog %. +%mode prog %in %out %. +%term _ prog z (vos/val value/z) %. +%term _ %pi (prog (s E) Dvos') %<- (prog E Dvos) %<- (prog/s Dvos Dvos') %. +%term _ + %pi (prog (ifz E E1 ([x] E2 x)) (vos/step Dstep)) + %<- (prog E Dvos) + %<- (prog/ifz Dvos _ _ Dstep) %. +%term _ prog (fun _ _ _) (vos/val value/fun) %. +%term _ + %pi (prog (app E1 E2) (vos/step Dstep)) + %<- (prog E1 Dvos1) + %<- (prog E2 Dvos2) + %<- (prog/app Dvos1 Dvos2 Dstep) %. +%term _ prog (comp M) (vos/val value/comp) %. +%worlds () (prog _ _) %. +%total Dof (prog Dof _) %. +%{! + +### Factoring lemmas for commands + +!}% +%sort mprog/ret %. +%mode mprog/ret %in %out %. +%term _ mprog/ret (vos/val Dval) (fos/fin (final/ret Dval)) %. +%term _ mprog/ret (vos/step Dstep) (fos/step (mstep/ret Dstep)) %. +%worlds () (mprog/ret _ _) %. +%total {} (mprog/ret _ _) %. +%sort mprog/let-comp %. +%mode mprog/let-comp %in %in %out %. +%term _ mprog/let-comp (fos/fin (%the (final (return E)) (final/ret Dval))) M2 (%the (mstep (let (comp (return E)) M2) (M2 E)) (mstep/let-beta (final/ret Dval))) %. +%term _ mprog/let-comp (fos/step (%the (mstep M1 M1') Dmstep)) M2 (%the (mstep (let (comp M1) M2) (let (comp M1') M2)) (mstep/let-m Dmstep)) %. +%worlds () (mprog/let-comp _ _ _) %. +%total {} (mprog/let-comp _ _ _) %. +%{! + +### Progress for commands + +The factoring lemma for ``let`` is is mutually recursive with the +main theorem---when the ``let``-bound expression is ``comp +M``, we need to recursively appeal to progress on ``M``. + +!}% +%sort mprog/let {E circle T exp} %. +%mode mprog/let %in %in %in %out %. +%sort mprog {M T command} %. +%mode mprog %in %out %. +%% mprog/let +%term _ + %pi (mprog/let (comp M) (vos/val value/comp) _ Dmstep) + %<- (mprog M Dfos) + %<- (mprog/let-comp Dfos _ Dmstep) %. +%term _ mprog/let _ (vos/step Dstep) _ (mstep/let-e Dstep) %. +%% mprog +%term _ %pi (mprog (return E) Dfos) %<- (prog E Dvos) %<- (mprog/ret Dvos Dfos) %. +%term _ + %pi (mprog (let (%the (circle T exp) E) (%the (%pi (T exp) %-> (U command)) M)) (fos/step Dmstep)) + %<- (prog E (%the (val-or-step E) Dvos)) + %<- (mprog/let E Dvos M (%the (mstep (let E M) M') Dmstep)) %. +%worlds () (mprog _ _) (mprog/let _ _ _ _) %. +%total (M E) (mprog M _) (mprog/let E _ _ _) %. +%{! +Because ``mprog`` and ``mprog/let`` are mutually recursive, we: +- Declare and give modes for both type families before declaring their constants. +- Declare their worlds simulataneously, by putting two call-patterns in the world declaration. +- Prove them total simultaneously, with the mutual termination order ``(M E)``. + +And thus we have proved type safety for MinML with monadic effects! + +\{\{Summer school 2008 + |prev=Exercises 3 + |prevname=Exercies 3 + |nonext= +\}\} + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008:Type_safety_for_MinML_with_monadic_effects). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008-type-safety-for-polymorphic-minml-intrinsic-encoding.lf b/new-tests/stelf-output/pages/summer-school-2008-type-safety-for-polymorphic-minml-intrinsic-encoding.lf new file mode 100644 index 0000000..589b5f3 --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008-type-safety-for-polymorphic-minml-intrinsic-encoding.lf @@ -0,0 +1,120 @@ +% +%{! +MinML: call-by-value, with recursive functions, using an intrinsic encoding +!}% +%{! +## Types +!}% +%sort tp %. +%name tp %. +%term nat tp %. +%term arr %pi tp %-> tp %-> tp %. +%term all %pi (%pi tp %-> tp) %-> tp %. +%{! +## Syntax / static semantics +!}% +%sort exp %. +%name exp %. +%term z exp nat %. +%term s %pi (exp nat) %-> (exp nat) %. +%term ifz %pi (exp nat) %-> (exp T) %-> (%pi (exp nat) %-> (exp T)) %-> (exp T) %. +%term fun {T1} {T2} %pi (%pi (exp (arr T1 T2)) %-> (exp T1) %-> (exp T2)) %-> (exp (arr T1 T2)) %. +%term app %pi (exp (arr T2 T)) %-> (exp T2) %-> (exp T) %. +%term tfun %pi ({a tp} exp (T a)) %-> (exp (all ([a] T a))) %. +%term tapp %pi (exp (all ([a] T a))) %-> ({T2 tp} exp (T T2)) %. +%{! +## Dynamic Semantics / Preservation +!}% +%sort value %. +%name value %. +%mode value %in %. +%term value/z value z %. +%term value/s %pi (value (s E)) %<- (value E) %. +%term value/fun value (fun _ _ _) %. +%term value/tfun value (tfun _) %. +%sort step %. +%name step %. +%mode step %in %out %. +%term step/s %pi (step (s E) (s E')) %<- (step E E') %. +%term step/ifz/arg %pi (step (ifz E E1 ([x] E2 x)) (ifz E' E1 ([x] E2 x))) %<- (step E E') %. +%term step/ifz/z step (ifz z E1 ([x] E2 x)) E1 %. +%term step/ifz/s %pi (step (ifz (s E) E1 ([x] E2 x)) (E2 E)) %<- (value E) %. +%term step/app/fun %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app/arg %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/app/beta-v + %pi (step (app (fun T1 T2 ([f] [x] E f x)) E2) (E (fun T1 T2 ([f] [x] E f x)) E2)) + %<- (value E2) %. +%term step/tapp/tfun %pi (step (tapp E T) (tapp E' T)) %<- (step E E') %. +%term step/tapp/beta step (tapp (tfun ([a] E a)) T) (E T) %. +%{! +## Progress +!}% +%sort val-or-step %. +%name val-or-step %. +%term vos/val %pi (val-or-step E) %<- (value E) %. +%term vos/step %pi (val-or-step E) %<- (step E _) %. +%{! +### Factoring lemmas + +These are necessary for case-analyzing the results of recursive calls. + +!}% +%sort prog/s %. +%mode prog/s %in %out %. +%term _ prog/s (vos/step Dstep) (vos/step (step/s Dstep)) %. +%term _ prog/s (vos/val Dval) (vos/val (value/s Dval)) %. +%worlds () (prog/s _ _) %. +%total {} (prog/s _ _) %. +%sort prog/ifz %. +%mode prog/ifz %in %in %in %out %. +%term _ prog/ifz (vos/step Dstep) _ _ (step/ifz/arg Dstep) %. +%term _ prog/ifz (vos/val value/z) _ _ step/ifz/z %. +%term _ prog/ifz (vos/val (value/s Dval)) _ _ (step/ifz/s Dval) %. +%worlds () (prog/ifz _ _ _ _) %. +%total {} (prog/ifz _ _ _ _) %. +%sort prog/app %. +%mode prog/app %in %in %out %. +%term _ prog/app (vos/step Dstep1) _ (step/app/fun Dstep1) %. +%term _ prog/app (vos/val Dval1) (vos/step Dstep2) (step/app/arg Dstep2 Dval1) %. +%term _ prog/app (vos/val Dval1) (vos/val Dval2) (step/app/beta-v Dval2) %. +%worlds () (prog/app _ _ _) %. +%total {} (prog/app _ _ _) %. +%sort prog/tapp %. +%mode prog/tapp %in %in %out %out %. +%term _ prog/tapp (vos/step Dstep1) _ _ (step/tapp/tfun Dstep1) %. +%term _ prog/tapp (vos/val Dval) _ _ step/tapp/beta %. +%worlds () (prog/tapp _ _ _ _) %. +%total {} (prog/tapp _ _ _ _) %. +%{! +## Main theorem +!}% +%sort prog {E exp T} %. +%mode prog %in %out %. +%term _ prog z (vos/val value/z) %. +%term _ %pi (prog (s E') Dvos') %<- (prog E' Dvos) %<- (prog/s Dvos Dvos') %. +%term _ + %pi (prog (ifz E E0 ([x] E1 x)) (vos/step Dstep)) + %<- (prog E Dvos) + %<- (prog/ifz Dvos _ _ Dstep) %. +%term _ prog (fun _ _ _) (vos/val value/fun) %. +%term _ + %pi (prog (app E1 E2) (vos/step Dstep)) + %<- (prog E1 Dvos1) + %<- (prog E2 Dvos2) + %<- (prog/app Dvos1 Dvos2 Dstep) %. +%term _ prog (tfun _) (vos/val value/tfun) %. +%term _ + %pi (prog (tapp E T) (vos/step Dstep)) + %<- (prog E Dvos) + %<- (prog/tapp Dvos _ _ Dstep) %. +%worlds () (prog _ _) %. +%total Dof (prog Dof _) %. +%{! +And thus we have proved type safety for MinML with polymorphism! +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008:Type_safety_for_polymorphic_MinML_(intrinsic_encoding)). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008-typed-arithmetic-expressions-extrinsic-encoding.lf b/new-tests/stelf-output/pages/summer-school-2008-typed-arithmetic-expressions-extrinsic-encoding.lf new file mode 100644 index 0000000..b467155 --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008-typed-arithmetic-expressions-extrinsic-encoding.lf @@ -0,0 +1,146 @@ +% +%{! + +\{\{Summer school 2008 + |prev=Typed arithmetic expressions + |prevname=Typed arithmetic expressions + |next=Exercises 1 + |nextname=Exercises 1 + \}\} + +Here we take a more traditional approach to representing syntax, by +first giving a definition of raw syntax, with a separate judgement +defining the well-typed expressions. This is called an [extrinsic encoding](/wiki/intrinsic-and-extrinsic-encodings/). + +Numbers and strings are as before. + +|hidden="true" !}% +%sort nat %. +%name nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort add %. +%mode add %in %in %out %. +%term add/z add z N N %. +%term add/s %pi (add (s M) N (s P)) %<- (add M N P) %. +%% addition is a total function on closed terms of type nat +%worlds () (add _ _ _) %. +%total M (add M _ _) %. +%sort char %. +%name char %. +%term a char %. +%term b char %. +%sort str %. +%name str %. +%term emp str %. +%term cons %pi char %-> str %-> str %. +%sort cat %. +%mode cat %in %in %out %. +%term cat/e cat emp S S %. +%term cat/c %pi (cat (cons X S1) S2 (cons X S3)) %<- (cat S1 S2 S3) %. +%worlds () (cat _ _ _) %. +%total S (cat S _ _) %. +%{! + +## Raw syntax + +!}% +%sort tp %. +%name tp %. +%term number tp %. +%term string tp %. +%sort val %. +%name val %. +%term num %pi nat %-> val %. +%term lit %pi str %-> val %. +%sort exp %. +%name exp %. +%term ret %pi val %-> exp %. +%term plus %pi exp %-> exp %-> exp %. +%term append %pi exp %-> exp %-> exp %. +%term let %pi exp %-> (%pi val %-> exp) %-> exp %. +%{! + +## Typing judgement on raw syntax + +!}% +%sort ofv %. +%name ofv %. +%mode ofv %in %out %. +%term _ ofv (num _) number %. +%term _ ofv (lit _) string %. +%sort of %. +%name of %. +%mode of %in %out %. +%term _ %pi (of (ret V) T) %<- (ofv V T) %. +%term _ %pi (of (plus E1 E2) number) %<- (of E1 number) %<- (of E2 number) %. +%term _ %pi (of (append E1 E2) string) %<- (of E1 string) %<- (of E2 string) %. +%term _ + %pi (of (let E1 ([x] E2 x)) T') + %<- (of E1 T) + %<- ({x val} %pi (ofv x T) %-> (of (E2 x) T')) %. +%{! + +## Evaluation + +### Answers + +!}% +%sort ans %. +%name ans %. +%term anum %pi nat %-> ans %. +%term astr %pi str %-> ans %. +%sort ofa %. +%name ofa %. +%mode ofa %in %out %. +%term _ ofa (anum _) number %. +%term _ ofa (astr _) string %. +%{! + +### Evaluation judgement on raw syntax + +!}% +%sort eval %. +%mode eval %in %out %. +%term eval/val/num eval (ret (num N)) (anum N) %. +%term eval/val/lit eval (ret (lit S)) (astr S) %. +%term eval/plus + %pi (eval (plus E1 E2) (anum N)) + %<- (eval E1 (anum N1)) + %<- (eval E2 (anum N2)) + %<- (add N1 N2 N) %. +%term eval/append + %pi (eval (append E1 E2) (astr S)) + %<- (eval E1 (astr S1)) + %<- (eval E2 (astr S2)) + %<- (cat S1 S2 S) %. +%term eval/let %pi (eval (let E1 ([x] E2 x)) A) %<- (eval E1 (anum N)) %<- (eval (E2 (num N)) A) %. +%{! + +Evaluation is not total because of run-time type errors! (e.g., try +appending two numbers). + +!}% +%worlds () (eval _ _) %. +%{! + +Twelf gives this error, refering to the first recursive call of ``eval/plus``. + +<twelf check="decl"> +%total E (eval E _). +</twelf> + +\{\{Summer school 2008 + |prev=Typed arithmetic expressions + |prevname=Typed arithmetic expressions + |next=Exercises 1 + |nextname=Exercises 1 + \}\} + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008:Typed_arithmetic_expressions_(extrinsic_encoding)). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008-typed-arithmetic-expressions-value.lf b/new-tests/stelf-output/pages/summer-school-2008-typed-arithmetic-expressions-value.lf new file mode 100644 index 0000000..957f1cc --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008-typed-arithmetic-expressions-value.lf @@ -0,0 +1,93 @@ +% +%{! + +\{\{Summer school 2008 + |prev=Typed arithmetic expressions + |prevname=Typed arithmetic expressions + |nonext= +\}\} + +Arithmetic expressions, where the answer type is values. + +!}% +%sort nat %. +%name nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort add %. +%mode add %in %in %out %. +%term add/z add z N N %. +%term add/s %pi (add (s M) N (s P)) %<- (add M N P) %. +%worlds () (add _ _ _) %. +%total M (add M _ _) %. +%sort char %. +%name char %. +%term a char %. +%term b char %. +%sort str %. +%name str %. +%term emp str %. +%term cons %pi char %-> str %-> str %. +%sort cat %. +%mode cat %in %in %out %. +%term cat/e cat emp S S %. +%term cat/c %pi (cat (cons X S1) S2 (cons X S3)) %<- (cat S1 S2 S3) %. +%worlds () (cat _ _ _) %. +%total S (cat S _ _) %. +%{! + +## Typed expressions + +!}% +%sort tp %. +%name tp %. +%term number tp %. +%term string tp %. +%sort val %. +%name val %. +%prec %postfix 1 val %. +%term num %pi nat %-> (number val) %. +%term lit %pi str %-> (string val) %. +%sort exp %. +%name exp %. +%prec %postfix 1 exp %. +%term ret %pi (T val) %-> (T exp) %. +%term plus %pi (number exp) %-> (number exp) %-> (number exp) %. +%term append %pi (string exp) %-> (string exp) %-> (string exp) %. +%term let %pi (T exp) %-> (%pi (T val) %-> (U exp)) %-> (U exp) %. +%{! + +For simplicity, take answers to be values. + +!}% +%sort eval %. +%mode eval %in %out %. +%term eval/val eval (ret V) V %. +%term eval/plus + %pi (eval (plus E1 E2) (num N)) + %<- (eval E1 (num N1)) + %<- (eval E2 (num N2)) + %<- (add N1 N2 N) %. +%term eval/append + %pi (eval (append E1 E2) (lit S)) + %<- (eval E1 (lit S1)) + %<- (eval E2 (lit S2)) + %<- (cat S1 S2 S) %. +%term eval/let %pi (eval (let E1 ([x] E2 x)) A) %<- (eval E1 V) %<- (eval (E2 V) A) %. +%worlds () (eval _ _) %. +%total E (eval E _) %. +%{! + +\{\{Summer school 2008 + |prev=Typed arithmetic expressions + |prevname=Typed arithmetic expressions + |nonext= +\}\} + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008:Typed_arithmetic_expressions_(value)). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008-typed-arithmetic-expressions-with-pairs.lf b/new-tests/stelf-output/pages/summer-school-2008-typed-arithmetic-expressions-with-pairs.lf new file mode 100644 index 0000000..fbaaf67 --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008-typed-arithmetic-expressions-with-pairs.lf @@ -0,0 +1,110 @@ +% +%{! + +\{\{Summer school 2008 + |prev=Exercises 1 + |prevname=Exercises 1 + |nonext= +\}\} + +Arithmetic expressions with pairs. + +Numbers and strings are unchanged. + +|hidden="true" !}% +%sort nat %. +%name nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort add %. +%mode add %in %in %out %. +%term add/z add z N N %. +%term add/s %pi (add (s M) N (s P)) %<- (add M N P) %. +%worlds () (add _ _ _) %. +%total M (add M _ _) %. +%sort char %. +%name char %. +%term a char %. +%term b char %. +%sort str %. +%name str %. +%term emp str %. +%term cons %pi char %-> str %-> str %. +%sort cat %. +%mode cat %in %in %out %. +%term cat/e cat emp S S %. +%term cat/c %pi (cat (cons X S1) S2 (cons X S3)) %<- (cat S1 S2 S3) %. +%worlds () (cat _ _ _) %. +%total S (cat S _ _) %. +%{! + +## Typed expressions + +Add a type for products: + +!}% +%sort tp %. +%name tp %. +%term number tp %. +%term string tp %. +%term prod %pi tp %-> tp %-> tp %. +%{! + +Add pairs as a value, projections as expressions: + +!}% +%sort val %. +%name val %. +%prec %postfix 1 val %. +%term num %pi nat %-> (number val) %. +%term lit %pi str %-> (string val) %. +%term pair %pi (T val) %-> (U val) %-> (prod T U val) %. +%sort exp %. +%name exp %. +%prec %postfix 1 exp %. +%term ret %pi (T val) %-> (T exp) %. +%term plus %pi (number exp) %-> (number exp) %-> (number exp) %. +%term append %pi (string exp) %-> (string exp) %-> (string exp) %. +%term let %pi (T exp) %-> (%pi (T val) %-> (U exp)) %-> (U exp) %. +%term fst %pi (prod T U exp) %-> (T exp) %. +%term snd %pi (prod T U exp) %-> (U exp) %. +%{! + +For simplicity, take answers to be values. + +And we add cases to eval: + +!}% +%sort eval %. +%mode eval %in %out %. +%term eval/val eval (ret V) V %. +%term eval/plus + %pi (eval (plus E1 E2) (num N)) + %<- (eval E1 (num N1)) + %<- (eval E2 (num N2)) + %<- (add N1 N2 N) %. +%term eval/append + %pi (eval (append E1 E2) (lit S)) + %<- (eval E1 (lit S1)) + %<- (eval E2 (lit S2)) + %<- (cat S1 S2 S) %. +%term eval/let %pi (eval (let E1 ([x] E2 x)) A) %<- (eval E1 V) %<- (eval (E2 V) A) %. +%term eval/fst %pi (eval (fst E) A) %<- (eval E (pair A _)) %. +%term eval/snd %pi (eval (snd E) B) %<- (eval E (pair _ B)) %. +%worlds () (eval _ _) %. +%total E (eval E _) %. +%{! + +\{\{Summer school 2008 + |prev=Exercises 1 + |prevname=Exercises 1 + |nonext= +\}\} + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008:Typed_arithmetic_expressions_with_pairs). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008-typed-arithmetic-expressions-with-sums-2.lf b/new-tests/stelf-output/pages/summer-school-2008-typed-arithmetic-expressions-with-sums-2.lf new file mode 100644 index 0000000..6b0ce87 --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008-typed-arithmetic-expressions-with-sums-2.lf @@ -0,0 +1,128 @@ +% +%{! + +\{\{Summer school 2008 + |prev=Typed arithmetic expressions + |prevname=Typed arithmetic expressions + |nonext= +\}\} + +Arithmetic expressions with pairs and sums. (This solution involves [output factoring](/wiki/output-factoring/), but no [identity types](/wiki/equality/). Dan posted a [solution involving identity types](/wiki/summer-school-2008-typed-arithmetic-expressions-with-sums/), and Chris posted a [solution that avoids output factoring](/wiki/summer-school-2008-alternate-typed-arithmetic-expressions-with-sums/).) + +Numbers and strings are unchanged. + +|hidden="true" !}% +%sort nat %. +%name nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort add %. +%mode add %in %in %out %. +%term add/z add z N N %. +%term add/s %pi (add (s M) N (s P)) %<- (add M N P) %. +%worlds () (add _ _ _) %. +%total M (add M _ _) %. +%sort char %. +%name char %. +%term a char %. +%term b char %. +%sort str %. +%name str %. +%term emp str %. +%term cons %pi char %-> str %-> str %. +%sort cat %. +%mode cat %in %in %out %. +%term cat/e cat emp S S %. +%term cat/c %pi (cat (cons X S1) S2 (cons X S3)) %<- (cat S1 S2 S3) %. +%worlds () (cat _ _ _) %. +%total S (cat S _ _) %. +%{! + +## Typed expressions + +Add a type for disjoint sums: + +!}% +%sort tp %. +%name tp %. +%term number tp %. +%term string tp %. +%term prod %pi tp %-> tp %-> tp %. +%term sum %pi tp %-> tp %-> tp %. +%{! + +Add injections as values, and case as an expression. + +!}% +%sort val %. +%name val %. +%prec %postfix 1 val %. +%term num %pi nat %-> (number val) %. +%term lit %pi str %-> (string val) %. +%term pair %pi (T val) %-> (U val) %-> (prod T U val) %. +%term inl %pi (T val) %-> (sum T U val) %. +%term inr %pi (U val) %-> (sum T U val) %. +%sort exp %. +%name exp %. +%prec %postfix 1 exp %. +%term ret %pi (T val) %-> (T exp) %. +%term plus %pi (number exp) %-> (number exp) %-> (number exp) %. +%term append %pi (string exp) %-> (string exp) %-> (string exp) %. +%term let %pi (T exp) %-> (%pi (T val) %-> (U exp)) %-> (U exp) %. +%term fst %pi (prod T U exp) %-> (T exp) %. +%term snd %pi (prod T U exp) %-> (U exp) %. +%{! +For technical reasons, we bundle the branches of the case into a separate term with type ``cases``; this makes the termination order more evident to Twelf. An element of "cases T U V" is the branches for a "sum T U" whose result has tp "V". +!}% +%sort cases %. +%term cases/i %pi (%pi (T val) %-> (V exp)) %-> (%pi (U val) %-> (V exp)) %-> (cases T U V) %. +%term case %pi (sum T U exp) %-> (cases T U V) %-> (V exp) %. +%{! + +Add eval cases for case. + +!}% +%sort eval %. +%mode eval %in %out %. +%term eval/val eval (ret V) V %. +%term eval/plus + %pi (eval (plus E1 E2) (num N)) + %<- (eval E1 (num N1)) + %<- (eval E2 (num N2)) + %<- (add N1 N2 N) %. +%term eval/append + %pi (eval (append E1 E2) (lit S)) + %<- (eval E1 (lit S1)) + %<- (eval E2 (lit S2)) + %<- (cat S1 S2 S) %. +%term eval/let %pi (eval (let E1 ([x] E2 x)) A) %<- (eval E1 V) %<- (eval (E2 V) A) %. +%term eval/fst %pi (eval (fst E) A) %<- (eval E (pair A _)) %. +%term eval/snd %pi (eval (snd E) B) %<- (eval E (pair _ B)) %. +%{! +For "case" evaluation, we need an auxiliary relation "evalcase" to branch on inl vs. inr ([output factoring](/wiki/output-factoring/)). +!}% +%sort evalcase %. +%mode evalcase %in %in %out %. +%term eval/case %pi (eval (case E Cases) A) %<- (eval E V) %<- (evalcase V Cases A) %. +%term evalcase/inl %pi (evalcase (inl V1) (cases/i ([x] F1 x) ([y] F2 y)) A) %<- (eval (F1 V1) A) %. +%term evalcase/inr %pi (evalcase (inr V2) (cases/i ([x] F1 x) ([y] F2 y)) A) %<- (eval (F2 V2) A) %. +%{! +Then we use [mutual induction](/wiki/mutual-induction/) to show evaluation total. +!}% +%worlds () (eval _ _) (evalcase _ _ _) %. +%total (E Cases) (eval E _) (evalcase _ Cases _) %. +%{! + +\{\{Summer school 2008 + |prev=Exercises 1 + |prevname=Exercises 1 + |nonext= +\}\} + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008:Typed_arithmetic_expressions_with_sums_2). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008-typed-arithmetic-expressions-with-sums.lf b/new-tests/stelf-output/pages/summer-school-2008-typed-arithmetic-expressions-with-sums.lf new file mode 100644 index 0000000..bf5fcfe --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008-typed-arithmetic-expressions-with-sums.lf @@ -0,0 +1,207 @@ +% +%{! + +\{\{Summer school 2008 + |prev=Typed arithmetic expressions + |prevname=Typed arithmetic expressions + |nonext= +\}\} + +Arithmetic expressions with pairs and sums, where the answer type is values. + +For an alternate version of sums that doesn't involve output factoring, see [Chris's solution](/wiki/summer-school-2008-alternate-typed-arithmetic-expressions-with-sums/). For a version that uses output factoring but avoids identity types, see [William's solution](/wiki/summer-school-2008-typed-arithmetic-expressions-with-sums-2/). + +!}% +%sort nat %. +%name nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort add %. +%mode add %in %in %out %. +%term add/z add z N N %. +%term add/s %pi (add (s M) N (s P)) %<- (add M N P) %. +%worlds () (add _ _ _) %. +%total M (add M _ _) %. +%sort char %. +%name char %. +%term a char %. +%term b char %. +%sort str %. +%name str %. +%term emp str %. +%term cons %pi char %-> str %-> str %. +%sort cat %. +%mode cat %in %in %out %. +%term cat/e cat emp S S %. +%term cat/c %pi (cat (cons X S1) S2 (cons X S3)) %<- (cat S1 S2 S3) %. +%worlds () (cat _ _ _) %. +%total S (cat S _ _) %. +%{! + +## Typed expressions + +!}% +%sort tp %. +%name tp %. +%term number tp %. +%term string tp %. +%term prod %pi tp %-> tp %-> tp %. +%term sum %pi tp %-> tp %-> tp %. +%sort val %. +%name val %. +%prec %postfix 1 val %. +%term num %pi nat %-> (number val) %. +%term lit %pi str %-> (string val) %. +%term pair %pi (U val) %-> (T val) %-> (prod U T val) %. +%term inl %pi (U val) %-> (sum U T val) %. +%term inr %pi (T val) %-> (sum U T val) %. +%sort exp %. +%name exp %. +%prec %postfix 1 exp %. +%term ret %pi (T val) %-> (T exp) %. +%term plus %pi (number exp) %-> (number exp) %-> (number exp) %. +%term append %pi (string exp) %-> (string exp) %-> (string exp) %. +%term let %pi (T exp) %-> (%pi (T val) %-> (U exp)) %-> (U exp) %. +%term fst %pi (prod U T exp) %-> (U exp) %. +%% (fst e) +%term snd %pi (prod U T exp) %-> (T exp) %. +%% (snd e) +%term case + %pi (sum T U exp) + %-> (%pi (T val) %-> (V exp)) + %-> (%pi (U val) %-> (V exp)) + %-> (V exp) %. +%{! + +## Evaluation + +For simplicity, take answers to be values. + +!}% +%sort eval %. +%mode eval %in %out %. +%% eval +%term eval/val eval (ret V) V %. +%term eval/plus + %pi (eval (plus E1 E2) (num N)) + %<- (eval E1 (num N1)) + %<- (eval E2 (num N2)) + %<- (add N1 N2 N) %. +%term eval/append + %pi (eval (append E1 E2) (lit S)) + %<- (eval E1 (lit S1)) + %<- (eval E2 (lit S2)) + %<- (cat S1 S2 S) %. +%term eval/let %pi (eval (let E1 ([x] E2 x)) A) %<- (eval E1 V) %<- (eval (E2 V) A) %. +%term eval/fst %pi (eval (fst E) A1) %<- (eval E (pair A1 A2)) %. +%term eval/snd %pi (eval (snd (%the (prod T1 T2 exp) E)) A2) %<- (eval E (pair A1 A2)) %. +%term eval/case/inl %pi (eval (case E E1 E2) A') %<- (eval E (inl A1)) %<- (eval (E1 A1) A') %. +%term eval/case/inr %pi (eval (case E E1 E2) A') %<- (eval E (inr A2)) %<- (eval (E2 A2) A') %. +%worlds () (eval _ _) %. +%{! + +We can write the obvious rules for evaluation, but Twelf can't prove +this definition total. The reason is that the [output coverage](/wiki/coverage-checking/) +checker looks at each constant in isolation, not at the whole set of +constants. In this case, it flags an error on the case +``eval/case/inl``: + +<twelf check="decl"> +%total E (eval E _). +</twelf> + +because it doesn't notice that the next constant, +``eval/case/inr``, covers the other possible output of evaluating +``E``. We can fix this using a technique called [output factoring](/wiki/output-factoring/). + +## Output factored evaluation + +!}% +%sort id %. +%term refl id E E %. +%sort eval'-case {E' V exp} {E sum T U exp} {E1} {E2} %. +%mode eval'-case %in %in %in %in %in %in %out %. +%sort eval' %. +%mode eval' %in %out %. +%% eval'-case +%term eval'-case/inl %pi (eval'-case _ E E1 E2 refl (inl A1) A) %<- (eval' (E1 A1) A) %. +%term eval'-case/inr %pi (eval'-case _ E E1 E2 refl (inr A2) A) %<- (eval' (E2 A2) A) %. +%% eval' +%term eval'/val eval' (ret V) V %. +%term eval'/plus + %pi (eval' (plus E1 E2) (num N)) + %<- (eval' E1 (num N1)) + %<- (eval' E2 (num N2)) + %<- (add N1 N2 N) %. +%term eval'/append + %pi (eval' (append E1 E2) (lit S)) + %<- (eval' E1 (lit S1)) + %<- (eval' E2 (lit S2)) + %<- (cat S1 S2 S) %. +%term eval'/let %pi (eval' (let E1 ([x] E2 x)) A) %<- (eval' E1 V) %<- (eval' (E2 V) A) %. +%term eval'/fst %pi (eval' (fst E) A1) %<- (eval' E (pair A1 A2)) %. +%term eval'/snd %pi (eval' (snd (%the (prod T1 T2 exp) E)) A2) %<- (eval' E (pair A1 A2)) %. +%term eval'/case + %pi (eval' (case E E1 E2) A') + %<- (eval' E A) + %<- (eval'-case (case E E1 E2) E E1 E2 refl A A') %. +%worlds () (eval' _ _) (eval'-case _ _ _ _ _ _ _) %. +%total (E E') (eval'-case E' _ _ _ _ _ _) (eval' E _) %. +%{! + +The constant ``eval'/case`` uses [output factoring](/wiki/output-factoring/): the +subsidiary case-analysis of the result of ``eval' E A`` is broken +out into a separate lemma, ``eval'-case``. This solves the above +problem, because while output coverage cannot be split across multiple +constants, input coverage certainly can. However, because +``eval'-case`` recursively calls ``eval'``, the two relations +must be proved total using [mutual induction](/wiki/mutual-induction/). This is why we +- write both type families and modes before writing any constants +- declare their worlds and modes at once +- use the mutual termination metric ``(E E')`` + +However, there is an additional subtlety that we need to address. You +might think ``eval'-case`` could be defined as follows: + +<twelf ignore="true"> +eval'-case : \{E : (sum T U) exp\} \{E1 : T val -> V exp\} \{E2 : U val -> V exp\} + (sum T U) val + -> A val + -> type. +%mode eval'-case +X0 +X1 +X2 +X3 -X4. + +eval'-case/inl : eval'-case E E1 E2 (inl A1) A + <- eval' (E1 A1) A. + +eval'-case/inr : eval'-case E E1 E2 (inr A2) A + <- eval' (E2 A2) A. +</twelf> + +The problem is that Twelf can't justify the recursive calls to +``eval'`` on ``E1`` and ``E2``. In fact, why does this +recursion pattern work at all? The reason is that we don't call +``eval'-case`` on just any ``E1`` and ``E2``, but on +subexpressions of the ``E`` that ``eval'`` was originally +called with. Thus, when we call back to ``eval'`` from +``eval'-case``, it's on a smaller expression. + +We make this invariant evident to Twelf as follows: +- when ``eval'`` calls ``eval'-case``, we pass in the original expression ``(case E E1 E2)`` that ``eval'`` was called with +- this argument is used as termination metric for ``eval-case'`` +- we constrain this position to be a case-analysis using an [identity type](/wiki/equality/), so that we don't have extraneous cases to consider. + +This explains the somewhat-convoluted code above. + +\{\{Summer school 2008 + |prev=Typed arithmetic expressions + |prevname=Typed arithmetic expressions + |nonext= +\}\} + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008:Typed_arithmetic_expressions_with_sums). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008-typed-arithmetic-expressions.lf b/new-tests/stelf-output/pages/summer-school-2008-typed-arithmetic-expressions.lf new file mode 100644 index 0000000..5857878 --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008-typed-arithmetic-expressions.lf @@ -0,0 +1,135 @@ +% +%{! + +\{\{Summer school 2008 + |prev=Arithmetic expressions with let-binding (hypothetical evaluation) + |prevname=Arithmetic expressions with let-binding (hypothetical evaluation) + |next=Typed arithmetic expressions (extrinsic encoding) + |nextname=Typed arithmetic expressions (extrinsic encoding) +\}\} + +Next, we add another base type besides numbers. + +## Numbers and addition + +Natural numbers and addition are the same as before: + +!}% +%sort nat %. +%name nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort add %. +%mode add %in %in %out %. +%term add/z add z N N %. +%term add/s %pi (add (s M) N (s P)) %<- (add M N P) %. +%worlds () (add _ _ _) %. +%total M (add M _ _) %. +%{! + +## Strings and append + +Strings and append are analogous + +!}% +%sort char %. +%name char %. +%term a char %. +%term b char %. +%sort str %. +%name str %. +%term emp str %. +%term cons %pi char %-> str %-> str %. +%sort cat %. +%mode cat %in %in %out %. +%term cat/e cat emp S S %. +%term cat/c %pi (cat (cons X S1) S2 (cons X S3)) %<- (cat S1 S2 S3) %. +%worlds () (cat _ _ _) %. +%total S (cat S _ _) %. +%{! + +## Typed expressions + +There are two types: + +!}% +%sort tp %. +%name tp %. +%term number tp %. +%term string tp %. +%{! + +We use an [intrinsic encoding](/wiki/intrinsic-and-extrinsic-encodings/), where only well-typed terms are +represented. Another way of looking at this is that we skip raw syntax +and work directly with typing derivations. This works well for +simply-typed languages, where the raw syntax for well-typed terms is +isomorphic to its typing derivations. + +!}% +%sort val %. +%name val %. +%prec %postfix 1 val %. +%term num %pi nat %-> (number val) %. +%term lit %pi str %-> (string val) %. +%sort exp %. +%name exp %. +%prec %postfix 1 exp %. +%term ret %pi (T val) %-> (T exp) %. +%term plus %pi (number exp) %-> (number exp) %-> (number exp) %. +%term append %pi (string exp) %-> (string exp) %-> (string exp) %. +%term let %pi (T exp) %-> (%pi (T val) %-> (U exp)) %-> (U exp) %. +%{! + +Answers are typed as well: + +!}% +%sort ans %. +%name ans %. +%prec %postfix 1 ans %. +%term anum %pi nat %-> (number ans) %. +%term astr %pi str %-> (string ans) %. +%{! + +Evaluation relates an expression to an answer of the same type, +guaranteeing type preservation. + +Because of Twelf's implicit argument mechanism, the cases for numbers +and let-binding are unchanged. + +!}% +%sort eval %. +%mode eval %in %out %. +%term eval/val/num eval (ret (num N)) (anum N) %. +%term eval/val/str eval (ret (lit S)) (astr S) %. +%term eval/plus + %pi (eval (plus E1 E2) (anum N)) + %<- (eval E1 (anum N1)) + %<- (eval E2 (anum N2)) + %<- (add N1 N2 N) %. +%term eval/append + %pi (eval (append E1 E2) (astr S)) + %<- (eval E1 (astr S1)) + %<- (eval E2 (astr S2)) + %<- (cat S1 S2 S) %. +%term eval/let/num %pi (eval (let E1 ([x] E2 x)) A) %<- (eval E1 (anum N)) %<- (eval (E2 (num N)) A) %. +%term eval/let/str %pi (eval (let E1 ([x] E2 x)) A) %<- (eval E1 (astr S)) %<- (eval (E2 (lit S)) A) %. +%worlds () (eval _ _) %. +%total E (eval E _) %. +%{! + +Note: in this example ``ans`` is isomorphic to ``val``, so we can simplify things slightly by not making the distinction. (See [Typed arithmetic expressions (value)](/wiki/summer-school-2008-typed-arithmetic-expressions-value/)). + +\{\{Summer school 2008 + |prev=Arithmetic expressions with let-binding (hypothetical evaluation) + |prevname=Arithmetic expressions with let-binding (hypothetical evaluation) + |next=Typed arithmetic expressions (extrinsic encoding) + |nextname=Typed arithmetic expressions (extrinsic encoding) +\}\} + +!}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008:Typed_arithmetic_expressions). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/summer-school-2008.lf b/new-tests/stelf-output/pages/summer-school-2008.lf new file mode 100644 index 0000000..8d3e176 --- /dev/null +++ b/new-tests/stelf-output/pages/summer-school-2008.lf @@ -0,0 +1,56 @@ +% +%{! __NOTOC__ +This page contains course materials for the Twelf course at [http://www.cs.uoregon.edu/research/summerschool/summer08/ the University of Oregon Summer School on Logic and Theorem Proving in Programming Languages, July 2008]. In this course, you will learn to represent [deductive system](/wiki/object-logic/)s in [LF](/wiki/lf/) and prove [metatheorem](/wiki/metatheorem/)s about them with [Twelf](/wiki/about-the-twelf-project/). + +## New: Feedback Form + +Please fill out our [http://gs4080.sp.cs.cmu.edu/servlets/summerschool2008/feedback.ss feedback form]! + +## Get Twelf + +Pre-built binaries of Twelf are available for most operating systems through the [http://twelf.plparty.org/builds Twelf Night(ly)]. +* [http://twelf.plparty.org/builds/twelf-linux.tar.gz Linux (tgz)] +* [http://twelf.plparty.org/builds/twelf.exe Windows (exe)] +* [http://twelf.plparty.org/builds/twelf-osx-intel.dmg Mac OS 10.4 or 10.5 (Intel only) (dmg)] + +Otherwise: +* you can build Twelf from the [http://twelf.plparty.org/builds/twelf-src.tar.gz source tarball]. You will need [http://www.mlton.org MLton] or [http://www.smlnj.org sml/nj]. +* you can make yourself an account on the wiki, and do the exercises on your User:<login> page (linked at the top after you log in). + +Then see [Twelf with Emacs](/wiki/twelf-with-emacs/) for the basics of interacting with Twelf. (You can also use [Twelf without Emacs](/wiki/twelf-without-emacs/), by interacting with the Twelf server directly.) + +## Lectures and Labs [http://www.cs.cmu.edu/~rwh/talks/UO_Summer08.pdf (slides)] + +### Class 1: Overview +* [Arithmetic expressions](/wiki/summer-school-2008-arithmetic-expressions/) +* [Arithmetic expressions with let-binding](/wiki/summer-school-2008-arithmetic-expressions-with-let-binding/) +** Variation: [Call-by-value let-binding syntax](/wiki/summer-school-2008-arithmetic-expressions-with-call-by-value-let-binding/) +** Variation: [Defining evaluation with a hypothetical judgement](/wiki/summer-school-2008-arithmetic-expressions-with-let-binding-hypothetical-evaluation/) +* [Typed arithmetic expressions](/wiki/summer-school-2008-typed-arithmetic-expressions/) +** Variation: [Typed arithmetic expressions (extrinsic encoding)](/wiki/summer-school-2008-typed-arithmetic-expressions-extrinsic-encoding/) + +* [Exercises 1](/wiki/summer-school-2008-exercises-1/) + +### Class 2: Representation +* [http://www.cs.cmu.edu/~drl/pubs/hl07mechanizing/hl07mechanizing.pdf Mechanizing Metatheory in a Logical Framework] discusses this material in detail. +* [Exercises 2](/wiki/summer-school-2008-exercises-2/) + +### Class 3: Mechanizing Metatheory + +* [Type safety for MinML (intrinsic encoding)](/wiki/summer-school-2008-type-safety-for-minml-intrinsic-encoding/) +* [Type safety for MinML (extrinsic encoding)](/wiki/summer-school-2008-type-safety-for-minml-extrinsic-encoding/) + +* [Exercises 3](/wiki/summer-school-2008-exercises-3/) + +## Additional reading + +* PFPL: We will use [http://www.cs.cmu.edu/~rwh/plbook/book.pdf Practical Foundations for Programming Languages] as a reference for basic PL concepts. +* MMLF: [http://www.cs.cmu.edu/~drl/pubs/hl07mechanizing/hl07mechanizing.pdf Mechanizing Metatheory in a Logical Framework] discusses LF, representation, and mechanized metatheory in technical detail. +* [Proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/) is a self-contained intro tutorial on this wiki. +* After you get spun up, there are lots of [tutorials](/wiki/tutorials/) and [case studies](/wiki/case-studies/). !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Summer_school_2008). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/syntax-object-logic.lf b/new-tests/stelf-output/pages/syntax-object-logic.lf new file mode 100644 index 0000000..1ad9e06 --- /dev/null +++ b/new-tests/stelf-output/pages/syntax-object-logic.lf @@ -0,0 +1,24 @@ +% +%{! The syntax of an [object logic](/wiki/object-logic/) is commonly encoded as an [LF](/wiki/lf/) [type family](/wiki/type-family/) of [[kind]] ``type``. + +For example, if the language of our object logic is the [natural numbers](/wiki/natural-numbers/), its syntax would be declared in the following way. !}% +% declare a type family of kind "type" for natural numbers. +%sort nat %. +% inhabit the nat with the appropriate constructors. +%term nat/z nat %. +%term nat/s %pi nat %-> nat %. +%{! ## See also + +* Read the [introductions to Twelf](/wiki/introductions-to-twelf/) to learn how to represent syntax in LF. +* [Judgment](/wiki/judgment/) +* [Judgments as types](/wiki/judgment/) +* [Higher-order abstract syntax](/wiki/higher-order-abstract-syntax/) + + +\{\{stub\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Syntax_(Object_logic)). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/tabled-logic-programming.lf b/new-tests/stelf-output/pages/tabled-logic-programming.lf new file mode 100644 index 0000000..1bcdeb6 --- /dev/null +++ b/new-tests/stelf-output/pages/tabled-logic-programming.lf @@ -0,0 +1,103 @@ +% +%{! +The operational semantics of Twelf are similar to those of Prolog, +a style known as _backward chaining_ or _goal-directed_ proof +search. **Tabled logic programming**, which uses Twelf's +[``%tabled``](/wiki/percent-tabled/) and [``%querytabled``](/wiki/percent-querytabled/) directives to +allow Twelf to prove theorems that might be time-consuming or +impossible to prove otherwise. The tabled logic programming +capabilities are a +part of Twelf's capabilities as a logical framework, but not as a +metalogical framework; in other worlds, tabled logic programming +cannot be used to prove [metatheorems](/wiki/metatheorem/). + +[[Image:Searchgraph-found.svg|thumb|right|325px|The graph used in this example, and the path that the tabled logic query finds to get from node a to node g.]] + +The simplest examples of the power of tabled logic programming in +practice involve judgments that involve transitive and/or symmetric +closures, such as searching for a path in a graph or formalizing a +language with subtyping. Using standard backward-chaining proof search, +it is almost impossible to write terminating programs that search for +paths, and using backward-chaining search for subtyping typically +requires a separate definition of "algorithmic subtyping" that must +be shown to be sound and complete with respect to the clearer, simpler +definition of subtyping that uses transitivity. + +This article will use as its example a path-finding algorithm +on an undirected graphs. + +!}% +%{! ## Defining the graph !}% +%{! ### Nodes !}% +%sort node %. +%term a node %. +%term b node %. +%term c node %. +%term d node %. +%term e node %. +%term f node %. +%term g node %. +%term h node %. +%{! ### Edges !}% +%{! While we will consider edges in our graph to be undirected, +we will only define the edge in one, arbitrary, direction. !}% +%sort edge %. +%term ab edge a b %. +%term ac edge a c %. +%term bc edge b c %. +%term bd edge b d %. +%term cd edge c d %. +%term ce edge c e %. +%term de edge d e %. +%term df edge d f %. +%term fg edge f g %. +%{! ### Paths in the graph !}% +%{! +This is normally where we would get into trouble; in standard +logic programming, it is difficult if not impossible to avoid +non-terminating behavior when writing a judgment defining a +transitive-symmetric closure like ``path``. However, by +adding the directive ``[%tabled](/wiki/percent-tabled/) path``, we will be able +to use [``%querytabled``](/wiki/percent-querytabled/) directives to cause each instance +of ``path A B`` to be derived at most once during the course +of a search: the result is a query that terminates rapidly, +instead of not at all. +!}% +%sort path %. +%tabled path %. +%term path/link %pi (edge A B) %-> (path A B) %. +%term path/refl %pi (path A B) %-> (path B A) %. +%term path/trans %pi (path A B) %-> (path B C) %-> (path A C) %. +%{! ## Searching the graph !}% +%{! +It is crucial that we use [``%querytabled``](/wiki/percent-querytabled/) rather than +[``%query``](/wiki/percent-query/) in our queries if we wish for them to terminate. +Even though we ask Twelf for as many solutions as it can find, the +tabled proof search ensures that we can only find one proof of any +given path, and the path that is found is by not necessarily the shortest. +While the path that is found is shown on the graph above, the edge ``ab`` +actually appears three times here - if we use parenthesis to show the order in which +tabled search connected paths, the path that is found is +``a`` - ((``b`` - ((``a`` - ``b`` - ``d``) - ``f``)) +- ``g``). +|check=decl discard=true !}% +%querytabled _ _ %the (path a g) D %. +%{! +We can also run a terminating search for a path that does not exist, +such as one from ``a`` to ``h``. +|check=decl discard=true !}% +%querytabled _ _ %the (path a h) D %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Tabled_logic_programming). +!}% +%. + +== See also == + +* [%tabled](/wiki/percent-tabled/) +* [%querytabled](/wiki/percent-querytabled/) +* {{guide|chapter=5|section=31|title=Tabled Logic Programming}} + diff --git a/new-tests/stelf-output/pages/tactical-theorem-proving.lf b/new-tests/stelf-output/pages/tactical-theorem-proving.lf new file mode 100644 index 0000000..3312fd6 --- /dev/null +++ b/new-tests/stelf-output/pages/tactical-theorem-proving.lf @@ -0,0 +1,121 @@ +% +%{! +Logics can be defined in Twelf in such a way that it may not be possible +to do proof search by the fixed search strategy of Twelf's +[logic programming](/wiki/logic-programming/) engine. In these cases, tactical theorem provers can +be written that may still be able to prove many theorems. This article +defines two approaches to writing these tactical theorem provers. +!}% +%{! ## Logic definition !}% +%{! +When we introduce numbers with addition on this wiki (see for example +[natural numbers](/wiki/natural-numbers/)), we usually define a judgmental definition of addition +and then prove that it has the properties we desire. This will be a rather +different presentation, more in line with the [Zermelo Frankel](/wiki/zermelo-frankel/) case study. +This is a signature for an object language (a logic) that has addition +as a primitive operation. + +We define a type for numbers ``num``, and then make addition a +primitive operation. We also define propositions; the only proposition we +define here is equality of two numbers. +!}% +%sort num %. +%name num %. +%term + %pi num %-> num %-> num %. +%prec %left 10 + %. +%term 0 num %. +%term 1 num %. +%sort prop %. +%term == %pi num %-> num %-> prop %. +%prec %none 5 == %. +%{! +We can create a valid proposition that may be obviously untrue; for instance, +``(0 == 1)`` is a valid object of type ``prop``. Therefore, +we create ``pf``, which is a particular little logic which +will allow us to prove a large number of theorems about addition based +on a small number of axioms. Eight axioms are defined below: +!}% +%sort pf %. +%term refl pf (N == N) %. +%term symm %pi (pf (N1 == N2)) %-> (pf (N2 == N1)) %. +%term trans %pi (pf (N1 == N2)) %-> (pf (N2 == N3)) %-> (pf (N1 == N3)) %. +%term plus_assoc pf (N1 + N2 + N3 == N1 + (N2 + N3)) %. +%term plus_comm pf (N1 + N2 == N2 + N1) %. +%term plus_zero pf (N1 + 0 == N1) %. +%term plus_cong %pi (pf (N1 == N1')) %-> (pf (N2 == N2')) %-> (pf (N1 + N2 == N1' + N2')) %. +%term plus_elim1 %pi (pf (N1 + N2 == N1 + N2')) %-> (pf (N2 == N2')) %. +%freeze pf %. +%{! +We [freeze](/wiki/percent-freeze/) the type family ``pf`` to prevent any more +axioms from being defined, +but we can still define (many!) more theorems using the axioms (for instance, +the complement to ``plus_elim1``. +!}% +%define plus_elim2 (%pi (pf (N1 + N2 == N1' + N2)) %-> (pf (N1 == N1'))) [p1 pf (N1 + N2 == N1' + N2)] plus_elim1 (trans (trans plus_comm p1) plus_comm) %. +%{! ## Tactical theorem proving !}% +%{! ### Motivation: "flattening" a numeric formula !}% +%{! +Say we want to define a predicate ``mklist`` that takes some numeric +formula ``num`` and applies associativity exhaustively to "flatten" +the formula into a list, for instance transforming ``(a + (b + c) + d)`` +into ``(a + b + c + d)`` - addition was defined to be left-associative, +so this is the same as ``(((a + b) + c) + d)``. + +The ``list`` type family below will do this when run as a logic program. +!}% +%sort list %. +%mode list %in %out %. +%term list-swap %pi (list (A + (B + C)) D) %<- (list (A + B + C) D) %. +%term list-step %pi (list (A + C) (B + C)) %<- (list A B) %. +%term list-stop list A A %. +%{! ### Returning a proof of equality !}% +%{! +The operation of ``list`` may seem a bit mysterious - how +do we know that the straightened out formula is equal to the old one? By using +Twelf's [dependent types](/wiki/dependent-types/), we can define a new type family ``mklist`` +which operates in the same way but returns a proof that the two numeric +formulas are equal. We do **not** prove that the second +formula is flattened, just that it is equal to the first formula. +!}% +%sort mklist {A} {B} %. +%mode mklist %in %out %out %. +%term mklist-swap + %pi (mklist (A + (B + C)) D (trans (symm plus_assoc) Pf)) + %<- (mklist (A + B + C) D (%the (pf (A + B + C == D)) Pf)) %. +%term mklist-step + %pi (mklist (A + C) (B + C) (plus_cong Pf refl)) + %<- (mklist A B (%the (pf (A == B)) Pf)) %. +%term mklist-stop mklist A A refl %. +%{! ### Using the tactical theorem prover !}% +%{! +We can then use [``%define``](/wiki/define-declaration/) and +[``%solve``](/wiki/percent-solve/) to create +a proof that (a + (b + c) + d == a + b + c + d). In order for ``mklist`` +to terminate, it must be given a [ground](/wiki/ground/) term, so we introduce a four +atomic terms ``a`` through ``d``. The proof ``Pf`` +must be explicitly allowed to rely on those terms, which is why +``(Pf a b c d)`` is written instead of just ``Pf``. +|check=decl !}% +%define p1 = Pf %solve _ : {a} {b} {c} {d} mklist (a + (b + c) + d) _ (Pf a b c d) %. +%{! ## Tactics with ``%clause`` !}% +%{! Another way to achieve the same goal is to _define_ ``(list A B)`` +as ``(pf (A == B))``, which we do for ``list'`` below. !}% +%define list' (%pi num %-> num %-> %type) [a] [b] pf (a == b) %. +%mode list %in %out %. +%{! Then, we have to _justify_ each clause of the logic in the same way +as we justified ``plus_elim2`` above. We have to write the +[``%clause``](/wiki/percent-clause/) because, if we do not, then Twelf will not use the +definitions in logic programming search. !}% +%clause list'-swap (%pi (list' (A + (B + C)) D) %<- (list' (A + B + C) D)) [Pf pf (A + B + C == D)] trans (symm plus_assoc) Pf %. +%clause list'-step (%pi (list' (A + C) (B + C)) %<- (list' A B)) [Pf pf (A == B)] plus_cong Pf refl %. +%clause list'-stop (list' A A) refl %. +%{! Now we do not need to use ``%define``; ``p2`` proves the +same thing as ``p1`` above. |check=decl !}% +%solve p2 : {a} {b} {c} {d} list' (a + (b + c) + d) _ %. +%{! This is not the recommended style for a number of reasons (metatheoretic parts of Twelf like [totality assertions](/wiki/totality-assertion/) won't work with ``%clause``), but a few large examples such as the [big algebraic solver](/wiki/big-algebraic-solver/) have been written in this style. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Tactical_theorem_proving). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-ad-hoc-binding-structures.lf b/new-tests/stelf-output/pages/talk-ad-hoc-binding-structures.lf new file mode 100644 index 0000000..e583399 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-ad-hoc-binding-structures.lf @@ -0,0 +1,8 @@ +% +%{! Would "Ad-hoc binding structures" be a better name? Maybe I'm just wrong, but "Ad hoc" looks weird to me. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 00:07, 19 October 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Ad_hoc_binding_structures). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-admissibility-of-cut.lf b/new-tests/stelf-output/pages/talk-admissibility-of-cut.lf new file mode 100644 index 0000000..83ee1b0 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-admissibility-of-cut.lf @@ -0,0 +1,18 @@ +% +%{! ## Questions on Sequent Calculus + +If init works at any prop, why do you need the two judgements when you do the encoding? + +If you give these two rules for and, is it right-asynch or left-asynch or both? And I thought making one connective both was bad. +[Drl](/wiki/user-drl/) 09:38, 26 October 2006 (EDT) + +## Sectioning looks off + +Are the other "... cuts" supposed to be indented an additional level? [Drl](/wiki/user-drl/) 17:45, 15 March 2007 (EDT) +* Nope. I fixed it. Thanks for spotting it!  — [Tom 7](/wiki/user-tom7/) 18:38, 15 March 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Admissibility_of_cut). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-alpha-equivalence.lf b/new-tests/stelf-output/pages/talk-alpha-equivalence.lf new file mode 100644 index 0000000..db8d17f --- /dev/null +++ b/new-tests/stelf-output/pages/talk-alpha-equivalence.lf @@ -0,0 +1,14 @@ +% +%{! The definition is not correct. +It's not enough to say that y is not a free variable of e. +You must also have y free for x in e. + +:: Not sure where you're taking issue here: a counterexample might be helpful, and you're welcome to edit the actual page to clarify if you'd like. Certainly it is the case that alpha equivalence is quite subtle, which is one of the reasons we like that Twelf takes care of all of it for us :-) — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 14:21, 27 February 2011 (EST) + +:::: The usual definition has the additional condition that y != x. Is that what you meant? [Drl](/wiki/user-drl/) 22:46, 15 March 2011 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Alpha-equivalence). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-ask-twelf-elf-archive-1.lf b/new-tests/stelf-output/pages/talk-ask-twelf-elf-archive-1.lf new file mode 100644 index 0000000..19bed68 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-ask-twelf-elf-archive-1.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-ask-twelf-elf-archive-2.lf b/new-tests/stelf-output/pages/talk-ask-twelf-elf-archive-2.lf new file mode 100644 index 0000000..19bed68 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-ask-twelf-elf-archive-2.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-ask-twelf-elf.lf b/new-tests/stelf-output/pages/talk-ask-twelf-elf.lf new file mode 100644 index 0000000..a47ed3a --- /dev/null +++ b/new-tests/stelf-output/pages/talk-ask-twelf-elf.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-canonical-forms-lemma.lf b/new-tests/stelf-output/pages/talk-canonical-forms-lemma.lf new file mode 100644 index 0000000..127fed3 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-canonical-forms-lemma.lf @@ -0,0 +1,13 @@ +% +%{! [canonical form](/wiki/canonical-form/) should be its own page (not a redirect) for the LF notion of canonical form. I deleted the redirect until we have such a page.  — [Tom 7](/wiki/user-tom7/) 17:32, 30 September 2006 (EDT) + + +I just realized after writing the examples, that to make canonical forms seem relevant, I had to do progress proofs anyway. I am considering re-doing this article as "Progress when you get canonical forms for free vs. progress when you need to prove canonical forms". --[DanielKLee](/wiki/user-danielklee/) 19:05, 30 September 2006 (EDT) + +I just also realized there is a way to do progress for the second type system and still get CFLs for free. GAH. Not sure whether I should do a third example to show it off, or just pretend it doesn't exist to make the second example still seem relevant. --[DanielKLee](/wiki/user-danielklee/) 19:05, 30 September 2006 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Canonical_forms_lemma). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-case-studies.lf b/new-tests/stelf-output/pages/talk-case-studies.lf new file mode 100644 index 0000000..6b843e5 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-case-studies.lf @@ -0,0 +1,14 @@ +% +%{! I don't really think [CPS conversion](/wiki/cps-conversion/) is a case study; it's meant to be a contrived tutorial. (I have a much larger code base from my thesis project that would be the "case study".) But I don't mind it being here until there's more to fill out this page...  — [Tom 7](/wiki/user-tom7/) 20:23, 11 October 2006 (EDT) +: I guess in my mental separation, a tutorial answers a question someone might ask or shows a technique someone might use directly, whereas a case study says "this is how you do something cool and non-trivial in Twelf." - to borrow Karl's separation from two ConCert meetings ago, tutorials are "let's think about how to do work in Twelf" and case studies are "used Twelf, got things done", and I saw CPS conversion more as "got things done." However, perhaps the best solution in these cases is to list it in both places, which I think would be entirely appropriate. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 20:49, 11 October 2006 (EDT) +:: Well, the CPS example is for a toy language specifically designed for pedagogical purposes, so it's not an example of getting things done—even though it was an offshoot of real work. But sure, let's put it in both places.  — [Tom 7](/wiki/user-tom7/) 20:59, 11 October 2006 (EDT) +::: Perhaps my problem is that we need a new category that isn't quite "Advanced Proof Techinques"... Trying to make a good sort of the things that are tutorials seems to get me in trouble every time. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 21:35, 11 October 2006 (EDT) +:::: Sounds like another Twelf coffee is in order!  — [Tom 7](/wiki/user-tom7/) 22:09, 11 October 2006 (EDT) +::::: Indeed, especially as I've now started the leak, apparently - Carsten, Brigette, and Kevin are aware of this project (intentional), Carsten may point some of his students to upload examples (encouraged), and Brigette may be sharing this with her McGill Computation and Deduction course (yipes!). — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 22:49, 11 October 2006 (EDT) +:: Might I suggest an "Illustrative Examples" category. Illustrative examples are minimal examples that highlight one or two particular encoding or proof techniques. The emphasis being on how a particular thing is done, rather than focusing on general techniques (tutorials) or larger proofs with their own agendas (case studies). --[DanielKLee](/wiki/user-danielklee/) 00:01, 12 October 2006 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Case_studies). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-classical-s5.lf b/new-tests/stelf-output/pages/talk-classical-s5.lf new file mode 100644 index 0000000..f8f5835 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-classical-s5.lf @@ -0,0 +1,10 @@ +% +%{! # > is getting turned into =&gt; in the ``<twelf>`` block. Any idea why? [Jcreed](/wiki/user-jcreed/) 18:11, 24 October 2006 (EDT) + +: It must be something that got changed in the syntax highlighting module, ugh. I am due for another rewriting of that stupid module soon, looks like. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 18:54, 24 October 2006 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Classical_S5). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-constraint-domains-and-coverage-checking.lf b/new-tests/stelf-output/pages/talk-constraint-domains-and-coverage-checking.lf new file mode 100644 index 0000000..f5b64ca --- /dev/null +++ b/new-tests/stelf-output/pages/talk-constraint-domains-and-coverage-checking.lf @@ -0,0 +1,9 @@ +% +%{! This page should start with a definition of what a constraint domain is. (Or else it is a page about constraint domains and metatheorems?))  — [Tom 7](/wiki/user-tom7/) 13:42, 3 November 2006 (EST) +: Yes, you had that thought 15 minutes before but independently of me - I just clicked on the redlink in [Tutorials](/wiki/tutorials/) and went crazy. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 13:56, 3 November 2006 (EST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Constraint_domains_and_coverage_checking). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-converting-between-implicit-and-explicit-parameters.lf b/new-tests/stelf-output/pages/talk-converting-between-implicit-and-explicit-parameters.lf new file mode 100644 index 0000000..9bfa005 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-converting-between-implicit-and-explicit-parameters.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-coverage-checking.lf b/new-tests/stelf-output/pages/talk-coverage-checking.lf new file mode 100644 index 0000000..e1eb3ee --- /dev/null +++ b/new-tests/stelf-output/pages/talk-coverage-checking.lf @@ -0,0 +1,10 @@ +% +%{! ## Freeness/coverage + +Do you consider output freeness to be an aspect of coverage checking? It is a position that makes sense; I suppose I'd just thought of it as an auxillary check that [``%total``](/wiki/percent-total/) performs. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 09:10, 21 March 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Coverage_checking). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-covers.lf b/new-tests/stelf-output/pages/talk-covers.lf new file mode 100644 index 0000000..0b10f6b --- /dev/null +++ b/new-tests/stelf-output/pages/talk-covers.lf @@ -0,0 +1,9 @@ +% +%{! ## ``%partial`` +If you're interested in a ``%covers`` that "does the right thing" (input, output, freeness, and checking of other type families in subgoals), I have a simple implementation of the ``%partial`` declaration. This declaration does the same thing as ``%total`` but does not perform the totality check. From it you get a "partial correctness" property: a ``%partial`` relation will either search forever or it will succeed.  — [Tom 7](/wiki/user-tom7/) 22:41, 17 October 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:%25covers). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-cps-conversion.lf b/new-tests/stelf-output/pages/talk-cps-conversion.lf new file mode 100644 index 0000000..97d8d9b --- /dev/null +++ b/new-tests/stelf-output/pages/talk-cps-conversion.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-debugging-coverage-errors.lf b/new-tests/stelf-output/pages/talk-debugging-coverage-errors.lf new file mode 100644 index 0000000..f263ff4 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-debugging-coverage-errors.lf @@ -0,0 +1,8 @@ +% +%{! When I have higher-order terms (in particular if I have a judgment that handles transitivity of higher-order contexts -- I can given an example) and do case analysis on them, I frequently get a nasty case of inactive splits AND nontermination. The only solution I have found is to split the metatheorem into two parts: one to handle the higher-order term in general and a second to do a case analysis between fixed terms. [Boyland](/wiki/user-boyland/) 09:31, 27 December 2009 (EST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Debugging_coverage_errors). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-define-declaration.lf b/new-tests/stelf-output/pages/talk-define-declaration.lf new file mode 100644 index 0000000..1753d5f --- /dev/null +++ b/new-tests/stelf-output/pages/talk-define-declaration.lf @@ -0,0 +1,10 @@ +% +%{! ## Name + +Wow, that's weird. As near as I can tell, it's because 'de' (and also maybe 'f') are hex characters, so %25de is being interpreted as malformed unicode or something like that. I'm going to move to [define declaration](/wiki/define-declaration/) though, because the space is pretty weird for an article title.  — [Tom 7](/wiki/user-tom7/) 09:42, 14 October 2006 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Define_declaration). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-documentation.lf b/new-tests/stelf-output/pages/talk-documentation.lf new file mode 100644 index 0000000..76d5d6b --- /dev/null +++ b/new-tests/stelf-output/pages/talk-documentation.lf @@ -0,0 +1,10 @@ +% +%{! ## includes + +Is there a better way to include the front page text on this page? They're both serving as portals, so I think the duplication is reasonable: the docs page is the "real" source for the list of intro material, but we happen to have all of it duplicated on the front page for convenience right now.[Drl](/wiki/user-drl/) 00:24, 24 October 2006 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Documentation). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-download.lf b/new-tests/stelf-output/pages/talk-download.lf new file mode 100644 index 0000000..f22a4ab --- /dev/null +++ b/new-tests/stelf-output/pages/talk-download.lf @@ -0,0 +1,60 @@ +% +%{! Perhaps this will be controversial, but I think the software page should have instructions about how to get/install Twelf CVS. %trustme is commonly used by myself, Karl, and Dan Licata in our Twelf code development, and occasionally Twelf code that requires it is publically released. --[DanielKLee](/wiki/user-danielklee/) 21:46, 5 September 2006 (MST) + +* I completely agree. In fact, that's a decent way of giving this page a purpose - to say "either go to the static software page over here, or get stuff from the CVS repository here". Does the repository have read-only anon. access? I'm actually using 1.5R1 :-P. [Rsimmons](/wiki/user-rsimmons/) 06:48, 6 September 2006 (MST) + +* Definitely.  — [Tom 7](/wiki/user-tom7/) 16:15, 9 September 2006 (MST) + +## Solved, yet confusing, problem + +I downloaded the CVS version, and it has the same problem with my SML as other versions have. I don't understand the source of this problem but can't believe I'm the only one who has been tripped up by it. The files ``twelf-server.sml`` looks like this, and ``twelf-sml.sml`` is similar. + +<code>(* comment out first line if undefined in your version of SMLofNJ *) +(* call sml-cm with @SMLdebug=/dev/null instead *) +SMLofNJ.Internals.GC.messages false; +CM.make' "server.cm"; +SMLofNJ.exportFn ("bin/.heap/twelf-server", Server.server);</code> + +No version of SML I've used in the last year supports ``CM.make'``, I've had to manually delete the apostrophe. If we include CVS information publicly, we should either fix this problem or explain the workaround. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 10:28, 8 September 2006 (MST) +: Does the mlton build work? mlton will produce code that is several times faster, so we ought recommend that in my opinion.  — [Tom 7](/wiki/user-tom7/) 16:15, 9 September 2006 (MST) +:: I've built the CVS version with MLton, but it requires changing around a .cm file (dependencies, I guess). I suppose I should inform the Twelf developers of that... --[DanielKLee](/wiki/user-danielklee/) 16:21, 9 September 2006 (MST) +::: Yeah - at the very least, the code should work for mlton *OR* SML/NJ. And we should explain the changes that need to be made on the installation page, if it only works for one or the other (as it seems currently to work "out of box" for neither...) — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 18:01, 9 September 2006 (MST) +:: And of course, if you're me and not Carsten you can't make mlton work on a 64-bit box... — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 18:02, 9 September 2006 (MST) +::: I sshed onto linux1.gp.cs.cmu.edu and used the MLton there to build my twelf-server, which then runs fine off my normal machine. Your afs home is the same on both machines. --[DanielKLee](/wiki/user-danielklee/) 19:34, 9 September 2006 (MST) + +## Protection + +Rob suggested the possibility of protecting this page, since it points to executable software and could be vandalized. Protection would mean that no users that have not been marked administrators by Rob or myself would be able to edit the page. They would still be able to leave messages on this talk page. Since we have so many administrators and all know each other, I think there's no reason not to protect the software page. Any objections?  — [Tom 7](/wiki/user-tom7/) 11:59, 22 September 2006 (MST) + +: **_Do**_ it. --[DanielKLee](/wiki/user-danielklee/) 12:18, 22 September 2006 (MST) + +Protecting the page might be give some sense of security, *but* I've just lost some time asking around for a solution of the make' problem; now I've created a wiki account to add a hint to the Download page about this, and find it protected. If you protect that page, I suggest you at least move such information to the protected page. Thanks. (BTW speaking for myself, I like if software is released with an OpenPGP (gpg) signature; this is more secure, and then wouldn't really require protecting the page.) --[[User:pflanze]] Tue Mar 17 17:25:44 EDT 2009 + +: I thought in the currently available version, running ``make -f smlnj/Makefile`` circumvented the ``make'`` bizarreness, does it not? You're right about the OpenPGP signature, and we should definitely do that, though that doesn't necessarily change the point that we want to have those links protected. Sorry you were forced to run in circles, though, can you give me a better idea what you need help with/wanted to fix? ---- + +## cvsnt? + +Doesn't cygwin have CVS? I think we should instruct them to install that version, which will simplify the process a little.  — [Tom 7](/wiki/user-tom7/) 14:14, 9 January 2007 (EST) + +: Cygwin is 1) heavyweight and 2) not available on my home computer, which I get in trouble for installing lots of things on. I agree, essentially. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 20:24, 9 January 2007 (EST) +:: Sorry, I misread the article. I thought it said to install cygwin and then install cvsnt.  — [Tom 7](/wiki/user-tom7/) 09:58, 10 January 2007 (EST) +::: I've removed most of said information; now that I can create new executible distributions in ~5minutes, I don't see the need as much. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 00:21, 28 January 2007 (EST) +:::: Good, I think that frequent binary releases are a good way to go. I think most potential Twelf users will know how to use CVS anyway.  — [Tom 7](/wiki/user-tom7/) 01:33, 30 January 2007 (EST) + +## Software links? + +Is it possible to get the "CVS build" links right on this page? I'd do it myself but the windows installer filename seems to be generated from the date (this might be a bad idea).  — [Tom 7](/wiki/user-tom7/) 11:11, 21 March 2007 (EDT) + +: I have to do some manual work to update windows anyway; I may change the PHP script in the future, but for now I've just created a symlink http://twelf.plparty.org/builds/twelf.exe to the most recent version. I'm not sure what you have in mind, so feel free to go wild :). — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 11:38, 21 March 2007 (EDT) + +## Building with MLTon + +Twelf cannot be built with latest MLTon, which complains about invalid file suffix, “.cm”. + +By the way, the binary archive cannot work on current Linux distributions, which does not provide libgmp 3 anymore, only libgmp 10. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Download). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-effectiveness-lemma.lf b/new-tests/stelf-output/pages/talk-effectiveness-lemma.lf new file mode 100644 index 0000000..c0c2bc2 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-effectiveness-lemma.lf @@ -0,0 +1,22 @@ +% +%{! Why "effectiveness" rather than "totality" ? As far as I can tell, proving "effectiveness" is the same +as proving totality at the meta-leval. [Boyland](/wiki/user-boyland/) 22:41, 25 February 2007 (EST) + +: I think it just dates back to some choices made in the Typed Assembly Language Two formalization. I think we thought that calling it a "totality lemma" would just cause confusion of a different sort, not that effectiveness is an obvious alternative by any means. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 23:40, 25 February 2007 (EST) + +This sentence is pretty confusing: "Proving an effectiveness lemma essentially establishes the same fact that a %total directive does, but does so in a way that can be used by other metatheorems." You use %total to prove any metatheorem, and any lemma so proved can be used by other metatheorems. + +An effectiveness lemma says that some relation is in fact an algorithm (hence "effective"), and returns for any inputs a derivation of the relation which witnesses that fact. [User:JakeD](/wiki/user-jaked/) 00:33, 26 February 2007 (EST) + +: I mean, that works for me too :) — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 01:04, 26 February 2007 (EST) +:: I edited it to try to simplify. We definitely should not lead with confusing stuff about metatheorems (since that is hard to wrap ones head around, especially given all the different takes on what a metatheorem is!), but a section in this article about its relationship to [``%total``](/wiki/percent-total/) is warranted, I think, because it is an obvious question.  — [Tom 7](/wiki/user-tom7/) 09:33, 26 February 2007 (EST) + +: I think language that connects "effectivenss" with totality is warranted. I don't see it in the article any more. There needs to be something on the order of "Proving an effectivessness lemma for a relation essentially establishes the same fact that a %total directive does for the relation, but does so in a way that the totality itself is named and can be used in proofs." Otherwise people will wonder. Also, personally (as an outsider), I find it annoying that one basically has to restate the entire definition of the relation in order to prove meta-totality ("effectivessness"), but Twelf's own wiki doesn't need to reflect my biased opinions. [Boyland](/wiki/user-boyland/) 21:03, 26 February 2007 (EST) + +:: I agree: effectiveness lemmas are just a workaround for limitations of Twelf. When you do a %total for, say, plus, Twelf proves "for all N1, N2, there exists an N3 and D:plus N1 N2 N3". When you do a %total for plus-effective, Twelf proves "for all N1, N2, there exist N3 and D:plus N1 N2 N3 and D':plus-effective N1 N2 N3 D D'". The only reason to do this is to work around the fact that Twelf won't let you name the output of a subgoal. [Drl](/wiki/user-drl/) 12:09, 1 March 2007 (EST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Effectiveness_lemma). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-equality.lf b/new-tests/stelf-output/pages/talk-equality.lf new file mode 100644 index 0000000..ffb1772 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-equality.lf @@ -0,0 +1,31 @@ +% +%{! ## Deep equality + +Not all equalities that are are _not_ syntactic equality are "defined inductively over the structure of terms". I know of some algorithmic equalities that are a combination of syntax directed and type directed. Some declarative equality systems aren't necessarily anything directed. + +Is "deep" equality meant to describe equalities that are defined as part of a logic? --[DanielKLee](/wiki/user-danielklee/) 18:23, 27 September 2006 (MST) + +## Equality vs. equivalence + +I don't really understand why people draw a distinction between "equality" and "equivalence". We often call things equal that are not the same; equality up to alpha-conversion is a great example. What purpose is the distinction supposed to serve? Originally, I had simply said: One can define equivalence relations in Twelf. You can choose to use LF's built-in notion of equality to do so, or you can define it yourself. To me, this is less confusing because it does not introduce distinctions between things that are not really different.  — [Tom 7](/wiki/user-tom7/) 10:39, 9 October 2006 (EDT) + +### Where I'm going + +In the (possibly if not probably non-standard) classification system in my head, "equivalence" is a special case of "equality" where equality is established by way of an equivalence relation rather than by way of being indistinguishable (i.e. "identity", which is then another special case of "equality"). Obviously I am tripping a bit over terminology here; I've also left the article in a series of inconsistant states - I'm going to shorten this article to something short but non-harmful and do work in User: space for a bit. + +I think this may need to turn into a disambiguation page going to two or three different tutorial-type articles. I don't know for sure what to *call* these articles. + +* _Two things are the same if they are indistinguishable_ - Equality via LF's built-in identity. This is what, at first, I saw this article being used for; essentially it is about what Dan Lee calls "respects lemmas" and Dan Licata calls "congruence lemmas," which are what make using this notion possible. +* _Two things are the same if they are related by an equivalence relation_ - Equality via some relation - this can use the tree example by creating a notion of trees that are equivalent modulo swapping left and right children of a pair node. +* _Two things are the same if they have the same canonical form_ - since Karl uses this approach successfully with the singleton kinds work I think it should be discussed - this uses the tree example by making a "canonical tree" one whose left branch is larger than the right (for some unambiguous value of "larger") - then I'd like to show that the canonical form definition is logically equivalent (yipes overloading!) to the canonical form usage. + +— [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 12:20, 9 October 2006 (EDT) +:* But you can make _respects_ lemmas for both deep and shallow equality (and sometimes you have to). They're just really easy if you define equality by way of LF's notion of equality. The fact that we have to write these lemmas at all is a demonstration that Twelf has no special support that would distinguish the first and second bullets as being really different. The third is just another example of defining an equivalence relation (in this case by quotienting), which is a good example but not a distinct category. (IMO) (Maybe I should put "(IMO)" in my sig. ;))  — [Tom 7](/wiki/user-tom7/) 14:45, 9 October 2006 (EDT) + +::* I think I now have something that makes sense based on your comments - Tom, care to weigh in? — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 16:03, 9 October 2006 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Equality). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-exchange-lemma.lf b/new-tests/stelf-output/pages/talk-exchange-lemma.lf new file mode 100644 index 0000000..6c009b1 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-exchange-lemma.lf @@ -0,0 +1,16 @@ +% +%{! ## Needs more information? + +I understand what's going on in this article, and think it's a good thing to show, but I think the article needs a descripition of how this would be encountered in practice - how the explicit exchange metatheorem would actually be avoided by rearranging the order of the lambdas... — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 19:03, 27 September 2006 (MST) + +: I editted the commentary on the code snippet to point to the substitution lemma example where it is being used. I'm not sure if there's a better/simpler example that could just be inlined. --[DanielKLee](/wiki/user-danielklee/) 19:45, 27 September 2006 (MST) + +:: The code doesn't have to make LOTS of sense out of context - for me, it helps to just see it. What do you think of these changes? — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 20:28, 27 September 2006 (MST) + +::: This snippet works for me. I re-worded it slightly and fixed a missing ``. --[DanielKLee](/wiki/user-danielklee/) 20:43, 27 September 2006 (MST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Exchange_lemma). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-freeze.lf b/new-tests/stelf-output/pages/talk-freeze.lf new file mode 100644 index 0000000..48ba16b --- /dev/null +++ b/new-tests/stelf-output/pages/talk-freeze.lf @@ -0,0 +1,15 @@ +% +%{! ## Version +Does anyone know what versions freeze and autofreeze are available in? I have the impression that they are new features (being undocumented).  — [Tom 7](/wiki/user-tom7/) 14:22, 1 December 2006 (EST) +: I think freeze has been available since around 1.5, but it wasn't in 1.4, I'm almost certain. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 18:53, 20 December 2006 (EST) + +## Soundness +Can unsoundness also result if there is no autofreeze? I'm not sure what Twelf's behavior was in this case.  — [Tom 7](/wiki/user-tom7/) 14:22, 1 December 2006 (EST) +: It should - you could prove ``eq (s N) N`` implied ``absurd``, and then add a case to ``eq`` to make that theorem no longer true... — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 18:52, 20 December 2006 (EST) +::Certainly if it continued to accept the totality of that theorem for the new extended ``eq``. But if any worlds/total/etc. declarations were invalidated by new declarations for a type family, I think that might still be sound. Did Twelf do that? Probably not.  — [Tom 7](/wiki/user-tom7/) 00:15, 25 December 2006 (EST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:%25freeze). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-general-description-of-twelf.lf b/new-tests/stelf-output/pages/talk-general-description-of-twelf.lf new file mode 100644 index 0000000..bb2ef35 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-general-description-of-twelf.lf @@ -0,0 +1,11 @@ +% +%{! Kepler conjecture link is broken and the Economist article is available to subscribers only... +Andrew Grey + +:: Thanks for leaving a note. I fixed the Kepler conjecture link, at least, it's too bad the Economist locks up their archives... — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 11:00, 5 August 2010 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:General_description_of_Twelf). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-glossary.lf b/new-tests/stelf-output/pages/talk-glossary.lf new file mode 100644 index 0000000..7400c78 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-glossary.lf @@ -0,0 +1,15 @@ +% +%{! TODO (delete from this page when done): + +basic: +* http://en.wikipedia.org/wiki/List_of_mathematical_symbols shows it as "much greater than." I see it in + + +high priority: +* [%total](/wiki/percent-total/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Glossary). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-implicit-and-explicit-parameters.lf b/new-tests/stelf-output/pages/talk-implicit-and-explicit-parameters.lf new file mode 100644 index 0000000..c4fa5d5 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-implicit-and-explicit-parameters.lf @@ -0,0 +1,35 @@ +% +%{! ## Universal quantification + +Is the implication supposed to be that only implicit parameters are universally quantified? In the type theory, both are exactly the same: Πs. I think this may be misleading. (Or is there some interpretation in logic programming that I'm missing?)  — [Tom 7](/wiki/user-tom7/) 13:35, 16 October 2006 (EDT) +:: No, you're right - seems like the right way to say it is that this is two different ways to do universal quantification, in fact. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 11:26, 12 November 2006 (EST) + +## This article or section needs a moral stand + +Is there a reason that this article doesn't say "don't use explicit parameters unless there is a good reason to; here are the only good reasons: \{termination, mode\}"? [Drl](/wiki/user-drl/) 20:29, 14 March 2007 (EDT) +* Well, I think there are good stylistic reasons to use explicit parameters (much like it is sometimes good style to use explicit type annotations or explicit quantification in SML) which are hard to circumscribe. I'd rather describe it in terms of pros and cons. (And actually, I don't think [%mode](/wiki/percent-mode/) is even necessarily a good reason, since it is possible to give modes to implicit arguments.)  — [Tom 7](/wiki/user-tom7/) 21:43, 14 March 2007 (EDT) +:* I would be inclined to agree, in that we should take "moral stands" where such are clear - don't start constants with capital letters, don't define an inductive notion of equality when what you mean is identity on LF terms. However, the article should probably say "we usually default to this style," which is what I tried to say. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 22:20, 14 March 2007 (EDT) + +## Always prints out the parameters + +I don't think it's true that Twelf always prints out the implicit parameters. For example, I think that implicit parameters to parameters are not always shown? This is why sometimes you see a ``\{A:typ\}`` where ``A`` appears nowhere else. Does anyone understand the actual rules? (Jason?)  — [Tom 7](/wiki/user-tom7/) 17:01, 26 October 2007 (EDT) + +It prints out implicit _parameters_ but not implicit _arguments._ I think this demonstrates the ``\{A:typ\}`` thing, I think I should change the example to something more like this. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 11:55, 28 October 2007 (EDT) !}% +%sort tp %. +%term tp/unit tp %. +%sort exp %. +%term unit exp tp/unit %. +%term abort/1 exp A %. +%term abort/2 {A} exp A %. +%{! !}% +%{! (options removed from twelftag: check=decl) !}% +%sort step %. +%term step/raise step unit abort/1 %. +%term step/raise step unit (abort/2 A) %. +%{! :: What's the difference between an implicit argument and an implicit parameter? Or you mean, it always prints out the implicit Π bindings, but never the implicit occurrences? Anyway, yes, I think this is a good example.  — [Tom 7](/wiki/user-tom7/) 18:47, 28 October 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Implicit_and_explicit_parameters). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-incremental-metatheorem-development.lf b/new-tests/stelf-output/pages/talk-incremental-metatheorem-development.lf new file mode 100644 index 0000000..d138281 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-incremental-metatheorem-development.lf @@ -0,0 +1,15 @@ +% +%{! ## Bottom-up vs. Top-down + +I'm not sure if the motivation for using %trustme to maintain a bottom-up development style is completely accurate here. Without %trustme, proofs must be done in a bottom-up way. That's just how math is built up. In practice, I've seen two modes of use of %trustme. The first is to work bottom-up until you hit something that will be pretty time-consuming/hard/impossible to do, and then you %trustme that particular lemma so you can keep working bottom-up. The other, which I've seen used by a particular Twelf master who happens to have an office right next to a stairwell in a corner on the 8th floor of Wean, is to %trustme a lot of boring/uninteresting lemmas which you are pretty sure are true, and then focus on the actual hard proof, and then fill in the boring holes at some point. The second strategy is very much a top-down approach. +--[DanielKLee](/wiki/user-danielklee/) 09:19, 22 September 2006 (MST) + +: I agree that there are multiple models of use, go ahead and put whatever you think is a more accurate motivation over what is there now (in general, I'd advocate a "change and explain why you did it on the Talk pages" as opposed to a "talk about changing on the Talk page" strategy here :) ). — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 09:26, 22 September 2006 (MST) +: I think that we are just using opposite definitions of "top-down" and "bottom-up"; for me, the end of the proof is the bottom. I guess it depends on how you think of it. (We had all sorts of similar terminological issues in automated theorem proving and computer graphics class..) We should just be more clear about what we mean.  — [Tom 7](/wiki/user-tom7/) 11:52, 22 September 2006 (MST) +:: Yeah, I think that's where the confusion is. I think of the main result being at the "top", and being built up from the smaller ones "below" it, like a tree. --[DanielKLee](/wiki/user-danielklee/) 10:30, 23 September 2006 (MST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Incremental_metatheorem_development). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-introductions-to-twelf.lf b/new-tests/stelf-output/pages/talk-introductions-to-twelf.lf new file mode 100644 index 0000000..f70ab00 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-introductions-to-twelf.lf @@ -0,0 +1,72 @@ +% +%{! If you have suggestions for an audience that is not addressed by the existing tutorials, please leave them here. + +I'd like to add a page with the following on how to run twelf programs. How can I do this? Note that this is for non-emacs users. If there are similar (quick and easy) instructions on how to run twelf on emacs, let me know and I'll link them from this page. [[User:Giselle|Giselle]] ([[User talk:Giselle|talk]]) 19:47, 12 December 2016 (UTC) + +Great! Thanks. I started making a new page and discovered we have a page already for this. +Take a look at [Twelf without Emacs](/wiki/twelf-without-emacs/). Perhaps you can use your examples to make it +more concrete? [Boyland](/wiki/user-boyland/) ([talk](/wiki/user-talk-boyland/)) 00:24, 13 December 2016 (UTC) + +---- + +## Running your first program + +Now that you have installed Twelf, it is time to run your first program. Open a file (in any text editor of your choice) and type the following: + + nat : type. + z : nat. + s : nat -> nat. <br> + add : nat -> nat -> nat -> type. + add/z : add z N N. + add/s : add M N P -> add (s M) N (s P). <br> + %mode add +M +N -P. + %worlds () (add _ _ _). + %total \{M N\} (add M N _). + +Save this file as <code>my_first_twelf.elf</code> (twelf files have the extension <code>.elf</code>). Now go to a terminal, preferably to the directory where you have the twelf file and start the twelf-server by running <code>twelf-server</code> if it is in your path or <code>/path/to/twelf/bin/twelf-server</code>. You should see something like this: + + Twelf 1.7.1+ (rUnversioned directory, built 12/05/16 at 12:27:57 on narnia) + %% OK %% + +This is twelf's [https://en.wikipedia.org/wiki/Read%E2%80%93eval%E2%80%93print_loop repl]. You can see a list of available commands by typing <code>help</code> following by _Enter_. To load the file you have just created, type: <code>loadFile my_first_twelf.elf</code> (if this is in a different directory, make sure to type with the right path). Twelf typechecks the code and returns successfully, which you can see by the <code>%% OK %%</code> at the end. + + %% OK %% + loadFile my_first_twelf.elf + [Opening file my_first_twelf.elf] + nat : type. + z : nat. + s : nat -> nat. + add : nat -> nat -> nat -> type. + add/z : \{N:nat\} add z N N. + add/s : \{M:nat\} \{N:nat\} \{P:nat\} add M N P -> add (s M) N (s P). + %mode +\{M:nat\} +\{N:nat\} -\{P:nat\} (add M N P). + %worlds () (add _ _ _). + %total \{M N\} (add M N _). + [Closing file my_first_twelf.elf] + %% OK %% + +You can enter in querying mode by typing <code>top</code> following by _Enter_ in the repl. This is indicated by the symbol <code>?-</code>. In this mode, you can ask twelf to compute things in a prolog-like manner. Note that this is independent of the <code>%mode</code> declared in the source file. Typing <code>Ctrl+c</code> exits the querying mode. Here's an example of querying twelf: + + %% OK %% + top + ?- add (s z) (s (s z)) X. + Solving... + X = s (s (s z)). + More? y + No more solutions + ?- add X (s z) (s (s z)). + Solving... + X = s z. + More? y + No more solutions + ?- ^C + interrupt + %% OK %% + +To understand better what just happened and what all the words in this file represent, move on to [Proving_metatheorems_with_Twelf](/wiki/proving-metatheorems-with-twelf/). !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Introductions_to_Twelf). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-letrec.lf b/new-tests/stelf-output/pages/talk-letrec.lf new file mode 100644 index 0000000..d3ef44c --- /dev/null +++ b/new-tests/stelf-output/pages/talk-letrec.lf @@ -0,0 +1,28 @@ +% +%{! ## Different formalism? + +The formalism on the current page is probably the right thing to do for "real-world binding", since it's closest to the actual syntax that people will have in mind. + +However, it looks a little funny to me to have projection from an object as a language construct, without the objects themselves being first class. Would it be better to do a version with objects and define letrec as syntactic sugar for an object that you immediately split? I guess that doesn't show off the binding issues as well, since an object only binds one name and the components are accessed by projection. + +Maybe at the end of this case study, we can say that letrec doesn't need to primitive, and elaborate it as +<code> +letrec x1 = e1 ... in e +</code> + +translates to + +<code> +let o = obj x is (e1[pi_1 o/x1], ...) in +let x1 = pi_1 o in + ... +in e +</code> + +[Drl](/wiki/user-drl/) 15:22, 16 October 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Letrec). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-lily.lf b/new-tests/stelf-output/pages/talk-lily.lf new file mode 100644 index 0000000..b609bb3 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-lily.lf @@ -0,0 +1,8 @@ +% +%{! This is obviously a huge and impressive body of code, but it could use a lot more descriptive text. In particular, there should be an introduction that says what Lily is (and point to a paper) along with some context of what will be proved. It would also make sense to describe in English the important theorems that you prove, when they come up. Otherwise I don't think there's any way anybody can get anything out of it.  — [Tom 7](/wiki/user-tom7/) 17:32, 19 October 2006 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Lily). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-lists.lf b/new-tests/stelf-output/pages/talk-lists.lf new file mode 100644 index 0000000..3e3fb8d --- /dev/null +++ b/new-tests/stelf-output/pages/talk-lists.lf @@ -0,0 +1,34 @@ +% +%{! ## Heterogeneous? + +I don't understand in what sense these are heterogeneous lists; it seems that they only hold terms of type 'tp'. Is this a mistake in terminology, or am I missing something?  — [Tom 7](/wiki/user-tom7/) 09:44, 20 October 2006 (EDT) +: You can define lists that all have to be the same 'tp,' I think is the distinction. However, I think that is a confusing distinction as well, because this is what we think of when we think of lists in Twelf and it's not what we think of when we think of heterogeneous lists. Move to move page to [Lists](/wiki/lists/) — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 09:53, 20 October 2006 (EDT) +:: In that interpretation homogeneous lists are trivial: they are just natural numbers saying how many of the one allowed element are in the list. That can't be right. I agree: move to [lists](/wiki/lists/).  — [Tom 7](/wiki/user-tom7/) 10:37, 20 October 2006 (EDT) + +## Tp + +Also, rather than defining one thing of type ``tp``, ``stuff``, wouldn't it be better to do the proofs in a world with arbitrary types? I'm still learning how worlds are used in practice, so I'm not certain; would it make the definitions and proofs less easily usable elsewhere, perhaps, because people would have to modify the worlds? — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 09:58, 20 October 2006 (EDT) + +: Eek - Carsten, ``tp`` is a confusing list object to use - this is a list of types, but to narrate you have to use "list of types" and "things that have [LF] type list" and it would be much less confusing if it was "list of nats" and "things that have type nat or type list." I probably should have argued this point with you better earlier on, but how would you feel about switching ``tp`` to something less ``tp``-ey? — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 10:26, 20 October 2006 (EDT) +:: Yes, agreed: ``elt`` for element?  — [Tom 7](/wiki/user-tom7/) 10:33, 20 October 2006 (EDT) + +## Math + +What do we think about the BNF notation introducing the object language? I think I like it but would like comments. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 10:28, 20 October 2006 (EDT) + +## Dependent lists + +Lists are a poster child for dependent types: the type ``list N`` with ``N : nat`` has as members lists of length ``N``. If this article is expanded to a general tutorial on lists, then this possibility should be considered. (It also has some drawbacks!)  — [Tom 7](/wiki/user-tom7/) 10:37, 20 October 2006 (EDT) +: That might be better as a second article - [Lists](/wiki/lists/) and [Indexed lists](/wiki/indexed-lists/) perhaps. But yes, totally in agreement, I was going to say it so I'm glad you did. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 10:47, 20 October 2006 (EDT) +:: I was thinking of doing an article on indexed lists. Apparently other people had similar thoughts! It's somewhere on my queue, so I'll end up doing it if no one beats me to it. --[DanielKLee](/wiki/user-danielklee/) 13:56, 20 October 2006 (EDT) + +## Polymorphic list type? + +Twelf doesn't like \{\{\{list : type -> type.\}\}\} very much. Is there a simple (no nasty encoding tricks like Google pointed me to) way to express polymorphic lists and list lemmata in Twelf? -- [[User:Stefan O'Rear|Stefan O'Rear]] 05:27, 10 August 2007 (EDT) +:: The short answer is no; Twelf is not polymorphic because making it so would complicate Twelf's ability to deal with [metatheorems](/wiki/metatheorem/). Depending on what it is you want to do, however, there are any number of techniques that can be used to do what you want to do (though some of them probably fall into the "nasty encoding tricks" category). — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 07:19, 10 August 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Lists). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-main-page.lf b/new-tests/stelf-output/pages/talk-main-page.lf new file mode 100644 index 0000000..3f012fe --- /dev/null +++ b/new-tests/stelf-output/pages/talk-main-page.lf @@ -0,0 +1,14 @@ +% +%{! ## How to edit the main page + +In rough order of how likely they are to need to be edited. If you want to edit a certain page, click on it and choose edit. + +*[What's new?](/wiki/whats-new/) +*[Template:The Twelf Project/Introduction](/wiki/template-the-twelf-project-introduction/) (About The Twelf Project) +*[Template:The Twelf Project/Getting Started](/wiki/template-the-twelf-project-getting-started/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Main_Page). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-meta-logic.lf b/new-tests/stelf-output/pages/talk-meta-logic.lf new file mode 100644 index 0000000..569a774 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-meta-logic.lf @@ -0,0 +1,10 @@ +% +%{! ## Relational metatheory redlink + +Does anyone think we need the redlink for relational metatheory? I removed the link, because I wasn't clear what it was going for. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 16:53, 29 December 2006 (EST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Meta-logic). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-metatheorem.lf b/new-tests/stelf-output/pages/talk-metatheorem.lf new file mode 100644 index 0000000..26fbc0a --- /dev/null +++ b/new-tests/stelf-output/pages/talk-metatheorem.lf @@ -0,0 +1,28 @@ +% +%{! ## Placeholder + +I'm going to work on this based on Dan's notes, with the idea that I'll fix it based on whatever total consensus arises with all interested parties. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 17:12, 20 October 2006 (EDT) + +## I agree, Tom + +The point about totality checking being *more* conservative than the fixed search strategy of logic programming is a good one - the need for factoring lemmas being the perfect example here - so I think that change is good. Does the article seem reasonable otherwise, because I'm not certain the approach is even the right one here. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 19:16, 9 March 2007 (EST) +::I think it's a hard subject but this is a reasonable take on it and a good start, yes.  — [Tom 7](/wiki/user-tom7/) 14:01, 12 March 2007 (EDT) + +## Why "meta" ? + +To an outsider, all the "meta"-theorem stuff seems like unnecessary baggage. Normally, I would think a meta-theorem is a theorem about theorems. Here's what I think the answer to my question is: In the olden days, one implemented XYZ logic in Twelf and wrote a theorem that XYZ theorems were valid. But now, we use Twelf to directly encode our theorems about object languages. The "meta" is not really appropriate any more. I suppose for historical reasons people are used to "metatheorem", but it might make more sense just to dispense with "meta" and just talk about theorems. Or if not, then a better explanation of "meta" is needed than "for historical reasons" [Boyland](/wiki/user-boyland/) 01:35, 13 March 2007 (EDT) + +: You're right that this needs to be explained more clearly. The point of the "meta" is to distinguish between +- a _theorem_ in an encoded deductive system, and +- a _metatheorem_ about an encoded deductive system. +: For example, if I encoded a logic with a judgement ``A true``, then ``(all [a:prop] impl a a) true`` might be a theorem in the encoded logic (if I can come up with an LF term of that type), whereas cut elimination for the logic is a metatheorem about it. [Drl](/wiki/user-drl/) 10:14, 13 March 2007 (EDT) +:: To answer another point (we edit-conflicted each other making the same first point), I don't see this as an issue of "historical resasons" at all, this is the way I understand the research community working now - the two-year-old workshop on doing this sort of thing is called the [http://www.cis.upenn.edu/~sweirich/wmm/ Worshop on Mechanizing Metatheory]. People still think of Prolog as a "theorem prover" for Horn clauses (see the [[w:Prolog|Wikipedia page on Prolog]]), and those kinds of theorems are the same kinds of "theorems" (``plus 3 4 7``, ``of (lam [x] E x) (arrow unit unit)``) that are discussed on this page - they are nothing like ∀∃-statement metatheorems. Dan/John/others, where do you think this article could explain this better? — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 10:30, 13 March 2007 (EDT) + +:: Reading the current article, I'm okay with the explanation of "theorem", but the transition to the Totality assertion section should rehash the discussion at the top about "metatheorem" being a statement about an object language and contrast it with theorem (I missed the definition of "metatheorem" at the top on the first read). Also, when I used the word "theorem" this way, Frank objected that he likes to reserve "theorem" for a judgement defining truth in a logic (so not all derivable object-language judgements are "theorems"). Not sure if we want to use the word that way or not. [Drl](/wiki/user-drl/) 12:39, 13 March 2007 (EDT) +::: I'll work on the first point - as for the second point, perhaps I could add in to the example something with a logic - just , and say "we could reserve 'theorem' for this kind of statement, but to avoid saying 'derivable object-language judgment' too many times we often blur the distinction by referring to both as theorems. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 08:41, 14 March 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Metatheorem). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-modes-of-use.lf b/new-tests/stelf-output/pages/talk-modes-of-use.lf new file mode 100644 index 0000000..ed27079 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-modes-of-use.lf @@ -0,0 +1,8 @@ +% +%{! Anyone think of a better name for this? I realized the best way to explain this was basically to write down Bob's lecture on the subject... — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 13:37, 9 April 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Modes_of_use). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-mutable-state.lf b/new-tests/stelf-output/pages/talk-mutable-state.lf new file mode 100644 index 0000000..a17cab9 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-mutable-state.lf @@ -0,0 +1,15 @@ +% +%{! Something looks screwy with the two families named "update-lemma", one of which's body gets escaped into normal text because it's block-commented out. Otherwise, cool! [Drl](/wiki/user-drl/) 11:51, 16 March 2007 (EDT) +:: Fixed. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 12:29, 16 March 2007 (EDT) + +## Rename? + +Somehow I think [language with references](/wiki/mutable-state/) is a little weird of a title for this page; can I suggest [mutable state](/wiki/mutable-state/) or encoding references or +encoding store or something like that? I think it would be more in line with our other tutorials/case studies.  — [Tom 7](/wiki/user-tom7/) 08:42, 19 March 2007 (EDT) +: Agreed, I couldn't think of a good name when I wrote the page. I'll use [mutable state](/wiki/mutable-state/). — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 11:21, 19 March 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Mutable_state). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-natural-numbers-with-inequality.lf b/new-tests/stelf-output/pages/talk-natural-numbers-with-inequality.lf new file mode 100644 index 0000000..040a817 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-natural-numbers-with-inequality.lf @@ -0,0 +1,10 @@ +% +%{! It would probably be a good idea to standardize on one name for the uninhabited type in our tutorials. My vote goes to "void" since that is its common name in type theory. I think "false" is not a good choice since we often have terms inhabiting a type called false (in an encoding of classical logic, for instance). "Absurd" seems a little hokey to me, but I don't have a good argument against it...  — [Tom 7](/wiki/user-tom7/) 13:28, 6 September 2006 (MST) + +: I'm all for void, I'll make a note of it in [Twelf style guide](/wiki/twelf-style-guide/), since that's exactly the sort of thing that goes in a twelf style guide. - [TheTwelfElf](/wiki/user-thetwelfelf/) 13:37, 6 September 2006 (MST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Natural_numbers_with_inequality). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-numeric-termination-metrics.lf b/new-tests/stelf-output/pages/talk-numeric-termination-metrics.lf new file mode 100644 index 0000000..7fbee6a --- /dev/null +++ b/new-tests/stelf-output/pages/talk-numeric-termination-metrics.lf @@ -0,0 +1,8 @@ +% +%{! I stole this code from Karl's Comp+Ded class. Does anyone remember off-hand which cases actually use the metric? I don't want to have to do them until I find one that does. [Drl](/wiki/user-drl/) 17:13, 15 March 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Numeric_termination_metrics). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-output-factoring.lf b/new-tests/stelf-output/pages/talk-output-factoring.lf new file mode 100644 index 0000000..3d18c29 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-output-factoring.lf @@ -0,0 +1,23 @@ +% +%{! I think we should just delete the discussion of the hypothetical auto-factorer and Karl's comment at the end. Thoughts? [Drl](/wiki/user-drl/) 00:01, 15 March 2007 (EDT) +: Agreed, that's just from the old article :) — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 01:03, 15 March 2007 (EDT) + +## typo: "progress" should be "preservation"? + +In the section of this page labeled "Progress", this text appears: + +"We prove progress for the STLC presented in Representing the judgements of the STLC in the tutorial Proving metatheorems with Twelf. For review, here is the LF signature for the STLC:" + +Actually, that section (currently) discusses only preservation. Should the word "progress" have instead been "preservation"? + +Hesitant to edit, + +[[User:Clements|Clements]] 16:16, 19 April 2007 (EDT) + +: Heh, English is fun. The parentheses I wanted were "We prove progress for (the STLC presented in Representing the judgements of the STLC in the tutorial Proving metatheorems with Twelf)", not the other association. New wording should be less ambiguous. Thanks for the suggestion! [Drl](/wiki/user-drl/) 17:57, 19 April 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Output_factoring). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-output-freeness.lf b/new-tests/stelf-output/pages/talk-output-freeness.lf new file mode 100644 index 0000000..b121900 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-output-freeness.lf @@ -0,0 +1,37 @@ +% +%{! This topic was on the other wiki as well: http://fp.logosphere.cs.cmu.edu/twelf/?n=Answers.Freeness . It's by Geoff Washburn, not me, so if you include a lot from that file you'll need to add a <nowiki>\{\{license-by-sa\}\}</nowiki> tag. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 16:57, 7 September 2006 (MST) +* Oops, I didn't notice that. Well, my MIT-license version seems to have all the same info, so I don't think there's any need to import it, though perhaps the example could be a little simpler.  — [Tom 7](/wiki/user-tom7/) 06:54, 11 September 2006 (MST) + +## How does output coverage differ from output freeness? + +That question deserves a thorough answer, but Jason and I came up with this program while screwing around: observe that ``fake-thm`` output covers, even though the subgoal invocation of ``ff`` in the ``ft/z`` case is not "output free" because the output argument is a ground term: !}% +%sort nat %. +%term s %pi nat %-> nat %. +%term z nat %. +%sort bool %. +%term t bool %. +%term f bool %. +%sort add %. +%mode add %in %in %out %. +%term add/z add M z M %. +%term add/s %pi (add M (s N) (s O')) %<- (add M N O') %. +%worlds () (add _ _ _) %. +%total N (add _ N _) %. +%sort boolcont %. +%term @ {B} boolcont B %. +%sort ff {B bool} {C boolcont B} %. +%mode ff %in %out %. +%term _ ff B (@ B) %. +%sort fake-thm {M nat} %. +%mode fake-thm %out %. +%term ft/z %pi (fake-thm z) %<- (ff t (@ t)) %. +%worlds () (ff _ _) (fake-thm _) %. +%total D (ff D _) %. +%total [] (fake-thm _) %. +%{! :::  — [Tom 7](/wiki/user-tom7/) 16:58, 26 October 2006 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Output_freeness). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-programming-language-theory-with-twelf.lf b/new-tests/stelf-output/pages/talk-programming-language-theory-with-twelf.lf new file mode 100644 index 0000000..8131a49 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-programming-language-theory-with-twelf.lf @@ -0,0 +1,8 @@ +% +%{! Hey Todd! Writing Twelf code for all of this material and assembling it into one coherent story is a great idea! Have you taught a course along these lines with Twelf (or are you going to?)? Or is it just for fun? [Drl](/wiki/user-drl/) 14:22, 3 July 2008 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Programming_language_theory_with_Twelf). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-proving-metatheorems-with-twelf.lf b/new-tests/stelf-output/pages/talk-proving-metatheorems-with-twelf.lf new file mode 100644 index 0000000..18da095 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-proving-metatheorems-with-twelf.lf @@ -0,0 +1,21 @@ +% +%{! The content is good, but I'm not sure if it's directed to the right +audience. It seems to be directed to someone who wants to +know what's _really_ going on in Twelf. If someone just wants to see +how to use Twelf, they might get bogged down in Simply-Typed LF, +wondering when the interesting content is going to come. +[Boyland](/wiki/user-boyland/) 22:30, 25 February 2007 (EST) + +: I've actually been looking at the [http://haskell.org Haskell Wiki] lately - I see this as one way to get an idea of what we're going for here (Twelf's not Haskell, but it's a good successful model). They have an article [[haskell:Haskell in 5 steps|Haskell in 5 steps]] that might be a model for something a little simpler than this; however, I'm not sure what audience we'd be directing towards such a tutorial. The thing we've tried to think about for the [introductions to Twelf](/wiki/introductions-to-twelf/) is how we want to direct various audiences, as the intro says this aims to be the "I've read Types and Programming Languages" audience - i.e. people that are interested in the type theory of what they're doing. What would your say is the audience we're missing here? — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 23:30, 25 February 2007 (EST) + +:: The audience you're missing are the people who have read TAPL, have written their own natural language proofs and want to write machine-checked proofs. I'm also assuming they understand pattern-matching and logic programming at a practical level. The article claims to address the audience but then goes on a long time about LF's type system. Their interested in _their own_ type system, not _yours._ Sure, if they find Twelf expressive, they'll want to learn more, but to expound upon the type system before you've sold them on the usefulness is probably counter-productive. And anyway, if they are type aficionados, they'll turn up their nose(s) at the lack of higher-order. [Boyland](/wiki/user-boyland/) 20:01, 26 February 2007 (EST) + +## The third layer + +Does this line "The third layer presents some more-interesting proofs and introduces one additional feature of Twelf, the ability to do proofs about open terms." need to be tweaked with the restructuring of the tutorial? — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 20:25, 14 March 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Proving_metatheorems_with_Twelf). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-reduces.lf b/new-tests/stelf-output/pages/talk-reduces.lf new file mode 100644 index 0000000..ea1e2d5 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-reduces.lf @@ -0,0 +1,18 @@ +% +%{! ## Splitting this up + +At this point, this article is pretty incomplete as there is still much to be said about the %reduces directive. I did want to get a code example in, though. The addition of the ``nat-less-reduces`` stuff makes the code example longer than necessary. We might want to just cut it off after ``nat-plus-reduces-s`` or ``nat-plus-reduces`` and have the entire proof of ``nat-less-reduces`` as its own code example/tutorial. --[DanielKLee](/wiki/user-danielklee/) 10:08, 27 September 2006 (MST) +* In my opinion it's important to be as brief as possible with code snippets; they should be only the minimal size needed to illustrate the relevant point. Perhaps breaking it up into two sections would be the right way to go.  — [Tom 7](/wiki/user-tom7/) 10:39, 27 September 2006 (MST) + +## Mutual recursion + +Sometimes when I prove an inversion lemma, I need to use recursion sometimes but not all always +and determining whether to use recursion requires a case analysis. This means we have a mutual +meta-theorem. I would like to show that the main inversion lemma has a '<' reduction, but the helper lemma +only has a '<=' recursion. I just figured out how to handle this: first prove the mutual reduction with '<=', and then separately prove the main reduction with '<'. [Boyland](/wiki/user-boyland/) 09:53, 3 June 2013 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:%25reduces). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-reformulating-languages-to-use-hypothetical-judgements.lf b/new-tests/stelf-output/pages/talk-reformulating-languages-to-use-hypothetical-judgements.lf new file mode 100644 index 0000000..b700c74 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-reformulating-languages-to-use-hypothetical-judgements.lf @@ -0,0 +1,40 @@ +% +%{! Thanks Dan, very useful. + +Here's two other inductive properties of complete development: + +First, ==> is not only deterministic, but total: for every M there is some X s.t. M ==> X. + +Second, ==> completes any =>: + + cd_pr_triangle: (M ==> N) -> (M => L) -> (L => N). + +These two properties give an easy proof that => has the diamond property: put two of these triangles together along their ==> side. + +[[User:Rpollack|Rpollack]] 17:04, 28 February 2008 (EST) + +: Cool. I know there's a proof of CR in the Twelf examples directory, but I think it proves diamond for parallel reduction directly. If I get a chance, I'll Twelf these theorems up and then maybe turn this page into a case study instead of a tutorial. [Drl](/wiki/user-drl/) 16:08, 3 March 2008 (EST) + +:: Done, but not commented yet: [Church-Rosser via Complete Development](/wiki/church-rosser-via-complete-development/). +[Drl](/wiki/user-drl/) 10:34, 4 March 2008 (EST) + +--------- +How about improving the rule ==>/beta to: + + ==>/beta : (app (lam M) N) ==> M' N' + <- (lam M) ==> (lam M') + <- N ==> N'. + +This simplifies the proof of "sound" too. + +[[User:Rpollack|Rpollack]] 15:34, 29 February 2008 (EST) + +: Good idea---it makes sense to localize the binding here. I've seen situations where this move actually gives you less information (e.g., a typed beta-reduction rule in a calculus with a definitional equality rule---you can't invert the derivation of (lam M : A -> B) to get a hypothetical), but for this tutorial it makes sense. [Drl](/wiki/user-drl/) 16:08, 3 March 2008 (EST) + +:: Actually, there's one downside to this reformulation: in the original version, Twelf can prove => total automatically, but in the new version, it gets tripped up by the inversion that if (lam M) => N then N is (lam N1). I changed the definition back because I want to let Twelf prove this totality for me in [Church-Rosser via Complete Development](/wiki/church-rosser-via-complete-development/), and I want the code here to stay synchronized with that. [Drl](/wiki/user-drl/) 10:34, 4 March 2008 (EST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Reformulating_languages_to_use_hypothetical_judgements). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-respects-lemma.lf b/new-tests/stelf-output/pages/talk-respects-lemma.lf new file mode 100644 index 0000000..97d8d9b --- /dev/null +++ b/new-tests/stelf-output/pages/talk-respects-lemma.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-strengthening.lf b/new-tests/stelf-output/pages/talk-strengthening.lf new file mode 100644 index 0000000..a47ed3a --- /dev/null +++ b/new-tests/stelf-output/pages/talk-strengthening.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-substitution-lemma.lf b/new-tests/stelf-output/pages/talk-substitution-lemma.lf new file mode 100644 index 0000000..7ee2572 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-substitution-lemma.lf @@ -0,0 +1,31 @@ +% +%{! ## Suggestions for points that require clarification + +Go wild. --[DanielKLee](/wiki/user-danielklee/) 21:07, 24 September 2006 (MST) + +## exchange can be subtle + +the current draft suggests that exchange is valid whenever a variable cannot occur in a subsequent type, but unfortunately this simple statement is not true in general. the reason is that the well-formation of a subsequent type may depend on the preceding variable, even if the type itself does not involve that variable! this would come up in situations where the equational theory of types is such as to permit deduction of equations from existence of variables of preceding types, eg x : False, y : Garbage wouldn't permit exchange if the formation of Garbage depended on knowing that x : False is a contradiction! + +a related point is strengthening: it is NOT true in general that if a variable doesn't occur, then the variable can be dropped. it is in general a deep property of a type theory that non-occurring variables can be dropped. this was a main complication for the metatheory of lf itself: it is quite non-obvious how to prove strengthening, but this is necessary for some proofs of decidability for beta-eta conversion. + +[[User:Rwh|Rwh]] 21:18, 24 September 2006 (MST) + +: Point noted. Language modified. --[DanielKLee](/wiki/user-danielklee/) 21:36, 24 September 2006 (MST) + +## on the use of catch-all cases + +I want to pick a nit about the use of catch-all cases in definitions like substitution. They are no doubt convenient, since they obviate the need to introduce a base case for the new variable when recurring under a binder and save some of the associated complexity in the context blocks, but they are (IMHO) objectionable for at least two reasons, one minor and one more serious: + +- they violate the aesthetic principle that all information about fresh variables should be contained in the context along with the variable, which is after all the purpose of a context block; and +- from a logic-programming standpoint, they introduce _exponential non-determinism_, since, in a vacuous substitution, the substitution goal will match both the catch-all case and the case for its outermost constructor, and the same will be true at _every one of its subterms_; thus a vacuous substitution into a term of size n will have 2^n different derivations. + +Of course, if we are just talking about truth, then something that is true 2^n different ways is not any more or less true than something that is true in just one way, but again, why not "code" things up more efficiently if we can? Anyway, just a thought. + +[Twilson](/wiki/user-twilson/) 22:32, 3 March 2015 (EST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Substitution_lemma). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-thaw.lf b/new-tests/stelf-output/pages/talk-thaw.lf new file mode 100644 index 0000000..4820287 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-thaw.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-theorem.lf b/new-tests/stelf-output/pages/talk-theorem.lf new file mode 100644 index 0000000..1b40918 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-theorem.lf @@ -0,0 +1,30 @@ +% +%{! This page seems too terse and erroneous too. +%theorem is equivalent to a type family definition and a mode definition +(and much more readable!). +It does NOT include %worlds. +I know that _real_ Twelf users don't ever use it, even +when it would make their meta-theorems much easier to read. +I think this is a shame. %theorem declarations are +perfectly compatible with manual theorem proving. !}% +%sort bool %. +%term true bool %. +%term false bool %. +%freeze bool %. +%sort eq %. +%term eq/ eq B B %. +%theorem eq-reflexive : forall {X} exists {E eq X X} true %. +%term _ eq-reflexive _ eq/ %. +%worlds () (eq-reflexive _ _) %. +%total {} (eq-reflexive _ _) %. +%{! [Boyland](/wiki/user-boyland/) 22:36, 24 January 2008 (EST) + +The confusion here, I believe, is that %theorem does perform the %worlds job when being used as part of the [theorem prover](/wiki/theorem-prover/), but not when being used in totality-declaration-style. I think the edit here is accurate, if you want to run with it it could use a simple example. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 16:08, 30 January 2008 (EST) + +Seems fine now [John](/wiki/user-boyland/) 11:33, 6 February 2008 (EST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:%25theorem). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-totality-assertion.lf b/new-tests/stelf-output/pages/talk-totality-assertion.lf new file mode 100644 index 0000000..f01441e --- /dev/null +++ b/new-tests/stelf-output/pages/talk-totality-assertion.lf @@ -0,0 +1,44 @@ +% +%{! ## Why I somewhat rudely deleted the whole page and redirected to something I wrote previously + +I found the previous version of this troubling, mainly just because it was confusing, but also because I'm not sure what a page on "totality assertion" is supposed to contain. I redirected to my summary at [%total](/wiki/percent-total/); that summary is brief and should be expanded (especially with examples). But when a sentence needs to be bullet-pointed, I think it's either saying way too much for one sentence or way too little for us to understand! I propose that we have a page about [metatheorem](/wiki/metatheorem/) that discusses why you'd like to make a [%total](/wiki/percent-total/) declaration and what it means for metatheory, and a page about [%total](/wiki/percent-total/) (still needs work obviously) that explains how you do it and specifically what it means in terms of logic programming. + +In case you disagree and recreate this, specifically: +* there are no infinite-sized terms; if we have a LF term then it is of finite size. +* Though a %total does prove it's "always possible to derive an object.." it is stronger than that: Twelf's logic program search procedure will always succeed. +* It's critical for correctness that the inputs are ground, and the outputs are also guaranteed to be ground. +* There may be no inputs or no outputs (or neither), so using the word "some" might be misleading (it can be read as "exists") +* I'd say (defined by [%worlds](/wiki/percent-worlds/) consisting of zero or more [%block](/wiki/percent-block/)s), since that's all that worlds are. +:::: — [Tom 7](/wiki/user-tom7/) 10:20, 9 March 2007 (EST) + +### Original +A **totality assertion** states the following: + +* Given one or more [LF](/wiki/lf/) type families indexed by some number of objects +* If some of those indices are treated as inputs and some are treated as outputs (as defined by a [%mode](/wiki/percent-mode/) for each family) +* If the relevant part of the LF context is formed in a certain way (defined by [``%worlds``](/wiki/percent-worlds/) and zero or more [blocks](/wiki/percent-block/)) +* Then given terms for each of the indices treated as inputs, it is always possible to derive an object of any one of the type families, which means that it is possible to construct an object for each of the output types. Furthermore, based on an induction metric (defined by [``%total``](/wiki/percent-total/)), this term has finite size. + +If a type family is indexed by judgments, then a totality assertion proves the [metatheorem](/wiki/metatheorem/) that for all proof terms witnessing the truth of inputs, there exist proof terms witnessing the truth of each of the outputs; therefore, the totality assertion can be seen as the statement of a theorem that has the form ∀A∃B for some sets of propositions A and B. + +## See also + +* [Metatheorem](/wiki/metatheorem/) +* [%mode](/wiki/percent-mode/) +* [%block](/wiki/percent-block/) +* [%worlds](/wiki/percent-worlds/) +* [%total](/wiki/percent-total/) + +## What it should have contained + +The page on totality assertions should have contained a general discussion of totality assertions for LF type families. This is a different concept than %total. The abstract notion of a totality assertion for a type family has nothing to do with the details of how Twelf proves some of them. You can write down type family and a true totality assertion for it that Twelf can't prove with %total, or even that we can't prove in Twelf at all. If %total changes (say, Carsten finishes his logic programming project on permitting more general orderings, or we switch to a functional meta-language), the notion of a totality assertion for a type family doesn't change. That's why I think they should be separate pages. Neither Rob nor I have had time to write more than the place-holder for TAs yet though. Is there an easy way to undo the redirect? [Drl](/wiki/user-drl/) 11:51, 9 March 2007 (EST) +:Yeah, it's really easy to undo by using history to navigate to an old version, then _edit_ing that version, and saving. I'm not sure I understand why [totality assertion](/wiki/totality-assertion/) shouldn't be a part of [metatheorem](/wiki/metatheorem/) but what you say makes sense. I think it's important though that this page doesn't appear to be documentation for Twelf's current notion of totality, since that could potentially be very confusing.  — [Tom 7](/wiki/user-tom7/) 15:25, 9 March 2007 (EST) +::In fact, [metatheorem](/wiki/metatheorem/) already has a section called [totality assertion](/wiki/metatheorem/); perhaps that would be a satisfactory redirect target?  — [Tom 7](/wiki/user-tom7/) 15:33, 9 March 2007 (EST) +:::The whole point about [metatheorem](/wiki/metatheorem/) as I wrote it was to be hopelessly general - I don't even use dependent types. Then there is a separate issue of how we _specify_ totality assertions over "Twelf things" - that was the idea for this article, which is basically what Dan said. While I don't see why you deleted rather than blanking, one might as well just start over, all I did last night was give the page complete sentences, which it did not have before. As you said in the edit to [metatheorem](/wiki/metatheorem/), you can specify true totality assertions that ``%total`` can't check, so a totality assertion is something different than a ``%total``. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 19:26, 9 March 2007 (EST) +::::I didn't delete it, I just redirected it. Also, feel free to revert if you don't think that was sensible... I'm just being bold and I have a lot of wiki ego hitpoints, don't worry! I'm not disagreeing that there are various levels at which we could discuss what a totality assertion is: clearly there is what is currently implemented in Twelf CVS; there is what ought to be implemented; there is what's in Carsten's thesis; there is what might be implemented due to current research; there is the general notion of totality on Twelf relations, both in the sense of totality assertions that are recursively enumerable and the sort of undecidable Platonic math set-of-all-total relations; there is the more general still notion of totality on not-necessarily-Twelf relations, etc. But we ought not have an article on all of these! So what's the pedagogical reason for an article on [totality assertion](/wiki/totality-assertion/) that isn't covered by [%total](/wiki/percent-total/) or [metatheorem](/wiki/metatheorem/)? My feeling is that it is an unnecessary middle ground, at least not nearly as important as most of the redlinks on [Twelf glossary](/wiki/glossary/).  — [Tom 7](/wiki/user-tom7/) 16:39, 12 March 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Totality_assertion). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-tutorials.lf b/new-tests/stelf-output/pages/talk-tutorials.lf new file mode 100644 index 0000000..49ad751 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-tutorials.lf @@ -0,0 +1,63 @@ +% +%{! ## Tutorials that need written + +* Double negation translation - _Jcreed_ +* Classical logic - _Will_ +* Lexographic induction - _Jcreed_ (it would be good to have something more direct than cut elimination, which certainly uses lexicographic induction, but that's not the main point of the article) +* Translating between logics - +* Reverse the polarity - +* Closure conversion - _tom_ if he does it for his thesis +* from svn repository: ipc-taut?, systemf-bidi? +* dklee's as-pattern trick + +## Output factoring/freeness + +Output factoring/freeness _are_ similar problems - output factoring is the way you solve the fact that Twelf thinks you have a problem with output freeness, to put it simply. Do you think it's more or less confusing to have the two different tutorials and just have them perhaps refer to each other? — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 01:05, 12 October 2006 (EDT) +* Factoring is more generally a problem with output _coverage_. Output freeness is part of output coverage, but we needed to do factoring even with the unsound Twelf before the freeness check was introduced. Freeness is not documented anywhere, which is why it's important to have a page about it on the wiki. The articles should definitely be separate, IMO.  — [Tom 7](/wiki/user-tom7/) 08:34, 12 October 2006 (EDT) + +## order? + +What does everyone think about moving the proof techniques and illustrative examples up and the troubleshooting down? Right now, when you go to this page it says to me "Twelf is hard to use; here's how you work around the trouble." +We can compensate by outlining a path through the tutorials. +[Drl](/wiki/user-drl/) 00:18, 24 October 2006 (EDT) + +: Agreed. Also, we can factor things out of this page whenever it gets too large, and "troubleshooting" is a good canidate for factoring whenever that point comes. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 10:00, 24 October 2006 (EDT) + +## What is the difference between troubleshooting and proof techniques? + +I don't understand the difference between troubleshooting and proof techniques. At what point does "a way to fix a common problem" become a proof technique? If we want to make any distinction at all, I propose that there should be an "Understanding error messages" section that contains articles about what went wrong, and a "proof techniques" section that tells you how to fix it (if the fix requires enough of an idiom to warrant a separate page). In particular, I moved output factoring to the "techniques" rather than "troubleshooting", and then either someone moved it back or I screwed up the edit and forgot to delete it. But I don't think it should be in both places, since this page should read like a table of contents, not like a list of tags. Thoughts? [Drl](/wiki/user-drl/) 16:39, 13 March 2007 (EDT) + +: So I think we've reached the point of needing to factor things out that I mention above... I propose we just have "Beginner" "Advanced" and "Illustrative examples," and we say in the preamble of "Illustrative examples" that there is significant overlap between 'illustrative examples' and [Case studies](/wiki/case-studies/). I would then say that "Data structures in Twelf" and "Error messages" get their own articles, but I don't know how to organize that in the greater structure of the wiki... — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 22:27, 14 March 2007 (EDT) + +:: I dunno, I think this is a good place for articles on understanding error messages. What about collapsing data structures into an appropriate category (illustrative examples?) That might make it feel less sprawling. I think we have about as many top-level places to look for documentation (i.e., what's in the sidebar) as we can handle already. Maybe I'll play with this page later to try some ideas. [Drl](/wiki/user-drl/) 23:34, 14 March 2007 (EDT) + +::: So collapse troubleshooting into proof techniques, or just some of the troubleshooting things into proof techniques (like [output factoring](/wiki/output-factoring/))? Also, I'd like to think about where we put other extended introductions (Computation and Deduction, Hints on Proving Theorems (appel), Boyland's tutorial, etc) - on this page or in a less-privlidged section in the "Introductions" page (my mind reacts negatively to all the whitespace on the introductions page, though I know this is a false reaction). — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 00:47, 15 March 2007 (EDT) + +## the problem with the organization of this page + +The problem with the organization of this page is that there are multiple ways of indexing this data: what we prove, and how we prove it. Right now, the breakdown seems to be +* proof techniques: here's **how** you prove this in twelf +* illustrative examples: here's how you prove **this** in twelf (as long as it's not too involved, in which case it's a case study instead) + +The current organization puts an article in the category that best describes it, but there are overlaps (e.g., output factoring could be an example of proving progress, but the focus is more on the technique than the theorem; cut elim could be a proof technique for lexicographic induction, but the focus is on the example, not the technique). Then troubleshooting gets used for debugging hints that don't require any particular devices (so output factoring is a proof technique, not a troubleshooting). + +Alternative organizations: +- list articles under both indexings on one page +- have two different "sortings" for the tutorials page +The downside is that these might appear more complicated, and it will be harder for the reader to figure out what he's read and what he hasn't. And we'd have to maintain multiple indices as we add new tutorials. Also, there are some tutorials that fit into one sort but not the other (e.g., proof techniques that only have a toy examples). + +Since this is relatively early in the learning-Twelf chain, I think the interface here needs to be particularly clean. + +Thoughts on what's best? I think I'm in favor of the current organization, where we put something into the best category for it, even though that makes it a little unclear where things should go. + +[Drl](/wiki/user-drl/) 11:04, 15 March 2007 (EDT) + +:: I think I'm currently questioning the "too involved" line between case studies and illustrative examples... maybe those we could factor out, but then where would data structures go? I'd argue "case studies," but I'm not sure. Or perhaps "Case studies" should just move to "Illustrative examples" — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 11:58, 16 March 2007 (EDT) + +::: I think you're right---let's move the whole illustrative examples section (including data structures) to the case studies. I do think we should then make a beginner/advanced distinction on the case studies page, since some of them are more accessible and more heavily commented than others. This will simplify the organization considerably (tutorial for **how**, case study for **this**). Just because of the length, I'm thinking we should maintain separate pages, rather than moving the case studies here. [Drl](/wiki/user-drl/) 17:40, 16 March 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Tutorials). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-twelf-live.lf b/new-tests/stelf-output/pages/talk-twelf-live.lf new file mode 100644 index 0000000..496e1c3 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-twelf-live.lf @@ -0,0 +1,22 @@ +% +%{! ## Sandboxing +Maybe we should suggest ``check="decl"`` for sandboxing so that people can actually see their code without linking - if you're sandboxing, you're going to put everything in one box anyway. Yes, I made up the word _sandboxing_. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 01:07, 12 October 2006 (EDT) + +## Front page + +And I'm not sure we should promote this off the front page - it's kind of an inconvienent way to try twelf unless you happen to be a MediaWiki hacker. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 01:20, 12 October 2006 (EDT) + +## Broken? + +I was really looking forward to this, great idea for giving people a chance to try twelf, but it seems to be broken. +:: Fixed (sort of, for the moment) — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 11:41, 9 September 2010 (EDT) +:::Thanks a lot! excellent. [[Special:Contributions/134.157.168.250|134.157.168.250]] 12:51, 9 September 2010 (EDT) + +I think one of my queries brought it down. +:: Don't do that! — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 17:08, 18 June 2012 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Twelf_Live). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-twelf-style-guide.lf b/new-tests/stelf-output/pages/talk-twelf-style-guide.lf new file mode 100644 index 0000000..a32f800 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-twelf-style-guide.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-twelf-with-emacs.lf b/new-tests/stelf-output/pages/talk-twelf-with-emacs.lf new file mode 100644 index 0000000..f6ef5c7 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-twelf-with-emacs.lf @@ -0,0 +1,20 @@ +% +%{! ## Emacs Mode maintenance + +I have some questions: + +- Are the changes suggested for use with frames useful for everyone? If so, are they added to the CVS/SVN repository for future release? + +- I defined a ``twelf-check-declaration-forward`` command for checking all declarations in the current buffer starting from the point. I have found it useful. I sent Rob a diff. Is that the right way to contribute changes? + +- Is the emacs code being maintained? It seems the syntax highlighting is worse than linear. Or at least has a high constant factor. Highlighting a file of 8000 lines can take a minute on a fast machine (MacBook Pro with dual-core, 2GB RAM). Since highlighting is done automatically when a twelf file is visited, this delay is frequent and unavoidable unless one turns off highlighting, + +- More on maintenance: the emacs mode in CVS (twelf -1.5 R3) used the wrong names for parameters (debugging etc). Has this been fixed? + +[Boyland](/wiki/user-boyland/) 17:31, 7 May 2008 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Twelf_with_Emacs). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-uniqueness-lemma.lf b/new-tests/stelf-output/pages/talk-uniqueness-lemma.lf new file mode 100644 index 0000000..5a81be7 --- /dev/null +++ b/new-tests/stelf-output/pages/talk-uniqueness-lemma.lf @@ -0,0 +1,12 @@ +% +%{! Revisions of this page should account for the fact that the thing we're showing unique doesn't have to be an output (in the %mode sense). [Drl](/wiki/user-drl/) 13:18, 9 March 2007 (EST) + +## Rewrite + +I just rewrote the top half of the article to set up some ideas that I needed in the bottom half. Let me know if you have any complaints about the changes. [Drl](/wiki/user-drl/) 16:58, 14 March 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:Uniqueness_lemma). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-users-guide.lf b/new-tests/stelf-output/pages/talk-users-guide.lf new file mode 100644 index 0000000..b3a36ca --- /dev/null +++ b/new-tests/stelf-output/pages/talk-users-guide.lf @@ -0,0 +1,8 @@ +% +%{! The official Twelf User Guide is out-of-date and inaccurate it some places. In addition, I've occasionally run into undocumented error messages. We should keep an eye out for potential articles that fill in gaps or correct errors in the User Guide. Ideally, the User Guide would be wikified, but who actually has time to do that? --[DanielKLee](/wiki/user-danielklee/) 18:46, 17 September 2006 (MST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:User%27s_Guide). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-using-nat-less-with-reduces.lf b/new-tests/stelf-output/pages/talk-using-nat-less-with-reduces.lf new file mode 100644 index 0000000..40d2dfe --- /dev/null +++ b/new-tests/stelf-output/pages/talk-using-nat-less-with-reduces.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-whats-new.lf b/new-tests/stelf-output/pages/talk-whats-new.lf new file mode 100644 index 0000000..4d6276c --- /dev/null +++ b/new-tests/stelf-output/pages/talk-whats-new.lf @@ -0,0 +1,8 @@ +% +%{! I like the archiving old posts idea! — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 17:18, 13 October 2006 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:What%27s_new). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/talk-worlds.lf b/new-tests/stelf-output/pages/talk-worlds.lf new file mode 100644 index 0000000..20dcbbd --- /dev/null +++ b/new-tests/stelf-output/pages/talk-worlds.lf @@ -0,0 +1,8 @@ +% +%{! This is a good article. Kudos.  — [Tom 7](/wiki/user-tom7/) 11:20, 21 March 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Talk:%25worlds). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/tat-church-elf.lf b/new-tests/stelf-output/pages/tat-church-elf.lf new file mode 100644 index 0000000..eeefd06 --- /dev/null +++ b/new-tests/stelf-output/pages/tat-church-elf.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/tat-plus-elf.lf b/new-tests/stelf-output/pages/tat-plus-elf.lf new file mode 100644 index 0000000..973e81d --- /dev/null +++ b/new-tests/stelf-output/pages/tat-plus-elf.lf @@ -0,0 +1,40 @@ +% +%{! Part 1 !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort plus %. +%mode plus %in %in %out %. +%term p/z plus z N N %. +%term p/s %pi (plus (s N) M (s P)) %<- (plus N M P) %. +%worlds () (plus _ _ _) %. +%total N (plus N _ _) %. +%sort plus/z {N nat} %. +%mode plus/z %in %out %. +%scope plus/z %term z plus/z z p/z %. +%term s %pi (plus/z (s N) (p/s Dplus)) %<- (plus/z N (%the (plus N z N) Dplus)) %. +%worlds () (plus/z _ _) %. +%total N (plus/z N _) %. +%sort plus/s %. +%mode plus/s %in %out %. +%scope plus/s %term z plus/s p/z p/z %. +%term s + %pi (plus/s (p/s (%the (plus N M P) Dplus)) (p/s Dplus')) + %<- (plus/s Dplus (%the (plus N (s M) (s P)) Dplus')) %. +%worlds () (plus/s _ _) %. +%total D (plus/s D _) %. +%sort plus/commutes %. +%mode plus/commutes %in %out %. +%scope plus/commutes %term z %pi (plus/commutes p/z D) %<- (plus/z _ D) %. +%term s + %pi (plus/commutes (p/s (%the (plus N M P) Dplus)) Dplus'') + %<- (plus/commutes Dplus (%the (plus M N P) Dplus')) + %<- (plus/s Dplus' (%the (plus M (s N) (s P)) Dplus'')) %. +%worlds () (plus/commutes _ _) %. +%total D (plus/commutes D _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/TAT/plus.elf). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-bibtex-af04dependent.lf b/new-tests/stelf-output/pages/template-bibtex-af04dependent.lf new file mode 100644 index 0000000..777e73b --- /dev/null +++ b/new-tests/stelf-output/pages/template-bibtex-af04dependent.lf @@ -0,0 +1,18 @@ +% +%{! <bibtex>@article\{af04dependent, + author = "Andrew W. Appel and Amy P. Felty", + title = "Dependent Types Ensure Partial Correctness of Theorem Provers", + journal = "Journal of Functional Programming", + volume = "14", + number = "1", + pages = "3--19", + year = "2004", + month = "January", + note = "Available from \verb|http://www.cs.princeton.edu/~appel/papers/|", +</bibtex> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Bibtex:af04dependent). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-bibtex-crary03talt.lf b/new-tests/stelf-output/pages/template-bibtex-crary03talt.lf new file mode 100644 index 0000000..0e9046d --- /dev/null +++ b/new-tests/stelf-output/pages/template-bibtex-crary03talt.lf @@ -0,0 +1,13 @@ +% +%{! <bibtex>@InProceedings\{crary03talt, + author = "Karl Crary", + title = "Toward a Foundational Typed Assembly Language", + booktitle = "2003 Symposium on Principles of Programming Languages", + year = "2003", +\}</bibtex> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Bibtex:crary03talt). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-bibtex-fm97lax.lf b/new-tests/stelf-output/pages/template-bibtex-fm97lax.lf new file mode 100644 index 0000000..4517fd5 --- /dev/null +++ b/new-tests/stelf-output/pages/template-bibtex-fm97lax.lf @@ -0,0 +1,20 @@ +% +%{! <bibtex>@article\{264412, + author = \{Matt Fairtlough and Michael Mendler\}, + title = \{Propositional lax logic\}, + journal = \{Inf. Comput.\}, + volume = \{137\}, + number = \{1\}, + year = \{1997\}, + issn = \{0890-5401\}, + pages = \{1--33\}, + doi = \{http://dx.doi.org/10.1006/inco.1997.2627\}, + publisher = \{Academic Press, Inc.\}, + address = \{Duluth, MN, USA\}, + \}</bibtex> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Bibtex:fm97lax). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-bibtex-hhp93lf.lf b/new-tests/stelf-output/pages/template-bibtex-hhp93lf.lf new file mode 100644 index 0000000..d9a6c83 --- /dev/null +++ b/new-tests/stelf-output/pages/template-bibtex-hhp93lf.lf @@ -0,0 +1,20 @@ +% +%{! <bibtex>@article\{hhp93lf, + author = \{Robert Harper and Furio Honsell and Gordon Plotkin\}, + title = \{A framework for defining logics\}, + journal = \{J. ACM\}, + volume = \{40\}, + number = \{1\}, + year = \{1993\}, + issn = \{0004-5411\}, + pages = \{143--184\}, + doi = \{http://doi.acm.org/10.1145/138027.138060\}, + publisher = \{ACM Press\}, + address = \{New York, NY, USA\}, + \}</bibtex> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Bibtex:hhp93lf). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-bibtex-hl07mechanizing.lf b/new-tests/stelf-output/pages/template-bibtex-hl07mechanizing.lf new file mode 100644 index 0000000..50fc8c0 --- /dev/null +++ b/new-tests/stelf-output/pages/template-bibtex-hl07mechanizing.lf @@ -0,0 +1,15 @@ +% +%{! <bibtex>@Article\{hl07mechanizing, + author = \{Robert Harper and Daniel R. Licata\}, + title = \{Mechanizing Metatheory in a Logical Framework\}, + journal = \{Journal of Functional Programming\}, + year = \{2007\}, + note = \{To appear. Available from \verb|http://www.cs.cmu.edu/~drl/|\}, +\} +</bibtex> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Bibtex:hl07mechanizing). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-bibtex-lch07ml.lf b/new-tests/stelf-output/pages/template-bibtex-lch07ml.lf new file mode 100644 index 0000000..330b487 --- /dev/null +++ b/new-tests/stelf-output/pages/template-bibtex-lch07ml.lf @@ -0,0 +1,13 @@ +% +%{! <bibtex>@inproceedings\{ lee07ml, + author = "Daniel K. Lee and Karl Crary and Robert Harper", + title = "Towards a mechanized metatheory of Standard ML", + booktitle = "ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages", + year = "2007", +\}</bibtex> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Bibtex:lch07ml). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-bibtex-pd01modal.lf b/new-tests/stelf-output/pages/template-bibtex-pd01modal.lf new file mode 100644 index 0000000..9f8423c --- /dev/null +++ b/new-tests/stelf-output/pages/template-bibtex-pd01modal.lf @@ -0,0 +1,20 @@ +% +%{! <bibtex>@article\{pd01modal, + author = \{Frank Pfenning and Rowan Davies\}, + title = \{A judgmental reconstruction of modal logic\}, + journal = \{Mathematical. Structures in Comp. Sci.\}, + volume = \{11\}, + number = \{4\}, + year = \{2001\}, + issn = \{0960-1295\}, + pages = \{511--540\}, + doi = \{http://dx.doi.org/10.1017/S0960129501003322\}, + publisher = \{Cambridge University Press\}, + address = \{New York, NY, USA\}, + \}</bibtex> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Bibtex:pd01modal). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-bibtex-pierce02tapl.lf b/new-tests/stelf-output/pages/template-bibtex-pierce02tapl.lf new file mode 100644 index 0000000..362230b --- /dev/null +++ b/new-tests/stelf-output/pages/template-bibtex-pierce02tapl.lf @@ -0,0 +1,15 @@ +% +%{! <bibtex>@book\{pierce02tapl, + author = \{Benjamin C. Pierce\}, + title = \{Types and Programming Languages\}, + publisher = \{MIT Press\}, + month = \{February\}, + isbn = \{978-0-262-16209-8\}, + year = \{2002\}, +\}</bibtex> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Bibtex:pierce02tapl). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-bottom.lf b/new-tests/stelf-output/pages/template-bottom.lf new file mode 100644 index 0000000..cb2c4b9 --- /dev/null +++ b/new-tests/stelf-output/pages/template-bottom.lf @@ -0,0 +1,12 @@ +% +%{! <font face="Lucida Sans Unicode">⊥</font><noinclude> + +This template renders the _bottom_ (latex \bot) symbol, with workarounds for Internet Explorer. Use it by typing <nowiki>⊥</nowiki> in an article. + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Bottom). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-box.lf b/new-tests/stelf-output/pages/template-box.lf new file mode 100644 index 0000000..f1854f3 --- /dev/null +++ b/new-tests/stelf-output/pages/template-box.lf @@ -0,0 +1,12 @@ +% +%{! <font face="Lucida Sans Unicode">□</font><noinclude> + +This template renders the box (latex \box) symbol from modal logic, with workarounds for Internet Explorer. Use it by typing <nowiki>□</nowiki> in an article. + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Box). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-case-study.lf b/new-tests/stelf-output/pages/template-case-study.lf new file mode 100644 index 0000000..1accdc1 --- /dev/null +++ b/new-tests/stelf-output/pages/template-case-study.lf @@ -0,0 +1,11 @@ +% +%{! <hr> +<small>Read more Twelf [case studies](/wiki/case-studies/) and other Twelf [documentation](/wiki/documentation/).</small> + +<includeonly><!-- template isn't in the category, only uses of it -->[Category:Case studies](/wiki/category-case-studies/)</includeonly> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Case_study). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-click-inline.lf b/new-tests/stelf-output/pages/template-click-inline.lf new file mode 100644 index 0000000..58b24fd --- /dev/null +++ b/new-tests/stelf-output/pages/template-click-inline.lf @@ -0,0 +1,19 @@ +% +%{! <includeonly><span style="position:relative; overflow:hidden; width:\{\{\{width\}\}\}; height:\{\{\{height\}\}\}; z-index:2;">[[Image:\{\{\{image\}\}\}|\{\{\{width\}\}\}]]<span style="position:absolute; top:0; left:0; padding-top:3px; z-index:3;">[[\{\{\{link\}\}\}|<span style="float:left; width:\{\{\{width\}\}\}; height:\{\{\{height\}\}\}; font-size:\{\{\{height\}\}\}; line-height:\{\{\{height\}\}\}; word-spacing:\{\{\{width\}\}\}; cursor:pointer;">   </span>]]</span></span></includeonly><noinclude> +Copy of the wikipedia template for clicky images of the same name. +Usage: + <nowiki>\{\{</nowiki>click-inline + | image = some icon of size 20x30.png + | link = destination page link + | width = 20px + | height = 30px + <nowiki>\}\}</nowiki> + +This template doesn't work in all browsers; perhaps the [http://www.mediawiki.org/wiki/Extension:ImageMap imagemap] extension is better. +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Click-inline). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-darrow.lf b/new-tests/stelf-output/pages/template-darrow.lf new file mode 100644 index 0000000..2440f15 --- /dev/null +++ b/new-tests/stelf-output/pages/template-darrow.lf @@ -0,0 +1,12 @@ +% +%{! <font face="Lucida Sans Unicode">⇒</font><noinclude> + +This template renders the double right arrow (latex \Rightarrow) symbol, with workarounds for Internet Explorer. Use it by typing <nowiki>⇒</nowiki> in an article. + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Darrow). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-dia.lf b/new-tests/stelf-output/pages/template-dia.lf new file mode 100644 index 0000000..84fc12f --- /dev/null +++ b/new-tests/stelf-output/pages/template-dia.lf @@ -0,0 +1,12 @@ +% +%{! <font face="Lucida Sans Unicode">⋄</font><noinclude> + +This template renders the diamond (latex \dia) symbol from modal logic, with workarounds for Internet Explorer. Use it by typing <nowiki>⋄</nowiki> in an article. + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Dia). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-discuss.lf b/new-tests/stelf-output/pages/template-discuss.lf new file mode 100644 index 0000000..9c0c082 --- /dev/null +++ b/new-tests/stelf-output/pages/template-discuss.lf @@ -0,0 +1,17 @@ +% +%{! [[Talk:\{\{PAGENAME\}\}#\{\{\{1\}\}\}|<sup>[?]</sup>]]<noinclude> + + +This template places a discussion link in an article. The link takes the user to a supplied section on the talk page for the same article. (It does not work for pages outside the article namespace). To use it, write: + +:<nowiki>\{\{discuss|</nowiki>_description of issue_<nowiki>\}\}</nowiki> + +where _description of issue_ is an informative section header for the talk page. You should then create that section on the talk page. The link looks like this.[<sup>[?]</sup>](/wiki/talk-main-page/) + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Discuss). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-draft.lf b/new-tests/stelf-output/pages/template-draft.lf new file mode 100644 index 0000000..63a1893 --- /dev/null +++ b/new-tests/stelf-output/pages/template-draft.lf @@ -0,0 +1,17 @@ +% +%{! <div style="float:right;border:solid #EEBBBB 1px;margin:1px;"> +\{| cellspacing="0" style="background:#FFE7E7;" +| style="font-weight:bold;font-size:24pt;padding:6pt;line-height:1.25em;margin:12pt;" | Draft of \{\{<includeonly>subst:</includeonly>CURRENTDAY\}\} \{\{<includeonly>subst:</includeonly>CURRENTMONTHNAME\}\} \{\{<includeonly>subst:</includeonly>CURRENTYEAR\}\}. +|\} +</div><noinclude> + + +This template marks a page as a draft. To use it, write <nowiki>\{\{draft\}\}</nowiki> at the top of an article. That produces the box (with date) shown at the right. + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Draft). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-exists.lf b/new-tests/stelf-output/pages/template-exists.lf new file mode 100644 index 0000000..6699ce7 --- /dev/null +++ b/new-tests/stelf-output/pages/template-exists.lf @@ -0,0 +1,12 @@ +% +%{! <font face="Lucida Sans Unicode">∃</font><noinclude> + +This template renders the _there exists_ (latex \exists) symbol, with workarounds for Internet Explorer. Use it by typing <nowiki>∃</nowiki> in an article. + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Exists). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-forall.lf b/new-tests/stelf-output/pages/template-forall.lf new file mode 100644 index 0000000..b70ce90 --- /dev/null +++ b/new-tests/stelf-output/pages/template-forall.lf @@ -0,0 +1,12 @@ +% +%{! <font face="Lucida Sans Unicode">∀</font><noinclude> + +This template renders the _for all_ (latex \forall) symbol, with workarounds for Internet Explorer. Use it by typing <nowiki>∀</nowiki> in an article. + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Forall). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-front-getting-started.lf b/new-tests/stelf-output/pages/template-front-getting-started.lf new file mode 100644 index 0000000..363abaf --- /dev/null +++ b/new-tests/stelf-output/pages/template-front-getting-started.lf @@ -0,0 +1,12 @@ +% +%{! [Try Twelf now](/wiki/twelf-live/) from your browser! No installation necessary. + +Twelf runs on Windows and varieties of Unix, including Linux and Mac OS X. + +The [software](/download/) page has downloads, as well as installation instructions. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Front-Getting_Started). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-front-release-history.lf b/new-tests/stelf-output/pages/template-front-release-history.lf new file mode 100644 index 0000000..2ad9aca --- /dev/null +++ b/new-tests/stelf-output/pages/template-front-release-history.lf @@ -0,0 +1,18 @@ +% +%{! * **March 1, 2005** - Twelf 1.5R1 is now available +:*twelf-1-5R1.tar.gz (source distribution only) +:*Significantly improved tabling over Twelf 1.5 +:*The documentation is not yet updated +:*Re-packaged on March 8 in a form suitable for SML/NJ 110.0.07, SML/NJ 110.53, Poly/ML 4.1.3 and MLton 20040227 +* **December 3, 2004** We have made a working version of Twelf 1.5 available. It fixes a significant bug in the totality checker of Twelf 1.4 which let some false meta-theorems pass. +:*twelf-1-5.tar.gz (source distribution only) +:*This is a source distribution only +:*It contains several undocumented features, among them some uniqueness checking and improved tabling. + +[Full Release History](/wiki/release-history/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Front-Release_History). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-front-whats-new.lf b/new-tests/stelf-output/pages/template-front-whats-new.lf new file mode 100644 index 0000000..f48d60e --- /dev/null +++ b/new-tests/stelf-output/pages/template-front-whats-new.lf @@ -0,0 +1,8 @@ +% +%{! * **August 31, 2006** - The ConCert group at CMU met to discuss summer projects and to brainstorm ideas for improving communication and information dissemination, including in the context of the Twelf Wiki. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Front-What%27s_new). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-guide.lf b/new-tests/stelf-output/pages/template-guide.lf new file mode 100644 index 0000000..126e9b8 --- /dev/null +++ b/new-tests/stelf-output/pages/template-guide.lf @@ -0,0 +1,22 @@ +% +%{! [http://www.cs.cmu.edu/~twelf/guide-1-4/twelf_\{\{\{chapter\}\}\}.html#SEC\{\{\{section\}\}\} \{\{\{title\}\}\}] in the [User's Guide](/wiki/users-guide/)<noinclude> + +---- + +**Documentation:** This template http-links to the Twelf [User's Guide](/wiki/users-guide/). It takes three parameters, the ``chapter`` number, ``section`` number, and the ``title`` to use. For example: + +::<nowiki>\{\{guide|chapter=8|sec=47|title=Reduction declaration\}\}</nowiki> + +yields + +::[http://www.cs.cmu.edu/~twelf/guide-1-4/twelf_8.html#SEC47 Reduction declaration] in the [User's Guide](/wiki/users-guide/) + +We should use this to link to sections in the guide, since the URL may change when the guide is updated. + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Guide). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-keyword.lf b/new-tests/stelf-output/pages/template-keyword.lf new file mode 100644 index 0000000..371afcb --- /dev/null +++ b/new-tests/stelf-output/pages/template-keyword.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-license-by-nc.lf b/new-tests/stelf-output/pages/template-license-by-nc.lf new file mode 100644 index 0000000..a32f800 --- /dev/null +++ b/new-tests/stelf-output/pages/template-license-by-nc.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-license-by-sa.lf b/new-tests/stelf-output/pages/template-license-by-sa.lf new file mode 100644 index 0000000..a32f800 --- /dev/null +++ b/new-tests/stelf-output/pages/template-license-by-sa.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-license-gfdl.lf b/new-tests/stelf-output/pages/template-license-gfdl.lf new file mode 100644 index 0000000..2283003 --- /dev/null +++ b/new-tests/stelf-output/pages/template-license-gfdl.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-license-gpl.lf b/new-tests/stelf-output/pages/template-license-gpl.lf new file mode 100644 index 0000000..08ae092 --- /dev/null +++ b/new-tests/stelf-output/pages/template-license-gpl.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-literatetwelf.lf b/new-tests/stelf-output/pages/template-literatetwelf.lf new file mode 100644 index 0000000..a32f800 --- /dev/null +++ b/new-tests/stelf-output/pages/template-literatetwelf.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-lolli.lf b/new-tests/stelf-output/pages/template-lolli.lf new file mode 100644 index 0000000..eeefd06 --- /dev/null +++ b/new-tests/stelf-output/pages/template-lolli.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-needs.lf b/new-tests/stelf-output/pages/template-needs.lf new file mode 100644 index 0000000..eeefd06 --- /dev/null +++ b/new-tests/stelf-output/pages/template-needs.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-needswork.lf b/new-tests/stelf-output/pages/template-needswork.lf new file mode 100644 index 0000000..c5afb53 --- /dev/null +++ b/new-tests/stelf-output/pages/template-needswork.lf @@ -0,0 +1,12 @@ +% +%{! \{| style="width:100%; margin-bottom:5px; border: 1px solid; background-color: #f2dfce; text-align:center; font-size:0.9em;" +|- +| style="align:center;" |This page needs some work: _\{\{\{1\}\}\}_ +|\} +<includeonly>[Category:Pages needing attention](/wiki/category-pages-needing-attention/)</includeonly> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Needswork). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-oplus.lf b/new-tests/stelf-output/pages/template-oplus.lf new file mode 100644 index 0000000..46fa94d --- /dev/null +++ b/new-tests/stelf-output/pages/template-oplus.lf @@ -0,0 +1,12 @@ +% +%{! <font face="Lucida Sans Unicode">⊕</font><noinclude> + +This template renders the _linear or_ (latex \oplus) symbol, with workarounds for Internet Explorer. Use it by typing <nowiki>⊕</nowiki> in an article. + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Oplus). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-otimes.lf b/new-tests/stelf-output/pages/template-otimes.lf new file mode 100644 index 0000000..920d0a8 --- /dev/null +++ b/new-tests/stelf-output/pages/template-otimes.lf @@ -0,0 +1,12 @@ +% +%{! <font face="Lucida Sans Unicode">⊗</font><noinclude> + +This template renders the tensor (latex \otimes) symbol, with workarounds for Internet Explorer. Use it by typing <nowiki>⊗</nowiki> in an article. + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Otimes). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-proving-metatheorems.lf b/new-tests/stelf-output/pages/template-proving-metatheorems.lf new file mode 100644 index 0000000..1bb8be8 --- /dev/null +++ b/new-tests/stelf-output/pages/template-proving-metatheorems.lf @@ -0,0 +1,18 @@ +% +%{! \{| style="width:100%; margin-bottom:5px; border: 1px solid #ADA; background-color: #E4F2E4; text-align:center; font-size:0.9em;" +|- +| style="width:35%; text-align:left; font-size:0.9em;" | \{\{\{noprevious|Previous:<br />[[Proving metatheorems:\{\{\{prev\}\}\}|\{\{\{prevname\}\}\}]]\}\}\} +| style="width:30%;" | <big>**[Proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)** </big> +| style="width:35%; text-align:right;font-size:0.9em;" | \{\{\{nonext|Next:<br />[[Proving metatheorems:\{\{\{next\}\}\}|\{\{\{nextname\}\}\}]]\}\}\} +|\} +<br />__NOTOC__<noinclude> + +This template is for use in the [Proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/) multi-page article. It should be given arguments prev (a wiki page) and prevname (the short name to use in the link), as well as arguments next and nextname. If the arguments noprevious or nonext are specified, then these links will not show at all. + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Proving_metatheorems). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-proving.lf b/new-tests/stelf-output/pages/template-proving.lf new file mode 100644 index 0000000..89d9480 --- /dev/null +++ b/new-tests/stelf-output/pages/template-proving.lf @@ -0,0 +1,20 @@ +% +%{! [[Proving metatheorems:\{\{\{title\}\}\}|\{\{\{title\}\}\}]] in the tutorial [Proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/)<noinclude> + +---- + +**Documentation:** This template wiki-links to the [User:Drl](/wiki/user-drl/)'s tutorial [Proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/). As an example,: + +::<nowiki>\{\{proving|title=Full LF\}\}</nowiki> + +yields + +::[Full LF](/wiki/proving-metatheorems-full-lf/) in the tutorial [Proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/) + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Proving). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-repository.lf b/new-tests/stelf-output/pages/template-repository.lf new file mode 100644 index 0000000..12d5ef7 --- /dev/null +++ b/new-tests/stelf-output/pages/template-repository.lf @@ -0,0 +1,8 @@ +% +%{! [https://cvs.concert.cs.cmu.edu/twelf-lib/\{\{\{1\}\}\} \{\{\{2\}\}\}] !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Repository). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-rule.lf b/new-tests/stelf-output/pages/template-rule.lf new file mode 100644 index 0000000..1911e00 --- /dev/null +++ b/new-tests/stelf-output/pages/template-rule.lf @@ -0,0 +1,20 @@ +% +%{! <strong>This template does not work because template arguments are not expanded inside the <nowiki><math></nowiki> tag in the current MediaWiki. Any workarounds?</strong> + + +<noinclude> + +<s>Render an inference rule. Use: \{<nowiki>\{</nowiki>rule|premise1 \qquad premise2 ...|conclusion|name<nowiki>\}</nowiki>\}. + +It produces: + + +</s> + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Rule). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-stub-tutorial.lf b/new-tests/stelf-output/pages/template-stub-tutorial.lf new file mode 100644 index 0000000..eda54c8 --- /dev/null +++ b/new-tests/stelf-output/pages/template-stub-tutorial.lf @@ -0,0 +1,9 @@ +% +%{! <hr><small>This is a tutorial that has been identified as needing to be extended, expanded, and or improved, and which does not yet a contain a full explanation of what it is meant to address. We're getting to it, leave a message for us on the discussion page or elsewhere if you need this information and we'll probably get to it faster.</small> +<includeonly>[Category:Stubs](/wiki/category-stubs/)[Category:Tutorials needing attention](/wiki/category-tutorials-needing-attention/)</includeonly> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Stub_tutorial). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-stub.lf b/new-tests/stelf-output/pages/template-stub.lf new file mode 100644 index 0000000..aedb076 --- /dev/null +++ b/new-tests/stelf-output/pages/template-stub.lf @@ -0,0 +1,9 @@ +% +%{! <hr><small>This page is incomplete. We should expand it.</small> +<includeonly>[Category:Stubs](/wiki/category-stubs/)[Category:Pages needing attention](/wiki/category-pages-needing-attention/)</includeonly> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Stub). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-summer-school-2008.lf b/new-tests/stelf-output/pages/template-summer-school-2008.lf new file mode 100644 index 0000000..a98ca2f --- /dev/null +++ b/new-tests/stelf-output/pages/template-summer-school-2008.lf @@ -0,0 +1,19 @@ +% +%{! \{| style="width:40%; margin-bottom:5px; border: 1px solid #ADA; background-color: #E4F2E4; text-align:left; font-size:0.9em;" +|<big>**[Summer school 2008](/wiki/summer-school-2008/)** </big> +|- +|style="text-align:left; font-size:0.9em;" |\{\{\{noprevious|Previous: [[Summer school 2008:\{\{\{prev\}\}\}|\{\{\{prevname\}\}\}]]\}\}\} +|- +|style="text-align:left;font-size:0.9em;" |\{\{\{nonext|Next: [[Summer school 2008:\{\{\{next\}\}\}|\{\{\{nextname\}\}\}]]\}\}\} +|\} +<br />__NOTOC__<noinclude> + +This template is for use in the [Summer school 2008](/wiki/summer-school-2008/) course notes. It should be given arguments prev (a wiki page) and prevname (the short name to use in the link), as well as arguments next and nextname. If the arguments noprevious or nonext are specified, then these links will not show at all. + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Summer_school_2008). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-talk-bibtex-hl07mechanizing.lf b/new-tests/stelf-output/pages/template-talk-bibtex-hl07mechanizing.lf new file mode 100644 index 0000000..6d44df0 --- /dev/null +++ b/new-tests/stelf-output/pages/template-talk-bibtex-hl07mechanizing.lf @@ -0,0 +1,9 @@ +% +%{! Cool!  — [Tom 7](/wiki/user-tom7/) 22:19, 14 March 2007 (EDT) +: Why thanks! At some point I'll need to translate the rest of the extension to English from French. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 22:21, 14 March 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template_talk:Bibtex:hl07mechanizing). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-talk-click-inline.lf b/new-tests/stelf-output/pages/template-talk-click-inline.lf new file mode 100644 index 0000000..68af787 --- /dev/null +++ b/new-tests/stelf-output/pages/template-talk-click-inline.lf @@ -0,0 +1,8 @@ +% +%{! History: This template is copied from the GFDL template of the same name [http://en.wikipedia.org/wiki/Template:Click-Inline at wikipedia]. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template_talk:Click-inline). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-talk-guide.lf b/new-tests/stelf-output/pages/template-talk-guide.lf new file mode 100644 index 0000000..93ffdd4 --- /dev/null +++ b/new-tests/stelf-output/pages/template-talk-guide.lf @@ -0,0 +1,10 @@ +% +%{! ## Guide gets updated and... + +Won't this template be a problem if the section numbers in the guide change? Ideally this template should be <nowiki>\{\{guide4\}\}</nowiki> or something, I would think... — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 17:35, 27 September 2006 (MST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template_talk:Guide). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-talk-the-twelf-project-introduction.lf b/new-tests/stelf-output/pages/template-talk-the-twelf-project-introduction.lf new file mode 100644 index 0000000..5092502 --- /dev/null +++ b/new-tests/stelf-output/pages/template-talk-the-twelf-project-introduction.lf @@ -0,0 +1,13 @@ +% +%{! Are piped links considered bad style on this wiki? +[Drl](/wiki/user-drl/) 15:12, 23 October 2006 (EDT) + +## length requirements + +I want this paragraph to be short enough that both introductions are on the first screenfull on my 1024x768 monitor. If someone could add _gracefulness_ that would be cool.[Drl](/wiki/user-drl/) 01:21, 24 October 2006 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template_talk:The_Twelf_Project/Introduction). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-the-twelf-project-getting-started.lf b/new-tests/stelf-output/pages/template-the-twelf-project-getting-started.lf new file mode 100644 index 0000000..a739ab6 --- /dev/null +++ b/new-tests/stelf-output/pages/template-the-twelf-project-getting-started.lf @@ -0,0 +1,18 @@ +% +%{! [Download](/download/) Twelf or [try it online](/wiki/twelf-live/). + +Learn Twelf: +* Read the [introductions to Twelf](/wiki/introductions-to-twelf/) first. +* The [tutorials](/wiki/tutorials/) explain common Twelf tricks and techniques. +* The [case studies](/wiki/case-studies/) present interesting applications of Twelf. +* The [glossary](/wiki/glossary/) defines Twelf terminology. +* Have a question? [Ask Twelf Elf](/wiki/ask-twelf-elf/)! +<!-- style guide will go here eventually, but it's not ready yet --> + +See the [documentation](/wiki/documentation/) page for more resources. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:The_Twelf_Project/Getting_Started). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-the-twelf-project-introduction.lf b/new-tests/stelf-output/pages/template-the-twelf-project-introduction.lf new file mode 100644 index 0000000..aee96ad --- /dev/null +++ b/new-tests/stelf-output/pages/template-the-twelf-project-introduction.lf @@ -0,0 +1,12 @@ +% +%{! [Twelf](/wiki/about-the-twelf-project/) is a language used to specify, implement, and prove properties of _deductive systems_ such as programming languages and logics. Large [research projects using Twelf](/wiki/research-projects-using-twelf/) include the TALT typed assembly language, a foundational proof-carrying-code system, and a type safety proof for Standard ML. + +Visitors without a technical background are encouraged to read the [general description of Twelf](/wiki/general-description-of-twelf/). + +<!-- (I need improvement! [http://twelf.plparty.org/w/index.php?title=Template:The_Twelf_Project/Introduction&action=edit edit me]!) --> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:The_Twelf_Project/Introduction). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-tl.lf b/new-tests/stelf-output/pages/template-tl.lf new file mode 100644 index 0000000..5b2a272 --- /dev/null +++ b/new-tests/stelf-output/pages/template-tl.lf @@ -0,0 +1,12 @@ +% +%{! <nowiki>\{\{</nowiki>[[Template:\{\{\{1\}\}\}|\{\{\{1\}\}\}]]<nowiki>\}\}</nowiki><noinclude> + +This template can be used for linking to other templates in expository text. For instance, to link to [Template:vdash](/wiki/template-vdash/), write <nowiki>\{\{tl|vdash\}\}</nowiki>, which renders as <nowiki>\{\{</nowiki>[vdash](/wiki/template-vdash/)<nowiki>\}\}</nowiki>. + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Tl). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-top.lf b/new-tests/stelf-output/pages/template-top.lf new file mode 100644 index 0000000..536cb6e --- /dev/null +++ b/new-tests/stelf-output/pages/template-top.lf @@ -0,0 +1,12 @@ +% +%{! <font face="Lucida Sans Unicode">⊤</font><noinclude> + +This template renders the top (or 'true') symbol, with workarounds for Internet Explorer. Use it by typing <nowiki>⊤</nowiki> in an article. + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Top). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-tutorial.lf b/new-tests/stelf-output/pages/template-tutorial.lf new file mode 100644 index 0000000..ce55055 --- /dev/null +++ b/new-tests/stelf-output/pages/template-tutorial.lf @@ -0,0 +1,11 @@ +% +%{! <hr> +<small>Read more Twelf [tutorial](/wiki/tutorials/)s and other Twelf [documentation](/wiki/documentation/).</small> + +<includeonly><!-- template isn't in the category, only uses of it -->[Category:Tutorials](/wiki/category-tutorials/)</includeonly> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Tutorial). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-twelf-code.lf b/new-tests/stelf-output/pages/template-twelf-code.lf new file mode 100644 index 0000000..e23d197 --- /dev/null +++ b/new-tests/stelf-output/pages/template-twelf-code.lf @@ -0,0 +1,9 @@ +% +%{! <hr><small>The above page contains an example of [[:Category:Twelf code|Twelf code]]. It may be attached to a [tutorial](/wiki/tutorials/) as an illustrative example, or it may stand on its own.</small> +<includeonly>[Category:Twelf code](/wiki/category-twelf-code/)</includeonly> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Twelf_code). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-undocumented.lf b/new-tests/stelf-output/pages/template-undocumented.lf new file mode 100644 index 0000000..9a069f0 --- /dev/null +++ b/new-tests/stelf-output/pages/template-undocumented.lf @@ -0,0 +1,16 @@ +% +%{! <div style="float:right;border:solid #BBBBEE 1px;margin:1em;"> +\{| cellspacing="0" style="background:#E7E7FF;" +| style="font-size:8pt;padding:4pt;line-height:1.25em;" | This \{\{\{1|article or section\}\}\} describes an [[:category:undocumented features|undocumented feature]] of Twelf.<br/>The information may be incomplete and subject to change. +|\} +</div><includeonly>[Category:Undocumented features](/wiki/category-undocumented-features/)</includeonly><noinclude><br clear="all" /> + +This template also lists the page in the [[:Category:undocumented features|undocumented features]] category. Use with an optional argument to change the text "article or section" (_e.g._ <nowiki>\{\{undocumented|article\}\}</nowiki>). + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Undocumented). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/template-vdash.lf b/new-tests/stelf-output/pages/template-vdash.lf new file mode 100644 index 0000000..116e9f9 --- /dev/null +++ b/new-tests/stelf-output/pages/template-vdash.lf @@ -0,0 +1,13 @@ +% +%{! <font face="Lucida Sans Unicode">⊦</font><noinclude> + +This template renders the turnstile (latex \vdash) symbol, with workarounds for Internet Explorer. Use it by typing <nowiki>⊦</nowiki> in an article. + +\{\{needs|more testing on versions of internet explorer. some systems have different behavior. any tips?\}\} +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Template:Vdash). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/tethered-modal-logic.lf b/new-tests/stelf-output/pages/tethered-modal-logic.lf new file mode 100644 index 0000000..0da948a --- /dev/null +++ b/new-tests/stelf-output/pages/tethered-modal-logic.lf @@ -0,0 +1,240 @@ +% +%{! Alex Simpson's thesis shows how to encode a variety of constructive +modal logics by making box and diamond into universal and existential +quantifiers over accessible Kripke worlds and orthogonally allowing +axiomatization of the accessibility relation. + +Here we encode and prove cut admissibility for a sequent calculus for +a similar spectrum of logics. The difference is that uses of +left rules are constrained (`tethered') to only fire when the label on the +conclusion is the same as the label on the hypothesis. + +The conjecture was that Pfenning-Davies judgmental modal logic is +exactly tethered modal logic where accessibility relation is +axiomatized by reflexivity and transitivity. However, this is refuted by considering , which is provable in Pfenning-Davies and (apparently?) not in the current system. The exact status of this logic relative to other constructive modal logics is still being determined. + +The standard litmus-test entailment that fails in this logic (and succeeds in Simpsons') +is + + + +(Author: Jason Reed, based on Pfenning's encoding of intuitionistic +logic with cut admissibility) +!}% +%sort w %. +% worlds +%name w %. +%sort <= %. +% accessibility relation +%prec %none 3 <= %. +%term refl P <= P %. +%term trans %pi (P <= Q) %-> (Q <= R) %-> (P <= R) %. +%term sym %pi (P <= Q) %-> (Q <= P) %. +%sort o %. +% formulas +%name o %. +%term and %pi o %-> o %-> o %. +%prec %right 11 and %. +%term imp %pi o %-> o %-> o %. +%prec %right 10 imp %. +%term or %pi o %-> o %-> o %. +%prec %right 11 or %. +%term true o %. +%term false o %. +%term box %pi o %-> o %. +%term dia %pi o %-> o %. +% Sequent Calculus +%sort hyp %. +% Hypotheses (left) +%sort ghyp %. +% Global-after-P Hypotheses (left) +%sort conc %. +% Conclusion (right) +%name hyp %. +%name conc %. +%term axiom %pi (hyp A P) %-> (conc A P) %. +%term andr %pi (conc A P) %-> (conc B P) %-> (conc (A and B) P) %. +%term andl1 %pi (%pi (hyp A P) %-> (conc C P)) %-> (hyp (A and B) P) %-> (conc C P) %. +%term andl2 %pi (%pi (hyp B P) %-> (conc C P)) %-> (hyp (A and B) P) %-> (conc C P) %. +%term impr %pi (%pi (hyp A P) %-> (conc B P)) %-> (conc (A imp B) P) %. +%term impl + %pi (conc A P) + %-> (%pi (hyp B P) %-> (conc C P)) + %-> (hyp (A imp B) P) + %-> (conc C P) %. +%term orr1 %pi (conc A P) %-> (conc (A or B) P) %. +%term orr2 %pi (conc B P) %-> (conc (A or B) P) %. +%term orl + %pi (%pi (hyp A P) %-> (conc C P)) + %-> (%pi (hyp B P) %-> (conc C P)) + %-> (hyp (A or B) P) + %-> (conc C P) %. +%term truer conc true P %. +%term falsel %pi (hyp false P) %-> (conc C P) %. +%term boxr %pi ({a w} %pi (P <= a) %-> (conc A a)) %-> (conc (box A) P) %. +%term boxl %pi (%pi (ghyp A P) %-> (conc C P)) %-> (hyp (box A) P) %-> (conc C P) %. +% Can copy to any world after P regardless of conc +%term copy %pi (%pi (hyp A Q) %-> (conc C R)) %-> (P <= Q) %-> (ghyp A P) %-> (conc C R) %. +%term diar %pi (conc A Q) %-> (P <= Q) %-> (conc (dia A) P) %. +%term dial + %pi ({a w} %pi (P <= a) %-> (hyp A a) %-> (conc C P)) + %-> (hyp (dia A) P) + %-> (conc C P) %. +%%% Termination Metric +%sort little %. +%sort big %. +%term little/ little %. +%term big/ %pi little %-> big %. +%%% Cut admissibility +%sort ca {M little} {A o} %. +%sort cag {M big} {A o} {a w} %. +%% Axioms +%term ca_axiom_d ca M A (axiom H) E (E H) %. +%term ca_axiom_e ca M A D axiom D %. +%% Principal Cases +%term ca_and1 + %pi (ca M (A1 and A2) (andr D1 D2) ([h] andl1 (E1 h) h) F) + %<- ({h1} ca M (A1 and A2) (andr D1 D2) ([h] E1 h h1) (E1' h1)) + %<- (ca M A1 D1 E1' F) %. +%term ca_and2 + %pi (ca M (A1 and A2) (andr D1 D2) ([h] andl2 (E2 h) h) F) + %<- ({h2} ca M (A1 and A2) (andr D1 D2) ([h] E2 h h2) (E2' h2)) + %<- (ca M A2 D2 E2' F) %. +%term ca_imp + %pi (ca M (A1 imp A2) (impr D2) ([h] impl (E1 h) (E2 h) h) F) + %<- (ca M (A1 imp A2) (impr D2) E1 E1') + %<- ({h2} ca M (A1 imp A2) (impr D2) ([h] E2 h h2) (E2' h2)) + %<- (ca M A1 E1' D2 D2') + %<- (ca M A2 D2' E2' F) %. +%term ca_or1 + %pi (ca M (A1 or A2) (orr1 D1) ([h] orl (E1 h) (E2 h) h) F) + %<- ({h1} ca M (A1 or A2) (orr1 D1) ([h] E1 h h1) (E1' h1)) + %<- (ca M A1 D1 E1' F) %. +%term ca_or2 + %pi (ca M (A1 or A2) (orr2 D2) ([h] orl (E1 h) (E2 h) h) F) + %<- ({h2} ca M (A1 or A2) (orr2 D2) ([h] E2 h h2) (E2' h2)) + %<- (ca M A2 D2 E2' F) %. +%term ca_box + %pi (ca M (box A) (boxr D1) ([h] boxl (E1 h) h) F) + %<- ({h2} ca M (box A) (boxr D1) ([h] E1 h h2) (F' h2)) + %<- (cag (big/ little/) A D1 F' F) %. +%term ca_dia + %pi (ca M (dia A) (diar D1 ACC) ([h] dial (E1 h) h) F) + %<- ({a w} {acc} {h2 hyp A a} ca M (dia A) (diar D1 ACC) ([h] E1 h a acc h2) (F' a acc h2)) + %<- (ca M A D1 ([h2] F' _ ACC h2) F) %. +%% D-Commutative Conversions +%term cad_andl1 %pi (ca M A (andl1 D1 H) E (andl1 D1' H)) %<- ({h1} ca M A (D1 h1) E (D1' h1)) %. +%term cad_andl2 %pi (ca M A (andl2 D2 H) E (andl2 D2' H)) %<- ({h2} ca M A (D2 h2) E (D2' h2)) %. +%term cad_impl + %pi (ca M A (impl D1 D2 H) E (impl D1 D2' H)) + %<- ({h2} ca M A (D2 h2) E (D2' h2)) %. +%term cad_orl + %pi (ca M A (orl D1 D2 H) E (orl D1' D2' H)) + %<- ({h1} ca M A (D1 h1) E (D1' h1)) + %<- ({h2} ca M A (D2 h2) E (D2' h2)) %. +%term cad_falsel ca M A (falsel H) E (falsel H) %. +%term cad_boxl %pi (ca M A (boxl D1 H) E (boxl D1' H)) %<- ({h} ca M A (D1 h) E (D1' h)) %. +%term cad_dial + %pi (ca M A (dial D1 H) E (dial D1' H)) + %<- ({a w} {acc} {h hyp B1 a} ca M A (D1 a acc h) E (D1' a acc h)) %. +%term cad_copy %pi (ca M A (copy D ACC H) E (copy D' ACC H)) %<- ({h} ca M A (D h) E (D' h)) %. +%% E-Commutative Conversions +%term cae_axiom ca M A D ([h] axiom H1) (axiom H1) %. +%term cae_andr + %pi (ca M A D ([h] andr (E1 h) (E2 h)) (andr E1' E2')) + %<- (ca M A D E1 E1') + %<- (ca M A D E2 E2') %. +%term cae_andl1 + %pi (ca M A D ([h] andl1 (E1 h) H) (andl1 E1' H)) + %<- ({h1} ca M A D ([h] E1 h h1) (E1' h1)) %. +%term cae_andl2 + %pi (ca M A D ([h] andl2 (E2 h) H) (andl2 E2' H)) + %<- ({h2} ca M A D ([h] E2 h h2) (E2' h2)) %. +%term cae_impr + %pi (ca M A D ([h] impr (E2 h)) (impr E2')) + %<- ({h1} ca M A D ([h] E2 h h1) (E2' h1)) %. +%term cae_impl + %pi (ca M A D ([h] impl (E1 h) (E2 h) H) (impl E1' E2' H)) + %<- (ca M A D E1 E1') + %<- ({h2} ca M A D ([h] E2 h h2) (E2' h2)) %. +%term cae_orr1 %pi (ca M A D ([h] orr1 (E1 h)) (orr1 E1')) %<- (ca M A D E1 E1') %. +%term cae_orr2 %pi (ca M A D ([h] orr2 (E2 h)) (orr2 E2')) %<- (ca M A D E2 E2') %. +%term cae_orl + %pi (ca M A D ([h] orl (E1 h) (E2 h) H) (orl E1' E2' H)) + %<- ({h1} ca M A D ([h] E1 h h1) (E1' h1)) + %<- ({h2} ca M A D ([h] E2 h h2) (E2' h2)) %. +%term cae_truer ca M A D ([h] truer) truer %. +%term cae_falsel ca M A D ([h] falsel H) (falsel H) %. +%term cae_boxr + %pi (ca M A D ([h] boxr (E1 h)) (boxr E1')) + %<- ({a w} {acc} ca M A D ([h] E1 h a acc) (E1' a acc)) %. +%term cae_boxl + %pi (ca M A D ([h] boxl (E1 h) H) (boxl E1' H)) + %<- ({gh1} ca M A D ([h] E1 h gh1) (E1' gh1)) %. +%term cae_diar %pi (ca M A D ([h] diar (E1 h) ACC) (diar E1' ACC)) %<- (ca M A D E1 E1') %. +%term cae_dial + %pi (ca M A D ([h] dial (E1 h) H) (dial E1' H)) + %<- ({a w} {acc} {h1 hyp B1 a} ca M A D ([h] E1 h a acc h1) (E1' a acc h1)) %. +%term cae_copy + %pi (ca M A D ([h] copy (E h) ACC H) (copy E' ACC H)) + %<- ({h2} ca M A D ([h] E h h2) (E' h2)) %. +%% E-Commutative Conversions for ca (global) +%term cage_axiom cag M A D ([h] axiom H1) (axiom H1) %. +%term cage_andr + %pi (cag M A D ([h] andr (E1 h) (E2 h)) (andr E1' E2')) + %<- (cag M A D E1 E1') + %<- (cag M A D E2 E2') %. +%term cage_andl1 + %pi (cag M A D ([h] andl1 (E1 h) H) (andl1 E1' H)) + %<- ({h1} cag M A D ([h] E1 h h1) (E1' h1)) %. +%term cage_andl2 + %pi (cag M A D ([h] andl2 (E2 h) H) (andl2 E2' H)) + %<- ({h2} cag M A D ([h] E2 h h2) (E2' h2)) %. +%term cage_impr + %pi (cag M A D ([h] impr (E2 h)) (impr E2')) + %<- ({h1} cag M A D ([h] E2 h h1) (E2' h1)) %. +%term cage_impl + %pi (cag M A D ([h] impl (E1 h) (E2 h) H) (impl E1' E2' H)) + %<- (cag M A D E1 E1') + %<- ({h2} cag M A D ([h] E2 h h2) (E2' h2)) %. +%term cage_orr1 %pi (cag M A D ([h] orr1 (E1 h)) (orr1 E1')) %<- (cag M A D E1 E1') %. +%term cage_orr2 %pi (cag M A D ([h] orr2 (E2 h)) (orr2 E2')) %<- (cag M A D E2 E2') %. +%term cage_orl + %pi (cag M A D ([h] orl (E1 h) (E2 h) H) (orl E1' E2' H)) + %<- ({h1} cag M A D ([h] E1 h h1) (E1' h1)) + %<- ({h2} cag M A D ([h] E2 h h2) (E2' h2)) %. +%term cage_truer cag M A D ([h] truer) truer %. +%term cage_falsel cag M A D ([h] falsel H) (falsel H) %. +%term cage_boxr + %pi (cag M A D ([h] boxr (E1 h)) (boxr E1')) + %<- ({a w} {acc} cag M A D ([h] E1 h a acc) (E1' a acc)) %. +%term cage_boxl + %pi (cag M A D ([h] boxl (E1 h) H) (boxl E1' H)) + %<- ({gh1} cag M A D ([h] E1 h gh1) (E1' gh1)) %. +%term cage_diar %pi (cag M A D ([h] diar (E1 h) ACC) (diar E1' ACC)) %<- (cag M A D E1 E1') %. +%term cage_dial + %pi (cag M A D ([h] dial (E1 h) H) (dial E1' H)) + %<- ({a w} {acc} {h1 hyp B1 a} cag M A D ([h] E1 h a acc h1) (E1' a acc h1)) %. +%term cage_copy + %pi (cag M A D ([h] copy (E h) ACC H) (copy E' ACC H)) + %<- ({h2} cag M A D ([h] E h h2) (E' h2)) %. +% Single principal case for ca (global) +%term cag_copy + %pi (cag (big/ M) A D ([h] copy (E h) ACC h) F) + %<- ({h2} cag (big/ M) A D ([h] E h h2) (E' h2)) + %<- (ca M A (D _ ACC) E' F) %. +%block bacc {Q w} {P w} [d P <= Q]%. +%block bhyp {A o} {P w} [h hyp A P]%. +%block bghyp {A o} {P w} [h ghyp A P]%. +%block bw [a w]%. +%block bo [p o]%. +%worlds (bhyp bghyp bw bo bacc) (ca M A D E F) (cag M A D E F) %. +%mode ca %in %in %in %in %out %. +%mode cag %in %in %in %in %out %. +%total {(A' A) (M' M) [(D' D) (E' E)]} (cag M' A' D' E' _) (ca M A D E _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Tethered_modal_logic). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-1-0.lf b/new-tests/stelf-output/pages/the-twelf-project-1-0.lf new file mode 100644 index 0000000..dca402a --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-1-0.lf @@ -0,0 +1,98 @@ +% +%{! This page is about plans for a **1.0** release of the Twelf Wiki. + +## Press Release + +Subject: New Twelf Wiki + +We are pleased to announce the Twelf Wiki, a major new source of documentation about Twelf: + +: http://twelf.plparty.org + +Twelf is a tool used to specify, implement, and prove properties of deductive systems. The wiki includes: +* A new introduction to LF and Twelf. +* Tutorials on common Twelf tricks and techniques. +* Case studies of larger applications of Twelf, including encodings of and proofs about linear logic, mutable state, and CPS conversion. +* Pre-compiled CVS builds of Twelf for Linux and Windows. + +We invite you to come share what you know, learn from what's there, and ask questions about what's not. + +The Twelf Wiki Team + + +:* I'm tempted to say we should say "a major _and growing_" or something to indicate that this is a work in progress, but perhaps [http://dictionary.oed.com/cgi/entry/50293088?query_type=word&queryword=wiki&first=1&max_to_show=10&single=1&sort_type=alpha with wiki being a real word and all] this isn't necessary... — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 03:47, 20 March 2007 (EDT) + +:: The thing I don't like about that is that it can give the impression that we're hedging on the "major"---that it will someday be major, but isn't yet (even though that's not what "major and growing" really means). Also, I think the last line makes it clear that people will be adding new stuff. [Drl](/wiki/user-drl/) 09:55, 20 March 2007 (EDT) + +: We should either say both "encodings of and proofs about" or something like that or junk them both. We don't want to give people the impression that it's just an LF signature. [Drl](/wiki/user-drl/) 09:26, 21 March 2007 (EDT) + +## PL Party Coffee - Notes + +The first two subsections, where things are assigned, are things that we'd like to have done by next Wednesday. Things that are unassigned will be claimed before then moved to the [[Project:To do|To do]] list at that point, if not before. + +Doing things off the [[Project:To do|To do]] list are also good things to do! + +### Come to the site, learn to use Twelf + +* Rewriting the pointers into these things: _Tom and Dan Licata to revise_ +:* [General description of Twelf](/wiki/general-description-of-twelf/) - _Will to revise_ +:* [[Reading a Twelf proof]] - _DKLee" +::* How to read declarations of proofs (type family + mode + worlds + total) +::* Why the cases in the middle don't matter at that point +::* Adequacy! +::* Minimal on syntax +:* [Proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/) - _Dan Licata_ +:* The "forest" - Case studies, Tutorials, Glossary - Redlinks to be attacked by _Aleffert_, _Spoons_ (?) and others (Also see [[Special:Wantedpages]]) + +### Neat Things + +* <strike>Cut elimination - _Tom_</strike> +* Double negation translation - _Jcreed_ +* Language with store - _Rob_ +* Classical logic - _Will_ +* <strike>False dependencies/strengthening - _Jake_</strike> +* Mutual and lexographic induction - _Jcreed_ +::(maybe handed in [admissibility of cut](/wiki/admissibility-of-cut/) already?) +* Translating between logics - +* Heriditary substitution - +* Canonical forms - +* Reverse the polarity - + +## Tom's demands + +: _DRL, on March 21, says: Peace in our time! I claim that the vast majority of Tom's demands have been appeased._ + +I think it is very important that we have a certain base level of content before we release this on the world. I have seen may projects like this die because they failed to gain sufficient momentum from the initial release. We have enough resources to make this wiki successful, so I will be the annoying one that insists that it meet a certain level of content and quality before we announce it. + +Here are some specific demands. If we already have these, or you address them, then please respond: + +* There needs to be a path, for a patient someone of undergraduate computer science education, to learn how to use Twelf in a basic way. They should be able to define, say, MinML and prove type safety for it. They should understand the meaning of the metatheorem, including the [%worlds](/wiki/percent-worlds/) declaration and adequacy (it should not simply be a _cut'n'paste_ understanding). By path, I mean that they should be able to start on some page, read a linear sequence of tutorials, and then have this knowledge. +:: [User:drl](/wiki/user-drl/)'s tutorial [Proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/) will satisfy this when complete, as long as the undergraduate education had a 15-312 like class.  — [Tom 7](/wiki/user-tom7/) 09:48, 26 February 2007 (EST) + +* There needs to be a stub article (at least) for each of the Twelf declarations. When reading a tutorial on the site, users need to be able to see documentation for the keywords. These can start by being short summaries with links into the User's Guide. +:: I'm satisfied on this.  — [Tom 7](/wiki/user-tom7/) 11:18, 21 March 2007 (EDT) + +* There needs to be a good article at [metatheorem](/wiki/metatheorem/), which could perhaps be a wikified _How to Believe a Twelf Proof_. This material proves to be a major stumbling block for getting even eminent type theorists to understand what Twelf is and how to use it. +: [Metatheorem](/wiki/metatheorem/) exists, and I think it's in a position where some criticism of it would be helpful; however, the DKLee-task-lisked HtBaTP-equivalent article is going to be [[Reading a Twelf proof]] - I'd be particularly happy to see this one by launch but I don't consider it a dealbreaker either. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 01:48, 28 February 2007 (EST) + +* There needs to be more content that will be new to, say, the POPLmark crowd. We should have tutorials that explain how to do some things in Twelf that people actually want to do in their research, like <s>[linear logic](/wiki/linear-logic/)</s>, normalization via <s>[cut elimination](/wiki/admissibility-of-cut/)</s>, and [[closure conversion]]. + +* We need to have good man pages for the most important declarations for mechanizing metatheory: [%worlds](/wiki/percent-worlds/) [%block](/wiki/percent-block/) [%total](/wiki/percent-total/). + +* We need at least minimal articles about the most important Twelf concepts, _e.g._ <s>[subordination](/wiki/subordination/)</s>, <s>[LF](/wiki/lf/)</s>, <s>[adequacy](/wiki/adequacy/)</s>, <s>[canonical form](/wiki/canonical-form/)</s>, <s>[unification](/wiki/unification/)</s>, <s>[ground](/wiki/ground/)</s>, [[kind]], [[subterm]]. I mean, really. If these are covered in the tutorials, we should extract those discussions to seed these pages. <s>Also, some other important articles are very incomplete, embarrassing stubs right now (_e.g._ [totality assertion](/wiki/totality-assertion/)).</s> + +::* Reading back over this list I don't disagree with most of these, and the half articles I have also identified as something I/we definitely need to address - but some of this is just the effect of us psyching ourselves out because we've held onto this too long already (i.e. "well, it's been long enough we should have...") which will lead to us never releasing the thing. An article on unification before the release would be great, but it absense won't prevent this from being a useful resource to some undergraduate somewhere. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 11:37, 26 February 2007 (EST) +:::* I think it will, because questions like "what is unification?" or "what is subordination?" are very natural when learning Twelf, and if unanswered, are a large impediment to understanding what's going on.  — [Tom 7](/wiki/user-tom7/) 16:01, 26 February 2007 (EST) +::::* I'm not arguing with your point, but at what point does the absense of _this wiki_ become a larger impediment than the absense of _something on this wiki_? :-) — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 16:46, 26 February 2007 (EST) + +* ((more soon..)) + +::::::: — [Tom 7](/wiki/user-tom7/) 13:43, 16 October 2006 (EDT) +::::::: — [Tom 7](/wiki/user-tom7/) 09:48, 26 February 2007 (EST) +::::::: — [Tom 7](/wiki/user-tom7/) 11:18, 21 March 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project:1.0). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-ajalf.lf b/new-tests/stelf-output/pages/the-twelf-project-ajalf.lf new file mode 100644 index 0000000..2be2d0b --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-ajalf.lf @@ -0,0 +1,24 @@ +% +%{! **AJALF** is the technology used to power [Twelf Live](/wiki/twelf-live/) and the [[Project:TwelfTag#Quick_check|Quick Twelf Check]] button. It stands for "Asynchronous Javascript and LF." + +## Quick check + +Quick check is implemented as a MediaWiki extension (extensions/AJALF.php) that introduces a new tag, <nowiki><ajalf>...</ajalf></nowiki>. The tag adds a button + +:<ajalf>(unused)</ajalf> + +which gleans twelf code from the edit window and passes it to the Twelf server (currently at ``/tomtest.php``). The button only works on edit pages, so there is no reason to ever use this tag outside the editor interface. It is included in the interface via a customization of the [MediaWiki:Copyrightwarning](/wiki/mediawiki-copyrightwarning/) string. + + +## Bugs and issues +* Quick check does not support any of the [[Project:TwelfTag|TwelfTag]] parameters. +* The output of Quick check appears to escape apostrophes for some reason. +* The layout of the close and info buttons is terrible. + +\{\{stub\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project:AJALF). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-by-nc.lf b/new-tests/stelf-output/pages/the-twelf-project-by-nc.lf new file mode 100644 index 0000000..23fbd8c --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-by-nc.lf @@ -0,0 +1,15 @@ +% +%{! ## Some Rights Reserved + +For any page marked with the tag <nowiki>\{\{license-by-nc\}\}</nowiki>, the content is available to you under the terms of the Creative Commons Attribution-Noncommercial license. + +* A summary of this license is available at http://creativecommons.org/licenses/by-nc/2.5/ +* The actual license is available at http://creativecommons.org/licenses/by-nc/2.5/legalcode + +See [[Project:Copyright]] for more information. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project:BY-NC). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-by-sa.lf b/new-tests/stelf-output/pages/the-twelf-project-by-sa.lf new file mode 100644 index 0000000..6edc9c8 --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-by-sa.lf @@ -0,0 +1,15 @@ +% +%{! ## Some Rights Reserved + +For any page marked with the tag <nowiki>\{\{license-by-sa\}\}</nowiki>, the content is available to you under the terms of the Creative Commons Attribution-ShareAlike license. + +* A summary of this license is available at http://creativecommons.org/licenses/by-sa/2.5/ +* The actual license is available at http://creativecommons.org/licenses/by-sa/2.5/legalcode + +See [[Project:Copyright]] for more information. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project:BY-SA). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-contributing.lf b/new-tests/stelf-output/pages/the-twelf-project-contributing.lf new file mode 100644 index 0000000..98c5cbc --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-contributing.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you should go to the [new version](/contributing/). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-contributors.lf b/new-tests/stelf-output/pages/the-twelf-project-contributors.lf new file mode 100644 index 0000000..9180338 --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-contributors.lf @@ -0,0 +1,28 @@ +% +%{! If you have a question or need assistance, your best bet is probably to [http://twelf.plparty.org/w/index.php?title=User_talk:Rsimmons&action=edit§ion=new leave a message for Rob] or [http://twelf.plparty.org/w/index.php?title=User_talk:Tom7&action=edit§ion=new leave a message for Tom]. + +* [[User:Aleffert]] - Akiva Leffert +* [User:Boyland](/wiki/user-boyland/) - John Boyland (Bureaucrat: can change user privileges) +* [User:Crary](/wiki/user-crary/) - Karl Crary +* [User:DanielKLee](/wiki/user-danielklee/) - Daniel K. Lee +* [[User:Drdreyer]] - Derek R. Dreyer +* [User:Drl](/wiki/user-drl/) - Daniel R. Licata (Bureaucrat: can change user privileges) +* [User:JakeD](/wiki/user-jaked/) - Jacob Donham +* [User:Jcreed](/wiki/user-jcreed/) - Jason C. Reed +* [[User:Neel]] - Neel Krishnaswami +* [User:Rsimmons](/wiki/user-rsimmons/) Robert J. Simmons (Bureaucrat: can change user privileges) +* [[User:Rwh]] - Robert Harper +* [User:Spoons](/wiki/user-spoons/) - Daniel Spoonhower +* [User:Tom7](/wiki/user-tom7/) - Tom Murphy VII (Bureaucrat: can change user privileges) +* [User:Twilson](/wiki/user-twilson/) - J. Todd Wilson +* [User:Varming](/wiki/user-varming/) - Carsten Varming +* [User:William Lovas](/wiki/user-william-lovas/) - William Lovas + +## See also +* [[Special:Listusers]] is always up-to-date, but has no explanatory text. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project:Contributors). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-copyright.lf b/new-tests/stelf-output/pages/the-twelf-project-copyright.lf new file mode 100644 index 0000000..09c5bce --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-copyright.lf @@ -0,0 +1,53 @@ +% +%{! **The following policy applies _only_ to code and content posted to this wiki, NOT to any version of the Twelf program itself or to documentation linked to from the wiki (for instance, the Twelf User's Guide.)** + +In general, all edits to The Twelf Project wiki are covered by the [[Project:MIT license | MIT License]] which allows essentially unlimited use for any reason. If a page on the wiki is not tagged with one of the license tags described in the section below, then it is under the MIT License, and if you edit an article with no tag and do not add one, you are agreeing to also licence your edits under the MIT License. + +## Restrictive licenses + +Some code and content is already licensed under different rules, and some authors do not wish to release their work to open use. While we suggest using the default open license, we want to allow the most content to be accessed from this wiki in the complicated world in which we live. + +In service of this goal, we support the following four licenses, which can be invoked by placing the corresponding tag at the bottom of a page. + +If you edit a page with one of these tags, you agree that your contributions are released under and controlled by that license. License tags may not be deleted unless +* you are the one that added the tag, and +* no one else has edited the article since you added the tag. +* you are only removing a subset of multiple tags on a page (effectively de-multi-licensing the page) + +Please contact the [[Project:Administrators | project administrators]] if you need a different license supported for some reason. + +### Adding a tag to an existing page + +A page in the Twelf Project wiki can either start out with one of the license tags, or one can be added after the page's creation. + +Legally, existing content under the default MIT license can be changed to any one of the four licenses below by adding a tag. **This is strongly discouraged and is only permitted in exceptional circumstances**, as the original author(s) intended to release the page under the MIT License. If you edit an existing page and add a license tag, [[Project:Administrators|administrators]] can and will delete all your changes (as well as the tag) if your changes are not substantial and positive. + +### GNU General Public License + +For GNU GPL content, add the <nowiki>\{\{license-gpl\}\}</nowiki> tag to the bottom of your page. See [[Project:GPL]] for more information. + +\{\{license-gpl\}\} + +### GNU Free Documentation License + +For GNU FDL content, add the <nowiki>\{\{license-gfd\}\}</nowiki> tag to the bottom of your page. See [[Project:GFDL]] for more information. + +\{\{license-gfdl\}\} + +### Creative Commons Attribution-ShareAlike License + +For CC BY-SA content, add the <nowiki>\{\{license-by-sa\}\}</nowiki> tag to the bottom of your page. + +\{\{license-by-sa\}\} + +### Creative Commons Attribution-Noncommercial License + +For CC BY-NC content, add the <nowiki>\{\{license-by-nc\}\}</nowiki> tag to the bottom of your page. _Note: we discourage use of this license because of its extreme incompatibility with other free licenses._ + +\{\{license-by-nc\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project:Copyright). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-geshi-twelf-extension.lf b/new-tests/stelf-output/pages/the-twelf-project-geshi-twelf-extension.lf new file mode 100644 index 0000000..8144d88 --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-geshi-twelf-extension.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-gfdl.lf b/new-tests/stelf-output/pages/the-twelf-project-gfdl.lf new file mode 100644 index 0000000..2dd5208 --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-gfdl.lf @@ -0,0 +1,410 @@ +% +%{! For any page marked with the tag <nowiki>\{\{license-gfdl\}\}</nowiki>, permission is granted to copy, distribute and/or modify the document under the terms of the GNU Free Documentation License, Version 1.1 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". + +See [[Project:Copyright]] for more information. + +## GNU Free Documentation License + +<code> GNU Free Documentation License + Version 1.2, November 2002 + + + Copyright (C) 2000,2001,2002 Free Software Foundation, Inc. + 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA + Everyone is permitted to copy and distribute verbatim copies + of this license document, but changing it is not allowed. + + +0. PREAMBLE + +The purpose of this License is to make a manual, textbook, or other +functional and useful document "free" in the sense of freedom: to +assure everyone the effective freedom to copy and redistribute it, +with or without modifying it, either commercially or noncommercially. +Secondarily, this License preserves for the author and publisher a way +to get credit for their work, while not being considered responsible +for modifications made by others. + +This License is a kind of "copyleft", which means that derivative +works of the document must themselves be free in the same sense. It +complements the GNU General Public License, which is a copyleft +license designed for free software. + +We have designed this License in order to use it for manuals for free +software, because free software needs free documentation: a free +program should come with manuals providing the same freedoms that the +software does. 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List on the Title Page, as authors, one or more persons or entities + responsible for authorship of the modifications in the Modified + Version, together with at least five of the principal authors of the + Document (all of its principal authors, if it has fewer than five), + unless they release you from this requirement. +C. State on the Title page the name of the publisher of the + Modified Version, as the publisher. +D. Preserve all the copyright notices of the Document. +E. Add an appropriate copyright notice for your modifications + adjacent to the other copyright notices. +F. Include, immediately after the copyright notices, a license notice + giving the public permission to use the Modified Version under the + terms of this License, in the form shown in the Addendum below. +G. Preserve in that license notice the full lists of Invariant Sections + and required Cover Texts given in the Document's license notice. +H. Include an unaltered copy of this License. +I. 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To do this, add their titles to the +list of Invariant Sections in the Modified Version's license notice. +These titles must be distinct from any other section titles. + +You may add a section Entitled "Endorsements", provided it contains +nothing but endorsements of your Modified Version by various +parties--for example, statements of peer review or that the text has +been approved by an organization as the authoritative definition of a +standard. + +You may add a passage of up to five words as a Front-Cover Text, and a +passage of up to 25 words as a Back-Cover Text, to the end of the list +of Cover Texts in the Modified Version. Only one passage of +Front-Cover Text and one of Back-Cover Text may be added by (or +through arrangements made by) any one entity. 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COMBINING DOCUMENTS + +You may combine the Document with other documents released under this +License, under the terms defined in section 4 above for modified +versions, provided that you include in the combination all of the +Invariant Sections of all of the original documents, unmodified, and +list them all as Invariant Sections of your combined work in its +license notice, and that you preserve all their Warranty Disclaimers. + +The combined work need only contain one copy of this License, and +multiple identical Invariant Sections may be replaced with a single +copy. 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TRANSLATION + +Translation is considered a kind of modification, so you may +distribute translations of the Document under the terms of section 4. +Replacing Invariant Sections with translations requires special +permission from their copyright holders, but you may include +translations of some or all Invariant Sections in addition to the +original versions of these Invariant Sections. You may include a +translation of this License, and all the license notices in the +Document, and any Warranty Disclaimers, provided that you also include +the original English version of this License and the original versions +of those notices and disclaimers. In case of a disagreement between +the translation and the original version of this License or a notice +or disclaimer, the original version will prevail. + +If a section in the Document is Entitled "Acknowledgements", +"Dedications", or "History", the requirement (section 4) to Preserve +its Title (section 1) will typically require changing the actual +title. + + +9. TERMINATION + +You may not copy, modify, sublicense, or distribute the Document except +as expressly provided for under this License. Any other attempt to +copy, modify, sublicense or distribute the Document is void, and will +automatically terminate your rights under this License. However, +parties who have received copies, or rights, from you under this +License will not have their licenses terminated so long as such +parties remain in full compliance. + + +10. FUTURE REVISIONS OF THIS LICENSE + +The Free Software Foundation may publish new, revised versions +of the GNU Free Documentation License from time to time. Such new +versions will be similar in spirit to the present version, but may +differ in detail to address new problems or concerns. See +http://www.gnu.org/copyleft/. + +Each version of the License is given a distinguishing version number. +If the Document specifies that a particular numbered version of this +License "or any later version" applies to it, you have the option of +following the terms and conditions either of that specified version or +of any later version that has been published (not as a draft) by the +Free Software Foundation. If the Document does not specify a version +number of this License, you may choose any version ever published (not +as a draft) by the Free Software Foundation. + + +ADDENDUM: How to use this License for your documents + +To use this License in a document you have written, include a copy of +the License in the document and put the following copyright and +license notices just after the title page: + + Copyright (c) YEAR YOUR NAME. + Permission is granted to copy, distribute and/or modify this document + under the terms of the GNU Free Documentation License, Version 1.2 + or any later version published by the Free Software Foundation; + with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. + A copy of the license is included in the section entitled "GNU + Free Documentation License". + +If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, +replace the "with...Texts." line with this: + + with the Invariant Sections being LIST THEIR TITLES, with the + Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST. + +If you have Invariant Sections without Cover Texts, or some other +combination of the three, merge those two alternatives to suit the +situation. + +If your document contains nontrivial examples of program code, we +recommend releasing these examples in parallel under your choice of +free software license, such as the GNU General Public License, +to permit their use in free software.</code> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project:GFDL). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-gpl.lf b/new-tests/stelf-output/pages/the-twelf-project-gpl.lf new file mode 100644 index 0000000..3b632c1 --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-gpl.lf @@ -0,0 +1,356 @@ +% +%{! For any page marked with the tag <nowiki>\{\{license-gpl\}\}</nowiki>, it is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. + +This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. 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(Exception: if the Program itself is interactive but + does not normally print such an announcement, your work based on + the Program is not required to print an announcement.) + +These requirements apply to the modified work as a whole. If +identifiable sections of that work are not derived from the Program, +and can be reasonably considered independent and separate works in +themselves, then this License, and its terms, do not apply to those +sections when you distribute them as separate works. But when you +distribute the same sections as part of a whole which is a work based +on the Program, the distribution of the whole must be on the terms of +this License, whose permissions for other licensees extend to the +entire whole, and thus to each and every part regardless of who wrote it. + +Thus, it is not the intent of this section to claim rights or contest +your rights to work written entirely by you; rather, the intent is to +exercise the right to control the distribution of derivative or +collective works based on the Program. + +In addition, mere aggregation of another work not based on the Program +with the Program (or with a work based on the Program) on a volume of +a storage or distribution medium does not bring the other work under +the scope of this License. + + 3. You may copy and distribute the Program (or a work based on it, +under Section 2) in object code or executable form under the terms of +Sections 1 and 2 above provided that you also do one of the following: + + a) Accompany it with the complete corresponding machine-readable + source code, which must be distributed under the terms of Sections + 1 and 2 above on a medium customarily used for software interchange; or, + + b) Accompany it with a written offer, valid for at least three + years, to give any third party, for a charge no more than your + cost of physically performing source distribution, a complete + machine-readable copy of the corresponding source code, to be + distributed under the terms of Sections 1 and 2 above on a medium + customarily used for software interchange; or, + + c) Accompany it with the information you received as to the offer + to distribute corresponding source code. (This alternative is + allowed only for noncommercial distribution and only if you + received the program in object code or executable form with such + an offer, in accord with Subsection b above.) + +The source code for a work means the preferred form of the work for +making modifications to it. For an executable work, complete source +code means all the source code for all modules it contains, plus any +associated interface definition files, plus the scripts used to +control compilation and installation of the executable. However, as a +special exception, the source code distributed need not include +anything that is normally distributed (in either source or binary +form) with the major components (compiler, kernel, and so on) of the +operating system on which the executable runs, unless that component +itself accompanies the executable. + +If distribution of executable or object code is made by offering +access to copy from a designated place, then offering equivalent +access to copy the source code from the same place counts as +distribution of the source code, even though third parties are not +compelled to copy the source along with the object code. + + 4. You may not copy, modify, sublicense, or distribute the Program +except as expressly provided under this License. Any attempt +otherwise to copy, modify, sublicense or distribute the Program is +void, and will automatically terminate your rights under this License. +However, parties who have received copies, or rights, from you under +this License will not have their licenses terminated so long as such +parties remain in full compliance. + + 5. You are not required to accept this License, since you have not +signed it. However, nothing else grants you permission to modify or +distribute the Program or its derivative works. These actions are +prohibited by law if you do not accept this License. Therefore, by +modifying or distributing the Program (or any work based on the +Program), you indicate your acceptance of this License to do so, and +all its terms and conditions for copying, distributing or modifying +the Program or works based on it. + + 6. Each time you redistribute the Program (or any work based on the +Program), the recipient automatically receives a license from the +original licensor to copy, distribute or modify the Program subject to +these terms and conditions. You may not impose any further +restrictions on the recipients' exercise of the rights granted herein. +You are not responsible for enforcing compliance by third parties to +this License. + + 7. If, as a consequence of a court judgment or allegation of patent +infringement or for any other reason (not limited to patent issues), +conditions are imposed on you (whether by court order, agreement or +otherwise) that contradict the conditions of this License, they do not +excuse you from the conditions of this License. If you cannot +distribute so as to satisfy simultaneously your obligations under this +License and any other pertinent obligations, then as a consequence you +may not distribute the Program at all. For example, if a patent +license would not permit royalty-free redistribution of the Program by +all those who receive copies directly or indirectly through you, then +the only way you could satisfy both it and this License would be to +refrain entirely from distribution of the Program. + +If any portion of this section is held invalid or unenforceable under +any particular circumstance, the balance of the section is intended to +apply and the section as a whole is intended to apply in other +circumstances. + +It is not the purpose of this section to induce you to infringe any +patents or other property right claims or to contest validity of any +such claims; this section has the sole purpose of protecting the +integrity of the free software distribution system, which is +implemented by public license practices. Many people have made +generous contributions to the wide range of software distributed +through that system in reliance on consistent application of that +system; it is up to the author/donor to decide if he or she is willing +to distribute software through any other system and a licensee cannot +impose that choice. + +This section is intended to make thoroughly clear what is believed to +be a consequence of the rest of this License. + + 8. If the distribution and/or use of the Program is restricted in +certain countries either by patents or by copyrighted interfaces, the +original copyright holder who places the Program under this License +may add an explicit geographical distribution limitation excluding +those countries, so that distribution is permitted only in or among +countries not thus excluded. In such case, this License incorporates +the limitation as if written in the body of this License. + + 9. The Free Software Foundation may publish revised and/or new versions +of the General Public License from time to time. Such new versions will +be similar in spirit to the present version, but may differ in detail to +address new problems or concerns. + +Each version is given a distinguishing version number. If the Program +specifies a version number of this License which applies to it and "any +later version", you have the option of following the terms and conditions +either of that version or of any later version published by the Free +Software Foundation. If the Program does not specify a version number of +this License, you may choose any version ever published by the Free Software +Foundation. + + 10. If you wish to incorporate parts of the Program into other free +programs whose distribution conditions are different, write to the author +to ask for permission. For software which is copyrighted by the Free +Software Foundation, write to the Free Software Foundation; we sometimes +make exceptions for this. Our decision will be guided by the two goals +of preserving the free status of all derivatives of our free software and +of promoting the sharing and reuse of software generally. + + NO WARRANTY + + 11. BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY +FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN +OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES +PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED +OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF +MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS +TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE +PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, +REPAIR OR CORRECTION. + + 12. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING +WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR +REDISTRIBUTE THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, +INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING +OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED +TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY +YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER +PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE +POSSIBILITY OF SUCH DAMAGES. + + END OF TERMS AND CONDITIONS + + How to Apply These Terms to Your New Programs + + If you develop a new program, and you want it to be of the greatest +possible use to the public, the best way to achieve this is to make it +free software which everyone can redistribute and change under these terms. + + To do so, attach the following notices to the program. It is safest +to attach them to the start of each source file to most effectively +convey the exclusion of warranty; and each file should have at least +the "copyright" line and a pointer to where the full notice is found. + + <one line to give the program's name and a brief idea of what it does.> + Copyright (C) <year> <name of author> + + This program is free software; you can redistribute it and/or modify + it under the terms of the GNU General Public License as published by + the Free Software Foundation; either version 2 of the License, or + (at your option) any later version. + + This program is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + GNU General Public License for more details. + + You should have received a copy of the GNU General Public License along + with this program; if not, write to the Free Software Foundation, Inc., + 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. + +Also add information on how to contact you by electronic and paper mail. + +If the program is interactive, make it output a short notice like this +when it starts in an interactive mode: + + Gnomovision version 69, Copyright (C) year name of author + Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type `show w'. + This is free software, and you are welcome to redistribute it + under certain conditions; type `show c' for details. + +The hypothetical commands `show w' and `show c' should show the appropriate +parts of the General Public License. Of course, the commands you use may +be called something other than `show w' and `show c'; they could even be +mouse-clicks or menu items--whatever suits your program. + +You should also get your employer (if you work as a programmer) or your +school, if any, to sign a "copyright disclaimer" for the program, if +necessary. Here is a sample; alter the names: + + Yoyodyne, Inc., hereby disclaims all copyright interest in the program + `Gnomovision' (which makes passes at compilers) written by James Hacker. + + <signature of Ty Coon>, 1 April 1989 + Ty Coon, President of Vice + +This General Public License does not permit incorporating your program into +proprietary programs. If your program is a subroutine library, you may +consider it more useful to permit linking proprietary applications with the +library. If this is what you want to do, use the GNU Lesser General +Public License instead of this License.</code> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project:GPL). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-literate-twelf-sample-page.lf b/new-tests/stelf-output/pages/the-twelf-project-literate-twelf-sample-page.lf new file mode 100644 index 0000000..0d1a268 --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-literate-twelf-sample-page.lf @@ -0,0 +1,29 @@ +% +%{! +Some natural numbers; a demo of [[Project:Literate Twelf|Literate Twelf]]. +!}% +%{! ## Syntax !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%{! ## Judgments !}% +%{! ### Equality !}% +%sort id-nat %. +%term id-nat/refl id-nat N N %. +%{! ### Addition === | !}% +%sort plus %. +%term plus/z plus z N N %. +%term plus/s %pi (plus N1 N2 N3) %-> (plus (s N1) N2 (s N3)) %. +%{! Now we can see what it looks like to run a query: |check=decl !}% +%solve _ : plus (s (s (s z))) (s (s z)) N %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project:Literate_Twelf/Sample_Page). +!}% +%. + +This is a pretty boring presentation of the natural numbers; +it is roughly like all the other ones, like the article on [natural numbers](/wiki/natural-numbers/). + diff --git a/new-tests/stelf-output/pages/the-twelf-project-literate-twelf.lf b/new-tests/stelf-output/pages/the-twelf-project-literate-twelf.lf new file mode 100644 index 0000000..bcdc69b --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-literate-twelf.lf @@ -0,0 +1,7 @@ +% +%{! +This page has been removed. The current Twelf Wiki uses the [Wiki Twelf syntax described here](/wiki-syntax/) for every page under the `/wiki/` prefix. + +You can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project:Literate_Twelf). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-mit-license.lf b/new-tests/stelf-output/pages/the-twelf-project-mit-license.lf new file mode 100644 index 0000000..54213d6 --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-mit-license.lf @@ -0,0 +1,19 @@ +% +%{! Copyright (c) 2006-2007 The Twelf Project. See [[Project:Copyright]] for more information. + +Unless otherwise indicated by a tag on an individual page, permission is hereby granted, free of charge, to any person obtaining a copy of +this software and associated documentation files published directly on this wiki (the "Software"), to deal in +the Software without restriction, including without limitation the rights to +use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of +the Software, and to permit persons to whom the Software is furnished to do so, +subject to the following conditions: + +The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. + +THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project:MIT_license). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-policy.lf b/new-tests/stelf-output/pages/the-twelf-project-policy.lf new file mode 100644 index 0000000..3072e91 --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-policy.lf @@ -0,0 +1,16 @@ +% +%{! A policy will go here when one is devised. + +Some highlights that I imagine will be included: + +* The Principles of Programming group at Carnegie Mellon University are essentally the maintainers of the wiki and project, and are final arbiters in terms of policy decisions. However, the wiki is open for anyone, including unregistered users, to embrace and extend, and the entire wiki is generally editable. + +* As the bottom of all pages states, unless an individual page states otherwise, The Twelf Wiki runs on the simple, permissive MIT license that allows anyone to use, modify, and extend things in the wiki for any purpose, including for commercial use. This is described in more detail at [[Project:Copyrights]]. + +* Blocks will be pursued agressively and without remorse. [Rob](/wiki/user-rsimmons/) spent way to much time reverting spam on the old wiki to not use available tools agressively. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project:Policy). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-protected-page.lf b/new-tests/stelf-output/pages/the-twelf-project-protected-page.lf new file mode 100644 index 0000000..11c331f --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-protected-page.lf @@ -0,0 +1,11 @@ +% +%{! Pages on the Twelf project wiki will be protected for one of the following reasons: + +* Software links - any pages that decribe download procedures or that link to downloadable resources must be protected to prevent anyone from tampering with the links. +* Vandalism - any pages subject to vandalism will be protected. Anyone tampering with or vandalizing the Twelf project wiki will be blocked, permanently and without warning. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project:Protected_page). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-style-guide.lf b/new-tests/stelf-output/pages/the-twelf-project-style-guide.lf new file mode 100644 index 0000000..d60618c --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-style-guide.lf @@ -0,0 +1,85 @@ +% +%{! _This page is a style guide for contributions to the Twelf Project Wiki. For a style guide for writing Twelf itself, see [Style guide](/wiki/twelf-style-guide/)_ + +## Basic style points + +* _Twelf_ is always capitalized. +* Always snippets of code more than one line long inside <nowiki> !}% +%{! </nowiki> tags. Constants, names of lemmas, and keywords like ``%total`` should be displayed in fixed-width fonts by inserting them into <nowiki>````</nowiki> tags. +* Use lowercase for the titles of pages and <nowiki>==Section headers==</nowiki>, except for the first character or where English otherwise requires capitalization. (Article titles are case-sensitive except for the first character.) This facilitates linking pages inline, such as "The %trustme declaration is often used to develop proofs with [holes in metatheorems](/wiki/incremental-metatheorem-development/)." + +## Word usage + +* A period-terminated line in Twelf is a _declaration_ or a _top-level declaration._ +* %-things are _keywords,_ declarations incorporating the keyword ``%foo`` are "``%foo`` declarations" ("``%foo`` directive" was used in some earlier articles). +* Use [metatheorem](/wiki/metatheorem/) instead of other meta-words (such as metalemma). +* The word [metatheorem](/wiki/metatheorem/) is ambiguous, as it can refer to a ∀∃-statement, a [totality assertion](/wiki/totality-assertion/), a logical relations argument, etc. It is best to use the word to mean ∀∃-statement. +* The thing which we believe to be true when we run [``%total``](/wiki/percent-total/) directives is a _[totality assertion](/wiki/totality-assertion/),_ rather than a ∀∃R-statement. Running a [``%total``](/wiki/percent-total/) directive causes Twelf to either _prove_ or _verfiy_ a totality assertion (the latter, "verify," is preferable). +* The person using Twelf is a _user,_ _Twelf user,_ or (if appropriate) _programmer._ + +## Editing pages + +The MediaWiki software does not come loaded with a style guide, becuase the existing guides are not available under general-user licenses. If you need an introduction to general Wiki markup, then checking out [[w:Wikipedia:How_to_edit_a_page | how to edit a page on Wikipedia]] might be helpful. + +### Tagging documents with templates + +You can leave templates on pages by adding the tag <nowiki>\{\{template\}\}</nowiki> to the text of the page, where "template" is replaced by what you want to add. Tags have a few uses on this wiki: + +* **Licensing** (see [[Project:Copyright]] for more information) - <nowiki>\{\{license-gdfl\}\} \{\{license-gpl\}\}, \{\{license-by-nc\}\}, or \{\{license-by-sa\}\}</nowiki> can be added to a page at the bottom. +* **Indicating problems** - the tag <nowiki>\{\{stub\}\} or \{\{stub tutorial\}\}</nowiki> should be added to the bottom of pages that are very short and need to be longer. If a page is not too short but has problems that need to be addressed, the tag <nowiki>\{\{needswork|Description of the problem\}\}</nowiki> should be added to the top (replacing "Description of the problem" with an actual description of the problem). +* **Classification** (see sections below) - the tag <nowiki>\{\{tutorial\}\}</nowiki> should be added to tutorials, the tag <nowiki>\{\{case study\}\}</nowiki> should be added to case studies. + +### Working with code (syntax highlighting) + +The Twelf Project has syntax highlighting enabled for most major programming languages, and has a custom module that can decently highlight most normal Twelf code. The short version is that you should surround any Twelf code you post with <nowiki> !}% +%{! </nowiki> tags. If you want things to be unhighlighted but still in monospaced, pre-formatted form, you can surround the content with <nowiki><code></code></nowiki> tags. + +The full story can be found at [[Project:Syntax Highlighting]]. + +Twelf code submitted to the site, when possible, should follow the [Twelf style guide](/wiki/twelf-style-guide/) when such a thing is written. + +## Adding new pages + +Adding new pages is relatively simple, and can be done in one of (at least) two ways. For this example, say you want to create a page titled "Not a page." + +* You can type "Not a page" into the search tool on the left side of the page. (You should do this anyway to make sure there's not a similar page already!) The first line on the page that you go to will say "**There is no page titled ":Not a page".** You can [[Not a page|create this page]]." Click on the link and you can create the page. +* You can edit (almost) any page by clicking the "edit" link at top. If you add the text <nowiki>[[Not a page]]</nowiki> to the page, it will show up like this: [[Not a page]]. Click on the link and you can create a page. + +The first letter of any page title can appear in uppercase or lowercase - the system always thinks of it as an uppercase letter. Other than the first word, titles of new pages should generally be in lowercase unless they are using a proper noun (like "Twelf"). + +### What goes where + +In general, there are two main "namespaces" in the wiki. This page is in the "Project" namespace, which is why it is called and linked to as "The Twelf Project:Style guide" or, equivalently, "Project:Style guide." Pages that start with "The Twelf Project:" or "Project:" are only for pages that _deal with the fact that this is a wiki._ + +Alternately, the "main namespace" (basically everything else) is for articles dealing with Twelf, the software, the user community, etc. Thus the page talking about The Twelf Project as funded ongoing research work is named [About The Twelf Project](/wiki/about-the-twelf-project/), and the style guide for writing Twelf code is [Twelf style guide](/wiki/twelf-style-guide/). The main namespace is relatively "flat," in that it has very little directory structure. Structure is preserved in other ways, such as adding categories or tags (which have associated categories) to pages. + +### Adding documentation + +The [Documentation](/wiki/documentation/) page indexes all the documentation on the wiki. If you're adding something new, here are rough guidelines for where it should go: +* [Tutorials](/wiki/tutorials/) - if it's an article that teaches a specific Twelf proof or troubleshooting technique. Tutorials are primarily focused on technique. +* [Case studies](/wiki/case-studies/) - if it's an article that show how an interesting theorem is proved in Twelf. Case studies are primarily focused on the application. +* [Glossary](/wiki/glossary/) - if it's an article that defines a Twelf term. + +The distinction between case studies and tutorials is largely a matter of emphasis, as an example in a tutorial may be an interesting theorem, and a proof in a case study may illustrate a technique. Put the article wherever you think it fits best. + +Completed tutorials should be tagged with <nowiki>\{\{tutorial\}\}</nowiki> at the bottom, and incomplete tutorials should either be tagged with <nowiki>\{\{stub tutorial\}\}</nowiki> at the bottom or with <nowiki>\{\{needswork\}\}</nowiki> at the top. Completed case studies should be tagged with <nowiki>\{\{case study\}\}</nowiki> at the bottom. + +### Adding code examples + +There is not currently a page listing all the code examples. The reason for this is that code here is currently primarily useful in terms of tutorials it is attached to. but all pages that are primarily a code example should include the tag <nowiki>\{\{twelf code\}\}</nowiki>, which adds the page to the [[:Category:Twelf code | category for twelf code]]. + +### Adding library code + +You can add code that can be thought of as "standard library"-style code and can be find in students [http://www.resumesplanet.com resume], but there is not, at this time, a framework for organizing that code on this site. The reason for this is that the organizers of the project wiki have not decided whether this code should go in a code repository, directly onto the wiki, or elsewhere. If you upload your contributions to the wiki and tag the file with the <nowiki>\{\{twelf code\}\}</nowiki> tag, your contributions will get added to this eventual resource. + +[http://customessaywritings.co.uk/samples.php essay samples] + +[http://cvresumewritingservices.org buy resume] + +[http://perfectessay.ca/ buy essay] !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project:Style_guide). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-subversion-transition.lf b/new-tests/stelf-output/pages/the-twelf-project-subversion-transition.lf new file mode 100644 index 0000000..4d9c3ce --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-subversion-transition.lf @@ -0,0 +1,105 @@ +% +%{! On Wednesday, April 23, 2008, the Twelf Project will switch from using CVS for its version control mechanism to using Subversion. + +## Primary checkout + +The last version in CVS was Revision Number 1276 when the transition was prepped on April 17, 2008. Revisions between April 17 and April 23 will be manually applied by [Rob](/wiki/user-rsimmons/). + +The trunk is be available from ``svn co https://cvs.concert.cmu.edu/twelf/trunk twelf``. As of April 17, it is only available for read-only access for people with login to the old Twelf repository; after April 23, it will be open to general access. + +### Tags + +Any tag is be accessible from ``svn co https://cvs.concert.cmu.edu/twelf/tags/<BRANCH NAME>``. The tags as of April 10 are the following: + +<code>delphin_last_merge_main +elphin +input-cover +output-cover1 +pre-roberto +twelf-1-2 +twelf-1-2pl1 +twelf-1-2pl2 +twelf-1-2pl3 +twelf-1-2pl4 +twelf-1-2pl5 +twelf-1-2R1 +twelf-1-2R2 +twelf-1-2R4 +twelf-1-2R5 +twelf-1-2R5pl1 +twelf-1-2R6 +twelf-1-2R7 +twelf-1-2R7pl1 +twelf-1-2R8 +twelf-1-2R8pl2 +twelf-1-2R9pl1 +twelf-1-2R9pl2 +twelf-1-3 +twelf-1-3pl1 +twelf-1-3R0 +twelf-1-3R1 +twelf-1-3R2 +twelf-1-3R3 +twelf-1-3R5 +twelf-1-3R6 +twelf-1-4 +twelf-1-4-carsten +twelf-1-4r1 +twelf-1-4-r2 +twelf-1-5-logosphere +twelf-1-5R2 +twelf-1-5R4 +twelf-branch-meta +twelf-poly</code> + +### Branches + +Any branch is be accessible from ``svn co https://cvs.concert.cmu.edu/twelf/branches/<BRANCH NAME>``. The branches as of April 10 are the following: + +<code>carsten-12-09 +delphin_last_merge_main +elphin +input-cover +output-cover1 +pre-roberto +twelf-1-2 +twelf-1-2pl1 +twelf-1-2pl2 +twelf-1-2pl3 +twelf-1-2pl4 +twelf-1-2pl5 +twelf-1-2R1 +twelf-1-2R2 +twelf-1-2R4 +twelf-1-2R5 +twelf-1-2R5pl1 +twelf-1-2R6 +twelf-1-2R7 +twelf-1-2R7pl1 +twelf-1-2R8 +twelf-1-2R8pl2 +twelf-1-2R9pl1 +twelf-1-2R9pl2 +twelf-1-3 +twelf-1-3pl1 +twelf-1-3R0 +twelf-1-3R1 +twelf-1-3R2 +twelf-1-3R3 +twelf-1-3R5 +twelf-1-3R6 +twelf-1-4 +twelf-1-4-carsten +twelf-1-4r1 +twelf-1-4-r2 +twelf-1-5-logosphere +twelf-1-5R2 +twelf-1-5R4 +twelf-branch-meta +twelf-poly</code> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project:Subversion_transition). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-syntax-highlighting.lf b/new-tests/stelf-output/pages/the-twelf-project-syntax-highlighting.lf new file mode 100644 index 0000000..35efe19 --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-syntax-highlighting.lf @@ -0,0 +1,56 @@ +% +%{! Syntax highlighting on this website is enabled by [[metawikipedia:GeSHiCodeTag Extension | GeSHiCodeTag]] running on the [http://qbnz.com/highlighter/ General Syntax Highlighter]. A prototype [syntax highlighting extension](/wiki/the-twelf-project-geshi-twelf-extension/) has been written for the Twelf Wiki that can be enabled by using the ``<code twelf></code>``, but this is depreciated; the syntax highlighter called when using the ``<twelf>`` tag of [[Project:TwelfTag|TwelfTag]] is written in Haskell. + +As a result, the GeSHi Twelf extensions has bugs that will not be fixed; however, the GeSHi code extension remains for backward-compatability and for highlighting code in languages besides Twelf. + +## Using Syntax Highlighting + +A full list of supported languages is below - GeSHi comes with support for the PHP, C, OCaml, and others. The best way to enter code is to use the ``<code language>`` tag, where "language" is the language you want to use - GeSHi will print out the list of languages it supports if the one you pick is not one it recognizes. Furthermore, a tag without a language option, i.e. ``<code>``, is a useful way to insert preformatted text into the pages. So this code: + +<pre><code twelf>plus_s : plus (s N1) N2 (s N3) + <- plus N1 N2 N3.</code></pre> + +would appear like this: + +<code twelf>plus_s : plus (s N1) N2 (s N3) + <- plus N1 N2 N3.</code> + +However, to restate the point made above, you shouldn't ever do this for _Twelf_ code, you should use [[Project:TwelfTag|TwelfTag]]. + +### Line numbers + +There's also an advaced code tag that adds in line numbers (which do _not_ get taken along for the ride if you copy and paste the text!) It would be nice to make this the common practice for large code snippiets, but because it makes the text so darn small in some browsers that it's not a good idea unless someone can figure out how to make it consistantly bigger. This code: + +<pre><code twelf n>nat : type. +z: nat. s: nat -> nat. + +plus : nat -> nat -> nat -> type. +%mode plus +N1 +N2 -N3. +plus_z : plus z N N. +plus_s : plus (s N1) N2 (s N3) + <- plus N1 N2 N3. +%worlds () (plus _ _ _). %total N (plus N _ _).</code></pre> + +will get displayed like this: + +<code twelf n>nat : type. +z: nat. s: nat -> nat. + +plus : nat -> nat -> nat -> type. +%mode plus +N1 +N2 -N3. +plus_z : plus z N N. +plus_s : plus (s N1) N2 (s N3) + <- plus N1 N2 N3. +%worlds () (plus _ _ _). %total N (plus N _ _).</code> + +## Supported Languages + +You can use the same structure for a wide variety of other languages, which are all listed below. + +<code list> echo "Hello World"; </php> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project:Syntax_highlighting). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-talk-ajalf.lf b/new-tests/stelf-output/pages/the-twelf-project-talk-ajalf.lf new file mode 100644 index 0000000..a8385d2 --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-talk-ajalf.lf @@ -0,0 +1,10 @@ +% +%{! AJALF deserves its own article. My guess is it's AJAX + Twelf server? This is also a good place to acknowledge the people who brought this sweet technology to us. It's been Jcreed, Tom7 and Rob, right? --[DanielKLee](/wiki/user-danielklee/) 20:29, 15 October 2006 (EDT) +: See, my only problem with this is that we need to recognize the fact that we're, on most levels, kidding - Tom advocated for AJALF instead of my suggestion of AJAT because AJALF sounded more fundamentally ridiculous. (and the meaning of the acronym is Async. Javascript and LF, as opposed to Async. Javascript and XML) — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 20:50, 15 October 2006 (EDT) +::Since it's only of use to developers, I moved it to the project namespace. But it's probably a good idea to leave notes on how it works, so that someone can maintain what we did.  — [Tom 7](/wiki/user-tom7/) 21:09, 15 October 2006 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project_talk:AJALF). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-talk-copyright.lf b/new-tests/stelf-output/pages/the-twelf-project-talk-copyright.lf new file mode 100644 index 0000000..e24ca69 --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-talk-copyright.lf @@ -0,0 +1,19 @@ +% +%{! ## CC-NC licenses + +Can we remove the option for noncommercial cc licenses? They are incompatible with the GPL and GFDL, and so are pretty much a dead end for free content. Is there any indication that people want to use them?  — [Tom 7](/wiki/user-tom7/) 13:41, 13 October 2006 (EDT) +: Some wikis are CC-NC, and it's not dead-end if there's a home for them here :). I'm not at all opposed to getting rid of some though, I just figured it was better to have the maximum number of options. There is no indication that anyone would want to use any of these, in all honesty, introducing them was merely futureproofing. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 17:15, 13 October 2006 (EDT) +:: OK, point taken. Well, I'd say we should at least maximally discourage noncommercial licenses.  — [Tom 7](/wiki/user-tom7/) 17:40, 13 October 2006 (EDT) + +## At most one tag + +Also, I don't see any reason why pages can't be dual- or tri-licensed. What's the idea with that restriction?  — [Tom 7](/wiki/user-tom7/) 13:43, 13 October 2006 (EDT) +: Two reasons - one is just simplicity and not having eight red tags on a single page. The other is that reading licenses I got the impression that there may actually be weird legal things involved in multi-licensing that I didn't quite want to understand. The subtyping relationship of multilicensing is quite complicated and might not be reasonably understood by all editors to a page. I think if someone _did_ strongly express an opinion about wanting to actually multilicense certain pages, the issue would be different, I just couldn't imagine such an odd situation and didn't want people doing it lightly. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 17:18, 13 October 2006 (EDT) +:: OK. The purpose of multi-licensing, actually, is to to solve the "subtyping" issues: since most licenses are not directly comparable, it is useful to essentially introduce "intersection types" by asserting that the content can be used under one of many licenses. (So if original content is licensed GPL, GFDL, and cc-by-sa, then someone can fork the content into a GPL codebase or onto Wikipedia or into some like Lessig Twelf remix song or something.) It's actually a pretty common thing on Wikipedia actually to license (especially images) GPL/GFDL/cc. But you're right, this is probably never going to come up. I just thought I'd mention it. (BTW, I do know a lot about copyright stuff in case you ever need an IANAL semi-expert.)  — [Tom 7](/wiki/user-tom7/) 17:40, 13 October 2006 (EDT) +::: Aah! Good to know. I am very interested in copyright issues as well... I think it's probably worth removing the one tag restriction, though really I'd rather people just license with in this particular case. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 18:20, 13 October 2006 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project_talk:Copyright). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-talk-geshi-twelf-extension.lf b/new-tests/stelf-output/pages/the-twelf-project-talk-geshi-twelf-extension.lf new file mode 100644 index 0000000..75019ea --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-talk-geshi-twelf-extension.lf @@ -0,0 +1,25 @@ +% +%{! ## Tom's suggestions + +Can I make a few suggestions? + +* Since red is used for links that don't exist, I think it ought to be avoided for the twelf syntax mode. When I'm editing wikis my eyes are immediately drawn to red as indicating a problem. +* Similarly, I think we should color comments and constants differently. +* Is there a way to have a regular expression for %[a-z] to catch all %commands? That way we don't have to have a keyword table. +::  — [Tom 7](/wiki/user-tom7/) 18:38, 23 September 2006 (MST) + +### Rob's replies + +* Not using red: what color do you think we should use? I think we'd confuse people that want to use syntax highlighting in Emacs, if we did that with constants. I'm open to suggestions, however. +:* I don't think it will be very confusing because many people customize their emacs colors anyway (the defaults are like eye poison to me). Anything else is fine; I don't think you're using anything on, say, the brown end of the spectrum.  — [Tom 7](/wiki/user-tom7/) 17:37, 30 September 2006 (EDT) +* This I've tried to do - making comments bold dark grey. What do you think? +:* Looks good to me.  — [Tom 7](/wiki/user-tom7/) 17:37, 30 September 2006 (EDT) +* That's what I tried at first, it didn't work so well for reasons I don't entirely understand - I think the problem is the interaction between it and other regexp searches. It's not the worst thing in the world to have a keyword table for, in any case. +:: — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 16:05, 30 September 2006 (EDT) +:* Okay, just wonderin'. =)  — [Tom 7](/wiki/user-tom7/) 17:37, 30 September 2006 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project_talk:GeSHi_Twelf_extension). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-talk-policy.lf b/new-tests/stelf-output/pages/the-twelf-project-talk-policy.lf new file mode 100644 index 0000000..34b098d --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-talk-policy.lf @@ -0,0 +1,9 @@ +% +%{! I don't think there's any reason to intimate that we'll protect editors' IP addresses unless ordered by a court; there should be no reason to edit the Twelf wiki anonymously.  — [Tom 7](/wiki/user-tom7/) 22:23, 24 February 2007 (EST) +: Sure. I said that a long time ago anyway. [[User:128.2.223.28|128.2.223.28]] 22:29, 24 February 2007 (EST) (That was me — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 22:30, 24 February 2007 (EST)) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project_talk:Policy). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-talk-style-guide.lf b/new-tests/stelf-output/pages/the-twelf-project-talk-style-guide.lf new file mode 100644 index 0000000..efbc4fa --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-talk-style-guide.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-talk-to-do.lf b/new-tests/stelf-output/pages/the-twelf-project-talk-to-do.lf new file mode 100644 index 0000000..fa876cc --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-talk-to-do.lf @@ -0,0 +1,20 @@ +% +%{! ## "Flying tour" + +One of the helpful things that ACL2 has is a [http://www.cs.utexas.edu/users/moore/acl2/v3-0/The_Tours.html flying tour] that takes people through the real basics of using ACL2 - they also have a walking tour which is much more in-depth. Right now we see the need for tutorials, and I think the glossary idea is a good one. Do you think Dan Licata's document might be effectively turned into a good "flying tour of twelf" through the wiki? — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 05:56, 25 September 2006 (MST) + +## Namespaces + +Do you think we need a new Tour: namespace that signals "this information is safe for n003s, don't put your crazy HOAS-filled coinductive hackbind here"? I'm realizing I'm creating tutorials like [equality](/wiki/equality/) that teach towards a number of levels of experience and sophistication, and might want a place to put an article, like, say, Tour:Equality that only covers shallow equality briefly, and then have forward/backward links that say _if you want \{more/less\} information you can go to the \{tutorial on shallow and deep [Equality](/wiki/equality/) / 'introductory tour' article Tour:Equality\}_ Thoughts? — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 18:57, 27 September 2006 (MST) + +: Tour namespaces sound like a reasonable idea. Deriving/using equality is somewhat of an intermediate topic in my opinion; I think we could do a fine n00best tour without it. --[DanielKLee](/wiki/user-danielklee/) 20:25, 27 September 2006 (MST) + +## Alt+D + +This wiki seems to have captured Alt+D as a "delete page" command. Other wikis I use don't seem to have this behavior. The default behavior for this in my browser is "go-to location bar", so I find myself regularly faced with the "Confirm delete" screen when really I just wanted to go somewhere else. Is there some preference to turn this off? I can't seem to find it. --[DanielKLee](/wiki/user-danielklee/) 20:25, 27 September 2006 (MST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project_talk:To_do). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-talk-twelftag.lf b/new-tests/stelf-output/pages/the-twelf-project-talk-twelftag.lf new file mode 100644 index 0000000..bb614e3 --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-talk-twelftag.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-to-do-syntax-highlighting.lf b/new-tests/stelf-output/pages/the-twelf-project-to-do-syntax-highlighting.lf new file mode 100644 index 0000000..704bba4 --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-to-do-syntax-highlighting.lf @@ -0,0 +1,21 @@ +% +%{! * An identifier within a lemma without a leading ``<-`` is interpreted as a new keyword (is this fixable?) + +### Done +* <strike>``*`` is not recognized as part of an identifier</strike> — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 13:21, 30 September 2006 (EDT) +* <strike>``%block`` is not recognized as a keyword</strike> — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 13:21, 30 September 2006 (EDT) +* <strike>[%infix](/wiki/fixity-declaration/), [%postfix](/wiki/fixity-declaration/), [%prefix](/wiki/fixity-declaration/) not keywords.  — [Tom 7](/wiki/user-tom7/) 10:49, 5 October 2006 (EDT)</strike> — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 19:06, 5 October 2006 (EDT) +* Doesn't dig many other characters in identifiers: > < &, etc.  — [Tom 7](/wiki/user-tom7/) 10:49, 5 October 2006 (EDT) +:* Made some fixes, other ones will be more complicated (because it's actually syntax higlighting <nowiki>></nowiki>, not >) — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 19:06, 5 October 2006 (EDT) +::* Whoa: [CPS conversion](/wiki/cps-conversion/) (scroll down)  — [Tom 7](/wiki/user-tom7/) 19:11, 5 October 2006 (EDT) +:::* I think I fixed all the crazy stuff. Did I fix all the crazy stuff? — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 00:51, 7 October 2006 (EDT) + +* !}% +%sort fuNNY %. +%{! looks bad.  — [Varming](/wiki/user-varming/) Sat Oct 7 00:52:14 EDT 2006 !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project:To_do/Syntax_highlighting). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-to-do.lf b/new-tests/stelf-output/pages/the-twelf-project-to-do.lf new file mode 100644 index 0000000..223c0df --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-to-do.lf @@ -0,0 +1,213 @@ +% +%{! __NOTOC__ + + +## Ongoing to-do items + +\{| border="1" cellpadding="5" cellspacing="0" +|- +| **Title** || **Description** || **Adopted By** || **Status** +|- +| **[[:Category:pages needing attention|Pages needing attention]]** +| Pages that use templates like \{\{tl|needs\}\}, or \{\{tl|stub\}\} are automatically placed in this [[:category:pages needing attention|category]]. Some jobs are small, some are large. +| Everyone +| _Ongoing_ +|- +| **[[Special:wantedpages|Pages that don't exist]]** +| [[Special:wantedpages]] contains a list of the pages that are linked to but don't exist yet. Sometimes the right thing to do is to adjust the target of the link to an existing page, but usually we need to make these pages. +| Everyone +| _Ongoing_ +|- +|\} + +## Active to-do items + +\{| border="1" cellpadding="5" cellspacing="0" +|- +| **Title** || **Description** || **Adopted By** || **Status** +|- +| **Set/map extension** +| A simplification of the issues Appel/Leroy seemed to have with CIVMark - from environments to simple sets of naturals, represented by lists. This exists at [[Sets of natural numbers]]. +| Rob & Dan +| _Almost ready?_ +|- +| **Twelf Standard Library** +| We have a lot of code examples, but very little code that can be used as "take this code and you can use it in your work; here's a list structure or a map structure or a tree that you can use in your proof easily" or "someone, by now, has really honestly proven what you want to prove about natural numbers. you don't have to write it again" The final nature of this will depend on how we decide to distribute code (a wish-list item below), but code development would be good. +| Rob (could use help) +| **Code development** - list, sets, trees, maps +|- +| **Tutorial level icons** +| I'm planning to make nice icons to mark the difficulty level (beginner / intermediate / advanced) of each tutorial. +| Tom +| _soon_ +|- +| **Understanding Twelf error messages** +| For novices, Twelf's error messages can be confusing, especially they're not familiar with the type theory. It might be useful to have an explanation of error messages in the context of how we are encouraging people to use Twelf (e.g. "Expected type family, found object constant" -- did you forget "-> type"? Have you read [judgments as types](/wiki/judgment/)?) +: That's a good idea. Some error messages already have pages, like [output freeness](/wiki/output-freeness/) and [ambiguous hyperkind](/wiki/ambiguous-hyperkind/) (heh). Having a page with a short summary of each error and common causes would be great. I suggest [error messages](/wiki/error-messages/) as a title?  — [Tom 7](/wiki/user-tom7/) 00:08, 18 November 2006 (EST) +: BTW, one really nice thing that Microsoft's compiler does (maybe the only nice thing) is that each error or warning has a short code number that can be used to search for that specific error message. I don't know how feasible it would be to get such a thing in Twelf, but it would make populating the list a lot easier!  — [Tom 7](/wiki/user-tom7/) 00:08, 18 November 2006 (EST) +| spoons +| _In progress_ +|- +|\} + +## Wish-list items + +Feel free to adopt a topic and leave the topic here if you'd like to work on it but may not have time for a bit. If you're about to start working on the project, move it to the Active items section. + +\{| border="1" cellpadding="5" cellspacing="0" +|- +| **Title** || **Description** || **Adopted By** || **Status** +|- +| **Non-locking [[Project:TwelfTag|TwelfTag]]** +| Currently, running an infinite-looping Twelf program through TwelfTag completely locks the wiki (just MediaWiki, not apache, and not the webserver) until I log in as root and kill the process. PHP-spawned processes are already set to much lower priority than httpd, and from there runaway processes can be killed at leisure, if not automatically. So the problem is just that it locks MediaWiki, which it absolutely can't be allowed to do, and I don't have enough apache/php-fu knowledge to know how to deal with the problem. +: Update, I've made it so that PHP disassociates itself from the process after 10 seconds; this is unsatisfying and reminds me of the way Emacs works, but it doesn't permanently lock up PHP, which is good enough for now. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 00:47, 11 October 2006 (EDT) +:: I suggest using bash and ``ulimit`` to limit the spawned twelf process to one or two seconds of CPU.  — [Tom 7](/wiki/user-tom7/) 10:59, 11 October 2006 (EDT) +| Rob? +| _Could use improvement_ +|- +| **Twelf.el** +| consolidate emacs hacks for new twelf.el? +* Karl has some, I know. I have a few too.  — [Tom 7](/wiki/user-tom7/) 18:02, 5 October 2006 (EDT) +| Tom? +| _Wishlist_ +|- +| **More tutorials** +| There is a list of tutorials that need to be written on the [talk page](/wiki/talk-tutorials/). +| ? +| _wishlist_ +|- +| **Offsite backup** +| I have the database do a full dump every day into [http://twelf.plparty.org/w/backup this folder] (the only privacy concern seems to be revealing the IP addresses of edits, which I don't think we care about - passwords are hashed against our usernames). Could someone automate a pull of this every couple of days to their home machine? +* I'm now doing it when I do my backup cycle, but that's not very often. I suggest that you bz2 it and copy it to AFS somewhere, since they do quality backups. (If your home dir is not big enough, we can usually get project space.)  — [Tom 7](/wiki/user-tom7/) 08:42, 16 October 2006 (EDT) +| +| _Partial_ +|- +| **Exercises** +| Does it make sense to have some suggested exercises for someone wanting to learn Twelf but who doesn't have an easy research project to try it out on? +| ? +| _?_ +|- +| **Code distribution** +| The wiki itself won't work for code development—it's way too high a cost to synchronize changes—but can work as an excellent index into some other code repository. My guess is that we're going to need to have a public SVN or CVS server running, preferably with a web interface that we can link to from the wiki. Either we roll our own, or we could go with SourceForge. +* ConCert has a subversion repository; we should use that IMO.  — [Tom 7](/wiki/user-tom7/) 15:21, 2 October 2006 (EDT) +:* Can we get that to allow public read-only access? If so, I agree. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 17:04, 2 October 2006 (EDT) +:** Yes, we can set up a repository on the ConCert SVN server with public read-only access. I just need to learn/remember how. — [Spoons](/wiki/user-spoons/) 13:33, 26 October 2006 (EDT) +| ? +| _Wishlist_ +|- +| **um.elf** +| Rob and Jake both did a Twelf UM implementation - we should make them pretty, possibly tweak them to run a _little_ better, and put them up. +| Rob? +| _Low priority_ +|- +| **Bibliography of LF** +| The [bibliography of LF](/bibliography/) needs cleaning up, finishing of the transwikificaition, and ultimately needs its links checked. It should also be updated with citations for more recent work (Is this second part its own wishlist item?). +| Rob +| _Low priority_ +|- +| **Negation as failure** +| Karl mentioned a new trick to do "negation as failure" using [%deterministic](/wiki/deterministic-declaration/) at the <s>ConCert</s> InCert meeting. Supposedly it's just a few lines, which would make a nice Twelf Wiki page. Maybe dklee knows how to do this and can post about it? +| +| _?_ +|- +| **Closure conversion** +| I know how to do closure conversion (Karl's technique), which was previously thought to be difficult in Twelf. It's worth a tutorial.  — [Tom 7](/wiki/user-tom7/) 13:07, 1 October 2007 (EDT) +| +| _?_ +|- + +|\} + +## Completed items + +\{| border="1" cellpadding="5" cellspacing="0" +|- +| **Title** || **Description** || **Adopted By** || **Status** +|- +| **[Glossary](/wiki/glossary/)** +| Glossary that contains the jargon terms that we see often. Nick Benton's WMM talk complained (rightfully) about the way that experts use jargon terms that _never appear anywhere in the manual_. There should be a single page for quickly answering the question, "What is a ______?" +* examples (please expand, anyone): adequacy, extensionality, shallow and deep encodings, shallow and deep equality, intrinsic and extrinsic encodings, hackbind, functionality, metatheorem, coverage, constant, judgment, relation, signature, regular worlds, world subsumption, subordination, canonical form, subderivation, total, ... +* rh: there's a difference (don't ask me to define it) between jargon and terminology. we shouldn't give the impression that technical terminology is mere jargon. for example, hackbind is jargon, but canonical form is terminology. one difference is that terminology is really defined somewhere, even if people don't know it. +** I agree and the distinction is important. Unfortunately for users, it's not easy to tell whether something is terminology or jargon, without first at least knowing what it means. Can a unified glossary keep this distinction? I think so.  — [Tom 7](/wiki/user-tom7/) 04:29, 24 September 2006 (MST) +| Dan, Rwh, Tom +| **Stable for now** +|- +| **Deep and shallow [equality](/wiki/equality/)** +| Dan explained the solution [here](/wiki/elves-talk-main-page/) to a question I had about how to relate the two obvious descriptions of [equality](/wiki/equality/) +| <strike>Rob</strike> DRL +| **Obsolete/Done** +|- +| **Software** +| The pages that hold or point to software have (mostly) now been protected. Protection should be completed, and the [software](/download/) and [release history](/release-history/) pages should be updated with correct information and links to the software itself. +| Rob +| **Done** +|- +| **Explanation of %trustme** +| The page [Holes in metatheorems](/wiki/incremental-metatheorem-development/) has a description of pre-1.5R1 holes, but currently no description of %trustme +| Dan Lee +| **Done** +|- +| **Debugging syntax highlighting** +| The syntax highlighting module is not, and isn't really built to be, perfect. If you find a quirk in the syntax highlighting, leave it at [[Project:To do/Syntax highlighting]]. +:: I rewrite the syntax highlighter in Haskell, it appears to be much faster to call out to Haskell than it was to parse in PHP. But because everything has been completely rewritten, there may be new and exciting bugs. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 21:05, 5 November 2006 (EST) +| Everyone (discovery), Rob (fixing) +| **Stable** +|- +| **AJALF** +| (from Dan Lee) The merit of this idea is probably some combination of ridiculous, cool, useful, and interesting exercise for an undergraduate. It would be neat if we had a webserver somewhere running a cgi script that would allow you to check Twelf code off the wiki/via a web interface. Being able to check code off the wiki would allow us to develop examples directly out of the wiki, instead of cutting and pasting from our workstations. The other advantage is that people could try Twelf hacking without going through any install procedure. +* According to Tom, Jason Reed has already written code for this. However, it will have to wait until the Wiki moves off my website to its real home, wherever that is. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 07:39, 22 September 2006 (MST) +** Note: We could use iframes or pop-up windows to host just that part off-site. I agree this is pretty low priority though.  — [Tom 7](/wiki/user-tom7/) 11:40, 22 September 2006 (MST) +*** New way to do this is almost in place.  — [Tom 7](/wiki/user-tom7/) 19:09, 12 October 2006 (EDT) +| Tom and Rob +| **Stable** +|- +| **Transwiki** +| Get things from the [http://fp.logosphere.cs.cmu.edu/twelf/ old twelf wiki] over here +| Rob +| **Done for now** +|- +| **Cute little icon** +| We should get one of those little icons that shows up in the navagation bar, like all the big grownup websites. +| Tom +| _Done_ +|- +| **Explanation of substitution lemmas** +| An explanation of how to permute the lambdas in the LF context to prove things like substitution in a language with a store. In particular, I want to show how you can do this for a substitution lemma for a dependently typed lambda calculus without resorting to explicit contexts. + +I've drafted the article, but I invite people to point out points that require clarification in [Talk:Substitution_lemma](/wiki/talk-substitution-lemma/) --[DanielKLee](/wiki/user-danielklee/) 21:08, 24 September 2006 (MST) +| Dan Lee +| **Done for now** +|- +| **Learning Twelf** +| A prominent page that leads prospective users to an appropriate tutorial, should they like to learn about or learn Twelf. (appears in the lower right corner of the [Main Page](/) +| Rob +| **Done (the portal part)** +|- +| **Debug Math** +| The math tag doesn't parse anymore, and I need to figure out what went wrong - Depot could have killed something that accidentally ended up in /user/local +:: For some reason /usr/local/bin dropped out of the server's search path. If someone figures out why, it would be good to fix this; for now, I have recompiled the math processor with built-in absolute filenames, which makes the missing $PATH irrelevant. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 14:58, 5 October 2006 (EDT) +| Rob +| **Done** +|- +| **CPS Conversion** +| CPS conversion is a little tricky, mainly because the natural algorithm is itself continuation based. I have done it several times, so I can write a tutorial using my best known technique. Relies on [[reversing the polarity]]. +| Tom +| **[done](/wiki/cps-conversion/)** +|- +| **[Converting between implicit and explicit parameters](/wiki/converting-between-implicit-and-explicit-parameters/)** +| Simple tutorial on this, referenced in [CPS conversion](/wiki/cps-conversion/) +| Tom +| _[done for now](/wiki/converting-between-implicit-and-explicit-parameters/)_ +|- +| **[Linear logic](/wiki/linear-logic/)** +| Someone should write a tutorial on linear logic, since we have a good way to do it but it is not obvious. +| Karl +| _[done](/wiki/linear-logic/)_ +|- +|\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project:To_do). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/the-twelf-project-twelftag.lf b/new-tests/stelf-output/pages/the-twelf-project-twelftag.lf new file mode 100644 index 0000000..f7bcd45 --- /dev/null +++ b/new-tests/stelf-output/pages/the-twelf-project-twelftag.lf @@ -0,0 +1,149 @@ +% +%{! TwelfTag is [Rob's](/wiki/user-rsimmons/) system for non-interactively checking Twelf code. This is a guide to TwelfTag for contributors; if you are a reader, then [this article](/wiki/twelftag/) may be more helpful. While [Twelf Live](/wiki/twelf-live/) allows people to interact with signatures, TwelfTag is designed for writing and checking code for use in wiki articles. + +Neither the (twelf) tag nor the <nowiki>``></nowiki> tag have any required arguments. The optional arguments for (twelf) tags are: +* ``check=\{true|false|decl\}`` (default ``false``) - cannot be used when hidden=true +* ``discard=\{true|false\}`` (default ``false``) +* ``hidden=\{true|false\}`` (default ``false``) +* ``import="..something.."`` (default ``"main"``) +* ``export="..something.."`` (default ``"main"``) + +The optional arguments for <nowiki>``></nowiki> tags are: +* ``check=\{true|false\}`` (default ``false``) +* ``import="..something.."`` (default ``"main"``) + +The following document discusses the function of each of these tags. + +## The ``(twelf)`` tag for syntax highlighting + +The ``(twelf)`` tag itself is simple enough, it works much like an HTML <nowiki><pre></nowiki> environment with [[Project:Syntax Highlighting|syntax highlighting]]. + +However, the ``(twelf)`` tag also does a number of other interesting and useful things. At the most basic level, it crops out any whitespace at the beginning or end of the environment so that this: + +<pre> !}% +%sort exp %. +%term abs %pi (%pi exp %-> exp) %-> exp %. +%{! </pre> + +Will look like this: !}% +%sort exp %. +%term abs %pi (%pi exp %-> exp) %-> exp %. +%{! ## The ``(twelf)`` and ````>`` tags + +Behind the scenes, the content of every ``(twelf)`` tag is grouped together. So if you have a series of Twelf tags that are defining a single signature, like this (continuing from the example above): !}% +%term app %pi exp %-> exp %-> exp %. +%{! and this: !}% +%sort step %. +%{! then TwelfTag, at this point in the process, has code for ``exp``, ``abs``, ``app``, and ``step`` gathered in a group (this group is named (``"main"``), though it hasn't done anything with it yet. + +### The ````>`` tag + +You can link to a collection of all the code that has been presented so far by using ````>`` tags in the same way you would use ``<a href=...>`` tags in normal HTML code. + +As an example, ````>OMG link!````, will appear like this: ``>OMG link!``, pointing to the four lines of code declared so far throughout the file. + +The filenames are just the MD5 hashes of the code which they contain; the [http://twelf.plparty.org/code code directory] contains all these stored files. + +### The ``check=true`` option + +We are not collecting code in the background just to link to it, we're creating code so we can check it in Twelf! If you use the twelf tag with the ``check=true`` option, Twelf will execute the collected segments in Twelf and link to the output. + +<pre> !}% +%{! (options removed from twelftag: check=true) !}% +%term step/app1 %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app2 %pi (step (app E1 E2) (app E1 E2')) %<- (step E2 E2') %. +%term step/appabs step (app (abs ([x] E x)) E') (E E') %. +%{! </pre> !}% +%{! (options removed from twelftag: check=true) !}% +%term step/app1 %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app2 %pi (step (app E1 E2) (app E1 E2')) %<- (step E2 E2') %. +%term step/appabs step (app (abs ([x] E x)) E') (E E') %. +%{! You may notice that the link above links to a file "twelf.plparty.org/code/<obvious md5 hash>.chk". The code that was checked is just at "twelf.plparty.org/code/<exact sameobvious md5 hash>" - that is, without the ".chk". + +You can use ```` check=true>`` to link to the Twelf output of checking the most recent code instead of just linking to that code code - so ```` check=true>See?```` just duplicates the link that was automatically entered in above. `` check=true>See?``. This works even if you haven't set ``check=true`` in any of your ``(twelf)`` tags; you could just avoid using the ``check=true`` option in your code in favor of causing checks only with ```` check=true>``. + +### The ``check=decl`` option + +If you want to concentrate on the output of the last code fragment, for instance to examine a particular case of type reconstruction, you can use the ``check=decl`` option for the Twelf tag. In the background, TwelfTag loads the previous portions of the signature on chatter 0, and then returns to the default chatter 3 to run the current code snippet. It then does some simple manipulations on Twelf's response to remove things like useless ``%% OK %%`` notices, and then inlines the result. + +<pre> !}% +%{! (options removed from twelftag: check=decl) !}% +%define test step/app1 (step/app2 step/appabs) %. +%{! </pre> !}% +%{! (options removed from twelftag: check=decl) !}% +%define test step/app1 (step/app2 step/appabs) %. +%{! ### The ``discard=true`` option + +If you add ``(twelf discard=true)`` then it will syntax highlight but will not incorporate the segment into the collected code. This is important when you want to demonstrate mistake, because otherwise Twelf will get caught up on that mistake in the future. As a result, this option is frequently used with ``check=decl``. + +<pre> !}% +%{! (options removed from twelftag: discard=true check=decl) !}% +% Don't do this! +%term step/appabs %pi (step E1 E2) %<- (step E1 E2) %. +%{! </pre> !}% +%{! (options removed from twelftag: discard=true check=decl) !}% +% Don't do this! +%term step/appabs %pi (step E1 E2) %<- (step E1 E2) %. +%{! ### The ``import=`` and ``export=`` options + +This is a tricky feature that allows you to run parallel segments of code on a single page. It should be used sparingly, as otherwise it can become extremely confusing to both readers and editors. Each ``(twelf)`` tag is associated with a value, ``import``, that defines where it gets its code _from_, and a value, ``export``, that defines where it binds the concatenation of the code the previous code and the new code within the tag. + +* If ``import=`` is not defined, it is assumed to be ``"main"``. +* If ``export=`` is not defined, it is assumed to be the value of ``import``. If you try to clobber an existing export, for instance by importing from ``"main"`` and exporting to ``"alt"`` twice, Twelf will complain. + +In most cases ``discard=true`` should be sufficient for one's purposes. + +This example is hopefully revealing: + +<pre> !}% +%{! (options removed from twelftag: import="nat") !}% +%sort nat %. +%{! This is a simple way of expressing nats: !}% +%{! (options removed from twelftag: import="nat" export="natsimple") !}% +%term z nat %. +%term s %pi nat %-> nat %. +%{! This is a verbose way of expressing nats: !}% +%{! (options removed from twelftag: import="nat" export="natverbose") !}% +%term nat/z nat %. +%term nat/s %pi nat %-> nat %. +%{! `` import="natsimple">See the simple code`` + +`` import="natverbose">See the verbose code``</pre> !}% +%{! (options removed from twelftag: import="nat") !}% +%sort nat %. +%{! This is a simple way of expressing nats: !}% +%{! (options removed from twelftag: import="nat" export="natsimple") !}% +%term z nat %. +%term s %pi nat %-> nat %. +%{! This is a verbose way of expressing nats: !}% +%{! (options removed from twelftag: import="nat" export="natverbose") !}% +%term nat/z nat %. +%term nat/s %pi nat %-> nat %. +%{! `` import="natsimple">See the simple code`` + +`` import="natverbose">See the verbose code`` + +### The ``hidden=true`` option + +You can use the ``hidden=true`` option to insert Twelf into a page without having it appear to the reader. You can't use the +``check`` option in conjunction with this setting, but it is quite useful in conjunction with ``import`` and ``export``. + +By inserting this code into the wiki you can add a definition of "plus" to the code group named "natsimple". + +<pre> !}% +%{! (options removed from twelftag: import="natsimple" hidden=true) !}% +%sort plus %. +%term plus/z plus z N N %. +%term plus/s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%{! </pre> !}% +%{! (options removed from twelftag: import="natsimple" hidden=true) !}% +%sort plus %. +%term plus/z plus z N N %. +%term plus/s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%{! `` import="natsimple">See the hidden code added to the existing natural number development.`` !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/The_Twelf_Project:TwelfTag). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/theorem-prover.lf b/new-tests/stelf-output/pages/theorem-prover.lf new file mode 100644 index 0000000..aaa3479 --- /dev/null +++ b/new-tests/stelf-output/pages/theorem-prover.lf @@ -0,0 +1,54 @@ +% +%{! The **Twelf theorem prover** is a way of specifying and automatically verifying [metatheorems](/wiki/metatheorem/) that have the form of ∀∃-statements. It uses the Twelf declarations [``%theorem``](/wiki/percent-theorem/), [``%prove``](/wiki/percent-prove/), [``%establish``](/wiki/percent-establish/), and [``%assert``](/wiki/percent-assert/). The last of these is an [unsafe](/wiki/unsafe-mode/) operation similar to [``%trustme``](/wiki/percent-trustme/). + +The current theorem prover is buggy, sometimes does not terminate, and does not output a total logic program that can be used to witness the ∀∃-statement as described in Lexical Conventions. Because of this, it cannot interact with the recommended technique of verifying metatheorems by writing [totality assertions](/wiki/totality-assertion/), and its use is not recommended. + +## Example - commutativity of addition + +### Object language + +We start with the standard presentation of the [natural numbers](/wiki/natural-numbers/) with addition. !}% +%sort nat %. +%term s %pi nat %-> nat %. +%term z nat %. +%sort plus %. +%term p-z plus z N N %. +%term p-s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%{! ### Establishing ∀∃-metatheorems with totality assertions + +A simple [metatheorem](/wiki/metatheorem/) about this object language is the ∀∃-statement that for all natural numbers , there exists a derivation of . This translates into LF as a ∀∃-statement that for all objects ``N`` with type ``nat``, there exists an object ``D`` of type ``plus N z N``. + +The "usual" way of proving this kind of metatheorem in Twelf is to define a type family which represents a relation, translate the cases of the inductive proof into LF objects in that type family, and use a [``%mode``](/wiki/percent-mode/) directive and [``%worlds``](/wiki/percent-worlds/) directive to specify a [totality assertion](/wiki/totality-assertion/) that corresponds to the ∀∃-statement. The totality assertion for type family ``plus-z-thm`` below establishes the ∀∃-statement that is stated above. + +```checkedtwelf +plus-z-thm : {N: nat} plus N z N -> type. + +- : plus-z-thm z p-z. +- : plus-z-thm (s N) (p-s D) + <- plus-z-thm N D. + +%mode plus-z-thm +N -D. +%worlds () (plus-z-thm _ _). +%total T (plus-z-thm T _). +``` + +### Establishing ∀∃-metatheorems with the theorem prover + +The equivalent of the type family definition for ``plus-z-thm`` and the ``%mode`` declaration from the above code can be stated with a [``%theorem``](/wiki/percent-theorem/). !}% +%theorem plus-z : forall {N nat} exists {D plus N z N} true %. +%{! (Side note: some versions of Twelf allow the ``%theorem`` declaration above to be used in lieu of the type family definition and ``%mode`` declaration; [John Boyland](/wiki/user-boyland/) utilizes this in his tutorial and code examples, but it is not a supported feature and may break in future versions of Twelf!) + +Given the ``%theorem`` declaration, we can ask Twelf to try to establish the metatheorem it states by using a [``%prove``](/wiki/percent-prove/) declaration. !}% +%{!! begin checked !!}% +%prove 5 N (plus-z N D) %. +%{!! end checked !!}% +%{! We can also include the equivalent of [implicit parameters](/wiki/implicit-and-explicit-parameters/) in our metatheorem statements, but unlike the standard method where we define a new type family, all our implicit parameters must be listed after the ``forall*`` keyword. We can then prove two more metatheorems (the second is the commutativity of addition). !}% +%theorem plus-s : forall* {N1 nat} {N2 nat} {N3 nat} forall {D1 plus N1 N2 N3} exists {D2 plus N1 (s N2) (s N3)} true %. +%prove 5 D1 (plus-s D1 D2) %. +%theorem plus-comm : forall* {N1 nat} {N2 nat} {N3 nat} forall {D1 plus N1 N2 N3} exists {D2 plus N2 N1 N3} true %. +%prove 5 D1 (plus-comm D1 D2) %. +%{! ## See also + +* Theorem Prover + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/totality-assertion.lf b/new-tests/stelf-output/pages/totality-assertion.lf new file mode 100644 index 0000000..d8cfe2f --- /dev/null +++ b/new-tests/stelf-output/pages/totality-assertion.lf @@ -0,0 +1,44 @@ +% +%{! A **totality assertion** is one type of [metatheorem](/wiki/metatheorem/) about [LF](/wiki/lf/) [type families](/wiki/type-family/). A totality assertion for a type family ``a`` is specified by designating some arguments of the type family as inputs and the remaining arguments as outputs, and by specifying a set of LF contexts. Then the totality assertion for the type family is the following proposition: + +: For all contexts Γ, for all inputs ``M`` in Γ, there exist outputs ``N`` in Γ such that the type ``a M N`` is inhabited in Γ. + +In Twelf, totality assertions are specified using [%mode](/wiki/percent-mode/) and [%worlds](/wiki/percent-worlds/) declarations and verified using [%total](/wiki/percent-total/) declarations. Totality assertions are useful for verifying properties of object-language [judgment](/wiki/judgment/)s. Additionally, Twelf's ability to verify totality assertions is used to prove general [metatheorem](/wiki/metatheorem/)s. + +## Example totality assertions + +Consider the following LF signature, which defined addition on natural numbers: !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%sort plus %. +%term plus-z plus z N2 N2 %. +%term plus-s %pi (plus (s N1) N2 (s N3)) %<- (plus N1 N2 N3) %. +%{! A type family such as ``plus`` defines a relation among its indices, where indices are related iff their instance of the family is inhabited by an LF term. For example, ``plus`` defines a relation between three terms of type ``nat``. The terms (``s (s z)``, ``s z``, ``s (s (s z))``) are related by this relation (because 2 + 1 = 3), but the terms (``z``, ``s z``, ``s (s (s z))``) are not. + +Totality assertions are one class of statements about LF type families. A totality assertion for a type family is specified by designating some index positions as inputs and others positions as outputs. We call this specification the [mode](/wiki/percent-mode/) of the totality assertion. Given a mode specification, the totality assertion asserts that for all inputs, there exist outputs that stand in the relation. Using ``plus`` as an example, if we designate the first two positions as inputs and the third position as an output, this specifies the following totality assertion: + +: For all ``N1:nat`` and ``N2:nat``, there exist ``N3:nat`` and ``D:plus N1 N2 N3``. + +That is, the sum of any two natural numbers exists. Designating instead the first position as output and second and third positions as inputs defines the following totality assertion: + +: For all ``N2:nat`` and ``N3:nat``, there exist ``N1:nat`` and ``D:plus N1 N2 N3``. + +Of course, this totality assertion is false, as ``N3 - N2`` might not be a natural number. + +A totality assertion for an LF type family corresponds with the standard notion of totality for the relation defined by the family. Proving a relation total is different from showing that the relation defines a function, as the outputs of a total relation are not necessarily unique—the relation may relate particular inputs to more than one collection of outputs. + +We may prove a totality assertion by induction on [canonical forms](/wiki/canonical-form/). For example, we can prove the first above totality assertion by induction on ``N1:nat``. + +The above examples consider only terms that are well-typed without mentioning LF variables. In general, it is useful to state totality assertions about all terms that are well-typed in any LF context in a particular [world](/wiki/percent-worlds/), which is a set of LF contexts. Thus, a totality assertion is specified by both a mode declaration and a [world declaration](/wiki/percent-worlds/). + +The page on [%worlds](/wiki/percent-worlds/) presents an example totality assertion in non-empty contexts. + +## See also +* This page's description of totality assertions was extracted from \{\{proving|title=Proving totality assertions about the natural numbers\}\}. Read that introduction to learn how to prove totality assertions using Twelf, and how to use totality assertions to mechanize proofs of more general [metatheorem](/wiki/metatheorem/)s. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Totality_assertion). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/tutorials.lf b/new-tests/stelf-output/pages/tutorials.lf new file mode 100644 index 0000000..8f0e819 --- /dev/null +++ b/new-tests/stelf-output/pages/tutorials.lf @@ -0,0 +1,53 @@ +% +%{! + +⚠️ All the tutorials on this page assume some familiarity with Twelf. If you are just getting started, check out [Proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/) or one of the other [introductions to Twelf](/wiki/introductions-to-twelf/). ⚠️ + + +## Proof techniques + +These tutorials cover helpful and common Twelf devices. Many of these proof techniques are also demonstrated by various [case studies](/wiki/case-studies/). + +### Beginner + +* [Reformulating languages to use hypothetical judgements](/wiki/reformulating-languages-to-use-hypothetical-judgements/) - how to present judgements in a way that's easy to encode in LF. +* [Holes in metatheorems](/wiki/incremental-metatheorem-development/) - how to assume lemmas while developing proofs. +* [Equality](/wiki/equality/) - how to represent equality of LF terms as a type family. +* [Respects lemma](/wiki/respects-lemma/)s - how to prove that other families and constants respect equality and other relations. +* [Uniqueness lemma](/wiki/uniqueness-lemma/)s - how to prove that the inputs to a relation determine an output uniquely. +* [Effectiveness lemma](/wiki/effectiveness-lemma/)s - how to prove [totality assertion](/wiki/totality-assertion/)s explicitly. +* [Output factoring](/wiki/output-factoring/) - how to reason from a disjunction. Illustrates proving the progress theorem for a programming language. +* [Reasoning from false](/wiki/reasoning-from-false/) - how to do proofs by contradiction. +* [Catch-all case](/wiki/catch-all-case/)s - how to avoid putting a theorem case in the LF context. +* [Mutual induction](/wiki/mutual-induction/) - how to prove mutually inductive theorems +* [Converting between implicit and explicit parameters](/wiki/converting-between-implicit-and-explicit-parameters/) - how to convert between implicit and explicit quantification of the parameters of a type family. + +### Advanced + +* [Strengthening](/wiki/strengthening/) - how to convince Twelf that a term does not depend on some assumptions. +* Explicit termination metrics - how to use a termination metric other than the subderivation ordering. +** [Numeric termination metrics](/wiki/numeric-termination-metrics/) - how to use numbers to induct on the size of some argument. +** [Structural termination metrics](/wiki/structural-metrics/) - how to use fancier termination metrics that capture the structure of an argument directly. +* [Simplifying dynamic clauses](/wiki/simplifying-dynamic-clauses/) - how to streamline certain proofs about relations that introduce hypotheses. +* [Canonical forms lemma](/wiki/canonical-forms-lemma/) for a progress theorem - how to get this lemma for free when you can, and how to prove it explicitly using [equality](/wiki/equality/) when you can't. +* Structural properties of a [hypothetical judgement](/wiki/hypothetical-judgment/) - how to get structural properties for free when you can, and how to prove them explicitly when you can't. +** [Weakening](/wiki/weakening-lemma/) +** [Exchange](/wiki/exchange-lemma/) +** [Substitution](/wiki/substitution-lemma/) +* [Evaluation contexts](/wiki/evaluation-contexts/) - how to represent evaluation contexts as LF functions +* [User-defined constraint domain](/wiki/user-defined-constraint-domain/)s - how to make encodings more [intrinsic](/wiki/intrinsic-and-extrinsic-encodings/) by replacing predicates with indexing, and how to use Church-encodings to create special index types that are similar to constraint domains in that they admit non-trivial equations on their elements. +For larger examples of Twelf in action, see see the [case studies](/wiki/case-studies/). + +## Troubleshooting + +These tutorials cover diagnosing and fixing Twelf errors. + + +* [Error messages](/wiki/error-messages/) - brief explanations of (and remedies for) common Twelf error messages. +* [Troubleshooting mode checking errors](/wiki/debugging-mode-checking-errors/) +* [Understanding output freeness](/wiki/output-freeness/) - understanding a subtle part of output coverage checking. +* [Debugging coverage errors](/wiki/debugging-coverage-errors/) - techniques for finding coverage errors. +* [Constraint domains and coverage checking](/wiki/constraint-domains-and-coverage-checking/) - in general, you can't use them in the same [signature](/wiki/twelf-signature/). + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/twelf-elf-rotation-charter.lf b/new-tests/stelf-output/pages/twelf-elf-rotation-charter.lf new file mode 100644 index 0000000..0be94f8 --- /dev/null +++ b/new-tests/stelf-output/pages/twelf-elf-rotation-charter.lf @@ -0,0 +1,29 @@ +% +%{! This is the charter for those who volunteer into the rotation of people making sure that Twelf related questions are answered in a timely fashion. + +* At any given time, there will be one person on the rotation designated as "possessed" by Twelf Elf. +* After a tenure two weeks (this figure may change), the next person in the rotation will become "possessed" by Twelf Elf. +* The "possessed" is responsible for monitoring the Twelf Elf e-mail account and [Ask Twelf Elf](/wiki/ask-twelf-elf/) wiki article for questions. +* The primary duty of the "possessed" is the ensure questions sent to the Twelf Elf are responded to in a timely fashion. +* "Timely fashion" is currently set at 48 hours. This may change in the future. +* The "possessed" should answer to any questions it is within his ability to do so. +* The "possessed" should arrange for someone on the rotation to fill in for him for periods he knows he will be unable to fulfill his duties (e.g. very sick, stuck on a cruise ship, at a conference, etc). +* If the "possessed" is unable to answer a question, he should attempt to refer the question to someone on the rotation who is able to do so. The question should get a "That one is tricky, but someone **will** get back to you in a few days!" response. If no such person exists, the question should get a "Sorry, we don't know!" response. +* If a question has been referred to someone on the rotation, that person should attempt to send an adequate response within a week. If the initial question was by e-mail, responses should either be sent from the Twelf Elf e-mail address or CCed to it for recording purposes. +* Questions/answers that are interesting and not covered by the wiki should become wiki articles. +* Valid questions/answers should not be deleted from the e-mail account. Feel free to nuke spam. +* Anyone on the rotation is free to monitor the e-mail address. +* Someone on the rotation but not currently "possessed" may volunteer to take over a question. + +## Current rotation + +Currently, we are tracking the rotation with google calendar. [http://www.google.com/calendar/render?cid=twelfelf%40gmail.com Subscribe]. + +*10/13/06 – 10/28/06 : [DanielKLee](/wiki/user-danielklee/) +*10/29/06 – 11/11/06 : [Rsimmons](/wiki/user-rsimmons/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Twelf_Elf_Rotation_Charter). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/twelf-live.lf b/new-tests/stelf-output/pages/twelf-live.lf new file mode 100644 index 0000000..172b02a --- /dev/null +++ b/new-tests/stelf-output/pages/twelf-live.lf @@ -0,0 +1,26 @@ +% +%{! You can **try Twelf now** from the wiki without installing any software! The same [TwelfTag](/wiki/the-twelf-project-twelftag/) system that allows code on the wiki to be automatically checked powers an online, interactive Twelf interface. + +http://twelf.plparty.org/live/ + +Feel free to develop bits of Twelf code in this manner. If you plan on doing serious development with Twelf, you will probably want to eventually [download](/download/) Twelf so you can develop your Twelf code in a more private and configurable environment. + +## Using Twelf Live + +### Regular view + +The interface for Twelf Live is fairly simple; text is entered into the "Input" window just as it would be entered into Emacs when using [Twelf with Emacs](/wiki/twelf-with-emacs/). Hitting the "check" button causes the "Output" window to display the output of Twelf from reading the file - the screenshot below shows Twelf Live immediately after the "check" button is clicked. + +[[Image:TWELFLIVE-2.png]] + +### Incremental view + +It can be difficult, when there is a great deal of output, to see just the part that you are interested in. In this situation it is useful to use the incremental view, which is opened by clicking "open incremental view." In incremental view, Twelf will not display anything while checking the first input window (assuming there are no errors), but will display output normally while checking the second input window. This can be seen below, where Twelf only outputs the result of checking the [``%query``](/wiki/percent-query/) declaration. + +[[Image:TWELFLIVE-3.png]] !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Twelf_Live). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/twelf-signature.lf b/new-tests/stelf-output/pages/twelf-signature.lf new file mode 100644 index 0000000..ad2f63d --- /dev/null +++ b/new-tests/stelf-output/pages/twelf-signature.lf @@ -0,0 +1,15 @@ +% +%{! A **Twelf signature** is the thing you are writing when you are writing Twelf code - when you create a [[configuration file]], all the Twelf code in all the files you are working with makes up a Twelf signature. + +\{\{needs|a simple description of how the term arises from the LF theory\}\} + +## See also +* See the [introductions to Twelf](/wiki/introductions-to-twelf/) to learn how to write Twelf signatures. + +\{\{stub\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Twelf_signature). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/twelf-style-guide.lf b/new-tests/stelf-output/pages/twelf-style-guide.lf new file mode 100644 index 0000000..b9d5c96 --- /dev/null +++ b/new-tests/stelf-output/pages/twelf-style-guide.lf @@ -0,0 +1,190 @@ +% +% +%{! +The style you use to write Twelf code should be consistent so that it is +readable by others now and by you in the future. However, the general +objectives of _consistency and readability of code_ is more important than +anything that is or possibly could be said on this page. Therefore, this page +should be seen as attempt to collect a set of good practices from experience +that can be used as recommendations for new users. + +The first two sections are about whitespace & comments, which are extremely +important aspects of readability. The final section is on conventions used for +writing identifiers, which is less important as long as consistency is +maintained within a project. The article on [naming +conventions](/wiki/naming-conventions/) also considers ways of standardizing +identifier names. + +## Whitespace + +Proper use of whitespace, line breaks, and spacing is very important for promoting the readability of Twelf code. + +### Object language syntax + +The declarations for the constants corresponding to a type for syntax of an object language should be grouped together. There should be an empty line between the declaration of the type and its constants. There should be a line break, but no empty lines, between each declaration. !}% +%sort color %. +%term color/red color %. +%term color/black color %. +%{! ### Judgments + +The declarations for the constants corresponding to the inference rules for a particular judgment should be grouped together. There should be an empty line between the declaration of the judgment and its inference rules, as well as one empty line between the declaration of two inference rules. If a [`%mode`](/wiki/percent-mode/) declaration is used, it should be on the line immediately following the declaration of the judgment. !}% +%sort color-neq %. +%term color-neq/red-black color-neq color/red color/black %. +%term color-neq/black-red color-neq color/black color/red %. +%sort color-swap %. +%mode color-swap %in %out %. +%term color-swap/red color-swap color/red color/black %. +%term color-swap/black color-swap color/black color/red %. +%{! ### Metatheorems + +The definitions for cases to a metatheorem should be grouped together. The ``%mode`` declaration should be on the line following the declaration of the type family for the metatheorem, with no line breaks in between. There should be an empty line following the mode declaration. There should be one empty line between each case. There should be an empty line preceding the ``%worlds`` declaration. The ``%total`` declaration should be on the following line. !}% +%sort can-color-neq {C color} %. +%mode color-neq %in %out %. +%term _ can-color-neq color/red color-neq/red-black %. +%term _ can-color-neq color/black color-neq/black-red %. +%worlds () (color-neq _ _) %. +%total {} (color-neq _ _) %. +%{! ### Additional guidelines + +* There should be at least 2 empty lines between any "groups", where a group is something on the scale of declarations for a type family or proof of a metatheorem. 4 or more empty lines should between super-groups of related groups, such as the proofs for a set of related lemmas. + +* The colons following the names of the constants for a group should all line up. + +* Directives such as [``%mode``](/wiki/percent-mode/), [``%reduces``](/wiki/percent-reduces/), [``%terminates``](/wiki/percent-terminates/), [``%worlds``](/wiki/percent-worlds/), [``%total``](/wiki/percent-total/) should be at the start of a new line. The one exception to this rule is the fixity directives such as [``%infix``](/wiki/fixity-declaration/) - it is often clearest to put the fixity declaration for a constructor on the same line as the declaration of that constructor. + +## Comments + +Comments are at least as important a consideration when writing Twelf code intended to be readable by others, especially when the other readers are not themselves Twelf users. The [CMU POPLMark solution](http://www.cis.upenn.edu/~plclub/wiki-static/cmu-twelf/1a.elf) contains a good representation of how comments can be used to aid comprehension of a proof. + +### Guidelines + +* Each file should begin with single-line comments establishing a title and authorship. If several more lines about the purpose of the file are needed, those should go into a multi-line comment immediately underneath the title/author information. + +* Single line comments should annotate most "groups" and "super-groups" explaining, briefly, their purpose. They should be associated with a relatively strong visual indicator that is consistent across the file, but there aren't any standards here, and the following are all used: !}% +%%%%% Syntax %%%%% +% ===== Type system ===== +% +% ******* The result is s(s(s nat-39)) ******* +%%%%% Metatheorems about nat +%%%%%% +%{ Type preservation }% +%{! Be conscious of the interaction of your comment style with the [Twelf Emacs mode](/wiki/twelf-with-emacs/) syntax highlighting – some comment styles which use percent signs at the end and the beginning of a line cause the syntax highlighting to behave unpredictably. + +* Comments are most useful as multi-line comments at the beginning of "groups" and "super-groups" as defined in the previous section. Listing the important metatheorems at the beginning of a long group of metatheorems is also helpful. + +* Multi-line comments should be as free of visual cruft (borders, horizontal lines) as possible. The %\{ \}% construct should be used for most multi-line comments. + +## Identifiers + +Due to the absence of a module system for Twelf, careful identifier choice is crucial to managing the namespace of a large project. Because small projects can often grow into something quite large, we suggest thinking about a system of conventions such as the one described here from the outset. + +The names of constants and [type families](/wiki/type-family/) should _always_ begin with a lower case letter. Type families should only contain lower-case letters, numbers, and dashes ("-") which are used to separate words in an identifier, as in ``this-is-my-neat-proof``. The other options using underscores (as in ``this_is_my_neat_proof``), an option that is often used but which is currently generally avoided. Using ``camelCase`` is generally to be avoided: it is easier to distinguish an [implicit parameter](/wiki/implicit-and-explicit-parameters/) from a constant if only implicit parameters use capital letters. + +[Implicit parameters](/wiki/implicit-and-explicit-parameters/) must be capitalized; when using [explicit parameters](/wiki/implicit-and-explicit-parameters/) it is sometimes clearer to capitalize them, but it is often more clear and less error-prone to make explicit parameters lower case. + +Twelf code can generally be sorted into one of three categories: **[object logic syntax](/wiki/syntax-object-logic/)**, **[judgments](/wiki/judgment/)** about the object logic syntax and other judgments, and **[metatheorem](/wiki/metatheorem/)s** about the object logic. Each of the three categories has distinct style conventions. + +In this section, conventions will only be given as to how to structure identifiers with separators such as "-" and "/" to make then more readable and organized. There is a separate guide for [naming conventions](/wiki/naming-conventions/) which suggests what words should go between such separators. + +The [natural numbers](/wiki/natural-numbers/) will be used as a running example. + +### Object language syntax !}% +%sort nat %. +%sort list-nat %. +%{! Type families are typically inhabited by constants that correspond to different pieces of syntax or inference rules. We use a slash ("/") between the name of the type family and the identifier for the individual rule. !}% +%term nat/z nat %. +%term nat/s %pi nat %-> nat %. +%term list-nat/nil list-nat %. +%term list-nat/cons %pi nat %-> list-nat %-> list-nat %. +%{! (Whether a list of natural numbers should be called ``list-nat`` or ``nat-list`` is a matter left to the [naming conventions](/wiki/naming-conventions/) article.) + +#### Abbreviations + +The disadvantage of this representation is that identifier names can get relatively long. + +There are two solutions for this. One is that, if you are defining a language or logic, and not a library to be used in another project, you may want to consider a small number of pieces of syntax your "leaf cases" and simply leave off the part before the forward slash. +Be careful if you are doing so; this will seem like a great idea until you realize that you need both an element of type and an element of syntax called ``cont`` or ``pair`` or ``tuple``. + +The other solution is to use an _abbreviation_ or a _notational definition_. !}% +% Example of abbreviation +%inline z nat/z %. +%inline s nat/s %. +% Example of notational definition +%define z nat/z %. +%define s nat/s %. +%{! Abbreviations are probably better for large projects, because even if the abbreviated identifier gets redefined, it appears in the unshadowed extended form in the Twelf buffer. Notational definitions are probably preferable for small projects - or projects which are being debugged - for the same reason: shorter, more informative identifiers will appear in the Twelf buffer. + +### Judgments + +The naming conventions for judgments and their inference rules are similar to the ones for object syntax. The part after the forward slash, if it is associated with a piece of relevant syntax (i.e. ``/cons`` or ``/s``) should use the same modifier. !}% +%sort plus-nat %. +%mode plus-nat %in %in %out %. +%term plus-nat/z plus-nat z N N %. +%term plus-nat/s %pi (plus-nat (s N1) N2 (s N3)) %<- (plus-nat N1 N2 N3) %. +%{! ### Metatheorems: Identifier names + + +Each metatheorem is given a name in Twelf. There are two ways in which someone might consider naming their metatheorems. If you are defining a metatheorem that defines an admissible rule, it may be reasonable to use the convention used for showing inference rules: !}% +%sort plus-nat/z-alt {N} %. +%mode plus-nat/z-alt %in %out %. +%{! In most other cases, however, it is more reasonable to simply name the lemma using words and hyphens: !}% +%sort plus-comm %. +%mode plus-comm %in %out %. +%{! ### Metatheorems: Case names + +The cases in a metatheoretic proof almost never need to be referred to, so it is common to name the labels for every individual case ``-``. This may, however, make it more difficult to correlate error messages with specific rules, and so should be done with care. !}% +%sort plus-nat/z-alt {N} %. +%mode plus-nat/z-alt %in %out %. +%term _ plus-nat/z-alt z plus-nat/z %. +%term _ %pi (plus-nat/z-alt (s N) (plus-nat/s D)) %<- (plus-nat/z-alt N D) %. +%worlds () (plus-nat/z-alt _ _) %. +%total T (plus-nat/z-alt T _) %. +%{! If it is preferable to give each case of a metatheorem a name, for debugging or clarity reasons, the convention is to put a ``-`` before an identifier for the case. For example, we could re-write the preceding theorem in the following way. !}% +%sort plus-nat/z-alt {N} %. +%mode plus-nat/z-alt %in %out %. +%scope plus-nat/z-alt %term z plus-nat/z-alt z plus-nat/z %. +%term s %pi (plus-nat/z-alt (s N) (plus-nat/s D)) %<- (plus-nat/z-alt N D) %. +%worlds () (plus-nat/z-alt _ _) %. +%total T (plus-nat/z-alt T _) %. +%{! ## Example signature !}% +%% Demonstration of good Twelf style +%% The Twelf Wiki authors, Febtober 19045 +%{ + +If we needed to say a bit more about what we were doing +here than just the line or two that fits at the top of the +page, then this would be about the right place to try +and put that in. + +}% +%%%%%%%% +%{ Syntax }% +%sort exp %. +%term exp/unit exp %. +%%%%%%%% +%{ Types }% +%sort tp %. +%term tp/unit tp %. +%%%%%%%% +%{ Judgments }% +%sort typed %. +%mode typed %in %out %. +%term typed/unit %type d exp/unit tp/unit %. +%sort isvalue %. +%mode isvalue %in %. +%term isvalue/unit isvalue exp/unit %. +%%%%%%%% +%{ Metatheorems }% +%{ + +We're only proving one, rather silly, methatheorem in this style example, +namely that if an expression is a value then it has a type. Of course, +we're relying on the fact that everything in our toy language is a value +and has a type. + +}% +%sort isvalue-implies-typed %. +%mode isvalue-implies-typed %in %out %. +%term _ isvalue-implies-typed isvalue/unit %type d/unit %. +%worlds () (isvalue-implies-typed _ _) %. +%total T (isvalue-implies-typed T _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/pages/twelf-with-emacs.lf b/new-tests/stelf-output/pages/twelf-with-emacs.lf new file mode 100644 index 0000000..c7f490d --- /dev/null +++ b/new-tests/stelf-output/pages/twelf-with-emacs.lf @@ -0,0 +1,137 @@ +% +%{! The preferred way to interact with Twelf is through its emacs mode, which is included in the Twelf distribution. The [Download](/download/) page has basic information for setting up Emacs mode with Twelf. + +## Prerequisites + +These instructions assume you've already [downloaded Twelf](/download/), and that either +* You used the Windows/OSX installer +* You downloaded a pre-compiled tarball +* You downloaded a source tarball and have already built Twelf + +These instructions should easily translate to using Subversion to get Twelf, however. + +## Installing the Twelf emacs mode + +If you put Twelf in the directory `/somewhere/twelf/` then you should have +these two lines (just replace `somewhere` with wherever you put your Twelf): + +Add the following to your Emacs configuration file, which should be the file .emacs in your home directory. + +``` +(setq twelf-root "/somewhere/twelf/") +(load (concat twelf-root "emacs/twelf-init.el")) +``` + +For instance, if the directory where you unpacked the tarball is ``/usr/local/twelf``, then you'll want those two lines to be: + +``` +(setq twelf-root "/usr/local/twelf/") +(load (concat twelf-root "emacs/twelf-init.el")) +``` + +If your Twelf directory is ``/home/foo/stuff/logic/twelf``, then you'll want those lines to be: + +``` +(setq twelf-root "/home/foo/stuff/logic/twelf/") +(load (concat twelf-root "emacs/twelf-init.el")) +``` + +If you used the default Windows installer, you'll want those to lines to be + +``` +(setq twelf-root "C:\\Program Files\\Twelf\\") +(load (concat twelf-root "emacs\\twelf-init.el")) +``` + +Exit Emacs and restart it. + +## How Twelf programs are divided up into files + +- Twelf programs should be stored in files with the ``.elf`` extension. +- All of the ``.elf`` files in a project are collected in a _configuration file_, typically called ``sources.cfg``. A configuration file lists (paths to) individual ``.elf`` files in dependency order. You can load a configuration file, which loads each ``.elf`` file in sequence. + +### Creating a .cfg file + +Start Emacs in some directory (for the purposes of this description, we'll assume it's ``~/tmp/twelf``. Create the **configuration file** ``sources.cfg`` with the following text in it: + +``` +sometwelf.elf +``` + +Save the file (by using CTRL-x CTRL-s in Emacs). + +### Creating a .elf file + +Then create a file ``sometwelf.elf`` (by using CTRL-x CTRL-f in Emacs) and enter the following text: !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%define ss (%pi nat %-> nat) [n] s (s n) %. +%sort plus %. +%term plus/z plus z N N %. +%term plus/s %pi (plus N M P) %-> (plus (s N) M (s P)) %. +%{! Save the file (by using CTRL-x CTRL-s in Emacs). + +## Using the Twelf emacs mode + +You use Twelf by asking it to process declarations. Twelf responds by either printing ``Server OK`` at the bottom of the screen if your declarations were processed successfully, or by popping up the twelf-server buffer if there was an error (ABORT). + +The emacs mode allows you to process declarations at a variety of granularities: +* A whole configuration file (CTRL-c CTRL-c loads a cfg file) +* A whole .elf file (CTRL-c CTRL-s loads the .elf file open in the current emacs window) +* A single declaration (thing ending in a period) (CTRL-c CTRL-d loads declaration the cursor is on in the in the current .elf file) + +For example, a typical use case is to load your entire project with CTRL-c CTRL-c when you start working, and then to process new Twelf code declaration-by-declaration as you write it, using CTRL-c CTRL-d. Sometimes you will go back and make edits in lots of different places in a file, in which case it's helpful to reload just the current file with CTRL-c CTRL-s. + +Only reloading the configuration clears the state of Twelf. Even reloading will not clear previous declarations from the file (though it will typically shadow them). This might give unexpected results if you remove a declaration and expect it to have disappeared. + +### Loading the configuration file: CTRL-c CTRL-c + +Assuming you still have ``sometwelf.elf`` open in emacs, hit CTRL-c CTRL-c. If ``sometwelf.elf`` has unsaved changes, you will be asked: + +``` +File not in current configuration. Save? (yes or no) +``` + +Type yes. + +You will then be asked: + +``` +Visit config file: (default sources.cfg) ~/tmp/twelf/ +``` + +This is asking for the location of the sources.cfg file you saved when following the instructions above; the default option is the correct one. Just hit ENTER. + +You will then be asked: + +``` +Twelf server: (default twelf-server) /something/bin/ +``` +This is asking which Twelf binary you want to run; the default should already be correct, just hit ENTER. + +This will process your configuration file, which in turn says to process ``sometwelf.elf``. + +### Checking an individual declaration: CTRL-c CTRL-d + +Now you can type new declarations into the emacs buffer and use CTRL-c CTRL-d to load them. + +### Checking an individual file: CTRL-c CTRL-s + +Or you can reload the file in the current window with CTRL-c CTRL-s. You must previously have loaded a configuration file. + +## Syntax highlighting + +Twelf does not automatically refresh its syntax highlighting as you type. You can manually refresh the highlighting as follows: + +- ``CTRL-c l`` fontifies the currently visible emacs buffer. E.g. in ``sometwelf.elf``m the constants (``nat``, ``plus/z``, etc.) should turn red, the bound variable ``n`` should turn green, and the logic-programming variables ``N``, ``M``, and ``P`` should turn blue. +- ``CTRL-c CTRL-l`` fontifies the current declaration only. This is useful if you have really big files. + +For more info on the emacs mode, see \{\{guide|chapter=13|section=74|title=Emacs Interface\}\}. + +## Hacks + +If you use multiple frames with Emacs, ``twelf-mode`` annoyingly pops up the Twelf server buffer in the current frame, even if it is already shown in another frame. To avoid this, set ``display-buffer-reuse-frames`` to ``t``. Now, you will still have the problem that although the buffer is shown, it does not scroll to the bottom; to fix this you need to edit ``twelf.el`` and change ``(get-buffer-window twelf-server-buffer)`` to ``(get-buffer-window twelf-server-buffer t)``. + +## See also +Emacs Interface !}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/twelf-without-emacs.lf b/new-tests/stelf-output/pages/twelf-without-emacs.lf new file mode 100644 index 0000000..1f2a066 --- /dev/null +++ b/new-tests/stelf-output/pages/twelf-without-emacs.lf @@ -0,0 +1,31 @@ +% +%{! First, you should read up on [Twelf with Emacs](/wiki/twelf-with-emacs/), to understand [how Twelf programs are divided up into files](/wiki/twelf-with-emacs/) and how one interacts with those files. Then, instead of using emacs, you can run the ``twelf-server`` command directly from your shell and interact with your files using the following commands. (You may find it convenient to run ``twelf-server`` under some readline wrapper program like ``rlwrap`` or ``ledit`` to get command-line editing and history.) + +### Loading the configuration file + +To read your current config from ``sources.cfg``, type <code>make</code> To read some other config ``other.cfg``, type <code>make other.cfg</code> + +### Checking an individual declaration + +To check an individual declaration, type <code>readDecl</code> Then, on a new line, type a Twelf declaration, terminated with a period. + +### Checking an individual file + +To load an individual file ``sometwelf.elf``, type <code>loadFile sometwelf.elf</code> + +### Logic programming proof search + +To animate your specifications by doing logic programming queries, type <code>top</code> You will be presented with a a prompt, <code>?-</code> at which you can type queries with existential metavariables, like <code>?- plus (s z) (s (s z)) N.</code> Assuming you've loaded the signature from the [Natural numbers](/wiki/natural-numbers/) article), the Twelf server responds with the following: <code>Solving... +N = s (s (s z)). +More?</code> +You can exit the logic programming top-level by pressing Ctrl-C. + +### Quitting + +To quit from the Twelf server, type <code>quit</code> or press Ctrl-D. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Twelf_without_Emacs). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/twelftag.lf b/new-tests/stelf-output/pages/twelftag.lf new file mode 100644 index 0000000..2792671 --- /dev/null +++ b/new-tests/stelf-output/pages/twelftag.lf @@ -0,0 +1,32 @@ +% +%{! The **TwelfTag** system is a way of adding Twelf code directly into the Twelf Project Wiki. This page gives an introduction to TwelfTag for readers. The page [[Project:TwelfTag]] has information about TwelfTag for editors, and the [Twelf Live](/wiki/twelf-live/) system provides a more interactive way of using Twelf with on this website. !}% +%{! ## Syntax highlighting !}% +%{! The most basic function of TwelfTag is to highlight Twelf code much the same way that [Emacs](/wiki/twelf-with-emacs/) does. Names are red, and metavariables (free variables) are in blue. +!}% +%sort elem %. +%sort list %. +%term nil list %. +%term cons %pi elem %-> list %-> list %. +%sort list-reverse %. +%term lr/nil list-reverse nil L L %. +%term lr/cons %pi (list-reverse (cons E L1) L2 L3) %<- (list-reverse L1 (cons E L2) L3) %. +%{! ## Showing Twelf's response !}% +%{! When Twelf checks a piece of code, it produces some output that represents Twelf's reconstruction of that code and a message (``%% OK %%`` or ``%% ABORT %%``) that signals whether it was successful. Sometimes it is helpful to show Twelf's response in an article, and TwelfTag can do this too, showing Twelf's response in green. |check=decl !}% +%solve test : {e1} {e2} {e3} {e4} list-reverse (cons e1 (cons e3 (cons e2 (cons e4 nil)))) nil _ %. +%% OK % +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/TwelfTag). +!}% +%. == Linking to Twelf code and Twelf output == + +TwelfTag has the ability to link, inside a page, to the code that has been shown highlighted on that page, and also to link to the response Twelf gives when checking that code. +*This link shows you the twelf code (the code that is shown with syntax ) for this page. +*This link shows you Twelf's response from reading the code on this page. + +== "Literate Twelf" == + +For some pages, including this one, the page itself is written in valid Twelf that is then automatically transformed into an article. Pages written in this way look exactly like other pages, but they have a note at the top of the article that reports on Twelf's status after reading the file - %% OK % + diff --git a/new-tests/stelf-output/pages/type-family.lf b/new-tests/stelf-output/pages/type-family.lf new file mode 100644 index 0000000..c0f743c --- /dev/null +++ b/new-tests/stelf-output/pages/type-family.lf @@ -0,0 +1,21 @@ +% +%{! We use the word **type family** to refer to the syntactic class of [LF](/wiki/lf/) that classifies terms and is classified by kinds. The type families include the types of LF, which are dependent function types and type family constants that are fully applied to arguments. The type families also include partially applied type family constants. We refer to the arguments of a type family constant as the **subjects** or **indices** of the family. + +The LF methodology specifies how [object-language judgements are represented as LF type families](/wiki/judgment/). + +## See also +* Read the [introductions to Twelf](/wiki/introductions-to-twelf/) to learn how to represent object-language entities as LF type families. +* [LF](/wiki/lf/) +* [Canonical form](/wiki/canonical-form/)s +* [Judgment](/wiki/judgment/) +* [Higher-order judgment](/wiki/higher-order-judgements/) +* [Syntax (Object logic)](/wiki/syntax-object-logic/) +* [Relation](/wiki/relation/) + +\{\{stub\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Type_family). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/typed-combinators-soundness-and-completeness.lf b/new-tests/stelf-output/pages/typed-combinators-soundness-and-completeness.lf new file mode 100644 index 0000000..3ca7b58 --- /dev/null +++ b/new-tests/stelf-output/pages/typed-combinators-soundness-and-completeness.lf @@ -0,0 +1,565 @@ +% +%{! + +This is a case study translating the simply-typed lambda calculus into the SKI +combinator calculus. The correctness of the translation is proven in the +following sense: if a two terms are beta-eta equal, then their translations +are beta-eta equal, and vice versa. + +[William Lovas](/wiki/user-william-lovas/) + +!}% +%{! + +## Syntax and static semantics + +!}% +%{! + +First we define the syntax of the two languages. ``term A`` is the +type of simply-typed lambda-calculus terms of type A; ``comb A`` is +the type of simply-typed combinator terms of type A. Combinators are of +interest primarily because they have no binding structure; despite this +apparent limitation, we can translate any lambda-calculus term to an +operationally and logically related combinator term. + +Note that since we're using an +[intrinsic encoding](/wiki/intrinsic-and-extrinsic-encodings/), these syntax +definitions double as definitions of the languages' static semantics. + +By working over intrinsically typed terms, we can regard our theorems as +theorems comparing a natural deduction presentation of propositional logic +with a Hilbert-style axiomatic presentation, but none of the theorems depend +on types, and would hold equally well of untyped terms. + +!}% +%sort tp %. +%name tp %. +%term o tp %. +%term => %pi tp %-> tp %-> tp %. +%prec %right 10 => %. +%sort term %. +%name term %. +%%% uncomment for untyped version: +%% tm : type. %name tm M x. +%% %abbrev term : tp -> type = [x] tm. +%term app %pi (term (A => B)) %-> (term A) %-> (term B) %. +%term lam %pi (%pi (term A) %-> (term B)) %-> (term (A => B)) %. +%sort comb %. +%name comb %. +%%% uncomment for untyped version: +%% cm : type. %name cm N y. +%% %abbrev comb : tp -> type = [x] cm. +%term s comb (A => B => C => (A => B) => A => C) %. +%term k comb (A => B => A) %. +%term i comb (A => A) %. +%term capp %pi (comb (A => B)) %-> (comb A) %-> (comb B) %. +%{! + +We can use Twelf's [abbreviation mechanism](/wiki/abbrev-declaration/) and +[fixity declarations](/wiki/fixity-declaration/) to specify convenient infix +syntax for application. + +!}% +%inline @ app %. +%prec %left 10 @ %. +%inline @@ capp %. +%prec %left 10 @@ %. +%{! + +## Equational theory + +!}% +%{! + +Then we define equality relations on both languages. ``teq`` is +definitional equality for lambda terms; ``ceq`` is definitional +equality for combinator terms. In both cases, the equality relation +amounts to beta-eta equivalence. (We use extensionality instead of a +rule based on eta-expansion.) + +!}% +% definitional equality on terms +%sort teq %. +% beta +%term eq/beta teq (app (lam ([x] M1 x)) M2) (M1 M2) %. +% eta +%term eq/ext %pi (teq M1 M2) %<- ({x} teq (app M1 x) (app M2 x)) %. +% compatibilities +%term eq/lam %pi (teq (lam ([x] M x)) (lam ([x] M' x))) %<- ({x} teq (M x) (M' x)) %. +%term eq/app %pi (teq (app M1 M2) (app M1' M2')) %<- (teq M1 M1') %<- (teq M2 M2') %. +% equivalence +%term eq/refl teq M M %. +%term eq/symm %pi (teq M M') %<- (teq M' M) %. +%term eq/trans %pi (teq M M'') %<- (teq M' M'') %<- (teq M M') %. +%{! + +Infix syntax for transitivity is convenient when building derivations. + +!}% +%inline ; (%pi (teq N N') %-> (teq N' N'') %-> (teq N N'')) [d1] [d2] eq/trans d1 d2 %. +%prec %right 5 ; %. +%block term-block {A tp} [x term A]%. +%worlds (term-block) (teq _ _) %. +%{! !}% +% definitional equality on combs +%sort ceq %. +% betas +%term ceq/i ceq (capp i N) N %. +%term ceq/k ceq (capp (capp k N1) N2) N1 %. +%term ceq/s ceq (capp (capp (capp s N1) N2) N3) (capp (capp N1 N3) (capp N2 N3)) %. +% eta +%term ceq/ext %pi (ceq N1 N2) %<- ({y} ceq (capp N1 y) (capp N2 y)) %. +% compatibility +%term ceq/app %pi (ceq (capp N1 N2) (capp N1' N2')) %<- (ceq N1 N1') %<- (ceq N2 N2') %. +% equivalence +%term ceq/refl ceq N N %. +%term ceq/symm %pi (ceq N N') %<- (ceq N' N) %. +%term ceq/trans %pi (ceq N N'') %<- (ceq N' N'') %<- (ceq N N') %. +%inline ;; (%pi (ceq N N') %-> (ceq N' N'') %-> (ceq N N'')) [d1] [d2] ceq/trans d1 d2 %. +%prec %right 5 ;; %. +%block comb-block {A tp} [y comb A]%. +%worlds (comb-block) (ceq _ _) %. +%{! + +## Translation + +!}% +%{! + +We can now define a compositional translation from lambda terms to combinator +terms in the standard way. The translation appeals to a function called +_bracket abstraction_ which simulates binding in the combinator calculus. + +Bracket abstraction is usually written where +is a combinator term, and is a variable potentially free in +. (Be careful not to confuse these brackets with Twelf's syntax +for lambda abstraction.) It is defined inductively over the term +: + +<blockquote> + <br> + + (where not free in )<br> + + <br> +</blockquote> + +Using it, we can define a translation on lambda-terms, , where + is a lambda-term, as follows: + +<blockquote> +<br> + = + <br> + +</blockquote> + +Note that in the definition of bracket abstraction, we need not consider a +case for lambda-abstraction since bracket abstraction works over combinator +terms, in which lambda-abstraction has already been eliminated. + +The main translation is represented in LF by a judgement (an LF type family) +``translate M N``. Bracket abstraction is represented by a judgement +relating LF-level abstractions in the combinator language to closed combinator +terms, ``bracket ([x] N x) N``. This definition is essentially an +instance of [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/). + +!}% +%sort translate %. +%mode translate %in %out %. +%name translate %. +%sort bracket %. +%mode bracket %in %out %. +%name bracket %. +%term t/app + %pi (translate (app M1 M2) (capp N1 N2)) + %<- (translate M1 N1) + %<- (translate M2 N2) %. +%{! + +The translation on lambda-abstractions has to work under an extended context +with a lambda-term variable, a combinator-term variable, and an assumption +that the one translates to the other. See the ``%worlds`` declaration +below. (The block ``trans-block`` will form the basis of later blocks.) + +!}% +%term t/lam + %pi (translate (lam ([x] M x)) N*) + %<- ({x term A} {y comb A} %pi (translate x y) %-> (translate (M x) (N y))) + %<- (bracket ([y] N y) N*) %. +%term b/i bracket ([y] y) i %. +%term b/k bracket ([y] N) (capp k N) %. +%term b/s + %pi (bracket ([y] capp (N1 y) (N2 y)) (capp (capp s N1*) N2*)) + %<- (bracket ([y] N1 y) N1*) + %<- (bracket ([y] N2 y) N2*) %. +%block trans-block {A tp} [x term A] [y comb A] [dtrans translate x y]%. +%worlds (trans-block) (bracket _ _) %. +%worlds (trans-block) (translate _ _) %. +%{! + +Both relations are effective, though we cannot use this fact in proofs. + +!}% +%total N (bracket N _) %. +%total M (translate M _) %. +%{! + +Instead, we need explicitly prove effectiveness lemmas. + +!}% +%sort can-bracket {N %pi (comb A) %-> (comb B)} %. +%mode can-bracket %in %out %. +%sort can-translate {M term A} %. +%mode can-translate %in %out %. +%term _ can-bracket ([x] x) b/i %. +%term _ can-bracket ([x] N) b/k %. +%term _ + %pi (can-bracket ([x] capp (N1 x) (N2 x)) (b/s Dbrack2 Dbrack1)) + %<- (can-bracket ([x] N1 x) Dbrack1) + %<- (can-bracket ([x] N2 x) Dbrack2) %. +%term _ + %pi (can-translate (app M1 M2) (t/app Dtrans2 Dtrans1)) + %<- (can-translate M1 (%the (translate M1 N1) Dtrans1)) + %<- (can-translate M2 (%the (translate M2 N2) Dtrans2)) %. +%term _ + %pi (can-translate (lam ([x] M x)) (t/lam Dbrack Dtrans)) + %<- ({x} {y} {dtrans translate x y} {thm can-translate x dtrans} can-translate (M x) (%the (translate (M x) (N y)) (Dtrans x y dtrans))) + %<- (can-bracket ([y] N y) (%the (bracket ([y] N y) N*) Dbrack)) %. +%block can-block {A tp} [x term A] [y comb A] [dtrans translate x y] [thm can-translate x dtrans]%. +%worlds (can-block) (can-bracket _ _) %. +%total N (can-bracket N _) %. +%worlds (can-block) (can-translate _ _) %. +%total M (can-translate M _) %. +%{! + +## Correctness of the translation + +!}% +%{! + +First, we prove the correctness of bracket abstraction itself: the +application of a bracket abstraction is equivalent to a substitution. + +!}% +%sort subst %. +%mode subst %in %in %out %. +%term _ subst (%the (bracket ([x] x) i) b/i) N0 (%the (ceq (i @@ N0) N0) ceq/i) %. +%term _ subst (%the (bracket ([x] Y) (k @@ Y)) b/k) N0 (%the (ceq (k @@ Y @@ N0) Y) ceq/k) %. +% developing incrementally, it's useful to write down the type of each output +%term _ + %pi (subst (%the (bracket ([x] N1 x @@ N2 x) (s @@ N1' @@ N2')) (b/s (%the (bracket ([x] N2 x) N2') B2) (%the (bracket ([x] N1 x) N1') B1))) N0 (ceq/trans (%the (ceq (s @@ N1' @@ N2' @@ N0) (N1' @@ N0 @@ (N2' @@ N0))) ceq/s) (%the (ceq (N1' @@ N0 @@ (N2' @@ N0)) (N1 N0 @@ N2 N0)) (ceq/app Dceq2 Dceq1)))) + %<- (subst B1 N0 (%the (ceq (N1' @@ N0) (N1 N0)) Dceq1)) + %<- (subst B2 N0 (%the (ceq (N2' @@ N0) (N2 N0)) Dceq2)) %. +%worlds (comb-block) (subst _ _ _) %. +%total {B} (subst B _ _) %. +%{! + +Next, we need to know that the translation of a term is unique up to +equivalence. + +!}% +%sort translate-unique %. +%mode translate-unique %in %in %out %. +%term _ + %pi (translate-unique (%the (translate (M1 @ M2) (N1 @@ N2)) (t/app (%the (translate M2 N2) Dtrans2) (%the (translate M1 N1) Dtrans1))) (%the (translate (M1 @ M2) (N1' @@ N2')) (t/app (%the (translate M2 N2') Dtrans2') (%the (translate M1 N1') Dtrans1'))) (ceq/app Dceq2 Dceq1)) + %<- (translate-unique Dtrans1 Dtrans1' (%the (ceq N1 N1') Dceq1)) + %<- (translate-unique Dtrans2 Dtrans2' (%the (ceq N2 N2') Dceq2)) %. +%term _ + %pi (translate-unique (%the (translate (lam ([x] M x)) N*) (t/lam (%the (bracket ([y] N y) N*) Dbrack) ([x] [y] [dtrans] %the (translate (M x) (N y)) (Dtrans x y dtrans)))) (%the (translate (lam ([x] M x)) N'*) (t/lam (%the (bracket ([y] N' y) N'*) Dbrack') ([x] [y] [dtrans] %the (translate (M x) (N' y)) (Dtrans' x y dtrans)))) (%the (ceq N* N'*) (ceq/ext ([y] %the (ceq (N* @@ y) (N'* @@ y)) ((%the (ceq (N* @@ y) (N y)) (Dceq y)) ;; (%the (ceq (N y) (N' y)) (Dceqtrans y)) ;; (%the (ceq (N' y) (N'* @@ y)) (ceq/symm (Dceq' y)))))))) + %<- ({x} {y} {dtrans translate x y} {thm-unique translate-unique dtrans dtrans ceq/refl} translate-unique (Dtrans x y dtrans) (Dtrans' x y dtrans) (%the (ceq (N y) (N' y)) (Dceqtrans y))) + %<- ({z} subst Dbrack z (%the (ceq (N* @@ z) (N z)) (Dceq z))) + %<- ({z} subst Dbrack' z (%the (ceq (N'* @@ z) (N' z)) (Dceq' z))) %. +%block unique-block {A tp} [x term A] [y comb A] [dtrans translate x y] [thm translate-unique dtrans dtrans ceq/refl]%. +%worlds (unique-block) (translate-unique _ _ _) %. +%total D (translate-unique D _ _) %. +%{! + +Then, we can prove simulation, the correctness of translation, by a +straightforward induction on equality derivations in the lambda-calculus, +using the correctness of bracket abstraction as a lemma in the case of a +beta-reduction. + +!}% +%sort simulate %. +%mode simulate %in %in %in %out %. +%term _ + %pi (simulate (%the (translate (app (lam ([x] M1 x)) M2) (capp N1* N2)) (t/app (%the (translate M2 N2) Dtrans2) (t/lam (%the (bracket ([y] N1 y) N1*) Dbrack) ([x term A2] [y comb A2] [dtrans translate x y] %the (translate (M1 x) (N1 y)) (Dtrans1 x y dtrans))))) (%the (translate (M1 M2) N3) Dtrans3) (%the (teq (app (lam ([x] M1 x)) M2) (M1 M2)) eq/beta) (%the (ceq (capp N1* N2) N3) (Dceq ;; Dceq3))) + %<- (subst Dbrack N2 (%the (ceq (capp N1* N2) (N1 N2)) Dceq)) + %<- (translate-unique (Dtrans1 M2 N2 Dtrans2) Dtrans3 (%the (ceq (N1 N2) N3) Dceq3)) %. +%term _ + %pi (simulate (%the (translate M1 N1) Dtrans1) (%the (translate M2 N2) Dtrans2) (%the (teq M1 M2) (eq/ext ([x] %the (teq (app M1 x) (app M2 x)) (Deq x)))) (%the (ceq N1 N2) (ceq/ext ([y] Dceq y)))) + %<- ({x} {y} {dtrans translate x y} {thm-can can-translate x dtrans} {thm-unique translate-unique dtrans dtrans ceq/refl} simulate (%the (translate (app M1 x) (capp N1 y)) (t/app dtrans Dtrans1)) (%the (translate (app M2 x) (capp N2 y)) (t/app dtrans Dtrans2)) (%the (teq (app M1 x) (app M2 x)) (Deq x)) (%the (ceq (capp N1 y) (capp N2 y)) (Dceq y))) %. +%term _ + %pi (simulate (%the (translate (lam ([x] M x)) N*) (t/lam (%the (bracket ([y] N y) N*) Dbrack) ([x] [y] [dtrans translate x y] %the (translate (M x) (N y)) (Dtrans x y dtrans)))) (%the (translate (lam ([x] M' x)) N*') (t/lam (%the (bracket ([y] N' y) N*') Dbrack') ([x] [y] [dtrans translate x y] %the (translate (M' x) (N' y)) (Dtrans' x y dtrans)))) (%the (teq (lam ([x] M x)) (lam ([x] M' x))) (eq/lam ([x] %the (teq (M x) (M' x)) (Deq x)))) (%the (ceq N* N*') (ceq/ext ([y] (%the (ceq (N* @@ y) (N y)) (Dceq* y)) ;; (%the (ceq (N y) (N' y)) (Dceq y)) ;; (%the (ceq (N' y) (N*' @@ y)) (ceq/symm (Dceq*' y))))))) + %<- ({x} {y} {dtrans translate x y} {thm-can can-translate x dtrans} {thm-unique translate-unique dtrans dtrans ceq/refl} simulate (%the (translate (M x) (N y)) (Dtrans x y dtrans)) (%the (translate (M' x) (N' y)) (Dtrans' x y dtrans)) (%the (teq (M x) (M' x)) (Deq x)) (%the (ceq (N y) (N' y)) (Dceq y))) + %<- ({y} subst Dbrack y (%the (ceq (N* @@ y) (N y)) (Dceq* y))) + %<- ({y} subst Dbrack' y (%the (ceq (N*' @@ y) (N' y)) (Dceq*' y))) %. +%term _ + %pi (simulate (%the (translate (M1 @ M2) (N1 @@ N2)) (t/app (%the (translate M2 N2) Dtrans2) (%the (translate M1 N1) Dtrans1))) (%the (translate (M1' @ M2') (N1' @@ N2')) (t/app (%the (translate M2' N2') Dtrans2') (%the (translate M1' N1') Dtrans1'))) (%the (teq (M1 @ M2) (M1' @ M2')) (eq/app (%the (teq M2 M2') Deq2) (%the (teq M1 M1') Deq1))) (%the (ceq (N1 @@ N2) (N1' @@ N2')) (ceq/app Dceq2 Dceq1))) + %<- (simulate Dtrans1 Dtrans1' Deq1 (%the (ceq N1 N1') Dceq1)) + %<- (simulate Dtrans2 Dtrans2' Deq2 (%the (ceq N2 N2') Dceq2)) %. +%term _ + %pi (simulate (%the (translate M N) Dtrans) (%the (translate M N') Dtrans') (%the (teq M M) eq/refl) Dceq) + %<- (translate-unique Dtrans Dtrans' (%the (ceq N N') Dceq)) %. +%term _ + %pi (simulate (%the (translate M N) Dtrans) (%the (translate M' N') Dtrans') (%the (teq M M') (eq/symm (%the (teq M' M) Deq))) (%the (ceq N N') (ceq/symm Dceq))) + %<- (simulate Dtrans' Dtrans Deq (%the (ceq N' N) Dceq)) %. +%term _ + %pi (simulate (%the (translate M N) Dtrans) (%the (translate M'' N'') Dtrans'') (%the (teq M M'') (eq/trans (%the (teq M M') Deq1) (%the (teq M' M'') Deq2))) (%the (ceq N N'') (ceq/trans Dceq1 Dceq2))) + %<- (can-translate M' (%the (translate M' N') Dtrans')) + %<- (simulate Dtrans Dtrans' Deq1 (%the (ceq N N') Dceq1)) + %<- (simulate Dtrans' Dtrans'' Deq2 (%the (ceq N' N'') Dceq2)) %. +%block simulate-block {A tp} [x term A] [y comb A] [dtrans translate x y] [thm-can can-translate x dtrans] [thm-unique translate-unique dtrans dtrans ceq/refl]%. +%worlds (simulate-block) (simulate _ _ _ _) %. +%total D (simulate _ _ D _) %. +%{! + +## Backward translation + +To show completeness we exhibit a backward translation---called +"reification"---taking combinators to lambda terms, and show that +for every translation, there is a reification to a term equivalent +to the original input term. + +Reification just replaces combinators with their definitions as +lambda terms. + +!}% +%sort reify %. +%name reify %. +%mode reify %in %out %. +%term r/s reify s (lam ([x] lam ([y] lam ([z] x @ z @ (y @ z))))) %. +%term r/k reify k (lam ([x] lam ([y] x))) %. +%term r/i reify i (lam ([x] x)) %. +%term r/capp %pi (reify (N1 @@ N2) (M1 @ M2)) %<- (reify N1 M1) %<- (reify N2 M2) %. +%{! + +We consider reification in any world with free variables as long as there's +a combinator variable for each lambda-term variable and an appropriate +correspondence between them. + +!}% +%block reify-block {A tp} [x term A] [y comb A] [dreify reify y x]%. +%worlds (reify-block) (reify _ _) %. +%{! + +Twelf can verify that reification is total and that the result is unique +up to identity, + +!}% +%total N (reify N _) %. +%unique reify %in %out %. +%{! + +but in order to use these facts in other metatheorems, we must prove them +ourselves. + +The theorem ``can-reify`` shows the effectiveness of reification by +constructing a ``reify`` derivation for any given combinator ``N``. + +!}% +%sort can-reify {N comb A} %. +%mode can-reify %in %out %. +%term _ can-reify s r/s %. +%term _ can-reify k r/k %. +%term _ can-reify i r/i %. +%term _ + %pi (can-reify (capp N1 N2) (r/capp Dreify2 Dreify1)) + %<- (can-reify N1 Dreify1) + %<- (can-reify N2 Dreify2) %. +%{! + +Note the use of a _theorem case_ in the block for ``can-reify``: +since we prove reification with respect to an open world, we have to include +proofs for the free variable cases. + +!}% +%block can-reify-block {A tp} [x term A] [y comb A] [dreify reify y x] [thm can-reify y dreify]%. +%worlds (can-reify-block) (can-reify _ _) %. +%total N (can-reify N _) %. +%{! + +The theorem ``reify-unique`` shows the uniqueness of reification. +For convenience, we demonstrate only that reifications are unique up to +term equivalence. + +!}% +%sort reify-unique %. +%mode reify-unique %in %in %out %. +%term _ reify-unique r/i r/i eq/refl %. +%term _ reify-unique r/k r/k eq/refl %. +%term _ reify-unique r/s r/s eq/refl %. +%term _ + %pi (reify-unique (r/capp Dreify2 Dreify1) (r/capp Dreify2' Dreify1') (eq/app Deq2 Deq1)) + %<- (reify-unique Dreify1 Dreify1' Deq1) + %<- (reify-unique Dreify2 Dreify2' Deq2) %. +%block reify-unique-block {A tp} [x term A] [y comb A] [dreify reify y x] [thm reify-unique dreify dreify eq/refl]%. +%worlds (reify-unique-block) (reify-unique _ _ _) %. +%total D (reify-unique D _ _) %. +%{! + +The reification of a closed combinator term is a closed lambda-term, but +we require the following strengthening lemma to show that this is the case. + +!}% +%sort strengthen-reify {x} {y} %. +%mode strengthen-reify %in %out %out %. +%term _ strengthen-reify ([x] [y] [dreify reify y x] r/s) r/s ([x] eq/refl) %. +%term _ strengthen-reify ([x] [y] [dreify reify y x] r/k) r/k ([x] eq/refl) %. +%term _ strengthen-reify ([x] [y] [dreify reify y x] r/i) r/i ([x] eq/refl) %. +%term _ + %pi (strengthen-reify ([x] [y] [dreify reify y x] %the (reify (N1 @@ N2) (M1! x @ (M2! x))) (r/capp (%the (reify N2 (M2! x)) (Dreify2! x y dreify)) (%the (reify N1 (M1! x)) (Dreify1! x y dreify)))) (r/capp Dreify2 Dreify1) ([x] eq/app (Deq2 x) (Deq1 x))) + %<- (strengthen-reify ([x] [y] [dreify] Dreify1! x y dreify) (%the (reify N1 M1) Dreify1) ([x] %the (teq M1 (M1! x)) (Deq1 x))) + %<- (strengthen-reify ([x] [y] [dreify] Dreify2! x y dreify) (%the (reify N2 M2) Dreify2) ([x] %the (teq M2 (M2! x)) (Deq2 x))) %. +%block strengthen-reify-block {A tp} [x term A] [y comb A] [dreify reify y x] [thm {B tp} strengthen-reify ([x1 term B] [y1 comb B] [dreify1] dreify) dreify ([x1] eq/refl)]%. +%worlds (strengthen-reify-block) (strengthen-reify _ _ _) %. +%total D (strengthen-reify D _ _) %. +%{! + +## Correctness of the backward translation + +Our main technique for showing completeness is to show that every translation +of ``M`` has a reification to an equivalent ``M'``. First, we show +the following lemma, which says that bracket abstractions reify to lambda +abstractions. + +!}% +%sort bracket-reify %. +%mode bracket-reify %in %in %out %out %. +%term _ bracket-reify (%the (bracket ([y] y) i) b/i) ([x] [y] [dreify reify y x] dreify) (%the (reify i (lam ([x] x))) r/i) (%the (teq (lam ([x] x)) (lam ([x] x))) eq/refl) %. +%{! + +In the case where bracket abstraction returns an application of the K +combinator to a closed term, we require our strengthening lemma to show +that the result of reification is also closed. + +!}% +%term _ + %pi (bracket-reify (%the (bracket ([y] N) (k @@ N)) b/k) ([x] [y] [dreify reify y x] %the (reify N (M! x)) (Dreify! x y dreify)) (%the (reify (k @@ N) (lam ([x] lam ([y] x)) @ M)) (r/capp Dreify r/k)) (%the (teq (lam ([x] lam ([y] x)) @ M) (lam ([y] M! y))) (eq/beta ; eq/lam ([x] Deq x)))) + %<- (strengthen-reify ([x] [y] [dreify reify y x] Dreify! x y dreify) (%the (reify N M) Dreify) ([x] %the (teq M (M! x)) (Deq x))) %. +%{! + +The case for reifying an application of the S combinator is tedious, but +straightforward. + +!}% +%term _ + %pi (bracket-reify (%the (bracket ([y] N1 y @@ (N2 y)) (s @@ N1* @@ N2*)) (b/s (%the (bracket ([y] N2 y) N2*) Dbrack2) (%the (bracket ([y] N1 y) N1*) Dbrack1))) ([x] [y] [dreify reify y x] %the (reify (N1 y @@ (N2 y)) (M1 x @ (M2 x))) (r/capp (%the (reify (N2 y) (M2 x)) (Dreify2 x y dreify)) (%the (reify (N1 y) (M1 x)) (Dreify1 x y dreify)))) (%the (reify (s @@ N1* @@ N2*) (lam ([x] lam ([y] lam ([z] x @ z @ (y @ z)))) @ M1* @ M2*)) (r/capp (%the (reify N2* M2*) Dreify2*) (r/capp (%the (reify N1* M1*) Dreify1*) r/s))) (%the (teq (lam ([x] lam ([y] lam ([z] x @ z @ (y @ z)))) @ M1* @ M2*) (lam ([z] M1 z @ (M2 z)))) (eq/app (%the (teq M2* M2*) eq/refl) eq/beta ; eq/beta ; eq/lam ([x] eq/app (%the (teq (M2* @ x) (M2 x)) (eq/app (%the (teq x x) eq/refl) Deq2* ; eq/beta)) (%the (teq (M1* @ x) (M1 x)) (eq/app (%the (teq x x) eq/refl) Deq1* ; eq/beta)))))) + %<- (bracket-reify Dbrack1 ([x] [y] [dreify reify y x] Dreify1 x y dreify) (%the (reify N1* M1*) Dreify1*) (%the (teq M1* (lam ([z] M1 z))) Deq1*)) + %<- (bracket-reify Dbrack2 ([x] [y] [dreify reify y x] Dreify2 x y dreify) (%the (reify N2* M2*) Dreify2*) (%the (teq M2* (lam ([z] M2 z))) Deq2*)) %. +%worlds (strengthen-reify-block) (bracket-reify _ _ _ _) %. +%total D (bracket-reify D _ _ _) %. +%{! + +Now we can prove that every translation reifies to an equivalent term, +appealing to the above bracket abstraction lemma in the ``lam`` case. + +!}% +%sort translate-reify %. +%mode translate-reify %in %out %out %. +%term _ + %pi (translate-reify (%the (translate (M1 @ M2) (N1 @@ N2)) (t/app (%the (translate M2 N2) Dtrans2) (%the (translate M1 N1) Dtrans1))) (%the (reify (N1 @@ N2) (M1' @ M2')) (r/capp (%the (reify N2 M2') Dreify2) (%the (reify N1 M1') Dreify1))) (%the (teq (M1 @ M2) (M1' @ M2')) (eq/app Deq2 Deq1))) + %<- (translate-reify Dtrans1 Dreify1 (%the (teq M1 M1') Deq1)) + %<- (translate-reify Dtrans2 Dreify2 (%the (teq M2 M2') Deq2)) %. +%term _ + %pi (translate-reify (%the (translate (lam ([x] M x)) N*) (t/lam (%the (bracket ([y] N y) N*) Dbrack) ([x] [y] [dtrans translate x y] %the (translate (M x) (N y)) (Dtrans x y dtrans)))) (%the (reify N* M*) Dreify*) (%the (teq (lam ([x] M x)) M*) (eq/lam ([x] Deq x) ; eq/symm Deq*))) + %<- ({x} {y} {dtrans translate x y} {dreify reify y x} {thm-closed {B} strengthen-reify ([x1 term B] [y1 comb B] [dreify1] dreify) dreify ([x1] eq/refl)} {thm-translate translate-reify dtrans dreify eq/refl} translate-reify (Dtrans x y dtrans) (%the (reify (N y) (M0 x)) (Dreify x y dreify)) (%the (teq (M x) (M0 x)) (Deq x))) + %<- (bracket-reify Dbrack ([x] [y] [dreify] Dreify x y dreify) (%the (reify N* M*) Dreify*) (%the (teq M* (lam ([x] M0 x))) Deq*)) %. +%block translate-reify-block {A tp} [x term A] [y comb A] [dtrans translate x y] [dreify reify y x] [thm-closed {B tp} strengthen-reify ([x1 term B] [y1 comb B] [dreify1] dreify) dreify ([x1] eq/refl)] [thm-translate translate-reify dtrans dreify eq/refl]%. +%worlds (translate-reify-block) (translate-reify _ _ _) %. +%total D (translate-reify D _ _) %. +%{! + +Next, we prove a theorem similar to our earlier ``simulate`` theorem, +but for reification: given two reification derivations from equivalent inputs, +their outputs are equivalent. + +!}% +%sort reify-simulate %. +%mode reify-simulate %in %in %in %out %. +%term _ + %pi (reify-simulate (%the (reify (i @@ N) (lam ([x] x) @ M)) (r/capp (%the (reify N M) Dreify) r/i)) (%the (reify N M') Dreify') (%the (ceq (i @@ N) N) ceq/i) (%the (teq (lam ([x] x) @ M) M') (eq/beta ; Deq))) + %<- (reify-unique Dreify Dreify' (%the (teq M M') Deq)) %. +%term _ + %pi (reify-simulate (%the (reify (k @@ N1 @@ N2) (lam ([x] lam ([y] x)) @ M1 @ M2)) (r/capp (%the (reify N2 M2) Dreify2) (r/capp (%the (reify N1 M1) Dreify1) r/k))) (%the (reify N1 M') Dreify') (%the (ceq (k @@ N1 @@ N2) N1) ceq/k) (%the (teq (lam ([x] lam ([y] x)) @ M1 @ M2) M') (eq/app eq/refl eq/beta ; eq/beta ; Deq))) + %<- (reify-unique Dreify1 Dreify' (%the (teq M1 M') Deq)) %. +%term _ + %pi (reify-simulate (%the (reify (s @@ N1 @@ N2 @@ N3) (lam ([x] lam ([y] lam ([z] x @ z @ (y @ z)))) @ M1 @ M2 @ M3)) (r/capp (%the (reify N3 M3) Dreify3) (r/capp (%the (reify N2 M2) Dreify2) (r/capp (%the (reify N1 M1) Dreify1) r/s)))) (%the (reify (N1 @@ N3 @@ (N2 @@ N3)) (M1' @ M3'' @ (M2' @ M3'))) (r/capp (r/capp (%the (reify N3 M3') Dreify3') (%the (reify N2 M2') Dreify2')) (r/capp (%the (reify N3 M3'') Dreify3'') (%the (reify N1 M1') Dreify1')))) (%the (ceq (s @@ N1 @@ N2 @@ N3) (N1 @@ N3 @@ (N2 @@ N3))) ceq/s) (%the (teq (lam ([x] lam ([y] lam ([z] x @ z @ (y @ z)))) @ M1 @ M2 @ M3) (M1' @ M3'' @ (M2' @ M3'))) (eq/app eq/refl (eq/app eq/refl eq/beta) ; eq/app eq/refl eq/beta ; eq/beta ; eq/app (eq/app Deq3 Deq2) (eq/app Deq3' Deq1)))) + %<- (reify-unique Dreify1 Dreify1' (%the (teq M1 M1') Deq1)) + %<- (reify-unique Dreify2 Dreify2' (%the (teq M2 M2') Deq2)) + %<- (reify-unique Dreify3 Dreify3' (%the (teq M3 M3') Deq3)) + %<- (reify-unique Dreify3 Dreify3'' (%the (teq M3 M3'') Deq3')) %. +%term _ + %pi (reify-simulate (%the (reify N1 M1) Dreify1) (%the (reify N2 M2) Dreify2) (%the (ceq N1 N2) (ceq/ext ([y] %the (ceq (N1 @@ y) (N2 @@ y)) (Dceq y)))) (%the (teq M1 M2) (eq/ext ([x] %the (teq (M1 @ x) (M2 @ x)) (Deq x))))) + %<- ({x} {y} {dreify reify y x} {thm-can can-reify y dreify} {thm-unique reify-unique dreify dreify eq/refl} reify-simulate (%the (reify (N1 @@ y) (M1 @ x)) (r/capp dreify Dreify1)) (%the (reify (N2 @@ y) (M2 @ x)) (r/capp dreify Dreify2)) (Dceq y) (%the (teq (M1 @ x) (M2 @ x)) (Deq x))) %. +%term _ + %pi (reify-simulate (%the (reify (N1 @@ N2) (M1 @ M2)) (r/capp (%the (reify N2 M2) Dreify2) (%the (reify N1 M1) Dreify1))) (%the (reify (N1' @@ N2') (M1' @ M2')) (r/capp (%the (reify N2' M2') Dreify2') (%the (reify N1' M1') Dreify1'))) (%the (ceq (N1 @@ N2) (N1' @@ N2')) (ceq/app (%the (ceq N2 N2') Dceq2) (%the (ceq N1 N1') Dceq1))) (%the (teq (M1 @ M2) (M1' @ M2')) (eq/app Deq2 Deq1))) + %<- (reify-simulate Dreify1 Dreify1' Dceq1 (%the (teq M1 M1') Deq1)) + %<- (reify-simulate Dreify2 Dreify2' Dceq2 (%the (teq M2 M2') Deq2)) %. +%term _ + %pi (reify-simulate (%the (reify N M) Dreify) (%the (reify N M') Dreify') (%the (ceq N N) ceq/refl) (%the (teq M M') Deq)) + %<- (reify-unique Dreify Dreify' (%the (teq M M') Deq)) %. +%term _ + %pi (reify-simulate (%the (reify N M) Dreify) (%the (reify N' M') Dreify') (%the (ceq N N') (ceq/symm (%the (ceq N' N) Dceq))) (%the (teq M M') (eq/symm Deq))) + %<- (reify-simulate Dreify' Dreify Dceq (%the (teq M' M) Deq)) %. +%term _ + %pi (reify-simulate (%the (reify N M) Dreify) (%the (reify N'' M'') Dreify'') (%the (ceq N N'') (ceq/trans (%the (ceq N N') Dceq1) (%the (ceq N' N'') Dceq2))) (%the (teq M M'') (eq/trans Deq1 Deq2))) + %<- (can-reify N' (%the (reify N' M') Dreify')) + %<- (reify-simulate Dreify Dreify' Dceq1 (%the (teq M M') Deq1)) + %<- (reify-simulate Dreify' Dreify'' Dceq2 (%the (teq M' M'') Deq2)) %. +%block reify-simulate-block {A tp} [x term A] [y comb A] [dreify reify y x] [thm-can can-reify y dreify] [thm-unique reify-unique dreify dreify eq/refl]%. +%worlds (reify-simulate-block) (reify-simulate _ _ _ _) %. +%total D (reify-simulate _ _ D _) %. +%{! + +Finally, we can now prove the translation complete: every combinator +equivalence hit by the translation comes from a lambda-calculus equivalence. + +Given two translations, we get reifications for both. By the above simulation +theorem, we show that the reifications must be equal. Then the desired result +follows by transitivity. + +!}% +%sort complete %. +%mode complete %in %in %in %out %. +%term _ + %pi (complete (%the (translate M N) Dtrans) (%the (translate M' N') Dtrans') (%the (ceq N N') Dceq) (%the (teq M M') (Deq ; Deq0 ; eq/symm Deq'))) + %<- (translate-reify Dtrans (%the (reify N M0) Dreify) (%the (teq M M0) Deq)) + %<- (translate-reify Dtrans' (%the (reify N' M0') Dreify') (%the (teq M' M0') Deq')) + %<- (reify-simulate Dreify Dreify' Dceq (%the (teq M0 M0') Deq0)) %. +%{! + +We can also make a copy of ``simulate`` called ``sound``, in order +to be able check it in the same world as ``complete``. + +!}% +%sort sound %. +%mode sound %in %in %in %out %. +%term _ + %pi (sound (%the (translate M N) Dtrans) (%the (translate M' N') Dtrans') (%the (teq M M') Deq) Dceq) + %<- (simulate Dtrans Dtrans' Deq (%the (ceq N N') Dceq)) %. +%{! + +We can make a master-block in which each of soundness and completeness hold. +Terms under translation can have any free variables x, as long as they are +matched by free combinator variables y such that x translates to y and y +reifies to x. The other variables in the block are theorem cases relating to +these new translation and reification assumptions. + +Note that for world subsumption to work correctly, it was necessary to always +state x and y in the same order, even in the reification theorems. + +!}% +%block master-block {A tp} [x term A] [y comb A] [dtrans translate x y] [thm-can-tr can-translate x dtrans] [thm-tr-unique translate-unique dtrans dtrans ceq/refl] [dreify reify y x] [thm-can-re can-reify y dreify] [thm-re-unique reify-unique dreify dreify eq/refl] [thm-re-closed {B tp} strengthen-reify ([x1 term B] [y1 comb B] [dreify1] dreify) dreify ([x1] eq/refl)] [thm-tr-re translate-reify dtrans dreify eq/refl]%. +%worlds (master-block) (complete _ _ _ _) %. +%total {} (complete _ _ _ _) %. +%worlds (master-block) (sound _ _ _ _) %. +%total {} (sound _ _ _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Typed_combinators_soundness_and_completeness). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/unification.lf b/new-tests/stelf-output/pages/unification.lf new file mode 100644 index 0000000..b52518a --- /dev/null +++ b/new-tests/stelf-output/pages/unification.lf @@ -0,0 +1,17 @@ +% +%{! **Unification** is an algorithm for comparing terms with holes (called **unification variables**) for equality, discovering substitutions for the unification variables which make those terms equal. Unification of LF terms is used in a variety of ways in Twelf: +- During type and term reconstruction, unification is used to infer omitted type and term information. +- In the [logic programming](/wiki/logic-programming/) operational semantics of LF, unification is used to find solutions to unification variables in queries. + +Unification for LF terms requires higher-order unification, which is undecidable. However, many instances of unification that arise in Twelf fall into the **pattern fragment**, which is decidable. Those that are not in the pattern fragment are postponed as **constraints**; unresolved constraints are reported to the user. + +\{\{needs|examples of "unresolved constrants" error messages.\}\} + + +\{\{stub\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Unification). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/uniqueness-lemma.lf b/new-tests/stelf-output/pages/uniqueness-lemma.lf new file mode 100644 index 0000000..58c40b2 --- /dev/null +++ b/new-tests/stelf-output/pages/uniqueness-lemma.lf @@ -0,0 +1,128 @@ +% +%{! Many judgements of an object language have the property that, given some subjects of the judgement, the other subjects are determined uniquely. For example: The [addition judgement](/wiki/proving-metatheorems-representing-the-judgements-of-the-natural-numbers/) on natural numbers has the property that the two addends determine the sum uniquely. A typing judgement for a programming language may have the property that a term has at most one type. The operational semantics of a programming language may have the property that a term steps to at most one other term. We refer to a metatheorem that establishes such a property a **uniqueness lemma**. + +The goal of this article is to teach you how to prove uniqueness lemmas in Twelf. The only subtlety is that the statement of a uniqueness lemma requires an auxiliary definition of a judgement defining equality at the relevant types. + +In this article, we refer to the "given" subjects of the judgement as _inputs_ and those subjects that are determined uniquely by the givens as _outputs_. However, these inputs and outputs do not need to correspond with the [``%mode``](/wiki/percent-mode/) of the judgement: it is meaningful to prove a uniqueness lemma for a judgement that has no mode specified in Twelf. Note that a judgement may satisfy a uniqueness lemma but not define a function: uniqueness says that the outputs are uniquely determined if they exist at all. To know that a judgement defines a function, it is necessary to prove a corresponding [effectiveness lemma](/wiki/effectiveness-lemma/) establishing that some instance of the judgement holds for each input. + +Sometimes a uniqueness lemma can be proven trivially by leveraging a [``%unique``](/wiki/percent-unique/) declaration. + +## Example: Flipping bits + +The following is a simple example of a uniqueness lemma. First we define bits and a simple relation ``bit-flip`` that negates a bit. !}% +%sort bit %. +%term bit/0 bit %. +%term bit/1 bit %. +%sort bit-flip %. +%term bit-flip/01 bit-flip bit/0 bit/1 %. +%term bit-flip/10 bit-flip bit/1 bit/0 %. +%{! We would like to prove that the second argument of ``bit-flip`` is determined uniquely by the first. How do we state this theorem? We assume two derivations of ``bit-flip`` for the same input that seem to have distinct outputs and we prove that the outputs are the same: + +: If ``D : bit-flip B1 B2`` and ``D' : bit-flip B1 B2'`` then ``B2 = B2'``. + +The only subtlety in proving this theorem in Twelf is that we need a judgement representing the equality used in the conclusion. If the output of the judgement is in fact unique (rather than unique modulo some equivalence relation), then the best judgement to use is the [identity](/wiki/equality/) relation at the appropriate type: !}% +%sort id-bit %. +%term id-bit/refl id-bit B B %. +%{! Identity internalizes syntactic equality of canonical forms as a judgement: ``id B B'`` is inhabited only when ``B`` an ``B'`` are the same LF term. + +Thus, we can prove the theorem by exhibiting the following total relation: !}% +%{! (options removed from twelftag: check="true" export="alt") !}% +%sort bit-flip-unique %. +%mode bit-flip-unique %in %in %out %. +%term _ bit-flip-unique bit-flip/01 bit-flip/01 id-bit/refl %. +%term _ bit-flip-unique bit-flip/10 bit-flip/10 id-bit/refl %. +%worlds () (bit-flip-unique _ _ _) %. +%total {} (bit-flip-unique _ _ _) %. +%{! ### Alternate formulation + +Uniqueness can also be expressed in a less-direct but sometimes more-convenient way that is nonetheless logically equivalent to the previous theorem. +In this theorem statement, we assume derivations whose _inputs_ are related by an identity relation, and show that the outputs are identical as well. This theorem statement applies when we have evidence that the inputs are identical but the two derivations of the judgement in question have different types. The proof is no more complicated than the previous formulation, because when we derive the precondition ``id-bit B1 B1'`` using ``id-bit/refl`` (the only rule we could use to derive that fact), ``B1`` and ``B1'`` are forced to be identical objects, just as they were in the previous proof. !}% +%{! (options removed from twelftag: check="true" discard="true") !}% +%sort bit-flip-unique-alt %. +%mode bit-flip-unique-alt %in %in %in %out %. +%term _ bit-flip-unique-alt id-bit/refl bit-flip/01 bit-flip/01 id-bit/refl %. +%term _ bit-flip-unique-alt id-bit/refl bit-flip/10 bit-flip/10 id-bit/refl %. +%worlds () (bit-flip-unique-alt _ _ _ _) %. +%total {} (bit-flip-unique-alt _ _ _ _) %. +%{! !}% +%{! (options removed from twelftag: hidden="true" import="alt") !}% +%sort bit-flip-resp %. +%mode bit-flip-resp %in %in %in %out %. +%term _ bit-flip-resp id-bit/refl id-bit/refl D D %. +%worlds () (bit-flip-resp _ _ _ _) %. +%total {} (bit-flip-resp _ _ _ _) %. +%sort bit-flip-unique-alt %. +%mode bit-flip-unique-alt %in %in %in %out %. +%term _ + %pi (bit-flip-unique-alt ID1 (%the (bit-flip B1 B2) BF) (%the (bit-flip B1' B2') BF') ID2) + %<- (bit-flip-resp ID1 id-bit/refl BF (%the (bit-flip B1' B2) BF'')) + %<- (bit-flip-unique BF'' BF' (%the (id-bit B2 B2') ID2)) %. +%worlds () (bit-flip-unique-alt _ _ _ _) %. +%total {} (bit-flip-unique-alt _ _ _ _) %. +%{! However, this alternate formulation is equivalent to the above uniqueness lemma. It is easy to see that this lemma implies the above, as we can always call this lemma with ``id-bit/refl``. To prove the other direction, we can compose the above uniqueness lemma with a [respects lemma](/wiki/respects-lemma/) showing that ``bit-flip`` respects equality. `` import=alt>Here is an example`` of proving ``bit-flip-unique-alt`` using a uniqueness lemma and a respects lemma (`` import=alt check=true>here is Twelf's output``). Because the [respects lemma](/wiki/respects-lemma/) will often be necessary for other reasons, it is usually preferable to prove it and the above uniqueness lemma, rather than the composed version presented here. + +## Bigger example: determinism of evaluation + +As a slightly more-involved example of a uniqueness lemma, we can prove that the operational semantics of the simply typed λ-calculus are deterministic: + +: If and then . + +Here, equality of terms means syntactic identity modulo α-equivalence + +This proof reuses the formulation of the STLC from [Proving metatheorems with Twelf](/wiki/proving-metatheorems-representing-the-syntax-of-the-stlc/); see that article for a description of the language and its operational semantics. We recap the signature for the STLC here: !}% +%% Syntax +%sort tp %. +%term unit tp %. +%term arrow %pi tp %-> tp %-> tp %. +%sort tm %. +%term empty tm %. +%term lam %pi tp %-> (%pi tm %-> tm) %-> tm %. +%term app %pi tm %-> tm %-> tm %. +%% Dynamic Semantics +%sort value %. +%term value_empty value empty %. +%term value_lam value (lam T2 ([x] E x)) %. +%sort step %. +%term step_app_1 %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step_app_2 %pi (step (app E1 E2) (app E1 E2')) %<- (step E2 E2') %<- (value E1) %. +%term step_app_beta %pi (step (app (lam T2 ([x] E x)) E2) (E E2)) %<- (value E2) %. +%{! To state the uniqueness lemma, we define an identity type for ``tm``: !}% +%sort id %. +%term refl id E E %. +%{! The judgement ``id`` represent object-language α-equivalence: α-equivalent STLC terms are represented by α-equivalent LF terms, and ``id E E'`` is inhabited exactly when ``E`` and ``E'`` are α-equivalent in LF. + +The proof of determinism requires a congruence lemma, which states that applications are equal if their subterms are: !}% +%sort id_app_cong %. +%mode id_app_cong %in %in %out %. +%term _ id_app_cong refl refl refl %. +%worlds () (id_app_cong _ _ _) %. +%total {} (id_app_cong _ _ _) %. +%{! Next, we present a complete proof of determinism: !}% +%{! (options removed from twelftag: check="true") !}% +%sort det %. +%mode det %in %in %out %. +%term det-1 + %pi (det (step_app_1 DstepE1') (step_app_1 DstepE1'') DidApp) + %<- (det DstepE1' DstepE1'' DidE1) + %<- (id_app_cong DidE1 refl DidApp) %. +%term det-2 + %pi (det (step_app_2 _ DstepE2') (step_app_2 _ DstepE2'') DidApp) + %<- (det DstepE2' DstepE2'' DidE2) + %<- (id_app_cong refl DidE2 DidApp) %. +%term det-b det (step_app_beta _) (step_app_beta _) refl %. +%worlds () (det _ _ _) %. +%total D (det D _ _) %. +%{! Observe that the [totality assertion](/wiki/totality-assertion/) for this type family implies the informal statement above. + +The proof consists of three cases, and in each case the two ``step`` derivations conclude with the same final rule. The cases ``det-1`` and ``det-2`` appeal to the inductive hypothesis on the subderivations and then use the congruence lemma for identity. The case ``det-b`` concludes equality by reflexivity because the left-hand term of ``step-app-beta`` uniquely determines the form of the right-hand term. + +Twelf successfully proves the totality of ``det``, even though the relation elides the "off-diagonal" cases where the two ``step`` derivations do not concludes with the same final rule. This is because the off-diagonal cases are all vacuously true---and moreover, Twelf's coverage checker +rules out these contradictory cases automatically. For example, if one derivation concluded with ``step-app-1`` and the other with ``step-app-2``, then there would be subderivations concluding both ``value E1`` and ``step E1 E1'``. These two types can never be simultaneously inhabited: ``step`` is only inhabited when ``E1`` is an application, and there is no rule inhabiting ``value`` for an application. Similarly, if one derivation concluded with ``step-app-1`` and the other with ``step-app-beta``, subderivations would give a ``step`` derivation whose left-hand side is a ``lam``, which cannot exist. The other off-diagonal cases can be contradicted in a similar manner. Twelf's coverage checker rules out cases like these where the inputs to a metatheorem result in an uninhabited instance of some type family. See also: [Reasoning from false](/wiki/reasoning-from-false/) + +\{\{tutorial\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Uniqueness_lemma). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/unsafe-mode.lf b/new-tests/stelf-output/pages/unsafe-mode.lf new file mode 100644 index 0000000..72eccdd --- /dev/null +++ b/new-tests/stelf-output/pages/unsafe-mode.lf @@ -0,0 +1,20 @@ +% +%{! In order for Twelf to be useful as a theorem verifier, Twelf should not accept untrue or unproven assumptions. However, it is sometimes helpful for the Twelf user to be able to state, and use, unproven assumptions, especially during proof development. Directives like [``%trustme``](/wiki/percent-trustme/) and [``%assert``](/wiki/percent-assert/) allow the Twelf user to do just this. + +When Twelf is set to not accept any untrue or unproven assumptions, it is in **safe mode** (this is the default setting). In order to use directives like [``%trustme``](/wiki/percent-trustme/), Twelf must be set to **unsafe mode.** + +## Setting Twelf in unsafe mode + +* In [Twelf with Emacs](/wiki/twelf-with-emacs/), select the **Twelf** menu, then select the **Options** submenu, then select the **unsafe** menu option. +* If using the Twelf server directly from the command line, the command <code>set unsafe true</code> will put Twelf in unsafe mode. +* In ML Twelf, the SML command <code>Twelf.unsafe := true;</code> will put Twelf in unsafe mode. + +## See also + +* [Holes in metatheorems](/wiki/incremental-metatheorem-development/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Unsafe_mode). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-boyland.lf b/new-tests/stelf-output/pages/user-boyland.lf new file mode 100644 index 0000000..fa0f9e5 --- /dev/null +++ b/new-tests/stelf-output/pages/user-boyland.lf @@ -0,0 +1,305 @@ +% +%{! John Boyland, +Professor, University of Wisconsin--Milwaukee. + +I'm interested in using Twelf and have +written a biased and under-informed [http://www.cs.uwm.edu/~boyland/proof/using-twelf.html tutorial] of Twelf. +I also have defined some [https://github.com/boyland/twelf-library library signatures] for Twelf. + +## Desired Features + +Here are some features that I would like to see in Twelf. +I start with the simplest to implement (in my opinion). + +### What free variables? + +In Twelf, if you neglect to declare all the free variables in a theorem, you +get the error message "Free variables In theorem." It would be nice +if the error message included the names of these free variables. Update: of course Control-C Control-L will tell you all the free variables that have names. Of course, the implicit ones don't show. A %block declaration will show you the reconstructed term; it would be nice if %theorem did too. + +### An ``exists*`` keyword + +In Twelf currently, a theorem has the form + +```twelf +my-theorem : + forall* {K1} {K2} {V1:term K1} {V2: term K2} + forall {D:rule V1 V2} + exists {K3} {V3:term K3} {DP:rule2 V1 V2 V3} + true. +``` + +Sometimes the existentially bound variables are as extraneous as the ``forall*`` +bound variables. One would like to write + +```twelf +my-theorem : + forall* {K1} {K2} {V1:term K1} {V2: term K2} + forall {D:rule V1 V2} + exists* {K3} + exists {V3:term K3} {DP:rule2 V1 V2 V3} + true. +``` + +Currently my workaround is to avoid the theorem syntax and use an explicit mode declaration: + +```twelf +my-theorem : (rule V1 V2) -> ({V3} (rule2 V1 V2 V3)) -> type. +%mode +{K1} +{K2} +{V1:term K1} +{V2: term K2} +{D:rule V1 V2} + -{K3} -{V3:term K3} -{DP:rule2 V1 V2 V3} + (my-theorem D DP). +``` + +An ``exists*`` keyword should be a trivial addition. + +### Multiple ``%reduces`` for a type family + +If one declares two ``%reduces`` for a type family, the first is forgotten. +Since '%reduces` is very important for later reduction/totality checks, it would be +very helpful to collect all that are attached. + +### Output coverage checking + +It would be desirable to have output coverage checking done at the same time as mode checking, or at least to have the ability to check output coverage before a total declaration. I have often found it frustrating to get an output coverage error late in the process of writing a proof. And the "covers" check doesn't include output coverage + +### Totality checker and theorem prover integration + +Currently the theorem prover and totality checker live in different worlds, +and neither trusts the other. This seems peculiar to a user of Twelf +who may wish to use the theorem prover for certain obvious lemmas but +the totality checker for more complex theorems. Currently this is impossible +except in unsafe mode with the addition of assertions and "trustme" declarations. + +I propose + +1. The theorem prover should accept proved theorems: If a totality flag is set on a metatheorem, it should accept it as proved. + +2. The totality checker should accept metatheorems proved automatically. If necessary, the theorem prover should deliver the proof to the totality checker internally (invisibly) for re-checking. + +3. The ``%assert`` and ``%trustme`` declarations should have the same effect: in unsafe mode make the metatheorem trusted by both the totality checker and the theorem prover. + +### Local Type Inference + +After using Twelf for about a year and writing 100K lines of Twelf, the most frustrating and time consuming process is "debugging" coverage errors. Because unification is bidirectional, any mistake in the middle of a theorem will cause the pattern taht one is trying to match to not be exhaustive, even though it appears to be. It would be nice to use a system with a slightly less powerful type inference system that would prevent this long-distance effect of unification. Pierce and Turner have defined "Local Type Inference". It is interesting to wonder what this would mean for Twelf. + +## Known Bugs + +In decreasing order of imporance. + +### Type Reconstruction Nontermination +Type reconstruction sometimes fails to terminate + +```twelf +xxtag : type. + +xxtag/a : xxtag. + + +xxterm : xxtag -> type. + +xxa : xxterm xxtag/a. + + +xxoutput : xxtag -> type. + +xxoutput/base : xxterm K -> xxoutput K. + +xxoutput/exists : (xxterm K -> xxoutput K') -> xxoutput K'. + + +xxtrans : xxoutput K -> xxoutput K -> type. + +xxtrans/transitive : xxtrans E1 E2 -> xxtrans E2 E3 -> xxtrans E1 E3. + +xxtrans/inside : ({v} xxtrans (FE1 v) (FE2 v)) -> + xxtrans (xxoutput/exists FE1) (xxoutput/exists FE2). + +xxtrans/gen-exists : {K1} {K2} {F:xxterm K1 -> xxoutput K2} {T} + xxtrans (F T) (xxoutput/exists F). + + +xxtrans-open-exists : + (xxtrans (xxoutput/exists FE1) E2) -> + ({v:xxterm K} xxtrans (FE1 v) E2) -> + type. + +- : {K1} {K2} {O} + xxtrans-open-exists + (xxtrans/gen-exists K1 K2 ([v] (xxoutput/exists ([v'] FF v v'))) O) + ([v'] (xxtrans/transitive + (xxtrans/gen-exists _ _ + ([v] (FF v v')) O) + (xxtrans/inside ([v] (xxtrans/gen-exists _ _ _ v'))))). +``` + +The last term is incorrectly typed because ``FF`` depends on K1 and K2, but is not declared explicitly. +But in my version of twelf (twelf 1.5R3, with trustme, allegedly from August 2005) doesn't +appear to terminate when attempting to type it. + +:Update: Apparently this is a known bug for which a fix was found by Jason Reed. http://en.scientificcommons.org/47856338 + +:The bug is still is in Twelf 1.7.1+ (r1887M) + +### Coverage Checker Nontermination + +The coverage checker sometimes fails to terminate. + +```twelf +%%%% coverage checking doesn't always terminate + +%%%% Definitions + +void : type. + + +expr : type. + +unit : expr. +add : expr -> expr -> expr. + + +%%% Nested for evaluation + + +expr-in-expr : (expr -> expr) -> type. + + +expr-in-expr/add1 : expr-in-expr ([E] (add E _)). + +expr-in-expr/add2 : expr-in-expr ([E] (add unit E)). + + +%%% ready for atomic evaluation + + +atomic-expr : expr -> type. + + +atomic-expr/add : atomic-expr (add unit unit). + + +%% this "theorem" is actually false + +%theorem expr-in-atomic-expr-contradiction : + forall* {E} {T} + forall {AE:atomic-expr (T E)} + {Tok:expr-in-expr T} + exists {F:void} + true. + +%worlds () (expr-in-atomic-expr-contradiction _ _ _). +%total { } (expr-in-atomic-expr-contradiction _ _ _). +``` + +### Broken Invariant in Coverage Checker + +Apparently (I learned of this problem from Rob Simmons), the coverage checker +sometimes reports a broken invariant when checking code that +uses the explicit-context idiom. This problem is not well understood, but manifests itself in several examples from the wiki, including [Concrete representation](/wiki/concrete-representation/) and [Tethered modal logic](/wiki/tethered-modal-logic/). + +### Coverage checker checking with functions of type A -> A when A is not subordinate to itself. + +If the coverage check, when expanding cases comes upon a term that might depend on a formal +parameter of the same type, it will require a case that handles such a function, even if the +type in question in not subordinate to itself. Such a case normally cannot be written because of freezing. +(Example to come later). + +### Higher-Order Blocks + +Again, I learned of this problem from Rob Simmons: +If a "block" includes a higher order function, then the coverage checker doesn't work. +No example yet. This bug is considered low-priority. + +### Not a block + +If you use the name of something that is not a block +where a block is expected: + + %worlds (nat) (foo _ _ _). + +then there is an "uncaught exception" rather than a helpful error message. + +### Theorem Prover Nontermination + +The prover sometimes fails to terminate. +See example in +[http://cs.nju.edu.cn/boyland/proof/using-twelf.html my tutorial] (Section 2.3) +I have copied it here for completeness: + +```twelf +%%% Example of Twelf from "using-twelf" paper + +term : type. + +true : term. +false : term. +if : term -> term -> term -> term. +zero : term. + +ty : type. + +bool : ty. +int : ty. + + +%% relations +is_value : term -> type. + +is_value/true : is_value true. +is_value/false : is_value false. +is_value/zero : is_value zero. + +eval : term -> term -> type. + +eval/if_true : eval (if true X _) X. +eval/if_false : eval (if false _ X) X. +eval/if : eval (if E E1 E2) (if E' E1 E2) + <- eval E E'. + +not_stuck : term -> type. + +not_stuck/value : not_stuck X <- is_value X. +not_stuck/eval : not_stuck X <- eval X X'. + +of : term -> ty -> type. + +of/true : of true bool. +of/false : of false bool. +of/zero : of zero int. +of/if : of X bool -> of Y T -> of Z T -> of (if X Y Z) T. + +%% theorems + +progress : (of X T) -> (not_stuck X) -> type. +%mode progress +D1 -D2. +%prove 2 T (progress T _). +``` + +The theorem prover is deprecated in Twelf, and fixing this bug is not required before producing +a new release of Twelf. + +## Challenges + +### Is HOAS countable? + +A challenge: define a mapping from HOAS to the natural numbers (or equivalently to a nameless term representation) and then prove that the mapping is isomorphic _using Twelf_. For details and sample mappings, see question on the Twelf Elf page: [Talk:Ask_Twelf_Elf#Mapping_HOAS_isomorphically_to_the_natural_numbers](/wiki/talk-ask-twelf-elf/). + +Update: [Rob](/wiki/user-rsimmons/) has posted a solution ([Concrete_representation](/wiki/concrete-representation/)) that defines a mapping between HOAS and nameless terms and proves that it is bijective. +(I asked for an "isomorphic" mapping between HOAS and natural numbers, but should have said "bijective mapping.") Thanks Rob! + +My personal challenge is to write a general technique for generating bijective mappings (with proof) between many kinds of terms and natural numbers. +In general, of course, this would be undecidable. I hope only to handle cases that are used as AST types. + +Update: I did it for a simple HOAS before POPL 2008 +(see [HOAS nat bijection](/wiki/hoas-nat-bijection/)), and in May 2008, finished a more complex +example [Indexed HOAS nat bijection](/wiki/indexed-hoas-nat-bijection/) that handle an indexed term language. + +Success: I have implemented a bijection generator. See http://www.cs.uwm.edu/~boyland/papers/map-natural.html + +:: Cool! — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 02:08, 12 December 2010 (EST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Boyland). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-carsten.lf b/new-tests/stelf-output/pages/user-carsten.lf new file mode 100644 index 0000000..ad3ce56 --- /dev/null +++ b/new-tests/stelf-output/pages/user-carsten.lf @@ -0,0 +1,8 @@ +% +%{! [CADE_Tutorial](/wiki/cade-tutorial/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Carsten). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-ccasin.lf b/new-tests/stelf-output/pages/user-ccasin.lf new file mode 100644 index 0000000..ad6b919 --- /dev/null +++ b/new-tests/stelf-output/pages/user-ccasin.lf @@ -0,0 +1,8 @@ +% +%{! I'm Chris Casinghino, a grad student at U Penn, formerly of CMU. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Ccasin). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-cmartens.lf b/new-tests/stelf-output/pages/user-cmartens.lf new file mode 100644 index 0000000..b79d5de --- /dev/null +++ b/new-tests/stelf-output/pages/user-cmartens.lf @@ -0,0 +1,12 @@ +% +%{! Tutorial Sunday, Oct 17, 2010. + +[TAT/plus.elf](/wiki/tat-plus-elf/) + +[TAT/church.elf](/wiki/tat-church-elf/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Cmartens). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-crary.lf b/new-tests/stelf-output/pages/user-crary.lf new file mode 100644 index 0000000..122083a --- /dev/null +++ b/new-tests/stelf-output/pages/user-crary.lf @@ -0,0 +1,8 @@ +% +%{! **Karl Crary**'s [http://www.cs.cmu.edu/~crary/ home page] is at CMU. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Crary). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-danielklee.lf b/new-tests/stelf-output/pages/user-danielklee.lf new file mode 100644 index 0000000..04b4254 --- /dev/null +++ b/new-tests/stelf-output/pages/user-danielklee.lf @@ -0,0 +1,8 @@ +% +%{! 2nd (as of Fall 2006) year PhD student at CMU. More info at [http://www.cs.cmu.edu/~dklee my homepage]. I'm lucky enough that my research heavily involves formalization and metatheorem proving in Twelf. Among other things, I try to contribute interesting/advanced examples to the Twelf wiki. With the help of the other wiki elves, these examples become informative _and_ human consumable. I've seen and fought a number of strange and challenging Twelf metatheorem proving related beasts in my travels, so if you've got questions or requests make a note in [User talk:DanielKLee](/wiki/user-talk-danielklee/) or [The Twelf Project: To do](/wiki/the-twelf-project-to-do/). !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:DanielKLee). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-defined-constraint-domain.lf b/new-tests/stelf-output/pages/user-defined-constraint-domain.lf new file mode 100644 index 0000000..7ca44cb --- /dev/null +++ b/new-tests/stelf-output/pages/user-defined-constraint-domain.lf @@ -0,0 +1,481 @@ +% +% +%{! This tutorial describes a technique for defining a limited form of **constraint domain** in Twelf, that is, a type with non-trivial equations among its terms. Unlike Twelf's built-in (but [currently problematic](/wiki/constraint-domains-and-coverage-checking/)) constraint domains of rationals, integers, and strings, these constraint domains are already part of the LF type theory and thus interact nicely with each other and with the rest of the theory. And, as we shall see, they can be used for many of the same purposes, not to mention a whole host of new ones, including a limited form of type refinement. + +Although this tutorial has (user-defined) constraint domains as its main theme, it touches on a lot of other topics and techniques, and so has some value as a general tutorial on Twelf. + +## Motivation + +We begin with some motivation (and review), which you can skip if you are interested in getting right to the examples that follow. + +LF is a representational framework: given a particular formal system, we construct LF signatures that determine sets of terms (and types and kinds) that can stand as representatives of the constituents of the formal system. We do this in such a way that the syntactic and proof-theoretic structures of the formal system, for example the binding structure of terms or the inductive construction of proofs, are mirrored in the type-theoretic structure of LF. The measure of success for such a representation is what we call [adequacy](/wiki/adequacy/). + +Equality of terms (and types and kinds) clearly plays a key role here, since adequacy depends on what terms are considered equal. Twelf's built-in notion of equality, which is much coarser than plain syntactic identity, is the combination of [beta](/wiki/beta-equivalence/)- and [eta-equivalence](/wiki/eta-equivalence/). The former, along with the notion of [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/), is what allows us to represent object-level substitution using meta-level function application. And the latter implies a kind of extensionality or behavioral equivalence that gets rid of some annoying duplication in our representations that would otherwise make adequacy much more difficult to achieve. Beta-eta equivalence has the very nice property that its equivalence classes have unique [canonical form](/wiki/canonical-form/)s, which are inductively defined; it is these canonical forms that are used to represent object-level entities. Equality also plays a fundamental role in the [operational semantics](/wiki/logic-programming/) of Twelf, since one of its main components, unification, is all about finding substitutions that make a set of terms equal up to beta-eta equivalence, which is a crucial but quite non-trivial endeavor. + +However, despite the importance of equality, it remains an _internal_ notion: there are no built-in "equality types" that we can use to reason about equality explicitly, for example to prove uniqueness of constructed objects or functionality of relations. Instead, we must [define](/wiki/equality/) equality for each type where it is needed, and prove [respects](/wiki/respects-lemma/) and [uniqueness](/wiki/uniqueness-lemma/) lemmas in order to make use of it. This approach has the advantage, though, that the same or closely-related techniques will work when the equality we define is _not_ the built-in equality but one that carries more information. + +Another crucial aspect of LF from a representational standpoint is that its terms are generated _freely_ from its signature. That is, no terms are present that are not built from the constructors in the signature and the variables in the context, and no identifications between terms are made, except those that follow from beta-eta equivalence. Without this, it would be difficult to represent aspects of our formal systems that are defined (freely) by induction. On the other hand, this freeness can also be an impediment, as when what we really want is a kind of _algebra_: a set endowed with certain operations that satisfy (non-trivial) equations. Think, for example, of the type ``bool``, which is freely generated by its two constants, ``true`` and ``false``, but which has operations ``and``, ``or``, ``not``, ``implies``, etc., that satisfy the equations of Boolean Algebra. + +We are therefore led to wonder whether we can represent such "algebras" as LF types. + +## Example 1: Even and Odd + +We borrow our first example from the tutorial on [Output factoring](/wiki/output-factoring/): + +```twelf +nat : type. +z : nat. +s : nat -> nat. + +odd : nat -> type. +even : nat -> type. + +z-e : even z. +s-o : odd (s X) <- even X. +s-e : even (s X) <- odd X. +``` + +Here, ``even`` and ``odd`` are _predicates_ on ``nat`` and are defined in terms of their relation to ``s``. The [output factoring tutorial](/wiki/output-factoring/) shows you how to prove that, under this definition, every natural number is either even or odd; the statement of the theorem requires a disjoint-union type, and the proof requires a lemma to take care of an output coverage problem. + +### An approach using indexing + +Since the subsets of even and odd natural numbers are disjoint, you might think of another approach to representing these subsets, namely, by introducing a new type of _natural number sorts_, and indexing ``nat`` by the sort: + +```twelf +nat_sort : type. +even : nat_sort. +odd : nat_sort. + +nat : nat_sort -> type. +z : nat even. +s : nat even -> nat odd. % Doesn't work: +s : nat odd -> nat even. % redefinition of s +``` + +We'd like for ``s`` to have both of the declared types at the same time, but that is not possible in LF (though some proposed extensions of LF to include _refinement types_ do allow something like this). Alternatively, we might try to write something like: + +```twelf +s : nat X -> nat (f X). +``` + +for some term ``f`` of type ``nat_sort -> nat_sort`` with the property that ``f even`` = ``odd`` and ``f odd`` = ``even``, but again this is not possible in LF, since terms are freely generated and functional terms are parametric in their arguments (i.e., there are no destructors or case-splitting on arguments). This approach seems to be stuck. Or is it? + +### A constraint domain of natural-number sorts + +The idea is to use a higher-order "Church-encoding" for our two-element type and to use abbreviations to "hide" the implementation of this type: !}% +%sort t %. +%inline nat_sort (%type) %pi t %-> t %-> t %. +%inline even (nat_sort) [even t] [odd t] even %. +%inline odd (nat_sort) [even t] [odd t] odd %. +%inline add1-mod2 (%pi nat_sort %-> nat_sort) [s nat_sort] [even t] [odd t] s odd even %. +%sort nat %. +%term z nat even %. +%term s %pi (nat X) %-> (nat (add1-mod2 X)) %. +%{! Why does this work? As long as the type ``t`` does not become [subordinate](/wiki/subordination/) to any other type, the only canonical terms of type ``nat_sort`` are ``even`` and ``odd``, and so the type looks "from the outside" just like the type ``nat_sort`` from the previous section. However, ``add1-mod2``, rather than being a constructor on this type that freely generates new terms, is instead a "pre-existing" canonical term of type ``nat_sort -> nat_sort``. In fact, there are _four_ canonical terms of this type, the other three being !}% +%inline const_even (%pi nat_sort %-> nat_sort) [s nat_sort] [even t] [odd t] s even even %. +% or: [s] even +%inline const_odd (%pi nat_sort %-> nat_sort) [s nat_sort] [even t] [odd t] s odd odd %. +% or: [s] odd +%inline identity (%pi nat_sort %-> nat_sort) [s nat_sort] [even t] [odd t] s even odd %. +% or: [s] s +%{! Similarly, there are canonical terms of type ``nat_sort -> nat_sort -> nat_sort``, and canonical terms of type ``(nat_sort -> nat_sort) -> nat_sort``, and so on. In fact, what we have here are all lambda-definable functions in the full type-hierarchy over \{``even``, ``odd``\}, which, since this set is finite, is the same as the full (set-theoretic) type-hierarchy over \{``even``, ``odd``\}. And these functions are just there waiting to be used! + +To see one such use, let's define the predicate ``plus X Y Z``. What should be its type? We'll give two definitions, the first with minimal type information: !}% +%sort plus_min %. +%mode plus_min %in %in %out %. +%term plus_min_z plus_min N1 z N1 %. +%term plus_min_s %pi (plus_min N1 (s N2) (s N3)) %<- (plus_min N1 N2 N3) %. +%worlds () (plus_min N1 N2 N3) %. +%total N2 (plus_min N1 N2 N3) %. +%{! This works fine, but we can do better. We know, for example, that if ``X`` and ``Y`` are the same, then ``Z`` will be ``even``, and if ``X`` and ``Y`` are different, then ``Z`` will be ``odd``. In other words, tracking the sorts of the elements involved in an addition requires being able to compute addition mod 2. But this is just a (finite) function in the type-hierarchy over \{``even``, ``odd``\}, and so to get a maximally precise type for ``plus``, we can just define the term of type ``nat_sort -> nat_sort -> nat_sort`` that represents addition mod 2, and then use that in the type: !}% +%{!! begin checked !!}% +%inline plus-mod2 (%pi nat_sort %-> nat_sort %-> nat_sort) [s1 nat_sort] [s2 nat_sort] [even t] [odd t] s1 (s2 even odd) (s2 odd even) %. +%sort plus %. +%mode plus %in %in %out %. +%term plus_z plus N1 z N1 %. +%term plus_s %pi (plus N1 (s N2) (s N3)) %<- (plus N1 N2 N3) %. +%worlds () (plus N1 N2 N3) %. +%total N2 (plus N1 N2 N3) %. +%{!! end checked !!}% +%{! You'll see in this output that all of our abbreviations have been expanded, which makes the output a bit harder to read. It would have been better to have used definitions rather than abbreviations, but (at present anyway) this is, for technical reasons, incompatible with Twelf's meta-checks. + +Also notice that, apart from the change in name from ``plus_min`` to ``plus``, the declarations of ``plus_z`` and ``plus_s`` are the same; it's just the type that has become more precise. As an exercise, you may like to check that if ``f`` is any of the other 15 possible canonical forms of type ``nat_sort -> nat_sort``, and you replace the declaration of ``plus`` above with + +```twelf +plus : nat X -> nat Y -> nat (f X Y) -> type. +``` + +then at least one of the two declarations for ``plus_z`` or ``plus_s`` will no longer type-check without generating unresolved constraints. Finally, we observe that, as compared to the definitions of even and odd in the tutorial on [output factoring](/wiki/output-factoring/), it is intrinsic in our encoding via the constraint domain ``nat_sort`` that every natural number is either even or odd, and so this is not something that needs to be proved. + +## Example 2: Polarities + +For our next example, we are going to use the same constraint domain that we used in Example 1, and for a similar purpose (although we will be renaming it to suit the current situation). Suppose that we have a language with a type system that includes a function-space operator, ``=>``, along possibly with some others, such as products, sums, and type constants, and we have represented these as canonical forms of an LF type ``tp``. We want to add an operator ``mu`` to represent _positive inductive types_. What this means is that if we have a type expression ``F : tp -> tp`` such that for any ``x : tp``, the type ``F x`` only has _positive_ occurrences of ``x`` (that is, every occurrence of ``x`` is on the left-hand side of an even number of ``=>`` operators), then we want to be able to form ``mu F : tp``. Furthermore, these are the _only_ ``mu``-types that we want to form; i.e., we want this to be an [intrinsic encoding](/wiki/intrinsic-and-extrinsic-encodings/). All of these types will have introduction and elimination forms, but for the purposes of this example, we will just focus on the types themselves. + +### An approach via predicates + +A natural approach to this problem is to try to represent the positivity condition as a predicate ``pos`` on ``tp -> tp`` and to define this predicate mutually with another predicate ``neg`` on ``tp -> tp``, as follows: !}% +%sort tp %. +%sort pos %. +%sort neg %. +%term un tp %. +% unit type +%term * %pi tp %-> tp %-> tp %. +%prec %right 11 * %. +% product type +%term => %pi tp %-> tp %-> tp %. +%prec %right 10 => %. +% function type +%term mu {F %pi tp %-> tp} %pi (pos F) %-> tp %. +% inductive type +%term pos_id pos ([a] a) %. +%term pos_un pos ([a] un) %. +%term pos_* %pi (pos ([a] F1 a)) %-> (pos ([a] F2 a)) %-> (pos ([a] F1 a * F2 a)) %. +%term pos_=> %pi (neg ([a] F1 a)) %-> (pos ([a] F2 a)) %-> (pos ([a] F1 a => F2 a)) %. +%term pos_mu + %pi ({b tp} %pi (pos ([a] b)) %-> (pos ([a] F a b))) + %-> (pos ([a] mu ([b] F a b) _)) %. +%term neg_un neg ([a] un) %. +%term neg_* %pi (neg ([a] F1 a)) %-> (neg ([a] F2 a)) %-> (neg ([a] F1 a * F2 a)) %. +%term neg_=> %pi (pos ([a] F1 a)) %-> (neg ([a] F2 a)) %-> (neg ([a] F1 a => F2 a)) %. +%term neg_mu + %pi ({b tp} %pi (neg ([a] b)) %-> (neg ([a] F a b))) + %-> (neg ([a] mu ([b] F a b) _)) %. +%{! Some comments about this code are in order: + +- We have chosen to the make the type constructor ``F`` an [explicit parameter](/wiki/implicit-and-explicit-parameters/) of ``mu`` in order to make ``mu``-terms appear more natural, with the positivity proof relegated to second position. However, we could just as easily have left this parameter implicit, declaring ``mu : pos F -> tp``, since we would still have access to ``F`` through unification with terms involving type ascriptions, as in ``mu (P : pos F)``. + +- We define ``pos F`` and ``neg F`` by induction on the body of ``F``, taking advantage of the inductive characterization of LF canonical forms of type ``tp -> tp`` as the canonical forms of type ``tp`` in a context that includes an extra assumption ``a : tp``. The cases for ``mu`` say that a type ``a`` occurs positively (or negatively) in a ``mu``-type if it occurs positively (or negatively) in its body, and are necessary if we want to allow nested ``mu``-types (but see below!). The proof components of the ``mu``-type are not used, hence the underscores ``_`` in these declarations. + +- We have chosen to define ``pos`` and ``neg`` using types involving the ``->`` arrow rather than the ``<-`` arrow that is typical in logic-programming-style definitions, e.g., + + ``` + pos_* : pos ([a] F1 a * F2 a) + <- pos ([a] F1 a) + <- pos ([a] F2 a). + ``` + +- There are two reasons. First, these declarations are not the same: using the earlier declaration, a term of the form ``pos_* P1 P2 : pos ([a] F1 a * F2 a)`` has ``P1`` as the positivity proof for ``F1`` and ``P2`` as the positivity proof for ``F2``, whereas in the declaration just given, it is reversed! Second, we prefer to use ``->`` when we are representing data (i.e., making object-level definitions) and ``<-`` when we are representing programs (i.e., proving metatheorems), but this is a matter of style. + +All of this appears to work, and we were able, in passing, to make some useful comments about the code, but unfortunately there is a problem here, and the problem has to do with nested ``mu``-types. As a simple example, take the μ-type μ``X``. μ``Y``. ``X * Y`` (which could be a considered the type of infinite streams of infinite streams of infinite streams of ...). In order for the outer ``mu`` in any representation of this type to be well-formed, it's body, the inner ``mu``, must have only positive occurrences of ``X`` in its body (which it seems to have) and similarly for the positivity of ``Y`` in the body of the inner ``mu``, which is ``X * Y``. However, you can see in the clause for ``pos_*`` above that, for ``Y`` to be positive in ``X * Y``, not only does it have to be positive in ``Y``, which it is by ``pos_id``, but it also has to be positive in ``X``, which we have not arranged to be the case. + +Now, since every occurrence of a variable is both positive and negative in an expression in which it does not occur—vacuously—we could add two [catch-all case](/wiki/catch-all-case/)s to our declarations to fix this problem: !}% +%term pos_vac pos ([a] F) %. +%term neg_vac neg ([a] F) %. +%{! We could then represent our μ-type μ``X``. μ``Y``. ``X * Y`` with the term !}% +%define streams tp mu ([x] mu ([y] x * y) (pos_* pos_vac pos_id)) (pos_mu ([y] [py] pos_* pos_id py)) %. +%{! However, as mentioned in the [tutorial on catch-all cases](/wiki/catch-all-case/), this creates _non-determinism_, since it is now possible to have multiple proofs of positivity for the same type constructor. For example, we have: !}% +%define proof1 (pos ([a] un * un)) pos_* pos_un pos_un %. +%define proof2 (pos ([a] un * un)) pos_* pos_vac pos_un %. +%define proof3 (pos ([a] un * un)) pos_* pos_un pos_vac %. +%define proof4 (pos ([a] un * un)) pos_* pos_vac pos_vac %. +%define proof5 (pos ([a] un * un)) pos_vac %. +%{! and so, filling in the hole labeled ``%{proof}%`` in the term + +``` +mu ([x] un * un) ( %{proof}% ) +``` + +with each of the five proofs above, we would get five different canonical terms of type ``tp`` representing the same μ-type, μ``X``. ``1 * 1`` ! This destroys the [adequacy](/wiki/adequacy/) of our representation, and is why it's best to avoid catch-all cases in object representations. On the other hand, the same can't be said, in general, of [metatheorem](/wiki/metatheorem/) representation, since there we are usually more concerned about the _existence_ of proofs rather than their unique representation (whatever that means: if you really figure out what it means for two proofs to be "equal", let me know!) Finally, just as we mentioned in connection with Example 1 that some proposed extensions to LF to include _refinement types_ could give a nice solution to the representation of even and odd natural numbers (among many others such "subsorts"), we should also mention here that some proposed extensions to LF to include _proof-irrelevance_ could address the problem we have described above, by identifying all elements, or "proofs", of certain types, restoring adequacy of such representations. + +### Is there an adequate intrinsic encoding? + +So how can we achieve an adequate intrinsic encoding of μ-types? The situation is worse than you might think: there doesn't seem to be any hope of getting such an encoding. We've seen that adding a catch-all case destroys adequacy, but yet we've also seen that, in representing the μ-type μ``X``. μ``Y``. ``X * Y``, we need a positivity assumption about ``X`` in order to show that ``Y`` occurs positively in ``X * Y``. So our only hope seems to be to add it to the argument of ``mu``, thus: + +```twelf +mu : {F:{a:tp} pos ([b] a) -> tp} pos F -> tp. +``` + +But this is not well-typed unless we also either + +1. change the type of ``pos`` so that it accepts arguments of the same type as ``F``, or +2. extract the underlying constructor of type ``tp -> tp`` from ``F`` and use that instead as the argument to ``pos``. + +But neither of these is possible: we can't do 1 because ``pos`` would then have to be declared as + +```twelf +pos : ({a:tp} pos ([b] a) -> tp) -> type +``` + +which is circular, and we can't do 2, because to get a ``tp`` output from ``F``, we'd need first to construct something of type ``pos ([b] a)``, which is exactly the problem we were trying to solve in the first place. So, it seems that our search for an adequate intrinsic encoding of μ-types is doomed. Or is it? + +### Using the even/odd constraint domain + +You may remember our original characterization of what it means for a type expression ``F : tp -> tp`` to have only positive occurrences of its bound variable, namely, that every occurrence of the variable is on the left-hand side of an _even_ number of ``=>`` operators. This suggests that, in the same way that we used the ``nat_sort`` constraint domain in Example 1 as an index to ``nat`` in order to capture the even (and odd) natural numbers, we can also use it here as an index to ``tp`` to capture occurrences on the left of an even (and odd) number of ``=>``s. When we follow this idea to its logical conclusion, this is what we get: !}% +%sort t %. +% polarities +%inline pol (%type) %pi t %-> t %-> t %. +% the opposite polarity +%inline opp (%pi pol %-> pol) [p pol] [pos t] [neg t] p neg pos %. +%sort tp %. +%term un tp P %. +% unit type +%term * %pi (tp P) %-> (tp P) %-> (tp P) %. +%prec %right 11 * %. +% product type +%term => %pi (tp (opp P)) %-> (tp P) %-> (tp P) %. +%prec %right 10 => %. +% function type +%term mu %pi (%pi (tp P) %-> (tp P)) %-> (tp P) %. +% inductive type +%{! That's it! You may find it hard to believe that this works, so let's look at it more closely. The constraint domain is the same one we used before, with ``nat_sort`` renamed to ``pol`` and ``add1-mod2`` renamed to ``opp``. In all of the type constructors the polarity index is implicitly quantified, so each will apply to both polarities, but the individual polarities themselves have no meaning—there isn't a "positive" polarity or a "negative" polarity, for example—so we haven't given them names. The main force of the polarities is to be found in the rules for ``=>`` and ``mu``. In the case of ``=>``, the rule says that, whatever polarity is to be assigned to the type ``A => B``, the same polarity must be assigned to ``B`` and the opposite polarity must be assigned to ``A``. And in the case of ``mu``, the fact that the functional argument to ``mu`` has domain and codomain types with the _same_ polarity means that, if we consider this functional argument to be an abstract syntax tree with instances of the bound variable at the leaves, then an even number of applications of ``opp`` must occur between the root of this tree and any occurrence of the bound variable at the leaves. + +In short, you can think of the rules as giving conditions for a particular simultaneous polarity assignment to all of the subterms of a given term to be _coherent_, and you'll notice that if one polarity assignment is coherent, then so is its opposite, reinforcing the idea that the individual polarities are meaningless. (We might say, following Einstein, that positivity and negativity are not _absolute_ notions, but are only meaningful relative to other instances of one another!) Thus, since each of the type constructors relates the polarity of its "output" to the polarities of its "inputs" (with, in the case of ``mu``, an assumption about the polarity of the bound type variable), a given term representing one of our types will have, overall, either zero or exactly two polarity assignments for its subterms that can be coherent, with only the latter case corresponding to the type being well-formed. It follows that the product type ``\{P:pol\} tp P`` adequately represents our informal notion of positive inductive types. + +Here are some sample encodings of μ-types, starting with the two examples we discussed before: !}% +%inline pos_ind (%type) {P pol} tp P %. +%define streams pos_ind [P] mu ([x] mu ([y] x * y)) %. +%define ones pos_ind [P] mu ([x] un * un) %. +%define ex1 pos_ind [P] mu ([x] un => x) %. +%define ex2 pos_ind [P] mu ([x] x => un => x) %. +%define ex3 pos_ind [P] mu ([x] x * x => un => x * x) %. +%define ex4 pos_ind [P] mu ([x] mu ([y] x => y) => x) %. +%{! You might also like to check that each of these definitions fails to type-check: + +```checkedtwelf +ex5 : pos_ind = [P] mu [x] x => un. +``` + +```checkedtwelf +ex6 : pos_ind = [P] mu [x] x => x. +``` + +```checkedtwelf +ex7 : pos_ind = [P] mu [x] x * (mu [y] x => y). +``` + +Curiously, if you remove the type ascriptions and polarity abstractions from these three ill-formed ``mu``-types, then they _do_ type-check, but only because Twelf, during type-reconstruction, abstracts a variable ``X`` of type ``t`` from these definitions and then assigns the "constant polarity" ``[t1][t2] X`` (which is its own opposite!) to the contradictory subterms. This illustrates again the reliance of this method on the fact that the basic type ``t`` cannot have any canonical elements. + +## Example 3: Uniform syntactic categories + +Looking over the examples from the previous two sections, you may have noticed a pattern in the method we used to achieve simplification in each case, a method that we might call _intrinsification by indexing_. That is, we were given a type ``A`` with some unary predicates ``P`` on that type and we "intrinsified" those predicates by turning ``A`` into a type family ``A'`` that was indexed by a particular constraint domain whose elements represented those predicates; for any predicate ``P``, the elements of ``A' P`` were exactly the original elements of ``A`` that satisfied ``P``. This opened up the possibility of giving more precise types to functions defined on ``A`` that respected the predicates, since we were are able to "track" their behavior on the indices by using definable functions in the type-hierarchy over the constraint domain. + +Thus, in Example 1, the type ``A`` was ``nat`` and the predicates were ``even`` and ``odd``. These predicates became elements of a constraint domain ``nat_sort`` that we used to index ``A``, and the functions ``s`` and (via its graph) ``plus`` could be tracked by functions ``add1-mod2`` and ``plus-mod2`` on ``nat_sort``. And in Example 2, the type ``A`` was ``tp -> tp`` and the predicates were ``pos`` and ``neg``. These predicates became elements of a constraint domain ``pol`` and the various type constructors were tracked with the help of (the identity function and) the function ``opp`` on ``pol``. + +We will continue to explore this theme by using a couple of examples from the typed lambda calculus and LF itself. But first, let's look a little more carefully at the conditions under which we can expect the methods of the previous two sections to work. Clearly, if we want there to be a bijection between ``A`` and the collection of types ``A' P`` indexed by the predicates ``P``, then the extensions of these predicates must form a partition of ``A`` (i.e., they must be pairwise disjoint and collectively exhaustive). This may seem like a severe restriction, but there are some ways around it, as we will see shortly. But another restriction of the method is that, if we want to give more informative types for functions defined on ``A``, these must must not only respect the predicates, but the induced functions on the indices must also be _uniform_ in the sense that they can be defined parametrically on the index. These restrictions will become clearer as we look at more examples. + +### Simply-typed lambda calculus with values + +Our next example won't actually require a constraint domain, but it nevertheless illustrates the intrinsification-by-indexing theme we are developing, and it shows how you can sometimes get around some of the limitations of the method that were discussed at the end of the previous section. Suppose we want to represent a simply-typed call-by-value lambda-calculus with a (big-step) evaluation relation that relates arbitrary terms with their values under evaluation, and we decide to represent typed terms intrinsically and values extrinsically: !}% +%sort tp %. +%term => %pi tp %-> tp %-> tp %. +%prec %right 10 => %. +%sort exp %. +%term lam %pi (%pi (exp A) %-> (exp B)) %-> (exp (A => B)) %. +%term app %pi (exp (A => B)) %-> (exp A) %-> (exp B) %. +%sort val %. +%term lam-val val (lam ([x] E x)) %. +%sort eval %. +%term eval-lam eval (lam ([x] E x)) (lam ([x] E x)) %. +%term eval-app + %pi (eval (app E1 E2) V) + %<- (eval E1 (lam ([x] E1' x))) + %<- (eval E2 V2) + %<- (eval (E1' V2) V) %. +%{! That is, the encoding of typed terms here is intrinsic in that we are only representing well-typed terms (along with their types), and not separate types and (not necessarily well-typed) terms that are related by a typing judgment. And our encoding of values is extrinisic in that we are representing values as a subset of terms through a separate judgment, instead of similarly building the notion into the terms. (See the tutorial on [Intrinsic and extrinsic encodings](/wiki/intrinsic-and-extrinsic-encodings/), which uses some similar examples.) + +Now, we'd like to "intrinsify" the representation of values in this encoding, but there is an obstacle here that wasn't present in our previous examples: we have two predicates here corresponding to values and general terms, but they aren't disjoint: every values is a general term. The solution is to introduce a separate predicate for _non-values_, which, along with values, partition the set of terms. Fortunately, this is possible here, since non-values have a simple inductive definition: they are precisely the applications. Here is the final result: !}% +%sort tp %. +%term => %pi tp %-> tp %-> tp %. +%prec %right 10 => %. +%sort stlc_sort %. +%term val stlc_sort %. +%term non-val stlc_sort %. +%sort exp %. +%term lam %pi (%pi (exp val A) %-> (exp S B)) %-> (exp val (A => B)) %. +%term app %pi (exp S1 (A => B)) %-> (exp S2 A) %-> (exp non-val B) %. +%sort eval %. +%term ev-lam eval (lam ([x] E x)) (lam ([x] E x)) %. +%term ev-app + %pi (eval (app E1 E2) V) + %<- (eval E1 (lam ([x] E1' x))) + %<- (eval E2 V2) + %<- (eval (E1' V2) V) %. +%{! Now we can see why a constraint domain wasn't required in this case: there are no special functions needed to track ``lam`` and ``app`` on the indices. Yet we still get an intrinsic encoding with more informative types for ``lam``, ``app``, and ``eval``. And whereas with the extrinisic encoding we would have to prove a metatheorem that if ``eval E V`` then ``val V``, with the intrinsic encoding this comes for free. + +### The syntax of Canonical LF + +In modern presentations of the LF type theory (see the glossary entry for [canonical form](/wiki/canonical-form/)s for more information), the set of syntactic objects is divided into five categories: kinds, canonical types, atomic type families, canonical terms, and atomic terms, often denoted by the letters K, A, P, M, and R. This stratification of syntax necessitates a similar stratification of the typing judgment (or, as we will call it, the classification judgment), the [hereditary substitution](/wiki/hereditary-substitution/) judgment, and the statements of theorems about these judgments, for example the substitution theorem. + +In this section, we will show how it is possible to define a constraint domain ``cat`` of _(syntactic) categories_ to allow an economical presentation of the syntax, judgments, and metatheorems of LF. In particular, we will have one type family of terms, one classification judgment, one substitution judgment, and one substitution theorem statement, each of which is indexed by a syntactic category. In order to achieve this unification, we will have to make use of some functions in the type hierarchy over ``cat``, much the way we did with ``add1-mod2``, ``plus-mod2``, and ``opp`` above. To simplify the presentation for this tutorial, we will not be representing signatures, (type family or term) constants, subordination, the simple-type subscripts to hereditary substitution, or the bi-directionality of the typing judgments and substitution theorem. + +#### The constrain domain of syntactic categories + +We first define the constraint domain, ``cat``. There are two subtleties here. First, because our unified classification judgment relates a syntactic object to its classifier, it will be necessary for the purpose of uniformity to introduce a category to classify kinds; we call it "unit" (letter: U) and give it a unique element ``*`` (this element might be called a "hyperkind"). Second, the hereditary substitution operation, when applied to an atomic term _R_, can produce either a canonical term or an atomic term, depending on whether the substitution variable occurs at the head of _R_ or not. Consequently, we introduce a syntactic category to represent the disjoint union of canonical and atomic terms (letter: M+R), and we introduce two injections from M and R into M+R as term constructors. + +Here is the definition of the constraint domain: !}% +%sort t %. +% seven element type +%inline cat (%type) %pi t %-> t %-> t %-> t %-> t %-> t %-> t %-> t %. +%inline k (cat) [k] [a] [p] [m] [r] [m+r] [u] k %. +%inline a (cat) [k] [a] [p] [m] [r] [m+r] [u] a %. +%inline p (cat) [k] [a] [p] [m] [r] [m+r] [u] p %. +%inline m (cat) [k] [a] [p] [m] [r] [m+r] [u] m %. +%inline r (cat) [k] [a] [p] [m] [r] [m+r] [u] r %. +%inline m+r (cat) [k] [a] [p] [m] [r] [m+r] [u] m+r %. +%inline u (cat) [k] [a] [p] [m] [r] [m+r] [u] u %. +%{! #### Terms and term constructors + +We can now introduce the type ``tm`` of terms, which is indexed by ``cat``, as well as the various term constructors: !}% +%sort tm %. +%term typ tm k %. +% type (the base kind) +%term pi_k %pi (tm a) %-> (%pi (tm r) %-> (tm k)) %-> (tm k) %. +% kind-level product +%term coer_pa %pi (tm p) %-> (tm a) %. +% type coercion +%term pi_a %pi (tm a) %-> (%pi (tm r) %-> (tm a)) %-> (tm a) %. +% type-level product +%term app_p %pi (tm p) %-> (tm m) %-> (tm p) %. +% atomic family application +%term coer_rm %pi (tm r) %-> (tm m) %. +% term coercion +%term lam_m %pi (%pi (tm r) %-> (tm m)) %-> (tm m) %. +% term-level lambda abstraction +%term app_r %pi (tm r) %-> (tm m) %-> (tm r) %. +% atomic term application +%term canon %pi (tm m) %-> (tm m+r) %. +% left injection +%term atom %pi (tm r) %-> (tm m+r) %. +% right injection +%term * tm u %. +% unit +%{! Here, we have used the standard [higher-order abstract syntax](/wiki/higher-order-abstract-syntax/) representations of binders, which in this case always abstract over atomic term variables. We have also included explicit coercions from atomic to canonical types and terms, since we really have disjoint syntactic categories here, instead of the usual overlapping ones, so some operation is needed to move from one category to another. + +#### Main judgments + +We come now to the main judgments, classification and substitution. We haven't yet made any special use of the constraint domain—for everything above, we could have used an ordinary type—but this is where we will need it. The reason is that, in order to state these judgments in a general enough way to include all of the usual stratified LF judgments, we will need to be able to track the functions that take a category to the category of objects that classifies it, as well as the function that takes a category to the category of objects that result from substitutions into it (recall the subtlety with M+R). These are provided by the functions ``class`` and ``subst``, respectively: !}% +%{! (options removed from twelftag: import="ex3d") !}% +%inline class (%pi cat %-> cat) [X cat] [k] [a] [p] [m] [r] [m+r] [u] X u u k a a a u %. +%inline subst (%pi cat %-> cat) [X cat] [k] [a] [p] [m] [r] [m+r] [u] X k a p m m+r m+r u %. +%sort classj %. +%sort substj %. +%{! Here, ``classj U V`` means that ``U`` is classified by (has type, has kind, has hyperkind) ``V``, and ``substj ([x] E x) M E'`` means that ``E'`` is the result of substituting ``M`` for ``x`` in ``E``. Also, notice that the category index of each of these judgments is _implicit_, so you won't see any categories mentioned in the rules that follow. + +#### Rules for classification + +We now give the encodings of the rules for classification. For the purposes of this tutorial, it is not necessary that you study these rules in detail (unless they interest you!); just notice that Twelf is, in each case, able to infer the syntactic categories of all syntactic constituents of these rules by the using the term constructors involved, as well as the ``class`` function used to track the category of the classifier. !}% +%term canon_kind_type classj typ * %. +%term canon_kind_pi + %pi (classj (pi_k A2 ([y] K y)) *) + %<- (classj A2 *) + %<- ({y} %pi (classj y A2) %-> (classj (K y) *)) %. +%term canon_fam_pi + %pi (classj (pi_a A2 ([y] A y)) *) + %<- (classj A2 *) + %<- ({y} %pi (classj y A2) %-> (classj (A y) *)) %. +%term canon_fam_atom %pi (classj (coer_pa P) *) %<- (classj P typ) %. +%term atom_fam_app + %pi (classj (app_p P1 M2) K) + %<- (classj P1 (pi_k A2 ([y] K1 y))) + %<- (classj M2 A2) + %<- (substj ([x] K1 x) M2 K) %. +%term canon_term_lam + %pi (classj (lam_m ([y] M y)) (pi_a A2 ([y] A y))) + %<- ({y} %pi (classj y A2) %-> (classj (M y) (A y))) %. +%term canon_term_atom %pi (classj (coer_rm R) (coer_pa P)) %<- (classj R (coer_pa P)) %. +% atom_term_var not necessary +%term atom_term_app + %pi (classj (app_r R1 M2) A) + %<- (classj R1 (pi_a A2 ([y] A1 y))) + %<- (classj M2 A2) + %<- (substj ([x] A1 x) M2 A) %. +%term canon_atom_can %pi (classj (canon M) A) %<- (classj M A) %. +%term canon_atom_atom %pi (classj (atom R) A) %<- (classj R A) %. +%term unit_unit classj * * %. +%{! #### Rules for substitution + +Similarly, we present the rules for substitution. Just notice here that categories are inferred in the same way as with classification, with the added subtlety of the disjoint union type returned by substitutions into atomic terms: !}% +%{! (options removed from twelftag: import="ex3d") !}% +%term subst_k_type substj ([x] typ) M0 typ %. +%term subst_k_pi + %pi (substj ([x] pi_k (A x) ([y] K x y)) M0 (pi_k A' ([y] K' y))) + %<- (substj ([x] A x) M0 A') + %<- ({y} %pi (substj ([x] y) M0 (atom y)) %-> (substj ([x] K x y) M0 (K' y))) %. +%term subst_a_p %pi (substj ([x] coer_pa (P x)) M0 (coer_pa P')) %<- (substj ([x] P x) M0 P') %. +%term subst_a_pi + %pi (substj ([x] pi_a (A2 x) ([y] A x y)) M0 (pi_a A2' ([y] A' y))) + %<- (substj ([x] A2 x) M0 A2') + %<- ({y} %pi (substj ([x] y) M0 (atom y)) %-> (substj ([x] A x y) M0 (A' y))) %. +%term subst_p_app + %pi (substj ([x] app_p (P x) (M x)) M0 (app_p P' M')) + %<- (substj ([x] P x) M0 P') + %<- (substj ([x] M x) M0 M') %. +%term subst_m_lam + %pi (substj ([x] lam_m ([y] M x y)) M0 (lam_m ([y] M' y))) + %<- ({y} %pi (substj ([x] y) M0 (atom y)) %-> (substj ([x] M x y) M0 (M' y))) %. +%term subst_m_rh %pi (substj ([x] coer_rm (R x)) M0 M') %<- (substj ([x] R x) M0 (canon M')) %. +%term subst_m_r + %pi (substj ([x] coer_rm (R x)) M0 (coer_rm R')) + %<- (substj ([x] R x) M0 (atom R')) %. +%term subst_rh_var substj ([x] x) M0 (canon M0) %. +% subst_r_var not necessary +%term subst_rh_app + %pi (substj ([x] app_r (R1 x) (M2 x)) M0 (canon M'')) + %<- (substj ([x] R1 x) M0 (canon (lam_m ([y] M' y)))) + %<- (substj ([x] M2 x) M0 M2') + %<- (substj ([y] M' y) M2' M'') %. +%term subst_r_app + %pi (substj ([x] app_r (R1 x) (M2 x)) M0 (atom (app_r R1' M2'))) + %<- (substj ([x] R1 x) M0 (atom R1')) + %<- (substj ([x] M2 x) M0 M2') %. +%term subst_m+r_canon %pi (substj ([x] canon (M x)) M0 (canon M')) %<- (substj ([x] M x) M0 M') %. +%term subst_m+r_atom %pi (substj ([x] atom (P x)) M0 (atom P')) %<- (substj ([x] P x) M0 (atom P')) %. +%term subst_unit_unit substj ([x] *) M0 * %. +%{! #### Statement of the substitution theorem + +We finish up this example with one more illustration of the unification afforded by our constraint domain: the statement of the substitution theorem for LF. The theorem statement involves an implicit quantification over the syntactic category. This means, in particular, that it isn't necessary to prove this by a mutual induction over the syntactic category, as is usually the case. (Note, however, that because of the simplifications to LF that we have used for this presentation, it won't be possible to prove the theorem in this form.) + +```checkedtwelf +sub_thm : {T:tm r -> tm X} classj M0 A0 + -> ({x:tm r} classj x A0 -> classj (T x) (C x)) + -> substj ([x] C x) M0 C' + %%--------------------------------------------- + -> {T':tm (subst X)} + substj ([x] T x) M0 T' + -> classj T' C' + -> type. +``` + +Expansion of abbreviations is really hurting us here in this output, since ``nat`` expands to a type with seven arrows! + +## Example 4: Lists of length _n_ + +All of the constraint domains that we have used so far have been finite. For our last example, we will use an infinite constraint domain to implement one of the examples from the Twelf Manual that was used to illustrate the built-in constraint domain of rational numbers. The example defines a dependent type of lists of items, where the index is the length of the list. It then defines an append function, using the integer addition function to track the index of the result. We can achieve that result here by using a Church encoding of the natural numbers: !}% +%sort t %. +%inline nat %pi (%pi t %-> t) %-> t %-> t %. +%inline 0 [s %pi t %-> t] [z t] z %. +%inline s [n nat] [s %pi t %-> t] [z t] s (n s z) %. +%inline + [n nat] [m nat] [s %pi t %-> t] [z t] n s (m s z) %. +%prec %right 10 + %. +%sort item %. +%sort list %. +% list N = item lists of length N +%term nil list 0 %. +%term cons %pi item %-> (list N) %-> (list (s N)) %. +%sort append %. +%term append_nil append nil L L %. +%term append_cons %pi (append (cons T L1) L2 (cons T L3)) %<- (append L1 L2 L3) %. +%{! It is interesting to note here that, in order to type-check the declarations for ``append_nil`` and ``append_cons``, Twelf needs to prove for itself that the natural numbers defined above satisfy, respectively, + +- 0 + ``X`` = ``X``, for all ``X`` in ``nat``, and +- ``X + Y = Z`` implies ``(s X) + Y = (s Z)``, for all ``X``, ``Y``, and ``Z`` in ``nat``. + +These are statements—an equation and an implication between two equations—that are universally quantified over an infinite set, and are normally the kind of statement that is proved by induction on the natural numbers. So, is Twelf secretly proving statements by induction during type-checking? No, but that would be nice! In fact, in the case of 1, if you apply the definition of ``+`` to ``0`` and a _free variable_ ``X`` of type ``nat`` and then (beta-eta) normalize the result, you will see that it is equal (i.e., beta-eta equivalent) to ``X``. Similarly, if you apply the definitions of ``+`` and ``s`` to ``(s X) + Y``, on the one hand, and ``s(X + Y)`` on the other, for free variables ``X`` and ``Y`` of type ``nat``, you will see that they are again equal. Thus, all Twelf needs in order to prove 1 and 2 above is beta-eta equivalence! + +## Conclusion + +The techniques described in this tutorial fit into the general category of using indexing of types instead of predicates to make encodings more intrinsic. Our so-called user-defined constraint domains were one variety of index type that we found to be useful in this regard, since they allow us to define "tracking functions" that would be impossible to define with a free representation. + +However, there are still some issues that we have not yet examined. For example, + +* All of our domains were defined in terms of a type ``t``, and we saw that it was necessary for the method that the type hierarchy over ``t`` (or at least the part of it that we are using) doesn't acquire any new canonical elements other than those that exist for all ``t``, but we didn't talk about the best way to insure this in our developments. + +* The examples we gave where certain ill-formed terms produced unresolved constraints when we tried to type-check them are in a kind of nether world between terms that type-check and terms that fail to type-check. We would like to believe that the set of constraints produced on these examples by Twelf's type-checker are unresolvable, but there is no part of Twelf that can check that. + +Nevertheless, these examples illustrate once again the power of LF as a representational framework. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User-defined_constraint_domain). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-drl.lf b/new-tests/stelf-output/pages/user-drl.lf new file mode 100644 index 0000000..d584604 --- /dev/null +++ b/new-tests/stelf-output/pages/user-drl.lf @@ -0,0 +1,15 @@ +% +%{! My name is Dan Licata. I'm a PhD student at CMU. Here's my [http://www.cs.cmu.edu/~drl/ Web site]. + +Sometimes, I get the urge to write introductions to LF and Twelf. The second time this happened, it resulted in a [http://www.cs.cmu.edu/~drl/pubs/hl07mechanizing/hl07mechanizing.pdf journal article]. The first time this happened, it resulted in a very chatty tutorial, which, the third time this happened, turned into [Proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/). + +I also use Twelf regularly in my research; see the page on [research projects using Twelf](/wiki/research-projects-using-twelf/) or visit my Web site for some links. + +Random Bits of Twelf Code: +* [Modally Propositional Logic](/wiki/modally-propositional-logic/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Drl). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-hdeyoung-monweakfoc-elf.lf b/new-tests/stelf-output/pages/user-hdeyoung-monweakfoc-elf.lf new file mode 100644 index 0000000..974b723 --- /dev/null +++ b/new-tests/stelf-output/pages/user-hdeyoung-monweakfoc-elf.lf @@ -0,0 +1,815 @@ +% +%{! +Completeness + +```twelf + +pol : type. %name pol S. + +pos : pol. +neg : pol. + + + +sort : type. %name sort S. + +prin : sort. + +%block sort-block : block {s : sort}. + + + +term : sort -> type. %name term T x. + +%block term-block : some {S : sort} block {x : term S}. + + + +atm : pol -> type. %name atm P. + +%block atm-block : some {S : pol} block {p : atm S}. + + + +atm-eq : atm S -> atm S -> type. + +atm-eq/refl : atm-eq P+- P+-. + + + +prop : pol -> type. %name prop A. + +atom : atm S -> prop S. +tensor : prop pos -> prop pos -> prop pos. +one : prop pos. +% exists : {S : sort} (term S -> prop pos) -> prop pos. +down : prop neg -> prop pos. +lolli : prop pos -> prop neg -> prop neg. +% forall : {S : sort} (term S -> prop neg) -> prop neg. +monad : prop pos -> prop neg. +says : term prin -> prop pos -> prop neg. + + + +prop-eq : prop S -> prop S -> type. + +prop-eq/refl : prop-eq A+- A+-. + + + +prop-resp-atm-eq : {A+- : atm S -> prop S'} atm-eq P1+- P2+- -> prop-eq (A+- P1+-) (A+- P2+-) -> type. +%mode prop-resp-atm-eq +A+- +Deq -Deq'. + +- : prop-resp-atm-eq A+- atm-eq/refl prop-eq/refl. + +%worlds (sort-block | term-block | atm-block) (prop-resp-atm-eq _ _ _). +%total {} (prop-resp-atm-eq _ _ _). + + + +prop-eq-trans : prop-eq A+- B+- -> prop-eq B+- C+- -> prop-eq A+- C+- -> type. +%mode prop-eq-trans +Deq1 +Deq2 -Deq3. + +- : prop-eq-trans prop-eq/refl prop-eq/refl prop-eq/refl. + +%worlds (sort-block | term-block | atm-block) (prop-eq-trans _ _ _). +%total {} (prop-eq-trans _ _ _). + + +prop-resp-prop-eq : {A+- : prop S -> prop S'} prop-eq B1+- B2+- -> prop-eq (A+- B1+-) (A+- B2+-) -> type. +%mode prop-resp-prop-eq +A+- +Deq -Deq'. + +- : prop-resp-prop-eq A+- prop-eq/refl prop-eq/refl. + +%worlds (sort-block | term-block | atm-block) (prop-resp-prop-eq _ _ _). +%total {} (prop-resp-prop-eq _ _ _). + + + +left : prop pos -> type. %name left L l. + +%block left-block : some {A+ : prop pos} block {l : left A+}. + + + +left-resp-prop-eq : left A1+- -> prop-eq A1+- A2+- -> left A2+- -> type. +%mode left-resp-prop-eq +L +Deq -L'. + +- : left-resp-prop-eq L prop-eq/refl L. + +%worlds (sort-block | term-block | atm-block | left-block) (left-resp-prop-eq _ _ _). +%total {} (left-resp-prop-eq _ _ _). + + + +judgment : type. %name judgment J. + +true : prop neg -> judgment. +lax : prop pos -> judgment. +affirms : term prin -> prop pos -> judgment. + + + +conclusion : type. %name conclusion C. + +conclusion/right : judgment -> conclusion. +conclusion/rfoc : prop pos -> conclusion. +conclusion/lfoc : prop neg -> judgment -> conclusion. + + + +conseq : conclusion -> type. %name conseq D. + +%abbrev right : judgment -> type = [j] conseq (conclusion/right j). +%abbrev rfoc : prop pos -> type = [a+] conseq (conclusion/rfoc a+). +%abbrev lfoc : prop neg -> judgment -> type = [a-] [j] conseq (conclusion/lfoc a- j). + +%block rfoc-block : some {A+ : prop pos} block {rf : rfoc A+}. + +init+ : left (atom P+) -> rfoc (atom P+). +init- : lfoc (atom P-) (true (atom P-)). +tensorR : rfoc A+ -> rfoc B+ -> rfoc (tensor A+ B+). +tensorL : (left A+ -> left B+ -> right J) -> (left (tensor A+ B+) -> right J). +oneR : rfoc one. +oneL : right J -> (left one -> right J). +% existsR : {T : term S} rfoc (A+ T) -> rfoc (exists S A+). +% existsL : ({a : term S} left (A+ a) -> right J) -> (left (exists S A+) -> right J). +downR : right (true A-) -> rfoc (down A-). +downL : lfoc A- J -> (left (down A-) -> right J). +lolliR : (left A+ -> right (true B-)) -> right (true (lolli A+ B-)). +lolliL : rfoc A+ -> lfoc B- J -> lfoc (lolli A+ B-) J. +% forallR : ({a : term S} right (true (A- a))) -> right (true (forall S A-)). +% forallL : {T : term S} lfoc (A- T) J -> lfoc (forall S A-) J. +laxR : rfoc A+ -> right (lax A+). +monadR : right (lax A+) -> right (true (monad A+)). +monadL : (left A+ -> right (lax C+)) -> lfoc (monad A+) (lax C+). +affirmsR : rfoc A+ -> right (affirms K A+). +saysR : right (affirms K A+) -> right (true (says K A+)). +saysL : (left A+ -> right (affirms K C+)) -> lfoc (says K A+) (affirms K C+). + + + +eta+ : {A+} (rfoc A+ -> right J) -> (left A+ -> right J) -> type. +%mode eta+ +A+ +D -E. +eta- : {A-} ({J} lfoc A- J -> right J) -> right (true A-) -> type. +%mode eta- +A- +D -E. + +- : eta+ (atom P+) + (D : rfoc (atom P+) -> right J) + ([l : left (atom P+)] D (init+ l)). + +- : eta+ (tensor A+ B+) + (D : rfoc (tensor A+ B+) -> right J) + ([l : left (tensor A+ B+)] tensorL E2 l) + <- ({rfb : rfoc B+} + eta+ A+ ([rfa : rfoc A+] D (tensorR rfa rfb)) (E1 rfb : left A+ -> right J)) + <- ({la : left A+} + eta+ B+ ([rfb : rfoc B+] E1 rfb la) (E2 la : left B+ -> right J)). + +- : eta+ one + (D : rfoc one -> right J) + ([l : left one] oneL (D oneR) l). + +- : eta+ (down A-) + (D : rfoc (down A-) -> right J) + ([l : left (down A-)] D (downR (E l))) + <- ({l : left (down A-)} + eta- A- ([j] [lfa : lfoc A- j] downL lfa l) (E l : right (true A-))). + +%worlds (sort-block | term-block | atm-block | left-block | rfoc-block) (eta+ _ _ _) (eta- _ _ _). +%trustme %total (A+ A-) (eta+ A+ _ _) (eta- A- _ _). + + + +cut+ : {A+} rfoc A+ -> (left A+ -> conseq C) -> conseq C -> type. +%mode cut+ +A+ +D +E -F. +cut- : {A-} right (true A-) -> lfoc A- J -> right J -> type. +%mode cut- +A- +D +E -F. +cut< : {A-} lfoc B- (true A-) -> lfoc A- J -> lfoc B- J -> type. +%mode cut< +A- +D +E -F. +cut+lax : {A+} right (lax A+) -> (left A+ -> right (lax C+)) -> right (lax C+) -> type. +%mode cut+lax +A+ +D +E -F. +cut+lax-lfoc : {A+} lfoc B- (lax A+) -> (left A+ -> right (lax C+)) -> lfoc B- (lax C+) -> type. +%mode cut+lax-lfoc +A+ +D +E -F. +cut+affirms : {A+} right (affirms K A+) -> (left A+ -> right (affirms K C+)) -> right (affirms K C+) -> type. +%mode cut+affirms +A+ +D +E -F. +cut+affirms-lfoc : {A+} lfoc B- (affirms K A+) -> (left A+ -> right (affirms K C+)) -> lfoc B- (affirms K C+) -> type. +%mode cut+affirms-lfoc +A+ +D +E -F. + +%worlds (sort-block | term-block | atm-block | left-block | rfoc-block) + (cut+ _ _ _ _) + (cut- _ _ _ _) + (cut< _ _ _ _) + (cut+lax _ _ _ _) (cut+lax-lfoc _ _ _ _) + (cut+affirms _ _ _ _) (cut+affirms-lfoc _ _ _ _). +%trustme %total {(A1+ A2- A3- A4+ A5+ A6+ A7+) [(D1 D2 D3 D4 D5 D6 D7) (E1 E2 E3 E4 E5 E6 E7)]} + (cut+ A1+ D1 E1 _) + (cut- A2- D2 E2 _) + (cut< A3- D3 E3 _) + (cut+lax A4+ D4 E4 _) (cut+lax-lfoc A5+ D5 E5 _) + (cut+affirms A6+ D6 E6 _) (cut+affirms-lfoc A7+ D7 E7 _). + + + + +atm' : type. %name atm' P'. + +%block atm'-block : block {p' : atm'}. + + + +prop' : type. %name prop' A'. + +atom' : atm' -> prop'. +tensor' : prop' -> prop' -> prop'. +one' : prop'. +% exists' : {S : sort} (term S -> prop') -> prop'. +lolli' : prop' -> prop' -> prop'. +% forall' : {S : sort} (term S -> prop') -> prop'. +monad' : prop' -> prop'. +says' : term prin -> prop' -> prop'. + + + +hyp : prop' -> type. %name hyp H h. + +%block hyp-block : some {A : prop'} block {h : hyp A}. + + + +judgment' : type. %name judgment' J'. + +judgment'/true : prop' -> judgment'. +judgment'/lax : prop' -> judgment'. +judgment'/affirms : term prin -> prop' -> judgment'. + + + +conc : judgment' -> type. %name conc D. + +%abbrev true' : prop' -> type = [a] conc (judgment'/true a). +%abbrev lax' : prop' -> type = [a] conc (judgment'/lax a). +%abbrev affirms' : term prin -> prop' -> type = [k] [a] conc (judgment'/affirms k a). + +init' : hyp (atom' P) -> true' (atom' P). +tensor'R : true' A -> true' B -> true' (tensor' A B). +tensor'L : (hyp A -> hyp B -> conc J') -> (hyp (tensor' A B) -> conc J'). +one'R : true' one'. +one'L : conc J' -> (hyp one' -> conc J'). +% exists'R : {T : term S} true' (A T) -> true' (exists' S A). +% exists'L : ({a : term S} hyp (A a) -> conc J') -> (hyp (exists' S A) -> conc J'). +lolli'R : (hyp A -> true' B) -> true' (lolli' A B). +lolli'L : true' A -> (hyp B -> conc J') -> (hyp (lolli' A B) -> conc J'). +% forall'R : ({a : term S} true' (A a)) -> true' (forall' S A). +% forall'L : {T : term S} (hyp (A T) -> conc J') -> (hyp (forall' S A) -> conc J'). +lax'R : true' A -> lax' A. +monad'R : lax' A -> true' (monad' A). +monad'L : (hyp A -> lax' C) -> (hyp (monad' A) -> lax' C). +affirms'R : true' A -> affirms' K A. +says'R : affirms' K A -> true' (says' K A). +says'L : (hyp A -> affirms' K C) -> (hyp (says' K A) -> affirms' K C). + + + + +erase-atm : atm S -> atm' -> type. + +%block erase-atm-block : some {S : pol} + block {p : atm S} {p' : atm'} {erase-atm/p : erase-atm p p'}. + +%worlds (erase-atm-block) (erase-atm _ _). + + +erase-atm-unique : erase-atm P1+- P -> erase-atm P2+- P -> atm-eq P1+- P2+- -> type. +%mode erase-atm-unique +E1 +E2 -Deq. + +- : erase-atm-unique E E atm-eq/refl. + +%worlds (erase-atm-block) (erase-atm-unique _ _ _). +%total {} (erase-atm-unique _ _ _). + + + +erase-prop : prop S -> prop' -> type. + +erase-prop/atom : erase-prop (atom P+-) (atom' P) + <- erase-atm P+- P. +erase-prop/tensor : erase-prop (tensor A+ B+) (tensor' A B) + <- erase-prop A+ A + <- erase-prop B+ B. +erase-prop/one : erase-prop one one'. +% erase-prop/exists : erase-prop (exists S A+) (exists' S A) +% <- ({a : term S} erase-prop (A+ a) (A a)). +erase-prop/down : erase-prop (down A-) A + <- erase-prop A- A. +erase-prop/lolli : erase-prop (lolli A+ B-) (lolli' A B) + <- erase-prop A+ A + <- erase-prop B- B. +% erase-prop/forall : erase-prop (forall S A-) (forall' S A) +% <- ({a : term S} erase-prop (A- a) (A a)). +erase-prop/monad : erase-prop (monad A+) (monad' A) + <- erase-prop A+ A. +erase-prop/says : erase-prop (says K A+) (says' K A) + <- erase-prop A+ A. + +%worlds (sort-block | term-block | erase-atm-block) (erase-prop _ _). + + + +erase-prop-unique : erase-prop A1+- A -> erase-prop A2+- A -> prop-eq A1+- A2+- -> type. +%mode erase-prop-unique +E1 +E2 -Deq. + +- : erase-prop-unique + (erase-prop/atom (E : erase-atm P+- P)) + (erase-prop/atom E) + prop-eq/refl. + +- : erase-prop-unique + (erase-prop/tensor (E12 : erase-prop B1+ B) (E11 : erase-prop A1+ A)) + (erase-prop/tensor (E22 : erase-prop B2+ B) (E21 : erase-prop A2+ A)) + Deq + <- erase-prop-unique E11 E21 (Deq1 : prop-eq A1+ A2+) + <- erase-prop-unique E12 E22 (Deq2 : prop-eq B1+ B2+) + <- prop-resp-prop-eq ([a1+] tensor a1+ B1+) Deq1 (Deq3 : prop-eq (tensor A1+ B1+) (tensor A2+ B1+)) + <- prop-resp-prop-eq ([b1+] tensor A2+ b1+) Deq2 (Deq4 : prop-eq (tensor A2+ B1+) (tensor A2+ B2+)) + <- prop-eq-trans Deq3 Deq4 (Deq : prop-eq (tensor A1+ B1+) (tensor A2+ B2+)). + +- : erase-prop-unique erase-prop/one erase-prop/one prop-eq/refl. + +- : erase-prop-unique + (erase-prop/down (E1 : erase-prop A1- A)) + (erase-prop/down (E2 : erase-prop A2- A)) + Deq' + <- erase-prop-unique E1 E2 (Deq : prop-eq A1- A2-) + <- prop-resp-prop-eq ([a-] down a-) Deq (Deq' : prop-eq (down A1-) (down A2-)). + +- : erase-prop-unique + (erase-prop/lolli (E12 : erase-prop B1- B) (E11 : erase-prop A1+ A)) + (erase-prop/lolli (E22 : erase-prop B2- B) (E21 : erase-prop A2+ A)) + Deq + <- erase-prop-unique E11 E21 (Deq1 : prop-eq A1+ A2+) + <- erase-prop-unique E12 E22 (Deq2 : prop-eq B1- B2-) + <- prop-resp-prop-eq ([a1+] lolli a1+ B1-) Deq1 (Deq3 : prop-eq (lolli A1+ B1-) (lolli A2+ B1-)) + <- prop-resp-prop-eq ([b1-] lolli A2+ b1-) Deq2 (Deq4 : prop-eq (lolli A2+ B1-) (lolli A2+ B2-)) + <- prop-eq-trans Deq3 Deq4 (Deq : prop-eq (lolli A1+ B1-) (lolli A2+ B2-)). + +- : erase-prop-unique + (erase-prop/monad (E1 : erase-prop A1+ A)) + (erase-prop/monad (E2 : erase-prop A2+ A)) + Deq' + <- erase-prop-unique E1 E2 (Deq : prop-eq A1+ A2+) + <- prop-resp-prop-eq ([a+] monad a+) Deq (Deq' : prop-eq (monad A1+) (monad A2+)). + +- : erase-prop-unique + (erase-prop/says (E1 : erase-prop A1+ A)) + (erase-prop/says (E2 : erase-prop A2+ A)) + Deq' + <- erase-prop-unique E1 E2 (Deq : prop-eq A1+ A2+) + <- prop-resp-prop-eq ([a+] says K a+) Deq (Deq' : prop-eq (says K A1+) (says K A2+)). + +%worlds (sort-block | term-block | erase-atm-block) (erase-prop-unique _ _ _). +%total E1 (erase-prop-unique E1 _ _). + + + +erase-judgment : judgment -> judgment' -> type. + +erase-judgment/true : erase-judgment (true A-) (judgment'/true A) + <- erase-prop A- A. +erase-judgment/lax : erase-judgment (lax A+) (judgment'/lax A) + <- erase-prop A+ A. +erase-judgment/affirms : erase-judgment (affirms K A+) (judgment'/affirms K A) + <- erase-prop A+ A. + + + + +% adm-forallL : (left (down (A- T)) -> right J) -> (left (down (forall S A-)) -> right J) -> type. +% %mode adm-forallL +D -E. + +% %worlds (sort-block | term-block | atm-block | left-block | rfoc-block) (adm-forallL _ _). +% %trustme %total D (adm-forallL D _). + + + +cmp-conc : conc J' -> erase-judgment J J' -> right J -> type. +%mode cmp-conc +D +E -F. +cmp-lax : true' A -> erase-prop A+ A -> right (lax A+) -> type. +%mode cmp-lax +D +E -F. +cmp-affirms : true' A -> erase-prop A+ A -> right (affirms K A+) -> type. +%mode +{A} +{A+} +{K} +{D : true' A} +{E : erase-prop A+ A} -{F : right (affirms K A+)} (cmp-affirms D E F). +cmp-lolli : true' A -> erase-prop A+ A -> (left (down B-) -> right J) -> (left (down (lolli A+ B-)) -> right J) -> type. +%mode cmp-lolli +D +E +F -G. +cmp-hyp : hyp A -> erase-prop A+ A -> left A+ -> type. +%mode cmp-hyp +H -E -L. +cmp-hyp+ : hyp (atom' P) -> erase-prop A+ (atom' P) -> left A+ -> type. +%mode cmp-hyp+ +H +E -L. + +- : cmp-hyp+ + (H : hyp (atom' P)) + (E : erase-prop A+ (atom' P)) + L' + <- cmp-hyp H (E' : erase-prop A'+ (atom' P)) (L : left A'+) + <- erase-prop-unique E' E (Deq : prop-eq A'+ A+) + <- left-resp-prop-eq L Deq (L' : left A+). + + +- : cmp-conc + (init' (H : hyp (atom' P))) + (erase-judgment/true (erase-prop/atom (E : erase-atm P- P))) + (downL init- L) + <- cmp-hyp+ H (erase-prop/down (erase-prop/atom E)) L. + +- : cmp-conc + (tensor'L (D : hyp A -> hyp B -> conc J') (H : hyp (tensor' A B))) + (E : erase-judgment J J') + (tensorL F L) + <- cmp-hyp H (erase-prop/tensor (E2 : erase-prop B+ B) (E1 : erase-prop A+ A)) (L : left (tensor A+ B+)) + <- ({ha : hyp A} {la : left A+} cmp-hyp ha E1 la + -> {hb : hyp B} {lb : left B+} cmp-hyp hb E2 lb + -> cmp-conc (D ha hb) E (F la lb : right J)). + +- : cmp-conc + (one'L (D : conc J') (H : hyp one')) + (E : erase-judgment J J') + (oneL F L) + <- cmp-hyp H erase-prop/one (L : left one) + <- cmp-conc D E (F : right J). + +% - : cmp-conc +% (exists'L (D : {a : term S} hyp (A a) -> conc J') (H : hyp (exists' S A))) +% (E : erase-judgment J J') +% (existsL F L) +% <- cmp-hyp H (erase-prop/exists (E1 : {x : term S} erase-prop (A+ x) (A x))) (L : left (exists S A+)) +% <- ({a : term S} +% {ha : hyp (A a)} {la : left (A+ a)} cmp-hyp ha (E1 a) la +% -> cmp-conc (D a ha) E (F a la : right J)). + +- : cmp-conc + (lolli'R (D : hyp A -> true' B)) + (erase-judgment/true (erase-prop/lolli (E2 : erase-prop B- B) (E1 : erase-prop A+ A))) + (lolliR F) + <- ({ha : hyp A} {la : left A+} cmp-hyp ha E1 la -> cmp-conc (D ha) (erase-judgment/true E2) (F la : right (true B-))). + +- : cmp-conc + (lolli'L (D1 : true' A) (D2 : hyp B -> conc J') (H : hyp (lolli' A B))) + (E : erase-judgment J J') + (F L) + <- cmp-hyp H (erase-prop/down (erase-prop/lolli (E2 : erase-prop B- B) (E1 : erase-prop A+ A))) (L : left (down (lolli A+ B-))) + <- ({hb : hyp B} {lb : left (down B-)} cmp-hyp hb (erase-prop/down E2) lb -> cmp-conc (D2 hb) E (F2 lb : right J)) + <- cmp-lolli D1 E1 F2 (F : left (down (lolli A+ B-)) -> right J). + +% - : cmp-conc +% (forall'R (D : {a: term S} true' (A a))) +% (erase-judgment/true (erase-prop/forall (E : {x : term S} erase-prop (A- x) (A x)))) +% (forallR F) +% <- ({a: term S} cmp-conc (D a) (erase-judgment/true (E a)) (F a)). + +% - : cmp-conc +% (forall'L (T : term S) (D : hyp (A T) -> conc J') (H : hyp (forall' S A))) +% (E : erase-judgment J J') +% (F' L) +% <- cmp-hyp H (erase-prop/down (erase-prop/forall (E1 : {x : term S} erase-prop (A- x) (A x)))) (L : left (down (forall S A-))) +% <- ({ha : hyp (A T)} {la : left (down (A- T))} cmp-hyp ha (erase-prop/down (E1 T)) la -> cmp-conc (D ha) E (F la : right J)) +% <- adm-forallL F (F' : left (down (forall S A-)) -> right J). + +- : cmp-conc (lax'R (D : true' A)) (erase-judgment/lax (E : erase-prop A+ A)) F + <- cmp-lax D E (F : right (lax A+)). + +- : cmp-conc + (monad'R (D : lax' A)) + (erase-judgment/true (erase-prop/monad (E : erase-prop A+ A))) + (monadR F) + <- cmp-conc D (erase-judgment/lax E) F. + +- : cmp-conc + (monad'L (D : hyp A -> lax' C) (H : hyp (monad' A))) + (erase-judgment/lax (E : erase-prop C+ C)) + (downL (monadL F) L) + <- cmp-hyp H (erase-prop/down (erase-prop/monad (E1 : erase-prop A+ A))) (L : left (down (monad A+))) + <- ({ha : hyp A} {la : left A+} cmp-hyp ha E1 la + -> cmp-conc (D ha) (erase-judgment/lax E) (F la : right (lax C+))). + +- : cmp-conc + (affirms'R (D : true' A)) + (erase-judgment/affirms (E : erase-prop A+ A)) + F + <- cmp-affirms D E (F : right (affirms K A+)). + +- : cmp-conc + (says'R (D : affirms' K A)) + (erase-judgment/true (erase-prop/says (E : erase-prop A+ A))) + (saysR F) + <- cmp-conc D (erase-judgment/affirms E) F. + +- : cmp-conc + (says'L (D : hyp A -> affirms' K C) (H : hyp (says' K A))) + + + + (erase-judgment/affirms (E : erase-prop C+ C)) + + + + (downL (saysL F) L) + + + + <- cmp-hyp H (erase-prop/down (erase-prop/says (E1 : erase-prop A+ A))) (L ) + + + + <- ({ha : hyp A} {la : left A+} cmp-hyp ha E1 la + + + + -> cmp-conc (D ha) (erase-judgment/affirms E) (F la : right (affirms K C+))). + + +- : cmp-lax + (init' (H : hyp (atom' P))) + (erase-prop/atom (E : erase-atm P+ P)) + (laxR (init+ L)) + <- cmp-hyp+ H (erase-prop/atom E) (L : left (atom P+)). + +- : cmp-lax + (tensor'R (D1 : true' A) (D2 : true' B)) + (erase-prop/tensor (E2 : erase-prop B+ B) (E1 : erase-prop A+ A)) + F6 + <- cmp-lax D1 E1 (F1 : right (lax A+)) + <- cmp-lax D2 E2 (F2 : right (lax B+)) + <- ({rfa : rfoc A+} eta+ B+ ([rfb : rfoc B+] laxR (tensorR rfa rfb)) (F3 rfa : left B+ -> right (lax (tensor A+ B+)))) + <- ({lb : left B+} eta+ A+ ([rfa : rfoc A+] F3 rfa lb) (F4 lb : left A+ -> right (lax (tensor A+ B+)))) + <- ({lb : left B+} cut+lax A+ F1 (F4 lb) (F5 lb : right (lax (tensor A+ B+)))) + <- cut+lax B+ F2 F5 (F6 : right (lax (tensor A+ B+))). + +- : cmp-lax + (tensor'L (D : hyp B1 -> hyp B2 -> true' A) (H : hyp (tensor' B1 B2))) + (E : erase-prop A+ A) + (tensorL F L) + <- cmp-hyp H (erase-prop/tensor (E2 : erase-prop B2+ B2) (E1 : erase-prop B1+ B1)) (L : left (tensor B1+ B2+)) + <- ({hb1 : hyp B1} {lb1 : left B1+} cmp-hyp hb1 E1 lb1 + -> {hb2 : hyp B2} {lb2 : left B2+} cmp-hyp hb2 E2 lb2 + -> cmp-lax (D hb1 hb2) E (F lb1 lb2 : right (lax A+))). + +- : cmp-lax one'R erase-prop/one (laxR oneR). + +- : cmp-lax + (one'L (D : true' A) (H : hyp one')) + (E : erase-prop A+ A) + (oneL F L) + <- cmp-hyp H erase-prop/one (L : left one) + <- cmp-lax D E (F : right (lax A+)). + +% - : cmp-lax +% (exists'R (T : term S) (D : true' (A T))) +% (erase-prop/exists (E : {x : term S} erase-prop (A+ x) (A x))) +% F3 +% <- cmp-lax D (E T) (F1 : right (lax (A+ T))) +% <- eta+ (A+ T) ([rfa : rfoc (A+ T)] laxR (existsR T rfa)) (F2 : left (A+ T) -> right (lax (exists S A+))) +% <- cut+lax (A+ T) F1 F2 (F3 : right (lax (exists S A+))). + +% - : cmp-lax +% (exists'L (D : {a : term S} hyp (B a) -> true' A) (H : hyp (exists' S B))) +% (E : erase-prop A+ A) +% (existsL F L) +% <- cmp-hyp H (erase-prop/exists (E1 : {x : term S} erase-prop (B+ x) (B x))) (L : left (exists S B+)) +% <- ({a : term S} +% {hb : hyp (B a)} {lb : left (B+ a)} cmp-hyp hb (E1 a) lb +% -> cmp-lax (D a hb) E (F a lb : right (lax A+))). + +- : cmp-lax + (D : true' A) + (erase-prop/down (E : erase-prop A- A)) + (laxR (downR F)) + <- cmp-conc D (erase-judgment/true E) (F : right (true A-)). + +- : cmp-lax + (lolli'L (D1 : true' B1) (D2 : hyp B2 -> true' A) (H : hyp (lolli' B1 B2))) + (E : erase-prop A+ A) + (F L) + <- cmp-hyp H (erase-prop/down (erase-prop/lolli (E2 : erase-prop B2- B2) (E1 : erase-prop B1+ B1))) (L : left (down (lolli B1+ B2-))) + <- ({hb2 : hyp B2} {lb2 : left (down B2-)} cmp-hyp hb2 (erase-prop/down E2) lb2 -> cmp-lax (D2 hb2) E (F2 lb2 : right (lax A+))) + <- cmp-lolli D1 E1 F2 (F : left (down (lolli B1+ B2-)) -> right (lax A+)). + +% - : cmp-lax +% (forall'L (T : term S) (D : hyp (B T) -> true' A) (H : hyp (forall' S B))) +% (E : erase-prop A+ A) +% (F' L) +% <- cmp-hyp H (erase-prop/down (erase-prop/forall (E1 : {x : term S} erase-prop (B- x) (B x)))) (L : left (down (forall S B-))) +% <- ({hb : hyp (B T)} {lb : left (down (B- T))} cmp-hyp hb (erase-prop/down (E1 T)) lb -> cmp-lax (D hb) E (F lb : right (lax A+))) +% <- adm-forallL F (F' : left (down (forall S B-)) -> right (lax A+)). + + +- : cmp-affirms + (init' (H : hyp (atom' P))) + (erase-prop/atom (E : erase-atm P+ P)) + (affirmsR (init+ L)) + <- cmp-hyp+ H (erase-prop/atom E) (L : left (atom P+)). + +- : cmp-affirms + (tensor'R (D1 : true' A) (D2 : true' B)) + (erase-prop/tensor (E2 : erase-prop B+ B) (E1 : erase-prop A+ A)) + F6 + <- cmp-affirms D1 E1 (F1 : right (affirms K A+)) + <- cmp-affirms D2 E2 (F2 : right (affirms K B+)) + <- ({rfa : rfoc A+} eta+ B+ ([rfb : rfoc B+] affirmsR (tensorR rfa rfb)) (F3 rfa : left B+ -> right (affirms K (tensor A+ B+)))) + <- ({lb : left B+} eta+ A+ ([rfa : rfoc A+] F3 rfa lb) (F4 lb : left A+ -> right (affirms K (tensor A+ B+)))) + <- ({lb : left B+} cut+affirms A+ F1 (F4 lb) (F5 lb : right (affirms K (tensor A+ B+)))) + <- cut+affirms B+ F2 F5 (F6 : right (affirms K (tensor A+ B+))). + +- : cmp-affirms + (tensor'L (D : hyp B1 -> hyp B2 -> true' A) (H : hyp (tensor' B1 B2))) + (E : erase-prop A+ A) + (tensorL F L) + <- cmp-hyp H (erase-prop/tensor (E2 : erase-prop B2+ B2) (E1 : erase-prop B1+ B1)) (L : left (tensor B1+ B2+)) + <- ({hb1 : hyp B1} {lb1 : left B1+} cmp-hyp hb1 E1 lb1 + -> {hb2 : hyp B2} {lb2 : left B2+} cmp-hyp hb2 E2 lb2 + -> cmp-affirms (D hb1 hb2) E (F lb1 lb2 : right (affirms K A+))). + +- : cmp-affirms one'R erase-prop/one (affirmsR oneR). + +- : cmp-affirms + (one'L (D : true' A) (H : hyp one')) + (E : erase-prop A+ A) + (oneL F L) + <- cmp-hyp H erase-prop/one (L : left one) + <- cmp-affirms D E (F : right (affirms K A+)). + +% - : cmp-affirms +% (exists'R (T : term S) (D : true' (A T))) +% (erase-prop/exists (E : {x : term S} erase-prop (A+ x) (A x))) +% F3 +% <- cmp-affirms D (E T) (F1 : right (affirms K (A+ T))) +% <- eta+ (A+ T) ([rfa : rfoc (A+ T)] affirmsR (existsR T rfa)) (F2 : left (A+ T) -> right (affirms K (exists S A+))) +% <- cut+affirms (A+ T) F1 F2 (F3 : right (affirms K (exists S A+))). + +% - : cmp-affirms +% (exists'L (D : {a : term S} hyp (B a) -> true' A) (H : hyp (exists' S B))) +% (E : erase-prop A+ A) +% (existsL F L) +% <- cmp-hyp H (erase-prop/exists (E1 : {x : term S} erase-prop (B+ x) (B x))) (L : left (exists S B+)) +% <- ({a : term S} +% {hb : hyp (B a)} {lb : left (B+ a)} cmp-hyp hb (E1 a) lb +% -> cmp-affirms (D a hb) E (F a lb : right (affirms K A+))). + +- : cmp-affirms + (D : true' A) + (erase-prop/down (E : erase-prop A- A)) + (affirmsR (downR F)) + <- cmp-conc D (erase-judgment/true E) (F : right (true A-)). + +- : cmp-affirms + (lolli'L (D1 : true' B1) (D2 : hyp B2 -> true' A) (H : hyp (lolli' B1 B2))) + (E : erase-prop A+ A) + (F L) + <- cmp-hyp H (erase-prop/down (erase-prop/lolli (E2 : erase-prop B2- B2) (E1 : erase-prop B1+ B1))) (L : left (down (lolli B1+ B2-))) + <- ({hb2 : hyp B2} {lb2 : left (down B2-)} cmp-hyp hb2 (erase-prop/down E2) lb2 -> cmp-affirms (D2 hb2) E (F2 lb2 : right (affirms K A+))) + <- cmp-lolli D1 E1 F2 (F : left (down (lolli B1+ B2-)) -> right (affirms K A+)). + +% - : cmp-affirms +% (forall'L (T : term S) (D : hyp (B T) -> true' A) (H : hyp (forall' S B))) +% (E : erase-prop A+ A) +% (F' L) +% <- cmp-hyp H (erase-prop/down (erase-prop/forall (E1 : {x : term S} erase-prop (B- x) (B x)))) (L : left (down (forall S B-))) +% <- ({hb : hyp (B T)} {lb : left (down (B- T))} cmp-hyp hb (erase-prop/down (E1 T)) lb -> cmp-affirms (D hb) E (F lb : right (affirms K A+))) +% <- adm-forallL F (F' : left (down (forall S B-)) -> right (affirms K A+)). + + +- : cmp-lolli + (init' (H : hyp (atom' P))) + (erase-prop/atom (E : erase-atm P+ P)) + (F : left (down B-) -> right J) + G2 + <- cmp-hyp+ H (erase-prop/atom E) (L : left (atom P+)) + <- ({l : left (down (lolli (atom P+) B-))} + eta- B- ([j] [lfb : lfoc B- j] downL (lolliL (init+ L) lfb) l) (G1 l : right (true B-))) + <- ({l : left (down (lolli (atom P+) B-))} + cut+ (down B-) (downR (G1 l)) F (G2 l : right J)). + +- : cmp-lolli + (tensor'R (D1 : true' A1) (D2 : true' A2)) + (erase-prop/tensor (E2 : erase-prop A2+ A2) (E1 : erase-prop A1+ A1)) + (F : left (down B-) -> right J) + G6 + <- cmp-lolli D2 E2 F (G1 : left (down (lolli A2+ B-)) -> right J) + <- cmp-lolli D1 E1 G1 (G2 : left (down (lolli A1+ (lolli A2+ B-))) -> right J) + <- ({l : left (down (lolli (tensor A1+ A2+) B-))} {rfa1 : rfoc A1+} {rfa2 : rfoc A2+} + eta- B- ([j] [lfb : lfoc B- j] downL (lolliL (tensorR rfa1 rfa2) lfb) l) (G3 l rfa1 rfa2 : right (true B-))) + <- ({l : left (down (lolli (tensor A1+ A2+) B-))} {rfa1 : rfoc A1+} + eta+ A2+ ([rfa2 : rfoc A2+] G3 l rfa1 rfa2) (G4 l rfa1 : left A2+ -> right (true B-))) + <- ({l : left (down (lolli (tensor A1+ A2+) B-))} {la2 : left A2+} + eta+ A1+ ([rfa1 : rfoc A1+] G4 l rfa1 la2) (G5 l la2 : left A1+ -> right (true B-))) + <- ({l : left (down (lolli (tensor A1+ A2+) B-))} + cut+ (down (lolli A1+ (lolli A2+ B-))) (downR (lolliR ([la1 : left A1+] lolliR ([la2 : left A2+] G5 l la2 la1)))) G2 (G6 l : right J)). + +- : cmp-lolli + (tensor'L (D : hyp B1 -> hyp B2 -> true' A) (H : hyp (tensor' B1 B2))) + (E : erase-prop A+ A) + (F : left (down B-) -> right J) + ([l : left (down (lolli A+ B-))] tensorL ([lb1 : left B1+] [lb2 : left B2+] G lb1 lb2 l) L) + <- cmp-hyp H (erase-prop/tensor (E2 : erase-prop B2+ B2) (E1 : erase-prop B1+ B1)) (L : left (tensor B1+ B2+)) + <- ({hb1 : hyp B1} {lb1 : left B1+} cmp-hyp hb1 E1 lb1 + -> {hb2 : hyp B2} {lb2 : left B2+} cmp-hyp hb2 E2 lb2 + -> cmp-lolli (D hb1 hb2) E F (G lb1 lb2 : left (down (lolli A+ B-)) -> right J)). + +- : cmp-lolli + one'R + erase-prop/one + (F : left (down B-) -> right J) + G2 + <- ({l : left (down (lolli one B-))} + eta- B- ([j] [lfb : lfoc B- j] downL (lolliL oneR lfb) l) (G1 l : right (true B-))) + <- ({l : left (down (lolli one B-))} + cut+ (down B-) (downR (G1 l)) F (G2 l : right J)). + +- : cmp-lolli + (one'L (D : true' A) (H : hyp one')) + (E : erase-prop A+ A) + (F : left (down B-) -> right J) + ([l : left (down (lolli A+ B-))] oneL (G l) L) + <- cmp-hyp H erase-prop/one (L : left one) + <- cmp-lolli D E F (G : left (down (lolli A+ B-)) -> right J). + +% - : cmp-lolli +% (exists'R (T : term S) (D : true' (A T))) +% (erase-prop/exists (E : {x : term S} erase-prop (A+ x) (A x))) +% (F : left (down B-) -> right J) +% G4 +% <- cmp-lolli D (E T) F (G1 : left (down (lolli (A+ T) B-)) -> right J) +% <- ({l : left (down (lolli (exists S A+) B-))} {rfa : rfoc (A+ T)} +% eta- B- ([j] [lfb : lfoc B- j] downL (lolliL (existsR T rfa) lfb) l) (G2 l rfa : right (true B-))) +% <- ({l : left (down (lolli (exists S A+) B-))} +% eta+ (A+ T) ([rfa : rfoc (A+ T)] G2 l rfa) (G3 l : left (A+ T) -> right (true B-))) +% <- ({l : left (down (lolli (exists S A+) B-))} +% cut+ (down (lolli (A+ T) B-)) (downR (lolliR ([la : left (A+ T)] G3 l la))) G1 (G4 l : right J)). + +% - : cmp-lolli +% (exists'L (D : {a : term S} hyp (B1 a) -> true' A) (H : hyp (exists' S B1))) +% (E : erase-prop A+ A) +% (F : left (down B-) -> right J) +% ([l : left (down (lolli A+ B-))] existsL ([a : term S] [lb1 : left (B1+ a)] G a lb1 l) L) +% <- cmp-hyp H (erase-prop/exists (E1 : {x : term S} erase-prop (B1+ x) (B1 x))) (L : left (exists S B1+)) +% <- ({a : term S} +% {hb1 : hyp (B1 a)} {lb1 : left (B1+ a)} cmp-hyp hb1 (E1 a) lb1 +% -> cmp-lolli (D a hb1) E F (G a lb1 : left (down (lolli A+ B-)) -> right J)). + +- : cmp-lolli + (D : true' A) + (erase-prop/down (E : erase-prop A- A)) + (F : left (down B-) -> right J) + G3 + <- cmp-conc D (erase-judgment/true E) (G1 : right (true A-)) + <- ({l : left (down (lolli (down A-) B-))} + eta- B- ([j] [lfb : lfoc B- j] downL (lolliL (downR G1) lfb) l) (G2 l : right (true B-))) + <- ({l : left (down (lolli (down A-) B-))} + cut+ (down B-) (downR (G2 l)) F (G3 l : right J)). + +- : cmp-lolli + (lolli'L (D1 : true' B1) (D2 : hyp B2 -> true' A) (H : hyp (lolli' B1 B2))) + (E : erase-prop A+ A) + (F : left (down B-) -> right J) + ([l : left (down (lolli A+ B-))] G2 l L) + <- cmp-hyp H (erase-prop/down (erase-prop/lolli (E2 : erase-prop B2- B2) (E1 : erase-prop B1+ B1))) (L : left (down (lolli B1+ B2-))) + <- ({hb2 : hyp B2} {lb2 : left (down B2-)} cmp-hyp hb2 (erase-prop/down E2) lb2 + -> cmp-lolli (D2 hb2) E F (G1 lb2 : left (down (lolli A+ B-)) -> right J)) + <- ({l : left (down (lolli A+ B-))} + cmp-lolli D1 E1 ([lb2 : left (down B2-)] G1 lb2 l) (G2 l : left (down (lolli B1+ B2-)) -> right J)). + +% - : cmp-lolli +% (forall'L (T : term S) (D : hyp (B1 T) -> true' A) (H : hyp (forall' S B1))) +% (E : erase-prop A+ A) +% (F : left (down B-) -> right J) +% ([l : left (down (lolli A+ B-))] G2 l L) +% <- cmp-hyp H (erase-prop/down (erase-prop/forall (E1 : {x : term S} erase-prop (B1- x) (B1 x)))) (L : left (down (forall S B1-))) +% <- ({hb1 : hyp (B1 T)} {lb1 : left (down (B1- T))} cmp-hyp hb1 (erase-prop/down (E1 T)) lb1 +% -> cmp-lolli (D hb1) E F (G1 lb1 : left (down (lolli A+ B-)) -> right J)) +% <- ({l : left (down (lolli A+ B-))} +% adm-forallL ([lb1 : left (down (B1- T))] G1 lb1 l) (G2 l : left (down (forall S B1-)) -> right J)). + + + +%block cmp-hyp-block : some {A : prop'} {A+ : prop pos} {E : erase-prop A+ A} + block {h : hyp A} {l : left A+} {cmp-hyp/h : cmp-hyp h E l}. +%worlds (sort-block | term-block | erase-atm-block | cmp-hyp-block | rfoc-block | left-block) + (cmp-conc _ _ _) + (cmp-lax _ _ _) + (cmp-affirms _ _ _) + (cmp-lolli _ _ _ _) + (cmp-hyp _ _ _) + (cmp-hyp+ _ _ _). + +%total (D1 D2 D3 D4 H5 H6) + (cmp-conc D1 _ _) + (cmp-lax D2 _ _) + (cmp-affirms D3 _ _) + (cmp-lolli D4 _ _ _) + (cmp-hyp H5 _ _) + (cmp-hyp+ H6 _ _). +``` + +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Hdeyoung/monweakfoc.elf). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-hdeyoung.lf b/new-tests/stelf-output/pages/user-hdeyoung.lf new file mode 100644 index 0000000..9891f05 --- /dev/null +++ b/new-tests/stelf-output/pages/user-hdeyoung.lf @@ -0,0 +1,10 @@ +% +%{! Henry's user page. + +* [User:Hdeyoung/monweakfoc.elf](/wiki/user-hdeyoung-monweakfoc-elf/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Hdeyoung). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-jaked.lf b/new-tests/stelf-output/pages/user-jaked.lf new file mode 100644 index 0000000..3dcc834 --- /dev/null +++ b/new-tests/stelf-output/pages/user-jaked.lf @@ -0,0 +1,8 @@ +% +%{! Hi. I am a 2nd year PhD student at CMU. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:JakeD). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-jcreed-sandbox.lf b/new-tests/stelf-output/pages/user-jcreed-sandbox.lf new file mode 100644 index 0000000..97d8d9b --- /dev/null +++ b/new-tests/stelf-output/pages/user-jcreed-sandbox.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-jcreed.lf b/new-tests/stelf-output/pages/user-jcreed.lf new file mode 100644 index 0000000..e974ded --- /dev/null +++ b/new-tests/stelf-output/pages/user-jcreed.lf @@ -0,0 +1,8 @@ +% +%{! Once upon a time I was a graduate student at CMU. I have a web page at [http://jcreed.org/ jcreed.org]. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Jcreed). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-newtestuser.lf b/new-tests/stelf-output/pages/user-newtestuser.lf new file mode 100644 index 0000000..7612c7a --- /dev/null +++ b/new-tests/stelf-output/pages/user-newtestuser.lf @@ -0,0 +1,8 @@ +% +%{! [Rob](/wiki/user-rsimmons/) created this user to have someone he could use to log in to who only had "generic user" permission. Leave Rob any messages you might have. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:NewTestUser). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-rsimmons-cr2.lf b/new-tests/stelf-output/pages/user-rsimmons-cr2.lf new file mode 100644 index 0000000..969dc0a --- /dev/null +++ b/new-tests/stelf-output/pages/user-rsimmons-cr2.lf @@ -0,0 +1,91 @@ +% +%{! ## Syntax !}% +%sort exp %. +%name exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%{! + +When we use this ``%block``, it expresses that we can be working in a +context with arbitrary expression variables. + +!}% +%block exps [x exp]%. +%worlds (exps) (exp) %. +%{! ## Reduction !}% +%{! + +We can reduce under binders and +reduce both sides of an application "in parallel." If we have a β-redex +``(λx.ea) eb``, then after reducing ``ea`` (with ``x`` free) to +``ea'`` (with ``x`` free) and reducing ``eb`` to ``eb'``, +we can return ``[eb'/x]ea'``, the substitution of ``eb'`` into +``ea'``. When we introduce a new variable, we always add in the fact +that it can evaluate to itself. + +The ``%block`` ``exps_id`` +explicitly states that we will be reducing in a setting +with free variables, with the invariant that every variable is added +with the invariant that it can evaluate to itself. + +!}% +%sort reduce %. +%mode reduce %in %out %. +%block exps_id [x exp] [d reduce x x]%. +%term reduce/lam + %pi (reduce (lam E) (lam E')) + %<- ({x exp} %pi (reduce x x) %-> (reduce (E x) (E' x))) %. +%term reduce/app %pi (reduce (app E1 E2) (app E1' E2')) %<- (reduce E1 E1') %<- (reduce E2 E2') %. +%term reduce/beta + %pi (reduce (app (lam E1) E2) (E1' E2')) + %<- ({x exp} %pi (reduce x x) %-> (reduce (E1 x) (E1' x))) + %<- (reduce E2 E2') %. +%worlds (exps_id) (reduce _ _) %. +%total E (reduce E _) %. +%{! ## Substitution !}% +%{! + +The substitution theorem says that if we have a term ``e`` with +``x`` free that reduces to ``e'`` (with ``x`` still free) +and ``e<sub>arg</sub>`` reduces to ``e'<sub>arg</sub>``, +then ``[e<sub>arg</sub>/x]e`` reduces to ``[e'<sub>arg</sub>/x]e'``. + +The proof is by induction on the structure of the term with the free variable. + +!}% +%sort substitute {x exp} %. +%mode substitute %in %in %out %. +%{! + +We actually need to think about what block this theorem will take place in. +The "variable case" of our lemma is one where the first argument is +``[x: exp] [idx: red x x] y``, where ``y`` is _not_ equal to +``x``. In this case, we will need to insert into the context the lemma +that, if + +!}% +%block exp_subst [x exp] [idx reduce x x] [subst {z exp} {z' exp} {red reduce z z'} substitute ([_] [_] idx) red idx]%. +%term _ substitute ([x] [idx reduce x x] idx) (%the (reduce Earg Earg') D) D %. +%term _ + %pi (substitute ([x] [idx reduce x x] reduce/lam (%the ({y} %pi (reduce y y) %-> (reduce (E x y) (E' x y))) (D x idx))) (%the (reduce Earg Earg') Darg) (%the (reduce (lam ([y] E Earg y)) (lam ([y] E' Earg' y))) (reduce/lam D'))) + %<- ({y} {idy reduce y y} + %pi ({z} {z'} {red_z reduce z z'} substitute ([_] [_] idy) red_z idy) + %-> (substitute ([x] [idx reduce x x] D x idx y idy) Darg (%the (reduce (E Earg y) (E' Earg' y)) (D' y idy)))) %. +%term _ + %pi (substitute ([x] [idx reduce x x] reduce/app (%the (reduce (Eb x) (Eb' x)) (Db x idx)) (%the (reduce (Ea x) (Ea' x)) (Da x idx))) (%the (reduce Earg Earg') Darg) (%the (reduce (app (Ea Earg) (Eb Earg)) (app (Ea' Earg') (Eb' Earg'))) (reduce/app Db' Da'))) + %<- (substitute Da Darg (%the (reduce (Ea Earg) (Ea' Earg')) Da')) + %<- (substitute Db Darg (%the (reduce (Eb Earg) (Eb' Earg')) Db')) %. +%term _ + %pi (substitute ([x] [idx reduce x x] reduce/beta (%the (reduce (Eb x) (Eb' x)) (Db x idx)) (%the ({y} %pi (reduce y y) %-> (reduce (Ea x y) (Ea' x y))) (Da x idx))) (%the (reduce Earg Earg') Darg) (%the (reduce (app (lam (Ea Earg)) (Eb Earg)) (Ea' Earg' (Eb' Earg'))) (reduce/beta Db' Da'))) + %<- ({y} {idy reduce y y} + %pi ({z} {z'} {red_z reduce z z'} substitute ([_] [_] idy) red_z idy) + %-> (substitute ([x] [idx reduce x x] Da x idx y idy) Darg (%the (reduce (Ea Earg y) (Ea' Earg' y)) (Da' y idy)))) + %<- (substitute Db Darg (%the (reduce (Eb Earg) (Eb' Earg')) Db')) %. +%worlds (exp_subst) (substitute _ _ _) %. +%total D (substitute D _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Rsimmons/CR2). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-rsimmons-fillholes-elf.lf b/new-tests/stelf-output/pages/user-rsimmons-fillholes-elf.lf new file mode 100644 index 0000000..fbc2f7d --- /dev/null +++ b/new-tests/stelf-output/pages/user-rsimmons-fillholes-elf.lf @@ -0,0 +1,127 @@ +% +%{! !}% +%sort elem %. +%term z elem %. +%sort list %. +%term nil list %. +%term cons %pi elem %-> list %-> list %. +%block bl_elem [x elem]%. +%sort id %. +%term refl id L L %. +%sort fillx %. +%term fillx1 fillx ([x] nil) nil %. +%term fillx2 %pi (fillx ([x] cons x (L₁ x)) L₂) %<- (fillx ([x] L₁ x) L₂) %. +%term fillx3 %pi (fillx ([x] cons Y (L₁ x)) (cons Y L₂)) %<- (fillx ([x] L₁ x) L₂) %. +%sort fill {D %pi elem %-> list} %. +%term _ fill ([x] nil) fillx1 %. +%term _ %pi (fill ([x] cons x (D x)) (fillx2 D⟦·⟧)) %<- (fill D D⟦·⟧) %. +%term _ %pi (fill ([x] cons Y (D x)) (fillx3 D⟦·⟧)) %<- (fill D D⟦·⟧) %. +%mode fill %in %out %. +%worlds (bl_elem) (fill _ _) %. +%total D (fill D _) %. +%sort fillR %. +%term fillR0 fillR ([x] L x) R (L R) %. +%sort fillR-cons-cong {Rh} %. +%mode fillR-cons-cong %in %in %out %. +%term _ fillR-cons-cong _ fillR0 fillR0 %. +%worlds (bl_elem) (fillR-cons-cong _ _ _) %. +%total {} (fillR-cons-cong _ _ _) %. +%sort void %. +%worlds () (void) %. +%sort fill-nil-contra %. +%mode fill-nil-contra %in %out %. +%worlds (bl_elem) (fill-nil-contra _ _) %. +%total {} (fill-nil-contra _ _) %. +%sort abort/fillR {L} {R} {LR} %. +%mode abort/fillR %in %in %in %in %out %. +%worlds (bl_elem) (abort/fillR _ _ _ _ _) %. +%total {} (abort/fillR _ _ _ _ _) %. +%sort abort/fillx {L} {Lx} %. +%mode abort/fillx %in %in %in %out %. +%worlds (bl_elem) (abort/fillx _ _ _ _) %. +%total {} (abort/fillx _ _ _ _) %. +%sort fillRx2 %. +%mode fillRx2 %in %out %. +%term _ fillRx2 fillR0 fillR0 %. +%worlds (bl_elem) (fillRx2 _ _) %. +%total {} (fillRx2 _ _) %. +%sort thm' {D□R %pi elem %-> list} {D·□ %pi elem %-> list} %. +%inline thm ({D□R %pi elem %-> list} {D·□ %pi elem %-> list} + %pi (fillx ([x] D□R x) D·R) + %-> (fillR ([x] D·□ x) R D·R) + %-> ({D□□ %pi elem %-> elem %-> list} + %pi ({y} fillx ([x] D□□ x y) (D·□ y)) + %-> ({x} fillR ([y] D□□ x y) R (D□R x)) + %-> %type)) [x1] [x2] [x3] [x4] [x5] [x6] [x7] thm' x1 x2 x4 x3 x5 x6 x7 %. +%term _ thm ([x] nil) ([y] nil) fillx1 F1 ([x] [y] nil) ([x] fillx1) ([x] fillR0) %. +%term _ + %pi (thm ([x] cons x (L1 x)) ([y] L2 y) (fillx2 F1) fillR0 ([x] [y] cons x (L x y)) ([y] fillx2 (B y)) ([x] %the (fillR ([y] cons x (L x y)) R (cons x (L1 x))) (C' x))) + %<- (thm L1 L2 F1 fillR0 ([x] [y] L x y) ([y] %the (fillx ([x] L x y) (L2 y)) (B y)) ([x] %the (fillR ([y] L x y) R (L1 x)) (C x))) + %<- ({x elem} fillR-cons-cong x (C x) (C' x)) %. +%term _ + %pi (thm ([x] cons R₁ (L1 x)) ([x] nil) (fillx3 F3) (%the (fillR ([x] nil) R₂ (cons R₁ L2')) F1) ([x] [y] nil) B C) + %<- (fill-nil-contra F1 XX) + %<- ({x} abort/fillx _ _ XX (B x)) + %<- ({x} abort/fillR _ _ _ XX (C x)) %. +%term _ + %pi (thm ([x] cons R₁ (L1 x)) ([x] cons R₁ (L2 x)) (fillx3 F1) (%the (fillR ([x] cons R₁ (L2 x)) R₂ (cons R₁ (L2 R₂))) fillR0) ([x] [y] cons R₁ (L x y)) ([y] fillx3 (F1' y)) ([x] %the (fillR ([y elem] cons R₁ (L x y)) R₂ (cons R₁ (L1 x))) (B' x))) + %<- (thm L1 L2 F1 fillR0 ([x] [y] L x y) ([y] %the (fillx ([x] L x y) (L2 y)) (F1' y)) ([x] %the (fillR ([x1 elem] L x x1) R₂ (L1 x)) (B x))) + %<- ({x elem} fillR-cons-cong R₁ (B x) (B' x)) %. +%term _ + %pi (thm ([x] cons R₁ (L1 x)) ([x] cons x (L2 x)) (fillx3 F1) (%the (fillR ([x] cons x (L2 x)) R₁ (cons R₁ (L2 R₁))) fillR0) ([x] [y] cons y (L x y)) ([y] fillx3 (F1' y)) ([x] %the (fillR ([y] cons y (L x y)) R₁ (cons R₁ (L1 x))) (B' x))) + %<- (thm L1 L2 F1 fillR0 ([x] [y] L x y) ([y] F1' y) ([x] %the (fillR ([y] L x y) R₁ (L1 x)) (B x))) + %<- ({x} fillRx2 (B x) (B' x)) %. +%worlds (bl_elem) (thm' _ _ _ _ _ _ _) %. +%covers thm' %in %in %in %in %out %out %out %. +%mode thm' %in %in %in %in %out %out %out %. +%total D (thm' D _ _ _ _ _ _) %. +%sort com·· {D□□ %pi elem %-> elem %-> list} {y} %. +%term _ com·· ([x] [y] nil) ([y] fillx1) ([x] fillx1) fillx1 fillx1 %. +%term _ + %pi (com·· ([x] [y] cons E (L x y)) ([y] fillx3 (Fx y)) ([x] fillx3 (Fy x)) (fillx3 Fxy) (fillx3 Fyx)) + %<- (com·· L Fx Fy Fxy Fyx) %. +%term _ + %pi (com·· ([x] [y] cons x (L x y)) ([y] fillx2 (Fx y)) ([x] fillx3 (Fy x)) Fxy (fillx2 Fyx)) + %<- (com·· L Fx Fy Fxy Fyx) %. +%term _ + %pi (com·· ([x] [y] cons y (L x y)) ([y] fillx3 (Fx y)) ([x] fillx2 (Fy x)) (fillx2 Fxy) Fyx) + %<- (com·· L Fx Fy Fxy Fyx) %. +%mode com·· %in %in %in %in %out %. +%worlds (bl_elem) (com·· _ _ _ _ _) %. +%total T (com·· T _ _ _ _) %. +%sort comR· {D□□ %pi elem %-> elem %-> list} {x} %. +%term _ comR· ([x] [y] nil) ([x] fillx1) fillx1 %. +%term _ + %pi (comR· ([x] [y] cons E (L x y)) ([x] fillx3 (Fy x)) (fillx3 Fxy)) + %<- (comR· L Fy Fxy) %. +%term _ + %pi (comR· ([x] [y] cons x (L x y)) ([x] fillx3 (Fy x)) (fillx3 Fxy)) + %<- (comR· L Fy Fxy) %. +%term _ + %pi (comR· ([x] [y] cons y (L x y)) ([x] fillx2 (Fy x)) (fillx2 Fxy)) + %<- (comR· L Fy Fxy) %. +%mode {%in D□· %pi elem %-> list} {%in A %pi elem %-> elem %-> list} {%in R elem} {%in C {x elem} fillx ([y elem] A x y) (D□· x)} {%out E fillx ([y elem] A R y) (D□· R)} comR· A C E %. +%worlds (bl_elem) (comR· _ _ _) %. +%total T (comR· T _ _) %. +%sort filleq {R₁} {x elem} %. +%term _ filleq _ ([x] fillR0) fillR0 %. +%worlds (bl_elem) (filleq _ _ _) %. +%mode filleq %in %in %out %. +%total {} (filleq _ _ _) %. +%sort holeswap {R₁} {R₂} {D□R %pi elem %-> list} {D·□ %pi elem %-> list} %. +%term _ + %pi (holeswap R₁ R₂ D□R D·□ fillR0 D□R⟦x⟧ fillR0 D·□⟦y⟧ D□· ([y] D□□ R₁ y) D□·⟦x⟧ fillR0 DR□⟦y⟧ D□R⟦R⟧) + %<- (thm' D□R D·□ fillR0 D□R⟦x⟧ D□□ ([y] %the (fillx ([x] D□□ x y) (D·□ y)) (D□□⟦x⟧ y)) ([x] %the (fillR ([y] D□□ x y) R₂ (D□R x)) (D□□⟦R⟧ x))) + %<- ({x} fill ([y] D□□ x y) (%the (fillx ([y] D□□ x y) (D□· x)) (D□□⟦y⟧ x))) + %<- (com·· D□□ D□□⟦x⟧ D□□⟦y⟧ D·□⟦y⟧ D□·⟦x⟧) + %<- (comR· D□□ D□□⟦y⟧ DR□⟦y⟧) + %<- (filleq _ D□□⟦R⟧ D□R⟦R⟧) %. +%worlds (bl_elem) (holeswap _ _ _ _ _ _ _ _ _ _ _ _ _ _) %. +%mode holeswap %in %in %in %in %in %in %in %in %out %out %out %out %out %out %. +%total {} (holeswap _ _ _ _ _ _ _ _ _ _ _ _ _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Rsimmons/fillholes.elf). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-rsimmons-fluid-binding.lf b/new-tests/stelf-output/pages/user-rsimmons-fluid-binding.lf new file mode 100644 index 0000000..497f0e3 --- /dev/null +++ b/new-tests/stelf-output/pages/user-rsimmons-fluid-binding.lf @@ -0,0 +1,755 @@ +% +%{! ## Fluid variable sets !}% +%sort uninhabited %. +%sort fluid %. +%sort fluids %. +%term # fluids %. +%term , %pi fluid %-> fluids %-> fluids %. +%prec %right 10 , %. +%{! ### Individual fluid variables !}% +%{! We treat typing rules as if there are *no* concrete fluid variables, +but we have to define specific fluid variables for the dynamic semantics. !}% +%term a fluid %. +%term b fluid %. +%term c fluid %. +%term d fluid %. +%block bfluid [f fluid]%. +%sort eqfluid %. +%term eqfluid/refl eqfluid A A %. +%sort neqfluid %. +%term neqfluid/ab neqfluid a b %. +%term neqfluid/ac neqfluid a c %. +%term neqfluid/ad neqfluid a d %. +%term neqfluid/ba neqfluid b a %. +%term neqfluid/bc neqfluid b c %. +%term neqfluid/bd neqfluid b d %. +%term neqfluid/ca neqfluid c a %. +%term neqfluid/cb neqfluid c b %. +%term neqfluid/cd neqfluid c d %. +%term neqfluid/da neqfluid d a %. +%term neqfluid/db neqfluid d b %. +%term neqfluid/dc neqfluid d c %. +%sort decidable-fluid %. +%term decidable-fluid/eq decidable-fluid A A %. +%term decidable-fluid/neq %pi (neqfluid A B) %-> (decidable-fluid A B) %. +%sort eq-or-neq-total {A} {B} %. +%term _ eq-or-neq-total a a decidable-fluid/eq %. +%term _ eq-or-neq-total a b (decidable-fluid/neq neqfluid/ab) %. +%term _ eq-or-neq-total a c (decidable-fluid/neq neqfluid/ac) %. +%term _ eq-or-neq-total a d (decidable-fluid/neq neqfluid/ad) %. +%term _ eq-or-neq-total b a (decidable-fluid/neq neqfluid/ba) %. +%term _ eq-or-neq-total b b decidable-fluid/eq %. +%term _ eq-or-neq-total b c (decidable-fluid/neq neqfluid/bc) %. +%term _ eq-or-neq-total b d (decidable-fluid/neq neqfluid/bd) %. +%term _ eq-or-neq-total c a (decidable-fluid/neq neqfluid/ca) %. +%term _ eq-or-neq-total c b (decidable-fluid/neq neqfluid/cb) %. +%term _ eq-or-neq-total c c decidable-fluid/eq %. +%term _ eq-or-neq-total c d (decidable-fluid/neq neqfluid/cd) %. +%term _ eq-or-neq-total d a (decidable-fluid/neq neqfluid/da) %. +%term _ eq-or-neq-total d b (decidable-fluid/neq neqfluid/db) %. +%term _ eq-or-neq-total d c (decidable-fluid/neq neqfluid/dc) %. +%term _ eq-or-neq-total d d decidable-fluid/eq %. +%mode eq-or-neq-total %in %in %out %. +%worlds () (eq-or-neq-total _ _ _) %. +%total {} (eq-or-neq-total _ _ _) %. +%sort exclusive-fluid %. +%mode exclusive-fluid %in %in %out %. +%worlds () (exclusive-fluid _ _ _) %. +%total {} (exclusive-fluid _ _ _) %. +%{! ### Operations on sets of fluid variables !}% +%sort in %. +%term in/z in A (A , As) %. +%term in/s %pi (in A As) %-> (in A (B , As)) %. +%sort subset %. +%term subset/z subset # As %. +%term subset/s %pi (in A Bs) %-> (subset As Bs) %-> (subset (A , As) Bs) %. +%sort merge %. +%term merge/z merge # Bs Bs %. +%term merge/s %pi (merge (A , As) Bs (A , Cs)) %<- (merge As Bs Cs) %. +%{! ### Structural properties !}% +%sort in-subset %. +%term _ in-subset in/z (%the (subset (A , Bs) Cs) (subset/s In Sub)) In %. +%term _ + %pi (in-subset (in/s (%the (in A Bs) In)) (%the (subset (B , Bs) Cs) (subset/s In' Sub)) In'') + %<- (in-subset In Sub (%the (in A Cs) In'')) %. +%mode in-subset %in %in %out %. +%worlds (bfluid) (in-subset _ _ _) %. +%total T (in-subset T _ _) %. +%sort in-merge1 %. +%term _ in-merge1 in/z (merge/s _) in/z %. +%term _ %pi (in-merge1 (in/s In) (merge/s Merge) (in/s In')) %<- (in-merge1 In Merge In') %. +%mode in-merge1 %in %in %out %. +%worlds () (in-merge1 _ _ _) %. +%total T (in-merge1 _ T _) %. +%sort in-merge2 %. +%term _ in-merge2 In merge/z In %. +%term _ %pi (in-merge2 In (merge/s Merge) (in/s In')) %<- (in-merge2 In Merge In') %. +%mode in-merge2 %in %in %out %. +%worlds () (in-merge2 _ _ _) %. +%total T (in-merge2 _ T _) %. +%sort eq-or-in %. +%term eq-or-in/eq eq-or-in A A Cs %. +%term eq-or-in/in %pi (in A Cs) %-> (eq-or-in A B Cs) %. +% exclude : {A}{B}{Cs} uninhabited -> eq-or-in A B Cs -> type. +% mode exclude +A +B +C +D -E. +% worlds (bfluid) (exclude _ _ _ _ _). +% total {} (exclude _ _ _ _ _). +% +% in-inversion' : decidable-fluid A B -> in A (B , Cs) -> eq-or-in A B Cs -> type. +% - : in-inversion' _ in/z eq-or-in/eq. +% - : in-inversion' decidable-fluid/eq in-s D +% <- exclusive-fluid +%sort in-inversion %. +%term _ in-inversion in/z eq-or-in/eq %. +%term _ in-inversion (in/s Sub) (eq-or-in/in Sub) %. +%mode in-inversion %in %out %. +%worlds (bfluid) (in-inversion _ _) %. +%total {} (in-inversion _ _) %. +%sort subset-trans %. +%term _ subset-trans subset/z _ subset/z %. +%term _ + %pi (subset-trans (subset/s In Sub) Sub' (subset/s In Sub'')) + %<- (subset-trans Sub Sub' (%the (subset FS HS) Sub'')) %. +%term _ + %pi (subset-trans (subset/s (%the (in A Bs) In) (%the (subset As Bs) Sub)) (%the (subset Bs Cs) Sub') (subset/s In' Sub'')) + %<- (subset-trans Sub Sub' (%the (subset As Cs) Sub'')) + %<- (in-subset In Sub' (%the (in A Cs) In')) %. +%mode subset-trans %in %in %out %. +%worlds (bfluid) (subset-trans _ _ _) %. +%total T (subset-trans T _ _) %. +%sort subset-weaken {C} %. +%term _ subset-weaken C subset/z subset/z %. +%term _ + %pi (subset-weaken C (%the (subset (B , Bs) Cs) (subset/s In Sub)) (%the (subset (B , Bs) (C , Cs)) (subset/s (in/s In) Sub'))) + %<- (subset-weaken C Sub (%the (subset Bs (C , Cs)) Sub')) %. +%mode subset-weaken %in %in %out %. +%worlds (bfluid) (subset-weaken _ _ _) %. +%total T (subset-weaken _ T _) %. +%sort subset-refl {FS} %. +%term _ subset-refl # subset/z %. +%term _ + %pi (subset-refl (A , As) (subset/s in/z Sub')) + %<- (subset-refl As (%the (subset As As) Sub)) + %<- (subset-weaken A Sub (%the (subset As (A , As)) Sub')) %. +%mode subset-refl %in %out %. +%worlds (bfluid) (subset-refl _ _) %. +%total T (subset-refl T _) %. +%sort merge-subset1 %. +%term _ merge-subset1 merge/z subset/z %. +%term _ + %pi (merge-subset1 (%the (merge (A , As) Bs (A , Cs)) (merge/s Merge)) (subset/s in/z Sub')) + %<- (merge-subset1 Merge Sub) + %<- (subset-weaken A Sub Sub') %. +%mode merge-subset1 %in %out %. +%worlds (bfluid) (merge-subset1 _ _) %. +%total T (merge-subset1 T _) %. +%{! ## Syntax !}% +%{! ### Types !}% +%sort ty %. +%term ty/nat ty %. +% nat +%term ty/arrow %pi ty %-> ty %-> ty %. +% tau1 tau2 +%term ty/mayuse %pi ty %-> fluids %-> ty %. +% [C] tau +%{! #### Type equality !}% +%sort eqty %. +%term eqty/refl eqty T T %. +%{! #### Individual fluid types !}% +%sort fluidty %. +%block ftyblock {T ty} [f fluid] [d fluidty f T]%. +%term fluidty/a fluidty a ty/nat %. +%term fluidty/b fluidty b (ty/arrow ty/nat (ty/mayuse ty/nat (a , c , #))) %. +%term fluidty/c fluidty c ty/nat %. +%term fluidty/d fluidty d (ty/arrow ty/nat ty/nat) %. +%sort fluidty-uniq %. +%term _ fluidty-uniq fluidty/a fluidty/a eqty/refl %. +%term _ fluidty-uniq fluidty/b fluidty/b eqty/refl %. +%term _ fluidty-uniq fluidty/c fluidty/c eqty/refl %. +%term _ fluidty-uniq fluidty/d fluidty/d eqty/refl %. +%mode fluidty-uniq %in %in %out %. +%worlds () (fluidty-uniq _ _ _) %. +%total {} (fluidty-uniq _ _ _) %. +%{! ### Terms !}% +%sort tm %. +%term tm/lam %pi ty %-> (%pi tm %-> tm) %-> tm %. +%term tm/app %pi tm %-> tm %-> tm %. +%term tm/z tm %. +%term tm/s %pi tm %-> tm %. +%term tm/if %pi tm %-> tm %-> (%pi tm %-> tm) %-> tm %. +%term tm/fix %pi ty %-> (%pi tm %-> tm) %-> tm %. +%term tm/access %pi fluid %-> tm %. +%term tm/fluid-let %pi fluid %-> tm %-> tm %-> tm %. +%term tm/susp %pi tm %-> tm %. +%term tm/unsusp %pi tm %-> tm %. +%define tm/let (%pi tm %-> ty %-> (%pi tm %-> tm) %-> tm) [e] [t] [f] tm/app (tm/lam t f) e %. +%{! #### Term equality !}% +%sort eqtm %. +%term eqtm/refl eqtm E E %. +%{! ### Values !}% +%sort v %. +%term v/lam v (tm/lam T ([x] E x)) %. +%term v/z v tm/z %. +%term v/s %pi (v E) %-> (v (tm/s E)) %. +%term v/susp v (tm/susp E) %. +%{! ### Frames and stacks !}% +%sort frame %. +%scope frame %sort - %. +%term -- -- %. +%term f/app1 %pi -- %-> tm %-> frame %. +%term f/app2 {t} %pi (v t) %-> -- %-> frame %. +%term f/s %pi -- %-> frame %. +%term f/if %pi -- %-> tm %-> (%pi tm %-> tm) %-> frame %. +%term f/fluid-let %pi fluid %-> -- %-> tm %-> frame %. +%term f/fluid-bind %pi fluid %-> ({t} %pi (v t) %-> frame) %. +%term f/unsusp %pi -- %-> frame %. +%sort stack %. +%term emp stack %. +%term ; %pi frame %-> stack %-> stack %. +%prec %right 10 ; %. +%{! ### Machine States !}% +%sort state %. +%term >> %pi stack %-> tm %-> state %. +%term << %pi stack %-> ({t} %pi (v t) %-> state) %. +%{! ## Static semantics !}% +%sort of %. +% May access the mentioned fluids +%term of/weaken %pi (of E T Bs) %-> (subset Bs Cs) %-> (of E T Cs) %. +%term of/lam + %pi ({x} %pi (of x T #) %-> (of (E x) T' #)) + %-> (of (tm/lam T E) (ty/arrow T T') #) %. +%term of/app %pi (of E1 (ty/arrow T T') Bs) %-> (of E2 T Bs) %-> (of (tm/app E1 E2) T' Bs) %. +%term of/z of tm/z ty/nat # %. +%term of/s %pi (of E ty/nat Bs) %-> (of (tm/s E) ty/nat Bs) %. +%term of/if + %pi (of E ty/nat Bs) + %-> (of E1 T Bs) + %-> ({x} %pi (of x ty/nat #) %-> (of (E2 x) T Bs)) + %-> (of (tm/if E E1 ([x] E2 x)) T Bs) %. +%term of/fix %pi ({x} %pi (of x T #) %-> (of (E x) T #)) %-> (of (tm/fix T ([x] E x)) T #) %. +%term of/access %pi (fluidty A T) %-> (of (tm/access A) T (A , #)) %. +%term of/fluid-let + %pi (fluidty A Tf) + %-> (of E1 Tf Bs) + %-> (of E2 T (A , Bs)) + %-> (of (tm/fluid-let A E1 E2) T Bs) %. +%term of/susp %pi (of E T Bs) %-> (of (tm/susp E) (ty/mayuse T Bs) #) %. +%term of/unsusp %pi (merge As Bs Cs) %-> (of E (ty/mayuse T As) Bs) %-> (of (tm/unsusp E) T Cs) %. +%sort offrame %. +%term offrame/app1 %pi (of E2 T' Bs) %-> (offrame (f/app1 --- E2) (ty/arrow T' T) Bs T Bs) %. +%term offrame/app2 %pi (of E1 (ty/arrow T' T) #) %-> (offrame (f/app2 E1 V ---) T' Bs T Bs) %. +%term offrame/s offrame (f/s ---) ty/nat Bs ty/nat Bs %. +%term offrame/if + %pi (of E1 T Bs) + %-> ({x} %pi (of x ty/nat #) %-> (of (E2 x) T Bs)) + %-> (offrame (f/if --- E1 ([x] E2 x)) ty/nat Bs T Bs) %. +%term offrame/fluid-let + %pi (fluidty A T) + %-> (of E T' (A , Bs)) + %-> (offrame (f/fluid-let A --- E) T Bs T' Bs) %. +%term offrame/fluid-bind + %pi (fluidty A Tf) + %-> (of E Tf #) + %-> (offrame (f/fluid-bind A E Dv) T (A , Bs) T Bs) %. +%term offrame/unsusp %pi (merge As Bs Cs) %-> (offrame (f/unsusp ---) (ty/mayuse T As) Bs T Cs) %. +%sort ofstack %. +%term ofstack/weaken %pi (ofstack K T Cs) %-> (subset Bs Cs) %-> (ofstack K T Bs) %. +%term ofstack/emp ofstack emp T # %. +%term ofstack/frame %pi (offrame F Bs T Bs' Cs') %-> (ofstack K Bs' Cs') %-> (ofstack (F ; K) Bs T) %. +%sort ok %. +%term ok>> %pi (ofstack K T Bs) %-> (of E T Bs) %-> (ok (>> K E)) %. +%term ok<< %pi (ofstack K T Bs) %-> (of E T Bs) %-> ({V v E} ok (<< K E V)) %. +%{! ## Dynamic semantics !}% +%{! ### Deep binding lookup !}% +%sort lookup %. +%term lookup/app1 %pi (lookup F K Val Dv) %-> (lookup F (f/app1 --- E2 ; K) Val Dv) %. +%term lookup/app2 %pi (lookup F K Val Dv) %-> (lookup F (f/app2 E1 DV --- ; K) Val Dv) %. +%term lookup/s %pi (lookup F K Val Dv) %-> (lookup F (f/s --- ; K) Val Dv) %. +%term lookup/if %pi (lookup F K Val Dv) %-> (lookup F (f/if --- E1 E2 ; K) Val Dv) %. +%term lookup/flet %pi (lookup F K Val Dv) %-> (lookup F (f/fluid-let A --- E2 ; K) Val Dv) %. +%term lookup/ususp %pi (lookup F K Val Dv) %-> (lookup F (f/unsusp --- ; K) Val Dv) %. +%term lookup/hit %pi (eqfluid A B) %-> (lookup A (f/fluid-bind B Val Dv ; K) Val Dv) %. +%term lookup/miss + %pi (neqfluid A B) + %-> (lookup A K Val Dv) + %-> (lookup A (f/fluid-bind B _ _ ; K) Val Dv) %. +%{! ### Small-step evaluation !}% +%sort step %. +%term step/lam step (>> K (tm/lam T ([x] E x))) (<< K (tm/lam T ([x] E x)) v/lam) %. +%term step/app1 step (>> K (tm/app E1 E2)) (>> (f/app1 --- E2 ; K) E1) %. +%term step/app2 step (<< (f/app1 --- E2 ; K) V DV) (>> (f/app2 V DV --- ; K) E2) %. +%term step/appr step (<< (f/app2 (tm/lam T E) DV --- ; K) V _) (>> K (E V)) %. +%term step/z step (>> K tm/z) (<< K tm/z v/z) %. +%term step/s step (>> K (tm/s E)) (>> (f/s --- ; K) E) %. +%term step/sr step (<< (f/s --- ; K) V DV) (<< K (tm/s V) (v/s DV)) %. +%term step/if step (>> K (tm/if E E1 E2)) (>> (f/if --- E1 E2 ; K) E) %. +%term step/ifz step (<< (f/if --- E1 E2 ; K) tm/z _) (>> K E1) %. +%term step/ifs step (<< (f/if --- E1 E2 ; K) (tm/s E) _) (>> K (E2 E)) %. +%term step/fix step (>> K (tm/fix T ([x] E x))) (>> K (E (tm/fix T ([x] E x)))) %. +%term step/access %pi (lookup A K V Dv) %-> (step (>> K (tm/access A)) (<< K V Dv)) %. +%term step/flet1 step (>> K (tm/fluid-let A E1 E2)) (>> (f/fluid-let A --- E2 ; K) E1) %. +%term step/flet2 step (<< (f/fluid-let A --- E2 ; K) V DV) (>> (f/fluid-bind A V DV ; K) E2) %. +%term step/fletr step (<< (f/fluid-bind A Vbound DVbound ; K) V DV) (<< K V DV) %. +%term step/susp step (>> K (tm/susp E)) (<< K (tm/susp E) v/susp) %. +%term step/ususp step (>> K (tm/unsusp E)) (>> (f/unsusp --- ; K) E) %. +%term step/ususpr step (<< (f/unsusp --- ; K) (tm/susp E) _) (>> K E) %. +%{! ## Progress !}% +%{! ### Not stuck !}% +%sort nstuck %. +%term nstuck/step %pi (step E E') %-> (nstuck E) %. +%term nstuck/final nstuck (<< emp V DV) %. +%{! ### First deep stack access theorem !}% +%sort access-stack %. +%mode access-stack %in %in %out %out %out %. +%sort read-stack %. +%mode read-stack %in %in %in %in %out %out %out %. +%term _ read-stack eq-or-in/eq _ (offrame/fluid-bind (%the (fluidty B Tf) Dfluid) (%the (of E Tf #) Dval)) _ (lookup/hit eqfluid/refl) Dfluid Dval %. +%term _ read-stack _ decidable-fluid/eq (offrame/fluid-bind (%the (fluidty B Tf) Dfluid) (%the (of E Tf #) Dval)) _ (lookup/hit eqfluid/refl) Dfluid Dval %. +%term _ + %pi (read-stack (eq-or-in/in In) (decidable-fluid/neq Neq) _ Dstack (lookup/miss Neq Look) Dfluid Dval) + %<- (access-stack In Dstack Look Dfluid Dval) %. +%term _ + %pi (access-stack (%the (in A Bs) In) (ofstack/weaken Dt (%the (subset Bs Cs) Sub)) Lookup Df Dvt) + %<- (in-subset In Sub In') + %<- (access-stack In' Dt Lookup Df Dvt) %. +%term _ + %pi (access-stack In (ofstack/frame (offrame/app1 _) Dt) (lookup/app1 Look) Df Dvt) + %<- (access-stack In Dt Look Df Dvt) %. +%term _ + %pi (access-stack In (ofstack/frame (offrame/app2 _) Dt) (lookup/app2 Look) Df Dvt) + %<- (access-stack In Dt Look Df Dvt) %. +%term _ + %pi (access-stack In (ofstack/frame offrame/s Dt) (lookup/s Look) Df Dvt) + %<- (access-stack In Dt Look Df Dvt) %. +%term _ + %pi (access-stack In (ofstack/frame (offrame/if _ _) Dt) (lookup/if Look) Df Dvt) + %<- (access-stack In Dt Look Df Dvt) %. +%term _ + %pi (access-stack In (ofstack/frame (offrame/fluid-let _ _) Dt) (lookup/flet Look) Df Dvt) + %<- (access-stack In Dt Look Df Dvt) %. +%term _ + %pi (access-stack (%the (in A (B , Cs)) In) (ofstack/frame Dframe Dstack) Look Df Dvt) + %<- (in-inversion In (%the (eq-or-in A B Cs) Deq-or-in)) + %<- (eq-or-neq-total A B (%the (decidable-fluid A B) Deq-or-neq)) + %<- (read-stack Deq-or-in Deq-or-neq Dframe Dstack (%the (lookup A (f/fluid-bind B _ _ ; _) _ _) Look) (%the (fluidty A Tf) Df) (%the (of E Tf #) Dvt)) %. +%term _ + %pi (access-stack In (ofstack/frame (offrame/unsusp (%the (merge As Bs Cs) Merge)) (%the (ofstack K T Cs) Dt)) (lookup/ususp Look) Df Dvt) + %<- (in-merge2 In Merge In') + %<- (access-stack In' (%the (ofstack K T Cs) Dt) Look Df Dvt) %. +%worlds () (access-stack _ _ _ _ _) (read-stack _ _ _ _ _ _ _) %. +%total (T S) (access-stack _ T _ _ _) (read-stack _ _ _ S _ _ _) %. +%{! ### Canonical Forms !}% +%sort canon-arrow %. +%term _ %pi (canon-arrow (of/weaken T _) Dv Eq) %<- (canon-arrow T Dv Eq) %. +%term _ canon-arrow (of/lam _) v/lam eqtm/refl %. +%mode canon-arrow %in %in %out %. +%worlds () (canon-arrow _ _ _) %. +%total T (canon-arrow T _ _) %. +%sort z-or-s %. +%term z-or-s/z z-or-s tm/z %. +%term z-or-s/s z-or-s (tm/s E) %. +%sort canon-nat %. +%term _ %pi (canon-nat (of/weaken T _) Dv Eq) %<- (canon-nat T Dv Eq) %. +%term _ canon-nat of/z v/z z-or-s/z %. +%term _ canon-nat (of/s _) (v/s _) z-or-s/s %. +%mode canon-nat %in %in %out %. +%worlds () (canon-nat _ _ _) %. +%total T (canon-nat T _ _) %. +%sort canon-mayuse %. +%term _ %pi (canon-mayuse (of/weaken T _) Dv Eq) %<- (canon-mayuse T Dv Eq) %. +%term _ canon-mayuse (of/susp _) v/susp eqtm/refl %. +%mode canon-mayuse %in %in %out %. +%worlds () (canon-mayuse _ _ _) %. +%total T (canon-mayuse T _ _) %. +%{! ### Progress !}% +%sort progress>> %. +%term _ + %pi (progress>> TK (of/weaken TE Sub) Step) + %<- (progress>> (ofstack/weaken TK Sub) TE Step) %. +%term _ progress>> TK (of/lam _) step/lam %. +%term _ progress>> TK (of/app _ _) step/app1 %. +%term _ progress>> TK of/z step/z %. +%term _ progress>> TK (of/s _) step/s %. +%term _ progress>> TK (of/if _ _ _) step/if %. +%term _ progress>> TK (of/fix _) step/fix %. +%term _ + %pi (progress>> TK (of/access (%the (fluidty A T) TA)) (step/access Look)) + %<- (access-stack in/z TK Look _ _) %. +%term _ progress>> TK (of/fluid-let _ _ _) step/flet1 %. +%term _ progress>> TK (of/susp _) step/susp %. +%term _ progress>> TK (of/unsusp _ _) step/ususp %. +%mode progress>> %in %in %out %. +%worlds () (progress>> _ _ _) %. +%total T (progress>> _ T _) %. +%sort progress<< %. +%mode progress<< %in %in %in %out %. +%term _ + %pi (progress<< (ofstack/weaken TK Sub) TE Dv NS) + %<- (progress<< TK (of/weaken TE Sub) Dv NS) %. +%term _ progress<< ofstack/emp TE Dv nstuck/final %. +%term _ progress<< (ofstack/frame (offrame/app1 TE2) TK) TE1 Dv (nstuck/step step/app2) %. +%sort p1 %. +%term _ p1 _ _ _ _ _ step/appr %. +%mode p1 %in %in %in %in %in %out %. +%worlds () (p1 _ _ _ _ _ _) %. +%total {} (p1 _ _ _ _ _ _) %. +%term _ + %pi (progress<< (ofstack/frame (%the (offrame (f/app2 E1 Dv ---) T' Bs T Bs) (offrame/app2 TE1)) TK) TE2 _ (nstuck/step Step)) + %<- (canon-arrow TE1 Dv Eq) + %<- (p1 Eq _ _ _ _ Step) %. +%term _ progress<< (ofstack/frame offrame/s TK) TE1 Dv (nstuck/step step/sr) %. +%sort p2 %. +%term _ p2 z-or-s/z _ _ _ _ step/ifz %. +%term _ p2 z-or-s/s _ _ _ _ step/ifs %. +%mode p2 %in %in %in %in %in %out %. +%worlds () (p2 _ _ _ _ _ _) %. +%total {} (p2 _ _ _ _ _ _) %. +%term _ + %pi (progress<< (ofstack/frame (offrame/if _ _) TK) TE Dv (nstuck/step Step)) + %<- (canon-nat TE Dv Eq) + %<- (p2 Eq _ _ _ _ Step) %. +%term _ progress<< (ofstack/frame (offrame/fluid-let _ _) TK) TE Dv (nstuck/step step/flet2) %. +%term _ progress<< (ofstack/frame (offrame/fluid-bind _ _) TK) TE Dv (nstuck/step step/fletr) %. +%{! Note: when I write in ``eqtm E (tm/susp E)`` here and then +try to check the case, the error I get is worrisome, occurs-check wise. !}% +%sort p3 %. +%term _ p3 eqtm/refl _ _ step/ususpr %. +%mode p3 %in %in %in %out %. +%worlds () (p3 _ _ _ _) %. +%total {} (p3 _ _ _ _) %. +%term _ + %pi (progress<< (ofstack/frame (offrame/unsusp _) TK) TE Dv (nstuck/step Step)) + %<- (canon-mayuse TE Dv Eq) + %<- (p3 Eq _ _ Step) %. +%worlds () (progress<< _ _ _ _) %. +%total T (progress<< T _ _ _) %. +%sort progress %. +%term _ %pi (progress (ok>> TK TE) (nstuck/step Step)) %<- (progress>> TK TE Step) %. +%term _ %pi (progress (ok<< TK TE Dv) NS) %<- (progress<< TK TE Dv NS) %. +%mode progress %in %out %. +%worlds () (progress _ _) %. +%total {} (progress _ _) %. +%{! ## Preservation !}% +%{! ### Second deep stack access theorem !}% +%sort access-stack' %. +%mode access-stack' %in %in %in %out %out %. +%sort read-stack' %. +%mode read-stack' %in %in %in %in %in %out %out %. +%term _ read-stack' eq-or-in/eq _ (offrame/fluid-bind (%the (fluidty B Tf) Dfluid) (%the (of E Tf #) Dval)) _ (lookup/hit eqfluid/refl) Dfluid Dval %. +%term _ read-stack' _ decidable-fluid/eq (offrame/fluid-bind (%the (fluidty B Tf) Dfluid) (%the (of E Tf #) Dval)) _ (lookup/hit eqfluid/refl) Dfluid Dval %. +%term _ + %pi (read-stack' (eq-or-in/in In) (decidable-fluid/neq Neq) _ Dstack (lookup/miss Neq Look) Dfluid Dval) + %<- (access-stack' In Dstack Look Dfluid Dval) %. +%term _ + %pi (access-stack' (%the (in A Bs) In) (ofstack/weaken Dt (%the (subset Bs Cs) Sub)) Lookup Df Dvt) + %<- (in-subset In Sub In') + %<- (access-stack' In' Dt Lookup Df Dvt) %. +%term _ + %pi (access-stack' In (ofstack/frame (offrame/app1 _) Dt) (lookup/app1 Look) Df Dvt) + %<- (access-stack' In Dt Look Df Dvt) %. +%term _ + %pi (access-stack' In (ofstack/frame (offrame/app2 _) Dt) (lookup/app2 Look) Df Dvt) + %<- (access-stack' In Dt Look Df Dvt) %. +%term _ + %pi (access-stack' In (ofstack/frame offrame/s Dt) (lookup/s Look) Df Dvt) + %<- (access-stack' In Dt Look Df Dvt) %. +%term _ + %pi (access-stack' In (ofstack/frame (offrame/if _ _) Dt) (lookup/if Look) Df Dvt) + %<- (access-stack' In Dt Look Df Dvt) %. +%term _ + %pi (access-stack' In (ofstack/frame (offrame/fluid-let _ _) Dt) (lookup/flet Look) Df Dvt) + %<- (access-stack' In Dt Look Df Dvt) %. +%term _ + %pi (access-stack' (%the (in A (B , Cs)) In) (ofstack/frame Dframe Dstack) Look Df Dvt) + %<- (in-inversion In (%the (eq-or-in A B Cs) Deq-or-in)) + %<- (eq-or-neq-total A B (%the (decidable-fluid A B) Deq-or-neq)) + %<- (read-stack' Deq-or-in Deq-or-neq Dframe Dstack (%the (lookup A (f/fluid-bind B _ _ ; _) _ _) Look) (%the (fluidty A Tf) Df) (%the (of E Tf #) Dvt)) %. +%term _ + %pi (access-stack' In (ofstack/frame (offrame/unsusp (%the (merge As Bs Cs) Merge)) (%the (ofstack K T Cs) Dt)) (lookup/ususp Look) Df Dvt) + %<- (in-merge2 In Merge In') + %<- (access-stack' In' (%the (ofstack K T Cs) Dt) Look Df Dvt) %. +%worlds () (access-stack' _ _ _ _ _) (read-stack' _ _ _ _ _ _ _) %. +%total (T S) (access-stack' _ T _ _ _) (read-stack' _ _ _ S _ _ _) %. +%{! ### Effect-free value theorem !}% +%{! Yipes! This is actually a complicated induction principle, because we need +to induct on two things at once - either the typing derivation or the +value derivation gets smaller with every recursive call. Perhaps this could +be factored out into two non-mutually-recursive lemmas? !}% +%sort effect-value %. +%term _ %pi (effect-value V (of/weaken T Sub) T') %<- (effect-value V T T') %. +%term _ effect-value v/lam T T %. +%term _ effect-value v/susp T T %. +%term _ effect-value v/z T T %. +%term _ %pi (effect-value (v/s V) (of/s T) (of/s T')) %<- (effect-value V T T') %. +%mode effect-value %in %in %out %. +%worlds () (effect-value _ _ _) %. +%total [S T] (effect-value S T _) %. +%{! ### Inversion !}% +%sort inv-lam %. +%term _ %pi (inv-lam (of/weaken T Sub) Eq Tf) %<- (inv-lam T Eq Tf) %. +%term _ inv-lam (of/lam F) eqty/refl F %. +%mode inv-lam %in %out %out %. +%worlds () (inv-lam _ _ _) %. +%total T (inv-lam T _ _) %. +%sort inv-app %. +%term _ + %pi (inv-app (of/weaken T Sub') Sub'' T1 T2) + %<- (inv-app T Sub T1 T2) + %<- (subset-trans Sub Sub' Sub'') %. +%term _ %pi (inv-app (of/app T1 T2) Sub T1 T2) %<- (subset-refl _ Sub) %. +%mode inv-app %in %out %out %out %. +%worlds () (inv-app _ _ _ _) %. +%total T (inv-app T _ _ _) %. +%sort inv-s %. +%term _ + %pi (inv-s (of/weaken T Sub') Sub'' Eq T') + %<- (inv-s T Sub Eq T') + %<- (subset-trans Sub Sub' Sub'') %. +%term _ %pi (inv-s (of/s T) Sub eqty/refl T) %<- (subset-refl _ Sub) %. +%mode inv-s %in %out %out %out %. +%worlds () (inv-s _ _ _ _) %. +%total T (inv-s T _ _ _) %. +%sort inv-if %. +%term _ + %pi (inv-if (of/weaken Tif Sub') Sub'' T T1 T2) + %<- (inv-if Tif Sub T T1 T2) + %<- (subset-trans Sub Sub' Sub'') %. +%term _ %pi (inv-if (of/if T T1 T2) Sub T T1 T2) %<- (subset-refl _ Sub) %. +%mode inv-if %in %out %out %out %out %. +%worlds () (inv-if _ _ _ _ _) %. +%total T (inv-if T _ _ _ _) %. +%sort inv-fix %. +%term _ %pi (inv-fix (of/weaken Tfix Sub) Eq Tfix' Tfn) %<- (inv-fix Tfix Eq Tfix' Tfn) %. +%term _ inv-fix (of/fix T) eqty/refl (of/fix T) T %. +%mode inv-fix %in %out %out %out %. +%worlds () (inv-fix _ _ _ _) %. +%total T (inv-fix T _ _ _) %. +%sort inv-access %. +%term _ + %pi (inv-access (of/weaken Taccess Sub) In' FT) + %<- (inv-access Taccess In FT) + %<- (in-subset In Sub In') %. +%term _ inv-access (of/access FT) in/z FT %. +%mode inv-access %in %out %out %. +%worlds () (inv-access _ _ _) %. +%total T (inv-access T _ _) %. +%sort inv-fluid-let %. +%term _ + %pi (inv-fluid-let (of/weaken Tfluid Sub') Sub'' FT T1 T2) + %<- (inv-fluid-let Tfluid Sub FT T1 T2) + %<- (subset-trans Sub Sub' Sub'') %. +%term _ %pi (inv-fluid-let (of/fluid-let FT T1 T2) Sub FT T1 T2) %<- (subset-refl _ Sub) %. +%mode inv-fluid-let %in %out %out %out %out %. +%worlds () (inv-fluid-let _ _ _ _ _) %. +%total T (inv-fluid-let T _ _ _ _) %. +%sort inv-unsusp %. +%term _ + %pi (inv-unsusp (of/weaken Tun Sub') Sub'' T M) + %<- (inv-unsusp Tun Sub T M) + %<- (subset-trans Sub Sub' Sub'') %. +%term _ %pi (inv-unsusp (of/unsusp Merge T) Sub T Merge) %<- (subset-refl _ Sub) %. +%mode inv-unsusp %in %out %out %out %. +%worlds () (inv-unsusp _ _ _ _) %. +%total T (inv-unsusp T _ _ _) %. +%sort invf-app1 %. +%term _ + %pi (invf-app1 (ofstack/weaken TK (%the (subset As Bs) Sub)) (%the (subset As Cs) Sub'') Eq TK' TE') + %<- (invf-app1 TK (%the (subset Bs Cs) Sub') Eq TK' TE') + %<- (subset-trans Sub Sub' Sub'') %. +%term _ + %pi (invf-app1 (ofstack/frame (offrame/app1 (%the (of E2 T1 Cs) TE)) (%the (ofstack K T2 Cs) TK)) Sub eqty/refl TK TE) + %<- (subset-refl _ Sub) %. +%mode invf-app1 %in %out %out %out %out %. +%worlds () (invf-app1 _ _ _ _ _) %. +%total T (invf-app1 T _ _ _ _) %. +%sort invf-app2 %. +%term _ + %pi (invf-app2 (ofstack/weaken TK Sub) Sub'' TE' TK') + %<- (invf-app2 TK Sub' TE' TK') + %<- (subset-trans Sub Sub' Sub'') %. +%term _ + %pi (invf-app2 (ofstack/frame (offrame/app2 TE) TK) Sub (of/weaken TE subset/z) TK) + %<- (subset-refl _ Sub) %. +%mode invf-app2 %in %out %out %out %. +%worlds () (invf-app2 _ _ _ _) %. +%total T (invf-app2 T _ _ _) %. +%sort invf-s %. +%term _ + %pi (invf-s (ofstack/weaken TK Sub) Sub'' Eq TK') + %<- (invf-s TK Sub' Eq TK') + %<- (subset-trans Sub Sub' Sub'') %. +%term _ + %pi (invf-s (ofstack/frame offrame/s TK) Sub eqty/refl TK) + %<- (subset-refl _ Sub) %. +%mode invf-s %in %out %out %out %. +%worlds () (invf-s _ _ _ _) %. +%total T (invf-s T _ _ _) %. +%sort invf-if %. +%term _ + %pi (invf-if (ofstack/weaken TK Sub) Sub'' Eq TE1 TE2 TK') + %<- (invf-if TK Sub' Eq TE1 TE2 TK') + %<- (subset-trans Sub Sub' Sub'') %. +%term _ + %pi (invf-if (ofstack/frame (offrame/if T1 T2) TK) Sub eqty/refl T1 T2 TK) + %<- (subset-refl _ Sub) %. +%mode invf-if %in %out %out %out %out %out %. +%worlds () (invf-if _ _ _ _ _ _) %. +%total T (invf-if T _ _ _ _ _) %. +%sort invf-fluid-let %. +%term _ + %pi (invf-fluid-let (ofstack/weaken TK Sub) Sub'' TF T2 TK') + %<- (invf-fluid-let TK Sub' TF T2 TK') + %<- (subset-trans Sub Sub' Sub'') %. +%term _ + %pi (invf-fluid-let (ofstack/frame (offrame/fluid-let FT T2) TK) Sub FT T2 TK) + %<- (subset-refl _ Sub) %. +%mode invf-fluid-let %in %out %out %out %out %. +%worlds () (invf-fluid-let _ _ _ _ _) %. +%total T (invf-fluid-let T _ _ _ _) %. +%sort invf-fluid-bind %. +%term _ + %pi (invf-fluid-bind (ofstack/weaken TK Sub) Sub'' TK') + %<- (invf-fluid-bind TK Sub' TK') + %<- (subset-trans Sub Sub' Sub'') %. +%term _ + %pi (invf-fluid-bind (ofstack/frame (offrame/fluid-bind _ _) TK) (subset/s in/z Sub') TK) + %<- (subset-refl _ Sub) + %<- (subset-weaken _ Sub Sub') %. +%mode invf-fluid-bind %in %out %out %. +%worlds () (invf-fluid-bind _ _ _) %. +%total T (invf-fluid-bind T _ _) %. +%{! ### Preservation !}% +%sort preservation %. +%mode preservation %in %in %out %. +%term _ preservation step/lam (ok>> TK TE) (ok<< TK TE v/lam) %. +%term _ + %pi (preservation step/app1 (ok>> TK TE) (ok>> (ofstack/frame (offrame/app1 T2) (ofstack/weaken TK Sub)) T1)) + %<- (inv-app TE Sub T1 T2) %. +%sort p1 %. +%term _ p1 eqty/refl T T %. +%mode p1 %in %in %out %. +%worlds () (p1 _ _ _) %. +%total {} (p1 _ _ _) %. +%term _ + %pi (preservation step/app2 (ok<< (%the (ofstack (f/app1 --- E2 ; K) T Bs) TK) TE Dv) (%the (ok (>> (f/app2 E1 Dv --- ; K) E2)) (ok>> (ofstack/frame (offrame/app2 TE1'') TK') TE2'))) + %<- (invf-app1 TK (%the (subset Bs Cs) Sub) (%the (eqty T (ty/arrow T1 T2)) Eq) (%the (ofstack K T2 Cs) TK') (%the (of E2 T1 Cs) TE2')) + %<- (p1 Eq TE TE1') + %<- (effect-value Dv TE1' (%the (of E1 (ty/arrow T1 T2) #) TE1'')) %. +%sort p3 %. +%term _ p3 eqty/refl TK TE1 TE2 (ok>> TK (of/weaken (TE1 _ TE2) subset/z)) %. +%mode p3 %in %in %in %in %out %. +%worlds () (p3 _ _ _ _ _) %. +%total {} (p3 _ _ _ _ _) %. +%term _ + %pi (preservation step/appr (ok<< TK TE2 Dv) (%the (ok (>> K (E E2))) Ans)) + %<- (effect-value Dv TE2 (%the (of E2 Ta #) TE2')) + %<- (invf-app2 (%the (ofstack (f/app2 (tm/lam T' E) V --- ; K) Ta Bs) TK) (%the (subset Bs Cs) Sub) (%the (of (tm/lam T' E) (ty/arrow Ta Tb) Cs) T1) (%the (ofstack K Tb Cs) TK'')) + %<- (inv-lam T1 (%the (eqty (ty/arrow Ta Tb) (ty/arrow T' T'')) Eq) (%the ({x} %pi (of x T' #) %-> (of (E x) T'' #)) Tfunc)) + %<- (p3 Eq TK'' Tfunc TE2' Ans) %. +%term _ preservation step/z (ok>> TK TE) (ok<< TK TE v/z) %. +%sort p2 %. +%term _ p2 eqty/refl T T %. +%mode p2 %in %in %out %. +%worlds () (p2 _ _ _) %. +%total {} (p2 _ _ _) %. +%term _ + %pi (preservation step/s (ok>> (%the (ofstack E T Bs) TK) TE) (ok>> (ofstack/frame offrame/s (ofstack/weaken TK' Sub)) TE')) + %<- (inv-s TE Sub Eq TE') + %<- (p2 Eq TK TK') %. +%sort p6 %. +%term _ p6 eqty/refl T T %. +%mode p6 %in %in %out %. +%worlds () (p6 _ _ _) %. +%total {} (p6 _ _ _) %. +%term _ + %pi (preservation step/sr (ok<< TK TE Dv) (ok<< TK' (of/weaken (of/s TE') Sub) (v/s Dv))) + %<- (invf-s (%the (ofstack (f/s --- ; K) T Bs) TK) Sub (%the (eqty T ty/nat) Eq) (%the (ofstack K ty/nat Cs) TK')) + %<- (p6 Eq TE TE') %. +%term _ + %pi (preservation step/if (ok>> TK TE) (ok>> (ofstack/frame (offrame/if T1 T2) (ofstack/weaken TK Sub)) T)) + %<- (inv-if TE Sub T T1 T2) %. +%term _ + %pi (preservation step/ifz (ok<< TK TE Dv) (ok>> TK' TE1)) + %<- (invf-if (%the (ofstack (f/if --- E1 E2 ; K) T Bs) TK) (%the (subset Bs Cs) Sub) (%the (eqty T ty/nat) Eq) (%the (of E1 T' Cs) TE1) _ (%the (ofstack K _ Cs) TK')) %. +%term _ + %pi (preservation step/ifs (ok<< TK TE Dv) (ok>> TK' (TE2 _ (of/weaken TN Sub#)))) + %<- (invf-if (%the (ofstack (f/if --- E1 E2 ; K) T Bs) TK) (%the (subset Bs Cs) Sub) (%the (eqty T ty/nat) Eq) _ (%the ({x} %pi (of x ty/nat #) %-> (of (E2 x) T' Cs)) TE2) (%the (ofstack K T' Cs) TK')) + %<- (effect-value Dv TE TE') + %<- (inv-s TE' (%the (subset As #) Sub#) _ (%the (of N ty/nat As) TN)) %. +%sort p6 %. +%term _ p6 eqty/refl T F T F %. +%mode p6 %in %in %in %out %out %. +%worlds () (p6 _ _ _ _ _) %. +%total {} (p6 _ _ _ _ _) %. +%term _ + %pi (preservation step/fix (ok>> TK TE) (ok>> TK (of/weaken (Tfn' _ Tfix') subset/z))) + %<- (inv-fix (%the (of (tm/fix T' F) T Bs) TE) (%the (eqty T T') Eq) (%the _ Tfix) Tfn) + %<- (p6 Eq Tfix Tfn Tfix' Tfn') %. +%sort p5 %. +%term _ p5 eqty/refl T T %. +%mode p5 %in %in %out %. +%worlds () (p5 _ _ _) %. +%total {} (p5 _ _ _) %. +%term _ + %pi (preservation (step/access (%the (lookup A K V Dv) Look)) (ok>> TK TE) (%the (ok (<< K V Dv)) (ok<< TK (of/weaken TE'' subset/z) Dv))) + %<- (inv-access TE (%the (in A Bs) In) (%the (fluidty A T) FT)) + %<- (access-stack' In TK Look (%the (fluidty A T') FT') (%the (of V T' #) TE')) + %<- (fluidty-uniq FT' FT Eq) + %<- (p5 Eq TE' TE'') %. +%term _ + %pi (preservation step/flet1 (ok>> TK TE) (ok>> (ofstack/frame (offrame/fluid-let FT T2) (ofstack/weaken TK Sub)) T1)) + %<- (inv-fluid-let (%the (of (tm/fluid-let A E1 E2) T Cs) TE) (%the (subset Bs Cs) Sub) (%the (fluidty A Tf) FT) (%the (of E1 Tf Bs) T1) (%the (of E2 T (A , Bs)) T2)) %. +%term _ + %pi (preservation step/flet2 (ok<< TK TE Dv) (ok>> (ofstack/frame (offrame/fluid-bind FT TE') TK') T2)) + %<- (invf-fluid-let (%the (ofstack (f/fluid-let A --- E2 ; K) Tf Bs) TK) (%the (subset Bs Cs) Sub) (%the (fluidty A Tf) FT) (%the (of E2 T (A , Cs)) T2) (%the (ofstack K T Cs) TK')) + %<- (effect-value Dv TE (%the (of E1 Tf #) TE')) %. +%term _ + %pi (preservation step/fletr (ok<< TK TE Dv) (ok<< TK' (of/weaken TE' subset/z) Dv)) + %<- (effect-value Dv TE TE') + %<- (invf-fluid-bind TK _ TK') %. +%term _ preservation step/susp (ok>> TK TE) (ok<< TK TE v/susp) %. +%term _ + %pi (preservation step/ususp (ok>> (%the (ofstack K T Ds) TK) (%the (of (tm/unsusp E) T Ds) TE)) (ok>> (ofstack/frame (offrame/unsusp (%the (merge As Bs Cs) Merge)) (ofstack/weaken TK (%the (subset Cs Ds) Sub))) (%the (of E (ty/mayuse T As) Bs) TE'))) + %<- (inv-unsusp TE Sub TE' Merge) %. +%sort inv-susp %. +%term _ %pi (inv-susp (of/weaken Ts Sub) Eq Ts' T) %<- (inv-susp Ts Eq Ts' T) %. +%term _ inv-susp (of/susp T) eqty/refl (of/susp T) T %. +%mode inv-susp %in %out %out %out %. +%worlds () (inv-susp _ _ _ _) %. +%total T (inv-susp T _ _ _) %. +%sort invf-unsusp %. +%term _ + %pi (invf-unsusp (ofstack/weaken TK Sub) Sub'' Eq Merge TK') + %<- (invf-unsusp TK Sub' Eq (%the (merge As Bs Cs) Merge) TK') + %<- (subset-trans Sub Sub' Sub'') %. +%term _ + %pi (invf-unsusp (ofstack/frame (offrame/unsusp (%the (merge As Bs Cs) Merge)) (%the (ofstack K T Cs) TK)) Sub eqty/refl Merge TK) + %<- (subset-refl Bs Sub) %. +%mode invf-unsusp %in %out %out %out %out %. +%worlds () (invf-unsusp _ _ _ _ _) %. +%total T (invf-unsusp T _ _ _ _) %. +%sort p7 %. +%term _ p7 eqty/refl eqty/refl T T %. +%mode p7 %in %in %in %out %. +%worlds () (p7 _ _ _ _) %. +%total {} (p7 _ _ _ _) %. +%sort p8 %. +%term _ p8 eqty/refl eqty/refl T T %. +%mode p8 %in %in %in %out %. +%worlds () (p8 _ _ _ _) %. +%total {} (p8 _ _ _ _) %. +%term _ + %pi (preservation step/ususpr (ok<< (%the (ofstack (f/unsusp --- ; K) T Bs) TK) (%the (of (tm/susp E) T Bs) TE) Dv) (ok>> (ofstack/weaken (%the (ofstack K T' Cs) TK'') (%the (subset As Cs) Sub)) TE')) + %<- (inv-susp TE (%the (eqty T (ty/mayuse T' As)) Eq) _ (%the (of E T' As) TE')) + %<- (invf-unsusp TK (%the (subset Bs B's) Sub') (%the (eqty T (ty/mayuse T'' As')) Eq2) (%the (merge As' B's Cs) Merge) (%the (ofstack K T'' Cs) TK')) + %<- (p7 Eq Eq2 TK' TK'') + %<- (p8 Eq Eq2 Merge Merge') + %<- (merge-subset1 Merge' Sub) %. +%worlds () (preservation _ _ _) %. +%total {} (preservation _ _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Rsimmons/Fluid_binding). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-rsimmons-handled-exceptions.lf b/new-tests/stelf-output/pages/user-rsimmons-handled-exceptions.lf new file mode 100644 index 0000000..9284773 --- /dev/null +++ b/new-tests/stelf-output/pages/user-rsimmons-handled-exceptions.lf @@ -0,0 +1,412 @@ +% +%{! ## Syntax !}% +%{! ### Types !}% +%sort ty %. +%term ty/nat ty %. +% nat +%term ty/arrow %pi ty %-> ty %-> ty %. +% tau1 tau2 +%term ty/mayraise %pi ty %-> ty %. +% [*] tau +%{! We have to pick some type associated with exceptions. +Here, we'll just use nat. !}% +%define ty/exn ty/nat %. +%{! #### Type equality !}% +%sort eqty %. +%term eqty/refl eqty T T %. +%{! ### Terms !}% +%sort tm %. +%term tm/lam %pi ty %-> (%pi tm %-> tm) %-> tm %. +%term tm/app %pi tm %-> tm %-> tm %. +%term tm/z tm %. +%term tm/s %pi tm %-> tm %. +%term tm/if %pi tm %-> tm %-> (%pi tm %-> tm) %-> tm %. +%term tm/fix %pi ty %-> (%pi tm %-> tm) %-> tm %. +%term tm/raise %pi tm %-> tm %. +%term tm/handle %pi tm %-> (%pi tm %-> tm) %-> tm %. +%term tm/susp %pi tm %-> tm %. +%term tm/unsusp %pi tm %-> tm %. +%define tm/let (%pi tm %-> ty %-> (%pi tm %-> tm) %-> tm) [e] [t] [f] tm/app (tm/lam t f) e %. +%{! ## Static semantics !}% +%sort of %. +% Will not throw exceptions +%sort of* %. +% May or may not throw exceptions +%term of*/weaken %pi (of E T) %-> (of* E T) %. +%term of/lam %pi ({x} %pi (of x T) %-> (of (E x) T')) %-> (of (tm/lam T E) (ty/arrow T T')) %. +%term of/app %pi (of E1 (ty/arrow T T')) %-> (of E2 T) %-> (of (tm/app E1 E2) T') %. +%term of/z of tm/z ty/nat %. +%term of/s %pi (of E ty/nat) %-> (of (tm/s E) ty/nat) %. +%term of/if + %pi (of E ty/nat) + %-> (of E1 T) + %-> ({x} %pi (of x ty/nat) %-> (of (E2 x) T)) + %-> (of (tm/if E E1 ([x] E2 x)) T) %. +%term of/fix %pi ({x} %pi (of x T) %-> (of (E x) T)) %-> (of (tm/fix T ([x] E x)) T) %. +%term of/handle + %pi (of* E T) + %-> ({x} %pi (of x ty/exn) %-> (of (E' x) T)) + %-> (of (tm/handle E ([x] E' x)) T) %. +%term of/susp %pi (of* E T) %-> (of (tm/susp E) (ty/mayraise T)) %. +%term of*/app %pi (of* E1 (ty/arrow T T')) %-> (of* E2 T) %-> (of* (tm/app E1 E2) T') %. +%term of*/s %pi (of* E ty/nat) %-> (of* (tm/s E) ty/nat) %. +%term of*/if + %pi (of* E ty/nat) + %-> (of* E1 T) + %-> ({x} %pi (of x ty/nat) %-> (of* (E2 x) T)) + %-> (of* (tm/if E E1 ([x] E2 x)) T) %. +%term of*/raise %pi (of* E ty/exn) %-> (of* (tm/raise E) T) %. +%term of*/handle + %pi (of* E T) + %-> ({x} %pi (of x ty/exn) %-> (of* (E' x) T)) + %-> (of* (tm/handle E ([x] E' x)) T) %. +%term of*/unsusp %pi (of* E (ty/mayraise T)) %-> (of* (tm/unsusp E) T) %. +%{! ## Dynamic semantics !}% +%{! ### Values !}% +%sort v %. +%term v/lam v (tm/lam T ([x] E x)) %. +%term v/z v tm/z %. +%term v/s %pi (v E) %-> (v (tm/s E)) %. +%term v/susp v (tm/susp E) %. +%{! ### Small-step evaluation !}% +%sort step %. +%term step/app1 %pi (step E1 E1') %-> (step (tm/app E1 E2) (tm/app E1' E2)) %. +%term step/app1x %pi (v E) %-> (step (tm/app (tm/raise E) _) (tm/raise E)) %. +%term step/app2 %pi (v E1) %-> (step E2 E2') %-> (step (tm/app E1 E2) (tm/app E1 E2')) %. +%term step/app2x %pi (v E) %-> (v E1) %-> (step (tm/app E1 (tm/raise E)) (tm/raise E)) %. +%term step/appr %pi (v E2) %-> (step (tm/app (tm/lam T ([x] E x)) E2) (E E2)) %. +%term step/s %pi (step E E') %-> (step (tm/s E) (tm/s E')) %. +%term step/sx %pi (v E) %-> (step (tm/s (tm/raise E)) (tm/raise E)) %. +%term step/if %pi (step E E') %-> (step (tm/if E E1 ([x] E2 x)) (tm/if E' E1 ([x] E2 x))) %. +%term step/ifz step (tm/if tm/z E1 ([x] E2 x)) E1 %. +%term step/ifs %pi (v E) %-> (step (tm/if (tm/s E) E1 ([x] E2 x)) (E2 E)) %. +%term step/ifx %pi (v E) %-> (step (tm/if (tm/raise E) E1 ([x] E2 x)) (tm/raise E)) %. +%term step/fix step (tm/fix T ([x] E x)) (E (tm/fix T ([x] E x))) %. +%term step/raise %pi (step E E') %-> (step (tm/raise E) (tm/raise E')) %. +%term step/raisex %pi (v E) %-> (step (tm/raise (tm/raise E)) (tm/raise E)) %. +%term step/handle %pi (step E1 E1') %-> (step (tm/handle E1 ([x] E2 x)) (tm/handle E1' E2)) %. +%term step/handler %pi (v E) %-> (step (tm/handle E _) E) %. +%term step/handlex %pi (v E) %-> (step (tm/handle (tm/raise E) ([x] E2 x)) (E2 E)) %. +%term step/unsusp %pi (step E E') %-> (step (tm/unsusp E) (tm/unsusp E')) %. +%term step/unsuspr step (tm/unsusp (tm/susp E)) E %. +%term step/unsuspx %pi (v E) %-> (step (tm/unsusp (tm/raise E)) (tm/raise E)) %. +%{! ## Progress !}% +%{! ### Not stuck !}% +%sort nstuck %. +%term nstuck/step %pi (step E E') %-> (nstuck E) %. +%term nstuck/value %pi (v E) %-> (nstuck E) %. +%sort nstuck* %. +%term nstuck*/step %pi (step E E') %-> (nstuck* E) %. +%term nstuck*/value %pi (v E) %-> (nstuck* E) %. +%term nstuck*/raise %pi (v E) %-> (nstuck* (tm/raise E)) %. +%sort ns-weaken %. +%term _ ns-weaken (nstuck/step E) (nstuck*/step E) %. +%term _ ns-weaken (nstuck/value V) (nstuck*/value V) %. +%mode ns-weaken %in %out %. +%worlds () (ns-weaken _ _) %. +%total {} (ns-weaken _ _) %. +%{! ### Mobile value !}% +%{! This theorem really isn't as interesting as its name makes +it sound, though it is necessary in one place. Otherwise, +all of the little factoring lemmas are only case analysis, not induction. !}% +%sort mobile %. +%term _ mobile v/lam (of*/weaken (of/lam F)) (of/lam F) %. +%term _ mobile v/z (of*/weaken of/z) of/z %. +%term _ mobile (v/s V) (of*/weaken (of/s S)) (of/s S) %. +%term _ %pi (mobile (v/s V) (of*/s S) (of/s S')) %<- (mobile V S S') %. +%term _ mobile v/susp (of*/weaken (of/susp T)) (of/susp T) %. +%mode mobile %in %in %out %. +%worlds () (mobile _ _ _) %. +%total T (mobile T _ _) %. +%{! ### Progress !}% +%sort progress %. +%sort progress* %. +%mode progress %in %out %. +%mode progress* %in %out %. +%term _ %pi (progress* (of*/weaken T) NS*) %<- (progress T NS) %<- (ns-weaken NS NS*) %. +%term _ progress (of/lam _) (nstuck/value v/lam) %. +%sort p0 %. +%term _ p0 _ (nstuck/step E) _ (nstuck/step (step/app1 E)) %. +%term _ p0 _ (nstuck/value V) (nstuck/step E) (nstuck/step (step/app2 V E)) %. +%term _ p0 (of/lam _) (nstuck/value _) (nstuck/value V) (nstuck/step (step/appr V)) %. +%mode p0 %in %in %in %out %. +%worlds () (p0 _ _ _ _) %. +%total {} (p0 _ _ _ _) %. +%term _ + %pi (progress (of/app T1 T2) NS) + %<- (progress T1 NS1) + %<- (progress T2 NS2) + %<- (p0 T1 NS1 NS2 NS) %. +%term _ progress of/z (nstuck/value v/z) %. +%sort p1 %. +%term _ p1 _ (nstuck/step E) (nstuck/step (step/s E)) %. +%term _ p1 _ (nstuck/value V) (nstuck/value (v/s V)) %. +%mode p1 %in %in %out %. +%worlds () (p1 _ _ _) %. +%total {} (p1 _ _ _) %. +%term _ %pi (progress (of/s T) NS) %<- (progress T NS1) %<- (p1 T NS1 NS) %. +%sort p2 {E1} {E2} %. +%term _ p2 _ _ _ (nstuck/step E) (nstuck/step (step/if E)) %. +%term _ p2 _ _ _ (nstuck/value v/z) (nstuck/step step/ifz) %. +%term _ p2 _ _ _ (nstuck/value (v/s V)) (nstuck/step (step/ifs V)) %. +%mode p2 %in %in %in %in %out %. +%worlds () (p2 _ _ _ _ _) %. +%total {} (p2 _ _ _ _ _) %. +%term _ %pi (progress (of/if T T1 T2) NS) %<- (progress T NS1) %<- (p2 _ _ T NS1 NS) %. +%term _ progress (of/fix T) (nstuck/step step/fix) %. +%sort p3 {E'} %. +%term _ p3 _ _ (nstuck*/step E) (nstuck/step (step/handle E)) %. +%term _ p3 _ _ (nstuck*/value V) (nstuck/step (step/handler V)) %. +%term _ p3 _ _ (nstuck*/raise V) (nstuck/step (step/handlex V)) %. +%mode p3 %in %in %in %out %. +%worlds () (p3 _ _ _ _) %. +%total {} (p3 _ _ _ _) %. +%term _ %pi (progress (of/handle T T2) NS) %<- (progress* T NS1) %<- (p3 _ T NS1 NS) %. +%term _ progress (of/susp _) (nstuck/value v/susp) %. +%sort p9 %. +%term _ p9 _ (nstuck*/step E) _ (nstuck*/step (step/app1 E)) %. +%term _ p9 _ (nstuck*/raise V) _ (nstuck*/step (step/app1x V)) %. +%term _ p9 _ (nstuck*/value V) (nstuck*/step E) (nstuck*/step (step/app2 V E)) %. +%term _ p9 _ (nstuck*/value V') (nstuck*/raise V) (nstuck*/step (step/app2x V V')) %. +%term _ + %pi (p9 (of*/weaken T) (nstuck*/value V1) (nstuck*/value V2) NS*) + %<- (p0 T (nstuck/value V1) (nstuck/value V2) NS) + %<- (ns-weaken NS NS*) %. +%mode p9 %in %in %in %out %. +%worlds () (p9 _ _ _ _) %. +%total {} (p9 _ _ _ _) %. +%term _ + %pi (progress* (of*/app T1 T2) NS) + %<- (progress* T1 NS1) + %<- (progress* T2 NS2) + %<- (p9 T1 NS1 NS2 NS) %. +%sort p5 %. +%term _ p5 _ (nstuck*/step E) (nstuck*/step (step/s E)) %. +%term _ p5 _ (nstuck*/value V) (nstuck*/value (v/s V)) %. +%term _ p5 _ (nstuck*/raise V) (nstuck*/step (step/sx V)) %. +%mode p5 %in %in %out %. +%worlds () (p5 _ _ _) %. +%total {} (p5 _ _ _) %. +%term _ %pi (progress* (of*/s T) NS) %<- (progress* T NS1) %<- (p5 T NS1 NS) %. +%sort p6 {E1} {E2} %. +%term _ p6 _ _ _ (nstuck*/step E) (nstuck*/step (step/if E)) %. +%term _ + %pi (p6 _ _ T* (nstuck*/value V) NS*) + %<- (mobile V T* T) + %<- (p2 _ _ T (nstuck/value V) NS) + %<- (ns-weaken NS NS*) %. +%term _ p6 _ _ _ (nstuck*/raise V) (nstuck*/step (step/ifx V)) %. +%mode p6 %in %in %in %in %out %. +%worlds () (p6 _ _ _ _ _) %. +%total {} (p6 _ _ _ _ _) %. +%term _ %pi (progress* (of*/if T T1 T2) NS) %<- (progress* T NS1) %<- (p6 _ _ T NS1 NS) %. +%sort pA %. +%term _ pA (nstuck*/step E) (nstuck*/step (step/raise E)) %. +%term _ pA (nstuck*/value V) (nstuck*/raise V) %. +%term _ pA (nstuck*/raise V) (nstuck*/step (step/raisex V)) %. +%mode pA %in %out %. +%worlds () (pA _ _) %. +%total {} (pA _ _) %. +%term _ %pi (progress* (of*/raise T) NS) %<- (progress* T NS1) %<- (pA NS1 NS) %. +%term _ + %pi (progress* (of*/handle T T2) NS) + %<- (progress* T NS1) + %<- (p3 _ T NS1 NS*) + %<- (ns-weaken NS* NS) %. +%sort p7 %. +%term _ p7 _ _ step/unsuspr %. +%mode p7 %in %in %out %. +%worlds () (p7 _ _ _) %. +%total {} (p7 _ _ _) %. +%sort p8 %. +%term _ p8 _ (nstuck*/step E) (nstuck*/step (step/unsusp E)) %. +%term _ %pi (p8 (of*/weaken T) (nstuck*/value V) (nstuck*/step E)) %<- (p7 V T E) %. +%term _ p8 _ (nstuck*/raise V) (nstuck*/step (step/unsuspx V)) %. +%mode p8 %in %in %out %. +%worlds () (p8 _ _ _) %. +%total {} (p8 _ _ _) %. +%term _ %pi (progress* (of*/unsusp T) NS) %<- (progress* T NS1) %<- (p8 T NS1 NS) %. +%worlds () (progress _ _) (progress* _ _) %. +%total (T T*) (progress T _) (progress* T* _) %. +%{! ## Preservation !}% +%{! ### Inversion !}% +%sort inv-lam %. +%term _ inv-lam (of*/weaken (of/lam F)) F eqty/refl %. +%mode inv-lam %in %out %out %. +%worlds () (inv-lam _ _ _) %. +%total {} (inv-lam _ _ _) %. +%sort inv-app %. +%term _ inv-app (of*/app T1 T2) T1 T2 %. +%term _ inv-app (of*/weaken (of/app T1 T2)) (of*/weaken T1) (of*/weaken T2) %. +%mode inv-app %in %out %out %. +%worlds () (inv-app _ _ _) %. +%total {} (inv-app _ _ _) %. +%sort inv-s %. +%term _ inv-s (of*/s T) T eqty/refl %. +%term _ inv-s (of*/weaken (of/s T)) (of*/weaken T) eqty/refl %. +%mode inv-s %in %out %out %. +%worlds () (inv-s _ _ _) %. +%total {} (inv-s _ _ _) %. +%sort inv-if %. +%term _ inv-if (of*/if T T1 T2) T T1 T2 %. +%term _ inv-if (of*/weaken (of/if T T1 T2)) (of*/weaken T) (of*/weaken T1) ([x] [d] of*/weaken (T2 x d)) %. +%mode inv-if %in %out %out %out %. +%worlds () (inv-if _ _ _ _) %. +%total {} (inv-if _ _ _ _) %. +%sort inv-raise %. +%term _ inv-raise (of*/raise T) T %. +%mode inv-raise %in %out %. +%worlds () (inv-raise _ _) %. +%total {} (inv-raise _ _) %. +%sort inv-handle %. +%term _ inv-handle (of*/handle T H) T H %. +%term _ inv-handle (of*/weaken (of/handle T H)) T ([x] [d] of*/weaken (H x d)) %. +%mode inv-handle %in %out %out %. +%worlds () (inv-handle _ _ _) %. +%total {} (inv-handle _ _ _) %. +%sort inv-susp %. +%term _ inv-susp (of*/weaken (of/susp T)) T eqty/refl %. +%mode inv-susp %in %out %out %. +%worlds () (inv-susp _ _ _) %. +%total {} (inv-susp _ _ _) %. +%sort inv-unsusp %. +%term _ inv-unsusp (of*/unsusp T) T %. +%mode inv-unsusp %in %out %. +%worlds () (inv-unsusp _ _) %. +%total {} (inv-unsusp _ _) %. +%{! ### Preservation !}% +%sort preservation %. +%sort preservation* %. +%mode preservation %in %in %out %. +%mode preservation* %in %in %out %. +%{! #### Application !}% +%term _ + %pi (preservation (step/app1 E) (of/app T1 T2) (of/app T1' T2)) + %<- (preservation E T1 T1') %. +%term _ + %pi (preservation (step/app2 V E) (of/app T1 T2) (of/app T1 T2')) + %<- (preservation E T2 T2') %. +%term _ preservation (step/appr V) (of/app (of/lam Tf) (%the (of E T) T2)) (Tf E T2) %. +%term _ + %pi (preservation* (step/app1 E) T (of*/app T1' T2)) + %<- (inv-app T T1 T2) + %<- (preservation* E T1 T1') %. +%term _ + %pi (preservation* (step/app1x V) T (of*/raise Texn)) + %<- (inv-app T T1 T2) + %<- (inv-raise T1 Texn) %. +%term _ + %pi (preservation* (step/app2 V E) T (of*/app T1 T2')) + %<- (inv-app T T1 T2) + %<- (preservation* E T2 T2') %. +%term _ + %pi (preservation* (step/app2x V V') T (of*/raise Texn)) + %<- (inv-app T T1 T2) + %<- (inv-raise T2 Texn) %. +%sort q2 %. +%term _ q2 eqty/refl F E (F _ E) %. +%mode q2 %in %in %in %out %. +%worlds () (q2 _ _ _ _) %. +%total {} (q2 _ _ _ _) %. +%term _ + %pi (preservation* (step/appr V) T (of*/weaken Tres)) + %<- (inv-app T T1 T2*) + %<- (mobile V T2* T2) + %<- (inv-lam T1 F Eq) + %<- (q2 Eq F T2 Tres) %. +%{! #### Natural numbers !}% +%term _ %pi (preservation (step/s E) (of/s T) (of/s T')) %<- (preservation E T T') %. +%term _ + %pi (preservation (step/if E) (of/if T T1 T2) (of/if T' T1 T2)) + %<- (preservation E T T') %. +%term _ preservation step/ifz (of/if of/z T1 _) T1 %. +%term _ preservation (step/ifs V) (of/if (of/s T) _ T2) (T2 _ T) %. +%sort q1 %. +%term _ q1 T _ (of*/s T) %. +%mode q1 %in %in %out %. +%worlds () (q1 _ _ _) %. +%total {} (q1 _ _ _) %. +%term _ + %pi (preservation* (step/s E) Ts Ts') + %<- (inv-s Ts T Eq) + %<- (preservation* E T T') + %<- (q1 T' Eq Ts') %. +%sort q3 %. +%term _ q3 T _ T %. +%mode q3 %in %in %out %. +%worlds () (q3 _ _ _) %. +%total {} (q3 _ _ _) %. +%term _ + %pi (preservation* (step/sx V) DTs DT') + %<- (inv-s DTs (%the (of* (tm/raise E) ty/nat) DT) (%the (eqty TN ty/nat) Eq)) + %<- (q3 DT Eq DT') %. +%term _ + %pi (preservation* (step/if E) Tif (of*/if T' T1 T2)) + %<- (inv-if Tif T T1 T2) + %<- (preservation* E T T') %. +%term _ + %pi (preservation* step/ifz Tif T1) + %<- (inv-if Tif (%the (of* tm/z ty/nat) DT) (%the (of* E1 T) T1) _) %. +%term _ + %pi (preservation* (step/ifs (%the (v N) V)) DTif (T2 N Tn)) + %<- (inv-if DTif Tsn* T1 (%the ({x} %pi (of x ty/nat) %-> (of* (E2 x) T)) T2)) + %<- (inv-s Tsn* (%the (of* N ty/nat) Tn*) _) + %<- (mobile V Tn* Tn) %. +%term _ + %pi (preservation* (step/ifx V) DTif (of*/raise DTexn)) + %<- (inv-if DTif DT _ _) + %<- (inv-raise DT DTexn) %. +%{! #### Fix/raise/handle !}% +%term _ preservation step/fix (of/fix F) (F _ (of/fix F)) %. +%term _ + %pi (preservation (step/handle E) (of/handle T* T) (of/handle T'* T)) + %<- (preservation* E T* T'*) %. +%term _ %pi (preservation (step/handler V) (of/handle T* T) T') %<- (mobile V T* T') %. +%term _ + %pi (preservation (step/handlex V) (of/handle (of*/raise Texn*) T) (T _ Texn)) + %<- (mobile V Texn* Texn) %. +%term _ + %pi (preservation* step/fix (of*/weaken T) (of*/weaken T')) + %<- (preservation step/fix T T') %. +%term _ + %pi (preservation* (step/raise E) (of*/raise T) (of*/raise T')) + %<- (preservation* E T T') %. +%term _ + %pi (preservation* (step/raisex E) (of*/raise T) (of*/raise Texn)) + %<- (inv-raise T Texn) %. +%term _ + %pi (preservation* (step/handle DE) (%the (of* (tm/handle E H) T) DT) (of*/handle DT1' DT2)) + %<- (inv-handle DT DT1 DT2) + %<- (preservation* DE DT1 DT1') %. +%term _ %pi (preservation* (step/handler V) DT DT1) %<- (inv-handle DT DT1 DT2) %. +%term _ + %pi (preservation* (step/handlex V) (%the (of* (tm/handle (tm/raise E) H) T) DT) (DT2 E DTexn')) + %<- (inv-handle DT (%the (of* (tm/raise E) T) DT1) DT2) + %<- (inv-raise DT1 (%the (of* E ty/exn) DTexn)) + %<- (mobile V DTexn DTexn') %. +%term _ + %pi (preservation* (step/unsusp DE) (%the (of* (tm/unsusp E) T) DT) (of*/unsusp DT'')) + %<- (inv-unsusp DT (%the (of* E (ty/mayraise T)) DT')) + %<- (preservation* DE DT' (%the (of* E' (ty/mayraise T)) DT'')) %. +%sort d4 %. +%term _ d4 _ T T %. +%mode d4 %in %in %out %. +%worlds () (d4 _ _ _) %. +%total {} (d4 _ _ _) %. +%term _ + %pi (preservation* step/unsuspr (%the (of* (tm/unsusp (tm/susp E)) T) DT) DT''') + %<- (inv-unsusp DT (%the (of* (tm/susp E) (ty/mayraise T)) DT')) + %<- (inv-susp DT' (%the (of* E T') DT'') (%the (eqty (ty/mayraise T) (ty/mayraise T')) Eq)) + %<- (d4 Eq DT'' DT''') %. +%term _ + %pi (preservation* (step/unsuspx V) DT (of*/raise DTexn)) + %<- (inv-unsusp DT (%the (of* (tm/raise E) (ty/mayraise T)) DT')) + %<- (inv-raise DT' (%the (of* E ty/exn) DTexn)) %. +%worlds () (preservation _ _ _) (preservation* _ _ _) %. +%total (E E*) (preservation E _ _) (preservation* E* _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Rsimmons/Handled_exceptions). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-rsimmons-homework-6.lf b/new-tests/stelf-output/pages/user-rsimmons-homework-6.lf new file mode 100644 index 0000000..3321756 --- /dev/null +++ b/new-tests/stelf-output/pages/user-rsimmons-homework-6.lf @@ -0,0 +1,169 @@ +% +%{! + +First: look to the right ----> + +Where it says "Code: here" click on that link. This will you to grab the code for this webpage in a form that you can cut and paste into a file and use in Twelf. + +Unless you want to do this whole assignment in [Twelf Live](/wiki/twelf-live/), then you will need to download Twelf on your machine. + +* [http://twelf.plparty.org/builds/twelf-linux.tar.gz Linux (tgz)] +* [http://twelf.plparty.org/builds/twelf.exe Windows (exe)] +* [http://twelf.plparty.org/builds/twelf-osx-intel.dmg Mac OS 10.4 or 10.5 (Intel only) (dmg)] +* [http://twelf.plparty.org/builds/twelf-src.tar.gz Source tarball (tgz)]. You will need [http://www.mlton.org MLton] or [http://www.smlnj.org sml/nj]. + +**_Handing in:**_ Place the file ``hw06.elf`` in your handin directory by midnight on **_Monday, November 24**_ + +!}% +%{! ## Part 1: Natural Numbers !}% +%{! In the last section of [http://www.andrew.cmu.edu/course/15-814/homeworks/hw05.html Homework 5], I developed natural numbers in System F and then made some inductive definitions. I will do that again, but in Twelf, not System F. + +Natural numbers are no longer an interesting System F type, they're just something I define: !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%{! Again, I can outright define the various natural numbers if I care to, and I can even define an LF term ``succ`` that takes a natural number and returns its successor, though this is kind of a dumb function, because it just applies the ``s`` constructor: !}% +%define 0 nat z %. +%define 1 nat s z %. +%define 2 nat s (s z) %. +%define 3 nat s (s (s z)) %. +%define 4 nat s (s (s (s z))) %. +%define 5 nat s (s (s (s (s z)))) %. +%define 6 nat s (s (s (s (s (s z))))) %. +%define succ (%pi nat %-> nat) [n nat] s n %. +%{! ### Inductive descriptions of functions... well, of total relations !}% +%{! As before, I want to inductively define functions on natural numbers, and I want to know they're _functions_ - that I'll get an +input for every output (actually, what we're showing here is that they are "total" or "effective" relations - that for every input there is an output - and not that that output is unique. We can treat a relation as a function if it satisfies uniqueness and effectiveness, but that is not something we care about in this assignment. + +Our approach to inductively defining an effective relation is to describe the logic program that actually does describe a function. !}% +%{! +**Addition:** _n+m=p_ is defined by induction on _n_. If _n = 0_ then _n+m=m_, and if _n = n'+1_ and _n'+m=p_, then _n+m=p+1_. +!}% +%sort sum %. +%mode sum %in %in %out %. +%term sum/z sum z M M %. +%term sum/s %pi (sum (s N') M (s P)) %<- (sum N' M P) %. +%worlds () (sum _ _ _) %. +%total N (sum N _ _) %. +%{! +**Multiplication:** _n×m=p_ is defined by induction on _n_. If _n = 0_, then _n×m=0_, and if _n = n'+1_ and _n'×m=p_, then _n×m=m+p_. +!}% +%sort mult %. +%mode mult %in %in %out %. +%term mult/z mult z M z %. +%term mult/s %pi (mult (s N') M P') %<- (mult N' M P) %<- (sum M P P') %. +%worlds () (mult _ _ _) %. +%total N (mult N _ _) %. +%{! +**Exponentiation:** _n^m=p_ is defined by induction on _m_ (**not** _n_). If _m = 0_, then _n^m=1_, and if _m = m'+1_ and _n^m'=p_, then _n^m=n*p_. +!}% +%sort pow %. +%mode pow %in %in %out %. +%term pow/z pow N z (s z) %. +%term pow/s %pi (pow N (s M) P') %<- (pow N M P) %<- (mult N P P') %. +%worlds () (pow _ _ _) %. +%total M (pow _ M _) %. +%{! ### Question 1: Specification from code !}% +%{! What function the following relations define? (Again, ``%total`` proves only that they are total relations, but these relations +also happen to be unique, meaning that they are functions.) + +A word or two should suffice in each case. !}% +%{! #### Relation 1 !}% +%sort rel1 %. +%mode rel1 %in %out %. +%term rel1/z rel1 z (s z) %. +%term rel1/s %pi (rel1 (s N') P') %<- (rel1 N' P) %<- (mult (s N') P P') %. +%worlds () (rel1 _ _) %. +%total N (rel1 N _) %. +%{! #### Relation 2 !}% +%sort rel2 %. +%mode rel2 %in %in %out %. +%term rel2/z1 rel2 N z N %. +%term rel2/z2 rel2 z M M %. +%term rel2/s %pi (rel2 (s N) (s M) (s P)) %<- (rel2 N M P) %. +%worlds () (rel2 _ _ _) %. +%total [N M] (rel2 N M _) %. +%{! #### Relation 3 !}% +%sort rel3 %. +%mode rel3 %in %out %. +%term rel3/z rel3 z (s z) %. +%term rel3/1 rel3 (s z) (s z) %. +%term rel3/s %pi (rel3 (s (s N')) P) %<- (rel3 N' P1) %<- (rel3 (s N') P2) %<- (sum P1 P2 P) %. +%worlds () (rel3 _ _) %. +%total N (rel3 N _) %. +%{! ## Part 2: Lists and Trees !}% +%{! We had lists and trees in Homework 5, and now we have them again. We don't have to give them System F types, we can just define the constructors in a straightforward manner: + +In BNF: + l ::= nil | cons n l + t ::= leaf | node n t t + +In Twelf: !}% +%sort list %. +%term nil list %. +%term cons %pi nat %-> list %-> list %. +%sort tree %. +%term leaf tree %. +%term node %pi nat %-> tree %-> tree %-> tree %. +%{! Remember this example? + 2 + / \ + * 1 + / \ + 6 * + / \ + / \ + 5 2 + / \ / \ + * * 3 * + / \ + * * +I gave a System F term for that tree in the previous assignment, and all I had to do was cut and paste here: !}% +%define mytree tree node 2 leaf (node 1 (node 6 (node 5 leaf leaf) (node 2 (node 3 leaf leaf) leaf)) leaf) %. +%{! ### Question 2: Code from specification !}% +%{! Define ``append`` and ``flip`` as defined in [http://www.andrew.cmu.edu/course/15-814/homeworks/hw05.html Homework 5]. The ``infix`` relation is defined for you, but you will need to define ``append`` before Twelf will accept that ``infix`` is total. !}% +%{! #### Append !}% +%sort append %. +%mode append %in %in %out %. +% Write code here +%worlds () (append _ _ _) %. +%% %total L (append L _ _). % Uncomment me to finish +%{! #### Flip !}% +%sort flip %. +%mode flip %in %out %. +% Write code here +%worlds () (flip _ _) %. +%% %total T (flip T _). % Uncomment me to finish +%{! #### Infix !}% +%sort infix %. +%mode infix %in %out %. +%term infix/z infix leaf nil %. +%term infix/s + %pi (infix (node N T1 T2) L) + %<- (infix T1 L1) + %<- (infix T2 L2) + %<- (append L1 (cons N L2) L) %. +%worlds () (infix _ _) %. +%% %total T (infix T _). % Uncomment me once append works. +%{! ### Test cases !}% +%% %query 1 * append (cons 4 (cons 5 nil)) (cons 3 (cons 1 nil)) (cons 4 (cons 5 (cons 3 (cons 1 nil)))). +%% %query 1 * append nil (cons 4 (cons 5 (cons 6 nil))) (cons 4 (cons 5 (cons 6 nil))). +%% %query 1 * append (cons 4 (cons 5 (cons 6 nil))) nil (cons 4 (cons 5 (cons 6 nil))). +%% %query 1 * flip (node 2 leaf leaf) (node 2 leaf leaf). +%% %query 1 * flip (node 3 (node 1 leaf leaf) leaf) (node 3 leaf (node 1 leaf leaf)). +%% %query 1 * flip +%% mytree +%% (node 2 +%% (node 1 +%% leaf +%% (node 6 +%% (node 2 leaf (node 3 leaf leaf)) +%% (node 5 leaf leaf))) +%% leaf). +%% %query 1 * infix mytree (cons 2 (cons 5 (cons 6 (cons 3 (cons 2 (cons 1 nil)))))). +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Rsimmons/Homework_6). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-rsimmons-homework-7-proofs-in-twelf.lf b/new-tests/stelf-output/pages/user-rsimmons-homework-7-proofs-in-twelf.lf new file mode 100644 index 0000000..bc83b9a --- /dev/null +++ b/new-tests/stelf-output/pages/user-rsimmons-homework-7-proofs-in-twelf.lf @@ -0,0 +1,160 @@ +% +%{! Due Wednesday, December 3, by 11:59pm. Turn in as hw07.elf in your handin directory. !}% +%{! ## Definition !}% +%{! ### Natural numbers !}% +%{! #### On paper + + n ::= z | s n + +!}% +%{! #### In Twelf !}% +%sort nat %. +%term z nat %. +%term s %pi nat %-> nat %. +%{! ### Addition !}% +%{! #### On paper + + ---------------- sum/z + sum(z,n,n) + + sum(n,m,p) + ---------------- sum/s + sum(s(n),m,s(p)) + +!}% +%{! #### In Twelf !}% +%sort sum %. +%term sum/z sum z N N %. +%term sum/s %pi (sum (s N) M (s P)) %<- (sum N M P) %. +%{! ### Less-than !}% +%{! #### On paper + +**_QUESTION 1:**_ Give the inductive definition for the judgment ``n < n'`` that corresponds to to what is defined below. + +!}% +%{! #### In Twelf !}% +%sort lt %. +%term lt/z lt z (s N) %. +%term lt/s %pi (lt (s N) (s M)) %<- (lt N M) %. +%{! ### Even/odd !}% +%{! #### On paper + + ---------------- even/z + even z + + odd n + ---------------- even/s + even s(n) + + even n + ---------------- odd/s + odd s(n) + +!}% +%{! #### In Twelf !}% +%sort even %. +%sort odd %. +%{! **_QUESTION 2:**_ Define in LF the rules for ``even`` and ``odd`` below. !}% +%{! ## Proof: Sum Commutes !}% +%{! ### Lemma: N + 0 = N !}% +%{! #### On paper + +For all natural numbers ``n``, ``sum(n,z,n)``. + +Proof by induction on ``n``. + +Case ``n=z``. We need to show ``sum(z,z,z)``, which we can show by rule ``sum/z``. + +Case ``n=s(n')``. We need to show ``sum(s(n'),z,s(n'))``. By the induction hypothesis, ``sum(n',z,n')``. By rule ``sum/s``, we have ``sum(s(n'),z,s(n'))``. + +!}% +%{! #### In Twelf !}% +%sort sum-ident {N nat} %. +%mode sum-ident %in %out %. +%term _ sum-ident z (%the (sum z z z) sum/z) %. +%term _ + %pi (sum-ident (s N) (%the (sum (s N) z (s N)) (sum/s D))) + %<- (sum-ident N (%the (sum N z N) D)) %. +%worlds () (sum-ident _ _) %. +%total N (sum-ident N _) %. +%{! ### Lemma: N + M = P implies N + (s M) = (s P) !}% +%{! #### On paper + +**_QUESTION 3:**_ State the "on paper" version of the lemma below. + +!}% +%{! #### In Twelf !}% +%sort sum-incr %. +%mode sum-incr %in %out %. +%term _ sum-incr sum/z (%the (sum z (s N) (s N)) sum/z) %. +%term _ + %pi (sum-incr (sum/s D) (%the (sum (s N) (s M) (s (s P))) (sum/s D'))) + %<- (sum-incr D (%the (sum N (s M) (s P)) D')) %. +%worlds () (sum-incr _ _) %. +%total D (sum-incr D _) %. +%{! ### Theorem: N + M = P implies M + N = P !}% +%{! #### On paper + +If ``sum(n,m,p)``, then ``sum(m,n,p)``. + +Proof by induction on the derivation of ``sum(n,m,p)``. + +Case sum/z: + | | n = z + | ---------- | m = p + | sum(z,m,m) | + +To show: ``sum(m,z,m)``. Immediate by the first lemma. + +Case sum/s: + | sum(n',m,p') | n = s(n') + | ----------------- | p = s(p') + | sum(s(n'),m,s(p') | + +To show: ``sum(m,s(n'),s(p'))``. + +By the induction hypothesis, ``sum(m,n',p')``. By the second lemma, ``sum(m,s(n'),s(p'))``. + +!}% +%{! #### In Twelf !}% +%sort sum-commutes %. +%mode sum-commutes %in %out %. +%term _ + %pi (sum-commutes (%the (sum z N N) sum/z) D) + %<- (sum-ident N (%the (sum N z N) D)) %. +%term _ + %pi (sum-commutes (%the (sum (s N) M (s P)) (sum/s D)) D'') + %<- (sum-commutes D (%the (sum M N P) D')) + %<- (sum-incr D' (%the (sum M (s N) (s P)) D'')) %. +%worlds () (sum-commutes _ _) %. +%total D (sum-commutes D _) %. +%{! ### Theorem : N + (s M) = P implies N < P !}% +%{! #### On paper + +If ``sum(n,s(m),p)``, then ``n < p``. Proof by induction on ``sum(n,s(m),p)``. + +Case ``sum/z``: + | | n = z + | ---------------- | p = s(m) + | sum(z,s(m),s(m)) | + +To show: ``z < s(m)``. Immediate by rule ``lt/z``. + +Case ``sum/s``: + | sum(n',s(m),p') | n = s(n') + | --------------------- | p = s(p') + | sum(s(n'),s(m),s(p')) | + +To show: ``s(n') < s(p')``. + +By the induction hypothesis, ``n' < p'``. By rule ``lt/s``, we have ``s(n') < s(p')``. + +!}% +%{! #### In Twelf !}% +%{! **_QUESTION 4:**_ State and prove the above theorem in Twelf. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Rsimmons/Homework_7:_Proofs_In_Twelf). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-rsimmons-lecture-20-logic-programming.lf b/new-tests/stelf-output/pages/user-rsimmons-lecture-20-logic-programming.lf new file mode 100644 index 0000000..642acdd --- /dev/null +++ b/new-tests/stelf-output/pages/user-rsimmons-lecture-20-logic-programming.lf @@ -0,0 +1,177 @@ +% +%{! +__NOTOC__ + +In this lecture, Karl uses Prolog notation. I'm going to skip a step and go ahead and start using Twelf, since I've got all this infrastructure set up to incorporate pretty Twelf into web pages. For the purpose of this exercise, this only means that we have to declare types for things and that we write ``connect a b`` instead of ``connect(a,b)`` +for facts representing a connection from ``a`` +to ``b``, and we say ``D <- A <- B <- C`` instead of ``D :- A, B, C`` for a three-premise rule. + +You'll notice there's a box on the upper-right hand corner that calls this page "Literate Twelf." This is related to the idea of [http://www.literateprogramming.com/ Literate Programming] - this wiki page is itself a page of valid Twelf code. + +If you'll forgive the ugly illustration, we're interesting in answering questions about rechability on this acyclic graph (we want it to be an acyclic graph!) + +<code> + -> b <- + / \ + / \ +a---------->c----->d +</code> + +!}% +%sort node %. +%term a node %. +%term b node %. +%term c node %. +%term d node %. +%sort connect %. +%sort reaches %. +%term connect/a-b connect a b %. +%term connect/a-c connect a c %. +%term connect/c-b connect c b %. +%term connect/c-d connect c d %. +%term reach/refl {X node} reaches X X %. +%term reach/trans {X node} {Y node} {Z node} %pi (reaches Y Z) %-> (connect X Y) %-> (reaches X Z) %. +%{! + +**NOTE:** You don't have to write the Π bound things, so you could have used: +<twelf name=dsfadsf> +reach/refl : reaches X X. +reach/trans : reaches Y Z -> connect X Y -> reaches X Z. +</twelf> +(Twelf assumes you meant to Π-bind capitalized things but were too lazy to write the Π, so it does it for you. + +I can ask the engine to ask questions about reachability, which will essentially perform a depth-first search on the graph. + +I can confirm that there is one way through the graph from ``a`` to +``d`` (the notation ``%query 1 *`` isn't important for you to remember, but it means that I want one answer +no matter how many times I try) + +|check=decl !}% +%query 1 _ _ reaches a d %. +%{! + +...I can ask the engine to verify that there are no paths through the graph from ``b`` to ``a``: + +|check=decl !}% +%query 0 _ _ reaches b a %. +%{! + +...finally, I can use a capital letter (metavariable or substitution variable) ``Z`` to ask, what is EVERY path, starting from +``a``, in the graph? Turns out there are 5 (``b`` can be reached in two different ways): + +|check=decl !}% +%query 5 _ _ reaches a Z %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Rsimmons/Lecture_20:_Logic_Programming). +!}% +%. + += The LF Logical Framework = + +== Syntax == + + +K ::= \texttt{type} \mid A \rightarrow K + + + +A ::= \Pi x{:}A.A \mid \lambda x{:}A.A \mid A~M \mid \texttt{a} + + + +M ::= x \mid \lambda x{:}A.M \mid M~M \mid \texttt{c} + + +== Typing Rules == + +=== Kinds === + +Kinds ''K'' classify types ''A''. + +
+ + +{} +\over +{\texttt{type} : \texttt{kind}} + + +
+ + +{A : \texttt{type} \qquad +K : \texttt{kind}} +\over +{A \rightarrow K : \texttt{kind}} + + +=== Types === + +Types ''A'' classify terms ''M'' and are classified by kinds ''K''. + +
+ + +{A : \texttt{type}\qquad +x : A \vdash B : \texttt{type}} +\over +{\Pi x{:}A.B : \texttt{type}} + + +
+ + +{A : \texttt{type} \qquad +x : A \vdash B : K} +\over +{\lambda x{:}A.B : A \rightarrow K} + + +
+ + +{A : A' \rightarrow K \qquad +M : A'} +\over +{A~M : K} + + +
+ + +{\texttt{a} : K \in \Sigma} +\over +{\texttt{a} : K} + + +=== Terms === + +Terms ''M'' are classified by types ''A''. + + +{A : \texttt{type} \qquad +x : A \vdash M : A'} +\over +{\lambda x{:}A . M : \Pi x {:} A . A'} + + +
+ + +{M_1 : \Pi x{:}A . A' \qquad +M_2 : A} +\over +{M_1~M_2 : A'[M_2/x]} + + +
+ + +{\texttt{c} : A \in \Sigma} +\over +{\texttt{c} : A} + + diff --git a/new-tests/stelf-output/pages/user-rsimmons-lecture-23-typed-lambda-calculus-in-twelf.lf b/new-tests/stelf-output/pages/user-rsimmons-lecture-23-typed-lambda-calculus-in-twelf.lf new file mode 100644 index 0000000..0b30692 --- /dev/null +++ b/new-tests/stelf-output/pages/user-rsimmons-lecture-23-typed-lambda-calculus-in-twelf.lf @@ -0,0 +1,155 @@ +% +%{! ## Syntax !}% +%{! + +We have a programming language that we started last time that we want to represent in Twelf. First the types "on paper:" + + τ ::= unit | τ → τ + +... and then, we can see those types in Twelf: + +!}% +%sort tp %. +%term unit tp %. +%term arrow %pi tp %-> tp %-> tp %. +%{! + +Next, the terms "on paper:" + + e ::= x | * | λx:τ.e | e e + +... and in Twelf: + +!}% +%sort exp %. +%term * exp %. +%term lam %pi tp %-> (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%{! + +Note that there is no case for "variables" ``x``. + +!}% +%{! ## Static semantics (typing rules) !}% +%{! + +The old judgment for typing was ``e : τ`` - in other words, it was a two-place relation that took an expression and a type. We will describe this in Twelf as ``of``. + +!}% +%sort of %. +%{! + +Okay - now we need to describe this rule: + + -------- of/star + * : unit + +!}% +%term of/star of * unit %. +%{! + +Next we describe the more interesting rule for application: + + e<sub>1</sub> : τ → τ' + e<sub>2</sub> : τ + ------------ of/app + e<sub>1</sub> e<sub>2</sub> : τ' +!}% +%term of/app %pi (of (app E1 E2) T) %<- (of E1 (arrow T T')) %<- (of E2 T) %. +%{! + +The rule for abstractions uses a hypothetical! We're going to pull this off by using a Pi-binding (represented in Twelf as ``\{x: exp\}`` to create a _fresh_ variable, and then using an LF arrow ``->`` to say that you are proving ``of (E x) T'`` under the _hypothetical assumption_ that ``of x T``, which matches exactly with how we've been using assumptions all semester. + + x : τ |- e : τ' + --------------- of/lam + λx:τ.e : τ → τ' + +!}% +%term of/lam + %pi (of (lam T ([x] E x)) (arrow T T')) + %<- ({x exp} %pi (of x T) %-> (of (E x) T')) %. +%{! ### Example derivation !}% +%{! On paper: + + -------------------- hyp + x : unit |- e : unit + ----------------------- of/lam + λx:unit.e : unit → unit + +In Twelf: + +| check=decl !}% +%define _ %the (of (lam unit ([x] x)) (arrow unit unit)) (of/lam _) %. +%{! Okay, it looks like we need something (``X1``) of type ``\{x:exp\} of x unit -> of x unit``, in other words, something of type ``Πx:exp. of x unit -> of x unit`` - this is the type of a (dependent) function that takes a term with type ``exp`` (call it ``x``), and returns a (not dependent) function that takes a term with type ``of x unit`` and then returns a term with type ``of x unit``. In other words, take an expression and then give me the identity function! + +You guys should know what to do from here (it's just like what you were doing in Homework 5, but for LF instead of System F!). We want ``λx:exp. λd:of x unit. d``. +| check=decl !}% +%define _ %the (of (lam unit ([x] x)) (arrow unit unit)) (of/lam ([x exp] [d of x unit] d)) %. +%{! The absolutely correct observation made about this in class was that **Hypotheticals in the object language are represented as functions in the metalanguage.** !}% +%{! ## Dynamic semantics (evaluation rules) !}% +%{! + +First, values ``e value``: + + ------------ value/star + * value + + ------------ value/lam + λx:τ.e value + +!}% +%sort value %. +%term value/star value * %. +%term value/lam value (lam T ([x] E x)) %. +%{! + +Next, evaluation rules ``e -> e`` + + e<sub>1</sub> -> e'<sub>1</sub> + --------------------- step/app1 + e<sub>1</sub> e<sub>2</sub> -> e'<sub>1</sub> e<sub>2</sub> + + e<sub>1</sub> value + e<sub>2</sub> -> e'<sub>2</sub> + --------------------- step/app2 + e<sub>1</sub> e<sub>2</sub> -> e<sub>1</sub> e'<sub>2</sub> + + e<sub>2</sub> value + --------------------- step/applam + (λx:τ.e) e<sub>2</sub> -> e<sub>1</sub> e'<sub>2</sub> + +!}% +%sort step %. +%term step/app1 %pi (step (app E1 E2) (app E1' E2)) %<- (step E1 E1') %. +%term step/app2 %pi (step (app E1 E2) (app E1 E2')) %<- (value E1) %<- (step E2 E2') %. +%term step/applam %pi (step (app (lam T ([x] E x)) E2) (E E2)) %<- (value E2) %. +%{! ## Coda: Type Safety !}% +%{! + +[Last time](/wiki/user-rsimmons-homework-7-proofs-in-twelf/), we used the commutativity of addition to write a logic program that took in a term of type ``sum N1 N2 N3</tT> and produced a term of type ``sum N2 N1 N3``. In other words, we proved this theorem: + +**_Sum commutes**_: if ``sum(n<sub>1</sub>, n<sub>2</sub>, n<sub>3</sub>)`` then ``sum(n<sub>1</sub>, n<sub>1</sub>, n<sub>1</sub>)`` + +...by writing this in Twelf and proving it to be a total relation: + +<twelf> +sum-commutes : sum N1 N2 N3 -> sum N2 N1 N3 -> type. +%mode sum-commutes +D1 -D2. +</twelf> + +So we an prove this theorem: + +**_Preservation**_: if ``e : τ`` and ``e -> e'``, then ``e' : τ`` + +...by writing this in Twelf and proving it to be a total relation: + +!}% +%sort preservation %. +%mode preservation %in %in %out %. +%{! Thus ends today's lesson. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Rsimmons/Lecture_23:_Typed_Lambda_Calculus_in_Twelf). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-rsimmons.lf b/new-tests/stelf-output/pages/user-rsimmons.lf new file mode 100644 index 0000000..41708b9 --- /dev/null +++ b/new-tests/stelf-output/pages/user-rsimmons.lf @@ -0,0 +1,28 @@ +% +%{! My name is Robert J. Simmons. I am a first-year graduate student at Carnegie Mellon University. + +* Web page: http://www.cs.cmu.edu/~rjsimmon/ +* Wikipedia: [[w:User:Sirmob]] +* Girard Reading Group Wiki: http://locuspocus.hyperkind.org/ + +I am the background maintainer for this website, and [http://moog.concert.cs.cmu.edu moog] lives in my office. I developed the the [[Project:TwelfTag|TwelfTag]] extension for syntax highlighting automatically checking included Twelf code, the [[Project:Literate Twelf|Literate Twelf]] extension to facilitate offline editing, and generally attempt to keep things working. Leave me a [http://twelf.plparty.org/w/index.php?title=User_talk:Rsimmons&action=edit§ion=new message] if anything needs to be done. + +## Scratch space + +Since writing [[Project:Literate Twelf|Literate Twelf]], I tend to incorporate the Wiki into my development process. Often I only use the preview feature, but sometimes I save drafts in my namespace. Some of these would probably make good case studies if I get around to it. + +* [[User:Rsimmons/primop.elf]] - Broken. +* [[User:Rsimmons/Subtype]] - Incomplete - an attempt at doing a reasonable encoding of (width,depth) subtyping for n-ary tuples. +* [User:Rsimmons/Handled exceptions](/wiki/user-rsimmons-handled-exceptions/) - An encoding of exceptions where the type system can statically determine code to be non-exception-raising. +* [User:Rsimmons/Fluid binding](/wiki/user-rsimmons-fluid-binding/) - A more organized example than handled exceptions. This encodes "deep binding," requiring the dynamic semantics to search the stack for the correct binding; the type system ensures that some binding will be found. I was worried that the unsatisfactory encoding of sets would run into trouble, but it's actually a very context-like encoding of sets of fluid variables (weakening and exchange need to be proven explicitly, etc etc). This is the first encoding where I've wanted "Twelf functors" to abstract over the variable names - I can't use hypotheticals because I need decidable equality/inequality. + +## Things to do + +* [``%terminates``](/wiki/percent-terminates/) +* [LF](/wiki/lf/) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Rsimmons). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-spoons.lf b/new-tests/stelf-output/pages/user-spoons.lf new file mode 100644 index 0000000..08a500a --- /dev/null +++ b/new-tests/stelf-output/pages/user-spoons.lf @@ -0,0 +1,8 @@ +% +%{! Hi, everyone! My name is Dan Spoonhower, and I am a graduate student at CMU. You can read more about me on my [http://www.cs.cmu.edu/~spoons/ homepage]. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Spoons). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-128-2-223-28.lf b/new-tests/stelf-output/pages/user-talk-128-2-223-28.lf new file mode 100644 index 0000000..08acab3 --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-128-2-223-28.lf @@ -0,0 +1,8 @@ +% +%{! you still should log in!  — [Tom 7](/wiki/user-tom7/) 15:47, 20 October 2006 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User_talk:128.2.223.28). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-128-237-241-108.lf b/new-tests/stelf-output/pages/user-talk-128-237-241-108.lf new file mode 100644 index 0000000..4b63d00 --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-128-237-241-108.lf @@ -0,0 +1,10 @@ +% +%{! You should log in! + +Yeah, I should have [Jcreed](/wiki/user-jcreed/) 15:38, 24 October 2006 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User_talk:128.237.241.108). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-aleffert.lf b/new-tests/stelf-output/pages/user-talk-aleffert.lf new file mode 100644 index 0000000..5740824 --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-aleffert.lf @@ -0,0 +1,8 @@ +% +%{! hi !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User_talk:Aleffert). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-alepa.lf b/new-tests/stelf-output/pages/user-talk-alepa.lf new file mode 100644 index 0000000..7016929 --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-alepa.lf @@ -0,0 +1,8 @@ +% +%{! Hello, user Alepa. I don't know why you created a random page with a link to Google, and would like to. Please let me know if you were just trying to learn about how wiki software works or something! Otherwise, because we've had a great deal of trouble with vandalism, I'm likely to diable your account. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 12:59, 16 July 2011 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User_talk:Alepa). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-boyland.lf b/new-tests/stelf-output/pages/user-talk-boyland.lf new file mode 100644 index 0000000..74a889c --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-boyland.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-ccasin.lf b/new-tests/stelf-output/pages/user-talk-ccasin.lf new file mode 100644 index 0000000..44fe756 --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-ccasin.lf @@ -0,0 +1,8 @@ +% +%{! Welcome to the Twelf Party!  — [Tom 7](/wiki/user-tom7/) 16:08, 2 October 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User_talk:Ccasin). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-crary.lf b/new-tests/stelf-output/pages/user-talk-crary.lf new file mode 100644 index 0000000..a5c808b --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-crary.lf @@ -0,0 +1,8 @@ +% +%{! You're now an [[Project:Administrators|administrator]].  — [Tom 7](/wiki/user-tom7/) 14:42, 16 October 2006 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User_talk:Crary). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-danielklee.lf b/new-tests/stelf-output/pages/user-talk-danielklee.lf new file mode 100644 index 0000000..991260f --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-danielklee.lf @@ -0,0 +1,58 @@ +% +%{! ## My To-Do/Wish List + +* Fill in the %trustme example for the [%trustme](/wiki/percent-trustme/) article +* Tutorial: Introductory series. Tutorials for people who start knowing nothing about Twelf +* Tutorial: Manually proving substitution lemmas [for dependently typed languages], i.e. the slide + +You should add that stuff to the page I made at [[Project:To do]] - I've already listed your adoption of the %trustme article — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 07:43, 22 September 2006 (MST) + +## Fixup + +I deleted the fictonal blog entries - until we know what we're doing, when we're doing it, and where things are permanently hosted, I don't think we should spend time on the blog - it's just something I wanted to make sure was there as part of the equation. In any case, my thought is that blog updates will be something along the lines of "we've now added a tutorial describing closure conversion," a paragraph about why that's important/hard/easy, and a link to a page on the wiki, where we have pretty things like syntax highlighting. + +You can find links that let you edit individual sections of the front page at [Talk:Main Page](/wiki/talk-main-page/), a fact which I have added to [[Project:Contributing]]. + +Also, was something like [[Project:Style guide]] what you were thinking about? We also should think about a [Twelf style guide](/wiki/twelf-style-guide/), which also might have been what you were thinking about. - [Rsimmons](/wiki/user-rsimmons/) 09:05, 6 September 2006 (MST) + +## CVS + +You were going to add information on anon CVS download when you got a chance, just a reminder :). — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 19:49, 7 September 2006 (MST) + +## Simpler version of "slide" + +Would a simpler version of your "slide" example be the storetype weakining issue I described at http://fp.logosphere.cs.cmu.edu/twelf/?n=Answers.WorldsAndWeakening ? One of my to-dos is to transwiki things from the old wiki; I can expand that example if it would be useful as a simple case of what you're describing. Alternatively, if what you're describing does what I did better, I could hold off on the transwiki. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 10:32, 22 September 2006 (MST) + +: That is the non-dependent types version of the technique. I was planning on doing a more generalized explanation of this (the current write-up is very Rob's thesis specific and fails to call the "extra induction" a substitution lemma), which would lead into the version for dependent types. The "slide" substitution for dependent types isn't as generally useful as explicit contexts, but it's among the most badass Twelf hacks I know. If you don't mind, I'd like to take this one over, but definitely do transwiki the other answers from the old Q&A. + +:: 'swhy I asked. Go for it, and horray for general solutions! [[User:128.2.223.28|128.2.223.28]] 11:57, 22 September 2006 (MST) + +## ``can`` vs. expressiveness + +Do you still call the ``can-`` lemmas expressiveness lemmas? That section should include a link to a page about expressiveness lemmas, or whatever we call them, even if it's a redlink for now. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 14:42, 23 September 2006 (MST) + +: I'm unaware of any category of lemmas called expressiveness lemmas. Is this something from the old wiki? I can't seem to find a reference to them. --[DanielKLee](/wiki/user-danielklee/) 14:51, 23 September 2006 (MST) + +:: Eek - susmit actually called them "Effectiveness lemmas" - whatever it is you call the ``can-`` lemmas, I feel like they have, or should have, some name. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 15:02, 23 September 2006 (MST) + +## Twelf coffee + +I'm at home today, so no coffee for me. Doesn't seem like anybody did anything since last week anyway. ;)  — [Tom 7](/wiki/user-tom7/) 16:41, 4 October 2006 (EDT) + +## "Group" doesn't work anymore + +Using include="..." works similarly - I had to change it to handle multiple inheritance. See [[Project:TwelfTag|the TwelfTag page]] for an explanation of the modified feature, I'm sorry about that. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 17:22, 11 October 2006 (EDT) + +: Oh I see... *cry* --[DanielKLee](/wiki/user-danielklee/) 17:34, 11 October 2006 (EDT) + +:: Hang on - I'm going to make "group" a silent alias for "include". Just don't use it anymore! — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 17:38, 11 October 2006 (EDT) + +::: Done. Don't say I never did anything for you :). — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 17:44, 11 October 2006 (EDT) + +::: Thanks!! --[DanielKLee](/wiki/user-danielklee/) 17:45, 11 October 2006 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User_talk:DanielKLee). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-drdreyer.lf b/new-tests/stelf-output/pages/user-talk-drdreyer.lf new file mode 100644 index 0000000..1498f64 --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-drdreyer.lf @@ -0,0 +1,8 @@ +% +%{! Yo! — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 15:02, 21 March 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User_talk:Drdreyer). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-drl.lf b/new-tests/stelf-output/pages/user-talk-drl.lf new file mode 100644 index 0000000..645b4f4 --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-drl.lf @@ -0,0 +1,117 @@ +% +%{! Hello! I actually like the monospaced presentation better: + + + +You've got admin rights, btw. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 16:02, 23 October 2006 (EDT) + + +## Sidebar + +You might be interested in editing the [sidebar](/wiki/mediawiki-sidebar/). — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 16:51, 23 October 2006 (EDT) + +## Oh yeah + +You did write one of those papers-about-twelf things, now didn't you? :-) — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 21:50, 20 March 2007 (EDT) + +## Aite + +I'm showering, driving to school, and emailing! — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 09:18, 21 March 2007 (EDT) + +## Some small feature requests (didn't want to lose track of these completely) + +* %reduces under a binder (comes up, e.g. in [strengthening](/wiki/strengthening/)). When you know (forall x. A x < B), you could at least conclude A < B when x doesn't occur, which as I recall is enough for this example. +* new emacs mode with karl's macros +* for world violations, move some of the info that gets printed on higher chatter to the default, so it's easier to debug + +## Example (which Adam did much <strike>better</strike> slower than me) !}% +%sort tp %. +%term unit tp %. +%term arrow %pi tp %-> tp %-> tp %. +%sort exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%term app %pi exp %-> exp %-> exp %. +%sort of %. +%term of/app %pi (of (app E1 E2) T) %<- (of E1 (arrow T2 T)) %<- (of E2 T2) %. +%term of/lam %pi (of (lam E) (arrow T2 T)) %<- ({x exp} %pi (of x T2) %-> (of (E x) T)) %. +%sort false %. +%block ofb {T tp} [x exp] [dx of x T]%. +%sort id %. +%term refl id T T %. +%sort lemma {x} {dx of x T} %. +%mode lemma %in %in %out %. +%worlds (ofb) (lemma _ _ _) %. +%total {} (lemma _ _ _) %. +%sort thm %. +%mode thm %in %out %. +%term _ %pi (thm (of/lam ([x] [dx] of/app (D x dx) (D' x dx))) XXX) %<- (lemma D D' XXX) %. +%worlds (ofb) (thm X Y) %. +%total {} (thm X Y) %. +%{! ### Any var, not just the last one !}% +%sort tp %. +%term ar %pi tp %-> tp %-> tp %. +%prec %right 10 ar %. +%term b tp %. +%sort exp %. +%term app %pi exp %-> exp %-> exp %. +%term lam %pi (%pi exp %-> exp) %-> exp %. +%sort of %. +%term of_app %pi (of E1 (T1 ar T2)) %-> (of E2 T1) %-> (of (app E1 E2) T2) %. +%term of_lam %pi ({x} %pi (of x T1) %-> (of (E x) T2)) %-> (of (lam E) (T1 ar T2)) %. +%block ofblock {T tp} [x exp] [u of x T]%. +%worlds (ofblock) (of _ _) %. +%sort false %. +%sort isvar %. +%block varBlock {T tp} [x exp] [u of x T] [v isvar x]%. +%sort contra %. +%mode contra %in %in %in %out %. +%worlds (varBlock) (contra _ _ _ _) %. +%total {E1 D1 D2} (contra E1 D1 D2 _) %. +%sort boo %. +%mode boo %in %in %out %. +%term boo_lam %pi (boo X (of_app D1 D2) B) %<- (contra X D2 D1 B) %. +%worlds (varBlock) (boo _ _ _) %. +%total {} (boo _ _ _) %. +%{! ### Delphin Version by Adam + +<code> +sig <tp : type> %name A t + <ar : tp -> tp -> tp> %infix right 10 + <b : tp> ; + +sig <exp : type> + <app : exp -> exp -> exp> + <lam : (exp -> exp) -> exp>; + +sig <of : exp -> tp -> type> + <of_app : of E1 (T1 ar T2) -> of E2 T1 -> of (app E1 E2) T2> + <of_lam : (\{x\}of x T1 -> of (E x) T2) -> of (lam E) (T1 ar T2)>; + +sig <false : type>; + +sig <eq : tp -> tp -> type> -> of X (T1 ar T2) -> false -> type. + <eqid : eq T1 T1>; + +params = <exp>, <of (E#) T>; + + +type contextProp = <x:exp#> -> <(of x T1)> -> <(of x T2)> -> <eq T1 T2>; + + +fun contraEq : <eq (T1 ar T2) T1> -> <false> = fn .; + +fun impossible1 : contextProp -> <of (lam [x] app x x) T> -> <false> + = fn C <of_lam ([x][d] of_app (D1 x d) (D2 x d))> => + (case \{<x>\}\{<d:of x T1>\} contraEq ((C with <x> <d> <d> => <eqid>) <x> <D1 x d> <D2 x d>) + of \{<x>\}\{<d>\}<bot> => <bot>); + d + +fun impossibleGeneral : contextProp -> <x:exp#> -> <of (app x x) T> -> <false> + = fn C <x> <of_app D1 D2> => contraEq (C <x> <D1> <D2>); +</code> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User_talk:Drl). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-hdeyoung.lf b/new-tests/stelf-output/pages/user-talk-hdeyoung.lf new file mode 100644 index 0000000..a574e06 --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-hdeyoung.lf @@ -0,0 +1,10 @@ +% +%{! ## Hey, you're on the wiki + +Made you a member of the "party-elf" group; now you should be able to make edits without CAPTCHA-ing! — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 00:12, 22 April 2009 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User_talk:Hdeyoung). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-jaked.lf b/new-tests/stelf-output/pages/user-talk-jaked.lf new file mode 100644 index 0000000..81c3e5b --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-jaked.lf @@ -0,0 +1,8 @@ +% +%{! hi !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User_talk:JakeD). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-jcreed.lf b/new-tests/stelf-output/pages/user-talk-jcreed.lf new file mode 100644 index 0000000..371afcb --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-jcreed.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-neel.lf b/new-tests/stelf-output/pages/user-talk-neel.lf new file mode 100644 index 0000000..91dee79 --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-neel.lf @@ -0,0 +1,8 @@ +% +%{! Welcome to Twelf (I don't like seeing red) — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 17:59, 19 March 2007 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User_talk:Neel). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-popik.lf b/new-tests/stelf-output/pages/user-talk-popik.lf new file mode 100644 index 0000000..7bf1f4d --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-popik.lf @@ -0,0 +1,8 @@ +% +%{! You are the weakest link. Goodbye! (indefinite block) — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 22:08, 14 February 2008 (EST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User_talk:Popik). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-rpollack.lf b/new-tests/stelf-output/pages/user-talk-rpollack.lf new file mode 100644 index 0000000..c2764aa --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-rpollack.lf @@ -0,0 +1,8 @@ +% +%{! Welcome! (You've been added to the "known humans" group so you don't have to fill in [[w:Captcha|Captcha]]s.) — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 14:27, 28 February 2008 (EST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User_talk:Rpollack). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-rsimmons.lf b/new-tests/stelf-output/pages/user-talk-rsimmons.lf new file mode 100644 index 0000000..6e0589e --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-rsimmons.lf @@ -0,0 +1,123 @@ +% +%{! ## Twelf Blog + +I think the environment subtyping stuff will make a great early entry into the Twelf Blog. On the announcement for it on the frontpage, I noticed that you referred to a "Dan" doing the entry. It takes a while for it to set it, but there are a lot of Dans in the POP group, and unfortunately both Dans who do Twelf related stuff are Dan Ls. I'm a wiki n00b, so I can't figure out how to edit that particular column (it seems locked down), but it might be less confusing to disambiguate that. Will I need to register for the blog seperately? --[DanielKLee](/wiki/user-danielklee/) 21:53, 5 September 2006 (MST) + +## linking to cats + +You can link to a category by writing <nowiki>[[:Category:tutorials|tutorials]]</nowiki>, which looks like [[:Category:tutorials|tutorials]]. [Tom7](/wiki/user-tom7/) 09:42, 6 September 2006 (MST) + +## most wanted articles + +My list of most-wanted articles: +* [[um.elf]] +* [[Image:Tom_Twelfelf2.jpg|thumb|left]] + — [Tom 7](/wiki/user-tom7/) 13:54, 14 September 2006 (MST) + + + + + + + + + + + + + + + + + + +## Tiny issue with syntax highlighting + +The syntax highlighter does not consider * to be part of an identifier. See the second half of [substitution lemma](/wiki/substitution-lemma/) to see an example of what I mean. This is a pretty minor issue, all things considered. --[DanielKLee](/wiki/user-danielklee/) 14:30, 24 September 2006 (MST) + +## Wiki +You keep breakin' it! ;)  — [Tom 7](/wiki/user-tom7/) 19:43, 4 October 2006 (EDT) + +## noinclude + +[http://twelf.plparty.org/w/index.php?title=User_talk%3ATom7&diff=2569&oldid=2384 re] My favorite would be if noinclude="true" were a derived form for name="_", meaning to bind the useless name _. If you want to rename it, I would vote for discard="true", but I can't think of any reason to do that...  — [Tom 7](/wiki/user-tom7/) 18:31, 10 October 2006 (EDT) +* Oh, sorry, I forgot the confusion between "include" and "noinclude". I support renaming it to discard="true", but maybe people can just do name="_"? Is that too weird?  — [Tom 7](/wiki/user-tom7/) 18:35, 10 October 2006 (EDT) + +## I LF Bees + +I started a page at [[Project:1.0]] where we can plan for a wide release, and seeded it with demands. Your thoughts are welcome...  — [Tom 7](/wiki/user-tom7/) 13:49, 16 October 2006 (EDT) + +## Sad + +I yearn for the era of daily Twelfwiki breakage! Thesisland is just not as much fun...  — [Tom 7](/wiki/user-tom7/) 13:09, 13 December 2006 (EST) +* Hooray, someone else is editing!  — [Tom 7](/wiki/user-tom7/) 00:18, 25 December 2006 (EST) + +## mlton builds + +Hey, I was just helping a student build Twelf at Dagstuhl; he used the experimental mlton release for x86-Darwin. Experimental builds of mlton return a version string of 'MLTONVERSION' which then fails in the build process. Can we fix this so that the report "version XXXX or greater required" is actually a warning? I think a large fraction of mlton users are using experimental or CVS releases since they are programmers themselves.  — [Tom 7](/wiki/user-tom7/) 05:29, 31 January 2007 (EST) +:: I believe that the change has been commited to Twelf CVS. If you can, check it out and verify that it works? I just added a new if-branch for MLTONVERSION, and if that does not work I'll change the error to a warning. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 10:13, 31 January 2007 (EST) + +## spam filter + +Also, can you disable the spam filter for admins?? It won't let me save my changes.  — [Tom 7](/wiki/user-tom7/) 05:50, 31 January 2007 (EST) +: It would be tricky to disable the spam filter regex just for admins, because if the forbidden string appears on the page then only admins would be able to edit it henceforth. However, if you need to disable some things, go ahead - just deleting style from the LocalSettings was the right thing to do; if it didn't work, go ahead and replace it with the more html-fu permissive setting from http://meta.wikimedia.org/wiki/Anti-spam_Features — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 10:11, 31 January 2007 (EST) +:: After a few minutes (I realize you aren't up at 5:50am) I found the place where the spam filter was defined and temporarily disabled it so that I could make the edits I need to. If it continues to be a problem I'll see about improving the filter...  — [Tom 7](/wiki/user-tom7/) 17:15, 31 January 2007 (EST) +::: I'm up then more than I should be; just make sure to leave the spam filter "weakened", like I said, pages with spam-filter-opposing content are effectively uneditable. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 17:57, 31 January 2007 (EST) +:::: Point taken, but this is for \{\{tl|click-inline\}\} which IMO should never be edited by non-admins (or really ever at all).  — [Tom 7](/wiki/user-tom7/) 06:11, 1 February 2007 (EST) + +## Tex broke + +The <nowiki><math></nowiki> tag ain't workin'; viz [lists](/wiki/lists/).  — [Tom 7](/wiki/user-tom7/) 23:32, 24 February 2007 (EST) +: This is why I keep the old mediawiki files around for awhile after an upgrade; we're back up and running, I'll have to add "make texvc work again" to my list of upgrade tasks. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 01:24, 25 February 2007 (EST) + +## SVG + +Thanks for making more keyword articles. Do you think it makes sense perhaps to separate them into "rarely used" and "frequently used" or perhaps by category of what kind of thing you want to do? A lot of these keywords I don't even know what they are because I've never used them, even though I've written like tens of thousands of lines of Twelf. + +: Instead of that distinction, as you use Twelf in a distinctive way like most here :), and I'm not sure how to clearly categorize things, I was imagining an additional Big Four template of some sort that described %mode %block %worlds %total - the keywords you need to specify [totality assertions](/wiki/totality-assertion/). This is a tricky distinction - what about %covers %reduces %terminates - but however you slice it, I was going to just specify that set, and leave the rest in the big jumble on the template - I think the idea is people find these by clicking on a link in an article, anyway. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 12:20, 28 February 2007 (EST) +:: OK, that sounds good! It's just that if I click on some of those keywords, I might get the sense that I need to understand them in order to use Twelf. As long as there is something to dispell that myth, I'm on board.  — [Tom 7](/wiki/user-tom7/) 14:21, 28 February 2007 (EST) + +Also, if you install SVG support I will make you some nice tree graphics.  — [Tom 7](/wiki/user-tom7/) 10:23, 28 February 2007 (EST) + +: I already have some trees in Adobe Illustrator I was going to use, I've been unable to track down clear and decisive information about SVG and mediawiki (are you sure it doesn't already support it?) but I'll try the IRC channel again today. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 12:20, 28 February 2007 (EST) +:: I tried uploading a SVG file and it wouldn't let me, so I'm pretty sure the extension isn't there. If you are using AI then SVG is the right format to upload in, so they can be resized and edited without losing the original form. It might be hard to get the SVG extension working though, I don't know.  — [Tom 7](/wiki/user-tom7/) 14:21, 28 February 2007 (EST) + +## done + +I made reasonable stubs for most of the remaining glossary entries. As far as I'm concerned, if you patch [%total](/wiki/percent-total/) a little, we can announce. [Drl](/wiki/user-drl/) 00:54, 21 March 2007 (EDT) + +## twelf sources with .cm + +FYI: the Twelf source tarball on your snazzy download page includes SML/NJ .cm directories, which are worse than useless because they can confuse/crash SML/NJ and make the tarball 3x the size it needs to be.  — [Tom 7](/wiki/user-tom7/) 16:50, 23 October 2007 (EDT) +: This broke because I ran a rogue compile a few weeks ago; I should be able to put it back together tomorrow. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 01:08, 24 October 2007 (EDT) +:: Should be fixed. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 12:09, 24 October 2007 (EDT) + +## front page news + +Hallo! +I'm not a twelf guy, and am just looking into logic programming. +I am unfamiliar with your specialized terms... which means that this doesn't parse for me - could you perhaps translate it back into a form of natural language? + +"Rob has a case study on lax logic that uses the admissibility of cut and identity show a sound and complete correspondence between two sequent calculus presentations of lax logic." + +1. Rob [presents] a case study on lax logic. + +2. The lax logic model used assumes the admissibility of "cut and identity". + +*** help! Brain overflow... does not parse! *** So we guess... +3. The lax logic model shows a "sound and complete correspondence" when tested using two sequent calculus presentations of lax logic? + +Is it self referential? Does lax logic mean something different each time it is used? + +I suspect there's a decent tutorial to be had just explaining that sentence. :) + +## Twelf Live broken? + +Hi Rob. It says on the front page to contact you if anything is broken. Well, Twelf Live doesn't seem to work. I was looking forward to trying it. Cheers. +:: Hi! Thanks for noticing that, I moved some stuff and deleted some code and forgot to update links and test that. Twelf Live is a bit rougher than it's supposed to be at the moment (lots of superfluous %% OK %%s) but it's the best I could do on short notice; I'll get back around to it. — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 11:41, 9 September 2010 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User_talk:Rsimmons). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-spoons.lf b/new-tests/stelf-output/pages/user-talk-spoons.lf new file mode 100644 index 0000000..8b89f15 --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-spoons.lf @@ -0,0 +1,11 @@ +% +%{! hi + +hello, everyone [Spoons](/wiki/user-spoons/) 21:58, 17 October 2006 (EDT) +* You're [red](/wiki/user-spoons/)!  — [Tom 7](/wiki/user-tom7/) 22:29, 17 October 2006 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User_talk:Spoons). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-twilson.lf b/new-tests/stelf-output/pages/user-talk-twilson.lf new file mode 100644 index 0000000..2473baa --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-twilson.lf @@ -0,0 +1,8 @@ +% +%{! Woo bugfixes! — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User_talk:Twilson). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-varming.lf b/new-tests/stelf-output/pages/user-talk-varming.lf new file mode 100644 index 0000000..70635a4 --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-varming.lf @@ -0,0 +1,10 @@ +% +%{! ## red be gone + +I hate seeing red so here is a message for you to make your talk page blue!  — [Tom 7](/wiki/user-tom7/) 09:33, 12 October 2006 (EDT) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User_talk:Varming). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-talk-william-lovas.lf b/new-tests/stelf-output/pages/user-talk-william-lovas.lf new file mode 100644 index 0000000..dcecaa0 --- /dev/null +++ b/new-tests/stelf-output/pages/user-talk-william-lovas.lf @@ -0,0 +1,13 @@ +% +%{! hi + +lo [William Lovas](/wiki/user-william-lovas/) 03:57, 6 October 2006 (EDT) +p.s. plz sign ur comments, n00b! + +⊦ ⋄ □ ⊸ ⊗ ⊤ ⇒ ∀ ∃ ⊕ ⊥ — [Rob](/wiki/user-rsimmons/) (and his [talk](/wiki/user-talk-rsimmons/)) 13:43, 9 November 2007 (EST) !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User_talk:William_Lovas). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-testuser.lf b/new-tests/stelf-output/pages/user-testuser.lf new file mode 100644 index 0000000..2191865 --- /dev/null +++ b/new-tests/stelf-output/pages/user-testuser.lf @@ -0,0 +1,10 @@ +% +%{! [Rob](/wiki/user-rsimmons/) created this user to have someone he could use to log in as someone who only had "generic user" permission. + +Leave Rob any messages you might have. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:TestUser). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-thetwelfelf.lf b/new-tests/stelf-output/pages/user-thetwelfelf.lf new file mode 100644 index 0000000..8fb9b3e --- /dev/null +++ b/new-tests/stelf-output/pages/user-thetwelfelf.lf @@ -0,0 +1,10 @@ +% +%{! The dude in the green hat. You may notice that every college's Facebook.com account has an account for "The Creator" that is presumably used when the admins at The Facebook want to look around as members of a school as members of that account. TheTwelfElf was the first account, and so has full control of the wiki, but so do [[Project:Administrators | other people]] so it's not really necessary for a development system. + +[Rob](/wiki/user-rsimmons/) has loose control over this account and sometimes forgets to log out and so edits as the Elf. That control could naturally change in the future. This is not a good place to leave a message, for that you should find an [[Project:Administrators | admin]] that actually exists. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:TheTwelfElf). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-tom7-test-elf.lf b/new-tests/stelf-output/pages/user-tom7-test-elf.lf new file mode 100644 index 0000000..a47ed3a --- /dev/null +++ b/new-tests/stelf-output/pages/user-tom7-test-elf.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-tom7.lf b/new-tests/stelf-output/pages/user-tom7.lf new file mode 100644 index 0000000..40e0295 --- /dev/null +++ b/new-tests/stelf-output/pages/user-tom7.lf @@ -0,0 +1,19 @@ +% +%{! Hi, I'm **Tom Murphy VII**, known colloquially as **Tom 7**. I am a graduate student at Carnegie Mellon University, nearing completion. My real home page is at [http://tom7.org tom7.org]. + +## Notes +* I always have trouble remembering [[Special:Allmessages]]. +* I also have trouble remembering [MediaWiki:Edittools](/wiki/mediawiki-edittools/). +* I want to customize the site so it doesn't look exactly like Wikipedia. [http://en.wikipedia.org/wiki/Help:User_style This page] has some explanation of [MediaWiki:Monobook.css](/wiki/mediawiki-monobook-css/) div IDs, etc. +* Here I can experiment with crazy [http://meta.wikimedia.org/wiki/ParserFunctions template features]. +* To do: add parameterized resp technique to [congruence lemmas](/wiki/respects-lemma/), along with semi-tasteless tricks + +## Photographs of me with the Twelf Elf + +[[Image:Tom_Twelfelf2.jpg|thumb|left|400px|Tom (left) with Twelf Elf, Halloween 2006.]] !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Tom7). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-twilson.lf b/new-tests/stelf-output/pages/user-twilson.lf new file mode 100644 index 0000000..541c255 --- /dev/null +++ b/new-tests/stelf-output/pages/user-twilson.lf @@ -0,0 +1,8 @@ +% +%{! Todd Wilson is a faculty member at California State University, Fresno, and was a visiting scholar at Carnegie Mellon in the Spring of 2007. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Twilson). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-varming.lf b/new-tests/stelf-output/pages/user-varming.lf new file mode 100644 index 0000000..ae29d88 --- /dev/null +++ b/new-tests/stelf-output/pages/user-varming.lf @@ -0,0 +1,14 @@ +% +%{! Carsten Varming<br /> +CMU CSD Wean Hall 4212<br /> +5000 Forbes Avenue<br /> +Pittsburgh PA 15213<br /> +USA + +[[http://www.cs.cmu.edu/~cvarming/ My web page]] !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:Varming). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/user-william-lovas.lf b/new-tests/stelf-output/pages/user-william-lovas.lf new file mode 100644 index 0000000..4fd48d9 --- /dev/null +++ b/new-tests/stelf-output/pages/user-william-lovas.lf @@ -0,0 +1,8 @@ +% +%{! Welcome to William Lovas's bog standard user page. !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/User:William_Lovas). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/users-guide.lf b/new-tests/stelf-output/pages/users-guide.lf new file mode 100644 index 0000000..3d1516e --- /dev/null +++ b/new-tests/stelf-output/pages/users-guide.lf @@ -0,0 +1,22 @@ +% +%{! +## Twelf User's Guide + +The current version of the Twelf **User's Guide** is for Twelf version 1.4. +While it does not describe all the features of Twelf 1.7, it is still the +most authoritative reference for the language's syntax. + +It is available in +* [HTML](https://www.cs.cmu.edu/~twelf/guide-1-4/twelf_toc.html) +* [Postscript](https://www.cs.cmu.edu/~twelf/guide-1-4/twelf.ps) +* [Gzipped Postscript](https://www.cs.cmu.edu/~twelf/guide-1-4/twelf.ps.gz) +* [PDF](https://www.cs.cmu.edu/~twelf/guide-1-4/twelf.pdf) + +## Documentation on twelf.org + +Most of Twelf's keywords are also documented on the wiki, including some that +are not covered in the User's Guide: + + + +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/using-nat-less-with-reduces.lf b/new-tests/stelf-output/pages/using-nat-less-with-reduces.lf new file mode 100644 index 0000000..19bed68 --- /dev/null +++ b/new-tests/stelf-output/pages/using-nat-less-with-reduces.lf @@ -0,0 +1,4 @@ +% +%{! +This page has been removed, you can refer to the wayback machine's version by following the link below. +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/verifications-and-uses-in-hlf.lf b/new-tests/stelf-output/pages/verifications-and-uses-in-hlf.lf new file mode 100644 index 0000000..3fcf68d --- /dev/null +++ b/new-tests/stelf-output/pages/verifications-and-uses-in-hlf.lf @@ -0,0 +1,127 @@ +% +%{! This example can only be run in the Hybrid LF implementation of Twelf +created by Jason Reed as part of his Ph.D. Thesis, +"A Hybrid Logical Framework." So that line above uses the [`%.`](/wiki/percent-eof/) notation to make sure that none of this file gets checked when the twelf wiki is built. + +The example on [verifications and uses with zippers](/wiki/verifications-and-uses-with-zippers/) has essentially the same content but actually works in non-HLF extended Twelf. + +This code is primarily here so it can be referenced by a [http://requestforlogic.blogspot.com/2010/09/natural-deduction-and-sequent-calculus.html blog post] by [Rob](/wiki/user-rsimmons/). +!}% +%{! ## Propositions and rules !}% +%{! ## Global completeness !}% +%{! ## Global soundness !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Verifications_and_uses_in_HLF). +!}% +%. + + + +%. + + + + +prop : type. +atom : type. +%block bl_atom : block {qp : atom}. + +a : atom -> prop. +⊃ : prop -> prop -> prop. %infix right 9 ⊃. +∧ : prop -> prop -> prop. %infix right 8 ∧. + +hyp : prop -> @type. +verif : prop -> @type. +use : prop -> @type. + +atm : hyp A -> (use A -o use (a Q)) -> verif (a Q). +⊃I : (hyp A₁ -> verif A₂) -> verif (A₁ ⊃ A₂). +⊃E : use (A₁ ⊃ A₂) -o verif A₁ -> use A₂. +∧I : verif A₁ -> verif A₂ -> verif (A₁ ∧ A₂). +∧E₁ : use (B₁ ∧ B₂) -o use B₁. +∧E₂ : use (B₁ ∧ B₂) -o use B₂. + +%block bl_hyp : some {A : prop} block {x : hyp A e}. +%block bl_use : some {A : prop} block {α}{x : use A α}. +%worlds (bl_atom | bl_hyp | bl_use) (verif _ _) (use _ _) (hyp _ _). + + + +eta : {A} ({B} hyp B -> (use B -o use A) -> verif A) -> type. +%mode eta +A -B. + +- : eta (a Q) ([B][x : hyp B][r : use B -o use (a Q)] + atm x ([u :^ (use B)] r ^ u)). +- : eta (A₁ ⊃ A₂) ([B][x : hyp B][r : use B -o use (A₁ ⊃ A₂)] + ⊃I ([y : hyp A₁] N₂ B x ([u :^ (use B)] + ⊃E ^ (r ^ u) (N₁ A₁ y ([u :^ (use A₁)] u))))) + <- eta A₁ ([B] N₁ B : hyp B -> (use B -o use A₁) -> verif A₁) + <- eta A₂ ([B] N₂ B : hyp B -> (use B -o use A₂) -> verif A₂). +- : eta (A₁ ∧ A₂) ([B][x : hyp B][r : use B -o use (A₁ ∧ A₂)] + ∧I (N₁ B x ([u :^ (use B)] ∧E₁ ^ (r ^ u))) + (N₂ B x ([u :^ (use B)] ∧E₂ ^ (r ^ u)))) + <- eta A₁ ([B] N₁ B : hyp B -> (use B -o use A₁) -> verif A₁) + <- eta A₂ ([B] N₂ B : hyp B -> (use B -o use A₂) -> verif A₂). + +%worlds (bl_atom | bl_hyp | bl_use) (eta _ _). +%total A (eta A _). + +%solve s : {q}{r}{s} eta (a q ⊃ a r ⊃ a s) (X q r s). +%solve s : {q}{r}{s} eta ((a q ⊃ a r) ⊃ a s) (X q r s). + + + +hsubst_n : {A} verif A -> (hyp A -> verif B) -> verif B -> type. +hsubst_rr : {A} verif A -> (hyp A -> use C -o use B) -> (use C -o use B) -> type. +hsubst_rn : {A}{B} verif A -> (hyp A -> use A -o use B) -> verif B -> type. +%mode hsubst_n +A +M₀ +M -N. +%mode hsubst_rr +A +M₀ +R -R'. +%mode hsubst_rn +A +B +M₀ +R -N. + +- : hsubst_n A M₀ ([x : hyp A] ⊃I [y : hyp B₁] M x y) (⊃I [y : hyp B₁] N y) + <- {y : hyp B₁} hsubst_n A M₀ ([x : hyp A] M x y) (N y : verif B₂). +- : hsubst_n A M₀ ([x : hyp A] ∧I (M₁ x) (M₂ x)) (∧I N₁ N₂) + <- hsubst_n A M₀ M₁ (N₁ : verif B₁) + <- hsubst_n A M₀ M₂ (N₂ : verif B₂). +- : hsubst_n A M₀ ([x : hyp A] atm x ([u :^ (use A)] R x ^ u)) N + <- hsubst_rn A (a Q) M₀ R (N : verif (a Q)). +- : hsubst_n A M₀ ([x : hyp A] atm Y (R x)) (atm Y R') + <- hsubst_rr A M₀ R (R' : use C -o use (a Q)). + +- : hsubst_rr A M₀ ([x : hyp A][u :^ (use C)] ⊃E ^ (R x ^ u) (M x)) + ([u :^ (use C)] ⊃E ^ (R' ^ u) N) + <- hsubst_rr A M₀ R (R' : use C -o use (B₁ ⊃ B₂)) + <- hsubst_n A M₀ M (N : verif B₁). +- : hsubst_rr A M₀ ([x : hyp A][u :^ (use C)] ∧E₁ ^ (R x ^ u)) + ([u :^ (use C)] ∧E₁ ^ (R' ^ u)) + <- hsubst_rr A M₀ R (R' : use C -o use (B₁ ∧ B₂)). +- : hsubst_rr A M₀ ([x : hyp A][u :^ (use C)] ∧E₂ ^ (R x ^ u)) + ([u :^ (use C)] ∧E₂ ^ (R' ^ u)) + <- hsubst_rr A M₀ R (R' : use C -o use (B₁ ∧ B₂)). +- : hsubst_rr A M₀ ([x : hyp A][u :^ (use C)] u) ([u :^ (use C)] u). + +- : hsubst_rn A B₂ M₀ ([x : hyp A][u :^ (use A)] ⊃E ^ (R x ^ u) (M x)) N' + <- hsubst_rn A (B₁ ⊃ B₂) M₀ R ((⊃I [y : hyp B₁] N y) : verif (B₁ ⊃ B₂)) + <- hsubst_n A M₀ M (M' : verif B₁) + <- hsubst_n B₁ M' N (N' : verif B₂). +- : hsubst_rn A B₁ M₀ ([x : hyp A][u :^ (use A)] ∧E₁ ^ (R x ^ u)) N₁ + <- hsubst_rn A (B₁ ∧ B₂) M₀ R (∧I N₁ N₂ : verif (B₁ ∧ B₂)). +- : hsubst_rn A B₂ M₀ ([x : hyp A][u :^ (use A)] ∧E₂ ^ (R x ^ u)) N₂ + <- hsubst_rn A (B₁ ∧ B₂) M₀ R (∧I N₁ N₂ : verif (B₁ ∧ B₂)). +- : hsubst_rn A A M₀ ([x : hyp A][u :^ (use A)] u) M₀. + +%worlds (bl_atom | bl_hyp) +(hsubst_n _ _ _ _) +(hsubst_rr _ _ _ _) +(hsubst_rn _ _ _ _ _). + +%reduces B <= A (hsubst_rn A B _ _ _). + +%total {(A B C) (M R S)} +(hsubst_n A _ M _) +(hsubst_rr B _ R _) +(hsubst_rn C _ _ S _). + diff --git a/new-tests/stelf-output/pages/verifications-and-uses-with-zippers.lf b/new-tests/stelf-output/pages/verifications-and-uses-with-zippers.lf new file mode 100644 index 0000000..2027eca --- /dev/null +++ b/new-tests/stelf-output/pages/verifications-and-uses-with-zippers.lf @@ -0,0 +1,102 @@ +% +%{! In the [verifications and uses](/wiki/verifications-and-uses/) article we saw a typical presentation of a logic of verifications and uses, and in article on [hereditary substitution with zippers](/wiki/hereditary-substitution-with-a-zipper/) we saw an attempt to clean up the "ugly part" of that proof by defining a zipper-like structure over terms. The reason that global soundness for this system did not hold is that Twelf could not verify (without the use of a [structural metric](/wiki/structural-metrics/)) that a "zipped" use was the same size as an "unzipped" use, so [termination checking](/wiki/percent-terminates/) failed when a recursive +call was made on a term that had been unzipped. + +In this article, we give a different solution. This system has something of the flavor of a [[spine form]] presentation; however, as we discussed in the +[hereditary substitution with zippers](/wiki/hereditary-substitution-with-a-zipper/), it is still much closer to a natural deduction system - the analogue of the natural deduction term ``atm (⊃E (⊃E (⊃E (var x) N<sub>1</sub>) N<sub>2</sub>) +N<sub>3</sub>)`` in the natural deduction system is +``atm x (⊃E (⊃E (⊃E end N<sub>1</sub>) N<sub>2</sub>) N<sub>3</sub>)`` in this version of "natural deduction" - the form of proofs stays the same +with the exception that the head variable ``x`` has been pulled out into the front. !}% +%{! ## Propositions and rules !}% +%sort prop %. +%sort atom %. +%block bl_atom [qp atom]%. +%term a %pi atom %-> prop %. +%term ⊃ %pi prop %-> prop %-> prop %. +%prec %right 9 ⊃ %. +%term ∧ %pi prop %-> prop %-> prop %. +%prec %right 8 ∧ %. +%sort hyp %. +%sort verif %. +%sort use %. +%block bl_hyp {A prop} [x hyp A]%. +%term end use A A %. +%term atm %pi (hyp A) %-> (use A (a Q)) %-> (verif (a Q)) %. +%term ⊃I %pi (%pi (hyp A) %-> (verif B)) %-> (verif (A ⊃ B)) %. +%term ⊃E %pi (use A (B₁ ⊃ B₂)) %-> (verif B₁) %-> (use A B₂) %. +%term ∧I %pi (verif A₁) %-> (verif A₂) %-> (verif (A₁ ∧ A₂)) %. +%term ∧E₁ %pi (use A (B₁ ∧ B₂)) %-> (use A B₁) %. +%term ∧E₂ %pi (use A (B₁ ∧ B₂)) %-> (use A B₂) %. +%{! ## Global completeness !}% +%{! Because we have changed the logic, the η-expansion theorem has changed somewhat. Recall that, in the pure system of [verifications and uses](/wiki/verifications-and-uses/), the statement of global completeness/η-expansion was that, for all ``A``, we can verify the truth of ``A`` in any context where we can use the fact that ``A`` is true. In this system, however, we don't ever use ``A`` in a vacuum - we always take a hypothesis of ``B`` and then prove that, given ``B``, we can use ``A``. +The eta expansion theorem reflects this: it says that, for any ``A``, given an arbitrary ``B`` that we have hypothesized to be true and a proof +that we can use ``A`` given ``B``, we can prove ``A``. + +The structure of the η-expansion theorem's proof is then mostly unchanged from the system of [verifications and uses](/wiki/verifications-and-uses/). !}% +%sort eta {A} {B} %. +%mode eta %in %out %. +%term _ eta (a Q) ([B] [x] [r] atm x r) %. +%term _ + %pi (eta (A₁ ⊃ A₂) ([B] [x] [r] ⊃I ([y] N₂ B x (⊃E r (N₁ A₁ y end))))) + %<- (eta A₁ ([B] %the (%pi (hyp B) %-> (use B A₁) %-> (verif A₁)) (N₁ B))) + %<- (eta A₂ ([B] %the (%pi (hyp B) %-> (use B A₂) %-> (verif A₂)) (N₂ B))) %. +%term _ + %pi (eta (A₁ ∧ A₂) ([B] [x] [r] ∧I (N₁ B x (∧E₁ r)) (N₂ B x (∧E₂ r)))) + %<- (eta A₁ ([B] %the (%pi (hyp B) %-> (use B A₁) %-> (verif A₁)) (N₁ B))) + %<- (eta A₂ ([B] %the (%pi (hyp B) %-> (use B A₂) %-> (verif A₂)) (N₂ B))) %. +%worlds (bl_atom bl_hyp) (eta _ _) %. +%total A (eta A _) %. +%{! Let's see some examples of η-expansions: | check = decl !}% +%solve _ : {q1} {q2} {q3} eta (a q1 ⊃ a q2 ⊃ a q3) (X q1 q2 q3) %. +%{! | check = decl !}% +%solve _ : {q1} {q2} {q3} eta (a q1 ⊃ a q2 ⊃ a q3) (X q1 q2 q3) %. +%{! ## Global soundness !}% +%{! The real benefit of this modified natural deduction system is that we no longer need to take any detours to prove the hereditary substitution theorem: when we reach +the ``atm`` case where we must verify an atomic proposition by using a proof of its truth, we know exactly what hypothesis we are using. If the +hypothesis is the one we're substituting for, then we call out to the ``hsubst_rn`` theorem where we do repeated reductions, and if the hypothesis +is not, we call out to the ``hsubst_rr`` theorem where the structure of the use stays essentially the same. !}% +%sort hsubst_n {A} %. +%sort hsubst_rr {A} %. +%sort hsubst_rn {A} {B} %. +%mode hsubst_n %in %in %in %out %. +%mode hsubst_rr %in %in %in %out %. +%mode hsubst_rn %in %in %in %in %out %. +%term _ + %pi (hsubst_n A M₀ ([x] ⊃I ([y] M x y)) (⊃I ([y] N y))) + %<- ({y hyp B₁} hsubst_n A M₀ ([x] M x y) (%the (verif B₂) (N y))) %. +%term _ + %pi (hsubst_n A M₀ ([x] ∧I (M₁ x) (M₂ x)) (∧I N₁ N₂)) + %<- (hsubst_n A M₀ ([x] M₁ x) (%the (verif B₁) N₁)) + %<- (hsubst_n A M₀ ([x] M₂ x) (%the (verif B₂) N₂)) %. +%term _ %pi (hsubst_n A M₀ ([x] atm x (R x)) N) %<- (hsubst_rn A (a Q) M₀ ([x] R x) N) %. +%term _ + %pi (hsubst_n A M₀ ([x] atm Y (R x)) (atm Y R')) + %<- (hsubst_rr A M₀ ([x] R x) R') %. +%term _ + %pi (hsubst_rr A M₀ ([x] ⊃E (R x) (M x)) (⊃E R' N)) + %<- (hsubst_rr A M₀ ([x] R x) R') + %<- (hsubst_n A M₀ ([x] M x) N) %. +%term _ %pi (hsubst_rr A M₀ ([x] ∧E₁ (R x)) (∧E₁ R')) %<- (hsubst_rr A M₀ ([x] R x) R') %. +%term _ %pi (hsubst_rr A M₀ ([x] ∧E₂ (R x)) (∧E₂ R')) %<- (hsubst_rr A M₀ ([x] R x) R') %. +%term _ hsubst_rr A M₀ ([x] end) end %. +%term _ + %pi (hsubst_rn A B₂ M₀ ([x] ⊃E (R x) (M x)) N') + %<- (hsubst_rn A (B₁ ⊃ B₂) M₀ ([x] R x) (%the (verif (B₁ ⊃ B₂)) (⊃I ([y] N y)))) + %<- (hsubst_n A M₀ ([x] M x) (%the (verif B₁) M')) + %<- (hsubst_n B₁ M' ([y] N y) (%the (verif B₂) N')) %. +%term _ + %pi (hsubst_rn A B₁ M₀ ([x] ∧E₁ (R x)) N₁) + %<- (hsubst_rn A (B₁ ∧ B₂) M₀ ([x] R x) (%the (verif (B₁ ∧ B₂)) (∧I N₁ N₂))) %. +%term _ + %pi (hsubst_rn A B₂ M₀ ([x] ∧E₂ (R x)) N₂) + %<- (hsubst_rn A _ M₀ ([x] R x) (%the (verif (B₁ ∧ B₂)) (∧I N₁ N₂))) %. +%term _ hsubst_rn A A M₀ ([x] end) M₀ %. +%worlds (bl_atom bl_hyp) (hsubst_n _ _ _ _) (hsubst_rr _ _ _ _) (hsubst_rn _ _ _ _ _) %. +%reduces <= B A (hsubst_rn A B _ _ _) %. +%total {(A B C) (M R S)} (hsubst_n A _ M _) (hsubst_rr B _ R _) (hsubst_rn C _ _ S _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Verifications_and_uses_with_zippers). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/verifications-and-uses.lf b/new-tests/stelf-output/pages/verifications-and-uses.lf new file mode 100644 index 0000000..3cdf342 --- /dev/null +++ b/new-tests/stelf-output/pages/verifications-and-uses.lf @@ -0,0 +1,221 @@ +% +%{! This article describes a Twelf formalization of +logic in terms of _verifications_ and _uses_ and a Twelf proof of global soundness and completeness +for this logic. For a full discussion of logics of verifications and uses, see +Frank Pfenning's [http://www.cs.cmu.edu/~fp/courses/15816-s10/lectures/01-judgments.pdf lecture notes] +from the spring 2010 course on Modal Logic. +Two critical properties of a logic are its _global completeness_ - that in any situation where we can +use a proof of ``A`` we can also verify ``A`` - +and its _global soundness_ - if we can verify the truth of proposition ``A``, and then use an assumption +that ``A`` is true to verify the truth of ``B``, then we can also verify the truth of +``B`` without the assumption that ``A`` is true. + +We can also think of the logic of verifications of uses, by way of the [[w:Curry-Howard correspondence|Curry-Howard correspondence]], as +a ("Church-style") [intrinsic encoding](/wiki/intrinsic-and-extrinsic-encodings/) of the [canonical forms](/wiki/canonical-form/) of the [simply-typed lambda calculus](/wiki/simply-typed-lambda-calculus/). In this view, the +constructive content of the global completeness theorem is η-expansion and the constructive content of +global soundness is [hereditary substitution](/wiki/hereditary-substitution/). Therefore, this article is closely connected to the case +study on [hereditary substitution for the STLC](/wiki/hereditary-substitution-for-the-stlc/). The case study is structured quite differently than this +article, however. That article defines a terminating partial function called "hereditary substitution" +that operates on untyped ("Curry-style") lambda-calculus terms, and then works through the process of showing that, +given well-typed inputs, the function is actually total. That view is helpful when thinking about dependent type +systems, [[bidirectional type checking]], or the implementation of a logical framework, but +certain problems can be greatly simplified when we think of terms as intrinsically typed. !}% +%{! ## Propositions and rules !}% +%{! The following is a straightforward representation of a natural deduction +system with verifications and uses, with one exception: in most presentations, +``use A`` and ``hyp A`` are conflated, which makes the +``var`` rule unnecessary. However, it is somewhat convenient +to do things this way, and also foreshadows a number of other interesting +developments. !}% +%sort prop %. +%sort atom %. +%block bl_atom [qp atom]%. +%term a %pi atom %-> prop %. +%term ⊃ %pi prop %-> prop %-> prop %. +%prec %right 9 ⊃ %. +%term ∧ %pi prop %-> prop %-> prop %. +%prec %right 8 ∧ %. +%sort hyp %. +%sort verif %. +%sort use %. +%block bl_hyp {A prop} [x hyp A]%. +%term var %pi (hyp A) %-> (use A) %. +%term atm %pi (use (a Q)) %-> (verif (a Q)) %. +%term ⊃I %pi (%pi (hyp A) %-> (verif B)) %-> (verif (A ⊃ B)) %. +%term ⊃E %pi (use (A ⊃ B)) %-> (verif A) %-> (use B) %. +%term ∧I %pi (verif A) %-> (verif B) %-> (verif (A ∧ B)) %. +%term ∧E₁ %pi (use (A ∧ B)) %-> (use A) %. +%term ∧E₂ %pi (use (A ∧ B)) %-> (use B) %. +%{! ## Global completeness !}% +%{! The global completeness (or η-expansion) theorem for the logic of verifications and uses +is expressed by the theorem: if we can use a proof that use ``A`` under certain assumptions, then +under the same assumptions we can verify that ``A`` is true. It is therefore critical to note that +the output of the global completeness theorem is a function ``(use A -> verif A)`` and not a function +``(hyp A -> verif A)``. Assumptions of ``hyp A`` alone are sufficient to describe the logic itself, +but in order to describe completeness we have to actually assume proofs of ``use A``. !}% +%sort eta {A prop} %. +%mode eta %in %out %. +%term _ eta (a Q) ([r] atm r) %. +%term _ + %pi (eta (A ⊃ B) ([r] ⊃I ([y] N₂ (⊃E r (N₁ (var y)))))) + %<- (eta A ([r₁] N₁ r₁)) + %<- (eta B ([r₂] N₂ r₂)) %. +%term _ + %pi (eta (A ∧ B) ([r] ∧I (N₁ (∧E₁ r)) (N₂ (∧E₂ r)))) + %<- (eta A ([r₁] N₁ r₁)) + %<- (eta B ([r₂] N₂ r₂)) %. +%worlds (bl_atom bl_hyp) (eta _ _) %. +%total A (eta A _) %. +%{! ## Detour !}% +%{! Before we proceed to global soundness, we +have to deal with the core annoyance of doing things +natural-deduction style as opposed to sequent-calculus style. +It is unavoidable that we must consider the case where we substitute +a verification into a use - if we associate the the metavariables +``M`` and ``N`` with derivations of ``verif A`` and +the metavariable ``R`` with +derivations of ``use B``, this looks like ``[N/x]R``. In these +cases, we have to do something very different depending on whether +the "head variable" (the variable all the way on the inside of ``R``) +is ``x`` (the variable we're substituting for) or +whether it is something else. +Specifically, if the variable is ``x`` we need to substitute +``N`` in and then perform a series of reductions, but +if it is something else we can leave the structure of the term basically the +same. + +The tutorial on [hereditary substitution for the STLC](/wiki/hereditary-substitution-for-the-stlc/) deals with this +problem by defining hereditary substitution in such a way that Twelf cannot +immediately establish that it is total, and then showing totality after the +fact. We want to immediately establish totality, and to do this we will +define a judgment that asks "am I in the case where I need to have a series +of reductions performed (``reduce_me``), or are such +reductions unnecessary (``just_fine``)? !}% +%sort just_fine %. +%term jfx just_fine ([x] var Y) %. +%term jf⊃ %pi (just_fine ([x] R x)) %-> (just_fine ([x] ⊃E (R x) (M x))) %. +%term jf∧₁ %pi (just_fine ([x] R x)) %-> (just_fine ([x] ∧E₁ (R x))) %. +%term jf∧₂ %pi (just_fine ([x] R x)) %-> (just_fine ([x] ∧E₂ (R x))) %. +%sort reduce_me %. +%term rmx reduce_me ([x] var x) %. +%term rm⊃ %pi (reduce_me ([x] R x)) %-> (reduce_me ([x] ⊃E (R x) (M x))) %. +%term rm∧₁ %pi (reduce_me ([x] R x)) %-> (reduce_me ([x] ∧E₁ (R x))) %. +%term rm∧₂ %pi (reduce_me ([x] R x)) %-> (reduce_me ([x] ∧E₂ (R x))) %. +%sort jf_or_rm %. +%term rm %pi (reduce_me ([x] R x)) %-> (jf_or_rm ([x] R x)) %. +%term jf %pi (just_fine ([x] R x)) %-> (jf_or_rm ([x] R x)) %. +%{! Now, we prove a metatheorem that _every_ possible substitution instance +either needs to ask hereditary substitution to ``reduce_me`` +or else is ``just_fine``. This has to use a bunch of +[output factoring](/wiki/output-factoring/) lemmas but is otherwise straightforward. !}% +%sort always_jf_or_rm {R %pi (hyp A) %-> (use B)} %. +%mode always_jf_or_rm %in %out %. +%term _ always_jf_or_rm ([x] var x) (rm rmx) %. +%term _ always_jf_or_rm ([x] var Y) (jf jfx) %. +%sort lem %. +%term _ lem (jf JF) _ (jf (jf⊃ JF)) %. +%term _ lem (rm RM) _ (rm (rm⊃ RM)) %. +%mode lem %in %in %out %. +%worlds (bl_atom bl_hyp) (lem _ _ _) %. +%total {} (lem _ _ _) %. +%term _ + %pi (always_jf_or_rm ([x] ⊃E (R x) (N x)) JFRM') + %<- (always_jf_or_rm ([x] R x) (%the (jf_or_rm ([x] R x)) JFRM)) + %<- (lem JFRM ([x] N x) (%the (jf_or_rm ([x] ⊃E (R x) (N x))) JFRM')) %. +%sort lem %. +%term _ lem (jf JF) (jf (jf∧₁ JF)) %. +%term _ lem (rm RM) (rm (rm∧₁ RM)) %. +%mode lem %in %out %. +%worlds (bl_atom bl_hyp) (lem _ _) %. +%total {} (lem _ _) %. +%term _ + %pi (always_jf_or_rm ([x] ∧E₁ (R x)) JFRM') + %<- (always_jf_or_rm ([x] R x) (%the (jf_or_rm ([x] R x)) JFRM)) + %<- (lem JFRM (%the (jf_or_rm ([x] ∧E₁ (R x))) JFRM')) %. +%sort lem %. +%term _ lem (jf JF) (jf (jf∧₂ JF)) %. +%term _ lem (rm RM) (rm (rm∧₂ RM)) %. +%mode lem %in %out %. +%worlds (bl_atom bl_hyp) (lem _ _) %. +%total {} (lem _ _) %. +%term _ + %pi (always_jf_or_rm ([x] ∧E₂ (R x)) JFRM') + %<- (always_jf_or_rm ([x] R x) (%the (jf_or_rm ([x] R x)) JFRM)) + %<- (lem JFRM (%the (jf_or_rm ([x] ∧E₂ (R x))) JFRM')) %. +%worlds (bl_atom bl_hyp) (always_jf_or_rm _ _) %. +%total R (always_jf_or_rm R _) %. +%{! One way to avoid this ugly detour is to use [[spine form]], another way is considered in the page on [verifications and uses with zippers](/wiki/verifications-and-uses-with-zippers/). !}% +%{! ## Global soundness !}% +%{! Showing that every (hyp A -> use B) always either is ``just_fine`` +or needs hereditary substitution to ``reduce_me`` is the entirety of the +"ugly" part of the hereditary substitution/global completeness, theorem. The +theorem is made up of four mutually inductive theorems. + +* ``hsubst_n`` represents substitutions ``[M<sub>0</sub>/x]M``. +* ``hsubst_r`` represents substitutions ``[M<sub>0</sub>/x]R`` where ``R`` has atomic type. It is basically an [output factoring](/wiki/output-factoring/) lemma that dispatches to ``hsubst_rr`` and ``hsubst_rn``. +* ``hsubst_rr`` represents substitutions ``[M<sub>0</sub>/x]R`` where the variable ``x`` is not the root of ``R``. +* ``hsubst_rn`` represents substitutions ``[M<sub>0</sub>/x]R`` where the variable ``x`` **is** the root of ``R``. This means that if we substitute ``M<sub>0</sub>`` for x and then start reducing the use ``R``, we will get back a verification ``N``. Luckily, the type of ``N`` is known ahead of time, and so we can use this to know something about the shape of ``N``. This case makes critical use of a [``%reduces``](/wiki/percent-reduces/) declaration, and is also the only case where the type ``A`` is different in a recursive call. !}% +%sort hsubst_n {A} {M₀ verif A} {M %pi (hyp A) %-> (verif B)} %. +%sort hsubst_r {A} {M₀ verif A} {R %pi (hyp A) %-> (use (a Q))} %. +%sort hsubst_rr {A} {M₀ verif A} {R %pi (hyp A) %-> (use B)} %. +%sort hsubst_rn {A} {B} {M₀ verif A} {R %pi (hyp A) %-> (use B)} %. +%mode hsubst_n %in %in %in %out %. +%mode hsubst_r %in %in %in %in %out %. +%mode hsubst_rr %in %in %in %in %out %. +%mode hsubst_rn %in %in %in %in %in %out %. +%term _ + %pi (hsubst_n A M₀ ([x] ⊃I ([y] M x y)) (⊃I ([y] N y))) + %<- ({y hyp B₁} hsubst_n A M₀ ([x] M x y) (%the (verif B₂) (N y))) %. +%term _ + %pi (hsubst_n A M₀ ([x] ∧I (M₁ x) (M₂ x)) (∧I N₁ N₂)) + %<- (hsubst_n A M₀ ([x] M₁ x) (%the (verif B₁) N₁)) + %<- (hsubst_n A M₀ ([x] M₂ x) (%the (verif B₂) N₂)) %. +%term _ + %pi (hsubst_n A M₀ ([x] atm (R x)) N) + %<- (always_jf_or_rm ([x] R x) (%the (jf_or_rm ([x] R x)) JFRM)) + %<- (hsubst_r A M₀ ([x] R x) JFRM N) %. +%term _ + %pi (hsubst_r A M₀ ([x] R x) (jf JF) (atm R')) + %<- (hsubst_rr A M₀ ([x] R x) JF R') %. +%term _ %pi (hsubst_r A M₀ ([x] R x) (rm RM) N) %<- (hsubst_rn A _ M₀ ([x] R x) RM N) %. +%term _ + %pi (hsubst_rr A M₀ ([x] ⊃E (R x) (M x)) (jf⊃ JF) (⊃E R' N)) + %<- (hsubst_rr A M₀ ([x] R x) JF R') + %<- (hsubst_n A M₀ ([x] M x) N) %. +%term _ + %pi (hsubst_rr A M₀ ([x] ∧E₁ (R x)) (jf∧₁ JF) (∧E₁ R')) + %<- (hsubst_rr A M₀ ([x] R x) JF R') %. +%term _ + %pi (hsubst_rr A M₀ ([x] ∧E₂ (R x)) (jf∧₂ JF) (∧E₂ R')) + %<- (hsubst_rr A M₀ ([x] R x) JF R') %. +%term _ hsubst_rr A M₀ ([x] var Y) jfx (var Y) %. +%term _ + %pi (hsubst_rn A B₂ M₀ ([x] ⊃E (R x) (M x)) (rm⊃ RM) N') + %<- (hsubst_rn A (B₁ ⊃ B₂) M₀ ([x] R x) RM (%the (verif (B₁ ⊃ B₂)) (⊃I ([y] N y)))) + %<- (hsubst_n A M₀ ([x] M x) (%the (verif B₁) M')) + %<- (hsubst_n B₁ M' N (%the (verif B₂) N')) %. +%term _ + %pi (hsubst_rn A B₁ M₀ ([x] ∧E₁ (R x)) (rm∧₁ RM) N₁) + %<- (hsubst_rn A (B₁ ∧ B₂) M₀ ([x] R x) RM (%the (verif (B₁ ∧ B₂)) (∧I N₁ N₂))) %. +%term _ + %pi (hsubst_rn A B₂ M₀ ([x] ∧E₂ (R x)) (rm∧₂ RM) N₂) + %<- (hsubst_rn A (B₁ ∧ B₂) M₀ ([x] R x) RM (%the (verif (B₁ ∧ B₂)) (∧I N₁ N₂))) %. +%term _ hsubst_rn A A M₀ ([x] var x) rmx M₀ %. +%worlds (bl_atom bl_hyp) (hsubst_n _ _ _ _) (hsubst_r _ _ _ _ _) (hsubst_rr _ _ _ _ _) (hsubst_rn _ _ _ _ _ _) %. +%{! A critical part of proving termination is showing that, during the hereditary substitution process, the second argument is no larger than the first - this is why +we are allowed to make the recursive call ``hsubst_n B<sub>1</sub> M' N (N': verif B<sub>2</sub>)`` in the ``⊃E`` case of ``hsubst_rn``: we know ``A`` is no larger than ``B<sub>1</sub> ⊃ B<sub>2</sub>``, and therefore ``B<sub>2</sub>`` is strictly smaller. +This is established by the following [``%reduces``](/wiki/percent-reduces/) declaration. !}% +%reduces <= B A (hsubst_rn A B _ _ _ _) %. +%{! The completeness of hereditary substitution is established first by lexicographic induction, +first on the type ``A`` and second on the term ``M`` +that we are substituting into. +The statement of ``hsubst_r`` has to come after ``hsubst_rr`` and +``hsubst_rn`` in order for termination checking to work, because the +former theorem calls the latter two theorems with all the same arguments. !}% +%total {(A B C D) (M R S T)} (hsubst_n A _ M _) (hsubst_rr C _ S _ _) (hsubst_rn D _ _ T _ _) (hsubst_r B _ R _ _) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Verifications_and_uses). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/weak-focusing.lf b/new-tests/stelf-output/pages/weak-focusing.lf new file mode 100644 index 0000000..740180b --- /dev/null +++ b/new-tests/stelf-output/pages/weak-focusing.lf @@ -0,0 +1,431 @@ +% +%{! ## Polarized logic !}% +%sort polarity %. +%name polarity %. +%term pos polarity %. +%term neg polarity %. +%sort atm %. +%name atm %. +%sort prop %. +%name prop %. +%term atom %pi (atm S) %-> (prop S) %. +%term up %pi (prop pos) %-> (prop neg) %. +%term down %pi (prop neg) %-> (prop pos) %. +%term * %pi (prop pos) %-> (prop pos) %-> (prop pos) %. +%prec %right 10 * %. +%term + %pi (prop pos) %-> (prop pos) %-> (prop pos) %. +%prec %right 9 + %. +%scope + %term o %pi (prop pos) %-> (prop neg) %-> (prop neg) %. +%prec %right 8 -o %. +%block bl_atmpos [qp atm pos]%. +%block bl_atmneg [qn atm neg]%. +%{! ## Sequent calculus for polarized logic !}% +%{! ### Logic definition !}% +%{! +A sequent is written + h1: hyp A1, ..., hn: hyp An |- conc A + A1, ..., An |- A +!}% +%sort hyp %. +%name hyp %. +%sort conc %. +%name conc %. +%block bl_hyp {S polarity} {A prop S} [h hyp A]%. +%term init %pi (hyp (atom Q)) %-> (conc (atom Q)) %. +%term upR %pi (conc P) %-> (conc (up P)) %. +%term upL %pi (%pi (hyp P) %-> (conc C)) %-> (hyp (up P)) %-> (conc C) %. +%term downL %pi (%pi (hyp N) %-> (conc C)) %-> (hyp (down N)) %-> (conc C) %. +%term downR %pi (conc N) %-> (conc (down N)) %. +%term *R %pi (conc P1) %-> (conc P2) %-> (conc (P1 * P2)) %. +%term *L %pi (%pi (hyp P1) %-> (hyp P2) %-> (conc C)) %-> (hyp (P1 * P2)) %-> (conc C) %. +%term +R1 %pi (conc P1) %-> (conc (P1 + P2)) %. +%term +R2 %pi (conc P2) %-> (conc (P1 + P2)) %. +%term +L + %pi (%pi (hyp P1) %-> (conc C)) + %-> (%pi (hyp P2) %-> (conc C)) + %-> (hyp (P1 + P2)) + %-> (conc C) %. +%scope +L %term oR %pi (%pi (hyp P) %-> (conc N)) %-> (conc (P -o N)) %. +%term oL %pi (conc P) %-> (%pi (hyp N) %-> (conc C)) %-> (hyp (P -o N)) %-> (conc C) %. +%worlds (bl_atmpos bl_atmneg bl_hyp) (hyp A) (conc C) %. +%{! ## Focused sequent calculus !}% +%{! ### Logic definition !}% +%{! +Neutral sequent: + l1:left P1, ..., ln:left Pn |- right N + P1, ..., Pn |- N + +Left-focused sequent: + l1:left P1, ..., ln:left Pn |- lfoc N1 N2 + P1, ..., Pn [N1] |- N2 + +Right-focused sequent: + l1:left P1, ..., ln:left Pn |- rfoc P + P1, ..., Pn |- [P] + +These can be generically expressed as: + l1:left P1, ..., ln:left Pn |- conseq C + P1, ..., Pn |- C +!}% +%sort left %. +%name left %. +%sort conclusion %. +%name conclusion %. +%sort conseq %. +%name conseq %. +%term conc-lfoc %pi (prop neg) %-> (prop neg) %-> conclusion %. +%term conc-rfoc %pi (prop pos) %-> conclusion %. +%term conc-right %pi (prop neg) %-> conclusion %. +%inline rfoc [P] conseq (conc-rfoc P) %. +%inline lfoc [N] [N'] conseq (conc-lfoc N N') %. +%inline right [N] conseq (conc-right N) %. +%block bl_left {P prop pos} [l left P]%. +%term initR' %pi (left (atom Qp)) %-> (rfoc (atom Qp)) %. +%term initL' lfoc (atom Qn) (atom Qn) %. +%term upR' %pi (rfoc P) %-> (right (up P)) %. +%term upL' %pi (%pi (left P) %-> (right N)) %-> (lfoc (up P) N) %. +%term downR' %pi (right N) %-> (rfoc (down N)) %. +%term downL' %pi (lfoc N N') %-> (left (down N)) %-> (right N') %. +%term *R' %pi (rfoc P1) %-> (rfoc P2) %-> (rfoc (P1 * P2)) %. +%term *L' + %pi (%pi (left P1) %-> (left P2) %-> (right N')) + %-> (left (P1 * P2)) + %-> (right N') %. +%term +R1' %pi (rfoc P1) %-> (rfoc (P1 + P2)) %. +%term +R2' %pi (rfoc P2) %-> (rfoc (P1 + P2)) %. +%term +L' + %pi (%pi (left P1) %-> (right N')) + %-> (%pi (left P2) %-> (right N')) + %-> (left (P1 + P2)) + %-> (right N') %. +%scope +L' %term oR' %pi (%pi (left P) %-> (right N)) %-> (right (P -o N)) %. +%term oL' %pi (rfoc P) %-> (lfoc N N') %-> (lfoc (P -o N) N') %. +%worlds (bl_atmpos bl_atmneg bl_left) (left P) (conseq C) %. +% = (right N) (left N N') (rfoc P). +%{! ### Cut admissibility !}% +%sort cut< {N} %. +%sort cut- {N} %. +%sort cut+ {P} %. +%mode cut< %in %in %in %out %. +%mode cut- %in %in %in %out %. +%mode cut+ %in %in %in %out %. +%{! #### Identity cuts !}% +%term _ cut+ (atom Qp) (initR' D) ([l] initR' l) (initR' D) %. +%term _ cut< (atom Qn) initL' initL' initL' %. +%{! #### Principal cuts !}% +%term _ + %pi (cut- (up P) (upR' (%the (rfoc P) D)) (upL' (%the (%pi (left P) %-> (right N)) E)) F) + %<- (cut+ P D E (%the (right N) F)) %. +%term _ + %pi (cut+ (down N) (downR' (%the (right N) D)) ([l left (down N)] downL' (%the (lfoc N NC) (E l)) l) G) + %<- (cut+ (down N) (downR' D) E (%the (lfoc N NC) F)) + %<- (cut- N D F (%the (right NC) G)) %. +%term _ + %pi (cut+ (P1 * P2) (*R' (%the (rfoc P1) D1) (%the (rfoc P2) D2)) ([l left (P1 * P2)] *L' (%the (%pi (left P1) %-> (left P2) %-> (right N)) (E l)) l) G) + %<- ({l1 left P1} {l2 left P2} cut+ (P1 * P2) (*R' D1 D2) ([l left (P1 * P2)] E l l1 l2) (%the (right N) (F l1 l2))) + %<- ({l1 left P1} cut+ P2 D2 ([l2 left P2] F l1 l2) (%the (right N) (F' l1))) + %<- (cut+ P1 D1 F' (%the (right N) G)) %. +%term _ + %pi (cut+ (P1 + P2) (+R1' (%the (rfoc P1) D1)) ([l left (P1 + P2)] +L' (%the (%pi (left P1) %-> (right N)) (E1 l)) (E2 l) l) F) + %<- ({l1 left P1} cut+ (P1 + P2) (+R1' D1) ([l left (P1 + P2)] E1 l l1) (%the (right N) (E1' l1))) + %<- (cut+ P1 D1 E1' (%the (right N) F)) %. +%term _ + %pi (cut+ (P1 + P2) (+R2' (%the (rfoc P2) D2)) ([l left (P1 + P2)] +L' (E1 l) (%the (%pi (left P2) %-> (right N)) (E2 l)) l) F) + %<- ({l2 left P2} cut+ (P1 + P2) (+R2' D2) ([l left (P1 + P2)] E2 l l2) (%the (right N) (E2' l2))) + %<- (cut+ P2 D2 E2' (%the (right N) F)) %. +%term _ + %pi (cut- (N -o P) (-oR' (%the (%pi (left N) %-> (right P)) D1)) (-oL' (%the (rfoc N) E1) (%the (lfoc P NC) E2)) F) + %<- (cut+ N E1 D1 (%the (right P) F1)) + %<- (cut- P F1 E2 (%the (right NC) F)) %. +%{! #### Left commutative cuts !}% +%term _ %pi (cut< N (upL' D) E (upL' F)) %<- ({l1} cut- N (D l1) E (F l1)) %. +%term _ %pi (cut- N (downL' D L) E (downL' F L)) %<- (cut< N D E F) %. +%term _ %pi (cut- N (*L' D L) E (*L' F L)) %<- ({l1} {l2} cut- N (D l1 l2) E (F l1 l2)) %. +%term _ + %pi (cut- N (+L' D1 D2 L) E (+L' F1 F2 L)) + %<- ({l1} cut- N (D1 l1) E (F1 l1)) + %<- ({l2} cut- N (D2 l2) E (F2 l2)) %. +%term _ %pi (cut< N (-oL' D1 D2) E (-oL' D1 F2)) %<- (cut< N D2 E F2) %. +%{! #### Right commutative cuts !}% +%term _ cut+ P D ([l] initR' L) (initR' L) %. +%term _ cut+ P D ([l] initL') initL' %. +%term _ %pi (cut+ P D ([l] upR' (E l)) (upR' F)) %<- (cut+ P D ([l] E l) F) %. +%term _ + %pi (cut+ P D ([l] upL' ([l1] E l l1)) (upL' F)) + %<- ({l1} cut+ P D ([l] E l l1) (F l1)) %. +%term _ %pi (cut+ P D ([l] downR' (E l)) (downR' F)) %<- (cut+ P D ([l] E l) F) %. +%term _ %pi (cut+ P D ([l] downL' (E l) L) (downL' F L)) %<- (cut+ P D ([l] E l) F) %. +%term _ + %pi (cut+ P D ([l] *R' (E1 l) (E2 l)) (*R' F1 F2)) + %<- (cut+ P D ([l] E1 l) F1) + %<- (cut+ P D ([l] E2 l) F2) %. +%term _ + %pi (cut+ P D ([l] *L' ([l1] [l2] E l l1 l2) L) (*L' ([l1] [l2] F l1 l2) L)) + %<- ({l1} {l2} cut+ P D ([l] E l l1 l2) (F l1 l2)) %. +%term _ %pi (cut+ P D ([l] +R1' (E l)) (+R1' F)) %<- (cut+ P D ([l] E l) F) %. +%term _ %pi (cut+ P D ([l] +R2' (E l)) (+R2' F)) %<- (cut+ P D ([l] E l) F) %. +%term _ + %pi (cut+ P D ([l] +L' ([l1] E1 l l1) ([l2] E2 l l2) L) (+L' F1 F2 L)) + %<- ({l1} cut+ P D ([l] E1 l l1) (F1 l1)) + %<- ({l2} cut+ P D ([l] E2 l l2) (F2 l2)) %. +%term _ %pi (cut+ P D ([l] -oR' (E l)) (-oR' F)) %<- ({l1} cut+ P D ([l] E l l1) (F l1)) %. +%term _ + %pi (cut+ P D ([l] -oL' (E1 l) (E2 l)) (-oL' F1 F2)) + %<- (cut+ P D ([l] E1 l) F1) + %<- (cut+ P D ([l] E2 l) F2) %. +%worlds (bl_atmpos bl_atmneg bl_left) (cut< _ _ _ _) (cut- _ _ _ _) (cut+ _ _ _ _) %. +%total {(N1 N2 P3) [(D1 D2 D3) (E1 E2 E3)]} (cut< N1 D1 E1 _) (cut+ N2 D2 E2 _) (cut- P3 D3 E3 _) %. +%{! #### Corollary - unfocused cut !}% +%sort cut-unfoc+ %. +%mode cut-unfoc+ %in %in %out %. +%term _ %pi (cut-unfoc+ D E F) %<- (cut- (up P) D (upL' E) F) %. +%worlds (bl_atmpos bl_atmneg bl_left) (cut-unfoc+ _ _ _) %. +%total [] (cut-unfoc+ _ _ _) %. +%sort cut-unfoc- %. +%mode cut-unfoc- %in %in %out %. +%term _ %pi (cut-unfoc- D E F) %<- (cut+ (down A) (downR' D) E F) %. +%worlds (bl_atmpos bl_atmneg bl_left) (cut-unfoc- _ _ _) %. +%total [] (cut-unfoc- _ _ _) %. +%{! ### Identity !}% +%{! #### Eta expansion lemmas !}% +%sort eta- {N prop neg} {N' prop neg} %. +%sort eta+ {P prop pos} %. +%mode eta- %in %in %out %. +%mode eta+ %in %in %out %. +%block bl_rfoc {P prop pos} [rf rfoc P]%. +%term _ eta+ (atom Qp) ([rf] R rf) ([l] R (initR' l)) %. +%term _ eta- (atom Qn) ([n'] [lf] R n' lf) (R (atom Qn) initL') %. +%term _ + %pi (eta- (up P) ([n'] [lf] R n' lf) (R (up P) (upL' ([l] R' l)))) + %<- (eta+ P ([rf] upR' rf) ([l] R' l)) %. +%term _ + %pi (eta+ (down N) ([rf] R rf) ([l left (down N)] R (downR' (R' l)))) + %<- ({l left (down N)} eta- N ([n'] [lf] downL' lf l) (R' l)) %. +%term _ + %pi (eta+ (P1 * P2) ([rf] R rf) ([l] *L' ([l1] [l2] R'' l1 l2) l)) + %<- ({rf1 rfoc P1} eta+ P2 ([rf2] R (*R' rf1 rf2)) ([l2] R' rf1 l2)) + %<- ({l2 left P2} eta+ P1 ([rf1] R' rf1 l2) ([l1] R'' l1 l2)) %. +%term _ + %pi (eta+ (P1 + P2) ([rf] R rf) (+L' R1 R2)) + %<- (eta+ P1 ([rf1] R (+R1' rf1)) ([l1 left P1] %the (right N) (R1 l1))) + %<- (eta+ P2 ([rf1] R (+R2' rf1)) ([l2 left P2] %the (right N) (R2 l2))) %. +%term _ + %pi (eta- (P1 -o N2) ([n'] [lf] R n' lf) (-oR' ([l1] R1 l1))) + %<- ({rf1 rfoc P1} eta- N2 ([n'] [lf2] R n' (-oL' rf1 lf2)) (R2 rf1)) + %<- (eta+ P1 ([rf1] R2 rf1) ([l1] R1 l1)) %. +%worlds (bl_atmpos bl_atmneg bl_left bl_rfoc) (eta- N R R') (eta+ P R R') %. +%total (N P) (eta- N _ _) (eta+ P _ _) %. +%{! #### Identity corollaries !}% +%sort idp {P prop pos} %. +%mode idp %in %out %. +%term _ %pi (idp P R) %<- (eta+ P ([rf] upR' rf) R) %. +%worlds (bl_atmpos bl_atmneg bl_left bl_rfoc) (idp _ _) %. +%total [] (idp _ _) %. +%sort idn {N prop neg} %. +%mode idn %in %out %. +%term _ + %pi (idn N ([l] R l)) + %<- ({l left (down N)} eta- N ([qn] [lf] downL' lf l) (R l)) %. +%worlds (bl_atmpos bl_atmneg bl_left bl_rfoc) (idn _ _) %. +%total [] (idn _ _) %. +%{! ## Soundness !}% +%sort sdR %. +%sort sd+ %. +%sort sd- %. +%sort sdL %. +%mode sdR %in %out %. +%mode sd+ %in %out %. +%mode sd- %in %out %. +%mode sdL %in %out %. +%term _ %pi (sd+ (initR' L) (init H)) %<- (sdL L H) %. +%term _ sd- initL' ([h] init h) %. +%term _ %pi (sdR (upR' R) (upR D)) %<- (sd+ R D) %. +%term _ + %pi (sd- (upL' ([l] R l)) ([h] upL ([hp] D hp) h)) + %<- ({l} {hp} %pi (sdL l hp) %-> (sdR (R l) (D hp))) %. +%term _ %pi (sd+ (downR' R) (downR D)) %<- (sdR R D) %. +%term _ %pi (sdR (downL' R L) (downL ([hn] D hn) H)) %<- (sd- R ([hn] D hn)) %<- (sdL L H) %. +%term _ %pi (sd+ (*R' R1 R2) (*R D1 D2)) %<- (sd+ R1 D1) %<- (sd+ R2 D2) %. +%term _ + %pi (sdR (*L' ([l1] [l2] R l1 l2) L12) (*L ([h1] [h2] D h1 h2) H12)) + %<- ({l1} {h1} %pi (sdL l1 h1) %-> ({l2} {h2} %pi (sdL l2 h2) %-> (sdR (R l1 l2) (D h1 h2)))) + %<- (sdL L12 H12) %. +%term _ %pi (sd+ (+R1' R1) (+R1 D1)) %<- (sd+ R1 D1) %. +%term _ %pi (sd+ (+R2' R2) (+R2 D2)) %<- (sd+ R2 D2) %. +%term _ + %pi (sdR (+L' ([l1] R1 l1) ([l2] R2 l2) L) (+L ([h1] D1 h1) ([h2] D2 h2) H)) + %<- ({l1} {h1} %pi (sdL l1 h1) %-> (sdR (R1 l1) (D1 h1))) + %<- ({l2} {h2} %pi (sdL l2 h2) %-> (sdR (R2 l2) (D2 h2))) + %<- (sdL L H) %. +%term _ + %pi (sdR (-oR' ([l] R l)) (-oR ([h] D h))) + %<- ({l} {h} %pi (sdL l h) %-> (sdR (R l) (D h))) %. +%term _ + %pi (sd- (-oL' R1 R2) ([h] -oL D1 ([h2] D2 h2) h)) + %<- (sd+ R1 D1) + %<- (sd- R2 ([h2] D2 h2)) %. +%block bl_sdL {P prop pos} [l left P] [h hyp P] [_ sdL l h]%. +%worlds (bl_atmpos bl_atmneg bl_sdL) (sdR _ _) (sd+ _ _) (sd- _ _) (sdL _ _) %. +%total [] (sdL _ _) %. +%total (R1 R2 R3) (sdR R1 _) (sd+ R2 _) (sd- R3 _) %. +%{! ## Completeness !}% +%{! ### Unfocused admissibility !}% +%sort adm*R %. +%mode adm*R %in %in %out %. +%term _ + %pi (adm*R (%the (right (up P1)) R1) (%the (right (up P2)) R2) R) + %<- ({rf1 rfoc P1} eta+ P2 ([rf2 rfoc P2] upR' (*R' rf1 rf2)) ([l2 left P2] R3 rf1 l2)) + %<- ({l2 left P2} eta+ P1 ([rf1 rfoc P1] R3 rf1 l2) ([l1] R4 l1 l2)) + %<- ({l2 left P2} cut-unfoc+ R1 ([l1 left P1] R4 l1 l2) (R5 l2)) + %<- (cut-unfoc+ R2 ([l2 left P2] R5 l2) R) %. +%worlds (bl_atmpos bl_atmneg bl_left) (adm*R _ _ _) %. +%total [] (adm*R _ _ _) %. +%sort adm+R1 %. +%mode {%in P1 prop pos} {%in P2 prop pos} {%in R1 right (up P1)} {%out R right (up (P1 + P2))} adm+R1 R1 R %. +%term _ + %pi (adm+R1 (%the (right (up P1)) R1) R) + %<- (eta+ P1 ([rf1 rfoc P1] upR' (+R1' rf1)) (%the (%pi (left P1) %-> (right (up (P1 + P2)))) R1')) + %<- (cut-unfoc+ R1 ([l1] R1' l1) R) %. +%worlds (bl_atmpos bl_atmneg bl_left) (adm+R1 _ _) %. +%total [] (adm+R1 _ _) %. +%sort adm+R2 %. +%mode {%in P1 prop pos} {%in P2 prop pos} {%in R2 right (up P2)} {%out R right (up (P1 + P2))} adm+R2 R2 R %. +%term _ + %pi (adm+R2 (%the (right (up P2)) R2) R) + %<- (eta+ P2 ([rf2 rfoc P2] upR' (+R2' rf2)) (%the (%pi (left P2) %-> (right (up (P1 + P2)))) R2')) + %<- (cut-unfoc+ R2 ([l2] R2' l2) R) %. +%worlds (bl_atmpos bl_atmneg bl_left) (adm+R2 _ _) %. +%total [] (adm+R2 _ _) %. +%sort adm-oL %. +%mode adm-oL %in %in %out %. +%term _ + %pi (adm-oL (%the (right (up P1)) R1) (%the (%pi (left (down N2)) %-> (right N')) R2) (%the (%pi (left (down (P1 -o N2))) %-> (right N')) R)) + %<- ({l left (down (P1 -o N2))} {rf1 rfoc P1} eta- N2 ([n] [lf2] downL' (-oL' rf1 lf2) l) (%the (right N2) (R3 rf1 l))) + %<- ({l} eta+ P1 ([rf1] R3 rf1 l) ([l1] R4 l l1)) + %<- ({l} {l1} cut-unfoc- (R4 l l1) ([l2] R2 l2) (R5 l l1)) + %<- ({l} cut-unfoc+ R1 ([l1] R5 l l1) (R l)) %. +%worlds (bl_atmpos bl_atmneg bl_left) (adm-oL _ _ _) %. +%total [] (adm-oL _ _ _) %. +%{! ### Completeness !}% +%sort cph+ %. +%sort cph- %. +%sort cp- %. +%sort cp+ %. +%mode cph+ %in %out %. +%mode cph- %in %out %. +%mode cp- %in %out %. +%mode cp+ %in %out %. +%term _ %pi (cp+ (init Hp) (upR' (initR' L))) %<- (cph+ Hp L) %. +%term _ %pi (cp- (init Hn) (downL' initL' L)) %<- (cph- Hn L) %. +%term _ %pi (cp- (upR Dp) R) %<- (cp+ Dp R) %. +%term _ + %pi (cp- (upL ([hp] Dn1 hp) Hn) (downL' (upL' ([l] R1 l)) L)) + %<- (cph- Hn L) + %<- ({hp hyp P} {l left P} %pi (cph+ hp l) %-> (cp- (Dn1 hp) (R1 l))) %. +%term _ + %pi (cp+ (upL ([hp] Dp1 hp) Hn) (downL' (upL' ([l] R1 l)) L)) + %<- (cph- Hn L) + %<- ({hp hyp P} {l left P} %pi (cph+ hp l) %-> (cp+ (Dp1 hp) (R1 l))) %. +%term _ %pi (cp+ (downR Dn) (upR' (downR' R))) %<- (cp- Dn R) %. +%term _ + %pi (cp- (downL ([hn] Dn hn) Hp) (R L)) + %<- (cph+ Hp L) + %<- ({hn hyp N} {l left (down N)} %pi (cph- hn l) %-> (cp- (Dn hn) (R l))) %. +%term _ + %pi (cp+ (downL ([hn] Dp hn) Hp) (R L)) + %<- (cph+ Hp L) + %<- ({hn hyp N} {l left (down N)} %pi (cph- hn l) %-> (cp+ (Dp hn) (R l))) %. +%term _ %pi (cp+ (*R Dp1 Dp2) R12) %<- (cp+ Dp1 R1) %<- (cp+ Dp2 R2) %<- (adm*R R1 R2 R12) %. +%term _ + %pi (cp- (*L ([hp1] [hp2] Dn hp1 hp2) Hp) (*L' ([l1] [l2] R l1 l2) L)) + %<- (cph+ Hp L) + %<- ({hp1 hyp P1} {l1 left P1} + %pi (cph+ hp1 l1) + %-> ({hp2 hyp P2} {l2 left P2} %pi (cph+ hp2 l2) %-> (cp- (Dn hp1 hp2) (R l1 l2)))) %. +%term _ + %pi (cp+ (*L ([hp1] [hp2] Dp hp1 hp2) Hp) (*L' ([l1] [l2] R l1 l2) L)) + %<- (cph+ Hp L) + %<- ({hp1 hyp P1} {l1 left P1} + %pi (cph+ hp1 l1) + %-> ({hp2 hyp P2} {l2 left P2} %pi (cph+ hp2 l2) %-> (cp+ (Dp hp1 hp2) (R l1 l2)))) %. +%term _ %pi (cp+ (+R1 Dp1) R12) %<- (cp+ Dp1 R1) %<- (adm+R1 R1 R12) %. +%term _ %pi (cp+ (+R2 Dp2) R12) %<- (cp+ Dp2 R2) %<- (adm+R2 R2 R12) %. +%term _ + %pi (cp- (+L ([h1] D1 h1) ([h2] D2 h2) H) (+L' R1 R2 L)) + %<- (cph+ H L) + %<- ({h1} {l1} %pi (cph+ h1 l1) %-> (cp- (D1 h1) (R1 l1))) + %<- ({h2} {l2} %pi (cph+ h2 l2) %-> (cp- (D2 h2) (R2 l2))) %. +%term _ + %pi (cp+ (+L ([h1] D1 h1) ([h2] D2 h2) H) (+L' R1 R2 L)) + %<- (cph+ H L) + %<- ({h1} {l1} %pi (cph+ h1 l1) %-> (cp+ (D1 h1) (R1 l1))) + %<- ({h2} {l2} %pi (cph+ h2 l2) %-> (cp+ (D2 h2) (R2 l2))) %. +%term _ + %pi (cp- (-oR ([hp] Dn hp)) (-oR' ([l] R l))) + %<- ({hp hyp P} {l left P} %pi (cph+ hp l) %-> (cp- (Dn hp) (R l))) %. +%term _ + %pi (cp- (-oL Dp1 ([hn2] Dn2 hn2) Hn) (R12 L)) + %<- (cph- Hn L) + %<- (cp+ Dp1 R1) + %<- ({hn2 hyp N2} {l2 left (down N2)} %pi (cph- hn2 l2) %-> (cp- (Dn2 hn2) (R2 l2))) + %<- (adm-oL R1 ([l2] R2 l2) ([l] R12 l)) %. +%term _ + %pi (cp+ (-oL Dp1 ([hn2] Dp2 hn2) Hn) (R12 L)) + %<- (cph- Hn L) + %<- (cp+ Dp1 R1) + %<- ({hn2 hyp N2} {l2 left (down N2)} %pi (cph- hn2 l2) %-> (cp+ (Dp2 hn2) (R2 l2))) + %<- (adm-oL R1 ([l2] R2 l2) ([l] R12 l)) %. +%block bl_cph+ {P prop pos} [hp hyp P] [l left P] [_ cph+ hp l]%. +%block bl_cph- {N prop neg} [hn hyp N] [l left (down N)] [_ cph- hn l]%. +%worlds (bl_atmpos bl_atmneg bl_cph+ bl_cph-) (cp- _ _) (cp+ _ _) (cph- _ _) (cph+ _ _) %. +%total [] (cph+ Hp _) %. +%total [] (cph- Hn _) %. +%total (Dn Dp) (cp- Dn _) (cp+ Dp _) %. +%{! ## Translating into focused logic !}% +%{! A simple proof which is isomorphic in the unfocused and +focused system, establishing ``|- a * b -o b * a`` +for positive a and b. !}% +%inline ' [q] atom q %. +%define r0 ({a atm pos} {b atm pos} right (' a * ' b -o up (' b * ' a))) [a] [b] -oR' ([lab] *L' ([la] [lb] upR' (*R' (initR' lb) (initR' la))) lab) %. +%define d0 ({a atm pos} {b atm pos} conc (' a * ' b -o up (' b * ' a))) [a] [b] -oR ([hab] *L ([ha] [hb] upR (*R (init hb) (init ha))) hab) %. +%{! |check=decl !}% +%query 1 _ _ {a} {b} sdR (r0 a b) (D0 a b) %. +%{! |check=decl !}% +%query 1 _ _ {a} {b} cp- (d0 a b) (R0 a b) %. +%{! ### Reducing the number of proofs !}% +%{! This first example is already in "focused form" - invertible rules +are applied only "before" upR is applied to prove +``|- (a + b) -> up (a + b)`` |check=decl !}% +%query 1 _ _ {a atm pos} {b atm pos} cp- (-oR ([h hyp (atom a + atom b)] +L ([ha hyp (atom a)] upR (+R1 (init ha))) ([hb hyp (atom b)] upR (+R2 (init hb))) h)) (D a b) %. +%{! This a proof of the same sequent, but upR is applied before +L, which +is fine in the unfocused system but not in the focused system, where upR +starts a focusing phase and makes it impossible to apply invertible rules like ++L. Translation produces the same focused proof from these two different +unfocused proofs. |check=decl !}% +%query 1 _ _ {a atm pos} {b atm pos} cp- (-oR ([h hyp (atom a + atom b)] upR (+L ([ha hyp (atom a)] +R1 (init ha)) ([hb hyp (atom b)] +R2 (init hb)) h))) (D a b) %. +%{! ### Forward and backward chaining !}% +%{! Finally, consder the following unfocused sequent proof. The proof +proceeds in what we generally consider a "forward-chaining" manner, using +the hypothesis (atom A) to obtain the hypothesis (atom B), which is then +used to obtain the hypothesis (atom C), which is then used to obtain +the hypothesis (atom D), which is finally used to finish. !}% +%define implies-proof [A atm neg] [B atm neg] [C atm neg] [D atm neg] -oR (downL ([ha hyp (atom A)] -oR (downL ([hab hyp (down (atom A) -o (atom B))] -oR (downL ([hbc hyp (down (atom B) -o (atom C))] -oR (downL ([hcd hyp (down (atom C) -o (atom D))] -oL (downR (init ha)) ([hb hyp (atom B)] -oL (downR (init hb)) ([hc hyp (atom C)] -oL (downR (init hc)) ([hd hyp (atom D)] init hd) hcd) hbc) hab)))))))) %. +%{! Because negative polarity forces search to be goal directed, translating +this proof into the focused system will flip the order in which the +implications are considered, and no facts (beyond the four in the implication) +will ever be added to the context. | check=decl !}% +%query 1 _ _ {a atm neg} {b atm neg} {c atm neg} {d atm neg} cp- (implies-proof a b c d) _ %. +%{! Finally, we can show the proof of a successful attempt to prove +``right (A -o (A -o B) -o (B -o C) -o C)`` in the weakly focused system. +In general, this could cause an infinite loop as Twelf has no problem +with deriving multiple copies of the same fact, but here we use all +negative atoms, and the resulting backward-chaining semantics +corresponds nicely to Twelf's built-in backward-chaining semantics. +|check=decl !}% +%solve p : {a atm neg} {b atm neg} {c atm neg} right (down (' a) -o down (down (' a) -o ' b) -o down (down (' b) -o ' c) -o ' c) %. +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Weak_focusing). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/weakening-lemma.lf b/new-tests/stelf-output/pages/weakening-lemma.lf new file mode 100644 index 0000000..b342721 --- /dev/null +++ b/new-tests/stelf-output/pages/weakening-lemma.lf @@ -0,0 +1,49 @@ +% +%{! **Weakening** is the property of a hypothetical judgment that if Γ ⊦ J, then Γ, A ⊦ J. That is, if a judgment holds in a context Γ, then it holds in Γ extended with additional hypotheses. Weakening holds for most logics, but its absence is central to the formulation of substructural logics such as [linear logic](/wiki/linear-logic/). + +Often, we represent an an object-language hypothetical judgement by [using LF binding to model hypotheses](/wiki/higher-order-judgements/). When a judgement is represented in such a fashion, weakening comes "for free" from the LF representation. In particular, we may weaken a derivation by wrapping LF lambdas around it. + +For example, consider the following simply typed λ-calculus: !}% +%% Syntax +%sort tp %. +%term tp/unit tp %. +%term tp/arrow %pi tp %-> tp %-> tp %. +%sort exp %. +%term exp/unit exp %. +%term exp/lam %pi tp %-> (%pi exp %-> exp) %-> exp %. +%term exp/app %pi exp %-> exp %-> exp %. +%% Typing +%sort of %. +%term of/unit of exp/unit tp/unit %. +%term of/lam + %pi (of (exp/lam T E) (tp/arrow T T')) + %<- ({x exp} %pi (of x T) %-> (of (E x) T')) %. +%term of/app %pi (of (exp/app E1 E2) T') %<- (of E2 T) %<- (of E1 (tp/arrow T T')) %. +%{! Object-language typing is a hypothetical judgement; we represent it using LF variables to model hypotheses (see \{\{proving|title=Representing the judgements of the STLC\}\} for more discussion of this representation). + +We can prove weakening for this hypothetical judgement as a metatheorem: !}% +%{! (options removed from twelftag: check="true") !}% +%sort wkn {T' tp} %. +%mode wkn %in %in %out %. +%term _ wkn T' D1 ([x] [dx of x T'] D1) %. +%block of-block {T} [x exp] [dx of x T]%. +%worlds (of-block) (wkn _ _ _) %. +%total {} (wkn _ _ _) %. +%{! The proof is direct: given any derivation ``D1``, we can wrap LF lambdas around it to create a derivation under the necessary hypotheses. The fact that this works corresponds to the fact that weakening is admissible for LF, which is why we say that the object language inherits the hypothetical structure of the meta-language. + +Although we proved weakening as a metatheorem here for illustrative purposes, in practice it is unnecessary to write such proofs. Uses of weakening can be "inlined" by simply introducing LF lambdas when necessary. + +\{\{needs|example of using weakening in another proof.\}\} + +## See also + +* [Exchange lemma](/wiki/exchange-lemma/) +* [Substitution lemma](/wiki/substitution-lemma/) + +\{\{tutorial\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Weakening_lemma). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/whats-new.lf b/new-tests/stelf-output/pages/whats-new.lf new file mode 100644 index 0000000..9279d02 --- /dev/null +++ b/new-tests/stelf-output/pages/whats-new.lf @@ -0,0 +1,126 @@ +% +%{! ;January 17, 2015 +*After a year of issues stemming from the age of our old server, twelf.org has been moved to a new machine. Some features, like Twelf Live and syntax highlighting, still only exist on the [http://twelf.plparty.org old site]. Let [Rob](/wiki/user-rsimmons/) know about any other deficiencies in the new site. + +;June 10, 2013 +* Taus Brock-Nannestad, Chris Martens, and Carsten Schürmann to hold a [http://www.twelf.org/wiki/CADE_Tutorial Twelf Tutorial at CADE]. + +;October 26, 2012 +* John Boyland's [https://github.com/boyland/twelf-library Twelf library] is available on Github. + +<includeonly>_To read about older updates, see the [What's new?](/wiki/whats-new/) page._</includeonly><noinclude> +## Archived posts +<!-- +| +| ================ BEGIN ARCHIVE SECTION ======================== +| +| Posts below this point will no longer be shown on the main page. To archive +| more posts move this notice along with the subhead and the +| < noinclude > and < include only > tags +| above it to a point higher on the page --> + +;September 2, 2011 +* We've finally moved from twelf.plparty.org to twelf.org! All old links will continue to work through at least August 2013 (and they should actually continue to work indefinitely). + +;March 19, 2011 +* After many years of new features being added only to the subversion branch, Twelf now has a new official point release! Twelf 1.7.1 contains many fixes and new features that are documented on this site, and using any version of Twelf prior to 1.7 is highly discouraged. The "development" version of Twelf in the subversion repository remains quite stable and is also recommended. Go to the [download](/download/) page to get Twelf. + +;September 1, 2010 +* The Twelf Wiki has undergone an upgrade, and in the process code underlying the syntax-highlighing [TwelfTag](/wiki/twelftag/) system has been substantially rewritten and simplified. However, this meant several deprecated-but-still-used options to the (twelf) tags now don't work and print error messages. Leave a note on [Rob's talk page](/wiki/user-talk-rsimmons/) if you see any weird error messages around Twelf code. + +;February 22, 2009 +* [Rob](/wiki/user-rsimmons/) has a case study on [lax logic](/wiki/lax-logic/) that uses the [admissibility of cut](/wiki/admissibility-of-cut/) and identity show a sound and complete correspondence between two sequent calculus presentations of lax logic. + +;October 1, 2008 +* [Carsten](/wiki/user-carsten/) says: As a result of the work of some overly active system administrators at the ITU, the twelf mailing list was accidentally erased a few weeks ago. Since then I have tried to reconstruct the subscriber list with more or less success, but there are still some that I have missed. Therefore, if you haven't received any mail from the list lately, but you expect to be on it, please resubscribe under [[http://mail.itu.dk/mailman/listinfo/twelf-list http://mail.itu.dk/mailman/listinfo/twelf-list]]. + +;September 19, 2008 +* **We are running a [Twelf Tutorial](/wiki/popl-tutorial/) co-located with POPL 2009.** + +;July 19, 2008 +* [Dan](/wiki/user-drl/) added [course materials for the UofO summer school](/wiki/summer-school-2008/). + +;January 28, 2008 +* [John](/wiki/user-boyland/) took Rob's example to the next step and directly provided a bijection between a [HOAS encoding of lambdas and natural numbers](/wiki/hoas-nat-bijection/). + +;October 4, 2007 +* [Rob](/wiki/user-rsimmons/) posted a page on [concrete representation](/wiki/concrete-representation/) based on a question by [John](/wiki/user-boyland/) about demonstrating a correspondence between HOAS and concrete term representations. + +;April 25, 2007 +* [Todd](/wiki/user-twilson/) posted a [tutorial](/wiki/tutorials/) on using Church encodings to create [user-defined constraint domain](/wiki/user-defined-constraint-domain/)s. + +;April 11, 2007 +* If you think up some exercises while you're learning Twelf, add them to the [intro tutorial](/wiki/proving-metatheorems-summary-the-natural-numbers/). + +;March 21, 2007 +* Official launch day! Thanks to [[Project:Contributors|everyone]] who has contributed so far, and welcome to new visitors. + +;March 16, 2007 +* [Dan](/wiki/user-drl/) has revised and expanded several of the [tutorials](/wiki/tutorials/), and written new ones on [catch-all case](/wiki/catch-all-case/)s, [numeric termination metrics](/wiki/numeric-termination-metrics/), and [hereditary substitution](/wiki/hereditary-substitution-for-the-stlc/). + +;March 14, 2007 +* [Dan](/wiki/user-drl/) has finished a draft of the introductory article [Proving metatheorems with Twelf](/wiki/proving-metatheorems-with-twelf/). + +;February 28, 2007 +* The Twelf Project wiki now supports uploading SVG images! Check out the article on [tabled logic programming](/wiki/tabled-logic-programming/) for an example of the unnecessarily beautiful illustrations this allows. + +**February 24, 2007** +* [Rob](/wiki/user-rsimmons/) has extended the [[Project:TwelfTag|TwelfTag]] feature to facilitate using [[Project:Literate Twelf|Literate Twelf]]. + +**January 25, 2007** +* [Rob](/wiki/user-rsimmons/) has developed a beta [http://twelf.plparty.org/builds build system] that has source, Linux binary, and Windows installer versions of "CVS Twelf." + +**December 1, 2006** +* [Tom](/wiki/user-tom7/) added a category for [[:category:undocumented features|undocumented features]] + +**October 30, 2006** +* [Tom](/wiki/user-tom7/) posted a [case study](/wiki/case-studies/) on [Admissibility of cut](/wiki/admissibility-of-cut/). + +**October 20, 2006** +* [Jake](/wiki/user-jaked/) posted a [tutorial](/wiki/tutorials/) on [Strengthening](/wiki/strengthening/). + +**October 19, 2006** +* [Carsten](/wiki/user-varming/) is developing a [case study](/wiki/case-studies/) on [lists](/wiki/lists/). + +**October 18, 2006** +* [Tom](/wiki/user-tom7/) posted a [case study](/wiki/case-studies/) for [Classical S5](/wiki/classical-s5/). + +**October 16, 2006** + +* [Karl](/wiki/user-crary/) posted a [case study](/wiki/case-studies/) on [Linear logic](/wiki/linear-logic/). + +**October 14, 2006** + +* An alpha version of "[[Project:AJALF|AJALF]]"-powered [Twelf Live](/wiki/twelf-live/) is online +* [Dan](/wiki/user-danielklee/) has started the [Ask Twelf Elf](/wiki/ask-twelf-elf/) project providing Twelf help over email + +**October 13, 2006** + +* The editor interface now has a [[Project:TwelfTag#Quick check|quick check]] button for Twelf code, thanks to "[[Project:AJALF|AJALF]]" technology. + +**October 9, 2006** + +* The [[Project:TwelfTag|TwelfTag]] system now allows for direct checking of code in the wiki. +* [Carsten](/wiki/user-varming/) has added code proving various properties of [Lily](/wiki/lily/) to the [case studies](/wiki/case-studies/). + +**October 5, 2006** + +* [Tom](/wiki/user-tom7/) posted an advanced [[:category:tutorials|tutorial]] on [CPS conversion](/wiki/cps-conversion/). + +**September 30, 2006** + +* The wiki has been moved to its real home on the [http://moog.concert.cs.cmu.edu/ Moog server], and given a permanent domain name, twelf.plparty.org. +* A first draft of the non-technical [quick introduction](/wiki/general-description-of-twelf/) has been introduced for consideration. + +**September 28, 2006** + +* [Substitution lemma](/wiki/substitution-lemma/) — [Dan Lee](/wiki/user-danielklee/)'s thorough explanation of the different ways substitution lemmas are dealt with by Twelf. +* [Twelf CVS](/download/) — We have instructions from the Software page on downloading the development version of Twelf from the CVS repository. The CVS version of Twelf has undocumented features which are being described on the wiki, such as its capacity for working with [holes in metatheorems](/wiki/incremental-metatheorem-development/). + +</noinclude> !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/What%27s_new). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/world-subsumption.lf b/new-tests/stelf-output/pages/world-subsumption.lf new file mode 100644 index 0000000..3dbd1c7 --- /dev/null +++ b/new-tests/stelf-output/pages/world-subsumption.lf @@ -0,0 +1,12 @@ +% +%{! **World subsumption** is a sufficient condition that a [metatheorem](/wiki/metatheorem/) proved for one set of LF contexts can be reused in another set of contexts. + +Eventually, this page will contain a self-contained explanation for world subsumption. For now, please read the discussion in \{\{proving|title=Proving totality assertions in non-empty contexts\}\} and on the [%worlds](/wiki/percent-worlds/) page. + +\{\{stub\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/World_subsumption). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/pages/zermelo-frankel.lf b/new-tests/stelf-output/pages/zermelo-frankel.lf new file mode 100644 index 0000000..c2b0537 --- /dev/null +++ b/new-tests/stelf-output/pages/zermelo-frankel.lf @@ -0,0 +1,71 @@ +% +%{! This [case study](/wiki/case-studies/) is an encoding of **Zermelo Frankel** (**ZFC**) set theory. + +\{\{needs|some explanatory text\}\} !}% +%% ZFC +%% by Daniel C. Wang +%% Transliterated from AUTOMATH definition +%% http://www.cs.ru.nl/~freek/zfc-etc/zfc.aut +%sort prop %. +%sort pf %. +%sort set %. +%% First Order Logic +%term false prop %. +%term imp %pi prop %-> prop %-> prop %. +%term all %pi (%pi set %-> prop) %-> prop %. +%term eq %pi set %-> set %-> prop %. +%term in %pi set %-> set %-> prop %. +%define not (%pi prop %-> prop) [a] imp a false %. +%define and (%pi prop %-> prop %-> prop) [a] [b] not (imp a (not b)) %. +%define or (%pi prop %-> prop %-> prop) [a] [b] imp (not a) b %. +%define iff (%pi prop %-> prop %-> prop) [a] [b] and (imp a b) (imp b a) %. +%define ex (%pi (%pi set %-> prop) %-> prop) [p] not (all ([z] not (p z))) %. +%define unique (%pi (%pi set %-> prop) %-> prop) [p] all ([z] imp (p z) (all ([z'] imp (p z') (eq z z')))) %. +%define ex_unique (%pi (%pi set %-> prop) %-> prop) [p] and (ex p) (unique p) %. +%term imp_i %pi (%pi (pf A) %-> (pf B)) %-> (pf (imp A B)) %. +%term imp_e %pi (pf (imp A B)) %-> (pf A) %-> (pf B) %. +%term all_i %pi ({z} pf (P z)) %-> (pf (all P)) %. +%term all_e %pi (pf (all P)) %-> ({z} pf (P z)) %. +%term classical %pi (pf (not (not A))) %-> (pf A) %. +%term eq_i pf (eq A A) %. +%term eq_e %pi (pf (eq A B)) %-> ({s %pi set %-> prop} %pi (pf (s A)) %-> (pf (s B))) %. +%term if %pi prop %-> set %-> set %-> set %. +%term if_then %pi (pf P) %-> (pf (eq (if P X Y) X)) %. +%term if_else %pi (pf (not P)) %-> (pf (eq (if P X Y) Y)) %. +%% Set Theory theory +%term empty set %. +%term double %pi set %-> set %-> set %. +% {x,y} +%term unions %pi set %-> set %. +% union sets in sets +%term powerset %pi set %-> set %. +%term replace %pi set %-> (%pi set %-> set) %-> set %. +%term omega set %. +%define single (%pi set %-> set) [x] double x x %. +%define restrict (%pi set %-> (%pi set %-> prop) %-> set) [x] [q] unions (replace x ([z] if (q z) (single z) empty)) %. +%define inter (%pi set %-> set %-> set) [x] [y] restrict x ([z] in z y) %. +%define union (%pi set %-> set %-> set) [x] [y] unions (double x y) %. +%define zero set empty %. +%define succ (%pi set %-> set) [x] union x (single x) %. +%define subset (%pi set %-> set %-> prop) [x] [y] all ([z] imp (in z x) (in z y)) %. +%define disjoint (%pi set %-> set %-> prop) [x] [y] eq (inter x y) empty %. +%define omega_closed (%pi set %-> prop) [x] and (in empty x) (all ([n] imp (in n x) (in (succ n) x))) %. +%% Axioms ZF +%term extensionality pf (iff (eq X Y) (all ([z] iff (in z X) (in z Y)))) %. +%term foundation pf (ex ([z] and (in z X) (disjoint z X))) %. +%term emtpy_ax pf (not (in X empty)) %. +%term double_ax pf (iff (in Z (double X Y)) (or (in Z X) (in Z Y))) %. +%term union_ax pf (iff (in Z (unions X)) (ex ([y] and (in Z y) (in y X)))) %. +%term powerset_ax pf (iff (in Z (powerset X)) (subset Z X)) %. +%term replace_ax pf (iff (in Z (replace X F)) (ex ([y] and (in y X) (eq Z (F y))))) %. +%term omega_ax pf (and (omega_closed omega) (all ([o] imp (omega_closed o) (subset omega o)))) %. +%% C +%term choice_ax pf (imp (all ([y1] imp (in y1 X) (all ([y2] imp (in y2 X) (disjoint y1 y2))))) (ex ([x'] all ([y] imp (in y X) (ex_unique ([y'] and (in y' x') (in y' y))))))) %. +%{! [Category:Twelf code](/wiki/category-twelf-code/) +\{\{case study\}\} !}% +%{! +----- +This page was copied from the MediaWiki version of the Twelf Wiki. +If anything looks wrong, you can refer to the +[wayback machine's version here](https://web.archive.org/web/20240303030303/http://twelf.org/wiki/Zermelo_Frankel). +!}% \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/aeq-completeness.lf b/new-tests/stelf-output/tslf/aeq-completeness.lf new file mode 100644 index 0000000..c112784 --- /dev/null +++ b/new-tests/stelf-output/tslf/aeq-completeness.lf @@ -0,0 +1,26 @@ +%%%% algorithmic equivalence for constructors a la SH +%sort aeq-complete %. +%mode aeq-complete %in %out %. +%worlds (ofkd+vdt-block) (aeq-complete _ _) %. +%total {} (aeq-complete _ _) %. +%sort cn-aeq-sound %. +%mode cn-aeq-sound %in %out %. +%worlds (ofkd-block) (cn-aeq-sound _ _) %. +%total {} (cn-aeq-sound _ _) %. +%sort kd-aeq-sound %. +%mode kd-aeq-sound %in %out %. +%term _ kd-aeq-sound kd-aeq/kd/unit kd-deq/kd/unit %. +%term _ kd-aeq-sound kd-aeq/kd/type kd-deq/kd/type %. +%term _ + %pi (kd-aeq-sound (kd-aeq/kd/sing D1) (kd-deq/kd/sing D2)) + %<- (cn-aeq-sound D1 D2) %. +%term _ + %pi (kd-aeq-sound (kd-aeq/kd/sgm D1 D2) (kd-deq/kd/sgm D1' D2')) + %<- (kd-aeq-sound D1 D1') + %<- ({a} {da} kd-aeq-sound (D2 a da) (D2' a da)) %. +%term _ + %pi (kd-aeq-sound (kd-aeq/kd/pi D1 D2) (kd-deq/kd/pi D1' D2')) + %<- (kd-aeq-sound D1 D1') + %<- ({a} {da} kd-aeq-sound (D2 a da) (D2' a da)) %. +%worlds (ofkd-block) (kd-aeq-sound _ _) %. +%total D1 (kd-aeq-sound D1 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/aeq.lf b/new-tests/stelf-output/tslf/aeq.lf new file mode 100644 index 0000000..60bde7a --- /dev/null +++ b/new-tests/stelf-output/tslf/aeq.lf @@ -0,0 +1,116 @@ +%%%% algorithmic equivalence for constructors a la SH +% cn-path : cn -> type. +%sort cn-nke %. +%mode cn-nke %in %out %. +%sort cn-redex %. +%mode cn-redex %in %out %. +%sort cn-whr %. +%mode cn-whr %in %out %. +%sort cn-whn %. +%mode cn-whn %in %out %. +%sort kd-aeq %. +%mode kd-aeq %in %in %. +%sort cn-aeq %. +%mode cn-aeq %in %in %star %. +%sort cn-peq %. +%mode cn-peq %in %in %star %. +%%%% +% cn-path/var : cn-path D +% cn-path/pj1 : cn-path +%%%% natural kind extraction +%{ +cn-nke/tp/unit : cn-nke tp/unit kd/type. + +cn-nke/tp/arrow : cn-nke (tp/arrow _ _) kd/type. + +cn-nke/tp/cross : cn-nke (tp/cross _ _) kd/type. + +cn-nke/tp/forall : cn-nke (tp/forall _ _) kd/type. + +cn-nke/tp/ref : cn-nke (tp/ref _) kd/type. + +cn-nke/tp/tag : cn-nke (tp/tag _) kd/type. + +cn-nke/cn/unit : cn-nke cn/unit kd/unit. +}% +%term cn-nke/cn/pj1 %pi (cn-nke (cn/pj1 P1) K') %<- (cn-nke P1 (kd/sgm K' K'')) %. +%term cn-nke/cn/pj2 %pi (cn-nke (cn/pj2 P1) (K'' (cn/pj1 P1))) %<- (cn-nke P1 (kd/sgm K' K'')) %. +%%%% +%term cn-redex/lam cn-redex (cn/app (cn/lam K C) C') (C C') %. +%term cn-redex/pj1 cn-redex (cn/pj1 (cn/pair C1 C2)) C1 %. +%term cn-redex/pj2 cn-redex (cn/pj2 (cn/pair C1 C2)) C2 %. +%%%% +%term cn-whr/app %pi (cn-whr (cn/app C1 C2) (cn/app C1' C2)) %<- (cn-whr C1 C1') %. +%term cn-whr/pj1 %pi (cn-whr (cn/pj1 C1) (cn/pj1 C1')) %<- (cn-whr C1 C1') %. +%term cn-whr/pj2 %pi (cn-whr (cn/pj2 C1) (cn/pj2 C1')) %<- (cn-whr C1 C1') %. +%term cn-whr/redex %pi (cn-whr C C') %<- (cn-redex C C') %. +%term cn-whr/def %pi (cn-whr P C) %<- (cn-nke P (kd/sing C)) %. +%%%% +%term cn-whn/step %pi (cn-whn C D) %<- (cn-whr C C') %<- (cn-whn C' D) %. +%term cn-whn/refl cn-whn C C %. +%%%% +%term kd-aeq/kd/unit kd-aeq kd/unit kd/unit %. +%term kd-aeq/kd/type kd-aeq kd/type kd/type %. +%term kd-aeq/kd/sing %pi (kd-aeq (kd/sing C1) (kd/sing C2)) %<- (cn-aeq C1 C2 kd/type) %. +%term kd-aeq/kd/sgm + %pi (kd-aeq (kd/sgm K1 K2) (kd/sgm K3 K4)) + %<- ({a cn} {da ofkd a K1} kd-aeq (K2 a) (K4 a)) + %<- (kd-aeq K1 K3) %. +%term kd-aeq/kd/pi + %pi (kd-aeq (kd/pi K1 K2) (kd/pi K3 K4)) + %<- ({a cn} {da ofkd a K3} kd-aeq (K2 a) (K4 a)) + %<- (kd-aeq K3 K1) %. +%%%% +%term cn-aeq/kd/unit cn-aeq C1 C2 kd/unit %. +%term cn-aeq/kd/type + %pi (cn-aeq C1 C2 kd/type) + %<- (cn-whn C1 P1) + %<- (cn-whn C2 P2) + %<- (cn-peq P1 P2 kd/type) %. +%term cn-aeq/kd/sing cn-aeq C1 C2 (kd/sing C) %. +%term cn-aeq/kd/pi + %pi (cn-aeq C1 C2 (kd/pi K1 K2)) + %<- ({a cn} {da ofkd a K1} cn-aeq (cn/app C1 a) (cn/app C2 a) (K2 a)) %. +%term cn-aeq/kd/sgm + %pi (cn-aeq C1 C2 (kd/sgm K1 K2)) + %<- (cn-aeq (cn/pj2 C1) (cn/pj2 C2) (K2 (cn/pj1 C1))) + %<- (cn-aeq (cn/pj1 C1) (cn/pj2 C2) K1) %. +%%%% path equivalence +%{ +cn-peq/var : cn-peq A A K + <- ofkd A K. +}% +%term cn-peq/tp/unit cn-peq tp/unit tp/unit kd/type %. +%term cn-peq/tp/tagged cn-peq tp/tagged tp/tagged kd/type %. +%term cn-peq/tp/cross + %pi (cn-peq (tp/cross C1 C1') (tp/cross C2 C2') kd/type) + %<- (cn-aeq C1' C2' kd/type) + %<- (cn-aeq C1 C2 kd/type) %. +%term cn-peq/tp/arrow + %pi (cn-peq (tp/arrow C1 C1') (tp/arrow C2 C2') kd/type) + %<- (cn-aeq C1' C2' kd/type) + %<- (cn-aeq C1 C2 kd/type) %. +%term cn-peq/tp/sum + %pi (cn-peq (tp/sum C1 C1') (tp/sum C2 C2') kd/type) + %<- (cn-aeq C1' C2' kd/type) + %<- (cn-aeq C1 C2 kd/type) %. +%term cn-peq/tp/forall + %pi (cn-peq (tp/forall K1 C1) (tp/forall K2 C2) kd/type) + %<- (kd-aeq K1 K2) + %<- ({a} {da ofkd a K1} cn-aeq (C1 a) (C2 a) kd/type) %. +%term cn-peq/cn/mu + %pi (cn-peq (cn/mu K1 C1) (cn/mu K2 C2) K) + %<- (kd-aeq K1 K2) + %<- ({a} {da ofkd a K} cn-aeq (C1 a) (C2 a) K) %. +%term cn-peq/tp/ref %pi (cn-peq (tp/ref C1) (tp/ref C2) kd/type) %<- (cn-aeq C1 C2 kd/type) %. +%term cn-peq/tp/tag %pi (cn-peq (tp/tag C1) (tp/tag C2) kd/type) %<- (cn-aeq C1 C2 kd/type) %. +%term cn-peq/cn/unit cn-peq cn/unit cn/unit kd/unit %. +%term cn-peq/cn/app + %pi (cn-peq (cn/app P1 C1) (cn/app P2 C2) (K'' C1)) + %<- (cn-peq P1 P2 (kd/pi K' K'')) + %<- (cn-aeq C1 C2 K') %. +%term cn-peq/cn/pj1 %pi (cn-peq (cn/pj1 P1) (cn/pj2 P2) K') %<- (cn-peq P1 P2 (kd/sgm K' K'')) %. +%term cn-peq/cn/pj2 + %pi (cn-peq (cn/pj2 P1) (cn/pj2 P2) (K'' (cn/pj1 P1))) + %<- (cn-peq P1 P2 (kd/sgm K' K'')) %. +%worlds (ofkd-block) (cn-aeq _ _ _) (cn-peq _ _ _) (cn-whr _ _) (cn-whn _ _) (kd-aeq _ _) (cn-nke _ _) (cn-redex _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/antisymmetry-kinds.lf b/new-tests/stelf-output/tslf/antisymmetry-kinds.lf new file mode 100644 index 0000000..4031008 --- /dev/null +++ b/new-tests/stelf-output/tslf/antisymmetry-kinds.lf @@ -0,0 +1,20 @@ +%%%% kd-anti +%sort kd-anti %. +%mode kd-anti %in %in %in %out %out %. +%term _ kd-anti kd-deq/kd/unit _ _ kd-sub/kd/unit kd-sub/kd/unit %. +%term _ kd-anti kd-deq/kd/type _ _ kd-sub/kd/type kd-sub/kd/type %. +%term _ kd-anti (kd-deq/kd/sing D1) _ _ (kd-sub/kd/sing-kd/sing D1) (kd-sub/kd/sing-kd/sing (cn-deq/sym D1)) %. +%term _ + %pi (kd-anti (kd-deq/kd/sgm D1 D2) (kd-wf/kd/sgm Do DL) (kd-wf/kd/sgm Dt DK) (kd-sub/kd/sgm D3 D4 DK) (kd-sub/kd/sgm D3' D4'' DL)) + %<- (kd-anti D1 Do Dt D3 D3') + %<- (kd-wkn/kd-wf DK D3 DKK) + %<- ({a cn} {da ofkd a K1} kd-anti (D2 a da) (DL a da) (DKK a da) (D4 a da) (D4' a da)) + %<- (kd-wkn/kd-sub D4' D3' D4'') %. +%term _ + %pi (kd-anti (kd-deq/kd/pi D1 D2) (kd-wf/kd/pi Do DL) (kd-wf/kd/pi Dt DK) (kd-sub/kd/pi D3 D4 DL) (kd-sub/kd/pi D3' D4'' DK)) + %<- (kd-anti D1 Dt Do D3 D3') + %<- (kd-wkn/kd-wf DL D3 DKK) + %<- ({a cn} {da ofkd a K2} kd-anti (D2 a da) (DKK a da) (DK a da) (D4 a da) (D4' a da)) + %<- (kd-wkn/kd-sub D4' D3' D4'') %. +%worlds (ofkd-block can-mofkd-block ofkd+vdt-block) (kd-anti _ _ _ _ _) %. +%total (D5) (kd-anti D5 _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/antisymmetry-sigs.lf b/new-tests/stelf-output/tslf/antisymmetry-sigs.lf new file mode 100644 index 0000000..0a7464a --- /dev/null +++ b/new-tests/stelf-output/tslf/antisymmetry-sigs.lf @@ -0,0 +1,40 @@ +%%%% anti-symmetry of signatures +%sort sg-anti %. +%mode sg-anti %in %in %in %out %out %out %out %. +%term _ sg-anti sg-deq/sg/unit fst-sg/unit fst-sg/unit sg-sub/sg/unit sg-sub/sg/unit kd-sub/kd/unit kd-sub/kd/unit %. +%term _ sg-anti (sg-deq/sg/cn D1) fst-sg/cn fst-sg/cn (sg-sub/sg/cn D1) (sg-sub/sg/cn (cn-deq/sym D1)) kd-sub/kd/unit kd-sub/kd/unit %. +%term _ + %pi (sg-anti (sg-deq/sg/kd D1) fst-sg/kd fst-sg/kd (sg-sub/sg/kd DS1) (sg-sub/sg/kd DS2) DS1 DS2) + %<- (vdt/kd-deq D1 DK1 DK2) + %<- (kd-anti D1 DK1 DK2 DS1 DS2) %. +%term _ + %pi (sg-anti (sg-deq/sg/sgm (%the (sg-deq S1 S2) D1) (%the ({a} %pi (ofkd a _) %-> (sg-deq (S3 a) (S4 a))) D2) (%the (fst-sg S1 K) FSo)) (fst-sg/sgm (%the (fst-sg S1 K1) FS1) FS2) (fst-sg/sgm FS1' FS2') (sg-sub/sg/sgm DS1 DS2o DWw FS1 FS1') (sg-sub/sg/sgm DS1' DS2'' DWo FS1' FS1) (kd-sub/kd/sgm DK1o DK1'o DKW') (kd-sub/kd/sgm DK2o DK2'' DW''o)) + %<- (fst-sg-seq FSo FS1 DQ) + %<- (sg-anti D1 FSo FS1' DS1 DS1' (%the (kd-sub K K2) DK1) (%the (kd-sub K2 K) DK2)) + %<- (vdt/kd-sub DK1 Dwk1 _) + %<- ({a cn} {da ofkd a K} {dm mofkd da met/unit} {_ can-mofkd da dm} {_ vdt/ofkd da Dwk1} sg-anti (D2 a da) (FS2 a) (FS2' a) (DS2 a da) (DS2' a da) (DK1' a da) (DK2' a da)) + %<- ({a cn} {da ofkd a K} {dm mofkd da met/unit} {_ can-mofkd da dm} {_ vdt/ofkd da Dwk1} vdt/sg-deq (D2 a da) (FS2 a) (FS2' a) (DW a da) (DW' a da) (DKQ' a da)) + %<- ({a cn} {da ofkd a K} {dm mofkd da met/unit} {_ can-mofkd da dm} {_ vdt/ofkd da Dwk1} vdt/kd-deq (DKQ' a da) (DW'' a da) (DKW a da)) + %<- (kd-wkn/kd-wf DKW DK2 DKW') + %<- (kd-wkn/kd-sub DK2' DK2 DK2'') + %<- (kd-wkn/sg-wf DW' DK2 DWw) + %<- (kd-wkn/sg-sub DS2' DK2 DS2'') + %<- (kd-sub/seq-l DQ DK1 DK1o) + %<- (kd-sub/seq-r DQ DK2 DK2o) + %<- (kd-sub/seq-a DQ DK1' DK1'o) + %<- (kd-wf/seq-a DQ DW'' DW''o) + %<- (sg-sub/seq-a DQ DS2 DS2o) + %<- (sg-wf/seq-a DQ DW DWo) %. +% there are ways to make this +% simpler? probably +%term _ + %pi (sg-anti (sg-deq/sg/pi D1 D2 FS) fst-sg/pi fst-sg/pi (sg-sub/sg/pi D1 D2 FS) (sg-sub/sg/pi D1' D2'' FS2) kd-sub/kd/unit kd-sub/kd/unit) + %<- (fst-sg-complete S2 FS2) + %<- (vdt/sg-deq D1 FS FS2 DW1 DW2 Dkeq) + %<- (vdt/kd-deq Dkeq DKW1 DKW2) + %<- (sg-sym D1 D1') + %<- ({a cn} {da ofkd a K} {dm mofkd da met/unit} {_ can-mofkd da dm} {_ vdt/ofkd da DKW1} sg-sym (D2 a da) (D2' a da)) + %<- (kd-anti Dkeq DKW1 DKW2 _ Dks') + %<- (kd-wkn/sg-deq D2' Dks' D2'') %. +%worlds (ofkd+vdt-block) (sg-anti _ _ _ _ _ _ _) %. +%total (D2) (sg-anti D2 _ _ _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/blocks-cnkd.lf b/new-tests/stelf-output/tslf/blocks-cnkd.lf new file mode 100644 index 0000000..3f6b997 --- /dev/null +++ b/new-tests/stelf-output/tslf/blocks-cnkd.lf @@ -0,0 +1,11 @@ +%block cn-block [a cn]%. +%block ofkd-block {k kd} [a cn] [da ofkd a k]%. +%block can-mofkd-block {k kd} [a] [da ofkd a k] [dm mofkd da met/unit] [_ can-mofkd da dm]%. +%{ +%block ofkd+vdt-block : some + {k : kd} {kwf : kd-wf k} + block + {a : cn} {da : ofkd a k} + {_ : vdt/ofkd da kwf}. +}% +%block ofkd+vdt-block {k kd} {kwf kd-wf k} [a cn] [da ofkd a k] [dm mofkd da met/unit] [_ can-mofkd da dm] [_ vdt/ofkd da kwf]%. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/blocks-predecls-cnkd.lf b/new-tests/stelf-output/tslf/blocks-predecls-cnkd.lf new file mode 100644 index 0000000..00f4366 --- /dev/null +++ b/new-tests/stelf-output/tslf/blocks-predecls-cnkd.lf @@ -0,0 +1,6 @@ +%sort vdt/ofkd %. +%mode vdt/ofkd %in %out %. +%sort mofkd %. +%mode mofkd %star %star %. +%sort can-mofkd {D ofkd C K} %. +%mode can-mofkd %in %out %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/blocks-predecls-tmmd.lf b/new-tests/stelf-output/tslf/blocks-predecls-tmmd.lf new file mode 100644 index 0000000..60865b5 --- /dev/null +++ b/new-tests/stelf-output/tslf/blocks-predecls-tmmd.lf @@ -0,0 +1,6 @@ +%sort vdt/assm/tm %. +%mode vdt/assm/tm %in %out %. +%sort vdt/assm/md %. +%mode vdt/assm/md %in %out %out %out %out %. +%sort fst-md-seq %. +%mode fst-md-seq %in %in %out %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/blocks-tmmd.lf b/new-tests/stelf-output/tslf/blocks-tmmd.lf new file mode 100644 index 0000000..632e441 --- /dev/null +++ b/new-tests/stelf-output/tslf/blocks-tmmd.lf @@ -0,0 +1,6 @@ +%block tm-block [x tm]%. +%block oftp-block {c cn} [x tm] [dx assm/tm x c]%. +%block oftp+vdt-block {c cn} {cwf ofkd c kd/type} [x tm] [dx assm/tm x c] [_ vdt/assm/tm dx cwf]%. +%block md-block [m md]%. +%block ofsg-block {S1 sg} {K1 kd} [s md] [ds assm/md s S1] [a cn] [da ofkd a K1] [dfm fst-md s a]%. +%block ofsg+vdt-block {S1 sg} {K1 kd} {DWK kd-wf K1} {DWS sg-wf S1} {DFS fst-sg S1 K1} [s md] [ds assm/md s S1] [a cn] [da ofkd a K1] [dm mofkd da met/unit] [_ can-mofkd da dm] [va vdt/ofkd da DWK] [dfm fst-md s a] [_ vdt/assm/md ds DWS DFS dfm da] [_ fst-md-seq dfm dfm seq/cn/refl]%. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/canonical-forms-mods.lf b/new-tests/stelf-output/tslf/canonical-forms-mods.lf new file mode 100644 index 0000000..48812d5 --- /dev/null +++ b/new-tests/stelf-output/tslf/canonical-forms-mods.lf @@ -0,0 +1,47 @@ +%%%% canonical forms for modules +%%%% this happens to be much easier than canonical forms for terms +%%%% in this language +%sort cfl/md/unit %. +%mode cfl/md/unit %in %in %out %. +%term _ cfl/md/unit val/md/unit ofsg/md/unit seq/md/refl %. +%term _ %pi (cfl/md/unit V (ofsg/sub D1 Dsub _) DS) %<- (cfl/md/unit V D1 DS) %. +%worlds () (cfl/md/unit _ _ _) %. +%total (D) (cfl/md/unit _ D _) %. +%sort cfl/sg/pi %. +%mode cfl/sg/pi %in %in %out %. +%term _ cfl/sg/pi val/md/lam (ofsg/md/lam _ _ _) seq/md/refl %. +%term _ %pi (cfl/sg/pi V (ofsg/sub D1 Dsub _) DS) %<- (cfl/sg/pi V D1 DS) %. +%worlds () (cfl/sg/pi _ _ _) %. +%total (D) (cfl/sg/pi _ D _) %. +%{ +cfl/md/cn : val/md M + -> ofsg L T Y M (sg/kd _) + -> seq/md M (md/cn _) + -> type. + +%mode cfl/md/cn +D1 +D2 -D3. + +- : cfl/md/cn val/md/cn (ofsg/md/cn _) seq/md/refl. + +- : cfl/md/cn V (ofsg/sub D1 Dsub _) DS + <- cfl/md/cn V D1 DS. + +%worlds () (cfl/md/cn _ _ _). +%total (D) (cfl/md/cn _ D _). +}% +%sort cfl/md/pair %. +%mode cfl/md/pair %in %in %out %. +%term _ cfl/md/pair (val/md/pair _ _) (ofsg/md/pair _ _) seq/md/refl %. +%term _ + %pi (cfl/md/pair V (ofsg/sgm-ext D1 _) DS) + %<- (subder/md/pj1 D1 D2 _) + %<- (cfl/md/pair V D2 DS) %. +%term _ %pi (cfl/md/pair V (ofsg/sub D1 Dsub _) DS) %<- (cfl/md/pair V D1 DS) %. +%worlds () (cfl/md/pair _ _ _) %. +%total (D) (cfl/md/pair _ D _) %. +%sort cfl/md/tm %. +%mode cfl/md/tm %in %in %out %. +%term _ cfl/md/tm (val/md/tm _) (ofsg/md/tm _) seq/md/refl %. +%term _ %pi (cfl/md/tm V (ofsg/sub D1 Dsub _) DS) %<- (cfl/md/tm V D1 DS) %. +%worlds () (cfl/md/tm _ _ _) %. +%total (D) (cfl/md/tm _ D _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/canonical-forms-terms.lf b/new-tests/stelf-output/tslf/canonical-forms-terms.lf new file mode 100644 index 0000000..8c9486c --- /dev/null +++ b/new-tests/stelf-output/tslf/canonical-forms-terms.lf @@ -0,0 +1,405 @@ +%%%% canonical forms lemmas +%sort cfl/tp/unit %. +%mode cfl/tp/unit %in %in %in %out %. +%term _ cfl/tp/unit _ oftp/tm/unit _ seq/tm/refl %. +%term _ + %pi (cfl/tp/unit V (oftp/deq DofE Deq) Deq' Dseq) + %<- (cfl/tp/unit V DofE (cn-deq/trans Deq Deq') Dseq) %. +%term _ + %pi (cfl/tp/unit (val/tm/pair _ _) (oftp/tm/pair _ _) Deq Dseq) + %<- (neq/deq/cross-unit Deq Dun) + %<- (uninhabited-seq/tm _ _ Dun Dseq) %. +%term _ + %pi (cfl/tp/unit val/tm/fun (oftp/tm/fun _ _) Deq Dseq) + %<- (neq/deq/arrow-unit Deq Dun) + %<- (uninhabited-seq/tm _ _ Dun Dseq) %. +%term _ + %pi (cfl/tp/unit _ (oftp/tm/cnabs _ _) Deq Dseq) + %<- (neq/deq/forall-unit Deq Dun) + %<- (uninhabited-seq/tm _ _ Dun Dseq) %. +%term _ + %pi (cfl/tp/unit val/tm/loc (oftp/tm/loc _ _) Deq Dseq) + %<- (neq/deq/ref-unit Deq Dun) + %<- (uninhabited-seq/tm _ _ Dun Dseq) %. +%term _ + %pi (cfl/tp/unit (val/tm/inl _) (oftp/tm/inl _ _) Deq Dseq) + %<- (neq/deq/sum-unit Deq Dun) + %<- (uninhabited-seq/tm _ _ Dun Dseq) %. +%term _ + %pi (cfl/tp/unit (val/tm/inr _) (oftp/tm/inr _ _) Deq Dseq) + %<- (neq/deq/sum-unit Deq Dun) + %<- (uninhabited-seq/tm _ _ Dun Dseq) %. +%term _ + %pi (cfl/tp/unit val/tm/tagloc (oftp/tm/tagloc _ _) Deq Dseq) + %<- (neq/deq/tag-unit Deq Dun) + %<- (uninhabited-seq/tm _ _ Dun Dseq) %. +%term _ + %pi (cfl/tp/unit _ (oftp/tm/tag _ _) Deq Dseq) + %<- (neq/deq/tagged-unit Deq Dun) + %<- (uninhabited-seq/tm _ _ Dun Dseq) %. +%term _ + %pi (cfl/tp/unit _ (oftp/tm/roll _ _) Deq Dseq) + %<- (neq/deq/mu-unit Deq Dun) + %<- (uninhabited-seq/tm _ _ Dun Dseq) %. +%worlds () (cfl/tp/unit _ _ _ _) %. +%total D1 (cfl/tp/unit _ D1 _ _) %. +%sort cfl/tp/arrow %. +%mode cfl/tp/arrow %in %in %in %out %. +%term _ cfl/tp/arrow _ (oftp/tm/fun _ _) _ seq/tm/refl %. +%term _ + %pi (cfl/tp/arrow V (oftp/deq DofE Deq) Deq' Dseq) + %<- (cfl/tp/arrow V DofE (cn-deq/trans Deq Deq') Dseq) %. +%term _ + %pi (cfl/tp/arrow _ (oftp/tm/pair _ _) Deq Dseq) + %<- (neq/deq/cross-arrow Deq Dun) + %<- (uninhabited-seq/tm _ (tm/fun tp/unit tp/unit ([x tm] [f tm] x)) Dun Dseq) %. +%term _ + %pi (cfl/tp/arrow _ oftp/tm/unit Deq Dseq) + %<- (neq/deq/unit-arrow Deq Dun) + %<- (uninhabited-seq/tm _ (tm/fun tp/unit tp/unit ([x tm] [f tm] x)) Dun Dseq) %. +%term _ + %pi (cfl/tp/arrow _ (oftp/tm/cnabs _ _) Deq Dseq) + %<- (neq/deq/forall-arrow Deq Dun) + %<- (uninhabited-seq/tm _ (tm/fun tp/unit tp/unit ([x tm] [f tm] x)) Dun Dseq) %. +%term _ + %pi (cfl/tp/arrow _ (oftp/tm/loc _ _) Deq Dseq) + %<- (neq/deq/ref-arrow Deq Dun) + %<- (uninhabited-seq/tm _ (tm/fun tp/unit tp/unit ([x tm] [f tm] x)) Dun Dseq) %. +%term _ + %pi (cfl/tp/arrow (val/tm/inl _) (oftp/tm/inl _ _) Deq Dseq) + %<- (neq/deq/sum-arrow Deq Dun) + %<- (uninhabited-seq/tm _ (tm/fun tp/unit tp/unit ([x tm] [f tm] x)) Dun Dseq) %. +%term _ + %pi (cfl/tp/arrow (val/tm/inr _) (oftp/tm/inr _ _) Deq Dseq) + %<- (neq/deq/sum-arrow Deq Dun) + %<- (uninhabited-seq/tm _ (tm/fun tp/unit tp/unit ([x tm] [f tm] x)) Dun Dseq) %. +%term _ + %pi (cfl/tp/arrow val/tm/tagloc (oftp/tm/tagloc _ _) Deq Dseq) + %<- (neq/deq/tag-arrow Deq Dun) + %<- (uninhabited-seq/tm _ (tm/fun tp/unit tp/unit ([x tm] [f tm] x)) Dun Dseq) %. +%term _ + %pi (cfl/tp/arrow _ (oftp/tm/tag _ _) Deq Dseq) + %<- (neq/deq/tagged-arrow Deq Dun) + %<- (uninhabited-seq/tm _ (tm/fun tp/unit tp/unit ([x tm] [f tm] x)) Dun Dseq) %. +%term _ + %pi (cfl/tp/arrow _ (oftp/tm/roll _ _) Deq Dseq) + %<- (neq/deq/mu-arrow Deq Dun) + %<- (uninhabited-seq/tm _ (tm/fun tp/unit tp/unit ([x tm] [f tm] x)) Dun Dseq) %. +%worlds () (cfl/tp/arrow _ _ _ _) %. +%total D1 (cfl/tp/arrow _ D1 _ _) %. +%sort cfl/tp/cross %. +%mode cfl/tp/cross %in %in %in %out %. +%term _ cfl/tp/cross _ (oftp/tm/pair _ _) _ seq/tm/refl %. +%term _ + %pi (cfl/tp/cross V (oftp/deq DofE Deq) Deq' Dseq) + %<- (cfl/tp/cross V DofE (cn-deq/trans Deq Deq') Dseq) %. +%term _ + %pi (cfl/tp/cross _ (oftp/tm/fun _ _) Deq Dseq) + %<- (neq/deq/arrow-cross Deq Dun) + %<- (uninhabited-seq/tm _ (tm/pair tm/unit tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/tp/cross _ oftp/tm/unit Deq Dseq) + %<- (neq/deq/unit-cross Deq Dun) + %<- (uninhabited-seq/tm _ (tm/pair tm/unit tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/tp/cross _ (oftp/tm/cnabs _ _) Deq Dseq) + %<- (neq/deq/forall-cross Deq Dun) + %<- (uninhabited-seq/tm _ (tm/pair tm/unit tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/tp/cross _ (oftp/tm/loc _ _) Deq Dseq) + %<- (neq/deq/ref-cross Deq Dun) + %<- (uninhabited-seq/tm _ (tm/pair tm/unit tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/tp/cross _ (oftp/tm/inl _ _) Deq Dseq) + %<- (neq/deq/sum-cross Deq Dun) + %<- (uninhabited-seq/tm _ (tm/pair tm/unit tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/tp/cross _ (oftp/tm/inr _ _) Deq Dseq) + %<- (neq/deq/sum-cross Deq Dun) + %<- (uninhabited-seq/tm _ (tm/pair tm/unit tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/tp/cross _ (oftp/tm/tagloc _ _) Deq Dseq) + %<- (neq/deq/tag-cross Deq Dun) + %<- (uninhabited-seq/tm _ (tm/pair tm/unit tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/tp/cross _ (oftp/tm/tag _ _) Deq Dseq) + %<- (neq/deq/tagged-cross Deq Dun) + %<- (uninhabited-seq/tm _ (tm/pair tm/unit tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/tp/cross _ (oftp/tm/roll _ _) Deq Dseq) + %<- (neq/deq/mu-cross Deq Dun) + %<- (uninhabited-seq/tm _ (tm/pair tm/unit tm/unit) Dun Dseq) %. +%worlds () (cfl/tp/cross _ _ _ _) %. +%total D1 (cfl/tp/cross _ D1 _ _) %. +%sort cfl/tp/forall %. +%mode cfl/tp/forall %in %in %in %out %. +%term _ cfl/tp/forall _ (oftp/tm/cnabs _ _) _ seq/tm/refl %. +%term _ + %pi (cfl/tp/forall V (oftp/deq DofE Deq) Deq' Dseq) + %<- (cfl/tp/forall V DofE (cn-deq/trans Deq Deq') Dseq) %. +%term _ + %pi (cfl/tp/forall _ (oftp/tm/fun _ _) Deq Dseq) + %<- (neq/deq/arrow-forall Deq Dun) + %<- (uninhabited-seq/tm _ (tm/cnabs kd/type ([a cn] tm/unit)) Dun Dseq) %. +%term _ + %pi (cfl/tp/forall _ oftp/tm/unit Deq Dseq) + %<- (neq/deq/unit-forall Deq Dun) + %<- (uninhabited-seq/tm _ (tm/cnabs kd/type ([a cn] tm/unit)) Dun Dseq) %. +%term _ + %pi (cfl/tp/forall _ (oftp/tm/pair _ _) Deq Dseq) + %<- (neq/deq/cross-forall Deq Dun) + %<- (uninhabited-seq/tm _ (tm/cnabs kd/type ([a cn] tm/unit)) Dun Dseq) %. +%term _ + %pi (cfl/tp/forall _ (oftp/tm/loc _ _) Deq Dseq) + %<- (neq/deq/ref-forall Deq Dun) + %<- (uninhabited-seq/tm _ (tm/cnabs kd/type ([a cn] tm/unit)) Dun Dseq) %. +%term _ + %pi (cfl/tp/forall _ (oftp/tm/inl _ _) Deq Dseq) + %<- (neq/deq/sum-forall Deq Dun) + %<- (uninhabited-seq/tm _ (tm/cnabs kd/type ([a cn] tm/unit)) Dun Dseq) %. +%term _ + %pi (cfl/tp/forall _ (oftp/tm/inr _ _) Deq Dseq) + %<- (neq/deq/sum-forall Deq Dun) + %<- (uninhabited-seq/tm _ (tm/cnabs kd/type ([a cn] tm/unit)) Dun Dseq) %. +%term _ + %pi (cfl/tp/forall _ (oftp/tm/tagloc _ _) Deq Dseq) + %<- (neq/deq/tag-forall Deq Dun) + %<- (uninhabited-seq/tm _ (tm/cnabs kd/type ([a cn] tm/unit)) Dun Dseq) %. +%term _ + %pi (cfl/tp/forall _ (oftp/tm/tag _ _) Deq Dseq) + %<- (neq/deq/tagged-forall Deq Dun) + %<- (uninhabited-seq/tm _ (tm/cnabs kd/type ([a cn] tm/unit)) Dun Dseq) %. +%term _ + %pi (cfl/tp/forall _ (oftp/tm/roll _ _) Deq Dseq) + %<- (neq/deq/mu-forall Deq Dun) + %<- (uninhabited-seq/tm _ (tm/cnabs kd/type ([a cn] tm/unit)) Dun Dseq) %. +%worlds () (cfl/tp/forall _ _ _ _) %. +%total D1 (cfl/tp/forall _ D1 _ _) %. +%sort cfl/tp/ref %. +%mode cfl/tp/ref %in %in %in %out %. +%term _ cfl/tp/ref _ (oftp/tm/loc _ _) _ seq/tm/refl %. +%term _ + %pi (cfl/tp/ref V (oftp/deq DofE Deq) Deq' Dseq) + %<- (cfl/tp/ref V DofE (cn-deq/trans Deq Deq') Dseq) %. +%term _ + %pi (cfl/tp/ref _ (oftp/tm/fun _ _) Deq Dseq) + %<- (neq/deq/arrow-ref Deq Dun) + %<- (uninhabited-seq/tm _ (tm/loc loc/z) Dun Dseq) %. +%term _ + %pi (cfl/tp/ref _ oftp/tm/unit Deq Dseq) + %<- (neq/deq/unit-ref Deq Dun) + %<- (uninhabited-seq/tm _ (tm/loc loc/z) Dun Dseq) %. +%term _ + %pi (cfl/tp/ref _ (oftp/tm/pair _ _) Deq Dseq) + %<- (neq/deq/cross-ref Deq Dun) + %<- (uninhabited-seq/tm _ (tm/loc loc/z) Dun Dseq) %. +%term _ + %pi (cfl/tp/ref _ (oftp/tm/cnabs _ _) Deq Dseq) + %<- (neq/deq/forall-ref Deq Dun) + %<- (uninhabited-seq/tm _ (tm/loc loc/z) Dun Dseq) %. +%term _ + %pi (cfl/tp/ref _ (oftp/tm/inl _ _) Deq Dseq) + %<- (neq/deq/sum-ref Deq Dun) + %<- (uninhabited-seq/tm _ (tm/loc loc/z) Dun Dseq) %. +%term _ + %pi (cfl/tp/ref _ (oftp/tm/inr _ _) Deq Dseq) + %<- (neq/deq/sum-ref Deq Dun) + %<- (uninhabited-seq/tm _ (tm/loc loc/z) Dun Dseq) %. +%term _ + %pi (cfl/tp/ref _ (oftp/tm/tagloc _ _) Deq Dseq) + %<- (neq/deq/tag-ref Deq Dun) + %<- (uninhabited-seq/tm _ (tm/loc loc/z) Dun Dseq) %. +%term _ + %pi (cfl/tp/ref _ (oftp/tm/tag _ _) Deq Dseq) + %<- (neq/deq/tagged-ref Deq Dun) + %<- (uninhabited-seq/tm _ (tm/loc loc/z) Dun Dseq) %. +%term _ + %pi (cfl/tp/ref _ (oftp/tm/roll _ _) Deq Dseq) + %<- (neq/deq/mu-ref Deq Dun) + %<- (uninhabited-seq/tm _ (tm/loc loc/z) Dun Dseq) %. +%worlds () (cfl/tp/ref _ _ _ _) %. +%total D1 (cfl/tp/ref _ D1 _ _) %. +%sort seq-in {E tm} %. +%term seq-inl seq-in (tm/inl C E) seq/tm/refl %. +%term seq-inr seq-in (tm/inr C E) seq/tm/refl %. +%sort cfl-neq-seq-in {E1 tm} %. +%mode cfl-neq-seq-in %in %in %out %. +%worlds (ofkd+vdt-block) (cfl-neq-seq-in _ _ _) %. +%total {} (cfl-neq-seq-in _ _ _) %. +%sort cfl/tp/sum %. +%mode cfl/tp/sum %in %in %in %out %. +%term _ cfl/tp/sum _ (oftp/tm/inl _ _) _ seq-inl %. +%term _ cfl/tp/sum _ (oftp/tm/inr _ _) _ seq-inr %. +%term _ + %pi (cfl/tp/sum V (oftp/deq DofE Deq) Deq' Dseq) + %<- (cfl/tp/sum V DofE (cn-deq/trans Deq Deq') Dseq) %. +%term _ + %pi (cfl/tp/sum _ (oftp/tm/fun _ _) Deq Dseq) + %<- (neq/deq/arrow-sum Deq Dun) + %<- (cfl-neq-seq-in _ Dun Dseq) %. +%term _ + %pi (cfl/tp/sum _ oftp/tm/unit Deq Dseq) + %<- (neq/deq/unit-sum Deq Dun) + %<- (cfl-neq-seq-in _ Dun Dseq) %. +%term _ + %pi (cfl/tp/sum _ (oftp/tm/pair _ _) Deq Dseq) + %<- (neq/deq/cross-sum Deq Dun) + %<- (cfl-neq-seq-in _ Dun Dseq) %. +%term _ + %pi (cfl/tp/sum _ (oftp/tm/cnabs _ _) Deq Dseq) + %<- (neq/deq/forall-sum Deq Dun) + %<- (cfl-neq-seq-in _ Dun Dseq) %. +%term _ + %pi (cfl/tp/sum _ (oftp/tm/loc _ _) Deq Dseq) + %<- (neq/deq/ref-sum Deq Dun) + %<- (cfl-neq-seq-in _ Dun Dseq) %. +%term _ + %pi (cfl/tp/sum _ (oftp/tm/tagloc _ _) Deq Dseq) + %<- (neq/deq/tag-sum Deq Dun) + %<- (cfl-neq-seq-in _ Dun Dseq) %. +%term _ + %pi (cfl/tp/sum _ (oftp/tm/tag _ _) Deq Dseq) + %<- (neq/deq/tagged-sum Deq Dun) + %<- (cfl-neq-seq-in _ Dun Dseq) %. +%term _ + %pi (cfl/tp/sum _ (oftp/tm/roll _ _) Deq Dseq) + %<- (neq/deq/mu-sum Deq Dun) + %<- (cfl-neq-seq-in _ Dun Dseq) %. +%worlds () (cfl/tp/sum _ _ _ _) %. +%total D1 (cfl/tp/sum _ D1 _ _) %. +%sort cfl/tp/tag %. +%mode cfl/tp/tag %in %in %in %out %. +%term _ cfl/tp/tag _ (oftp/tm/tagloc _ _) _ seq/tm/refl %. +%term _ + %pi (cfl/tp/tag V (oftp/deq DofE Deq) Deq' Dseq) + %<- (cfl/tp/tag V DofE (cn-deq/trans Deq Deq') Dseq) %. +%term _ + %pi (cfl/tp/tag _ (oftp/tm/fun _ _) Deq Dseq) + %<- (neq/deq/arrow-tag Deq Dun) + %<- (uninhabited-seq/tm _ (tm/tagloc loc/z) Dun Dseq) %. +%term _ + %pi (cfl/tp/tag _ oftp/tm/unit Deq Dseq) + %<- (neq/deq/unit-tag Deq Dun) + %<- (uninhabited-seq/tm _ (tm/tagloc loc/z) Dun Dseq) %. +%term _ + %pi (cfl/tp/tag _ (oftp/tm/pair _ _) Deq Dseq) + %<- (neq/deq/cross-tag Deq Dun) + %<- (uninhabited-seq/tm _ (tm/tagloc loc/z) Dun Dseq) %. +%term _ + %pi (cfl/tp/tag _ (oftp/tm/cnabs _ _) Deq Dseq) + %<- (neq/deq/forall-tag Deq Dun) + %<- (uninhabited-seq/tm _ (tm/tagloc loc/z) Dun Dseq) %. +%term _ + %pi (cfl/tp/tag _ (oftp/tm/inl _ _) Deq Dseq) + %<- (neq/deq/sum-tag Deq Dun) + %<- (uninhabited-seq/tm _ (tm/tagloc loc/z) Dun Dseq) %. +%term _ + %pi (cfl/tp/tag _ (oftp/tm/inr _ _) Deq Dseq) + %<- (neq/deq/sum-tag Deq Dun) + %<- (uninhabited-seq/tm _ (tm/tagloc loc/z) Dun Dseq) %. +%term _ + %pi (cfl/tp/tag _ (oftp/tm/loc _ _) Deq Dseq) + %<- (neq/deq/ref-tag Deq Dun) + %<- (uninhabited-seq/tm _ (tm/tagloc loc/z) Dun Dseq) %. +%term _ + %pi (cfl/tp/tag _ (oftp/tm/tag _ _) Deq Dseq) + %<- (neq/deq/tagged-tag Deq Dun) + %<- (uninhabited-seq/tm _ (tm/tagloc loc/z) Dun Dseq) %. +%term _ + %pi (cfl/tp/tag _ (oftp/tm/roll _ _) Deq Dseq) + %<- (neq/deq/mu-tag Deq Dun) + %<- (uninhabited-seq/tm _ (tm/tagloc loc/z) Dun Dseq) %. +%worlds () (cfl/tp/tag _ _ _ _) %. +%total D1 (cfl/tp/tag _ D1 _ _) %. +%sort cfl/tp/tagged %. +%mode cfl/tp/tagged %in %in %in %out %. +%term _ + %pi (cfl/tp/tagged (val/tm/tag V _) (oftp/tm/tag D1 _) _ DQ') + %<- (vdt/oftp D1 DC) + %<- (cfl/tp/tag V D1 (cn-deq/refl DC) DQ) + %<- (seq/tm/tag _ DQ DQ') %. +%term _ + %pi (cfl/tp/tagged V (oftp/deq DofE Deq) Deq' Dseq) + %<- (cfl/tp/tagged V DofE (cn-deq/trans Deq Deq') Dseq) %. +%term _ + %pi (cfl/tp/tagged _ (oftp/tm/fun _ _) Deq Dseq) + %<- (neq/deq/arrow-tagged Deq Dun) + %<- (uninhabited-seq/tm _ (tm/tag (tm/tagloc loc/z) tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/tp/tagged _ oftp/tm/unit Deq Dseq) + %<- (neq/deq/unit-tagged Deq Dun) + %<- (uninhabited-seq/tm _ (tm/tag (tm/tagloc loc/z) tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/tp/tagged _ (oftp/tm/cnabs _ _) Deq Dseq) + %<- (neq/deq/forall-tagged Deq Dun) + %<- (uninhabited-seq/tm _ (tm/tag (tm/tagloc loc/z) tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/tp/tagged _ (oftp/tm/loc _ _) Deq Dseq) + %<- (neq/deq/ref-tagged Deq Dun) + %<- (uninhabited-seq/tm _ (tm/tag (tm/tagloc loc/z) tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/tp/tagged _ (oftp/tm/inl _ _) Deq Dseq) + %<- (neq/deq/sum-tagged Deq Dun) + %<- (uninhabited-seq/tm _ (tm/tag (tm/tagloc loc/z) tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/tp/tagged _ (oftp/tm/inr _ _) Deq Dseq) + %<- (neq/deq/sum-tagged Deq Dun) + %<- (uninhabited-seq/tm _ (tm/tag (tm/tagloc loc/z) tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/tp/tagged _ (oftp/tm/tagloc _ _) Deq Dseq) + %<- (neq/deq/tag-tagged Deq Dun) + %<- (uninhabited-seq/tm _ (tm/tag (tm/tagloc loc/z) tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/tp/tagged _ (oftp/tm/pair _ _) Deq Dseq) + %<- (neq/deq/cross-tagged Deq Dun) + %<- (uninhabited-seq/tm _ (tm/tag (tm/tagloc loc/z) tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/tp/tagged _ (oftp/tm/roll _ _) Deq Dseq) + %<- (neq/deq/mu-tagged Deq Dun) + %<- (uninhabited-seq/tm _ (tm/tag (tm/tagloc loc/z) tm/unit) Dun Dseq) %. +%worlds () (cfl/tp/tagged _ _ _ _) %. +%total D1 (cfl/tp/tagged _ D1 _ _) %. +%sort cfl/cn/mu %. +%mode cfl/cn/mu %in %in %in %out %. +%term _ cfl/cn/mu _ (oftp/tm/roll _ _) _ seq/tm/refl %. +%term _ + %pi (cfl/cn/mu V (oftp/deq DofE Deq) Deq' Dseq) + %<- (cfl/cn/mu V DofE (cn-deq/trans Deq Deq') Dseq) %. +%term _ + %pi (cfl/cn/mu _ (oftp/tm/fun _ _) Deq Dseq) + %<- (neq/deq/arrow-mu Deq Dun) + %<- (uninhabited-seq/tm _ (tm/roll tp/unit tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/cn/mu _ oftp/tm/unit Deq Dseq) + %<- (neq/deq/unit-mu Deq Dun) + %<- (uninhabited-seq/tm _ (tm/roll tp/unit tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/cn/mu _ (oftp/tm/pair _ _) Deq Dseq) + %<- (neq/deq/cross-mu Deq Dun) + %<- (uninhabited-seq/tm _ (tm/roll tp/unit tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/cn/mu _ (oftp/tm/loc _ _) Deq Dseq) + %<- (neq/deq/ref-mu Deq Dun) + %<- (uninhabited-seq/tm _ (tm/roll tp/unit tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/cn/mu _ (oftp/tm/inl _ _) Deq Dseq) + %<- (neq/deq/sum-mu Deq Dun) + %<- (uninhabited-seq/tm _ (tm/roll tp/unit tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/cn/mu _ (oftp/tm/inr _ _) Deq Dseq) + %<- (neq/deq/sum-mu Deq Dun) + %<- (uninhabited-seq/tm _ (tm/roll tp/unit tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/cn/mu _ (oftp/tm/tagloc _ _) Deq Dseq) + %<- (neq/deq/tag-mu Deq Dun) + %<- (uninhabited-seq/tm _ (tm/roll tp/unit tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/cn/mu _ (oftp/tm/tag _ _) Deq Dseq) + %<- (neq/deq/tagged-mu Deq Dun) + %<- (uninhabited-seq/tm _ (tm/roll tp/unit tm/unit) Dun Dseq) %. +%term _ + %pi (cfl/cn/mu _ (oftp/tm/cnabs _ _) Deq Dseq) + %<- (neq/deq/forall-mu Deq Dun) + %<- (uninhabited-seq/tm _ (tm/roll tp/unit tm/unit) Dun Dseq) %. +%worlds () (cfl/cn/mu _ _ _ _) %. +%total D1 (cfl/cn/mu _ D1 _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/constructor-beta.lf b/new-tests/stelf-output/tslf/constructor-beta.lf new file mode 100644 index 0000000..ed660b1 --- /dev/null +++ b/new-tests/stelf-output/tslf/constructor-beta.lf @@ -0,0 +1,111 @@ +%sort psi-cn %. +%term psi-cn/eps psi-cn ([a] a) %. +%term psi-cn/pj1 %pi (psi-cn ([a] cn/pj1 (PS a))) %<- (psi-cn PS) %. +%term psi-cn/pj2 %pi (psi-cn ([a] cn/pj2 (PS a))) %<- (psi-cn PS) %. +%term psi-cn/app {C cn} %pi (psi-cn ([a] cn/app (PS a) C)) %<- (psi-cn PS) %. +%%%% beta rules from SH +%sort inv/ofkd/cn/pair %. +%mode inv/ofkd/cn/pair %in %out %out %out %. +%sort cn-beta-pj1 %. +%mode cn-beta-pj1 %in %in %out %. +%sort cn-beta-pj2 %. +%mode cn-beta-pj2 %in %in %out %. +%term _ + %pi (inv/ofkd/cn/pair (ofkd/cn/pair D1 D2) D1 D2 DKS) + %<- (vdt/ofkd (ofkd/cn/pair D1 D2) DK) + %<- (kd-refl/sub DK DKS) %. +%term _ + %pi (inv/ofkd/cn/pair (ofkd/sgm-ext D1 D2) D1' D2' DKS) + %<- (vdt/ofkd (ofkd/sgm-ext D1 D2) DK) + %<- (kd-refl/sub DK DKS) + %<- (cn-beta-pj1 psi-cn/eps D1 DQ1) + %<- (vdt/cn-deq DQ1 _ D1' _) + %<- (cn-beta-pj2 psi-cn/eps D2 DQ2) + %<- (vdt/cn-deq DQ2 _ D2' _) %. +%term _ + %pi (inv/ofkd/cn/pair (ofkd/sub DP DS) D1 D2 DSS) + %<- (inv/ofkd/cn/pair DP D1 D2 DKS) + %<- (kd-trans/sub DKS DS DSS) %. +%term _ + %pi (inv/ofkd/cn/pair (ofkd/deq DP DQ) D1 D2 DSS) + %<- (vdt/kd-deq DQ DQ1 DQ2) + %<- (kd-anti DQ DQ1 DQ2 DS _) + %<- (inv/ofkd/cn/pair DP D1 D2 DKS) + %<- (kd-trans/sub DKS DS DSS) %. +%term _ + %pi (cn-beta-pj1 psi-cn/eps (ofkd/cn/pj1 DP) (cn-deq/sub DCQ DS1)) + %<- (inv/ofkd/cn/pair DP D1 D2 (kd-sub/kd/sgm DS1 DS2 _)) + %<- (stone-2/108 D1 D2 DCQ) %. +%term _ + %pi (cn-beta-pj1 (%the (psi-cn P) PS) (%the (ofkd (P (cn/pj1 (cn/pair C1 C2))) _) (ofkd/sgm-ext D1 D2)) (cn-deq/sgm-ext DCQ1 DCQ2)) + %<- (cn-beta-pj1 (psi-cn/pj1 PS) D1 DCQ1) + %<- (cn-beta-pj1 (psi-cn/pj2 PS) D2 DCQ2) %. +%term _ + %pi (cn-beta-pj1 (%the (psi-cn P) PS) (ofkd/sub (%the (ofkd (P (cn/pj1 (cn/pair C1 C2))) _) D1) DS) (cn-deq/sub DCQ DS)) + %<- (cn-beta-pj1 PS D1 DCQ) %. +%term _ + %pi (cn-beta-pj1 (psi-cn/pj1 (%the (psi-cn P) PS)) (ofkd/cn/pj1 D1) (cn-deq/cn/pj1 DCQ)) + %<- (cn-beta-pj1 PS D1 DCQ) %. +%term _ + %pi (cn-beta-pj1 (psi-cn/pj2 (%the (psi-cn P) PS)) (ofkd/cn/pj2 D1) (cn-deq/cn/pj2 DCQ)) + %<- (cn-beta-pj1 PS D1 DCQ) %. +%term _ + %pi (cn-beta-pj1 (psi-cn/app _ (%the (psi-cn P) PS)) (ofkd/cn/app D1 D2) (cn-deq/cn/app DCQ (cn-deq/refl D2))) + %<- (cn-beta-pj1 PS D1 DCQ) %. +%term _ + %pi (cn-beta-pj1 (%the (psi-cn P) PS) (%the (ofkd (P (cn/pj1 (cn/pair C1 C2))) _) (ofkd/pi-ext D1 D2)) (cn-deq/pi-ext DCA DCB DCQ2)) + %<- (cn-beta-pj1 PS D1 DCQ1) + %<- (vdt/cn-deq DCQ1 DCA DCB (kd-wf/kd/pi DW _)) + %<- ({a} {da} {dm mofkd da met/unit} {_ can-mofkd da dm} {_ vdt/ofkd da DW} cn-beta-pj1 (psi-cn/app a PS) (D2 a da) (DCQ2 a da)) %. +%term _ + %pi (cn-beta-pj1 (%the (psi-cn P) PS) (%the (ofkd (P (cn/pj1 (cn/pair C1 C2))) _) (ofkd/kd/sing D1)) DCQ1') + %<- (cn-beta-pj1 PS D1 DCQ1) + %<- (stone-2/120 DCQ1 DCQ1') %. +%term _ + %pi (cn-beta-pj1 psi-cn/eps (ofkd/cn/pj1 DP) (cn-deq/sub DCQ DS1)) + %<- (inv/ofkd/cn/pair DP D1 D2 (kd-sub/kd/sgm DS1 DS2 _)) + %<- (stone-2/108 D1 D2 DCQ) %. +%term _ + %pi (cn-beta-pj1 (%the (psi-cn P) PS) (%the (ofkd (P (cn/pj1 (cn/pair C1 C2))) _) (ofkd/sgm-ext D1 D2)) (cn-deq/sgm-ext DCQ1 DCQ2)) + %<- (cn-beta-pj1 (psi-cn/pj1 PS) D1 DCQ1) + %<- (cn-beta-pj1 (psi-cn/pj2 PS) D2 DCQ2) %. +%term _ + %pi (cn-beta-pj1 (%the (psi-cn P) PS) (ofkd/sub (%the (ofkd (P (cn/pj1 (cn/pair C1 C2))) _) D1) DS) (cn-deq/sub DCQ DS)) + %<- (cn-beta-pj1 PS D1 DCQ) %. +%term _ + %pi (cn-beta-pj1 (%the (psi-cn P) PS) (ofkd/deq (%the (ofkd (P (cn/pj1 (cn/pair C1 C2))) _) D1) DS) (cn-deq/deq DCQ DS)) + %<- (cn-beta-pj1 PS D1 DCQ) %. +%term _ + %pi (cn-beta-pj2 (psi-cn/pj1 (%the (psi-cn P) PS)) (ofkd/cn/pj1 D1) (cn-deq/cn/pj1 DCQ)) + %<- (cn-beta-pj2 PS D1 DCQ) %. +%term _ + %pi (cn-beta-pj2 (psi-cn/pj2 (%the (psi-cn P) PS)) (ofkd/cn/pj2 D1) (cn-deq/cn/pj2 DCQ)) + %<- (cn-beta-pj2 PS D1 DCQ) %. +%term _ + %pi (cn-beta-pj2 (psi-cn/app _ (%the (psi-cn P) PS)) (ofkd/cn/app D1 D2) (cn-deq/cn/app DCQ (cn-deq/refl D2))) + %<- (cn-beta-pj2 PS D1 DCQ) %. +%term _ + %pi (cn-beta-pj2 (%the (psi-cn P) PS) (%the (ofkd (P (cn/pj2 (cn/pair C1 C2))) _) (ofkd/pi-ext D1 D2)) (cn-deq/pi-ext DCA DCB DCQ2)) + %<- (cn-beta-pj2 PS D1 DCQ1) + %<- (vdt/cn-deq DCQ1 DCA DCB (kd-wf/kd/pi DW _)) + %<- ({a} {da} {dm mofkd da met/unit} {_ can-mofkd da dm} {_ vdt/ofkd da DW} cn-beta-pj2 (psi-cn/app a PS) (D2 a da) (DCQ2 a da)) %. +%term _ + %pi (cn-beta-pj2 (%the (psi-cn P) PS) (%the (ofkd (P (cn/pj2 (cn/pair C1 C2))) _) (ofkd/kd/sing D1)) DCQ1') + %<- (cn-beta-pj2 PS D1 DCQ1) + %<- (stone-2/120 DCQ1 DCQ1') %. +%term _ + %pi (cn-beta-pj2 psi-cn/eps (ofkd/cn/pj2 DP) (cn-deq/sub DCQ (DS2 _ (ofkd/cn/pj1 (ofkd/cn/pair D1 D2))))) + %<- (inv/ofkd/cn/pair DP D1 D2 (kd-sub/kd/sgm DS1 DS2 _)) + %<- (stone-2/109 D1 D2 DCQ) %. +%term _ + %pi (cn-beta-pj2 (%the (psi-cn P) PS) (%the (ofkd (P (cn/pj2 (cn/pair C1 C2))) _) (ofkd/sgm-ext D1 D2)) (cn-deq/sgm-ext DCQ1 DCQ2)) + %<- (cn-beta-pj2 (psi-cn/pj1 PS) D1 DCQ1) + %<- (cn-beta-pj2 (psi-cn/pj2 PS) D2 DCQ2) %. +%term _ + %pi (cn-beta-pj2 (%the (psi-cn P) PS) (ofkd/sub (%the (ofkd (P (cn/pj2 (cn/pair C1 C2))) _) D1) DS) (cn-deq/sub DCQ DS)) + %<- (cn-beta-pj2 PS D1 DCQ) %. +%term _ + %pi (cn-beta-pj2 (%the (psi-cn P) PS) (ofkd/deq (%the (ofkd (P (cn/pj2 (cn/pair C1 C2))) _) D1) DS) (cn-deq/deq DCQ DS)) + %<- (cn-beta-pj2 PS D1 DCQ) %. +%worlds (ofkd+vdt-block) (cn-beta-pj2 _ _ _) (cn-beta-pj1 _ _ _) (inv/ofkd/cn/pair _ _ _ _) %. +%total (D1 D2 D3) (inv/ofkd/cn/pair D1 _ _ _) (cn-beta-pj1 _ D3 _) (cn-beta-pj2 _ D2 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/correspondence-thm.lf b/new-tests/stelf-output/tslf/correspondence-thm.lf new file mode 100644 index 0000000..aaa169d --- /dev/null +++ b/new-tests/stelf-output/tslf/correspondence-thm.lf @@ -0,0 +1,1251 @@ +%%%%% Map into the singleton language %%%%% +%sort map %. +%sort tmap %. +%mode map %star %star %. +%mode tmap %star %star %. +%term tmap/t tmap kd/type et %. +%term tmap/sing %pi (tmap (kd/sing C) (esing M)) %<- (map C M) %. +%term tmap/pi + %pi (tmap (kd/pi K1 K2) (epi A1 A2)) + %<- (tmap K1 A1) + %<- ({a} {x} %pi (map a x) %-> (tmap (K2 a) (A2 x))) %. +%term tmap/sigma + %pi (tmap (kd/sgm K1 K2) (esigma A1 A2)) + %<- (tmap K1 A1) + %<- ({a} {x} %pi (map a x) %-> (tmap (K2 a) (A2 x))) %. +%term map/app %pi (map (cn/app C1 C2) (eapp M1 M2)) %<- (map C1 M1) %<- (map C2 M2) %. +%term map/pi1 %pi (map (cn/pj1 C) (epi1 M)) %<- (map C M) %. +%term map/pi2 %pi (map (cn/pj2 C) (epi2 M)) %<- (map C M) %. +%term map/lam + %pi (map (cn/lam K C) (elam A M)) + %<- (tmap K A) + %<- ({a} {x} %pi (map a x) %-> (map (C a) (M x))) %. +%term map/pair %pi (map (cn/pair C1 C2) (epair M1 M2)) %<- (map C1 M1) %<- (map C2 M2) %. +%term map/unit map tp/unit (econst unit) %. +%term map/tagged map tp/tagged (econst tagged) %. +%term map/ref %pi (map (tp/ref C) (eapp (econst ref) M)) %<- (map C M) %. +%term map/tag %pi (map (tp/tag C) (eapp (econst tag) M)) %<- (map C M) %. +%term map/mu + %pi (map (cn/mu _ C) (eapp (econst mu) (elam et M))) + %<- ({a} {x} %pi (map a x) %-> (map (C a) (M x))) %. +%term map/arrow + %pi (map (tp/arrow C1 C2) (eapp (eapp (econst arrow) M1) M2)) + %<- (map C1 M1) + %<- (map C2 M2) %. +%term map/cross + %pi (map (tp/cross C1 C2) (eapp (eapp (econst cross) M1) M2)) + %<- (map C1 M1) + %<- (map C2 M2) %. +%term map/sum + %pi (map (tp/sum C1 C2) (eapp (eapp (econst dsum) M1) M2)) + %<- (map C1 M1) + %<- (map C2 M2) %. +%term map/forall + %pi (map (tp/forall K C) (eapp (eforall A) (elam A M))) + %<- (tmap K A) + %<- ({a} {x} %pi (map a x) %-> (map (C a) (M x))) %. +%%%%% Map from the singleton language %%%%% +%sort unmap %. +%sort tunmap %. +%mode unmap %star %star %. +%mode tunmap %star %star %. +%term tunmap/t tunmap et kd/type %. +%term tunmap/sing %pi (tunmap (esing M) (kd/sing C)) %<- (unmap M C) %. +%term tunmap/pi + %pi (tunmap (epi A1 A2) (kd/pi K1 K2)) + %<- (tunmap A1 K1) + %<- ({x} {a} %pi (unmap x a) %-> (tunmap (A2 x) (K2 a))) %. +%term tunmap/sigma + %pi (tunmap (esigma A1 A2) (kd/sgm K1 K2)) + %<- (tunmap A1 K1) + %<- ({x} {a} %pi (unmap x a) %-> (tunmap (A2 x) (K2 a))) %. +%term unmap/app %pi (unmap (eapp M1 M2) (cn/app C1 C2)) %<- (unmap M1 C1) %<- (unmap M2 C2) %. +%term unmap/pi1 %pi (unmap (epi1 M) (cn/pj1 C)) %<- (unmap M C) %. +%term unmap/pi2 %pi (unmap (epi2 M) (cn/pj2 C)) %<- (unmap M C) %. +%term unmap/lam + %pi (unmap (elam A M) (cn/lam K C)) + %<- (tunmap A K) + %<- ({x} {a} %pi (unmap x a) %-> (unmap (M x) (C a))) %. +%term unmap/pair %pi (unmap (epair M1 M2) (cn/pair C1 C2)) %<- (unmap M1 C1) %<- (unmap M2 C2) %. +%term unmap/unit unmap (econst unit) tp/unit %. +%term unmap/tagged unmap (econst tagged) tp/tagged %. +%term unmap/ref unmap (econst ref) (cn/lam kd/type ([a] tp/ref a)) %. +%term unmap/tag unmap (econst tag) (cn/lam kd/type ([a] tp/tag a)) %. +%term unmap/mu unmap (econst mu) (cn/lam (kd/pi kd/type ([_] kd/type)) ([a] cn/mu kd/type ([b] cn/app a b))) %. +%term unmap/arrow unmap (econst arrow) (cn/lam kd/type ([a] cn/lam kd/type ([b] tp/arrow a b))) %. +%term unmap/cross unmap (econst cross) (cn/lam kd/type ([a] cn/lam kd/type ([b] tp/cross a b))) %. +%term unmap/sum unmap (econst dsum) (cn/lam kd/type ([a] cn/lam kd/type ([b] tp/sum a b))) %. +%term unmap/forall + %pi (unmap (eforall A) (cn/lam (kd/pi K ([_] kd/type)) ([a] tp/forall K ([b] cn/app a b)))) + %<- (tunmap A K) %. +%%%%% Equality %%%%% +%sort cn-eq %. +%term cn-eq/i cn-eq C C %. +%sort kd-eq %. +%term kd-eq/i kd-eq K K %. +%%% cons +%sort cn-resp-bind {C %pi kd %-> (%pi cn %-> cn) %-> cn} %. +%mode cn-resp-bind %in %in %in %out %. +%term _ cn-resp-bind _ kd-eq/i ([_] cn-eq/i) cn-eq/i %. +%worlds (cn-block) (cn-resp-bind _ _ _ _) %. +%total {} (cn-resp-bind _ _ _ _) %. +%sort cn-resp-cn {C %pi cn %-> cn} %. +%mode cn-resp-cn %in %in %out %. +%term _ cn-resp-cn _ cn-eq/i cn-eq/i %. +%worlds (cn-block) (cn-resp-cn _ _ _) %. +%total {} (cn-resp-cn _ _ _) %. +%sort cn-resp-cn2 {C %pi cn %-> cn %-> cn} %. +%mode cn-resp-cn2 %in %in %in %out %. +%term _ cn-resp-cn2 _ cn-eq/i cn-eq/i cn-eq/i %. +%worlds (cn-block) (cn-resp-cn2 _ _ _ _) %. +%total {} (cn-resp-cn2 _ _ _ _) %. +%sort cn-resp-kd {C %pi kd %-> cn} %. +%mode cn-resp-kd %in %in %out %. +%term _ cn-resp-kd _ kd-eq/i cn-eq/i %. +%worlds (cn-block) (cn-resp-kd _ _ _) %. +%total {} (cn-resp-kd _ _ _) %. +%sort kd-resp-bind {K %pi kd %-> (%pi cn %-> kd) %-> kd} %. +%mode kd-resp-bind %in %in %in %out %. +%term _ kd-resp-bind _ kd-eq/i ([_] kd-eq/i) kd-eq/i %. +%worlds (cn-block) (kd-resp-bind _ _ _ _) %. +%total {} (kd-resp-bind _ _ _ _) %. +%sort kd-resp-cn {K %pi cn %-> kd} %. +%mode kd-resp-cn %in %in %out %. +%term _ kd-resp-cn _ cn-eq/i kd-eq/i %. +%worlds (cn-block) (kd-resp-cn _ _ _) %. +%total {} (kd-resp-cn _ _ _) %. +%%% respection +%sort cn-deq-resp %. +%mode cn-deq-resp %in %in %in %in %out %. +%term _ cn-deq-resp cn-eq/i cn-eq/i kd-eq/i D D %. +%worlds (ofkd+vdt-block ofkd-block) (cn-deq-resp _ _ _ _ _) %. +%total {} (cn-deq-resp _ _ _ _ _) %. +%sort kd-deq-resp %. +%mode kd-deq-resp %in %in %in %out %. +%term _ kd-deq-resp kd-eq/i kd-eq/i D D %. +%worlds (ofkd+vdt-block ofkd-block) (kd-deq-resp _ _ _ _) %. +%total {} (kd-deq-resp _ _ _ _) %. +%sort kd-wf-resp %. +%mode kd-wf-resp %in %in %out %. +%term _ kd-wf-resp kd-eq/i D D %. +%worlds (ofkd+vdt-block ofkd-block) (kd-wf-resp _ _ _) %. +%total {} (kd-wf-resp _ _ _) %. +%sort ofkd-resp %. +%mode ofkd-resp %in %in %in %out %. +%term _ ofkd-resp cn-eq/i kd-eq/i D D %. +%worlds (ofkd+vdt-block ofkd-block) (ofkd-resp _ _ _ _) %. +%total {} (ofkd-resp _ _ _ _) %. +%sort cn-deq-resp %. +%mode cn-deq-resp %in %in %in %in %out %. +%term _ cn-deq-resp cn-eq/i cn-eq/i kd-eq/i D D %. +%worlds (ofkd+vdt-block ofkd-block) (cn-deq-resp _ _ _ _ _) %. +%total {} (cn-deq-resp _ _ _ _ _) %. +%%%%% Unmap Functionality %%%%% +%sort can-unmap {M} %. +%mode can-unmap %in %out %. +%sort can-tunmap {A} %. +%mode can-tunmap %in %out %. +%block can-unmap-bind {a cn} [x eterm] [xt unmap x a] [thm can-unmap x xt]%. +%block can-unmap-bind-for-unmap {a cn} {b etp} [x eterm] [d evof x b] [xt unmap x a] [thm can-unmap x xt]%. +%term _ %pi (can-unmap _ (unmap/app D2 D1)) %<- (can-unmap _ D1) %<- (can-unmap _ D2) %. +%term _ %pi (can-unmap _ (unmap/pi1 D)) %<- (can-unmap _ D) %. +%term _ %pi (can-unmap _ (unmap/pi2 D)) %<- (can-unmap _ D) %. +%term _ + %pi (can-unmap _ (unmap/lam D2 D1)) + %<- (can-tunmap _ D1) + %<- ({a} {x} {xt} %pi (can-unmap x xt) %-> (can-unmap _ (D2 x a xt))) %. +%term _ %pi (can-unmap _ (unmap/pair D2 D1)) %<- (can-unmap _ D1) %<- (can-unmap _ D2) %. +%term _ can-unmap _ unmap/ref %. +%term _ can-unmap _ unmap/tag %. +%term _ can-unmap _ unmap/unit %. +%term _ can-unmap _ unmap/tagged %. +%term _ can-unmap _ unmap/arrow %. +%term _ can-unmap _ unmap/cross %. +%term _ can-unmap _ unmap/sum %. +%term _ can-unmap _ unmap/mu %. +%term _ %pi (can-unmap _ (unmap/forall D)) %<- (can-tunmap _ D) %. +%term _ can-tunmap _ tunmap/t %. +%term _ %pi (can-tunmap _ (tunmap/sing D)) %<- (can-unmap _ D) %. +%term _ + %pi (can-tunmap _ (tunmap/pi D2 D1)) + %<- (can-tunmap _ D1) + %<- ({a} {x} {xt} %pi (can-unmap x xt) %-> (can-tunmap _ (D2 x a xt))) %. +%term _ + %pi (can-tunmap _ (tunmap/sigma D2 D1)) + %<- (can-tunmap _ D1) + %<- ({a} {x} {xt} %pi (can-unmap x xt) %-> (can-tunmap _ (D2 x a xt))) %. +%worlds (cn-block can-unmap-bind can-unmap-bind-for-unmap) (can-unmap _ _) (can-tunmap _ _) %. +%total (D1 D2) (can-unmap D1 _) (can-tunmap D2 _) %. +%sort unmap-fun %. +%mode unmap-fun %in %in %out %. +%sort tunmap-fun %. +%mode tunmap-fun %in %in %out %. +%block unmap-fun-bind {a cn} [x eterm] [xt unmap x a]%. +%block unmap-fun-bind-for-invert {a cn} [x eterm] [at map a x] [xt unmap x a]%. +%block unmap-fun-bind-for-unmap {a cn} {b etp} [x eterm] [d evof x b] [xt unmap x a]%. +%term _ unmap-fun D D cn-eq/i %. +%term _ + %pi (unmap-fun (unmap/app D2 D1) (unmap/app D2' D1') Deq) + %<- (unmap-fun D1 D1' Deq1) + %<- (unmap-fun D2 D2' Deq2) + %<- (cn-resp-cn2 cn/app Deq1 Deq2 Deq) %. +%term _ + %pi (unmap-fun (unmap/pi1 D) (unmap/pi1 D') Deq) + %<- (unmap-fun D D' Deq1) + %<- (cn-resp-cn cn/pj1 Deq1 Deq) %. +%term _ + %pi (unmap-fun (unmap/pi2 D) (unmap/pi2 D') Deq) + %<- (unmap-fun D D' Deq1) + %<- (cn-resp-cn cn/pj2 Deq1 Deq) %. +%term _ + %pi (unmap-fun (unmap/lam D2 D1) (unmap/lam D2' D1') Deq) + %<- (tunmap-fun D1 D1' Deq1) + %<- ({a} {x} {xt} unmap-fun (D2 x a xt) (D2' x a xt) (Deq2 a)) + %<- (cn-resp-bind cn/lam Deq1 Deq2 Deq) %. +%term _ + %pi (unmap-fun (unmap/pair D2 D1) (unmap/pair D2' D1') Deq) + %<- (unmap-fun D1 D1' Deq1) + %<- (unmap-fun D2 D2' Deq2) + %<- (cn-resp-cn2 cn/pair Deq1 Deq2 Deq) %. +%term _ + %pi (unmap-fun (unmap/forall D) (unmap/forall D') Deq) + %<- (tunmap-fun D D' Deq1) + %<- (cn-resp-kd ([k] cn/lam (kd/pi k ([_] kd/type)) ([a] tp/forall k ([b] cn/app a b))) Deq1 Deq) %. +%term _ tunmap-fun tunmap/t tunmap/t kd-eq/i %. +%term _ + %pi (tunmap-fun (tunmap/sing D) (tunmap/sing D') Deq) + %<- (unmap-fun D D' Deq1) + %<- (kd-resp-cn kd/sing Deq1 Deq) %. +%term _ + %pi (tunmap-fun (tunmap/pi D2 D1) (tunmap/pi D2' D1') Deq) + %<- (tunmap-fun D1 D1' Deq1) + %<- ({a} {x} {xt} tunmap-fun (D2 x a xt) (D2' x a xt) (Deq2 a)) + %<- (kd-resp-bind kd/pi Deq1 Deq2 Deq) %. +%term _ + %pi (tunmap-fun (tunmap/sigma D2 D1) (tunmap/sigma D2' D1') Deq) + %<- (tunmap-fun D1 D1' Deq1) + %<- ({a} {x} {xt} tunmap-fun (D2 x a xt) (D2' x a xt) (Deq2 a)) + %<- (kd-resp-bind kd/sgm Deq1 Deq2 Deq) %. +%sort _ %. +%worlds (cn-block unmap-fun-bind unmap-fun-bind-for-invert unmap-fun-bind-for-unmap) (unmap-fun _ _ _) (tunmap-fun _ _ _) %. +%total (D1 D2) (unmap-fun _ D1 _) (tunmap-fun _ D2 _) %. +%%%%% Unmap Inverts Map %%%%% +%sort invert-map %. +%mode invert-map %in %in %out %out %. +%sort invert-tmap %. +%mode invert-tmap %in %in %out %out %. +%block invert-bind {k kd} {d_wf kd-wf k} [a cn] [d ofkd a k] [dm mofkd d met/unit] [thm can-mofkd d dm] [thm vdt/ofkd d d_wf] [x eterm] [at map a x] [xt unmap x a] [thm invert-map d at xt (cn-deq/refl d)]%. +%scope invert-tmap %term app + %pi (invert-map (ofkd/cn/app Dof1 Dof2) (map/app D2 D1) (unmap/app D2' D1') (cn-deq/cn/app Dequiv1 Dequiv2)) + %<- (invert-map Dof1 D1 D1' Dequiv1) + %<- (invert-map Dof2 D2 D2' Dequiv2) %. +%term pi1 + %pi (invert-map (ofkd/cn/pj1 Dof) (map/pi1 D) (unmap/pi1 D') (cn-deq/cn/pj1 Dequiv)) + %<- (invert-map Dof D D' Dequiv) %. +%scope invert-tmap %term pi2 + %pi (invert-map (ofkd/cn/pj2 Dof) (map/pi2 D) (unmap/pi2 D') (cn-deq/cn/pj2 Dequiv)) + %<- (invert-map Dof D D' Dequiv) %. +%term lam + %pi (invert-map (ofkd/cn/lam Dof2 Dof1) (map/lam D2 D1) (unmap/lam D2' D1') (cn-deq/cn/lam Dequiv2 Dequiv1)) + %<- (invert-tmap Dof1 D1 D1' Dequiv1) + %<- ({a} {d} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (vdt/ofkd d Dof1) + %-> ({x} {at} {xt} + %pi (invert-map d at xt (cn-deq/refl d)) + %-> (invert-map (Dof2 a d) (D2 a x at) (D2' x a xt) (Dequiv2 a d)))) %. +%scope invert-tmap %term pair + %pi (invert-map (ofkd/cn/pair Dof1 Dof2) (map/pair D2 D1) (unmap/pair D2' D1') (cn-deq/cn/pair Dequiv1 Dequiv2)) + %<- (invert-map Dof1 D1 D1' Dequiv1) + %<- (invert-map Dof2 D2 D2' Dequiv2) %. +%term sing + %pi (invert-map (ofkd/kd/sing Dof) D D' (cn-deq/kd/sing2 Dequiv Dof')) + %<- (invert-map Dof D D' Dequiv) + %<- (vdt/cn-deq Dequiv _ Dof' _) %. +%scope invert-tmap %term piext + %pi (invert-map (ofkd/pi-ext Dof1 Dof2) D D' (cn-deq/pi-ext-2 Dequiv1 Dequiv2')) + %<- (invert-map Dof1 D D' Dequiv1) + %<- (vdt/ofkd Dof1 (kd-wf/kd/pi (%the (kd-wf K1) Dwf1) _)) + %<- ({a} {d} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (vdt/ofkd d Dwf1) + %-> ({x} {at map a x} {xt} + %pi (invert-map d at xt (cn-deq/refl d)) + %-> (invert-map (Dof2 a d) (map/app at D) (unmap/app xt (D'' x a xt)) (Dequiv2 a d)))) + %<- ({a} {x} {at map a x} {xt unmap x a} unmap-fun (D'' x a xt) D' (Deq a)) + %<- ({a} cn-resp-cn ([b] cn/app b a) (Deq a) (Deq' a)) + %<- ({a} {d} cn-deq-resp cn-eq/i (Deq' a) kd-eq/i (Dequiv2 a d) (Dequiv2' a d)) %. +%term sigext + %pi (invert-map (ofkd/sgm-ext Dof1 Dof2) D D' (cn-deq/sgm-ext Dequiv1 Dequiv2')) + %<- (invert-map Dof1 (map/pi1 D) (unmap/pi1 D') Dequiv1) + %<- (invert-map Dof2 (map/pi2 D) (unmap/pi2 D'') Dequiv2) + %<- (unmap-fun D'' D' Deq) + %<- (cn-resp-cn cn/pj2 Deq Deq') + %<- (cn-deq-resp cn-eq/i Deq' kd-eq/i Dequiv2 Dequiv2') %. +%scope invert-tmap %term sub + %pi (invert-map (ofkd/sub Dof Dsub) D D' (cn-deq/sub Dequiv Dsub)) + %<- (invert-map Dof D D' Dequiv) %. +%term deq + %pi (invert-map (ofkd/deq Dof Dtequiv) D D' (cn-deq/deq Dequiv Dtequiv)) + %<- (invert-map Dof D D' Dequiv) %. +%scope invert-tmap %term unit invert-map ofkd/tp/unit map/unit unmap/unit (cn-deq/refl ofkd/tp/unit) %. +%term tagged invert-map ofkd/tp/tagged map/tagged unmap/tagged (cn-deq/refl ofkd/tp/tagged) %. +%scope invert-tmap %term ref + %pi (invert-map (ofkd/tp/ref Dof) (map/ref D) (unmap/app D' unmap/ref) (cn-deq/trans (cn-deq/tp/ref Dequiv) (cn-deq/sym Dequiv'))) + %<- (invert-map Dof D D' Dequiv) + %<- (vdt/cn-deq Dequiv _ Dof' _) + %<- (cn-deq-beta ([a] [d] ofkd/tp/ref d) Dof' Dequiv') %. +%term tag + %pi (invert-map (ofkd/tp/tag Dof) (map/tag D) (unmap/app D' unmap/tag) (cn-deq/trans (cn-deq/tp/tag Dequiv) (cn-deq/sym Dequiv'))) + %<- (invert-map Dof D D' Dequiv) + %<- (vdt/cn-deq Dequiv _ Dof' _) + %<- (cn-deq-beta ([a] [d] ofkd/tp/tag d) Dof' Dequiv') %. +%scope invert-tmap %term arrow + %pi (invert-map (ofkd/tp/arrow Dof1 Dof2) (map/arrow D2 D1) (unmap/app D2' (unmap/app D1' unmap/arrow)) (cn-deq/trans (cn-deq/tp/arrow Dequiv1 Dequiv2) (cn-deq/sym (cn-deq/trans (cn-deq/cn/app Dequiv (cn-deq/refl Dof2')) Dequiv')))) + %<- (invert-map Dof1 D1 D1' Dequiv1) + %<- (invert-map Dof2 D2 D2' Dequiv2) + %<- (vdt/cn-deq Dequiv1 _ Dof1' _) + %<- (vdt/cn-deq Dequiv2 _ Dof2' _) + %<- (cn-deq-beta ([a] [d] ofkd/cn/lam ([b] [e] ofkd/tp/arrow d e) kd-wf/kd/type) Dof1' Dequiv) + %<- (cn-deq-beta ([b] [e] ofkd/tp/arrow Dof1' e) Dof2' Dequiv') %. +%term cross + %pi (invert-map (ofkd/tp/cross Dof1 Dof2) (map/cross D2 D1) (unmap/app D2' (unmap/app D1' unmap/cross)) (cn-deq/trans (cn-deq/tp/cross Dequiv1 Dequiv2) (cn-deq/sym (cn-deq/trans (cn-deq/cn/app Dequiv (cn-deq/refl Dof2')) Dequiv')))) + %<- (invert-map Dof1 D1 D1' Dequiv1) + %<- (invert-map Dof2 D2 D2' Dequiv2) + %<- (vdt/cn-deq Dequiv1 _ Dof1' _) + %<- (vdt/cn-deq Dequiv2 _ Dof2' _) + %<- (cn-deq-beta ([a] [d] ofkd/cn/lam ([b] [e] ofkd/tp/cross d e) kd-wf/kd/type) Dof1' Dequiv) + %<- (cn-deq-beta ([b] [e] ofkd/tp/cross Dof1' e) Dof2' Dequiv') %. +%scope invert-tmap %term sum + %pi (invert-map (ofkd/tp/sum Dof1 Dof2) (map/sum D2 D1) (unmap/app D2' (unmap/app D1' unmap/sum)) (cn-deq/trans (cn-deq/tp/sum Dequiv1 Dequiv2) (cn-deq/sym (cn-deq/trans (cn-deq/cn/app Dequiv (cn-deq/refl Dof2')) Dequiv')))) + %<- (invert-map Dof1 D1 D1' Dequiv1) + %<- (invert-map Dof2 D2 D2' Dequiv2) + %<- (vdt/cn-deq Dequiv1 _ Dof1' _) + %<- (vdt/cn-deq Dequiv2 _ Dof2' _) + %<- (cn-deq-beta ([a] [d] ofkd/cn/lam ([b] [e] ofkd/tp/sum d e) kd-wf/kd/type) Dof1' Dequiv) + %<- (cn-deq-beta ([b] [e] ofkd/tp/sum Dof1' e) Dof2' Dequiv') %. +%term mu + %pi (invert-map (ofkd/cn/mu Dof _) (map/mu D) (unmap/app (unmap/lam D' tunmap/t) unmap/mu) (cn-deq/trans (cn-deq/cn/mu Dequiv kd-deq/kd/type) (cn-deq/sym (cn-deq/trans Dequiv' (cn-deq/cn/mu Dequiv'' kd-deq/kd/type))))) + %<- ({a} {d} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (vdt/ofkd d kd-wf/kd/type) + %-> ({x} {at} {xt} + %pi (invert-map d at xt (cn-deq/refl d)) + %-> (invert-map (Dof a d) (D a x at) (D' x a xt) (Dequiv a d)))) + %<- ({a} {d} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (vdt/ofkd d kd-wf/kd/type) + %-> (vdt/cn-deq (Dequiv a d) (Dof'' a d) (Dof' a d) (Dwf a d))) + %<- (cn-deq-beta ([a] [d ofkd a (kd/pi kd/type ([_] kd/type))] ofkd/cn/mu ([b] [e ofkd b kd/type] ofkd/cn/app d e) kd-wf/kd/type) (ofkd/cn/lam Dof' kd-wf/kd/type) Dequiv') + %<- ({a} {d ofkd a kd/type} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (vdt/ofkd d kd-wf/kd/type) + %-> (cn-deq-beta Dof' d (Dequiv'' a d))) %. +%scope invert-tmap %term forall + %pi (invert-map (ofkd/tp/forall Dof Dwf) (map/forall D2 D1) (unmap/app (unmap/lam D2' D1') (unmap/forall D1')) (cn-deq/trans (cn-deq/tp/forall Dequiv2 Dequiv1) (cn-deq/sym (cn-deq/trans Dequiv' (cn-deq/tp/forall Dequiv'' Dequiv'''))))) + %<- (invert-tmap Dwf D1 D1' (%the (kd-deq K K') Dequiv1)) + %<- (vdt/kd-deq Dequiv1 _ Dwf') + %<- ({a} {d} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (vdt/ofkd d Dwf) + %-> ({x} {at map a x} {xt} + %pi (invert-map d at xt (cn-deq/refl d)) + %-> (invert-map (Dof a d) (D2 a x at) (D2' x a xt) (Dequiv2 a d)))) + %<- ({a} {d ofkd a K} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (vdt/ofkd d Dwf) + %-> (vdt/cn-deq (Dequiv2 a d) (Dof'' a d) (Dof' a d) (Dwf'' a d))) + %<- (kd-sym Dequiv1 Dequiv1') + %<- (cn-deq-beta ([a] [d ofkd a (kd/pi K' ([_] kd/type))] ofkd/tp/forall ([b] [e ofkd b K'] ofkd/cn/app d e) Dwf') (ofkd/cn/lam ([a] [d] Dof' a (ofkd/deq d Dequiv1')) Dwf') Dequiv') + %<- ({a} {d ofkd a K'} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (vdt/ofkd d Dwf') + %-> (cn-deq-beta ([a] [d] Dof' a (ofkd/deq d Dequiv1')) d (Dequiv'' a d))) + %<- (kd-refl/deq Dwf' Dequiv''') %. +%term t invert-tmap kd-wf/kd/type tmap/t tunmap/t kd-deq/kd/type %. +%scope invert-tmap %term sing + %pi (invert-tmap (kd-wf/kd/sing Dof) (tmap/sing D) (tunmap/sing D') (kd-deq/kd/sing Dequiv)) + %<- (invert-map Dof D D' Dequiv) %. +%term pi + %pi (invert-tmap (kd-wf/kd/pi Dof1 Dof2) (tmap/pi D2 D1) (tunmap/pi D2' D1') (kd-deq/kd/pi Dequiv1' ([a] [d] Dequiv2 a (ofkd/deq d Dequiv1')))) + %<- (invert-tmap Dof1 D1 D1' (%the (kd-deq K1 K1') Dequiv1)) + %<- (kd-sym Dequiv1 Dequiv1') + %<- ({a} {d ofkd a K1} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (vdt/ofkd d Dof1) + %-> ({x} {at} {xt} + %pi (invert-map d at xt (cn-deq/refl d)) + %-> (invert-tmap (Dof2 a d) (D2 a x at) (D2' x a xt) (Dequiv2 a d)))) %. +%scope invert-tmap %term sigma + %pi (invert-tmap (kd-wf/kd/sgm Dof1 Dof2) (tmap/sigma D2 D1) (tunmap/sigma D2' D1') (kd-deq/kd/sgm Dequiv1 Dequiv2)) + %<- (invert-tmap Dof1 D1 D1' (%the (kd-deq K1 K1') Dequiv1)) + %<- ({a} {d} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (vdt/ofkd d Dof1) + %-> ({x} {at} {xt} + %pi (invert-map d at xt (cn-deq/refl d)) + %-> (invert-tmap (Dof2 a d) (D2 a x at) (D2' x a xt) (Dequiv2 a d)))) %. +%worlds (invert-bind) (invert-map _ _ _ _) (invert-tmap _ _ _ _) %. +%total (D1 D2) (invert-map D1 _ _ _) (invert-tmap D2 _ _ _) %. +%%%%% Unmap Preserves Typing %%%%% +%sort unmap-vof %. +%mode unmap-vof %in %in %out %out %. +%sort unmap-of %. +%mode unmap-of %in %in %out %out %. +%sort unmap-of' %. +%mode unmap-of' %in %in %in %out %. +%sort unmap-wf %. +%mode unmap-wf %in %out %out %. +%sort unmap-wf' %. +%mode unmap-wf' %in %in %out %. +%sort unmap-subtp %. +%mode unmap-subtp %in %in %in %out %. +%sort unmap-tequiv %. +%mode unmap-tequiv %in %in %in %out %. +%% need the equiv argument last so that Twelf will split it +%sort unmap-equiv %. +%mode unmap-equiv %in %in %in %out %out %. +%sort unmap-equiv' %. +%mode unmap-equiv' %in %in %in %in %out %. +%block unmap-bind {k kd} {d_wf kd-wf k} {b etp} {d_unmap tunmap b k} [a cn] [e ofkd a k] [em mofkd e met/unit] [thm1 can-mofkd e em] [thm2 vdt/ofkd e d_wf] [x eterm] [d evof x b] [xt unmap x a] [thm can-unmap x xt] [thm unmap-vof d xt d_unmap e]%. +%term _ %pi (unmap-of (eof/var _ Dvof) D Dt Dof) %<- (unmap-vof Dvof D Dt Dof) %. +%term _ + %pi (unmap-of (eof/app Dof2 Dof1) (unmap/app Dmap2 Dmap1) (DmapB _ _ Dmap2) (ofkd/cn/app Dof1' Dof2')) + %<- (unmap-of Dof1 Dmap1 (tunmap/pi DmapB DmapA) Dof1') + %<- (unmap-of' Dof2 Dmap2 DmapA Dof2') %. +%term _ + %pi (unmap-of (eof/pi1 Dof) (unmap/pi1 Dmap) DmapA (ofkd/cn/pj1 Dof')) + %<- (unmap-of Dof Dmap (tunmap/sigma DmapB DmapA) Dof') %. +%term _ + %pi (unmap-of (eof/pi2 Dof) (unmap/pi2 Dmap) (DmapB _ _ (unmap/pi1 Dmap)) (ofkd/cn/pj2 Dof')) + %<- (unmap-of Dof Dmap (tunmap/sigma DmapB DmapA) Dof') %. +%term _ + %pi (unmap-of (eof/lam Dof Dwf) (unmap/lam Dmap DmapA) (tunmap/pi DmapB DmapA) (ofkd/cn/lam Dof' Dwf')) + %<- (unmap-wf' Dwf DmapA Dwf') + %<- ({a} {e ofkd a K1} {em mofkd e met/unit} + %pi (can-mofkd e em) + %-> (vdt/ofkd e Dwf') + %-> ({x} {d} {xt} + %pi (can-unmap x xt) + %-> (unmap-vof d xt DmapA e) + %-> (unmap-of (Dof x d) (Dmap x a xt) (DmapB x a xt) (Dof' a e)))) %. +%term _ + %pi (unmap-of (eof/pair Dwf2 Dof2 Dof1) (unmap/pair Dmap2 Dmap1) (tunmap/sigma DmapB DmapA) Dof') + %<- (unmap-of Dof1 Dmap1 DmapA Dof1') + %<- (vdt/ofkd Dof1' Dwf1') + %<- ({a} {e ofkd a K1} {em mofkd e met/unit} + %pi (can-mofkd e em) + %-> (vdt/ofkd e Dwf1') + %-> ({x} {d} {xt} + %pi (can-unmap x xt) + %-> (unmap-vof d xt DmapA e) + %-> (unmap-wf (Dwf2 x d) (DmapB x a xt) (Dwf2' a e)))) + %<- (unmap-of' Dof2 Dmap2 (DmapB _ _ Dmap1) Dof2') + %<- (ofkd-sgm-intro Dwf2' Dof1' Dof2' Dof') %. +%term _ + %pi (unmap-of (eof/sing Dof) Dmap (tunmap/sing Dmap) (ofkd/kd/sing Dof')) + %<- (unmap-of' Dof Dmap tunmap/t Dof') %. +%term _ + %pi (unmap-of (eof/subsume Dsub Dof) Dmap DmapB (ofkd/sub Dof' Dsub')) + %<- (unmap-of Dof Dmap DmapA Dof') + %<- (can-tunmap _ DmapB) + %<- (unmap-subtp Dsub DmapA DmapB Dsub') %. +%term _ + %pi (unmap-of (eof/extpi Dof2 Dof1) Dmap (tunmap/pi DmapB DmapA) (ofkd/pi-ext Dof1' Dof2')) + %<- (unmap-of Dof1 Dmap (tunmap/pi _ DmapA) Dof1') + %<- (vdt/ofkd Dof1' (kd-wf/kd/pi Dwf1' _)) + %<- ({a} {e ofkd a K1} {em mofkd e met/unit} + %pi (can-mofkd e em) + %-> (vdt/ofkd e Dwf1') + %-> ({x} {d} {xt} + %pi (can-unmap x xt) + %-> (unmap-vof d xt DmapA e) + %-> (unmap-of (Dof2 x d) (unmap/app xt Dmap) (DmapB x a xt) (Dof2' a e)))) %. +%term _ + %pi (unmap-of (eof/extsigma Dwf2 Dof2 Dof1) Dmap (tunmap/sigma DmapB DmapA) Dof') + %<- (unmap-of Dof1 (unmap/pi1 Dmap) DmapA Dof1') + %<- (vdt/ofkd Dof1' Dwf1') + %<- ({a} {e ofkd a K1} {em mofkd e met/unit} + %pi (can-mofkd e em) + %-> (vdt/ofkd e Dwf1') + %-> ({x} {d} {xt} + %pi (can-unmap x xt) + %-> (unmap-vof d xt DmapA e) + %-> (unmap-wf (Dwf2 x d) (DmapB x a xt) (Dwf2' a e)))) + %<- (unmap-of' Dof2 (unmap/pi2 Dmap) (DmapB _ _ (unmap/pi1 Dmap)) Dof2') + %<- (ofkd-sgm-ext-intro Dwf2' Dof1' Dof2' Dof') %. +%term _ unmap-of (eof/const _ (ekof/i etopen/t ckof/unit)) unmap/unit tunmap/t ofkd/tp/unit %. +%term _ unmap-of (eof/const _ (ekof/i etopen/t ckof/tagged)) unmap/tagged tunmap/t ofkd/tp/tagged %. +%term _ unmap-of (eof/const _ (ekof/i (etopen/pi etopen/t etopen/t) ckof/ref)) unmap/ref (tunmap/pi ([_] [_] [_] tunmap/t) tunmap/t) (ofkd/cn/lam ([a] [e] ofkd/tp/ref e) kd-wf/kd/type) %. +%term _ unmap-of (eof/const _ (ekof/i (etopen/pi etopen/t etopen/t) ckof/tag)) unmap/tag (tunmap/pi ([_] [_] [_] tunmap/t) tunmap/t) (ofkd/cn/lam ([a] [e] ofkd/tp/tag e) kd-wf/kd/type) %. +%term _ unmap-of (eof/const _ (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/arrow)) unmap/arrow (tunmap/pi ([_] [_] [_] tunmap/pi ([_] [_] [_] tunmap/t) tunmap/t) tunmap/t) (ofkd/cn/lam ([a] [e] ofkd/cn/lam ([b] [f] ofkd/tp/arrow e f) kd-wf/kd/type) kd-wf/kd/type) %. +%term _ unmap-of (eof/const _ (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/cross)) unmap/cross (tunmap/pi ([_] [_] [_] tunmap/pi ([_] [_] [_] tunmap/t) tunmap/t) tunmap/t) (ofkd/cn/lam ([a] [e] ofkd/cn/lam ([b] [f] ofkd/tp/cross e f) kd-wf/kd/type) kd-wf/kd/type) %. +%term _ unmap-of (eof/const _ (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/dsum)) unmap/sum (tunmap/pi ([_] [_] [_] tunmap/pi ([_] [_] [_] tunmap/t) tunmap/t) tunmap/t) (ofkd/cn/lam ([a] [e] ofkd/cn/lam ([b] [f] ofkd/tp/sum e f) kd-wf/kd/type) kd-wf/kd/type) %. +%term _ + %pi (unmap-of (eof/forall Dwf) (unmap/forall Dmap) (tunmap/pi ([_] [_] [_] tunmap/t) (tunmap/pi ([_] [_] [_] tunmap/t) Dmap)) (ofkd/cn/lam ([a] [e] ofkd/tp/forall ([b] [f] ofkd/cn/app e f) Dwf') (kd-wf/kd/pi Dwf' ([_] [_] kd-wf/kd/type)))) + %<- (unmap-wf' Dwf Dmap Dwf') %. +%term _ unmap-of (eof/const _ (ekof/i (etopen/pi etopen/t (etopen/pi etopen/t etopen/t)) ckof/mu)) unmap/mu (tunmap/pi ([_] [_] [_] tunmap/t) (tunmap/pi ([_] [_] [_] tunmap/t) tunmap/t)) (ofkd/cn/lam ([a] [e] ofkd/cn/mu ([b] [f] ofkd/cn/app e f) kd-wf/kd/type) (kd-wf/kd/pi kd-wf/kd/type ([_] [_] kd-wf/kd/type))) %. +%%% +%term _ + %pi (unmap-of' Dof Dmap DmapA Dof'') + %<- (unmap-of Dof Dmap DmapA' Dof') + %<- (tunmap-fun DmapA' DmapA Deq) + %<- (ofkd-resp cn-eq/i Deq Dof' Dof'') %. +%%% +%term _ unmap-wf ewf/t tunmap/t kd-wf/kd/type %. +%term _ + %pi (unmap-wf (ewf/pi Dwf2 Dwf1) (tunmap/pi Dmap2 Dmap1) (kd-wf/kd/pi Dwf1' Dwf2')) + %<- (unmap-wf Dwf1 Dmap1 Dwf1') + %<- ({a} {e ofkd a K1} {em mofkd e met/unit} + %pi (can-mofkd e em) + %-> (vdt/ofkd e Dwf1') + %-> ({x} {d} {xt} + %pi (can-unmap x xt) + %-> (unmap-vof d xt Dmap1 e) + %-> (unmap-wf (Dwf2 x d) (Dmap2 x a xt) (Dwf2' a e)))) %. +%term _ + %pi (unmap-wf (ewf/sigma Dwf2 Dwf1) (tunmap/sigma Dmap2 Dmap1) (kd-wf/kd/sgm Dwf1' Dwf2')) + %<- (unmap-wf Dwf1 Dmap1 Dwf1') + %<- ({a} {e ofkd a K1} {em mofkd e met/unit} + %pi (can-mofkd e em) + %-> (vdt/ofkd e Dwf1') + %-> ({x} {d} {xt} + %pi (can-unmap x xt) + %-> (unmap-vof d xt Dmap1 e) + %-> (unmap-wf (Dwf2 x d) (Dmap2 x a xt) (Dwf2' a e)))) %. +%term _ + %pi (unmap-wf (ewf/sing Dof) (tunmap/sing Dmap) (kd-wf/kd/sing Dof')) + %<- (can-unmap _ Dmap) + %<- (unmap-of Dof Dmap tunmap/t Dof') %. +%%% +%term _ + %pi (unmap-wf' Dwf Dmap Dwf'') + %<- (unmap-wf Dwf Dmap' Dwf') + %<- (tunmap-fun Dmap' Dmap Deq) + %<- (kd-wf-resp Deq Dwf' Dwf'') %. +%%% +%term _ + %pi (unmap-subtp (subtp/reflex Dequiv) Dmap1 Dmap2 Dsub') + %<- (unmap-tequiv Dequiv Dmap1 Dmap2 Dequiv') + %<- (vdt/kd-deq Dequiv' Dwf1' Dwf2') + %<- (kd-anti Dequiv' Dwf1' Dwf2' Dsub' _) %. +%term _ + %pi (unmap-subtp (subtp/trans Dsub23 Dsub12) Dmap1 Dmap3 Dsub13') + %<- (can-tunmap _ Dmap2) + %<- (unmap-subtp Dsub12 Dmap1 Dmap2 Dsub12') + %<- (unmap-subtp Dsub23 Dmap2 Dmap3 Dsub23') + %<- (kd-trans/sub Dsub12' Dsub23' Dsub13') %. +%term _ + %pi (unmap-subtp (subtp/sing_t Dof) (tunmap/sing Dmap) tunmap/t (kd-sub/kd/sing-kd/type Dof')) + %<- (unmap-of' Dof Dmap tunmap/t Dof') %. +%term _ + %pi (unmap-subtp (subtp/pi DwfB1 DsubB DsubA) (tunmap/pi DmapB1 DmapA1) (tunmap/pi DmapB2 DmapA2) (kd-sub/kd/pi DsubA' DsubB' DwfB1')) + %<- (unmap-subtp DsubA DmapA2 DmapA1 DsubA') + %<- (vdt/kd-sub DsubA' DwfA2' DwfA1') + %<- ({a} {e} {em mofkd e met/unit} + %pi (can-mofkd e em) + %-> (vdt/ofkd e DwfA2') + %-> ({x} {d} {xt} + %pi (can-unmap x xt) + %-> (unmap-vof d xt DmapA2 e) + %-> (unmap-subtp (DsubB x d) (DmapB1 x a xt) (DmapB2 x a xt) (DsubB' a e)))) + %<- ({a} {e} {em mofkd e met/unit} + %pi (can-mofkd e em) + %-> (vdt/ofkd e DwfA1') + %-> ({x} {d} {xt} + %pi (can-unmap x xt) + %-> (unmap-vof d xt DmapA1 e) + %-> (unmap-wf' (DwfB1 x d) (DmapB1 x a xt) (DwfB1' a e)))) %. +%term _ + %pi (unmap-subtp (subtp/sigma DwfB2 DsubB DsubA) (tunmap/sigma DmapB1 DmapA1) (tunmap/sigma DmapB2 DmapA2) (kd-sub/kd/sgm DsubA' DsubB' DwfB2')) + %<- (unmap-subtp DsubA DmapA1 DmapA2 DsubA') + %<- (vdt/kd-sub DsubA' DwfA1' DwfA2') + %<- ({a} {e} {em mofkd e met/unit} + %pi (can-mofkd e em) + %-> (vdt/ofkd e DwfA1') + %-> ({x} {d} {xt} + %pi (can-unmap x xt) + %-> (unmap-vof d xt DmapA1 e) + %-> (unmap-subtp (DsubB x d) (DmapB1 x a xt) (DmapB2 x a xt) (DsubB' a e)))) + %<- ({a} {e} {em mofkd e met/unit} + %pi (can-mofkd e em) + %-> (vdt/ofkd e DwfA2') + %-> ({x} {d} {xt} + %pi (can-unmap x xt) + %-> (unmap-vof d xt DmapA2 e) + %-> (unmap-wf' (DwfB2 x d) (DmapB2 x a xt) (DwfB2' a e)))) %. +%%% +%term _ + %pi (unmap-tequiv (tequiv/reflex Dwf) Dmap Dmap' Dequiv'') + %<- (tunmap-fun Dmap Dmap' Deq) + %<- (unmap-wf' Dwf Dmap Dwf') + %<- (kd-refl/deq Dwf' Dequiv') + %<- (kd-deq-resp kd-eq/i Deq Dequiv' Dequiv'') %. +%term _ + %pi (unmap-tequiv (tequiv/symm Dequiv) Dmap Dmap' Dequiv'') + %<- (unmap-tequiv Dequiv Dmap' Dmap Dequiv') + %<- (kd-sym Dequiv' Dequiv'') %. +%term _ + %pi (unmap-tequiv (tequiv/trans Dsub23 Dsub12) Dmap1 Dmap3 Dsub13') + %<- (can-tunmap _ Dmap2) + %<- (unmap-tequiv Dsub12 Dmap1 Dmap2 Dsub12') + %<- (unmap-tequiv Dsub23 Dmap2 Dmap3 Dsub23') + %<- (kd-trans/deq Dsub12' Dsub23' Dsub13') %. +%term _ + %pi (unmap-tequiv (tequiv/sing Dequiv) (tunmap/sing Dmap1) (tunmap/sing Dmap2) (kd-deq/kd/sing Dequiv')) + %<- (unmap-equiv' Dmap1 Dmap2 Dequiv tunmap/t Dequiv') %. +%term _ + %pi (unmap-tequiv (tequiv/pi DequivB DequivA) (tunmap/pi DmapB1 DmapA1) (tunmap/pi DmapB2 DmapA2) (kd-deq/kd/pi DequivA'' ([a] [e] DequivB' a (ofkd/deq e DequivA'')))) + %<- (unmap-tequiv DequivA DmapA1 DmapA2 DequivA') + %<- (vdt/kd-deq DequivA' DwfA1' DwfA2') + %<- (kd-sym DequivA' DequivA'') + %<- ({a} {e} {em mofkd e met/unit} + %pi (can-mofkd e em) + %-> (vdt/ofkd e DwfA1') + %-> ({x} {d} {xt} + %pi (can-unmap x xt) + %-> (unmap-vof d xt DmapA1 e) + %-> (unmap-tequiv (DequivB x d) (DmapB1 x a xt) (DmapB2 x a xt) (DequivB' a e)))) %. +%term _ + %pi (unmap-tequiv (tequiv/sigma DequivB DequivA) (tunmap/sigma DmapB1 DmapA1) (tunmap/sigma DmapB2 DmapA2) (kd-deq/kd/sgm DequivA' DequivB')) + %<- (unmap-tequiv DequivA DmapA1 DmapA2 DequivA') + %<- (vdt/kd-deq DequivA' DwfA1' DwfA2') + %<- ({a} {e} {em mofkd e met/unit} + %pi (can-mofkd e em) + %-> (vdt/ofkd e DwfA1') + %-> ({x} {d} {xt} + %pi (can-unmap x xt) + %-> (unmap-vof d xt DmapA1 e) + %-> (unmap-tequiv (DequivB x d) (DmapB1 x a xt) (DmapB2 x a xt) (DequivB' a e)))) %. +%%% +%term _ + %pi (unmap-equiv Dmap Dmap' (equiv/reflex Dof) DmapA Dequiv') + %<- (unmap-fun Dmap Dmap' Deq) + %<- (unmap-of Dof Dmap DmapA Dof') + %<- (cn-deq-resp cn-eq/i Deq kd-eq/i (cn-deq/refl Dof') Dequiv') %. +%term _ + %pi (unmap-equiv Dmap1 Dmap2 (equiv/symm Dequiv) DmapA (cn-deq/sym Dequiv')) + %<- (unmap-equiv Dmap2 Dmap1 Dequiv DmapA Dequiv') %. +%term _ + %pi (unmap-equiv Dmap1 Dmap3 (equiv/trans Dsub23 Dsub12) DmapA (cn-deq/trans Dsub12' Dsub23')) + %<- (can-unmap _ Dmap2) + %<- (unmap-equiv Dmap1 Dmap2 Dsub12 DmapA Dsub12') + %<- (unmap-equiv' Dmap2 Dmap3 Dsub23 DmapA Dsub23') %. +%term _ + %pi (unmap-equiv (unmap/app Dmap2 Dmap1) (unmap/app Dmap2' Dmap1') (equiv/app Dequiv2 Dequiv1) (DmapB _ _ Dmap2) (cn-deq/cn/app Dequiv1' Dequiv2')) + %<- (unmap-equiv Dmap1 Dmap1' Dequiv1 (tunmap/pi DmapB DmapA) Dequiv1') + %<- (unmap-equiv' Dmap2 Dmap2' Dequiv2 DmapA Dequiv2') %. +%term _ + %pi (unmap-equiv (unmap/pi1 Dmap1) (unmap/pi1 Dmap2) (equiv/pi1 Dequiv) DmapA (cn-deq/cn/pj1 Dequiv')) + %<- (unmap-equiv Dmap1 Dmap2 Dequiv (tunmap/sigma DmapB DmapA) Dequiv') %. +%term _ + %pi (unmap-equiv (unmap/pi2 Dmap1) (unmap/pi2 Dmap2) (equiv/pi2 Dequiv) (DmapB _ _ (unmap/pi1 Dmap1)) (cn-deq/cn/pj2 Dequiv')) + %<- (unmap-equiv Dmap1 Dmap2 Dequiv (tunmap/sigma DmapB DmapA) Dequiv') %. +%term _ + %pi (unmap-equiv (unmap/lam Dmap1 DmapA1) (unmap/lam Dmap2 DmapA2) (equiv/lam Dequiv2 Dequiv1) (tunmap/pi DmapB DmapA1) (cn-deq/cn/lam Dequiv2' Dequiv1')) + %<- (unmap-tequiv Dequiv1 DmapA1 DmapA2 Dequiv1') + %<- (vdt/kd-deq Dequiv1' DwfA1' DwfA2') + %<- ({a} {e} {em mofkd e met/unit} + %pi (can-mofkd e em) + %-> (vdt/ofkd e DwfA1') + %-> ({x} {d} {xt} + %pi (can-unmap x xt) + %-> (unmap-vof d xt DmapA1 e) + %-> (unmap-equiv (Dmap1 x a xt) (Dmap2 x a xt) (Dequiv2 x d) (DmapB x a xt) (Dequiv2' a e)))) %. +%term _ + %pi (unmap-equiv (unmap/pair Dmap2 Dmap1) (unmap/pair Dmap2' Dmap1') (equiv/pair Dwf2 Dequiv2 Dequiv1) (tunmap/sigma DmapB DmapA) Dequiv') + %<- (unmap-equiv Dmap1 Dmap1' Dequiv1 DmapA Dequiv1') + %<- (vdt/cn-deq Dequiv1' _ _ Dwf1') + %<- ({a} {e ofkd a K1} {em mofkd e met/unit} + %pi (can-mofkd e em) + %-> (vdt/ofkd e Dwf1') + %-> ({x} {d} {xt} + %pi (can-unmap x xt) + %-> (unmap-vof d xt DmapA e) + %-> (unmap-wf (Dwf2 x d) (DmapB x a xt) (Dwf2' a e)))) + %<- (unmap-equiv' Dmap2 Dmap2' Dequiv2 (DmapB _ _ Dmap1) Dequiv2') + %<- (cn-deq-sgm-intro Dwf2' Dequiv1' Dequiv2' Dequiv') %. +%term _ + %pi (unmap-equiv Dmap1 Dmap2 (equiv/sing Dequiv) (tunmap/sing Dmap1) (cn-deq/kd/sing2 Dequiv' Dof2')) + %<- (unmap-equiv' Dmap1 Dmap2 Dequiv tunmap/t Dequiv') + %<- (vdt/cn-deq Dequiv' _ Dof2' _) %. +%term _ + %pi (unmap-equiv Dmap1 Dmap2 (equiv/singelim Dof) tunmap/t (cn-deq/kd/singelim Dwf' Dof')) + %<- (unmap-of' Dof Dmap1 (tunmap/sing Dmap2) Dof') + %<- (vdt/ofkd Dof' (kd-wf/kd/sing Dwf')) %. +%term _ + %pi (unmap-equiv Dmap1 Dmap2 (equiv/extpi Dequiv Dof2 Dof1) (tunmap/pi DmapB DmapA) (cn-deq/pi-ext Dof1' Dof2' Dequiv')) + %<- (unmap-of Dof1 Dmap1 (tunmap/pi DmapB' DmapA) Dof1') + %<- (vdt/ofkd Dof1' (kd-wf/kd/pi DwfA' _)) + %<- ({a} {x} {xt unmap x a} %pi (can-unmap x xt) %-> (can-tunmap _ (DmapB'' x a xt))) + %<- (unmap-of' Dof2 Dmap2 (tunmap/pi DmapB'' DmapA) Dof2') + %<- ({a} {e ofkd a K1} {em mofkd e met/unit} + %pi (can-mofkd e em) + %-> (vdt/ofkd e DwfA') + %-> ({x} {d} {xt} + %pi (can-unmap x xt) + %-> (unmap-vof d xt DmapA e) + %-> (unmap-equiv (unmap/app xt Dmap1) (unmap/app xt Dmap2) (Dequiv x d) (DmapB x a xt) (Dequiv' a e)))) %. +%term _ + %pi (unmap-equiv Dmap1 Dmap2 (equiv/extpiw Dequiv DequivPre) (tunmap/pi DmapB DmapA) (cn-deq/pi-ext-2 DequivPre' Dequiv')) + %<- (unmap-equiv Dmap1 Dmap2 DequivPre (tunmap/pi _ DmapA) DequivPre') + %<- (vdt/cn-deq DequivPre' _ _ (kd-wf/kd/pi DwfA' _)) + %<- ({a} {e ofkd a K1} {em mofkd e met/unit} + %pi (can-mofkd e em) + %-> (vdt/ofkd e DwfA') + %-> ({x} {d} {xt} + %pi (can-unmap x xt) + %-> (unmap-vof d xt DmapA e) + %-> (unmap-equiv (unmap/app xt Dmap1) (unmap/app xt Dmap2) (Dequiv x d) (DmapB x a xt) (Dequiv' a e)))) %. +%term _ + %pi (unmap-equiv Dmap Dmap' (equiv/extsigma Dwf2 Dequiv2 Dequiv1) (tunmap/sigma DmapB DmapA) Dequiv') + %<- (unmap-equiv (unmap/pi1 Dmap) (unmap/pi1 Dmap') Dequiv1 DmapA Dequiv1') + %<- (vdt/cn-deq Dequiv1' _ _ Dwf1') + %<- ({a} {e ofkd a K1} {em mofkd e met/unit} + %pi (can-mofkd e em) + %-> (vdt/ofkd e Dwf1') + %-> ({x} {d} {xt} + %pi (can-unmap x xt) + %-> (unmap-vof d xt DmapA e) + %-> (unmap-wf (Dwf2 x d) (DmapB x a xt) (Dwf2' a e)))) + %<- (unmap-equiv' (unmap/pi2 Dmap) (unmap/pi2 Dmap') Dequiv2 (DmapB _ _ (unmap/pi1 Dmap)) Dequiv2') + %<- (cn-deq-sgm-ext-intro Dwf2' Dequiv1' Dequiv2' Dequiv') %. +%term _ + %pi (unmap-equiv Dmap1 Dmap2 (equiv/subsume Dsub Dequiv) DmapB (cn-deq/sub Dequiv' Dsub')) + %<- (unmap-equiv Dmap1 Dmap2 Dequiv DmapA Dequiv') + %<- (can-tunmap _ DmapB) + %<- (unmap-subtp Dsub DmapA DmapB Dsub') %. +%term _ + %pi (unmap-equiv (unmap/app Dmap2 (unmap/lam Dmap1 DmapA)) Dmap (equiv/beta Dof2 Dof1) (DmapB _ _ Dmap2) Dequiv'') + %<- (unmap-fun (Dmap1 _ _ Dmap2) Dmap Deq) + %<- (unmap-of' Dof2 Dmap2 DmapA Dof2') + %<- (vdt/ofkd Dof2' DwfA') + %<- ({a} {e} {em mofkd e met/unit} + %pi (can-mofkd e em) + %-> (vdt/ofkd e DwfA') + %-> ({x} {d} {xt} + %pi (can-unmap x xt) + %-> (unmap-vof d xt DmapA e) + %-> (unmap-of (Dof1 x d) (Dmap1 x a xt) (DmapB x a xt) (Dof1' a e)))) + %<- (cn-deq-beta Dof1' Dof2' Dequiv') + %<- (cn-deq-resp cn-eq/i Deq kd-eq/i Dequiv' Dequiv'') %. +%term _ + %pi (unmap-equiv (unmap/pi1 (unmap/pair Dmap2 Dmap1)) Dmap1' (equiv/beta1 Dof2 Dof1) DmapA Dequiv'') + %<- (unmap-fun Dmap1 Dmap1' Deq) + %<- (unmap-of Dof1 Dmap1 DmapA Dof1') + %<- (unmap-of Dof2 Dmap2 DmapB Dof2') + %<- (stone-2/108 Dof1' Dof2' Dequiv') + %<- (cn-deq-resp cn-eq/i Deq kd-eq/i Dequiv' Dequiv'') %. +%term _ + %pi (unmap-equiv (unmap/pi2 (unmap/pair Dmap2 Dmap1)) Dmap2' (equiv/beta2 Dof2 Dof1) DmapB Dequiv'') + %<- (unmap-fun Dmap2 Dmap2' Deq) + %<- (unmap-of Dof1 Dmap1 DmapA Dof1') + %<- (unmap-of Dof2 Dmap2 DmapB Dof2') + %<- (stone-2/109 Dof1' Dof2' Dequiv') + %<- (cn-deq-resp cn-eq/i Deq kd-eq/i Dequiv' Dequiv'') %. +%term _ + %pi (unmap-equiv (unmap/forall Dmap1) (unmap/forall Dmap2) (equiv/forall Dequiv) (tunmap/pi ([_] [_] [_] tunmap/t) (tunmap/pi ([_] [_] [_] tunmap/t) Dmap1)) (cn-deq/cn/lam ([a] [e] cn-deq/tp/forall ([b] [f] cn-deq/refl (ofkd/cn/app e f)) Dequiv') (kd-deq/kd/pi Dequiv'' ([_] [_] kd-deq/kd/type)))) + %<- (unmap-tequiv Dequiv Dmap1 Dmap2 Dequiv') + %<- (kd-sym Dequiv' Dequiv'') %. +%%% +%term _ + %pi (unmap-equiv' Dmap1 Dmap2 Dequiv DmapA Dequiv'') + %<- (unmap-equiv Dmap1 Dmap2 Dequiv DmapA' Dequiv') + %<- (tunmap-fun DmapA' DmapA Deq) + %<- (cn-deq-resp cn-eq/i cn-eq/i Deq Dequiv' Dequiv'') %. +%worlds (unmap-bind) (unmap-vof _ _ _ _) (unmap-of _ _ _ _) (unmap-of' _ _ _ _) (unmap-wf _ _ _) (unmap-wf' _ _ _) (unmap-subtp _ _ _ _) (unmap-tequiv _ _ _ _) (unmap-equiv _ _ _ _ _) (unmap-equiv' _ _ _ _ _) %. +%total (D1 D2 D3 D4 D5 D6 D7 D8 D9) (unmap-vof D1 _ _ _) (unmap-of D2 _ _ _) (unmap-of' D3 _ _ _) (unmap-wf D4 _ _) (unmap-wf' D5 _ _) (unmap-subtp D6 _ _ _) (unmap-tequiv D7 _ _ _) (unmap-equiv _ _ D8 _ _) (unmap-equiv' _ _ D9 _ _) %. +%sort invert-map-equiv %. +%mode invert-map-equiv %in %in %in %in %in %in %out %. +%term _ + %pi (invert-map-equiv Dmap1 Dmap2 Dmap Dof1 Dof2 Dequiv (cn-deq/trans Dequiv1' (cn-deq/trans (cn-deq/deq Dequiv' Dtequiv'') (cn-deq/sym Dequiv2')))) + %<- (invert-map Dof1 Dmap1 Dunmap1 Dequiv1') + %<- (invert-map Dof2 Dmap2 Dunmap2 Dequiv2') + %<- (vdt/ofkd Dof1 Dwf) + %<- (invert-tmap Dwf Dmap Dunmap Dtequiv') + %<- (unmap-equiv' Dunmap1 Dunmap2 Dequiv Dunmap Dequiv') + %<- (kd-sym Dtequiv' Dtequiv'') %. +%worlds () (invert-map-equiv _ _ _ _ _ _ _) %. +%total {} (invert-map-equiv _ _ _ _ _ _ _) %. +%sort invert-map-equiv1 %. +%mode invert-map-equiv1 %in %in %in %in %in %in %in %in %out %. +%term _ + %pi (invert-map-equiv1 DmapB Dmap1 Dmap2 Dmap DofB Dof1 Dof2 Dequiv ([a] [d] cn-deq/trans (Dequiv1' a d) (cn-deq/trans (cn-deq/deq (Dequiv' a (ofkd/deq d DequivB)) (Dtequiv'' a d)) (cn-deq/sym (Dequiv2' a d))))) + %<- ({a} {d} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (vdt/ofkd d DofB) + %-> ({x} {at} {xt} + %pi (invert-map d at xt (cn-deq/refl d)) + %-> (invert-map (Dof1 a d) (Dmap1 a x at) (Dunmap1 x a xt) (Dequiv1' a d)))) + %<- ({a} {d} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (vdt/ofkd d DofB) + %-> ({x} {at} {xt} + %pi (invert-map d at xt (cn-deq/refl d)) + %-> (invert-map (Dof2 a d) (Dmap2 a x at) (Dunmap2 x a xt) (Dequiv2' a d)))) + %<- ({a} {d} {dm mofkd d met/unit} %pi (can-mofkd d dm) %-> (vdt/ofkd d DofB) %-> (vdt/ofkd (Dof1 a d) (Dwf a d))) + %<- ({a} {d} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (vdt/ofkd d DofB) + %-> ({x} {at} {xt} + %pi (invert-map d at xt (cn-deq/refl d)) + %-> (invert-tmap (Dwf a d) (Dmap a x at) (Dunmap x a xt) (Dtequiv' a d)))) + %<- (invert-tmap DofB DmapB DunmapB (%the (kd-deq K K') DequivB)) + %<- (vdt/kd-deq DequivB _ DofB') + %<- ({a} {d} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (vdt/ofkd d DofB') + %-> ({x} {d'} {xt} + %pi (can-unmap x xt) + %-> (unmap-vof d' xt DunmapB d) + %-> (unmap-equiv' (Dunmap1 x a xt) (Dunmap2 x a xt) (Dequiv x d') (Dunmap x a xt) (Dequiv' a d)))) + %<- ({a} {d} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (vdt/ofkd d DofB) + %-> (kd-sym (Dtequiv' a d) (Dtequiv'' a d))) %. +%worlds () (invert-map-equiv1 _ _ _ _ _ _ _ _ _) %. +%total {} (invert-map-equiv1 _ _ _ _ _ _ _ _ _) %. +%sort invert-map-tequiv %. +%mode invert-map-tequiv %in %in %in %in %in %out %. +%term _ + %pi (invert-map-tequiv Dmap1 Dmap2 Dwf1 Dwf2 Dequiv Dequiv'') + %<- (invert-tmap Dwf1 Dmap1 Dunmap1 Dequiv1') + %<- (invert-tmap Dwf2 Dmap2 Dunmap2 Dequiv2') + %<- (unmap-tequiv Dequiv Dunmap1 Dunmap2 Dequiv') + %<- (kd-trans/deq Dequiv1' Dequiv' Dequiv1'') + %<- (kd-sym Dequiv2' Dequiv2'') + %<- (kd-trans/deq Dequiv1'' Dequiv2'' Dequiv'') %. +%worlds () (invert-map-tequiv _ _ _ _ _ _) %. +%total {} (invert-map-tequiv _ _ _ _ _ _) %. +%sort false-implies-uninhabited %. +%mode false-implies-uninhabited %in %out %. +%worlds () (false-implies-uninhabited _ _) %. +%total {} (false-implies-uninhabited _ _) %. +%%%%% Map Functionality %%%%% +%sort can-map {C} %. +%mode can-map %in %out %. +%sort can-tmap {K} %. +%mode can-tmap %in %out %. +%block can-map-bind {m eterm} [a cn] [at map a m] [thm can-map a at]%. +%block can-map-bind-for-map {x eterm} {k kd} [a cn] [e ofkd a k] [at map a x] [thm can-map a at]%. +%term _ %pi (can-map _ (map/app D2 D1)) %<- (can-map _ D1) %<- (can-map _ D2) %. +%term _ %pi (can-map _ (map/pi1 D)) %<- (can-map _ D) %. +%term _ %pi (can-map _ (map/pi2 D)) %<- (can-map _ D) %. +%term _ + %pi (can-map _ (map/lam D2 D1)) + %<- (can-tmap _ D1) + %<- ({a} {x} {xt} %pi (can-map x xt) %-> (can-map _ (D2 x a xt))) %. +%term _ %pi (can-map _ (map/pair D2 D1)) %<- (can-map _ D1) %<- (can-map _ D2) %. +%term _ can-map _ map/unit %. +%term _ can-map _ map/tagged %. +%term _ %pi (can-map _ (map/ref D)) %<- (can-map _ D) %. +%term _ %pi (can-map _ (map/tag D)) %<- (can-map _ D) %. +%term _ %pi (can-map _ (map/arrow D2 D1)) %<- (can-map _ D1) %<- (can-map _ D2) %. +%term _ %pi (can-map _ (map/cross D2 D1)) %<- (can-map _ D1) %<- (can-map _ D2) %. +%term _ %pi (can-map _ (map/sum D2 D1)) %<- (can-map _ D1) %<- (can-map _ D2) %. +%term _ + %pi (can-map _ (map/forall D2 D1)) + %<- (can-tmap _ D1) + %<- ({a} {x} {xt} %pi (can-map x xt) %-> (can-map _ (D2 x a xt))) %. +%term _ + %pi (can-map _ (map/mu D2)) + %<- ({a} {x} {xt} %pi (can-map x xt) %-> (can-map _ (D2 x a xt))) %. +%term _ can-tmap _ tmap/t %. +%term _ %pi (can-tmap _ (tmap/sing D)) %<- (can-map _ D) %. +%term _ + %pi (can-tmap _ (tmap/pi D2 D1)) + %<- (can-tmap _ D1) + %<- ({a} {x} {xt} %pi (can-map x xt) %-> (can-tmap _ (D2 x a xt))) %. +%term _ + %pi (can-tmap _ (tmap/sigma D2 D1)) + %<- (can-tmap _ D1) + %<- ({a} {x} {xt} %pi (can-map x xt) %-> (can-tmap _ (D2 x a xt))) %. +%worlds (evar can-map-bind can-map-bind-for-map) (can-map _ _) (can-tmap _ _) %. +%total (D1 D2) (can-map D1 _) (can-tmap D2 _) %. +%sort map-fun %. +%mode map-fun %in %in %out %. +%sort tmap-fun %. +%mode tmap-fun %in %in %out %. +%block map-fun-bind {x eterm} [a cn] [at map a x]%. +%block map-fun-bind-for-map {x eterm} {k kd} [a cn] [e ofkd a k] [at map a x]%. +%term _ map-fun D D eterm-eq/i %. +%term _ + %pi (map-fun (map/app D2 D1) (map/app D2' D1') Deq) + %<- (map-fun D1 D1' Deq1) + %<- (map-fun D2 D2' Deq2) + %<- (eterm-resp-eterm2 eapp Deq1 Deq2 Deq) %. +%term _ + %pi (map-fun (map/pi1 D) (map/pi1 D') Deq) + %<- (map-fun D D' Deq1) + %<- (eterm-resp-eterm epi1 Deq1 Deq) %. +%term _ + %pi (map-fun (map/pi2 D) (map/pi2 D') Deq) + %<- (map-fun D D' Deq1) + %<- (eterm-resp-eterm epi2 Deq1 Deq) %. +%term _ + %pi (map-fun (map/lam D2 D1) (map/lam D2' D1') Deq) + %<- (tmap-fun D1 D1' Deq1) + %<- ({a} {x} {xt} map-fun (D2 x a xt) (D2' x a xt) (Deq2 a)) + %<- (elam-resp Deq1 Deq2 Deq) %. +%term _ + %pi (map-fun (map/pair D2 D1) (map/pair D2' D1') Deq) + %<- (map-fun D1 D1' Deq1) + %<- (map-fun D2 D2' Deq2) + %<- (eterm-resp-eterm2 epair Deq1 Deq2 Deq) %. +%term _ + %pi (map-fun (map/ref D) (map/ref D') Deq) + %<- (map-fun D D' Deq1) + %<- (eterm-resp-eterm ([m] eapp (econst ref) m) Deq1 Deq) %. +%term _ + %pi (map-fun (map/tag D) (map/tag D') Deq) + %<- (map-fun D D' Deq1) + %<- (eterm-resp-eterm ([m] eapp (econst tag) m) Deq1 Deq) %. +%term _ + %pi (map-fun (map/arrow D2 D1) (map/arrow D2' D1') Deq) + %<- (map-fun D1 D1' Deq1) + %<- (map-fun D2 D2' Deq2) + %<- (eterm-resp-eterm2 ([m] [n] eapp (eapp (econst arrow) m) n) Deq1 Deq2 Deq) %. +%term _ + %pi (map-fun (map/cross D2 D1) (map/cross D2' D1') Deq) + %<- (map-fun D1 D1' Deq1) + %<- (map-fun D2 D2' Deq2) + %<- (eterm-resp-eterm2 ([m] [n] eapp (eapp (econst cross) m) n) Deq1 Deq2 Deq) %. +%term _ + %pi (map-fun (map/sum D2 D1) (map/sum D2' D1') Deq) + %<- (map-fun D1 D1' Deq1) + %<- (map-fun D2 D2' Deq2) + %<- (eterm-resp-eterm2 ([m] [n] eapp (eapp (econst dsum) m) n) Deq1 Deq2 Deq) %. +%term _ + %pi (map-fun (map/forall D2 D1) (map/forall D2' D1') Deq) + %<- (tmap-fun D1 D1' Deq1) + %<- ({a} {x} {xt} map-fun (D2 x a xt) (D2' x a xt) (Deq2 a)) + %<- (eterm-resp-bind ([a] [m] eapp (eforall a) (elam a m)) Deq1 Deq2 Deq) %. +%term _ + %pi (map-fun (map/mu D) (map/mu D') Deq) + %<- ({a} {x} {xt} map-fun (D x a xt) (D' x a xt) (Deq1 a)) + %<- (eterm-resp-halfbind ([m] eapp (econst mu) (elam et m)) Deq1 Deq) %. +%term _ tmap-fun tmap/t tmap/t etp-eq/i %. +%term _ + %pi (tmap-fun (tmap/sing D) (tmap/sing D') Deq) + %<- (map-fun D D' Deq1) + %<- (etp-resp-eterm esing Deq1 Deq) %. +%term _ + %pi (tmap-fun (tmap/pi D2 D1) (tmap/pi D2' D1') Deq) + %<- (tmap-fun D1 D1' Deq1) + %<- ({a} {x} {xt} tmap-fun (D2 x a xt) (D2' x a xt) (Deq2 a)) + %<- (etp-resp-bind epi Deq1 Deq2 Deq) %. +%term _ + %pi (tmap-fun (tmap/sigma D2 D1) (tmap/sigma D2' D1') Deq) + %<- (tmap-fun D1 D1' Deq1) + %<- ({a} {x} {xt} tmap-fun (D2 x a xt) (D2' x a xt) (Deq2 a)) + %<- (etp-resp-bind esigma Deq1 Deq2 Deq) %. +%worlds (evar map-fun-bind map-fun-bind-for-map) (map-fun _ _ _) (tmap-fun _ _ _) %. +%total (D1 D2) (map-fun _ D1 _) (tmap-fun _ D2 _) %. +%%%%% Map Preserves Typing %%%%% +%sort map-of %. +%mode map-of %in %in %out %out %. +%sort map-of' %. +%mode map-of' %in %in %in %out %. +%sort map-wf %. +%mode map-wf %in %out %out %. +%sort map-wf' %. +%mode map-wf' %in %in %out %. +%sort map-subtp %. +%mode map-subtp %in %in %in %out %. +%sort map-tequiv %. +%mode map-tequiv %in %in %in %out %. +%sort map-equiv' %. +%mode map-equiv' %in %in %in %out %out %. +%sort map-equiv'' %. +%mode map-equiv'' %in %in %in %in %out %. +%block map-bind {b etp} {k kd} {d_map tmap k b} {d_wf ewf b} [x eterm] [d evof x b] [a cn] [e ofkd a k] [at map a x] [thm can-map a at] [thm map-of e at d_map (eof/var d_wf d)]%. +%term _ + %pi (map-of (ofkd/cn/app Dof1 Dof2) (map/app Dmap2 Dmap1) (DmapB _ _ Dmap2) (eof/app Dof2' Dof1')) + %<- (map-of Dof1 Dmap1 (tmap/pi DmapB DmapA) Dof1') + %<- (map-of' Dof2 Dmap2 DmapA Dof2') %. +%term _ + %pi (map-of (ofkd/cn/pj1 Dof) (map/pi1 Dmap) DmapA (eof/pi1 Dof')) + %<- (map-of Dof Dmap (tmap/sigma DmapB DmapA) Dof') %. +%term _ + %pi (map-of (ofkd/cn/pj2 Dof) (map/pi2 Dmap) (DmapB _ _ (map/pi1 Dmap)) (eof/pi2 Dof')) + %<- (map-of Dof Dmap (tmap/sigma DmapB DmapA) Dof') %. +%term _ + %pi (map-of (ofkd/cn/lam Dof Dwf) (map/lam Dmap DmapA) (tmap/pi DmapB DmapA) (eof/lam Dof' Dwf')) + %<- (map-wf' Dwf DmapA Dwf') + %<- ({x} {d} {a} {e} {at} + %pi (can-map a at) + %-> (map-of e at DmapA (eof/var Dwf' d)) + %-> (map-of (Dof a e) (Dmap a x at) (DmapB a x at) (Dof' x d))) %. +%term _ + %pi (map-of (ofkd/cn/pair Dof1 Dof2) (map/pair Dmap2 Dmap1) (tmap/sigma ([_] [_] [_] DmapB) DmapA) (eof/pair ([_] [_] Dwf2) Dof2' Dof1')) + %<- (map-of Dof1 Dmap1 DmapA Dof1') + %<- (map-of Dof2 Dmap2 DmapB Dof2') + %<- (eof-reg Dof2' Dwf2) %. +%term _ + %pi (map-of (ofkd/kd/sing Dof) Dmap (tmap/sing Dmap) (eof/sing Dof')) + %<- (map-of' Dof Dmap tmap/t Dof') %. +%term _ + %pi (map-of (ofkd/sgm-ext Dof1 Dof2) Dmap (tmap/sigma ([_] [_] [_] DmapB) DmapA) (eof/extsigma ([_] [_] Dwf2') Dof2' Dof1')) + %<- (map-of Dof1 (map/pi1 Dmap) DmapA Dof1') + %<- (map-of Dof2 (map/pi2 Dmap) DmapB Dof2') + %<- (eof-reg Dof2' Dwf2') %. +%term _ + %pi (map-of (ofkd/pi-ext DofPre Dof) Dmap (tmap/pi DmapB DmapA) (eof/extpi Dof' DofPre')) + %<- (map-of DofPre Dmap (tmap/pi _ DmapA) DofPre') + %<- (eof-reg DofPre' (ewf/pi _ Dwf')) + %<- ({x} {d} {a} {e} {at} + %pi (can-map a at) + %-> (map-of e at DmapA (eof/var Dwf' d)) + %-> (map-of (Dof a e) (map/app at Dmap) (DmapB a x at) (Dof' x d))) %. +%term _ + %pi (map-of (ofkd/sub Dof Dsub) Dmap DmapB (eof/subsume Dsub' Dof')) + %<- (map-of Dof Dmap DmapA Dof') + %<- (can-tmap _ DmapB) + %<- (map-subtp Dsub DmapA DmapB Dsub') %. +%term _ + %pi (map-of (ofkd/deq Dof Dsub) Dmap DmapB (eof/equiv Dsub' Dof')) + %<- (map-of Dof Dmap DmapA Dof') + %<- (can-tmap _ DmapB) + %<- (map-tequiv Dsub DmapA DmapB Dsub') %. +%term _ map-of ofkd/tp/unit map/unit tmap/t (eof/const ewf/t (ekof/i etopen/t ckof/unit)) %. +%term _ map-of ofkd/tp/tagged map/tagged tmap/t (eof/const ewf/t (ekof/i etopen/t ckof/tagged)) %. +%term _ + %pi (map-of (ofkd/tp/ref Dof) (map/ref Dmap) tmap/t (eof/app Dof' (eof/const (ewf/pi ([_] [_] ewf/t) ewf/t) (ekof/i (etopen/pi etopen/t etopen/t) ckof/ref)))) + %<- (map-of' Dof Dmap tmap/t Dof') %. +%term _ + %pi (map-of (ofkd/tp/tag Dof) (map/tag Dmap) tmap/t (eof/app Dof' (eof/const (ewf/pi ([_] [_] ewf/t) ewf/t) (ekof/i (etopen/pi etopen/t etopen/t) ckof/tag)))) + %<- (map-of' Dof Dmap tmap/t Dof') %. +%term _ + %pi (map-of (ofkd/tp/arrow Dof1 Dof2) (map/arrow Dmap2 Dmap1) tmap/t (eof/app Dof2' (eof/app Dof1' (eof/const (ewf/pi ([_] [_] ewf/pi ([_] [_] ewf/t) ewf/t) ewf/t) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/arrow))))) + %<- (map-of' Dof1 Dmap1 tmap/t Dof1') + %<- (map-of' Dof2 Dmap2 tmap/t Dof2') %. +%term _ + %pi (map-of (ofkd/tp/cross Dof1 Dof2) (map/cross Dmap2 Dmap1) tmap/t (eof/app Dof2' (eof/app Dof1' (eof/const (ewf/pi ([_] [_] ewf/pi ([_] [_] ewf/t) ewf/t) ewf/t) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/cross))))) + %<- (map-of' Dof1 Dmap1 tmap/t Dof1') + %<- (map-of' Dof2 Dmap2 tmap/t Dof2') %. +%term _ + %pi (map-of (ofkd/tp/sum Dof1 Dof2) (map/sum Dmap2 Dmap1) tmap/t (eof/app Dof2' (eof/app Dof1' (eof/const (ewf/pi ([_] [_] ewf/pi ([_] [_] ewf/t) ewf/t) ewf/t) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/dsum))))) + %<- (map-of' Dof1 Dmap1 tmap/t Dof1') + %<- (map-of' Dof2 Dmap2 tmap/t Dof2') %. +%term _ + %pi (map-of (ofkd/tp/forall Dof Dwf) (map/forall Dmap2 Dmap1) tmap/t (eof/app (eof/lam Dof' Dwf') (eof/forall Dwf'))) + %<- (map-wf' Dwf Dmap1 Dwf') + %<- ({x} {d} {a} {e} {at} + %pi (can-map a at) + %-> (map-of e at Dmap1 (eof/var Dwf' d)) + %-> (map-of' (Dof a e) (Dmap2 a x at) tmap/t (Dof' x d))) %. +%term _ + %pi (map-of (ofkd/cn/mu Dof _) (map/mu Dmap) tmap/t (eof/app (eof/lam Dof' ewf/t) (eof/const (ewf/pi ([_] [_] ewf/t) (ewf/pi ([_] [_] ewf/t) ewf/t)) (ekof/i (etopen/pi etopen/t (etopen/pi etopen/t etopen/t)) ckof/mu)))) + %<- ({x} {d} {a} {e} {at} + %pi (can-map a at) + %-> (map-of e at tmap/t (eof/var ewf/t d)) + %-> (map-of' (Dof a e) (Dmap a x at) tmap/t (Dof' x d))) %. +%%% +%term _ + %pi (map-of' Dof Dmap DmapA Dof'') + %<- (map-of Dof Dmap DmapA' Dof') + %<- (tmap-fun DmapA' DmapA Deq) + %<- (eof-resp eterm-eq/i Deq Dof' Dof'') %. +%%% +%term _ map-wf kd-wf/kd/type tmap/t ewf/t %. +%term _ + %pi (map-wf (kd-wf/kd/sing Dof) (tmap/sing Dmap) (ewf/sing Dof')) + %<- (can-map _ Dmap) + %<- (map-of' Dof Dmap tmap/t Dof') %. +%term _ + %pi (map-wf (kd-wf/kd/pi Dwf1 Dwf2) (tmap/pi Dmap2 Dmap1) (ewf/pi Dwf2' Dwf1')) + %<- (map-wf Dwf1 Dmap1 Dwf1') + %<- ({x} {d} {a} {e} {at} + %pi (can-map a at) + %-> (map-of e at Dmap1 (eof/var Dwf1' d)) + %-> (map-wf (Dwf2 a e) (Dmap2 a x at) (Dwf2' x d))) %. +%term _ + %pi (map-wf (kd-wf/kd/sgm Dwf1 Dwf2) (tmap/sigma Dmap2 Dmap1) (ewf/sigma Dwf2' Dwf1')) + %<- (map-wf Dwf1 Dmap1 Dwf1') + %<- ({x} {d} {a} {e} {at} + %pi (can-map a at) + %-> (map-of e at Dmap1 (eof/var Dwf1' d)) + %-> (map-wf (Dwf2 a e) (Dmap2 a x at) (Dwf2' x d))) %. +%%% +%term _ + %pi (map-wf' Dwf Dmap Dwf'') + %<- (map-wf Dwf Dmap' Dwf') + %<- (tmap-fun Dmap' Dmap Deq) + %<- (ewf-resp Deq Dwf' Dwf'') %. +%%% +%term _ map-subtp kd-sub/kd/type tmap/t tmap/t subtp/t %. +%term _ + %pi (map-subtp (kd-sub/kd/sing-kd/sing Dequiv) (tmap/sing Dmap) (tmap/sing Dmap') (subtp/sing Dequiv')) + %<- (map-equiv'' Dmap Dmap' Dequiv tmap/t Dequiv') %. +%term _ + %pi (map-subtp (kd-sub/kd/sing-kd/type Dof) (tmap/sing Dmap) tmap/t (subtp/sing_t Dof')) + %<- (map-of' Dof Dmap tmap/t Dof') %. +%term _ + %pi (map-subtp (kd-sub/kd/sgm Dsub1 Dsub2 Dwf2) (tmap/sigma Dmap2 Dmap1) (tmap/sigma Dmap2' Dmap1') (subtp/sigma Dwf2' Dsub2' Dsub1')) + %<- (map-subtp Dsub1 Dmap1 Dmap1' Dsub1') + %<- (subtp-reg Dsub1' Dwf1 Dwf1') + %<- ({x} {d} {a} {e} {at} + %pi (can-map a at) + %-> (map-of e at Dmap1 (eof/var Dwf1 d)) + %-> (map-subtp (Dsub2 a e) (Dmap2 a x at) (Dmap2' a x at) (Dsub2' x d))) + %<- ({x} {d} {a} {e} {at} + %pi (can-map a at) + %-> (map-of e at Dmap1' (eof/var Dwf1' d)) + %-> (map-wf' (Dwf2 a e) (Dmap2' a x at) (Dwf2' x d))) %. +%term _ + %pi (map-subtp (kd-sub/kd/pi Dsub1 Dsub2 Dwf2) (tmap/pi Dmap2 Dmap1) (tmap/pi Dmap2' Dmap1') (subtp/pi Dwf2' Dsub2' Dsub1')) + %<- (map-subtp Dsub1 Dmap1' Dmap1 Dsub1') + %<- (subtp-reg Dsub1' Dwf1' Dwf1) + %<- ({x} {d} {a} {e} {at} + %pi (can-map a at) + %-> (map-of e at Dmap1' (eof/var Dwf1' d)) + %-> (map-subtp (Dsub2 a e) (Dmap2 a x at) (Dmap2' a x at) (Dsub2' x d))) + %<- ({x} {d} {a} {e} {at} + %pi (can-map a at) + %-> (map-of e at Dmap1 (eof/var Dwf1 d)) + %-> (map-wf' (Dwf2 a e) (Dmap2 a x at) (Dwf2' x d))) %. +%%% +%term _ map-tequiv kd-deq/kd/type tmap/t tmap/t tequiv/t %. +%term _ + %pi (map-tequiv (kd-deq/kd/sing Dequiv) (tmap/sing Dmap) (tmap/sing Dmap') (tequiv/sing Dequiv')) + %<- (map-equiv'' Dmap Dmap' Dequiv tmap/t Dequiv') %. +%term _ + %pi (map-tequiv (kd-deq/kd/sgm Dequiv1 Dequiv2) (tmap/sigma Dmap2 Dmap1) (tmap/sigma Dmap2' Dmap1') (tequiv/sigma Dequiv2' Dequiv1')) + %<- (map-tequiv Dequiv1 Dmap1 Dmap1' Dequiv1') + %<- (tequiv-reg Dequiv1' Dwf1' _) + %<- ({x} {d} {a} {e} {at} + %pi (can-map a at) + %-> (map-of e at Dmap1 (eof/var Dwf1' d)) + %-> (map-tequiv (Dequiv2 a e) (Dmap2 a x at) (Dmap2' a x at) (Dequiv2' x d))) %. +%term _ + %pi (map-tequiv (kd-deq/kd/pi Dequiv1 Dequiv2) (tmap/pi Dmap2 Dmap1) (tmap/pi Dmap2' Dmap1') (tequiv/symm (tequiv/pi ([x] [d] tequiv/symm (Dequiv2' x d)) Dequiv1'))) + %<- (map-tequiv Dequiv1 Dmap1' Dmap1 Dequiv1') + %<- (tequiv-reg Dequiv1' Dwf1'' _) + %<- ({x} {d} {a} {e} {at} + %pi (can-map a at) + %-> (map-of e at Dmap1' (eof/var Dwf1'' d)) + %-> (map-tequiv (Dequiv2 a e) (Dmap2 a x at) (Dmap2' a x at) (Dequiv2' x d))) %. +%%% +%term _ + %pi (map-equiv' Dmap Dmap' (cn-deq/refl Dof) DmapA Dequiv') + %<- (map-fun Dmap Dmap' Deq) + %<- (map-of Dof Dmap DmapA Dof') + %<- (equiv-resp eterm-eq/i Deq etp-eq/i (equiv/reflex Dof') Dequiv') %. +%term _ + %pi (map-equiv' Dmap Dmap' (cn-deq/sym Dequiv) DmapA (equiv/symm Dequiv')) + %<- (map-equiv' Dmap' Dmap Dequiv DmapA Dequiv') %. +%term _ + %pi (map-equiv' Dmap1 Dmap3 (cn-deq/trans D12 D23) DmapA (equiv/trans D23' D12')) + %<- (can-map _ Dmap2) + %<- (map-equiv' Dmap1 Dmap2 D12 DmapA D12') + %<- (map-equiv'' Dmap2 Dmap3 D23 DmapA D23') %. +%term _ + %pi (map-equiv' (map/app Dmap2 Dmap1) (map/app Dmap2' Dmap1') (cn-deq/cn/app Dequiv1 Dequiv2) (DmapB _ _ Dmap2) (equiv/app Dequiv2' Dequiv1')) + %<- (map-equiv' Dmap1 Dmap1' Dequiv1 (tmap/pi DmapB DmapA) Dequiv1') + %<- (map-equiv'' Dmap2 Dmap2' Dequiv2 DmapA Dequiv2') %. +%term _ + %pi (map-equiv' (map/pi1 Dmap) (map/pi1 Dmap') (cn-deq/cn/pj1 Dequiv) DmapA (equiv/pi1 Dequiv')) + %<- (map-equiv' Dmap Dmap' Dequiv (tmap/sigma DmapB DmapA) Dequiv') %. +%term _ + %pi (map-equiv' (map/pi2 Dmap) (map/pi2 Dmap') (cn-deq/cn/pj2 Dequiv) (DmapB _ _ (map/pi1 Dmap)) (equiv/pi2 Dequiv')) + %<- (map-equiv' Dmap Dmap' Dequiv (tmap/sigma DmapB DmapA) Dequiv') %. +%term _ + %pi (map-equiv' (map/lam Dmap DmapA) (map/lam Dmap' DmapA') (cn-deq/cn/lam Dequiv2 Dequiv1) (tmap/pi DmapB DmapA) (equiv/lam Dequiv2' Dequiv1')) + %<- (map-tequiv Dequiv1 DmapA DmapA' Dequiv1') + %<- (tequiv-reg Dequiv1' Dwf' _) + %<- ({x} {d} {a} {e} {at} + %pi (can-map a at) + %-> (map-of e at DmapA (eof/var Dwf' d)) + %-> (map-equiv' (Dmap a x at) (Dmap' a x at) (Dequiv2 a e) (DmapB a x at) (Dequiv2' x d))) %. +%term _ + %pi (map-equiv' (map/pair Dmap2 Dmap1) (map/pair Dmap2' Dmap1') (cn-deq/cn/pair Dequiv1 Dequiv2) (tmap/sigma ([_] [_] [_] DmapB) DmapA) (equiv/pair ([_] [_] Dwf2) Dequiv2' Dequiv1')) + %<- (map-equiv' Dmap1 Dmap1' Dequiv1 DmapA Dequiv1') + %<- (map-equiv' Dmap2 Dmap2' Dequiv2 DmapB Dequiv2') + %<- (equiv-reg Dequiv2' _ _ Dwf2) %. +%term _ + %pi (map-equiv' Dmap1 Dmap2 (cn-deq/kd/sing Dof) (tmap/sing Dmap2) (equiv/symm (equiv/sing (equiv/symm (equiv/singelim Dof'))))) + %<- (map-of' Dof Dmap1 (tmap/sing Dmap2) Dof') %. +%term _ + %pi (map-equiv' Dmap Dmap' (cn-deq/sgm-ext Dequiv1 Dequiv2) (tmap/sigma ([_] [_] [_] DmapB) DmapA) (equiv/extsigma ([_] [_] Dwf2') Dequiv2' Dequiv1')) + %<- (map-equiv' (map/pi1 Dmap) (map/pi1 Dmap') Dequiv1 DmapA Dequiv1') + %<- (map-equiv' (map/pi2 Dmap) (map/pi2 Dmap') Dequiv2 DmapB Dequiv2') + %<- (equiv-reg Dequiv2' _ _ Dwf2') %. +%term _ + %pi (map-equiv' Dmap Dmap' (cn-deq/pi-ext DofPre1 DofPre2 Dequiv) (tmap/pi DmapB DmapA) (equiv/extpi Dequiv' DofPre2' DofPre1')) + %<- (map-of DofPre1 Dmap (tmap/pi _ DmapA) DofPre1') + %<- ({x} {a} {at} %pi (can-map a at) %-> (can-tmap _ (DmapB2 a x at))) + %<- (map-of' DofPre2 Dmap' (tmap/pi DmapB2 DmapA) DofPre2') + %<- (eof-reg DofPre1' (ewf/pi _ Dwf')) + %<- ({x} {d} {a} {e} {at} + %pi (can-map a at) + %-> (map-of e at DmapA (eof/var Dwf' d)) + %-> (map-equiv' (map/app at Dmap) (map/app at Dmap') (Dequiv a e) (DmapB a x at) (Dequiv' x d))) %. +%term _ + %pi (map-equiv' Dmap Dmap' (cn-deq/pi-ext-2 DequivPre Dequiv) (tmap/pi DmapB DmapA) (equiv/extpiw Dequiv' DequivPre')) + %<- (map-equiv' Dmap Dmap' DequivPre (tmap/pi _ DmapA) DequivPre') + %<- (equiv-reg DequivPre' _ _ (ewf/pi _ Dwf')) + %<- ({x} {d} {a} {e} {at} + %pi (can-map a at) + %-> (map-of e at DmapA (eof/var Dwf' d)) + %-> (map-equiv' (map/app at Dmap) (map/app at Dmap') (Dequiv a e) (DmapB a x at) (Dequiv' x d))) %. +%term _ + %pi (map-equiv' Dmap Dmap' (cn-deq/sub Dequiv Dsub) DmapB (equiv/subsume Dsub' Dequiv')) + %<- (map-equiv' Dmap Dmap' Dequiv DmapA Dequiv') + %<- (can-tmap _ DmapB) + %<- (map-subtp Dsub DmapA DmapB Dsub') %. +%term _ + %pi (map-equiv' Dmap Dmap' (cn-deq/deq Dequiv Dsub) DmapB (equiv/equiv Dsub' Dequiv')) + %<- (map-equiv' Dmap Dmap' Dequiv DmapA Dequiv') + %<- (can-tmap _ DmapB) + %<- (map-tequiv Dsub DmapA DmapB Dsub') %. +%term _ map-equiv' map/unit map/unit cn-deq/tp/unit tmap/t (equiv/reflex (eof/const ewf/t (ekof/i etopen/t ckof/unit))) %. +%term _ map-equiv' map/tagged map/tagged cn-deq/tp/tagged tmap/t (equiv/reflex (eof/const ewf/t (ekof/i etopen/t ckof/tagged))) %. +%term _ + %pi (map-equiv' (map/ref Dmap) (map/ref Dmap') (cn-deq/tp/ref Dequiv) tmap/t (equiv/app Dequiv' (equiv/reflex (eof/const (ewf/pi ([_] [_] ewf/t) ewf/t) (ekof/i (etopen/pi etopen/t etopen/t) ckof/ref))))) + %<- (map-equiv'' Dmap Dmap' Dequiv tmap/t Dequiv') %. +%term _ + %pi (map-equiv' (map/tag Dmap) (map/tag Dmap') (cn-deq/tp/tag Dequiv) tmap/t (equiv/app Dequiv' (equiv/reflex (eof/const (ewf/pi ([_] [_] ewf/t) ewf/t) (ekof/i (etopen/pi etopen/t etopen/t) ckof/tag))))) + %<- (map-equiv'' Dmap Dmap' Dequiv tmap/t Dequiv') %. +%term _ + %pi (map-equiv' (map/arrow Dmap2 Dmap1) (map/arrow Dmap2' Dmap1') (cn-deq/tp/arrow Dequiv1 Dequiv2) tmap/t (equiv/app Dequiv2' (equiv/app Dequiv1' (equiv/reflex (eof/const (ewf/pi ([_] [_] ewf/pi ([_] [_] ewf/t) ewf/t) ewf/t) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/arrow)))))) + %<- (map-equiv'' Dmap1 Dmap1' Dequiv1 tmap/t Dequiv1') + %<- (map-equiv'' Dmap2 Dmap2' Dequiv2 tmap/t Dequiv2') %. +%term _ + %pi (map-equiv' (map/cross Dmap2 Dmap1) (map/cross Dmap2' Dmap1') (cn-deq/tp/cross Dequiv1 Dequiv2) tmap/t (equiv/app Dequiv2' (equiv/app Dequiv1' (equiv/reflex (eof/const (ewf/pi ([_] [_] ewf/pi ([_] [_] ewf/t) ewf/t) ewf/t) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/cross)))))) + %<- (map-equiv'' Dmap1 Dmap1' Dequiv1 tmap/t Dequiv1') + %<- (map-equiv'' Dmap2 Dmap2' Dequiv2 tmap/t Dequiv2') %. +%term _ + %pi (map-equiv' (map/sum Dmap2 Dmap1) (map/sum Dmap2' Dmap1') (cn-deq/tp/sum Dequiv1 Dequiv2) tmap/t (equiv/app Dequiv2' (equiv/app Dequiv1' (equiv/reflex (eof/const (ewf/pi ([_] [_] ewf/pi ([_] [_] ewf/t) ewf/t) ewf/t) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/dsum)))))) + %<- (map-equiv'' Dmap1 Dmap1' Dequiv1 tmap/t Dequiv1') + %<- (map-equiv'' Dmap2 Dmap2' Dequiv2 tmap/t Dequiv2') %. +%term _ + %pi (map-equiv' (map/forall Dmap2 Dmap1) (map/forall Dmap2' Dmap1') (cn-deq/tp/forall Dequiv2 Dequiv1) tmap/t (equiv/app (equiv/lam Dequiv2' Dequiv1') (equiv/forall Dequiv1'))) + %<- (map-tequiv Dequiv1 Dmap1 Dmap1' Dequiv1') + %<- (tequiv-reg Dequiv1' Dwf' _) + %<- ({x} {d} {a} {e} {at} + %pi (can-map a at) + %-> (map-of e at Dmap1 (eof/var Dwf' d)) + %-> (map-equiv'' (Dmap2 a x at) (Dmap2' a x at) (Dequiv2 a e) tmap/t (Dequiv2' x d))) %. +%term _ + %pi (map-equiv' (map/mu Dmap) (map/mu Dmap') (cn-deq/cn/mu Dequiv _) tmap/t (equiv/app (equiv/lam Dequiv' tequiv/t) (equiv/reflex (eof/const (ewf/pi ([_] [_] ewf/t) (ewf/pi ([_] [_] ewf/t) ewf/t)) (ekof/i (etopen/pi etopen/t (etopen/pi etopen/t etopen/t)) ckof/mu))))) + %<- ({x} {d} {a} {e} {at} + %pi (can-map a at) + %-> (map-of e at tmap/t (eof/var ewf/t d)) + %-> (map-equiv'' (Dmap a x at) (Dmap' a x at) (Dequiv a e) tmap/t (Dequiv' x d))) %. +%%% +%term _ + %pi (map-equiv'' Dmap Dmap' Dequiv DmapA Dequiv'') + %<- (map-equiv' Dmap Dmap' Dequiv DmapA' Dequiv') + %<- (tmap-fun DmapA' DmapA Deq) + %<- (equiv-resp eterm-eq/i eterm-eq/i Deq Dequiv' Dequiv'') %. +%worlds (map-bind) (map-of _ _ _ _) (map-of' _ _ _ _) (map-wf _ _ _) (map-wf' _ _ _) (map-subtp _ _ _ _) (map-tequiv _ _ _ _) (map-equiv' _ _ _ _ _) (map-equiv'' _ _ _ _ _) %. +%total (D1 D2 D3 D4 D5 D6 D7 D8) (map-of D1 _ _ _) (map-of' D2 _ _ _) (map-wf D3 _ _) (map-wf' D4 _ _) (map-subtp D5 _ _ _) (map-tequiv D6 _ _ _) (map-equiv' _ _ D7 _ _) (map-equiv'' _ _ D8 _ _) %. +%sort map-equiv %. +%mode map-equiv %in %out %out %out %out %. +%term _ + %pi (map-equiv Dequiv Dmap1 Dmap2 DmapA Dequiv') + %<- (can-map _ Dmap1) + %<- (can-map _ Dmap2) + %<- (map-equiv' Dmap1 Dmap2 Dequiv DmapA Dequiv') %. +%worlds () (map-equiv _ _ _ _ _) %. +%total {} (map-equiv _ _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/dynamic-semantics.lf b/new-tests/stelf-output/tslf/dynamic-semantics.lf new file mode 100644 index 0000000..04ff029 --- /dev/null +++ b/new-tests/stelf-output/tslf/dynamic-semantics.lf @@ -0,0 +1,145 @@ +%%%% dynamic semantics for terms +%sort step/tm %. +%mode step/tm %star %star %star %star %star %star %. +%sort step/md %. +%mode step/md %star %star %star %star %star %star %. +%term step/tm/pair-1 + %pi (step/tm (tm/pair E1 E2) S1 T1 (tm/pair E1' E2) S2 T2) + %<- (step/tm E1 S1 T1 E1' S2 T2) %. +%term step/tm/pair-2 + %pi (step/tm (tm/pair E1 E2) S1 T1 (tm/pair E1 E2') S2 T2) + %<- (step/tm E2 S1 T1 E2' S2 T2) + %<- (val/tm E1) %. +%term step/tm/pj1 %pi (step/tm (tm/pj1 E) S1 T1 (tm/pj1 E') S2 T2) %<- (step/tm E S1 T1 E' S2 T2) %. +%term step/tm/pj1-beta {S1 st} + %pi (step/tm (tm/pj1 (tm/pair E1 E2)) S1 T1 E1 S1 T1) + %<- (val/tm (tm/pair E1 E2)) %. +%term step/tm/pj2 %pi (step/tm (tm/pj2 E) S1 T1 (tm/pj2 E') S2 T2) %<- (step/tm E S1 T1 E' S2 T2) %. +%term step/tm/pj2-beta {S1 st} + %pi (step/tm (tm/pj2 (tm/pair E1 E2)) S1 T1 E2 S1 T1) + %<- (val/tm (tm/pair E1 E2)) %. +%term step/tm/tmapp-1 + %pi (step/tm (tm/tmapp E1 E2) S1 T1 (tm/tmapp E1' E2) S2 T2) + %<- (step/tm E1 S1 T1 E1' S2 T2) %. +%term step/tm/tmapp-2 + %pi (step/tm (tm/tmapp E1 E2) S1 T1 (tm/tmapp E1 E2') S2 T2) + %<- (step/tm E2 S1 T1 E2' S2 T2) + %<- (val/tm E1) %. +%term step/tm/tmapp-beta {S1 st} + %pi (step/tm (tm/tmapp (tm/fun C1 C2 E) E') S1 T1 (E E' (tm/fun C1 C2 E)) S1 T1) + %<- (val/tm E') %. +%term step/tm/cnapp + %pi (step/tm (tm/cnapp E1 C) S1 T1 (tm/cnapp E1' C) S2 T2) + %<- (step/tm E1 S1 T1 E1' S2 T2) %. +%term step/tm/cnapp-beta {S1 st} step/tm (tm/cnapp (tm/cnabs K E) C) S1 T1 (E C) S1 T1 %. +%term step/tm/term %pi (step/tm (tm/term M) S1 T1 (tm/term M') S2 T2) %<- (step/md M S1 T1 M' S2 T2) %. +%term step/tm/term-beta {S1 st} %pi (step/tm (tm/term (md/tm E)) S1 T1 E S1 T1) %<- (val/md (md/tm E)) %. +%term step/tm/ref %pi (step/tm (tm/ref E) S T1 (tm/ref E') S' T2) %<- (step/tm E S T1 E' S' T2) %. +%term step/tm/ref-beta + %pi (step/tm (tm/ref E) S T1 (tm/loc LC) S' T1) + %<- (st-alloc S E S' LC) + %<- (val/tm E) %. +%term step/tm/set-1 + %pi (step/tm (tm/set E1 E2) S T1 (tm/set E1' E2) S' T2) + %<- (step/tm E1 S T1 E1' S' T2) %. +%term step/tm/set-2 + %pi (step/tm (tm/set E1 E2) S T1 (tm/set E1 E2') S' T2) + %<- (step/tm E2 S T1 E2' S' T2) + %<- (val/tm E1) %. +%term step/tm/set-beta + %pi (step/tm (tm/set (tm/loc L) E2) S T1 tm/unit S' T1) + %<- (st-update S L E2 S') + %<- (val/tm E2) + %<- (val/tm (tm/loc L)) %. +%term step/tm/get %pi (step/tm (tm/get E) S T1 (tm/get E') S' T2) %<- (step/tm E S T1 E' S' T2) %. +%term step/tm/get-beta + %pi (step/tm (tm/get (tm/loc L)) S T E S T) + %<- (st-look S L E) + %<- (val/tm (tm/loc L)) %. +%term step/tm/inl + %pi (step/tm (tm/inl (tp/sum C1 C2) E) S1 T1 (tm/inl (tp/sum C1 C2) E') S2 T2) + %<- (step/tm E S1 T1 E' S2 T2) %. +%term step/tm/inr + %pi (step/tm (tm/inr (tp/sum C1 C2) E) S1 T1 (tm/inr (tp/sum C1 C2) E') S2 T2) + %<- (step/tm E S1 T1 E' S2 T2) %. +%term step/tm/case + %pi (step/tm (tm/case E E1 E2) S1 T1 (tm/case E' E1 E2) S2 T2) + %<- (step/tm E S1 T1 E' S2 T2) %. +%term step/tm/case-beta-l + %pi (step/tm (tm/case (tm/inl (tp/sum C1 C2) E) E1 E2) S1 T1 (E1 E) S1 T1) + %<- (val/tm (tm/inl (tp/sum C1 C2) E)) %. +%term step/tm/case-beta-r + %pi (step/tm (tm/case (tm/inr (tp/sum C1 C2) E) E1 E2) S1 T1 (E2 E) S1 T1) + %<- (val/tm (tm/inr (tp/sum C1 C2) E)) %. +%term step/tm/try + %pi (step/tm (tm/try E1 E2) S1 T1 (tm/try E1' E2) S2 T2) + %<- (step/tm E1 S1 T1 E1' S2 T2) %. +%term step/tm/try-beta %pi (step/tm (tm/try E1 E2) S1 T1 E1 S1 T1) %<- (val/tm E1) %. +%term step/tm/try-handle %pi (step/tm (tm/try E1 E2) S1 T1 (E2 V) S1 T1) %<- (raises/tm E1 V) %. +%term step/tm/new-tag-beta + %pi (step/tm (tm/new-tag C1) S1 T1 (tm/tagloc LC) S1 T1') + %<- (lt-extend T1 C1 T1' LC) %. +%term step/tm/raise %pi (step/tm (tm/raise E) S T1 (tm/raise E') S' T2) %<- (step/tm E S T1 E' S' T2) %. +%term step/tm/tag-1 + %pi (step/tm (tm/tag E1 E2) S1 T1 (tm/tag E1' E2) S2 T2) + %<- (step/tm E1 S1 T1 E1' S2 T2) %. +%term step/tm/tag-2 + %pi (step/tm (tm/tag E1 E2) S1 T1 (tm/tag E1 E2') S2 T2) + %<- (step/tm E2 S1 T1 E2' S2 T2) + %<- (val/tm E1) %. +%term step/tm/iftag-1 + %pi (step/tm (tm/iftag E E1 E2 E3) S1 T1 (tm/iftag E' E1 E2 E3) S2 T2) + %<- (step/tm E S1 T1 E' S2 T2) %. +%term step/tm/iftag-2 + %pi (step/tm (tm/iftag E E1 E2 E3) S1 T1 (tm/iftag E E1' E2 E3) S2 T2) + %<- (step/tm E1 S1 T1 E1' S2 T2) + %<- (val/tm E) %. +%term step/tm/iftag-beta-1 + %pi (step/tm (tm/iftag (tm/tag (tm/tagloc L) E) (tm/tagloc L) E2 E3) S1 T1 (E2 E) S1 T1) + %<- (val/tm (tm/tag (tm/tagloc L) E)) %. +%term step/tm/iftag-beta-2 + %pi (step/tm (tm/iftag (tm/tag (tm/tagloc L) E) (tm/tagloc L') E2 E3) S1 T1 E3 S1 T1) + %<- (loc-neq L L') + %<- (val/tm (tm/tag (tm/tagloc L) E)) %. +%term step/tm/roll + %pi (step/tm (tm/roll (cn/mu K1 C2) E) S1 T1 (tm/roll (cn/mu K1 C2) E') S2 T2) + %<- (step/tm E S1 T1 E' S2 T2) %. +%term step/tm/unroll + %pi (step/tm (tm/unroll E) S1 T1 (tm/unroll E') S2 T2) + %<- (step/tm E S1 T1 E' S2 T2) %. +%term step/tm/unroll-beta %pi (step/tm (tm/unroll (tm/roll C E)) S1 T1 E S1 T1) %<- (val/tm (tm/roll C E)) %. +%term step/md/pair-1 + %pi (step/md (md/pair M1 M2) S1 T1 (md/pair M1' M2) S2 T2) + %<- (step/md M1 S1 T1 M1' S2 T2) %. +%term step/md/pair-2 + %pi (step/md (md/pair M1 M2) S1 T1 (md/pair M1 M2') S2 T2) + %<- (step/md M2 S1 T1 M2' S2 T2) + %<- (val/md M1) %. +%term step/md/pj1 %pi (step/md (md/pj1 M) S1 T1 (md/pj1 M') S2 T2) %<- (step/md M S1 T1 M' S2 T2) %. +%term step/md/pj1-beta {S1 st} + %pi (step/md (md/pj1 (md/pair M1 M2)) S1 T1 M1 S1 T1) + %<- (val/md (md/pair M1 M2)) %. +%term step/md/pj2 %pi (step/md (md/pj2 M) S1 T1 (md/pj2 M') S2 T2) %<- (step/md M S1 T1 M' S2 T2) %. +%term step/md/pj2-beta {S1 st} + %pi (step/md (md/pj2 (md/pair M1 M2)) S1 T1 M2 S1 T1) + %<- (val/md (md/pair M1 M2)) %. +%term step/md/app-1 + %pi (step/md (md/app M1 M2) S1 T1 (md/app M1' M2) S2 T2) + %<- (step/md M1 S1 T1 M1' S2 T2) %. +%term step/md/app-2 + %pi (step/md (md/app M1 M2) S1 T1 (md/app M1 M2') S2 T2) + %<- (step/md M2 S1 T1 M2' S2 T2) + %<- (val/md M1) %. +%term step/md/app-beta {ST st} + %pi (step/md (md/app (md/lam S1 S2 M) M') ST TT (M M' C) ST TT) + %<- (fst-md M' C) + %<- (val/md M') %. +%term step/md/tm %pi (step/md (md/tm E) S1 T1 (md/tm E') S2 T2) %<- (step/tm E S1 T1 E' S2 T2) %. +%term step/md/seal-beta {S1 st} step/md (md/seal M S) S1 T1 M S1 T1 %. +%term step/md/let + %pi (step/md (md/let M1 M2 S) S1 T1 (md/let M1' M2 S) S2 T2) + %<- (step/md M1 S1 T1 M1' S2 T2) %. +%term step/md/let-beta {S1 st} + %pi (step/md (md/let M1 M2 S) S1 T1 (M2 M1 C) S1 T1) + %<- (fst-md M1 C) + %<- (val/md M1) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/explicate-lemmas-sigs.lf b/new-tests/stelf-output/tslf/explicate-lemmas-sigs.lf new file mode 100644 index 0000000..f936d3c --- /dev/null +++ b/new-tests/stelf-output/tslf/explicate-lemmas-sigs.lf @@ -0,0 +1,116 @@ +%sort esg-wf-ordered %. +%mode esg-wf-ordered %in %out %. +%term _ esg-wf-ordered (esg-wf/sg/unit DO) DO %. +%term _ %pi (esg-wf-ordered (esg-wf/sg/cn D1) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (esg-wf-ordered (esg-wf/sg/kd D1) DO) %<- (ekd-wf-ordered D1 DO) %. +%term _ %pi (esg-wf-ordered (esg-wf/sg/sgm D1 _ _) DO) %<- (esg-wf-ordered D1 DO) %. +%term _ %pi (esg-wf-ordered (esg-wf/sg/pi D1 _ _) DO) %<- (esg-wf-ordered D1 DO) %. +%worlds (cn-block ovar-block bind-block wbind-block ofkd-block) (esg-wf-ordered _ _) %. +%total (D1) (esg-wf-ordered D1 _) %. +%sort esg-deq-ordered %. +%mode esg-deq-ordered %in %out %. +%term _ esg-deq-ordered (esg-deq/sg/unit DO) DO %. +%term _ %pi (esg-deq-ordered (esg-deq/sg/cn D1) DO) %<- (ecn-deq-ordered D1 DO) %. +%term _ %pi (esg-deq-ordered (esg-deq/sg/kd D1) DO) %<- (ekd-deq-ordered D1 DO) %. +%term _ %pi (esg-deq-ordered (esg-deq/sg/sgm D1 _ _) DO) %<- (esg-deq-ordered D1 DO) %. +%term _ %pi (esg-deq-ordered (esg-deq/sg/pi D1 _ _) DO) %<- (esg-deq-ordered D1 DO) %. +%worlds (cn-block ovar-block wbind-block ofkd-block) (esg-deq-ordered _ _) %. +%total (D1) (esg-deq-ordered D1 _) %. +%sort cut-sg-wf {Cm} {D %pi (ofkd C K) %-> (sg-wf K')} {d} %. +%mode cut-sg-wf %in %in %in %in %out %out %. +%sort cut-sg-deq {Cm} {D %pi (ofkd C K) %-> (sg-deq K' K'')} {d} %. +%mode cut-sg-deq %in %in %in %in %out %out %. +%sort cut-esg-wf {Cm} {D %pi (ofkd C K) %-> (esg-wf G K')} {d} %. +%mode cut-esg-wf %in %in %in %in %out %out %. +%sort cut-esg-deq {Cm} {D %pi (ofkd C K) %-> (esg-deq G K' K'')} {d} %. +%mode cut-esg-deq %in %in %in %in %out %out %. +%term _ + %pi (cut-sg-wf _ ([d] sg-wf/sg/unit) ([d'] [dm] msg-wf/sg/unit) DL (esg-wf/sg/unit DO) mesg-wf/sg/unit) + %<- (cxt-lookup-ordered DL DO) %. +%term _ + %pi (cut-sg-wf (met/sing Mm) ([d] sg-wf/sg/cn (D1 d)) ([d'] [dm] msg-wf/sg/cn (D2 d' dm)) DL (esg-wf/sg/cn D1') (mesg-wf/sg/cn D2')) + %<- (cut-ofkd Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-sg-wf (met/sing Mm) ([d] sg-wf/sg/kd (D1 d)) ([d'] [dm] msg-wf/sg/kd (D2 d' dm)) DL (esg-wf/sg/kd D1') (mesg-wf/sg/kd D2')) + %<- (cut-kd-wf Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-sg-wf (met/pair M1 M2) ([d] sg-wf/sg/sgm (D1 d) (D2 d) DFS) ([d'] [dm] msg-wf/sg/sgm (D3 d' dm) (D4 d' dm)) DL (esg-wf/sg/sgm D1' D2'' DFS) (mesg-wf/sg/sgm D3' D4'')) + %<- (cxt-lookup-ordered DL DO) + %<- (cxt-extend-ordered DO DB) + %<- (cut-sg-wf M1 D1 D3 DL D1' D3') + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (%the (cxt-lookup (cxt/cons G y B) X A) (DL' y e'))) + %<- ({d ofkd X A} {dm mofkd d met/unit} {y} {e' isvar y J} cut-sg-wf M2 ([e] D2 d y e) ([e] [em] D4 d dm y e em) (cxt-lookup/hit (DB y e')) (%the (esg-wf (cxt/cons G y B) (C y)) (D2' d y e')) (%the (mesg-wf (D2' d y e') M2) (D4' d dm y e'))) + %<- ({y} {e' isvar y J} cut-esg-wf M2 ([d] D2' d y e') ([d] [dm] D4' d dm y e') (DL' y e') (D2'' y e') (D4'' y e')) %. +%term _ + %pi (cut-sg-wf (met/pair M1 M2) ([d] sg-wf/sg/pi (D1 d) (D2 d) DFS) ([d'] [dm] msg-wf/sg/pi (D3 d' dm) (D4 d' dm)) DL (esg-wf/sg/pi D1' D2'' DFS) (mesg-wf/sg/pi D3' D4'')) + %<- (cxt-lookup-ordered DL DO) + %<- (cxt-extend-ordered DO DB) + %<- (cut-sg-wf M1 D1 D3 DL D1' D3') + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (%the (cxt-lookup (cxt/cons G y B) X A) (DL' y e'))) + %<- ({d ofkd X A} {dm mofkd d met/unit} {y} {e' isvar y J} cut-sg-wf M2 ([e] D2 d y e) ([e] [em] D4 d dm y e em) (cxt-lookup/hit (DB y e')) (%the (esg-wf (cxt/cons G y B) (C y)) (D2' d y e')) (%the (mesg-wf (D2' d y e') M2) (D4' d dm y e'))) + %<- ({y} {e' isvar y J} cut-esg-wf M2 ([d] D2' d y e') ([d] [dm] D4' d dm y e') (DL' y e') (D2'' y e') (D4'' y e')) %. +%term _ + %pi (cut-sg-deq _ ([d] sg-deq/sg/unit) ([d'] [dm] msg-deq/sg/unit) DL (esg-deq/sg/unit DO) mesg-deq/sg/unit) + %<- (cxt-lookup-ordered DL DO) %. +%term _ + %pi (cut-sg-deq (met/sing Mm) ([d] sg-deq/sg/cn (D1 d)) ([d'] [dm] msg-deq/sg/cn (D2 d' dm)) DL (esg-deq/sg/cn D1') (mesg-deq/sg/cn D2')) + %<- (cut-cn-deq Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-sg-deq (met/sing Mm) ([d] sg-deq/sg/kd (D1 d)) ([d'] [dm] msg-deq/sg/kd (D2 d' dm)) DL (esg-deq/sg/kd D1') (mesg-deq/sg/kd D2')) + %<- (cut-kd-deq Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-sg-deq (met/pair M1 M2) ([d] sg-deq/sg/sgm (D1 d) (D2 d) DFS) ([d'] [dm] msg-deq/sg/sgm (D3 d' dm) (D4 d' dm)) DL (esg-deq/sg/sgm D1' D2'' DFS) (mesg-deq/sg/sgm D3' D4'')) + %<- (cxt-lookup-ordered DL DO) + %<- (cxt-extend-ordered DO DB) + %<- (cut-sg-deq M1 D1 D3 DL D1' D3') + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (DL' y e')) + %<- ({d ofkd X A} {dm mofkd d met/unit} {y} {e' isvar y J} cut-sg-deq M2 ([e] D2 d y e) ([e] [em] D4 d dm y e em) (cxt-lookup/hit (DB y e')) (D2' d y e') (D4' d dm y e')) + %<- ({y} {e' isvar y J} cut-esg-deq M2 ([d] D2' d y e') ([d] [dm] D4' d dm y e') (DL' y e') (D2'' y e') (D4'' y e')) %. +%term _ + %pi (cut-sg-deq (met/pair M1 M2) ([d] sg-deq/sg/pi (D1 d) (D2 d) DFS) ([d'] [dm] msg-deq/sg/pi (D3 d' dm) (D4 d' dm)) DL (esg-deq/sg/pi D1' D2'' DFS) (mesg-deq/sg/pi D3' D4'')) + %<- (cxt-lookup-ordered DL DO) + %<- (cxt-extend-ordered DO DB) + %<- (cut-sg-deq M1 D1 D3 DL D1' D3') + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (DL' y e')) + %<- ({d ofkd X A} {dm mofkd d met/unit} {y} {e' isvar y J} cut-sg-deq M2 ([e] D2 d y e) ([e] [em] D4 d dm y e em) (cxt-lookup/hit (DB y e')) (D2' d y e') (D4' d dm y e')) + %<- ({y} {e' isvar y J} cut-esg-deq M2 ([d] D2' d y e') ([d] [dm] D4' d dm y e') (DL' y e') (D2'' y e') (D4'' y e')) %. +%term _ cut-esg-wf met/unit ([d] esg-wf/sg/unit DO) ([d] [dm] mesg-wf/sg/unit) DL (esg-wf/sg/unit DO) mesg-wf/sg/unit %. +%term _ + %pi (cut-esg-wf (met/sing Mm) ([d] esg-wf/sg/cn (D1 d)) ([d] [dm] mesg-wf/sg/cn (D2 d dm)) DL (esg-wf/sg/cn D1') (mesg-wf/sg/cn D2')) + %<- (cut-eofkd Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-esg-wf (met/sing Mm) ([d] esg-wf/sg/kd (D1 d)) ([d] [dm] mesg-wf/sg/kd (D2 d dm)) DL (esg-wf/sg/kd D1') (mesg-wf/sg/kd D2')) + %<- (cut-ekd-wf Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-esg-wf (met/pair M1 M2) ([d] esg-wf/sg/sgm (D1 d) (D2 d) DFS) ([d] [dm] mesg-wf/sg/sgm (D3 d dm) (D4 d dm)) DL (esg-wf/sg/sgm D1' D2' DFS) (mesg-wf/sg/sgm D3' D4')) + %<- (cut-esg-wf M1 D1 D3 DL D1' D3') + %<- ({d ofkd _ _} {y} {e isvar y J} esg-wf-ordered (D2 d y e) (cxt-ordered/cons (DB y e))) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL' y e)) + %<- ({y} {e isvar y J} cut-esg-wf M2 ([d] D2 d y e) ([d] [dm] D4 d dm y e) (DL' y e) (D2' y e) (D4' y e)) %. +%term _ + %pi (cut-esg-wf (met/pair M1 M2) ([d] esg-wf/sg/pi (D1 d) (D2 d) DFS) ([d] [dm] mesg-wf/sg/pi (D3 d dm) (D4 d dm)) DL (esg-wf/sg/pi D1' D2' DFS) (mesg-wf/sg/pi D3' D4')) + %<- (cut-esg-wf M1 D1 D3 DL D1' D3') + %<- ({d ofkd _ _} {y} {e isvar y J} esg-wf-ordered (D2 d y e) (cxt-ordered/cons (DB y e))) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL' y e)) + %<- ({y} {e isvar y J} cut-esg-wf M2 ([d] D2 d y e) ([d] [dm] D4 d dm y e) (DL' y e) (D2' y e) (D4' y e)) %. +%term _ cut-esg-deq met/unit ([d] esg-deq/sg/unit DO) ([d] [dm] mesg-deq/sg/unit) DL (esg-deq/sg/unit DO) mesg-deq/sg/unit %. +%term _ + %pi (cut-esg-deq (met/sing Mm) ([d] esg-deq/sg/cn (D1 d)) ([d] [dm] mesg-deq/sg/cn (D2 d dm)) DL (esg-deq/sg/cn D1') (mesg-deq/sg/cn D2')) + %<- (cut-ecn-deq Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-esg-deq (met/sing Mm) ([d] esg-deq/sg/kd (D1 d)) ([d] [dm] mesg-deq/sg/kd (D2 d dm)) DL (esg-deq/sg/kd D1') (mesg-deq/sg/kd D2')) + %<- (cut-ekd-deq Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-esg-deq (met/pair M1 M2) ([d] esg-deq/sg/sgm (D1 d) (D2 d) DFS) ([d] [dm] mesg-deq/sg/sgm (D3 d dm) (D4 d dm)) DL (esg-deq/sg/sgm D1' D2' DFS) (mesg-deq/sg/sgm D3' D4')) + %<- (cut-esg-deq M1 D1 D3 DL D1' D3') + %<- ({d ofkd _ _} {y} {e isvar y J} esg-deq-ordered (D2 d y e) (cxt-ordered/cons (DB y e))) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL' y e)) + %<- ({y} {e isvar y J} cut-esg-deq M2 ([d] D2 d y e) ([d] [dm] D4 d dm y e) (DL' y e) (D2' y e) (D4' y e)) %. +%term _ + %pi (cut-esg-deq (met/pair M1 M2) ([d] esg-deq/sg/pi (D1 d) (D2 d) DFS) ([d] [dm] mesg-deq/sg/pi (D3 d dm) (D4 d dm)) DL (esg-deq/sg/pi D1' D2' DFS) (mesg-deq/sg/pi D3' D4')) + %<- (cut-esg-deq M1 D1 D3 DL D1' D3') + %<- ({d ofkd _ _} {y} {e isvar y J} esg-deq-ordered (D2 d y e) (cxt-ordered/cons (DB y e))) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL' y e)) + %<- ({y} {e isvar y J} cut-esg-deq M2 ([d] D2 d y e) ([d] [dm] D4 d dm y e) (DL' y e) (D2' y e) (D4' y e)) %. +%worlds (can-mofkd-block ofkd-block ovar-block bind-block wbind-block) (cut-sg-wf _ _ _ _ _ _) (cut-sg-deq _ _ _ _ _ _) (cut-esg-wf _ _ _ _ _ _) (cut-esg-deq _ _ _ _ _ _) %. +%total (M1 M2 M3 M4) (cut-sg-wf M1 _ _ _ _ _) (cut-sg-deq M2 _ _ _ _ _) (cut-esg-wf M3 _ _ _ _ _) (cut-esg-deq M4 _ _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/explicate-lemmas.lf b/new-tests/stelf-output/tslf/explicate-lemmas.lf new file mode 100644 index 0000000..cab59a9 --- /dev/null +++ b/new-tests/stelf-output/tslf/explicate-lemmas.lf @@ -0,0 +1,621 @@ +%block bind-block {c cn} {k kd} [da ofkd c k] [dm mofkd da met/unit]%. +%block wbind-block {c cn} {k kd} [da ofkd c k]%. +%sort ekd-wf-ordered %. +%mode ekd-wf-ordered %in %out %. +%sort eofkd-ordered %. +%mode eofkd-ordered %in %out %. +%term _ ekd-wf-ordered (ekd-wf/kd/unit DO) DO %. +%term _ ekd-wf-ordered (ekd-wf/kd/type DO) DO %. +%term _ %pi (ekd-wf-ordered (ekd-wf/kd/sing D1) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (ekd-wf-ordered (ekd-wf/kd/sgm D1 _) DO) %<- (ekd-wf-ordered D1 DO) %. +%term _ %pi (ekd-wf-ordered (ekd-wf/kd/pi D1 _) DO) %<- (ekd-wf-ordered D1 DO) %. +%term _ eofkd-ordered (eofkd/cn/unit DO) DO %. +%term _ eofkd-ordered (eofkd/tp/unit DO) DO %. +%term _ eofkd-ordered (eofkd/tp/tagged DO) DO %. +%term _ %pi (eofkd-ordered (eofkd/tp/ref D1) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (eofkd-ordered (eofkd/tp/tag D1) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (eofkd-ordered (eofkd/cn/pj1 D1) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (eofkd-ordered (eofkd/cn/pj2 D1) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (eofkd-ordered (eofkd/kd/sing D1) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (eofkd-ordered (eofkd/cn/pair D1 _) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (eofkd-ordered (eofkd/cn/app D1 _) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (eofkd-ordered (eofkd/sgm-ext D1 _) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (eofkd-ordered (eofkd/pi-ext D1 _) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (eofkd-ordered (eofkd/tp/cross D1 _) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (eofkd-ordered (eofkd/tp/arrow D1 _) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (eofkd-ordered (eofkd/tp/sum D1 _) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (eofkd-ordered (eofkd/sub D1 _) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (eofkd-ordered (eofkd/deq D1 _) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (eofkd-ordered (eofkd/cn/lam _ D1) DO) %<- (ekd-wf-ordered D1 DO) %. +%term _ %pi (eofkd-ordered (eofkd/tp/forall _ D1) DO) %<- (ekd-wf-ordered D1 DO) %. +%term _ %pi (eofkd-ordered (eofkd/cn/mu _ D1) DO) %<- (ekd-wf-ordered D1 DO) %. +%term _ %pi (eofkd-ordered (eofkd/var DL) DO) %<- (cxt-lookup-ordered DL DO) %. +%term _ eofkd-ordered (eofkd/closed _ DO) DO %. +%worlds (cn-block ovar-block bind-block wbind-block ofkd-block) (ekd-wf-ordered _ _) (eofkd-ordered _ _) %. +%total (D1 D2) (ekd-wf-ordered D1 _) (eofkd-ordered D2 _) %. +%sort ekd-deq-ordered %. +%mode ekd-deq-ordered %in %out %. +%sort ecn-deq-ordered %. +%mode ecn-deq-ordered %in %out %. +%term _ ekd-deq-ordered (ekd-deq/kd/unit DO) DO %. +%term _ ekd-deq-ordered (ekd-deq/kd/type DO) DO %. +%term _ %pi (ekd-deq-ordered (ekd-deq/kd/sing D1) DO) %<- (ecn-deq-ordered D1 DO) %. +%term _ %pi (ekd-deq-ordered (ekd-deq/kd/sgm D1 _) DO) %<- (ekd-deq-ordered D1 DO) %. +%term _ %pi (ekd-deq-ordered (ekd-deq/kd/pi D1 _) DO) %<- (ekd-deq-ordered D1 DO) %. +%term _ ecn-deq-ordered (ecn-deq/cn/unit DO) DO %. +%term _ ecn-deq-ordered (ecn-deq/tp/unit DO) DO %. +%term _ ecn-deq-ordered (ecn-deq/tp/tagged DO) DO %. +%term _ %pi (ecn-deq-ordered (ecn-deq/tp/ref D1) DO) %<- (ecn-deq-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/tp/tag D1) DO) %<- (ecn-deq-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/cn/pj1 D1) DO) %<- (ecn-deq-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/cn/pj2 D1) DO) %<- (ecn-deq-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/sym D1) DO) %<- (ecn-deq-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/kd/sing D1) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/refl D1) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/trans D1 _) DO) %<- (ecn-deq-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/cn/pair D1 _) DO) %<- (ecn-deq-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/cn/app D1 _) DO) %<- (ecn-deq-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/sgm-ext D1 _) DO) %<- (ecn-deq-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/pi-ext D1 _ _) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/pi-ext-2 D1 _) DO) %<- (ecn-deq-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/kd/unit D1 _) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/tp/cross D1 _) DO) %<- (ecn-deq-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/tp/arrow D1 _) DO) %<- (ecn-deq-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/tp/sum D1 _) DO) %<- (ecn-deq-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/sub D1 _) DO) %<- (ecn-deq-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/deq D1 _) DO) %<- (ecn-deq-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/cn/lam _ D1) DO) %<- (ekd-deq-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/tp/forall _ D1) DO) %<- (ekd-deq-ordered D1 DO) %. +%term _ %pi (ecn-deq-ordered (ecn-deq/cn/mu _ D1) DO) %<- (ekd-deq-ordered D1 DO) %. +%worlds (cn-block ovar-block wbind-block ofkd-block) (ekd-deq-ordered _ _) (ecn-deq-ordered _ _) %. +%total (D1 D2) (ekd-deq-ordered D1 _) (ecn-deq-ordered D2 _) %. +%sort ekd-sub-ordered %. +%mode ekd-sub-ordered %in %out %. +%term _ ekd-sub-ordered (ekd-sub/kd/unit DO) DO %. +%term _ ekd-sub-ordered (ekd-sub/kd/type DO) DO %. +%term _ %pi (ekd-sub-ordered (ekd-sub/kd/sing-kd/sing D1) DO) %<- (ecn-deq-ordered D1 DO) %. +%term _ %pi (ekd-sub-ordered (ekd-sub/kd/sing-kd/type D1) DO) %<- (eofkd-ordered D1 DO) %. +%term _ %pi (ekd-sub-ordered (ekd-sub/kd/sgm D1 _ _) DO) %<- (ekd-sub-ordered D1 DO) %. +%term _ %pi (ekd-sub-ordered (ekd-sub/kd/pi D1 _ _) DO) %<- (ekd-sub-ordered D1 DO) %. +%worlds (cn-block ovar-block wbind-block ofkd-block) (ekd-sub-ordered _ _) %. +%total (D1) (ekd-sub-ordered D1 _) %. +%sort cut-ofkd {Cm} {D %pi (ofkd C K) %-> (ofkd C' K')} {d} %. +%mode cut-ofkd %in %in %in %in %out %out %. +%sort cut-kd-wf {Cm} {D %pi (ofkd C K) %-> (kd-wf K')} {d} %. +%mode cut-kd-wf %in %in %in %in %out %out %. +%sort cut-kd-sub {Cm} {D %pi (ofkd C K) %-> (kd-sub K' K'')} {d} %. +%mode cut-kd-sub %in %in %in %in %out %out %. +%sort cut-kd-deq {Cm} {D %pi (ofkd C K) %-> (kd-deq K' K'')} {d} %. +%mode cut-kd-deq %in %in %in %in %out %out %. +%sort cut-cn-deq {Cm} {D %pi (ofkd C K) %-> (cn-deq C' C'' K')} {d} %. +%mode cut-cn-deq %in %in %in %in %out %out %. +%sort cut-ekd-wf {Cm} {D %pi (ofkd C K) %-> (ekd-wf G K')} {d} %. +%mode cut-ekd-wf %in %in %in %in %out %out %. +%sort cut-eofkd {Cm} {D %pi (ofkd C K) %-> (eofkd G C' K')} {d} %. +%mode cut-eofkd %in %in %in %in %out %out %. +%sort cut-ekd-sub {Cm} {D %pi (ofkd C K) %-> (ekd-sub G K' K'')} {d} %. +%mode cut-ekd-sub %in %in %in %in %out %out %. +%sort cut-ekd-deq {Cm} {D %pi (ofkd C K) %-> (ekd-deq G K' K'')} {d} %. +%mode cut-ekd-deq %in %in %in %in %out %out %. +%sort cut-ecn-deq {Cm} {D %pi (ofkd C K) %-> (ecn-deq G C' C'' K')} {d} %. +%mode cut-ecn-deq %in %in %in %in %out %out %. +%term _ + %pi (cut-kd-wf _ ([d] kd-wf/kd/unit) ([d'] [dm] mkd-wf/kd/unit) DL (ekd-wf/kd/unit DO) mekd-wf/kd/unit) + %<- (cxt-lookup-ordered DL DO) %. +%term _ + %pi (cut-kd-wf _ ([d] kd-wf/kd/type) ([d'] [dm] mkd-wf/kd/type) DL (ekd-wf/kd/type DO) mekd-wf/kd/type) + %<- (cxt-lookup-ordered DL DO) %. +%term _ + %pi (cut-kd-wf (met/sing Mm) ([d] kd-wf/kd/sing (D1 d)) ([d'] [dm] mkd-wf/kd/sing (D2 d' dm)) DL (ekd-wf/kd/sing D1') (mekd-wf/kd/sing D2')) + %<- (cut-ofkd Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-kd-wf (met/pair M1 M2) ([d] kd-wf/kd/sgm (D1 d) (D2 d)) ([d'] [dm] mkd-wf/kd/sgm (D3 d' dm) (D4 d' dm)) DL (ekd-wf/kd/sgm D1' D2'') (mekd-wf/kd/sgm D3' D4'')) + %<- (cxt-lookup-ordered DL DO) + %<- (cxt-extend-ordered DO DB) + %<- (cut-kd-wf M1 D1 D3 DL D1' D3') + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (%the (cxt-lookup (cxt/cons G y B) X A) (DL' y e'))) + %<- ({d ofkd X A} {dm mofkd d met/unit} {y} {e' isvar y J} cut-kd-wf M2 ([e] D2 d y e) ([e] [em] D4 d dm y e em) (cxt-lookup/hit (DB y e')) (%the (ekd-wf (cxt/cons G y B) (C y)) (D2' d y e')) (%the (mekd-wf (D2' d y e') M2) (D4' d dm y e'))) + %<- ({y} {e' isvar y J} cut-ekd-wf M2 ([d] D2' d y e') ([d] [dm] D4' d dm y e') (DL' y e') (D2'' y e') (D4'' y e')) %. +%term _ + %pi (cut-kd-wf (met/pair M1 M2) ([d] kd-wf/kd/pi (D1 d) (D2 d)) ([d'] [dm] mkd-wf/kd/pi (D3 d' dm) (D4 d' dm)) DL (ekd-wf/kd/pi D1' D2'') (mekd-wf/kd/pi D3' D4'')) + %<- (cxt-lookup-ordered DL DO) + %<- (cxt-extend-ordered DO DB) + %<- (cut-kd-wf M1 D1 D3 DL D1' D3') + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (%the (cxt-lookup (cxt/cons G y B) X A) (DL' y e'))) + %<- ({d ofkd X A} {dm mofkd d met/unit} {y} {e' isvar y J} cut-kd-wf M2 ([e] D2 d y e) ([e] [em] D4 d dm y e em) (cxt-lookup/hit (DB y e')) (%the (ekd-wf (cxt/cons G y B) (C y)) (D2' d y e')) (%the (mekd-wf (D2' d y e') M2) (D4' d dm y e'))) + %<- ({y} {e' isvar y J} cut-ekd-wf M2 ([d] D2' d y e') ([d] [dm] D4' d dm y e') (DL' y e') (D2'' y e') (D4'' y e')) %. +%term _ + %pi (cut-ofkd _ ([d] ofkd/cn/unit) ([d'] [dm] mofkd/cn/unit) DL (eofkd/cn/unit DO) meofkd/cn/unit) + %<- (cxt-lookup-ordered DL DO) %. +%term _ + %pi (cut-ofkd _ ([d] ofkd/tp/unit) ([d'] [dm] mofkd/tp/unit) DL (eofkd/tp/unit DO) meofkd/tp/unit) + %<- (cxt-lookup-ordered DL DO) %. +%term _ + %pi (cut-ofkd _ ([d] ofkd/tp/tagged) ([d'] [dm] mofkd/tp/tagged) DL (eofkd/tp/tagged DO) meofkd/tp/tagged) + %<- (cxt-lookup-ordered DL DO) %. +%term _ + %pi (cut-ofkd (met/sing Mm) ([d] ofkd/tp/ref (D1 d)) ([d'] [dm] mofkd/tp/ref (D2 d' dm)) DL (eofkd/tp/ref D1') (meofkd/tp/ref D2')) + %<- (cut-ofkd Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-ofkd (met/sing Mm) ([d] ofkd/tp/tag (D1 d)) ([d'] [dm] mofkd/tp/tag (D2 d' dm)) DL (eofkd/tp/tag D1') (meofkd/tp/tag D2')) + %<- (cut-ofkd Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-ofkd (met/sing Mm) ([d] ofkd/kd/sing (D1 d)) ([d'] [dm] mofkd/kd/sing (D2 d' dm)) DL (eofkd/kd/sing D1') (meofkd/kd/sing D2')) + %<- (cut-ofkd Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-ofkd (met/sing Mm) ([d] ofkd/cn/pj1 (D1 d)) ([d'] [dm] mofkd/cn/pj1 (D2 d' dm)) DL (eofkd/cn/pj1 D1') (meofkd/cn/pj1 D2')) + %<- (cut-ofkd Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-ofkd (met/sing Mm) ([d] ofkd/cn/pj2 (D1 d)) ([d'] [dm] mofkd/cn/pj2 (D2 d' dm)) DL (eofkd/cn/pj2 D1') (meofkd/cn/pj2 D2')) + %<- (cut-ofkd Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-ofkd (met/pair M1 M2) ([d] ofkd/cn/pair (D1 d) (D3 d)) ([d'] [dm] mofkd/cn/pair (D2 d' dm) (D4 d' dm)) DL (eofkd/cn/pair D1' D3') (meofkd/cn/pair D2' D4')) + %<- (cut-ofkd M1 D1 D2 DL D1' D2') + %<- (cut-ofkd M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-ofkd (met/pair M1 M2) ([d] ofkd/tp/cross (D1 d) (D3 d)) ([d'] [dm] mofkd/tp/cross (D2 d' dm) (D4 d' dm)) DL (eofkd/tp/cross D1' D3') (meofkd/tp/cross D2' D4')) + %<- (cut-ofkd M1 D1 D2 DL D1' D2') + %<- (cut-ofkd M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-ofkd (met/pair M1 M2) ([d] ofkd/tp/arrow (D1 d) (D3 d)) ([d'] [dm] mofkd/tp/arrow (D2 d' dm) (D4 d' dm)) DL (eofkd/tp/arrow D1' D3') (meofkd/tp/arrow D2' D4')) + %<- (cut-ofkd M1 D1 D2 DL D1' D2') + %<- (cut-ofkd M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-ofkd (met/pair M1 M2) ([d] ofkd/tp/sum (D1 d) (D3 d)) ([d'] [dm] mofkd/tp/sum (D2 d' dm) (D4 d' dm)) DL (eofkd/tp/sum D1' D3') (meofkd/tp/sum D2' D4')) + %<- (cut-ofkd M1 D1 D2 DL D1' D2') + %<- (cut-ofkd M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-ofkd (met/pair M1 M2) ([d] ofkd/cn/app (D1 d) (D3 d)) ([d'] [dm] mofkd/cn/app (D2 d' dm) (D4 d' dm)) DL (eofkd/cn/app D1' D3') (meofkd/cn/app D2' D4')) + %<- (cut-ofkd M1 D1 D2 DL D1' D2') + %<- (cut-ofkd M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-ofkd (met/pair M1 M2) ([d] ofkd/sgm-ext (D1 d) (D3 d)) ([d'] [dm] mofkd/sgm-ext (D2 d' dm) (D4 d' dm)) DL (eofkd/sgm-ext D1' D3') (meofkd/sgm-ext D2' D4')) + %<- (cut-ofkd M1 D1 D2 DL D1' D2') + %<- (cut-ofkd M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-ofkd (met/pair M1 M2) ([d] ofkd/sub (D1 d) (D3 d)) ([d'] [dm] mofkd/sub (D2 d' dm) (D4 d' dm)) DL (eofkd/sub D1' D3') (meofkd/sub D2' D4')) + %<- (cut-ofkd M1 D1 D2 DL D1' D2') + %<- (cut-kd-sub M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-ofkd (met/pair M1 M2) ([d] ofkd/deq (D1 d) (D3 d)) ([d'] [dm] mofkd/deq (D2 d' dm) (D4 d' dm)) DL (eofkd/deq D1' D3') (meofkd/deq D2' D4')) + %<- (cut-ofkd M1 D1 D2 DL D1' D2') + %<- (cut-kd-deq M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-ofkd (met/pair M1 M2) ([d] ofkd/cn/lam (D1 d) (D2 d)) ([d'] [dm] mofkd/cn/lam (D3 d' dm) (D4 d' dm)) DL (eofkd/cn/lam D1'' D2') (meofkd/cn/lam D3'' D4')) + %<- (cxt-lookup-ordered DL DO) + %<- (cxt-extend-ordered DO DB) + %<- (cut-kd-wf M2 D2 D4 DL D2' D4') + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (%the (cxt-lookup (cxt/cons G y B) X A) (DL' y e'))) + %<- ({d ofkd X A} {dm mofkd d met/unit} {y} {e' isvar y J} cut-ofkd M1 ([e] D1 d y e) ([e] [em] D3 d dm y e em) (cxt-lookup/hit (DB y e')) (D1' d y e') (D3' d dm y e')) + %<- ({y} {e' isvar y J} cut-eofkd M1 ([d] D1' d y e') ([d] [dm] D3' d dm y e') (DL' y e') (D1'' y e') (D3'' y e')) %. +%term _ + %pi (cut-ofkd (met/pair M1 M2) ([d] ofkd/tp/forall (D1 d) (D2 d)) ([d'] [dm] mofkd/tp/forall (D3 d' dm) (D4 d' dm)) DL (eofkd/tp/forall D1'' D2') (meofkd/tp/forall D3'' D4')) + %<- (cxt-lookup-ordered DL DO) + %<- (cxt-extend-ordered DO DB) + %<- (cut-kd-wf M2 D2 D4 DL D2' D4') + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (%the (cxt-lookup (cxt/cons G y B) X A) (DL' y e'))) + %<- ({d ofkd X A} {dm mofkd d met/unit} {y} {e' isvar y J} cut-ofkd M1 ([e] D1 d y e) ([e] [em] D3 d dm y e em) (cxt-lookup/hit (DB y e')) (D1' d y e') (D3' d dm y e')) + %<- ({y} {e' isvar y J} cut-eofkd M1 ([d] D1' d y e') ([d] [dm] D3' d dm y e') (DL' y e') (D1'' y e') (D3'' y e')) %. +%term _ + %pi (cut-ofkd (met/pair M1 M2) ([d] ofkd/cn/mu (D1 d) (D2 d)) ([d'] [dm] mofkd/cn/mu (D3 d' dm) (D4 d' dm)) DL (eofkd/cn/mu D1'' D2') (meofkd/cn/mu D3'' D4')) + %<- (cxt-lookup-ordered DL DO) + %<- (cxt-extend-ordered DO DB) + %<- (cut-kd-wf M2 D2 D4 DL D2' D4') + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (DL' y e')) + %<- ({d ofkd X A} {dm mofkd d met/unit} {y} {e' isvar y J} cut-ofkd M1 ([e] D1 d y e) ([e] [em] D3 d dm y e em) (cxt-lookup/hit (DB y e')) (D1' d y e') (D3' d dm y e')) + %<- ({y} {e' isvar y J} cut-eofkd M1 ([d] D1' d y e') ([d] [dm] D3' d dm y e') (DL' y e') (D1'' y e') (D3'' y e')) %. +%term _ + %pi (cut-ofkd (met/pair M1 M2) ([d] ofkd/pi-ext (D1 d) (D2 d)) ([d'] [dm] mofkd/pi-ext (D3 d' dm) (D4 d' dm)) DL (eofkd/pi-ext D1' D2'') (meofkd/pi-ext D3' D4'')) + %<- (cxt-lookup-ordered DL DO) + %<- (cxt-extend-ordered DO DB) + %<- (cut-ofkd M1 D1 D3 DL D1' D3') + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (DL' y e')) + %<- ({d ofkd X A} {dm mofkd d met/unit} {y} {e' isvar y J} cut-ofkd M2 ([e] D2 d y e) ([e] [em] D4 d dm y e em) (cxt-lookup/hit (DB y e')) (D2' d y e') (D4' d dm y e')) + %<- ({y} {e' isvar y J} cut-eofkd M2 ([d] D2' d y e') ([d] [dm] D4' d dm y e') (DL' y e') (D2'' y e') (D4'' y e')) %. +%term _ + %pi (cut-ofkd met/unit ([d] D) ([d'] [dm] DM) DL (eofkd/closed D DO) (meofkd/closed DM)) + %<- (cxt-lookup-ordered DL DO) %. +%term _ cut-ofkd met/unit ([d] d) ([d] [dm] dm) DL (eofkd/var DL) meofkd/var %. +%term _ + %pi (cut-kd-deq _ ([d] kd-deq/kd/unit) ([d'] [dm] mkd-deq/kd/unit) DL (ekd-deq/kd/unit DO) mekd-deq/kd/unit) + %<- (cxt-lookup-ordered DL DO) %. +%term _ + %pi (cut-kd-deq _ ([d] kd-deq/kd/type) ([d'] [dm] mkd-deq/kd/type) DL (ekd-deq/kd/type DO) mekd-deq/kd/type) + %<- (cxt-lookup-ordered DL DO) %. +%term _ + %pi (cut-kd-deq (met/sing Mm) ([d] kd-deq/kd/sing (D1 d)) ([d'] [dm] mkd-deq/kd/sing (D2 d' dm)) DL (ekd-deq/kd/sing D1') (mekd-deq/kd/sing D2')) + %<- (cut-cn-deq Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-kd-deq (met/pair M1 M2) ([d] kd-deq/kd/sgm (D1 d) (D2 d)) ([d'] [dm] mkd-deq/kd/sgm (D3 d' dm) (D4 d' dm)) DL (ekd-deq/kd/sgm D1' D2'') (mekd-deq/kd/sgm D3' D4'')) + %<- (cxt-lookup-ordered DL DO) + %<- (cxt-extend-ordered DO DB) + %<- (cut-kd-deq M1 D1 D3 DL D1' D3') + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (DL' y e')) + %<- ({d ofkd X A} {dm mofkd d met/unit} {y} {e' isvar y J} cut-kd-deq M2 ([e] D2 d y e) ([e] [em] D4 d dm y e em) (cxt-lookup/hit (DB y e')) (D2' d y e') (D4' d dm y e')) + %<- ({y} {e' isvar y J} cut-ekd-deq M2 ([d] D2' d y e') ([d] [dm] D4' d dm y e') (DL' y e') (D2'' y e') (D4'' y e')) %. +%term _ + %pi (cut-kd-deq (met/pair M1 M2) ([d] kd-deq/kd/pi (D1 d) (D2 d)) ([d'] [dm] mkd-deq/kd/pi (D3 d' dm) (D4 d' dm)) DL (ekd-deq/kd/pi D1' D2'') (mekd-deq/kd/pi D3' D4'')) + %<- (cxt-lookup-ordered DL DO) + %<- (cxt-extend-ordered DO DB) + %<- (cut-kd-deq M1 D1 D3 DL D1' D3') + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (DL' y e')) + %<- ({d ofkd X A} {dm mofkd d met/unit} {y} {e' isvar y J} cut-kd-deq M2 ([e] D2 d y e) ([e] [em] D4 d dm y e em) (cxt-lookup/hit (DB y e')) (D2' d y e') (D4' d dm y e')) + %<- ({y} {e' isvar y J} cut-ekd-deq M2 ([d] D2' d y e') ([d] [dm] D4' d dm y e') (DL' y e') (D2'' y e') (D4'' y e')) %. +%term _ + %pi (cut-kd-sub _ ([d] kd-sub/kd/unit) ([d'] [dm] mkd-sub/kd/unit) DL (ekd-sub/kd/unit DO) mekd-sub/kd/unit) + %<- (cxt-lookup-ordered DL DO) %. +%term _ + %pi (cut-kd-sub _ ([d] kd-sub/kd/type) ([d'] [dm] mkd-sub/kd/type) DL (ekd-sub/kd/type DO) mekd-sub/kd/type) + %<- (cxt-lookup-ordered DL DO) %. +%term _ + %pi (cut-kd-sub (met/sing Mm) ([d] kd-sub/kd/sing-kd/sing (D1 d)) ([d'] [dm] mkd-sub/kd/sing-kd/sing (D2 d' dm)) DL (ekd-sub/kd/sing-kd/sing D1') (mekd-sub/kd/sing-kd/sing D2')) + %<- (cut-cn-deq Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-kd-sub (met/sing Mm) ([d] kd-sub/kd/sing-kd/type (D1 d)) ([d'] [dm] mkd-sub/kd/sing-kd/type (D2 d' dm)) DL (ekd-sub/kd/sing-kd/type D1') (mekd-sub/kd/sing-kd/type D2')) + %<- (cut-ofkd Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-kd-sub (met/pair M1 (met/pair M2 M3)) ([d] kd-sub/kd/sgm (D1 d) (D2 d) (D5 d)) ([d'] [dm] mkd-sub/kd/sgm (D3 d' dm) (D4 d' dm) (D6 d' dm)) DL (ekd-sub/kd/sgm D1' D2'' D5'') (mekd-sub/kd/sgm D3' D4'' D6'')) + %<- (cxt-lookup-ordered DL DO) + %<- (cxt-extend-ordered DO DB) + %<- (cut-kd-sub M1 D1 D3 DL D1' D3') + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (DL' y e')) + %<- ({d ofkd _ _} {dm mofkd d met/unit} {y} {e' isvar y J} cut-kd-sub M2 ([e] D2 d y e) ([e] [em] D4 d dm y e em) (cxt-lookup/hit (DB y e')) (D2' d y e') (D4' d dm y e')) + %<- ({y} {e' isvar y J} cut-ekd-sub M2 ([d] D2' d y e') ([d] [dm] D4' d dm y e') (DL' y e') (D2'' y e') (D4'' y e')) + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (DL'' y e')) + %<- ({d ofkd _ _} {dm mofkd d met/unit} {y} {e' isvar y J} cut-kd-wf M3 ([e] D5 d y e) ([e] [em] D6 d dm y e em) (cxt-lookup/hit (DB y e')) (D5' d y e') (D6' d dm y e')) + %<- ({y} {e' isvar y J} cut-ekd-wf M3 ([d] D5' d y e') ([d] [dm] D6' d dm y e') (DL'' y e') (D5'' y e') (D6'' y e')) %. +%term _ + %pi (cut-kd-sub (met/pair M1 (met/pair M2 M3)) ([d] kd-sub/kd/pi (D1 d) (D2 d) (D5 d)) ([d'] [dm] mkd-sub/kd/pi (D3 d' dm) (D4 d' dm) (D6 d' dm)) DL (ekd-sub/kd/pi D1' D2'' D5'') (mekd-sub/kd/pi D3' D4'' D6'')) + %<- (cxt-lookup-ordered DL DO) + %<- (cxt-extend-ordered DO DB) + %<- (cut-kd-sub M1 D1 D3 DL D1' D3') + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (DL' y e')) + %<- ({d ofkd _ _} {dm mofkd d met/unit} {y} {e' isvar y J} cut-kd-sub M2 ([e] D2 d y e) ([e] [em] D4 d dm y e em) (cxt-lookup/hit (DB y e')) (D2' d y e') (D4' d dm y e')) + %<- ({y} {e' isvar y J} cut-ekd-sub M2 ([d] D2' d y e') ([d] [dm] D4' d dm y e') (DL' y e') (D2'' y e') (D4'' y e')) + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (DL'' y e')) + %<- ({d ofkd _ _} {dm mofkd d met/unit} {y} {e' isvar y J} cut-kd-wf M3 ([e] D5 d y e) ([e] [em] D6 d dm y e em) (cxt-lookup/hit (DB y e')) (D5' d y e') (D6' d dm y e')) + %<- ({y} {e' isvar y J} cut-ekd-wf M3 ([d] D5' d y e') ([d] [dm] D6' d dm y e') (DL'' y e') (D5'' y e') (D6'' y e')) %. +%term _ + %pi (cut-cn-deq _ ([d] cn-deq/cn/unit) ([d'] [dm] mcn-deq/cn/unit) DL (ecn-deq/cn/unit DO) mecn-deq/cn/unit) + %<- (cxt-lookup-ordered DL DO) %. +%term _ + %pi (cut-cn-deq _ ([d] cn-deq/tp/unit) ([d'] [dm] mcn-deq/tp/unit) DL (ecn-deq/tp/unit DO) mecn-deq/tp/unit) + %<- (cxt-lookup-ordered DL DO) %. +%term _ + %pi (cut-cn-deq _ ([d] cn-deq/tp/tagged) ([d'] [dm] mcn-deq/tp/tagged) DL (ecn-deq/tp/tagged DO) mecn-deq/tp/tagged) + %<- (cxt-lookup-ordered DL DO) %. +%term _ + %pi (cut-cn-deq (met/sing Mm) ([d] cn-deq/tp/ref (D1 d)) ([d'] [dm] mcn-deq/tp/ref (D2 d' dm)) DL (ecn-deq/tp/ref D1') (mecn-deq/tp/ref D2')) + %<- (cut-cn-deq Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-cn-deq (met/sing Mm) ([d] cn-deq/tp/tag (D1 d)) ([d'] [dm] mcn-deq/tp/tag (D2 d' dm)) DL (ecn-deq/tp/tag D1') (mecn-deq/tp/tag D2')) + %<- (cut-cn-deq Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-cn-deq (met/pair Mm Mm') ([d] cn-deq/kd/unit (D1 d) (D3 d)) ([d'] [dm] mcn-deq/kd/unit (D2 d' dm) (D4 d' dm)) DL (ecn-deq/kd/unit D1' D3') (mecn-deq/kd/unit D2' D4')) + %<- (cut-ofkd Mm D1 D2 DL D1' D2') + %<- (cut-ofkd Mm' D3 D4 DL D3' D4') %. +%term _ + %pi (cut-cn-deq (met/sing Mm) ([d] cn-deq/kd/sing (D1 d)) ([d'] [dm] mcn-deq/kd/sing (D2 d' dm)) DL (ecn-deq/kd/sing D1') (mecn-deq/kd/sing D2')) + %<- (cut-ofkd Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-cn-deq (met/sing Mm) ([d] cn-deq/refl (D1 d)) ([d'] [dm] mcn-deq/refl (D2 d' dm)) DL (ecn-deq/refl D1') (mecn-deq/refl D2')) + %<- (cut-ofkd Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-cn-deq (met/sing Mm) ([d] cn-deq/sym (D1 d)) ([d'] [dm] mcn-deq/sym (D2 d' dm)) DL (ecn-deq/sym D1') (mecn-deq/sym D2')) + %<- (cut-cn-deq Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-cn-deq (met/sing Mm) ([d] cn-deq/cn/pj1 (D1 d)) ([d'] [dm] mcn-deq/cn/pj1 (D2 d' dm)) DL (ecn-deq/cn/pj1 D1') (mecn-deq/cn/pj1 D2')) + %<- (cut-cn-deq Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-cn-deq (met/sing Mm) ([d] cn-deq/cn/pj2 (D1 d)) ([d'] [dm] mcn-deq/cn/pj2 (D2 d' dm)) DL (ecn-deq/cn/pj2 D1') (mecn-deq/cn/pj2 D2')) + %<- (cut-cn-deq Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-cn-deq (met/pair M1 M2) ([d] cn-deq/trans (D1 d) (D3 d)) ([d'] [dm] mcn-deq/trans (D2 d' dm) (D4 d' dm)) DL (ecn-deq/trans D1' D3') (mecn-deq/trans D2' D4')) + %<- (cut-cn-deq M1 D1 D2 DL D1' D2') + %<- (cut-cn-deq M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-cn-deq (met/pair M1 M2) ([d] cn-deq/cn/pair (D1 d) (D3 d)) ([d'] [dm] mcn-deq/cn/pair (D2 d' dm) (D4 d' dm)) DL (ecn-deq/cn/pair D1' D3') (mecn-deq/cn/pair D2' D4')) + %<- (cut-cn-deq M1 D1 D2 DL D1' D2') + %<- (cut-cn-deq M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-cn-deq (met/pair M1 M2) ([d] cn-deq/tp/cross (D1 d) (D3 d)) ([d'] [dm] mcn-deq/tp/cross (D2 d' dm) (D4 d' dm)) DL (ecn-deq/tp/cross D1' D3') (mecn-deq/tp/cross D2' D4')) + %<- (cut-cn-deq M1 D1 D2 DL D1' D2') + %<- (cut-cn-deq M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-cn-deq (met/pair M1 M2) ([d] cn-deq/tp/arrow (D1 d) (D3 d)) ([d'] [dm] mcn-deq/tp/arrow (D2 d' dm) (D4 d' dm)) DL (ecn-deq/tp/arrow D1' D3') (mecn-deq/tp/arrow D2' D4')) + %<- (cut-cn-deq M1 D1 D2 DL D1' D2') + %<- (cut-cn-deq M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-cn-deq (met/pair M1 M2) ([d] cn-deq/tp/sum (D1 d) (D3 d)) ([d'] [dm] mcn-deq/tp/sum (D2 d' dm) (D4 d' dm)) DL (ecn-deq/tp/sum D1' D3') (mecn-deq/tp/sum D2' D4')) + %<- (cut-cn-deq M1 D1 D2 DL D1' D2') + %<- (cut-cn-deq M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-cn-deq (met/pair M1 M2) ([d] cn-deq/cn/app (D1 d) (D3 d)) ([d'] [dm] mcn-deq/cn/app (D2 d' dm) (D4 d' dm)) DL (ecn-deq/cn/app D1' D3') (mecn-deq/cn/app D2' D4')) + %<- (cut-cn-deq M1 D1 D2 DL D1' D2') + %<- (cut-cn-deq M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-cn-deq (met/pair M1 M2) ([d] cn-deq/sgm-ext (D1 d) (D3 d)) ([d'] [dm] mcn-deq/sgm-ext (D2 d' dm) (D4 d' dm)) DL (ecn-deq/sgm-ext D1' D3') (mecn-deq/sgm-ext D2' D4')) + %<- (cut-cn-deq M1 D1 D2 DL D1' D2') + %<- (cut-cn-deq M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-cn-deq (met/pair M1 M2) ([d] cn-deq/sub (D1 d) (D3 d)) ([d'] [dm] mcn-deq/sub (D2 d' dm) (D4 d' dm)) DL (ecn-deq/sub D1' D3') (mecn-deq/sub D2' D4')) + %<- (cut-cn-deq M1 D1 D2 DL D1' D2') + %<- (cut-kd-sub M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-cn-deq (met/pair M1 M2) ([d] cn-deq/deq (D1 d) (D3 d)) ([d'] [dm] mcn-deq/deq (D2 d' dm) (D4 d' dm)) DL (ecn-deq/deq D1' D3') (mecn-deq/deq D2' D4')) + %<- (cut-cn-deq M1 D1 D2 DL D1' D2') + %<- (cut-kd-deq M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-cn-deq (met/pair M1 M2) ([d] cn-deq/cn/lam (D1 d) (D2 d)) ([d'] [dm] mcn-deq/cn/lam (D3 d' dm) (D4 d' dm)) DL (ecn-deq/cn/lam D1'' D2') (mecn-deq/cn/lam D3'' D4')) + %<- (cxt-lookup-ordered DL DO) + %<- (cxt-extend-ordered DO DB) + %<- (cut-kd-deq M2 D2 D4 DL D2' D4') + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (%the (cxt-lookup (cxt/cons G y B) X A) (DL' y e'))) + %<- ({d ofkd X A} {dm mofkd d met/unit} {y} {e' isvar y J} cut-cn-deq M1 ([e] D1 d y e) ([e] [em] D3 d dm y e em) (cxt-lookup/hit (DB y e')) (D1' d y e') (D3' d dm y e')) + %<- ({y} {e' isvar y J} cut-ecn-deq M1 ([d] D1' d y e') ([d] [dm] D3' d dm y e') (DL' y e') (D1'' y e') (D3'' y e')) %. +%term _ + %pi (cut-cn-deq (met/pair M1 M2) ([d] cn-deq/tp/forall (D1 d) (D2 d)) ([d'] [dm] mcn-deq/tp/forall (D3 d' dm) (D4 d' dm)) DL (ecn-deq/tp/forall D1'' D2') (mecn-deq/tp/forall D3'' D4')) + %<- (cxt-lookup-ordered DL DO) + %<- (cxt-extend-ordered DO DB) + %<- (cut-kd-deq M2 D2 D4 DL D2' D4') + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (%the (cxt-lookup (cxt/cons G y B) X A) (DL' y e'))) + %<- ({d ofkd X A} {dm mofkd d met/unit} {y} {e' isvar y J} cut-cn-deq M1 ([e] D1 d y e) ([e] [em] D3 d dm y e em) (cxt-lookup/hit (DB y e')) (D1' d y e') (D3' d dm y e')) + %<- ({y} {e' isvar y J} cut-ecn-deq M1 ([d] D1' d y e') ([d] [dm] D3' d dm y e') (DL' y e') (D1'' y e') (D3'' y e')) %. +%term _ + %pi (cut-cn-deq (met/pair M1 M2) ([d] cn-deq/cn/mu (D1 d) (D2 d)) ([d'] [dm] mcn-deq/cn/mu (D3 d' dm) (D4 d' dm)) DL (ecn-deq/cn/mu D1'' D2') (mecn-deq/cn/mu D3'' D4')) + %<- (cxt-lookup-ordered DL DO) + %<- (cxt-extend-ordered DO DB) + %<- (cut-kd-deq M2 D2 D4 DL D2' D4') + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (DL' y e')) + %<- ({d ofkd X A} {dm mofkd d met/unit} {y} {e' isvar y J} cut-cn-deq M1 ([e] D1 d y e) ([e] [em] D3 d dm y e em) (cxt-lookup/hit (DB y e')) (D1' d y e') (D3' d dm y e')) + %<- ({y} {e' isvar y J} cut-ecn-deq M1 ([d] D1' d y e') ([d] [dm] D3' d dm y e') (DL' y e') (D1'' y e') (D3'' y e')) %. +%term _ + %pi (cut-cn-deq (met/pair M1 (met/pair M2 M3)) ([d] cn-deq/pi-ext (D1 d) (D2 d) (D5 d)) ([d'] [dm] mcn-deq/pi-ext (D3 d' dm) (D4 d' dm) (D6 d' dm)) DL (ecn-deq/pi-ext D1' D2' D5'') (mecn-deq/pi-ext D3' D4' D6'')) + %<- (cxt-lookup-ordered DL DO) + %<- (cxt-extend-ordered DO DB) + %<- (cut-ofkd M1 D1 D3 DL D1' D3') + %<- (cut-ofkd M2 D2 D4 DL D2' D4') + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (DL' y e')) + %<- ({d ofkd X A} {dm mofkd d met/unit} {y} {e' isvar y J} cut-cn-deq M3 ([e] D5 d y e) ([e] [em] D6 d dm y e em) (cxt-lookup/hit (DB y e')) (D5' d y e') (D6' d dm y e')) + %<- ({y} {e' isvar y J} cut-ecn-deq M3 ([d] D5' d y e') ([d] [dm] D6' d dm y e') (DL' y e') (D5'' y e') (D6'' y e')) %. +%term _ + %pi (cut-cn-deq (met/pair M1 M3) ([d] cn-deq/pi-ext-2 (D1 d) (D5 d)) ([d'] [dm] mcn-deq/pi-ext-2 (D3 d' dm) (D6 d' dm)) DL (ecn-deq/pi-ext-2 D1' D5'') (mecn-deq/pi-ext-2 D3' D6'')) + %<- (cxt-lookup-ordered DL DO) + %<- (cxt-extend-ordered DO DB) + %<- (cut-cn-deq M1 D1 D3 DL D1' D3') + %<- ({y} {e' isvar y J} cxt-weaken-lookup (DB y e') DL _ (DL' y e')) + %<- ({d ofkd X A} {dm mofkd d met/unit} {y} {e' isvar y J} cut-cn-deq M3 ([e] D5 d y e) ([e] [em] D6 d dm y e em) (cxt-lookup/hit (DB y e')) (D5' d y e') (D6' d dm y e')) + %<- ({y} {e' isvar y J} cut-ecn-deq M3 ([d] D5' d y e') ([d] [dm] D6' d dm y e') (DL' y e') (D5'' y e') (D6'' y e')) %. +%term _ cut-ekd-wf met/unit ([d] ekd-wf/kd/unit DO) ([d] [dm] mekd-wf/kd/unit) DL (ekd-wf/kd/unit DO) mekd-wf/kd/unit %. +%term _ cut-ekd-wf met/unit ([d] ekd-wf/kd/type DO) ([d] [dm] mekd-wf/kd/type) DL (ekd-wf/kd/type DO) mekd-wf/kd/type %. +%term _ + %pi (cut-ekd-wf (met/sing Mm) ([d] ekd-wf/kd/sing (D1 d)) ([d] [dm] mekd-wf/kd/sing (D2 d dm)) DL (ekd-wf/kd/sing D1') (mekd-wf/kd/sing D2')) + %<- (cut-eofkd Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-ekd-wf (met/pair M1 M2) ([d] ekd-wf/kd/sgm (D1 d) (D2 d)) ([d] [dm] mekd-wf/kd/sgm (D3 d dm) (D4 d dm)) DL (ekd-wf/kd/sgm D1' D2') (mekd-wf/kd/sgm D3' D4')) + %<- (cut-ekd-wf M1 D1 D3 DL D1' D3') + %<- ({d ofkd _ _} {y} {e isvar y J} ekd-wf-ordered (D2 d y e) (cxt-ordered/cons (DB y e))) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL' y e)) + %<- ({y} {e isvar y J} cut-ekd-wf M2 ([d] D2 d y e) ([d] [dm] D4 d dm y e) (DL' y e) (D2' y e) (D4' y e)) %. +%term _ + %pi (cut-ekd-wf (met/pair M1 M2) ([d] ekd-wf/kd/pi (D1 d) (D2 d)) ([d] [dm] mekd-wf/kd/pi (D3 d dm) (D4 d dm)) DL (ekd-wf/kd/pi D1' D2') (mekd-wf/kd/pi D3' D4')) + %<- (cut-ekd-wf M1 D1 D3 DL D1' D3') + %<- ({d ofkd _ _} {y} {e isvar y J} ekd-wf-ordered (D2 d y e) (cxt-ordered/cons (DB y e))) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL' y e)) + %<- ({y} {e isvar y J} cut-ekd-wf M2 ([d] D2 d y e) ([d] [dm] D4 d dm y e) (DL' y e) (D2' y e) (D4' y e)) %. +%term _ cut-ekd-deq met/unit ([d] ekd-deq/kd/unit DO) ([d] [dm] mekd-deq/kd/unit) DL (ekd-deq/kd/unit DO) mekd-deq/kd/unit %. +%term _ cut-ekd-deq met/unit ([d] ekd-deq/kd/type DO) ([d] [dm] mekd-deq/kd/type) DL (ekd-deq/kd/type DO) mekd-deq/kd/type %. +%term _ + %pi (cut-ekd-deq (met/sing Mm) ([d] ekd-deq/kd/sing (D1 d)) ([d] [dm] mekd-deq/kd/sing (D2 d dm)) DL (ekd-deq/kd/sing D1') (mekd-deq/kd/sing D2')) + %<- (cut-ecn-deq Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-ekd-deq (met/pair M1 M2) ([d] ekd-deq/kd/sgm (D1 d) (D2 d)) ([d] [dm] mekd-deq/kd/sgm (D3 d dm) (D4 d dm)) DL (ekd-deq/kd/sgm D1' D2') (mekd-deq/kd/sgm D3' D4')) + %<- (cut-ekd-deq M1 D1 D3 DL D1' D3') + %<- ({d ofkd _ _} {y} {e isvar y J} ekd-deq-ordered (D2 d y e) (cxt-ordered/cons (DB y e))) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL' y e)) + %<- ({y} {e isvar y J} cut-ekd-deq M2 ([d] D2 d y e) ([d] [dm] D4 d dm y e) (DL' y e) (D2' y e) (D4' y e)) %. +%term _ + %pi (cut-ekd-deq (met/pair M1 M2) ([d] ekd-deq/kd/pi (D1 d) (D2 d)) ([d] [dm] mekd-deq/kd/pi (D3 d dm) (D4 d dm)) DL (ekd-deq/kd/pi D1' D2') (mekd-deq/kd/pi D3' D4')) + %<- (cut-ekd-deq M1 D1 D3 DL D1' D3') + %<- ({d ofkd _ _} {y} {e isvar y J} ekd-deq-ordered (D2 d y e) (cxt-ordered/cons (DB y e))) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL' y e)) + %<- ({y} {e isvar y J} cut-ekd-deq M2 ([d] D2 d y e) ([d] [dm] D4 d dm y e) (DL' y e) (D2' y e) (D4' y e)) %. +%term _ cut-ekd-sub met/unit ([d] ekd-sub/kd/unit DO) ([d] [dm] mekd-sub/kd/unit) DL (ekd-sub/kd/unit DO) mekd-sub/kd/unit %. +%term _ cut-ekd-sub met/unit ([d] ekd-sub/kd/type DO) ([d] [dm] mekd-sub/kd/type) DL (ekd-sub/kd/type DO) mekd-sub/kd/type %. +%term _ + %pi (cut-ekd-sub (met/sing Mm) ([d] ekd-sub/kd/sing-kd/sing (D1 d)) ([d] [dm] mekd-sub/kd/sing-kd/sing (D2 d dm)) DL (ekd-sub/kd/sing-kd/sing D1') (mekd-sub/kd/sing-kd/sing D2')) + %<- (cut-ecn-deq Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-ekd-sub (met/sing Mm) ([d] ekd-sub/kd/sing-kd/type (D1 d)) ([d] [dm] mekd-sub/kd/sing-kd/type (D2 d dm)) DL (ekd-sub/kd/sing-kd/type D1') (mekd-sub/kd/sing-kd/type D2')) + %<- (cut-eofkd Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-ekd-sub (met/pair M1 (met/pair M2 M3)) ([d] ekd-sub/kd/pi (D1 d) (D2 d) (D5 d)) ([d] [dm] mekd-sub/kd/pi (D3 d dm) (D4 d dm) (D6 d dm)) DL (ekd-sub/kd/pi D1' D2' D5') (mekd-sub/kd/pi D3' D4' D6')) + %<- (cut-ekd-sub M1 D1 D3 DL D1' D3') + %<- ({d ofkd _ _} {y} {e isvar y J} ekd-sub-ordered (D2 d y e) (cxt-ordered/cons (DB y e))) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL' y e)) + %<- ({y} {e isvar y J} cut-ekd-sub M2 ([d] D2 d y e) ([d] [dm] D4 d dm y e) (DL' y e) (D2' y e) (D4' y e)) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL'' y e)) + %<- ({y} {e isvar y J} cut-ekd-wf M3 ([d] D5 d y e) ([d] [dm] D6 d dm y e) (DL'' y e) (D5' y e) (D6' y e)) %. +%term _ + %pi (cut-ekd-sub (met/pair M1 (met/pair M2 M3)) ([d] ekd-sub/kd/sgm (D1 d) (D2 d) (D5 d)) ([d] [dm] mekd-sub/kd/sgm (D3 d dm) (D4 d dm) (D6 d dm)) DL (ekd-sub/kd/sgm D1' D2' D5') (mekd-sub/kd/sgm D3' D4' D6')) + %<- (cut-ekd-sub M1 D1 D3 DL D1' D3') + %<- ({d ofkd _ _} {y} {e isvar y J} ekd-sub-ordered (D2 d y e) (cxt-ordered/cons (DB y e))) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL' y e)) + %<- ({y} {e isvar y J} cut-ekd-sub M2 ([d] D2 d y e) ([d] [dm] D4 d dm y e) (DL' y e) (D2' y e) (D4' y e)) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL'' y e)) + %<- ({y} {e isvar y J} cut-ekd-wf M3 ([d] D5 d y e) ([d] [dm] D6 d dm y e) (DL'' y e) (D5' y e) (D6' y e)) %. +%term _ cut-eofkd met/unit ([d] eofkd/cn/unit DO) ([d] [dm] meofkd/cn/unit) DL (eofkd/cn/unit DO) meofkd/cn/unit %. +%term _ cut-eofkd met/unit ([d] eofkd/tp/unit DO) ([d] [dm] meofkd/tp/unit) DL (eofkd/tp/unit DO) meofkd/tp/unit %. +%term _ cut-eofkd met/unit ([d] eofkd/tp/tagged DO) ([d] [dm] meofkd/tp/tagged) DL (eofkd/tp/tagged DO) meofkd/tp/tagged %. +%term _ + %pi (cut-eofkd (met/sing Mm) ([d] eofkd/kd/sing (D1 d)) ([d] [dm] meofkd/kd/sing (D2 d dm)) DL (eofkd/kd/sing D1') (meofkd/kd/sing D2')) + %<- (cut-eofkd Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-eofkd (met/sing Mm) ([d] eofkd/tp/ref (D1 d)) ([d] [dm] meofkd/tp/ref (D2 d dm)) DL (eofkd/tp/ref D1') (meofkd/tp/ref D2')) + %<- (cut-eofkd Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-eofkd (met/sing Mm) ([d] eofkd/tp/tag (D1 d)) ([d] [dm] meofkd/tp/tag (D2 d dm)) DL (eofkd/tp/tag D1') (meofkd/tp/tag D2')) + %<- (cut-eofkd Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-eofkd (met/sing Mm) ([d] eofkd/cn/pj1 (D1 d)) ([d] [dm] meofkd/cn/pj1 (D2 d dm)) DL (eofkd/cn/pj1 D1') (meofkd/cn/pj1 D2')) + %<- (cut-eofkd Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-eofkd (met/sing Mm) ([d] eofkd/cn/pj2 (D1 d)) ([d] [dm] meofkd/cn/pj2 (D2 d dm)) DL (eofkd/cn/pj2 D1') (meofkd/cn/pj2 D2')) + %<- (cut-eofkd Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-eofkd (met/pair M1 M2) ([d] eofkd/cn/pair (D1 d) (D3 d)) ([d] [dm] meofkd/cn/pair (D2 d dm) (D4 d dm)) DL (eofkd/cn/pair D1' D3') (meofkd/cn/pair D2' D4')) + %<- (cut-eofkd M1 D1 D2 DL D1' D2') + %<- (cut-eofkd M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-eofkd (met/pair M1 M2) ([d] eofkd/tp/cross (D1 d) (D3 d)) ([d] [dm] meofkd/tp/cross (D2 d dm) (D4 d dm)) DL (eofkd/tp/cross D1' D3') (meofkd/tp/cross D2' D4')) + %<- (cut-eofkd M1 D1 D2 DL D1' D2') + %<- (cut-eofkd M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-eofkd (met/pair M1 M2) ([d] eofkd/cn/app (D1 d) (D3 d)) ([d] [dm] meofkd/cn/app (D2 d dm) (D4 d dm)) DL (eofkd/cn/app D1' D3') (meofkd/cn/app D2' D4')) + %<- (cut-eofkd M1 D1 D2 DL D1' D2') + %<- (cut-eofkd M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-eofkd (met/pair M1 M2) ([d] eofkd/sgm-ext (D1 d) (D3 d)) ([d] [dm] meofkd/sgm-ext (D2 d dm) (D4 d dm)) DL (eofkd/sgm-ext D1' D3') (meofkd/sgm-ext D2' D4')) + %<- (cut-eofkd M1 D1 D2 DL D1' D2') + %<- (cut-eofkd M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-eofkd (met/pair M1 M2) ([d] eofkd/tp/arrow (D1 d) (D3 d)) ([d] [dm] meofkd/tp/arrow (D2 d dm) (D4 d dm)) DL (eofkd/tp/arrow D1' D3') (meofkd/tp/arrow D2' D4')) + %<- (cut-eofkd M1 D1 D2 DL D1' D2') + %<- (cut-eofkd M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-eofkd (met/pair M1 M2) ([d] eofkd/tp/sum (D1 d) (D3 d)) ([d] [dm] meofkd/tp/sum (D2 d dm) (D4 d dm)) DL (eofkd/tp/sum D1' D3') (meofkd/tp/sum D2' D4')) + %<- (cut-eofkd M1 D1 D2 DL D1' D2') + %<- (cut-eofkd M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-eofkd (met/pair M1 M2) ([d] eofkd/sub (D1 d) (D3 d)) ([d] [dm] meofkd/sub (D2 d dm) (D4 d dm)) DL (eofkd/sub D1' D3') (meofkd/sub D2' D4')) + %<- (cut-eofkd M1 D1 D2 DL D1' D2') + %<- (cut-ekd-sub M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-eofkd (met/pair M1 M2) ([d] eofkd/deq (D1 d) (D3 d)) ([d] [dm] meofkd/deq (D2 d dm) (D4 d dm)) DL (eofkd/deq D1' D3') (meofkd/deq D2' D4')) + %<- (cut-eofkd M1 D1 D2 DL D1' D2') + %<- (cut-ekd-deq M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-eofkd (met/pair M1 M2) ([d] eofkd/cn/lam (D1 d) (D2 d)) ([d] [dm] meofkd/cn/lam (D3 d dm) (D4 d dm)) DL (eofkd/cn/lam D1' D2') (meofkd/cn/lam D3' D4')) + %<- (cut-ekd-wf M2 D2 D4 DL D2' D4') + %<- ({d ofkd _ _} {y} {e isvar y J} eofkd-ordered (D1 d y e) (cxt-ordered/cons (DB y e))) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL' y e)) + %<- ({y} {e isvar y J} cut-eofkd M1 ([d] D1 d y e) ([d] [dm] D3 d dm y e) (DL' y e) (D1' y e) (D3' y e)) %. +%term _ + %pi (cut-eofkd (met/pair M1 M2) ([d] eofkd/tp/forall (D1 d) (D2 d)) ([d] [dm] meofkd/tp/forall (D3 d dm) (D4 d dm)) DL (eofkd/tp/forall D1' D2') (meofkd/tp/forall D3' D4')) + %<- (cut-ekd-wf M2 D2 D4 DL D2' D4') + %<- ({d ofkd _ _} {y} {e isvar y J} eofkd-ordered (D1 d y e) (cxt-ordered/cons (DB y e))) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL' y e)) + %<- ({y} {e isvar y J} cut-eofkd M1 ([d] D1 d y e) ([d] [dm] D3 d dm y e) (DL' y e) (D1' y e) (D3' y e)) %. +%term _ + %pi (cut-eofkd (met/pair M1 M2) ([d] eofkd/cn/mu (D1 d) (D2 d)) ([d] [dm] meofkd/cn/mu (D3 d dm) (D4 d dm)) DL (eofkd/cn/mu D1' D2') (meofkd/cn/mu D3' D4')) + %<- (cut-ekd-wf M2 D2 D4 DL D2' D4') + %<- ({d ofkd _ _} {y} {e isvar y J} eofkd-ordered (D1 d y e) (cxt-ordered/cons (DB y e))) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL' y e)) + %<- ({y} {e isvar y J} cut-eofkd M1 ([d] D1 d y e) ([d] [dm] D3 d dm y e) (DL' y e) (D1' y e) (D3' y e)) %. +%term _ + %pi (cut-eofkd (met/pair M1 M2) ([d] eofkd/pi-ext (D1 d) (D2 d)) ([d] [dm] meofkd/pi-ext (D3 d dm) (D4 d dm)) DL (eofkd/pi-ext D1' D2') (meofkd/pi-ext D3' D4')) + %<- (cut-eofkd M1 D1 D3 DL D1' D3') + %<- ({d ofkd _ _} {y} {e isvar y J} eofkd-ordered (D2 d y e) (cxt-ordered/cons (DB y e))) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL' y e)) + %<- ({y} {e isvar y J} cut-eofkd M2 ([d] D2 d y e) ([d] [dm] D4 d dm y e) (DL' y e) (D2' y e) (D4' y e)) %. +%term _ cut-eofkd met/unit ([d] eofkd/var DL) ([d] [dm] meofkd/var) _ (eofkd/var DL) meofkd/var %. +%term _ + %pi (cut-eofkd Mm ([d] eofkd/closed (D1 d) DO) ([d] [dm] meofkd/closed (D2 d dm)) DL D1' D2') + %<- (cut-ofkd Mm D1 D2 DL D1' D2') %. +%term _ cut-ecn-deq met/unit ([d] ecn-deq/cn/unit DO) ([d] [dm] mecn-deq/cn/unit) DL (ecn-deq/cn/unit DO) mecn-deq/cn/unit %. +%term _ cut-ecn-deq met/unit ([d] ecn-deq/tp/unit DO) ([d] [dm] mecn-deq/tp/unit) DL (ecn-deq/tp/unit DO) mecn-deq/tp/unit %. +%term _ cut-ecn-deq met/unit ([d] ecn-deq/tp/tagged DO) ([d] [dm] mecn-deq/tp/tagged) DL (ecn-deq/tp/tagged DO) mecn-deq/tp/tagged %. +%term _ + %pi (cut-ecn-deq (met/sing Mm) ([d] ecn-deq/kd/sing (D1 d)) ([d] [dm] mecn-deq/kd/sing (D2 d dm)) DL (ecn-deq/kd/sing D1') (mecn-deq/kd/sing D2')) + %<- (cut-eofkd Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-ecn-deq (met/sing Mm) ([d] ecn-deq/refl (D1 d)) ([d] [dm] mecn-deq/refl (D2 d dm)) DL (ecn-deq/refl D1') (mecn-deq/refl D2')) + %<- (cut-eofkd Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-ecn-deq (met/sing Mm) ([d] ecn-deq/sym (D1 d)) ([d] [dm] mecn-deq/sym (D2 d dm)) DL (ecn-deq/sym D1') (mecn-deq/sym D2')) + %<- (cut-ecn-deq Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-ecn-deq (met/sing Mm) ([d] ecn-deq/tp/ref (D1 d)) ([d] [dm] mecn-deq/tp/ref (D2 d dm)) DL (ecn-deq/tp/ref D1') (mecn-deq/tp/ref D2')) + %<- (cut-ecn-deq Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-ecn-deq (met/sing Mm) ([d] ecn-deq/tp/tag (D1 d)) ([d] [dm] mecn-deq/tp/tag (D2 d dm)) DL (ecn-deq/tp/tag D1') (mecn-deq/tp/tag D2')) + %<- (cut-ecn-deq Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-ecn-deq (met/sing Mm) ([d] ecn-deq/cn/pj1 (D1 d)) ([d] [dm] mecn-deq/cn/pj1 (D2 d dm)) DL (ecn-deq/cn/pj1 D1') (mecn-deq/cn/pj1 D2')) + %<- (cut-ecn-deq Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-ecn-deq (met/sing Mm) ([d] ecn-deq/cn/pj2 (D1 d)) ([d] [dm] mecn-deq/cn/pj2 (D2 d dm)) DL (ecn-deq/cn/pj2 D1') (mecn-deq/cn/pj2 D2')) + %<- (cut-ecn-deq Mm D1 D2 DL D1' D2') %. +%term _ + %pi (cut-ecn-deq (met/pair M1 M2) ([d] ecn-deq/kd/unit (D1 d) (D3 d)) ([d] [dm] mecn-deq/kd/unit (D2 d dm) (D4 d dm)) DL (ecn-deq/kd/unit D1' D3') (mecn-deq/kd/unit D2' D4')) + %<- (cut-eofkd M1 D1 D2 DL D1' D2') + %<- (cut-eofkd M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-ecn-deq (met/pair M1 M2) ([d] ecn-deq/trans (D1 d) (D3 d)) ([d] [dm] mecn-deq/trans (D2 d dm) (D4 d dm)) DL (ecn-deq/trans D1' D3') (mecn-deq/trans D2' D4')) + %<- (cut-ecn-deq M1 D1 D2 DL D1' D2') + %<- (cut-ecn-deq M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-ecn-deq (met/pair M1 M2) ([d] ecn-deq/cn/pair (D1 d) (D3 d)) ([d] [dm] mecn-deq/cn/pair (D2 d dm) (D4 d dm)) DL (ecn-deq/cn/pair D1' D3') (mecn-deq/cn/pair D2' D4')) + %<- (cut-ecn-deq M1 D1 D2 DL D1' D2') + %<- (cut-ecn-deq M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-ecn-deq (met/pair M1 M2) ([d] ecn-deq/tp/cross (D1 d) (D3 d)) ([d] [dm] mecn-deq/tp/cross (D2 d dm) (D4 d dm)) DL (ecn-deq/tp/cross D1' D3') (mecn-deq/tp/cross D2' D4')) + %<- (cut-ecn-deq M1 D1 D2 DL D1' D2') + %<- (cut-ecn-deq M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-ecn-deq (met/pair M1 M2) ([d] ecn-deq/cn/app (D1 d) (D3 d)) ([d] [dm] mecn-deq/cn/app (D2 d dm) (D4 d dm)) DL (ecn-deq/cn/app D1' D3') (mecn-deq/cn/app D2' D4')) + %<- (cut-ecn-deq M1 D1 D2 DL D1' D2') + %<- (cut-ecn-deq M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-ecn-deq (met/pair M1 M2) ([d] ecn-deq/sgm-ext (D1 d) (D3 d)) ([d] [dm] mecn-deq/sgm-ext (D2 d dm) (D4 d dm)) DL (ecn-deq/sgm-ext D1' D3') (mecn-deq/sgm-ext D2' D4')) + %<- (cut-ecn-deq M1 D1 D2 DL D1' D2') + %<- (cut-ecn-deq M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-ecn-deq (met/pair M1 M2) ([d] ecn-deq/tp/arrow (D1 d) (D3 d)) ([d] [dm] mecn-deq/tp/arrow (D2 d dm) (D4 d dm)) DL (ecn-deq/tp/arrow D1' D3') (mecn-deq/tp/arrow D2' D4')) + %<- (cut-ecn-deq M1 D1 D2 DL D1' D2') + %<- (cut-ecn-deq M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-ecn-deq (met/pair M1 M2) ([d] ecn-deq/tp/sum (D1 d) (D3 d)) ([d] [dm] mecn-deq/tp/sum (D2 d dm) (D4 d dm)) DL (ecn-deq/tp/sum D1' D3') (mecn-deq/tp/sum D2' D4')) + %<- (cut-ecn-deq M1 D1 D2 DL D1' D2') + %<- (cut-ecn-deq M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-ecn-deq (met/pair M1 M2) ([d] ecn-deq/sub (D1 d) (D3 d)) ([d] [dm] mecn-deq/sub (D2 d dm) (D4 d dm)) DL (ecn-deq/sub D1' D3') (mecn-deq/sub D2' D4')) + %<- (cut-ecn-deq M1 D1 D2 DL D1' D2') + %<- (cut-ekd-sub M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-ecn-deq (met/pair M1 M2) ([d] ecn-deq/deq (D1 d) (D3 d)) ([d] [dm] mecn-deq/deq (D2 d dm) (D4 d dm)) DL (ecn-deq/deq D1' D3') (mecn-deq/deq D2' D4')) + %<- (cut-ecn-deq M1 D1 D2 DL D1' D2') + %<- (cut-ekd-deq M2 D3 D4 DL D3' D4') %. +%term _ + %pi (cut-ecn-deq (met/pair M1 M2) ([d] ecn-deq/cn/lam (D1 d) (D2 d)) ([d] [dm] mecn-deq/cn/lam (D3 d dm) (D4 d dm)) DL (ecn-deq/cn/lam D1' D2') (mecn-deq/cn/lam D3' D4')) + %<- (cut-ekd-deq M2 D2 D4 DL D2' D4') + %<- ({d ofkd _ _} {y} {e isvar y J} ecn-deq-ordered (D1 d y e) (cxt-ordered/cons (DB y e))) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL' y e)) + %<- ({y} {e isvar y J} cut-ecn-deq M1 ([d] D1 d y e) ([d] [dm] D3 d dm y e) (DL' y e) (D1' y e) (D3' y e)) %. +%term _ + %pi (cut-ecn-deq (met/pair M1 M2) ([d] ecn-deq/tp/forall (D1 d) (D2 d)) ([d] [dm] mecn-deq/tp/forall (D3 d dm) (D4 d dm)) DL (ecn-deq/tp/forall D1' D2') (mecn-deq/tp/forall D3' D4')) + %<- (cut-ekd-deq M2 D2 D4 DL D2' D4') + %<- ({d ofkd _ _} {y} {e isvar y J} ecn-deq-ordered (D1 d y e) (cxt-ordered/cons (DB y e))) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL' y e)) + %<- ({y} {e isvar y J} cut-ecn-deq M1 ([d] D1 d y e) ([d] [dm] D3 d dm y e) (DL' y e) (D1' y e) (D3' y e)) %. +%term _ + %pi (cut-ecn-deq (met/pair M1 M2) ([d] ecn-deq/cn/mu (D1 d) (D2 d)) ([d] [dm] mecn-deq/cn/mu (D3 d dm) (D4 d dm)) DL (ecn-deq/cn/mu D1' D2') (mecn-deq/cn/mu D3' D4')) + %<- (cut-ekd-deq M2 D2 D4 DL D2' D4') + %<- ({d ofkd _ _} {y} {e isvar y J} ecn-deq-ordered (D1 d y e) (cxt-ordered/cons (DB y e))) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL' y e)) + %<- ({y} {e isvar y J} cut-ecn-deq M1 ([d] D1 d y e) ([d] [dm] D3 d dm y e) (DL' y e) (D1' y e) (D3' y e)) %. +%term _ + %pi (cut-ecn-deq (met/pair M1 (met/pair M2 M3)) ([d] ecn-deq/pi-ext (D1 d) (D2 d) (D5 d)) ([d] [dm] mecn-deq/pi-ext (D3 d dm) (D4 d dm) (D6 d dm)) DL (ecn-deq/pi-ext D1' D2' D5') (mecn-deq/pi-ext D3' D4' D6')) + %<- (cut-eofkd M1 D1 D3 DL D1' D3') + %<- (cut-eofkd M2 D2 D4 DL D2' D4') + %<- ({d ofkd _ _} {y} {e isvar y J} ecn-deq-ordered (D5 d y e) (cxt-ordered/cons (DB y e))) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL' y e)) + %<- ({y} {e isvar y J} cut-ecn-deq M3 ([d] D5 d y e) ([d] [dm] D6 d dm y e) (DL' y e) (D5' y e) (D6' y e)) %. +%term _ + %pi (cut-ecn-deq (met/pair M1 M3) ([d] ecn-deq/pi-ext-2 (D1 d) (D5 d)) ([d] [dm] mecn-deq/pi-ext-2 (D3 d dm) (D6 d dm)) DL (ecn-deq/pi-ext-2 D1' D5') (mecn-deq/pi-ext-2 D3' D6')) + %<- (cut-ecn-deq M1 D1 D3 DL D1' D3') + %<- ({d ofkd _ _} {y} {e isvar y J} ecn-deq-ordered (D5 d y e) (cxt-ordered/cons (DB y e))) + %<- ({y} {e isvar y J} cxt-weaken-lookup (DB y e) DL _ (DL' y e)) + %<- ({y} {e isvar y J} cut-ecn-deq M3 ([d] D5 d y e) ([d] [dm] D6 d dm y e) (DL' y e) (D5' y e) (D6' y e)) %. +%worlds (can-mofkd-block ofkd-block ovar-block bind-block wbind-block) (cut-ofkd _ _ _ _ _ _) (cut-kd-wf _ _ _ _ _ _) (cut-kd-sub _ _ _ _ _ _) (cut-kd-deq _ _ _ _ _ _) (cut-ekd-wf _ _ _ _ _ _) (cut-eofkd _ _ _ _ _ _) (cut-cn-deq _ _ _ _ _ _) (cut-ecn-deq _ _ _ _ _ _) (cut-ekd-deq _ _ _ _ _ _) (cut-ekd-sub _ _ _ _ _ _) %. +%total (M1 M2 M3 M4 M5 M6 M7 M8 M9 M10) (cut-kd-wf M2 _ _ _ _ _) (cut-ofkd M1 _ _ _ _ _) (cut-kd-sub M4 _ _ _ _ _) (cut-kd-deq M9 _ _ _ _ _) (cut-cn-deq M6 _ _ _ _ _) (cut-ekd-wf M3 _ _ _ _ _) (cut-ekd-deq M10 _ _ _ _ _) (cut-ekd-sub M7 _ _ _ _ _) (cut-eofkd M5 _ _ _ _ _) (cut-ecn-deq M8 _ _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/explicate-sigs.lf b/new-tests/stelf-output/tslf/explicate-sigs.lf new file mode 100644 index 0000000..ebdd603 --- /dev/null +++ b/new-tests/stelf-output/tslf/explicate-sigs.lf @@ -0,0 +1,55 @@ +%%%% explicate +%sort explicate-open-sg-wf {I loc} {x} %. +%mode explicate-open-sg-wf %in %in %out %. +%term _ + %pi (explicate-open-sg-wf I Dof Dofe) + %<- ({x} {d ofkd x A} {dm mofkd d met/unit} {_ can-mofkd d dm} can-msg-wf (Dof x d) (%the (msg-wf _ Mm) (Dmof x d dm))) + %<- ({x} {d' isvar x I} cut-sg-wf Mm ([d] Dof x d) ([d] [dm] Dmof x d dm) (cxt-lookup/hit (cxt-bounded/nil d')) (Dofe x d') _) %. +%worlds (can-mofkd-block ovar-block) (explicate-open-sg-wf _ _ _) %. +%total {} (explicate-open-sg-wf _ _ _) %. +%sort explicate-open-sg-deq {I loc} {x} %. +%mode explicate-open-sg-deq %in %in %out %. +%term _ + %pi (explicate-open-sg-deq I Dof Dofe) + %<- ({x} {d ofkd x A} {dm mofkd d met/unit} {_ can-mofkd d dm} can-msg-deq (Dof x d) (%the (msg-deq _ Mm) (Dmof x d dm))) + %<- ({x} {d' isvar x I} cut-sg-deq Mm ([d] Dof x d) ([d] [dm] Dmof x d dm) (cxt-lookup/hit (cxt-bounded/nil d')) (Dofe x d') _) %. +%worlds (can-mofkd-block ovar-block) (explicate-open-sg-deq _ _ _) %. +%total {} (explicate-open-sg-deq _ _ _) %. +%sort explicate-closed-sg-wf %. +%mode explicate-closed-sg-wf %in %out %. +%term _ explicate-closed-sg-wf sg-wf/sg/unit (esg-wf/sg/unit cxt-ordered/nil) %. +%term _ + %pi (explicate-closed-sg-wf (sg-wf/sg/cn D1) (esg-wf/sg/cn D1')) + %<- (explicate-closed-ofkd D1 D1') %. +%term _ + %pi (explicate-closed-sg-wf (sg-wf/sg/kd D1) (esg-wf/sg/kd D1')) + %<- (explicate-closed-kd-wf D1 D1') %. +%term _ + %pi (explicate-closed-sg-wf (sg-wf/sg/sgm D1 D2 DFS) (esg-wf/sg/sgm D1' D2' DFS)) + %<- (explicate-closed-sg-wf D1 D1') + %<- (explicate-open-sg-wf loc/z D2 D2') %. +%term _ + %pi (explicate-closed-sg-wf (sg-wf/sg/pi D1 D2 DFS) (esg-wf/sg/pi D1' D2' DFS)) + %<- (explicate-closed-sg-wf D1 D1') + %<- (explicate-open-sg-wf loc/z D2 D2') %. +%worlds (can-mofkd-block) (explicate-closed-sg-wf _ _) %. +%total {D1} (explicate-closed-sg-wf D1 _) %. +%sort explicate-closed-sg-deq %. +%mode explicate-closed-sg-deq %in %out %. +%term _ explicate-closed-sg-deq sg-deq/sg/unit (esg-deq/sg/unit cxt-ordered/nil) %. +%term _ + %pi (explicate-closed-sg-deq (sg-deq/sg/cn D1) (esg-deq/sg/cn D1')) + %<- (explicate-closed-cn-deq D1 D1') %. +%term _ + %pi (explicate-closed-sg-deq (sg-deq/sg/kd D1) (esg-deq/sg/kd D1')) + %<- (explicate-closed-kd-deq D1 D1') %. +%term _ + %pi (explicate-closed-sg-deq (sg-deq/sg/sgm D1 D2 DFS) (esg-deq/sg/sgm D1' D2' DFS)) + %<- (explicate-closed-sg-deq D1 D1') + %<- (explicate-open-sg-deq loc/z D2 D2') %. +%term _ + %pi (explicate-closed-sg-deq (sg-deq/sg/pi D1 D2 DFS) (esg-deq/sg/pi D1' D2' DFS)) + %<- (explicate-closed-sg-deq D1 D1') + %<- (explicate-open-sg-deq loc/z D2 D2') %. +%worlds (can-mofkd-block) (explicate-closed-sg-deq _ _) %. +%total (D1) (explicate-closed-sg-deq D1 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/explicate.lf b/new-tests/stelf-output/tslf/explicate.lf new file mode 100644 index 0000000..261130a --- /dev/null +++ b/new-tests/stelf-output/tslf/explicate.lf @@ -0,0 +1,187 @@ +%%%% explicate +%sort explicate-open-ofkd {I loc} {x} %. +%mode explicate-open-ofkd %in %in %out %. +%term _ + %pi (explicate-open-ofkd I Dof Dofe) + %<- ({x} {d ofkd x A} {dm mofkd d met/unit} {_ can-mofkd d dm} can-mofkd (Dof x d) (%the (mofkd _ Mm) (Dmof x d dm))) + %<- ({x} {d' isvar x I} cut-ofkd Mm ([d] Dof x d) ([d] [dm] Dmof x d dm) (cxt-lookup/hit (cxt-bounded/nil d')) (Dofe x d') _) %. +%worlds (can-mofkd-block ovar-block) (explicate-open-ofkd _ _ _) %. +%total {} (explicate-open-ofkd _ _ _) %. +%sort explicate-open-kd-wf {I loc} {x} %. +%mode explicate-open-kd-wf %in %in %out %. +%term _ + %pi (explicate-open-kd-wf I Dof Dofe) + %<- ({x} {d ofkd x A} {dm mofkd d met/unit} {_ can-mofkd d dm} can-mkd-wf (Dof x d) (%the (mkd-wf _ Mm) (Dmof x d dm))) + %<- ({x} {d' isvar x I} cut-kd-wf Mm ([d] Dof x d) ([d] [dm] Dmof x d dm) (cxt-lookup/hit (cxt-bounded/nil d')) (Dofe x d') _) %. +%worlds (can-mofkd-block ovar-block) (explicate-open-kd-wf _ _ _) %. +%total {} (explicate-open-kd-wf _ _ _) %. +%sort explicate-open-kd-deq {I loc} {x} %. +%mode explicate-open-kd-deq %in %in %out %. +%term _ + %pi (explicate-open-kd-deq I Dof Dofe) + %<- ({x} {d ofkd x A} {dm mofkd d met/unit} {_ can-mofkd d dm} can-mkd-deq (Dof x d) (%the (mkd-deq _ Mm) (Dmof x d dm))) + %<- ({x} {d' isvar x I} cut-kd-deq Mm ([d] Dof x d) ([d] [dm] Dmof x d dm) (cxt-lookup/hit (cxt-bounded/nil d')) (Dofe x d') _) %. +%worlds (can-mofkd-block ovar-block) (explicate-open-kd-deq _ _ _) %. +%total {} (explicate-open-kd-deq _ _ _) %. +%sort explicate-open-kd-sub {I loc} {x} %. +%mode explicate-open-kd-sub %in %in %out %. +%term _ + %pi (explicate-open-kd-sub I Dof Dofe) + %<- ({x} {d ofkd x A} {dm mofkd d met/unit} {_ can-mofkd d dm} can-mkd-sub (Dof x d) (%the (mkd-sub _ Mm) (Dmof x d dm))) + %<- ({x} {d' isvar x I} cut-kd-sub Mm ([d] Dof x d) ([d] [dm] Dmof x d dm) (cxt-lookup/hit (cxt-bounded/nil d')) (Dofe x d') _) %. +%worlds (can-mofkd-block ovar-block) (explicate-open-kd-sub _ _ _) %. +%total {} (explicate-open-kd-sub _ _ _) %. +%sort explicate-open-cn-deq {I loc} {x} %. +%mode explicate-open-cn-deq %in %in %out %. +%term _ + %pi (explicate-open-cn-deq I Dof Dofe) + %<- ({x} {d ofkd x A} {dm mofkd d met/unit} {_ can-mofkd d dm} can-mcn-deq (Dof x d) (%the (mcn-deq _ Mm) (Dmof x d dm))) + %<- ({x} {d' isvar x I} cut-cn-deq Mm ([d] Dof x d) ([d] [dm] Dmof x d dm) (cxt-lookup/hit (cxt-bounded/nil d')) (Dofe x d') _) %. +%worlds (can-mofkd-block ovar-block) (explicate-open-cn-deq _ _ _) %. +%total {} (explicate-open-cn-deq _ _ _) %. +%sort explicate-closed-ofkd %. +%mode explicate-closed-ofkd %in %out %. +%term _ explicate-closed-ofkd D1 (eofkd/closed D1 cxt-ordered/nil) %. +%worlds (can-mofkd-block) (explicate-closed-ofkd _ _) %. +%total {} (explicate-closed-ofkd _ _) %. +%sort explicate-closed-kd-wf %. +%mode explicate-closed-kd-wf %in %out %. +%term _ explicate-closed-kd-wf kd-wf/kd/unit (ekd-wf/kd/unit cxt-ordered/nil) %. +%term _ explicate-closed-kd-wf kd-wf/kd/type (ekd-wf/kd/type cxt-ordered/nil) %. +%term _ + %pi (explicate-closed-kd-wf (kd-wf/kd/sing D1) (ekd-wf/kd/sing D1')) + %<- (explicate-closed-ofkd D1 D1') %. +%term _ + %pi (explicate-closed-kd-wf (kd-wf/kd/sgm D1 D2) (ekd-wf/kd/sgm D1' D2')) + %<- (explicate-closed-kd-wf D1 D1') + %<- (explicate-open-kd-wf loc/z D2 D2') %. +%term _ + %pi (explicate-closed-kd-wf (kd-wf/kd/pi D1 D2) (ekd-wf/kd/pi D1' D2')) + %<- (explicate-closed-kd-wf D1 D1') + %<- (explicate-open-kd-wf loc/z D2 D2') %. +%worlds (can-mofkd-block) (explicate-closed-kd-wf _ _) %. +%total {D1} (explicate-closed-kd-wf D1 _) %. +%sort explicate-closed-kd-sub %. +%mode explicate-closed-kd-sub %in %out %. +%sort explicate-closed-kd-deq %. +%mode explicate-closed-kd-deq %in %out %. +%sort explicate-closed-cn-deq %. +%mode explicate-closed-cn-deq %in %out %. +%term _ explicate-closed-kd-sub kd-sub/kd/unit (ekd-sub/kd/unit cxt-ordered/nil) %. +%term _ explicate-closed-kd-sub kd-sub/kd/type (ekd-sub/kd/type cxt-ordered/nil) %. +%term _ + %pi (explicate-closed-kd-sub (kd-sub/kd/sing-kd/sing D1) (ekd-sub/kd/sing-kd/sing D1')) + %<- (explicate-closed-cn-deq D1 D1') %. +%term _ + %pi (explicate-closed-kd-sub (kd-sub/kd/sing-kd/type D1) (ekd-sub/kd/sing-kd/type D1')) + %<- (explicate-closed-ofkd D1 D1') %. +%term _ + %pi (explicate-closed-kd-sub (kd-sub/kd/sgm D1 D2 D3) (ekd-sub/kd/sgm D1' D2' D3')) + %<- (explicate-closed-kd-sub D1 D1') + %<- (explicate-open-kd-sub loc/z D2 D2') + %<- (explicate-open-kd-wf loc/z D3 D3') %. +%term _ + %pi (explicate-closed-kd-sub (kd-sub/kd/pi D1 D2 D3) (ekd-sub/kd/pi D1' D2' D3')) + %<- (explicate-closed-kd-sub D1 D1') + %<- (explicate-open-kd-sub loc/z D2 D2') + %<- (explicate-open-kd-wf loc/z D3 D3') %. +%term _ explicate-closed-kd-deq kd-deq/kd/unit (ekd-deq/kd/unit cxt-ordered/nil) %. +%term _ explicate-closed-kd-deq kd-deq/kd/type (ekd-deq/kd/type cxt-ordered/nil) %. +%term _ + %pi (explicate-closed-kd-deq (kd-deq/kd/sing D1) (ekd-deq/kd/sing D1')) + %<- (explicate-closed-cn-deq D1 D1') %. +%term _ + %pi (explicate-closed-kd-deq (kd-deq/kd/sgm D1 D2) (ekd-deq/kd/sgm D1' D2')) + %<- (explicate-closed-kd-deq D1 D1') + %<- (explicate-open-kd-deq loc/z D2 D2') %. +%term _ + %pi (explicate-closed-kd-deq (kd-deq/kd/pi D1 D2) (ekd-deq/kd/pi D1' D2')) + %<- (explicate-closed-kd-deq D1 D1') + %<- (explicate-open-kd-deq loc/z D2 D2') %. +%term _ explicate-closed-cn-deq cn-deq/cn/unit (ecn-deq/cn/unit cxt-ordered/nil) %. +%term _ explicate-closed-cn-deq cn-deq/tp/unit (ecn-deq/tp/unit cxt-ordered/nil) %. +%term _ explicate-closed-cn-deq cn-deq/tp/tagged (ecn-deq/tp/tagged cxt-ordered/nil) %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/kd/sing D1) (ecn-deq/kd/sing D1')) + %<- (explicate-closed-ofkd D1 D1') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/refl D1) (ecn-deq/refl D1')) + %<- (explicate-closed-ofkd D1 D1') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/sym D1) (ecn-deq/sym D1')) + %<- (explicate-closed-cn-deq D1 D1') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/tp/ref D1) (ecn-deq/tp/ref D1')) + %<- (explicate-closed-cn-deq D1 D1') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/tp/tag D1) (ecn-deq/tp/tag D1')) + %<- (explicate-closed-cn-deq D1 D1') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/cn/pj1 D1) (ecn-deq/cn/pj1 D1')) + %<- (explicate-closed-cn-deq D1 D1') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/cn/pj2 D1) (ecn-deq/cn/pj2 D1')) + %<- (explicate-closed-cn-deq D1 D1') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/trans D1 D2) (ecn-deq/trans D1' D2')) + %<- (explicate-closed-cn-deq D1 D1') + %<- (explicate-closed-cn-deq D2 D2') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/tp/cross D1 D2) (ecn-deq/tp/cross D1' D2')) + %<- (explicate-closed-cn-deq D1 D1') + %<- (explicate-closed-cn-deq D2 D2') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/tp/arrow D1 D2) (ecn-deq/tp/arrow D1' D2')) + %<- (explicate-closed-cn-deq D1 D1') + %<- (explicate-closed-cn-deq D2 D2') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/tp/sum D1 D2) (ecn-deq/tp/sum D1' D2')) + %<- (explicate-closed-cn-deq D1 D1') + %<- (explicate-closed-cn-deq D2 D2') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/cn/pair D1 D2) (ecn-deq/cn/pair D1' D2')) + %<- (explicate-closed-cn-deq D1 D1') + %<- (explicate-closed-cn-deq D2 D2') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/cn/app D1 D2) (ecn-deq/cn/app D1' D2')) + %<- (explicate-closed-cn-deq D1 D1') + %<- (explicate-closed-cn-deq D2 D2') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/sgm-ext D1 D2) (ecn-deq/sgm-ext D1' D2')) + %<- (explicate-closed-cn-deq D1 D1') + %<- (explicate-closed-cn-deq D2 D2') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/sub D1 D2) (ecn-deq/sub D1' D2')) + %<- (explicate-closed-cn-deq D1 D1') + %<- (explicate-closed-kd-sub D2 D2') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/deq D1 D2) (ecn-deq/deq D1' D2')) + %<- (explicate-closed-cn-deq D1 D1') + %<- (explicate-closed-kd-deq D2 D2') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/kd/unit D1 D2) (ecn-deq/kd/unit D1' D2')) + %<- (explicate-closed-ofkd D1 D1') + %<- (explicate-closed-ofkd D2 D2') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/cn/lam D1 D2) (ecn-deq/cn/lam D1' D2')) + %<- (explicate-open-cn-deq loc/z D1 D1') + %<- (explicate-closed-kd-deq D2 D2') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/tp/forall D1 D2) (ecn-deq/tp/forall D1' D2')) + %<- (explicate-open-cn-deq loc/z D1 D1') + %<- (explicate-closed-kd-deq D2 D2') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/cn/mu D1 D2) (ecn-deq/cn/mu D1' D2')) + %<- (explicate-open-cn-deq loc/z D1 D1') + %<- (explicate-closed-kd-deq D2 D2') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/pi-ext D1 D2 D3) (ecn-deq/pi-ext D1' D2' D3')) + %<- (explicate-closed-ofkd D1 D1') + %<- (explicate-closed-ofkd D2 D2') + %<- (explicate-open-cn-deq loc/z D3 D3') %. +%term _ + %pi (explicate-closed-cn-deq (cn-deq/pi-ext-2 D1 D3) (ecn-deq/pi-ext-2 D1' D3')) + %<- (explicate-closed-cn-deq D1 D1') + %<- (explicate-open-cn-deq loc/z D3 D3') %. +%worlds (can-mofkd-block) (explicate-closed-kd-sub D1 _) (explicate-closed-kd-deq _ _) (explicate-closed-cn-deq _ _) %. +%total (D1 D2 D3) (explicate-closed-kd-sub D3 _) (explicate-closed-kd-deq D1 _) (explicate-closed-cn-deq D2 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/explicit-context-lemmas.lf b/new-tests/stelf-output/tslf/explicit-context-lemmas.lf new file mode 100644 index 0000000..4c7f695 --- /dev/null +++ b/new-tests/stelf-output/tslf/explicit-context-lemmas.lf @@ -0,0 +1,204 @@ +%sort cxt-append-complete {G1} {G2} {G3} %. +%mode cxt-append-complete %in %in %out %out %. +%term _ cxt-append-complete G cxt/nil G cxt-append/nil %. +%term _ + %pi (cxt-append-complete G (cxt/cons G' C K) (cxt/cons G'' C K) (cxt-append/cons D)) + %<- (cxt-append-complete G G' G'' D) %. +%worlds (cn-block) (cxt-append-complete _ _ _ _) %. +%total G (cxt-append-complete _ G _ _) %. +%sort cxt-precedes-trans %. +%mode cxt-precedes-trans %in %in %out %. +%term _ + %pi (cxt-precedes-trans (cxt-precedes/i DX _ DL1) (cxt-precedes/i _ DZ DL2) (cxt-precedes/i DX DZ DL3)) + %<- (loc-less-trans DL1 DL2 DL3) %. +%worlds (cn-block ovar-block) (cxt-precedes-trans _ _ _) %. +%total {} (cxt-precedes-trans _ _ _) %. +%sort cxt-lookup-ordered %. +%mode cxt-lookup-ordered %in %out %. +%term _ cxt-lookup-ordered (cxt-lookup/hit (%the (cxt-bounded G1 C2) DB)) (%the (cxt-ordered (cxt/cons G1 C2 K1)) (cxt-ordered/cons DB)) %. +%term _ cxt-lookup-ordered (cxt-lookup/miss (%the (cxt-lookup (cxt/cons G1 C1 K1) CA K3) DL) (%the (cxt-precedes CA C2) DP) DB) (cxt-ordered/cons DB) %. +%worlds (ofkd-block ovar-block cn-block) (cxt-lookup-ordered _ _) %. +%total D1 (cxt-lookup-ordered D1 _) %. +%sort cxt-bounded-isvar %. +%mode cxt-bounded-isvar %in %out %. +%term _ cxt-bounded-isvar (cxt-bounded/nil D) D %. +%term _ cxt-bounded-isvar (cxt-bounded/cons _ (cxt-precedes/i _ D _)) D %. +%worlds (ofkd-block ovar-block cn-block) (cxt-bounded-isvar _ _) %. +%total {} (cxt-bounded-isvar _ _) %. +%sort cxt-extend-ordered %. +%mode cxt-extend-ordered %in %out %. +%term _ cxt-extend-ordered cxt-ordered/nil ([x] [d isvar x loc/z] cxt-bounded/nil d) %. +%term _ + %pi (cxt-extend-ordered (cxt-ordered/cons DB) ([x] [d isvar x (loc/s I)] cxt-bounded/cons DB (cxt-precedes/i Disvar d Dlt))) + %<- (cxt-bounded-isvar DB Disvar) + %<- (loc-less-immsucc I Dlt) %. +%worlds (cn-block ovar-block) (cxt-extend-ordered _ _) %. +%total {} (cxt-extend-ordered _ _) %. +%sort cxt-bounded-lookup %. +%mode cxt-bounded-lookup %in %in %out %. +%term _ cxt-bounded-lookup (cxt-bounded/cons _ DP) (cxt-lookup/hit _) DP %. +%term _ + %pi (cxt-bounded-lookup (cxt-bounded/cons DB DP) (cxt-lookup/miss DL _ _) DP'') + %<- (cxt-bounded-lookup DB DL DP') + %<- (cxt-precedes-trans DP' DP DP'') %. +%worlds (cn-block ovar-block) (cxt-bounded-lookup _ _ _) %. +%total {DB} (cxt-bounded-lookup DB _ _) %. +%sort cxt-weaken-lookup %. +%mode cxt-weaken-lookup %in %in %in %out %. +%term _ + %pi (cxt-weaken-lookup DB DL _ (cxt-lookup/miss DL DP DB)) + %<- (cxt-bounded-lookup DB DL DP) %. +%worlds (cn-block ovar-block) (cxt-weaken-lookup _ _ _ _) %. +%total {} (cxt-weaken-lookup _ _ _ _) %. +%sort id-or-cl %. +%term id-or-cl/id id-or-cl ([x] x) ([x] K) %. +%term id-or-cl/cl id-or-cl ([x] C) _ %. +%sort cxt-lookup-ioc-c {x} %. +%mode cxt-lookup-ioc-c %in %out %. +%term _ cxt-lookup-ioc-c ([x] [di] cxt-lookup/hit _) id-or-cl/cl %. +%term _ + %pi (cxt-lookup-ioc-c ([x] [di] cxt-lookup/miss (DL x di) _ _) IOC) + %<- (cxt-lookup-ioc-c DL IOC) %. +%worlds (cn-block ovar-block) (cxt-lookup-ioc-c _ _) %. +%total D1 (cxt-lookup-ioc-c D1 _) %. +%sort cxt-append-precedes {x cn} %. +%mode cxt-append-precedes %in %in %out %. +%term _ cxt-append-precedes ([x] cxt-append/nil) ([x] [di] %the (cxt-bounded _ (X x)) (cxt-bounded/cons _ (DP x di))) (%the ({x} %pi (isvar x I) %-> (cxt-precedes x (X x))) DP) %. +%term _ + %pi (cxt-append-precedes ([x] cxt-append/cons (DA x)) ([x] [di] cxt-bounded/cons (DB x di) (DP x di)) DP'') + %<- (cxt-append-precedes DA DB DP') + %<- ({x} {di isvar x I} cxt-precedes-trans (DP' x di) (DP x di) (DP'' x di)) %. +%worlds (cn-block ovar-block) (cxt-append-precedes _ _ _) %. +%total D1 (cxt-append-precedes D1 _ _) %. +%sort cxt-precedes-not-refl {x cn} %. +%mode cxt-precedes-not-refl %in %out %. +%term _ + %pi (cxt-precedes-not-refl ([x] [di] cxt-precedes/i di di D11) DU) + %<- (loc-less-not-refl D11 DU) %. +%worlds (cn-block ovar-block) (cxt-precedes-not-refl _ _) %. +%total {} (cxt-precedes-not-refl _ _) %. +%sort isvar-fun %. +%mode isvar-fun %in %in %out %. +%block isvar-fun-block {n loc} [x cn] [d isvar x n] [thm isvar-fun d d seq/loc/refl]%. +%worlds (cn-block isvar-fun-block) (isvar-fun _ _ _) %. +%total {} (isvar-fun _ _ _) %. +%sort seq/loc/less-r %. +%mode seq/loc/less-r %in %in %out %. +%term _ seq/loc/less-r DL seq/loc/refl DL %. +%worlds () (seq/loc/less-r _ _ _) %. +%total {} (seq/loc/less-r _ _ _) %. +%sort cxt-bounded-strengthen {G cxt} {x cn} {di isvar x I} %. +%mode cxt-bounded-strengthen %in %in %out %. +%term _ cxt-bounded-strengthen cxt/nil ([x] [di] cxt-bounded/nil di) cxt-ordered/nil %. +%term _ cxt-bounded-strengthen (cxt/cons cxt/nil _ _) ([x] [di] cxt-bounded/cons (cxt-bounded/nil D1) (cxt-precedes/i D1 di DL)) (cxt-ordered/cons (cxt-bounded/nil D1)) %. +%term _ + %pi (cxt-bounded-strengthen (cxt/cons (cxt/cons G C K) _ _) ([x] [di] cxt-bounded/cons (cxt-bounded/cons (DB x di) (cxt-precedes/i D1 D2 DL12)) (cxt-precedes/i D2' di DL2'3)) (cxt-ordered/cons (cxt-bounded/cons DB' (cxt-precedes/i D1 D2 DL12)))) + %<- (isvar-fun D2 D2' DQ) + %<- (seq/loc/less-r DL12 DQ DL12') + %<- (loc-less-trans DL12' DL2'3 DL13) + %<- (cxt-bounded-strengthen (cxt/cons G C K) ([x] [di] cxt-bounded/cons (DB x di) (cxt-precedes/i D1 di DL13)) (cxt-ordered/cons DB')) %. +%worlds (cn-block isvar-fun-block) (cxt-bounded-strengthen _ _ _) %. +%total G (cxt-bounded-strengthen G _ _) %. +%sort cxt-bounded-strengthen-c {x cn} {di isvar x I} %. +%mode cxt-bounded-strengthen-c %in %out %. +%term _ cxt-bounded-strengthen-c ([x] [di] cxt-bounded/nil DI) (cxt-bounded/nil DI) %. +%term _ + %pi (cxt-bounded-strengthen-c ([x] [di] cxt-bounded/cons (DB x di) DP) (cxt-bounded/cons DB' DP)) + %<- (cxt-bounded-strengthen-c DB DB') %. +%worlds (cn-block isvar-fun-block) (cxt-bounded-strengthen-c _ _) %. +%total G (cxt-bounded-strengthen-c G _) %. +%sort cxt-lookup-strengthen {C' cn} {x cn} {di isvar x I} %. +%mode cxt-lookup-strengthen %in %in %out %. +%term _ + %pi (cxt-lookup-strengthen _ ([x] [di] cxt-lookup/hit (DB x di)) (cxt-lookup/hit DB')) + %<- (cxt-bounded-strengthen-c DB DB') %. +%term _ + %pi (cxt-lookup-strengthen C ([x] [di] cxt-lookup/miss (DL x di) DP (DB x di)) (cxt-lookup/miss DL' DP DB')) + %<- (cxt-lookup-strengthen C DL DL') + %<- (cxt-bounded-strengthen-c DB DB') %. +%worlds (cn-block isvar-fun-block) (cxt-lookup-strengthen _ _ _) %. +%total G (cxt-lookup-strengthen _ G _) %. +%sort uninhabited-seq/kd {S1 kd} {S2 kd} %. +%mode uninhabited-seq/kd %in %in %in %out %. +%worlds (cn-block isvar-fun-block) (uninhabited-seq/kd _ _ _ _) %. +%total {} (uninhabited-seq/kd _ _ _ _) %. +%sort cxt-lookup-unique {x cn} %. +%mode cxt-lookup-unique %in %in %out %. +%term _ cxt-lookup-unique ([x] cxt-append/nil) _ seq/kd/refl %. +%term _ + %pi (cxt-lookup-unique ([x] cxt-append/nil) ([x] [di] cxt-lookup/miss _ (cxt-precedes/i di di D11) _) DQ) + %<- (loc-less-not-refl D11 DU) + %<- (uninhabited-seq/kd _ _ DU DQ) %. +%term _ + %pi (cxt-lookup-unique ([x] cxt-append/cons (DA x)) ([x] [di] cxt-lookup/miss (DL x di) _ _) DQ) + %<- (cxt-lookup-unique DA DL DQ) %. +%term _ + %pi (cxt-lookup-unique ([x] cxt-append/cons (DA x)) ([x] [di isvar x I] cxt-lookup/hit (DB x di)) DQ) + %<- (cxt-append-precedes DA DB DP) + %<- (cxt-precedes-not-refl DP DU) + %<- (uninhabited-seq/kd _ _ DU DQ) %. +%worlds (cn-block ovar-block) (cxt-lookup-unique _ _ _) %. +%total D1 (cxt-lookup-unique D1 _ _) %. +%sort cxt-inc {K kd} %. +%mode cxt-inc %in %in %out %. +%term _ + %pi (cxt-inc _ DO ([x] [di] cxt-ordered/cons (DB x di))) + %<- (cxt-extend-ordered DO DB) %. +%worlds (cn-block ovar-block) (cxt-inc _ _ _) %. +%total {} (cxt-inc _ _ _) %. +%sort cxt-inc-duece {K kd} {K' kd} %. +%mode cxt-inc-duece %in %in %in %out %out %. +%term _ cxt-inc-duece _ _ cxt-ordered/nil ([y] [dy isvar y loc/z] cxt-ordered/cons (cxt-bounded/nil dy)) ([y] [dy isvar y loc/z] cxt-ordered/cons (cxt-bounded/nil dy)) %. +%term _ + %pi (cxt-inc-duece _ _ (cxt-ordered/cons (cxt-bounded/nil DI)) ([x] [dx] cxt-ordered/cons (cxt-bounded/cons (cxt-bounded/nil DI) (cxt-precedes/i DI dx DL))) ([x] [dx] cxt-ordered/cons (cxt-bounded/cons (cxt-bounded/nil DI) (cxt-precedes/i DI dx DL)))) + %<- (loc-less-immsucc _ DL) %. +%term _ + %pi (cxt-inc-duece _ _ (cxt-ordered/cons (cxt-bounded/cons DB (cxt-precedes/i D1 D2 D12))) ([x] [dx] cxt-ordered/cons (cxt-bounded/cons (cxt-bounded/cons DB (cxt-precedes/i D1 D2 D12)) (cxt-precedes/i D2 dx DL))) ([x] [dx] cxt-ordered/cons (cxt-bounded/cons (cxt-bounded/cons DB (cxt-precedes/i D1 D2 D12)) (cxt-precedes/i D2 dx DL)))) + %<- (loc-less-immsucc _ DL) %. +%worlds (cn-block ovar-block) (cxt-inc-duece _ _ _ _ _) %. +%total {} (cxt-inc-duece _ _ _ _ _) %. +%sort cxt-inc-precedes {x} %. +%mode cxt-inc-precedes %in %in %out %out %. +%term _ + %pi (cxt-inc-precedes ([x] [di] cxt-precedes/i di di Di) ([x] [di] cxt-precedes/i di di Dj) ([x] [di] [y] [dy] cxt-precedes/i di dy DL1) ([x] [di] [y] [dy] cxt-precedes/i di dy DL2)) + %<- (loc-less-bound _ _ DL1 DL2) %. +%term _ + %pi (cxt-inc-precedes ([x] [di] cxt-precedes/i (DI x) di Di) ([x] [di] cxt-precedes/i (DJ x) di Dj) ([x] [di] [y] [dy] cxt-precedes/i di dy DL1) ([x] [di] [y] [dy] cxt-precedes/i di dy DL2)) + %<- (loc-less-bound _ _ DL1 DL2) %. +%term _ + %pi (cxt-inc-precedes ([x] [di] cxt-precedes/i di (DI x) Di) ([x] [di] cxt-precedes/i di (DJ x) Dj) ([x] [di] [y] [dy] cxt-precedes/i (DI x) dy DL1) ([x] [di] [y] [dy] cxt-precedes/i (DJ x) dy DL2)) + %<- (loc-less-bound _ _ DL1 DL2) %. +%term _ + %pi (cxt-inc-precedes ([x] [di] cxt-precedes/i (DI' x) (DI x) Di) ([x] [di] cxt-precedes/i (DJ' x) (DJ x) Dj) ([x] [di] [y] [dy] cxt-precedes/i (DI x) dy DL1) ([x] [di] [y] [dy] cxt-precedes/i (DJ x) dy DL2)) + %<- (loc-less-bound _ _ DL1 DL2) %. +%worlds (cn-block ovar-block) (cxt-inc-precedes _ _ _ _) %. +%total {} (cxt-inc-precedes _ _ _ _) %. +%sort cxt-inc-2-b {K %pi cn %-> kd} {x} %. +%mode cxt-inc-2-b %in %in %in %out %out %. +%term _ + %pi (cxt-inc-2-b _ ([x] [di] cxt-bounded/nil di) ([x] [di] cxt-bounded/nil di) ([x] [di] [y] [dy] cxt-bounded/cons (cxt-bounded/nil di) (cxt-precedes/i di dy DL1)) ([x] [di] [y] [dy] cxt-bounded/cons (cxt-bounded/nil di) (cxt-precedes/i di dy DL2))) + %<- (loc-less-bound _ _ DL1 DL2) %. +%term _ + %pi (cxt-inc-2-b _ ([x] [di] cxt-bounded/nil (D1 x)) ([x] [di] cxt-bounded/nil (D2 x)) ([x] [di] [y] [dy] cxt-bounded/cons (cxt-bounded/nil (D1 x)) (cxt-precedes/i (D1 x) dy DL1)) ([x] [di] [y] [dy] cxt-bounded/cons (cxt-bounded/nil (D2 x)) (cxt-precedes/i (D2 x) dy DL2))) + %<- (loc-less-bound _ _ DL1 DL2) %. +%term _ + %pi (cxt-inc-2-b _ ([x] [di] cxt-bounded/cons (DB1 x di) (DP1 x di)) ([x] [di] cxt-bounded/cons (DB2 x di) (DP2 x di)) ([x] [di] [y] [dy] cxt-bounded/cons (cxt-bounded/cons (DB1 x di) (DP1 x di)) (DP1' x di y dy)) ([x] [di] [y] [dy] cxt-bounded/cons (cxt-bounded/cons (DB2 x di) (DP2 x di)) (DP2' x di y dy))) + %<- (cxt-inc-precedes DP1 DP2 DP1' DP2') %. +%worlds (cn-block ovar-block) (cxt-inc-2-b _ _ _ _ _) %. +%total {} (cxt-inc-2-b _ _ _ _ _) %. +%sort cxt-inc-2 {x} %. +%mode cxt-inc-2 %in %in %out %out %. +%term _ cxt-inc-2 ([x] [di] cxt-ordered/nil) _ ([x] [di] [y] [dy isvar y loc/z] cxt-bounded/nil dy) ([x] [di] [y] [dy isvar y loc/z] cxt-bounded/nil dy) %. +%term _ + %pi (cxt-inc-2 ([x] [dx] cxt-ordered/cons (DB1 x dx)) ([x] [dx] cxt-ordered/cons (DB2 x dx)) DB1' DB2') + %<- (cxt-inc-2-b _ DB1 DB2 DB1' DB2') %. +%worlds (cn-block ovar-block) (cxt-inc-2 _ _ _ _) %. +%total {} (cxt-inc-2 _ _ _ _) %. +%sort cxt-lookup-precedes %. +%mode cxt-lookup-precedes %in %in %out %. +%term _ cxt-lookup-precedes (cxt-lookup/hit _) (cxt-bounded/cons _ DP) DP %. +%term _ + %pi (cxt-lookup-precedes (cxt-lookup/miss _ DP _) (cxt-bounded/cons _ DP') DP'') + %<- (cxt-precedes-trans DP DP' DP'') %. +%worlds (cn-block ovar-block) (cxt-lookup-precedes _ _ _) %. +%total {} (cxt-lookup-precedes _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/explicit-context.lf b/new-tests/stelf-output/tslf/explicit-context.lf new file mode 100644 index 0000000..4ef4be9 --- /dev/null +++ b/new-tests/stelf-output/tslf/explicit-context.lf @@ -0,0 +1,38 @@ +%%%% Explicit contexts +%sort cxt %. +%name cxt %. +%term cxt/nil cxt %. +%term cxt/cons %pi cxt %-> cn %-> kd %-> cxt %. +%sort cxt-append %. +%mode cxt-append %star %star %star %. +%term cxt-append/nil cxt-append G cxt/nil G %. +%term cxt-append/cons %pi (cxt-append G (cxt/cons G' C K) (cxt/cons G'' C K)) %<- (cxt-append G G' G'') %. +%sort isvar %. +%mode isvar %star %star %. +%sort cxt-precedes %. +%mode cxt-precedes %star %star %. +%term cxt-precedes/i %pi (cxt-precedes C D) %<- (loc-less I J) %<- (isvar D J) %<- (isvar C I) %. +%sort cxt-bounded %. +%mode cxt-bounded %star %star %. +%term cxt-bounded/nil %pi (cxt-bounded cxt/nil C) %<- (isvar C _) %. +%term cxt-bounded/cons %pi (cxt-bounded (cxt/cons G Y _) X) %<- (cxt-precedes Y X) %<- (cxt-bounded G Y) %. +%sort cxt-ordered %. +%mode cxt-ordered %star %. +%term cxt-ordered/nil cxt-ordered cxt/nil %. +%term cxt-ordered/cons %pi (cxt-ordered (cxt/cons G X _)) %<- (cxt-bounded G X) %. +%block ovar-block {l loc} [a cn] [do isvar a l]%. +%%%% context lookup for explicit contexts +%sort cxt-lookup %. +%mode cxt-lookup %star %star %star %. +%term cxt-lookup/hit %pi (cxt-lookup (cxt/cons G A K) A K) %<- (cxt-bounded G A) %. +%term cxt-lookup/miss + %pi (cxt-lookup (cxt/cons G B L) A K) + %<- (cxt-bounded G B) + %<- (cxt-precedes A B) + %<- (cxt-lookup G A K) %. +%sort cxt-append-sub %. +%mode cxt-append-sub %star %star %star %star %. +%term cxt-append-sub/nil cxt-append-sub G ([x] cxt/nil) _ G %. +%term cxt-append-sub/cons + %pi (cxt-append-sub G ([x] cxt/cons (G' x) C' (K x)) C (cxt/cons G'' C' (K C))) + %<- (cxt-append-sub G G' C G'') %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/explicit-metrics-sigs.lf b/new-tests/stelf-output/tslf/explicit-metrics-sigs.lf new file mode 100644 index 0000000..9c87454 --- /dev/null +++ b/new-tests/stelf-output/tslf/explicit-metrics-sigs.lf @@ -0,0 +1,79 @@ +%%%% metrics on typing derivations for the explicit system +%sort msg-wf %. +%mode msg-wf %star %star %. +%sort msg-deq %. +%mode msg-deq %star %star %. +%term msg-wf/sg/unit msg-wf sg-wf/sg/unit met/unit %. +%term msg-wf/sg/cn %pi (msg-wf (sg-wf/sg/cn D1) (met/sing D1')) %<- (mofkd D1 D1') %. +%term msg-wf/sg/kd %pi (msg-wf (sg-wf/sg/kd D1) (met/sing D1')) %<- (mkd-wf D1 D1') %. +%term msg-wf/sg/sgm + %pi (msg-wf (sg-wf/sg/sgm D1 D2 _) (met/pair D1' D2')) + %<- ({a} {da} {dm mofkd da met/unit} msg-wf (D2 a da) D2') + %<- (msg-wf D1 D1') %. +%term msg-wf/sg/pi + %pi (msg-wf (sg-wf/sg/pi D1 D2 _) (met/pair D1' D2')) + %<- ({a} {da} {dm mofkd da met/unit} msg-wf (D2 a da) D2') + %<- (msg-wf D1 D1') %. +%term msg-deq/sg/unit msg-deq sg-deq/sg/unit met/unit %. +%term msg-deq/sg/cn %pi (msg-deq (sg-deq/sg/cn D1) (met/sing D1')) %<- (mcn-deq D1 D1') %. +%term msg-deq/sg/kd %pi (msg-deq (sg-deq/sg/kd D1) (met/sing D1')) %<- (mkd-deq D1 D1') %. +%term msg-deq/sg/sgm + %pi (msg-deq (sg-deq/sg/sgm D1 D2 _) (met/pair D1' D2')) + %<- ({a} {da} {dm mofkd da met/unit} msg-deq (D2 a da) D2') + %<- (msg-deq D1 D1') %. +%term msg-deq/sg/pi + %pi (msg-deq (sg-deq/sg/pi D1 D2 _) (met/pair D1' D2')) + %<- ({a} {da} {dm mofkd da met/unit} msg-deq (D2 a da) D2') + %<- (msg-deq D1 D1') %. +%sort can-msg-wf {D sg-wf K} %. +%mode can-msg-wf %in %out %. +%sort can-msg-deq {D sg-deq K K'} %. +%mode can-msg-deq %in %out %. +%term _ can-msg-wf sg-wf/sg/unit msg-wf/sg/unit %. +%term _ %pi (can-msg-wf (sg-wf/sg/cn D1) (msg-wf/sg/cn D1')) %<- (can-mofkd D1 D1') %. +%term _ %pi (can-msg-wf (sg-wf/sg/kd D1) (msg-wf/sg/kd D1')) %<- (can-mkd-wf D1 D1') %. +%term _ + %pi (can-msg-wf (sg-wf/sg/sgm D1 D2 _) (msg-wf/sg/sgm D1' D2')) + %<- (can-msg-wf D1 D1') + %<- ({a} {da} {dm mofkd da _} {dcm can-mofkd da dm} can-msg-wf (D2 a da) (D2' a da dm)) %. +%term _ + %pi (can-msg-wf (sg-wf/sg/pi D1 D2 _) (msg-wf/sg/pi D1' D2')) + %<- (can-msg-wf D1 D1') + %<- ({a} {da} {dm mofkd da _} {dcm can-mofkd da dm} can-msg-wf (D2 a da) (D2' a da dm)) %. +%term _ can-msg-deq sg-deq/sg/unit msg-deq/sg/unit %. +%term _ %pi (can-msg-deq (sg-deq/sg/cn D1) (msg-deq/sg/cn D1')) %<- (can-mcn-deq D1 D1') %. +%term _ %pi (can-msg-deq (sg-deq/sg/kd D1) (msg-deq/sg/kd D1')) %<- (can-mkd-deq D1 D1') %. +%term _ + %pi (can-msg-deq (sg-deq/sg/sgm D1 D2 _) (msg-deq/sg/sgm D1' D2')) + %<- (can-msg-deq D1 D1') + %<- ({a} {da} {dm mofkd da _} {dcm can-mofkd da dm} can-msg-deq (D2 a da) (D2' a da dm)) %. +%term _ + %pi (can-msg-deq (sg-deq/sg/pi D1 D2 _) (msg-deq/sg/pi D1' D2')) + %<- (can-msg-deq D1 D1') + %<- ({a} {da} {dm mofkd da _} {dcm can-mofkd da dm} can-msg-deq (D2 a da) (D2' a da dm)) %. +%worlds (cn-block can-mofkd-block ovar-block) (can-msg-wf _ _) (can-msg-deq _ _) %. +%total (D1 D2) (can-msg-wf D1 _) (can-msg-deq D2 _) %. +%sort mesg-wf %. +%sort mesg-deq %. +%term mesg-wf/sg/unit mesg-wf (esg-wf/sg/unit _) met/unit %. +%term mesg-wf/sg/cn %pi (mesg-wf (esg-wf/sg/cn D1) (met/sing D1')) %<- (meofkd D1 D1') %. +%term mesg-wf/sg/kd %pi (mesg-wf (esg-wf/sg/kd D1) (met/sing D1')) %<- (mekd-wf D1 D1') %. +%term mesg-wf/sg/sgm + %pi (mesg-wf (esg-wf/sg/sgm D1 D2 _) (met/pair D1' D2')) + %<- ({a} {da} mesg-wf (D2 a da) D2') + %<- (mesg-wf D1 D1') %. +%term mesg-wf/sg/pi + %pi (mesg-wf (esg-wf/sg/pi D1 D2 _) (met/pair D1' D2')) + %<- ({a} {da} mesg-wf (D2 a da) D2') + %<- (mesg-wf D1 D1') %. +%term mesg-deq/sg/unit mesg-deq (esg-deq/sg/unit _) met/unit %. +%term mesg-deq/sg/cn %pi (mesg-deq (esg-deq/sg/cn D1) (met/sing D1')) %<- (mecn-deq D1 D1') %. +%term mesg-deq/sg/kd %pi (mesg-deq (esg-deq/sg/kd D1) (met/sing D1')) %<- (mekd-deq D1 D1') %. +%term mesg-deq/sg/sgm + %pi (mesg-deq (esg-deq/sg/sgm D1 D2 _) (met/pair D1' D2')) + %<- ({a} {da} mesg-deq (D2 a da) D2') + %<- (mesg-deq D1 D1') %. +%term mesg-deq/sg/pi + %pi (mesg-deq (esg-deq/sg/pi D1 D2 _) (met/pair D1' D2')) + %<- ({a} {da} mesg-deq (D2 a da) D2') + %<- (mesg-deq D1 D1') %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/explicit-metrics.lf b/new-tests/stelf-output/tslf/explicit-metrics.lf new file mode 100644 index 0000000..92a6bff --- /dev/null +++ b/new-tests/stelf-output/tslf/explicit-metrics.lf @@ -0,0 +1,726 @@ +%%%% metrics on typing derivations for the explicit system +%sort mkd-wf %. +%mode mkd-wf %star %star %. +%sort mkd-sub %. +%mode mkd-sub %star %star %. +%sort mkd-deq %. +%mode mkd-deq %star %star %. +%sort mcn-deq %. +%mode mcn-deq %star %star %. +%term mofkd/cn/unit mofkd ofkd/cn/unit met/unit %. +%term mofkd/tp/unit mofkd ofkd/tp/unit met/unit %. +%term mofkd/tp/tagged mofkd ofkd/tp/tagged met/unit %. +%term mofkd/tp/ref %pi (mofkd (ofkd/tp/ref D1) (met/sing D1')) %<- (mofkd D1 D1') %. +%term mofkd/tp/tag %pi (mofkd (ofkd/tp/tag D1) (met/sing D1')) %<- (mofkd D1 D1') %. +%term mofkd/cn/pj1 %pi (mofkd (ofkd/cn/pj1 D1) (met/sing D1')) %<- (mofkd D1 D1') %. +%term mofkd/cn/pj2 %pi (mofkd (ofkd/cn/pj2 D1) (met/sing D1')) %<- (mofkd D1 D1') %. +%term mofkd/kd/sing %pi (mofkd (ofkd/kd/sing D1) (met/sing D1')) %<- (mofkd D1 D1') %. +%term mofkd/tp/cross + %pi (mofkd (ofkd/tp/cross D1 D2) (met/pair D1' D2')) + %<- (mofkd D2 D2') + %<- (mofkd D1 D1') %. +%term mofkd/tp/arrow + %pi (mofkd (ofkd/tp/arrow D1 D2) (met/pair D1' D2')) + %<- (mofkd D2 D2') + %<- (mofkd D1 D1') %. +%term mofkd/tp/sum + %pi (mofkd (ofkd/tp/sum D1 D2) (met/pair D1' D2')) + %<- (mofkd D2 D2') + %<- (mofkd D1 D1') %. +%term mofkd/cn/pair + %pi (mofkd (ofkd/cn/pair D1 D2) (met/pair D1' D2')) + %<- (mofkd D2 D2') + %<- (mofkd D1 D1') %. +%term mofkd/cn/app + %pi (mofkd (ofkd/cn/app D1 D2) (met/pair D1' D2')) + %<- (mofkd D2 D2') + %<- (mofkd D1 D1') %. +%term mofkd/sgm-ext + %pi (mofkd (ofkd/sgm-ext D1 D2) (met/pair D1' D2')) + %<- (mofkd D2 D2') + %<- (mofkd D1 D1') %. +%term mofkd/tp/forall + %pi (mofkd (ofkd/tp/forall D1 D2) (met/pair D1' D2')) + %<- (mkd-wf D2 D2') + %<- ({a} {da} {dm mofkd da met/unit} mofkd (D1 a da) D1') %. +%term mofkd/cn/mu + %pi (mofkd (ofkd/cn/mu D1 D2) (met/pair D1' D2')) + %<- (mkd-wf D2 D2') + %<- ({a} {da} {dm mofkd da met/unit} mofkd (D1 a da) D1') %. +%term mofkd/cn/lam + %pi (mofkd (ofkd/cn/lam D1 D2) (met/pair D1' D2')) + %<- (mkd-wf D2 D2') + %<- ({a} {da} {dm mofkd da met/unit} mofkd (D1 a da) D1') %. +%term mofkd/pi-ext + %pi (mofkd (ofkd/pi-ext D1 D2) (met/pair D1' D2')) + %<- ({a} {da} {dm mofkd da met/unit} mofkd (D2 a da) D2') + %<- (mofkd D1 D1') %. +%term mofkd/sub + %pi (mofkd (ofkd/sub D1 DS) (met/pair D1' DS')) + %<- (mkd-sub DS DS') + %<- (mofkd D1 D1') %. +%term mofkd/deq + %pi (mofkd (ofkd/deq D1 DS) (met/pair D1' DS')) + %<- (mkd-deq DS DS') + %<- (mofkd D1 D1') %. +%term mkd-wf/kd/unit mkd-wf kd-wf/kd/unit met/unit %. +%term mkd-wf/kd/type mkd-wf kd-wf/kd/type met/unit %. +%term mkd-wf/kd/sing %pi (mkd-wf (kd-wf/kd/sing D1) (met/sing D1')) %<- (mofkd D1 D1') %. +%term mkd-wf/kd/sgm + %pi (mkd-wf (kd-wf/kd/sgm D1 D2) (met/pair D1' D2')) + %<- ({a} {da} {dm mofkd da met/unit} mkd-wf (D2 a da) D2') + %<- (mkd-wf D1 D1') %. +%term mkd-wf/kd/pi + %pi (mkd-wf (kd-wf/kd/pi D1 D2) (met/pair D1' D2')) + %<- ({a} {da} {dm mofkd da met/unit} mkd-wf (D2 a da) D2') + %<- (mkd-wf D1 D1') %. +%term mkd-deq/kd/unit mkd-deq kd-deq/kd/unit met/unit %. +%term mkd-deq/kd/type mkd-deq kd-deq/kd/type met/unit %. +%term mkd-deq/kd/sing %pi (mkd-deq (kd-deq/kd/sing D1) (met/sing D1')) %<- (mcn-deq D1 D1') %. +%term mkd-deq/kd/sgm + %pi (mkd-deq (kd-deq/kd/sgm D1 D2) (met/pair D1' D2')) + %<- ({a} {da} {dm mofkd da met/unit} mkd-deq (D2 a da) D2') + %<- (mkd-deq D1 D1') %. +%term mkd-deq/kd/pi + %pi (mkd-deq (kd-deq/kd/pi D1 D2) (met/pair D1' D2')) + %<- ({a} {da} {dm mofkd da met/unit} mkd-deq (D2 a da) D2') + %<- (mkd-deq D1 D1') %. +%term mkd-sub/kd/unit mkd-sub kd-sub/kd/unit met/unit %. +%term mkd-sub/kd/type mkd-sub kd-sub/kd/type met/unit %. +%term mkd-sub/kd/sing-kd/sing %pi (mkd-sub (kd-sub/kd/sing-kd/sing D1) (met/sing D1')) %<- (mcn-deq D1 D1') %. +%term mkd-sub/kd/sing-kd/type %pi (mkd-sub (kd-sub/kd/sing-kd/type D1) (met/sing D1')) %<- (mofkd D1 D1') %. +%term mkd-sub/kd/sgm + %pi (mkd-sub (kd-sub/kd/sgm D1 D2 D3) (met/pair D1' (met/pair D2' D3'))) + %<- ({a} {da} {dm mofkd da met/unit} mkd-wf (D3 a da) D3') + %<- ({a} {da} {dm mofkd da met/unit} mkd-sub (D2 a da) D2') + %<- (mkd-sub D1 D1') %. +%term mkd-sub/kd/pi + %pi (mkd-sub (kd-sub/kd/pi D1 D2 D3) (met/pair D1' (met/pair D2' D3'))) + %<- ({a} {da} {dm mofkd da met/unit} mkd-wf (D3 a da) D3') + %<- ({a} {da} {dm mofkd da met/unit} mkd-sub (D2 a da) D2') + %<- (mkd-sub D1 D1') %. +%term mcn-deq/refl %pi (mcn-deq (cn-deq/refl D1) (met/sing D1')) %<- (mofkd D1 D1') %. +%term mcn-deq/sym %pi (mcn-deq (cn-deq/sym D1) (met/sing D1')) %<- (mcn-deq D1 D1') %. +%term mcn-deq/trans + %pi (mcn-deq (cn-deq/trans D1 D2) (met/pair D1' D2')) + %<- (mcn-deq D2 D2') + %<- (mcn-deq D1 D1') %. +%term mcn-deq/cn/unit mcn-deq cn-deq/cn/unit met/unit %. +%term mcn-deq/tp/unit mcn-deq cn-deq/tp/unit met/unit %. +%term mcn-deq/tp/tagged mcn-deq cn-deq/tp/tagged met/unit %. +%term mcn-deq/tp/ref %pi (mcn-deq (cn-deq/tp/ref D1) (met/sing D1')) %<- (mcn-deq D1 D1') %. +%term mcn-deq/tp/tag %pi (mcn-deq (cn-deq/tp/tag D1) (met/sing D1')) %<- (mcn-deq D1 D1') %. +%term mcn-deq/cn/pj1 %pi (mcn-deq (cn-deq/cn/pj1 D1) (met/sing D1')) %<- (mcn-deq D1 D1') %. +%term mcn-deq/cn/pj2 %pi (mcn-deq (cn-deq/cn/pj2 D1) (met/sing D1')) %<- (mcn-deq D1 D1') %. +%term mcn-deq/kd/sing %pi (mcn-deq (cn-deq/kd/sing D1) (met/sing D1')) %<- (mofkd D1 D1') %. +%term mcn-deq/tp/cross + %pi (mcn-deq (cn-deq/tp/cross D1 D2) (met/pair D1' D2')) + %<- (mcn-deq D2 D2') + %<- (mcn-deq D1 D1') %. +%term mcn-deq/tp/arrow + %pi (mcn-deq (cn-deq/tp/arrow D1 D2) (met/pair D1' D2')) + %<- (mcn-deq D2 D2') + %<- (mcn-deq D1 D1') %. +%term mcn-deq/tp/sum + %pi (mcn-deq (cn-deq/tp/sum D1 D2) (met/pair D1' D2')) + %<- (mcn-deq D2 D2') + %<- (mcn-deq D1 D1') %. +%term mcn-deq/cn/pair + %pi (mcn-deq (cn-deq/cn/pair D1 D2) (met/pair D1' D2')) + %<- (mcn-deq D2 D2') + %<- (mcn-deq D1 D1') %. +%term mcn-deq/cn/app + %pi (mcn-deq (cn-deq/cn/app D1 D2) (met/pair D1' D2')) + %<- (mcn-deq D2 D2') + %<- (mcn-deq D1 D1') %. +%term mcn-deq/sgm-ext + %pi (mcn-deq (cn-deq/sgm-ext D1 D2) (met/pair D1' D2')) + %<- (mcn-deq D2 D2') + %<- (mcn-deq D1 D1') %. +%term mcn-deq/kd/unit + %pi (mcn-deq (cn-deq/kd/unit D1 D2) (met/pair D1' D2')) + %<- (mofkd D2 D2') + %<- (mofkd D1 D1') %. +%term mcn-deq/tp/forall + %pi (mcn-deq (cn-deq/tp/forall D1 D2) (met/pair D1' D2')) + %<- (mkd-deq D2 D2') + %<- ({a} {da} {dm mofkd da met/unit} mcn-deq (D1 a da) D1') %. +%term mcn-deq/cn/mu + %pi (mcn-deq (cn-deq/cn/mu D1 D2) (met/pair D1' D2')) + %<- (mkd-deq D2 D2') + %<- ({a} {da} {dm mofkd da met/unit} mcn-deq (D1 a da) D1') %. +%term mcn-deq/cn/lam + %pi (mcn-deq (cn-deq/cn/lam D1 D2) (met/pair D1' D2')) + %<- (mkd-deq D2 D2') + %<- ({a} {da} {dm mofkd da met/unit} mcn-deq (D1 a da) D1') %. +%term mcn-deq/pi-ext + %pi (mcn-deq (cn-deq/pi-ext D1 D2 D3) (met/pair D1' (met/pair D2' D3'))) + %<- ({a} {da} {dm mofkd da met/unit} mcn-deq (D3 a da) D3') + %<- (mofkd D2 D2') + %<- (mofkd D1 D1') %. +%term mcn-deq/pi-ext-2 + %pi (mcn-deq (cn-deq/pi-ext-2 D1 D3) (met/pair D1' D3')) + %<- ({a} {da} {dm mofkd da met/unit} mcn-deq (D3 a da) D3') + %<- (mcn-deq D1 D1') %. +%term mcn-deq/sub + %pi (mcn-deq (cn-deq/sub D1 DS) (met/pair D1' DS')) + %<- (mkd-sub DS DS') + %<- (mcn-deq D1 D1') %. +%term mcn-deq/deq + %pi (mcn-deq (cn-deq/deq D1 DS) (met/pair D1' DS')) + %<- (mkd-deq DS DS') + %<- (mcn-deq D1 D1') %. +%sort can-mkd-wf {D kd-wf K} %. +%mode can-mkd-wf %in %out %. +%sort can-mkd-sub {D kd-sub K K'} %. +%mode can-mkd-sub %in %out %. +%sort can-mkd-deq {D kd-deq K K'} %. +%mode can-mkd-deq %in %out %. +%sort can-mcn-deq {D cn-deq C C' K'} %. +%mode can-mcn-deq %in %out %. +%term _ can-mofkd ofkd/cn/unit mofkd/cn/unit %. +%term _ can-mofkd ofkd/tp/unit mofkd/tp/unit %. +%term _ can-mofkd ofkd/tp/tagged mofkd/tp/tagged %. +%term _ %pi (can-mofkd (ofkd/tp/ref D1) (mofkd/tp/ref D1')) %<- (can-mofkd D1 D1') %. +%term _ %pi (can-mofkd (ofkd/tp/tag D1) (mofkd/tp/tag D1')) %<- (can-mofkd D1 D1') %. +%term _ %pi (can-mofkd (ofkd/kd/sing D1) (mofkd/kd/sing D1')) %<- (can-mofkd D1 D1') %. +%term _ %pi (can-mofkd (ofkd/cn/pj1 D1) (mofkd/cn/pj1 D1')) %<- (can-mofkd D1 D1') %. +%term _ %pi (can-mofkd (ofkd/cn/pj2 D1) (mofkd/cn/pj2 D1')) %<- (can-mofkd D1 D1') %. +%term _ + %pi (can-mofkd (ofkd/cn/pair D1 D2) (mofkd/cn/pair D1' D2')) + %<- (can-mofkd D1 D1') + %<- (can-mofkd D2 D2') %. +%term _ + %pi (can-mofkd (ofkd/tp/cross D1 D2) (mofkd/tp/cross D1' D2')) + %<- (can-mofkd D1 D1') + %<- (can-mofkd D2 D2') %. +%term _ + %pi (can-mofkd (ofkd/tp/arrow D1 D2) (mofkd/tp/arrow D1' D2')) + %<- (can-mofkd D1 D1') + %<- (can-mofkd D2 D2') %. +%term _ + %pi (can-mofkd (ofkd/tp/sum D1 D2) (mofkd/tp/sum D1' D2')) + %<- (can-mofkd D1 D1') + %<- (can-mofkd D2 D2') %. +%term _ + %pi (can-mofkd (ofkd/sgm-ext D1 D2) (mofkd/sgm-ext D1' D2')) + %<- (can-mofkd D1 D1') + %<- (can-mofkd D2 D2') %. +%term _ + %pi (can-mofkd (ofkd/cn/app D1 D2) (mofkd/cn/app D1' D2')) + %<- (can-mofkd D1 D1') + %<- (can-mofkd D2 D2') %. +%term _ + %pi (can-mofkd (ofkd/pi-ext D1 D2) (mofkd/pi-ext D1' D2')) + %<- (can-mofkd D1 D1') + %<- ({a} {da} {dm mofkd da met/unit} {dcm can-mofkd da dm} can-mofkd (D2 a da) (D2' a da dm)) %. +%term _ + %pi (can-mofkd (ofkd/cn/lam D1 D2) (mofkd/cn/lam D1' D2')) + %<- ({a} {da} {dm mofkd da met/unit} {dcm can-mofkd da dm} can-mofkd (D1 a da) (D1' a da dm)) + %<- (can-mkd-wf D2 D2') %. +%term _ + %pi (can-mofkd (ofkd/tp/forall D1 D2) (mofkd/tp/forall D1' D2')) + %<- ({a} {da} {dm mofkd da met/unit} {dcm can-mofkd da dm} can-mofkd (D1 a da) (D1' a da dm)) + %<- (can-mkd-wf D2 D2') %. +%term _ + %pi (can-mofkd (ofkd/cn/mu D1 D2) (mofkd/cn/mu D1' D2')) + %<- ({a} {da} {dm mofkd da met/unit} {dcm can-mofkd da dm} can-mofkd (D1 a da) (D1' a da dm)) + %<- (can-mkd-wf D2 D2') %. +%term _ + %pi (can-mofkd (ofkd/sub D1 DS) (mofkd/sub D1' DS')) + %<- (can-mofkd D1 D1') + %<- (can-mkd-sub DS DS') %. +%term _ + %pi (can-mofkd (ofkd/deq D1 DS) (mofkd/deq D1' DS')) + %<- (can-mofkd D1 D1') + %<- (can-mkd-deq DS DS') %. +%term _ can-mkd-wf kd-wf/kd/unit mkd-wf/kd/unit %. +%term _ can-mkd-wf kd-wf/kd/type mkd-wf/kd/type %. +%term _ %pi (can-mkd-wf (kd-wf/kd/sing D1) (mkd-wf/kd/sing D1')) %<- (can-mofkd D1 D1') %. +%term _ + %pi (can-mkd-wf (kd-wf/kd/sgm D1 D2) (mkd-wf/kd/sgm D1' D2')) + %<- (can-mkd-wf D1 D1') + %<- ({a} {da} {dm mofkd da _} {dcm can-mofkd da dm} can-mkd-wf (D2 a da) (D2' a da dm)) %. +%term _ + %pi (can-mkd-wf (kd-wf/kd/pi D1 D2) (mkd-wf/kd/pi D1' D2')) + %<- (can-mkd-wf D1 D1') + %<- ({a} {da} {dm mofkd da _} {dcm can-mofkd da dm} can-mkd-wf (D2 a da) (D2' a da dm)) %. +%term _ can-mkd-deq kd-deq/kd/unit mkd-deq/kd/unit %. +%term _ can-mkd-deq kd-deq/kd/type mkd-deq/kd/type %. +%term _ + %pi (can-mkd-deq (kd-deq/kd/sing D1) (mkd-deq/kd/sing D1')) + %<- (can-mcn-deq D1 D1') %. +%term _ + %pi (can-mkd-deq (kd-deq/kd/sgm D1 D2) (mkd-deq/kd/sgm D1' D2')) + %<- (can-mkd-deq D1 D1') + %<- ({a} {da} {dm mofkd da _} {dcm can-mofkd da dm} can-mkd-deq (D2 a da) (D2' a da dm)) %. +%term _ + %pi (can-mkd-deq (kd-deq/kd/pi D1 D2) (mkd-deq/kd/pi D1' D2')) + %<- (can-mkd-deq D1 D1') + %<- ({a} {da} {dm mofkd da _} {dcm can-mofkd da dm} can-mkd-deq (D2 a da) (D2' a da dm)) %. +%term _ can-mkd-sub kd-sub/kd/unit mkd-sub/kd/unit %. +%term _ can-mkd-sub kd-sub/kd/type mkd-sub/kd/type %. +%term _ + %pi (can-mkd-sub (kd-sub/kd/sing-kd/sing D1) (mkd-sub/kd/sing-kd/sing D1')) + %<- (can-mcn-deq D1 D1') %. +%term _ + %pi (can-mkd-sub (kd-sub/kd/sing-kd/type D1) (mkd-sub/kd/sing-kd/type D1')) + %<- (can-mofkd D1 D1') %. +%term _ + %pi (can-mkd-sub (kd-sub/kd/sgm D1 D2 D3) (mkd-sub/kd/sgm D1' D2' D3')) + %<- (can-mkd-sub D1 D1') + %<- ({a} {da} {dm mofkd da _} {dcm can-mofkd da dm} can-mkd-sub (D2 a da) (D2' a da dm)) + %<- ({a} {da} {dm mofkd da _} {dcm can-mofkd da dm} can-mkd-wf (D3 a da) (D3' a da dm)) %. +%term _ + %pi (can-mkd-sub (kd-sub/kd/pi D1 D2 D3) (mkd-sub/kd/pi D1' D2' D3')) + %<- (can-mkd-sub D1 D1') + %<- ({a} {da} {dm mofkd da _} {dcm can-mofkd da dm} can-mkd-sub (D2 a da) (D2' a da dm)) + %<- ({a} {da} {dm mofkd da _} {dcm can-mofkd da dm} can-mkd-wf (D3 a da) (D3' a da dm)) %. +%term _ can-mcn-deq cn-deq/cn/unit mcn-deq/cn/unit %. +%term _ can-mcn-deq cn-deq/tp/unit mcn-deq/tp/unit %. +%term _ can-mcn-deq cn-deq/tp/tagged mcn-deq/tp/tagged %. +%term _ %pi (can-mcn-deq (cn-deq/sym D1) (mcn-deq/sym D1')) %<- (can-mcn-deq D1 D1') %. +%term _ + %pi (can-mcn-deq (cn-deq/tp/ref D1) (mcn-deq/tp/ref D1')) + %<- (can-mcn-deq D1 D1') %. +%term _ + %pi (can-mcn-deq (cn-deq/tp/tag D1) (mcn-deq/tp/tag D1')) + %<- (can-mcn-deq D1 D1') %. +%term _ + %pi (can-mcn-deq (cn-deq/kd/sing D1) (mcn-deq/kd/sing D1')) + %<- (can-mofkd D1 D1') %. +%term _ %pi (can-mcn-deq (cn-deq/refl D1) (mcn-deq/refl D1')) %<- (can-mofkd D1 D1') %. +%term _ + %pi (can-mcn-deq (cn-deq/cn/pj1 D1) (mcn-deq/cn/pj1 D1')) + %<- (can-mcn-deq D1 D1') %. +%term _ + %pi (can-mcn-deq (cn-deq/cn/pj2 D1) (mcn-deq/cn/pj2 D1')) + %<- (can-mcn-deq D1 D1') %. +%term _ + %pi (can-mcn-deq (cn-deq/kd/unit D1 D2) (mcn-deq/kd/unit D1' D2')) + %<- (can-mofkd D1 D1') + %<- (can-mofkd D2 D2') %. +%term _ + %pi (can-mcn-deq (cn-deq/trans D1 D2) (mcn-deq/trans D1' D2')) + %<- (can-mcn-deq D1 D1') + %<- (can-mcn-deq D2 D2') %. +%term _ + %pi (can-mcn-deq (cn-deq/cn/pair D1 D2) (mcn-deq/cn/pair D1' D2')) + %<- (can-mcn-deq D1 D1') + %<- (can-mcn-deq D2 D2') %. +%term _ + %pi (can-mcn-deq (cn-deq/tp/cross D1 D2) (mcn-deq/tp/cross D1' D2')) + %<- (can-mcn-deq D1 D1') + %<- (can-mcn-deq D2 D2') %. +%term _ + %pi (can-mcn-deq (cn-deq/tp/arrow D1 D2) (mcn-deq/tp/arrow D1' D2')) + %<- (can-mcn-deq D1 D1') + %<- (can-mcn-deq D2 D2') %. +%term _ + %pi (can-mcn-deq (cn-deq/tp/sum D1 D2) (mcn-deq/tp/sum D1' D2')) + %<- (can-mcn-deq D1 D1') + %<- (can-mcn-deq D2 D2') %. +%term _ + %pi (can-mcn-deq (cn-deq/sgm-ext D1 D2) (mcn-deq/sgm-ext D1' D2')) + %<- (can-mcn-deq D1 D1') + %<- (can-mcn-deq D2 D2') %. +%term _ + %pi (can-mcn-deq (cn-deq/cn/app D1 D2) (mcn-deq/cn/app D1' D2')) + %<- (can-mcn-deq D1 D1') + %<- (can-mcn-deq D2 D2') %. +%term _ + %pi (can-mcn-deq (cn-deq/pi-ext D1 D2 D3) (mcn-deq/pi-ext D1' D2' D3')) + %<- (can-mofkd D1 D1') + %<- (can-mofkd D2 D2') + %<- ({a} {da} {dm mofkd da met/unit} {dcm can-mofkd da dm} can-mcn-deq (D3 a da) (D3' a da dm)) %. +%term _ + %pi (can-mcn-deq (cn-deq/pi-ext-2 D1 D3) (mcn-deq/pi-ext-2 D1' D3')) + %<- (can-mcn-deq D1 D1') + %<- ({a} {da} {dm mofkd da met/unit} {dcm can-mofkd da dm} can-mcn-deq (D3 a da) (D3' a da dm)) %. +%term _ + %pi (can-mcn-deq (cn-deq/cn/lam D1 D2) (mcn-deq/cn/lam D1' D2')) + %<- ({a} {da} {dm mofkd da met/unit} {dcm can-mofkd da dm} can-mcn-deq (D1 a da) (D1' a da dm)) + %<- (can-mkd-deq D2 D2') %. +%term _ + %pi (can-mcn-deq (cn-deq/tp/forall D1 D2) (mcn-deq/tp/forall D1' D2')) + %<- ({a} {da} {dm mofkd da met/unit} {dcm can-mofkd da dm} can-mcn-deq (D1 a da) (D1' a da dm)) + %<- (can-mkd-deq D2 D2') %. +%term _ + %pi (can-mcn-deq (cn-deq/cn/mu D1 D2) (mcn-deq/cn/mu D1' D2')) + %<- ({a} {da} {dm mofkd da met/unit} {dcm can-mofkd da dm} can-mcn-deq (D1 a da) (D1' a da dm)) + %<- (can-mkd-deq D2 D2') %. +%term _ + %pi (can-mcn-deq (cn-deq/sub D1 DS) (mcn-deq/sub D1' DS')) + %<- (can-mcn-deq D1 D1') + %<- (can-mkd-sub DS DS') %. +%term _ + %pi (can-mcn-deq (cn-deq/deq D1 DS) (mcn-deq/deq D1' DS')) + %<- (can-mcn-deq D1 D1') + %<- (can-mkd-deq DS DS') %. +%worlds (cn-block can-mofkd-block ovar-block) (can-mofkd _ _) (can-mkd-wf _ _) (can-mkd-deq _ _) (can-mkd-sub _ _) (can-mcn-deq _ _) %. +%total (D1 D2 D3 D4 D5) (can-mofkd D1 _) (can-mkd-wf D2 _) (can-mkd-deq D4 _) (can-mkd-sub D3 _) (can-mcn-deq D5 _) %. +%sort meofkd %. +%sort mekd-wf %. +%sort mekd-sub %. +%sort mekd-deq %. +%sort mecn-deq %. +%term mekd-wf/kd/unit mekd-wf (ekd-wf/kd/unit _) met/unit %. +%term mekd-wf/kd/type mekd-wf (ekd-wf/kd/type _) met/unit %. +%term mekd-wf/kd/sing %pi (mekd-wf (ekd-wf/kd/sing D1) (met/sing D1')) %<- (meofkd D1 D1') %. +%term mekd-wf/kd/sgm + %pi (mekd-wf (ekd-wf/kd/sgm D1 D2) (met/pair D1' D2')) + %<- ({a} {da} mekd-wf (D2 a da) D2') + %<- (mekd-wf D1 D1') %. +%term mekd-wf/kd/pi + %pi (mekd-wf (ekd-wf/kd/pi D1 D2) (met/pair D1' D2')) + %<- ({a} {da} mekd-wf (D2 a da) D2') + %<- (mekd-wf D1 D1') %. +%term meofkd/cn/unit meofkd (eofkd/cn/unit _) met/unit %. +%term meofkd/tp/unit meofkd (eofkd/tp/unit _) met/unit %. +%term meofkd/tp/tagged meofkd (eofkd/tp/tagged _) met/unit %. +%term meofkd/tp/ref %pi (meofkd (eofkd/tp/ref D1) (met/sing D1')) %<- (meofkd D1 D1') %. +%term meofkd/tp/tag %pi (meofkd (eofkd/tp/tag D1) (met/sing D1')) %<- (meofkd D1 D1') %. +%term meofkd/cn/pj1 %pi (meofkd (eofkd/cn/pj1 D1) (met/sing D1')) %<- (meofkd D1 D1') %. +%term meofkd/cn/pj2 %pi (meofkd (eofkd/cn/pj2 D1) (met/sing D1')) %<- (meofkd D1 D1') %. +%term meofkd/kd/sing %pi (meofkd (eofkd/kd/sing D1) (met/sing D1')) %<- (meofkd D1 D1') %. +%term meofkd/tp/cross + %pi (meofkd (eofkd/tp/cross D1 D2) (met/pair D1' D2')) + %<- (meofkd D2 D2') + %<- (meofkd D1 D1') %. +%term meofkd/tp/arrow + %pi (meofkd (eofkd/tp/arrow D1 D2) (met/pair D1' D2')) + %<- (meofkd D2 D2') + %<- (meofkd D1 D1') %. +%term meofkd/tp/sum + %pi (meofkd (eofkd/tp/sum D1 D2) (met/pair D1' D2')) + %<- (meofkd D2 D2') + %<- (meofkd D1 D1') %. +%term meofkd/cn/pair + %pi (meofkd (eofkd/cn/pair D1 D2) (met/pair D1' D2')) + %<- (meofkd D2 D2') + %<- (meofkd D1 D1') %. +%term meofkd/cn/app + %pi (meofkd (eofkd/cn/app D1 D2) (met/pair D1' D2')) + %<- (meofkd D2 D2') + %<- (meofkd D1 D1') %. +%term meofkd/sgm-ext + %pi (meofkd (eofkd/sgm-ext D1 D2) (met/pair D1' D2')) + %<- (meofkd D2 D2') + %<- (meofkd D1 D1') %. +%term meofkd/tp/forall + %pi (meofkd (eofkd/tp/forall D1 D2) (met/pair D1' D2')) + %<- (mekd-wf D2 D2') + %<- ({a} {da} meofkd (D1 a da) D1') %. +%term meofkd/cn/mu + %pi (meofkd (eofkd/cn/mu D1 D2) (met/pair D1' D2')) + %<- (mekd-wf D2 D2') + %<- ({a} {da} meofkd (D1 a da) D1') %. +%term meofkd/cn/lam + %pi (meofkd (eofkd/cn/lam D1 D2) (met/pair D1' D2')) + %<- (mekd-wf D2 D2') + %<- ({a} {da} meofkd (D1 a da) D1') %. +%term meofkd/pi-ext + %pi (meofkd (eofkd/pi-ext D1 D2) (met/pair D1' D2')) + %<- ({a} {da} meofkd (D2 a da) D2') + %<- (meofkd D1 D1') %. +%term meofkd/sub + %pi (meofkd (eofkd/sub D1 DS) (met/pair D1' DS')) + %<- (mekd-sub DS DS') + %<- (meofkd D1 D1') %. +%term meofkd/deq + %pi (meofkd (eofkd/deq D1 DS) (met/pair D1' DS')) + %<- (mekd-deq DS DS') + %<- (meofkd D1 D1') %. +%term meofkd/var meofkd (eofkd/var _) met/unit %. +%term meofkd/closed %pi (meofkd (eofkd/closed D1 _) DM) %<- (mofkd D1 DM) %. +%term mekd-deq/kd/unit mekd-deq (ekd-deq/kd/unit _) met/unit %. +%term mekd-deq/kd/type mekd-deq (ekd-deq/kd/type _) met/unit %. +%term mekd-deq/kd/sing %pi (mekd-deq (ekd-deq/kd/sing D1) (met/sing D1')) %<- (mecn-deq D1 D1') %. +%term mekd-deq/kd/sgm + %pi (mekd-deq (ekd-deq/kd/sgm D1 D2) (met/pair D1' D2')) + %<- ({a} {da} mekd-deq (D2 a da) D2') + %<- (mekd-deq D1 D1') %. +%term mekd-deq/kd/pi + %pi (mekd-deq (ekd-deq/kd/pi D1 D2) (met/pair D1' D2')) + %<- ({a} {da} mekd-deq (D2 a da) D2') + %<- (mekd-deq D1 D1') %. +%term mekd-sub/kd/unit mekd-sub (ekd-sub/kd/unit _) met/unit %. +%term mekd-sub/kd/type mekd-sub (ekd-sub/kd/type _) met/unit %. +%term mekd-sub/kd/sing-kd/sing %pi (mekd-sub (ekd-sub/kd/sing-kd/sing D1) (met/sing D1')) %<- (mecn-deq D1 D1') %. +%term mekd-sub/kd/sing-kd/type %pi (mekd-sub (ekd-sub/kd/sing-kd/type D1) (met/sing D1')) %<- (meofkd D1 D1') %. +%term mekd-sub/kd/sgm + %pi (mekd-sub (ekd-sub/kd/sgm D1 D2 D3) (met/pair D1' (met/pair D2' D3'))) + %<- ({a} {da} mekd-wf (D3 a da) D3') + %<- ({a} {da} mekd-sub (D2 a da) D2') + %<- (mekd-sub D1 D1') %. +%term mekd-sub/kd/pi + %pi (mekd-sub (ekd-sub/kd/pi D1 D2 D3) (met/pair D1' (met/pair D2' D3'))) + %<- ({a} {da} mekd-wf (D3 a da) D3') + %<- ({a} {da} mekd-sub (D2 a da) D2') + %<- (mekd-sub D1 D1') %. +%term mecn-deq/refl %pi (mecn-deq (ecn-deq/refl D1) (met/sing D1')) %<- (meofkd D1 D1') %. +%term mecn-deq/sym %pi (mecn-deq (ecn-deq/sym D1) (met/sing D1')) %<- (mecn-deq D1 D1') %. +%term mecn-deq/trans + %pi (mecn-deq (ecn-deq/trans D1 D2) (met/pair D1' D2')) + %<- (mecn-deq D2 D2') + %<- (mecn-deq D1 D1') %. +%term mecn-deq/cn/unit mecn-deq (ecn-deq/cn/unit _) met/unit %. +%term mecn-deq/tp/unit mecn-deq (ecn-deq/tp/unit _) met/unit %. +%term mecn-deq/tp/tagged mecn-deq (ecn-deq/tp/tagged _) met/unit %. +%term mecn-deq/tp/ref %pi (mecn-deq (ecn-deq/tp/ref D1) (met/sing D1')) %<- (mecn-deq D1 D1') %. +%term mecn-deq/tp/tag %pi (mecn-deq (ecn-deq/tp/tag D1) (met/sing D1')) %<- (mecn-deq D1 D1') %. +%term mecn-deq/cn/pj1 %pi (mecn-deq (ecn-deq/cn/pj1 D1) (met/sing D1')) %<- (mecn-deq D1 D1') %. +%term mecn-deq/cn/pj2 %pi (mecn-deq (ecn-deq/cn/pj2 D1) (met/sing D1')) %<- (mecn-deq D1 D1') %. +%term mecn-deq/kd/sing %pi (mecn-deq (ecn-deq/kd/sing D1) (met/sing D1')) %<- (meofkd D1 D1') %. +%term mecn-deq/tp/cross + %pi (mecn-deq (ecn-deq/tp/cross D1 D2) (met/pair D1' D2')) + %<- (mecn-deq D2 D2') + %<- (mecn-deq D1 D1') %. +%term mecn-deq/tp/arrow + %pi (mecn-deq (ecn-deq/tp/arrow D1 D2) (met/pair D1' D2')) + %<- (mecn-deq D2 D2') + %<- (mecn-deq D1 D1') %. +%term mecn-deq/tp/sum + %pi (mecn-deq (ecn-deq/tp/sum D1 D2) (met/pair D1' D2')) + %<- (mecn-deq D2 D2') + %<- (mecn-deq D1 D1') %. +%term mecn-deq/cn/pair + %pi (mecn-deq (ecn-deq/cn/pair D1 D2) (met/pair D1' D2')) + %<- (mecn-deq D2 D2') + %<- (mecn-deq D1 D1') %. +%term mecn-deq/cn/app + %pi (mecn-deq (ecn-deq/cn/app D1 D2) (met/pair D1' D2')) + %<- (mecn-deq D2 D2') + %<- (mecn-deq D1 D1') %. +%term mecn-deq/sgm-ext + %pi (mecn-deq (ecn-deq/sgm-ext D1 D2) (met/pair D1' D2')) + %<- (mecn-deq D2 D2') + %<- (mecn-deq D1 D1') %. +%term mecn-deq/kd/unit + %pi (mecn-deq (ecn-deq/kd/unit D1 D2) (met/pair D1' D2')) + %<- (meofkd D2 D2') + %<- (meofkd D1 D1') %. +%term mecn-deq/tp/forall + %pi (mecn-deq (ecn-deq/tp/forall D1 D2) (met/pair D1' D2')) + %<- (mekd-deq D2 D2') + %<- ({a} {da} mecn-deq (D1 a da) D1') %. +%term mecn-deq/cn/mu + %pi (mecn-deq (ecn-deq/cn/mu D1 D2) (met/pair D1' D2')) + %<- (mekd-deq D2 D2') + %<- ({a} {da} mecn-deq (D1 a da) D1') %. +%term mecn-deq/cn/lam + %pi (mecn-deq (ecn-deq/cn/lam D1 D2) (met/pair D1' D2')) + %<- (mekd-deq D2 D2') + %<- ({a} {da} mecn-deq (D1 a da) D1') %. +%term mecn-deq/pi-ext + %pi (mecn-deq (ecn-deq/pi-ext D1 D2 D3) (met/pair D1' (met/pair D2' D3'))) + %<- ({a} {da} mecn-deq (D3 a da) D3') + %<- (meofkd D2 D2') + %<- (meofkd D1 D1') %. +%term mecn-deq/pi-ext-2 + %pi (mecn-deq (ecn-deq/pi-ext-2 D1 D3) (met/pair D1' D3')) + %<- ({a} {da} mecn-deq (D3 a da) D3') + %<- (mecn-deq D1 D1') %. +%term mecn-deq/sub + %pi (mecn-deq (ecn-deq/sub D1 DS) (met/pair D1' DS')) + %<- (mekd-sub DS DS') + %<- (mecn-deq D1 D1') %. +%term mecn-deq/deq + %pi (mecn-deq (ecn-deq/deq D1 DS) (met/pair D1' DS')) + %<- (mekd-deq DS DS') + %<- (mecn-deq D1 D1') %. +%sort can-mekd-wf {D ekd-wf G K} %. +%mode can-mekd-wf %in %out %. +%sort can-mekd-sub {D ekd-sub G K K'} %. +%mode can-mekd-sub %in %out %. +%sort can-mekd-deq {D ekd-deq G K K'} %. +%mode can-mekd-deq %in %out %. +%sort can-mecn-deq {D ecn-deq G C1 C2 K} %. +%mode can-mecn-deq %in %out %. +%sort can-meofkd {D eofkd G C2 K} %. +%mode can-meofkd %in %out %. +%term _ can-meofkd (eofkd/cn/unit _) meofkd/cn/unit %. +%term _ can-meofkd (eofkd/tp/unit _) meofkd/tp/unit %. +%term _ can-meofkd (eofkd/tp/tagged _) meofkd/tp/tagged %. +%term _ %pi (can-meofkd (eofkd/tp/ref D1) (meofkd/tp/ref D1')) %<- (can-meofkd D1 D1') %. +%term _ %pi (can-meofkd (eofkd/tp/tag D1) (meofkd/tp/tag D1')) %<- (can-meofkd D1 D1') %. +%term _ %pi (can-meofkd (eofkd/kd/sing D1) (meofkd/kd/sing D1')) %<- (can-meofkd D1 D1') %. +%term _ %pi (can-meofkd (eofkd/cn/pj1 D1) (meofkd/cn/pj1 D1')) %<- (can-meofkd D1 D1') %. +%term _ %pi (can-meofkd (eofkd/cn/pj2 D1) (meofkd/cn/pj2 D1')) %<- (can-meofkd D1 D1') %. +%term _ + %pi (can-meofkd (eofkd/cn/pair D1 D2) (meofkd/cn/pair D1' D2')) + %<- (can-meofkd D1 D1') + %<- (can-meofkd D2 D2') %. +%term _ + %pi (can-meofkd (eofkd/tp/cross D1 D2) (meofkd/tp/cross D1' D2')) + %<- (can-meofkd D1 D1') + %<- (can-meofkd D2 D2') %. +%term _ + %pi (can-meofkd (eofkd/tp/arrow D1 D2) (meofkd/tp/arrow D1' D2')) + %<- (can-meofkd D1 D1') + %<- (can-meofkd D2 D2') %. +%term _ + %pi (can-meofkd (eofkd/tp/sum D1 D2) (meofkd/tp/sum D1' D2')) + %<- (can-meofkd D1 D1') + %<- (can-meofkd D2 D2') %. +%term _ + %pi (can-meofkd (eofkd/sgm-ext D1 D2) (meofkd/sgm-ext D1' D2')) + %<- (can-meofkd D1 D1') + %<- (can-meofkd D2 D2') %. +%term _ + %pi (can-meofkd (eofkd/cn/app D1 D2) (meofkd/cn/app D1' D2')) + %<- (can-meofkd D1 D1') + %<- (can-meofkd D2 D2') %. +%term _ + %pi (can-meofkd (eofkd/pi-ext D1 D2) (meofkd/pi-ext D1' D2')) + %<- (can-meofkd D1 D1') + %<- ({a} {da} can-meofkd (D2 a da) (D2' a da)) %. +%term _ + %pi (can-meofkd (eofkd/cn/lam D1 D2) (meofkd/cn/lam D1' D2')) + %<- ({a} {da} can-meofkd (D1 a da) (D1' a da)) + %<- (can-mekd-wf D2 D2') %. +%term _ + %pi (can-meofkd (eofkd/tp/forall D1 D2) (meofkd/tp/forall D1' D2')) + %<- ({a} {da} can-meofkd (D1 a da) (D1' a da)) + %<- (can-mekd-wf D2 D2') %. +%term _ + %pi (can-meofkd (eofkd/cn/mu D1 D2) (meofkd/cn/mu D1' D2')) + %<- ({a} {da} can-meofkd (D1 a da) (D1' a da)) + %<- (can-mekd-wf D2 D2') %. +%term _ + %pi (can-meofkd (eofkd/sub D1 DS) (meofkd/sub D1' DS')) + %<- (can-meofkd D1 D1') + %<- (can-mekd-sub DS DS') %. +%term _ + %pi (can-meofkd (eofkd/deq D1 DS) (meofkd/deq D1' DS')) + %<- (can-meofkd D1 D1') + %<- (can-mekd-deq DS DS') %. +%term _ can-meofkd (eofkd/var _) meofkd/var %. +%term _ %pi (can-meofkd (eofkd/closed DC _) (meofkd/closed DM)) %<- (can-mofkd DC DM) %. +%term _ can-mekd-wf (ekd-wf/kd/unit _) mekd-wf/kd/unit %. +%term _ can-mekd-wf (ekd-wf/kd/type _) mekd-wf/kd/type %. +%term _ + %pi (can-mekd-wf (ekd-wf/kd/sing D1) (mekd-wf/kd/sing D1')) + %<- (can-meofkd D1 D1') %. +%term _ + %pi (can-mekd-wf (ekd-wf/kd/sgm D1 D2) (mekd-wf/kd/sgm D1' D2')) + %<- (can-mekd-wf D1 D1') + %<- ({a} {da} can-mekd-wf (D2 a da) (D2' a da)) %. +%term _ + %pi (can-mekd-wf (ekd-wf/kd/pi D1 D2) (mekd-wf/kd/pi D1' D2')) + %<- (can-mekd-wf D1 D1') + %<- ({a} {da} can-mekd-wf (D2 a da) (D2' a da)) %. +%term _ can-mekd-deq (ekd-deq/kd/unit _) mekd-deq/kd/unit %. +%term _ can-mekd-deq (ekd-deq/kd/type _) mekd-deq/kd/type %. +%term _ + %pi (can-mekd-deq (ekd-deq/kd/sing D1) (mekd-deq/kd/sing D1')) + %<- (can-mecn-deq D1 D1') %. +%term _ + %pi (can-mekd-deq (ekd-deq/kd/sgm D1 D2) (mekd-deq/kd/sgm D1' D2')) + %<- (can-mekd-deq D1 D1') + %<- ({a} {da} can-mekd-deq (D2 a da) (D2' a da)) %. +%term _ + %pi (can-mekd-deq (ekd-deq/kd/pi D1 D2) (mekd-deq/kd/pi D1' D2')) + %<- (can-mekd-deq D1 D1') + %<- ({a} {da} can-mekd-deq (D2 a da) (D2' a da)) %. +%term _ can-mekd-sub (ekd-sub/kd/unit _) mekd-sub/kd/unit %. +%term _ can-mekd-sub (ekd-sub/kd/type _) mekd-sub/kd/type %. +%term _ + %pi (can-mekd-sub (ekd-sub/kd/sing-kd/sing D1) (mekd-sub/kd/sing-kd/sing D1')) + %<- (can-mecn-deq D1 D1') %. +%term _ + %pi (can-mekd-sub (ekd-sub/kd/sing-kd/type D1) (mekd-sub/kd/sing-kd/type D1')) + %<- (can-meofkd D1 D1') %. +%term _ + %pi (can-mekd-sub (ekd-sub/kd/sgm D1 D2 D3) (mekd-sub/kd/sgm D1' D2' D3')) + %<- (can-mekd-sub D1 D1') + %<- ({a} {da} can-mekd-sub (D2 a da) (D2' a da)) + %<- ({a} {da} can-mekd-wf (D3 a da) (D3' a da)) %. +%term _ + %pi (can-mekd-sub (ekd-sub/kd/pi D1 D2 D3) (mekd-sub/kd/pi D1' D2' D3')) + %<- (can-mekd-sub D1 D1') + %<- ({a} {da} can-mekd-sub (D2 a da) (D2' a da)) + %<- ({a} {da} can-mekd-wf (D3 a da) (D3' a da)) %. +%term _ can-mecn-deq (ecn-deq/cn/unit _) mecn-deq/cn/unit %. +%term _ can-mecn-deq (ecn-deq/tp/unit _) mecn-deq/tp/unit %. +%term _ can-mecn-deq (ecn-deq/tp/tagged _) mecn-deq/tp/tagged %. +%term _ %pi (can-mecn-deq (ecn-deq/sym D1) (mecn-deq/sym D1')) %<- (can-mecn-deq D1 D1') %. +%term _ + %pi (can-mecn-deq (ecn-deq/tp/ref D1) (mecn-deq/tp/ref D1')) + %<- (can-mecn-deq D1 D1') %. +%term _ + %pi (can-mecn-deq (ecn-deq/tp/tag D1) (mecn-deq/tp/tag D1')) + %<- (can-mecn-deq D1 D1') %. +%term _ + %pi (can-mecn-deq (ecn-deq/kd/sing D1) (mecn-deq/kd/sing D1')) + %<- (can-meofkd D1 D1') %. +%term _ %pi (can-mecn-deq (ecn-deq/refl D1) (mecn-deq/refl D1')) %<- (can-meofkd D1 D1') %. +%term _ + %pi (can-mecn-deq (ecn-deq/cn/pj1 D1) (mecn-deq/cn/pj1 D1')) + %<- (can-mecn-deq D1 D1') %. +%term _ + %pi (can-mecn-deq (ecn-deq/cn/pj2 D1) (mecn-deq/cn/pj2 D1')) + %<- (can-mecn-deq D1 D1') %. +%term _ + %pi (can-mecn-deq (ecn-deq/kd/unit D1 D2) (mecn-deq/kd/unit D1' D2')) + %<- (can-meofkd D1 D1') + %<- (can-meofkd D2 D2') %. +%term _ + %pi (can-mecn-deq (ecn-deq/trans D1 D2) (mecn-deq/trans D1' D2')) + %<- (can-mecn-deq D1 D1') + %<- (can-mecn-deq D2 D2') %. +%term _ + %pi (can-mecn-deq (ecn-deq/cn/pair D1 D2) (mecn-deq/cn/pair D1' D2')) + %<- (can-mecn-deq D1 D1') + %<- (can-mecn-deq D2 D2') %. +%term _ + %pi (can-mecn-deq (ecn-deq/tp/cross D1 D2) (mecn-deq/tp/cross D1' D2')) + %<- (can-mecn-deq D1 D1') + %<- (can-mecn-deq D2 D2') %. +%term _ + %pi (can-mecn-deq (ecn-deq/tp/arrow D1 D2) (mecn-deq/tp/arrow D1' D2')) + %<- (can-mecn-deq D1 D1') + %<- (can-mecn-deq D2 D2') %. +%term _ + %pi (can-mecn-deq (ecn-deq/tp/sum D1 D2) (mecn-deq/tp/sum D1' D2')) + %<- (can-mecn-deq D1 D1') + %<- (can-mecn-deq D2 D2') %. +%term _ + %pi (can-mecn-deq (ecn-deq/sgm-ext D1 D2) (mecn-deq/sgm-ext D1' D2')) + %<- (can-mecn-deq D1 D1') + %<- (can-mecn-deq D2 D2') %. +%term _ + %pi (can-mecn-deq (ecn-deq/cn/app D1 D2) (mecn-deq/cn/app D1' D2')) + %<- (can-mecn-deq D1 D1') + %<- (can-mecn-deq D2 D2') %. +%term _ + %pi (can-mecn-deq (ecn-deq/pi-ext D1 D2 D3) (mecn-deq/pi-ext D1' D2' D3')) + %<- (can-meofkd D1 D1') + %<- (can-meofkd D2 D2') + %<- ({a} {da} can-mecn-deq (D3 a da) (D3' a da)) %. +%term _ + %pi (can-mecn-deq (ecn-deq/pi-ext-2 D1 D3) (mecn-deq/pi-ext-2 D1' D3')) + %<- (can-mecn-deq D1 D1') + %<- ({a} {da} can-mecn-deq (D3 a da) (D3' a da)) %. +%term _ + %pi (can-mecn-deq (ecn-deq/cn/lam D1 D2) (mecn-deq/cn/lam D1' D2')) + %<- ({a} {da} can-mecn-deq (D1 a da) (D1' a da)) + %<- (can-mekd-deq D2 D2') %. +%term _ + %pi (can-mecn-deq (ecn-deq/tp/forall D1 D2) (mecn-deq/tp/forall D1' D2')) + %<- ({a} {da} can-mecn-deq (D1 a da) (D1' a da)) + %<- (can-mekd-deq D2 D2') %. +%term _ + %pi (can-mecn-deq (ecn-deq/cn/mu D1 D2) (mecn-deq/cn/mu D1' D2')) + %<- ({a} {da} can-mecn-deq (D1 a da) (D1' a da)) + %<- (can-mekd-deq D2 D2') %. +%term _ + %pi (can-mecn-deq (ecn-deq/sub D1 DS) (mecn-deq/sub D1' DS')) + %<- (can-mecn-deq D1 D1') + %<- (can-mekd-sub DS DS') %. +%term _ + %pi (can-mecn-deq (ecn-deq/deq D1 DS) (mecn-deq/deq D1' DS')) + %<- (can-mecn-deq D1 D1') + %<- (can-mekd-deq DS DS') %. +%worlds (cn-block can-mofkd-block ovar-block) (can-meofkd _ _) (can-mekd-wf _ _) (can-mekd-deq _ _) (can-mekd-sub _ _) (can-mecn-deq _ _) %. +%total (D1 D2 D3 D4 D5) (can-meofkd D1 _) (can-mekd-wf D2 _) (can-mekd-deq D4 _) (can-mekd-sub D3 _) (can-mecn-deq D5 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/fst-md-lemmas.lf b/new-tests/stelf-output/tslf/fst-md-lemmas.lf new file mode 100644 index 0000000..c0d3ed1 --- /dev/null +++ b/new-tests/stelf-output/tslf/fst-md-lemmas.lf @@ -0,0 +1,27 @@ +%%%% lemmas about fst (it turns out much of what I had to prove about +%%%% fst on paper were part of validity. +%{ fst-md-seq : fst-md M C + -> fst-md M C' + -> seq/cn C C' + -> type. + +%mode fst-md-seq +D1 +D2 -D3. }% +%term _ fst-md-seq fst-md/unit fst-md/unit seq/cn/refl %. +%term _ fst-md-seq fst-md/cn fst-md/cn seq/cn/refl %. +%term _ fst-md-seq fst-md/tm fst-md/tm seq/cn/refl %. +%term _ + %pi (fst-md-seq (fst-md/pair D1 D2) (fst-md/pair D3 D4) DQ) + %<- (fst-md-seq D1 D3 DQ1) + %<- (fst-md-seq D2 D4 DQ2) + %<- (seq/cn/pair DQ1 DQ2 DQ) %. +%term _ + %pi (fst-md-seq (fst-md/pj1 D1) (fst-md/pj1 D2) DQ') + %<- (fst-md-seq D1 D2 DQ) + %<- (seq/cn/pj1 DQ DQ') %. +%term _ + %pi (fst-md-seq (fst-md/pj2 D1) (fst-md/pj2 D2) DQ') + %<- (fst-md-seq D1 D2 DQ) + %<- (seq/cn/pj2 DQ DQ') %. +%term _ fst-md-seq fst-md/lam fst-md/lam seq/cn/refl %. +%worlds (ofkd+vdt-block ofsg+vdt-block oftp+vdt-block) (fst-md-seq _ _ _) %. +%total {D1 D2} (fst-md-seq D1 D2 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/fst-sg-lemmas.lf b/new-tests/stelf-output/tslf/fst-sg-lemmas.lf new file mode 100644 index 0000000..e12a940 --- /dev/null +++ b/new-tests/stelf-output/tslf/fst-sg-lemmas.lf @@ -0,0 +1,28 @@ +%%%% lemmas about fst (it turns out much of what I had to prove about +%%%% fst on paper were part of validity. +%%%%% first on signatures +%sort fst-sg-complete {S sg} %. +%mode fst-sg-complete %in %out %. +%term _ fst-sg-complete _ fst-sg/unit %. +%term _ fst-sg-complete _ fst-sg/kd %. +%term _ fst-sg-complete _ fst-sg/cn %. +%term _ + %pi (fst-sg-complete (sg/sgm S1 S2) (fst-sg/sgm D1 D2)) + %<- (fst-sg-complete S1 D1) + %<- ({a cn} fst-sg-complete (S2 a) (D2 a)) %. +%term _ fst-sg-complete _ fst-sg/pi %. +%worlds (cn-block) (fst-sg-complete _ _) %. +%total {S} (fst-sg-complete S _) %. +%sort fst-sg-seq %. +%mode fst-sg-seq %in %in %out %. +%term _ fst-sg-seq fst-sg/unit fst-sg/unit seq/kd/refl %. +%term _ fst-sg-seq fst-sg/kd fst-sg/kd seq/kd/refl %. +%term _ fst-sg-seq fst-sg/cn fst-sg/cn seq/kd/refl %. +%term _ + %pi (fst-sg-seq (fst-sg/sgm D1 D2) (fst-sg/sgm D3 D4) D7) + %<- (fst-sg-seq D1 D3 D5) + %<- ({a cn} fst-sg-seq (D2 a) (D4 a) (D6 a)) + %<- (seq/kd/sgm D5 D6 D7) %. +%term _ fst-sg-seq fst-sg/pi fst-sg/pi seq/kd/refl %. +%worlds (cn-block) (fst-sg-seq _ _ _) %. +%total {D1 D2} (fst-sg-seq D1 D2 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/functionality-cnkd-e.lf b/new-tests/stelf-output/tslf/functionality-cnkd-e.lf new file mode 100644 index 0000000..5aeb0f0 --- /dev/null +++ b/new-tests/stelf-output/tslf/functionality-cnkd-e.lf @@ -0,0 +1,140 @@ +% + +%sort ecn-deq/seq/kd %. +%mode ecn-deq/seq/kd %in %in %out %. +%term _ ecn-deq/seq/kd seq/kd/refl D1 D1 %. +%worlds (cn-block isvar-fun-block can-mofkd-block) (ecn-deq/seq/kd _ _ _) %. +%total D1 (ecn-deq/seq/kd D1 _ _) %. +%sort funct-cxt-lookup {x cn} %. +%mode funct-cxt-lookup %in %in %in %in %in %out %. +%term _ + %pi (funct-cxt-lookup DA DA' DC DL id-or-cl/cl (ecn-deq/refl (eofkd/var DL'))) + %<- (subst-cxt-lookup-c DA DA' DL DL') %. +%term _ + %pi (funct-cxt-lookup DA DA' DC DL id-or-cl/id DC'') + %<- (cxt-lookup-unique DA DL DQ) + %<- ({x} {di isvar x I} cxt-lookup-ordered (DL x di) (DO x di)) + %<- (subst-cxt-ordered DA DA' DO DO') + %<- (wkn-ecn-deq-append DC DA' DO' DC') + %<- (ecn-deq/seq/kd DQ DC' DC'') %. +%worlds (cn-block isvar-fun-block can-mofkd-block) (funct-cxt-lookup _ _ _ _ _ _) %. +%total D1 (funct-cxt-lookup _ _ _ _ D1 _) %. +%sort funct-closed-ofkd^ {C1 cn} {C2 cn} %. +%mode funct-closed-ofkd^ %in %in %in %in %in %in %out %. +%term _ funct-closed-ofkd^ _ _ D1 closed-cn/i closed-kd/i DO (ecn-deq/refl (eofkd/closed D1 DO)) %. +%worlds (cn-block isvar-fun-block can-mofkd-block) (funct-closed-ofkd^ _ _ _ _ _ _ _) %. +%total {} (funct-closed-ofkd^ _ _ _ _ _ _ _) %. +%sort funct-closed-ofkd {C1 cn} {C2 cn} {x cn} %. +%mode funct-closed-ofkd %in %in %in %in %out %. +%term _ + %pi (funct-closed-ofkd C1 C2 D1 DO DQ) + %<- (strengthen-ofkd D1 DC1 DC2 D1') + %<- (funct-closed-ofkd^ C1 C2 D1' DC1 DC2 DO DQ) %. +%worlds (cn-block isvar-fun-block can-mofkd-block) (funct-closed-ofkd _ _ _ _ _) %. +%total {} (funct-closed-ofkd _ _ _ _ _) %. +%sort funct-eofkd {x cn} %. +%mode funct-eofkd %in %in %in %in %in %in %out %. +%sort funct-ekd-wf {x cn} %. +%mode funct-ekd-wf %in %in %in %in %in %in %out %out %. +%term _ + %pi (funct-ekd-wf DA DA' DC DCC DQ ([x] [di] ekd-wf/kd/unit (DO x di)) (ekd-deq/kd/unit DO') (ekd-deq/kd/unit DO')) + %<- (subst-cxt-ordered DA DA' DO DO') %. +%term _ + %pi (funct-ekd-wf DA DA' DC DCC DQ ([x] [di] ekd-wf/kd/type (DO x di)) (ekd-deq/kd/type DO') (ekd-deq/kd/type DO')) + %<- (subst-cxt-ordered DA DA' DO DO') %. +%term _ + %pi (funct-ekd-wf DA DA' DC DCCz DQ ([x] [di] ekd-wf/kd/sing (DCC x di)) (ekd-deq/kd/sing DCQ) (ekd-deq/kd/sing (ecn-deq/sym DCQ))) + %<- (funct-eofkd DA DA' DC DCCz DQ DCC DCQ) %. +%term _ + %pi (funct-ekd-wf DA DA' DC DCC DQ ([x] [di] ekd-wf/kd/sgm (D1 x di) (D2 x di)) (ekd-deq/kd/sgm D1' D2') (ekd-deq/kd/sgm D1'' D2''w)) + %<- (funct-ekd-wf DA DA' DC DCC DQ D1 D1' D1'') + %<- ({y} {dy} {_ isvar-fun dy dy seq/loc/refl} funct-ekd-wf ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC DCC DQ ([x] [di] D2 x di y dy) (D2' y dy) (D2'' y dy)) + %<- (wkn-deq-ekd-deq ([x] cxt-append/nil) ([x] cxt-append/nil) D1'' D2'' D2''w) %. +%term _ + %pi (funct-ekd-wf DA DA' DC DCC DQ ([x] [di] ekd-wf/kd/pi (D1 x di) (D2 x di)) (ekd-deq/kd/pi D1'' D2'w) (ekd-deq/kd/pi D1' D2'')) + %<- (funct-ekd-wf DA DA' DC DCC DQ D1 D1' D1'') + %<- ({y} {dy} {_ isvar-fun dy dy seq/loc/refl} funct-ekd-wf ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC DCC DQ ([x] [di] D2 x di y dy) (D2' y dy) (D2'' y dy)) + %<- (wkn-deq-ekd-deq ([x] cxt-append/nil) ([x] cxt-append/nil) D1'' D2' D2'w) %. +%term _ + %pi (funct-eofkd DA DA' DC DCC DQ ([x] [di] eofkd/cn/unit (DO x di)) (ecn-deq/cn/unit DO')) + %<- (subst-cxt-ordered DA DA' DO DO') %. +%term _ + %pi (funct-eofkd DA DA' DC DCC DQ ([x] [di] eofkd/tp/unit (DO x di)) (ecn-deq/tp/unit DO')) + %<- (subst-cxt-ordered DA DA' DO DO') %. +%term _ + %pi (funct-eofkd DA DA' DC DCC DQ ([x] [di] eofkd/tp/tagged (DO x di)) (ecn-deq/tp/tagged DO')) + %<- (subst-cxt-ordered DA DA' DO DO') %. +%term _ + %pi (funct-eofkd DA DA' DC DCCz DQ ([x] [di] eofkd/tp/ref (DCC x di)) (ecn-deq/tp/ref DCQ)) + %<- (funct-eofkd DA DA' DC DCCz DQ DCC DCQ) %. +%term _ + %pi (funct-eofkd DA DA' DC DCCz DQ ([x] [di] eofkd/tp/tag (DCC x di)) (ecn-deq/tp/tag DCQ)) + %<- (funct-eofkd DA DA' DC DCCz DQ DCC DCQ) %. +%term _ + %pi (funct-eofkd DA DA' DC DCCz DQ ([x] [di] eofkd/cn/pj1 (DCC x di)) (ecn-deq/cn/pj1 DCQ)) + %<- (funct-eofkd DA DA' DC DCCz DQ DCC DCQ) %. +%term _ + %pi (funct-eofkd DA DA' DC DCCz DQ ([x] [di] eofkd/cn/pj2 (DCC x di)) (ecn-deq/cn/pj2 DCQ)) + %<- (funct-eofkd DA DA' DC DCCz DQ DCC DCQ) %. +%term _ + %pi (funct-eofkd DA DA' DC DCCz DQ ([x] [di] eofkd/kd/sing (D1 x di)) (ecn-deq/sub (ecn-deq/kd/sing (eofkd/sub (eofkd/kd/sing D1s) (ekd-sub/kd/sing-kd/sing D1Q))) (ekd-sub/kd/sing-kd/sing (ecn-deq/sym D1Q)))) + %<- (funct-eofkd DA DA' DC DCCz DQ D1 D1Q) + %<- (subst-eofkd DA DA' DC D1 D1s) %. +%term _ + %pi (funct-eofkd DA DA' DC DCCz DQ ([x] [di] eofkd/cn/pair (D1 x di) (D2 x di)) (ecn-deq/cn/pair D1' D2')) + %<- (funct-eofkd DA DA' DC DCCz DQ D1 D1') + %<- (funct-eofkd DA DA' DC DCCz DQ D2 D2') %. +%term _ + %pi (funct-eofkd DA DA' DC DCCz DQ ([x] [di] eofkd/cn/app (D1 x di) (D2 x di)) (ecn-deq/cn/app D1' D2')) + %<- (funct-eofkd DA DA' DC DCCz DQ D1 D1') + %<- (funct-eofkd DA DA' DC DCCz DQ D2 D2') %. +%term _ + %pi (funct-eofkd DA DA' DC DCCz DQ ([x] [di] eofkd/sgm-ext (D1 x di) (D2 x di)) (ecn-deq/sgm-ext D1' D2')) + %<- (funct-eofkd DA DA' DC DCCz DQ D1 D1') + %<- (funct-eofkd DA DA' DC DCCz DQ D2 D2') %. +%term _ + %pi (funct-eofkd DA DA' DC DCCz DQ ([x] [di] eofkd/tp/cross (D1 x di) (D2 x di)) (ecn-deq/tp/cross D1' D2')) + %<- (funct-eofkd DA DA' DC DCCz DQ D1 D1') + %<- (funct-eofkd DA DA' DC DCCz DQ D2 D2') %. +%term _ + %pi (funct-eofkd DA DA' DC DCCz DQ ([x] [di] eofkd/tp/arrow (D1 x di) (D2 x di)) (ecn-deq/tp/arrow D1' D2')) + %<- (funct-eofkd DA DA' DC DCCz DQ D1 D1') + %<- (funct-eofkd DA DA' DC DCCz DQ D2 D2') %. +%term _ + %pi (funct-eofkd DA DA' DC DCCz DQ ([x] [di] eofkd/tp/sum (D1 x di) (D2 x di)) (ecn-deq/tp/sum D1' D2')) + %<- (funct-eofkd DA DA' DC DCCz DQ D1 D1') + %<- (funct-eofkd DA DA' DC DCCz DQ D2 D2') %. +%term _ + %pi (funct-eofkd DA DA' DC DCCz DQ ([x] [di] eofkd/sub (D1 x di) (D2 x di)) (ecn-deq/sub D1' D2')) + %<- (funct-eofkd DA DA' DC DCCz DQ D1 D1') + %<- (subst-ekd-sub DA DA' DC D2 D2') %. +%term _ + %pi (funct-eofkd DA DA' DC DCCz DQ ([x] [di] eofkd/deq (D1 x di) (D2 x di)) (ecn-deq/deq D1' D2')) + %<- (funct-eofkd DA DA' DC DCCz DQ D1 D1') + %<- (subst-ekd-deq DA DA' DC D2 D2') %. +%term _ + %pi (funct-eofkd DA DA' DC DCCz DQ ([x] [di] eofkd/cn/lam (D1 x di) (D2 x di)) (ecn-deq/cn/lam D1' D2')) + %<- ({y} {dy} {_ isvar-fun dy dy seq/loc/refl} funct-eofkd ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC DCCz DQ ([x] [di] D1 x di y dy) (D1' y dy)) + %<- (funct-ekd-wf DA DA' DC DCCz DQ D2 D2' _) %. +%term _ + %pi (funct-eofkd DA DA' DC DCCz DQ ([x] [di] eofkd/tp/forall (D1 x di) (D2 x di)) (ecn-deq/tp/forall D1' D2')) + %<- ({y} {dy} {_ isvar-fun dy dy seq/loc/refl} funct-eofkd ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC DCCz DQ ([x] [di] D1 x di y dy) (D1' y dy)) + %<- (funct-ekd-wf DA DA' DC DCCz DQ D2 D2' _) %. +%term _ + %pi (funct-eofkd DA DA' DC DCCz DQ ([x] [di] eofkd/cn/mu (D1 x di) (D2 x di)) (ecn-deq/cn/mu D1' D2')) + %<- ({y} {dy} {_ isvar-fun dy dy seq/loc/refl} funct-eofkd ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC DCCz DQ ([x] [di] D1 x di y dy) (D1' y dy)) + %<- (funct-ekd-wf DA DA' DC DCCz DQ D2 D2' _) %. +%term _ + %pi (funct-eofkd DA DA' DC DCCz DQ ([x] [di] eofkd/pi-ext (D2 x di) (D1 x di)) (ecn-deq/pi-ext-2 D2' D1')) + %<- ({y} {dy} {_ isvar-fun dy dy seq/loc/refl} funct-eofkd ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC DCCz DQ ([x] [di] D1 x di y dy) (D1' y dy)) + %<- (funct-eofkd DA DA' DC DCCz DQ D2 D2') %. +%term _ + %pi (funct-eofkd DA DA' DC DCCz DQ ([x] [di] eofkd/var (DL x di)) DC') + %<- (cxt-lookup-ioc DA DA' DL IOC) + %<- (funct-cxt-lookup DA DA' DQ DL IOC DC') %. +%term _ + %pi (funct-eofkd DA DA' _ _ (%the (ecn-deq _ C1 C2 _) _) ([x] [di] eofkd/closed (DC x) (DO x di)) DQ) + %<- (subst-cxt-ordered DA DA' DO DO') + %<- (funct-closed-ofkd C1 C2 DC DO' DQ) %. +%worlds (cn-block isvar-fun-block can-mofkd-block) (funct-ekd-wf _ _ _ _ _ _ _ _) (funct-eofkd _ _ _ _ _ _ _) %. +%total (D1 D3) (funct-ekd-wf _ _ _ _ _ D1 _ _) (funct-eofkd _ _ _ _ _ D3 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/functionality-cnkd.lf b/new-tests/stelf-output/tslf/functionality-cnkd.lf new file mode 100644 index 0000000..3196f3c --- /dev/null +++ b/new-tests/stelf-output/tslf/functionality-cnkd.lf @@ -0,0 +1,22 @@ +%sort funct/kd-wf {a cn} {da ofkd a K} %. +%mode funct/kd-wf %in %in %in %in %out %. +%sort funct/ofkd {a cn} {da ofkd a K} %. +%mode funct/ofkd %in %in %in %in %out %. +%term _ + %pi (funct/kd-wf DO DQ D1 D2 DQI) + %<- (explicate-open-kd-wf loc/z DO DO') + %<- (explicate-closed-cn-deq DQ DQ') + %<- (explicate-closed-ofkd D1 D1') + %<- (explicate-closed-ofkd D2 D2') + %<- (funct-ekd-wf ([x] cxt-append/nil) (%the (cxt-append-sub _ _ _ _) cxt-append-sub/nil) D1' D2' DQ' DO' DQE _) + %<- (implicate-closed/kd-deq DQE DQI) %. +%term _ + %pi (funct/ofkd DO DQ D1 D2 DQI) + %<- (explicate-open-ofkd loc/z DO DO') + %<- (explicate-closed-cn-deq DQ DQ') + %<- (explicate-closed-ofkd D1 D1') + %<- (explicate-closed-ofkd D2 D2') + %<- (funct-eofkd ([x] cxt-append/nil) (%the (cxt-append-sub _ _ _ _) cxt-append-sub/nil) D1' D2' DQ' DO' DQE) + %<- (implicate-closed/cn-deq DQE DQI) %. +%worlds (can-mofkd-block ofkd+vdt-block) (funct/ofkd _ _ _ _ _) (funct/kd-wf _ _ _ _ _) %. +%total {} (funct/kd-wf _ _ _ _ _) (funct/ofkd _ _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/functionality-full.lf b/new-tests/stelf-output/tslf/functionality-full.lf new file mode 100644 index 0000000..96ac571 --- /dev/null +++ b/new-tests/stelf-output/tslf/functionality-full.lf @@ -0,0 +1,10 @@ +%%%% SH proposition 3.14, full functionality +%sort functf/cn-deq %. +%mode functf/cn-deq %in %in %out %. +%term _ + %pi (functf/cn-deq Deq Deq' (cn-deq/trans (Deq' C1 D1) Deq2)) + %<- (vdt/cn-deq Deq D1 D2 DK) + %<- ({a} {da ofkd a K} {dm mofkd da met/unit} {_ can-mofkd da dm} {dw vdt/ofkd da DK} vdt/cn-deq (Deq' a da) (D1' a da) (D2' a da) (DK' a da)) + %<- (funct/ofkd D2' Deq D1 D2 Deq2) %. +%worlds (ofkd+vdt-block) (functf/cn-deq _ _ _) %. +%total {} (functf/cn-deq _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/functionality-sigs-e.lf b/new-tests/stelf-output/tslf/functionality-sigs-e.lf new file mode 100644 index 0000000..63f37c0 --- /dev/null +++ b/new-tests/stelf-output/tslf/functionality-sigs-e.lf @@ -0,0 +1,72 @@ +%sort ekd-wf/seq-a %. +%mode ekd-wf/seq-a %in %in %out %. +%term _ ekd-wf/seq-a seq/kd/refl D1 D1 %. +%worlds (cn-block ovar-block ofkd-block) (ekd-wf/seq-a _ _ _) %. +%total {} (ekd-wf/seq-a _ _ _) %. +%sort vdt/esg-wf %. +%mode vdt/esg-wf %in %in %out %. +%term _ vdt/esg-wf (esg-wf/sg/unit DO) fst-sg/unit (ekd-wf/kd/unit DO) %. +%term _ vdt/esg-wf (esg-wf/sg/kd DK) fst-sg/kd DK %. +%term _ + %pi (vdt/esg-wf (esg-wf/sg/cn DC) fst-sg/cn (ekd-wf/kd/unit DO)) + %<- (eofkd-ordered DC DO) %. +%term _ + %pi (vdt/esg-wf (esg-wf/sg/sgm D1 D2 DFS') (fst-sg/sgm DFS DFS2) (ekd-wf/kd/sgm DW DW2)) + %<- (fst-sg-seq DFS' DFS (%the (seq/kd K3 K1) DQ)) + %<- (vdt/esg-wf D1 DFS DW) + %<- (vdt/esg-wf D1 DFS' DW') + %<- ({a cn} {da} vdt/esg-wf (D2 a da) (DFS2 a) (DW2' a da)) + %<- (ekd-wf/seq-a DQ DW2' DW2) %. +%term _ + %pi (vdt/esg-wf (esg-wf/sg/pi D1 _ _) fst-sg/pi (ekd-wf/kd/unit DO)) + %<- (esg-wf-ordered D1 DO) %. +%worlds (cn-block ovar-block ofkd-block) (vdt/esg-wf _ _ _) %. +%total D1 (vdt/esg-wf D1 _ _) %. +%sort wkn-deq-esg-deq {x} %. +%mode wkn-deq-esg-deq %in %in %in %in %out %. +%term _ + %pi (wkn-deq-esg-deq DA DA' _ ([x] [di] esg-deq/sg/unit (DO x di)) ([x] [di] esg-deq/sg/unit (DO' x di))) + %<- ({x} {di} wkn-deq-cxt-ordered (DA x) (DA' x) (DO x di) (DO' x di)) %. +%term _ + %pi (wkn-deq-esg-deq DA DA' DQ ([x] [di] esg-deq/sg/cn (D1 x di)) ([x] [di] esg-deq/sg/cn (D1' x di))) + %<- (wkn-deq-ecn-deq DA DA' DQ D1 D1') %. +%term _ + %pi (wkn-deq-esg-deq DA DA' DQ ([x] [di] esg-deq/sg/kd (D1 x di)) ([x] [di] esg-deq/sg/kd (D1' x di))) + %<- (wkn-deq-ekd-deq DA DA' DQ D1 D1') %. +%term _ + %pi (wkn-deq-esg-deq DA DA' DQ ([x] [di] esg-deq/sg/sgm (D1 x di) (D2 x di) (DFS x)) ([x] [di] esg-deq/sg/sgm (D1' x di) (D2' x di) (DFS x))) + %<- (wkn-deq-esg-deq DA DA' DQ D1 D1') + %<- ({y} {dy} wkn-deq-esg-deq ([x] cxt-append/cons (DA x)) ([x] cxt-append/cons (DA' x)) DQ ([x] [di] D2 x di y dy) ([x] [di] D2' x di y dy)) %. +%term _ + %pi (wkn-deq-esg-deq DA DA' DQ ([x] [di] esg-deq/sg/pi (D1 x di) (D2 x di) (DFS x)) ([x] [di] esg-deq/sg/pi (D1' x di) (D2' x di) (DFS x))) + %<- (wkn-deq-esg-deq DA DA' DQ D1 D1') + %<- ({y} {dy} wkn-deq-esg-deq ([x] cxt-append/cons (DA x)) ([x] cxt-append/cons (DA' x)) DQ ([x] [di] D2 x di y dy) ([x] [di] D2' x di y dy)) %. +%worlds (cn-block ovar-block can-mofkd-block) (wkn-deq-esg-deq _ _ _ _ _) %. +%total (D1) (wkn-deq-esg-deq _ _ _ D1 _) %. +%sort funct-esg-wf {x cn} %. +%mode funct-esg-wf %in %in %in %in %in %in %out %out %. +%term _ + %pi (funct-esg-wf DA DA' DC DCC DQ ([x] [di] esg-wf/sg/unit (DO x di)) (esg-deq/sg/unit DO') (esg-deq/sg/unit DO')) + %<- (subst-cxt-ordered DA DA' DO DO') %. +%term _ + %pi (funct-esg-wf DA DA' DC DCCz DQ ([x] [di] esg-wf/sg/cn (DCC x di)) (esg-deq/sg/cn DCQ) (esg-deq/sg/cn (ecn-deq/sym DCQ))) + %<- (funct-eofkd DA DA' DC DCCz DQ DCC DCQ) %. +%term _ + %pi (funct-esg-wf DA DA' DC DCCz DQ ([x] [di] esg-wf/sg/kd (DCC x di)) (esg-deq/sg/kd DQ1) (esg-deq/sg/kd DQ2)) + %<- (funct-ekd-wf DA DA' DC DCCz DQ DCC DQ1 DQ2) %. +%term _ + %pi (funct-esg-wf DA DA' DC DCC (%the (ecn-deq _ C1 C2 _) DQ) ([x] [di] esg-wf/sg/sgm (D1 x di) (D2 x di) (DFS x)) (esg-deq/sg/sgm D1' D2' (DFS C1)) (esg-deq/sg/sgm D1'' D2''w (DFS C2))) + %<- (funct-esg-wf DA DA' DC DCC DQ D1 D1' D1'') + %<- ({y} {dy} {_ isvar-fun dy dy seq/loc/refl} funct-esg-wf ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC DCC DQ ([x] [di] D2 x di y dy) (D2' y dy) (D2'' y dy)) + %<- ({x} {di} vdt/esg-wf (D1 x di) (DFS x) (D1c x di)) + %<- (funct-ekd-wf DA DA' DC DCC DQ D1c _ D1k'') + %<- (wkn-deq-esg-deq ([x] cxt-append/nil) ([x] cxt-append/nil) D1k'' D2'' D2''w) %. +%term _ + %pi (funct-esg-wf DA DA' DC DCC (%the (ecn-deq _ C1 C2 _) DQ) ([x] [di] esg-wf/sg/pi (D1 x di) (D2 x di) (DFS x)) (esg-deq/sg/pi D1'' D2'w (DFS C2)) (esg-deq/sg/pi D1' D2'' (DFS C1))) + %<- (funct-esg-wf DA DA' DC DCC DQ D1 D1' D1'') + %<- ({y} {dy} {_ isvar-fun dy dy seq/loc/refl} funct-esg-wf ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC DCC DQ ([x] [di] D2 x di y dy) (D2' y dy) (D2'' y dy)) + %<- ({x} {di} vdt/esg-wf (D1 x di) (DFS x) (D1c x di)) + %<- (funct-ekd-wf DA DA' DC DCC DQ D1c _ D1k'') + %<- (wkn-deq-esg-deq ([x] cxt-append/nil) ([x] cxt-append/nil) D1k'' D2' D2'w) %. +%worlds (cn-block isvar-fun-block can-mofkd-block) (funct-esg-wf _ _ _ _ _ _ _ _) %. +%total (D1) (funct-esg-wf _ _ _ _ _ D1 _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/functionality-sigs.lf b/new-tests/stelf-output/tslf/functionality-sigs.lf new file mode 100644 index 0000000..a8f6119 --- /dev/null +++ b/new-tests/stelf-output/tslf/functionality-sigs.lf @@ -0,0 +1,13 @@ +%sort funct/sg-wf {a cn} {da ofkd a K} %. +%mode funct/sg-wf %in %in %in %out %. +%term _ + %pi (funct/sg-wf DO DQ D1 DQI) + %<- (vdt/cn-deq DQ _ D2 _) + %<- (explicate-open-sg-wf loc/z DO DO') + %<- (explicate-closed-cn-deq DQ DQ') + %<- (explicate-closed-ofkd D1 D1') + %<- (explicate-closed-ofkd D2 D2') + %<- (funct-esg-wf ([x] cxt-append/nil) cxt-append-sub/nil D1' D2' DQ' DO' DQE _) + %<- (implicate-closed/sg-deq DQE DQI) %. +%worlds (ofkd+vdt-block) (funct/sg-wf _ _ _ _) %. +%total (D1) (funct/sg-wf D1 _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/implicate-lemmas-sigs.lf b/new-tests/stelf-output/tslf/implicate-lemmas-sigs.lf new file mode 100644 index 0000000..15273d0 --- /dev/null +++ b/new-tests/stelf-output/tslf/implicate-lemmas-sigs.lf @@ -0,0 +1,88 @@ +%%%% lemmas needed for implcate +% "halfway" explicit semantics? +%sort isg-wf %. +%term isg-wf/nil %pi (isg-wf cxt/nil K) %<- (sg-wf K) %. +%term isg-wf/cons %pi (isg-wf (cxt/cons G C K) K') %<- (%pi (ofkd C K) %-> (isg-wf G K')) %. +%sort isg-deq %. +%term isg-deq/nil %pi (isg-deq cxt/nil K K') %<- (sg-deq K K') %. +%term isg-deq/cons %pi (isg-deq (cxt/cons G C K) K1 K2) %<- (%pi (ofkd C K) %-> (isg-deq G K1 K2)) %. +%sort sg-wf-to-isg-wf {G cxt} %. +%mode sg-wf-to-isg-wf %in %in %out %. +%term _ sg-wf-to-isg-wf cxt/nil D (isg-wf/nil D) %. +%term _ + %pi (sg-wf-to-isg-wf (cxt/cons G C K) D (isg-wf/cons ([d] D'))) + %<- (sg-wf-to-isg-wf G D D') %. +%worlds (ovar-block iofkd-block ofkd-block) (sg-wf-to-isg-wf _ _ _) %. +%total G (sg-wf-to-isg-wf G _ _) %. +%sort sg-deq-to-isg-deq {G cxt} %. +%mode sg-deq-to-isg-deq %in %in %out %. +%term _ sg-deq-to-isg-deq cxt/nil D (isg-deq/nil D) %. +%term _ + %pi (sg-deq-to-isg-deq (cxt/cons G C K) D (isg-deq/cons ([d] D'))) + %<- (sg-deq-to-isg-deq G D D') %. +%worlds (ovar-block iofkd-block ofkd-block) (sg-deq-to-isg-deq _ _ _) %. +%total G (sg-deq-to-isg-deq G _ _) %. +%sort isg-wf/sg/cn %. +%mode isg-wf/sg/cn %in %out %. +%term _ isg-wf/sg/cn (iofkd/nil D) (isg-wf/nil (sg-wf/sg/cn D)) %. +%term _ + %pi (isg-wf/sg/cn (iofkd/cons D) (isg-wf/cons D')) + %<- ({d ofkd C K} isg-wf/sg/cn (D d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (isg-wf/sg/cn _ _) %. +%total D (isg-wf/sg/cn D _) %. +%sort isg-wf/sg/kd %. +%mode isg-wf/sg/kd %in %out %. +%term _ isg-wf/sg/kd (ikd-wf/nil D) (isg-wf/nil (sg-wf/sg/kd D)) %. +%term _ + %pi (isg-wf/sg/kd (ikd-wf/cons D) (isg-wf/cons D')) + %<- ({d ofkd C K} isg-wf/sg/kd (D d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (isg-wf/sg/kd _ _) %. +%total D (isg-wf/sg/kd D _) %. +%sort isg-wf/sg/sgm %. +%mode isg-wf/sg/sgm %in %in %in %out %. +%term _ isg-wf/sg/sgm (isg-wf/nil D) DFS ([c] [d] isg-wf/nil (D' c d)) (isg-wf/nil (sg-wf/sg/sgm D D' DFS)) %. +%term _ + %pi (isg-wf/sg/sgm (isg-wf/cons D1) DFS ([c cn] [d ofkd c _] isg-wf/cons ([d'] D2 c d d')) (isg-wf/cons D')) + %<- ({d ofkd C K} isg-wf/sg/sgm (D1 d) DFS ([c] [d'] D2 c d' d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (isg-wf/sg/sgm _ _ _ _) %. +%total D (isg-wf/sg/sgm D _ _ _) %. +%sort isg-wf/sg/pi %. +%mode isg-wf/sg/pi %in %in %in %out %. +%term _ isg-wf/sg/pi (isg-wf/nil D) DFS ([c] [d] isg-wf/nil (D' c d)) (isg-wf/nil (sg-wf/sg/pi D D' DFS)) %. +%term _ + %pi (isg-wf/sg/pi (isg-wf/cons D1) DFS ([c cn] [d ofkd c _] isg-wf/cons ([d'] D2 c d d')) (isg-wf/cons D')) + %<- ({d ofkd C K} isg-wf/sg/pi (D1 d) DFS ([c] [d'] D2 c d' d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (isg-wf/sg/pi _ _ _ _) %. +%total D (isg-wf/sg/pi D _ _ _) %. +%sort isg-deq/sg/cn %. +%mode isg-deq/sg/cn %in %out %. +%term _ isg-deq/sg/cn (icn-deq/nil D) (isg-deq/nil (sg-deq/sg/cn D)) %. +%term _ + %pi (isg-deq/sg/cn (icn-deq/cons D) (isg-deq/cons D')) + %<- ({d ofkd C K} isg-deq/sg/cn (D d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (isg-deq/sg/cn _ _) %. +%total D (isg-deq/sg/cn D _) %. +%sort isg-deq/sg/kd %. +%mode isg-deq/sg/kd %in %out %. +%term _ isg-deq/sg/kd (ikd-deq/nil D) (isg-deq/nil (sg-deq/sg/kd D)) %. +%term _ + %pi (isg-deq/sg/kd (ikd-deq/cons D) (isg-deq/cons D')) + %<- ({d ofkd C K} isg-deq/sg/kd (D d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (isg-deq/sg/kd _ _) %. +%total D (isg-deq/sg/kd D _) %. +%sort isg-deq/sg/sgm %. +%mode isg-deq/sg/sgm %in %in %in %out %. +%term _ isg-deq/sg/sgm (isg-deq/nil D) DFS ([c] [d] isg-deq/nil (D' c d)) (isg-deq/nil (sg-deq/sg/sgm D D' DFS)) %. +%term _ + %pi (isg-deq/sg/sgm (isg-deq/cons D1) DFS ([c cn] [d ofkd c _] isg-deq/cons ([d'] D2 c d d')) (isg-deq/cons D')) + %<- ({d ofkd C K} isg-deq/sg/sgm (D1 d) DFS ([c] [d'] D2 c d' d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (isg-deq/sg/sgm _ _ _ _) %. +%total D (isg-deq/sg/sgm D _ _ _) %. +%sort isg-deq/sg/pi %. +%mode isg-deq/sg/pi %in %in %in %out %. +%term _ isg-deq/sg/pi (isg-deq/nil D) DFS ([c] [d] isg-deq/nil (D' c d)) (isg-deq/nil (sg-deq/sg/pi D D' DFS)) %. +%term _ + %pi (isg-deq/sg/pi (isg-deq/cons D1) DFS ([c cn] [d ofkd c _] isg-deq/cons ([d'] D2 c d d')) (isg-deq/cons D')) + %<- ({d ofkd C K} isg-deq/sg/pi (D1 d) DFS ([c] [d'] D2 c d' d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (isg-deq/sg/pi _ _ _ _) %. +%total D (isg-deq/sg/pi D _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/implicate-lemmas.lf b/new-tests/stelf-output/tslf/implicate-lemmas.lf new file mode 100644 index 0000000..396a556 --- /dev/null +++ b/new-tests/stelf-output/tslf/implicate-lemmas.lf @@ -0,0 +1,464 @@ +%%%% lemmas needed for implcate +% "halfway" explicit semantics? +%sort ikd-wf %. +%term ikd-wf/nil %pi (ikd-wf cxt/nil K) %<- (kd-wf K) %. +%term ikd-wf/cons %pi (ikd-wf (cxt/cons G C K) K') %<- (%pi (ofkd C K) %-> (ikd-wf G K')) %. +%sort iofkd %. +%term iofkd/nil %pi (iofkd cxt/nil C K) %<- (ofkd C K) %. +%term iofkd/cons %pi (iofkd (cxt/cons G C K) C' K') %<- (%pi (ofkd C K) %-> (iofkd G C' K')) %. +%sort ikd-sub %. +%term ikd-sub/nil %pi (ikd-sub cxt/nil K K') %<- (kd-sub K K') %. +%term ikd-sub/cons %pi (ikd-sub (cxt/cons G C K) K1 K2) %<- (%pi (ofkd C K) %-> (ikd-sub G K1 K2)) %. +%sort ikd-deq %. +%term ikd-deq/nil %pi (ikd-deq cxt/nil K K') %<- (kd-deq K K') %. +%term ikd-deq/cons %pi (ikd-deq (cxt/cons G C K) K1 K2) %<- (%pi (ofkd C K) %-> (ikd-deq G K1 K2)) %. +%sort icn-deq %. +%term icn-deq/nil %pi (icn-deq cxt/nil C1 C2 K) %<- (cn-deq C1 C2 K) %. +%term icn-deq/cons + %pi (icn-deq (cxt/cons G C K) C1 C2 K') + %<- (%pi (ofkd C K) %-> (icn-deq G C1 C2 K')) %. +%block iofkd-block {c cn} {k kd} [d ofkd c k]%. +%sort kd-wf-to-ikd-wf {G cxt} %. +%mode kd-wf-to-ikd-wf %in %in %out %. +%term _ kd-wf-to-ikd-wf cxt/nil D (ikd-wf/nil D) %. +%term _ + %pi (kd-wf-to-ikd-wf (cxt/cons G C K) D (ikd-wf/cons ([d] D'))) + %<- (kd-wf-to-ikd-wf G D D') %. +%worlds (ovar-block iofkd-block ofkd-block) (kd-wf-to-ikd-wf _ _ _) %. +%total G (kd-wf-to-ikd-wf G _ _) %. +%sort ofkd-to-iofkd {G cxt} %. +%mode ofkd-to-iofkd %in %in %out %. +%term _ ofkd-to-iofkd cxt/nil D (iofkd/nil D) %. +%term _ + %pi (ofkd-to-iofkd (cxt/cons G C K) D (iofkd/cons ([d] D'))) + %<- (ofkd-to-iofkd G D D') %. +%worlds (ovar-block iofkd-block ofkd-block) (ofkd-to-iofkd _ _ _) %. +%total G (ofkd-to-iofkd G _ _) %. +%sort kd-sub-to-ikd-sub {G cxt} %. +%mode kd-sub-to-ikd-sub %in %in %out %. +%term _ kd-sub-to-ikd-sub cxt/nil D (ikd-sub/nil D) %. +%term _ + %pi (kd-sub-to-ikd-sub (cxt/cons G C K) D (ikd-sub/cons ([d] D'))) + %<- (kd-sub-to-ikd-sub G D D') %. +%worlds (ovar-block iofkd-block ofkd-block) (kd-sub-to-ikd-sub _ _ _) %. +%total G (kd-sub-to-ikd-sub G _ _) %. +%sort kd-deq-to-ikd-deq {G cxt} %. +%mode kd-deq-to-ikd-deq %in %in %out %. +%term _ kd-deq-to-ikd-deq cxt/nil D (ikd-deq/nil D) %. +%term _ + %pi (kd-deq-to-ikd-deq (cxt/cons G C K) D (ikd-deq/cons ([d] D'))) + %<- (kd-deq-to-ikd-deq G D D') %. +%worlds (ovar-block iofkd-block ofkd-block) (kd-deq-to-ikd-deq _ _ _) %. +%total G (kd-deq-to-ikd-deq G _ _) %. +%sort cn-deq-to-icn-deq {G cxt} %. +%mode cn-deq-to-icn-deq %in %in %out %. +%term _ cn-deq-to-icn-deq cxt/nil D (icn-deq/nil D) %. +%term _ + %pi (cn-deq-to-icn-deq (cxt/cons G C K) D (icn-deq/cons ([d] D'))) + %<- (cn-deq-to-icn-deq G D D') %. +%worlds (ovar-block iofkd-block ofkd-block) (cn-deq-to-icn-deq _ _ _) %. +%total G (cn-deq-to-icn-deq G _ _) %. +%sort implicate-cxt-lookup %. +%mode implicate-cxt-lookup %in %out %. +%term _ + %pi (implicate-cxt-lookup (cxt-lookup/hit _) (iofkd/cons D)) + %<- ({d ofkd C K} ofkd-to-iofkd G d (D d)) %. +%term _ + %pi (implicate-cxt-lookup (cxt-lookup/miss D _ _) (iofkd/cons ([d] D'))) + %<- (implicate-cxt-lookup D D') %. +%worlds (ovar-block iofkd-block ofkd-block) (implicate-cxt-lookup _ _) %. +%total D (implicate-cxt-lookup D _) %. +%sort ikd-wf/kd/unit {G cxt} %. +%mode ikd-wf/kd/unit %in %out %. +%sort ikd-wf/kd/sing %. +%mode ikd-wf/kd/sing %in %out %. +%term _ ikd-wf/kd/sing (iofkd/nil D) (ikd-wf/nil (kd-wf/kd/sing D)) %. +%term _ + %pi (ikd-wf/kd/sing (iofkd/cons D) (ikd-wf/cons D')) + %<- ({d ofkd C K} ikd-wf/kd/sing (D d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (ikd-wf/kd/sing _ _) %. +%total D (ikd-wf/kd/sing D _) %. +%sort ikd-wf/kd/sgm %. +%mode ikd-wf/kd/sgm %in %in %out %. +%term _ ikd-wf/kd/sgm (ikd-wf/nil D) ([c] [d] ikd-wf/nil (D' c d)) (ikd-wf/nil (kd-wf/kd/sgm D D')) %. +%term _ + %pi (ikd-wf/kd/sgm (ikd-wf/cons D1) ([c cn] [d ofkd c _] ikd-wf/cons ([d'] D2 c d d')) (ikd-wf/cons D')) + %<- ({d ofkd C K} ikd-wf/kd/sgm (D1 d) ([c] [d'] D2 c d' d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (ikd-wf/kd/sgm _ _ _) %. +%total D (ikd-wf/kd/sgm D _ _) %. +%sort ikd-wf/kd/pi %. +%mode ikd-wf/kd/pi %in %in %out %. +%term _ ikd-wf/kd/pi (ikd-wf/nil D) ([c] [d] ikd-wf/nil (D' c d)) (ikd-wf/nil (kd-wf/kd/pi D D')) %. +%term _ + %pi (ikd-wf/kd/pi (ikd-wf/cons D1) ([c cn] [d ofkd c _] ikd-wf/cons ([d'] D2 c d d')) (ikd-wf/cons D')) + %<- ({d ofkd C K} ikd-wf/kd/pi (D1 d) ([c] [d'] D2 c d' d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (ikd-wf/kd/pi _ _ _) %. +%total D (ikd-wf/kd/pi D _ _) %. +%sort ikd-deq/kd/sing %. +%mode ikd-deq/kd/sing %in %out %. +%term _ ikd-deq/kd/sing (icn-deq/nil D) (ikd-deq/nil (kd-deq/kd/sing D)) %. +%term _ + %pi (ikd-deq/kd/sing (icn-deq/cons D) (ikd-deq/cons D')) + %<- ({d ofkd C K} ikd-deq/kd/sing (D d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (ikd-deq/kd/sing _ _) %. +%total D (ikd-deq/kd/sing D _) %. +%sort ikd-deq/kd/sgm %. +%mode ikd-deq/kd/sgm %in %in %out %. +%term _ ikd-deq/kd/sgm (ikd-deq/nil D) ([c] [d] ikd-deq/nil (D' c d)) (ikd-deq/nil (kd-deq/kd/sgm D D')) %. +%term _ + %pi (ikd-deq/kd/sgm (ikd-deq/cons D1) ([c cn] [d ofkd c _] ikd-deq/cons ([d'] D2 c d d')) (ikd-deq/cons D')) + %<- ({d ofkd C K} ikd-deq/kd/sgm (D1 d) ([c] [d'] D2 c d' d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (ikd-deq/kd/sgm _ _ _) %. +%total D (ikd-deq/kd/sgm D _ _) %. +%sort ikd-deq/kd/pi %. +%mode ikd-deq/kd/pi %in %in %out %. +%term _ ikd-deq/kd/pi (ikd-deq/nil D) ([c] [d] ikd-deq/nil (D' c d)) (ikd-deq/nil (kd-deq/kd/pi D D')) %. +%term _ + %pi (ikd-deq/kd/pi (ikd-deq/cons D1) ([c cn] [d ofkd c _] ikd-deq/cons ([d'] D2 c d d')) (ikd-deq/cons D')) + %<- ({d ofkd C K} ikd-deq/kd/pi (D1 d) ([c] [d'] D2 c d' d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (ikd-deq/kd/pi _ _ _) %. +%total D (ikd-deq/kd/pi D _ _) %. +%sort ikd-sub/kd/sing-kd/sing %. +%mode ikd-sub/kd/sing-kd/sing %in %out %. +%term _ ikd-sub/kd/sing-kd/sing (icn-deq/nil D) (ikd-sub/nil (kd-sub/kd/sing-kd/sing D)) %. +%term _ + %pi (ikd-sub/kd/sing-kd/sing (icn-deq/cons D) (ikd-sub/cons D')) + %<- ({d ofkd C K} ikd-sub/kd/sing-kd/sing (D d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (ikd-sub/kd/sing-kd/sing _ _) %. +%total D (ikd-sub/kd/sing-kd/sing D _) %. +%sort ikd-sub/kd/sing-kd/type %. +%mode ikd-sub/kd/sing-kd/type %in %out %. +%term _ ikd-sub/kd/sing-kd/type (iofkd/nil D) (ikd-sub/nil (kd-sub/kd/sing-kd/type D)) %. +%term _ + %pi (ikd-sub/kd/sing-kd/type (iofkd/cons D) (ikd-sub/cons D')) + %<- ({d ofkd C K} ikd-sub/kd/sing-kd/type (D d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (ikd-sub/kd/sing-kd/type _ _) %. +%total D (ikd-sub/kd/sing-kd/type D _) %. +%sort ikd-sub/kd/sgm %. +%mode ikd-sub/kd/sgm %in %in %in %out %. +%term _ ikd-sub/kd/sgm (ikd-sub/nil D) ([c] [d] ikd-sub/nil (D' c d)) ([c] [d] ikd-wf/nil (D'' c d)) (ikd-sub/nil (kd-sub/kd/sgm D D' D'')) %. +%term _ + %pi (ikd-sub/kd/sgm (ikd-sub/cons D1) ([c cn] [d ofkd c _] ikd-sub/cons ([d'] D2 c d d')) ([c cn] [d ofkd c _] ikd-wf/cons ([d'] D3 c d d')) (ikd-sub/cons D')) + %<- ({d ofkd C K} ikd-sub/kd/sgm (D1 d) ([c] [d'] D2 c d' d) ([c] [d'] D3 c d' d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (ikd-sub/kd/sgm _ _ _ _) %. +%total D (ikd-sub/kd/sgm D _ _ _) %. +%sort ikd-sub/kd/pi %. +%mode ikd-sub/kd/pi %in %in %in %out %. +%term _ ikd-sub/kd/pi (ikd-sub/nil D) ([c] [d] ikd-sub/nil (D' c d)) ([c] [d] ikd-wf/nil (D'' c d)) (ikd-sub/nil (kd-sub/kd/pi D D' D'')) %. +%term _ + %pi (ikd-sub/kd/pi (ikd-sub/cons D1) ([c cn] [d ofkd c _] ikd-sub/cons ([d'] D2 c d d')) ([c cn] [d ofkd c _] ikd-wf/cons ([d'] D3 c d d')) (ikd-sub/cons D')) + %<- ({d ofkd C K} ikd-sub/kd/pi (D1 d) ([c] [d'] D2 c d' d) ([c] [d'] D3 c d' d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (ikd-sub/kd/pi _ _ _ _) %. +%total D (ikd-sub/kd/pi D _ _ _) %. +%sort iofkd/tp/cross %. +%mode iofkd/tp/cross %in %in %out %. +%term _ iofkd/tp/cross (iofkd/nil D) (iofkd/nil D') (iofkd/nil (ofkd/tp/cross D D')) %. +%term _ + %pi (iofkd/tp/cross (iofkd/cons D1) (iofkd/cons D2) (iofkd/cons D')) + %<- ({d ofkd C K} iofkd/tp/cross (D1 d) (D2 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (iofkd/tp/cross _ _ _) %. +%total D (iofkd/tp/cross D _ _) %. +%sort iofkd/tp/arrow %. +%mode iofkd/tp/arrow %in %in %out %. +%term _ iofkd/tp/arrow (iofkd/nil D) (iofkd/nil D') (iofkd/nil (ofkd/tp/arrow D D')) %. +%term _ + %pi (iofkd/tp/arrow (iofkd/cons D1) (iofkd/cons D2) (iofkd/cons D')) + %<- ({d ofkd C K} iofkd/tp/arrow (D1 d) (D2 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (iofkd/tp/arrow _ _ _) %. +%total D (iofkd/tp/arrow D _ _) %. +%sort iofkd/tp/sum %. +%mode iofkd/tp/sum %in %in %out %. +%term _ iofkd/tp/sum 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(iofkd/cons D1) (ikd-deq/cons D2) (iofkd/cons D')) + %<- ({d ofkd C K} iofkd/deq (D1 d) (D2 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (iofkd/deq _ _ _) %. +%total D (iofkd/deq D _ _) %. +%sort iofkd/tp/ref %. +%mode iofkd/tp/ref %in %out %. +%term _ iofkd/tp/ref (iofkd/nil D) (iofkd/nil (ofkd/tp/ref D)) %. +%term _ + %pi (iofkd/tp/ref (iofkd/cons D1) (iofkd/cons D')) + %<- ({d ofkd C K} iofkd/tp/ref (D1 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (iofkd/tp/ref _ _) %. +%total D (iofkd/tp/ref D _) %. +%sort iofkd/tp/tag %. +%mode iofkd/tp/tag %in %out %. +%term _ iofkd/tp/tag (iofkd/nil D) (iofkd/nil (ofkd/tp/tag D)) %. +%term _ + %pi (iofkd/tp/tag (iofkd/cons D1) (iofkd/cons D')) + %<- ({d ofkd C K} iofkd/tp/tag (D1 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (iofkd/tp/tag _ _) %. +%total D (iofkd/tp/tag D _) %. +%sort iofkd/cn/pj1 %. +%mode iofkd/cn/pj1 %in %out %. +%term _ iofkd/cn/pj1 (iofkd/nil D) (iofkd/nil (ofkd/cn/pj1 D)) %. +%term 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(iofkd/cn/mu ([c cn] [d ofkd c _] iofkd/cons ([d'] D2 c d d')) (ikd-wf/cons D1) (iofkd/cons D')) + %<- ({d ofkd C K} iofkd/cn/mu ([c] [d'] D2 c d' d) (D1 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (iofkd/cn/mu _ _ _) %. +%total D (iofkd/cn/mu D _ _) %. +%sort iofkd/pi-ext %. +%mode iofkd/pi-ext %in %in %out %. +%term _ iofkd/pi-ext (iofkd/nil D) ([c] [d] iofkd/nil (D' c d)) (iofkd/nil (ofkd/pi-ext D D')) %. +%term _ + %pi (iofkd/pi-ext (iofkd/cons D1) ([c cn] [d ofkd c _] iofkd/cons ([d'] D2 c d d')) (iofkd/cons D')) + %<- ({d ofkd C K} iofkd/pi-ext (D1 d) ([c] [d'] D2 c d' d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (iofkd/pi-ext _ _ _) %. +%total D (iofkd/pi-ext D _ _) %. +%sort icn-deq/tp/cross %. +%mode icn-deq/tp/cross %in %in %out %. +%term _ icn-deq/tp/cross (icn-deq/nil D) (icn-deq/nil D') (icn-deq/nil (cn-deq/tp/cross D D')) %. +%term _ + %pi (icn-deq/tp/cross (icn-deq/cons D1) (icn-deq/cons D2) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/tp/cross (D1 d) (D2 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/tp/cross _ _ _) %. +%total D (icn-deq/tp/cross D _ _) %. +%sort icn-deq/tp/arrow %. +%mode icn-deq/tp/arrow %in %in %out %. +%term _ icn-deq/tp/arrow (icn-deq/nil D) (icn-deq/nil D') (icn-deq/nil (cn-deq/tp/arrow D D')) %. +%term _ + %pi (icn-deq/tp/arrow (icn-deq/cons D1) (icn-deq/cons D2) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/tp/arrow (D1 d) (D2 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/tp/arrow _ _ _) %. +%total D (icn-deq/tp/arrow D _ _) %. +%sort icn-deq/tp/sum %. +%mode icn-deq/tp/sum %in %in %out %. +%term _ icn-deq/tp/sum (icn-deq/nil D) (icn-deq/nil D') (icn-deq/nil (cn-deq/tp/sum D D')) %. +%term _ + %pi (icn-deq/tp/sum (icn-deq/cons D1) (icn-deq/cons D2) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/tp/sum (D1 d) (D2 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/tp/sum _ _ _) %. +%total D (icn-deq/tp/sum D _ _) %. +%sort icn-deq/cn/pair %. +%mode icn-deq/cn/pair %in %in %out %. +%term _ icn-deq/cn/pair (icn-deq/nil D) (icn-deq/nil D') (icn-deq/nil (cn-deq/cn/pair D D')) %. +%term _ + %pi (icn-deq/cn/pair (icn-deq/cons D1) (icn-deq/cons D2) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/cn/pair (D1 d) (D2 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/cn/pair _ _ _) %. +%total D (icn-deq/cn/pair D _ _) %. +%sort icn-deq/cn/app %. +%mode icn-deq/cn/app %in %in %out %. +%term _ icn-deq/cn/app (icn-deq/nil D) (icn-deq/nil D') (icn-deq/nil (cn-deq/cn/app D D')) %. +%term _ + %pi (icn-deq/cn/app (icn-deq/cons D1) (icn-deq/cons D2) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/cn/app (D1 d) (D2 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/cn/app _ _ _) %. +%total D (icn-deq/cn/app D _ _) %. +%sort icn-deq/sgm-ext %. +%mode icn-deq/sgm-ext %in %in %out %. +%term _ icn-deq/sgm-ext (icn-deq/nil D) (icn-deq/nil D') (icn-deq/nil (cn-deq/sgm-ext D D')) %. +%term _ + %pi (icn-deq/sgm-ext (icn-deq/cons D1) (icn-deq/cons D2) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/sgm-ext (D1 d) (D2 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/sgm-ext _ _ _) %. +%total D (icn-deq/sgm-ext D _ _) %. +%sort icn-deq/trans %. +%mode icn-deq/trans %in %in %out %. +%term _ icn-deq/trans (icn-deq/nil D) (icn-deq/nil D') (icn-deq/nil (cn-deq/trans D D')) %. +%term _ + %pi (icn-deq/trans (icn-deq/cons D1) (icn-deq/cons D2) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/trans (D1 d) (D2 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/trans _ _ _) %. +%total D (icn-deq/trans D _ _) %. +%sort icn-deq/sub %. +%mode icn-deq/sub %in %in %out %. +%term _ icn-deq/sub (icn-deq/nil D) (ikd-sub/nil D') (icn-deq/nil (cn-deq/sub D D')) %. +%term _ + %pi (icn-deq/sub (icn-deq/cons D1) (ikd-sub/cons D2) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/sub (D1 d) (D2 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/sub _ _ _) %. +%total D (icn-deq/sub D _ _) %. +%sort icn-deq/deq %. +%mode icn-deq/deq %in %in %out %. +%term _ icn-deq/deq (icn-deq/nil D) (ikd-deq/nil D') (icn-deq/nil (cn-deq/deq D D')) %. +%term _ + %pi (icn-deq/deq (icn-deq/cons D1) (ikd-deq/cons D2) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/deq (D1 d) (D2 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/deq _ _ _) %. +%total D (icn-deq/deq D _ _) %. +%sort icn-deq/tp/ref %. +%mode icn-deq/tp/ref %in %out %. +%term _ icn-deq/tp/ref (icn-deq/nil D) (icn-deq/nil (cn-deq/tp/ref D)) %. +%term _ + %pi (icn-deq/tp/ref (icn-deq/cons D1) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/tp/ref (D1 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/tp/ref _ _) %. +%total D (icn-deq/tp/ref D _) %. +%sort icn-deq/tp/tag %. +%mode icn-deq/tp/tag %in %out %. +%term _ icn-deq/tp/tag (icn-deq/nil D) (icn-deq/nil (cn-deq/tp/tag D)) %. +%term _ + %pi (icn-deq/tp/tag (icn-deq/cons D1) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/tp/tag (D1 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/tp/tag _ _) %. +%total D (icn-deq/tp/tag D _) %. +%sort icn-deq/cn/pj1 %. +%mode icn-deq/cn/pj1 %in %out %. +%term _ icn-deq/cn/pj1 (icn-deq/nil D) (icn-deq/nil (cn-deq/cn/pj1 D)) %. +%term _ + %pi (icn-deq/cn/pj1 (icn-deq/cons D1) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/cn/pj1 (D1 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/cn/pj1 _ _) %. +%total D (icn-deq/cn/pj1 D _) %. +%sort icn-deq/cn/pj2 %. +%mode icn-deq/cn/pj2 %in %out %. +%term _ icn-deq/cn/pj2 (icn-deq/nil D) (icn-deq/nil (cn-deq/cn/pj2 D)) %. +%term _ + %pi (icn-deq/cn/pj2 (icn-deq/cons D1) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/cn/pj2 (D1 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/cn/pj2 _ _) %. +%total D (icn-deq/cn/pj2 D _) %. +%sort icn-deq/kd/sing %. +%mode icn-deq/kd/sing %in %out %. +%term _ icn-deq/kd/sing (iofkd/nil D) (icn-deq/nil (cn-deq/kd/sing D)) %. +%term _ + %pi (icn-deq/kd/sing (iofkd/cons D1) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/kd/sing (D1 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/kd/sing _ _) %. +%total D (icn-deq/kd/sing D _) %. +%sort icn-deq/kd/unit %. +%mode icn-deq/kd/unit %in %in %out %. +%term _ icn-deq/kd/unit (iofkd/nil D) (iofkd/nil D') (icn-deq/nil (cn-deq/kd/unit D D')) %. +%term _ + %pi (icn-deq/kd/unit (iofkd/cons D1) (iofkd/cons D2) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/kd/unit (D1 d) (D2 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/kd/unit _ _ _) %. +%total D (icn-deq/kd/unit D _ _) %. +%sort icn-deq/refl %. +%mode icn-deq/refl %in %out %. +%term _ icn-deq/refl (iofkd/nil D) (icn-deq/nil (cn-deq/refl D)) %. +%term _ + %pi (icn-deq/refl (iofkd/cons D1) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/refl (D1 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/refl _ _) %. +%total D (icn-deq/refl D _) %. +%sort icn-deq/sym %. +%mode icn-deq/sym %in %out %. +%term _ icn-deq/sym (icn-deq/nil D) (icn-deq/nil (cn-deq/sym D)) %. +%term _ + %pi (icn-deq/sym (icn-deq/cons D1) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/sym (D1 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/sym _ _) %. +%total D (icn-deq/sym D _) %. +%sort icn-deq/tp/forall {c cn} %. +%mode icn-deq/tp/forall %in %in %out %. +%term _ icn-deq/tp/forall ([c] [d] icn-deq/nil (D' c d)) (ikd-deq/nil D) (icn-deq/nil (cn-deq/tp/forall D' D)) %. +%term _ + %pi (icn-deq/tp/forall ([c cn] [d ofkd c _] icn-deq/cons ([d'] D2 c d d')) (ikd-deq/cons D1) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/tp/forall ([c] [d'] D2 c d' d) (D1 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/tp/forall _ _ _) %. +%total D (icn-deq/tp/forall D _ _) %. +%sort icn-deq/cn/lam {c cn} %. +%mode icn-deq/cn/lam %in %in %out %. +%term _ icn-deq/cn/lam ([c] [d] icn-deq/nil (D' c d)) (ikd-deq/nil D) (icn-deq/nil (cn-deq/cn/lam D' D)) %. +%term _ + %pi (icn-deq/cn/lam ([c cn] [d ofkd c _] icn-deq/cons ([d'] D2 c d d')) (ikd-deq/cons D1) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/cn/lam ([c] [d'] D2 c d' d) (D1 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/cn/lam _ _ _) %. +%total D (icn-deq/cn/lam D _ _) %. +%sort icn-deq/cn/mu {c cn} %. +%mode icn-deq/cn/mu %in %in %out %. +%term _ icn-deq/cn/mu ([c] [d] icn-deq/nil (D' c d)) (ikd-deq/nil D) (icn-deq/nil (cn-deq/cn/mu D' D)) %. +%term _ + %pi (icn-deq/cn/mu ([c cn] [d ofkd c _] icn-deq/cons ([d'] D2 c d d')) (ikd-deq/cons D1) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/cn/mu ([c] [d'] D2 c d' d) (D1 d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/cn/mu _ _ _) %. +%total D (icn-deq/cn/mu D _ _) %. +%sort icn-deq/pi-ext %. +%mode icn-deq/pi-ext %in %in %in %out %. +%term _ icn-deq/pi-ext (iofkd/nil D) (iofkd/nil D'') ([c] [d] icn-deq/nil (D' c d)) (icn-deq/nil (cn-deq/pi-ext D D'' D')) %. +%term _ + %pi (icn-deq/pi-ext (iofkd/cons D1) (iofkd/cons D1') ([c cn] [d ofkd c _] icn-deq/cons ([d'] D2 c d d')) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/pi-ext (D1 d) (D1' d) ([c] [d'] D2 c d' d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/pi-ext _ _ _ _) %. +%total D (icn-deq/pi-ext D _ _ _) %. +%sort icn-deq/pi-ext-2 %. +%mode icn-deq/pi-ext-2 %in %in %out %. +%term _ icn-deq/pi-ext-2 (icn-deq/nil D) ([c] [d] icn-deq/nil (D' c d)) (icn-deq/nil (cn-deq/pi-ext-2 D D')) %. +%term _ + %pi (icn-deq/pi-ext-2 (icn-deq/cons D1) ([c cn] [d ofkd c _] icn-deq/cons ([d'] D2 c d d')) (icn-deq/cons D')) + %<- ({d ofkd C K} icn-deq/pi-ext-2 (D1 d) ([c] [d'] D2 c d' d) (D' d)) %. +%worlds (ovar-block iofkd-block ofkd-block) (icn-deq/pi-ext-2 _ _ _) %. +%total D (icn-deq/pi-ext-2 D _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/implicate-sigs.lf b/new-tests/stelf-output/tslf/implicate-sigs.lf new file mode 100644 index 0000000..438e1b8 --- /dev/null +++ b/new-tests/stelf-output/tslf/implicate-sigs.lf @@ -0,0 +1,59 @@ +%%%% implicate lemma. explicit system to implicit system. +%sort implicate/sg-wf %. +%mode implicate/sg-wf %in %out %. +%sort implicate/sg-deq %. +%mode implicate/sg-deq %in %out %. +%term _ + %pi (implicate/sg-wf (esg-wf/sg/unit _) D) + %<- (sg-wf-to-isg-wf G sg-wf/sg/unit D) %. +%term _ + %pi (implicate/sg-wf (esg-wf/sg/cn D) D'') + %<- (implicate/ofkd D D') + %<- (isg-wf/sg/cn D' D'') %. +%term _ + %pi (implicate/sg-wf (esg-wf/sg/kd D) D'') + %<- (implicate/kd-wf D D') + %<- (isg-wf/sg/kd D' D'') %. +%term _ + %pi (implicate/sg-wf (esg-wf/sg/sgm D1 D2 DFS) D'') + %<- (implicate/sg-wf D1 D1') + %<- ({c cn} {d isvar c I} implicate/sg-wf (D2 c d) (isg-wf/cons (D2' c))) + %<- (isg-wf/sg/sgm D1' DFS D2' D'') %. +%term _ + %pi (implicate/sg-wf (esg-wf/sg/pi D1 D2 DFS) D'') + %<- (implicate/sg-wf D1 D1') + %<- ({c cn} {d isvar c I} implicate/sg-wf (D2 c d) (isg-wf/cons (D2' c))) + %<- (isg-wf/sg/pi D1' DFS D2' D'') %. +%term _ + %pi (implicate/sg-deq (esg-deq/sg/unit _) D) + %<- (sg-deq-to-isg-deq G sg-deq/sg/unit D) %. +%term _ + %pi (implicate/sg-deq (esg-deq/sg/cn D) D'') + %<- (implicate/cn-deq D D') + %<- (isg-deq/sg/cn D' D'') %. +%term _ + %pi (implicate/sg-deq (esg-deq/sg/kd D) D'') + %<- (implicate/kd-deq D D') + %<- (isg-deq/sg/kd D' D'') %. +%term _ + %pi (implicate/sg-deq (esg-deq/sg/sgm D1 D2 DFS) D'') + %<- (implicate/sg-deq D1 D1') + %<- ({c cn} {d isvar c I} implicate/sg-deq (D2 c d) (isg-deq/cons (D2' c))) + %<- (isg-deq/sg/sgm D1' DFS D2' D'') %. +%term _ + %pi (implicate/sg-deq (esg-deq/sg/pi D1 D2 DFS) D'') + %<- (implicate/sg-deq D1 D1') + %<- ({c cn} {d isvar c I} implicate/sg-deq (D2 c d) (isg-deq/cons (D2' c))) + %<- (isg-deq/sg/pi D1' DFS D2' D'') %. +%worlds (ovar-block ofkd-block) (implicate/sg-wf _ _) (implicate/sg-deq _ _) %. +%total (D1 D2) (implicate/sg-wf D1 _) (implicate/sg-deq D2 _) %. +%sort implicate-closed/sg-wf %. +%mode implicate-closed/sg-wf %in %out %. +%term _ %pi (implicate-closed/sg-wf D D') %<- (implicate/sg-wf D (isg-wf/nil D')) %. +%worlds (ofkd-block) (implicate-closed/sg-wf _ _) %. +%total {} (implicate-closed/sg-wf _ _) %. +%sort implicate-closed/sg-deq %. +%mode implicate-closed/sg-deq %in %out %. +%term _ %pi (implicate-closed/sg-deq D D') %<- (implicate/sg-deq D (isg-deq/nil D')) %. +%worlds (ofkd-block) (implicate-closed/sg-deq _ _) %. +%total {} (implicate-closed/sg-deq _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/implicate.lf b/new-tests/stelf-output/tslf/implicate.lf new file mode 100644 index 0000000..8e2f2b1 --- /dev/null +++ b/new-tests/stelf-output/tslf/implicate.lf @@ -0,0 +1,302 @@ +%%%% implicate lemma. explicit system to implicit system. +%sort implicate/kd-wf %. +%mode implicate/kd-wf %in %out %. +%sort implicate/kd-deq %. +%mode implicate/kd-deq %in %out %. +%sort implicate/kd-sub %. +%mode implicate/kd-sub %in %out %. +%sort implicate/ofkd %. +%mode implicate/ofkd %in %out %. +%sort implicate/cn-deq %. +%mode implicate/cn-deq %in %out %. +%term _ + %pi (implicate/kd-wf (ekd-wf/kd/unit _) D) + %<- (kd-wf-to-ikd-wf G kd-wf/kd/unit D) %. +%term _ + %pi (implicate/kd-wf (ekd-wf/kd/type _) D) + %<- (kd-wf-to-ikd-wf G kd-wf/kd/type D) %. +%term _ + %pi (implicate/kd-wf (ekd-wf/kd/sing D) D'') + %<- (implicate/ofkd D D') + %<- (ikd-wf/kd/sing D' D'') %. +%term _ + %pi (implicate/kd-wf (ekd-wf/kd/sgm D1 D2) D'') + %<- (implicate/kd-wf D1 D1') + %<- ({c cn} {d isvar c I} implicate/kd-wf (D2 c d) (ikd-wf/cons (D2' c))) + %<- (ikd-wf/kd/sgm D1' D2' D'') %. +%term _ + %pi (implicate/kd-wf (ekd-wf/kd/pi D1 D2) D'') + %<- (implicate/kd-wf D1 D1') + %<- ({c cn} {d isvar c I} implicate/kd-wf (D2 c d) (ikd-wf/cons (D2' c))) + %<- (ikd-wf/kd/pi D1' D2' D'') %. +%term _ + %pi (implicate/kd-deq (ekd-deq/kd/unit _) D) + %<- (kd-deq-to-ikd-deq G kd-deq/kd/unit D) %. +%term _ + %pi (implicate/kd-deq (ekd-deq/kd/type _) D) + %<- (kd-deq-to-ikd-deq G kd-deq/kd/type D) %. +%term _ + %pi (implicate/kd-deq (ekd-deq/kd/sing D) D'') + %<- (implicate/cn-deq D D') + %<- (ikd-deq/kd/sing D' D'') %. +%term _ + %pi (implicate/kd-deq (ekd-deq/kd/sgm D1 D2) D'') + %<- (implicate/kd-deq D1 D1') + %<- ({c cn} {d isvar c I} implicate/kd-deq (D2 c d) (ikd-deq/cons (D2' c))) + %<- (ikd-deq/kd/sgm D1' D2' D'') %. +%term _ + %pi (implicate/kd-deq (ekd-deq/kd/pi D1 D2) D'') + %<- (implicate/kd-deq D1 D1') + %<- ({c cn} {d isvar c I} implicate/kd-deq (D2 c d) (ikd-deq/cons (D2' c))) + %<- (ikd-deq/kd/pi D1' D2' D'') %. +%term _ + %pi (implicate/kd-sub (ekd-sub/kd/unit _) D) + %<- (kd-sub-to-ikd-sub G kd-sub/kd/unit D) %. +%term _ + %pi (implicate/kd-sub (ekd-sub/kd/type _) D) + %<- (kd-sub-to-ikd-sub G kd-sub/kd/type D) %. +%term _ + %pi (implicate/kd-sub (ekd-sub/kd/sing-kd/sing D) D'') + %<- (implicate/cn-deq D D') + %<- (ikd-sub/kd/sing-kd/sing D' D'') %. +%term _ + %pi (implicate/kd-sub (ekd-sub/kd/sing-kd/type D) D'') + %<- (implicate/ofkd D D') + %<- (ikd-sub/kd/sing-kd/type D' D'') %. +%term _ + %pi (implicate/kd-sub (ekd-sub/kd/sgm D1 D2 D3) D'') + %<- (implicate/kd-sub D1 D1') + %<- ({c cn} {d isvar c I} implicate/kd-sub (D2 c d) (ikd-sub/cons (D2' c))) + %<- ({c cn} {d isvar c I} implicate/kd-wf (D3 c d) (ikd-wf/cons (D3' c))) + %<- (ikd-sub/kd/sgm D1' D2' D3' D'') %. +%term _ + %pi (implicate/kd-sub (ekd-sub/kd/pi D1 D2 D3) D'') + %<- (implicate/kd-sub D1 D1') + %<- ({c cn} {d isvar c I} implicate/kd-sub (D2 c d) (ikd-sub/cons (D2' c))) + %<- ({c cn} {d isvar c I} implicate/kd-wf (D3 c d) (ikd-wf/cons (D3' c))) + %<- (ikd-sub/kd/pi D1' D2' D3' D'') %. +%term _ %pi (implicate/ofkd (eofkd/cn/unit _) D) %<- (ofkd-to-iofkd _ ofkd/cn/unit D) %. +%term _ %pi (implicate/ofkd (eofkd/tp/unit _) D) %<- (ofkd-to-iofkd _ ofkd/tp/unit D) %. +%term _ %pi (implicate/ofkd (eofkd/tp/tagged _) D) %<- (ofkd-to-iofkd _ ofkd/tp/tagged D) %. +%term _ + %pi (implicate/ofkd (eofkd/tp/cross D1 D2) D') + %<- (implicate/ofkd D1 D1') + %<- (implicate/ofkd D2 D2') + %<- (iofkd/tp/cross D1' D2' D') %. +%term _ + %pi (implicate/ofkd (eofkd/tp/arrow D1 D2) D') + %<- (implicate/ofkd D1 D1') + %<- (implicate/ofkd D2 D2') + %<- (iofkd/tp/arrow D1' D2' D') %. +%term _ + %pi (implicate/ofkd (eofkd/tp/sum D1 D2) D') + %<- (implicate/ofkd D1 D1') + %<- (implicate/ofkd D2 D2') + %<- (iofkd/tp/sum D1' D2' D') %. +%term _ + %pi (implicate/ofkd (eofkd/tp/forall D1 DW) D') + %<- ({c cn} {d isvar c I} implicate/ofkd (D1 c d) (iofkd/cons (D1' c))) + %<- (implicate/kd-wf DW DW') + %<- (iofkd/tp/forall D1' DW' D') %. +%term _ + %pi (implicate/ofkd (eofkd/tp/ref D1) D') + %<- (implicate/ofkd D1 D1') + %<- (iofkd/tp/ref D1' D') %. +%term _ + %pi (implicate/ofkd (eofkd/tp/tag D1) D') + %<- (implicate/ofkd D1 D1') + %<- (iofkd/tp/tag D1' D') %. +%term _ + %pi (implicate/ofkd (eofkd/cn/pair D1 D2) D') + %<- (implicate/ofkd D1 D1') + %<- (implicate/ofkd D2 D2') + %<- (iofkd/cn/pair D1' D2' D') %. +%term _ + %pi (implicate/ofkd (eofkd/cn/pj1 D1) D') + %<- (implicate/ofkd D1 D1') + %<- (iofkd/cn/pj1 D1' D') %. +%term _ + %pi (implicate/ofkd (eofkd/cn/pj2 D1) D') + %<- (implicate/ofkd D1 D1') + %<- (iofkd/cn/pj2 D1' D') %. +%term _ + %pi (implicate/ofkd (eofkd/cn/lam D1 DW) D') + %<- ({c cn} {d isvar c I} implicate/ofkd (D1 c d) (iofkd/cons (D1' c))) + %<- (implicate/kd-wf DW DW') + %<- (iofkd/cn/lam D1' DW' D') %. +%term _ + %pi (implicate/ofkd (eofkd/cn/mu D1 DW) D') + %<- ({c cn} {d isvar c I} implicate/ofkd (D1 c d) (iofkd/cons (D1' c))) + %<- (implicate/kd-wf DW DW') + %<- (iofkd/cn/mu D1' DW' D') %. +%term _ + %pi (implicate/ofkd (eofkd/cn/app D1 D2) D') + %<- (implicate/ofkd D1 D1') + %<- (implicate/ofkd D2 D2') + %<- (iofkd/cn/app D1' D2' D') %. +%term _ + %pi (implicate/ofkd (eofkd/kd/sing D1) D') + %<- (implicate/ofkd D1 D1') + %<- (iofkd/kd/sing D1' D') %. +%term _ + %pi (implicate/ofkd (eofkd/sgm-ext D1 D2) D') + %<- (implicate/ofkd D1 D1') + %<- (implicate/ofkd D2 D2') + %<- (iofkd/sgm-ext D1' D2' D') %. +%term _ + %pi (implicate/ofkd (eofkd/pi-ext D1 D2) D') + %<- ({c cn} {d isvar c I} implicate/ofkd (D2 c d) (iofkd/cons (D2' c))) + %<- (implicate/ofkd D1 D1') + %<- (iofkd/pi-ext D1' D2' D') %. +%term _ + %pi (implicate/ofkd (eofkd/sub D1 D2) D') + %<- (implicate/ofkd D1 D1') + %<- (implicate/kd-sub D2 D2') + %<- (iofkd/sub D1' D2' D') %. +%term _ + %pi (implicate/ofkd (eofkd/deq D1 D2) D') + %<- (implicate/ofkd D1 D1') + %<- (implicate/kd-deq D2 D2') + %<- (iofkd/deq D1' D2' D') %. +%term _ %pi (implicate/ofkd (eofkd/var D) D') %<- (implicate-cxt-lookup D D') %. +%term _ %pi (implicate/ofkd (eofkd/closed D _) D') %<- (ofkd-to-iofkd _ D D') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/refl D) D'') + %<- (implicate/ofkd D D') + %<- (icn-deq/refl D' D'') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/sym D) D'') + %<- (implicate/cn-deq D D') + %<- (icn-deq/sym D' D'') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/trans D1 D2) D') + %<- (implicate/cn-deq D1 D1') + %<- (implicate/cn-deq D2 D2') + %<- (icn-deq/trans D1' D2' D') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/cn/unit _) D) + %<- (cn-deq-to-icn-deq _ cn-deq/cn/unit D) %. +%term _ + %pi (implicate/cn-deq (ecn-deq/tp/unit _) D) + %<- (cn-deq-to-icn-deq _ cn-deq/tp/unit D) %. +%term _ + %pi (implicate/cn-deq (ecn-deq/tp/tagged _) D) + %<- (cn-deq-to-icn-deq _ cn-deq/tp/tagged D) %. +%term _ + %pi (implicate/cn-deq (ecn-deq/tp/cross D1 D2) D') + %<- (implicate/cn-deq D1 D1') + %<- (implicate/cn-deq D2 D2') + %<- (icn-deq/tp/cross D1' D2' D') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/tp/arrow D1 D2) D') + %<- (implicate/cn-deq D1 D1') + %<- (implicate/cn-deq D2 D2') + %<- (icn-deq/tp/arrow D1' D2' D') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/tp/sum D1 D2) D') + %<- (implicate/cn-deq D1 D1') + %<- (implicate/cn-deq D2 D2') + %<- (icn-deq/tp/sum D1' D2' D') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/tp/forall D1 DW) D') + %<- ({c cn} {d isvar c I} implicate/cn-deq (D1 c d) (icn-deq/cons (D1' c))) + %<- (implicate/kd-deq DW DW') + %<- (icn-deq/tp/forall D1' DW' D') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/tp/ref D1) D') + %<- (implicate/cn-deq D1 D1') + %<- (icn-deq/tp/ref D1' D') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/tp/tag D1) D') + %<- (implicate/cn-deq D1 D1') + %<- (icn-deq/tp/tag D1' D') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/cn/pair D1 D2) D') + %<- (implicate/cn-deq D1 D1') + %<- (implicate/cn-deq D2 D2') + %<- (icn-deq/cn/pair D1' D2' D') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/cn/pj1 D1) D') + %<- (implicate/cn-deq D1 D1') + %<- (icn-deq/cn/pj1 D1' D') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/cn/pj2 D1) D') + %<- (implicate/cn-deq D1 D1') + %<- (icn-deq/cn/pj2 D1' D') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/cn/lam D1 DW) D') + %<- ({c cn} {d isvar c I} implicate/cn-deq (D1 c d) (icn-deq/cons (D1' c))) + %<- (implicate/kd-deq DW DW') + %<- (icn-deq/cn/lam D1' DW' D') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/cn/mu D1 DW) D') + %<- ({c cn} {d isvar c I} implicate/cn-deq (D1 c d) (icn-deq/cons (D1' c))) + %<- (implicate/kd-deq DW DW') + %<- (icn-deq/cn/mu D1' DW' D') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/cn/app D1 D2) D') + %<- (implicate/cn-deq D1 D1') + %<- (implicate/cn-deq D2 D2') + %<- (icn-deq/cn/app D1' D2' D') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/kd/unit D1 D2) D') + %<- (implicate/ofkd D1 D1') + %<- (implicate/ofkd D2 D2') + %<- (icn-deq/kd/unit D1' D2' D') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/kd/sing D1) D') + %<- (implicate/ofkd D1 D1') + %<- (icn-deq/kd/sing D1' D') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/sgm-ext D1 D2) D') + %<- (implicate/cn-deq D1 D1') + %<- (implicate/cn-deq D2 D2') + %<- (icn-deq/sgm-ext D1' D2' D') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/pi-ext D1 D3 D2) D') + %<- ({c cn} {d isvar c I} implicate/cn-deq (D2 c d) (icn-deq/cons (D2' c))) + %<- (implicate/ofkd D1 D1') + %<- (implicate/ofkd D3 D3') + %<- (icn-deq/pi-ext D1' D3' D2' D') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/pi-ext-2 D1 D2) D') + %<- ({c cn} {d isvar c I} implicate/cn-deq (D2 c d) (icn-deq/cons (D2' c))) + %<- (implicate/cn-deq D1 D1') + %<- (icn-deq/pi-ext-2 D1' D2' D') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/sub D1 D2) D') + %<- (implicate/cn-deq D1 D1') + %<- (implicate/kd-sub D2 D2') + %<- (icn-deq/sub D1' D2' D') %. +%term _ + %pi (implicate/cn-deq (ecn-deq/deq D1 D2) D') + %<- (implicate/cn-deq D1 D1') + %<- (implicate/kd-deq D2 D2') + %<- (icn-deq/deq D1' D2' D') %. +%worlds (ovar-block ofkd-block) (implicate/kd-wf _ _) (implicate/kd-deq _ _) (implicate/kd-sub _ _) (implicate/ofkd _ _) (implicate/cn-deq _ _) %. +%total (D1 D2 D3 D4 D5) (implicate/kd-wf D1 _) (implicate/kd-deq D4 _) (implicate/kd-sub D3 _) (implicate/ofkd D2 _) (implicate/cn-deq D5 _) %. +%sort implicate-closed/kd-wf %. +%mode implicate-closed/kd-wf %in %out %. +%term _ %pi (implicate-closed/kd-wf D D') %<- (implicate/kd-wf D (ikd-wf/nil D')) %. +%worlds (ofkd-block) (implicate-closed/kd-wf _ _) %. +%total {} (implicate-closed/kd-wf _ _) %. +%sort implicate-closed/kd-deq %. +%mode implicate-closed/kd-deq %in %out %. +%term _ %pi (implicate-closed/kd-deq D D') %<- (implicate/kd-deq D (ikd-deq/nil D')) %. +%worlds (ofkd-block) (implicate-closed/kd-deq _ _) %. +%total {} (implicate-closed/kd-deq _ _) %. +%sort implicate-closed/kd-sub %. +%mode implicate-closed/kd-sub %in %out %. +%term _ %pi (implicate-closed/kd-sub D D') %<- (implicate/kd-sub D (ikd-sub/nil D')) %. +%worlds (ofkd-block) (implicate-closed/kd-sub _ _) %. +%total {} (implicate-closed/kd-sub _ _) %. +%sort implicate-closed/ofkd %. +%mode implicate-closed/ofkd %in %out %. +%term _ %pi (implicate-closed/ofkd D D') %<- (implicate/ofkd D (iofkd/nil D')) %. +%worlds (ofkd-block) (implicate-closed/ofkd _ _) %. +%total {} (implicate-closed/ofkd _ _) %. +%sort implicate-closed/cn-deq %. +%mode implicate-closed/cn-deq %in %out %. +%term _ %pi (implicate-closed/cn-deq D D') %<- (implicate/cn-deq D (icn-deq/nil D')) %. +%worlds (ofkd-block) (implicate-closed/cn-deq _ _) %. +%total {} (implicate-closed/cn-deq _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/inequalities.lf b/new-tests/stelf-output/tslf/inequalities.lf new file mode 100644 index 0000000..b763329 --- /dev/null +++ b/new-tests/stelf-output/tslf/inequalities.lf @@ -0,0 +1,604 @@ +%sort neq/deq/arrow-unit %. +%mode neq/deq/arrow-unit %in %out %. +%term _ + %pi (neq/deq/arrow-unit Deq Dun) + %<- (map-equiv Deq (map/arrow _ _) map/unit tmap/t Deq') + %<- (nullary-binary-equiv (ekof/i etopen/t ckof/unit) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/arrow) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/arrow-unit _ _) %. +%total {} (neq/deq/arrow-unit _ _) %. +%sort neq/deq/cross-unit %. +%mode neq/deq/cross-unit %in %out %. +%term _ + %pi (neq/deq/cross-unit Deq Dun) + %<- (map-equiv Deq (map/cross _ _) map/unit tmap/t Deq') + %<- (nullary-binary-equiv (ekof/i etopen/t ckof/unit) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/cross) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/cross-unit _ _) %. +%total {} (neq/deq/cross-unit _ _) %. +%sort neq/deq/forall-unit %. +%mode neq/deq/forall-unit %in %out %. +%term _ + %pi (neq/deq/forall-unit Deq Dun) + %<- (map-equiv Deq (map/forall _ _) map/unit tmap/t Deq') + %<- (nullary-forall-equiv (ekof/i etopen/t ckof/unit) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/forall-unit _ _) %. +%total {} (neq/deq/forall-unit _ _) %. +%sort neq/deq/ref-unit %. +%mode neq/deq/ref-unit %in %out %. +%term _ + %pi (neq/deq/ref-unit Deq Dun) + %<- (map-equiv Deq (map/ref _) map/unit tmap/t Deq') + %<- (nullary-unary-equiv (ekof/i etopen/t ckof/unit) (ekof/i (etopen/pi etopen/t etopen/t) ckof/ref) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/ref-unit _ _) %. +%total {} (neq/deq/ref-unit _ _) %. +%sort neq/deq/tag-unit %. +%mode neq/deq/tag-unit %in %out %. +%term _ + %pi (neq/deq/tag-unit Deq Dun) + %<- (map-equiv Deq (map/tag _) map/unit tmap/t Deq') + %<- (nullary-unary-equiv (ekof/i etopen/t ckof/unit) (ekof/i (etopen/pi etopen/t etopen/t) ckof/tag) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/tag-unit _ _) %. +%total {} (neq/deq/tag-unit _ _) %. +%sort neq/deq/sum-unit %. +%mode neq/deq/sum-unit %in %out %. +%term _ + %pi (neq/deq/sum-unit Deq Dun) + %<- (map-equiv Deq (map/sum _ _) map/unit tmap/t Deq') + %<- (nullary-binary-equiv (ekof/i etopen/t ckof/unit) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/dsum) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/sum-unit _ _) %. +%total {} (neq/deq/sum-unit _ _) %. +%sort neq/deq/tagged-unit %. +%mode neq/deq/tagged-unit %in %out %. +%sort neq/deq/tagged-unit! %. +%mode neq/deq/tagged-unit! %in %out %. +%worlds () (neq/deq/tagged-unit! _ _) %. +%total {} (neq/deq/tagged-unit! _ _) %. +%term _ + %pi (neq/deq/tagged-unit Deq Dun) + %<- (map-equiv Deq map/tagged map/unit tmap/t Deq') + %<- (nullary-equiv-head (ekof/i etopen/t ckof/tagged) (ekof/i etopen/t ckof/unit) Deq' D) + %<- (neq/deq/tagged-unit! D Dun) %. +%worlds () (neq/deq/tagged-unit _ _) %. +%total {} (neq/deq/tagged-unit _ _) %. +%sort neq/deq/mu-unit %. +%mode neq/deq/mu-unit %in %out %. +%term _ + %pi (neq/deq/mu-unit Deq Dun) + %<- (map-equiv Deq (map/mu _) map/unit tmap/t Deq') + %<- (nullary-unary-equiv (ekof/i etopen/t ckof/unit) (ekof/i (etopen/pi etopen/t (etopen/pi etopen/t etopen/t)) ckof/mu) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/mu-unit _ _) %. +%total {} (neq/deq/mu-unit _ _) %. +%sort neq/deq/unit-arrow %. +%mode neq/deq/unit-arrow %in %out %. +%term _ %pi (neq/deq/unit-arrow Deq Dun) %<- (neq/deq/arrow-unit (cn-deq/sym Deq) Dun) %. +%worlds () (neq/deq/unit-arrow _ _) %. +%total {} (neq/deq/unit-arrow _ _) %. +%sort neq/deq/tagged-arrow %. +%mode neq/deq/tagged-arrow %in %out %. +%term _ + %pi (neq/deq/tagged-arrow Deq Dun) + %<- (map-equiv Deq map/tagged (map/arrow _ _) tmap/t Deq') + %<- (nullary-binary-equiv (ekof/i etopen/t ckof/tagged) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/arrow) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/tagged-arrow _ _) %. +%total {} (neq/deq/tagged-arrow _ _) %. +%sort neq/deq/cross-arrow %. +%mode neq/deq/cross-arrow %in %out %. +%sort neq/deq/cross-arrow! %. +%mode neq/deq/cross-arrow! %in %out %. +%worlds () (neq/deq/cross-arrow! _ _) %. +%total {} (neq/deq/cross-arrow! _ _) %. +%term _ + %pi (neq/deq/cross-arrow Deq Dun) + %<- (map-equiv Deq (map/cross _ _) (map/arrow _ _) tmap/t Deq') + %<- (binary-equiv-head (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/cross) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/arrow) Deq' D) + %<- (neq/deq/cross-arrow! D Dun) %. +%worlds () (neq/deq/cross-arrow _ _) %. +%total {} (neq/deq/cross-arrow _ _) %. +%sort neq/deq/forall-arrow %. +%mode neq/deq/forall-arrow %in %out %. +%term _ + %pi (neq/deq/forall-arrow Deq Dun) + %<- (map-equiv Deq (map/forall _ _) (map/arrow _ _) tmap/t Deq') + %<- (binary-forall-equiv (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/arrow) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/forall-arrow _ _) %. +%total {} (neq/deq/forall-arrow _ _) %. +%sort neq/deq/ref-arrow %. +%mode neq/deq/ref-arrow %in %out %. +%term _ + %pi (neq/deq/ref-arrow Deq Dun) + %<- (map-equiv Deq (map/ref _) (map/arrow _ _) tmap/t Deq') + %<- (unary-binary-equiv (ekof/i (etopen/pi etopen/t etopen/t) ckof/ref) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/arrow) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/ref-arrow _ _) %. +%total {} (neq/deq/ref-arrow _ _) %. +%sort neq/deq/tag-arrow %. +%mode neq/deq/tag-arrow %in %out %. +%term _ + %pi (neq/deq/tag-arrow Deq Dun) + %<- (map-equiv Deq (map/tag _) (map/arrow _ _) tmap/t Deq') + %<- (unary-binary-equiv (ekof/i (etopen/pi etopen/t etopen/t) ckof/tag) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/arrow) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/tag-arrow _ _) %. +%total {} (neq/deq/tag-arrow _ _) %. +%sort neq/deq/sum-arrow %. +%mode neq/deq/sum-arrow %in %out %. +%sort neq/deq/sum-arrow! %. +%mode neq/deq/sum-arrow! %in %out %. +%worlds () (neq/deq/sum-arrow! _ _) %. +%total {} (neq/deq/sum-arrow! _ _) %. +%term _ + %pi (neq/deq/sum-arrow Deq Dun) + %<- (map-equiv Deq (map/sum _ _) (map/arrow _ _) tmap/t Deq') + %<- (binary-equiv-head (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/dsum) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/arrow) Deq' D) + %<- (neq/deq/sum-arrow! D Dun) %. +%worlds () (neq/deq/sum-arrow _ _) %. +%total {} (neq/deq/sum-arrow _ _) %. +%sort neq/deq/mu-arrow %. +%mode neq/deq/mu-arrow %in %out %. +%term _ + %pi (neq/deq/mu-arrow Deq Dun) + %<- (map-equiv Deq (map/mu _) (map/arrow _ _) tmap/t Deq') + %<- (unary-binary-equiv (ekof/i (etopen/pi etopen/t (etopen/pi etopen/t etopen/t)) ckof/mu) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/arrow) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/mu-arrow _ _) %. +%total {} (neq/deq/mu-arrow _ _) %. +%sort neq/deq/unit-cross %. +%mode neq/deq/unit-cross %in %out %. +%term _ %pi (neq/deq/unit-cross Deq Dun) %<- (neq/deq/cross-unit (cn-deq/sym Deq) Dun) %. +%worlds () (neq/deq/unit-cross _ _) %. +%total {} (neq/deq/unit-cross _ _) %. +%sort neq/deq/tagged-cross %. +%mode neq/deq/tagged-cross %in %out %. +%term _ + %pi (neq/deq/tagged-cross Deq Dun) + %<- (map-equiv Deq map/tagged (map/cross _ _) tmap/t Deq') + %<- (nullary-binary-equiv (ekof/i etopen/t ckof/tagged) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/cross) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/tagged-cross _ _) %. +%total {} (neq/deq/tagged-cross _ _) %. +%sort neq/deq/arrow-cross %. +%mode neq/deq/arrow-cross %in %out %. +%term _ %pi (neq/deq/arrow-cross Deq Dun) %<- (neq/deq/cross-arrow (cn-deq/sym Deq) Dun) %. +%worlds () (neq/deq/arrow-cross _ _) %. +%total {} (neq/deq/arrow-cross _ _) %. +%sort neq/deq/forall-cross %. +%mode neq/deq/forall-cross %in %out %. +%term _ + %pi (neq/deq/forall-cross Deq Dun) + %<- (map-equiv Deq (map/forall _ _) (map/cross _ _) tmap/t Deq') + %<- (binary-forall-equiv (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/cross) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/forall-cross _ _) %. +%total {} (neq/deq/forall-cross _ _) %. +%sort neq/deq/ref-cross %. +%mode neq/deq/ref-cross %in %out %. +%term _ + %pi (neq/deq/ref-cross Deq Dun) + %<- (map-equiv Deq (map/ref _) (map/cross _ _) tmap/t Deq') + %<- (unary-binary-equiv (ekof/i (etopen/pi etopen/t etopen/t) ckof/ref) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/cross) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/ref-cross _ _) %. +%total {} (neq/deq/ref-cross _ _) %. +%sort neq/deq/tag-cross %. +%mode neq/deq/tag-cross %in %out %. +%term _ + %pi (neq/deq/tag-cross Deq Dun) + %<- (map-equiv Deq (map/tag _) (map/cross _ _) tmap/t Deq') + %<- (unary-binary-equiv (ekof/i (etopen/pi etopen/t etopen/t) ckof/tag) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/cross) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/tag-cross _ _) %. +%total {} (neq/deq/tag-cross _ _) %. +%sort neq/deq/sum-cross %. +%mode neq/deq/sum-cross %in %out %. +%sort neq/deq/sum-cross! %. +%mode neq/deq/sum-cross! %in %out %. +%worlds () (neq/deq/sum-cross! _ _) %. +%total {} (neq/deq/sum-cross! _ _) %. +%term _ + %pi (neq/deq/sum-cross Deq Dun) + %<- (map-equiv Deq (map/sum _ _) (map/cross _ _) tmap/t Deq') + %<- (binary-equiv-head (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/dsum) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/cross) Deq' D) + %<- (neq/deq/sum-cross! D Dun) %. +%worlds () (neq/deq/sum-cross _ _) %. +%total {} (neq/deq/sum-cross _ _) %. +%sort neq/deq/mu-cross %. +%mode neq/deq/mu-cross %in %out %. +%term _ + %pi (neq/deq/mu-cross Deq Dun) + %<- (map-equiv Deq (map/mu _) (map/cross _ _) tmap/t Deq') + %<- (unary-binary-equiv (ekof/i (etopen/pi etopen/t (etopen/pi etopen/t etopen/t)) ckof/mu) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/cross) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/mu-cross _ _) %. +%total {} (neq/deq/mu-cross _ _) %. +%sort neq/deq/unit-forall %. +%mode neq/deq/unit-forall %in %out %. +%term _ + %pi (neq/deq/unit-forall Deq Dun) + %<- (map-equiv Deq map/unit (map/forall _ _) tmap/t Deq') + %<- (nullary-forall-equiv (ekof/i etopen/t ckof/unit) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/unit-forall _ _) %. +%total {} (neq/deq/unit-forall _ _) %. +%sort neq/deq/tagged-forall %. +%mode neq/deq/tagged-forall %in %out %. +%term _ + %pi (neq/deq/tagged-forall Deq Dun) + %<- (map-equiv Deq map/tagged (map/forall _ _) tmap/t Deq') + %<- (nullary-forall-equiv (ekof/i etopen/t ckof/tagged) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/tagged-forall _ _) %. +%total {} (neq/deq/tagged-forall _ _) %. +%sort neq/deq/arrow-forall %. +%mode neq/deq/arrow-forall %in %out %. +%term _ + %pi (neq/deq/arrow-forall Deq Dun) + %<- (map-equiv Deq (map/arrow _ _) (map/forall _ _) tmap/t Deq') + %<- (binary-forall-equiv (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/arrow) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/arrow-forall _ _) %. +%total {} (neq/deq/arrow-forall _ _) %. +%sort neq/deq/cross-forall %. +%mode neq/deq/cross-forall %in %out %. +%term _ + %pi (neq/deq/cross-forall Deq Dun) + %<- (map-equiv Deq (map/cross _ _) (map/forall _ _) tmap/t Deq') + %<- (binary-forall-equiv (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/cross) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/cross-forall _ _) %. +%total {} (neq/deq/cross-forall _ _) %. +%sort neq/deq/ref-forall %. +%mode neq/deq/ref-forall %in %out %. +%term _ + %pi (neq/deq/ref-forall Deq Dun) + %<- (map-equiv Deq (map/ref _) (map/forall _ _) tmap/t Deq') + %<- (unary-forall-equiv (ekof/i (etopen/pi etopen/t etopen/t) ckof/ref) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/ref-forall _ _) %. +%total {} (neq/deq/ref-forall _ _) %. +%sort neq/deq/tag-forall %. +%mode neq/deq/tag-forall %in %out %. +%term _ + %pi (neq/deq/tag-forall Deq Dun) + %<- (map-equiv Deq (map/tag _) (map/forall _ _) tmap/t Deq') + %<- (unary-forall-equiv (ekof/i (etopen/pi etopen/t etopen/t) ckof/tag) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/tag-forall _ _) %. +%total {} (neq/deq/tag-forall _ _) %. +%sort neq/deq/sum-forall %. +%mode neq/deq/sum-forall %in %out %. +%term _ + %pi (neq/deq/sum-forall Deq Dun) + %<- (map-equiv Deq (map/sum _ _) (map/forall _ _) tmap/t Deq') + %<- (binary-forall-equiv (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/dsum) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/sum-forall _ _) %. +%total {} (neq/deq/sum-forall _ _) %. +%sort neq/deq/mu-forall %. +%mode neq/deq/mu-forall %in %out %. +%term _ + %pi (neq/deq/mu-forall Deq Dun) + %<- (map-equiv Deq (map/mu _) (map/forall _ _) tmap/t Deq') + %<- (unary-forall-equiv (ekof/i (etopen/pi etopen/t (etopen/pi etopen/t etopen/t)) ckof/mu) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/mu-forall _ _) %. +%total {} (neq/deq/mu-forall _ _) %. +%sort neq/deq/unit-ref %. +%mode neq/deq/unit-ref %in %out %. +%term _ %pi (neq/deq/unit-ref Deq Dun) %<- (neq/deq/ref-unit (cn-deq/sym Deq) Dun) %. +%worlds () (neq/deq/unit-ref _ _) %. +%total {} (neq/deq/unit-ref _ _) %. +%sort neq/deq/tagged-ref %. +%mode neq/deq/tagged-ref %in %out %. +%term _ + %pi (neq/deq/tagged-ref Deq Dun) + %<- (map-equiv Deq map/tagged (map/ref _) tmap/t Deq') + %<- (nullary-unary-equiv (ekof/i etopen/t ckof/tagged) (ekof/i (etopen/pi etopen/t etopen/t) ckof/ref) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/tagged-ref _ _) %. +%total {} (neq/deq/tagged-ref _ _) %. +%sort neq/deq/arrow-ref %. +%mode neq/deq/arrow-ref %in %out %. +%term _ + %pi (neq/deq/arrow-ref Deq Dun) + %<- (map-equiv Deq (map/arrow _ _) (map/ref _) tmap/t Deq') + %<- (unary-binary-equiv (ekof/i (etopen/pi etopen/t etopen/t) ckof/ref) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/arrow) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/arrow-ref _ _) %. +%total {} (neq/deq/arrow-ref _ _) %. +%sort neq/deq/cross-ref %. +%mode neq/deq/cross-ref %in %out %. +%term _ + %pi (neq/deq/cross-ref Deq Dun) + %<- (map-equiv Deq (map/cross _ _) (map/ref _) tmap/t Deq') + %<- (unary-binary-equiv (ekof/i (etopen/pi etopen/t etopen/t) ckof/ref) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/cross) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/cross-ref _ _) %. +%total {} (neq/deq/cross-ref _ _) %. +%sort neq/deq/forall-ref %. +%mode neq/deq/forall-ref %in %out %. +%term _ %pi (neq/deq/forall-ref Deq Dun) %<- (neq/deq/ref-forall (cn-deq/sym Deq) Dun) %. +%worlds () (neq/deq/forall-ref _ _) %. +%total {} (neq/deq/forall-ref _ _) %. +%sort neq/deq/sum-ref %. +%mode neq/deq/sum-ref %in %out %. +%term _ + %pi (neq/deq/sum-ref Deq Dun) + %<- (map-equiv Deq (map/sum _ _) (map/ref _) tmap/t Deq') + %<- (unary-binary-equiv (ekof/i (etopen/pi etopen/t etopen/t) ckof/ref) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/dsum) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/sum-ref _ _) %. +%total {} (neq/deq/sum-ref _ _) %. +%sort neq/deq/mu-ref %. +%mode neq/deq/mu-ref %in %out %. +%sort neq/deq/mu-ref! %. +%mode neq/deq/mu-ref! %in %out %. +%worlds () (neq/deq/mu-ref! _ _) %. +%total {} (neq/deq/mu-ref! _ _) %. +%term _ + %pi (neq/deq/mu-ref Deq Dun) + %<- (map-equiv Deq (map/mu _) (map/ref _) tmap/t Deq') + %<- (unary-equiv-head (ekof/i (etopen/pi etopen/t (etopen/pi etopen/t etopen/t)) ckof/mu) (ekof/i (etopen/pi etopen/t etopen/t) ckof/ref) Deq' D) + %<- (neq/deq/mu-ref! D Dun) %. +%worlds () (neq/deq/mu-ref _ _) %. +%total {} (neq/deq/mu-ref _ _) %. +%sort neq/deq/tag-ref %. +%mode neq/deq/tag-ref %in %out %. +%sort neq/deq/tag-ref! %. +%mode neq/deq/tag-ref! %in %out %. +%worlds () (neq/deq/tag-ref! _ _) %. +%total {} (neq/deq/tag-ref! _ _) %. +%term _ + %pi (neq/deq/tag-ref Deq Dun) + %<- (map-equiv Deq (map/tag _) (map/ref _) tmap/t Deq') + %<- (unary-equiv-head (ekof/i (etopen/pi etopen/t etopen/t) ckof/tag) (ekof/i (etopen/pi etopen/t etopen/t) ckof/ref) Deq' D) + %<- (neq/deq/tag-ref! D Dun) %. +%worlds () (neq/deq/tag-ref _ _) %. +%total {} (neq/deq/tag-ref _ _) %. +%sort neq/deq/unit-tag %. +%mode neq/deq/unit-tag %in %out %. +%term _ %pi (neq/deq/unit-tag Deq Dun) %<- (neq/deq/tag-unit (cn-deq/sym Deq) Dun) %. +%worlds () (neq/deq/unit-tag _ _) %. +%total {} (neq/deq/unit-tag _ _) %. +%sort neq/deq/tagged-tag %. +%mode neq/deq/tagged-tag %in %out %. +%term _ + %pi (neq/deq/tagged-tag Deq Dun) + %<- (map-equiv Deq map/tagged (map/tag _) tmap/t Deq') + %<- (nullary-unary-equiv (ekof/i etopen/t ckof/tagged) (ekof/i (etopen/pi etopen/t etopen/t) ckof/tag) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/tagged-tag _ _) %. +%total {} (neq/deq/tagged-tag _ _) %. +%sort neq/deq/arrow-tag %. +%mode neq/deq/arrow-tag %in %out %. +%term _ + %pi (neq/deq/arrow-tag Deq Dun) + %<- (map-equiv Deq (map/arrow _ _) (map/tag _) tmap/t Deq') + %<- (unary-binary-equiv (ekof/i (etopen/pi etopen/t etopen/t) ckof/tag) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/arrow) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/arrow-tag _ _) %. +%total {} (neq/deq/arrow-tag _ _) %. +%sort neq/deq/cross-tag %. +%mode neq/deq/cross-tag %in %out %. +%term _ + %pi (neq/deq/cross-tag Deq Dun) + %<- (map-equiv Deq (map/cross _ _) (map/tag _) tmap/t Deq') + %<- (unary-binary-equiv (ekof/i (etopen/pi etopen/t etopen/t) ckof/tag) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/cross) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/cross-tag _ _) %. +%total {} (neq/deq/cross-tag _ _) %. +%sort neq/deq/forall-tag %. +%mode neq/deq/forall-tag %in %out %. +%term _ %pi (neq/deq/forall-tag Deq Dun) %<- (neq/deq/tag-forall (cn-deq/sym Deq) Dun) %. +%worlds () (neq/deq/forall-tag _ _) %. +%total {} (neq/deq/forall-tag _ _) %. +%sort neq/deq/sum-tag %. +%mode neq/deq/sum-tag %in %out %. +%term _ + %pi (neq/deq/sum-tag Deq Dun) + %<- (map-equiv Deq (map/sum _ _) (map/tag _) tmap/t Deq') + %<- (unary-binary-equiv (ekof/i (etopen/pi etopen/t etopen/t) ckof/tag) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/dsum) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/sum-tag _ _) %. +%total {} (neq/deq/sum-tag _ _) %. +%sort neq/deq/mu-tag %. +%mode neq/deq/mu-tag %in %out %. +%sort neq/deq/mu-tag! %. +%mode neq/deq/mu-tag! %in %out %. +%worlds () (neq/deq/mu-tag! _ _) %. +%total {} (neq/deq/mu-tag! _ _) %. +%term _ + %pi (neq/deq/mu-tag Deq Dun) + %<- (map-equiv Deq (map/mu _) (map/tag _) tmap/t Deq') + %<- (unary-equiv-head (ekof/i (etopen/pi etopen/t (etopen/pi etopen/t etopen/t)) ckof/mu) (ekof/i (etopen/pi etopen/t etopen/t) ckof/tag) Deq' D) + %<- (neq/deq/mu-tag! D Dun) %. +%worlds () (neq/deq/mu-tag _ _) %. +%total {} (neq/deq/mu-tag _ _) %. +%sort neq/deq/ref-tag %. +%mode neq/deq/ref-tag %in %out %. +%term _ %pi (neq/deq/ref-tag Deq Dun) %<- (neq/deq/tag-ref (cn-deq/sym Deq) Dun) %. +%worlds () (neq/deq/ref-tag _ _) %. +%total {} (neq/deq/ref-tag _ _) %. +%sort neq/deq/unit-sum %. +%mode neq/deq/unit-sum %in %out %. +%term _ %pi (neq/deq/unit-sum Deq Dun) %<- (neq/deq/sum-unit (cn-deq/sym Deq) Dun) %. +%worlds () (neq/deq/unit-sum _ _) %. +%total {} (neq/deq/unit-sum _ _) %. +%sort neq/deq/tagged-sum %. +%mode neq/deq/tagged-sum %in %out %. +%term _ + %pi (neq/deq/tagged-sum Deq Dun) + %<- (map-equiv Deq map/tagged (map/sum _ _) tmap/t Deq') + %<- (nullary-binary-equiv (ekof/i etopen/t ckof/tagged) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/dsum) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/tagged-sum _ _) %. +%total {} (neq/deq/tagged-sum _ _) %. +%sort neq/deq/cross-sum %. +%mode neq/deq/cross-sum %in %out %. +%sort neq/deq/cross-sum! %. +%mode neq/deq/cross-sum! %in %out %. +%worlds () (neq/deq/cross-sum! _ _) %. +%total {} (neq/deq/cross-sum! _ _) %. +%term _ + %pi (neq/deq/cross-sum Deq Dun) + %<- (map-equiv Deq (map/cross _ _) (map/sum _ _) tmap/t Deq') + %<- (binary-equiv-head (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/cross) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/dsum) Deq' D) + %<- (neq/deq/cross-sum! D Dun) %. +%worlds () (neq/deq/cross-sum _ _) %. +%total {} (neq/deq/cross-sum _ _) %. +%sort neq/deq/forall-sum %. +%mode neq/deq/forall-sum %in %out %. +%term _ + %pi (neq/deq/forall-sum Deq Dun) + %<- (map-equiv Deq (map/forall _ _) (map/sum _ _) tmap/t Deq') + %<- (binary-forall-equiv (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/dsum) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/forall-sum _ _) %. +%total {} (neq/deq/forall-sum _ _) %. +%sort neq/deq/ref-sum %. +%mode neq/deq/ref-sum %in %out %. +%term _ + %pi (neq/deq/ref-sum Deq Dun) + %<- (map-equiv Deq (map/ref _) (map/sum _ _) tmap/t Deq') + %<- (unary-binary-equiv (ekof/i (etopen/pi etopen/t etopen/t) ckof/ref) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/dsum) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/ref-sum _ _) %. +%total {} (neq/deq/ref-sum _ _) %. +%sort neq/deq/tag-sum %. +%mode neq/deq/tag-sum %in %out %. +%term _ + %pi (neq/deq/tag-sum Deq Dun) + %<- (map-equiv Deq (map/tag _) (map/sum _ _) tmap/t Deq') + %<- (unary-binary-equiv (ekof/i (etopen/pi etopen/t etopen/t) ckof/tag) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/dsum) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/tag-sum _ _) %. +%total {} (neq/deq/tag-sum _ _) %. +%sort neq/deq/arrow-sum %. +%mode neq/deq/arrow-sum %in %out %. +%term _ %pi (neq/deq/arrow-sum Deq Dun) %<- (neq/deq/sum-arrow (cn-deq/sym Deq) Dun) %. +%worlds () (neq/deq/arrow-sum _ _) %. +%total {} (neq/deq/arrow-sum _ _) %. +%sort neq/deq/mu-sum %. +%mode neq/deq/mu-sum %in %out %. +%term _ + %pi (neq/deq/mu-sum Deq Dun) + %<- (map-equiv Deq (map/mu _) (map/sum _ _) tmap/t Deq') + %<- (unary-binary-equiv (ekof/i (etopen/pi etopen/t (etopen/pi etopen/t etopen/t)) ckof/mu) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/dsum) Deq' D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/mu-sum _ _) %. +%total {} (neq/deq/mu-sum _ _) %. +%sort neq/deq/arrow-tagged %. +%mode neq/deq/arrow-tagged %in %out %. +%term _ + %pi (neq/deq/arrow-tagged Deq Dun) + %<- (map-equiv Deq (map/arrow _ _) map/tagged tmap/t Deq') + %<- (nullary-binary-equiv (ekof/i etopen/t ckof/tagged) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/arrow) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/arrow-tagged _ _) %. +%total {} (neq/deq/arrow-tagged _ _) %. +%sort neq/deq/cross-tagged %. +%mode neq/deq/cross-tagged %in %out %. +%term _ + %pi (neq/deq/cross-tagged Deq Dun) + %<- (map-equiv Deq (map/cross _ _) map/tagged tmap/t Deq') + %<- (nullary-binary-equiv (ekof/i etopen/t ckof/tagged) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/cross) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/cross-tagged _ _) %. +%total {} (neq/deq/cross-tagged _ _) %. +%sort neq/deq/forall-tagged %. +%mode neq/deq/forall-tagged %in %out %. +%term _ + %pi (neq/deq/forall-tagged Deq Dun) + %<- (map-equiv Deq (map/forall _ _) map/tagged tmap/t Deq') + %<- (nullary-forall-equiv (ekof/i etopen/t ckof/tagged) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/forall-tagged _ _) %. +%total {} (neq/deq/forall-tagged _ _) %. +%sort neq/deq/ref-tagged %. +%mode neq/deq/ref-tagged %in %out %. +%term _ + %pi (neq/deq/ref-tagged Deq Dun) + %<- (map-equiv Deq (map/ref _) map/tagged tmap/t Deq') + %<- (nullary-unary-equiv (ekof/i etopen/t ckof/tagged) (ekof/i (etopen/pi etopen/t etopen/t) ckof/ref) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/ref-tagged _ _) %. +%total {} (neq/deq/ref-tagged _ _) %. +%sort neq/deq/tag-tagged %. +%mode neq/deq/tag-tagged %in %out %. +%term _ + %pi (neq/deq/tag-tagged Deq Dun) + %<- (map-equiv Deq (map/tag _) map/tagged tmap/t Deq') + %<- (nullary-unary-equiv (ekof/i etopen/t ckof/tagged) (ekof/i (etopen/pi etopen/t etopen/t) ckof/tag) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/tag-tagged _ _) %. +%total {} (neq/deq/tag-tagged _ _) %. +%sort neq/deq/sum-tagged %. +%mode neq/deq/sum-tagged %in %out %. +%term _ + %pi (neq/deq/sum-tagged Deq Dun) + %<- (map-equiv Deq (map/sum _ _) map/tagged tmap/t Deq') + %<- (nullary-binary-equiv (ekof/i etopen/t ckof/tagged) (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/dsum) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/sum-tagged _ _) %. +%total {} (neq/deq/sum-tagged _ _) %. +%sort neq/deq/unit-tagged %. +%mode neq/deq/unit-tagged %in %out %. +%term _ %pi (neq/deq/unit-tagged Deq Dun) %<- (neq/deq/tagged-unit (cn-deq/sym Deq) Dun) %. +%worlds () (neq/deq/unit-tagged _ _) %. +%total {} (neq/deq/unit-tagged _ _) %. +%sort neq/deq/mu-tagged %. +%mode neq/deq/mu-tagged %in %out %. +%term _ + %pi (neq/deq/mu-tagged Deq Dun) + %<- (map-equiv Deq (map/mu _) map/tagged tmap/t Deq') + %<- (nullary-unary-equiv (ekof/i etopen/t ckof/tagged) (ekof/i (etopen/pi etopen/t (etopen/pi etopen/t etopen/t)) ckof/mu) (equiv/symm Deq') D) + %<- (false-implies-uninhabited D Dun) %. +%worlds () (neq/deq/mu-tagged _ _) %. +%total {} (neq/deq/mu-tagged _ _) %. +%sort neq/deq/unit-mu %. +%mode neq/deq/unit-mu %in %out %. +%term _ %pi (neq/deq/unit-mu Deq Dun) %<- (neq/deq/mu-unit (cn-deq/sym Deq) Dun) %. +%worlds () (neq/deq/unit-mu _ _) %. +%total {} (neq/deq/unit-mu _ _) %. +%sort neq/deq/tagged-mu %. +%mode neq/deq/tagged-mu %in %out %. +%term _ %pi (neq/deq/tagged-mu Deq Dun) %<- (neq/deq/mu-tagged (cn-deq/sym Deq) Dun) %. +%worlds () (neq/deq/tagged-mu _ _) %. +%total {} (neq/deq/tagged-mu _ _) %. +%sort neq/deq/arrow-mu %. +%mode neq/deq/arrow-mu %in %out %. +%term _ %pi (neq/deq/arrow-mu Deq Dun) %<- (neq/deq/mu-arrow (cn-deq/sym Deq) Dun) %. +%worlds () (neq/deq/arrow-mu _ _) %. +%total {} (neq/deq/arrow-mu _ _) %. +%sort neq/deq/cross-mu %. +%mode neq/deq/cross-mu %in %out %. +%term _ %pi (neq/deq/cross-mu Deq Dun) %<- (neq/deq/mu-cross (cn-deq/sym Deq) Dun) %. +%worlds () (neq/deq/cross-mu _ _) %. +%total {} (neq/deq/cross-mu _ _) %. +%sort neq/deq/ref-mu %. +%mode neq/deq/ref-mu %in %out %. +%term _ %pi (neq/deq/ref-mu Deq Dun) %<- (neq/deq/mu-ref (cn-deq/sym Deq) Dun) %. +%worlds () (neq/deq/ref-mu _ _) %. +%total {} (neq/deq/ref-mu _ _) %. +%sort neq/deq/tag-mu %. +%mode neq/deq/tag-mu %in %out %. +%term _ %pi (neq/deq/tag-mu Deq Dun) %<- (neq/deq/mu-tag (cn-deq/sym Deq) Dun) %. +%worlds () (neq/deq/tag-mu _ _) %. +%total {} (neq/deq/tag-mu _ _) %. +%sort neq/deq/sum-mu %. +%mode neq/deq/sum-mu %in %out %. +%term _ %pi (neq/deq/sum-mu Deq Dun) %<- (neq/deq/mu-sum (cn-deq/sym Deq) Dun) %. +%worlds () (neq/deq/sum-mu _ _) %. +%total {} (neq/deq/sum-mu _ _) %. +%sort neq/deq/forall-mu %. +%mode neq/deq/forall-mu %in %out %. +%term _ %pi (neq/deq/forall-mu Deq Dun) %<- (neq/deq/mu-forall (cn-deq/sym Deq) Dun) %. +%worlds () (neq/deq/forall-mu _ _) %. +%total {} (neq/deq/forall-mu _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/inversions-cons.lf b/new-tests/stelf-output/tslf/inversions-cons.lf new file mode 100644 index 0000000..fd46fa2 --- /dev/null +++ b/new-tests/stelf-output/tslf/inversions-cons.lf @@ -0,0 +1,127 @@ +%%%% simple inversion lemmas +%sort inv/ofkd/tp/cross %. +%mode inv/ofkd/tp/cross %in %in %out %out %. +%term _ inv/ofkd/tp/cross (ofkd/tp/cross D1 D2) _ D1 D2 %. +%term _ + %pi (inv/ofkd/tp/cross (ofkd/kd/sing DC) _ D1 D2) + %<- (inv/ofkd/tp/cross DC kd-sub/kd/type D1 D2) %. +%term _ + %pi (inv/ofkd/tp/cross (ofkd/sub DC DS) KS D1 D2) + %<- (kd-trans/sub DS KS DS') + %<- (inv/ofkd/tp/cross DC DS' D1 D2) %. +%term _ + %pi (inv/ofkd/tp/cross (ofkd/deq DC DQ) KS D1 D2) + %<- (vdt/kd-deq DQ DQ1 DQ2) + %<- (kd-anti DQ DQ1 DQ2 DS _) + %<- (kd-trans/sub DS KS DS') + %<- (inv/ofkd/tp/cross DC DS' D1 D2) %. +%worlds (ofkd+vdt-block) (inv/ofkd/tp/cross _ _ _ _) %. +%total D1 (inv/ofkd/tp/cross D1 _ _ _) %. +%sort inv/ofkd/tp/arrow %. +%mode inv/ofkd/tp/arrow %in %in %out %out %. +%term _ inv/ofkd/tp/arrow (ofkd/tp/arrow D1 D2) _ D1 D2 %. +%term _ + %pi (inv/ofkd/tp/arrow (ofkd/kd/sing DC) _ D1 D2) + %<- (inv/ofkd/tp/arrow DC kd-sub/kd/type D1 D2) %. +%term _ + %pi (inv/ofkd/tp/arrow (ofkd/sub DC DS) KS D1 D2) + %<- (kd-trans/sub DS KS DS') + %<- (inv/ofkd/tp/arrow DC DS' D1 D2) %. +%term _ + %pi (inv/ofkd/tp/arrow (ofkd/deq DC DQ) KS D1 D2) + %<- (vdt/kd-deq DQ DQ1 DQ2) + %<- (kd-anti DQ DQ1 DQ2 DS _) + %<- (kd-trans/sub DS KS DS') + %<- (inv/ofkd/tp/arrow DC DS' D1 D2) %. +%worlds (ofkd+vdt-block) (inv/ofkd/tp/arrow _ _ _ _) %. +%total D1 (inv/ofkd/tp/arrow D1 _ _ _) %. +%sort inv/ofkd/tp/sum %. +%mode inv/ofkd/tp/sum %in %in %out %out %. +%term _ inv/ofkd/tp/sum (ofkd/tp/sum D1 D2) _ D1 D2 %. +%term _ + %pi (inv/ofkd/tp/sum (ofkd/kd/sing DC) _ D1 D2) + %<- (inv/ofkd/tp/sum DC kd-sub/kd/type D1 D2) %. +%term _ + %pi (inv/ofkd/tp/sum (ofkd/sub DC DS) KS D1 D2) + %<- (kd-trans/sub DS KS DS') + %<- (inv/ofkd/tp/sum DC DS' D1 D2) %. +%term _ + %pi (inv/ofkd/tp/sum (ofkd/deq DC DQ) KS D1 D2) + %<- (vdt/kd-deq DQ DQ1 DQ2) + %<- (kd-anti DQ DQ1 DQ2 DS _) + %<- (kd-trans/sub DS KS DS') + %<- (inv/ofkd/tp/sum DC DS' D1 D2) %. +%worlds (ofkd+vdt-block) (inv/ofkd/tp/sum _ _ _ _) %. +%total D1 (inv/ofkd/tp/sum D1 _ _ _) %. +%sort inv/ofkd/tp/forall %. +%mode inv/ofkd/tp/forall %in %in %out %out %. +%term _ inv/ofkd/tp/forall (ofkd/tp/forall D2 D1) _ D1 D2 %. +%term _ + %pi (inv/ofkd/tp/forall (ofkd/kd/sing DC) _ D1 D2) + %<- (inv/ofkd/tp/forall DC kd-sub/kd/type D1 D2) %. +%term _ + %pi (inv/ofkd/tp/forall (ofkd/sub DC DS) KS D1 D2) + %<- (kd-trans/sub DS KS DS') + %<- (inv/ofkd/tp/forall DC DS' D1 D2) %. +%term _ + %pi (inv/ofkd/tp/forall (ofkd/deq DC DQ) KS D1 D2) + %<- (vdt/kd-deq DQ DQ1 DQ2) + %<- (kd-anti DQ DQ1 DQ2 DS _) + %<- (kd-trans/sub DS KS DS') + %<- (inv/ofkd/tp/forall DC DS' D1 D2) %. +%worlds (ofkd+vdt-block) (inv/ofkd/tp/forall _ _ _ _) %. +%total D1 (inv/ofkd/tp/forall D1 _ _ _) %. +%sort inv/ofkd/tp/ref %. +%mode inv/ofkd/tp/ref %in %in %out %. +%term _ inv/ofkd/tp/ref (ofkd/tp/ref D1) _ D1 %. +%term _ + %pi (inv/ofkd/tp/ref (ofkd/kd/sing DC) _ D1) + %<- (inv/ofkd/tp/ref DC kd-sub/kd/type D1) %. +%term _ + %pi (inv/ofkd/tp/ref (ofkd/sub DC DS) KS D1) + %<- (kd-trans/sub DS KS DS') + %<- (inv/ofkd/tp/ref DC DS' D1) %. +%term _ + %pi (inv/ofkd/tp/ref (ofkd/deq DC DQ) KS D1) + %<- (vdt/kd-deq DQ DQ1 DQ2) + %<- (kd-anti DQ DQ1 DQ2 DS _) + %<- (kd-trans/sub DS KS DS') + %<- (inv/ofkd/tp/ref DC DS' D1) %. +%worlds (ofkd+vdt-block) (inv/ofkd/tp/ref _ _ _) %. +%total D1 (inv/ofkd/tp/ref D1 _ _) %. +%sort inv/ofkd/tp/tag %. +%mode inv/ofkd/tp/tag %in %in %out %. +%term _ inv/ofkd/tp/tag (ofkd/tp/tag D1) _ D1 %. +%term _ + %pi (inv/ofkd/tp/tag (ofkd/kd/sing DC) _ D1) + %<- (inv/ofkd/tp/tag DC kd-sub/kd/type D1) %. +%term _ + %pi (inv/ofkd/tp/tag (ofkd/sub DC DS) KS D1) + %<- (kd-trans/sub DS KS DS') + %<- (inv/ofkd/tp/tag DC DS' D1) %. +%term _ + %pi (inv/ofkd/tp/tag (ofkd/deq DC DQ) KS D1) + %<- (vdt/kd-deq DQ DQ1 DQ2) + %<- (kd-anti DQ DQ1 DQ2 DS _) + %<- (kd-trans/sub DS KS DS') + %<- (inv/ofkd/tp/tag DC DS' D1) %. +%worlds (ofkd+vdt-block) (inv/ofkd/tp/tag _ _ _) %. +%total D1 (inv/ofkd/tp/tag D1 _ _) %. +%sort inv/ofkd/cn/mu %. +%mode inv/ofkd/cn/mu %in %in %out %. +%term _ %pi (inv/ofkd/cn/mu (ofkd/cn/mu D1 D2) DS D1') %<- (kd-wkn/ofkd D1 DS D1') %. +%term _ + %pi (inv/ofkd/cn/mu (ofkd/kd/sing DC) DS D1) + %<- (inv/ofkd/cn/mu DC kd-sub/kd/type D1) %. +%term _ + %pi (inv/ofkd/cn/mu (ofkd/sub DC DS) DS' D1) + %<- (kd-trans/sub DS DS' DS'') + %<- (inv/ofkd/cn/mu DC DS'' D1) %. +%term _ + %pi (inv/ofkd/cn/mu (ofkd/deq DC DQ) DS' D1) + %<- (vdt/kd-deq DQ DQ1 DQ2) + %<- (kd-anti DQ DQ1 DQ2 DS _) + %<- (kd-trans/sub DS DS' DS'') + %<- (inv/ofkd/cn/mu DC DS'' D1) %. +%worlds (ofkd+vdt-block) (inv/ofkd/cn/mu _ _ _) %. +%total D1 (inv/ofkd/cn/mu D1 _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/inversions-deq.lf b/new-tests/stelf-output/tslf/inversions-deq.lf new file mode 100644 index 0000000..3c9fe4c --- /dev/null +++ b/new-tests/stelf-output/tslf/inversions-deq.lf @@ -0,0 +1,90 @@ +%sort inv/cn-deq/tp/cross %. +%mode inv/cn-deq/tp/cross %in %out %out %. +%term _ + %pi (inv/cn-deq/tp/cross D D1 D2) + %<- (map-equiv D (map/cross Dmap2 Dmap1) (map/cross Dmap2' Dmap1') tmap/t Deq) + %<- (binary-equiv-inversion (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/cross) Deq Deq1 Deq2) + %<- (vdt/cn-deq D Dof Dof' _) + %<- (inv/ofkd/tp/cross Dof kd-sub/kd/type Dof1 Dof2) + %<- (inv/ofkd/tp/cross Dof' kd-sub/kd/type Dof1' Dof2') + %<- (invert-map-equiv Dmap1 Dmap1' tmap/t Dof1 Dof1' Deq1 D1) + %<- (invert-map-equiv Dmap2 Dmap2' tmap/t Dof2 Dof2' Deq2 D2) %. +%worlds () (inv/cn-deq/tp/cross _ _ _) %. +%total {} (inv/cn-deq/tp/cross _ _ _) %. +%sort inv/cn-deq/tp/ref %. +%mode inv/cn-deq/tp/ref %in %out %. +%term _ + %pi (inv/cn-deq/tp/ref D D1) + %<- (map-equiv D (map/ref Dmap1) (map/ref Dmap1') tmap/t Deq) + %<- (unary-equiv-inversion (ekof/i (etopen/pi etopen/t etopen/t) ckof/ref) Deq Deq1) + %<- (vdt/cn-deq D Dof Dof' _) + %<- (inv/ofkd/tp/ref Dof kd-sub/kd/type Dof1) + %<- (inv/ofkd/tp/ref Dof' kd-sub/kd/type Dof1') + %<- (invert-map-equiv Dmap1 Dmap1' tmap/t Dof1 Dof1' Deq1 D1) %. +%worlds () (inv/cn-deq/tp/ref _ _) %. +%total {} (inv/cn-deq/tp/ref _ _) %. +%sort inv/cn-deq/tp/forall %. +%mode inv/cn-deq/tp/forall %in %out %out %. +%term _ + %pi (inv/cn-deq/tp/forall D D1 D2) + %<- (map-equiv D (map/forall Dmap2 Dmap1) (map/forall Dmap2' Dmap1') tmap/t Deq) + %<- (forall-equiv-inversion Deq Deq1 Deq2) + %<- (vdt/cn-deq D Dof Dof' _) + %<- (inv/ofkd/tp/forall Dof kd-sub/kd/type Dwf Dof1) + %<- (inv/ofkd/tp/forall Dof' kd-sub/kd/type Dwf' Dof1') + %<- (invert-map-tequiv Dmap1 Dmap1' Dwf Dwf' Deq1 D1) + %<- (elam-equiv-invert Deq2 _ _ Deq2') + %<- (invert-map-equiv1 Dmap1 Dmap2 Dmap2' ([_] [_] [_] tmap/t) Dwf Dof1 ([a] [d] Dof1' a (ofkd/deq d D1)) Deq2' D2) %. +%worlds () (inv/cn-deq/tp/forall _ _ _) %. +%total {} (inv/cn-deq/tp/forall _ _ _) %. +%sort inv/cn-deq/tp/arrow %. +%mode inv/cn-deq/tp/arrow %in %out %out %. +%term _ + %pi (inv/cn-deq/tp/arrow D D1 D2) + %<- (map-equiv D (map/arrow Dmap2 Dmap1) (map/arrow Dmap2' Dmap1') tmap/t Deq) + %<- (binary-equiv-inversion (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/arrow) Deq Deq1 Deq2) + %<- (vdt/cn-deq D Dof Dof' _) + %<- (inv/ofkd/tp/arrow Dof kd-sub/kd/type Dof1 Dof2) + %<- (inv/ofkd/tp/arrow Dof' kd-sub/kd/type Dof1' Dof2') + %<- (invert-map-equiv Dmap1 Dmap1' tmap/t Dof1 Dof1' Deq1 D1) + %<- (invert-map-equiv Dmap2 Dmap2' tmap/t Dof2 Dof2' Deq2 D2) %. +%worlds () (inv/cn-deq/tp/arrow _ _ _) %. +%total {} (inv/cn-deq/tp/arrow _ _ _) %. +%sort inv/cn-deq/tp/sum %. +%mode inv/cn-deq/tp/sum %in %out %out %. +%term _ + %pi (inv/cn-deq/tp/sum D D1 D2) + %<- (map-equiv D (map/sum Dmap2 Dmap1) (map/sum Dmap2' Dmap1') tmap/t Deq) + %<- (binary-equiv-inversion (ekof/i (etopen/pi (etopen/pi etopen/t etopen/t) etopen/t) ckof/dsum) Deq Deq1 Deq2) + %<- (vdt/cn-deq D Dof Dof' _) + %<- (inv/ofkd/tp/sum Dof kd-sub/kd/type Dof1 Dof2) + %<- (inv/ofkd/tp/sum Dof' kd-sub/kd/type Dof1' Dof2') + %<- (invert-map-equiv Dmap1 Dmap1' tmap/t Dof1 Dof1' Deq1 D1) + %<- (invert-map-equiv Dmap2 Dmap2' tmap/t Dof2 Dof2' Deq2 D2) %. +%worlds () (inv/cn-deq/tp/sum _ _ _) %. +%total {} (inv/cn-deq/tp/sum _ _ _) %. +%sort inv/cn-deq/cn/mu %. +%mode inv/cn-deq/cn/mu %in %out %. +%term _ + %pi (inv/cn-deq/cn/mu D D2) + %<- (map-equiv D (map/mu Dmap2) (map/mu Dmap2') tmap/t Deq) + %<- (unary-equiv-inversion (ekof/i (etopen/pi etopen/t (etopen/pi etopen/t etopen/t)) ckof/mu) Deq Deq2) + %<- (vdt/cn-deq D Dof Dof' _) + %<- (inv/ofkd/cn/mu Dof kd-sub/kd/type Dof1) + %<- (inv/ofkd/cn/mu Dof' kd-sub/kd/type Dof1') + %<- (elam-equiv-invert Deq2 _ _ Deq2') + %<- (invert-map-equiv1 tmap/t Dmap2 Dmap2' ([_] [_] [_] tmap/t) kd-wf/kd/type Dof1 Dof1' Deq2' D2) %. +%worlds () (inv/cn-deq/cn/mu _ _) %. +%total {} (inv/cn-deq/cn/mu _ _) %. +%sort inv/cn-deq/tp/tag %. +%mode inv/cn-deq/tp/tag %in %out %. +%term _ + %pi (inv/cn-deq/tp/tag D D1) + %<- (map-equiv D (map/tag Dmap1) (map/tag Dmap1') tmap/t Deq) + %<- (unary-equiv-inversion (ekof/i (etopen/pi etopen/t etopen/t) ckof/tag) Deq Deq1) + %<- (vdt/cn-deq D Dof Dof' _) + %<- (inv/ofkd/tp/tag Dof kd-sub/kd/type Dof1) + %<- (inv/ofkd/tp/tag Dof' kd-sub/kd/type Dof1') + %<- (invert-map-equiv Dmap1 Dmap1' tmap/t Dof1 Dof1' Deq1 D1) %. +%worlds () (inv/cn-deq/tp/tag _ _) %. +%total {} (inv/cn-deq/tp/tag _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/inversions-mods.lf b/new-tests/stelf-output/tslf/inversions-mods.lf new file mode 100644 index 0000000..7077061 --- /dev/null +++ b/new-tests/stelf-output/tslf/inversions-mods.lf @@ -0,0 +1,418 @@ +%%%% inversion lemmas for modules +%sort inv/ofsg/md/tm %. +%mode inv/ofsg/md/tm %in %out %. +%term _ inv/ofsg/md/tm (ofsg/md/tm D1) D1 %. +%term _ + %pi (inv/ofsg/md/tm (ofsg/sub D1 (sg-sub/sg/cn DQ) _) (oftp/deq D1' DQ)) + %<- (inv/ofsg/md/tm D1 D1') %. +%worlds (ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (inv/ofsg/md/tm _ _) %. +%total D1 (inv/ofsg/md/tm D1 _) %. +%sort md/tm-not-sg/sgm %. +%mode md/tm-not-sg/sgm %in %out %. +%term _ %pi (md/tm-not-sg/sgm (ofsg/sub D1 _ pty-sub/pp) DU) %<- (md/tm-not-sg/sgm D1 DU) %. +%term _ + %pi (md/tm-not-sg/sgm (ofsg/sgm-ext D1 _) DU) + %<- (subder/md/pj1 D1 D1' pty-sub/pp) + %<- (md/tm-not-sg/sgm D1' DU) %. +%worlds (ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (md/tm-not-sg/sgm _ _) %. +%total D1 (md/tm-not-sg/sgm D1 _) %. +%sort md/tm-not-sg/kd %. +%mode md/tm-not-sg/kd %in %out %. +%term _ %pi (md/tm-not-sg/kd (ofsg/sub D1 _ pty-sub/pp) DU) %<- (md/tm-not-sg/kd D1 DU) %. +%term _ %pi (md/tm-not-sg/kd (ofsg/kd-ext D1 _ _) DU) %<- (md/tm-not-sg/kd D1 DU) %. +%worlds (ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (md/tm-not-sg/kd _ _) %. +%total D1 (md/tm-not-sg/kd D1 _) %. +%sort inv/ofsg/md/tm %. +%mode inv/ofsg/md/tm %in %out %out %. +%term _ + %pi (inv/ofsg/md/tm (ofsg/md/tm D1) D1 DS) + %<- (vdt/ofsg (ofsg/md/tm D1) DW) + %<- (sg-refl/sub DW DS) %. +%term _ + %pi (inv/ofsg/md/tm (ofsg/sub D1 DS1 _) D1' DS'') + %<- (inv/ofsg/md/tm D1 D1' DS) + %<- (sg-trans/sub DS DS1 DS'') %. +%term _ + %pi (inv/ofsg/md/tm (ofsg/sgm-ext D1 D2) DE DS) + %<- (md/tm-not-sg/sgm (ofsg/sgm-ext D1 D2) (%the uninhabited DU)) + %<- (uninhabited-oftp DU _ _ _ cn/unit DE) + %<- (uninhabited-sg-sub (sg/cn cn/unit) _ DU DS) %. +%term _ + %pi (inv/ofsg/md/tm (ofsg/kd-ext D1 _ _) DE DS) + %<- (md/tm-not-sg/kd D1 (%the uninhabited DU)) + %<- (uninhabited-oftp DU _ _ _ cn/unit DE) + %<- (uninhabited-sg-sub (sg/cn cn/unit) _ DU DS) %. +%worlds () (inv/ofsg/md/tm _ _ _) %. +%total D1 (inv/ofsg/md/tm D1 _ _) %. +%sort inv/ofsg/md/pj1 %. +%mode inv/ofsg/md/pj1 %in %out %out %. +%term _ + %pi (inv/ofsg/md/pj1 (ofsg/md/pj1 D1) D1 DS) + %<- (vdt/ofsg (ofsg/md/pj1 D1) DW) + %<- (sg-refl/sub DW DS) %. +%term _ + %pi (inv/ofsg/md/pj1 (ofsg/sub D1 (%the (sg-sub S2 S3) DS1) DP) (ofsg/sub D1' DR DP) DSS) + %<- (inv/ofsg/md/pj1 D1 D1' DS1') + %<- (sg-trans/sub (%the (sg-sub S1 S2) DS1') DS1 (%the (sg-sub S1 S3) DSS)) + %<- (vdt/ofsg D1' DW) + %<- (sg-refl/sub DW DR) %. +%term _ + %pi (inv/ofsg/md/pj1 (ofsg/sgm-ext D1 D2) (ofsg/sgm-ext (ofsg/sgm-ext D1 D2) (ofsg/md/pj2 D1'' DM)) DS) + %<- (subder/md/pj1 D1 D1' pty-sub/pp) + %<- (subder/md/pj1 D1' D1'' pty-sub/pp) + %<- (vdt/ofsg/p D1'' _ _ DM _) + %<- (vdt/ofsg (ofsg/sgm-ext D1 D2) DW) + %<- (sg-refl/sub DW DS) %. +%term _ + %pi (inv/ofsg/md/pj1 (ofsg/kd-ext D1 (fst-md/pj1 DM) DC) (ofsg/sgm-ext (ofsg/kd-ext D1 (fst-md/pj1 DM) DC) (ofsg/md/pj2 D1' DM)) DS) + %<- (subder/md/pj1 D1 D1' pty-sub/pp) + %<- (vdt/ofsg (ofsg/kd-ext D1 (fst-md/pj1 DM) DC) DW) + %<- (sg-refl/sub DW DS) %. +%worlds (ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (inv/ofsg/md/pj1 _ _ _) %. +%total D1 (inv/ofsg/md/pj1 D1 _ _) %. +%sort inv/ofsg/md/pj2 %. +%mode inv/ofsg/md/pj2 %in %out %out %out %out %. +%term _ + %pi (inv/ofsg/md/pj2 (ofsg/md/pj2 D1 DF) D1 DF DS pty-sub/pp) + %<- (vdt/ofsg (ofsg/md/pj2 D1 DF) DW) + %<- (sg-refl/sub DW DS) %. +%term _ + %pi (inv/ofsg/md/pj2 (ofsg/sub D1 (%the (sg-sub S2 S3) DS1) DP) (ofsg/sub D1' DR pty-sub/pp) DM DSS DPt) + %<- (inv/ofsg/md/pj2 D1 D1' DM DS1' DPl) + %<- (sg-trans/sub (%the (sg-sub _ S2) DS1') DS1 (%the (sg-sub _ S3) DSS)) + %<- (vdt/ofsg D1' DW) + %<- (sg-refl/sub DW DR) + %<- (pty-sub-trans DPl DP DPt) %. +%term _ + %pi (inv/ofsg/md/pj2 (%the (ofsg _ _ _ _ _) (ofsg/sgm-ext D1 D2)) (ofsg/sgm-ext (ofsg/md/pj1 D1'') (ofsg/sgm-ext D1 D2)) DM DS pty-sub/pp) + %<- (subder/md/pj1 D1 D1' pty-sub/pp) + %<- (subder/md/pj2 D1' D1'' pty-sub/pp) + %<- (vdt/ofsg/p D1'' _ _ DM _) + %<- (vdt/ofsg (ofsg/sgm-ext D1 D2) DW2) + %<- (sg-refl/sub DW2 DS) %. +%term _ + %pi (inv/ofsg/md/pj2 (ofsg/kd-ext D1 (fst-md/pj2 DM) DC) (ofsg/sgm-ext (ofsg/md/pj1 D1') (ofsg/kd-ext D1 (fst-md/pj2 DM) DC)) DM DS pty-sub/pp) + %<- (subder/md/pj2 D1 D1' pty-sub/pp) + %<- (vdt/ofsg (ofsg/kd-ext D1 (fst-md/pj2 DM) DC) DW) + %<- (sg-refl/sub DW DS) %. +%worlds (ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (inv/ofsg/md/pj2 _ _ _ _ _) %. +%total D1 (inv/ofsg/md/pj2 D1 _ _ _ _) %. +%sort md/lam-not-sg/sgm %. +%mode md/lam-not-sg/sgm %in %out %. +%term _ + %pi (md/lam-not-sg/sgm (ofsg/sub D1 _ pty-sub/pp) DU) + %<- (md/lam-not-sg/sgm D1 DU) %. +%term _ + %pi (md/lam-not-sg/sgm (ofsg/sgm-ext D1 _) DU) + %<- (subder/md/pj1 D1 D1' pty-sub/pp) + %<- (md/lam-not-sg/sgm D1' DU) %. +%worlds (ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (md/lam-not-sg/sgm _ _) %. +%total D1 (md/lam-not-sg/sgm D1 _) %. +%sort md/lam-not-sg/kd %. +%mode md/lam-not-sg/kd %in %out %. +%term _ %pi (md/lam-not-sg/kd (ofsg/sub D1 _ pty-sub/pp) DU) %<- (md/lam-not-sg/kd D1 DU) %. +%term _ %pi (md/lam-not-sg/kd (ofsg/kd-ext D1 _ _) DU) %<- (md/lam-not-sg/kd D1 DU) %. +%worlds (ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (md/lam-not-sg/kd _ _) %. +%total D1 (md/lam-not-sg/kd D1 _) %. +%sort inv/ofsg/md/lam %. +%mode inv/ofsg/md/lam %in %out %out %out %out %. +%term _ + %pi (inv/ofsg/md/lam (ofsg/md/lam D1 DW DF) D1 DW DF Dsub) + %<- (vdt/ofsg (ofsg/md/lam D1 DW DF) DS) + %<- (sg-refl/sub DS Dsub) %. +%term _ + %pi (inv/ofsg/md/lam (ofsg/sub D1 DSS1 _) D1' DW' DF DSS') + %<- (inv/ofsg/md/lam D1 D1' DW' DF DSS2) + %<- (sg-trans/sub DSS2 DSS1 DSS') %. +%term _ + %pi (inv/ofsg/md/lam (ofsg/sgm-ext D1 D2) D2'' DW DF DSS) + %<- (md/lam-not-sg/sgm (ofsg/sgm-ext D1 D2) DU) + %<- (fst-sg-complete S1 (%the (fst-sg S1 K1) DF)) + %<- ({s md} {dm assm/md s S1} {a cn} {da ofkd a K1} {df fst-md s a} uninhabited-ofsg L T pty/p (M s a) (S2 a) DU (D2'' s dm a da df)) + %<- (uninhabited-sg-wf S1 DU DW) + %<- (uninhabited-sg-sub (sg/pi S1 S2) _ DU DSS) %. +%term _ + %pi (inv/ofsg/md/lam (ofsg/kd-ext D1 _ _) D2'' DW DF DSS) + %<- (md/lam-not-sg/kd D1 DU) + %<- (fst-sg-complete S1 (%the (fst-sg S1 K1) DF)) + %<- ({s md} {dm assm/md s S1} {a cn} {da ofkd a K1} {df fst-md s a} uninhabited-ofsg L T pty/p (M s a) (S2 a) DU (D2'' s dm a da df)) + %<- (uninhabited-sg-wf S1 DU DW) + %<- (uninhabited-sg-sub (sg/pi S1 S2) _ DU DSS) %. +%worlds () (inv/ofsg/md/lam _ _ _ _ _) %. +%total D1 (inv/ofsg/md/lam D1 _ _ _ _) %. +%sort ofsg/md/seal-pty/i %. +%mode ofsg/md/seal-pty/i %in %out %. +%term _ + %pi (ofsg/md/seal-pty/i (ofsg/sub D1 _ pty-sub/pp) DU) + %<- (ofsg/md/seal-pty/i D1 DU) %. +%term _ + %pi (ofsg/md/seal-pty/i (ofsg/sgm-ext D1 _) DU) + %<- (subder/md/pj1 D1 D1' pty-sub/pp) + %<- (ofsg/md/seal-pty/i D1' DU) %. +%worlds (ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (ofsg/md/seal-pty/i _ _) %. +%total D1 (ofsg/md/seal-pty/i D1 _) %. +%sort inv/ofsg/md/seal %. +%mode inv/ofsg/md/seal %in %out %. +%term _ + %pi (inv/ofsg/md/seal (ofsg/md/seal D1) (ofsg/sub D1 DSub pty-sub/pi)) + %<- (vdt/ofsg D1 DS) + %<- (sg-refl/sub DS DSub) %. +%term _ inv/ofsg/md/seal (ofsg/md/seal D1) D1 %. +%term _ + %pi (inv/ofsg/md/seal (ofsg/sub D1 DS DP) (ofsg/sub D1' DS DP)) + %<- (inv/ofsg/md/seal D1 D1') %. +%term _ + %pi (inv/ofsg/md/seal (ofsg/sub D1 DS (%the (pty-sub pty/p _) DP)) D1') + %<- (ofsg/md/seal-pty/i D1 DU) + %<- (uninhabited-ofsg _ _ _ _ _ DU D1') %. +%term _ + %pi (inv/ofsg/md/seal (ofsg/sgm-ext D1 D2) D1'') + %<- (ofsg/md/seal-pty/i (ofsg/sgm-ext D1 D2) DU) + %<- (uninhabited-ofsg _ _ _ _ _ DU D1'') %. +%worlds (ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (inv/ofsg/md/seal _ _) %. +%total D1 (inv/ofsg/md/seal D1 _) %. +%sort ofsg/md/let-pty/i %. +%mode ofsg/md/let-pty/i %in %out %. +%term _ + %pi (ofsg/md/let-pty/i (ofsg/sub D1 _ pty-sub/pp) DU) + %<- (ofsg/md/let-pty/i D1 DU) %. +%term _ + %pi (ofsg/md/let-pty/i (ofsg/sgm-ext D1 _) DU) + %<- (subder/md/pj1 D1 D1' pty-sub/pp) + %<- (ofsg/md/let-pty/i D1' DU) %. +%worlds (ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (ofsg/md/let-pty/i _ _) %. +%total D1 (ofsg/md/let-pty/i D1 _) %. +%sort inv/ofsg/md/let %. +%mode inv/ofsg/md/let %in %out %out %out %out %out %. +%term _ + %pi (inv/ofsg/md/let (ofsg/md/let D1 D2 DW DF) D1 D2 DW DF DSS) + %<- (sg-refl/sub DW DSS) %. +%term _ + %pi (inv/ofsg/md/let (ofsg/sub D1 DSS1 _) D1' D2' DW DF DSS') + %<- (inv/ofsg/md/let D1 D1' D2' DW DF DSS2) + %<- (sg-trans/sub DSS2 DSS1 DSS') %. +%term _ + %pi (inv/ofsg/md/let (%the (ofsg L T _ _ _) (ofsg/sgm-ext D1 D2)) D1'' D2'' DW fst-sg/unit DSS) + %<- (ofsg/md/let-pty/i (ofsg/sgm-ext D1 D2) DU) + %<- (uninhabited-ofsg _ _ pty/i M1 sg/unit DU D1'') + %<- ({s md} {dm assm/md s sg/unit} {a cn} {da ofkd a kd/unit} {df fst-md s a} uninhabited-ofsg L T pty/p (M2 s a) S' DU (D2'' s dm a da df)) + %<- (uninhabited-sg-wf S' DU DW) + %<- (uninhabited-sg-sub S' _ DU DSS) %. +%worlds () (inv/ofsg/md/let _ _ _ _ _ _) %. +%total D1 (inv/ofsg/md/let D1 _ _ _ _ _) %. +%sort ofsg/md/app-pty/i %. +%mode ofsg/md/app-pty/i %in %out %. +%term _ + %pi (ofsg/md/app-pty/i (ofsg/sub D1 _ pty-sub/pp) DU) + %<- (ofsg/md/app-pty/i D1 DU) %. +%term _ + %pi (ofsg/md/app-pty/i (ofsg/sgm-ext D1 _) DU) + %<- (subder/md/pj1 D1 D1' pty-sub/pp) + %<- (ofsg/md/app-pty/i D1' DU) %. +%worlds (ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (ofsg/md/app-pty/i _ _) %. +%total D1 (ofsg/md/app-pty/i D1 _) %. +%sort inv/ofsg/md/app %. +%mode inv/ofsg/md/app %in %out %out %out %out %. +%term _ + %pi (inv/ofsg/md/app (ofsg/md/app D1 D2 DM) D1 D2 DM DSS) + %<- (vdt/ofsg (ofsg/md/app D1 D2 DM) DS) + %<- (sg-refl/sub DS DSS) %. +%term _ + %pi (inv/ofsg/md/app (ofsg/sub (%the (ofsg _ _ _ _ S) D1) DSS1 _) D1' D2' DM DSS') + %<- (inv/ofsg/md/app D1 D1' D2' DM DSS2) + %<- (sg-trans/sub DSS2 DSS1 DSS') %. +%term _ + %pi (inv/ofsg/md/app (ofsg/sgm-ext D1 D2) D1'' D2'' DM DSS) + %<- (ofsg/md/app-pty/i (ofsg/sgm-ext D1 D2) DU) + %<- (uninhabited-ofsg _ _ pty/p M1 (sg/pi sg/unit ([c] sg/unit)) DU D1'') + %<- (uninhabited-ofsg _ _ pty/p M2 sg/unit DU D2'') + %<- (uninhabited-fst-md M2 cn/unit DU DM) + %<- (uninhabited-sg-sub sg/unit _ DU DSS) %. +%worlds () (inv/ofsg/md/app _ _ _ _ _) %. +%total D1 (inv/ofsg/md/app D1 _ _ _ _) %. +%sort md/pair-not-sg/kd %. +%mode md/pair-not-sg/kd %in %out %. +%term _ + %pi (md/pair-not-sg/kd (ofsg/sub D1 _ pty-sub/pp) DU) + %<- (md/pair-not-sg/kd D1 DU) %. +%term _ %pi (md/pair-not-sg/kd (ofsg/kd-ext D1 _ _) DU) %<- (md/pair-not-sg/kd D1 DU) %. +%worlds (ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (md/pair-not-sg/kd _ _) %. +%total D1 (md/pair-not-sg/kd D1 _) %. +%sort psi-md %. +%term psi-md/eps psi-md ([a] a) %. +%term psi-md/pj1 %pi (psi-md ([a] md/pj1 (P a))) %<- (psi-md P) %. +%term psi-md/pj2 %pi (psi-md ([a] md/pj2 (P a))) %<- (psi-md P) %. +%sort psi-md-subder %. +%mode psi-md-subder %in %in %out %out %. +%term _ %pi (psi-md-subder psi-md/eps D1 D1 P) %<- (pty-sub-refl _ P) %. +%term _ + %pi (psi-md-subder (psi-md/pj1 PC) D1 D1'' P'') + %<- (subder/md/pj1 D1 D1' P) + %<- (psi-md-subder PC D1' D1'' P') + %<- (pty-sub-trans P' P P'') %. +%term _ + %pi (psi-md-subder (psi-md/pj2 PC) D1 D1'' P'') + %<- (subder/md/pj2 D1 D1' P) + %<- (psi-md-subder PC D1' D1'' P') + %<- (pty-sub-trans P' P P'') %. +%worlds (ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (psi-md-subder _ _ _ _) %. +%reduces <= D2 D1 (psi-md-subder _ D1 D2 _) %. +%total (D1) (psi-md-subder D1 _ _ _) %. +%sort psi-pj1-fst-beta %. +%mode psi-pj1-fst-beta %in %in %in %out %out %out %. +%term _ + %pi (psi-pj1-fst-beta D1 D2 psi-md/eps psi-cn/eps (fst-md/pj1 (fst-md/pair DM1 DM2)) DM1) + %<- (vdt/ofsg/p D1 _ _ DM1 _) + %<- (vdt/ofsg/p D2 _ _ DM2 _) %. +%term _ + %pi (psi-pj1-fst-beta D1 D2 (psi-md/pj1 (%the (psi-md PMC) PM)) (psi-cn/pj1 (%the (psi-cn PCC) PC)) (fst-md/pj1 DM1) (fst-md/pj1 DM2)) + %<- (psi-pj1-fst-beta D1 D2 PM PC DM1 DM2) %. +%term _ + %pi (psi-pj1-fst-beta D1 D2 (psi-md/pj2 (%the (psi-md PMC) PM)) (psi-cn/pj2 (%the (psi-cn PCC) PC)) (fst-md/pj2 DM1) (fst-md/pj2 DM2)) + %<- (psi-pj1-fst-beta D1 D2 PM PC DM1 DM2) %. +%worlds (ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (psi-pj1-fst-beta _ _ _ _ _ _) %. +%total (D1) (psi-pj1-fst-beta _ _ D1 _ _ _) %. +%sort psi-pj2-fst-beta %. +%mode psi-pj2-fst-beta %in %in %in %out %out %out %. +%term _ + %pi (psi-pj2-fst-beta D1 D2 psi-md/eps psi-cn/eps (fst-md/pj2 (fst-md/pair DM1 DM2)) DM2) + %<- (vdt/ofsg/p D1 _ _ DM1 _) + %<- (vdt/ofsg/p D2 _ _ DM2 _) %. +%term _ + %pi (psi-pj2-fst-beta D1 D2 (psi-md/pj1 (%the (psi-md PMC) PM)) (psi-cn/pj1 (%the (psi-cn PCC) PC)) (fst-md/pj1 DM1) (fst-md/pj1 DM2)) + %<- (psi-pj2-fst-beta D1 D2 PM PC DM1 DM2) %. +%term _ + %pi (psi-pj2-fst-beta D1 D2 (psi-md/pj2 (%the (psi-md PMC) PM)) (psi-cn/pj2 (%the (psi-cn PCC) PC)) (fst-md/pj2 DM1) (fst-md/pj2 DM2)) + %<- (psi-pj2-fst-beta D1 D2 PM PC DM1 DM2) %. +%worlds (ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (psi-pj2-fst-beta _ _ _ _ _ _) %. +%total (D1) (psi-pj2-fst-beta _ _ D1 _ _ _) %. +%sort inv/ofsg/md/pair %. +%mode inv/ofsg/md/pair %in %out %out %out %. +%sort module-beta/pj1 %. +%mode module-beta/pj1 %in %in %out %. +%sort module-beta/pj2 %. +%mode module-beta/pj2 %in %in %out %. +%term _ + %pi (inv/ofsg/md/pair (ofsg/md/pair D1 D2) D1 D2 DS) + %<- (vdt/ofsg (ofsg/md/pair D1 D2) DW) + %<- (sg-refl/sub DW DS) %. +%term _ + %pi (inv/ofsg/md/pair (ofsg/sub DP DS P) (ofsg/sub D1 DS1 P) (ofsg/sub D2 DS2 P) DSS) + %<- (inv/ofsg/md/pair DP D1 D2 DS') + %<- (vdt/ofsg D1 DW1) + %<- (sg-refl/sub DW1 DS1) + %<- (vdt/ofsg D2 DW2) + %<- (sg-refl/sub DW2 DS2) + %<- (sg-trans/sub DS' DS DSS) %. +%term _ + %pi (inv/ofsg/md/pair (ofsg/sgm-ext D1 D2) DA DB DS) + %<- (vdt/ofsg (ofsg/sgm-ext D1 D2) DW) + %<- (sg-refl/sub DW DS) + %<- (module-beta/pj1 psi-md/eps D1 DA) + %<- (module-beta/pj2 psi-md/eps D2 DB) %. +%term _ + %pi (inv/ofsg/md/pair (ofsg/kd-ext D1 _ _) DL DR DS) + %<- (md/pair-not-sg/kd D1 DU) + %<- (uninhabited-ofsg _ _ _ _ _ DU DL) + %<- (uninhabited-ofsg _ _ _ _ _ DU DR) + %<- (uninhabited-sg-sub (sg/sgm sg/unit ([x] sg/unit)) _ DU DS) %. +%term _ + %pi (module-beta/pj1 psi-md/eps (ofsg/md/pj1 DP) (ofsg/sub D1 DS1 P)) + %<- (inv/ofsg/md/pair DP D1 _ (sg-sub/sg/sgm DS1 _ _ _ _)) + %<- (pty-sub-refl _ P) %. +%term _ + %pi (module-beta/pj1 (psi-md/pj1 (%the (psi-md P) PS)) (ofsg/md/pj1 DP) (ofsg/md/pj1 DP')) + %<- (module-beta/pj1 PS DP (%the (ofsg _ _ _ (P M1) _) DP')) %. +%term _ + %pi (module-beta/pj1 (psi-md/pj2 (%the (psi-md P) PS)) (ofsg/md/pj2 (%the (ofsg _ _ _ (P (md/pj1 (md/pair M1 _))) (sg/sgm _ S1)) DP) (%the (fst-md (P (md/pj1 (md/pair M1 _))) CC) DM1C)) (ofsg/sub (ofsg/md/pj2 D2 DM2) DSS pty-sub/pp)) + %<- (psi-md-subder PS DP DPP pty-sub/pp) + %<- (subder/md/pj1 DPP DPP' pty-sub/pp) + %<- (inv/ofsg/md/pair DPP' DL DR _) + %<- (psi-pj1-fst-beta DL DR (%the (psi-md P) PS) (%the (psi-cn PCC) PC) (%the (fst-md (P _) (PCC (cn/pj1 (cn/pair C1 _)))) DM1A) (%the (fst-md (P M1) (PCC C1)) DM2)) + %<- (module-beta/pj1 (%the (psi-md P) PS) DP (%the (ofsg _ _ _ (P M1) _) D2)) + %<- (vdt/ofsg/p DP (sg-wf/sg/sgm _ DWF DFS1G) (fst-sg/sgm DFS1H DFS2H) (%the (fst-md _ CB) DM1B) (%the (ofkd CB _) DCB)) + %<- (fst-sg-seq DFS1H DFS1G DQKHG) + %<- (fst-md-seq DM1B DM1A (%the (seq/cn CB (PCC (cn/pj1 (cn/pair C1 _)))) DQMBA)) + %<- (fst-md-seq (%the (fst-md _ (PCC (cn/pj1 (cn/pair C1 _)))) DM1A) DM1C (%the (seq/cn (PCC (cn/pj1 (cn/pair C1 _))) CC) DQMAC)) + %<- (ofkd/seq-c DQMBA DCB (%the (ofkd (PCC (cn/pj1 (cn/pair C1 _))) _) DCA)) + %<- (cn-beta-pj1 PC DCA (%the (cn-deq (PCC (cn/pj1 (cn/pair C1 _))) (PCC C1) _) DCQA)) + %<- (cn-deq/seq-l DQMAC (%the (cn-deq (PCC (cn/pj1 (cn/pair C1 _))) (PCC C1) _) DCQA) (%the (cn-deq CC _ _) DCQC)) + %<- (cn-deq/seq-k DQKHG (cn-deq/cn/pj1 DCQC) (%the (cn-deq (cn/pj1 CC) (cn/pj1 (PCC C1)) _) DCQCG)) + %<- (vdt/cn-deq DCQCG DC1 _ _) + %<- (funct/sg-wf DWF DCQCG DC1 DSQ) + %<- (fst-sg-complete _ DFF1) + %<- (fst-sg-complete _ DFF2) + %<- (sg-anti DSQ DFF1 DFF2 _ (%the (sg-sub (S1 (cn/pj1 (PCC C1))) (S1 (cn/pj1 CC))) DSS) _ _) %. +%term _ + %pi (module-beta/pj1 (%the (psi-md P) PS) (ofsg/sub D1 DS DPS) (ofsg/sub D1' DS DPS)) + %<- (module-beta/pj1 PS D1 D1') %. +%term _ + %pi (module-beta/pj1 (%the (psi-md P) PS) (ofsg/sgm-ext D1 D2) (ofsg/sgm-ext DA DB)) + %<- (module-beta/pj1 (psi-md/pj1 PS) D1 DA) + %<- (module-beta/pj1 (psi-md/pj2 PS) D2 DB) %. +%term _ + %pi (module-beta/pj1 (%the (psi-md P) PS) (%the (ofsg _ _ _ (P (md/pj1 (md/pair M1 _))) _) (ofsg/kd-ext D1 DM DC)) (ofsg/kd-ext D1' DM2 DC2)) + %<- (module-beta/pj1 PS D1 (%the (ofsg _ _ _ (P M1) _) D1')) + %<- (psi-md-subder PS D1 D1p pty-sub/pp) + %<- (subder/md/pj1 D1p D1p' pty-sub/pp) + %<- (inv/ofsg/md/pair D1p' DL DR _) + %<- (psi-pj1-fst-beta DL DR PS (%the (psi-cn PCC) PC) DM1 (%the (fst-md (P M1) (PCC C1)) DM2)) + %<- (fst-md-seq DM DM1 DQM) + %<- (ofkd/seq-c DQM DC DC1) + %<- (cn-beta-pj1 PC DC1 DCQ) + %<- (vdt/cn-deq DCQ _ DC2 _) %. +%term _ + %pi (module-beta/pj2 psi-md/eps (ofsg/md/pj2 D1 (fst-md/pair DMa' DMb')) (ofsg/sub DB DS' pty-sub/pp)) + %<- (inv/ofsg/md/pair D1 (%the (ofsg _ _ _ MA SA) DA) (%the (ofsg _ _ _ MB SB) DB) (sg-sub/sg/sgm DS1 (%the ({a} {da} sg-sub _ (S1 a)) DS2) _ FSa' _)) + %<- (vdt/ofsg/p DA _ FSa DMa DCa) + %<- (vdt/ofsg/p DB _ _ (%the (fst-md MB _) DMb) DCb) + %<- (fst-sg-seq FSa FSa' DQfa) + %<- (fst-md-seq DMa DMa' DQa) + %<- (fst-md-seq DMb DMb' DQb) + %<- (seq/cn/pair DQa DQb DQab) + %<- (seq/cn/pj1 DQab DQabp) + %<- (ofkd/seq-k DQfa (ofkd/cn/pj1 (ofkd/cn/pair DCa DCb)) DCP') + %<- (seq/cn/sg-sub-o ([a] S1 a) DQabp (DS2 _ DCP') DS') %. +%term _ + %pi (module-beta/pj2 (psi-md/pj1 (%the (psi-md P) PS)) (ofsg/md/pj1 DP) (ofsg/md/pj1 DP')) + %<- (module-beta/pj2 PS DP (%the (ofsg _ _ _ (P M1) _) DP')) %. +%term _ + %pi (module-beta/pj2 (psi-md/pj2 (%the (psi-md P) PS)) (ofsg/md/pj2 (%the (ofsg _ _ _ (P (md/pj2 (md/pair _ M2))) (sg/sgm _ S1)) DP) (%the (fst-md (P (md/pj2 (md/pair M1 M2))) CC) DM1C)) (ofsg/sub (ofsg/md/pj2 D2 DM2) DSS pty-sub/pp)) + %<- (psi-md-subder PS DP DPP pty-sub/pp) + %<- (subder/md/pj2 DPP DPP' pty-sub/pp) + %<- (inv/ofsg/md/pair DPP' DL DR _) + %<- (psi-pj2-fst-beta DL DR (%the (psi-md P) PS) (%the (psi-cn PCC) PC) (%the (fst-md (P (md/pj2 (md/pair M1 M2))) (PCC (cn/pj2 (cn/pair C1 C2)))) DM1A) (%the (fst-md (P M2) (PCC C2)) DM2)) + %<- (module-beta/pj2 (%the (psi-md P) PS) DP (%the (ofsg _ _ _ (P M2) _) D2)) + %<- (vdt/ofsg/p DP (sg-wf/sg/sgm _ DWF DFS1G) (fst-sg/sgm DFS1H DFS2H) (%the (fst-md _ CB) DM1B) (%the (ofkd CB _) DCB)) + %<- (fst-sg-seq DFS1H DFS1G DQKHG) + %<- (fst-md-seq DM1B DM1A (%the (seq/cn CB (PCC (cn/pj2 (cn/pair C1 C2)))) DQMBA)) + %<- (fst-md-seq (%the (fst-md _ (PCC (cn/pj2 (cn/pair C1 C2)))) DM1A) DM1C (%the (seq/cn (PCC (cn/pj2 (cn/pair C1 C2))) CC) DQMAC)) + %<- (ofkd/seq-c DQMBA DCB (%the (ofkd (PCC (cn/pj2 (cn/pair C1 C2))) _) DCA)) + %<- (cn-beta-pj2 PC DCA (%the (cn-deq (PCC (cn/pj2 (cn/pair C1 C2))) (PCC C2) _) DCQA)) + %<- (cn-deq/seq-l DQMAC (%the (cn-deq (PCC (cn/pj2 (cn/pair C1 C2))) (PCC C2) _) DCQA) (%the (cn-deq CC _ _) DCQC)) + %<- (cn-deq/seq-k DQKHG (cn-deq/cn/pj1 DCQC) (%the (cn-deq (cn/pj1 CC) (cn/pj1 (PCC C2)) _) DCQCG)) + %<- (vdt/cn-deq DCQCG DC1 _ _) + %<- (funct/sg-wf DWF DCQCG DC1 DSQ) + %<- (fst-sg-complete _ DFF1) + %<- (fst-sg-complete _ DFF2) + %<- (sg-anti DSQ DFF1 DFF2 _ (%the (sg-sub (S1 (cn/pj1 (PCC C2))) (S1 (cn/pj1 CC))) DSS) _ _) %. +%term _ + %pi (module-beta/pj2 (%the (psi-md P) PS) (ofsg/sub D1 DS DPS) (ofsg/sub D1' DS DPS)) + %<- (module-beta/pj2 PS D1 D1') %. +%term _ + %pi (module-beta/pj2 (%the (psi-md P) PS) (ofsg/sgm-ext D1 D2) (ofsg/sgm-ext DA DB)) + %<- (module-beta/pj2 (psi-md/pj1 PS) D1 DA) + %<- (module-beta/pj2 (psi-md/pj2 PS) D2 DB) %. +%term _ + %pi (module-beta/pj2 (%the (psi-md P) PS) (%the (ofsg _ _ _ (P (md/pj2 (md/pair _ _))) _) (ofsg/kd-ext D1 DM DC)) (ofsg/kd-ext D1' DM2 DC2)) + %<- (module-beta/pj2 PS D1 (%the (ofsg _ _ _ (P M1) _) D1')) + %<- (psi-md-subder PS D1 D1p pty-sub/pp) + %<- (subder/md/pj2 D1p D1p' pty-sub/pp) + %<- (inv/ofsg/md/pair D1p' DL DR _) + %<- (psi-pj2-fst-beta DL DR PS (%the (psi-cn PCC) PC) DM1 (%the (fst-md (P M1) (PCC C1)) DM2)) + %<- (fst-md-seq DM DM1 DQM) + %<- (ofkd/seq-c DQM DC DC1) + %<- (cn-beta-pj2 PC DC1 DCQ) + %<- (vdt/cn-deq DCQ _ DC2 _) %. +%worlds () (inv/ofsg/md/pair _ _ _ _) (module-beta/pj1 _ _ _) (module-beta/pj2 _ _ _) %. +%total (D1 D2 D3) (inv/ofsg/md/pair D1 _ _ _) (module-beta/pj1 _ D2 _) (module-beta/pj2 _ D3 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/inversions-terms.lf b/new-tests/stelf-output/tslf/inversions-terms.lf new file mode 100644 index 0000000..8056477 --- /dev/null +++ b/new-tests/stelf-output/tslf/inversions-terms.lf @@ -0,0 +1,208 @@ +%%%% inversion lemmas for typing of terms +%sort inv/oftp/tm/pj1 %. +%mode inv/oftp/tm/pj1 %in %out %. +%term _ inv/oftp/tm/pj1 (oftp/tm/pj1 D) D %. +%term _ + %pi (inv/oftp/tm/pj1 (oftp/deq DE DC) (oftp/deq DE' (cn-deq/tp/cross DC (cn-deq/refl DC2)))) + %<- (inv/oftp/tm/pj1 DE DE') + %<- (vdt/oftp DE' DC') + %<- (inv/ofkd/tp/cross DC' kd-sub/kd/type DC1' DC2) %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/pj1 _ _) %. +%total D (inv/oftp/tm/pj1 D _) %. +%sort inv/oftp/tm/pj2 %. +%mode inv/oftp/tm/pj2 %in %out %. +%term _ inv/oftp/tm/pj2 (oftp/tm/pj2 D) D %. +%term _ + %pi (inv/oftp/tm/pj2 (oftp/deq DE DC) (oftp/deq DE' (cn-deq/tp/cross (cn-deq/refl DC1') DC))) + %<- (inv/oftp/tm/pj2 DE DE') + %<- (vdt/oftp DE' DC') + %<- (inv/ofkd/tp/cross DC' kd-sub/kd/type DC1' DC2) %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/pj2 _ _) %. +%total D (inv/oftp/tm/pj2 D _) %. +%sort inv/oftp/tm/pair %. +%mode inv/oftp/tm/pair %in %out %out %out %. +%term _ + %pi (inv/oftp/tm/pair (oftp/tm/pair D1 D2) (cn-deq/refl DC) D1 D2) + %<- (vdt/oftp (oftp/tm/pair D1 D2) DC) %. +%term _ + %pi (inv/oftp/tm/pair (oftp/deq DE DC) (cn-deq/trans DC' DC) DE1 DE2) + %<- (inv/oftp/tm/pair DE DC' DE1 DE2) %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/pair _ _ _ _) %. +%total D (inv/oftp/tm/pair D _ _ _) %. +%sort inv/oftp/tm/tmapp %. +%mode inv/oftp/tm/tmapp %in %out %out %out %. +%term _ + %pi (inv/oftp/tm/tmapp (oftp/tm/tmapp D1 D2) (cn-deq/refl DC') D1 D2) + %<- (vdt/oftp D1 DC) + %<- (inv/ofkd/tp/arrow DC kd-sub/kd/type _ DC') %. +%term _ + %pi (inv/oftp/tm/tmapp (oftp/deq DE DC) (cn-deq/trans DC' DC) DE1 DE2) + %<- (inv/oftp/tm/tmapp DE DC' DE1 DE2) %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/tmapp _ _ _ _) %. +%total D (inv/oftp/tm/tmapp D _ _ _) %. +%sort inv/oftp/tm/cnapp %. +%mode inv/oftp/tm/cnapp %in %out %out %out %. +%term _ + %pi (inv/oftp/tm/cnapp (oftp/tm/cnapp D1 D2) (cn-deq/refl (DC' C2 D2)) D1 D2) + %<- (vdt/oftp D1 DC) + %<- (inv/ofkd/tp/forall DC kd-sub/kd/type _ DC') %. +%term _ + %pi (inv/oftp/tm/cnapp (oftp/deq DE DC) (cn-deq/trans DC' DC) DE1 DE2) + %<- (inv/oftp/tm/cnapp DE DC' DE1 DE2) %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/cnapp _ _ _ _) %. +%total D (inv/oftp/tm/cnapp D _ _ _) %. +%sort inv/oftp/tm/fun %. +%mode inv/oftp/tm/fun %in %out %out %out %. +%term _ inv/oftp/tm/fun (oftp/tm/fun D1 DC) (cn-deq/refl DC) D1 DC %. +%term _ + %pi (inv/oftp/tm/fun (oftp/deq DE DQ) (cn-deq/trans DQ' DQ) D1 DC) + %<- (inv/oftp/tm/fun DE DQ' D1 DC) %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/fun _ _ _ _) %. +%total D (inv/oftp/tm/fun D _ _ _) %. +%sort inv/oftp/tm/cnabs %. +%mode inv/oftp/tm/cnabs %in %out %out %out %. +%term _ + %pi (inv/oftp/tm/cnabs (oftp/tm/cnabs D1 DW) (cn-deq/refl DC) D1 DW) + %<- (vdt/oftp (oftp/tm/cnabs D1 DW) DC) %. +%term _ + %pi (inv/oftp/tm/cnabs (oftp/deq DE DQ) (cn-deq/trans DQ' DQ) D1 DW) + %<- (inv/oftp/tm/cnabs DE DQ' D1 DW) %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/cnabs _ _ _ _) %. +%total D (inv/oftp/tm/cnabs D _ _ _) %. +%sort inv/oftp/tm/set %. +%mode inv/oftp/tm/set %in %out %out %out %. +%term _ inv/oftp/tm/set (oftp/tm/set D1 D2) (cn-deq/refl ofkd/tp/unit) D1 D2 %. +%term _ + %pi (inv/oftp/tm/set (oftp/deq DE DC) (cn-deq/trans DC' DC) DE1 DE2) + %<- (inv/oftp/tm/set DE DC' DE1 DE2) %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/set _ _ _ _) %. +%total D (inv/oftp/tm/set D _ _ _) %. +%sort inv/oftp/tm/get %. +%mode inv/oftp/tm/get %in %out %. +%term _ inv/oftp/tm/get (oftp/tm/get D1) D1 %. +%term _ + %pi (inv/oftp/tm/get (oftp/deq DE DC) (oftp/deq D1 (cn-deq/tp/ref DC))) + %<- (inv/oftp/tm/get DE D1) %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/get _ _) %. +%total D (inv/oftp/tm/get D _) %. +%sort inv/oftp/tm/ref %. +%mode inv/oftp/tm/ref %in %out %out %. +%term _ + %pi (inv/oftp/tm/ref (oftp/tm/ref D1) (cn-deq/refl (ofkd/tp/ref DC)) D1) + %<- (vdt/oftp D1 DC) %. +%term _ + %pi (inv/oftp/tm/ref (oftp/deq DE DC) (cn-deq/trans DC' DC) D1) + %<- (inv/oftp/tm/ref DE DC' D1) %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/ref _ _ _) %. +%total D (inv/oftp/tm/ref D _ _) %. +%sort inv/oftp/tm/term %. +%mode inv/oftp/tm/term %in %out %. +%term _ inv/oftp/tm/term (oftp/tm/term D) D %. +%term _ + %pi (inv/oftp/tm/term (oftp/deq DE DC) (ofsg/sub DM (sg-sub/sg/cn DC) P)) + %<- (inv/oftp/tm/term DE DM) + %<- (pty-sub-refl _ P) %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/term _ _) %. +%total D (inv/oftp/tm/term D _) %. +%sort inv/oftp/tm/loc %. +%mode inv/oftp/tm/loc %in %out %out %out %. +%term _ inv/oftp/tm/loc (oftp/tm/loc LL DC) (cn-deq/refl (ofkd/tp/ref DC)) LL DC %. +%term _ + %pi (inv/oftp/tm/loc (oftp/deq D1 DQ) (cn-deq/trans DQ' DQ) LL DC) + %<- (inv/oftp/tm/loc D1 DQ' LL DC) %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/loc _ _ _ _) %. +%total D1 (inv/oftp/tm/loc D1 _ _ _) %. +%sort inv/oftp/tm/inl %. +%mode inv/oftp/tm/inl %in %out %out %out %. +%term _ + %pi (inv/oftp/tm/inl (oftp/tm/inl D1 DCR) D1 DCR (cn-deq/refl (ofkd/tp/sum DCL DCR))) + %<- (vdt/oftp D1 DCL) %. +%term _ + %pi (inv/oftp/tm/inl (oftp/deq DE DQ) D1 DC (cn-deq/trans DQ' DQ)) + %<- (inv/oftp/tm/inl DE D1 DC DQ') %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/inl _ _ _ _) %. +%total D1 (inv/oftp/tm/inl D1 _ _ _) %. +%sort inv/oftp/tm/inr %. +%mode inv/oftp/tm/inr %in %out %out %out %. +%term _ + %pi (inv/oftp/tm/inr (oftp/tm/inr D1 DCL) D1 DCL (cn-deq/refl (ofkd/tp/sum DCL DCR))) + %<- (vdt/oftp D1 DCR) %. +%term _ + %pi (inv/oftp/tm/inr (oftp/deq DE DQ) D1 DC (cn-deq/trans DQ' DQ)) + %<- (inv/oftp/tm/inr DE D1 DC DQ') %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/inr _ _ _ _) %. +%total D1 (inv/oftp/tm/inr D1 _ _ _) %. +%sort inv/oftp/tm/case %. +%mode inv/oftp/tm/case %in %out %out %out %. +%term _ inv/oftp/tm/case (oftp/tm/case D1 D2 D3) D1 D2 D3 %. +%term _ + %pi (inv/oftp/tm/case (oftp/deq DE DQ) D1 ([x] [dx] oftp/deq (D2 x dx) DQ) ([x] [dx] oftp/deq (D3 x dx) DQ)) + %<- (inv/oftp/tm/case DE D1 ([x] [dx] D2 x dx) ([x] [dx] D3 x dx)) %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/case _ _ _ _) %. +%total (D1) (inv/oftp/tm/case D1 _ _ _) %. +%sort inv/oftp/tm/try %. +%mode inv/oftp/tm/try %in %out %out %. +%term _ inv/oftp/tm/try (oftp/tm/try D1 D2) D1 D2 %. +%term _ + %pi (inv/oftp/tm/try (oftp/deq DE DQ) (oftp/deq D1 DQ) ([x] [dx] oftp/deq (D2 x dx) DQ)) + %<- (inv/oftp/tm/try DE D1 D2) %. +%worlds () (inv/oftp/tm/try _ _ _) %. +%total (D1) (inv/oftp/tm/try D1 _ _) %. +%sort inv/oftp/tm/new-tag %. +%mode inv/oftp/tm/new-tag %in %out %out %. +%term _ inv/oftp/tm/new-tag (oftp/tm/new-tag DC) (cn-deq/refl (ofkd/tp/tag DC)) DC %. +%term _ + %pi (inv/oftp/tm/new-tag (oftp/deq DE DC) (cn-deq/trans DC' DC) D1) + %<- (inv/oftp/tm/new-tag DE DC' D1) %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/new-tag _ _ _) %. +%total D (inv/oftp/tm/new-tag D _ _) %. +%sort inv/oftp/tm/tag %. +%mode inv/oftp/tm/tag %in %out %out %out %. +%term _ inv/oftp/tm/tag (oftp/tm/tag D1 D2) (cn-deq/refl ofkd/tp/tagged) D1 D2 %. +%term _ + %pi (inv/oftp/tm/tag (oftp/deq DE DC) (cn-deq/trans DC' DC) DE1 DE2) + %<- (inv/oftp/tm/tag DE DC' DE1 DE2) %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/tag _ _ _ _) %. +%total D (inv/oftp/tm/tag D _ _ _) %. +%sort inv/oftp/tm/raise %. +%mode inv/oftp/tm/raise %in %out %. +%term _ inv/oftp/tm/raise (oftp/tm/raise D1 _) D1 %. +%term _ %pi (inv/oftp/tm/raise (oftp/deq DE DC) D1) %<- (inv/oftp/tm/raise DE D1) %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/raise _ _) %. +%total D (inv/oftp/tm/raise D _) %. +%sort inv/oftp/tm/iftag %. +%mode inv/oftp/tm/iftag %in %out %out %out %out %. +%term _ inv/oftp/tm/iftag (oftp/tm/iftag D1 D2 D3 D4) D1 D2 D3 D4 %. +%term _ + %pi (inv/oftp/tm/iftag (oftp/deq DE DQ) D1 D2 ([x] [dx] oftp/deq (D3 x dx) DQ) (oftp/deq D4 DQ)) + %<- (inv/oftp/tm/iftag DE D1 D2 ([x] [dx] D3 x dx) D4) %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/iftag _ _ _ _ _) %. +%total (D1) (inv/oftp/tm/iftag D1 _ _ _ _) %. +%sort inv/oftp/tm/tagloc %. +%mode inv/oftp/tm/tagloc %in %out %out %out %. +%term _ inv/oftp/tm/tagloc (oftp/tm/tagloc LL DC) (cn-deq/refl (ofkd/tp/tag DC)) LL DC %. +%term _ + %pi (inv/oftp/tm/tagloc (oftp/deq D1 DQ) (cn-deq/trans DQ' DQ) LL DC) + %<- (inv/oftp/tm/tagloc D1 DQ' LL DC) %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/tagloc _ _ _ _) %. +%total D1 (inv/oftp/tm/tagloc D1 _ _ _) %. +%sort inv/oftp/tm/roll %. +%mode inv/oftp/tm/roll %in %out %out %out %. +%term _ + %pi (inv/oftp/tm/roll (oftp/tm/roll D1 DC) (cn-deq/refl DC') D1 DC) + %<- (vdt/oftp (oftp/tm/roll D1 DC) DC') %. +%term _ + %pi (inv/oftp/tm/roll (oftp/deq DE DC) (cn-deq/trans DC' DC) D1 DCC) + %<- (inv/oftp/tm/roll DE DC' D1 DCC) %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/roll _ _ _ _) %. +%total D (inv/oftp/tm/roll D _ _ _) %. +%sort inv/oftp/tm/unroll %. +%mode inv/oftp/tm/unroll %in %out %out %. +%term _ + %pi (inv/oftp/tm/unroll (oftp/tm/unroll D1) (cn-deq/refl DC') D1) + %<- (vdt/oftp (oftp/tm/unroll D1) DC') %. +%term _ + %pi (inv/oftp/tm/unroll (oftp/deq DE DC) (cn-deq/trans DC' DC) D1) + %<- (inv/oftp/tm/unroll DE DC' D1) %. +%worlds (ofkd+vdt-block) (inv/oftp/tm/unroll _ _ _) %. +%total D (inv/oftp/tm/unroll D _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/judgments-loc.lf b/new-tests/stelf-output/tslf/judgments-loc.lf new file mode 100644 index 0000000..358f35e --- /dev/null +++ b/new-tests/stelf-output/tslf/judgments-loc.lf @@ -0,0 +1,12 @@ +%%%% +%sort loc-less %. +%mode loc-less %star %star %. +%term loc-less/z loc-less loc/z (loc/s L) %. +%term loc-less/s %pi (loc-less (loc/s L) (loc/s L')) %<- (loc-less L L') %. +%sort loc-neq %. +%mode loc-neq %star %star %. +%term loc-neq/l %pi (loc-neq L L') %<- (loc-less L L') %. +%term loc-neq/r %pi (loc-neq L L') %<- (loc-less L' L) %. +%sort loc-seq-or-neq %. +%term loc-seq-or-neq/seq %pi (loc-seq-or-neq L1 L2) %<- (seq/loc L1 L2) %. +%term loc-seq-or-neq/neq %pi (loc-seq-or-neq L1 L2) %<- (loc-neq L1 L2) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/judgments-lt.lf b/new-tests/stelf-output/tslf/judgments-lt.lf new file mode 100644 index 0000000..472227e --- /dev/null +++ b/new-tests/stelf-output/tslf/judgments-lt.lf @@ -0,0 +1,22 @@ +%sort lt-look %. +%mode lt-look %star %star %star %. +%term lt-look/hit lt-look (lt/cons _ LC C) LC C %. +%term lt-look/miss %pi (lt-look (lt/cons L _ _) LC C) %<- (lt-look L LC C) %. +%worlds (cn-block) (lt-look _ _ _) %. +%sort lt-extend %. +%mode lt-extend %star %star %star %star %. +%term lt-extend/nil lt-extend lt/nil C (lt/cons lt/nil loc/z C) loc/z %. +%term lt-extend/cons lt-extend (lt/cons LT LC C') C (lt/cons (lt/cons LT LC C') (loc/s LC) C) (loc/s LC) %. +%sort lt-wf %. +%mode lt-wf %star %. +%term lt-wf/nil lt-wf lt/nil %. +%term lt-wf/cons + %pi (lt-wf (lt/cons L1 LC C)) + %<- (lt-extend L1 C (lt/cons L1 LC C) LC) + %<- (lt-wf L1) %. +%sort lt-sub %. +%term lt-sub/refl lt-sub L L %. +%term lt-sub/trans + %pi (lt-sub L (lt/cons L' LC C')) + %<- (lt-extend L' C' (lt/cons L' LC C') LC) + %<- (lt-sub L L') %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/loc-lemmas.lf b/new-tests/stelf-output/tslf/loc-lemmas.lf new file mode 100644 index 0000000..9743a6f --- /dev/null +++ b/new-tests/stelf-output/tslf/loc-lemmas.lf @@ -0,0 +1,54 @@ +%%%% lemmas about location judgments +%sort loc-less-pred %. +%mode loc-less-pred %in %out %. +%term _ loc-less-pred _ loc-less/z %. +%term _ %pi (loc-less-pred (loc-less/s LL) (loc-less/s LL')) %<- (loc-less-pred LL LL') %. +%worlds () (loc-less-pred _ _) %. +%total (D1) (loc-less-pred D1 _) %. +%sort loc-less-immsucc {L loc} %. +%mode loc-less-immsucc %in %out %. +%term _ loc-less-immsucc _ loc-less/z %. +%term _ %pi (loc-less-immsucc (loc/s L) (loc-less/s LL')) %<- (loc-less-immsucc L LL') %. +%worlds () (loc-less-immsucc _ _) %. +%total (D1) (loc-less-immsucc D1 _) %. +%sort loc-seq-or-neq-s %. +%mode loc-seq-or-neq-s %in %out %. +%term _ loc-seq-or-neq-s (loc-seq-or-neq/seq _) (loc-seq-or-neq/seq seq/loc/refl) %. +%term _ loc-seq-or-neq-s (loc-seq-or-neq/neq (loc-neq/l LS)) (loc-seq-or-neq/neq (loc-neq/l (loc-less/s LS))) %. +%term _ loc-seq-or-neq-s (loc-seq-or-neq/neq (loc-neq/r LS)) (loc-seq-or-neq/neq (loc-neq/r (loc-less/s LS))) %. +%worlds () (loc-seq-or-neq-s _ _) %. +%total {} (loc-seq-or-neq-s _ _) %. +%sort loc-seq-or-neq-total {L1 loc} {L2 loc} %. +%mode loc-seq-or-neq-total %in %in %out %. +%term _ loc-seq-or-neq-total _ _ (loc-seq-or-neq/seq seq/loc/refl) %. +%term _ loc-seq-or-neq-total loc/z (loc/s _) (loc-seq-or-neq/neq (loc-neq/l loc-less/z)) %. +%term _ loc-seq-or-neq-total (loc/s _) loc/z (loc-seq-or-neq/neq (loc-neq/r loc-less/z)) %. +%term _ + %pi (loc-seq-or-neq-total (loc/s LC1) (loc/s LC2) LCS') + %<- (loc-seq-or-neq-total LC1 LC2 LCS) + %<- (loc-seq-or-neq-s LCS LCS') %. +%worlds () (loc-seq-or-neq-total _ _ _) %. +%total (D1) (loc-seq-or-neq-total D1 _ _) %. +%sort loc-less-not-refl %. +%mode loc-less-not-refl %in %out %. +%term _ %pi (loc-less-not-refl (loc-less/s LG) DU) %<- (loc-less-not-refl LG DU) %. +%worlds () (loc-less-not-refl _ _) %. +%total (D1) (loc-less-not-refl D1 _) %. +%sort loc-less-trans %. +%mode loc-less-trans %in %in %out %. +%term _ loc-less-trans loc-less/z _ loc-less/z %. +%term _ + %pi (loc-less-trans (loc-less/s L1) (loc-less/s L2) (loc-less/s L3)) + %<- (loc-less-trans L1 L2 L3) %. +%worlds () (loc-less-trans _ _ _) %. +%total D1 (loc-less-trans D1 _ _) %. +%sort loc-less-bound {I loc} {J loc} %. +%mode loc-less-bound %in %in %out %out %. +%term _ loc-less-bound loc/z loc/z (%the (loc-less loc/z (loc/s loc/z)) loc-less/z) loc-less/z %. +%term _ %pi (loc-less-bound loc/z (loc/s _) loc-less/z DL) %<- (loc-less-immsucc _ DL) %. +%term _ %pi (loc-less-bound (loc/s _) loc/z DL loc-less/z) %<- (loc-less-immsucc _ DL) %. +%term _ + %pi (loc-less-bound (loc/s L1) (loc/s L2) (loc-less/s DL1) (loc-less/s DL2)) + %<- (loc-less-bound L1 L2 DL1 DL2) %. +%worlds () (loc-less-bound _ _ _ _) %. +%total {I} (loc-less-bound I _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/location-lemmas.lf b/new-tests/stelf-output/tslf/location-lemmas.lf new file mode 100644 index 0000000..45c1166 --- /dev/null +++ b/new-tests/stelf-output/tslf/location-lemmas.lf @@ -0,0 +1,73 @@ +%%%% lemmas about location judgments +%sort lt-look-seq {L lt} %. +%mode lt-look-seq %in %in %in %out %. +%term _ lt-look-seq _ (%the (lt-look (lt/cons L LC _) LC _) lt-look/hit) (%the (lt-look (lt/cons L LC _) LC _) lt-look/hit) seq/cn/refl %. +%term _ + %pi (lt-look-seq (lt/cons L _ _) (lt-look/miss L1) (lt-look/miss L2) DQ) + %<- (lt-look-seq L L1 L2 DQ) %. +%worlds () (lt-look-seq _ _ _ _) %. +% total {D1} (lt-look-seq D1 _ _ _). +%sort loc-less-pred %. +%mode loc-less-pred %in %out %. +%term _ loc-less-pred _ loc-less/z %. +%term _ %pi (loc-less-pred (loc-less/s LL) (loc-less/s LL')) %<- (loc-less-pred LL LL') %. +%worlds () (loc-less-pred _ _) %. +%total (D1) (loc-less-pred D1 _) %. +%sort loc-less-immsucc {L loc} %. +%mode loc-less-immsucc %in %out %. +%term _ loc-less-immsucc _ loc-less/z %. +%term _ %pi (loc-less-immsucc (loc/s L) (loc-less/s LL')) %<- (loc-less-immsucc L LL') %. +%worlds () (loc-less-immsucc _ _) %. +%total (D1) (loc-less-immsucc D1 _) %. +%sort loc-seq-or-neq-s %. +%mode loc-seq-or-neq-s %in %out %. +%term _ loc-seq-or-neq-s (loc-seq-or-neq/seq _) (loc-seq-or-neq/seq seq/loc/refl) %. +%term _ loc-seq-or-neq-s (loc-seq-or-neq/neq (loc-neq/l LS)) (loc-seq-or-neq/neq (loc-neq/l (loc-less/s LS))) %. +%term _ loc-seq-or-neq-s (loc-seq-or-neq/neq (loc-neq/r LS)) (loc-seq-or-neq/neq (loc-neq/r (loc-less/s LS))) %. +%worlds () (loc-seq-or-neq-s _ _) %. +%total {} (loc-seq-or-neq-s _ _) %. +%sort loc-seq-or-neq-total {L1 loc} {L2 loc} %. +%mode loc-seq-or-neq-total %in %in %out %. +%term _ loc-seq-or-neq-total _ _ (loc-seq-or-neq/seq seq/loc/refl) %. +%term _ loc-seq-or-neq-total loc/z (loc/s _) (loc-seq-or-neq/neq (loc-neq/l loc-less/z)) %. +%term _ loc-seq-or-neq-total (loc/s _) loc/z (loc-seq-or-neq/neq (loc-neq/r loc-less/z)) %. +%term _ + %pi (loc-seq-or-neq-total (loc/s LC1) (loc/s LC2) LCS') + %<- (loc-seq-or-neq-total LC1 LC2 LCS) + %<- (loc-seq-or-neq-s LCS LCS') %. +%worlds () (loc-seq-or-neq-total _ _ _) %. +%total (D1) (loc-seq-or-neq-total D1 _ _) %. +%sort loc-less-not-refl %. +%mode loc-less-not-refl %in %out %. +%term _ %pi (loc-less-not-refl (loc-less/s LG) DU) %<- (loc-less-not-refl LG DU) %. +%worlds () (loc-less-not-refl _ _) %. +%total (D1) (loc-less-not-refl D1 _) %. +%sort lt-wf-look-sane %. +%mode lt-wf-look-sane %in %in %in %out %. +%term _ + %pi (lt-wf-look-sane (lt-look/miss LL) (lt-wf/cons LCS _) LG DU) + %<- (loc-less-pred LG LG') + %<- (lt-wf-look-sane LL LCS LG' DU) %. +%term _ + %pi (lt-wf-look-sane lt-look/hit (lt-wf/cons _ _) DG DU) + %<- (loc-less-not-refl DG DU) %. +%worlds () (lt-wf-look-sane _ _ _ _) %. +%total D1 (lt-wf-look-sane _ D1 _ _) %. +%sort lt-wf-look-unique %. +%mode lt-wf-look-unique %in %in %in %out %. +%term _ lt-wf-look-unique lt-look/hit lt-look/hit _ seq/cn/refl %. +%term _ + %pi (lt-wf-look-unique (lt-look/miss LL1) (lt-look/miss LL2) (lt-wf/cons LW _) DQ) + %<- (lt-wf-look-unique LL1 LL2 LW DQ) %. +%term _ + %pi (lt-wf-look-unique lt-look/hit (lt-look/miss LL) (lt-wf/cons LW _) DQ) + %<- (loc-less-immsucc _ LG) + %<- (lt-wf-look-sane LL LW LG DU) + %<- (uninhabited-seq/cn _ _ DU DQ) %. +%term _ + %pi (lt-wf-look-unique (lt-look/miss LL) lt-look/hit (lt-wf/cons LW _) DQ) + %<- (loc-less-immsucc _ LG) + %<- (lt-wf-look-sane LL LW LG DU) + %<- (uninhabited-seq/cn _ _ DU DQ) %. +%worlds () (lt-wf-look-unique _ _ _ _) %. +%total D1 (lt-wf-look-unique _ _ D1 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/location.lf b/new-tests/stelf-output/tslf/location.lf new file mode 100644 index 0000000..9acf439 --- /dev/null +++ b/new-tests/stelf-output/tslf/location.lf @@ -0,0 +1,38 @@ +%%%% location typing for terms +%sort lt %. +%name lt %. +%term lt/nil lt %. +%term lt/cons %pi lt %-> loc %-> cn %-> lt %. +%sort lt-look %. +%mode lt-look %star %star %star %. +%term lt-look/hit lt-look (lt/cons _ LC C) LC C %. +%term lt-look/miss %pi (lt-look (lt/cons L _ _) LC C) %<- (lt-look L LC C) %. +%sort lt-extend %. +%mode lt-extend %star %star %star %star %. +%term lt-extend/nil lt-extend lt/nil C (lt/cons lt/nil loc/z C) loc/z %. +%term lt-extend/cons lt-extend (lt/cons LT LC C') C (lt/cons (lt/cons LT LC C') (loc/s LC) C) (loc/s LC) %. +%sort lt-wf %. +%mode lt-wf %star %. +%term lt-wf/nil lt-wf lt/nil %. +%term lt-wf/cons + %pi (lt-wf (lt/cons L1 LC C)) + %<- (lt-extend L1 C (lt/cons L1 LC C) LC) + %<- (lt-wf L1) %. +%sort lt-sub %. +%term lt-sub/refl lt-sub L L %. +%term lt-sub/trans + %pi (lt-sub L (lt/cons L' LC C')) + %<- (lt-extend L' C' (lt/cons L' LC C') LC) + %<- (lt-sub L L') %. +%sort loc-less %. +%mode loc-less %star %star %. +%term loc-less/z loc-less loc/z (loc/s L) %. +%term loc-less/s %pi (loc-less (loc/s L) (loc/s L')) %<- (loc-less L L') %. +%% wts loc-greater (loc/s L) L' -> loc-greater L L' +%sort loc-neq %. +%mode loc-neq %star %star %. +%term loc-neq/l %pi (loc-neq L L') %<- (loc-less L L') %. +%term loc-neq/r %pi (loc-neq L L') %<- (loc-less L' L) %. +%sort loc-seq-or-neq %. +%term loc-seq-or-neq/seq %pi (loc-seq-or-neq L1 L2) %<- (seq/loc L1 L2) %. +%term loc-seq-or-neq/neq %pi (loc-seq-or-neq L1 L2) %<- (loc-neq L1 L2) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/lookup-context.lf b/new-tests/stelf-output/tslf/lookup-context.lf new file mode 100644 index 0000000..e69de29 diff --git a/new-tests/stelf-output/tslf/lt-lemmas.lf b/new-tests/stelf-output/tslf/lt-lemmas.lf new file mode 100644 index 0000000..c57727d --- /dev/null +++ b/new-tests/stelf-output/tslf/lt-lemmas.lf @@ -0,0 +1,30 @@ +%%%% lemmas about location judgments +%sort lt-wf-look-sane %. +%mode lt-wf-look-sane %in %in %in %out %. +%term _ + %pi (lt-wf-look-sane (lt-look/miss LL) (lt-wf/cons LCS _) LG DU) + %<- (loc-less-pred LG LG') + %<- (lt-wf-look-sane LL LCS LG' DU) %. +%term _ + %pi (lt-wf-look-sane lt-look/hit (lt-wf/cons _ _) DG DU) + %<- (loc-less-not-refl DG DU) %. +%worlds () (lt-wf-look-sane _ _ _ _) %. +%total D1 (lt-wf-look-sane _ D1 _ _) %. +%sort lt-wf-look-unique %. +%mode lt-wf-look-unique %in %in %in %out %. +%term _ lt-wf-look-unique lt-look/hit lt-look/hit _ seq/cn/refl %. +%term _ + %pi (lt-wf-look-unique (lt-look/miss LL1) (lt-look/miss LL2) (lt-wf/cons LW _) DQ) + %<- (lt-wf-look-unique LL1 LL2 LW DQ) %. +%term _ + %pi (lt-wf-look-unique lt-look/hit (lt-look/miss LL) (lt-wf/cons LW _) DQ) + %<- (loc-less-immsucc _ LG) + %<- (lt-wf-look-sane LL LW LG DU) + %<- (uninhabited-seq/cn _ _ DU DQ) %. +%term _ + %pi (lt-wf-look-unique (lt-look/miss LL) lt-look/hit (lt-wf/cons LW _) DQ) + %<- (loc-less-immsucc _ LG) + %<- (lt-wf-look-sane LL LW LG DU) + %<- (uninhabited-seq/cn _ _ DU DQ) %. +%worlds () (lt-wf-look-unique _ _ _ _) %. +%total D1 (lt-wf-look-unique _ _ D1 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/metrics.lf b/new-tests/stelf-output/tslf/metrics.lf new file mode 100644 index 0000000..5cab724 --- /dev/null +++ b/new-tests/stelf-output/tslf/metrics.lf @@ -0,0 +1,4 @@ +%sort met %. +%term met/unit met %. +%term met/pair %pi met %-> met %-> met %. +%term met/sing %pi met %-> met %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/notstuck.lf b/new-tests/stelf-output/tslf/notstuck.lf new file mode 100644 index 0000000..f7b129f --- /dev/null +++ b/new-tests/stelf-output/tslf/notstuck.lf @@ -0,0 +1,11 @@ +%%%% definition of not stuck +%sort notstuck/tm %. +%mode notstuck/tm %star %star %star %. +%term notstuck/tm/val %pi (notstuck/tm E ST TT) %<- (val/tm E) %. +%term notstuck/tm/step %pi (notstuck/tm E ST TT) %<- (step/tm E ST TT E' ST' TT') %. +%term notstuck/tm/raises %pi (notstuck/tm E ST TT) %<- (raises/tm E _) %. +%sort notstuck/md %. +%mode notstuck/md %star %star %star %. +%term notstuck/md/val %pi (notstuck/md E ST TT) %<- (val/md E) %. +%term notstuck/md/step %pi (notstuck/md E ST TT) %<- (step/md E ST TT E' ST' TT') %. +%term notstuck/md/raises %pi (notstuck/md M ST TT) %<- (raises/md M _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/preservation-fst.lf b/new-tests/stelf-output/tslf/preservation-fst.lf new file mode 100644 index 0000000..c62fd92 --- /dev/null +++ b/new-tests/stelf-output/tslf/preservation-fst.lf @@ -0,0 +1,54 @@ +%sort preserve/fst-md %. +%mode preserve/fst-md %in %in %in %out %out %out %. +%term _ + %pi (preserve/fst-md DE fst-md/tm (step/md/tm S) fst-md/tm DFS (cn-deq/refl DC)) + %<- (vdt/ofsg/p DE _ DFS fst-md/tm DC) %. +%term _ + %pi (preserve/fst-md DE (fst-md/pair DM1 DM2) (step/md/pair-1 S) (fst-md/pair DM1' DM2) DFS (cn-deq/sub (cn-deq/cn/pair DCQ (cn-deq/refl DC2')) KS)) + %<- (vdt/ofsg/p DE _ DFS _ _) + %<- (inv/ofsg/md/pair DE D1 D2 DS) + %<- (preserve/fst-md D1 DM1 S DM1' DFS1 DCQ) + %<- (vdt/ofsg/p D2 _ DFS2 DM2' DC2) + %<- (fst-md-seq DM2' DM2 DQ2) + %<- (ofkd/seq-c DQ2 DC2 DC2') + %<- (vdt/sg-sub DS (fst-sg/sgm DFS1 ([a] DFS2)) DFS _ _ KS) %. +%term _ + %pi (preserve/fst-md DE (fst-md/pair DM1 DM2) (step/md/pair-2 _ S) (fst-md/pair DM1 DM2') DFS (cn-deq/sub (cn-deq/cn/pair (cn-deq/refl DC1') DCQ) KS)) + %<- (vdt/ofsg/p DE _ DFS _ _) + %<- (inv/ofsg/md/pair DE D1 D2 DS) + %<- (preserve/fst-md D2 DM2 S DM2' DFS2 DCQ) + %<- (vdt/ofsg/p D1 _ DFS1 DM1' DC1) + %<- (fst-md-seq DM1' DM1 DQ1) + %<- (ofkd/seq-c DQ1 DC1 DC1') + %<- (vdt/sg-sub DS (fst-sg/sgm DFS1 ([a] DFS2)) DFS _ _ KS) %. +%term _ + %pi (preserve/fst-md DE (fst-md/pj1 DM1) (step/md/pj1 S) (fst-md/pj1 DM1') DFS (cn-deq/sub (cn-deq/cn/pj1 DCQ) KS)) + %<- (vdt/ofsg/p DE _ DFS _ _) + %<- (inv/ofsg/md/pj1 DE D1 DS) + %<- (preserve/fst-md D1 DM1 S DM1' (fst-sg/sgm DFS1 _) DCQ) + %<- (vdt/sg-sub DS DFS1 DFS _ _ KS) %. +%term _ + %pi (preserve/fst-md DE (fst-md/pj2 DM1) (step/md/pj2 S) (fst-md/pj2 DM1') DFS (cn-deq/sub (cn-deq/cn/pj2 DCQ) KS)) + %<- (inv/ofsg/md/pj2 DE D1 DMp (%the (sg-sub (S1 (cn/pj1 Cp)) SG) DS) _) + %<- (fst-md-seq DMp DM1 (%the (seq/cn Cp C1) DMQ)) + %<- (seq/cn/pj1 DMQ DMQp) + %<- (preserve/fst-md D1 DM1 S DM1' (fst-sg/sgm _ (%the ({a} fst-sg (S1 a) (K a)) DFS2)) DCQ) + %<- (seq/cn/sg-sub-ol S1 DMQp (%the (sg-sub (S1 (cn/pj1 Cp)) SG) DS) (%the (sg-sub (S1 (cn/pj1 C1)) SG) DS')) + %<- (fst-sg-complete SG (%the (fst-sg SG _) DFS)) + %<- (vdt/sg-sub DS' (DFS2 (cn/pj1 C1)) DFS _ _ KS) %. +%term _ + %pi (preserve/fst-md DE (fst-md/pj1 (fst-md/pair DM1 DM2)) (step/md/pj1-beta _ _) DM1 DFS DCQ') + %<- (vdt/ofsg/p DE _ DFS (fst-md/pj1 (fst-md/pair DM1' DM2')) DC) + %<- (cn-beta-pj1 psi-cn/eps DC DCQ) + %<- (fst-md-seq DM1' DM1 DQ1) + %<- (fst-md-seq DM2' DM2 DQ2) + %<- (seq/cn/beta-pj1 DQ1 DQ2 DCQ DCQ') %. +%term _ + %pi (preserve/fst-md DE (fst-md/pj2 (fst-md/pair DM1 DM2)) (step/md/pj2-beta _ _) DM2 DFS DCQ') + %<- (vdt/ofsg/p DE _ DFS (fst-md/pj2 (fst-md/pair DM1' DM2')) DC) + %<- (cn-beta-pj2 psi-cn/eps DC DCQ) + %<- (fst-md-seq DM1' DM1 DQ1) + %<- (fst-md-seq DM2' DM2 DQ2) + %<- (seq/cn/beta-pj2 DQ2 DQ1 DCQ DCQ') %. +%worlds () (preserve/fst-md _ _ _ _ _ _) %. +%total (D3) (preserve/fst-md _ _ D3 _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/preservation-lemmas.lf b/new-tests/stelf-output/tslf/preservation-lemmas.lf new file mode 100644 index 0000000..15df242 --- /dev/null +++ b/new-tests/stelf-output/tslf/preservation-lemmas.lf @@ -0,0 +1,71 @@ +%%%% preservation lemmas for terms and modules +%sort st-wf-can-look {SL st-look ST LC E} {LL lt-look L LC C} %. +%mode st-wf-can-look %in %in %in %out %. +%term _ st-wf-can-look st-look/hit lt-look/hit (st-wf/cons _ DC _ _) DC %. +%term _ + %pi (st-wf-can-look (st-look/miss LS) (lt-look/miss LL) (st-wf/cons LCS _ _ _) DC) + %<- (st-wf-can-look LS LL LCS DC) %. +%term _ + %pi (st-wf-can-look st-look/hit (lt-look/miss LL) (st-wf/cons LCS _ _ _) DC) + %<- (loc-less-immsucc _ LG) + %<- (lt-look-wf LL LCS LG DU) + %<- (uninhabited-oftp (%the uninhabited DU) _ _ _ _ DC) %. +%term _ + %pi (st-wf-can-look (st-look/miss LS) lt-look/hit (st-wf/cons LCS _ _ _) DC) + %<- (loc-less-immsucc _ LG) + %<- (st-look-wf LS LCS LG DU) + %<- (uninhabited-oftp (%the uninhabited DU) _ _ _ _ DC) %. +%worlds () (st-wf-can-look _ _ _ _) %. +%total D1 (st-wf-can-look _ _ D1 _) %. +%sort lt-wkn/st-wf %. +%mode lt-wkn/st-wf %in %in %out %. +%term _ lt-wkn/st-wf st-wf/nil _ st-wf/nil %. +%term _ + %pi (lt-wkn/st-wf (st-wf/cons DV DE DG DSG) DX (st-wf/cons DV' DE' DG DSG)) + %<- (lt-wkn/st-wf DV DX DV') + %<- (lt-wkn/oftp DX DE DE') %. +%worlds () (lt-wkn/st-wf _ _ _) %. +%total D1 (lt-wkn/st-wf D1 _ _) %. +%sort tt-wkn/st-wf %. +%mode tt-wkn/st-wf %in %in %out %. +%term _ tt-wkn/st-wf st-wf/nil _ st-wf/nil %. +%term _ + %pi (tt-wkn/st-wf (st-wf/cons DV DE DG DSG) DX (st-wf/cons DV' DE' DG DSG)) + %<- (tt-wkn/st-wf DV DX DV') + %<- (tt-wkn/oftp DX DE DE') %. +%worlds () (tt-wkn/st-wf _ _ _) %. +%total D1 (tt-wkn/st-wf D1 _ _) %. +%sort preserve/st-alloc %. +%mode preserve/st-alloc %in %in %in %in %out %. +%term _ + %pi (preserve/st-alloc DE DA DCVR DX (st-wf/cons DCVR' DE' lt-extend/nil st-alloc/nil)) + %<- (lt-wkn/oftp DX DE DE') + %<- (lt-wkn/st-wf DCVR DX DCVR') %. +%term _ + %pi (preserve/st-alloc DE DA DCVR DX (st-wf/cons DCVR' DE' lt-extend/cons st-alloc/cons)) + %<- (lt-wkn/oftp DX DE DE') + %<- (lt-wkn/st-wf DCVR DX DCVR') %. +%worlds () (preserve/st-alloc _ _ _ _ _) %. +%total D1 (preserve/st-alloc _ D1 _ _ _) %. +%sort preserve/st-update %. +%mode preserve/st-update %in %in %in %in %out %. +%term _ preserve/st-update st-update/hit lt-look/hit DE (st-wf/cons LCS DE' DG DSG) (st-wf/cons LCS DE DG st-alloc/nil) %. +%term _ preserve/st-update st-update/hit lt-look/hit DE (st-wf/cons LCS DE' DG DSG) (st-wf/cons LCS DE DG st-alloc/cons) %. +%term _ + %pi (preserve/st-update st-update/hit (lt-look/miss LL) DE (st-wf/cons LCS DE' DG DSG) LCS') + %<- (loc-less-immsucc _ LG) + %<- (lt-look-wf LL LCS LG DU) + %<- (uninhabited-st-wf (%the uninhabited DU) _ _ _ _ LCS') %. +%term _ + %pi (preserve/st-update (st-update/miss LS) lt-look/hit DE (st-wf/cons LCS DE' DG DSG) LCS') + %<- (loc-less-immsucc _ LG) + %<- (st-update-wf LS LCS LG DU) + %<- (uninhabited-st-wf (%the uninhabited DU) _ _ _ _ LCS') %. +%term _ + %pi (preserve/st-update (st-update/miss st-update/hit) (lt-look/miss LL) DE (st-wf/cons LCS DE' DG DSG) (st-wf/cons LCS' DE' DG st-alloc/cons)) + %<- (preserve/st-update st-update/hit LL DE LCS LCS') %. +%term _ + %pi (preserve/st-update (st-update/miss (st-update/miss LS)) (lt-look/miss LL) DE (st-wf/cons LCS DE' DG DSG) (st-wf/cons LCS' DE' DG st-alloc/cons)) + %<- (preserve/st-update (st-update/miss LS) LL DE LCS LCS') %. +%worlds () (preserve/st-update _ _ _ _ _) %. +%total D1 (preserve/st-update _ _ _ D1 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/preservation-raises.lf b/new-tests/stelf-output/tslf/preservation-raises.lf new file mode 100644 index 0000000..429de29 --- /dev/null +++ b/new-tests/stelf-output/tslf/preservation-raises.lf @@ -0,0 +1,133 @@ +%sort preserve-raises/tm %. +%mode preserve-raises/tm %in %in %out %. +%sort preserve-raises/md %. +%mode preserve-raises/md %in %in %out %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/pair-1 S) D1') + %<- (inv/oftp/tm/pair DE DQ D1 D2) + %<- (preserve-raises/tm D1 S D1') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/pair-2 V S) D2') + %<- (inv/oftp/tm/pair DE DQ D1 D2) + %<- (preserve-raises/tm D2 S D2') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/pj1 S) D1') + %<- (inv/oftp/tm/pj1 DE D1) + %<- (preserve-raises/tm D1 S D1') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/pj2 S) D1') + %<- (inv/oftp/tm/pj2 DE D1) + %<- (preserve-raises/tm D1 S D1') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/tmapp-1 S) D1') + %<- (inv/oftp/tm/tmapp DE DQ D1 D2) + %<- (preserve-raises/tm D1 S D1') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/tmapp-2 V S) D2') + %<- (inv/oftp/tm/tmapp DE DQ D1 D2) + %<- (preserve-raises/tm D2 S D2') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/cnapp S) D1') + %<- (inv/oftp/tm/cnapp DE DQ D1 DC) + %<- (preserve-raises/tm D1 S D1') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/set-1 S) D1') + %<- (inv/oftp/tm/set DE DQ D1 D2) + %<- (preserve-raises/tm D1 S D1') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/set-2 V S) D2') + %<- (inv/oftp/tm/set DE DQ D1 D2) + %<- (preserve-raises/tm D2 S D2') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/get S) D1') + %<- (inv/oftp/tm/get DE D1) + %<- (preserve-raises/tm D1 S D1') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/ref S) D1') + %<- (inv/oftp/tm/ref DE DQ D1) + %<- (preserve-raises/tm D1 S D1') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/raise-1 S) D1) + %<- (inv/oftp/tm/raise DE D1) %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/raise-2 S) D1') + %<- (inv/oftp/tm/raise DE D1) + %<- (preserve-raises/tm D1 S D1') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/term S) D1') + %<- (inv/oftp/tm/term DE D1) + %<- (preserve-raises/md D1 S D1') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/inl S) D1') + %<- (inv/oftp/tm/inl DE D1 DC DQ) + %<- (preserve-raises/tm D1 S D1') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/inr S) D1') + %<- (inv/oftp/tm/inr DE D1 DC DQ) + %<- (preserve-raises/tm D1 S D1') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/case S) D1') + %<- (inv/oftp/tm/case DE D1 D2 D3) + %<- (preserve-raises/tm D1 S D1') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/tag-1 S) D1') + %<- (inv/oftp/tm/tag DE DQ D1 D2) + %<- (preserve-raises/tm D1 S D1') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/tag-2 V S) D2') + %<- (inv/oftp/tm/tag DE DQ D1 D2) + %<- (preserve-raises/tm D2 S D2') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/iftag-1 S) D1') + %<- (inv/oftp/tm/iftag DE D1 D2 D3 D4) + %<- (preserve-raises/tm D1 S D1') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/iftag-2 _ S) D1') + %<- (inv/oftp/tm/iftag DE D1 D2 D3 D4) + %<- (preserve-raises/tm D2 S D1') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/roll S) D1') + %<- (inv/oftp/tm/roll DE DQ D1 _) + %<- (preserve-raises/tm D1 S D1') %. +%term _ + %pi (preserve-raises/tm DE (raises/tm/unroll S) D1') + %<- (inv/oftp/tm/unroll DE DQ D1) + %<- (preserve-raises/tm D1 S D1') %. +%term _ + %pi (preserve-raises/md DE (raises/md/pair-1 S) D1') + %<- (inv/ofsg/md/pair DE D1 D2 DS) + %<- (preserve-raises/md D1 S D1') %. +%term _ + %pi (preserve-raises/md DE (raises/md/pair-2 _ S) D2') + %<- (inv/ofsg/md/pair DE D1 D2 DS) + %<- (preserve-raises/md D2 S D2') %. +%term _ + %pi (preserve-raises/md DE (raises/md/pj1 S) D1') + %<- (inv/ofsg/md/pj1 DE D1 DS) + %<- (preserve-raises/md D1 S D1') %. +%term _ + %pi (preserve-raises/md DE (raises/md/pj2 S) D1') + %<- (inv/ofsg/md/pj2 DE D1 _ _ DS) + %<- (preserve-raises/md D1 S D1') %. +%term _ + %pi (preserve-raises/md DE (raises/md/app-1 S) D1') + %<- (inv/ofsg/md/app DE D1 D2 DM DS) + %<- (preserve-raises/md D1 S D1') %. +%term _ + %pi (preserve-raises/md DE (raises/md/app-2 _ S) D2') + %<- (inv/ofsg/md/app DE D1 D2 DM DS) + %<- (preserve-raises/md D2 S D2') %. +%term _ + %pi (preserve-raises/md DE (raises/md/tm S) D1') + %<- (inv/ofsg/md/tm DE D1 DS) + %<- (preserve-raises/tm D1 S D1') %. +%term _ + %pi (preserve-raises/md DE (raises/md/seal S) D1') + %<- (inv/ofsg/md/seal DE D1) + %<- (preserve-raises/md D1 S D1') %. +%term _ + %pi (preserve-raises/md DE (raises/md/let S) D1') + %<- (inv/ofsg/md/let DE D1 D2 DW DF DS) + %<- (preserve-raises/md D1 S D1') %. +%worlds () (preserve-raises/tm _ _ _) (preserve-raises/md _ _ _) %. +%total (D4 D5) (preserve-raises/tm _ D4 _) (preserve-raises/md _ D5 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/preservation.lf b/new-tests/stelf-output/tslf/preservation.lf new file mode 100644 index 0000000..65fd336 --- /dev/null +++ b/new-tests/stelf-output/tslf/preservation.lf @@ -0,0 +1,324 @@ +%%%% preservation for terms and modules +%sort preserve/tm %. +%mode preserve/tm %in %in %in %out %out %out %out %. +%sort preserve/md %. +%mode preserve/md %in %in %in %out %out %out %out %. +%term _ + %pi (preserve/tm DE DL (step/tm/pair-1 S) DS TS (oftp/deq (oftp/tm/pair D1' D2') DQ) DL') + %<- (inv/oftp/tm/pair DE DQ D1 D2) + %<- (preserve/tm D1 DL S DS TS D1' DL') + %<- (lt-tt-wkn-sub/oftp DS TS D2 D2') %. +%term _ + %pi (preserve/tm DE DL (step/tm/pair-2 V S) DS TS (oftp/deq (oftp/tm/pair D1' D2') DQ) DL') + %<- (inv/oftp/tm/pair DE DQ D1 D2) + %<- (preserve/tm D2 DL S DS TS D2' DL') + %<- (lt-tt-wkn-sub/oftp DS TS D1 D1') %. +%term _ + %pi (preserve/tm DE DL (step/tm/pj1 S) DS TS (oftp/tm/pj1 D1') DL') + %<- (inv/oftp/tm/pj1 DE D1) + %<- (preserve/tm D1 DL S DS TS D1' DL') %. +%term _ + %pi (preserve/tm DE DL' (step/tm/pj1-beta _ _) lt-sub/refl lt-sub/refl (oftp/deq DL DQL) DL') + %<- (inv/oftp/tm/pj1 DE D1) + %<- (vdt/oftp D1 DC) + %<- (inv/oftp/tm/pair D1 DQ DL DR) + %<- (inv/cn-deq/tp/cross DQ DQL _) %. +%term _ + %pi (preserve/tm DE DL (step/tm/pj2 S) DS TS (oftp/tm/pj2 D1') DL') + %<- (inv/oftp/tm/pj2 DE D1) + %<- (preserve/tm D1 DL S DS TS D1' DL') %. +%term _ + %pi (preserve/tm DE DL (step/tm/pj2-beta _ _) lt-sub/refl lt-sub/refl (oftp/deq DR DQR) DL) + %<- (inv/oftp/tm/pj2 DE D1) + %<- (vdt/oftp D1 DC) + %<- (inv/oftp/tm/pair D1 DQ _ DR) + %<- (inv/cn-deq/tp/cross DQ _ DQR) %. +%term _ + %pi (preserve/tm DE DL (step/tm/tmapp-1 S) DS TS (oftp/deq (oftp/tm/tmapp D1' D2') DQ) DL') + %<- (inv/oftp/tm/tmapp DE DQ D1 D2) + %<- (preserve/tm D1 DL S DS TS D1' DL') + %<- (lt-tt-wkn-sub/oftp DS TS D2 D2') %. +%term _ + %pi (preserve/tm DE DL (step/tm/tmapp-2 V S) DS TS (oftp/deq (oftp/tm/tmapp D1' D2') DQ) DL') + %<- (inv/oftp/tm/tmapp DE DQ D1 D2) + %<- (preserve/tm D2 DL S DS TS D2' DL') + %<- (lt-tt-wkn-sub/oftp DS TS D1 D1') %. +%term _ + %pi (preserve/tm DE DL (step/tm/tmapp-beta _ V) lt-sub/refl lt-sub/refl (oftp/deq D1''' (cn-deq/trans DQB2 DQ)) DL) + %<- (inv/oftp/tm/tmapp DE (%the (cn-deq C2 C kd/type) DQ) (%the (oftp L _ _ (tp/arrow C1 C2)) D1) D2) + %<- (inv/oftp/tm/fun D1 (%the (cn-deq (tp/arrow CA CB) (tp/arrow C1 C2) kd/type) DQ') D1' DC) + %<- (inv/cn-deq/tp/arrow DQ' DQA1 DQB2) + %<- ({x} {dx} subst/tm-oftp ([f] [df] D1' x dx f df) (oftp/deq D1 (cn-deq/sym DQ')) (D1'' x dx)) + %<- (subst/tm-oftp D1'' (oftp/deq D2 (cn-deq/sym DQA1)) D1''') %. +%term _ + %pi (preserve/tm DE DL (step/tm/cnapp S) DS TS (oftp/deq (oftp/tm/cnapp D1' DC) DQ) DL') + %<- (inv/oftp/tm/cnapp DE DQ D1 DC) + %<- (preserve/tm D1 DL S DS TS D1' DL') %. +%term _ + %pi (preserve/tm DE DL (step/tm/cnapp-beta V) lt-sub/refl lt-sub/refl (oftp/deq (D1' C' (ofkd/sub DC DKS12)) (cn-deq/trans (DQo C' (ofkd/sub DC DKS12)) DQ)) DL) + %<- (inv/oftp/tm/cnapp DE (%the (cn-deq (C1 C') C kd/type) DQ) D1 DC) + %<- (inv/oftp/tm/cnabs D1 (%the (cn-deq (tp/forall K2 C2) (tp/forall K1 C1) kd/type) DQ') D1' _) + %<- (inv/cn-deq/tp/forall DQ' (%the (kd-deq K2 K1) DQk) DQo) + %<- (vdt/kd-deq DQk DK2 DK1) + %<- (kd-anti DQk DK2 DK1 _ DKS12) %. +%term _ + %pi (preserve/tm DE DL (step/tm/set-1 S) DS TS (oftp/deq (oftp/tm/set D1' D2') DQ) DL') + %<- (inv/oftp/tm/set DE DQ D1 D2) + %<- (preserve/tm D1 DL S DS TS D1' DL') + %<- (lt-tt-wkn-sub/oftp DS TS D2 D2') %. +%term _ + %pi (preserve/tm DE DL (step/tm/set-2 V S) DS TS (oftp/deq (oftp/tm/set D1' D2') DQ) DL') + %<- (inv/oftp/tm/set DE DQ D1 D2) + %<- (preserve/tm D2 DL S DS TS D2' DL') + %<- (lt-tt-wkn-sub/oftp DS TS D1 D1') %. +%term _ + %pi (preserve/tm DE DL (step/tm/get S) DS TS (oftp/tm/get D1') DL') + %<- (inv/oftp/tm/get DE D1) + %<- (preserve/tm D1 DL S DS TS D1' DL') %. +%term _ + %pi (preserve/tm DE DL (step/tm/ref S) DS TS (oftp/deq (oftp/tm/ref D1') DQ) DL') + %<- (inv/oftp/tm/ref DE DQ D1) + %<- (preserve/tm D1 DL S DS TS D1' DL') %. +%term _ + %pi (preserve/tm DE DL (step/tm/raise S) DS TS (oftp/tm/raise D1' DC) DL') + %<- (vdt/oftp DE DC) + %<- (inv/oftp/tm/raise DE D1) + %<- (preserve/tm D1 DL S DS TS D1' DL') %. +%term _ + %pi (preserve/tm DE DL (step/tm/term S) DS TS (oftp/tm/term D1') DL') + %<- (inv/oftp/tm/term DE D1) + %<- (preserve/md D1 DL S DS TS D1' DL') %. +%term _ + %pi (preserve/tm DE DL (step/tm/term-beta _ V) lt-sub/refl lt-sub/refl (oftp/deq D1' DQ) DL) + %<- (inv/oftp/tm/term DE D1) + %<- (inv/ofsg/md/tm D1 D1' (sg-sub/sg/cn DQ)) %. +%term _ + %pi (preserve/tm DE DL (step/tm/inl S) DS TS (oftp/deq (oftp/tm/inl D1' DC) DQ) DL') + %<- (inv/oftp/tm/inl DE D1 DC DQ) + %<- (preserve/tm D1 DL S DS TS D1' DL') %. +%term _ + %pi (preserve/tm DE DL (step/tm/inr S) DS TS (oftp/deq (oftp/tm/inr D1' DC) DQ) DL') + %<- (inv/oftp/tm/inr DE D1 DC DQ) + %<- (preserve/tm D1 DL S DS TS D1' DL') %. +%term _ + %pi (preserve/tm DE DL (step/tm/case S) DS TS (oftp/tm/case D1' D2' D3') DL') + %<- (inv/oftp/tm/case DE D1 D2 D3) + %<- (preserve/tm D1 DL S DS TS D1' DL') + %<- ({x} {dx} lt-tt-wkn-sub/oftp DS TS (D2 x dx) (D2' x dx)) + %<- ({x} {dx} lt-tt-wkn-sub/oftp DS TS (D3 x dx) (D3' x dx)) %. +%term _ + %pi (preserve/tm DE DL (step/tm/case-beta-l V) lt-sub/refl lt-sub/refl D2' DL) + %<- (inv/oftp/tm/case DE D1 D2 D3) + %<- (inv/oftp/tm/inl D1 D1' _ DQ) + %<- (inv/cn-deq/tp/sum DQ DLe _) + %<- (subst/tm-oftp D2 (oftp/deq D1' DLe) D2') %. +%term _ + %pi (preserve/tm DE DL (step/tm/case-beta-r V) lt-sub/refl lt-sub/refl D3' DL) + %<- (inv/oftp/tm/case DE D1 D2 D3) + %<- (inv/oftp/tm/inr D1 D1' _ DQ) + %<- (inv/cn-deq/tp/sum DQ _ DR) + %<- (subst/tm-oftp D3 (oftp/deq D1' DR) D3') %. +%term _ + %pi (preserve/tm DE (oflt/c DCVR DWF) (step/tm/ref-beta V st-alloc/nil) (lt-sub/trans lt-sub/refl lt-extend/nil) lt-sub/refl (oftp/deq (oftp/tm/loc lt-look/hit DC) DQ) (oflt/c DCVR' DWF)) + %<- (inv/oftp/tm/ref DE DQ D1) + %<- (vdt/oftp D1 DC) + %<- (preserve/st-alloc D1 st-alloc/nil DCVR lt-extend/nil DCVR') %. +%term _ + %pi (preserve/tm DE (oflt/c (st-wf/cons DCVR DT DG DSG) DWF) (step/tm/ref-beta V st-alloc/cons) (lt-sub/trans lt-sub/refl lt-extend/cons) lt-sub/refl (oftp/deq (oftp/tm/loc lt-look/hit DC) DQ) (oflt/c DCVR' DWF)) + %<- (inv/oftp/tm/ref DE DQ D1) + %<- (vdt/oftp D1 DC) + %<- (preserve/st-alloc D1 st-alloc/cons (st-wf/cons DCVR DT DG DSG) lt-extend/cons DCVR') %. +%term _ + %pi (preserve/tm DE (oflt/c DL DWF) (step/tm/get-beta V SL) lt-sub/refl lt-sub/refl (oftp/deq DE' DQ) (oflt/c DL DWF)) + %<- (inv/oftp/tm/get DE DLoc) + %<- (inv/oftp/tm/loc DLoc DQr LL _) + %<- (inv/cn-deq/tp/ref DQr DQ) + %<- (st-wf-can-look SL LL DL DE') %. +%term _ + %pi (preserve/tm DE (oflt/c DCVR DWF) (step/tm/set-beta V1 V2 SU) lt-sub/refl lt-sub/refl (oftp/deq oftp/tm/unit DQ) (oflt/c DCVR' DWF)) + %<- (inv/oftp/tm/set DE (%the (cn-deq tp/unit Cu kd/type) DQ) DEL (%the (oftp _ _ _ C) DEA)) + %<- (inv/oftp/tm/loc (%the (oftp _ _ _ (tp/ref C)) DEL) DQR (%the (lt-look L LC C') LL) (%the (ofkd C' kd/type) DC)) + %<- (inv/cn-deq/tp/ref DQR DQQ) + %<- (preserve/st-update SU LL (oftp/deq DEA (cn-deq/sym DQQ)) DCVR DCVR') %. +%term _ + %pi (preserve/tm DE DL (step/tm/try S) LS TS (oftp/tm/try D1' D2') DL') + %<- (inv/oftp/tm/try DE D1 D2) + %<- (preserve/tm D1 DL S LS TS D1' DL') + %<- ({x} {dx} lt-tt-wkn-sub/oftp LS TS (D2 x dx) (D2' x dx)) %. +%term _ + %pi (preserve/tm DE DL (step/tm/try-beta _) lt-sub/refl lt-sub/refl D1 DL) + %<- (inv/oftp/tm/try DE D1 D2) %. +%term _ + %pi (preserve/tm DE DL (step/tm/try-handle R) lt-sub/refl lt-sub/refl D2' DL) + %<- (inv/oftp/tm/try DE D1 D2) + %<- (preserve-raises/tm D1 R D1') + %<- (subst/tm-oftp D2 D1' D2') %. +%term _ + %pi (preserve/tm DE (oflt/c DL DWF) (step/tm/new-tag-beta LX) lt-sub/refl (lt-sub/trans lt-sub/refl LX) (oftp/deq (oftp/tm/tagloc lt-look/hit DC) DQ) (oflt/c DL' (lt-wf/cons DWF LX))) + %<- (inv/oftp/tm/new-tag DE DQ DC) + %<- (tt-wkn/st-wf DL LX DL') %. +%term _ + %pi (preserve/tm DE DL (step/tm/tag-1 S) DS TS (oftp/deq (oftp/tm/tag D1' D2') DQ) DL') + %<- (inv/oftp/tm/tag DE DQ D1 D2) + %<- (preserve/tm D1 DL S DS TS D1' DL') + %<- (lt-tt-wkn-sub/oftp DS TS D2 D2') %. +%term _ + %pi (preserve/tm DE DL (step/tm/tag-2 V S) DS TS (oftp/deq (oftp/tm/tag D1' D2') DQ) DL') + %<- (inv/oftp/tm/tag DE DQ D1 D2) + %<- (preserve/tm D2 DL S DS TS D2' DL') + %<- (lt-tt-wkn-sub/oftp DS TS D1 D1') %. +%term _ + %pi (preserve/tm DE DL (step/tm/iftag-1 S) DS TS (oftp/tm/iftag D1' D2' D3' D4') DL') + %<- (inv/oftp/tm/iftag DE D1 D2 D3 D4) + %<- (preserve/tm D1 DL S DS TS D1' DL') + %<- (lt-tt-wkn-sub/oftp DS TS D2 D2') + %<- ({x} {dx} lt-tt-wkn-sub/oftp DS TS (D3 x dx) (D3' x dx)) + %<- (lt-tt-wkn-sub/oftp DS TS D4 D4') %. +%term _ + %pi (preserve/tm DE DL (step/tm/iftag-2 _ S) DS TS (oftp/tm/iftag D1' D2' D3' D4') DL') + %<- (inv/oftp/tm/iftag DE D1 D2 D3 D4) + %<- (preserve/tm D2 DL S DS TS D2' DL') + %<- (lt-tt-wkn-sub/oftp DS TS D1 D1') + %<- ({x} {dx} lt-tt-wkn-sub/oftp DS TS (D3 x dx) (D3' x dx)) + %<- (lt-tt-wkn-sub/oftp DS TS D4 D4') %. +%term _ + %pi (preserve/tm DE (oflt/c DL DWF) (step/tm/iftag-beta-1 _) lt-sub/refl lt-sub/refl D3' (oflt/c DL DWF)) + %<- (inv/oftp/tm/iftag DE D1 (%the (oftp _ _ _ (tp/tag C)) D2) D3 D4) + %<- (inv/oftp/tm/tag D1 DQ1 DA (%the (oftp _ _ _ CA) DB)) + %<- (inv/oftp/tm/tagloc DA (%the (cn-deq (tp/tag CA') (tp/tag CA) kd/type) DCQA) (%the (lt-look _ _ CA') LLA) _) + %<- (inv/oftp/tm/tagloc D2 (%the (cn-deq (tp/tag C2') (tp/tag C) kd/type) DCQ2) (%the (lt-look _ _ C2') LL2) _) + %<- (lt-wf-look-unique LLA LL2 DWF DQ) + %<- (seq/cn/tag DQ DQ') + %<- (cn-deq/seq-l DQ' DCQA DCQA') + %<- (inv/cn-deq/tp/tag (cn-deq/trans (cn-deq/sym DCQA') DCQ2) DQQ) + %<- (subst/tm-oftp D3 (oftp/deq DB DQQ) D3') %. +%term _ + %pi (preserve/tm DE DL (step/tm/iftag-beta-2 _ _) lt-sub/refl lt-sub/refl D4 DL) + %<- (inv/oftp/tm/iftag DE D1 D2 D3 D4) %. +%term _ + %pi (preserve/tm DE DL (step/tm/roll S) DS TS (oftp/deq (oftp/tm/roll D1' DC) DQ) DL') + %<- (inv/oftp/tm/roll DE DQ D1 DC) + %<- (preserve/tm D1 DL S DS TS D1' DL') %. +%term _ + %pi (preserve/tm DE DL (step/tm/unroll S) DS TS (oftp/deq (oftp/tm/unroll D1') DQ) DL') + %<- (inv/oftp/tm/unroll DE DQ D1) + %<- (preserve/tm D1 DL S DS TS D1' DL') %. +%term _ + %pi (preserve/tm (%the (oftp _ _ (tm/unroll (tm/roll (cn/mu K1 C1) E)) _) DE) DL (step/tm/unroll-beta V) lt-sub/refl lt-sub/refl (%the (oftp _ _ _ CC) (oftp/deq D1'' (cn-deq/trans DQQ DQ))) DL) + %<- (inv/oftp/tm/unroll DE (%the (cn-deq (C2 (cn/mu K2 C2)) CC kd/type) DQ) (%the (oftp _ _ _ (cn/mu K2 C2)) D1')) + %<- (inv/oftp/tm/roll D1' (%the (cn-deq (cn/mu K1 C1) (cn/mu K2 C2) kd/type) DQ') (%the (oftp _ _ _ (C1 (cn/mu K1 C1))) D1'') _) + %<- (inv/cn-deq/cn/mu DQ' DQo) + %<- (functf/cn-deq DQ' DQo (%the (cn-deq (C1 (cn/mu K1 C1)) (C2 (cn/mu K2 C2)) kd/type) DQQ)) %. +%term _ + %pi (preserve/md DE DL (step/md/pair-1 S) LS TS (ofsg/sub (ofsg/md/pair D1' D2') DS P) DL') + %<- (inv/ofsg/md/pair DE D1 D2 DS) + %<- (preserve/md D1 DL S LS TS D1' DL') + %<- (pty-sub-refl _ P) + %<- (lt-tt-wkn-sub/ofsg LS TS D2 D2') %. +%term _ + %pi (preserve/md DE DL (step/md/pair-2 _ S) LS TS (ofsg/sub (ofsg/md/pair D1' D2') DS P) DL') + %<- (inv/ofsg/md/pair DE D1 D2 DS) + %<- (preserve/md D2 DL S LS TS D2' DL') + %<- (pty-sub-refl _ P) + %<- (lt-tt-wkn-sub/ofsg LS TS D1 D1') %. +%term _ + %pi (preserve/md DE DL (step/md/pj1 S) LS TS (ofsg/sub (ofsg/md/pj1 D1') DS P) DL') + %<- (inv/ofsg/md/pj1 DE D1 DS) + %<- (preserve/md D1 DL S LS TS D1' DL') + %<- (pty-sub-refl _ P) %. +%term _ + %pi (preserve/md DE DL (step/md/pj1-beta _ _) lt-sub/refl lt-sub/refl DE' DL) + %<- (module-beta/pj1 psi-md/eps DE DE') %. +%term _ + %pi (preserve/md DE DL (step/md/app-1 S) LS TS (ofsg/sub (ofsg/md/app D1' D2' DM) DS pty-sub/ii) DL') + %<- (inv/ofsg/md/app DE D1 D2 DM DS) + %<- (preserve/md D1 DL S LS TS D1' DL') + %<- (lt-tt-wkn-sub/ofsg LS TS D2 D2') %. +%term _ + %pi (preserve/md DE DL _ lt-sub/refl DS D1' DL') + %<- (ofsg/md/app-pty/i DE DU) + %<- (uninhabited-ofsg L _ pty/p _ _ DU D1') + %<- (uninhabited-oflt _ L _ DU DL') + %<- (uninhabited-lt-sub _ _ DU DS) %. +%term _ + %pi (preserve/md DE DL (step/md/tm S) LS TS (ofsg/sub (ofsg/md/tm D1') DS pty-sub/pp) DL') + %<- (inv/ofsg/md/tm DE D1 DS) + %<- (preserve/tm D1 DL S LS TS D1' DL') %. +%term _ + %pi (preserve/md DE DL (step/md/tm S) LS TS (ofsg/sub (ofsg/md/tm D1') DS pty-sub/pi) DL') + %<- (inv/ofsg/md/tm DE D1 DS) + %<- (preserve/tm D1 DL S LS TS D1' DL') %. +%term _ + %pi (preserve/md DE DL (step/md/seal-beta _) lt-sub/refl lt-sub/refl DE' DL) + %<- (inv/ofsg/md/seal DE DE') %. +%term _ + %pi (preserve/md DE DL (step/md/let S) LS TS (ofsg/sub (ofsg/md/let D1' D2' DW DF) DS pty-sub/ii) DL') + %<- (inv/ofsg/md/let DE D1 D2 DW DF DS) + %<- (preserve/md D1 DL S LS TS D1' DL') + %<- ({s} {ds} {a} {da} {dm} lt-tt-wkn-sub/ofsg LS TS (D2 s ds a da dm) (D2' s ds a da dm)) %. +%term _ + %pi (preserve/md DE DL _ lt-sub/refl DS D1' DL') + %<- (ofsg/md/let-pty/i DE DU) + %<- (uninhabited-ofsg L _ pty/p _ _ DU D1') + %<- (uninhabited-oflt _ L _ DU DL') + %<- (uninhabited-lt-sub _ _ DU DS) %. +%term _ + %pi (preserve/md DE DL (step/md/let-beta _ V DF) lt-sub/refl lt-sub/refl (ofsg/sub D2' DS P) DL) + %<- (inv/ofsg/md/let DE D1 D2 DW DFS DS) + %<- (val/md-pty/p V (%the (ofsg _ _ _ M' _) D1) (%the (ofsg _ _ _ M' _) D1p)) + %<- (vdt/ofsg/p D1p _ DS' DF' DC') + %<- (subst/md-ofsg (%the ({s} {ds} {a} {da} {dm} ofsg _ _ _ (M s a) _) D2) (%the (ofsg _ _ _ M' _) D1p) DFS DF (%the (ofsg _ _ _ (M M' C') _) D2')) + %<- (pty-sub-pty/i-top _ P) %. +%term _ + %pi (preserve/md DE DL (step/md/pj2-beta _ _) lt-sub/refl lt-sub/refl DE' DL) + %<- (module-beta/pj2 psi-md/eps DE DE') %. +%term _ + %pi (preserve/md DE DL (step/md/pj2 S) LS TS (ofsg/sub (ofsg/md/pj2 D1' DM') DSS DP) DL') + %<- (inv/ofsg/md/pj2 DE D1 DM DS DP) + %<- (preserve/md D1 DL S LS TS D1' DL') + %<- (preserve/fst-md D1 DM S DM' (fst-sg/sgm FSa' _) DQ) + %<- (vdt/cn-deq DQ DK1 DK2 _) + %<- (vdt/ofsg (%the (ofsg _ _ _ _ (sg/sgm _ S1)) D1') (sg-wf/sg/sgm _ DSo (%the (fst-sg _ Ka) FSa))) + %<- (fst-sg-seq FSa FSa' DQFS) + %<- (sg-wf/seq-a DQFS DSo DSo') + %<- (funct/sg-wf DSo' (cn-deq/cn/pj1 DQ) (ofkd/cn/pj1 DK1) DSQ) + %<- (fst-sg-complete (S1 (cn/pj1 CM)) FSM) + %<- (fst-sg-complete (S1 (cn/pj1 C')) FS') + %<- (sg-anti DSQ FSM FS' _ DS21 _ _) + %<- (sg-trans/sub DS21 DS DSS) %. +%term _ + %pi (preserve/md (%the (ofsg _ _ _ _ SG) DE) DL (step/md/app-2 _ S) LS TS (ofsg/sub (%the (ofsg _ _ _ _ (S1 C')) (ofsg/md/app (%the (ofsg _ _ _ _ (sg/pi _ S1)) D1') D2' DM')) (%the (sg-sub (S1 C') SG) DSS) pty-sub/ii) DL') + %<- (inv/ofsg/md/app (%the (ofsg _ _ _ _ SG) DE) (%the (ofsg _ _ _ _ (sg/pi SA S1)) D1) (%the (ofsg _ _ _ _ SA) D2) DM DS) + %<- (preserve/md (%the (ofsg _ _ _ _ SA) D2) DL S LS TS (%the (ofsg _ _ _ _ SA) D2') DL') + %<- (vdt/ofsg (%the (ofsg _ _ _ _ (sg/pi SA S1)) D1) (sg-wf/sg/pi (%the (sg-wf SA) _) (%the ({a} {da ofkd a K} sg-wf (S1 a)) DW) (%the (fst-sg SA K) DFS))) + %<- (preserve/fst-md (%the (ofsg _ _ _ _ SA) D2) DM S DM' (%the (fst-sg SA K') DFS') (%the (cn-deq CM C' K') DQ)) + %<- (fst-sg-seq DFS DFS' DQFS) + %<- (sg-wf/seq-a DQFS (%the ({a} {da ofkd a K} sg-wf (S1 a)) DW) DW') + %<- (vdt/cn-deq DQ DK1 _ _) + %<- (funct/sg-wf DW' (%the (cn-deq CM C' K') DQ) DK1 (%the (sg-deq (S1 CM) (S1 C')) DSQ)) + %<- (fst-sg-complete (S1 CM) FSM) + %<- (fst-sg-complete (S1 C') FS') + %<- (sg-anti DSQ FSM FS' (%the (sg-sub (S1 CM) (S1 C')) DS12) (%the (sg-sub (S1 C') (S1 CM)) DS21) _ _) + %<- (sg-trans/sub DS21 DS DSS) + %<- (lt-tt-wkn-sub/ofsg LS TS D1 D1') %. +%term _ + %pi (preserve/md (%the (ofsg _ _ _ _ SG) DE) DL (step/md/app-beta _ _ DM') lt-sub/refl lt-sub/refl (ofsg/sub Dc DSS P) DL) + %<- (inv/ofsg/md/app (%the (ofsg _ _ _ _ SG) DE) (%the (ofsg _ _ _ _ (sg/pi SA S1)) D1) (%the (ofsg _ _ _ _ SA) D2) (%the (fst-md _ C) DM) (%the (sg-sub (S1 C) SG) DS)) + %<- (vdt/ofsg/p D2 _ FSAA' DMv DvC) + %<- (fst-md-seq DM' DM DMQ') + %<- (fst-md-seq DMv (%the (fst-md M' C') DM') DMQv) + %<- (inv/ofsg/md/lam D1 D1o _ DFFS (sg-sub/sg/pi (%the (sg-deq SA _) DS1) DS2 (%the (fst-sg _ KK) FSAA))) + %<- (fst-sg-complete _ Df2) + %<- (sg-anti DS1 FSAA Df2 DSA' _ DKA' _) + %<- (ofkd/seq-c DMQv DvC DCv) + %<- (fst-sg-seq FSAA' FSAA DQFSAA) + %<- (ofkd/seq-k DQFSAA DCv (%the (ofkd C' KK) DCv')) + %<- (subst/md-ofsg D1o (ofsg/sub D2 DSA' pty-sub/pp) DFFS DM' (%the (ofsg _ _ Y _ _) Dc)) + %<- (fst-sg-complete _ DF1) + %<- (fst-sg-complete _ DF2) + %<- (sg-anti (DS2 _ DCv') DF1 DF2 DSla _ _ _) + %<- (seq/cn/sg-sub-o ([a] S1 a) DMQ' DSla DS') + %<- (sg-trans/sub DS' DS DSS) + %<- (pty-sub-pty/i-top Y (%the (pty-sub Y pty/i) P)) %. +%worlds () (preserve/tm _ _ _ _ _ _ _) (preserve/md _ _ _ _ _ _ _) %. +%total (D1 D2) (preserve/md _ _ D2 _ _ _ _) (preserve/tm _ _ D1 _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/progress-lemmas-mods.lf b/new-tests/stelf-output/tslf/progress-lemmas-mods.lf new file mode 100644 index 0000000..8c0417a --- /dev/null +++ b/new-tests/stelf-output/tslf/progress-lemmas-mods.lf @@ -0,0 +1,65 @@ +%%%% progress factoring lemmas for modules +%sort progress/md/pair %. +%mode progress/md/pair %in %in %out %. +%term _ progress/md/pair (notstuck/md/step S) _ (notstuck/md/step (step/md/pair-1 S)) %. +%term _ progress/md/pair (notstuck/md/val V) (notstuck/md/step S) (notstuck/md/step (step/md/pair-2 V S)) %. +%term _ progress/md/pair (notstuck/md/raises S) _ (notstuck/md/raises (raises/md/pair-1 S)) %. +%term _ progress/md/pair (notstuck/md/val V) (notstuck/md/raises S) (notstuck/md/raises (raises/md/pair-2 V S)) %. +%term _ progress/md/pair (notstuck/md/val V1) (notstuck/md/val V2) (notstuck/md/val (val/md/pair V1 V2)) %. +%worlds () (progress/md/pair _ _ _) %. +%total {} (progress/md/pair _ _ _) %. +%sort progress/md/pj1 %. +%mode progress/md/pj1 %in %in %out %. +%term _ progress/md/pj1 (notstuck/md/step S) _ (notstuck/md/step (step/md/pj1 S)) %. +%term _ progress/md/pj1 (notstuck/md/raises S) _ (notstuck/md/raises (raises/md/pj1 S)) %. +%term _ + %pi (progress/md/pj1 (%the (notstuck/md _ ST1 _) (notstuck/md/val V)) D (notstuck/md/step S)) + %<- (cfl/md/pair V D SQ) + %<- (seq/md/pj1 SQ SQf) + %<- (seq/val/md SQ V V') + %<- (step/md/seq SQf (step/md/pj1-beta ST1 V') S) %. +%worlds () (progress/md/pj1 _ _ _) %. +%total {} (progress/md/pj1 _ _ _) %. +%sort progress/md/pj2 %. +%mode progress/md/pj2 %in %in %out %. +%term _ progress/md/pj2 (notstuck/md/step S) _ (notstuck/md/step (step/md/pj2 S)) %. +%term _ progress/md/pj2 (notstuck/md/raises S) _ (notstuck/md/raises (raises/md/pj2 S)) %. +%term _ + %pi (progress/md/pj2 (%the (notstuck/md _ ST1 _) (notstuck/md/val V)) D (notstuck/md/step S)) + %<- (cfl/md/pair V D SQ) + %<- (seq/md/pj2 SQ SQf) + %<- (seq/val/md SQ V V') + %<- (step/md/seq SQf (step/md/pj2-beta ST1 V') S) %. +%worlds () (progress/md/pj2 _ _ _) %. +%total {} (progress/md/pj2 _ _ _) %. +%sort progress/md/app %. +%mode progress/md/app %in %in %in %in %out %. +%term _ progress/md/app (notstuck/md/step S) _ _ _ (notstuck/md/step (step/md/app-1 S)) %. +%term _ progress/md/app (notstuck/md/val V) (notstuck/md/step S) _ _ (notstuck/md/step (step/md/app-2 V S)) %. +%term _ progress/md/app (notstuck/md/raises S) _ _ _ (notstuck/md/raises (raises/md/app-1 S)) %. +%term _ progress/md/app (notstuck/md/val V) (notstuck/md/raises S) _ _ (notstuck/md/raises (raises/md/app-2 V S)) %. +%term _ + %pi (progress/md/app (notstuck/md/val V1) (notstuck/md/val V2) D1 DMF (notstuck/md/step S)) + %<- (cfl/sg/pi V1 D1 SQ) + %<- (seq/md/app _ SQ SQf) + %<- (seq/val/md SQ V1 V') + %<- (step/md/seq SQf (step/md/app-beta ST V2 DMF) S) %. +%worlds () (progress/md/app _ _ _ _ _) %. +%total {} (progress/md/app _ _ _ _ _) %. +%sort progress/md/let {M2 %pi md %-> cn %-> md} {S sg} %. +%mode progress/md/let %in %in %in %in %out %. +%term _ progress/md/let _ _ (notstuck/md/step S) _ (notstuck/md/step (step/md/let S)) %. +%term _ progress/md/let _ _ (notstuck/md/raises S) _ (notstuck/md/raises (raises/md/let S)) %. +%term _ + %pi (progress/md/let _ _ (notstuck/md/val V) DM1 (notstuck/md/step (step/md/let-beta ST V DC))) + %<- (val/md-pty/p V DM1 DM1') + %<- (vdt/ofsg/p DM1' _ _ DC _) %. +%worlds () (progress/md/let _ _ _ _ _) %. +%total {} (progress/md/let _ _ _ _ _) %. +%sort progress/md/tm %. +%mode progress/md/tm %in %out %. +%term _ progress/md/tm (notstuck/tm/val V) (notstuck/md/val (val/md/tm V)) %. +%term _ progress/md/tm (notstuck/tm/step S) (notstuck/md/step (step/md/tm S)) %. +%term _ progress/md/tm (notstuck/tm/raises S) (notstuck/md/raises (raises/md/tm S)) %. +%worlds () (progress/md/tm _ _) %. +%total {} (progress/md/tm _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/progress-lemmas-terms.lf b/new-tests/stelf-output/tslf/progress-lemmas-terms.lf new file mode 100644 index 0000000..69ce00c --- /dev/null +++ b/new-tests/stelf-output/tslf/progress-lemmas-terms.lf @@ -0,0 +1,238 @@ +%%%% progress factoring lemmas +%sort progress/tm/pair %. +%mode progress/tm/pair %in %in %out %. +%term _ progress/tm/pair (notstuck/tm/step S) _ (notstuck/tm/step (step/tm/pair-1 S)) %. +%term _ progress/tm/pair (notstuck/tm/val V) (notstuck/tm/step S) (notstuck/tm/step (step/tm/pair-2 V S)) %. +%term _ progress/tm/pair (notstuck/tm/raises R) _ (notstuck/tm/raises (raises/tm/pair-1 R)) %. +%term _ progress/tm/pair (notstuck/tm/val V) (notstuck/tm/raises R) (notstuck/tm/raises (raises/tm/pair-2 V R)) %. +%term _ progress/tm/pair (notstuck/tm/val V1) (notstuck/tm/val V2) (notstuck/tm/val (val/tm/pair V1 V2)) %. +%worlds () (progress/tm/pair _ _ _) %. +%total {} (progress/tm/pair _ _ _) %. +%sort progress/tm/tag %. +%mode progress/tm/tag %in %in %out %. +%term _ progress/tm/tag (notstuck/tm/step S) _ (notstuck/tm/step (step/tm/tag-1 S)) %. +%term _ progress/tm/tag (notstuck/tm/val V) (notstuck/tm/step S) (notstuck/tm/step (step/tm/tag-2 V S)) %. +%term _ progress/tm/tag (notstuck/tm/raises R) _ (notstuck/tm/raises (raises/tm/tag-1 R)) %. +%term _ progress/tm/tag (notstuck/tm/val V) (notstuck/tm/raises R) (notstuck/tm/raises (raises/tm/tag-2 V R)) %. +%term _ progress/tm/tag (notstuck/tm/val V1) (notstuck/tm/val V2) (notstuck/tm/val (val/tm/tag V1 V2)) %. +%worlds () (progress/tm/tag _ _ _) %. +%total {} (progress/tm/tag _ _ _) %. +%sort progress/tm/pj1 %. +%mode progress/tm/pj1 %in %in %out %. +%term _ progress/tm/pj1 (notstuck/tm/step S) _ (notstuck/tm/step (step/tm/pj1 S)) %. +%term _ progress/tm/pj1 (notstuck/tm/raises S) _ (notstuck/tm/raises (raises/tm/pj1 S)) %. +%term _ + %pi (progress/tm/pj1 (notstuck/tm/val V) D (notstuck/tm/step (%the (step/tm _ ST1 _ _ _ _) S))) + %<- (vdt/oftp D DC) + %<- (cfl/tp/cross V D (cn-deq/refl DC) SQ) + %<- (seq/tm/pj1 SQ SQf) + %<- (seq/val/tm SQ V V') + %<- (step/tm/seq SQf (step/tm/pj1-beta ST1 V') S) %. +%worlds () (progress/tm/pj1 _ _ _) %. +%total {} (progress/tm/pj1 _ _ _) %. +%sort progress/tm/pj2 %. +%mode progress/tm/pj2 %in %in %out %. +%term _ progress/tm/pj2 (notstuck/tm/step S) _ (notstuck/tm/step (step/tm/pj2 S)) %. +%term _ progress/tm/pj2 (notstuck/tm/raises S) _ (notstuck/tm/raises (raises/tm/pj2 S)) %. +%term _ + %pi (progress/tm/pj2 (notstuck/tm/val V) D (notstuck/tm/step S)) + %<- (vdt/oftp D DC) + %<- (cfl/tp/cross V D (cn-deq/refl DC) SQ) + %<- (seq/tm/pj2 SQ SQf) + %<- (seq/val/tm SQ V V') + %<- (step/tm/seq SQf (step/tm/pj2-beta ST1 V') S) %. +%worlds () (progress/tm/pj2 _ _ _) %. +%total {} (progress/tm/pj2 _ _ _) %. +%sort progress/tm/tmapp %. +%mode progress/tm/tmapp %in %in %in %out %. +%term _ progress/tm/tmapp (notstuck/tm/step S) _ _ (notstuck/tm/step (step/tm/tmapp-1 S)) %. +%term _ progress/tm/tmapp (notstuck/tm/val V) (notstuck/tm/step S) _ (notstuck/tm/step (step/tm/tmapp-2 V S)) %. +%term _ progress/tm/tmapp (notstuck/tm/raises S) _ _ (notstuck/tm/raises (raises/tm/tmapp-1 S)) %. +%term _ progress/tm/tmapp (notstuck/tm/val V) (notstuck/tm/raises S) _ (notstuck/tm/raises (raises/tm/tmapp-2 V S)) %. +%term _ + %pi (progress/tm/tmapp (%the (notstuck/tm _ ST1 _) (notstuck/tm/val V1)) (notstuck/tm/val V2) D1 (notstuck/tm/step S)) + %<- (vdt/oftp D1 DC) + %<- (cfl/tp/arrow V1 D1 (cn-deq/refl DC) SQ) + %<- (seq/tm/tmapp _ SQ SQf) + %<- (seq/val/tm SQ V1 V') + %<- (step/tm/seq SQf (step/tm/tmapp-beta ST1 V2) S) %. +%worlds () (progress/tm/tmapp _ _ _ _) %. +%total {} (progress/tm/tmapp _ _ _ _) %. +%sort progress/tm/cnapp {C' cn} %. +%mode progress/tm/cnapp %in %in %in %out %. +%term _ progress/tm/cnapp _ (notstuck/tm/step S) _ (notstuck/tm/step (step/tm/cnapp S)) %. +%term _ progress/tm/cnapp _ (notstuck/tm/raises S) _ (notstuck/tm/raises (raises/tm/cnapp S)) %. +%term _ + %pi (progress/tm/cnapp _ (%the (notstuck/tm _ ST1 _) (notstuck/tm/val V1)) D1 (notstuck/tm/step S)) + %<- (vdt/oftp D1 DC) + %<- (cfl/tp/forall V1 D1 (cn-deq/refl DC) SQ) + %<- (seq/tm/cnapp _ SQ SQf) + %<- (seq/val/tm SQ V1 V') + %<- (step/tm/seq SQf (step/tm/cnapp-beta ST1) S) %. +%worlds () (progress/tm/cnapp _ _ _ _) %. +%total {} (progress/tm/cnapp _ _ _ _) %. +%sort progress/tm/term %. +%mode progress/tm/term %in %in %out %. +%term _ progress/tm/term (notstuck/md/step S) _ (notstuck/tm/step (step/tm/term S)) %. +%term _ progress/tm/term (notstuck/md/raises S) _ (notstuck/tm/raises (raises/tm/term S)) %. +%term _ + %pi (progress/tm/term (%the (notstuck/md _ ST1 _) (notstuck/md/val Vt)) D (notstuck/tm/step S)) + %<- (cfl/md/tm Vt D DQ) + %<- (seq/val/md DQ Vt V') + %<- (seq/tm/term DQ DQ') + %<- (step/tm/seq DQ' (step/tm/term-beta ST1 V') S) %. +%worlds () (progress/tm/term _ _ _) %. +%total {} (progress/tm/term _ _ _) %. +% lemma that states it is always possible to allocate new memory +%sort progress/st {ST st} {E tm} %. +%mode progress/st %in %in %out %. +%term _ progress/st st/nil E st-alloc/nil %. +%term _ progress/st (st/cons _ _ _) E st-alloc/cons %. +%worlds () (progress/st _ _ _) %. +%total {} (progress/st _ _ _) %. +%sort progress/tm/ref %. +%mode progress/tm/ref %in %in %out %. +%term _ progress/tm/ref (notstuck/tm/step S) _ (notstuck/tm/step (step/tm/ref S)) %. +%term _ progress/tm/ref (notstuck/tm/raises S) _ (notstuck/tm/raises (raises/tm/ref S)) %. +%term _ + %pi (progress/tm/ref (notstuck/tm/val V) (%the (oftp L T E1 C) D1) (notstuck/tm/step (step/tm/ref-beta V SA))) + %<- (progress/st ST E1 SA) %. +%worlds () (progress/tm/ref _ _ _) %. +%total {} (progress/tm/ref _ _ _) %. +%sort progress/st-look %. +%mode progress/st-look %in %in %out %. +%term _ progress/st-look lt-look/hit (st-wf/cons _ _ _ _) st-look/hit %. +%term _ + %pi (progress/st-look (lt-look/miss LL) (st-wf/cons LCS _ _ _) (st-look/miss LS)) + %<- (progress/st-look LL LCS LS) %. +%worlds () (progress/st-look _ _ _) %. +%total D1 (progress/st-look D1 _ _) %. +%sort progress/tm/get %. +%mode progress/tm/get %in %in %in %out %. +%term _ progress/tm/get (notstuck/tm/step S) _ _ (notstuck/tm/step (step/tm/get S)) %. +%term _ progress/tm/get (notstuck/tm/raises S) _ _ (notstuck/tm/raises (raises/tm/get S)) %. +%term _ + %pi (progress/tm/get (notstuck/tm/val (%the (val/tm E) V)) (%the (oftp L T E (tp/ref C)) D) (oflt/c LCS _) (notstuck/tm/step S)) + %<- (vdt/oftp D DC) + %<- (cfl/tp/ref V D (cn-deq/refl DC) DQ) + %<- (seq/val/tm DQ V (%the (val/tm (tm/loc LC)) V')) + %<- (oftp/seq-t DQ D D') + %<- (inv/oftp/tm/loc D' _ LL _) + %<- (progress/st-look LL LCS LS) + %<- (seq/tm/get DQ DQ') + %<- (step/tm/seq DQ' (step/tm/get-beta V' LS) S) %. +%worlds () (progress/tm/get _ _ _ _) %. +%total {} (progress/tm/get _ _ _ _) %. +%sort progress/st-update {E tm} %. +%mode progress/st-update %in %in %in %out %. +%term _ progress/st-update _ lt-look/hit (st-wf/cons _ _ _ _) st-update/hit %. +%term _ + %pi (progress/st-update E (lt-look/miss LL) (st-wf/cons LCS _ _ _) (st-update/miss LS)) + %<- (progress/st-update E LL LCS LS) %. +%worlds () (progress/st-update _ _ _ _) %. +%total D1 (progress/st-update _ D1 _ _) %. +%sort progress/tm/set %. +%mode progress/tm/set %in %in %in %in %in %out %. +%term _ progress/tm/set (notstuck/tm/step S) NS _ _ _ (notstuck/tm/step (step/tm/set-1 S)) %. +%term _ progress/tm/set (notstuck/tm/val V) (notstuck/tm/step S) _ _ _ (notstuck/tm/step (step/tm/set-2 V S)) %. +%term _ progress/tm/set (notstuck/tm/raises S) NS _ _ _ (notstuck/tm/raises (raises/tm/set-1 S)) %. +%term _ progress/tm/set (notstuck/tm/val V) (notstuck/tm/raises S) _ _ _ (notstuck/tm/raises (raises/tm/set-2 V S)) %. +%term _ + %pi (progress/tm/set (notstuck/tm/val (%the (val/tm E1) V1)) (notstuck/tm/val (%the (val/tm E2) V2)) D1 D2 (oflt/c LCS _) (notstuck/tm/step S)) + %<- (vdt/oftp D1 DC) + %<- (cfl/tp/ref V1 D1 (cn-deq/refl DC) DQ) + %<- (seq/val/tm DQ V1 V') + %<- (oftp/seq-t DQ D1 D') + %<- (inv/oftp/tm/loc D' _ LL _) + %<- (progress/st-update _ LL LCS SU) + %<- (seq/tm/set E2 DQ DQ') + %<- (step/tm/seq DQ' (step/tm/set-beta V' V2 SU) S) %. +%worlds () (progress/tm/set _ _ _ _ _ _) %. +%total {} (progress/tm/set _ _ _ _ _ _) %. +%sort progress/tm/inl {C1 cn} {C2 cn} %. +%mode progress/tm/inl %in %in %in %out %. +%term _ progress/tm/inl _ _ (notstuck/tm/step S) (notstuck/tm/step (step/tm/inl S)) %. +%term _ progress/tm/inl _ _ (notstuck/tm/raises S) (notstuck/tm/raises (raises/tm/inl S)) %. +%term _ progress/tm/inl _ _ (notstuck/tm/val V1) (notstuck/tm/val (val/tm/inl V1)) %. +%worlds () (progress/tm/inl _ _ _ _) %. +%total {} (progress/tm/inl _ _ _ _) %. +%sort progress/tm/inr {C1 cn} {C2 cn} %. +%mode progress/tm/inr %in %in %in %out %. +%term _ progress/tm/inr _ _ (notstuck/tm/step S) (notstuck/tm/step (step/tm/inr S)) %. +%term _ progress/tm/inr _ _ (notstuck/tm/raises S) (notstuck/tm/raises (raises/tm/inr S)) %. +%term _ progress/tm/inr _ _ (notstuck/tm/val V1) (notstuck/tm/val (val/tm/inr V1)) %. +%worlds () (progress/tm/inr _ _ _ _) %. +%total {} (progress/tm/inr _ _ _ _) %. +%sort progress/tm/case-beta {E1 tm} {E2 %pi tm %-> tm} {E3 %pi tm %-> tm} {ST st} {T lt} {SQ seq/tm E1 E'} %. +%mode progress/tm/case-beta %in %in %in %in %in %in %in %in %out %. +%term _ progress/tm/case-beta _ _ _ _ _ _ V seq-inl (step/tm/case-beta-l V) %. +%term _ progress/tm/case-beta _ _ _ _ _ _ V seq-inr (step/tm/case-beta-r V) %. +%worlds () (progress/tm/case-beta _ _ _ _ _ _ _ _ _) %. +%total {} (progress/tm/case-beta _ _ _ _ _ _ _ _ _) %. +%sort progress/tm/case {E2 %pi tm %-> tm} {E3 %pi tm %-> tm} %. +%mode progress/tm/case %in %in %in %in %out %. +%term _ progress/tm/case _ _ (notstuck/tm/step S) _ (notstuck/tm/step (step/tm/case S)) %. +%term _ progress/tm/case _ _ (notstuck/tm/raises S) _ (notstuck/tm/raises (raises/tm/case S)) %. +%term _ + %pi (progress/tm/case _ _ (notstuck/tm/val V) D (notstuck/tm/step S)) + %<- (vdt/oftp D DC) + %<- (cfl/tp/sum V D (cn-deq/refl DC) SQ) + %<- (progress/tm/case-beta _ _ _ _ _ _ V SQ S) %. +%worlds () (progress/tm/case _ _ _ _ _) %. +%total {} (progress/tm/case _ _ _ _ _) %. +%sort progress/tm/try {E2 %pi tm %-> tm} %. +%mode progress/tm/try %in %in %in %out %. +%term _ progress/tm/try _ (notstuck/tm/val V) _ (notstuck/tm/step (step/tm/try-beta V)) %. +%term _ progress/tm/try _ (notstuck/tm/step S) _ (notstuck/tm/step (step/tm/try S)) %. +%term _ progress/tm/try _ (notstuck/tm/raises R) _ (notstuck/tm/step (step/tm/try-handle R)) %. +%worlds () (progress/tm/try _ _ _ _) %. +%total {} (progress/tm/try _ _ _ _) %. +%sort progress/tm/raise %. +%mode progress/tm/raise %in %in %out %. +%term _ progress/tm/raise (notstuck/tm/step S) _ (notstuck/tm/step (step/tm/raise S)) %. +%term _ progress/tm/raise (notstuck/tm/raises S) _ (notstuck/tm/raises (raises/tm/raise-2 S)) %. +%term _ progress/tm/raise (notstuck/tm/val V) _ (notstuck/tm/raises (raises/tm/raise-1 V)) %. +%worlds () (progress/tm/raise _ _ _) %. +%total {} (progress/tm/raise _ _ _) %. +%sort progress/tm/iftag-beta {E3 %pi tm %-> tm} {E4 tm} {ST st} {T lt} %. +%mode progress/tm/iftag-beta %in %in %in %in %in %in %in %in %out %. +%term _ progress/tm/iftag-beta _ _ _ _ V seq/tm/refl seq/tm/refl (loc-seq-or-neq/seq seq/loc/refl) (step/tm/iftag-beta-1 V) %. +%term _ progress/tm/iftag-beta _ _ _ _ V seq/tm/refl seq/tm/refl (loc-seq-or-neq/neq N) (step/tm/iftag-beta-2 V N) %. +%worlds () (progress/tm/iftag-beta _ _ _ _ _ _ _ _ _) %. +%total {} (progress/tm/iftag-beta _ _ _ _ _ _ _ _ _) %. +%sort progress/tm/iftag {E3 %pi tm %-> tm} {E4 tm} %. +%mode progress/tm/iftag %in %in %in %in %in %in %out %. +%term _ progress/tm/iftag _ _ (notstuck/tm/step S) _ _ _ (notstuck/tm/step (step/tm/iftag-1 S)) %. +%term _ progress/tm/iftag _ _ (notstuck/tm/val V) (notstuck/tm/step S) _ _ (notstuck/tm/step (step/tm/iftag-2 V S)) %. +%term _ progress/tm/iftag _ _ (notstuck/tm/raises S) _ _ _ (notstuck/tm/raises (raises/tm/iftag-1 S)) %. +%term _ progress/tm/iftag _ _ (notstuck/tm/val V) (notstuck/tm/raises S) _ _ (notstuck/tm/raises (raises/tm/iftag-2 V S)) %. +%term _ + %pi (progress/tm/iftag _ _ (notstuck/tm/val V1) (notstuck/tm/val V2) D1 D2 (notstuck/tm/step S)) + %<- (vdt/oftp D1 DC1) + %<- (cfl/tp/tagged V1 D1 (cn-deq/refl DC1) SQ1) + %<- (vdt/oftp D2 DC2) + %<- (cfl/tp/tag V2 D2 (cn-deq/refl DC2) SQ2) + %<- (loc-seq-or-neq-total _ _ LS) + %<- (progress/tm/iftag-beta _ _ _ _ V1 SQ1 SQ2 LS S) %. +%worlds () (progress/tm/iftag _ _ _ _ _ _ _) %. +%total {} (progress/tm/iftag _ _ _ _ _ _ _) %. +%sort progress/tm/roll {K kd} {C' %pi cn %-> cn} %. +%mode progress/tm/roll %in %in %in %out %. +%term _ progress/tm/roll _ _ (notstuck/tm/step S) (notstuck/tm/step (step/tm/roll S)) %. +%term _ progress/tm/roll _ _ (notstuck/tm/raises S) (notstuck/tm/raises (raises/tm/roll S)) %. +%term _ progress/tm/roll _ _ (notstuck/tm/val V) (notstuck/tm/val (val/tm/roll V)) %. +%worlds () (progress/tm/roll _ _ _ _) %. +%total {} (progress/tm/roll _ _ _ _) %. +%sort progress/tm/unroll %. +%mode progress/tm/unroll %in %in %out %. +%term _ progress/tm/unroll (notstuck/tm/step S) _ (notstuck/tm/step (step/tm/unroll S)) %. +%term _ progress/tm/unroll (notstuck/tm/raises S) _ (notstuck/tm/raises (raises/tm/unroll S)) %. +%term _ + %pi (progress/tm/unroll (notstuck/tm/val V) D1 (notstuck/tm/step S)) + %<- (vdt/oftp D1 DC) + %<- (cfl/cn/mu V D1 (cn-deq/refl DC) DQ) + %<- (seq/val/tm DQ V V') + %<- (seq/tm/unroll DQ DQ') + %<- (step/tm/seq DQ' (step/tm/unroll-beta V') S) %. +%worlds () (progress/tm/unroll _ _ _) %. +%total {} (progress/tm/unroll _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/progress.lf b/new-tests/stelf-output/tslf/progress.lf new file mode 100644 index 0000000..0be4891 --- /dev/null +++ b/new-tests/stelf-output/tslf/progress.lf @@ -0,0 +1,128 @@ +%%%% Progress Theorem +%sort progress/tm %. +%mode progress/tm %in %in %out %. +%sort progress/md %. +%mode progress/md %in %in %out %. +%term _ progress/tm oftp/tm/unit _ (notstuck/tm/val val/tm/unit) %. +%term _ + %pi (progress/tm (oftp/tm/pair D1 D2) DS NS3) + %<- (progress/tm D1 DS NS1) + %<- (progress/tm D2 DS NS2) + %<- (progress/tm/pair NS1 NS2 NS3) %. +%term _ + %pi (progress/tm (oftp/tm/tag D1 D2) DS NS3) + %<- (progress/tm D1 DS NS1) + %<- (progress/tm D2 DS NS2) + %<- (progress/tm/tag NS1 NS2 NS3) %. +%term _ + %pi (progress/tm (oftp/tm/pj1 D1) DS NS2) + %<- (progress/tm D1 DS NS1) + %<- (progress/tm/pj1 NS1 D1 NS2) %. +%term _ + %pi (progress/tm (oftp/tm/pj2 D1) DS NS2) + %<- (progress/tm D1 DS NS1) + %<- (progress/tm/pj2 NS1 D1 NS2) %. +%term _ progress/tm (oftp/tm/fun _ _) _ (notstuck/tm/val val/tm/fun) %. +%term _ + %pi (progress/tm (oftp/tm/tmapp D1 D2) DS NS3) + %<- (progress/tm D1 DS NS1) + %<- (progress/tm D2 DS NS2) + %<- (progress/tm/tmapp NS1 NS2 D1 NS3) %. +%term _ progress/tm (oftp/tm/cnabs _ _) _ (notstuck/tm/val val/tm/cnabs) %. +%term _ + %pi (progress/tm (oftp/tm/cnapp D1 DC) DS NS2) + %<- (progress/tm D1 DS NS1) + %<- (progress/tm/cnapp _ NS1 D1 NS2) %. +%term _ %pi (progress/tm (oftp/deq D1 DC) DS NS) %<- (progress/tm D1 DS NS) %. +%term _ + %pi (progress/tm (oftp/tm/term D1) DS NS) + %<- (progress/md D1 DS NSM) + %<- (progress/tm/term NSM D1 NS) %. +%term _ progress/tm (oftp/tm/loc _ _) _ (notstuck/tm/val val/tm/loc) %. +%term _ + %pi (progress/tm (oftp/tm/ref D1) DS NS') + %<- (progress/tm D1 DS NS) + %<- (progress/tm/ref NS D1 NS') %. +%term _ + %pi (progress/tm (oftp/tm/set D1 D2) DS NS') + %<- (progress/tm D1 DS NS1) + %<- (progress/tm D2 DS NS2) + %<- (progress/tm/set NS1 NS2 D1 D2 DS NS') %. +%term _ + %pi (progress/tm (oftp/tm/get D1) DS NS') + %<- (progress/tm D1 DS NS) + %<- (progress/tm/get NS D1 DS NS') %. +%term _ + %pi (progress/tm (oftp/tm/inl D1 _) DS NS2) + %<- (progress/tm D1 DS NS1) + %<- (progress/tm/inl _ _ NS1 NS2) %. +%term _ + %pi (progress/tm (oftp/tm/inr D1 _) DS NS2) + %<- (progress/tm D1 DS NS1) + %<- (progress/tm/inr _ _ NS1 NS2) %. +%term _ + %pi (progress/tm (oftp/tm/case D1 _ _) DS NS2) + %<- (progress/tm D1 DS NS1) + %<- (progress/tm/case _ _ NS1 D1 NS2) %. +%term _ + %pi (progress/tm (oftp/tm/raise D1 _) DS NS') + %<- (progress/tm D1 DS NS) + %<- (progress/tm/raise NS D1 NS') %. +%term _ + %pi (progress/tm (oftp/tm/try D1 D2) DS NS2) + %<- (progress/tm D1 DS NS1) + %<- (progress/tm/try _ NS1 (oftp/tm/try D1 D2) NS2) %. +%term _ progress/tm (oftp/tm/tagloc _ _) DS (notstuck/tm/val val/tm/tagloc) %. +%term _ progress/tm (oftp/tm/new-tag _) DS (notstuck/tm/step (step/tm/new-tag-beta lt-extend/nil)) %. +%term _ progress/tm (oftp/tm/new-tag _) DS (notstuck/tm/step (step/tm/new-tag-beta lt-extend/cons)) %. +%term _ + %pi (progress/tm (oftp/tm/iftag D1 D2 _ _) DS NS') + %<- (progress/tm D1 DS NS1) + %<- (progress/tm D2 DS NS2) + %<- (progress/tm/iftag _ _ NS1 NS2 D1 D2 NS') %. +%term _ + %pi (progress/tm (oftp/tm/roll D1 _) DS NS') + %<- (progress/tm D1 DS NS) + %<- (progress/tm/roll _ _ NS NS') %. +%term _ + %pi (progress/tm (oftp/tm/unroll D1) DS NS') + %<- (progress/tm D1 DS NS) + %<- (progress/tm/unroll NS D1 NS') %. +%term _ progress/md ofsg/md/unit DS (notstuck/md/val val/md/unit) %. +%term _ progress/md (ofsg/md/cn _) DS (notstuck/md/val val/md/cn) %. +%term _ + %pi (progress/md (ofsg/md/tm D) DS NS') + %<- (progress/tm D DS NS) + %<- (progress/md/tm NS NS') %. +%term _ + %pi (progress/md (ofsg/md/pair D1 D2) DS NS) + %<- (progress/md D1 DS NS1) + %<- (progress/md D2 DS NS2) + %<- (progress/md/pair NS1 NS2 NS) %. +%term _ + %pi (progress/md (ofsg/md/pj1 D1) DS NS') + %<- (progress/md D1 DS NS) + %<- (progress/md/pj1 NS D1 NS') %. +%term _ + %pi (progress/md (ofsg/md/pj2 D1 DC) DS NS') + %<- (progress/md D1 DS NS) + %<- (progress/md/pj2 NS D1 NS') %. +%term _ progress/md (ofsg/md/lam _ _ _) _ (notstuck/md/val val/md/lam) %. +%term _ + %pi (progress/md (ofsg/md/app D1 D2 DC) DS NS') + %<- (progress/md D1 DS NS1) + %<- (progress/md D2 DS NS2) + %<- (progress/md/app NS1 NS2 D1 DC NS') %. +%term _ + %pi (progress/md (ofsg/md/let D1 D2 DW _) DS NS') + %<- (progress/md D1 DS NS) + %<- (progress/md/let _ _ NS D1 NS') %. +%term _ progress/md (ofsg/md/seal D1) (%the (oflt ST _ _) DS) (notstuck/md/step (step/md/seal-beta ST)) %. +%term _ + %pi (progress/md (ofsg/sgm-ext D1 _) DS NS) + %<- (subder/md/pj1 D1 D1' _) + %<- (progress/md D1' DS NS) %. +%term _ %pi (progress/md (ofsg/kd-ext D1 _ _) DS NS) %<- (progress/md D1 DS NS) %. +%term _ %pi (progress/md (ofsg/sub D1 _ _) DS NS) %<- (progress/md D1 DS NS) %. +%worlds () (progress/tm _ _ _) (progress/md _ _ _) %. +%total (D1 D2) (progress/tm D1 _ _) (progress/md D2 _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/proj-val-lemma.lf b/new-tests/stelf-output/tslf/proj-val-lemma.lf new file mode 100644 index 0000000..941eac1 --- /dev/null +++ b/new-tests/stelf-output/tslf/proj-val-lemma.lf @@ -0,0 +1,19 @@ +%%%% projectible value lemma +%%%% if a module is well-formed and an val/md, then it is projectible (pty/p) +%%%% when we have an inversion lemma for pairs, this proof is simpler +%%%% because we can induct over the derivation of val/md +%%%% but since we don't, for now, we can induct over the typing derivation +%%%% for a less elegant, but equally accurate proof. +%%%% Actually, this one isn't so bad. Just 2 more cases... +%sort val/md-pty/p %. +%mode val/md-pty/p %in %in %out %. +%term _ val/md-pty/p V D1 D1 %. +%term _ + %pi (val/md-pty/p (val/md/pair V1 V2) (ofsg/md/pair D1 D2) (ofsg/md/pair D1' D2')) + %<- (val/md-pty/p V1 D1 D1') + %<- (val/md-pty/p V2 D2 D2') %. +%term _ + %pi (val/md-pty/p V (ofsg/sub D1 DS _) (ofsg/sub D1' DS pty-sub/pp)) + %<- (val/md-pty/p V D1 D1') %. +%worlds () (val/md-pty/p _ _ _) %. +%total D1 (val/md-pty/p _ D1 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/projectibility-lemmas.lf b/new-tests/stelf-output/tslf/projectibility-lemmas.lf new file mode 100644 index 0000000..57e6ba5 --- /dev/null +++ b/new-tests/stelf-output/tslf/projectibility-lemmas.lf @@ -0,0 +1,20 @@ +%sort pty-sub-refl {P pty} %. +%mode pty-sub-refl %in %out %. +%term _ pty-sub-refl pty/p pty-sub/pp %. +%term _ pty-sub-refl pty/i pty-sub/ii %. +%worlds () (pty-sub-refl _ _) %. +%total {} (pty-sub-refl _ _) %. +%sort pty-sub-trans %. +%mode pty-sub-trans %in %in %out %. +%term _ pty-sub-trans pty-sub/pp pty-sub/pp pty-sub/pp %. +%term _ pty-sub-trans pty-sub/pp pty-sub/pi pty-sub/pi %. +%term _ pty-sub-trans pty-sub/pi pty-sub/ii pty-sub/pi %. +%term _ pty-sub-trans pty-sub/ii pty-sub/ii pty-sub/ii %. +%worlds () (pty-sub-trans _ _ _) %. +%total {} (pty-sub-trans _ _ _) %. +%sort pty-sub-pty/i-top {P pty} %. +%mode pty-sub-pty/i-top %in %out %. +%term _ pty-sub-pty/i-top _ pty-sub/pi %. +%term _ pty-sub-pty/i-top _ pty-sub/ii %. +%worlds () (pty-sub-pty/i-top _ _) %. +%total {} (pty-sub-pty/i-top _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/projectibility.lf b/new-tests/stelf-output/tslf/projectibility.lf new file mode 100644 index 0000000..8a7a643 --- /dev/null +++ b/new-tests/stelf-output/tslf/projectibility.lf @@ -0,0 +1,16 @@ +%%%% projectibility. this is stricter than purity. +%%%% this is a slight divergence from Dreyer's thesis, where P and I +%%%% denoted purity and impurity, where all projectible modules were pure +%%%% but not all pure modules were projectible +%%%% for reference, projectible modules are denoted by the "translucent" +%%%% type-face in Dreyer's thesis +%sort pty %. +%name pty %. +%term pty/p pty %. +%term pty/i pty %. +%sort pty-sub %. +%mode pty-sub %star %star %. +%term pty-sub/pp pty-sub pty/p pty/p %. +%term pty-sub/pi pty-sub pty/p pty/i %. +%term pty-sub/ii pty-sub pty/i pty/i %. +%worlds () (pty-sub _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/raises.lf b/new-tests/stelf-output/tslf/raises.lf new file mode 100644 index 0000000..0874f7c --- /dev/null +++ b/new-tests/stelf-output/tslf/raises.lf @@ -0,0 +1,37 @@ +%%%% raises judgment +%sort raises/tm %. +%mode raises/tm %star %star %. +%sort raises/md %. +%mode raises/md %star %star %. +%term raises/tm/pair-1 %pi (raises/tm (tm/pair E1 E2) EX) %<- (raises/tm E1 EX) %. +%term raises/tm/pair-2 %pi (raises/tm (tm/pair E1 E2) EX) %<- (raises/tm E2 EX) %<- (val/tm E1) %. +%term raises/tm/pj1 %pi (raises/tm (tm/pj1 E) EX) %<- (raises/tm E EX) %. +%term raises/tm/pj2 %pi (raises/tm (tm/pj2 E) EX) %<- (raises/tm E EX) %. +%term raises/tm/tmapp-1 %pi (raises/tm (tm/tmapp E1 E2) EX) %<- (raises/tm E1 EX) %. +%term raises/tm/tmapp-2 %pi (raises/tm (tm/tmapp E1 E2) EX) %<- (raises/tm E2 EX) %<- (val/tm E1) %. +%term raises/tm/cnapp %pi (raises/tm (tm/cnapp E1 C) EX) %<- (raises/tm E1 EX) %. +%term raises/tm/term %pi (raises/tm (tm/term M) EX) %<- (raises/md M EX) %. +%term raises/tm/ref %pi (raises/tm (tm/ref E) EX) %<- (raises/tm E EX) %. +%term raises/tm/set-1 %pi (raises/tm (tm/set E1 E2) EX) %<- (raises/tm E1 EX) %. +%term raises/tm/set-2 %pi (raises/tm (tm/set E1 E2) EX) %<- (raises/tm E2 EX) %<- (val/tm E1) %. +%term raises/tm/get %pi (raises/tm (tm/get E) EX) %<- (raises/tm E EX) %. +%term raises/tm/inl %pi (raises/tm (tm/inl (tp/sum C1 C2) E) EX) %<- (raises/tm E EX) %. +%term raises/tm/inr %pi (raises/tm (tm/inr (tp/sum C1 C2) E) EX) %<- (raises/tm E EX) %. +%term raises/tm/case %pi (raises/tm (tm/case E E1 E2) EX) %<- (raises/tm E EX) %. +%term raises/tm/raise-1 %pi (raises/tm (tm/raise E) E) %<- (val/tm E) %. +%term raises/tm/raise-2 %pi (raises/tm (tm/raise E) V) %<- (raises/tm E V) %. +%term raises/tm/tag-1 %pi (raises/tm (tm/tag E1 E2) EX) %<- (raises/tm E1 EX) %. +%term raises/tm/tag-2 %pi (raises/tm (tm/tag E1 E2) EX) %<- (raises/tm E2 EX) %<- (val/tm E1) %. +%term raises/tm/iftag-1 %pi (raises/tm (tm/iftag E E1 E2 E3) EX) %<- (raises/tm E EX) %. +%term raises/tm/iftag-2 %pi (raises/tm (tm/iftag E E1 E2 E3) EX) %<- (raises/tm E1 EX) %<- (val/tm E) %. +%term raises/tm/roll %pi (raises/tm (tm/roll (cn/mu K1 C2) E) V) %<- (raises/tm E V) %. +%term raises/tm/unroll %pi (raises/tm (tm/unroll E) V) %<- (raises/tm E V) %. +%term raises/md/pair-1 %pi (raises/md (md/pair M1 M2) M1') %<- (raises/md M1 M1') %. +%term raises/md/pair-2 %pi (raises/md (md/pair M1 M2) M2') %<- (raises/md M2 M2') %<- (val/md M1) %. +%term raises/md/pj1 %pi (raises/md (md/pj1 M) M') %<- (raises/md M M') %. +%term raises/md/pj2 %pi (raises/md (md/pj2 M) M') %<- (raises/md M M') %. +%term raises/md/app-1 %pi (raises/md (md/app M1 M2) M1') %<- (raises/md M1 M1') %. +%term raises/md/app-2 %pi (raises/md (md/app M1 M2) M2') %<- (raises/md M2 M2') %<- (val/md M1) %. +%term raises/md/tm %pi (raises/md (md/tm E) E') %<- (raises/tm E E') %. +%term raises/md/seal %pi (raises/md (md/seal M S) M') %<- (raises/md M M') %. +%term raises/md/let %pi (raises/md (md/let M1 M2 S) M1') %<- (raises/md M1 M1') %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/reflexivity-kinds.lf b/new-tests/stelf-output/tslf/reflexivity-kinds.lf new file mode 100644 index 0000000..49fbeea --- /dev/null +++ b/new-tests/stelf-output/tslf/reflexivity-kinds.lf @@ -0,0 +1,31 @@ +%%%% reflexivity of kinds +%sort kd-refl/deq %. +%mode kd-refl/deq %in %out %. +%term _ kd-refl/deq kd-wf/kd/unit kd-deq/kd/unit %. +%term _ kd-refl/deq kd-wf/kd/type kd-deq/kd/type %. +%term _ kd-refl/deq (kd-wf/kd/sing D1) (kd-deq/kd/sing (cn-deq/refl D1)) %. +%term _ + %pi (kd-refl/deq (kd-wf/kd/sgm D1 D2) (kd-deq/kd/sgm D3 D4)) + %<- (kd-refl/deq D1 D3) + %<- ({a cn} {da ofkd a _} kd-refl/deq (D2 a da) (D4 a da)) %. +%term _ + %pi (kd-refl/deq (kd-wf/kd/pi D1 D2) (kd-deq/kd/pi D3 D4)) + %<- (kd-refl/deq D1 D3) + %<- ({a cn} {da ofkd a _} kd-refl/deq (D2 a da) (D4 a da)) %. +%worlds (ofkd-block) (kd-refl/deq _ _) %. +%total (D1) (kd-refl/deq D1 _) %. +%sort kd-refl/sub %. +%mode kd-refl/sub %in %out %. +%term _ kd-refl/sub kd-wf/kd/unit kd-sub/kd/unit %. +%term _ kd-refl/sub kd-wf/kd/type kd-sub/kd/type %. +%term _ kd-refl/sub (kd-wf/kd/sing D1) (kd-sub/kd/sing-kd/sing (cn-deq/refl D1)) %. +%term _ + %pi (kd-refl/sub (kd-wf/kd/sgm D1 D2) (kd-sub/kd/sgm D3 D4 D2)) + %<- (kd-refl/sub D1 D3) + %<- ({a cn} {da ofkd a _} kd-refl/sub (D2 a da) (D4 a da)) %. +%term _ + %pi (kd-refl/sub (kd-wf/kd/pi D1 D2) (kd-sub/kd/pi D3 D4 D2)) + %<- (kd-refl/sub D1 D3) + %<- ({a cn} {da ofkd a _} kd-refl/sub (D2 a da) (D4 a da)) %. +%worlds (ofkd-block can-mofkd-block ofkd+vdt-block) (kd-refl/sub _ _) %. +%total (D1) (kd-refl/sub D1 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/reflexivity-sigs.lf b/new-tests/stelf-output/tslf/reflexivity-sigs.lf new file mode 100644 index 0000000..10f1ed9 --- /dev/null +++ b/new-tests/stelf-output/tslf/reflexivity-sigs.lf @@ -0,0 +1,31 @@ +%%%% reflexivity of kinds +%sort sg-refl/deq %. +%mode sg-refl/deq %in %out %. +%term _ sg-refl/deq sg-wf/sg/unit sg-deq/sg/unit %. +%term _ %pi (sg-refl/deq (sg-wf/sg/kd DK) (sg-deq/sg/kd DKS)) %<- (kd-refl/deq DK DKS) %. +%term _ sg-refl/deq (sg-wf/sg/cn DC) (sg-deq/sg/cn (cn-deq/refl DC)) %. +%term _ + %pi (sg-refl/deq (sg-wf/sg/sgm D1 D2 DFS) (sg-deq/sg/sgm D3 D4 DFS)) + %<- (sg-refl/deq D1 D3) + %<- ({a cn} {da ofkd a _} sg-refl/deq (D2 a da) (D4 a da)) %. +%term _ + %pi (sg-refl/deq (sg-wf/sg/pi D1 D2 DFS) (sg-deq/sg/pi D3 D4 DFS)) + %<- (sg-refl/deq D1 D3) + %<- ({a cn} {da ofkd a _} sg-refl/deq (D2 a da) (D4 a da)) %. +%worlds (ofkd-block) (sg-refl/deq _ _) %. +%total (D1) (sg-refl/deq D1 _) %. +%sort sg-refl/sub %. +%mode sg-refl/sub %in %out %. +%term _ sg-refl/sub sg-wf/sg/unit sg-sub/sg/unit %. +%term _ %pi (sg-refl/sub (sg-wf/sg/kd DK) (sg-sub/sg/kd DKS)) %<- (kd-refl/sub DK DKS) %. +%term _ sg-refl/sub (sg-wf/sg/cn DC) (sg-sub/sg/cn (cn-deq/refl DC)) %. +%term _ + %pi (sg-refl/sub (sg-wf/sg/sgm D1 D2 DFS) (sg-sub/sg/sgm D3 D4 D2 DFS DFS)) + %<- (sg-refl/sub D1 D3) + %<- ({a cn} {da ofkd a _} sg-refl/sub (D2 a da) (D4 a da)) %. +%term _ + %pi (sg-refl/sub (sg-wf/sg/pi D1 D2 DFS) (sg-sub/sg/pi D3 D4 DFS)) + %<- (sg-refl/deq D1 D3) + %<- ({a cn} {da ofkd a _} sg-refl/deq (D2 a da) (D4 a da)) %. +%worlds (ofkd-block) (sg-refl/sub _ _) %. +%total (D1) (sg-refl/sub D1 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/seq-lemmas-cnkd.lf b/new-tests/stelf-output/tslf/seq-lemmas-cnkd.lf new file mode 100644 index 0000000..f9ce9a6 --- /dev/null +++ b/new-tests/stelf-output/tslf/seq-lemmas-cnkd.lf @@ -0,0 +1,110 @@ +%sort seq/cn/sym %. +%mode seq/cn/sym %in %out %. +%term _ seq/cn/sym _ seq/cn/refl %. +%worlds (cn-block) (seq/cn/sym _ _) %. +%total {} (seq/cn/sym _ _) %. +%sort seq/cn/pair %. +%mode seq/cn/pair %in %in %out %. +%term _ seq/cn/pair _ _ seq/cn/refl %. +%worlds (cn-block) (seq/cn/pair _ _ _) %. +%total {} (seq/cn/pair _ _ _) %. +%sort seq/cn/pj1 %. +%mode seq/cn/pj1 %in %out %. +%term _ seq/cn/pj1 _ seq/cn/refl %. +%worlds (cn-block) (seq/cn/pj1 _ _) %. +%total {} (seq/cn/pj1 _ _) %. +%sort seq/cn/pj2 %. +%mode seq/cn/pj2 %in %out %. +%term _ seq/cn/pj2 _ seq/cn/refl %. +%worlds (cn-block) (seq/cn/pj2 _ _) %. +%total {} (seq/cn/pj2 _ _) %. +%sort seq/cn/tag %. +%mode seq/cn/tag %in %out %. +%term _ seq/cn/tag _ seq/cn/refl %. +%worlds (cn-block) (seq/cn/tag _ _) %. +%total {} (seq/cn/tag _ _) %. +%sort ofkd/seq-k %. +%mode ofkd/seq-k %in %in %out %. +%term _ ofkd/seq-k seq/kd/refl D1 D1 %. +%worlds (cn-block ofkd-block) (ofkd/seq-k _ _ _) %. +%total {} (ofkd/seq-k _ _ _) %. +%sort ofkd/seq-k-pi-l %. +%mode ofkd/seq-k-pi-l %in %in %out %. +%term _ ofkd/seq-k-pi-l seq/kd/refl D1 D1 %. +%worlds (cn-block ofkd-block) (ofkd/seq-k-pi-l _ _ _) %. +%total {} (ofkd/seq-k-pi-l _ _ _) %. +%sort ofkd/seq-c %. +%mode ofkd/seq-c %in %in %out %. +%term _ ofkd/seq-c seq/cn/refl D1 D1 %. +%worlds (cn-block ofkd-block) (ofkd/seq-c _ _ _) %. +%total {} (ofkd/seq-c _ _ _) %. +%sort kd-wf/seq %. +%mode kd-wf/seq %in %in %out %. +%term _ kd-wf/seq seq/kd/refl D1 D1 %. +%worlds (cn-block ofkd-block) (kd-wf/seq _ _ _) %. +%total {} (kd-wf/seq _ _ _) %. +%sort seq/kd/sgm %. +%mode seq/kd/sgm %in %in %out %. +%term _ seq/kd/sgm _ _ seq/kd/refl %. +%worlds (cn-block cn-block) (seq/kd/sgm _ _ _) %. +%total {} (seq/kd/sgm _ _ _) %. +%sort kd-wf/seq-a %. +%mode kd-wf/seq-a %in %in %out %. +%term _ kd-wf/seq-a seq/kd/refl D1 D1 %. +%worlds (cn-block ofkd+vdt-block ofkd-block) (kd-wf/seq-a _ _ _) %. +%total {} (kd-wf/seq-a _ _ _) %. +%sort kd-sub/seq-l %. +%mode kd-sub/seq-l %in %in %out %. +%term _ kd-sub/seq-l seq/kd/refl D1 D1 %. +%worlds (cn-block ofkd-block) (kd-sub/seq-l _ _ _) %. +%total {} (kd-sub/seq-l _ _ _) %. +%sort kd-sub/seq-r %. +%mode kd-sub/seq-r %in %in %out %. +%term _ kd-sub/seq-r seq/kd/refl D1 D1 %. +%worlds (cn-block ofkd-block) (kd-sub/seq-r _ _ _) %. +%total {} (kd-sub/seq-r _ _ _) %. +%sort kd-sub/seq-a %. +%mode kd-sub/seq-a %in %in %out %. +%term _ kd-sub/seq-a seq/kd/refl D1 D1 %. +%worlds (cn-block ofkd-block) (kd-sub/seq-a _ _ _) %. +%total {} (kd-sub/seq-a _ _ _) %. +%sort kd-deq/seq-l %. +%mode kd-deq/seq-l %in %in %out %. +%term _ kd-deq/seq-l seq/kd/refl D1 D1 %. +%worlds (cn-block ofkd-block) (kd-deq/seq-l _ _ _) %. +%total {} (kd-deq/seq-l _ _ _) %. +%sort kd-deq/seq-r %. +%mode kd-deq/seq-r %in %in %out %. +%term _ kd-deq/seq-r seq/kd/refl D1 D1 %. +%worlds (cn-block ofkd-block) (kd-deq/seq-r _ _ _) %. +%total {} (kd-deq/seq-r _ _ _) %. +%sort kd-deq/seq-a %. +%mode kd-deq/seq-a %in %in %out %. +%term _ kd-deq/seq-a seq/kd/refl D1 D1 %. +%worlds (cn-block ofkd-block) (kd-deq/seq-a _ _ _) %. +%total {} (kd-deq/seq-a _ _ _) %. +%sort seq/cn/beta-pj1 %. +%mode seq/cn/beta-pj1 %in %in %in %out %. +%term _ seq/cn/beta-pj1 seq/cn/refl seq/cn/refl D1 D1 %. +%worlds () (seq/cn/beta-pj1 _ _ _ _) %. +%total {} (seq/cn/beta-pj1 _ _ _ _) %. +%sort seq/cn/beta-pj2 %. +%mode seq/cn/beta-pj2 %in %in %in %out %. +%term _ seq/cn/beta-pj2 seq/cn/refl seq/cn/refl D1 D1 %. +%worlds () (seq/cn/beta-pj2 _ _ _ _) %. +%total {} (seq/cn/beta-pj2 _ _ _ _) %. +%sort cn-deq/seq-l %. +%mode cn-deq/seq-l %in %in %out %. +%term _ cn-deq/seq-l seq/cn/refl D1 D1 %. +%worlds (cn-block ofkd-block) (cn-deq/seq-l _ _ _) %. +%total {} (cn-deq/seq-l _ _ _) %. +%sort cn-deq/seq-r %. +%mode cn-deq/seq-r %in %in %out %. +%term _ cn-deq/seq-r seq/cn/refl D1 D1 %. +%worlds (cn-block ofkd-block) (cn-deq/seq-r _ _ _) %. +%total {} (cn-deq/seq-r _ _ _) %. +%sort cn-deq/seq-k %. +%mode cn-deq/seq-k %in %in %out %. +%term _ cn-deq/seq-k seq/kd/refl D1 D1 %. +%worlds (cn-block ofkd-block) (cn-deq/seq-k _ _ _) %. +%total {} (cn-deq/seq-k _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/seq-lemmas-dyn.lf b/new-tests/stelf-output/tslf/seq-lemmas-dyn.lf new file mode 100644 index 0000000..7800ed8 --- /dev/null +++ b/new-tests/stelf-output/tslf/seq-lemmas-dyn.lf @@ -0,0 +1,23 @@ +%%%% lemmas about syntactic equality +%%%% this file could benefit from better organization. perhaps splitting it +%%%% into different files, etc.. +%sort seq/val/tm %. +%mode seq/val/tm %in %in %out %. +%term _ seq/val/tm seq/tm/refl D1 D1 %. +%worlds (ofkd+vdt-block) (seq/val/tm _ _ _) %. +%total {} (seq/val/tm _ _ _) %. +%sort seq/val/md %. +%mode seq/val/md %in %in %out %. +%term _ seq/val/md seq/md/refl D1 D1 %. +%worlds (ofkd+vdt-block) (seq/val/md _ _ _) %. +%total {} (seq/val/md _ _ _) %. +%sort step/tm/seq %. +%mode step/tm/seq %in %in %out %. +%term _ step/tm/seq seq/tm/refl S1 S1 %. +%worlds () (step/tm/seq _ _ _) %. +%total {} (step/tm/seq _ _ _) %. +%sort step/md/seq %. +%mode step/md/seq %in %in %out %. +%term _ step/md/seq seq/md/refl S1 S1 %. +%worlds () (step/md/seq _ _ _) %. +%total {} (step/md/seq _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/seq-lemmas-sigs.lf b/new-tests/stelf-output/tslf/seq-lemmas-sigs.lf new file mode 100644 index 0000000..a6f52c8 --- /dev/null +++ b/new-tests/stelf-output/tslf/seq-lemmas-sigs.lf @@ -0,0 +1,30 @@ +%sort sg-wf/seq %. +%mode sg-wf/seq %in %in %out %. +%term _ sg-wf/seq seq/cn/refl D1 D1 %. +%worlds (ofkd+vdt-block) (sg-wf/seq _ _ _) %. +%total {} (sg-wf/seq _ _ _) %. +%sort fst-sg/seq %. +%mode fst-sg/seq %in %in %out %. +%term _ fst-sg/seq seq/cn/refl D1 D1 %. +%worlds (ofkd+vdt-block) (fst-sg/seq _ _ _) %. +%total {} (fst-sg/seq _ _ _) %. +%sort sg-sub/seq-a %. +%mode sg-sub/seq-a %in %in %out %. +%term _ sg-sub/seq-a seq/kd/refl D1 D1 %. +%worlds (ofkd+vdt-block) (sg-sub/seq-a _ _ _) %. +%total {} (sg-sub/seq-a _ _ _) %. +%sort sg-wf/seq-a %. +%mode sg-wf/seq-a %in %in %out %. +%term _ sg-wf/seq-a seq/kd/refl D1 D1 %. +%worlds (ofkd+vdt-block) (sg-wf/seq-a _ _ _) %. +%total {} (sg-wf/seq-a _ _ _) %. +%sort seq/cn/sg-sub-o {S1 %pi cn %-> sg} %. +%mode seq/cn/sg-sub-o %in %in %in %out %. +%term _ seq/cn/sg-sub-o _ seq/cn/refl D1 D1 %. +%worlds (ofkd+vdt-block) (seq/cn/sg-sub-o _ _ _ _) %. +%total {} (seq/cn/sg-sub-o _ _ _ _) %. +%sort seq/cn/sg-sub-ol {S1 %pi cn %-> sg} %. +%mode seq/cn/sg-sub-ol %in %in %in %out %. +%term _ seq/cn/sg-sub-ol _ seq/cn/refl D1 D1 %. +%worlds () (seq/cn/sg-sub-ol _ _ _ _) %. +%total {} (seq/cn/sg-sub-ol _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/seq-lemmas-tmmd.lf b/new-tests/stelf-output/tslf/seq-lemmas-tmmd.lf new file mode 100644 index 0000000..1343c93 --- /dev/null +++ b/new-tests/stelf-output/tslf/seq-lemmas-tmmd.lf @@ -0,0 +1,83 @@ +%%%% lemmas about syntactic equality +%%%% this file could benefit from better organization. perhaps splitting it +%%%% into different files, etc.. +%sort seq/tm/pj1 %. +%mode seq/tm/pj1 %in %out %. +%term _ seq/tm/pj1 _ seq/tm/refl %. +%worlds (ofkd+vdt-block) (seq/tm/pj1 _ _) %. +%total {} (seq/tm/pj1 _ _) %. +%sort seq/tm/pj2 %. +%mode seq/tm/pj2 %in %out %. +%term _ seq/tm/pj2 _ seq/tm/refl %. +%worlds (ofkd+vdt-block) (seq/tm/pj2 _ _) %. +%total {} (seq/tm/pj2 _ _) %. +%sort seq/tm/tmapp {E3 tm} %. +%mode seq/tm/tmapp %in %in %out %. +%term _ seq/tm/tmapp _ _ seq/tm/refl %. +%worlds (ofkd+vdt-block) (seq/tm/tmapp _ _ _) %. +%total {} (seq/tm/tmapp _ _ _) %. +%sort seq/tm/tag {E3 tm} %. +%mode seq/tm/tag %in %in %out %. +%term _ seq/tm/tag _ _ seq/tm/refl %. +%worlds (ofkd+vdt-block) (seq/tm/tag _ _ _) %. +%total {} (seq/tm/tag _ _ _) %. +%sort seq/tm/cnapp {E3 cn} %. +%mode seq/tm/cnapp %in %in %out %. +%term _ seq/tm/cnapp _ _ seq/tm/refl %. +%worlds (ofkd+vdt-block) (seq/tm/cnapp _ _ _) %. +%total {} (seq/tm/cnapp _ _ _) %. +%sort seq/tm/term %. +%mode seq/tm/term %in %out %. +%term _ seq/tm/term _ seq/tm/refl %. +%worlds (ofkd+vdt-block) (seq/tm/term _ _) %. +%total {} (seq/tm/term _ _) %. +%sort seq/tm/unroll %. +%mode seq/tm/unroll %in %out %. +%term _ seq/tm/unroll _ seq/tm/refl %. +%worlds (ofkd+vdt-block) (seq/tm/unroll _ _) %. +%total {} (seq/tm/unroll _ _) %. +%sort seq/tm/sym %. +%mode seq/tm/sym %in %out %. +%term _ seq/tm/sym _ seq/tm/refl %. +%worlds (ofkd+vdt-block) (seq/tm/sym _ _) %. +%total {} (seq/tm/sym _ _) %. +%sort seq/tm/term-a %. +%mode seq/tm/term-a %in %out %. +%term _ seq/tm/term-a _ seq/md/refl %. +%worlds (ofkd+vdt-block ofsg+vdt-block) (seq/tm/term-a _ _) %. +%total {} (seq/tm/term-a _ _) %. +%sort seq/md/app {M3 md} %. +%mode seq/md/app %in %in %out %. +%term _ seq/md/app _ _ seq/md/refl %. +%worlds (ofkd+vdt-block) (seq/md/app _ _ _) %. +%total {} (seq/md/app _ _ _) %. +%sort fst-md/seq-m %. +%mode fst-md/seq-m %in %in %out %. +%term _ fst-md/seq-m seq/md/refl D1 D1 %. +%worlds (ofkd+vdt-block ofsg+vdt-block) (fst-md/seq-m _ _ _) %. +%total {} (fst-md/seq-m _ _ _) %. +%sort seq/md/pj1 %. +%mode seq/md/pj1 %in %out %. +%term _ seq/md/pj1 _ seq/md/refl %. +%worlds (ofkd+vdt-block) (seq/md/pj1 _ _) %. +%total {} (seq/md/pj1 _ _) %. +%sort seq/md/pj2 %. +%mode seq/md/pj2 %in %out %. +%term _ seq/md/pj2 _ seq/md/refl %. +%worlds (ofkd+vdt-block) (seq/md/pj2 _ _) %. +%total {} (seq/md/pj2 _ _) %. +%sort oftp/seq-t %. +%mode oftp/seq-t %in %in %out %. +%term _ oftp/seq-t seq/tm/refl D1 D1 %. +%worlds (ofkd+vdt-block ofsg+vdt-block) (oftp/seq-t _ _ _) %. +%total {} (oftp/seq-t _ _ _) %. +%sort seq/tm/get %. +%mode seq/tm/get %in %out %. +%term _ seq/tm/get _ seq/tm/refl %. +%worlds (ofkd+vdt-block) (seq/tm/get _ _) %. +%total {} (seq/tm/get _ _) %. +%sort seq/tm/set {E3 tm} %. +%mode seq/tm/set %in %in %out %. +%term _ seq/tm/set _ _ seq/tm/refl %. +%worlds (ofkd+vdt-block) (seq/tm/set _ _ _) %. +%total {} (seq/tm/set _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/algorithm.lf b/new-tests/stelf-output/tslf/sing/algorithm.lf new file mode 100644 index 0000000..a656767 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/algorithm.lf @@ -0,0 +1,146 @@ +%%%%% Variable Identification %%%%% +%sort variable %. +%mode variable %in %. +%block evvar [x eterm] [dv variable x]%. +%block evbind {a etp} [x eterm] [d evof x a] [dv variable x]%. +%block evblock {x eterm} [dv variable x]%. +%%%%% Beta Normalization %%%%% +%sort oa %. +%% outermost atomic +%mode oa %in %. +%term oa/const oa (econst C) %. +%term oa/forall oa (eforall A) %. +%term oa/var %pi (oa X) %<- (variable X) %. +%term oa/app oa (eapp M N) %. +%term oa/pi1 oa (epi1 M) %. +%term oa/pi2 oa (epi2 M) %. +%sort reduce %. +%mode reduce %in %out %. +%sort treduce %. +%mode treduce %in %out %. +%sort reduce-app %. +%mode reduce-app %in %in %out %. +%sort reduce-pi1 %. +%mode reduce-pi1 %in %out %. +%sort reduce-pi2 %. +%mode reduce-pi2 %in %out %. +%term reduce/const reduce (econst C) (econst C) %. +%term reduce/forall %pi (reduce (eforall A) (eforall A')) %<- (treduce A A') %. +%term reduce/var %pi (reduce X X) %<- (variable X) %. +%term reduce/lam + %pi (reduce (elam A M) (elam A' M')) + %<- (treduce A A') + %<- ({x} %pi (variable x) %-> (reduce (M x) (M' x))) %. +%term reduce/app + %pi (reduce (eapp M N) O) + %<- (reduce M M') + %<- (reduce N N') + %<- (reduce-app M' N' O) %. +%term reduce-app/atom %pi (reduce-app M N (eapp M N)) %<- (oa M) %. +%term reduce-app/beta %pi (reduce-app (elam A M) N O) %<- (reduce (M N) O) %. +%term reduce/pair %pi (reduce (epair M N) (epair M' N')) %<- (reduce M M') %<- (reduce N N') %. +%term reduce/pi1 %pi (reduce (epi1 M) N) %<- (reduce M M') %<- (reduce-pi1 M' N) %. +%term reduce-pi1/atom %pi (reduce-pi1 M (epi1 M)) %<- (oa M) %. +%term reduce-pi1/beta reduce-pi1 (epair M N) M %. +%term reduce/pi2 %pi (reduce (epi2 M) N) %<- (reduce M M') %<- (reduce-pi2 M' N) %. +%term reduce-pi2/atom %pi (reduce-pi2 M (epi2 M)) %<- (oa M) %. +%term reduce-pi2/beta reduce-pi2 (epair M N) N %. +%term treduce/t treduce et et %. +%term treduce/pi + %pi (treduce (epi A B) (epi A' B')) + %<- (treduce A A') + %<- ({x} %pi (variable x) %-> (treduce (B x) (B' x))) %. +%term treduce/sigma + %pi (treduce (esigma A B) (esigma A' B')) + %<- (treduce A A') + %<- ({x} %pi (variable x) %-> (treduce (B x) (B' x))) %. +%term treduce/sing %pi (treduce (esing M) (esing M')) %<- (reduce M M') %. +%%%%% Eta Expansion %%%%% +%sort selfify %. +%mode selfify %in %in %out %out %. +%term selfify/t selfify M et M (esing M) %. +%term selfify/pi + %pi (selfify M (epi A B) (elam A O) (epi A Bs)) + %<- ({x} selfify x A (N x) _) + %<- ({x} selfify (eapp M (N x)) (B x) (O x) (Bs x)) %. +%term selfify/sigma + %pi (selfify M (esigma A B) (epair N O) (esigma As ([_] Bs))) + %<- (selfify (epi1 M) A N As) + %<- (selfify (epi2 M) (B (epi1 M)) O Bs) %. +%term selfify/sing selfify M (esing N) N (esing N) %. +%%%%% Coercion %%%%% +%sort coerce %. +%mode coerce %in %in %out %. +%term coerce/t coerce et et ([x] x) %. +%term coerce/pi + %pi (coerce (epi A1 A2) (epi B1 B2) ([f] elam B1 ([x] M2 x (eapp f (M1 x))))) + %<- (coerce B1 A1 ([x] M1 x)) + %<- ({x} %pi (variable x) %-> (treduce (A2 (M1 x)) (A2' x))) + %<- ({x} %pi (variable x) %-> (coerce (A2' x) (B2 x) ([y] M2 x y))) %. +%term coerce/sigma + %pi (coerce (esigma A1 A2) (esigma B1 B2) ([p] epair (M1 (epi1 p)) (M2 (epi1 p) (epi2 p)))) + %<- (coerce A1 B1 ([x] M1 x)) + %<- ({x} %pi (variable x) %-> (treduce (B2 (M1 x)) (B2' x))) + %<- ({x} %pi (variable x) %-> (coerce (A2 x) (B2' x) ([y] M2 x y))) %. +%term coerce/sing_t coerce (esing M) et ([x] M) %. +%term coerce/sing coerce (esing M) (esing M) ([x] M) %. +%%%%% Canonization %%%%% +%sort canonize %. +%sort tcanonize %. +%mode ekof %in %out %. +%mode evof %in %out %. +%mode canonize %in %out %. +%mode tcanonize %in %out %. +%term canonize/const %pi (canonize (econst K) As) %<- (ekof K A) %<- (selfify (econst K) A M As) %. +%term canonize/forall + %pi (canonize (eforall A) B) + %<- (tcanonize A A') + %<- (selfify (eforall A') (qetp A') M B) %. +%term canonize/var %pi (canonize X As) %<- (evof X A) %<- (selfify X A M As) %. +%term canonize/app + %pi (canonize (eapp M N) D) + %<- (canonize M (epi A B)) + %<- (canonize N C) + %<- (coerce C A ([_] N')) + %<- (treduce (B N') D) %. +%term canonize/pi1 %pi (canonize (epi1 M) A) %<- (canonize M (esigma A ([_] B))) %. +%term canonize/pi2 %pi (canonize (epi2 M) B) %<- (canonize M (esigma A ([_] B))) %. +%term canonize/lam + %pi (canonize (elam A M) (epi A' B)) + %<- (tcanonize A A') + %<- ({x} %pi (evof x A') %-> (variable x) %-> (canonize (M x) (B x))) %. +%term canonize/pair + %pi (canonize (epair M N) (esigma A ([_] B))) + %<- (canonize M A) + %<- (canonize N B) %. +%term tcanonize/t tcanonize et et %. +%term tcanonize/pi + %pi (tcanonize (epi A B) (epi A' B')) + %<- (tcanonize A A') + %<- ({x} %pi (evof x A') %-> (variable x) %-> (tcanonize (B x) (B' x))) %. +%term tcanonize/sigma + %pi (tcanonize (esigma A B) (esigma A' B')) + %<- (tcanonize A A') + %<- ({x} %pi (evof x A') %-> (variable x) %-> (tcanonize (B x) (B' x))) %. +%term tcanonize/sing %pi (tcanonize (esing M) (esing M')) %<- (canonize M (esing M')) %. +%%%%% Entry Points %%%%% +%sort check-eof %. +%term check-eof/i %pi (check-eof M A) %<- (canonize M B) %<- (tcanonize A A') %<- (coerce B A' _) %. +%sort check-equiv %. +%term check-equiv/i + %pi (check-equiv M N C) + %<- (canonize M A) + %<- (canonize N B) + %<- (tcanonize C C') + %<- (coerce A C' ([_] O)) + %<- (coerce B C' ([_] O)) %. +%sort check-ewf %. +%term check-ewf/i %pi (check-ewf A) %<- (tcanonize A _) %. +%sort check-subtp %. +%term check-subtp/i + %pi (check-subtp A B) + %<- (tcanonize A A') + %<- (tcanonize B B') + %<- (coerce A' B' _) %. +%sort check-tequiv %. +%term check-tequiv/i %pi (check-tequiv A B) %<- (tcanonize A C) %<- (tcanonize B C) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/complete-thm.lf b/new-tests/stelf-output/tslf/sing/complete-thm.lf new file mode 100644 index 0000000..443db0e --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/complete-thm.lf @@ -0,0 +1,703 @@ +%%%%% Completeness %%%%% +%sort etopen-fun %. +%mode etopen-fun %in %in %out %. +%term _ etopen-fun etopen/t etopen/t etp-eq/i %. +%term _ + %pi (etopen-fun (etopen/pi D2 D1) (etopen/pi D2' D1') Deq) + %<- (etopen-fun D1 D1' Deq1) + %<- (etopen-fun D2 D2' Deq2) + %<- (epi-resp Deq1 ([_] Deq2) Deq) %. +%worlds (evar) (etopen-fun _ _ _) %. +%total D (etopen-fun D _ _) %. +%sort trans-topen %. +%mode trans-topen %in %out %out %. +%scope trans-topen %term t trans-topen etopen/t ttrans/t topen/t %. +%term pi + %pi (trans-topen (etopen/pi Dopen2 Dopen1) (ttrans/pi ([_] [_] [_] [_] Dtrans2) Dtrans1) (topen/pi Dopen2' Dopen1')) + %<- (trans-topen Dopen1 Dtrans1 Dopen1') + %<- (trans-topen Dopen2 Dtrans2 Dopen2') %. +%worlds (bind tbind) (trans-topen _ _ _) %. +%total D (trans-topen D _ _) %. +%sort kof-comp %. +%mode kof-comp %in %in %out %. +%term _ + %pi (kof-comp (ekof/i (%the (etopen As EA) Deopen) (%the (ckof K As) Dckof)) (%the (ttrans EA A) Dtrans) (kof/i Dopen' Dckof)) + %<- (trans-topen Deopen Dtrans' Dopen) + %<- (ttrans-fun Dtrans' Dtrans Deq) + %<- (topen-resp ctp-eq/i Deq Dopen Dopen') %. +%worlds (bind tbind) (kof-comp _ _ _) %. +%total {} (kof-comp _ _ _) %. +%sort vof-comp %. +%mode vof-comp %in %out %out %out %. +%sort of-comp %. +%mode of-comp %in %out %out %out %. +%sort wf-comp %. +%mode wf-comp %in %out %. +%sort subtp-comp %. +%mode subtp-comp %in %out %out %out %. +%sort tequiv-comp %. +%mode tequiv-comp %in %out %out %. +%sort equiv-comp %. +%mode equiv-comp %in %out %out %out %out %out %. +%block cbind {x atom} {a tp} {ea etp} {d_ttrans ttrans ea a} {d_vof vof x a} [ex eterm] [xt vtrans ex x] [ed evof ex ea] [thm1 vof-comp ed d_ttrans xt d_vof]%. +%% combine sbind & cbind +%block scbind {a tp} {ea etp} {d_tconvert tconvert a ea} {d_ttrans ttrans ea a} {d_wf wf a} [x atom] [d vof x a] [ex eterm] [xt vtrans ex x] [ed evof ex ea] [dv variable ex] [thm1 vtrans-variable xt dv] [thm2 vsound d xt d_tconvert ed] [thm3 vof-comp ed d_ttrans xt d]%. +%scope equiv-comp %term var + %pi (of-comp (eof/var _ (%the (evof EX EA) Devof)) DtransA (trans/var Dself Dexpand DwfA Dvof DtransX) Dsubtype) + %<- (vof-comp Devof (%the (ttrans EA A) DtransA) (%the (vtrans EX X) DtransX) (%the (vof X A) Dvof)) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (can-expand X A M (%the (expand X A M) Dexpand)) + %<- (expand-reg (aof/var DwfA Dvof) Dexpand (%the (of M A) DofM)) + %<- (can-self DofM (%the (self M A B) Dself)) + %<- (self-reg Dself DofM (%the (of M B) DofM') (%the (subtype B A ([_] M)) Dsubtype)) %. +%term const + %pi (of-comp (eof/const (%the (ewf EA) Dewf) (%the (ekof C EA) Dekof)) DtransA (trans/const Dself Dexpand DwfA Dkof) Dsubtype) + %<- (wf-comp Dewf (%the (ttrans EA A) DtransA)) + %<- (kof-comp Dekof DtransA (%the (kof C A) Dkof)) + %<- (can-expand (const C) A M (%the (expand (const C) A M) Dexpand)) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (expand-reg (aof/const DwfA Dkof) Dexpand (%the (of M A) DofM)) + %<- (can-self DofM (%the (self M A A') Dself)) + %<- (self-reg Dself DofM _ (%the (subtype A' A ([_] M)) Dsubtype)) %. +%scope equiv-comp %term forall + %pi (of-comp (eof/forall (%the (ewf EA) Dewf)) (ttrans/pi ([_] [_] [_] [_] ttrans/t) (ttrans/pi ([_] [_] [_] [_] ttrans/t) DtransA)) (trans/forall Dself Dexpand DtransA) Dsubtype) + %<- (wf-comp Dewf (%the (ttrans EA A) DtransA)) + %<- (can-expand (forall A) (qtp A) M (%the (expand (forall A) (qtp A) M) Dexpand)) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (expand-reg (aof/forall DwfA) Dexpand (%the (of M (qtp A)) DofM)) + %<- (can-self DofM (%the (self M (qtp A) B) Dself)) + %<- (self-reg Dself DofM _ (%the (subtype B (qtp A) ([_] M)) Dsubtype)) %. +%term lam + %pi (of-comp (eof/lam (%the ({ex} %pi (evof ex EA) %-> (eof (EM ex) (EB ex))) Deof) (%the (ewf EA) Dewf)) (ttrans/pi DtransB DtransA) (trans/lam Dtrans DtransA) (subtype/pi DsubtypeB' Dtsub DsubtypeA)) + %<- (wf-comp Dewf (%the (ttrans EA A) DtransA)) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- ({x} {d vof x A} {ex} {tr vtrans ex x} {ed evof ex EA} + %pi (vof-comp ed DtransA tr d) + %-> (of-comp (Deof ex ed) (%the (ttrans (EB ex) (B x)) (DtransB x d ex tr)) (%the (trans (EM ex) (C x)) (Dtrans x d ex tr)) (%the (subtype (C x) (B x) _) (DsubtypeB' x)))) + %<- ({x} {d vof x A} {ex} {tr vtrans ex x} trans-reg (Dtrans x d ex tr) (%the (wf (C x)) (DwfC x d))) + %<- (subtype-refl DwfA (%the (subtype A A X) DsubtypeA) (%the ({x} expand x A (X x)) Dexpand)) + %<- (tsub-expand-var DwfA DwfC Dexpand (%the ({x} tsub ([x] C x) (X x) (C x)) Dtsub)) %. +%scope equiv-comp %term app + %pi (of-comp (eof/app (%the (eof EN EA) DeofN) (%the (eof EM (epi EA EB)) DeofM)) DtransBN (trans/app DsubDx DsubtypeFC DtransN DtransM) DsubtypeDxBx) + %<- (of-comp DeofM (ttrans/pi (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (ttrans (EB ex) (B x)))) DtransB) (%the (ttrans EA A) DtransA)) (%the (trans EM (pi C D)) DtransM) (subtype/pi (%the ({x} subtype (E x) (B x) _) DsubtypeEB) (%the ({x} tsub D (Lac x) (E x)) DsubE) (%the (subtype A C Lac) DsubtypeAC))) + %<- (of-comp DeofN (%the (ttrans EA A') DtransA') (%the (trans EN F) DtransN) (%the (subtype F A' N') DsubtypeFA')) + %<- (ttrans-fun DtransA' DtransA (%the (tp-eq A' A) DeqA)) + %<- (trans-principal DtransN (%the (principal F) DprincipalF)) + %<- (principal-subtype DprincipalF DsubtypeFA' (%the ({y} term-eq (N' y) N) DeqN')) + %<- (subtype-resp tp-eq/i DeqA DeqN' DsubtypeFA' (%the (subtype F A ([_] N)) DsubtypeFA)) + %<- (trans-reg DtransN (%the (wf F) DwfF)) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (trans-reg DtransM (wf/pi (%the ({x} %pi (vof x C) %-> (wf (D x))) DwfD) (%the (wf C) DwfC))) + %<- (subtype-reg DsubtypeAC DwfA DwfC (%the ({x} %pi (vof x A) %-> (of (Lac x) C)) DofLac)) + %<- (subtype-reg' DsubtypeFA DwfF DwfA (%the (of N A) DofN)) + %<- (can-sub DofLac DofN (%the (sub Lac N O) DsubO)) + %<- (subst' DsubO DofLac DofN (%the (of O C) DofO)) + %<- (subtype-trans DwfF DwfA DwfC DsubtypeFA DsubtypeAC ([_] DsubO) (%the (subtype F C ([_] O)) DsubtypeFC)) + %<- (can-tsub DwfD DofO (%the (tsub D O Dx) DsubDx)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} ttrans-reg (DtransB x d ex xt) (%the (wf (B x)) (DwfB x d))) + %<- (can-tsub DwfB DofN (%the (tsub B N Bx) DsubBx)) + %<- (ttrans-sub DtransB DtransN DwfA DsubtypeFA DsubBx (%the (ttrans (EB EN) Bx) DtransBN)) + %<- ({x} {d vof x A} tsubst (DsubE x) DwfD (DofLac x d) (%the (wf (E x)) (DwfE x d))) + %<- (can-tsub DwfE DofN (%the (tsub E N Ex) DsubEx)) + %<- (subtype-sub DsubtypeEB DwfE DwfB DofN DsubEx DsubBx _ (%the (subtype Ex Bx _) DsubtypeExBx)) + %<- ({y} tsub-absent (D y) N (%the (tsub ([x] D y) N (D y)) (DsubDabs y))) + %<- (tsub-permute DofN DofLac ([x] [d vof x A] [y] [e vof y C] DwfD y e) DsubDabs DsubO DsubDx DsubE DsubEx (%the (tp-eq Dx Ex) Deq)) + %<- (tp-eq-symm Deq (%the (tp-eq Ex Dx) Deq')) + %<- (subtype-resp Deq' tp-eq/i ([_] term-eq/i) DsubtypeExBx (%the (subtype Dx Bx _) DsubtypeDxBx)) %. +%term pair + %pi (of-comp (eof/pair (%the ({ex} %pi (evof ex EA) %-> (ewf (EB ex))) DewfB) (%the (eof EN (EB EM)) DeofN) (%the (eof EM EA) DeofM)) (ttrans/sigma DtransB DtransA) (trans/pair DtransN DtransM) (subtype/sigma ([_] DsubtypeDB) ([_] DsubBx) DsubtypeCA')) + %<- (of-comp DeofM (%the (ttrans EA A) DtransA) (%the (trans EM C) DtransM) (%the (subtype C A Lca) DsubtypeCA)) + %<- (of-comp DeofN (%the (ttrans (EB EM) E) DtransE) (%the (trans EN D) DtransN) (%the (subtype D E _) DsubtypeDE)) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} + %pi (vof-comp ed DtransA xt d) + %-> (wf-comp (DewfB ex ed) (%the (ttrans (EB ex) (B x)) (DtransB x d ex xt)))) + %<- (trans-principal DtransM (%the (principal C) DprincipalC)) + %<- (principal-subtype DprincipalC DsubtypeCA (%the ({x} term-eq (Lca x) M) DeqLca)) + %<- (subtype-resp tp-eq/i tp-eq/i DeqLca DsubtypeCA (%the (subtype C A ([_] M)) DsubtypeCA')) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} ttrans-reg (DtransB x d ex xt) (%the (wf (B x)) (DwfB x d))) + %<- (trans-reg DtransM (%the (wf C) DwfC)) + %<- (subtype-reg' DsubtypeCA' DwfC DwfA (%the (of M A) DofM)) + %<- (can-tsub DwfB DofM (%the (tsub B M Bx) DsubBx)) + %<- (ttrans-sub DtransB DtransM DwfA DsubtypeCA' DsubBx (%the (ttrans (EB EM) Bx) DtransBx)) + %<- (ttrans-fun DtransE DtransBx (%the (tp-eq E Bx) Deq)) + %<- (subtype-resp tp-eq/i Deq ([_] term-eq/i) DsubtypeDE (%the (subtype D Bx _) DsubtypeDB)) %. +%scope equiv-comp %term pi1 + %pi (of-comp (eof/pi1 (%the (eof EM (esigma EA EB)) Deof)) DtransA (trans/pi1 DtransM') DsubtypeCA) + %<- (of-comp Deof (ttrans/sigma (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (ttrans (EB ex) (B x)))) DtransB) (%the (ttrans EA A) DtransA)) (%the (trans EM CD) DtransM) (%the (subtype CD (sigma A B) _) DsubtypeCDAB)) + %<- (trans-principal DtransM (%the (principal CD) DprincipalCD)) + %<- (principal-subtype-sigma-invert DprincipalCD DsubtypeCDAB (%the (tp-eq CD (sigma C ([_] D))) DeqCD)) + %<- (trans-resp eterm-eq/i DeqCD DtransM (%the (trans EM (sigma C ([_] D))) DtransM')) + %<- (subtype-resp DeqCD tp-eq/i ([_] term-eq/i) DsubtypeCDAB (%the (subtype (sigma C ([_] D)) (sigma A B) _) DsubtypeCDAB')) + %<- (subtype-sigma-invert DsubtypeCDAB' (%the (subtype C A _) DsubtypeCA) _ _ _) %. +%term pi2 + %pi (of-comp (eof/pi2 (%the (eof EM (esigma EA EB)) Deof)) DtransBx (trans/pi2 DtransM') (DsubtypeDB aca)) + %<- (of-comp Deof (ttrans/sigma (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (ttrans (EB ex) (B x)))) DtransB) (%the (ttrans EA A) DtransA)) (%the (trans EM CD) DtransM) (%the (subtype CD (sigma A B) _) DsubtypeCDAB)) + %<- (trans-principal DtransM (%the (principal CD) DprincipalCD)) + %<- (principal-subtype-sigma-invert DprincipalCD DsubtypeCDAB (%the (tp-eq CD (sigma C ([_] D))) DeqCD)) + %<- (trans-resp eterm-eq/i DeqCD DtransM (%the (trans EM (sigma C ([_] D))) DtransM')) + %<- (subtype-resp DeqCD tp-eq/i ([_] term-eq/i) DsubtypeCDAB (%the (subtype (sigma C ([_] D)) (sigma A B) _) DsubtypeCDAB')) + %<- (subtype-sigma-invert DsubtypeCDAB' (%the (subtype C A Lca) DsubtypeCA) (%the ({x} tsub B (Lca x) (E x)) DsubE) (%the ({x} subtype D (E x) _) DsubtypeDE) _) + %<- (principal-resp DeqCD DprincipalCD (principal/sigma (%the (principal D) DprincipalD) (%the (principal C) DprincipalC))) + %<- (principal-subtype DprincipalC DsubtypeCA (%the ({x} term-eq (Lca x) M) DeqLca)) + %<- (subtype-resp tp-eq/i tp-eq/i DeqLca DsubtypeCA (%the (subtype C A ([_] M)) DsubtypeCA')) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (trans-reg DtransM' (wf/sigma _ (%the (wf C) DwfC))) + %<- (subtype-reg' DsubtypeCA' DwfC DwfA (%the (of M A) DofM)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} ttrans-reg (DtransB x d ex xt) (%the (wf (B x)) (DwfB x d))) + %<- (can-tsub DwfB DofM (%the (tsub B M Bx) DsubBx)) + %<- (ttrans-sub DtransB (trans/pi1 DtransM') DwfA DsubtypeCA' DsubBx (%the (ttrans (EB (epi1 EM)) Bx) DtransBx)) + %<- ({x} tsub-resp ([_] tp-eq/i) (DeqLca x) tp-eq/i (DsubE x) (%the (tsub B M (E x)) (DsubE' x))) + %<- ({x} tsub-fun (DsubE' x) DsubBx (%the (tp-eq (E x) Bx) (DeqE x))) + %<- ({x} subtype-resp tp-eq/i (DeqE x) ([_] term-eq/i) (DsubtypeDE x) (%the (subtype D Bx _) (DsubtypeDB x))) %. +%scope equiv-comp %term sing + %pi (of-comp (eof/sing (%the (eof EM et) Deof)) (ttrans/sing Dtrans') Dtrans' subtype/sing) + %<- (of-comp Deof ttrans/t (%the (trans EM A) Dtrans) (%the (subtype A t _) Dsubtype)) + %<- (trans-principal Dtrans (%the (principal A) Dprincipal)) + %<- (principal-subtype-t-invert Dprincipal Dsubtype (%the (tp-eq A (sing R)) Deq) _) + %<- (trans-resp eterm-eq/i Deq Dtrans (%the (trans EM (sing R)) Dtrans')) %. +%term extpi + %pi (of-comp (eof/extpi (%the ({ex} %pi (evof ex EA) %-> (eof (eapp EM ex) (EB ex))) Deof) (%the (eof EM (epi EA EBoth)) DeofM)) (ttrans/pi DtransB DtransA) DtransM (subtype/pi DsubtypeDB DsubDx DsubtypeAC)) + %<- (of-comp DeofM (ttrans/pi (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (ttrans (EBoth ex) (Both x)))) DtransH) (%the (ttrans EA A) DtransA)) (%the (trans EM (pi C D)) DtransM) (%the (subtype (pi C D) (pi A Both) _) (subtype/pi (%the ({x} subtype (Dx x) (Both x) _) DsubtypeDBoth) (%the ({x} tsub D (Lac x) (Dx x)) DsubDx) (%the (subtype A C Lac) DsubtypeAC)))) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} + %pi (vof-comp ed DtransA xt d) + %-> (of-comp (Deof ex ed) (%the (ttrans (EB ex) (B x)) (DtransB x d ex xt)) (%the (trans (eapp EM ex) (Fy x)) (trans/app (%the (tsub ([y] F x y) (M x) (Fy x)) (DsubFy x)) (%the (subtype (H x) (E x) ([_] M x)) (DsubtypeHE x)) (%the (trans ex (H x)) (DtransX x d ex xt)) (%the (trans EM (pi (E x) ([y] F x y))) (DtransM' x d ex xt)))) (%the (subtype (Fy x) (B x) _) (DsubtypeFB x)))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} trans-fun (DtransM' x d ex xt) DtransM (%the (tp-eq (pi (E x) ([y] F x y)) (pi C D)) (DeqEFCD x))) + %<- ({x} tp-eq-cdr-pi (DeqEFCD x) (%the (tp-eq (E x) C) (DeqEC x)) (%the ({y} tp-eq (F x y) (D y)) (DeqFD x))) + %<- ({x} can-expand x A (X x) (%the (expand x A (X x)) (Dexpand x))) + %<- ({x} {d vof x A} expand-reg (aof/var DwfA d) (Dexpand x) (%the (of (X x) A) (DofX x d))) + %<- ({x} {d vof x A} can-self (DofX x d) (%the (self (X x) A (As x)) (Dself x))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} trans-fun (DtransX x d ex xt) (trans/var (Dself x) (Dexpand x) DwfA d xt) (%the (tp-eq (H x) (As x)) (DeqHAs x))) + %<- ({x} subtype-resp (DeqHAs x) (DeqEC x) ([_] term-eq/i) (DsubtypeHE x) (%the (subtype (As x) C ([_] M x)) (DsubtypeAsC x))) + %<- ({x} {d vof x A} self-reg (Dself x) (DofX x d) (%the (of (X x) (As x)) (DofX' x d)) (%the (subtype (As x) A ([_] X x)) (DsubtypeAsA x))) + %<- (trans-reg DtransM (wf/pi (%the ({x} %pi (vof x C) %-> (wf (D x))) DwfD) (%the (wf C) DwfC))) + %<- (subtype-reg DsubtypeAC DwfA DwfC (%the ({x} %pi (vof x A) %-> (of (Lac x) C)) DofLac)) + %<- ({x} {d vof x A} can-sub DofLac (DofX x d) (%the (sub Lac (X x) (X' x)) (DsubX' x))) + %<- ({x} {d vof x A} sub-expand (aof/var DwfA d) DofLac (Dexpand x) (DsubX' x) (%the (term-eq (X' x) (Lac x)) (DeqX x))) + %<- ({x} sub-resp ([_] term-eq/i) term-eq/i (DeqX x) (DsubX' x) (%the (sub Lac (X x) (Lac x)) (DsubLac x))) + %<- ({x} {d vof x A} of-reg (DofX' x d) (%the (wf (As x)) (DwfAs x d))) + %<- ({x} {d vof x A} subtype-trans (DwfAs x d) DwfA DwfC (DsubtypeAsA x) DsubtypeAC ([_] DsubLac x) (%the (subtype (As x) C ([_] Lac x)) (DsubtypeAsC' x))) + %<- ({x} subtype-fun (DsubtypeAsC x) (DsubtypeAsC' x) (%the ({y} term-eq (M x) (Lac x)) (DeqM x))) + %<- ({x} tsub-resp ([y] DeqFD x y) (DeqM x aca) tp-eq/i (DsubFy x) (%the (tsub ([y] D y) (Lac x) (Fy x)) (DsubFy' x))) + %<- ({x} tsub-fun (DsubFy' x) (DsubDx x) (%the (tp-eq (Fy x) (Dx x)) (DeqDx x))) + %<- ({x} subtype-resp (DeqDx x) tp-eq/i ([_] term-eq/i) (DsubtypeFB x) (%the (subtype (Dx x) (B x) _) (DsubtypeDB x))) %. +%scope equiv-comp %term extsig + %pi (of-comp (eof/extsigma (%the ({ex} %pi (evof ex EA) %-> (ewf (EB ex))) DewfB) (%the (eof (epi2 EM) (EB (epi1 EM))) Deof2) (%the (eof (epi1 EM) EA) Deof1)) (ttrans/sigma DtransB DtransA) DtransM (subtype/sigma ([_] DsubtypeDB) ([_] DsubBx) DsubtypeCA')) + %<- (of-comp Deof1 (%the (ttrans EA A) DtransA) (trans/pi1 (%the (trans EM (sigma C ([_] D))) DtransM)) (%the (subtype C A Lca) DsubtypeCA)) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} + %pi (vof-comp ed DtransA xt d) + %-> (wf-comp (DewfB ex ed) (%the (ttrans (EB ex) (B x)) (DtransB x d ex xt)))) + %<- (trans-principal DtransM (principal/sigma (%the (principal D) DprincipalD) (%the (principal C) DprincipalC))) + %<- (principal-subtype DprincipalC DsubtypeCA (%the ({x} term-eq (Lca x) M) DeqLca)) + %<- (subtype-resp tp-eq/i tp-eq/i DeqLca DsubtypeCA (%the (subtype C A ([_] M)) DsubtypeCA')) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} ttrans-reg (DtransB x d ex xt) (%the (wf (B x)) (DwfB x d))) + %<- (trans-reg DtransM (wf/sigma _ (%the (wf C) DwfC))) + %<- (subtype-reg' DsubtypeCA' DwfC DwfA (%the (of M A) DofM)) + %<- (can-tsub DwfB DofM (%the (tsub B M Bx) DsubBx)) + %<- (ttrans-sub DtransB (trans/pi1 DtransM) DwfA DsubtypeCA' DsubBx (%the (ttrans (EB (epi1 EM)) Bx) DtransBx)) + %<- (of-comp Deof2 (%the (ttrans (EB (epi1 EM)) H) DtransH) (trans/pi2 (%the (trans EM (sigma E ([_] F))) DtransM')) (%the (subtype F H _) DsubtypeFH)) + %<- (ttrans-fun DtransH DtransBx (%the (tp-eq H Bx) DeqHB)) + %<- (trans-fun DtransM' DtransM (%the (tp-eq (sigma E ([_] F)) (sigma C ([_] D))) DeqEFCD)) + %<- (tp-eq-cdr-sigma DeqEFCD _ (%the ({x} tp-eq F D) DeqFD)) + %<- (subtype-resp (DeqFD aca) DeqHB ([_] term-eq/i) DsubtypeFH (%the (subtype D Bx _) DsubtypeDB)) %. +%term sub + %pi (of-comp (eof/subsume (%the (subtp EA EB) Dsubtp) (%the (eof EM EA) Deof)) DtransB Dtrans DsubtypeCB) + %<- (of-comp Deof (%the (ttrans EA A) DtransA) (%the (trans EM C) Dtrans) (%the (subtype C A _) DsubtypeCA)) + %<- (subtp-comp Dsubtp (%the (ttrans EA A') DtransA') (%the (ttrans EB B) DtransB) (%the (subtype A' B _) DsubtypeA'B)) + %<- (ttrans-fun DtransA' DtransA (%the (tp-eq A' A) Deq)) + %<- (subtype-resp Deq tp-eq/i ([_] term-eq/i) DsubtypeA'B (%the (subtype A B _) DsubtypeAB)) + %<- (trans-reg Dtrans (%the (wf C) DwfC)) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (ttrans-reg DtransB (%the (wf B) DwfB)) + %<- (subtype-trans' DwfC DwfA DwfB DsubtypeCA DsubtypeAB _ (%the (subtype C B _) DsubtypeCB)) %. +%%% +%scope equiv-comp %term t wf-comp ewf/t ttrans/t %. +%term pi + %pi (wf-comp (ewf/pi (%the ({ex} %pi (evof ex EA) %-> (ewf (EB ex))) DewfB) (%the (ewf EA) DewfA)) (ttrans/pi DtransB DtransA)) + %<- (wf-comp DewfA (%the (ttrans EA A) DtransA)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} + %pi (vof-comp ed DtransA xt d) + %-> (wf-comp (DewfB ex ed) (%the (ttrans (EB ex) (B x)) (DtransB x d ex xt)))) %. +%scope equiv-comp %term sigma + %pi (wf-comp (ewf/sigma (%the ({ex} %pi (evof ex EA) %-> (ewf (EB ex))) DewfB) (%the (ewf EA) DewfA)) (ttrans/sigma DtransB DtransA)) + %<- (wf-comp DewfA (%the (ttrans EA A) DtransA)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} + %pi (vof-comp ed DtransA xt d) + %-> (wf-comp (DewfB ex ed) (%the (ttrans (EB ex) (B x)) (DtransB x d ex xt)))) %. +%term sing + %pi (wf-comp (ewf/sing (%the (eof EM et) DofM)) (ttrans/sing Dtrans')) + %<- (of-comp DofM ttrans/t (%the (trans EM A) Dtrans) (%the (subtype A t _) Dsubtype)) + %<- (trans-principal Dtrans (%the (principal A) Dprincipal)) + %<- (principal-subtype-t-invert Dprincipal Dsubtype (%the (tp-eq A (sing R)) Deq) _) + %<- (trans-resp eterm-eq/i Deq Dtrans (%the (trans EM (sing R)) Dtrans')) %. +%%% +%scope equiv-comp %term reflex + %pi (subtp-comp (subtp/reflex (%the (tequiv EA EB) Dequiv)) DtransA DtransB Dsubtype) + %<- (tequiv-comp Dequiv (%the (ttrans EA A) DtransA) (%the (ttrans EB A) DtransB)) + %<- (ttrans-reg DtransA (%the (wf A) Dwf)) + %<- (subtype-refl Dwf (%the (subtype A A _) Dsubtype) _) %. +%term trans + %pi (subtp-comp (subtp/trans (%the (subtp EB EC) DsubtpBC) (%the (subtp EA EB) DsubtpAB)) DtransA DtransC DsubtypeAC) + %<- (subtp-comp DsubtpAB (%the (ttrans EA A) DtransA) (%the (ttrans EB B) DtransB) (%the (subtype A B _) DsubtypeAB)) + %<- (subtp-comp DsubtpBC (%the (ttrans EB B') DtransB') (%the (ttrans EC C) DtransC) (%the (subtype B' C _) DsubtypeBC)) + %<- (ttrans-fun DtransB' DtransB (%the (tp-eq B' B) Deq)) + %<- (subtype-resp Deq tp-eq/i ([_] term-eq/i) DsubtypeBC (%the (subtype B C _) DsubtypeBC')) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (ttrans-reg DtransB (%the (wf B) DwfB)) + %<- (ttrans-reg DtransC (%the (wf C) DwfC)) + %<- (subtype-trans' DwfA DwfB DwfC DsubtypeAB DsubtypeBC' _ (%the (subtype A C _) DsubtypeAC)) %. +%scope equiv-comp %term sing_t + %pi (subtp-comp (subtp/sing_t (%the (eof EM et) Deof)) (ttrans/sing Dtrans') ttrans/t subtype/sing_t) + %<- (of-comp Deof ttrans/t (%the (trans EM A) Dtrans) (%the (subtype A t _) Dsubtype)) + %<- (trans-principal Dtrans (%the (principal A) Dprincipal)) + %<- (principal-subtype-t-invert Dprincipal Dsubtype (%the (tp-eq A (sing R)) Deq) _) + %<- (trans-resp eterm-eq/i Deq Dtrans (%the (trans EM (sing R)) Dtrans')) %. +%term pi + %pi (subtp-comp (subtp/pi (%the ({ex} %pi (evof ex EA) %-> (ewf (EB ex))) DewfB) (%the ({ex} %pi (evof ex EC) %-> (subtp (EB ex) (ED ex))) DsubtpBD) (%the (subtp EC EA) DsubtpCA)) (ttrans/pi DtransB DtransA) (ttrans/pi DtransD DtransC) (subtype/pi DsubtypeBD Dsub DsubtypeCA)) + %<- (subtp-comp DsubtpCA (%the (ttrans EC C) DtransC) (%the (ttrans EA A) DtransA) (%the (subtype C A M) DsubtypeCA)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} + %pi (vof-comp ed DtransA xt d) + %-> (wf-comp (DewfB ex ed) (%the (ttrans (EB ex) (B x)) (DtransB x d ex xt)))) + %<- ({x} {d vof x C} {ex} {xt vtrans ex x} {ed evof ex EC} + %pi (vof-comp ed DtransC xt d) + %-> (subtp-comp (DsubtpBD ex ed) (%the (ttrans (EB ex) (B' x)) (DtransB' x d ex xt)) (%the (ttrans (ED ex) (D x)) (DtransD x d ex xt)) (%the (subtype (B' x) (D x) ([y] N x y)) (DsubtypeBD x)))) + %<- (trans-var-coerce DtransC DtransA DsubtypeCA DtransB' DtransB (%the ({x} tsub B (M x) (B' x)) Dsub)) %. +%scope equiv-comp %term sigma + %pi (subtp-comp (subtp/sigma (%the ({ex} %pi (evof ex EC) %-> (ewf (ED ex))) DewfD) (%the ({ex} %pi (evof ex EA) %-> (subtp (EB ex) (ED ex))) DsubtpBD) (%the (subtp EA EC) DsubtpAC)) (ttrans/sigma DtransB DtransA) (ttrans/sigma DtransD DtransC) (subtype/sigma DsubtypeBD Dsub DsubtypeAC)) + %<- (subtp-comp DsubtpAC (%the (ttrans EA A) DtransA) (%the (ttrans EC C) DtransC) (%the (subtype A C M) DsubtypeAC)) + %<- ({x} {d vof x C} {ex} {xt vtrans ex x} {ed evof ex EC} + %pi (vof-comp ed DtransC xt d) + %-> (wf-comp (DewfD ex ed) (%the (ttrans (ED ex) (D x)) (DtransD x d ex xt)))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} + %pi (vof-comp ed DtransA xt d) + %-> (subtp-comp (DsubtpBD ex ed) (%the (ttrans (EB ex) (B x)) (DtransB x d ex xt)) (%the (ttrans (ED ex) (D' x)) (DtransD' x d ex xt)) (%the (subtype (B x) (D' x) ([y] N x y)) (DsubtypeBD x)))) + %<- (trans-var-coerce DtransA DtransC DsubtypeAC DtransD' DtransD (%the ({x} tsub D (M x) (D' x)) Dsub)) %. +%term refl + %pi (tequiv-comp (tequiv/reflex (%the (ewf EA) Dewf)) DtransA DtransA) + %<- (wf-comp Dewf (%the (ttrans EA A) DtransA)) %. +%%% +%scope equiv-comp %term symm + %pi (tequiv-comp (tequiv/symm (%the (tequiv EB EA) Dequiv)) DtransA DtransB) + %<- (tequiv-comp Dequiv (%the (ttrans EB A) DtransB) (%the (ttrans EA A) DtransA)) %. +%term trans + %pi (tequiv-comp (tequiv/trans (%the (tequiv EB EC) DequivBC) (%the (tequiv EA EB) DequivAB)) DtransA DtransC') + %<- (tequiv-comp DequivAB (%the (ttrans EA A) DtransA) (%the (ttrans EB A) DtransB)) + %<- (tequiv-comp DequivBC (%the (ttrans EB B) DtransB') (%the (ttrans EC B) DtransC)) + %<- (ttrans-fun DtransB' DtransB (%the (tp-eq B A) Deq)) + %<- (ttrans-resp etp-eq/i Deq DtransC (%the (ttrans EC A) DtransC')) %. +%scope equiv-comp %term pi + %pi (tequiv-comp (tequiv/pi (%the ({ex} %pi (evof ex EA) %-> (tequiv (EB ex) (EB' ex))) DequivB) (%the (tequiv EA EA') DequivA)) (ttrans/pi DtransB DtransA) (ttrans/pi DtransB' DtransA')) + %<- (tequiv-comp DequivA (%the (ttrans EA A) DtransA) (%the (ttrans EA' A) DtransA')) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} + %pi (vof-comp ed DtransA xt d) + %-> (tequiv-comp (DequivB ex ed) (%the (ttrans (EB ex) (B x)) (DtransB x d ex xt)) (%the (ttrans (EB' ex) (B x)) (DtransB' x d ex xt)))) %. +%term sigma + %pi (tequiv-comp (tequiv/sigma (%the ({ex} %pi (evof ex EA) %-> (tequiv (EB ex) (EB' ex))) DequivB) (%the (tequiv EA EA') DequivA)) (ttrans/sigma DtransB DtransA) (ttrans/sigma DtransB' DtransA')) + %<- (tequiv-comp DequivA (%the (ttrans EA A) DtransA) (%the (ttrans EA' A) DtransA')) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} + %pi (vof-comp ed DtransA xt d) + %-> (tequiv-comp (DequivB ex ed) (%the (ttrans (EB ex) (B x)) (DtransB x d ex xt)) (%the (ttrans (EB' ex) (B x)) (DtransB' x d ex xt)))) %. +%scope equiv-comp %term sing + %pi (tequiv-comp (tequiv/sing (%the (equiv EM EN et) Dequiv)) (ttrans/sing DtransM') (ttrans/sing DtransN')) + %<- (equiv-comp Dequiv ttrans/t (%the (trans EM A) DtransM) (%the (trans EN B) DtransN) (%the (subtype A t M) DsubtypeA) (%the (subtype B t M) DsubtypeB)) + %<- (trans-principal DtransM (%the (principal A) DprincipalA)) + %<- (trans-principal DtransN (%the (principal B) DprincipalB)) + %<- (principal-subtype-t-invert DprincipalA DsubtypeA (%the (tp-eq A (sing R)) DeqA) (%the ({x} term-eq (M x) (at R)) DeqM)) + %<- (principal-subtype-t-invert' DprincipalB DsubtypeB DeqM (%the (tp-eq B (sing R)) DeqB)) + %<- (trans-resp eterm-eq/i DeqA DtransM (%the (trans EM (sing R)) DtransM')) + %<- (trans-resp eterm-eq/i DeqB DtransN (%the (trans EN (sing R)) DtransN')) %. +%term refl + %pi (equiv-comp (equiv/reflex (%the (eof EM EA) Deof)) DtransA DtransM DtransM Dsubtype Dsubtype) + %<- (of-comp Deof (%the (ttrans EA A) DtransA) (%the (trans EM B) DtransM) (%the (subtype B A M) Dsubtype)) %. +%%% +%scope equiv-comp %term symm + %pi (equiv-comp (equiv/symm (%the (equiv EN EM EA) Dequiv)) DtransA DtransM DtransN DsubtypeBA DsubtypeCA) + %<- (equiv-comp Dequiv (%the (ttrans EA A) DtransA) (%the (trans EN C) DtransN) (%the (trans EM B) DtransM) (%the (subtype C A N) DsubtypeCA) (%the (subtype B A N) DsubtypeBA)) %. +%term trans + %pi (equiv-comp (equiv/trans (%the (equiv EN EO EA) DequivNO) (%the (equiv EM EN EA) DequivMN)) DtransA DtransM DtransO DsubtypeBA DsubtypeDA') + %<- (equiv-comp DequivMN (%the (ttrans EA A) DtransA) (%the (trans EM B) DtransM) (%the (trans EN C) DtransN) (%the (subtype B A P) DsubtypeBA) (%the (subtype C A P) DsubtypeCA)) + %<- (equiv-comp DequivNO (%the (ttrans EA A') DtransA') (%the (trans EN C') DtransN') (%the (trans EO D) DtransO) (%the (subtype C' A' P') DsubtypeCA') (%the (subtype D A' P') DsubtypeDA)) + %<- (ttrans-fun DtransA' DtransA (%the (tp-eq A' A) DeqA)) + %<- (trans-fun DtransN' DtransN (%the (tp-eq C' C) DeqC)) + %<- (subtype-resp DeqC DeqA ([_] term-eq/i) DsubtypeCA' (%the (subtype C A P') DsubtypeCA'')) + %<- (subtype-fun DsubtypeCA'' DsubtypeCA (%the ({x} term-eq (P' x) (P x)) DeqP)) + %<- (subtype-resp tp-eq/i DeqA DeqP DsubtypeDA (%the (subtype D A P) DsubtypeDA')) %. +%scope equiv-comp %term forall + %pi (equiv-comp (equiv/forall (%the (tequiv EA EA') DequivA)) (ttrans/pi ([_] [_] [_] [_] ttrans/t) (ttrans/pi ([_] [_] [_] [_] ttrans/t) DtransA)) (trans/forall Dself Dexpand DtransA) (trans/forall Dself Dexpand DtransA') Dsubtype Dsubtype) + %<- (tequiv-comp DequivA (%the (ttrans EA A) DtransA) (%the (ttrans EA' A) DtransA')) + %<- (can-expand (forall A) (qtp A) M (%the (expand (forall A) (qtp A) M) Dexpand)) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (expand-reg (aof/forall DwfA) Dexpand (%the (of M (qtp A)) DofM)) + %<- (can-self DofM (%the (self M (qtp A) B) Dself)) + %<- (self-reg Dself DofM _ (%the (subtype B (qtp A) ([_] M)) Dsubtype)) %. +%term app + %pi (equiv-comp (equiv/app (%the (equiv EN1 EN2 EA) DequivN) (%the (equiv EM1 EM2 (epi EA EB)) DequivM)) DtransBN (trans/app DsubD1x DsubtypeF1C1 DtransN1 DtransM1) (trans/app DsubD2x DsubtypeF2C2 DtransN2 DtransM2) DsubtypeD1xBx DsubtypeD2xBx) + %<- (equiv-comp DequivM (ttrans/pi (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (ttrans (EB ex) (B x)))) DtransB) (%the (ttrans EA A) DtransA)) (%the (trans EM1 (pi C1 D1)) DtransM1) (%the (trans EM2 (pi C2 D2)) DtransM2) (%the (subtype (pi C1 D1) (pi A B) M) DsubtypeCD1) (%the (subtype (pi C2 D2) (pi A B) M) DsubtypeCD2)) + %<- (equiv-comp DequivN (%the (ttrans EA A') DtransA') (%the (trans EN1 F1) DtransN1) (%the (trans EN2 F2) DtransN2) (%the (subtype F1 A' N') DsubtypeF1A') (%the (subtype F2 A' N') DsubtypeF2A')) + %<- (subtype-pi-invert DsubtypeCD1 (%the (subtype A C1 Lac1) DsubtypeAC1) (%the ({x} tsub D1 (Lac1 x) (E1 x)) DsubE1) (%the ({x} subtype (E1 x) (B x) (Leb1 x)) DsubtypeE1B) (%the ({f} term-eq (M f) (lam ([x] Leb1 x (app f (Lac1 x))))) DeqM1)) + %<- (subtype-pi-invert DsubtypeCD2 (%the (subtype A C2 Lac2) DsubtypeAC2) (%the ({x} tsub D2 (Lac2 x) (E2 x)) DsubE2) (%the ({x} subtype (E2 x) (B x) (Leb2 x)) DsubtypeE2B) (%the ({f} term-eq (M f) (lam ([x] Leb2 x (app f (Lac2 x))))) DeqM2)) + %<- (ttrans-fun DtransA' DtransA (%the (tp-eq A' A) DeqA)) + %<- (trans-principal DtransN1 (%the (principal F1) DprincipalF1)) + %<- (principal-subtype DprincipalF1 DsubtypeF1A' (%the ({y} term-eq (N' y) N) DeqN')) + %<- (subtype-resp tp-eq/i DeqA DeqN' DsubtypeF1A' (%the (subtype F1 A ([_] N)) DsubtypeF1A)) + %<- (subtype-resp tp-eq/i DeqA DeqN' DsubtypeF2A' (%the (subtype F2 A ([_] N)) DsubtypeF2A)) + %<- (trans-reg DtransN1 (%the (wf F1) DwfF1)) + %<- (trans-reg DtransN2 (%the (wf F2) DwfF2)) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (trans-reg DtransM1 (wf/pi (%the ({x} %pi (vof x C1) %-> (wf (D1 x))) DwfD1) (%the (wf C1) DwfC1))) + %<- (trans-reg DtransM2 (wf/pi (%the ({x} %pi (vof x C2) %-> (wf (D2 x))) DwfD2) (%the (wf C2) DwfC2))) + %<- (subtype-reg DsubtypeAC1 DwfA DwfC1 (%the ({x} %pi (vof x A) %-> (of (Lac1 x) C1)) DofLac1)) + %<- (subtype-reg DsubtypeAC2 DwfA DwfC2 (%the ({x} %pi (vof x A) %-> (of (Lac2 x) C2)) DofLac2)) + %<- (subtype-reg' DsubtypeF1A DwfF1 DwfA (%the (of N A) DofN)) + %<- (can-sub DofLac1 DofN (%the (sub Lac1 N O1) DsubO1)) + %<- (can-sub DofLac2 DofN (%the (sub Lac2 N O2) DsubO2)) + %<- (subst' DsubO1 DofLac1 DofN (%the (of O1 C1) DofO1)) + %<- (subst' DsubO2 DofLac2 DofN (%the (of O2 C2) DofO2)) + %<- (subtype-trans DwfF1 DwfA DwfC1 DsubtypeF1A DsubtypeAC1 ([_] DsubO1) (%the (subtype F1 C1 ([_] O1)) DsubtypeF1C1)) + %<- (subtype-trans DwfF2 DwfA DwfC2 DsubtypeF2A DsubtypeAC2 ([_] DsubO2) (%the (subtype F2 C2 ([_] O2)) DsubtypeF2C2)) + %<- (can-tsub DwfD1 DofO1 (%the (tsub D1 O1 D1x) DsubD1x)) + %<- (can-tsub DwfD2 DofO2 (%the (tsub D2 O2 D2x) DsubD2x)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} ttrans-reg (DtransB x d ex xt) (%the (wf (B x)) (DwfB x d))) + %<- (can-tsub DwfB DofN (%the (tsub B N Bx) DsubBx)) + %<- (ttrans-sub DtransB DtransN1 DwfA DsubtypeF1A DsubBx (%the (ttrans (EB EN1) Bx) DtransBN)) + %<- ({x} {d vof x A} tsubst (DsubE1 x) DwfD1 (DofLac1 x d) (%the (wf (E1 x)) (DwfE1 x d))) + %<- ({x} {d vof x A} tsubst (DsubE2 x) DwfD2 (DofLac2 x d) (%the (wf (E2 x)) (DwfE2 x d))) + %<- (can-tsub DwfE1 DofN (%the (tsub E1 N E1x) DsubE1x)) + %<- (can-tsub DwfE2 DofN (%the (tsub E2 N E2x) DsubE2x)) + %<- (trans-principal DtransM1 (principal/pi (%the ({x} principal (D1 x)) DprincipalD1))) + %<- (trans-principal DtransM2 (principal/pi (%the ({x} principal (D2 x)) DprincipalD2))) + %<- ({x} principal-sub DprincipalD1 (DsubE1 x) (%the (principal (E1 x)) (DprincipalE1 x))) + %<- ({x} principal-sub DprincipalD2 (DsubE2 x) (%the (principal (E2 x)) (DprincipalE2 x))) + %<- ({x} principal-subtype (DprincipalE1 x) (DsubtypeE1B x) (%the ({y} term-eq (Leb1 x y) (P1 x)) (DeqLeb1 x))) + %<- ({x} principal-subtype (DprincipalE2 x) (DsubtypeE2B x) (%the ({y} term-eq (Leb2 x y) (P2 x)) (DeqLeb2 x))) + %<- ({x} subtype-resp tp-eq/i tp-eq/i (DeqLeb1 x) (DsubtypeE1B x) (%the (subtype (E1 x) (B x) ([_] P1 x)) (DsubtypeE1B' x))) + %<- ({x} subtype-resp tp-eq/i tp-eq/i (DeqLeb2 x) (DsubtypeE2B x) (%the (subtype (E2 x) (B x) ([_] P2 x)) (DsubtypeE2B' x))) + %<- ({f} term-eq-symm (DeqM1 f) (%the (term-eq (lam ([x] Leb1 x (app f (Lac1 x)))) (M f)) (DeqM1' f))) + %<- ({f} term-eq-trans (DeqM1' f) (DeqM2 f) (%the (term-eq (lam ([x] Leb1 x (app f (Lac1 x)))) (lam ([x] Leb2 x (app f (Lac2 x))))) (DeqLam f))) + %<- ({f} term-eq-cdr-lam (DeqLam f) (%the ({x} term-eq (Leb1 x (app f (Lac1 x))) (Leb2 x (app f (Lac2 x)))) (DeqLeby f))) + %<- ({x} {y} term-eq-symm (DeqLeb1 x y) (%the (term-eq (P1 x) (Leb1 x y)) (DeqLeb1' x y))) + %<- ({f} {x} term-eq-trans (DeqLeb1' x _) (DeqLeby f x) (%the (term-eq (P1 x) (Leb2 x (app f (Lac2 x)))) (DeqLeby' f x))) + %<- ({f} {x} term-eq-trans (DeqLeby' f x) (DeqLeb2 x _) (%the (term-eq (P1 x) (P2 x)) (DeqP f x))) + %<- ({x} {d vof x A} subtype-reg' (DsubtypeE1B' x) (DwfE1 x d) (DwfB x d) (%the (of (P1 x) (B x)) (DofP1 x d))) + %<- (can-sub DofP1 DofN (%the (sub P1 N Px) DsubP1x)) + %<- (sub-resp ([x] DeqP aca x) term-eq/i term-eq/i DsubP1x (%the (sub P2 N Px) DsubP2x)) + %<- (subtype-sub' DsubtypeE1B' DwfE1 DwfB DofN DsubE1x DsubBx ([y] %the (sub ([x] P1 x) N Px) DsubP1x) (%the (subtype E1x Bx ([_] Px)) DsubtypeE1xBx)) + %<- (subtype-sub' DsubtypeE2B' DwfE2 DwfB DofN DsubE2x DsubBx ([y] %the (sub ([x] P2 x) N Px) DsubP2x) (%the (subtype E2x Bx ([_] Px)) DsubtypeE2xBx)) + %<- ({y} tsub-absent (D1 y) N (%the (tsub ([x] D1 y) N (D1 y)) (DsubD1abs y))) + %<- ({y} tsub-absent (D2 y) N (%the (tsub ([x] D2 y) N (D2 y)) (DsubD2abs y))) + %<- (tsub-permute DofN DofLac1 ([x] [d vof x A] [y] [e vof y C1] DwfD1 y e) DsubD1abs DsubO1 DsubD1x DsubE1 DsubE1x (%the (tp-eq D1x E1x) Deq1)) + %<- (tp-eq-symm Deq1 (%the (tp-eq E1x D1x) Deq1')) + %<- (tsub-permute DofN DofLac2 ([x] [d vof x A] [y] [e vof y C2] DwfD2 y e) DsubD2abs DsubO2 DsubD2x DsubE2 DsubE2x (%the (tp-eq D2x E2x) Deq2)) + %<- (tp-eq-symm Deq2 (%the (tp-eq E2x D2x) Deq2')) + %<- (subtype-resp Deq1' tp-eq/i ([_] term-eq/i) DsubtypeE1xBx (%the (subtype D1x Bx ([_] Px)) DsubtypeD1xBx)) + %<- (subtype-resp Deq2' tp-eq/i ([_] term-eq/i) DsubtypeE2xBx (%the (subtype D2x Bx ([_] Px)) DsubtypeD2xBx)) %. +%scope equiv-comp %term lam + %pi (equiv-comp (equiv/lam (%the ({ex} %pi (evof ex EA1) %-> (equiv (EM1 ex) (EM2 ex) (EB ex))) DequivM) (%the (tequiv EA1 EA2) DequivA)) (ttrans/pi DtransB DtransA1) (trans/lam DtransM1 DtransA1) (trans/lam DtransM2 DtransA2) (subtype/pi DsubtypeC1B DsubC1 DsubtypeA) (subtype/pi DsubtypeC2B DsubC2 DsubtypeA)) + %<- (tequiv-comp DequivA (%the (ttrans EA1 A) DtransA1) (%the (ttrans EA2 A) DtransA2)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA1} + %pi (vof-comp ed DtransA1 xt d) + %-> (equiv-comp (DequivM ex ed) (%the (ttrans (EB ex) (B x)) (DtransB x d ex xt)) (%the (trans (EM1 ex) (C1 x)) (DtransM1 x d ex xt)) (%the (trans (EM2 ex) (C2 x)) (DtransM2 x d ex xt)) (%the (subtype (C1 x) (B x) (M x)) (DsubtypeC1B x)) (%the (subtype (C2 x) (B x) (M x)) (DsubtypeC2B x)))) + %<- (ttrans-reg DtransA1 (%the (wf A) DwfA)) + %<- (subtype-refl DwfA (%the (subtype A A X) DsubtypeA) (%the ({x} expand x A (X x)) Dexpand)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} trans-reg (DtransM1 x d ex xt) (%the (wf (C1 x)) (DwfC1 x d))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} trans-reg (DtransM2 x d ex xt) (%the (wf (C2 x)) (DwfC2 x d))) + %<- (tsub-expand-var DwfA DwfC1 Dexpand (%the ({x} tsub C1 (X x) (C1 x)) DsubC1)) + %<- (tsub-expand-var DwfA DwfC2 Dexpand (%the ({x} tsub C2 (X x) (C2 x)) DsubC2)) %. +%term pair + %pi (equiv-comp (equiv/pair (%the ({ex} %pi (evof ex EA) %-> (ewf (EB ex))) DewfB) (%the (equiv EN1 EN2 (EB EM1)) DequivN) (%the (equiv EM1 EM2 EA) DequivM)) (ttrans/sigma DtransB DtransA) (trans/pair DtransN1 DtransM1) (trans/pair DtransN2 DtransM2) (subtype/sigma ([_] DsubtypeD1Bx) ([_] DsubBx) DsubtypeC1A') (subtype/sigma ([_] DsubtypeD2Bx) ([_] DsubBx) DsubtypeC2A')) + %<- (equiv-comp DequivM (%the (ttrans EA A) DtransA) (%the (trans EM1 C1) DtransM1) (%the (trans EM2 C2) DtransM2) (%the (subtype C1 A M') DsubtypeC1A) (%the (subtype C2 A M') DsubtypeC2A)) + %<- (equiv-comp DequivN (%the (ttrans (EB EM1) E) DtransE) (%the (trans EN1 D1) DtransN1) (%the (trans EN2 D2) DtransN2) (%the (subtype D1 E N') DsubtypeD1E) (%the (subtype D2 E N') DsubtypeD2E)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} + %pi (vof-comp ed DtransA xt d) + %-> (wf-comp (DewfB ex ed) (%the (ttrans (EB ex) (B x)) (DtransB x d ex xt)))) + %<- (trans-principal DtransM1 (%the (principal C1) DprincipalC1)) + %<- (principal-subtype DprincipalC1 DsubtypeC1A (%the ({x} term-eq (M' x) M) DeqM')) + %<- (subtype-resp tp-eq/i tp-eq/i DeqM' DsubtypeC1A (%the (subtype C1 A ([_] M)) DsubtypeC1A')) + %<- (subtype-resp tp-eq/i tp-eq/i DeqM' DsubtypeC2A (%the (subtype C2 A ([_] M)) DsubtypeC2A')) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} ttrans-reg (DtransB x d ex xt) (%the (wf (B x)) (DwfB x d))) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (trans-reg DtransM1 (%the (wf C1) DwfC1)) + %<- (subtype-reg' DsubtypeC1A' DwfC1 DwfA (%the (of M A) DofM)) + %<- (can-tsub DwfB DofM (%the (tsub B M Bx) DsubBx)) + %<- (ttrans-sub DtransB DtransM1 DwfA DsubtypeC1A' DsubBx (%the (ttrans (EB EM1) Bx) DtransBx)) + %<- (ttrans-fun DtransE DtransBx (%the (tp-eq E Bx) DeqE)) + %<- (subtype-resp tp-eq/i DeqE ([_] term-eq/i) DsubtypeD1E (%the (subtype D1 Bx N') DsubtypeD1Bx)) + %<- (subtype-resp tp-eq/i DeqE ([_] term-eq/i) DsubtypeD2E (%the (subtype D2 Bx N') DsubtypeD2Bx)) %. +%scope equiv-comp %term pi1 + %pi (equiv-comp (equiv/pi1 (%the (equiv EM1 EM2 (esigma EA EB)) Dequiv)) DtransA (trans/pi1 DtransM1') (trans/pi1 DtransM2') DsubtypeC1A' DsubtypeC2A') + %<- (equiv-comp Dequiv (ttrans/sigma (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (ttrans (EB ex) (B x)))) DtransB) (%the (ttrans EA A) DtransA)) (%the (trans EM1 CD1) DtransM1) (%the (trans EM2 CD2) DtransM2) (%the (subtype CD1 (sigma A B) O) DsubtypeCD1AB) (%the (subtype CD2 (sigma A B) O) DsubtypeCD2AB)) + %<- (trans-principal DtransM1 (%the (principal CD1) DprincipalCD1)) + %<- (trans-principal DtransM2 (%the (principal CD2) DprincipalCD2)) + %<- (principal-subtype-sigma-invert DprincipalCD1 DsubtypeCD1AB (%the (tp-eq CD1 (sigma C1 ([_] D1))) DeqCD1)) + %<- (principal-subtype-sigma-invert DprincipalCD2 DsubtypeCD2AB (%the (tp-eq CD2 (sigma C2 ([_] D2))) DeqCD2)) + %<- (trans-resp eterm-eq/i DeqCD1 DtransM1 (%the (trans EM1 (sigma C1 ([_] D1))) DtransM1')) + %<- (trans-resp eterm-eq/i DeqCD2 DtransM2 (%the (trans EM2 (sigma C2 ([_] D2))) DtransM2')) + %<- (subtype-resp DeqCD1 tp-eq/i ([_] term-eq/i) DsubtypeCD1AB (%the (subtype (sigma C1 ([_] D1)) (sigma A B) O) DsubtypeCD1AB')) + %<- (subtype-resp DeqCD2 tp-eq/i ([_] term-eq/i) DsubtypeCD2AB (%the (subtype (sigma C2 ([_] D2)) (sigma A B) O) DsubtypeCD2AB')) + %<- (subtype-sigma-invert DsubtypeCD1AB' (%the (subtype C1 A Lca1) DsubtypeC1A) (%the ({x} tsub B (Lca1 x) (E1 x)) DsubE1) (%the ({x} subtype D1 (E1 x) (Lde1 x)) DsubtypeD1E1) (%the ({p} term-eq (O p) (pair (Lca1 (pi1 p)) (Lde1 (pi1 p) (pi2 p)))) DeqO1)) + %<- (subtype-sigma-invert DsubtypeCD2AB' (%the (subtype C2 A Lca2) DsubtypeC2A) (%the ({x} tsub B (Lca2 x) (E2 x)) DsubE2) (%the ({x} subtype D2 (E2 x) (Lde2 x)) DsubtypeD2E2) (%the ({p} term-eq (O p) (pair (Lca2 (pi1 p)) (Lde2 (pi1 p) (pi2 p)))) DeqO2)) + %<- (trans-principal DtransM1' (principal/sigma (%the (principal D1) DprincipalD1) (%the (principal C1) DprincipalC1))) + %<- (trans-principal DtransM2' (principal/sigma (%the (principal D2) DprincipalD2) (%the (principal C2) DprincipalC2))) + %<- (principal-subtype DprincipalC1 DsubtypeC1A (%the ({x} term-eq (Lca1 x) M) DeqLca1)) + %<- (principal-subtype DprincipalC2 DsubtypeC2A (%the ({x} term-eq (Lca2 x) M') DeqLca2)) + %<- ({p} term-eq-symm (DeqO1 p) (%the (term-eq (pair (Lca1 (pi1 p)) (Lde1 (pi1 p) (pi2 p))) (O p)) (DeqO1' p))) + %<- ({p} term-eq-trans (DeqO1' p) (DeqO2 p) (%the (term-eq (pair (Lca1 (pi1 p)) (Lde1 (pi1 p) (pi2 p))) (pair (Lca2 (pi1 p)) (Lde2 (pi1 p) (pi2 p)))) (DeqPair p))) + %<- ({p} term-eq-cdr-pair (DeqPair p) (%the (term-eq (Lca1 (pi1 p)) (Lca2 (pi1 p))) (DeqLca p)) (%the (term-eq (Lde1 (pi1 p) (pi2 p)) (Lde2 (pi1 p) (pi2 p))) (DeqLde p))) + %<- ({x} term-eq-symm (DeqLca2 x) (%the (term-eq M' (Lca2 x)) (DeqLca2' x))) + %<- ({p} term-eq-symm (DeqLca p) (%the (term-eq (Lca2 (pi1 p)) (Lca1 (pi1 p))) (DeqLca' p))) + %<- ({p} term-eq-trans (DeqLca2' (pi1 p)) (DeqLca' p) (%the (term-eq M' (Lca1 (pi1 p))) (DeqLcay p))) + %<- ({p} term-eq-trans (DeqLcay p) (DeqLca1 (pi1 p)) (%the (term-eq M' M) (DeqM p))) + %<- ({x} term-eq-trans (DeqLca2 x) (DeqM aca) (%the (term-eq (Lca2 x) M) (DeqLca2'' x))) + %<- (subtype-resp tp-eq/i tp-eq/i DeqLca1 DsubtypeC1A (%the (subtype C1 A ([_] M)) DsubtypeC1A')) + %<- (subtype-resp tp-eq/i tp-eq/i DeqLca2'' DsubtypeC2A (%the (subtype C2 A ([_] M)) DsubtypeC2A')) %. +%term pi2 + %pi (equiv-comp (equiv/pi2 (%the (equiv EM1 EM2 (esigma EA EB)) Dequiv)) DtransBx (trans/pi2 DtransM1') (trans/pi2 DtransM2') (DsubtypeD1Bx' aca) (DsubtypeD2Bx' aca)) + %<- (equiv-comp Dequiv (ttrans/sigma (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (ttrans (EB ex) (B x)))) DtransB) (%the (ttrans EA A) DtransA)) (%the (trans EM1 CD1) DtransM1) (%the (trans EM2 CD2) DtransM2) (%the (subtype CD1 (sigma A B) O) DsubtypeCD1AB) (%the (subtype CD2 (sigma A B) O) DsubtypeCD2AB)) + %<- (trans-principal DtransM1 (%the (principal CD1) DprincipalCD1)) + %<- (trans-principal DtransM2 (%the (principal CD2) DprincipalCD2)) + %<- (principal-subtype-sigma-invert DprincipalCD1 DsubtypeCD1AB (%the (tp-eq CD1 (sigma C1 ([_] D1))) DeqCD1)) + %<- (principal-subtype-sigma-invert DprincipalCD2 DsubtypeCD2AB (%the (tp-eq CD2 (sigma C2 ([_] D2))) DeqCD2)) + %<- (trans-resp eterm-eq/i DeqCD1 DtransM1 (%the (trans EM1 (sigma C1 ([_] D1))) DtransM1')) + %<- (trans-resp eterm-eq/i DeqCD2 DtransM2 (%the (trans EM2 (sigma C2 ([_] D2))) DtransM2')) + %<- (subtype-resp DeqCD1 tp-eq/i ([_] term-eq/i) DsubtypeCD1AB (%the (subtype (sigma C1 ([_] D1)) (sigma A B) O) DsubtypeCD1AB')) + %<- (subtype-resp DeqCD2 tp-eq/i ([_] term-eq/i) DsubtypeCD2AB (%the (subtype (sigma C2 ([_] D2)) (sigma A B) O) DsubtypeCD2AB')) + %<- (subtype-sigma-invert DsubtypeCD1AB' (%the (subtype C1 A Lca1) DsubtypeC1A) (%the ({x} tsub B (Lca1 x) (E1 x)) DsubE1) (%the ({x} subtype D1 (E1 x) (Lde1 x)) DsubtypeD1E1) (%the ({p} term-eq (O p) (pair (Lca1 (pi1 p)) (Lde1 (pi1 p) (pi2 p)))) DeqO1)) + %<- (subtype-sigma-invert DsubtypeCD2AB' (%the (subtype C2 A Lca2) DsubtypeC2A) (%the ({x} tsub B (Lca2 x) (E2 x)) DsubE2) (%the ({x} subtype D2 (E2 x) (Lde2 x)) DsubtypeD2E2) (%the ({p} term-eq (O p) (pair (Lca2 (pi1 p)) (Lde2 (pi1 p) (pi2 p)))) DeqO2)) + %<- (trans-principal DtransM1' (principal/sigma (%the (principal D1) DprincipalD1) (%the (principal C1) DprincipalC1))) + %<- (trans-principal DtransM2' (principal/sigma (%the (principal D2) DprincipalD2) (%the (principal C2) DprincipalC2))) + %<- (principal-subtype DprincipalC1 DsubtypeC1A (%the ({x} term-eq (Lca1 x) M) DeqLca1)) + %<- (principal-subtype DprincipalC2 DsubtypeC2A (%the ({x} term-eq (Lca2 x) M') DeqLca2)) + %<- ({p} term-eq-symm (DeqO1 p) (%the (term-eq (pair (Lca1 (pi1 p)) (Lde1 (pi1 p) (pi2 p))) (O p)) (DeqO1' p))) + %<- ({p} term-eq-trans (DeqO1' p) (DeqO2 p) (%the (term-eq (pair (Lca1 (pi1 p)) (Lde1 (pi1 p) (pi2 p))) (pair (Lca2 (pi1 p)) (Lde2 (pi1 p) (pi2 p)))) (DeqPair p))) + %<- ({p} term-eq-cdr-pair (DeqPair p) (%the (term-eq (Lca1 (pi1 p)) (Lca2 (pi1 p))) (DeqLca p)) (%the (term-eq (Lde1 (pi1 p) (pi2 p)) (Lde2 (pi1 p) (pi2 p))) (DeqLde p))) + %<- ({x} term-eq-symm (DeqLca2 x) (%the (term-eq M' (Lca2 x)) (DeqLca2' x))) + %<- ({p} term-eq-symm (DeqLca p) (%the (term-eq (Lca2 (pi1 p)) (Lca1 (pi1 p))) (DeqLca' p))) + %<- ({p} term-eq-trans (DeqLca2' (pi1 p)) (DeqLca' p) (%the (term-eq M' (Lca1 (pi1 p))) (DeqLcay p))) + %<- ({p} term-eq-trans (DeqLcay p) (DeqLca1 (pi1 p)) (%the (term-eq M' M) (DeqM p))) + %<- ({x} term-eq-trans (DeqLca2 x) (DeqM aca) (%the (term-eq (Lca2 x) M) (DeqLca2'' x))) + %<- (subtype-resp tp-eq/i tp-eq/i DeqLca1 DsubtypeC1A (%the (subtype C1 A ([_] M)) DsubtypeC1A')) + %<- (subtype-resp tp-eq/i tp-eq/i DeqLca2'' DsubtypeC2A (%the (subtype C2 A ([_] M)) DsubtypeC2A')) + %<- (trans-reg DtransM1' (wf/sigma (%the ({x} %pi (vof x C1) %-> (wf D1)) DwfD1) (%the (wf C1) DwfC1))) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (subtype-reg' DsubtypeC1A' DwfC1 DwfA (%the (of M A) DofM)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} ttrans-reg (DtransB x d ex xt) (%the (wf (B x)) (DwfB x d))) + %<- (can-tsub DwfB DofM (%the (tsub B M Bx) DsubBx)) + %<- (ttrans-sub DtransB (trans/pi1 DtransM1') DwfA DsubtypeC1A' DsubBx (%the (ttrans (EB (epi1 EM1)) Bx) DtransBx)) + %<- ({x} tsub-resp ([_] tp-eq/i) (DeqLca1 x) tp-eq/i (DsubE1 x) (%the (tsub B M (E1 x)) (DsubE1' x))) + %<- ({x} tsub-resp ([_] tp-eq/i) (DeqLca2'' x) tp-eq/i (DsubE2 x) (%the (tsub B M (E2 x)) (DsubE2' x))) + %<- ({x} tsub-fun (DsubE1' x) DsubBx (%the (tp-eq (E1 x) Bx) (DeqE1 x))) + %<- ({x} tsub-fun (DsubE2' x) DsubBx (%the (tp-eq (E2 x) Bx) (DeqE2 x))) + %<- ({x} subtype-resp tp-eq/i (DeqE1 x) ([_] term-eq/i) (DsubtypeD1E1 x) (%the (subtype D1 Bx (Lde1 x)) (DsubtypeD1Bx x))) + %<- ({x} subtype-resp tp-eq/i (DeqE2 x) ([_] term-eq/i) (DsubtypeD2E2 x) (%the (subtype D2 Bx (Lde2 x)) (DsubtypeD2Bx x))) + %<- ({x} subtype-fun (DsubtypeD1Bx x) (DsubtypeD1Bx aca) (%the ({y} term-eq (Lde1 x y) (Lde1 aca y)) (DeqLde1x x))) + %<- ({x} subtype-fun (DsubtypeD2Bx x) (DsubtypeD2Bx aca) (%the ({y} term-eq (Lde2 x y) (Lde2 aca y)) (DeqLde2x x))) + %<- (principal-subtype DprincipalD1 (DsubtypeD1Bx aca) (%the ({y} term-eq (Lde1 aca y) N) DeqLde1y)) + %<- (principal-subtype DprincipalD2 (DsubtypeD2Bx aca) (%the ({y} term-eq (Lde2 aca y) N') DeqLde2y)) + %<- ({x} {y} term-eq-trans (DeqLde1x x y) (DeqLde1y y) (%the (term-eq (Lde1 x y) N) (DeqLde1 x y))) + %<- ({x} {y} term-eq-trans (DeqLde2x x y) (DeqLde2y y) (%the (term-eq (Lde2 x y) N') (DeqLde2 x y))) + %<- ({x} {y} term-eq-symm (DeqLde2 x y) (%the (term-eq N' (Lde2 x y)) (DeqLde2' x y))) + %<- ({p} term-eq-symm (DeqLde p) (%the (term-eq (Lde2 (pi1 p) (pi2 p)) (Lde1 (pi1 p) (pi2 p))) (DeqLde' p))) + %<- ({p} term-eq-trans (DeqLde2' (pi1 p) (pi2 p)) (DeqLde' p) (%the (term-eq N' (Lde1 (pi1 p) (pi2 p))) (DeqLdey p))) + %<- ({p} term-eq-trans (DeqLdey p) (DeqLde1 (pi1 p) (pi2 p)) (%the (term-eq N' N) (DeqN p))) + %<- ({x} {y} term-eq-trans (DeqLde2 x y) (DeqN aca) (%the (term-eq (Lde2 x y) N) (DeqLde2'' x y))) + %<- ({x} subtype-resp tp-eq/i tp-eq/i (DeqLde1 x) (DsubtypeD1Bx x) (%the (subtype D1 Bx ([_] N)) (DsubtypeD1Bx' x))) + %<- ({x} subtype-resp tp-eq/i tp-eq/i (DeqLde2'' x) (DsubtypeD2Bx x) (%the (subtype D2 Bx ([_] N)) (DsubtypeD2Bx' x))) %. +%scope equiv-comp %term sing + %pi (equiv-comp (equiv/sing (%the (equiv EM EN et) Dequiv)) (ttrans/sing DtransM') DtransM' DtransN' subtype/sing subtype/sing) + %<- (equiv-comp Dequiv ttrans/t (%the (trans EM A) DtransM) (%the (trans EN B) DtransN) (%the (subtype A t M) DsubtypeA) (%the (subtype B t M) DsubtypeB)) + %<- (trans-principal DtransM (%the (principal A) DprincipalA)) + %<- (trans-principal DtransN (%the (principal B) DprincipalB)) + %<- (principal-subtype-t-invert DprincipalA DsubtypeA (%the (tp-eq A (sing R)) DeqA) (%the ({x} term-eq (M x) (at R)) DeqM)) + %<- (principal-subtype-t-invert' DprincipalB DsubtypeB DeqM (%the (tp-eq B (sing R)) DeqB)) + %<- (trans-resp eterm-eq/i DeqA DtransM (%the (trans EM (sing R)) DtransM')) + %<- (trans-resp eterm-eq/i DeqB DtransN (%the (trans EN (sing R)) DtransN')) %. +%term singel + %pi (equiv-comp (equiv/singelim (%the (eof EM (esing EN)) Deof)) ttrans/t DtransM DtransN subtype/sing_t subtype/sing_t) + %<- (of-comp Deof (ttrans/sing (%the (trans EN (sing R)) DtransN)) (%the (trans EM (sing R)) DtransM) subtype/sing) %. +%scope equiv-comp %term extpi + %pi (equiv-comp (equiv/extpi (%the ({ex} %pi (evof ex EA) %-> (equiv (eapp EM1 ex) (eapp EM2 ex) (EB ex))) Dequiv) (%the (eof EM2 (epi EA EB2)) Deof2) (%the (eof EM1 (epi EA EB1)) Deof1)) (ttrans/pi DtransB DtransA) Dtrans1 Dtrans2 (subtype/pi DsubtypeD1xB' DsubD1x DsubtypeAC1) (subtype/pi DsubtypeD2xB' DsubD2x DsubtypeAC2')) + %<- (of-comp Deof1 (ttrans/pi _ (%the (ttrans EA A) DtransA)) (%the (trans EM1 (pi C1 D1)) Dtrans1) (subtype/pi _ (%the ({x} tsub D1 (Lac1 x) (D1x x)) DsubD1x) (%the (subtype A C1 Lac1) DsubtypeAC1))) + %<- (of-comp Deof2 (ttrans/pi _ (%the (ttrans EA A') DtransA')) (%the (trans EM2 (pi C2 D2)) Dtrans2) (subtype/pi _ (%the ({x} tsub D2 (Lac2 x) (D2x x)) DsubD2x) (%the (subtype A' C2 Lac2) DsubtypeAC2))) + %<- (ttrans-fun DtransA' DtransA (%the (tp-eq A' A) DeqA)) + %<- (subtype-resp DeqA tp-eq/i ([_] term-eq/i) DsubtypeAC2 (%the (subtype A C2 Lac2) DsubtypeAC2')) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} + %pi (vof-comp ed DtransA xt d) + %-> (equiv-comp (Dequiv ex ed) (%the (ttrans (EB ex) (B x)) (DtransB x d ex xt)) (trans/app (%the (tsub ([y] F1 x y) (M1 x) (F1y x)) (DsubF1y x)) (%the (subtype (H1 x) (E1 x) ([_] M1 x)) (DsubtypeH1E1 x)) (%the (trans ex (H1 x)) (DtransH1 x d ex xt)) (%the (trans EM1 (pi (E1 x) ([y] F1 x y))) (Dtrans1' x d ex xt))) (trans/app (%the (tsub ([y] F2 x y) (M2 x) (F2y x)) (DsubF2y x)) (%the (subtype (H2 x) (E2 x) ([_] M2 x)) (DsubtypeH2E2 x)) (%the (trans ex (H2 x)) (DtransH2 x d ex xt)) (%the (trans EM2 (pi (E2 x) ([y] F2 x y))) (Dtrans2' x d ex xt))) (%the (subtype (F1y x) (B x) (O x)) (DsubtypeF1yB x)) (%the (subtype (F2y x) (B x) (O x)) (DsubtypeF2yB x)))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} trans-fun (Dtrans1' x d ex xt) Dtrans1 (%the (tp-eq (pi (E1 x) (F1 x)) (pi C1 D1)) (DeqEFCD1 x))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} trans-fun (Dtrans2' x d ex xt) Dtrans2 (%the (tp-eq (pi (E2 x) (F2 x)) (pi C2 D2)) (DeqEFCD2 x))) + %<- ({x} tp-eq-cdr-pi (DeqEFCD1 x) (%the (tp-eq (E1 x) C1) (DeqE1C1 x)) (%the ({y} tp-eq (F1 x y) (D1 y)) (DeqF1D1 x))) + %<- ({x} tp-eq-cdr-pi (DeqEFCD2 x) (%the (tp-eq (E2 x) C2) (DeqE2C2 x)) (%the ({y} tp-eq (F2 x y) (D2 y)) (DeqF2D2 x))) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- ({x} can-expand x A (X x) (%the (expand x A (X x)) (Dexpand x))) + %<- ({x} {d vof x A} expand-reg (aof/var DwfA d) (Dexpand x) (%the (of (X x) A) (DofX x d))) + %<- ({x} {d vof x A} can-self (DofX x d) (%the (self (X x) A (As x)) (Dself x))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} trans-fun (DtransH1 x d ex xt) (trans/var (Dself x) (Dexpand x) DwfA d xt) (%the (tp-eq (H1 x) (As x)) (DeqH1 x))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} trans-fun (DtransH2 x d ex xt) (trans/var (Dself x) (Dexpand x) DwfA d xt) (%the (tp-eq (H2 x) (As x)) (DeqH2 x))) + %<- ({x} subtype-resp (DeqH1 x) (DeqE1C1 x) ([_] term-eq/i) (DsubtypeH1E1 x) (%the (subtype (As x) C1 ([_] M1 x)) (DsubtypeAsC1 x))) + %<- ({x} subtype-resp (DeqH2 x) (DeqE2C2 x) ([_] term-eq/i) (DsubtypeH2E2 x) (%the (subtype (As x) C2 ([_] M2 x)) (DsubtypeAsC2 x))) + %<- ({x} {d vof x A} self-reg (Dself x) (DofX x d) (%the (of (X x) (As x)) (DofX' x d)) (%the (subtype (As x) A ([_] X x)) (DsubtypeAsA x))) + %<- (trans-reg Dtrans1 (wf/pi (%the ({x} %pi (vof x C1) %-> (wf (D1 x))) DwfD1) (%the (wf C1) DwfC1))) + %<- (trans-reg Dtrans2 (wf/pi (%the ({x} %pi (vof x C2) %-> (wf (D2 x))) DwfD2) (%the (wf C2) DwfC2))) + %<- (subtype-reg DsubtypeAC1 DwfA DwfC1 (%the ({x} %pi (vof x A) %-> (of (Lac1 x) C1)) DofLac1)) + %<- (subtype-reg DsubtypeAC2' DwfA DwfC2 (%the ({x} %pi (vof x A) %-> (of (Lac2 x) C2)) DofLac2)) + %<- (sub-expand-var DwfA DofLac1 Dexpand (%the ({x} sub Lac1 (X x) (Lac1 x)) DsubLac1)) + %<- (sub-expand-var DwfA DofLac2 Dexpand (%the ({x} sub Lac2 (X x) (Lac2 x)) DsubLac2)) + %<- ({x} {d vof x A} of-reg (DofX' x d) (%the (wf (As x)) (DwfAs x d))) + %<- ({x} {d vof x A} subtype-trans (DwfAs x d) DwfA DwfC1 (DsubtypeAsA x) DsubtypeAC1 ([_] DsubLac1 x) (%the (subtype (As x) C1 ([_] Lac1 x)) (DsubtypeAsC1' x))) + %<- ({x} {d vof x A} subtype-trans (DwfAs x d) DwfA DwfC2 (DsubtypeAsA x) DsubtypeAC2' ([_] DsubLac2 x) (%the (subtype (As x) C2 ([_] Lac2 x)) (DsubtypeAsC2' x))) + %<- ({x} subtype-fun (DsubtypeAsC1 x) (DsubtypeAsC1' x) (%the ({y} term-eq (M1 x) (Lac1 x)) (DeqM1 x))) + %<- ({x} subtype-fun (DsubtypeAsC2 x) (DsubtypeAsC2' x) (%the ({y} term-eq (M2 x) (Lac2 x)) (DeqM2 x))) + %<- ({x} tsub-resp ([y] DeqF1D1 x y) (DeqM1 x aca) tp-eq/i (DsubF1y x) (%the (tsub D1 (Lac1 x) (F1y x)) (DsubF1y' x))) + %<- ({x} tsub-resp ([y] DeqF2D2 x y) (DeqM2 x aca) tp-eq/i (DsubF2y x) (%the (tsub D2 (Lac2 x) (F2y x)) (DsubF2y' x))) + %<- ({x} tsub-fun (DsubF1y' x) (DsubD1x x) (%the (tp-eq (F1y x) (D1x x)) (DeqD1x x))) + %<- ({x} tsub-fun (DsubF2y' x) (DsubD2x x) (%the (tp-eq (F2y x) (D2x x)) (DeqD2x x))) + %<- (trans-principal Dtrans1 (principal/pi (%the ({x} principal (D1 x)) DprincipalD1))) + %<- ({x} principal-sub DprincipalD1 (DsubD1x x) (%the (principal (D1x x)) (DprincipalD1x x))) + %<- ({x} subtype-resp (DeqD1x x) tp-eq/i ([_] term-eq/i) (DsubtypeF1yB x) (%the (subtype (D1x x) (B x) (O x)) (DsubtypeD1xB x))) + %<- ({x} subtype-resp (DeqD2x x) tp-eq/i ([_] term-eq/i) (DsubtypeF2yB x) (%the (subtype (D2x x) (B x) (O x)) (DsubtypeD2xB x))) + %<- ({x} principal-subtype (DprincipalD1x x) (DsubtypeD1xB x) (%the ({y} term-eq (O x y) (O' x)) (DeqO x))) + %<- ({x} subtype-resp tp-eq/i tp-eq/i (DeqO x) (DsubtypeD1xB x) (%the (subtype (D1x x) (B x) ([_] O' x)) (DsubtypeD1xB' x))) + %<- ({x} subtype-resp tp-eq/i tp-eq/i (DeqO x) (DsubtypeD2xB x) (%the (subtype (D2x x) (B x) ([_] O' x)) (DsubtypeD2xB' x))) %. +%term extpiw + %pi (equiv-comp (equiv/extpiw (%the ({ex} %pi (evof ex EA) %-> (equiv (eapp EM1 ex) (eapp EM2 ex) (EB ex))) Dequiv) (%the (equiv EM1 EM2 (epi EA EB1)) Dequiv12)) (ttrans/pi DtransB DtransA) Dtrans1 Dtrans2 (subtype/pi DsubtypeD1xB' DsubD1x DsubtypeAC1) (subtype/pi DsubtypeD2xB' DsubD2x DsubtypeAC2')) + %<- (equiv-comp Dequiv12 (ttrans/pi _ (%the (ttrans EA A) DtransA)) (%the (trans EM1 (pi C1 D1)) Dtrans1) (%the (trans EM2 (pi C2 D2)) Dtrans2) (%the (subtype (pi C1 D1) (pi A B') _) DsubtypeCDAB1) (%the (subtype (pi C2 D2) (pi A B') _) DsubtypeCDAB2)) + %<- (subtype-pi-invert DsubtypeCDAB1 (%the (subtype A C1 Lac1) DsubtypeAC1) (%the ({x} tsub D1 (Lac1 x) (D1x x)) DsubD1x) _ _) + %<- (subtype-pi-invert DsubtypeCDAB2 (%the (subtype A C2 Lac2) DsubtypeAC2') (%the ({x} tsub D2 (Lac2 x) (D2x x)) DsubD2x) _ _) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} + %pi (vof-comp ed DtransA xt d) + %-> (equiv-comp (Dequiv ex ed) (%the (ttrans (EB ex) (B x)) (DtransB x d ex xt)) (trans/app (%the (tsub ([y] F1 x y) (M1 x) (F1y x)) (DsubF1y x)) (%the (subtype (H1 x) (E1 x) ([_] M1 x)) (DsubtypeH1E1 x)) (%the (trans ex (H1 x)) (DtransH1 x d ex xt)) (%the (trans EM1 (pi (E1 x) ([y] F1 x y))) (Dtrans1' x d ex xt))) (trans/app (%the (tsub ([y] F2 x y) (M2 x) (F2y x)) (DsubF2y x)) (%the (subtype (H2 x) (E2 x) ([_] M2 x)) (DsubtypeH2E2 x)) (%the (trans ex (H2 x)) (DtransH2 x d ex xt)) (%the (trans EM2 (pi (E2 x) ([y] F2 x y))) (Dtrans2' x d ex xt))) (%the (subtype (F1y x) (B x) (O x)) (DsubtypeF1yB x)) (%the (subtype (F2y x) (B x) (O x)) (DsubtypeF2yB x)))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} trans-fun (Dtrans1' x d ex xt) Dtrans1 (%the (tp-eq (pi (E1 x) (F1 x)) (pi C1 D1)) (DeqEFCD1 x))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} trans-fun (Dtrans2' x d ex xt) Dtrans2 (%the (tp-eq (pi (E2 x) (F2 x)) (pi C2 D2)) (DeqEFCD2 x))) + %<- ({x} tp-eq-cdr-pi (DeqEFCD1 x) (%the (tp-eq (E1 x) C1) (DeqE1C1 x)) (%the ({y} tp-eq (F1 x y) (D1 y)) (DeqF1D1 x))) + %<- ({x} tp-eq-cdr-pi (DeqEFCD2 x) (%the (tp-eq (E2 x) C2) (DeqE2C2 x)) (%the ({y} tp-eq (F2 x y) (D2 y)) (DeqF2D2 x))) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- ({x} can-expand x A (X x) (%the (expand x A (X x)) (Dexpand x))) + %<- ({x} {d vof x A} expand-reg (aof/var DwfA d) (Dexpand x) (%the (of (X x) A) (DofX x d))) + %<- ({x} {d vof x A} can-self (DofX x d) (%the (self (X x) A (As x)) (Dself x))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} trans-fun (DtransH1 x d ex xt) (trans/var (Dself x) (Dexpand x) DwfA d xt) (%the (tp-eq (H1 x) (As x)) (DeqH1 x))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} trans-fun (DtransH2 x d ex xt) (trans/var (Dself x) (Dexpand x) DwfA d xt) (%the (tp-eq (H2 x) (As x)) (DeqH2 x))) + %<- ({x} subtype-resp (DeqH1 x) (DeqE1C1 x) ([_] term-eq/i) (DsubtypeH1E1 x) (%the (subtype (As x) C1 ([_] M1 x)) (DsubtypeAsC1 x))) + %<- ({x} subtype-resp (DeqH2 x) (DeqE2C2 x) ([_] term-eq/i) (DsubtypeH2E2 x) (%the (subtype (As x) C2 ([_] M2 x)) (DsubtypeAsC2 x))) + %<- ({x} {d vof x A} self-reg (Dself x) (DofX x d) (%the (of (X x) (As x)) (DofX' x d)) (%the (subtype (As x) A ([_] X x)) (DsubtypeAsA x))) + %<- (trans-reg Dtrans1 (wf/pi (%the ({x} %pi (vof x C1) %-> (wf (D1 x))) DwfD1) (%the (wf C1) DwfC1))) + %<- (trans-reg Dtrans2 (wf/pi (%the ({x} %pi (vof x C2) %-> (wf (D2 x))) DwfD2) (%the (wf C2) DwfC2))) + %<- (subtype-reg DsubtypeAC1 DwfA DwfC1 (%the ({x} %pi (vof x A) %-> (of (Lac1 x) C1)) DofLac1)) + %<- (subtype-reg DsubtypeAC2' DwfA DwfC2 (%the ({x} %pi (vof x A) %-> (of (Lac2 x) C2)) DofLac2)) + %<- (sub-expand-var DwfA DofLac1 Dexpand (%the ({x} sub Lac1 (X x) (Lac1 x)) DsubLac1)) + %<- (sub-expand-var DwfA DofLac2 Dexpand (%the ({x} sub Lac2 (X x) (Lac2 x)) DsubLac2)) + %<- ({x} {d vof x A} of-reg (DofX' x d) (%the (wf (As x)) (DwfAs x d))) + %<- ({x} {d vof x A} subtype-trans (DwfAs x d) DwfA DwfC1 (DsubtypeAsA x) DsubtypeAC1 ([_] DsubLac1 x) (%the (subtype (As x) C1 ([_] Lac1 x)) (DsubtypeAsC1' x))) + %<- ({x} {d vof x A} subtype-trans (DwfAs x d) DwfA DwfC2 (DsubtypeAsA x) DsubtypeAC2' ([_] DsubLac2 x) (%the (subtype (As x) C2 ([_] Lac2 x)) (DsubtypeAsC2' x))) + %<- ({x} subtype-fun (DsubtypeAsC1 x) (DsubtypeAsC1' x) (%the ({y} term-eq (M1 x) (Lac1 x)) (DeqM1 x))) + %<- ({x} subtype-fun (DsubtypeAsC2 x) (DsubtypeAsC2' x) (%the ({y} term-eq (M2 x) (Lac2 x)) (DeqM2 x))) + %<- ({x} tsub-resp ([y] DeqF1D1 x y) (DeqM1 x aca) tp-eq/i (DsubF1y x) (%the (tsub D1 (Lac1 x) (F1y x)) (DsubF1y' x))) + %<- ({x} tsub-resp ([y] DeqF2D2 x y) (DeqM2 x aca) tp-eq/i (DsubF2y x) (%the (tsub D2 (Lac2 x) (F2y x)) (DsubF2y' x))) + %<- ({x} tsub-fun (DsubF1y' x) (DsubD1x x) (%the (tp-eq (F1y x) (D1x x)) (DeqD1x x))) + %<- ({x} tsub-fun (DsubF2y' x) (DsubD2x x) (%the (tp-eq (F2y x) (D2x x)) (DeqD2x x))) + %<- (trans-principal Dtrans1 (principal/pi (%the ({x} principal (D1 x)) DprincipalD1))) + %<- ({x} principal-sub DprincipalD1 (DsubD1x x) (%the (principal (D1x x)) (DprincipalD1x x))) + %<- ({x} subtype-resp (DeqD1x x) tp-eq/i ([_] term-eq/i) (DsubtypeF1yB x) (%the (subtype (D1x x) (B x) (O x)) (DsubtypeD1xB x))) + %<- ({x} subtype-resp (DeqD2x x) tp-eq/i ([_] term-eq/i) (DsubtypeF2yB x) (%the (subtype (D2x x) (B x) (O x)) (DsubtypeD2xB x))) + %<- ({x} principal-subtype (DprincipalD1x x) (DsubtypeD1xB x) (%the ({y} term-eq (O x y) (O' x)) (DeqO x))) + %<- ({x} subtype-resp tp-eq/i tp-eq/i (DeqO x) (DsubtypeD1xB x) (%the (subtype (D1x x) (B x) ([_] O' x)) (DsubtypeD1xB' x))) + %<- ({x} subtype-resp tp-eq/i tp-eq/i (DeqO x) (DsubtypeD2xB x) (%the (subtype (D2x x) (B x) ([_] O' x)) (DsubtypeD2xB' x))) %. +%scope equiv-comp %term extsig + %pi (equiv-comp (equiv/extsigma (%the ({ex} %pi (evof ex EA) %-> (ewf (EB ex))) Dewf) (%the (equiv (epi2 EM1) (epi2 EM2) (EB (epi1 EM1))) Dequiv2) (%the (equiv (epi1 EM1) (epi1 EM2) EA) Dequiv1)) (ttrans/sigma DtransB DtransA) Dtrans1 Dtrans2 (subtype/sigma ([_] DsubtypeD1Bx') ([_] DsubBx) DsubtypeC1A') (subtype/sigma ([_] DsubtypeD2Bx') ([_] DsubBx) DsubtypeC2A')) + %<- (equiv-comp Dequiv1 (%the (ttrans EA A) DtransA) (trans/pi1 (%the (trans EM1 (sigma C1 ([_] D1))) Dtrans1)) (trans/pi1 (%the (trans EM2 (sigma C2 ([_] D2))) Dtrans2)) (%the (subtype C1 A Lca) DsubtypeC1A) (%the (subtype C2 A Lca) DsubtypeC2A)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} + %pi (vof-comp ed DtransA xt d) + %-> (wf-comp (Dewf ex ed) (%the (ttrans (EB ex) (B x)) (DtransB x d ex xt)))) + %<- (trans-principal Dtrans1 (principal/sigma (%the (principal D1) DprincipalD1) (%the (principal C1) DprincipalC1))) + %<- (principal-subtype DprincipalC1 DsubtypeC1A (%the ({x} term-eq (Lca x) M) DeqLca)) + %<- (subtype-resp tp-eq/i tp-eq/i DeqLca DsubtypeC1A (%the (subtype C1 A ([_] M)) DsubtypeC1A')) + %<- (subtype-resp tp-eq/i tp-eq/i DeqLca DsubtypeC2A (%the (subtype C2 A ([_] M)) DsubtypeC2A')) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} ttrans-reg (DtransB x d ex xt) (%the (wf (B x)) (DwfB x d))) + %<- (trans-reg Dtrans1 (wf/sigma _ (%the (wf C1) DwfC1))) + %<- (subtype-reg' DsubtypeC1A' DwfC1 DwfA (%the (of M A) DofM)) + %<- (can-tsub DwfB DofM (%the (tsub B M Bx) DsubBx)) + %<- (ttrans-sub DtransB (trans/pi1 Dtrans1) DwfA DsubtypeC1A' DsubBx (%the (ttrans (EB (epi1 EM1)) Bx) DtransBx)) + %<- (equiv-comp Dequiv2 (%the (ttrans (EB (epi1 EM1)) Bx') DtransBx') (trans/pi2 (%the (trans EM1 (sigma C1' ([_] D1'))) Dtrans1')) (trans/pi2 (%the (trans EM2 (sigma C2' ([_] D2'))) Dtrans2')) (%the (subtype D1' Bx' Ldb) DsubtypeD1'Bx') (%the (subtype D2' Bx' Ldb) DsubtypeD2'Bx')) + %<- (ttrans-fun DtransBx' DtransBx (%the (tp-eq Bx' Bx) DeqBx)) + %<- (trans-fun Dtrans1' Dtrans1 (%the (tp-eq (sigma C1' ([_] D1')) (sigma C1 ([_] D1))) DeqCD1)) + %<- (trans-fun Dtrans2' Dtrans2 (%the (tp-eq (sigma C2' ([_] D2')) (sigma C2 ([_] D2))) DeqCD2)) + %<- (tp-eq-cdr-sigma DeqCD1 _ (%the ({x} tp-eq D1' D1) DeqD1)) + %<- (tp-eq-cdr-sigma DeqCD2 _ (%the ({x} tp-eq D2' D2) DeqD2)) + %<- (subtype-resp (DeqD1 aca) DeqBx ([_] term-eq/i) DsubtypeD1'Bx' (%the (subtype D1 Bx Ldb) DsubtypeD1Bx)) + %<- (subtype-resp (DeqD2 aca) DeqBx ([_] term-eq/i) DsubtypeD2'Bx' (%the (subtype D2 Bx Ldb) DsubtypeD2Bx)) + %<- (principal-subtype DprincipalD1 DsubtypeD1Bx (%the ({x} term-eq (Ldb x) N) DeqLdb)) + %<- (subtype-resp tp-eq/i tp-eq/i DeqLdb DsubtypeD1Bx (%the (subtype D1 Bx ([_] N)) DsubtypeD1Bx')) + %<- (subtype-resp tp-eq/i tp-eq/i DeqLdb DsubtypeD2Bx (%the (subtype D2 Bx ([_] N)) DsubtypeD2Bx')) %. +%term subsum + %pi (equiv-comp (equiv/subsume (%the (subtp EA EB) Dsubtp) (%the (equiv EM EN EA) Dequiv)) DtransB DtransM DtransN DsubtypeCB DsubtypeDB) + %<- (equiv-comp Dequiv (%the (ttrans EA A) DtransA) (%the (trans EM C) DtransM) (%the (trans EN D) DtransN) (%the (subtype C A O) DsubtypeCA) (%the (subtype D A O) DsubtypeDA)) + %<- (subtp-comp Dsubtp (%the (ttrans EA A') DtransA') (%the (ttrans EB B) DtransB) (%the (subtype A' B P) DsubtypeA'B)) + %<- (ttrans-fun DtransA' DtransA (%the (tp-eq A' A) DeqA)) + %<- (subtype-resp DeqA tp-eq/i ([_] term-eq/i) DsubtypeA'B (%the (subtype A B P) DsubtypeAB)) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (ttrans-reg DtransB (%the (wf B) DwfB)) + %<- (trans-reg DtransM (%the (wf C) DwfC)) + %<- (trans-reg DtransN (%the (wf D) DwfD)) + %<- (subtype-reg DsubtypeCA DwfC DwfA (%the ({x} %pi (vof x C) %-> (of (O x) A)) DofO)) + %<- (subtype-reg DsubtypeAB DwfA DwfB (%the ({x} %pi (vof x A) %-> (of (P x) B)) DofP)) + %<- ({x} {d vof x C} can-sub DofP (DofO x d) (%the (sub P (O x) (Px x)) (DsubPx x))) + %<- (subtype-trans DwfC DwfA DwfB DsubtypeCA DsubtypeAB DsubPx (%the (subtype C B Px) DsubtypeCB)) + %<- (subtype-trans DwfD DwfA DwfB DsubtypeDA DsubtypeAB DsubPx (%the (subtype D B Px) DsubtypeDB)) %. +%scope equiv-comp %term beta + %pi (equiv-comp (equiv/beta (%the (eof EN EA) DeofN) (%the ({ex} %pi (evof ex EA) %-> (eof (EM ex) (EB ex))) DeofM)) DtransBx (trans/app DsubDx DsubtypeCA' DtransN (trans/lam DtransM DtransA)) DtransMN DsubtypeDxBx DsubtypeEBx) + %<- (of-comp DeofN (%the (ttrans EA A) DtransA) (%the (trans EN C) DtransN) (%the (subtype C A Lca) DsubtypeCA)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} + %pi (vof-comp ed DtransA xt d) + %-> (of-comp (DeofM ex ed) (%the (ttrans (EB ex) (B x)) (DtransB x d ex xt)) (%the (trans (EM ex) (D x)) (DtransM x d ex xt)) (%the (subtype (D x) (B x) (Ldb x)) (DsubtypeDB x)))) + %<- (trans-principal DtransN (%the (principal C) DprincipalC)) + %<- (principal-subtype DprincipalC DsubtypeCA (%the ({x} term-eq (Lca x) N) DeqLca)) + %<- (subtype-resp tp-eq/i tp-eq/i DeqLca DsubtypeCA (%the (subtype C A ([_] N)) DsubtypeCA')) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} ttrans-reg (DtransB x d ex xt) (%the (wf (B x)) (DwfB x d))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} trans-reg (DtransM x d ex xt) (%the (wf (D x)) (DwfD x d))) + %<- (trans-reg DtransN (%the (wf C) DwfC)) + %<- (subtype-reg' DsubtypeCA' DwfC DwfA (%the (of N A) DofN)) + %<- (can-tsub DwfB DofN (%the (tsub B N Bx) DsubBx)) + %<- (can-tsub DwfD DofN (%the (tsub D N Dx) DsubDx)) + %<- (ttrans-sub DtransB DtransN DwfA DsubtypeCA' DsubBx (%the (ttrans (EB EN) Bx) DtransBx)) + %<- (trans-sub DtransM DtransN DwfA DsubtypeCA' DsubDx (%the (trans (EM EN) E) DtransMN) (%the (subtype E Dx Led) DsubtypeEDx)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} trans-principal (DtransM x d ex xt) (%the (principal (D x)) (DprincipalD x))) + %<- ({x} principal-subtype (DprincipalD x) (DsubtypeDB x) (%the ({y} term-eq (Ldb x y) (M x)) (DeqLdb x))) + %<- ({x} subtype-resp tp-eq/i tp-eq/i (DeqLdb x) (DsubtypeDB x) (%the (subtype (D x) (B x) ([_] M x)) (DsubtypeDB' x))) + %<- ({x} {d vof x A} subtype-reg' (DsubtypeDB' x) (DwfD x d) (DwfB x d) (%the (of (M x) (B x)) (DofM x d))) + %<- (can-sub DofM DofN (%the (sub M N Mx) DsubMx)) + %<- (subtype-sub' DsubtypeDB' DwfD DwfB DofN DsubDx DsubBx ([_] DsubMx) (%the (subtype Dx Bx ([_] Mx)) DsubtypeDxBx)) + %<- ({x} sub-absent Mx (Led x) (%the (sub ([_] Mx) (Led x) Mx) (DsubMxabs x))) + %<- (trans-reg DtransMN (%the (wf E) DwfE)) + %<- (tsubst DsubDx DwfD DofN (%the (wf Dx) DwfDx)) + %<- (tsubst DsubBx DwfB DofN (%the (wf Bx) DwfBx)) + %<- (subtype-trans DwfE DwfDx DwfBx DsubtypeEDx DsubtypeDxBx DsubMxabs (%the (subtype E Bx ([_] Mx)) DsubtypeEBx)) %. +%term beta1 + %pi (equiv-comp (equiv/beta1 (%the (eof EN EB) DeofN) (%the (eof EM EA) DeofM)) DtransA (trans/pi1 (trans/pair DtransN DtransM)) DtransM DsubtypeCA DsubtypeCA) + %<- (of-comp DeofM (%the (ttrans EA A) DtransA) (%the (trans EM C) DtransM) (%the (subtype C A _) DsubtypeCA)) + %<- (of-comp DeofN (%the (ttrans EB B) DtransB) (%the (trans EN D) DtransN) (%the (subtype D B _) DsubtypeDB)) %. +%scope equiv-comp %term beta2 + %pi (equiv-comp (equiv/beta2 (%the (eof EN EB) DeofN) (%the (eof EM EA) DeofM)) DtransB (trans/pi2 (trans/pair DtransN DtransM)) DtransN DsubtypeDB DsubtypeDB) + %<- (of-comp DeofM (%the (ttrans EA A) DtransA) (%the (trans EM C) DtransM) (%the (subtype C A _) DsubtypeCA)) + %<- (of-comp DeofN (%the (ttrans EB B) DtransB) (%the (trans EN D) DtransN) (%the (subtype D B _) DsubtypeDB)) %. +%worlds (bind cbind scbind) (vof-comp _ _ _ _) (of-comp _ _ _ _) (wf-comp _ _) (subtp-comp _ _ _ _) (tequiv-comp _ _ _) (equiv-comp _ _ _ _ _ _) %. +%total (D0 D1 D2 D3 D4 D5) (vof-comp D0 _ _ _) (of-comp D1 _ _ _) (wf-comp D2 _) (subtp-comp D3 _ _ _) (tequiv-comp D4 _ _) (equiv-comp D5 _ _ _ _ _) %. +%sort of-comp' %. +%mode of-comp' %in %out %out %out %. +%term _ + %pi (of-comp' Dof DtransA DtransM Dsubtype') + %<- (of-comp Dof DtransA DtransM Dsubtype) + %<- (trans-subtype DtransM Dsubtype Dsubtype' _) %. +%worlds (bind cbind scbind) (of-comp' _ _ _ _) %. +%total {} (of-comp' _ _ _ _) %. +%sort equiv-comp' %. +%mode equiv-comp' %in %out %out %out %out %out %. +%term _ + %pi (equiv-comp' Dequiv DtransA DtransM DtransN DsubtypeBA' DsubtypeCA') + %<- (equiv-comp Dequiv DtransA DtransM DtransN DsubtypeBA DsubtypeCA) + %<- (trans-subtype DtransM DsubtypeBA DsubtypeBA' Deq) + %<- (subtype-resp tp-eq/i tp-eq/i Deq DsubtypeCA DsubtypeCA') %. +%worlds (bind cbind scbind) (equiv-comp' _ _ _ _ _ _) %. +%total {} (equiv-comp' _ _ _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/constant.lf b/new-tests/stelf-output/tslf/sing/constant.lf new file mode 100644 index 0000000..280d6a9 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/constant.lf @@ -0,0 +1,24 @@ +%sort constant %. +%name constant %. +%sort ctp %. +%name ctp %. +%term ct ctp %. +%term cpi %pi ctp %-> ctp %-> ctp %. +%sort ckof %. +%mode ckof %in %out %. +%term arrow constant %. +%term ckof/arrow ckof arrow (cpi ct (cpi ct ct)) %. +%term cross constant %. +%term ckof/cross ckof cross (cpi ct (cpi ct ct)) %. +%term unit constant %. +%term ckof/unit ckof unit ct %. +%term ref constant %. +%term ckof/ref ckof ref (cpi ct ct) %. +%term dsum constant %. +%term ckof/dsum ckof dsum (cpi ct (cpi ct ct)) %. +%term tag constant %. +%term ckof/tag ckof tag (cpi ct ct) %. +%term tagged constant %. +%term ckof/tagged ckof tagged ct %. +%term mu constant %. +%term ckof/mu ckof mu (cpi (cpi ct ct) ct) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/convert-effect-thm.lf b/new-tests/stelf-output/tslf/sing/convert-effect-thm.lf new file mode 100644 index 0000000..045fa56 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/convert-effect-thm.lf @@ -0,0 +1,86 @@ +%%%%% Converting Variables and Worlds %%%%% +%define eaca eterm econst unit %. +%sort vof-const-contra %. +%mode vof-const-contra %in %out %. +%worlds (bind) (vof-const-contra _ _) %. +%total {} (vof-const-contra _ _) %. +%sort vsound %. +%mode vsound %in %out %out %out %. +%sort vtrans-variable %. +%mode vtrans-variable %in %out %. +%block sbind {a tp} {ea etp} {d_tconvert tconvert a ea} [x atom] [d vof x a] [ex eterm] [xt vtrans ex x] [ed evof ex ea] [dv variable ex] [thm1 vtrans-variable xt dv] [thm2 vsound d xt d_tconvert ed]%. +%% Without the d_tconvert assumption, Twelf will split this. +%block sbind' {a tp} {ea etp} [x atom] [d vof x a] [ex eterm] [xt vtrans ex x] [ed evof ex ea] [dv variable ex] [thm1 vtrans-variable xt dv]%. +%block tvbind {r atom} [ex eterm] [tr vtrans ex r] [dv variable ex] [thm vtrans-variable tr dv]%. +%worlds (sbind' ovar tvbind evar) (vtrans-variable _ _) %. +%total {} (vtrans-variable _ _) %. +%% force Twelf to split X +%term _ + %pi (vsound (%the (vof aca _) Dvof) D1 D2 D3) + %<- (vof-const-contra Dvof Dfalse) + %<- (false-implies-vtrans Dfalse D1) + %<- (false-implies-tconvert Dfalse D2) + %<- (false-implies-evof Dfalse (%the (evof eaca et) D3)) %. +%worlds (sbind) (vsound _ _ _ _) %. +%total {} (vsound _ _ _ _) %. +%%%%% Conversion Effectiveness %%%%% +%sort can-tconvert %. +%mode can-tconvert %in %out %. +%sort can-convert %. +%mode can-convert %in %out %. +%sort can-aconvert %. +%mode can-aconvert %in %out %. +%scope can-aconvert %term const + %pi (can-aconvert (aof/const Dwf Dcof) (aconvert/const Dconv Dwf Dcof)) + %<- (can-tconvert Dwf Dconv) %. +%term forall + %pi (can-aconvert (aof/forall Dwf) (aconvert/forall Dconv)) + %<- (can-tconvert Dwf Dconv) %. +%scope can-aconvert %term var + %pi (can-aconvert (aof/var Dwf Dvof) (aconvert/var Dvtrans Dconvert Dwf Dvof)) + %<- (vsound Dvof Dvtrans _ _) + %<- (can-tconvert Dwf Dconvert) %. +%term app + %pi (can-aconvert (aof/app _ Dsub DofM DofR) (aconvert/app Dsub DconvM DconvR)) + %<- (can-aconvert DofR DconvR) + %<- (can-convert DofM DconvM) %. +%scope can-aconvert %term pi1 + %pi (can-aconvert (aof/pi1 Dof) (aconvert/pi1 Dconv)) + %<- (can-aconvert Dof Dconv) %. +%term pi2 + %pi (can-aconvert (aof/pi2 Dof) (aconvert/pi2 Dconv)) + %<- (can-aconvert Dof Dconv) %. +%scope can-aconvert %term at %pi (can-convert (of/at Dof) (convert/at Dconv)) %<- (can-aconvert Dof Dconv) %. +%term lam + %pi (can-convert (of/lam Dof Dwf) (convert/lam Dconv Dtconv)) + %<- (can-tconvert Dwf Dtconv) + %<- ({x} {d} {ex} {xt} {ed} {dv} + %pi (vtrans-variable xt dv) + %-> (vsound d xt Dtconv ed) + %-> (can-convert (Dof x d) (Dconv x d ex xt))) %. +%scope can-aconvert %term pair + %pi (can-convert (of/pair Dwf Dof2 Dsub Dof1) (convert/pair Dwf Dconv2 Dsub Dconv1)) + %<- (can-convert Dof1 Dconv1) + %<- (can-convert Dof2 Dconv2) %. +%term at %pi (can-convert (of/sing Dof) (convert/sing Dconv)) %<- (can-aconvert Dof Dconv) %. +%scope can-aconvert %term t can-tconvert wf/t tconvert/t %. +%term pi + %pi (can-tconvert (wf/pi Dwf' Dwf) (tconvert/pi Dtconv' Dtconv)) + %<- (can-tconvert Dwf Dtconv) + %<- ({x} {d} {ex} {xt} {ed} {dv} + %pi (vtrans-variable xt dv) + %-> (vsound d xt Dtconv ed) + %-> (can-tconvert (Dwf' x d) (Dtconv' x d ex xt))) %. +%scope can-aconvert %term sigma + %pi (can-tconvert (wf/sigma Dwf' Dwf) (tconvert/sigma Dtconv' Dtconv)) + %<- (can-tconvert Dwf Dtconv) + %<- ({x} {d} {ex} {xt} {ed} {dv} + %pi (vtrans-variable xt dv) + %-> (vsound d xt Dtconv ed) + %-> (can-tconvert (Dwf' x d) (Dtconv' x d ex xt))) %. +%term sing + %pi (can-tconvert (wf/sing Dof) (tconvert/sing Dconv)) + %<- (can-aconvert Dof Dconv) %. +%sort _ %. +%worlds (sbind) (can-aconvert _ _) (can-convert _ _) (can-tconvert _ _) %. +%total (D1 D2 D3) (can-aconvert D1 _) (can-convert D2 _) (can-tconvert D3 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/convert-explicit-thm.lf b/new-tests/stelf-output/tslf/sing/convert-explicit-thm.lf new file mode 100644 index 0000000..8696efb --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/convert-explicit-thm.lf @@ -0,0 +1,678 @@ +%%%%% Worlds %%%%% +%block fbind {a tp} [x atom] [d vof x a] [ex eterm] [xt vtrans ex x]%. +%block fobind {n nat} [x atom] [d isvar x n] [ex eterm] [xt vtrans ex x]%. +%%%%% Context Regularity %%%%% +%sort aconverte-context %. +%mode aconverte-context %in %out %. +%sort converte-context %. +%mode converte-context %in %out %. +%sort tconverte-context %. +%mode tconverte-context %in %out %. +%scope tconverte-context %term const %pi (aconverte-context (aconverte/const D _ _) D') %<- (tconverte-context D D') %. +%term forall %pi (aconverte-context (aconverte/forall D) D') %<- (tconverte-context D D') %. +%scope tconverte-context %term var %pi (aconverte-context (aconverte/var _ _ Dlook) D) %<- (lookup-context Dlook D) %. +%term vari + %pi (aconverte-context (aconverte/vari _ Dconv _ _) D) + %<- (tconverte-context Dconv D) %. +%scope tconverte-context %term app + %pi (aconverte-context (aconverte/app _ _ Dconv) D) + %<- (aconverte-context Dconv D) %. +%term pi1 %pi (aconverte-context (aconverte/pi1 Dconv) D) %<- (aconverte-context Dconv D) %. +%scope tconverte-context %term pi2 %pi (aconverte-context (aconverte/pi2 Dconv) D) %<- (aconverte-context Dconv D) %. +%term at %pi (converte-context (converte/at Dconv) D) %<- (aconverte-context Dconv D) %. +%scope tconverte-context %term lam %pi (converte-context (converte/lam _ Dconv) D) %<- (tconverte-context Dconv D) %. +%term pair + %pi (converte-context (converte/pair _ _ _ Dconv) D) + %<- (converte-context Dconv D) %. +%scope tconverte-context %term sing %pi (converte-context (converte/sing Dconv) D) %<- (aconverte-context Dconv D) %. +%term t tconverte-context (tconverte/t D) D %. +%scope tconverte-context %term pi %pi (tconverte-context (tconverte/pi _ Dconv) D) %<- (tconverte-context Dconv D) %. +%term sigma + %pi (tconverte-context (tconverte/sigma _ Dconv) D) + %<- (tconverte-context Dconv D) %. +%scope tconverte-context %term sing %pi (tconverte-context (tconverte/sing Dconv) D) %<- (aconverte-context Dconv D) %. +%worlds (fbind fobind bind ovar obind tbind evar) (aconverte-context _ _) (converte-context _ _) (tconverte-context _ _) %. +%total (D1 D2 D3) (aconverte-context D1 _) (converte-context D2 _) (tconverte-context D3 _) %. +%%%%% Conversion Metric %%%%% +%sort mconvert %. +%sort maconvert %. +%sort mtconvert %. +%term mconvert/at %pi (mconvert (convert/at D) (mat X)) %<- (maconvert D X) %. +%term mconvert/lam + %pi (mconvert (convert/lam D2 D1) (mlam X2 X1)) + %<- (mtconvert D1 X1) + %<- ({x} {d} {ex} {xt} mconvert (D2 x d ex xt) X2) %. +%term mconvert/pair + %pi (mconvert (convert/pair _ D2 _ D1) (mpair mt X2 X1)) + %<- (mconvert D1 X1) + %<- (mconvert D2 X2) %. +%term mconvert/sing %pi (mconvert (convert/sing D) (msingtm X)) %<- (maconvert D X) %. +%term maconvert/const %pi (maconvert (aconvert/const D _ _) (mvar X)) %<- (mtconvert D X) %. +%term maconvert/forall %pi (maconvert (aconvert/forall D) (mvar X)) %<- (mtconvert D X) %. +%term maconvert/var %pi (maconvert (aconvert/var _ D _ _) (mvar X)) %<- (mtconvert D X) %. +%term maconvert/app + %pi (maconvert (aconvert/app _ D2 D1) (mapp mt X2 X1)) + %<- (maconvert D1 X1) + %<- (mconvert D2 X2) %. +%term maconvert/pi1 %pi (maconvert (aconvert/pi1 D) (mpi1 X)) %<- (maconvert D X) %. +%term maconvert/pi2 %pi (maconvert (aconvert/pi2 D) (mpi2 X)) %<- (maconvert D X) %. +%term mtconvert/t mtconvert tconvert/t mt %. +%term mtconvert/pi + %pi (mtconvert (tconvert/pi D2 D1) (mpi X2 X1)) + %<- (mtconvert D1 X1) + %<- ({x} {d} {ex} {xt} mtconvert (D2 x d ex xt) X2) %. +%term mtconvert/sigma + %pi (mtconvert (tconvert/sigma D2 D1) (msigma X2 X1)) + %<- (mtconvert D1 X1) + %<- ({x} {d} {ex} {xt} mtconvert (D2 x d ex xt) X2) %. +%term mtconvert/sing %pi (mtconvert (tconvert/sing D) (msing X)) %<- (maconvert D X) %. +%sort mconverte %. +%sort maconverte %. +%sort mtconverte %. +%term mconverte/at %pi (mconverte (converte/at D) (mat X)) %<- (maconverte D X) %. +%term mconverte/lam + %pi (mconverte (converte/lam D2 D1) (mlam X2 X1)) + %<- (mtconverte D1 X1) + %<- ({x} {d} {ex} {xt} mconverte (D2 x d ex xt) X2) %. +%term mconverte/pair + %pi (mconverte (converte/pair _ D2 _ D1) (mpair mt X2 X1)) + %<- (mconverte D1 X1) + %<- (mconverte D2 X2) %. +%term mconverte/sing %pi (mconverte (converte/sing D) (msingtm X)) %<- (maconverte D X) %. +%term maconverte/const %pi (maconverte (aconverte/const D _ _) (mvar X)) %<- (mtconverte D X) %. +%term maconverte/forall %pi (maconverte (aconverte/forall D) (mvar X)) %<- (mtconverte D X) %. +%term maconverte/var %pi (maconverte (aconverte/var _ D _) (mvar X)) %<- (mtconverte D X) %. +%term maconverte/vari %pi (maconverte (aconverte/vari _ D _ _) (mvar X)) %<- (mtconverte D X) %. +%term maconverte/app + %pi (maconverte (aconverte/app _ D2 D1) (mapp mt X2 X1)) + %<- (maconverte D1 X1) + %<- (mconverte D2 X2) %. +%term maconverte/pi1 %pi (maconverte (aconverte/pi1 D) (mpi1 X)) %<- (maconverte D X) %. +%term maconverte/pi2 %pi (maconverte (aconverte/pi2 D) (mpi2 X)) %<- (maconverte D X) %. +%term mtconverte/t mtconverte (tconverte/t _) mt %. +%term mtconverte/pi + %pi (mtconverte (tconverte/pi D2 D1) (mpi X2 X1)) + %<- (mtconverte D1 X1) + %<- ({x} {d} {ex} {xt} mtconverte (D2 x d ex xt) X2) %. +%term mtconverte/sigma + %pi (mtconverte (tconverte/sigma D2 D1) (msigma X2 X1)) + %<- (mtconverte D1 X1) + %<- ({x} {d} {ex} {xt} mtconverte (D2 x d ex xt) X2) %. +%term mtconverte/sing %pi (mtconverte (tconverte/sing D) (msing X)) %<- (maconverte D X) %. +%sort can-mconvert {D convert _ _ _} %. +%mode can-mconvert %in %out %. +%sort can-maconvert {D aconvert _ _ _} %. +%mode can-maconvert %in %out %. +%sort can-mtconvert {D tconvert _ _} %. +%mode can-mtconvert %in %out %. +%scope can-mtconvert %term at %pi (can-mconvert (convert/at D) (mconvert/at D')) %<- (can-maconvert D D') %. +%term lam + %pi (can-mconvert (convert/lam D2 D1) (mconvert/lam D2' D1')) + %<- (can-mtconvert D1 D1') + %<- ({x} {d} {ex} {xt} can-mconvert (D2 x d ex xt) (D2' x d ex xt)) %. +%scope can-mtconvert %term pair + %pi (can-mconvert (convert/pair _ D2 _ D1) (mconvert/pair D2' D1')) + %<- (can-mconvert D1 D1') + %<- (can-mconvert D2 D2') %. +%term sing %pi (can-mconvert (convert/sing D) (mconvert/sing D')) %<- (can-maconvert D D') %. +%scope can-mtconvert %term const + %pi (can-maconvert (aconvert/const D _ _) (maconvert/const D')) + %<- (can-mtconvert D D') %. +%term forall + %pi (can-maconvert (aconvert/forall D) (maconvert/forall D')) + %<- (can-mtconvert D D') %. +%scope can-mtconvert %term var + %pi (can-maconvert (aconvert/var _ D _ _) (maconvert/var D')) + %<- (can-mtconvert D D') %. +%term app + %pi (can-maconvert (aconvert/app _ D2 D1) (maconvert/app D2' D1')) + %<- (can-maconvert D1 D1') + %<- (can-mconvert D2 D2') %. +%scope can-mtconvert %term pi1 %pi (can-maconvert (aconvert/pi1 D) (maconvert/pi1 D')) %<- (can-maconvert D D') %. +%term pi2 %pi (can-maconvert (aconvert/pi2 D) (maconvert/pi2 D')) %<- (can-maconvert D D') %. +%scope can-mtconvert %term t can-mtconvert tconvert/t mtconvert/t %. +%term pi + %pi (can-mtconvert (tconvert/pi D2 D1) (mtconvert/pi D2' D1')) + %<- (can-mtconvert D1 D1') + %<- ({x} {d} {ex} {xt} can-mtconvert (D2 x d ex xt) (D2' x d ex xt)) %. +%scope can-mtconvert %term sigma + %pi (can-mtconvert (tconvert/sigma D2 D1) (mtconvert/sigma D2' D1')) + %<- (can-mtconvert D1 D1') + %<- ({x} {d} {ex} {xt} can-mtconvert (D2 x d ex xt) (D2' x d ex xt)) %. +%term sing + %pi (can-mtconvert (tconvert/sing D) (mtconvert/sing D')) + %<- (can-maconvert D D') %. +%worlds (fbind tbind evar bind var) (can-mconvert _ _) (can-maconvert _ _) (can-mtconvert _ _) %. +%total (D1 D2 D3) (can-mconvert D1 _) (can-maconvert D2 _) (can-mtconvert D3 _) %. +%sort can-mconverte {D converte _ _ _ _} %. +%mode can-mconverte %in %out %. +%sort can-maconverte {D aconverte _ _ _ _} %. +%mode can-maconverte %in %out %. +%sort can-mtconverte {D tconverte _ _ _} %. +%mode can-mtconverte %in %out %. +%scope can-mtconverte %term at %pi (can-mconverte (converte/at D) (mconverte/at D')) %<- (can-maconverte D D') %. +%term lam + %pi (can-mconverte (converte/lam D2 D1) (mconverte/lam D2' D1')) + %<- (can-mtconverte D1 D1') + %<- ({x} {d} {ex} {xt} can-mconverte (D2 x d ex xt) (D2' x d ex xt)) %. +%scope can-mtconverte %term pair + %pi (can-mconverte (converte/pair _ D2 _ D1) (mconverte/pair D2' D1')) + %<- (can-mconverte D1 D1') + %<- (can-mconverte D2 D2') %. +%term sing + %pi (can-mconverte (converte/sing D) (mconverte/sing D')) + %<- (can-maconverte D D') %. +%scope can-mtconverte %term const + %pi (can-maconverte (aconverte/const D _ _) (maconverte/const D')) + %<- (can-mtconverte D D') %. +%term forall + %pi (can-maconverte (aconverte/forall D) (maconverte/forall D')) + %<- (can-mtconverte D D') %. +%scope can-mtconverte %term var + %pi (can-maconverte (aconverte/var _ D _) (maconverte/var D')) + %<- (can-mtconverte D D') %. +%term vari + %pi (can-maconverte (aconverte/vari _ D _ _) (maconverte/vari D')) + %<- (can-mtconverte D D') %. +%scope can-mtconverte %term app + %pi (can-maconverte (aconverte/app _ D2 D1) (maconverte/app D2' D1')) + %<- (can-maconverte D1 D1') + %<- (can-mconverte D2 D2') %. +%term pi1 + %pi (can-maconverte (aconverte/pi1 D) (maconverte/pi1 D')) + %<- (can-maconverte D D') %. +%scope can-mtconverte %term pi2 + %pi (can-maconverte (aconverte/pi2 D) (maconverte/pi2 D')) + %<- (can-maconverte D D') %. +%term t can-mtconverte (tconverte/t _) mtconverte/t %. +%scope can-mtconverte %term pi + %pi (can-mtconverte (tconverte/pi D2 D1) (mtconverte/pi D2' D1')) + %<- (can-mtconverte D1 D1') + %<- ({x} {d} {ex} {xt} can-mtconverte (D2 x d ex xt) (D2' x d ex xt)) %. +%term sigma + %pi (can-mtconverte (tconverte/sigma D2 D1) (mtconverte/sigma D2' D1')) + %<- (can-mtconverte D1 D1') + %<- ({x} {d} {ex} {xt} can-mtconverte (D2 x d ex xt) (D2' x d ex xt)) %. +%scope can-mtconverte %term sing + %pi (can-mtconverte (tconverte/sing D) (mtconverte/sing D')) + %<- (can-maconverte D D') %. +%worlds (fbind tbind evar bind var fobind ovar) (can-mconverte _ _) (can-maconverte _ _) (can-mtconverte _ _) %. +%total (D1 D2 D3) (can-mconverte D1 _) (can-maconverte D2 _) (can-mtconverte D3 _) %. +%%%%% Translation to Explicit Form %%%%% +%sort cut-convert {Mm} {D {x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (convert (M x) (B x) (EM ex)))} {x} {d} {ex} {xt} %. +%mode cut-convert %in %in %in %in %out %out %. +%sort cut-aconvert {Rm} {D {x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (aconvert (R x) (B x) (EM ex)))} {x} {d} {ex} {xt} %. +%mode cut-aconvert %in %in %in %in %out %out %. +%sort cut-tconvert {Am} {D {x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))} {x} {d} {ex} {xt} %. +%mode cut-tconvert %in %in %in %in %out %out %. +%sort cut-converte {Mm} {D {x} + %pi (vof x A) + %-> (isvar x I) + %-> ({ex} %pi (vtrans ex x) %-> (converte (G x) (M x) (B x) (EM ex)))} {x} {d} {d'} {ex} {xt} %. +%mode cut-converte %in %in %in %in %out %out %. +%sort cut-aconverte {Rm} {D {x} + %pi (vof x A) + %-> (isvar x I) + %-> ({ex} %pi (vtrans ex x) %-> (aconverte (G x) (R x) (B x) (EM ex)))} {x} {d} {d'} {ex} {xt} %. +%mode cut-aconverte %in %in %in %in %out %out %. +%sort cut-tconverte {Am} {D {x} + %pi (vof x A) + %-> (isvar x I) + %-> ({ex} %pi (vtrans ex x) %-> (tconverte (G x) (B x) (EM ex)))} {x} {d} {d'} {ex} {xt} %. +%mode cut-tconverte %in %in %in %in %out %out %. +%scope cut-tconverte %term const + %pi (cut-aconvert _ ([x] [d] [ex] [xt] aconvert/const (Dconv x d ex xt) (Dwf x d) (D x)) ([x] [d] [ex] [xt] maconvert/const (X3 x d ex xt)) Dlook ([x] [d'] [ex] [xt] aconverte/const (Dconv' x d' ex xt) (Dwf''' x) (D x)) ([x] [d'] [ex] [xt] maconverte/const (X3' x d' ex xt))) + %<- (kof-strengthen D Deq) + %<- ({x} tp-eq-symm (Deq x) (Deq' x)) + %<- ({x} {d} wf-resp (Deq x) (Dwf x d) (Dwf' x d)) + %<- (strengthen-wf Dwf' Dwf'') + %<- ({x} wf-resp (Deq' x) Dwf'' (Dwf''' x)) + %<- (cut-tconvert _ Dconv X3 Dlook Dconv' X3') %. +%term forall + %pi (cut-aconvert _ ([x] [d] [ex] [xt] aconvert/forall (D1 x d ex xt)) ([x] [d] [ex] [xt] maconvert/forall (X1 x d ex xt)) Dlook ([x] [d'] [ex] [xt] aconverte/forall (D1' x d' ex xt)) ([x] [d'] [ex] [xt] maconverte/forall (X1' x d' ex xt))) + %<- (cut-tconvert _ D1 X1 Dlook D1' X1') %. +%scope cut-tconverte %term varsam + %pi (cut-aconvert _ ([x] [d] [ex] [xt] aconvert/var (D3 x ex xt) (D2 x d ex xt) (Dwf x d) d) ([x] [d] [ex] [xt] maconvert/var (X2 x d ex xt)) Dlook ([x] [d'] [ex] [xt] aconverte/var (D3 x ex xt) (D2' x d' ex xt) (Dlook x d')) ([x] [d'] [ex] [xt] maconverte/var (X2' x d' ex xt))) + %<- (cut-tconvert _ D2 X2 Dlook D2' X2') %. +%term varoth + %pi (cut-aconvert _ ([x] [d] [ex] [xt] aconvert/var (D3 x ex xt) (D2 x d ex xt) (Dwf x d) D1) ([x] [d] [ex] [xt] maconvert/var (X2 x d ex xt)) Dlook ([x] [d'] [ex] [xt] aconverte/vari (D3 x ex xt) (D2' x d' ex xt) Dwf' D1) ([x] [d'] [ex] [xt] maconverte/vari (X2' x d' ex xt))) + %<- (cut-tconvert _ D2 X2 Dlook D2' X2') + %<- (strengthen-wf Dwf Dwf') %. +%scope cut-tconverte %term app + %pi (cut-aconvert _ ([x] [d] [ex] [xt] aconvert/app (D3 x) (D2 x d ex xt) (D1 x d ex xt)) ([x] [d] [ex] [xt] maconvert/app (X2 x d ex xt) (X1 x d ex xt)) Dlook ([x] [d'] [ex] [xt] aconverte/app (D3 x) (D2' x d' ex xt) (D1' x d' ex xt)) ([x] [d'] [ex] [xt] maconverte/app (X2' x d' ex xt) (X1' x d' ex xt))) + %<- (cut-aconvert _ D1 X1 Dlook D1' X1') + %<- (cut-convert _ D2 X2 Dlook D2' X2') %. +%term pi1 + %pi (cut-aconvert _ ([x] [d] [ex] [xt] aconvert/pi1 (D1 x d ex xt)) ([x] [d] [ex] [xt] maconvert/pi1 (X1 x d ex xt)) Dlook ([x] [d'] [ex] [xt] aconverte/pi1 (D1' x d' ex xt)) ([x] [d'] [ex] [xt] maconverte/pi1 (X1' x d' ex xt))) + %<- (cut-aconvert _ D1 X1 Dlook D1' X1') %. +%scope cut-tconverte %term pi2 + %pi (cut-aconvert _ ([x] [d] [ex] [xt] aconvert/pi2 (D1 x d ex xt)) ([x] [d] [ex] [xt] maconvert/pi2 (X1 x d ex xt)) Dlook ([x] [d'] [ex] [xt] aconverte/pi2 (D1' x d' ex xt)) ([x] [d'] [ex] [xt] maconverte/pi2 (X1' x d' ex xt))) + %<- (cut-aconvert _ D1 X1 Dlook D1' X1') %. +%term at + %pi (cut-convert _ ([x] [d] [ex] [xt] convert/at (D1 x d ex xt)) ([x] [d] [ex] [xt] mconvert/at (X1 x d ex xt)) Dlook ([x] [d'] [ex] [xt] converte/at (D1' x d' ex xt)) ([x] [d'] [ex] [xt] mconverte/at (X1' x d' ex xt))) + %<- (cut-aconvert _ D1 X1 Dlook D1' X1') %. +%%% +%scope cut-tconverte %term lam + %pi (cut-convert _ ([x] [d] [ex] [xt] convert/lam (D2 x d ex xt) (D1 x d ex xt)) ([x] [d] [ex] [xt] mconvert/lam (X2 x d ex xt) (X1 x d ex xt)) Dlook ([x] [d'] [ex] [xt] converte/lam (D2'' x d' ex xt) (D1' x d' ex xt)) ([x] [d'] [ex] [xt] mconverte/lam (X2'' x d' ex xt) (X1' x d' ex xt))) + %<- (cut-tconvert _ D1 X1 Dlook D1' X1') + %<- ({x} {d'} lookup-context (Dlook x d') (Dord x d')) + %<- ({x} {d'} extend-context (Dord x d') (%the ({y} %pi (isvar y J) %-> (bounded (G x) y)) (Dbound x d'))) + %<- ({x} {d} {d'} {ex} {xt} cut-convert _ ([y] [e] [ey] [yt] D2 x d ex xt y e ey yt) ([y] [e] [ey] [yt] X2 x d ex xt y e ey yt) ([y] [e'] lookup/hit (Dbound x d' y e')) ([y] [e'] [ey] [yt] D2' x d d' ex xt y e' ey yt) ([y] [e'] [ey] [yt] X2' x d d' ex xt y e' ey yt)) + %<- ({y} {e'} {ey} {yt} cut-converte _ ([x] [d] [d'] [ex] [xt] D2' x d d' ex xt y e' ey yt) ([x] [d] [d'] [ex] [xt] X2' x d d' ex xt y e' ey yt) ([x] [d'] lookup/miss (Dlook x d') (Dbound x d' y e')) ([x] [d'] [ex] [xt] D2'' x d' ex xt y e' ey yt) ([x] [d'] [ex] [xt] X2'' x d' ex xt y e' ey yt)) %. +%term pair + %pi (cut-convert _ ([x] [d] [ex] [xt] convert/pair (Dwf x d) (D3 x d ex xt) (D2 x) (D1 x d ex xt)) ([x] [d] [ex] [xt] mconvert/pair (X3 x d ex xt) (X1 x d ex xt)) Dlook ([x] [d'] [ex] [xt] converte/pair (Dwfe' x d') (D3' x d' ex xt) (D2 x) (D1' x d' ex xt)) ([x] [d'] [ex] [xt] mconverte/pair (X3' x d' ex xt) (X1' x d' ex xt))) + %<- (cut-convert _ D1 X1 Dlook D1' X1') + %<- (cut-convert _ D3 X3 Dlook D3' X3') + %<- ({x} {d'} lookup-context (Dlook x d') (Dord x d')) + %<- ({x} {d'} extend-context (Dord x d') (%the ({y} %pi (isvar y J) %-> (bounded (G x) y)) (Dbound x d'))) + %<- ({x} {d} {y} {e} can-mwf (Dwf x d y e) (Dmwf x d y e)) + %<- ({x} {d} {d'} cut-wf _ ([y] [e] Dwf x d y e) ([y] [e] Dmwf x d y e) ([y] [e'] lookup/hit (Dbound x d' y e')) ([y] [e'] Dwfe x d d' y e') ([y] [e'] Dmwfe x d d' y e')) + %<- ({y} {e'} cut-wfe _ ([x] [d] [d'] Dwfe x d d' y e') ([x] [d] [d'] Dmwfe x d d' y e') ([x] [d'] lookup/miss (Dlook x d') (Dbound x d' y e')) ([x] [d'] Dwfe' x d' y e') _) %. +%scope cut-tconverte %term sing + %pi (cut-convert _ ([x] [d] [ex] [xt] convert/sing (D1 x d ex xt)) ([x] [d] [ex] [xt] mconvert/sing (X1 x d ex xt)) Dlook ([x] [d'] [ex] [xt] converte/sing (D1' x d' ex xt)) ([x] [d'] [ex] [xt] mconverte/sing (X1' x d' ex xt))) + %<- (cut-aconvert _ D1 X1 Dlook D1' X1') %. +%term t + %pi (cut-tconvert _ ([x] [d] [ex] [xt] tconvert/t) ([x] [d] [ex] [xt] mtconvert/t) Dlook ([x] [d'] [ex] [xt] tconverte/t (Dord x d')) ([x] [d'] [ex] [xt] mtconverte/t)) + %<- ({x} {d'} lookup-context (Dlook x d') (Dord x d')) %. +%%% +%scope cut-tconverte %term pi + %pi (cut-tconvert _ ([x] [d] [ex] [xt] tconvert/pi (D2 x d ex xt) (D1 x d ex xt)) ([x] [d] [ex] [xt] mtconvert/pi (X2 x d ex xt) (X1 x d ex xt)) Dlook ([x] [d'] [ex] [xt] tconverte/pi (D2'' x d' ex xt) (D1' x d' ex xt)) ([x] [d'] [ex] [xt] mtconverte/pi (X2'' x d' ex xt) (X1' x d' ex xt))) + %<- (cut-tconvert _ D1 X1 Dlook D1' X1') + %<- ({x} {d'} lookup-context (Dlook x d') (Dord x d')) + %<- ({x} {d'} extend-context (Dord x d') (%the ({y} %pi (isvar y J) %-> (bounded (G x) y)) (Dbound x d'))) + %<- ({x} {d} {d'} {ex} {xt} cut-tconvert _ ([y] [e] [ey] [yt] D2 x d ex xt y e ey yt) ([y] [e] [ey] [yt] X2 x d ex xt y e ey yt) ([y] [e'] lookup/hit (Dbound x d' y e')) ([y] [e'] [ey] [yt] D2' x d d' ex xt y e' ey yt) ([y] [e'] [ey] [yt] X2' x d d' ex xt y e' ey yt)) + %<- ({y} {e'} {ey} {yt} cut-tconverte _ ([x] [d] [d'] [ex] [xt] D2' x d d' ex xt y e' ey yt) ([x] [d] [d'] [ex] [xt] X2' x d d' ex xt y e' ey yt) ([x] [d'] lookup/miss (Dlook x d') (Dbound x d' y e')) ([x] [d'] [ex] [xt] D2'' x d' ex xt y e' ey yt) ([x] [d'] [ex] [xt] X2'' x d' ex xt y e' ey yt)) %. +%term sigma + %pi (cut-tconvert _ ([x] [d] [ex] [xt] tconvert/sigma (D2 x d ex xt) (D1 x d ex xt)) ([x] [d] [ex] [xt] mtconvert/sigma (X2 x d ex xt) (X1 x d ex xt)) Dlook ([x] [d'] [ex] [xt] tconverte/sigma (D2'' x d' ex xt) (D1' x d' ex xt)) ([x] [d'] [ex] [xt] mtconverte/sigma (X2'' x d' ex xt) (X1' x d' ex xt))) + %<- (cut-tconvert _ D1 X1 Dlook D1' X1') + %<- ({x} {d'} lookup-context (Dlook x d') (Dord x d')) + %<- ({x} {d'} extend-context (Dord x d') (%the ({y} %pi (isvar y J) %-> (bounded (G x) y)) (Dbound x d'))) + %<- ({x} {d} {d'} {ex} {xt} cut-tconvert _ ([y] [e] [ey] [yt] D2 x d ex xt y e ey yt) ([y] [e] [ey] [yt] X2 x d ex xt y e ey yt) ([y] [e'] lookup/hit (Dbound x d' y e')) ([y] [e'] [ey] [yt] D2' x d d' ex xt y e' ey yt) ([y] [e'] [ey] [yt] X2' x d d' ex xt y e' ey yt)) + %<- ({y} {e'} {ey} {yt} cut-tconverte _ ([x] [d] [d'] [ex] [xt] D2' x d d' ex xt y e' ey yt) ([x] [d] [d'] [ex] [xt] X2' x d d' ex xt y e' ey yt) ([x] [d'] lookup/miss (Dlook x d') (Dbound x d' y e')) ([x] [d'] [ex] [xt] D2'' x d' ex xt y e' ey yt) ([x] [d'] [ex] [xt] X2'' x d' ex xt y e' ey yt)) %. +%scope cut-tconverte %term sing + %pi (cut-tconvert _ ([x] [d] [ex] [xt] tconvert/sing (D1 x d ex xt)) ([x] [d] [ex] [xt] mtconvert/sing (X1 x d ex xt)) Dlook ([x] [d'] [ex] [xt] tconverte/sing (D1' x d' ex xt)) ([x] [d'] [ex] [xt] mtconverte/sing (X1' x d' ex xt))) + %<- (cut-aconvert _ D1 X1 Dlook D1' X1') %. +%term const + %pi (cut-aconverte _ ([x] [d] [d'] [ex] [xt] aconverte/const (D2 x d d' ex xt) (Dwf x d) (D1 x)) ([x] [d] [d'] [ex] [xt] maconverte/const (X2 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] aconverte/const (D2' x d' ex xt) (Dwf''' x) (D1 x)) ([x] [d'] [ex] [xt] maconverte/const (X2' x d' ex xt))) + %<- (kof-strengthen D1 Deq) + %<- ({x} tp-eq-symm (Deq x) (Deq' x)) + %<- ({x} {d} wf-resp (Deq x) (Dwf x d) (Dwf' x d)) + %<- (strengthen-wf Dwf' Dwf'') + %<- ({x} wf-resp (Deq' x) Dwf'' (Dwf''' x)) + %<- (cut-tconverte _ D2 X2 Dlook D2' X2') %. +%%% +%scope cut-tconverte %term forall + %pi (cut-aconverte _ ([x] [d] [d'] [ex] [xt] aconverte/forall (D1 x d d' ex xt)) ([x] [d] [d'] [ex] [xt] maconverte/forall (X1 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] aconverte/forall (D1' x d' ex xt)) ([x] [d'] [ex] [xt] maconverte/forall (X1' x d' ex xt))) + %<- (cut-tconverte _ D1 X1 Dlook D1' X1') %. +%term var + %pi (cut-aconverte _ ([x] [d] [d'] [ex] [xt] aconverte/var (D3 x ex xt) (D2 x d d' ex xt) (D1 x d')) ([x] [d] [d'] [ex] [xt] maconverte/var (X2 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] aconverte/var (D3 x ex xt) (D2' x d' ex xt) (D1 x d')) ([x] [d'] [ex] [xt] maconverte/var (X2' x d' ex xt))) + %<- (cut-tconverte _ D2 X2 Dlook D2' X2') %. +%scope cut-tconverte %term varsam + %pi (cut-aconverte _ ([x] [d] [d'] [ex] [xt] aconverte/vari (D3 x ex xt) (D2 x d d' ex xt) (Dwf x d) d) ([x] [d] [d'] [ex] [xt] maconverte/vari (X2 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] aconverte/var (D3 x ex xt) (D2' x d' ex xt) (Dlook x d')) ([x] [d'] [ex] [xt] maconverte/var (X2' x d' ex xt))) + %<- (cut-tconverte _ D2 X2 Dlook D2' X2') %. +%term varoth + %pi (cut-aconverte _ ([x] [d] [d'] [ex] [xt] aconverte/vari (D3 x ex xt) (D2 x d d' ex xt) (Dwf x d) D1) ([x] [d] [d'] [ex] [xt] maconverte/vari (X2 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] aconverte/vari (D3 x ex xt) (D2' x d' ex xt) Dwf' D1) ([x] [d'] [ex] [xt] maconverte/vari (X2' x d' ex xt))) + %<- (cut-tconverte _ D2 X2 Dlook D2' X2') + %<- (strengthen-wf Dwf Dwf') %. +%scope cut-tconverte %term app + %pi (cut-aconverte _ ([x] [d] [d'] [ex] [xt] aconverte/app (D3 x) (D2 x d d' ex xt) (D1 x d d' ex xt)) ([x] [d] [d'] [ex] [xt] maconverte/app (X2 x d d' ex xt) (X1 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] aconverte/app (D3 x) (D2' x d' ex xt) (D1' x d' ex xt)) ([x] [d'] [ex] [xt] maconverte/app (X2' x d' ex xt) (X1' x d' ex xt))) + %<- (cut-aconverte _ D1 X1 Dlook D1' X1') + %<- (cut-converte _ D2 X2 Dlook D2' X2') %. +%term pi1 + %pi (cut-aconverte _ ([x] [d] [d'] [ex] [xt] aconverte/pi1 (D1 x d d' ex xt)) ([x] [d] [d'] [ex] [xt] maconverte/pi1 (X1 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] aconverte/pi1 (D1' x d' ex xt)) ([x] [d'] [ex] [xt] maconverte/pi1 (X1' x d' ex xt))) + %<- (cut-aconverte _ D1 X1 Dlook D1' X1') %. +%scope cut-tconverte %term pi2 + %pi (cut-aconverte _ ([x] [d] [d'] [ex] [xt] aconverte/pi2 (D1 x d d' ex xt)) ([x] [d] [d'] [ex] [xt] maconverte/pi2 (X1 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] aconverte/pi2 (D1' x d' ex xt)) ([x] [d'] [ex] [xt] maconverte/pi2 (X1' x d' ex xt))) + %<- (cut-aconverte _ D1 X1 Dlook D1' X1') %. +%term at + %pi (cut-converte _ ([x] [d] [d'] [ex] [xt] converte/at (D1 x d d' ex xt)) ([x] [d] [d'] [ex] [xt] mconverte/at (X1 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] converte/at (D1' x d' ex xt)) ([x] [d'] [ex] [xt] mconverte/at (X1' x d' ex xt))) + %<- (cut-aconverte _ D1 X1 Dlook D1' X1') %. +%%% +%scope cut-tconverte %term lam + %pi (cut-converte _ ([x] [d] [d'] [ex] [xt] converte/lam (D2 x d d' ex xt) (D1 x d d' ex xt)) ([x] [d] [d'] [ex] [xt] mconverte/lam (X2 x d d' ex xt) (X1 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] converte/lam (D2' x d' ex xt) (D1' x d' ex xt)) ([x] [d'] [ex] [xt] mconverte/lam (X2' x d' ex xt) (X1' x d' ex xt))) + %<- (cut-tconverte _ D1 X1 Dlook D1' X1') + %<- ({x} {d} {d'} {ex} {xt} {y} {e'} {ey} {yt} converte-context (D2 x d d' ex xt y e' ey yt) (ordered/cons (Dbound x d' y e'))) + %<- ({y} {e'} {ey} {yt} cut-converte _ ([x] [d] [d'] [ex] [xt] D2 x d d' ex xt y e' ey yt) ([x] [d] [d'] [ex] [xt] X2 x d d' ex xt y e' ey yt) ([x] [d'] lookup/miss (Dlook x d') (Dbound x d' y e')) ([x] [d'] [ex] [xt] D2' x d' ex xt y e' ey yt) ([x] [d'] [ex] [xt] X2' x d' ex xt y e' ey yt)) %. +%term pair + %pi (cut-converte _ ([x] [d] [d'] [ex] [xt] converte/pair (Dwfe x d d') (D3 x d d' ex xt) (D2 x) (D1 x d d' ex xt)) ([x] [d] [d'] [ex] [xt] mconverte/pair (X3 x d d' ex xt) (X1 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] converte/pair (Dwfe' x d') (D3' x d' ex xt) (D2 x) (D1' x d' ex xt)) ([x] [d'] [ex] [xt] mconverte/pair (X3' x d' ex xt) (X1' x d' ex xt))) + %<- (cut-converte _ D1 X1 Dlook D1' X1') + %<- (cut-converte _ D3 X3 Dlook D3' X3') + %<- ({x} {d} {d'} {y} {e'} wfe-context (Dwfe x d d' y e') (ordered/cons (Dbound x d' y e'))) + %<- ({x} {d} {d'} {y} {e} can-mwfe (Dwfe x d d' y e) (%the (mwfe _ Mm) (Dmwfe x d d' y e))) + %<- ({y} {e'} cut-wfe _ ([x] [d] [d'] Dwfe x d d' y e') ([x] [d] [d'] Dmwfe x d d' y e') ([x] [d'] lookup/miss (Dlook x d') (Dbound x d' y e')) ([x] [d'] Dwfe' x d' y e') _) %. +%scope cut-tconverte %term sing + %pi (cut-converte _ ([x] [d] [d'] [ex] [xt] converte/sing (D1 x d d' ex xt)) ([x] [d] [d'] [ex] [xt] mconverte/sing (X1 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] converte/sing (D1' x d' ex xt)) ([x] [d'] [ex] [xt] mconverte/sing (X1' x d' ex xt))) + %<- (cut-aconverte _ D1 X1 Dlook D1' X1') %. +%term t cut-tconverte _ ([x] [d] [d'] [ex] [xt] tconverte/t (D1 x d')) ([x] [d] [d'] [ex] [xt] mtconverte/t) _ ([x] [d'] [ex] [xt] tconverte/t (D1 x d')) ([x] [d'] [ex] [xt] mtconverte/t) %. +%%% +%scope cut-tconverte %term pi + %pi (cut-tconverte _ ([x] [d] [d'] [ex] [xt] tconverte/pi (D2 x d d' ex xt) (D1 x d d' ex xt)) ([x] [d] [d'] [ex] [xt] mtconverte/pi (X2 x d d' ex xt) (X1 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] tconverte/pi (D2' x d' ex xt) (D1' x d' ex xt)) ([x] [d'] [ex] [xt] mtconverte/pi (X2' x d' ex xt) (X1' x d' ex xt))) + %<- (cut-tconverte _ D1 X1 Dlook D1' X1') + %<- ({x} {d} {d'} {ex} {xt} {y} {e'} {ey} {yt} tconverte-context (D2 x d d' ex xt y e' ey yt) (ordered/cons (Dbound x d' y e'))) + %<- ({y} {e'} {ey} {yt} cut-tconverte _ ([x] [d] [d'] [ex] [xt] D2 x d d' ex xt y e' ey yt) ([x] [d] [d'] [ex] [xt] X2 x d d' ex xt y e' ey yt) ([x] [d'] lookup/miss (Dlook x d') (Dbound x d' y e')) ([x] [d'] [ex] [xt] D2' x d' ex xt y e' ey yt) ([x] [d'] [ex] [xt] X2' x d' ex xt y e' ey yt)) %. +%term sigma + %pi (cut-tconverte _ ([x] [d] [d'] [ex] [xt] tconverte/sigma (D2 x d d' ex xt) (D1 x d d' ex xt)) ([x] [d] [d'] [ex] [xt] mtconverte/sigma (X2 x d d' ex xt) (X1 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] tconverte/sigma (D2' x d' ex xt) (D1' x d' ex xt)) ([x] [d'] [ex] [xt] mtconverte/sigma (X2' x d' ex xt) (X1' x d' ex xt))) + %<- (cut-tconverte _ D1 X1 Dlook D1' X1') + %<- ({x} {d} {d'} {ex} {xt} {y} {e'} {ey} {yt} tconverte-context (D2 x d d' ex xt y e' ey yt) (ordered/cons (Dbound x d' y e'))) + %<- ({y} {e'} {ey} {yt} cut-tconverte _ ([x] [d] [d'] [ex] [xt] D2 x d d' ex xt y e' ey yt) ([x] [d] [d'] [ex] [xt] X2 x d d' ex xt y e' ey yt) ([x] [d'] lookup/miss (Dlook x d') (Dbound x d' y e')) ([x] [d'] [ex] [xt] D2' x d' ex xt y e' ey yt) ([x] [d'] [ex] [xt] X2' x d' ex xt y e' ey yt)) %. +%scope cut-tconverte %term sing + %pi (cut-tconverte _ ([x] [d] [d'] [ex] [xt] tconverte/sing (D1 x d d' ex xt)) ([x] [d] [d'] [ex] [xt] mtconverte/sing (X1 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] tconverte/sing (D1' x d' ex xt)) ([x] [d'] [ex] [xt] mtconverte/sing (X1' x d' ex xt))) + %<- (cut-aconverte _ D1 X1 Dlook D1' X1') %. +%worlds (fbind ovar obind tbind evar bind) (cut-convert _ _ _ _ _ _) (cut-aconvert _ _ _ _ _ _) (cut-tconvert _ _ _ _ _ _) (cut-converte _ _ _ _ _ _) (cut-aconverte _ _ _ _ _ _) (cut-tconverte _ _ _ _ _ _) %. +%total (X1 X2 X3 X4 X5 X6) (cut-aconvert X1 _ _ _ _ _) (cut-convert X2 _ _ _ _ _) (cut-tconvert X3 _ _ _ _ _) (cut-aconverte X4 _ _ _ _ _) (cut-converte X5 _ _ _ _ _) (cut-tconverte X6 _ _ _ _ _) %. +%sort convert1-to-converte {I} {x} %. +%mode convert1-to-converte %in %in %out %. +%term _ + %pi (convert1-to-converte I Dconvert Dconverte) + %<- ({x} {d} {ex} {xt} can-mconvert (Dconvert x d ex xt) (Dmconvert x d ex xt)) + %<- (cut-convert _ Dconvert Dmconvert ([x] [d] lookup/hit (bounded/nil d)) Dconverte _) %. +%worlds (fbind ovar obind tbind evar bind) (convert1-to-converte _ _ _) %. +%total {} (convert1-to-converte _ _ _) %. +%sort aconvert1-to-aconverte {I} {x} %. +%mode aconvert1-to-aconverte %in %in %out %. +%term _ + %pi (aconvert1-to-aconverte I Dconvert Dconverte) + %<- ({x} {d} {ex} {xt} can-maconvert (Dconvert x d ex xt) (Dmconvert x d ex xt)) + %<- (cut-aconvert _ Dconvert Dmconvert ([x] [d] lookup/hit (bounded/nil d)) Dconverte _) %. +%worlds (fbind ovar obind tbind evar bind) (aconvert1-to-aconverte _ _ _) %. +%total {} (aconvert1-to-aconverte _ _ _) %. +%sort tconvert1-to-tconverte {I} {x} %. +%mode tconvert1-to-tconverte %in %in %out %. +%term _ + %pi (tconvert1-to-tconverte I Dconvert Dconverte) + %<- ({x} {d} {ex} {xt} can-mtconvert (Dconvert x d ex xt) (Dmconvert x d ex xt)) + %<- (cut-tconvert _ Dconvert Dmconvert ([x] [d] lookup/hit (bounded/nil d)) Dconverte _) %. +%worlds (fbind ovar obind tbind evar bind) (tconvert1-to-tconverte _ _ _) %. +%total {} (tconvert1-to-tconverte _ _ _) %. +%sort convert2-to-converte %. +%mode convert2-to-converte %in %in %out %. +%term _ + %pi (convert2-to-converte (%the (lt I J) Dlt) Dconvert Dconvert'') + %<- ({x} {d} {ex} {xt} {y} {e} {ey} {yt} can-mconvert (Dconvert x d ex xt y e ey yt) (%the (mconvert _ Mm) (Dmconvert x d ex xt y e ey yt))) + %<- ({x} {d} {d' isvar x I} {ex} {xt} cut-convert _ ([y] [e] [ey] [yt] Dconvert x d ex xt y e ey yt) ([y] [e] [ey] [yt] Dmconvert x d ex xt y e ey yt) ([y] [e'] lookup/hit (bounded/cons (bounded/nil d') (precedes/i Dlt e' d'))) ([y] [e'] [ey] [yt] Dconvert' x d d' ex xt y e' ey yt) ([y] [e'] [ey] [yt] Dmconvert' x d d' ex xt y e' ey yt)) + %<- ({y} {e' isvar y J} {ey} {yt} cut-converte _ ([x] [d] [d'] [ex] [xt] Dconvert' x d d' ex xt y e' ey yt) ([x] [d] [d'] [ex] [xt] Dmconvert' x d d' ex xt y e' ey yt) ([x] [d'] lookup/miss (lookup/hit (bounded/nil d')) (bounded/cons (bounded/nil d') (precedes/i Dlt e' d'))) ([x] [d'] [ex] [xt] Dconvert'' x d' ex xt y e' ey yt) ([x] [d'] [ex] [xt] Dmconvert'' x d' ex xt y e' ey yt)) %. +%worlds (fbind ovar obind tbind evar bind) (convert2-to-converte _ _ _) %. +%total {} (convert2-to-converte _ _ _) %. +%sort convert-to-converte %. +%mode convert-to-converte %in %out %. +%sort aconvert-to-aconverte %. +%mode aconvert-to-aconverte %in %out %. +%sort tconvert-to-tconverte %. +%mode tconvert-to-tconverte %in %out %. +%scope tconvert-to-tconverte %term t tconvert-to-tconverte tconvert/t (tconverte/t ordered/nil) %. +%term pi + %pi (tconvert-to-tconverte (tconvert/pi D2 D1) (tconverte/pi D2' D1')) + %<- (tconvert-to-tconverte D1 D1') + %<- (tconvert1-to-tconverte 0 D2 D2') %. +%scope tconvert-to-tconverte %term sigma + %pi (tconvert-to-tconverte (tconvert/sigma D2 D1) (tconverte/sigma D2' D1')) + %<- (tconvert-to-tconverte D1 D1') + %<- (tconvert1-to-tconverte 0 D2 D2') %. +%term sing + %pi (tconvert-to-tconverte (tconvert/sing D1) (tconverte/sing D1')) + %<- (aconvert-to-aconverte D1 D1') %. +%scope tconvert-to-tconverte %term at + %pi (convert-to-converte (convert/at D1) (converte/at D1')) + %<- (aconvert-to-aconverte D1 D1') %. +%term pi + %pi (convert-to-converte (convert/lam D2 D1) (converte/lam D2' D1')) + %<- (tconvert-to-tconverte D1 D1') + %<- (convert1-to-converte 0 D2 D2') %. +%scope tconvert-to-tconverte %term pair + %pi (convert-to-converte (convert/pair D4 D3 D2 D1) (converte/pair D4' D3' D2 D1')) + %<- (convert-to-converte D1 D1') + %<- (convert-to-converte D3 D3') + %<- (wf1-to-wfe 0 D4 D4') %. +%term sing + %pi (convert-to-converte (convert/sing D1) (converte/sing D1')) + %<- (aconvert-to-aconverte D1 D1') %. +%scope tconvert-to-tconverte %term const + %pi (aconvert-to-aconverte (aconvert/const D3 D2 D1) (aconverte/const D3' D2 D1)) + %<- (tconvert-to-tconverte D3 D3') %. +%term forall + %pi (aconvert-to-aconverte (aconvert/forall D1) (aconverte/forall D1')) + %<- (tconvert-to-tconverte D1 D1') %. +%scope tconvert-to-tconverte %term var + %pi (aconvert-to-aconverte (aconvert/var D4 D3 D2 D1) (aconverte/vari D4 D3' D2 D1)) + %<- (tconvert-to-tconverte D3 D3') %. +%term app + %pi (aconvert-to-aconverte (aconvert/app D3 D2 D1) (aconverte/app D3 D2' D1')) + %<- (aconvert-to-aconverte D1 D1') + %<- (convert-to-converte D2 D2') %. +%scope tconvert-to-tconverte %term pi1 + %pi (aconvert-to-aconverte (aconvert/pi1 D1) (aconverte/pi1 D1')) + %<- (aconvert-to-aconverte D1 D1') %. +%term pi2 + %pi (aconvert-to-aconverte (aconvert/pi2 D1) (aconverte/pi2 D1')) + %<- (aconvert-to-aconverte D1 D1') %. +%worlds (fbind fobind ovar tbind evar bind) (convert-to-converte _ _) (aconvert-to-aconverte _ _) (tconvert-to-tconverte _ _) %. +%total (D1 D2 D3) (convert-to-converte D1 _) (aconvert-to-aconverte D2 _) (tconvert-to-tconverte D3 _) %. +%%%%% Bump %%%%% +%sort bump-converte-m {Mm} {D {x} + %pi (isvar x I) + %-> ({ex} %pi (vtrans ex x) %-> (converte (G x) (M x) (A x) (EM ex)))} {x} {d} {ex} {xt} %. +%mode bump-converte-m %in %in %in %in %out %out %. +%sort bump-aconverte-m {Rm} {D {x} + %pi (isvar x I) + %-> ({ex} %pi (vtrans ex x) %-> (aconverte (G x) (R x) (A x) (EM ex)))} {x} {d} {ex} {xt} %. +%mode bump-aconverte-m %in %in %in %in %out %out %. +%sort bump-tconverte-m {Am} {D {x} %pi (isvar x I) %-> ({ex} %pi (vtrans ex x) %-> (tconverte (G x) (A x) (EA ex)))} {x} {d} {ex} {xt} %. +%mode bump-tconverte-m %in %in %in %in %out %out %. +%scope bump-tconverte-m %term const + %pi (bump-aconverte-m _ ([x] [d] [ex] [xt] aconverte/const (Dconv x d ex xt) (Dwf x) (Dof x)) ([x] [d] [ex] [xt] maconverte/const (DM x d ex xt)) Dord ([x] [d] [ex] [xt] aconverte/const (Dconv' x d ex xt) (Dwf x) (Dof x)) ([x] [d] [ex] [xt] maconverte/const (DM' x d ex xt))) + %<- (bump-tconverte-m _ Dconv DM Dord Dconv' DM') %. +%term forall + %pi (bump-aconverte-m _ ([x] [d] [ex] [xt] aconverte/forall (Dconv x d ex xt)) ([x] [d] [ex] [xt] maconverte/forall (DM x d ex xt)) Dord ([x] [d] [ex] [xt] aconverte/forall (Dconv' x d ex xt)) ([x] [d] [ex] [xt] maconverte/forall (DM' x d ex xt))) + %<- (bump-tconverte-m _ Dconv DM Dord Dconv' DM') %. +%scope bump-tconverte-m %term var + %pi (bump-aconverte-m _ ([x] [d] [ex] [xt] aconverte/var (Dtrans x ex xt) (Dconv x d ex xt) (Dlook x d)) ([x] [d] [ex] [xt] maconverte/var (DM x d ex xt)) Dord ([x] [d] [ex] [xt] aconverte/var (Dtrans x ex xt) (Dconv' x d ex xt) (Dlook' x d)) ([x] [d] [ex] [xt] maconverte/var (DM' x d ex xt))) + %<- (bump-lookup Dlook Dord Dlook') + %<- (bump-tconverte-m _ Dconv DM Dord Dconv' DM') %. +%term vari + %pi (bump-aconverte-m _ ([x] [d] [ex] [xt] aconverte/vari (Dtrans x ex xt) (Dconv x d ex xt) (Dwf x) (Dvof x)) ([x] [d] [ex] [xt] maconverte/vari (DM x d ex xt)) Dord ([x] [d] [ex] [xt] aconverte/vari (Dtrans x ex xt) (Dconv' x d ex xt) (Dwf x) (Dvof x)) ([x] [d] [ex] [xt] maconverte/vari (DM' x d ex xt))) + %<- (bump-tconverte-m _ Dconv DM Dord Dconv' DM') %. +%scope bump-tconverte-m %term app + %pi (bump-aconverte-m _ ([x] [d] [ex] [xt] aconverte/app (Dsub x) (Dconv2 x d ex xt) (Dconv1 x d ex xt)) ([x] [d] [ex] [xt] maconverte/app (DM2 x d ex xt) (DM1 x d ex xt)) Dord ([x] [d] [ex] [xt] aconverte/app (Dsub x) (Dconv2' x d ex xt) (Dconv1' x d ex xt)) ([x] [d] [ex] [xt] maconverte/app (DM2' x d ex xt) (DM1' x d ex xt))) + %<- (bump-aconverte-m _ Dconv1 DM1 Dord Dconv1' DM1') + %<- (bump-converte-m _ Dconv2 DM2 Dord Dconv2' DM2') %. +%term pi1 + %pi (bump-aconverte-m _ ([x] [d] [ex] [xt] aconverte/pi1 (Dconv x d ex xt)) ([x] [d] [ex] [xt] maconverte/pi1 (DM x d ex xt)) Dord ([x] [d] [ex] [xt] aconverte/pi1 (Dconv' x d ex xt)) ([x] [d] [ex] [xt] maconverte/pi1 (DM' x d ex xt))) + %<- (bump-aconverte-m _ Dconv DM Dord Dconv' DM') %. +%scope bump-tconverte-m %term pi2 + %pi (bump-aconverte-m _ ([x] [d] [ex] [xt] aconverte/pi2 (%the (aconverte _ _ (sigma (A x) (B x)) _) (Dconv x d ex xt))) ([x] [d] [ex] [xt] maconverte/pi2 (DM x d ex xt)) Dord ([x] [d] [ex] [xt] aconverte/pi2 (Dconv' x d ex xt)) ([x] [d] [ex] [xt] maconverte/pi2 (DM' x d ex xt))) + %<- (bump-aconverte-m _ Dconv DM Dord Dconv' DM') %. +%term at + %pi (bump-converte-m _ ([x] [d] [ex] [xt] converte/at (Dconv x d ex xt)) ([x] [d] [ex] [xt] mconverte/at (DM x d ex xt)) Dord ([x] [d] [ex] [xt] converte/at (Dconv' x d ex xt)) ([x] [d] [ex] [xt] mconverte/at (DM' x d ex xt))) + %<- (bump-aconverte-m _ Dconv DM Dord Dconv' DM') %. +%scope bump-tconverte-m %term lam + %pi (bump-converte-m _ ([x] [d] [ex] [xt] converte/lam (Dconv2 x d ex xt) (Dconv1 x d ex xt)) ([x] [d] [ex] [xt] mconverte/lam (DM2 x d ex xt) (DM1 x d ex xt)) Dord ([x] [d] [ex] [xt] converte/lam (Dconv2'' x d ex xt) (Dconv1' x d ex xt)) ([x] [d] [ex] [xt] mconverte/lam (DM2'' x d ex xt) (DM1' x d ex xt))) + %<- (bump-tconverte-m _ Dconv1 DM1 Dord Dconv1' DM1') + %<- ({x} {d isvar x I} {ex} {xt} {y} {e isvar y J1} {ey} {yt} converte-context (Dconv2 x d ex xt y e ey yt) (ordered/cons (%the (bounded (G x) y) (Dbound1 x d y e)))) + %<- ({x} {d isvar x I'} extend-context (Dord x d) (%the ({y} %pi (isvar y J2) %-> (bounded (G x) y)) (Dbound2 x d))) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- ({x} {d isvar x I} bump-bounded Dleq1 ([y] [e isvar y J1] Dbound1 x d y e) ([y] [e isvar y J] %the (bounded (G x) y) (Dbound1' x d y e))) + %<- ({x} {d isvar x I'} bump-bounded Dleq2 ([y] [e isvar y J2] Dbound2 x d y e) ([y] [e isvar y J] %the (bounded (G x) y) (Dbound2' x d y e))) + %<- ({x} {d} {ex} {xt} bump-converte-m _ ([y] [e] [ey] [yt] Dconv2 x d ex xt y e ey yt) ([y] [e] [ey] [yt] DM2 x d ex xt y e ey yt) ([y] [e] ordered/cons (Dbound1' x d y e)) ([y] [e] [ey] [yt] Dconv2' x d ex xt y e ey yt) ([y] [e] [ey] [yt] DM2' x d ex xt y e ey yt)) + %<- ({y} {e} {ey} {yt} bump-converte-m _ ([x] [d] [ex] [xt] Dconv2' x d ex xt y e ey yt) ([x] [d] [ex] [xt] DM2' x d ex xt y e ey yt) ([x] [d] ordered/cons (Dbound2' x d y e)) ([x] [d] [ex] [xt] Dconv2'' x d ex xt y e ey yt) ([x] [d] [ex] [xt] DM2'' x d ex xt y e ey yt)) %. +%term pair + %pi (bump-converte-m _ ([x] [d] [ex] [xt] converte/pair (Dwf x d) (Dconv2 x d ex xt) (Dsub x) (Dconv1 x d ex xt)) ([x] [d] [ex] [xt] mconverte/pair (DM2 x d ex xt) (DM1 x d ex xt)) Dord ([x] [d] [ex] [xt] converte/pair (Dwf' x d) (Dconv2' x d ex xt) (Dsub x) (Dconv1' x d ex xt)) ([x] [d] [ex] [xt] mconverte/pair (DM2' x d ex xt) (DM1' x d ex xt))) + %<- (bump-converte-m _ Dconv1 DM1 Dord Dconv1' DM1') + %<- (bump-converte-m _ Dconv2 DM2 Dord Dconv2' DM2') + %<- (bump-wfe-under Dwf Dord Dwf') %. +%scope bump-tconverte-m %term sing + %pi (bump-converte-m _ ([x] [d] [ex] [xt] converte/sing (Dconv x d ex xt)) ([x] [d] [ex] [xt] mconverte/sing (DM x d ex xt)) Dord ([x] [d] [ex] [xt] converte/sing (Dconv' x d ex xt)) ([x] [d] [ex] [xt] mconverte/sing (DM' x d ex xt))) + %<- (bump-aconverte-m _ Dconv DM Dord Dconv' DM') %. +%term t bump-tconverte-m _ ([x] [d] [ex] [xt] tconverte/t (Dord' x d)) ([x] [d] [ex] [xt] mtconverte/t) Dord ([x] [d] [ex] [xt] tconverte/t (Dord x d)) ([x] [d] [ex] [xt] mtconverte/t) %. +%scope bump-tconverte-m %term pi + %pi (bump-tconverte-m _ ([x] [d] [ex] [xt] tconverte/pi (Dconv2 x d ex xt) (Dconv1 x d ex xt)) ([x] [d] [ex] [xt] mtconverte/pi (DM2 x d ex xt) (DM1 x d ex xt)) Dord ([x] [d] [ex] [xt] tconverte/pi (Dconv2'' x d ex xt) (Dconv1' x d ex xt)) ([x] [d] [ex] [xt] mtconverte/pi (DM2'' x d ex xt) (DM1' x d ex xt))) + %<- (bump-tconverte-m _ Dconv1 DM1 Dord Dconv1' DM1') + %<- ({x} {d isvar x I} {ex} {xt} {y} {e isvar y J1} {ey} {yt} tconverte-context (Dconv2 x d ex xt y e ey yt) (ordered/cons (%the (bounded (G x) y) (Dbound1 x d y e)))) + %<- ({x} {d isvar x I'} extend-context (Dord x d) (%the ({y} %pi (isvar y J2) %-> (bounded (G x) y)) (Dbound2 x d))) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- ({x} {d isvar x I} bump-bounded Dleq1 ([y] [e isvar y J1] Dbound1 x d y e) ([y] [e isvar y J] %the (bounded (G x) y) (Dbound1' x d y e))) + %<- ({x} {d isvar x I'} bump-bounded Dleq2 ([y] [e isvar y J2] Dbound2 x d y e) ([y] [e isvar y J] %the (bounded (G x) y) (Dbound2' x d y e))) + %<- ({x} {d} {ex} {xt} bump-tconverte-m _ ([y] [e] [ey] [yt] Dconv2 x d ex xt y e ey yt) ([y] [e] [ey] [yt] DM2 x d ex xt y e ey yt) ([y] [e] ordered/cons (Dbound1' x d y e)) ([y] [e] [ey] [yt] Dconv2' x d ex xt y e ey yt) ([y] [e] [ey] [yt] DM2' x d ex xt y e ey yt)) + %<- ({y} {e} {ey} {yt} bump-tconverte-m _ ([x] [d] [ex] [xt] Dconv2' x d ex xt y e ey yt) ([x] [d] [ex] [xt] DM2' x d ex xt y e ey yt) ([x] [d] ordered/cons (Dbound2' x d y e)) ([x] [d] [ex] [xt] Dconv2'' x d ex xt y e ey yt) ([x] [d] [ex] [xt] DM2'' x d ex xt y e ey yt)) %. +%term sigma + %pi (bump-tconverte-m _ ([x] [d] [ex] [xt] tconverte/sigma (Dconv2 x d ex xt) (Dconv1 x d ex xt)) ([x] [d] [ex] [xt] mtconverte/sigma (DM2 x d ex xt) (DM1 x d ex xt)) Dord ([x] [d] [ex] [xt] tconverte/sigma (Dconv2'' x d ex xt) (Dconv1' x d ex xt)) ([x] [d] [ex] [xt] mtconverte/sigma (DM2'' x d ex xt) (DM1' x d ex xt))) + %<- (bump-tconverte-m _ Dconv1 DM1 Dord Dconv1' DM1') + %<- ({x} {d isvar x I} {ex} {xt} {y} {e isvar y J1} {ey} {yt} tconverte-context (Dconv2 x d ex xt y e ey yt) (ordered/cons (%the (bounded (G x) y) (Dbound1 x d y e)))) + %<- ({x} {d isvar x I'} extend-context (Dord x d) (%the ({y} %pi (isvar y J2) %-> (bounded (G x) y)) (Dbound2 x d))) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- ({x} {d isvar x I} bump-bounded Dleq1 ([y] [e isvar y J1] Dbound1 x d y e) ([y] [e isvar y J] %the (bounded (G x) y) (Dbound1' x d y e))) + %<- ({x} {d isvar x I'} bump-bounded Dleq2 ([y] [e isvar y J2] Dbound2 x d y e) ([y] [e isvar y J] %the (bounded (G x) y) (Dbound2' x d y e))) + %<- ({x} {d} {ex} {xt} bump-tconverte-m _ ([y] [e] [ey] [yt] Dconv2 x d ex xt y e ey yt) ([y] [e] [ey] [yt] DM2 x d ex xt y e ey yt) ([y] [e] ordered/cons (Dbound1' x d y e)) ([y] [e] [ey] [yt] Dconv2' x d ex xt y e ey yt) ([y] [e] [ey] [yt] DM2' x d ex xt y e ey yt)) + %<- ({y} {e} {ey} {yt} bump-tconverte-m _ ([x] [d] [ex] [xt] Dconv2' x d ex xt y e ey yt) ([x] [d] [ex] [xt] DM2' x d ex xt y e ey yt) ([x] [d] ordered/cons (Dbound2' x d y e)) ([x] [d] [ex] [xt] Dconv2'' x d ex xt y e ey yt) ([x] [d] [ex] [xt] DM2'' x d ex xt y e ey yt)) %. +%scope bump-tconverte-m %term sing + %pi (bump-tconverte-m _ ([x] [d] [ex] [xt] tconverte/sing (Dconv x d ex xt)) ([x] [d] [ex] [xt] mtconverte/sing (DM x d ex xt)) Dord ([x] [d] [ex] [xt] tconverte/sing (Dconv' x d ex xt)) ([x] [d] [ex] [xt] mtconverte/sing (DM' x d ex xt))) + %<- (bump-aconverte-m _ Dconv DM Dord Dconv' DM') %. +%worlds (fbind fobind ovar tbind evar) (bump-aconverte-m _ _ _ _ _ _) (bump-converte-m _ _ _ _ _ _) (bump-tconverte-m _ _ _ _ _ _) %. +%total (Rm Mm Am) (bump-aconverte-m Rm _ _ _ _ _) (bump-converte-m Am _ _ _ _ _) (bump-tconverte-m Mm _ _ _ _ _) %. +%sort bump-converte {x} %. +%mode bump-converte %in %in %out %. +%term _ + %pi (bump-converte Dconv Dord Dconv') + %<- ({x} {d} {ex} {xt} can-mconverte (Dconv x d ex xt) (%the (mconverte _ Mm) (DM x d ex xt))) + %<- (bump-converte-m _ Dconv DM Dord Dconv' _) %. +%worlds (fbind fobind) (bump-converte _ _ _) %. +%total {} (bump-converte _ _ _) %. +%sort bump-tconverte {x} %. +%mode bump-tconverte %in %in %out %. +%term _ + %pi (bump-tconverte Dconv Dord Dconv') + %<- ({x} {d} {ex} {xt} can-mtconverte (Dconv x d ex xt) (%the (mtconverte _ Am) (DM x d ex xt))) + %<- (bump-tconverte-m _ Dconv DM Dord Dconv' _) %. +%worlds (fbind fobind) (bump-tconverte _ _ _) %. +%total {} (bump-tconverte _ _ _) %. +%%%%% Weakening %%%%% +%sort weakeng-aconverte {Rm} %. +%mode weakeng-aconverte %in %in %in %in %in %in %out %. +%sort weakeng-converte {Mm} %. +%mode weakeng-converte %in %in %in %in %in %in %out %. +%sort weakeng-tconverte {Am} %. +%mode weakeng-tconverte %in %in %in %in %in %in %out %. +%scope weakeng-tconverte %term const + %pi (weakeng-aconverte _ Dappend Dappend' Dord (aconverte/const Dconv Dwf Dcof) (maconverte/const DM) (aconverte/const Dconv' Dwf Dcof)) + %<- (weakeng-tconverte _ Dappend Dappend' Dord Dconv DM Dconv') %. +%term forall + %pi (weakeng-aconverte _ Dappend Dappend' Dord (aconverte/forall Dconv) (maconverte/forall DM) (aconverte/forall Dconv')) + %<- (weakeng-tconverte _ Dappend Dappend' Dord Dconv DM Dconv') %. +%scope weakeng-tconverte %term var + %pi (weakeng-aconverte _ Dappend Dappend' Dord (aconverte/var Dtrans Dconv Dlook) (maconverte/var DM) (aconverte/var Dtrans Dconv' Dlook')) + %<- (weakeng-tconverte _ Dappend Dappend' Dord Dconv DM Dconv') + %<- (weakeng-lookup Dappend Dappend' Dord Dlook Dlook') %. +%term vari + %pi (weakeng-aconverte _ Dappend Dappend' Dord (aconverte/vari Dtrans Dconv Dwf Dvof) (maconverte/vari DM) (aconverte/vari Dtrans Dconv' Dwf Dvof)) + %<- (weakeng-tconverte _ Dappend Dappend' Dord Dconv DM Dconv') %. +%scope weakeng-tconverte %term app + %pi (weakeng-aconverte _ Dappend Dappend' Dord (aconverte/app Dsub Dconv2 Dconv1) (maconverte/app DM2 DM1) (aconverte/app Dsub Dconv2' Dconv1')) + %<- (weakeng-aconverte _ Dappend Dappend' Dord Dconv1 DM1 Dconv1') + %<- (weakeng-converte _ Dappend Dappend' Dord Dconv2 DM2 Dconv2') %. +%term pi1 + %pi (weakeng-aconverte _ Dappend Dappend' Dord (aconverte/pi1 Dconv) (maconverte/pi1 DM) (aconverte/pi1 Dconv')) + %<- (weakeng-aconverte _ Dappend Dappend' Dord Dconv DM Dconv') %. +%scope weakeng-tconverte %term pi2 + %pi (weakeng-aconverte _ Dappend Dappend' Dord (aconverte/pi2 Dconv) (maconverte/pi2 DM) (aconverte/pi2 Dconv')) + %<- (weakeng-aconverte _ Dappend Dappend' Dord Dconv DM Dconv') %. +%term at + %pi (weakeng-converte _ Dappend Dappend' Dord (converte/at Dconv) (mconverte/at DM) (converte/at Dconv')) + %<- (weakeng-aconverte _ Dappend Dappend' Dord Dconv DM Dconv') %. +%scope weakeng-tconverte %term lam + %pi (weakeng-converte _ Dappend Dappend' Dord (converte/lam Dconv2 Dconv1) (mconverte/lam DM2 DM1) (converte/lam Dconv2'' Dconv1')) + %<- (weakeng-tconverte _ Dappend Dappend' Dord Dconv1 DM1 Dconv1') + %<- ({y} {e isvar y J1} {ey} {yt} converte-context (Dconv2 y e ey yt) (ordered/cons (%the (bounded G y) (Dbound1 y e)))) + %<- (extend-context Dord (%the ({y} %pi (isvar y J2) %-> (bounded G' y)) Dbound2)) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- (bump-bounded Dleq1 ([y] [e isvar y J1] Dbound1 y e) ([y] [e isvar y J] %the (bounded G y) (Dbound1' y e))) + %<- (bump-bounded Dleq2 ([y] [e isvar y J2] Dbound2 y e) ([y] [e isvar y J] %the (bounded G' y) (Dbound2' y e))) + %<- (bump-converte-m _ Dconv2 DM2 ([y] [e] ordered/cons (Dbound1' y e)) Dconv2' DM2') + %<- ({y} {e} {ey} {yt} weakeng-converte _ (append/cons Dappend) (append/cons Dappend') (ordered/cons (Dbound2' y e)) (Dconv2' y e ey yt) (DM2' y e ey yt) (Dconv2'' y e ey yt)) %. +%term pair + %pi (weakeng-converte _ Dappend Dappend' Dord (converte/pair Dwf Dconv2 Dsub Dconv1) (mconverte/pair DM2 DM1) (converte/pair Dwf'' Dconv2' Dsub Dconv1')) + %<- (weakeng-converte _ Dappend Dappend' Dord Dconv1 DM1 Dconv1') + %<- (weakeng-converte _ Dappend Dappend' Dord Dconv2 DM2 Dconv2') + %<- (extend-context Dord Dbound) + %<- ({x} {d} bounded-drop Dappend' Dappend (Dbound x d) (Dbound' x d)) + %<- (bump-wfe Dwf ([x] [d] ordered/cons (Dbound' x d)) Dwf') + %<- ({x} {d} weakeng-wfe (append/cons Dappend) (append/cons Dappend') (ordered/cons (Dbound x d)) (Dwf' x d) (Dwf'' x d)) %. +%scope weakeng-tconverte %term sing + %pi (weakeng-converte _ Dappend Dappend' Dord (converte/sing Dconv) (mconverte/sing DM) (converte/sing Dconv')) + %<- (weakeng-aconverte _ Dappend Dappend' Dord Dconv DM Dconv') %. +%term t weakeng-tconverte _ Dappend Dappend' Dord (tconverte/t _) mtconverte/t (tconverte/t Dord) %. +%scope weakeng-tconverte %term pi + %pi (weakeng-tconverte _ Dappend Dappend' Dord (tconverte/pi Dconv2 Dconv1) (mtconverte/pi DM2 DM1) (tconverte/pi Dconv2'' Dconv1')) + %<- (weakeng-tconverte _ Dappend Dappend' Dord Dconv1 DM1 Dconv1') + %<- ({y} {e isvar y J1} {ey} {yt} tconverte-context (Dconv2 y e ey yt) (ordered/cons (%the (bounded G y) (Dbound1 y e)))) + %<- (extend-context Dord (%the ({y} %pi (isvar y J2) %-> (bounded G' y)) Dbound2)) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- (bump-bounded Dleq1 ([y] [e isvar y J1] Dbound1 y e) ([y] [e isvar y J] %the (bounded G y) (Dbound1' y e))) + %<- (bump-bounded Dleq2 ([y] [e isvar y J2] Dbound2 y e) ([y] [e isvar y J] %the (bounded G' y) (Dbound2' y e))) + %<- (bump-tconverte-m _ Dconv2 DM2 ([y] [e] ordered/cons (Dbound1' y e)) Dconv2' DM2') + %<- ({y} {e} {ey} {yt} weakeng-tconverte _ (append/cons Dappend) (append/cons Dappend') (ordered/cons (Dbound2' y e)) (Dconv2' y e ey yt) (DM2' y e ey yt) (Dconv2'' y e ey yt)) %. +%term sigma + %pi (weakeng-tconverte _ Dappend Dappend' Dord (tconverte/sigma Dconv2 Dconv1) (mtconverte/sigma DM2 DM1) (tconverte/sigma Dconv2'' Dconv1')) + %<- (weakeng-tconverte _ Dappend Dappend' Dord Dconv1 DM1 Dconv1') + %<- ({y} {e isvar y J1} {ey} {yt} tconverte-context (Dconv2 y e ey yt) (ordered/cons (%the (bounded G y) (Dbound1 y e)))) + %<- (extend-context Dord (%the ({y} %pi (isvar y J2) %-> (bounded G' y)) Dbound2)) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- (bump-bounded Dleq1 ([y] [e isvar y J1] Dbound1 y e) ([y] [e isvar y J] %the (bounded G y) (Dbound1' y e))) + %<- (bump-bounded Dleq2 ([y] [e isvar y J2] Dbound2 y e) ([y] [e isvar y J] %the (bounded G' y) (Dbound2' y e))) + %<- (bump-tconverte-m _ Dconv2 DM2 ([y] [e] ordered/cons (Dbound1' y e)) Dconv2' DM2') + %<- ({y} {e} {ey} {yt} weakeng-tconverte _ (append/cons Dappend) (append/cons Dappend') (ordered/cons (Dbound2' y e)) (Dconv2' y e ey yt) (DM2' y e ey yt) (Dconv2'' y e ey yt)) %. +%scope weakeng-tconverte %term sing + %pi (weakeng-tconverte _ Dappend Dappend' Dord (tconverte/sing Dconv) (mtconverte/sing DM) (tconverte/sing Dconv')) + %<- (weakeng-aconverte _ Dappend Dappend' Dord Dconv DM Dconv') %. +%worlds (fbind fobind ovar tbind evar) (weakeng-aconverte _ _ _ _ _ _ _) (weakeng-converte _ _ _ _ _ _ _) (weakeng-tconverte _ _ _ _ _ _ _) %. +%total (Rm Mm Am) (weakeng-aconverte Rm _ _ _ _ _ _) (weakeng-converte Mm _ _ _ _ _ _) (weakeng-tconverte Am _ _ _ _ _ _) %. +%sort weaken-converte %. +%mode weaken-converte %in %in %in %out %. +%term _ + %pi (weaken-converte Dbound Dconv _ Dconv') + %<- (can-mconverte Dconv DM) + %<- (weakeng-converte _ append/nil append/nil (ordered/cons Dbound) Dconv DM Dconv') %. +%worlds (fbind fobind ovar tbind evar) (weaken-converte _ _ _ _) %. +%total {} (weaken-converte _ _ _ _) %. +%sort weaken-aconverte %. +%mode weaken-aconverte %in %in %in %out %. +%term _ + %pi (weaken-aconverte Dbound Daconv _ Daconv') + %<- (can-maconverte Daconv DM) + %<- (weakeng-aconverte _ append/nil append/nil (ordered/cons Dbound) Daconv DM Daconv') %. +%worlds (fbind fobind ovar tbind evar) (weaken-aconverte _ _ _ _) %. +%total {} (weaken-aconverte _ _ _ _) %. +%sort weaken-converte' %. +%mode weaken-converte' %in %in %in %out %. +%sort weaken-converte'' %. +%mode weaken-converte'' %in %in %in %out %. +%scope weaken-converte'' %term nil weaken-converte' append/nil _ D D %. +%term cons + %pi (weaken-converte' (append/cons Dapp) (ordered/cons Dbound) Dconv Dconv'') + %<- (weaken-converte'' Dapp Dbound Dconv Dconv') + %<- (weaken-converte Dbound Dconv' _ Dconv'') %. +%term _ + %pi (weaken-converte'' Dapp Dbound Dconv Dconv') + %<- (bounded-is-ordered Dbound Dord) + %<- (weaken-converte' Dapp Dord Dconv Dconv') %. +%worlds (sbind fobind ovar tbind evar) (weaken-converte' _ _ _ _) (weaken-converte'' _ _ _ _) %. +%total (D1 D2) (weaken-converte' D1 _ _ _) (weaken-converte'' D2 _ _ _) %. +%sort weaken-aconverte' %. +%mode weaken-aconverte' %in %in %in %out %. +%sort weaken-aconverte'' %. +%mode weaken-aconverte'' %in %in %in %out %. +%scope weaken-aconverte'' %term nil weaken-aconverte' append/nil _ D D %. +%term cons + %pi (weaken-aconverte' (append/cons Dapp) (ordered/cons Dbound) Daconv Daconv'') + %<- (weaken-aconverte'' Dapp Dbound Daconv Daconv') + %<- (weaken-aconverte Dbound Daconv' _ Daconv'') %. +%term _ + %pi (weaken-aconverte'' Dapp Dbound Daconv Daconv') + %<- (bounded-is-ordered Dbound Dord) + %<- (weaken-aconverte' Dapp Dord Daconv Daconv') %. +%worlds (sbind fobind ovar tbind evar) (weaken-aconverte' _ _ _ _) (weaken-aconverte'' _ _ _ _) %. +%total (D1 D2) (weaken-aconverte' D1 _ _ _) (weaken-aconverte'' D2 _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/convert-fun-thm.lf b/new-tests/stelf-output/tslf/sing/convert-fun-thm.lf new file mode 100644 index 0000000..ef47d79 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/convert-fun-thm.lf @@ -0,0 +1,292 @@ +%%%%% Conversion Functionality (Explicit Context) %%%%% +%sort tconverte-fun %. +%mode tconverte-fun %in %in %out %. +%sort converte-fun %. +%mode converte-fun %in %in %out %. +%sort aconverte-fun %. +%mode aconverte-fun %in %in %out %out %. +%scope aconverte-fun %term const + %pi (aconverte-fun (aconverte/const _ _ Dkof) (aconverte/const _ _ Dkof') Deq eterm-eq/i) + %<- (kof-fun Dkof Dkof' Deq) %. +%term forall + %pi (aconverte-fun (aconverte/forall Dconv) (aconverte/forall Dconv') tp-eq/i Deq') + %<- (tconverte-fun Dconv Dconv' Deq) + %<- (eterm-resp-etp eforall Deq Deq') %. +%scope aconverte-fun %term var + %pi (aconverte-fun (aconverte/var Dvtrans _ Dlookup) (aconverte/var Dvtrans _ Dlookup') Deq eterm-eq/i) + %<- (lookup-fun Dlookup Dlookup' Deq) %. +%term vari aconverte-fun (aconverte/vari Dvtrans _ _ Dvof) (aconverte/vari Dvtrans _ _ Dvof) tp-eq/i eterm-eq/i %. +%scope aconverte-fun %term varbad + %pi (aconverte-fun (aconverte/vari Dvtrans _ _ Dvof) (aconverte/var Dvtrans _ Dlookup) Deq eterm-eq/i) + %<- (lookup-isvar Dlookup Disvar) + %<- (vof-isvar-contra Dvof Disvar Dfalse) + %<- (false-implies-tp-eq Dfalse Deq) %. +%term varbad + %pi (aconverte-fun (aconverte/var Dvtrans _ Dlookup) (aconverte/vari Dvtrans _ _ Dvof) Deq eterm-eq/i) + %<- (lookup-isvar Dlookup Disvar) + %<- (vof-isvar-contra Dvof Disvar Dfalse) + %<- (false-implies-tp-eq Dfalse Deq) %. +%scope aconverte-fun %term app + %pi (aconverte-fun (aconverte/app Dsub Dconv Daconv) (aconverte/app Dsub' Dconv' Daconv') DeqBx DeqMN) + %<- (aconverte-fun Daconv Daconv' DeqAB DeqM) + %<- (tp-eq-cdr-pi DeqAB DeqA DeqB) + %<- (converte-resp ctx-eq/i term-eq/i DeqA eterm-eq/i Dconv Dconv'') + %<- (converte-fun Dconv'' Dconv' DeqN) + %<- (eterm-resp-eterm2 eapp DeqM DeqN DeqMN) + %<- (tsub-resp DeqB term-eq/i tp-eq/i Dsub Dsub'') + %<- (tsub-fun Dsub'' Dsub' DeqBx) %. +%term pi1 + %pi (aconverte-fun (aconverte/pi1 Dconv) (aconverte/pi1 Dconv') DeqA Deq) + %<- (aconverte-fun Dconv Dconv' DeqAB DeqM) + %<- (tp-eq-cdr-sigma DeqAB DeqA DeqB) + %<- (eterm-resp-eterm epi1 DeqM Deq) %. +%scope aconverte-fun %term pi2 + %pi (aconverte-fun (aconverte/pi2 Dconv) (aconverte/pi2 Dconv') (DeqB (pi1 R)) Deq) + %<- (aconverte-fun (%the (aconverte _ R _ _) Dconv) Dconv' DeqAB DeqM) + %<- (tp-eq-cdr-sigma DeqAB DeqA DeqB) + %<- (eterm-resp-eterm epi2 DeqM Deq) %. +%term at + %pi (converte-fun (converte/at Dconv) (converte/at Dconv') Deq) + %<- (aconverte-fun Dconv Dconv' _ Deq) %. +%scope aconverte-fun %term lam + %pi (converte-fun (converte/lam Dconv DconvA) (converte/lam Dconv' DconvA') Deq') + %<- (tconverte-fun DconvA DconvA' DeqA) + %<- ({x} {d} {ex} {xt} converte-context (Dconv' x d ex xt) (Dord x d)) + %<- (bump-converte Dconv Dord Dconv'') + %<- ({x} {d} {ex} {xt} converte-fun (Dconv'' x d ex xt) (Dconv' x d ex xt) (Deq ex)) + %<- (elam-resp DeqA Deq Deq') %. +%term pair + %pi (converte-fun (converte/pair _ DconvN Dsub DconvM) (converte/pair _ DconvN' Dsub' DconvM') Deq) + %<- (converte-fun DconvM DconvM' DeqM) + %<- (tsub-fun Dsub Dsub' DeqBx) + %<- (converte-resp ctx-eq/i term-eq/i DeqBx eterm-eq/i DconvN DconvN'') + %<- (converte-fun DconvN'' DconvN' DeqN) + %<- (epair-resp DeqM DeqN Deq) %. +%scope aconverte-fun %term sing + %pi (converte-fun (converte/sing Dconv) (converte/sing Dconv') Deq) + %<- (aconverte-fun Dconv Dconv' _ Deq) %. +%term t tconverte-fun (tconverte/t _) (tconverte/t _) etp-eq/i %. +%scope aconverte-fun %term pi + %pi (tconverte-fun (tconverte/pi Dconv DconvA) (tconverte/pi Dconv' DconvA') Deq') + %<- (tconverte-fun DconvA DconvA' DeqA) + %<- ({x} {d} {ex} {xt} tconverte-context (Dconv' x d ex xt) (Dord x d)) + %<- (bump-tconverte Dconv Dord Dconv'') + %<- ({x} {d} {ex} {xt} tconverte-fun (Dconv'' x d ex xt) (Dconv' x d ex xt) (Deq ex)) + %<- (epi-resp DeqA Deq Deq') %. +%term sigma + %pi (tconverte-fun (tconverte/sigma Dconv DconvA) (tconverte/sigma Dconv' DconvA') Deq') + %<- (tconverte-fun DconvA DconvA' DeqA) + %<- ({x} {d} {ex} {xt} tconverte-context (Dconv' x d ex xt) (Dord x d)) + %<- (bump-tconverte Dconv Dord Dconv'') + %<- ({x} {d} {ex} {xt} tconverte-fun (Dconv'' x d ex xt) (Dconv' x d ex xt) (Deq ex)) + %<- (esigma-resp DeqA Deq Deq') %. +%scope aconverte-fun %term sing + %pi (tconverte-fun (tconverte/sing Dconv) (tconverte/sing Dconv') Deq') + %<- (aconverte-fun Dconv Dconv' _ Deq) + %<- (etp-resp-eterm esing Deq Deq') %. +%worlds (fbind fobind) (aconverte-fun _ _ _ _) (converte-fun _ _ _) (tconverte-fun _ _ _) %. +%total (D1 D2 D3) (aconverte-fun _ D1 _ _) (converte-fun _ D2 _) (tconverte-fun _ D3 _) %. +%%%%% Conversion Functionality %%%%% +%sort tconvert-fun %. +%mode tconvert-fun %in %in %out %. +%term _ + %pi (tconvert-fun Dconv Dconv' Deq) + %<- (tconvert-to-tconverte Dconv Dconve) + %<- (tconvert-to-tconverte Dconv' Dconve') + %<- (tconverte-fun Dconve Dconve' Deq) %. +%worlds (fbind) (tconvert-fun _ _ _) %. +%total {} (tconvert-fun _ _ _) %. +%sort convert-fun %. +%mode convert-fun %in %in %out %. +%term _ + %pi (convert-fun Dconv Dconv' Deq) + %<- (convert-to-converte Dconv Dconve) + %<- (convert-to-converte Dconv' Dconve') + %<- (converte-fun Dconve Dconve' Deq) %. +%worlds (fbind) (convert-fun _ _ _) %. +%total {} (convert-fun _ _ _) %. +%sort aconvert-fun %. +%mode aconvert-fun %in %in %out %out %. +%term _ + %pi (aconvert-fun Dconv Dconv' Deq Deq') + %<- (aconvert-to-aconverte Dconv Dconve) + %<- (aconvert-to-aconverte Dconv' Dconve') + %<- (aconverte-fun Dconve Dconve' Deq Deq') %. +%worlds (fbind) (aconvert-fun _ _ _ _) %. +%total {} (aconvert-fun _ _ _ _) %. +%sort vconvert-fun %. +%mode vconvert-fun %in %in %out %. +%term _ vconvert-fun D D eterm-eq/i %. +%worlds (fbind) (vconvert-fun _ _ _) %. +%total {} (vconvert-fun _ _ _) %. +%sort vsound' %. +%mode vsound' %in %in %out %out %. +%term _ + %pi (vsound' Dvof Dvtrans Dconv Devof') + %<- (vsound Dvof Dvtrans' Dconv Devof) + %<- (vconvert-fun Dvtrans' Dvtrans Deq) + %<- (evof-resp Deq etp-eq/i Devof Devof') %. +%worlds (sbind) (vsound' _ _ _ _) %. +%total {} (vsound' _ _ _ _) %. +%%%%% Conversion Functionality in Reverse %%%%% +%sort aconvert-fun2 %. +%mode aconvert-fun2 %in %in %out %out %. +%sort convert-fun2 %. +%mode convert-fun2 %in %in %out %. +%sort tconvert-fun2 %. +%mode tconvert-fun2 %in %in %out %. +%term _ + %pi (aconvert-fun2 (aconvert/const _ _ D) (aconvert/const _ _ D') atom-eq/i Deq) + %<- (kof-fun D D' Deq) %. +%term _ + %pi (aconvert-fun2 (aconvert/forall D) (aconvert/forall D') DeqF DeqQ) + %<- (tconvert-fun2 D D' DeqA) + %<- (tp-resp-tp qtp DeqA DeqQ) + %<- (atom-resp-tp forall DeqA DeqF) %. +%term _ aconvert-fun2 (aconvert/var Dvtrans _ _ Dvof) (aconvert/var Dvtrans _ _ Dvof) atom-eq/i tp-eq/i %. +%term _ + %pi (aconvert-fun2 (aconvert/app Dsub Dconv Daconv) (aconvert/app Dsub' Dconv' Daconv') DeqRM DeqBx) + %<- (aconvert-fun2 Daconv Daconv' DeqR DeqAB) + %<- (tp-eq-cdr-pi DeqAB DeqA DeqB) + %<- (convert-resp term-eq/i DeqA eterm-eq/i Dconv Dconv'') + %<- (convert-fun2 Dconv'' Dconv' DeqM) + %<- (app-resp DeqR DeqM DeqRM) + %<- (tsub-resp DeqB DeqM tp-eq/i Dsub Dsub'') + %<- (tsub-fun Dsub'' Dsub' DeqBx) %. +%term _ + %pi (aconvert-fun2 (aconvert/pi1 D) (aconvert/pi1 D') DeqR1 DeqA) + %<- (aconvert-fun2 D D' DeqR DeqAB) + %<- (atom-resp-atom pi1 DeqR DeqR1) + %<- (tp-eq-cdr-sigma DeqAB DeqA DeqB) %. +%term _ + %pi (aconvert-fun2 (aconvert/pi2 D) (aconvert/pi2 D') DeqR2 Deq) + %<- (aconvert-fun2 D D' DeqR DeqAB) + %<- (atom-resp-atom pi1 DeqR DeqR1) + %<- (atom-resp-atom pi2 DeqR DeqR2) + %<- (tp-eq-cdr-sigma DeqAB DeqA DeqB) + %<- (tp-resp-atom-fun DeqB DeqR1 Deq) %. +%term _ + %pi (convert-fun2 (convert/at D) (convert/at D') Deq') + %<- (aconvert-fun2 D D' Deq _) + %<- (term-resp-atom at Deq Deq') %. +%term _ + %pi (convert-fun2 (convert/lam D2 D1) (convert/lam D2' D1') Deq) + %<- (tconvert-fun2 D1 D1' Deq1) + %<- (convert-resp-underbind Deq1 D2 D2'') + %<- ({x} {d} {ex} {xt} convert-fun2 (D2'' x d ex xt) (D2' x d ex xt) (Deq2 x)) + %<- (lam-resp Deq2 Deq) %. +%term _ + %pi (convert-fun2 (convert/pair _ D3 D2 D1) (convert/pair _ D3' D2' D1') Deq) + %<- (convert-fun2 D1 D1' Deq1) + %<- (tsub-resp ([_] tp-eq/i) Deq1 tp-eq/i D2 D2'') + %<- (tsub-fun D2'' D2' Deq2) + %<- (convert-resp term-eq/i Deq2 eterm-eq/i D3 D3'') + %<- (convert-fun2 D3'' D3' Deq3) + %<- (pair-resp Deq1 Deq3 Deq) %. +%term _ + %pi (convert-fun2 (convert/sing D) (convert/sing D') Deq') + %<- (aconvert-fun2 D D' Deq _) + %<- (term-resp-atom at Deq Deq') %. +%term _ tconvert-fun2 tconvert/t tconvert/t tp-eq/i %. +%term _ + %pi (tconvert-fun2 (tconvert/pi D2 D1) (tconvert/pi D2' D1') Deq) + %<- (tconvert-fun2 D1 D1' Deq1) + %<- (tconvert-resp-underbind Deq1 D2 D2'') + %<- ({x} {d} {ex} {xt} tconvert-fun2 (D2'' x d ex xt) (D2' x d ex xt) (Deq2 x)) + %<- (pi-resp Deq1 Deq2 Deq) %. +%term _ + %pi (tconvert-fun2 (tconvert/sigma D2 D1) (tconvert/sigma D2' D1') Deq) + %<- (tconvert-fun2 D1 D1' Deq1) + %<- (tconvert-resp-underbind Deq1 D2 D2'') + %<- ({x} {d} {ex} {xt} tconvert-fun2 (D2'' x d ex xt) (D2' x d ex xt) (Deq2 x)) + %<- (sigma-resp Deq1 Deq2 Deq) %. +%term _ + %pi (tconvert-fun2 (tconvert/sing D) (tconvert/sing D') Deq') + %<- (aconvert-fun2 D D' Deq _) + %<- (tp-resp-atom sing Deq Deq') %. +%worlds (bind tbind) (aconvert-fun2 _ _ _ _) (convert-fun2 _ _ _) (tconvert-fun2 _ _ _) %. +%total (D1 D2 D3) (aconvert-fun2 _ D1 _ _) (convert-fun2 _ D2 _) (tconvert-fun2 _ D3 _) %. +%%%%% Translation Inverts Conversion %%%%% +%sort invert-tconvert %. +%mode invert-tconvert %in %out %. +%sort invert-convert %. +%mode invert-convert %in %in %out %. +%sort invert-aconvert %. +%mode invert-aconvert %in %in %in %out %. +%scope invert-aconvert %term const invert-aconvert (%the (expand (const K) A M) Dexpand) (%the (self M A As) Dself) (aconvert/const _ (%the (wf A) Dwf) (%the (kof K A) Dkof)) (trans/const Dself Dexpand Dwf Dkof) %. +%term forall + %pi (invert-aconvert (%the (expand (forall A) (qtp A) M) Dexpand) (%the (self M (qtp A) B) Dself) (aconvert/forall Dconv) (trans/forall Dself Dexpand Dtrans)) + %<- (invert-tconvert Dconv Dtrans) %. +%scope invert-aconvert %term var invert-aconvert (%the (expand X A M) Dexpand) (%the (self M A As) Dself) (aconvert/var (%the (vtrans EX X) Dvtrans) _ (%the (wf A) Dwf) (%the (vof X A) Dvof)) (trans/var Dself Dexpand Dwf Dvof Dvtrans) %. +%term app + %pi (invert-aconvert (%the (expand (app R N) Bx M) Dexpand) (%the (self M Bx Bsx) Dself) (aconvert/app (%the (tsub B N Bx) DsubBx) (%the (convert N A EN) DconvN) (%the (aconvert R (pi A B) ER) DconvR)) (trans/app DsubBsx Dsubtype DtransN DtransR)) + %<- (aconvert-reg-il DconvR (%the (aof R (pi A B)) DofR)) + %<- (convert-reg-il DconvN (%the (of N A) DofN)) + %<- (aof-reg DofR (wf/pi (%the ({x} %pi (vof x A) %-> (wf (B x))) DwfB) (%the (wf A) DwfA))) + %<- ({x} can-expand x A (X x) (%the (expand x A (X x)) (DexpandX x))) + %<- ({x} can-expand (app R (X x)) (B x) (O x) (%the (expand (app R (X x)) (B x) (O x)) (DexpandO x))) + %<- (expand-reg DofR (expand/pi DexpandO DexpandX) (of/lam (%the ({x} %pi (vof x A) %-> (of (O x) (B x))) DofO) _)) + %<- ({x} {d} can-self (DofO x d) (%the (self (O x) (B x) (Bs x)) (DselfBs x))) + %<- (invert-aconvert (expand/pi DexpandO DexpandX) (self/pi DselfBs) DconvR (%the (trans ER (pi A Bs)) DtransR)) + %<- (can-self DofN (%the (self N A As) DselfAs)) + %<- (invert-convert DselfAs DconvN (%the (trans EN As) DtransN)) + %<- (self-reg DselfAs DofN _ (%the (subtype As A ([_] N)) Dsubtype)) + %<- (sub-into-expand-var DexpandX DofN (%the (sub X N N) DsubN)) + %<- (tsub-expand-var DwfA DwfB DexpandX (%the ({x} tsub B (X x) (B x)) DsubB)) + %<- ({x} {d} expand-reg (aof/var DwfA d) (DexpandX x) (%the (of (X x) A) (DofX x d))) + %<- (expand-aasub DofN ([x] [d] aof/app (DwfB x d) (DsubB x) (DofX x d) DofR) DexpandO (aasub/app DsubN aasub/closed) DsubBx Dexpand (%the (sub O N M) DsubOx)) + %<- (self-sub' DofN DofO DselfBs DsubOx DsubBx Dself (%the (tsub Bs N Bsx) DsubBsx)) %. +%scope invert-aconvert %term pi1 + %pi (invert-aconvert (%the (expand (pi1 R) A M1) Dexpand1) (%the (self M1 A As) Dself1) (aconvert/pi1 (%the (aconvert R (sigma A B) ER) Dconv)) (trans/pi1 Dtrans)) + %<- (aconvert-reg-il Dconv (%the (aof R (sigma A B)) DofR)) + %<- (aof-reg DofR (wf/sigma (%the ({x} %pi (vof x A) %-> (wf (B x))) DwfB) _)) + %<- (can-expand (pi2 R) (B (pi1 R)) _ (%the (expand (pi2 R) (B (pi1 R)) M2) Dexpand2)) + %<- (expand-reg (aof/pi1 DofR) Dexpand1 (%the (of M1 A) Dof1)) + %<- (expand-reg (aof/pi2 DofR) Dexpand2 (%the (of M2 (B (pi1 R))) Dof2)) + %<- (can-self Dof2 (%the (self M2 (B (pi1 R)) Bs) Dself2)) + %<- (can-tsub DwfB Dof1 (%the (tsub B M1 Bx) DsubBx)) + %<- (tsub-expand (aof/pi1 DofR) DwfB Dexpand1 DsubBx (%the (tp-eq Bx (B (pi1 R))) Deq)) + %<- (tsub-resp ([_] tp-eq/i) term-eq/i Deq DsubBx (%the (tsub B M1 (B (pi1 R))) DsubBx')) + %<- (invert-aconvert (expand/sigma Dexpand2 Dexpand1) (self/sigma Dself2 DsubBx' Dself1) Dconv (%the (trans ER (sigma As ([_] Bs))) Dtrans)) %. +%term pi2 + %pi (invert-aconvert (%the (expand (pi2 R) (B (pi1 R)) M2) Dexpand2) (%the (self M2 (B (pi1 R)) Bs) Dself2) (aconvert/pi2 (%the (aconvert R (sigma A B) ER) Dconv)) (trans/pi2 Dtrans)) + %<- (aconvert-reg-il Dconv (%the (aof R (sigma A B)) DofR)) + %<- (aof-reg DofR (wf/sigma (%the ({x} %pi (vof x A) %-> (wf (B x))) DwfB) _)) + %<- (can-expand (pi1 R) A _ (%the (expand (pi1 R) A M1) Dexpand1)) + %<- (expand-reg (aof/pi1 DofR) Dexpand1 (%the (of M1 A) Dof1)) + %<- (expand-reg (aof/pi2 DofR) Dexpand2 (%the (of M2 (B (pi1 R))) Dof2)) + %<- (can-self Dof1 (%the (self M1 A As) Dself1)) + %<- (can-tsub DwfB Dof1 (%the (tsub B M1 Bx) DsubBx)) + %<- (tsub-expand (aof/pi1 DofR) DwfB Dexpand1 DsubBx (%the (tp-eq Bx (B (pi1 R))) Deq)) + %<- (tsub-resp ([_] tp-eq/i) term-eq/i Deq DsubBx (%the (tsub B M1 (B (pi1 R))) DsubBx')) + %<- (invert-aconvert (expand/sigma Dexpand2 Dexpand1) (self/sigma Dself2 DsubBx' Dself1) Dconv (%the (trans ER (sigma As ([_] Bs))) Dtrans)) %. +%scope invert-aconvert %term at + %pi (invert-convert self/t (convert/at Dconv) Dtrans) + %<- (invert-aconvert expand/t self/t Dconv Dtrans) %. +%term sing + %pi (invert-convert self/sing (convert/sing Dconv) Dtrans) + %<- (invert-aconvert expand/t self/t Dconv Dtrans) %. +%scope invert-aconvert %term lam + %pi (invert-convert (self/pi Dself) (convert/lam DconvM DconvA) (trans/lam DtransM DtransA)) + %<- (invert-tconvert DconvA DtransA) + %<- ({x} {d} {ex} {xt} invert-convert (Dself x) (DconvM x d ex xt) (DtransM x d ex xt)) %. +%term pair + %pi (invert-convert (self/sigma Dself2 DsubBx Dself1) (convert/pair DwfB DconvN DsubBx' DconvM) (trans/pair DtransN DtransM)) + %<- (invert-convert Dself1 DconvM DtransM) + %<- (tsub-fun DsubBx DsubBx' Deq) + %<- (self-resp term-eq/i Deq tp-eq/i Dself2 Dself2') + %<- (invert-convert Dself2' DconvN DtransN) %. +%scope invert-aconvert %term t invert-tconvert tconvert/t ttrans/t %. +%term sing + %pi (invert-tconvert (tconvert/sing Dconv) (ttrans/sing Dtrans)) + %<- (invert-aconvert expand/t self/t Dconv Dtrans) %. +%scope invert-aconvert %term pi + %pi (invert-tconvert (tconvert/pi DconvB DconvA) (ttrans/pi DtransB DtransA)) + %<- (invert-tconvert DconvA DtransA) + %<- ({x} {d} {ex} {xt} invert-tconvert (DconvB x d ex xt) (DtransB x d ex xt)) %. +%term sigma + %pi (invert-tconvert (tconvert/sigma DconvB DconvA) (ttrans/sigma DtransB DtransA)) + %<- (invert-tconvert DconvA DtransA) + %<- ({x} {d} {ex} {xt} invert-tconvert (DconvB x d ex xt) (DtransB x d ex xt)) %. +%worlds (bind tbind) (invert-aconvert _ _ _ _) (invert-convert _ _ _) (invert-tconvert _ _) %. +%total (D1 D2 D3) (invert-aconvert _ _ D1 _) (invert-convert _ D2 _) (invert-tconvert D3 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/convert-normal-thm.lf b/new-tests/stelf-output/tslf/sing/convert-normal-thm.lf new file mode 100644 index 0000000..84f6b83 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/convert-normal-thm.lf @@ -0,0 +1,145 @@ +%%%% Normal Terms %%%%% +%sort atomic %. +%sort normal %. +%sort tnormal %. +%term atomic/const atomic (econst C) %. +%term atomic/forall %pi (atomic (eforall A)) %<- (tnormal A) %. +%term atomic/var %pi (atomic X) %<- (variable X) %. +%term atomic/app %pi (atomic (eapp M N)) %<- (atomic M) %<- (normal N) %. +%term atomic/pi1 %pi (atomic (epi1 M)) %<- (atomic M) %. +%term atomic/pi2 %pi (atomic (epi2 M)) %<- (atomic M) %. +%term normal/atom %pi (normal M) %<- (atomic M) %. +%term normal/lam + %pi (normal (elam A M)) + %<- (tnormal A) + %<- ({x} %pi (variable x) %-> (normal (M x))) %. +%term normal/pair %pi (normal (epair M N)) %<- (normal M) %<- (normal N) %. +%term tnormal/t tnormal et %. +%term tnormal/pi + %pi (tnormal (epi A B)) + %<- (tnormal A) + %<- ({x} %pi (variable x) %-> (tnormal (B x))) %. +%term tnormal/sigma + %pi (tnormal (esigma A B)) + %<- (tnormal A) + %<- ({x} %pi (variable x) %-> (tnormal (B x))) %. +%term tnormal/sing %pi (tnormal (esing M)) %<- (atomic M) %. +%%%%% Converted Terms are Normal %%%%% +%sort aconverte-atomic %. +%mode aconverte-atomic %in %out %. +%sort converte-normal %. +%mode converte-normal %in %out %. +%sort tconverte-normal %. +%mode tconverte-normal %in %out %. +%term _ aconverte-atomic (aconverte/const _ _ _) atomic/const %. +%term _ + %pi (aconverte-atomic (aconverte/forall Dconv) (atomic/forall Dnorm)) + %<- (tconverte-normal Dconv Dnorm) %. +%term _ + %pi (aconverte-atomic (aconverte/var Dvtrans _ _) (atomic/var Dvar)) + %<- (vtrans-variable Dvtrans Dvar) %. +%term _ + %pi (aconverte-atomic (aconverte/vari Dvtrans _ _ _) (atomic/var Dvar)) + %<- (vtrans-variable Dvtrans Dvar) %. +%term _ + %pi (aconverte-atomic (aconverte/app _ DconvM DconvR) (atomic/app Dnorm Datom)) + %<- (aconverte-atomic DconvR Datom) + %<- (converte-normal DconvM Dnorm) %. +%term _ + %pi (aconverte-atomic (aconverte/pi1 Dconv) (atomic/pi1 Datom)) + %<- (aconverte-atomic Dconv Datom) %. +%term _ + %pi (aconverte-atomic (aconverte/pi2 Dconv) (atomic/pi2 Datom)) + %<- (aconverte-atomic Dconv Datom) %. +%term _ + %pi (converte-normal (converte/at Dconv) (normal/atom Datom)) + %<- (aconverte-atomic Dconv Datom) %. +%term _ + %pi (converte-normal (converte/lam DconvM DconvA) (normal/lam DnormM DnormA)) + %<- (tconverte-normal DconvA DnormA) + %<- ({x} {d} {ex} {xt} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (converte-normal (DconvM x d ex xt) (DnormM ex dv))) %. +%term _ + %pi (converte-normal (converte/pair _ DconvN _ DconvM) (normal/pair DnormN DnormM)) + %<- (converte-normal DconvM DnormM) + %<- (converte-normal DconvN DnormN) %. +%term _ + %pi (converte-normal (converte/sing Dconv) (normal/atom Datom)) + %<- (aconverte-atomic Dconv Datom) %. +%term _ tconverte-normal (tconverte/t _) tnormal/t %. +%term _ + %pi (tconverte-normal (tconverte/pi DconvB DconvA) (tnormal/pi DnormB DnormA)) + %<- (tconverte-normal DconvA DnormA) + %<- ({x} {d} {ex} {xt} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (tconverte-normal (DconvB x d ex xt) (DnormB ex dv))) %. +%term _ + %pi (tconverte-normal (tconverte/sigma DconvB DconvA) (tnormal/sigma DnormB DnormA)) + %<- (tconverte-normal DconvA DnormA) + %<- ({x} {d} {ex} {xt} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (tconverte-normal (DconvB x d ex xt) (DnormB ex dv))) %. +%term _ + %pi (tconverte-normal (tconverte/sing Dconv) (tnormal/sing Datom)) + %<- (aconverte-atomic Dconv Datom) %. +%worlds (sbind tvbind ovar evar) (aconverte-atomic _ _) (converte-normal _ _) (tconverte-normal _ _) %. +%total (D1 D2 D3) (aconverte-atomic D1 _) (converte-normal D2 _) (tconverte-normal D3 _) %. +%%%%% Reduction is Reflexive on Normal Terms %%%%% +%sort atomic-implies-oa %. +%mode atomic-implies-oa %in %out %. +%term _ atomic-implies-oa atomic/const oa/const %. +%term _ atomic-implies-oa (atomic/forall _) oa/forall %. +%term _ atomic-implies-oa (atomic/var D) (oa/var D) %. +%term _ atomic-implies-oa (atomic/app _ _) oa/app %. +%term _ atomic-implies-oa (atomic/pi1 _) oa/pi1 %. +%term _ atomic-implies-oa (atomic/pi2 _) oa/pi2 %. +%worlds (sbind tvbind evar evvar) (atomic-implies-oa _ _) %. +%total {} (atomic-implies-oa _ _) %. +%sort atomic-reduce-refl %. +%mode atomic-reduce-refl %in %out %. +%sort normal-reduce-refl %. +%mode normal-reduce-refl %in %out %. +%sort tnormal-reduce-refl %. +%mode tnormal-reduce-refl %in %out %. +%term _ atomic-reduce-refl atomic/const reduce/const %. +%term _ + %pi (atomic-reduce-refl (atomic/forall Dnorm) (reduce/forall D)) + %<- (tnormal-reduce-refl Dnorm D) %. +%term _ atomic-reduce-refl (atomic/var D) (reduce/var D) %. +%term _ + %pi (atomic-reduce-refl (atomic/app Dnorm Datom) (reduce/app (reduce-app/atom Doa) D2 D1)) + %<- (atomic-reduce-refl Datom D1) + %<- (normal-reduce-refl Dnorm D2) + %<- (atomic-implies-oa Datom Doa) %. +%term _ + %pi (atomic-reduce-refl (atomic/pi1 Datom) (reduce/pi1 (reduce-pi1/atom Doa) D)) + %<- (atomic-reduce-refl Datom D) + %<- (atomic-implies-oa Datom Doa) %. +%term _ + %pi (atomic-reduce-refl (atomic/pi2 Datom) (reduce/pi2 (reduce-pi2/atom Doa) D)) + %<- (atomic-reduce-refl Datom D) + %<- (atomic-implies-oa Datom Doa) %. +%term _ %pi (normal-reduce-refl (normal/atom Datom) D) %<- (atomic-reduce-refl Datom D) %. +%term _ + %pi (normal-reduce-refl (normal/lam DnormM DnormA) (reduce/lam D2 D1)) + %<- (tnormal-reduce-refl DnormA D1) + %<- ({x} {dv variable x} normal-reduce-refl (DnormM x dv) (D2 x dv)) %. +%term _ + %pi (normal-reduce-refl (normal/pair DnormN DnormM) (reduce/pair D2 D1)) + %<- (normal-reduce-refl DnormM D1) + %<- (normal-reduce-refl DnormN D2) %. +%term _ tnormal-reduce-refl tnormal/t treduce/t %. +%term _ + %pi (tnormal-reduce-refl (tnormal/pi DnormB DnormA) (treduce/pi D2 D1)) + %<- (tnormal-reduce-refl DnormA D1) + %<- ({x} {dv} tnormal-reduce-refl (DnormB x dv) (D2 x dv)) %. +%term _ + %pi (tnormal-reduce-refl (tnormal/sigma DnormB DnormA) (treduce/sigma D2 D1)) + %<- (tnormal-reduce-refl DnormA D1) + %<- ({x} {dv} tnormal-reduce-refl (DnormB x dv) (D2 x dv)) %. +%term _ + %pi (tnormal-reduce-refl (tnormal/sing Dnorm) (treduce/sing D)) + %<- (atomic-reduce-refl Dnorm D) %. +%worlds (sbind tvbind evar evvar) (atomic-reduce-refl _ _) (normal-reduce-refl _ _) (tnormal-reduce-refl _ _) %. +%total (D1 D2 D3) (atomic-reduce-refl D1 _) (normal-reduce-refl D2 _) (tnormal-reduce-refl D3 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/convert-reg-il-thm.lf b/new-tests/stelf-output/tslf/sing/convert-reg-il-thm.lf new file mode 100644 index 0000000..679e9e3 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/convert-reg-il-thm.lf @@ -0,0 +1,87 @@ +%%%%% Conversion Regularity -- IL side (Explicit Context) %%%%% +%sort aconverte-reg-il %. +%mode aconverte-reg-il %in %out %. +%sort converte-reg-il %. +%mode converte-reg-il %in %out %. +%sort tconverte-reg-il %. +%mode tconverte-reg-il %in %out %. +%scope tconverte-reg-il %term const + %pi (aconverte-reg-il (aconverte/const Dconv Dwf Dkof) (aofe/closed Dord (aof/const Dwf Dkof))) + %<- (tconverte-context Dconv Dord) %. +%term forall + %pi (aconverte-reg-il (aconverte/forall Dconv) (aofe/forall Dwf)) + %<- (tconverte-reg-il Dconv Dwf) %. +%scope tconverte-reg-il %term var + %pi (aconverte-reg-il (aconverte/var _ Dconv Dlook) (aofe/var Dwf Dlook)) + %<- (tconverte-reg-il Dconv Dwf) %. +%term vari + %pi (aconverte-reg-il (aconverte/vari _ Dconv Dwf Dvof) (aofe/closed Dord (aof/var Dwf Dvof))) + %<- (tconverte-context Dconv Dord) %. +%scope tconverte-reg-il %term app + %pi (aconverte-reg-il (aconverte/app Dsub DconvM DconvR) (aofe/app DwfBx Dsub DofM DofR)) + %<- (aconverte-reg-il DconvR DofR) + %<- (converte-reg-il DconvM DofM) + %<- (aofe-reg DofR (wfe/pi DwfB _)) + %<- (tsubst-e ([_] append/nil) csub/base DofM Dsub DwfB DwfBx) %. +%term pi1 + %pi (aconverte-reg-il (aconverte/pi1 Dconv) (aofe/pi1 Dof)) + %<- (aconverte-reg-il Dconv Dof) %. +%scope tconverte-reg-il %term pi2 + %pi (aconverte-reg-il (aconverte/pi2 Dconv) (aofe/pi2 Dof)) + %<- (aconverte-reg-il Dconv Dof) %. +%term at + %pi (converte-reg-il (converte/at Dconv) (ofe/at Dof)) + %<- (aconverte-reg-il Dconv Dof) %. +%scope tconverte-reg-il %term lam + %pi (converte-reg-il (converte/lam Dconv2 Dconv1) (ofe/lam Dof Dwf)) + %<- (tconverte-reg-il Dconv1 Dwf) + %<- ({x} {d} {ex} {xt} converte-reg-il (Dconv2 x d ex xt) (Dof x d)) %. +%term pair + %pi (converte-reg-il (converte/pair Dwf Dconv2 Dsub Dconv1) (ofe/pair Dwf Dof2 Dsub Dof1)) + %<- (converte-reg-il Dconv1 Dof1) + %<- (converte-reg-il Dconv2 Dof2) %. +%scope tconverte-reg-il %term sing + %pi (converte-reg-il (converte/sing Dconv) (ofe/sing Dof)) + %<- (aconverte-reg-il Dconv Dof) %. +%term t tconverte-reg-il (tconverte/t Dord) (wfe/t Dord) %. +%scope tconverte-reg-il %term pi + %pi (tconverte-reg-il (tconverte/pi Dconv2 Dconv1) (wfe/pi Dwf2 Dwf1)) + %<- (tconverte-reg-il Dconv1 Dwf1) + %<- ({x} {d} {ex} {xt} tconverte-reg-il (Dconv2 x d ex xt) (Dwf2 x d)) %. +%term sigma + %pi (tconverte-reg-il (tconverte/sigma Dconv2 Dconv1) (wfe/sigma Dwf2 Dwf1)) + %<- (tconverte-reg-il Dconv1 Dwf1) + %<- ({x} {d} {ex} {xt} tconverte-reg-il (Dconv2 x d ex xt) (Dwf2 x d)) %. +%scope tconverte-reg-il %term sing + %pi (tconverte-reg-il (tconverte/sing Dconv) (wfe/sing Dof)) + %<- (aconverte-reg-il Dconv Dof) %. +%worlds (fbind fobind bind ovar tbind evar) (aconverte-reg-il _ _) (converte-reg-il _ _) (tconverte-reg-il _ _) %. +%total (D1 D2 D3) (aconverte-reg-il D1 _) (converte-reg-il D2 _) (tconverte-reg-il D3 _) %. +%%%%% Conversion Regularity -- IL side %%%%% +%sort tconvert-reg-il %. +%mode tconvert-reg-il %in %out %. +%term _ + %pi (tconvert-reg-il Dconv Dwf) + %<- (tconvert-to-tconverte Dconv Dconve) + %<- (tconverte-reg-il Dconve Dwfe) + %<- (wfe-to-wf Dwfe Dwf) %. +%worlds (fbind bind tbind) (tconvert-reg-il _ _) %. +%total {} (tconvert-reg-il _ _) %. +%sort convert-reg-il %. +%mode convert-reg-il %in %out %. +%term _ + %pi (convert-reg-il Dconv Dof) + %<- (convert-to-converte Dconv Dconve) + %<- (converte-reg-il Dconve Dofe) + %<- (ofe-to-of Dofe Dof) %. +%worlds (fbind bind tbind eofblock) (convert-reg-il _ _) %. +%total {} (convert-reg-il _ _) %. +%sort aconvert-reg-il %. +%mode aconvert-reg-il %in %out %. +%term _ + %pi (aconvert-reg-il Dconv Dof) + %<- (aconvert-to-aconverte Dconv Dconve) + %<- (aconverte-reg-il Dconve Dofe) + %<- (aofe-to-aof Dofe Dof) %. +%worlds (fbind bind tbind) (aconvert-reg-il _ _) %. +%total {} (aconvert-reg-il _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/convert-reg-thm.lf b/new-tests/stelf-output/tslf/sing/convert-reg-thm.lf new file mode 100644 index 0000000..8aba613 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/convert-reg-thm.lf @@ -0,0 +1,596 @@ +%%%%% Strong Typing %%%%% +%% No extensionality, and limited subsumption. +%sort ewfs %. +%sort eofs %. +%term ewfs/t ewfs et %. +%term ewfs/pi %pi (ewfs (epi A B)) %<- (ewfs A) %<- ({x} %pi (evof x A) %-> (ewfs (B x))) %. +%term ewfs/sigma %pi (ewfs (esigma A B)) %<- (ewfs A) %<- ({x} %pi (evof x A) %-> (ewfs (B x))) %. +%term ewfs/sing %pi (ewfs (esing M)) %<- (eofs M et) %. +%term eofs/var %pi (eofs X A) %<- (evof X A) %<- (ewf A) %. +%term eofs/forall %pi (eofs (eforall A) (qetp A)) %<- (ewfs A) %. +%term eofs/const %pi (eofs (econst C) A) %<- (ekof C A) %<- (ewf A) %. +%term eofs/lam + %pi (eofs (elam A M) (epi A B)) + %<- (ewfs A) + %<- ({x} %pi (evof x A) %-> (eofs (M x) (B x))) %. +%term eofs/app %pi (eofs (eapp M N) (B N)) %<- (eofs M (epi A B)) %<- (eofs N A) %. +%term eofs/pair + %pi (eofs (epair M N) (esigma A B)) + %<- (eofs M A) + %<- (eofs N (B M)) + %<- ({x} %pi (evof x A) %-> (ewf (B x))) %. +%term eofs/pi1 %pi (eofs (epi1 M) A) %<- (eofs M (esigma A B)) %. +%term eofs/pi2 %pi (eofs (epi2 M) (B (epi1 M))) %<- (eofs M (esigma A B)) %. +%term eofs/sing %pi (eofs M (esing M)) %<- (eofs M et) %. +%term eofs/equiv %pi (eofs M B) %<- (eofs M A) %<- (tequiv A B) %. +%sort eofs-resp %. +%mode eofs-resp %in %in %in %out %. +%term _ eofs-resp eterm-eq/i etp-eq/i D D %. +%worlds (evar eofblock) (eofs-resp _ _ _ _) %. +%total {} (eofs-resp _ _ _ _) %. +%reduces = D D' (eofs-resp _ _ D D') %. +%sort ewfs-resp %. +%mode ewfs-resp %in %in %out %. +%term _ ewfs-resp etp-eq/i D D %. +%worlds (evar eofblock) (ewfs-resp _ _ _) %. +%total {} (ewfs-resp _ _ _) %. +%%%%% Strong and Weak Typing %%%%% +%sort eofs-to-eof %. +%mode eofs-to-eof %in %out %. +%sort ewfs-to-ewf %. +%mode ewfs-to-ewf %in %out %. +%term _ eofs-to-eof (eofs/var D2 D1) (eof/var D2 D1) %. +%term _ %pi (eofs-to-eof (eofs/forall D) (eof/forall D')) %<- (ewfs-to-ewf D D') %. +%term _ eofs-to-eof (eofs/const D2 D1) (eof/const D2 D1) %. +%term _ + %pi (eofs-to-eof (eofs/lam D2 D1) (eof/lam D2' D1')) + %<- (ewfs-to-ewf D1 D1') + %<- ({x} {d} eofs-to-eof (D2 x d) (D2' x d)) %. +%term _ + %pi (eofs-to-eof (eofs/app D2 D1) (eof/app D2' D1')) + %<- (eofs-to-eof D1 D1') + %<- (eofs-to-eof D2 D2') %. +%term _ + %pi (eofs-to-eof (eofs/pair D3 D2 D1) (eof/pair D3 D2' D1')) + %<- (eofs-to-eof D1 D1') + %<- (eofs-to-eof D2 D2') %. +%term _ %pi (eofs-to-eof (eofs/pi1 D) (eof/pi1 D')) %<- (eofs-to-eof D D') %. +%term _ %pi (eofs-to-eof (eofs/pi2 D) (eof/pi2 D')) %<- (eofs-to-eof D D') %. +%term _ %pi (eofs-to-eof (eofs/sing D) (eof/sing D')) %<- (eofs-to-eof D D') %. +%term _ %pi (eofs-to-eof (eofs/equiv D2 D1) (eof/equiv D2 D1')) %<- (eofs-to-eof D1 D1') %. +%term _ ewfs-to-ewf ewfs/t ewf/t %. +%term _ + %pi (ewfs-to-ewf (ewfs/pi D2 D1) (ewf/pi D2' D1')) + %<- (ewfs-to-ewf D1 D1') + %<- ({x} {d} ewfs-to-ewf (D2 x d) (D2' x d)) %. +%term _ + %pi (ewfs-to-ewf (ewfs/sigma D2 D1) (ewf/sigma D2' D1')) + %<- (ewfs-to-ewf D1 D1') + %<- ({x} {d} ewfs-to-ewf (D2 x d) (D2' x d)) %. +%term _ %pi (ewfs-to-ewf (ewfs/sing D) (ewf/sing D')) %<- (eofs-to-eof D D') %. +%worlds (sbind evar eofblock ebind) (eofs-to-eof _ _) (ewfs-to-ewf _ _) %. +%total (D1 D2) (eofs-to-eof D1 _) (ewfs-to-ewf D2 _) %. +%%%%% Substitution, Strong Typing %%%%% +%sort esubsts-of-gen %. +%mode esubsts-of-gen %in %out %. +%sort esubsts-wf-gen %. +%mode esubsts-wf-gen %in %out %. +%scope esubsts-wf-gen %term varsam esubsts-of-gen ([d] eofs/var (D d) d) ([d] [d'] d') %. +%term varoth + %pi (esubsts-of-gen ([d] eofs/var (D2 d) D) ([d] [d'] eofs/var (D2' d) D)) + %<- (esubst-wf-gen D2 D2') %. +%scope esubsts-wf-gen %term const + %pi (esubsts-of-gen ([d] eofs/const (D2 d) D) ([d] [d'] eofs/const (D2' d) D)) + %<- (esubst-wf-gen D2 D2') %. +%term forall + %pi (esubsts-of-gen ([d] eofs/forall (D d)) ([d] [d'] eofs/forall (D' d d'))) + %<- (esubsts-wf-gen D D') %. +%scope esubsts-wf-gen %term lam + %pi (esubsts-of-gen ([d] eofs/lam (D2 d) (D1 d)) ([d] [d'] eofs/lam (D2' d d') (D1' d d'))) + %<- (esubsts-wf-gen D1 D1') + %<- ({y} {e} esubsts-of-gen ([d] D2 d y e) ([d] [d'] D2' d d' y e)) %. +%term app + %pi (esubsts-of-gen ([d] eofs/app (D2 d) (D1 d)) ([d] [d'] eofs/app (D2' d d') (D1' d d'))) + %<- (esubsts-of-gen D1 D1') + %<- (esubsts-of-gen D2 D2') %. +%scope esubsts-wf-gen %term pair + %pi (esubsts-of-gen ([d] eofs/pair (%the ({y} %pi (evof y A) %-> (ewf (B y))) (D3 d)) (D2 d) (D1 d)) ([d] [d'] eofs/pair (D3' d) (D2' d d') (D1' d d'))) + %<- (esubsts-of-gen D1 D1') + %<- (esubsts-of-gen D2 D2') + %<- ({y} {e} esubst-wf-gen ([d] D3 d y e) ([d] D3' d y e)) %. +%term pi1 + %pi (esubsts-of-gen ([d] eofs/pi1 (D d)) ([d] [d'] eofs/pi1 (D' d d'))) + %<- (esubsts-of-gen D D') %. +%scope esubsts-wf-gen %term pi2 + %pi (esubsts-of-gen ([d] eofs/pi2 (D d)) ([d] [d'] eofs/pi2 (D' d d'))) + %<- (esubsts-of-gen D D') %. +%term sing + %pi (esubsts-of-gen ([d] eofs/sing (D d)) ([d] [d'] eofs/sing (D' d d'))) + %<- (esubsts-of-gen D D') %. +%scope esubsts-wf-gen %term subsum + %pi (esubsts-of-gen ([d] eofs/equiv (D2 d) (D1 d)) ([d] [d'] eofs/equiv (D2' d) (D1' d d'))) + %<- (esubsts-of-gen D1 D1') + %<- (esubst-tequiv-gen D2 D2') %. +%term t esubsts-wf-gen ([d] ewfs/t) ([d] [d'] ewfs/t) %. +%scope esubsts-wf-gen %term pi + %pi (esubsts-wf-gen ([d] ewfs/pi (D2 d) (D1 d)) ([d] [d'] ewfs/pi (D2' d d') (D1' d d'))) + %<- (esubsts-wf-gen D1 D1') + %<- ({y} {e} esubsts-wf-gen ([d] D2 d y e) ([d] [d'] D2' d d' y e)) %. +%term sigma + %pi (esubsts-wf-gen ([d] ewfs/sigma (D2 d) (D1 d)) ([d] [d'] ewfs/sigma (D2' d d') (D1' d d'))) + %<- (esubsts-wf-gen D1 D1') + %<- ({y} {e} esubsts-wf-gen ([d] D2 d y e) ([d] [d'] D2' d d' y e)) %. +%scope esubsts-wf-gen %term sing + %pi (esubsts-wf-gen ([d] ewfs/sing (D d)) ([d] [d'] ewfs/sing (D' d d'))) + %<- (esubsts-of-gen D D') %. +%worlds (sbind evar eofblock ebind) (esubsts-of-gen _ _) (esubsts-wf-gen _ _) %. +%total (D1 D2) (esubsts-of-gen D1 _) (esubsts-wf-gen D2 _) %. +%sort esubsts {x} %. +%mode esubsts %in %in %out %. +%term _ + %pi (esubsts D1 D2 (D1' D2' D2)) + %<- (esubsts-of-gen (D1 _) D1') + %<- (eofs-to-eof D2 D2') %. +%worlds (evar eofblock ebind) (esubsts _ _ _) %. +%total {} (esubsts _ _ _) %. +%sort esubsts-wf {x} %. +%mode esubsts-wf %in %in %out %. +%term _ + %pi (esubsts-wf D1 D2 (D1' D2' D2)) + %<- (esubsts-wf-gen (D1 _) D1') + %<- (eofs-to-eof D2 D2') %. +%worlds (evar eofblock ebind) (esubsts-wf _ _ _) %. +%total {} (esubsts-wf _ _ _) %. +%%%%% Reduction implies Equivalence %%%%% +%sort tequiv-t-invert %. +%mode tequiv-t-invert %in %out %. +%sort tequiv-t-invert' %. +%mode tequiv-t-invert' %in %out %. +%term _ tequiv-t-invert (tequiv/reflex ewf/t) etp-eq/i %. +%term _ %pi (tequiv-t-invert (tequiv/symm D) D1) %<- (tequiv-t-invert' D D1) %. +%term _ + %pi (tequiv-t-invert (tequiv/trans D2 D1) Deq') + %<- (tequiv-t-invert D2 Deq) + %<- (tequiv-resp etp-eq/i Deq D1 D1') + %<- (tequiv-t-invert D1' Deq') %. +%term _ tequiv-t-invert tequiv/t etp-eq/i %. +%term _ tequiv-t-invert' (tequiv/reflex ewf/t) etp-eq/i %. +%term _ %pi (tequiv-t-invert' (tequiv/symm D) D1) %<- (tequiv-t-invert D D1) %. +%term _ + %pi (tequiv-t-invert' (tequiv/trans D2 D1) Deq') + %<- (tequiv-t-invert' D1 Deq) + %<- (tequiv-resp Deq etp-eq/i D2 D2') + %<- (tequiv-t-invert' D2' Deq') %. +%term _ tequiv-t-invert' tequiv/t etp-eq/i %. +%worlds (ebind) (tequiv-t-invert _ _) (tequiv-t-invert' _ _) %. +%total (D1 D2) (tequiv-t-invert D1 _) (tequiv-t-invert' D2 _) %. +%sort tequiv-sing-invert %. +%mode tequiv-sing-invert %in %out %out %. +%sort tequiv-sing-invert' %. +%mode tequiv-sing-invert' %in %out %out %. +%term _ tequiv-sing-invert (tequiv/reflex (ewf/sing D)) etp-eq/i (equiv/reflex D) %. +%term _ %pi (tequiv-sing-invert (tequiv/symm D) D1 D2) %<- (tequiv-sing-invert' D D1 D2) %. +%term _ + %pi (tequiv-sing-invert (tequiv/trans D2 D1) Deq' (equiv/trans Dequiv Dequiv')) + %<- (tequiv-sing-invert D2 Deq Dequiv) + %<- (tequiv-resp etp-eq/i Deq D1 D1') + %<- (tequiv-sing-invert D1' Deq' Dequiv') %. +%term _ tequiv-sing-invert (tequiv/sing D) etp-eq/i D %. +%term _ tequiv-sing-invert' (tequiv/reflex (ewf/sing D)) etp-eq/i (equiv/reflex D) %. +%term _ %pi (tequiv-sing-invert' (tequiv/symm D) D1 D2) %<- (tequiv-sing-invert D D1 D2) %. +%term _ + %pi (tequiv-sing-invert' (tequiv/trans D2 D1) Deq' (equiv/trans Dequiv Dequiv')) + %<- (tequiv-sing-invert' D1 Deq Dequiv) + %<- (tequiv-resp Deq etp-eq/i D2 D2') + %<- (tequiv-sing-invert' D2' Deq' Dequiv') %. +%term _ tequiv-sing-invert' (tequiv/sing D) etp-eq/i (equiv/symm D) %. +%worlds (ebind) (tequiv-sing-invert _ _ _) (tequiv-sing-invert' _ _ _) %. +%total (D1 D2) (tequiv-sing-invert D1 _ _) (tequiv-sing-invert' D2 _ _) %. +%sort tequiv-pi-invert %. +%mode tequiv-pi-invert %in %out %out %out %. +%sort tequiv-pi-invert' %. +%mode tequiv-pi-invert' %in %out %out %out %. +%term _ tequiv-pi-invert (tequiv/reflex (ewf/pi DwfB DwfA)) etp-eq/i (tequiv/reflex DwfA) ([x] [d] tequiv/reflex (DwfB x d)) %. +%term _ + %pi (tequiv-pi-invert (tequiv/symm D) D1 D2 D3) + %<- (tequiv-pi-invert' D D1 D2 D3) %. +%term _ + %pi (tequiv-pi-invert (tequiv/trans D2 D1) Deq' (tequiv/trans DequivA DequivA') ([x] [d] tequiv/trans (DequivB'' x d) (DequivB' x d))) + %<- (tequiv-pi-invert D2 Deq DequivA DequivB) + %<- (tequiv-resp etp-eq/i Deq D1 D1') + %<- (tequiv-pi-invert D1' Deq' DequivA' DequivB') + %<- (tequiv-reg DequivA' DwfA' _) + %<- ({x} {d evof x A'} esubst-tequiv DequivB (eof/equiv DequivA' (eof/var DwfA' d)) (DequivB'' x d)) %. +%term _ tequiv-pi-invert (tequiv/pi D2 D1) etp-eq/i D1 D2 %. +%term _ tequiv-pi-invert' (tequiv/reflex (ewf/pi DwfB DwfA)) etp-eq/i (tequiv/reflex DwfA) ([x] [d] tequiv/reflex (DwfB x d)) %. +%term _ + %pi (tequiv-pi-invert' (tequiv/symm D) D1 D2 D3) + %<- (tequiv-pi-invert D D1 D2 D3) %. +%term _ + %pi (tequiv-pi-invert' (tequiv/trans D2 D1) Deq' (tequiv/trans DequivA DequivA') ([x] [d] tequiv/trans (DequivB'' x d) (DequivB' x d))) + %<- (tequiv-pi-invert' D1 Deq DequivA DequivB) + %<- (tequiv-resp Deq etp-eq/i D2 D2') + %<- (tequiv-pi-invert' D2' Deq' DequivA' DequivB') + %<- (tequiv-reg DequivA' DwfA' _) + %<- ({x} {d evof x A'} esubst-tequiv DequivB (eof/equiv DequivA' (eof/var DwfA' d)) (DequivB'' x d)) %. +%term _ + %pi (tequiv-pi-invert' (tequiv/pi D2 D1) etp-eq/i (tequiv/symm D1) ([x] [d] tequiv/symm (D2' x d))) + %<- (tequiv-reg D1 _ Dwf) + %<- ({x} {d evof x A} esubst-tequiv D2 (eof/subsume (subtp/reflex (tequiv/symm D1)) (eof/var Dwf d)) (D2' x d)) %. +%worlds (sbind ebind) (tequiv-pi-invert _ _ _ _) (tequiv-pi-invert' _ _ _ _) %. +%total (D1 D2) (tequiv-pi-invert D1 _ _ _) (tequiv-pi-invert' D2 _ _ _) %. +%sort tequiv-sigma-invert %. +%mode tequiv-sigma-invert %in %out %out %out %. +%sort tequiv-sigma-invert' %. +%mode tequiv-sigma-invert' %in %out %out %out %. +%term _ tequiv-sigma-invert (tequiv/reflex (ewf/sigma DwfB DwfA)) etp-eq/i (tequiv/reflex DwfA) ([x] [d] tequiv/reflex (DwfB x d)) %. +%term _ + %pi (tequiv-sigma-invert (tequiv/symm D) D1 D2 D3) + %<- (tequiv-sigma-invert' D D1 D2 D3) %. +%term _ + %pi (tequiv-sigma-invert (tequiv/trans D2 D1) Deq' (tequiv/trans DequivA DequivA') ([x] [d] tequiv/trans (DequivB'' x d) (DequivB' x d))) + %<- (tequiv-sigma-invert D2 Deq DequivA DequivB) + %<- (tequiv-resp etp-eq/i Deq D1 D1') + %<- (tequiv-sigma-invert D1' Deq' DequivA' DequivB') + %<- (tequiv-reg DequivA' DwfA' _) + %<- ({x} {d evof x A'} esubst-tequiv DequivB (eof/equiv DequivA' (eof/var DwfA' d)) (DequivB'' x d)) %. +%term _ tequiv-sigma-invert (tequiv/sigma D2 D1) etp-eq/i D1 D2 %. +%term _ tequiv-sigma-invert' (tequiv/reflex (ewf/sigma DwfB DwfA)) etp-eq/i (tequiv/reflex DwfA) ([x] [d] tequiv/reflex (DwfB x d)) %. +%term _ + %pi (tequiv-sigma-invert' (tequiv/symm D) D1 D2 D3) + %<- (tequiv-sigma-invert D D1 D2 D3) %. +%term _ + %pi (tequiv-sigma-invert' (tequiv/trans D2 D1) Deq' (tequiv/trans DequivA DequivA') ([x] [d] tequiv/trans (DequivB'' x d) (DequivB' x d))) + %<- (tequiv-sigma-invert' D1 Deq DequivA DequivB) + %<- (tequiv-resp Deq etp-eq/i D2 D2') + %<- (tequiv-sigma-invert' D2' Deq' DequivA' DequivB') + %<- (tequiv-reg DequivA' DwfA' _) + %<- ({x} {d evof x A'} esubst-tequiv DequivB (eof/equiv DequivA' (eof/var DwfA' d)) (DequivB'' x d)) %. +%term _ + %pi (tequiv-sigma-invert' (tequiv/sigma D2 D1) etp-eq/i (tequiv/symm D1) ([x] [d] tequiv/symm (D2' x d))) + %<- (tequiv-reg D1 _ Dwf) + %<- ({x} {d evof x A} esubst-tequiv D2 (eof/subsume (subtp/reflex (tequiv/symm D1)) (eof/var Dwf d)) (D2' x d)) %. +%worlds (sbind ebind) (tequiv-sigma-invert _ _ _ _) (tequiv-sigma-invert' _ _ _ _) %. +%total (D1 D2) (tequiv-sigma-invert D1 _ _ _) (tequiv-sigma-invert' D2 _ _ _) %. +%sort eofs-lam-invert %. +%mode eofs-lam-invert %in %out %out %. +%term _ + %pi (eofs-lam-invert (eofs/lam D2 D1) (tequiv/reflex D1') D2) + %<- (ewfs-to-ewf D1 D1') %. +%term _ + %pi (eofs-lam-invert (eofs/equiv (%the (tequiv D (epi A B)) Dequiv) (%the (eofs (elam C M) D) Deof)) (tequiv/trans DequivAC (tequiv/symm DequivA)) ([x] [d] eofs/equiv (DequivB' x d) (DeofM' x d))) + %<- (tequiv-pi-invert Dequiv (%the (etp-eq D (epi A' B')) Deq) (%the (tequiv A' A) DequivA) (%the ({x} %pi (evof x A') %-> (tequiv (B' x) (B x))) DequivB)) + %<- (eofs-resp eterm-eq/i Deq Deof (%the (eofs (elam C M) (epi A' B')) Deof')) + %<- (eofs-lam-invert Deof' (%the (tequiv A' C) DequivAC) (%the ({x} %pi (evof x A') %-> (eofs (M x) (B' x))) DeofM)) + %<- (tequiv-reg DequivA _ (%the (ewf A) DwfA)) + %<- ({x} {d evof x A} esubsts DeofM (eofs/equiv (tequiv/symm DequivA) (eofs/var DwfA d)) (%the (eofs (M x) (B' x)) (DeofM' x d))) + %<- ({x} {d evof x A} esubst-tequiv DequivB (eof/subsume (subtp/reflex (tequiv/symm DequivA)) (eof/var DwfA d)) (%the (tequiv (B' x) (B x)) (DequivB' x d))) %. +%worlds (sbind ebind) (eofs-lam-invert _ _ _) %. +%total D (eofs-lam-invert D _ _) %. +%sort eofs-pair-invert %. +%mode eofs-pair-invert %in %out %out %out %. +%term _ eofs-pair-invert (eofs/pair D3 D2 D1) D1 D2 D3 %. +%term _ + %pi (eofs-pair-invert (eofs/equiv (%the (tequiv D (esigma A B)) Dequiv) (%the (eofs (epair M N) D) Deof)) (eofs/equiv DequivA DeofM) (eofs/equiv DequivBx DeofN) DwfB') + %<- (tequiv-sigma-invert Dequiv (%the (etp-eq D (esigma A' B')) Deq) (%the (tequiv A' A) DequivA) (%the ({x} %pi (evof x A') %-> (tequiv (B' x) (B x))) DequivB)) + %<- (eofs-resp eterm-eq/i Deq Deof (%the (eofs (epair M N) (esigma A' B')) Deof')) + %<- (eofs-pair-invert Deof' (%the (eofs M A') DeofM) (%the (eofs N (B' M)) DeofN) _) + %<- (eofs-to-eof DeofM (%the (eof M A') DeofM')) + %<- (esubst-tequiv DequivB DeofM' (%the (tequiv (B' M) (B M)) DequivBx)) + %<- ({x} {d evof x A'} tequiv-reg (DequivB x d) _ (%the (ewf (B x)) (DwfB x d))) + %<- (tequiv-reg DequivA _ (%the (ewf A) DwfA)) + %<- ({x} {d evof x A} esubst-wf DwfB (eof/equiv (tequiv/symm DequivA) (eof/var DwfA d)) (%the (ewf (B x)) (DwfB' x d))) %. +%worlds (sbind ebind) (eofs-pair-invert _ _ _ _) %. +%total D (eofs-pair-invert D _ _ _) %. +%sort reduce-equiv %. +%mode reduce-equiv %in %in %out %out %. +%sort treduce-equiv %. +%mode treduce-equiv %in %in %out %out %. +%sort reduce-app-equiv %. +%mode reduce-app-equiv %in %in %in %out %out %. +%sort reduce-pi1-equiv %. +%mode reduce-pi1-equiv %in %in %out %out %. +%sort reduce-pi2-equiv %. +%mode reduce-pi2-equiv %in %in %out %out %. +%scope reduce-pi2-equiv %term sing + %pi (reduce-equiv (eofs/sing (%the (eofs M et) Deof)) (%the (reduce M N) Dreduce) (eofs/equiv (tequiv/sing (equiv/symm Dequiv)) (eofs/sing Deof')) (equiv/sing Dequiv)) + %<- (reduce-equiv Deof Dreduce (%the (eofs N et) Deof') (%the (equiv M N et) Dequiv)) %. +%term equiv + %pi (reduce-equiv (eofs/equiv (%the (tequiv A B) Dtequiv) (%the (eofs M A) Deof)) (%the (reduce M N) Dreduce) (eofs/equiv Dtequiv Deof') (equiv/equiv Dtequiv Dequiv)) + %<- (reduce-equiv Deof Dreduce (%the (eofs N A) Deof') (%the (equiv M N A) Dequiv)) %. +%scope reduce-pi2-equiv %term const %pi (reduce-equiv D reduce/const D (equiv/reflex D')) %<- (eofs-to-eof D D') %. +%term forall + %pi (reduce-equiv (eofs/forall Dwf) (reduce/forall Dreduce) (eofs/equiv (tequiv/pi ([_] [_] tequiv/t) (tequiv/pi ([_] [_] tequiv/t) (tequiv/symm Dequiv))) (eofs/forall Dwf')) (equiv/forall Dequiv)) + %<- (treduce-equiv Dwf Dreduce Dwf' Dequiv) %. +%scope reduce-pi2-equiv %term var %pi (reduce-equiv D (reduce/var _) D (equiv/reflex D')) %<- (eofs-to-eof D D') %. +%term lam + %pi (reduce-equiv (eofs/lam (%the ({x} %pi (evof x A) %-> (eofs (M x) (B x))) Deofs) (%the (ewfs A) Dewfs)) (reduce/lam (%the ({x} %pi (variable x) %-> (reduce (M x) (M' x))) DreduceM) (%the (treduce A A') DreduceA)) (eofs/equiv (tequiv/pi ([x] [d] tequiv/reflex (DwfB' x d)) (tequiv/symm DequivA)) (eofs/lam Deofs'' Dewfs')) (equiv/lam DequivM DequivA)) + %<- (treduce-equiv Dewfs DreduceA (%the (ewfs A') Dewfs') (%the (tequiv A A') DequivA)) + %<- ({x} {d evof x A} {dv variable x} reduce-equiv (Deofs x d) (DreduceM x dv) (%the (eofs (M' x) (B x)) (Deofs' x d)) (%the (equiv (M x) (M' x) (B x)) (DequivM x d))) + %<- (tequiv-reg DequivA _ (%the (ewf A') DwfA')) + %<- ({x} {d evof x A'} esubsts Deofs' (eofs/equiv (tequiv/symm DequivA) (eofs/var DwfA' d)) (%the (eofs (M' x) (B x)) (Deofs'' x d))) + %<- ({x} {d evof x A} eofs-to-eof (Deofs x d) (%the (eof (M x) (B x)) (Deof x d))) + %<- ({x} {d evof x A} eof-reg (Deof x d) (%the (ewf (B x)) (DwfB x d))) + %<- ({x} {d evof x A'} esubst-wf DwfB (eof/equiv (tequiv/symm DequivA) (eof/var DwfA' d)) (%the (ewf (B x)) (DwfB' x d))) %. +%scope reduce-pi2-equiv %term app + %pi (reduce-equiv (eofs/app (%the (eofs N A) DeofsN) (%the (eofs M (epi A B)) DeofsM)) (reduce/app (%the (reduce-app M' N' O) DreduceApp) (%the (reduce N N') DreduceN) (%the (reduce M M') DreduceM)) (eofs/equiv DequivBx DeofsO) (equiv/trans (equiv/equiv DequivBx Dequiv) (equiv/app DequivN DequivM))) + %<- (reduce-equiv DeofsM DreduceM (%the (eofs M' (epi A B)) DeofsM') (%the (equiv M M' (epi A B)) DequivM)) + %<- (reduce-equiv DeofsN DreduceN (%the (eofs N' A) DeofsN') (%the (equiv N N' A) DequivN)) + %<- (reduce-app-equiv DeofsM' DeofsN' DreduceApp (%the (eofs O (B N')) DeofsO) (%the (equiv (eapp M' N') O (B N')) Dequiv)) + %<- (eofs-to-eof DeofsM' (%the (eof M' (epi A B)) DeofM')) + %<- (eof-reg DeofM' (ewf/pi (%the ({x} %pi (evof x A) %-> (ewf (B x))) DwfB) (%the (ewf A) DwfA))) + %<- (tfunctionality DwfB (equiv/symm DequivN) (%the (tequiv (B N') (B N)) DequivBx)) %. +%term pair + %pi (reduce-equiv (eofs/pair (%the ({x} %pi (evof x A) %-> (ewf (B x))) DwfB) (%the (eofs N (B M)) DeofsN) (%the (eofs M A) DeofsM)) (reduce/pair (%the (reduce N N') DreduceN) (%the (reduce M M') DreduceM)) (eofs/pair DwfB (eofs/equiv DequivBx DeofsN') DeofsM') (equiv/pair DwfB DequivN DequivM)) + %<- (reduce-equiv DeofsM DreduceM (%the (eofs M' A) DeofsM') (%the (equiv M M' A) DequivM)) + %<- (reduce-equiv DeofsN DreduceN (%the (eofs N' (B M)) DeofsN') (%the (equiv N N' (B M)) DequivN)) + %<- (tfunctionality DwfB DequivM (%the (tequiv (B M) (B M')) DequivBx)) %. +%scope reduce-pi2-equiv %term pi1 + %pi (reduce-equiv (eofs/pi1 (%the (eofs M (esigma A B)) Deofs)) (reduce/pi1 (%the (reduce-pi1 M' N) DreducePi) (%the (reduce M M') Dreduce)) DeofsN (equiv/trans Dequiv (equiv/pi1 DequivM))) + %<- (reduce-equiv Deofs Dreduce (%the (eofs M' (esigma A B)) Deofs') (%the (equiv M M' (esigma A B)) DequivM)) + %<- (reduce-pi1-equiv Deofs' DreducePi (%the (eofs N A) DeofsN) (%the (equiv (epi1 M') N A) Dequiv)) %. +%term pi2 + %pi (reduce-equiv (eofs/pi2 (%the (eofs M (esigma A B)) Deofs)) (reduce/pi2 (%the (reduce-pi2 M' N) DreducePi) (%the (reduce M M') Dreduce)) (eofs/equiv DequivBx DeofsN) (equiv/trans (equiv/equiv DequivBx Dequiv) (equiv/pi2 DequivM))) + %<- (reduce-equiv Deofs Dreduce (%the (eofs M' (esigma A B)) Deofs') (%the (equiv M M' (esigma A B)) DequivM)) + %<- (reduce-pi2-equiv Deofs' DreducePi (%the (eofs N (B (epi1 M'))) DeofsN) (%the (equiv (epi2 M') N (B (epi1 M'))) Dequiv)) + %<- (eofs-to-eof Deofs (%the (eof M (esigma A B)) Deof)) + %<- (eof-reg Deof (ewf/sigma (%the ({x} %pi (evof x A) %-> (ewf (B x))) DwfB) (%the (ewf A) DwfA))) + %<- (tfunctionality DwfB (equiv/symm (equiv/pi1 DequivM)) (%the (tequiv (B (epi1 M')) (B (epi1 M))) DequivBx)) %. +%scope reduce-pi2-equiv %term t %pi (treduce-equiv D treduce/t D (tequiv/reflex D')) %<- (ewfs-to-ewf D D') %. +%term pi + %pi (treduce-equiv (ewfs/pi (%the ({x} %pi (evof x A) %-> (ewfs (B x))) DwfsB) (%the (ewfs A) DwfsA)) (treduce/pi (%the ({x} %pi (variable x) %-> (treduce (B x) (B' x))) DreduceB) (%the (treduce A A') DreduceA)) (ewfs/pi DwfsB'' DwfsA') (tequiv/pi DequivB DequivA)) + %<- (treduce-equiv DwfsA DreduceA (%the (ewfs A') DwfsA') (%the (tequiv A A') DequivA)) + %<- ({x} {d evof x A} {dv variable x} treduce-equiv (DwfsB x d) (DreduceB x dv) (%the (ewfs (B' x)) (DwfsB' x d)) (%the (tequiv (B x) (B' x)) (DequivB x d))) + %<- (tequiv-reg DequivA _ DwfA') + %<- ({x} {d evof x A'} esubsts-wf DwfsB' (eofs/equiv (tequiv/symm DequivA) (eofs/var DwfA' d)) (%the (ewfs (B' x)) (DwfsB'' x d))) %. +%scope reduce-pi2-equiv %term sigma + %pi (treduce-equiv (ewfs/sigma (%the ({x} %pi (evof x A) %-> (ewfs (B x))) DwfsB) (%the (ewfs A) DwfsA)) (treduce/sigma (%the ({x} %pi (variable x) %-> (treduce (B x) (B' x))) DreduceB) (%the (treduce A A') DreduceA)) (ewfs/sigma DwfsB'' DwfsA') (tequiv/sigma DequivB DequivA)) + %<- (treduce-equiv DwfsA DreduceA (%the (ewfs A') DwfsA') (%the (tequiv A A') DequivA)) + %<- ({x} {d evof x A} {dv variable x} treduce-equiv (DwfsB x d) (DreduceB x dv) (%the (ewfs (B' x)) (DwfsB' x d)) (%the (tequiv (B x) (B' x)) (DequivB x d))) + %<- (tequiv-reg DequivA _ DwfA') + %<- ({x} {d evof x A'} esubsts-wf DwfsB' (eofs/equiv (tequiv/symm DequivA) (eofs/var DwfA' d)) (%the (ewfs (B' x)) (DwfsB'' x d))) %. +%term sing + %pi (treduce-equiv (ewfs/sing (%the (eofs M et) Deofs)) (treduce/sing (%the (reduce M M') Dreduce)) (ewfs/sing Deofs') (tequiv/sing Dequiv)) + %<- (reduce-equiv Deofs Dreduce (%the (eofs M' et) Deofs') (%the (equiv M M' et) Dequiv)) %. +%scope reduce-pi2-equiv %term equiv + %pi (reduce-app-equiv (eofs/equiv (%the (tequiv C (epi A B)) Dtequiv) (%the (eofs M C) DofsM)) (%the (eofs N A) DofsN) (%the (reduce-app M N O) Dreduce) (eofs/equiv DequivBx DofsO) (equiv/equiv DequivBx Dequiv)) + %<- (tequiv-pi-invert Dtequiv (%the (etp-eq C (epi A' B')) Deq) (%the (tequiv A' A) DequivA) (%the ({x} %pi (evof x A') %-> (tequiv (B' x) (B x))) DequivB)) + %<- (eofs-resp eterm-eq/i Deq DofsM (%the (eofs M (epi A' B')) DofsM')) + %<- (reduce-app-equiv DofsM' (eofs/equiv (tequiv/symm DequivA) DofsN) Dreduce (%the (eofs O (B' N)) DofsO) (%the (equiv (eapp M N) O (B' N)) Dequiv)) + %<- (eofs-to-eof DofsN (%the (eof N A) DofN)) + %<- (esubst-tequiv DequivB (eof/equiv (tequiv/symm DequivA) DofN) (%the (tequiv (B' N) (B N)) DequivBx)) %. +%term atom + %pi (reduce-app-equiv (%the (eofs M (epi A B)) DeofsM) (%the (eofs N A) DeofsN) (reduce-app/atom _) (eofs/app DeofsN DeofsM) (equiv/reflex D)) + %<- (eofs-to-eof (eofs/app DeofsN DeofsM) D) %. +%scope reduce-pi2-equiv %term beta + %pi (reduce-app-equiv (eofs/lam (%the ({x} %pi (evof x A) %-> (eofs (M x) (B x))) DeofsM) (%the (ewfs A) DewfA)) (%the (eofs N A) DeofsN) (reduce-app/beta (%the (reduce (M N) O) Dreduce)) DeofsO (equiv/trans Dequiv (equiv/beta DeofN DeofM))) + %<- (esubsts DeofsM DeofsN (%the (eofs (M N) (B N)) DeofsMx)) + %<- (reduce-equiv DeofsMx Dreduce (%the (eofs O (B N)) DeofsO) (%the (equiv (M N) O (B N)) Dequiv)) + %<- ({x} {d} eofs-to-eof (DeofsM x d) (%the (eof (M x) (B x)) (DeofM x d))) + %<- (eofs-to-eof DeofsN (%the (eof N A) DeofN)) %. +%term equiv + %pi (reduce-pi1-equiv (eofs/equiv (%the (tequiv C (esigma A B)) Dtequiv) (%the (eofs M C) DofsM)) (%the (reduce-pi1 M N) Dreduce) (eofs/equiv DequivA DofsN) (equiv/equiv DequivA Dequiv)) + %<- (tequiv-sigma-invert Dtequiv (%the (etp-eq C (esigma A' B')) Deq) (%the (tequiv A' A) DequivA) (%the ({x} %pi (evof x A') %-> (tequiv (B' x) (B x))) DequivB)) + %<- (eofs-resp eterm-eq/i Deq DofsM (%the (eofs M (esigma A' B')) DofsM')) + %<- (reduce-pi1-equiv DofsM' Dreduce (%the (eofs N A') DofsN) (%the (equiv (epi1 M) N A') Dequiv)) %. +%scope reduce-pi2-equiv %term atom + %pi (reduce-pi1-equiv (%the (eofs M (esigma A B)) DofsM) (reduce-pi1/atom _) (eofs/pi1 DofsM) (equiv/reflex (eof/pi1 DofM))) + %<- (eofs-to-eof DofsM DofM) %. +%term beta + %pi (reduce-pi1-equiv (eofs/pair (%the ({x} %pi (evof x A) %-> (ewf (B x))) DwfB) (%the (eofs N (B M)) DofsN) (%the (eofs M A) DofsM)) reduce-pi1/beta DofsM (equiv/beta1 DofN DofM)) + %<- (eofs-to-eof DofsM DofM) + %<- (eofs-to-eof DofsN DofN) %. +%scope reduce-pi2-equiv %term equiv + %pi (reduce-pi2-equiv (eofs/equiv (%the (tequiv C (esigma A B)) Dtequiv) (%the (eofs M C) DofsM)) (%the (reduce-pi2 M N) Dreduce) (eofs/equiv DequivBx DofsN) (equiv/equiv DequivBx Dequiv)) + %<- (tequiv-sigma-invert Dtequiv (%the (etp-eq C (esigma A' B')) Deq) (%the (tequiv A' A) DequivA) (%the ({x} %pi (evof x A') %-> (tequiv (B' x) (B x))) DequivB)) + %<- (eofs-resp eterm-eq/i Deq DofsM (%the (eofs M (esigma A' B')) DofsM')) + %<- (reduce-pi2-equiv DofsM' Dreduce (%the (eofs N (B' (epi1 M))) DofsN) (%the (equiv (epi2 M) N (B' (epi1 M))) Dequiv)) + %<- (eofs-to-eof DofsM' (%the (eof M (esigma A' B')) DofM)) + %<- (esubst-tequiv DequivB (eof/pi1 DofM) (%the (tequiv (B' (epi1 M)) (B (epi1 M))) DequivBx)) %. +%term atom + %pi (reduce-pi2-equiv (%the (eofs M (esigma A B)) DofsM) (reduce-pi2/atom _) (eofs/pi2 DofsM) (equiv/reflex (eof/pi2 DofM))) + %<- (eofs-to-eof DofsM DofM) %. +%scope reduce-pi2-equiv %term beta + %pi (reduce-pi2-equiv (eofs/pair (%the ({x} %pi (evof x A) %-> (ewf (B x))) DwfB) (%the (eofs N (B M)) DofsN) (%the (eofs M A) DofsM)) reduce-pi2/beta (eofs/equiv (tequiv/symm Dtequiv) DofsN) (equiv/equiv (tequiv/symm Dtequiv) (equiv/beta2 DofN DofM))) + %<- (eofs-to-eof DofsM DofM) + %<- (eofs-to-eof DofsN DofN) + %<- (tfunctionality DwfB (equiv/beta1 DofN DofM) (%the (tequiv (B (epi1 (epair M N))) (B M)) Dtequiv)) %. +%worlds (sbind ebind evbind) (reduce-equiv _ _ _ _) (treduce-equiv _ _ _ _) (reduce-app-equiv _ _ _ _ _) (reduce-pi1-equiv _ _ _ _) (reduce-pi2-equiv _ _ _ _) %. +%total {(D1 D2 D3 D4 D5) (E1 E2 E3 E4 E5)} (reduce-equiv E1 D1 _ _) (treduce-equiv E2 D2 _ _) (reduce-app-equiv E3 _ D3 _ _) (reduce-pi1-equiv E4 D4 _ _) (reduce-pi2-equiv E5 D5 _ _) %. +%%%%% Conversion Regularity, Strong %%%%% +%sort convert-topen %. +%mode convert-topen %in %out %out %. +%term _ convert-topen topen/t tconvert/t etopen/t %. +%term _ + %pi (convert-topen (topen/pi Dtopen2 Dtopen1) (tconvert/pi ([_] [_] [_] [_] Dconv2) Dconv1) (etopen/pi Detopen2 Detopen1)) + %<- (convert-topen Dtopen1 Dconv1 Detopen1) + %<- (convert-topen Dtopen2 Dconv2 Detopen2) %. +%worlds (bind tbind) (convert-topen _ _ _) %. +%total D (convert-topen D _ _) %. +%sort convert-kof %. +%mode convert-kof %in %in %out %. +%term _ + %pi (convert-kof (kof/i (%the (topen Ac A) Dopen) (%the (ckof C Ac) Dckof)) (%the (tconvert A EA) Dconvert) (ekof/i Deopen' Dckof)) + %<- (convert-topen Dopen (%the (tconvert A EA') Dconvert') (%the (etopen Ac EA') Deopen)) + %<- (tconvert-fun Dconvert' Dconvert (%the (etp-eq EA' EA) Deq)) + %<- (etopen-resp ctp-eq/i Deq Deopen (%the (etopen Ac EA) Deopen')) %. +%worlds (fbind) (convert-kof _ _ _) %. +%total {} (convert-kof _ _ _) %. +%sort vconvert-fun %. +%mode vconvert-fun %in %in %out %. +%term _ vconvert-fun D D eterm-eq/i %. +%worlds (nbind) (vconvert-fun _ _ _) %. +%total {} (vconvert-fun _ _ _) %. +%sort convert-reg-strong %. +%mode convert-reg-strong %in %in %in %out %. +%sort aconvert-reg-strong %. +%mode aconvert-reg-strong %in %out %out %out %. +%sort tconvert-reg-strong %. +%mode tconvert-reg-strong %in %out %. +%scope tconvert-reg-strong %term t tconvert-reg-strong tconvert/t ewfs/t %. +%term pi + %pi (tconvert-reg-strong (tconvert/pi (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))) DconvertB) (%the (tconvert A EA) DconvertA)) (ewfs/pi DwfB DwfA)) + %<- (tconvert-reg-strong DconvertA (%the (ewfs EA) DwfA)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (tconvert-reg-strong (DconvertB x d ex xt) (%the (ewfs (EB ex)) (DwfB ex ed)))) %. +%scope tconvert-reg-strong %term sigma + %pi (tconvert-reg-strong (tconvert/sigma (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))) DconvertB) (%the (tconvert A EA) DconvertA)) (ewfs/sigma DwfB DwfA)) + %<- (tconvert-reg-strong DconvertA (%the (ewfs EA) DwfA)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (tconvert-reg-strong (DconvertB x d ex xt) (%the (ewfs (EB ex)) (DwfB ex ed)))) %. +%term sing + %pi (tconvert-reg-strong (tconvert/sing (%the (aconvert R t EM) Dconvert)) (ewfs/sing Dof)) + %<- (aconvert-reg-strong Dconvert tconvert/t _ (%the (eofs EM et) Dof)) %. +%scope tconvert-reg-strong %term const + %pi (aconvert-reg-strong (aconvert/const (%the (tconvert A EA) Dconvert) _ (%the (kof K A) Dkof)) Dconvert Dewfs (eofs/const Dewf Dekof)) + %<- (convert-kof Dkof Dconvert (%the (ekof K EA) Dekof)) + %<- (tconvert-reg-strong Dconvert (%the (ewfs EA) Dewfs)) + %<- (ewfs-to-ewf Dewfs Dewf) %. +%term forall + %pi (aconvert-reg-strong (aconvert/forall (%the (tconvert A EA) Dconv)) (tconvert/pi ([_] [_] [_] [_] tconvert/t) (tconvert/pi ([_] [_] [_] [_] tconvert/t) Dconv)) (ewfs/pi ([_] [_] ewfs/t) (ewfs/pi ([_] [_] ewfs/t) Dwf)) (eofs/forall Dwf)) + %<- (tconvert-reg-strong Dconv Dwf) %. +%scope tconvert-reg-strong %term var + %pi (aconvert-reg-strong (aconvert/var (%the (vtrans EX X) Dvtrans) (%the (tconvert A EA') Dconvert') _ (%the (vof X A) Dvof)) Dconvert Dewf (eofs/var Dewf'' Devof')) + %<- (vsound Dvof (%the (vtrans EX' X) Dvtrans') (%the (tconvert A EA) Dconvert) (%the (evof EX' EA) Devof)) + %<- (vconvert-fun Dvtrans' Dvtrans (%the (eterm-eq EX' EX) Deq)) + %<- (evof-resp Deq etp-eq/i Devof (%the (evof EX EA) Devof')) + %<- (tconvert-reg-strong Dconvert' (%the (ewfs EA') Dewf')) + %<- (tconvert-fun Dconvert' Dconvert (%the (etp-eq EA' EA) DeqA)) + %<- (ewfs-resp DeqA Dewf' (%the (ewfs EA) Dewf)) + %<- (ewfs-to-ewf Dewf (%the (ewf EA) Dewf'')) %. +%term app + %pi (aconvert-reg-strong (aconvert/app (%the (tsub B N Bx) DsubBx) (%the (convert N A EN) DconvertN) (%the (aconvert R (pi A B) EM) DconvertM)) DconvertBx DwfBx' (eofs/equiv Dequiv (eofs/app DeofN DeofM))) + %<- (aconvert-reg-strong DconvertM (tconvert/pi (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))) DconvertB) (%the (tconvert A EA) DconvertA)) (ewfs/pi (%the ({ex} %pi (evof ex EA) %-> (ewfs (EB ex))) DwfB) (%the (ewfs EA) DwfA)) (%the (eofs EM (epi EA EB)) DeofM)) + %<- (convert-reg-strong DconvertN DconvertA DwfA (%the (eofs EN EA) DeofN)) + %<- (convert-tsub-reduce DconvertB DconvertN DsubBx (%the (tconvert Bx EBx) DconvertBx) (%the (treduce (EB EN) EBx) Dreduce)) + %<- (esubsts-wf DwfB DeofN (%the (ewfs (EB EN)) DwfBx)) + %<- (treduce-equiv DwfBx Dreduce (%the (ewfs EBx) DwfBx') (%the (tequiv (EB EN) EBx) Dequiv)) %. +%scope tconvert-reg-strong %term pi1 + %pi (aconvert-reg-strong (aconvert/pi1 (%the (aconvert R (sigma A B) EM) DconvertM)) DconvertA DwfA (eofs/pi1 Dof)) + %<- (aconvert-reg-strong DconvertM (tconvert/sigma (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))) DconvertB) (%the (tconvert A EA) DconvertA)) (ewfs/sigma (%the ({ex} %pi (evof ex EA) %-> (ewfs (EB ex))) DwfB) (%the (ewfs EA) DwfA)) (%the (eofs EM (esigma EA EB)) Dof)) %. +%term pi2 + %pi (aconvert-reg-strong (aconvert/pi2 (%the (aconvert R (sigma A B) EM) DconvertM)) DconvertBx DwfBx (eofs/pi2 Dof)) + %<- (aconvert-reg-strong DconvertM (tconvert/sigma (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))) DconvertB) (%the (tconvert A EA) DconvertA)) (ewfs/sigma (%the ({ex} %pi (evof ex EA) %-> (ewfs (EB ex))) DwfB) (%the (ewfs EA) DwfA)) (%the (eofs EM (esigma EA EB)) Dof)) + %<- (convert-stsub DconvertB (aconvert/pi1 DconvertM) (%the (tconvert (B (pi1 R)) (EB (epi1 EM))) DconvertBx)) + %<- (esubsts-wf DwfB (eofs/pi1 Dof) (%the (ewfs (EB (epi1 EM))) DwfBx)) %. +%scope tconvert-reg-strong %term at + %pi (convert-reg-strong (convert/at (%the (aconvert R t EM) DconvertM)) tconvert/t ewfs/t Dof) + %<- (aconvert-reg-strong DconvertM tconvert/t _ (%the (eofs EM et) Dof)) %. +%sort convert-reg-strong! %. +%mode convert-reg-strong! %in %in %out %. +%term _ convert-reg-strong! etp-eq/i D D %. +%worlds (sbind) (convert-reg-strong! _ _ _) %. +%total {} (convert-reg-strong! _ _ _) %. +%term lam + %pi (convert-reg-strong (convert/lam (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (convert (M x) (B x) (EM ex)))) DconvertM) (%the (tconvert A EA) DconvertA)) (tconvert/pi (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))) DconvertB) (%the (tconvert A EA') DconvertA')) (ewfs/pi (%the ({ex} %pi (evof ex EA') %-> (ewfs (EB ex))) DwfB) (%the (ewfs EA') DwfA')) Dof') + %<- (tconvert-fun DconvertA DconvertA' (%the (etp-eq EA EA') Deq)) + %<- (etp-eq-symm Deq (%the (etp-eq EA' EA) Deq')) + %<- (convert-reg-strong! Deq' DwfB (%the ({ex} %pi (evof ex EA) %-> (ewfs (EB ex))) DwfB')) + %<- (etp-resp-etp ([a] epi a EB) Deq (%the (etp-eq (epi EA EB) (epi EA' EB)) Deq'')) + %<- (ewfs-resp Deq' DwfA' (%the (ewfs EA) DwfA)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (convert-reg-strong (DconvertM x d ex xt) (DconvertB x d ex xt) (DwfB' ex ed) (%the (eofs (EM ex) (EB ex)) (Dof ex ed)))) + %<- (eofs-resp eterm-eq/i Deq'' (%the (eofs (elam EA EM) (epi EA EB)) (eofs/lam Dof DwfA)) (%the (eofs (elam EA EM) (epi EA' EB)) Dof')) %. +%term pair + %pi (convert-reg-strong (convert/pair (%the ({x} %pi (vof x A) %-> (wf (B x))) Dwf) (%the (convert N Bx EN) DconvertN) (%the (tsub B M Bx) DsubBx) (%the (convert M A EM) DconvertM)) (tconvert/sigma (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))) DconvertB) (%the (tconvert A EA) DconvertA)) (ewfs/sigma (%the ({ex} %pi (evof ex EA) %-> (ewfs (EB ex))) DwfB) (%the (ewfs EA) DwfA)) (eofs/pair DwfB' (eofs/equiv (tequiv/symm Dequiv) DeofN) DeofM)) + %<- (convert-reg-strong DconvertM DconvertA DwfA (%the (eofs EM EA) DeofM)) + %<- (esubsts-wf DwfB DeofM (%the (ewfs (EB EM)) DeofBM)) + %<- (convert-tsub-reduce DconvertB DconvertM DsubBx (%the (tconvert Bx EBx) DconvertBx) (%the (treduce (EB EM) EBx) Dreduce)) + %<- (treduce-equiv DeofBM Dreduce (%the (ewfs EBx) DwfBx) (%the (tequiv (EB EM) EBx) Dequiv)) + %<- (convert-reg-strong DconvertN DconvertBx DwfBx (%the (eofs EN EBx) DeofN)) + %<- ({ex} {ed evof ex EA} ewfs-to-ewf (DwfB ex ed) (%the (ewf (EB ex)) (DwfB' ex ed))) %. +%term at + %pi (convert-reg-strong (convert/sing (%the (aconvert R t EM) DconvertM)) (tconvert/sing (%the (aconvert R t EM') DconvertM')) (ewfs/sing (%the (eofs EM' et) Dof)) Dof') + %<- (aconvert-fun DconvertM' DconvertM _ (%the (eterm-eq EM' EM) Deq)) + %<- (eofs-resp Deq etp-eq/i (eofs/sing Dof) (%the (eofs EM (esing EM')) Dof')) %. +%worlds (sbind) (convert-reg-strong _ _ _ _) (aconvert-reg-strong _ _ _ _) (tconvert-reg-strong _ _) %. +%total (D1 D2 D3) (aconvert-reg-strong D2 _ _ _) (tconvert-reg-strong D3 _) (convert-reg-strong D1 _ _ _) %. +%%%%% Conversion Regularity -- EL side %%%%% +%sort tconvert-reg %. +%mode tconvert-reg %in %out %. +%term _ + %pi (tconvert-reg Dconvert Dwf) + %<- (tconvert-reg-strong Dconvert Dwfs) + %<- (ewfs-to-ewf Dwfs Dwf) %. +%worlds (sbind) (tconvert-reg _ _) %. +%total {} (tconvert-reg _ _) %. +%sort convert-reg %. +%mode convert-reg %in %in %out %. +%term _ + %pi (convert-reg Dconvert Dtconvert Dof) + %<- (tconvert-reg-strong Dtconvert Dwfs) + %<- (convert-reg-strong Dconvert Dtconvert Dwfs Dofs) + %<- (eofs-to-eof Dofs Dof) %. +%worlds (sbind) (convert-reg _ _ _) %. +%total {} (convert-reg _ _ _) %. +%sort aconvert-reg %. +%mode aconvert-reg %in %out %out %. +%term _ + %pi (aconvert-reg Dconvert Dtconvert Dof) + %<- (aconvert-reg-strong Dconvert Dtconvert _ Dofs) + %<- (eofs-to-eof Dofs Dof) %. +%worlds (sbind) (aconvert-reg _ _ _) %. +%total {} (aconvert-reg _ _ _) %. +%sort aconvert-reg' %. +%mode aconvert-reg' %in %in %out %. +%term _ + %pi (aconvert-reg' Dconvert Dtconvert Dof) + %<- (aconvert-reg Dconvert Dtconvert' Dof') + %<- (tconvert-fun Dtconvert' Dtconvert Deq) + %<- (eof-resp eterm-eq/i Deq Dof' Dof) %. +%worlds (sbind) (aconvert-reg' _ _ _) %. +%total {} (aconvert-reg' _ _ _) %. +%%%%% Subsitution Revisited %%%%% +%sort convert-tsub {x} %. +%mode convert-tsub %in %in %in %out %out %. +%term _ + %pi (convert-tsub DconvertB DconvertN Dsub DconvertBx Dequiv) + %<- (convert-tsub-reduce DconvertB DconvertN Dsub DconvertBx Dreduce) + %<- (convert-reg-il DconvertN DofN) + %<- (of-reg DofN DwfA) + %<- (can-tconvert DwfA DconvertA) + %<- ({x} {d} {ex} {xt} {ed} {dv} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (tconvert-reg-strong (DconvertB x d ex xt) (DewfB ex ed))) + %<- (tconvert-reg-strong DconvertA DewfA) + %<- (convert-reg-strong DconvertN DconvertA DewfA DeofN) + %<- (esubsts-wf DewfB DeofN DewfBx) + %<- (treduce-equiv DewfBx Dreduce _ Dequiv) %. +%worlds (sbind) (convert-tsub _ _ _ _ _) %. +%total {} (convert-tsub _ _ _ _ _) %. +%sort convert-sub {x} %. +%mode convert-sub %in %in %in %in %out %out %out %. +%term _ + %pi (convert-sub (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (convert (M x) (B x) (EM ex)))) DconvertM) (%the (convert N A EN) DconvertN) (%the (sub M N Mx) DsubM) (%the (tsub B N Bx) DsubB) DconvertMx DconvertBx (equiv/equiv Dtequiv Dequiv)) + %<- (convert-reg-il DconvertN (%the (of N A) DofN)) + %<- (of-reg DofN (%the (wf A) DwfA)) + %<- (can-tconvert DwfA (%the (tconvert A EA) DconvertA)) + %<- ({x} {d} {ex} {xt} convert-reg-il (DconvertM x d ex xt) (%the (of (M x) (B x)) (DofM x d))) + %<- ({x} {d} of-reg (DofM x d) (%the (wf (B x)) (DwfB x d))) + %<- ({x} {d} {ex} {xt} {ed evof ex EA} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (can-tconvert (DwfB x d) (%the (tconvert (B x) (EB ex)) (DconvertB x d ex xt)))) + %<- (convert-sub-reduce DconvertM DconvertN DsubM DsubB (%the (convert Mx Bx EO) DconvertMx) (%the (reduce (EM EN) EO) DreduceM)) + %<- ({x} {d} {ex} {xt} {ed evof ex EA} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (tconvert-reg-strong (DconvertB x d ex xt) (%the (ewfs (EB ex)) (DewfB ex ed)))) + %<- ({x} {d} {ex} {xt} {ed evof ex EA} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (convert-reg-strong (DconvertM x d ex xt) (DconvertB x d ex xt) (DewfB ex ed) (%the (eofs (EM ex) (EB ex)) (DeofM ex ed)))) + %<- (tconvert-reg-strong DconvertA (%the (ewfs EA) DewfA)) + %<- (convert-reg-strong DconvertN DconvertA DewfA (%the (eofs EN EA) DeofN)) + %<- (esubsts DeofM DeofN (%the (eofs (EM EN) (EB EN)) DeofMx)) + %<- (reduce-equiv DeofMx DreduceM (%the (eofs EO (EB EN)) DeofO) (%the (equiv (EM EN) EO (EB EN)) Dequiv)) + %<- (convert-tsub DconvertB DconvertN DsubB (%the (tconvert Bx EC) DconvertBx) (%the (tequiv (EB EN) EC) Dtequiv)) %. +%worlds (sbind) (convert-sub _ _ _ _ _ _ _) %. +%total {} (convert-sub _ _ _ _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/convert-sub-thm.lf b/new-tests/stelf-output/tslf/sing/convert-sub-thm.lf new file mode 100644 index 0000000..3985864 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/convert-sub-thm.lf @@ -0,0 +1,499 @@ +%%%%% Conversion Commutes with Simple Substitution (Explicit Context) %%%%% +%sort strengthen-vtrans {x} {ex} %. +%mode strengthen-vtrans %in %out %out %. +%term _ strengthen-vtrans ([x] [ex] [d] D) ([ex] eterm-eq/i) D %. +%worlds (bind ovar tbind evar) (strengthen-vtrans _ _ _) %. +%total {} (strengthen-vtrans _ _ _) %. +%sort convert-ssub-e {x} %. +%mode convert-ssub-e %in %in %in %in %out %. +%sort aconvert-ssub-e {x} %. +%mode aconvert-ssub-e %in %in %in %in %out %. +%sort tconvert-ssub-e {x} %. +%mode tconvert-ssub-e %in %in %in %in %out %. +%scope tconvert-ssub-e %term const + %pi (aconvert-ssub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) Dappend' ([x] [d] [ex] [xt] aconverte/const (DconvB x d ex xt) (Dwf x) (%the (kof _ (A' x)) (Dkof x))) Dconv (aconverte/const DconvB' (Dwf _) (Dkof _))) + %<- (tconvert-ssub-e Dappend Dappend' DconvB Dconv DconvB') %. +%term forall + %pi (aconvert-ssub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) Dappend' ([x] [d] [ex] [xt] aconverte/forall (DconvB x d ex xt)) Dconv (aconverte/forall DconvB')) + %<- (tconvert-ssub-e Dappend Dappend' DconvB Dconv DconvB') %. +%scope tconvert-ssub-e %term varsam + %pi (aconvert-ssub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) Dappend' ([x] [d] [ex] [xt] aconverte/var xt (%the (tconverte _ _ (B ex)) _) (Dlookup x d)) Dconv Dconv'') + %<- ({x} {d} lookup-binder-fun (Dappend x) (Dlookup x d) (%the (tp-eq _ (A' x)) (Deq x))) + %<- ({x} {d} lookup-context (Dlookup x d) (Dord x d)) + %<- (scsub-ordered Dappend Dappend' Dord Dord') + %<- (weaken-aconverte' Dappend' Dord' Dconv Dconv') + %<- (aconverte-resp ctx-eq/i atom-eq/i (Deq _) eterm-eq/i Dconv' Dconv'') %. +%term varoth + %pi (aconvert-ssub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) Dappend' ([x] [d] [ex] [xt] aconverte/var Dvtrans (DconvB x d ex xt) (Dlookup x d)) Dconv (aconverte/var Dvtrans DconvB' Dlookup')) + %<- (tconvert-ssub-e Dappend Dappend' DconvB Dconv DconvB') + %<- (scsub-lookup Dappend Dappend' Dlookup Dlookup') %. +%scope tconvert-ssub-e %term vari + %pi (aconvert-ssub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (append G1 (G2 R) G') Dappend') ([x] [d] [ex] [xt] aconverte/vari (%the (vtrans (EY ex) (Y x)) (Dvtrans x ex xt)) (%the (tconverte (G x) (B x) (EB ex)) (DconvB x d ex xt)) (%the (wf (B x)) (Dwf x)) (%the (vof (Y x) (B x)) (Dvof x))) (%the (aconverte G1 R _ EM) Dconv) (aconverte/vari (Dvtrans''' R EM) DconvB' (Dwf R) (Dvof R))) + %<- (tconvert-ssub-e Dappend Dappend' DconvB Dconv (%the (tconverte G' (B R) (EB EM)) DconvB')) + %<- (vof-noassm' Dvof (%the ({x} atom-eq (Y x) Y') DeqY) (%the ({x} tp-eq (B x) B') DeqB)) + %<- ({x} atom-eq-symm (DeqY x) (%the (atom-eq Y' (Y x)) (DeqY' x))) + %<- ({x} {ex} {xt} vtrans-resp eterm-eq/i (DeqY x) (Dvtrans x ex xt) (%the (vtrans (EY ex) Y') (Dvtrans' x ex xt))) + %<- (strengthen-vtrans Dvtrans' (%the ({ex} eterm-eq (EY ex) EY') DeqEY) (%the (vtrans EY' Y') Dvtrans'')) + %<- ({ex} eterm-eq-symm (DeqEY ex) (%the (eterm-eq EY' (EY ex)) (DeqEY' ex))) + %<- ({x} {ex} vtrans-resp (DeqEY' ex) (DeqY' x) Dvtrans'' (%the (vtrans (EY ex) (Y x)) (Dvtrans''' x ex))) %. +%term app + %pi (aconvert-ssub-e Dappend Dappend' ([x] [d] [ex] [xt] aconverte/app (Dsub x) (DconvM x d ex xt) (DconvR x d ex xt)) (%the (aconverte _ R _ _) Dconv) (aconverte/app (Dsub R) DconvM' DconvR')) + %<- (aconvert-ssub-e Dappend Dappend' DconvR Dconv DconvR') + %<- (convert-ssub-e Dappend Dappend' DconvM Dconv DconvM') %. +%scope tconvert-ssub-e %term pi1 + %pi (aconvert-ssub-e Dappend Dappend' ([x] [d] [ex] [xt] aconverte/pi1 (DconvR x d ex xt)) Dconv (aconverte/pi1 DconvR')) + %<- (aconvert-ssub-e Dappend Dappend' DconvR Dconv DconvR') %. +%term pi2 + %pi (aconvert-ssub-e Dappend Dappend' ([x] [d] [ex] [xt] aconverte/pi2 (DconvR x d ex xt)) Dconv (aconverte/pi2 DconvR')) + %<- (aconvert-ssub-e Dappend Dappend' DconvR Dconv DconvR') %. +%scope tconvert-ssub-e %term at + %pi (convert-ssub-e Dappend Dappend' ([x] [d] [ex] [xt] converte/at (D x d ex xt)) Dconv (converte/at D')) + %<- (aconvert-ssub-e Dappend Dappend' D Dconv D') %. +%term lam + %pi (convert-ssub-e Dappend Dappend' ([x] [d] [ex] [xt] converte/lam (DconvM x d ex xt) (DconvA x d ex xt)) Dconv (converte/lam DconvM' DconvA')) + %<- (tconvert-ssub-e Dappend Dappend' DconvA Dconv DconvA') + %<- ({y} {e} {ey} {yt} convert-ssub-e ([x] append/cons (Dappend x)) (append/cons Dappend') ([x] [d] [ex] [xt] DconvM x d ex xt y e ey yt) Dconv (DconvM' y e ey yt)) %. +%scope tconvert-ssub-e %term pair + %pi (convert-ssub-e Dappend Dappend' ([x] [d] [ex] [xt] converte/pair (Dwf x d) (Dconv2 x d ex xt) (Dsub x) (Dconv1 x d ex xt)) (%the (aconverte _ R _ _) Dconv) (converte/pair Dwf' Dconv2' (Dsub R) Dconv1')) + %<- (convert-ssub-e Dappend Dappend' Dconv1 Dconv Dconv1') + %<- (convert-ssub-e Dappend Dappend' Dconv2 Dconv Dconv2') + %<- (aconverte-reg-il Dconv Daof) + %<- ({y} {e} stsubst-e ([x] append/cons (Dappend x)) (append/cons Dappend') ([x] [d] Dwf x d y e) Daof (Dwf' y e)) %. +%term sing + %pi (convert-ssub-e Dappend Dappend' ([x] [d] [ex] [xt] converte/sing (D x d ex xt)) Dconv (converte/sing D')) + %<- (aconvert-ssub-e Dappend Dappend' D Dconv D') %. +%scope tconvert-ssub-e %term t + %pi (tconvert-ssub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) Dappend' ([x] [d] [ex] [xt] tconverte/t (%the (ordered (G x)) (Dord x d))) (%the (aconverte _ R _ _) _) (tconverte/t Dord')) + %<- (scsub-ordered Dappend Dappend' Dord Dord') %. +%term pi + %pi (tconvert-ssub-e Dappend Dappend' ([x] [d] [ex] [xt] tconverte/pi (DconvB x d ex xt) (DconvA x d ex xt)) Dconv (tconverte/pi DconvB' DconvA')) + %<- (tconvert-ssub-e Dappend Dappend' DconvA Dconv DconvA') + %<- ({y} {e} {ey} {yt} tconvert-ssub-e ([x] append/cons (Dappend x)) (append/cons Dappend') ([x] [d] [ex] [xt] DconvB x d ex xt y e ey yt) Dconv (DconvB' y e ey yt)) %. +%scope tconvert-ssub-e %term sigma + %pi (tconvert-ssub-e Dappend Dappend' ([x] [d] [ex] [xt] tconverte/sigma (DconvB x d ex xt) (DconvA x d ex xt)) Dconv (tconverte/sigma DconvB' DconvA')) + %<- (tconvert-ssub-e Dappend Dappend' DconvA Dconv DconvA') + %<- ({y} {e} {ey} {yt} tconvert-ssub-e ([x] append/cons (Dappend x)) (append/cons Dappend') ([x] [d] [ex] [xt] DconvB x d ex xt y e ey yt) Dconv (DconvB' y e ey yt)) %. +%term sing + %pi (tconvert-ssub-e Dappend Dappend' ([x] [d] [ex] [xt] tconverte/sing (D x d ex xt)) Dconv (tconverte/sing D')) + %<- (aconvert-ssub-e Dappend Dappend' D Dconv D') %. +%worlds (sbind ovar tbind) (convert-ssub-e _ _ _ _ _) (aconvert-ssub-e _ _ _ _ _) (tconvert-ssub-e _ _ _ _ _) %. +%total (D1 D2 D3) (aconvert-ssub-e _ _ D1 _ _) (convert-ssub-e _ _ D2 _ _) (tconvert-ssub-e _ _ D3 _ _) %. +%%%%% Conversion Commutes with Substitution (Explicit Context) %%%%% +%sort convert-sub-em {T} %. +%mode convert-sub-em %in %in %in %in %in %in %in %in %out %out %. +%sort convert-aasub-em {T} %. +%mode convert-aasub-em %in %in %in %in %in %in %in %in %out %out %. +%sort convert-aosub-em {T} %. +%mode convert-aosub-em %in %in %in %in %in %in %in %in %out %out %. +%sort convert-tsub-em {T} %. +%mode convert-tsub-em %in %in %in %in %in %in %in %out %out %. +%scope convert-tsub-em %term var + %pi (convert-aosub-em _ _ (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G N Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] aconverte/var xt (%the (tconverte (G x) (A' x) (EA' ex)) (DconvertA x d ex xt)) (%the (lookup (G x) x (A' x)) (Dlookup x d))) (%the (converte G1 N A EN) Dconvert) aosub/var (%the (tsub A' N Ax) Dtsub) Dconvert'' Dreduce) + %<- (converte-normal Dconvert (%the (normal EN) Dnorm)) + %<- (normal-reduce-refl Dnorm (%the (reduce EN EN) Dreduce)) + %<- ({x} {d} lookup-binder-fun (Dappend x) (Dlookup x d) (%the (tp-eq A (A' x)) (DeqA x))) + %<- (tsub-absent A N (%the (tsub ([_] A) N A) Dtsub')) + %<- (tsub-resp DeqA term-eq/i tp-eq/i Dtsub' (%the (tsub A' N A) Dtsub'')) + %<- (tsub-fun Dtsub'' Dtsub (%the (tp-eq A Ax) DeqAx)) + %<- (csub-append Dcsub Dappend (%the (append G1 G2x Gx) Dappend')) + %<- ({x} {d} lookup-context (Dlookup x d) (%the (ordered (G x)) (Dordered x d))) + %<- (csub-ordered Dordered Dcsub (%the (ordered Gx) Dordered')) + %<- (weaken-converte' Dappend' Dordered' Dconvert (%the (converte Gx N A EN) Dconvert')) + %<- (converte-resp ctx-eq/i term-eq/i DeqAx eterm-eq/i Dconvert' (%the (converte Gx N Ax EN) Dconvert'')) %. +%term app + %pi (convert-aosub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] aconverte/app (%the (tsub ([y] C x y) (N x) (Cy x)) (DsubCy x)) (%the (converte (G x) (N x) (B x) (EN ex)) (DconvertN x d ex xt)) (%the (aconverte (G x) (R x) (pi (B x) ([y] C x y)) (EO ex)) (DconvertR x d ex xt))) (%the (converte G1 M A EM) DconvertM) (aosub/app (%the (sub P Nx Py) DsubPy) (%the (sub N M Nx) DsubNx) (%the (aosub R M (lam P)) DsubRx)) (%the (tsub Cy M Cyx) DsubCyx) DconvertPy' (reduce/app (reduce-app/beta DreducePy) DreduceBx DreduceP)) + %<- (converte-reg-il DconvertM (%the (ofe G1 M A) DofM)) + %<- ({x} {d} {ex} {xt} aconverte-reg-il (DconvertR x d ex xt) (%the (aofe (G x) (R x) (pi (B x) ([y] C x y))) (DofR x d))) + %<- ({x} {d} aofe-reg (DofR x d) (wfe/pi (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- (can-tsub-e Dappend DwfB DofM (%the (tsub B M Bx) DsubBx)) + %<- ({y} {e isvar y J} can-tsub-e ([x] append/cons (Dappend x)) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DsubCx y))) + %<- (convert-aosub-em _ Dsimp Dappend Dcsub DconvertR DconvertM DsubRx (tsub/pi DsubCx DsubBx) (converte/lam (%the ({y} + %pi (isvar y K) + %-> ({ey} %pi (vtrans ey y) %-> (converte (cons Gx y Bx) (P y) (Cx y) (EP ey)))) DconvertP) (%the (tconverte Gx Bx EBx) DconvertBx)) (%the (reduce (EO EM) (elam EBx EP)) DreduceP)) + %<- (convert-sub-em _ Dsimp Dappend Dcsub DconvertN DconvertM DsubNx DsubBx (%the (converte Gx Nx Bx ENx) DconvertNx) (%the (reduce (EN EM) ENx) DreduceBx)) + %<- ({y} {e isvar y J} tsubst-e ([x] append/cons (Dappend x)) (csub/cons DsubBx Dcsub) DofM (DsubCx y) ([x] [d] DwfC x d y e) (%the (wfe (cons Gx y Bx) (Cx y)) (DwfCx y e))) + %<- ({x} {d} {ex} {xt} converte-reg-il (DconvertN x d ex xt) (%the (ofe (G x) (N x) (B x)) (DofN x d))) + %<- (subst-e Dappend Dcsub DofM DsubNx DsubBx DofN (%the (ofe Gx Nx Bx) DofNx)) + %<- (can-tsub-e ([_] append/nil) DwfCx DofNx (%the (tsub ([y] Cx y) Nx Cxy) DsubCxy)) + %<- (aosub-headvar DsubRx (%the (headvar R) Dheadvar)) + %<- (headvar-tp-size Dappend DofR Dsimp Dheadvar ([x] simp/pi (%the ({y} simp (C x y) U2) (DsimpC x)) (%the (simp (B x) U1) (DsimpB x))) (%the (stp-leq (spi U1 U2) T) Dleq)) + %<- (tsub-preserves-simp DsubBx DsimpB (%the (simp Bx U1) DsimpBx)) + %<- (employ-stp-leq _ _ Dleq) + %<- (convert-sub-em _ DsimpBx ([_] append/nil) csub/base DconvertP DconvertNx DsubPy DsubCxy (%the (converte Gx Py Cxy EPy) DconvertPy) (%the (reduce (EP ENx) EPy) DreducePy)) + %<- (tsub-permute-e Dappend DofM DofN DwfC DsubCx DsubNx DsubCxy DsubCy DsubCyx (%the (tp-eq Cxy Cyx) Deq)) + %<- (converte-resp ctx-eq/i term-eq/i Deq eterm-eq/i DconvertPy (%the (converte Gx Py Cyx EPy) DconvertPy')) %. +%scope convert-tsub-em %term appbad + %pi (convert-aosub-em _ _ _ _ ([x] [d] [ex] [xt] aconverte/var (Dvtrans x ex xt) (%the (tconverte _ _ (EA ex)) (Dconvert x d ex xt)) (Dlookup x d)) _ (aosub/app _ _ _) _ D1 (%the (reduce _ eaca) D2)) + %<- ({x} {d} lookup-isvar (Dlookup x d) (%the (isvar (app _ _) J) (Disvar x))) + %<- ({x} isvar-app-contra (Disvar x) Dfalse) + %<- (false-implies-converte Dfalse D1) + %<- (false-implies-reduce Dfalse D2) %. +%term pi1 + %pi (convert-aosub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] aconverte/pi1 (%the (aconverte (G x) (R x) (sigma (B x) ([y] C x y)) (EN ex)) (DconvertR x d ex xt))) (%the (converte G1 M A EM) DconvertM) (aosub/pi1 (%the (aosub R M (pair O P)) DsubRx)) (%the (tsub B M Bx) DsubBx) DconvertO (reduce/pi1 reduce-pi1/beta Dreduce)) + %<- (converte-reg-il DconvertM (%the (ofe G1 M A) DofM)) + %<- ({x} {d} {ex} {xt} aconverte-reg-il (DconvertR x d ex xt) (%the (aofe (G x) (R x) (sigma (B x) ([y] C x y))) (DofR x d))) + %<- ({x} {d} aofe-reg (DofR x d) (wfe/sigma (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- ({y} {e isvar y J} can-tsub-e ([x] append/cons (Dappend x)) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DsubCx y))) + %<- (convert-aosub-em _ Dsimp Dappend Dcsub DconvertR DconvertM DsubRx (tsub/sigma DsubCx DsubBx) (converte/pair _ (%the (converte Gx P Cxy EP) DconvertP) (%the (tsub Cx O Cxy) DsubCxy) (%the (converte Gx O Bx EO) DconvertO)) (%the (reduce (EN EM) (epair EO EP)) Dreduce)) %. +%scope convert-tsub-em %term pi1bad + %pi (convert-aosub-em _ _ _ _ ([x] [d] [ex] [xt] aconverte/var (Dvtrans x ex xt) (%the (tconverte _ _ (EA ex)) (Dconvert x d ex xt)) (Dlookup x d)) _ (aosub/pi1 _) _ D1 (%the (reduce _ eaca) D2)) + %<- ({x} {d} lookup-isvar (Dlookup x d) (%the (isvar (pi1 _) J) (Disvar x))) + %<- ({x} isvar-pi1-contra (Disvar x) Dfalse) + %<- (false-implies-converte Dfalse D1) + %<- (false-implies-reduce Dfalse D2) %. +%term pi2 + %pi (convert-aosub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] aconverte/pi2 (%the (aconverte (G x) (R x) (sigma (B x) ([y] C x y)) (EN ex)) (DconvertR x d ex xt))) (%the (converte G1 M A EM) DconvertM) (aosub/pi2 (%the (aosub R M (pair O P)) DsubRx)) (%the (tsub ([x] C x (pi1 (R x))) M Cyx) DsubCyx) DconvertP' (reduce/pi2 reduce-pi2/beta Dreduce)) + %<- (converte-reg-il DconvertM (%the (ofe G1 M A) DofM)) + %<- ({x} {d} {ex} {xt} aconverte-reg-il (DconvertR x d ex xt) (%the (aofe (G x) (R x) (sigma (B x) ([y] C x y))) (DofR x d))) + %<- ({x} {d} aofe-reg (DofR x d) (wfe/sigma (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- (can-tsub-e Dappend DwfB DofM (%the (tsub ([x] B x) M Bx) DsubBx)) + %<- ({y} {e isvar y J} can-tsub-e ([x] append/cons (Dappend x)) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DsubCx y))) + %<- ({y} {e isvar y J} tsubst-e ([x] append/cons (Dappend x)) (csub/cons DsubBx Dcsub) DofM (DsubCx y) ([x] [d] DwfC x d y e) (%the (wfe (cons Gx y Bx) (Cx y)) (DwfCx y e))) + %<- (convert-aosub-em _ Dsimp Dappend Dcsub DconvertR DconvertM DsubRx (tsub/sigma DsubCx DsubBx) (converte/pair _ (%the (converte Gx P Cxy EP) DconvertP) (%the (tsub Cx O Cxy) DsubCxy) (%the (converte Gx O Bx EO) DconvertO)) (%the (reduce (EN EM) (epair EO EP)) Dreduce)) + %<- (stsub-ao-permute-e Dappend DofM ([x] [d] aofe/pi1 (DofR x d)) DwfC DsubCx (aosub/pi1 DsubRx) DsubCxy DsubCyx (%the (tp-eq Cxy Cyx) Deq)) + %<- (converte-resp ctx-eq/i term-eq/i Deq eterm-eq/i DconvertP (%the (converte Gx P Cyx EP) DconvertP')) %. +%scope convert-tsub-em %term pi2bad + %pi (convert-aosub-em _ _ _ _ ([x] [d] [ex] [xt] aconverte/var (Dvtrans x ex xt) (%the (tconverte _ _ (EA ex)) (Dconvert x d ex xt)) (Dlookup x d)) _ (aosub/pi2 _) _ D1 (%the (reduce _ eaca) D2)) + %<- ({x} {d} lookup-isvar (Dlookup x d) (%the (isvar (pi2 _) J) (Disvar x))) + %<- ({x} isvar-pi2-contra (Disvar x) Dfalse) + %<- (false-implies-converte Dfalse D1) + %<- (false-implies-reduce Dfalse D2) %. +%term const + %pi (convert-aasub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G N Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] aconverte/const (%the (tconverte (G x) (B x) (EB ex)) (Dconv x d ex xt)) (%the (wf (B x)) (Dwf x)) (%the (kof C (B x)) (Dkof x))) (%the (converte G1 N A EN) DconvertN) aasub/closed (%the (tsub B N Bx) DsubBx) (aconverte/const DconvertBx Dwf'' Dkof') reduce/const) + %<- (subst-kof Dkof DsubBx (%the (kof C Bx) Dkof')) + %<- (wf-noassm Dwf (%the ({x} tp-eq (B x) B') Deq)) + %<- (tsub-resp Deq term-eq/i tp-eq/i DsubBx (%the (tsub ([_] B') N Bx) DsubBx')) + %<- (tsub-absent-fun DsubBx' (%the (tp-eq Bx B') DeqBx)) + %<- (tp-eq-symm DeqBx DeqBx') + %<- (wf-resp (Deq aca) (Dwf aca) Dwf') + %<- (wf-resp DeqBx' Dwf' Dwf'') + %<- (convert-tsub-em _ Dsimp Dappend Dcsub Dconv DconvertN DsubBx (%the (tconverte Gx Bx EBx) DconvertBx) _) %. +%%% +%scope convert-tsub-em %term forall + %pi (convert-aasub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G N Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] aconverte/forall (%the (tconverte (G x) (B x) (EB ex)) (Dconv x d ex xt))) (%the (converte G1 N A EN) DconvertN) (aasub/forall (%the (tsub B N Bx) DsubBx)) (%the (tsub ([x] qtp (B x)) N Qx) DsubQx) Dconv' (reduce/forall Dreduce)) + %<- (convert-tsub-em _ Dsimp Dappend Dcsub Dconv DconvertN DsubBx (%the (tconverte Gx Bx EBx) DconvBx) (%the (treduce (EB EN) EBx) Dreduce)) + %<- (tsub-fun (tsub/pi ([_] tsub/t) (tsub/pi ([_] tsub/t) DsubBx)) DsubQx (%the (tp-eq (qtp Bx) Qx) Deq)) + %<- (aconverte-resp ctx-eq/i atom-eq/i Deq eterm-eq/i (aconverte/forall DconvBx) (%the (aconverte Gx (forall Bx) Qx (eforall EBx)) Dconv')) %. +%term allclo + %pi (convert-aasub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G N Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] aconverte/forall (%the (tconverte (G x) B (EB ex)) (Dconv x d ex xt))) (%the (converte G1 N A EN) DconvertN) aasub/closed (%the (tsub ([x] qtp B) N Qx) DsubQx) Dconv' (reduce/forall Dreduce)) + %<- (tsub-absent B N (%the (tsub ([_] B) N B) DsubBx)) + %<- (convert-tsub-em _ Dsimp Dappend Dcsub Dconv DconvertN DsubBx (%the (tconverte Gx B EBx) DconvBx) (%the (treduce (EB EN) EBx) Dreduce)) + %<- (tsub-fun (tsub/pi ([_] tsub/t) (tsub/pi ([_] tsub/t) DsubBx)) DsubQx (%the (tp-eq (qtp B) Qx) Deq)) + %<- (aconverte-resp ctx-eq/i atom-eq/i Deq eterm-eq/i (aconverte/forall DconvBx) (%the (aconverte Gx (forall B) Qx (eforall EBx)) Dconv')) %. +%scope convert-tsub-em %term allbad + %pi (convert-aasub-em _ _ _ _ ([x] [d] [ex] [xt] aconverte/var (Dvtrans x ex xt) (%the (tconverte _ _ (EA ex)) (Dconvert x d ex xt)) (Dlookup x d)) _ (aasub/forall _) _ D1 (%the (reduce _ eaca) D2)) + %<- ({x} {d} lookup-isvar (Dlookup x d) (%the (isvar (forall _) J) (Disvar x))) + %<- ({x} isvar-forall-contra (Disvar x) Dfalse) + %<- (false-implies-aconverte Dfalse D1) + %<- (false-implies-reduce Dfalse D2) %. +%term var + %pi (convert-aasub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G N Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] aconverte/var (%the (vtrans (EX ex) X) (Dvtrans x ex xt)) (%the (tconverte (G x) (B x) (EB ex)) (DconvertB x d ex xt)) (%the (lookup (G x) X (B x)) (Dlookup x d))) (%the (converte G1 N A EN) DconvertN) aasub/closed (%the (tsub B N Bx) DsubBx) (aconverte/var Dvtrans' DconvertBx Dlookup') Dreduce) + %<- (csub-lookup Dcsub Dlookup DsubBx (%the (lookup Gx X Bx) Dlookup')) + %<- (strengthen-vtrans Dvtrans (%the ({ex} eterm-eq (EX ex) EX') Deq) (%the (vtrans EX' X) Dvtrans')) + %<- (convert-tsub-em _ Dsimp Dappend Dcsub DconvertB DconvertN DsubBx (%the (tconverte Gx Bx EBx) DconvertBx) _) + %<- ({ex} eterm-eq-symm (Deq ex) (%the (eterm-eq EX' (EX ex)) (Deq' ex))) + %<- (vtrans-variable Dvtrans' (%the (variable EX') Dvar)) + %<- (reduce-resp (Deq' EN) eterm-eq/i (reduce/var Dvar) (%the (reduce (EX EN) EX') Dreduce)) %. +%scope convert-tsub-em %term vari + %pi (convert-aasub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G N Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] aconverte/vari (%the (vtrans (EX ex) X) (Dvtrans x ex xt)) (%the (tconverte (G x) (B x) (EB ex)) (DconvertB x d ex xt)) (%the (wf (B x)) (Dwf x)) (%the (vof X (B x)) (Dvof x))) (%the (converte G1 N A EN) DconvertN) aasub/closed (%the (tsub B N Bx) DsubBx) (aconverte/vari Dvtrans' DconvertBx Dwf'' Dvof'') Dreduce) + %<- (vof-noassm Dvof (%the ({x} tp-eq (B x) B') DeqB)) + %<- ({x} vof-resp atom-eq/i (DeqB x) (Dvof x) (%the (vof X B') (Dvof' x))) + %<- (tsub-resp DeqB term-eq/i tp-eq/i DsubBx (%the (tsub ([_] B') N Bx) DsubBx')) + %<- (tsub-absent B' N (%the (tsub ([_] B') N B') DsubBabs)) + %<- (tsub-fun DsubBx' DsubBabs (%the (tp-eq Bx B') DeqBx)) + %<- (tp-eq-symm DeqBx (%the (tp-eq B' Bx) DeqBx')) + %<- (vof-resp atom-eq/i DeqBx' (Dvof' aca) (%the (vof X Bx) Dvof'')) + %<- ({x} wf-resp (DeqB x) (Dwf x) (%the (wf B') (Dwf' x))) + %<- (wf-resp DeqBx' (Dwf' aca) (%the (wf Bx) Dwf'')) + %<- (convert-tsub-em _ Dsimp Dappend Dcsub DconvertB DconvertN DsubBx (%the (tconverte Gx Bx EBx) DconvertBx) _) + %<- (strengthen-vtrans Dvtrans (%the ({ex} eterm-eq (EX ex) EX') DeqEX) (%the (vtrans EX' X) Dvtrans')) + %<- ({ex} eterm-eq-symm (DeqEX ex) (%the (eterm-eq EX' (EX ex)) (DeqEX' ex))) + %<- (vtrans-variable Dvtrans' (%the (variable EX') Dvar)) + %<- (reduce-resp (DeqEX' EN) eterm-eq/i (reduce/var Dvar) (%the (reduce (EX EN) EX') Dreduce)) %. +%term app + %pi (convert-aasub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] aconverte/app (%the (tsub ([y] C x y) (N x) (Cy x)) (DsubCy x)) (%the (converte (G x) (N x) (B x) (EN ex)) (DconvertN x d ex xt)) (%the (aconverte (G x) (R x) (pi (B x) ([y] C x y)) (EO ex)) (DconvertR x d ex xt))) (%the (converte G1 M A EM) DconvertM) (aasub/app (%the (sub N M Nx) DsubNx) (%the (aasub R M Rx) DsubRx)) (%the (tsub Cy M Cyx) DsubCyx) (aconverte/app DsubCxy' DconvertNx DconvertRx) (reduce/app (reduce-app/atom Doa) DreduceNx DreduceRx)) + %<- ({x} {d} {ex} {xt} aconverte-reg-il (DconvertR x d ex xt) (%the (aofe (G x) (R x) (pi (B x) ([y] C x y))) (DofR x d))) + %<- ({x} {d} aofe-reg (DofR x d) (wfe/pi (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- (converte-reg-il DconvertM (%the (ofe G1 M A) DofM)) + %<- (can-tsub-e Dappend DwfB DofM (%the (tsub B M Bx) DsubBx)) + %<- ({y} {e} can-tsub-e ([x] append/cons (Dappend x)) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DsubCx y))) + %<- (convert-aasub-em _ Dsimp Dappend Dcsub DconvertR DconvertM DsubRx (tsub/pi DsubCx DsubBx) (%the (aconverte Gx Rx (pi Bx Cx) ERx) DconvertRx) (%the (reduce (EO EM) ERx) DreduceRx)) + %<- (convert-sub-em _ Dsimp Dappend Dcsub DconvertN DconvertM DsubNx DsubBx (%the (converte Gx Nx Bx ENx) DconvertNx) (%the (reduce (EN EM) ENx) DreduceNx)) + %<- ({y} {e} tsubst-e ([x] append/cons (Dappend x)) (csub/cons DsubBx Dcsub) DofM (DsubCx y) ([x] [d] DwfC x d y e) (%the (wfe (cons Gx y Bx) (Cx y)) (DwfCx y e))) + %<- ({x} {d} {ex} {xt} converte-reg-il (DconvertN x d ex xt) (%the (ofe (G x) (N x) (B x)) (DofN x d))) + %<- (subst-e Dappend Dcsub DofM DsubNx DsubBx DofN (%the (ofe Gx Nx Bx) DofNx)) + %<- (can-tsub-e ([_] append/nil) DwfCx DofNx (%the (tsub Cx Nx Cxy) DsubCxy)) + %<- (tsub-permute-e Dappend DofM DofN DwfC DsubCx DsubNx DsubCxy DsubCy DsubCyx (%the (tp-eq Cxy Cyx) Deq)) + %<- (tsub-resp ([_] tp-eq/i) term-eq/i Deq DsubCxy (%the (tsub Cx Nx Cyx) DsubCxy')) + %<- (aconverte-atomic DconvertRx (%the (atomic ERx) Datom)) + %<- (atomic-implies-oa Datom (%the (oa ERx) Doa)) %. +%scope convert-tsub-em %term appclo + %pi (convert-aasub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] aconverte/app (%the (tsub ([y] C x y) N (Cy x)) (DsubCy x)) (%the (converte (G x) N (B x) (EN ex)) (DconvertN x d ex xt)) (%the (aconverte (G x) R (pi (B x) ([y] C x y)) (EO ex)) (DconvertR x d ex xt))) (%the (converte G1 M A EM) DconvertM) aasub/closed (%the (tsub Cy M Cyx) DsubCyx) (aconverte/app DsubCxy' DconvertNx DconvertRx) (reduce/app (reduce-app/atom Doa) DreduceNx DreduceRx)) + %<- ({x} {d} {ex} {xt} aconverte-reg-il (DconvertR x d ex xt) (%the (aofe (G x) R (pi (B x) ([y] C x y))) (DofR x d))) + %<- ({x} {d} aofe-reg (DofR x d) (wfe/pi (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- (converte-reg-il DconvertM (%the (ofe G1 M A) DofM)) + %<- (can-tsub-e Dappend DwfB DofM (%the (tsub B M Bx) DsubBx)) + %<- ({y} {e} can-tsub-e ([x] append/cons (Dappend x)) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DsubCx y))) + %<- (convert-aasub-em _ Dsimp Dappend Dcsub DconvertR DconvertM aasub/closed (tsub/pi DsubCx DsubBx) (%the (aconverte Gx R (pi Bx Cx) ERx) DconvertRx) (%the (reduce (EO EM) ERx) DreduceRx)) + %<- (sub-absent N M (%the (sub ([_] N) M N) DsubNx)) + %<- (convert-sub-em _ Dsimp Dappend Dcsub DconvertN DconvertM DsubNx DsubBx (%the (converte Gx N Bx ENx) DconvertNx) (%the (reduce (EN EM) ENx) DreduceNx)) + %<- ({y} {e} tsubst-e ([x] append/cons (Dappend x)) (csub/cons DsubBx Dcsub) DofM (DsubCx y) ([x] [d] DwfC x d y e) (%the (wfe (cons Gx y Bx) (Cx y)) (DwfCx y e))) + %<- ({x} {d} {ex} {xt} converte-reg-il (DconvertN x d ex xt) (%the (ofe (G x) N (B x)) (DofN x d))) + %<- (subst-e Dappend Dcsub DofM DsubNx DsubBx DofN (%the (ofe Gx N Bx) DofNx)) + %<- (can-tsub-e ([_] append/nil) DwfCx DofNx (%the (tsub Cx N Cxy) DsubCxy)) + %<- (tsub-permute-e Dappend DofM DofN DwfC DsubCx DsubNx DsubCxy DsubCy DsubCyx (%the (tp-eq Cxy Cyx) Deq)) + %<- (tsub-resp ([_] tp-eq/i) term-eq/i Deq DsubCxy (%the (tsub Cx N Cyx) DsubCxy')) + %<- (aconverte-atomic DconvertRx (%the (atomic ERx) Datom)) + %<- (atomic-implies-oa Datom (%the (oa ERx) Doa)) %. +%term appbad + %pi (convert-aasub-em _ _ _ _ ([x] [d] [ex] [xt] aconverte/var (Dvtrans x ex xt) (%the (tconverte _ _ (EA ex)) (Dconvert x d ex xt)) (Dlookup x d)) _ (aasub/app _ _) _ D1 (%the (reduce _ eaca) D2)) + %<- ({x} {d} lookup-isvar (Dlookup x d) (%the (isvar (app _ _) J) (Disvar x))) + %<- ({x} isvar-app-contra (Disvar x) Dfalse) + %<- (false-implies-aconverte Dfalse D1) + %<- (false-implies-reduce Dfalse D2) %. +%scope convert-tsub-em %term pi1 + %pi (convert-aasub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] aconverte/pi1 (%the (aconverte (G x) (R x) (sigma (B x) ([y] C x y)) (EN ex)) (DconvertR x d ex xt))) (%the (converte G1 M A EM) DconvertM) (aasub/pi1 (%the (aasub R M Rx) DsubRx)) (%the (tsub B M Bx) DsubBx) (aconverte/pi1 DconvertRx) (reduce/pi1 (reduce-pi1/atom Doa) Dreduce)) + %<- ({x} {d} {ex} {xt} aconverte-reg-il (DconvertR x d ex xt) (%the (aofe (G x) (R x) (sigma (B x) ([y] C x y))) (DofR x d))) + %<- ({x} {d} aofe-reg (DofR x d) (wfe/sigma (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- (converte-reg-il DconvertM (%the (ofe G1 M A) DofM)) + %<- ({y} {e} can-tsub-e ([x] append/cons (Dappend x)) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DsubCx y))) + %<- (convert-aasub-em _ Dsimp Dappend Dcsub DconvertR DconvertM DsubRx (tsub/sigma DsubCx DsubBx) (%the (aconverte Gx Rx (sigma Bx Cx) ERx) DconvertRx) (%the (reduce (EN EM) ERx) Dreduce)) + %<- (aconverte-atomic DconvertRx (%the (atomic ERx) Datom)) + %<- (atomic-implies-oa Datom (%the (oa ERx) Doa)) %. +%term pi1clo + %pi (convert-aasub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] aconverte/pi1 (%the (aconverte (G x) R (sigma (B x) ([y] C x y)) (EN ex)) (DconvertR x d ex xt))) (%the (converte G1 M A EM) DconvertM) aasub/closed (%the (tsub B M Bx) DsubBx) (aconverte/pi1 DconvertRx) (reduce/pi1 (reduce-pi1/atom Doa) Dreduce)) + %<- ({x} {d} {ex} {xt} aconverte-reg-il (DconvertR x d ex xt) (%the (aofe (G x) R (sigma (B x) ([y] C x y))) (DofR x d))) + %<- ({x} {d} aofe-reg (DofR x d) (wfe/sigma (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- (converte-reg-il DconvertM (%the (ofe G1 M A) DofM)) + %<- ({y} {e} can-tsub-e ([x] append/cons (Dappend x)) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DsubCx y))) + %<- (convert-aasub-em _ Dsimp Dappend Dcsub DconvertR DconvertM aasub/closed (tsub/sigma DsubCx DsubBx) (%the (aconverte Gx R (sigma Bx Cx) ERx) DconvertRx) (%the (reduce (EN EM) ERx) Dreduce)) + %<- (aconverte-atomic DconvertRx (%the (atomic ERx) Datom)) + %<- (atomic-implies-oa Datom (%the (oa ERx) Doa)) %. +%scope convert-tsub-em %term pi1bad + %pi (convert-aasub-em _ _ _ _ ([x] [d] [ex] [xt] aconverte/var (Dvtrans x ex xt) (%the (tconverte _ _ (EA ex)) (Dconvert x d ex xt)) (Dlookup x d)) _ (aasub/pi1 _) _ D1 (%the (reduce _ eaca) D2)) + %<- ({x} {d} lookup-isvar (Dlookup x d) (%the (isvar (pi1 _) J) (Disvar x))) + %<- ({x} isvar-pi1-contra (Disvar x) Dfalse) + %<- (false-implies-aconverte Dfalse D1) + %<- (false-implies-reduce Dfalse D2) %. +%term pi2 + %pi (convert-aasub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] aconverte/pi2 (%the (aconverte (G x) (R x) (sigma (B x) ([y] C x y)) (EN ex)) (DconvertR x d ex xt))) (%the (converte G1 M A EM) DconvertM) (aasub/pi2 (%the (aasub R M Rx) DsubRx)) (%the (tsub ([x] C x (pi1 (R x))) M Cyx) DsubCyx) Dconvert' (reduce/pi2 (reduce-pi2/atom Doa) Dreduce)) + %<- ({x} {d} {ex} {xt} aconverte-reg-il (DconvertR x d ex xt) (%the (aofe (G x) (R x) (sigma (B x) ([y] C x y))) (DofR x d))) + %<- ({x} {d} aofe-reg (DofR x d) (wfe/sigma (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- (converte-reg-il DconvertM (%the (ofe G1 M A) DofM)) + %<- (can-tsub-e Dappend DwfB DofM (%the (tsub B M Bx) DsubBx)) + %<- ({y} {e} can-tsub-e ([x] append/cons (Dappend x)) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DsubCx y))) + %<- (convert-aasub-em _ Dsimp Dappend Dcsub DconvertR DconvertM DsubRx (tsub/sigma DsubCx DsubBx) (%the (aconverte Gx Rx (sigma Bx Cx) ERx) DconvertRx) (%the (reduce (EN EM) ERx) Dreduce)) + %<- (aconverte-reg-il DconvertRx (%the (aofe Gx Rx (sigma Bx Cx)) DofRx)) + %<- (stsub-aa-permute-e Dappend DofM ([x] [d] aofe/pi1 (DofR x d)) DwfC DsubCx (aasub/pi1 DsubRx) DsubCyx (%the (tp-eq (Cx (pi1 Rx)) Cyx) Deq)) + %<- (aconverte-resp ctx-eq/i atom-eq/i Deq eterm-eq/i (aconverte/pi2 DconvertRx) (%the (aconverte Gx (pi2 Rx) Cyx (epi2 ERx)) Dconvert')) + %<- (aconverte-atomic DconvertRx (%the (atomic ERx) Datom)) + %<- (atomic-implies-oa Datom (%the (oa ERx) Doa)) %. +%scope convert-tsub-em %term pi2clo + %pi (convert-aasub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] aconverte/pi2 (%the (aconverte (G x) R (sigma (B x) ([y] C x y)) (EN ex)) (DconvertR x d ex xt))) (%the (converte G1 M A EM) DconvertM) aasub/closed (%the (tsub ([x] C x (pi1 R)) M Cyx) DsubCyx) Dconvert' (reduce/pi2 (reduce-pi2/atom Doa) Dreduce)) + %<- ({x} {d} {ex} {xt} aconverte-reg-il (DconvertR x d ex xt) (%the (aofe (G x) R (sigma (B x) ([y] C x y))) (DofR x d))) + %<- ({x} {d} aofe-reg (DofR x d) (wfe/sigma (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- (converte-reg-il DconvertM (%the (ofe G1 M A) DofM)) + %<- (can-tsub-e Dappend DwfB DofM (%the (tsub B M Bx) DsubBx)) + %<- ({y} {e} can-tsub-e ([x] append/cons (Dappend x)) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DsubCx y))) + %<- (convert-aasub-em _ Dsimp Dappend Dcsub DconvertR DconvertM aasub/closed (tsub/sigma DsubCx DsubBx) (%the (aconverte Gx R (sigma Bx Cx) ERx) DconvertRx) (%the (reduce (EN EM) ERx) Dreduce)) + %<- (aconverte-reg-il DconvertRx (%the (aofe Gx R (sigma Bx Cx)) DofRx)) + %<- (stsub-aa-permute-e Dappend DofM ([x] [d] aofe/pi1 (DofR x d)) DwfC DsubCx aasub/closed DsubCyx (%the (tp-eq (Cx (pi1 R)) Cyx) Deq)) + %<- (aconverte-resp ctx-eq/i atom-eq/i Deq eterm-eq/i (aconverte/pi2 DconvertRx) (%the (aconverte Gx (pi2 R) Cyx (epi2 ERx)) Dconvert')) + %<- (aconverte-atomic DconvertRx (%the (atomic ERx) Datom)) + %<- (atomic-implies-oa Datom (%the (oa ERx) Doa)) %. +%term pi2bad + %pi (convert-aasub-em _ _ _ _ ([x] [d] [ex] [xt] aconverte/var (Dvtrans x ex xt) (%the (tconverte _ _ (EA ex)) (Dconvert x d ex xt)) (Dlookup x d)) _ (aasub/pi2 _) _ D1 (%the (reduce _ eaca) D2)) + %<- ({x} {d} lookup-isvar (Dlookup x d) (%the (isvar (pi2 _) J) (Disvar x))) + %<- ({x} isvar-pi2-contra (Disvar x) Dfalse) + %<- (false-implies-aconverte Dfalse D1) + %<- (false-implies-reduce Dfalse D2) %. +%%% +%scope convert-tsub-em %term aa + %pi (convert-sub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] converte/at (%the (aconverte (G x) (R x) t (EN ex)) (DconvertR x d ex xt))) (%the (converte G1 M A EM) DconvertM) (sub/aa (%the (aasub R M Rx) DsubRx)) tsub/t (converte/at DconvertRx) Dreduce) + %<- (convert-aasub-em _ Dsimp Dappend Dcsub DconvertR DconvertM DsubRx tsub/t (%the (aconverte Gx Rx t ENx) DconvertRx) (%the (reduce (EN EM) ENx) Dreduce)) %. +%term ao + %pi (convert-sub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] converte/at (%the (aconverte (G x) (R x) t (EN ex)) (DconvertR x d ex xt))) (%the (converte G1 M A EM) DconvertM) (sub/ao (%the (aosub R M O) DsubRx)) tsub/t DconvertO Dreduce) + %<- (convert-aosub-em _ Dsimp Dappend Dcsub DconvertR DconvertM DsubRx tsub/t (%the (converte Gx O t EO) DconvertO) (%the (reduce (EN EM) EO) Dreduce)) %. +%scope convert-tsub-em %term lam + %pi (convert-sub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] converte/lam (%the ({y} {e isvar y J} {ey} {yt vtrans ey y} converte (cons (G x) y (B x)) (N x y) (C x y) (EN ex ey)) (DconvertN x d ex xt)) (%the (tconverte (G x) (B x) (EB ex)) (DconvertB x d ex xt))) (%the (converte G1 M A EM) DconvertM) (sub/lam (%the ({y} sub ([x] N x y) M (Nx y)) DsubNx)) (tsub/pi (%the ({y} tsub ([x] C x y) M (Cx y)) DsubCx) (%the (tsub B M Bx) DsubBx)) (converte/lam DconvertNx DconvertBx) (reduce/lam DreduceNx DreduceBx)) + %<- (convert-tsub-em _ Dsimp Dappend Dcsub DconvertB DconvertM DsubBx (%the (tconverte Gx Bx EBx) DconvertBx) (%the (treduce (EB EM) EBx) DreduceBx)) + %<- ({y} {e} {ey} {yt} {ev} + %pi (vtrans-variable yt ev) + %-> (convert-sub-em _ Dsimp ([x] append/cons (Dappend x)) (csub/cons DsubBx Dcsub) ([x] [d] [ex] [xt] DconvertN x d ex xt y e ey yt) DconvertM (DsubNx y) (DsubCx y) (%the (converte (cons Gx y Bx) (Nx y) (Cx y) (ENx ey)) (DconvertNx y e ey yt)) (%the (reduce (EN EM ey) (ENx ey)) (DreduceNx ey ev)))) %. +%term pair + %pi (convert-sub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] converte/pair (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (converte (G x) (O x) (Cy x) (EO ex)) (DconvertO x d ex xt)) (%the (tsub ([y] C x y) (N x) (Cy x)) (DsubCy x)) (%the (converte (G x) (N x) (B x) (EN ex)) (DconvertN x d ex xt))) (%the (converte G1 M A EM) DconvertM) (sub/pair (%the (sub O M Ox) DsubOx) (%the (sub N M Nx) DsubNx)) (tsub/sigma (%the ({y} tsub ([x] C x y) M (Cx y)) DsubCx) (%the (tsub B M Bx) DsubBx)) (converte/pair DwfCx DconvertOx' DsubCxy DconvertNx) (reduce/pair DreduceOx DreduceNx)) + %<- (convert-sub-em _ Dsimp Dappend Dcsub DconvertN DconvertM DsubNx DsubBx (%the (converte Gx Nx Bx ENx) DconvertNx) (%the (reduce (EN EM) ENx) DreduceNx)) + %<- ({x} {d} {ex} {xt} converte-reg-il (DconvertN x d ex xt) (%the (ofe (G x) (N x) (B x)) (DofN x d))) + %<- ({x} {d} tsubst-e ([_] append/nil) csub/base (DofN x d) (DsubCy x) ([y] [e] DwfC x d y e) (%the (wfe (G x) (Cy x)) (DwfCy x d))) + %<- (converte-reg-il DconvertM (%the (ofe G1 M A) DofM)) + %<- (can-tsub-e Dappend DwfCy DofM (%the (tsub Cy M Cyx) DsubCyx)) + %<- (convert-sub-em _ Dsimp Dappend Dcsub DconvertO DconvertM DsubOx DsubCyx (%the (converte Gx Ox Cyx EOx) DconvertOx) (%the (reduce (EO EM) EOx) DreduceOx)) + %<- (subst-e Dappend Dcsub DofM DsubNx DsubBx DofN (%the (ofe Gx Nx Bx) DofNx)) + %<- ({y} {e} tsubst-e ([x] append/cons (Dappend x)) (csub/cons DsubBx Dcsub) DofM (DsubCx y) ([x] [d] DwfC x d y e) (%the (wfe (cons Gx y Bx) (Cx y)) (DwfCx y e))) + %<- (can-tsub-e ([_] append/nil) DwfCx DofNx (%the (tsub Cx Nx Cxy) DsubCxy)) + %<- (tsub-permute-e Dappend DofM DofN DwfC DsubCx DsubNx DsubCxy DsubCy DsubCyx (%the (tp-eq Cxy Cyx) Deq)) + %<- (tp-eq-symm Deq (%the (tp-eq Cyx Cxy) Deq')) + %<- (converte-resp ctx-eq/i term-eq/i Deq' eterm-eq/i DconvertOx (%the (converte Gx Ox Cxy EOx) DconvertOx')) %. +%scope convert-tsub-em %term singa + %pi (convert-sub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] converte/sing (%the (aconverte (G x) (R x) t (EN ex)) (DconvertR x d ex xt))) (%the (converte G1 M A EM) DconvertM) (sub/aa (%the (aasub R M Rx) DsubRx)) (tsub/singa (%the (aasub R M Rx') DsubRx')) DconvertRx' Dreduce) + %<- (convert-aasub-em _ Dsimp Dappend Dcsub DconvertR DconvertM DsubRx tsub/t (%the (aconverte Gx Rx t ENx) DconvertRx) (%the (reduce (EN EM) ENx) Dreduce)) + %<- (aasub-fun DsubRx DsubRx' (%the (atom-eq Rx Rx') Deq)) + %<- (tp-resp-atom sing Deq (%the (tp-eq (sing Rx) (sing Rx')) Deq')) + %<- (converte-resp ctx-eq/i term-eq/i Deq' eterm-eq/i (converte/sing DconvertRx) (%the (converte Gx (at Rx) (sing Rx') ENx) DconvertRx')) %. +%term singo + %pi (convert-sub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] converte/sing (%the (aconverte (G x) (R x) t (EN ex)) (DconvertR x d ex xt))) (%the (converte G1 M A EM) DconvertM) (sub/ao (%the (aosub R M O) DsubO)) (tsub/singo (%the (aosub R M (at Rx)) DsubRx)) DconvertO' Dreduce) + %<- (convert-aosub-em _ Dsimp Dappend Dcsub DconvertR DconvertM DsubO tsub/t (%the (converte Gx O t EO) DconvertO) (%the (reduce (EN EM) EO) Dreduce)) + %<- (aosub-fun DsubO DsubRx (%the (term-eq O (at Rx)) Deq)) + %<- (term-eq-symm Deq (%the (term-eq (at Rx) O) Deq')) + %<- (converte-resp ctx-eq/i Deq tp-eq/i eterm-eq/i DconvertO (converte/at (%the (aconverte Gx Rx t EO) DconvertRx))) + %<- (converte-resp ctx-eq/i Deq' tp-eq/i eterm-eq/i (converte/sing DconvertRx) (%the (converte Gx O (sing Rx) EO) DconvertO')) %. +%scope convert-tsub-em %term aaao + %pi (convert-sub-em _ _ _ _ ([x] [d isvar x I] [ex] [xt] converte/sing _) _ (sub/aa (%the (aasub R M Rx) DsubRx)) (tsub/singo (%the (aosub R M (at Rx')) DsubRx')) D1 D2) + %<- (aasub-aosub-contra DsubRx DsubRx' Dfalse) + %<- (false-implies-converte Dfalse (%the (converte _ _ _ eaca) D1)) + %<- (false-implies-reduce Dfalse D2) %. +%term aoaa + %pi (convert-sub-em _ _ _ _ ([x] [d isvar x I] [ex] [xt] converte/sing _) _ (sub/ao (%the (aosub R M O) DsubRx)) (tsub/singa (%the (aasub R M Rx) DsubRx')) D1 D2) + %<- (aasub-aosub-contra DsubRx' DsubRx Dfalse) + %<- (false-implies-converte Dfalse (%the (converte _ _ _ eaca) D1)) + %<- (false-implies-reduce Dfalse D2) %. +%%% +%scope convert-tsub-em %term t + %pi (convert-tsub-em _ _ _ (%the (csub G M Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] tconverte/t (%the (ordered (G x)) (Dordered x d))) _ tsub/t (tconverte/t Dordered') treduce/t) + %<- (csub-ordered Dordered Dcsub Dordered') %. +%term pi + %pi (convert-tsub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] tconverte/pi (%the ({y} {e isvar y J} {ey} {yt vtrans ey y} tconverte (cons (G x) y (B x)) (C x y) (EC ex ey)) (DconvertC x d ex xt)) (%the (tconverte (G x) (B x) (EB ex)) (DconvertB x d ex xt))) (%the (converte G1 M A EM) DconvertM) (tsub/pi (%the ({y} tsub ([x] C x y) M (Cx y)) DsubCx) (%the (tsub B M Bx) DsubBx)) (tconverte/pi DconvertCx DconvertBx) (treduce/pi DreduceCx DreduceBx)) + %<- (convert-tsub-em _ Dsimp Dappend Dcsub DconvertB DconvertM DsubBx (%the (tconverte Gx Bx EBx) DconvertBx) (%the (treduce (EB EM) EBx) DreduceBx)) + %<- ({y} {e} {ey} {yt} {ev} + %pi (vtrans-variable yt ev) + %-> (convert-tsub-em _ Dsimp ([x] append/cons (Dappend x)) (csub/cons DsubBx Dcsub) ([x] [d] [ex] [xt] DconvertC x d ex xt y e ey yt) DconvertM (DsubCx y) (%the (tconverte (cons Gx y Bx) (Cx y) (ECx ey)) (DconvertCx y e ey yt)) (%the (treduce (EC EM ey) (ECx ey)) (DreduceCx ey ev)))) %. +%scope convert-tsub-em %term sigma + %pi (convert-tsub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] tconverte/sigma (%the ({y} {e isvar y J} {ey} {yt vtrans ey y} tconverte (cons (G x) y (B x)) (C x y) (EC ex ey)) (DconvertC x d ex xt)) (%the (tconverte (G x) (B x) (EB ex)) (DconvertB x d ex xt))) (%the (converte G1 M A EM) DconvertM) (tsub/sigma (%the ({y} tsub ([x] C x y) M (Cx y)) DsubCx) (%the (tsub B M Bx) DsubBx)) (tconverte/sigma DconvertCx DconvertBx) (treduce/sigma DreduceCx DreduceBx)) + %<- (convert-tsub-em _ Dsimp Dappend Dcsub DconvertB DconvertM DsubBx (%the (tconverte Gx Bx EBx) DconvertBx) (%the (treduce (EB EM) EBx) DreduceBx)) + %<- ({y} {e} {ey} {yt} {ev} + %pi (vtrans-variable yt ev) + %-> (convert-tsub-em _ Dsimp ([x] append/cons (Dappend x)) (csub/cons DsubBx Dcsub) ([x] [d] [ex] [xt] DconvertC x d ex xt y e ey yt) DconvertM (DsubCx y) (%the (tconverte (cons Gx y Bx) (Cx y) (ECx ey)) (DconvertCx y e ey yt)) (%the (treduce (EC EM ey) (ECx ey)) (DreduceCx ey ev)))) %. +%term singa + %pi (convert-tsub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] tconverte/sing (%the (aconverte (G x) (R x) t (EN ex)) (DconvertR x d ex xt))) (%the (converte G1 M A EM) DconvertM) (tsub/singa (%the (aasub R M Rx) DsubRx)) (tconverte/sing DconvertRx) (treduce/sing Dreduce)) + %<- (convert-aasub-em _ Dsimp Dappend Dcsub DconvertR DconvertM DsubRx tsub/t (%the (aconverte Gx Rx t ENx) DconvertRx) (%the (reduce (EN EM) ENx) Dreduce)) %. +%scope convert-tsub-em %term singo + %pi (convert-tsub-em _ (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) ([x] [d isvar x I] [ex] [xt] tconverte/sing (%the (aconverte (G x) (R x) t (EN ex)) (DconvertR x d ex xt))) (%the (converte G1 M A EM) DconvertM) (tsub/singo (%the (aosub R M (at Rx)) DsubRx)) (tconverte/sing DconvertRx) (treduce/sing Dreduce)) + %<- (convert-aosub-em _ Dsimp Dappend Dcsub DconvertR DconvertM DsubRx tsub/t (converte/at (%the (aconverte Gx Rx t ENx) DconvertRx)) (%the (reduce (EN EM) ENx) Dreduce)) %. +%worlds (sbind ovar tvbind evar) (convert-sub-em _ _ _ _ _ _ _ _ _ _) (convert-aasub-em _ _ _ _ _ _ _ _ _ _) (convert-aosub-em _ _ _ _ _ _ _ _ _ _) (convert-tsub-em _ _ _ _ _ _ _ _ _) %. +%total {(T1 T2 T3 T4) (D1 D2 D3 D4)} (convert-aosub-em T1 _ _ _ D1 _ _ _ _ _) (convert-aasub-em T2 _ _ _ D2 _ _ _ _ _) (convert-sub-em T3 _ _ _ D3 _ _ _ _ _) (convert-tsub-em T4 _ _ _ D4 _ _ _ _) %. +%%%%% Conversion Commutes with Simple Substitution (Implicit Context) %%%%% +%sort convert-ssub {x} %. +%mode convert-ssub %in %in %out %. +%term _ + %pi (convert-ssub (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (convert (M x) (B x) (EM ex)))) DconvM) (%the (aconvert R A EN) DconvR) DconvMR') + %<- (convert1-to-converte 0 DconvM (%the ({x} + %pi (isvar x 0) + %-> ({ex} %pi (vtrans ex x) %-> (converte (cons nil x A) (M x) (B x) (EM ex)))) DconveM)) + %<- (aconvert-to-aconverte DconvR (%the (aconverte nil R A EN) DconveR)) + %<- (convert-ssub-e ([_] append/nil) append/nil DconveM DconveR (%the (converte nil (M R) (B R) (EM EN)) DconveMR)) + %<- ({x} {d} {ex} {xt} convert-reg-il (DconvM x d ex xt) (%the (of (M x) (B x)) (DofM x d))) + %<- (aconvert-reg-il DconvR (%the (aof R A) DofR)) + %<- (ssubst DofM DofR (%the (of (M R) (B R)) DofMR)) + %<- (can-convert DofMR (%the (convert (M R) (B R) EM') DconvMR)) + %<- (convert-to-converte DconvMR (%the (converte nil (M R) (B R) EM') DconveMR')) + %<- (converte-fun DconveMR' DconveMR (%the (eterm-eq EM' (EM EN)) Deq)) + %<- (convert-resp term-eq/i tp-eq/i Deq DconvMR (%the (convert (M R) (B R) (EM EN)) DconvMR')) %. +%sort _ %. +%worlds (sbind) (convert-ssub _ _ _) %. +%total {} (convert-ssub _ _ _) %. +%sort convert-stsub {x} %. +%mode convert-stsub %in %in %out %. +%term _ + %pi (convert-stsub (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))) DconvB) (%the (aconvert R A EN) DconvR) DconvBR') + %<- (tconvert1-to-tconverte 0 DconvB (%the ({x} + %pi (isvar x 0) + %-> ({ex} %pi (vtrans ex x) %-> (tconverte (cons nil x A) (B x) (EB ex)))) DconveB)) + %<- (aconvert-to-aconverte DconvR (%the (aconverte nil R A EN) DconveR)) + %<- (tconvert-ssub-e ([_] append/nil) append/nil DconveB DconveR (%the (tconverte nil (B R) (EB EN)) DconveBR)) + %<- ({x} {d} {ex} {xt} tconvert-reg-il (DconvB x d ex xt) (%the (wf (B x)) (DwfB x d))) + %<- (aconvert-reg-il DconvR (%the (aof R A) DofR)) + %<- (stsubst DwfB DofR (%the (wf (B R)) DwfBR)) + %<- (can-tconvert DwfBR (%the (tconvert (B R) EB') DconvBR)) + %<- (tconvert-to-tconverte DconvBR (%the (tconverte nil (B R) EB') DconveBR')) + %<- (tconverte-fun DconveBR' DconveBR (%the (etp-eq EB' (EB EN)) Deq)) + %<- (tconvert-resp tp-eq/i Deq DconvBR (%the (tconvert (B R) (EB EN)) DconvBR')) %. +%sort _ %. +%worlds (sbind) (convert-stsub _ _ _) %. +%total {} (convert-stsub _ _ _) %. +%sort convert-ssub1 {x} %. +%mode convert-ssub1 %in %in %in %out %. +%term _ + %pi (convert-ssub1 (%the ({x} + %pi (vof x A) + %-> ({ex} + %pi (vtrans ex x) + %-> ({y} + %pi (vof y (B x)) + %-> ({ey} %pi (vtrans ey y) %-> (convert (M x y) (C x y) (EM ex ey)))))) DconvM) (%the (aconvert R A EN) DconvR) (%the ({x} %pi (vof x A) %-> (wf (B x))) DwfB) DconvMR') + %<- (convert2-to-converte (%the (lt 0 (s 0)) lt/z) DconvM (%the ({x} + %pi (isvar x 0) + %-> ({ex} + %pi (vtrans ex x) + %-> ({y} + %pi (isvar y (s 0)) + %-> ({ey} + %pi (vtrans ey y) + %-> (converte (cons (cons nil x A) y (B x)) (M x y) (C x y) (EM ex ey)))))) DconveM)) + %<- (aconvert-to-aconverte DconvR (%the (aconverte nil R A EN) DconveR)) + %<- ({y} {e} {ey} {yt} convert-ssub-e ([_] append/cons append/nil) (append/cons append/nil) ([x] [d] [ex] [xt] DconveM x d ex xt y e ey yt) DconveR (%the (converte (cons nil y (B R)) (M R y) (C R y) (EM EN ey)) (DconveMR y e ey yt))) + %<- ({x} {d} {ex} {xt} {y} {e} {ey} {yt} convert-reg-il (DconvM x d ex xt y e ey yt) (%the (of (M x y) (C x y)) (DofM x d y e))) + %<- (aconvert-reg-il DconvR (%the (aof R A) DofR)) + %<- (ssubst1 DofM DofR (%the ({y} %pi (vof y (B R)) %-> (of (M R y) (C R y))) DofMR)) + %<- (stsubst DwfB DofR (%the (wf (B R)) DwfBR)) + %<- (can-tconvert DwfBR (%the (tconvert (B R) EB') DconvBR)) + %<- ({y} {e} {ey} {yt vtrans ey y} {ee evof ey EB'} {ev variable ey} + %pi (vtrans-variable yt ev) + %-> (vsound e yt DconvBR ee) + %-> (can-convert (DofMR y e) (%the (convert (M R y) (C R y) (EM' ey)) (DconvMR y e ey yt)))) + %<- (convert1-to-converte (s 0) DconvMR (%the ({y} + %pi (isvar y (s 0)) + %-> ({ey} %pi (vtrans ey y) %-> (converte (cons nil y (B R)) (M R y) (C R y) (EM' ey)))) DconveMR')) + %<- ({y} {e} {ey} {yt} converte-fun (DconveMR' y e ey yt) (DconveMR y e ey yt) (%the (eterm-eq (EM' ey) (EM EN ey)) (Deq ey))) + %<- ({y} {e} {ey} {yt} convert-resp term-eq/i tp-eq/i (Deq ey) (DconvMR y e ey yt) (%the (convert (M R y) (C R y) (EM EN ey)) (DconvMR' y e ey yt))) %. +%sort _ %. +%worlds (sbind) (convert-ssub1 _ _ _ _) %. +%total {} (convert-ssub1 _ _ _ _) %. +%%%%% Conversion Commutes with Substitution (Implicit Context) %%%%% +%sort convert-tsub-reduce {x} %. +%mode convert-tsub-reduce %in %in %in %out %out %. +%term _ + %pi (convert-tsub-reduce (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))) DconvertB) (%the (convert N A EN) DconvertN) (%the (tsub B N Bx) DsubBx) DconvertBx' Dreduce) + %<- ({x} {d} {ex} {xt} tconvert-reg-il (DconvertB x d ex xt) (%the (wf (B x)) (DwfB x d))) + %<- (convert-reg-il DconvertN (%the (of N A) DofN)) + %<- (tsubst DsubBx DwfB DofN (%the (wf Bx) DwfBx)) + %<- (can-tconvert DwfBx (%the (tconvert Bx EBx) DconvertBx)) + %<- (tconvert1-to-tconverte 0 DconvertB (%the ({x} + %pi (isvar x 0) + %-> ({ex} %pi (vtrans ex x) %-> (tconverte (cons nil x A) (B x) (EB ex)))) DconverteB)) + %<- (convert-to-converte DconvertN (%the (converte nil N A EN) DconverteN)) + %<- (tconvert-to-tconverte DconvertBx (%the (tconverte nil Bx EBx) DconverteBx)) + %<- (can-simp A (%the (simp A T) Dsimp)) + %<- (convert-tsub-em _ Dsimp ([_] append/nil) csub/base DconverteB DconverteN DsubBx (%the (tconverte nil Bx EC) DconverteBx') (%the (treduce (EB EN) EC) Dreduce)) + %<- (tconverte-fun DconverteBx DconverteBx' (%the (etp-eq EBx EC) Deq)) + %<- (tconvert-resp tp-eq/i Deq DconvertBx (%the (tconvert Bx EC) DconvertBx')) %. +%worlds (sbind) (convert-tsub-reduce _ _ _ _ _) %. +%total {} (convert-tsub-reduce _ _ _ _ _) %. +%sort convert-sub-reduce {x} %. +%mode convert-sub-reduce %in %in %in %in %out %out %. +%term _ + %pi (convert-sub-reduce (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (convert (M x) (B x) (EM ex)))) DconvertM) (%the (convert N A EN) DconvertN) (%the (sub M N Mx) DsubMx) (%the (tsub B N Bx) DsubBx) DconvertMx' Dreduce) + %<- ({x} {d} {ex} {xt} convert-reg-il (DconvertM x d ex xt) (%the (of (M x) (B x)) (DofM x d))) + %<- (convert-reg-il DconvertN (%the (of N A) DofN)) + %<- (subst DsubMx DsubBx DofM DofN (%the (of Mx Bx) DofMx)) + %<- (can-convert DofMx (%the (convert Mx Bx EMx) DconvertMx)) + %<- (convert1-to-converte 0 DconvertM (%the ({x} + %pi (isvar x 0) + %-> ({ex} %pi (vtrans ex x) %-> (converte (cons nil x A) (M x) (B x) (EM ex)))) DconverteM)) + %<- (convert-to-converte DconvertN (%the (converte nil N A EN) DconverteN)) + %<- (convert-to-converte DconvertMx (%the (converte nil Mx Bx EMx) DconverteMx)) + %<- (can-simp A (%the (simp A T) Dsimp)) + %<- (convert-sub-em _ Dsimp ([_] append/nil) csub/base DconverteM DconverteN DsubMx DsubBx (%the (converte nil Mx Bx EO) DconverteMx') (%the (reduce (EM EN) EO) Dreduce)) + %<- (converte-fun DconverteMx DconverteMx' (%the (eterm-eq EMx EO) Deq)) + %<- (convert-resp term-eq/i tp-eq/i Deq DconvertMx (%the (convert Mx Bx EO) DconvertMx')) %. +%worlds (sbind) (convert-sub-reduce _ _ _ _ _ _) %. +%total {} (convert-sub-reduce _ _ _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/convert.lf b/new-tests/stelf-output/tslf/sing/convert.lf new file mode 100644 index 0000000..0b1c70e --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/convert.lf @@ -0,0 +1,89 @@ +%sort tconvert %. +%sort convert %. +%sort aconvert %. +%term tconvert/t tconvert t et %. +%term tconvert/pi + %pi (tconvert (pi A B) (epi EA EB)) + %<- (tconvert A EA) + %<- ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))) %. +%term tconvert/sigma + %pi (tconvert (sigma A B) (esigma EA EB)) + %<- (tconvert A EA) + %<- ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))) %. +%term tconvert/sing %pi (tconvert (sing R) (esing EM)) %<- (aconvert R t EM) %. +%term convert/at %pi (convert (at R) t EM) %<- (aconvert R t EM) %. +%term convert/lam + %pi (convert (lam M) (pi A B) (elam EA EM)) + %<- (tconvert A EA) + %<- ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (convert (M x) (B x) (EM ex)))) %. +%term convert/pair + %pi (convert (pair M N) (sigma A B) (epair EM EN)) + %<- (convert M A EM) + %<- (tsub B M Bx) + %<- (convert N Bx EN) + %<- ({x} %pi (vof x A) %-> (wf (B x))) %. +%term convert/sing %pi (convert (at R) (sing R) EM) %<- (aconvert R t EM) %. +%term aconvert/const %pi (aconvert (const C) A (econst C)) %<- (kof C A) %<- (wf A) %<- (tconvert A _) %. +%term aconvert/forall %pi (aconvert (forall A) (qtp A) (eforall EA)) %<- (tconvert A EA) %. +%term aconvert/var %pi (aconvert X A EX) %<- (vof X A) %<- (wf A) %<- (tconvert A _) %<- (vtrans EX X) %. +%term aconvert/app + %pi (aconvert (app R N) Bx (eapp EM EN)) + %<- (aconvert R (pi A B) EM) + %<- (convert N A EN) + %<- (tsub B N Bx) %. +%term aconvert/pi1 %pi (aconvert (pi1 R) A (epi1 EM)) %<- (aconvert R (sigma A B) EM) %. +%term aconvert/pi2 %pi (aconvert (pi2 R) (B (pi1 R)) (epi2 EM)) %<- (aconvert R (sigma A B) EM) %. +%%%%% Explicit Contexts %%%%% +%sort tconverte %. +%sort converte %. +%sort aconverte %. +%term tconverte/t %pi (tconverte G t et) %<- (ordered G) %. +%term tconverte/pi + %pi (tconverte G (pi A B) (epi EA EB)) + %<- (tconverte G A EA) + %<- ({x} + %pi (isvar x I) + %-> ({ex} %pi (vtrans ex x) %-> (tconverte (cons G x A) (B x) (EB ex)))) %. +%term tconverte/sigma + %pi (tconverte G (sigma A B) (esigma EA EB)) + %<- (tconverte G A EA) + %<- ({x} + %pi (isvar x I) + %-> ({ex} %pi (vtrans ex x) %-> (tconverte (cons G x A) (B x) (EB ex)))) %. +%term tconverte/sing %pi (tconverte G (sing R) (esing EM)) %<- (aconverte G R t EM) %. +%term converte/at %pi (converte G (at R) t EM) %<- (aconverte G R t EM) %. +%term converte/lam + %pi (converte G (lam M) (pi A B) (elam EA EM)) + %<- (tconverte G A EA) + %<- ({x} + %pi (isvar x I) + %-> ({ex} %pi (vtrans ex x) %-> (converte (cons G x A) (M x) (B x) (EM ex)))) %. +%term converte/pair + %pi (converte G (pair M N) (sigma A B) (epair EM EN)) + %<- (converte G M A EM) + %<- (tsub B M Bx) + %<- (converte G N Bx EN) + %<- ({x} %pi (isvar x I) %-> (wfe (cons G x A) (B x))) %. +%term converte/sing %pi (converte G (at R) (sing R) EM) %<- (aconverte G R t EM) %. +%term aconverte/const + %pi (aconverte G (const C) A (econst C)) + %<- (kof C A) + %<- (wf A) + %<- (tconverte G A _) %. +%term aconverte/forall %pi (aconverte G (forall A) (qtp A) (eforall EA)) %<- (tconverte G A EA) %. +%term aconverte/var %pi (aconverte G X A EX) %<- (lookup G X A) %<- (tconverte G A _) %<- (vtrans EX X) %. +%term aconverte/vari + %pi (aconverte G X A EX) + %<- (vof X A) + %<- (wf A) + %<- (tconverte G A _) + %<- (vtrans EX X) %. +%term aconverte/app + %pi (aconverte G (app R N) Bx (eapp EM EN)) + %<- (aconverte G R (pi A B) EM) + %<- (converte G N A EN) + %<- (tsub B N Bx) %. +%term aconverte/pi1 %pi (aconverte G (pi1 R) A (epi1 EM)) %<- (aconverte G R (sigma A B) EM) %. +%term aconverte/pi2 + %pi (aconverte G (pi2 R) (B (pi1 R)) (epi2 EM)) + %<- (aconverte G R (sigma A B) EM) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/correct-thm.lf b/new-tests/stelf-output/tslf/sing/correct-thm.lf new file mode 100644 index 0000000..7272638 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/correct-thm.lf @@ -0,0 +1,752 @@ +%%%%% Reduction %%%%% +%sort reduce-fun %. +%mode reduce-fun %in %in %out %. +%sort treduce-fun %. +%mode treduce-fun %in %in %out %. +%sort reduce-app-fun %. +%mode reduce-app-fun %in %in %out %. +%sort reduce-pi1-fun %. +%mode reduce-pi1-fun %in %in %out %. +%sort reduce-pi2-fun %. +%mode reduce-pi2-fun %in %in %out %. +%term _ reduce-fun reduce/const reduce/const eterm-eq/i %. +%term _ + %pi (reduce-fun (reduce/forall D) (reduce/forall D') Deq') + %<- (treduce-fun D D' Deq) + %<- (eterm-resp-etp eforall Deq Deq') %. +%term _ reduce-fun (reduce/var _) (reduce/var _) eterm-eq/i %. +%term _ + %pi (reduce-fun (reduce/lam D2 D1) (reduce/lam D2' D1') Deq) + %<- (treduce-fun D1 D1' Deq1) + %<- ({x} {dv} reduce-fun (D2 x dv) (D2' x dv) (Deq2 x)) + %<- (elam-resp Deq1 Deq2 Deq) %. +%term _ + %pi (reduce-fun (reduce/app D3 D2 D1) (reduce/app D3' D2' D1') Deq) + %<- (reduce-fun D1 D1' Deq1) + %<- (reduce-fun D2 D2' Deq2) + %<- (reduce-app-resp Deq1 Deq2 eterm-eq/i D3 D3'') + %<- (reduce-app-fun D3'' D3' Deq) %. +%term _ + %pi (reduce-fun (reduce/pair D2 D1) (reduce/pair D2' D1') Deq) + %<- (reduce-fun D1 D1' Deq1) + %<- (reduce-fun D2 D2' Deq2) + %<- (epair-resp Deq1 Deq2 Deq) %. +%term _ + %pi (reduce-fun (reduce/pi1 D2 D1) (reduce/pi1 D2' D1') Deq) + %<- (reduce-fun D1 D1' Deq1) + %<- (reduce-pi1-resp Deq1 eterm-eq/i D2 D2'') + %<- (reduce-pi1-fun D2'' D2' Deq) %. +%term _ + %pi (reduce-fun (reduce/pi2 D2 D1) (reduce/pi2 D2' D1') Deq) + %<- (reduce-fun D1 D1' Deq1) + %<- (reduce-pi2-resp Deq1 eterm-eq/i D2 D2'') + %<- (reduce-pi2-fun D2'' D2' Deq) %. +%term _ treduce-fun treduce/t treduce/t etp-eq/i %. +%term _ + %pi (treduce-fun (treduce/pi D2 D1) (treduce/pi D2' D1') Deq) + %<- (treduce-fun D1 D1' Deq1) + %<- ({x} {dv} treduce-fun (D2 x dv) (D2' x dv) (Deq2 x)) + %<- (epi-resp Deq1 Deq2 Deq) %. +%term _ + %pi (treduce-fun (treduce/sigma D2 D1) (treduce/sigma D2' D1') Deq) + %<- (treduce-fun D1 D1' Deq1) + %<- ({x} {dv} treduce-fun (D2 x dv) (D2' x dv) (Deq2 x)) + %<- (esigma-resp Deq1 Deq2 Deq) %. +%term _ + %pi (treduce-fun (treduce/sing D1) (treduce/sing D1') Deq) + %<- (reduce-fun D1 D1' Deq1) + %<- (etp-resp-eterm esing Deq1 Deq) %. +%term _ reduce-app-fun (reduce-app/atom _) (reduce-app/atom _) eterm-eq/i %. +%term _ + %pi (reduce-app-fun (reduce-app/beta D1) (reduce-app/beta D2) Deq) + %<- (reduce-fun D1 D2 Deq) %. +%term _ reduce-pi1-fun (reduce-pi1/atom _) (reduce-pi1/atom _) eterm-eq/i %. +%term _ reduce-pi1-fun reduce-pi1/beta reduce-pi1/beta eterm-eq/i %. +%term _ reduce-pi2-fun (reduce-pi2/atom _) (reduce-pi2/atom _) eterm-eq/i %. +%term _ reduce-pi2-fun reduce-pi2/beta reduce-pi2/beta eterm-eq/i %. +%worlds (evvar evbind sbind) (reduce-fun _ _ _) (treduce-fun _ _ _) (reduce-app-fun _ _ _) (reduce-pi1-fun _ _ _) (reduce-pi2-fun _ _ _) %. +%total (D1 D2 D3 D4 D5) (reduce-fun _ D1 _) (treduce-fun _ D2 _) (reduce-app-fun _ D3 _) (reduce-pi1-fun _ D4 _) (reduce-pi2-fun _ D5 _) %. +%%%%% Selfification %%%%% +%sort can-selfify %. +%mode can-selfify %in %in %in %in %out %out %out %. +%scope can-selfify %term t can-selfify expand/t self/t tconvert/t (%the (aconvert R t ER) Dconv) (convert/at Dconv) (tconvert/sing Dconv) selfify/t %. +%term sing can-selfify expand/sing self/sing (tconvert/sing (%the (aconvert R t ER) DconvR)) _ (convert/sing DconvR) (tconvert/sing DconvR) selfify/sing %. +%scope can-selfify %term pi + %pi (can-selfify (expand/pi (%the ({x} expand (app R (X x)) (B x) (N x)) Dexpand2) (%the ({x} expand x A (X x)) Dexpand1)) (self/pi (%the ({x} self (N x) (B x) (Bs x)) DselfB)) (tconvert/pi (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))) DconvB) (%the (tconvert A EA) DconvA)) (%the (aconvert R (pi A B) ER) DconvR) (convert/lam DconvN DconvA) (tconvert/pi DconvBs DconvA) (selfify/pi Dselfify2 Dselfify1)) + %<- (aconvert-reg-il DconvR (%the (aof R (pi A B)) DofR)) + %<- (aof-reg DofR (wf/pi (%the ({x} %pi (vof x A) %-> (wf (B x))) DwfB) (%the (wf A) DwfA))) + %<- ({x} {d} expand-reg (aof/var DwfA d) (Dexpand1 x) (%the (of (X x) A) (DofX x d))) + %<- ({x} {d} can-self (DofX x d) (%the (self (X x) A (As x)) (DselfA x))) + %<- ({x} {d} {ex} {xt vtrans ex x} {ed} {dv} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA ed) + %-> (can-selfify (Dexpand1 x) (DselfA x) DconvA (aconvert/var xt DconvA DwfA d) (%the (convert (X x) A (EX ex)) (DconvX x d ex xt)) (%the (tconvert (As x) (EAs ex)) (DconvAs x d ex xt)) (%the (selfify ex EA (EX ex) (EAs ex)) (Dselfify1 ex)))) + %<- (tsub-expand-var DwfA DwfB Dexpand1 (%the ({x} tsub B (X x) (B x)) DsubB)) + %<- ({x} {d} {ex} {xt} {ed} {dv} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA ed) + %-> (can-selfify (Dexpand2 x) (DselfB x) (DconvB x d ex xt) (aconvert/app (DsubB x) (DconvX x d ex xt) DconvR) (%the (convert (N x) (B x) (EN ex)) (DconvN x d ex xt)) (%the (tconvert (Bs x) (EBs ex)) (DconvBs x d ex xt)) (%the (selfify (eapp ER (EX ex)) (EB ex) (EN ex) (EBs ex)) (Dselfify2 ex)))) %. +%term sigma + %pi (can-selfify (expand/sigma (%the (expand (pi2 R) (B (pi1 R)) N) Dexpand2) (%the (expand (pi1 R) A M) Dexpand1)) (self/sigma (%the (self N Bx Bs) DselfB) (%the (tsub B M Bx) DsubBx) (%the (self M A As) DselfA)) (tconvert/sigma (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))) DconvB) (%the (tconvert A EA) DconvA)) (%the (aconvert R (sigma A B) ER) DconvR) (convert/pair DwfB DconvN' DsubBx DconvM) (tconvert/sigma ([_] [_] [_] [_] DconvBs) DconvAs) (selfify/sigma Dselfify2 Dselfify1)) + %<- (aconvert-reg-il DconvR (%the (aof R (sigma A B)) DofR)) + %<- (aof-reg DofR (wf/sigma (%the ({x} %pi (vof x A) %-> (wf (B x))) DwfB) (%the (wf A) DwfA))) + %<- (can-selfify Dexpand1 DselfA DconvA (aconvert/pi1 DconvR) (%the (convert M A EM) DconvM) (%the (tconvert As EAs) DconvAs) (%the (selfify (epi1 ER) EA EM EAs) Dselfify1)) + %<- (tsub-expand (aof/pi1 DofR) DwfB Dexpand1 DsubBx (%the (tp-eq Bx (B (pi1 R))) Deq)) + %<- (self-resp term-eq/i Deq tp-eq/i DselfB (%the (self N (B (pi1 R)) Bs) DselfB')) + %<- (convert-stsub DconvB (aconvert/pi1 DconvR) (%the (tconvert (B (pi1 R)) (EB (epi1 ER))) DconvBx)) + %<- (can-selfify Dexpand2 DselfB' DconvBx (aconvert/pi2 DconvR) (%the (convert N (B (pi1 R)) EN) DconvN) (%the (tconvert Bs EBs) DconvBs) (%the (selfify (epi2 ER) (EB (epi1 ER)) EN EBs) Dselfify2)) + %<- (tp-eq-symm Deq (%the (tp-eq (B (pi1 R)) Bx) Deq')) + %<- (convert-resp term-eq/i Deq' eterm-eq/i DconvN (%the (convert N Bx EN) DconvN')) %. +%worlds (sbind) (can-selfify _ _ _ _ _ _ _) %. +%total D (can-selfify D _ _ _ _ _ _) %. +%sort selfify-fun %. +%mode selfify-fun %in %in %out %out %. +%term _ selfify-fun selfify/t selfify/t eterm-eq/i etp-eq/i %. +%term _ + %pi (selfify-fun (selfify/pi Dself2 Dself1) (selfify/pi Dself2' Dself1') DeqO' DeqB') + %<- ({x} selfify-fun (Dself1 x) (Dself1' x) (DeqN x) _) + %<- ({x} eterm-resp-eterm ([m] eapp M m) (DeqN x) (DeqN' x)) + %<- ({x} selfify-resp (DeqN' x) etp-eq/i eterm-eq/i etp-eq/i (Dself2 x) (Dself2'' x)) + %<- ({x} selfify-fun (Dself2'' x) (Dself2' x) (DeqO x) (DeqB x)) + %<- (elam-resp etp-eq/i DeqO DeqO') + %<- (epi-resp etp-eq/i DeqB DeqB') %. +%term _ + %pi (selfify-fun (selfify/sigma Dself2 Dself1) (selfify/sigma Dself2' Dself1') DeqN DeqB) + %<- (selfify-fun Dself1 Dself1' DeqN1 DeqB1) + %<- (selfify-fun Dself2 Dself2' DeqN2 DeqB2) + %<- (epair-resp DeqN1 DeqN2 DeqN) + %<- (esigma-resp DeqB1 ([_] DeqB2) DeqB) %. +%term _ selfify-fun selfify/sing selfify/sing eterm-eq/i etp-eq/i %. +%worlds (evar) (selfify-fun _ _ _ _) %. +%total D (selfify-fun _ D _ _) %. +%%%%% Coercion %%%%% +%sort can-coerce %. +%mode can-coerce %in %in %in %out %out %. +%scope can-coerce %term t can-coerce tconvert/t tconvert/t subtype/t ([x] [d] [ex] [xt] convert/at (aconvert/var xt tconvert/t wf/t d)) coerce/t %. +%term pi + %pi (can-coerce (tconvert/pi (%the ({x} %pi (vof x A1) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (A2 x) (EA2 ex)))) DconvA2) (%the (tconvert A1 EA1) DconvA1)) (tconvert/pi (%the ({x} %pi (vof x B1) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B2 x) (EB2 ex)))) DconvB2) (%the (tconvert B1 EB1) DconvB1)) (subtype/pi (%the ({x} subtype (A2' x) (B2 x) ([y] M2 x y)) Dsubtype2) (%the ({x} tsub A2 (M1 x) (A2' x)) Dsub) (%the (subtype B1 A1 M1) Dsubtype1)) ([z] [f] [ez] [zt] convert/lam ([x] [d] [ex] [xt] Dconv z f ez zt x d ex xt) DconvB1) (coerce/pi Dcoerce2 Dreduce Dcoerce1)) + %<- (can-coerce DconvB1 DconvA1 Dsubtype1 (%the ({x} %pi (vof x B1) %-> ({ex} %pi (vtrans ex x) %-> (convert (M1 x) A1 (EM1 ex)))) Dconv1) (%the (coerce EB1 EA1 EM1) Dcoerce1)) + %<- ({x} {d vof x B1} {ex} {xt} {ed evof ex EB1} {dv} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvB1 ed) + %-> (convert-tsub-reduce DconvA2 (Dconv1 x d ex xt) (Dsub x) (%the (tconvert (A2' x) (EA2' ex)) (DconvA2' x d ex xt)) (%the (treduce (EA2 (EM1 ex)) (EA2' ex)) (Dreduce ex dv)))) + %<- ({x} {d vof x B1} {ex} {xt} {ed evof ex EB1} {dv} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvB1 ed) + %-> (can-coerce (DconvA2' x d ex xt) (DconvB2 x d ex xt) (Dsubtype2 x) (%the ({y} + %pi (vof y (A2' x)) + %-> ({ey} %pi (vtrans ey y) %-> (convert (M2 x y) (B2 x) (EM2 ex ey)))) (Dconv2 x d ex xt)) (%the (coerce (EA2' ex) (EB2 ex) ([ey] EM2 ex ey)) (Dcoerce2 ex dv)))) + %<- (tconvert-reg-il DconvA1 (%the (wf A1) DwfA1)) + %<- ({x} {d} {ex} {xt} tconvert-reg-il (DconvA2 x d ex xt) (%the (wf (A2 x)) (DwfA2 x d))) + %<- ({z} {f vof z (pi A1 A2)} {ez} {zt} {ef evof ez (epi EA1 EA2)} {fv} + %pi (vtrans-variable zt fv) + %-> (vsound f zt (tconvert/pi DconvA2 DconvA1) ef) + %-> ({x} {d vof x B1} {ex} {xt} {ed evof ex EB1} {dv} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvB1 ed) + %-> (convert-ssub (Dconv2 x d ex xt) (%the (aconvert (app z (M1 x)) (A2' x) (eapp ez (EM1 ex))) (aconvert/app (Dsub x) (Dconv1 x d ex xt) (aconvert/var zt (tconvert/pi DconvA2 DconvA1) (wf/pi DwfA2 DwfA1) f))) (%the (convert (M2 x (app z (M1 x))) (B2 x) (EM2 ex (eapp ez (EM1 ex)))) (Dconv z f ez zt x d ex xt))))) %. +%scope can-coerce %term sigma + %pi (can-coerce (tconvert/sigma (%the ({x} %pi (vof x A1) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (A2 x) (EA2 ex)))) DconvA2) (%the (tconvert A1 EA1) DconvA1)) (tconvert/sigma (%the ({x} %pi (vof x B1) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B2 x) (EB2 ex)))) DconvB2) (%the (tconvert B1 EB1) DconvB1)) (subtype/sigma (%the ({x} subtype (A2 x) (B2' x) ([y] M2 x y)) Dsubtype2) (%the ({x} tsub B2 (M1 x) (B2' x)) Dsub) (%the (subtype A1 B1 M1) Dsubtype1)) ([z] [f] [ez] [zt] convert/pair DwfB2 (Dconv2'' z f ez zt) (Dsub (pi1 z)) (Dconv1' z f ez zt)) (coerce/sigma Dcoerce2 Dreduce Dcoerce1)) + %<- (can-coerce DconvA1 DconvB1 Dsubtype1 (%the ({x} %pi (vof x A1) %-> ({ex} %pi (vtrans ex x) %-> (convert (M1 x) B1 (EM1 ex)))) Dconv1) (%the (coerce EA1 EB1 EM1) Dcoerce1)) + %<- (tconvert-reg-il DconvA1 (%the (wf A1) DwfA1)) + %<- ({x} {d} {ex} {xt} tconvert-reg-il (DconvA2 x d ex xt) (%the (wf (A2 x)) (DwfA2 x d))) + %<- ({x} {d} {ex} {xt} tconvert-reg-il (DconvB2 x d ex xt) (%the (wf (B2 x)) (DwfB2 x d))) + %<- ({z} {f vof z (sigma A1 A2)} {ez} {zt} {ef evof ez (esigma EA1 EA2)} {fv} + %pi (vtrans-variable zt fv) + %-> (vsound f zt (tconvert/sigma DconvA2 DconvA1) ef) + %-> (convert-ssub Dconv1 (%the (aconvert (pi1 z) A1 (epi1 ez)) (aconvert/pi1 (aconvert/var zt (tconvert/sigma DconvA2 DconvA1) (wf/sigma DwfA2 DwfA1) f))) (%the (convert (M1 (pi1 z)) B1 (EM1 (epi1 ez))) (Dconv1' z f ez zt)))) + %<- ({x} {d vof x A1} {ex} {xt} {ed evof ex EA1} {dv} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA1 ed) + %-> (convert-tsub-reduce DconvB2 (Dconv1 x d ex xt) (Dsub x) (%the (tconvert (B2' x) (EB2' ex)) (DconvB2' x d ex xt)) (%the (treduce (EB2 (EM1 ex)) (EB2' ex)) (Dreduce ex dv)))) + %<- ({x} {d vof x A1} {ex} {xt} {ed evof ex EA1} {dv} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA1 ed) + %-> (can-coerce (DconvA2 x d ex xt) (DconvB2' x d ex xt) (Dsubtype2 x) (%the ({y} + %pi (vof y (A2 x)) + %-> ({ey} %pi (vtrans ey y) %-> (convert (M2 x y) (B2' x) (EM2 ex ey)))) (Dconv2 x d ex xt)) (%the (coerce (EA2 ex) (EB2' ex) ([ey] EM2 ex ey)) (Dcoerce2 ex dv)))) + %<- ({z} {f vof z (sigma A1 A2)} {ez} {zt} {ef evof ez (esigma EA1 EA2)} {fv} + %pi (vtrans-variable zt fv) + %-> (vsound f zt (tconvert/sigma DconvA2 DconvA1) ef) + %-> (convert-ssub1 Dconv2 (%the (aconvert (pi1 z) A1 (epi1 ez)) (aconvert/pi1 (aconvert/var zt (tconvert/sigma DconvA2 DconvA1) (wf/sigma DwfA2 DwfA1) f))) DwfA2 (%the ({y} + %pi (vof y (A2 (pi1 z))) + %-> ({ey} %pi (vtrans ey y) %-> (convert (M2 (pi1 z) y) (B2' (pi1 z)) (EM2 (epi1 ez) ey)))) (Dconv2' z f ez zt)))) + %<- ({z} {f vof z (sigma A1 A2)} {ez} {zt} {ef evof ez (esigma EA1 EA2)} {fv} + %pi (vtrans-variable zt fv) + %-> (vsound f zt (tconvert/sigma DconvA2 DconvA1) ef) + %-> (convert-ssub (Dconv2' z f ez zt) (%the (aconvert (pi2 z) (A2 (pi1 z)) (epi2 ez)) (aconvert/pi2 (aconvert/var zt (tconvert/sigma DconvA2 DconvA1) (wf/sigma DwfA2 DwfA1) f))) (%the (convert (M2 (pi1 z) (pi2 z)) (B2' (pi1 z)) (EM2 (epi1 ez) (epi2 ez))) (Dconv2'' z f ez zt)))) %. +%term sing_t can-coerce (tconvert/sing (%the (aconvert R t ER) DconvR)) tconvert/t subtype/sing_t ([_] [_] [_] [_] convert/at DconvR) coerce/sing_t %. +%scope can-coerce %term sing + %pi (can-coerce (tconvert/sing (%the (aconvert R t ER) DconvR)) (tconvert/sing (%the (aconvert R t ER') DconvR')) subtype/sing ([_] [_] [_] [_] convert/sing DconvR) Dcoerce) + %<- (aconvert-fun DconvR DconvR' _ (%the (eterm-eq ER ER') Deq)) + %<- (etp-resp-eterm esing Deq (%the (etp-eq (esing ER) (esing ER')) Deq')) + %<- (coerce-resp etp-eq/i Deq' ([_] eterm-eq/i) coerce/sing Dcoerce) %. +%worlds (sbind) (can-coerce _ _ _ _ _) %. +%total D (can-coerce _ _ D _ _) %. +%sort can-coerce' %. +%mode can-coerce' %in %in %in %in %out %. +%term _ + %pi (can-coerce' DconvA DconvB Dsubtype DconvM Dcoerce') + %<- (can-coerce DconvA DconvB Dsubtype DconvM' Dcoerce) + %<- ({x} {d} {ex} {xt} convert-fun (DconvM' x d ex xt) (DconvM x d ex xt) (Deq ex)) + %<- (coerce-resp etp-eq/i etp-eq/i Deq Dcoerce Dcoerce') %. +%worlds (sbind) (can-coerce' _ _ _ _ _) %. +%total {} (can-coerce' _ _ _ _ _) %. +%sort coerce-fun %. +%mode coerce-fun %in %in %out %. +%term _ coerce-fun coerce/t coerce/t ([_] eterm-eq/i) %. +%term _ + %pi (coerce-fun (%the (coerce (epi A1 A2) _ _) (coerce/pi D3 D2 D1)) (coerce/pi D3' D2' D1') Deq') + %<- (coerce-fun D1 D1' Deq1) + %<- ({x} etp-resp-eterm A2 (Deq1 x) (Deq1' x)) + %<- ({x} {dv} treduce-resp (Deq1' x) etp-eq/i (D2 x dv) (D2'' x dv)) + %<- ({x} {dv} treduce-fun (D2'' x dv) (D2' x dv) (Deq2 x)) + %<- ({x} {dv} coerce-resp (Deq2 x) etp-eq/i ([_] eterm-eq/i) (D3 x dv) (D3'' x dv)) + %<- ({x} {dv} coerce-fun (D3'' x dv) (D3' x dv) ([y] Deq3 x y)) + %<- ({z} {x} eterm-resp-eterm ([m] eapp z m) (Deq1 x) (Deq1'' z x)) + %<- ({z} {x} eterm-resp-eterm-fun ([y] Deq3 x y) (Deq1'' z x) (Deq z x)) + %<- ({z} elam-resp etp-eq/i (Deq z) (Deq' z)) %. +%term _ + %pi (coerce-fun (%the (coerce _ (esigma B1 B2) _) (coerce/sigma D3 D2 D1)) (coerce/sigma D3' D2' D1') Deq) + %<- (coerce-fun D1 D1' Deq1) + %<- ({x} etp-resp-eterm B2 (Deq1 x) (Deq1' x)) + %<- ({x} {dv} treduce-resp (Deq1' x) etp-eq/i (D2 x dv) (D2'' x dv)) + %<- ({x} {dv} treduce-fun (D2'' x dv) (D2' x dv) (Deq2 x)) + %<- ({x} {dv} coerce-resp etp-eq/i (Deq2 x) ([_] eterm-eq/i) (D3 x dv) (D3'' x dv)) + %<- ({x} {dv} coerce-fun (D3'' x dv) (D3' x dv) ([y] Deq3 x y)) + %<- ({z} epair-resp (Deq1 (epi1 z)) (Deq3 (epi1 z) (epi2 z)) (Deq z)) %. +%term _ coerce-fun coerce/sing_t coerce/sing_t ([_] eterm-eq/i) %. +%term _ coerce-fun coerce/sing coerce/sing ([_] eterm-eq/i) %. +%worlds (evvar evbind sbind) (coerce-fun _ _ _) %. +%total D (coerce-fun _ D _) %. +%sort coerce-convert %. +%mode coerce-convert %in %in %in %out %. +%scope coerce-convert %term t coerce-convert tconvert/t tconvert/t coerce/t subtype/t %. +%term pi + %pi (coerce-convert (tconvert/pi (%the ({x} %pi (vof x A1) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (A2 x) (EA2 ex)))) DconvA2) (%the (tconvert A1 EA1) DconvA1)) (tconvert/pi (%the ({x} %pi (vof x B1) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B2 x) (EB2 ex)))) DconvB2) (%the (tconvert B1 EB1) DconvB1)) (%the (coerce (epi EA1 EA2) (epi EB1 EB2) _) (coerce/pi (%the ({ex} %pi (variable ex) %-> (coerce (EA2' ex) (EB2 ex) ([ey] EM2 ex ey))) Dcoerce2) (%the ({ex} %pi (variable ex) %-> (treduce (EA2 (EM1 ex)) (EA2' ex))) Dreduce) (%the (coerce EB1 EA1 EM1) Dcoerce1))) (subtype/pi Dsubtype2 DsubA2' Dsubtype1)) + %<- (coerce-convert DconvB1 DconvA1 Dcoerce1 (%the (subtype B1 A1 M1) Dsubtype1)) + %<- (can-coerce DconvB1 DconvA1 Dsubtype1 (%the ({x} %pi (vof x B1) %-> ({ex} %pi (vtrans ex x) %-> (convert (M1 x) A1 (EM1' ex)))) DconvM1') (%the (coerce EB1 EA1 EM1') Dcoerce1')) + %<- (coerce-fun Dcoerce1' Dcoerce1 (%the ({ex} eterm-eq (EM1' ex) (EM1 ex)) DeqEM1)) + %<- ({x} {d vof x B1} {ex} {xt} convert-resp term-eq/i tp-eq/i (DeqEM1 ex) (DconvM1' x d ex xt) (%the (convert (M1 x) A1 (EM1 ex)) (DconvM1 x d ex xt))) + %<- (tconvert-reg-il DconvA1 (%the (wf A1) DwfA1)) + %<- (tconvert-reg-il DconvB1 (%the (wf B1) DwfB1)) + %<- (subtype-reg Dsubtype1 DwfB1 DwfA1 (%the ({x} %pi (vof x B1) %-> (of (M1 x) A1)) DofM1)) + %<- ({x} {d vof x A1} {ex} {xt} tconvert-reg-il (DconvA2 x d ex xt) (%the (wf (A2 x)) (DwfA2 x d))) + %<- ({x} {d vof x B1} can-tsub DwfA2 (DofM1 x d) (%the (tsub A2 (M1 x) (A2' x)) (DsubA2' x))) + %<- ({x} {d vof x B1} {ex} {xt} {ed} {dv} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvB1 ed) + %-> (convert-tsub-reduce DconvA2 (DconvM1 x d ex xt) (DsubA2' x) (%the (tconvert (A2' x) (EA2'' ex)) (DconvA2'' x d ex xt)) (%the (treduce (EA2 (EM1 ex)) (EA2'' ex)) (Dreduce' ex dv)))) + %<- ({ex} {dv} treduce-fun (Dreduce' ex dv) (Dreduce ex dv) (%the (etp-eq (EA2'' ex) (EA2' ex)) (DeqEA2 ex))) + %<- ({x} {d vof x B1} {ex} {xt} tconvert-resp tp-eq/i (DeqEA2 ex) (DconvA2'' x d ex xt) (%the (tconvert (A2' x) (EA2' ex)) (DconvA2' x d ex xt))) + %<- ({x} {d vof x B1} {ex} {xt} {ed} {dv} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvB1 ed) + %-> (coerce-convert (DconvA2' x d ex xt) (DconvB2 x d ex xt) (Dcoerce2 ex dv) (%the (subtype (A2' x) (B2 x) (M2 x)) (Dsubtype2 x)))) %. +%scope coerce-convert %term sigma + %pi (coerce-convert (tconvert/sigma (%the ({x} %pi (vof x A1) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (A2 x) (EA2 ex)))) DconvA2) (%the (tconvert A1 EA1) DconvA1)) (tconvert/sigma (%the ({x} %pi (vof x B1) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B2 x) (EB2 ex)))) DconvB2) (%the (tconvert B1 EB1) DconvB1)) (%the (coerce (esigma EA1 EA2) (esigma EB1 EB2) _) (coerce/sigma (%the ({ex} %pi (variable ex) %-> (coerce (EA2 ex) (EB2' ex) ([ey] EM2 ex ey))) Dcoerce2) (%the ({ex} %pi (variable ex) %-> (treduce (EB2 (EM1 ex)) (EB2' ex))) Dreduce) (%the (coerce EA1 EB1 EM1) Dcoerce1))) (subtype/sigma Dsubtype2 DsubB2' Dsubtype1)) + %<- (coerce-convert DconvA1 DconvB1 Dcoerce1 (%the (subtype A1 B1 M1) Dsubtype1)) + %<- (can-coerce DconvA1 DconvB1 Dsubtype1 (%the ({x} %pi (vof x A1) %-> ({ex} %pi (vtrans ex x) %-> (convert (M1 x) B1 (EM1' ex)))) DconvM1') (%the (coerce EA1 EB1 EM1') Dcoerce1')) + %<- (coerce-fun Dcoerce1' Dcoerce1 (%the ({ex} eterm-eq (EM1' ex) (EM1 ex)) DeqEM1)) + %<- ({x} {d vof x A1} {ex} {xt} convert-resp term-eq/i tp-eq/i (DeqEM1 ex) (DconvM1' x d ex xt) (%the (convert (M1 x) B1 (EM1 ex)) (DconvM1 x d ex xt))) + %<- (tconvert-reg-il DconvA1 (%the (wf A1) DwfA1)) + %<- (tconvert-reg-il DconvB1 (%the (wf B1) DwfB1)) + %<- (subtype-reg Dsubtype1 DwfA1 DwfB1 (%the ({x} %pi (vof x A1) %-> (of (M1 x) B1)) DofM1)) + %<- ({x} {d vof x B1} {ex} {xt} tconvert-reg-il (DconvB2 x d ex xt) (%the (wf (B2 x)) (DwfB2 x d))) + %<- ({x} {d vof x A1} can-tsub DwfB2 (DofM1 x d) (%the (tsub B2 (M1 x) (B2' x)) (DsubB2' x))) + %<- ({x} {d vof x A1} {ex} {xt} {ed} {dv} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA1 ed) + %-> (convert-tsub-reduce DconvB2 (DconvM1 x d ex xt) (DsubB2' x) (%the (tconvert (B2' x) (EB2'' ex)) (DconvB2'' x d ex xt)) (%the (treduce (EB2 (EM1 ex)) (EB2'' ex)) (Dreduce' ex dv)))) + %<- ({ex} {dv} treduce-fun (Dreduce' ex dv) (Dreduce ex dv) (%the (etp-eq (EB2'' ex) (EB2' ex)) (DeqEB2 ex))) + %<- ({x} {d vof x A1} {ex} {xt} tconvert-resp tp-eq/i (DeqEB2 ex) (DconvB2'' x d ex xt) (%the (tconvert (B2' x) (EB2' ex)) (DconvB2' x d ex xt))) + %<- ({x} {d vof x A1} {ex} {xt} {ed} {dv} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA1 ed) + %-> (coerce-convert (DconvA2 x d ex xt) (DconvB2' x d ex xt) (Dcoerce2 ex dv) (%the (subtype (A2 x) (B2' x) (M2 x)) (Dsubtype2 x)))) %. +%term sing_t coerce-convert (tconvert/sing _) tconvert/t coerce/sing_t subtype/sing_t %. +%scope coerce-convert %term sing + %pi (coerce-convert (tconvert/sing Dconv) (tconvert/sing Dconv') coerce/sing D) + %<- (aconvert-fun2 Dconv Dconv' Deq _) + %<- (tp-resp-atom sing Deq Deq') + %<- (subtype-resp tp-eq/i Deq' ([_] term-eq/i) subtype/sing D) %. +%worlds (sbind) (coerce-convert _ _ _ _) %. +%total D (coerce-convert _ _ D _) %. +%sort coerce-convert' %. +%mode coerce-convert' %in %in %in %out %out %. +%term _ + %pi (coerce-convert' DconvA DconvB Dcoerce Dsubtype DconvM') + %<- (coerce-convert DconvA DconvB Dcoerce Dsubtype) + %<- (can-coerce DconvA DconvB Dsubtype DconvM Dcoerce') + %<- (coerce-fun Dcoerce' Dcoerce Deq) + %<- ({x} {d} {ex} {xt} convert-resp term-eq/i tp-eq/i (Deq ex) (DconvM x d ex xt) (DconvM' x d ex xt)) %. +%worlds (sbind) (coerce-convert' _ _ _ _ _) %. +%total {} (coerce-convert' _ _ _ _ _) %. +%%%%% Canonization %%%%% +%sort canonize-comp %. +%mode canonize-comp %in %out %out %. +%sort tcanonize-comp %. +%mode tcanonize-comp %in %out %out %. +%scope tcanonize-comp %term const + %pi (canonize-comp (trans/const (%the (self M A As) Dself) (%the (expand (const C) A M) Dexpand) (%the (wf A) DwfA) (%the (kof C A) Dkof)) DconvAs (canonize/const Dselfify Dekof)) + %<- (can-tconvert DwfA (%the (tconvert A EA) DconvA)) + %<- (convert-kof Dkof DconvA (%the (ekof C EA) Dekof)) + %<- (can-selfify Dexpand Dself DconvA (aconvert/const DconvA DwfA Dkof) (%the (convert M A EM) DconvM) (%the (tconvert As EAs) DconvAs) (%the (selfify (econst C) EA EM EAs) Dselfify)) %. +%term forall + %pi (canonize-comp (trans/forall (%the (self M (qtp A) B) Dself) (%the (expand (forall A) (qtp A) M) Dexpand) (%the (ttrans EA A) DtransA)) DconvB (canonize/forall Dselfify Dcanon)) + %<- (tcanonize-comp DtransA (%the (tconvert A EA') DconvA) (%the (tcanonize EA EA') Dcanon)) + %<- (can-selfify Dexpand Dself (tconvert/pi ([_] [_] [_] [_] tconvert/t) (tconvert/pi ([_] [_] [_] [_] tconvert/t) DconvA)) (aconvert/forall DconvA) (%the (convert M (qtp A) EM) DconvM) (%the (tconvert B EB) DconvB) (%the (selfify (eforall EA') (qetp EA') EM EB) Dselfify)) %. +%scope tcanonize-comp %term var + %pi (canonize-comp (trans/var (%the (self M A As) Dself) (%the (expand X A M) Dexpand) (%the (wf A) Dwf) (%the (vof X A) Dvof) (%the (vtrans EX X) Dtrans)) DconvAs (canonize/var Dselfify Devof)) + %<- (vsound' Dvof Dtrans (%the (tconvert A EA) DconvA) (%the (evof EX EA) Devof)) + %<- (can-selfify Dexpand Dself DconvA (aconvert/var Dtrans DconvA Dwf Dvof) (%the (convert M A EM) DconvM) (%the (tconvert As EAs) DconvAs) (%the (selfify EX EA EM EAs) Dselfify)) %. +%term app + %pi (canonize-comp (trans/app (%the (tsub B N Bx) DsubBx) (%the (subtype C A ([_] N)) Dsubtype) (%the (trans EN C) DtransN) (%the (trans EM (pi A B)) DtransM)) DconvBx (canonize/app Dreduce Dcoerce DcanonN DcanonM)) + %<- (canonize-comp DtransM (tconvert/pi (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))) DconvB) (%the (tconvert A EA) DconvA)) (%the (canonize EM (epi EA EB)) DcanonM)) + %<- (canonize-comp DtransN (%the (tconvert C EC) DconvC) (%the (canonize EN EC) DcanonN)) + %<- (trans-reg DtransM (wf/pi (%the ({x} %pi (vof x A) %-> (wf (B x))) DwfB) (%the (wf A) DwfA))) + %<- (trans-reg DtransN (%the (wf C) DwfC)) + %<- (subtype-reg' Dsubtype DwfC DwfA (%the (of N A) DofN)) + %<- (can-convert DofN (%the (convert N A EN') DconvN)) + %<- (can-coerce' DconvC DconvA Dsubtype ([_] [_] [_] [_] DconvN) (%the (coerce EC EA ([_] EN')) Dcoerce)) + %<- (convert-tsub-reduce DconvB DconvN DsubBx (%the (tconvert Bx EBx) DconvBx) (%the (treduce (EB EN') EBx) Dreduce)) %. +%scope tcanonize-comp %term pi1 + %pi (canonize-comp (trans/pi1 (%the (trans EM (sigma A ([_] B))) Dtrans)) DconvA (canonize/pi1 Dcanon')) + %<- (canonize-comp Dtrans (tconvert/sigma (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert B (EB' ex)))) DconvB') (%the (tconvert A EA) DconvA)) (%the (canonize EM (esigma EA EB')) Dcanon)) + %<- ({x} {d} {ex} {xt} tconvert-reg-il (DconvB' x d ex xt) (%the (wf B) (DwfB x d))) + %<- (strengthen-wf DwfB (%the (wf B) DwfB')) + %<- (can-tconvert DwfB' (%the (tconvert B EB) DconvB)) + %<- ({x} {d} {ex} {xt} tconvert-fun (DconvB' x d ex xt) DconvB (%the (etp-eq (EB' ex) EB) (DeqB ex))) + %<- (esigma-resp etp-eq/i DeqB (%the (etp-eq (esigma EA EB') (esigma EA ([_] EB))) Deq)) + %<- (canonize-resp eterm-eq/i Deq Dcanon (%the (canonize EM (esigma EA ([_] EB))) Dcanon')) %. +%term pi2 + %pi (canonize-comp (trans/pi2 (%the (trans EM (sigma A ([_] B))) Dtrans)) DconvB (canonize/pi2 Dcanon')) + %<- (canonize-comp Dtrans (tconvert/sigma (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert B (EB' ex)))) DconvB') (%the (tconvert A EA) DconvA)) (%the (canonize EM (esigma EA EB')) Dcanon)) + %<- ({x} {d} {ex} {xt} tconvert-reg-il (DconvB' x d ex xt) (%the (wf B) (DwfB x d))) + %<- (strengthen-wf DwfB (%the (wf B) DwfB')) + %<- (can-tconvert DwfB' (%the (tconvert B EB) DconvB)) + %<- ({x} {d} {ex} {xt} tconvert-fun (DconvB' x d ex xt) DconvB (%the (etp-eq (EB' ex) EB) (DeqB ex))) + %<- (esigma-resp etp-eq/i DeqB (%the (etp-eq (esigma EA EB') (esigma EA ([_] EB))) Deq)) + %<- (canonize-resp eterm-eq/i Deq Dcanon (%the (canonize EM (esigma EA ([_] EB))) Dcanon')) %. +%scope tcanonize-comp %term lam + %pi (canonize-comp (trans/lam (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (trans (EM ex) (B x)))) DtransM) (%the (ttrans EA A) DtransA)) (tconvert/pi DconvB DconvA) (canonize/lam DcanonM DcanonA)) + %<- (tcanonize-comp DtransA (%the (tconvert A EA') DconvA) (%the (tcanonize EA EA') DcanonA)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA'} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA ed) + %-> (canonize-comp (DtransM x d ex xt) (%the (tconvert (B x) (EB ex)) (DconvB x d ex xt)) (%the (canonize (EM ex) (EB ex)) (DcanonM ex ed dv)))) %. +%term pair + %pi (canonize-comp (trans/pair (%the (trans EN B) DtransN) (%the (trans EM A) DtransM)) (tconvert/sigma ([_] [_] [_] [_] DconvB) DconvA) (canonize/pair DcanonN DcanonM)) + %<- (canonize-comp DtransM (%the (tconvert A EA) DconvA) (%the (canonize EM EA) DcanonM)) + %<- (canonize-comp DtransN (%the (tconvert B EB) DconvB) (%the (canonize EN EB) DcanonN)) %. +%scope tcanonize-comp %term t tcanonize-comp ttrans/t tconvert/t tcanonize/t %. +%term pi + %pi (tcanonize-comp (ttrans/pi (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (ttrans (EB ex) (B x)))) DtransB) (%the (ttrans EA A) DtransA)) (tconvert/pi DconvB DconvA) (tcanonize/pi DcanonB DcanonA)) + %<- (tcanonize-comp DtransA (%the (tconvert A EA') DconvA) (%the (tcanonize EA EA') DcanonA)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA'} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA ed) + %-> (tcanonize-comp (DtransB x d ex xt) (%the (tconvert (B x) (EB' ex)) (DconvB x d ex xt)) (%the (tcanonize (EB ex) (EB' ex)) (DcanonB ex ed dv)))) %. +%scope tcanonize-comp %term sigma + %pi (tcanonize-comp (ttrans/sigma (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (ttrans (EB ex) (B x)))) DtransB) (%the (ttrans EA A) DtransA)) (tconvert/sigma DconvB DconvA) (tcanonize/sigma DcanonB DcanonA)) + %<- (tcanonize-comp DtransA (%the (tconvert A EA') DconvA) (%the (tcanonize EA EA') DcanonA)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA'} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA ed) + %-> (tcanonize-comp (DtransB x d ex xt) (%the (tconvert (B x) (EB' ex)) (DconvB x d ex xt)) (%the (tcanonize (EB ex) (EB' ex)) (DcanonB ex ed dv)))) %. +%term sing + %pi (tcanonize-comp (ttrans/sing (%the (trans EM (sing R)) Dtrans)) (tconvert/sing Dconv) (tcanonize/sing Dcanon)) + %<- (canonize-comp Dtrans (tconvert/sing (%the (aconvert R t ER) Dconv)) (%the (canonize EM (esing ER)) Dcanon)) %. +%worlds (sbind) (canonize-comp _ _ _) (tcanonize-comp _ _ _) %. +%total (D1 D2) (canonize-comp D1 _ _) (tcanonize-comp D2 _ _) %. +%sort canonize-comp' %. +%mode canonize-comp' %in %in %out %. +%term _ + %pi (canonize-comp' Dtrans Dconv Dcanon') + %<- (canonize-comp Dtrans Dconv' Dcanon) + %<- (tconvert-fun Dconv' Dconv Deq) + %<- (canonize-resp eterm-eq/i Deq Dcanon Dcanon') %. +%worlds (sbind) (canonize-comp' _ _ _) %. +%total {} (canonize-comp' _ _ _) %. +%sort tcanonize-comp' %. +%mode tcanonize-comp' %in %in %out %. +%term _ + %pi (tcanonize-comp' Dtrans Dconv Dcanon') + %<- (tcanonize-comp Dtrans Dconv' Dcanon) + %<- (tconvert-fun Dconv' Dconv Deq) + %<- (tcanonize-resp etp-eq/i Deq Dcanon Dcanon') %. +%worlds (sbind) (tcanonize-comp' _ _ _) %. +%total {} (tcanonize-comp' _ _ _) %. +%sort ekof-fun %. +%mode ekof-fun %in %in %out %. +%term _ + %pi (ekof-fun (ekof/i Detopen Dckof) (ekof/i Detopen' Dckof) Deq) + %<- (etopen-fun Detopen Detopen' Deq) %. +%worlds (evbind sbind) (ekof-fun _ _ _) %. +%total {} (ekof-fun _ _ _) %. +%sort canonize-fun %. +%mode canonize-fun %in %in %out %. +%sort tcanonize-fun %. +%mode tcanonize-fun %in %in %out %. +%term _ + %pi (canonize-fun (canonize/const D Dkof) (canonize/const D' Dkof') Deq') + %<- (ekof-fun Dkof Dkof' Deq) + %<- (selfify-resp eterm-eq/i Deq eterm-eq/i etp-eq/i D D'') + %<- (selfify-fun D'' D' _ Deq') %. +%term _ + %pi (canonize-fun (canonize/forall D2 D1) (canonize/forall D2' D1') Deq) + %<- (tcanonize-fun D1 D1' Deq1) + %<- (eterm-resp-etp eforall Deq1 Deq1') + %<- (etp-resp-etp qetp Deq1 Deq1'') + %<- (selfify-resp Deq1' Deq1'' eterm-eq/i etp-eq/i D2 D2'') + %<- (selfify-fun D2'' D2' _ Deq) %. +%term _ + %pi (canonize-fun (canonize/var D Dvof) (canonize/var D' Dvof) Deq) + %<- (selfify-fun D D' _ Deq) %. +%term _ + %pi (canonize-fun (canonize/app D4 D3 D2 D1) (canonize/app D4' D3' D2' D1') Deq) + %<- (canonize-fun D1 D1' Deq1) + %<- (etp-eq-cdr-epi Deq1 Deq1a Deq1b) + %<- (canonize-fun D2 D2' Deq2) + %<- (coerce-resp Deq2 Deq1a ([_] eterm-eq/i) D3 D3'') + %<- (coerce-fun D3'' D3' Deq3) + %<- (etp-resp-eterm-fun Deq1b (Deq3 eaca) Deq3') + %<- (treduce-resp Deq3' etp-eq/i D4 D4'') + %<- (treduce-fun D4'' D4' Deq) %. +%term _ + %pi (canonize-fun (canonize/pi1 D) (canonize/pi1 D') Deq1) + %<- (canonize-fun D D' Deq) + %<- (etp-eq-cdr-esigma Deq Deq1 Deq2) %. +%term _ + %pi (canonize-fun (canonize/pi2 D) (canonize/pi2 D') (Deq2 eaca)) + %<- (canonize-fun D D' Deq) + %<- (etp-eq-cdr-esigma Deq Deq1 Deq2) %. +%term _ + %pi (canonize-fun (canonize/lam D2 D1) (canonize/lam D2' D1') Deq) + %<- (tcanonize-fun D1 D1' Deq1) + %<- (canonize-resp-underbind Deq1 D2 D2'') + %<- ({x} {d} {dv} canonize-fun (D2'' x d dv) (D2' x d dv) (Deq2 x)) + %<- (epi-resp Deq1 Deq2 Deq) %. +%term _ + %pi (canonize-fun (canonize/pair D2 D1) (canonize/pair D2' D1') Deq) + %<- (canonize-fun D1 D1' Deq1) + %<- (canonize-fun D2 D2' Deq2) + %<- (esigma-resp Deq1 ([_] Deq2) Deq) %. +%term _ tcanonize-fun tcanonize/t tcanonize/t etp-eq/i %. +%term _ + %pi (tcanonize-fun (tcanonize/pi D2 D1) (tcanonize/pi D2' D1') Deq) + %<- (tcanonize-fun D1 D1' Deq1) + %<- (tcanonize-resp-underbind Deq1 D2 D2'') + %<- ({x} {d} {dv} tcanonize-fun (D2'' x d dv) (D2' x d dv) (Deq2 x)) + %<- (epi-resp Deq1 Deq2 Deq) %. +%term _ + %pi (tcanonize-fun (tcanonize/sigma D2 D1) (tcanonize/sigma D2' D1') Deq) + %<- (tcanonize-fun D1 D1' Deq1) + %<- (tcanonize-resp-underbind Deq1 D2 D2'') + %<- ({x} {d} {dv} tcanonize-fun (D2'' x d dv) (D2' x d dv) (Deq2 x)) + %<- (esigma-resp Deq1 Deq2 Deq) %. +%term _ + %pi (tcanonize-fun (tcanonize/sing D) (tcanonize/sing D') Deq) + %<- (canonize-fun D D' Deq) %. +%worlds (evbind sbind) (canonize-fun _ _ _) (tcanonize-fun _ _ _) %. +%total (D1 D2) (canonize-fun _ D1 _) (tcanonize-fun _ D2 _) %. +%sort etopen-reg %. +%mode etopen-reg %in %out %. +%term _ etopen-reg etopen/t ewf/t %. +%term _ + %pi (etopen-reg (etopen/pi D2 D1) (ewf/pi ([_] [_] D2') D1')) + %<- (etopen-reg D1 D1') + %<- (etopen-reg D2 D2') %. +%worlds (scbind) (etopen-reg _ _) %. +%total D (etopen-reg D _) %. +%sort ekof-reg %. +%mode ekof-reg %in %out %. +%term _ %pi (ekof-reg (ekof/i Dopen _) Dwf) %<- (etopen-reg Dopen Dwf) %. +%worlds (scbind) (ekof-reg _ _) %. +%total {} (ekof-reg _ _) %. +%sort canonize-trans %. +%mode canonize-trans %in %out %. +%sort canonize-trans' %. +%mode canonize-trans' %in %out %out %. +%sort tcanonize-trans %. +%mode tcanonize-trans %in %out %. +%sort tcanonize-trans' %. +%mode tcanonize-trans' %in %out %out %. +%scope tcanonize-trans' %term const + %pi (canonize-trans (canonize/const _ (%the (ekof C EA) Dekof)) (trans/const Dself Dexpand Dwf Dkof)) + %<- (ekof-reg Dekof Dewf) + %<- (wf-comp Dewf (%the (ttrans EA A) Dtrans)) + %<- (kof-comp Dekof Dtrans (%the (kof C A) Dkof)) + %<- (ttrans-reg Dtrans (%the (wf A) Dwf)) + %<- (can-expand (const C) A M (%the (expand (const C) A M) Dexpand)) + %<- (expand-reg (aof/const Dwf Dkof) Dexpand (%the (of M A) Dof)) + %<- (can-self Dof (%the (self M A As) Dself)) %. +%term forall + %pi (canonize-trans (canonize/forall _ (%the (tcanonize EA EA') Dcanon)) (trans/forall Dself Dexpand Dtrans)) + %<- (tcanonize-trans Dcanon (%the (ttrans EA A) Dtrans)) + %<- (ttrans-reg Dtrans (%the (wf A) Dwf)) + %<- (can-expand (forall A) (qtp A) M (%the (expand (forall A) (qtp A) M) Dexpand)) + %<- (expand-reg (aof/forall Dwf) Dexpand (%the (of M (qtp A)) Dof)) + %<- (can-self Dof (%the (self M (qtp A) B) Dself)) %. +%scope tcanonize-trans' %term var + %pi (canonize-trans (canonize/var _ (%the (evof EX EA) Devof)) (trans/var Dself Dexpand DwfA Dvof Dvtrans)) + %<- (vof-comp Devof (%the (ttrans EA A) Dttrans) (%the (vtrans EX X) Dvtrans) (%the (vof X A) Dvof)) + %<- (ttrans-reg Dttrans (%the (wf A) DwfA)) + %<- (can-expand X A M (%the (expand X A M) Dexpand)) + %<- (expand-reg (aof/var DwfA Dvof) Dexpand (%the (of M A) Dof)) + %<- (can-self Dof (%the (self M A As) Dself)) %. +%term app + %pi (canonize-trans (canonize/app (%the (treduce (EB EN') ED) Dreduce) (%the (coerce EC EA ([_] EN')) Dcoerce) (%the (canonize EN EC) DcanonN) (%the (canonize EM (epi EA EB)) DcanonM)) (trans/app DsubBx Dsubtype' DtransN DtransM)) + %<- (canonize-trans' DcanonM (%the (trans EM (pi A B)) DtransM) (tconvert/pi (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))) DconvB) (%the (tconvert A EA) DconvA))) + %<- (canonize-trans' DcanonN (%the (trans EN C) DtransN) (%the (tconvert C EC) DconvC)) + %<- (coerce-convert' DconvC DconvA Dcoerce (%the (subtype C A N) Dsubtype) (%the ({x} %pi (vof x C) %-> ({ex} %pi (vtrans ex x) %-> (convert (N x) A EN'))) DconvN)) + %<- (trans-subtype DtransN Dsubtype (%the (subtype C A ([_] N')) Dsubtype') (%the ({x} term-eq (N x) N') DeqN)) + %<- (trans-reg DtransM (wf/pi (%the ({x} %pi (vof x A) %-> (wf (B x))) DwfB) (%the (wf A) DwfA))) + %<- (trans-reg DtransN (%the (wf C) DwfC)) + %<- (subtype-reg' Dsubtype' DwfC DwfA (%the (of N' A) DofN')) + %<- (can-tsub DwfB DofN' (%the (tsub B N' Bx) DsubBx)) %. +%scope tcanonize-trans' %term pi1 + %pi (canonize-trans (canonize/pi1 (%the (canonize EM (esigma EA ([_] EB))) Dcanon)) (trans/pi1 Dtrans')) + %<- (canonize-trans' Dcanon (%the (trans EM (sigma A B)) Dtrans) (tconvert/sigma (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) EB))) DconvB) (%the (tconvert A EA) DconvA))) + %<- (trans-principal Dtrans (%the (principal (sigma A B)) Dprincipal)) + %<- (principal-sigma-invert Dprincipal (%the ({x} tp-eq (B x) B') DeqB)) + %<- (sigma-resp tp-eq/i DeqB (%the (tp-eq (sigma A B) (sigma A ([_] B'))) Deq)) + %<- (trans-resp eterm-eq/i Deq Dtrans (%the (trans EM (sigma A ([_] B'))) Dtrans')) %. +%term pi2 + %pi (canonize-trans (canonize/pi2 (%the (canonize EM (esigma EA ([_] EB))) Dcanon)) (trans/pi2 Dtrans')) + %<- (canonize-trans' Dcanon (%the (trans EM (sigma A B)) Dtrans) (tconvert/sigma (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) EB))) DconvB) (%the (tconvert A EA) DconvA))) + %<- (trans-principal Dtrans (%the (principal (sigma A B)) Dprincipal)) + %<- (principal-sigma-invert Dprincipal (%the ({x} tp-eq (B x) B') DeqB)) + %<- (sigma-resp tp-eq/i DeqB (%the (tp-eq (sigma A B) (sigma A ([_] B'))) Deq)) + %<- (trans-resp eterm-eq/i Deq Dtrans (%the (trans EM (sigma A ([_] B'))) Dtrans')) %. +%scope tcanonize-trans' %term lam + %pi (canonize-trans (canonize/lam (%the ({ex} %pi (evof ex EA') %-> (variable ex) %-> (canonize (EM ex) (EB ex))) DcanonM) (%the (tcanonize EA EA') DcanonA)) (trans/lam DtransM DtransA)) + %<- (tcanonize-trans' DcanonA (%the (ttrans EA A) DtransA) (%the (tconvert A EA') DconvA)) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (invert-tconvert DconvA (%the (ttrans EA' A) DtransA')) + %<- ({x} {d vof x A} {ex} {xt} {ed evof ex EA'} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA ed) + %-> (vof-comp ed DtransA' xt d) + %-> (canonize-trans (DcanonM ex ed dv) (%the (trans (EM ex) (B x)) (DtransM x d ex xt)))) %. +%term pair + %pi (canonize-trans (canonize/pair (%the (canonize EN EB) DcanonN) (%the (canonize EM EA) DcanonM)) (trans/pair DtransN DtransM)) + %<- (canonize-trans DcanonM DtransM) + %<- (canonize-trans DcanonN DtransN) %. +%scope tcanonize-trans' %term t tcanonize-trans tcanonize/t ttrans/t %. +%term pi + %pi (tcanonize-trans (tcanonize/pi (%the ({ex} %pi (evof ex EA') %-> (variable ex) %-> (tcanonize (EB ex) (EB' ex))) DcanonB) (%the (tcanonize EA EA') DcanonA)) (ttrans/pi DtransB DtransA)) + %<- (tcanonize-trans' DcanonA (%the (ttrans EA A) DtransA) (%the (tconvert A EA') DconvA)) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (invert-tconvert DconvA (%the (ttrans EA' A) DtransA')) + %<- ({x} {d vof x A} {ex} {xt} {ed evof ex EA'} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA ed) + %-> (vof-comp ed DtransA' xt d) + %-> (tcanonize-trans (DcanonB ex ed dv) (%the (ttrans (EB ex) (B x)) (DtransB x d ex xt)))) %. +%scope tcanonize-trans' %term sigma + %pi (tcanonize-trans (tcanonize/sigma (%the ({ex} %pi (evof ex EA') %-> (variable ex) %-> (tcanonize (EB ex) (EB' ex))) DcanonB) (%the (tcanonize EA EA') DcanonA)) (ttrans/sigma DtransB DtransA)) + %<- (tcanonize-trans' DcanonA (%the (ttrans EA A) DtransA) (%the (tconvert A EA') DconvA)) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (invert-tconvert DconvA (%the (ttrans EA' A) DtransA')) + %<- ({x} {d vof x A} {ex} {xt} {ed evof ex EA'} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA ed) + %-> (vof-comp ed DtransA' xt d) + %-> (tcanonize-trans (DcanonB ex ed dv) (%the (ttrans (EB ex) (B x)) (DtransB x d ex xt)))) %. +%term sing + %pi (tcanonize-trans (tcanonize/sing (%the (canonize EM (esing EM')) Dcanon)) (ttrans/sing Dtrans)) + %<- (canonize-trans' Dcanon (%the (trans EM (sing R)) Dtrans) (tconvert/sing (%the (aconvert R t EM') Dconv))) %. +%term _ + %pi (canonize-trans' Dcanon Dtrans Dconv') + %<- (canonize-trans Dcanon Dtrans) + %<- (canonize-comp Dtrans Dconv Dcanon') + %<- (canonize-fun Dcanon' Dcanon Deq) + %<- (tconvert-resp tp-eq/i Deq Dconv Dconv') %. +%term _ + %pi (tcanonize-trans' Dcanon Dtrans Dconv') + %<- (tcanonize-trans Dcanon Dtrans) + %<- (tcanonize-comp Dtrans Dconv Dcanon') + %<- (tcanonize-fun Dcanon' Dcanon Deq) + %<- (tconvert-resp tp-eq/i Deq Dconv Dconv') %. +%worlds (scbind) (canonize-trans _ _) (tcanonize-trans _ _) (canonize-trans' _ _ _) (tcanonize-trans' _ _ _) %. +%total (D1 D2 D3 D4) (canonize-trans D1 _) (tcanonize-trans D2 _) (canonize-trans' D3 _ _) (tcanonize-trans' D4 _ _) %. +%sort canonize-sound %. +%mode canonize-sound %in %out %. +%term _ + %pi (canonize-sound (%the (canonize EM EA) Dcanon) Dof) + %<- (canonize-trans' Dcanon (%the (trans EM A) Dtrans) (%the (tconvert A EA) Dconv)) + %<- (sound-trans' Dtrans Dconv (%the (eof EM EA) Dof)) %. +%worlds (scbind) (canonize-sound _ _) %. +%total {} (canonize-sound _ _) %. +%sort tcanonize-sound %. +%mode tcanonize-sound %in %out %. +%term _ + %pi (tcanonize-sound (%the (tcanonize EA EB) Dcanon) Dequiv) + %<- (tcanonize-trans' Dcanon (%the (ttrans EA B) Dtrans) (%the (tconvert B EB) Dconv)) + %<- (sound-ttrans' Dtrans Dconv (%the (tequiv EA EB) Dequiv)) %. +%worlds (scbind) (tcanonize-sound _ _) %. +%total {} (tcanonize-sound _ _) %. +%sort canonize-idem %. +%mode canonize-idem %in %out %. +%term _ + %pi (canonize-idem (%the (canonize EM EA) Dcanon) Dcanon') + %<- (canonize-trans' Dcanon _ (%the (tconvert A EA) Dconv)) + %<- (invert-tconvert Dconv (%the (ttrans EA A) Dtrans)) + %<- (tcanonize-comp' Dtrans Dconv (%the (tcanonize EA EA) Dcanon')) %. +%worlds (scbind) (canonize-idem _ _) %. +%total {} (canonize-idem _ _) %. +%sort tcanonize-idem %. +%mode tcanonize-idem %in %out %. +%term _ + %pi (tcanonize-idem Dcanon Dcanon') + %<- (tcanonize-trans' Dcanon _ Dconv) + %<- (invert-tconvert Dconv Dtrans) + %<- (tcanonize-comp' Dtrans Dconv Dcanon') %. +%worlds (scbind) (tcanonize-idem _ _) %. +%total {} (tcanonize-idem _ _) %. +%%%%% Check-Eof %%%%% +%sort check-eof-comp %. +%mode check-eof-comp %in %out %. +%term _ + %pi (check-eof-comp (%the (eof EM EA) Deof) (check-eof/i Dcoerce DcanonA DcanonM)) + %<- (of-comp Deof (%the (ttrans EA A) DtransA) (%the (trans EM B) DtransM) (%the (subtype B A _) Dsubtype)) + %<- (canonize-comp DtransM (%the (tconvert B EB) DconvB) (%the (canonize EM EB) DcanonM)) + %<- (tcanonize-comp DtransA (%the (tconvert A EA') DconvA) (%the (tcanonize EA EA') DcanonA)) + %<- (can-coerce DconvB DconvA Dsubtype _ (%the (coerce EB EA' _) Dcoerce)) %. +%worlds (scbind) (check-eof-comp _ _) %. +%total {} (check-eof-comp _ _) %. +%sort check-eof-sound %. +%mode check-eof-sound %in %out %. +%term _ + %pi (check-eof-sound (check-eof/i (%the (coerce EB EA' _) Dcoerce) (%the (tcanonize EA EA') DcanonA) (%the (canonize EM EB) DcanonM)) (eof/equiv (tequiv/symm Dequiv) (eof/subsume Dsubtp Deof))) + %<- (canonize-trans' DcanonM (%the (trans EM B) DtransM) (%the (tconvert B EB) DconvB)) + %<- (tcanonize-trans' DcanonA (%the (ttrans EA A) DtransA) (%the (tconvert A EA') DconvA)) + %<- (sound-trans' DtransM DconvB (%the (eof EM EB) Deof)) + %<- (sound-ttrans' DtransA DconvA (%the (tequiv EA EA') Dequiv)) + %<- (coerce-convert DconvB DconvA Dcoerce (%the (subtype B A M) Dsubtype)) + %<- (convert-subtype Dsubtype DconvB DconvA (%the (subtp EB EA') Dsubtp) _ _) %. +%worlds (scbind) (check-eof-sound _ _) %. +%total {} (check-eof-sound _ _) %. +%%%%% Check-Equiv %%%%% +%sort check-equiv-comp %. +%mode check-equiv-comp %in %out %. +%term _ + %pi (check-equiv-comp (%the (equiv EM EN EA) Dequiv) (check-equiv/i DcoerceCA DcoerceBA DcanonA DcanonN DcanonM)) + %<- (equiv-comp Dequiv (%the (ttrans EA A) DtransA) (%the (trans EM B) DtransM) (%the (trans EN C) DtransN) (%the (subtype B A O) DsubtypeBA) (%the (subtype C A O) DsubtypeCA)) + %<- (tcanonize-comp DtransA (%the (tconvert A EA') DconvA) (%the (tcanonize EA EA') DcanonA)) + %<- (canonize-comp DtransM (%the (tconvert B EB) DconvB) (%the (canonize EM EB) DcanonM)) + %<- (canonize-comp DtransN (%the (tconvert C EC) DconvC) (%the (canonize EN EC) DcanonN)) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (trans-reg DtransM (%the (wf B) DwfB)) + %<- (trans-subtype DtransM DsubtypeBA (%the (subtype B A ([_] P)) DsubtypeBA') (%the ({x} term-eq (O x) P) DeqO)) + %<- (subtype-resp tp-eq/i tp-eq/i DeqO DsubtypeCA (%the (subtype C A ([_] P)) DsubtypeCA')) + %<- (subtype-reg' DsubtypeBA' DwfB DwfA (%the (of P A) DofP)) + %<- (can-convert DofP (%the (convert P A EP) DconvertP)) + %<- (can-coerce' DconvB DconvA DsubtypeBA' ([_] [_] [_] [_] DconvertP) (%the (coerce EB EA' ([_] EP)) DcoerceBA)) + %<- (can-coerce' DconvC DconvA DsubtypeCA' ([_] [_] [_] [_] DconvertP) (%the (coerce EC EA' ([_] EP)) DcoerceCA)) %. +%worlds (scbind) (check-equiv-comp _ _) %. +%total {} (check-equiv-comp _ _) %. +%sort canonize-coerce-equiv %. +%mode canonize-coerce-equiv %in %in %in %in %out %. +%term _ + %pi (canonize-coerce-equiv (%the (eof EM EA) DeofM) (%the (tcanonize EA EB) DcanonA) (%the (canonize EM EC) DcanonM) (%the (coerce EC EB ([_] EN)) Dcoerce) (equiv/equiv (tequiv/symm Dtequiv) Dequiv)) + %<- (of-comp' DeofM (%the (ttrans EA A) DtransA) (%the (trans EM C) DtransM) (%the (subtype C A ([_] N)) Dsubtype)) + %<- (canonize-comp DtransM (%the (tconvert C EC') DconvC) (%the (canonize EM EC') DcanonM')) + %<- (canonize-fun DcanonM' DcanonM (%the (etp-eq EC' EC) DeqC')) + %<- (tconvert-resp tp-eq/i DeqC' DconvC (%the (tconvert C EC) DconvC')) + %<- (tcanonize-comp DtransA (%the (tconvert A EB') DconvA) (%the (tcanonize EA EB') DcanonA')) + %<- (tcanonize-fun DcanonA' DcanonA (%the (etp-eq EB' EB) DeqB')) + %<- (tconvert-resp tp-eq/i DeqB' DconvA (%the (tconvert A EB) DconvA')) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (trans-reg DtransM (%the (wf C) DwfC)) + %<- (subtype-reg' Dsubtype DwfC DwfA (%the (of N A) DofN)) + %<- (can-convert DofN (%the (convert N A EN') DconvN)) + %<- (can-coerce' DconvC' DconvA' Dsubtype ([_] [_] [_] [_] DconvN) (%the (coerce EC EB ([_] EN')) Dcoerce')) + %<- (coerce-fun Dcoerce' Dcoerce (%the (%pi eterm %-> (eterm-eq EN' EN)) DeqN)) + %<- (convert-resp term-eq/i tp-eq/i (DeqN eaca) DconvN (%the (convert N A EN) DconvN')) + %<- (sound-trans' DtransM DconvC' (%the (eof EM EC) DeofM')) + %<- (convert-subtype-equiv Dsubtype DconvC' DconvA' DconvN' DeofM' (%the (equiv EM EN EB) Dequiv)) + %<- (sound-ttrans' DtransA DconvA' (%the (tequiv EA EB) Dtequiv)) %. +%worlds (scbind) (canonize-coerce-equiv _ _ _ _ _) %. +%total {} (canonize-coerce-equiv _ _ _ _ _) %. +%sort check-equiv-sound %. +%mode check-equiv-sound %in %out %. +%term _ + %pi (check-equiv-sound (check-equiv/i (%the (coerce EC EA' ([_] EO)) DcoerceCA) (%the (coerce EB EA' ([_] EO)) DcoerceBA) (%the (tcanonize EA EA') DcanonA) (%the (canonize EN EC) DcanonN) (%the (canonize EM EB) DcanonM)) (equiv/trans (equiv/symm Dequiv') Dequiv)) + %<- (check-eof-sound (check-eof/i DcoerceBA DcanonA DcanonM) (%the (eof EM EA) DeofM)) + %<- (check-eof-sound (check-eof/i DcoerceCA DcanonA DcanonN) (%the (eof EN EA) DeofN)) + %<- (canonize-coerce-equiv DeofM DcanonA DcanonM DcoerceBA (%the (equiv EM EO EA) Dequiv)) + %<- (canonize-coerce-equiv DeofN DcanonA DcanonN DcoerceCA (%the (equiv EN EO EA) Dequiv')) %. +%worlds (scbind) (check-equiv-sound _ _) %. +%total {} (check-equiv-sound _ _) %. +%%%%% Check-Ewf %%%%% +%sort check-ewf-comp %. +%mode check-ewf-comp %in %out %. +%term _ + %pi (check-ewf-comp (%the (ewf EA) Dewf) (check-ewf/i Dcanon)) + %<- (wf-comp Dewf (%the (ttrans EA A) Dtrans)) + %<- (tcanonize-comp Dtrans (%the (tconvert A EA') Dconv) (%the (tcanonize EA EA') Dcanon)) %. +%worlds (scbind) (check-ewf-comp _ _) %. +%total {} (check-ewf-comp _ _) %. +%sort check-ewf-sound %. +%mode check-ewf-sound %in %out %. +%term _ + %pi (check-ewf-sound (check-ewf/i (%the (tcanonize EA EA') Dcanon)) Dwf) + %<- (tcanonize-trans Dcanon (%the (ttrans EA A) Dtrans)) + %<- (sound-ttrans Dtrans (%the (ewf EA) Dwf) _ _) %. +%worlds (scbind) (check-ewf-sound _ _) %. +%total {} (check-ewf-sound _ _) %. +%%%%% Check-Subtp %%%%% +%sort check-subtp-comp %. +%mode check-subtp-comp %in %out %. +%term _ + %pi (check-subtp-comp (%the (subtp EA EB) Dsubtp) (check-subtp/i Dcoerce DcanonB DcanonA)) + %<- (subtp-comp Dsubtp (%the (ttrans EA A) DtransA) (%the (ttrans EB B) DtransB) (%the (subtype A B _) Dsubtype)) + %<- (tcanonize-comp DtransA (%the (tconvert A EA') DconvA) (%the (tcanonize EA EA') DcanonA)) + %<- (tcanonize-comp DtransB (%the (tconvert B EB') DconvB) (%the (tcanonize EB EB') DcanonB)) + %<- (can-coerce DconvA DconvB Dsubtype _ (%the (coerce EA' EB' _) Dcoerce)) %. +%worlds (scbind) (check-subtp-comp _ _) %. +%total {} (check-subtp-comp _ _) %. +%sort check-subtp-sound %. +%mode check-subtp-sound %in %out %. +%term _ + %pi (check-subtp-sound (check-subtp/i (%the (coerce EA' EB' _) Dcoerce) (%the (tcanonize EB EB') DcanonB) (%the (tcanonize EA EA') DcanonA)) (subtp/trans (subtp/trans (subtp/reflex (tequiv/symm DequivB')) Dsubtp) (subtp/reflex DequivA'))) + %<- (tcanonize-trans' DcanonA (%the (ttrans EA A) DtransA) (%the (tconvert A EA') DconvA)) + %<- (tcanonize-trans' DcanonB (%the (ttrans EB B) DtransB) (%the (tconvert B EB') DconvB)) + %<- (coerce-convert DconvA DconvB Dcoerce (%the (subtype A B _) Dsubtype)) + %<- (convert-subtype Dsubtype DconvA DconvB (%the (subtp EA' EB') Dsubtp) _ _) + %<- (sound-ttrans DtransA _ (%the (tconvert A EA'') DconvA') (%the (tequiv EA EA'') DequivA)) + %<- (sound-ttrans DtransB _ (%the (tconvert B EB'') DconvB') (%the (tequiv EB EB'') DequivB)) + %<- (tconvert-fun DconvA' DconvA (%the (etp-eq EA'' EA') DeqA)) + %<- (tconvert-fun DconvB' DconvB (%the (etp-eq EB'' EB') DeqB)) + %<- (tequiv-resp etp-eq/i DeqA DequivA (%the (tequiv EA EA') DequivA')) + %<- (tequiv-resp etp-eq/i DeqB DequivB (%the (tequiv EB EB') DequivB')) %. +%worlds (scbind) (check-subtp-sound _ _) %. +%total {} (check-subtp-sound _ _) %. +%%%%% Check-Tequiv %%%%% +%sort check-tequiv-comp %. +%mode check-tequiv-comp %in %out %. +%term _ + %pi (check-tequiv-comp (%the (tequiv EA EB) Dequiv) (check-tequiv/i DcanonB' DcanonA)) + %<- (tequiv-comp Dequiv (%the (ttrans EA C) DtransA) (%the (ttrans EB C) DtransB)) + %<- (tcanonize-comp DtransA (%the (tconvert C EC) Dconv) (%the (tcanonize EA EC) DcanonA)) + %<- (tcanonize-comp DtransB (%the (tconvert C EC') Dconv') (%the (tcanonize EB EC') DcanonB)) + %<- (tconvert-fun Dconv' Dconv (%the (etp-eq EC' EC) Deq)) + %<- (tcanonize-resp etp-eq/i Deq DcanonB (%the (tcanonize EB EC) DcanonB')) %. +%worlds (scbind) (check-tequiv-comp _ _) %. +%total {} (check-tequiv-comp _ _) %. +%sort check-tequiv-sound %. +%mode check-tequiv-sound %in %out %. +%term _ + %pi (check-tequiv-sound (check-tequiv/i (%the (tcanonize EB EC) DcanonB) (%the (tcanonize EA EC) DcanonA)) (tequiv/trans (tequiv/symm DequivB') DequivA')) + %<- (tcanonize-trans' DcanonA (%the (ttrans EA A) DtransA) (%the (tconvert A EC) DconvA)) + %<- (tcanonize-trans' DcanonB (%the (ttrans EB B) DtransB) (%the (tconvert B EC) DconvB)) + %<- (sound-ttrans DtransA _ (%the (tconvert A EA') DconvA') (%the (tequiv EA EA') DequivA)) + %<- (sound-ttrans DtransB _ (%the (tconvert B EB') DconvB') (%the (tequiv EB EB') DequivB)) + %<- (tconvert-fun DconvA' DconvA (%the (etp-eq EA' EC) DeqA)) + %<- (tconvert-fun DconvB' DconvB (%the (etp-eq EB' EC) DeqB)) + %<- (tequiv-resp etp-eq/i DeqA DequivA (%the (tequiv EA EC) DequivA')) + %<- (tequiv-resp etp-eq/i DeqB DequivB (%the (tequiv EB EC) DequivB')) %. +%worlds (scbind) (check-tequiv-sound _ _) %. +%total {} (check-tequiv-sound _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/el-inversion-thm.lf b/new-tests/stelf-output/tslf/sing/el-inversion-thm.lf new file mode 100644 index 0000000..dbe188f --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/el-inversion-thm.lf @@ -0,0 +1,510 @@ +%sort epair-invert %. +%mode epair-invert %in %out %out %. +%term _ + %pi (epair-invert (%the (eof (epair EM EN) (esigma EA EB)) DeofMN) (eof/equiv (tequiv/symm DequivA) (eof/subsume DsubtpCA DeofM)) (eof/subsume (subtp/trans (subtp/reflex (tequiv/trans (tequiv/symm Dequiv) (tequiv/symm DequivBx))) DsubtpDB) DeofN)) + %<- (of-comp DeofMN (ttrans/sigma (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (ttrans (EB ex) (B x)))) DtransB) (%the (ttrans EA A) DtransA)) (trans/pair (%the (trans EN D) DtransN) (%the (trans EM C) DtransM)) (subtype/sigma (%the ({x} subtype D (B' x) (N' x)) DsubtypeDB) (%the ({x} tsub B (M' x) (B' x)) DsubB') (%the (subtype C A M') DsubtypeCA))) + %<- (sound-trans DtransM (%the (tconvert C EC) DconvC) (%the (eof EM EC) DeofM)) + %<- (sound-trans DtransN (%the (tconvert D ED) DconvD) (%the (eof EN ED) DeofN)) + %<- (sound-ttrans DtransA _ (%the (tconvert A EA') DconvA) (%the (tequiv EA EA') DequivA)) + %<- (convert-subtype' DsubtypeCA DconvC DconvA (%the (subtp EC EA') DsubtpCA)) + %<- ({x} {d vof x A} {ex} {xt} {ed evof ex EA'} {dv} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA ed) + %-> (sound-ttrans (DtransB x d ex xt) (%the (ewf (EB ex)) (DewfB ex ed)) (%the (tconvert (B x) (EB' ex)) (DconvB x d ex xt)) (%the (tequiv (EB ex) (EB' ex)) (DequivB ex ed)))) + %<- (trans-subtype DtransM DsubtypeCA (%the (subtype C A ([_] M)) DsubtypeCA') (%the ({x} term-eq (M' x) M) DeqM)) + %<- (trans-reg DtransM (%the (wf C) DwfC)) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (subtype-reg' DsubtypeCA' DwfC DwfA (%the (of M A) DofM)) + %<- ({x} {d} {ex} {xt} ttrans-reg (DtransB x d ex xt) (%the (wf (B x)) (DwfB x d))) + %<- (can-tsub DwfB DofM (%the (tsub B M Bx) DsubBx)) + %<- ({x} tsub-resp ([_] tp-eq/i) (DeqM x) tp-eq/i (DsubB' x) (%the (tsub B M (B' x)) (DsubB'' x))) + %<- ({x} tsub-fun (DsubB'' x) DsubBx (%the (tp-eq (B' x) Bx) (DeqB x))) + %<- ({x} subtype-resp tp-eq/i (DeqB x) ([_] term-eq/i) (DsubtypeDB x) (%the (subtype D Bx (N' x)) (DsubtypeDB' x))) + %<- (can-convert DofM (%the (convert M A EM') DconvM)) + %<- (convert-subtype-equiv DsubtypeCA' DconvC DconvA DconvM DeofM (%the (equiv EM EM' EA') DequivM)) + %<- (convert-tsub DconvB DconvM DsubBx (%the (tconvert Bx EBx) DconvBx) (%the (tequiv (EB' EM') EBx) DequivBx)) + %<- (convert-subtype' (DsubtypeDB' aca) DconvD DconvBx (%the (subtp ED EBx) DsubtpDB)) + %<- (tequiv-functionality DequivB DequivM (%the (tequiv (EB EM) (EB' EM')) Dequiv)) %. +%worlds (scbind) (epair-invert _ _ _) %. +%total {} (epair-invert _ _ _) %. +%sort elam-equiv-invert %. +%mode elam-equiv-invert %in %out %out %out %. +%term _ + %pi (elam-equiv-invert (%the (equiv (elam EA1 EM1) (elam EA2 EM2) (epi EA EB)) Dequiv) (subtp/trans (subtp/reflex (tequiv/symm DequivA1)) (subtp/trans Dsubtp1 (subtp/reflex DequivA))) (subtp/trans (subtp/reflex (tequiv/symm DequivA2)) (subtp/trans Dsubtp2 (subtp/reflex DequivA))) Dequiv') + %<- (equiv-comp Dequiv (ttrans/pi (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (ttrans (EB ex) (B x)))) DtransB) (%the (ttrans EA A) DtransA)) (trans/lam (%the ({x} %pi (vof x A1) %-> ({ex} %pi (vtrans ex x) %-> (trans (EM1 ex) (C1 x)))) DtransM1) (%the (ttrans EA1 A1) DtransA1)) (trans/lam (%the ({x} %pi (vof x A2) %-> ({ex} %pi (vtrans ex x) %-> (trans (EM2 ex) (C2 x)))) DtransM2) (%the (ttrans EA2 A2) DtransA2)) (%the (subtype (pi A1 C1) (pi A B) O') Dsubtype1) (%the (subtype (pi A2 C2) (pi A B) O') Dsubtype2)) + %<- (subtype-pi-invert Dsubtype1 (%the (subtype A A1 N1) DsubtypeA1) (%the ({x} tsub C1 (N1 x) (C1x x)) DsubC1x) (%the ({x} subtype (C1x x) (B x) (O1 x)) DsubtypeC1') (%the ({f} term-eq (O' f) (lam ([x] O1 x (app f (N1 x))))) DeqO1)) + %<- (subtype-pi-invert Dsubtype2 (%the (subtype A A2 N2) DsubtypeA2) (%the ({x} tsub C2 (N2 x) (C2x x)) DsubC2x) (%the ({x} subtype (C2x x) (B x) (O2 x)) DsubtypeC2') (%the ({f} term-eq (O' f) (lam ([x] O2 x (app f (N2 x))))) DeqO2)) + %<- ({x} {d} {ex} {xt} trans-principal (DtransM1 x d ex xt) (%the (principal (C1 x)) (DprinC1 x))) + %<- ({x} principal-sub DprinC1 (DsubC1x x) (%the (principal (C1x x)) (DprinC1x x))) + %<- ({x} principal-subtype (DprinC1x x) (DsubtypeC1' x) (%the ({y} term-eq (O1 x y) (O x)) (DeqO1' x))) + %<- ({f} lam-resp ([x] DeqO1' x (app f (N1 x))) (%the (term-eq (lam ([x] O1 x (app f (N1 x)))) (lam O)) (DeqLam1 f))) + %<- ({x} {d} {ex} {xt} trans-principal (DtransM2 x d ex xt) (%the (principal (C2 x)) (DprinC2 x))) + %<- ({x} principal-sub DprinC2 (DsubC2x x) (%the (principal (C2x x)) (DprinC2x x))) + %<- ({x} principal-subtype (DprinC2x x) (DsubtypeC2' x) (%the ({y} term-eq (O2 x y) (O2' x)) (DeqO2' x))) + %<- ({f} lam-resp ([x] DeqO2' x (app f (N2 x))) (%the (term-eq (lam ([x] O2 x (app f (N2 x)))) (lam O2')) (DeqLam2 f))) + %<- ({f} term-eq-trans (DeqO1 f) (DeqLam1 f) (DeqLam1' f)) + %<- ({f} term-eq-trans (DeqO2 f) (DeqLam2 f) (DeqLam2' f)) + %<- ({f} term-eq-symm (DeqLam2' f) (DeqLam2'' f)) + %<- ({f} term-eq-trans (DeqLam2'' f) (DeqLam1' f) (%the (term-eq (lam O2') (lam O)) (DeqLam f))) + %<- (term-eq-cdr-lam (DeqLam aca) (%the ({x} term-eq (O2' x) (O x)) DeqO2'')) + %<- ({x} {y} term-eq-trans (DeqO2' x y) (DeqO2'' x) (DeqO2''' x y)) + %<- ({x} subtype-resp tp-eq/i tp-eq/i (DeqO1' x) (DsubtypeC1' x) (%the (subtype (C1x x) (B x) ([_] O x)) (DsubtypeC1 x))) + %<- ({x} subtype-resp tp-eq/i tp-eq/i (DeqO2''' x) (DsubtypeC2' x) (%the (subtype (C2x x) (B x) ([_] O x)) (DsubtypeC2 x))) + %<- (sound-ttrans DtransA _ (%the (tconvert A EA') DconvA) (%the (tequiv EA EA') DequivA)) + %<- (tequiv-reg DequivA (%the (ewf EA) DwfEA) (%the (ewf EA') DwfEA')) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA'} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA ed) + %-> (sound-ttrans (DtransB x d ex xt) (DewfB ex ed) (%the (tconvert (B x) (EB' ex)) (DconvB x d ex xt)) (%the (tequiv (EB ex) (EB' ex)) (DequivB ex ed)))) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (ttrans-reg DtransA1 (%the (wf A1) DwfA1)) + %<- (subtype-reg DsubtypeA1 DwfA DwfA1 (%the ({x} %pi (vof x A) %-> (of (N1 x) A1)) DofN1)) + %<- ({x} {d vof x A1} {ex} {xt} trans-reg (DtransM1 x d ex xt) (%the (wf (C1 x)) (DwfC1 x d))) + %<- ({x} {d vof x A} tsubst (DsubC1x x) DwfC1 (DofN1 x d) (%the (wf (C1x x)) (DwfC1x x d))) + %<- ({x} {d vof x A} {ex} {xt} ttrans-reg (DtransB x d ex xt) (%the (wf (B x)) (DwfB x d))) + %<- ({x} {d vof x A} subtype-reg' (DsubtypeC1 x) (DwfC1x x d) (DwfB x d) (%the (of (O x) (B x)) (DofO x d))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA'} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA ed) + %-> (can-convert (DofO x d) (%the (convert (O x) (B x) (EO ex)) (DconvO x d ex xt)))) + %<- (sound-ttrans DtransA1 _ (%the (tconvert A1 EA1') DconvA1) (%the (tequiv EA1 EA1') DequivA1)) + %<- (convert-subtype DsubtypeA1 DconvA DconvA1 (%the (subtp EA' EA1') Dsubtp1) (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (convert (N1 x) A1 (EN1 ex)))) DconvN1) (%the ({ex} %pi (evof ex EA') %-> (equiv ex (EN1 ex) EA1')) DequivN1)) + %<- ({x} {d vof x A1} {ex} {xt vtrans ex x} {ed evof ex EA1'} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA1 ed) + %-> (sound-trans (DtransM1 x d ex xt) (%the (tconvert (C1 x) (EC1 ex)) (DconvC1 x d ex xt)) (%the (eof (EM1 ex) (EC1 ex)) (DeofM1 ex ed)))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA'} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA ed) + %-> (convert-tsub DconvC1 (DconvN1 x d ex xt) (DsubC1x x) (%the (tconvert (C1x x) (EC1x ex)) (DconvC1x x d ex xt)) (%the (tequiv (EC1 (EN1 ex)) (EC1x ex)) (DequivC1x ex ed)))) + %<- ({ex} {ed evof ex EA1'} eof-reg (DeofM1 ex ed) (%the (ewf (EC1 ex)) (DewfC1 ex ed))) + %<- ({ex} {ed evof ex EA'} tfunctionality DewfC1 (DequivN1 ex ed) (%the (tequiv (EC1 ex) (EC1 (EN1 ex))) (DequivC1 ex ed))) + %<- ({ex} {ed evof ex EA'} esubst DeofM1 (eof/subsume Dsubtp1 (eof/var DwfEA' ed)) (%the (eof (EM1 ex) (EC1 ex)) (DeofM1' ex ed))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA'} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA ed) + %-> (convert-subtype-equiv (DsubtypeC1 x) (DconvC1x x d ex xt) (DconvB x d ex xt) (DconvO x d ex xt) (%the (eof (EM1 ex) (EC1x ex)) (eof/equiv (DequivC1x ex ed) (eof/equiv (DequivC1 ex ed) (DeofM1' ex ed)))) (%the (equiv (EM1 ex) (EO ex) (EB' ex)) (Dequiv1 ex ed)))) + %<- (sound-ttrans DtransA2 _ (%the (tconvert A2 EA2') DconvA2) (%the (tequiv EA2 EA2') DequivA2)) + %<- (convert-subtype DsubtypeA2 DconvA DconvA2 (%the (subtp EA' EA2') Dsubtp2) (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (convert (N2 x) A2 (EN2 ex)))) DconvN2) (%the ({ex} %pi (evof ex EA') %-> (equiv ex (EN2 ex) EA2')) DequivN2)) + %<- ({x} {d vof x A2} {ex} {xt vtrans ex x} {ed evof ex EA2'} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA2 ed) + %-> (sound-trans (DtransM2 x d ex xt) (%the (tconvert (C2 x) (EC2 ex)) (DconvC2 x d ex xt)) (%the (eof (EM2 ex) (EC2 ex)) (DeofM2 ex ed)))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA'} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA ed) + %-> (convert-tsub DconvC2 (DconvN2 x d ex xt) (DsubC2x x) (%the (tconvert (C2x x) (EC2x ex)) (DconvC2x x d ex xt)) (%the (tequiv (EC2 (EN2 ex)) (EC2x ex)) (DequivC2x ex ed)))) + %<- ({ex} {ed evof ex EA2'} eof-reg (DeofM2 ex ed) (%the (ewf (EC2 ex)) (DewfC2 ex ed))) + %<- ({ex} {ed evof ex EA'} tfunctionality DewfC2 (DequivN2 ex ed) (%the (tequiv (EC2 ex) (EC2 (EN2 ex))) (DequivC2 ex ed))) + %<- ({ex} {ed evof ex EA'} esubst DeofM2 (eof/subsume Dsubtp2 (eof/var DwfEA' ed)) (%the (eof (EM2 ex) (EC2 ex)) (DeofM2' ex ed))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA'} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvA ed) + %-> (convert-subtype-equiv (DsubtypeC2 x) (DconvC2x x d ex xt) (DconvB x d ex xt) (DconvO x d ex xt) (%the (eof (EM2 ex) (EC2x ex)) (eof/equiv (DequivC2x ex ed) (eof/equiv (DequivC2 ex ed) (DeofM2' ex ed)))) (%the (equiv (EM2 ex) (EO ex) (EB' ex)) (Dequiv2 ex ed)))) + %<- ({ex'} {ed' evof ex' EA} esubst-equiv ([ex] [ed evof ex EA'] %the (equiv (EM1 ex) (EM2 ex) (EB ex)) (equiv/equiv (tequiv/symm (DequivB ex ed)) (equiv/trans (equiv/symm (Dequiv2 ex ed)) (Dequiv1 ex ed)))) (%the (eof ex' EA') (eof/equiv DequivA (eof/var DwfEA ed'))) (%the (equiv (EM1 ex') (EM2 ex') (EB ex')) (Dequiv' ex' ed'))) %. +%worlds (scbind) (elam-equiv-invert _ _ _ _) %. +%total {} (elam-equiv-invert _ _ _ _) %. +%sort nullary-trans %. +%mode nullary-trans %in %in %in %out %. +%term _ + %pi (nullary-trans (%the (kof K t) Dkof) (%the (trans (econst K) B) DtransK) (%the (subtype B t ([_] N)) DsubtypeB) (Deq aca)) + %<- (trans-fun DtransK (%the (trans (econst K) (sing (const K))) (trans/const self/t expand/t wf/t Dkof)) (%the (tp-eq B (sing (const K))) Deq1)) + %<- (subtype-resp Deq1 tp-eq/i ([_] term-eq/i) DsubtypeB (%the (subtype (sing (const K)) t ([_] N)) DsubtypeB')) + %<- (subtype-fun DsubtypeB' subtype/sing_t (%the ({x} term-eq N (at (const K))) Deq)) %. +%worlds (bind tbind) (nullary-trans _ _ _ _) %. +%total {} (nullary-trans _ _ _ _) %. +%sort trans-kof-invert %. +%mode trans-kof-invert %in %in %out %. +%term _ + %pi (trans-kof-invert (trans/const _ _ Dwf Dkof) Dkof' Dwf') + %<- (kof-fun Dkof Dkof' Deq) + %<- (wf-resp Deq Dwf Dwf') %. +%worlds (bind tbind) (trans-kof-invert _ _ _) %. +%total {} (trans-kof-invert _ _ _) %. +%sort unary-trans %. +%mode unary-trans %in %in %in %out %out %out %. +%term _ + %pi (unary-trans (%the (kof K (pi A ([_] t))) Dkof) (trans/app (%the (tsub B M Bx) DsubBx) (%the (subtype C A' ([_] M)) DsubtypeCA') (%the (trans EM C) DtransM) (%the (trans (econst K) (pi A' B)) DtransK)) (%the (subtype Bx t ([_] N)) DsubtypeBx) DtransM DsubtypeCA (Deq aca)) + %<- ({x} can-expand x A (X x) (%the (expand x A (X x)) (DexpandX x))) + %<- (trans-kof-invert DtransK Dkof (wf/pi _ (%the (wf A) DwfA))) + %<- (trans-fun DtransK (%the (trans (econst K) (pi A ([x] sing (app (const K) (X x))))) (trans/const (%the (self _ (pi A ([_] t)) (pi A ([x] sing (app (const K) (X x))))) (self/pi ([_] self/t))) (%the (expand (const K) (pi A ([_] t)) (lam ([x] at (app (const K) (X x))))) (expand/pi ([_] expand/t) DexpandX)) (wf/pi ([_] [_] wf/t) DwfA) Dkof)) (%the (tp-eq (pi A' B) (pi A ([x] sing (app (const K) (X x))))) Deq1)) + %<- (tp-eq-cdr-pi Deq1 (%the (tp-eq A' A) DeqA) (%the ({x} tp-eq (B x) (sing (app (const K) (X x)))) DeqB)) + %<- (subtype-resp tp-eq/i DeqA ([_] term-eq/i) DsubtypeCA' (%the (subtype C A ([_] M)) DsubtypeCA)) + %<- (trans-reg DtransM (%the (wf C) DwfC)) + %<- (subtype-reg' DsubtypeCA DwfC DwfA (%the (of M A) DofM)) + %<- (sub-into-expand-var DexpandX DofM (%the (sub X M M) Dsub)) + %<- ({x} tp-eq-symm (DeqB x) (%the (tp-eq (sing (app (const K) (X x))) (B x)) (DeqB' x))) + %<- (tsub-resp DeqB' term-eq/i tp-eq/i (%the (tsub ([x] sing (app (const K) (X x))) M (sing (app (const K) M))) (tsub/singa (aasub/app Dsub aasub/closed))) (%the (tsub B M (sing (app (const K) M))) DsubBx')) + %<- (tsub-fun DsubBx DsubBx' (%the (tp-eq Bx (sing (app (const K) M))) DeqBx)) + %<- (subtype-resp DeqBx tp-eq/i ([_] term-eq/i) DsubtypeBx (%the (subtype (sing (app (const K) M)) t ([_] N)) DsubtypeBx')) + %<- (subtype-fun DsubtypeBx' subtype/sing_t (%the ({x} term-eq N (at (app (const K) M))) Deq)) %. +%worlds (bind tbind) (unary-trans _ _ _ _ _ _) %. +%total {} (unary-trans _ _ _ _ _ _) %. +%sort trans-forall-invert %. +%mode trans-forall-invert %in %out %. +%term _ trans-forall-invert (trans/forall _ _ Dtrans) Dtrans %. +%worlds (bind tbind) (trans-forall-invert _ _) %. +%total {} (trans-forall-invert _ _) %. +%sort forall-trans %. +%mode forall-trans %in %in %out %out %out %out %. +%term _ + %pi (forall-trans (trans/app (%the (tsub C M Cx) DsubCx) (%the (subtype D B ([_] M)) DsubtypeDB) (%the (trans EM D) DtransM) (%the (trans (eforall EA) (pi B C)) DtransQ)) (%the (subtype Cx t ([_] N)) DsubtypeCx) DtransA DtransM DsubtypeDB' (DeqN aca)) + %<- (trans-forall-invert DtransQ (%the (ttrans EA A) DtransA)) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (trans-reg DtransM (%the (wf D) DwfD)) + %<- ({f} can-expand f (pi A ([_] t)) (F f) (%the (expand f (pi A ([_] t)) (F f)) (DexpandF f))) + %<- (trans-fun DtransQ (trans/forall (%the (self _ (pi (pi A ([_] t)) ([_] t)) (pi (pi A ([_] t)) ([f] sing (app (forall A) (F f))))) (self/pi ([f] self/t))) (%the (expand (forall A) (pi (pi A ([_] t)) ([_] t)) (lam ([f] at (app (forall A) (F f))))) (expand/pi ([_] expand/t) DexpandF)) DtransA) (%the (tp-eq (pi B C) (pi (pi A ([_] t)) ([f] sing (app (forall A) (F f))))) DeqQ)) + %<- (tp-eq-cdr-pi DeqQ (%the (tp-eq B (pi A ([_] t))) DeqB) (%the ({f} tp-eq (C f) (sing (app (forall A) (F f)))) DeqC)) + %<- (subtype-resp tp-eq/i DeqB ([_] term-eq/i) DsubtypeDB (%the (subtype D (pi A ([_] t)) ([_] M)) DsubtypeDB')) + %<- (tsub-resp DeqC term-eq/i tp-eq/i DsubCx (%the (tsub ([f] sing (app (forall A) (F f))) M Cx) DsubCx')) + %<- (subtype-reg' DsubtypeDB' DwfD (wf/pi ([_] [_] wf/t) DwfA) (%the (of M (pi A ([_] t))) DofM)) + %<- (sub-into-expand-var DexpandF DofM (%the (sub F M M) DsubF)) + %<- (tsub-fun DsubCx' (tsub/singa (aasub/app DsubF aasub/closed)) (%the (tp-eq Cx (sing (app (forall A) M))) DeqCx)) + %<- (subtype-resp DeqCx tp-eq/i ([_] term-eq/i) DsubtypeCx (%the (subtype (sing (app (forall A) M)) t ([_] N)) DsubtypeCx')) + %<- (subtype-fun DsubtypeCx' subtype/sing_t (%the ({x} term-eq N (at (app (forall A) M))) DeqN)) %. +%worlds (bind tbind) (forall-trans _ _ _ _ _ _) %. +%total {} (forall-trans _ _ _ _ _ _) %. +%sort binary-trans %. +%mode binary-trans %in %in %in %out %out %out %out %out %out %. +%term _ + %pi (binary-trans (%the (kof K (pi A ([x] pi (B x) ([_] t)))) Dkof) (trans/app (%the (tsub Cx N Cxy) DsubCxy) (%the (subtype E Bx' ([_] N)) DsubtypeEB') (%the (trans EN E) DtransN) (trans/app (%the (tsub BC M (pi Bx' Cx)) DsubBCx) (%the (subtype D A' ([_] M)) DsubtypeDA') (%the (trans EM D) DtransM) (%the (trans (econst K) (pi A' BC)) DtransK))) (%the (subtype Cxy t ([_] O)) DsubtypeCxy) DtransM DsubtypeDA DsubBx DtransN DsubtypeEB (Deq aca)) + %<- ({x} can-expand x A (X x) (%the (expand x A (X x)) (DexpandX x))) + %<- ({x} {y} can-expand y (B x) (Y x y) (%the (expand y (B x) (Y x y)) (DexpandY x y))) + %<- (trans-kof-invert DtransK Dkof (wf/pi ([x] [d vof x A] wf/pi _ (%the (wf (B x)) (DwfB x d))) (%the (wf A) DwfA))) + %<- (trans-fun DtransK (%the (trans (econst K) (pi A ([x] pi (B x) ([y] sing (app (app (const K) (X x)) (Y x y)))))) (trans/const (%the (self _ (pi A ([x] pi (B x) ([y] t))) (pi A ([x] pi (B x) ([y] sing (app (app (const K) (X x)) (Y x y)))))) (self/pi ([x] self/pi ([y] self/t)))) (%the (expand (const K) (pi A ([x] pi (B x) ([_] t))) (lam ([x] lam ([y] at (app (app (const K) (X x)) (Y x y)))))) (expand/pi ([x] expand/pi ([y] expand/t) ([y] DexpandY x y)) DexpandX)) (wf/pi ([x] [d] wf/pi ([_] [_] wf/t) (DwfB x d)) DwfA) Dkof)) (%the (tp-eq (pi A' BC) (pi A ([x] pi (B x) ([y] sing (app (app (const K) (X x)) (Y x y)))))) Deq1)) + %<- (tp-eq-cdr-pi Deq1 (%the (tp-eq A' A) DeqA) (%the ({x} tp-eq (BC x) (pi (B x) ([y] sing (app (app (const K) (X x)) (Y x y))))) DeqBC)) + %<- (subtype-resp tp-eq/i DeqA ([_] term-eq/i) DsubtypeDA' (%the (subtype D A ([_] M)) DsubtypeDA)) + %<- (trans-reg DtransM (%the (wf D) DwfD)) + %<- (subtype-reg' DsubtypeDA DwfD DwfA (%the (of M A) DofM)) + %<- (sub-into-expand-var DexpandX DofM (%the (sub X M M) DsubX)) + %<- (can-tsub DwfB DofM (%the (tsub B M Bx) DsubBx)) + %<- ({y} can-expand y Bx (Y' y) (%the (expand y Bx (Y' y)) (DexpandY' y))) + %<- (expand-aasub1 DofM ([x] [d] [y] [e] aof/var (DwfB x d) e) DexpandY ([y] aasub/closed) ([_] DsubBx) DexpandY' (%the ({y} sub ([x] Y x y) M (Y' y)) DsubY)) + %<- ({x} tp-eq-symm (DeqBC x) (%the (tp-eq (pi (B x) ([y] sing (app (app (const K) (X x)) (Y x y)))) (BC x)) (DeqBC' x))) + %<- (tsub-resp DeqBC' term-eq/i tp-eq/i (%the (tsub ([x] pi (B x) ([y] sing (app (app (const K) (X x)) (Y x y)))) M (pi Bx ([y] sing (app (app (const K) M) (Y' y))))) (tsub/pi ([y] tsub/singa (aasub/app (DsubY y) (aasub/app DsubX aasub/closed))) DsubBx)) (%the (tsub BC M (pi Bx ([y] sing (app (app (const K) M) (Y' y))))) DsubBCx')) + %<- (tsub-fun DsubBCx DsubBCx' (%the (tp-eq (pi Bx' Cx) (pi Bx ([y] sing (app (app (const K) M) (Y' y))))) Deq2)) + %<- (tp-eq-cdr-pi Deq2 (%the (tp-eq Bx' Bx) DeqBx) (%the ({y} tp-eq (Cx y) (sing (app (app (const K) M) (Y' y)))) DeqCx)) + %<- (subtype-resp tp-eq/i DeqBx ([_] term-eq/i) DsubtypeEB' (%the (subtype E Bx ([_] N)) DsubtypeEB)) + %<- (trans-reg DtransN (%the (wf E) DwfE)) + %<- (tsubst DsubBx DwfB DofM (%the (wf Bx) DwfBx)) + %<- (subtype-reg' DsubtypeEB DwfE DwfBx (%the (of N Bx) DofN)) + %<- (sub-into-expand-var DexpandY' DofN (%the (sub Y' N N) DsubY')) + %<- ({y} tp-eq-symm (DeqCx y) (%the (tp-eq (sing (app (app (const K) M) (Y' y))) (Cx y)) (DeqCx' y))) + %<- (sub-absent M N (%the (sub ([_] M) N M) DsubMabs)) + %<- (tsub-resp DeqCx' term-eq/i tp-eq/i (%the (tsub ([y] sing (app (app (const K) M) (Y' y))) N (sing (app (app (const K) M) N))) (tsub/singa (aasub/app DsubY' (aasub/app DsubMabs aasub/closed)))) (%the (tsub Cx N (sing (app (app (const K) M) N))) DsubCxy')) + %<- (tsub-fun DsubCxy DsubCxy' (%the (tp-eq Cxy (sing (app (app (const K) M) N))) DeqCxy)) + %<- (subtype-resp DeqCxy tp-eq/i ([_] term-eq/i) DsubtypeCxy (%the (subtype (sing (app (app (const K) M) N)) t ([_] O)) DsubtypeCxy')) + %<- (subtype-fun DsubtypeCxy' subtype/sing_t (%the ({x} term-eq O (at (app (app (const K) M) N))) Deq)) %. +%worlds (bind tbind) (binary-trans _ _ _ _ _ _ _ _ _) %. +%total {} (binary-trans _ _ _ _ _ _ _ _ _) %. +%sort eof-const-invert %. +%mode eof-const-invert %in %in %out %. +%term _ + %pi (eof-const-invert Dkof Dof Dwf') + %<- (check-eof-comp Dof (check-eof/i _ _ (canonize/const _ Dkof'))) + %<- (ekof-reg Dkof' Dwf) + %<- (ekof-fun Dkof' Dkof Deq) + %<- (ewf-resp Deq Dwf Dwf') %. +%worlds (scbind) (eof-const-invert _ _ _) %. +%total {} (eof-const-invert _ _ _) %. +%sort eof-app-subterms %. +%mode eof-app-subterms %in %out %out %. +%term _ + %pi (eof-app-subterms Dof DofM DofN) + %<- (check-eof-comp Dof (check-eof/i _ _ (canonize/app _ _ DcanonN DcanonM))) + %<- (canonize-sound DcanonM DofM) + %<- (canonize-sound DcanonN DofN) %. +%worlds (scbind) (eof-app-subterms _ _ _) %. +%total {} (eof-app-subterms _ _ _) %. +%sort unary-equiv-inversion %. +%mode unary-equiv-inversion %in %in %out %. +%term _ + %pi (unary-equiv-inversion (%the (ekof K (epi EA ([_] et))) Dekof) (%the (equiv (eapp (econst K) EM1) (eapp (econst K) EM2) et) DequivApp) DequivM) + %<- (equiv-reg DequivApp DofApp _ _) + %<- (eof-app-subterms DofApp DofConst _) + %<- (eof-const-invert Dekof DofConst (ewf/pi _ (%the (ewf EA) DewfA))) + %<- (wf-comp DewfA (%the (ttrans EA A) DtransA)) + %<- (equiv-comp' DequivApp ttrans/t (%the (trans (eapp (econst K) EM1) B1) DtransApp1) (%the (trans (eapp (econst K) EM2) B2) DtransApp2) (%the (subtype B1 t ([_] N)) DsubtypeB1) (%the (subtype B2 t ([_] N)) DsubtypeB2)) + %<- (kof-comp Dekof (ttrans/pi ([_] [_] [_] [_] ttrans/t) (%the (ttrans EA A) DtransA)) (%the (kof K (pi A ([_] t))) Dkof)) + %<- (unary-trans Dkof DtransApp1 DsubtypeB1 (%the (trans EM1 C1) DtransM1) (%the (subtype C1 A ([_] M1)) DsubtypeC1) (%the (term-eq N (at (app (const K) M1))) DeqN1)) + %<- (unary-trans Dkof DtransApp2 DsubtypeB2 (%the (trans EM2 C2) DtransM2) (%the (subtype C2 A ([_] M2)) DsubtypeC2) (%the (term-eq N (at (app (const K) M2))) DeqN2)) + %<- (term-eq-symm DeqN1 DeqN1') + %<- (term-eq-trans DeqN1' DeqN2 (%the (term-eq (at (app (const K) M1)) (at (app (const K) M2))) Deq)) + %<- (term-eq-cdr-at Deq (%the (atom-eq (app (const K) M1) (app (const K) M2)) Deq')) + %<- (atom-eq-cdr-app Deq' _ (%the (term-eq M1 M2) DeqM)) + %<- (subtype-resp tp-eq/i tp-eq/i ([_] DeqM) DsubtypeC1 (%the (subtype C1 A ([_] M2)) DsubtypeC1')) + %<- (il-soundness DtransM1 DtransM2 DtransA DsubtypeC1' DsubtypeC2 (%the (equiv EM1 EM2 EA) DequivM)) %. +%worlds (scbind) (unary-equiv-inversion _ _ _) %. +%total {} (unary-equiv-inversion _ _ _) %. +%sort binary-equiv-inversion %. +%mode binary-equiv-inversion %in %in %out %out %. +%term _ + %pi (binary-equiv-inversion (%the (ekof K (epi EA ([ex] epi (EB ex) ([_] et)))) Dekof) (%the (equiv (eapp (eapp (econst K) EM1) EN1) (eapp (eapp (econst K) EM2) EN2) et) DequivApp) DequivM DequivN) + %<- (equiv-reg DequivApp DofApp _ _) + %<- (eof-app-subterms DofApp DofApp' _) + %<- (eof-app-subterms DofApp' DofConst _) + %<- (eof-const-invert Dekof DofConst (ewf/pi ([ex] [ed] ewf/pi _ (%the (ewf (EB ex)) (DewfB ex ed))) (%the (ewf EA) DewfA))) + %<- (wf-comp DewfA (%the (ttrans EA A) DtransA)) + %<- ({x} {d} {ex} {xt} {ed} + %pi (vof-comp ed DtransA xt d) + %-> (wf-comp (DewfB ex ed) (%the (ttrans (EB ex) (B x)) (DtransB x d ex xt)))) + %<- (equiv-comp' DequivApp ttrans/t (%the (trans (eapp (eapp (econst K) EM1) EN1) C1) DtransApp1) (%the (trans (eapp (eapp (econst K) EM2) EN2) C2) DtransApp2) (%the (subtype C1 t ([_] O)) DsubtypeC1) (%the (subtype C2 t ([_] O)) DsubtypeC2)) + %<- (kof-comp Dekof (ttrans/pi ([x] [d vof x A] [ex] [xt] ttrans/pi ([_] [_] [_] [_] ttrans/t) (%the (ttrans (EB ex) (B x)) (DtransB x d ex xt))) (%the (ttrans EA A) DtransA)) (%the (kof K (pi A ([x] pi (B x) ([y] t)))) Dkof)) + %<- (binary-trans Dkof DtransApp1 DsubtypeC1 (%the (trans EM1 D1) DtransM1) (%the (subtype D1 A ([_] M1)) DsubtypeD1) (%the (tsub B M1 Bx1) DsubBx1) (%the (trans EN1 E1) DtransN1) (%the (subtype E1 Bx1 ([_] N1)) DsubtypeE1) (%the (term-eq O (at (app (app (const K) M1) N1))) DeqO1)) + %<- (binary-trans Dkof DtransApp2 DsubtypeC2 (%the (trans EM2 D2) DtransM2) (%the (subtype D2 A ([_] M2)) DsubtypeD2) (%the (tsub B M2 Bx2) DsubBx2) (%the (trans EN2 E2) DtransN2) (%the (subtype E2 Bx2 ([_] N2)) DsubtypeE2) (%the (term-eq O (at (app (app (const K) M2) N2))) DeqO2)) + %<- (term-eq-symm DeqO1 DeqO1') + %<- (term-eq-trans DeqO1' DeqO2 (%the (term-eq (at (app (app (const K) M1) N1)) (at (app (app (const K) M2) N2))) Deq)) + %<- (term-eq-cdr-at Deq (%the (atom-eq (app (app (const K) M1) N1) (app (app (const K) M2) N2)) Deq')) + %<- (atom-eq-cdr-app Deq' (%the (atom-eq (app (const K) M1) (app (const K) M2)) Deq'') (%the (term-eq N1 N2) DeqN)) + %<- (atom-eq-cdr-app Deq'' _ (%the (term-eq M1 M2) DeqM)) + %<- (subtype-resp tp-eq/i tp-eq/i ([_] DeqM) DsubtypeD1 (%the (subtype D1 A ([_] M2)) DsubtypeD1')) + %<- (il-soundness DtransM1 DtransM2 DtransA DsubtypeD1' DsubtypeD2 (%the (equiv EM1 EM2 EA) DequivM)) + %<- (tsub-resp ([_] tp-eq/i) DeqM tp-eq/i DsubBx1 (%the (tsub B M2 Bx1) DsubBx1')) + %<- (tsub-fun DsubBx1' DsubBx2 (%the (tp-eq Bx1 Bx2) DeqBx)) + %<- (subtype-resp tp-eq/i DeqBx ([_] DeqN) DsubtypeE1 (%the (subtype E1 Bx2 ([_] N2)) DsubtypeE1')) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (ttrans-sub DtransB DtransM1 DwfA DsubtypeD1 DsubBx1 (%the (ttrans (EB EM1) Bx1) DtransBx)) + %<- (ttrans-resp etp-eq/i DeqBx DtransBx (%the (ttrans (EB EM1) Bx2) DtransBx')) + %<- (il-soundness DtransN1 DtransN2 DtransBx' DsubtypeE1' DsubtypeE2 (%the (equiv EN1 EN2 (EB EM1)) DequivN)) %. +%worlds (scbind) (binary-equiv-inversion _ _ _ _) %. +%total {} (binary-equiv-inversion _ _ _ _) %. +%sort eof-forall-invert %. +%mode eof-forall-invert %in %out %. +%term _ + %pi (eof-forall-invert Dof Dwf) + %<- (check-eof-comp Dof (check-eof/i _ _ (canonize/forall _ Dcanon))) + %<- (tcanonize-sound Dcanon Dequiv) + %<- (tequiv-reg Dequiv Dwf _) %. +%worlds (scbind) (eof-forall-invert _ _) %. +%total {} (eof-forall-invert _ _) %. +%sort forall-equiv-inversion %. +%mode forall-equiv-inversion %in %out %out %. +%term _ + %pi (forall-equiv-inversion (%the (equiv (eapp (eforall EA1) EM1) (eapp (eforall EA2) EM2) et) DequivApp) DequivA DequivM) + %<- (equiv-reg DequivApp DofApp _ _) + %<- (equiv-comp' DequivApp ttrans/t (%the (trans (eapp (eforall EA1) EM1) B1) DtransApp1) (%the (trans (eapp (eforall EA2) EM2) B2) DtransApp2) (%the (subtype B1 t ([_] N)) DsubtypeB1) (%the (subtype B2 t ([_] N)) DsubtypeB2)) + %<- (forall-trans DtransApp1 DsubtypeB1 (%the (ttrans EA1 A1) DtransA1) (%the (trans EM1 C1) DtransM1) (%the (subtype C1 (pi A1 ([_] t)) ([_] M1)) DsubtypeC1) (%the (term-eq N (at (app (forall A1) M1))) DeqN1)) + %<- (forall-trans DtransApp2 DsubtypeB2 (%the (ttrans EA2 A2) DtransA2) (%the (trans EM2 C2) DtransM2) (%the (subtype C2 (pi A2 ([_] t)) ([_] M2)) DsubtypeC2) (%the (term-eq N (at (app (forall A2) M2))) DeqN2)) + %<- (term-eq-symm DeqN1 DeqN1') + %<- (term-eq-trans DeqN1' DeqN2 (%the (term-eq (at (app (forall A1) M1)) (at (app (forall A2) M2))) Deq)) + %<- (term-eq-cdr-at Deq (%the (atom-eq (app (forall A1) M1) (app (forall A2) M2)) Deq')) + %<- (atom-eq-cdr-app Deq' (%the (atom-eq (forall A1) (forall A2)) DeqQ) (%the (term-eq M1 M2) DeqM)) + %<- (atom-eq-cdr-forall DeqQ (%the (tp-eq A1 A2) DeqA)) + %<- (term-eq-symm DeqM DeqM') + %<- (tp-eq-symm DeqA DeqA') + %<- (tp-resp-tp ([a] pi a ([_] t)) DeqA' DeqA'') + %<- (subtype-resp tp-eq/i DeqA'' ([_] DeqM') DsubtypeC2 (%the (subtype C2 (pi A1 ([_] t)) ([_] M1)) DsubtypeC2')) + %<- (il-soundness DtransM1 DtransM2 (ttrans/pi ([_] [_] [_] [_] ttrans/t) DtransA1) DsubtypeC1 DsubtypeC2' (%the (equiv EM1 EM2 (epi EA1 ([_] et))) DequivM)) + %<- (ttrans-resp etp-eq/i DeqA DtransA1 (%the (ttrans EA1 A2) DtransA1')) + %<- (il-soundness-tp DtransA1' DtransA2 (%the (tequiv EA1 EA2) DequivA)) %. +%worlds (scbind) (forall-equiv-inversion _ _ _) %. +%total {} (forall-equiv-inversion _ _ _) %. +%sort nullary-equiv-head %. +%mode nullary-equiv-head %in %in %in %out %. +%term _ + %pi (nullary-equiv-head (%the (ekof K1 et) Dekof1) (%the (ekof K2 et) Dekof2) (%the (equiv (econst K1) (econst K2) et) DequivApp) DeqK) + %<- (equiv-comp' DequivApp ttrans/t (%the (trans (econst K1) B1) DtransApp1) (%the (trans (econst K2) B2) DtransApp2) (%the (subtype B1 t ([_] N)) DsubtypeB1) (%the (subtype B2 t ([_] N)) DsubtypeB2)) + %<- (kof-comp Dekof1 ttrans/t (%the (kof K1 t) Dkof1)) + %<- (kof-comp Dekof2 ttrans/t (%the (kof K2 t) Dkof2)) + %<- (nullary-trans Dkof1 DtransApp1 DsubtypeB1 (%the (term-eq N (at (const K1))) DeqN1)) + %<- (nullary-trans Dkof2 DtransApp2 DsubtypeB2 (%the (term-eq N (at (const K2))) DeqN2)) + %<- (term-eq-symm DeqN1 DeqN1') + %<- (term-eq-trans DeqN1' DeqN2 (%the (term-eq (at (const K1)) (at (const K2))) Deq)) + %<- (term-eq-cdr-at Deq (%the (atom-eq (const K1) (const K2)) Deq')) + %<- (atom-eq-cdr-const Deq' (%the (constant-eq K1 K2) DeqK)) %. +%worlds (scbind) (nullary-equiv-head _ _ _ _) %. +%total {} (nullary-equiv-head _ _ _ _) %. +%sort unary-equiv-head %. +%mode unary-equiv-head %in %in %in %out %. +%term _ + %pi (unary-equiv-head (%the (ekof K1 (epi EA1 ([_] et))) Dekof1) (%the (ekof K2 (epi EA2 ([_] et))) Dekof2) (%the (equiv (eapp (econst K1) EM1) (eapp (econst K2) EM2) et) DequivApp) DeqK) + %<- (equiv-reg DequivApp DofApp1 DofApp2 _) + %<- (eof-app-subterms DofApp1 DofConst1 _) + %<- (eof-const-invert Dekof1 DofConst1 (ewf/pi _ (%the (ewf EA1) DewfA1))) + %<- (wf-comp DewfA1 (%the (ttrans EA1 A1) DtransA1)) + %<- (eof-app-subterms DofApp2 DofConst2 _) + %<- (eof-const-invert Dekof2 DofConst2 (ewf/pi _ (%the (ewf EA2) DewfA2))) + %<- (wf-comp DewfA2 (%the (ttrans EA2 A2) DtransA2)) + %<- (equiv-comp' DequivApp ttrans/t (%the (trans (eapp (econst K1) EM1) B1) DtransApp1) (%the (trans (eapp (econst K2) EM2) B2) DtransApp2) (%the (subtype B1 t ([_] N)) DsubtypeB1) (%the (subtype B2 t ([_] N)) DsubtypeB2)) + %<- (kof-comp Dekof1 (ttrans/pi ([_] [_] [_] [_] ttrans/t) (%the (ttrans EA1 A1) DtransA1)) (%the (kof K1 (pi A1 ([_] t))) Dkof1)) + %<- (kof-comp Dekof2 (ttrans/pi ([_] [_] [_] [_] ttrans/t) (%the (ttrans EA2 A2) DtransA2)) (%the (kof K2 (pi A2 ([_] t))) Dkof2)) + %<- (unary-trans Dkof1 DtransApp1 DsubtypeB1 _ _ (%the (term-eq N (at (app (const K1) M1))) DeqN1)) + %<- (unary-trans Dkof2 DtransApp2 DsubtypeB2 _ _ (%the (term-eq N (at (app (const K2) M2))) DeqN2)) + %<- (term-eq-symm DeqN1 DeqN1') + %<- (term-eq-trans DeqN1' DeqN2 (%the (term-eq (at (app (const K1) M1)) (at (app (const K2) M2))) Deq)) + %<- (term-eq-cdr-at Deq (%the (atom-eq (app (const K1) M1) (app (const K2) M2)) Deq')) + %<- (atom-eq-cdr-app Deq' (%the (atom-eq (const K1) (const K2)) Deq'') _) + %<- (atom-eq-cdr-const Deq'' (%the (constant-eq K1 K2) DeqK)) %. +%worlds (scbind) (unary-equiv-head _ _ _ _) %. +%total {} (unary-equiv-head _ _ _ _) %. +%sort binary-equiv-head %. +%mode binary-equiv-head %in %in %in %out %. +%term _ + %pi (binary-equiv-head (%the (ekof K1 (epi EA1 ([ex] epi (EB1 ex) ([_] et)))) Dekof1) (%the (ekof K2 (epi EA2 ([ex] epi (EB2 ex) ([_] et)))) Dekof2) (%the (equiv (eapp (eapp (econst K1) EM1) EN1) (eapp (eapp (econst K2) EM2) EN2) et) DequivApp) DeqK) + %<- (equiv-reg DequivApp DofApp1 DofApp2 _) + %<- (eof-app-subterms DofApp1 DofApp1' _) + %<- (eof-app-subterms DofApp1' DofConst1 _) + %<- (eof-const-invert Dekof1 DofConst1 (ewf/pi ([ex] [ed] ewf/pi _ (%the (ewf (EB1 ex)) (DewfB1 ex ed))) (%the (ewf EA1) DewfA1))) + %<- (wf-comp DewfA1 (%the (ttrans EA1 A1) DtransA1)) + %<- ({x} {d} {ex} {xt} {ed} + %pi (vof-comp ed DtransA1 xt d) + %-> (wf-comp (DewfB1 ex ed) (%the (ttrans (EB1 ex) (B1 x)) (DtransB1 x d ex xt)))) + %<- (eof-app-subterms DofApp2 DofApp2' _) + %<- (eof-app-subterms DofApp2' DofConst2 _) + %<- (eof-const-invert Dekof2 DofConst2 (ewf/pi ([ex] [ed] ewf/pi _ (%the (ewf (EB2 ex)) (DewfB2 ex ed))) (%the (ewf EA2) DewfA2))) + %<- (wf-comp DewfA2 (%the (ttrans EA2 A2) DtransA2)) + %<- ({x} {d} {ex} {xt} {ed} + %pi (vof-comp ed DtransA2 xt d) + %-> (wf-comp (DewfB2 ex ed) (%the (ttrans (EB2 ex) (B2 x)) (DtransB2 x d ex xt)))) + %<- (equiv-comp' DequivApp ttrans/t (%the (trans (eapp (eapp (econst K1) EM1) EN1) C1) DtransApp1) (%the (trans (eapp (eapp (econst K2) EM2) EN2) C2) DtransApp2) (%the (subtype C1 t ([_] O)) DsubtypeC1) (%the (subtype C2 t ([_] O)) DsubtypeC2)) + %<- (kof-comp Dekof1 (ttrans/pi ([x] [d vof x A1] [ex] [xt] ttrans/pi ([_] [_] [_] [_] ttrans/t) (%the (ttrans (EB1 ex) (B1 x)) (DtransB1 x d ex xt))) (%the (ttrans EA1 A1) DtransA1)) (%the (kof K1 (pi A1 ([x] pi (B1 x) ([y] t)))) Dkof1)) + %<- (kof-comp Dekof2 (ttrans/pi ([x] [d vof x A2] [ex] [xt] ttrans/pi ([_] [_] [_] [_] ttrans/t) (%the (ttrans (EB2 ex) (B2 x)) (DtransB2 x d ex xt))) (%the (ttrans EA2 A2) DtransA2)) (%the (kof K2 (pi A2 ([x] pi (B2 x) ([y] t)))) Dkof2)) + %<- (binary-trans Dkof1 DtransApp1 DsubtypeC1 _ _ _ _ _ (%the (term-eq O (at (app (app (const K1) M1) N1))) DeqO1)) + %<- (binary-trans Dkof2 DtransApp2 DsubtypeC2 _ _ _ _ _ (%the (term-eq O (at (app (app (const K2) M2) N2))) DeqO2)) + %<- (term-eq-symm DeqO1 DeqO1') + %<- (term-eq-trans DeqO1' DeqO2 Deq) + %<- (term-eq-cdr-at Deq Deq') + %<- (atom-eq-cdr-app Deq' Deq'' _) + %<- (atom-eq-cdr-app Deq'' Deq''' _) + %<- (atom-eq-cdr-const Deq''' DeqK) %. +%worlds (scbind) (binary-equiv-head _ _ _ _) %. +%total {} (binary-equiv-head _ _ _ _) %. +%sort eq-const-app-contra %. +%mode eq-const-app-contra %in %out %. +%worlds (scbind) (eq-const-app-contra _ _) %. +%total {} (eq-const-app-contra _ _) %. +%sort nullary-unary-equiv %. +%mode nullary-unary-equiv %in %in %in %out %. +%term _ + %pi (nullary-unary-equiv Dekof1 Dekof2 DequivApp D) + %<- (equiv-reg DequivApp DofApp1 DofApp2 _) + %<- (eof-app-subterms DofApp2 DofConst2 _) + %<- (eof-const-invert Dekof2 DofConst2 (ewf/pi _ (%the (ewf EA2) DewfA2))) + %<- (wf-comp DewfA2 (%the (ttrans EA2 A2) DtransA2)) + %<- (equiv-comp' DequivApp ttrans/t DtransApp1 DtransApp2 Dsubtype1 Dsubtype2) + %<- (kof-comp Dekof1 ttrans/t Dkof1) + %<- (kof-comp Dekof2 (ttrans/pi ([_] [_] [_] [_] ttrans/t) DtransA2) Dkof2) + %<- (nullary-trans Dkof1 DtransApp1 Dsubtype1 DeqN1) + %<- (unary-trans Dkof2 DtransApp2 Dsubtype2 _ _ DeqN2) + %<- (term-eq-symm DeqN1 DeqN1') + %<- (term-eq-trans DeqN1' DeqN2 Deq) + %<- (term-eq-cdr-at Deq Deq') + %<- (eq-const-app-contra Deq' D) %. +%worlds (scbind) (nullary-unary-equiv _ _ _ _) %. +%total {} (nullary-unary-equiv _ _ _ _) %. +%sort nullary-binary-equiv %. +%mode nullary-binary-equiv %in %in %in %out %. +%term _ + %pi (nullary-binary-equiv Dekof1 Dekof2 DequivApp D) + %<- (equiv-reg DequivApp DofApp1 DofApp2 _) + %<- (eof-app-subterms DofApp2 DofApp2' _) + %<- (eof-app-subterms DofApp2' DofConst2 _) + %<- (eof-const-invert Dekof2 DofConst2 (ewf/pi ([ex] [ed] ewf/pi _ (%the (ewf (EB2 ex)) (DewfB2 ex ed))) (%the (ewf EA2) DewfA2))) + %<- (wf-comp DewfA2 (%the (ttrans EA2 A2) DtransA2)) + %<- ({x} {d} {ex} {xt} {ed} + %pi (vof-comp ed DtransA2 xt d) + %-> (wf-comp (DewfB2 ex ed) (%the (ttrans (EB2 ex) (B2 x)) (DtransB2 x d ex xt)))) + %<- (equiv-comp' DequivApp ttrans/t DtransApp1 DtransApp2 Dsubtype1 Dsubtype2) + %<- (kof-comp Dekof1 ttrans/t Dkof1) + %<- (kof-comp Dekof2 (ttrans/pi ([x] [d] [ex] [xt] ttrans/pi ([_] [_] [_] [_] ttrans/t) (DtransB2 x d ex xt)) DtransA2) Dkof2) + %<- (nullary-trans Dkof1 DtransApp1 Dsubtype1 DeqN1) + %<- (binary-trans Dkof2 DtransApp2 Dsubtype2 _ _ _ _ _ DeqN2) + %<- (term-eq-symm DeqN1 DeqN1') + %<- (term-eq-trans DeqN1' DeqN2 Deq) + %<- (term-eq-cdr-at Deq Deq') + %<- (eq-const-app-contra Deq' D) %. +%worlds (scbind) (nullary-binary-equiv _ _ _ _) %. +%total {} (nullary-binary-equiv _ _ _ _) %. +%sort unary-binary-equiv %. +%mode unary-binary-equiv %in %in %in %out %. +%term _ + %pi (unary-binary-equiv Dekof1 Dekof2 DequivApp D) + %<- (equiv-reg DequivApp DofApp1 DofApp2 _) + %<- (eof-app-subterms DofApp1 DofConst1 _) + %<- (eof-const-invert Dekof1 DofConst1 (ewf/pi _ (%the (ewf EA1) DewfA1))) + %<- (wf-comp DewfA1 (%the (ttrans EA1 A1) DtransA1)) + %<- (eof-app-subterms DofApp2 DofApp2' _) + %<- (eof-app-subterms DofApp2' DofConst2 _) + %<- (eof-const-invert Dekof2 DofConst2 (ewf/pi ([ex] [ed] ewf/pi _ (%the (ewf (EB2 ex)) (DewfB2 ex ed))) (%the (ewf EA2) DewfA2))) + %<- (wf-comp DewfA2 (%the (ttrans EA2 A2) DtransA2)) + %<- ({x} {d} {ex} {xt} {ed} + %pi (vof-comp ed DtransA2 xt d) + %-> (wf-comp (DewfB2 ex ed) (%the (ttrans (EB2 ex) (B2 x)) (DtransB2 x d ex xt)))) + %<- (equiv-comp' DequivApp ttrans/t DtransApp1 DtransApp2 Dsubtype1 Dsubtype2) + %<- (kof-comp Dekof1 (ttrans/pi ([_] [_] [_] [_] ttrans/t) DtransA1) Dkof1) + %<- (kof-comp Dekof2 (ttrans/pi ([x] [d] [ex] [xt] ttrans/pi ([_] [_] [_] [_] ttrans/t) (DtransB2 x d ex xt)) DtransA2) Dkof2) + %<- (unary-trans Dkof1 DtransApp1 Dsubtype1 _ _ DeqN1) + %<- (binary-trans Dkof2 DtransApp2 Dsubtype2 _ _ _ _ _ DeqN2) + %<- (term-eq-symm DeqN1 DeqN1') + %<- (term-eq-trans DeqN1' DeqN2 Deq) + %<- (term-eq-cdr-at Deq Deq') + %<- (atom-eq-cdr-app Deq' Deq'' _) + %<- (eq-const-app-contra Deq'' D) %. +%worlds (scbind) (unary-binary-equiv _ _ _ _) %. +%total {} (unary-binary-equiv _ _ _ _) %. +%sort nullary-forall-equiv %. +%mode nullary-forall-equiv %in %in %out %. +%term _ + %pi (nullary-forall-equiv Dekof1 DequivApp D) + %<- (equiv-reg DequivApp DofApp1 DofApp2 _) + %<- (eof-app-subterms DofApp2 DofConst2 _) + %<- (equiv-comp' DequivApp ttrans/t DtransApp1 DtransApp2 Dsubtype1 Dsubtype2) + %<- (kof-comp Dekof1 ttrans/t Dkof1) + %<- (nullary-trans Dkof1 DtransApp1 Dsubtype1 DeqN1) + %<- (forall-trans DtransApp2 Dsubtype2 _ _ _ DeqN2) + %<- (term-eq-symm DeqN1 DeqN1') + %<- (term-eq-trans DeqN1' DeqN2 Deq) + %<- (term-eq-cdr-at Deq Deq') + %<- (eq-const-app-contra Deq' D) %. +%worlds (scbind) (nullary-forall-equiv _ _ _) %. +%total {} (nullary-forall-equiv _ _ _) %. +%sort eq-const-forall-contra %. +%mode eq-const-forall-contra %in %out %. +%worlds (scbind) (eq-const-forall-contra _ _) %. +%total {} (eq-const-forall-contra _ _) %. +%sort unary-forall-equiv %. +%mode unary-forall-equiv %in %in %out %. +%term _ + %pi (unary-forall-equiv (%the (ekof K1 (epi EA1 ([_] et))) Dekof1) (%the (equiv (eapp (econst K1) EM1) (eapp (eforall EQ) EM2) et) DequivApp) Dfalse) + %<- (equiv-reg DequivApp DofApp1 DofApp2 _) + %<- (eof-app-subterms DofApp1 DofConst1 _) + %<- (eof-const-invert Dekof1 DofConst1 (ewf/pi _ (%the (ewf EA1) DewfA1))) + %<- (wf-comp DewfA1 (%the (ttrans EA1 A1) DtransA1)) + %<- (equiv-comp' DequivApp ttrans/t (%the (trans (eapp (econst K1) EM1) B1) DtransApp1) (%the (trans (eapp (eforall EQ) EM2) B2) DtransApp2) (%the (subtype B1 t ([_] N)) DsubtypeB1) (%the (subtype B2 t ([_] N)) DsubtypeB2)) + %<- (kof-comp Dekof1 (ttrans/pi ([_] [_] [_] [_] ttrans/t) (%the (ttrans EA1 A1) DtransA1)) (%the (kof K1 (pi A1 ([_] t))) Dkof1)) + %<- (unary-trans Dkof1 DtransApp1 DsubtypeB1 _ _ (%the (term-eq N (at (app (const K1) M1))) DeqN1)) + %<- (forall-trans DtransApp2 DsubtypeB2 _ _ _ DeqN2) + %<- (term-eq-symm DeqN1 DeqN1') + %<- (term-eq-trans DeqN1' DeqN2 (%the (term-eq (at (app (const K1) M1)) (at (app (forall Q) M2))) Deq)) + %<- (term-eq-cdr-at Deq (%the (atom-eq (app (const K1) M1) (app (forall Q) M2)) Deq')) + %<- (atom-eq-cdr-app Deq' (%the (atom-eq (const K1) (forall Q)) Deq'') _) + %<- (eq-const-forall-contra Deq'' Dfalse) %. +%worlds (scbind) (unary-forall-equiv _ _ _) %. +%total {} (unary-forall-equiv _ _ _) %. +%sort eq-app-forall-contra %. +%mode eq-app-forall-contra %in %out %. +%worlds (scbind) (eq-app-forall-contra _ _) %. +%total {} (eq-app-forall-contra _ _) %. +%sort binary-forall-equiv %. +%mode binary-forall-equiv %in %in %out %. +%term _ + %pi (binary-forall-equiv Dekof1 DequivApp Dfalse) + %<- (equiv-reg DequivApp DofApp1 DofApp2 _) + %<- (eof-app-subterms DofApp1 DofApp1' _) + %<- (eof-app-subterms DofApp1' DofConst1 _) + %<- (eof-const-invert Dekof1 DofConst1 (ewf/pi ([ex] [ed] ewf/pi _ (%the (ewf (EB1 ex)) (DewfB1 ex ed))) (%the (ewf EA1) DewfA1))) + %<- (wf-comp DewfA1 (%the (ttrans EA1 A1) DtransA1)) + %<- ({x} {d} {ex} {xt} {ed} + %pi (vof-comp ed DtransA1 xt d) + %-> (wf-comp (DewfB1 ex ed) (%the (ttrans (EB1 ex) (B1 x)) (DtransB1 x d ex xt)))) + %<- (equiv-comp' DequivApp ttrans/t DtransApp1 DtransApp2 Dsubtype1 Dsubtype2) + %<- (kof-comp Dekof1 (ttrans/pi ([x] [d] [ex] [xt] ttrans/pi ([_] [_] [_] [_] ttrans/t) (DtransB1 x d ex xt)) DtransA1) Dkof1) + %<- (binary-trans Dkof1 DtransApp1 Dsubtype1 _ _ _ _ _ DeqN1) + %<- (forall-trans DtransApp2 Dsubtype2 _ _ _ DeqN2) + %<- (term-eq-symm DeqN1 DeqN1') + %<- (term-eq-trans DeqN1' DeqN2 Deq) + %<- (term-eq-cdr-at Deq Deq') + %<- (atom-eq-cdr-app Deq' Deq'' _) + %<- (eq-app-forall-contra Deq'' Dfalse) %. +%worlds (scbind) (binary-forall-equiv _ _ _) %. +%total {} (binary-forall-equiv _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/el.lf b/new-tests/stelf-output/tslf/sing/el.lf new file mode 100644 index 0000000..6c68d61 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/el.lf @@ -0,0 +1,145 @@ +%%%%% Syntax %%%%% +%sort etp %. +%name etp %. +%sort eterm %. +%name eterm %. +%term et etp %. +%term epi %pi etp %-> (%pi eterm %-> etp) %-> etp %. +%term esigma %pi etp %-> (%pi eterm %-> etp) %-> etp %. +%term esing %pi eterm %-> etp %. +%term econst %pi constant %-> eterm %. +%term eforall %pi etp %-> eterm %. +%term eapp %pi eterm %-> eterm %-> eterm %. +%term epi1 %pi eterm %-> eterm %. +%term epi2 %pi eterm %-> eterm %. +%term elam %pi etp %-> (%pi eterm %-> eterm) %-> eterm %. +%term epair %pi eterm %-> eterm %-> eterm %. +%%%%% Rules %%%%% +%sort ekof %. +%sort evof %. +%sort ewf %. +%sort subtp %. +%sort eof %. +%sort equiv %. +%sort tequiv %. +%%% type formation +%term ewf/t ewf et %. +%term ewf/pi %pi (ewf (epi A B)) %<- (ewf A) %<- ({x} %pi (evof x A) %-> (ewf (B x))) %. +%term ewf/sigma %pi (ewf (esigma A B)) %<- (ewf A) %<- ({x} %pi (evof x A) %-> (ewf (B x))) %. +%term ewf/sing %pi (ewf (esing M)) %<- (eof M et) %. +%%% typing +%term eof/var %pi (eof X A) %<- (evof X A) %<- (ewf A) %. +%term eof/const %pi (eof (econst K) A) %<- (ekof K A) %<- (ewf A) %. +%define qetp (%pi etp %-> etp) [a] epi (epi a ([_] et)) ([_] et) %. +%term eof/forall %pi (eof (eforall A) (qetp A)) %<- (ewf A) %. +%term eof/lam + %pi (eof (elam A M) (epi A B)) + %<- (ewf A) + %<- ({x} %pi (evof x A) %-> (eof (M x) (B x))) %. +%term eof/app %pi (eof (eapp M N) (B N)) %<- (eof M (epi A B)) %<- (eof N A) %. +%term eof/pair + %pi (eof (epair M N) (esigma A B)) + %<- (eof M A) + %<- (eof N (B M)) + %<- ({x} %pi (evof x A) %-> (ewf (B x))) %. +%term eof/pi1 %pi (eof (epi1 M) A) %<- (eof M (esigma A B)) %. +%term eof/pi2 %pi (eof (epi2 M) (B (epi1 M))) %<- (eof M (esigma A B)) %. +%term eof/sing %pi (eof M (esing M)) %<- (eof M et) %. +%term eof/extpi + %pi (eof M (epi A B)) + %<- (eof M (epi A B')) + %<- ({x} %pi (evof x A) %-> (eof (eapp M x) (B x))) %. +%term eof/extsigma + %pi (eof M (esigma A B)) + %<- (eof (epi1 M) A) + %<- (eof (epi2 M) (B (epi1 M))) + %<- ({x} %pi (evof x A) %-> (ewf (B x))) %. +%term eof/subsume %pi (eof M A') %<- (eof M A) %<- (subtp A A') %. +%%% term equivalence +%term equiv/reflex %pi (equiv M M A) %<- (eof M A) %. +%term equiv/symm %pi (equiv M N A) %<- (equiv N M A) %. +%term equiv/trans %pi (equiv M P A) %<- (equiv M N A) %<- (equiv N P A) %. +%term equiv/forall + %pi (equiv (eforall A) (eforall A') (epi (epi A ([_] et)) ([_] et))) + %<- (tequiv A A') %. +%term equiv/lam + %pi (equiv (elam A M) (elam A' M') (epi A B)) + %<- (tequiv A A') + %<- ({x} %pi (evof x A) %-> (equiv (M x) (M' x) (B x))) %. +%term equiv/app + %pi (equiv (eapp M N) (eapp M' N') (B N)) + %<- (equiv M M' (epi A B)) + %<- (equiv N N' A) %. +%term equiv/pair + %pi (equiv (epair M N) (epair M' N') (esigma A B)) + %<- (equiv M M' A) + %<- (equiv N N' (B M)) + %<- ({x} %pi (evof x A) %-> (ewf (B x))) %. +%term equiv/pi1 %pi (equiv (epi1 M) (epi1 M') A) %<- (equiv M M' (esigma A B)) %. +%term equiv/pi2 %pi (equiv (epi2 M) (epi2 M') (B (epi1 M))) %<- (equiv M M' (esigma A B)) %. +%term equiv/sing %pi (equiv M N (esing M)) %<- (equiv M N et) %. +%term equiv/singelim %pi (equiv M N et) %<- (eof M (esing N)) %. +%term equiv/extpi + %pi (equiv M N (epi A B)) + %<- (eof M (epi A B')) + %<- (eof N (epi A B'')) + %<- ({x} %pi (evof x A) %-> (equiv (eapp M x) (eapp N x) (B x))) %. +%term equiv/extpiw + %pi (equiv M N (epi A B)) + %<- (equiv M N (epi A B')) + %<- ({x} %pi (evof x A) %-> (equiv (eapp M x) (eapp N x) (B x))) %. +%term equiv/extsigma + %pi (equiv M N (esigma A B)) + %<- (equiv (epi1 M) (epi1 N) A) + %<- (equiv (epi2 M) (epi2 N) (B (epi1 M))) + %<- ({x} %pi (evof x A) %-> (ewf (B x))) %. +%term equiv/subsume %pi (equiv M N B) %<- (equiv M N A) %<- (subtp A B) %. +%term equiv/beta + %pi (equiv (eapp (elam A M) N) (M N) (B N)) + %<- ({x} %pi (evof x A) %-> (eof (M x) (B x))) + %<- (eof N A) %. +%term equiv/beta1 %pi (equiv (epi1 (epair M N)) M A) %<- (eof M A) %<- (eof N B) %. +%term equiv/beta2 %pi (equiv (epi2 (epair M N)) N B) %<- (eof M A) %<- (eof N B) %. +%%% subtyping +%term subtp/reflex %pi (subtp A B) %<- (tequiv A B) %. +%term subtp/trans %pi (subtp A C) %<- (subtp A B) %<- (subtp B C) %. +%term subtp/sing_t %pi (subtp (esing M) et) %<- (eof M et) %. +%term subtp/pi + %pi (subtp (epi A B) (epi A' B')) + %<- (subtp A' A) + %<- ({x} %pi (evof x A') %-> (subtp (B x) (B' x))) + %<- ({x} %pi (evof x A) %-> (ewf (B x))) %. +%term subtp/sigma + %pi (subtp (esigma A B) (esigma A' B')) + %<- (subtp A A') + %<- ({x} %pi (evof x A) %-> (subtp (B x) (B' x))) + %<- ({x} %pi (evof x A') %-> (ewf (B' x))) %. +%%% type equivalence +%term tequiv/reflex %pi (tequiv A A) %<- (ewf A) %. +%term tequiv/symm %pi (tequiv A B) %<- (tequiv B A) %. +%term tequiv/trans %pi (tequiv A C) %<- (tequiv A B) %<- (tequiv B C) %. +%term tequiv/pi + %pi (tequiv (epi A B) (epi A' B')) + %<- (tequiv A A') + %<- ({x} %pi (evof x A) %-> (tequiv (B x) (B' x))) %. +%term tequiv/sigma + %pi (tequiv (esigma A B) (esigma A' B')) + %<- (tequiv A A') + %<- ({x} %pi (evof x A) %-> (tequiv (B x) (B' x))) %. +%term tequiv/sing %pi (tequiv (esing M) (esing M')) %<- (equiv M M' et) %. +%%% derived rules +%define eof/equiv (%pi (eof M B) %<- (eof M A) %<- (tequiv A B)) [d2] [d1] eof/subsume (subtp/reflex d2) d1 %. +%define equiv/equiv (%pi (equiv M N B) %<- (equiv M N A) %<- (tequiv A B)) [d2] [d1] equiv/subsume (subtp/reflex d2) d1 %. +%define subtp/t (subtp et et) subtp/reflex (tequiv/reflex ewf/t) %. +%define subtp/sing (%pi (subtp (esing M) (esing M')) %<- (equiv M M' et)) [d] subtp/reflex (tequiv/sing d) %. +%define tequiv/t (tequiv et et) tequiv/reflex ewf/t %. +%%%%% Constants %%%%% +%sort etopen %. +%mode etopen %in %out %. +%term etopen/t etopen ct et %. +%term etopen/pi %pi (etopen (cpi Ac Bc) (epi A ([_] B))) %<- (etopen Ac A) %<- (etopen Bc B) %. +%term ekof/i %pi (ekof K A') %<- (ckof K A) %<- (etopen A A') %. +%%%%% Worlds %%%%% +%block evar [ex eterm]%. +%block ebind {ea etp} [ex eterm] [ed evof ex ea]%. +%block eofblock {ex eterm} {ea etp} [ed evof ex ea]%. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/equality-thm.lf b/new-tests/stelf-output/tslf/sing/equality-thm.lf new file mode 100644 index 0000000..dbb20ca --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/equality-thm.lf @@ -0,0 +1,723 @@ +%%%%% Extra Worlds %%%%% +%worlds () (constant) %. +%worlds (evar) (etp) (eterm) %. +%%%%% Equalities %%%%% +%sort atom-eq %. +%term atom-eq/i atom-eq R R %. +%sort constant-eq %. +%term constant-eq/i constant-eq C C %. +%sort ctp-eq %. +%term ctp-eq/i ctp-eq A A %. +%sort ctx-eq %. +%term ctx-eq/i ctx-eq G G %. +%sort eterm-eq %. +%term eterm-eq/i eterm-eq M M %. +%sort etp-eq %. +%term etp-eq/i etp-eq A A %. +%sort sctx-eq %. +%term sctx-eq/i sctx-eq G G %. +%sort stp-eq %. +%term stp-eq/i stp-eq T T %. +%sort term-eq %. +%term term-eq/i term-eq M M %. +%sort tp-eq %. +%term tp-eq/i tp-eq T T %. +%%%%% A Closed Atom %%%%% +%define aca atom const unit %. +%%%%% Symmetry %%%%% +%sort atom-eq-symm %. +%mode atom-eq-symm %in %out %. +%term _ atom-eq-symm atom-eq/i atom-eq/i %. +%worlds (var) (atom-eq-symm _ _) %. +%total {} (atom-eq-symm _ _) %. +%sort eterm-eq-symm %. +%mode eterm-eq-symm %in %out %. +%term _ eterm-eq-symm eterm-eq/i eterm-eq/i %. +%worlds (evar) (eterm-eq-symm _ _) %. +%total {} (eterm-eq-symm _ _) %. +%sort etp-eq-symm %. +%mode etp-eq-symm %in %out %. +%term _ etp-eq-symm etp-eq/i etp-eq/i %. +%worlds (evar) (etp-eq-symm _ _) %. +%total {} (etp-eq-symm _ _) %. +%sort term-eq-symm %. +%mode term-eq-symm %in %out %. +%term _ term-eq-symm term-eq/i term-eq/i %. +%worlds (var) (term-eq-symm _ _) %. +%total {} (term-eq-symm _ _) %. +%sort tp-eq-symm %. +%mode tp-eq-symm %in %out %. +%term _ tp-eq-symm tp-eq/i tp-eq/i %. +%worlds (var) (tp-eq-symm _ _) %. +%total {} (tp-eq-symm _ _) %. +%%%%% Transitivity %%%%% +%sort atom-eq-trans %. +%mode atom-eq-trans %in %in %out %. +%term _ atom-eq-trans atom-eq/i atom-eq/i atom-eq/i %. +%worlds (var) (atom-eq-trans _ _ _) %. +%total {} (atom-eq-trans _ _ _) %. +%sort stp-eq-trans %. +%mode stp-eq-trans %in %in %out %. +%term _ stp-eq-trans stp-eq/i stp-eq/i stp-eq/i %. +%worlds () (stp-eq-trans _ _ _) %. +%total {} (stp-eq-trans _ _ _) %. +%sort term-eq-trans %. +%mode term-eq-trans %in %in %out %. +%term _ term-eq-trans term-eq/i term-eq/i term-eq/i %. +%worlds (var) (term-eq-trans _ _ _) %. +%total {} (term-eq-trans _ _ _) %. +%sort tp-eq-trans %. +%mode tp-eq-trans %in %in %out %. +%term _ tp-eq-trans tp-eq/i tp-eq/i tp-eq/i %. +%worlds (var) (tp-eq-trans _ _ _) %. +%total {} (tp-eq-trans _ _ _) %. +%%%%% Cons Lemmas %%%%% +%sort app-resp %. +%mode app-resp %in %in %out %. +%term _ app-resp atom-eq/i term-eq/i atom-eq/i %. +%worlds (var) (app-resp _ _ _) %. +%total {} (app-resp _ _ _) %. +%sort atom-resp-atom {R %pi atom %-> atom} %. +%mode atom-resp-atom %in %in %out %. +%term _ atom-resp-atom _ atom-eq/i atom-eq/i %. +%worlds (var) (atom-resp-atom _ _ _) %. +%total {} (atom-resp-atom _ _ _) %. +%sort atom-resp-term {R %pi term %-> atom} %. +%mode atom-resp-term %in %in %out %. +%term _ atom-resp-term _ term-eq/i atom-eq/i %. +%worlds (var) (atom-resp-term _ _ _) %. +%total {} (atom-resp-term _ _ _) %. +%sort atom-resp-tp {R %pi tp %-> atom} %. +%mode atom-resp-tp %in %in %out %. +%term _ atom-resp-tp _ tp-eq/i atom-eq/i %. +%worlds (var) (atom-resp-tp _ _ _) %. +%total {} (atom-resp-tp _ _ _) %. +%sort cons-resp %. +%mode cons-resp %in %in %in %out %. +%term _ cons-resp ctx-eq/i atom-eq/i tp-eq/i ctx-eq/i %. +%worlds (var) (cons-resp _ _ _ _) %. +%total {} (cons-resp _ _ _ _) %. +%sort elam-resp %. +%mode elam-resp %in %in %out %. +%term _ elam-resp etp-eq/i ([_] eterm-eq/i) eterm-eq/i %. +%worlds (evar) (elam-resp _ _ _) %. +%total {} (elam-resp _ _ _) %. +%sort epair-resp %. +%mode epair-resp %in %in %out %. +%term _ epair-resp eterm-eq/i eterm-eq/i eterm-eq/i %. +%worlds (evar) (epair-resp _ _ _) %. +%total {} (epair-resp _ _ _) %. +%sort epi-resp %. +%mode epi-resp %in %in %out %. +%term _ epi-resp etp-eq/i ([_] etp-eq/i) etp-eq/i %. +%worlds (evar) (epi-resp _ _ _) %. +%total {} (epi-resp _ _ _) %. +%sort equiv-resp %. +%mode equiv-resp %in %in %in %in %out %. +%term _ equiv-resp eterm-eq/i eterm-eq/i etp-eq/i D D %. +%worlds (evar ebind eofblock) (equiv-resp _ _ _ _ _) %. +%total {} (equiv-resp _ _ _ _ _) %. +%sort esigma-resp %. +%mode esigma-resp %in %in %out %. +%term _ esigma-resp etp-eq/i ([_] etp-eq/i) etp-eq/i %. +%worlds (evar) (esigma-resp _ _ _) %. +%total {} (esigma-resp _ _ _) %. +%sort eterm-resp-bind {M %pi etp %-> (%pi eterm %-> eterm) %-> eterm} %. +%mode eterm-resp-bind %in %in %in %out %. +%term _ eterm-resp-bind _ etp-eq/i ([_] eterm-eq/i) eterm-eq/i %. +%worlds (evar) (eterm-resp-bind _ _ _ _) %. +%total {} (eterm-resp-bind _ _ _ _) %. +%sort eterm-resp-eterm {M %pi eterm %-> eterm} %. +%mode eterm-resp-eterm %in %in %out %. +%term _ eterm-resp-eterm _ eterm-eq/i eterm-eq/i %. +%worlds (evar) (eterm-resp-eterm _ _ _) %. +%total {} (eterm-resp-eterm _ _ _) %. +%sort eterm-resp-eterm2 {M %pi eterm %-> eterm %-> eterm} %. +%mode eterm-resp-eterm2 %in %in %in %out %. +%term _ eterm-resp-eterm2 _ eterm-eq/i eterm-eq/i eterm-eq/i %. +%worlds (evar) (eterm-resp-eterm2 _ _ _ _) %. +%total {} (eterm-resp-eterm2 _ _ _ _) %. +%sort eterm-resp-eterm-fun {x} %. +%mode eterm-resp-eterm-fun %in %in %out %. +%term _ eterm-resp-eterm-fun ([_] eterm-eq/i) eterm-eq/i eterm-eq/i %. +%worlds (evar) (eterm-resp-eterm-fun _ _ _) %. +%total {} (eterm-resp-eterm-fun _ _ _) %. +%sort eterm-resp-etp {M %pi etp %-> eterm} %. +%mode eterm-resp-etp %in %in %out %. +%term _ eterm-resp-etp _ etp-eq/i eterm-eq/i %. +%worlds (evar) (eterm-resp-etp _ _ _) %. +%total {} (eterm-resp-etp _ _ _) %. +%sort eterm-resp-halfbind {M %pi (%pi eterm %-> eterm) %-> eterm} {x} %. +%mode eterm-resp-halfbind %in %in %out %. +%term _ eterm-resp-halfbind _ ([_] eterm-eq/i) eterm-eq/i %. +%worlds (evar) (eterm-resp-halfbind _ _ _) %. +%total {} (eterm-resp-halfbind _ _ _) %. +%sort etp-resp-bind {A %pi etp %-> (%pi eterm %-> etp) %-> etp} %. +%mode etp-resp-bind %in %in %in %out %. +%term _ etp-resp-bind _ etp-eq/i ([_] etp-eq/i) etp-eq/i %. +%worlds (evar) (etp-resp-bind _ _ _ _) %. +%total {} (etp-resp-bind _ _ _ _) %. +%sort etp-resp-eterm {A %pi eterm %-> etp} %. +%mode etp-resp-eterm %in %in %out %. +%term _ etp-resp-eterm _ eterm-eq/i etp-eq/i %. +%worlds (evar) (etp-resp-eterm _ _ _) %. +%total {} (etp-resp-eterm _ _ _) %. +%sort etp-resp-eterm-fun {x} %. +%mode etp-resp-eterm-fun %in %in %out %. +%term _ etp-resp-eterm-fun ([_] etp-eq/i) eterm-eq/i etp-eq/i %. +%worlds (evar) (etp-resp-eterm-fun _ _ _) %. +%total {} (etp-resp-eterm-fun _ _ _) %. +%sort etp2-resp-eterm {A %pi (%pi eterm %-> eterm) %-> etp} {x} %. +%mode etp2-resp-eterm %in %in %out %. +%term _ etp2-resp-eterm _ ([_] eterm-eq/i) etp-eq/i %. +%worlds (evar) (etp2-resp-eterm _ _ _) %. +%total {} (etp2-resp-eterm _ _ _) %. +%sort etp-resp-etp {A %pi etp %-> etp} %. +%mode etp-resp-etp %in %in %out %. +%term _ etp-resp-etp _ etp-eq/i etp-eq/i %. +%worlds (evar) (etp-resp-etp _ _ _) %. +%total {} (etp-resp-etp _ _ _) %. +%sort lam-resp {x} %. +%mode lam-resp %in %out %. +%term _ lam-resp ([_] term-eq/i) term-eq/i %. +%worlds (var) (lam-resp _ _) %. +%total {} (lam-resp _ _) %. +%sort pair-resp %. +%mode pair-resp %in %in %out %. +%term _ pair-resp term-eq/i term-eq/i term-eq/i %. +%worlds (var) (pair-resp _ _ _) %. +%total {} (pair-resp _ _ _) %. +%sort pi-resp %. +%mode pi-resp %in %in %out %. +%term _ pi-resp tp-eq/i ([_] tp-eq/i) tp-eq/i %. +%worlds (var) (pi-resp _ _ _) %. +%total {} (pi-resp _ _ _) %. +%sort scons-resp %. +%mode scons-resp %in %in %in %out %. +%term _ scons-resp sctx-eq/i atom-eq/i stp-eq/i sctx-eq/i %. +%worlds (var ovar bind) (scons-resp _ _ _ _) %. +%total {} (scons-resp _ _ _ _) %. +%sort sigma-resp %. +%mode sigma-resp %in %in %out %. +%term _ sigma-resp tp-eq/i ([_] tp-eq/i) tp-eq/i %. +%worlds (var) (sigma-resp _ _ _) %. +%total {} (sigma-resp _ _ _) %. +%sort stp-resp-stp {T %pi stp %-> stp} %. +%mode stp-resp-stp %in %in %out %. +%term _ stp-resp-stp _ stp-eq/i stp-eq/i %. +%worlds () (stp-resp-stp _ _ _) %. +%total {} (stp-resp-stp _ _ _) %. +%sort term-resp-atom {M %pi atom %-> term} %. +%mode term-resp-atom %in %in %out %. +%term _ term-resp-atom _ atom-eq/i term-eq/i %. +%worlds (var) (term-resp-atom _ _ _) %. +%total {} (term-resp-atom _ _ _) %. +%sort term-resp-term {M %pi term %-> term} %. +%mode term-resp-term %in %in %out %. +%term _ term-resp-term _ term-eq/i term-eq/i %. +%worlds (var) (term-resp-term _ _ _) %. +%total {} (term-resp-term _ _ _) %. +%sort topen-resp %. +%mode topen-resp %in %in %in %out %. +%term _ topen-resp ctp-eq/i tp-eq/i D D %. +%worlds (var) (topen-resp _ _ _ _) %. +%total {} (topen-resp _ _ _ _) %. +%sort tp-resp-atom {A %pi atom %-> tp} %. +%mode tp-resp-atom %in %in %out %. +%term _ tp-resp-atom _ atom-eq/i tp-eq/i %. +%worlds (var) (tp-resp-atom _ _ _) %. +%total {} (tp-resp-atom _ _ _) %. +%sort tp-resp-term {A %pi term %-> tp} %. +%mode tp-resp-term %in %in %out %. +%term _ tp-resp-term _ term-eq/i tp-eq/i %. +%worlds (var) (tp-resp-term _ _ _) %. +%total {} (tp-resp-term _ _ _) %. +%sort tp-resp-tp {A %pi tp %-> tp} %. +%mode tp-resp-tp %in %in %out %. +%term _ tp-resp-tp _ tp-eq/i tp-eq/i %. +%worlds (var) (tp-resp-tp _ _ _) %. +%total {} (tp-resp-tp _ _ _) %. +%sort tp-resp-atom-fun {x} %. +%mode tp-resp-atom-fun %in %in %out %. +%term _ tp-resp-atom-fun ([_] tp-eq/i) atom-eq/i tp-eq/i %. +%worlds (var) (tp-resp-atom-fun _ _ _) %. +%total {} (tp-resp-atom-fun _ _ _) %. +%%%%% Cdr Lemmas %%%%% +%sort atom-eq-cdr-app %. +%mode atom-eq-cdr-app %in %out %out %. +%term _ atom-eq-cdr-app atom-eq/i atom-eq/i term-eq/i %. +%worlds (var) (atom-eq-cdr-app _ _ _) %. +%total {} (atom-eq-cdr-app _ _ _) %. +%sort atom-eq-cdr-const %. +%mode atom-eq-cdr-const %in %out %. +%term _ atom-eq-cdr-const atom-eq/i constant-eq/i %. +%worlds (var) (atom-eq-cdr-const _ _) %. +%total {} (atom-eq-cdr-const _ _) %. +%sort atom-eq-cdr-forall %. +%mode atom-eq-cdr-forall %in %out %. +%term _ atom-eq-cdr-forall atom-eq/i tp-eq/i %. +%worlds (var) (atom-eq-cdr-forall _ _) %. +%total {} (atom-eq-cdr-forall _ _) %. +%sort etp-eq-cdr-epi %. +%mode etp-eq-cdr-epi %in %out %out %. +%term _ etp-eq-cdr-epi etp-eq/i etp-eq/i ([_] etp-eq/i) %. +%worlds (evar) (etp-eq-cdr-epi _ _ _) %. +%total {} (etp-eq-cdr-epi _ _ _) %. +%sort etp-eq-cdr-esigma %. +%mode etp-eq-cdr-esigma %in %out %out %. +%term _ etp-eq-cdr-esigma etp-eq/i etp-eq/i ([_] etp-eq/i) %. +%worlds (evar) (etp-eq-cdr-esigma _ _ _) %. +%total {} (etp-eq-cdr-esigma _ _ _) %. +%sort term-eq-cdr-at %. +%mode term-eq-cdr-at %in %out %. +%term _ term-eq-cdr-at term-eq/i atom-eq/i %. +%worlds (var) (term-eq-cdr-at _ _) %. +%total {} (term-eq-cdr-at _ _) %. +%sort term-eq-cdr-lam %. +%mode term-eq-cdr-lam %in %out %. +%term _ term-eq-cdr-lam term-eq/i ([_] term-eq/i) %. +%worlds (var) (term-eq-cdr-lam _ _) %. +%total {} (term-eq-cdr-lam _ _) %. +%sort term-eq-cdr-pair %. +%mode term-eq-cdr-pair %in %out %out %. +%term _ term-eq-cdr-pair term-eq/i term-eq/i term-eq/i %. +%worlds (var) (term-eq-cdr-pair _ _ _) %. +%total {} (term-eq-cdr-pair _ _ _) %. +%sort tp-eq-cdr-pi %. +%mode tp-eq-cdr-pi %in %out %out %. +%term _ tp-eq-cdr-pi tp-eq/i tp-eq/i ([_] tp-eq/i) %. +%worlds (var) (tp-eq-cdr-pi _ _ _) %. +%total {} (tp-eq-cdr-pi _ _ _) %. +%sort tp-eq-cdr-sigma %. +%mode tp-eq-cdr-sigma %in %out %out %. +%term _ tp-eq-cdr-sigma tp-eq/i tp-eq/i ([_] tp-eq/i) %. +%worlds (var) (tp-eq-cdr-sigma _ _ _) %. +%total {} (tp-eq-cdr-sigma _ _ _) %. +%sort tp-eq-pi-strengthen {x atom} %. +%mode tp-eq-pi-strengthen %in %out %. +%term _ tp-eq-pi-strengthen ([x] tp-eq/i) tp-eq/i %. +%worlds (var) (tp-eq-pi-strengthen _ _) %. +%total {} (tp-eq-pi-strengthen _ _) %. +%sort tp-eq-sigma-strengthen {x atom} %. +%mode tp-eq-sigma-strengthen %in %out %. +%term _ tp-eq-sigma-strengthen ([x] tp-eq/i) tp-eq/i %. +%worlds (var) (tp-eq-sigma-strengthen _ _) %. +%total {} (tp-eq-sigma-strengthen _ _) %. +%%%%% Equality Respection %%%%% +%sort aasub-resp {x} %. +%mode aasub-resp %in %in %in %in %out %. +%term _ aasub-resp ([_] atom-eq/i) term-eq/i atom-eq/i D D %. +%worlds (var) (aasub-resp _ _ _ _ _) %. +%total {} (aasub-resp _ _ _ _ _) %. +%sort aconvert-resp %. +%mode aconvert-resp %in %in %in %in %out %. +%term _ aconvert-resp atom-eq/i tp-eq/i eterm-eq/i D D %. +%worlds (bind tbind var ofblock) (aconvert-resp _ _ _ _ _) %. +%total {} (aconvert-resp _ _ _ _ _) %. +%sort aconverte-resp %. +%mode aconverte-resp %in %in %in %in %in %out %. +%term _ aconverte-resp ctx-eq/i atom-eq/i tp-eq/i eterm-eq/i D D %. +%worlds (var bind tbind ovar evar) (aconverte-resp _ _ _ _ _ _) %. +%total {} (aconverte-resp _ _ _ _ _ _) %. +%sort aof-resp %. +%mode aof-resp %in %in %in %out %. +%term _ aof-resp atom-eq/i tp-eq/i D D %. +%worlds (var bind) (aof-resp _ _ _ _) %. +%total {} (aof-resp _ _ _ _) %. +%sort aofe-resp %. +%mode aofe-resp %in %in %in %in %out %. +%term _ aofe-resp ctx-eq/i atom-eq/i tp-eq/i D D %. +%worlds (bind ovar var) (aofe-resp _ _ _ _ _) %. +%total {} (aofe-resp _ _ _ _ _) %. +%sort aofes-resp %. +%mode aofes-resp %in %in %in %in %out %. +%term _ aofes-resp sctx-eq/i atom-eq/i stp-eq/i D D %. +%worlds (var bind ovar) (aofes-resp _ _ _ _ _) %. +%total {} (aofes-resp _ _ _ _ _) %. +%sort aosub-resp {x} %. +%mode aosub-resp %in %in %in %in %out %. +%term _ aosub-resp ([_] atom-eq/i) term-eq/i term-eq/i D D %. +%worlds (var) (aosub-resp _ _ _ _ _) %. +%total {} (aosub-resp _ _ _ _ _) %. +%sort canonize-resp %. +%mode canonize-resp %in %in %in %out %. +%term _ canonize-resp eterm-eq/i etp-eq/i D D %. +%worlds (evar evvar evbind bind tbind eofblock evblock) (canonize-resp _ _ _ _) %. +%total {} (canonize-resp _ _ _ _) %. +%sort canonize-resp-underbind %. +%mode canonize-resp-underbind %in %in %out %. +%term _ canonize-resp-underbind etp-eq/i D D %. +%worlds (evar evvar evbind bind tbind eofblock evblock) (canonize-resp-underbind _ _ _) %. +%total {} (canonize-resp-underbind _ _ _) %. +%sort coerce-resp %. +%mode coerce-resp %in %in %in %in %out %. +%term _ coerce-resp etp-eq/i etp-eq/i ([_] eterm-eq/i) D D %. +%worlds (evar evvar evbind bind tbind eofblock evblock) (coerce-resp _ _ _ _ _) %. +%total {} (coerce-resp _ _ _ _ _) %. +%sort convert-resp %. +%mode convert-resp %in %in %in %in %out %. +%term _ convert-resp term-eq/i tp-eq/i eterm-eq/i D D %. +%worlds (bind tbind) (convert-resp _ _ _ _ _) %. +%total {} (convert-resp _ _ _ _ _) %. +%sort convert-resp-underbind %. +%mode convert-resp-underbind %in %in %out %. +%term _ convert-resp-underbind tp-eq/i D D %. +%worlds (bind tbind) (convert-resp-underbind _ _ _) %. +%total {} (convert-resp-underbind _ _ _) %. +%sort converte-resp %. +%mode converte-resp %in %in %in %in %in %out %. +%term _ converte-resp ctx-eq/i term-eq/i tp-eq/i eterm-eq/i D D %. +%worlds (var bind tbind ovar evar) (converte-resp _ _ _ _ _ _) %. +%total {} (converte-resp _ _ _ _ _ _) %. +%sort ekof-resp %. +%mode ekof-resp %in %in %out %. +%term _ ekof-resp etp-eq/i D D %. +%worlds (evar) (ekof-resp _ _ _) %. +%total {} (ekof-resp _ _ _) %. +%sort eof-resp %. +%mode eof-resp %in %in %in %out %. +%term _ eof-resp eterm-eq/i etp-eq/i D D %. +%worlds (evar eofblock ebind) (eof-resp _ _ _ _) %. +%total {} (eof-resp _ _ _ _) %. +%sort etopen-resp %. +%mode etopen-resp %in %in %in %out %. +%term _ etopen-resp ctp-eq/i etp-eq/i D D %. +%worlds (evar) (etopen-resp _ _ _ _) %. +%total {} (etopen-resp _ _ _ _) %. +%sort evof-resp %. +%mode evof-resp %in %in %in %out %. +%term _ evof-resp eterm-eq/i etp-eq/i D D %. +%worlds (evar eofblock) (evof-resp _ _ _ _) %. +%total {} (evof-resp _ _ _ _) %. +%sort ewf-resp %. +%mode ewf-resp %in %in %out %. +%term _ ewf-resp etp-eq/i D D %. +%worlds (evar eofblock) (ewf-resp _ _ _) %. +%total {} (ewf-resp _ _ _) %. +%sort expand-resp %. +%mode expand-resp %in %in %in %in %out %. +%term _ expand-resp atom-eq/i tp-eq/i term-eq/i D D %. +%worlds (var) (expand-resp _ _ _ _ _) %. +%total {} (expand-resp _ _ _ _ _) %. +%reduces = D D' (expand-resp _ _ _ D D') %. +%sort kof-resp %. +%mode kof-resp %in %in %out %. +%term _ kof-resp tp-eq/i D D %. +%worlds (var) (kof-resp _ _ _) %. +%total {} (kof-resp _ _ _) %. +%sort lookup-resp %. +%mode lookup-resp %in %in %in %in %out %. +%term _ lookup-resp ctx-eq/i atom-eq/i tp-eq/i D D %. +%worlds (var bind ovar) (lookup-resp _ _ _ _ _) %. +%total {} (lookup-resp _ _ _ _ _) %. +%sort of-resp %. +%mode of-resp %in %in %in %out %. +%term _ of-resp term-eq/i tp-eq/i D D %. +%worlds (var bind) (of-resp _ _ _ _) %. +%total {} (of-resp _ _ _ _) %. +%sort of-bind-resp %. +%mode of-bind-resp %in %in %out %. +%term _ of-bind-resp tp-eq/i D D %. +%worlds (var bind) (of-bind-resp _ _ _) %. +%total {} (of-bind-resp _ _ _) %. +%reduces = D D' (of-bind-resp _ D D') %. +%sort of-bind-resp1 {y atom} %. +%mode of-bind-resp1 %in %in %out %. +%term _ of-bind-resp1 ([_] tp-eq/i) D D %. +%worlds (var bind) (of-bind-resp1 _ _ _) %. +%total {} (of-bind-resp1 _ _ _) %. +%reduces = D D' (of-bind-resp1 _ D D') %. +%sort ofe-resp %. +%mode ofe-resp %in %in %in %in %out %. +%term _ ofe-resp ctx-eq/i term-eq/i tp-eq/i D D %. +%worlds (bind ovar var) (ofe-resp _ _ _ _ _) %. +%total {} (ofe-resp _ _ _ _ _) %. +%sort ofes-resp %. +%mode ofes-resp %in %in %in %in %out %. +%term _ ofes-resp sctx-eq/i term-eq/i stp-eq/i D D %. +%worlds (var ovar bind) (ofes-resp _ _ _ _ _) %. +%total {} (ofes-resp _ _ _ _ _) %. +%sort principal-resp %. +%mode principal-resp %in %in %out %. +%term _ principal-resp tp-eq/i D D %. +%worlds (var) (principal-resp _ _ _) %. +%total {} (principal-resp _ _ _) %. +%sort reduce-resp %. +%mode reduce-resp %in %in %in %out %. +%term _ reduce-resp eterm-eq/i eterm-eq/i D D %. +%worlds (bind tbind ovar eofblock evblock evar) (reduce-resp _ _ _ _) %. +%total {} (reduce-resp _ _ _ _) %. +%sort reduce-app-resp %. +%mode reduce-app-resp %in %in %in %in %out %. +%term _ reduce-app-resp eterm-eq/i eterm-eq/i eterm-eq/i D D %. +%worlds (evvar evbind bind tbind evblock) (reduce-app-resp _ _ _ _ _) %. +%total {} (reduce-app-resp _ _ _ _ _) %. +%sort reduce-pi1-resp %. +%mode reduce-pi1-resp %in %in %in %out %. +%term _ reduce-pi1-resp eterm-eq/i eterm-eq/i D D %. +%worlds (evvar evbind bind tbind evblock) (reduce-pi1-resp _ _ _ _) %. +%total {} (reduce-pi1-resp _ _ _ _) %. +%sort reduce-pi2-resp %. +%mode reduce-pi2-resp %in %in %in %out %. +%term _ reduce-pi2-resp eterm-eq/i eterm-eq/i D D %. +%worlds (evvar evbind bind tbind evblock) (reduce-pi2-resp _ _ _ _) %. +%total {} (reduce-pi2-resp _ _ _ _) %. +%sort self-resp %. +%mode self-resp %in %in %in %in %out %. +%term _ self-resp term-eq/i tp-eq/i tp-eq/i D D %. +%worlds (var bind) (self-resp _ _ _ _ _) %. +%total {} (self-resp _ _ _ _ _) %. +%sort selfify-resp %. +%mode selfify-resp %in %in %in %in %in %out %. +%term _ selfify-resp eterm-eq/i etp-eq/i eterm-eq/i etp-eq/i D D %. +%worlds (evar) (selfify-resp _ _ _ _ _ _) %. +%total {} (selfify-resp _ _ _ _ _ _) %. +%sort simp-resp %. +%mode simp-resp %in %in %in %out %. +%term _ simp-resp tp-eq/i stp-eq/i D D %. +%worlds (var) (simp-resp _ _ _ _) %. +%total {} (simp-resp _ _ _ _) %. +%sort slookup-resp %. +%mode slookup-resp %in %in %in %in %out %. +%term _ slookup-resp sctx-eq/i atom-eq/i stp-eq/i D D %. +%worlds (var ovar) (slookup-resp _ _ _ _ _) %. +%total {} (slookup-resp _ _ _ _ _) %. +%sort stp-leq-resp %. +%mode stp-leq-resp %in %in %in %out %. +%term _ stp-leq-resp stp-eq/i stp-eq/i D D %. +%worlds () (stp-leq-resp _ _ _ _) %. +%total {} (stp-leq-resp _ _ _ _) %. +%sort sub-resp {x} %. +%mode sub-resp %in %in %in %in %out %. +%term _ sub-resp ([_] term-eq/i) term-eq/i term-eq/i D D %. +%worlds (var) (sub-resp _ _ _ _ _) %. +%total {} (sub-resp _ _ _ _ _) %. +%reduces = D D' (sub-resp _ _ _ D D') %. +%sort sub-resp1 {x atom} {y} %. +%mode sub-resp1 %in %in %in %in %out %. +%term _ sub-resp1 ([_] [_] term-eq/i) ([_] term-eq/i) ([_] term-eq/i) D D %. +%worlds (var) (sub-resp1 _ _ _ _ _) %. +%total {} (sub-resp1 _ _ _ _ _) %. +%reduces = D D' (sub-resp1 _ _ _ D D') %. +%sort sub-resp-strengthen {y atom} {x} %. +%mode sub-resp-strengthen %in %in %in %in %out %. +%term _ sub-resp-strengthen ([_] [_] term-eq/i) ([_] term-eq/i) ([_] term-eq/i) Dsub (Dsub aca) %. +%worlds (var) (sub-resp-strengthen _ _ _ _ _) %. +%total {} (sub-resp-strengthen _ _ _ _ _) %. +%sort subtp-resp %. +%mode subtp-resp %in %in %in %out %. +%term _ subtp-resp etp-eq/i etp-eq/i D D %. +%worlds (evar eofblock) (subtp-resp _ _ _ _) %. +%total {} (subtp-resp _ _ _ _) %. +%sort subtype-resp %. +%mode subtype-resp %in %in %in %in %out %. +%term _ subtype-resp tp-eq/i tp-eq/i ([_] term-eq/i) D D %. +%worlds (var bind ovar) (subtype-resp _ _ _ _ _) %. +%total {} (subtype-resp _ _ _ _ _) %. +%sort sum-resp %. +%mode sum-resp %in %in %in %in %out %. +%term _ sum-resp nat-eq/i nat-eq/i nat-eq/i D D %. +%worlds () (sum-resp _ _ _ _ _) %. +%total {} (sum-resp _ _ _ _ _) %. +%sort tcanonize-resp %. +%mode tcanonize-resp %in %in %in %out %. +%term _ tcanonize-resp etp-eq/i etp-eq/i D D %. +%worlds (evar evvar evbind bind tbind eofblock evblock) (tcanonize-resp _ _ _ _) %. +%total {} (tcanonize-resp _ _ _ _) %. +%sort tcanonize-resp-underbind %. +%mode tcanonize-resp-underbind %in %in %out %. +%term _ tcanonize-resp-underbind etp-eq/i D D %. +%worlds (evar evvar evbind bind tbind eofblock evblock) (tcanonize-resp-underbind _ _ _) %. +%total {} (tcanonize-resp-underbind _ _ _) %. +%sort tconvert-resp %. +%mode tconvert-resp %in %in %in %out %. +%term _ tconvert-resp tp-eq/i etp-eq/i D D %. +%worlds (bind tbind) (tconvert-resp _ _ _ _) %. +%total {} (tconvert-resp _ _ _ _) %. +%sort tconvert-resp-underbind %. +%mode tconvert-resp-underbind %in %in %out %. +%term _ tconvert-resp-underbind tp-eq/i D D %. +%worlds (bind tbind) (tconvert-resp-underbind _ _ _) %. +%total {} (tconvert-resp-underbind _ _ _) %. +%sort tequiv-resp %. +%mode tequiv-resp %in %in %in %out %. +%term _ tequiv-resp etp-eq/i etp-eq/i D D %. +%worlds (evar eofblock) (tequiv-resp _ _ _ _) %. +%total {} (tequiv-resp _ _ _ _) %. +%reduces = D D' (tequiv-resp _ _ D D') %. +%sort trans-resp %. +%mode trans-resp %in %in %in %out %. +%term _ trans-resp eterm-eq/i tp-eq/i D D %. +%worlds (var bind ovar tbind) (trans-resp _ _ _ _) %. +%total {} (trans-resp _ _ _ _) %. +%sort trans-resp-underbind %. +%mode trans-resp-underbind %in %in %out %. +%term _ trans-resp-underbind tp-eq/i D D %. +%worlds (var bind tbind) (trans-resp-underbind _ _ _) %. +%total {} (trans-resp-underbind _ _ _) %. +%sort transe-resp %. +%mode transe-resp %in %in %in %in %out %. +%term _ transe-resp ctx-eq/i eterm-eq/i tp-eq/i D D %. +%worlds (bind ovar tbind) (transe-resp _ _ _ _ _) %. +%total {} (transe-resp _ _ _ _ _) %. +%sort treduce-resp %. +%mode treduce-resp %in %in %in %out %. +%term _ treduce-resp etp-eq/i etp-eq/i D D %. +%worlds (evar evvar evbind bind tbind evblock) (treduce-resp _ _ _ _) %. +%total {} (treduce-resp _ _ _ _) %. +%sort tsub-resp {x} %. +%mode tsub-resp %in %in %in %in %out %. +%term _ tsub-resp ([_] tp-eq/i) term-eq/i tp-eq/i D D %. +%worlds (var) (tsub-resp _ _ _ _ _) %. +%total {} (tsub-resp _ _ _ _ _) %. +%sort ttrans-resp %. +%mode ttrans-resp %in %in %in %out %. +%term _ ttrans-resp etp-eq/i tp-eq/i D D %. +%worlds (var bind tbind) (ttrans-resp _ _ _ _) %. +%total {} (ttrans-resp _ _ _ _) %. +%sort ttrans-resp-underbind %. +%mode ttrans-resp-underbind %in %in %out %. +%term _ ttrans-resp-underbind tp-eq/i D D %. +%worlds (var bind tbind) (ttrans-resp-underbind _ _ _) %. +%total {} (ttrans-resp-underbind _ _ _) %. +%sort vof-resp %. +%mode vof-resp %in %in %in %out %. +%term _ vof-resp atom-eq/i tp-eq/i D D %. +%worlds (var bind) (vof-resp _ _ _ _) %. +%total {} (vof-resp _ _ _ _) %. +%sort vtrans-resp %. +%mode vtrans-resp %in %in %in %out %. +%term _ vtrans-resp eterm-eq/i atom-eq/i D D %. +%worlds (var evar tbind) (vtrans-resp _ _ _ _) %. +%total {} (vtrans-resp _ _ _ _) %. +%sort wf-bind-resp1 {y atom} %. +%mode wf-bind-resp1 %in %in %out %. +%term _ wf-bind-resp1 ([_] tp-eq/i) D D %. +%worlds (var bind) (wf-bind-resp1 _ _ _) %. +%total {} (wf-bind-resp1 _ _ _) %. +%reduces = D D' (wf-bind-resp1 _ D D') %. +%sort wf-resp %. +%mode wf-resp %in %in %out %. +%term _ wf-resp tp-eq/i D D %. +%worlds (var bind) (wf-resp _ _ _) %. +%total {} (wf-resp _ _ _) %. +%reduces = D D' (wf-resp _ D D') %. +%sort wfe-resp %. +%mode wfe-resp %in %in %in %out %. +%term _ wfe-resp ctx-eq/i tp-eq/i D D %. +%worlds (var bind ovar) (wfe-resp _ _ _ _) %. +%total {} (wfe-resp _ _ _ _) %. +%sort wfes-resp %. +%mode wfes-resp %in %in %in %out %. +%term _ wfes-resp sctx-eq/i tp-eq/i D D %. +%worlds (var ovar bind) (wfes-resp _ _ _ _) %. +%total {} (wfes-resp _ _ _ _) %. +%%%%% Reductio Ad Absurdio %%%%% +%sort false-implies-aconverte %. +%mode {%in G ctx} {%in R atom} {%in A tp} {%in EM eterm} {%in X1 false} {%out X2 aconverte G R A EM} false-implies-aconverte X1 X2 %. +%worlds (var bind tbind ovar evar) (false-implies-aconverte _ _) %. +%total {} (false-implies-aconverte _ _) %. +%sort false-implies-aofe %. +%mode {%in G ctx} {%in R atom} {%in A tp} {%in X1 false} {%out X2 aofe G R A} false-implies-aofe X1 X2 %. +%worlds (bind ovar var) (false-implies-aofe _ _) %. +%total {} (false-implies-aofe _ _) %. +%sort false-implies-aofes %. +%mode {%in G sctx} {%in R atom} {%in T stp} {%in X1 false} {%out X2 aofes G R T} false-implies-aofes X1 X2 %. +%worlds (bind ovar var) (false-implies-aofes _ _) %. +%total {} (false-implies-aofes _ _) %. +%sort false-implies-aosub %. +%mode {%in R %pi atom %-> atom} {%in M term} {%in N term} {%in X1 false} {%out X2 aosub ([r atom] R r) M N} false-implies-aosub X1 X2 %. +%worlds (var bind ovar) (false-implies-aosub _ _) %. +%total {} (false-implies-aosub _ _) %. +%sort false-implies-bounded %. +%mode {%in G ctx} {%in X atom} {%in X1 false} {%out X2 bounded G X} false-implies-bounded X1 X2 %. +%worlds (var bind ovar) (false-implies-bounded _ _) %. +%total {} (false-implies-bounded _ _) %. +%sort false-implies-converte %. +%mode {%in G ctx} {%in M term} {%in A tp} {%in EM eterm} {%in X1 false} {%out X2 converte G M A EM} false-implies-converte X1 X2 %. +%worlds (var bind tbind ovar evar) (false-implies-converte _ _) %. +%total {} (false-implies-converte _ _) %. +%sort false-implies-eof %. +%mode {%in EM eterm} {%in EA etp} {%in X1 false} {%out X2 eof EM EA} false-implies-eof X1 X2 %. +%worlds (evar eofblock ebind) (false-implies-eof _ _) %. +%total {} (false-implies-eof _ _) %. +%sort false-implies-evof %. +%mode {%in EM eterm} {%in EA etp} {%in X1 false} {%out X2 evof EM EA} false-implies-evof X1 X2 %. +%worlds (evar eofblock ebind) (false-implies-evof _ _) %. +%total {} (false-implies-evof _ _) %. +%sort false-implies-eterm-eq %. +%mode {%in EM eterm} {%in EM' eterm} {%in X1 false} {%out X2 eterm-eq EM EM'} false-implies-eterm-eq X1 X2 %. +%worlds (evar) (false-implies-eterm-eq _ _) %. +%total {} (false-implies-eterm-eq _ _) %. +%sort false-implies-lookup %. +%mode {%in G ctx} {%in X atom} {%in A tp} {%in X1 false} {%out X2 lookup G X A} false-implies-lookup X1 X2 %. +%worlds (var bind ovar) (false-implies-lookup _ _) %. +%total {} (false-implies-lookup _ _) %. +%sort false-implies-ofe %. +%mode {%in G ctx} {%in M term} {%in A tp} {%in X1 false} {%out X2 ofe G M A} false-implies-ofe X1 X2 %. +%worlds (bind ovar var) (false-implies-ofe _ _) %. +%total {} (false-implies-ofe _ _) %. +%sort false-implies-ofes %. +%mode {%in G sctx} {%in M term} {%in T stp} {%in X1 false} {%out X2 ofes G M T} false-implies-ofes X1 X2 %. +%worlds (var bind ovar) (false-implies-ofes _ _) %. +%total {} (false-implies-ofes _ _) %. +%sort false-implies-ordered %. +%mode {%in G ctx} {%in X1 false} {%out X2 ordered G} false-implies-ordered X1 X2 %. +%worlds (var bind ovar) (false-implies-ordered _ _) %. +%total {} (false-implies-ordered _ _) %. +%sort false-implies-reduce %. +%mode {%in M eterm} {%in N eterm} {%in X1 false} {%out X2 reduce M N} false-implies-reduce X1 X2 %. +%worlds (bind tbind ovar eofblock evblock evar) (false-implies-reduce _ _) %. +%total {} (false-implies-reduce _ _) %. +%sort false-implies-self %. +%mode {%in M term} {%in A tp} {%in B tp} {%in X1 false} {%out X2 self M A B} false-implies-self X1 X2 %. +%worlds (var) (false-implies-self _ _) %. +%total {} (false-implies-self _ _) %. +%sort false-implies-simp %. +%mode {%in A tp} {%in T stp} {%in X1 false} {%out X2 simp A T} false-implies-simp X1 X2 %. +%worlds (var) (false-implies-simp _ _) %. +%total {} (false-implies-simp _ _) %. +%sort false-implies-stp-eq %. +%mode {%in T stp} {%in T' stp} {%in X1 false} {%out X2 stp-eq T T'} false-implies-stp-eq X1 X2 %. +%worlds () (false-implies-stp-eq _ _) %. +%total {} (false-implies-stp-eq _ _) %. +%sort false-implies-stp-leq %. +%mode {%in T stp} {%in T' stp} {%in X1 false} {%out X2 stp-leq T T'} false-implies-stp-leq X1 X2 %. +%worlds () (false-implies-stp-leq _ _) %. +%total {} (false-implies-stp-leq _ _) %. +%sort false-implies-sub %. +%mode {%in M %pi atom %-> term} {%in N term} {%in O term} {%in X1 false} {%out X2 sub ([r atom] M r) N O} false-implies-sub X1 X2 %. +%worlds (var) (false-implies-sub _ _) %. +%total {} (false-implies-sub _ _) %. +%sort false-implies-subtype %. +%mode {%in A tp} {%in B tp} {%in O %pi atom %-> term} {%in X1 false} {%out X2 subtype A B ([r atom] O r)} false-implies-subtype X1 X2 %. +%worlds (var) (false-implies-subtype _ _) %. +%total {} (false-implies-subtype _ _) %. +%sort false-implies-tconvert %. +%mode {%in A tp} {%in EA etp} {%in X1 false} {%out X2 tconvert A EA} false-implies-tconvert X1 X2 %. +%worlds (bind tbind) (false-implies-tconvert _ _) %. +%total {} (false-implies-tconvert _ _) %. +%sort false-implies-term-eq %. +%mode {%in M term} {%in N term} {%in X1 false} {%out X2 term-eq M N} false-implies-term-eq X1 X2 %. +%worlds (var) (false-implies-term-eq _ _) %. +%total {} (false-implies-term-eq _ _) %. +%sort false-implies-tp-eq %. +%mode {%in A tp} {%in B tp} {%in X1 false} {%out X2 tp-eq A B} false-implies-tp-eq X1 X2 %. +%worlds (var) (false-implies-tp-eq _ _) %. +%total {} (false-implies-tp-eq _ _) %. +%sort false-implies-tsub %. +%mode {%in A %pi atom %-> tp} {%in M term} {%in B tp} {%in X1 false} {%out X2 tsub ([r atom] A r) M B} false-implies-tsub X1 X2 %. +%worlds (var) (false-implies-tsub _ _) %. +%total {} (false-implies-tsub _ _) %. +%sort false-implies-vtrans %. +%mode {%in EX eterm} {%in X atom} {%in X1 false} {%out X2 vtrans EX X} false-implies-vtrans X1 X2 %. +%worlds (bind tbind) (false-implies-vtrans _ _) %. +%total {} (false-implies-vtrans _ _) %. +%sort false-implies-wfe %. +%mode {%in G ctx} {%in A tp} {%in X1 false} {%out X2 wfe G A} false-implies-wfe X1 X2 %. +%worlds (var bind ovar) (false-implies-wfe _ _) %. +%total {} (false-implies-wfe _ _) %. +%sort false-implies-wfes %. +%mode {%in G sctx} {%in A tp} {%in X1 false} {%out X2 wfes G A} false-implies-wfes X1 X2 %. +%worlds (var bind ovar) (false-implies-wfes _ _) %. +%total {} (false-implies-wfes _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/eregularity-thm.lf b/new-tests/stelf-output/tslf/sing/eregularity-thm.lf new file mode 100644 index 0000000..dee6341 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/eregularity-thm.lf @@ -0,0 +1,174 @@ +%sort eof-reg %. +%mode eof-reg %in %out %. +%sort subtp-reg %. +%mode subtp-reg %in %out %out %. +%sort tequiv-reg %. +%mode tequiv-reg %in %out %out %. +%sort equiv-reg %. +%mode equiv-reg %in %out %out %out %. +%term _ eof-reg (eof/var D _) D %. +%term _ eof-reg (eof/const D _) D %. +%term _ eof-reg (eof/forall D) (ewf/pi ([_] [_] ewf/t) (ewf/pi ([_] [_] ewf/t) D)) %. +%term _ + %pi (eof-reg (eof/lam Dof DwfA) (ewf/pi DwfB DwfA)) + %<- ({x} {d} eof-reg (Dof x d) (DwfB x d)) %. +%term _ + %pi (eof-reg (eof/app DofN DofM) DwfBx) + %<- (eof-reg DofM (ewf/pi DwfB DwfA)) + %<- (esubst-wf DwfB DofN DwfBx) %. +%term _ %pi (eof-reg (eof/pair DwfB _ Dof) (ewf/sigma DwfB DwfA)) %<- (eof-reg Dof DwfA) %. +%term _ %pi (eof-reg (eof/pi1 Dof) DwfA) %<- (eof-reg Dof (ewf/sigma DwfB DwfA)) %. +%term _ + %pi (eof-reg (eof/pi2 Dof) DwfBx) + %<- (eof-reg Dof (ewf/sigma DwfB DwfA)) + %<- (esubst-wf DwfB (eof/pi1 Dof) DwfBx) %. +%term _ eof-reg (eof/sing Dof) (ewf/sing Dof) %. +%term _ + %pi (eof-reg (eof/extpi Dof2 Dof1) (ewf/pi DwfB DwfA)) + %<- (eof-reg Dof1 (ewf/pi _ DwfA)) + %<- ({x} {d} eof-reg (Dof2 x d) (DwfB x d)) %. +%term _ + %pi (eof-reg (eof/extsigma DwfB _ Dof) (ewf/sigma DwfB DwfA)) + %<- (eof-reg Dof DwfA) %. +%term _ %pi (eof-reg (eof/subsume Dsub _) Dwf) %<- (subtp-reg Dsub _ Dwf) %. +%term _ %pi (subtp-reg (subtp/reflex D) D1 D2) %<- (tequiv-reg D D1 D2) %. +%term _ + %pi (subtp-reg (subtp/trans D23 D12) D1 D3) + %<- (subtp-reg D12 D1 _) + %<- (subtp-reg D23 _ D3) %. +%term _ subtp-reg (subtp/sing_t Dof) (ewf/sing Dof) ewf/t %. +%term _ + %pi (subtp-reg (subtp/pi DwfB DsubB DsubA) (ewf/pi DwfB DwfA) (ewf/pi DwfB' DwfA')) + %<- (subtp-reg DsubA DwfA' DwfA) + %<- ({x} {d} subtp-reg (DsubB x d) _ (DwfB' x d)) %. +%term _ + %pi (subtp-reg (subtp/sigma DwfB' DsubB DsubA) (ewf/sigma DwfB DwfA) (ewf/sigma DwfB' DwfA')) + %<- (subtp-reg DsubA DwfA DwfA') + %<- ({x} {d} subtp-reg (DsubB x d) (DwfB x d) _) %. +%term _ tequiv-reg (tequiv/reflex D) D D %. +%term _ + %pi (tequiv-reg (tequiv/trans D23 D12) D1 D3) + %<- (tequiv-reg D12 D1 _) + %<- (tequiv-reg D23 _ D3) %. +%term _ %pi (tequiv-reg (tequiv/symm D) D1 D2) %<- (tequiv-reg D D2 D1) %. +%term _ + %pi (tequiv-reg (tequiv/pi DequivB DequivA) (ewf/pi DwfB DwfA) (ewf/pi DwfB'' DwfA')) + %<- (tequiv-reg DequivA DwfA DwfA') + %<- ({x} {d} tequiv-reg (DequivB x d) (DwfB x d) (DwfB' x d)) + %<- ({x} {d} esubst-wf DwfB' (eof/equiv (tequiv/symm DequivA) (eof/var DwfA' d)) (DwfB'' x d)) %. +%term _ + %pi (tequiv-reg (tequiv/sigma DequivB DequivA) (ewf/sigma DwfB DwfA) (ewf/sigma DwfB'' DwfA')) + %<- (tequiv-reg DequivA DwfA DwfA') + %<- ({x} {d} tequiv-reg (DequivB x d) (DwfB x d) (DwfB' x d)) + %<- ({x} {d} esubst-wf DwfB' (eof/equiv (tequiv/symm DequivA) (eof/var DwfA' d)) (DwfB'' x d)) %. +%term _ + %pi (tequiv-reg (tequiv/sing Dequiv) (ewf/sing D1) (ewf/sing D2)) + %<- (equiv-reg Dequiv D1 D2 _) %. +%term _ %pi (equiv-reg (equiv/reflex Dof) Dof Dof Dwf) %<- (eof-reg Dof Dwf) %. +%term _ + %pi (equiv-reg (equiv/symm Dequiv) Dof' Dof Dwf) + %<- (equiv-reg Dequiv Dof Dof' Dwf) %. +%term _ + %pi (equiv-reg (equiv/trans D23 D12) D1 D3 Dwf) + %<- (equiv-reg D12 D1 _ Dwf) + %<- (equiv-reg D23 _ D3 _) %. +%term _ + %pi (equiv-reg (equiv/forall D) (eof/forall D1) (eof/equiv (tequiv/pi ([_] [_] tequiv/t) (tequiv/pi ([_] [_] tequiv/t) (tequiv/symm D))) (eof/forall D2)) (ewf/pi ([_] [_] ewf/t) (ewf/pi ([_] [_] ewf/t) D1))) + %<- (tequiv-reg D D1 D2) %. +%term _ + %pi (equiv-reg (equiv/lam DequivM DequivA) (eof/lam DofM DwfA) (eof/equiv (tequiv/symm (tequiv/pi ([x] [d] tequiv/reflex (DwfB x d)) DequivA)) (eof/lam DofM'' DwfA')) (ewf/pi DwfB DwfA)) + %<- (tequiv-reg DequivA DwfA DwfA') + %<- ({x} {d} equiv-reg (DequivM x d) (DofM x d) (DofM' x d) (DwfB x d)) + %<- ({x} {d} esubst DofM' (eof/equiv (tequiv/symm DequivA) (eof/var DwfA' d)) (DofM'' x d)) %. +%term _ + %pi (equiv-reg (equiv/app DequivN DequivM) (eof/app DofN DofM) (eof/equiv Dtequiv (eof/app DofN' DofM')) DwfBx) + %<- (equiv-reg DequivM DofM DofM' (ewf/pi DwfB DwfA)) + %<- (equiv-reg DequivN DofN DofN' _) + %<- (esubst-wf DwfB DofN DwfBx) + %<- (tfunctionality* DwfB (equiv/symm DequivN) DofN' Dtequiv) %. +%term _ + %pi (equiv-reg (equiv/pair DwfB DequivN DequivM) (eof/pair DwfB DofN DofM) (eof/pair DwfB (eof/equiv Dtequiv DofN') DofM') (ewf/sigma DwfB DwfA)) + %<- (equiv-reg DequivM DofM DofM' DwfA) + %<- (equiv-reg DequivN DofN DofN' _) + %<- (tfunctionality* DwfB DequivM DofM Dtequiv) %. +%term _ + %pi (equiv-reg (equiv/pi1 Dequiv) (eof/pi1 Dof) (eof/pi1 Dof') DwfA) + %<- (equiv-reg Dequiv Dof Dof' (ewf/sigma DwfB DwfA)) %. +%term _ + %pi (equiv-reg (equiv/pi2 Dequiv) (eof/pi2 Dof) (eof/equiv Dtequiv (eof/pi2 Dof')) DwfBx) + %<- (equiv-reg Dequiv Dof Dof' (ewf/sigma DwfB DwfA)) + %<- (esubst-wf DwfB (eof/pi1 Dof) DwfBx) + %<- (tfunctionality* DwfB (equiv/pi1 (equiv/symm Dequiv)) (eof/pi1 Dof') Dtequiv) %. +%term _ + %pi (equiv-reg (equiv/sing Dequiv) (eof/sing Dof) (eof/equiv (tequiv/symm (tequiv/sing Dequiv)) (eof/sing Dof')) (ewf/sing Dof)) + %<- (equiv-reg Dequiv Dof Dof' _) %. +%term _ + %pi (equiv-reg (equiv/singelim Dof) (eof/subsume (subtp/sing_t Dof') Dof) Dof' ewf/t) + %<- (eof-reg Dof (ewf/sing Dof')) %. +%term _ + %pi (equiv-reg (equiv/extpi Dequiv Dof' Dof) (eof/extpi DofApp Dof) (eof/extpi DofApp' Dof') (ewf/pi DwfB DwfA)) + %<- (eof-reg Dof (ewf/pi _ DwfA)) + %<- ({x} {d} equiv-reg (Dequiv x d) (DofApp x d) (DofApp' x d) (DwfB x d)) %. +%term _ + %pi (equiv-reg (equiv/extpiw Dequiv Dequiv') (eof/extpi DofApp Dof) (eof/extpi DofApp' Dof') (ewf/pi DwfB DwfA)) + %<- (equiv-reg Dequiv' Dof Dof' (ewf/pi _ DwfA)) + %<- ({x} {d} equiv-reg (Dequiv x d) (DofApp x d) (DofApp' x d) (DwfB x d)) %. +%term _ + %pi (equiv-reg (equiv/extsigma DwfB Dequiv2 Dequiv1) (eof/extsigma DwfB DofM2 DofM1) (eof/extsigma DwfB (eof/equiv Dtequiv DofN2) DofN1) (ewf/sigma DwfB DwfA)) + %<- (equiv-reg Dequiv1 DofM1 DofN1 DwfA) + %<- (equiv-reg Dequiv2 DofM2 DofN2 _) + %<- (tfunctionality* DwfB Dequiv1 DofM1 Dtequiv) %. +%term _ + %pi (equiv-reg (equiv/subsume Dsub Dequiv) (eof/subsume Dsub Dof) (eof/subsume Dsub Dof') Dwf) + %<- (equiv-reg Dequiv Dof Dof' _) + %<- (subtp-reg Dsub _ Dwf) %. +%term _ + %pi (equiv-reg (equiv/beta DofN DofM) (eof/app DofN (eof/lam DofM DwfA)) DofMx DwfBx) + %<- ({x} {d} eof-reg (DofM x d) (DwfB x d)) + %<- (eof-reg DofN DwfA) + %<- (esubst-wf DwfB DofN DwfBx) + %<- (esubst DofM DofN DofMx) %. +%term _ + %pi (equiv-reg (equiv/beta1 DofN DofM) (eof/pi1 (eof/pair ([_] [_] DwfB) DofN DofM)) DofM DwfA) + %<- (eof-reg DofM DwfA) + %<- (eof-reg DofN DwfB) %. +%term _ + %pi (equiv-reg (equiv/beta2 DofN DofM) (eof/pi2 (eof/pair ([_] [_] DwfB) DofN DofM)) DofN DwfB) + %<- (eof-reg DofM DwfA) + %<- (eof-reg DofN DwfB) %. +%worlds (nbind ebind) (eof-reg _ _) (subtp-reg _ _ _) (tequiv-reg _ _ _) (equiv-reg _ _ _ _) %. +%total (D1 D2 D3 D4) (eof-reg D1 _) (subtp-reg D2 _ _) (tequiv-reg D3 _ _) (equiv-reg D4 _ _ _) %. +%sort tfunctionality {x} %. +%mode tfunctionality %in %in %out %. +%term _ + %pi (tfunctionality Dwf Dequiv Dtequiv) + %<- (equiv-reg Dequiv Dof _ _) + %<- (tfunctionality* Dwf Dequiv Dof Dtequiv) %. +%worlds (nbind ebind) (tfunctionality _ _ _) %. +%total {} (tfunctionality _ _ _) %. +%sort tequiv-functionality {x} %. +%mode tequiv-functionality %in %in %out %. +%term _ + %pi (tequiv-functionality DequivBC DequivMN (tequiv/trans DequivCMCN DequivBMCM)) + %<- (equiv-reg DequivMN DofM _ _) + %<- ({x} {d} tequiv-reg (DequivBC x d) (DwfB x d) (DwfC x d)) + %<- (esubst-tequiv DequivBC DofM DequivBMCM) + %<- (tfunctionality DwfC DequivMN DequivCMCN) %. +%worlds (nbind ebind) (tequiv-functionality _ _ _) %. +%total {} (tequiv-functionality _ _ _) %. +%sort tfunctionality1 {x} %. +%mode tfunctionality1 %in %in %out %. +%term _ + %pi (tfunctionality1 Dwf Dequiv Dtequiv) + %<- (equiv-reg Dequiv Dof _ _) + %<- (tfunctionality1* Dwf Dequiv Dof Dtequiv) %. +%worlds (nbind ebind) (tfunctionality1 _ _ _) %. +%total {} (tfunctionality1 _ _ _) %. +%sort tfunctionality2 {x} %. +%mode tfunctionality2 %in %in %out %. +%term _ + %pi (tfunctionality2 Dwf Dequiv Dtequiv) + %<- (equiv-reg Dequiv Dof _ _) + %<- (tfunctionality2* Dwf Dequiv Dof Dtequiv) %. +%worlds (nbind ebind) (tfunctionality2 _ _ _) %. +%total {} (tfunctionality2 _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/esubstitution-thm.lf b/new-tests/stelf-output/tslf/sing/esubstitution-thm.lf new file mode 100644 index 0000000..3fda862 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/esubstitution-thm.lf @@ -0,0 +1,206 @@ +%%%%% Worlds %%%%% +%block nbind {a tp} {ea etp} [x atom] [d vof x a] [ex eterm] [xt vtrans ex x] [ed evof ex ea]%. +%%%%% Substitution %%%%% +%sort esubst-of-gen %. +%mode esubst-of-gen %in %out %. +%sort esubst-wf-gen %. +%mode esubst-wf-gen %in %out %. +%sort esubst-subtp-gen %. +%mode esubst-subtp-gen %in %out %. +%sort esubst-tequiv-gen %. +%mode esubst-tequiv-gen %in %out %. +%sort esubst-equiv-gen %. +%mode esubst-equiv-gen %in %out %. +%scope esubst-equiv-gen %term varsam esubst-of-gen ([d] eof/var (D d) d) ([d] d) %. +%term varoth + %pi (esubst-of-gen ([d] eof/var (D2 d) D1) ([d] eof/var (D2' d) D1)) + %<- (esubst-wf-gen D2 D2') %. +%scope esubst-equiv-gen %term forall + %pi (esubst-of-gen ([d] eof/forall (D d)) ([d] eof/forall (D' d))) + %<- (esubst-wf-gen D D') %. +%term const + %pi (esubst-of-gen ([d] eof/const (D2 d) D) ([d] eof/const (D2' d) D)) + %<- (esubst-wf-gen D2 D2') %. +%scope esubst-equiv-gen %term lam + %pi (esubst-of-gen ([d] eof/lam (D2 d) (D1 d)) ([d] eof/lam (D2' d) (D1' d))) + %<- (esubst-wf-gen D1 D1') + %<- ({y} {e} esubst-of-gen ([d] D2 d y e) ([d] D2' d y e)) %. +%term app + %pi (esubst-of-gen ([d] eof/app (D2 d) (D1 d)) ([d] eof/app (D2' d) (D1' d))) + %<- (esubst-of-gen D1 D1') + %<- (esubst-of-gen D2 D2') %. +%scope esubst-equiv-gen %term pair + %pi (esubst-of-gen ([d] eof/pair (%the ({y} %pi (evof y A) %-> (ewf (B y))) (D3 d)) (D2 d) (D1 d)) ([d] eof/pair (D3' d) (D2' d) (D1' d))) + %<- (esubst-of-gen D1 D1') + %<- (esubst-of-gen D2 D2') + %<- ({y} {e} esubst-wf-gen ([d] D3 d y e) ([d] D3' d y e)) %. +%term pi1 + %pi (esubst-of-gen ([d] eof/pi1 (D d)) ([d] eof/pi1 (D' d))) + %<- (esubst-of-gen D D') %. +%scope esubst-equiv-gen %term pi2 + %pi (esubst-of-gen ([d] eof/pi2 (D d)) ([d] eof/pi2 (D' d))) + %<- (esubst-of-gen D D') %. +%term sing + %pi (esubst-of-gen ([d] eof/sing (D d)) ([d] eof/sing (D' d))) + %<- (esubst-of-gen D D') %. +%scope esubst-equiv-gen %term extpi + %pi (esubst-of-gen ([d] eof/extpi (D2 d) (D1 d)) ([d] eof/extpi (D2' d) (D1' d))) + %<- (esubst-of-gen D1 D1') + %<- ({y} {e} esubst-of-gen ([d] D2 d y e) ([d] D2' d y e)) %. +%term extsig + %pi (esubst-of-gen ([d] eof/extsigma (D3 d) (D2 d) (D1 d)) ([d] eof/extsigma (D3' d) (D2' d) (D1' d))) + %<- (esubst-of-gen D1 D1') + %<- (esubst-of-gen D2 D2') + %<- ({y} {e} esubst-wf-gen ([d] D3 d y e) ([d] D3' d y e)) %. +%scope esubst-equiv-gen %term subsum + %pi (esubst-of-gen ([d] eof/subsume (D2 d) (D1 d)) ([d] eof/subsume (D2' d) (D1' d))) + %<- (esubst-of-gen D1 D1') + %<- (esubst-subtp-gen D2 D2') %. +%term t esubst-wf-gen ([d] ewf/t) ([d] ewf/t) %. +%scope esubst-equiv-gen %term pi + %pi (esubst-wf-gen ([d] ewf/pi (D2 d) (D1 d)) ([d] ewf/pi (D2' d) (D1' d))) + %<- (esubst-wf-gen D1 D1') + %<- ({y} {e} esubst-wf-gen ([d] D2 d y e) ([d] D2' d y e)) %. +%term sigma + %pi (esubst-wf-gen ([d] ewf/sigma (D2 d) (D1 d)) ([d] ewf/sigma (D2' d) (D1' d))) + %<- (esubst-wf-gen D1 D1') + %<- ({y} {e} esubst-wf-gen ([d] D2 d y e) ([d] D2' d y e)) %. +%scope esubst-equiv-gen %term sing + %pi (esubst-wf-gen ([d] ewf/sing (D d)) ([d] ewf/sing (D' d))) + %<- (esubst-of-gen D D') %. +%term refl + %pi (esubst-subtp-gen ([d] subtp/reflex (D d)) ([d] subtp/reflex (D' d))) + %<- (esubst-tequiv-gen D D') %. +%scope esubst-equiv-gen %term trans + %pi (esubst-subtp-gen ([d] subtp/trans (D2 d) (D1 d)) ([d] subtp/trans (D2' d) (D1' d))) + %<- (esubst-subtp-gen D1 D1') + %<- (esubst-subtp-gen D2 D2') %. +%term sing_t + %pi (esubst-subtp-gen ([d] subtp/sing_t (D d)) ([d] subtp/sing_t (D' d))) + %<- (esubst-of-gen D D') %. +%scope esubst-equiv-gen %term pi + %pi (esubst-subtp-gen ([d] subtp/pi (D3 d) (D2 d) (D1 d)) ([d] subtp/pi (D3' d) (D2' d) (D1' d))) + %<- (esubst-subtp-gen D1 D1') + %<- ({y} {e} esubst-subtp-gen ([d] D2 d y e) ([d] D2' d y e)) + %<- ({y} {e} esubst-wf-gen ([d] D3 d y e) ([d] D3' d y e)) %. +%term sigma + %pi (esubst-subtp-gen ([d] subtp/sigma (D3 d) (D2 d) (D1 d)) ([d] subtp/sigma (D3' d) (D2' d) (D1' d))) + %<- (esubst-subtp-gen D1 D1') + %<- ({y} {e} esubst-subtp-gen ([d] D2 d y e) ([d] D2' d y e)) + %<- ({y} {e} esubst-wf-gen ([d] D3 d y e) ([d] D3' d y e)) %. +%scope esubst-equiv-gen %term refl + %pi (esubst-tequiv-gen ([d] tequiv/reflex (D d)) ([d] tequiv/reflex (D' d))) + %<- (esubst-wf-gen D D') %. +%term symm + %pi (esubst-tequiv-gen ([d] tequiv/symm (D d)) ([d] tequiv/symm (D' d))) + %<- (esubst-tequiv-gen D D') %. +%scope esubst-equiv-gen %term trans + %pi (esubst-tequiv-gen ([d] tequiv/trans (D2 d) (D1 d)) ([d] tequiv/trans (D2' d) (D1' d))) + %<- (esubst-tequiv-gen D1 D1') + %<- (esubst-tequiv-gen D2 D2') %. +%term pi + %pi (esubst-tequiv-gen ([d] tequiv/pi (D2 d) (D1 d)) ([d] tequiv/pi (D2' d) (D1' d))) + %<- (esubst-tequiv-gen D1 D1') + %<- ({y} {e} esubst-tequiv-gen ([d] D2 d y e) ([d] D2' d y e)) %. +%scope esubst-equiv-gen %term sigma + %pi (esubst-tequiv-gen ([d] tequiv/sigma (D2 d) (D1 d)) ([d] tequiv/sigma (D2' d) (D1' d))) + %<- (esubst-tequiv-gen D1 D1') + %<- ({y} {e} esubst-tequiv-gen ([d] D2 d y e) ([d] D2' d y e)) %. +%term sing + %pi (esubst-tequiv-gen ([d] tequiv/sing (D d)) ([d] tequiv/sing (D' d))) + %<- (esubst-equiv-gen D D') %. +%scope esubst-equiv-gen %term refl + %pi (esubst-equiv-gen ([d] equiv/reflex (D d)) ([d] equiv/reflex (D' d))) + %<- (esubst-of-gen D D') %. +%term symm + %pi (esubst-equiv-gen ([d] equiv/symm (D d)) ([d] equiv/symm (D' d))) + %<- (esubst-equiv-gen D D') %. +%scope esubst-equiv-gen %term trans + %pi (esubst-equiv-gen ([d] equiv/trans (D2 d) (D1 d)) ([d] equiv/trans (D2' d) (D1' d))) + %<- (esubst-equiv-gen D1 D1') + %<- (esubst-equiv-gen D2 D2') %. +%term forall + %pi (esubst-equiv-gen ([d] equiv/forall (D d)) ([d] equiv/forall (D' d))) + %<- (esubst-tequiv-gen D D') %. +%scope esubst-equiv-gen %term lam + %pi (esubst-equiv-gen ([d] equiv/lam (D2 d) (D1 d)) ([d] equiv/lam (D2' d) (D1' d))) + %<- (esubst-tequiv-gen D1 D1') + %<- ({y} {e} esubst-equiv-gen ([d] D2 d y e) ([d] D2' d y e)) %. +%term app + %pi (esubst-equiv-gen ([d] equiv/app (D2 d) (D1 d)) ([d] equiv/app (D2' d) (D1' d))) + %<- (esubst-equiv-gen D1 D1') + %<- (esubst-equiv-gen D2 D2') %. +%scope esubst-equiv-gen %term pair + %pi (esubst-equiv-gen ([d] equiv/pair (%the ({y} %pi (evof y A) %-> (ewf (B y))) (D3 d)) (D2 d) (D1 d)) ([d] equiv/pair (D3' d) (D2' d) (D1' d))) + %<- (esubst-equiv-gen D1 D1') + %<- (esubst-equiv-gen D2 D2') + %<- ({y} {e} esubst-wf-gen ([d] D3 d y e) ([d] D3' d y e)) %. +%term pi1 + %pi (esubst-equiv-gen ([d] equiv/pi1 (D d)) ([d] equiv/pi1 (D' d))) + %<- (esubst-equiv-gen D D') %. +%scope esubst-equiv-gen %term pi2 + %pi (esubst-equiv-gen ([d] equiv/pi2 (D d)) ([d] equiv/pi2 (D' d))) + %<- (esubst-equiv-gen D D') %. +%term sing + %pi (esubst-equiv-gen ([d] equiv/sing (D d)) ([d] equiv/sing (D' d))) + %<- (esubst-equiv-gen D D') %. +%scope esubst-equiv-gen %term singel + %pi (esubst-equiv-gen ([d] equiv/singelim (D d)) ([d] equiv/singelim (D' d))) + %<- (esubst-of-gen D D') %. +%term extpi + %pi (esubst-equiv-gen ([d] equiv/extpi (D3 d) (D2 d) (D1 d)) ([d] equiv/extpi (D3' d) (D2' d) (D1' d))) + %<- (esubst-of-gen D1 D1') + %<- (esubst-of-gen D2 D2') + %<- ({y} {e} esubst-equiv-gen ([d] D3 d y e) ([d] D3' d y e)) %. +%scope esubst-equiv-gen %term extpiw + %pi (esubst-equiv-gen ([d] equiv/extpiw (D2 d) (D1 d)) ([d] equiv/extpiw (D2' d) (D1' d))) + %<- (esubst-equiv-gen D1 D1') + %<- ({y} {e} esubst-equiv-gen ([d] D2 d y e) ([d] D2' d y e)) %. +%term extsig + %pi (esubst-equiv-gen ([d] equiv/extsigma (D3 d) (D2 d) (D1 d)) ([d] equiv/extsigma (D3' d) (D2' d) (D1' d))) + %<- (esubst-equiv-gen D1 D1') + %<- (esubst-equiv-gen D2 D2') + %<- ({y} {e} esubst-wf-gen ([d] D3 d y e) ([d] D3' d y e)) %. +%scope esubst-equiv-gen %term subsum + %pi (esubst-equiv-gen ([d] equiv/subsume (D2 d) (D1 d)) ([d] equiv/subsume (D2' d) (D1' d))) + %<- (esubst-equiv-gen D1 D1') + %<- (esubst-subtp-gen D2 D2') %. +%term beta + %pi (esubst-equiv-gen ([d] equiv/beta (D2 d) (D1 d)) ([d] equiv/beta (D2' d) (D1' d))) + %<- ({y} {e} esubst-of-gen ([d] D1 d y e) ([d] D1' d y e)) + %<- (esubst-of-gen D2 D2') %. +%scope esubst-equiv-gen %term beta1 + %pi (esubst-equiv-gen ([d] equiv/beta1 (D2 d) (D1 d)) ([d] equiv/beta1 (D2' d) (D1' d))) + %<- (esubst-of-gen D1 D1') + %<- (esubst-of-gen D2 D2') %. +%term beta2 + %pi (esubst-equiv-gen ([d] equiv/beta2 (D2 d) (D1 d)) ([d] equiv/beta2 (D2' d) (D1' d))) + %<- (esubst-of-gen D1 D1') + %<- (esubst-of-gen D2 D2') %. +%worlds (evar eofblock ebind) (esubst-of-gen _ _) (esubst-wf-gen _ _) (esubst-subtp-gen _ _) (esubst-tequiv-gen _ _) (esubst-equiv-gen _ _) %. +%total (D1 D2 D3 D4 D5) (esubst-of-gen D1 _) (esubst-wf-gen D2 _) (esubst-subtp-gen D3 _) (esubst-tequiv-gen D4 _) (esubst-equiv-gen D5 _) %. +%sort esubst {x} %. +%mode esubst %in %in %out %. +%term _ %pi (esubst D1 D2 (D1' D2)) %<- (esubst-of-gen (D1 _) D1') %. +%worlds (evar eofblock ebind) (esubst _ _ _) %. +%total {} (esubst _ _ _) %. +%sort esubst-subtp {x} %. +%mode esubst-subtp %in %in %out %. +%term _ %pi (esubst-subtp D1 D2 (D1' D2)) %<- (esubst-subtp-gen (D1 _) D1') %. +%worlds (evar eofblock ebind) (esubst-subtp _ _ _) %. +%total {} (esubst-subtp _ _ _) %. +%sort esubst-wf {x} %. +%mode esubst-wf %in %in %out %. +%term _ %pi (esubst-wf D1 D2 (D1' D2)) %<- (esubst-wf-gen (D1 _) D1') %. +%worlds (evar eofblock ebind) (esubst-wf _ _ _) %. +%total {} (esubst-wf _ _ _) %. +%sort esubst-equiv {x} %. +%mode esubst-equiv %in %in %out %. +%term _ %pi (esubst-equiv D1 D2 (D1' D2)) %<- (esubst-equiv-gen (D1 _) D1') %. +%worlds (evar eofblock ebind) (esubst-equiv _ _ _) %. +%total {} (esubst-equiv _ _ _) %. +%sort esubst-tequiv {x} %. +%mode esubst-tequiv %in %in %out %. +%term _ %pi (esubst-tequiv D1 D2 (D1' D2)) %<- (esubst-tequiv-gen (D1 _) D1') %. +%worlds (evar eofblock ebind) (esubst-tequiv _ _ _) %. +%total {} (esubst-tequiv _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/expand-thm.lf b/new-tests/stelf-output/tslf/sing/expand-thm.lf new file mode 100644 index 0000000..915394c --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/expand-thm.lf @@ -0,0 +1,533 @@ +%%%%% Expansion Effectiveness/Functionality %%%%% +%sort can-expand* {T} {R atom} {A tp} %. +%mode can-expand* %in %in %in %in %out %out %. +%term _ can-expand* _ R t simp/t (at R) expand/t %. +%term _ + %pi (can-expand* _ R (pi A B) (simp/pi Dsimp2 Dsimp1) (lam N) (expand/pi Dexpand2 Dexpand1)) + %<- ({x} can-expand* _ x A Dsimp1 (M x) (Dexpand1 x)) + %<- ({x} can-expand* _ (app R (M x)) (B x) (Dsimp2 x) (N x) (Dexpand2 x)) %. +%term _ + %pi (can-expand* _ R (sigma A B) (simp/sigma Dsimp2 Dsimp1) (pair M N) (expand/sigma Dexpand2 Dexpand1)) + %<- (can-expand* _ (pi1 R) A Dsimp1 M Dexpand1) + %<- (can-expand* _ (pi2 R) (B (pi1 R)) (Dsimp2 (pi1 R)) N Dexpand2) %. +%term _ can-expand* _ R (sing R') simp/sing (at R') expand/sing %. +%worlds (var bind) (can-expand* T _ _ _ _ _) %. +%total T (can-expand* T _ _ _ _ _) %. +%sort can-expand {R atom} {A tp} {M term} %. +%mode can-expand %in %in %out %out %. +%term _ %pi (can-expand R A M D) %<- (can-simp A Dsimp) %<- (can-expand* _ R A Dsimp M D) %. +%worlds (var bind) (can-expand _ _ _ _) %. +%total {} (can-expand _ _ _ _) %. +%sort expand-fun %. +%mode expand-fun %in %in %out %. +%term _ expand-fun expand/t expand/t term-eq/i %. +%term _ + %pi (expand-fun (expand/pi Dexpand2 Dexpand1) (expand/pi Dexpand2' Dexpand1') Deq) + %<- ({x} expand-fun (Dexpand1 x) (Dexpand1' x) (Deq1 x)) + %<- ({x} atom-resp-term ([m] app R m) (Deq1 x) (Deq1' x)) + %<- ({x} expand-resp (Deq1' x) tp-eq/i term-eq/i (Dexpand2 x) (Dexpand2'' x)) + %<- ({x} expand-fun (Dexpand2'' x) (Dexpand2' x) (Deq2 x)) + %<- (lam-resp Deq2 Deq) %. +%term _ + %pi (expand-fun (expand/sigma Dexpand2 Dexpand1) (expand/sigma Dexpand2' Dexpand1') Deq) + %<- (expand-fun Dexpand1 Dexpand1' Deq1) + %<- (expand-fun Dexpand2 Dexpand2' Deq2) + %<- (pair-resp Deq1 Deq2 Deq) %. +%term _ expand-fun expand/sing expand/sing term-eq/i %. +%worlds (var bind) (expand-fun _ _ _) %. +%total D (expand-fun _ D _) %. +%sort expand-t-invert %. +%mode expand-t-invert %in %out %. +%term _ expand-t-invert expand/t term-eq/i %. +%worlds (var) (expand-t-invert _ _) %. +%total {} (expand-t-invert _ _) %. +%sort expand-pi-invert %. +%mode expand-pi-invert %in %out %out %. +%term _ expand-pi-invert (expand/pi D2 D1) D1 D2 %. +%worlds (var) (expand-pi-invert _ _ _) %. +%total {} (expand-pi-invert _ _ _) %. +%sort expand-sigma-invert %. +%mode expand-sigma-invert %in %out %out %. +%term _ expand-sigma-invert (expand/sigma D2 D1) D1 D2 %. +%worlds (var) (expand-sigma-invert _ _ _) %. +%total {} (expand-sigma-invert _ _ _) %. +%%%%% Expansion (Explicit Context) %%%%% +%sort expand-reg-em {Ncase} {T} %. +%mode expand-reg-em %in %in %in %in %in %in %out %. +%sort expand-aasub-em {Ncase} {T} %. +%mode expand-aasub-em %in %in %in %in %in %in %in %in %in %in %in %out %. +%sort expand-aosub-em {Ncase} {T} %. +%mode expand-aosub-em %in %in %in %in %in %in %in %in %in %in %out %. +%sort aasub-expand-em {Ncase} {T} %. +%mode aasub-expand-em %in %in %in %in %in %in %in %in %in %out %. +%sort aosub-expand-em {Ncase} {T} %. +%mode aosub-expand-em %in %in %in %in %in %in %in %in %in %out %. +%sort sub-expand-em {Ncase} {T} %. +%mode sub-expand-em %in %in %in %in %in %in %in %in %in %out %. +%sort tsub-expand-em {Ncase} {T} %. +%mode tsub-expand-em %in %in %in %in %in %in %in %in %in %out %. +%sort tsub-expand-var-em {Ncase} {T} %. +%mode tsub-expand-var-em %in %in %in %in %in %in %in %out %. +%sort sub-expand-var-em {Ncase} {T} %. +%mode sub-expand-var-em %in %in %in %in %in %in %in %out %. +%scope sub-expand-var-em %term t expand-reg-em _ _ _ _ Daof expand/t (ofe/at Daof) %. +%term pi + %pi (expand-reg-em _ _ nat-eq/i (simp/pi (%the ({x} simp (B x) T) DsimpB) (%the (simp A S) DsimpA)) (%the (aofe G R (pi A B)) DofR) (expand/pi (%the ({x} expand (app R (M x)) (B x) (N x)) Dexpand2) (%the ({x} expand x A (M x)) Dexpand1)) (ofe/lam DofN DwfA)) + %<- (aofe-reg DofR (wfe/pi (%the ({x} %pi (isvar x I) %-> (wfe (cons G x A) (B x))) DwfB) (%the (wfe G A) DwfA))) + %<- ({x} {d isvar x I} wfe-context (DwfB x d) (ordered/cons (%the (bounded G x) (Dbounded x d)))) + %<- ({x} {d isvar x I} weaken-wfe (Dbounded x d) DwfA _ (%the (wfe (cons G x A) A) (DwfA' x d))) + %<- ({x} {d isvar x I} weaken-aofe (Dbounded x d) DofR _ (%the (aofe (cons G x A) R (pi A B)) (DofR' x d))) + %<- ({x} {d isvar x I} expand-reg-em _ _ nat-eq/i DsimpA (aofe/var (DwfA' x d) (lookup/hit (Dbounded x d))) (Dexpand1 x) (%the (ofe (cons G x A) (M x) A) (DofM x d))) + %<- (weaken-wfe-insert1 DwfB Dbounded _ (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (wfe (cons (cons G x A) y A) (B y)))) DwfB')) + %<- (can-tsub-context-e DofM DwfB (%the ({x} tsub B (M x) (B' x)) Dtsub)) + %<- ({x} {d isvar x I} tsub-expand-em _ _ nat-eq/i DsimpA ([_] append/nil) (aofe/var (DwfA' x d) (lookup/hit (Dbounded x d))) (DwfB' x d) (Dexpand1 x) (Dtsub x) (%the (tp-eq (B' x) (B x)) (Deq x))) + %<- ({x} tsub-resp ([_] tp-eq/i) term-eq/i (Deq x) (Dtsub x) (%the (tsub B (M x) (B x)) (Dtsub' x))) + %<- ({x} {d isvar x I} expand-reg-em _ _ nat-eq/i (DsimpB x) (aofe/app (DwfB x d) (Dtsub' x) (DofM x d) (DofR' x d)) (Dexpand2 x) (%the (ofe (cons G x A) (N x) (B x)) (DofN x d))) %. +%scope sub-expand-var-em %term sigma + %pi (expand-reg-em _ _ nat-eq/i (simp/sigma (%the ({x} simp (B x) T) DsimpB) (%the (simp A S) DsimpA)) (%the (aofe G R (sigma A B)) DofR) (expand/sigma (%the (expand (pi2 R) (B (pi1 R)) N) Dexpand2) (%the (expand (pi1 R) A M) Dexpand1)) (ofe/pair DwfB DofN DsubBx' DofM)) + %<- (aofe-reg DofR (wfe/sigma (%the ({x} %pi (isvar x I) %-> (wfe (cons G x A) (B x))) DwfB) (%the (wfe G A) DwfA))) + %<- (expand-reg-em _ _ nat-eq/i DsimpA (aofe/pi1 DofR) Dexpand1 (%the (ofe G M A) DofM)) + %<- (can-tsub-e ([_] append/nil) DwfB DofM (%the (tsub B M Bx) DsubBx)) + %<- (tsub-expand-em _ _ nat-eq/i DsimpA ([_] append/nil) (aofe/pi1 DofR) DwfB Dexpand1 DsubBx (%the (tp-eq Bx (B (pi1 R))) Deq)) + %<- (tsub-resp ([_] tp-eq/i) term-eq/i Deq DsubBx (%the (tsub B M (B (pi1 R))) DsubBx')) + %<- (expand-reg-em _ _ nat-eq/i (DsimpB (pi1 R)) (aofe/pi2 DofR) Dexpand2 (%the (ofe G N (B (pi1 R))) DofN)) %. +%term sing + %pi (expand-reg-em _ _ _ _ (%the (aofe G R (sing R')) DofR) expand/sing (ofe/sing DofR')) + %<- (aofe-reg DofR (wfe/sing (%the (aofe G R' t) DofR'))) %. +%%% +%scope sub-expand-var-em %term t expand-aasub-em _ _ _ _ _ _ _ ([_] expand/t) Dsub tsub/t expand/t (sub/aa Dsub) %. +%term pi + %pi (expand-aasub-em _ _ nat-eq/i ([x] simp/pi (%the ({y} simp (C x y) T) (DsimpC x)) (%the (simp (B x) S) (DsimpB x))) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (ofe G1 M A) DofM) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (pi (B x) ([y] C x y)))) DofR) ([x] %the (expand (R x) (pi (B x) ([y] C x y)) (lam ([y] N x y))) (expand/pi (%the ({y} expand (app (R x) (Y x y)) (C x y) (N x y)) (DexpandRY x)) (%the ({y} expand y (B x) (Y x y)) (DexpandY x)))) (%the (aasub ([x] R x) M Rx) DsubRx) (%the (tsub ([x] pi (B x) ([y] C x y)) M (pi Bx ([y] Cx y))) (tsub/pi (%the ({y} tsub ([x] C x y) M (Cx y)) DtsubC) (%the (tsub ([x] B x) M Bx) DtsubB))) (%the (expand Rx (pi Bx ([y] Cx y)) (lam ([y] N' y))) (expand/pi (%the ({y} expand (app Rx (Y' y)) (Cx y) (N' y)) DexpandRY') (%the ({y} expand y Bx (Y' y)) DexpandY'))) (sub/lam DsubRYx)) + %<- ({x} {d isvar x I} aofe-reg (DofR x d) (wfe/pi (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- ({x} {d isvar x I} {y} {e isvar y J} wfe-context (DwfC x d y e) (ordered/cons (%the (bounded (G x) y) (Dbounded x d y e)))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-aofe (Dbounded x d y e) (DofR x d) _ (%the (aofe (cons (G x) y (B x)) (R x) (pi (B x) ([y] C x y))) (DofR' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-wfe (Dbounded x d y e) (DwfB x d) _ (%the (wfe (cons (G x) y (B x)) (B x)) (DwfB' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} expand-reg-em _ _ nat-eq/i (DsimpB x) (aofe/var (DwfB' x d y e) (lookup/hit (Dbounded x d y e))) (DexpandY x y) (%the (ofe (cons (G x) y (B x)) (Y x y) (B x)) (DofY x d y e))) + %<- ({y} aasub-absent y M (%the (aasub ([x] y) M y) (Dsuby y))) + %<- ({y} {e isvar y J} expand-aasub-em _ _ nat-eq/i ([x] DsimpB x) ([x] append/cons (Dappend x)) DofM ([x] [d] aofe/var (DwfB' x d y e) (lookup/hit (Dbounded x d y e))) ([x] DexpandY x y) (Dsuby y) DtsubB (DexpandY' y) (%the (sub ([x] Y x y) M (Y' y)) (DsubYx y))) + %<- ({x} {d isvar x I} tsub-expand-var-em _ _ nat-eq/i (DsimpB x) (DwfB x d) ([y] [e] DwfC x d y e) (DexpandY x) (%the ({y} tsub ([y] C x y) (Y x y) (C x y)) (DsubCy x))) + %<- ({y} {e isvar y J} expand-aasub-em _ _ nat-eq/i ([x] DsimpC x y) ([x] append/cons (Dappend x)) DofM ([x] [d] aofe/app (DwfC x d y e) (DsubCy x y) (DofY x d y e) (DofR' x d y e)) ([x] DexpandRY x y) (aasub/app (DsubYx y) DsubRx) (DtsubC y) (DexpandRY' y) (%the (sub ([x] N x y) M (N' y)) (DsubRYx y))) %. +%scope sub-expand-var-em %term sigma + %pi (expand-aasub-em _ _ nat-eq/i ([x] simp/sigma (%the ({y} simp (C x y) T) (DsimpC x)) (%the (simp (B x) S) (DsimpB x))) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (ofe G1 M A) DofM) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (sigma (B x) ([y] C x y)))) DofR) ([x] %the (expand (R x) (sigma (B x) ([y] C x y)) (pair (N1 x) (N2 x))) (expand/sigma (%the (expand (pi2 (R x)) (C x (pi1 (R x))) (N2 x)) (DexpandR2 x)) (%the (expand (pi1 (R x)) (B x) (N1 x)) (DexpandR1 x)))) (%the (aasub ([x] R x) M Rx) DsubRx) (%the (tsub ([x] sigma (B x) ([y] C x y)) M (sigma Bx ([y] Cx y))) (tsub/sigma (%the ({y} tsub ([x] C x y) M (Cx y)) DtsubC) (%the (tsub ([x] B x) M Bx) DtsubB))) (%the (expand Rx (sigma Bx ([y] Cx y)) (pair N1' N2')) (expand/sigma (%the (expand (pi2 Rx) (Cx (pi1 Rx)) N2') DexpandR2') (%the (expand (pi1 Rx) Bx N1') DexpandR1'))) (sub/pair DsubN2x DsubN1x)) + %<- ({x} {d isvar x I} aofe-reg (DofR x d) (wfe/sigma (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- ({x} {d isvar x I} {y} {e isvar y J} wfe-context (DwfC x d y e) (ordered/cons (%the (bounded (G x) y) (Dbounded x d y e)))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-aofe (Dbounded x d y e) (DofR x d) _ (%the (aofe (cons (G x) y (B x)) (R x) (sigma (B x) ([y] C x y))) (DofR' x d y e))) + %<- (expand-aasub-em _ _ nat-eq/i ([x] DsimpB x) Dappend DofM ([x] [d] aofe/pi1 (DofR x d)) DexpandR1 (aasub/pi1 DsubRx) DtsubB DexpandR1' (%the (sub ([x] N1 x) M N1') DsubN1x)) + %<- ({x} {d isvar x I} stsubst-e ([_] append/nil) append/nil ([y] [e] DwfC x d y e) (aofe/pi1 (DofR x d)) (%the (wfe (G x) (C x (pi1 (R x)))) (DwfCy x d))) + %<- (can-tsub-e Dappend DwfCy DofM (%the (tsub ([x] C x (pi1 (R x))) M Cyx) DtsubCyx)) + %<- (stsub-aa-permute-e Dappend DofM ([x] [d] aofe/pi1 (DofR x d)) ([x] [d] [y] [e] DwfC x d y e) DtsubC (aasub/pi1 DsubRx) DtsubCyx (%the (tp-eq (Cx (pi1 Rx)) Cyx) DeqC)) + %<- (tp-eq-symm DeqC (%the (tp-eq Cyx (Cx (pi1 Rx))) DeqC')) + %<- (tsub-resp ([_] tp-eq/i) term-eq/i DeqC' DtsubCyx (%the (tsub ([x] C x (pi1 (R x))) M (Cx (pi1 Rx))) DtsubCyx')) + %<- (expand-aasub-em _ _ nat-eq/i ([x] DsimpC x _) Dappend DofM ([x] [d] aofe/pi2 (DofR x d)) DexpandR2 (aasub/pi2 DsubRx) DtsubCyx' DexpandR2' (%the (sub ([x] N2 x) M N2') DsubN2x)) %. +%term sing expand-aasub-em _ _ _ _ _ _ _ ([_] expand/sing) _ (tsub/singa Dsub) expand/sing (sub/aa Dsub) %. +%scope sub-expand-var-em %term sing expand-aasub-em _ _ _ _ _ _ _ ([_] expand/sing) _ (tsub/singo Dsub) expand/sing (sub/ao Dsub) %. +%term t expand-aosub-em _ _ _ _ _ _ _ _ ([x] expand/t) Daosub (sub/ao Daosub) %. +%%% +%scope sub-expand-var-em %term pi + %pi (expand-aosub-em _ _ nat-eq/i ([x] simp/pi (%the ({y} simp (C x y) T) (DsimpC x)) (%the (simp (B x) S) (DsimpB x))) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) (%the (ofe G1 M A) DofM) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (pi (B x) ([y] C x y)))) DofR) ([x] %the (expand (R x) (pi (B x) ([y] C x y)) (lam ([y] O x y))) (expand/pi (%the ({y} expand (app (R x) (N x y)) (C x y) (O x y)) (Dexpand2 x)) (%the ({y} expand y (B x) (N x y)) (Dexpand1 x)))) (%the (aosub ([x] R x) M (lam ([y] P y))) Daosub) (sub/lam Dsub)) + %<- ({x} {d isvar x I} aofe-reg (DofR x d) (wfe/pi (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- ({x} {d isvar x I} {y} {e isvar y J} wfe-context (DwfC x d y e) (ordered/cons (%the (bounded (G x) y) (Dbounded x d y e)))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-wfe (Dbounded x d y e) (DwfB x d) _ (%the (wfe (cons (G x) y (B x)) (B x)) (DwfB' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-aofe (Dbounded x d y e) (DofR x d) _ (%the (aofe (cons (G x) y (B x)) (R x) (pi (B x) ([y] C x y))) (DofR' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} expand-reg-em _ _ nat-eq/i (DsimpB x) (aofe/var (DwfB' x d y e) (lookup/hit (Dbounded x d y e))) (Dexpand1 x y) (%the (ofe (cons (G x) y (B x)) (N x y) (B x)) (DofN x d y e))) + %<- ({x} {d isvar x I} weaken-wfe-insert1 ([y] [e] DwfC x d y e) (Dbounded x d) _ (%the ({y} + %pi (isvar y J) + %-> ({z} %pi (isvar z K) %-> (wfe (cons (cons (G x) y (B x)) z (B x)) (C x z)))) (DwfC' x d))) + %<- ({x} {d isvar x I} can-tsub-context-e ([y] [e] DofN x d y e) ([y] [e] DwfC x d y e) (%the ({y} tsub ([y] C x y) (N x y) (C' x y)) (Dtsub x))) + %<- ({x} {d isvar x I} {y} {e isvar y J} tsub-expand-em _ _ nat-eq/i (DsimpB x) ([_] append/nil) (aofe/var (DwfB' x d y e) (lookup/hit (Dbounded x d y e))) ([z] [f] DwfC' x d y e z f) (Dexpand1 x y) (Dtsub x y) (%the (tp-eq (C' x y) (C x y)) (Deq x y))) + %<- ({x} {y} tsub-resp ([_] tp-eq/i) term-eq/i (Deq x y) (Dtsub x y) (%the (tsub ([y] C x y) (N x y) (C x y)) (Dtsub' x y))) + %<- (can-tsub-e Dappend DwfB DofM (%the (tsub ([x] B x) M Bx) DsubBx)) + %<- ({y} {e isvar y J} can-tsub-e ([x] append/cons (Dappend x)) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DsubCx y))) + %<- ({y} can-expand y Bx (Nx y) (%the (expand y Bx (Nx y)) (Dexpand1' y))) + %<- ({y} {e isvar y J} expand-aasub-em _ _ nat-eq/i DsimpB ([x] append/cons (Dappend x)) DofM ([x] [d] aofe/var (DwfB' x d y e) (lookup/hit (Dbounded x d y e))) ([x] Dexpand1 x y) aasub/closed DsubBx (Dexpand1' y) (%the (sub ([x] N x y) M (Nx y)) (DsubNx y))) + %<- (tsubst-e Dappend Dcsub DofM DsubBx DwfB (%the (wfe Gx Bx) DwfBx)) + %<- (aosubst-e Dappend Dcsub DofM Daosub (tsub/pi DsubCx DsubBx) DofR (ofe/lam (%the ({y} %pi (isvar y J') %-> (ofe (cons Gx y Bx) (P y) (Cx y))) DofP) _)) + %<- (tsub-preserves-simp DsubBx DsimpB DsimpBx) + %<- (sub-expand-var-em _ _ nat-eq/i DsimpBx DwfBx DofP Dexpand1' (%the ({y} sub ([y] P y) (Nx y) (P y)) DsubPy)) + %<- ({y} {e isvar y J} expand-aosub-em _ _ nat-eq/i ([x] DsimpC x y) ([x] append/cons (Dappend x)) (csub/cons DsubBx Dcsub) DofM ([x] [d] aofe/app (DwfC x d y e) (Dtsub' x y) (DofN x d y e) (DofR' x d y e)) ([x] Dexpand2 x y) (%the (aosub ([x] app (R x) (N x y)) M (P y)) (aosub/app (DsubPy y) (DsubNx y) Daosub)) (%the (sub ([x] O x y) M (P y)) (Dsub y))) %. +%term pibad + %pi (expand-aosub-em _ _ _ _ (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) (%the (ofe G1 M A) DofM) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (pi (B x) ([y] C x y)))) DofR) _ (%the (aosub ([x] R x) M (at Q)) Daosub) Dsub) + %<- ({x} {d isvar x I} aofe-reg (DofR x d) (%the (wfe (G x) (pi (B x) ([y] C x y))) (Dwf x d))) + %<- (can-tsub-e Dappend Dwf DofM (%the (tsub ([x] pi (B x) ([y] C x y)) M (pi Bx ([y] Cx y))) Dtsub)) + %<- (aosubst-e Dappend Dcsub DofM Daosub Dtsub DofR (%the (ofe Gx (at Q) (pi Bx Cx)) Dof)) + %<- (ofe-at-pi-contra Dof Dfalse) + %<- (false-implies-sub Dfalse Dsub) %. +%scope sub-expand-var-em %term pibad + %pi (expand-aosub-em _ _ _ _ (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) (%the (ofe G1 M A) DofM) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (pi (B x) ([y] C x y)))) DofR) _ (%the (aosub ([x] R x) M (pair P1 P2)) Daosub) Dsub) + %<- ({x} {d isvar x I} aofe-reg (DofR x d) (%the (wfe (G x) (pi (B x) ([y] C x y))) (Dwf x d))) + %<- (can-tsub-e Dappend Dwf DofM (%the (tsub ([x] pi (B x) ([y] C x y)) M (pi Bx ([y] Cx y))) Dtsub)) + %<- (aosubst-e Dappend Dcsub DofM Daosub Dtsub DofR (%the (ofe Gx (pair P1 P2) (pi Bx Cx)) Dof)) + %<- (ofe-pair-pi-contra Dof Dfalse) + %<- (false-implies-sub Dfalse Dsub) %. +%term sigma + %pi (expand-aosub-em _ _ nat-eq/i ([x] simp/sigma (%the ({y} simp (C x y) T) (DsimpC x)) (%the (simp (B x) S) (DsimpB x))) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) (%the (ofe G1 M A) DofM) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (sigma (B x) ([y] C x y)))) DofR) ([x] %the (expand (R x) (sigma (B x) ([y] C x y)) (pair (N1 x) (N2 x))) (expand/sigma (%the (expand (pi2 (R x)) (C x (pi1 (R x))) (N2 x)) (Dexpand2 x)) (%the (expand (pi1 (R x)) (B x) (N1 x)) (Dexpand1 x)))) (%the (aosub ([x] R x) M (pair P1 P2)) Daosub) (sub/pair Dsub2 Dsub1)) + %<- (expand-aosub-em _ _ nat-eq/i DsimpB Dappend Dcsub DofM ([x] [d] aofe/pi1 (DofR x d)) Dexpand1 (aosub/pi1 Daosub) (%the (sub N1 M P1) Dsub1)) + %<- (expand-aosub-em _ _ nat-eq/i ([x] DsimpC x _) Dappend Dcsub DofM ([x] [d] aofe/pi2 (DofR x d)) Dexpand2 (aosub/pi2 Daosub) (%the (sub N2 M P2) Dsub2)) %. +%scope sub-expand-var-em %term sigmab + %pi (expand-aosub-em _ _ _ _ (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) (%the (ofe G1 M A) DofM) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (sigma (B x) ([y] C x y)))) DofR) _ (%the (aosub ([x] R x) M (at Q)) Daosub) Dsub) + %<- ({x} {d isvar x I} aofe-reg (DofR x d) (%the (wfe (G x) (sigma (B x) ([y] C x y))) (Dwf x d))) + %<- (can-tsub-e Dappend Dwf DofM (%the (tsub ([x] sigma (B x) ([y] C x y)) M (sigma Bx ([y] Cx y))) Dtsub)) + %<- (aosubst-e Dappend Dcsub DofM Daosub Dtsub DofR (%the (ofe Gx (at Q) (sigma Bx Cx)) Dof)) + %<- (ofe-at-sigma-contra Dof Dfalse) + %<- (false-implies-sub Dfalse Dsub) %. +%term sigmab + %pi (expand-aosub-em _ _ _ _ (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) (%the (ofe G1 M A) DofM) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (sigma (B x) ([y] C x y)))) DofR) _ (%the (aosub ([x] R x) M (lam P)) Daosub) Dsub) + %<- ({x} {d isvar x I} aofe-reg (DofR x d) (%the (wfe (G x) (sigma (B x) ([y] C x y))) (Dwf x d))) + %<- (can-tsub-e Dappend Dwf DofM (%the (tsub ([x] sigma (B x) ([y] C x y)) M (sigma Bx ([y] Cx y))) Dtsub)) + %<- (aosubst-e Dappend Dcsub DofM Daosub Dtsub DofR (%the (ofe Gx (lam P) (sigma Bx Cx)) Dof)) + %<- (ofe-lam-sigma-contra Dof Dfalse) + %<- (false-implies-sub Dfalse Dsub) %. +%scope sub-expand-var-em %term sing + %pi (expand-aosub-em _ _ _ _ (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) (%the (ofe G1 M A) DofM) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (sing (R' x)))) DofR) ([x] %the (expand (R x) (sing (R' x)) (at (R' x))) expand/sing) (%the (aosub R M N) Daosub) Dsub') + %<- ({x} {d isvar x I} aofe-reg (DofR x d) (wfe/sing (%the (aofe (G x) (R' x) t) (DofR' x d)))) + %<- (can-tsub-e Dappend ([x] [d] wfe/sing (DofR' x d)) DofM (%the (tsub ([x] sing (R' x)) M (sing R'')) Dtsub)) + %<- (tsub-sing-sub-at Dtsub (%the (sub ([x] at (R' x)) M (at R'')) Dsub)) + %<- (aosubst-e Dappend Dcsub DofM Daosub Dtsub DofR (%the (ofe Gx N (sing R'')) DofN)) + %<- (ofe-invert-sing DofN (%the (term-eq N (at R'')) Deq)) + %<- (term-eq-symm Deq (%the (term-eq (at R'') N) Deq')) + %<- (sub-resp ([_] term-eq/i) term-eq/i Deq' Dsub (%the (sub ([x] at (R' x)) M N) Dsub')) %. +%term closed aasub-expand-em _ _ _ _ _ _ _ _ aasub/closed atom-eq/i %. +%%% +%scope sub-expand-var-em %term forall + %pi (aasub-expand-em _ _ nat-eq/i (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (aofe G1 R A) DofR) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (forall (B x)) _)) DaofeQuant) (%the (expand R A XR) DexpandR) (aasub/forall (%the (tsub B XR Bx) DsubBx)) Deq') + %<- ({x} {d} aofe-invert-forall (DaofeQuant x d) (%the (wfe (G x) (B x)) (Dwf x d)) (DeqT x)) + %<- (tsub-expand-em _ _ nat-eq/i Dsimp Dappend DofR Dwf DexpandR DsubBx Deq) + %<- (atom-resp-tp forall Deq Deq') %. +%term app + %pi (aasub-expand-em _ _ nat-eq/i (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (aofe G1 R A) DofR) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (app (Q x) (M x)) (Cy x))) DaofeApp) (%the (expand R A XR) DexpandR) (aasub/app (%the (sub M XR Mx) DsubMx) (%the (aasub Q XR Qx) DsubQx)) Deq) + %<- ({x} {d isvar x I} aofe-invert-app (DaofeApp x d) (%the (aofe (G x) (Q x) (pi (B x) ([y] C x y))) (DofQ x d)) (%the (ofe (G x) (M x) (B x)) (DofM x d)) (%the (tsub ([y] C x y) (M x) (Cy x)) (DsubCy x))) + %<- (aasub-expand-em _ _ nat-eq/i Dsimp Dappend DofR DofQ DexpandR DsubQx (%the (atom-eq Qx (Q R)) DeqQ)) + %<- (sub-expand-em _ _ nat-eq/i Dsimp Dappend DofR DofM DexpandR DsubMx (%the (term-eq Mx (M R)) DeqM)) + %<- (atom-resp-atom ([r] app r Mx) DeqQ (%the (atom-eq (app Qx Mx) (app (Q R) Mx)) Deq1)) + %<- (atom-resp-term ([m] app (Q R) m) DeqM (%the (atom-eq (app (Q R) Mx) (app (Q R) (M R))) Deq2)) + %<- (atom-eq-trans Deq1 Deq2 (%the (atom-eq (app Qx Mx) (app (Q R) (M R))) Deq)) %. +%scope sub-expand-var-em %term pi1 + %pi (aasub-expand-em _ _ nat-eq/i (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (aofe G1 R A) DofR) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (pi1 (Q x)) (B x))) DaofeProj) (%the (expand R A XR) DexpandR) (aasub/pi1 (%the (aasub Q XR Qx) DsubQx)) Deq) + %<- ({x} {d isvar x I} aofe-invert-pi1 (DaofeProj x d) (%the (aofe (G x) (Q x) (sigma (B x) ([y] C x y))) (DofQ x d))) + %<- (aasub-expand-em _ _ nat-eq/i Dsimp Dappend DofR DofQ DexpandR DsubQx (%the (atom-eq Qx (Q R)) DeqQ)) + %<- (atom-resp-atom pi1 DeqQ (%the (atom-eq (pi1 Qx) (pi1 (Q R))) Deq)) %. +%term pi2 + %pi (aasub-expand-em _ _ nat-eq/i (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (aofe G1 R A) DofR) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (pi2 (Q x)) (Cy x))) DaofeProj) (%the (expand R A XR) DexpandR) (aasub/pi2 (%the (aasub Q XR Qx) DsubQx)) Deq) + %<- ({x} {d isvar x I} aofe-invert-pi2 (DaofeProj x d) (%the (aofe (G x) (Q x) (sigma (B x) ([y] C x y))) (DofQ x d)) (%the (tp-eq (Cy x) (C x (pi1 (Q x)))) (DeqCy x))) + %<- (aasub-expand-em _ _ nat-eq/i Dsimp Dappend DofR DofQ DexpandR DsubQx (%the (atom-eq Qx (Q R)) DeqQ)) + %<- (atom-resp-atom pi2 DeqQ (%the (atom-eq (pi2 Qx) (pi2 (Q R))) Deq)) %. +%%% +%scope sub-expand-var-em %term var + %pi (aosub-expand-em _ _ nat-eq/i (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (aofe G1 R A) DofR) (%the ({x} %pi (isvar x I) %-> (aofe (G x) x (B x))) DaofeVar) (%the (expand R A XR) DexpandR) aosub/var DexpandR') + %<- (aofe-invert-var DaofeVar (%the ({x} %pi (isvar x I) %-> (lookup (G x) x (B x))) Dlookup)) + %<- ({x} {d isvar x I} lookup-binder-fun (Dappend x) (Dlookup x d) (%the (tp-eq A (B x)) (Deq x))) + %<- (expand-resp atom-eq/i (Deq R) term-eq/i DexpandR (%the (expand R (B R) XR) DexpandR')) %. +%term app + %pi (aosub-expand-em _ _ nat-eq/i (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (aofe G1 R A) DofR) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (app (Q x) (M x)) (Cy x))) DaofeApp) (%the (expand R A XR) DexpandR) (aosub/app (%the (sub P Mx Py) DsubPy) (%the (sub M XR Mx) DsubMx) (%the (aosub Q XR (lam ([y] P y))) DsubP)) Dexpand') + %<- ({x} {d isvar x I} aofe-invert-app (DaofeApp x d) (%the (aofe (G x) (Q x) (pi (B x) ([y] C x y))) (DofQ x d)) (%the (ofe (G x) (M x) (B x)) (DofM x d)) (%the (tsub ([y] C x y) (M x) (Cy x)) (DsubCy x))) + %<- ({x} {d isvar x I} aofe-reg (DofQ x d) (wfe/pi (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- ({x} {d isvar x I} {y} {e isvar y J} wfe-context (DwfC x d y e) (ordered/cons (%the (bounded (G x) y) (Dbounded x d y e)))) + %<- (aosub-expand-em _ _ nat-eq/i Dsimp Dappend DofR DofQ DexpandR DsubP (%the (expand (Q R) (pi (B R) ([y] C R y)) (lam ([y] P y))) DexpandQx)) + %<- (expand-pi-invert DexpandQx (%the ({y} expand y (B R) (Y y)) Dexpand1) (%the ({y} expand (app (Q R) (Y y)) (C R y) (P y)) Dexpand2)) + %<- (sub-expand-em _ _ nat-eq/i Dsimp Dappend DofR DofM DexpandR DsubMx (%the (term-eq Mx (M R)) DeqM)) + %<- (can-expand (app (Q R) (M R)) (Cy R) XQM (%the (expand (app (Q R) (M R)) (Cy R) XQM) Dexpand)) + %<- (can-append G1 (G2 R) G' (%the (append G1 (G2 R) G') Dappend')) + %<- ({y} {e isvar y J} scsub-bounded Dappend Dappend' ([x] [d] Dbounded x d y e) (%the (bounded G' y) (Dbounded' y e))) + %<- (ssubst-e Dappend Dappend' DofM DofR (%the (ofe G' (M R) (B R)) DofMx)) + %<- (ofe-reg DofMx (%the (wfe G' (B R)) DwfBx)) + %<- ({y} {e isvar y J} weaken-wfe (Dbounded' y e) DwfBx _ (%the (wfe (cons G' y (B R)) (B R)) (DwfBx' y e))) + %<- ({y} {e isvar y J} stsubst-e ([x] append/cons (Dappend x)) (append/cons Dappend') ([x] [d] DwfC x d y e) DofR (%the (wfe (cons G' y (B R)) (C R y)) (DwfCx y e))) + %<- (aosub-headvar DsubP (%the (headvar ([x] Q x)) Dheadvar)) + %<- (headvar-tp-size Dappend DofQ Dsimp Dheadvar ([x] simp/pi (%the ({y} simp (C x y) U2) (DsimpC x)) (%the (simp (B x) U1) (DsimpB x))) (%the (stp-leq (spi U1 U2) T) Dleq)) + %<- (employ-stp-leq (spi U1 U2) T Dleq) + %<- (expand-aosub-em _ _ nat-eq/i ([_] DsimpB R) ([_] append/nil) csub/base DofMx ([y] [e isvar y J] aofe/var (DwfBx' y e) (lookup/hit (Dbounded' y e))) Dexpand1 aosub/var (%the (sub ([y] Y y) (M R) (M R)) DsubYy)) + %<- (aasub-absent (Q R) (M R) (%the (aasub ([y] Q R) (M R) (Q R)) DsubQxy)) + %<- (sasubst-e Dappend Dappend' DofQ DofR (%the (aofe G' (Q R) (pi (B R) ([y] C R y))) DofQx)) + %<- ({y} {e isvar y J} weaken-aofe (Dbounded' y e) DofQx _ (%the (aofe (cons G' y (B R)) (Q R) (pi (B R) ([y] C R y))) (DofQx' y e))) + %<- ({y} {e isvar y J} expand-reg-em _ _ nat-eq/i (DsimpB R) (aofe/var (DwfBx' y e) (lookup/hit (Dbounded' y e))) (Dexpand1 y) (%the (ofe (cons G' y (B R)) (Y y) (B R)) (DofY y e))) + %<- (tsub-expand-var-em _ _ nat-eq/i (DsimpB R) DwfBx DwfCx Dexpand1 (%the ({y} tsub ([y] C R y) (Y y) (C R y)) Dtsub)) + %<- (expand-aasub-em _ _ nat-eq/i ([y] DsimpC R y) ([y] append/nil) DofMx ([y] [e isvar y J] aofe/app (DwfCx y e) (Dtsub y) (DofY y e) (DofQx' y e)) Dexpand2 (%the (aasub ([y] app (Q R) (Y y)) (M R) (app (Q R) (M R))) (aasub/app DsubYy DsubQxy)) (%the (tsub ([y] C R y) (M R) (Cy R)) (DsubCy R)) Dexpand (%the (sub ([y] P y) (M R) XQM) DsubPy')) + %<- (sub-resp ([_] term-eq/i) DeqM term-eq/i DsubPy (%the (sub ([y] P y) (M R) Py) DsubPy'')) + %<- (sub-fun DsubPy' DsubPy'' (%the (term-eq XQM Py) DeqPy)) + %<- (expand-resp atom-eq/i tp-eq/i DeqPy Dexpand (%the (expand (app (Q R) (M R)) (Cy R) Py) Dexpand')) %. +%scope sub-expand-var-em %term pi1 + %pi (aosub-expand-em _ _ nat-eq/i (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (aofe G1 R A) DofR) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (pi1 (Q x)) (B x))) DaofeProj) (%the (expand R A XR) DexpandR) (aosub/pi1 (%the (aosub Q XR (pair P1 P2)) DsubP)) Dexpand1) + %<- ({x} {d isvar x I} aofe-invert-pi1 (DaofeProj x d) (%the (aofe (G x) (Q x) (sigma (B x) ([y] C x y))) (DofQ x d))) + %<- (aosub-expand-em _ _ nat-eq/i Dsimp Dappend DofR DofQ DexpandR DsubP (%the (expand (Q R) (sigma (B R) ([y] C R y)) (pair P1 P2)) DexpandQx)) + %<- (expand-sigma-invert DexpandQx (%the (expand (pi1 (Q R)) (B R) P1) Dexpand1) (%the (expand (pi2 (Q R)) (C R (pi1 (Q R))) P2) Dexpand2)) %. +%term pi2 + %pi (aosub-expand-em _ _ nat-eq/i (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (aofe G1 R A) DofR) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (pi2 (Q x)) (Cy x))) DaofeProj) (%the (expand R A XR) DexpandR) (aosub/pi2 (%the (aosub Q XR (pair P1 P2)) DsubP)) Dexpand2') + %<- ({x} {d isvar x I} aofe-invert-pi2 (DaofeProj x d) (%the (aofe (G x) (Q x) (sigma (B x) ([y] C x y))) (DofQ x d)) (%the (tp-eq (Cy x) (C x (pi1 (Q x)))) (DeqCy x))) + %<- ({x} tp-eq-symm (DeqCy x) (%the (tp-eq (C x (pi1 (Q x))) (Cy x)) (DeqCy' x))) + %<- (aosub-expand-em _ _ nat-eq/i Dsimp Dappend DofR DofQ DexpandR DsubP (%the (expand (Q R) (sigma (B R) ([y] C R y)) (pair P1 P2)) DexpandQx)) + %<- (expand-sigma-invert DexpandQx (%the (expand (pi1 (Q R)) (B R) P1) Dexpand1) (%the (expand (pi2 (Q R)) (C R (pi1 (Q R))) P2) Dexpand2)) + %<- (expand-resp atom-eq/i (DeqCy' R) term-eq/i Dexpand2 (%the (expand (pi2 (Q R)) (Cy R) P2) Dexpand2')) %. +%%% +%scope sub-expand-var-em %term aa + %pi (sub-expand-em _ _ nat-eq/i (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (aofe G1 R A) DofR) (%the ({x} %pi (isvar x I) %-> (ofe (G x) (at (Q x)) (B x))) DofAt) (%the (expand R A XR) DexpandR) (sub/aa (%the (aasub Q XR Qx) DsubQx)) Deq) + %<- ({x} {d isvar x I} ofe-invert-at (DofAt x d) (%the (aofe (G x) (Q x) t) (DofQ x d))) + %<- (aasub-expand-em _ _ nat-eq/i Dsimp Dappend DofR DofQ DexpandR DsubQx (%the (atom-eq Qx (Q R)) DeqQ)) + %<- (term-resp-atom at DeqQ (%the (term-eq (at Qx) (at (Q R))) Deq)) %. +%term ao + %pi (sub-expand-em _ _ nat-eq/i (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (aofe G1 R A) DofR) (%the ({x} %pi (isvar x I) %-> (ofe (G x) (at (Q x)) (B x))) DofAt) (%the (expand R A XR) DexpandR) (sub/ao (%the (aosub Q XR LQx) DsubQx)) Deq) + %<- ({x} {d isvar x I} ofe-invert-at (DofAt x d) (%the (aofe (G x) (Q x) t) (DofQ x d))) + %<- (aosub-expand-em _ _ nat-eq/i Dsimp Dappend DofR DofQ DexpandR DsubQx (%the (expand (Q R) t LQx) DexpandQ)) + %<- (expand-t-invert DexpandQ (%the (term-eq LQx (at (Q R))) Deq)) %. +%scope sub-expand-var-em %term lam + %pi (sub-expand-em _ _ nat-eq/i (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (aofe G1 R A) DofR) ([x] [d isvar x I] ofe/lam (%the ({y} %pi (isvar y J) %-> (ofe (cons (G x) y (B x)) (M x y) (C x y))) (DofM x d)) (%the (wfe (G x) (B x)) (DwfB x d))) (%the (expand R A XR) DexpandR) (sub/lam (%the ({y} sub ([x] M x y) XR (Mx y)) DsubMx)) Deq) + %<- ({y} {e isvar y J} sub-expand-em _ _ nat-eq/i Dsimp ([x] append/cons (Dappend x)) DofR ([x] [d] DofM x d y e) DexpandR (DsubMx y) (%the (term-eq (Mx y) (M R y)) (DeqM y))) + %<- (lam-resp DeqM (%the (term-eq (lam ([y] Mx y)) (lam ([y] M R y))) Deq)) %. +%term pair + %pi (sub-expand-em _ _ nat-eq/i (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (aofe G1 R A) DofR) ([x] [d isvar x I] ofe/pair (%the ({y} %pi (isvar y J) %-> _) _) (%the (ofe (G x) (N x) (Cy x)) (DofN x d)) (%the (tsub ([y] C x y) (M x) (Cy x)) (DsubCy x)) (%the (ofe (G x) (M x) (B x)) (DofM x d))) (%the (expand R A XR) DexpandR) (sub/pair (%the (sub ([x] N x) XR Nx) DsubNx) (%the (sub ([x] M x) XR Mx) DsubMx)) Deq) + %<- (sub-expand-em _ _ nat-eq/i Dsimp Dappend DofR DofM DexpandR DsubMx (%the (term-eq Mx (M R)) DeqM)) + %<- (sub-expand-em _ _ nat-eq/i Dsimp Dappend DofR DofN DexpandR DsubNx (%the (term-eq Nx (N R)) DeqN)) + %<- (term-resp-term ([m] pair m Nx) DeqM (%the (term-eq (pair Mx Nx) (pair (M R) Nx)) Deq1)) + %<- (term-resp-term ([m] pair (M R) m) DeqN (%the (term-eq (pair (M R) Nx) (pair (M R) (N R))) Deq2)) + %<- (term-eq-trans Deq1 Deq2 (%the (term-eq (pair Mx Nx) (pair (M R) (N R))) Deq)) %. +%%% +%scope sub-expand-var-em %term t tsub-expand-em _ _ _ _ _ _ _ _ tsub/t tp-eq/i %. +%term pi + %pi (tsub-expand-em _ _ nat-eq/i (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (aofe G1 R A) DofR) ([x] [d isvar x I] wfe/pi (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d))) (%the (expand R A XR) DexpandR) (tsub/pi (%the ({y} tsub ([x] C x y) XR (Cx y)) DsubCx) (%the (tsub B XR Bx) DsubBx)) Deq) + %<- (tsub-expand-em _ _ nat-eq/i Dsimp Dappend DofR DwfB DexpandR DsubBx (%the (tp-eq Bx (B R)) DeqB)) + %<- ({y} {e isvar y J} tsub-expand-em _ _ nat-eq/i Dsimp ([x] append/cons (Dappend x)) DofR ([x] [d] DwfC x d y e) DexpandR (DsubCx y) (%the (tp-eq (Cx y) (C R y)) (DeqC y))) + %<- (pi-resp DeqB DeqC (%the (tp-eq (pi Bx Cx) (pi (B R) ([y] C R y))) Deq)) %. +%scope sub-expand-var-em %term sigma + %pi (tsub-expand-em _ _ nat-eq/i (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (aofe G1 R A) DofR) ([x] [d isvar x I] wfe/sigma (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d))) (%the (expand R A XR) DexpandR) (tsub/sigma (%the ({y} tsub ([x] C x y) XR (Cx y)) DsubCx) (%the (tsub B XR Bx) DsubBx)) Deq) + %<- (tsub-expand-em _ _ nat-eq/i Dsimp Dappend DofR DwfB DexpandR DsubBx (%the (tp-eq Bx (B R)) DeqB)) + %<- ({y} {e isvar y J} tsub-expand-em _ _ nat-eq/i Dsimp ([x] append/cons (Dappend x)) DofR ([x] [d] DwfC x d y e) DexpandR (DsubCx y) (%the (tp-eq (Cx y) (C R y)) (DeqC y))) + %<- (sigma-resp DeqB DeqC (%the (tp-eq (sigma Bx Cx) (sigma (B R) ([y] C R y))) Deq)) %. +%term singa + %pi (tsub-expand-em _ _ nat-eq/i (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (aofe G1 R A) DofR) ([x] [d isvar x I] wfe/sing (%the (aofe (G x) (Q x) t) (DofQ x d))) (%the (expand R A XR) DexpandR) (tsub/singa (%the (aasub Q XR Qx) DsubQx)) Deq) + %<- (aasub-expand-em _ _ nat-eq/i Dsimp Dappend DofR DofQ DexpandR DsubQx (%the (atom-eq Qx (Q R)) DeqQ)) + %<- (tp-resp-atom sing DeqQ (%the (tp-eq (sing Qx) (sing (Q R))) Deq)) %. +%scope sub-expand-var-em %term singo + %pi (tsub-expand-em _ _ nat-eq/i (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (aofe G1 R A) DofR) ([x] [d isvar x I] wfe/sing (%the (aofe (G x) (Q x) t) (DofQ x d))) (%the (expand R A XR) DexpandR) (tsub/singo (%the (aosub Q XR (at Q')) DsubQx)) Deq) + %<- (aosub-expand-em _ _ nat-eq/i Dsimp Dappend DofR DofQ DexpandR DsubQx (%the (expand (Q R) t (at Q')) DexpandQ)) + %<- (expand-t-invert DexpandQ (%the (term-eq (at Q') (at (Q R))) DeqAtQ)) + %<- (term-eq-cdr-at DeqAtQ (%the (atom-eq Q' (Q R)) DeqQ)) + %<- (tp-resp-atom sing DeqQ (%the (tp-eq (sing Q') (sing (Q R))) Deq)) %. +%%% +%term _ + %pi (tsub-expand-var-em _ _ nat-eq/i (%the (simp A T) Dsimp) (%the (wfe G A) DwfA) (%the ({x} %pi (isvar x I) %-> (wfe (cons G x A) (B x))) DwfB) (%the ({x} expand x A (X x)) Dexpand) Dtsub') + %<- ({x} {d isvar x I} wfe-context (DwfB x d) (ordered/cons (%the (bounded G x) (Dbounded x d)))) + %<- ({x} {d isvar x I} weaken-wfe (Dbounded x d) DwfA _ (%the (wfe (cons G x A) A) (DwfA' x d))) + %<- ({x} {d isvar x I} expand-reg-em _ _ nat-eq/i Dsimp (aofe/var (DwfA' x d) (lookup/hit (Dbounded x d))) (Dexpand x) (%the (ofe (cons G x A) (X x) A) (DofX x d))) + %<- (can-tsub-context-e DofX DwfB (%the ({x} tsub ([x] B x) (X x) (B' x)) Dtsub)) + %<- (weaken-wfe-insert1 DwfB Dbounded _ (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (wfe (cons (cons G x A) y A) (B y)))) DwfB')) + %<- ({x} {d isvar x I} tsub-expand-em _ _ nat-eq/i Dsimp ([_] append/nil) (aofe/var (DwfA' x d) (lookup/hit (Dbounded x d))) ([y] [e] DwfB' x d y e) (Dexpand x) (Dtsub x) (%the (tp-eq (B' x) (B x)) (Deq x))) + %<- ({x} tsub-resp ([_] tp-eq/i) term-eq/i (Deq x) (Dtsub x) (%the (tsub ([x] B x) (X x) (B x)) (Dtsub' x))) %. +%%% +%term _ + %pi (sub-expand-var-em _ _ nat-eq/i (%the (simp A T) Dsimp) (%the (wfe G A) DwfA) (%the ({x} %pi (isvar x I) %-> (ofe (cons G x A) (M x) (B x))) DofM) (%the ({x} expand x A (X x)) Dexpand) Dsub') + %<- ({x} {d isvar x I} ofe-context (DofM x d) (ordered/cons (%the (bounded G x) (Dbounded x d)))) + %<- ({x} {d isvar x I} weaken-wfe (Dbounded x d) DwfA _ (%the (wfe (cons G x A) A) (DwfA' x d))) + %<- ({x} {d isvar x I} expand-reg-em _ _ nat-eq/i Dsimp (aofe/var (DwfA' x d) (lookup/hit (Dbounded x d))) (Dexpand x) (%the (ofe (cons G x A) (X x) A) (DofX x d))) + %<- (can-sub-context-e DofX DofM (%the ({x} sub ([x] M x) (X x) (M' x)) Dsub)) + %<- (weaken-ofe-insert1 DofM Dbounded _ (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y J) %-> (ofe (cons (cons G x A) y A) (M y) (B y)))) DofM')) + %<- ({x} {d isvar x I} sub-expand-em _ _ nat-eq/i Dsimp ([_] append/nil) (aofe/var (DwfA' x d) (lookup/hit (Dbounded x d))) ([y] [e] DofM' x d y e) (Dexpand x) (Dsub x) (%the (term-eq (M' x) (M x)) (Deq x))) + %<- ({x} sub-resp ([_] term-eq/i) term-eq/i (Deq x) (Dsub x) (%the (sub ([x] M x) (X x) (M x)) (Dsub' x))) %. +%worlds (bind ovar var) (expand-reg-em _ _ _ _ _ _ _) (expand-aasub-em _ _ _ _ _ _ _ _ _ _ _ _) (expand-aosub-em _ _ _ _ _ _ _ _ _ _ _) (aasub-expand-em _ _ _ _ _ _ _ _ _ _) (aosub-expand-em _ _ _ _ _ _ _ _ _ _) (sub-expand-em _ _ _ _ _ _ _ _ _ _) (tsub-expand-em _ _ _ _ _ _ _ _ _ _) (tsub-expand-var-em _ _ _ _ _ _ _ _) (sub-expand-var-em _ _ _ _ _ _ _ _) %. +%total {(T1 T2 T3 T4 T5 T6 T7 T8 T9) (N1 N2 N3 N4 N5 N6 N7 N8 N9) (D1 D2 D3 D4 D5 D6 D7 D8 D9)} (expand-reg-em N1 T1 _ _ _ D1 _) (expand-aasub-em N2 T2 _ _ _ _ _ _ _ _ D2 _) (expand-aosub-em N3 T3 _ _ _ _ _ _ _ D3 _) (aasub-expand-em N4 T4 _ _ _ _ _ _ D4 _) (aosub-expand-em N5 T5 _ _ _ _ _ _ D5 _) (sub-expand-em N6 T6 _ _ _ _ _ _ D6 _) (tsub-expand-em N7 T7 _ _ _ _ _ _ D7 _) (tsub-expand-var-em N8 T8 _ _ _ _ D8 _) (sub-expand-var-em N9 T9 _ _ _ _ D9 _) %. +%sort expand-reg-e %. +%mode expand-reg-e %in %in %out %. +%term _ + %pi (expand-reg-e Daof Dexpand Dof) + %<- (can-simp _ Dsimp) + %<- (expand-reg-em _ _ nat-eq/i Dsimp Daof Dexpand Dof) %. +%worlds (bind ovar var) (expand-reg-e _ _ _) %. +%total {} (expand-reg-e _ _ _) %. +%sort expand-aasub-e {x} %. +%mode expand-aasub-e %in %in %in %in %in %in %in %out %. +%term _ + %pi (expand-aasub-e D1 D2 D3 D4 D5 D6 D7 D8) + %<- ({x} can-simp _ (Dsimp x)) + %<- (expand-aasub-em _ _ nat-eq/i Dsimp D1 D2 D3 D4 D5 D6 D7 D8) %. +%worlds (bind ovar var) (expand-aasub-e _ _ _ _ _ _ _ _) %. +%total {} (expand-aasub-e _ _ _ _ _ _ _ _) %. +%sort expand-aosub-e {x} %. +%mode expand-aosub-e %in %in %in %in %in %in %out %. +%term _ + %pi (expand-aosub-e D1 D2 D3 D4 D5 D6 D7) + %<- ({x} can-simp _ (Dsimp x)) + %<- (expand-aosub-em _ _ nat-eq/i Dsimp D1 D2 D3 D4 D5 D6 D7) %. +%worlds (bind ovar var) (expand-aosub-e _ _ _ _ _ _ _) %. +%total {} (expand-aosub-e _ _ _ _ _ _ _) %. +%sort aasub-expand-e {x} %. +%mode aasub-expand-e %in %in %in %in %in %out %. +%term _ + %pi (aasub-expand-e D1 D2 D3 D4 D5 D6) + %<- (can-simp _ Dsimp) + %<- (aasub-expand-em _ _ nat-eq/i Dsimp D1 D2 D3 D4 D5 D6) %. +%worlds (bind ovar var) (aasub-expand-e _ _ _ _ _ _) %. +%total {} (aasub-expand-e _ _ _ _ _ _) %. +%sort aosub-expand-e {x} %. +%mode aosub-expand-e %in %in %in %in %in %out %. +%term _ + %pi (aosub-expand-e D1 D2 D3 D4 D5 D6) + %<- (can-simp _ Dsimp) + %<- (aosub-expand-em _ _ nat-eq/i Dsimp D1 D2 D3 D4 D5 D6) %. +%worlds (bind ovar var) (aosub-expand-e _ _ _ _ _ _) %. +%total {} (aosub-expand-e _ _ _ _ _ _) %. +%sort sub-expand-e {x} %. +%mode sub-expand-e %in %in %in %in %in %out %. +%term _ + %pi (sub-expand-e D1 D2 D3 D4 D5 D6) + %<- (can-simp _ Dsimp) + %<- (sub-expand-em _ _ nat-eq/i Dsimp D1 D2 D3 D4 D5 D6) %. +%worlds (bind ovar var) (sub-expand-e _ _ _ _ _ _) %. +%total {} (sub-expand-e _ _ _ _ _ _) %. +%sort tsub-expand-e {x} %. +%mode tsub-expand-e %in %in %in %in %in %out %. +%term _ + %pi (tsub-expand-e D1 D2 D3 D4 D5 D6) + %<- (can-simp _ Dsimp) + %<- (tsub-expand-em _ _ nat-eq/i Dsimp D1 D2 D3 D4 D5 D6) %. +%worlds (bind ovar var) (tsub-expand-e _ _ _ _ _ _) %. +%total {} (tsub-expand-e _ _ _ _ _ _) %. +%sort tsub-expand-var-e %. +%mode tsub-expand-var-e %in %in %in %out %. +%term _ + %pi (tsub-expand-var-e D1 D2 D3 D4) + %<- (can-simp _ Dsimp) + %<- (tsub-expand-var-em _ _ nat-eq/i Dsimp D1 D2 D3 D4) %. +%worlds (bind ovar var) (tsub-expand-var-e _ _ _ _) %. +%total {} (tsub-expand-var-e _ _ _ _) %. +%sort sub-expand-var-e %. +%mode sub-expand-var-e %in %in %in %out %. +%term _ + %pi (sub-expand-var-e D1 D2 D3 D4) + %<- (can-simp _ Dsimp) + %<- (sub-expand-var-em _ _ nat-eq/i Dsimp D1 D2 D3 D4) %. +%worlds (bind ovar var) (sub-expand-var-e _ _ _ _) %. +%total {} (sub-expand-var-e _ _ _ _) %. +%%%%% Expansion (Implicit Context) %%%%% +%sort expand-reg %. +%mode expand-reg %in %in %out %. +%term _ + %pi (expand-reg Daof Dexpand Dof) + %<- (aof-to-aofe Daof Daofe) + %<- (expand-reg-e Daofe Dexpand Dofe) + %<- (ofe-to-of Dofe Dof) %. +%worlds (bind var ovar) (expand-reg _ _ _) %. +%total {} (expand-reg _ _ _) %. +%sort expand-aasub %. +%mode expand-aasub %in %in %in %in %in %in %out %. +%term _ + %pi (expand-aasub DofN DaofR DexpandR Daasub Dtsub DexpandRx Dsub) + %<- (of-to-ofe DofN DofeN) + %<- (aof1-to-aofe 0 DaofR DaofeR) + %<- (expand-aasub-e ([_] append/nil) DofeN DaofeR DexpandR Daasub Dtsub DexpandRx Dsub) %. +%worlds (bind var ovar) (expand-aasub _ _ _ _ _ _ _) %. +%total {} (expand-aasub _ _ _ _ _ _ _) %. +%sort expand-aasub1 %. +%mode expand-aasub1 %in %in %in %in %in %in %out %. +%term _ + %pi (expand-aasub1 DofN DofR DexpandR Daasub DsubCx DexpandRx Dsub) + %<- (of-to-ofe DofN DofeN) + %<- (aof2-to-aofe (%the (lt 0 (s 0)) lt/z) DofR DofeR) + %<- ({y} {e} expand-aasub-e ([_] append/cons append/nil) DofeN ([x] [d] DofeR x d y e) ([x] DexpandR x y) (Daasub y) (DsubCx y) (DexpandRx y) (Dsub y)) %. +%worlds (bind var ovar) (expand-aasub1 _ _ _ _ _ _ _) %. +%total {} (expand-aasub1 _ _ _ _ _ _ _) %. +%sort expand-aosub %. +%mode expand-aosub %in %in %in %in %out %. +%term _ + %pi (expand-aosub DofN DaofR Dexpand Daosub Dsub) + %<- (of-to-ofe DofN DofeN) + %<- (aof1-to-aofe 0 DaofR DaofeR) + %<- (expand-aosub-e ([_] append/nil) csub/base DofeN DaofeR Dexpand Daosub Dsub) %. +%worlds (bind var ovar) (expand-aosub _ _ _ _ _) %. +%total {} (expand-aosub _ _ _ _ _) %. +%sort expand-aosub1 %. +%mode expand-aosub1 %in %in %in %in %in %out %. +%term _ + %pi (expand-aosub1 DofN DwfB DofR Dexpand Daosub Dsub) + %<- (can-tsub DwfB DofN DsubBx) + %<- (of-to-ofe DofN DofeN) + %<- (aof2-to-aofe (%the (lt 0 (s 0)) lt/z) DofR DofeR) + %<- ({y} {e} expand-aosub-e ([_] append/cons append/nil) (csub/cons DsubBx csub/base) DofeN ([x] [d] DofeR x d y e) ([x] Dexpand x y) (Daosub y) (Dsub y)) %. +%worlds (bind var ovar) (expand-aosub1 _ _ _ _ _ _) %. +%total {} (expand-aosub1 _ _ _ _ _ _) %. +%sort aasub-expand %. +%mode aasub-expand %in %in %in %in %out %. +%term _ + %pi (aasub-expand DaofR DaofQ Dexpand Daasub Deq) + %<- (aof-to-aofe DaofR DaofeR) + %<- (aof1-to-aofe 0 DaofQ DaofeQ) + %<- (aasub-expand-e ([_] append/nil) DaofeR DaofeQ Dexpand Daasub Deq) %. +%worlds (bind var ovar) (aasub-expand _ _ _ _ _) %. +%total {} (aasub-expand _ _ _ _ _) %. +%sort aosub-expand %. +%mode aosub-expand %in %in %in %in %out %. +%term _ + %pi (aosub-expand DaofR DaofQ Dexpand Daosub Dexpand') + %<- (aof-to-aofe DaofR DaofeR) + %<- (aof1-to-aofe 0 DaofQ DaofeQ) + %<- (aosub-expand-e ([_] append/nil) DaofeR DaofeQ Dexpand Daosub Dexpand') %. +%worlds (bind var ovar) (aosub-expand _ _ _ _ _) %. +%total {} (aosub-expand _ _ _ _ _) %. +%sort sub-expand %. +%mode sub-expand %in %in %in %in %out %. +%term _ + %pi (sub-expand DaofR DofM Dexpand Daasub Deq) + %<- (aof-to-aofe DaofR DaofeR) + %<- (of1-to-ofe 0 DofM DofeM) + %<- (sub-expand-e ([_] append/nil) DaofeR DofeM Dexpand Daasub Deq) %. +%worlds (bind var ovar) (sub-expand _ _ _ _ _) %. +%total {} (sub-expand _ _ _ _ _) %. +%sort tsub-expand %. +%mode tsub-expand %in %in %in %in %out %. +%term _ + %pi (tsub-expand DaofR DwfB Dexpand Daasub Deq) + %<- (aof-to-aofe DaofR DaofeR) + %<- (wf1-to-wfe 0 DwfB DwfeB) + %<- (tsub-expand-e ([_] append/nil) DaofeR DwfeB Dexpand Daasub Deq) %. +%worlds (bind var ovar) (tsub-expand _ _ _ _ _) %. +%total {} (tsub-expand _ _ _ _ _) %. +%%%%% Corollaries %%%%% +%sort sub-expand-var %. +%mode sub-expand-var %in %in %in %out %. +%term _ + %pi (sub-expand-var DwfA DofM Dexpand Dsub') + %<- ({x} {d vof x A} expand-reg (aof/var DwfA d) (Dexpand x) (DofX x d)) + %<- ({x} {d vof x A} can-sub DofM (DofX x d) (%the (sub ([y] M y) (X x) (M' x)) (Dsub x))) + %<- ({x} {d vof x A} sub-expand (aof/var DwfA d) DofM (Dexpand x) (Dsub x) (%the (term-eq (M' x) (M x)) (Deq x))) + %<- ({x} sub-resp ([_] term-eq/i) term-eq/i (Deq x) (Dsub x) (%the (sub ([y] M y) (X x) (M x)) (Dsub' x))) %. +%worlds (bind) (sub-expand-var _ _ _ _) %. +%total {} (sub-expand-var _ _ _ _) %. +%sort tsub-expand-var-e %. +%mode tsub-expand-var-e %in %in %in %out %. +%term _ + %pi (tsub-expand-var-e DwfA DwfB Dexpand Dtsub') + %<- ({x} {d} wfe-context (DwfB x d) (ordered/cons (Dord x d))) + %<- ({x} {d} weaken-wfe (Dord x d) DwfA _ (DwfA' x d)) + %<- ({x} {d} expand-reg-e (aofe/var (DwfA' x d) (lookup/hit (Dord x d))) (Dexpand x) (DofX x d)) + %<- (can-tsub-context-e DofX DwfB (%the ({x} tsub ([y] B y) (X x) (B' x)) Dtsub)) + %<- (weaken-wfe-insert1 DwfB Dord _ DwfB') + %<- ({x} {d} tsub-expand-e ([_] append/nil) (aofe/var (DwfA' x d) (lookup/hit (Dord x d))) ([y] [e] DwfB' x d y e) (Dexpand x) (Dtsub x) (%the (tp-eq (B' x) (B x)) (Deq x))) + %<- ({x} tsub-resp ([_] tp-eq/i) term-eq/i (Deq x) (Dtsub x) (%the (tsub ([y] B y) (X x) (B x)) (Dtsub' x))) %. +%worlds (bind var ovar) (tsub-expand-var-e _ _ _ _) %. +%total {} (tsub-expand-var-e _ _ _ _) %. +%sort tsub-expand-var %. +%mode tsub-expand-var %in %in %in %out %. +%term _ + %pi (tsub-expand-var DwfA DwfB Dexpand Dtsub') + %<- ({x} {d vof x A} expand-reg (aof/var DwfA d) (Dexpand x) (DofX x d)) + %<- ({x} {d vof x A} can-tsub DwfB (DofX x d) (%the (tsub ([y] B y) (X x) (B' x)) (Dtsub x))) + %<- ({x} {d vof x A} tsub-expand (aof/var DwfA d) DwfB (Dexpand x) (Dtsub x) (%the (tp-eq (B' x) (B x)) (Deq x))) + %<- ({x} tsub-resp ([_] tp-eq/i) term-eq/i (Deq x) (Dtsub x) (%the (tsub ([y] B y) (X x) (B x)) (Dtsub' x))) %. +%worlds (bind var ovar) (tsub-expand-var _ _ _ _) %. +%total {} (tsub-expand-var _ _ _ _) %. +%sort sub-into-expand-var {x} %. +%mode sub-into-expand-var %in %in %out %. +%term _ + %pi (sub-into-expand-var Dexpand DofM Dsub) + %<- (of-reg DofM DwfA) + %<- (expand-aosub DofM ([x] [d] aof/var DwfA d) Dexpand aosub/var (%the (sub X M M) Dsub)) %. +%worlds (bind var) (sub-into-expand-var _ _ _) %. +%total {} (sub-into-expand-var _ _ _) %. +%%%%% Miscellaneous %%%%% +%sort strengthen-expand {x atom} %. +%mode strengthen-expand %in %out %. +%term _ + %pi (strengthen-expand Dexpand Deq) + %<- (can-expand R A M' Dexpand') + %<- ({x} expand-fun (Dexpand x) Dexpand' (Deq x)) %. +%worlds (var var) (strengthen-expand _ _) %. +%total {} (strengthen-expand _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/expand.lf b/new-tests/stelf-output/tslf/sing/expand.lf new file mode 100644 index 0000000..4d293b2 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/expand.lf @@ -0,0 +1,13 @@ +%%%%% Expansion %%%%% +%sort expand %. +%mode expand %in %in %out %. +%term expand/t expand R t (at R) %. +%term expand/pi + %pi (expand R (pi A B) (lam N)) + %<- ({x} expand x A (M x)) + %<- ({x} expand (app R (M x)) (B x) (N x)) %. +%term expand/sigma + %pi (expand R (sigma A B) (pair M N)) + %<- (expand (pi1 R) A M) + %<- (expand (pi2 R) (B (pi1 R)) N) %. +%term expand/sing expand R (sing R') (at R') %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/explicit-lemmas-thm.lf b/new-tests/stelf-output/tslf/sing/explicit-lemmas-thm.lf new file mode 100644 index 0000000..ed5cb1e --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/explicit-lemmas-thm.lf @@ -0,0 +1,2094 @@ +%%%%% Ordered Variables %%%%% +%sort isvar-fun %. +%mode isvar-fun %in %in %out %. +%term _ isvar-fun D D nat-eq/i %. +%worlds (var ovar obind) (isvar-fun _ _ _) %. +%total {} (isvar-fun _ _ _) %. +%sort precedes-trans %. +%mode precedes-trans %in %in %out %. +%term _ + %pi (precedes-trans (precedes/i DltIJ (%the (isvar Y J) DisvarY) (%the (isvar X I) DisvarX)) (precedes/i DltJ'K (%the (isvar Z K) DisvarZ) (%the (isvar Y J') DisvarY')) (precedes/i DltIK DisvarZ DisvarX)) + %<- (isvar-fun DisvarY DisvarY' (%the (nat-eq J J') Deq)) + %<- (lt-resp nat-eq/i Deq DltIJ DltIJ') + %<- (lt-trans DltIJ' DltJ'K DltIK) %. +%worlds (var ovar) (precedes-trans _ _ _) %. +%total {} (precedes-trans _ _ _) %. +%sort precedes-trans-abs {x} %. +%mode precedes-trans-abs %in %in %out %. +%term _ + %pi (precedes-trans-abs ([x] [d] [y] [e] precedes/i DltIJ e d) ([y] [e] [z] [f] precedes/i DltJK f e) ([x] [d] [z] [f] precedes/i DltIK f d)) + %<- (lt-trans DltIJ DltJK DltIK) %. +%worlds (var ovar) (precedes-trans-abs _ _ _) %. +%total {} (precedes-trans-abs _ _ _) %. +%sort precedes-antisymm %. +%mode precedes-antisymm %in %out %. +%term _ + %pi (precedes-antisymm (precedes/i (%the (lt I J) Dlt) (%the (isvar X J) Disvar') (%the (isvar X I) Disvar)) Dfalse) + %<- (isvar-fun Disvar Disvar' (%the (nat-eq I J) Deq)) + %<- (lt-resp Deq nat-eq/i Dlt (%the (lt J J) Dlt')) + %<- (lt-antisymm Dlt' Dfalse) %. +%worlds (var ovar) (precedes-antisymm _ _) %. +%total {} (precedes-antisymm _ _) %. +%sort bounded-isvar %. +%mode bounded-isvar %in %out %. +%term _ bounded-isvar (bounded/nil D) D %. +%term _ bounded-isvar (bounded/cons _ (precedes/i _ D _)) D %. +%worlds (var ovar) (bounded-isvar _ _) %. +%total {} (bounded-isvar _ _) %. +%sort sbounded-isvar %. +%mode sbounded-isvar %in %out %. +%term _ sbounded-isvar (sbounded/nil D) D %. +%term _ sbounded-isvar (sbounded/cons _ (precedes/i _ D _)) D %. +%worlds (var ovar) (sbounded-isvar _ _) %. +%total {} (sbounded-isvar _ _) %. +%sort lookup-isvar %. +%mode lookup-isvar %in %out %. +%term _ %pi (lookup-isvar (lookup/hit D) D') %<- (bounded-isvar D D') %. +%term _ %pi (lookup-isvar (lookup/miss D _) D') %<- (lookup-isvar D D') %. +%worlds (var ovar) (lookup-isvar _ _) %. +%total D (lookup-isvar D _) %. +%sort slookup-isvar %. +%mode slookup-isvar %in %out %. +%term _ %pi (slookup-isvar (slookup/hit D) D') %<- (sbounded-isvar D D') %. +%term _ %pi (slookup-isvar (slookup/miss D _) D') %<- (slookup-isvar D D') %. +%worlds (var ovar) (slookup-isvar _ _) %. +%total D (slookup-isvar D _) %. +%sort isvar-app-contra %. +%mode isvar-app-contra %in %out %. +%worlds (bind ovar var) (isvar-app-contra _ _) %. +%total {} (isvar-app-contra _ _) %. +%sort isvar-pi1-contra %. +%mode isvar-pi1-contra %in %out %. +%worlds (bind ovar var) (isvar-pi1-contra _ _) %. +%total {} (isvar-pi1-contra _ _) %. +%sort isvar-pi2-contra %. +%mode isvar-pi2-contra %in %out %. +%worlds (bind ovar var) (isvar-pi2-contra _ _) %. +%total {} (isvar-pi2-contra _ _) %. +%sort isvar-forall-contra %. +%mode isvar-forall-contra %in %out %. +%worlds (bind ovar var) (isvar-forall-contra _ _) %. +%total {} (isvar-forall-contra _ _) %. +%sort vof-isvar-contra %. +%mode vof-isvar-contra %in %in %out %. +%worlds (var bind ovar) (vof-isvar-contra _ _ _) %. +%total {} (vof-isvar-contra _ _ _) %. +%%%%% Context Manipulation %%%%% +%sort can-append {G1} {G2} {G} %. +%mode can-append %in %in %out %out %. +%term _ can-append _ _ _ append/nil %. +%term _ %pi (can-append _ _ _ (append/cons D)) %<- (can-append _ _ _ D) %. +%worlds (var) (can-append _ _ _ _) %. +%total G (can-append _ G _ _) %. +%sort bounded-is-ordered %. +%mode bounded-is-ordered %in %out %. +%term _ bounded-is-ordered (bounded/nil _) ordered/nil %. +%term _ bounded-is-ordered (bounded/cons D _) (ordered/cons D) %. +%worlds (var bind ovar obind) (bounded-is-ordered _ _) %. +%total {} (bounded-is-ordered _ _) %. +%sort sbounded-is-sordered %. +%mode sbounded-is-sordered %in %out %. +%term _ sbounded-is-sordered (sbounded/nil _) sordered/nil %. +%term _ sbounded-is-sordered (sbounded/cons D _) (sordered/cons D) %. +%worlds (var bind ovar obind) (sbounded-is-sordered _ _) %. +%total {} (sbounded-is-sordered _ _) %. +%sort bounded-increase-bound %. +%mode bounded-increase-bound %in %in %out %. +%term _ bounded-increase-bound (bounded/nil _) (precedes/i _ D _) (bounded/nil D) %. +%term _ + %pi (bounded-increase-bound (bounded/cons Dbounded Dprecedes) Dprecedes' (bounded/cons Dbounded Dprecedes'')) + %<- (precedes-trans Dprecedes Dprecedes' Dprecedes'') %. +%worlds (bind ovar var) (bounded-increase-bound _ _ _) %. +%total {} (bounded-increase-bound _ _ _) %. +%sort sbounded-increase-bound %. +%mode sbounded-increase-bound %in %in %out %. +%term _ sbounded-increase-bound (sbounded/nil _) (precedes/i _ D _) (sbounded/nil D) %. +%term _ + %pi (sbounded-increase-bound (sbounded/cons Dbounded Dprecedes) Dprecedes' (sbounded/cons Dbounded Dprecedes'')) + %<- (precedes-trans Dprecedes Dprecedes' Dprecedes'') %. +%worlds (bind ovar var) (sbounded-increase-bound _ _ _) %. +%total {} (sbounded-increase-bound _ _ _) %. +%sort append-bounded %. +%mode append-bounded %in %in %out %. +%term _ + %pi (append-bounded (append/cons Dappend) (bounded/cons Dbounded Dprecedes) D) + %<- (bounded-increase-bound Dbounded Dprecedes Dbounded') + %<- (append-bounded Dappend Dbounded' D) %. +%term _ append-bounded append/nil D D %. +%worlds (bind ovar var) (append-bounded _ _ _) %. +%total D (append-bounded D _ _) %. +%sort append-ordered %. +%mode append-ordered %in %in %out %. +%term _ + %pi (append-ordered (append/cons Dappend) (ordered/cons Dbounded) D') + %<- (append-bounded Dappend Dbounded D) + %<- (bounded-is-ordered D D') %. +%term _ append-ordered append/nil D D %. +%worlds (var bind ovar) (append-ordered _ _ _) %. +%total {} (append-ordered _ _ _) %. +%{ +sappend-sbounded : sappend G1 G2 G -> sbounded G X -> sbounded G1 X -> type. +%mode sappend-sbounded +X1 +X2 -X3. + +- : sappend-sbounded (sappend/cons Dsappend) (sbounded/cons Dsbounded Dprecedes) D + <- sbounded-increase-bound Dsbounded Dprecedes Dsbounded' + <- sappend-sbounded Dsappend Dsbounded' D. + +- : sappend-sbounded sappend/nil D D. + +%worlds (bind | ovar | var) (sappend-sbounded _ _ _). +%total D (sappend-sbounded D _ _). + + + +sappend-sordered : sappend G1 G2 G -> sordered G -> sordered G1 -> type. +%mode sappend-sordered +X1 +X2 -X3. + +- : sappend-sordered (sappend/cons Dsappend) (sordered/cons Dsbounded) D' + <- sappend-sbounded Dsappend Dsbounded D + <- sbounded-is-sordered D D'. + +- : sappend-sordered sappend/nil D D. + +%worlds (var | bind | ovar) (sappend-sordered _ _ _). +%total {} (sappend-sordered _ _ _). +}% +%{ +append-lookup-bounded : append (cons G1 X A) G2 G -> lookup G _ _ -> bounded G1 X -> type. +%mode append-lookup-bounded +X1 +X2 -X3. + +- : append-lookup-bounded Dappend Dlookup Dbounded + <- lookup-context Dlookup Dordered + <- append-ordered Dappend Dordered + (ordered/cons Dbounded). + +%worlds (var | ovar) (append-lookup-bounded _ _ _). +%total {} (append-lookup-bounded _ _ _). +}% +%% this can be proved more elegantly using append-bounded +%sort append-bounded-contra %. +%mode append-bounded-contra %in %in %out %. +%term _ + %pi (append-bounded-contra append/nil (bounded/cons _ Dprecedes) Dfalse) + %<- (precedes-antisymm Dprecedes Dfalse) %. +%term _ + %pi (append-bounded-contra (append/cons Dapp) (bounded/cons Dbounded Dprecedes) Dfalse) + %<- (bounded-increase-bound Dbounded Dprecedes Dbounded') + %<- (append-bounded-contra Dapp Dbounded' Dfalse) %. +%worlds (bind ovar var) (append-bounded-contra _ _ _) %. +%total D (append-bounded-contra D _ _) %. +%sort strengthen-bounded {x} %. +%mode strengthen-bounded %in %out %. +%term _ strengthen-bounded ([x] [d] bounded/nil D) (bounded/nil D) %. +%term _ + %pi (strengthen-bounded ([x] [d] bounded/cons (Dbound x d) Dprec) (bounded/cons Dbound' Dprec)) + %<- (strengthen-bounded Dbound Dbound') %. +%worlds (var bind ovar obind) (strengthen-bounded _ _) %. +%total D (strengthen-bounded D _) %. +%sort strengthen-ordered {x} %. +%mode strengthen-ordered %in %out %. +%term _ strengthen-ordered ([x] [d] ordered/nil) ordered/nil %. +%term _ + %pi (strengthen-ordered ([x] [d] ordered/cons (Dbound x d)) (ordered/cons Dbound')) + %<- (strengthen-bounded Dbound Dbound') %. +%worlds (var ovar) (strengthen-ordered _ _) %. +%total {} (strengthen-ordered _ _) %. +%sort bounded-is-ordered-strengthen {x} %. +%mode bounded-is-ordered-strengthen %in %out %. +%term _ bounded-is-ordered-strengthen ([x] [d] bounded/nil d) ordered/nil %. +%term _ + %pi (bounded-is-ordered-strengthen ([x] [d] bounded/cons (D x d) (precedes/i Dlt d Disvar)) (ordered/cons D')) + %<- (strengthen-bounded D D') %. +%worlds (var bind ovar obind) (bounded-is-ordered-strengthen _ _) %. +%total {} (bounded-is-ordered-strengthen _ _) %. +%sort strengthen-sbounded {x} %. +%mode strengthen-sbounded %in %out %. +%term _ strengthen-sbounded ([x] [d] sbounded/nil D) (sbounded/nil D) %. +%term _ + %pi (strengthen-sbounded ([x] [d] sbounded/cons (Dbound x d) Dprec) (sbounded/cons Dbound' Dprec)) + %<- (strengthen-sbounded Dbound Dbound') %. +%worlds (var bind ovar obind) (strengthen-sbounded _ _) %. +%total D (strengthen-sbounded D _) %. +%sort strengthen-sordered {x} %. +%mode strengthen-sordered %in %out %. +%term _ strengthen-sordered ([x] [d] sordered/nil) sordered/nil %. +%term _ + %pi (strengthen-sordered ([x] [d] sordered/cons (Dbound x d)) (sordered/cons Dbound')) + %<- (strengthen-sbounded Dbound Dbound') %. +%worlds (var ovar) (strengthen-sordered _ _) %. +%total {} (strengthen-sordered _ _) %. +%sort sbounded-is-sordered-strengthen {x} %. +%mode sbounded-is-sordered-strengthen %in %out %. +%term _ sbounded-is-sordered-strengthen ([x] [d] sbounded/nil d) sordered/nil %. +%term _ + %pi (sbounded-is-sordered-strengthen ([x] [d] sbounded/cons (D x d) (precedes/i Dlt d Disvar)) (sordered/cons D')) + %<- (strengthen-sbounded D D') %. +%worlds (var bind ovar obind) (sbounded-is-sordered-strengthen _ _) %. +%total {} (sbounded-is-sordered-strengthen _ _) %. +%sort strengthen-lookup {x} %. +%mode strengthen-lookup %in %out %out %. +%term _ + %pi (strengthen-lookup ([x] [d] lookup/hit (Dbounded x d)) (lookup/hit Dbounded') ([_] tp-eq/i)) + %<- (strengthen-bounded Dbounded Dbounded') %. +%term _ + %pi (strengthen-lookup ([x] [d] lookup/miss (Dlookup x d) (Dbounded x d)) (lookup/miss Dlookup' Dbounded') Deq) + %<- (strengthen-lookup Dlookup Dlookup' Deq) + %<- (strengthen-bounded Dbounded Dbounded') %. +%worlds (var bind ovar) (strengthen-lookup _ _ _) %. +%total D (strengthen-lookup D _ _) %. +%sort strengthen-slookup {x} %. +%mode strengthen-slookup %in %out %. +%term _ + %pi (strengthen-slookup ([x] [d] slookup/hit (Dbounded x d)) (slookup/hit Dbounded')) + %<- (strengthen-sbounded Dbounded Dbounded') %. +%term _ + %pi (strengthen-slookup ([x] [d] slookup/miss (Dlookup x d) (Dbounded x d)) (slookup/miss Dlookup' Dbounded')) + %<- (strengthen-slookup Dlookup Dlookup') + %<- (strengthen-sbounded Dbounded Dbounded') %. +%worlds (var bind ovar) (strengthen-slookup _ _) %. +%total D (strengthen-slookup D _) %. +%sort bounded-increase-bound-abs {x} %. +%mode bounded-increase-bound-abs %in %in %out %. +%term _ + %pi (bounded-increase-bound-abs Dprecedes D D'') + %<- ({x} {d isvar x I} {y} {e isvar y J} bounded-increase-bound (D x d) (Dprecedes x d y e) (D' x d y e)) + %<- ({y} {e isvar y J} strengthen-bounded ([x] [d] D' x d y e) (D'' y e)) %. +%worlds (var bind ovar) (bounded-increase-bound-abs _ _ _) %. +%total {} (bounded-increase-bound-abs _ _ _) %. +%sort strengthen-precedes {x} %. +%mode strengthen-precedes %in %out %. +%term _ strengthen-precedes ([x] [d] precedes/i Dlt D1 D2) (precedes/i Dlt D1 D2) %. +%worlds (var bind ovar obind) (strengthen-precedes _ _) %. +%total {} (strengthen-precedes _ _) %. +%sort csub-bounded {x} %. +%mode csub-bounded %in %in %out %. +%term _ + %pi (csub-bounded ([x] [d] bounded/cons (Dbound x d) (Dprec x d)) csub/base Dbound'') + %<- ({x} {d} bounded-increase-bound (Dbound x d) (Dprec x d) (Dbound' x d)) + %<- (strengthen-bounded Dbound' Dbound'') %. +%term _ + %pi (csub-bounded ([x] [d] bounded/cons (Dbound x d) Dprec) (csub/cons Dsub Dcsub) (bounded/cons Dbound' Dprec)) + %<- (csub-bounded Dbound Dcsub Dbound') %. +%worlds (bind ovar var) (csub-bounded _ _ _) %. +%total D (csub-bounded D _ _) %. +%sort csub-ordered {x} %. +%mode csub-ordered %in %in %out %. +%term _ + %pi (csub-ordered ([x] [d] ordered/cons (Dbound x d)) csub/base Dord) + %<- (bounded-is-ordered-strengthen Dbound Dord) %. +%term _ + %pi (csub-ordered ([x] [d] ordered/cons (Dbound x d)) (csub/cons Dsub Dcsub) (ordered/cons Dbound')) + %<- (csub-bounded Dbound Dcsub Dbound') %. +%worlds (bind ovar var) (csub-ordered _ _ _) %. +%total {} (csub-ordered _ _ _) %. +%sort scsub-bounded {x} %. +%mode scsub-bounded %in %in %in %out %. +%term _ + %pi (scsub-bounded ([x] append/cons (%the (append (cons G1 x A) (G2 x) (G x)) (Dappend x))) (append/cons Dappend') ([x] [d] bounded/cons (Dbounded x d) Dprecedes) (bounded/cons Dbounded' Dprecedes)) + %<- (scsub-bounded Dappend Dappend' Dbounded Dbounded') %. +%term _ + %pi (scsub-bounded ([x] append/nil) append/nil ([x] [d] bounded/cons (Dbounded x d) (Dprecedes x d)) Dbounded'') + %<- ({x} {d} bounded-increase-bound (Dbounded x d) (Dprecedes x d) (Dbounded' x d)) + %<- (strengthen-bounded Dbounded' Dbounded'') %. +%term _ + %pi (scsub-bounded ([x] append/cons (Dappend x)) (append/cons Dappend') ([x] [d] bounded/cons (%the (bounded (G x) x) (Dbounded x d)) _) D) + %<- ({x} {d} append-bounded-contra (Dappend x) (Dbounded x d) Dfalse) + %<- (false-implies-bounded Dfalse D) %. +%worlds (bind ovar var) (scsub-bounded _ _ _ _) %. +%total D (scsub-bounded D _ _ _) %. +%sort scsub-ordered {x} %. +%mode scsub-ordered %in %in %in %out %. +%term _ + %pi (scsub-ordered ([x] append/cons (%the (append (cons G1 x A) (G2 x) (G x)) (Dappend x))) (append/cons Dappend') ([x] [d] ordered/cons (Dbounded x d)) (ordered/cons Dbounded')) + %<- (scsub-bounded Dappend Dappend' Dbounded Dbounded') %. +%term _ + %pi (scsub-ordered ([x] append/nil) append/nil ([x] [d] ordered/cons (Dbounded x d)) D) + %<- (bounded-is-ordered-strengthen Dbounded D) %. +%term _ + %pi (scsub-ordered ([x] append/cons (Dappend x)) (append/cons Dappend') ([x] [d] ordered/cons (%the (bounded (G x) x) (Dbounded x d))) D) + %<- ({x} {d} append-bounded-contra (Dappend x) (Dbounded x d) Dfalse) + %<- (false-implies-ordered Dfalse D) %. +%term _ + %pi (scsub-ordered _ _ ([x] [d] ordered/cons (Dbounded x d)) D) + %<- ({x} {d} bounded-isvar (Dbounded x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-app-contra (Disvar x d) Dfalse) + %<- (false-implies-ordered Dfalse D) %. +%term _ + %pi (scsub-ordered _ _ ([x] [d] ordered/cons (Dbounded x d)) D) + %<- ({x} {d} bounded-isvar (Dbounded x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-pi1-contra (Disvar x d) Dfalse) + %<- (false-implies-ordered Dfalse D) %. +%term _ + %pi (scsub-ordered _ _ ([x] [d] ordered/cons (Dbounded x d)) D) + %<- ({x} {d} bounded-isvar (Dbounded x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-pi2-contra (Disvar x d) Dfalse) + %<- (false-implies-ordered Dfalse D) %. +%term _ + %pi (scsub-ordered _ _ ([x] [d] ordered/cons (Dbounded x d)) D) + %<- ({x} {d} bounded-isvar (Dbounded x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-forall-contra (Disvar x d) Dfalse) + %<- (false-implies-ordered Dfalse D) %. +%worlds (bind ovar var) (scsub-ordered _ _ _ _) %. +%total {} (scsub-ordered _ _ _ _) %. +%sort bounded-lookup %. +%mode bounded-lookup %in %in %out %. +%term _ bounded-lookup (bounded/cons _ Dprecedes) (lookup/hit _) Dprecedes %. +%term _ + %pi (bounded-lookup (bounded/cons Dbounded Dprecedes) (lookup/miss Dlookup _) Dprecedes'') + %<- (bounded-lookup Dbounded Dlookup Dprecedes') + %<- (precedes-trans Dprecedes' Dprecedes Dprecedes'') %. +%worlds (var ovar bind) (bounded-lookup _ _ _) %. +%total D (bounded-lookup D _ _) %. +%sort sbounded-slookup %. +%mode sbounded-slookup %in %in %out %. +%term _ sbounded-slookup (sbounded/cons _ Dprecedes) (slookup/hit _) Dprecedes %. +%term _ + %pi (sbounded-slookup (sbounded/cons Dbounded Dprecedes) (slookup/miss Dlookup _) Dprecedes'') + %<- (sbounded-slookup Dbounded Dlookup Dprecedes') + %<- (precedes-trans Dprecedes' Dprecedes Dprecedes'') %. +%worlds (var ovar bind) (sbounded-slookup _ _ _) %. +%total D (sbounded-slookup D _ _) %. +%sort lookup-binder-fun %. +%mode lookup-binder-fun %in %in %out %. +%term _ + %pi (lookup-binder-fun (append/cons Dapp) (lookup/miss Dlook _) Deq) + %<- (lookup-binder-fun Dapp Dlook Deq) %. +%term _ lookup-binder-fun append/nil (lookup/hit _) tp-eq/i %. +%term _ + %pi (lookup-binder-fun (append/cons Dapp) (lookup/hit Dbounded) Deq) + %<- (append-bounded-contra Dapp Dbounded Dfalse) + %<- (false-implies-tp-eq Dfalse Deq) %. +%term _ + %pi (lookup-binder-fun append/nil (lookup/miss Dlookup Dbounded) Deq) + %<- (bounded-lookup Dbounded Dlookup Dprecedes) + %<- (precedes-antisymm Dprecedes Dfalse) + %<- (false-implies-tp-eq Dfalse Deq) %. +%worlds (bind ovar var) (lookup-binder-fun _ _ _) %. +%total D (lookup-binder-fun _ D _) %. +%sort append-all {G} %. +%mode append-all %in %out %. +%term _ append-all _ append/nil %. +%term _ %pi (append-all _ (append/cons D)) %<- (append-all _ D) %. +%worlds (var) (append-all _ _) %. +%total G (append-all G _) %. +%sort sappend-all {G} %. +%mode sappend-all %in %out %. +%term _ sappend-all _ sappend/nil %. +%term _ %pi (sappend-all _ (sappend/cons D)) %<- (sappend-all _ D) %. +%worlds (var) (sappend-all _ _) %. +%total G (sappend-all G _) %. +%sort csub-append %. +%mode csub-append %in %in %out %. +%term _ csub-append csub/base ([x] append/nil) append/nil %. +%term _ + %pi (csub-append (csub/cons _ Dsub) ([x] append/cons (Dappend x)) (append/cons Dappend')) + %<- (csub-append Dsub Dappend Dappend') %. +%sort csub-append! %. +%mode csub-append! %in %out %. +%term _ csub-append! append/nil (append/cons append/nil) %. +%term _ %pi (csub-append! (append/cons D) (append/cons D')) %<- (csub-append! D D') %. +%worlds (var ovar) (csub-append! _ _) %. +%total D (csub-append! D _) %. +%term _ + %pi (csub-append csub/base ([x] append/cons (Dappend x)) D) + %<- (csub-append! (Dappend aca) D) %. +%worlds (var ovar) (csub-append _ _ _) %. +%total D (csub-append D _ _) %. +%sort csub-lookup %. +%mode csub-lookup %in %in %in %out %. +%term _ + %pi (csub-lookup (csub/cons Dtsub Dcsub) ([x] [d] lookup/hit (Dbounded x d)) Dtsub' D) + %<- (tsub-fun Dtsub Dtsub' Deq) + %<- (csub-bounded Dbounded Dcsub Dbounded') + %<- (lookup-resp ctx-eq/i atom-eq/i Deq (lookup/hit Dbounded') D) %. +%term _ + %pi (csub-lookup (csub/cons _ Dcsub) ([x] [d] lookup/miss (Dlookup x d) (Dbounded x d)) Dtsub (lookup/miss Dlookup' Dbounded')) + %<- (csub-lookup Dcsub Dlookup Dtsub Dlookup') + %<- (csub-bounded Dbounded Dcsub Dbounded') %. +%term _ + %pi (csub-lookup csub/base ([x] [d] lookup/miss (Dlookup x d) (Dbounded x d)) Dtsub Dlookup'') + %<- (strengthen-lookup Dlookup Dlookup' Deq) + %<- (tsub-resp Deq term-eq/i tp-eq/i Dtsub Dtsub') + %<- (tsub-absent _ _ Dtsub'') + %<- (tsub-fun Dtsub'' Dtsub' Deq') + %<- (lookup-resp ctx-eq/i atom-eq/i Deq' Dlookup' Dlookup'') %. +%worlds (var bind ovar) (csub-lookup _ _ _ _) %. +%total D (csub-lookup _ D _ _) %. +%sort scsub-lookup {x} %. +%mode scsub-lookup %in %in %in %out %. +%term _ + %pi (scsub-lookup ([x] append/cons (%the (append _ (G2 x) _) (Dappend x))) (append/cons Dappend') ([x] [d] lookup/hit (Dbounded x d)) (lookup/hit Dbounded')) + %<- (scsub-bounded Dappend Dappend' Dbounded Dbounded') %. +%term _ + %pi (scsub-lookup ([x] append/cons (%the (append _ (G2 x) _) (Dappend x))) (append/cons Dappend') ([x] [d] lookup/miss (Dlookup x d) (Dbounded x d)) (lookup/miss Dlookup' Dbounded')) + %<- (scsub-lookup Dappend Dappend' Dlookup Dlookup') + %<- (scsub-bounded Dappend Dappend' Dbounded Dbounded') %. +%term _ + %pi (scsub-lookup ([x] append/nil) append/nil ([x] [d] lookup/miss (Dlookup x d) (Dbounded x d)) Dlookup'') + %<- (strengthen-lookup Dlookup Dlookup' (%the ({x} tp-eq (A x) A') Deq)) + %<- ({x} tp-eq-symm (Deq x) (Deq' x)) + %<- (lookup-resp ctx-eq/i atom-eq/i (Deq' R) Dlookup' Dlookup'') %. +%term _ + %pi (scsub-lookup ([x] append/cons (Dappend x)) (append/cons Dappend') ([x] [d] lookup/miss _ (%the (bounded (G x) x) (Dbounded x d))) D) + %<- ({x} {d} append-bounded-contra (Dappend x) (Dbounded x d) Dfalse) + %<- (false-implies-lookup Dfalse D) %. +%term _ + %pi (scsub-lookup _ _ ([x] [d] lookup/miss _ (Dbounded x d)) D) + %<- ({x} {d} bounded-isvar (Dbounded x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-app-contra (Disvar x d) Dfalse) + %<- (false-implies-lookup Dfalse D) %. +%term _ + %pi (scsub-lookup _ _ ([x] [d] lookup/miss _ (Dbounded x d)) D) + %<- ({x} {d} bounded-isvar (Dbounded x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-pi1-contra (Disvar x d) Dfalse) + %<- (false-implies-lookup Dfalse D) %. +%term _ + %pi (scsub-lookup _ _ ([x] [d] lookup/miss _ (Dbounded x d)) D) + %<- ({x} {d} bounded-isvar (Dbounded x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-pi2-contra (Disvar x d) Dfalse) + %<- (false-implies-lookup Dfalse D) %. +%term _ + %pi (scsub-lookup _ _ ([x] [d] lookup/miss _ (Dbounded x d)) D) + %<- ({x} {d} bounded-isvar (Dbounded x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-forall-contra (Disvar x d) Dfalse) + %<- (false-implies-lookup Dfalse D) %. +%worlds (var bind ovar) (scsub-lookup _ _ _ _) %. +%total D (scsub-lookup D _ _ _) %. +%sort lookup-bounded-contra %. +%mode lookup-bounded-contra %in %in %out %. +%term _ + %pi (lookup-bounded-contra (lookup/hit _) (bounded/cons _ Dprec) Dfalse) + %<- (precedes-antisymm Dprec Dfalse) %. +%term _ + %pi (lookup-bounded-contra (lookup/miss Dlook _) (bounded/cons Dbound Dprec) Dfalse) + %<- (bounded-increase-bound Dbound Dprec Dbound') + %<- (lookup-bounded-contra Dlook Dbound' Dfalse) %. +%worlds (var ovar) (lookup-bounded-contra _ _ _) %. +%total D (lookup-bounded-contra D _ _) %. +%sort lookup-fun %. +%mode lookup-fun %in %in %out %. +%term _ lookup-fun (lookup/hit _) (lookup/hit _) tp-eq/i %. +%term _ + %pi (lookup-fun (lookup/miss Dlook _) (lookup/miss Dlook' _) Deq) + %<- (lookup-fun Dlook Dlook' Deq) %. +%term _ + %pi (lookup-fun (lookup/miss Dlook Dbound) (lookup/hit _) D) + %<- (lookup-bounded-contra Dlook Dbound Dfalse) + %<- (false-implies-tp-eq Dfalse D) %. +%term _ + %pi (lookup-fun (lookup/hit _) (lookup/miss Dlook Dbound) D) + %<- (lookup-bounded-contra Dlook Dbound Dfalse) + %<- (false-implies-tp-eq Dfalse D) %. +%worlds (var ovar) (lookup-fun _ _ _) %. +%total D (lookup-fun D _ _) %. +%sort slookup-sbounded-contra %. +%mode slookup-sbounded-contra %in %in %out %. +%term _ + %pi (slookup-sbounded-contra (slookup/hit _) (sbounded/cons _ Dprec) Dfalse) + %<- (precedes-antisymm Dprec Dfalse) %. +%term _ + %pi (slookup-sbounded-contra (slookup/miss Dlook _) (sbounded/cons Dbound Dprec) Dfalse) + %<- (sbounded-increase-bound Dbound Dprec Dbound') + %<- (slookup-sbounded-contra Dlook Dbound' Dfalse) %. +%worlds (var ovar) (slookup-sbounded-contra _ _ _) %. +%total D (slookup-sbounded-contra D _ _) %. +%sort slookup-fun %. +%mode slookup-fun %in %in %out %. +%term _ slookup-fun (slookup/hit _) (slookup/hit _) stp-eq/i %. +%term _ + %pi (slookup-fun (slookup/miss Dlook _) (slookup/miss Dlook' _) Deq) + %<- (slookup-fun Dlook Dlook' Deq) %. +%term _ + %pi (slookup-fun (slookup/miss Dlook Dbound) (slookup/hit _) D) + %<- (slookup-sbounded-contra Dlook Dbound Dfalse) + %<- (false-implies-stp-eq Dfalse D) %. +%term _ + %pi (slookup-fun (slookup/hit _) (slookup/miss Dlook Dbound) D) + %<- (slookup-sbounded-contra Dlook Dbound Dfalse) + %<- (false-implies-stp-eq Dfalse D) %. +%worlds (var ovar) (slookup-fun _ _ _) %. +%total D (slookup-fun D _ _) %. +%sort bounded-drop %. +%mode bounded-drop %in %in %in %out %. +%term _ + %pi (bounded-drop append/nil append/nil (bounded/cons Dbound Dprec) Dbound') + %<- (bounded-increase-bound Dbound Dprec Dbound') %. +%term _ + %pi (bounded-drop (append/cons Dapp) (append/cons Dapp') (bounded/cons Dbound Dprec) (bounded/cons Dbound' Dprec)) + %<- (bounded-drop Dapp Dapp' Dbound Dbound') %. +%worlds (var ovar) (bounded-drop _ _ _ _) %. +%total D (bounded-drop D _ _ _) %. +%sort ordered-drop %. +%mode ordered-drop %in %in %in %out %. +%term _ + %pi (ordered-drop append/nil append/nil (ordered/cons Dbound) Dord) + %<- (bounded-is-ordered Dbound Dord) %. +%term _ + %pi (ordered-drop (append/cons Dapp) (append/cons Dapp') (ordered/cons Dbound) (ordered/cons Dbound')) + %<- (bounded-drop Dapp Dapp' Dbound Dbound') %. +%worlds (var ovar) (ordered-drop _ _ _ _) %. +%total {} (ordered-drop _ _ _ _) %. +%sort sbounded-drop %. +%mode sbounded-drop %in %in %in %out %. +%term _ + %pi (sbounded-drop sappend/nil sappend/nil (sbounded/cons Dbound Dprec) Dbound') + %<- (sbounded-increase-bound Dbound Dprec Dbound') %. +%term _ + %pi (sbounded-drop (sappend/cons Dapp) (sappend/cons Dapp') (sbounded/cons Dbound Dprec) (sbounded/cons Dbound' Dprec)) + %<- (sbounded-drop Dapp Dapp' Dbound Dbound') %. +%worlds (var ovar) (sbounded-drop _ _ _ _) %. +%total D (sbounded-drop D _ _ _) %. +%sort sordered-drop %. +%mode sordered-drop %in %in %in %out %. +%term _ + %pi (sordered-drop sappend/nil sappend/nil (sordered/cons Dbound) Dord) + %<- (sbounded-is-sordered Dbound Dord) %. +%term _ + %pi (sordered-drop (sappend/cons Dapp) (sappend/cons Dapp') (sordered/cons Dbound) (sordered/cons Dbound')) + %<- (sbounded-drop Dapp Dapp' Dbound Dbound') %. +%worlds (var ovar) (sordered-drop _ _ _ _) %. +%total {} (sordered-drop _ _ _ _) %. +%sort slookup-drop {x} %. +%mode slookup-drop %in %in %in %out %. +%term _ + %pi (slookup-drop ([x] sappend/cons (Dappend x)) (sappend/cons Dappend') ([x] [d] slookup/hit (Dbound x d)) (slookup/hit Dbound'')) + %<- ({x} {d} sbounded-drop (Dappend x) Dappend' (Dbound x d) (Dbound' x d)) + %<- (strengthen-sbounded Dbound' Dbound'') %. +%term _ + %pi (slookup-drop ([_] sappend/nil) sappend/nil ([x] [d] slookup/miss (Dlook x d) (Dprec x d)) Dlook') + %<- (strengthen-slookup Dlook Dlook') %. +%term _ + %pi (slookup-drop ([x] sappend/cons (Dappend x)) (sappend/cons Dappend') ([x] [d] slookup/miss (Dlook x d) (Dbound x d)) (slookup/miss Dlook' Dbound'')) + %<- (slookup-drop Dappend Dappend' Dlook Dlook') + %<- ({x} {d} sbounded-drop (Dappend x) Dappend' (Dbound x d) (Dbound' x d)) + %<- (strengthen-sbounded Dbound' Dbound'') %. +%worlds (var ovar) (slookup-drop _ _ _ _) %. +%total D (slookup-drop D _ _ _) %. +%%%%% Context Regularity %%%%% +%sort lookup-context %. +%mode lookup-context %in %out %. +%term _ lookup-context (lookup/hit D) (ordered/cons D) %. +%term _ lookup-context (lookup/miss _ D) (ordered/cons D) %. +%worlds (bind ovar var obind) (lookup-context _ _) %. +%total {} (lookup-context _ _) %. +%sort aofe-context %. +%mode aofe-context %in %out %. +%sort ofe-context %. +%mode ofe-context %in %out %. +%sort wfe-context %. +%mode wfe-context %in %out %. +%scope wfe-context %term closed aofe-context (aofe/closed D _) D %. +%term var %pi (aofe-context (aofe/var _ D) D') %<- (lookup-context D D') %. +%scope wfe-context %term forall %pi (aofe-context (aofe/forall D) D') %<- (wfe-context D D') %. +%term app %pi (aofe-context (aofe/app _ _ _ D) D') %<- (aofe-context D D') %. +%scope wfe-context %term pi1 %pi (aofe-context (aofe/pi1 D) D') %<- (aofe-context D D') %. +%term pi2 %pi (aofe-context (aofe/pi2 D) D') %<- (aofe-context D D') %. +%scope wfe-context %term at %pi (ofe-context (ofe/at D) D') %<- (aofe-context D D') %. +%term lam %pi (ofe-context (ofe/lam _ D) D') %<- (wfe-context D D') %. +%scope wfe-context %term pair %pi (ofe-context (ofe/pair _ _ _ D) D') %<- (ofe-context D D') %. +%term sing %pi (ofe-context (ofe/sing D) D') %<- (aofe-context D D') %. +%scope wfe-context %term t wfe-context (wfe/t D) D %. +%term pi %pi (wfe-context (wfe/pi _ D) D') %<- (wfe-context D D') %. +%scope wfe-context %term sigma %pi (wfe-context (wfe/sigma _ D) D') %<- (wfe-context D D') %. +%term sing %pi (wfe-context (wfe/sing D) D') %<- (aofe-context D D') %. +%worlds (bind ovar var obind) (aofe-context _ _) (ofe-context _ _) (wfe-context _ _) %. +%total (D1 D2 D3) (aofe-context D1 _) (ofe-context D2 _) (wfe-context D3 _) %. +%sort ofe-context-precedes %. +%mode ofe-context-precedes %in %out %. +%term _ + %pi (ofe-context-precedes D D') + %<- (ofe-context D (ordered/cons (bounded/cons _ D'))) %. +%worlds (bind ovar var obind) (ofe-context-precedes _ _) %. +%total {} (ofe-context-precedes _ _) %. +%sort slookup-context %. +%mode slookup-context %in %out %. +%term _ slookup-context (slookup/hit D) (sordered/cons D) %. +%term _ slookup-context (slookup/miss _ D) (sordered/cons D) %. +%worlds (bind ovar var obind) (slookup-context _ _) %. +%total {} (slookup-context _ _) %. +%sort aofes-context %. +%mode aofes-context %in %out %. +%sort ofes-context %. +%mode ofes-context %in %out %. +%sort wfes-context %. +%mode wfes-context %in %out %. +%scope wfes-context %term closed aofes-context (aofes/closed D _ _) D %. +%term var %pi (aofes-context (aofes/var D) D') %<- (slookup-context D D') %. +%scope wfes-context %term forall %pi (aofes-context (aofes/forall _ D) D') %<- (wfes-context D D') %. +%term app %pi (aofes-context (aofes/app _ D) D') %<- (aofes-context D D') %. +%scope wfes-context %term pi1 %pi (aofes-context (aofes/pi1 D) D') %<- (aofes-context D D') %. +%term pi2 %pi (aofes-context (aofes/pi2 D) D') %<- (aofes-context D D') %. +%scope wfes-context %term at %pi (ofes-context (ofes/at D) D') %<- (aofes-context D D') %. +%term lam + %pi (ofes-context (ofes/lam D) D'') + %<- ({x} {d} ofes-context (D x d) (sordered/cons (D' x d))) + %<- (sbounded-is-sordered-strengthen D' D'') %. +%scope wfes-context %term pair %pi (ofes-context (ofes/pair _ D) D') %<- (ofes-context D D') %. +%term t wfes-context (wfes/t D) D %. +%scope wfes-context %term pi %pi (wfes-context (wfes/pi _ _ D) D') %<- (wfes-context D D') %. +%term sigma %pi (wfes-context (wfes/sigma _ _ D) D') %<- (wfes-context D D') %. +%scope wfes-context %term sing %pi (wfes-context (wfes/sing D) D') %<- (aofes-context D D') %. +%worlds (bind ovar var) (aofes-context _ _) (ofes-context _ _) (wfes-context _ _) %. +%total (D1 D2 D3) (aofes-context D1 _) (ofes-context D2 _) (wfes-context D3 _) %. +%%%%% Context Extension %%%%% +%sort bound-isvar %. +%mode bound-isvar %in %out %. +%term _ bound-isvar (bounded/nil D) D %. +%term _ bound-isvar (bounded/cons _ (precedes/i _ D _)) D %. +%worlds (var bind ovar obind) (bound-isvar _ _) %. +%total {} (bound-isvar _ _) %. +%sort following-var %. +%mode following-var %in %out %. +%term _ + %pi (following-var (%the (isvar X I) D) ([y] [e isvar y (s I)] precedes/i Dlt e D)) + %<- (lt-succ I (%the (lt I (s I)) Dlt)) %. +%worlds (var bind ovar obind) (following-var _ _) %. +%total {} (following-var _ _) %. +%sort extend-context %. +%mode extend-context %in %out %. +%term _ extend-context ordered/nil ([x] [d isvar x 0] bounded/nil d) %. +%term _ + %pi (extend-context (%the (ordered (cons G Y A)) (ordered/cons (%the (bounded G Y) Dbounded))) ([x] [d isvar x J] bounded/cons Dbounded (Dprecedes x d))) + %<- (bound-isvar Dbounded (%the (isvar Y I) Disvar)) + %<- (following-var Disvar (%the ({x} %pi (isvar x J) %-> (precedes Y x)) Dprecedes)) %. +%worlds (var bind ovar obind) (extend-context _ _) %. +%total {} (extend-context _ _) %. +%sort sbound-isvar %. +%mode sbound-isvar %in %out %. +%term _ sbound-isvar (sbounded/nil D) D %. +%term _ sbound-isvar (sbounded/cons _ (precedes/i _ D _)) D %. +%worlds (var bind ovar obind) (sbound-isvar _ _) %. +%total {} (sbound-isvar _ _) %. +%sort extend-scontext %. +%mode extend-scontext %in %out %. +%term _ extend-scontext sordered/nil ([x] [d isvar x 0] sbounded/nil d) %. +%term _ + %pi (extend-scontext (%the (sordered (scons G Y A)) (sordered/cons (%the (sbounded G Y) Dbounded))) ([x] [d isvar x (s I)] sbounded/cons Dbounded (precedes/i Dlt d Disvar))) + %<- (sbound-isvar Dbounded (%the (isvar Y I) Disvar)) + %<- (lt-succ I (%the (lt I (s I)) Dlt)) %. +%worlds (var bind ovar) (extend-scontext _ _) %. +%total {} (extend-scontext _ _) %. +%%%%% Metrics on typing derivations %%%%% +%sort mtp %. +%sort matom %. +%sort mterm %. +%term mt mtp %. +%term mpi %pi mtp %-> mtp %-> mtp %. +%term msigma %pi mtp %-> mtp %-> mtp %. +%term msing %pi matom %-> mtp %. +%term mconst matom %. +%term mvar %pi mtp %-> matom %. +%term mapp %pi mtp %-> mterm %-> matom %-> matom %. +%term mpi1 %pi matom %-> matom %. +%term mpi2 %pi matom %-> matom %. +%term mat %pi matom %-> mterm %. +%term mlam %pi mterm %-> mtp %-> mterm %. +%term mpair %pi mtp %-> mterm %-> mterm %-> mterm %. +%term msingtm %pi matom %-> mterm %. +%sort mof %. +%sort maof %. +%sort mwf %. +%term mof/at %pi (mof (of/at Daof) (mat Daof')) %<- (maof Daof Daof') %. +%term mof/lam + %pi (mof (of/lam Dof Dwf) (mlam Dof' Dwf')) + %<- (mwf Dwf Dwf') + %<- ({x} {d} mof (Dof x d) Dof') %. +%term mof/pair + %pi (mof (of/pair Dwf Dof2 _ Dof1) (mpair Dwf' Dof2' Dof1')) + %<- (mof Dof1 Dof1') + %<- (mof Dof2 Dof2') + %<- ({x} {d} mwf (Dwf x d) Dwf') %. +%term mof/sing %pi (mof (of/sing Daof) (msingtm Daof')) %<- (maof Daof Daof') %. +%term maof/const %pi (maof (aof/const Dwf _) (mvar Dwf')) %<- (mwf Dwf Dwf') %. +%term maof/var %pi (maof (aof/var Dwf _) (mvar Dwf')) %<- (mwf Dwf Dwf') %. +%term maof/forall %pi (maof (aof/forall Dwf) (mvar Dwf')) %<- (mwf Dwf Dwf') %. +%term maof/app + %pi (maof (aof/app Dwf _ Dof Daof) (mapp Dwf' Dof' Daof')) + %<- (maof Daof Daof') + %<- (mof Dof Dof') + %<- (mwf Dwf Dwf') %. +%term maof/pi1 %pi (maof (aof/pi1 Daof) (mpi1 Daof')) %<- (maof Daof Daof') %. +%term maof/pi2 %pi (maof (aof/pi2 Daof) (mpi2 Daof')) %<- (maof Daof Daof') %. +%term mwf/t mwf wf/t mt %. +%term mwf/pi + %pi (mwf (wf/pi Dwf2 Dwf1) (mpi Dwf2' Dwf1')) + %<- (mwf Dwf1 Dwf1') + %<- ({x} {d} mwf (Dwf2 x d) Dwf2') %. +%term mwf/sigma + %pi (mwf (wf/sigma Dwf2 Dwf1) (msigma Dwf2' Dwf1')) + %<- (mwf Dwf1 Dwf1') + %<- ({x} {d} mwf (Dwf2 x d) Dwf2') %. +%term mwf/sing %pi (mwf (wf/sing Daof) (msing Daof')) %<- (maof Daof Daof') %. +%sort mofe %. +%sort maofe %. +%sort mwfe %. +%term mofe/at %pi (mofe (ofe/at Daofe) (mat Daofe')) %<- (maofe Daofe Daofe') %. +%term mofe/lam + %pi (mofe (ofe/lam Dofe Dwfe) (mlam Dofe' Dwfe')) + %<- (mwfe Dwfe Dwfe') + %<- ({x} {d} mofe (Dofe x d) Dofe') %. +%term mofe/pair + %pi (mofe (ofe/pair Dwfe Dofe2 _ Dofe1) (mpair Dwfe' Dofe2' Dofe1')) + %<- (mofe Dofe1 Dofe1') + %<- (mofe Dofe2 Dofe2') + %<- ({x} {d} mwfe (Dwfe x d) Dwfe') %. +%term mofe/sing %pi (mofe (ofe/sing Daofe) (msingtm Daofe')) %<- (maofe Daofe Daofe') %. +%term maofe/closed %pi (maofe (aofe/closed _ Daof) Daof') %<- (maof Daof Daof') %. +%term maofe/var %pi (maofe (aofe/var Dwf _) (mvar Dwf')) %<- (mwfe Dwf Dwf') %. +%term maofe/forall %pi (maofe (aofe/forall Dwf) (mvar Dwf')) %<- (mwfe Dwf Dwf') %. +%term maofe/app + %pi (maofe (aofe/app Dwfe _ Dofe Daofe) (mapp Dwfe' Dofe' Daofe')) + %<- (maofe Daofe Daofe') + %<- (mofe Dofe Dofe') + %<- (mwfe Dwfe Dwfe') %. +%term maofe/pi1 %pi (maofe (aofe/pi1 Daofe) (mpi1 Daofe')) %<- (maofe Daofe Daofe') %. +%term maofe/pi2 %pi (maofe (aofe/pi2 Daofe) (mpi2 Daofe')) %<- (maofe Daofe Daofe') %. +%term mwfe/t mwfe (wfe/t _) mt %. +%term mwfe/pi + %pi (mwfe (wfe/pi Dwfe2 Dwfe1) (mpi Dwfe2' Dwfe1')) + %<- (mwfe Dwfe1 Dwfe1') + %<- ({x} {d} mwfe (Dwfe2 x d) Dwfe2') %. +%term mwfe/sigma + %pi (mwfe (wfe/sigma Dwfe2 Dwfe1) (msigma Dwfe2' Dwfe1')) + %<- (mwfe Dwfe1 Dwfe1') + %<- ({x} {d} mwfe (Dwfe2 x d) Dwfe2') %. +%term mwfe/sing %pi (mwfe (wfe/sing Daofe) (msing Daofe')) %<- (maofe Daofe Daofe') %. +%sort can-maof {D aof R A} %. +%mode can-maof %in %out %. +%sort can-mof {D of M A} %. +%mode can-mof %in %out %. +%sort can-mwf {D wf A} %. +%mode can-mwf %in %out %. +%term _ %pi (can-maof (aof/const D _) (maof/const D')) %<- (can-mwf D D') %. +%term _ %pi (can-maof (aof/var D _) (maof/var D')) %<- (can-mwf D D') %. +%term _ %pi (can-maof (aof/forall D) (maof/forall D')) %<- (can-mwf D D') %. +%term _ + %pi (can-maof (aof/app D3 _ D2 D1) (maof/app D3' D2' D1')) + %<- (can-maof D1 D1') + %<- (can-mof D2 D2') + %<- (can-mwf D3 D3') %. +%term _ %pi (can-maof (aof/pi1 D) (maof/pi1 D')) %<- (can-maof D D') %. +%term _ %pi (can-maof (aof/pi2 D) (maof/pi2 D')) %<- (can-maof D D') %. +%term _ %pi (can-mof (of/at D) (mof/at D')) %<- (can-maof D D') %. +%term _ + %pi (can-mof (of/lam D2 D1) (mof/lam D2' D1')) + %<- (can-mwf D1 D1') + %<- ({x} {d} can-mof (D2 x d) (D2' x d)) %. +%term _ + %pi (can-mof (of/pair D3 D2 _ D1) (mof/pair D3' D2' D1')) + %<- (can-mof D1 D1') + %<- (can-mof D2 D2') + %<- ({x} {d} can-mwf (D3 x d) (D3' x d)) %. +%term _ %pi (can-mof (of/sing D) (mof/sing D')) %<- (can-maof D D') %. +%term _ can-mwf wf/t mwf/t %. +%term _ + %pi (can-mwf (wf/pi D2 D1) (mwf/pi D2' D1')) + %<- (can-mwf D1 D1') + %<- ({x} {d} can-mwf (D2 x d) (D2' x d)) %. +%term _ + %pi (can-mwf (wf/sigma D2 D1) (mwf/sigma D2' D1')) + %<- (can-mwf D1 D1') + %<- ({x} {d} can-mwf (D2 x d) (D2' x d)) %. +%term _ %pi (can-mwf (wf/sing D) (mwf/sing D')) %<- (can-maof D D') %. +%worlds (var bind ovar obind) (can-maof _ _) (can-mof _ _) (can-mwf _ _) %. +%total (D1 D2 D3) (can-maof D1 _) (can-mof D2 _) (can-mwf D3 _) %. +%sort can-maofe {D aofe G R A} %. +%mode can-maofe %in %out %. +%sort can-mofe {D ofe G M A} %. +%mode can-mofe %in %out %. +%sort can-mwfe {D wfe G A} %. +%mode can-mwfe %in %out %. +%term _ %pi (can-maofe (aofe/closed _ D) (maofe/closed D')) %<- (can-maof D D') %. +%term _ %pi (can-maofe (aofe/var D _) (maofe/var D')) %<- (can-mwfe D D') %. +%term _ %pi (can-maofe (aofe/forall D) (maofe/forall D')) %<- (can-mwfe D D') %. +%term _ + %pi (can-maofe (aofe/app D3 _ D2 D1) (maofe/app D3' D2' D1')) + %<- (can-maofe D1 D1') + %<- (can-mofe D2 D2') + %<- (can-mwfe D3 D3') %. +%term _ %pi (can-maofe (aofe/pi1 D) (maofe/pi1 D')) %<- (can-maofe D D') %. +%term _ %pi (can-maofe (aofe/pi2 D) (maofe/pi2 D')) %<- (can-maofe D D') %. +%term _ %pi (can-mofe (ofe/at D) (mofe/at D')) %<- (can-maofe D D') %. +%term _ + %pi (can-mofe (ofe/lam D2 D1) (mofe/lam D2' D1')) + %<- (can-mwfe D1 D1') + %<- ({x} {d} can-mofe (D2 x d) (D2' x d)) %. +%term _ + %pi (can-mofe (ofe/pair D3 D2 _ D1) (mofe/pair D3' D2' D1')) + %<- (can-mofe D1 D1') + %<- (can-mofe D2 D2') + %<- ({x} {d} can-mwfe (D3 x d) (D3' x d)) %. +%term _ %pi (can-mofe (ofe/sing D) (mofe/sing D')) %<- (can-maofe D D') %. +%term _ can-mwfe (wfe/t _) mwfe/t %. +%term _ + %pi (can-mwfe (wfe/pi D2 D1) (mwfe/pi D2' D1')) + %<- (can-mwfe D1 D1') + %<- ({x} {d} can-mwfe (D2 x d) (D2' x d)) %. +%term _ + %pi (can-mwfe (wfe/sigma D2 D1) (mwfe/sigma D2' D1')) + %<- (can-mwfe D1 D1') + %<- ({x} {d} can-mwfe (D2 x d) (D2' x d)) %. +%term _ %pi (can-mwfe (wfe/sing D) (mwfe/sing D')) %<- (can-maofe D D') %. +%worlds (var bind ovar obind) (can-maofe _ _) (can-mofe _ _) (can-mwfe _ _) %. +%total (D1 D2 D3) (can-maofe D1 _) (can-mofe D2 _) (can-mwfe D3 _) %. +%sort maof-resp %. +%mode maof-resp %in %in %in %in %out %out %. +%term _ maof-resp atom-eq/i tp-eq/i D Dm D Dm %. +%worlds (var bind) (maof-resp _ _ _ _ _ _) %. +%total {} (maof-resp _ _ _ _ _ _) %. +%sort mof-resp %. +%mode mof-resp %in %in %in %in %out %out %. +%term _ mof-resp term-eq/i tp-eq/i D Dm D Dm %. +%worlds (var bind) (mof-resp _ _ _ _ _ _) %. +%total {} (mof-resp _ _ _ _ _ _) %. +%sort mwf-resp %. +%mode mwf-resp %in %in %in %out %out %. +%term _ mwf-resp tp-eq/i D Dm D Dm %. +%worlds (var bind) (mwf-resp _ _ _ _ _) %. +%total {} (mwf-resp _ _ _ _ _) %. +%%%%% Bump Lemma %%%%% +%sort bump-precedes %. +%mode bump-precedes %in %in %out %. +%term _ + %pi (bump-precedes Dleq ([x] [d] precedes/i Dlt d Disvar) ([x] [d] precedes/i Dlt' d Disvar)) + %<- (lt-leq-trans Dlt Dleq Dlt') %. +%worlds (var bind ovar obind) (bump-precedes _ _ _) %. +%total {} (bump-precedes _ _ _) %. +%sort bump-bounded %. +%mode bump-bounded %in %in %out %. +%term _ bump-bounded _ ([x] [d] bounded/nil d) ([x] [d] bounded/nil d) %. +%term _ + %pi (bump-bounded Dleq ([x] [d] bounded/cons (Dbound x d) (precedes/i Dlt d Disvar)) ([x] [d] bounded/cons Dbound' (precedes/i Dlt' d Disvar))) + %<- (lt-leq-trans Dlt Dleq Dlt') + %<- (strengthen-bounded Dbound Dbound') %. +%worlds (var bind ovar obind) (bump-bounded _ _ _) %. +%total {} (bump-bounded _ _ _) %. +%sort bump-lookup {x} %. +%mode bump-lookup %in %in %out %. +%scope bump-lookup %term hit bump-lookup ([x] [d] lookup/hit _) ([x] [d] ordered/cons (Dbound x d)) ([x] [d] lookup/hit (Dbound x d)) %. +%term miss + %pi (bump-lookup ([x] [d] lookup/miss (Dlook x d) _) ([x] [d] ordered/cons (Dbound x d)) ([x] [d] lookup/miss (Dlook' x d) (Dbound x d))) + %<- ({x} {d} bounded-is-ordered (Dbound x d) (Dord x d)) + %<- (bump-lookup Dlook Dord Dlook') %. +%worlds (var bind ovar obind) (bump-lookup _ _ _) %. +%total D (bump-lookup D _ _) %. +%sort bump-aofe-m {Rm} {D {x} %pi (isvar x I) %-> (aofe (G x) (R x) (A x))} {x} {d} %. +%mode bump-aofe-m %in %in %in %in %out %out %. +%sort bump-ofe-m {Mm} {D {x} %pi (isvar x I) %-> (ofe (G x) (M x) (A x))} {x} {d} %. +%mode bump-ofe-m %in %in %in %in %out %out %. +%sort bump-wfe-m {Am} {D {x} %pi (isvar x I) %-> (wfe (G x) (A x))} {x} {d} %. +%mode bump-wfe-m %in %in %in %in %out %out %. +%scope bump-wfe-m %term closed bump-aofe-m _ ([x] [d] aofe/closed _ (Daof x)) ([x] [d] maofe/closed (Dmaof x)) Dord ([x] [d] aofe/closed (Dord x d) (Daof x)) ([x] [d] maofe/closed (Dmaof x)) %. +%term var + %pi (bump-aofe-m _ ([x] [d] aofe/var (Dwf x d) (Dlook x d)) ([x] [d] maofe/var (Dmwf x d)) Dord ([x] [d] aofe/var (Dwf' x d) (Dlook' x d)) ([x] [d] maofe/var (Dmwf' x d))) + %<- (bump-wfe-m _ Dwf Dmwf Dord Dwf' Dmwf') + %<- (bump-lookup Dlook Dord Dlook') %. +%scope bump-wfe-m %term forall + %pi (bump-aofe-m _ ([x] [d] aofe/forall (Dwf x d)) ([x] [d] maofe/forall (Dmwf x d)) Dord ([x] [d] aofe/forall (Dwf' x d)) ([x] [d] maofe/forall (Dmwf' x d))) + %<- (bump-wfe-m _ Dwf Dmwf Dord Dwf' Dmwf') %. +%term app + %pi (bump-aofe-m _ ([x] [d] aofe/app (Dwf x d) (Dsub x) (Dof x d) (Daof x d)) ([x] [d] maofe/app (Dmwf x d) (Dmof x d) (Dmaof x d)) Dord ([x] [d] aofe/app (Dwf' x d) (Dsub x) (Dof' x d) (Daof' x d)) ([x] [d] maofe/app (Dmwf' x d) (Dmof' x d) (Dmaof' x d))) + %<- (bump-aofe-m _ Daof Dmaof Dord Daof' Dmaof') + %<- (bump-ofe-m _ Dof Dmof Dord Dof' Dmof') + %<- (bump-wfe-m _ Dwf Dmwf Dord Dwf' Dmwf') %. +%scope bump-wfe-m %term pi1 + %pi (bump-aofe-m _ ([x] [d] aofe/pi1 (Daof x d)) ([x] [d] maofe/pi1 (Dmaof x d)) Dord ([x] [d] aofe/pi1 (Daof' x d)) ([x] [d] maofe/pi1 (Dmaof' x d))) + %<- (bump-aofe-m _ Daof Dmaof Dord Daof' Dmaof') %. +%term pi2 + %pi (bump-aofe-m _ ([x] [d] aofe/pi2 (Daof x d)) ([x] [d] maofe/pi2 (Dmaof x d)) Dord ([x] [d] aofe/pi2 (Daof' x d)) ([x] [d] maofe/pi2 (Dmaof' x d))) + %<- (bump-aofe-m _ Daof Dmaof Dord Daof' Dmaof') %. +%scope bump-wfe-m %term at + %pi (bump-ofe-m _ ([x] [d] ofe/at (Daof x d)) ([x] [d] mofe/at (Dmaof x d)) Dord ([x] [d] ofe/at (Daof' x d)) ([x] [d] mofe/at (Dmaof' x d))) + %<- (bump-aofe-m _ Daof Dmaof Dord Daof' Dmaof') %. +%term lam + %pi (bump-ofe-m _ ([x] [d isvar x I] ofe/lam (%the ({y} %pi (isvar y J1) %-> (ofe (cons (G x) y (A x)) (M x y) (B x y))) (Dof x d)) (%the (wfe (G x) (A x)) (Dwf x d))) ([x] [d] mofe/lam (%the ({y} {e} mofe (Dof x d y e) Mm) (Dmof x d)) (%the (mwfe (Dwf x d) Am) (Dmwf x d))) (%the ({x} %pi (isvar x I') %-> (ordered (G x))) Dord) ([x] [d] ofe/lam (Dof'' x d) (Dwf' x d)) ([x] [d] mofe/lam (Dmof'' x d) (Dmwf' x d))) + %<- (bump-wfe-m _ Dwf Dmwf Dord Dwf' Dmwf') + %<- ({x} {d isvar x I} {y} {e isvar y J1} ofe-context (Dof x d y e) (ordered/cons (%the (bounded (G x) y) (Dbound1 x d y e)))) + %<- ({x} {d isvar x I'} extend-context (Dord x d) (%the ({y} %pi (isvar y J2) %-> (bounded (G x) y)) (Dbound2 x d))) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- ({x} {d isvar x I} bump-bounded Dleq1 ([y] [e isvar y J1] Dbound1 x d y e) ([y] [e isvar y J] %the (bounded (G x) y) (Dbound1' x d y e))) + %<- ({x} {d isvar x I'} bump-bounded Dleq2 ([y] [e isvar y J2] Dbound2 x d y e) ([y] [e isvar y J] %the (bounded (G x) y) (Dbound2' x d y e))) + %<- ({x} {d isvar x I} bump-ofe-m _ ([y] [e isvar y J1] Dof x d y e) ([y] [e] Dmof x d y e) ([y] [e isvar y J] ordered/cons (Dbound1' x d y e)) ([y] [e isvar y J] Dof' x d y e) ([y] [e isvar y J] Dmof' x d y e)) + %<- ({y} {e isvar y J} bump-ofe-m _ ([x] [d isvar x I] Dof' x d y e) ([x] [d] Dmof' x d y e) ([x] [d isvar x I'] ordered/cons (Dbound2' x d y e)) ([x] [d isvar x I'] Dof'' x d y e) ([x] [d isvar x I'] Dmof'' x d y e)) %. +%scope bump-wfe-m %term pair + %pi (bump-ofe-m _ ([x] [d isvar x I] ofe/pair (%the ({y} %pi (isvar y J1) %-> (wfe (cons (G x) y (A x)) (B x y))) (Dwf x d)) (%the (ofe (G x) (N x) (B' x)) (Dof2 x d)) (%the (tsub ([y] B x y) (M x) (B' x)) (Dsub x)) (%the (ofe (G x) (M x) (A x)) (Dof1 x d))) ([x] [d] mofe/pair (%the ({y} {e} mwfe (Dwf x d y e) Bm) (Dmwf x d)) (%the (mofe (Dof2 x d) Nm) (Dmof2 x d)) (%the (mofe (Dof1 x d) Mm) (Dmof1 x d))) (%the ({x} %pi (isvar x I') %-> (ordered (G x))) Dord) ([x] [d] ofe/pair (Dwf'' x d) (Dof2' x d) (Dsub x) (Dof1' x d)) ([x] [d] mofe/pair (Dmwf'' x d) (Dmof2' x d) (Dmof1' x d))) + %<- (bump-ofe-m _ Dof1 Dmof1 Dord Dof1' Dmof1') + %<- (bump-ofe-m _ Dof2 Dmof2 Dord Dof2' Dmof2') + %<- ({x} {d isvar x I} {y} {e isvar y J1} wfe-context (Dwf x d y e) (ordered/cons (%the (bounded (G x) y) (Dbound1 x d y e)))) + %<- ({x} {d isvar x I'} extend-context (Dord x d) (%the ({y} %pi (isvar y J2) %-> (bounded (G x) y)) (Dbound2 x d))) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- ({x} {d isvar x I} bump-bounded Dleq1 ([y] [e isvar y J1] Dbound1 x d y e) ([y] [e isvar y J] %the (bounded (G x) y) (Dbound1' x d y e))) + %<- ({x} {d isvar x I'} bump-bounded Dleq2 ([y] [e isvar y J2] Dbound2 x d y e) ([y] [e isvar y J] %the (bounded (G x) y) (Dbound2' x d y e))) + %<- ({x} {d isvar x I} bump-wfe-m _ ([y] [e isvar y J1] Dwf x d y e) ([y] [e] Dmwf x d y e) ([y] [e isvar y J] ordered/cons (Dbound1' x d y e)) ([y] [e isvar y J] Dwf' x d y e) ([y] [e isvar y J] Dmwf' x d y e)) + %<- ({y} {e isvar y J} bump-wfe-m _ ([x] [d isvar x I] Dwf' x d y e) ([x] [d] Dmwf' x d y e) ([x] [d isvar x I'] ordered/cons (Dbound2' x d y e)) ([x] [d isvar x I'] Dwf'' x d y e) ([x] [d isvar x I'] Dmwf'' x d y e)) %. +%term sing + %pi (bump-ofe-m _ ([x] [d] ofe/sing (Daof x d)) ([x] [d] mofe/sing (Dmaof x d)) Dord ([x] [d] ofe/sing (Daof' x d)) ([x] [d] mofe/sing (Dmaof' x d))) + %<- (bump-aofe-m _ Daof Dmaof Dord Daof' Dmaof') %. +%scope bump-wfe-m %term t bump-wfe-m _ ([x] [d] wfe/t _) ([x] [d] mwfe/t) Dord ([x] [d] wfe/t (Dord x d)) ([x] [d] mwfe/t) %. +%term pi + %pi (bump-wfe-m _ ([x] [d isvar x I] wfe/pi (%the ({y} %pi (isvar y J1) %-> (wfe (cons (G x) y (A x)) (B x y))) (Dwf2 x d)) (%the (wfe (G x) (A x)) (Dwf1 x d))) ([x] [d] mwfe/pi (%the ({y} {e} mwfe (Dwf2 x d y e) Bm) (Dmwf2 x d)) (%the (mwfe (Dwf1 x d) Am) (Dmwf1 x d))) (%the ({x} %pi (isvar x I') %-> (ordered (G x))) Dord) ([x] [d] wfe/pi (Dwf2'' x d) (Dwf1' x d)) ([x] [d] mwfe/pi (Dmwf2'' x d) (Dmwf1' x d))) + %<- (bump-wfe-m _ Dwf1 Dmwf1 Dord Dwf1' Dmwf1') + %<- ({x} {d isvar x I} {y} {e isvar y J1} wfe-context (Dwf2 x d y e) (ordered/cons (%the (bounded (G x) y) (Dbound1 x d y e)))) + %<- ({x} {d isvar x I'} extend-context (Dord x d) (%the ({y} %pi (isvar y J2) %-> (bounded (G x) y)) (Dbound2 x d))) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- ({x} {d isvar x I} bump-bounded Dleq1 ([y] [e isvar y J1] Dbound1 x d y e) ([y] [e isvar y J] %the (bounded (G x) y) (Dbound1' x d y e))) + %<- ({x} {d isvar x I'} bump-bounded Dleq2 ([y] [e isvar y J2] Dbound2 x d y e) ([y] [e isvar y J] %the (bounded (G x) y) (Dbound2' x d y e))) + %<- ({x} {d isvar x I} bump-wfe-m _ ([y] [e isvar y J1] Dwf2 x d y e) ([y] [e] Dmwf2 x d y e) ([y] [e isvar y J] ordered/cons (Dbound1' x d y e)) ([y] [e isvar y J] Dwf2' x d y e) ([y] [e isvar y J] Dmwf2' x d y e)) + %<- ({y} {e isvar y J} bump-wfe-m _ ([x] [d isvar x I] Dwf2' x d y e) ([x] [d] Dmwf2' x d y e) ([x] [d isvar x I'] ordered/cons (Dbound2' x d y e)) ([x] [d isvar x I'] Dwf2'' x d y e) ([x] [d isvar x I'] Dmwf2'' x d y e)) %. +%scope bump-wfe-m %term sigma + %pi (bump-wfe-m _ ([x] [d isvar x I] wfe/sigma (%the ({y} %pi (isvar y J1) %-> (wfe (cons (G x) y (A x)) (B x y))) (Dwf2 x d)) (%the (wfe (G x) (A x)) (Dwf1 x d))) ([x] [d] mwfe/sigma (%the ({y} {e} mwfe (Dwf2 x d y e) Bm) (Dmwf2 x d)) (%the (mwfe (Dwf1 x d) Am) (Dmwf1 x d))) (%the ({x} %pi (isvar x I') %-> (ordered (G x))) Dord) ([x] [d] wfe/sigma (Dwf2'' x d) (Dwf1' x d)) ([x] [d] mwfe/sigma (Dmwf2'' x d) (Dmwf1' x d))) + %<- (bump-wfe-m _ Dwf1 Dmwf1 Dord Dwf1' Dmwf1') + %<- ({x} {d isvar x I} {y} {e isvar y J1} wfe-context (Dwf2 x d y e) (ordered/cons (%the (bounded (G x) y) (Dbound1 x d y e)))) + %<- ({x} {d isvar x I'} extend-context (Dord x d) (%the ({y} %pi (isvar y J2) %-> (bounded (G x) y)) (Dbound2 x d))) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- ({x} {d isvar x I} bump-bounded Dleq1 ([y] [e isvar y J1] Dbound1 x d y e) ([y] [e isvar y J] %the (bounded (G x) y) (Dbound1' x d y e))) + %<- ({x} {d isvar x I'} bump-bounded Dleq2 ([y] [e isvar y J2] Dbound2 x d y e) ([y] [e isvar y J] %the (bounded (G x) y) (Dbound2' x d y e))) + %<- ({x} {d isvar x I} bump-wfe-m _ ([y] [e isvar y J1] Dwf2 x d y e) ([y] [e] Dmwf2 x d y e) ([y] [e isvar y J] ordered/cons (Dbound1' x d y e)) ([y] [e isvar y J] Dwf2' x d y e) ([y] [e isvar y J] Dmwf2' x d y e)) + %<- ({y} {e isvar y J} bump-wfe-m _ ([x] [d isvar x I] Dwf2' x d y e) ([x] [d] Dmwf2' x d y e) ([x] [d isvar x I'] ordered/cons (Dbound2' x d y e)) ([x] [d isvar x I'] Dwf2'' x d y e) ([x] [d isvar x I'] Dmwf2'' x d y e)) %. +%term sing + %pi (bump-wfe-m _ ([x] [d] wfe/sing (Daof x d)) ([x] [d] mwfe/sing (Dmaof x d)) Dord ([x] [d] wfe/sing (Daof' x d)) ([x] [d] mwfe/sing (Dmaof' x d))) + %<- (bump-aofe-m _ Daof Dmaof Dord Daof' Dmaof') %. +%worlds (var bind ovar obind) (bump-aofe-m _ _ _ _ _ _) (bump-ofe-m _ _ _ _ _ _) (bump-wfe-m _ _ _ _ _ _) %. +%total (Rm Mm Am) (bump-aofe-m Rm _ _ _ _ _) (bump-ofe-m Mm _ _ _ _ _) (bump-wfe-m Am _ _ _ _ _) %. +%sort bump-aofe {x} %. +%mode bump-aofe %in %in %out %. +%term _ + %pi (bump-aofe Daof Dordered Daof') + %<- ({x} {d isvar x I} can-maofe (Daof x d) (Dmaof x d)) + %<- (bump-aofe-m _ Daof Dmaof Dordered Daof' _) %. +%worlds (var bind ovar obind) (bump-aofe _ _ _) %. +%total {} (bump-aofe _ _ _) %. +%sort bump-ofe {x} %. +%mode bump-ofe %in %in %out %. +%term _ + %pi (bump-ofe Dof Dordered Dof') + %<- ({x} {d isvar x I} can-mofe (Dof x d) (Dmof x d)) + %<- (bump-ofe-m _ Dof Dmof Dordered Dof' _) %. +%worlds (var bind ovar obind) (bump-ofe _ _ _) %. +%total {} (bump-ofe _ _ _) %. +%sort bump-wfe {x} %. +%mode bump-wfe %in %in %out %. +%term _ + %pi (bump-wfe Dwf Dordered Dwf') + %<- ({x} {d isvar x I} can-mwfe (Dwf x d) (Dmwf x d)) + %<- (bump-wfe-m _ Dwf Dmwf Dordered Dwf' _) %. +%worlds (var bind ovar obind) (bump-wfe _ _ _) %. +%total {} (bump-wfe _ _ _) %. +%sort bump2-ofe {x} %. +%mode bump2-ofe %in %in %in %out %. +%term _ + %pi (bump2-ofe (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofe (G x y) (M x y) (A x y)))) Dof1) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J') %-> (ordered (G x y)))) Dordered1) (%the ({x} %pi (isvar x I') %-> ({y} %pi (isvar y J') %-> (ordered (G x y)))) Dordered2) Dof3) + %<- ({x} {d isvar x I} bump-ofe ([y] [e] Dof1 x d y e) ([y] [e] Dordered1 x d y e) ([y] [e isvar y J'] %the (ofe (G x y) (M x y) (A x y)) (Dof2 x d y e))) + %<- ({y} {e isvar y J'} bump-ofe ([x] [d] Dof2 x d y e) ([x] [d] Dordered2 x d y e) ([x] [d isvar x I'] %the (ofe (G x y) (M x y) (A x y)) (Dof3 x d y e))) %. +%worlds (var bind ovar obind) (bump2-ofe _ _ _ _) %. +%total {} (bump2-ofe _ _ _ _) %. +%sort bump2-wfe {x} %. +%mode bump2-wfe %in %in %in %out %. +%term _ + %pi (bump2-wfe (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (wfe (G x y) (A x y)))) Dof1) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J') %-> (ordered (G x y)))) Dordered1) (%the ({x} %pi (isvar x I') %-> ({y} %pi (isvar y J') %-> (ordered (G x y)))) Dordered2) Dof3) + %<- ({x} {d isvar x I} bump-wfe ([y] [e] Dof1 x d y e) ([y] [e] Dordered1 x d y e) ([y] [e isvar y J'] %the (wfe (G x y) (A x y)) (Dof2 x d y e))) + %<- ({y} {e isvar y J'} bump-wfe ([x] [d] Dof2 x d y e) ([x] [d] Dordered2 x d y e) ([x] [d isvar x I'] %the (wfe (G x y) (A x y)) (Dof3 x d y e))) %. +%worlds (var bind ovar obind) (bump2-wfe _ _ _ _) %. +%total {} (bump2-wfe _ _ _ _) %. +%sort bump-ofe-under {x} %. +%mode bump-ofe-under %in %in %out %. +%term _ + %pi (bump-ofe-under (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J1) %-> _)) Dof) (%the ({x} %pi (isvar x I') %-> _) Dord) Dof') + %<- ({x} {d isvar x I} {y} {e isvar y J1} ofe-context (Dof x d y e) (ordered/cons (%the (bounded (G x) y) (DboundJ1 x d y e)))) + %<- ({x} {d isvar x I'} extend-context (Dord x d) (%the ({y} %pi (isvar y J2) %-> (bounded (G x) y)) (DboundJ2 x d))) + %<- (can-max J1 J2 J' (%the (max J1 J2 J') Dmax)) + %<- (max-leq Dmax (%the (leq J1 J') Dleq1) (%the (leq J2 J') Dleq2)) + %<- ({x} {d isvar x I} bump-bounded Dleq1 (DboundJ1 x d) (%the ({y} %pi (isvar y J') %-> (bounded (G x) y)) (DboundJ x d))) + %<- ({x} {d isvar x I'} bump-bounded Dleq2 (DboundJ2 x d) (%the ({y} %pi (isvar y J') %-> (bounded (G x) y)) (DboundJ' x d))) + %<- (bump2-ofe Dof ([x] [d isvar x I] [y] [e isvar y J'] ordered/cons (DboundJ x d y e)) ([x] [d isvar x I'] [y] [e isvar y J'] ordered/cons (DboundJ' x d y e)) Dof') %. +%worlds (var bind ovar) (bump-ofe-under _ _ _) %. +%total {} (bump-ofe-under _ _ _) %. +%sort bump-wfe-under {x} %. +%mode bump-wfe-under %in %in %out %. +%term _ + %pi (bump-wfe-under (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J1) %-> _)) Dof) (%the ({x} %pi (isvar x I') %-> _) Dord) Dof') + %<- ({x} {d isvar x I} {y} {e isvar y J1} wfe-context (Dof x d y e) (ordered/cons (%the (bounded (G x) y) (DboundJ1 x d y e)))) + %<- ({x} {d isvar x I'} extend-context (Dord x d) (%the ({y} %pi (isvar y J2) %-> (bounded (G x) y)) (DboundJ2 x d))) + %<- (can-max J1 J2 J' (%the (max J1 J2 J') Dmax)) + %<- (max-leq Dmax (%the (leq J1 J') Dleq1) (%the (leq J2 J') Dleq2)) + %<- ({x} {d isvar x I} bump-bounded Dleq1 (DboundJ1 x d) (%the ({y} %pi (isvar y J') %-> (bounded (G x) y)) (DboundJ x d))) + %<- ({x} {d isvar x I'} bump-bounded Dleq2 (DboundJ2 x d) (%the ({y} %pi (isvar y J') %-> (bounded (G x) y)) (DboundJ' x d))) + %<- (bump2-wfe Dof ([x] [d isvar x I] [y] [e isvar y J'] ordered/cons (DboundJ x d y e)) ([x] [d isvar x I'] [y] [e isvar y J'] ordered/cons (DboundJ' x d y e)) Dof') %. +%worlds (var bind ovar) (bump-wfe-under _ _ _) %. +%total {} (bump-wfe-under _ _ _) %. +%%%%% Bump Lemma (Simple Types) %%%%% +%sort bump-sbounded %. +%mode bump-sbounded %in %in %out %. +%term _ bump-sbounded _ ([x] [d] sbounded/nil d) ([x] [d] sbounded/nil d) %. +%term _ + %pi (bump-sbounded Dleq ([x] [d] sbounded/cons (Dbound x d) (precedes/i Dlt d Disvar)) ([x] [d] sbounded/cons Dbound' (precedes/i Dlt' d Disvar))) + %<- (lt-leq-trans Dlt Dleq Dlt') + %<- (strengthen-sbounded Dbound Dbound') %. +%worlds (var bind ovar obind) (bump-sbounded _ _ _) %. +%total {} (bump-sbounded _ _ _) %. +%sort bump-slookup {x} %. +%mode bump-slookup %in %in %out %. +%scope bump-slookup %term hit bump-slookup ([x] [d] slookup/hit _) ([x] [d] sordered/cons (Dbound x d)) ([x] [d] slookup/hit (Dbound x d)) %. +%term miss + %pi (bump-slookup ([x] [d] %the (slookup (scons _ _ T) _ _) (slookup/miss (Dlook x d) _)) ([x] [d] sordered/cons (Dbound x d)) ([x] [d] slookup/miss (Dlook' x d) (Dbound x d))) + %<- ({x} {d} sbounded-is-sordered (Dbound x d) (Dord x d)) + %<- (bump-slookup Dlook Dord Dlook') %. +%worlds (var bind ovar obind) (bump-slookup _ _ _) %. +%total D (bump-slookup D _ _) %. +%sort bump-aofes {R} {x} %. +%mode bump-aofes %in %in %in %out %. +%sort bump-ofes {M} {x} %. +%mode bump-ofes %in %in %in %out %. +%sort bump-wfes {A} {x} %. +%mode bump-wfes %in %in %in %out %. +%scope bump-wfes %term closed bump-aofes _ ([x] [d] aofes/closed _ (Dsimp x) (Daof x)) Dordered ([x] [d] aofes/closed (Dordered x d) (Dsimp x) (Daof x)) %. +%term forall + %pi (bump-aofes _ ([x] [d] aofes/forall (Dsimp x) (Dwf x d)) Dordered ([x] [d] aofes/forall (Dsimp x) (Dwf' x d))) + %<- (bump-wfes _ Dwf Dordered Dwf') %. +%scope bump-wfes %term var + %pi (bump-aofes _ ([x] [d] aofes/var (Dlookup x d)) Dordered ([x] [d] aofes/var (Dlookup' x d))) + %<- (bump-slookup Dlookup Dordered Dlookup') %. +%term app + %pi (bump-aofes _ ([x] [d] aofes/app (D2 x d) (D1 x d)) Dord ([x] [d] aofes/app (D2' x d) (D1' x d))) + %<- (bump-aofes _ D1 Dord D1') + %<- (bump-ofes _ D2 Dord D2') %. +%scope bump-wfes %term pi1 + %pi (bump-aofes _ ([x] [d] aofes/pi1 (D x d)) Dord ([x] [d] aofes/pi1 (D' x d))) + %<- (bump-aofes _ D Dord D') %. +%term pi2 + %pi (bump-aofes _ ([x] [d] aofes/pi2 (D x d)) Dord ([x] [d] aofes/pi2 (D' x d))) + %<- (bump-aofes _ D Dord D') %. +%scope bump-wfes %term at + %pi (bump-ofes _ ([x] [d] ofes/at (D x d)) Dord ([x] [d] ofes/at (D' x d))) + %<- (bump-aofes _ D Dord D') %. +%term lam + %pi (bump-ofes _ ([x] [d isvar x I] ofes/lam (%the ({y} %pi (isvar y J1) %-> (ofes (scons (G x) y S) (M x y) T)) (Dof x d))) (%the ({x} %pi (isvar x I') %-> (sordered (G x))) Dord) ([x] [d] ofes/lam (Dof'' x d))) + %<- ({x} {d isvar x I} {y} {e isvar y J1} ofes-context (Dof x d y e) (sordered/cons (%the (sbounded (G x) y) (Dbound1 x d y e)))) + %<- ({x} {d isvar x I'} extend-scontext (Dord x d) (%the ({y} %pi (isvar y J2) %-> (sbounded (G x) y)) (Dbound2 x d))) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- ({x} {d isvar x I} bump-sbounded Dleq1 ([y] [e isvar y J1] Dbound1 x d y e) ([y] [e isvar y J] %the (sbounded (G x) y) (Dbound1' x d y e))) + %<- ({x} {d isvar x I'} bump-sbounded Dleq2 ([y] [e isvar y J2] Dbound2 x d y e) ([y] [e isvar y J] %the (sbounded (G x) y) (Dbound2' x d y e))) + %<- ({x} {d isvar x I} bump-ofes _ ([y] [e isvar y J1] Dof x d y e) ([y] [e isvar y J] sordered/cons (Dbound1' x d y e)) ([y] [e isvar y J] Dof' x d y e)) + %<- ({y} {e isvar y J} bump-ofes _ ([x] [d isvar x I] Dof' x d y e) ([x] [d isvar x I'] sordered/cons (Dbound2' x d y e)) ([x] [d isvar x I'] Dof'' x d y e)) %. +%scope bump-wfes %term pair + %pi (bump-ofes _ ([x] [d] ofes/pair (D2 x d) (D1 x d)) Dord ([x] [d] ofes/pair (D2' x d) (D1' x d))) + %<- (bump-ofes _ D1 Dord D1') + %<- (bump-ofes _ D2 Dord D2') %. +%term t bump-wfes _ ([x] [d] wfes/t _) Dord ([x] [d] wfes/t (Dord x d)) %. +%scope bump-wfes %term pi + %pi (bump-wfes _ ([x] [d isvar x I] wfes/pi (%the ({y} %pi (isvar y J1) %-> (wfes (scons (G x) y S) (B x y))) (D2 x d)) (%the (simp (A x) S) (Dsimp x)) (%the (wfes (G x) (A x)) (D1 x d))) (%the ({x} %pi (isvar x I') %-> (sordered (G x))) Dord) ([x] [d] wfes/pi (D2'' x d) (Dsimp x) (D1' x d))) + %<- (bump-wfes _ D1 Dord D1') + %<- ({x} {d isvar x I} {y} {e isvar y J1} wfes-context (D2 x d y e) (sordered/cons (%the (sbounded (G x) y) (Dbound1 x d y e)))) + %<- ({x} {d isvar x I'} extend-scontext (Dord x d) (%the ({y} %pi (isvar y J2) %-> (sbounded (G x) y)) (Dbound2 x d))) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- ({x} {d isvar x I} bump-sbounded Dleq1 ([y] [e isvar y J1] Dbound1 x d y e) ([y] [e isvar y J] %the (sbounded (G x) y) (Dbound1' x d y e))) + %<- ({x} {d isvar x I'} bump-sbounded Dleq2 ([y] [e isvar y J2] Dbound2 x d y e) ([y] [e isvar y J] %the (sbounded (G x) y) (Dbound2' x d y e))) + %<- ({x} {d isvar x I} bump-wfes _ ([y] [e isvar y J1] D2 x d y e) ([y] [e isvar y J] sordered/cons (Dbound1' x d y e)) ([y] [e isvar y J] D2' x d y e)) + %<- ({y} {e isvar y J} bump-wfes _ ([x] [d isvar x I] D2' x d y e) ([x] [d isvar x I'] sordered/cons (Dbound2' x d y e)) ([x] [d isvar x I'] D2'' x d y e)) %. +%term sigma + %pi (bump-wfes _ ([x] [d isvar x I] wfes/sigma (%the ({y} %pi (isvar y J1) %-> (wfes (scons (G x) y S) (B x y))) (D2 x d)) (%the (simp (A x) S) (Dsimp x)) (%the (wfes (G x) (A x)) (D1 x d))) (%the ({x} %pi (isvar x I') %-> (sordered (G x))) Dord) ([x] [d] wfes/sigma (D2'' x d) (Dsimp x) (D1' x d))) + %<- (bump-wfes _ D1 Dord D1') + %<- ({x} {d isvar x I} {y} {e isvar y J1} wfes-context (D2 x d y e) (sordered/cons (%the (sbounded (G x) y) (Dbound1 x d y e)))) + %<- ({x} {d isvar x I'} extend-scontext (Dord x d) (%the ({y} %pi (isvar y J2) %-> (sbounded (G x) y)) (Dbound2 x d))) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- ({x} {d isvar x I} bump-sbounded Dleq1 ([y] [e isvar y J1] Dbound1 x d y e) ([y] [e isvar y J] %the (sbounded (G x) y) (Dbound1' x d y e))) + %<- ({x} {d isvar x I'} bump-sbounded Dleq2 ([y] [e isvar y J2] Dbound2 x d y e) ([y] [e isvar y J] %the (sbounded (G x) y) (Dbound2' x d y e))) + %<- ({x} {d isvar x I} bump-wfes _ ([y] [e isvar y J1] D2 x d y e) ([y] [e isvar y J] sordered/cons (Dbound1' x d y e)) ([y] [e isvar y J] D2' x d y e)) + %<- ({y} {e isvar y J} bump-wfes _ ([x] [d isvar x I] D2' x d y e) ([x] [d isvar x I'] sordered/cons (Dbound2' x d y e)) ([x] [d isvar x I'] D2'' x d y e)) %. +%scope bump-wfes %term sing + %pi (bump-wfes _ ([x] [d] wfes/sing (D x d)) Dord ([x] [d] wfes/sing (D' x d))) + %<- (bump-aofes _ D Dord D') %. +%worlds (var bind ovar) (bump-aofes _ _ _ _) (bump-ofes _ _ _ _) (bump-wfes _ _ _ _) %. +%total (R M A) (bump-aofes R _ _ _) (bump-ofes M _ _ _) (bump-wfes A _ _ _) %. +%sort bump-wfes {A} {x} %. +%mode bump-wfes %in %in %in %out %. +%scope bump-wfes %term t bump-wfes _ ([x] [d] wfes/t _) Dord ([x] [d] wfes/t (Dord x d)) %. +%term pi + %pi (bump-wfes _ ([x] [d isvar x I] wfes/pi (%the ({y} %pi (isvar y J1) %-> (wfes (scons (G x) y S) (B x y))) (D2 x d)) (%the (simp (A x) S) (Dsimp x)) (%the (wfes (G x) (A x)) (D1 x d))) (%the ({x} %pi (isvar x I') %-> (sordered (G x))) Dord) ([x] [d] wfes/pi (D2'' x d) (Dsimp x) (D1' x d))) + %<- (bump-wfes _ D1 Dord D1') + %<- ({x} {d isvar x I} {y} {e isvar y J1} wfes-context (D2 x d y e) (sordered/cons (%the (sbounded (G x) y) (Dbound1 x d y e)))) + %<- ({x} {d isvar x I'} extend-scontext (Dord x d) (%the ({y} %pi (isvar y J2) %-> (sbounded (G x) y)) (Dbound2 x d))) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- ({x} {d isvar x I} bump-sbounded Dleq1 ([y] [e isvar y J1] Dbound1 x d y e) ([y] [e isvar y J] %the (sbounded (G x) y) (Dbound1' x d y e))) + %<- ({x} {d isvar x I'} bump-sbounded Dleq2 ([y] [e isvar y J2] Dbound2 x d y e) ([y] [e isvar y J] %the (sbounded (G x) y) (Dbound2' x d y e))) + %<- ({x} {d isvar x I} bump-wfes _ ([y] [e isvar y J1] D2 x d y e) ([y] [e isvar y J] sordered/cons (Dbound1' x d y e)) ([y] [e isvar y J] D2' x d y e)) + %<- ({y} {e isvar y J} bump-wfes _ ([x] [d isvar x I] D2' x d y e) ([x] [d isvar x I'] sordered/cons (Dbound2' x d y e)) ([x] [d isvar x I'] D2'' x d y e)) %. +%scope bump-wfes %term sigma + %pi (bump-wfes _ ([x] [d isvar x I] wfes/sigma (%the ({y} %pi (isvar y J1) %-> (wfes (scons (G x) y S) (B x y))) (D2 x d)) (%the (simp (A x) S) (Dsimp x)) (%the (wfes (G x) (A x)) (D1 x d))) (%the ({x} %pi (isvar x I') %-> (sordered (G x))) Dord) ([x] [d] wfes/sigma (D2'' x d) (Dsimp x) (D1' x d))) + %<- (bump-wfes _ D1 Dord D1') + %<- ({x} {d isvar x I} {y} {e isvar y J1} wfes-context (D2 x d y e) (sordered/cons (%the (sbounded (G x) y) (Dbound1 x d y e)))) + %<- ({x} {d isvar x I'} extend-scontext (Dord x d) (%the ({y} %pi (isvar y J2) %-> (sbounded (G x) y)) (Dbound2 x d))) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- ({x} {d isvar x I} bump-sbounded Dleq1 ([y] [e isvar y J1] Dbound1 x d y e) ([y] [e isvar y J] %the (sbounded (G x) y) (Dbound1' x d y e))) + %<- ({x} {d isvar x I'} bump-sbounded Dleq2 ([y] [e isvar y J2] Dbound2 x d y e) ([y] [e isvar y J] %the (sbounded (G x) y) (Dbound2' x d y e))) + %<- ({x} {d isvar x I} bump-wfes _ ([y] [e isvar y J1] D2 x d y e) ([y] [e isvar y J] sordered/cons (Dbound1' x d y e)) ([y] [e isvar y J] D2' x d y e)) + %<- ({y} {e isvar y J} bump-wfes _ ([x] [d isvar x I] D2' x d y e) ([x] [d isvar x I'] sordered/cons (Dbound2' x d y e)) ([x] [d isvar x I'] D2'' x d y e)) %. +%term sing + %pi (bump-wfes _ ([x] [d] wfes/sing (D x d)) Dord ([x] [d] wfes/sing (D' x d))) + %<- (bump-aofes _ D Dord D') %. +%worlds (var bind ovar) (bump-wfes _ _ _ _) %. +%total A (bump-wfes A _ _ _) %. +%%%%% Weakening %%%%% +%sort weakeng-lookup %. +%mode weakeng-lookup %in %in %in %in %out %. +%scope weakeng-lookup %term hit weakeng-lookup (append/cons _) (append/cons _) (ordered/cons D) (lookup/hit _) (lookup/hit D) %. +%term miss + %pi (weakeng-lookup (append/cons Dapp1) (append/cons Dapp2) (ordered/cons Dbound) (lookup/miss Dlook _) (lookup/miss Dlook' Dbound)) + %<- (bounded-is-ordered Dbound Dord) + %<- (weakeng-lookup Dapp1 Dapp2 Dord Dlook Dlook') %. +%scope weakeng-lookup %term nil weakeng-lookup append/nil append/nil (ordered/cons Dbound) Dlook (lookup/miss Dlook Dbound) %. +%worlds (var bind ovar obind) (weakeng-lookup _ _ _ _ _) %. +%total D (weakeng-lookup _ _ _ D _) %. +%sort weakeng-aofe-m {Rm} %. +%mode weakeng-aofe-m %in %in %in %in %in %in %out %out %. +%sort weakeng-ofe-m {Mm} %. +%mode weakeng-ofe-m %in %in %in %in %in %in %out %out %. +%sort weakeng-wfe-m {Am} %. +%mode weakeng-wfe-m %in %in %in %in %in %in %out %out %. +%scope weakeng-wfe-m %term closed weakeng-aofe-m _ Dapp Dapp' Dord (aofe/closed _ Daof) (maofe/closed Dmaof) (aofe/closed Dord Daof) (maofe/closed Dmaof) %. +%term var + %pi (weakeng-aofe-m _ Dapp Dapp' Dord (aofe/var Dwf Dlook) (maofe/var Dmwf) (aofe/var Dwf' Dlook') (maofe/var Dmwf')) + %<- (weakeng-lookup Dapp Dapp' Dord Dlook Dlook') + %<- (weakeng-wfe-m _ Dapp Dapp' Dord Dwf Dmwf Dwf' Dmwf') %. +%scope weakeng-wfe-m %term forall + %pi (weakeng-aofe-m _ Dapp Dapp' Dord (aofe/forall Dwf) (maofe/forall Dmwf) (aofe/forall Dwf') (maofe/forall Dmwf')) + %<- (weakeng-wfe-m _ Dapp Dapp' Dord Dwf Dmwf Dwf' Dmwf') %. +%term app + %pi (weakeng-aofe-m _ Dapp Dapp' Dord (aofe/app Dwf Dsub Dof Daof) (maofe/app Dmwf Dmof Dmaof) (aofe/app Dwf' Dsub Dof' Daof') (maofe/app Dmwf' Dmof' Dmaof')) + %<- (weakeng-aofe-m _ Dapp Dapp' Dord Daof Dmaof Daof' Dmaof') + %<- (weakeng-ofe-m _ Dapp Dapp' Dord Dof Dmof Dof' Dmof') + %<- (weakeng-wfe-m _ Dapp Dapp' Dord Dwf Dmwf Dwf' Dmwf') %. +%scope weakeng-wfe-m %term pi1 + %pi (weakeng-aofe-m _ Dapp Dapp' Dord (aofe/pi1 Daof) (maofe/pi1 Dmaof) (aofe/pi1 Daof') (maofe/pi1 Dmaof')) + %<- (weakeng-aofe-m _ Dapp Dapp' Dord Daof Dmaof Daof' Dmaof') %. +%term pi2 + %pi (weakeng-aofe-m _ Dapp Dapp' Dord (aofe/pi2 Daof) (maofe/pi2 Dmaof) (aofe/pi2 Daof') (maofe/pi2 Dmaof')) + %<- (weakeng-aofe-m _ Dapp Dapp' Dord Daof Dmaof Daof' Dmaof') %. +%scope weakeng-wfe-m %term at + %pi (weakeng-ofe-m _ Dapp Dapp' Dord (ofe/at Daof) (mofe/at Dmaof) (ofe/at Daof') (mofe/at Dmaof')) + %<- (weakeng-aofe-m _ Dapp Dapp' Dord Daof Dmaof Daof' Dmaof') %. +%term lam + %pi (weakeng-ofe-m _ (%the (append G1 G2 G) Dapp) (%the (append (cons G1 X B) G2 G') Dapp') (%the (ordered G') Dord) (ofe/lam (%the ({y} %pi (isvar y J1) %-> (ofe (cons G y A) (M y) (C y))) Dof) (%the (wfe G A) Dwf)) (mofe/lam (%the ({y} {e} mofe (Dof y e) Mm) Dmof) (%the (mwfe Dwf Am) Dmwf)) (ofe/lam Dof'' Dwf') (mofe/lam Dmof'' Dmwf')) + %<- (weakeng-wfe-m _ Dapp Dapp' Dord Dwf Dmwf Dwf' Dmwf') + %<- ({y} {e isvar y J1} ofe-context (Dof y e) (ordered/cons (%the (bounded G y) (Dbound1 y e)))) + %<- (extend-context Dord (%the ({y} %pi (isvar y J2) %-> (bounded G' y)) Dbound2)) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- (bump-bounded Dleq1 ([y] [e isvar y J1] Dbound1 y e) ([y] [e isvar y J] %the (bounded G y) (Dbound1' y e))) + %<- (bump-bounded Dleq2 ([y] [e isvar y J2] Dbound2 y e) ([y] [e isvar y J] %the (bounded G' y) (Dbound2' y e))) + %<- (bump-ofe-m _ Dof Dmof ([y] [e isvar y J] ordered/cons (Dbound1' y e)) (%the ({y} %pi (isvar y J) %-> (ofe (cons G y A) (M y) (C y))) Dof') (%the ({y} {e} mofe (Dof' y e) Mm) Dmof')) + %<- ({y} {e isvar y J} weakeng-ofe-m _ (append/cons Dapp) (append/cons Dapp') (ordered/cons (Dbound2' y e)) (Dof' y e) (Dmof' y e) (%the (ofe (cons G' y A) (M y) (C y)) (Dof'' y e)) (%the (mofe (Dof'' y e) Mm) (Dmof'' y e))) %. +%scope weakeng-wfe-m %term pair + %pi (weakeng-ofe-m _ (%the (append G1 G2 G) Dapp) (%the (append (cons G1 X B) G2 G') Dapp') (%the (ordered G') Dord) (ofe/pair (%the ({y} %pi (isvar y J1) %-> (wfe (cons G y A) (C y))) Dwf) (%the (ofe G N C') Dof2) (%the (tsub C M C') Dsub) (%the (ofe G M A) Dof1)) (mofe/pair (%the ({y} {e} mwfe (Dwf y e) Cm) Dmwf) (%the (mofe Dof2 Nm) Dmof2) (%the (mofe Dof1 Mm) Dmof1)) (ofe/pair Dwf'' Dof2' Dsub Dof1') (mofe/pair Dmwf'' Dmof2' Dmof1')) + %<- (weakeng-ofe-m _ Dapp Dapp' Dord Dof1 Dmof1 Dof1' Dmof1') + %<- (weakeng-ofe-m _ Dapp Dapp' Dord Dof2 Dmof2 Dof2' Dmof2') + %<- ({y} {e isvar y J1} wfe-context (Dwf y e) (ordered/cons (%the (bounded G y) (Dbound1 y e)))) + %<- (extend-context Dord (%the ({y} %pi (isvar y J2) %-> (bounded G' y)) Dbound2)) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- (bump-bounded Dleq1 ([y] [e isvar y J1] Dbound1 y e) ([y] [e isvar y J] %the (bounded G y) (Dbound1' y e))) + %<- (bump-bounded Dleq2 ([y] [e isvar y J2] Dbound2 y e) ([y] [e isvar y J] %the (bounded G' y) (Dbound2' y e))) + %<- (bump-wfe-m _ Dwf Dmwf ([y] [e isvar y J] ordered/cons (Dbound1' y e)) (%the ({y} %pi (isvar y J) %-> (wfe (cons G y A) (C y))) Dwf') (%the ({y} {e} mwfe (Dwf' y e) Cm) Dmwf')) + %<- ({y} {e isvar y J} weakeng-wfe-m _ (append/cons Dapp) (append/cons Dapp') (ordered/cons (Dbound2' y e)) (Dwf' y e) (Dmwf' y e) (%the (wfe (cons G' y A) (C y)) (Dwf'' y e)) (%the (mwfe (Dwf'' y e) Cm) (Dmwf'' y e))) %. +%term sing + %pi (weakeng-ofe-m _ Dapp Dapp' Dord (ofe/sing Daof) (mofe/sing Dmaof) (ofe/sing Daof') (mofe/sing Dmaof')) + %<- (weakeng-aofe-m _ Dapp Dapp' Dord Daof Dmaof Daof' Dmaof') %. +%scope weakeng-wfe-m %term t weakeng-wfe-m _ Dapp Dapp' Dord (wfe/t _) mwfe/t (wfe/t Dord) mwfe/t %. +%term pi + %pi (weakeng-wfe-m _ (%the (append G1 G2 G) Dapp) (%the (append (cons G1 X B) G2 G') Dapp') (%the (ordered G') Dord) (wfe/pi (%the ({y} %pi (isvar y J1) %-> (wfe (cons G y A) (C y))) Dwf2) (%the (wfe G A) Dwf1)) (mwfe/pi (%the ({y} {e} mwfe (Dwf2 y e) Cm) Dmwf2) (%the (mwfe Dwf1 Am) Dmwf1)) (wfe/pi Dwf2'' Dwf1') (mwfe/pi Dmwf2'' Dmwf1')) + %<- (weakeng-wfe-m _ Dapp Dapp' Dord Dwf1 Dmwf1 Dwf1' Dmwf1') + %<- ({y} {e isvar y J1} wfe-context (Dwf2 y e) (ordered/cons (%the (bounded G y) (Dbound1 y e)))) + %<- (extend-context Dord (%the ({y} %pi (isvar y J2) %-> (bounded G' y)) Dbound2)) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- (bump-bounded Dleq1 ([y] [e isvar y J1] Dbound1 y e) ([y] [e isvar y J] %the (bounded G y) (Dbound1' y e))) + %<- (bump-bounded Dleq2 ([y] [e isvar y J2] Dbound2 y e) ([y] [e isvar y J] %the (bounded G' y) (Dbound2' y e))) + %<- (bump-wfe-m _ Dwf2 Dmwf2 ([y] [e isvar y J] ordered/cons (Dbound1' y e)) (%the ({y} %pi (isvar y J) %-> (wfe (cons G y A) (C y))) Dwf2') (%the ({y} {e} mwfe (Dwf2' y e) Cm) Dmwf2')) + %<- ({y} {e isvar y J} weakeng-wfe-m _ (append/cons Dapp) (append/cons Dapp') (ordered/cons (Dbound2' y e)) (Dwf2' y e) (Dmwf2' y e) (%the (wfe (cons G' y A) (C y)) (Dwf2'' y e)) (%the (mwfe (Dwf2'' y e) Cm) (Dmwf2'' y e))) %. +%scope weakeng-wfe-m %term sigma + %pi (weakeng-wfe-m _ (%the (append G1 G2 G) Dapp) (%the (append (cons G1 X B) G2 G') Dapp') (%the (ordered G') Dord) (wfe/sigma (%the ({y} %pi (isvar y J1) %-> (wfe (cons G y A) (C y))) Dwf2) (%the (wfe G A) Dwf1)) (mwfe/sigma (%the ({y} {e} mwfe (Dwf2 y e) Cm) Dmwf2) (%the (mwfe Dwf1 Am) Dmwf1)) (wfe/sigma Dwf2'' Dwf1') (mwfe/sigma Dmwf2'' Dmwf1')) + %<- (weakeng-wfe-m _ Dapp Dapp' Dord Dwf1 Dmwf1 Dwf1' Dmwf1') + %<- ({y} {e isvar y J1} wfe-context (Dwf2 y e) (ordered/cons (%the (bounded G y) (Dbound1 y e)))) + %<- (extend-context Dord (%the ({y} %pi (isvar y J2) %-> (bounded G' y)) Dbound2)) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- (bump-bounded Dleq1 ([y] [e isvar y J1] Dbound1 y e) ([y] [e isvar y J] %the (bounded G y) (Dbound1' y e))) + %<- (bump-bounded Dleq2 ([y] [e isvar y J2] Dbound2 y e) ([y] [e isvar y J] %the (bounded G' y) (Dbound2' y e))) + %<- (bump-wfe-m _ Dwf2 Dmwf2 ([y] [e isvar y J] ordered/cons (Dbound1' y e)) (%the ({y} %pi (isvar y J) %-> (wfe (cons G y A) (C y))) Dwf2') (%the ({y} {e} mwfe (Dwf2' y e) Cm) Dmwf2')) + %<- ({y} {e isvar y J} weakeng-wfe-m _ (append/cons Dapp) (append/cons Dapp') (ordered/cons (Dbound2' y e)) (Dwf2' y e) (Dmwf2' y e) (%the (wfe (cons G' y A) (C y)) (Dwf2'' y e)) (%the (mwfe (Dwf2'' y e) Cm) (Dmwf2'' y e))) %. +%term sing + %pi (weakeng-wfe-m _ Dapp Dapp' Dord (wfe/sing Daof) (mwfe/sing Dmaof) (wfe/sing Daof') (mwfe/sing Dmaof')) + %<- (weakeng-aofe-m _ Dapp Dapp' Dord Daof Dmaof Daof' Dmaof') %. +%worlds (var bind ovar obind) (weakeng-aofe-m _ _ _ _ _ _ _ _) (weakeng-ofe-m _ _ _ _ _ _ _ _) (weakeng-wfe-m _ _ _ _ _ _ _ _) %. +%total (Rm Mm Am) (weakeng-aofe-m Rm _ _ _ _ _ _ _) (weakeng-ofe-m Mm _ _ _ _ _ _ _) (weakeng-wfe-m Am _ _ _ _ _ _ _) %. +%sort weakeng-aofe %. +%mode weakeng-aofe %in %in %in %in %out %. +%term _ + %pi (weakeng-aofe Dapp Dapp' Dordered Daof Daof') + %<- (can-maofe Daof Dmaof) + %<- (weakeng-aofe-m _ Dapp Dapp' Dordered Daof Dmaof Daof' _) %. +%worlds (var bind ovar obind) (weakeng-aofe _ _ _ _ _) %. +%total {} (weakeng-aofe _ _ _ _ _) %. +%sort weakeng-ofe %. +%mode weakeng-ofe %in %in %in %in %out %. +%term _ + %pi (weakeng-ofe Dapp Dapp' Dordered Dof Dof') + %<- (can-mofe Dof Dmof) + %<- (weakeng-ofe-m _ Dapp Dapp' Dordered Dof Dmof Dof' _) %. +%worlds (var bind ovar obind) (weakeng-ofe _ _ _ _ _) %. +%total {} (weakeng-ofe _ _ _ _ _) %. +%sort weakeng-wfe %. +%mode weakeng-wfe %in %in %in %in %out %. +%term _ + %pi (weakeng-wfe Dapp Dapp' Dordered Dwf Dwf') + %<- (can-mwfe Dwf Dmwf) + %<- (weakeng-wfe-m _ Dapp Dapp' Dordered Dwf Dmwf Dwf' _) %. +%worlds (var bind ovar obind) (weakeng-wfe _ _ _ _ _) %. +%total {} (weakeng-wfe _ _ _ _ _) %. +%sort weaken-lookup %. +%mode weaken-lookup %in %in %in %out %. +%term _ weaken-lookup Dbounded Dlook _ (lookup/miss Dlook Dbounded) %. +%worlds (bind ovar var obind) (weaken-lookup _ _ _ _) %. +%total {} (weaken-lookup _ _ _ _) %. +%sort weaken-ofe %. +%mode weaken-ofe %in %in %in %out %. +%term _ + %pi (weaken-ofe Dbound Dofe _ Dofe') + %<- (weakeng-ofe append/nil append/nil (ordered/cons Dbound) Dofe Dofe') %. +%worlds (var bind ovar obind) (weaken-ofe _ _ _ _) %. +%total {} (weaken-ofe _ _ _ _) %. +%sort weaken-aofe %. +%mode weaken-aofe %in %in %in %out %. +%term _ + %pi (weaken-aofe Dbound Daofe _ Daofe') + %<- (weakeng-aofe append/nil append/nil (ordered/cons Dbound) Daofe Daofe') %. +%worlds (var bind ovar obind) (weaken-aofe _ _ _ _) %. +%total {} (weaken-aofe _ _ _ _) %. +%sort weaken-wfe %. +%mode weaken-wfe %in %in %in %out %. +%term _ + %pi (weaken-wfe Dbound Dwfe _ Dwfe') + %<- (weakeng-wfe append/nil append/nil (ordered/cons Dbound) Dwfe Dwfe') %. +%worlds (var bind ovar obind) (weaken-wfe _ _ _ _) %. +%total {} (weaken-wfe _ _ _ _) %. +%sort weaken-ofe' %. +%mode weaken-ofe' %in %in %in %out %. +%sort weaken-ofe'' %. +%mode weaken-ofe'' %in %in %in %out %. +%scope weaken-ofe'' %term nil weaken-ofe' append/nil _ D D %. +%term cons + %pi (weaken-ofe' (append/cons Dapp) (ordered/cons Dbound) Dofe Dofe'') + %<- (weaken-ofe'' Dapp Dbound Dofe Dofe') + %<- (weaken-ofe Dbound Dofe' _ Dofe'') %. +%term _ + %pi (weaken-ofe'' Dapp Dbound Dofe Dofe') + %<- (bounded-is-ordered Dbound Dord) + %<- (weaken-ofe' Dapp Dord Dofe Dofe') %. +%worlds (var bind ovar obind) (weaken-ofe' _ _ _ _) (weaken-ofe'' _ _ _ _) %. +%total (D1 D2) (weaken-ofe' D1 _ _ _) (weaken-ofe'' D2 _ _ _) %. +%sort weaken-aofe' %. +%mode weaken-aofe' %in %in %in %out %. +%sort weaken-aofe'' %. +%mode weaken-aofe'' %in %in %in %out %. +%scope weaken-aofe'' %term nil weaken-aofe' append/nil _ D D %. +%term cons + %pi (weaken-aofe' (append/cons Dapp) (ordered/cons Dbound) Dofe Dofe'') + %<- (weaken-aofe'' Dapp Dbound Dofe Dofe') + %<- (weaken-aofe Dbound Dofe' _ Dofe'') %. +%term _ + %pi (weaken-aofe'' Dapp Dbound Dofe Dofe') + %<- (bounded-is-ordered Dbound Dord) + %<- (weaken-aofe' Dapp Dord Dofe Dofe') %. +%worlds (var bind ovar obind) (weaken-aofe' _ _ _ _) (weaken-aofe'' _ _ _ _) %. +%total (D1 D2) (weaken-aofe' D1 _ _ _) (weaken-aofe'' D2 _ _ _) %. +%sort weaken-wfe' %. +%mode weaken-wfe' %in %in %in %out %. +%sort weaken-wfe'' %. +%mode weaken-wfe'' %in %in %in %out %. +%scope weaken-wfe'' %term nil weaken-wfe' append/nil _ D D %. +%term cons + %pi (weaken-wfe' (append/cons Dapp) (ordered/cons Dbound) Dwfe Dwfe'') + %<- (weaken-wfe'' Dapp Dbound Dwfe Dwfe') + %<- (weaken-wfe Dbound Dwfe' _ Dwfe'') %. +%term _ + %pi (weaken-wfe'' Dapp Dbound Dwfe Dwfe') + %<- (bounded-is-ordered Dbound Dord) + %<- (weaken-wfe' Dapp Dord Dwfe Dwfe') %. +%worlds (var bind ovar obind) (weaken-wfe' _ _ _ _) (weaken-wfe'' _ _ _ _) %. +%total (D1 D2) (weaken-wfe' D1 _ _ _) (weaken-wfe'' D2 _ _ _) %. +%sort weaken-ofe-csub {x} %. +%mode weaken-ofe-csub %in %in %in %in %out %. +%scope weaken-ofe-csub %term nil weaken-ofe-csub ([x] append/nil) _ csub/base D D %. +%term cons + %pi (weaken-ofe-csub ([x] append/cons (Dapp x)) ([x] [d] ordered/cons (Dbound x d)) (csub/cons _ Dsub) Dof Dof'') + %<- ({x} {d isvar x I} bounded-is-ordered (Dbound x d) (Dord x d)) + %<- (weaken-ofe-csub Dapp Dord Dsub Dof Dof') + %<- (csub-bounded Dbound Dsub Dbound') + %<- (weaken-ofe Dbound' Dof' _ Dof'') %. +%scope weaken-ofe-csub %term bad + %pi (weaken-ofe-csub ([x] append/cons (Dapp x)) ([x] [d] ordered/cons (Dbound x d)) csub/base _ D) + %<- ({x} {d isvar x I} append-bounded-contra (Dapp x) (Dbound x d) Dfalse) + %<- (false-implies-ofe Dfalse D) %. +%worlds (bind ovar var) (weaken-ofe-csub _ _ _ _ _) %. +%total D (weaken-ofe-csub D _ _ _ _) %. +%sort weaken-aofe-insert1 {y} %. +%mode weaken-aofe-insert1 %in %in %in %out %. +%term _ + %pi (weaken-aofe-insert1 (%the ({y} %pi (isvar y J1) %-> (aofe (cons G y B) (R y) (C y))) Daof) (%the ({x} %pi (isvar x I) %-> (bounded G x)) DboundX) A Daof'') + %<- ({y} {e isvar y J1} aofe-context (Daof y e) (ordered/cons (%the (bounded G y) (DboundY y e)))) + %<- ({x} {d isvar x I} following-var d (%the ({y} %pi (isvar y J2) %-> (precedes x y)) (Dprec x d))) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- (bump-bounded Dleq1 DboundY (%the ({y} %pi (isvar y J) %-> (bounded G y)) DboundY')) + %<- ({x} {d isvar x I} bump-precedes Dleq2 (Dprec x d) (%the ({y} %pi (isvar y J) %-> (precedes x y)) (Dprec' x d))) + %<- (bump-aofe Daof ([y] [e isvar y J] ordered/cons (DboundY' y e)) (%the ({y} %pi (isvar y J) %-> (aofe (cons G y B) (R y) (C y))) Daof')) + %<- ({x} {d isvar x I} {y} {e isvar y J} weakeng-aofe (append/cons append/nil) (append/cons append/nil) (ordered/cons (bounded/cons (DboundX x d) (Dprec' x d y e))) (Daof' y e) (%the (aofe (cons (cons G x A) y B) (R y) (C y)) (Daof'' x d y e))) %. +%worlds (var bind ovar) (weaken-aofe-insert1 _ _ _ _) %. +%total {} (weaken-aofe-insert1 _ _ _ _) %. +%sort weaken-ofe-insert1 {y} %. +%mode weaken-ofe-insert1 %in %in %in %out %. +%term _ + %pi (weaken-ofe-insert1 (%the ({y} %pi (isvar y J1) %-> (ofe (cons G y B) (M y) (C y))) Dof) (%the ({x} %pi (isvar x I) %-> (bounded G x)) DboundX) A Dof'') + %<- ({y} {e isvar y J1} ofe-context (Dof y e) (ordered/cons (%the (bounded G y) (DboundY y e)))) + %<- ({x} {d isvar x I} following-var d (%the ({y} %pi (isvar y J2) %-> (precedes x y)) (Dprec x d))) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- (bump-bounded Dleq1 DboundY (%the ({y} %pi (isvar y J) %-> (bounded G y)) DboundY')) + %<- ({x} {d isvar x I} bump-precedes Dleq2 (Dprec x d) (%the ({y} %pi (isvar y J) %-> (precedes x y)) (Dprec' x d))) + %<- (bump-ofe Dof ([y] [e isvar y J] ordered/cons (DboundY' y e)) (%the ({y} %pi (isvar y J) %-> (ofe (cons G y B) (M y) (C y))) Dof')) + %<- ({x} {d isvar x I} {y} {e isvar y J} weakeng-ofe (append/cons append/nil) (append/cons append/nil) (ordered/cons (bounded/cons (DboundX x d) (Dprec' x d y e))) (Dof' y e) (%the (ofe (cons (cons G x A) y B) (M y) (C y)) (Dof'' x d y e))) %. +%worlds (bind ovar var) (weaken-ofe-insert1 _ _ _ _) %. +%total {} (weaken-ofe-insert1 _ _ _ _) %. +%sort weaken-ofe-insert2 {y} %. +%mode weaken-ofe-insert2 %in %in %in %out %. +%term _ + %pi (weaken-ofe-insert2 (%the ({y} + %pi (isvar y J) + %-> ({z} %pi (isvar z K1) %-> (ofe (cons (cons G y B) z (C y)) (M y z) (D y z)))) Dof) (%the ({x} %pi (isvar x I) %-> (bounded G x)) DboundI) A Dof'') + %<- ({y} {e isvar y J} {z} {f isvar z K1} ofe-context (Dof y e z f) (ordered/cons (bounded/cons (%the (bounded G y) (DboundJ_z y e z f)) (%the (precedes y z) (DprecJK1 y e z f))))) + %<- ({y} {e isvar y J} strengthen-bounded (DboundJ_z y e) (%the (bounded G y) (DboundJ y e))) + %<- ({x} {d isvar x I} following-var d (%the ({y} %pi (isvar y J') %-> (precedes x y)) (DprecIJ' x d))) + %<- ({x} {d isvar x I} {y} {e isvar y J'} bounded-increase-bound (DboundI x d) (DprecIJ' x d y e) (%the (bounded G y) (DboundJ'_x x d y e))) + %<- ({y} {e isvar y J'} strengthen-bounded ([x] [d] DboundJ'_x x d y e) (%the (bounded G y) (DboundJ' y e))) + %<- ({y} {e isvar y J'} following-var e (%the ({z} %pi (isvar z K2) %-> (precedes y z)) (DprecJ'K2 y e))) + %<- (can-max K1 K2 K' (%the (max K1 K2 K') Dmax)) + %<- (max-leq Dmax (%the (leq K1 K') Dleq1) (%the (leq K2 K') Dleq2)) + %<- ({y} {e isvar y J} bump-precedes Dleq1 (DprecJK1 y e) (%the ({z} %pi (isvar z K') %-> (precedes y z)) (DprecJK' y e))) + %<- ({y} {e isvar y J'} bump-precedes Dleq2 (DprecJ'K2 y e) (%the ({z} %pi (isvar z K') %-> (precedes y z)) (DprecJ'K' y e))) + %<- (bump2-ofe Dof ([y] [e isvar y J] [z] [f isvar z K'] ordered/cons (bounded/cons (DboundJ y e) (DprecJK' y e z f))) ([y] [e isvar y J'] [z] [f isvar z K'] ordered/cons (bounded/cons (DboundJ' y e) (DprecJ'K' y e z f))) (%the ({y} + %pi (isvar y J') + %-> ({z} %pi (isvar z K') %-> (ofe (cons (cons G y B) z (C y)) (M y z) (D y z)))) Dof')) + %<- ({x} {d isvar x I} {y} {e isvar y J'} {z} {f isvar z K'} weakeng-ofe (append/cons (append/cons append/nil)) (append/cons (append/cons append/nil)) (ordered/cons (bounded/cons (bounded/cons (DboundI x d) (DprecIJ' x d y e)) (DprecJ'K' y e z f))) (Dof' y e z f) (%the (ofe (cons (cons (cons G x A) y B) z (C y)) (M y z) (D y z)) (Dof'' x d y e z f))) %. +%worlds (var bind ovar) (weaken-ofe-insert2 _ _ _ _) %. +%total {} (weaken-ofe-insert2 _ _ _ _) %. +%sort weaken-wfe-insert1 {y} %. +%mode weaken-wfe-insert1 %in %in %in %out %. +%term _ + %pi (weaken-wfe-insert1 (%the ({y} %pi (isvar y J1) %-> (wfe (cons G y B) (C y))) Dwf) (%the ({x} %pi (isvar x I) %-> (bounded G x)) DboundX) A Dwf'') + %<- ({y} {e isvar y J1} wfe-context (Dwf y e) (ordered/cons (%the (bounded G y) (DboundY y e)))) + %<- ({x} {d isvar x I} following-var d (%the ({y} %pi (isvar y J2) %-> (precedes x y)) (Dprec x d))) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- (bump-bounded Dleq1 DboundY (%the ({y} %pi (isvar y J) %-> (bounded G y)) DboundY')) + %<- ({x} {d isvar x I} bump-precedes Dleq2 (Dprec x d) (%the ({y} %pi (isvar y J) %-> (precedes x y)) (Dprec' x d))) + %<- (bump-wfe Dwf ([y] [e isvar y J] ordered/cons (DboundY' y e)) (%the ({y} %pi (isvar y J) %-> (wfe (cons G y B) (C y))) Dwf')) + %<- ({x} {d isvar x I} {y} {e isvar y J} weakeng-wfe (append/cons append/nil) (append/cons append/nil) (ordered/cons (bounded/cons (DboundX x d) (Dprec' x d y e))) (Dwf' y e) (%the (wfe (cons (cons G x A) y B) (C y)) (Dwf'' x d y e))) %. +%worlds (var bind ovar) (weaken-wfe-insert1 _ _ _ _) %. +%total {} (weaken-wfe-insert1 _ _ _ _) %. +%sort weaken-ofe-insert1-bump {y} %. +%mode weaken-ofe-insert1-bump %in %in %in %in %out %. +%term _ + %pi (weaken-ofe-insert1-bump Dof Dbound Dprec A Dof'') + %<- (weaken-ofe-insert1 Dof Dbound A Dof') + %<- ({x} {d} bump-ofe (Dof' x d) ([y] [e] ordered/cons (bounded/cons (Dbound x d) (Dprec x d y e))) (Dof'' x d)) %. +%worlds (var bind ovar) (weaken-ofe-insert1-bump _ _ _ _ _) %. +%total {} (weaken-ofe-insert1-bump _ _ _ _ _) %. +%sort weaken-wfe-insert1-bump {y} %. +%mode weaken-wfe-insert1-bump %in %in %in %in %out %. +%term _ + %pi (weaken-wfe-insert1-bump Dwf Dbound Dprec A Dwf'') + %<- (weaken-wfe-insert1 Dwf Dbound A Dwf') + %<- ({x} {d} bump-wfe (Dwf' x d) ([y] [e] ordered/cons (bounded/cons (Dbound x d) (Dprec x d y e))) (Dwf'' x d)) %. +%worlds (var bind ovar) (weaken-wfe-insert1-bump _ _ _ _ _) %. +%total {} (weaken-wfe-insert1-bump _ _ _ _ _) %. +%%%%% Weakening for Simple Types %%%%% +%sort weakeng-slookup %. +%mode weakeng-slookup %in %in %in %in %out %. +%scope weakeng-slookup %term hit weakeng-slookup (sappend/cons _) (sappend/cons _) (sordered/cons D) (slookup/hit _) (slookup/hit D) %. +%term miss + %pi (weakeng-slookup (sappend/cons Dapp1) (sappend/cons Dapp2) (sordered/cons Dbound) (slookup/miss Dlook _) (slookup/miss Dlook' Dbound)) + %<- (sbounded-is-sordered Dbound Dord) + %<- (weakeng-slookup Dapp1 Dapp2 Dord Dlook Dlook') %. +%scope weakeng-slookup %term snil weakeng-slookup sappend/nil sappend/nil (sordered/cons Dbound) Dlook (slookup/miss Dlook Dbound) %. +%worlds (var bind ovar obind) (weakeng-slookup _ _ _ _ _) %. +%total D (weakeng-slookup _ _ _ D _) %. +%sort weakeng-aofes {R} %. +%mode weakeng-aofes %in %in %in %in %in %out %. +%sort weakeng-ofes {M} %. +%mode weakeng-ofes %in %in %in %in %in %out %. +%sort weakeng-wfes {A} %. +%mode weakeng-wfes %in %in %in %in %in %out %. +%scope weakeng-wfes %term closed weakeng-aofes _ _ _ Dord (aofes/closed _ Dsimp Daof) (aofes/closed Dord Dsimp Daof) %. +%term forall + %pi (weakeng-aofes _ Dappend Dappend' Dord (aofes/forall Dsimp Dwf) (aofes/forall Dsimp Dwf')) + %<- (weakeng-wfes _ Dappend Dappend' Dord Dwf Dwf') %. +%scope weakeng-wfes %term var + %pi (weakeng-aofes _ Dappend Dappend' Dord (aofes/var Dlookup) (aofes/var Dlookup')) + %<- (weakeng-slookup Dappend Dappend' Dord Dlookup Dlookup') %. +%term app + %pi (weakeng-aofes _ Dappend Dappend' Dord (aofes/app D2 D1) (aofes/app D2' D1')) + %<- (weakeng-aofes _ Dappend Dappend' Dord D1 D1') + %<- (weakeng-ofes _ Dappend Dappend' Dord D2 D2') %. +%scope weakeng-wfes %term pi1 + %pi (weakeng-aofes _ Dappend Dappend' Dord (aofes/pi1 D1) (aofes/pi1 D1')) + %<- (weakeng-aofes _ Dappend Dappend' Dord D1 D1') %. +%term pi2 + %pi (weakeng-aofes _ Dappend Dappend' Dord (aofes/pi2 D1) (aofes/pi2 D1')) + %<- (weakeng-aofes _ Dappend Dappend' Dord D1 D1') %. +%scope weakeng-wfes %term at + %pi (weakeng-ofes _ Dappend Dappend' Dord (ofes/at D1) (ofes/at D1')) + %<- (weakeng-aofes _ Dappend Dappend' Dord D1 D1') %. +%term lam + %pi (weakeng-ofes _ Dappend Dappend' Dord (ofes/lam (%the ({y} %pi (isvar y J1) %-> (ofes (scons G y S) (M y) T)) Dof)) (ofes/lam Dof'')) + %<- ({y} {e isvar y J1} ofes-context (Dof y e) (sordered/cons (%the (sbounded G y) (Dbound1 y e)))) + %<- (extend-scontext Dord (%the ({y} %pi (isvar y J2) %-> (sbounded G' y)) Dbound2)) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- (bump-sbounded Dleq1 ([y] [e isvar y J1] Dbound1 y e) ([y] [e isvar y J] %the (sbounded G y) (Dbound1' y e))) + %<- (bump-sbounded Dleq2 ([y] [e isvar y J2] Dbound2 y e) ([y] [e isvar y J] %the (sbounded G' y) (Dbound2' y e))) + %<- (bump-ofes _ Dof ([y] [e isvar y J] sordered/cons (Dbound1' y e)) (%the ({y} %pi (isvar y J) %-> (ofes (scons G y S) (M y) T)) Dof')) + %<- ({y} {e isvar y J} weakeng-ofes _ (sappend/cons Dappend) (sappend/cons Dappend') (sordered/cons (Dbound2' y e)) (Dof' y e) (%the (ofes (scons G' y S) (M y) T) (Dof'' y e))) %. +%scope weakeng-wfes %term pair + %pi (weakeng-ofes _ Dappend Dappend' Dord (ofes/pair D2 D1) (ofes/pair D2' D1')) + %<- (weakeng-ofes _ Dappend Dappend' Dord D1 D1') + %<- (weakeng-ofes _ Dappend Dappend' Dord D2 D2') %. +%term t weakeng-wfes _ _ _ Dord (wfes/t _) (wfes/t Dord) %. +%scope weakeng-wfes %term pi + %pi (weakeng-wfes _ Dappend Dappend' Dord (wfes/pi (%the ({y} %pi (isvar y J1) %-> (wfes (scons G y S) (B y))) D2) (%the (simp A S) Dsimp) (%the (wfes G A) D1)) (wfes/pi D2'' Dsimp D1')) + %<- (weakeng-wfes _ Dappend Dappend' Dord D1 D1') + %<- ({y} {e isvar y J1} wfes-context (D2 y e) (sordered/cons (%the (sbounded G y) (Dbound1 y e)))) + %<- (extend-scontext Dord (%the ({y} %pi (isvar y J2) %-> (sbounded G' y)) Dbound2)) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- (bump-sbounded Dleq1 ([y] [e isvar y J1] Dbound1 y e) ([y] [e isvar y J] %the (sbounded G y) (Dbound1' y e))) + %<- (bump-sbounded Dleq2 ([y] [e isvar y J2] Dbound2 y e) ([y] [e isvar y J] %the (sbounded G' y) (Dbound2' y e))) + %<- (bump-wfes _ D2 ([y] [e isvar y J] sordered/cons (Dbound1' y e)) (%the ({y} %pi (isvar y J) %-> (wfes (scons G y S) (B y))) D2')) + %<- ({y} {e isvar y J} weakeng-wfes _ (sappend/cons Dappend) (sappend/cons Dappend') (sordered/cons (Dbound2' y e)) (D2' y e) (%the (wfes (scons G' y S) (B y)) (D2'' y e))) %. +%term sigma + %pi (weakeng-wfes _ Dappend Dappend' Dord (wfes/sigma (%the ({y} %pi (isvar y J1) %-> (wfes (scons G y S) (B y))) D2) (%the (simp A S) Dsimp) (%the (wfes G A) D1)) (wfes/sigma D2'' Dsimp D1')) + %<- (weakeng-wfes _ Dappend Dappend' Dord D1 D1') + %<- ({y} {e isvar y J1} wfes-context (D2 y e) (sordered/cons (%the (sbounded G y) (Dbound1 y e)))) + %<- (extend-scontext Dord (%the ({y} %pi (isvar y J2) %-> (sbounded G' y)) Dbound2)) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- (bump-sbounded Dleq1 ([y] [e isvar y J1] Dbound1 y e) ([y] [e isvar y J] %the (sbounded G y) (Dbound1' y e))) + %<- (bump-sbounded Dleq2 ([y] [e isvar y J2] Dbound2 y e) ([y] [e isvar y J] %the (sbounded G' y) (Dbound2' y e))) + %<- (bump-wfes _ D2 ([y] [e isvar y J] sordered/cons (Dbound1' y e)) (%the ({y} %pi (isvar y J) %-> (wfes (scons G y S) (B y))) D2')) + %<- ({y} {e isvar y J} weakeng-wfes _ (sappend/cons Dappend) (sappend/cons Dappend') (sordered/cons (Dbound2' y e)) (D2' y e) (%the (wfes (scons G' y S) (B y)) (D2'' y e))) %. +%scope weakeng-wfes %term sing + %pi (weakeng-wfes _ Dappend Dappend' Dord (wfes/sing D1) (wfes/sing D1')) + %<- (weakeng-aofes _ Dappend Dappend' Dord D1 D1') %. +%worlds (var bind ovar) (weakeng-aofes _ _ _ _ _ _) (weakeng-ofes _ _ _ _ _ _) (weakeng-wfes _ _ _ _ _ _) %. +%total (R M A) (weakeng-aofes R _ _ _ _ _) (weakeng-ofes M _ _ _ _ _) (weakeng-wfes A _ _ _ _ _) %. +%sort weaken-slookup %. +%mode weaken-slookup %in %in %in %out %. +%term _ weaken-slookup Dbounded Dlook _ (slookup/miss Dlook Dbounded) %. +%worlds (bind ovar var) (weaken-slookup _ _ _ _) %. +%total {} (weaken-slookup _ _ _ _) %. +%sort weaken-aofes %. +%mode weaken-aofes %in %in %in %out %. +%term _ + %pi (weaken-aofes Dbound D _ D') + %<- (weakeng-aofes _ sappend/nil sappend/nil (sordered/cons Dbound) D D') %. +%worlds (bind var ovar) (weaken-aofes _ _ _ _) %. +%total {} (weaken-aofes _ _ _ _) %. +%sort weaken-ofes %. +%mode weaken-ofes %in %in %in %out %. +%term _ + %pi (weaken-ofes Dbound D _ D') + %<- (weakeng-ofes _ sappend/nil sappend/nil (sordered/cons Dbound) D D') %. +%worlds (bind var ovar) (weaken-ofes _ _ _ _) %. +%total {} (weaken-ofes _ _ _ _) %. +%sort weaken-wfes %. +%mode weaken-wfes %in %in %in %out %. +%term _ + %pi (weaken-wfes Dbound D _ D') + %<- (weakeng-wfes _ sappend/nil sappend/nil (sordered/cons Dbound) D D') %. +%worlds (bind var ovar) (weaken-wfes _ _ _ _) %. +%total {} (weaken-wfes _ _ _ _) %. +%sort weaken-ofes' %. +%mode weaken-ofes' %in %in %in %out %. +%sort weaken-ofes'' %. +%mode weaken-ofes'' %in %in %in %out %. +%scope weaken-ofes'' %term nil weaken-ofes' sappend/nil _ D D %. +%term cons + %pi (weaken-ofes' (sappend/cons Dapp) (sordered/cons Dbound) Dofes Dofes'') + %<- (weaken-ofes'' Dapp Dbound Dofes Dofes') + %<- (weaken-ofes Dbound Dofes' _ Dofes'') %. +%term _ + %pi (weaken-ofes'' Dapp Dbound Dofes Dofes') + %<- (sbounded-is-sordered Dbound Dord) + %<- (weaken-ofes' Dapp Dord Dofes Dofes') %. +%worlds (var bind ovar) (weaken-ofes' _ _ _ _) (weaken-ofes'' _ _ _ _) %. +%total (D1 D2) (weaken-ofes' D1 _ _ _) (weaken-ofes'' D2 _ _ _) %. +%sort weaken-wfes' %. +%mode weaken-wfes' %in %in %in %out %. +%sort weaken-wfes'' %. +%mode weaken-wfes'' %in %in %in %out %. +%scope weaken-wfes'' %term nil weaken-wfes' sappend/nil _ D D %. +%term cons + %pi (weaken-wfes' (sappend/cons Dapp) (sordered/cons Dbound) Dwfes Dwfes'') + %<- (weaken-wfes'' Dapp Dbound Dwfes Dwfes') + %<- (weaken-wfes Dbound Dwfes' _ Dwfes'') %. +%term _ + %pi (weaken-wfes'' Dapp Dbound Dwfes Dwfes') + %<- (sbounded-is-sordered Dbound Dord) + %<- (weaken-wfes' Dapp Dord Dwfes Dwfes') %. +%worlds (var bind ovar) (weaken-wfes' _ _ _ _) (weaken-wfes'' _ _ _ _) %. +%total (D1 D2) (weaken-wfes' D1 _ _ _) (weaken-wfes'' D2 _ _ _) %. +%%%%% Translation to Implicit Form %%%%% +%sort wfi %. +%term wfi/nil %pi (wfi nil A) %<- (wf A) %. +%term wfi/cons %pi (wfi (cons G X A) B) %<- (%pi (vof X A) %-> (wfi G B)) %. +%sort aofi %. +%term aofi/nil %pi (aofi nil R A) %<- (aof R A) %. +%term aofi/cons %pi (aofi (cons G X A) R B) %<- (%pi (vof X A) %-> (aofi G R B)) %. +%sort ofi %. +%term ofi/nil %pi (ofi nil M A) %<- (of M A) %. +%term ofi/cons %pi (ofi (cons G X A) M B) %<- (%pi (vof X A) %-> (ofi G M B)) %. +%sort aof-to-aofi {G} %. +%mode aof-to-aofi %in %in %out %. +%term _ aof-to-aofi nil D (aofi/nil D) %. +%term _ %pi (aof-to-aofi (cons G X A) D (aofi/cons ([_] D'))) %<- (aof-to-aofi G D D') %. +%worlds (bind var ovar ofblock) (aof-to-aofi _ _ _) %. +%total G (aof-to-aofi G _ _) %. +%sort aofi-var %. +%mode aofi-var %in %in %out %. +%term _ aofi-var Daof (wfi/nil Dwf) (aofi/nil (Daof Dwf)) %. +%term _ + %pi (aofi-var (%the (%pi (wf A) %-> (aof R A)) Daof) (wfi/cons (%the (%pi (vof Y B) %-> (wfi G A)) Dwf)) (aofi/cons Daofi)) + %<- ({d vof Y B} aofi-var Daof (Dwf d) (%the (aofi G R A) (Daofi d))) %. +%worlds (bind var ovar ofblock) (aofi-var _ _ _) %. +%total D (aofi-var _ D _) %. +%sort aofi-lookup %. +%mode aofi-lookup %in %in %out %. +%term _ + %pi (aofi-lookup (%the (lookup (cons G X A) X A) (lookup/hit _)) (%the (wfi (cons G X A) A) (wfi/cons (%the (%pi (vof X A) %-> (wfi G A)) Dwf))) (aofi/cons Daofi)) + %<- ({d vof X A} aofi-var ([d_wf] aof/var d_wf d) (Dwf d) (%the (aofi G X A) (Daofi d))) %. +%term _ + %pi (aofi-lookup (%the (lookup (cons G Y B) X A) (lookup/miss (%the (lookup G X A) Dlook) _)) (%the (wfi (cons G Y B) A) (wfi/cons (%the (%pi (vof Y B) %-> (wfi G A)) Dwf))) (aofi/cons D)) + %<- ({d vof Y B} aofi-lookup Dlook (Dwf d) (%the (aofi G X A) (D d))) %. +%worlds (bind var ovar ofblock obind) (aofi-lookup _ _ _) %. +%total D (aofi-lookup D _ _) %. +%sort aofi-forall %. +%mode aofi-forall %in %out %. +%term _ aofi-forall (wfi/nil D) (aofi/nil (aof/forall D)) %. +%term _ %pi (aofi-forall (wfi/cons D) (aofi/cons D')) %<- ({d} aofi-forall (D d) (D' d)) %. +%worlds (bind var ovar ofblock obind) (aofi-forall _ _) %. +%total D (aofi-forall D _) %. +%sort aofi-app %. +%mode aofi-app %in %in %in %in %out %. +%term _ aofi-app (aofi/nil D1) (ofi/nil D2) D3 (wfi/nil D4) (aofi/nil (aof/app D4 D3 D2 D1)) %. +%term _ + %pi (aofi-app (aofi/cons D1) (ofi/cons D2) D3 (wfi/cons D4) (aofi/cons D)) + %<- ({d} aofi-app (D1 d) (D2 d) D3 (D4 d) (D d)) %. +%worlds (bind var ovar ofblock) (aofi-app _ _ _ _ _) %. +%total D (aofi-app D _ _ _ _) %. +%sort aofi-pi1 %. +%mode aofi-pi1 %in %out %. +%term _ aofi-pi1 (aofi/nil D) (aofi/nil (aof/pi1 D)) %. +%term _ %pi (aofi-pi1 (aofi/cons D) (aofi/cons D')) %<- ({d} aofi-pi1 (D d) (D' d)) %. +%worlds (bind var ovar ofblock) (aofi-pi1 _ _) %. +%total D (aofi-pi1 D _) %. +%sort aofi-pi2 %. +%mode aofi-pi2 %in %out %. +%term _ aofi-pi2 (aofi/nil D) (aofi/nil (aof/pi2 D)) %. +%term _ %pi (aofi-pi2 (aofi/cons D) (aofi/cons D')) %<- ({d} aofi-pi2 (D d) (D' d)) %. +%worlds (bind var ovar ofblock) (aofi-pi2 _ _) %. +%total D (aofi-pi2 D _) %. +%sort ofi-at %. +%mode ofi-at %in %out %. +%term _ ofi-at (aofi/nil D) (ofi/nil (of/at D)) %. +%term _ %pi (ofi-at (aofi/cons D) (ofi/cons D')) %<- ({d} ofi-at (D d) (D' d)) %. +%worlds (bind var ovar ofblock) (ofi-at _ _) %. +%total D (ofi-at D _) %. +%sort ofi-lam %. +%mode ofi-lam %in %in %out %. +%term _ ofi-lam (wfi/nil D1) ([x] ofi/cons ([d] ofi/nil (D2 x d))) (ofi/nil (of/lam D2 D1)) %. +%term _ + %pi (ofi-lam (wfi/cons D1) ([x] ofi/cons ([d] ofi/cons ([e] D2 x d e))) (ofi/cons D)) + %<- ({e} ofi-lam (D1 e) ([x] ofi/cons ([d] D2 x d e)) (D e)) %. +%worlds (bind var ovar ofblock) (ofi-lam _ _ _) %. +%total D (ofi-lam D _ _) %. +%sort ofi-pair %. +%mode ofi-pair %in %in %in %in %out %. +%term _ ofi-pair (ofi/nil D1) D2 (ofi/nil D3) ([x] wfi/cons ([d] wfi/nil (D4 x d))) (ofi/nil (of/pair D4 D3 D2 D1)) %. +%term _ + %pi (ofi-pair (ofi/cons D1) D2 (ofi/cons D3) ([x] wfi/cons ([d] wfi/cons ([e] D4 x d e))) (ofi/cons D)) + %<- ({e} ofi-pair (D1 e) D2 (D3 e) ([x] wfi/cons ([d] D4 x d e)) (D e)) %. +%worlds (bind var ovar ofblock) (ofi-pair _ _ _ _ _) %. +%total D (ofi-pair D _ _ _ _) %. +%sort ofi-sing %. +%mode ofi-sing %in %out %. +%term _ ofi-sing (aofi/nil D) (ofi/nil (of/sing D)) %. +%term _ %pi (ofi-sing (aofi/cons D) (ofi/cons D')) %<- ({d} ofi-sing (D d) (D' d)) %. +%worlds (bind var ovar ofblock) (ofi-sing _ _) %. +%total D (ofi-sing D _) %. +%sort wfi-t {G} %. +%mode wfi-t %in %out %. +%term _ wfi-t nil (wfi/nil wf/t) %. +%term _ %pi (wfi-t (cons G X A) (wfi/cons D)) %<- ({d} wfi-t G (D d)) %. +%worlds (bind var ovar ofblock) (wfi-t _ _) %. +%total G (wfi-t G _) %. +%sort wfi-pi %. +%mode wfi-pi %in %in %out %. +%term _ wfi-pi (wfi/nil D1) ([x] wfi/cons ([d] wfi/nil (D2 x d))) (wfi/nil (wf/pi D2 D1)) %. +%term _ + %pi (wfi-pi (wfi/cons D1) ([x] wfi/cons ([d] wfi/cons ([e] D2 x d e))) (wfi/cons D)) + %<- ({e} wfi-pi (D1 e) ([x] wfi/cons ([d] D2 x d e)) (D e)) %. +%worlds (bind var ovar ofblock) (wfi-pi _ _ _) %. +%total D (wfi-pi D _ _) %. +%sort wfi-sigma %. +%mode wfi-sigma %in %in %out %. +%term _ wfi-sigma (wfi/nil D1) ([x] wfi/cons ([d] wfi/nil (D2 x d))) (wfi/nil (wf/sigma D2 D1)) %. +%term _ + %pi (wfi-sigma (wfi/cons D1) ([x] wfi/cons ([d] wfi/cons ([e] D2 x d e))) (wfi/cons D)) + %<- ({e} wfi-sigma (D1 e) ([x] wfi/cons ([d] D2 x d e)) (D e)) %. +%worlds (bind var ovar ofblock) (wfi-sigma _ _ _) %. +%total D (wfi-sigma D _ _) %. +%sort wfi-sing %. +%mode wfi-sing %in %out %. +%term _ wfi-sing (aofi/nil D) (wfi/nil (wf/sing D)) %. +%term _ %pi (wfi-sing (aofi/cons D) (wfi/cons D')) %<- ({d} wfi-sing (D d) (D' d)) %. +%worlds (bind var ovar ofblock) (wfi-sing _ _) %. +%total D (wfi-sing D _) %. +%sort aofe-to-aofi %. +%mode aofe-to-aofi %in %out %. +%sort ofe-to-ofi %. +%mode ofe-to-ofi %in %out %. +%sort wfe-to-wfi %. +%mode wfe-to-wfi %in %out %. +%scope wfe-to-wfi %term closed %pi (aofe-to-aofi (aofe/closed _ D) D') %<- (aof-to-aofi _ D D') %. +%term var + %pi (aofe-to-aofi (aofe/var Dwfe Dlook) Daofi) + %<- (wfe-to-wfi Dwfe Dwfi) + %<- (aofi-lookup Dlook Dwfi Daofi) %. +%scope wfe-to-wfi %term forall + %pi (aofe-to-aofi (aofe/forall D) D'') + %<- (wfe-to-wfi D D') + %<- (aofi-forall D' D'') %. +%term app + %pi (aofe-to-aofi (aofe/app Dwfe Dsub Dofe Daofe) D) + %<- (aofe-to-aofi Daofe Daofi) + %<- (ofe-to-ofi Dofe Dofi) + %<- (wfe-to-wfi Dwfe Dwfi) + %<- (aofi-app Daofi Dofi Dsub Dwfi D) %. +%scope wfe-to-wfi %term pi1 %pi (aofe-to-aofi (aofe/pi1 D) D'') %<- (aofe-to-aofi D D') %<- (aofi-pi1 D' D'') %. +%term pi2 %pi (aofe-to-aofi (aofe/pi2 D) D'') %<- (aofe-to-aofi D D') %<- (aofi-pi2 D' D'') %. +%scope wfe-to-wfi %term at %pi (ofe-to-ofi (ofe/at D) D'') %<- (aofe-to-aofi D D') %<- (ofi-at D' D'') %. +%term lam + %pi (ofe-to-ofi (ofe/lam ([x] [d] Dofe x d) Dwfe) D) + %<- (wfe-to-wfi Dwfe Dwfi) + %<- ({x} {d isvar x I} ofe-to-ofi (Dofe x d) (Dofi x)) + %<- (ofi-lam Dwfi Dofi D) %. +%scope wfe-to-wfi %term pair + %pi (ofe-to-ofi (ofe/pair ([x] [d] Dwfe x d) Dofe2 Dsub Dofe1) D) + %<- (ofe-to-ofi Dofe1 Dofi1) + %<- (ofe-to-ofi Dofe2 Dofi2) + %<- ({x} {d isvar x I} wfe-to-wfi (Dwfe x d) (Dwfi x)) + %<- (ofi-pair Dofi1 Dsub Dofi2 Dwfi D) %. +%term sing + %pi (ofe-to-ofi (ofe/sing Daofe) D) + %<- (aofe-to-aofi Daofe Daofi) + %<- (ofi-sing Daofi D) %. +%scope wfe-to-wfi %term t %pi (wfe-to-wfi (wfe/t _) D) %<- (wfi-t _ D) %. +%term pi + %pi (wfe-to-wfi (wfe/pi ([x] [d] Dwfe2 x d) Dwfe1) D) + %<- (wfe-to-wfi Dwfe1 Dwfi1) + %<- ({x} {d isvar x I} wfe-to-wfi (Dwfe2 x d) (Dwfi2 x)) + %<- (wfi-pi Dwfi1 Dwfi2 D) %. +%scope wfe-to-wfi %term sigma + %pi (wfe-to-wfi (wfe/sigma ([x] [d] Dwfe2 x d) Dwfe1) D) + %<- (wfe-to-wfi Dwfe1 Dwfi1) + %<- ({x} {d isvar x I} wfe-to-wfi (Dwfe2 x d) (Dwfi2 x)) + %<- (wfi-sigma Dwfi1 Dwfi2 D) %. +%term sing + %pi (wfe-to-wfi (wfe/sing Daofe) D) + %<- (aofe-to-aofi Daofe Daofi) + %<- (wfi-sing Daofi D) %. +%worlds (var bind ovar ofblock obind) (ofe-to-ofi _ _) (aofe-to-aofi _ _) (wfe-to-wfi _ _) %. +%total (D1 D2 D3) (aofe-to-aofi D1 _) (ofe-to-ofi D2 _) (wfe-to-wfi D3 _) %. +%sort ofe-to-of %. +%mode ofe-to-of %in %out %. +%term _ %pi (ofe-to-of D D') %<- (ofe-to-ofi D (ofi/nil D')) %. +%worlds (var bind ovar) (ofe-to-of _ _) %. +%total {} (ofe-to-of _ _) %. +%sort aofe-to-aof %. +%mode aofe-to-aof %in %out %. +%term _ %pi (aofe-to-aof D D') %<- (aofe-to-aofi D (aofi/nil D')) %. +%worlds (var bind ovar) (aofe-to-aof _ _) %. +%total {} (aofe-to-aof _ _) %. +%sort wfe-to-wf %. +%mode wfe-to-wf %in %out %. +%term _ %pi (wfe-to-wf D D') %<- (wfe-to-wfi D (wfi/nil D')) %. +%worlds (var bind ovar obind) (wfe-to-wf _ _) %. +%total {} (wfe-to-wf _ _) %. +%sort ofe-to-of1 {x} %. +%mode ofe-to-of1 %in %out %. +%term _ + %pi (ofe-to-of1 Dofe Dof) + %<- ({x} {d' isvar x I} ofe-to-ofi (Dofe x d') (ofi/cons ([d] ofi/nil (Dof x d)))) %. +%worlds (var bind) (ofe-to-of1 _ _) %. +%total {} (ofe-to-of1 _ _) %. +%sort wfe-to-wf1 {x} %. +%mode wfe-to-wf1 %in %out %. +%term _ + %pi (wfe-to-wf1 Dwfe Dof) + %<- ({x} {d' isvar x I} wfe-to-wfi (Dwfe x d') (wfi/cons ([d] wfi/nil (Dof x d)))) %. +%worlds (var bind) (wfe-to-wf1 _ _) %. +%total {} (wfe-to-wf1 _ _) %. +%%%%% Regularity (Explicit Context Version) %%%%% +%% We can't prove regularity of aofe until we've shown that wf implies wfe. +%sort ofe-reg %. +%mode ofe-reg %in %out %. +%scope ofe-reg %term at %pi (ofe-reg (ofe/at Daofe) (wfe/t Dordered)) %<- (aofe-context Daofe Dordered) %. +%term lam + %pi (ofe-reg (ofe/lam Dof Dwf1) (wfe/pi Dwf2 Dwf1)) + %<- ({x} {d} ofe-reg (Dof x d) (Dwf2 x d)) %. +%scope ofe-reg %term pair + %pi (ofe-reg (ofe/pair Dwf2 _ _ Dof) (wfe/sigma Dwf2 Dwf1)) + %<- (ofe-reg Dof Dwf1) %. +%term sing ofe-reg (ofe/sing Daof) (wfe/sing Daof) %. +%worlds (bind ovar var obind) (ofe-reg _ _) %. +%total D (ofe-reg D _) %. +%%%%% Translation to Explicit Form %%%%% +%sort strengthen-for-cut-aof {Rm} {D {x} %pi (vof x A) %-> (aof R (B x))} {x} {d} %. +%mode strengthen-for-cut-aof %in %in %in %out %out %out %. +%sort strengthen-for-cut-of {Mm} {D {x} %pi (vof x A) %-> (of M B)} {x} {d} %. +%mode strengthen-for-cut-of %in %in %in %out %out %. +%sort strengthen-for-cut-wf {Am} {D {x} %pi (vof x A) %-> (wf B)} {x} {d} %. +%mode strengthen-for-cut-wf %in %in %in %out %out %. +%scope strengthen-for-cut-wf %term const + %pi (strengthen-for-cut-aof _ ([x] [d] aof/const (Dwf x d) (Dkof x)) ([x] [d] maof/const (Dmwf x d)) Deq (aof/const Dwf'' (Dkof' aca)) (maof/const Dmwf'')) + %<- (kof-strengthen Dkof Deq) + %<- ({x} kof-resp (Deq x) (Dkof x) (Dkof' x)) + %<- ({x} {d} mwf-resp (Deq x) (Dwf x d) (Dmwf x d) (Dwf' x d) (Dmwf' x d)) + %<- (strengthen-for-cut-wf _ Dwf' Dmwf' Dwf'' Dmwf'') %. +%term forall + %pi (strengthen-for-cut-aof _ ([x] [d] aof/forall (Dwf x d)) ([x] [d] maof/forall (Dmwf x d)) ([x] tp-eq/i) (aof/forall Dwf') (maof/forall Dmwf')) + %<- (strengthen-for-cut-wf _ Dwf Dmwf Dwf' Dmwf') %. +%scope strengthen-for-cut-wf %term var + %pi (strengthen-for-cut-aof _ ([x] [d] aof/var (Dwf x d) Dvof) ([x] [d] maof/var (Dmwf x d)) ([x] tp-eq/i) (aof/var Dwf' Dvof) (maof/var Dmwf')) + %<- (strengthen-for-cut-wf _ Dwf Dmwf Dwf' Dmwf') %. +%term app + %pi (strengthen-for-cut-aof _ ([x] [d] aof/app (Dwf x d) (Dsub x) (Dof x d) (Daof x d)) ([x] [d] maof/app (Dmwf x d) (Dmof x d) (Dmaof x d)) DeqRes (aof/app Dwf'' (Dsub'' aca) Dof'' Daof'') (maof/app Dmwf'' Dmof'' Dmaof'')) + %<- (strengthen-for-cut-aof _ Daof Dmaof DeqPi Daof' Dmaof') + %<- (tp-eq-pi-strengthen DeqPi DeqPi') + %<- ({x} tp-eq-trans (DeqPi x) DeqPi' (DeqPi'' x)) + %<- ({x} tp-eq-cdr-pi (DeqPi'' x) (DeqDom x) (DeqCod x)) + %<- (maof-resp atom-eq/i DeqPi' Daof' Dmaof' Daof'' Dmaof'') + %<- ({x} {d} mof-resp term-eq/i (DeqDom x) (Dof x d) (Dmof x d) (Dof' x d) (Dmof' x d)) + %<- (strengthen-for-cut-of _ Dof' Dmof' Dof'' Dmof'') + %<- ({x} tsub-resp ([y] DeqCod x y) term-eq/i tp-eq/i (Dsub x) (Dsub' x)) + %<- (tsub-closed Dsub' DeqRes) + %<- ({x} tsub-resp ([_] tp-eq/i) term-eq/i (DeqRes x) (Dsub' x) (Dsub'' x)) + %<- ({x} {d} mwf-resp (DeqRes x) (Dwf x d) (Dmwf x d) (Dwf' x d) (Dmwf' x d)) + %<- (strengthen-for-cut-wf _ Dwf' Dmwf' Dwf'' Dmwf'') %. +%scope strengthen-for-cut-wf %term pi1 + %pi (strengthen-for-cut-aof _ ([x] [d] aof/pi1 (Daof x d)) ([x] [d] maof/pi1 (Dmaof x d)) Deq1 (aof/pi1 Daof'') (maof/pi1 Dmaof'')) + %<- (strengthen-for-cut-aof _ Daof Dmaof Deq Daof' Dmaof') + %<- (tp-eq-sigma-strengthen Deq Deq') + %<- ({x} tp-eq-trans (Deq x) Deq' (Deq'' x)) + %<- ({x} tp-eq-cdr-sigma (Deq'' x) (Deq1 x) (Deq2 x)) + %<- (maof-resp atom-eq/i Deq' Daof' Dmaof' Daof'' Dmaof'') %. +%term pi2 + %pi (strengthen-for-cut-aof _ ([x] [d] aof/pi2 (%the (aof R _) (Daof x d))) ([x] [d] maof/pi2 (Dmaof x d)) ([x] Deq2 x (pi1 R)) (aof/pi2 Daof'') (maof/pi2 Dmaof'')) + %<- (strengthen-for-cut-aof _ Daof Dmaof Deq Daof' Dmaof') + %<- (tp-eq-sigma-strengthen Deq Deq') + %<- ({x} tp-eq-trans (Deq x) Deq' (Deq'' x)) + %<- ({x} tp-eq-cdr-sigma (Deq'' x) (Deq1 x) (Deq2 x)) + %<- (maof-resp atom-eq/i Deq' Daof' Dmaof' Daof'' Dmaof'') %. +%scope strengthen-for-cut-wf %term at + %pi (strengthen-for-cut-of _ ([x] [d] of/at (Daof x d)) ([x] [d] mof/at (Dmaof x d)) (of/at Daof'') (mof/at Dmaof'')) + %<- (strengthen-for-cut-aof _ Daof Dmaof Deq Daof' Dmaof') + %<- (tp-eq-symm (Deq aca) Deq') + %<- (maof-resp atom-eq/i Deq' Daof' Dmaof' Daof'' Dmaof'') %. +%term lam + %pi (strengthen-for-cut-of _ ([x] [d] of/lam (Dof x d) (Dwf x d)) ([x] [d] mof/lam (Dmof x d) (Dmwf x d)) (of/lam Dof' Dwf') (mof/lam Dmof' Dmwf')) + %<- (strengthen-for-cut-wf _ Dwf Dmwf Dwf' Dmwf') + %<- ({y} {e} strengthen-for-cut-of _ ([x] [d] Dof x d y e) ([x] [d] Dmof x d y e) (Dof' y e) (Dmof' y e)) %. +%scope strengthen-for-cut-wf %term pair + %pi (strengthen-for-cut-of _ ([x] [d] of/pair (Dwf x d) (Dof2 x d) (Dsub x) (Dof1 x d)) ([x] [d] mof/pair (Dmwf x d) (Dmof2 x d) (Dmof1 x d)) (of/pair Dwf' Dof2'' (Dsub' aca) Dof1') (mof/pair Dmwf' Dmof2'' Dmof1')) + %<- (strengthen-for-cut-of _ Dof1 Dmof1 Dof1' Dmof1') + %<- (tsub-closed Dsub Deq) + %<- ({x} tsub-resp ([_] tp-eq/i) term-eq/i (Deq x) (Dsub x) (Dsub' x)) + %<- ({x} {d} mof-resp term-eq/i (Deq x) (Dof2 x d) (Dmof2 x d) (Dof2' x d) (Dmof2' x d)) + %<- (strengthen-for-cut-of _ Dof2' Dmof2' Dof2'' Dmof2'') + %<- ({y} {e} strengthen-for-cut-wf _ ([x] [d] Dwf x d y e) ([x] [d] Dmwf x d y e) (Dwf' y e) (Dmwf' y e)) %. +%term sing + %pi (strengthen-for-cut-of _ ([x] [d] of/sing (Daof x d)) ([x] [d] mof/sing (Dmaof x d)) (of/sing Daof'') (mof/sing Dmaof'')) + %<- (strengthen-for-cut-aof _ Daof Dmaof Deq Daof' Dmaof') + %<- (tp-eq-symm (Deq aca) Deq') + %<- (maof-resp atom-eq/i Deq' Daof' Dmaof' Daof'' Dmaof'') %. +%scope strengthen-for-cut-wf %term t strengthen-for-cut-wf _ ([x] [d] wf/t) ([x] [d] mwf/t) wf/t mwf/t %. +%term pi + %pi (strengthen-for-cut-wf _ ([x] [d] wf/pi (Dwf2 x d) (Dwf1 x d)) ([x] [d] mwf/pi (Dmwf2 x d) (Dmwf1 x d)) (wf/pi Dwf2' Dwf1') (mwf/pi Dmwf2' Dmwf1')) + %<- (strengthen-for-cut-wf _ Dwf1 Dmwf1 Dwf1' Dmwf1') + %<- ({y} {e} strengthen-for-cut-wf _ ([x] [d] Dwf2 x d y e) ([x] [d] Dmwf2 x d y e) (Dwf2' y e) (Dmwf2' y e)) %. +%scope strengthen-for-cut-wf %term sigma + %pi (strengthen-for-cut-wf _ ([x] [d] wf/sigma (Dwf2 x d) (Dwf1 x d)) ([x] [d] mwf/sigma (Dmwf2 x d) (Dmwf1 x d)) (wf/sigma Dwf2' Dwf1') (mwf/sigma Dmwf2' Dmwf1')) + %<- (strengthen-for-cut-wf _ Dwf1 Dmwf1 Dwf1' Dmwf1') + %<- ({y} {e} strengthen-for-cut-wf _ ([x] [d] Dwf2 x d y e) ([x] [d] Dmwf2 x d y e) (Dwf2' y e) (Dmwf2' y e)) %. +%term sing + %pi (strengthen-for-cut-wf _ ([x] [d] wf/sing (Daof x d)) ([x] [d] mwf/sing (Dmaof x d)) (wf/sing Daof'') (mwf/sing Dmaof'')) + %<- (strengthen-for-cut-aof _ Daof Dmaof Deq Daof' Dmaof') + %<- (tp-eq-symm (Deq aca) Deq') + %<- (maof-resp atom-eq/i Deq' Daof' Dmaof' Daof'' Dmaof'') %. +%worlds (var bind ovar obind) (strengthen-for-cut-aof _ _ _ _ _ _) (strengthen-for-cut-of _ _ _ _ _) (strengthen-for-cut-wf _ _ _ _ _) %. +%total (M1 M2 M3) (strengthen-for-cut-aof M1 _ _ _ _ _) (strengthen-for-cut-of M2 _ _ _ _) (strengthen-for-cut-wf M3 _ _ _ _) %. +%sort strengthen-wf {x} %. +%mode strengthen-wf %in %out %. +%term _ + %pi (strengthen-wf Dwf D) + %<- ({x} {d} can-mwf (Dwf x d) (Dmwf x d)) + %<- (strengthen-for-cut-wf _ Dwf Dmwf D _) %. +%worlds (var bind ovar) (strengthen-wf _ _) %. +%total {} (strengthen-wf _ _) %. +%sort cut-aof {Rm} {D {x} %pi (vof x A) %-> (aof (R x) (B x))} {x} {d} %. +%mode cut-aof %in %in %in %in %out %out %. +%sort cut-of {Mm} {D {x} %pi (vof x A) %-> (of (M x) (B x))} {x} {d} %. +%mode cut-of %in %in %in %in %out %out %. +%sort cut-wf {Bm} {D {x} %pi (vof x A) %-> (wf (B x))} {x} {d} %. +%mode cut-wf %in %in %in %in %out %out %. +%sort cut-aofe {Rm} {D {x} %pi (vof x A) %-> (isvar x I) %-> (aofe (G x) (R x) (B x))} {x} {d} {d'} %. +%mode cut-aofe %in %in %in %in %out %out %. +%sort cut-ofe {Mm} {D {x} %pi (vof x A) %-> (isvar x I) %-> (ofe (G x) (M x) (B x))} {x} {d} {d'} %. +%mode cut-ofe %in %in %in %in %out %out %. +%sort cut-wfe {Bm} {D {x} %pi (vof x A) %-> (isvar x I) %-> (wfe (G x) (B x))} {x} {d} {d'} %. +%mode cut-wfe %in %in %in %in %out %out %. +%scope cut-wfe %term const + %pi (cut-aof (mvar Am) ([x] [d vof x A] aof/const (Dwf x d) (%the (kof C (B x)) (Dkof x))) ([x] [d] maof/const (Dmwf x d)) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] aofe/closed (Dordered x d') (aof/const (Dwf''' x) (Dkof x))) ([x] [d'] maofe/closed (maof/const (Dmwf''' x)))) + %<- (kof-strengthen Dkof Deq) + %<- ({x} tp-eq-symm (Deq x) (Deq' x)) + %<- ({x} {d} mwf-resp (Deq x) (Dwf x d) (Dmwf x d) (Dwf' x d) (Dmwf' x d)) + %<- (strengthen-for-cut-wf _ Dwf' Dmwf' Dwf'' Dmwf'') + %<- ({x} mwf-resp (Deq' x) Dwf'' Dmwf'' (Dwf''' x) (Dmwf''' x)) + %<- ({x} {d' isvar x I} lookup-context (Dlookup x d') (%the (ordered (G x)) (Dordered x d'))) %. +%term forall + %pi (cut-aof (mvar Am) ([x] [d vof x A] aof/forall (%the (wf (B x)) (Dwf x d))) ([x] [d] maof/forall (%the (mwf (Dwf x d) Am) (Dmwf x d))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] aofe/forall (Dwfe x d')) ([x] [d'] maofe/forall (Dmwfe x d'))) + %<- (cut-wf Am Dwf Dmwf Dlookup (%the ({x} %pi (isvar x I) %-> (wfe (G x) (B x))) Dwfe) (%the ({x} {d} mwfe (Dwfe x d) Am) Dmwfe)) %. +%scope cut-wfe %term varsam + %pi (cut-aof (mvar Am) ([x] [d vof x A] aof/var (%the (wf A) (Dwf x d)) d) ([x] [d] maof/var (%the (mwf (Dwf x d) Am) (Dmwf x d))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] aofe/var (Dwfe x d') (Dlookup x d')) ([x] [d'] maofe/var (Dmwfe x d'))) + %<- (cut-wf Am Dwf Dmwf Dlookup (%the ({x} %pi (isvar x I) %-> (wfe (G x) A)) Dwfe) (%the ({x} {d} mwfe (Dwfe x d) Am) Dmwfe)) %. +%term varoth + %pi (cut-aof (mvar Bm) ([x] [d vof x A] aof/var (%the (wf B) (Dwf x d)) (%the (vof Y B) Dvof)) ([x] [d] maof/var (%the (mwf (Dwf x d) Bm) (Dmwf x d))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] aofe/closed (Dordered x d') (aof/var Dwf' Dvof)) ([x] [d'] maofe/closed (maof/var Dmwf'))) + %<- (strengthen-for-cut-wf _ Dwf Dmwf Dwf' Dmwf') + %<- ({x} {d' isvar x I} lookup-context (Dlookup x d') (%the (ordered (G x)) (Dordered x d'))) %. +%scope cut-wfe %term app + %pi (cut-aof (mapp Cm Mm Rm) ([x] [d vof x A] aof/app (%the (wf (C' x)) (Dwf x d)) (%the (tsub ([y] C x y) (M x) (C' x)) (Dsub x)) (%the (of (M x) (B x)) (Dof x d)) (%the (aof (R x) (pi (B x) ([y] C x y))) (Daof x d))) ([x] [d] maof/app (%the (mwf (Dwf x d) Cm) (Dmwf x d)) (%the (mof (Dof x d) Mm) (Dmof x d)) (%the (maof (Daof x d) Rm) (Dmaof x d))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] aofe/app (Dwfe x d') (Dsub x) (Dofe x d') (Daofe x d')) ([x] [d'] maofe/app (Dmwfe x d') (Dmofe x d') (Dmaofe x d'))) + %<- (cut-aof Rm Daof Dmaof Dlookup (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (pi (B x) ([y] C x y)))) Daofe) (%the ({x} {d} maofe (Daofe x d) Rm) Dmaofe)) + %<- (cut-of Mm Dof Dmof Dlookup (%the ({x} %pi (isvar x I) %-> (ofe (G x) (M x) (B x))) Dofe) (%the ({x} {d} mofe (Dofe x d) Mm) Dmofe)) + %<- (cut-wf Cm Dwf Dmwf Dlookup (%the ({x} %pi (isvar x I) %-> (wfe (G x) (C' x))) Dwfe) (%the ({x} {d'} mwfe (Dwfe x d') Cm) Dmwfe)) %. +%term pi1 + %pi (cut-aof (mpi1 Rm) ([x] [d vof x A] aof/pi1 (%the (aof (R x) (sigma (B x) ([y] C x y))) (Daof x d))) ([x] [d] maof/pi1 (%the (maof (Daof x d) Rm) (Dmaof x d))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] aofe/pi1 (Daofe x d')) ([x] [d'] maofe/pi1 (Dmaofe x d'))) + %<- (cut-aof Rm Daof Dmaof Dlookup (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (sigma (B x) ([y] C x y)))) Daofe) (%the ({x} {d'} maofe (Daofe x d') Rm) Dmaofe)) %. +%scope cut-wfe %term pi2 + %pi (cut-aof (mpi2 Rm) ([x] [d vof x A] aof/pi2 (%the (aof (R x) (sigma (B x) ([y] C x y))) (Daof x d))) ([x] [d] maof/pi2 (%the (maof (Daof x d) Rm) (Dmaof x d))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] aofe/pi2 (Daofe x d')) ([x] [d'] maofe/pi2 (Dmaofe x d'))) + %<- (cut-aof Rm Daof Dmaof Dlookup (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (sigma (B x) ([y] C x y)))) Daofe) (%the ({x} {d'} maofe (Daofe x d') Rm) Dmaofe)) %. +%term at + %pi (cut-of (mat Rm) ([x] [d vof x A] of/at (%the (aof (R x) t) (Daof x d))) ([x] [d] mof/at (%the (maof (Daof x d) Rm) (Dmaof x d))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] ofe/at (Daofe x d')) ([x] [d'] mofe/at (Dmaofe x d'))) + %<- (cut-aof Rm Daof Dmaof Dlookup (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) t)) Daofe) (%the ({x} {d'} maofe (Daofe x d') Rm) Dmaofe)) %. +%scope cut-wfe %term lam + %pi (cut-of (mlam Mm Bm) ([x] [d vof x A] of/lam (%the ({y} %pi (vof y (B x)) %-> (of (M x y) (C x y))) (Dof x d)) (%the (wf (B x)) (Dwf x d))) ([x] [d] mof/lam (%the ({y} {e} mof (Dof x d y e) Mm) (Dmof x d)) (%the (mwf (Dwf x d) Bm) (Dmwf x d))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] ofe/lam (Dofe' x d') (Dwfe x d')) ([x] [d'] mofe/lam (Dmofe' x d') (Dmwfe x d'))) + %<- ({x} {d' isvar x I} lookup-context (Dlookup x d') (%the (ordered (G x)) (Dordered x d'))) + %<- ({x} {d' isvar x I} extend-context (Dordered x d') (%the ({y} %pi (isvar y J) %-> (bounded (G x) y)) (Dbounded x d'))) + %<- (cut-wf Bm Dwf Dmwf Dlookup (%the ({x} %pi (isvar x I) %-> (wfe (G x) (B x))) Dwfe) (%the ({x} {d'} mwfe (Dwfe x d') Bm) Dmwfe)) + %<- ({x} {d' isvar x I} {y} {e' isvar y J} weaken-lookup (Dbounded x d' y e') (Dlookup x d') _ (%the (lookup (cons (G x) y (B x)) x A) (Dlookup' x d' y e'))) + %<- ({x} {d vof x A} {d' isvar x I} cut-of Mm ([y] [e] Dof x d y e) ([y] [e] Dmof x d y e) ([y] [e'] lookup/hit (Dbounded x d' y e')) (%the ({y} %pi (isvar y J) %-> (ofe (cons (G x) y (B x)) (M x y) (C x y))) (Dofe x d d')) (%the ({y} {e'} mofe (Dofe x d d' y e') Mm) (Dmofe x d d'))) + %<- ({y} {e' isvar y J} cut-ofe Mm ([x] [d] [d'] Dofe x d d' y e') ([x] [d] [d'] Dmofe x d d' y e') ([x] [d'] Dlookup' x d' y e') ([x] [d'] %the (ofe (cons (G x) y (B x)) (M x y) (C x y)) (Dofe' x d' y e')) ([x] [d'] %the (mofe (Dofe' x d' y e') Mm) (Dmofe' x d' y e'))) %. +%term pair + %pi (cut-of (mpair Cm Nm Mm) ([x] [d vof x A] of/pair (%the ({y} %pi (vof y (B x)) %-> (wf (C x y))) (Dwf x d)) (%the (of (N x) (C' x)) (DofN x d)) (%the (tsub ([y] C x y) (M x) (C' x)) (Dsub x)) (%the (of (M x) (B x)) (DofM x d))) ([x] [d] mof/pair (%the ({y} {e} mwf (Dwf x d y e) Cm) (Dmwf x d)) (%the (mof (DofN x d) Nm) (DmofN x d)) (%the (mof (DofM x d) Mm) (DmofM x d))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] ofe/pair (Dwfe' x d') (DofeN x d') (Dsub x) (DofeM x d')) ([x] [d'] mofe/pair (Dmwfe' x d') (DmofeN x d') (DmofeM x d'))) + %<- ({x} {d' isvar x I} lookup-context (Dlookup x d') (%the (ordered (G x)) (Dordered x d'))) + %<- ({x} {d' isvar x I} extend-context (Dordered x d') (%the ({y} %pi (isvar y J) %-> (bounded (G x) y)) (Dbounded x d'))) + %<- (cut-of Mm DofM DmofM Dlookup (%the ({x} %pi (isvar x I) %-> (ofe (G x) (M x) (B x))) DofeM) (%the ({x} {d'} mofe (DofeM x d') Mm) DmofeM)) + %<- ({x} {d' isvar x I} ofe-reg (DofeM x d') (%the (wfe (G x) (B x)) (DwfeB x d'))) + %<- (cut-of Nm DofN DmofN Dlookup (%the ({x} %pi (isvar x I) %-> (ofe (G x) (N x) (C' x))) DofeN) (%the ({x} {d'} mofe (DofeN x d') Nm) DmofeN)) + %<- ({x} {d' isvar x I} {y} {e' isvar y J} weaken-lookup (Dbounded x d' y e') (Dlookup x d') _ (%the (lookup (cons (G x) y (B x)) x A) (Dlookup' x d' y e'))) + %<- ({x} {d vof x A} {d' isvar x I} cut-wf Cm ([y] [e] Dwf x d y e) ([y] [e] Dmwf x d y e) ([y] [e'] lookup/hit (Dbounded x d' y e')) ([y] [e'] %the (wfe (cons (G x) y (B x)) (C x y)) (Dwfe x d d' y e')) ([y] [e'] %the (mwfe (Dwfe x d d' y e') Cm) (Dmwfe x d d' y e'))) + %<- ({y} {e' isvar y J} cut-wfe Cm ([x] [d] [d'] Dwfe x d d' y e') ([x] [d] [d'] Dmwfe x d d' y e') ([x] [d'] Dlookup' x d' y e') ([x] [d'] %the (wfe (cons (G x) y (B x)) (C x y)) (Dwfe' x d' y e')) ([x] [d'] %the (mwfe (Dwfe' x d' y e') Cm) (Dmwfe' x d' y e'))) %. +%scope cut-wfe %term sing + %pi (cut-of (msingtm Rm) ([x] [d vof x A] of/sing (%the (aof (R x) t) (Daof x d))) ([x] [d] mof/sing (%the (maof (Daof x d) Rm) (Dmaof x d))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] ofe/sing (Daofe x d')) ([x] [d'] mofe/sing (Dmaofe x d'))) + %<- (cut-aof Rm Daof Dmaof Dlookup (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) t)) Daofe) (%the ({x} {d'} maofe (Daofe x d') Rm) Dmaofe)) %. +%term t + %pi (cut-wf mt ([x] [d vof x A] wf/t) ([x] [d] mwf/t) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] wfe/t (Dordered x d')) ([x] [d'] mwfe/t)) + %<- ({x} {d' isvar x I} lookup-context (Dlookup x d') (%the (ordered (G x)) (Dordered x d'))) %. +%scope cut-wfe %term pi + %pi (cut-wf (mpi Cm Bm) ([x] [d vof x A] wf/pi (%the ({y} %pi (vof y (B x)) %-> (wf (C x y))) (DwfC x d)) (%the (wf (B x)) (DwfB x d))) ([x] [d] mwf/pi (%the ({y} {e} mwf (DwfC x d y e) Cm) (DmwfC x d)) (%the (mwf (DwfB x d) Bm) (DmwfB x d))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] wfe/pi (DwfeC' x d') (DwfeB x d')) ([x] [d'] mwfe/pi (DmwfeC' x d') (DmwfeB x d'))) + %<- ({x} {d' isvar x I} lookup-context (Dlookup x d') (%the (ordered (G x)) (Dordered x d'))) + %<- ({x} {d' isvar x I} extend-context (Dordered x d') (%the ({y} %pi (isvar y J) %-> (bounded (G x) y)) (Dbounded x d'))) + %<- (cut-wf Bm DwfB DmwfB Dlookup (%the ({x} %pi (isvar x I) %-> (wfe (G x) (B x))) DwfeB) (%the ({x} {d'} mwfe (DwfeB x d') Bm) DmwfeB)) + %<- ({x} {d' isvar x I} {y} {e' isvar y J} weaken-lookup (Dbounded x d' y e') (Dlookup x d') _ (%the (lookup (cons (G x) y (B x)) x A) (Dlookup' x d' y e'))) + %<- ({x} {d vof x A} {d' isvar x I} cut-wf Cm ([y] [e] DwfC x d y e) ([y] [e] DmwfC x d y e) ([y] [e'] lookup/hit (Dbounded x d' y e')) ([y] [e'] %the (wfe (cons (G x) y (B x)) (C x y)) (DwfeC x d d' y e')) ([y] [e'] %the (mwfe (DwfeC x d d' y e') Cm) (DmwfeC x d d' y e'))) + %<- ({y} {e' isvar y J} cut-wfe Cm ([x] [d] [d'] DwfeC x d d' y e') ([x] [d] [d'] DmwfeC x d d' y e') ([x] [d'] Dlookup' x d' y e') ([x] [d'] %the (wfe (cons (G x) y (B x)) (C x y)) (DwfeC' x d' y e')) ([x] [d'] %the (mwfe (DwfeC' x d' y e') Cm) (DmwfeC' x d' y e'))) %. +%term sigma + %pi (cut-wf (msigma Cm Bm) ([x] [d vof x A] wf/sigma (%the ({y} %pi (vof y (B x)) %-> (wf (C x y))) (DwfC x d)) (%the (wf (B x)) (DwfB x d))) ([x] [d] mwf/sigma (%the ({y} {e} mwf (DwfC x d y e) Cm) (DmwfC x d)) (%the (mwf (DwfB x d) Bm) (DmwfB x d))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] wfe/sigma (DwfeC' x d') (DwfeB x d')) ([x] [d'] mwfe/sigma (DmwfeC' x d') (DmwfeB x d'))) + %<- ({x} {d' isvar x I} lookup-context (Dlookup x d') (%the (ordered (G x)) (Dordered x d'))) + %<- ({x} {d' isvar x I} extend-context (Dordered x d') (%the ({y} %pi (isvar y J) %-> (bounded (G x) y)) (Dbounded x d'))) + %<- (cut-wf Bm DwfB DmwfB Dlookup (%the ({x} %pi (isvar x I) %-> (wfe (G x) (B x))) DwfeB) (%the ({x} {d'} mwfe (DwfeB x d') Bm) DmwfeB)) + %<- ({x} {d' isvar x I} {y} {e' isvar y J} weaken-lookup (Dbounded x d' y e') (Dlookup x d') _ (%the (lookup (cons (G x) y (B x)) x A) (Dlookup' x d' y e'))) + %<- ({x} {d vof x A} {d' isvar x I} cut-wf Cm ([y] [e] DwfC x d y e) ([y] [e] DmwfC x d y e) ([y] [e'] lookup/hit (Dbounded x d' y e')) ([y] [e'] %the (wfe (cons (G x) y (B x)) (C x y)) (DwfeC x d d' y e')) ([y] [e'] %the (mwfe (DwfeC x d d' y e') Cm) (DmwfeC x d d' y e'))) + %<- ({y} {e' isvar y J} cut-wfe Cm ([x] [d] [d'] DwfeC x d d' y e') ([x] [d] [d'] DmwfeC x d d' y e') ([x] [d'] Dlookup' x d' y e') ([x] [d'] %the (wfe (cons (G x) y (B x)) (C x y)) (DwfeC' x d' y e')) ([x] [d'] %the (mwfe (DwfeC' x d' y e') Cm) (DmwfeC' x d' y e'))) %. +%scope cut-wfe %term sing + %pi (cut-wf (msing Rm) ([x] [d vof x A] wf/sing (%the (aof (R x) t) (Daof x d))) ([x] [d] mwf/sing (%the (maof (Daof x d) Rm) (Dmaof x d))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] wfe/sing (Daofe x d')) ([x] [d'] mwfe/sing (Dmaofe x d'))) + %<- (cut-aof Rm Daof Dmaof Dlookup (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) t)) Daofe) (%the ({x} {d'} maofe (Daofe x d') Rm) Dmaofe)) %. +%term closed + %pi (cut-aofe Rm ([x] [d vof x A] [d' isvar x I] aofe/closed (%the (ordered (G x)) (Dordered x d')) (%the (aof (R x) (B x)) (Daof x d))) ([x] [d] [d'] maofe/closed (%the (maof (Daof x d) Rm) (Dmaof x d))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) Daofe Dmaofe) + %<- (cut-aof Rm Daof Dmaof Dlookup (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (B x))) Daofe) (%the ({x} {d'} maofe (Daofe x d') Rm) Dmaofe)) %. +%scope cut-wfe %term forall + %pi (cut-aofe (mvar Am) ([x] [d vof x A] [d' isvar x I] aofe/forall (%the (wfe (G x) (B x)) (Dwfe x d d'))) ([x] [d] [d'] maofe/forall (%the (mwfe (Dwfe x d d') Am) (Dmwfe x d d'))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] aofe/forall (Dwfe' x d')) ([x] [d'] maofe/forall (Dmwfe' x d'))) + %<- (cut-wfe Am Dwfe Dmwfe Dlookup (%the ({x} %pi (isvar x I) %-> (wfe (G x) (B x))) Dwfe') (%the ({x} {d} mwfe (Dwfe' x d) Am) Dmwfe')) %. +%term var + %pi (cut-aofe (mvar Bm) ([x] [d vof x A] [d' isvar x I] aofe/var (%the (wfe (G x) (B x)) (DwfeB x d d')) (%the (lookup (G x) (Y x) (B x)) (DlookupY x d'))) ([x] [d] [d'] maofe/var (%the (mwfe (DwfeB x d d') Bm) (DmwfeB x d d'))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) DlookupX) ([x] [d'] aofe/var (DwfeB' x d') (DlookupY x d')) ([x] [d'] maofe/var (DmwfeB' x d'))) + %<- (cut-wfe Bm DwfeB DmwfeB DlookupX (%the ({x} %pi (isvar x I) %-> (wfe (G x) (B x))) DwfeB') (%the ({x} {d'} mwfe (DwfeB' x d') Bm) DmwfeB')) %. +%scope cut-wfe %term app + %pi (cut-aofe (mapp Cm Mm Rm) ([x] [d vof x A] [d' isvar x I] aofe/app (%the (wfe (G x) (C' x)) (Dwfe x d d')) (%the (tsub ([y] C x y) (M x) (C' x)) (Dsub x)) (%the (ofe (G x) (M x) (B x)) (Dofe x d d')) (%the (aofe (G x) (R x) (pi (B x) ([y] C x y))) (Daofe x d d'))) ([x] [d] [d'] maofe/app (%the (mwfe (Dwfe x d d') Cm) (Dmwfe x d d')) (%the (mofe (Dofe x d d') Mm) (Dmofe x d d')) (%the (maofe (Daofe x d d') Rm) (Dmaofe x d d'))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] aofe/app (Dwfe' x d') (Dsub x) (Dofe' x d') (Daofe' x d')) ([x] [d'] maofe/app (Dmwfe' x d') (Dmofe' x d') (Dmaofe' x d'))) + %<- (cut-aofe Rm Daofe Dmaofe Dlookup (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (pi (B x) ([y] C x y)))) Daofe') (%the ({x} {d'} maofe (Daofe' x d') Rm) Dmaofe')) + %<- (cut-ofe Mm Dofe Dmofe Dlookup (%the ({x} %pi (isvar x I) %-> (ofe (G x) (M x) (B x))) Dofe') (%the ({x} {d'} mofe (Dofe' x d') Mm) Dmofe')) + %<- (cut-wfe Cm Dwfe Dmwfe Dlookup (%the ({x} %pi (isvar x I) %-> (wfe (G x) (C' x))) Dwfe') (%the ({x} {d'} mwfe (Dwfe' x d') Cm) Dmwfe')) %. +%term pi1 + %pi (cut-aofe (mpi1 Rm) ([x] [d vof x A] [d' isvar x I] aofe/pi1 (%the (aofe (G x) (R x) (sigma (B x) ([y] C x y))) (Daofe x d d'))) ([x] [d] [d'] maofe/pi1 (%the (maofe (Daofe x d d') Rm) (Dmaofe x d d'))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] aofe/pi1 (Daofe' x d')) ([x] [d'] maofe/pi1 (Dmaofe' x d'))) + %<- (cut-aofe Rm Daofe Dmaofe Dlookup (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (sigma (B x) ([y] C x y)))) Daofe') (%the ({x} {d'} maofe (Daofe' x d') Rm) Dmaofe')) %. +%scope cut-wfe %term pi2 + %pi (cut-aofe (mpi2 Rm) ([x] [d vof x A] [d' isvar x I] aofe/pi2 (%the (aofe (G x) (R x) (sigma (B x) ([y] C x y))) (Daofe x d d'))) ([x] [d] [d'] maofe/pi2 (%the (maofe (Daofe x d d') Rm) (Dmaofe x d d'))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] aofe/pi2 (Daofe' x d')) ([x] [d'] maofe/pi2 (Dmaofe' x d'))) + %<- (cut-aofe Rm Daofe Dmaofe Dlookup (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (sigma (B x) ([y] C x y)))) Daofe') (%the ({x} {d'} maofe (Daofe' x d') Rm) Dmaofe')) %. +%term at + %pi (cut-ofe (mat Rm) ([x] [d vof x A] [d' isvar x I] ofe/at (%the (aofe (G x) (R x) t) (Daofe x d d'))) ([x] [d] [d'] mofe/at (%the (maofe (Daofe x d d') Rm) (Dmaofe x d d'))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] ofe/at (Daofe' x d')) ([x] [d'] mofe/at (Dmaofe' x d'))) + %<- (cut-aofe Rm Daofe Dmaofe Dlookup (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) t)) Daofe') (%the ({x} {d'} maofe (Daofe' x d') Rm) Dmaofe')) %. +%scope cut-wfe %term lam + %pi (cut-ofe (mlam Mm Bm) ([x] [d vof x A] [d' isvar x I] ofe/lam (%the ({y} %pi (isvar y J) %-> (ofe (cons (G x) y (B x)) (M x y) (C x y))) (Dofe x d d')) (%the (wfe (G x) (B x)) (Dwfe x d d'))) ([x] [d] [d'] mofe/lam (%the ({y} {e} mofe (Dofe x d d' y e) Mm) (Dmofe x d d')) (%the (mwfe (Dwfe x d d') Bm) (Dmwfe x d d'))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] ofe/lam (Dofe' x d') (Dwfe' x d')) ([x] [d'] mofe/lam (Dmofe' x d') (Dmwfe' x d'))) + %<- ({x} {d vof x A} {d' isvar x I} {y} {e' isvar y J} ofe-context (Dofe x d d' y e') (%the (ordered (cons (G x) y (B x))) (ordered/cons (%the (bounded (G x) y) (Dbounded x d' y e'))))) + %<- (cut-wfe Bm Dwfe Dmwfe Dlookup (%the ({x} %pi (isvar x I) %-> (wfe (G x) (B x))) Dwfe') (%the ({x} {d'} mwfe (Dwfe' x d') Bm) Dmwfe')) + %<- ({x} {d' isvar x I} {y} {e' isvar y J} weaken-lookup (Dbounded x d' y e') (Dlookup x d') _ (%the (lookup (cons (G x) y (B x)) x A) (Dlookup' x d' y e'))) + %<- ({y} {e' isvar y J} cut-ofe Mm ([x] [d] [d'] Dofe x d d' y e') ([x] [d] [d'] Dmofe x d d' y e') ([x] [d'] Dlookup' x d' y e') ([x] [d' isvar x I] %the (ofe (cons (G x) y (B x)) (M x y) (C x y)) (Dofe' x d' y e')) ([x] [d'] %the (mofe (Dofe' x d' y e') Mm) (Dmofe' x d' y e'))) %. +%term pair + %pi (cut-ofe (mpair Cm Nm Mm) ([x] [d vof x A] [d' isvar x I] ofe/pair (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (Dwfe x d d')) (%the (ofe (G x) (N x) (C' x)) (DofeN x d d')) (%the (tsub ([y] C x y) (M x) (C' x)) (Dsub x)) (%the (ofe (G x) (M x) (B x)) (DofeM x d d'))) ([x] [d] [d'] mofe/pair (%the ({y} {e} mwfe (Dwfe x d d' y e) Cm) (Dmwfe x d d')) (%the (mofe (DofeN x d d') Nm) (DmofeN x d d')) (%the (mofe (DofeM x d d') Mm) (DmofeM x d d'))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] ofe/pair (Dwfe' x d') (DofeN' x d') (Dsub x) (DofeM' x d')) ([x] [d'] mofe/pair (Dmwfe' x d') (DmofeN' x d') (DmofeM' x d'))) + %<- ({x} {d vof x A} {d' isvar x I} {y} {e' isvar y J} wfe-context (Dwfe x d d' y e') (%the (ordered (cons (G x) y (B x))) (ordered/cons (%the (bounded (G x) y) (Dbounded x d' y e'))))) + %<- (cut-ofe Mm DofeM DmofeM Dlookup (%the ({x} %pi (isvar x I) %-> (ofe (G x) (M x) (B x))) DofeM') (%the ({x} {d'} mofe (DofeM' x d') Mm) DmofeM')) + %<- (cut-ofe Nm DofeN DmofeN Dlookup (%the ({x} %pi (isvar x I) %-> (ofe (G x) (N x) (C' x))) DofeN') (%the ({x} {d'} mofe (DofeN' x d') Nm) DmofeN')) + %<- ({x} {d' isvar x I} {y} {e' isvar y J} weaken-lookup (Dbounded x d' y e') (Dlookup x d') _ (%the (lookup (cons (G x) y (B x)) x A) (Dlookup' x d' y e'))) + %<- ({y} {e' isvar y J} cut-wfe Cm ([x] [d] [d'] Dwfe x d d' y e') ([x] [d] [d'] Dmwfe x d d' y e') ([x] [d'] Dlookup' x d' y e') ([x] [d'] %the (wfe (cons (G x) y (B x)) (C x y)) (Dwfe' x d' y e')) ([x] [d'] %the (mwfe (Dwfe' x d' y e') Cm) (Dmwfe' x d' y e'))) %. +%scope cut-wfe %term sing + %pi (cut-ofe (msingtm Rm) ([x] [d vof x A] [d' isvar x I] ofe/sing (%the (aofe (G x) (R x) t) (Daofe x d d'))) ([x] [d] [d'] mofe/sing (%the (maofe (Daofe x d d') Rm) (Dmaofe x d d'))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] ofe/sing (Daofe' x d')) ([x] [d'] mofe/sing (Dmaofe' x d'))) + %<- (cut-aofe Rm Daofe Dmaofe Dlookup (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) t)) Daofe') (%the ({x} {d'} maofe (Daofe' x d') Rm) Dmaofe')) %. +%term t cut-wfe mt ([x] [d vof x A] [d' isvar x I] wfe/t (%the (ordered (G x)) (Dordered x d'))) ([x] [d] [d'] mwfe/t) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] wfe/t (Dordered x d')) ([x] [d'] mwfe/t) %. +%scope cut-wfe %term pi + %pi (cut-wfe (mpi Cm Bm) ([x] [d vof x A] [d' isvar x I] wfe/pi (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfeC x d d')) (%the (wfe (G x) (B x)) (DwfeB x d d'))) ([x] [d] [d'] mwfe/pi (%the ({y} {e'} mwfe (DwfeC x d d' y e') Cm) (DmwfeC x d d')) (%the (mwfe (DwfeB x d d') Bm) (DmwfeB x d d'))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] wfe/pi (DwfeC' x d') (DwfeB' x d')) ([x] [d'] mwfe/pi (DmwfeC' x d') (DmwfeB' x d'))) + %<- ({x} {d vof x A} {d' isvar x I} {y} {e' isvar y J} wfe-context (DwfeC x d d' y e') (%the (ordered (cons (G x) y (B x))) (ordered/cons (%the (bounded (G x) y) (Dbounded x d' y e'))))) + %<- (cut-wfe Bm DwfeB DmwfeB Dlookup (%the ({x} %pi (isvar x I) %-> (wfe (G x) (B x))) DwfeB') (%the ({x} {d'} mwfe (DwfeB' x d') Bm) DmwfeB')) + %<- ({x} {d' isvar x I} {y} {e' isvar y J} weaken-lookup (Dbounded x d' y e') (Dlookup x d') _ (%the (lookup (cons (G x) y (B x)) x A) (Dlookup' x d' y e'))) + %<- ({y} {e' isvar y J} cut-wfe Cm ([x] [d] [d'] DwfeC x d d' y e') ([x] [d] [d'] DmwfeC x d d' y e') ([x] [d'] Dlookup' x d' y e') ([x] [d'] %the (wfe (cons (G x) y (B x)) (C x y)) (DwfeC' x d' y e')) ([x] [d'] %the (mwfe (DwfeC' x d' y e') Cm) (DmwfeC' x d' y e'))) %. +%term sigma + %pi (cut-wfe (msigma Cm Bm) ([x] [d vof x A] [d' isvar x I] wfe/sigma (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfeC x d d')) (%the (wfe (G x) (B x)) (DwfeB x d d'))) ([x] [d] [d'] mwfe/sigma (%the ({y} {e'} mwfe (DwfeC x d d' y e') Cm) (DmwfeC x d d')) (%the (mwfe (DwfeB x d d') Bm) (DmwfeB x d d'))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] wfe/sigma (DwfeC' x d') (DwfeB' x d')) ([x] [d'] mwfe/sigma (DmwfeC' x d') (DmwfeB' x d'))) + %<- ({x} {d vof x A} {d' isvar x I} {y} {e' isvar y J} wfe-context (DwfeC x d d' y e') (%the (ordered (cons (G x) y (B x))) (ordered/cons (%the (bounded (G x) y) (Dbounded x d' y e'))))) + %<- (cut-wfe Bm DwfeB DmwfeB Dlookup (%the ({x} %pi (isvar x I) %-> (wfe (G x) (B x))) DwfeB') (%the ({x} {d'} mwfe (DwfeB' x d') Bm) DmwfeB')) + %<- ({x} {d' isvar x I} {y} {e' isvar y J} weaken-lookup (Dbounded x d' y e') (Dlookup x d') _ (%the (lookup (cons (G x) y (B x)) x A) (Dlookup' x d' y e'))) + %<- ({y} {e' isvar y J} cut-wfe Cm ([x] [d] [d'] DwfeC x d d' y e') ([x] [d] [d'] DmwfeC x d d' y e') ([x] [d'] Dlookup' x d' y e') ([x] [d'] %the (wfe (cons (G x) y (B x)) (C x y)) (DwfeC' x d' y e')) ([x] [d'] %the (mwfe (DwfeC' x d' y e') Cm) (DmwfeC' x d' y e'))) %. +%scope cut-wfe %term sing + %pi (cut-wfe (msing Rm) ([x] [d vof x A] [d' isvar x I] wfe/sing (%the (aofe (G x) (R x) t) (Daofe x d d'))) ([x] [d] [d'] mwfe/sing (%the (maofe (Daofe x d d') Rm) (Dmaofe x d d'))) (%the ({x} %pi (isvar x I) %-> (lookup (G x) x A)) Dlookup) ([x] [d'] wfe/sing (Daofe' x d')) ([x] [d'] mwfe/sing (Dmaofe' x d'))) + %<- (cut-aofe Rm Daofe Dmaofe Dlookup (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) t)) Daofe') (%the ({x} {d'} maofe (Daofe' x d') Rm) Dmaofe')) %. +%worlds (var bind ovar obind) (cut-aof _ _ _ _ _ _) (cut-of _ _ _ _ _ _) (cut-wf _ _ _ _ _ _) (cut-aofe _ _ _ _ _ _) (cut-ofe _ _ _ _ _ _) (cut-wfe _ _ _ _ _ _) %. +%total (M1 M2 M3 M4 M5 M6) (cut-aof M1 _ _ _ _ _) (cut-of M2 _ _ _ _ _) (cut-wf M3 _ _ _ _ _) (cut-aofe M4 _ _ _ _ _) (cut-ofe M5 _ _ _ _ _) (cut-wfe M6 _ _ _ _ _) %. +%sort of1-to-ofe {I nat} {x} %. +%mode of1-to-ofe %in %in %out %. +%term _ + %pi (of1-to-ofe I (%the ({x} %pi (vof x A) %-> (of (M x) (B x))) Dof) Dofe) + %<- ({x} {d vof x A} can-mof (Dof x d) (%the (mof (Dof x d) Mm) (Dmof x d))) + %<- (cut-of Mm Dof Dmof ([x] [d' isvar x I] lookup/hit (bounded/nil d')) (%the ({x} %pi (isvar x I) %-> (ofe (cons nil x A) (M x) (B x))) Dofe) _) %. +%worlds (var bind ovar) (of1-to-ofe _ _ _) %. +%total {} (of1-to-ofe _ _ _) %. +%sort aof1-to-aofe {I nat} {x} %. +%mode aof1-to-aofe %in %in %out %. +%term _ + %pi (aof1-to-aofe I (%the ({x} %pi (vof x A) %-> (aof (M x) (B x))) Daof) Daofe) + %<- ({x} {d vof x A} can-maof (Daof x d) (%the (maof (Daof x d) Mm) (Dmaof x d))) + %<- (cut-aof Mm Daof Dmaof ([x] [d'] lookup/hit (bounded/nil d')) (%the ({x} %pi (isvar x I) %-> (aofe (cons nil x A) (M x) (B x))) Daofe) _) %. +%worlds (var bind ovar) (aof1-to-aofe _ _ _) %. +%total {} (aof1-to-aofe _ _ _) %. +%sort wf1-to-wfe {I nat} {x} %. +%mode wf1-to-wfe %in %in %out %. +%term _ + %pi (wf1-to-wfe I (%the ({x} %pi (vof x A) %-> (wf (B x))) Dwf) Dwfe) + %<- ({x} {d vof x A} can-mwf (Dwf x d) (%the (mwf (Dwf x d) Bm) (Dmwf x d))) + %<- (cut-wf Bm Dwf Dmwf ([x] [d'] lookup/hit (bounded/nil d')) (%the ({x} %pi (isvar x I) %-> (wfe (cons nil x A) (B x))) Dwfe) _) %. +%worlds (var bind ovar) (wf1-to-wfe _ _ _) %. +%total {} (wf1-to-wfe _ _ _) %. +%sort aof-to-aofe %. +%mode aof-to-aofe %in %out %. +%term _ aof-to-aofe Daof (aofe/closed ordered/nil Daof) %. +%worlds (var bind ovar) (aof-to-aofe _ _) %. +%total {} (aof-to-aofe _ _) %. +%sort wf-to-wfe %. +%mode wf-to-wfe %in %out %. +%scope wf-to-wfe %term t wf-to-wfe wf/t (wfe/t ordered/nil) %. +%term pi + %pi (wf-to-wfe (wf/pi D2 D1) (wfe/pi D2' D1')) + %<- (wf-to-wfe D1 D1') + %<- (wf1-to-wfe 0 D2 D2') %. +%scope wf-to-wfe %term sigma + %pi (wf-to-wfe (wf/sigma D2 D1) (wfe/sigma D2' D1')) + %<- (wf-to-wfe D1 D1') + %<- (wf1-to-wfe 0 D2 D2') %. +%term sing %pi (wf-to-wfe (wf/sing D) (wfe/sing D')) %<- (aof-to-aofe D D') %. +%worlds (var bind ovar) (wf-to-wfe _ _) %. +%total D (wf-to-wfe D _) %. +%sort of-to-ofe %. +%mode of-to-ofe %in %out %. +%scope of-to-ofe %term at of-to-ofe (of/at Daof) (ofe/at (aofe/closed ordered/nil Daof)) %. +%term lam + %pi (of-to-ofe (of/lam Dof Dwf) (ofe/lam Dofe Dwfe)) + %<- (of1-to-ofe 0 Dof Dofe) + %<- (wf-to-wfe Dwf Dwfe) %. +%scope of-to-ofe %term pair + %pi (of-to-ofe (of/pair Dwf Dof2 Dsub Dof1) (ofe/pair Dwfe Dofe2 Dsub Dofe1)) + %<- (of-to-ofe Dof1 Dofe1) + %<- (of-to-ofe Dof2 Dofe2) + %<- (wf1-to-wfe 0 Dwf Dwfe) %. +%term sing of-to-ofe (of/sing Daof) (ofe/sing (aofe/closed ordered/nil Daof)) %. +%worlds (var bind ovar) (of-to-ofe _ _) %. +%total D (of-to-ofe D _) %. +%sort of2-to-ofe %. +%mode of2-to-ofe %in %in %out %. +%term _ + %pi (of2-to-ofe (%the (lt I J) Dlt) (%the ({x} %pi (vof x A) %-> ({y} %pi (vof y (B x)) %-> (of (M x y) (C x y)))) Dof) Dofe') + %<- ({x} {d vof x A} {y} {e vof y (B x)} can-mof (Dof x d y e) (%the (mof (Dof x d y e) Mm) (Dmof x d y e))) + %<- ({x} {d vof x A} {d' isvar x I} cut-of Mm ([y] [e] Dof x d y e) ([y] [e] Dmof x d y e) ([y] [e'] lookup/hit (bounded/cons (bounded/nil d') (precedes/i Dlt e' d'))) ([y] [e'] %the (ofe (cons (cons nil x A) y (B x)) (M x y) (C x y)) (Dofe x d d' y e')) ([y] [e'] %the (mofe (Dofe x d d' y e') Mm) (Dmofe x d d' y e'))) + %<- ({y} {e' isvar y J} cut-ofe Mm ([x] [d] [d'] Dofe x d d' y e') ([x] [d] [d'] Dmofe x d d' y e') ([x] [d'] lookup/miss (lookup/hit (bounded/nil d')) (bounded/cons (bounded/nil d') (precedes/i Dlt e' d'))) ([x] [d'] %the (ofe (cons (cons nil x A) y (B x)) (M x y) (C x y)) (Dofe' x d' y e')) _) %. +%worlds (var bind ovar) (of2-to-ofe _ _ _) %. +%total {} (of2-to-ofe _ _ _) %. +%sort aof2-to-aofe %. +%mode aof2-to-aofe %in %in %out %. +%term _ + %pi (aof2-to-aofe (%the (lt I J) Dlt) (%the ({x} %pi (vof x A) %-> ({y} %pi (vof y (B x)) %-> (aof (R x y) (C x y)))) Daof) Daofe') + %<- ({x} {d vof x A} {y} {e vof y (B x)} can-maof (Daof x d y e) (%the (maof (Daof x d y e) Rm) (Dmaof x d y e))) + %<- ({x} {d vof x A} {d' isvar x I} cut-aof Rm ([y] [e] Daof x d y e) ([y] [e] Dmaof x d y e) ([y] [e'] lookup/hit (bounded/cons (bounded/nil d') (precedes/i Dlt e' d'))) ([y] [e'] %the (aofe (cons (cons nil x A) y (B x)) (R x y) (C x y)) (Daofe x d d' y e')) ([y] [e'] %the (maofe (Daofe x d d' y e') Rm) (Dmaofe x d d' y e'))) + %<- ({y} {e' isvar y J} cut-aofe Rm ([x] [d] [d'] Daofe x d d' y e') ([x] [d] [d'] Dmaofe x d d' y e') ([x] [d'] lookup/miss (lookup/hit (bounded/nil d')) (bounded/cons (bounded/nil d') (precedes/i Dlt e' d'))) ([x] [d'] %the (aofe (cons (cons nil x A) y (B x)) (R x y) (C x y)) (Daofe' x d' y e')) _) %. +%worlds (var bind ovar) (aof2-to-aofe _ _ _) %. +%total {} (aof2-to-aofe _ _ _) %. +%sort wf2-to-wfe %. +%mode wf2-to-wfe %in %in %out %. +%term _ + %pi (wf2-to-wfe (%the (lt I J) Dlt) (%the ({x} %pi (vof x A) %-> ({y} %pi (vof y (B x)) %-> (wf (C x y)))) Dwf) Dwfe') + %<- ({x} {d vof x A} {y} {e vof y (B x)} can-mwf (Dwf x d y e) (%the (mwf (Dwf x d y e) Cm) (Dmwf x d y e))) + %<- ({x} {d vof x A} {d' isvar x I} cut-wf Cm ([y] [e] Dwf x d y e) ([y] [e] Dmwf x d y e) ([y] [e'] lookup/hit (bounded/cons (bounded/nil d') (precedes/i Dlt e' d'))) ([y] [e'] %the (wfe (cons (cons nil x A) y (B x)) (C x y)) (Dwfe x d d' y e')) ([y] [e'] %the (mwfe (Dwfe x d d' y e') Cm) (Dmwfe x d d' y e'))) + %<- ({y} {e' isvar y J} cut-wfe Cm ([x] [d] [d'] Dwfe x d d' y e') ([x] [d] [d'] Dmwfe x d d' y e') ([x] [d'] lookup/miss (lookup/hit (bounded/nil d')) (bounded/cons (bounded/nil d') (precedes/i Dlt e' d'))) ([x] [d'] %the (wfe (cons (cons nil x A) y (B x)) (C x y)) (Dwfe' x d' y e')) _) %. +%worlds (var bind ovar) (wf2-to-wfe _ _ _) %. +%total {} (wf2-to-wfe _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/functionality-thm.lf b/new-tests/stelf-output/tslf/sing/functionality-thm.lf new file mode 100644 index 0000000..b3d0215 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/functionality-thm.lf @@ -0,0 +1,682 @@ +%%%%% Contexts %%%%% +%sort eisvar %. +%block eovar {n nat} [x eterm] [d eisvar x]%. +%sort ectx %. +%name ectx %. +%term enil ectx %. +%term econs %pi ectx %-> eterm %-> etp %-> ectx %. +%sort clean %. +%term clean/nil clean enil %. +%term clean/cons %pi (clean (econs G X A)) %<- (clean G) %<- (eisvar X) %. +%sort elookup %. +%term elookup/hit elookup (econs G X A) X A %. +%term elookup/miss %pi (elookup (econs G Y _) X A) %<- (elookup G X A) %. +%%%%% Context Lemmas %%%%% +%sort eisvar-closed {x eterm} %. +%mode eisvar-closed %in %out %. +%term _ eisvar-closed ([_] D) ([_] eterm-eq/i) %. +%worlds (nbind ebind evar eovar eofblock) (eisvar-closed _ _) %. +%total {} (eisvar-closed _ _) %. +%sort lookup-clean {x eterm} %. +%mode lookup-clean %in %in %out %. +%term _ + %pi (lookup-clean ([x] elookup/hit) ([x] clean/cons (Disvar x) _) Deq) + %<- (eisvar-closed Disvar Deq) %. +%term _ + %pi (lookup-clean ([x] elookup/miss (Dlook x)) ([x] clean/cons _ (Dclean x)) Deq) + %<- (lookup-clean Dlook Dclean Deq) %. +%worlds (nbind ebind evar eovar eofblock) (lookup-clean _ _ _) %. +%total D (lookup-clean D _ _) %. +%%%%% Explicit/Implicit Context System %%%%% +%sort ewfi %. +%sort eofi %. +%sort subtpi %. +%sort tequivi %. +%sort equivi %. +%term ewfi/nil %pi (ewfi enil A) %<- (ewf A) %. +%term ewfi/cons %pi (ewfi (econs G X A) B) %<- (%pi (evof X A) %-> (ewfi G B)) %. +%term eofi/nil %pi (eofi enil M A) %<- (eof M A) %. +%term eofi/cons %pi (eofi (econs G X A) M B) %<- (%pi (evof X A) %-> (eofi G M B)) %. +%term subtpi/nil %pi (subtpi enil B C) %<- (subtp B C) %. +%term subtpi/cons %pi (subtpi (econs G X A) B C) %<- (%pi (evof X A) %-> (subtpi G B C)) %. +%term tequivi/nil %pi (tequivi enil B C) %<- (tequiv B C) %. +%term tequivi/cons %pi (tequivi (econs G X A) B C) %<- (%pi (evof X A) %-> (tequivi G B C)) %. +%term equivi/nil %pi (equivi enil M N B) %<- (equiv M N B) %. +%term equivi/cons %pi (equivi (econs G X A) M N B) %<- (%pi (evof X A) %-> (equivi G M N B)) %. +%%%%% Equality %%%%% +%sort ectx-eq %. +%term ectx-eq/i ectx-eq G G %. +%sort eofi-resp %. +%mode eofi-resp %in %in %in %in %out %. +%term _ eofi-resp ectx-eq/i eterm-eq/i etp-eq/i D D %. +%worlds (nbind ebind evar eovar eofblock) (eofi-resp _ _ _ _ _) %. +%total {} (eofi-resp _ _ _ _ _) %. +%sort equivi-resp %. +%mode equivi-resp %in %in %in %in %in %out %. +%term _ equivi-resp ectx-eq/i eterm-eq/i eterm-eq/i etp-eq/i D D %. +%worlds (nbind ebind evar eovar eofblock) (equivi-resp _ _ _ _ _ _) %. +%total {} (equivi-resp _ _ _ _ _ _) %. +%sort tequivi-resp %. +%mode tequivi-resp %in %in %in %in %out %. +%term _ tequivi-resp ectx-eq/i etp-eq/i etp-eq/i D D %. +%worlds (nbind ebind evar eofblock eovar) (tequivi-resp _ _ _ _ _) %. +%total {} (tequivi-resp _ _ _ _ _) %. +%%%%% Proof Skeletons %%%%% +%sort metp %. +%sort meterm %. +%term metp/t metp %. +%term metp/pi %pi metp %-> metp %-> metp %. +%term metp/sigma %pi metp %-> metp %-> metp %. +%term metp/sing %pi meterm %-> metp %. +%term meterm/var meterm %. +%term meterm/const meterm %. +%term meterm/forall %pi metp %-> meterm %. +%term meterm/lam %pi meterm %-> metp %-> meterm %. +%term meterm/app %pi meterm %-> meterm %-> meterm %. +%term meterm/pair %pi meterm %-> meterm %-> meterm %. +%term meterm/pi1 %pi meterm %-> meterm %. +%term meterm/pi2 %pi meterm %-> meterm %. +%term meterm/sing %pi meterm %-> meterm %. +%term meterm/extpi %pi meterm %-> meterm %-> meterm %. +%term meterm/extsigma %pi meterm %-> meterm %-> meterm %. +%term meterm/subsume %pi meterm %-> meterm %. +%sort mewf %. +%sort meof %. +%term mewf/t mewf ewf/t metp/t %. +%term mewf/pi + %pi (mewf (ewf/pi D2 D1) (metp/pi A2 A1)) + %<- (mewf D1 A1) + %<- ({x} {d} mewf (D2 x d) A2) %. +%term mewf/sigma + %pi (mewf (ewf/sigma D2 D1) (metp/sigma A2 A1)) + %<- (mewf D1 A1) + %<- ({x} {d} mewf (D2 x d) A2) %. +%term mewf/sing %pi (mewf (ewf/sing D) (metp/sing M)) %<- (meof D M) %. +%term meof/var meof (eof/var _ _) meterm/var %. +%term meof/const meof (eof/const _ _) meterm/const %. +%term meof/forall %pi (meof (eof/forall D) (meterm/forall A)) %<- (mewf D A) %. +%term meof/lam + %pi (meof (eof/lam D2 D1) (meterm/lam M A)) + %<- (mewf D1 A) + %<- ({x} {d} meof (D2 x d) M) %. +%term meof/app %pi (meof (eof/app D2 D1) (meterm/app M2 M1)) %<- (meof D1 M1) %<- (meof D2 M2) %. +%term meof/pair + %pi (meof (eof/pair _ D2 D1) (meterm/pair M2 M1)) + %<- (meof D1 M1) + %<- (meof D2 M2) %. +%term meof/pi1 %pi (meof (eof/pi1 D) (meterm/pi1 M)) %<- (meof D M) %. +%term meof/pi2 %pi (meof (eof/pi2 D) (meterm/pi2 M)) %<- (meof D M) %. +%term meof/sing %pi (meof (eof/sing D) (meterm/sing M)) %<- (meof D M) %. +%term meof/extpi + %pi (meof (eof/extpi D2 D1) (meterm/extpi M2 M1)) + %<- (meof D1 M1) + %<- ({x} {d} meof (D2 x d) M2) %. +%term meof/extsigma + %pi (meof (eof/extsigma _ D2 D1) (meterm/extsigma M2 M1)) + %<- (meof D1 M1) + %<- (meof D2 M2) %. +%term meof/subsume %pi (meof (eof/subsume _ D) (meterm/subsume M)) %<- (meof D M) %. +%sort mewfi %. +%term mewfi/nil %pi (mewfi (ewfi/nil D) M) %<- (mewf D M) %. +%term mewfi/cons %pi (mewfi (ewfi/cons D) M) %<- ({d} mewfi (D d) M) %. +%sort meofi %. +%term meofi/nil %pi (meofi (eofi/nil D) M) %<- (meof D M) %. +%term meofi/cons %pi (meofi (eofi/cons D) M) %<- ({d} meofi (D d) M) %. +%%%%% Substitution %%%%% +%sort eofi-subst-gen %. +%mode eofi-subst-gen %in %out %. +%term _ + %pi (eofi-subst-gen ([d] eofi/nil (D d)) ([d] eofi/nil (D' d))) + %<- (esubst-of-gen D D') %. +%term _ + %pi (eofi-subst-gen ([d] eofi/cons (D d)) ([d] eofi/cons (D' d))) + %<- ({e} eofi-subst-gen ([d] D d e) ([d] D' d e)) %. +%worlds (nbind ebind evar eofblock eovar) (eofi-subst-gen _ _) %. +%total D (eofi-subst-gen D _) %. +%sort ewfi-subst-gen %. +%mode ewfi-subst-gen %in %out %. +%term _ + %pi (ewfi-subst-gen ([d] ewfi/nil (D d)) ([d] ewfi/nil (D' d))) + %<- (esubst-wf-gen D D') %. +%term _ + %pi (ewfi-subst-gen ([d] ewfi/cons (D d)) ([d] ewfi/cons (D' d))) + %<- ({e} ewfi-subst-gen ([d] D d e) ([d] D' d e)) %. +%worlds (nbind ebind evar eofblock eovar) (ewfi-subst-gen _ _) %. +%total D (ewfi-subst-gen D _) %. +%sort subtpi-subst-gen %. +%mode subtpi-subst-gen %in %out %. +%term _ + %pi (subtpi-subst-gen ([d] subtpi/nil (D d)) ([d] subtpi/nil (D' d))) + %<- (esubst-subtp-gen D D') %. +%term _ + %pi (subtpi-subst-gen ([d] subtpi/cons (D d)) ([d] subtpi/cons (D' d))) + %<- ({e} subtpi-subst-gen ([d] D d e) ([d] D' d e)) %. +%worlds (nbind ebind evar eofblock eovar) (subtpi-subst-gen _ _) %. +%total D (subtpi-subst-gen D _) %. +%sort eofi-subst {x} %. +%mode eofi-subst %in %in %out %. +%term _ %pi (eofi-subst D1 D2 (D1' D2)) %<- (eofi-subst-gen (D1 _) D1') %. +%worlds (nbind ebind evar eovar eofblock) (eofi-subst _ _ _) %. +%total {} (eofi-subst _ _ _) %. +%sort ewfi-subst {x} %. +%mode ewfi-subst %in %in %out %. +%term _ %pi (ewfi-subst D1 D2 (D1' D2)) %<- (ewfi-subst-gen (D1 _) D1') %. +%worlds (nbind ebind evar eovar eofblock) (ewfi-subst _ _ _) %. +%total {} (ewfi-subst _ _ _) %. +%sort subtpi-subst {x} %. +%mode subtpi-subst %in %in %out %. +%term _ %pi (subtpi-subst D1 D2 (D1' D2)) %<- (subtpi-subst-gen (D1 _) D1') %. +%worlds (nbind ebind evar eofblock eovar) (subtpi-subst _ _ _) %. +%total {} (subtpi-subst _ _ _) %. +%%%%% Destructors %%%%% +%sort elookup? %. +%term elookup?/look %pi (elookup? G X A) %<- (elookup G X A) %. +%term elookup?/not %pi (elookup? G X A) %<- (evof X A) %. +%sort elookup?-cons %. +%mode elookup?-cons %in %out %. +%term _ elookup?-cons ([d] elookup?/look D) (elookup?/look (elookup/miss D)) %. +%term _ elookup?-cons ([d] elookup?/not D) (elookup?/not D) %. +%term _ elookup?-cons ([d] elookup?/not d) (elookup?/look elookup/hit) %. +%worlds (nbind ebind evar eovar eofblock) (elookup?-cons _ _) %. +%total {} (elookup?-cons _ _) %. +%sort eofi-var-invert {D eofi G M A} %. +%mode eofi-var-invert %in %in %out %out %. +%term _ eofi-var-invert (eofi/nil (eof/var Dwf Dvof)) (meofi/nil meof/var) (elookup?/not Dvof) (ewfi/nil Dwf) %. +%term _ + %pi (eofi-var-invert (eofi/cons Dof) (meofi/cons Dmof) Dlookup?' (ewfi/cons Dewfi)) + %<- ({d} eofi-var-invert (Dof d) (Dmof d) (Dlookup? d) (Dewfi d)) + %<- (elookup?-cons Dlookup? Dlookup?') %. +%worlds (nbind ebind evar eofblock eovar) (eofi-var-invert _ _ _ _) %. +%total D (eofi-var-invert D _ _ _) %. +%sort eofi-const-invert {D eofi G M A} %. +%mode eofi-const-invert %in %in %out %out %out %. +%term _ eofi-const-invert (eofi/nil (eof/const D2 D1)) _ eterm-eq/i D1 (ewfi/nil D2) %. +%term _ + %pi (eofi-const-invert (eofi/cons D) (meofi/cons Dm) Deq Dkof (ewfi/cons Dwf)) + %<- ({d} eofi-const-invert (D d) (Dm d) Deq Dkof (Dwf d)) %. +%worlds (nbind ebind evar eofblock eovar) (eofi-const-invert _ _ _ _ _) %. +%total D (eofi-const-invert D _ _ _ _) %. +%sort eofi-forall-invert {D eofi G M C} %. +%mode eofi-forall-invert %in %in %out %out %out %out %. +%term _ eofi-forall-invert (eofi/nil (eof/forall D)) (meofi/nil (meof/forall DM)) eterm-eq/i etp-eq/i (ewfi/nil D) (mewfi/nil DM) %. +%term _ + %pi (eofi-forall-invert (eofi/cons D) (meofi/cons DM) Deq Deq' (ewfi/cons D1) (mewfi/cons DM1)) + %<- ({d} eofi-forall-invert (D d) (DM d) Deq Deq' (D1 d) (DM1 d)) %. +%worlds (nbind ebind evar eofblock eovar) (eofi-forall-invert _ _ _ _ _ _) %. +%total D (eofi-forall-invert D _ _ _ _ _) %. +%sort eofi-lam-invert {D eofi G O C} %. +%mode eofi-lam-invert %in %in %out %out %out %out %out %out %. +%term _ eofi-lam-invert (eofi/nil (eof/lam D2 D1)) (meofi/nil (meof/lam DM2 DM1)) eterm-eq/i etp-eq/i (ewfi/nil D1) (mewfi/nil DM1) ([x] eofi/cons ([d] eofi/nil (D2 x d))) ([x] meofi/cons ([d] meofi/nil (DM2 x d))) %. +%term _ + %pi (eofi-lam-invert (eofi/cons D) (meofi/cons DM) Deq Deq' (ewfi/cons D1) (mewfi/cons DM1) ([y] eofi/cons ([e] eofi/cons ([d] D2 d y e))) ([y] meofi/cons ([e] meofi/cons ([d] DM2 d y e)))) + %<- ({d} eofi-lam-invert (D d) (DM d) Deq Deq' (D1 d) (DM1 d) ([y] eofi/cons ([e] D2 d y e)) ([y] meofi/cons ([e] DM2 d y e))) %. +%worlds (nbind ebind evar eofblock eovar) (eofi-lam-invert _ _ _ _ _ _ _ _) %. +%total D (eofi-lam-invert D _ _ _ _ _ _ _) %. +%sort eofi-app-invert {D eofi G O C} %. +%mode eofi-app-invert %in %in %out %out %out %out %out %out %. +%term _ eofi-app-invert (eofi/nil (eof/app D2 D1)) (meofi/nil (meof/app DM2 DM1)) eterm-eq/i etp-eq/i (eofi/nil D1) (meofi/nil DM1) (eofi/nil D2) (meofi/nil DM2) %. +%term _ + %pi (eofi-app-invert (eofi/cons D) (meofi/cons DM) Deq Deq' (eofi/cons D1) (meofi/cons DM1) (eofi/cons D2) (meofi/cons DM2)) + %<- ({d} eofi-app-invert (D d) (DM d) Deq Deq' (D1 d) (DM1 d) (D2 d) (DM2 d)) %. +%worlds (nbind ebind evar eofblock eovar) (eofi-app-invert _ _ _ _ _ _ _ _) %. +%total D (eofi-app-invert D _ _ _ _ _ _ _) %. +%sort eofi-pair-invert {D eofi G O C} %. +%mode eofi-pair-invert %in %in %out %out %out %out %out %out %out %. +%term _ eofi-pair-invert (eofi/nil (eof/pair D3 D2 D1)) (meofi/nil (meof/pair DM2 DM1)) eterm-eq/i etp-eq/i (eofi/nil D1) (meofi/nil DM1) (eofi/nil D2) (meofi/nil DM2) ([x] ewfi/cons ([d] ewfi/nil (D3 x d))) %. +%term _ + %pi (eofi-pair-invert (eofi/cons D) (meofi/cons DM) Deq Deq' (eofi/cons D1) (meofi/cons DM1) (eofi/cons D2) (meofi/cons DM2) ([y] ewfi/cons ([e] ewfi/cons ([d] D3 d y e)))) + %<- ({d} eofi-pair-invert (D d) (DM d) Deq Deq' (D1 d) (DM1 d) (D2 d) (DM2 d) ([y] ewfi/cons ([e] D3 d y e))) %. +%worlds (nbind ebind evar eofblock eovar) (eofi-pair-invert _ _ _ _ _ _ _ _ _) %. +%total D (eofi-pair-invert D _ _ _ _ _ _ _ _) %. +%sort eofi-pi1-invert {D eofi G O A} %. +%mode eofi-pi1-invert %in %in %out %out %out %. +%term _ eofi-pi1-invert (eofi/nil (eof/pi1 D)) (meofi/nil (meof/pi1 DM)) eterm-eq/i (eofi/nil D) (meofi/nil DM) %. +%term _ + %pi (eofi-pi1-invert (eofi/cons D) (meofi/cons DM) Deq (eofi/cons D1) (meofi/cons DM1)) + %<- ({d} eofi-pi1-invert (D d) (DM d) Deq (D1 d) (DM1 d)) %. +%worlds (nbind ebind evar eofblock eovar) (eofi-pi1-invert _ _ _ _ _) %. +%total D (eofi-pi1-invert D _ _ _ _) %. +%sort eofi-pi2-invert {D eofi G O C} %. +%mode eofi-pi2-invert %in %in %out %out %out %out %. +%term _ eofi-pi2-invert (eofi/nil (eof/pi2 D)) (meofi/nil (meof/pi2 DM)) eterm-eq/i etp-eq/i (eofi/nil D) (meofi/nil DM) %. +%term _ + %pi (eofi-pi2-invert (eofi/cons D) (meofi/cons DM) Deq Deq' (eofi/cons D1) (meofi/cons DM1)) + %<- ({d} eofi-pi2-invert (D d) (DM d) Deq Deq' (D1 d) (DM1 d)) %. +%worlds (nbind ebind evar eofblock eovar) (eofi-pi2-invert _ _ _ _ _ _) %. +%total D (eofi-pi2-invert D _ _ _ _ _) %. +%sort eofi-sing-invert {D eofi G M C} %. +%mode eofi-sing-invert %in %in %out %out %out %. +%term _ eofi-sing-invert (eofi/nil (eof/sing D)) (meofi/nil (meof/sing DM)) etp-eq/i (eofi/nil D) (meofi/nil DM) %. +%term _ + %pi (eofi-sing-invert (eofi/cons D) (meofi/cons DM) Deq (eofi/cons D1) (meofi/cons DM1)) + %<- ({d} eofi-sing-invert (D d) (DM d) Deq (D1 d) (DM1 d)) %. +%worlds (nbind ebind evar eofblock eovar) (eofi-sing-invert _ _ _ _ _) %. +%total D (eofi-sing-invert D _ _ _ _) %. +%sort eofi-extpi-invert {D eofi G M C} %. +%mode eofi-extpi-invert %in %in %out %out %out %out %out %. +%term _ eofi-extpi-invert (eofi/nil (eof/extpi D2 D1)) (meofi/nil (meof/extpi DM2 DM1)) etp-eq/i (eofi/nil D1) (meofi/nil DM1) ([x] eofi/cons ([d] eofi/nil (D2 x d))) ([x] meofi/cons ([d] meofi/nil (DM2 x d))) %. +%term _ + %pi (eofi-extpi-invert (eofi/cons D) (meofi/cons DM) Deq (eofi/cons D1) (meofi/cons DM1) ([y] eofi/cons ([e] eofi/cons ([d] D2 d y e))) ([y] meofi/cons ([e] meofi/cons ([d] DM2 d y e)))) + %<- ({d} eofi-extpi-invert (D d) (DM d) Deq (D1 d) (DM1 d) ([y] eofi/cons ([e] D2 d y e)) ([y] meofi/cons ([e] DM2 d y e))) %. +%worlds (nbind ebind evar eofblock eovar) (eofi-extpi-invert _ _ _ _ _ _ _) %. +%total D (eofi-extpi-invert D _ _ _ _ _ _) %. +%sort eofi-extsigma-invert {D eofi G M C} %. +%mode eofi-extsigma-invert %in %in %out %out %out %out %out %out %. +%term _ eofi-extsigma-invert (eofi/nil (eof/extsigma D3 D2 D1)) (meofi/nil (meof/extsigma DM2 DM1)) etp-eq/i (eofi/nil D1) (meofi/nil DM1) (eofi/nil D2) (meofi/nil DM2) ([x] ewfi/cons ([d] ewfi/nil (D3 x d))) %. +%term _ + %pi (eofi-extsigma-invert (eofi/cons D) (meofi/cons DM) Deq (eofi/cons D1) (meofi/cons DM1) (eofi/cons D2) (meofi/cons DM2) ([y] ewfi/cons ([e] ewfi/cons ([d] D3 d y e)))) + %<- ({d} eofi-extsigma-invert (D d) (DM d) Deq (D1 d) (DM1 d) (D2 d) (DM2 d) ([y] ewfi/cons ([e] D3 d y e))) %. +%worlds (nbind ebind evar eofblock eovar) (eofi-extsigma-invert _ _ _ _ _ _ _ _) %. +%total D (eofi-extsigma-invert D _ _ _ _ _ _ _) %. +%sort eofi-subsume-invert {D eofi G M A} %. +%mode eofi-subsume-invert %in %in %out %out %out %. +%term _ eofi-subsume-invert (eofi/nil (eof/subsume D2 D1)) (meofi/nil (meof/subsume DM1)) (eofi/nil D1) (meofi/nil DM1) (subtpi/nil D2) %. +%term _ + %pi (eofi-subsume-invert (eofi/cons D) (meofi/cons DM) (eofi/cons D1) (meofi/cons DM1) (subtpi/cons D2)) + %<- ({d} eofi-subsume-invert (D d) (DM d) (D1 d) (DM1 d) (D2 d)) %. +%worlds (nbind ebind evar eofblock eovar) (eofi-subsume-invert _ _ _ _ _) %. +%total D (eofi-subsume-invert D _ _ _ _) %. +%sort ewfi-t-invert {D ewfi G C} %. +%mode ewfi-t-invert %in %in %out %. +%term _ ewfi-t-invert (ewfi/nil ewf/t) _ etp-eq/i %. +%term _ + %pi (ewfi-t-invert (ewfi/cons D) (mewfi/cons DM) Deq) + %<- ({d} ewfi-t-invert (D d) (DM d) Deq) %. +%worlds (nbind ebind evar eofblock eovar) (ewfi-t-invert _ _ _) %. +%total D (ewfi-t-invert D _ _) %. +%sort ewfi-pi-invert {D ewfi G C} %. +%mode ewfi-pi-invert %in %in %out %out %out %out %out %. +%term _ ewfi-pi-invert (ewfi/nil (ewf/pi D2 D1)) (mewfi/nil (mewf/pi DM2 DM1)) etp-eq/i (ewfi/nil D1) (mewfi/nil DM1) ([x] ewfi/cons ([d] ewfi/nil (D2 x d))) ([x] mewfi/cons ([d] mewfi/nil (DM2 x d))) %. +%term _ + %pi (ewfi-pi-invert (ewfi/cons D) (mewfi/cons DM) Deq (ewfi/cons D1) (mewfi/cons DM1) ([y] ewfi/cons ([e] ewfi/cons ([d] D2 d y e))) ([y] mewfi/cons ([e] mewfi/cons ([d] DM2 d y e)))) + %<- ({d} ewfi-pi-invert (D d) (DM d) Deq (D1 d) (DM1 d) ([y] ewfi/cons ([e] D2 d y e)) ([y] mewfi/cons ([e] DM2 d y e))) %. +%worlds (nbind ebind evar eofblock eovar) (ewfi-pi-invert _ _ _ _ _ _ _) %. +%total D (ewfi-pi-invert D _ _ _ _ _ _) %. +%sort ewfi-sigma-invert {D ewfi G C} %. +%mode ewfi-sigma-invert %in %in %out %out %out %out %out %. +%term _ ewfi-sigma-invert (ewfi/nil (ewf/sigma D2 D1)) (mewfi/nil (mewf/sigma DM2 DM1)) etp-eq/i (ewfi/nil D1) (mewfi/nil DM1) ([x] ewfi/cons ([d] ewfi/nil (D2 x d))) ([x] mewfi/cons ([d] mewfi/nil (DM2 x d))) %. +%term _ + %pi (ewfi-sigma-invert (ewfi/cons D) (mewfi/cons DM) Deq (ewfi/cons D1) (mewfi/cons DM1) ([y] ewfi/cons ([e] ewfi/cons ([d] D2 d y e))) ([y] mewfi/cons ([e] mewfi/cons ([d] DM2 d y e)))) + %<- ({d} ewfi-sigma-invert (D d) (DM d) Deq (D1 d) (DM1 d) ([y] ewfi/cons ([e] D2 d y e)) ([y] mewfi/cons ([e] DM2 d y e))) %. +%worlds (nbind ebind evar eofblock eovar) (ewfi-sigma-invert _ _ _ _ _ _ _) %. +%total D (ewfi-sigma-invert D _ _ _ _ _ _) %. +%sort ewfi-sing-invert {D ewfi G C} %. +%mode ewfi-sing-invert %in %in %out %out %out %. +%term _ ewfi-sing-invert (ewfi/nil (ewf/sing D)) (mewfi/nil (mewf/sing DM)) etp-eq/i (eofi/nil D) (meofi/nil DM) %. +%term _ + %pi (ewfi-sing-invert (ewfi/cons D) (mewfi/cons DM) Deq (eofi/cons D1) (meofi/cons DM1)) + %<- ({d} ewfi-sing-invert (D d) (DM d) Deq (D1 d) (DM1 d)) %. +%worlds (nbind ebind evar eofblock eovar) (ewfi-sing-invert _ _ _ _ _) %. +%total D (ewfi-sing-invert D _ _ _ _) %. +%%%%% Constructors %%%%% +%sort equiv-to-equivi {G} %. +%mode equiv-to-equivi %in %in %out %. +%term _ equiv-to-equivi enil D (equivi/nil D) %. +%term _ + %pi (equiv-to-equivi (econs G X A) D (equivi/cons D')) + %<- ({d} equiv-to-equivi G D (D' d)) %. +%worlds (nbind ebind evar eovar eofblock) (equiv-to-equivi _ _ _) %. +%total G (equiv-to-equivi G _ _) %. +%sort tequiv-to-tequivi {G} %. +%mode tequiv-to-tequivi %in %in %out %. +%term _ tequiv-to-tequivi enil D (tequivi/nil D) %. +%term _ + %pi (tequiv-to-tequivi (econs G X A) D (tequivi/cons D')) + %<- ({d} tequiv-to-tequivi G D (D' d)) %. +%worlds (nbind ebind evar eofblock eovar) (tequiv-to-tequivi _ _ _) %. +%total G (tequiv-to-tequivi G _ _) %. +%sort eofi-evof %. +%mode eofi-evof %in %in %out %. +%term _ eofi-evof Dvof (ewfi/nil Dwf) (eofi/nil (eof/var Dwf Dvof)) %. +%term _ + %pi (eofi-evof Dvof (ewfi/cons Dwf) (eofi/cons Dof)) + %<- ({d} eofi-evof Dvof (Dwf d) (Dof d)) %. +%worlds (nbind ebind evar eovar eofblock) (eofi-evof _ _ _) %. +%total D (eofi-evof _ D _) %. +%sort eofi-lookup %. +%mode eofi-lookup %in %in %out %. +%term _ + %pi (eofi-lookup elookup/hit (ewfi/cons Dwf) (eofi/cons Dof)) + %<- ({d} eofi-evof d (Dwf d) (Dof d)) %. +%term _ + %pi (eofi-lookup (elookup/miss Dlook) (ewfi/cons Dwf) (eofi/cons Dof)) + %<- ({d} eofi-lookup Dlook (Dwf d) (Dof d)) %. +%worlds (nbind ebind evar eovar eofblock) (eofi-lookup _ _ _) %. +%total D (eofi-lookup D _ _) %. +%sort eofi-const %. +%mode eofi-const %in %in %out %. +%term _ eofi-const Dkof (ewfi/nil Dwf) (eofi/nil (eof/const Dwf Dkof)) %. +%term _ + %pi (eofi-const Dkof (ewfi/cons Dwf) (eofi/cons Dof)) + %<- ({d} eofi-const Dkof (Dwf d) (Dof d)) %. +%worlds (nbind ebind evar eovar eofblock) (eofi-const _ _ _) %. +%total D (eofi-const _ D _) %. +%sort equivi-reflex %. +%mode equivi-reflex %in %out %. +%term _ equivi-reflex (eofi/nil D) (equivi/nil (equiv/reflex D)) %. +%term _ + %pi (equivi-reflex (eofi/cons D) (equivi/cons D')) + %<- ({d} equivi-reflex (D d) (D' d)) %. +%worlds (nbind ebind evar eovar eofblock) (equivi-reflex _ _) %. +%total D (equivi-reflex D _) %. +%sort equivi-forall %. +%mode equivi-forall %in %out %. +%term _ equivi-forall (tequivi/nil D) (equivi/nil (equiv/forall D)) %. +%term _ + %pi (equivi-forall (tequivi/cons D) (equivi/cons D')) + %<- ({d} equivi-forall (D d) (D' d)) %. +%worlds (nbind ebind evar eovar eofblock) (equivi-forall _ _) %. +%total D (equivi-forall D _) %. +%sort equivi-lam %. +%mode equivi-lam %in %in %out %. +%term _ equivi-lam (tequivi/nil D1) ([x] equivi/cons ([d] equivi/nil (D2 x d))) (equivi/nil (equiv/lam D2 D1)) %. +%term _ + %pi (equivi-lam (tequivi/cons D1) ([x] equivi/cons ([d] equivi/cons ([e] D2 x d e))) (equivi/cons D)) + %<- ({e} equivi-lam (D1 e) ([x] equivi/cons ([d] D2 x d e)) (D e)) %. +%worlds (nbind ebind evar eofblock eovar) (equivi-lam _ _ _) %. +%total D (equivi-lam D _ _) %. +%sort equivi-app %. +%mode equivi-app %in %in %out %. +%term _ equivi-app (equivi/nil D1) (equivi/nil D2) (equivi/nil (equiv/app D2 D1)) %. +%term _ + %pi (equivi-app (equivi/cons D1) (equivi/cons D2) (equivi/cons D)) + %<- ({d} equivi-app (D1 d) (D2 d) (D d)) %. +%worlds (nbind ebind evar eofblock eovar) (equivi-app _ _ _) %. +%total D (equivi-app D _ _) %. +%sort equivi-pair %. +%mode equivi-pair %in %in %in %out %. +%term _ equivi-pair (equivi/nil D1) (equivi/nil D2) ([x] ewfi/cons ([d] ewfi/nil (D3 x d))) (equivi/nil (equiv/pair D3 D2 D1)) %. +%term _ + %pi (equivi-pair (equivi/cons D1) (equivi/cons D2) ([x] ewfi/cons ([d] ewfi/cons ([e] D3 x d e))) (equivi/cons D)) + %<- ({e} equivi-pair (D1 e) (D2 e) ([x] ewfi/cons ([d] D3 x d e)) (D e)) %. +%worlds (nbind ebind evar eofblock eovar) (equivi-pair _ _ _ _) %. +%total D (equivi-pair D _ _ _) %. +%sort equivi-pi1 %. +%mode equivi-pi1 %in %out %. +%term _ equivi-pi1 (equivi/nil D) (equivi/nil (equiv/pi1 D)) %. +%term _ + %pi (equivi-pi1 (equivi/cons D1) (equivi/cons D)) + %<- ({d} equivi-pi1 (D1 d) (D d)) %. +%worlds (nbind ebind evar eofblock eovar) (equivi-pi1 _ _) %. +%total D (equivi-pi1 D _) %. +%sort equivi-pi2 %. +%mode equivi-pi2 %in %out %. +%term _ equivi-pi2 (equivi/nil D) (equivi/nil (equiv/pi2 D)) %. +%term _ + %pi (equivi-pi2 (equivi/cons D1) (equivi/cons D)) + %<- ({d} equivi-pi2 (D1 d) (D d)) %. +%worlds (nbind ebind evar eofblock eovar) (equivi-pi2 _ _) %. +%total D (equivi-pi2 D _) %. +%sort equivi-sing %. +%mode equivi-sing %in %out %. +%term _ equivi-sing (equivi/nil D) (equivi/nil (equiv/sing D)) %. +%term _ + %pi (equivi-sing (equivi/cons D1) (equivi/cons D)) + %<- ({d} equivi-sing (D1 d) (D d)) %. +%worlds (nbind ebind evar eofblock eovar) (equivi-sing _ _) %. +%total D (equivi-sing D _) %. +%sort equivi-extpiw %. +%mode equivi-extpiw %in %in %out %. +%term _ equivi-extpiw (equivi/nil D1) ([x] equivi/cons ([d] equivi/nil (D2 x d))) (equivi/nil (equiv/extpiw D2 D1)) %. +%term _ + %pi (equivi-extpiw (equivi/cons D1) ([x] equivi/cons ([d] equivi/cons ([e] D2 x d e))) (equivi/cons D)) + %<- ({e} equivi-extpiw (D1 e) ([x] equivi/cons ([d] D2 x d e)) (D e)) %. +%worlds (nbind ebind evar eofblock eovar) (equivi-extpiw _ _ _) %. +%total D (equivi-extpiw D _ _) %. +%sort equivi-extsigma %. +%mode equivi-extsigma %in %in %in %out %. +%term _ equivi-extsigma (equivi/nil D1) (equivi/nil D2) ([x] ewfi/cons ([d] ewfi/nil (D3 x d))) (equivi/nil (equiv/extsigma D3 D2 D1)) %. +%term _ + %pi (equivi-extsigma (equivi/cons D1) (equivi/cons D2) ([x] ewfi/cons ([d] ewfi/cons ([e] D3 x d e))) (equivi/cons D)) + %<- ({e} equivi-extsigma (D1 e) (D2 e) ([x] ewfi/cons ([d] D3 x d e)) (D e)) %. +%worlds (nbind ebind evar eofblock eovar) (equivi-extsigma _ _ _ _) %. +%total D (equivi-extsigma D _ _ _) %. +%sort equivi-subsume %. +%mode equivi-subsume %in %in %out %. +%term _ equivi-subsume (equivi/nil D1) (subtpi/nil D2) (equivi/nil (equiv/subsume D2 D1)) %. +%term _ + %pi (equivi-subsume (equivi/cons D1) (subtpi/cons D2) (equivi/cons D)) + %<- ({d} equivi-subsume (D1 d) (D2 d) (D d)) %. +%worlds (nbind ebind evar eofblock eovar) (equivi-subsume _ _ _) %. +%total D (equivi-subsume D _ _) %. +%sort tequivi-pi %. +%mode tequivi-pi %in %in %out %. +%term _ tequivi-pi (tequivi/nil D1) ([x] tequivi/cons ([d] tequivi/nil (D2 x d))) (tequivi/nil (tequiv/pi D2 D1)) %. +%term _ + %pi (tequivi-pi (tequivi/cons D1) ([x] tequivi/cons ([d] tequivi/cons ([e] D2 x d e))) (tequivi/cons D)) + %<- ({e} tequivi-pi (D1 e) ([x] tequivi/cons ([d] D2 x d e)) (D e)) %. +%worlds (nbind ebind evar eofblock eovar) (tequivi-pi _ _ _) %. +%total D (tequivi-pi D _ _) %. +%sort tequivi-sigma %. +%mode tequivi-sigma %in %in %out %. +%term _ tequivi-sigma (tequivi/nil D1) ([x] tequivi/cons ([d] tequivi/nil (D2 x d))) (tequivi/nil (tequiv/sigma D2 D1)) %. +%term _ + %pi (tequivi-sigma (tequivi/cons D1) ([x] tequivi/cons ([d] tequivi/cons ([e] D2 x d e))) (tequivi/cons D)) + %<- ({e} tequivi-sigma (D1 e) ([x] tequivi/cons ([d] D2 x d e)) (D e)) %. +%worlds (nbind ebind evar eofblock eovar) (tequivi-sigma _ _ _) %. +%total D (tequivi-sigma D _ _) %. +%sort tequivi-sing %. +%mode tequivi-sing %in %out %. +%term _ tequivi-sing (equivi/nil D1) (tequivi/nil (tequiv/sing D1)) %. +%term _ + %pi (tequivi-sing (equivi/cons D1) (tequivi/cons D)) + %<- ({d} tequivi-sing (D1 d) (D d)) %. +%worlds (nbind ebind evar eofblock eovar) (tequivi-sing _ _) %. +%total D (tequivi-sing D _) %. +%%%%% Functionality (Explicit/Implicit Context) %%%%% +%sort tfunctionality-e {Am} {D {x} %pi (evof x A) %-> (ewfi (G x) (B x))} {x} {d} %. +%mode tfunctionality-e %in %in %in %in %in %in %out %. +%sort functionality-e {Mm} {D {x} %pi (evof x A) %-> (eofi (G x) (O x) (B x))} {x} {d} %. +%mode functionality-e %in %in %in %in %in %in %out %. +%sort functionality-e! {x} %. +%mode functionality-e! %in %in %in %in %in %out %. +%scope functionality-e! %term var + %pi (functionality-e _ (%the ({x} %pi (evof x A) %-> (eofi (G x) (Y x) (B x))) Dof) (%the ({x} {d} meofi (Dof x d) meterm/var) Dmof) (%the ({x} clean (G x)) Dclean) (%the (equiv M N A) Dequiv) (%the (eof M A) DofM) D) + %<- ({x} {d} eofi-var-invert (Dof x d) (Dmof x d) (%the (elookup? (G x) (Y x) (B x)) (Dlookup x d)) (%the (ewfi (G x) (B x)) (Dwf x d))) + %<- (functionality-e! Dlookup Dwf Dclean Dequiv DofM D) %. +%term varsam + %pi (functionality-e! ([x] [d] elookup?/not d) (%the ({x} %pi (evof x A) %-> (ewfi (G x) A)) Dwf) (%the ({x} clean (G x)) Dclean) (%the (equiv M N A) DequivMN) (%the (eof M A) DofM) Dequiv') + %<- (equiv-to-equivi _ DequivMN Dequiv') %. +%scope functionality-e! %term varoth + %pi (functionality-e! ([x] [d] elookup?/not (%the (evof X B) Dvof)) (%the ({x} %pi (evof x A) %-> (ewfi (G x) B)) Dwf) (%the ({x} clean (G x)) Dclean) (%the (equiv M N A) DequivMN) (%the (eof M A) DofM) Dequiv) + %<- (ewfi-subst Dwf DofM (%the (ewfi (G M) B) Dwf')) + %<- (eofi-evof Dvof Dwf' (%the (eofi (G M) X B) Dof)) + %<- (equivi-reflex Dof (%the (equivi (G M) X X B) Dequiv)) %. +%term varoth + %pi (functionality-e! ([x] [d] elookup?/look (%the (elookup (G x) (Y x) (B x)) (Dlookup x))) (%the ({x} %pi (evof x A) %-> (ewfi (G x) (B x))) Dwf) (%the ({x} clean (G x)) Dclean) (%the (equiv M N A) DequivMN) (%the (eof M A) DofM) Dequiv'') + %<- (lookup-clean Dlookup Dclean (%the ({x} eterm-eq (Y x) Y') Deq)) + %<- ({x} eterm-eq-symm (Deq x) (%the (eterm-eq Y' (Y x)) (Deq' x))) + %<- ({x} {d} eofi-lookup (Dlookup x) (Dwf x d) (%the (eofi (G x) (Y x) (B x)) (Dof x d))) + %<- ({x} {d} eofi-resp ectx-eq/i (Deq x) etp-eq/i (Dof x d) (%the (eofi (G x) Y' (B x)) (Dof' x d))) + %<- (eofi-subst Dof' DofM (%the (eofi (G M) Y' (B M)) Dof'')) + %<- (equivi-reflex Dof'' (%the (equivi (G M) Y' Y' (B M)) Dequiv')) + %<- (equivi-resp ectx-eq/i (Deq' M) (Deq' N) etp-eq/i Dequiv' (%the (equivi (G M) (Y M) (Y N) (B M)) Dequiv'')) %. +%worlds (nbind ebind evar eovar eofblock) (functionality-e! _ _ _ _ _ _) %. +%total {} (functionality-e! _ _ _ _ _ _) %. +%scope functionality-e! %term const + %pi (functionality-e _ (%the ({x} %pi (evof x A) %-> (eofi (G x) (O x) (B x))) Dof) (%the ({x} {d} meofi (Dof x d) meterm/const) Dmof) (%the ({x} clean (G x)) Dclean) (%the (equiv M N A) DequivMN) (%the (eof M A) DofM) Dequiv') + %<- ({x} {d} eofi-const-invert (Dof x d) (Dmof x d) (%the (eterm-eq (econst K) (O x)) (Deq x)) (%the (ekof K (B x)) (Dkof x)) (%the (ewfi (G x) (B x)) (Dwf x d))) + %<- (ewfi-subst Dwf DofM (%the (ewfi (G M) (B M)) Dwf')) + %<- (eofi-const (Dkof M) Dwf' (%the (eofi (G M) (econst K) (B M)) Dof')) + %<- (equivi-reflex Dof' Dequiv) + %<- (equivi-resp ectx-eq/i (Deq M) (Deq N) etp-eq/i Dequiv Dequiv') %. +%term forall + %pi (functionality-e _ (%the ({x} %pi (evof x A) %-> (eofi (G x) (O x) (B x))) Dof) (%the ({x} {d} meofi (Dof x d) (meterm/forall Cm)) Dmof) (%the ({x} clean (G x)) Dclean) (%the (equiv M N A) DequivMN) (%the (eof M A) DofM) Dequiv'') + %<- ({x} {d} eofi-forall-invert (Dof x d) (Dmof x d) (%the (eterm-eq (eforall (C x)) (O x)) (DeqO x)) (%the (etp-eq (epi (epi (C x) ([_] et)) ([_] et)) (B x)) (DeqB x)) (%the (ewfi (G x) (C x)) (DwfC x d)) (%the (mewfi (DwfC x d) Cm) (DmwfC x d))) + %<- (tfunctionality-e _ DwfC DmwfC Dclean DequivMN DofM (%the (tequivi (G M) (C M) (C N)) Dequiv)) + %<- (equivi-forall Dequiv (%the (equivi (G M) (eforall (C M)) (eforall (C N)) (epi (epi (C M) ([_] et)) ([_] et))) Dequiv')) + %<- (equivi-resp ectx-eq/i (DeqO M) (DeqO N) (DeqB M) Dequiv' Dequiv'') %. +%scope functionality-e! %term lam + %pi (functionality-e _ (%the ({x} %pi (evof x A) %-> (eofi (G x) (O x) (B x))) Dof) (%the ({x} {d} meofi (Dof x d) (meterm/lam Mm Cm)) Dmof) (%the ({x} clean (G x)) Dclean) (%the (equiv P P' A) DequivP) (%the (eof P A) DofP) Dequiv') + %<- ({x} {d} eofi-lam-invert (Dof x d) (Dmof x d) (%the (eterm-eq (elam (C x) ([y] M x y)) (O x)) (DeqO x)) (%the (etp-eq (epi (C x) (D x)) (B x)) (DeqB x)) (%the (ewfi (G x) (C x)) (DwfC x d)) (%the (mewfi (DwfC x d) Cm) (DmwfC x d)) (%the ({y} eofi (econs (G x) y (C x)) (M x y) (D x y)) (DofM x d)) (%the ({y} meofi (DofM x d y) Mm) (DmofM x d))) + %<- (tfunctionality-e _ DwfC DmwfC Dclean DequivP DofP (%the (tequivi (G P) (C P) (C P')) DequivC)) + %<- ({y} {e eisvar y} functionality-e _ ([x] [d] DofM x d y) ([x] [d] DmofM x d y) ([x] clean/cons e (Dclean x)) DequivP DofP (%the (equivi (econs (G P) y (C P)) (M P y) (M P' y) (D P y)) (DequivM y))) + %<- (equivi-lam DequivC DequivM (%the (equivi (G P) (elam (C P) (M P)) (elam (C P') (M P')) (epi (C P) (D P))) Dequiv)) + %<- (equivi-resp ectx-eq/i (DeqO P) (DeqO P') (DeqB P) Dequiv (%the (equivi (G P) (O P) (O P') (B P)) Dequiv')) %. +%term app + %pi (functionality-e _ (%the ({x} %pi (evof x D) %-> (eofi (G x) (O x) (C x))) Dof) (%the ({x} {d} meofi (Dof x d) (meterm/app Nm Mm)) Dmof) (%the ({x} clean (G x)) Dclean) (%the (equiv P P' D) DequivP) (%the (eof P D) DofP) Dequiv') + %<- ({x} {d} eofi-app-invert (Dof x d) (Dmof x d) (%the (eterm-eq (eapp (M x) (N x)) (O x)) (DeqO x)) (%the (etp-eq (B x (N x)) (C x)) (DeqC x)) (%the (eofi (G x) (M x) (epi (A x) (B x))) (D1 x d)) (%the (meofi (D1 x d) Mm) (DM1 x d)) (%the (eofi (G x) (N x) (A x)) (D2 x d)) (%the (meofi (D2 x d) Nm) (DM2 x d))) + %<- (functionality-e _ D1 DM1 Dclean DequivP DofP (%the (equivi (G P) (M P) (M P') (epi (A P) (B P))) DequivM)) + %<- (functionality-e _ D2 DM2 Dclean DequivP DofP (%the (equivi (G P) (N P) (N P') (A P)) DequivN)) + %<- (equivi-app DequivM DequivN (%the (equivi (G P) (eapp (M P) (N P)) (eapp (M P') (N P')) (B P (N P))) Dequiv)) + %<- (equivi-resp ectx-eq/i (DeqO P) (DeqO P') (DeqC P) Dequiv (%the (equivi (G P) (O P) (O P') (C P)) Dequiv')) %. +%scope functionality-e! %term pair + %pi (functionality-e _ (%the ({x} %pi (evof x D) %-> (eofi (G x) (O x) (C x))) Dof) (%the ({x} {d} meofi (Dof x d) (meterm/pair Nm Mm)) Dmof) (%the ({x} clean (G x)) Dclean) (%the (equiv P P' D) DequivP) (%the (eof P D) DofP) Dequiv') + %<- ({x} {d} eofi-pair-invert (Dof x d) (Dmof x d) (%the (eterm-eq (epair (M x) (N x)) (O x)) (DeqO x)) (%the (etp-eq (esigma (A x) (B x)) (C x)) (DeqC x)) (%the (eofi (G x) (M x) (A x)) (D1 x d)) (%the (meofi (D1 x d) Mm) (DM1 x d)) (%the (eofi (G x) (N x) (B x (M x))) (D2 x d)) (%the (meofi (D2 x d) Nm) (DM2 x d)) (%the ({y} ewfi (econs (G x) y (A x)) (B x y)) (D3 x d))) + %<- (functionality-e _ D1 DM1 Dclean DequivP DofP (%the (equivi (G P) (M P) (M P') (A P)) DequivM)) + %<- (functionality-e _ D2 DM2 Dclean DequivP DofP (%the (equivi (G P) (N P) (N P') (B P (M P))) DequivN)) + %<- ({y} ewfi-subst-gen ([d] D3 P d y) ([d eof P D] %the (ewfi (econs (G P) y (A P)) (B P y)) (D3' d y))) + %<- (equivi-pair DequivM DequivN ([x] D3' DofP x) (%the (equivi (G P) (epair (M P) (N P)) (epair (M P') (N P')) (esigma (A P) (B P))) Dequiv)) + %<- (equivi-resp ectx-eq/i (DeqO P) (DeqO P') (DeqC P) Dequiv (%the (equivi (G P) (O P) (O P') (C P)) Dequiv')) %. +%term pi1 + %pi (functionality-e _ (%the ({x} %pi (evof x D) %-> (eofi (G x) (O x) (A x))) Dof) (%the ({x} {d} meofi (Dof x d) (meterm/pi1 Mm)) Dmof) (%the ({x} clean (G x)) Dclean) (%the (equiv P P' D) DequivP) (%the (eof P D) DofP) Dequiv') + %<- ({x} {d} eofi-pi1-invert (Dof x d) (Dmof x d) (%the (eterm-eq (epi1 (M x)) (O x)) (DeqO x)) (%the (eofi (G x) (M x) (esigma (A x) (B x))) (D1 x d)) (%the (meofi (D1 x d) Mm) (DM1 x d))) + %<- (functionality-e _ D1 DM1 Dclean DequivP DofP (%the (equivi (G P) (M P) (M P') (esigma (A P) (B P))) DequivM)) + %<- (equivi-pi1 DequivM (%the (equivi (G P) (epi1 (M P)) (epi1 (M P')) (A P)) Dequiv)) + %<- (equivi-resp ectx-eq/i (DeqO P) (DeqO P') etp-eq/i Dequiv (%the (equivi (G P) (O P) (O P') (A P)) Dequiv')) %. +%scope functionality-e! %term pi2 + %pi (functionality-e _ (%the ({x} %pi (evof x D) %-> (eofi (G x) (O x) (C x))) Dof) (%the ({x} {d} meofi (Dof x d) (meterm/pi2 Mm)) Dmof) (%the ({x} clean (G x)) Dclean) (%the (equiv P P' D) DequivP) (%the (eof P D) DofP) Dequiv') + %<- ({x} {d} eofi-pi2-invert (Dof x d) (Dmof x d) (%the (eterm-eq (epi2 (M x)) (O x)) (DeqO x)) (%the (etp-eq (B x (epi1 (M x))) (C x)) (DeqC x)) (%the (eofi (G x) (M x) (esigma (A x) (B x))) (D1 x d)) (%the (meofi (D1 x d) Mm) (DM1 x d))) + %<- (functionality-e _ D1 DM1 Dclean DequivP DofP (%the (equivi (G P) (M P) (M P') (esigma (A P) (B P))) DequivM)) + %<- (equivi-pi2 DequivM (%the (equivi (G P) (epi2 (M P)) (epi2 (M P')) (B P (epi1 (M P)))) Dequiv)) + %<- (equivi-resp ectx-eq/i (DeqO P) (DeqO P') (DeqC P) Dequiv (%the (equivi (G P) (O P) (O P') (C P)) Dequiv')) %. +%term sing + %pi (functionality-e _ (%the ({x} %pi (evof x D) %-> (eofi (G x) (M x) (C x))) Dof) (%the ({x} {d} meofi (Dof x d) (meterm/sing Mm)) Dmof) (%the ({x} clean (G x)) Dclean) (%the (equiv P P' D) DequivP) (%the (eof P D) DofP) Dequiv') + %<- ({x} {d} eofi-sing-invert (Dof x d) (Dmof x d) (%the (etp-eq (esing (M x)) (C x)) (DeqC x)) (%the (eofi (G x) (M x) et) (D1 x d)) (%the (meofi (D1 x d) Mm) (DM1 x d))) + %<- (functionality-e _ D1 DM1 Dclean DequivP DofP (%the (equivi (G P) (M P) (M P') et) DequivM)) + %<- (equivi-sing DequivM (%the (equivi (G P) (M P) (M P') (esing (M P))) Dequiv)) + %<- (equivi-resp ectx-eq/i eterm-eq/i eterm-eq/i (DeqC P) Dequiv (%the (equivi (G P) (M P) (M P') (C P)) Dequiv')) %. +%scope functionality-e! %term extpi + %pi (functionality-e _ (%the ({x} %pi (evof x D) %-> (eofi (G x) (M x) (C x))) Dof) (%the ({x} {d} meofi (Dof x d) (meterm/extpi Nm Mm)) Dmof) (%the ({x} clean (G x)) Dclean) (%the (equiv P P' D) DequivP) (%the (eof P D) DofP) Dequiv') + %<- ({x} {d} eofi-extpi-invert (Dof x d) (Dmof x d) (%the (etp-eq (epi (A x) (B x)) (C x)) (DeqC x)) (%the (eofi (G x) (M x) (epi (A x) (B' x))) (D1 x d)) (%the (meofi (D1 x d) Mm) (DM1 x d)) (%the ({y} eofi (econs (G x) y (A x)) (eapp (M x) y) (B x y)) (D2 x d)) (%the ({y} meofi (D2 x d y) Nm) (DM2 x d))) + %<- (functionality-e _ D1 DM1 Dclean DequivP DofP (%the (equivi (G P) (M P) (M P') (epi (A P) (B' P))) Dequiv1)) + %<- ({y} {e eisvar y} functionality-e _ ([x] [d] D2 x d y) ([x] [d] DM2 x d y) ([x] clean/cons e (Dclean x)) DequivP DofP (%the (equivi (econs (G P) y (A P)) (eapp (M P) y) (eapp (M P') y) (B P y)) (Dequiv2 y))) + %<- (equivi-extpiw Dequiv1 Dequiv2 (%the (equivi (G P) (M P) (M P') (epi (A P) (B P))) Dequiv)) + %<- (equivi-resp ectx-eq/i eterm-eq/i eterm-eq/i (DeqC P) Dequiv (%the (equivi (G P) (M P) (M P') (C P)) Dequiv')) %. +%term extsig + %pi (functionality-e _ (%the ({x} %pi (evof x D) %-> (eofi (G x) (M x) (C x))) Dof) (%the ({x} {d} meofi (Dof x d) (meterm/extsigma Nm Mm)) Dmof) (%the ({x} clean (G x)) Dclean) (%the (equiv P P' D) DequivP) (%the (eof P D) DofP) Dequiv') + %<- ({x} {d} eofi-extsigma-invert (Dof x d) (Dmof x d) (%the (etp-eq (esigma (A x) (B x)) (C x)) (DeqC x)) (%the (eofi (G x) (epi1 (M x)) (A x)) (D1 x d)) (%the (meofi (D1 x d) Mm) (DM1 x d)) (%the (eofi (G x) (epi2 (M x)) (B x (epi1 (M x)))) (D2 x d)) (%the (meofi (D2 x d) Nm) (DM2 x d)) (%the ({y} ewfi (econs (G x) y (A x)) (B x y)) (Dwf x d))) + %<- (functionality-e _ D1 DM1 Dclean DequivP DofP (%the (equivi (G P) (epi1 (M P)) (epi1 (M P')) (A P)) Dequiv1)) + %<- (functionality-e _ D2 DM2 Dclean DequivP DofP (%the (equivi (G P) (epi2 (M P)) (epi2 (M P')) (B P (epi1 (M P)))) Dequiv2)) + %<- ({y} ewfi-subst ([x] [d] Dwf x d y) DofP (%the (ewfi (econs (G P) y (A P)) (B P y)) (Dwf' y))) + %<- (equivi-extsigma Dequiv1 Dequiv2 Dwf' (%the (equivi (G P) (M P) (M P') (esigma (A P) (B P))) Dequiv)) + %<- (equivi-resp ectx-eq/i eterm-eq/i eterm-eq/i (DeqC P) Dequiv (%the (equivi (G P) (M P) (M P') (C P)) Dequiv')) %. +%scope functionality-e! %term subsum + %pi (functionality-e _ (%the ({x} %pi (evof x D) %-> (eofi (G x) (M x) (A x))) Dof) (%the ({x} {d} meofi (Dof x d) (meterm/subsume Mm)) Dmof) (%the ({x} clean (G x)) Dclean) (%the (equiv P P' D) DequivP) (%the (eof P D) DofP) Dequiv') + %<- ({x} {d} eofi-subsume-invert (Dof x d) (Dmof x d) (%the (eofi (G x) (M x) (B x)) (D1 x d)) (%the (meofi (D1 x d) Mm) (DM1 x d)) (%the (subtpi (G x) (B x) (A x)) (Dsub x d))) + %<- (functionality-e _ D1 DM1 Dclean DequivP DofP (%the (equivi (G P) (M P) (M P') (B P)) Dequiv)) + %<- (subtpi-subst Dsub DofP (%the (subtpi (G P) (B P) (A P)) Dsub')) + %<- (equivi-subsume Dequiv Dsub' (%the (equivi (G P) (M P) (M P') (A P)) Dequiv')) %. +%term t + %pi (tfunctionality-e _ (%the ({x} %pi (evof x D) %-> (ewfi (G x) (C x))) Dwf) (%the ({x} {d} mewfi (Dwf x d) metp/t) Dmwf) (%the ({x} clean (G x)) Dclean) (%the (equiv P P' D) DequivP) (%the (eof P D) DofP) Dequiv') + %<- ({x} {d} ewfi-t-invert (Dwf x d) (Dmwf x d) (%the (etp-eq et (C x)) (DeqC x))) + %<- (tequiv-to-tequivi (G P) (tequiv/reflex ewf/t) (%the (tequivi (G P) et et) Dequiv)) + %<- (tequivi-resp ectx-eq/i (DeqC P) (DeqC P') Dequiv (%the (tequivi (G P) (C P) (C P')) Dequiv')) %. +%scope functionality-e! %term pi + %pi (tfunctionality-e _ (%the ({x} %pi (evof x D) %-> (ewfi (G x) (C x))) Dwf) (%the ({x} {d} mewfi (Dwf x d) (metp/pi Bm Am)) Dmwf) (%the ({x} clean (G x)) Dclean) (%the (equiv P P' D) DequivP) (%the (eof P D) DofP) Dequiv') + %<- ({x} {d} ewfi-pi-invert (Dwf x d) (Dmwf x d) (%the (etp-eq (epi (A x) (B x)) (C x)) (DeqC x)) (%the (ewfi (G x) (A x)) (DwfA x d)) (%the (mewfi (DwfA x d) Am) (DmwfA x d)) (%the ({y} ewfi (econs (G x) y (A x)) (B x y)) (DwfB x d)) (%the ({y} mewfi (DwfB x d y) Bm) (DmwfB x d))) + %<- (tfunctionality-e _ DwfA DmwfA Dclean DequivP DofP (%the (tequivi (G P) (A P) (A P')) DequivA)) + %<- ({y} {e eisvar y} tfunctionality-e _ ([x] [d] DwfB x d y) ([x] [d] DmwfB x d y) ([x] clean/cons e (Dclean x)) DequivP DofP (%the (tequivi (econs (G P) y (A P)) (B P y) (B P' y)) (DequivB y))) + %<- (tequivi-pi DequivA DequivB (%the (tequivi (G P) (epi (A P) (B P)) (epi (A P') (B P'))) Dequiv)) + %<- (tequivi-resp ectx-eq/i (DeqC P) (DeqC P') Dequiv (%the (tequivi (G P) (C P) (C P')) Dequiv')) %. +%term sigma + %pi (tfunctionality-e _ (%the ({x} %pi (evof x D) %-> (ewfi (G x) (C x))) Dwf) (%the ({x} {d} mewfi (Dwf x d) (metp/sigma Bm Am)) Dmwf) (%the ({x} clean (G x)) Dclean) (%the (equiv P P' D) DequivP) (%the (eof P D) DofP) Dequiv') + %<- ({x} {d} ewfi-sigma-invert (Dwf x d) (Dmwf x d) (%the (etp-eq (esigma (A x) (B x)) (C x)) (DeqC x)) (%the (ewfi (G x) (A x)) (DwfA x d)) (%the (mewfi (DwfA x d) Am) (DmwfA x d)) (%the ({y} ewfi (econs (G x) y (A x)) (B x y)) (DwfB x d)) (%the ({y} mewfi (DwfB x d y) Bm) (DmwfB x d))) + %<- (tfunctionality-e _ DwfA DmwfA Dclean DequivP DofP (%the (tequivi (G P) (A P) (A P')) DequivA)) + %<- ({y} {e eisvar y} tfunctionality-e _ ([x] [d] DwfB x d y) ([x] [d] DmwfB x d y) ([x] clean/cons e (Dclean x)) DequivP DofP (%the (tequivi (econs (G P) y (A P)) (B P y) (B P' y)) (DequivB y))) + %<- (tequivi-sigma DequivA DequivB (%the (tequivi (G P) (esigma (A P) (B P)) (esigma (A P') (B P'))) Dequiv)) + %<- (tequivi-resp ectx-eq/i (DeqC P) (DeqC P') Dequiv (%the (tequivi (G P) (C P) (C P')) Dequiv')) %. +%scope functionality-e! %term sing + %pi (tfunctionality-e _ (%the ({x} %pi (evof x D) %-> (ewfi (G x) (C x))) Dwf) (%the ({x} {d} mewfi (Dwf x d) (metp/sing Mm)) Dmwf) (%the ({x} clean (G x)) Dclean) (%the (equiv P P' D) DequivP) (%the (eof P D) DofP) Dequiv') + %<- ({x} {d} ewfi-sing-invert (Dwf x d) (Dmwf x d) (%the (etp-eq (esing (M x)) (C x)) (DeqC x)) (%the (eofi (G x) (M x) et) (DofM x d)) (%the (meofi (DofM x d) Mm) (DmofM x d))) + %<- (functionality-e _ DofM DmofM Dclean DequivP DofP (%the (equivi (G P) (M P) (M P') et) DequivM)) + %<- (tequivi-sing DequivM (%the (tequivi (G P) (esing (M P)) (esing (M P'))) Dequiv)) + %<- (tequivi-resp ectx-eq/i (DeqC P) (DeqC P') Dequiv (%the (tequivi (G P) (C P) (C P')) Dequiv')) %. +%worlds (nbind ebind evar eovar) (tfunctionality-e _ _ _ _ _ _ _) (functionality-e _ _ _ _ _ _ _) %. +%total (D1 D2) (functionality-e D1 _ _ _ _ _ _) (tfunctionality-e D2 _ _ _ _ _ _) %. +%sort can-meof {D eof M A} %. +%mode can-meof %in %out %. +%sort can-mewf {D ewf A} %. +%mode can-mewf %in %out %. +%term _ can-meof (eof/var _ _) meof/var %. +%term _ can-meof (eof/const _ _) meof/const %. +%term _ %pi (can-meof (eof/forall D) (meof/forall D')) %<- (can-mewf D D') %. +%term _ + %pi (can-meof (eof/lam D2 D1) (meof/lam D2' D1')) + %<- (can-mewf D1 D1') + %<- ({x} {d} can-meof (D2 x d) (D2' x d)) %. +%term _ + %pi (can-meof (eof/app D2 D1) (meof/app D2' D1')) + %<- (can-meof D1 D1') + %<- (can-meof D2 D2') %. +%term _ + %pi (can-meof (eof/pair _ D2 D1) (meof/pair D2' D1')) + %<- (can-meof D1 D1') + %<- (can-meof D2 D2') %. +%term _ %pi (can-meof (eof/pi1 D) (meof/pi1 D')) %<- (can-meof D D') %. +%term _ %pi (can-meof (eof/pi2 D) (meof/pi2 D')) %<- (can-meof D D') %. +%term _ %pi (can-meof (eof/sing D) (meof/sing D')) %<- (can-meof D D') %. +%term _ + %pi (can-meof (eof/extpi D2 D1) (meof/extpi D2' D1')) + %<- (can-meof D1 D1') + %<- ({x} {d} can-meof (D2 x d) (D2' x d)) %. +%term _ + %pi (can-meof (eof/extsigma _ D2 D1) (meof/extsigma D2' D1')) + %<- (can-meof D1 D1') + %<- (can-meof D2 D2') %. +%term _ %pi (can-meof (eof/subsume _ D) (meof/subsume D')) %<- (can-meof D D') %. +%term _ can-mewf ewf/t mewf/t %. +%term _ + %pi (can-mewf (ewf/pi D2 D1) (mewf/pi D2' D1')) + %<- (can-mewf D1 D1') + %<- ({x} {d} can-mewf (D2 x d) (D2' x d)) %. +%term _ + %pi (can-mewf (ewf/sigma D2 D1) (mewf/sigma D2' D1')) + %<- (can-mewf D1 D1') + %<- ({x} {d} can-mewf (D2 x d) (D2' x d)) %. +%term _ %pi (can-mewf (ewf/sing D) (mewf/sing D')) %<- (can-meof D D') %. +%worlds (nbind ebind evar eofblock eovar) (can-meof _ _) (can-mewf _ _) %. +%total (D1 D2) (can-meof D1 _) (can-mewf D2 _) %. +%%%%% Functionality (Implicit Context) %%%%% +%sort tequivi-nil-invert %. +%mode tequivi-nil-invert %in %out %. +%term _ tequivi-nil-invert (tequivi/nil D) D %. +%worlds (nbind ebind evar eofblock eovar) (tequivi-nil-invert _ _) %. +%total {} (tequivi-nil-invert _ _) %. +%sort tfunctionality* {x} %. +%mode tfunctionality* %in %in %in %out %. +%term _ + %pi (tfunctionality* Dwf Dequiv Dof Dtequiv) + %<- ({x} {d} can-mewf (Dwf x d) (Dmwf x d)) + %<- (tfunctionality-e _ ([x] [d] ewfi/nil (Dwf x d)) ([x] [d] mewfi/nil (Dmwf x d)) ([x] clean/nil) Dequiv Dof Dtequivi) + %<- (tequivi-nil-invert Dtequivi Dtequiv) %. +%worlds (nbind ebind) (tfunctionality* _ _ _ _) %. +%total {} (tfunctionality* _ _ _ _) %. +%sort tequivi-cons1-invert {x} %. +%mode tequivi-cons1-invert %in %out %. +%term _ tequivi-cons1-invert ([x] tequivi/cons ([d] tequivi/nil (D x d))) D %. +%worlds (nbind ebind) (tequivi-cons1-invert _ _) %. +%total {} (tequivi-cons1-invert _ _) %. +%sort tfunctionality1* {x} %. +%mode tfunctionality1* %in %in %in %out %. +%term _ + %pi (tfunctionality1* Dwf Dequiv Dof Dtequiv) + %<- ({x} {d} {y} {e} can-mewf (Dwf x d y e) (%the (mewf _ Mm) (Dmwf x d y e))) + %<- ({y} {ev eisvar y} tfunctionality-e _ ([x] [d] ewfi/cons ([e] ewfi/nil (Dwf x d y e))) ([x] [d] mewfi/cons ([e] mewfi/nil (Dmwf x d y e))) ([x] clean/cons ev clean/nil) Dequiv Dof (Dtequivi y)) + %<- (tequivi-cons1-invert Dtequivi Dtequiv) %. +%worlds (nbind ebind) (tfunctionality1* _ _ _ _) %. +%total {} (tfunctionality1* _ _ _ _) %. +%sort tequivi-cons2-invert {x} {y} %. +%mode tequivi-cons2-invert %in %out %. +%term _ tequivi-cons2-invert ([x] [y] tequivi/cons ([e] tequivi/cons ([d] tequivi/nil (D x d y e)))) D %. +%worlds (nbind ebind) (tequivi-cons2-invert _ _) %. +%total {} (tequivi-cons2-invert _ _) %. +%sort tfunctionality2* {x} %. +%mode tfunctionality2* %in %in %in %out %. +%term _ + %pi (tfunctionality2* Dwf Dequiv Dof Dtequiv) + %<- ({x} {d} {y} {e} {z} {f} can-mewf (Dwf x d y e z f) (%the (mewf _ Mm) (Dmwf x d y e z f))) + %<- ({y} {ev eisvar y} {z} {fv eisvar z} tfunctionality-e _ ([x] [d] ewfi/cons ([f] ewfi/cons ([e] ewfi/nil (Dwf x d y e z f)))) ([x] [d] mewfi/cons ([f] mewfi/cons ([e] mewfi/nil (Dmwf x d y e z f)))) ([x] clean/cons fv (clean/cons ev clean/nil)) Dequiv Dof (Dtequivi y z)) + %<- (tequivi-cons2-invert Dtequivi Dtequiv) %. +%worlds (nbind ebind) (tfunctionality2* _ _ _ _) %. +%total {} (tfunctionality2* _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/il-simp.lf b/new-tests/stelf-output/tslf/sing/il-simp.lf new file mode 100644 index 0000000..63cf053 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/il-simp.lf @@ -0,0 +1,61 @@ +%%%%% Simple Types %%%%% +%sort stp %. +%name stp %. +%term st stp %. +%term spi %pi stp %-> stp %-> stp %. +%term ssigma %pi stp %-> stp %-> stp %. +%sort simp %. +%term simp/t simp t st %. +%term simp/pi %pi (simp (pi A B) (spi S T)) %<- (simp A S) %<- ({x} simp (B x) T) %. +%term simp/sigma %pi (simp (sigma A B) (ssigma S T)) %<- (simp A S) %<- ({x} simp (B x) T) %. +%term simp/sing simp (sing R) st %. +%sort stp-leq %. +%term stp-leq/eq stp-leq T T %. +%term stp-leq/pi1 %pi (stp-leq S (spi T1 T2)) %<- (stp-leq S T1) %. +%term stp-leq/pi2 %pi (stp-leq S (spi T1 T2)) %<- (stp-leq S T2) %. +%term stp-leq/sigma1 %pi (stp-leq S (ssigma T1 T2)) %<- (stp-leq S T1) %. +%term stp-leq/sigma2 %pi (stp-leq S (ssigma T1 T2)) %<- (stp-leq S T2) %. +%%%%% Simple Contexts %%%%% +%sort sctx %. +%name sctx %. +%term snil sctx %. +%term scons %pi sctx %-> atom %-> stp %-> sctx %. +%sort sbounded %. +%term sbounded/nil %pi (sbounded snil X) %<- (isvar X _) %. +%term sbounded/cons %pi (sbounded (scons G Y _) X) %<- (precedes Y X) %<- (sbounded G Y) %. +%sort sordered %. +%term sordered/nil sordered snil %. +%term sordered/cons %pi (sordered (scons G X _)) %<- (sbounded G X) %. +%sort slookup %. +%term slookup/hit %pi (slookup (scons G X T) X T) %<- (sbounded G X) %. +%term slookup/miss %pi (slookup (scons G Y _) X T) %<- (sbounded G Y) %<- (slookup G X T) %. +%sort sappend %. +%term sappend/nil sappend G snil G %. +%term sappend/cons %pi (sappend G1 (scons G2 X A) (scons G X A)) %<- (sappend G1 G2 G) %. +%%%%% Simple Typing with Explicit Contexts %%%%% +%sort wfes %. +%sort aofes %. +%sort ofes %. +%term aofes/closed %pi (aofes G R T) %<- (aof R A) %<- (simp A T) %<- (sordered G) %. +%term aofes/var %pi (aofes G X T) %<- (slookup G X T) %. +%term aofes/forall %pi (aofes G (forall A) (spi (spi T st) st)) %<- (wfes G A) %<- (simp A T) %. +%term aofes/app %pi (aofes G (app R M) T) %<- (aofes G R (spi S T)) %<- (ofes G M S) %. +%term aofes/pi1 %pi (aofes G (pi1 R) S) %<- (aofes G R (ssigma S T)) %. +%term aofes/pi2 %pi (aofes G (pi2 R) T) %<- (aofes G R (ssigma S T)) %. +%term ofes/at %pi (ofes G (at R) st) %<- (aofes G R st) %. +%term ofes/lam + %pi (ofes G (lam M) (spi S T)) + %<- ({x} %pi (isvar x I) %-> (ofes (scons G x S) (M x) T)) %. +%term ofes/pair %pi (ofes G (pair M N) (ssigma S T)) %<- (ofes G M S) %<- (ofes G N T) %. +%term wfes/t %pi (wfes G t) %<- (sordered G) %. +%term wfes/pi + %pi (wfes G (pi A B)) + %<- (wfes G A) + %<- (simp A T) + %<- ({x} %pi (isvar x I) %-> (wfes (scons G x T) (B x))) %. +%term wfes/sigma + %pi (wfes G (sigma A B)) + %<- (wfes G A) + %<- (simp A T) + %<- ({x} %pi (isvar x I) %-> (wfes (scons G x T) (B x))) %. +%term wfes/sing %pi (wfes G (sing R)) %<- (aofes G R st) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/il.lf b/new-tests/stelf-output/tslf/sing/il.lf new file mode 100644 index 0000000..e8200ac --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/il.lf @@ -0,0 +1,110 @@ +%%%%% Syntax %%%%% +%sort tp %. +%name tp %. +%sort atom %. +%name atom %. +%sort term %. +%name term %. +%term t tp %. +%term pi %pi tp %-> (%pi atom %-> tp) %-> tp %. +%term sigma %pi tp %-> (%pi atom %-> tp) %-> tp %. +%term sing %pi atom %-> tp %. +%term const %pi constant %-> atom %. +%term forall %pi tp %-> atom %. +%term app %pi atom %-> term %-> atom %. +%term pi1 %pi atom %-> atom %. +%term pi2 %pi atom %-> atom %. +%term at %pi atom %-> term %. +%term lam %pi (%pi atom %-> term) %-> term %. +%term pair %pi term %-> term %-> term %. +%%%%% Substitution %%%%% +%sort aasub %. +%mode aasub %in %in %out %. +%sort aosub %. +%mode aosub %in %in %out %. +%sort sub %. +%mode sub %in %in %out %. +%sort tsub %. +%mode tsub %in %in %out %. +%term aasub/closed aasub ([_] R) _ R %. +%term aasub/forall %pi (aasub ([x] forall (A x)) M (forall A')) %<- (tsub A M A') %. +%term aosub/var aosub ([x] x) M M %. +%term aasub/app + %pi (aasub ([x] app (R x) (M x)) N (app R' M')) + %<- (aasub R N R') + %<- (sub M N M') %. +%term aosub/app + %pi (aosub ([x] app (R x) (M x)) N M1') + %<- (aosub R N (lam M1)) + %<- (sub M N M') + %<- (sub M1 M' M1') %. +%term aasub/pi1 %pi (aasub ([x] pi1 (R x)) M (pi1 R')) %<- (aasub R M R') %. +%term aosub/pi1 %pi (aosub ([x] pi1 (R x)) M M1) %<- (aosub R M (pair M1 M2)) %. +%term aasub/pi2 %pi (aasub ([x] pi2 (R x)) M (pi2 R')) %<- (aasub R M R') %. +%term aosub/pi2 %pi (aosub ([x] pi2 (R x)) M M2) %<- (aosub R M (pair M1 M2)) %. +%term sub/aa %pi (sub ([x] at (R x)) M (at R')) %<- (aasub R M R') %. +%term sub/ao %pi (sub ([x] at (R x)) M M') %<- (aosub R M M') %. +%term sub/lam + %pi (sub ([x] lam ([y] M x y)) N (lam ([y] M' y))) + %<- ({y} sub ([x] M x y) N (M' y)) %. +%term sub/pair + %pi (sub ([x] pair (M1 x) (M2 x)) N (pair M1' M2')) + %<- (sub M1 N M1') + %<- (sub M2 N M2') %. +%term tsub/t tsub ([_] t) _ t %. +%term tsub/pi + %pi (tsub ([x] pi (A x) ([y] B x y)) M (pi A' ([y] B' y))) + %<- (tsub A M A') + %<- ({y} tsub ([x] B x y) M (B' y)) %. +%term tsub/sigma + %pi (tsub ([x] sigma (A x) ([y] B x y)) M (sigma A' ([y] B' y))) + %<- (tsub A M A') + %<- ({y} tsub ([x] B x y) M (B' y)) %. +%term tsub/singa %pi (tsub ([x] sing (R x)) M (sing R')) %<- (aasub R M R') %. +%term tsub/singo %pi (tsub ([x] sing (R x)) M (sing R')) %<- (aosub R M (at R')) %. +%%%%% Typing Rules %%%%% +%sort kof %. +%mode kof %in %out %. +%sort wf %. +%mode wf %in %. +%sort vof %. +%mode vof %in %out %. +%sort aof %. +%mode aof %in %out %. +%sort of %. +%mode of %in %in %. +%term wf/t wf t %. +%term wf/pi %pi (wf (pi A B)) %<- (wf A) %<- ({x} %pi (vof x A) %-> (wf (B x))) %. +%term wf/sigma %pi (wf (sigma A B)) %<- (wf A) %<- ({x} %pi (vof x A) %-> (wf (B x))) %. +%term wf/sing %pi (wf (sing R)) %<- (aof R t) %. +%term aof/const %pi (aof (const C) A) %<- (kof C A) %<- (wf A) %. +%define qtp (%pi tp %-> tp) [a] pi (pi a ([_] t)) ([_] t) %. +%term aof/forall %pi (aof (forall A) (qtp A)) %<- (wf A) %. +%term aof/var %pi (aof X A) %<- (vof X A) %<- (wf A) %. +%term aof/app + %pi (aof (app R M) B') + %<- (aof R (pi A B)) + %<- (of M A) + %<- (tsub B M B') + %<- (wf B') %. +%term aof/pi1 %pi (aof (pi1 R) A) %<- (aof R (sigma A B)) %. +%term aof/pi2 %pi (aof (pi2 R) (B (pi1 R))) %<- (aof R (sigma A B)) %. +%term of/at %pi (of (at R) t) %<- (aof R t) %. +%term of/lam %pi (of (lam M) (pi A B)) %<- (wf A) %<- ({x} %pi (vof x A) %-> (of (M x) (B x))) %. +%term of/pair + %pi (of (pair M N) (sigma A B)) + %<- (of M A) + %<- (tsub B M B') + %<- (of N B') + %<- ({x} %pi (vof x A) %-> (wf (B x))) %. +%term of/sing %pi (of (at R) (sing R)) %<- (aof R t) %. +%%%%% Constants %%%%% +%sort topen %. +%mode topen %in %out %. +%term topen/t topen ct t %. +%term topen/pi %pi (topen (cpi Ac Bc) (pi A ([_] B))) %<- (topen Ac A) %<- (topen Bc B) %. +%term kof/i %pi (kof C A') %<- (ckof C A) %<- (topen A A') %. +%%%%% Worlds %%%%% +%block var [x atom]%. +%block bind {a tp} [x atom] [d vof x a]%. +%block ofblock {x atom} {a tp} [d vof x a]%. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/ile.lf b/new-tests/stelf-output/tslf/sing/ile.lf new file mode 100644 index 0000000..cb69dd3 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/ile.lf @@ -0,0 +1,68 @@ +%%%%% Ordered Variables %%%%% +%sort isvar %. +%sort _ %. +%sort precedes %. +%term precedes/i %pi (precedes X Y) %<- (isvar X I) %<- (isvar Y J) %<- (lt I J) %. +%%%%% Contexts %%%%% +%sort ctx %. +%name ctx %. +%term nil ctx %. +%term cons %pi ctx %-> atom %-> tp %-> ctx %. +%sort bounded %. +%term bounded/nil %pi (bounded nil X) %<- (isvar X _) %. +%term bounded/cons %pi (bounded (cons G Y _) X) %<- (precedes Y X) %<- (bounded G Y) %. +%sort ordered %. +%term ordered/nil ordered nil %. +%term ordered/cons %pi (ordered (cons G X _)) %<- (bounded G X) %. +%sort lookup %. +%term lookup/hit %pi (lookup (cons G X A) X A) %<- (bounded G X) %. +%term lookup/miss %pi (lookup (cons G Y _) X A) %<- (bounded G Y) %<- (lookup G X A) %. +%sort csub %. +%term csub/base csub ([x] cons G x A) M G %. +%term csub/cons + %pi (csub ([x] cons (G x) Y (A x)) M (cons G' Y A')) + %<- (csub G M G') + %<- (tsub A M A') %. +%sort append %. +%term append/nil append G nil G %. +%term append/cons %pi (append G1 (cons G2 X A) (cons G X A)) %<- (append G1 G2 G) %. +%%%%% Typing with Explicit Contexts %%%%% +%sort wfe %. +%sort aofe %. +%sort ofe %. +%term wfe/t %pi (wfe G t) %<- (ordered G) %. +%term wfe/pi + %pi (wfe G (pi A B)) + %<- (wfe G A) + %<- ({x} %pi (isvar x I) %-> (wfe (cons G x A) (B x))) %. +%term wfe/sigma + %pi (wfe G (sigma A B)) + %<- (wfe G A) + %<- ({x} %pi (isvar x I) %-> (wfe (cons G x A) (B x))) %. +%term wfe/sing %pi (wfe G (sing R)) %<- (aofe G R t) %. +%term aofe/closed %pi (aofe G R A) %<- (aof R A) %<- (ordered G) %. +%term aofe/forall %pi (aofe G (forall A) (pi (pi A ([_] t)) ([_] t))) %<- (wfe G A) %. +%term aofe/var %pi (aofe G X A) %<- (lookup G X A) %<- (wfe G A) %. +%term aofe/app + %pi (aofe G (app R M) B') + %<- (aofe G R (pi A B)) + %<- (ofe G M A) + %<- (tsub B M B') + %<- (wfe G B') %. +%term aofe/pi1 %pi (aofe G (pi1 R) A) %<- (aofe G R (sigma A B)) %. +%term aofe/pi2 %pi (aofe G (pi2 R) (B (pi1 R))) %<- (aofe G R (sigma A B)) %. +%term ofe/at %pi (ofe G (at R) t) %<- (aofe G R t) %. +%term ofe/lam + %pi (ofe G (lam M) (pi A B)) + %<- (wfe G A) + %<- ({x} %pi (isvar x I) %-> (ofe (cons G x A) (M x) (B x))) %. +%term ofe/pair + %pi (ofe G (pair M N) (sigma A B)) + %<- (ofe G M A) + %<- (tsub B M B') + %<- (ofe G N B') + %<- ({x} %pi (isvar x I) %-> (wfe (cons G x A) (B x))) %. +%term ofe/sing %pi (ofe G (at R) (sing R)) %<- (aofe G R t) %. +%%%%% Worlds %%%%% +%block ovar {n nat} [x atom] [d isvar x n]%. +%block obind {a tp} {n nat} [x atom] [d vof x a] [d' isvar x n]%. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/inversion-thm.lf b/new-tests/stelf-output/tslf/sing/inversion-thm.lf new file mode 100644 index 0000000..086132b --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/inversion-thm.lf @@ -0,0 +1,181 @@ +%%%%% Inversion Lemmas (Simply Typed, Explicit Context) %%%%% +%sort aofes-invert-app %. +%mode aofes-invert-app %in %out %out %. +%term _ aofes-invert-app (aofes/app D2 D1) D1 D2 %. +%term _ + %pi (aofes-invert-app (aofes/closed Dordered Dsimp (aof/app _ Dsub Dof Daof)) (aofes/closed Dordered (simp/pi DsimpB' DsimpA) Daof) Dofes') + %<- (can-simp A (%the (simp A S) DsimpA)) + %<- ({x atom} can-simp (B x) (%the (simp (B x) T') (DsimpB x))) + %<- (tsub-preserves-simp Dsub DsimpB DsimpBx) + %<- (simp-fun DsimpBx Dsimp (%the (stp-eq T' T) Deq)) + %<- ({x} simp-resp tp-eq/i Deq (DsimpB x) (%the (simp (B x) T) (DsimpB' x))) + %<- (of-to-ofe Dof Dofe) + %<- (ofe-simp simpctx/nil DsimpA Dofe Dofes) + %<- (sappend-all _ Dappend) + %<- (weaken-ofes' Dappend Dordered Dofes Dofes') %. +%term _ + %pi (aofes-invert-app (aofes/var Dlookup) D1 D2) + %<- (slookup-isvar Dlookup Disvar) + %<- (isvar-app-contra Disvar Dfalse) + %<- (false-implies-aofes Dfalse D1) + %<- (false-implies-ofes Dfalse (%the (ofes _ _ st) D2)) %. +%sort _ %. +%worlds (var ovar bind) (aofes-invert-app _ _ _) %. +%total {} (aofes-invert-app _ _ _) %. +%sort aofes-invert-pi1 %. +%mode aofes-invert-pi1 %in %out %. +%term _ aofes-invert-pi1 (aofes/pi1 D) D %. +%term _ + %pi (aofes-invert-pi1 (aofes/closed Dordered Dsimp (aof/pi1 Daof)) (aofes/closed Dordered (simp/sigma DsimpB Dsimp) Daof)) + %<- ({x} can-simp (B x) (%the (simp (B x) T) (DsimpB x))) %. +%term _ + %pi (aofes-invert-pi1 (aofes/var Dlookup) D) + %<- (slookup-isvar Dlookup Disvar) + %<- (isvar-pi1-contra Disvar Dfalse) + %<- (false-implies-aofes Dfalse (%the (aofes _ _ (ssigma _ st)) D)) %. +%sort _ %. +%worlds (var ovar bind) (aofes-invert-pi1 _ _) %. +%total {} (aofes-invert-pi1 _ _) %. +%sort aofes-invert-pi2 %. +%mode aofes-invert-pi2 %in %out %. +%term _ aofes-invert-pi2 (aofes/pi2 D) D %. +%term _ + %pi (aofes-invert-pi2 (aofes/closed Dordered Dsimp (aof/pi2 Daof)) (aofes/closed Dordered (simp/sigma DsimpB' DsimpA) Daof)) + %<- (can-simp A (%the (simp A S) DsimpA)) + %<- ({x atom} can-simp (B x) (%the (simp (B x) T') (DsimpB x))) + %<- (simp-fun (DsimpB _) Dsimp (%the (stp-eq T' T) Deq)) + %<- ({x} simp-resp tp-eq/i Deq (DsimpB x) (%the (simp (B x) T) (DsimpB' x))) %. +%term _ + %pi (aofes-invert-pi2 (aofes/var Dlookup) D) + %<- (slookup-isvar Dlookup Disvar) + %<- (isvar-pi2-contra Disvar Dfalse) + %<- (false-implies-aofes Dfalse (%the (aofes _ _ (ssigma st _)) D)) %. +%sort _ %. +%worlds (var ovar bind) (aofes-invert-pi2 _ _) %. +%total {} (aofes-invert-pi2 _ _) %. +%sort aofes-invert-forall %. +%mode aofes-invert-forall %in %out %. +%term _ aofes-invert-forall (aofes/forall _ D) D %. +%term _ + %pi (aofes-invert-forall (aofes/closed Dordered _ (aof/forall Dwf)) Dwfes') + %<- (wf-to-wfe Dwf Dwfe) + %<- (wfe-simp simpctx/nil Dwfe Dwfes) + %<- (sappend-all _ Dappend) + %<- (weaken-wfes' Dappend Dordered Dwfes Dwfes') %. +%term _ + %pi (aofes-invert-forall (aofes/var Dlookup) D) + %<- (slookup-isvar Dlookup Disvar) + %<- (isvar-forall-contra Disvar Dfalse) + %<- (false-implies-wfes Dfalse D) %. +%sort _ %. +%worlds (var bind ovar) (aofes-invert-forall _ _) %. +%total {} (aofes-invert-forall _ _) %. +%sort ofes-invert-st %. +%mode ofes-invert-st %in %out %. +%term _ ofes-invert-st (ofes/at _) term-eq/i %. +%sort _ %. +%worlds (var ovar bind) (ofes-invert-st _ _) %. +%total {} (ofes-invert-st _ _) %. +%%%%% Inversion Lemmas (Explicit Context) %%%%% +%sort aofe-invert-var {x} %. +%mode aofe-invert-var %in %out %. +%term _ aofe-invert-var ([x] [d] aofe/var _ (Dlookup x d)) Dlookup %. +%term _ + %pi (aofe-invert-var ([x] [d] aofe/closed _ (Daof x)) ([x] [_] Dlookup x)) + %<- (aof-noassm-var Daof Dfalse) + %<- ({x} false-implies-lookup Dfalse (Dlookup x)) %. +%worlds (var ovar bind) (aofe-invert-var _ _) %. +%total {} (aofe-invert-var _ _) %. +%sort aofe-invert-forall %. +%mode aofe-invert-forall %in %out %out %. +%term _ aofe-invert-forall (aofe/forall D) D tp-eq/i %. +%term _ + %pi (aofe-invert-forall (aofe/closed Dord (aof/forall Dwf)) Dwfe' tp-eq/i) + %<- (wf-to-wfe Dwf Dwfe) + %<- (append-all _ Dappend) + %<- (weaken-wfe' Dappend Dord Dwfe Dwfe') %. +%term _ + %pi (aofe-invert-forall (aofe/var _ Dlookup) Dwf Deq) + %<- (lookup-isvar Dlookup Disvar) + %<- (isvar-forall-contra Disvar Dfalse) + %<- (false-implies-wfe Dfalse Dwf) + %<- (false-implies-tp-eq Dfalse Deq) %. +%worlds (var bind ovar) (aofe-invert-forall _ _ _) %. +%total {} (aofe-invert-forall _ _ _) %. +%sort aofe-invert-app %. +%mode aofe-invert-app %in %out %out %out %. +%term _ aofe-invert-app (aofe/app _ D3 D2 D1) D1 D2 D3 %. +%term _ + %pi (aofe-invert-app (aofe/closed Dordered (aof/app _ Dsub Dof Daof)) (aofe/closed Dordered Daof) Dofe' Dsub) + %<- (of-to-ofe Dof Dofe) + %<- (append-all _ Dappend) + %<- (weaken-ofe' Dappend Dordered Dofe Dofe') %. +%term _ + %pi (aofe-invert-app (aofe/var _ Dlookup) Daofe Dofe Dsub) + %<- (lookup-isvar Dlookup Disvar) + %<- (isvar-app-contra Disvar Dfalse) + %<- (false-implies-aofe Dfalse (%the (aofe _ _ (pi t ([_] t))) Daofe)) + %<- (false-implies-ofe Dfalse Dofe) + %<- (false-implies-tsub Dfalse Dsub) %. +%sort _ %. +%worlds (var ovar bind) (aofe-invert-app _ _ _ _) %. +%total {} (aofe-invert-app _ _ _ _) %. +%sort aofe-invert-pi1 %. +%mode aofe-invert-pi1 %in %out %. +%term _ aofe-invert-pi1 (aofe/pi1 D) D %. +%term _ aofe-invert-pi1 (aofe/closed Dordered (aof/pi1 D)) (aofe/closed Dordered D) %. +%term _ + %pi (aofe-invert-pi1 (aofe/var _ Dlookup) Daofe) + %<- (lookup-isvar Dlookup Disvar) + %<- (isvar-pi1-contra Disvar Dfalse) + %<- (false-implies-aofe Dfalse (%the (aofe _ _ (sigma _ ([_] t))) Daofe)) %. +%worlds (var ovar bind) (aofe-invert-pi1 _ _) %. +%total {} (aofe-invert-pi1 _ _) %. +%sort aofe-invert-pi2 %. +%mode aofe-invert-pi2 %in %out %out %. +%term _ aofe-invert-pi2 (aofe/pi2 D) D tp-eq/i %. +%term _ aofe-invert-pi2 (aofe/closed Dordered (aof/pi2 D)) (aofe/closed Dordered D) tp-eq/i %. +%term _ + %pi (aofe-invert-pi2 (aofe/var _ Dlookup) Daofe Deq) + %<- (lookup-isvar Dlookup Disvar) + %<- (isvar-pi2-contra Disvar Dfalse) + %<- (false-implies-aofe Dfalse (%the (aofe _ _ (sigma t ([_] t))) Daofe)) + %<- (false-implies-tp-eq Dfalse Deq) %. +%sort _ %. +%worlds (var ovar bind) (aofe-invert-pi2 _ _ _) %. +%total {} (aofe-invert-pi2 _ _ _) %. +%sort ofe-invert-at %. +%mode ofe-invert-at %in %out %. +%term _ ofe-invert-at (ofe/at D) D %. +%term _ ofe-invert-at (ofe/sing D) D %. +%sort _ %. +%worlds (var bind ovar) (ofe-invert-at _ _) %. +%total {} (ofe-invert-at _ _) %. +%sort ofe-invert-sing %. +%mode ofe-invert-sing %in %out %. +%term _ ofe-invert-sing (ofe/sing _) term-eq/i %. +%worlds (bind ovar var) (ofe-invert-sing _ _) %. +%total {} (ofe-invert-sing _ _) %. +%sort ofe-at-pi-contra %. +%mode ofe-at-pi-contra %in %out %. +%worlds (bind ovar var) (ofe-at-pi-contra _ _) %. +%total {} (ofe-at-pi-contra _ _) %. +%sort ofe-at-sigma-contra %. +%mode ofe-at-sigma-contra %in %out %. +%worlds (bind ovar var) (ofe-at-sigma-contra _ _) %. +%total {} (ofe-at-sigma-contra _ _) %. +%sort ofe-lam-sigma-contra %. +%mode ofe-lam-sigma-contra %in %out %. +%worlds (bind ovar var) (ofe-lam-sigma-contra _ _) %. +%total {} (ofe-lam-sigma-contra _ _) %. +%sort ofe-pair-pi-contra %. +%mode ofe-pair-pi-contra %in %out %. +%worlds (bind ovar var) (ofe-pair-pi-contra _ _) %. +%total {} (ofe-pair-pi-contra _ _) %. +%%%%% Miscellaneous %%%%% +%sort tsub-sing-sub-at %. +%mode tsub-sing-sub-at %in %out %. +%scope tsub-sing-sub-at %term a tsub-sing-sub-at (tsub/singa D) (sub/aa D) %. +%term o tsub-sing-sub-at (tsub/singo D) (sub/ao D) %. +%worlds (var) (tsub-sing-sub-at _ _) %. +%total {} (tsub-sing-sub-at _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/nat-thm.lf b/new-tests/stelf-output/tslf/sing/nat-thm.lf new file mode 100644 index 0000000..95aab98 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/nat-thm.lf @@ -0,0 +1,290 @@ +%%%%% Equality %%%%% +%sort lt-resp %. +%mode lt-resp %in %in %in %out %. +%term _ lt-resp nat-eq/i nat-eq/i D D %. +%worlds () (lt-resp _ _ _ _) %. +%total {} (lt-resp _ _ _ _) %. +%sort leq-resp %. +%mode leq-resp %in %in %in %out %. +%term _ leq-resp nat-eq/i nat-eq/i D D %. +%worlds () (leq-resp _ _ _ _) %. +%total {} (leq-resp _ _ _ _) %. +%sort sum-resp %. +%mode sum-resp %in %in %in %in %out %. +%term _ sum-resp nat-eq/i nat-eq/i nat-eq/i D D %. +%worlds () (sum-resp _ _ _ _ _) %. +%total {} (sum-resp _ _ _ _ _) %. +%sort nat-resp {N %pi nat %-> nat} %. +%mode nat-resp %in %in %out %. +%term _ nat-resp _ nat-eq/i nat-eq/i %. +%worlds () (nat-resp _ _ _) %. +%total {} (nat-resp _ _ _) %. +%sort nat-eq-inc %. +%mode nat-eq-inc %in %out %. +%term _ nat-eq-inc nat-eq/i nat-eq/i %. +%worlds () (nat-eq-inc _ _) %. +%total {} (nat-eq-inc _ _) %. +%%%%% Sum %%%%% +%sort sum-ident {N nat} %. +%mode sum-ident %in %out %. +%term _ sum-ident 0 sum/z %. +%term _ %pi (sum-ident (s N) (sum/s D)) %<- (sum-ident N D) %. +%worlds () (sum-ident _ _) %. +%total N (sum-ident N _) %. +%sort sum-inc %. +%mode sum-inc %in %out %. +%term _ sum-inc sum/z sum/z %. +%term _ %pi (sum-inc (sum/s D) (sum/s D')) %<- (sum-inc D D') %. +%worlds () (sum-inc _ _) %. +%total D (sum-inc D _) %. +%sort sum-commute %. +%mode sum-commute %in %out %. +%term _ %pi (sum-commute (%the (sum 0 N N) sum/z) D) %<- (sum-ident N D) %. +%term _ + %pi (sum-commute (%the (sum (s N1) N2 (s N3)) (sum/s D)) D'') + %<- (sum-commute D (%the (sum N2 N1 N3) D')) + %<- (sum-inc D' D'') %. +%worlds () (sum-commute _ _) %. +%total D (sum-commute D _) %. +%sort sum-assoc %. +%mode sum-assoc %in %in %out %out %. +%term _ sum-assoc sum/z D D sum/z %. +%term _ + %pi (sum-assoc (sum/s D12) (sum/s D12_3) D23 (sum/s D1_23)) + %<- (sum-assoc D12 D12_3 D23 D1_23) %. +%worlds () (sum-assoc _ _ _ _) %. +%total D (sum-assoc D _ _ _) %. +%% same as above but different mode +%sort sum-assoc' %. +%mode sum-assoc' %out %out %in %in %. +%term _ sum-assoc' sum/z D D sum/z %. +%term _ + %pi (sum-assoc' (sum/s D12) (sum/s D12-3) D23 (sum/s D1-23)) + %<- (sum-assoc' D12 D12-3 D23 D1-23) %. +%worlds () (sum-assoc' _ _ _ _) %. +%total D (sum-assoc' _ _ _ D) %. +%sort can-sum {N1 nat} {N2 nat} {N3 nat} %. +%mode can-sum %in %in %out %out %. +%term _ can-sum 0 N N sum/z %. +%term _ %pi (can-sum (s N1) N2 (s N3) (sum/s D)) %<- (can-sum N1 N2 N3 D) %. +%worlds () (can-sum _ _ _ _) %. +%total N (can-sum N _ _ _) %. +%sort sum-fun2 %. +%mode sum-fun2 %in %in %out %. +%term _ sum-fun2 sum/z sum/z nat-eq/i %. +%term _ %pi (sum-fun2 (sum/s D1) (sum/s D2) D3) %<- (sum-fun2 D1 D2 D3) %. +%worlds () (sum-fun2 _ _ _) %. +%total D (sum-fun2 D _ _) %. +%sort sum-fun1 %. +%mode sum-fun1 %in %in %out %. +%term _ + %pi (sum-fun1 D1 D2 D) + %<- (sum-commute D1 D1') + %<- (sum-commute D2 D2') + %<- (sum-fun2 D1' D2' D) %. +%worlds () (sum-fun1 _ _ _) %. +%total D (sum-fun1 D _ _) %. +%sort sum-fun3 %. +%mode sum-fun3 %in %in %out %. +%term _ sum-fun3 sum/z sum/z nat-eq/i %. +%term _ + %pi (sum-fun3 (sum/s D1) (sum/s D2) D') + %<- (sum-fun3 D1 D2 D) + %<- (nat-eq-inc D D') %. +%worlds () (sum-fun3 _ _ _) %. +%total D (sum-fun3 D _ _) %. +%%%%% Leq %%%%% +%sort leq-trans %. +%mode leq-trans %in %in %out %. +%term _ leq-trans leq/z _ leq/z %. +%term _ %pi (leq-trans (leq/s D1) (leq/s D2) (leq/s D3)) %<- (leq-trans D1 D2 D3) %. +%worlds () (leq-trans _ _ _) %. +%total D (leq-trans D _ _) %. +%sort leq-reflex {N nat} %. +%mode leq-reflex %in %out %. +%term _ leq-reflex 0 leq/z %. +%term _ %pi (leq-reflex (s N) (leq/s D)) %<- (leq-reflex N D) %. +%worlds () (leq-reflex _ _) %. +%total N (leq-reflex N _) %. +%sort leq-reflex' %. +%mode leq-reflex' %in %out %. +%term _ %pi (leq-reflex' nat-eq/i D) %<- (leq-reflex _ D) %. +%worlds () (leq-reflex' _ _) %. +%total {} (leq-reflex' _ _) %. +%sort sum-implies-leq %. +%mode sum-implies-leq %in %out %. +%term _ sum-implies-leq sum/z leq/z %. +%term _ %pi (sum-implies-leq (sum/s D) (leq/s D')) %<- (sum-implies-leq D D') %. +%worlds () (sum-implies-leq _ _) %. +%total D (sum-implies-leq D _) %. +%sort sum-implies-leq' %. +%mode sum-implies-leq' %in %out %. +%term _ + %pi (sum-implies-leq' Dsum Dleq) + %<- (sum-commute Dsum Dsum') + %<- (sum-implies-leq Dsum' Dleq) %. +%worlds () (sum-implies-leq' _ _) %. +%total {} (sum-implies-leq' _ _) %. +%sort leq-implies-sum %. +%mode leq-implies-sum %in %out %. +%term _ leq-implies-sum leq/z sum/z %. +%term _ %pi (leq-implies-sum (leq/s D) (sum/s D')) %<- (leq-implies-sum D D') %. +%worlds () (leq-implies-sum _ _) %. +%total D (leq-implies-sum D _) %. +%sort sum-leq %. +%mode sum-leq %in %in %in %in %out %. +%term _ + %pi (sum-leq Dsum Dsum' Dleq1 Dleq2 Dleq3) + %<- (leq-implies-sum Dleq1 (%the (sum N1 N1'' N1') Dsum1)) + %<- (leq-implies-sum Dleq2 (%the (sum N2 N2'' N2') Dsum2)) + %<- (sum-assoc Dsum1 Dsum' (%the (sum N1'' N2' N4) Dsum4) (%the (sum N1 N4 N3') Dsum5)) + %<- (sum-commute Dsum4 (%the (sum N2' N1'' N4) Dsum6)) + %<- (sum-assoc Dsum2 Dsum6 (%the (sum N2'' N1'' N5) Dsum7) (%the (sum N2 N5 N4) Dsum8)) + %<- (sum-assoc' (%the (sum N1 N2 N6) Dsum9) (%the (sum N6 N5 N3') Dsum10) Dsum8 Dsum5) + %<- (sum-fun3 Dsum9 Dsum (%the (nat-eq N6 N3) Deq)) + %<- (sum-resp Deq nat-eq/i nat-eq/i Dsum10 (%the (sum N3 N5 N3') Dsum3)) + %<- (sum-implies-leq Dsum3 Dleq3) %. +%worlds () (sum-leq _ _ _ _ _) %. +%total {} (sum-leq _ _ _ _ _) %. +%%%%% Lt %%%%% +%sort leq-lt-trans %. +%mode leq-lt-trans %in %in %out %. +%term _ leq-lt-trans leq/z lt/z lt/z %. +%term _ leq-lt-trans leq/z (lt/s _) lt/z %. +%term _ %pi (leq-lt-trans (leq/s D1) (lt/s D2) (lt/s D3)) %<- (leq-lt-trans D1 D2 D3) %. +%worlds () (leq-lt-trans _ _ _) %. +%total D (leq-lt-trans D _ _) %. +%sort lt-leq-trans %. +%mode lt-leq-trans %in %in %out %. +%term _ lt-leq-trans lt/z (leq/s _) lt/z %. +%term _ %pi (lt-leq-trans (lt/s D1) (leq/s D2) (lt/s D3)) %<- (lt-leq-trans D1 D2 D3) %. +%worlds () (lt-leq-trans _ _ _) %. +%total D (lt-leq-trans D _ _) %. +%sort lt-trans %. +%mode lt-trans %in %in %out %. +%term _ lt-trans lt/z (lt/s _) lt/z %. +%term _ %pi (lt-trans (lt/s D1) (lt/s D2) (lt/s D3)) %<- (lt-trans D1 D2 D3) %. +%worlds () (lt-trans _ _ _) %. +%total D (lt-trans D _ _) %. +%sort lt-antisymm %. +%mode lt-antisymm %in %out %. +%term _ %pi (lt-antisymm (lt/s D) D') %<- (lt-antisymm D D') %. +%worlds () (lt-antisymm _ _) %. +%total D (lt-antisymm D _) %. +%sort sum-implies-lt %. +%mode sum-implies-lt %in %in %out %. +%term _ sum-implies-lt sum/z lt/z lt/z %. +%term _ %pi (sum-implies-lt (sum/s D) Dlt (lt/s D')) %<- (sum-implies-lt D Dlt D') %. +%worlds () (sum-implies-lt _ _ _) %. +%total D (sum-implies-lt D _ _) %. +%sort sum-implies-lt' %. +%mode sum-implies-lt' %in %in %out %. +%term _ + %pi (sum-implies-lt' Dsum Dlt Dlt') + %<- (sum-commute Dsum Dsum') + %<- (sum-implies-lt Dsum' Dlt Dlt') %. +%worlds () (sum-implies-lt' _ _ _) %. +%total {} (sum-implies-lt' _ _ _) %. +%sort lt-succ {N} %. +%mode lt-succ %in %out %. +%term _ %pi (lt-succ N Dlt) %<- (sum-implies-lt' (sum/s sum/z) lt/z Dlt) %. +%worlds () (lt-succ _ _) %. +%total {} (lt-succ _ _) %. +%sort lt-implies-sum %. +%mode lt-implies-sum %in %out %. +%scope lt-implies-sum %term z lt-implies-sum lt/z sum/z %. +%term s %pi (lt-implies-sum (lt/s D) (sum/s D')) %<- (lt-implies-sum D D') %. +%worlds () (lt-implies-sum _ _) %. +%total D (lt-implies-sum D _) %. +%sort sum-leq-lt %. +%mode sum-leq-lt %in %in %in %in %out %. +%term _ + %pi (sum-leq-lt (%the (sum N1 N2 N3) Dsum) (%the (sum N1' N2' N3') Dsum') (%the (leq N1 N1') Dleq) (%the (lt N2 N2') Dlt) D) + %<- (leq-implies-sum Dleq (%the (sum N1 N1'' N1') Dsum1)) + %<- (lt-implies-sum Dlt (%the (sum N2 (s N2'') N2') Dsum2)) + %<- (sum-assoc Dsum1 Dsum' (%the (sum N1'' N2' N4) Dsum4) (%the (sum N1 N4 N3') Dsum5)) + %<- (sum-commute Dsum4 (%the (sum N2' N1'' N4) Dsum6)) + %<- (sum-assoc Dsum2 Dsum6 (sum/s (%the (sum N2'' N1'' N5) Dsum7)) (%the (sum N2 (s N5) N4) Dsum8)) + %<- (sum-assoc' (%the (sum N1 N2 N6) Dsum9) (%the (sum N6 (s N5) N3') Dsum10) Dsum8 Dsum5) + %<- (sum-fun3 Dsum9 Dsum (%the (nat-eq N6 N3) Deq)) + %<- (sum-resp Deq nat-eq/i nat-eq/i Dsum10 (%the (sum N3 (s N5) N3') Dsum3)) + %<- (sum-implies-lt Dsum3 lt/z D) %. +%worlds () (sum-leq-lt _ _ _ _ _) %. +%total {} (sum-leq-lt _ _ _ _ _) %. +%sort sum-lt-leq %. +%mode sum-lt-leq %in %in %in %in %out %. +%term _ + %pi (sum-lt-leq DsumA DsumB Dlt Dleq D) + %<- (sum-commute DsumA DsumA') + %<- (sum-commute DsumB DsumB') + %<- (sum-leq-lt DsumA' DsumB' Dleq Dlt D) %. +%worlds () (sum-lt-leq _ _ _ _ _) %. +%total {} (sum-lt-leq _ _ _ _ _) %. +%%%%% Maximum %%%%% +%sort can-max {N1} {N2} {N3} %. +%mode can-max %in %in %out %out %. +%term _ can-max 0 N N max/z* %. +%term _ can-max N 0 N max/*z %. +%term _ %pi (can-max (s N1) (s N2) (s N3) (max/s D)) %<- (can-max N1 N2 N3 D) %. +%worlds () (can-max _ _ _ _) %. +%total N (can-max N _ _ _) %. +%sort max-lt %. +%mode max-lt %in %in %in %out %. +%term _ max-lt max/z* _ D D %. +%term _ max-lt max/*z D _ D %. +%term _ max-lt (max/s D) lt/z lt/z lt/z %. +%term _ %pi (max-lt (max/s D) (lt/s D1) (lt/s D2) (lt/s D')) %<- (max-lt D D1 D2 D') %. +%worlds () (max-lt _ _ _ _) %. +%total D (max-lt D _ _ _) %. +%sort max-leq %. +%mode max-leq %in %out %out %. +%term _ %pi (max-leq max/z* leq/z D) %<- (leq-reflex _ D) %. +%term _ %pi (max-leq max/*z D leq/z) %<- (leq-reflex _ D) %. +%term _ %pi (max-leq (max/s D) (leq/s D1) (leq/s D2)) %<- (max-leq D D1 D2) %. +%worlds () (max-leq _ _ _) %. +%total D (max-leq D _ _) %. +%%%%% Subterm Ordering %%%%% +%sort sum-subterm2 {N1} {N2} {N3} %. +%mode sum-subterm2 %in %in %in %in %. +%term _ sum-subterm2 _ _ _ sum/z %. +%term _ %pi (sum-subterm2 _ _ _ (sum/s D)) %<- (sum-subterm2 _ _ _ D) %. +%worlds () (sum-subterm2 _ _ _ _) %. +%total D (sum-subterm2 _ _ _ D) %. +%reduces <= N2 N3 (sum-subterm2 _ N2 N3 _) %. +%sort sum-subterm1 {N1} {N2} {N3} %. +%mode sum-subterm1 %in %in %in %in %. +%term _ %pi (sum-subterm1 _ _ _ D) %<- (sum-commute D D') %<- (sum-subterm2 _ _ _ D') %. +%worlds () (sum-subterm1 _ _ _ _) %. +%total D (sum-subterm1 _ _ _ D) %. +%reduces <= N1 N3 (sum-subterm1 N1 _ N3 _) %. +%sort sum-subterm2' {N1} {N2} {N3} %. +%mode sum-subterm2' %in %in %in %in %. +%term _ sum-subterm2' _ _ _ (sum/s sum/z) %. +%term _ %pi (sum-subterm2' _ _ _ (sum/s D)) %<- (sum-subterm2' _ _ _ D) %. +%worlds () (sum-subterm2' _ _ _ _) %. +%total D (sum-subterm2' _ _ _ D) %. +%reduces < N2 N3 (sum-subterm2' _ N2 N3 _) %. +%sort sum-subterm1' {N1} {N2} {N3} %. +%mode sum-subterm1' %in %in %in %in %. +%term _ %pi (sum-subterm1' _ _ _ D) %<- (sum-commute D D') %<- (sum-subterm2' _ _ _ D') %. +%worlds () (sum-subterm1' _ _ _ _) %. +%total D (sum-subterm1' _ _ _ D) %. +%reduces < N1 N3 (sum-subterm1' N1 _ N3 _) %. +%sort leq-subterm {N1} {N2} %. +%mode leq-subterm %in %in %in %. +%term _ + %pi (leq-subterm _ _ Dleq) + %<- (leq-implies-sum Dleq Dsum) + %<- (sum-subterm1 _ _ _ Dsum) %. +%worlds () (leq-subterm _ _ _) %. +%total {} (leq-subterm _ _ _) %. +%reduces <= N1 N2 (leq-subterm N1 N2 _) %. +%sort lt-subterm {N1} {N2} %. +%mode lt-subterm %in %in %in %. +%term _ + %pi (lt-subterm _ _ Dlt) + %<- (lt-implies-sum Dlt Dsum) + %<- (sum-subterm1' _ _ _ Dsum) %. +%worlds () (lt-subterm _ _ _) %. +%total {} (lt-subterm _ _ _) %. +%reduces < N1 N2 (lt-subterm N1 N2 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/nat.lf b/new-tests/stelf-output/tslf/sing/nat.lf new file mode 100644 index 0000000..c58d682 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/nat.lf @@ -0,0 +1,22 @@ +%%%%% Defs %%%%% +%sort nat %. +%name nat %. +%term 0 nat %. +%term s %pi nat %-> nat %. +%sort false %. +%sort nat-eq %. +%term nat-eq/i nat-eq N N %. +%sort sum %. +%term sum/z sum 0 N N %. +%term sum/s %pi (sum (s N1) N2 (s N3)) %<- (sum N1 N2 N3) %. +%sort leq %. +%term leq/z leq 0 _ %. +%term leq/s %pi (leq (s N1) (s N2)) %<- (leq N1 N2) %. +%sort lt %. +%term lt/z lt 0 (s _) %. +%term lt/s %pi (lt (s N1) (s N2)) %<- (lt N1 N2) %. +%worlds () (lt _ _) %. +%sort max %. +%term max/z* max 0 N N %. +%term max/*z max N 0 N %. +%term max/s %pi (max (s N1) (s N2) (s N3)) %<- (max N1 N2 N3) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/regularity-thm.lf b/new-tests/stelf-output/tslf/sing/regularity-thm.lf new file mode 100644 index 0000000..acc0f46 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/regularity-thm.lf @@ -0,0 +1,223 @@ +%%%%% Simple Substitition %%%%% +%sort ssubst-gen %. +%mode ssubst-gen %in %out %. +%sort sasubst-gen %. +%mode sasubst-gen %in %out %. +%sort stsubst-gen %. +%mode stsubst-gen %in %out %. +%scope stsubst-gen %term at %pi (ssubst-gen ([d] of/at (D d)) ([d] of/at (D' d))) %<- (sasubst-gen D D') %. +%term lam + %pi (ssubst-gen ([d] of/lam (D2 d) (D1 d)) ([d] of/lam (D2' d) (D1' d))) + %<- (stsubst-gen D1 D1') + %<- ({y} {e} ssubst-gen ([d] D2 d y e) ([d] D2' d y e)) %. +%scope stsubst-gen %term pair + %pi (ssubst-gen ([d] of/pair (D4 d) (D3 d) D2 (D1 d)) ([d] of/pair (D4' d) (D3' d) D2 (D1' d))) + %<- (ssubst-gen D1 D1') + %<- (ssubst-gen D3 D3') + %<- ({y} {e} stsubst-gen ([d] D4 d y e) ([d] D4' d y e)) %. +%term sing %pi (ssubst-gen ([d] of/sing (D d)) ([d] of/sing (D' d))) %<- (sasubst-gen D D') %. +%scope stsubst-gen %term closed sasubst-gen ([d] D) ([d] D) %. +%term const + %pi (sasubst-gen ([d] aof/const (D2 d) D1) ([d] aof/const (D2' d) D1)) + %<- (stsubst-gen D2 D2') %. +%scope stsubst-gen %term forall + %pi (sasubst-gen ([d] aof/forall (D1 d)) ([d] aof/forall (D1' d))) + %<- (stsubst-gen D1 D1') %. +%term varsam sasubst-gen ([d] aof/var _ d) ([d] d) %. +%scope stsubst-gen %term varoth + %pi (sasubst-gen ([d] aof/var (D2 d) D1) ([d] aof/var (D2' d) D1)) + %<- (stsubst-gen D2 D2') %. +%term app + %pi (sasubst-gen ([d] aof/app (D4 d) D3 (D2 d) (D1 d)) ([d] aof/app (D4' d) D3 (D2' d) (D1' d))) + %<- (sasubst-gen D1 D1') + %<- (ssubst-gen D2 D2') + %<- (stsubst-gen D4 D4') %. +%scope stsubst-gen %term pi1 + %pi (sasubst-gen ([d] aof/pi1 (D1 d)) ([d] aof/pi1 (D1' d))) + %<- (sasubst-gen D1 D1') %. +%term pi2 + %pi (sasubst-gen ([d] aof/pi2 (D1 d)) ([d] aof/pi2 (D1' d))) + %<- (sasubst-gen D1 D1') %. +%scope stsubst-gen %term t stsubst-gen ([d] wf/t) ([d] wf/t) %. +%term pi + %pi (stsubst-gen ([d] wf/pi (D2 d) (D1 d)) ([d] wf/pi (D2' d) (D1' d))) + %<- (stsubst-gen D1 D1') + %<- ({y} {e} stsubst-gen ([d] D2 d y e) ([d] D2' d y e)) %. +%scope stsubst-gen %term sigma + %pi (stsubst-gen ([d] wf/sigma (D2 d) (D1 d)) ([d] wf/sigma (D2' d) (D1' d))) + %<- (stsubst-gen D1 D1') + %<- ({y} {e} stsubst-gen ([d] D2 d y e) ([d] D2' d y e)) %. +%term sing + %pi (stsubst-gen ([d] wf/sing (D1 d)) ([d] wf/sing (D1' d))) + %<- (sasubst-gen D1 D1') %. +%worlds (var bind) (ssubst-gen _ _) (sasubst-gen _ _) (stsubst-gen _ _) %. +%total (D1 D2 D3) (ssubst-gen D1 _) (sasubst-gen D2 _) (stsubst-gen D3 _) %. +%sort ssubst {x} %. +%mode ssubst %in %in %out %. +%term _ %pi (ssubst D1 D2 (D1' D2)) %<- (ssubst-gen (D1 _) D1') %. +%worlds (var bind) (ssubst _ _ _) %. +%total {} (ssubst _ _ _) %. +%sort ssubst1 {x} %. +%mode ssubst1 %in %in %out %. +%term _ + %pi (ssubst1 Dof Daof (Dof' Daof)) + %<- ({y} {e} ssubst-gen ([d] Dof R d y e) ([d] Dof' d y e)) %. +%worlds (var bind) (ssubst1 _ _ _) %. +%total {} (ssubst1 _ _ _) %. +%sort sasubst {x} %. +%mode sasubst %in %in %out %. +%term _ %pi (sasubst D1 D2 (D1' D2)) %<- (sasubst-gen (D1 _) D1') %. +%worlds (var bind) (sasubst _ _ _) %. +%total {} (sasubst _ _ _) %. +%sort stsubst {x} %. +%mode stsubst %in %in %out %. +%term _ %pi (stsubst D1 D2 (D1' D2)) %<- (stsubst-gen (D1 _) D1') %. +%worlds (var bind) (stsubst _ _ _) %. +%total {} (stsubst _ _ _) %. +%sort stsubst2 {x} %. +%mode stsubst2 %in %in %out %. +%term _ + %pi (stsubst2 Dwf Daof (Dwf' Daof)) + %<- ({y} {e} {z} {f} stsubst-gen ([d] Dwf R d y e z f) ([d] Dwf' d y e z f)) %. +%worlds (var bind) (stsubst2 _ _ _) %. +%total {} (stsubst2 _ _ _) %. +%sort stsubst1 {x} %. +%mode stsubst1 %in %in %out %. +%term _ + %pi (stsubst1 Dwf Daof (Dwf' Daof)) + %<- ({y} {e} stsubst-gen ([d] Dwf R d y e) ([d] Dwf' d y e)) %. +%worlds (var bind) (stsubst1 _ _ _) %. +%total {} (stsubst1 _ _ _) %. +%%%%% Simple Substitution (Explicit Context) %%%%% +%sort ssubst-e {x} %. +%mode ssubst-e %in %in %in %in %out %. +%sort sasubst-e {x} %. +%mode sasubst-e %in %in %in %in %out %. +%sort stsubst-e {x} %. +%mode stsubst-e %in %in %in %in %out %. +%scope stsubst-e %term at + %pi (ssubst-e Dappend Dappend' ([x] [d] ofe/at (D1 x d)) DofR (ofe/at D1')) + %<- (sasubst-e Dappend Dappend' D1 DofR D1') %. +%term lam + %pi (ssubst-e Dappend Dappend' ([x] [d] ofe/lam (D2 x d) (D1 x d)) DofR (ofe/lam D2' D1')) + %<- (stsubst-e Dappend Dappend' D1 DofR D1') + %<- ({y} {e} ssubst-e ([x] append/cons (Dappend x)) (append/cons Dappend') ([x] [d] D2 x d y e) DofR (D2' y e)) %. +%scope stsubst-e %term sing + %pi (ssubst-e Dappend Dappend' ([x] [d] ofe/sing (D1 x d)) DofR (ofe/sing D1')) + %<- (sasubst-e Dappend Dappend' D1 DofR D1') %. +%term pair + %pi (ssubst-e Dappend Dappend' ([x] [d] ofe/pair (D4 x d) (D3 x d) (D2 x) (D1 x d)) (%the (aofe _ R _) DofR) (ofe/pair D4' D3' (D2 R) D1')) + %<- (ssubst-e Dappend Dappend' D1 DofR D1') + %<- (ssubst-e Dappend Dappend' D3 DofR D3') + %<- ({y} {e} stsubst-e ([x] append/cons (Dappend x)) (append/cons Dappend') ([x] [d] D4 x d y e) DofR (D4' y e)) %. +%scope stsubst-e %term closed + %pi (sasubst-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (append G1 (G2 R) G') Dappend') ([x] [d] aofe/closed (%the (ordered (G x)) (Dordered x d)) (%the (aof (Q x) (B x)) (Daof x))) (%the (aofe G1 R A) DofR) (aofe/closed Dordered' (Daof R))) + %<- (scsub-ordered Dappend Dappend' Dordered Dordered') %. +%term forall + %pi (sasubst-e Dappend Dappend' ([x] [d] aofe/forall (D1 x d)) DofR (aofe/forall D1')) + %<- (stsubst-e Dappend Dappend' D1 DofR D1') %. +%scope stsubst-e %term forbad + %pi (sasubst-e _ _ ([x] [d] aofe/var _ (D x d)) _ D'') + %<- ({x} {d} lookup-isvar (D x d) (%the (isvar (forall (B x)) I) (D' x d))) + %<- ({x} {d} isvar-forall-contra (D' x d) Dfalse) + %<- (false-implies-aofe Dfalse D'') %. +%term varsam + %pi (sasubst-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (append G1 (G2 R) G') Dappend') ([x] [d] aofe/var (%the (wfe (G x) (B x)) (Dwf x d)) (%the (lookup (G x) x (B x)) (Dlookup x d))) (%the (aofe G1 R A) DofR) DofR'') + %<- ({x} {d} lookup-binder-fun (Dappend x) (Dlookup x d) (Deq x)) + %<- (aofe-resp ctx-eq/i atom-eq/i (Deq R) DofR (%the (aofe G1 R (B R)) DofR')) + %<- ({x} {d} lookup-context (Dlookup x d) (%the (ordered (G x)) (Dordered x d))) + %<- (scsub-ordered Dappend Dappend' Dordered (%the (ordered G') Dordered')) + %<- (weaken-aofe' Dappend' Dordered' DofR' (%the (aofe G' R (B R)) DofR'')) %. +%scope stsubst-e %term varoth + %pi (sasubst-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (append G1 (G2 R) G') Dappend') ([x] [d] aofe/var (%the (wfe (G x) (B x)) (Dwf x d)) (%the (lookup (G x) Y (B x)) (Dlookup x d))) (%the (aofe G1 R A) DofR) (aofe/var Dwf' Dlookup')) + %<- (scsub-lookup Dappend Dappend' Dlookup (%the (lookup G' Y (B R)) Dlookup')) + %<- (stsubst-e Dappend Dappend' Dwf DofR (%the (wfe G' (B R)) Dwf')) %. +%term app + %pi (sasubst-e Dappend Dappend' ([x] [d] aofe/app (D4 x d) (D3 x) (D2 x d) (D1 x d)) (%the (aofe _ R _) DofR) (aofe/app D4' (D3 R) D2' D1')) + %<- (sasubst-e Dappend Dappend' D1 DofR D1') + %<- (ssubst-e Dappend Dappend' D2 DofR D2') + %<- (stsubst-e Dappend Dappend' D4 DofR D4') %. +%scope stsubst-e %term appbad + %pi (sasubst-e _ _ ([x] [d] aofe/var _ (D x d)) _ D'') + %<- ({x} {d} lookup-isvar (D x d) (%the (isvar (app (R x) (M x)) I) (D' x d))) + %<- ({x} {d} isvar-app-contra (D' x d) Dfalse) + %<- (false-implies-aofe Dfalse D'') %. +%term pi1 + %pi (sasubst-e Dappend Dappend' ([x] [d] aofe/pi1 (D1 x d)) DofR (aofe/pi1 D1')) + %<- (sasubst-e Dappend Dappend' D1 DofR D1') %. +%scope stsubst-e %term pi1bad + %pi (sasubst-e _ _ ([x] [d] aofe/var _ (D x d)) _ D'') + %<- ({x} {d} lookup-isvar (D x d) (%the (isvar (pi1 (R x)) I) (D' x d))) + %<- ({x} {d} isvar-pi1-contra (D' x d) Dfalse) + %<- (false-implies-aofe Dfalse D'') %. +%term pi2 + %pi (sasubst-e Dappend Dappend' ([x] [d] aofe/pi2 (D1 x d)) DofR (aofe/pi2 D1')) + %<- (sasubst-e Dappend Dappend' D1 DofR D1') %. +%scope stsubst-e %term pi2bad + %pi (sasubst-e _ _ ([x] [d] aofe/var _ (D x d)) _ D'') + %<- ({x} {d} lookup-isvar (D x d) (%the (isvar (pi2 (R x)) I) (D' x d))) + %<- ({x} {d} isvar-pi2-contra (D' x d) Dfalse) + %<- (false-implies-aofe Dfalse D'') %. +%term t + %pi (stsubst-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (append G1 (G2 R) G') Dappend') ([x] [d] wfe/t (%the (ordered (G x)) (Dordered x d))) (%the (aofe G1 R A) DofR) (wfe/t Dordered')) + %<- (scsub-ordered Dappend Dappend' Dordered Dordered') %. +%scope stsubst-e %term pi + %pi (stsubst-e Dappend Dappend' ([x] [d] wfe/pi (D2 x d) (D1 x d)) DofR (wfe/pi D2' D1')) + %<- (stsubst-e Dappend Dappend' D1 DofR D1') + %<- ({y} {e} stsubst-e ([x] append/cons (Dappend x)) (append/cons Dappend') ([x] [d] D2 x d y e) DofR (D2' y e)) %. +%term sigma + %pi (stsubst-e Dappend Dappend' ([x] [d] wfe/sigma (D2 x d) (D1 x d)) DofR (wfe/sigma D2' D1')) + %<- (stsubst-e Dappend Dappend' D1 DofR D1') + %<- ({y} {e} stsubst-e ([x] append/cons (Dappend x)) (append/cons Dappend') ([x] [d] D2 x d y e) DofR (D2' y e)) %. +%scope stsubst-e %term at + %pi (stsubst-e Dappend Dappend' ([x] [d] wfe/sing (D1 x d)) DofR (wfe/sing D1')) + %<- (sasubst-e Dappend Dappend' D1 DofR D1') %. +%worlds (var bind ovar) (ssubst-e _ _ _ _ _) (sasubst-e _ _ _ _ _) (stsubst-e _ _ _ _ _) %. +%total (D1 D2 D3) (ssubst-e _ _ D1 _ _) (sasubst-e _ _ D2 _ _) (stsubst-e _ _ D3 _ _) %. +%%%%% Regularity (Implicit Context Version) %%%%% +%sort aof-reg %. +%mode aof-reg %in %out %. +%scope aof-reg %term const aof-reg (aof/const D _) D %. +%term forall aof-reg (aof/forall D) (wf/pi ([_] [_] wf/t) (wf/pi ([_] [_] wf/t) D)) %. +%scope aof-reg %term var aof-reg (aof/var D _) D %. +%term app aof-reg (aof/app D _ _ _) D %. +%scope aof-reg %term pi1 %pi (aof-reg (aof/pi1 Daof) D) %<- (aof-reg Daof (wf/sigma _ D)) %. +%term pi2 + %pi (aof-reg (aof/pi2 Daof) D') + %<- (aof-reg Daof (wf/sigma D _)) + %<- (stsubst D (aof/pi1 Daof) D') %. +%worlds (var ovar bind obind) (aof-reg _ _) %. +%total D (aof-reg D _) %. +%sort of-reg %. +%mode of-reg %in %out %. +%scope of-reg %term at of-reg (of/at _) wf/t %. +%term lam + %pi (of-reg (of/lam Dof Dwf1) (wf/pi Dwf2 Dwf1)) + %<- ({x} {d} of-reg (Dof x d) (Dwf2 x d)) %. +%scope of-reg %term pair %pi (of-reg (of/pair Dwf2 _ _ Dof) (wf/sigma Dwf2 Dwf1)) %<- (of-reg Dof Dwf1) %. +%term sing of-reg (of/sing Daof) (wf/sing Daof) %. +%worlds (var ovar bind obind) (of-reg _ _) %. +%total D (of-reg D _) %. +%%%%% Regularity (Explicit Context) %%%%% +%% ofe-reg appears in explicit-lemmas.thm +%sort aofe-reg %. +%mode aofe-reg %in %out %. +%scope aofe-reg %term closed + %pi (aofe-reg (aofe/closed Dordered Daof) Dwfe') + %<- (aof-reg Daof Dwf) + %<- (wf-to-wfe Dwf Dwfe) + %<- (append-all G (%the (append nil G G) Dappend)) + %<- (weaken-wfe' Dappend Dordered Dwfe Dwfe') %. +%term forall + %pi (aofe-reg (aofe/forall D) (wfe/pi ([x] [d] wfe/t (ordered/cons (Dbound x d))) (wfe/pi ([x] [d] wfe/t (ordered/cons (Dbound x d))) D))) + %<- (wfe-context D Dord) + %<- (extend-context Dord Dbound) %. +%scope aofe-reg %term var aofe-reg (aofe/var D _) D %. +%term app aofe-reg (aofe/app D _ _ _) D %. +%scope aofe-reg %term pi1 %pi (aofe-reg (aofe/pi1 D) D') %<- (aofe-reg D (wfe/sigma _ D')) %. +%term pi2 + %pi (aofe-reg (aofe/pi2 Daof) D) + %<- (aofe-reg Daof (wfe/sigma Dwf _)) + %<- (stsubst-e ([_] append/nil) append/nil Dwf (aofe/pi1 Daof) D) %. +%worlds (bind ovar var) (aofe-reg _ _) %. +%total D (aofe-reg D _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/sound-thm.lf b/new-tests/stelf-output/tslf/sing/sound-thm.lf new file mode 100644 index 0000000..819ef9b --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/sound-thm.lf @@ -0,0 +1,388 @@ +%%%%% Conversion of Subtypes %%%%% +%sort convert-subtype %. +%mode convert-subtype %in %in %in %out %out %out %. +%scope convert-subtype %term t convert-subtype subtype/t tconvert/t tconvert/t subtp/t ([x] [d] [ex] [xt] convert/at (aconvert/var xt tconvert/t wf/t d)) ([ex] [ed] equiv/reflex (eof/var ewf/t ed)) %. +%term pi + %pi (convert-subtype (subtype/pi (%the ({x} subtype (Bx x) (D x) ([y] N x y)) DsubtypeBD) (%the ({x} tsub B (M x) (Bx x)) DsubBx) (%the (subtype C A ([x] M x)) DsubtypeCA)) (tconvert/pi (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))) DconvertB) (%the (tconvert A EA) DconvertA)) (tconvert/pi (%the ({x} %pi (vof x C) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (D x) (ED ex)))) DconvertD) (%the (tconvert C EC) DconvertC)) (subtp/pi DewfB ([ex] [ed] subtp/trans (subtp/trans (DsubtpBD ex ed) (subtp/reflex (Dtequiv ex ed))) (subtp/reflex (DequivBM ex ed))) DsubtpCA) ([z] [f] [ez] [zt] convert/lam ([x] [d] [ex] [xt] Dconvert z f ez zt x d ex xt) DconvertC) ([ez] [ef] equiv/extpi ([ex] [ed] equiv/trans (equiv/symm (equiv/beta (eof/var DewfC ed) ([ex] [ed] Deof ez ef ex ed))) (equiv/trans (DequivN' ez ef ex ed) (equiv/subsume (DsubtpBD ex ed) (equiv/subsume (subtp/reflex (Dtequiv ex ed)) (equiv/symm (equiv/app (equiv/symm (DequivM ex ed)) (equiv/reflex (eof/var (ewf/pi DewfB DewfA) ef)))))))) (eof/lam ([ex] [ed] Deof ez ef ex ed) DewfC) (eof/subsume (subtp/pi DewfB ([ex] [ed] subtp/reflex (tequiv/reflex (DewfB' ex ed))) DsubtpCA) (eof/var (ewf/pi DewfB DewfA) ef)))) + %<- (convert-subtype DsubtypeCA DconvertC DconvertA (%the (subtp EC EA) DsubtpCA) (%the ({x} %pi (vof x C) %-> ({ex} %pi (vtrans ex x) %-> (convert (M x) A (EM ex)))) DconvertM) (%the ({ex} %pi (evof ex EC) %-> (equiv ex (EM ex) EA)) DequivM)) + %<- ({x} {d vof x C} {ex} {xt vtrans ex x} {ed evof ex EC} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertC ed) + %-> (convert-tsub DconvertB (DconvertM x d ex xt) (DsubBx x) (%the (tconvert (Bx x) (EBx ex)) (DconvertBx x d ex xt)) (%the (tequiv (EB (EM ex)) (EBx ex)) (Dtequiv ex ed)))) + %<- ({x} {d vof x C} {ex} {xt vtrans ex x} {ed evof ex EC} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertC ed) + %-> (convert-subtype (DsubtypeBD x) (DconvertBx x d ex xt) (DconvertD x d ex xt) (%the (subtp (EBx ex) (ED ex)) (DsubtpBD ex ed)) (%the ({y} + %pi (vof y (Bx x)) + %-> ({ey} %pi (vtrans ey y) %-> (convert (N x y) (D x) (EN ex ey)))) (DconvertN x d ex xt)) (%the ({ey} %pi (evof ey (EBx ex)) %-> (equiv ey (EN ex ey) (ED ex))) (DequivN ex ed)))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (tconvert-reg (DconvertB x d ex xt) (%the (ewf (EB ex)) (DewfB ex ed)))) + %<- (tconvert-reg-il DconvertA (%the (wf A) DwfA)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} tconvert-reg-il (DconvertB x d ex xt) (%the (wf (B x)) (DwfB x d))) + %<- ({z} {f vof z (pi A B)} {ez} {zt vtrans ez z} {ef evof ez (epi EA EB)} {fv variable ez} + %pi (vtrans-variable zt fv) + %-> (vsound f zt (tconvert/pi DconvertB DconvertA) ef) + %-> ({x} {d vof x C} {ex} {xt vtrans ex x} {ed evof ex EC} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertC ed) + %-> (convert-ssub (DconvertN x d ex xt) (%the (aconvert (app z (M x)) (Bx x) (eapp ez (EM ex))) (aconvert/app (DsubBx x) (DconvertM x d ex xt) (aconvert/var zt (tconvert/pi DconvertB DconvertA) (wf/pi DwfB DwfA) f))) (%the (convert (N x (app z (M x))) (D x) (EN ex (eapp ez (EM ex)))) (Dconvert z f ez zt x d ex xt))))) + %<- ({z} {f vof z (pi A B)} {ez} {zt vtrans ez z} {ef evof ez (epi EA EB)} {fv variable ez} + %pi (vtrans-variable zt fv) + %-> (vsound f zt (tconvert/pi DconvertB DconvertA) ef) + %-> ({x} {d vof x C} {ex} {xt vtrans ex x} {ed evof ex EC} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertC ed) + %-> (convert-reg (Dconvert z f ez zt x d ex xt) (DconvertD x d ex xt) (%the (eof (EN ex (eapp ez (EM ex))) (ED ex)) (Deof ez ef ex ed))))) + %<- ({x} {d vof x C} {ex} {xt vtrans ex x} {ed evof ex EC} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertC ed) + %-> (convert-reg (DconvertM x d ex xt) DconvertA (%the (eof (EM ex) EA) (DeofM ex ed)))) + %<- (subtp-reg DsubtpCA (%the (ewf EC) DewfC) (%the (ewf EA) DewfA)) + %<- ({ez} {ef evof ez (epi EA EB)} {ex} {ed evof ex EC} esubst-equiv (DequivN ex ed) (%the (eof (eapp ez (EM ex)) (EBx ex)) (eof/subsume (subtp/reflex (Dtequiv ex ed)) (eof/app (DeofM ex ed) (eof/var (ewf/pi DewfB DewfA) ef)))) (%the (equiv (eapp ez (EM ex)) (EN ex (eapp ez (EM ex))) (ED ex)) (DequivN' ez ef ex ed))) + %<- ({ex} {ed evof ex EC} esubst-wf DewfB (eof/subsume DsubtpCA (eof/var DewfC ed)) (%the (ewf (EB ex)) (DewfB' ex ed))) + %<- ({ex} {ed} tfunctionality DewfB (DequivM ex ed) (%the (tequiv (EB ex) (EB (EM ex))) (DequivBM ex ed))) %. +%scope convert-subtype %term sigma + %pi (convert-subtype (subtype/sigma (%the ({x} subtype (B x) (Dx x) ([y] N x y)) DsubtypeBD) (%the ({x} tsub D (M x) (Dx x)) DsubDx) (%the (subtype A C ([x] M x)) DsubtypeAC)) (tconvert/sigma (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))) DconvertB) (%the (tconvert A EA) DconvertA)) (tconvert/sigma (%the ({x} %pi (vof x C) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (D x) (ED ex)))) DconvertD) (%the (tconvert C EC) DconvertC)) (subtp/sigma DewfD ([ex] [ed] subtp/trans (subtp/trans (subtp/reflex (tequiv/symm (DequivDM ex ed))) (subtp/reflex (tequiv/symm (Dtequiv ex ed)))) (DsubtpBD ex ed)) DsubtpAC) ([z] [f] [zt] [ef] convert/pair DwfD (Dconvert2' z f zt ef) (DsubDx (pi1 z)) (Dconvert1 z f zt ef)) ([ez] [ef] equiv/extsigma DewfD (equiv/trans (equiv/symm (equiv/beta2 (eof/subsume (subtp/trans (subtp/reflex (DequivDMx ez ef)) (subtp/reflex (tequiv/symm (Dtequiv' ez ef)))) (DeofNxy ez ef)) (DeofMx ez ef))) (equiv/subsume (subtp/trans (subtp/reflex (DequivDMx ez ef)) (subtp/reflex (tequiv/symm (Dtequiv' ez ef)))) (DequivNxy ez ef))) (equiv/trans (equiv/symm (equiv/beta1 (DeofNxy ez ef) (DeofMx ez ef))) (DequivMx ez ef)))) + %<- (convert-subtype DsubtypeAC DconvertA DconvertC (%the (subtp EA EC) DsubtpAC) (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (convert (M x) C (EM ex)))) DconvertM) (%the ({ex} %pi (evof ex EA) %-> (equiv ex (EM ex) EC)) DequivM)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (convert-tsub DconvertD (DconvertM x d ex xt) (DsubDx x) (%the (tconvert (Dx x) (EDx ex)) (DconvertDx x d ex xt)) (%the (tequiv (ED (EM ex)) (EDx ex)) (Dtequiv ex ed)))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (convert-subtype (DsubtypeBD x) (DconvertB x d ex xt) (DconvertDx x d ex xt) (%the (subtp (EB ex) (EDx ex)) (DsubtpBD ex ed)) (%the ({y} + %pi (vof y (B x)) + %-> ({ey} %pi (vtrans ey y) %-> (convert (N x y) (Dx x) (EN ex ey)))) (DconvertN x d ex xt)) (%the ({ey} %pi (evof ey (EB ex)) %-> (equiv ey (EN ex ey) (EDx ex))) (DequivN ex ed)))) + %<- ({x} {d vof x C} {ex} {xt vtrans ex x} {ed evof ex EC} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertC ed) + %-> (tconvert-reg (DconvertD x d ex xt) (%the (ewf (ED ex)) (DewfD ex ed)))) + %<- (tconvert-reg-il DconvertA (%the (wf A) DwfA)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} tconvert-reg-il (DconvertB x d ex xt) (%the (wf (B x)) (DwfB x d))) + %<- ({z} {f vof z (sigma A B)} {ez} {zt vtrans ez z} {ef evof ez (esigma EA EB)} {fv variable ez} + %pi (vtrans-variable zt fv) + %-> (vsound f zt (tconvert/sigma DconvertB DconvertA) ef) + %-> (convert-ssub DconvertM (%the (aconvert (pi1 z) A (epi1 ez)) (aconvert/pi1 (aconvert/var zt (tconvert/sigma DconvertB DconvertA) (wf/sigma DwfB DwfA) f))) (%the (convert (M (pi1 z)) C (EM (epi1 ez))) (Dconvert1 z f ez zt)))) + %<- ({z} {f vof z (sigma A B)} {ez} {zt vtrans ez z} {ef evof ez (esigma EA EB)} {fv variable ez} + %pi (vtrans-variable zt fv) + %-> (vsound f zt (tconvert/sigma DconvertB DconvertA) ef) + %-> (convert-stsub DconvertB (aconvert/pi1 (aconvert/var zt (tconvert/sigma DconvertB DconvertA) (wf/sigma DwfB DwfA) f)) (%the (tconvert (B (pi1 z)) (EB (epi1 ez))) (DconvertBx z f ez zt)))) + %<- ({z} {f vof z (sigma A B)} {ez} {zt vtrans ez z} {ef evof ez (esigma EA EB)} {fv variable ez} + %pi (vtrans-variable zt fv) + %-> (vsound f zt (tconvert/sigma DconvertB DconvertA) ef) + %-> (convert-ssub1 DconvertN (%the (aconvert (pi1 z) A (epi1 ez)) (aconvert/pi1 (aconvert/var zt (tconvert/sigma DconvertB DconvertA) (wf/sigma DwfB DwfA) f))) DwfB (%the ({y} + %pi (vof y (B (pi1 z))) + %-> ({ey} %pi (vtrans ey y) %-> (convert (N (pi1 z) y) (Dx (pi1 z)) (EN (epi1 ez) ey)))) (Dconvert2 z f ez zt)))) + %<- ({z} {f vof z (sigma A B)} {ez} {zt vtrans ez z} {ef evof ez (esigma EA EB)} {fv variable ez} + %pi (vtrans-variable zt fv) + %-> (vsound f zt (tconvert/sigma DconvertB DconvertA) ef) + %-> (convert-ssub (Dconvert2 z f ez zt) (%the (aconvert (pi2 z) (B (pi1 z)) (epi2 ez)) (aconvert/pi2 (aconvert/var zt (tconvert/sigma DconvertB DconvertA) (wf/sigma DwfB DwfA) f))) (%the (convert (N (pi1 z) (pi2 z)) (Dx (pi1 z)) (EN (epi1 ez) (epi2 ez))) (Dconvert2' z f ez zt)))) + %<- ({x} {d vof x C} {ex} {xt vtrans ex x} tconvert-reg-il (DconvertD x d ex xt) (%the (wf (D x)) (DwfD x d))) + %<- ({z} {f vof z (sigma A B)} {ez} {zt vtrans ez z} {ef evof ez (esigma EA EB)} {fv variable ez} + %pi (vtrans-variable zt fv) + %-> (vsound f zt (tconvert/sigma DconvertB DconvertA) ef) + %-> (convert-reg (Dconvert1 z f ez zt) DconvertC (%the (eof (EM (epi1 ez)) EC) (DeofMx ez ef)))) + %<- ({z} {f vof z (sigma A B)} {ez} {zt vtrans ez z} {ef evof ez (esigma EA EB)} {fv variable ez} + %pi (vtrans-variable zt fv) + %-> (vsound f zt (tconvert/sigma DconvertB DconvertA) ef) + %-> (convert-stsub DconvertDx (aconvert/pi1 (aconvert/var zt (tconvert/sigma DconvertB DconvertA) (wf/sigma DwfB DwfA) f)) (%the (tconvert (Dx (pi1 z)) (EDx (epi1 ez))) (DconvertDxx z f ez zt)))) + %<- ({z} {f vof z (sigma A B)} {ez} {zt vtrans ez z} {ef evof ez (esigma EA EB)} {fv variable ez} + %pi (vtrans-variable zt fv) + %-> (vsound f zt (tconvert/sigma DconvertB DconvertA) ef) + %-> (convert-reg (Dconvert2' z f ez zt) (DconvertDxx z f ez zt) (%the (eof (EN (epi1 ez) (epi2 ez)) (EDx (epi1 ez))) (DeofNxy ez ef)))) + %<- (tconvert-reg DconvertA (%the (ewf EA) DewfA)) + %<- ({x} {d} {ex} {xt} {ed} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (tconvert-reg (DconvertB x d ex xt) (%the (ewf (EB ex)) (DewfB ex ed)))) + %<- ({ez} {ef evof ez (esigma EA EB)} esubst-equiv DequivM (eof/pi1 (eof/var (ewf/sigma DewfB DewfA) ef)) (%the (equiv (epi1 ez) (EM (epi1 ez)) EC) (DequivMx ez ef))) + %<- ({ez} {ef evof ez (esigma EA EB)} esubst-tequiv Dtequiv (eof/pi1 (eof/var (ewf/sigma DewfB DewfA) ef)) (%the (tequiv (ED (EM (epi1 ez))) (EDx (epi1 ez))) (Dtequiv' ez ef))) + %<- ({ez} {ef evof ez (esigma EA EB)} {ey} {ee evof ey (EB (epi1 ez))} esubst-equiv-gen ([ed] DequivN (epi1 ez) ed ey ee) (%the (%pi (eof (epi1 ez) EA) %-> (equiv ey (EN (epi1 ez) ey) (EDx (epi1 ez)))) (DequivNx ez ef ey ee))) + %<- ({ez} {ef evof ez (esigma EA EB)} esubst-equiv ([ey] [ee] DequivNx ez ef ey ee (eof/pi1 (eof/var (ewf/sigma DewfB DewfA) ef))) (eof/pi2 (eof/var (ewf/sigma DewfB DewfA) ef)) (%the (equiv (epi2 ez) (EN (epi1 ez) (epi2 ez)) (EDx (epi1 ez))) (DequivNxy ez ef))) + %<- ({ex} {ed} tfunctionality DewfD (DequivM ex ed) (%the (tequiv (ED ex) (ED (EM ex))) (DequivDM ex ed))) + %<- ({ez} {ef} tfunctionality DewfD (equiv/symm (DequivMx ez ef)) (%the (tequiv (ED (EM (epi1 ez))) (ED (epi1 ez))) (DequivDMx ez ef))) %. +%term sing_t + %pi (convert-subtype subtype/sing_t (tconvert/sing (%the (aconvert R t EM) Dconvert)) tconvert/t (subtp/sing_t Deof) ([x] [d] [ex] [xt] convert/at Dconvert) ([ex] [ed] equiv/singelim (eof/var (ewf/sing Deof) ed))) + %<- (aconvert-reg' Dconvert tconvert/t (%the (eof EM et) Deof)) %. +%scope convert-subtype %term sing + %pi (convert-subtype subtype/sing (tconvert/sing (%the (aconvert R t EM) Dconvert)) (tconvert/sing (%the (aconvert R t EM') Dconvert')) Dsubtp ([x] [d] [ex] [xt] convert/sing Dconvert) Dequiv) + %<- (aconvert-reg' Dconvert tconvert/t (%the (eof EM et) Deof)) + %<- (aconvert-fun Dconvert Dconvert' _ (%the (eterm-eq EM EM') DeqM)) + %<- (etp-resp-eterm esing DeqM (%the (etp-eq (esing EM) (esing EM')) Deq)) + %<- (subtp-resp etp-eq/i Deq (subtp/sing (equiv/reflex Deof)) (%the (subtp (esing EM) (esing EM')) Dsubtp)) + %<- ({ex} {ed evof ex (esing EM)} equiv-resp eterm-eq/i eterm-eq/i Deq (%the (equiv ex EM (esing EM)) (equiv/symm (equiv/sing (equiv/symm (equiv/singelim (eof/var (ewf/sing Deof) ed)))))) (%the (equiv ex EM (esing EM')) (Dequiv ex ed))) %. +%worlds (sbind) (convert-subtype _ _ _ _ _ _) %. +%total D (convert-subtype D _ _ _ _ _) %. +%sort convert-subtype' %. +%mode convert-subtype' %in %in %in %out %. +%term _ %pi (convert-subtype' D1 D2 D3 D) %<- (convert-subtype D1 D2 D3 D _ _) %. +%worlds (sbind) (convert-subtype' _ _ _ _) %. +%total {} (convert-subtype' _ _ _ _) %. +%sort convert-subtype-equiv %. +%mode convert-subtype-equiv %in %in %in %in %in %out %. +%term _ + %pi (convert-subtype-equiv (%the (subtype A B ([_] N)) Dsubtype) (%the (tconvert A EA) DconvertA) (%the (tconvert B EB) DconvertB) (%the (convert N B EN) DconvertN) (%the (eof EM EA) DeofM) DequivM) + %<- (convert-subtype Dsubtype DconvertA DconvertB _ (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (convert N B (EN' ex)))) DconvertN') (%the ({ex} %pi (evof ex EA) %-> (equiv ex (EN' ex) EB)) Dequiv)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} convert-fun (DconvertN' x d ex xt) DconvertN (%the (eterm-eq (EN' ex) EN) (Deq ex))) + %<- ({ex} {ed evof ex EA} equiv-resp eterm-eq/i (Deq ex) etp-eq/i (Dequiv ex ed) (%the (equiv ex EN EB) (Dequiv' ex ed))) + %<- (esubst-equiv Dequiv' DeofM (%the (equiv EM EN EB) DequivM)) %. +%worlds (sbind) (convert-subtype-equiv _ _ _ _ _ _) %. +%total {} (convert-subtype-equiv _ _ _ _ _ _) %. +%%%%% Soundness of Eta-expansion %%%%% +%sort expand-convert %. +%mode expand-convert %in %in %in %out %out %. +%scope expand-convert %term t + %pi (expand-convert expand/t (%the (aconvert R t EM) Dconvert) (convert/at (%the (aconvert R t EM') Dconvert')) tconvert/t Dequiv) + %<- (aconvert-fun Dconvert Dconvert' _ (%the (eterm-eq EM EM') Deq)) + %<- (aconvert-reg Dconvert tconvert/t (%the (eof EM et) Deof)) + %<- (equiv-resp eterm-eq/i Deq etp-eq/i (equiv/reflex Deof) (%the (equiv EM EM' et) Dequiv)) %. +%term pi + %pi (expand-convert (%the (expand R (pi A B) (lam M)) (expand/pi (%the ({x} expand (app R (X x)) (B x) (M x)) Dexpand2) (%the ({x} expand x A (X x)) Dexpand1))) (%the (aconvert R (pi A B) EO) Daconvert) (convert/lam (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (convert (M x) (B x) (EM ex)))) DconvertM) (%the (tconvert A EA) DconvertA)) (tconvert/pi DconvertB DconvertA) (equiv/extpi ([ex] [ed evof ex EA] equiv/trans (equiv/trans (equiv/symm (equiv/beta (eof/var DewfA ed) DeofM)) (Dequiv2 ex ed)) (equiv/app (Dequiv1' ex ed) (equiv/reflex DeofO))) (eof/lam DeofM DewfA) DeofO)) + %<- (aconvert-reg-il Daconvert (%the (aof R (pi A B)) DofR)) + %<- (aof-reg DofR (wf/pi (%the ({x} %pi (vof x A) %-> (wf (B x))) DwfB) (%the (wf A) DwfA))) + %<- ({x} {d vof x A} expand-reg (aof/var DwfA d) (Dexpand1 x) (%the (of (X x) A) (DofX x d))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (can-convert (DofX x d) (%the (convert (X x) A (EN ex)) (DconvertN x d ex xt)))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (expand-convert (Dexpand1 x) (aconvert/var xt DconvertA DwfA d) (DconvertN x d ex xt) (%the (tconvert A (EA' ex)) (DconvertA' x d ex xt)) (%the (equiv ex (EN ex) (EA' ex)) (Dequiv1 ex ed)))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} tconvert-fun (DconvertA' x d ex xt) DconvertA (%the (etp-eq (EA' ex) EA) (DeqEA ex))) + %<- ({ex} {ed} equiv-resp eterm-eq/i eterm-eq/i (DeqEA ex) (Dequiv1 ex ed) (%the (equiv ex (EN ex) EA) (Dequiv1' ex ed))) + %<- ({x} {d vof x A} can-tsub DwfB (DofX x d) (%the (tsub B (X x) (B' x)) (DsubB' x))) + %<- ({x} {d vof x A} tsub-expand (aof/var DwfA d) DwfB (Dexpand1 x) (DsubB' x) (%the (tp-eq (B' x) (B x)) (DeqB x))) + %<- ({x} tsub-resp ([_] tp-eq/i) term-eq/i (DeqB x) (DsubB' x) (%the (tsub B (X x) (B x)) (DsubB x))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (expand-convert (Dexpand2 x) (%the (aconvert (app R (X x)) (B x) (eapp EO (EN ex))) (aconvert/app (DsubB x) (DconvertN x d ex xt) Daconvert)) (DconvertM x d ex xt) (%the (tconvert (B x) (EB ex)) (DconvertB x d ex xt)) (%the (equiv (eapp EO (EN ex)) (EM ex) (EB ex)) (Dequiv2 ex ed)))) + %<- (tconvert-reg DconvertA (%the (ewf EA) DewfA)) + %<- (aconvert-reg' Daconvert (tconvert/pi DconvertB DconvertA) (%the (eof EO (epi EA EB)) DeofO)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (convert-reg (DconvertM x d ex xt) (DconvertB x d ex xt) (%the (eof (EM ex) (EB ex)) (DeofM ex ed)))) %. +%scope expand-convert %term sigma + %pi (expand-convert (%the (expand R (sigma A B) (pair M N)) (expand/sigma (%the (expand (pi2 R) (B (pi1 R)) N) Dexpand2) (%the (expand (pi1 R) A M) Dexpand1))) (%the (aconvert R (sigma A B) EO) Daconvert) (convert/pair _ (%the (convert N Bx EN) DconvertN) (%the (tsub B M Bx) DsubBx) (%the (convert M A EM) DconvertM)) (tconvert/sigma DconvertB DconvertA) (equiv/extsigma DewfB (equiv/trans (equiv/symm (equiv/beta2 DeofN DeofM)) Dequiv2') (equiv/trans (equiv/symm (equiv/beta1 DeofN DeofM)) Dequiv1))) + %<- (expand-convert Dexpand1 (aconvert/pi1 Daconvert) DconvertM (%the (tconvert A EA) DconvertA) (%the (equiv (epi1 EO) EM EA) Dequiv1)) + %<- (aconvert-reg-il Daconvert (%the (aof R (sigma A B)) DofR)) + %<- (aof-reg DofR (wf/sigma (%the ({x} %pi (vof x A) %-> (wf (B x))) DwfB) (%the (wf A) DwfA))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (can-tconvert (DwfB x d) (%the (tconvert (B x) (EB ex)) (DconvertB x d ex xt)))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (tconvert-reg (DconvertB x d ex xt) (%the (ewf (EB ex)) (DewfB ex ed)))) + %<- (tsub-expand (aof/pi1 DofR) DwfB Dexpand1 DsubBx (%the (tp-eq Bx (B (pi1 R))) DeqBx)) + %<- (convert-stsub DconvertB (aconvert/pi1 Daconvert) (%the (tconvert (B (pi1 R)) (EB (epi1 EO))) DconvertBx)) + %<- (convert-resp term-eq/i DeqBx eterm-eq/i DconvertN (%the (convert N (B (pi1 R)) EN) DconvertN')) + %<- (expand-convert Dexpand2 (aconvert/pi2 Daconvert) DconvertN' (%the (tconvert (B (pi1 R)) EBx) DconvertBx') (%the (equiv (epi2 EO) EN EBx) Dequiv2)) + %<- (tconvert-fun DconvertBx' DconvertBx (%the (etp-eq EBx (EB (epi1 EO))) DeqEBx)) + %<- (equiv-resp eterm-eq/i eterm-eq/i DeqEBx Dequiv2 (%the (equiv (epi2 EO) EN (EB (epi1 EO))) Dequiv2')) + %<- (convert-reg DconvertM DconvertA (%the (eof EM EA) DeofM)) + %<- (convert-reg DconvertN' DconvertBx (%the (eof EN (EB (epi1 EO))) DeofN)) %. +%term sing + %pi (expand-convert expand/sing (%the (aconvert R (sing R') EM) Dconvert) (convert/sing (%the (aconvert R' t EM') Dconvert')) (tconvert/sing Dconvert') (equiv/symm (equiv/sing (equiv/symm (equiv/singelim Deof))))) + %<- (aconvert-reg' Dconvert (tconvert/sing Dconvert') (%the (eof EM (esing EM')) Deof)) %. +%worlds (sbind) (expand-convert _ _ _ _ _) %. +%total D (expand-convert D _ _ _ _) %. +%sort self-convert %. +%mode self-convert %in %in %in %in %in %out %out %. +%scope self-convert %term t self-convert self/t (convert/at (%the (aconvert R t EM) Dconvert)) tconvert/t (%the (eof EN et) DeofN) (%the (equiv EN EM et) Dequiv) (tconvert/sing Dconvert) (eof/subsume (subtp/sing Dequiv) (eof/sing DeofN)) %. +%term pi + %pi (self-convert (self/pi (%the ({x} self (M x) (B x) (Bs x)) Dself)) (convert/lam (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (convert (M x) (B x) (EM ex)))) DconvertM) (%the (tconvert A EA) DconvertA)) (tconvert/pi (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))) DconvertB) (%the (tconvert A EA') DconvertA')) (%the (eof EN (epi EA' EB)) DeofN) (%the (equiv EN (elam EA EM) (epi EA' EB)) DequivN) (tconvert/pi DconvertBs DconvertA) (eof/extpi Deof DeofN')) + %<- (tconvert-fun DconvertA' DconvertA (%the (etp-eq EA' EA) DeqA)) + %<- (etp-resp-etp ([a] epi a EB) DeqA (%the (etp-eq (epi EA' EB) (epi EA EB)) Deq)) + %<- (eof-resp eterm-eq/i Deq DeofN (%the (eof EN (epi EA EB)) DeofN')) + %<- (equiv-resp eterm-eq/i eterm-eq/i Deq DequivN (%the (equiv EN (elam EA EM) (epi EA EB)) DequivN')) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (convert-reg (DconvertM x d ex xt) (DconvertB x d ex xt) (%the (eof (EM ex) (EB ex)) (DeofM ex ed)))) + %<- (tconvert-reg DconvertA (%the (ewf EA) DewfA)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (self-convert (Dself x) (DconvertM x d ex xt) (DconvertB x d ex xt) (%the (eof (eapp EN ex) (EB ex)) (eof/app (eof/var DewfA ed) DeofN')) (%the (equiv (eapp EN ex) (EM ex) (EB ex)) (equiv/trans (equiv/beta (eof/var DewfA ed) DeofM) (equiv/app (equiv/reflex (eof/var DewfA ed)) DequivN'))) (%the (tconvert (Bs x) (EBs ex)) (DconvertBs x d ex xt)) (%the (eof (eapp EN ex) (EBs ex)) (Deof ex ed)))) %. +%scope self-convert %term sigma + %pi (self-convert (self/sigma (%the (self N Bx Bs) Dself2) (%the (tsub B M Bx) DsubBx) (%the (self M A As) Dself1)) (convert/pair _ (%the (convert N Bx' EN) DconvertN) (%the (tsub B M Bx') DsubBx') (%the (convert M A EM) DconvertM)) (tconvert/sigma (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))) DconvertB) (%the (tconvert A EA) DconvertA)) (%the (eof EO (esigma EA EB)) DeofO) (%the (equiv EO (epair EM EN) (esigma EA EB)) DequivO) (tconvert/sigma ([_] [_] [_] [_] DconvertBs) DconvertAs) (eof/extsigma ([_] [_] DewfBs) Deof2 Deof1)) + %<- (convert-reg DconvertM DconvertA (%the (eof EM EA) DeofM)) + %<- (tsub-fun DsubBx' DsubBx (%the (tp-eq Bx' Bx) DeqBx)) + %<- (convert-resp term-eq/i DeqBx eterm-eq/i DconvertN (%the (convert N Bx EN) DconvertN')) + %<- (convert-tsub DconvertB DconvertM DsubBx (%the (tconvert Bx EBx) DconvertBx) (%the (tequiv (EB EM) EBx) Dtequiv)) + %<- (convert-reg DconvertN' DconvertBx (%the (eof EN EBx) DeofN)) + %<- (self-convert Dself1 DconvertM DconvertA (%the (eof (epi1 EO) EA) (eof/pi1 DeofO)) (%the (equiv (epi1 EO) EM EA) (equiv/trans (equiv/beta1 DeofN DeofM) (equiv/pi1 DequivO))) (%the (tconvert As EAs) DconvertAs) (%the (eof (epi1 EO) EAs) Deof1)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (tconvert-reg (DconvertB x d ex xt) (%the (ewf (EB ex)) (DewfB ex ed)))) + %<- (tfunctionality DewfB (equiv/trans (equiv/beta1 DeofN DeofM) (equiv/pi1 DequivO)) (%the (tequiv (EB (epi1 EO)) (EB EM)) DequivBx)) + %<- (self-convert Dself2 DconvertN' DconvertBx (%the (eof (epi2 EO) EBx) (eof/subsume (subtp/reflex Dtequiv) (eof/equiv DequivBx (eof/pi2 DeofO)))) (%the (equiv (epi2 EO) EN EBx) (equiv/subsume (subtp/reflex Dtequiv) (equiv/trans (equiv/beta2 (eof/subsume (subtp/reflex (tequiv/symm Dtequiv)) DeofN) DeofM) (equiv/equiv DequivBx (equiv/pi2 DequivO))))) (%the (tconvert Bs EBs) DconvertBs) (%the (eof (epi2 EO) EBs) Deof2)) + %<- (tconvert-reg DconvertBs (%the (ewf EBs) DewfBs)) %. +%term sing + %pi (self-convert self/sing (convert/sing (%the (aconvert R t EM) Dconvert)) (tconvert/sing (%the (aconvert R t EM') Dconvert')) (%the (eof EN (esing EM')) Deof) (%the (equiv EN EM (esing EM')) Dequiv) (tconvert/sing Dconvert) Deof') + %<- (aconvert-fun Dconvert' Dconvert _ (%the (eterm-eq EM' EM) Deq)) + %<- (etp-resp-eterm esing Deq (%the (etp-eq (esing EM') (esing EM)) Deq')) + %<- (eof-resp eterm-eq/i Deq' Deof (%the (eof EN (esing EM)) Deof')) %. +%worlds (sbind) (self-convert _ _ _ _ _ _ _) %. +%total D (self-convert D _ _ _ _ _ _) %. +%%%%% Soundness %%%%% +%sort sound-trans %. +%mode sound-trans %in %out %out %. +%sort sound-ttrans %. +%mode sound-ttrans %in %out %out %out %. +%scope sound-ttrans %term const + %pi (sound-trans (trans/const (%the (self M A As) Dself) (%the (expand (const C) A M) Dexpand) (%the (wf A) Dwf) (%the (kof C A) Dkof)) DconvertAs Deof) + %<- (expand-reg (aof/const Dwf Dkof) Dexpand (%the (of M A) DofM)) + %<- (can-tconvert Dwf (%the (tconvert A EA) DconvertA)) + %<- (can-convert DofM (%the (convert M A EM) DconvertM)) + %<- (expand-convert Dexpand (aconvert/const DconvertA Dwf Dkof) DconvertM (%the (tconvert A EA') DconvertA') (%the (equiv (econst C) EM EA') Dequiv)) + %<- (aconvert-reg' (aconvert/const DconvertA Dwf Dkof) DconvertA (%the (eof (econst C) EA) DeofX)) + %<- (tconvert-fun DconvertA' DconvertA (%the (etp-eq EA' EA) DeqA)) + %<- (equiv-resp eterm-eq/i eterm-eq/i DeqA Dequiv (%the (equiv (econst C) EM EA) Dequiv')) + %<- (self-convert Dself DconvertM DconvertA DeofX Dequiv' (%the (tconvert As EAs) DconvertAs) (%the (eof (econst C) EAs) Deof)) %. +%term forall + %pi (sound-trans (trans/forall (%the (self M (qtp A) B) Dself) (%the (expand (forall A) (qtp A) M) Dexpand) (%the (ttrans EA A) DtransA)) DconvertB Deof') + %<- (sound-ttrans DtransA (%the (ewf EA) DewfA) (%the (tconvert A EA') DconvA) (%the (tequiv EA EA') DequivA)) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (expand-reg (aof/forall DwfA) Dexpand (%the (of M (qtp A)) DofM)) + %<- (can-convert DofM (%the (convert M (qtp A) EM) DconvertM)) + %<- (expand-convert Dexpand (aconvert/forall DconvA) DconvertM (%the (tconvert (qtp A) EQ) DconvertQ) (%the (equiv (eforall EA') EM EQ) Dequiv)) + %<- (tconvert-fun DconvertQ (tconvert/pi ([_] [_] [_] [_] tconvert/t) (tconvert/pi ([_] [_] [_] [_] tconvert/t) DconvA)) (%the (etp-eq EQ (qetp EA')) DeqQ)) + %<- (equiv-resp eterm-eq/i eterm-eq/i DeqQ Dequiv (%the (equiv (eforall EA') EM (qetp EA')) Dequiv')) + %<- (self-convert Dself DconvertM (tconvert/pi ([_] [_] [_] [_] tconvert/t) (tconvert/pi ([_] [_] [_] [_] tconvert/t) DconvA)) (eof/equiv (tequiv/pi ([_] [_] tequiv/t) (tequiv/pi ([_] [_] tequiv/t) DequivA)) (eof/forall DewfA)) (equiv/trans Dequiv' (%the (equiv (eforall EA) (eforall EA') (qetp EA')) (equiv/symm (equiv/forall (tequiv/symm DequivA))))) (%the (tconvert B EB) DconvertB) (%the (eof (eforall EA) EB) Deof')) %. +%scope sound-ttrans %term var + %pi (sound-trans (trans/var (%the (self M A As) Dself) (%the (expand X A M) Dexpand) (%the (wf A) DwfA) (%the (vof X A) Dvof) (%the (vtrans EX X) Dvtrans)) DconvertAs DeofX') + %<- (expand-reg (aof/var DwfA Dvof) Dexpand (%the (of M A) DofM)) + %<- (can-convert DofM (%the (convert M A EM) DconvertM)) + %<- (can-tconvert DwfA (%the (tconvert A _) DconvertA')) + %<- (expand-convert Dexpand (aconvert/var Dvtrans DconvertA' DwfA Dvof) DconvertM (%the (tconvert A EA) DconvertA) (%the (equiv EX EM EA) Dequiv)) + %<- (aconvert-reg' (aconvert/var Dvtrans DconvertA' DwfA Dvof) DconvertA (%the (eof EX EA) DeofX)) + %<- (self-convert Dself DconvertM DconvertA DeofX Dequiv (%the (tconvert As EAs) DconvertAs) (%the (eof EX EAs) DeofX')) %. +%term app + %pi (sound-trans (trans/app (%the (tsub B O Bx) DsubBx) (%the (subtype C A ([_] O)) DsubtypeCA) (%the (trans EN C) DtransN) (%the (trans EM (pi A B)) DtransM)) DconvertBx (eof/subsume (subtp/reflex Dtequiv) (eof/subsume (subtp/reflex DequivBx) (eof/app (eof/subsume Dsubtp DeofN) DeofM)))) + %<- (trans-reg DtransM (wf/pi (%the ({x} %pi (vof x A) %-> (wf (B x))) DwfB) (%the (wf A) DwfA))) + %<- (trans-reg DtransN (%the (wf C) DwfC)) + %<- (sound-trans DtransM (tconvert/pi (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert (B x) (EB ex)))) DconvertB) (%the (tconvert A EA) DconvertA)) (%the (eof EM (epi EA EB)) DeofM)) + %<- (sound-trans DtransN (%the (tconvert C EC) DconvertC) (%the (eof EN EC) DeofN)) + %<- (subtype-reg' DsubtypeCA DwfC DwfA (%the (of O A) DofO)) + %<- (can-convert DofO (%the (convert O A EO) DconvertO)) + %<- (convert-tsub DconvertB DconvertO DsubBx (%the (tconvert Bx EBx) DconvertBx) (%the (tequiv (EB EO) EBx) Dtequiv)) + %<- (convert-subtype' DsubtypeCA DconvertC DconvertA (%the (subtp EC EA) Dsubtp)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (tconvert-reg (DconvertB x d ex xt) (%the (ewf (EB ex)) (DewfB ex ed)))) + %<- (convert-subtype-equiv DsubtypeCA DconvertC DconvertA DconvertO DeofN (%the (equiv EN EO EA) Dequiv)) + %<- (tfunctionality DewfB Dequiv (%the (tequiv (EB EN) (EB EO)) DequivBx)) %. +%scope sound-ttrans %term pi1 + %pi (sound-trans (trans/pi1 (%the (trans EM (sigma A ([_] B))) DtransM)) DconvertA (eof/pi1 Deof')) + %<- (trans-reg DtransM (wf/sigma (%the ({x} %pi (vof x A) %-> (wf B)) DwfB) (%the (wf A) DwfA))) + %<- (tinhabitsub DwfA DwfB (%the (wf B) DwfB')) + %<- (can-tconvert DwfB' (%the (tconvert B EB) DconvertB)) + %<- (sound-trans DtransM (tconvert/sigma (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert B (EB' ex)))) DconvertB') (%the (tconvert A EA) DconvertA)) (%the (eof EM (esigma EA EB')) Deof)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} tconvert-fun (DconvertB' x d ex xt) DconvertB (%the (etp-eq (EB' ex) EB) (DeqB ex))) + %<- (esigma-resp etp-eq/i DeqB (%the (etp-eq (esigma EA EB') (esigma EA ([_] EB))) Deq)) + %<- (eof-resp eterm-eq/i Deq Deof (%the (eof EM (esigma EA ([_] EB))) Deof')) %. +%term pi2 + %pi (sound-trans (trans/pi2 (%the (trans EM (sigma A ([_] B))) DtransM)) DconvertB (eof/pi2 Deof')) + %<- (trans-reg DtransM (wf/sigma (%the ({x} %pi (vof x A) %-> (wf B)) DwfB) (%the (wf A) DwfA))) + %<- (tinhabitsub DwfA DwfB (%the (wf B) DwfB')) + %<- (can-tconvert DwfB' (%the (tconvert B EB) DconvertB)) + %<- (sound-trans DtransM (tconvert/sigma (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (tconvert B (EB' ex)))) DconvertB') (%the (tconvert A EA) DconvertA)) (%the (eof EM (esigma EA EB')) Deof)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} tconvert-fun (DconvertB' x d ex xt) DconvertB (%the (etp-eq (EB' ex) EB) (DeqB ex))) + %<- (esigma-resp etp-eq/i DeqB (%the (etp-eq (esigma EA EB') (esigma EA ([_] EB))) Deq)) + %<- (eof-resp eterm-eq/i Deq Deof (%the (eof EM (esigma EA ([_] EB))) Deof')) %. +%scope sound-ttrans %term lam + %pi (sound-trans (trans/lam (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (trans (EM ex) (B x)))) DtransM) (%the (ttrans EC A) DtransA)) (tconvert/pi DconvertB DconvertA) (eof/subsume (subtp/reflex (tequiv/symm (tequiv/pi ([ex] [ed] tequiv/reflex (DewfB ex ed)) (tequiv/symm Dtequiv)))) (eof/lam DeofM' DewfC))) + %<- (sound-ttrans DtransA (%the (ewf EC) DewfC) (%the (tconvert A EA) DconvertA) (%the (tequiv EC EA) Dtequiv)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (sound-trans (DtransM x d ex xt) (%the (tconvert (B x) (EB ex)) (DconvertB x d ex xt)) (%the (eof (EM ex) (EB ex)) (DeofM ex ed)))) + %<- ({ex} {ed evof ex EC} esubst DeofM (eof/subsume (subtp/reflex Dtequiv) (eof/var DewfC ed)) (%the (eof (EM ex) (EB ex)) (DeofM' ex ed))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (tconvert-reg (DconvertB x d ex xt) (%the (ewf (EB ex)) (DewfB ex ed)))) %. +%term pair + %pi (sound-trans (trans/pair (%the (trans EN B) DtransN) (%the (trans EM A) DtransM)) (tconvert/sigma ([_] [_] [_] [_] DconvertB) DconvertA) (eof/pair ([_] [_] DewfB) DeofN DeofM)) + %<- (sound-trans DtransM (%the (tconvert A EA) DconvertA) (%the (eof EM EA) DeofM)) + %<- (sound-trans DtransN (%the (tconvert B EB) DconvertB) (%the (eof EN EB) DeofN)) + %<- (tconvert-reg DconvertB (%the (ewf EB) DewfB)) %. +%scope sound-ttrans %term t sound-ttrans ttrans/t ewf/t tconvert/t (tequiv/reflex ewf/t) %. +%term pi + %pi (sound-ttrans (ttrans/pi (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (ttrans (EB ex) (B x)))) DtransB) (%the (ttrans EA A) DtransA)) (ewf/pi DewfB' DewfA) (tconvert/pi DconvertB DconvertA) (tequiv/symm (tequiv/pi ([ex] [ed] tequiv/symm (DequivB ex ed)) (tequiv/symm DequivA)))) + %<- (sound-ttrans DtransA (%the (ewf EA) DewfA) (%the (tconvert A EA') DconvertA) (%the (tequiv EA EA') DequivA)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA'} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (sound-ttrans (DtransB x d ex xt) (%the (ewf (EB ex)) (DewfB ex ed)) (%the (tconvert (B x) (EB' ex)) (DconvertB x d ex xt)) (%the (tequiv (EB ex) (EB' ex)) (DequivB ex ed)))) + %<- ({ex} {ed evof ex EA} esubst-wf DewfB (eof/subsume (subtp/reflex DequivA) (eof/var DewfA ed)) (%the (ewf (EB ex)) (DewfB' ex ed))) %. +%scope sound-ttrans %term sigma + %pi (sound-ttrans (ttrans/sigma (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (ttrans (EB ex) (B x)))) DtransB) (%the (ttrans EA A) DtransA)) (ewf/sigma DewfB' DewfA) (tconvert/sigma DconvertB DconvertA) (tequiv/symm (tequiv/sigma ([ex] [ed] tequiv/symm (DequivB ex ed)) (tequiv/symm DequivA)))) + %<- (sound-ttrans DtransA (%the (ewf EA) DewfA) (%the (tconvert A EA') DconvertA) (%the (tequiv EA EA') DequivA)) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} {ed evof ex EA'} {dv variable ex} + %pi (vtrans-variable xt dv) + %-> (vsound d xt DconvertA ed) + %-> (sound-ttrans (DtransB x d ex xt) (%the (ewf (EB ex)) (DewfB ex ed)) (%the (tconvert (B x) (EB' ex)) (DconvertB x d ex xt)) (%the (tequiv (EB ex) (EB' ex)) (DequivB ex ed)))) + %<- ({ex} {ed evof ex EA} esubst-wf DewfB (eof/subsume (subtp/reflex DequivA) (eof/var DewfA ed)) (%the (ewf (EB ex)) (DewfB' ex ed))) %. +%term sing + %pi (sound-ttrans (ttrans/sing (%the (trans EM (sing R)) DtransM)) (ewf/sing (eof/subsume (subtp/sing_t DeofN) DeofM)) (tconvert/sing Dconvert) (tequiv/sing (equiv/singelim DeofM))) + %<- (sound-trans DtransM (tconvert/sing (%the (aconvert R t EN) Dconvert)) (%the (eof EM (esing EN)) DeofM)) + %<- (aconvert-reg' Dconvert tconvert/t (%the (eof EN et) DeofN)) %. +%worlds (sbind) (sound-trans _ _ _) (sound-ttrans _ _ _ _) %. +%total (D1 D2) (sound-trans D1 _ _) (sound-ttrans D2 _ _ _) %. +%sort sound-trans' %. +%mode sound-trans' %in %in %out %. +%term _ + %pi (sound-trans' Dtrans Dconv Dof) + %<- (sound-trans Dtrans Dconv' Dof') + %<- (tconvert-fun Dconv' Dconv Deq) + %<- (eof-resp eterm-eq/i Deq Dof' Dof) %. +%worlds (sbind) (sound-trans' _ _ _) %. +%total {} (sound-trans' _ _ _) %. +%sort sound-ttrans' %. +%mode sound-ttrans' %in %in %out %. +%term _ + %pi (sound-ttrans' Dtrans Dconv Dequiv) + %<- (sound-ttrans Dtrans _ Dconv' Dequiv') + %<- (tconvert-fun Dconv' Dconv Deq) + %<- (tequiv-resp etp-eq/i Deq Dequiv' Dequiv) %. +%worlds (sbind) (sound-ttrans' _ _ _) %. +%total {} (sound-ttrans' _ _ _) %. +%%%%% Wrapping Up %%%%% +%sort il-soundness %. +%mode il-soundness %in %in %in %in %in %out %. +%term _ + %pi (il-soundness (%the (trans EM A) DtransM) (%the (trans EN B) DtransN) (%the (ttrans EC C) DtransC) (%the (subtype A C O) DsubtypeAC) (%the (subtype B C O) DsubtypeBC) (equiv/subsume (subtp/reflex (tequiv/symm DequivCD)) (equiv/trans (equiv/symm DequivN) DequivM))) + %<- (sound-trans DtransM (%the (tconvert A EA) DconvertA) (%the (eof EM EA) DeofM)) + %<- (sound-trans DtransN (%the (tconvert B EB) DconvertB) (%the (eof EN EB) DeofN)) + %<- (sound-ttrans DtransC _ (%the (tconvert C ED) DconvertD) (%the (tequiv EC ED) DequivCD)) + %<- (trans-subtype DtransM DsubtypeAC (%the (subtype A C ([_] P)) DsubtypeAC') (%the ({x} term-eq (O x) P) DeqO)) + %<- (subtype-resp tp-eq/i tp-eq/i DeqO DsubtypeBC (%the (subtype B C ([_] P)) DsubtypeBC')) + %<- (trans-reg DtransM (%the (wf A) DwfA)) + %<- (ttrans-reg DtransC (%the (wf C) DwfC)) + %<- (subtype-reg' DsubtypeAC' DwfA DwfC (%the (of P C) DofP)) + %<- (can-convert DofP (%the (convert P C EP) DconvertP)) + %<- (convert-subtype-equiv DsubtypeAC' DconvertA DconvertD DconvertP DeofM (%the (equiv EM EP ED) DequivM)) + %<- (convert-subtype-equiv DsubtypeBC' DconvertB DconvertD DconvertP DeofN (%the (equiv EN EP ED) DequivN)) %. +%worlds (sbind) (il-soundness _ _ _ _ _ _) %. +%total {} (il-soundness _ _ _ _ _ _) %. +%sort il-soundness-tp %. +%mode il-soundness-tp %in %in %out %. +%term _ + %pi (il-soundness-tp DtransA DtransA' (tequiv/trans (tequiv/symm DequivA') DequivA'')) + %<- (sound-ttrans DtransA _ DconvA DequivA) + %<- (sound-ttrans DtransA' _ DconvA' DequivA') + %<- (tconvert-fun DconvA DconvA' Deq) + %<- (tequiv-resp etp-eq/i Deq DequivA DequivA'') %. +%worlds (sbind) (il-soundness-tp _ _ _) %. +%total {} (il-soundness-tp _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/sources.lf b/new-tests/stelf-output/tslf/sing/sources.lf new file mode 100644 index 0000000..edb67bc --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/sources.lf @@ -0,0 +1,39 @@ +%require %[ constant %] %. +%require %[ il %] %. +%require %[ nat %] %. +%require %[ ile %] %. +%require %[ il-simp %] %. +%require %[ expand %] %. +%require %[ subtype %] %. +%require %[ el %] %. +%require %[ algorithm %] %. +%require %[ translate %] %. +%require %[ convert %] %. +%require %[ nat-thm %] %. +%require %[ equality-thm %] %. +%require %[ subst-fun-thm %] %. +%require %[ strengthen-thm %] %. +%require %[ explicit-lemmas-thm %] %. +%require %[ regularity-thm %] %. +%require %[ subst-effect-thm %] %. +%require %[ inversion-thm %] %. +%require %[ substitution-thm %] %. +%require %[ expand-thm %] %. +%require %[ subtype-lemmas-thm %] %. +%require %[ trans-fun-thm %] %. +%require %[ trans-reg-thm %] %. +%require %[ trans-sub-thm %] %. +%require %[ esubstitution-thm %] %. +%require %[ functionality-thm %] %. +%require %[ eregularity-thm %] %. +%require %[ convert-effect-thm %] %. +%require %[ convert-explicit-thm %] %. +%require %[ convert-reg-il-thm %] %. +%require %[ convert-fun-thm %] %. +%require %[ convert-normal-thm %] %. +%require %[ convert-sub-thm %] %. +%require %[ convert-reg-thm %] %. +%require %[ sound-thm %] %. +%require %[ complete-thm %] %. +%require %[ correct-thm %] %. +%require %[ el-inversion-thm %] %. diff --git a/new-tests/stelf-output/tslf/sing/stelf.toml b/new-tests/stelf-output/tslf/sing/stelf.toml new file mode 100644 index 0000000..03cd2c1 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] diff --git a/new-tests/stelf-output/tslf/sing/strengthen-thm.lf b/new-tests/stelf-output/tslf/sing/strengthen-thm.lf new file mode 100644 index 0000000..1dcbe5a --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/strengthen-thm.lf @@ -0,0 +1,254 @@ +%%%%% Substitution for Absent Variables %%%%% +%sort aasub-absent {R} {M} %. +%mode aasub-absent %in %in %out %. +%term _ aasub-absent R M aasub/closed %. +%worlds (var) (aasub-absent _ _ _) %. +%total {} (aasub-absent _ _ _) %. +%sort sub-absent {N} {M} %. +%mode sub-absent %in %in %out %. +%scope sub-absent %term at sub-absent (at R) M (sub/aa aasub/closed) %. +%term lam %pi (sub-absent (lam N) M (sub/lam D)) %<- ({y} sub-absent (N y) M (D y)) %. +%scope sub-absent %term pair + %pi (sub-absent (pair N1 N2) M (sub/pair D2 D1)) + %<- (sub-absent N1 M D1) + %<- (sub-absent N2 M D2) %. +%worlds (var) (sub-absent _ _ _) %. +%total N (sub-absent N _ _) %. +%sort tsub-absent {A} {M} %. +%mode tsub-absent %in %in %out %. +%scope tsub-absent %term t tsub-absent t M tsub/t %. +%term pi + %pi (tsub-absent (pi A1 A2) M (tsub/pi D2 D1)) + %<- (tsub-absent A1 M D1) + %<- ({y} tsub-absent (A2 y) M (D2 y)) %. +%scope tsub-absent %term sigma + %pi (tsub-absent (sigma A1 A2) M (tsub/sigma D2 D1)) + %<- (tsub-absent A1 M D1) + %<- ({y} tsub-absent (A2 y) M (D2 y)) %. +%term sing tsub-absent (sing R) M (tsub/singa aasub/closed) %. +%worlds (var) (tsub-absent _ _ _) %. +%total A (tsub-absent A _ _) %. +%%%%% Substitution Strengthening %%%%% +%sort aasub-closed {x atom} %. +%mode aasub-closed %in %out %. +%sort aosub-closed {x atom} %. +%mode aosub-closed %in %out %. +%sort sub-closed {x atom} %. +%mode sub-closed %in %out %. +%sort tsub-closed {x atom} %. +%mode tsub-closed %in %out %. +%scope tsub-closed %term closed aasub-closed ([_] aasub/closed) ([_] atom-eq/i) %. +%term forall + %pi (aasub-closed ([x] aasub/forall (D x)) Deq') + %<- (tsub-closed D Deq) + %<- ({x} atom-resp-tp forall (Deq x) (Deq' x)) %. +%scope tsub-closed %term app + %pi (aasub-closed ([x] aasub/app (D2 x) (D1 x)) Deq) + %<- (aasub-closed D1 Deq1) + %<- (sub-closed D2 Deq2) + %<- ({x} app-resp (Deq1 x) (Deq2 x) (Deq x)) %. +%term pi1 + %pi (aasub-closed ([x] aasub/pi1 (D1 x)) Deq') + %<- (aasub-closed D1 Deq) + %<- ({x} atom-resp-atom pi1 (Deq x) (Deq' x)) %. +%scope tsub-closed %term pi2 + %pi (aasub-closed ([x] aasub/pi2 (D1 x)) Deq') + %<- (aasub-closed D1 Deq) + %<- ({x} atom-resp-atom pi2 (Deq x) (Deq' x)) %. +%term var aosub-closed ([_] aosub/var) ([_] term-eq/i) %. +%scope tsub-closed %term app + %pi (aosub-closed ([x] aosub/app (D3 x) (D2 x) (D1 x)) Deq) + %<- (aosub-closed D1 Deq1) + %<- ({x} term-eq-cdr-lam (Deq1 x) (Deq1' x)) + %<- (sub-closed D2 Deq2) + %<- (sub-resp1 Deq1' Deq2 ([_] term-eq/i) D3 D3') + %<- (sub-closed D3' Deq) %. +%term pi1 + %pi (aosub-closed ([x] aosub/pi1 (D1 x)) Deq1) + %<- (aosub-closed D1 Deq) + %<- ({x} term-eq-cdr-pair (Deq x) (Deq1 x) (Deq2 x)) %. +%scope tsub-closed %term pi2 + %pi (aosub-closed ([x] aosub/pi2 (D1 x)) Deq2) + %<- (aosub-closed D1 Deq) + %<- ({x} term-eq-cdr-pair (Deq x) (Deq1 x) (Deq2 x)) %. +%term aa + %pi (sub-closed ([x] sub/aa (D1 x)) Deq') + %<- (aasub-closed D1 Deq) + %<- ({x} term-resp-atom at (Deq x) (Deq' x)) %. +%scope tsub-closed %term ao %pi (sub-closed ([x] sub/ao (D1 x)) Deq) %<- (aosub-closed D1 Deq) %. +%term lam + %pi (sub-closed ([x] sub/lam (D1 x)) Deq') + %<- ({y} sub-closed ([x] D1 x y) ([x] Deq x y)) + %<- ({x} lam-resp (Deq x) (Deq' x)) %. +%scope tsub-closed %term pair + %pi (sub-closed ([x] sub/pair (D2 x) (D1 x)) Deq) + %<- (sub-closed D1 Deq1) + %<- (sub-closed D2 Deq2) + %<- ({x} pair-resp (Deq1 x) (Deq2 x) (Deq x)) %. +%term t tsub-closed ([_] tsub/t) ([_] tp-eq/i) %. +%scope tsub-closed %term pi + %pi (tsub-closed ([x] tsub/pi (D2 x) (D1 x)) Deq) + %<- (tsub-closed D1 Deq1) + %<- ({y} tsub-closed ([x] D2 x y) ([x] Deq2 x y)) + %<- ({x} pi-resp (Deq1 x) (Deq2 x) (Deq x)) %. +%term sigma + %pi (tsub-closed ([x] tsub/sigma (D2 x) (D1 x)) Deq) + %<- (tsub-closed D1 Deq1) + %<- ({y} tsub-closed ([x] D2 x y) ([x] Deq2 x y)) + %<- ({x} sigma-resp (Deq1 x) (Deq2 x) (Deq x)) %. +%scope tsub-closed %term singa + %pi (tsub-closed ([x] tsub/singa (D1 x)) Deq') + %<- (aasub-closed D1 Deq) + %<- ({x} tp-resp-atom sing (Deq x) (Deq' x)) %. +%term singo + %pi (tsub-closed ([x] tsub/singo (D1 x)) Deq'') + %<- (aosub-closed D1 Deq) + %<- ({x} term-eq-cdr-at (Deq x) (Deq' x)) + %<- ({x} tp-resp-atom sing (Deq' x) (Deq'' x)) %. +%worlds (var) (aasub-closed _ _) (aosub-closed _ _) (sub-closed _ _) (tsub-closed _ _) %. +%total (D1 D2 D3 D4) (aasub-closed D1 _) (aosub-closed D2 _) (sub-closed D3 _) (tsub-closed D4 _) %. +%sort sub-closed' {x atom} %. +%mode sub-closed' %in %out %. +%term _ + %pi (sub-closed' Dsub (Dsub' aca)) + %<- (sub-closed Dsub Deq) + %<- ({x} sub-resp ([_] term-eq/i) term-eq/i (Deq x) (Dsub x) (Dsub' x)) %. +%worlds (var) (sub-closed' _ _) %. +%total {} (sub-closed' _ _) %. +%%%%% Kof %%%%% +%sort topen-fun %. +%mode topen-fun %in %in %out %. +%term _ topen-fun topen/t topen/t tp-eq/i %. +%term _ + %pi (topen-fun (topen/pi D2 D1) (topen/pi D2' D1') Deq) + %<- (topen-fun D1 D1' Deq1) + %<- (topen-fun D2 D2' Deq2) + %<- (pi-resp Deq1 ([_] Deq2) Deq) %. +%worlds (var) (topen-fun _ _ _) %. +%total D (topen-fun D _ _) %. +%sort kof-strengthen {x atom} %. +%mode kof-strengthen %in %out %. +%term _ + %pi (kof-strengthen ([x] kof/i (%the (topen Ac (A x)) (Dopen x)) (%the (ckof K Ac) Dckof)) Deq) + %<- ({x} topen-fun (Dopen x) (Dopen aca) (%the (tp-eq (A x) (A aca)) (Deq x))) %. +%worlds (var) (kof-strengthen _ _) %. +%total {} (kof-strengthen _ _) %. +%sort subst-kof {x atom} %. +%mode subst-kof %in %in %out %. +%term _ + %pi (subst-kof Dkof Dsub Dkof'') + %<- (kof-strengthen Dkof Deq) + %<- ({x} kof-resp (Deq x) (Dkof x) (Dkof' x)) + %<- (tsub-absent A' M Dsub') + %<- (tsub-resp Deq term-eq/i tp-eq/i Dsub Dsub'') + %<- (tsub-fun Dsub' Dsub'' Deq') + %<- (kof-resp Deq' (Dkof' aca) Dkof'') %. +%worlds (var) (subst-kof _ _ _) %. +%total {} (subst-kof _ _ _) %. +%%%%% Noassm %%%%% +%sort vof-noassm {x atom} %. +%mode vof-noassm %in %out %. +%term _ vof-noassm ([_] D) ([_] tp-eq/i) %. +%worlds (var bind) (vof-noassm _ _) %. +%total {} (vof-noassm _ _) %. +%sort vof-noassm' {x atom} %. +%mode vof-noassm' %in %out %out %. +%term _ vof-noassm' ([_] D) ([_] atom-eq/i) ([_] tp-eq/i) %. +%worlds (var bind) (vof-noassm' _ _ _) %. +%total {} (vof-noassm' _ _ _) %. +%sort aof-noassm {x atom} %. +%mode aof-noassm %in %out %out %. +%sort of-noassm {x atom} %. +%mode of-noassm %in %out %. +%sort wf-noassm {x atom} %. +%mode wf-noassm %in %out %. +%scope wf-noassm %term const + %pi (aof-noassm ([x] aof/const _ (Dkof x)) ([_] atom-eq/i) Deq) + %<- (kof-strengthen Dkof Deq) %. +%term var aof-noassm ([x] aof/var (Dwf x) D) ([_] atom-eq/i) ([_] tp-eq/i) %. +%scope wf-noassm %term forall + %pi (aof-noassm ([x] aof/forall (Dwf x)) Deqa Deqt) + %<- (wf-noassm Dwf Deq) + %<- ({x} atom-resp-tp forall (Deq x) (Deqa x)) + %<- ({x} tp-resp-tp ([a] pi (pi a ([_] t)) ([_] t)) (Deq x) (Deqt x)) %. +%term app + %pi (aof-noassm ([x] aof/app _ (D3 x) (D2 x) (D1 x)) Deqa Deqt) + %<- (aof-noassm D1 Deq1 DeqAB) + %<- (of-noassm D2 Deq2) + %<- ({x} tp-eq-cdr-pi (DeqAB x) _ (DeqB x)) + %<- ({x} app-resp (Deq1 x) (Deq2 x) (Deqa x)) + %<- ({x} tsub-resp (DeqB x) (Deq2 x) tp-eq/i (D3 x) (D3' x)) + %<- (tsub-closed D3' Deqt) %. +%scope wf-noassm %term pi1 + %pi (aof-noassm ([x] aof/pi1 (D1 x)) Deqa' DeqA) + %<- (aof-noassm D1 Deqa DeqAB) + %<- ({x} atom-resp-atom pi1 (Deqa x) (Deqa' x)) + %<- ({x} tp-eq-cdr-sigma (DeqAB x) (DeqA x) _) %. +%term pi2 + %pi (aof-noassm ([x] aof/pi2 (D1 x)) Deqa' DeqB'') + %<- (aof-noassm D1 (%the ({x} atom-eq (R x) R') Deqa) DeqAB) + %<- ({x} atom-resp-atom pi2 (Deqa x) (Deqa' x)) + %<- ({x} tp-eq-cdr-sigma (DeqAB x) _ (%the ({y} tp-eq (B x y) (B' y)) (DeqB x))) + %<- ({x} tp-resp-atom ([r] B' (pi1 r)) (Deqa x) (DeqB' x)) + %<- ({x} tp-eq-trans (DeqB x (pi1 (R x))) (DeqB' x) (DeqB'' x)) %. +%scope wf-noassm %term at + %pi (of-noassm ([x] of/at (D1 x)) Deq') + %<- (aof-noassm D1 Deq _) + %<- ({x} term-resp-atom at (Deq x) (Deq' x)) %. +%term lam + %pi (of-noassm ([x] of/lam (D2 x) (D1 x)) Deq') + %<- (wf-noassm D1 Deq1) + %<- (of-bind-resp1 Deq1 D2 D2') + %<- ({y} {e} of-noassm ([x] D2' x y e) ([x] Deq2 x y)) + %<- ({x} lam-resp (Deq2 x) (Deq' x)) %. +%scope wf-noassm %term pair + %pi (of-noassm ([x] of/pair _ (D2 x) _ (D1 x)) Deq) + %<- (of-noassm D1 Deq1) + %<- (of-noassm D2 Deq2) + %<- ({x} pair-resp (Deq1 x) (Deq2 x) (Deq x)) %. +%term sing + %pi (of-noassm ([x] of/sing (D1 x)) Deq') + %<- (aof-noassm D1 Deq _) + %<- ({x} term-resp-atom at (Deq x) (Deq' x)) %. +%scope wf-noassm %term t wf-noassm ([x] wf/t) ([_] tp-eq/i) %. +%term pi + %pi (wf-noassm ([x] wf/pi (D2 x) (D1 x)) Deq') + %<- (wf-noassm D1 Deq1) + %<- (wf-bind-resp1 Deq1 D2 D2') + %<- ({y} {e} wf-noassm ([x] D2' x y e) ([x] Deq2 x y)) + %<- ({x} pi-resp (Deq1 x) (Deq2 x) (Deq' x)) %. +%scope wf-noassm %term sigma + %pi (wf-noassm ([x] wf/sigma (D2 x) (D1 x)) Deq') + %<- (wf-noassm D1 Deq1) + %<- (wf-bind-resp1 Deq1 D2 D2') + %<- ({y} {e} wf-noassm ([x] D2' x y e) ([x] Deq2 x y)) + %<- ({x} sigma-resp (Deq1 x) (Deq2 x) (Deq' x)) %. +%term sing + %pi (wf-noassm ([x] wf/sing (D1 x)) Deq') + %<- (aof-noassm D1 Deq _) + %<- ({x} tp-resp-atom sing (Deq x) (Deq' x)) %. +%worlds (bind var) (aof-noassm _ _ _) (of-noassm _ _) (wf-noassm _ _) %. +%total (D1 D2 D3) (aof-noassm D1 _ _) (of-noassm D2 _) (wf-noassm D3 _) %. +%sort vof-noassm-var {x} %. +%mode vof-noassm-var %in %out %. +%worlds (bind var) (vof-noassm-var _ _) %. +%total {} (vof-noassm-var _ _) %. +%sort aof-noassm-var {x} %. +%mode aof-noassm-var %in %out %. +%term _ + %pi (aof-noassm-var ([x] aof/var _ (Dvof x)) Dfalse) + %<- (vof-noassm-var Dvof Dfalse) %. +%worlds (bind var) (aof-noassm-var _ _) %. +%total {} (aof-noassm-var _ _) %. +%sort vof-noassm-tsub {x} %. +%mode vof-noassm-tsub %in %in %out %. +%term _ + %pi (vof-noassm-tsub Dvof Dsub Dvof'') + %<- (vof-noassm Dvof Deq) + %<- (tsub-resp Deq term-eq/i tp-eq/i Dsub Dsub') + %<- (tsub-absent A' M Dsub'') + %<- (tsub-fun Dsub'' Dsub' Deq') + %<- ({x} vof-resp atom-eq/i (Deq x) (Dvof x) (Dvof' x)) + %<- (vof-resp atom-eq/i Deq' (Dvof' aca) Dvof'') %. +%worlds (bind var) (vof-noassm-tsub _ _ _) %. +%total {} (vof-noassm-tsub _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/subst-effect-thm.lf b/new-tests/stelf-output/tslf/sing/subst-effect-thm.lf new file mode 100644 index 0000000..24e1643 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/subst-effect-thm.lf @@ -0,0 +1,862 @@ +%%%%% Head Variables %%%%% +%sort headvar %. +%term headvar/var headvar ([x] x) %. +%term headvar/app %pi (headvar ([x] app (R x) (M x))) %<- (headvar R) %. +%term headvar/pi1 %pi (headvar ([x] pi1 (R x))) %<- (headvar R) %. +%term headvar/pi2 %pi (headvar ([x] pi2 (R x))) %<- (headvar R) %. +%%%%% Equality %%%%% +%sort headvar-resp {x atom} %. +%mode headvar-resp %in %in %out %. +%term _ headvar-resp ([_] atom-eq/i) D D %. +%worlds (var) (headvar-resp _ _ _) %. +%total {} (headvar-resp _ _ _) %. +%%%%% Simple Type Lemmas %%%%% +%sort simp-fun %. +%mode simp-fun %in %in %out %. +%scope simp-fun %term t simp-fun simp/t simp/t stp-eq/i %. +%term pi + %pi (simp-fun (simp/pi Dsimp2 Dsimp1) (simp/pi Dsimp2' Dsimp1') Deq) + %<- (simp-fun Dsimp1 Dsimp1' Deq1) + %<- ({x} simp-fun (Dsimp2 x) (Dsimp2' x) Deq2) + %<- (stp-resp-stp ([t] spi t T2) Deq1 Deq1') + %<- (stp-resp-stp ([t] spi T1' t) Deq2 Deq2') + %<- (stp-eq-trans Deq1' Deq2' Deq) %. +%scope simp-fun %term sigma + %pi (simp-fun (simp/sigma Dsimp2 Dsimp1) (simp/sigma Dsimp2' Dsimp1') Deq) + %<- (simp-fun Dsimp1 Dsimp1' Deq1) + %<- ({x} simp-fun (Dsimp2 x) (Dsimp2' x) Deq2) + %<- (stp-resp-stp ([t] ssigma t T2) Deq1 Deq1') + %<- (stp-resp-stp ([t] ssigma T1' t) Deq2 Deq2') + %<- (stp-eq-trans Deq1' Deq2' Deq) %. +%term sing simp-fun simp/sing simp/sing stp-eq/i %. +%worlds (var) (simp-fun _ _ _) %. +%total D (simp-fun D _ _) %. +%sort employ-stp-leq {S} {T} %. +%mode employ-stp-leq %in %in %in %. +%term _ employ-stp-leq _ _ stp-leq/eq %. +%term _ %pi (employ-stp-leq _ _ (stp-leq/pi1 D)) %<- (employ-stp-leq _ _ D) %. +%term _ %pi (employ-stp-leq _ _ (stp-leq/pi2 D)) %<- (employ-stp-leq _ _ D) %. +%term _ %pi (employ-stp-leq _ _ (stp-leq/sigma1 D)) %<- (employ-stp-leq _ _ D) %. +%term _ %pi (employ-stp-leq _ _ (stp-leq/sigma2 D)) %<- (employ-stp-leq _ _ D) %. +%worlds () (employ-stp-leq _ _ _) %. +%total D (employ-stp-leq _ _ D) %. +%reduces <= S T (employ-stp-leq S T _) %. +%sort stp-leq-trans %. +%mode stp-leq-trans %in %in %out %. +%term _ stp-leq-trans D stp-leq/eq D %. +%term _ + %pi (stp-leq-trans D (stp-leq/pi1 D') (stp-leq/pi1 D'')) + %<- (stp-leq-trans D D' D'') %. +%term _ + %pi (stp-leq-trans D (stp-leq/pi2 D') (stp-leq/pi2 D'')) + %<- (stp-leq-trans D D' D'') %. +%term _ + %pi (stp-leq-trans D (stp-leq/sigma1 D') (stp-leq/sigma1 D'')) + %<- (stp-leq-trans D D' D'') %. +%term _ + %pi (stp-leq-trans D (stp-leq/sigma2 D') (stp-leq/sigma2 D'')) + %<- (stp-leq-trans D D' D'') %. +%worlds () (stp-leq-trans _ _ _) %. +%total D (stp-leq-trans _ D _) %. +%sort tsub-preserves-simp %. +%mode tsub-preserves-simp %in %in %out %. +%term _ tsub-preserves-simp tsub/t ([x] simp/t) simp/t %. +%term _ + %pi (tsub-preserves-simp (tsub/pi Dsub2 Dsub1) ([x] simp/pi ([y] Dsimp2 x y) (Dsimp1 x)) (simp/pi Dsimp2' Dsimp1')) + %<- (tsub-preserves-simp Dsub1 Dsimp1 Dsimp1') + %<- ({y} tsub-preserves-simp (Dsub2 y) ([x] Dsimp2 x y) (Dsimp2' y)) %. +%term _ + %pi (tsub-preserves-simp (tsub/sigma Dsub2 Dsub1) ([x] simp/sigma ([y] Dsimp2 x y) (Dsimp1 x)) (simp/sigma Dsimp2' Dsimp1')) + %<- (tsub-preserves-simp Dsub1 Dsimp1 Dsimp1') + %<- ({y} tsub-preserves-simp (Dsub2 y) ([x] Dsimp2 x y) (Dsimp2' y)) %. +%term _ tsub-preserves-simp (tsub/singa _) ([_] simp/sing) simp/sing %. +%term _ tsub-preserves-simp (tsub/singo _) ([_] simp/sing) simp/sing %. +%worlds (var) (tsub-preserves-simp _ _ _) %. +%total D (tsub-preserves-simp _ D _) %. +%sort can-simp {A} %. +%mode can-simp %in %out %. +%scope can-simp %term t can-simp t simp/t %. +%term pi + %pi (can-simp (pi A B) (simp/pi D2 D1)) + %<- (can-simp A D1) + %<- ({x} can-simp (B x) (D2 x)) %. +%scope can-simp %term sigma + %pi (can-simp (sigma A B) (simp/sigma D2 D1)) + %<- (can-simp A D1) + %<- ({x} can-simp (B x) (D2 x)) %. +%term sing can-simp (sing R) simp/sing %. +%worlds (var) (can-simp _ _) %. +%total A (can-simp A _) %. +%%%%% Simplifying Typings %%%%% +%sort simpctx %. +%term simpctx/nil simpctx nil snil %. +%term simpctx/cons %pi (simpctx (cons G X A) (scons G' X T)) %<- (simpctx G G') %<- (simp A T) %. +%sort can-simpctx {G} %. +%mode can-simpctx %in %out %. +%term _ can-simpctx nil simpctx/nil %. +%term _ + %pi (can-simpctx (cons G X A) (simpctx/cons D2 D1)) + %<- (can-simpctx G D1) + %<- (can-simp A D2) %. +%worlds (var) (can-simpctx _ _) %. +%total G (can-simpctx G _) %. +%sort bounded-simpctx %. +%mode bounded-simpctx %in %in %out %. +%term _ bounded-simpctx simpctx/nil (bounded/nil D) (sbounded/nil D) %. +%term _ + %pi (bounded-simpctx (simpctx/cons _ Dsimp) (bounded/cons Dbound Dprec) (sbounded/cons Dbound' Dprec)) + %<- (bounded-simpctx Dsimp Dbound Dbound') %. +%worlds (var ovar) (bounded-simpctx _ _ _) %. +%total D (bounded-simpctx D _ _) %. +%sort ordered-simpctx %. +%mode ordered-simpctx %in %in %out %. +%term _ ordered-simpctx simpctx/nil ordered/nil sordered/nil %. +%term _ + %pi (ordered-simpctx (simpctx/cons _ Dsimp) (ordered/cons Dbound) (sordered/cons Dbound')) + %<- (bounded-simpctx Dsimp Dbound Dbound') %. +%worlds (var ovar) (ordered-simpctx _ _ _) %. +%total {} (ordered-simpctx _ _ _) %. +%sort lookup-simp %. +%mode lookup-simp %in %in %in %out %. +%term _ + %pi (lookup-simp (simpctx/cons Dsimp Dsimpctx) Dsimp' (lookup/hit Dbound) D) + %<- (bounded-simpctx Dsimpctx Dbound Dbound') + %<- (simp-fun Dsimp Dsimp' Deq) + %<- (slookup-resp sctx-eq/i atom-eq/i Deq (slookup/hit Dbound') D) %. +%term _ + %pi (lookup-simp (simpctx/cons _ Dsimpctx) Dsimp (lookup/miss Dlookup Dbound) (slookup/miss Dlookup' Dbound')) + %<- (lookup-simp Dsimpctx Dsimp Dlookup Dlookup') + %<- (bounded-simpctx Dsimpctx Dbound Dbound') %. +%worlds (var ovar) (lookup-simp _ _ _ _) %. +%total D (lookup-simp _ _ D _) %. +%sort aofe-simp %. +%mode aofe-simp %in %in %in %out %. +%sort ofe-simp %. +%mode ofe-simp %in %in %in %out %. +%sort wfe-simp %. +%mode wfe-simp %in %in %out %. +%term _ + %pi (aofe-simp Dsimpctx Dsimp (aofe/closed Dord Daof) (aofes/closed Dord' Dsimp Daof)) + %<- (ordered-simpctx Dsimpctx Dord Dord') %. +%term _ + %pi (aofe-simp Dsimpctx (simp/pi ([_] simp/t) (simp/pi ([_] simp/t) Dsimp)) (aofe/forall Dwf) (aofes/forall Dsimp Dwf')) + %<- (wfe-simp Dsimpctx Dwf Dwf') %. +%term _ + %pi (aofe-simp Dsimpctx Dsimp (aofe/var _ Dlookup) (aofes/var Dlookup')) + %<- (lookup-simp Dsimpctx Dsimp Dlookup Dlookup') %. +%term _ + %pi (aofe-simp Dsimpctx Dsimp (aofe/app _ Dsub Dof Daof) D) + %<- (can-simp _ DsimpA) + %<- ({x} can-simp _ (DsimpB x)) + %<- (aofe-simp Dsimpctx (simp/pi DsimpB DsimpA) Daof Daof') + %<- (ofe-simp Dsimpctx DsimpA Dof Dof') + %<- (tsub-preserves-simp Dsub DsimpB Dsimp') + %<- (simp-fun Dsimp' Dsimp Deq) + %<- (aofes-resp sctx-eq/i atom-eq/i Deq (aofes/app Dof' Daof') D) %. +%term _ + %pi (aofe-simp Dsimpctx Dsimp (aofe/pi1 Daof) (aofes/pi1 Daof')) + %<- ({x} can-simp _ (DsimpB x)) + %<- (aofe-simp Dsimpctx (simp/sigma DsimpB Dsimp) Daof Daof') %. +%term _ + %pi (aofe-simp Dsimpctx Dsimp (aofe/pi2 Daof) D) + %<- (can-simp _ DsimpA) + %<- ({x} can-simp _ (%the (simp (B x) _) (DsimpB x))) + %<- (aofe-simp Dsimpctx (simp/sigma DsimpB DsimpA) Daof Daof') + %<- (simp-fun (DsimpB _) Dsimp Deq) + %<- (aofes-resp sctx-eq/i atom-eq/i Deq (aofes/pi2 Daof') D) %. +%term _ + %pi (ofe-simp Dsimpctx simp/t (ofe/at Daof) (ofes/at Daof')) + %<- (aofe-simp Dsimpctx simp/t Daof Daof') %. +%term _ + %pi (ofe-simp Dsimpctx (simp/pi DsimpB DsimpA) (ofe/lam Dof _) (ofes/lam Dof')) + %<- ({x} {d} ofe-simp (simpctx/cons DsimpA Dsimpctx) (DsimpB x) (Dof x d) (Dof' x d)) %. +%term _ + %pi (ofe-simp Dsimpctx (simp/sigma DsimpB DsimpA) (ofe/pair _ Dof2 Dsub Dof1) (ofes/pair Dof2' Dof1')) + %<- (ofe-simp Dsimpctx DsimpA Dof1 Dof1') + %<- (tsub-preserves-simp Dsub DsimpB DsimpB') + %<- (ofe-simp Dsimpctx DsimpB' Dof2 Dof2') %. +%term _ + %pi (ofe-simp Dsimpctx simp/sing (ofe/sing Daof) (ofes/at Daof')) + %<- (aofe-simp Dsimpctx simp/t Daof Daof') %. +%term _ + %pi (wfe-simp Dsimpctx (wfe/t Dord) (wfes/t Dord')) + %<- (ordered-simpctx Dsimpctx Dord Dord') %. +%term _ + %pi (wfe-simp Dsimpctx (wfe/pi D2 D1) (wfes/pi D2' Dsimp D1')) + %<- (wfe-simp Dsimpctx D1 D1') + %<- (can-simp _ Dsimp) + %<- ({x} {d} wfe-simp (simpctx/cons Dsimp Dsimpctx) (D2 x d) (D2' x d)) %. +%term _ + %pi (wfe-simp Dsimpctx (wfe/sigma D2 D1) (wfes/sigma D2' Dsimp D1')) + %<- (wfe-simp Dsimpctx D1 D1') + %<- (can-simp _ Dsimp) + %<- ({x} {d} wfe-simp (simpctx/cons Dsimp Dsimpctx) (D2 x d) (D2' x d)) %. +%term _ + %pi (wfe-simp Dsimpctx (wfe/sing D) (wfes/sing D')) + %<- (aofe-simp Dsimpctx simp/t D D') %. +%worlds (bind var ovar) (aofe-simp _ _ _ _) (ofe-simp _ _ _ _) (wfe-simp _ _ _) %. +%total (D1 D2 D3) (aofe-simp _ _ D1 _) (ofe-simp _ _ D2 _) (wfe-simp _ D3 _) %. +%sort context-append-simp-lookup {x} %. +%mode context-append-simp-lookup %in %in %in %out %out %out %. +%term _ + %pi (context-append-simp-lookup ([x] append/nil) ([x] [d] bounded/cons (Dbounded x d) (Dprecedes x d)) Dsimp ([x] simpctx/cons Dsimp Dsimpctx) ([x] [d] sbounded/cons (Dsbounded x d) (Dprecedes x d)) ([x] [d] slookup/hit (Dsbounded x d))) + %<- (can-simpctx G Dsimpctx) + %<- ({x} {d isvar x I} bounded-simpctx Dsimpctx (Dbounded x d) (Dsbounded x d)) %. +%term _ + %pi (context-append-simp-lookup ([x] append/cons (Dappend x)) ([x] [d] bounded/cons (Dbounded x d) (Dprecedes x d)) Dsimp ([x] simpctx/cons (Dsimp' x) (Dsimpctx x)) ([x] [d] sbounded/cons (Dsbounded x d) (Dprecedes x d)) ([x] [d] slookup/miss (Dlookup x d) (Dsbounded x d))) + %<- ({x} can-simp _ (%the (simp _ T) (Dsimp' x))) + %<- (context-append-simp-lookup Dappend Dbounded Dsimp Dsimpctx Dsbounded Dlookup) %. +%worlds (var ovar) (context-append-simp-lookup _ _ _ _ _ _) %. +%total D (context-append-simp-lookup D _ _ _ _ _) %. +%sort context-append-simp-lookup' {x} %. +%mode context-append-simp-lookup' %in %in %in %out %out %. +%term _ + %pi (context-append-simp-lookup' ([x] append/nil) ([x] [d] ordered/cons (Dbounded x d)) Dsimp ([x] simpctx/cons Dsimp Dsimpctx) ([x] [d] slookup/hit (Dsbounded x d))) + %<- (can-simpctx G Dsimpctx) + %<- ({x} {d isvar x I} bounded-simpctx Dsimpctx (Dbounded x d) (Dsbounded x d)) %. +%term _ + %pi (context-append-simp-lookup' ([x] append/cons (Dappend x)) ([x] [d] ordered/cons (Dbounded x d)) Dsimp ([x] simpctx/cons (Dsimp' x) (Dsimpctx x)) ([x] [d] slookup/miss (Dlookup x d) (Dsbounded x d))) + %<- ({x} can-simp _ (%the (simp _ T) (Dsimp' x))) + %<- (context-append-simp-lookup Dappend Dbounded Dsimp Dsimpctx Dsbounded Dlookup) %. +%worlds (var ovar) (context-append-simp-lookup' _ _ _ _ _) %. +%total {} (context-append-simp-lookup' _ _ _ _ _) %. +%%%%% Head Variable Lemmas %%%%% +%sort aosub-headvar %. +%mode aosub-headvar %in %out %. +%scope aosub-headvar %term var aosub-headvar aosub/var headvar/var %. +%term app %pi (aosub-headvar (aosub/app _ _ D) (headvar/app D')) %<- (aosub-headvar D D') %. +%scope aosub-headvar %term pi1 %pi (aosub-headvar (aosub/pi1 D) (headvar/pi1 D')) %<- (aosub-headvar D D') %. +%term pi2 %pi (aosub-headvar (aosub/pi2 D) (headvar/pi2 D')) %<- (aosub-headvar D D') %. +%worlds (var) (aosub-headvar _ _) %. +%total D (aosub-headvar D _) %. +%sort headvar-closed %. +%mode headvar-closed %in %out %. +%term _ %pi (headvar-closed (headvar/app D) D') %<- (headvar-closed D D') %. +%term _ %pi (headvar-closed (headvar/pi1 D) D') %<- (headvar-closed D D') %. +%term _ %pi (headvar-closed (headvar/pi2 D) D') %<- (headvar-closed D D') %. +%worlds (var) (headvar-closed _ _) %. +%total D (headvar-closed D _) %. +%%%%% Head Variable Size (Explicit Context) %%%%% +%sort headvar-tp-size {x} %. +%mode headvar-tp-size %in %in %in %in %out %out %. +%scope headvar-tp-size %term var + %pi (headvar-tp-size (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) ([x] [d] aofe/var _ (%the (lookup (G x) x (A' x)) (Dlook x d))) (%the (simp A T) Dsimp) headvar/var Dsimp' stp-leq/eq) + %<- ({x} {d} lookup-binder-fun (Dapp x) (Dlook x d) (%the (tp-eq A (A' x)) (Deq x))) + %<- ({x} simp-resp (Deq x) stp-eq/i Dsimp (%the (simp (A' x) T) (Dsimp' x))) %. +%term closed + %pi (headvar-tp-size _ ([x] [d] aofe/closed (%the (ordered (G x)) (Dordered x d)) (%the (aof (R x) (B x)) (Daof x))) _ (%the (headvar R) Dheadvar) Dsimp stp-leq/eq) + %<- (aof-noassm Daof (%the ({x} atom-eq (R x) R') Deq) _) + %<- (headvar-resp Deq Dheadvar (%the (headvar ([x] R')) Dheadvar')) + %<- (headvar-closed Dheadvar' Dfalse) + %<- ({x} false-implies-simp Dfalse (Dsimp x)) %. +%scope headvar-tp-size %term app + %pi (headvar-tp-size (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) ([x] [d] aofe/app _ (%the (tsub ([y] C x y) (M x) (Cy x)) (Dtsub x)) _ (%the (aofe (G x) (R x) (pi (B x) ([y] C x y))) (Daof x d))) (%the (simp A S) Dsimp) (headvar/app (%the (headvar R) Dheadvar)) DsimpCy Dleq') + %<- (headvar-tp-size Dapp Daof Dsimp Dheadvar ([x] simp/pi (%the ({y} simp (C x y) T2) (DsimpC x)) (%the (simp (B x) T1) (DsimpB x))) (%the (stp-leq (spi T1 T2) S) Dleq)) + %<- ({x} tsub-preserves-simp (Dtsub x) (DsimpC x) (%the (simp (Cy x) T2) (DsimpCy x))) + %<- (stp-leq-trans (stp-leq/pi2 stp-leq/eq) Dleq (%the (stp-leq T2 S) Dleq')) %. +%term appbad + %pi (headvar-tp-size _ ([x] [d isvar x I] aofe/var _ (Dlookup x d)) _ (headvar/app _) Dsimp Dleq) + %<- ({x} {d isvar x I} lookup-isvar (Dlookup x d) (%the (isvar (app _ _) J) (Disvar x))) + %<- ({x} isvar-app-contra (Disvar x) Dfalse) + %<- ({x} false-implies-simp Dfalse (%the (simp _ st) (Dsimp x))) + %<- (false-implies-stp-leq Dfalse Dleq) %. +%scope headvar-tp-size %term pi1 + %pi (headvar-tp-size (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) ([x] [d] aofe/pi1 (%the (aofe (G x) (R x) (sigma (B x) ([y] C x y))) (Daof x d))) (%the (simp A S) Dsimp) (headvar/pi1 (%the (headvar R) Dheadvar)) DsimpB Dleq') + %<- (headvar-tp-size Dapp Daof Dsimp Dheadvar ([x] simp/sigma (%the ({y} simp (C x y) T2) (DsimpC x)) (%the (simp (B x) T1) (DsimpB x))) (%the (stp-leq (ssigma T1 T2) S) Dleq)) + %<- (stp-leq-trans (stp-leq/sigma1 stp-leq/eq) Dleq (%the (stp-leq T1 S) Dleq')) %. +%term pi1bad + %pi (headvar-tp-size _ ([x] [d isvar x I] aofe/var _ (Dlookup x d)) _ (headvar/pi1 _) Dsimp Dleq) + %<- ({x} {d isvar x I} lookup-isvar (Dlookup x d) (%the (isvar (pi1 _) J) (Disvar x))) + %<- ({x} isvar-pi1-contra (Disvar x) Dfalse) + %<- ({x} false-implies-simp Dfalse (%the (simp _ st) (Dsimp x))) + %<- (false-implies-stp-leq Dfalse Dleq) %. +%scope headvar-tp-size %term pi2 + %pi (headvar-tp-size (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) ([x] [d] aofe/pi2 (%the (aofe (G x) (R x) (sigma (B x) ([y] C x y))) (Daof x d))) (%the (simp A S) Dsimp) (headvar/pi2 (%the (headvar R) Dheadvar)) ([x] DsimpC x (pi1 (R x))) Dleq') + %<- (headvar-tp-size Dapp Daof Dsimp Dheadvar ([x] simp/sigma (%the ({y} simp (C x y) T2) (DsimpC x)) (%the (simp (B x) T1) (DsimpB x))) (%the (stp-leq (ssigma T1 T2) S) Dleq)) + %<- (stp-leq-trans (stp-leq/sigma2 stp-leq/eq) Dleq (%the (stp-leq T2 S) Dleq')) %. +%term pi2bad + %pi (headvar-tp-size _ ([x] [d isvar x I] aofe/var _ (Dlookup x d)) _ (headvar/pi2 _) Dsimp Dleq) + %<- ({x} {d isvar x I} lookup-isvar (Dlookup x d) (%the (isvar (pi2 _) J) (Disvar x))) + %<- ({x} isvar-pi2-contra (Disvar x) Dfalse) + %<- ({x} false-implies-simp Dfalse (%the (simp _ st) (Dsimp x))) + %<- (false-implies-stp-leq Dfalse Dleq) %. +%worlds (bind ovar var) (headvar-tp-size _ _ _ _ _ _) %. +%total D (headvar-tp-size _ _ _ D _ _) %. +%sort closed-headvar-contra {x} %. +%mode closed-headvar-contra %in %in %out %. +%term _ + %pi (closed-headvar-contra Daof headvar/var Dfalse) + %<- (aof-noassm-var Daof Dfalse) %. +%term _ + %pi (closed-headvar-contra ([x] aof/app _ _ _ (D x)) (headvar/app Dheadvar) Dfalse) + %<- (closed-headvar-contra D Dheadvar Dfalse) %. +%term _ + %pi (closed-headvar-contra ([x] aof/pi1 (D x)) (headvar/pi1 Dheadvar) Dfalse) + %<- (closed-headvar-contra D Dheadvar Dfalse) %. +%term _ + %pi (closed-headvar-contra ([x] aof/pi2 (D x)) (headvar/pi2 Dheadvar) Dfalse) + %<- (closed-headvar-contra D Dheadvar Dfalse) %. +%worlds (var bind) (closed-headvar-contra _ _ _) %. +%total D (closed-headvar-contra D _ _) %. +%sort headvar-stp-size {x} %. +%mode headvar-stp-size %in %in %in %out %. +%scope headvar-stp-size %term closed + %pi (headvar-stp-size _ ([x] [d] aofes/closed _ (Dsimp x) (Daof x)) Dheadvar D) + %<- (closed-headvar-contra Daof Dheadvar Dfalse) + %<- (false-implies-stp-leq Dfalse D) %. +%term var + %pi (headvar-stp-size Dlookup ([x] [d] aofes/var (Dlookup' x d)) headvar/var D) + %<- ({x} {d} slookup-fun (Dlookup x d) (Dlookup' x d) Deq) + %<- (stp-leq-resp Deq stp-eq/i stp-leq/eq D) %. +%scope headvar-stp-size %term app + %pi (headvar-stp-size Dlookup ([x] [d] aofes/app _ (Daof x d)) (headvar/app Dheadvar) Dleq') + %<- (headvar-stp-size Dlookup Daof Dheadvar Dleq) + %<- (stp-leq-trans (stp-leq/pi2 stp-leq/eq) Dleq Dleq') %. +%term appbad + %pi (headvar-stp-size _ ([x] [d] aofes/var (Dlookup x d)) (headvar/app _) D) + %<- ({x} {d} slookup-isvar (Dlookup x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-app-contra (Disvar x d) Dfalse) + %<- (false-implies-stp-leq Dfalse D) %. +%scope headvar-stp-size %term pi1 + %pi (headvar-stp-size Dlookup ([x] [d] aofes/pi1 (Daof x d)) (headvar/pi1 Dheadvar) Dleq') + %<- (headvar-stp-size Dlookup Daof Dheadvar Dleq) + %<- (stp-leq-trans (stp-leq/sigma1 stp-leq/eq) Dleq Dleq') %. +%term pi1bad + %pi (headvar-stp-size _ ([x] [d] aofes/var (Dlookup x d)) (headvar/pi1 _) D) + %<- ({x} {d} slookup-isvar (Dlookup x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-pi1-contra (Disvar x d) Dfalse) + %<- (false-implies-stp-leq Dfalse D) %. +%scope headvar-stp-size %term pi2 + %pi (headvar-stp-size Dlookup ([x] [d] aofes/pi2 (Daof x d)) (headvar/pi2 Dheadvar) Dleq') + %<- (headvar-stp-size Dlookup Daof Dheadvar Dleq) + %<- (stp-leq-trans (stp-leq/sigma2 stp-leq/eq) Dleq Dleq') %. +%term pi2bad + %pi (headvar-stp-size _ ([x] [d] aofes/var (Dlookup x d)) (headvar/pi2 _) D) + %<- ({x} {d} slookup-isvar (Dlookup x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-pi2-contra (Disvar x d) Dfalse) + %<- (false-implies-stp-leq Dfalse D) %. +%worlds (bind var ovar) (headvar-stp-size _ _ _ _) %. +%total D (headvar-stp-size _ D _ _) %. +%%%%% Substitution Lemma (Simply Typed, Explicit Context Version) %%%%% +%sort sasubst-es {x} %. +%mode sasubst-es %in %in %in %out %. +%sort ssubst-es {x} %. +%mode ssubst-es %in %in %in %out %. +%sort stsubst-es {x} %. +%mode stsubst-es %in %in %in %out %. +%term _ sasubst-es Dlook DofR ([x] [d] aofes/closed (Dord x d) (Dsimp x) (%the (aof (Q x) _) (Daof x))) ([x] [d] aofes/closed (Dord x d) (Dsimp R) (Daof R)) %. +%term _ + %pi (sasubst-es Dlook (%the ({x} %pi _ %-> (aofes _ R _)) DofR) ([x] [d] aofes/forall (Dsimp x) (D x d)) ([x] [d] aofes/forall (Dsimp R) (D' x d))) + %<- (stsubst-es Dlook DofR D D') %. +%term _ + %pi (sasubst-es _ _ ([x] [d] aofes/var (Dlook x d)) D) + %<- ({x} {d} slookup-isvar (Dlook x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-forall-contra (Disvar x d) Dfalse) + %<- ({x} {d} false-implies-aofes Dfalse (D x d)) %. +%term _ + %pi (sasubst-es Dlook DofR ([x] [d] aofes/var (Dlook' x d)) DofR') + %<- ({x} {d} slookup-fun (Dlook x d) (Dlook' x d) Deq) + %<- ({x} {d} aofes-resp sctx-eq/i atom-eq/i Deq (DofR x d) (DofR' x d)) %. +%term _ sasubst-es _ _ ([x] [d] aofes/var (%the (slookup _ Y _) (Dlook x d))) ([x] [d] aofes/var (Dlook x d)) %. +%term _ + %pi (sasubst-es Dlook DofR ([x] [d] aofes/app (DofM x d) (DofQ x d)) ([x] [d] aofes/app (DofM' x d) (DofQ' x d))) + %<- (sasubst-es Dlook DofR DofQ DofQ') + %<- (ssubst-es Dlook DofR DofM DofM') %. +%term _ + %pi (sasubst-es _ _ ([x] [d] aofes/var (Dlook x d)) D) + %<- ({x} {d} slookup-isvar (Dlook x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-app-contra (Disvar x d) Dfalse) + %<- ({x} {d} false-implies-aofes Dfalse (D x d)) %. +%term _ + %pi (sasubst-es Dlook DofR ([x] [d] aofes/pi1 (DofQ x d)) ([x] [d] aofes/pi1 (DofQ' x d))) + %<- (sasubst-es Dlook DofR DofQ DofQ') %. +%term _ + %pi (sasubst-es _ _ ([x] [d] aofes/var (Dlook x d)) D) + %<- ({x} {d} slookup-isvar (Dlook x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-pi1-contra (Disvar x d) Dfalse) + %<- ({x} {d} false-implies-aofes Dfalse (D x d)) %. +%term _ + %pi (sasubst-es Dlook DofR ([x] [d] aofes/pi2 (DofQ x d)) ([x] [d] aofes/pi2 (DofQ' x d))) + %<- (sasubst-es Dlook DofR DofQ DofQ') %. +%term _ + %pi (sasubst-es _ _ ([x] [d] aofes/var (Dlook x d)) D) + %<- ({x} {d} slookup-isvar (Dlook x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-pi2-contra (Disvar x d) Dfalse) + %<- ({x} {d} false-implies-aofes Dfalse (D x d)) %. +%term _ + %pi (ssubst-es Dlook DofR ([x] [d] ofes/at (D x d)) ([x] [d] ofes/at (D' x d))) + %<- (sasubst-es Dlook DofR D D') %. +%term _ + %pi (ssubst-es Dlook DofR ([x] [d] ofes/lam (D x d)) ([x] [d] ofes/lam (D' x d))) + %<- ({x} {d} {y} {e} ofes-context (D x d y e) (sordered/cons (Dbound x d y e))) + %<- ({x} {d} {y} {e} weaken-slookup (Dbound x d y e) (Dlook x d) _ (Dlook' x d y e)) + %<- ({x} {d} {y} {e} weaken-aofes (Dbound x d y e) (DofR x d) _ (DofR' x d y e)) + %<- ({y} {e} ssubst-es ([x] [d] Dlook' x d y e) ([x] [d] DofR' x d y e) ([x] [d] D x d y e) ([x] [d] D' x d y e)) %. +%term _ + %pi (ssubst-es Dlook DofR ([x] [d] ofes/pair (D2 x d) (D1 x d)) ([x] [d] ofes/pair (D2' x d) (D1' x d))) + %<- (ssubst-es Dlook DofR D1 D1') + %<- (ssubst-es Dlook DofR D2 D2') %. +%term _ stsubst-es _ _ ([x] [d] wfes/t (D x d)) ([x] [d] wfes/t (D x d)) %. +%term _ + %pi (stsubst-es Dlook DofR ([x] [d] wfes/pi (D2 x d) (%the (simp _ T) (Dsimp x)) (%the (wfes _ (A x)) (D1 x d))) ([x] [d] wfes/pi (D2' x d) (Dsimp R) (D1' x d))) + %<- (stsubst-es Dlook DofR D1 D1') + %<- ({x} {d} {y} {e} wfes-context (D2 x d y e) (sordered/cons (Dbound x d y e))) + %<- ({x} {d} {y} {e} weaken-slookup (Dbound x d y e) (Dlook x d) _ (Dlook' x d y e)) + %<- ({x} {d} {y} {e} weaken-aofes (Dbound x d y e) (DofR x d) _ (DofR' x d y e)) + %<- ({y} {e} stsubst-es ([x] [d] Dlook' x d y e) ([x] [d] DofR' x d y e) ([x] [d] D2 x d y e) ([x] [d] D2' x d y e)) %. +%term _ + %pi (stsubst-es Dlook DofR ([x] [d] wfes/sigma (D2 x d) (%the (simp _ T) (Dsimp x)) (%the (wfes _ (A x)) (D1 x d))) ([x] [d] wfes/sigma (D2' x d) (Dsimp R) (D1' x d))) + %<- (stsubst-es Dlook DofR D1 D1') + %<- ({x} {d} {y} {e} wfes-context (D2 x d y e) (sordered/cons (Dbound x d y e))) + %<- ({x} {d} {y} {e} weaken-slookup (Dbound x d y e) (Dlook x d) _ (Dlook' x d y e)) + %<- ({x} {d} {y} {e} weaken-aofes (Dbound x d y e) (DofR x d) _ (DofR' x d y e)) + %<- ({y} {e} stsubst-es ([x] [d] Dlook' x d y e) ([x] [d] DofR' x d y e) ([x] [d] D2 x d y e) ([x] [d] D2' x d y e)) %. +%term _ + %pi (stsubst-es Dlook DofR ([x] [d] wfes/sing (D x d)) ([x] [d] wfes/sing (D' x d))) + %<- (sasubst-es Dlook DofR D D') %. +%worlds (var bind ovar) (sasubst-es _ _ _ _) (ssubst-es _ _ _ _) (stsubst-es _ _ _ _) %. +%total (D1 D2 D3) (sasubst-es _ _ D1 _) (ssubst-es _ _ D2 _) (stsubst-es _ _ D3 _) %. +%sort strengthen-aofes {x} %. +%mode strengthen-aofes %in %in %in %out %. +%sort strengthen-ofes {x} %. +%mode strengthen-ofes %in %in %in %out %. +%sort strengthen-wfes {x} %. +%mode strengthen-wfes %in %in %in %out %. +%scope strengthen-wfes %term closed + %pi (strengthen-aofes (%the ({x} sappend (scons G1 x S) G2 (G x)) Dappend) (%the (sappend G1 G2 G') Dappend') ([x] [d] aofes/closed (%the (sordered (G x)) (Dord x d)) (%the (simp (A x) T) (Dsimp x)) (%the (aof R (A x)) (Daof x))) (aofes/closed Dord'' (Dsimp' aca) (Daof' aca))) + %<- (aof-noassm Daof _ (%the ({x} tp-eq (A x) A') Deq)) + %<- ({x} {d} sordered-drop (Dappend x) Dappend' (Dord x d) (Dord' x d)) + %<- (strengthen-sordered Dord' Dord'') + %<- ({x} aof-resp atom-eq/i (Deq x) (Daof x) (%the (aof R A') (Daof' x))) + %<- ({x} simp-resp (Deq x) stp-eq/i (Dsimp x) (%the (simp A' T) (Dsimp' x))) %. +%term forall + %pi (strengthen-aofes (%the ({x} sappend (scons G1 x S) G2 (G x)) Dappend) (%the (sappend G1 G2 G') Dappend') ([x] [d] aofes/forall (%the (simp A T) (Dsimp x)) (%the (wfes (G x) A) (DwfA x d))) (aofes/forall (Dsimp aca) DwfA')) + %<- (strengthen-wfes Dappend Dappend' DwfA DwfA') %. +%scope strengthen-wfes %term var + %pi (strengthen-aofes (%the ({x} sappend (scons G1 x S) G2 (G x)) Dappend) (%the (sappend G1 G2 G') Dappend') ([x] [d] aofes/var (%the (slookup (G x) Y T) (Dlookup x d))) (aofes/var Dlookup')) + %<- (slookup-drop Dappend Dappend' Dlookup Dlookup') %. +%term app + %pi (strengthen-aofes (%the ({x} sappend (scons G1 x S) G2 (G x)) Dappend) (%the (sappend G1 G2 G') Dappend') ([x] [d] aofes/app (%the (ofes (G x) M T) (DofM x d)) (%the (aofes (G x) R (spi T U)) (DofR x d))) (aofes/app DofM' DofR')) + %<- (strengthen-aofes Dappend Dappend' DofR DofR') + %<- (strengthen-ofes Dappend Dappend' DofM DofM') %. +%scope strengthen-wfes %term pi1 + %pi (strengthen-aofes (%the ({x} sappend (scons G1 x S) G2 (G x)) Dappend) (%the (sappend G1 G2 G') Dappend') ([x] [d] aofes/pi1 (%the (aofes (G x) R (ssigma T U)) (DofR x d))) (aofes/pi1 DofR')) + %<- (strengthen-aofes Dappend Dappend' DofR DofR') %. +%term pi2 + %pi (strengthen-aofes (%the ({x} sappend (scons G1 x S) G2 (G x)) Dappend) (%the (sappend G1 G2 G') Dappend') ([x] [d] aofes/pi2 (%the (aofes (G x) R (ssigma T U)) (DofR x d))) (aofes/pi2 DofR')) + %<- (strengthen-aofes Dappend Dappend' DofR DofR') %. +%scope strengthen-wfes %term at + %pi (strengthen-ofes (%the ({x} sappend (scons G1 x S) G2 (G x)) Dappend) (%the (sappend G1 G2 G') Dappend') ([x] [d] ofes/at (%the (aofes (G x) R st) (DofR x d))) (ofes/at DofR')) + %<- (strengthen-aofes Dappend Dappend' DofR DofR') %. +%term lam + %pi (strengthen-ofes (%the ({x} sappend (scons G1 x S) G2 (G x)) Dappend) (%the (sappend G1 G2 G') Dappend') ([x] [d] ofes/lam (%the ({y} %pi (isvar y J) %-> (ofes (scons (G x) y T) (M y) U)) (DofM x d))) (ofes/lam DofM')) + %<- ({y} {e} strengthen-ofes ([x] sappend/cons (Dappend x)) (sappend/cons Dappend') ([x] [d] DofM x d y e) (DofM' y e)) %. +%scope strengthen-wfes %term pair + %pi (strengthen-ofes (%the ({x} sappend (scons G1 x S) G2 (G x)) Dappend) (%the (sappend G1 G2 G') Dappend') ([x] [d] ofes/pair (%the (ofes (G x) N U) (DofN x d)) (%the (ofes (G x) M T) (DofM x d))) (ofes/pair DofN' DofM')) + %<- (strengthen-ofes Dappend Dappend' DofM DofM') + %<- (strengthen-ofes Dappend Dappend' DofN DofN') %. +%term t + %pi (strengthen-wfes (%the ({x} sappend (scons G1 x S) G2 (G x)) Dappend) (%the (sappend G1 G2 G') Dappend') ([x] [d] wfes/t (%the (sordered (G x)) (Dord x d))) (wfes/t Dord'')) + %<- ({x} {d} sordered-drop (Dappend x) Dappend' (Dord x d) (Dord' x d)) + %<- (strengthen-sordered Dord' Dord'') %. +%scope strengthen-wfes %term pi + %pi (strengthen-wfes (%the ({x} sappend (scons G1 x S) G2 (G x)) Dappend) (%the (sappend G1 G2 G') Dappend') ([x] [d] wfes/pi (%the ({y} %pi (isvar y J) %-> (wfes (scons (G x) y T) (B y))) (DwfB x d)) (%the (simp A T) (Dsimp x)) (%the (wfes (G x) A) (DwfA x d))) (wfes/pi DwfB' (Dsimp aca) DwfA')) + %<- (strengthen-wfes Dappend Dappend' DwfA DwfA') + %<- ({y} {e} strengthen-wfes ([x] sappend/cons (Dappend x)) (sappend/cons Dappend') ([x] [d] DwfB x d y e) (DwfB' y e)) %. +%term sigma + %pi (strengthen-wfes (%the ({x} sappend (scons G1 x S) G2 (G x)) Dappend) (%the (sappend G1 G2 G') Dappend') ([x] [d] wfes/sigma (%the ({y} %pi (isvar y J) %-> (wfes (scons (G x) y T) (B y))) (DwfB x d)) (%the (simp A T) (Dsimp x)) (%the (wfes (G x) A) (DwfA x d))) (wfes/sigma DwfB' (Dsimp aca) DwfA')) + %<- (strengthen-wfes Dappend Dappend' DwfA DwfA') + %<- ({y} {e} strengthen-wfes ([x] sappend/cons (Dappend x)) (sappend/cons Dappend') ([x] [d] DwfB x d y e) (DwfB' y e)) %. +%scope strengthen-wfes %term sing + %pi (strengthen-wfes (%the ({x} sappend (scons G1 x S) G2 (G x)) Dappend) (%the (sappend G1 G2 G') Dappend') ([x] [d] wfes/sing (%the (aofes (G x) R st) (DofR x d))) (wfes/sing DofR')) + %<- (strengthen-aofes Dappend Dappend' DofR DofR') %. +%worlds (var bind ovar) (strengthen-aofes _ _ _ _) (strengthen-ofes _ _ _ _) (strengthen-wfes _ _ _ _) %. +%total (D1 D2 D3) (strengthen-aofes _ _ D1 _) (strengthen-ofes _ _ D2 _) (strengthen-wfes _ _ D3 _) %. +%sort aasubst-es {x} %. +%mode aasubst-es %in %in %in %in %out %. +%sort aosubst-es {x} %. +%mode aosubst-es %in %in %in %in %out %. +%sort subst-es {x} %. +%mode subst-es %in %in %in %in %out %. +%sort tsubst-es {x} %. +%mode tsubst-es %in %in %in %in %out %. +%scope tsubst-es %term closed aasubst-es (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) (%the ({x} %pi (isvar x I) %-> (aofes (G x) R T)) DofR) aasub/closed DofR %. +%term closed + %pi (aasubst-es (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) ([x] [d] aofes/closed (%the (sordered (G x)) (Dord x d)) (%the (simp (A x) T) (Dsimp x)) (%the (aof (R x) (A x)) (Daof x))) (%the (aasub R M Rx) Dsub) ([x] [d] aofes/closed (Dord x d) (Dsimp' aca) Daof'')) + %<- (aof-noassm Daof (%the ({x} atom-eq (R x) R') DeqR) (%the ({x} tp-eq (A x) A') DeqA)) + %<- (aasub-resp DeqR term-eq/i atom-eq/i Dsub (%the (aasub ([_] R') M Rx) Dsub')) + %<- (aasub-absent-fun Dsub' (%the (atom-eq Rx R') DeqRx)) + %<- (atom-eq-symm DeqRx (%the (atom-eq R' Rx) DeqRx')) + %<- ({x} aof-resp (DeqR x) (DeqA x) (Daof x) (%the (aof R' A') (Daof' x))) + %<- (aof-resp DeqRx' tp-eq/i (Daof' aca) (%the (aof Rx A') Daof'')) + %<- ({x} simp-resp (DeqA x) stp-eq/i (Dsimp x) (%the (simp A' T) (Dsimp' x))) %. +%scope tsubst-es %term forall + %pi (aasubst-es (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) ([x] [d] aofes/forall (Dsimp x) (%the (wfes (G x) (A x)) (DwfA x d))) (aasub/forall DsubA) ([x] [d] aofes/forall Dsimp' (DwfA' x d))) + %<- (tsubst-es Dlookup DofM DwfA DsubA DwfA') + %<- (tsub-preserves-simp DsubA Dsimp Dsimp') %. +%term _ + %pi (aasubst-es _ _ ([x] [d] aofes/var (Dlookup x d)) _ D) + %<- ({x} {d} slookup-isvar (Dlookup x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-forall-contra (Disvar x d) Dfalse) + %<- ({x} {d} false-implies-aofes Dfalse (D x d)) %. +%term app + %pi (aasubst-es (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) ([x] [d] aofes/app (%the (ofes (G x) (N x) T) (DofN x d)) (%the (aofes (G x) (R x) (spi T U)) (DofR x d))) (aasub/app DsubN DsubR) ([x] [d] aofes/app (DofN' x d) (DofR' x d))) + %<- (aasubst-es Dlookup DofM DofR DsubR DofR') + %<- (subst-es Dlookup DofM DofN DsubN DofN') %. +%term _ + %pi (aasubst-es _ _ ([x] [d] aofes/var (Dlookup x d)) _ D) + %<- ({x} {d} slookup-isvar (Dlookup x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-app-contra (Disvar x d) Dfalse) + %<- ({x} {d} false-implies-aofes Dfalse (D x d)) %. +%term pi1 + %pi (aasubst-es (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) ([x] [d] aofes/pi1 (%the (aofes (G x) (R x) (ssigma T U)) (DofR x d))) (aasub/pi1 DsubR) ([x] [d] aofes/pi1 (DofR' x d))) + %<- (aasubst-es Dlookup DofM DofR DsubR DofR') %. +%term _ + %pi (aasubst-es _ _ ([x] [d] aofes/var (Dlookup x d)) _ D) + %<- ({x} {d} slookup-isvar (Dlookup x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-pi1-contra (Disvar x d) Dfalse) + %<- ({x} {d} false-implies-aofes Dfalse (D x d)) %. +%term pi2 + %pi (aasubst-es (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) ([x] [d] aofes/pi2 (%the (aofes (G x) (R x) (ssigma T U)) (DofR x d))) (aasub/pi2 DsubR) ([x] [d] aofes/pi2 (DofR' x d))) + %<- (aasubst-es Dlookup DofM DofR DsubR DofR') %. +%term _ + %pi (aasubst-es _ _ ([x] [d] aofes/var (Dlookup x d)) _ D) + %<- ({x} {d} slookup-isvar (Dlookup x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-pi2-contra (Disvar x d) Dfalse) + %<- ({x} {d} false-implies-aofes Dfalse (D x d)) %. +%%% +%term closed + %pi (aosubst-es (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) ([x] [d] aofes/closed (%the (sordered (G x)) (Dord x d)) (%the (simp (A x) T) (Dsimp x)) (%the (aof (R x) (A x)) (Daof x))) (%the (aosub R M LRx) Dsub) D) + %<- (aof-noassm Daof (%the ({x} atom-eq (R x) R') DeqR) (%the ({x} tp-eq (A x) A') DeqA)) + %<- (aosub-resp DeqR term-eq/i term-eq/i Dsub (%the (aosub ([_] R') M LRx) Dsub')) + %<- (aosub-absent Dsub' Dfalse) + %<- ({x} {d} false-implies-ofes Dfalse (D x d)) %. +%term var + %pi (aosubst-es (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) ([x] [d] aofes/var (%the (slookup (G x) x T) (Dlookup' x d))) aosub/var DofM') + %<- ({x} {d} slookup-fun (Dlookup x d) (Dlookup' x d) (%the (stp-eq S T) Deq)) + %<- ({x} {d} ofes-resp sctx-eq/i term-eq/i Deq (DofM x d) (%the (ofes (G x) M T) (DofM' x d))) %. +%term app + %pi (aosubst-es (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) ([x] [d] aofes/app (%the (ofes (G x) (N x) T) (DofN x d)) (%the (aofes (G x) (R x) (spi T U)) (DofR x d))) (aosub/app (%the (sub P Nx Py) DsubP) (%the (sub N M Nx) DsubN) (%the (aosub R M (lam P)) DsubR)) DofP'') + %<- (aosubst-es Dlookup DofM DofR DsubR ([x] [d] ofes/lam (%the ({y} %pi (isvar y J) %-> (ofes (scons (G x) y T) (P y) U)) (DofP x d)))) + %<- (subst-es Dlookup DofM DofN DsubN (%the ({x} %pi (isvar x I) %-> (ofes (G x) Nx T)) DofN')) + %<- ({x} {d} {y} {e} ofes-context (DofP x d y e) (sordered/cons (%the (sbounded (G x) y) (Dbound x d y e)))) + %<- ({x} {d} {y} {e} weaken-ofes (Dbound x d y e) (DofN' x d) _ (%the (ofes (scons (G x) y T) Nx T) (DofN'' x d y e))) + %<- ({x} {d} subst-es ([y] [e] slookup/hit (Dbound x d y e)) ([y] [e] DofN'' x d y e) ([y] [e] DofP x d y e) DsubP ([y] [e] %the (ofes (scons (G x) y T) Py U) (DofP' x d y e))) + %<- ({x} {d} strengthen-ofes ([y] sappend/nil) sappend/nil ([y] [e] DofP' x d y e) (DofP'' x d)) %. +%term _ + %pi (aosubst-es _ _ ([x] [d] aofes/var (Dlookup x d)) _ D) + %<- ({x} {d} slookup-isvar (Dlookup x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-app-contra (Disvar x d) Dfalse) + %<- ({x} {d} false-implies-ofes Dfalse (D x d)) %. +%term pi1 + %pi (aosubst-es (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) ([x] [d] aofes/pi1 (%the (aofes (G x) (R x) (ssigma T U)) (DofR x d))) (aosub/pi1 (%the (aosub R M (pair P1 P2)) Dsub)) DofP1) + %<- (aosubst-es Dlookup DofM DofR Dsub ([x] [d] ofes/pair (%the (ofes (G x) P2 U) (DofP2 x d)) (%the (ofes (G x) P1 T) (DofP1 x d)))) %. +%term _ + %pi (aosubst-es _ _ ([x] [d] aofes/var (Dlookup x d)) _ D) + %<- ({x} {d} slookup-isvar (Dlookup x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-pi1-contra (Disvar x d) Dfalse) + %<- ({x} {d} false-implies-ofes Dfalse (D x d)) %. +%term pi2 + %pi (aosubst-es (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) ([x] [d] aofes/pi2 (%the (aofes (G x) (R x) (ssigma T U)) (DofR x d))) (aosub/pi2 (%the (aosub R M (pair P1 P2)) Dsub)) DofP2) + %<- (aosubst-es Dlookup DofM DofR Dsub ([x] [d] ofes/pair (%the (ofes (G x) P2 U) (DofP2 x d)) (%the (ofes (G x) P1 T) (DofP1 x d)))) %. +%term _ + %pi (aosubst-es _ _ ([x] [d] aofes/var (Dlookup x d)) _ D) + %<- ({x} {d} slookup-isvar (Dlookup x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-pi2-contra (Disvar x d) Dfalse) + %<- ({x} {d} false-implies-ofes Dfalse (D x d)) %. +%%% +%term aa + %pi (subst-es (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) ([x] [d] ofes/at (%the (aofes (G x) (R x) st) (DofR x d))) (sub/aa Dsub) ([x] [d] ofes/at (DofR' x d))) + %<- (aasubst-es Dlookup DofM DofR Dsub DofR') %. +%term ao + %pi (subst-es (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) ([x] [d] ofes/at (%the (aofes (G x) (R x) st) (DofR x d))) (sub/ao Dsub) D) + %<- (aosubst-es Dlookup DofM DofR Dsub D) %. +%term lam + %pi (subst-es (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) ([x] [d] ofes/lam (%the ({y} %pi (isvar y J) %-> (ofes (scons (G x) y T) (N x y) U)) (DofN x d))) (sub/lam (%the ({y} sub ([x] N x y) M (Nx y)) Dsub)) ([x] [d] ofes/lam (DofN' x d))) + %<- ({x} {d} {y} {e} ofes-context (DofN x d y e) (sordered/cons (%the (sbounded (G x) y) (Dbound x d y e)))) + %<- ({x} {d} {y} {e} weaken-ofes (Dbound x d y e) (DofM x d) _ (%the (ofes (scons (G x) y T) M S) (DofM' x d y e))) + %<- ({y} {e} subst-es ([x] [d] slookup/miss (Dlookup x d) (Dbound x d y e)) ([x] [d] DofM' x d y e) ([x] [d] DofN x d y e) (Dsub y) ([x] [d] %the (ofes (scons (G x) y T) (Nx y) U) (DofN' x d y e))) %. +%term pair + %pi (subst-es (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) ([x] [d] ofes/pair (%the (ofes (G x) (O x) U) (DofO x d)) (%the (ofes (G x) (N x) T) (DofN x d))) (sub/pair (%the (sub O M Ox) DsubO) (%the (sub N M Nx) DsubN)) ([x] [d] ofes/pair (DofO' x d) (DofN' x d))) + %<- (subst-es Dlookup DofM DofN DsubN DofN') + %<- (subst-es Dlookup DofM DofO DsubO DofO') %. +%%% +%term t tsubst-es (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) ([x] [d] wfes/t (Dord x d)) tsub/t ([x] [d] wfes/t (Dord x d)) %. +%term pi + %pi (tsubst-es (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) ([x] [d] wfes/pi (%the ({y} %pi (isvar y J) %-> (wfes (scons (G x) y T) (B x y))) (DwfB x d)) (%the (simp (A x) T) (Dsimp x)) (%the (wfes (G x) (A x)) (DwfA x d))) (tsub/pi (%the ({y} tsub ([x] B x y) M (Bx y)) DsubB) (%the (tsub A M Ax) DsubA)) ([x] [d] wfes/pi (DwfB' x d) Dsimp' (DwfA' x d))) + %<- (tsubst-es Dlookup DofM DwfA DsubA DwfA') + %<- ({x} {d} {y} {e} wfes-context (DwfB x d y e) (sordered/cons (%the (sbounded (G x) y) (Dbound x d y e)))) + %<- ({x} {d} {y} {e} weaken-ofes (Dbound x d y e) (DofM x d) _ (%the (ofes (scons (G x) y T) M S) (DofM' x d y e))) + %<- ({y} {e} tsubst-es ([x] [d] slookup/miss (Dlookup x d) (Dbound x d y e)) ([x] [d] DofM' x d y e) ([x] [d] DwfB x d y e) (DsubB y) ([x] [d] %the (wfes (scons (G x) y T) (Bx y)) (DwfB' x d y e))) + %<- (tsub-preserves-simp DsubA Dsimp Dsimp') %. +%term sigma + %pi (tsubst-es (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) ([x] [d] wfes/sigma (%the ({y} %pi (isvar y J) %-> (wfes (scons (G x) y T) (B x y))) (DwfB x d)) (%the (simp (A x) T) (Dsimp x)) (%the (wfes (G x) (A x)) (DwfA x d))) (tsub/sigma (%the ({y} tsub ([x] B x y) M (Bx y)) DsubB) (%the (tsub A M Ax) DsubA)) ([x] [d] wfes/sigma (DwfB' x d) Dsimp' (DwfA' x d))) + %<- (tsubst-es Dlookup DofM DwfA DsubA DwfA') + %<- ({x} {d} {y} {e} wfes-context (DwfB x d y e) (sordered/cons (%the (sbounded (G x) y) (Dbound x d y e)))) + %<- ({x} {d} {y} {e} weaken-ofes (Dbound x d y e) (DofM x d) _ (%the (ofes (scons (G x) y T) M S) (DofM' x d y e))) + %<- ({y} {e} tsubst-es ([x] [d] slookup/miss (Dlookup x d) (Dbound x d y e)) ([x] [d] DofM' x d y e) ([x] [d] DwfB x d y e) (DsubB y) ([x] [d] %the (wfes (scons (G x) y T) (Bx y)) (DwfB' x d y e))) + %<- (tsub-preserves-simp DsubA Dsimp Dsimp') %. +%term singa + %pi (tsubst-es (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) ([x] [d] wfes/sing (%the (aofes (G x) (R x) st) (DofR x d))) (tsub/singa Dsub) ([x] [d] wfes/sing (DofR' x d))) + %<- (aasubst-es Dlookup DofM DofR Dsub DofR') %. +%term singo + %pi (tsubst-es (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) ([x] [d] wfes/sing (%the (aofes (G x) (R x) st) (DofR x d))) (tsub/singo Dsub) ([x] [d] wfes/sing (D x d))) + %<- (aosubst-es Dlookup DofM DofR Dsub ([x] [d] ofes/at (D x d))) %. +%worlds (var ovar bind) (aasubst-es _ _ _ _ _) (aosubst-es _ _ _ _ _) (subst-es _ _ _ _ _) (tsubst-es _ _ _ _ _) %. +%total (D1 D2 D3 D4) (aasubst-es _ _ _ D1 _) (aosubst-es _ _ _ D2 _) (subst-es _ _ _ D3 _) (tsubst-es _ _ _ D4 _) %. +%%%%% Substitution Effectiveness (Simply Typed, Explicit Context Version) %%%%% +%sort ofes-pi-invert-underbind {x} %. +%mode ofes-pi-invert-underbind %in %out %out %. +%term _ ofes-pi-invert-underbind ([x] [d] ofes/lam (D x d)) term-eq/i D %. +%worlds (var bind ovar) (ofes-pi-invert-underbind _ _ _) %. +%total {} (ofes-pi-invert-underbind _ _ _) %. +%sort ofes-sigma-invert-underbind {x} %. +%mode ofes-sigma-invert-underbind %in %out %out %out %. +%term _ ofes-sigma-invert-underbind ([x] [d] ofes/pair (D2 x d) (D1 x d)) term-eq/i D1 D2 %. +%worlds (var bind ovar) (ofes-sigma-invert-underbind _ _ _ _) %. +%total {} (ofes-sigma-invert-underbind _ _ _ _) %. +%sort ofes-t-invert-underbind {x} %. +%mode ofes-t-invert-underbind %in %out %. +%term _ ofes-t-invert-underbind ([x] [d] ofes/at _) term-eq/i %. +%worlds (var bind ovar) (ofes-t-invert-underbind _ _) %. +%total {} (ofes-t-invert-underbind _ _) %. +%sort asub %. +%term asub/aasub %pi (asub R M) %<- (aasub R M _) %. +%term asub/aosub %pi (asub R M) %<- (aosub R M _) %. +%sort can-asub-pi1 {x} %. +%mode can-asub-pi1 %in %in %in %in %out %. +%scope can-asub-pi1 %term aa can-asub-pi1 _ _ _ (asub/aasub Daasub) (asub/aasub (aasub/pi1 Daasub)) %. +%term ao + %pi (can-asub-pi1 Dlookup DofR DofM (asub/aosub Daosub) (asub/aosub (aosub/pi1 Daosub'))) + %<- (aosubst-es Dlookup DofM DofR Daosub (%the ({x} %pi (isvar x I) %-> (ofes (G x) LRx (ssigma T U))) DofLRx)) + %<- (ofes-sigma-invert-underbind DofLRx (%the (term-eq LRx (pair P1 P2)) Deq) _ _) + %<- (aosub-resp ([_] atom-eq/i) term-eq/i Deq Daosub (%the (aosub R M (pair P1 P2)) Daosub')) %. +%worlds (var bind ovar) (can-asub-pi1 _ _ _ _ _) %. +%total {} (can-asub-pi1 _ _ _ _ _) %. +%sort can-asub-pi2 {x} %. +%mode can-asub-pi2 %in %in %in %in %out %. +%scope can-asub-pi2 %term aa can-asub-pi2 _ _ _ (asub/aasub Daasub) (asub/aasub (aasub/pi2 Daasub)) %. +%term ao + %pi (can-asub-pi2 Dlookup DofR DofM (asub/aosub Daosub) (asub/aosub (aosub/pi2 Daosub'))) + %<- (aosubst-es Dlookup DofM DofR Daosub (%the ({x} %pi (isvar x I) %-> (ofes (G x) LRx (ssigma T U))) DofLRx)) + %<- (ofes-sigma-invert-underbind DofLRx (%the (term-eq LRx (pair P1 P2)) Deq) _ _) + %<- (aosub-resp ([_] atom-eq/i) term-eq/i Deq Daosub (%the (aosub R M (pair P1 P2)) Daosub')) %. +%worlds (var bind ovar) (can-asub-pi2 _ _ _ _ _) %. +%total {} (can-asub-pi2 _ _ _ _ _) %. +%sort asub-sub-at %. +%mode asub-sub-at %in %out %. +%term _ asub-sub-at (asub/aasub D) (sub/aa D) %. +%term _ asub-sub-at (asub/aosub D) (sub/ao D) %. +%worlds (var) (asub-sub-at _ _) %. +%total {} (asub-sub-at _ _) %. +%sort asub-tsub-sing {x} %. +%mode asub-tsub-sing %in %in %in %in %out %. +%term _ asub-tsub-sing _ _ _ (asub/aasub D) (tsub/singa D) %. +%term _ + %pi (asub-tsub-sing Dlookup DofR DofM (asub/aosub Daosub) (tsub/singo Daosub')) + %<- (aosubst-es Dlookup DofM DofR Daosub (%the ({x} %pi (isvar x I) %-> (ofes (G x) LRx st)) DofLRx)) + %<- (ofes-t-invert-underbind DofLRx (%the (term-eq LRx (at R')) Deq)) + %<- (aosub-resp ([_] atom-eq/i) term-eq/i Deq Daosub (%the (aosub R M (at R')) Daosub')) %. +%worlds (var bind ovar) (asub-tsub-sing _ _ _ _ _) %. +%total {} (asub-tsub-sing _ _ _ _ _) %. +%sort can-asub-esm {S} {x} %. +%mode can-asub-esm %in %in %in %in %out %. +%sort can-sub-esm {S} {x} %. +%mode can-sub-esm %in %in %in %in %out %. +%sort can-asub-app {S} {x} %. +%mode can-asub-app %in %in %in %in %in %in %in %out %. +%sort can-tsub-esm {S} {x} %. +%mode can-tsub-esm %in %in %in %in %out %. +%scope can-tsub-esm %term closed + %pi (can-asub-esm S (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) ([x] [d] aofes/closed _ (%the (simp (A x) T) (Dsimp x)) (%the (aof (R x) (A x)) (Daof x))) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) (asub/aasub Daasub)) + %<- (aof-noassm Daof (%the ({x} atom-eq (R x) R') Deq) _) + %<- ({x} atom-eq-symm (Deq x) (%the (atom-eq R' (R x)) (Deq' x))) + %<- (aasub-resp Deq' term-eq/i atom-eq/i aasub/closed (%the (aasub R M R') Daasub)) %. +%term closed can-asub-esm _ _ _ _ (asub/aasub aasub/closed) %. +%scope can-tsub-esm %term var can-asub-esm _ _ _ _ (asub/aosub aosub/var) %. +%term forall + %pi (can-asub-esm S (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) ([x] [d] aofes/forall (%the (simp (A x) T) (Dsimp x)) (%the (wfes (G x) (A x)) (DofA x d))) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) (asub/aasub (aasub/forall Dsub))) + %<- (can-tsub-esm S Dlookup DofA DofM (%the (tsub A M A') Dsub)) %. +%scope can-tsub-esm %term allbad + %pi (can-asub-esm _ _ ([x] [d] aofes/var (Dlookup x d)) _ (asub/aosub Daosub)) + %<- ({x} {d} slookup-isvar (Dlookup x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-forall-contra (Disvar x d) Dfalse) + %<- (false-implies-aosub Dfalse (%the (aosub _ _ (at aca)) Daosub)) %. +%term app + %pi (can-asub-esm S (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) ([x] [d] aofes/app (%the (ofes (G x) (N x) T) (DofN x d)) (%the (aofes (G x) (R x) (spi T U)) (DofR x d))) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) Dasub') + %<- (can-asub-esm S Dlookup DofR DofM Dasub) + %<- (can-sub-esm S Dlookup DofN DofM Dsub) + %<- (can-asub-app S Dlookup DofR DofN DofM Dasub Dsub Dasub') %. +%scope can-tsub-esm %term ap-aa can-asub-app S (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (aofes (G x) (R x) (spi T U))) DofR) (%the ({x} %pi (isvar x I) %-> (ofes (G x) (N x) T)) DofN) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) (asub/aasub (%the (aasub R M Rx) Daasub)) (%the (sub N M Nx) Dsub) (asub/aasub (aasub/app Dsub Daasub)) %. +%term app-ao + %pi (can-asub-app S (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) (%the ({x} %pi (isvar x I) %-> (aofes (G x) (R x) (spi T U))) DofR) (%the ({x} %pi (isvar x I) %-> (ofes (G x) (N x) T)) DofN) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) (asub/aosub (%the (aosub R M LRx) Daosub)) (%the (sub N M Nx) DsubNx) (asub/aosub (aosub/app (DsubPy_x aca) DsubNx Daosub'))) + %<- (aosub-headvar Daosub (%the (headvar R) Dheadvar)) + %<- (headvar-stp-size Dlookup DofR Dheadvar (%the (stp-leq (spi T U) S) Dleq)) + %<- (aosubst-es Dlookup DofM DofR Daosub (%the ({x} %pi (isvar x I) %-> (ofes (G x) LRx (spi T U))) DofLRx)) + %<- (subst-es Dlookup DofM DofN DsubNx (%the ({x} %pi (isvar x I) %-> (ofes (G x) Nx T)) DofNx)) + %<- (ofes-pi-invert-underbind DofLRx (%the (term-eq LRx (lam P)) DeqLRx) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (scons (G x) y T) (P y) U))) DofP)) + %<- (aosub-resp ([_] atom-eq/i) term-eq/i DeqLRx Daosub (%the (aosub R M (lam P)) Daosub')) + %<- ({x} {d} {y} {e} ofes-context (DofP x d y e) (sordered/cons (%the (sbounded (G x) y) (Dbound x d y e)))) + %<- ({x} {d} {y} {e} weaken-ofes (Dbound x d y e) (DofNx x d) _ (%the (ofes (scons (G x) y T) Nx T) (DofNx' x d y e))) + %<- (employ-stp-leq (spi T U) S Dleq) + %<- ({x} {d} can-sub-esm T ([y] [e] slookup/hit (Dbound x d y e)) ([y] [e] DofP x d y e) ([y] [e] DofNx' x d y e) (%the (sub P Nx (Py_x x)) (DsubPy_x x))) %. +%scope can-tsub-esm %term appbad + %pi (can-asub-esm _ _ ([x] [d] aofes/var (Dlookup x d)) _ (asub/aosub Daosub)) + %<- ({x} {d} slookup-isvar (Dlookup x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-app-contra (Disvar x d) Dfalse) + %<- (false-implies-aosub Dfalse (%the (aosub _ _ (at aca)) Daosub)) %. +%term pi1 + %pi (can-asub-esm S (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) ([x] [d] aofes/pi1 (%the (aofes (G x) (R x) (ssigma T U)) (DofR x d))) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) Dasub') + %<- (can-asub-esm S Dlookup DofR DofM (%the (asub R M) Dasub)) + %<- (can-asub-pi1 Dlookup DofR DofM Dasub (%the (asub ([x] pi1 (R x)) M) Dasub')) %. +%scope can-tsub-esm %term pi1bad + %pi (can-asub-esm _ _ ([x] [d] aofes/var (Dlookup x d)) _ (asub/aosub Daosub)) + %<- ({x} {d} slookup-isvar (Dlookup x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-pi1-contra (Disvar x d) Dfalse) + %<- (false-implies-aosub Dfalse (%the (aosub _ _ (at aca)) Daosub)) %. +%term pi2 + %pi (can-asub-esm S (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) ([x] [d] aofes/pi2 (%the (aofes (G x) (R x) (ssigma T U)) (DofR x d))) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) Dasub') + %<- (can-asub-esm S Dlookup DofR DofM (%the (asub R M) Dasub)) + %<- (can-asub-pi2 Dlookup DofR DofM Dasub (%the (asub ([x] pi2 (R x)) M) Dasub')) %. +%scope can-tsub-esm %term pi2bad + %pi (can-asub-esm _ _ ([x] [d] aofes/var (Dlookup x d)) _ (asub/aosub Daosub)) + %<- ({x} {d} slookup-isvar (Dlookup x d) (%the (isvar _ I) (Disvar x d))) + %<- ({x} {d} isvar-pi2-contra (Disvar x d) Dfalse) + %<- (false-implies-aosub Dfalse (%the (aosub _ _ (at aca)) Daosub)) %. +%term at + %pi (can-sub-esm S (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) ([x] [d isvar x I] ofes/at (%the (aofes (G x) (R x) st) (DofR x d))) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) Dsub) + %<- (can-asub-esm S Dlookup DofR DofM (%the (asub R M) Dasub)) + %<- (asub-sub-at Dasub (%the (sub ([x] at (R x)) M _) Dsub)) %. +%%% +%scope can-tsub-esm %term lam + %pi (can-sub-esm S (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) ([x] [d isvar x I] ofes/lam (%the ({y} %pi (isvar y J) %-> (ofes (scons (G x) y T) (N x y) U)) (DofN x d))) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) (sub/lam Dsub)) + %<- ({x} {d} {y} {e} ofes-context (DofN x d y e) (sordered/cons (%the (sbounded (G x) y) (Dbound x d y e)))) + %<- ({x} {d} {y} {e} weaken-slookup (Dbound x d y e) (Dlookup x d) _ (%the (slookup (scons (G x) y T) x S) (Dlookup' x d y e))) + %<- ({x} {d} {y} {e} weaken-ofes (Dbound x d y e) (DofM x d) _ (%the (ofes (scons (G x) y T) M S) (DofM' x d y e))) + %<- ({y} {e} can-sub-esm S ([x] [d] Dlookup' x d y e) ([x] [d] DofN x d y e) ([x] [d] DofM' x d y e) (%the (sub ([x] N x y) M (Nx y)) (Dsub y))) %. +%term pair + %pi (can-sub-esm S (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) ([x] [d isvar x I] ofes/pair (%the (ofes (G x) (O x) U) (DofO x d)) (%the (ofes (G x) (N x) T) (DofN x d))) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) (sub/pair DsubOx DsubNx)) + %<- (can-sub-esm S Dlookup DofN DofM (%the (sub N M Nx) DsubNx)) + %<- (can-sub-esm S Dlookup DofO DofM (%the (sub O M Ox) DsubOx)) %. +%%% +%scope can-tsub-esm %term t can-tsub-esm _ _ ([x] [d] wfes/t _) _ tsub/t %. +%term pi + %pi (can-tsub-esm _ (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) ([x] [d isvar x I] wfes/pi (%the ({y} %pi (isvar y J) %-> (wfes (scons (G x) y T) (B x y))) (DwfB x d)) (%the (simp (A x) T) (DsimpA x)) (%the (wfes (G x) (A x)) (DwfA x d))) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) (tsub/pi DsubB DsubA)) + %<- (can-tsub-esm _ Dlookup DwfA DofM (%the (tsub A M Ax) DsubA)) + %<- ({x} {d} {y} {e} wfes-context (DwfB x d y e) (sordered/cons (%the (sbounded (G x) y) (Dbound x d y e)))) + %<- ({x} {d} {y} {e} weaken-slookup (Dbound x d y e) (Dlookup x d) _ (%the (slookup (scons (G x) y T) x S) (Dlookup' x d y e))) + %<- ({x} {d} {y} {e} weaken-ofes (Dbound x d y e) (DofM x d) _ (%the (ofes (scons (G x) y T) M S) (DofM' x d y e))) + %<- ({y} {e} can-tsub-esm _ ([x] [d] Dlookup' x d y e) ([x] [d] DwfB x d y e) ([x] [d] DofM' x d y e) (%the (tsub ([x] B x y) M (Bx y)) (DsubB y))) %. +%scope can-tsub-esm %term sigma + %pi (can-tsub-esm _ (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) ([x] [d isvar x I] wfes/sigma (%the ({y} %pi (isvar y J) %-> (wfes (scons (G x) y T) (B x y))) (DwfB x d)) (%the (simp (A x) T) (DsimpA x)) (%the (wfes (G x) (A x)) (DwfA x d))) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) (tsub/sigma DsubB DsubA)) + %<- (can-tsub-esm _ Dlookup DwfA DofM (%the (tsub A M Ax) DsubA)) + %<- ({x} {d} {y} {e} wfes-context (DwfB x d y e) (sordered/cons (%the (sbounded (G x) y) (Dbound x d y e)))) + %<- ({x} {d} {y} {e} weaken-slookup (Dbound x d y e) (Dlookup x d) _ (%the (slookup (scons (G x) y T) x S) (Dlookup' x d y e))) + %<- ({x} {d} {y} {e} weaken-ofes (Dbound x d y e) (DofM x d) _ (%the (ofes (scons (G x) y T) M S) (DofM' x d y e))) + %<- ({y} {e} can-tsub-esm _ ([x] [d] Dlookup' x d y e) ([x] [d] DwfB x d y e) ([x] [d] DofM' x d y e) (%the (tsub ([x] B x y) M (Bx y)) (DsubB y))) %. +%term sing + %pi (can-tsub-esm _ (%the ({x} %pi (isvar x I) %-> (slookup (G x) x S)) Dlookup) ([x] [d isvar x I] wfes/sing (%the (aofes (G x) (R x) st) (DofR x d))) (%the ({x} %pi (isvar x I) %-> (ofes (G x) M S)) DofM) Dtsub) + %<- (can-asub-esm S Dlookup DofR DofM (%the (asub R M) Dasub)) + %<- (asub-tsub-sing Dlookup DofR DofM Dasub (%the (tsub ([x] sing (R x)) M A) Dtsub)) %. +%worlds (bind ovar var) (can-asub-esm _ _ _ _ _) (can-sub-esm _ _ _ _ _) (can-asub-app _ _ _ _ _ _ _ _) (can-tsub-esm _ _ _ _ _) %. +%total {(S1 S2 S3 S4) (D1 D2 D3 D4)} (can-asub-esm S1 _ D1 _ _) (can-sub-esm S2 _ D2 _ _) (can-asub-app S3 _ D3 _ _ _ _ _) (can-tsub-esm S4 _ D4 _ _) %. +%sort can-sub-es {x} %. +%mode can-sub-es %in %in %in %out %. +%term _ + %pi (can-sub-es Dlookup DofN DofM Dsub) + %<- (can-sub-esm _ Dlookup DofN DofM Dsub) %. +%worlds (var bind ovar) (can-sub-es _ _ _ _) %. +%total {} (can-sub-es _ _ _ _) %. +%sort can-tsub-es {x} %. +%mode can-tsub-es %in %in %in %out %. +%term _ %pi (can-tsub-es Dlookup Dwf Dof Dsub) %<- (can-tsub-esm _ Dlookup Dwf Dof Dsub) %. +%worlds (var bind ovar) (can-tsub-es _ _ _ _) %. +%total D (can-tsub-es _ D _ _) %. +%%%%% Substitution Effectiveness (Explicit Context Version) %%%%% +%sort can-sub-e {x} %. +%mode can-sub-e %in %in %in %out %. +%term _ + %pi (can-sub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the ({x} %pi (isvar x I) %-> (ofe (G x) (N x) (B x))) DofN) (%the (ofe G1 M A) DofM) Dsub) + %<- ({x} {d} ofe-context (DofN x d) (%the (ordered (G x)) (Dordered x d))) + %<- ({x} {d} append-ordered (Dappend x) (Dordered x d) (ordered/cons (%the (bounded G1 x) (Dbounded x d)))) + %<- ({x} {d} weaken-ofe (Dbounded x d) DofM _ (%the (ofe (cons G1 x A) M A) (DofM' x d))) + %<- ({x} {d} weaken-ofe' (Dappend x) (Dordered x d) (DofM' x d) (%the (ofe (G x) M A) (DofM'' x d))) + %<- (can-simp A (%the (simp A S) DsimpA)) + %<- ({x} can-simp (B x) (%the (simp (B x) T) (DsimpB x))) + %<- (context-append-simp-lookup' Dappend Dordered DsimpA (%the ({x} simpctx (G x) (G' x)) Dsimpctx) (%the ({x} %pi (isvar x I) %-> (slookup (G' x) x S)) Dlookup)) + %<- ({x} {d} ofe-simp (Dsimpctx x) DsimpA (DofM'' x d) (%the (ofes (G' x) M S) (DofM''' x d))) + %<- ({x} {d} ofe-simp (Dsimpctx x) (DsimpB x) (DofN x d) (%the (ofes (G' x) (N x) T) (DofN' x d))) + %<- (can-sub-es Dlookup DofN' DofM''' (%the (sub N M Nx) Dsub)) %. +%worlds (var bind ovar) (can-sub-e _ _ _ _) %. +%total {} (can-sub-e _ _ _ _) %. +%sort can-tsub-e {x} %. +%mode can-tsub-e %in %in %in %out %. +%term _ + %pi (can-tsub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the ({x} %pi (isvar x I) %-> (wfe (G x) (B x))) DwfB) (%the (ofe G1 M A) DofM) Dsub) + %<- ({x} {d} wfe-context (DwfB x d) (%the (ordered (G x)) (Dordered x d))) + %<- ({x} {d} append-ordered (Dappend x) (Dordered x d) (ordered/cons (%the (bounded G1 x) (Dbounded x d)))) + %<- ({x} {d} weaken-ofe (Dbounded x d) DofM _ (%the (ofe (cons G1 x A) M A) (DofM' x d))) + %<- ({x} {d} weaken-ofe' (Dappend x) (Dordered x d) (DofM' x d) (%the (ofe (G x) M A) (DofM'' x d))) + %<- (can-simp A (%the (simp A S) DsimpA)) + %<- ({x} can-simp (B x) (%the (simp (B x) T) (DsimpB x))) + %<- (context-append-simp-lookup' Dappend Dordered DsimpA (%the ({x} simpctx (G x) (G' x)) Dsimpctx) (%the ({x} %pi (isvar x I) %-> (slookup (G' x) x S)) Dlookup)) + %<- ({x} {d} ofe-simp (Dsimpctx x) DsimpA (DofM'' x d) (%the (ofes (G' x) M S) (DofM''' x d))) + %<- ({x} {d} wfe-simp (Dsimpctx x) (DwfB x d) (%the (wfes (G' x) (B x)) (DwfB' x d))) + %<- (can-tsub-es Dlookup DwfB' DofM''' (%the (tsub B M Bx) Dsub)) %. +%worlds (bind ovar var) (can-tsub-e _ _ _ _) %. +%total {} (can-tsub-e _ _ _ _) %. +%%%%% Substitution Effectiveness %%%%% +%sort can-sub {x} %. +%mode can-sub %in %in %out %. +%term _ + %pi (can-sub DofN DofM Dsub) + %<- (of1-to-ofe 0 DofN DofeN) + %<- (of-to-ofe DofM DofeM) + %<- (can-sub-e ([_] append/nil) DofeN DofeM Dsub) %. +%worlds (bind var) (can-sub _ _ _) %. +%total {} (can-sub _ _ _) %. +%sort can-tsub {x} %. +%mode can-tsub %in %in %out %. +%term _ + %pi (can-tsub DwfB DofM Dsub) + %<- (wf1-to-wfe 0 DwfB DwfeB) + %<- (of-to-ofe DofM DofeM) + %<- (can-tsub-e ([_] append/nil) DwfeB DofeM Dsub) %. +%worlds (bind var ovar) (can-tsub _ _ _) %. +%total {} (can-tsub _ _ _) %. +%%%%% Corollaries %%%%% +%sort can-sub-context-e {x} %. +%mode can-sub-context-e %in %in %out %. +%term _ + %pi (can-sub-context-e (%the ({x} %pi (isvar x I) %-> (ofe (cons G x A) (M x) B)) Dof) (%the ({y} %pi (isvar y J) %-> (ofe (cons G y B) (N y) (C y))) DofN) Dsub) + %<- ({x} {d isvar x I} ofe-context (Dof x d) (%the (ordered (cons G x A)) (ordered/cons (%the (bounded G x) (DboundedGx x d))))) + %<- ({x} {d isvar x I} following-var d ([y] [e isvar y J'] %the (precedes x y) (Dprec x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J'} bounded-increase-bound (DboundedGx x d) (Dprec x d y e) (%the (bounded G y) (DboundedGy x d y e))) + %<- ({x} {d isvar x I} bump-ofe DofN ([y] [e isvar y J'] ordered/cons (DboundedGy x d y e)) ([y] [e isvar y J'] %the (ofe (cons G y B) (N y) (C y)) (DofN' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J'} weakeng-ofe (append/cons append/nil) (append/cons append/nil) (ordered/cons (bounded/cons (DboundedGx x d) (Dprec x d y e))) (DofN' x d y e) (%the (ofe (cons (cons G x A) y B) (N y) (C y)) (DofN'' x d y e))) + %<- ({x} {d isvar x I} can-sub-e ([_] append/nil) ([y] [e] DofN'' x d y e) (Dof x d) (%the (sub ([y] N y) (M x) (Ny x)) (Dsub x))) %. +%worlds (bind ovar var) (can-sub-context-e _ _ _) %. +%total {} (can-sub-context-e _ _ _) %. +%sort can-tsub-context-e {x} %. +%mode can-tsub-context-e %in %in %out %. +%term _ + %pi (can-tsub-context-e (%the ({x} %pi (isvar x I) %-> (ofe (cons G x A) (M x) B)) Dof) (%the ({y} %pi (isvar y J) %-> (wfe (cons G y B) (C y))) Dwf) Dtsub) + %<- ({x} {d isvar x I} ofe-context (Dof x d) (%the (ordered (cons G x A)) (ordered/cons (%the (bounded G x) (DboundedGx x d))))) + %<- ({x} {d isvar x I} following-var d ([y] [e isvar y J'] %the (precedes x y) (Dprec x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J'} bounded-increase-bound (DboundedGx x d) (Dprec x d y e) (%the (bounded G y) (DboundedGy x d y e))) + %<- ({x} {d isvar x I} bump-wfe Dwf ([y] [e isvar y J'] ordered/cons (DboundedGy x d y e)) ([y] [e isvar y J'] %the (wfe (cons G y B) (C y)) (Dwf' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J'} weakeng-wfe (append/cons append/nil) (append/cons append/nil) (ordered/cons (bounded/cons (DboundedGx x d) (Dprec x d y e))) (Dwf' x d y e) (%the (wfe (cons (cons G x A) y B) (C y)) (Dwf'' x d y e))) + %<- ({x} {d isvar x I} can-tsub-e ([_] append/nil) ([y] [e] Dwf'' x d y e) (Dof x d) (%the (tsub ([y] C y) (M x) (Cy x)) (Dtsub x))) %. +%worlds (bind var ovar) (can-tsub-context-e _ _ _) %. +%total {} (can-tsub-context-e _ _ _) %. +%sort can-tsub1 %. +%mode can-tsub1 %in %in %in %out %. +%term _ + %pi (can-tsub1 (%the (tsub ([x] B x) M Bx) DsubBx) (%the ({x} %pi (vof x A) %-> ({y} %pi (vof y (B x)) %-> (wf (C x y)))) DwfC) (%the (of M A) DofM) DsubCx) + %<- (wf2-to-wfe (%the (lt 0 (s 0)) lt/z) DwfC (%the ({x} + %pi (isvar x 0) + %-> ({y} %pi (isvar y (s 0)) %-> (wfe (cons (cons nil x A) y (B x)) (C x y)))) DwfeC)) + %<- (of-to-ofe DofM (%the (ofe nil M A) DofeM)) + %<- ({y} {e isvar y (s 0)} can-tsub-e ([_] append/cons append/nil) ([x] [d] DwfeC x d y e) DofeM (%the (tsub ([x] C x y) M (Cx y)) (DsubCx y))) %. +%worlds (bind var) (can-tsub1 _ _ _ _) %. +%total {} (can-tsub1 _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/subst-fun-thm.lf b/new-tests/stelf-output/tslf/sing/subst-fun-thm.lf new file mode 100644 index 0000000..dfe4c3d --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/subst-fun-thm.lf @@ -0,0 +1,188 @@ +%%%%% Substitution Functionality %%%%% +%sort aasub-aosub-contra %. +%mode aasub-aosub-contra %in %in %out %. +%scope aasub-aosub-contra %term app + %pi (aasub-aosub-contra (aasub/app _ Daasub) (aosub/app _ _ Daosub) D) + %<- (aasub-aosub-contra Daasub Daosub D) %. +%term appcl + %pi (aasub-aosub-contra aasub/closed (aosub/app _ _ Daosub) D) + %<- (aasub-aosub-contra aasub/closed Daosub D) %. +%scope aasub-aosub-contra %term pi1 + %pi (aasub-aosub-contra (aasub/pi1 D1) (aosub/pi1 D2) D) + %<- (aasub-aosub-contra D1 D2 D) %. +%term pi1 + %pi (aasub-aosub-contra aasub/closed (aosub/pi1 D2) D) + %<- (aasub-aosub-contra aasub/closed D2 D) %. +%scope aasub-aosub-contra %term pi2 + %pi (aasub-aosub-contra (aasub/pi2 D1) (aosub/pi2 D2) D) + %<- (aasub-aosub-contra D1 D2 D) %. +%term pi2 + %pi (aasub-aosub-contra aasub/closed (aosub/pi2 D2) D) + %<- (aasub-aosub-contra aasub/closed D2 D) %. +%worlds (var) (aasub-aosub-contra _ _ _) %. +%total D (aasub-aosub-contra _ D _) %. +%sort aosub-absent %. +%mode aosub-absent %in %out %. +%term _ %pi (aosub-absent Daosub D) %<- (aasub-aosub-contra aasub/closed Daosub D) %. +%worlds (var) (aosub-absent _ _) %. +%total {} (aosub-absent _ _) %. +%sort aasub-absent-fun %. +%mode aasub-absent-fun %in %out %. +%sort sub-absent-fun %. +%mode sub-absent-fun %in %out %. +%sort tsub-absent-fun %. +%mode tsub-absent-fun %in %out %. +%term _ aasub-absent-fun aasub/closed atom-eq/i %. +%term _ + %pi (aasub-absent-fun (aasub/forall D) Deq') + %<- (tsub-absent-fun D Deq) + %<- (atom-resp-tp forall Deq Deq') %. +%term _ + %pi (aasub-absent-fun (aasub/app D2 D1) Deq) + %<- (aasub-absent-fun D1 Deq1) + %<- (sub-absent-fun D2 Deq2) + %<- (app-resp Deq1 Deq2 Deq) %. +%term _ + %pi (aasub-absent-fun (aasub/pi1 D) Deq') + %<- (aasub-absent-fun D Deq) + %<- (atom-resp-atom pi1 Deq Deq') %. +%term _ + %pi (aasub-absent-fun (aasub/pi2 D) Deq') + %<- (aasub-absent-fun D Deq) + %<- (atom-resp-atom pi2 Deq Deq') %. +%term _ + %pi (sub-absent-fun (sub/aa D) Deq') + %<- (aasub-absent-fun D Deq) + %<- (term-resp-atom at Deq Deq') %. +%term _ + %pi (sub-absent-fun (sub/ao D) Deq) + %<- (aosub-absent D Dfalse) + %<- (false-implies-term-eq Dfalse Deq) %. +%term _ + %pi (sub-absent-fun (sub/lam D) Deq') + %<- ({y} sub-absent-fun (D y) (Deq y)) + %<- (lam-resp Deq Deq') %. +%term _ + %pi (sub-absent-fun (sub/pair D2 D1) Deq) + %<- (sub-absent-fun D1 Deq1) + %<- (sub-absent-fun D2 Deq2) + %<- (pair-resp Deq1 Deq2 Deq) %. +%term _ tsub-absent-fun tsub/t tp-eq/i %. +%term _ + %pi (tsub-absent-fun (tsub/pi D2 D1) Deq) + %<- (tsub-absent-fun D1 Deq1) + %<- ({y} tsub-absent-fun (D2 y) (Deq2 y)) + %<- (pi-resp Deq1 Deq2 Deq) %. +%term _ + %pi (tsub-absent-fun (tsub/sigma D2 D1) Deq) + %<- (tsub-absent-fun D1 Deq1) + %<- ({y} tsub-absent-fun (D2 y) (Deq2 y)) + %<- (sigma-resp Deq1 Deq2 Deq) %. +%term _ + %pi (tsub-absent-fun (tsub/singa D) Deq') + %<- (aasub-absent-fun D Deq) + %<- (tp-resp-atom sing Deq Deq') %. +%term _ + %pi (tsub-absent-fun (tsub/singo D) Deq) + %<- (aosub-absent D Dfalse) + %<- (false-implies-tp-eq Dfalse Deq) %. +%worlds (var) (aasub-absent-fun _ _) (sub-absent-fun _ _) (tsub-absent-fun _ _) %. +%total (D1 D2 D3) (aasub-absent-fun D1 _) (sub-absent-fun D2 _) (tsub-absent-fun D3 _) %. +%sort aasub-fun %. +%mode aasub-fun %in %in %out %. +%sort aosub-fun %. +%mode aosub-fun %in %in %out %. +%sort sub-fun %. +%mode sub-fun %in %in %out %. +%sort tsub-fun %. +%mode tsub-fun %in %in %out %. +%scope tsub-fun %term closed + %pi (aasub-fun aasub/closed D Deq') + %<- (aasub-absent-fun D Deq) + %<- (atom-eq-symm Deq Deq') %. +%term closed %pi (aasub-fun D aasub/closed Deq) %<- (aasub-absent-fun D Deq) %. +%scope tsub-fun %term forall + %pi (aasub-fun (aasub/forall D1) (aasub/forall D2) Deq') + %<- (tsub-fun D1 D2 Deq) + %<- (atom-resp-tp forall Deq Deq') %. +%term app + %pi (aasub-fun (aasub/app D2 D1) (aasub/app D2' D1') D) + %<- (aasub-fun D1 D1' Deq1) + %<- (sub-fun D2 D2' Deq2) + %<- (app-resp Deq1 Deq2 D) %. +%scope tsub-fun %term pi1 + %pi (aasub-fun (aasub/pi1 D1) (aasub/pi1 D2) Deq') + %<- (aasub-fun D1 D2 Deq) + %<- (atom-resp-atom pi1 Deq Deq') %. +%term pi1 + %pi (aasub-fun (aasub/pi2 D1) (aasub/pi2 D2) Deq') + %<- (aasub-fun D1 D2 Deq) + %<- (atom-resp-atom pi2 Deq Deq') %. +%scope tsub-fun %term var aosub-fun aosub/var aosub/var term-eq/i %. +%term app + %pi (aosub-fun (aosub/app D3 D2 D1) (aosub/app D3' D2' D1') Deq) + %<- (aosub-fun D1 D1' Deq1) + %<- (term-eq-cdr-lam Deq1 Deq1') + %<- (sub-fun D2 D2' Deq2) + %<- (sub-resp Deq1' Deq2 term-eq/i D3 D3'') + %<- (sub-fun D3'' D3' Deq) %. +%scope tsub-fun %term pi1 + %pi (aosub-fun (aosub/pi1 D1) (aosub/pi1 D2) Deq1) + %<- (aosub-fun D1 D2 Deq) + %<- (term-eq-cdr-pair Deq Deq1 Deq2) %. +%term pi2 + %pi (aosub-fun (aosub/pi2 D1) (aosub/pi2 D2) Deq2) + %<- (aosub-fun D1 D2 Deq) + %<- (term-eq-cdr-pair Deq Deq1 Deq2) %. +%scope tsub-fun %term aa + %pi (sub-fun (sub/aa D1) (sub/aa D2) Deq') + %<- (aasub-fun D1 D2 Deq) + %<- (term-resp-atom at Deq Deq') %. +%term ao %pi (sub-fun (sub/ao D1) (sub/ao D2) Deq) %<- (aosub-fun D1 D2 Deq) %. +%scope tsub-fun %term aaao + %pi (sub-fun (sub/aa D1) (sub/ao D2) Deq) + %<- (aasub-aosub-contra D1 D2 D) + %<- (false-implies-term-eq D Deq) %. +%term aoaa + %pi (sub-fun (sub/ao D2) (sub/aa D1) Deq) + %<- (aasub-aosub-contra D1 D2 D) + %<- (false-implies-term-eq D Deq) %. +%scope tsub-fun %term lam + %pi (sub-fun (sub/lam D1) (sub/lam D2) Deq') + %<- ({y} sub-fun (D1 y) (D2 y) (Deq y)) + %<- (lam-resp Deq Deq') %. +%term pair + %pi (sub-fun (sub/pair D2 D1) (sub/pair D2' D1') Deq) + %<- (sub-fun D1 D1' Deq1) + %<- (sub-fun D2 D2' Deq2) + %<- (pair-resp Deq1 Deq2 Deq) %. +%scope tsub-fun %term t tsub-fun tsub/t tsub/t tp-eq/i %. +%term pi + %pi (tsub-fun (tsub/pi D2 D1) (tsub/pi D2' D1') Deq) + %<- (tsub-fun D1 D1' Deq1) + %<- ({y} tsub-fun (D2 y) (D2' y) (Deq2 y)) + %<- (pi-resp Deq1 Deq2 Deq) %. +%scope tsub-fun %term sigma + %pi (tsub-fun (tsub/sigma D2 D1) (tsub/sigma D2' D1') Deq) + %<- (tsub-fun D1 D1' Deq1) + %<- ({y} tsub-fun (D2 y) (D2' y) (Deq2 y)) + %<- (sigma-resp Deq1 Deq2 Deq) %. +%term singa + %pi (tsub-fun (tsub/singa D1) (tsub/singa D2) Deq') + %<- (aasub-fun D1 D2 Deq) + %<- (tp-resp-atom sing Deq Deq') %. +%scope tsub-fun %term singo + %pi (tsub-fun (tsub/singo D1) (tsub/singo D2) Deq'') + %<- (aosub-fun D1 D2 Deq) + %<- (term-eq-cdr-at Deq Deq') + %<- (tp-resp-atom sing Deq' Deq'') %. +%term singao + %pi (tsub-fun (tsub/singa D1) (tsub/singo D2) Deq) + %<- (aasub-aosub-contra D1 D2 Dfalse) + %<- (false-implies-tp-eq Dfalse Deq) %. +%scope tsub-fun %term singoa + %pi (tsub-fun (tsub/singo D2) (tsub/singa D1) Deq) + %<- (aasub-aosub-contra D1 D2 Dfalse) + %<- (false-implies-tp-eq Dfalse Deq) %. +%worlds (var) (aasub-fun _ _ _) (aosub-fun _ _ _) (sub-fun _ _ _) (tsub-fun _ _ _) %. +%total (D1 D2 D3 D4) (aasub-fun D1 _ _) (aosub-fun D2 _ _) (sub-fun D3 _ _) (tsub-fun D4 _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/substitution-thm.lf b/new-tests/stelf-output/tslf/sing/substitution-thm.lf new file mode 100644 index 0000000..249d35e --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/substitution-thm.lf @@ -0,0 +1,1682 @@ +%%%%% Simple Type Sizes %%%%% +%sort stp-size %. +%term stp-size/t stp-size st (s 0) %. +%term stp-size/pi + %pi (stp-size (spi S T) M+N) + %<- (stp-size S M) + %<- (stp-size T N) + %<- (sum M N M+N) %. +%term stp-size/sigma + %pi (stp-size (ssigma S T) M+N) + %<- (stp-size S M) + %<- (stp-size T N) + %<- (sum M N M+N) %. +%sort stp-size-fun %. +%mode stp-size-fun %in %in %out %. +%scope stp-size-fun %term t stp-size-fun stp-size/t stp-size/t nat-eq/i %. +%term pi + %pi (stp-size-fun (stp-size/pi Dsum DsizeT DsizeS) (stp-size/pi Dsum' DsizeT' DsizeS') Deq) + %<- (stp-size-fun DsizeS DsizeS' DeqS) + %<- (stp-size-fun DsizeT DsizeT' DeqT) + %<- (sum-resp DeqS DeqT nat-eq/i Dsum Dsum'') + %<- (sum-fun3 Dsum'' Dsum' Deq) %. +%scope stp-size-fun %term sigma + %pi (stp-size-fun (stp-size/sigma Dsum DsizeT DsizeS) (stp-size/sigma Dsum' DsizeT' DsizeS') Deq) + %<- (stp-size-fun DsizeS DsizeS' DeqS) + %<- (stp-size-fun DsizeT DsizeT' DeqT) + %<- (sum-resp DeqS DeqT nat-eq/i Dsum Dsum'') + %<- (sum-fun3 Dsum'' Dsum' Deq) %. +%worlds () (stp-size-fun _ _ _) %. +%total D (stp-size-fun D _ _) %. +%sort can-stp-size {T stp} %. +%mode can-stp-size %in %out %. +%term _ can-stp-size st stp-size/t %. +%term _ + %pi (can-stp-size (spi S T) (stp-size/pi Dsum DsizeT DsizeS)) + %<- (can-stp-size S DsizeS) + %<- (can-stp-size T DsizeT) + %<- (can-sum _ _ _ Dsum) %. +%term _ + %pi (can-stp-size (ssigma S T) (stp-size/sigma Dsum DsizeT DsizeS)) + %<- (can-stp-size S DsizeS) + %<- (can-stp-size T DsizeT) + %<- (can-sum _ _ _ Dsum) %. +%worlds () (can-stp-size _ _) %. +%total T (can-stp-size T _) %. +%sort stp-size-leq %. +%mode stp-size-leq %in %in %in %out %. +%scope stp-size-leq %term eq + %pi (stp-size-leq D1 D2 stp-leq/eq Dleq) + %<- (stp-size-fun D1 D2 Deq) + %<- (leq-reflex' Deq Dleq) %. +%term pi1 + %pi (stp-size-leq DsizeS (stp-size/pi Dsum DsizeT2 DsizeT1) (stp-leq/pi1 Dleqt) Dleq'') + %<- (stp-size-leq DsizeS DsizeT1 Dleqt Dleq) + %<- (sum-implies-leq Dsum Dleq') + %<- (leq-trans Dleq Dleq' Dleq'') %. +%scope stp-size-leq %term pi2 + %pi (stp-size-leq DsizeS (stp-size/pi Dsum DsizeT2 DsizeT1) (stp-leq/pi2 Dleqt) Dleq'') + %<- (stp-size-leq DsizeS DsizeT2 Dleqt Dleq) + %<- (sum-implies-leq' Dsum Dleq') + %<- (leq-trans Dleq Dleq' Dleq'') %. +%term sigma1 + %pi (stp-size-leq DsizeS (stp-size/sigma Dsum DsizeT2 DsizeT1) (stp-leq/sigma1 Dleqt) Dleq'') + %<- (stp-size-leq DsizeS DsizeT1 Dleqt Dleq) + %<- (sum-implies-leq Dsum Dleq') + %<- (leq-trans Dleq Dleq' Dleq'') %. +%scope stp-size-leq %term sigma2 + %pi (stp-size-leq DsizeS (stp-size/sigma Dsum DsizeT2 DsizeT1) (stp-leq/sigma2 Dleqt) Dleq'') + %<- (stp-size-leq DsizeS DsizeT2 Dleqt Dleq) + %<- (sum-implies-leq' Dsum Dleq') + %<- (leq-trans Dleq Dleq' Dleq'') %. +%worlds () (stp-size-leq _ _ _ _) %. +%total D (stp-size-leq _ _ D _) %. +%sort stp-size-pos %. +%mode stp-size-pos %in %out %. +%term _ stp-size-pos stp-size/t lt/z %. +%term _ + %pi (stp-size-pos (stp-size/pi Dsum _ Dsize) Dlt'') + %<- (stp-size-pos Dsize Dlt) + %<- (leq-reflex _ Dleq) + %<- (sum-lt-leq sum/z Dsum Dlt Dleq Dlt') + %<- (leq-lt-trans leq/z Dlt' Dlt'') %. +%term _ + %pi (stp-size-pos (stp-size/sigma Dsum _ Dsize) Dlt'') + %<- (stp-size-pos Dsize Dlt) + %<- (leq-reflex _ Dleq) + %<- (sum-lt-leq sum/z Dsum Dlt Dleq Dlt') + %<- (leq-lt-trans leq/z Dlt' Dlt'') %. +%worlds () (stp-size-pos _ _) %. +%total D (stp-size-pos D _) %. +%%%%% Substitution Permutation (Simply Typed, Explicit Context) %%%%% +%sort sub-at-tsub %. +%mode sub-at-tsub %in %out %. +%term _ sub-at-tsub (sub/aa Dsub) (tsub/singa Dsub) %. +%term _ sub-at-tsub (sub/ao Dsub) (tsub/singo Dsub) %. +%worlds (var) (sub-at-tsub _ _) %. +%total {} (sub-at-tsub _ _) %. +%sort aosub-permute-esm {Size} %. +%mode aosub-permute-esm %in %in %in %in %in %in %in %in %in %in %in %out %out %out %. +%sort aasub-aosub-permute-esm {Size} %. +%mode aasub-aosub-permute-esm %in %in %in %in %in %in %in %in %in %in %in %out %out %out %. +%sort aasub-permute-esm {Size} %. +%mode aasub-permute-esm %in %in %in %in %in %in %in %in %in %in %in %out %out %out %. +%sort sub-permute-esm {Size} %. +%mode sub-permute-esm %in %in %in %in %in %in %in %in %in %in %in %out %out %out %. +%sort tsub-permute-esm {Size} %. +%mode tsub-permute-esm %in %in %in %in %in %in %in %in %in %in %in %out %out %out %. +%scope tsub-permute-esm %term var aosub-permute-esm _ _ _ _ _ _ _ _ _ ([x] aosub/var) (%the (sub ([x] N x) M O) Dsub) ([y] aasub/closed) Dsub aosub/var %. +%term app + %pi (aosub-permute-esm _ (%the (stp-size S SizeS) DsizeS) (%the (stp-size T SizeT) DsizeT) (%the (sum SizeS SizeT Size) Dsum) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (app (R x y) (O x y)) U2))) DofApp) ([x] %the (aosub ([y] app (R x y) (O x y)) (N x) (Pz x)) (aosub/app (%the (sub ([z] P x z) (Oy x) (Pz x)) (DsubPz x)) (%the (sub ([y] O x y) (N x) (Oy x)) (DsubOy x)) (%the (aosub ([y] R x y) (N x) (lam ([z] P x z))) (Daosub x)))) (%the (sub ([x] Pz x) M Pzx) DsubPzx) ([y] %the (aasub ([x] app (R x y) (O x y)) M (app (Rx y) (Ox y))) (aasub/app (DsubOx y) (DsubRx y))) DsubNx (%the (aosub ([y] app (Rx y) (Ox y)) Nx Pzx) (aosub/app (DsubPxz aca) DsubOxy DsubLam'))) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-app (DofApp x d y e) (%the (aofes (G x y) (R x y) (spi U1 U2)) (DofR x d y e)) (%the (ofes (G x y) (O x y) U1) (DofO x d y e))) + %<- ({x} {d isvar x I} subst-es ([y] [e] DlookupY x d y e) ([y] [e] DofN x d y e) ([y] [e] DofO x d y e) (DsubOy x) (%the ({y} %pi (isvar y J) %-> (ofes (G x y) (Oy x) U1)) (DofOy x d))) + %<- ({x} {d isvar x I} aosubst-es ([y] [e] DlookupY x d y e) ([y] [e] DofN x d y e) ([y] [e] DofR x d y e) (Daosub x) ([y] [e] %the (ofes (G x y) (lam ([z] P x z)) (spi U1 U2)) (ofes/lam (%the ({z} %pi (isvar z K) %-> (ofes (scons (G x y) z U1) (P x z) U2)) (DofP x d y e))))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} ofes-context (DofP x d y e z f) (%the (sordered (scons (G x y) z U1)) (sordered/cons (%the (sbounded (G x y) z) (Dbounded x d y e z f))))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-ofes (Dbounded x d y e z f) (DofM x d y e) _ (%the (ofes (scons (G x y) z U1) M S) (DofM' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-ofes (Dbounded x d y e z f) (DofOy x d y e) _ (%the (ofes (scons (G x y) z U1) (Oy x) U1) (DofOy' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupX x d y e) _ (%the (slookup (scons (G x y) z U1) x S) (DlookupX' x d y e z f))) + %<- (can-stp-size U1 (%the (stp-size U1 SizeU1) DsizeU1)) + %<- (can-stp-size U2 (%the (stp-size U2 SizeU2) DsizeU2)) + %<- (can-sum SizeU1 SizeU2 SizeU1->U2 (%the (sum SizeU1 SizeU2 SizeU1->U2) DsumU1+U2)) + %<- (can-sum SizeS SizeU1 SizeS+U1 (%the (sum SizeS SizeU1 SizeS+U1) DsumS+U1)) + %<- ({x} aosub-headvar (Daosub x) (%the (headvar ([y] R x y)) (Dheadvar x))) + %<- ({x} {d isvar x I} headvar-stp-size ([y] [e] DlookupY x d y e) ([y] [e] DofR x d y e) (Dheadvar x) (%the (stp-leq (spi U1 U2) T) Dleqt)) + %<- (stp-size-leq (stp-size/pi DsumU1+U2 DsizeU2 DsizeU1) DsizeT Dleqt (%the (leq SizeU1->U2 SizeT) Dleq)) + %<- (stp-size-pos DsizeU2 (%the (lt 0 SizeU2) DltU2)) + %<- (sum-implies-lt DsumU1+U2 DltU2 (%the (lt SizeU1 SizeU1->U2) DltU1)) + %<- (lt-leq-trans DltU1 Dleq (%the (lt SizeU1 SizeT) DltU1T)) + %<- (leq-reflex SizeS (%the (leq SizeS SizeS) DleqS)) + %<- (sum-leq-lt DsumS+U1 Dsum DleqS DltU1T (%the (lt SizeS+U1 Size) Dlt)) + %<- (lt-subterm SizeS+U1 Size Dlt) + %<- ({y} {e isvar y J} sub-permute-esm SizeS+U1 DsizeS DsizeU1 DsumS+U1 ([x] [d] [z] [f] DlookupX' x d y e z f) ([x] [d] [z] [f] slookup/hit (Dbounded x d y e z f)) ([x] [d] [z] [f] DofM' x d y e z f) ([x] [d] [z] [f] DofOy' x d y e z f) ([x] [d isvar x I] [z] [f isvar z K] DofP x d y e z f) DsubPz DsubPzx (%the ({z} sub ([x] P x z) M (Px_y y z)) (DsubPx_y y)) (%the (sub Oy M (Oyx_y y)) (DsubOyx_y y)) (%the (sub ([z] Px_y y z) (Oyx_y y) Pzx) (DsubPxz_y y))) + %<- ({z} sub-closed ([y] DsubPx_y y z) ([y] %the (term-eq (Px_y y z) (Px z)) (DeqPx_y y z))) + %<- (sub-closed DsubOyx_y ([y] %the (term-eq (Oyx_y y) Oyx) (DeqOyx_y y))) + %<- ({y} {z} sub-resp ([_] term-eq/i) term-eq/i (DeqPx_y y z) (DsubPx_y y z) (%the (sub ([x] P x z) M (Px z)) (DsubPx y z))) + %<- ({y} sub-resp ([_] term-eq/i) term-eq/i (DeqOyx_y y) (DsubOyx_y y) (%the (sub Oy M Oyx) (DsubOyx y))) + %<- ({y} sub-resp ([z] DeqPx_y y z) (DeqOyx_y y) term-eq/i (DsubPxz_y y) (%the (sub ([z] Px z) Oyx Pzx) (DsubPxz y))) + %<- (sub-permute-esm _ DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofO DsubOy (DsubOyx aca) (%the ({y} sub ([x] O x y) M (Ox y)) DsubOx) (%the (sub N M Nx) DsubNx) (%the (sub Ox Nx Oyx) DsubOxy)) + %<- (aosub-permute-esm _ DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR Daosub (%the (sub ([x] lam ([z] P x z)) M (lam ([z] Px z))) (sub/lam (DsubPx aca))) (%the ({y} aasub ([x] R x y) M (Rx y)) DsubRx) (%the (sub N M Nx') DsubNx') (%the (aosub ([y] Rx y) Nx' (lam ([z] Px z))) DsubLam)) + %<- (sub-fun DsubNx' DsubNx (%the (term-eq Nx' Nx) Deq)) + %<- (aosub-resp ([_] atom-eq/i) Deq term-eq/i DsubLam (%the (aosub ([y] Rx y) Nx (lam ([z] Px z))) DsubLam')) %. +%scope tsub-permute-esm %term pi1 + %pi (aosub-permute-esm _ DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (pi1 (R x y)) U1))) DofPi) ([x] %the (aosub ([y] pi1 (R x y)) (N x) (P1 x)) (aosub/pi1 (%the (aosub ([y] R x y) (N x) (pair (P1 x) (P2 x))) (Daosub x)))) (%the (sub ([x] P1 x) M P1x) DsubP1x) ([y] %the (aasub ([x] pi1 (R x y)) M (pi1 (Rx y))) (aasub/pi1 (DsubRx y))) DsubNx (%the (aosub ([y] pi1 (Rx y)) Nx P1x) (aosub/pi1 DsubPair))) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-pi1 (DofPi x d y e) (%the (aofes (G x y) (R x y) (ssigma U1 U2)) (DofR x d y e))) + %<- ({x} {d isvar x I} aosubst-es ([y] [e] DlookupY x d y e) ([y] [e] DofN x d y e) ([y] [e] DofR x d y e) (Daosub x) ([y] [e] ofes/pair (%the (ofes (G x y) (P2 x) U2) (DofP2 x d y e)) (%the (ofes (G x y) (P1 x) U1) (DofP1 x d y e)))) + %<- ({y} {e isvar y J} can-sub-es ([x] [d] DlookupX x d y e) ([x] [d] DofP2 x d y e) ([x] [d] DofM x d y e) (%the (sub ([x] P2 x) M (P2x_y y)) (DsubP2x_y y))) + %<- (sub-closed DsubP2x_y ([y] %the (term-eq (P2x_y y) P2x) (DeqP2x_y y))) + %<- ({y} sub-resp ([_] term-eq/i) term-eq/i (DeqP2x_y y) (DsubP2x_y y) (%the (sub ([x] P2 x) M P2x) (DsubP2x y))) + %<- (aosub-permute-esm _ DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR Daosub (%the (sub ([x] pair (P1 x) (P2 x)) M (pair P1x P2x)) (sub/pair (DsubP2x aca) DsubP1x)) (%the ({y} aasub ([x] R x y) M (Rx y)) DsubRx) (%the (sub ([x] N x) M Nx) DsubNx) (%the (aosub ([y] Rx y) Nx (pair P1x P2x)) DsubPair)) %. +%term pi2 + %pi (aosub-permute-esm _ DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (pi2 (R x y)) U2))) DofPi) ([x] %the (aosub ([y] pi2 (R x y)) (N x) (P2 x)) (aosub/pi2 (%the (aosub ([y] R x y) (N x) (pair (P1 x) (P2 x))) (Daosub x)))) (%the (sub ([x] P2 x) M P2x) DsubP2x) ([y] %the (aasub ([x] pi2 (R x y)) M (pi2 (Rx y))) (aasub/pi2 (DsubRx y))) DsubNx (%the (aosub ([y] pi2 (Rx y)) Nx P2x) (aosub/pi2 DsubPair))) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-pi2 (DofPi x d y e) (%the (aofes (G x y) (R x y) (ssigma U1 U2)) (DofR x d y e))) + %<- ({x} {d isvar x I} aosubst-es ([y] [e] DlookupY x d y e) ([y] [e] DofN x d y e) ([y] [e] DofR x d y e) (Daosub x) ([y] [e] ofes/pair (%the (ofes (G x y) (P2 x) U2) (DofP2 x d y e)) (%the (ofes (G x y) (P1 x) U1) (DofP1 x d y e)))) + %<- ({y} {e isvar y J} can-sub-es ([x] [d] DlookupX x d y e) ([x] [d] DofP1 x d y e) ([x] [d] DofM x d y e) (%the (sub ([x] P1 x) M (P1x_y y)) (DsubP1x_y y))) + %<- (sub-closed DsubP1x_y ([y] %the (term-eq (P1x_y y) P1x) (DeqP1x_y y))) + %<- ({y} sub-resp ([_] term-eq/i) term-eq/i (DeqP1x_y y) (DsubP1x_y y) (%the (sub ([x] P1 x) M P1x) (DsubP1x y))) + %<- (aosub-permute-esm _ DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR Daosub (%the (sub ([x] pair (P1 x) (P2 x)) M (pair P1x P2x)) (sub/pair DsubP2x (DsubP1x aca))) (%the ({y} aasub ([x] R x y) M (Rx y)) DsubRx) (%the (sub ([x] N x) M Nx) DsubNx) (%the (aosub ([y] Rx y) Nx (pair P1x P2x)) DsubPair)) %. +%scope tsub-permute-esm %term closed + %pi (aasub-aosub-permute-esm _ _ _ _ (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) _ (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) _ ([x] aasub/closed) (%the (aosub R M O) Daosub) ([y] Daosub) DsubNx Dsub) + %<- ({y} {e isvar y J} can-sub-es ([x] [d] DlookupX x d y e) ([x] [d] DofN x d y e) ([x] [d] DofM x d y e) (%the (sub N M (Nx_y y)) (DsubNx_y y))) + %<- (sub-closed' DsubNx_y (%the (sub N M Nx) DsubNx)) + %<- (sub-absent O Nx (%the (sub ([_] O) Nx O) Dsub)) %. +%sort ofes-spi-invert {x} %. +%mode ofes-spi-invert %in %out %. +%term _ ofes-spi-invert ([x] [d] [y] [e] ofes/lam ([z] [f isvar z K] D x d y e z f)) ([y] term-eq/i) %. +%worlds (var ovar bind) (ofes-spi-invert _ _) %. +%total {} (ofes-spi-invert _ _) %. +%term app + %pi (aasub-aosub-permute-esm _ (%the (stp-size S SizeS) DsizeS) (%the (stp-size T SizeT) DsizeT) (%the (sum SizeS SizeT Size) Dsum) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (app (R x y) (O x y)) U2))) DofApp) ([x] aasub/app (%the (sub ([y] O x y) (N x) (Oy x)) (DsubOy x)) (%the (aasub ([y] R x y) (N x) (Ry x)) (DsubRy x))) (aosub/app (%the (sub P Oyx Pz) DsubPz) (%the (sub Oy M Oyx) DsubOyx) (%the (aosub Ry M (lam ([z] P z))) Daosub)) ([y] aosub/app (DsubQz y) (DsubOx y) (Daosub'' y)) DsubNx DsubQzy') + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-app (DofApp x d y e) (%the (aofes (G x y) (R x y) (spi U1 U2)) (DofR x d y e)) (%the (ofes (G x y) (O x y) U1) (DofO x d y e))) + %<- (aasub-aosub-permute-esm _ DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR DsubRy Daosub (%the ({y} aosub ([x] R x y) M (Ql y)) Daosub') (%the (sub N M Nx) DsubNx) (%the (sub ([y] Ql y) Nx (lam ([z] P z))) DsubQly)) + %<- ({y} {e isvar y J} subst-es ([x] [d] DlookupX x d y e) ([x] [d] DofM x d y e) ([x] [d] DofN x d y e) DsubNx ([x] [d isvar x I] %the (ofes (G x y) Nx T) (DofNx x d y e))) + %<- ({y} {e isvar y J} aosubst-es ([x] [d] DlookupX x d y e) ([x] [d] DofM x d y e) ([x] [d] DofR x d y e) (Daosub' y) ([x] [d isvar x I] %the (ofes (G x y) (Ql y) (spi U1 U2)) (DofQl x d y e))) + %<- (ofes-spi-invert DofQl (%the ({y} term-eq (Ql y) (lam ([z] Q y z))) DeqQl)) + %<- (sub-resp DeqQl term-eq/i term-eq/i DsubQly (%the (sub ([y] lam ([z] Q y z)) Nx (lam ([z] P z))) (sub/lam (%the ({z} sub ([y] Q y z) Nx (P z)) DsubQy)))) + %<- ({y} aosub-resp ([_] atom-eq/i) term-eq/i (DeqQl y) (Daosub' y) (%the (aosub ([x] R x y) M (lam ([z] Q y z))) (Daosub'' y))) + %<- (sub-permute-esm _ DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofO DsubOy DsubOyx (%the ({y} sub ([x] O x y) M (Ox y)) DsubOx) (%the (sub N M Nx') DsubNx') (%the (sub ([y] Ox y) Nx' Oyx) DsubOxy)) + %<- (sub-fun DsubNx' DsubNx (%the (term-eq Nx' Nx) DeqNx)) + %<- (sub-resp ([_] term-eq/i) DeqNx term-eq/i DsubOxy (%the (sub ([y] Ox y) Nx Oyx) DsubOxy')) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofes-resp sctx-eq/i (DeqQl y) stp-eq/i (DofQl x d y e) (ofes/lam ([z] [f isvar z K] %the (ofes (scons (G x y) z U1) (Q y z) U2) (DofQ x d y e z f)))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} ofes-context (DofQ x d y e z f) (sordered/cons (%the (sbounded (G x y) z) (Dbounded x d y e z f)))) + %<- ({y} {e isvar y J} subst-es ([x] [d] DlookupX x d y e) ([x] [d] DofM x d y e) ([x] [d] DofO x d y e) (DsubOx y) ([x] [d] %the (ofes (G x y) (Ox y) U1) (DofOx x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-ofes (Dbounded x d y e z f) (DofOx x d y e) _ (%the (ofes (scons (G x y) z U1) (Ox y) U1) (DofOx' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} can-sub-es ([z] [f isvar z K] slookup/hit (Dbounded x d y e z f)) ([z] [f] DofQ x d y e z f) ([z] [f] DofOx' x d y e z f) (%the (sub ([z] Q y z) (Ox y) (Qz_x x y)) (DsubQz_x x y))) + %<- ({y} sub-closed' ([x] DsubQz_x x y) (%the (sub ([z] Q y z) (Ox y) (Qz y)) (DsubQz y))) + %<- ({x} {d isvar x I} {y} {e isvar y J} subst-es ([z] [f isvar z K] slookup/hit (Dbounded x d y e z f)) ([z] [f] DofOx' x d y e z f) ([z] [f] DofQ x d y e z f) (DsubQz y) ([z] [f isvar z K] %the (ofes (scons (G x y) z U1) (Qz y) U2) (DofQz x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupY x d y e) _ (%the (slookup (scons (G x y) z U1) y T) (DlookupY' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-ofes (Dbounded x d y e z f) (DofNx x d y e) _ (%the (ofes (scons (G x y) z U1) Nx T) (DofNx' x d y e z f))) + %<- ({x} {d isvar x I} {z} {f isvar z K} can-sub-es ([y] [e] DlookupY' x d y e z f) ([y] [e] DofQz x d y e z f) ([y] [e isvar y J] DofNx' x d y e z f) (%the (sub ([y] Qz y) Nx (Qzy_xz x z)) (DsubQzy_xz x z))) + %<- ({z} sub-closed' ([x] DsubQzy_xz x z) (%the (sub ([y] Qz y) Nx (Qzy_z z)) (DsubQzy_z z))) + %<- (sub-closed' DsubQzy_z (%the (sub ([y] Qz y) Nx Qzy) DsubQzy)) + %<- (can-stp-size U1 (%the (stp-size U1 SizeU1) DsizeU1)) + %<- (can-stp-size U2 (%the (stp-size U2 SizeU2) DsizeU2)) + %<- (can-sum SizeU1 SizeU2 SizeU1->U2 (%the (sum SizeU1 SizeU2 SizeU1->U2) DsumU1+U2)) + %<- (can-sum SizeT SizeU1 SizeT+U1 (%the (sum SizeT SizeU1 SizeT+U1) DsumT+U1)) + %<- ({y} aosub-headvar (Daosub' y) (%the (headvar ([x] R x y)) (Dheadvar y))) + %<- ({y} {e isvar y J} headvar-stp-size ([x] [d] DlookupX x d y e) ([x] [d] DofR x d y e) (Dheadvar y) (%the (stp-leq (spi U1 U2) S) Dleqt)) + %<- (stp-size-leq (stp-size/pi DsumU1+U2 DsizeU2 DsizeU1) DsizeS Dleqt (%the (leq SizeU1->U2 SizeS) Dleq)) + %<- (stp-size-pos DsizeU2 (%the (lt 0 SizeU2) DltU2)) + %<- (sum-implies-lt DsumU1+U2 DltU2 (%the (lt SizeU1 SizeU1->U2) DltU1)) + %<- (lt-leq-trans DltU1 Dleq (%the (lt SizeU1 SizeS) DltU1S)) + %<- (leq-reflex SizeT (%the (leq SizeT SizeT) DleqT)) + %<- (sum-commute Dsum (%the (sum SizeT SizeS Size) Dsum')) + %<- (sum-leq-lt DsumT+U1 Dsum' DleqT DltU1S (%the (lt SizeT+U1 Size) Dlt)) + %<- (lt-subterm SizeT+U1 Size Dlt) + %<- ({x} {d isvar x I} sub-permute-esm SizeT+U1 DsizeT DsizeU1 DsumT+U1 ([y] [e] [z] [f] DlookupY' x d y e z f) ([y] [e] [z] [f] slookup/hit (Dbounded x d y e z f)) ([y] [e] [z] [f] DofNx' x d y e z f) ([y] [e] [z] [f] DofOx' x d y e z f) ([y] [e] [z] [f] DofQ x d y e z f) DsubQz DsubQzy (%the ({z} sub ([y] Q y z) Nx (Qy_x x z)) (DsubQy'_x x)) (%the (sub ([y] Ox y) Nx (Oxy_x x)) (DsubOxy''_x x)) (%the (sub ([z] Qy_x x z) (Oxy_x x) Qzy) (DsubQyz_x x))) + %<- ({z} sub-closed ([x] DsubQy'_x x z) ([x] %the (term-eq (Qy_x x z) (Qy z)) (DeqQy_x x z))) + %<- (sub-closed DsubOxy''_x (%the ({x} term-eq (Oxy_x x) Oxy) DeqOxy_x)) + %<- ({z} sub-resp-strengthen ([_] [_] term-eq/i) ([_] term-eq/i) ([x] DeqQy_x x z) ([x] DsubQy'_x x z) (%the (sub ([y] Q y z) Nx (Qy z)) (DsubQy' z))) + %<- (sub-resp-strengthen ([_] [_] term-eq/i) ([_] term-eq/i) DeqOxy_x DsubOxy''_x (%the (sub ([y] Ox y) Nx Oxy) DsubOxy'')) + %<- (sub-resp-strengthen ([x] [z] DeqQy_x x z) DeqOxy_x ([_] term-eq/i) DsubQyz_x (%the (sub ([z] Qy z) Oxy Qzy) DsubQyz)) + %<- ({z} sub-fun (DsubQy' z) (DsubQy z) (%the (term-eq (Qy z) (P z)) (DeqQy z))) + %<- (sub-fun DsubOxy'' DsubOxy' (%the (term-eq Oxy Oyx) DeqOxy)) + %<- (sub-resp DeqQy DeqOxy term-eq/i DsubQyz (%the (sub ([z] P z) Oyx Qzy) DsubPz')) + %<- (sub-fun DsubPz' DsubPz (%the (term-eq Qzy Pz) DeqQzy)) + %<- (sub-resp ([_] term-eq/i) term-eq/i DeqQzy DsubQzy (%the (sub ([y] Qz y) Nx Pz) DsubQzy')) %. +%sort ofes-ssigma-invert {x} %. +%mode ofes-ssigma-invert %in %out %. +%term _ ofes-ssigma-invert ([x] [d] [y] [e] ofes/pair (D2 x d y e) (D1 x d y e)) ([y] term-eq/i) %. +%worlds (var ovar bind) (ofes-ssigma-invert _ _) %. +%total {} (ofes-ssigma-invert _ _) %. +%term pi1 + %pi (aasub-aosub-permute-esm _ DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (pi1 (R x y)) U1))) DofPi) ([x] aasub/pi1 (%the (aasub ([y] R x y) (N x) (Ry x)) (DsubRy x))) (aosub/pi1 (%the (aosub ([x] Ry x) M (pair O1 O2)) Daosub)) ([y] aosub/pi1 (Daosub'' y)) DsubNx DsubQ1y) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-pi1 (DofPi x d y e) (%the (aofes (G x y) (R x y) (ssigma U1 U2)) (DofR x d y e))) + %<- (aasub-aosub-permute-esm _ DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR DsubRy Daosub (%the ({y} aosub ([x] R x y) M (Qp y)) Daosub') (%the (sub N M Nx) DsubNx) (%the (sub ([y] Qp y) Nx (pair O1 O2)) DsubQpy)) + %<- ({y} {e isvar y J} aosubst-es ([x] [d] DlookupX x d y e) ([x] [d] DofM x d y e) ([x] [d] DofR x d y e) (Daosub' y) ([x] [d isvar x I] %the (ofes (G x y) (Qp y) (ssigma U1 U2)) (DofQp x d y e))) + %<- (ofes-ssigma-invert DofQp (%the ({y} term-eq (Qp y) (pair (Q1 y) (Q2 y))) DeqQp)) + %<- (sub-resp DeqQp term-eq/i term-eq/i DsubQpy (%the (sub ([y] pair (Q1 y) (Q2 y)) Nx (pair O1 O2)) (sub/pair (%the (sub Q2 Nx O2) DsubQ2y) (%the (sub Q1 Nx O1) DsubQ1y)))) + %<- ({y} aosub-resp ([_] atom-eq/i) term-eq/i (DeqQp y) (Daosub' y) (%the (aosub ([x] R x y) M (pair (Q1 y) (Q2 y))) (Daosub'' y))) %. +%term pi2 + %pi (aasub-aosub-permute-esm _ DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (pi2 (R x y)) U2))) DofPi) ([x] aasub/pi2 (%the (aasub ([y] R x y) (N x) (Ry x)) (DsubRy x))) (aosub/pi2 (%the (aosub ([x] Ry x) M (pair O1 O2)) Daosub)) ([y] aosub/pi2 (Daosub'' y)) DsubNx DsubQ2y) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-pi2 (DofPi x d y e) (%the (aofes (G x y) (R x y) (ssigma U1 U2)) (DofR x d y e))) + %<- (aasub-aosub-permute-esm _ DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR DsubRy Daosub (%the ({y} aosub ([x] R x y) M (Qp y)) Daosub') (%the (sub N M Nx) DsubNx) (%the (sub ([y] Qp y) Nx (pair O1 O2)) DsubQpy)) + %<- ({y} {e isvar y J} aosubst-es ([x] [d] DlookupX x d y e) ([x] [d] DofM x d y e) ([x] [d] DofR x d y e) (Daosub' y) ([x] [d isvar x I] %the (ofes (G x y) (Qp y) (ssigma U1 U2)) (DofQp x d y e))) + %<- (ofes-ssigma-invert DofQp (%the ({y} term-eq (Qp y) (pair (Q1 y) (Q2 y))) DeqQp)) + %<- (sub-resp DeqQp term-eq/i term-eq/i DsubQpy (%the (sub ([y] pair (Q1 y) (Q2 y)) Nx (pair O1 O2)) (sub/pair (%the (sub Q2 Nx O2) DsubQ2y) (%the (sub Q1 Nx O1) DsubQ1y)))) + %<- ({y} aosub-resp ([_] atom-eq/i) term-eq/i (DeqQp y) (Daosub' y) (%the (aosub ([x] R x y) M (pair (Q1 y) (Q2 y))) (Daosub'' y))) %. +%term at-o + %pi (sub-permute-esm _ DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) ([x] [d isvar x I] [y] [e isvar y J] ofes/at (%the (aofes (G x y) (R x y) st) (DofR x d y e))) ([x] %the (sub ([y] at (R x y)) (N x) (Oy x)) (sub/ao (%the (aosub ([y] R x y) (N x) (Oy x)) (DsubOy x)))) (%the (sub ([x] Oy x) M Oyx) DsubOyx) ([y] %the (sub ([x] at (R x y)) M (at (Rx y))) (sub/aa (DsubRx y))) DsubNx (%the (sub ([y] at (Rx y)) Nx Oyx) (sub/ao DsubOxy))) + %<- (aosub-permute-esm _ DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR DsubOy DsubOyx (%the ({y} aasub ([x] R x y) M (Rx y)) DsubRx) (%the (sub ([x] N x) M Nx) DsubNx) (%the (aosub ([y] Rx y) Nx Oyx) DsubOxy)) %. +%term at-ao + %pi (sub-permute-esm _ DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) ([x] [d isvar x I] [y] [e isvar y J] ofes/at (%the (aofes (G x y) (R x y) st) (DofR x d y e))) ([x] %the (sub ([y] at (R x y)) (N x) (at (Ry x))) (sub/aa (%the (aasub ([y] R x y) (N x) (Ry x)) (DsubRy x)))) (%the (sub ([x] at (Ry x)) M Oyx) (sub/ao (%the (aosub ([x] Ry x) M Oyx) DsubOyx))) ([y] sub/ao (DsubOx y)) DsubNx DsubOxy) + %<- (aasub-aosub-permute-esm _ DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR DsubRy DsubOyx (%the ({y} aosub ([x] R x y) M (Ox y)) DsubOx) (%the (sub N M Nx) DsubNx) (%the (sub ([y] Ox y) Nx Oyx) DsubOxy)) %. +%term at-a + %pi (sub-permute-esm _ DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) ([x] [d isvar x I] [y] [e isvar y J] ofes/at (%the (aofes (G x y) (R x y) st) (DofR x d y e))) ([x] %the (sub ([y] at (R x y)) (N x) (at (Ry x))) (sub/aa (%the (aasub ([y] R x y) (N x) (Ry x)) (DsubRy x)))) (%the (sub ([x] at (Ry x)) M (at Ryx)) (sub/aa (%the (aasub ([x] Ry x) M Ryx) DsubRyx))) ([y] sub/aa (DsubRx y)) DsubNx (sub/aa DsubRxy)) + %<- (aasub-permute-esm _ DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR DsubRy DsubRyx (%the ({y} aasub ([x] R x y) M (Rx y)) DsubRx) (%the (sub N M Nx) DsubNx) (%the (aasub ([y] Rx y) Nx Ryx) DsubRxy)) %. +%scope at-a %term lam + %pi (sub-permute-esm _ DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) ([x] [d isvar x I] [y] [e isvar y J] %the (ofes (G x y) (lam ([z] O x y z)) (spi U1 U2)) (ofes/lam (%the ({z} %pi (isvar z K) %-> (ofes (scons (G x y) z U1) (O x y z) U2)) (DofO x d y e)))) ([x] %the (sub ([y] lam ([z] O x y z)) (N x) (lam ([z] Oy x z))) (sub/lam (%the ({z} sub ([y] O x y z) (N x) (Oy x z)) (DsubOy x)))) (%the (sub ([x] lam ([z] Oy x z)) M (lam ([z] Oyx z))) (sub/lam (%the ({z} sub ([x] Oy x z) M (Oyx z)) DsubOyx))) ([y] sub/lam ([z] DsubOx z y)) DsubNx (sub/lam ([z] DsubOxy' z))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} ofes-context (DofO x d y e z f) (%the (sordered (scons (G x y) z U1)) (sordered/cons (%the (sbounded (G x y) z) (Dbounded x d y e z f))))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-ofes (Dbounded x d y e z f) (DofM x d y e) _ (%the (ofes (scons (G x y) z U1) M S) (DofM' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-ofes (Dbounded x d y e z f) (DofN x d y e) _ (%the (ofes (scons (G x y) z U1) (N x) T) (DofN' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupX x d y e) _ (%the (slookup (scons (G x y) z U1) x S) (DlookupX' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupY x d y e) _ (%the (slookup (scons (G x y) z U1) y T) (DlookupY' x d y e z f))) + %<- ({z} {f isvar z K} sub-permute-esm _ DsizeS DsizeT Dsum ([x] [d] [y] [e] DlookupX' x d y e z f) ([x] [d] [y] [e] DlookupY' x d y e z f) ([x] [d] [y] [e] DofM' x d y e z f) ([x] [d] [y] [e] DofN' x d y e z f) ([x] [d] [y] [e] DofO x d y e z f) ([x] DsubOy x z) (DsubOyx z) (%the ({y} sub ([x] O x y z) M (Ox y z)) (DsubOx z)) (%the (sub N M (Nx_z z)) (DsubNx_z z)) (%the (sub ([y] Ox y z) (Nx_z z) (Oyx z)) (DsubOxy z))) + %<- (sub-closed DsubNx_z (%the ({z} term-eq (Nx_z z) Nx) DeqNx_z)) + %<- (sub-resp-strengthen ([_] [_] term-eq/i) ([_] term-eq/i) DeqNx_z DsubNx_z (%the (sub N M Nx) DsubNx)) + %<- ({z} sub-resp ([_] term-eq/i) (DeqNx_z z) term-eq/i (DsubOxy z) (DsubOxy' z)) %. +%term pair + %pi (sub-permute-esm _ DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) ([x] [d isvar x I] [y] [e isvar y J] ofes/pair (%the (ofes (G x y) (O2 x y) U2) (DofO2 x d y e)) (%the (ofes (G x y) (O1 x y) U1) (DofO1 x d y e))) ([x] sub/pair (%the (sub ([y] O2 x y) (N x) (O2y x)) (DsubO2y x)) (%the (sub ([y] O1 x y) (N x) (O1y x)) (DsubO1y x))) (sub/pair (%the (sub ([x] O2y x) M O2yx) DsubO2yx) (%the (sub ([x] O1y x) M O1yx) DsubO1yx)) ([y] sub/pair (DsubO2x y) (DsubO1x y)) DsubNx (sub/pair DsubO2xy' DsubO1xy)) + %<- (sub-permute-esm _ DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofO1 DsubO1y DsubO1yx (%the ({y} sub ([x] O1 x y) M (O1x y)) DsubO1x) (%the (sub N M Nx) DsubNx) (%the (sub ([y] O1x y) Nx O1yx) DsubO1xy)) + %<- (sub-permute-esm _ DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofO2 DsubO2y DsubO2yx (%the ({y} sub ([x] O2 x y) M (O2x y)) DsubO2x) (%the (sub N M Nx') DsubNx') (%the (sub ([y] O2x y) Nx' O2yx) DsubO2xy)) + %<- (sub-fun DsubNx' DsubNx (%the (term-eq Nx' Nx) Deq)) + %<- (sub-resp ([_] term-eq/i) Deq term-eq/i DsubO2xy (%the (sub ([y] O2x y) Nx O2yx) DsubO2xy')) %. +%scope at-a %term closed + %pi (aasub-permute-esm _ _ _ _ (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) _ ([_] aasub/closed) Daasub ([_] Daasub) DsubNx aasub/closed) + %<- ({y} {e isvar y J} can-sub-es ([x] [d] DlookupX x d y e) ([x] [d] DofN x d y e) ([x] [d] DofM x d y e) (%the (sub N M (Nx_y y)) (DsubNx_y y))) + %<- (sub-closed' DsubNx_y (%the (sub N M Nx) DsubNx)) %. +%term forall + %pi (aasub-permute-esm Size DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (forall (A x y)) U1))) DofQuant) ([x] aasub/forall (%the (tsub ([y] A x y) (N x) (Ay x)) (DsubAy x))) (aasub/forall (%the (tsub ([x] Ay x) M Ayx) DsubAyx)) ([y] aasub/forall (DsubAx y)) DsubNx (aasub/forall DsubAxy)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-forall (DofQuant x d y e) (%the (wfes (G x y) (A x y)) (DofA x d y e))) + %<- (tsub-permute-esm Size DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofA DsubAy DsubAyx (%the ({y} tsub ([x] A x y) M (Ax y)) DsubAx) (%the (sub N M Nx) DsubNx) (%the (tsub ([y] Ax y) Nx Ayx) DsubAxy)) %. +%scope at-a %term all-c + %pi (aasub-permute-esm Size DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (forall (A x y)) U1))) DofQuant) ([x] aasub/forall (%the (tsub ([y] A x y) (N x) Ay) (DsubAy x))) aasub/closed ([y] aasub/forall (DsubAx y)) DsubNx (aasub/forall DsubAxy)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-forall (DofQuant x d y e) (%the (wfes (G x y) (A x y)) (DofA x d y e))) + %<- (tsub-absent Ay M DsubAyx) + %<- (tsub-permute-esm Size DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofA DsubAy DsubAyx (%the ({y} tsub ([x] A x y) M (Ax y)) DsubAx) (%the (sub N M Nx) DsubNx) (%the (tsub ([y] Ax y) Nx Ay) DsubAxy)) %. +%term app + %pi (aasub-permute-esm Size DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (app (R x y) (O x y)) U2))) DofApp) ([x] aasub/app (%the (sub ([y] O x y) (N x) (Oy x)) (DsubOy x)) (%the (aasub ([y] R x y) (N x) (Ry x)) (DsubRy x))) (aasub/app (%the (sub ([x] Oy x) M Oyx) DsubOyx) (%the (aasub ([x] Ry x) M Ryx) DsubRyx)) ([y] aasub/app (DsubOx y) (DsubRx y)) DsubNx (aasub/app DsubOxy' DsubRxy)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-app (DofApp x d y e) (%the (aofes (G x y) (R x y) (spi U1 U2)) (DofR x d y e)) (%the (ofes (G x y) (O x y) U1) (DofO x d y e))) + %<- (aasub-permute-esm Size DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR DsubRy DsubRyx (%the ({y} aasub ([x] R x y) M (Rx y)) DsubRx) (%the (sub N M Nx) DsubNx) (%the (aasub ([y] Rx y) Nx Ryx) DsubRxy)) + %<- (sub-permute-esm Size DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofO DsubOy DsubOyx (%the ({y} sub ([x] O x y) M (Ox y)) DsubOx) (%the (sub N M Nx') DsubNx') (%the (sub ([y] Ox y) Nx' Oyx) DsubOxy)) + %<- (sub-fun DsubNx' DsubNx (%the (term-eq Nx' Nx) DeqNx)) + %<- (sub-resp ([_] term-eq/i) DeqNx term-eq/i DsubOxy (%the (sub ([y] Ox y) Nx Oyx) DsubOxy')) %. +%scope at-a %term app-c + %pi (aasub-permute-esm Size DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (app (R x y) (O x y)) U2))) DofApp) ([x] aasub/app (%the (sub ([y] O x y) (N x) Oy) (DsubOy x)) (%the (aasub ([y] R x y) (N x) Ry) (DsubRy x))) aasub/closed ([y] aasub/app (DsubOx y) (DsubRx y)) DsubNx (aasub/app DsubOxy' DsubRxy)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-app (DofApp x d y e) (%the (aofes (G x y) (R x y) (spi U1 U2)) (DofR x d y e)) (%the (ofes (G x y) (O x y) U1) (DofO x d y e))) + %<- (aasub-permute-esm Size DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR DsubRy aasub/closed (%the ({y} aasub ([x] R x y) M (Rx y)) DsubRx) (%the (sub N M Nx) DsubNx) (%the (aasub ([y] Rx y) Nx Ry) DsubRxy)) + %<- (sub-absent Oy M (%the (sub ([x] Oy) M Oy) DsubOyx)) + %<- (sub-permute-esm Size DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofO DsubOy DsubOyx (%the ({y} sub ([x] O x y) M (Ox y)) DsubOx) (%the (sub N M Nx') DsubNx') (%the (sub ([y] Ox y) Nx' Oy) DsubOxy)) + %<- (sub-fun DsubNx' DsubNx (%the (term-eq Nx' Nx) DeqNx)) + %<- (sub-resp ([_] term-eq/i) DeqNx term-eq/i DsubOxy (%the (sub ([y] Ox y) Nx Oy) DsubOxy')) %. +%term pi1 + %pi (aasub-permute-esm Size DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (pi1 (R x y)) U1))) DofPi) ([x] aasub/pi1 (%the (aasub ([y] R x y) (N x) (Ry x)) (DsubRy x))) (aasub/pi1 (%the (aasub ([x] Ry x) M Ryx) DsubRyx)) ([y] aasub/pi1 (DsubRx y)) DsubNx (aasub/pi1 DsubRxy)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-pi1 (DofPi x d y e) (%the (aofes (G x y) (R x y) (ssigma U1 U2)) (DofR x d y e))) + %<- (aasub-permute-esm Size DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR DsubRy DsubRyx (%the ({y} aasub ([x] R x y) M (Rx y)) DsubRx) (%the (sub N M Nx) DsubNx) (%the (aasub ([y] Rx y) Nx Ryx) DsubRxy)) %. +%scope at-a %term pi1-c + %pi (aasub-permute-esm Size DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (pi1 (R x y)) U1))) DofPi) ([x] aasub/pi1 (%the (aasub ([y] R x y) (N x) Ry) (DsubRy x))) aasub/closed ([y] aasub/pi1 (DsubRx y)) DsubNx (aasub/pi1 DsubRxy)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-pi1 (DofPi x d y e) (%the (aofes (G x y) (R x y) (ssigma U1 U2)) (DofR x d y e))) + %<- (aasub-permute-esm Size DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR DsubRy aasub/closed (%the ({y} aasub ([x] R x y) M (Rx y)) DsubRx) (%the (sub N M Nx) DsubNx) (%the (aasub ([y] Rx y) Nx Ry) DsubRxy)) %. +%term pi2 + %pi (aasub-permute-esm Size DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (pi2 (R x y)) U2))) DofPi) ([x] aasub/pi2 (%the (aasub ([y] R x y) (N x) (Ry x)) (DsubRy x))) (aasub/pi2 (%the (aasub ([x] Ry x) M Ryx) DsubRyx)) ([y] aasub/pi2 (DsubRx y)) DsubNx (aasub/pi2 DsubRxy)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-pi2 (DofPi x d y e) (%the (aofes (G x y) (R x y) (ssigma U1 U2)) (DofR x d y e))) + %<- (aasub-permute-esm Size DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR DsubRy DsubRyx (%the ({y} aasub ([x] R x y) M (Rx y)) DsubRx) (%the (sub N M Nx) DsubNx) (%the (aasub ([y] Rx y) Nx Ryx) DsubRxy)) %. +%scope at-a %term pi2-c + %pi (aasub-permute-esm Size DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (pi2 (R x y)) U2))) DofPi) ([x] aasub/pi2 (%the (aasub ([y] R x y) (N x) Ry) (DsubRy x))) aasub/closed ([y] aasub/pi2 (DsubRx y)) DsubNx (aasub/pi2 DsubRxy)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-pi2 (DofPi x d y e) (%the (aofes (G x y) (R x y) (ssigma U1 U2)) (DofR x d y e))) + %<- (aasub-permute-esm Size DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR DsubRy aasub/closed (%the ({y} aasub ([x] R x y) M (Rx y)) DsubRx) (%the (sub N M Nx) DsubNx) (%the (aasub ([y] Rx y) Nx Ry) DsubRxy)) %. +%term t + %pi (tsub-permute-esm _ DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) _ ([_] tsub/t) tsub/t ([_] tsub/t) DsubNx tsub/t) + %<- ({y} {e isvar y J} can-sub-es ([x] [d] DlookupX x d y e) ([x] [d] DofN x d y e) ([x] [d] DofM x d y e) (%the (sub ([x] N x) M (Nx_y y)) (DsubNx_y y))) + %<- (sub-closed' DsubNx_y (%the (sub ([x] N x) M Nx) DsubNx)) %. +%% +%scope at-a %term pi + %pi (tsub-permute-esm _ DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) ([x] [d isvar x I] [y] [e isvar y J] wfes/pi ([z] [f isvar z K] %the (wfes (scons (G x y) z U) (B x y z)) (DofB x d y e z f)) (%the (simp (A x y) U) (Dsimp x y)) (%the (wfes (G x y) (A x y)) (DofA x d y e))) ([x] tsub/pi ([z] %the (tsub ([y] B x y z) (N x) (By x z)) (DsubBy x z)) (%the (tsub ([y] A x y) (N x) (Ay x)) (DsubAy x))) (tsub/pi ([z] %the (tsub ([x] By x z) M (Byx z)) (DsubByx z)) (%the (tsub ([x] Ay x) M Ayx) DsubAyx)) ([y] tsub/pi ([z] DsubBx y z) (DsubAx y)) DsubNx (tsub/pi ([z] DsubBxy' z) DsubAxy)) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} wfes-context (DofB x d y e z f) (%the (sordered (scons (G x y) z U)) (sordered/cons (%the (sbounded (G x y) z) (Dbounded x d y e z f))))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupX x d y e) _ (%the (slookup (scons (G x y) z U) x S) (DlookupX' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupY x d y e) _ (%the (slookup (scons (G x y) z U) y T) (DlookupY' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-ofes (Dbounded x d y e z f) (DofM x d y e) _ (%the (ofes (scons (G x y) z U) M S) (DofM' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-ofes (Dbounded x d y e z f) (DofN x d y e) _ (%the (ofes (scons (G x y) z U) (N x) T) (DofN' x d y e z f))) + %<- (tsub-permute-esm _ DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofA DsubAy DsubAyx (%the ({y} tsub ([x] A x y) M (Ax y)) DsubAx) (%the (sub ([x] N x) M Nx) DsubNx) (%the (tsub ([y] Ax y) Nx Ayx) DsubAxy)) + %<- ({z} {f isvar z K} tsub-permute-esm _ DsizeS DsizeT Dsum ([x] [d] [y] [e] DlookupX' x d y e z f) ([x] [d] [y] [e] DlookupY' x d y e z f) ([x] [d] [y] [e] DofM' x d y e z f) ([x] [d] [y] [e] DofN' x d y e z f) ([x] [d] [y] [e] DofB x d y e z f) ([x] DsubBy x z) (DsubByx z) ([y] %the (tsub ([x] B x y z) M (Bx y z)) (DsubBx y z)) (%the (sub ([x] N x) M (Nx' z)) (DsubNx' z)) (%the (tsub ([y] Bx y z) (Nx' z) (Byx z)) (DsubBxy z))) + %<- ({z} sub-fun (DsubNx' z) DsubNx (%the (term-eq (Nx' z) Nx) (DeqNx z))) + %<- ({z} tsub-resp ([_] tp-eq/i) (DeqNx z) tp-eq/i (DsubBxy z) (%the (tsub ([y] Bx y z) Nx (Byx z)) (DsubBxy' z))) %. +%term sigma + %pi (tsub-permute-esm _ DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) ([x] [d isvar x I] [y] [e isvar y J] wfes/sigma ([z] [f isvar z K] %the (wfes (scons (G x y) z U) (B x y z)) (DofB x d y e z f)) (%the (simp (A x y) U) (Dsimp x y)) (%the (wfes (G x y) (A x y)) (DofA x d y e))) ([x] tsub/sigma ([z] %the (tsub ([y] B x y z) (N x) (By x z)) (DsubBy x z)) (%the (tsub ([y] A x y) (N x) (Ay x)) (DsubAy x))) (tsub/sigma ([z] %the (tsub ([x] By x z) M (Byx z)) (DsubByx z)) (%the (tsub ([x] Ay x) M Ayx) DsubAyx)) ([y] tsub/sigma ([z] DsubBx y z) (DsubAx y)) DsubNx (tsub/sigma ([z] DsubBxy' z) DsubAxy)) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} wfes-context (DofB x d y e z f) (%the (sordered (scons (G x y) z U)) (sordered/cons (%the (sbounded (G x y) z) (Dbounded x d y e z f))))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupX x d y e) _ (%the (slookup (scons (G x y) z U) x S) (DlookupX' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupY x d y e) _ (%the (slookup (scons (G x y) z U) y T) (DlookupY' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-ofes (Dbounded x d y e z f) (DofM x d y e) _ (%the (ofes (scons (G x y) z U) M S) (DofM' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-ofes (Dbounded x d y e z f) (DofN x d y e) _ (%the (ofes (scons (G x y) z U) (N x) T) (DofN' x d y e z f))) + %<- (tsub-permute-esm _ DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofA DsubAy DsubAyx (%the ({y} tsub ([x] A x y) M (Ax y)) DsubAx) (%the (sub ([x] N x) M Nx) DsubNx) (%the (tsub ([y] Ax y) Nx Ayx) DsubAxy)) + %<- ({z} {f isvar z K} tsub-permute-esm _ DsizeS DsizeT Dsum ([x] [d] [y] [e] DlookupX' x d y e z f) ([x] [d] [y] [e] DlookupY' x d y e z f) ([x] [d] [y] [e] DofM' x d y e z f) ([x] [d] [y] [e] DofN' x d y e z f) ([x] [d] [y] [e] DofB x d y e z f) ([x] DsubBy x z) (DsubByx z) ([y] %the (tsub ([x] B x y z) M (Bx y z)) (DsubBx y z)) (%the (sub ([x] N x) M (Nx' z)) (DsubNx' z)) (%the (tsub ([y] Bx y z) (Nx' z) (Byx z)) (DsubBxy z))) + %<- ({z} sub-fun (DsubNx' z) DsubNx (%the (term-eq (Nx' z) Nx) (DeqNx z))) + %<- ({z} tsub-resp ([_] tp-eq/i) (DeqNx z) tp-eq/i (DsubBxy z) (%the (tsub ([y] Bx y z) Nx (Byx z)) (DsubBxy' z))) %. +%scope at-a %term singaa + %pi (tsub-permute-esm _ DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) ([x] [d isvar x I] [y] [e isvar y J] wfes/sing (%the (aofes (G x y) (R x y) st) (DofR x d y e))) ([x] tsub/singa (%the (aasub ([y] R x y) (N x) (Ry x)) (DsubRy x))) (tsub/singa (%the (aasub ([x] Ry x) M Ryx) DsubRyx)) ([y] tsub/singa (DsubRx y)) DsubNx (tsub/singa DsubRxy)) + %<- (aasub-permute-esm _ DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR DsubRy DsubRyx (%the ({y} aasub ([x] R x y) M (Rx y)) DsubRx) (%the (sub ([x] N x) M Nx) DsubNx) (%the (aasub ([y] Rx y) Nx Ryx) DsubRxy)) %. +%term singao + %pi (tsub-permute-esm _ DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) ([x] [d isvar x I] [y] [e isvar y J] wfes/sing (%the (aofes (G x y) (R x y) st) (DofR x d y e))) ([x] tsub/singa (%the (aasub ([y] R x y) (N x) (Ry x)) (DsubRy x))) (tsub/singo (%the (aosub ([x] Ry x) M (at Ryx)) DsubRyx)) ([y] tsub/singo (DsubRx' aca y)) DsubNx Dtsub) + %<- (aasub-aosub-permute-esm _ DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR DsubRy DsubRyx (%the ({y} aosub ([x] R x y) M (LRx y)) DsubRx) (%the (sub ([x] N x) M Nx) DsubNx) (%the (sub ([y] LRx y) Nx (at Ryx)) DsubRxy)) + %<- ({y} {e isvar y J} aosubst-es ([x] [d] DlookupX x d y e) ([x] [d] DofM x d y e) ([x] [d] DofR x d y e) (DsubRx y) ([x] [d] %the (ofes (G x y) (LRx y) st) (DofRx x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofes-invert-st (DofRx x d y e) (%the (term-eq (LRx y) (at (Rx_x x y))) (DeqRx x y))) + %<- ({x} {y} aosub-resp ([_] atom-eq/i) term-eq/i (DeqRx x y) (DsubRx y) (%the (aosub ([x] R x y) M (at (Rx_x x y))) (DsubRx' x y))) + %<- (sub-resp ([y] DeqRx aca y) term-eq/i term-eq/i DsubRxy (%the (sub ([y] at (Rx_x aca y)) Nx (at Ryx)) DsubRxy')) + %<- (sub-at-tsub DsubRxy' (%the (tsub ([y] sing (Rx_x aca y)) Nx (sing Ryx)) Dtsub)) %. +%scope at-a %term singoa + %pi (tsub-permute-esm _ DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) ([x] [d isvar x I] [y] [e isvar y J] wfes/sing (%the (aofes (G x y) (R x y) st) (DofR x d y e))) ([x] tsub/singo (%the (aosub ([y] R x y) (N x) (at (Ry x))) (DsubRy x))) (tsub/singa (%the (aasub ([x] Ry x) M Ryx) DsubRyx)) ([y] tsub/singa (DsubRx y)) DsubNx (tsub/singo DsubRxy)) + %<- (aosub-permute-esm _ DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR DsubRy (sub/aa DsubRyx) (%the ({y} aasub ([x] R x y) M (Rx y)) DsubRx) (%the (sub ([x] N x) M Nx) DsubNx) (%the (aosub ([y] Rx y) Nx (at Ryx)) DsubRxy)) %. +%term singoo + %pi (tsub-permute-esm _ DsizeS DsizeT Dsum (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) (N x) T))) DofN) ([x] [d isvar x I] [y] [e isvar y J] wfes/sing (%the (aofes (G x y) (R x y) st) (DofR x d y e))) ([x] tsub/singo (%the (aosub ([y] R x y) (N x) (at (Ry x))) (DsubRy x))) (tsub/singo (%the (aosub ([x] Ry x) M (at Ryx)) DsubRyx)) ([y] tsub/singa (DsubRx y)) DsubNx (tsub/singo DsubRxy)) + %<- (aosub-permute-esm _ DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR DsubRy (sub/ao DsubRyx) (%the ({y} aasub ([x] R x y) M (Rx y)) DsubRx) (%the (sub ([x] N x) M Nx) DsubNx) (%the (aosub ([y] Rx y) Nx (at Ryx)) DsubRxy)) %. +%worlds (var ovar bind) (aosub-permute-esm _ _ _ _ _ _ _ _ _ _ _ _ _ _) (aasub-aosub-permute-esm _ _ _ _ _ _ _ _ _ _ _ _ _ _) (aasub-permute-esm _ _ _ _ _ _ _ _ _ _ _ _ _ _) (sub-permute-esm _ _ _ _ _ _ _ _ _ _ _ _ _ _) (tsub-permute-esm _ _ _ _ _ _ _ _ _ _ _ _ _ _) %. +%total {(N1 N2 N3 N4 N5) (D1 D2 D3 D4 D5)} (sub-permute-esm N4 _ _ _ _ _ _ _ _ D4 _ _ _ _) (aosub-permute-esm N1 _ _ _ _ _ _ _ _ D1 _ _ _ _) (aasub-aosub-permute-esm N2 _ _ _ _ _ _ _ _ D2 _ _ _ _) (aasub-permute-esm N3 _ _ _ _ _ _ _ _ D3 _ _ _ _) (tsub-permute-esm N5 _ _ _ _ _ _ _ _ D5 _ _ _ _) %. +%sort sub-permute-es {x} %. +%mode sub-permute-es %in %in %in %in %in %in %in %out %out %out %. +%term _ + %pi (sub-permute-es DlookupX DlookupY DofM DofN DofO DsubOy DsubOyx DsubOx DsubNx DsubOxy) + %<- (can-stp-size S (%the (stp-size S SizeS) DsizeS)) + %<- (can-stp-size T (%the (stp-size T SizeT) DsizeT)) + %<- (can-sum SizeS SizeT Size (%the (sum SizeS SizeT Size) Dsum)) + %<- (sub-permute-esm Size DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofO DsubOy DsubOyx DsubOx DsubNx DsubOxy) %. +%worlds (var ovar bind) (sub-permute-es _ _ _ _ _ _ _ _ _ _) %. +%total {} (sub-permute-es _ _ _ _ _ _ _ _ _ _) %. +%sort aasub-aosub-permute-es {x} %. +%mode aasub-aosub-permute-es %in %in %in %in %in %in %in %out %out %out %. +%term _ + %pi (aasub-aosub-permute-es DlookupX DlookupY DofM DofN DofR DsubRy DsubRyx DsubRx DsubNx DsubRxy) + %<- (can-stp-size S (%the (stp-size S SizeS) DsizeS)) + %<- (can-stp-size T (%the (stp-size T SizeT) DsizeT)) + %<- (can-sum SizeS SizeT Size (%the (sum SizeS SizeT Size) Dsum)) + %<- (aasub-aosub-permute-esm Size DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR DsubRy DsubRyx DsubRx DsubNx DsubRxy) %. +%worlds (var ovar bind) (aasub-aosub-permute-es _ _ _ _ _ _ _ _ _ _) %. +%total {} (aasub-aosub-permute-es _ _ _ _ _ _ _ _ _ _) %. +%sort aasub-permute-es {x} %. +%mode aasub-permute-es %in %in %in %in %in %in %in %out %out %out %. +%term _ + %pi (aasub-permute-es DlookupX DlookupY DofM DofN DofR DsubRy DsubRyx DsubRx DsubNx DsubRxy) + %<- (can-stp-size S (%the (stp-size S SizeS) DsizeS)) + %<- (can-stp-size T (%the (stp-size T SizeT) DsizeT)) + %<- (can-sum SizeS SizeT Size (%the (sum SizeS SizeT Size) Dsum)) + %<- (aasub-permute-esm Size DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR DsubRy DsubRyx DsubRx DsubNx DsubRxy) %. +%worlds (var ovar bind) (aasub-permute-es _ _ _ _ _ _ _ _ _ _) %. +%total {} (aasub-permute-es _ _ _ _ _ _ _ _ _ _) %. +%sort aosub-permute-es {x} %. +%mode aosub-permute-es %in %in %in %in %in %in %in %out %out %out %. +%term _ + %pi (aosub-permute-es DlookupX DlookupY DofM DofN DofR DsubRy DsubRyx DsubRx DsubNx DsubRxy) + %<- (can-stp-size S (%the (stp-size S SizeS) DsizeS)) + %<- (can-stp-size T (%the (stp-size T SizeT) DsizeT)) + %<- (can-sum SizeS SizeT Size (%the (sum SizeS SizeT Size) Dsum)) + %<- (aosub-permute-esm Size DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DofR DsubRy DsubRyx DsubRx DsubNx DsubRxy) %. +%worlds (var ovar bind) (aosub-permute-es _ _ _ _ _ _ _ _ _ _) %. +%total {} (aosub-permute-es _ _ _ _ _ _ _ _ _ _) %. +%sort tsub-permute-es {x} %. +%mode tsub-permute-es %in %in %in %in %in %in %in %out %out %out %. +%term _ + %pi (tsub-permute-es DlookupX DlookupY DofM DofN DwfA DsubAy DsubAyx DsubAx DsubNx DsubAxy) + %<- (can-stp-size S (%the (stp-size S SizeS) DsizeS)) + %<- (can-stp-size T (%the (stp-size T SizeT) DsizeT)) + %<- (can-sum SizeS SizeT Size (%the (sum SizeS SizeT Size) Dsum)) + %<- (tsub-permute-esm Size DsizeS DsizeT Dsum DlookupX DlookupY DofM DofN DwfA DsubAy DsubAyx DsubAx DsubNx DsubAxy) %. +%worlds (var ovar bind) (tsub-permute-es _ _ _ _ _ _ _ _ _ _) %. +%total {} (tsub-permute-es _ _ _ _ _ _ _ _ _ _) %. +%%%%% Substitution Permutation (Explicit Context) %%%%% +%sort sub-permute-e {x} %. +%mode sub-permute-e %in %in %in %in %in %in %in %in %in %out %. +%term _ + %pi (sub-permute-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (ofe G1 M A) DofM) (%the ({x} %pi (isvar x I) %-> (ofe (G x) (N x) (B x))) DofN) (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y J) %-> (ofe (cons (G x) y (B x)) (O x y) (C x y)))) DofO) (%the ({y} sub ([x] O x y) M (Ox y)) DsubOx) (%the (sub N M Nx) DsubNx) (%the (sub ([y] Ox y) Nx Oxy) DsubOxy) (%the ({x} sub ([y] O x y) (N x) (Oy x)) DsubOy) (%the (sub ([x] Oy x) M Oyx) DsubOyx) Deq) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-context (DofO x d y e) (ordered/cons (%the (bounded (G x) y) (Dbounded x d y e)))) + %<- ({x} {d isvar x I} {y} {e isvar y J} append-bounded (Dappend x) (Dbounded x d y e) (bounded/cons (%the (bounded G1 x) (Dbounded' x d y e)) (%the (precedes x y) (DprecedesXY x d y e)))) + %<- (can-simp A (%the (simp A S) DsimpA)) + %<- ({x} can-simp (B x) (%the (simp (B x) T) (DsimpB x))) + %<- ({x} {y} can-simp (C x y) (%the (simp (C x y) U) (DsimpC x y))) + %<- ({y} {e isvar y J} context-append-simp-lookup Dappend ([x] [d] Dbounded x d y e) DsimpA ([x] %the (simpctx (G x) (G' x y)) (Dsimpctx x y)) ([x] [d] %the (sbounded (G' x y) y) (Dsbounded x d y e)) ([x] [d] %the (slookup (G' x y) x S) (Dlookup x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded' x d y e) DofM _ (%the (ofe (cons G1 x A) M A) (DofM' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe'' (Dappend x) (Dbounded x d y e) (DofM' x d y e) (%the (ofe (G x) M A) (DofM'' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded x d y e) (DofM'' x d y e) _ (%the (ofe (cons (G x) y (B x)) M A) (DofM''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) DsimpA (DofM''' x d y e) (%the (ofes (scons (G' x y) y T) M S) (DofM'''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded x d y e) (DofN x d) _ (%the (ofe (cons (G x) y (B x)) (N x) (B x)) (DofN' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) (DsimpB x) (DofN' x d y e) (%the (ofes (scons (G' x y) y T) (N x) T) (DofN'' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) (DsimpC x y) (DofO x d y e) (%the (ofes (scons (G' x y) y T) (O x y) U) (DofO' x d y e))) + %<- (sub-permute-es ([x] [d isvar x I] [y] [e isvar y J] slookup/miss (Dlookup x d y e) (Dsbounded x d y e)) ([x] [d] [y] [e] slookup/hit (Dsbounded x d y e)) DofM'''' DofN'' DofO' DsubOy DsubOyx (%the ({y} sub ([x] O x y) M (Ox' y)) DsubOx') (%the (sub N M Nx') DsubNx') (%the (sub Ox' Nx' Oyx) DsubOxy')) + %<- ({y} sub-fun (DsubOx' y) (DsubOx y) (%the (term-eq (Ox' y) (Ox y)) (DeqOx y))) + %<- (sub-fun DsubNx' DsubNx (%the (term-eq Nx' Nx) DeqNx)) + %<- (sub-resp DeqOx DeqNx term-eq/i DsubOxy' (%the (sub Ox Nx Oyx) DsubOxy'')) + %<- (sub-fun DsubOxy DsubOxy'' (%the (term-eq Oxy Oyx) Deq)) %. +%worlds (var ovar bind) (sub-permute-e _ _ _ _ _ _ _ _ _ _) %. +%total {} (sub-permute-e _ _ _ _ _ _ _ _ _ _) %. +%sort aasub-permute-e {x} %. +%mode aasub-permute-e %in %in %in %in %in %in %in %in %in %out %. +%term _ + %pi (aasub-permute-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (ofe G1 M A) DofM) (%the ({x} %pi (isvar x I) %-> (ofe (G x) (N x) (B x))) DofN) (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y J) %-> (aofe (cons (G x) y (B x)) (R x y) (C x y)))) DaofR) (%the ({y} aasub ([x] R x y) M (Rx y)) DsubRx) (%the (sub N M Nx) DsubNx) (%the (aasub ([y] Rx y) Nx Rxy) DsubRxy) (%the ({x} aasub ([y] R x y) (N x) (Ry x)) DsubRy) (%the (aasub ([x] Ry x) M Ryx) DsubRyx) Deq) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofe-context (DaofR x d y e) (ordered/cons (%the (bounded (G x) y) (Dbounded x d y e)))) + %<- ({x} {d isvar x I} {y} {e isvar y J} append-bounded (Dappend x) (Dbounded x d y e) (bounded/cons (%the (bounded G1 x) (Dbounded' x d y e)) (%the (precedes x y) (DprecedesXY x d y e)))) + %<- (can-simp A (%the (simp A S) DsimpA)) + %<- ({x} can-simp (B x) (%the (simp (B x) T) (DsimpB x))) + %<- ({x} {y} can-simp (C x y) (%the (simp (C x y) U) (DsimpC x y))) + %<- ({y} {e isvar y J} context-append-simp-lookup Dappend ([x] [d] Dbounded x d y e) DsimpA ([x] %the (simpctx (G x) (G' x y)) (Dsimpctx x y)) ([x] [d] %the (sbounded (G' x y) y) (Dsbounded x d y e)) ([x] [d] %the (slookup (G' x y) x S) (Dlookup x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded' x d y e) DofM _ (%the (ofe (cons G1 x A) M A) (DofM' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe'' (Dappend x) (Dbounded x d y e) (DofM' x d y e) (%the (ofe (G x) M A) (DofM'' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded x d y e) (DofM'' x d y e) _ (%the (ofe (cons (G x) y (B x)) M A) (DofM''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) DsimpA (DofM''' x d y e) (%the (ofes (scons (G' x y) y T) M S) (DofM'''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded x d y e) (DofN x d) _ (%the (ofe (cons (G x) y (B x)) (N x) (B x)) (DofN' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) (DsimpB x) (DofN' x d y e) (%the (ofes (scons (G' x y) y T) (N x) T) (DofN'' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) (DsimpC x y) (DaofR x d y e) (%the (aofes (scons (G' x y) y T) (R x y) U) (DaofR' x d y e))) + %<- (aasub-permute-es ([x] [d isvar x I] [y] [e isvar y J] slookup/miss (Dlookup x d y e) (Dsbounded x d y e)) ([x] [d] [y] [e] slookup/hit (Dsbounded x d y e)) DofM'''' DofN'' DaofR' DsubRy DsubRyx (%the ({y} aasub ([x] R x y) M (Rx' y)) DsubRx') (%the (sub N M Nx') DsubNx') (%the (aasub Rx' Nx' Ryx) DsubRxy')) + %<- ({y} aasub-fun (DsubRx' y) (DsubRx y) (%the (atom-eq (Rx' y) (Rx y)) (DeqRx y))) + %<- (sub-fun DsubNx' DsubNx (%the (term-eq Nx' Nx) DeqNx)) + %<- (aasub-resp DeqRx DeqNx atom-eq/i DsubRxy' (%the (aasub Rx Nx Ryx) DsubRxy'')) + %<- (aasub-fun DsubRxy DsubRxy'' (%the (atom-eq Rxy Ryx) Deq)) %. +%worlds (var ovar bind) (aasub-permute-e _ _ _ _ _ _ _ _ _ _) %. +%total {} (aasub-permute-e _ _ _ _ _ _ _ _ _ _) %. +%sort aasub-aosub-permute-e {x} %. +%mode aasub-aosub-permute-e %in %in %in %in %in %in %in %in %in %out %. +%term _ + %pi (aasub-aosub-permute-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (ofe G1 M A) DofM) (%the ({x} %pi (isvar x I) %-> (ofe (G x) (N x) (B x))) DofN) (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y J) %-> (aofe (cons (G x) y (B x)) (R x y) (C x y)))) DaofR) (%the ({y} aosub ([x] R x y) M (LRx y)) DsubRx) (%the (sub N M Nx) DsubNx) (%the (sub ([y] LRx y) Nx LRxy) DsubRxy) (%the ({x} aasub ([y] R x y) (N x) (Ry x)) DsubRy) (%the (aosub ([x] Ry x) M LRyx) DsubRyx) Deq) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofe-context (DaofR x d y e) (ordered/cons (%the (bounded (G x) y) (Dbounded x d y e)))) + %<- ({x} {d isvar x I} {y} {e isvar y J} append-bounded (Dappend x) (Dbounded x d y e) (bounded/cons (%the (bounded G1 x) (Dbounded' x d y e)) (%the (precedes x y) (DprecedesXY x d y e)))) + %<- (can-simp A (%the (simp A S) DsimpA)) + %<- ({x} can-simp (B x) (%the (simp (B x) T) (DsimpB x))) + %<- ({x} {y} can-simp (C x y) (%the (simp (C x y) U) (DsimpC x y))) + %<- ({y} {e isvar y J} context-append-simp-lookup Dappend ([x] [d] Dbounded x d y e) DsimpA ([x] %the (simpctx (G x) (G' x y)) (Dsimpctx x y)) ([x] [d] %the (sbounded (G' x y) y) (Dsbounded x d y e)) ([x] [d] %the (slookup (G' x y) x S) (Dlookup x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded' x d y e) DofM _ (%the (ofe (cons G1 x A) M A) (DofM' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe'' (Dappend x) (Dbounded x d y e) (DofM' x d y e) (%the (ofe (G x) M A) (DofM'' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded x d y e) (DofM'' x d y e) _ (%the (ofe (cons (G x) y (B x)) M A) (DofM''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) DsimpA (DofM''' x d y e) (%the (ofes (scons (G' x y) y T) M S) (DofM'''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded x d y e) (DofN x d) _ (%the (ofe (cons (G x) y (B x)) (N x) (B x)) (DofN' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) (DsimpB x) (DofN' x d y e) (%the (ofes (scons (G' x y) y T) (N x) T) (DofN'' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) (DsimpC x y) (DaofR x d y e) (%the (aofes (scons (G' x y) y T) (R x y) U) (DaofR' x d y e))) + %<- (aasub-aosub-permute-es ([x] [d isvar x I] [y] [e isvar y J] slookup/miss (Dlookup x d y e) (Dsbounded x d y e)) ([x] [d] [y] [e] slookup/hit (Dsbounded x d y e)) DofM'''' DofN'' DaofR' DsubRy DsubRyx (%the ({y} aosub ([x] R x y) M (LRx' y)) DsubRx') (%the (sub N M Nx') DsubNx') (%the (sub LRx' Nx' LRyx) DsubRxy')) + %<- ({y} aosub-fun (DsubRx' y) (DsubRx y) (%the (term-eq (LRx' y) (LRx y)) (DeqLRx y))) + %<- (sub-fun DsubNx' DsubNx (%the (term-eq Nx' Nx) DeqNx)) + %<- (sub-resp DeqLRx DeqNx term-eq/i DsubRxy' (%the (sub LRx Nx LRyx) DsubRxy'')) + %<- (sub-fun DsubRxy DsubRxy'' (%the (term-eq LRxy LRyx) Deq)) %. +%worlds (var ovar bind) (aasub-aosub-permute-e _ _ _ _ _ _ _ _ _ _) %. +%total {} (aasub-aosub-permute-e _ _ _ _ _ _ _ _ _ _) %. +%sort aosub-permute-e {x} %. +%mode aosub-permute-e %in %in %in %in %in %in %in %in %in %out %. +%term _ + %pi (aosub-permute-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (ofe G1 M A) DofM) (%the ({x} %pi (isvar x I) %-> (ofe (G x) (N x) (B x))) DofN) (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y J) %-> (aofe (cons (G x) y (B x)) (R x y) (C x y)))) DaofR) (%the ({y} aasub ([x] R x y) M (Rx y)) DsubRx) (%the (sub N M Nx) DsubNx) (%the (aosub ([y] Rx y) Nx LRxy) DsubRxy) (%the ({x} aosub ([y] R x y) (N x) (LRy x)) DsubRy) (%the (sub ([x] LRy x) M LRyx) DsubRyx) Deq) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofe-context (DaofR x d y e) (ordered/cons (%the (bounded (G x) y) (Dbounded x d y e)))) + %<- ({x} {d isvar x I} {y} {e isvar y J} append-bounded (Dappend x) (Dbounded x d y e) (bounded/cons (%the (bounded G1 x) (Dbounded' x d y e)) (%the (precedes x y) (DprecedesXY x d y e)))) + %<- (can-simp A (%the (simp A S) DsimpA)) + %<- ({x} can-simp (B x) (%the (simp (B x) T) (DsimpB x))) + %<- ({x} {y} can-simp (C x y) (%the (simp (C x y) U) (DsimpC x y))) + %<- ({y} {e isvar y J} context-append-simp-lookup Dappend ([x] [d] Dbounded x d y e) DsimpA ([x] %the (simpctx (G x) (G' x y)) (Dsimpctx x y)) ([x] [d] %the (sbounded (G' x y) y) (Dsbounded x d y e)) ([x] [d] %the (slookup (G' x y) x S) (Dlookup x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded' x d y e) DofM _ (%the (ofe (cons G1 x A) M A) (DofM' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe'' (Dappend x) (Dbounded x d y e) (DofM' x d y e) (%the (ofe (G x) M A) (DofM'' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded x d y e) (DofM'' x d y e) _ (%the (ofe (cons (G x) y (B x)) M A) (DofM''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) DsimpA (DofM''' x d y e) (%the (ofes (scons (G' x y) y T) M S) (DofM'''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded x d y e) (DofN x d) _ (%the (ofe (cons (G x) y (B x)) (N x) (B x)) (DofN' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) (DsimpB x) (DofN' x d y e) (%the (ofes (scons (G' x y) y T) (N x) T) (DofN'' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) (DsimpC x y) (DaofR x d y e) (%the (aofes (scons (G' x y) y T) (R x y) U) (DaofR' x d y e))) + %<- (aosub-permute-es ([x] [d isvar x I] [y] [e isvar y J] slookup/miss (Dlookup x d y e) (Dsbounded x d y e)) ([x] [d] [y] [e] slookup/hit (Dsbounded x d y e)) DofM'''' DofN'' DaofR' DsubRy DsubRyx (%the ({y} aasub ([x] R x y) M (Rx' y)) DsubRx') (%the (sub N M Nx') DsubNx') (%the (aosub Rx' Nx' LRyx) DsubRxy')) + %<- ({y} aasub-fun (DsubRx' y) (DsubRx y) (%the (atom-eq (Rx' y) (Rx y)) (DeqRx y))) + %<- (sub-fun DsubNx' DsubNx (%the (term-eq Nx' Nx) DeqNx)) + %<- (aosub-resp DeqRx DeqNx term-eq/i DsubRxy' (%the (aosub Rx Nx LRyx) DsubRxy'')) + %<- (aosub-fun DsubRxy DsubRxy'' (%the (term-eq LRxy LRyx) Deq)) %. +%worlds (var ovar bind) (aosub-permute-e _ _ _ _ _ _ _ _ _ _) %. +%total {} (aosub-permute-e _ _ _ _ _ _ _ _ _ _) %. +%sort tsub-permute-e {x} %. +%mode tsub-permute-e %in %in %in %in %in %in %in %in %in %out %. +%term _ + %pi (tsub-permute-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (ofe G1 M A) DofM) (%the ({x} %pi (isvar x I) %-> (ofe (G x) (N x) (B x))) DofN) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y)))) DwfC) (%the ({y} tsub ([x] C x y) M (Cx y)) DsubCx) (%the (sub N M Nx) DsubNx) (%the (tsub ([y] Cx y) Nx Cxy) DsubCxy) (%the ({x} tsub ([y] C x y) (N x) (Cy x)) DsubCy) (%the (tsub ([x] Cy x) M Cyx) DsubCyx) Deq) + %<- ({x} {d isvar x I} {y} {e isvar y J} wfe-context (DwfC x d y e) (ordered/cons (%the (bounded (G x) y) (Dbounded x d y e)))) + %<- ({x} {d isvar x I} {y} {e isvar y J} append-bounded (Dappend x) (Dbounded x d y e) (bounded/cons (%the (bounded G1 x) (Dbounded' x d y e)) (%the (precedes x y) (DprecedesXY x d y e)))) + %<- (can-simp A (%the (simp A S) DsimpA)) + %<- ({x} can-simp (B x) (%the (simp (B x) T) (DsimpB x))) + %<- ({y} {e isvar y J} context-append-simp-lookup Dappend ([x] [d] Dbounded x d y e) DsimpA ([x] %the (simpctx (G x) (G' x y)) (Dsimpctx x y)) ([x] [d] %the (sbounded (G' x y) y) (Dsbounded x d y e)) ([x] [d] %the (slookup (G' x y) x S) (Dlookup x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded' x d y e) DofM _ (%the (ofe (cons G1 x A) M A) (DofM' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe'' (Dappend x) (Dbounded x d y e) (DofM' x d y e) (%the (ofe (G x) M A) (DofM'' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded x d y e) (DofM'' x d y e) _ (%the (ofe (cons (G x) y (B x)) M A) (DofM''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) DsimpA (DofM''' x d y e) (%the (ofes (scons (G' x y) y T) M S) (DofM'''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded x d y e) (DofN x d) _ (%the (ofe (cons (G x) y (B x)) (N x) (B x)) (DofN' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) (DsimpB x) (DofN' x d y e) (%the (ofes (scons (G' x y) y T) (N x) T) (DofN'' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} wfe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) (DwfC x d y e) (%the (wfes (scons (G' x y) y T) (C x y)) (DwfC' x d y e))) + %<- (tsub-permute-es ([x] [d isvar x I] [y] [e isvar y J] slookup/miss (Dlookup x d y e) (Dsbounded x d y e)) ([x] [d] [y] [e] slookup/hit (Dsbounded x d y e)) DofM'''' DofN'' DwfC' DsubCy DsubCyx (%the ({y} tsub ([x] C x y) M (Cx' y)) DsubCx') (%the (sub N M Nx') DsubNx') (%the (tsub Cx' Nx' Cyx) DsubCxy')) + %<- ({y} tsub-fun (DsubCx' y) (DsubCx y) (%the (tp-eq (Cx' y) (Cx y)) (DeqCx y))) + %<- (sub-fun DsubNx' DsubNx (%the (term-eq Nx' Nx) DeqNx)) + %<- (tsub-resp DeqCx DeqNx tp-eq/i DsubCxy' (%the (tsub Cx Nx Cyx) DsubCxy'')) + %<- (tsub-fun DsubCxy DsubCxy'' (%the (tp-eq Cxy Cyx) Deq)) %. +%worlds (var ovar bind) (tsub-permute-e _ _ _ _ _ _ _ _ _ _) %. +%total {} (tsub-permute-e _ _ _ _ _ _ _ _ _ _) %. +%%%%% Simple (ie, atom for atom) Substitution Permutations (Simply Typed, Explicit Context) %%%%% +%sort aasub-into-term {N %pi atom %-> term} %. +%mode aasub-into-term %in %in %out %. +%sort aasub-into-atom {Q %pi atom %-> atom} %. +%mode aasub-into-atom %in %in %out %. +%sort aasub-into-tp {A %pi atom %-> tp} %. +%mode aasub-into-tp %in %in %out %. +%term _ aasub-into-atom ([y] y) (%the (aasub ([x] R x) M Rx) DsubRx) DsubRx %. +%term _ + %pi (aasub-into-atom ([_] Q) _ Daasub) + %<- (aasub-absent Q M (%the (aasub ([_] Q) M Q) Daasub)) %. +%term _ + %pi (aasub-into-atom ([y] app (Q y) (N y)) (%the (aasub ([x] R x) M Rx) DsubRx) (aasub/app DsubNyx DsubQyx)) + %<- (aasub-into-atom Q DsubRx (%the (aasub ([x] Q (R x)) M (Q Rx)) DsubQyx)) + %<- (aasub-into-term N DsubRx (%the (sub ([x] N (R x)) M (N Rx)) DsubNyx)) %. +%term _ + %pi (aasub-into-atom ([y] pi1 (Q y)) (%the (aasub ([x] R x) M Rx) DsubRx) (aasub/pi1 DsubQyx)) + %<- (aasub-into-atom Q DsubRx (%the (aasub ([x] Q (R x)) M (Q Rx)) DsubQyx)) %. +%term _ + %pi (aasub-into-atom ([y] forall (A y)) (%the (aasub ([x] R x) M Rx) DsubRx) (aasub/forall DsubAyx)) + %<- (aasub-into-tp A DsubRx (%the (tsub ([x] A (R x)) M (A Rx)) DsubAyx)) %. +%term _ + %pi (aasub-into-atom ([y] pi2 (Q y)) (%the (aasub ([x] R x) M Rx) DsubRx) (aasub/pi2 DsubQyx)) + %<- (aasub-into-atom Q DsubRx (%the (aasub ([x] Q (R x)) M (Q Rx)) DsubQyx)) %. +%term at + %pi (aasub-into-term ([y] at (Q y)) (%the (aasub ([x] R x) M Rx) DsubRx) (sub/aa DsubQyx)) + %<- (aasub-into-atom Q DsubRx (%the (aasub ([x] Q (R x)) M (Q Rx)) DsubQyx)) %. +%term lam + %pi (aasub-into-term ([y] lam ([z] N y z)) (%the (aasub ([x] R x) M Rx) DsubRx) (sub/lam DsubNyx)) + %<- ({z} aasub-into-term ([y] N y z) DsubRx (%the (sub ([x] N (R x) z) M (N Rx z)) (DsubNyx z))) %. +%term pair + %pi (aasub-into-term ([y] pair (N1 y) (N2 y)) (%the (aasub ([x] R x) M Rx) DsubRx) (sub/pair DsubN2yx DsubN1yx)) + %<- (aasub-into-term N1 DsubRx (%the (sub ([x] N1 (R x)) M (N1 Rx)) DsubN1yx)) + %<- (aasub-into-term N2 DsubRx (%the (sub ([x] N2 (R x)) M (N2 Rx)) DsubN2yx)) %. +%term t aasub-into-tp ([y] t) _ tsub/t %. +%term pi + %pi (aasub-into-tp ([y] pi (A y) ([z] B y z)) (%the (aasub ([x] R x) M Rx) DsubRx) (tsub/pi DsubNyx DsubAyx)) + %<- (aasub-into-tp A DsubRx DsubAyx) + %<- ({z} aasub-into-tp ([y] B y z) DsubRx (%the (tsub ([x] B (R x) z) M (B Rx z)) (DsubNyx z))) %. +%term sigma + %pi (aasub-into-tp ([y] sigma (A y) ([z] B y z)) (%the (aasub ([x] R x) M Rx) DsubRx) (tsub/sigma DsubNyx DsubAyx)) + %<- (aasub-into-tp A DsubRx DsubAyx) + %<- ({z} aasub-into-tp ([y] B y z) DsubRx (%the (tsub ([x] B (R x) z) M (B Rx z)) (DsubNyx z))) %. +%term sing + %pi (aasub-into-tp ([y] sing (Q y)) (%the (aasub ([x] R x) M Rx) DsubRx) (tsub/singa DsubQyx)) + %<- (aasub-into-atom Q DsubRx (%the (aasub ([x] Q (R x)) M (Q Rx)) DsubQyx)) %. +%worlds (var ovar bind) (aasub-into-atom _ _ _) (aasub-into-term _ _ _) (aasub-into-tp _ _ _) %. +%total (R N A) (aasub-into-atom R _ _) (aasub-into-term N _ _) (aasub-into-tp A _ _) %. +%sort aosub-into-aasub %. +%mode aosub-into-aasub %in %in %out %. +%sort aosub-into-aosub %. +%mode aosub-into-aosub %in %in %out %. +%sort aosub-into-sub %. +%mode aosub-into-sub %in %in %out %. +%sort aosub-into-tsub %. +%mode aosub-into-tsub %in %in %out %. +%scope aosub-into-tsub %term closed + %pi (aosub-into-aasub (%the (aosub ([x] R x) M LRx) DsubRx) aasub/closed Daasub) + %<- (aasub-absent Q M (%the (aasub ([_] Q) M Q) Daasub)) %. +%term forall + %pi (aosub-into-aasub (%the (aosub ([x] R x) M LRx) DsubRx) (aasub/forall (%the (tsub ([y] A y) LRx Ay) DsubAy)) (aasub/forall DsubAyx)) + %<- (aosub-into-tsub DsubRx DsubAy (%the (tsub ([x] A (R x)) M Ay) DsubAyx)) %. +%scope aosub-into-tsub %term app + %pi (aosub-into-aasub (%the (aosub ([x] R x) M LRx) DsubRx) (aasub/app (%the (sub ([y] N y) LRx Ny) DsubNy) (%the (aasub ([y] Q y) LRx Qy) DsubQy)) (aasub/app DsubNyx DsubQyx)) + %<- (aosub-into-aasub DsubRx DsubQy (%the (aasub ([x] Q (R x)) M Qy) DsubQyx)) + %<- (aosub-into-sub DsubRx DsubNy (%the (sub ([x] N (R x)) M Ny) DsubNyx)) %. +%term pi1 + %pi (aosub-into-aasub (%the (aosub ([x] R x) M LRx) DsubRx) (aasub/pi1 (%the (aasub ([y] Q y) LRx Qy) DsubQy)) (aasub/pi1 DsubQyx)) + %<- (aosub-into-aasub DsubRx DsubQy (%the (aasub ([x] Q (R x)) M Qy) DsubQyx)) %. +%scope aosub-into-tsub %term pi2 + %pi (aosub-into-aasub (%the (aosub ([x] R x) M LRx) DsubRx) (aasub/pi2 (%the (aasub ([y] Q y) LRx Qy) DsubQy)) (aasub/pi2 DsubQyx)) + %<- (aosub-into-aasub DsubRx DsubQy (%the (aasub ([x] Q (R x)) M Qy) DsubQyx)) %. +%term var aosub-into-aosub (%the (aosub ([x] R x) M LRx) DsubRx) aosub/var DsubRx %. +%scope aosub-into-tsub %term app + %pi (aosub-into-aosub (%the (aosub ([x] R x) M LRx) DsubRx) (aosub/app (%the (sub ([z] P z) Ny Pz) DsubPz) (%the (sub ([y] N y) LRx Ny) DsubNy) (%the (aosub ([y] Q y) LRx (lam ([z] P z))) DsubQy)) (aosub/app DsubPz DsubNyx DsubQyx)) + %<- (aosub-into-aosub DsubRx DsubQy (%the (aosub ([x] Q (R x)) M (lam ([z] P z))) DsubQyx)) + %<- (aosub-into-sub DsubRx DsubNy (%the (sub ([x] N (R x)) M Ny) DsubNyx)) %. +%term pi1 + %pi (aosub-into-aosub (%the (aosub ([x] R x) M LRx) DsubRx) (aosub/pi1 (%the (aosub ([y] Q y) LRx (pair P1 P2)) DsubQy)) (aosub/pi1 DsubQyx)) + %<- (aosub-into-aosub DsubRx DsubQy (%the (aosub ([x] Q (R x)) M (pair P1 P2)) DsubQyx)) %. +%scope aosub-into-tsub %term pi2 + %pi (aosub-into-aosub (%the (aosub ([x] R x) M LRx) DsubRx) (aosub/pi2 (%the (aosub ([y] Q y) LRx (pair P1 P2)) DsubQy)) (aosub/pi2 DsubQyx)) + %<- (aosub-into-aosub DsubRx DsubQy (%the (aosub ([x] Q (R x)) M (pair P1 P2)) DsubQyx)) %. +%term aa + %pi (aosub-into-sub (%the (aosub ([x] R x) M LRx) DsubRx) (sub/aa (%the (aasub ([y] Q y) LRx Qy) DsubQy)) (sub/aa DsubQyx)) + %<- (aosub-into-aasub DsubRx DsubQy (%the (aasub ([x] Q (R x)) M Qy) DsubQyx)) %. +%scope aosub-into-tsub %term ao + %pi (aosub-into-sub (%the (aosub ([x] R x) M LRx) DsubRx) (sub/ao (%the (aosub ([y] Q y) LRx LQy) DsubQy)) (sub/ao DsubQyx)) + %<- (aosub-into-aosub DsubRx DsubQy (%the (aosub ([x] Q (R x)) M LQy) DsubQyx)) %. +%term lam + %pi (aosub-into-sub (%the (aosub ([x] R x) M LRx) DsubRx) (sub/lam ([z] %the (sub ([y] N y z) LRx (Ny z)) (DsubNy z))) (sub/lam DsubNyx)) + %<- ({z} aosub-into-sub DsubRx (DsubNy z) (%the (sub ([x] N (R x) z) M (Ny z)) (DsubNyx z))) %. +%scope aosub-into-tsub %term pair + %pi (aosub-into-sub (%the (aosub ([x] R x) M LRx) DsubRx) (sub/pair (%the (sub ([y] N2 y) LRx N2y) DsubN2y) (%the (sub ([y] N1 y) LRx N1y) DsubN1y)) (sub/pair DsubN2yx DsubN1yx)) + %<- (aosub-into-sub DsubRx DsubN1y (%the (sub ([x] N1 (R x)) M N1y) DsubN1yx)) + %<- (aosub-into-sub DsubRx DsubN2y (%the (sub ([x] N2 (R x)) M N2y) DsubN2yx)) %. +%term t aosub-into-tsub _ tsub/t tsub/t %. +%scope aosub-into-tsub %term pi + %pi (aosub-into-tsub (%the (aosub ([x] R x) M LRx) DsubRx) (tsub/pi ([z] %the (tsub ([y] B y z) LRx (By z)) (DsubBy z)) (%the (tsub A LRx Ay) DsubAy)) (tsub/pi DsubByx DsubAyx)) + %<- (aosub-into-tsub DsubRx DsubAy (%the (tsub ([x] A (R x)) M Ay) DsubAyx)) + %<- ({z} aosub-into-tsub DsubRx (DsubBy z) (%the (tsub ([x] B (R x) z) M (By z)) (DsubByx z))) %. +%term sigma + %pi (aosub-into-tsub (%the (aosub ([x] R x) M LRx) DsubRx) (tsub/sigma ([z] %the (tsub ([y] B y z) LRx (By z)) (DsubBy z)) (%the (tsub A LRx Ay) DsubAy)) (tsub/sigma DsubByx DsubAyx)) + %<- (aosub-into-tsub DsubRx DsubAy (%the (tsub ([x] A (R x)) M Ay) DsubAyx)) + %<- ({z} aosub-into-tsub DsubRx (DsubBy z) (%the (tsub ([x] B (R x) z) M (By z)) (DsubByx z))) %. +%scope aosub-into-tsub %term aa + %pi (aosub-into-tsub (%the (aosub ([x] R x) M LRx) DsubRx) (tsub/singa (%the (aasub ([y] Q y) LRx Qy) DsubQy)) (tsub/singa DsubQyx)) + %<- (aosub-into-aasub DsubRx DsubQy (%the (aasub ([x] Q (R x)) M Qy) DsubQyx)) %. +%term ao + %pi (aosub-into-tsub (%the (aosub ([x] R x) M LRx) DsubRx) (tsub/singo (%the (aosub ([y] Q y) LRx (at LQy)) DsubQy)) (tsub/singo DsubQyx)) + %<- (aosub-into-aosub DsubRx DsubQy (%the (aosub ([x] Q (R x)) M (at LQy)) DsubQyx)) %. +%worlds (var ovar bind) (aosub-into-aasub _ _ _) (aosub-into-aosub _ _ _) (aosub-into-sub _ _ _) (aosub-into-tsub _ _ _) %. +%total (D1 D2 D3 D4) (aosub-into-aasub _ D1 _) (aosub-into-aosub _ D2 _) (aosub-into-sub _ D3 _) (aosub-into-tsub _ D4 _) %. +%% hd(R) = z (ie, something other than x) +%sort ssub-z-permute-es {x} %. +%mode ssub-z-permute-es %in %in %in %in %in %in %in %out %. +%% hd(R) = x +%sort ssub-x-permute-es {x} %. +%mode ssub-x-permute-es %in %in %in %in %in %in %in %in %out %. +%% hd(R) = z, hd(Q) = z +%sort ssub-zz-permute-es {x} %. +%mode ssub-zz-permute-es %in %in %in %in %in %in %in %out %. +%% hd(R) = z, hd(Q) = x +%sort ssub-zx-permute-es {x} %. +%mode ssub-zx-permute-es %in %in %in %in %in %in %in %out %. +%% hd(R) = x, hd(Q) = z +%sort ssub-xz-permute-es {x} %. +%mode ssub-xz-permute-es %in %in %in %in %in %in %in %in %out %. +%% hd(R) = x, hd(Q) = y +%sort ssub-xy-permute-es {x} %. +%mode ssub-xy-permute-es %in %in %in %in %in %in %in %in %out %. +%% hd(R) = x, hd(Q) = x +%sort ssub-xx-permute-es {x} %. +%mode ssub-xx-permute-es %in %in %in %in %in %in %in %in %out %. +%% hd(R) = z +%sort stsub-z-permute-es {x} %. +%mode stsub-z-permute-es %in %in %in %in %in %in %in %out %. +%% hd(R) = x +%sort stsub-x-permute-es {x} %. +%mode stsub-x-permute-es %in %in %in %in %in %in %in %in %out %. +%scope stsub-x-permute-es %term aa + %pi (ssub-z-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] %the (ofes (G x y) (at (Q x y)) st) (ofes/at (%the (aofes (G x y) (Q x y) st) (DofQ x d y e)))) ([y] sub/aa (%the (aasub ([x] Q x y) M (Qx y)) (DsubQx y))) (%the (aasub R M Rx) DsubRx) (sub/aa DsubQyx)) + %<- (ssub-zz-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx (%the (aasub ([x] Q x (R x)) M (Qx Rx)) DsubQyx)) %. +%term ao + %pi (ssub-z-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] %the (ofes (G x y) (at (Q x y)) st) (ofes/at (%the (aofes (G x y) (Q x y) st) (DofQ x d y e)))) ([y] sub/ao (%the (aosub ([x] Q x y) M (LQx y)) (DsubQx y))) (%the (aasub R M Rx) DsubRx) (sub/ao DsubQyx)) + %<- (ssub-zx-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx (%the (aosub ([x] Q x (R x)) M (LQx Rx)) DsubQyx)) %. +%scope stsub-x-permute-es %term lam + %pi (ssub-z-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] %the (ofes (G x y) (lam ([z] O x y z)) (spi U1 U2)) (ofes/lam (%the ({z} %pi (isvar z K) %-> (ofes (scons (G x y) z U1) (O x y z) U2)) (DofO x d y e)))) ([y] sub/lam ([z] %the (sub ([x] O x y z) M (Ox y z)) (DsubOx y z))) (%the (aasub R M Rx) DsubRx) (sub/lam DsubOyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} ofes-context (DofO x d y e z f) (%the (sordered (scons (G x y) z U1)) (sordered/cons (%the (sbounded (G x y) z) (Dbounded x d y e z f))))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-ofes (Dbounded x d y e z f) (DofM x d y e) _ (%the (ofes (scons (G x y) z U1) M S) (DofM' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-aofes (Dbounded x d y e z f) (DofR x d y e) _ (%the (aofes (scons (G x y) z U1) (R x) T) (DofR' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupX x d y e) _ (%the (slookup (scons (G x y) z U1) x S) (DlookupX' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupY x d y e) _ (%the (slookup (scons (G x y) z U1) y T) (DlookupY' x d y e z f))) + %<- ({z} {f isvar z K} ssub-z-permute-es ([x] [d] [y] [e] DlookupX' x d y e z f) ([x] [d] [y] [e] DlookupY' x d y e z f) ([x] [d] [y] [e] DofM' x d y e z f) ([x] [d] [y] [e] DofR' x d y e z f) ([x] [d] [y] [e] DofO x d y e z f) ([y] DsubOx y z) DsubRx (%the (sub ([x] O x (R x) z) M (Ox Rx z)) (DsubOyx z))) %. +%term pair + %pi (ssub-z-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] %the (ofes (G x y) (pair (O1 x y) (O2 x y)) (ssigma U1 U2)) (ofes/pair (%the (ofes (G x y) (O2 x y) U2) (DofO2 x d y e)) (%the (ofes (G x y) (O1 x y) U1) (DofO1 x d y e)))) ([y] sub/pair (%the (sub ([x] O2 x y) M (O2x y)) (DsubO2x y)) (%the (sub ([x] O1 x y) M (O1x y)) (DsubO1x y))) (%the (aasub R M Rx) DsubRx) (sub/pair DsubO2yx DsubO1yx)) + %<- (ssub-z-permute-es DlookupX DlookupY DofM DofR DofO1 DsubO1x DsubRx (%the (sub ([x] O1 x (R x)) M (O1x Rx)) DsubO1yx)) + %<- (ssub-z-permute-es DlookupX DlookupY DofM DofR DofO2 DsubO2x DsubRx (%the (sub ([x] O2 x (R x)) M (O2x Rx)) DsubO2yx)) %. +%scope stsub-x-permute-es %term aa-aa + %pi (ssub-x-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] %the (ofes (G x y) (at (Q x y)) st) (ofes/at (%the (aofes (G x y) (Q x y) st) (DofQ x d y e)))) ([y] sub/aa (%the (aasub ([x] Q x y) M (Qx y)) (DsubQx y))) (%the (aosub R M LRx) DsubRx) (%the (sub ([y] at (Qx y)) LRx (at Qxy)) (sub/aa (%the (aasub ([y] Qx y) LRx Qxy) DsubQxy))) (sub/aa DsubQyx)) + %<- (ssub-xz-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx DsubQxy (%the (aasub ([x] Q x (R x)) M Qxy) DsubQyx)) %. +%term aa-ao + %pi (ssub-x-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] %the (ofes (G x y) (at (Q x y)) st) (ofes/at (%the (aofes (G x y) (Q x y) st) (DofQ x d y e)))) ([y] sub/aa (%the (aasub ([x] Q x y) M (Qx y)) (DsubQx y))) (%the (aosub R M LRx) DsubRx) (%the (sub ([y] at (Qx y)) LRx LQxy) (sub/ao (%the (aosub ([y] Qx y) LRx LQxy) DsubQxy))) (sub/ao DsubQyx)) + %<- (ssub-xy-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx DsubQxy (%the (aosub ([x] Q x (R x)) M LQxy) DsubQyx)) %. +%scope stsub-x-permute-es %term ao + %pi (ssub-x-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] %the (ofes (G x y) (at (Q x y)) st) (ofes/at (%the (aofes (G x y) (Q x y) st) (DofQ x d y e)))) ([y] sub/ao (%the (aosub ([x] Q x y) M (LQx y)) (DsubQx y))) (%the (aosub R M LRx) DsubRx) (%the (sub ([y] LQx y) LRx LQxy) DsubQxy) (sub/ao DsubQyx)) + %<- (ssub-xx-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx DsubQxy (%the (aosub ([x] Q x (R x)) M LQxy) DsubQyx)) %. +%term lam + %pi (ssub-x-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] %the (ofes (G x y) (lam ([z] O x y z)) (spi U1 U2)) (ofes/lam (%the ({z} %pi (isvar z K) %-> (ofes (scons (G x y) z U1) (O x y z) U2)) (DofO x d y e)))) ([y] sub/lam ([z] %the (sub ([x] O x y z) M (Ox y z)) (DsubOx y z))) (%the (aosub R M LRx) DsubRx) (sub/lam ([z] %the (sub ([y] Ox y z) LRx (Oxy z)) (DsubOxy z))) (sub/lam DsubOyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} ofes-context (DofO x d y e z f) (%the (sordered (scons (G x y) z U1)) (sordered/cons (%the (sbounded (G x y) z) (Dbounded x d y e z f))))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-ofes (Dbounded x d y e z f) (DofM x d y e) _ (%the (ofes (scons (G x y) z U1) M S) (DofM' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-aofes (Dbounded x d y e z f) (DofR x d y e) _ (%the (aofes (scons (G x y) z U1) (R x) T) (DofR' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupX x d y e) _ (%the (slookup (scons (G x y) z U1) x S) (DlookupX' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupY x d y e) _ (%the (slookup (scons (G x y) z U1) y T) (DlookupY' x d y e z f))) + %<- ({z} {f isvar z K} ssub-x-permute-es ([x] [d] [y] [e] DlookupX' x d y e z f) ([x] [d] [y] [e] DlookupY' x d y e z f) ([x] [d] [y] [e] DofM' x d y e z f) ([x] [d] [y] [e] DofR' x d y e z f) ([x] [d] [y] [e] DofO x d y e z f) ([y] DsubOx y z) DsubRx (DsubOxy z) (%the (sub ([x] O x (R x) z) M (Oxy z)) (DsubOyx z))) %. +%scope stsub-x-permute-es %term pair + %pi (ssub-x-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] %the (ofes (G x y) (pair (O1 x y) (O2 x y)) (ssigma U1 U2)) (ofes/pair (%the (ofes (G x y) (O2 x y) U2) (DofO2 x d y e)) (%the (ofes (G x y) (O1 x y) U1) (DofO1 x d y e)))) ([y] sub/pair (%the (sub ([x] O2 x y) M (O2x y)) (DsubO2x y)) (%the (sub ([x] O1 x y) M (O1x y)) (DsubO1x y))) (%the (aosub R M LRx) DsubRx) (sub/pair (%the (sub ([y] O2x y) LRx O2xy) DsubO2xy) (%the (sub ([y] O1x y) LRx O1xy) DsubO1xy)) (sub/pair DsubO2yx DsubO1yx)) + %<- (ssub-x-permute-es DlookupX DlookupY DofM DofR DofO1 DsubO1x DsubRx DsubO1xy (%the (sub ([x] O1 x (R x)) M O1xy) DsubO1yx)) + %<- (ssub-x-permute-es DlookupX DlookupY DofM DofR DofO2 DsubO2x DsubRx DsubO2xy (%the (sub ([x] O2 x (R x)) M O2xy) DsubO2yx)) %. +%term closed + %pi (ssub-zz-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (Q y) U))) DofQ) ([_] aasub/closed) (%the (aasub ([x] R x) M Rx) DsubRx) DsubQyx) + %<- (aasub-into-atom Q DsubRx (%the (aasub ([x] Q (R x)) M (Q Rx)) DsubQyx)) %. +%scope stsub-x-permute-es %term forall + %pi (ssub-zz-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (forall (A x y)) U1))) DofQuant) ([y] aasub/forall (%the (tsub ([x] A x y) M (Ax y)) (DsubAx y))) (%the (aasub ([x] R x) M Rx) DsubRx) (aasub/forall DsubAyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-forall (DofQuant x d y e) (%the (wfes (G x y) (A x y)) (DofA x d y e))) + %<- (stsub-z-permute-es DlookupX DlookupY DofM DofR DofA DsubAx DsubRx (%the (tsub ([x] A x (R x)) M (Ax Rx)) DsubAyx)) %. +%term app + %pi (ssub-zz-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (app (Q x y) (N x y)) U2))) DofApp) ([y] aasub/app (%the (sub ([x] N x y) M (Nx y)) (DsubNx y)) (%the (aasub ([x] Q x y) M (Qx y)) (DsubQx y))) (%the (aasub ([x] R x) M Rx) DsubRx) (aasub/app DsubNyx DsubQyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-app (DofApp x d y e) (%the (aofes (G x y) (Q x y) (spi U1 U2)) (DofQ x d y e)) (%the (ofes (G x y) (N x y) U1) (DofN x d y e))) + %<- (ssub-zz-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx (%the (aasub ([x] Q x (R x)) M (Qx Rx)) DsubQyx)) + %<- (ssub-z-permute-es DlookupX DlookupY DofM DofR DofN DsubNx DsubRx (%the (sub ([x] N x (R x)) M (Nx Rx)) DsubNyx)) %. +%scope stsub-x-permute-es %term pi1 + %pi (ssub-zz-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (pi1 (Q x y)) U1))) DofPi) ([y] aasub/pi1 (%the (aasub ([x] Q x y) M (Qx y)) (DsubQx y))) (%the (aasub ([x] R x) M Rx) DsubRx) (aasub/pi1 DsubQyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-pi1 (DofPi x d y e) (%the (aofes (G x y) (Q x y) (ssigma U1 U2)) (DofQ x d y e))) + %<- (ssub-zz-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx (%the (aasub ([x] Q x (R x)) M (Qx Rx)) DsubQyx)) %. +%term pi2 + %pi (ssub-zz-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (pi2 (Q x y)) U2))) DofPi) ([y] aasub/pi2 (%the (aasub ([x] Q x y) M (Qx y)) (DsubQx y))) (%the (aasub ([x] R x) M Rx) DsubRx) (aasub/pi2 DsubQyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-pi2 (DofPi x d y e) (%the (aofes (G x y) (Q x y) (ssigma U1 U2)) (DofQ x d y e))) + %<- (ssub-zz-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx (%the (aasub ([x] Q x (R x)) M (Qx Rx)) DsubQyx)) %. +%scope stsub-x-permute-es %term var ssub-zx-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) _ ([_] aosub/var) (%the (aasub ([x] R x) M Rx) DsubRx) aosub/var %. +%term forall + %pi (ssub-xz-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (forall (A x y)) U1))) DofQuant) ([y] aasub/forall (%the (tsub ([x] A x y) M (Ax y)) (DsubAx y))) (%the (aosub ([x] R x) M LRx) DsubRx) (aasub/forall (%the (tsub ([y] Ax y) LRx Axy) DsubAxy)) (aasub/forall DsubAyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-forall (DofQuant x d y e) (%the (wfes (G x y) (A x y)) (DofA x d y e))) + %<- (stsub-x-permute-es DlookupX DlookupY DofM DofR DofA DsubAx DsubRx DsubAxy (%the (tsub ([x] A x (R x)) M Axy) DsubAyx)) %. +%scope stsub-x-permute-es %term allcl + %pi (ssub-xz-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (forall (A x y)) U1))) DofQuant) ([y] aasub/forall (%the (tsub ([x] A x y) M Ax) (DsubAx y))) (%the (aosub ([x] R x) M LRx) DsubRx) aasub/closed (aasub/forall DsubAyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-forall (DofQuant x d y e) (%the (wfes (G x y) (A x y)) (DofA x d y e))) + %<- (tsub-absent Ax LRx (%the (tsub ([_] Ax) LRx Ax) DsubAxy)) + %<- (stsub-x-permute-es DlookupX DlookupY DofM DofR DofA DsubAx DsubRx DsubAxy (%the (tsub ([x] A x (R x)) M Ax) DsubAyx)) %. +%term app + %pi (ssub-zx-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (app (Q x y) (N x y)) U2))) DofApp) ([y] aosub/app (%the (sub ([z] P y z) (Nx y) (Pz y)) (DsubPz y)) (%the (sub ([x] N x y) M (Nx y)) (DsubNx y)) (%the (aosub ([x] Q x y) M (lam ([z] P y z))) (DsubQx y))) (%the (aasub ([x] R x) M Rx) DsubRx) (aosub/app (DsubPz Rx) DsubNyx DsubQyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-app (DofApp x d y e) (%the (aofes (G x y) (Q x y) (spi U1 U2)) (DofQ x d y e)) (%the (ofes (G x y) (N x y) U1) (DofN x d y e))) + %<- (ssub-zx-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx (%the (aosub ([x] Q x (R x)) M (lam ([z] P Rx z))) DsubQyx)) + %<- (ssub-z-permute-es DlookupX DlookupY DofM DofR DofN DsubNx DsubRx (%the (sub ([x] N x (R x)) M (Nx Rx)) DsubNyx)) %. +%scope stsub-x-permute-es %term pi1 + %pi (ssub-zx-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (pi1 (Q x y)) U1))) DofPi) ([y] aosub/pi1 (%the (aosub ([x] Q x y) M (pair (P1 y) (P2 y))) (DsubQx y))) (%the (aasub ([x] R x) M Rx) DsubRx) (aosub/pi1 DsubQyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-pi1 (DofPi x d y e) (%the (aofes (G x y) (Q x y) (ssigma U1 U2)) (DofQ x d y e))) + %<- (ssub-zx-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx (%the (aosub ([x] Q x (R x)) M (pair (P1 Rx) (P2 Rx))) DsubQyx)) %. +%term pi2 + %pi (ssub-zx-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (pi2 (Q x y)) U2))) DofPi) ([y] aosub/pi2 (%the (aosub ([x] Q x y) M (pair (P1 y) (P2 y))) (DsubQx y))) (%the (aasub ([x] R x) M Rx) DsubRx) (aosub/pi2 DsubQyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-pi2 (DofPi x d y e) (%the (aofes (G x y) (Q x y) (ssigma U1 U2)) (DofQ x d y e))) + %<- (ssub-zx-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx (%the (aosub ([x] Q x (R x)) M (pair (P1 Rx) (P2 Rx))) DsubQyx)) %. +%scope stsub-x-permute-es %term closed + %pi (ssub-xz-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (Q y) U))) DofQ) ([_] aasub/closed) (%the (aosub ([x] R x) M LRx) DsubRx) (%the (aasub ([y] Q y) LRx Qy) DsubQxy) DsubQyx) + %<- (aosub-into-aasub DsubRx DsubQxy (%the (aasub ([x] Q (R x)) M Qy) DsubQyx)) %. +%term app + %pi (ssub-xz-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (app (Q x y) (N x y)) U2))) DofApp) ([y] aasub/app (%the (sub ([x] N x y) M (Nx y)) (DsubNx y)) (%the (aasub ([x] Q x y) M (Qx y)) (DsubQx y))) (%the (aosub ([x] R x) M LRx) DsubRx) (aasub/app (%the (sub ([y] Nx y) LRx Nxy) DsubNxy) (%the (aasub ([y] Qx y) LRx Qxy) DsubQxy)) (aasub/app DsubNyx DsubQyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-app (DofApp x d y e) (%the (aofes (G x y) (Q x y) (spi U1 U2)) (DofQ x d y e)) (%the (ofes (G x y) (N x y) U1) (DofN x d y e))) + %<- (ssub-xz-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx DsubQxy (%the (aasub ([x] Q x (R x)) M Qxy) DsubQyx)) + %<- (ssub-x-permute-es DlookupX DlookupY DofM DofR DofN DsubNx DsubRx DsubNxy (%the (sub ([x] N x (R x)) M Nxy) DsubNyx)) %. +%scope stsub-x-permute-es %term app-cl + %pi (ssub-xz-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (app (Q x y) (N x y)) U2))) DofApp) ([y] aasub/app (%the (sub ([x] N x y) M Nx) (DsubNx y)) (%the (aasub ([x] Q x y) M Qx) (DsubQx y))) (%the (aosub ([x] R x) M LRx) DsubRx) aasub/closed (aasub/app DsubNyx DsubQyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-app (DofApp x d y e) (%the (aofes (G x y) (Q x y) (spi U1 U2)) (DofQ x d y e)) (%the (ofes (G x y) (N x y) U1) (DofN x d y e))) + %<- (ssub-xz-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx aasub/closed (%the (aasub ([x] Q x (R x)) M Qx) DsubQyx)) + %<- (sub-absent Nx LRx (%the (sub ([y] Nx) LRx Nx) DsubNxy)) + %<- (ssub-x-permute-es DlookupX DlookupY DofM DofR DofN DsubNx DsubRx DsubNxy (%the (sub ([x] N x (R x)) M Nx) DsubNyx)) %. +%term pi1 + %pi (ssub-xz-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (pi1 (Q x y)) U1))) DofPi) ([y] aasub/pi1 (%the (aasub ([x] Q x y) M (Qx y)) (DsubQx y))) (%the (aosub ([x] R x) M LRx) DsubRx) (aasub/pi1 (%the (aasub ([y] Qx y) LRx Qxy) DsubQxy)) (aasub/pi1 DsubQyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-pi1 (DofPi x d y e) (%the (aofes (G x y) (Q x y) (ssigma U1 U2)) (DofQ x d y e))) + %<- (ssub-xz-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx DsubQxy (%the (aasub ([x] Q x (R x)) M Qxy) DsubQyx)) %. +%scope stsub-x-permute-es %term pi1-cl + %pi (ssub-xz-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (pi1 (Q x y)) U1))) DofPi) ([y] aasub/pi1 (%the (aasub ([x] Q x y) M Qx) (DsubQx y))) (%the (aosub ([x] R x) M LRx) DsubRx) aasub/closed (aasub/pi1 DsubQyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-pi1 (DofPi x d y e) (%the (aofes (G x y) (Q x y) (ssigma U1 U2)) (DofQ x d y e))) + %<- (ssub-xz-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx aasub/closed (%the (aasub ([x] Q x (R x)) M Qx) DsubQyx)) %. +%term pi2 + %pi (ssub-xz-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (pi2 (Q x y)) U2))) DofPi) ([y] aasub/pi2 (%the (aasub ([x] Q x y) M (Qx y)) (DsubQx y))) (%the (aosub ([x] R x) M LRx) DsubRx) (aasub/pi2 (%the (aasub ([y] Qx y) LRx Qxy) DsubQxy)) (aasub/pi2 DsubQyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-pi2 (DofPi x d y e) (%the (aofes (G x y) (Q x y) (ssigma U1 U2)) (DofQ x d y e))) + %<- (ssub-xz-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx DsubQxy (%the (aasub ([x] Q x (R x)) M Qxy) DsubQyx)) %. +%scope stsub-x-permute-es %term pi2-cl + %pi (ssub-xz-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (pi2 (Q x y)) U2))) DofPi) ([y] aasub/pi2 (%the (aasub ([x] Q x y) M Qx) (DsubQx y))) (%the (aosub ([x] R x) M LRx) DsubRx) aasub/closed (aasub/pi2 DsubQyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-pi2 (DofPi x d y e) (%the (aofes (G x y) (Q x y) (ssigma U1 U2)) (DofQ x d y e))) + %<- (ssub-xz-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx aasub/closed (%the (aasub ([x] Q x (R x)) M Qx) DsubQyx)) %. +%term closed + %pi (ssub-xy-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (Q y) U))) DofQ) ([_] aasub/closed) (%the (aosub ([x] R x) M LRx) DsubRx) (%the (aosub ([y] Q y) LRx LQy) DsubQxy) DsubQyx) + %<- (aosub-into-aosub DsubRx DsubQxy (%the (aosub ([x] Q (R x)) M LQy) DsubQyx)) %. +%scope stsub-x-permute-es %term app + %pi (ssub-xy-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (app (Q x y) (N x y)) U2))) DofApp) ([y] aasub/app (%the (sub ([x] N x y) M (Nx y)) (DsubNx y)) (%the (aasub ([x] Q x y) M (Qx y)) (DsubQx y))) (%the (aosub ([x] R x) M LRx) DsubRx) (aosub/app (%the (sub ([z] P z) Nxy Pz) DsubPz) (%the (sub ([y] Nx y) LRx Nxy) DsubNxy) (%the (aosub ([y] Qx y) LRx (lam ([z] P z))) DsubQxy)) (aosub/app DsubPz DsubNyx DsubQyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-app (DofApp x d y e) (%the (aofes (G x y) (Q x y) (spi U1 U2)) (DofQ x d y e)) (%the (ofes (G x y) (N x y) U1) (DofN x d y e))) + %<- (ssub-xy-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx DsubQxy (%the (aosub ([x] Q x (R x)) M (lam ([z] P z))) DsubQyx)) + %<- (ssub-x-permute-es DlookupX DlookupY DofM DofR DofN DsubNx DsubRx DsubNxy (%the (sub ([x] N x (R x)) M Nxy) DsubNyx)) %. +%term pi1 + %pi (ssub-xy-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (pi1 (Q x y)) U1))) DofPi) ([y] aasub/pi1 (%the (aasub ([x] Q x y) M (Qx y)) (DsubQx y))) (%the (aosub ([x] R x) M LRx) DsubRx) (aosub/pi1 (%the (aosub ([y] Qx y) LRx (pair P1 P2)) DsubQxy)) (aosub/pi1 DsubQyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-pi1 (DofPi x d y e) (%the (aofes (G x y) (Q x y) (ssigma U1 U2)) (DofQ x d y e))) + %<- (ssub-xy-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx DsubQxy (%the (aosub ([x] Q x (R x)) M (pair P1 P2)) DsubQyx)) %. +%scope stsub-x-permute-es %term pi2 + %pi (ssub-xy-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (pi2 (Q x y)) U2))) DofPi) ([y] aasub/pi2 (%the (aasub ([x] Q x y) M (Qx y)) (DsubQx y))) (%the (aosub ([x] R x) M LRx) DsubRx) (aosub/pi2 (%the (aosub ([y] Qx y) LRx (pair P1 P2)) DsubQxy)) (aosub/pi2 DsubQyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-pi2 (DofPi x d y e) (%the (aofes (G x y) (Q x y) (ssigma U1 U2)) (DofQ x d y e))) + %<- (ssub-xy-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx DsubQxy (%the (aosub ([x] Q x (R x)) M (pair P1 P2)) DsubQyx)) %. +%term var + %pi (ssub-xx-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) _ ([_] aosub/var) (%the (aosub ([x] R x) M LRx) DsubRx) (%the (sub ([y] M) LRx LQxy) Dsub) Daosub) + %<- (sub-absent M LRx (%the (sub ([y] M) LRx M) Dsub')) + %<- (sub-fun Dsub' Dsub (%the (term-eq M LQxy) Deq)) + %<- (aosub-resp ([_] atom-eq/i) term-eq/i Deq aosub/var (%the (aosub ([y] y) M LQxy) Daosub)) %. +%scope stsub-x-permute-es %term app + %pi (ssub-xx-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (app (Q x y) (N x y)) U2))) DofApp) ([y] aosub/app (%the (sub ([z] P y z) (Nx y) (Pz y)) (DsubPz y)) (%the (sub ([x] N x y) M (Nx y)) (DsubNx y)) (%the (aosub ([x] Q x y) M (lam ([z] P y z))) (DsubQx y))) (%the (aosub ([x] R x) M LRx) DsubRx) (%the (sub ([y] Pz y) LRx Pzy) DsubPzy) (aosub/app DsubPyz DsubNyx DsubQyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-app (DofApp x d y e) (%the (aofes (G x y) (Q x y) (spi U1 U2)) (DofQ x d y e)) (%the (ofes (G x y) (N x y) U1) (DofN x d y e))) + %<- ({y} {e isvar y J} aosubst-es ([x] [d] DlookupX x d y e) ([x] [d] DofM x d y e) ([x] [d] DofR x d y e) DsubRx ([x] [d] %the (ofes (G x y) LRx T) (DofLRx x d y e))) + %<- ({y} {e isvar y J} subst-es ([x] [d] DlookupX x d y e) ([x] [d] DofM x d y e) ([x] [d] DofN x d y e) (DsubNx y) ([x] [d] %the (ofes (G x y) (Nx y) U1) (DofNx x d y e))) + %<- ({y} {e isvar y J} aosubst-es ([x] [d] DlookupX x d y e) ([x] [d] DofM x d y e) ([x] [d] DofQ x d y e) (DsubQx y) ([x] [d isvar x I] ofes/lam ([z] [f isvar z K] %the (ofes (scons (G x y) z U1) (P y z) U2) (DofP x d y e z f)))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} ofes-context (DofP x d y e z f) (%the (sordered (scons (G x y) z U1)) (sordered/cons (%the (sbounded (G x y) z) (Dbounded x d y e z f))))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupY x d y e) _ (%the (slookup (scons (G x y) z U1) y T) (DlookupY' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-ofes (Dbounded x d y e z f) (DofLRx x d y e) _ (%the (ofes (scons (G x y) z U1) LRx T) (DofLRx' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-ofes (Dbounded x d y e z f) (DofNx x d y e) _ (%the (ofes (scons (G x y) z U1) (Nx y) U1) (DofNx' x d y e z f))) + %<- ({x} {d isvar x I} sub-permute-es ([y] [e] [z] [f] DlookupY' x d y e z f) ([y] [e] [z] [f] slookup/hit (Dbounded x d y e z f)) ([y] [e] [z] [f] DofLRx' x d y e z f) ([y] [e] [z] [f] DofNx' x d y e z f) ([y] [e] [z] [f] DofP x d y e z f) DsubPz DsubPzy (%the ({z} sub ([y] P y z) LRx (Py_x x z)) (DsubPy_x x)) (%the (sub ([y] Nx y) LRx (Nxy_x x)) (DsubNxy_x x)) (%the (sub ([z] Py_x x z) (Nxy_x x) Pzy) (DsubPyz_x x))) + %<- ({z} sub-closed ([x] DsubPy_x x z) ([x] %the (term-eq (Py_x x z) (Py z)) (DeqPy_x x z))) + %<- (sub-closed ([x] DsubNxy_x x) ([x] %the (term-eq (Nxy_x x) Nxy) (DeqNxy_x x))) + %<- ({z} sub-resp-strengthen ([_] [_] term-eq/i) ([_] term-eq/i) ([x] DeqPy_x x z) ([x] DsubPy_x x z) (%the (sub ([y] P y z) LRx (Py z)) (DsubPy z))) + %<- (sub-resp-strengthen ([_] [_] term-eq/i) ([_] term-eq/i) ([x] DeqNxy_x x) ([x] DsubNxy_x x) (%the (sub ([y] Nx y) LRx Nxy) DsubNxy)) + %<- (sub-resp-strengthen ([x] [z] DeqPy_x x z) ([x] DeqNxy_x x) ([_] term-eq/i) ([x] DsubPyz_x x) (%the (sub ([z] Py z) Nxy Pzy) DsubPyz)) + %<- (ssub-xx-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx (sub/lam DsubPy) (%the (aosub ([x] Q x (R x)) M (lam ([z] Py z))) DsubQyx)) + %<- (ssub-x-permute-es DlookupX DlookupY DofM DofR DofN DsubNx DsubRx (%the (sub ([y] Nx y) LRx Nxy) DsubNxy) (%the (sub ([x] N x (R x)) M Nxy) DsubNyx)) %. +%term pi1 + %pi (ssub-xx-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (pi1 (Q x y)) U1))) DofPi) ([y] aosub/pi1 (%the (aosub ([x] Q x y) M (pair (P1 y) (P2 y))) (DsubQx y))) (%the (aosub ([x] R x) M LRx) DsubRx) (%the (sub ([y] P1 y) LRx P1y) DsubP1y) (aosub/pi1 DsubQyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-pi1 (DofPi x d y e) (%the (aofes (G x y) (Q x y) (ssigma U1 U2)) (DofQ x d y e))) + %<- ({y} {e isvar y J} aosubst-es ([x] [d] DlookupX x d y e) ([x] [d] DofM x d y e) ([x] [d] DofR x d y e) DsubRx ([x] [d] %the (ofes (G x y) LRx T) (DofLRx x d y e))) + %<- ({y} {e isvar y J} aosubst-es ([x] [d] DlookupX x d y e) ([x] [d] DofM x d y e) ([x] [d] DofQ x d y e) (DsubQx y) ([x] [d] ofes/pair (DofP2 x d y e) (DofP1 x d y e))) + %<- ({x} {d isvar x I} can-sub-es ([y] [e] DlookupY x d y e) ([y] [e] DofP2 x d y e) ([y] [e] DofLRx x d y e) (%the (sub ([y] P2 y) LRx (P2y_x x)) (DsubP2y_x x))) + %<- (sub-closed' DsubP2y_x (%the (sub ([y] P2 y) LRx P2y) DsubP2y)) + %<- (ssub-xx-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx (sub/pair DsubP2y DsubP1y) (%the (aosub ([x] Q x (R x)) M (pair P1y P2y)) DsubQyx)) %. +%scope stsub-x-permute-es %term pi2 + %pi (ssub-xx-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (pi2 (Q x y)) U2))) DofPi) ([y] aosub/pi2 (%the (aosub ([x] Q x y) M (pair (P1 y) (P2 y))) (DsubQx y))) (%the (aosub ([x] R x) M LRx) DsubRx) (%the (sub ([y] P2 y) LRx P2y) DsubP2y) (aosub/pi2 DsubQyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofes-invert-pi2 (DofPi x d y e) (%the (aofes (G x y) (Q x y) (ssigma U1 U2)) (DofQ x d y e))) + %<- ({y} {e isvar y J} aosubst-es ([x] [d] DlookupX x d y e) ([x] [d] DofM x d y e) ([x] [d] DofR x d y e) DsubRx ([x] [d] %the (ofes (G x y) LRx T) (DofLRx x d y e))) + %<- ({y} {e isvar y J} aosubst-es ([x] [d] DlookupX x d y e) ([x] [d] DofM x d y e) ([x] [d] DofQ x d y e) (DsubQx y) ([x] [d] ofes/pair (DofP2 x d y e) (DofP1 x d y e))) + %<- ({x} {d isvar x I} can-sub-es ([y] [e] DlookupY x d y e) ([y] [e] DofP1 x d y e) ([y] [e] DofLRx x d y e) (%the (sub ([y] P1 y) LRx (P1y_x x)) (DsubP1y_x x))) + %<- (sub-closed' DsubP1y_x (%the (sub ([y] P1 y) LRx P1y) DsubP1y)) + %<- (ssub-xx-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx (sub/pair DsubP2y DsubP1y) (%the (aosub ([x] Q x (R x)) M (pair P1y P2y)) DsubQyx)) %. +%term t stsub-z-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] wfes/t _) ([_] tsub/t) _ tsub/t %. +%scope stsub-x-permute-es %term pi + %pi (stsub-z-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] wfes/pi (%the ({z} %pi (isvar z K) %-> (wfes (scons (G x y) z U) (B x y z))) (DwfB x d y e)) (%the (simp (A x y) U) (Dsimp x y)) (%the (wfes (G x y) (A x y)) (DwfA x d y e))) ([y] tsub/pi (%the ({z} tsub ([x] B x y z) M (Bx y z)) (DsubBx y)) (%the (tsub ([x] A x y) M (Ax y)) (DsubAx y))) (%the (aasub ([x] R x) M Rx) DsubRx) (tsub/pi DsubByx DsubAyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} wfes-context (DwfB x d y e z f) (%the (sordered (scons (G x y) z U)) (sordered/cons (%the (sbounded (G x y) z) (Dbounded x d y e z f))))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-ofes (Dbounded x d y e z f) (DofM x d y e) _ (%the (ofes (scons (G x y) z U) M S) (DofM' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-aofes (Dbounded x d y e z f) (DofR x d y e) _ (%the (aofes (scons (G x y) z U) (R x) T) (DofR' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupX x d y e) _ (%the (slookup (scons (G x y) z U) x S) (DlookupX' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupY x d y e) _ (%the (slookup (scons (G x y) z U) y T) (DlookupY' x d y e z f))) + %<- (stsub-z-permute-es DlookupX DlookupY DofM DofR DwfA DsubAx DsubRx (%the (tsub ([x] A x (R x)) M (Ax Rx)) DsubAyx)) + %<- ({z} {f} stsub-z-permute-es ([x] [d] [y] [e] DlookupX' x d y e z f) ([x] [d] [y] [e] DlookupY' x d y e z f) ([x] [d] [y] [e] DofM' x d y e z f) ([x] [d] [y] [e] DofR' x d y e z f) ([x] [d] [y] [e] DwfB x d y e z f) ([y] DsubBx y z) DsubRx (%the (tsub ([x] B x (R x) z) M (Bx Rx z)) (DsubByx z))) %. +%term sigma + %pi (stsub-z-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] wfes/sigma (%the ({z} %pi (isvar z K) %-> (wfes (scons (G x y) z U) (B x y z))) (DwfB x d y e)) (%the (simp (A x y) U) (Dsimp x y)) (%the (wfes (G x y) (A x y)) (DwfA x d y e))) ([y] tsub/sigma (%the ({z} tsub ([x] B x y z) M (Bx y z)) (DsubBx y)) (%the (tsub ([x] A x y) M (Ax y)) (DsubAx y))) (%the (aasub ([x] R x) M Rx) DsubRx) (tsub/sigma DsubByx DsubAyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} wfes-context (DwfB x d y e z f) (%the (sordered (scons (G x y) z U)) (sordered/cons (%the (sbounded (G x y) z) (Dbounded x d y e z f))))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-ofes (Dbounded x d y e z f) (DofM x d y e) _ (%the (ofes (scons (G x y) z U) M S) (DofM' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-aofes (Dbounded x d y e z f) (DofR x d y e) _ (%the (aofes (scons (G x y) z U) (R x) T) (DofR' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupX x d y e) _ (%the (slookup (scons (G x y) z U) x S) (DlookupX' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupY x d y e) _ (%the (slookup (scons (G x y) z U) y T) (DlookupY' x d y e z f))) + %<- (stsub-z-permute-es DlookupX DlookupY DofM DofR DwfA DsubAx DsubRx (%the (tsub ([x] A x (R x)) M (Ax Rx)) DsubAyx)) + %<- ({z} {f} stsub-z-permute-es ([x] [d] [y] [e] DlookupX' x d y e z f) ([x] [d] [y] [e] DlookupY' x d y e z f) ([x] [d] [y] [e] DofM' x d y e z f) ([x] [d] [y] [e] DofR' x d y e z f) ([x] [d] [y] [e] DwfB x d y e z f) ([y] DsubBx y z) DsubRx (%the (tsub ([x] B x (R x) z) M (Bx Rx z)) (DsubByx z))) %. +%scope stsub-x-permute-es %term singa + %pi (stsub-z-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] wfes/sing (%the (aofes (G x y) (Q x y) st) (DofQ x d y e))) ([y] tsub/singa (%the (aasub ([x] Q x y) M (Qx y)) (DsubQx y))) (%the (aasub ([x] R x) M Rx) DsubRx) (tsub/singa DsubQyx)) + %<- (ssub-zz-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx (%the (aasub ([x] Q x (R x)) M (Qx Rx)) DsubQyx)) %. +%term singo + %pi (stsub-z-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] wfes/sing (%the (aofes (G x y) (Q x y) st) (DofQ x d y e))) ([y] tsub/singo (%the (aosub ([x] Q x y) M (at (Qx y))) (DsubQx y))) (%the (aasub ([x] R x) M Rx) DsubRx) (tsub/singo DsubQyx)) + %<- (ssub-zx-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx (%the (aosub ([x] Q x (R x)) M (at (Qx Rx))) DsubQyx)) %. +%scope stsub-x-permute-es %term t stsub-x-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] wfes/t _) ([_] tsub/t) _ _ tsub/t %. +%term pi + %pi (stsub-x-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] wfes/pi (%the ({z} %pi (isvar z K) %-> (wfes (scons (G x y) z U) (B x y z))) (DwfB x d y e)) (%the (simp (A x y) U) (Dsimp x y)) (%the (wfes (G x y) (A x y)) (DwfA x d y e))) ([y] tsub/pi (%the ({z} tsub ([x] B x y z) M (Bx y z)) (DsubBx y)) (%the (tsub ([x] A x y) M (Ax y)) (DsubAx y))) (%the (aosub ([x] R x) M LRx) DsubRx) (tsub/pi (%the ({z} tsub ([y] Bx y z) LRx (Bxy z)) DsubBxy) (%the (tsub ([y] Ax y) LRx Axy) DsubAxy)) (tsub/pi DsubByx DsubAyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} wfes-context (DwfB x d y e z f) (%the (sordered (scons (G x y) z U)) (sordered/cons (%the (sbounded (G x y) z) (Dbounded x d y e z f))))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-ofes (Dbounded x d y e z f) (DofM x d y e) _ (%the (ofes (scons (G x y) z U) M S) (DofM' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-aofes (Dbounded x d y e z f) (DofR x d y e) _ (%the (aofes (scons (G x y) z U) (R x) T) (DofR' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupX x d y e) _ (%the (slookup (scons (G x y) z U) x S) (DlookupX' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupY x d y e) _ (%the (slookup (scons (G x y) z U) y T) (DlookupY' x d y e z f))) + %<- (stsub-x-permute-es DlookupX DlookupY DofM DofR DwfA DsubAx DsubRx DsubAxy (%the (tsub ([x] A x (R x)) M Axy) DsubAyx)) + %<- ({z} {f} stsub-x-permute-es ([x] [d] [y] [e] DlookupX' x d y e z f) ([x] [d] [y] [e] DlookupY' x d y e z f) ([x] [d] [y] [e] DofM' x d y e z f) ([x] [d] [y] [e] DofR' x d y e z f) ([x] [d] [y] [e] DwfB x d y e z f) ([y] DsubBx y z) DsubRx (DsubBxy z) (%the (tsub ([x] B x (R x) z) M (Bxy z)) (DsubByx z))) %. +%scope stsub-x-permute-es %term sigma + %pi (stsub-x-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] wfes/sigma (%the ({z} %pi (isvar z K) %-> (wfes (scons (G x y) z U) (B x y z))) (DwfB x d y e)) (%the (simp (A x y) U) (Dsimp x y)) (%the (wfes (G x y) (A x y)) (DwfA x d y e))) ([y] tsub/sigma (%the ({z} tsub ([x] B x y z) M (Bx y z)) (DsubBx y)) (%the (tsub ([x] A x y) M (Ax y)) (DsubAx y))) (%the (aosub ([x] R x) M LRx) DsubRx) (tsub/sigma (%the ({z} tsub ([y] Bx y z) LRx (Bxy z)) DsubBxy) (%the (tsub ([y] Ax y) LRx Axy) DsubAxy)) (tsub/sigma DsubByx DsubAyx)) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} wfes-context (DwfB x d y e z f) (%the (sordered (scons (G x y) z U)) (sordered/cons (%the (sbounded (G x y) z) (Dbounded x d y e z f))))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-ofes (Dbounded x d y e z f) (DofM x d y e) _ (%the (ofes (scons (G x y) z U) M S) (DofM' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-aofes (Dbounded x d y e z f) (DofR x d y e) _ (%the (aofes (scons (G x y) z U) (R x) T) (DofR' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupX x d y e) _ (%the (slookup (scons (G x y) z U) x S) (DlookupX' x d y e z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} weaken-slookup (Dbounded x d y e z f) (DlookupY x d y e) _ (%the (slookup (scons (G x y) z U) y T) (DlookupY' x d y e z f))) + %<- (stsub-x-permute-es DlookupX DlookupY DofM DofR DwfA DsubAx DsubRx DsubAxy (%the (tsub ([x] A x (R x)) M Axy) DsubAyx)) + %<- ({z} {f} stsub-x-permute-es ([x] [d] [y] [e] DlookupX' x d y e z f) ([x] [d] [y] [e] DlookupY' x d y e z f) ([x] [d] [y] [e] DofM' x d y e z f) ([x] [d] [y] [e] DofR' x d y e z f) ([x] [d] [y] [e] DwfB x d y e z f) ([y] DsubBx y z) DsubRx (DsubBxy z) (%the (tsub ([x] B x (R x) z) M (Bxy z)) (DsubByx z))) %. +%term singaa + %pi (stsub-x-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] wfes/sing (%the (aofes (G x y) (Q x y) st) (DofQ x d y e))) ([y] tsub/singa (%the (aasub ([x] Q x y) M (Qx y)) (DsubQx y))) (%the (aosub ([x] R x) M LRx) DsubRx) (tsub/singa (%the (aasub ([y] Qx y) LRx Qxy) DsubQxy)) (tsub/singa DsubQyx)) + %<- (ssub-xz-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx DsubQxy (%the (aasub ([x] Q x (R x)) M Qxy) DsubQyx)) %. +%scope stsub-x-permute-es %term singao + %pi (stsub-x-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] wfes/sing (%the (aofes (G x y) (Q x y) st) (DofQ x d y e))) ([y] tsub/singa (%the (aasub ([x] Q x y) M (Qx y)) (DsubQx y))) (%the (aosub ([x] R x) M LRx) DsubRx) (tsub/singo (%the (aosub ([y] Qx y) LRx (at Qxy)) DsubQxy)) (tsub/singo DsubQyx)) + %<- (ssub-xy-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx DsubQxy (%the (aosub ([x] Q x (R x)) M (at Qxy)) DsubQyx)) %. +%term singoa + %pi (stsub-x-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] wfes/sing (%the (aofes (G x y) (Q x y) st) (DofQ x d y e))) ([y] tsub/singo (%the (aosub ([x] Q x y) M (at (Qx y))) (DsubQx y))) (%the (aosub ([x] R x) M LRx) DsubRx) (tsub/singa (%the (aasub ([y] Qx y) LRx Qxy) DsubQxy)) (tsub/singo DsubQyx)) + %<- (ssub-xx-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx (sub/aa DsubQxy) (%the (aosub ([x] Q x (R x)) M (at Qxy)) DsubQyx)) %. +%scope stsub-x-permute-es %term singoo + %pi (stsub-x-permute-es (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) x S))) DlookupX) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (slookup (G x y) y T))) DlookupY) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (ofes (G x y) M S))) DofM) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (aofes (G x y) (R x) T))) DofR) ([x] [d] [y] [e] wfes/sing (%the (aofes (G x y) (Q x y) st) (DofQ x d y e))) ([y] tsub/singo (%the (aosub ([x] Q x y) M (at (Qx y))) (DsubQx y))) (%the (aosub ([x] R x) M LRx) DsubRx) (tsub/singo (%the (aosub ([y] Qx y) LRx (at Qxy)) DsubQxy)) (tsub/singo DsubQyx)) + %<- (ssub-xx-permute-es DlookupX DlookupY DofM DofR DofQ DsubQx DsubRx (sub/ao DsubQxy) (%the (aosub ([x] Q x (R x)) M (at Qxy)) DsubQyx)) %. +%worlds (var ovar bind) (ssub-z-permute-es _ _ _ _ _ _ _ _) (ssub-x-permute-es _ _ _ _ _ _ _ _ _) (ssub-zz-permute-es _ _ _ _ _ _ _ _) (ssub-zx-permute-es _ _ _ _ _ _ _ _) (ssub-xz-permute-es _ _ _ _ _ _ _ _ _) (ssub-xy-permute-es _ _ _ _ _ _ _ _ _) (ssub-xx-permute-es _ _ _ _ _ _ _ _ _) (stsub-x-permute-es _ _ _ _ _ _ _ _ _) (stsub-z-permute-es _ _ _ _ _ _ _ _) %. +%total (D1 D2 D3 D4 D5 D6 D7 D8 D9) (ssub-z-permute-es _ _ _ _ _ D1 _ _) (ssub-x-permute-es _ _ _ _ _ D2 _ _ _) (ssub-zz-permute-es _ _ _ _ _ D3 _ _) (ssub-zx-permute-es _ _ _ _ _ D4 _ _) (ssub-xz-permute-es _ _ _ _ _ D5 _ _ _) (ssub-xy-permute-es _ _ _ _ _ D6 _ _ _) (ssub-xx-permute-es _ _ _ _ _ D7 _ _ _) (stsub-x-permute-es _ _ _ _ _ D8 _ _ _) (stsub-z-permute-es _ _ _ _ _ D9 _ _) %. +%%%%% Simple (ie, atom for atom) Substitution Permutations (Explicit Context) %%%%% +%sort ssub-aa-permute-e {x} %. +%mode ssub-aa-permute-e %in %in %in %in %in %in %in %out %. +%term _ + %pi (ssub-aa-permute-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (ofe G1 M A) DofM) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (B x))) DofR) (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y J) %-> (ofe (cons (G x) y (B x)) (N x y) (C x y)))) DofN) (%the ({y} sub ([x] N x y) M (Nx y)) DsubNx) (%the (aasub R M Rx) DsubRx) (%the (sub ([x] N x (R x)) M Nyx) DsubNyx) Deq) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-context (DofN x d y e) (ordered/cons (%the (bounded (G x) y) (Dbounded x d y e)))) + %<- ({x} {d isvar x I} {y} {e isvar y J} append-bounded (Dappend x) (Dbounded x d y e) (bounded/cons (%the (bounded G1 x) (Dbounded' x d y e)) (%the (precedes x y) (DprecedesXY x d y e)))) + %<- (can-simp A (%the (simp A S) DsimpA)) + %<- ({x} can-simp (B x) (%the (simp (B x) T) (DsimpB x))) + %<- ({x} {y} can-simp (C x y) (%the (simp (C x y) U) (DsimpC x y))) + %<- ({y} {e isvar y J} context-append-simp-lookup Dappend ([x] [d] Dbounded x d y e) DsimpA ([x] %the (simpctx (G x) (G' x y)) (Dsimpctx x y)) ([x] [d] %the (sbounded (G' x y) y) (Dsbounded x d y e)) ([x] [d] %the (slookup (G' x y) x S) (Dlookup x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded' x d y e) DofM _ (%the (ofe (cons G1 x A) M A) (DofM' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe'' (Dappend x) (Dbounded x d y e) (DofM' x d y e) (%the (ofe (G x) M A) (DofM'' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded x d y e) (DofM'' x d y e) _ (%the (ofe (cons (G x) y (B x)) M A) (DofM''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) DsimpA (DofM''' x d y e) (%the (ofes (scons (G' x y) y T) M S) (DofM'''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-aofe (Dbounded x d y e) (DofR x d) _ (%the (aofe (cons (G x) y (B x)) (R x) (B x)) (DofR' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) (DsimpB x) (DofR' x d y e) (%the (aofes (scons (G' x y) y T) (R x) T) (DofR'' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) (DsimpC x y) (DofN x d y e) (%the (ofes (scons (G' x y) y T) (N x y) U) (DofN' x d y e))) + %<- (ssub-z-permute-es ([x] [d isvar x I] [y] [e isvar y J] slookup/miss (Dlookup x d y e) (Dsbounded x d y e)) ([x] [d] [y] [e] slookup/hit (Dsbounded x d y e)) DofM'''' DofR'' DofN' DsubNx DsubRx (%the (sub ([x] N x (R x)) M (Nx Rx)) DsubNyx')) + %<- (sub-fun DsubNyx' DsubNyx (%the (term-eq (Nx Rx) Nyx) Deq)) %. +%worlds (var ovar bind) (ssub-aa-permute-e _ _ _ _ _ _ _ _) %. +%total {} (ssub-aa-permute-e _ _ _ _ _ _ _ _) %. +%sort ssub-ao-permute-e {x} %. +%mode ssub-ao-permute-e %in %in %in %in %in %in %in %in %out %. +%term _ + %pi (ssub-ao-permute-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (ofe G1 M A) DofM) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (B x))) DofR) (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y J) %-> (ofe (cons (G x) y (B x)) (N x y) (C x y)))) DofN) (%the ({y} sub ([x] N x y) M (Nx y)) DsubNx) (%the (aosub R M LRx) DsubRx) (%the (sub ([y] Nx y) LRx Nxy) DsubNxy) (%the (sub ([x] N x (R x)) M Nyx) DsubNyx) Deq) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-context (DofN x d y e) (ordered/cons (%the (bounded (G x) y) (Dbounded x d y e)))) + %<- ({x} {d isvar x I} {y} {e isvar y J} append-bounded (Dappend x) (Dbounded x d y e) (bounded/cons (%the (bounded G1 x) (Dbounded' x d y e)) (%the (precedes x y) (DprecedesXY x d y e)))) + %<- (can-simp A (%the (simp A S) DsimpA)) + %<- ({x} can-simp (B x) (%the (simp (B x) T) (DsimpB x))) + %<- ({x} {y} can-simp (C x y) (%the (simp (C x y) U) (DsimpC x y))) + %<- ({y} {e isvar y J} context-append-simp-lookup Dappend ([x] [d] Dbounded x d y e) DsimpA ([x] %the (simpctx (G x) (G' x y)) (Dsimpctx x y)) ([x] [d] %the (sbounded (G' x y) y) (Dsbounded x d y e)) ([x] [d] %the (slookup (G' x y) x S) (Dlookup x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded' x d y e) DofM _ (%the (ofe (cons G1 x A) M A) (DofM' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe'' (Dappend x) (Dbounded x d y e) (DofM' x d y e) (%the (ofe (G x) M A) (DofM'' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded x d y e) (DofM'' x d y e) _ (%the (ofe (cons (G x) y (B x)) M A) (DofM''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) DsimpA (DofM''' x d y e) (%the (ofes (scons (G' x y) y T) M S) (DofM'''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-aofe (Dbounded x d y e) (DofR x d) _ (%the (aofe (cons (G x) y (B x)) (R x) (B x)) (DofR' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) (DsimpB x) (DofR' x d y e) (%the (aofes (scons (G' x y) y T) (R x) T) (DofR'' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) (DsimpC x y) (DofN x d y e) (%the (ofes (scons (G' x y) y T) (N x y) U) (DofN' x d y e))) + %<- (ssub-x-permute-es ([x] [d isvar x I] [y] [e isvar y J] slookup/miss (Dlookup x d y e) (Dsbounded x d y e)) ([x] [d] [y] [e] slookup/hit (Dsbounded x d y e)) DofM'''' DofR'' DofN' DsubNx DsubRx DsubNxy (%the (sub ([x] N x (R x)) M Nxy) DsubNyx')) + %<- (sub-fun DsubNyx' DsubNyx (%the (term-eq Nxy Nyx) Deq)) %. +%worlds (var ovar bind) (ssub-ao-permute-e _ _ _ _ _ _ _ _ _) %. +%total {} (ssub-ao-permute-e _ _ _ _ _ _ _ _ _) %. +%sort ssub-ao-permute2-e {x} %. +%mode ssub-ao-permute2-e %in %in %in %in %in %in %in %in %out %. +%term _ + %pi (ssub-ao-permute2-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (ofe G1 M A) DofM) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (B x))) DofR) (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y J) %-> (ofe (cons (cons (G x) y (B x)) Z (D x y)) (N x y) (C x y)))) DofN) (%the ({y} sub ([x] N x y) M (Nx y)) DsubNx) (%the (aosub R M LRx) DsubRx) (%the (sub ([y] Nx y) LRx Nxy) DsubNxy) (%the (sub ([x] N x (R x)) M Nyx) DsubNyx) Deq) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-context (DofN x d y e) (ordered/cons (bounded/cons (%the (bounded (G x) y) (Dbounded x d y e)) (%the (precedes y Z) (DprecedesYZ x d y e))))) + %<- ({x} {d isvar x I} {y} {e isvar y J} append-bounded (Dappend x) (Dbounded x d y e) (bounded/cons (%the (bounded G1 x) (Dbounded' x d y e)) (%the (precedes x y) (DprecedesXY x d y e)))) + %<- (can-simp A (%the (simp A S) DsimpA)) + %<- ({x} can-simp (B x) (%the (simp (B x) T) (DsimpB x))) + %<- ({x} {y} can-simp (C x y) (%the (simp (C x y) U) (DsimpC x y))) + %<- ({x} {y} can-simp (D x y) (%the (simp (D x y) V) (DsimpD x y))) + %<- ({y} {e isvar y J} context-append-simp-lookup Dappend ([x] [d] Dbounded x d y e) DsimpA ([x] %the (simpctx (G x) (G' x y)) (Dsimpctx x y)) ([x] [d] %the (sbounded (G' x y) y) (Dsbounded x d y e)) ([x] [d] %the (slookup (G' x y) x S) (Dlookup x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded' x d y e) DofM _ (%the (ofe (cons G1 x A) M A) (DofM' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe'' (Dappend x) (Dbounded x d y e) (DofM' x d y e) (%the (ofe (G x) M A) (DofM'' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded x d y e) (DofM'' x d y e) _ (%the (ofe (cons (G x) y (B x)) M A) (DofM''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (bounded/cons (Dbounded x d y e) (DprecedesYZ x d y e)) (DofM''' x d y e) _ (%the (ofe (cons (cons (G x) y (B x)) Z (D x y)) M A) (DofM'''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-simp (simpctx/cons (DsimpD x y) (simpctx/cons (DsimpB x) (Dsimpctx x y))) DsimpA (DofM'''' x d y e) (%the (ofes (scons (scons (G' x y) y T) Z V) M S) (DofM''''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-aofe (Dbounded x d y e) (DofR x d) _ (%the (aofe (cons (G x) y (B x)) (R x) (B x)) (DofR' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-aofe (bounded/cons (Dbounded x d y e) (DprecedesYZ x d y e)) (DofR' x d y e) _ (%the (aofe (cons (cons (G x) y (B x)) Z (D x y)) (R x) (B x)) (DofR'' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofe-simp (simpctx/cons (DsimpD x y) (simpctx/cons (DsimpB x) (Dsimpctx x y))) (DsimpB x) (DofR'' x d y e) (%the (aofes (scons (scons (G' x y) y T) Z V) (R x) T) (DofR''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-simp (simpctx/cons (DsimpD x y) (simpctx/cons (DsimpB x) (Dsimpctx x y))) (DsimpC x y) (DofN x d y e) (%the (ofes (scons (scons (G' x y) y T) Z V) (N x y) U) (DofN' x d y e))) + %<- (ssub-x-permute-es ([x] [d isvar x I] [y] [e isvar y J] slookup/miss (slookup/miss (Dlookup x d y e) (Dsbounded x d y e)) (sbounded/cons (Dsbounded x d y e) (DprecedesYZ x d y e))) ([x] [d] [y] [e] slookup/miss (slookup/hit (Dsbounded x d y e)) (sbounded/cons (Dsbounded x d y e) (DprecedesYZ x d y e))) DofM''''' DofR''' DofN' DsubNx DsubRx DsubNxy (%the (sub ([x] N x (R x)) M Nxy) DsubNyx')) + %<- (sub-fun DsubNyx' DsubNyx (%the (term-eq Nxy Nyx) Deq)) %. +%worlds (var ovar bind) (ssub-ao-permute2-e _ _ _ _ _ _ _ _ _) %. +%total {} (ssub-ao-permute2-e _ _ _ _ _ _ _ _ _) %. +%sort stsub-aa-permute-e {x} %. +%mode stsub-aa-permute-e %in %in %in %in %in %in %in %out %. +%term _ + %pi (stsub-aa-permute-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (ofe G1 M A) DofM) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (B x))) DofR) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y)))) DwfC) (%the ({y} tsub ([x] C x y) M (Cx y)) DsubCx) (%the (aasub R M Rx) DsubRx) (%the (tsub ([x] C x (R x)) M Cyx) DsubCyx) Deq) + %<- ({x} {d isvar x I} {y} {e isvar y J} wfe-context (DwfC x d y e) (ordered/cons (%the (bounded (G x) y) (Dbounded x d y e)))) + %<- ({x} {d isvar x I} {y} {e isvar y J} append-bounded (Dappend x) (Dbounded x d y e) (bounded/cons (%the (bounded G1 x) (Dbounded' x d y e)) (%the (precedes x y) (DprecedesXY x d y e)))) + %<- (can-simp A (%the (simp A S) DsimpA)) + %<- ({x} can-simp (B x) (%the (simp (B x) T) (DsimpB x))) + %<- ({y} {e isvar y J} context-append-simp-lookup Dappend ([x] [d] Dbounded x d y e) DsimpA ([x] %the (simpctx (G x) (G' x y)) (Dsimpctx x y)) ([x] [d] %the (sbounded (G' x y) y) (Dsbounded x d y e)) ([x] [d] %the (slookup (G' x y) x S) (Dlookup x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded' x d y e) DofM _ (%the (ofe (cons G1 x A) M A) (DofM' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe'' (Dappend x) (Dbounded x d y e) (DofM' x d y e) (%the (ofe (G x) M A) (DofM'' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded x d y e) (DofM'' x d y e) _ (%the (ofe (cons (G x) y (B x)) M A) (DofM''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) DsimpA (DofM''' x d y e) (%the (ofes (scons (G' x y) y T) M S) (DofM'''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-aofe (Dbounded x d y e) (DofR x d) _ (%the (aofe (cons (G x) y (B x)) (R x) (B x)) (DofR' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) (DsimpB x) (DofR' x d y e) (%the (aofes (scons (G' x y) y T) (R x) T) (DofR'' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} wfe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) (DwfC x d y e) (%the (wfes (scons (G' x y) y T) (C x y)) (DwfC' x d y e))) + %<- (stsub-z-permute-es ([x] [d isvar x I] [y] [e isvar y J] slookup/miss (Dlookup x d y e) (Dsbounded x d y e)) ([x] [d] [y] [e] slookup/hit (Dsbounded x d y e)) DofM'''' DofR'' DwfC' DsubCx DsubRx (%the (tsub ([x] C x (R x)) M (Cx Rx)) DsubCyx')) + %<- (tsub-fun DsubCyx' DsubCyx (%the (tp-eq (Cx Rx) Cyx) Deq)) %. +%worlds (var ovar bind) (stsub-aa-permute-e _ _ _ _ _ _ _ _) %. +%total {} (stsub-aa-permute-e _ _ _ _ _ _ _ _) %. +%sort stsub-ao-permute-e {x} %. +%mode stsub-ao-permute-e %in %in %in %in %in %in %in %in %out %. +%term _ + %pi (stsub-ao-permute-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (ofe G1 M A) DofM) (%the ({x} %pi (isvar x I) %-> (aofe (G x) (R x) (B x))) DofR) (%the ({x} %pi (isvar x I) %-> ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y)))) DwfC) (%the ({y} tsub ([x] C x y) M (Cx y)) DsubCx) (%the (aosub R M LRx) DsubRx) (%the (tsub ([y] Cx y) LRx Cxy) DsubCxy) (%the (tsub ([x] C x (R x)) M Cyx) DsubCyx) Deq) + %<- ({x} {d isvar x I} {y} {e isvar y J} wfe-context (DwfC x d y e) (ordered/cons (%the (bounded (G x) y) (Dbounded x d y e)))) + %<- ({x} {d isvar x I} {y} {e isvar y J} append-bounded (Dappend x) (Dbounded x d y e) (bounded/cons (%the (bounded G1 x) (Dbounded' x d y e)) (%the (precedes x y) (DprecedesXY x d y e)))) + %<- (can-simp A (%the (simp A S) DsimpA)) + %<- ({x} can-simp (B x) (%the (simp (B x) T) (DsimpB x))) + %<- ({y} {e isvar y J} context-append-simp-lookup Dappend ([x] [d] Dbounded x d y e) DsimpA ([x] %the (simpctx (G x) (G' x y)) (Dsimpctx x y)) ([x] [d] %the (sbounded (G' x y) y) (Dsbounded x d y e)) ([x] [d] %the (slookup (G' x y) x S) (Dlookup x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded' x d y e) DofM _ (%the (ofe (cons G1 x A) M A) (DofM' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe'' (Dappend x) (Dbounded x d y e) (DofM' x d y e) (%the (ofe (G x) M A) (DofM'' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-ofe (Dbounded x d y e) (DofM'' x d y e) _ (%the (ofe (cons (G x) y (B x)) M A) (DofM''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) DsimpA (DofM''' x d y e) (%the (ofes (scons (G' x y) y T) M S) (DofM'''' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} weaken-aofe (Dbounded x d y e) (DofR x d) _ (%the (aofe (cons (G x) y (B x)) (R x) (B x)) (DofR' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} aofe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) (DsimpB x) (DofR' x d y e) (%the (aofes (scons (G' x y) y T) (R x) T) (DofR'' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} wfe-simp (simpctx/cons (DsimpB x) (Dsimpctx x y)) (DwfC x d y e) (%the (wfes (scons (G' x y) y T) (C x y)) (DwfC' x d y e))) + %<- (stsub-x-permute-es ([x] [d isvar x I] [y] [e isvar y J] slookup/miss (Dlookup x d y e) (Dsbounded x d y e)) ([x] [d] [y] [e] slookup/hit (Dsbounded x d y e)) DofM'''' DofR'' DwfC' DsubCx DsubRx DsubCxy (%the (tsub ([x] C x (R x)) M Cxy) DsubCyx')) + %<- (tsub-fun DsubCyx' DsubCyx (%the (tp-eq Cxy Cyx) Deq)) %. +%worlds (var ovar bind) (stsub-ao-permute-e _ _ _ _ _ _ _ _ _) %. +%total {} (stsub-ao-permute-e _ _ _ _ _ _ _ _ _) %. +%%%%% Substitution Lemma (Explicit Context) %%%%% +%sort aasubst-em {T stp} %. +%mode aasubst-em %in %in %in %in %in %in %in %in %out %. +%sort aosubst-em {T stp} %. +%mode aosubst-em %in %in %in %in %in %in %in %in %out %. +%sort subst-em {T stp} %. +%mode subst-em %in %in %in %in %in %in %in %in %out %. +%sort tsubst-em {T stp} %. +%mode tsubst-em %in %in %in %in %in %in %in %out %. +%scope tsubst-em %term closed + %pi (aasubst-em _ _ (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) (%the (aasub R M Rx) Dsub) (%the (tsub B M Bx) Dtsub) ([x] [d isvar x I] aofe/closed (%the (ordered (G x)) (Dordered x d)) (%the (aof (R x) (B x)) (Dof x))) (aofe/closed Dordered' Dof'')) + %<- (csub-ordered Dordered Dcsub (%the (ordered G') Dordered')) + %<- (aof-noassm Dof (%the ({x} atom-eq (R x) R') DeqR) (%the ({x} tp-eq (B x) B') DeqB)) + %<- (aasub-resp DeqR term-eq/i atom-eq/i Dsub (%the (aasub ([_] R') M Rx) Dsub')) + %<- (tsub-resp DeqB term-eq/i tp-eq/i Dtsub (%the (tsub ([_] B') M Bx) Dtsub')) + %<- (aasub-absent R' M (%the (aasub ([_] R') M R') Dsub'')) + %<- (tsub-absent B' M (%the (tsub ([_] B') M B') Dtsub'')) + %<- (aasub-fun Dsub'' Dsub' (%the (atom-eq R' Rx) DeqR')) + %<- (tsub-fun Dtsub'' Dtsub' (%the (tp-eq B' Bx) DeqB')) + %<- ({x} aof-resp (DeqR x) (DeqB x) (Dof x) (%the (aof R' B') (Dof' x))) + %<- (aof-resp DeqR' DeqB' (Dof' aca) (%the (aof Rx Bx) Dof'')) %. +%term closed + %pi (aosubst-em _ _ (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) (%the (aosub R M N) Dsub) (%the (tsub B M Bx) Dtsub) ([x] [d isvar x I] aofe/closed (%the (ordered (G x)) _) (%the (aof (R x) (B x)) (Dof x))) D) + %<- (aof-noassm Dof (%the ({x} atom-eq (R x) R') DeqR) _) + %<- (aosub-resp DeqR term-eq/i term-eq/i Dsub (%the (aosub ([_] R') M N) Dsub')) + %<- (aosub-absent Dsub' Dfalse) + %<- (false-implies-ofe Dfalse D) %. +%scope tsubst-em %term forall + %pi (aasubst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) (aasub/forall (%the (tsub B M Bx) Dsub)) (tsub/pi ([_] tsub/t) (tsub/pi ([_] tsub/t) (%the (tsub B M Bx') Dsub'))) ([x] [d isvar x I] aofe/forall (%the (wfe (G x) (B x)) (Dwf x d))) D) + %<- (tsubst-em T Dsimp Dapp Dcsub DofM Dsub Dwf (%the (wfe G' Bx) Dwf')) + %<- (tsub-fun Dsub Dsub' (%the (tp-eq Bx Bx') Deq)) + %<- (tp-resp-tp ([a] pi (pi a ([_] t)) ([_] t)) Deq Deq') + %<- (aofe-resp ctx-eq/i atom-eq/i Deq' (aofe/forall Dwf') D) %. +%term allcl + %pi (aasubst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) aasub/closed (tsub/pi ([_] tsub/t) (tsub/pi ([_] tsub/t) (%the (tsub ([_] B) M Bx) Dsub))) ([x] [d isvar x I] aofe/forall (%the (wfe (G x) B) (Dwf x d))) D) + %<- (tsubst-em T Dsimp Dapp Dcsub DofM Dsub Dwf (%the (wfe G' Bx) Dwf')) + %<- (tsub-absent-fun Dsub (%the (tp-eq Bx B) Deq)) + %<- (atom-resp-tp forall Deq Deq') + %<- (aofe-resp ctx-eq/i Deq' tp-eq/i (aofe/forall Dwf') D) %. +%scope tsubst-em %term allbad + %pi (aasubst-em _ _ _ _ _ (aasub/forall _) _ ([x] [d] aofe/var _ (Dlookup x d)) D) + %<- ({x} {d isvar x I} lookup-isvar (Dlookup x d) (%the (isvar (forall _) J) (Disvar x))) + %<- ({x} isvar-forall-contra (Disvar x) Dfalse) + %<- (false-implies-aofe Dfalse D) %. +%term var + %pi (aasubst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) aasub/closed (%the (tsub B M Bx) DsubBx) ([x] [d] aofe/var (%the (wfe (G x) (B x)) (Dwf x d)) (%the (lookup (G x) X (B x)) (Dlookup x d))) (aofe/var DwfBx Dlookup')) + %<- (csub-lookup Dcsub Dlookup DsubBx (%the (lookup G' X Bx) Dlookup')) + %<- (tsubst-em T Dsimp Dapp Dcsub DofM DsubBx Dwf (%the (wfe G' Bx) DwfBx)) %. +%scope tsubst-em %term var + %pi (aosubst-em _ _ (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) aosub/var (%the (tsub B M Bx) Dtsub) ([x] [d isvar x I] aofe/var _ (%the (lookup (G x) x (B x)) (Dlook x d))) DofM'') + %<- ({x} {d isvar x I} lookup-binder-fun (Dapp x) (Dlook x d) (%the (tp-eq A (B x)) (Deq x))) + %<- (tsub-absent A M (%the (tsub ([_] A) M A) Dtsub')) + %<- (tsub-resp Deq term-eq/i tp-eq/i Dtsub' (%the (tsub B M A) Dtsub'')) + %<- (tsub-fun Dtsub'' Dtsub (%the (tp-eq A Bx) Deq')) + %<- ({x} {d isvar x I} lookup-context (Dlook x d) (%the (ordered (G x)) (Dordered x d))) + %<- (weaken-ofe-csub Dapp Dordered Dcsub DofM (%the (ofe G' M A) DofM')) + %<- (ofe-resp ctx-eq/i term-eq/i Deq' DofM' (%the (ofe G' M Bx) DofM'')) %. +%term app + %pi (aasubst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) (aasub/app (%the (sub N M N') DsubN') (%the (aasub R M R') DsubR')) (%the (tsub Cy M Cyx) DtsubCyx) ([x] [d isvar x I] %the (aofe (G x) (app (R x) (N x)) (Cy x)) (aofe/app (%the (wfe (G x) (Cy x)) (DwfCy x d)) (%the (tsub ([y] C x y) (N x) (Cy x)) (DtsubCy x)) (%the (ofe (G x) (N x) (B x)) (DofN x d)) (%the (aofe (G x) (R x) (pi (B x) ([y] C x y))) (DofR x d)))) (aofe/app DwfCyx DtsubCxy' DofN' DofR')) + %<- ({x} {d isvar x I} aofe-reg (DofR x d) (wfe/pi (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- (can-tsub-e Dapp DwfB DofM (%the (tsub ([x] B x) M Bx) DtsubBx)) + %<- ({y} {e isvar y J} can-tsub-e ([x] append/cons (Dapp x)) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DtsubCx y))) + %<- (aasubst-em T Dsimp Dapp Dcsub DofM DsubR' (%the (tsub ([x] pi (B x) ([y] C x y)) M (pi Bx ([y] Cx y))) (tsub/pi DtsubCx DtsubBx)) DofR (%the (aofe G' R' (pi Bx Cx)) DofR')) + %<- (subst-em T Dsimp Dapp Dcsub DofM DsubN' DtsubBx DofN (%the (ofe G' N' Bx) DofN')) + %<- (aofe-reg DofR' (wfe/pi (%the ({y} %pi (isvar y J') %-> (wfe (cons G' y Bx) (Cx y))) DwfCx) (%the (wfe G' Bx) DwfBx))) + %<- (can-tsub-e ([y] append/nil) DwfCx DofN' (%the (tsub ([y] Cx y) N' Cxy) DtsubCxy)) + %<- (tsub-permute-e Dapp DofM DofN DwfC DtsubCx DsubN' DtsubCxy DtsubCy DtsubCyx (%the (tp-eq Cxy Cyx) Deq)) + %<- (tsub-resp ([_] tp-eq/i) term-eq/i Deq DtsubCxy (%the (tsub Cx N' Cyx) DtsubCxy')) + %<- (tsubst-em T Dsimp Dapp Dcsub DofM DtsubCyx DwfCy (%the (wfe G' Cyx) DwfCyx)) %. +%scope tsubst-em %term appclo + %pi (aasubst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) aasub/closed (%the (tsub Cy M Cyx) DtsubCyx) ([x] [d isvar x I] %the (aofe (G x) (app R N) (Cy x)) (aofe/app (%the (wfe (G x) (Cy x)) (DwfCy x d)) (%the (tsub ([y] C x y) N (Cy x)) (DtsubCy x)) (%the (ofe (G x) N (B x)) (DofN x d)) (%the (aofe (G x) R (pi (B x) ([y] C x y))) (DofR x d)))) (aofe/app DwfCyx DtsubCxy' DofN' DofR')) + %<- ({x} {d isvar x I} aofe-reg (DofR x d) (wfe/pi (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- (can-tsub-e Dapp DwfB DofM (%the (tsub ([x] B x) M Bx) DtsubBx)) + %<- ({y} {e isvar y J} can-tsub-e ([x] append/cons (Dapp x)) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DtsubCx y))) + %<- (aasubst-em T Dsimp Dapp Dcsub DofM aasub/closed (%the (tsub ([x] pi (B x) ([y] C x y)) M (pi Bx ([y] Cx y))) (tsub/pi DtsubCx DtsubBx)) DofR (%the (aofe G' R (pi Bx Cx)) DofR')) + %<- (sub-absent N M (%the (sub ([_] N) M N) DsubN')) + %<- (subst-em T Dsimp Dapp Dcsub DofM DsubN' DtsubBx DofN (%the (ofe G' N Bx) DofN')) + %<- (aofe-reg DofR' (wfe/pi (%the ({y} %pi (isvar y J') %-> (wfe (cons G' y Bx) (Cx y))) DwfCx) (%the (wfe G' Bx) DwfBx))) + %<- (can-tsub-e ([y] append/nil) DwfCx DofN' (%the (tsub ([y] Cx y) N Cxy) DtsubCxy)) + %<- (tsub-permute-e Dapp DofM DofN DwfC DtsubCx DsubN' DtsubCxy DtsubCy DtsubCyx (%the (tp-eq Cxy Cyx) Deq)) + %<- (tsub-resp ([_] tp-eq/i) term-eq/i Deq DtsubCxy (%the (tsub Cx N Cyx) DtsubCxy')) + %<- (tsubst-em T Dsimp Dapp Dcsub DofM DtsubCyx DwfCy (%the (wfe G' Cyx) DwfCyx)) %. +%term app + %pi (aosubst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) (aosub/app (%the (sub N1 Nx N2) DsubN2) (%the (sub N M Nx) DsubNx) (%the (aosub R M (lam N1)) DsubN1)) (%the (tsub Cy M Cyx) DtsubCyx) ([x] [d isvar x I] %the (aofe (G x) (app (R x) (N x)) (Cy x)) (aofe/app (%the (wfe (G x) (Cy x)) (DwfCy x d)) (%the (tsub ([y] C x y) (N x) (Cy x)) (DtsubCy x)) (%the (ofe (G x) (N x) (B x)) (DofN x d)) (%the (aofe (G x) (R x) (pi (B x) ([y] C x y))) (DofR x d)))) DofN2') + %<- ({x} {d isvar x I} aofe-reg (DofR x d) (wfe/pi (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- (can-tsub-e Dapp DwfB DofM (%the (tsub ([x] B x) M Bx) DtsubBx)) + %<- ({y} {e isvar y J} can-tsub-e ([x] append/cons (Dapp x)) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DtsubCx y))) + %<- (aosubst-em T Dsimp Dapp Dcsub DofM DsubN1 (%the (tsub ([x] pi (B x) ([y] C x y)) M (pi Bx ([y] Cx y))) (tsub/pi DtsubCx DtsubBx)) DofR (ofe/lam (%the ({y} %pi (isvar y J') %-> (ofe (cons G' y Bx) (N1 y) (Cx y))) DofN1) (%the (wfe G' Bx) DwfBx))) + %<- (subst-em T Dsimp Dapp Dcsub DofM DsubNx DtsubBx DofN (%the (ofe G' Nx Bx) DofNx)) + %<- ({y} {e isvar y J'} ofe-reg (DofN1 y e) (%the (wfe (cons G' y Bx) (Cx y)) (DwfCx y e))) + %<- (can-tsub-e ([_] append/nil) DwfCx DofNx (%the (tsub Cx Nx Cxy) DtsubCxy)) + %<- (aosub-headvar DsubN1 (%the (headvar R) Dheadvar)) + %<- (headvar-tp-size Dapp DofR Dsimp Dheadvar ([x] simp/pi ([y] %the (simp (C x y) T2) (Dsimp2 x y)) (%the (simp (B x) T1) (Dsimp1 x))) (%the (stp-leq (spi T1 T2) T) Dleq)) + %<- (tsub-preserves-simp DtsubBx Dsimp1 (%the (simp Bx T1) Dsimp1')) + %<- (employ-stp-leq (spi T1 T2) T Dleq) + %<- (subst-em T1 Dsimp1' ([_] append/nil) csub/base DofNx DsubN2 DtsubCxy DofN1 (%the (ofe G' N2 Cxy) DofN2)) + %<- (tsub-permute-e Dapp DofM DofN DwfC DtsubCx DsubNx DtsubCxy DtsubCy DtsubCyx (%the (tp-eq Cxy Cyx) Deq)) + %<- (ofe-resp ctx-eq/i term-eq/i Deq DofN2 (%the (ofe G' N2 Cyx) DofN2')) %. +%scope tsubst-em %term appbad + %pi (aasubst-em _ _ _ _ _ (aasub/app _ _) _ ([x] [d] aofe/var _ (Dlookup x d)) D) + %<- ({x} {d isvar x I} lookup-isvar (Dlookup x d) (%the (isvar (app _ _) J) (Disvar x))) + %<- ({x} isvar-app-contra (Disvar x) Dfalse) + %<- (false-implies-aofe Dfalse D) %. +%term appbad + %pi (aosubst-em _ _ _ _ _ (aosub/app _ _ _) _ ([x] [d] aofe/var _ (Dlookup x d)) D) + %<- ({x} {d isvar x I} lookup-isvar (Dlookup x d) (%the (isvar (app _ _) J) (Disvar x))) + %<- ({x} isvar-app-contra (Disvar x) Dfalse) + %<- (false-implies-ofe Dfalse D) %. +%scope tsubst-em %term pi1 + %pi (aasubst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) (aasub/pi1 (%the (aasub R M R') Dsub)) (%the (tsub B M Bx) DtsubBx) ([x] [d isvar x I] aofe/pi1 (%the (aofe (G x) (R x) (sigma (B x) ([y] C x y))) (Dof x d))) (aofe/pi1 Dof')) + %<- ({x} {d isvar x I} aofe-reg (Dof x d) (wfe/sigma (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- ({y} {e isvar y J} can-tsub-e ([x] append/cons (Dapp x)) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DtsubCx y))) + %<- (aasubst-em T Dsimp Dapp Dcsub DofM Dsub (%the (tsub ([x] sigma (B x) ([y] C x y)) M (sigma Bx ([y] Cx y))) (tsub/sigma DtsubCx DtsubBx)) Dof (%the (aofe G' R' (sigma Bx ([y] Cx y))) Dof')) %. +%term pi1clo + %pi (aasubst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) aasub/closed (%the (tsub B M Bx) DtsubBx) ([x] [d isvar x I] aofe/pi1 (%the (aofe (G x) R (sigma (B x) ([y] C x y))) (Dof x d))) (aofe/pi1 Dof')) + %<- ({x} {d isvar x I} aofe-reg (Dof x d) (wfe/sigma (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- ({y} {e isvar y J} can-tsub-e ([x] append/cons (Dapp x)) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DtsubCx y))) + %<- (aasubst-em T Dsimp Dapp Dcsub DofM aasub/closed (%the (tsub ([x] sigma (B x) ([y] C x y)) M (sigma Bx ([y] Cx y))) (tsub/sigma DtsubCx DtsubBx)) Dof (%the (aofe G' R (sigma Bx ([y] Cx y))) Dof')) %. +%scope tsubst-em %term pi1 + %pi (aosubst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) (aosub/pi1 (%the (aosub R M (pair N1 N2)) Dsub)) (%the (tsub B M Bx) DtsubBx) ([x] [d isvar x I] aofe/pi1 (%the (aofe (G x) (R x) (sigma (B x) ([y] C x y))) (Dof x d))) Dof1) + %<- ({x} {d isvar x I} aofe-reg (Dof x d) (wfe/sigma (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- ({y} {e isvar y J} can-tsub-e ([x] append/cons (Dapp x)) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DtsubCx y))) + %<- (aosubst-em T Dsimp Dapp Dcsub DofM Dsub (%the (tsub ([x] sigma (B x) ([y] C x y)) M (sigma Bx ([y] Cx y))) (tsub/sigma DtsubCx DtsubBx)) Dof (ofe/pair _ _ _ (%the (ofe G' N1 Bx) Dof1))) %. +%term pi1bad + %pi (aasubst-em _ _ _ _ _ (aasub/pi1 _) _ ([x] [d] aofe/var _ (Dlookup x d)) D) + %<- ({x} {d isvar x I} lookup-isvar (Dlookup x d) (%the (isvar (pi1 _) J) (Disvar x))) + %<- ({x} isvar-pi1-contra (Disvar x) Dfalse) + %<- (false-implies-aofe Dfalse D) %. +%scope tsubst-em %term pi1bad + %pi (aosubst-em _ _ _ _ _ (aosub/pi1 _) _ ([x] [d] aofe/var _ (Dlookup x d)) D) + %<- ({x} {d isvar x I} lookup-isvar (Dlookup x d) (%the (isvar (pi1 _) J) (Disvar x))) + %<- ({x} isvar-pi1-contra (Disvar x) Dfalse) + %<- (false-implies-ofe Dfalse D) %. +%term pi2 + %pi (aasubst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) (aasub/pi2 (%the (aasub R M R') Dsub)) (%the (tsub ([x] C x (pi1 (R x))) M Cyx) DtsubCyx) ([x] [d isvar x I] aofe/pi2 (%the (aofe (G x) (R x) (sigma (B x) ([y] C x y))) (Dof x d))) Dof'') + %<- ({x} {d isvar x I} aofe-reg (Dof x d) (wfe/sigma (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- (can-tsub-e Dapp DwfB DofM (%the (tsub ([x] B x) M Bx) DtsubBx)) + %<- ({y} {e isvar y J} can-tsub-e ([x] append/cons (Dapp x)) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DtsubCx y))) + %<- (aasubst-em T Dsimp Dapp Dcsub DofM Dsub (%the (tsub ([x] sigma (B x) ([y] C x y)) M (sigma Bx ([y] Cx y))) (tsub/sigma DtsubCx DtsubBx)) Dof (%the (aofe G' R' (sigma Bx ([y] Cx y))) Dof')) + %<- (stsub-aa-permute-e Dapp DofM ([x] [d] aofe/pi1 (Dof x d)) DwfC DtsubCx (aasub/pi1 Dsub) DtsubCyx (%the (tp-eq (Cx (pi1 R')) Cyx) Deq)) + %<- (aofe-resp ctx-eq/i atom-eq/i Deq (aofe/pi2 Dof') (%the (aofe G' (pi2 R') Cyx) Dof'')) %. +%scope tsubst-em %term pi2clo + %pi (aasubst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) aasub/closed (%the (tsub ([x] C x (pi1 R)) M Cyx) DtsubCyx) ([x] [d isvar x I] aofe/pi2 (%the (aofe (G x) R (sigma (B x) ([y] C x y))) (Dof x d))) Dof'') + %<- ({x} {d isvar x I} aofe-reg (Dof x d) (wfe/sigma (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- (can-tsub-e Dapp DwfB DofM (%the (tsub ([x] B x) M Bx) DtsubBx)) + %<- ({y} {e isvar y J} can-tsub-e ([x] append/cons (Dapp x)) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DtsubCx y))) + %<- (aasubst-em T Dsimp Dapp Dcsub DofM aasub/closed (%the (tsub ([x] sigma (B x) ([y] C x y)) M (sigma Bx ([y] Cx y))) (tsub/sigma DtsubCx DtsubBx)) Dof (%the (aofe G' R (sigma Bx ([y] Cx y))) Dof')) + %<- (stsub-aa-permute-e Dapp DofM ([x] [d] aofe/pi1 (Dof x d)) DwfC DtsubCx aasub/closed DtsubCyx (%the (tp-eq (Cx (pi1 R)) Cyx) Deq)) + %<- (aofe-resp ctx-eq/i atom-eq/i Deq (aofe/pi2 Dof') (%the (aofe G' (pi2 R) Cyx) Dof'')) %. +%term pi2 + %pi (aosubst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) (aosub/pi2 (%the (aosub R M (pair N1 N2)) Dsub)) (%the (tsub ([x] C x (pi1 (R x))) M Cyx) DtsubCyx) ([x] [d isvar x I] aofe/pi2 (%the (aofe (G x) (R x) (sigma (B x) ([y] C x y))) (Dof x d))) DofN2') + %<- ({x} {d isvar x I} aofe-reg (Dof x d) (wfe/sigma (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- (can-tsub-e Dapp DwfB DofM (%the (tsub ([x] B x) M Bx) DtsubBx)) + %<- ({y} {e isvar y J} can-tsub-e ([x] append/cons (Dapp x)) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DtsubCx y))) + %<- (aosubst-em T Dsimp Dapp Dcsub DofM Dsub (%the (tsub ([x] sigma (B x) ([y] C x y)) M (sigma Bx ([y] Cx y))) (tsub/sigma DtsubCx DtsubBx)) Dof (%the (ofe G' (pair N1 N2) (sigma Bx ([y] Cx y))) (ofe/pair (%the ({y} %pi (isvar y J') %-> (wfe (cons G' y Bx) (Cx y))) DwfCx) (%the (ofe G' N2 Cxy) DofN2) (%the (tsub Cx N1 Cxy) DtsubCxy) (%the (ofe G' N1 Bx) DofN1)))) + %<- (stsub-ao-permute-e Dapp DofM ([x] [d] aofe/pi1 (Dof x d)) DwfC DtsubCx (aosub/pi1 Dsub) DtsubCxy DtsubCyx (%the (tp-eq Cxy Cyx) Deq)) + %<- (ofe-resp ctx-eq/i term-eq/i Deq DofN2 (%the (ofe G' N2 Cyx) DofN2')) %. +%scope tsubst-em %term pi2bad + %pi (aasubst-em _ _ _ _ _ (aasub/pi2 _) _ ([x] [d] aofe/var _ (Dlookup x d)) D) + %<- ({x} {d isvar x I} lookup-isvar (Dlookup x d) (%the (isvar (pi2 _) J) (Disvar x))) + %<- ({x} isvar-pi2-contra (Disvar x) Dfalse) + %<- (false-implies-aofe Dfalse D) %. +%term pi2bad + %pi (aosubst-em _ _ _ _ _ (aosub/pi2 _) _ ([x] [d] aofe/var _ (Dlookup x d)) D) + %<- ({x} {d isvar x I} lookup-isvar (Dlookup x d) (%the (isvar (pi2 _) J) (Disvar x))) + %<- ({x} isvar-pi2-contra (Disvar x) Dfalse) + %<- (false-implies-ofe Dfalse D) %. +%scope tsubst-em %term at-a + %pi (subst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) (sub/aa (%the (aasub R M R') Dsub)) tsub/t ([x] [d isvar x I] ofe/at (%the (aofe (G x) (R x) t) (Dof x d))) (ofe/at Dof')) + %<- (aasubst-em T Dsimp Dapp Dcsub DofM Dsub tsub/t Dof (%the (aofe G' R' t) Dof')) %. +%term at-o + %pi (subst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) (sub/ao (%the (aosub R M N) Dsub)) tsub/t ([x] [d isvar x I] ofe/at (%the (aofe (G x) (R x) t) (Dof x d))) Dof') + %<- (aosubst-em T Dsimp Dapp Dcsub DofM Dsub tsub/t Dof (%the (ofe G' N t) Dof')) %. +%scope tsubst-em %term lam + %pi (subst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) (sub/lam (%the ({y} sub ([x] N x y) M (N' y)) Dsub)) (tsub/pi (%the ({y} tsub ([x] B2 x y) M (B2' y)) Dtsub2) (%the (tsub B1 M B1') Dtsub1)) ([x] [d isvar x I] ofe/lam ([y] [e isvar y J] %the (ofe (cons (G x) y (B1 x)) (N x y) (B2 x y)) (Dof x d y e)) (%the (wfe (G x) (B1 x)) (Dwf x d))) (ofe/lam Dof' Dwf')) + %<- (tsubst-em T Dsimp Dapp Dcsub DofM Dtsub1 Dwf (%the (wfe G' B1') Dwf')) + %<- ({y} {e isvar y J} subst-em T Dsimp ([x] append/cons (Dapp x)) (csub/cons Dtsub1 Dcsub) DofM (Dsub y) (Dtsub2 y) ([x] [d] Dof x d y e) (%the (ofe (cons G' y B1') (N' y) (B2' y)) (Dof' y e))) %. +%term pair + %pi (subst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M Gx) Dcsub) (%the (ofe G1 M A) DofM) (sub/pair (%the (sub N2 M N2x) Dsub2) (%the (sub N1 M N1x) Dsub1)) (tsub/sigma (%the ({y} tsub ([x] B2 x y) M (B2x y)) Dtsub2x) (%the (tsub B1 M B1x) Dtsub1)) ([x] [d isvar x I] ofe/pair ([y] [e isvar y J] %the (wfe (cons (G x) y (B1 x)) (B2 x y)) (Dwf x d y e)) (%the (ofe (G x) (N2 x) (B2y x)) (Dof2 x d)) (%the (tsub ([y] B2 x y) (N1 x) (B2y x)) (Dtsub2y x)) (%the (ofe (G x) (N1 x) (B1 x)) (Dof1 x d))) (ofe/pair Dwf-x Dof2x' Dtsub2xy Dof1x)) + %<- (subst-em T Dsimp Dapp Dcsub DofM Dsub1 Dtsub1 Dof1 (%the (ofe Gx N1x B1x) Dof1x)) + %<- ({y} {e isvar y J} tsubst-em T Dsimp ([x] append/cons (Dapp x)) (csub/cons Dtsub1 Dcsub) DofM (Dtsub2x y) ([x] [d] Dwf x d y e) (%the (wfe (cons Gx y B1x) (B2x y)) (Dwf-x y e))) + %<- (can-tsub-e ([x] append/nil) Dwf-x Dof1x (%the (tsub B2x N1x B2xy) Dtsub2xy)) + %<- ({x} {d isvar x I} ofe-reg (Dof2 x d) (%the (wfe (G x) (B2y x)) (Dwf-y x d))) + %<- (can-tsub-e Dapp Dwf-y DofM (%the (tsub B2y M B2yx) Dtsub2yx)) + %<- (subst-em T Dsimp Dapp Dcsub DofM Dsub2 Dtsub2yx Dof2 (%the (ofe Gx N2x B2yx) Dof2x)) + %<- (tsub-permute-e Dapp DofM Dof1 Dwf Dtsub2x Dsub1 Dtsub2xy Dtsub2y Dtsub2yx (%the (tp-eq B2xy B2yx) Deq)) + %<- (tp-eq-symm Deq (%the (tp-eq B2yx B2xy) Deq')) + %<- (ofe-resp ctx-eq/i term-eq/i Deq' Dof2x (%the (ofe Gx N2x B2xy) Dof2x')) %. +%scope tsubst-em %term sing-a + %pi (subst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) (sub/aa (%the (aasub R M R') Dsub)) (tsub/singa (%the (aasub R M R'') Dsub')) ([x] [d isvar x I] ofe/sing (%the (aofe (G x) (R x) t) (Dof x d))) Dof'') + %<- (aasubst-em T Dsimp Dapp Dcsub DofM Dsub tsub/t Dof (%the (aofe G' R' t) Dof')) + %<- (aasub-fun Dsub Dsub' (%the (atom-eq R' R'') Deq)) + %<- (tp-resp-atom sing Deq (%the (tp-eq (sing R') (sing R'')) Deq')) + %<- (ofe-resp ctx-eq/i term-eq/i Deq' (ofe/sing Dof') (%the (ofe G' (at R') (sing R'')) Dof'')) %. +%term sing-o + %pi (subst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) (sub/ao (%the (aosub R M L) Dsub)) (tsub/singo (%the (aosub R M (at R')) Dsub')) ([x] [d isvar x I] ofe/sing (%the (aofe (G x) (R x) t) (Dof x d))) Dof'') + %<- (aosubst-em T Dsimp Dapp Dcsub DofM Dsub' tsub/t Dof (ofe/at (%the (aofe G' R' t) Dof'))) + %<- (aosub-fun Dsub' Dsub (%the (term-eq (at R') L) Deq)) + %<- (ofe-resp ctx-eq/i Deq tp-eq/i (ofe/sing Dof') (%the (ofe G' L (sing R')) Dof'')) %. +%scope tsubst-em %term sing-x + %pi (subst-em _ _ _ _ _ (sub/ao (%the (aosub R M _) Dsub)) (tsub/singa (%the (aasub R M _) Dsub')) _ D) + %<- (aasub-aosub-contra Dsub' Dsub Dfalse) + %<- (false-implies-ofe Dfalse D) %. +%term sing-x + %pi (subst-em _ _ _ _ _ (sub/aa (%the (aasub R M _) Dsub)) (tsub/singo (%the (aosub R M _) Dsub')) _ D) + %<- (aasub-aosub-contra Dsub Dsub' Dfalse) + %<- (false-implies-ofe Dfalse D) %. +%scope tsubst-em %term t + %pi (tsubst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) tsub/t ([x] [d] wfe/t (Dordered x d)) (wfe/t Dordered')) + %<- (csub-ordered Dordered Dcsub Dordered') %. +%term pi + %pi (tsubst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) (tsub/pi (%the ({y} tsub ([x] B2 x y) M (B2' y)) Dtsub2) (%the (tsub B1 M B1') Dtsub1)) ([x] [d isvar x I] wfe/pi (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B1 x)) (B2 x y))) (Dwf2 x d)) (%the (wfe (G x) (B1 x)) (Dwf1 x d))) (wfe/pi Dwf2' Dwf1')) + %<- (tsubst-em T Dsimp Dapp Dcsub DofM Dtsub1 Dwf1 (%the (wfe G' B1') Dwf1')) + %<- ({y} {e isvar y J} tsubst-em T Dsimp ([x] append/cons (Dapp x)) (csub/cons Dtsub1 Dcsub) DofM (Dtsub2 y) ([x] [d] Dwf2 x d y e) (%the (wfe (cons G' y B1') (B2' y)) (Dwf2' y e))) %. +%scope tsubst-em %term sigma + %pi (tsubst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) (tsub/sigma (%the ({y} tsub ([x] B2 x y) M (B2' y)) Dtsub2) (%the (tsub B1 M B1') Dtsub1)) ([x] [d isvar x I] wfe/sigma (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B1 x)) (B2 x y))) (Dwf2 x d)) (%the (wfe (G x) (B1 x)) (Dwf1 x d))) (wfe/sigma Dwf2' Dwf1')) + %<- (tsubst-em T Dsimp Dapp Dcsub DofM Dtsub1 Dwf1 (%the (wfe G' B1') Dwf1')) + %<- ({y} {e isvar y J} tsubst-em T Dsimp ([x] append/cons (Dapp x)) (csub/cons Dtsub1 Dcsub) DofM (Dtsub2 y) ([x] [d] Dwf2 x d y e) (%the (wfe (cons G' y B1') (B2' y)) (Dwf2' y e))) %. +%term singa + %pi (tsubst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) (tsub/singa (%the (aasub R M R') Daasub)) ([x] [d] wfe/sing (%the (aofe (G x) (R x) t) (Daof x d))) (wfe/sing Daof')) + %<- (aasubst-em T Dsimp Dapp Dcsub DofM Daasub tsub/t Daof (%the (aofe G' R' t) Daof')) %. +%scope tsubst-em %term singo + %pi (tsubst-em T (%the (simp A T) Dsimp) (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dapp) (%the (csub G M G') Dcsub) (%the (ofe G1 M A) DofM) (tsub/singo (%the (aosub R M (at R')) Daosub)) ([x] [d] wfe/sing (%the (aofe (G x) (R x) t) (Daof x d))) (wfe/sing Daof')) + %<- (aosubst-em T Dsimp Dapp Dcsub DofM Daosub tsub/t Daof (ofe/at (%the (aofe G' R' t) Daof'))) %. +%worlds (bind ovar var) (aasubst-em _ _ _ _ _ _ _ _ _) (aosubst-em _ _ _ _ _ _ _ _ _) (subst-em _ _ _ _ _ _ _ _ _) (tsubst-em _ _ _ _ _ _ _ _) %. +%total {(T1 T2 T3 T4) (D1 D2 D3 D4)} (aasubst-em T1 _ _ _ _ _ _ D1 _) (aosubst-em T2 _ _ _ _ _ _ D2 _) (subst-em T3 _ _ _ _ _ _ D3 _) (tsubst-em T4 _ _ _ _ _ D4 _) %. +%sort aasubst-e {x} %. +%mode aasubst-e %in %in %in %in %in %in %out %. +%term _ + %pi (aasubst-e Dapp Dcsub Dof Dsub Dtsub Daof D) + %<- (can-simp _ Dsimp) + %<- (aasubst-em _ Dsimp Dapp Dcsub Dof Dsub Dtsub Daof D) %. +%worlds (bind ovar var) (aasubst-e _ _ _ _ _ _ _) %. +%total {} (aasubst-e _ _ _ _ _ _ _) %. +%sort aosubst-e {x} %. +%mode aosubst-e %in %in %in %in %in %in %out %. +%term _ + %pi (aosubst-e Dapp Dcsub Dof Dsub Dtsub Daof D) + %<- (can-simp _ Dsimp) + %<- (aosubst-em _ Dsimp Dapp Dcsub Dof Dsub Dtsub Daof D) %. +%worlds (bind ovar var) (aosubst-e _ _ _ _ _ _ _) %. +%total {} (aosubst-e _ _ _ _ _ _ _) %. +%sort subst-e {x} %. +%mode subst-e %in %in %in %in %in %in %out %. +%term _ + %pi (subst-e Dapp Dcsub Dof Dsub Dtsub Dof' D) + %<- (can-simp _ Dsimp) + %<- (subst-em _ Dsimp Dapp Dcsub Dof Dsub Dtsub Dof' D) %. +%worlds (bind ovar var) (subst-e _ _ _ _ _ _ _) %. +%total {} (subst-e _ _ _ _ _ _ _) %. +%sort tsubst-e {x} %. +%mode tsubst-e %in %in %in %in %in %out %. +%term _ + %pi (tsubst-e Dapp Dcsub Dof Dsub Dwf D) + %<- (can-simp _ Dsimp) + %<- (tsubst-em _ Dsimp Dapp Dcsub Dof Dsub Dwf D) %. +%worlds (bind ovar var) (tsubst-e _ _ _ _ _ _) %. +%total {} (tsubst-e _ _ _ _ _ _) %. +%%%%% Substitution Lemma (Implicit Context) %%%%% +%sort subst %. +%mode subst %in %in %in %in %out %. +%term _ + %pi (subst (%the (sub ([x] M x) N Mx) Dsub) (%the (tsub ([x] B x) N Bx) Dtsub) (%the ({x} %pi (vof x A) %-> (of (M x) (B x))) DofM) (%the (of N A) DofN) Dof) + %<- (of1-to-ofe 0 DofM (%the ({x} %pi (isvar x 0) %-> (ofe (cons nil x A) (M x) (B x))) DofeM)) + %<- (of-to-ofe DofN (%the (ofe nil N A) DofeN)) + %<- (subst-e ([x] append/nil) csub/base DofeN Dsub Dtsub DofeM (%the (ofe nil Mx Bx) Dofe)) + %<- (ofe-to-of Dofe (%the (of Mx Bx) Dof)) %. +%worlds (bind var) (subst _ _ _ _ _) %. +%total {} (subst _ _ _ _ _) %. +%sort subst' %. +%mode subst' %in %in %in %out %. +%term _ + %pi (subst' Dsub DofM DofN DofMx) + %<- (tsub-absent B N (%the (tsub ([_] B) N B) Dtsub)) + %<- (subst Dsub Dtsub DofM DofN DofMx) %. +%worlds (bind var) (subst' _ _ _ _) %. +%total {} (subst' _ _ _ _) %. +%sort aosubst %. +%mode aosubst %in %in %in %in %out %. +%term _ + %pi (aosubst (%the (aosub ([x] R x) N LRx) Dsub) (%the (tsub ([x] B x) N Bx) Dtsub) (%the ({x} %pi (vof x A) %-> (aof (R x) (B x))) DofR) (%the (of N A) DofN) Dof) + %<- (aof1-to-aofe 0 DofR (%the ({x} %pi (isvar x 0) %-> (aofe (cons nil x A) (R x) (B x))) DaofeR)) + %<- (of-to-ofe DofN (%the (ofe nil N A) DofeN)) + %<- (aosubst-e ([x] append/nil) csub/base DofeN Dsub Dtsub DaofeR (%the (ofe nil LRx Bx) Dofe)) + %<- (ofe-to-of Dofe (%the (of LRx Bx) Dof)) %. +%worlds (bind var) (aosubst _ _ _ _ _) %. +%total {} (aosubst _ _ _ _ _) %. +%sort aasubst %. +%mode aasubst %in %in %in %in %out %. +%term _ + %pi (aasubst (%the (aasub ([x] R x) N Rx) Dsub) (%the (tsub ([x] B x) N Bx) Dtsub) (%the ({x} %pi (vof x A) %-> (aof (R x) (B x))) DofR) (%the (of N A) DofN) Daof) + %<- (aof1-to-aofe 0 DofR (%the ({x} %pi (isvar x 0) %-> (aofe (cons nil x A) (R x) (B x))) DaofeR)) + %<- (of-to-ofe DofN (%the (ofe nil N A) DofeN)) + %<- (aasubst-e ([x] append/nil) csub/base DofeN Dsub Dtsub DaofeR (%the (aofe nil Rx Bx) Daofe)) + %<- (aofe-to-aof Daofe (%the (aof Rx Bx) Daof)) %. +%worlds (bind var) (aasubst _ _ _ _ _) %. +%total {} (aasubst _ _ _ _ _) %. +%sort tsubst %. +%mode tsubst %in %in %in %out %. +%term _ + %pi (tsubst (%the (tsub ([x] B x) N Bx) Dsub) (%the ({x} %pi (vof x A) %-> (wf (B x))) DwfB) (%the (of N A) DofN) Dwf) + %<- (wf1-to-wfe 0 DwfB (%the ({x} %pi (isvar x 0) %-> (wfe (cons nil x A) (B x))) DwfeB)) + %<- (of-to-ofe DofN (%the (ofe nil N A) DofeN)) + %<- (tsubst-e ([x] append/nil) csub/base DofeN Dsub DwfeB (%the (wfe nil Bx) Dwfe)) + %<- (wfe-to-wf Dwfe (%the (wf Bx) Dwf)) %. +%worlds (bind var ovar) (tsubst _ _ _ _) %. +%total {} (tsubst _ _ _ _) %. +%%%%% Substitution Permutation (Implicit Context) %%%%% +%sort sub-permute %. +%mode sub-permute %in %in %in %in %in %in %in %in %out %. +%term _ + %pi (sub-permute DofM DofN DofO DsubOx DsubNx DsubOxy DsubOy DsubOyx Deq) + %<- (of-to-ofe DofM (%the (ofe nil M A) DofeM)) + %<- (of1-to-ofe 0 DofN (%the ({x} %pi (isvar x 0) %-> (ofe (cons nil x A) (N x) (B x))) DofeN)) + %<- (of2-to-ofe lt/z DofO (%the ({x} + %pi (isvar x 0) + %-> ({y} %pi (isvar y (s 0)) %-> (ofe (cons (cons nil x A) y (B x)) (O x y) (C x y)))) DofeO)) + %<- (sub-permute-e ([_] append/nil) DofeM DofeN DofeO DsubOx DsubNx DsubOxy DsubOy DsubOyx Deq) %. +%worlds (bind) (sub-permute _ _ _ _ _ _ _ _ _) %. +%total {} (sub-permute _ _ _ _ _ _ _ _ _) %. +%sort aasub-permute %. +%mode aasub-permute %in %in %in %in %in %in %in %in %out %. +%term _ + %pi (aasub-permute DofM DofN DaofR DsubRx DsubNx DsubRxy DsubRy DsubRyx Deq) + %<- (of-to-ofe DofM (%the (ofe nil M A) DofeM)) + %<- (of1-to-ofe 0 DofN (%the ({x} %pi (isvar x 0) %-> (ofe (cons nil x A) (N x) (B x))) DofeN)) + %<- (aof2-to-aofe lt/z DaofR (%the ({x} + %pi (isvar x 0) + %-> ({y} %pi (isvar y (s 0)) %-> (aofe (cons (cons nil x A) y (B x)) (R x y) (C x y)))) DaofeR)) + %<- (aasub-permute-e ([_] append/nil) DofeM DofeN DaofeR DsubRx DsubNx DsubRxy DsubRy DsubRyx Deq) %. +%worlds (bind) (aasub-permute _ _ _ _ _ _ _ _ _) %. +%total {} (aasub-permute _ _ _ _ _ _ _ _ _) %. +%sort aasub-aosub-permute %. +%mode aasub-aosub-permute %in %in %in %in %in %in %in %in %out %. +%term _ + %pi (aasub-aosub-permute DofM DofN DaofR DsubRx DsubNx DsubRxy DsubRy DsubRyx Deq) + %<- (of-to-ofe DofM (%the (ofe nil M A) DofeM)) + %<- (of1-to-ofe 0 DofN (%the ({x} %pi (isvar x 0) %-> (ofe (cons nil x A) (N x) (B x))) DofeN)) + %<- (aof2-to-aofe lt/z DaofR (%the ({x} + %pi (isvar x 0) + %-> ({y} %pi (isvar y (s 0)) %-> (aofe (cons (cons nil x A) y (B x)) (R x y) (C x y)))) DaofeR)) + %<- (aasub-aosub-permute-e ([_] append/nil) DofeM DofeN DaofeR DsubRx DsubNx DsubRxy DsubRy DsubRyx Deq) %. +%worlds (bind) (aasub-aosub-permute _ _ _ _ _ _ _ _ _) %. +%total {} (aasub-aosub-permute _ _ _ _ _ _ _ _ _) %. +%sort aosub-permute %. +%mode aosub-permute %in %in %in %in %in %in %in %in %out %. +%term _ + %pi (aosub-permute DofM DofN DaofR DsubRx DsubNx DsubRxy DsubRy DsubRyx Deq) + %<- (of-to-ofe DofM (%the (ofe nil M A) DofeM)) + %<- (of1-to-ofe 0 DofN (%the ({x} %pi (isvar x 0) %-> (ofe (cons nil x A) (N x) (B x))) DofeN)) + %<- (aof2-to-aofe lt/z DaofR (%the ({x} + %pi (isvar x 0) + %-> ({y} %pi (isvar y (s 0)) %-> (aofe (cons (cons nil x A) y (B x)) (R x y) (C x y)))) DaofeR)) + %<- (aosub-permute-e ([_] append/nil) DofeM DofeN DaofeR DsubRx DsubNx DsubRxy DsubRy DsubRyx Deq) %. +%worlds (bind) (aosub-permute _ _ _ _ _ _ _ _ _) %. +%total {} (aosub-permute _ _ _ _ _ _ _ _ _) %. +%sort tsub-permute %. +%mode tsub-permute %in %in %in %in %in %in %in %in %out %. +%term _ + %pi (tsub-permute DofM DofN DwfC DsubOx DsubNx DsubOxy DsubOy DsubOyx Deq) + %<- (of-to-ofe DofM (%the (ofe nil M A) DofeM)) + %<- (of1-to-ofe 0 DofN (%the ({x} %pi (isvar x 0) %-> (ofe (cons nil x A) (N x) (B x))) DofeN)) + %<- (wf2-to-wfe lt/z DwfC (%the ({x} + %pi (isvar x 0) + %-> ({y} %pi (isvar y (s 0)) %-> (wfe (cons (cons nil x A) y (B x)) (C x y)))) DwfeC)) + %<- (tsub-permute-e ([_] append/nil) DofeM DofeN DwfeC DsubOx DsubNx DsubOxy DsubOy DsubOyx Deq) %. +%worlds (bind) (tsub-permute _ _ _ _ _ _ _ _ _) %. +%total {} (tsub-permute _ _ _ _ _ _ _ _ _) %. +%sort ssub-ao-permute %. +%mode ssub-ao-permute %in %in %in %in %in %in %in %out %. +%term _ + %pi (ssub-ao-permute DofM DofR DofN DsubNx DsubRx DsubNxy DsubNyx Deq) + %<- (of-to-ofe DofM DofeM) + %<- (aof1-to-aofe 0 DofR DofeR) + %<- (of2-to-ofe (%the (lt 0 (s 0)) lt/z) DofN DofeN) + %<- (ssub-ao-permute-e ([_] append/nil) DofeM DofeR DofeN DsubNx DsubRx DsubNxy DsubNyx Deq) %. +%worlds (bind) (ssub-ao-permute _ _ _ _ _ _ _ _) %. +%total {} (ssub-ao-permute _ _ _ _ _ _ _ _) %. +%%%%% Inhabitation %%%%% +%sort inhabitation-e %. +%mode inhabitation-e %in %out %. +%scope inhabitation-e %term t inhabitation-e (wfe/t Dordered) (ofe/at (aofe/closed Dordered (aof/const wf/t (kof/i topen/t ckof/unit)))) %. +%term pi + %pi (inhabitation-e (wfe/pi DwfB DwfA) (ofe/lam DofM DwfA)) + %<- ({x} {d isvar x I} inhabitation-e (DwfB x d) (DofM x d)) %. +%scope inhabitation-e %term sigma + %pi (inhabitation-e (wfe/sigma DwfB DwfA) (ofe/pair DwfB DofNx DsubBx DofM)) + %<- (inhabitation-e DwfA DofM) + %<- ({x} {d isvar x I} inhabitation-e (DwfB x d) (DofN x d)) + %<- (can-tsub-e ([_] append/nil) DwfB DofM DsubBx) + %<- (can-sub-e ([_] append/nil) DofN DofM DsubNx) + %<- (subst-e ([_] append/nil) csub/base DofM DsubNx DsubBx DofN DofNx) %. +%term sing inhabitation-e (wfe/sing Daof) (ofe/sing Daof) %. +%worlds (var bind ovar) (inhabitation-e _ _) %. +%total D (inhabitation-e D _) %. +%sort inhabitation %. +%mode inhabitation %in %out %. +%term _ + %pi (inhabitation Dwf Dof) + %<- (wf-to-wfe Dwf Dwfe) + %<- (inhabitation-e Dwfe Dofe) + %<- (ofe-to-of Dofe Dof) %. +%worlds (var bind) (inhabitation _ _) %. +%total {} (inhabitation _ _) %. +%sort inhabitsub-e %. +%mode inhabitsub-e %in %in %out %. +%term _ + %pi (inhabitsub-e DwfA DofM DofM') + %<- (inhabitation-e DwfA DofN) + %<- (sub-absent M N (%the (sub ([_] M) N M) DsubM)) + %<- (tsub-absent B N (%the (tsub ([_] B) N B) DsubB)) + %<- (subst-e ([_] append/nil) csub/base DofN DsubM DsubB DofM (%the (ofe G M B) DofM')) %. +%worlds (var bind ovar) (inhabitsub-e _ _ _) %. +%total {} (inhabitsub-e _ _ _) %. +%sort tinhabitsub-e %. +%mode tinhabitsub-e %in %in %out %. +%term _ + %pi (tinhabitsub-e DwfA DwfB DwfB') + %<- (inhabitation-e DwfA DofN) + %<- (tsub-absent B N (%the (tsub ([_] B) N B) DsubB)) + %<- (tsubst-e ([_] append/nil) csub/base DofN DsubB DwfB (%the (wfe G B) DwfB')) %. +%worlds (var bind ovar) (tinhabitsub-e _ _ _) %. +%total {} (tinhabitsub-e _ _ _) %. +%sort tinhabitsub %. +%mode tinhabitsub %in %in %out %. +%term _ + %pi (tinhabitsub DwfA DwfB D) + %<- (inhabitation DwfA DofM) + %<- (tsub-absent B N DsubN) + %<- (tsubst DsubN DwfB DofM D) %. +%worlds (bind) (tinhabitsub _ _ _) %. +%total {} (tinhabitsub _ _ _) %. +%%%%% Corollaries %%%%% +%sort compose-sub %. +%mode compose-sub %in %in %in %in %in %in %out %. +%term _ + %pi (compose-sub (%the (of MA A) DofMA) (%the ({x} %pi (vof x A) %-> (of (OAB x) B)) DofOAB) (%the ({x} %pi (vof x B) %-> (of (OBC x) C)) DofOBC) (%the (sub OAB MA MB) DsubAB) (%the (sub OBC MB MC) DsubBC) (%the ({x} sub OBC (OAB x) (OAC x)) DsubOAC) DsubAC') + %<- ({x} {d vof x A} subst' (DsubOAC x) DofOBC (DofOAB x d) (%the (of (OAC x) C) (DofOAC x d))) + %<- (can-sub DofOAC DofMA (%the (sub OAC MA MC') DsubAC)) + %<- ({x} sub-absent (OBC x) MA (%the (sub ([_] OBC x) MA (OBC x)) (DsubOBCabs x))) + %<- (sub-permute DofMA DofOAB ([_] [_] DofOBC) DsubOBCabs DsubAB DsubBC DsubOAC DsubAC (%the (term-eq MC MC') Deq)) + %<- (term-eq-symm Deq (%the (term-eq MC' MC) Deq')) + %<- (sub-resp ([_] term-eq/i) term-eq/i Deq' DsubAC (%the (sub OAC MA MC) DsubAC')) %. +%worlds (bind) (compose-sub _ _ _ _ _ _ _) %. +%total {} (compose-sub _ _ _ _ _ _ _) %. +%sort subst1 {y} %. +%mode subst1 %in %in %in %in %in %out %. +%term _ + %pi (subst1 (%the ({y} sub ([x] N x y) M (Nx y)) DsubNx) (%the ({y} tsub ([x] C x y) M (Cx y)) DsubCx) (%the (tsub ([x] B x) M Bx) DsubBx) (%the ({x} %pi (vof x A) %-> ({y} %pi (vof y (B x)) %-> (of (N x y) (C x y)))) DofN) (%the (of M A) DofM) DofNx) + %<- (of2-to-ofe (%the (lt 0 (s 0)) lt/z) DofN (%the ({x} + %pi (isvar x 0) + %-> ({y} %pi (isvar y (s 0)) %-> (ofe (cons (cons nil x A) y (B x)) (N x y) (C x y)))) DofeN)) + %<- (of-to-ofe DofM (%the (ofe nil M A) DofeM)) + %<- ({y} {e isvar y (s 0)} subst-e ([_] append/cons append/nil) (csub/cons DsubBx csub/base) DofeM (DsubNx y) (DsubCx y) ([x] [d] DofeN x d y e) (%the (ofe (cons nil y Bx) (Nx y) (Cx y)) (DofeNx y e))) + %<- (ofe-to-of1 DofeNx (%the ({y} %pi (vof y Bx) %-> (of (Nx y) (Cx y))) DofNx)) %. +%worlds (bind) (subst1 _ _ _ _ _ _) %. +%total {} (subst1 _ _ _ _ _ _) %. +%sort tsubst1 {y} %. +%mode tsubst1 %in %in %in %in %out %. +%term _ + %pi (tsubst1 (%the ({y} tsub ([x] C x y) M (Cx y)) DsubCx) (%the (tsub ([x] B x) M Bx) DsubBx) (%the ({x} %pi (vof x A) %-> ({y} %pi (vof y (B x)) %-> (wf (C x y)))) DwfC) (%the (of M A) DofM) DwfCx) + %<- (wf2-to-wfe (%the (lt 0 (s 0)) lt/z) DwfC (%the ({x} + %pi (isvar x 0) + %-> ({y} %pi (isvar y (s 0)) %-> (wfe (cons (cons nil x A) y (B x)) (C x y)))) DwfeC)) + %<- (of-to-ofe DofM (%the (ofe nil M A) DofeM)) + %<- ({y} {e isvar y (s 0)} tsubst-e ([_] append/cons append/nil) (csub/cons DsubBx csub/base) DofeM (DsubCx y) ([x] [d] DwfeC x d y e) (%the (wfe (cons nil y Bx) (Cx y)) (DwfeCx y e))) + %<- (wfe-to-wf1 DwfeCx (%the ({y} %pi (vof y Bx) %-> (wf (Cx y))) DwfCx)) %. +%worlds (bind) (tsubst1 _ _ _ _ _) %. +%total {} (tsubst1 _ _ _ _ _) %. +%sort subst-context-e {x} %. +%mode subst-context-e %in %in %in %in %out %. +%term _ + %pi (subst-context-e (%the ({x} %pi (isvar x I) %-> (ofe (cons G x A) (M x) B)) DofM) (%the ({y} %pi (isvar y J) %-> (ofe (cons G y B) (N y) (C y))) DofN) (%the ({x} sub ([y] N y) (M x) (Ny x)) DsubNy) (%the ({x} tsub ([y] C y) (M x) (Cy x)) DsubCy) DofN''') + %<- ({x} {d isvar x I} ofe-context (DofM x d) (%the (ordered (cons G x A)) (ordered/cons (%the (bounded G x) (DboundedGx x d))))) + %<- ({x} {d isvar x I} following-var d ([y] [e isvar y J'] %the (precedes x y) (Dprec x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J'} bounded-increase-bound (DboundedGx x d) (Dprec x d y e) (%the (bounded G y) (DboundedGy x d y e))) + %<- ({x} {d isvar x I} bump-ofe DofN ([y] [e isvar y J'] ordered/cons (DboundedGy x d y e)) ([y] [e isvar y J'] %the (ofe (cons G y B) (N y) (C y)) (DofN' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J'} weakeng-ofe (append/cons append/nil) (append/cons append/nil) (ordered/cons (bounded/cons (DboundedGx x d) (Dprec x d y e))) (DofN' x d y e) (%the (ofe (cons (cons G x A) y B) (N y) (C y)) (DofN'' x d y e))) + %<- ({x} {d isvar x I} subst-e ([_] append/nil) csub/base (DofM x d) (DsubNy x) (DsubCy x) ([y] [e isvar y J'] DofN'' x d y e) (%the (ofe (cons G x A) (Ny x) (Cy x)) (DofN''' x d))) %. +%worlds (var bind ovar) (subst-context-e _ _ _ _ _) %. +%total {} (subst-context-e _ _ _ _ _) %. +%sort tsubst-context-e {x} %. +%mode tsubst-context-e %in %in %in %out %. +%term _ + %pi (tsubst-context-e (%the ({x} %pi (isvar x I) %-> (ofe (cons G x A) (M x) B)) Dof) (%the ({y} %pi (isvar y J) %-> (wfe (cons G y B) (C y))) Dwf) (%the ({x} tsub ([y] C y) (M x) (Cy x)) Dsub) Dwf''') + %<- ({x} {d isvar x I} ofe-context (Dof x d) (%the (ordered (cons G x A)) (ordered/cons (%the (bounded G x) (DboundedGx x d))))) + %<- ({x} {d isvar x I} following-var d ([y] [e isvar y J'] %the (precedes x y) (Dprec x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J'} bounded-increase-bound (DboundedGx x d) (Dprec x d y e) (%the (bounded G y) (DboundedGy x d y e))) + %<- ({x} {d isvar x I} bump-wfe Dwf ([y] [e isvar y J'] ordered/cons (DboundedGy x d y e)) ([y] [e isvar y J'] %the (wfe (cons G y B) (C y)) (Dwf' x d y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J'} weakeng-wfe (append/cons append/nil) (append/cons append/nil) (ordered/cons (bounded/cons (DboundedGx x d) (Dprec x d y e))) (Dwf' x d y e) (%the (wfe (cons (cons G x A) y B) (C y)) (Dwf'' x d y e))) + %<- ({x} {d isvar x I} tsubst-e ([_] append/nil) csub/base (Dof x d) (Dsub x) ([y] [e isvar y J'] Dwf'' x d y e) (%the (wfe (cons G x A) (Cy x)) (Dwf''' x d))) %. +%worlds (bind ovar var) (tsubst-context-e _ _ _ _) %. +%total {} (tsubst-context-e _ _ _ _) %. +%sort ssubst-context-e {x} %. +%mode ssubst-context-e %in %in %out %. +%term _ + %pi (ssubst-context-e Daof Dof D) + %<- ({x} {d} aofe-context (Daof x d) (ordered/cons (Dbound x d))) + %<- (weaken-ofe-insert1 Dof Dbound _ Dof') + %<- ({x} {d} ssubst-e ([_] append/nil) append/nil (Dof' x d) (Daof x d) (D x d)) %. +%worlds (var bind ovar) (ssubst-context-e _ _ _) %. +%total {} (ssubst-context-e _ _ _) %. +%sort ssubst-context2-e {x} %. +%mode ssubst-context2-e %in %in %in %out %. +%term _ + %pi (ssubst-context2-e Daof1 Daof2 Dof Dof''') + %<- ({x} {d} aofe-context (Daof1 x d) (ordered/cons (Dbound x d))) + %<- (weaken-ofe-insert2 Dof Dbound _ Dof') + %<- ({x} {d} {z} {f} ssubst-e ([_] append/cons append/nil) (append/cons append/nil) ([y] [e] Dof' x d y e z f) (Daof1 x d) (Dof'' x d z f)) + %<- ({x} {d} ssubst-e ([_] append/nil) append/nil ([z] [f] Dof'' x d z f) (Daof2 x d) (Dof''' x d)) %. +%worlds (var bind ovar) (ssubst-context2-e _ _ _ _) %. +%total {} (ssubst-context2-e _ _ _ _) %. +%sort subst-simult-pair-e {x} %. +%mode subst-simult-pair-e %in %in %in %in %in %out %. +%term _ + %pi (subst-simult-pair-e (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y J) %-> (ofe (cons (cons G x A) y B) (O x y) (C x y)))) DofO) (%the (ofe G M A) DofM) (%the (ofe G N B) DofN) (%the (sub ([z] O (pi1 z) (pi2 z)) (pair M N) P) Dsub) (%the ({y} sub ([x] O x y) M (Ox y)) DsubOx) DsubOxy') + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-context (DofO x d y e) (ordered/cons (bounded/cons (%the (bounded G x) (Dbounded x d y e)) (%the (precedes x y) (DprecIJ x d y e))))) + %<- ({x} {d isvar x I} strengthen-bounded ([y] [e] Dbounded x d y e) (%the (bounded G x) (Dbounded' x d))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-reg (DofO x d y e) (%the (wfe (cons (cons G x A) y B) (C x y)) (DwfC x d y e))) + %<- ({y} {e isvar y J} can-tsub-e ([_] append/cons append/nil) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DsubCx y))) + %<- (tsub-absent B M (%the (tsub ([_] B) M B) DsubB)) + %<- ({y} {e isvar y J} subst-e ([_] append/cons append/nil) (csub/cons DsubB csub/base) DofM (DsubOx y) (DsubCx y) ([x] [d] DofO x d y e) (%the (ofe (cons G y B) (Ox y) (Cx y)) (DofOx y e))) + %<- (can-sub-e ([_] append/nil) DofOx DofN (%the (sub ([y] Ox y) N Oxy) DsubOxy)) + %<- ({x} {d isvar x I} weaken-ofe (Dbounded' x d) DofN _ (%the (ofe (cons G x A) N B) (DofN' x d))) + %<- ({x} {d isvar x I} can-sub-e ([_] append/nil) ([y] [e] DofO x d y e) (DofN' x d) (%the (sub ([y] O x y) N (Oy x)) (DsubOy x))) + %<- ({x} {d isvar x I} can-tsub-e ([_] append/nil) ([y] [e] DwfC x d y e) (DofN' x d) (%the (tsub ([y] C x y) N (Cy x)) (DsubCy x))) + %<- ({x} {d isvar x I} subst-e ([_] append/nil) csub/base (DofN' x d) (DsubOy x) (DsubCy x) ([y] [e] DofO x d y e) (%the (ofe (cons G x A) (Oy x) (Cy x)) (DofOy x d))) + %<- (can-sub-e ([_] append/nil) DofOy DofM (%the (sub ([x] Oy x) M Oyx) DsubOyx)) + %<- ({x} {y} sub-absent (O x y) (pair M N) (%the (sub ([z] O x y) (pair M N) (O x y)) (DsubOz x y))) + %<- ({y} {e isvar y J} following-var e (%the ({z} %pi (isvar z K) %-> (precedes y z)) (DprecJK y e))) + %<- ({x} {d isvar x I} {y} {e isvar y J} {z} {f isvar z K} precedes-trans (DprecIJ x d y e) (DprecJK y e z f) (%the (precedes x z) (DprecIK x d y e z f))) + %<- ({x} {d isvar x I} {z} {f isvar z K} strengthen-precedes ([y] [e] DprecIK x d y e z f) (%the (precedes x z) (DprecIK' x d z f))) + %<- ({x} {d isvar x I} {y} {e isvar y J} bounded-increase-bound (Dbounded' x d) (DprecIJ x d y e) (%the (bounded G y) (DboundedJ x d y e))) + %<- ({y} {e isvar y J} strengthen-bounded ([x] [d] DboundedJ x d y e) (%the (bounded G y) (DboundedJ' y e))) + %<- (bump2-ofe DofO ([x] [d isvar x I] [y] [e isvar y K] ordered/cons (bounded/cons (Dbounded' x d) (DprecIK' x d y e))) ([x] [d isvar x J] [y] [e isvar y K] ordered/cons (bounded/cons (DboundedJ' x d) (DprecJK x d y e))) (%the ({x} + %pi (isvar x J) + %-> ({y} %pi (isvar y K) %-> (ofe (cons (cons G x A) y B) (O x y) (C x y)))) DofO')) + %<- ({z} {f isvar z I} {x} {d isvar x J} {y} {e isvar y K} weakeng-ofe (append/cons (append/cons append/nil)) (append/cons (append/cons append/nil)) (ordered/cons (bounded/cons (bounded/cons (Dbounded' z f) (DprecIJ z f x d)) (DprecJK x d y e))) (DofO' x d y e) (%the (ofe (cons (cons (cons G z (sigma A ([_] B))) x A) y B) (O x y) (C x y)) (DofO'' z f x d y e))) + %<- (ofe-reg DofM (%the (wfe G A) DwfA)) + %<- (ofe-reg DofN (%the (wfe G B) DwfB)) + %<- ({x} {d isvar x I} weaken-wfe (Dbounded' x d) DwfB _ (%the (wfe (cons G x A) B) (DwfB' x d))) + %<- ({z} {f isvar z I} weaken-wfe (Dbounded' z f) (wfe/sigma DwfB' DwfA) _ (%the (wfe (cons G z (sigma A ([_] B))) (sigma A ([_] B))) (DwfAB z f))) + %<- ({z} {f isvar z I} {x} {d isvar x J} weaken-wfe (bounded/cons (Dbounded' z f) (DprecIJ z f x d)) (DwfAB z f) _ (%the (wfe (cons (cons G z (sigma A ([_] B))) x A) (sigma A ([_] B))) (DwfAB' z f x d))) + %<- ({z} {f isvar z I} {x} {d isvar x J} ssubst-e ([_] append/nil) append/nil ([y] [e] DofO'' z f x d y e) (%the (aofe (cons (cons G z (sigma A ([_] B))) x A) (pi2 z) B) (aofe/pi2 (aofe/var (DwfAB' z f x d) (lookup/miss (lookup/hit (Dbounded' z f)) (bounded/cons (Dbounded' z f) (DprecIJ z f x d)))))) (%the (ofe (cons (cons G z (sigma A ([_] B))) x A) (O x (pi2 z)) (C x (pi2 z))) (DofOypi2 z f x d))) + %<- ({x} {d isvar x J} can-sub-e ([_] append/cons append/nil) ([z] [f] DofOypi2 z f x d) (ofe/pair DwfB' DofN DsubB DofM) (%the (sub ([z] O x (pi2 z)) (pair M N) (Oyz x)) (DsubOyz x))) + %<- ({x} {d isvar x J} ssub-ao-permute-e ([_] append/cons append/nil) (%the (ofe G (pair M N) (sigma A ([_] B))) (ofe/pair DwfB' DofN DsubB DofM)) ([z] [f isvar z I] %the (aofe (cons (cons G z (sigma A ([_] B))) x A) (pi2 z) B) (aofe/pi2 (aofe/var (DwfAB' z f x d) (lookup/miss (lookup/hit (Dbounded' z f)) (bounded/cons (Dbounded' z f) (DprecIJ z f x d)))))) ([z] [f isvar z I] [y] [e isvar y K] DofO'' z f x d y e) ([y] %the (sub ([z] O x y) (pair M N) (O x y)) (DsubOz x y)) (%the (aosub ([z] pi2 z) (pair M N) N) (aosub/pi2 aosub/var)) (DsubOy x) (DsubOyz x) (%the (term-eq (Oy x) (Oyz x)) (DeqOyz x))) + %<- ({x} term-eq-symm (DeqOyz x) (%the (term-eq (Oyz x) (Oy x)) (DeqOyz' x))) + %<- ({x} sub-resp ([_] term-eq/i) term-eq/i (DeqOyz' x) (DsubOyz x) (%the (sub ([z] O x (pi2 z)) (pair M N) (Oy x)) (DsubOyz' x))) + %<- (ssub-ao-permute-e ([_] append/nil) (%the (ofe G (pair M N) (sigma A ([_] B))) (ofe/pair DwfB' DofN DsubB DofM)) ([z] [f isvar z I] %the (aofe (cons G z (sigma A ([_] B))) (pi1 z) A) (aofe/pi1 (aofe/var (DwfAB z f) (lookup/hit (Dbounded' z f))))) ([z] [f isvar z I] [x] [d isvar x J] DofOypi2 z f x d) DsubOyz' (%the (aosub ([z] pi1 z) (pair M N) M) (aosub/pi1 aosub/var)) DsubOyx Dsub (%the (term-eq Oyx P) DeqOyx)) + %<- (sub-resp ([_] term-eq/i) term-eq/i DeqOyx DsubOyx (%the (sub ([x] Oy x) M P) DsubOyx')) + %<- (sub-absent N M (%the (sub ([_] N) M N) DsubNx)) + %<- (sub-permute-e ([_] append/nil) DofM DofN' DofO DsubOx DsubNx DsubOxy DsubOy DsubOyx' (%the (term-eq Oxy P) DeqOxy)) + %<- (sub-resp ([_] term-eq/i) term-eq/i DeqOxy DsubOxy (%the (sub ([y] Ox y) N P) DsubOxy')) %. +%worlds (bind ovar) (subst-simult-pair-e _ _ _ _ _ _) %. +%total {} (subst-simult-pair-e _ _ _ _ _ _) %. +%sort tsubst-context-permute-e {x} %. +%mode tsubst-context-permute-e %in %in %in %in %in %in %out %. +%term _ + %pi (tsubst-context-permute-e (%the ({x} %pi (isvar x I) %-> (ofe (cons G x A) (M x) B)) DofM) (%the ({y} %pi (isvar y J) %-> (ofe (cons G y B) (N y) C)) DofN) (%the ({z} %pi (isvar z K) %-> (wfe (cons G z C) (D z))) DwfD) (%the ({x} sub N (M x) (NM x)) DsubNM) (%the ({y} tsub D (N y) (Db y)) DsubDb) (%the ({x} tsub D (NM x) (Dc x)) DsubDc) DsubDc'') + %<- (tsubst-context-e DofN DwfD DsubDb (%the ({y} %pi (isvar y J) %-> (wfe (cons G y B) (Db y))) DwfDb)) + %<- (can-tsub-context-e DofM DwfDb (%the ({x} tsub Db (M x) (Dc' x)) DsubDc')) + %<- ({x} {z} tsub-absent (D z) (M x) (%the (tsub ([y] D z) (M x) (D z)) (DsubDabs x z))) + %<- ({x} {d isvar x I} ofe-context (DofM x d) (ordered/cons (%the (bounded G x) (DboundI x d)))) + %<- (weaken-ofe-insert1 DofN DboundI _ (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y J') %-> (ofe (cons (cons G x A) y B) (N y) C))) DofN')) + %<- ({x} {d isvar x I} {y} {e isvar y J'} ofe-context (DofN' x d y e) (ordered/cons (bounded/cons _ (%the (precedes x y) (DprecIJ x d y e))))) + %<- (weaken-wfe-insert1 DwfD DboundI _ (%the ({x} + %pi (isvar x I) + %-> ({z} %pi (isvar z K') %-> (wfe (cons (cons G x A) z C) (D z)))) DwfD')) + %<- ({x} {d} weaken-wfe-insert1 (DwfD' x d) ([y] [e] bounded/cons (DboundI x d) (DprecIJ x d y e)) _ (%the ({y} + %pi (isvar y J') + %-> ({z} %pi (isvar z K'') %-> (wfe (cons (cons (cons G x A) y B) z C) (D z)))) (DwfD'' x d))) + %<- ({x} {d} tsub-permute-e ([_] append/nil) (DofM x d) (DofN' x d) (DwfD'' x d) (DsubDabs x) (DsubNM x) (DsubDc x) DsubDb (DsubDc' x) (%the (tp-eq (Dc x) (Dc' x)) (Deq x))) + %<- ({x} tp-eq-symm (Deq x) (%the (tp-eq (Dc' x) (Dc x)) (Deq' x))) + %<- ({x} tsub-resp ([_] tp-eq/i) term-eq/i (Deq' x) (DsubDc' x) (%the (tsub Db (M x) (Dc x)) (DsubDc'' x))) %. +%worlds (var bind ovar) (tsubst-context-permute-e _ _ _ _ _ _ _) %. +%total {} (tsubst-context-permute-e _ _ _ _ _ _ _) %. +%sort aasub-into-sub-e %. +%mode aasub-into-sub-e %in %in %in %in %in %out %. +%term _ + %pi (aasub-into-sub-e DofM DofR DofO DsubOx DsubRx DsubOyx') + %<- ({x} {d} aofe-context (DofR x d) (Dord x d)) + %<- (bump-ofe-under DofO Dord DofO') + %<- ({x} {d} ssubst-e ([_] append/nil) append/nil (DofO' x d) (DofR x d) (DofOy x d)) + %<- (can-sub-e ([_] append/nil) DofOy DofM DsubOyx) + %<- (ssub-aa-permute-e ([_] append/nil) DofM DofR DofO' DsubOx DsubRx DsubOyx Deq) + %<- (term-eq-symm Deq Deq') + %<- (sub-resp ([_] term-eq/i) term-eq/i Deq' DsubOyx DsubOyx') %. +%worlds (var bind ovar) (aasub-into-sub-e _ _ _ _ _ _) %. +%total {} (aasub-into-sub-e _ _ _ _ _ _) %. +%sort aosub-into-ssub-e %. +%mode aosub-into-ssub-e %in %in %in %in %in %out %. +%term _ + %pi (aosub-into-ssub-e DofM DofR DofN DsubRx DsubNxy DsubNyx') + %<- (ssubst-context-e DofR DofN DofNy) + %<- (can-sub-e ([_] append/nil) DofNy DofM DsubNyx) + %<- ({x} {d} aofe-context (DofR x d) (ordered/cons (Dbound x d))) + %<- (weaken-ofe-insert1 DofN Dbound _ DofN') + %<- ({y} sub-absent (N y) M (DsubNabs y)) + %<- (ssub-ao-permute-e ([_] append/nil) DofM DofR DofN' DsubNabs DsubRx DsubNxy DsubNyx Deq) + %<- (term-eq-symm Deq Deq') + %<- (sub-resp ([_] term-eq/i) term-eq/i Deq' DsubNyx DsubNyx') %. +%worlds (var bind ovar) (aosub-into-ssub-e _ _ _ _ _ _) %. +%total {} (aosub-into-ssub-e _ _ _ _ _ _) %. +%sort aosub-into-ssub-e' {z} %. +%mode aosub-into-ssub-e' %in %in %in %in %in %out %. +%term _ + %pi (aosub-into-ssub-e' DofM DofR DofN DsubRx DsubNxy DsubNyx) + %<- ({z} {f} ofe-context (DofM z f) (ordered/cons (Dbound z f))) + %<- (weaken-aofe-insert1 DofR Dbound _ DofR') + %<- (weaken-ofe-insert1 DofN Dbound _ DofN') + %<- ({z} {f} aosub-into-ssub-e (DofM z f) (DofR' z f) (DofN' z f) (DsubRx z) (DsubNxy z) (DsubNyx z)) %. +%worlds (var bind ovar) (aosub-into-ssub-e' _ _ _ _ _ _) %. +%total {} (aosub-into-ssub-e' _ _ _ _ _ _) %. +%sort aosub-into-ssub-e'' %. +%mode aosub-into-ssub-e'' %in %in %in %in %in %in %in %in %out %. +%term _ + %pi (aosub-into-ssub-e'' (%the (ofe G M A) DofM) (%the ({x} %pi (isvar x I) %-> (aofe (cons G x A) (Q x) B)) DofQ) (%the ({x} %pi (isvar x I') %-> (aofe (cons G x A) (R x) (C (Q x)))) DofR) (%the ({y} + %pi (isvar y J) + %-> ({z} %pi (isvar z K) %-> (ofe (cons (cons G y B) z (C y)) (N y z) (D y z)))) DofN) (%the (aosub Q M LQx) DsubQx) (%the (aosub R M LRx) DsubRx) (%the ({z} sub ([y] N y z) LQx (Ny z)) DsubNy) (%the (sub Ny LRx Nyz) DsubNyz) DsubNyzx') + %<- ({x} {d isvar x I} aofe-context (DofQ x d) (ordered/cons (%the (bounded G x) (Dbound x d)))) + %<- (weaken-ofe-insert2 DofN Dbound _ (%the ({x} + %pi (isvar x I) + %-> ({y} + %pi (isvar y J') + %-> ({z} %pi (isvar z K') %-> (ofe (cons (cons (cons G x A) y B) z (C y)) (N y z) (D y z))))) DofN')) + %<- ({x} {d isvar x I} {z} {f isvar z K'} ssubst-e ([_] append/cons append/nil) (append/cons append/nil) ([y] [e] DofN' x d y e z f) (DofQ x d) (%the (ofe (cons (cons G x A) z (C (Q x))) (N (Q x) z) (D (Q x) z)) (DofNy x d z f))) + %<- ({z} {f} can-sub-e ([_] append/cons append/nil) ([x] [d] DofNy x d z f) DofM (%the (sub ([x] N (Q x) z) M (Nyx z)) (DsubNyx z))) + %<- ({y} {z} sub-absent (N y z) M (%the (sub ([x] N y z) M (N y z)) (DsubNabs y z))) + %<- ({z} {f} ssub-ao-permute2-e ([_] append/nil) DofM DofQ ([x] [d] [y] [e] DofN' x d y e z f) ([y] DsubNabs y z) DsubQx (DsubNy z) (DsubNyx z) (%the (term-eq (Ny z) (Nyx z)) (Deq1 z))) + %<- ({z} term-eq-symm (Deq1 z) (%the (term-eq (Nyx z) (Ny z)) (Deq1' z))) + %<- ({z} sub-resp ([_] term-eq/i) term-eq/i (Deq1' z) (DsubNyx z) (%the (sub ([x] N (Q x) z) M (Ny z)) (DsubNyx' z))) + %<- (bump-aofe DofR ([x] [d] ordered/cons (Dbound x d)) (%the ({x} %pi (isvar x I) %-> (aofe (cons G x A) (R x) (C (Q x)))) DofR')) + %<- ({x} {d isvar x I} ssubst-e ([_] append/nil) append/nil ([z] [f] DofNy x d z f) (DofR' x d) (%the (ofe (cons G x A) (N (Q x) (R x)) (D (Q x) (R x))) (DofNyz x d))) + %<- (can-sub-e ([_] append/nil) DofNyz DofM (%the (sub ([x] N (Q x) (R x)) M Nyzx) DsubNyzx)) + %<- (ssub-ao-permute-e ([_] append/nil) DofM DofR' DofNy DsubNyx' DsubRx DsubNyz DsubNyzx (%the (term-eq Nyz Nyzx) Deq2)) + %<- (term-eq-symm Deq2 (%the (term-eq Nyzx Nyz) Deq2')) + %<- (sub-resp ([_] term-eq/i) term-eq/i Deq2' DsubNyzx (%the (sub ([x] N (Q x) (R x)) M Nyz) DsubNyzx')) %. +%worlds (var bind ovar) (aosub-into-ssub-e'' _ _ _ _ _ _ _ _ _) %. +%total {} (aosub-into-ssub-e'' _ _ _ _ _ _ _ _ _) %. +%sort aosub-into-ssub-e''' {z} %. +%mode aosub-into-ssub-e''' %in %in %in %in %in %in %in %in %out %. +%term _ + %pi (aosub-into-ssub-e''' DofM DofQ DofR DofN DsubQx DsubRx DsubNxy1 DsubNxy2 DsubNyx) + %<- ({z} {f} ofe-context (DofM z f) (ordered/cons (Dbound z f))) + %<- (weaken-aofe-insert1 DofQ Dbound _ DofQ') + %<- (weaken-aofe-insert1 DofR Dbound _ DofR') + %<- (weaken-ofe-insert2 DofN Dbound _ DofN') + %<- ({z} {f} aosub-into-ssub-e'' (DofM z f) (DofQ' z f) (DofR' z f) (DofN' z f) (DsubQx z) (DsubRx z) (DsubNxy1 z) (DsubNxy2 z) (DsubNyx z)) %. +%worlds (var bind ovar) (aosub-into-ssub-e''' _ _ _ _ _ _ _ _ _) %. +%total {} (aosub-into-ssub-e''' _ _ _ _ _ _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/subtype-lemmas-thm.lf b/new-tests/stelf-output/tslf/sing/subtype-lemmas-thm.lf new file mode 100644 index 0000000..406bfbf --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/subtype-lemmas-thm.lf @@ -0,0 +1,902 @@ +%%%%% Subtyping Lemmas %%%%% +%sort subtype-fun %. +%mode subtype-fun %in %in %out %. +%scope subtype-fun %term t subtype-fun subtype/t subtype/t ([_] term-eq/i) %. +%term pi + %pi (subtype-fun (subtype/pi Dsubtype2 Dsub Dsubtype1) (subtype/pi Dsubtype2' Dsub' Dsubtype1') Deq') + %<- (subtype-fun Dsubtype1 Dsubtype1' (%the ({x} term-eq (M1 x) (M1' x)) Deq1)) + %<- ({x} tsub-resp ([_] tp-eq/i) (Deq1 x) tp-eq/i (Dsub x) (Dsub'' x)) + %<- ({x} tsub-fun (Dsub'' x) (Dsub' x) (DeqA2 x)) + %<- ({x} subtype-resp (DeqA2 x) tp-eq/i ([_] term-eq/i) (Dsubtype2 x) (Dsubtype2'' x)) + %<- ({x} subtype-fun (Dsubtype2'' x) (Dsubtype2' x) (%the ({y} term-eq (M2 x y) (M2' x y)) (Deq2 x))) + %<- ({z} {x} term-resp-term ([m] M2 x (app z m)) (Deq1 x) (%the (term-eq (M2 x (app z (M1 x))) (M2 x (app z (M1' x)))) (Deq2' z x))) + %<- ({z} {x} term-eq-trans (Deq2' z x) (Deq2 x (app z (M1' x))) (%the (term-eq (M2 x (app z (M1 x))) (M2' x (app z (M1' x)))) (Deq z x))) + %<- ({z} lam-resp ([x] Deq z x) (Deq' z)) %. +%scope subtype-fun %term sigma + %pi (subtype-fun (subtype/sigma Dsubtype2 Dsub Dsubtype1) (subtype/sigma Dsubtype2' Dsub' Dsubtype1') Deq) + %<- (subtype-fun Dsubtype1 Dsubtype1' (%the ({x} term-eq (M1 x) (M1' x)) Deq1)) + %<- ({x} tsub-resp ([_] tp-eq/i) (Deq1 x) tp-eq/i (Dsub x) (Dsub'' x)) + %<- ({x} tsub-fun (Dsub'' x) (Dsub' x) (DeqB2 x)) + %<- ({x} subtype-resp tp-eq/i (DeqB2 x) ([_] term-eq/i) (Dsubtype2 x) (Dsubtype2'' x)) + %<- ({x} subtype-fun (Dsubtype2'' x) (Dsubtype2' x) (%the ({y} term-eq (M2 x y) (M2' x y)) (Deq2 x))) + %<- ({z} pair-resp (Deq1 (pi1 z)) (Deq2 (pi1 z) (pi2 z)) (Deq z)) %. +%term sing subtype-fun subtype/sing subtype/sing ([_] term-eq/i) %. +%scope subtype-fun %term sing_t subtype-fun subtype/sing_t subtype/sing_t ([_] term-eq/i) %. +%worlds (var) (subtype-fun _ _ _) %. +%total D (subtype-fun _ D _) %. +%sort subtype-reg-e %. +%mode subtype-reg-e %in %in %in %out %. +%scope subtype-reg-e %term t + %pi (subtype-reg-e subtype/t (wfe/t (%the (ordered G) Dordered)) (wfe/t _) ([x] [d isvar x I] ofe/at (aofe/var (wfe/t (ordered/cons (Dbounded x d))) (lookup/hit (Dbounded x d))))) + %<- (extend-context Dordered (%the ({x} %pi (isvar x I) %-> (bounded G x)) Dbounded)) %. +%term pi + %pi (subtype-reg-e (%the (subtype (pi A1 A2) (pi B1 B2) ([f] lam ([x] M2 x (app f (M1 x))))) (subtype/pi (%the ({x} subtype (A2' x) (B2 x) ([y] M2 x y)) Dsubtype2) (%the ({x} tsub A2 (M1 x) (A2' x)) Dsub) (%the (subtype B1 A1 M1) Dsubtype1))) (wfe/pi (%the ({x} %pi (isvar x I') %-> (wfe (cons G x A1) (A2 x))) DwfA2) (%the (wfe G A1) DwfA1)) (wfe/pi (%the ({x} %pi (isvar x I) %-> (wfe (cons G x B1) (B2 x))) DwfB2) (%the (wfe G B1) DwfB1)) ([z] [f isvar z I] ofe/lam ([x] [d isvar x J] DofM2y z f x d) (DwfB1' z f))) + %<- ({x} {d isvar x I} wfe-context (DwfB2 x d) (ordered/cons (%the (bounded G x) (Dbounded x d)))) + %<- (subtype-reg-e Dsubtype1 DwfB1 DwfA1 (%the ({x} %pi (isvar x I'') %-> (ofe (cons G x B1) (M1 x) A1)) Dof1)) + %<- (tsubst-context-e Dof1 DwfA2 Dsub (%the ({x} %pi (isvar x I'') %-> (wfe (cons G x B1) (A2' x))) DwfA2')) + %<- (bump-wfe DwfA2' ([x] [d] ordered/cons (Dbounded x d)) (%the ({x} %pi (isvar x I) %-> (wfe (cons G x B1) (A2' x))) DwfA2'')) + %<- ({x} {d isvar x I} subtype-reg-e (Dsubtype2 x) (DwfA2'' x d) (DwfB2 x d) (%the ({y} %pi (isvar y J) %-> (ofe (cons (cons G x B1) y (A2' x)) (M2 x y) (B2 x))) (Dof2 x d))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-context-precedes (Dof2 x d y e) (%the (precedes x y) (DprecedesIJ x d y e))) + %<- ({x} {d isvar x J} following-var d (%the ({y} %pi (isvar y K) %-> (precedes x y)) (DprecedesJK x d))) + %<- ({z} {f isvar z I} weaken-wfe (Dbounded z f) DwfB1 _ (%the (wfe (cons G z (pi A1 A2)) B1) (DwfB1' z f))) + %<- (bounded-increase-bound-abs DprecedesIJ Dbounded (%the ({x} %pi (isvar x J) %-> (bounded G x)) Dbounded')) + %<- (bump-ofe Dof1 ([x] [d isvar x J] ordered/cons (Dbounded' x d)) (%the ({x} %pi (isvar x J) %-> (ofe (cons G x B1) (M1 x) A1)) Dof1')) + %<- ({z} {f isvar z I} {x} {d isvar x J} weakeng-ofe (append/cons append/nil) (append/cons append/nil) (ordered/cons (bounded/cons (Dbounded z f) (DprecedesIJ z f x d))) (Dof1' x d) (%the (ofe (cons (cons G z (pi A1 A2)) x B1) (M1 x) A1) (Dof1'' z f x d))) + %<- (precedes-trans-abs DprecedesIJ DprecedesJK (%the ({z} %pi (isvar z I) %-> ({y} %pi (isvar y K) %-> (precedes z y))) DprecedesIK)) + %<- (bump2-ofe Dof2 ([x] [d isvar x I] [y] [e isvar y K] ordered/cons (bounded/cons (Dbounded x d) (DprecedesIK x d y e))) ([x] [d isvar x J] [y] [e isvar y K] ordered/cons (bounded/cons (Dbounded' x d) (DprecedesJK x d y e))) (%the ({x} + %pi (isvar x J) + %-> ({y} %pi (isvar y K) %-> (ofe (cons (cons G x B1) y (A2' x)) (M2 x y) (B2 x)))) Dof2')) + %<- ({z} {f isvar z I} {x} {d isvar x J} {y} {e isvar y K} weakeng-ofe (append/cons (append/cons append/nil)) (append/cons (append/cons append/nil)) (ordered/cons (bounded/cons (bounded/cons (Dbounded z f) (DprecedesIJ z f x d)) (DprecedesJK x d y e))) (Dof2' x d y e) (%the (ofe (cons (cons (cons G z (pi A1 A2)) x B1) y (A2' x)) (M2 x y) (B2 x)) (Dof2'' z f x d y e))) + %<- ({z} {f isvar z I} {x} {d isvar x J} weaken-wfe' (append/cons (append/cons append/nil)) (ordered/cons (bounded/cons (Dbounded z f) (DprecedesIJ z f x d))) (wfe/pi DwfA2 DwfA1) (%the (wfe (cons (cons G z (pi A1 A2)) x B1) (pi A1 A2)) (DwfA z f x d))) + %<- (bump-wfe DwfA2'' ([x] [d isvar x J] ordered/cons (Dbounded' x d)) (%the ({x} %pi (isvar x J) %-> (wfe (cons G x B1) (A2' x))) DwfA2''')) + %<- ({z} {f isvar z I} {x} {d isvar x J} weakeng-wfe (append/cons append/nil) (append/cons append/nil) (ordered/cons (bounded/cons (Dbounded z f) (DprecedesIJ z f x d))) (DwfA2''' x d) (%the (wfe (cons (cons G z (pi A1 A2)) x B1) (A2' x)) (DwfA2'''' z f x d))) + %<- ({z} {f isvar z I} {x} {d isvar x J} ssubst-e ([_] append/nil) append/nil ([y] [e isvar y K] Dof2'' z f x d y e) (%the (aofe _ (app z (M1 x)) (A2' x)) (aofe/app (DwfA2'''' z f x d) (Dsub x) (Dof1'' z f x d) (aofe/var (DwfA z f x d) (lookup/miss (lookup/hit (Dbounded z f)) (bounded/cons (Dbounded z f) (DprecedesIJ z f x d)))))) (%the (ofe (cons (cons G z (pi A1 A2)) x B1) (M2 x (app z (M1 x))) (B2 x)) (DofM2y z f x d))) %. +%scope subtype-reg-e %term sigma + %pi (subtype-reg-e (%the (subtype (sigma A1 A2) (sigma B1 B2) ([p] pair (M1 (pi1 p)) (M2 (pi1 p) (pi2 p)))) (subtype/sigma (%the ({x} subtype (A2 x) (B2' x) ([y] M2 x y)) Dsubtype2) (%the ({x} tsub B2 (M1 x) (B2' x)) Dsub) (%the (subtype A1 B1 M1) Dsubtype1))) (wfe/sigma (%the ({x} %pi (isvar x I) %-> (wfe (cons G x A1) (A2 x))) DwfA2) (%the (wfe G A1) DwfA1)) (wfe/sigma (%the ({x} %pi (isvar x I') %-> (wfe (cons G x B1) (B2 x))) DwfB2) (%the (wfe G B1) DwfB1)) ([z] [f isvar z I] ofe/pair ([x] [d isvar x J] DwfB2'''' z f x d) (Dof2'''' z f) (Dsub (pi1 z)) (Dof1''' z f))) + %<- ({x} {d isvar x I} wfe-context (DwfA2 x d) (ordered/cons (%the (bounded G x) (Dbounded x d)))) + %<- (subtype-reg-e Dsubtype1 DwfA1 DwfB1 (%the ({x} %pi (isvar x I'') %-> (ofe (cons G x A1) (M1 x) B1)) Dof1)) + %<- (tsubst-context-e Dof1 DwfB2 Dsub (%the ({x} %pi (isvar x I'') %-> (wfe (cons G x A1) (B2' x))) DwfB2')) + %<- (bump-wfe DwfB2' ([x] [d] ordered/cons (Dbounded x d)) (%the ({x} %pi (isvar x I) %-> (wfe (cons G x A1) (B2' x))) DwfB2'')) + %<- ({x} {d isvar x I} subtype-reg-e (Dsubtype2 x) (DwfA2 x d) (DwfB2'' x d) (%the ({y} %pi (isvar y J) %-> (ofe (cons (cons G x A1) y (A2 x)) (M2 x y) (B2' x))) (Dof2 x d))) + %<- ({x} {d isvar x I} {y} {e isvar y J} ofe-context-precedes (Dof2 x d y e) (%the (precedes x y) (DprecedesIJ x d y e))) + %<- ({x} {d isvar x J} following-var d (%the ({y} %pi (isvar y K) %-> (precedes x y)) (DprecedesJK x d))) + %<- ({z} {f isvar z I} weaken-wfe (Dbounded z f) DwfB1 _ (%the (wfe (cons G z (pi A1 A2)) B1) (DwfB1' z f))) + %<- (bounded-increase-bound-abs DprecedesIJ Dbounded (%the ({x} %pi (isvar x J) %-> (bounded G x)) Dbounded')) + %<- (bump-ofe Dof1 ([x] [d isvar x J] ordered/cons (Dbounded' x d)) (%the ({x} %pi (isvar x J) %-> (ofe (cons G x A1) (M1 x) B1)) Dof1')) + %<- ({z} {f isvar z I} {x} {d isvar x J} weakeng-ofe (append/cons append/nil) (append/cons append/nil) (ordered/cons (bounded/cons (Dbounded z f) (DprecedesIJ z f x d))) (Dof1' x d) (%the (ofe (cons (cons G z (sigma A1 A2)) x A1) (M1 x) B1) (Dof1'' z f x d))) + %<- (precedes-trans-abs DprecedesIJ DprecedesJK (%the ({z} %pi (isvar z I) %-> ({y} %pi (isvar y K) %-> (precedes z y))) DprecedesIK)) + %<- (bump2-ofe Dof2 ([x] [d isvar x I] [y] [e isvar y K] ordered/cons (bounded/cons (Dbounded x d) (DprecedesIK x d y e))) ([x] [d isvar x J] [y] [e isvar y K] ordered/cons (bounded/cons (Dbounded' x d) (DprecedesJK x d y e))) (%the ({x} + %pi (isvar x J) + %-> ({y} %pi (isvar y K) %-> (ofe (cons (cons G x A1) y (A2 x)) (M2 x y) (B2' x)))) Dof2')) + %<- ({z} {f isvar z I} {x} {d isvar x J} {y} {e isvar y K} weakeng-ofe (append/cons (append/cons append/nil)) (append/cons (append/cons append/nil)) (ordered/cons (bounded/cons (bounded/cons (Dbounded z f) (DprecedesIJ z f x d)) (DprecedesJK x d y e))) (Dof2' x d y e) (%the (ofe (cons (cons (cons G z (sigma A1 A2)) x A1) y (A2 x)) (M2 x y) (B2' x)) (Dof2'' z f x d y e))) + %<- ({z} {f isvar z I} weaken-wfe (Dbounded z f) (wfe/sigma DwfA2 DwfA1) _ (%the (wfe (cons G z (sigma A1 A2)) (sigma A1 A2)) (DwfA z f))) + %<- ({z} {f isvar z I} ssubst-e ([_] append/nil) append/nil ([x] [d isvar x J] Dof1'' z f x d) (aofe/pi1 (aofe/var (DwfA z f) (lookup/hit (Dbounded z f)))) (%the (ofe (cons G z (sigma A1 A2)) (M1 (pi1 z)) B1) (Dof1''' z f))) + %<- ({z} {f isvar z I} {y} {e isvar y K} ssubst-e ([_] append/cons append/nil) (append/cons append/nil) ([x] [d isvar x J] Dof2'' z f x d y e) (aofe/pi1 (aofe/var (DwfA z f) (lookup/hit (Dbounded z f)))) (%the (ofe (cons (cons G z (sigma A1 A2)) y (A2 (pi1 z))) (M2 (pi1 z) y) (B2' (pi1 z))) (Dof2''' z f y e))) + %<- ({z} {f isvar z I} ssubst-e ([_] append/nil) append/nil ([y] [e isvar y K] Dof2''' z f y e) (aofe/pi2 (aofe/var (DwfA z f) (lookup/hit (Dbounded z f)))) (%the (ofe (cons G z (sigma A1 A2)) (M2 (pi1 z) (pi2 z)) (B2' (pi1 z))) (Dof2'''' z f))) + %<- (bump-wfe DwfB2 ([x] [d isvar x J] ordered/cons (Dbounded' x d)) (%the ({x} %pi (isvar x J) %-> (wfe (cons G x B1) (B2 x))) DwfB2''')) + %<- ({z} {f isvar z I} {x} {d isvar x J} weakeng-wfe (append/cons append/nil) (append/cons append/nil) (ordered/cons (bounded/cons (Dbounded z f) (DprecedesIJ z f x d))) (DwfB2''' x d) (%the (wfe (cons (cons G z (sigma A1 A2)) x B1) (B2 x)) (DwfB2'''' z f x d))) %. +%term singt + %pi (subtype-reg-e subtype/sing_t (wfe/sing (%the (aofe G R t) Daof)) (wfe/t _) ([x] [d] ofe/at (Daof' x d))) + %<- (aofe-context Daof (%the (ordered G) Dordered)) + %<- (extend-context Dordered (%the ({x} %pi (isvar x I) %-> (bounded G x)) Dbounded)) + %<- ({x} {d isvar x I} weaken-aofe (Dbounded x d) Daof _ (%the (aofe (cons G x (sing R)) R t) (Daof' x d))) %. +%scope subtype-reg-e %term sing + %pi (subtype-reg-e subtype/sing (wfe/sing (%the (aofe G R t) Daof)) (wfe/sing _) ([x] [d] ofe/sing (Daof' x d))) + %<- (aofe-context Daof (%the (ordered G) Dordered)) + %<- (extend-context Dordered (%the ({x} %pi (isvar x I) %-> (bounded G x)) Dbounded)) + %<- ({x} {d isvar x I} weaken-aofe (Dbounded x d) Daof _ (%the (aofe (cons G x (sing R)) R t) (Daof' x d))) %. +%worlds (bind ovar var) (subtype-reg-e _ _ _ _) %. +%total D (subtype-reg-e D _ _ _) %. +%sort subtype-reg-e-bump %. +%mode subtype-reg-e-bump %in %in %in %in %out %. +%term _ + %pi (subtype-reg-e-bump Dsubtype DwfA DwfB Dbound Dof') + %<- (subtype-reg-e Dsubtype DwfA DwfB Dof) + %<- (bump-ofe Dof ([x] [d] ordered/cons (Dbound x d)) Dof') %. +%worlds (var bind ovar) (subtype-reg-e-bump _ _ _ _ _) %. +%total {} (subtype-reg-e-bump _ _ _ _ _) %. +%sort subtype-reg-e' %. +%mode subtype-reg-e' %in %in %in %out %. +%term _ + %pi (subtype-reg-e' Dsubtype DwfA DwfB DofM') + %<- (subtype-reg-e Dsubtype DwfA DwfB (%the ({x} %pi (isvar x I) %-> (ofe (cons G x A) M B)) DofM)) + %<- (inhabitation-e DwfA (%the (ofe G N A) DofN)) + %<- (sub-absent M N (%the (sub ([_] M) N M) DsubM)) + %<- (tsub-absent B N (%the (tsub ([_] B) N B) DsubB)) + %<- (subst-e ([_] append/nil) csub/base DofN DsubM DsubB DofM (%the (ofe G M B) DofM')) %. +%worlds (bind ovar var) (subtype-reg-e' _ _ _ _) %. +%total {} (subtype-reg-e' _ _ _ _) %. +%sort subtype-reg %. +%mode subtype-reg %in %in %in %out %. +%scope subtype-reg %term t subtype-reg subtype/t wf/t wf/t ([x] [d] of/at (aof/var wf/t d)) %. +%term pi + %pi (subtype-reg (%the (subtype (pi A1 A2) (pi B1 B2) ([f] lam ([x] M2 x (app f (M1 x))))) (subtype/pi (%the ({x} subtype (A2' x) (B2 x) ([y] M2 x y)) Dsubtype2) (%the ({x} tsub A2 (M1 x) (A2' x)) Dsub) (%the (subtype B1 A1 M1) Dsubtype1))) (wf/pi (%the ({x} %pi (vof x A1) %-> (wf (A2 x))) DwfA2) (%the (wf A1) DwfA1)) (wf/pi (%the ({x} %pi (vof x B1) %-> (wf (B2 x))) DwfB2) (%the (wf B1) DwfB1)) ([z] [f vof z (pi A1 A2)] of/lam ([x] [d vof x B1] DofM2y z f x d) DwfB1)) + %<- (subtype-reg Dsubtype1 DwfB1 DwfA1 (%the ({x} %pi (vof x B1) %-> (of (M1 x) A1)) Dof1)) + %<- ({x} {d vof x B1} tsubst (Dsub x) DwfA2 (Dof1 x d) (%the (wf (A2' x)) (DwfA2' x d))) + %<- ({x} {d vof x B1} subtype-reg (Dsubtype2 x) (DwfA2' x d) (DwfB2 x d) (%the ({y} %pi (vof y (A2' x)) %-> (of (M2 x y) (B2 x))) (Dof2 x d))) + %<- ({z} {f vof z (pi A1 A2)} {x} {d vof x B1} ssubst (Dof2 x d) (aof/app (DwfA2' x d) (Dsub x) (Dof1 x d) (aof/var (wf/pi DwfA2 DwfA1) f)) (%the (of (M2 x (app z (M1 x))) (B2 x)) (DofM2y z f x d))) %. +%scope subtype-reg %term sigma + %pi (subtype-reg (%the (subtype (sigma A1 A2) (sigma B1 B2) ([p] pair (M1 (pi1 p)) (M2 (pi1 p) (pi2 p)))) (subtype/sigma (%the ({x} subtype (A2 x) (B2' x) ([y] M2 x y)) Dsubtype2) (%the ({x} tsub B2 (M1 x) (B2' x)) Dsub) (%the (subtype A1 B1 M1) Dsubtype1))) (wf/sigma (%the ({x} %pi (vof x A1) %-> (wf (A2 x))) DwfA2) (%the (wf A1) DwfA1)) (wf/sigma (%the ({x} %pi (vof x B1) %-> (wf (B2 x))) DwfB2) (%the (wf B1) DwfB1)) ([z] [f vof z (sigma A1 A2)] of/pair DwfB2 (DofM2xy z f) (Dsub (pi1 z)) (DofM1x z f))) + %<- (subtype-reg Dsubtype1 DwfA1 DwfB1 (%the ({x} %pi (vof x A1) %-> (of (M1 x) B1)) Dof1)) + %<- ({x} {d vof x A1} tsubst (Dsub x) DwfB2 (Dof1 x d) (%the (wf (B2' x)) (DwfB2' x d))) + %<- ({x} {d vof x A1} subtype-reg (Dsubtype2 x) (DwfA2 x d) (DwfB2' x d) (%the ({y} %pi (vof y (A2 x)) %-> (of (M2 x y) (B2' x))) (Dof2 x d))) + %<- ({z} {f vof z (sigma A1 A2)} ssubst Dof1 (aof/pi1 (aof/var (wf/sigma DwfA2 DwfA1) f)) (%the (of (M1 (pi1 z)) B1) (DofM1x z f))) + %<- ({x} {y} {e vof y (A2 x)} ssubst-gen ([d vof x A1] Dof2 x d y e) ([d aof x A1] %the (of (M2 x y) (B2' x)) (Dof2' x d y e))) + %<- ({z} {f vof z (sigma A1 A2)} ssubst ([y] [e] Dof2' (pi1 z) (aof/pi1 (aof/var (wf/sigma DwfA2 DwfA1) f)) y e) (aof/pi2 (aof/var (wf/sigma DwfA2 DwfA1) f)) (DofM2xy z f)) %. +%term singt subtype-reg subtype/sing_t (wf/sing (%the (aof R t) Daof)) wf/t ([x] [d] of/at Daof) %. +%scope subtype-reg %term sing subtype-reg subtype/sing (wf/sing (%the (aof R t) Daof)) (wf/sing _) ([x] [d] of/sing Daof) %. +%worlds (bind) (subtype-reg _ _ _ _) %. +%total D (subtype-reg D _ _ _) %. +%sort subtype-reg' %. +%mode subtype-reg' %in %in %in %out %. +%term _ + %pi (subtype-reg' (%the (subtype A B ([_] M)) Dsubtype) (%the (wf A) DwfA) (%the (wf B) DwfB) DofM') + %<- (subtype-reg Dsubtype DwfA DwfB (%the ({x} %pi (vof x A) %-> (of M B)) DofM)) + %<- (inhabitation DwfA (%the (of N A) DofN)) + %<- (sub-absent M N (%the (sub ([_] M) N M) DsubM)) + %<- (tsub-absent B N (%the (tsub ([_] B) N B) DsubB)) + %<- (subst DsubM DsubB DofM DofN (%the (of M B) DofM')) %. +%worlds (bind) (subtype-reg' _ _ _ _) %. +%total {} (subtype-reg' _ _ _ _) %. +%%%%% Subtyping Commutes with Substitution %%%%% +%sort subtype-sub-e {x} %. +%mode subtype-sub-e %in %in %in %in %in %in %in %in %out %out %. +%scope subtype-sub-e %term t subtype-sub-e _ _ _ _ _ ([x] subtype/t) tsub/t tsub/t ([y] sub/aa aasub/closed) subtype/t %. +%term pi + %pi (subtype-sub-e (%the ({x} append (cons G1 x C) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) (%the (ofe G1 M C) DofM) ([x] [d isvar x I] wfe/pi (%the ({y} %pi (isvar y J1) %-> (wfe (cons (G x) y (A1 x)) (A2 x y))) (DwfA2 x d)) (%the (wfe (G x) (A1 x)) (DwfA1 x d))) ([x] [d isvar x I] wfe/pi (%the ({y} %pi (isvar y J2) %-> (wfe (cons (G x) y (B1 x)) (B2 x y))) (DwfB2 x d)) (%the (wfe (G x) (B1 x)) (DwfB1 x d))) ([x] subtype/pi (%the ({z} subtype (A2y x z) (B2 x z) ([w] O2 x z w)) (Dsubtype2 x)) (%the ({z} tsub ([y] A2 x y) (O1 x z) (A2y x z)) (DsubA2y x)) (%the (subtype (B1 x) (A1 x) ([y] O1 x y)) (Dsubtype1 x))) (tsub/pi (%the ({y} tsub ([x] A2 x y) M (A2x y)) DsubA2x) (%the (tsub A1 M A1x) DsubA1x)) (tsub/pi (%the ({y} tsub ([x] B2 x y) M (B2x y)) DsubB2x) (%the (tsub B1 M B1x) DsubB1x)) ([v] sub/lam ([z] DsubO2wx' z v)) (subtype/pi Dsubtype2'' DsubA2xy Dsubtype1')) + %<- (subtype-sub-e Dappend Dcsub DofM DwfB1 DwfA1 Dsubtype1 DsubB1x DsubA1x (%the ({z} sub ([x] O1 x z) M (O1x z)) DsubO1x) (%the (subtype B1x A1x O1x) Dsubtype1')) + %<- ({x} {d isvar x I} subtype-reg-e (Dsubtype1 x) (DwfB1 x d) (DwfA1 x d) (%the ({z} %pi (isvar z K) %-> (ofe (cons (G x) z (B1 x)) (O1 x z) (A1 x))) (DofO1 x d))) + %<- ({x} {d isvar x I} {z} {f isvar z K} ofe-context (DofO1 x d z f) (ordered/cons (%the (bounded (G x) z) (DboundGZ x d z f)))) + %<- ({z} {f isvar z K} subst-e ([x] append/cons (Dappend x)) (csub/cons DsubB1x Dcsub) DofM (DsubO1x z) DsubA1x ([x] [d] DofO1 x d z f) (%the (ofe (cons Gx z B1x) (O1x z) A1x) (DofO1x z f))) + %<- ({y} {e isvar y J1} tsubst-e ([x] append/cons (Dappend x)) (csub/cons DsubA1x Dcsub) DofM (DsubA2x y) ([x] [d] DwfA2 x d y e) (%the (wfe (cons Gx y A1x) (A2x y)) (DwfA2x y e))) + %<- ({z} {f isvar z K} ofe-context (DofO1x z f) (ordered/cons (%the (bounded Gx z) (DboundGxZ z f)))) + %<- (weaken-wfe-insert1 DwfA2x DboundGxZ _ (%the ({z} + %pi (isvar z K) + %-> ({y} %pi (isvar y J1') %-> (wfe (cons (cons Gx z B1x) y A1x) (A2x y)))) DwfA2x')) + %<- ({z} {f isvar z K} can-tsub-e ([y] append/nil) ([y] [e isvar y J1'] DwfA2x' z f y e) (DofO1x z f) (%the (tsub ([y] A2x y) (O1x z) (A2xy z)) (DsubA2xy z))) + %<- ({x} {d isvar x I} weaken-wfe-insert1 ([y] [e isvar y J1] DwfA2 x d y e) ([z] [f isvar z K] DboundGZ x d z f) _ ([z] [f isvar z K] [y] [e isvar y J] %the (wfe (cons (cons (G x) z (B1 x)) y (A1 x)) (A2 x y)) (DwfA2' x d z f y e))) + %<- ({x} {d isvar x I} {z} {f isvar z K} tsubst-e ([y] append/nil) csub/base (DofO1 x d z f) (DsubA2y x z) ([y] [e isvar y J] DwfA2' x d z f y e) (%the (wfe (cons (G x) z (B1 x)) (A2y x z)) (DwfA2y x d z f))) + %<- ({x} {d isvar x I} bump-wfe ([y] [e isvar y J2] DwfB2 x d y e) ([z] [f isvar z K] ordered/cons (DboundGZ x d z f)) (%the ({z} %pi (isvar z K) %-> (wfe (cons (G x) z (B1 x)) (B2 x z))) (DwfB2' x d))) + %<- ({z} {f isvar z K} can-tsub-e ([x] append/cons (Dappend x)) ([x] [d isvar x I] DwfA2y x d z f) DofM (%the (tsub ([x] A2y x z) M (A2yx z)) (DsubA2yx z))) + %<- ({z} {f isvar z K} subtype-sub-e ([x] append/cons (Dappend x)) (csub/cons DsubB1x Dcsub) DofM ([x] [d isvar x I] DwfA2y x d z f) ([x] [d isvar x I] DwfB2' x d z f) ([x] Dsubtype2 x z) (DsubA2yx z) (DsubB2x z) ([w] %the (sub ([x] O2 x z w) M (O2x z w)) (DsubO2x z w)) (%the (subtype (A2yx z) (B2x z) ([w] O2x z w)) (Dsubtype2' z))) + %<- ({z} {f isvar z K} tsub-permute-e ([x] append/cons (Dappend x)) DofM ([x] [d isvar x I] DofO1 x d z f) ([x] [d isvar x I] [y] [e isvar y J] DwfA2' x d z f y e) DsubA2x (DsubO1x z) (DsubA2xy z) ([x] DsubA2y x z) (DsubA2yx z) (%the (tp-eq (A2xy z) (A2yx z)) (DeqA2 z))) + %<- ({z} tp-eq-symm (DeqA2 z) (%the (tp-eq (A2yx z) (A2xy z)) (DeqA2' z))) + %<- ({z} subtype-resp (DeqA2' z) tp-eq/i ([_] term-eq/i) (Dsubtype2' z) (%the (subtype (A2xy z) (B2x z) ([w] O2x z w)) (Dsubtype2'' z))) + %<- ({x} {d isvar x I} {z} {f isvar z K} subtype-reg-e (Dsubtype2 x z) (DwfA2y x d z f) (DwfB2' x d z f) (%the ({w} + %pi (isvar w L) + %-> (ofe (cons (cons (G x) z (B1 x)) w (A2y x z)) (O2 x z w) (B2 x z))) (DofO2 x d z f))) + %<- ({x} {d isvar x I} {z} {f isvar z K} {v} {h isvar v L} ofe-context (DofO2 x d z f v h) (ordered/cons (%the (bounded (cons (G x) z (B1 x)) v) (DboundedGZV x d z f v h)))) + %<- ({x} {d isvar x I} {z} {f isvar z K} weaken-ofe-insert1 (DofO2 x d z f) (DboundedGZV x d z f) _ (%the ({v} + %pi (isvar v L) + %-> ({w} + %pi (isvar w L') + %-> (ofe (cons (cons (cons (G x) z (B1 x)) v (pi (A1 x) ([y] A2 x y))) w (A2y x z)) (O2 x z w) (B2 x z)))) (DofO2' x d z f))) + %<- ({x} {d isvar x I} {z} {f isvar z K} {v} {h isvar v L} weaken-ofe (DboundedGZV x d z f v h) (DofO1 x d z f) _ (%the (ofe (cons (cons (G x) z (B1 x)) v (pi (A1 x) ([y] A2 x y))) (O1 x z) (A1 x)) (DofO1' x d z f v h))) + %<- ({x} {d isvar x I} {z} {f isvar z K} {v} {h isvar v L} weaken-wfe (DboundedGZV x d z f v h) (DwfA2y x d z f) _ (%the (wfe (cons (cons (G x) z (B1 x)) v (pi (A1 x) ([y] A2 x y))) (A2y x z)) (DwfA2y' x d z f v h))) + %<- ({x} {d isvar x I} {z} {f isvar z K} {v} {h isvar v L} weaken-wfe' (append/cons (append/cons append/nil)) (ordered/cons (DboundedGZV x d z f v h)) (wfe/pi (DwfA2 x d) (DwfA1 x d)) (%the (wfe (cons (cons (G x) z (B1 x)) v (pi (A1 x) ([y] A2 x y))) (pi (A1 x) ([y] A2 x y))) (DwfA1->A2 x d z f v h))) + %<- ({x} {d isvar x I} {z} {f isvar z K} {v} {h isvar v L} ssubst-e ([_] append/nil) append/nil ([w] [g isvar w L'] DofO2' x d z f v h w g) (%the (aofe (cons (cons (G x) z (B1 x)) v (pi (A1 x) ([y] A2 x y))) (app v (O1 x z)) (A2y x z)) (aofe/app (DwfA2y' x d z f v h) (DsubA2y x z) (DofO1' x d z f v h) (aofe/var (DwfA1->A2 x d z f v h) (lookup/hit (DboundedGZV x d z f v h))))) (%the (ofe (cons (cons (G x) z (B1 x)) v (pi (A1 x) ([y] A2 x y))) (O2 x z (app v (O1 x z))) (B2 x z)) (DofO2w x d z f v h))) + %<- ({z} {f isvar z K} {v} {h isvar v L} can-sub-e ([x] append/cons (append/cons (Dappend x))) ([x] [d isvar x I] DofO2w x d z f v h) DofM (%the (sub ([x] O2 x z (app v (O1 x z))) M (O2wx z v)) (DsubO2wx z v))) + %<- ({z} {f isvar z K} {v} {h isvar v L} ssub-aa-permute-e ([x] append/cons (append/cons (Dappend x))) DofM ([x] [d isvar x I] %the (aofe (cons (cons (G x) z (B1 x)) v (pi (A1 x) ([y] A2 x y))) (app v (O1 x z)) (A2y x z)) (aofe/app (DwfA2y' x d z f v h) (DsubA2y x z) (DofO1' x d z f v h) (aofe/var (DwfA1->A2 x d z f v h) (lookup/hit (DboundedGZV x d z f v h))))) ([x] [d isvar x I] [w] [g isvar w L'] DofO2' x d z f v h w g) ([w] DsubO2x z w) (aasub/app (DsubO1x z) aasub/closed) (DsubO2wx z v) (%the (term-eq (O2x z (app v (O1x z))) (O2wx z v)) (DeqO2 z v))) + %<- ({z} {v} term-eq-symm (DeqO2 z v) (%the (term-eq (O2wx z v) (O2x z (app v (O1x z)))) (DeqO2' z v))) + %<- ({z} {v} sub-resp ([_] term-eq/i) term-eq/i (DeqO2' z v) (DsubO2wx z v) (%the (sub ([x] O2 x z (app v (O1 x z))) M (O2x z (app v (O1x z)))) (DsubO2wx' z v))) %. +%scope subtype-sub-e %term sigma + %pi (subtype-sub-e (%the ({x} append (cons G1 x C) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) (%the (ofe G1 M C) DofM) ([x] [d isvar x I] wfe/sigma (%the ({y} %pi (isvar y J1) %-> (wfe (cons (G x) y (A1 x)) (A2 x y))) (DwfA2 x d)) (%the (wfe (G x) (A1 x)) (DwfA1 x d))) ([x] [d isvar x I] wfe/sigma (%the ({y} %pi (isvar y J2) %-> (wfe (cons (G x) y (B1 x)) (B2 x y))) (DwfB2 x d)) (%the (wfe (G x) (B1 x)) (DwfB1 x d))) ([x] subtype/sigma (%the ({z} subtype (A2 x z) (B2y x z) ([w] O2 x z w)) (Dsubtype2 x)) (%the ({z} tsub ([y] B2 x y) (O1 x z) (B2y x z)) (DsubB2y x)) (%the (subtype (A1 x) (B1 x) ([y] O1 x y)) (Dsubtype1 x))) (tsub/sigma (%the ({y} tsub ([x] A2 x y) M (A2x y)) DsubA2x) (%the (tsub A1 M A1x) DsubA1x)) (tsub/sigma (%the ({y} tsub ([x] B2 x y) M (B2x y)) DsubB2x) (%the (tsub B1 M B1x) DsubB1x)) ([v] sub/pair (DsubO2x (pi1 v) (pi2 v)) (DsubO1x (pi1 v))) (subtype/sigma Dsubtype2'' DsubB2xy Dsubtype1')) + %<- (subtype-sub-e Dappend Dcsub DofM DwfA1 DwfB1 Dsubtype1 DsubA1x DsubB1x (%the ({z} sub ([x] O1 x z) M (O1x z)) DsubO1x) (%the (subtype A1x B1x O1x) Dsubtype1')) + %<- ({x} {d isvar x I} subtype-reg-e (Dsubtype1 x) (DwfA1 x d) (DwfB1 x d) (%the ({z} %pi (isvar z K) %-> (ofe (cons (G x) z (A1 x)) (O1 x z) (B1 x))) (DofO1 x d))) + %<- ({x} {d isvar x I} {z} {f isvar z K} ofe-context (DofO1 x d z f) (ordered/cons (%the (bounded (G x) z) (DboundGZ x d z f)))) + %<- ({z} {f isvar z K} subst-e ([x] append/cons (Dappend x)) (csub/cons DsubA1x Dcsub) DofM (DsubO1x z) DsubB1x ([x] [d] DofO1 x d z f) (%the (ofe (cons Gx z A1x) (O1x z) B1x) (DofO1x z f))) + %<- ({y} {e isvar y J2} tsubst-e ([x] append/cons (Dappend x)) (csub/cons DsubB1x Dcsub) DofM (DsubB2x y) ([x] [d] DwfB2 x d y e) (%the (wfe (cons Gx y B1x) (B2x y)) (DwfB2x y e))) + %<- ({z} {f isvar z K} ofe-context (DofO1x z f) (ordered/cons (%the (bounded Gx z) (DboundGxZ z f)))) + %<- (weaken-wfe-insert1 DwfB2x DboundGxZ _ (%the ({z} + %pi (isvar z K) + %-> ({y} %pi (isvar y J2') %-> (wfe (cons (cons Gx z A1x) y B1x) (B2x y)))) DwfB2x')) + %<- ({z} {f isvar z K} can-tsub-e ([y] append/nil) ([y] [e isvar y J2'] DwfB2x' z f y e) (DofO1x z f) (%the (tsub ([y] B2x y) (O1x z) (B2xy z)) (DsubB2xy z))) + %<- ({x} {d isvar x I} weaken-wfe-insert1 ([y] [e isvar y J2] DwfB2 x d y e) ([z] [f isvar z K] DboundGZ x d z f) _ ([z] [f isvar z K] [y] [e isvar y J] %the (wfe (cons (cons (G x) z (A1 x)) y (B1 x)) (B2 x y)) (DwfB2' x d z f y e))) + %<- ({x} {d isvar x I} {z} {f isvar z K} tsubst-e ([y] append/nil) csub/base (DofO1 x d z f) (DsubB2y x z) ([y] [e isvar y J] DwfB2' x d z f y e) (%the (wfe (cons (G x) z (A1 x)) (B2y x z)) (DwfB2y x d z f))) + %<- ({x} {d isvar x I} bump-wfe ([y] [e isvar y J1] DwfA2 x d y e) ([z] [f isvar z K] ordered/cons (DboundGZ x d z f)) (%the ({z} %pi (isvar z K) %-> (wfe (cons (G x) z (A1 x)) (A2 x z))) (DwfA2' x d))) + %<- ({z} {f isvar z K} can-tsub-e ([x] append/cons (Dappend x)) ([x] [d isvar x I] DwfB2y x d z f) DofM (%the (tsub ([x] B2y x z) M (B2yx z)) (DsubB2yx z))) + %<- ({z} {f isvar z K} subtype-sub-e ([x] append/cons (Dappend x)) (csub/cons DsubA1x Dcsub) DofM ([x] [d isvar x I] DwfA2' x d z f) ([x] [d isvar x I] DwfB2y x d z f) ([x] Dsubtype2 x z) (DsubA2x z) (DsubB2yx z) ([w] %the (sub ([x] O2 x z w) M (O2x z w)) (DsubO2x z w)) (%the (subtype (A2x z) (B2yx z) ([w] O2x z w)) (Dsubtype2' z))) + %<- ({z} {f isvar z K} tsub-permute-e ([x] append/cons (Dappend x)) DofM ([x] [d isvar x I] DofO1 x d z f) ([x] [d isvar x I] [y] [e isvar y J] DwfB2' x d z f y e) DsubB2x (DsubO1x z) (DsubB2xy z) ([x] DsubB2y x z) (DsubB2yx z) (%the (tp-eq (B2xy z) (B2yx z)) (DeqB2 z))) + %<- ({z} tp-eq-symm (DeqB2 z) (%the (tp-eq (B2yx z) (B2xy z)) (DeqB2' z))) + %<- ({z} subtype-resp tp-eq/i (DeqB2' z) ([_] term-eq/i) (Dsubtype2' z) (%the (subtype (A2x z) (B2xy z) ([w] O2x z w)) (Dsubtype2'' z))) %. +%term singta subtype-sub-e _ _ _ _ _ ([x] subtype/sing_t) (tsub/singa (%the (aasub R M Rx) Daasub)) tsub/t ([y] sub/aa Daasub) subtype/sing_t %. +%scope subtype-sub-e %term singto subtype-sub-e _ _ _ _ _ ([x] subtype/sing_t) (tsub/singo (%the (aosub R M (at Rx)) Daosub)) tsub/t ([y] sub/ao Daosub) subtype/sing_t %. +%term singa + %pi (subtype-sub-e _ _ _ _ _ ([x] subtype/sing) (tsub/singa (%the (aasub R M Rx) Daasub)) (tsub/singa (%the (aasub R M Rx') Daasub')) ([y] sub/aa Daasub) Dsubtype) + %<- (aasub-fun Daasub Daasub' (%the (atom-eq Rx Rx') Deq)) + %<- (tp-resp-atom sing Deq (%the (tp-eq (sing Rx) (sing Rx')) Deq')) + %<- (subtype-resp tp-eq/i Deq' ([_] term-eq/i) subtype/sing (%the (subtype (sing Rx) (sing Rx') ([_] at Rx)) Dsubtype)) %. +%scope subtype-sub-e %term singo + %pi (subtype-sub-e _ _ _ _ _ ([x] subtype/sing) (tsub/singo (%the (aosub R M (at Rx)) Daosub)) (tsub/singo (%the (aosub R M (at Rx')) Daosub')) ([y] sub/ao Daosub) Dsubtype) + %<- (aosub-fun Daosub Daosub' (%the (term-eq (at Rx) (at Rx')) Deq)) + %<- (term-eq-cdr-at Deq (%the (atom-eq Rx Rx') Deq')) + %<- (tp-resp-atom sing Deq' (%the (tp-eq (sing Rx) (sing Rx')) Deq'')) + %<- (subtype-resp tp-eq/i Deq'' ([_] term-eq/i) subtype/sing (%the (subtype (sing Rx) (sing Rx') ([_] at Rx)) Dsubtype)) %. +%term singao + %pi (subtype-sub-e _ _ _ _ _ ([x] subtype/sing) (tsub/singa Daasub) (tsub/singo Daosub) ([y] sub/aa Daasub) Dsubtype) + %<- (aasub-aosub-contra Daasub Daosub Dfalse) + %<- (false-implies-subtype Dfalse Dsubtype) %. +%scope subtype-sub-e %term singoa + %pi (subtype-sub-e _ _ _ _ _ ([x] subtype/sing) (tsub/singo Daosub) (tsub/singa Daasub) ([y] sub/aa Daasub) Dsubtype) + %<- (aasub-aosub-contra Daasub Daosub Dfalse) + %<- (false-implies-subtype Dfalse Dsubtype) %. +%worlds (bind ovar var) (subtype-sub-e _ _ _ _ _ _ _ _ _ _) %. +%total D (subtype-sub-e _ _ _ _ _ D _ _ _ _) %. +%sort subtype-sub {x} %. +%mode subtype-sub %in %in %in %in %in %in %out %out %. +%term _ + %pi (subtype-sub Dsubtype DwfA DwfB DofM DsubAx DsubBx DsubOx Dsubtype') + %<- (of-to-ofe DofM (%the (ofe nil M C) DofeM)) + %<- (wf1-to-wfe 0 DwfA (%the ({x} %pi (isvar x 0) %-> (wfe (cons nil x C) (A x))) DwfeA)) + %<- (wf1-to-wfe 0 DwfB (%the ({x} %pi (isvar x 0) %-> (wfe (cons nil x C) (B x))) DwfeB)) + %<- (subtype-sub-e ([x] append/nil) csub/base DofeM DwfeA DwfeB Dsubtype DsubAx DsubBx DsubOx Dsubtype') %. +%worlds (bind) (subtype-sub _ _ _ _ _ _ _ _) %. +%total {} (subtype-sub _ _ _ _ _ _ _ _) %. +%sort subtype-sub' {x} %. +%mode subtype-sub' %in %in %in %in %in %in %in %out %. +%term _ + %pi (subtype-sub' DsubtypeAB DwfA DwfB DofM DsubAx DsubBx DsubOx DsubtypeAxBx') + %<- (subtype-sub DsubtypeAB DwfA DwfB DofM DsubAx DsubBx DsubOx' DsubtypeAxBx) + %<- ({y} sub-fun (DsubOx' y) (DsubOx y) (Deq y)) + %<- (subtype-resp tp-eq/i tp-eq/i Deq DsubtypeAxBx DsubtypeAxBx') %. +%worlds (bind) (subtype-sub' _ _ _ _ _ _ _ _) %. +%total {} (subtype-sub' _ _ _ _ _ _ _ _) %. +%%%%% Reflexivity and Transitivity %%%%% +%sort subtype-refl-e %. +%mode subtype-refl-e %in %out %out %. +%term _ subtype-refl-e _ subtype/t ([x] expand/t) %. +%term _ + %pi (subtype-refl-e (wfe/pi (%the ({x} %pi (isvar x I) %-> (wfe (cons G x A) (B x))) DwfB) (%the (wfe G A) DwfA)) (subtype/pi DsubtypeB DsubB DsubtypeA) ([z] expand/pi ([x] DexpandY x (app z (X x))) DexpandX)) + %<- (subtype-refl-e DwfA (%the (subtype A A X) DsubtypeA) (%the ({x} expand x A (X x)) DexpandX)) + %<- (tsub-expand-var-e DwfA DwfB DexpandX (%the ({x} tsub B (X x) (B x)) DsubB)) + %<- ({x} {d} subtype-refl-e (DwfB x d) (%the (subtype (B x) (B x) ([y] Y x y)) (DsubtypeB x)) (%the ({y} expand y (B x) (Y x y)) (DexpandY x))) %. +%term _ + %pi (subtype-refl-e (wfe/sigma (%the ({x} %pi (isvar x I) %-> (wfe (cons G x A) (B x))) DwfB) (%the (wfe G A) DwfA)) (subtype/sigma DsubtypeB DsubB DsubtypeA) ([z] expand/sigma (DexpandY (pi1 z) (pi2 z)) (DexpandX (pi1 z)))) + %<- (subtype-refl-e DwfA (%the (subtype A A X) DsubtypeA) (%the ({x} expand x A (X x)) DexpandX)) + %<- (tsub-expand-var-e DwfA DwfB DexpandX (%the ({x} tsub B (X x) (B x)) DsubB)) + %<- ({x} {d} subtype-refl-e (DwfB x d) (%the (subtype (B x) (B x) ([y] Y x y)) (DsubtypeB x)) (%the ({y} expand y (B x) (Y x y)) (DexpandY x))) %. +%term _ subtype-refl-e _ subtype/sing ([_] expand/sing) %. +%worlds (bind ovar var) (subtype-refl-e _ _ _) %. +%total D (subtype-refl-e D _ _) %. +%sort subtype-refl %. +%mode subtype-refl %in %out %out %. +%term _ + %pi (subtype-refl Dwf Dsubtype Dexpand) + %<- (wf-to-wfe Dwf Dwfe) + %<- (subtype-refl-e Dwfe Dsubtype Dexpand) %. +%worlds (bind var ovar) (subtype-refl _ _ _) %. +%total {} (subtype-refl _ _ _) %. +%sort subtype-trans-em {T} %. +%mode subtype-trans-em %in %in %in %in %in %in %in %out %out %. +%scope subtype-trans-em %term ttt subtype-trans-em _ _ _ _ _ subtype/t subtype/t ([x] sub/ao aosub/var) subtype/t %. +%term stt subtype-trans-em _ _ _ _ _ subtype/sing_t subtype/t ([x] sub/ao aosub/var) subtype/sing_t %. +%scope subtype-trans-em %term sst subtype-trans-em _ _ _ _ _ subtype/sing subtype/sing_t ([x] sub/aa aasub/closed) subtype/sing_t %. +%term sss subtype-trans-em _ _ _ _ _ subtype/sing subtype/sing ([x] sub/aa aasub/closed) subtype/sing %. +%scope subtype-trans-em %term pi + %pi (subtype-trans-em (spi S T) (simp/pi (%the ({x} simp (B2 x) T) DsimpB) (%the (simp A2 S) DsimpA)) (wfe/pi (%the ({x} %pi (isvar x I1) %-> (wfe (cons G x A1) (B1 x))) DwfB1) (%the (wfe G A1) DwfA1)) (wfe/pi (%the ({x} %pi (isvar x I2) %-> (wfe (cons G x A2) (B2 x))) DwfB2) (%the (wfe G A2) DwfA2)) (wfe/pi (%the ({x} %pi (isvar x I3) %-> (wfe (cons G x A3) (B3 x))) DwfB3) (%the (wfe G A3) DwfA3)) (subtype/pi (%the ({x} subtype (B1x x) (B2 x) ([y] N12 x y)) DsubtypeB12) (%the ({x} tsub B1 (M21 x) (B1x x)) DsubB1x) (%the (subtype A2 A1 M21) DsubtypeA21)) (subtype/pi (%the ({x} subtype (B2x x) (B3 x) ([y] N23 x y)) DsubtypeB23) (%the ({x} tsub B2 (M32 x) (B2x x)) DsubB2x) (%the (subtype A3 A2 M32) DsubtypeA32)) ([z] sub/lam ([x] DsubLam z x)) (subtype/pi DsubtypeB13 DsubB1x3 DsubA31)) + %<- (subtype-reg-e DsubtypeA21 DwfA2 DwfA1 (%the ({x} %pi (isvar x J') %-> (ofe (cons G x A2) (M21 x) A1)) DofM21)) + %<- (subtype-reg-e DsubtypeA32 DwfA3 DwfA2 (%the ({x} %pi (isvar x J) %-> (ofe (cons G x A3) (M32 x) A2)) DofM32)) + %<- ({x} {d isvar x J} ofe-context (DofM32 x d) (ordered/cons (%the (bounded G x) (DboundJ x d)))) + %<- (bump-ofe DofM21 ([x] [d isvar x J] ordered/cons (DboundJ x d)) (%the ({x} %pi (isvar x J) %-> (ofe (cons G x A2) (M21 x) A1)) DofM21')) + %<- (subtype-trans-em S DsimpA DwfA3 DwfA2 DwfA1 DsubtypeA32 DsubtypeA21 (%the ({x} sub M21 (M32 x) (M31 x)) DsubM31) (%the (subtype A3 A1 M31) DsubA31)) + %<- ({x} tsub-absent A1 (M32 x) (%the (tsub ([_] A1) (M32 x) A1) (DsubA1abs x))) + %<- (subst-context-e DofM32 DofM21 DsubM31 DsubA1abs (%the ({x} %pi (isvar x J) %-> (ofe (cons G x A3) (M31 x) A1)) DofM31)) + %<- (can-tsub-context-e DofM31 DwfB1 (%the ({x} tsub B1 (M31 x) (B1x3 x)) DsubB1x3)) + %<- (tsubst-context-e DofM21' DwfB1 DsubB1x (%the ({x} %pi (isvar x J) %-> (wfe (cons G x A2) (B1x x))) DwfB1x)) + %<- (weaken-wfe-insert1 DwfB1x DboundJ _ (%the ({x} + %pi (isvar x J) + %-> ({y} %pi (isvar y K) %-> (wfe (cons (cons G x A3) y A2) (B1x y)))) DwfB1x')) + %<- ({x} {d isvar x J} {y} {e isvar y K} wfe-context (DwfB1x' x d y e) (ordered/cons (bounded/cons _ (%the (precedes x y) (DprecJK x d y e))))) + %<- (weaken-wfe-insert1 DwfB2 DboundJ _ (%the ({x} + %pi (isvar x J) + %-> ({y} %pi (isvar y K') %-> (wfe (cons (cons G x A3) y A2) (B2 y)))) DwfB2')) + %<- ({x} {d} bump-wfe (DwfB2' x d) ([y] [e] ordered/cons (bounded/cons (DboundJ x d) (DprecJK x d y e))) (%the ({y} %pi (isvar y K) %-> (wfe (cons (cons G x A3) y A2) (B2 y))) (DwfB2'' x d))) + %<- (tsubst-context-permute-e DofM32 DofM21 DwfB1 DsubM31 DsubB1x DsubB1x3 (%the ({x} tsub B1x (M32 x) (B1x3 x)) DsubB1x3')) + %<- ({x} {d} subtype-sub-e ([_] append/nil) csub/base (DofM32 x d) (DwfB1x' x d) (DwfB2'' x d) DsubtypeB12 (DsubB1x3' x) (DsubB2x x) (%the ({y} sub ([x] N12 x y) (M32 x) (N12x x y)) (DsubN12x x)) (%the (subtype (B1x3 x) (B2x x) ([y] N12x x y)) (DsubtypeB12x x))) + %<- ({x} tsub-preserves-simp (DsubB2x x) DsimpB (%the (simp (B2x x) T) (DsimpB' x))) + %<- (tsubst-context-e DofM31 DwfB1 DsubB1x3 (%the ({x} %pi (isvar x J) %-> (wfe (cons G x A3) (B1x3 x))) DwfB1x3)) + %<- (tsubst-context-e DofM32 DwfB2 DsubB2x (%the ({x} %pi (isvar x J) %-> (wfe (cons G x A3) (B2x x))) DwfB2x)) + %<- (bump-wfe DwfB3 ([x] [d] ordered/cons (DboundJ x d)) (%the ({x} %pi (isvar x J) %-> (wfe (cons G x A3) (B3 x))) DwfB3')) + %<- ({x} {d isvar x J} subtype-trans-em T (DsimpB' x) (DwfB1x3 x d) (DwfB2x x d) (DwfB3' x d) (DsubtypeB12x x) (DsubtypeB23 x) (%the ({y} sub ([y] N23 x y) (N12x x y) (N13 x y)) (DsubN13 x)) (%the (subtype (B1x3 x) (B3 x) ([y] N13 x y)) (DsubtypeB13 x))) + %<- ({x} {d isvar x J} {z} {e isvar z K} weaken-ofe (bounded/cons (DboundJ x d) (DprecJK x d z e)) (DofM32 x d) _ (%the (ofe (cons (cons G x A3) z (pi A1 B1)) (M32 x) A2) (DofM32' x d z e))) + %<- (weaken-ofe-insert1 DofM21 DboundJ _ (%the ({x} + %pi (isvar x J) + %-> ({x'} %pi (isvar x' L') %-> (ofe (cons (cons G x A3) x' A2) (M21 x') A1))) DofM21'')) + %<- ({x} {d} weaken-ofe-insert1 (DofM21'' x d) ([y] [e] bounded/cons (DboundJ x d) (DprecJK x d y e)) _ (%the ({z} + %pi (isvar z K) + %-> ({x'} + %pi (isvar x' L) + %-> (ofe (cons (cons (cons G x A3) z (pi A1 B1)) x' A2) (M21 x') A1))) (DofM21''' x d))) + %<- ({x} {d} {z} {e} {x'} {f} ofe-context (DofM21''' x d z e x' f) (ordered/cons (bounded/cons _ (DprecKL x d z e x' f)))) + %<- ({x} {d} weaken-wfe-insert1-bump (DwfB1x' x d) ([y] [e] bounded/cons (DboundJ x d) (DprecJK x d y e)) ([z] [e] [x'] [f] DprecKL x d z e x' f) _ (%the ({z} + %pi (isvar z K) + %-> ({y} %pi (isvar y L) %-> (wfe (cons (cons (cons G x A3) z (pi A1 B1)) y A2) (B1x y)))) (DwfB1x'' x d))) + %<- (weaken-wfe-insert1 DwfB2 DboundJ _ (%the ({x} + %pi (isvar x J) + %-> ({y} %pi (isvar y F) %-> (wfe (cons (cons G x A3) y A2) (B2 y)))) DwfB2''')) + %<- ({x} {d} weaken-wfe-insert1-bump (DwfB2''' x d) ([z] [e] bounded/cons (DboundJ x d) (DprecJK x d z e)) ([z] [e] [x'] [f] DprecKL x d z e x' f) _ (%the ({z} + %pi (isvar z K) + %-> ({y} %pi (isvar y L) %-> (wfe (cons (cons (cons G x A3) z (pi A1 B1)) y A2) (B2 y)))) (DwfB2'''' x d))) + %<- ({x} {d} {z} {e} {x'} {f isvar x' L} subtype-reg-e (DsubtypeB12 x') (DwfB1x'' x d z e x' f) (DwfB2'''' x d z e x' f) (%the ({y} + %pi (isvar y H') + %-> (ofe (cons (cons (cons (cons G x A3) z (pi A1 B1)) x' A2) y (B1x x')) (N12 x' y) (B2 x'))) (DofN12 x d z e x' f))) + %<- ({x} {d} {z} {e} {x'} {f} ofe-context (DofM21''' x d z e x' f) (%the (ordered (cons (cons (cons G x A3) z (pi A1 B1)) x' A2)) (Dordfull x d z e x' f))) + %<- ({x} {d} {z} {e} {x'} {f} weaken-wfe' (append/cons (append/cons (append/cons append/nil))) (Dordfull x d z e x' f) (wfe/pi DwfB1 DwfA1) (%the (wfe (cons (cons (cons G x A3) z (pi A1 B1)) x' A2) (pi A1 B1)) (DwfA1B1 x d z e x' f))) + %<- ({x} {d} {z} {e} aasub-into-sub-e (DofM32' x d z e) ([x'] [f isvar x' L] %the (aofe (cons (cons (cons G x A3) z (pi A1 B1)) x' A2) (app z (M21 x')) (B1x x')) (aofe/app (DwfB1x'' x d z e x' f) (DsubB1x x') (DofM21''' x d z e x' f) (aofe/var (DwfA1B1 x d z e x' f) (lookup/miss (lookup/hit (bounded/cons (DboundJ x d) (DprecJK x d z e))) (bounded/cons (bounded/cons (DboundJ x d) (DprecJK x d z e)) (DprecKL x d z e x' f)))))) ([x'] [f isvar x' L] [y] [g isvar y H'] DofN12 x d z e x' f y g) ([y] DsubN12x x y) (%the (aasub ([x] app z (M21 x)) (M32 x) (app z (M31 x))) (aasub/app (DsubM31 x) aasub/closed)) (%the (sub ([x] N12 x (app z (M21 x))) (M32 x) (N12x x (app z (M31 x)))) (DsubN12yx z x))) + %<- ({z} {x} sub-absent (M32 x) (lam ([x] N12 x (app z (M21 x)))) (%the (sub ([y] M32 x) (lam ([x] N12 x (app z (M21 x)))) (M32 x)) (DsubM32abs z x))) + %<- ({z} {e} weaken-wfe (DboundJ z e) DwfA2 _ (%the (wfe (cons G z (pi A1 B1)) A2) (DwfA2' z e))) + %<- ({x} {d isvar x J} subtype-reg-e (DsubtypeB23 x) (DwfB2x x d) (DwfB3' x d) (%the ({y} %pi (isvar y E) %-> (ofe (cons (cons G x A3) y (B2x x)) (N23 x y) (B3 x))) (DofN23 x d))) + %<- ({x} {d isvar x J} {z} {e isvar z K} weaken-ofe (bounded/cons (DboundJ x d) (DprecJK x d z e)) (DofM32 x d) _ (%the (ofe (cons (cons G x A3) z (pi A2 B2)) (M32 x) A2) (DofM32'' x d z e))) + %<- ({x} {d isvar x J} {z} {e isvar z K} ofe-context (DofM32'' x d z e) (%the (ordered (cons (cons G x A3) z (pi A2 B2))) (Dordfull' x d z e))) + %<- ({x} {d isvar x J} {z} {e isvar z K} weaken-wfe' (append/cons (append/cons append/nil)) (Dordfull' x d z e) (wfe/pi DwfB2 DwfA2) (%the (wfe (cons (cons G x A3) z (pi A2 B2)) (pi A2 B2)) (DwfA2B2 x d z e))) + %<- ({x} {d isvar x J} {z} {e isvar z K} weaken-wfe (bounded/cons (DboundJ x d) (DprecJK x d z e)) (DwfB2x x d) _ (%the (wfe (cons (cons G x A3) z (pi A2 B2)) (B2x x)) (DwfB2x' x d z e))) + %<- (weaken-wfe-insert1-bump DwfB1x DboundJ DprecJK _ (%the ({z} + %pi (isvar z J) + %-> ({x} %pi (isvar x K) %-> (wfe (cons (cons G z (pi A1 B1)) x A2) (B1x x)))) DwfB1x''')) + %<- (weaken-wfe-insert1-bump DwfB2 DboundJ DprecJK _ (%the ({z} + %pi (isvar z J) + %-> ({x} %pi (isvar x K) %-> (wfe (cons (cons G z (pi A1 B1)) x A2) (B2 x)))) DwfB2''''')) + %<- ({z} {e isvar z J} {x} {d isvar x K} subtype-reg-e (DsubtypeB12 x) (DwfB1x''' z e x d) (DwfB2''''' z e x d) (%the ({y} + %pi (isvar y F'') + %-> (ofe (cons (cons (cons G z (pi A1 B1)) x A2) y (B1x x)) (N12 x y) (B2 x))) (DofN12' z e x d))) + %<- (weaken-ofe-insert1-bump DofM21 DboundJ DprecJK _ (%the ({z} + %pi (isvar z J) + %-> ({x} %pi (isvar x K) %-> (ofe (cons (cons G z (pi A1 B1)) x A2) (M21 x) A1))) DofM21'''')) + %<- ({z} {e isvar z J} {x} {d isvar x K} weaken-wfe' (append/cons (append/cons append/nil)) (ordered/cons (bounded/cons (DboundJ z e) (DprecJK z e x d))) (wfe/pi DwfB1 DwfA1) (%the (wfe (cons (cons G z (pi A1 B1)) x A2) (pi A1 B1)) (DwfA1B1' z e x d))) + %<- ({z} {e isvar z J} {x} {d isvar x K} ssubst-e ([_] append/nil) append/nil (DofN12' z e x d) (%the (aofe (cons (cons G z (pi A1 B1)) x A2) (app z (M21 x)) (B1x x)) (aofe/app (DwfB1x''' z e x d) (DsubB1x x) (DofM21'''' z e x d) (aofe/var (DwfA1B1' z e x d) (lookup/miss (lookup/hit (DboundJ z e)) (bounded/cons (DboundJ z e) (DprecJK z e x d)))))) (%the (ofe (cons (cons G z (pi A1 B1)) x A2) (N12 x (app z (M21 x))) (B2 x)) (DofN2y z e x d))) + %<- (weaken-ofe-insert1 ([z] [e isvar z J] %the (ofe (cons G z (pi A1 B1)) (lam ([x] N12 x (app z (M21 x)))) (pi A2 B2)) (ofe/lam ([x] [d isvar x K] DofN2y z e x d) (DwfA2' z e))) DboundJ _ (%the ({x} + %pi (isvar x J) + %-> ({z} + %pi (isvar z K'') + %-> (ofe (cons (cons G x A3) z (pi A1 B1)) (lam ([x] N12 x (app z (M21 x)))) (pi A2 B2)))) DofLam')) + %<- ({x} {d isvar x J} aosub-into-ssub-e' (DofLam' x d) ([z] [e isvar z K] aofe/app (DwfB2x' x d z e) (DsubB2x x) (DofM32'' x d z e) (aofe/var (DwfA2B2 x d z e) (lookup/hit (bounded/cons (DboundJ x d) (DprecJK x d z e))))) (DofN23 x d) ([z] %the (aosub ([y] app y (M32 x)) (lam ([x] N12 x (app z (M21 x)))) (N12x x (app z (M31 x)))) (aosub/app (DsubN12yx z x) (DsubM32abs z x) aosub/var)) ([z] DsubN13 x (app z (M31 x))) ([z] %the (sub ([y] N23 x (app y (M32 x))) (lam ([x] N12 x (app z (M21 x)))) (N13 x (app z (M31 x)))) (DsubLam z x))) %. +%term sigma + %pi (subtype-trans-em (ssigma S T) (simp/sigma (%the ({x} simp (B2 x) T) DsimpB) (%the (simp A2 S) DsimpA)) (wfe/sigma (%the ({x} %pi (isvar x I1) %-> (wfe (cons G x A1) (B1 x))) DwfB1) (%the (wfe G A1) DwfA1)) (wfe/sigma (%the ({x} %pi (isvar x I2) %-> (wfe (cons G x A2) (B2 x))) DwfB2) (%the (wfe G A2) DwfA2)) (wfe/sigma (%the ({x} %pi (isvar x I3) %-> (wfe (cons G x A3) (B3 x))) DwfB3) (%the (wfe G A3) DwfA3)) (subtype/sigma (%the ({x} subtype (B1 x) (B2x x) ([y] N12 x y)) DsubtypeB12) (%the ({x} tsub B2 (M12 x) (B2x x)) DsubB2x) (%the (subtype A1 A2 M12) DsubtypeA12)) (subtype/sigma (%the ({x} subtype (B2 x) (B3x x) ([y] N23 x y)) DsubtypeB23) (%the ({x} tsub B3 (M23 x) (B3x x)) DsubB3x) (%the (subtype A2 A3 M23) DsubtypeA23)) ([z] sub/pair (Dsub2 z) (Dsub1 z)) (subtype/sigma DsubtypeB13 DsubB3x1 DsubtypeA13)) + %<- (subtype-trans-em S DsimpA DwfA1 DwfA2 DwfA3 DsubtypeA12 DsubtypeA23 (%the ({x} sub M23 (M12 x) (M13 x)) DsubM13) (%the (subtype A1 A3 M13) DsubtypeA13)) + %<- (subtype-reg-e DsubtypeA13 DwfA1 DwfA3 (%the ({x} %pi (isvar x J) %-> (ofe (cons G x A1) (M13 x) A3)) DofM13)) + %<- ({x} {d isvar x J} ofe-context (DofM13 x d) (ordered/cons (%the (bounded G x) (DboundJ x d)))) + %<- (can-tsub-context-e DofM13 DwfB3 (%the ({x} tsub B3 (M13 x) (B3x1 x)) DsubB3x1)) + %<- (subtype-reg-e-bump DsubtypeA12 DwfA1 DwfA2 DboundJ (%the ({x} %pi (isvar x J) %-> (ofe (cons G x A1) (M12 x) A2)) DofM12)) + %<- (subtype-reg-e-bump DsubtypeA23 DwfA2 DwfA3 DboundJ (%the ({x} %pi (isvar x J) %-> (ofe (cons G x A2) (M23 x) A3)) DofM23)) + %<- (tsubst-context-permute-e DofM12 DofM23 DwfB3 DsubM13 DsubB3x DsubB3x1 (%the ({x} tsub B3x (M12 x) (B3x1 x)) DsubB3x1')) + %<- ({x} {d isvar x J} following-var d (%the ({y} %pi (isvar y K) %-> (precedes x y)) (DprecJK x d))) + %<- (weaken-wfe-insert1-bump DwfB2 DboundJ DprecJK _ (%the ({x} + %pi (isvar x J) + %-> ({y} %pi (isvar y K) %-> (wfe (cons (cons G x A1) y A2) (B2 y)))) DwfB2')) + %<- (tsubst-context-e DofM23 DwfB3 DsubB3x (%the ({x} %pi (isvar x J) %-> (wfe (cons G x A2) (B3x x))) DwfB3x)) + %<- (weaken-wfe-insert1-bump DwfB3x DboundJ DprecJK _ (%the ({x} + %pi (isvar x J) + %-> ({y} %pi (isvar y K) %-> (wfe (cons (cons G x A1) y A2) (B3x y)))) DwfB3x')) + %<- ({x} {d isvar x J} subtype-sub-e ([_] append/nil) csub/base (DofM12 x d) (DwfB2' x d) (DwfB3x' x d) DsubtypeB23 (DsubB2x x) (DsubB3x1' x) (%the ({y} sub ([x] N23 x y) (M12 x) (N23x x y)) (DsubN23x x)) (%the (subtype (B2x x) (B3x1 x) (N23x x)) (DsubtypeB23x x))) + %<- ({x} tsub-preserves-simp (DsubB2x x) DsimpB (%the (simp (B2x x) T) (DsimpB' x))) + %<- (bump-wfe DwfB1 ([x] [d] ordered/cons (DboundJ x d)) (%the ({x} %pi (isvar x J) %-> (wfe (cons G x A1) (B1 x))) DwfB1')) + %<- (tsubst-context-e DofM12 DwfB2 DsubB2x (%the ({x} %pi (isvar x J) %-> (wfe (cons G x A1) (B2x x))) DwfB2x)) + %<- (tsubst-context-e DofM12 DwfB3x DsubB3x1' (%the ({x} %pi (isvar x J) %-> (wfe (cons G x A1) (B3x1 x))) DwfB3x1)) + %<- ({x} {d isvar x J} subtype-trans-em T (DsimpB' x) (DwfB1' x d) (DwfB2x x d) (DwfB3x1 x d) (DsubtypeB12 x) (DsubtypeB23x x) (%the ({y} sub ([y] N23x x y) (N12 x y) (N13 x y)) (DsubN13 x)) (%the (subtype (B1 x) (B3x1 x) (N13 x)) (DsubtypeB13 x))) + %<- ({z} {e isvar z J} weaken-wfe (DboundJ z e) (wfe/sigma DwfB1 DwfA1) _ (%the (wfe (cons G z (sigma A1 B1)) (sigma A1 B1)) (DwfA1B1 z e))) + %<- ({z} {e isvar z J} weaken-wfe (DboundJ z e) (wfe/sigma DwfB2 DwfA2) _ (%the (wfe (cons G z (sigma A2 B2)) (sigma A2 B2)) (DwfA2B2 z e))) + %<- (ssubst-context-e ([z] [e] aofe/pi1 (aofe/var (DwfA1B1 z e) (lookup/hit (DboundJ z e)))) DofM12 (%the ({z} %pi (isvar z J) %-> (ofe (cons G z (sigma A1 B1)) (M12 (pi1 z)) A2)) DofM12x)) + %<- (bump-wfe DwfB1 ([x] [d] ordered/cons (DboundJ x d)) (%the ({x} %pi (isvar x J) %-> (wfe (cons G x A1) (B1 x))) DwfB1'')) + %<- ({x} {d isvar x J} subtype-reg-e-bump (DsubtypeB12 x) (DwfB1'' x d) (DwfB2x x d) ([y] [e] bounded/cons (DboundJ x d) (DprecJK x d y e)) (%the ({y} %pi (isvar y K) %-> (ofe (cons (cons G x A1) y (B1 x)) (N12 x y) (B2x x))) (DofN12 x d))) + %<- (ssubst-context2-e ([z] [e] aofe/pi1 (aofe/var (DwfA1B1 z e) (lookup/hit (DboundJ z e)))) ([z] [e] aofe/pi2 (aofe/var (DwfA1B1 z e) (lookup/hit (DboundJ z e)))) DofN12 (%the ({z} + %pi (isvar z J) + %-> (ofe (cons G z (sigma A1 B1)) (N12 (pi1 z) (pi2 z)) (B2x (pi1 z)))) DofN12xy)) + %<- (weaken-wfe-insert1-bump DwfB2 DboundJ DprecJK _ (%the ({z} + %pi (isvar z J) + %-> ({x} %pi (isvar x K) %-> (wfe (cons (cons G z (sigma A1 B1)) x A2) (B2 x)))) DwfB2'')) + %<- (aosub-into-ssub-e' ([z] [e] ofe/pair (DwfB2'' z e) (DofN12xy z e) (DsubB2x (pi1 z)) (DofM12x z e)) ([z] [e] aofe/pi1 (aofe/var (DwfA2B2 z e) (lookup/hit (DboundJ z e)))) DofM23 ([z] aosub/pi1 aosub/var) ([z] DsubM13 (pi1 z)) ([z] %the (sub ([z] M23 (pi1 z)) (pair (M12 (pi1 z)) (N12 (pi1 z) (pi2 z))) (M13 (pi1 z))) (Dsub1 z))) + %<- (bump-wfe DwfB2 ([x] [d] ordered/cons (DboundJ x d)) (%the ({x} %pi (isvar x J) %-> (wfe (cons G x A2) (B2 x))) DwfB2''')) + %<- ({x} {d isvar x J} subtype-reg-e (DsubtypeB23 x) (DwfB2''' x d) (DwfB3x x d) (%the ({y} %pi (isvar y L) %-> (ofe (cons (cons G x A2) y (B2 x)) (N23 x y) (B3x x))) (DofN23 x d))) + %<- (aosub-into-ssub-e''' ([z] [e] ofe/pair (DwfB2'' z e) (DofN12xy z e) (DsubB2x (pi1 z)) (DofM12x z e)) ([z] [e] aofe/pi1 (aofe/var (DwfA2B2 z e) (lookup/hit (DboundJ z e)))) ([z] [e] aofe/pi2 (aofe/var (DwfA2B2 z e) (lookup/hit (DboundJ z e)))) DofN23 ([z] %the (aosub ([z] pi1 z) (pair (M12 (pi1 z)) (N12 (pi1 z) (pi2 z))) (M12 (pi1 z))) (aosub/pi1 aosub/var)) ([z] %the (aosub ([z] pi2 z) (pair (M12 (pi1 z)) (N12 (pi1 z) (pi2 z))) (N12 (pi1 z) (pi2 z))) (aosub/pi2 aosub/var)) ([z] [y] %the (sub ([x] N23 x y) (M12 (pi1 z)) (N23x (pi1 z) y)) (DsubN23x (pi1 z) y)) ([z] %the (sub ([y] N23x (pi1 z) y) (N12 (pi1 z) (pi2 z)) (N13 (pi1 z) (pi2 z))) (DsubN13 (pi1 z) (pi2 z))) ([z] %the (sub ([z] N23 (pi1 z) (pi2 z)) (pair (M12 (pi1 z)) (N12 (pi1 z) (pi2 z))) (N13 (pi1 z) (pi2 z))) (Dsub2 z))) %. +%worlds (var bind ovar) (subtype-trans-em _ _ _ _ _ _ _ _ _) %. +%total T (subtype-trans-em T _ _ _ _ _ _ _ _) %. +%sort subtype-trans-e %. +%mode subtype-trans-e %in %in %in %in %in %in %out %. +%term _ + %pi (subtype-trans-e DwfA DwfB DwfC DsubtypeAB DsubtypeBC Dsub DsubtypeAC') + %<- (can-simp _ Dsimp) + %<- (subtype-trans-em _ Dsimp DwfA DwfB DwfC DsubtypeAB DsubtypeBC Dsub' DsubtypeAC) + %<- ({x} sub-fun (Dsub' x) (Dsub x) (Deq x)) + %<- (subtype-resp tp-eq/i tp-eq/i Deq DsubtypeAC DsubtypeAC') %. +%worlds (var bind ovar) (subtype-trans-e _ _ _ _ _ _ _) %. +%total {} (subtype-trans-e _ _ _ _ _ _ _) %. +%sort subtype-trans %. +%mode subtype-trans %in %in %in %in %in %in %out %. +%term _ + %pi (subtype-trans DwfA DwfB DwfC DsubtypeAB DsubtypeBC Dsub DsubtypeAC) + %<- (wf-to-wfe DwfA DwfeA) + %<- (wf-to-wfe DwfB DwfeB) + %<- (wf-to-wfe DwfC DwfeC) + %<- (subtype-trans-e DwfeA DwfeB DwfeC DsubtypeAB DsubtypeBC Dsub DsubtypeAC) %. +%worlds (bind) (subtype-trans _ _ _ _ _ _ _) %. +%total {} (subtype-trans _ _ _ _ _ _ _) %. +%sort subtype-trans-e' %. +%mode subtype-trans-e' %in %in %in %in %in %out %out %. +%term _ + %pi (subtype-trans-e' DwfA DwfB DwfC DsubtypeAB DsubtypeBC Dsub DsubtypeAC) + %<- (can-simp _ Dsimp) + %<- (subtype-trans-em _ Dsimp DwfA DwfB DwfC DsubtypeAB DsubtypeBC Dsub DsubtypeAC) %. +%worlds (bind) (subtype-trans-e' _ _ _ _ _ _ _) %. +%total {} (subtype-trans-e' _ _ _ _ _ _ _) %. +%sort subtype-trans' %. +%mode subtype-trans' %in %in %in %in %in %out %out %. +%term _ + %pi (subtype-trans' DwfA DwfB DwfC DsubtypeAB DsubtypeBC Dsub DsubtypeAC) + %<- (subtype-reg DsubtypeAB DwfA DwfB DofM) + %<- (subtype-reg DsubtypeBC DwfB DwfC DofN) + %<- ({x} {d vof x A} can-sub DofN (DofM x d) (Dsub x)) + %<- (subtype-trans DwfA DwfB DwfC DsubtypeAB DsubtypeBC Dsub DsubtypeAC) %. +%worlds (bind) (subtype-trans' _ _ _ _ _ _ _) %. +%total {} (subtype-trans' _ _ _ _ _ _ _) %. +%%%%% Subtyping Inversions %%%%% +%sort subtype-sing-invert %. +%mode subtype-sing-invert %in %out %out %. +%term _ subtype-sing-invert subtype/sing tp-eq/i ([x] term-eq/i) %. +%worlds (bind ovar) (subtype-sing-invert _ _ _) %. +%total {} (subtype-sing-invert _ _ _) %. +%sort subtype-pi-invert %. +%mode subtype-pi-invert %in %out %out %out %out %. +%term _ subtype-pi-invert (subtype/pi D3 D2 D1) D1 D2 D3 ([_] term-eq/i) %. +%worlds (var) (subtype-pi-invert _ _ _ _ _) %. +%total {} (subtype-pi-invert _ _ _ _ _) %. +%sort subtype-sigma-invert %. +%mode subtype-sigma-invert %in %out %out %out %out %. +%term _ subtype-sigma-invert (subtype/sigma D3 D2 D1) D1 D2 D3 ([_] term-eq/i) %. +%worlds (var) (subtype-sigma-invert _ _ _ _ _) %. +%total {} (subtype-sigma-invert _ _ _ _ _) %. +%sort subtype-closed {x} %. +%mode subtype-closed %in %out %. +%scope subtype-closed %term t subtype-closed ([x] subtype/t) ([x] [y] term-eq/i) %. +%sort subtype-closed! {x atom} {y atom} %. +%mode subtype-closed! %in %in %in %out %. +%term _ subtype-closed! D ([_] [_] tp-eq/i) ([_] [_] tp-eq/i) D %. +%worlds (var) (subtype-closed! _ _ _ _) %. +%total {} (subtype-closed! _ _ _ _) %. +%reduces = D D' (subtype-closed! D _ _ D') %. +%term pi + %pi (subtype-closed ([x] subtype/pi (%the ({y} subtype (C x y) (B2 y) ([z] M2 x y z)) (Dsubtype2 x)) (%the ({y} tsub ([y] A2 y) (M1 x y) (C x y)) (Dsub x)) (%the (subtype B1 A1 ([y] M1 x y)) (Dsubtype1 x))) Deq') + %<- (subtype-closed Dsubtype1 (%the ({x} {y} term-eq (M1 x y) (M1x y)) DeqM1)) + %<- ({x} {y} tsub-resp ([_] tp-eq/i) (DeqM1 x y) tp-eq/i (Dsub x y) (%the (tsub ([y] A2 y) (M1x y) (C x y)) (Dsub' x y))) + %<- ({y} tsub-closed ([x] Dsub' x y) ([x] %the (tp-eq (C x y) (Cx y)) (DeqC x y))) + %<- (subtype-closed! Dsubtype2 DeqC ([_] [_] tp-eq/i) (%the ({x} {y} subtype (Cx y) (B2 y) ([z] M2 x y z)) Dsubtype2')) + %<- ({y} subtype-closed ([x] Dsubtype2' x y) ([x] [z] %the (term-eq (M2 x y z) (M2x y z)) (DeqM2 x y z))) + %<- ({f} {x} {y} term-resp-term ([m] M2 x y (app f m)) (DeqM1 x y) (%the (term-eq (M2 x y (app f (M1 x y))) (M2 x y (app f (M1x y)))) (DeqM2M1 f x y))) + %<- ({f} {x} {y} term-eq-trans (DeqM2M1 f x y) (DeqM2 x y (app f (M1x y))) (%the (term-eq (M2 x y (app f (M1 x y))) (M2x y (app f (M1x y)))) (Deq f x y))) + %<- ({f} {x} lam-resp ([y] Deq f x y) (%the (term-eq (lam ([y] M2 x y (app f (M1 x y)))) (lam ([y] M2x y (app f (M1x y))))) (Deq' x f))) %. +%term sigma + %pi (subtype-closed ([x] subtype/sigma (%the ({y} subtype (A2 y) (C x y) ([z] M2 x y z)) (Dsubtype2 x)) (%the ({y} tsub ([y] B2 y) (M1 x y) (C x y)) (Dsub x)) (%the (subtype A1 B1 ([y] M1 x y)) (Dsubtype1 x))) Deq) + %<- (subtype-closed Dsubtype1 (%the ({x} {y} term-eq (M1 x y) (M1x y)) DeqM1)) + %<- ({x} {y} tsub-resp ([_] tp-eq/i) (DeqM1 x y) tp-eq/i (Dsub x y) (%the (tsub ([y] B2 y) (M1x y) (C x y)) (Dsub' x y))) + %<- ({y} tsub-closed ([x] Dsub' x y) ([x] %the (tp-eq (C x y) (Cx y)) (DeqC x y))) + %<- (subtype-closed! Dsubtype2 ([_] [_] tp-eq/i) DeqC (%the ({x} {y} subtype (A2 y) (Cx y) ([z] M2 x y z)) Dsubtype2')) + %<- ({y} subtype-closed ([x] Dsubtype2' x y) ([x] [z] %the (term-eq (M2 x y z) (M2x y z)) (DeqM2 x y z))) + %<- ({p} {x} pair-resp (DeqM1 x (pi1 p)) (DeqM2 x (pi1 p) (pi2 p)) (%the (term-eq (pair (M1 x (pi1 p)) (M2 x (pi1 p) (pi2 p))) (pair (M1x (pi1 p)) (M2x (pi1 p) (pi2 p)))) (Deq x p))) %. +%term sing_t subtype-closed ([x] subtype/sing_t) ([x] [y] term-eq/i) %. +%term sing subtype-closed ([x] subtype/sing) ([x] [y] term-eq/i) %. +%worlds (var) (subtype-closed _ _) %. +%total D (subtype-closed D _) %. +%%%%% Selfification Lemmas %%%%% +%sort self-reg-e %. +%mode self-reg-e %in %in %out %out %. +%scope self-reg-e %term t self-reg-e self/t (ofe/at Daof) (ofe/sing Daof) subtype/sing_t %. +%term pi + %pi (self-reg-e (self/pi (%the ({x} self (M x) (B x) (B' x)) Dself)) (ofe/lam (%the ({x} %pi (isvar x I) %-> (ofe (cons G x A) (M x) (B x))) Dof) (%the (wfe G A) DwfA)) (ofe/lam Dof' DwfA) (subtype/pi DsubtypeB Dsub DsubtypeA)) + %<- (subtype-refl-e DwfA (%the (subtype A A X) DsubtypeA) (%the ({x} expand x A (X x)) Dexpand)) + %<- ({x} {d} self-reg-e (Dself x) (Dof x d) (%the (ofe (cons G x A) (M x) (B' x)) (Dof' x d)) (%the (subtype (B' x) (B x) ([_] M x)) (DsubtypeB x))) + %<- ({x} {d} ofe-reg (Dof' x d) (%the (wfe (cons G x A) (B' x)) (DwfB' x d))) + %<- (tsub-expand-var-e DwfA DwfB' Dexpand (%the ({x} tsub B' (X x) (B' x)) Dsub)) %. +%scope self-reg-e %term sigma + %pi (self-reg-e (self/sigma (%the (self N Bx Bx_self) DselfN) (%the (tsub B M Bx) Dsub) (%the (self M A Aself) DselfM)) (ofe/pair (%the ({x} %pi (isvar x I) %-> (wfe (cons G x A) (B x))) DwfB) (%the (ofe G N Bx') DofN) (%the (tsub B M Bx') Dsub') (%the (ofe G M A) DofM)) (ofe/pair Dwf' DofNself DsubTriv (%the (ofe G M Aself) DofMself)) (subtype/sigma ([_] DsubtypeB) ([_] Dsub) DsubtypeA)) + %<- (self-reg-e DselfM DofM (%the (ofe G M Aself) DofMself) (%the (subtype Aself A ([_] M)) DsubtypeA)) + %<- (tsub-absent Bx_self M (%the (tsub ([_] Bx_self) M Bx_self) DsubTriv)) + %<- (tsub-fun Dsub' Dsub (%the (tp-eq Bx' Bx) Deq)) + %<- (ofe-resp ctx-eq/i term-eq/i Deq DofN (%the (ofe G N Bx) DofN')) + %<- (self-reg-e DselfN DofN' (%the (ofe G N Bx_self) DofNself) (%the (subtype Bx_self Bx ([_] N)) DsubtypeB)) + %<- (ofe-reg DofNself (%the (wfe G Bx_self) Dwf)) + %<- ({x} {d} wfe-context (DwfB x d) (ordered/cons (Dbounded x d))) + %<- ({x} {d} weaken-wfe (Dbounded x d) Dwf _ (%the (wfe (cons G x Aself) Bx_self) (Dwf' x d))) %. +%term sing self-reg-e self/sing (ofe/sing Daof) (ofe/sing Daof) subtype/sing %. +%worlds (var bind ovar) (self-reg-e _ _ _ _) %. +%total D (self-reg-e D _ _ _) %. +%sort self-reg-e' %. +%mode self-reg-e' %in %in %out %. +%term _ + %pi (self-reg-e' Dself Dof Dwf) + %<- (self-reg-e Dself Dof Dof' _) + %<- (ofe-reg Dof' Dwf) %. +%worlds (var bind ovar) (self-reg-e' _ _ _) %. +%total {} (self-reg-e' _ _ _) %. +%sort self-reg %. +%mode self-reg %in %in %out %out %. +%scope self-reg %term t self-reg self/t (of/at Daof) (of/sing Daof) subtype/sing_t %. +%term pi + %pi (self-reg (self/pi (%the ({x} self (M x) (B x) (B' x)) Dself)) (of/lam (%the ({x} %pi (vof x A) %-> (of (M x) (B x))) Dof) (%the (wf A) DwfA)) (of/lam Dof' DwfA) (subtype/pi DsubtypeB Dtsub DsubtypeA)) + %<- (subtype-refl DwfA (%the (subtype A A ([x] X x)) DsubtypeA) (%the ({x} expand x A (X x)) DexpandA)) + %<- ({x} {d vof x A} self-reg (Dself x) (Dof x d) (%the (of (M x) (B' x)) (Dof' x d)) (%the (subtype (B' x) (B x) ([_] M x)) (DsubtypeB x))) + %<- ({x} {d vof x A} of-reg (Dof' x d) (%the (wf (B' x)) (DwfB' x d))) + %<- (tsub-expand-var DwfA DwfB' DexpandA (%the ({x} tsub ([x] B' x) (X x) (B' x)) Dtsub)) %. +%scope self-reg %term sigma + %pi (self-reg (self/sigma (%the (self N Bx Bx_self) DselfN) (%the (tsub B M Bx) Dsub) (%the (self M A Aself) DselfM)) (of/pair (%the ({x} %pi (vof x A) %-> (wf (B x))) DwfB) (%the (of N Bx') DofN) (%the (tsub B M Bx') Dsub') (%the (of M A) DofM)) (of/pair ([_] [_] Dwf) DofNself DsubTriv (%the (of M Aself) DofMself)) (subtype/sigma ([_] DsubtypeB) ([_] Dsub) DsubtypeA)) + %<- (self-reg DselfM DofM (%the (of M Aself) DofMself) (%the (subtype Aself A ([_] M)) DsubtypeA)) + %<- (tsub-absent Bx_self M (%the (tsub ([_] Bx_self) M Bx_self) DsubTriv)) + %<- (tsub-fun Dsub' Dsub (%the (tp-eq Bx' Bx) Deq)) + %<- (of-resp term-eq/i Deq DofN (%the (of N Bx) DofN')) + %<- (self-reg DselfN DofN' (%the (of N Bx_self) DofNself) (%the (subtype Bx_self Bx ([_] N)) DsubtypeB)) + %<- (of-reg DofNself (%the (wf Bx_self) Dwf)) %. +%term sing self-reg self/sing (of/sing Daof) (of/sing Daof) subtype/sing %. +%worlds (bind var ovar) (self-reg _ _ _ _) %. +%total D (self-reg D _ _ _) %. +%sort self-reg' %. +%mode self-reg' %in %in %out %. +%term _ + %pi (self-reg' Dself Dof Dwf) + %<- (self-reg Dself Dof Dof' _) + %<- (of-reg Dof' Dwf) %. +%worlds (bind var ovar) (self-reg' _ _ _) %. +%total {} (self-reg' _ _ _) %. +%sort can-self-e %. +%mode can-self-e %in %out %. +%scope can-self-e %term t can-self-e _ self/t %. +%term pi + %pi (can-self-e (ofe/lam Dof _) (self/pi Dself)) + %<- ({x} {d} can-self-e (Dof x d) (Dself x)) %. +%scope can-self-e %term sigma + %pi (can-self-e (ofe/pair _ Dof2 Dsub Dof1) (self/sigma Dself2 Dsub Dself1)) + %<- (can-self-e Dof1 Dself1) + %<- (can-self-e Dof2 Dself2) %. +%term sing can-self-e (ofe/sing _) self/sing %. +%worlds (var bind ovar) (can-self-e _ _) %. +%total D (can-self-e D _) %. +%sort can-self %. +%mode can-self %in %out %. +%term _ %pi (can-self Dof Dself) %<- (of-to-ofe Dof Dofe) %<- (can-self-e Dofe Dself) %. +%worlds (var bind ovar) (can-self _ _) %. +%total {} (can-self _ _) %. +%sort self-fun %. +%mode self-fun %in %in %out %. +%scope self-fun %term t self-fun self/t self/t tp-eq/i %. +%term pi + %pi (self-fun (self/pi Dself) (self/pi Dself') Deq') + %<- ({x} self-fun (Dself x) (Dself' x) (Deq x)) + %<- (pi-resp tp-eq/i Deq Deq') %. +%scope self-fun %term sigma + %pi (self-fun (self/sigma Dself2 Dsub1 Dself1) (self/sigma Dself2' Dsub2 Dself1') Deq) + %<- (self-fun Dself1 Dself1' Deq1) + %<- (tsub-fun Dsub2 Dsub1 Deqs) + %<- (self-resp term-eq/i Deqs tp-eq/i Dself2' Dself2'') + %<- (self-fun Dself2 Dself2'' Deq2) + %<- (sigma-resp Deq1 ([_] Deq2) Deq) %. +%term sing self-fun self/sing self/sing tp-eq/i %. +%worlds (var bind) (self-fun _ _ _) %. +%total D (self-fun D _ _) %. +%%%%% Self Commutes with Substitution %%%%% +%sort self-sub-e {x} %. +%mode self-sub-e %in %in %in %in %in %in %in %in %out %. +%scope self-sub-e %term t + %pi (self-sub-e _ _ _ _ ([_] self/t) (sub/ao Daosub) tsub/t (tsub/singo Daosub') D) + %<- (aosub-fun Daosub' Daosub Deq) + %<- (self-resp Deq tp-eq/i tp-eq/i self/t D) %. +%term t + %pi (self-sub-e _ _ _ _ ([_] self/t) (sub/aa Daasub) tsub/t (tsub/singa Daasub') D) + %<- (aasub-fun Daasub' Daasub Deq) + %<- (term-resp-atom at Deq Deq') + %<- (self-resp Deq' tp-eq/i tp-eq/i self/t D) %. +%scope self-sub-e %term t + %pi (self-sub-e _ _ _ _ ([_] self/t) (sub/aa Daasub) tsub/t (tsub/singo Daosub) D) + %<- (aasub-aosub-contra Daasub Daosub Dfalse) + %<- (false-implies-self Dfalse D) %. +%term t + %pi (self-sub-e _ _ _ _ ([_] self/t) (sub/ao Daosub) tsub/t (tsub/singa Daasub) D) + %<- (aasub-aosub-contra Daasub Daosub Dfalse) + %<- (false-implies-self Dfalse D) %. +%scope self-sub-e %term pi + %pi (self-sub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) (%the (ofe G1 M A) DofM) ([x] [d isvar x I] ofe/lam (%the ({y} %pi (isvar y J) %-> (ofe (cons (G x) y (B x)) (N x y) (C x y))) (DofN x d)) (%the (wfe (G x) (B x)) (DwfB x d))) ([x] self/pi (%the ({y} self (N x y) (C x y) (C' x y)) (Dself x))) (sub/lam (%the ({y} sub ([x] N x y) M (Nx y)) DsubNx)) (tsub/pi (%the ({y} tsub ([x] C x y) M (Cx y)) DsubCx) (%the (tsub B M Bx) DsubBx)) (tsub/pi (%the ({y} tsub ([x] C' x y) M (Cx' y)) DsubCx') (%the (tsub B M Bx') DsubBx')) Dself'') + %<- ({y} {e isvar y J} self-sub-e ([x] append/cons (Dappend x)) (csub/cons DsubBx Dcsub) DofM ([x] [d] DofN x d y e) ([x] Dself x y) (DsubNx y) (DsubCx y) (DsubCx' y) (%the (self (Nx y) (Cx y) (Cx' y)) (Dself' y))) + %<- (tsub-fun DsubBx DsubBx' (%the (tp-eq Bx Bx') DeqBx)) + %<- (tp-resp-tp ([a] pi a ([y] Cx' y)) DeqBx (%the (tp-eq (pi Bx Cx') (pi Bx' Cx')) Deq)) + %<- (self-resp term-eq/i tp-eq/i Deq (self/pi Dself') (%the (self (lam Nx) (pi Bx Cx) (pi Bx' Cx')) Dself'')) %. +%term sigma + %pi (self-sub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) (%the (csub G M Gx) Dcsub) (%the (ofe G1 M A) DofM) ([x] [d isvar x I] ofe/pair (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (ofe (G x) (O x) (Cy x)) (DofO x d)) (%the (tsub ([y] C x y) (N x) (Cy x)) (DsubCy x)) (%the (ofe (G x) (N x) (B x)) (DofN x d))) ([x] self/sigma (%the (self (O x) (Cyalt x) (C' x)) (Dself2 x)) (%the (tsub ([y] C x y) (N x) (Cyalt x)) (DsubCyalt x)) (%the (self (N x) (B x) (B' x)) (Dself1 x))) (sub/pair (%the (sub O M Ox) DsubOx) (%the (sub N M Nx) DsubNx)) (tsub/sigma (%the ({y} tsub ([x] C x y) M (Cx y)) DsubCx) (%the (tsub B M Bx) DsubBx)) (tsub/sigma (%the ({y} tsub ([x] C' x) M (Cyx' y)) DsubCyx') (%the (tsub B' M Bx') DsubBx')) D) + %<- ({x} tsub-fun (DsubCyalt x) (DsubCy x) (%the (tp-eq (Cyalt x) (Cy x)) (DeqCy x))) + %<- ({x} self-resp term-eq/i (DeqCy x) tp-eq/i (Dself2 x) (%the (self (O x) (Cy x) (C' x)) (Dself2' x))) + %<- (tsub-closed DsubCyx' (%the ({y} tp-eq (Cyx' y) Cx') DeqCyx')) + %<- ({y} tsub-resp ([_] tp-eq/i) term-eq/i (DeqCyx' y) (DsubCyx' y) (%the (tsub ([x] C' x) M Cx') (DsubCx' y))) + %<- ({y} tp-eq-symm (DeqCyx' y) (%the (tp-eq Cx' (Cyx' y)) (DeqCyx'' y))) + %<- (self-sub-e Dappend Dcsub DofM DofN Dself1 DsubNx DsubBx DsubBx' (%the (self Nx Bx Bx') Dself1')) + %<- ({x} {d isvar x I} tsubst-e ([_] append/nil) csub/base (DofN x d) (DsubCy x) ([y] [e] DwfC x d y e) (%the (wfe (G x) (Cy x)) (DwfCy x d))) + %<- (can-tsub-e Dappend DwfCy DofM (%the (tsub Cy M Cyx) DsubCyx)) + %<- (subst-e Dappend Dcsub DofM DsubNx DsubBx DofN (%the (ofe Gx Nx Bx) DofNx)) + %<- ({y} {e isvar y J} tsubst-e ([x] append/cons (Dappend x)) (csub/cons DsubBx Dcsub) DofM (DsubCx y) ([x] [d] DwfC x d y e) (%the (wfe (cons Gx y Bx) (Cx y)) (DofCx y e))) + %<- (can-tsub-e ([_] append/nil) DofCx DofNx (%the (tsub Cx Nx Cxy) DsubCxy)) + %<- (tsub-permute-e Dappend DofM DofN DwfC DsubCx DsubNx DsubCxy DsubCy DsubCyx (%the (tp-eq Cxy Cyx) DeqCxy)) + %<- (tp-eq-symm DeqCxy (%the (tp-eq Cyx Cxy) DeqCxy')) + %<- (tsub-resp ([_] tp-eq/i) term-eq/i DeqCxy' DsubCyx (%the (tsub Cy M Cxy) DsubCyx_xy)) + %<- (self-sub-e Dappend Dcsub DofM DofO Dself2' DsubOx DsubCyx_xy (DsubCx' aca) (%the (self Ox Cxy Cx') Dself2'')) + %<- (sigma-resp tp-eq/i DeqCyx'' (%the (tp-eq (sigma Bx' ([_] Cx')) (sigma Bx' ([y] Cyx' y))) Deq)) + %<- (self-resp term-eq/i tp-eq/i Deq (%the (self (pair Nx Ox) (sigma Bx Cx) (sigma Bx' ([_] Cx'))) (self/sigma Dself2'' DsubCxy Dself1')) (%the (self (pair Nx Ox) (sigma Bx Cx) (sigma Bx' Cyx')) D)) %. +%scope self-sub-e %term sing + %pi (self-sub-e _ _ _ _ ([_] self/sing) (sub/ao Daosub1) (tsub/singo Daosub2) (tsub/singo Daosub3) D) + %<- (aosub-fun Daosub2 Daosub1 Deq21) + %<- (aosub-fun Daosub2 Daosub3 Deq23) + %<- (term-eq-cdr-at Deq23 Deq23') + %<- (tp-resp-atom sing Deq23' Deq23'') + %<- (self-resp Deq21 tp-eq/i Deq23'' self/sing D) %. +%term sing + %pi (self-sub-e _ _ _ _ ([_] self/sing) (sub/aa Daasub1) (tsub/singa Daasub2) (tsub/singa Daasub3) D) + %<- (aasub-fun Daasub2 Daasub1 Deq21) + %<- (aasub-fun Daasub2 Daasub3 Deq23) + %<- (tp-resp-atom sing Deq23 Deq23') + %<- (term-resp-atom at Deq21 Deq21') + %<- (self-resp Deq21' tp-eq/i Deq23' self/sing D) %. +%scope self-sub-e %term sing + %pi (self-sub-e _ _ _ _ ([_] self/sing) (sub/aa Daasub) (tsub/singo Daosub) _ D) + %<- (aasub-aosub-contra Daasub Daosub Dfalse) + %<- (false-implies-self Dfalse D) %. +%term sing + %pi (self-sub-e _ _ _ _ ([_] self/sing) (sub/ao Daosub) (tsub/singa Daasub) _ D) + %<- (aasub-aosub-contra Daasub Daosub Dfalse) + %<- (false-implies-self Dfalse D) %. +%scope self-sub-e %term sing + %pi (self-sub-e _ _ _ _ ([_] self/sing) _ (tsub/singo Daosub) (tsub/singa Daasub) D) + %<- (aasub-aosub-contra Daasub Daosub Dfalse) + %<- (false-implies-self Dfalse D) %. +%term sing + %pi (self-sub-e _ _ _ _ ([_] self/sing) _ (tsub/singa Daasub) (tsub/singo Daosub) D) + %<- (aasub-aosub-contra Daasub Daosub Dfalse) + %<- (false-implies-self Dfalse D) %. +%worlds (bind ovar) (self-sub-e _ _ _ _ _ _ _ _ _) %. +%total D (self-sub-e _ _ _ D _ _ _ _ _) %. +%sort self-sub %. +%mode self-sub %in %in %in %in %in %in %out %. +%term _ + %pi (self-sub DofM DofN Dself DsubNx DsubBx DsubBsx Dself') + %<- (of-to-ofe DofM DofeM) + %<- (of1-to-ofe 0 DofN DofeN) + %<- (self-sub-e ([_] append/nil) csub/base DofeM DofeN Dself DsubNx DsubBx DsubBsx Dself') %. +%worlds (bind ovar) (self-sub _ _ _ _ _ _ _) %. +%total {} (self-sub _ _ _ _ _ _ _) %. +%sort self-sub' %. +%mode self-sub' %in %in %in %in %in %in %out %. +%term _ + %pi (self-sub' DofM DofN DselfB DsubNx DsubBx DselfBs DsubBsx') + %<- ({x} {d} self-reg' (DselfB x) (DofN x d) (%the (wf (Bs x)) (DwfBs x d))) + %<- (can-tsub DwfBs DofM (%the (tsub Bs M Bsx) DsubBsx)) + %<- (self-sub DofM DofN DselfB DsubNx DsubBx DsubBsx DselfBs') + %<- (self-fun DselfBs' DselfBs Deq) + %<- (tsub-resp ([_] tp-eq/i) term-eq/i Deq DsubBsx DsubBsx') %. +%worlds (bind ovar) (self-sub' _ _ _ _ _ _ _) %. +%total {} (self-sub' _ _ _ _ _ _ _) %. +%%%%% Principality Lemmas %%%%% +%sort self-principal %. +%mode self-principal %in %out %. +%scope self-principal %term t self-principal self/t principal/sing %. +%term pi + %pi (self-principal (self/pi Dself) (principal/pi Dprin)) + %<- ({x} self-principal (Dself x) (Dprin x)) %. +%scope self-principal %term sigma + %pi (self-principal (self/sigma Dself2 _ Dself1) (principal/sigma Dprin2 Dprin1)) + %<- (self-principal Dself1 Dprin1) + %<- (self-principal Dself2 Dprin2) %. +%term sing self-principal self/sing principal/sing %. +%worlds (var) (self-principal _ _) %. +%total D (self-principal D _) %. +%sort principal-sub {x} %. +%mode principal-sub %in %in %out %. +%term _ + %pi (principal-sub ([x] principal/pi ([y] Dprin x y)) (tsub/pi Dsub _) (principal/pi Dprin')) + %<- ({y} principal-sub ([x] Dprin x y) (Dsub y) (Dprin' y)) %. +%term _ + %pi (principal-sub ([x] principal/sigma (Dprin2 x) (Dprin1 x)) (tsub/sigma Dsub2 Dsub1) D) + %<- (principal-sub Dprin1 Dsub1 Dprin1') + %<- (tsub-closed Dsub2 Deq) + %<- ({y} tsub-resp ([_] tp-eq/i) term-eq/i (Deq y) (Dsub2 y) (Dsub2' y)) + %<- (principal-sub Dprin2 (Dsub2' aca) Dprin2') + %<- ({y} tp-eq-symm (Deq y) (Deq' y)) + %<- (sigma-resp tp-eq/i Deq' (%the (tp-eq (sigma A1 ([_] A2')) (sigma A1 A2)) Deqsig)) + %<- (principal-resp Deqsig (principal/sigma Dprin2' Dprin1') D) %. +%term _ principal-sub ([x] principal/sing) (tsub/singa _) principal/sing %. +%term _ principal-sub ([x] principal/sing) (tsub/singo _) principal/sing %. +%worlds (var) (principal-sub _ _ _) %. +%total D (principal-sub D _ _) %. +%sort principal-self-e %. +%mode principal-self-e %in %in %out %. +%scope principal-self-e %term pi + %pi (principal-self-e (principal/pi (%the ({x} principal (B x)) DprincipalB)) (ofe/lam (%the ({x} %pi (isvar x I) %-> (ofe (cons G x A) (M x) (B x))) DofM) (%the (wfe G A) DwfA)) (self/pi Dself)) + %<- ({x} {d isvar x I} principal-self-e (DprincipalB x) (DofM x d) (%the (self (M x) (B x) (B x)) (Dself x))) %. +%term sigma + %pi (principal-self-e (principal/sigma (%the (principal B) DprincipalB) (%the (principal A) DprincipalA)) (ofe/pair _ (%the (ofe G N Bx) DofN) (%the (tsub ([_] B) M Bx) DsubBx) (%the (ofe G M A) DofM)) (self/sigma Dself2 DsubB Dself1)) + %<- (principal-self-e DprincipalA DofM (%the (self M A A) Dself1)) + %<- (tsub-absent B M (%the (tsub ([_] B) M B) DsubB)) + %<- (tsub-fun DsubBx DsubB (%the (tp-eq Bx B) Deq)) + %<- (ofe-resp ctx-eq/i term-eq/i Deq DofN (%the (ofe G N B) DofN')) + %<- (principal-self-e DprincipalB DofN' (%the (self N B B) Dself2)) %. +%scope principal-self-e %term sing principal-self-e principal/sing (ofe/sing _) self/sing %. +%worlds (var bind ovar) (principal-self-e _ _ _) %. +%total D (principal-self-e D _ _) %. +%sort self-singleton-e %. +%mode self-singleton-e %in %in %out %. +%term _ self-singleton-e self/t (ofe/sing _) term-eq/i %. +%term _ + %pi (self-singleton-e (self/pi Dself) (ofe/lam Dof _) Deq') + %<- ({x} {d isvar x I} self-singleton-e (Dself x) (Dof x d) (Deq x)) + %<- (lam-resp Deq Deq') %. +%term _ + %pi (self-singleton-e (self/sigma Dself2 _ Dself1) (ofe/pair _ Dof2 Dtsub Dof1) Deq) + %<- (self-singleton-e Dself1 Dof1 Deq1) + %<- (tsub-absent _ _ Dtsub') + %<- (tsub-fun Dtsub Dtsub' Deqt) + %<- (ofe-resp ctx-eq/i term-eq/i Deqt Dof2 Dof2') + %<- (self-singleton-e Dself2 Dof2' Deq2) + %<- (term-resp-term ([m] pair m M2) Deq1 Deq1') + %<- (term-resp-term ([m] pair M1' m) Deq2 Deq2') + %<- (term-eq-trans Deq1' Deq2' Deq) %. +%term _ self-singleton-e self/sing (ofe/sing _) term-eq/i %. +%worlds (bind ovar var) (self-singleton-e _ _ _) %. +%total D (self-singleton-e D _ _) %. +%sort principal-singleton-e %. +%mode principal-singleton-e %in %in %in %out %. +%term _ + %pi (principal-singleton-e Dprincipal DofeM DofeN Deq) + %<- (principal-self-e Dprincipal DofeM Dself) + %<- (self-singleton-e Dself DofeN Deq) %. +%worlds (bind ovar var) (principal-singleton-e _ _ _ _) %. +%total {} (principal-singleton-e _ _ _ _) %. +%sort subtype-refl-principal-e %. +%mode subtype-refl-principal-e %in %in %out %. +%term _ + %pi (subtype-refl-principal-e (%the (principal A) Dprincipal) (%the (ofe G M A) DofM) Dsubtype') + %<- (ofe-reg DofM (%the (wfe G A) DwfA)) + %<- (subtype-refl-e DwfA (%the (subtype A A ([x] N x)) Dsubtype) (%the ({x} expand x A (N x)) Dexpand)) + %<- (wfe-context DwfA (%the (ordered G) Dordered)) + %<- (extend-context Dordered (%the ({x} %pi (isvar x I) %-> (bounded G x)) Dbounded)) + %<- ({x} {d isvar x I} weaken-ofe (Dbounded x d) DofM _ (%the (ofe (cons G x A) M A) (DofM' x d))) + %<- ({x} {d isvar x I} ofe-reg (DofM' x d) (%the (wfe (cons G x A) A) (DwfA' x d))) + %<- ({x} {d isvar x I} expand-reg-e (aofe/var (DwfA' x d) (lookup/hit (Dbounded x d))) (Dexpand x) (%the (ofe (cons G x A) (N x) A) (DofN x d))) + %<- ({x} {d isvar x I} principal-singleton-e Dprincipal (DofN x d) (DofM' x d) (%the (term-eq (N x) M) (Deq x))) + %<- (subtype-resp tp-eq/i tp-eq/i Deq Dsubtype (%the (subtype A A ([_] M)) Dsubtype')) %. +%worlds (ovar bind var) (subtype-refl-principal-e _ _ _) %. +%total {} (subtype-refl-principal-e _ _ _) %. +%sort subtype-refl-principal %. +%mode subtype-refl-principal %in %in %out %. +%term _ + %pi (subtype-refl-principal Dprincipal Dof Dsub) + %<- (of-to-ofe Dof Dofe) + %<- (subtype-refl-principal-e Dprincipal Dofe Dsub) %. +%worlds (bind var ovar) (subtype-refl-principal _ _ _) %. +%total {} (subtype-refl-principal _ _ _) %. +%sort principal-subtype-t-invert %. +%mode principal-subtype-t-invert %in %in %out %out %. +%term _ principal-subtype-t-invert principal/sing subtype/sing_t tp-eq/i ([_] term-eq/i) %. +%worlds (var) (principal-subtype-t-invert _ _ _ _) %. +%total {} (principal-subtype-t-invert _ _ _ _) %. +%sort principal-subtype-t-invert' %. +%mode principal-subtype-t-invert' %in %in %in %out %. +%term _ principal-subtype-t-invert' principal/sing subtype/sing_t ([_] term-eq/i) tp-eq/i %. +%worlds (var) (principal-subtype-t-invert' _ _ _ _) %. +%total {} (principal-subtype-t-invert' _ _ _ _) %. +%sort principal-subtype-sigma-invert %. +%mode principal-subtype-sigma-invert %in %in %out %. +%term _ principal-subtype-sigma-invert (principal/sigma _ _) (subtype/sigma _ _ _) tp-eq/i %. +%worlds (var) (principal-subtype-sigma-invert _ _ _) %. +%total {} (principal-subtype-sigma-invert _ _ _) %. +%sort principal-sigma-invert %. +%mode principal-sigma-invert %in %out %. +%term _ principal-sigma-invert (principal/sigma _ _) ([_] tp-eq/i) %. +%worlds (var) (principal-sigma-invert _ _) %. +%total {} (principal-sigma-invert _ _) %. +%sort principal-subtype %. +%mode principal-subtype %in %in %out %. +%scope principal-subtype %term pi + %pi (principal-subtype (principal/pi (%the ({x} principal (A2 x)) Dprincipal)) (subtype/pi (%the ({x} subtype (A2' x) (B2 x) ([y] L2 x y)) Dsubtype2) (%the ({x} tsub A2 (L1 x) (A2' x)) DsubA2') (%the (subtype B1 A1 L1) Dsubtype1)) Deq') + %<- ({x} principal-sub Dprincipal (DsubA2' x) (%the (principal (A2' x)) (Dprincipal' x))) + %<- ({x} principal-subtype (Dprincipal' x) (Dsubtype2 x) ([y] %the (term-eq (L2 x y) (L2' x)) (Deq x y))) + %<- ({f} lam-resp ([x] Deq x (app f (L1 x))) (%the (term-eq (lam ([x] L2 x (app f (L1 x)))) (lam ([x] L2' x))) (Deq' f))) %. +%sort principal-subtype! {x atom} {y atom} %. +%mode principal-subtype! %in %in %out %. +%term _ principal-subtype! ([_] [_] term-eq/i) ([_] [_] term-eq/i) ([_] [_] term-eq/i) %. +%worlds (var) (principal-subtype! _ _ _) %. +%total {} (principal-subtype! _ _ _) %. +%term sigma + %pi (principal-subtype (principal/sigma (%the (principal A2) Dprincipal2) (%the (principal A1) Dprincipal1)) (subtype/sigma (%the ({x} subtype A2 (B2' x) ([y] L2 x y)) Dsubtype2) (%the ({x} tsub B2 (L1 x) (B2' x)) DsubB2') (%the (subtype A1 B1 L1) Dsubtype1)) Deq) + %<- (principal-subtype Dprincipal1 Dsubtype1 (%the ({y} term-eq (L1 y) L1') Deq1)) + %<- ({x} tsub-resp ([_] tp-eq/i) (Deq1 x) tp-eq/i (DsubB2' x) (%the (tsub B2 L1' (B2' x)) (DsubB2'' x))) + %<- (tsub-closed DsubB2'' (%the ({x} tp-eq (B2' x) B2'') DeqB2')) + %<- ({x} subtype-resp tp-eq/i (DeqB2' x) ([_] term-eq/i) (Dsubtype2 x) (%the (subtype A2 B2'' ([y] L2 x y)) (Dsubtype2' x))) + %<- (subtype-closed Dsubtype2' (%the ({x} {y} term-eq (L2 x y) (L2x y)) Deq2x)) + %<- ({x} principal-subtype Dprincipal2 (Dsubtype2 x) (%the ({y} term-eq (L2 x y) (L2y x)) (Deq2y x))) + %<- (principal-subtype! Deq2x Deq2y (%the ({x} {y} term-eq (L2 x y) L2') Deq2)) + %<- ({p} pair-resp (Deq1 (pi1 p)) (Deq2 (pi1 p) (pi2 p)) (%the (term-eq (pair (L1 (pi1 p)) (L2 (pi1 p) (pi2 p))) (pair L1' L2')) (Deq p))) %. +%term sing_t principal-subtype principal/sing subtype/sing_t ([_] term-eq/i) %. +%term sing principal-subtype principal/sing subtype/sing ([_] term-eq/i) %. +%worlds (var) (principal-subtype _ _ _) %. +%total D (principal-subtype _ D _) %. +%%%%%% Strengthening subtype RHS %%%%% +%% This is stated a bit clumsily. It turns out that O must be ([_] N), +%% and that M need not be considered at all. The formulation you really +%% want appears below, as subtype-self-e. We use subtype-self-e' to help +%% prove subtype-self-e. It's probably possible to come up with a clean +%% proof of subtype-self-e without recourse to subtype-self-e', but the +%% latter is already done, so why bother. +%sort subtype-self-e' %. +%mode subtype-self-e' %in %in %in %in %in %in %out %. +%% Naming conventions are not so good in this case. +%scope subtype-self-e' %term pi + %pi (subtype-self-e' (%the (subtype (pi A1 A2) (pi B1 B2) ([f] lam ([x] O2 x (app f (O1 x))))) (subtype/pi (%the ({x} subtype (A2' x) (B2 x) ([y] O2 x y)) Dsubtype2) (%the ({x} tsub A2 (O1 x) (A2' x)) Dtsub) (%the (subtype B1 A1 O1) Dsubtype1))) (%the (ofe G (lam M) (pi A1 A2)) (ofe/lam (%the ({x} %pi (isvar x I) %-> (ofe (cons G x A1) (M x) (A2 x))) DofM) (%the (wfe G A1) DwfA1))) (wfe/pi (%the ({x} %pi (isvar x K) %-> (wfe (cons G x B1) (B2 x))) DwfB2) (%the (wfe G B1) DwfB1)) (%the (sub ([f] lam ([x] O2 x (app f (O1 x)))) (lam M) (lam N)) (sub/lam (%the ({x} sub ([f] O2 x (app f (O1 x))) (lam M) (N x)) Dsub))) (principal/pi (%the ({x} principal (A2 x)) Dprincipal)) (%the (self (lam N) (pi B1 B2) (pi B1 B2')) (self/pi (%the ({x} self (N x) (B2 x) (B2' x)) Dself))) (%the (subtype (pi A1 A2) (pi B1 B2') ([_] lam N)) (subtype/pi Dsubtype2' Dtsub Dsubtype1))) + %<- ({x} principal-sub Dprincipal (Dtsub x) (%the (principal (A2' x)) (Dprincipal' x))) + %<- (subtype-reg-e Dsubtype1 DwfB1 DwfA1 (%the ({y} %pi (isvar y J) %-> (ofe (cons G y B1) (O1 y) A1)) DofO1)) + %<- ({y} {e isvar y J} ofe-context (DofO1 y e) (ordered/cons (%the (bounded G y) (Dbounded y e)))) + %<- (bump-wfe DwfB2 ([x] [d] ordered/cons (Dbounded x d)) (%the ({x} %pi (isvar x J) %-> (wfe (cons G x B1) (B2 x))) DwfB2bump)) + %<- (can-sub-context-e DofO1 DofM (%the ({y} sub ([x] M x) (O1 y) (Mx y)) DsubMx)) + %<- (subst-context-e DofO1 DofM DsubMx Dtsub (%the ({y} %pi (isvar y J) %-> (ofe (cons G y B1) (Mx y) (A2' y))) DofMx)) + %<- ({x} {y} sub-absent (O2 x y) (lam M) (%the (sub ([f] O2 x y) (lam M) (O2 x y)) (DsubO2f x y))) + %<- ({x} {d isvar x I} ofe-reg (DofM x d) (%the (wfe (cons G x A1) (A2 x)) (DwfA2 x d))) + %<- (tsubst-context-e DofO1 DwfA2 Dtsub (%the ({x} %pi (isvar x J) %-> (wfe (cons G x B1) (A2' x))) DwfA2')) + %<- ({x} {d isvar x J} subtype-reg-e (Dsubtype2 x) (DwfA2' x d) (DwfB2bump x d) (%the ({y} %pi (isvar y L) %-> (ofe (cons (cons G x B1) y (A2' x)) (O2 x y) (B2 x))) (DofO2 x d))) + %<- ({x} {d isvar x J} {y} {e isvar y L} ofe-context (DofO2 x d y e) (ordered/cons (bounded/cons _ (%the (precedes x y) (Dprec x d y e))))) + %<- ({x} {d isvar x J} can-sub-e ([_] append/nil) (DofO2 x d) (DofMx x d) (%the (sub ([y] O2 x y) (Mx x) (O2y x)) (DsubO2y x))) + %<- ({x} sub-absent (O1 x) (lam M) (%the (sub ([y] O1 x) (lam M) (O1 x)) (DsubO1y x))) + %<- ({x} {d isvar x J} weaken-ofe (Dbounded x d) (ofe/lam DofM DwfA1) _ (%the (ofe (cons G x B1) (lam M) (pi A1 A2)) (DofLam x d))) + %<- ({x} {d isvar x J} {z} {f isvar z L} weaken-wfe (bounded/cons (Dbounded x d) (Dprec x d z f)) (DwfA2' x d) _ (%the (wfe (cons (cons G x B1) z (pi A1 A2)) (A2' x)) (DwfA2'' x d z f))) + %<- ({x} {d isvar x J} {z} {f isvar z L} weaken-ofe (bounded/cons (Dbounded x d) (Dprec x d z f)) (DofO1 x d) _ (%the (ofe (cons (cons G x B1) z (pi A1 A2)) (O1 x) A1) (DofO1' x d z f))) + %<- ({x} {d isvar x J} {z} {f isvar z L} weaken-wfe' (append/cons (append/cons append/nil)) (ordered/cons (bounded/cons (Dbounded x d) (Dprec x d z f))) (wfe/pi DwfA2 DwfA1) (%the (wfe (cons (cons G x B1) z (pi A1 A2)) (pi A1 A2)) (DwfPi x d z f))) + %<- ({x} {d isvar x J} weaken-ofe-insert1 ([y] [e] DofO2 x d y e) ([z] [f isvar z L] bounded/cons (Dbounded x d) (Dprec x d z f)) (pi A1 A2) (%the ({z} + %pi (isvar z L) + %-> ({y} + %pi (isvar y H) + %-> (ofe (cons (cons (cons G x B1) z (pi A1 A2)) y (A2' x)) (O2 x y) (B2 x)))) (DofO2' x d))) + %<- ({x} {d isvar x J} ssub-ao-permute-e ([_] append/nil) (DofLam x d) ([z] [f] aofe/app (DwfA2'' x d z f) (Dtsub x) (DofO1' x d z f) (aofe/var (DwfPi x d z f) (lookup/hit (bounded/cons (Dbounded x d) (Dprec x d z f))))) ([z] [f] [y] [e] DofO2' x d z f y e) (DsubO2f x) (%the (aosub ([f] app f (O1 x)) (lam M) (Mx x)) (aosub/app (DsubMx x) (DsubO1y x) aosub/var)) (DsubO2y x) (Dsub x) (%the (term-eq (O2y x) (N x)) (Deq x))) + %<- ({x} sub-resp ([_] term-eq/i) term-eq/i (Deq x) (DsubO2y x) (%the (sub ([y] O2 x y) (Mx x) (N x)) (DsubN x))) + %<- ({x} {d isvar x J} subtype-self-e' (Dsubtype2 x) (DofMx x d) (DwfB2bump x d) (DsubN x) (Dprincipal' x) (Dself x) (%the (subtype (A2' x) (B2' x) ([_] N x)) (Dsubtype2' x))) %. +%term sigma + %pi (subtype-self-e' (%the (subtype (sigma A1 ([_] A2)) (sigma B1 B2) ([p] pair (O1 (pi1 p)) (O2 (pi1 p) (pi2 p)))) (subtype/sigma (%the ({x} subtype A2 (B2' x) ([y] O2 x y)) Dsubtype2) (%the ({x} tsub B2 (O1 x) (B2' x)) DtsubB2') (%the (subtype A1 B1 O1) Dsubtype1))) (%the (ofe G (pair M1 M2) (sigma A1 ([_] A2))) (ofe/pair (%the ({x} %pi (isvar x I) %-> (wfe (cons G x A1) A2)) DwfA2) (%the (ofe G M2 A2x) DofM2) (%the (tsub ([_] A2) M1 A2x) DsubA2x) (%the (ofe G M1 A1) DofM1))) (wfe/sigma (%the ({x} %pi (isvar x K) %-> (wfe (cons G x B1) (B2 x))) DwfB2) (%the (wfe G B1) DwfB1)) (%the (sub ([p] pair (O1 (pi1 p)) (O2 (pi1 p) (pi2 p))) (pair M1 M2) (pair N1 N2)) (sub/pair (%the (sub ([p] O2 (pi1 p) (pi2 p)) (pair M1 M2) N2) Dsub2) (%the (sub ([p] O1 (pi1 p)) (pair M1 M2) N1) Dsub1))) (%the (principal (sigma A1 ([_] A2))) (principal/sigma (%the (principal A2) Dprincipal2) (%the (principal A1) Dprincipal1))) (self/sigma (%the (self N2 B2'' B2self) Dself2) (%the (tsub B2 N1 B2'') Dsub) (%the (self N1 B1 B1self) Dself1)) (subtype/sigma ([_] Dsubtype2'') ([_] DtsubB2self) Dsubtype1')) + %<- (ofe-reg DofM1 (%the (wfe G A1) DwfA1)) + %<- (tsub-absent A2 M1 (%the (tsub ([_] A2) M1 A2) DsubA2x')) + %<- (tsub-fun DsubA2x DsubA2x' (%the (tp-eq A2x A2) DeqA2)) + %<- (ofe-resp ctx-eq/i term-eq/i DeqA2 DofM2 (%the (ofe G M2 A2) DofM2')) + %<- ({x} {d isvar x I} wfe-context (DwfA2 x d) (ordered/cons (%the (bounded G x) (Dbounded x d)))) + %<- ({x} sub-absent (O1 x) (pair M1 M2) (%the (sub ([p] O1 x) (pair M1 M2) (O1 x)) (DsubO1p x))) + %<- (subtype-reg-e Dsubtype1 DwfA1 DwfB1 (%the ({y} %pi (isvar y J) %-> (ofe (cons G y A1) (O1 y) B1)) DofO1)) + %<- (bump-ofe DofO1 ([x] [d] ordered/cons (Dbounded x d)) (%the ({x} %pi (isvar x I) %-> (ofe (cons G x A1) (O1 x) B1)) DofO1')) + %<- (can-sub-e ([_] append/nil) DofO1 DofM1 (%the (sub ([x] O1 x) M1 O1x) DsubO1x)) + %<- ({z} {f isvar z I} weaken-wfe (Dbounded z f) (wfe/sigma DwfA2 DwfA1) _ (%the (wfe (cons G z (sigma A1 ([_] A2))) (sigma A1 ([_] A2))) (DwfA1A2' z f))) + %<- (weaken-ofe-insert1 DofO1 Dbounded _ (%the ({z} + %pi (isvar z I) + %-> ({y} %pi (isvar y E) %-> (ofe (cons (cons G z (sigma A1 ([_] A2))) y A1) (O1 y) B1))) DofO1'')) + %<- (ssub-ao-permute-e ([_] append/nil) (ofe/pair DwfA2 DofM2 DsubA2x DofM1) ([z] [f] aofe/pi1 (aofe/var (DwfA1A2' z f) (lookup/hit (Dbounded z f)))) DofO1'' DsubO1p (%the (aosub ([p] pi1 p) (pair M1 M2) M1) (aosub/pi1 aosub/var)) DsubO1x Dsub1 (%the (term-eq O1x N1) Deq1)) + %<- (sub-resp ([_] term-eq/i) term-eq/i Deq1 DsubO1x (%the (sub ([x] O1 x) M1 N1) Dsub1')) + %<- (subtype-self-e' Dsubtype1 DofM1 DwfB1 Dsub1' Dprincipal1 Dself1 (%the (subtype A1 B1self ([_] N1)) Dsubtype1')) + %<- (tsub-absent B2self N1 (%the (tsub ([_] B2self) N1 B2self) DtsubB2self)) + %<- (weaken-wfe-insert1 DwfB2 Dbounded _ (%the ({x} + %pi (isvar x I) + %-> ({y} %pi (isvar y H) %-> (wfe (cons (cons G x A1) y B1) (B2 y)))) DwfB2weak)) + %<- ({x} {d isvar x I} tsubst-e ([_] append/nil) csub/base (DofO1' x d) (DtsubB2' x) (DwfB2weak x d) (%the (wfe (cons G x A1) (B2' x)) (DwfB2' x d))) + %<- ({x} {d isvar x I} subtype-reg-e (Dsubtype2 x) (DwfA2 x d) (DwfB2' x d) (%the ({y} %pi (isvar y L) %-> (ofe (cons (cons G x A1) y A2) (O2 x y) (B2' x))) (DofO2 x d))) + %<- (can-tsub-e ([_] append/nil) ([x] [d] DwfB2' x d) DofM1 (%the (tsub B2' M1 B2'x) DsubB2'x)) + %<- ({x} tsub-absent (B2 x) M1 (%the (tsub ([_] B2 x) M1 (B2 x)) (DsubB2y x))) + %<- (tsub-permute-e ([_] append/nil) DofM1 DofO1' DwfB2weak DsubB2y Dsub1' Dsub DtsubB2' DsubB2'x (%the (tp-eq B2'' B2'x) Deq2)) + %<- (tp-eq-symm Deq2 (%the (tp-eq B2'x B2'') Deq2')) + %<- (tsub-resp ([_] tp-eq/i) term-eq/i Deq2' DsubB2'x (%the (tsub B2' M1 B2'') DsubB2'')) + %<- (subtype-sub-e ([_] append/nil) csub/base DofM1 DwfA2 DwfB2' Dsubtype2 DsubA2x' DsubB2'' (%the ({y} sub ([x] O2 x y) M1 (O2x y)) DsubO2x) (%the (subtype A2 B2'' O2x) Dsubtype2')) + %<- (subst-simult-pair-e DofO2 DofM1 DofM2' Dsub2 DsubO2x (%the (sub ([y] O2x y) M2 N2) Dsub2')) + %<- (tsubst-e ([_] append/nil) csub/base DofM1 DsubB2'' DwfB2' (%the (wfe G B2'') DwfB2'')) + %<- (subtype-self-e' Dsubtype2' DofM2' DwfB2'' (%the (sub O2x M2 N2) Dsub2') Dprincipal2 Dself2 (%the (subtype A2 B2self ([_] N2)) Dsubtype2'')) %. +%scope subtype-self-e' %term sing_t + %pi (subtype-self-e' subtype/sing_t (ofe/sing _) _ Dsub principal/sing self/t Dsubtype) + %<- (sub-absent _ _ Dsub') + %<- (sub-fun Dsub Dsub' Deq) + %<- (term-eq-cdr-at Deq Deq') + %<- (tp-resp-atom sing Deq' Deq'') + %<- (subtype-resp Deq'' tp-eq/i ([_] term-eq/i) subtype/sing Dsubtype) %. +%term sing subtype-self-e' subtype/sing (ofe/sing _) _ Dsub principal/sing self/sing subtype/sing %. +%worlds (bind ovar) (subtype-self-e' _ _ _ _ _ _ _) %. +%total D (subtype-self-e' _ _ _ _ _ D _) %. +%sort subtype-self-e %. +%mode subtype-self-e %in %in %in %in %in %out %. +%term _ + %pi (subtype-self-e (%the (subtype A B ([_] M)) Dsubtype) (%the (wfe G A) DwfA) (%the (wfe G B) DwfB) (%the (principal A) DprincipalA) (%the (self M B B') Dself) Dsubtype') + %<- (inhabitation-e DwfA (%the (ofe G N A) DofN)) + %<- (subtype-reg-e' Dsubtype DwfA DwfB (%the (ofe G M B) DofM)) + %<- (sub-absent M N (%the (sub ([_] M) N M) DsubM)) + %<- (subtype-self-e' Dsubtype DofN DwfB DsubM DprincipalA Dself (%the (subtype A B' ([_] M)) Dsubtype')) %. +%worlds (bind ovar) (subtype-self-e _ _ _ _ _ _) %. +%total {} (subtype-self-e _ _ _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/subtype.lf b/new-tests/stelf-output/tslf/sing/subtype.lf new file mode 100644 index 0000000..a671b93 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/subtype.lf @@ -0,0 +1,32 @@ +%%%%% Subtyping and Coercions %%%%% +%sort subtype %. +%mode subtype %in %in %out %. +%term subtype/t subtype t t ([x] at x) %. +%term subtype/pi + %pi (subtype (pi A1 A2) (pi B1 B2) ([f] lam ([x] M2 x (app f (M1 x))))) + %<- (subtype B1 A1 ([x] M1 x)) + %<- ({x} tsub A2 (M1 x) (A2' x)) + %<- ({x} subtype (A2' x) (B2 x) ([y] M2 x y)) %. +%term subtype/sigma + %pi (subtype (sigma A1 A2) (sigma B1 B2) ([p] pair (M1 (pi1 p)) (M2 (pi1 p) (pi2 p)))) + %<- (subtype A1 B1 ([x] M1 x)) + %<- ({x} tsub B2 (M1 x) (B2' x)) + %<- ({x} subtype (A2 x) (B2' x) ([y] M2 x y)) %. +%term subtype/sing_t subtype (sing R) t ([x] at R) %. +%term subtype/sing subtype (sing R) (sing R) ([x] at R) %. +%%%%% Selfification %%%%% +%sort self %. +%term self/t self (at R) t (sing R) %. +%term self/pi + %pi (self (lam ([x] M x)) (pi A ([x] B x)) (pi A ([x] B' x))) + %<- ({x} self (M x) (B x) (B' x)) %. +%term self/sigma + %pi (self (pair M N) (sigma A ([x] B x)) (sigma A' ([_] B''))) + %<- (self M A A') + %<- (tsub B M B') + %<- (self N B' B'') %. +%term self/sing self (at R) (sing R) (sing R) %. +%sort principal %. +%term principal/sing principal (sing R) %. +%term principal/pi %pi (principal (pi A B)) %<- ({x} principal (B x)) %. +%term principal/sigma %pi (principal (sigma A ([_] B))) %<- (principal A) %<- (principal B) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/trans-fun-thm.lf b/new-tests/stelf-output/tslf/sing/trans-fun-thm.lf new file mode 100644 index 0000000..fc10892 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/trans-fun-thm.lf @@ -0,0 +1,77 @@ +%%%%% Translation Functionality Proof %%%%% +%sort kof-fun %. +%mode kof-fun %in %in %out %. +%term _ %pi (kof-fun (kof/i D Dckof) (kof/i D' Dckof) Deq) %<- (topen-fun D D' Deq) %. +%worlds (var) (kof-fun _ _ _) %. +%total {} (kof-fun _ _ _) %. +%sort trans-fun %. +%mode trans-fun %in %in %out %. +%sort ttrans-fun %. +%mode ttrans-fun %in %in %out %. +%scope ttrans-fun %term const + %pi (trans-fun (trans/const Dself Dexpand _ Dkof) (trans/const Dself' Dexpand' _ Dkof') Deq) + %<- (kof-fun Dkof Dkof' Deqt) + %<- (expand-resp atom-eq/i Deqt term-eq/i Dexpand Dexpand'') + %<- (expand-fun Dexpand'' Dexpand' Deqm) + %<- (self-resp Deqm Deqt tp-eq/i Dself Dself'') + %<- (self-fun Dself'' Dself' Deq) %. +%term var + %pi (trans-fun (trans/var (%the (self M A B) Dself) (%the (expand X A M) Dexpand) _ (%the (vof X A) Dvof) (%the (vtrans EM X) Dvtrans)) (trans/var (%the (self M' A B') Dself') (%the (expand X A M') Dexpand') _ (%the (vof X A) Dvof) (%the (vtrans EM X) Dvtrans)) DeqB) + %<- (expand-fun Dexpand Dexpand' (%the (term-eq M M') DeqM)) + %<- (self-resp DeqM tp-eq/i tp-eq/i Dself (%the (self M' A B) Dself'')) + %<- (self-fun Dself'' Dself' (%the (tp-eq B B') DeqB)) %. +%scope ttrans-fun %term forall + %pi (trans-fun (trans/forall (%the (self M (qtp A) B) Dself) (%the (expand (forall A) (qtp A) M) Dexpand) (%the (ttrans EA A) Dtrans)) (trans/forall (%the (self M' (qtp A') B') Dself') (%the (expand (forall A') (qtp A') M') Dexpand') (%the (ttrans EA A') Dtrans')) DeqB) + %<- (ttrans-fun Dtrans Dtrans' (%the (tp-eq A A') DeqA)) + %<- (atom-resp-tp forall DeqA (%the (atom-eq (forall A) (forall A')) DeqF)) + %<- (tp-resp-tp qtp DeqA (%the (tp-eq (qtp A) (qtp A')) DeqQ)) + %<- (expand-resp DeqF DeqQ term-eq/i Dexpand Dexpand'') + %<- (expand-fun Dexpand'' Dexpand' (%the (term-eq M M') DeqM)) + %<- (self-resp DeqM DeqQ tp-eq/i Dself (%the (self M' (qtp A') B) Dself'')) + %<- (self-fun Dself'' Dself' (%the (tp-eq B B') DeqB)) %. +%term app + %pi (trans-fun (trans/app Dsub Dsubtype DtransN DtransM) (trans/app Dsub' Dsubtype' DtransN' DtransM') Deq) + %<- (trans-fun DtransM DtransM' DeqPi) + %<- (tp-eq-cdr-pi DeqPi DeqA DeqB) + %<- (trans-fun DtransN DtransN' DeqC) + %<- (subtype-resp DeqC DeqA ([_] term-eq/i) Dsubtype Dsubtype'') + %<- (subtype-fun Dsubtype'' Dsubtype' (%the (%pi atom %-> (term-eq N N')) DeqN)) + %<- (tsub-resp DeqB (DeqN aca) tp-eq/i Dsub Dsub'') + %<- (tsub-fun Dsub'' Dsub' Deq) %. +%scope ttrans-fun %term pi1 + %pi (trans-fun (trans/pi1 Dtrans) (trans/pi1 Dtrans') D) + %<- (trans-fun Dtrans Dtrans' DeqSig) + %<- (tp-eq-cdr-sigma DeqSig D _) %. +%term pi2 + %pi (trans-fun (trans/pi2 Dtrans) (trans/pi2 Dtrans') (D aca)) + %<- (trans-fun Dtrans Dtrans' DeqSig) + %<- (tp-eq-cdr-sigma DeqSig _ D) %. +%scope ttrans-fun %term lam + %pi (trans-fun (trans/lam Dtrans Dttrans) (trans/lam Dtrans' Dttrans') Deq) + %<- (ttrans-fun Dttrans Dttrans' Deqt) + %<- (trans-resp-underbind Deqt Dtrans Dtrans'') + %<- ({x} {d} {ex} {xt} trans-fun (Dtrans'' x d ex xt) (Dtrans' x d ex xt) (Deqm x)) + %<- (pi-resp Deqt Deqm Deq) %. +%term pair + %pi (trans-fun (trans/pair Dtrans2 Dtrans1) (trans/pair Dtrans2' Dtrans1') Deq) + %<- (trans-fun Dtrans1 Dtrans1' Deq1) + %<- (trans-fun Dtrans2 Dtrans2' Deq2) + %<- (sigma-resp Deq1 ([_] Deq2) Deq) %. +%scope ttrans-fun %term t ttrans-fun ttrans/t ttrans/t tp-eq/i %. +%term pi + %pi (ttrans-fun (ttrans/pi Dtrans Dttrans) (ttrans/pi Dtrans' Dttrans') Deq) + %<- (ttrans-fun Dttrans Dttrans' Deqt) + %<- (ttrans-resp-underbind Deqt Dtrans Dtrans'') + %<- ({x} {d} {ex} {xt} ttrans-fun (Dtrans'' x d ex xt) (Dtrans' x d ex xt) (Deqm x)) + %<- (pi-resp Deqt Deqm Deq) %. +%scope ttrans-fun %term sigma + %pi (ttrans-fun (ttrans/sigma Dtrans Dttrans) (ttrans/sigma Dtrans' Dttrans') Deq) + %<- (ttrans-fun Dttrans Dttrans' Deqt) + %<- (ttrans-resp-underbind Deqt Dtrans Dtrans'') + %<- ({x} {d} {ex} {xt} ttrans-fun (Dtrans'' x d ex xt) (Dtrans' x d ex xt) (Deqm x)) + %<- (sigma-resp Deqt Deqm Deq) %. +%term sing + %pi (ttrans-fun (ttrans/sing Dtrans) (ttrans/sing Dtrans') Deq) + %<- (trans-fun Dtrans Dtrans' Deq) %. +%worlds (var bind tbind) (ttrans-fun _ _ _) (trans-fun _ _ _) %. +%total (D1 D2) (trans-fun _ D1 _) (ttrans-fun _ D2 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/trans-reg-thm.lf b/new-tests/stelf-output/tslf/sing/trans-reg-thm.lf new file mode 100644 index 0000000..de24d2f --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/trans-reg-thm.lf @@ -0,0 +1,651 @@ +%%%%% Translation Regularity (Implicit Context) %%%%% +%sort transe-context %. +%mode transe-context %in %out %. +%sort ttranse-context %. +%mode ttranse-context %in %out %. +%scope ttranse-context %term t ttranse-context (ttranse/t D) D %. +%term pi %pi (ttranse-context (ttranse/pi _ D) D') %<- (ttranse-context D D') %. +%scope ttranse-context %term sigma %pi (ttranse-context (ttranse/sigma _ D) D') %<- (ttranse-context D D') %. +%term sing %pi (ttranse-context (ttranse/sing D) D') %<- (transe-context D D') %. +%scope ttranse-context %term const transe-context (transe/const D _ _ _ _) D %. +%term forall %pi (transe-context (transe/forall _ _ D) D') %<- (ttranse-context D D') %. +%scope ttranse-context %term var %pi (transe-context (transe/var _ _ _ D _) D') %<- (lookup-context D D') %. +%term vari transe-context (transe/vari D _ _ _ _ _) D %. +%scope ttranse-context %term app %pi (transe-context (transe/app _ _ _ D) D') %<- (transe-context D D') %. +%term pi1 %pi (transe-context (transe/pi1 D) D') %<- (transe-context D D') %. +%scope ttranse-context %term pi2 %pi (transe-context (transe/pi2 D) D') %<- (transe-context D D') %. +%term lam %pi (transe-context (transe/lam _ D) D') %<- (ttranse-context D D') %. +%scope ttranse-context %term pair %pi (transe-context (transe/pair _ D) D') %<- (transe-context D D') %. +%worlds (var bind ovar obind tbind) (ttranse-context _ _) (transe-context _ _) %. +%total (D1 D2) (ttranse-context D1 _) (transe-context D2 _) %. +%%%%% Translation to Explicit Form %%%%% +%sort cut-ttrans {EB} {x} %. +%mode cut-ttrans %in %in %in %out %. +%sort cut-trans {EM} {x} %. +%mode cut-trans %in %in %in %out %. +%sort cut-ttranse {EB} {x} %. +%mode cut-ttranse %in %in %in %out %. +%sort cut-transe {EM} {x} %. +%mode cut-transe %in %in %in %out %. +%scope cut-transe %term t + %pi (cut-ttrans _ ([x] [d] [ex] [xt] ttrans/t) Dlook ([x] [d'] [ex] [xt] ttranse/t (Dord x d'))) + %<- ({x} {d'} lookup-context (Dlook x d') (Dord x d')) %. +%term pi + %pi (cut-ttrans _ ([x] [d] [ex] [xt] ttrans/pi (D2 x d ex xt) (D1 x d ex xt)) Dlook ([x] [d'] [ex] [xt] ttranse/pi (D2'' x d' ex xt) (D1' x d' ex xt))) + %<- (cut-ttrans _ D1 Dlook D1') + %<- ({x} {d'} lookup-context (Dlook x d') (Dord x d')) + %<- ({x} {d'} extend-context (Dord x d') (%the ({y} %pi (isvar y J) %-> (bounded (G x) y)) (Dbound x d'))) + %<- ({x} {d} {d'} {ex} {xt} cut-ttrans _ ([y] [e] [ey] [yt] D2 x d ex xt y e ey yt) ([y] [e'] lookup/hit (Dbound x d' y e')) ([y] [e'] [ey] [yt] D2' x d d' ex xt y e' ey yt)) + %<- ({y} {e'} {ey} {yt} cut-ttranse _ ([x] [d] [d'] [ex] [xt] D2' x d d' ex xt y e' ey yt) ([x] [d'] lookup/miss (Dlook x d') (Dbound x d' y e')) ([x] [d'] [ex] [xt] D2'' x d' ex xt y e' ey yt)) %. +%scope cut-transe %term sigma + %pi (cut-ttrans _ ([x] [d] [ex] [xt] ttrans/sigma (D2 x d ex xt) (D1 x d ex xt)) Dlook ([x] [d'] [ex] [xt] ttranse/sigma (D2'' x d' ex xt) (D1' x d' ex xt))) + %<- (cut-ttrans _ D1 Dlook D1') + %<- ({x} {d'} lookup-context (Dlook x d') (Dord x d')) + %<- ({x} {d'} extend-context (Dord x d') (%the ({y} %pi (isvar y J) %-> (bounded (G x) y)) (Dbound x d'))) + %<- ({x} {d} {d'} {ex} {xt} cut-ttrans _ ([y] [e] [ey] [yt] D2 x d ex xt y e ey yt) ([y] [e'] lookup/hit (Dbound x d' y e')) ([y] [e'] [ey] [yt] D2' x d d' ex xt y e' ey yt)) + %<- ({y} {e'} {ey} {yt} cut-ttranse _ ([x] [d] [d'] [ex] [xt] D2' x d d' ex xt y e' ey yt) ([x] [d'] lookup/miss (Dlook x d') (Dbound x d' y e')) ([x] [d'] [ex] [xt] D2'' x d' ex xt y e' ey yt)) %. +%term sing + %pi (cut-ttrans _ ([x] [d] [ex] [xt] ttrans/sing (D1 x d ex xt)) Dlook ([x] [d'] [ex] [xt] ttranse/sing (D1' x d' ex xt))) + %<- (cut-trans _ D1 Dlook D1') %. +%%% +%scope cut-transe %term const + %pi (cut-trans _ ([x] [d] [ex] [xt] trans/const (D3 x) (D2 x) (Dwf x d) (D1 x)) Dlook ([x] [d'] [ex] [xt] transe/const (Dord x d') (D3 x) (D2 x) (Dwf''' x) (D1 x))) + %<- (kof-strengthen D1 Deq) + %<- ({x} tp-eq-symm (Deq x) (Deq' x)) + %<- ({x} {d} wf-resp (Deq x) (Dwf x d) (Dwf' x d)) + %<- (strengthen-wf Dwf' Dwf'') + %<- ({x} wf-resp (Deq' x) Dwf'' (Dwf''' x)) + %<- ({x} {d'} lookup-context (Dlook x d') (Dord x d')) %. +%term forall + %pi (cut-trans _ ([x] [d] [ex] [xt] trans/forall (D3 x) (D2 x) (D1 x d ex xt)) Dlook ([x] [d'] [ex] [xt] transe/forall (D3 x) (D2 x) (D1' x d' ex xt))) + %<- (cut-ttrans _ D1 Dlook D1') %. +%scope cut-transe %term varsam + %pi (cut-trans _ ([x] [d] [ex] [xt] trans/var (D5 x) (D4 x) (Dwf x d) d (D1 x ex xt)) Dlook ([x] [d'] [ex] [xt] transe/var (D5 x) (D4 x) (Dwfe x d') (Dlook x d') (D1 x ex xt))) + %<- ({x} {d} can-mwf (Dwf x d) (Dmwf x d)) + %<- (cut-wf _ Dwf Dmwf Dlook Dwfe _) %. +%term varoth + %pi (cut-trans _ ([x] [d] [ex] [xt] trans/var (D5 x) (D4 x) (Dwf x d) D2 D1) Dlook ([x] [d'] [ex] [xt] transe/vari (Dord x d') (D5 x) (D4 x) Dwf' D2 D1)) + %<- ({x} {d'} lookup-context (Dlook x d') (Dord x d')) + %<- ({x} {d} can-mwf (Dwf x d) (Dmwf x d)) + %<- (strengthen-for-cut-wf _ Dwf Dmwf Dwf' _) %. +%scope cut-transe %term lam + %pi (cut-trans _ ([x] [d] [ex] [xt] trans/lam (D2 x d ex xt) (D1 x d ex xt)) Dlook ([x] [d'] [ex] [xt] transe/lam (D2'' x d' ex xt) (D1' x d' ex xt))) + %<- (cut-ttrans _ D1 Dlook D1') + %<- ({x} {d'} lookup-context (Dlook x d') (Dord x d')) + %<- ({x} {d'} extend-context (Dord x d') (%the ({y} %pi (isvar y J) %-> (bounded (G x) y)) (Dbound x d'))) + %<- ({x} {d} {d'} {ex} {xt} cut-trans _ ([y] [e] [ey] [yt] D2 x d ex xt y e ey yt) ([y] [e'] lookup/hit (Dbound x d' y e')) ([y] [e'] [ey] [yt] D2' x d d' ex xt y e' ey yt)) + %<- ({y} {e'} {ey} {yt} cut-transe _ ([x] [d] [d'] [ex] [xt] D2' x d d' ex xt y e' ey yt) ([x] [d'] lookup/miss (Dlook x d') (Dbound x d' y e')) ([x] [d'] [ex] [xt] D2'' x d' ex xt y e' ey yt)) %. +%term app + %pi (cut-trans _ ([x] [d] [ex] [xt] trans/app (D4 x) (D3 x) (D2 x d ex xt) (D1 x d ex xt)) Dlook ([x] [d] [ex] [xt] transe/app (D4 x) (D3 x) (D2' x d ex xt) (D1' x d ex xt))) + %<- (cut-trans _ D1 Dlook D1') + %<- (cut-trans _ D2 Dlook D2') %. +%scope cut-transe %term pi1 + %pi (cut-trans _ ([x] [d] [ex] [xt] trans/pi1 (D1 x d ex xt)) Dlook ([x] [d] [ex] [xt] transe/pi1 (D1' x d ex xt))) + %<- (cut-trans _ D1 Dlook D1') %. +%term pi2 + %pi (cut-trans _ ([x] [d] [ex] [xt] trans/pi2 (D1 x d ex xt)) Dlook ([x] [d] [ex] [xt] transe/pi2 (D1' x d ex xt))) + %<- (cut-trans _ D1 Dlook D1') %. +%scope cut-transe %term pair + %pi (cut-trans _ ([x] [d] [ex] [xt] trans/pair (D2 x d ex xt) (D1 x d ex xt)) Dlook ([x] [d] [ex] [xt] transe/pair (D2' x d ex xt) (D1' x d ex xt))) + %<- (cut-trans _ D1 Dlook D1') + %<- (cut-trans _ D2 Dlook D2') %. +%term t cut-ttranse _ ([x] [d] [d'] [ex] [xt] ttranse/t (Dord x d')) Dlook ([x] [d'] [ex] [xt] ttranse/t (Dord x d')) %. +%%% +%scope cut-transe %term pi + %pi (cut-ttranse _ ([x] [d] [d'] [ex] [xt] ttranse/pi (D2 x d d' ex xt) (D1 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] ttranse/pi (D2'' x d' ex xt) (D1' x d' ex xt))) + %<- (cut-ttranse _ D1 Dlook D1') + %<- ({x} {d} {d'} {ex} {xt} {y} {e'} {ey} {yt} ttranse-context (D2 x d d' ex xt y e' ey yt) (ordered/cons (Dbound x d' y e'))) + %<- ({x} {d} {d'} {ex} {xt} cut-ttranse _ ([y] [e] [e'] [ey] [yt] D2 x d d' ex xt y e' ey yt) ([y] [e'] lookup/hit (Dbound x d' y e')) ([y] [e'] [ey] [yt] D2' x d d' ex xt y e' ey yt)) + %<- ({y} {e'} {ey} {yt} cut-ttranse _ ([x] [d] [d'] [ex] [xt] D2' x d d' ex xt y e' ey yt) ([x] [d'] lookup/miss (Dlook x d') (Dbound x d' y e')) ([x] [d'] [ex] [xt] D2'' x d' ex xt y e' ey yt)) %. +%term sigma + %pi (cut-ttranse _ ([x] [d] [d'] [ex] [xt] ttranse/sigma (D2 x d d' ex xt) (D1 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] ttranse/sigma (D2'' x d' ex xt) (D1' x d' ex xt))) + %<- (cut-ttranse _ D1 Dlook D1') + %<- ({x} {d} {d'} {ex} {xt} {y} {e'} {ey} {yt} ttranse-context (D2 x d d' ex xt y e' ey yt) (ordered/cons (Dbound x d' y e'))) + %<- ({x} {d} {d'} {ex} {xt} cut-ttranse _ ([y] [e] [e'] [ey] [yt] D2 x d d' ex xt y e' ey yt) ([y] [e'] lookup/hit (Dbound x d' y e')) ([y] [e'] [ey] [yt] D2' x d d' ex xt y e' ey yt)) + %<- ({y} {e'} {ey} {yt} cut-ttranse _ ([x] [d] [d'] [ex] [xt] D2' x d d' ex xt y e' ey yt) ([x] [d'] lookup/miss (Dlook x d') (Dbound x d' y e')) ([x] [d'] [ex] [xt] D2'' x d' ex xt y e' ey yt)) %. +%scope cut-transe %term sing + %pi (cut-ttranse _ ([x] [d] [d'] [ex] [xt] ttranse/sing (D1 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] ttranse/sing (D1' x d' ex xt))) + %<- (cut-transe _ D1 Dlook D1') %. +%term const + %pi (cut-transe _ ([x] [d] [d'] [ex] [xt] transe/const (Dord x d') (D3 x) (D2 x) (Dwf x d) (D1 x)) Dlook ([x] [d'] [ex] [xt] transe/const (Dord x d') (D3 x) (D2 x) (Dwf''' x) (D1 x))) + %<- (kof-strengthen D1 Deq) + %<- ({x} tp-eq-symm (Deq x) (Deq' x)) + %<- ({x} {d} wf-resp (Deq x) (Dwf x d) (Dwf' x d)) + %<- (strengthen-wf Dwf' Dwf'') + %<- ({x} wf-resp (Deq' x) Dwf'' (Dwf''' x)) %. +%%% +%scope cut-transe %term var + %pi (cut-transe _ ([x] [d] [d'] [ex] [xt] transe/var (D5 x) (D4 x) (Dwfe x d d') (D2 x d') (D1 x ex xt)) Dlook ([x] [d'] [ex] [xt] transe/var (D5 x) (D4 x) (Dwfe' x d') (D2 x d') (D1 x ex xt))) + %<- ({x} {d} {d'} can-mwfe (Dwfe x d d') (Dmwfe x d d')) + %<- (cut-wfe _ Dwfe Dmwfe Dlook Dwfe' _) %. +%term forall + %pi (cut-transe _ ([x] [d] [d'] [ex] [xt] transe/forall (D3 x) (D2 x) (D1 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] transe/forall (D3 x) (D2 x) (D1' x d' ex xt))) + %<- (cut-ttranse _ D1 Dlook D1') %. +%scope cut-transe %term varsam + %pi (cut-transe _ ([x] [d] [d'] [ex] [xt] transe/vari (D6 x d') (D5 x) (D4 x) (Dwf x d) d (D1 x ex xt)) Dlook ([x] [d'] [ex] [xt] transe/var (D5 x) (D4 x) (Dwfe x d') (Dlook x d') (D1 x ex xt))) + %<- ({x} {d} can-mwf (Dwf x d) (Dmwf x d)) + %<- (cut-wf _ Dwf Dmwf Dlook Dwfe _) %. +%term varoth + %pi (cut-transe _ ([x] [d] [d'] [ex] [xt] transe/vari (D6 x d') (D5 x) (D4 x) (Dwf x d) D2 D1) Dlook ([x] [d'] [ex] [xt] transe/vari (D6 x d') (D5 x) (D4 x) Dwf' D2 D1)) + %<- ({x} {d} can-mwf (Dwf x d) (Dmwf x d)) + %<- (strengthen-for-cut-wf _ Dwf Dmwf Dwf' _) %. +%scope cut-transe %term lam + %pi (cut-transe _ ([x] [d] [d'] [ex] [xt] transe/lam (D2 x d d' ex xt) (D1 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] transe/lam (D2'' x d' ex xt) (D1' x d' ex xt))) + %<- (cut-ttranse _ D1 Dlook D1') + %<- ({x} {d} {d'} {ex} {xt} {y} {e'} {ey} {yt} transe-context (D2 x d d' ex xt y e' ey yt) (ordered/cons (Dbound x d' y e'))) + %<- ({x} {d} {d'} {ex} {xt} cut-transe _ ([y] [e] [e'] [ey] [yt] D2 x d d' ex xt y e' ey yt) ([y] [e'] lookup/hit (Dbound x d' y e')) ([y] [e'] [ey] [yt] D2' x d d' ex xt y e' ey yt)) + %<- ({y} {e'} {ey} {yt} cut-transe _ ([x] [d] [d'] [ex] [xt] D2' x d d' ex xt y e' ey yt) ([x] [d'] lookup/miss (Dlook x d') (Dbound x d' y e')) ([x] [d'] [ex] [xt] D2'' x d' ex xt y e' ey yt)) %. +%term app + %pi (cut-transe _ ([x] [d] [d'] [ex] [xt] transe/app (D4 x) (D3 x) (D2 x d d' ex xt) (D1 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] transe/app (D4 x) (D3 x) (D2' x d' ex xt) (D1' x d' ex xt))) + %<- (cut-transe _ D1 Dlook D1') + %<- (cut-transe _ D2 Dlook D2') %. +%scope cut-transe %term pi1 + %pi (cut-transe _ ([x] [d] [d'] [ex] [xt] transe/pi1 (D1 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] transe/pi1 (D1' x d' ex xt))) + %<- (cut-transe _ D1 Dlook D1') %. +%term pi2 + %pi (cut-transe _ ([x] [d] [d'] [ex] [xt] transe/pi2 (D1 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] transe/pi2 (D1' x d' ex xt))) + %<- (cut-transe _ D1 Dlook D1') %. +%scope cut-transe %term pair + %pi (cut-transe _ ([x] [d] [d'] [ex] [xt] transe/pair (D2 x d d' ex xt) (D1 x d d' ex xt)) Dlook ([x] [d'] [ex] [xt] transe/pair (D2' x d' ex xt) (D1' x d' ex xt))) + %<- (cut-transe _ D1 Dlook D1') + %<- (cut-transe _ D2 Dlook D2') %. +%worlds (var bind ovar obind tbind) (cut-ttrans _ _ _ _) (cut-trans _ _ _ _) (cut-transe _ _ _ _) (cut-ttranse _ _ _ _) %. +%total (A1 M1 A2 M2) (cut-ttrans A1 _ _ _) (cut-trans M1 _ _ _) (cut-transe A2 _ _ _) (cut-ttranse M2 _ _ _) %. +%sort trans1-to-transe {I} {x} %. +%mode trans1-to-transe %in %in %out %. +%term _ + %pi (trans1-to-transe I Dtrans Dtranse) + %<- (cut-trans EM Dtrans ([x] [d] lookup/hit (bounded/nil d)) Dtranse) %. +%worlds (var bind ovar obind tbind) (trans1-to-transe _ _ _) %. +%total {} (trans1-to-transe _ _ _) %. +%sort ttrans1-to-ttranse {I} {x} %. +%mode ttrans1-to-ttranse %in %in %out %. +%term _ + %pi (ttrans1-to-ttranse I Dtrans Dtranse) + %<- (cut-ttrans EM Dtrans ([x] [d] lookup/hit (bounded/nil d)) Dtranse) %. +%worlds (var bind ovar obind tbind) (ttrans1-to-ttranse _ _ _) %. +%total {} (ttrans1-to-ttranse _ _ _) %. +%sort trans-to-transe %. +%mode trans-to-transe %in %out %. +%sort ttrans-to-ttranse %. +%mode ttrans-to-ttranse %in %out %. +%scope ttrans-to-ttranse %term t ttrans-to-ttranse ttrans/t (ttranse/t ordered/nil) %. +%term pi + %pi (ttrans-to-ttranse (ttrans/pi D2 D1) (ttranse/pi D2' D1')) + %<- (ttrans-to-ttranse D1 D1') + %<- (ttrans1-to-ttranse 0 D2 D2') %. +%scope ttrans-to-ttranse %term sigma + %pi (ttrans-to-ttranse (ttrans/sigma D2 D1) (ttranse/sigma D2' D1')) + %<- (ttrans-to-ttranse D1 D1') + %<- (ttrans1-to-ttranse 0 D2 D2') %. +%term sing + %pi (ttrans-to-ttranse (ttrans/sing D) (ttranse/sing D')) + %<- (trans-to-transe D D') %. +%scope ttrans-to-ttranse %term const trans-to-transe (trans/const D4 D3 D2 D1) (transe/const ordered/nil D4 D3 D2 D1) %. +%term forall + %pi (trans-to-transe (trans/forall D3 D2 D1) (transe/forall D3 D2 D1')) + %<- (ttrans-to-ttranse D1 D1') %. +%scope ttrans-to-ttranse %term var trans-to-transe (trans/var D5 D4 D3 D2 D1) (transe/vari ordered/nil D5 D4 D3 D2 D1) %. +%term app + %pi (trans-to-transe (trans/app D4 D3 D2 D1) (transe/app D4 D3 D2' D1')) + %<- (trans-to-transe D1 D1') + %<- (trans-to-transe D2 D2') %. +%scope ttrans-to-ttranse %term pi1 %pi (trans-to-transe (trans/pi1 D) (transe/pi1 D')) %<- (trans-to-transe D D') %. +%term pi2 %pi (trans-to-transe (trans/pi2 D) (transe/pi2 D')) %<- (trans-to-transe D D') %. +%scope ttrans-to-ttranse %term lam + %pi (trans-to-transe (trans/lam D2 D1) (transe/lam D2' D1')) + %<- (ttrans-to-ttranse D1 D1') + %<- (trans1-to-transe 0 D2 D2') %. +%term pair + %pi (trans-to-transe (trans/pair D2 D1) (transe/pair D2' D1')) + %<- (trans-to-transe D1 D1') + %<- (trans-to-transe D2 D2') %. +%worlds (var bind ovar tbind) (trans-to-transe _ _) (ttrans-to-ttranse _ _) %. +%total (D1 D2) (trans-to-transe D1 _) (ttrans-to-ttranse D2 _) %. +%%%%% Translation to Implicit Form %%%%% +%sort transi %. +%term transi/nil %pi (transi nil EM B) %<- (trans EM B) %. +%term transi/cons %pi (transi (cons G X A) EM B) %<- (%pi (vof X A) %-> (transi G EM B)) %. +%sort ttransi %. +%term ttransi/nil %pi (ttransi nil EB B) %<- (ttrans EB B) %. +%term ttransi/cons %pi (ttransi (cons G X A) EB B) %<- (%pi (vof X A) %-> (ttransi G EB B)) %. +%sort ttransi-t {G} %. +%mode ttransi-t %in %out %. +%term _ ttransi-t nil (ttransi/nil ttrans/t) %. +%term _ %pi (ttransi-t (cons G X A) (ttransi/cons D)) %<- ({d} ttransi-t G (D d)) %. +%worlds (bind var ovar tbind ofblock) (ttransi-t _ _) %. +%total G (ttransi-t G _) %. +%sort ttransi-pi %. +%mode ttransi-pi %in %in %out %. +%term _ ttransi-pi (ttransi/nil D1) ([x] [ex] [xt] ttransi/cons ([d] ttransi/nil (D2 x d ex xt))) (ttransi/nil (ttrans/pi D2 D1)) %. +%term _ + %pi (ttransi-pi (ttransi/cons D1) ([x] [ex] [xt] ttransi/cons ([d] ttransi/cons ([e] D2 x d ex xt e))) (ttransi/cons D)) + %<- ({e} ttransi-pi (D1 e) ([x] [ex] [xt] ttransi/cons ([d] D2 x d ex xt e)) (D e)) %. +%worlds (bind var ovar tbind ofblock) (ttransi-pi _ _ _) %. +%total D (ttransi-pi D _ _) %. +%sort ttransi-sigma %. +%mode ttransi-sigma %in %in %out %. +%term _ ttransi-sigma (ttransi/nil D1) ([x] [ex] [xt] ttransi/cons ([d] ttransi/nil (D2 x d ex xt))) (ttransi/nil (ttrans/sigma D2 D1)) %. +%term _ + %pi (ttransi-sigma (ttransi/cons D1) ([x] [ex] [xt] ttransi/cons ([d] ttransi/cons ([e] D2 x d ex xt e))) (ttransi/cons D)) + %<- ({e} ttransi-sigma (D1 e) ([x] [ex] [xt] ttransi/cons ([d] D2 x d ex xt e)) (D e)) %. +%worlds (bind var ovar tbind ofblock) (ttransi-sigma _ _ _) %. +%total D (ttransi-sigma D _ _) %. +%sort ttransi-sing %. +%mode ttransi-sing %in %out %. +%term _ ttransi-sing (transi/nil D) (ttransi/nil (ttrans/sing D)) %. +%term _ + %pi (ttransi-sing (transi/cons D) (ttransi/cons D')) + %<- ({d} ttransi-sing (D d) (D' d)) %. +%worlds (bind var ovar tbind ofblock) (ttransi-sing _ _) %. +%total D (ttransi-sing D _) %. +%sort transi-forall %. +%mode transi-forall %in %in %in %out %. +%term _ transi-forall (ttransi/nil D1) D2 D3 (transi/nil (trans/forall D3 D2 D1)) %. +%term _ + %pi (transi-forall (ttransi/cons D1) D2 D3 (transi/cons D)) + %<- ({d} transi-forall (D1 d) D2 D3 (D d)) %. +%worlds (bind var ovar tbind ofblock) (transi-forall _ _ _ _) %. +%total D (transi-forall D _ _ _) %. +%sort transi-app %. +%mode transi-app %in %in %in %in %out %. +%term _ transi-app (transi/nil D1) (transi/nil D2) D3 D4 (transi/nil (trans/app D4 D3 D2 D1)) %. +%term _ + %pi (transi-app (transi/cons D1) (transi/cons D2) D3 D4 (transi/cons D)) + %<- ({d} transi-app (D1 d) (D2 d) D3 D4 (D d)) %. +%worlds (bind var ovar tbind ofblock) (transi-app _ _ _ _ _) %. +%total D (transi-app D _ _ _ _) %. +%sort transi-pi1 %. +%mode transi-pi1 %in %out %. +%term _ transi-pi1 (transi/nil D) (transi/nil (trans/pi1 D)) %. +%term _ + %pi (transi-pi1 (transi/cons D) (transi/cons D')) + %<- ({d} transi-pi1 (D d) (D' d)) %. +%worlds (bind var ovar tbind ofblock) (transi-pi1 _ _) %. +%total D (transi-pi1 D _) %. +%sort transi-pi2 %. +%mode transi-pi2 %in %out %. +%term _ transi-pi2 (transi/nil D) (transi/nil (trans/pi2 D)) %. +%term _ + %pi (transi-pi2 (transi/cons D) (transi/cons D')) + %<- ({d} transi-pi2 (D d) (D' d)) %. +%worlds (bind var ovar tbind ofblock) (transi-pi2 _ _) %. +%total D (transi-pi2 D _) %. +%sort transi-lam %. +%mode transi-lam %in %in %out %. +%term _ transi-lam (ttransi/nil D1) ([x] [ex] [xt] transi/cons ([d] transi/nil (D2 x d ex xt))) (transi/nil (trans/lam D2 D1)) %. +%term _ + %pi (transi-lam (ttransi/cons D1) ([x] [ex] [xt] transi/cons ([d] transi/cons ([e] D2 x d ex xt e))) (transi/cons D)) + %<- ({e} transi-lam (D1 e) ([x] [ex] [xt] transi/cons ([d] D2 x d ex xt e)) (D e)) %. +%worlds (bind var ovar tbind ofblock) (transi-lam _ _ _) %. +%total D (transi-lam D _ _) %. +%sort transi-pair %. +%mode transi-pair %in %in %out %. +%term _ transi-pair (transi/nil D1) (transi/nil D2) (transi/nil (trans/pair D2 D1)) %. +%term _ + %pi (transi-pair (transi/cons D1) (transi/cons D2) (transi/cons D)) + %<- ({d} transi-pair (D1 d) (D2 d) (D d)) %. +%worlds (bind var ovar tbind ofblock) (transi-pair _ _ _) %. +%total D (transi-pair D _ _) %. +%sort transi-var* %. +%mode transi-var* %in %in %out %. +%term _ transi-var* D (wfi/nil D') (transi/nil (D D')) %. +%term _ + %pi (transi-var* D (wfi/cons D') (transi/cons D'')) + %<- ({d} transi-var* D (D' d) (D'' d)) %. +%worlds (bind var ovar tbind ofblock) (transi-var* _ _ _) %. +%total D (transi-var* _ D _) %. +%sort transi-var %. +%mode transi-var %in %in %in %in %in %out %. +%term _ + %pi (transi-var Dvtrans (lookup/hit _) (wfi/cons Dwfi) Dexpand Dself (transi/cons D)) + %<- ({d} transi-var* ([d_wf] trans/var Dself Dexpand d_wf d Dvtrans) (Dwfi d) (D d)) %. +%term _ + %pi (transi-var D1 (lookup/miss D2 _) (wfi/cons D3) D4 D5 (transi/cons D)) + %<- ({d} transi-var D1 D2 (D3 d) D4 D5 (D d)) %. +%worlds (bind var ovar tbind ofblock) (transi-var _ _ _ _ _ _) %. +%total D (transi-var _ D _ _ _ _) %. +%sort trans-to-transi {G} %. +%mode trans-to-transi %in %in %out %. +%term _ trans-to-transi _ D (transi/nil D) %. +%term _ %pi (trans-to-transi _ D (transi/cons ([d] D'))) %<- (trans-to-transi _ D D') %. +%worlds (bind var ovar tbind ofblock) (trans-to-transi _ _ _) %. +%total G (trans-to-transi G _ _) %. +%sort transe-to-transi %. +%mode transe-to-transi %in %out %. +%sort ttranse-to-ttransi %. +%mode ttranse-to-ttransi %in %out %. +%scope ttranse-to-ttransi %term const + %pi (transe-to-transi (transe/const _ D4 D3 D2 D1) D) + %<- (trans-to-transi _ (trans/const D4 D3 D2 D1) D) %. +%term forall + %pi (transe-to-transi (transe/forall D3 D2 D1) D) + %<- (ttranse-to-ttransi D1 D1') + %<- (transi-forall D1' D2 D3 D) %. +%scope ttranse-to-ttransi %term var + %pi (transe-to-transi (transe/var D5 D4 D3 D2 D1) D) + %<- (wfe-to-wfi D3 D3') + %<- (transi-var D1 D2 D3' D4 D5 D) %. +%term vari + %pi (transe-to-transi (transe/vari _ D5 D4 D3 D2 D1) D) + %<- (trans-to-transi _ (trans/var D5 D4 D3 D2 D1) D) %. +%scope ttranse-to-ttransi %term app + %pi (transe-to-transi (transe/app D4 D3 D2 D1) D) + %<- (transe-to-transi D1 D1') + %<- (transe-to-transi D2 D2') + %<- (transi-app D1' D2' D3 D4 D) %. +%term pi1 + %pi (transe-to-transi (transe/pi1 D1) D) + %<- (transe-to-transi D1 D1') + %<- (transi-pi1 D1' D) %. +%scope ttranse-to-ttransi %term pi2 + %pi (transe-to-transi (transe/pi2 D1) D) + %<- (transe-to-transi D1 D1') + %<- (transi-pi2 D1' D) %. +%term lam + %pi (transe-to-transi (transe/lam D2 D1) D) + %<- (ttranse-to-ttransi D1 D1') + %<- ({x} {d} {ex} {xt} transe-to-transi (D2 x d ex xt) (D2' x ex xt)) + %<- (transi-lam D1' D2' D) %. +%scope ttranse-to-ttransi %term pair + %pi (transe-to-transi (transe/pair D2 D1) D) + %<- (transe-to-transi D1 D1') + %<- (transe-to-transi D2 D2') + %<- (transi-pair D1' D2' D) %. +%term t %pi (ttranse-to-ttransi (ttranse/t _) D) %<- (ttransi-t _ D) %. +%scope ttranse-to-ttransi %term pi + %pi (ttranse-to-ttransi (ttranse/pi D2 D1) D) + %<- (ttranse-to-ttransi D1 D1') + %<- ({x} {d} {ex} {xt} ttranse-to-ttransi (D2 x d ex xt) (D2' x ex xt)) + %<- (ttransi-pi D1' D2' D) %. +%term sigma + %pi (ttranse-to-ttransi (ttranse/sigma D2 D1) D) + %<- (ttranse-to-ttransi D1 D1') + %<- ({x} {d} {ex} {xt} ttranse-to-ttransi (D2 x d ex xt) (D2' x ex xt)) + %<- (ttransi-sigma D1' D2' D) %. +%scope ttranse-to-ttransi %term sing + %pi (ttranse-to-ttransi (ttranse/sing D1) D) + %<- (transe-to-transi D1 D1') + %<- (ttransi-sing D1' D) %. +%worlds (bind var ovar tbind ofblock) (transe-to-transi _ _) (ttranse-to-ttransi _ _) %. +%total (D1 D2) (transe-to-transi D1 _) (ttranse-to-ttransi D2 _) %. +%sort ttranse-to-ttrans %. +%mode ttranse-to-ttrans %in %out %. +%term _ %pi (ttranse-to-ttrans D D') %<- (ttranse-to-ttransi D (ttransi/nil D')) %. +%worlds (var bind ovar tbind) (ttranse-to-ttrans _ _) %. +%total {} (ttranse-to-ttrans _ _) %. +%sort transe-to-trans %. +%mode transe-to-trans %in %out %. +%term _ %pi (transe-to-trans D D') %<- (transe-to-transi D (transi/nil D')) %. +%worlds (var bind ovar tbind) (transe-to-trans _ _) %. +%total {} (transe-to-trans _ _) %. +%%%%% Translation Weakening %%%%% +%sort bump-transe {EM} {x} %. +%mode bump-transe %in %in %in %out %. +%sort bump-ttranse {EA} {x} %. +%mode bump-ttranse %in %in %in %out %. +%scope bump-ttranse %term const bump-transe _ ([x] [d] [ex] [xt] transe/const (Dord' x d) (Dself x) (Dexpand x) (Dwf x) (Dkof x)) Dord ([x] [d] [ex] [xt] transe/const (Dord x d) (Dself x) (Dexpand x) (Dwf x) (Dkof x)) %. +%term forall + %pi (bump-transe _ ([x] [d] [ex] [xt] transe/forall (D3 x) (D2 x) (D1 x d ex xt)) Dord ([x] [d] [ex] [xt] transe/forall (D3 x) (D2 x) (D1' x d ex xt))) + %<- (bump-ttranse _ D1 Dord D1') %. +%scope bump-ttranse %term var + %pi (bump-transe _ ([x] [d] [ex] [xt] transe/var (Dself x) (Dexpand x) (Dwf x d) (Dlookup x d) (Dvtrans x ex xt)) Dord ([x] [d] [ex] [xt] transe/var (Dself x) (Dexpand x) (Dwf' x d) (Dlookup' x d) (Dvtrans x ex xt))) + %<- (bump-wfe Dwf Dord Dwf') + %<- (bump-lookup Dlookup Dord Dlookup') %. +%term vari bump-transe _ ([x] [d] [ex] [xt] transe/vari (Dord' x d) (Dself x) (Dexpand x) (Dwf x) (Dvof x) (Dvtrans x ex xt)) Dord ([x] [d] [ex] [xt] transe/vari (Dord x d) (Dself x) (Dexpand x) (Dwf x) (Dvof x) (Dvtrans x ex xt)) %. +%scope bump-ttranse %term app + %pi (bump-transe _ ([x] [d] [ex] [xt] transe/app (Dsub x) (Dsubtype x) (DtransN x d ex xt) (DtransM x d ex xt)) Dord ([x] [d] [ex] [xt] transe/app (Dsub x) (Dsubtype x) (DtransN' x d ex xt) (DtransM' x d ex xt))) + %<- (bump-transe _ DtransM Dord DtransM') + %<- (bump-transe _ DtransN Dord DtransN') %. +%term pi1 + %pi (bump-transe _ ([x] [d] [ex] [xt] transe/pi1 (Dtrans x d ex xt)) Dord ([x] [d] [ex] [xt] transe/pi1 (Dtrans' x d ex xt))) + %<- (bump-transe _ Dtrans Dord Dtrans') %. +%scope bump-ttranse %term pi2 + %pi (bump-transe _ ([x] [d] [ex] [xt] transe/pi2 (Dtrans x d ex xt)) Dord ([x] [d] [ex] [xt] transe/pi2 (Dtrans' x d ex xt))) + %<- (bump-transe _ Dtrans Dord Dtrans') %. +%term lam + %pi (bump-transe _ ([x] [d isvar x I] [ex] [xt] transe/lam (Dtrans x d ex xt) (Dttrans x d ex xt)) (%the ({x} %pi (isvar x I') %-> (ordered (G x))) Dord) ([x] [d] [ex] [xt] transe/lam (Dtrans'' x d ex xt) (Dttrans' x d ex xt))) + %<- (bump-ttranse _ Dttrans Dord Dttrans') + %<- ({x} {d isvar x I} {ex} {xt} {y} {e isvar y J1} {ey} {yt} transe-context (Dtrans x d ex xt y e ey yt) (ordered/cons (%the (bounded (G x) y) (Dbound1 x d y e)))) + %<- ({x} {d isvar x I'} extend-context (Dord x d) (%the ({y} %pi (isvar y J2) %-> (bounded (G x) y)) (Dbound2 x d))) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- ({x} {d isvar x I} bump-bounded Dleq1 ([y] [e isvar y J1] Dbound1 x d y e) ([y] [e isvar y J] %the (bounded (G x) y) (Dbound1' x d y e))) + %<- ({x} {d isvar x I'} bump-bounded Dleq2 ([y] [e isvar y J2] Dbound2 x d y e) ([y] [e isvar y J] %the (bounded (G x) y) (Dbound2' x d y e))) + %<- ({x} {d isvar x I} {ex} {xt} bump-transe _ ([y] [e] [ey] [yt] Dtrans x d ex xt y e ey yt) ([y] [e] ordered/cons (Dbound1' x d y e)) ([y] [e] [ey] [yt] Dtrans' x d ex xt y e ey yt)) + %<- ({y} {e isvar y J} {ey} {yt} bump-transe _ ([x] [d] [ex] [xt] Dtrans' x d ex xt y e ey yt) ([x] [d] ordered/cons (Dbound2' x d y e)) ([x] [d] [ex] [xt] Dtrans'' x d ex xt y e ey yt)) %. +%scope bump-ttranse %term pair + %pi (bump-transe _ ([x] [d] [ex] [xt] transe/pair (DtransN x d ex xt) (DtransM x d ex xt)) Dord ([x] [d] [ex] [xt] transe/pair (DtransN' x d ex xt) (DtransM' x d ex xt))) + %<- (bump-transe _ DtransM Dord DtransM') + %<- (bump-transe _ DtransN Dord DtransN') %. +%term t bump-ttranse _ ([x] [d] [ex] [xt] ttranse/t (Dord' x d)) Dord ([x] [d] [ex] [xt] ttranse/t (Dord x d)) %. +%scope bump-ttranse %term pi + %pi (bump-ttranse _ ([x] [d isvar x I] [ex] [xt] ttranse/pi (Dtrans x d ex xt) (Dttrans x d ex xt)) (%the ({x} %pi (isvar x I') %-> (ordered (G x))) Dord) ([x] [d] [ex] [xt] ttranse/pi (Dtrans'' x d ex xt) (Dttrans' x d ex xt))) + %<- (bump-ttranse _ Dttrans Dord Dttrans') + %<- ({x} {d isvar x I} {ex} {xt} {y} {e isvar y J1} {ey} {yt} ttranse-context (Dtrans x d ex xt y e ey yt) (ordered/cons (%the (bounded (G x) y) (Dbound1 x d y e)))) + %<- ({x} {d isvar x I'} extend-context (Dord x d) (%the ({y} %pi (isvar y J2) %-> (bounded (G x) y)) (Dbound2 x d))) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- ({x} {d isvar x I} bump-bounded Dleq1 ([y] [e isvar y J1] Dbound1 x d y e) ([y] [e isvar y J] %the (bounded (G x) y) (Dbound1' x d y e))) + %<- ({x} {d isvar x I'} bump-bounded Dleq2 ([y] [e isvar y J2] Dbound2 x d y e) ([y] [e isvar y J] %the (bounded (G x) y) (Dbound2' x d y e))) + %<- ({x} {d isvar x I} {ex} {xt} bump-ttranse _ ([y] [e] [ey] [yt] Dtrans x d ex xt y e ey yt) ([y] [e] ordered/cons (Dbound1' x d y e)) ([y] [e] [ey] [yt] Dtrans' x d ex xt y e ey yt)) + %<- ({y} {e isvar y J} {ey} {yt} bump-ttranse _ ([x] [d] [ex] [xt] Dtrans' x d ex xt y e ey yt) ([x] [d] ordered/cons (Dbound2' x d y e)) ([x] [d] [ex] [xt] Dtrans'' x d ex xt y e ey yt)) %. +%term sigma + %pi (bump-ttranse _ ([x] [d isvar x I] [ex] [xt] ttranse/sigma (Dtrans x d ex xt) (Dttrans x d ex xt)) (%the ({x} %pi (isvar x I') %-> (ordered (G x))) Dord) ([x] [d] [ex] [xt] ttranse/sigma (Dtrans'' x d ex xt) (Dttrans' x d ex xt))) + %<- (bump-ttranse _ Dttrans Dord Dttrans') + %<- ({x} {d isvar x I} {ex} {xt} {y} {e isvar y J1} {ey} {yt} ttranse-context (Dtrans x d ex xt y e ey yt) (ordered/cons (%the (bounded (G x) y) (Dbound1 x d y e)))) + %<- ({x} {d isvar x I'} extend-context (Dord x d) (%the ({y} %pi (isvar y J2) %-> (bounded (G x) y)) (Dbound2 x d))) + %<- (can-max J1 J2 J (%the (max J1 J2 J) Dmax)) + %<- (max-leq Dmax (%the (leq J1 J) Dleq1) (%the (leq J2 J) Dleq2)) + %<- ({x} {d isvar x I} bump-bounded Dleq1 ([y] [e isvar y J1] Dbound1 x d y e) ([y] [e isvar y J] %the (bounded (G x) y) (Dbound1' x d y e))) + %<- ({x} {d isvar x I'} bump-bounded Dleq2 ([y] [e isvar y J2] Dbound2 x d y e) ([y] [e isvar y J] %the (bounded (G x) y) (Dbound2' x d y e))) + %<- ({x} {d isvar x I} {ex} {xt} bump-ttranse _ ([y] [e] [ey] [yt] Dtrans x d ex xt y e ey yt) ([y] [e] ordered/cons (Dbound1' x d y e)) ([y] [e] [ey] [yt] Dtrans' x d ex xt y e ey yt)) + %<- ({y} {e isvar y J} {ey} {yt} bump-ttranse _ ([x] [d] [ex] [xt] Dtrans' x d ex xt y e ey yt) ([x] [d] ordered/cons (Dbound2' x d y e)) ([x] [d] [ex] [xt] Dtrans'' x d ex xt y e ey yt)) %. +%scope bump-ttranse %term sing + %pi (bump-ttranse _ ([x] [d] [ex] [xt] ttranse/sing (Dtrans x d ex xt)) Dord ([x] [d] [ex] [xt] ttranse/sing (Dtrans' x d ex xt))) + %<- (bump-transe _ Dtrans Dord Dtrans') %. +%worlds (bind ovar tbind) (bump-transe _ _ _ _) (bump-ttranse _ _ _ _) %. +%total (EM EA) (bump-transe EM _ _ _) (bump-ttranse EA _ _ _) %. +%sort weakeng-transe {EM} %. +%mode weakeng-transe %in %in %in %in %in %out %. +%sort weakeng-ttranse {EA} %. +%mode weakeng-ttranse %in %in %in %in %in %out %. +%scope weakeng-ttranse %term const weakeng-transe _ _ _ Dord (transe/const _ Dself Dexpand Dwf Dkof) (transe/const Dord Dself Dexpand Dwf Dkof) %. +%term forall + %pi (weakeng-transe _ Dappend Dappend' Dord (transe/forall Dself Dexpand D) (transe/forall Dself Dexpand D')) + %<- (weakeng-ttranse _ Dappend Dappend' Dord D D') %. +%scope weakeng-ttranse %term var + %pi (weakeng-transe _ Dappend Dappend' Dord (transe/var Dself Dexpand Dwf Dlookup Dvtrans) (transe/var Dself Dexpand Dwf' Dlookup' Dvtrans)) + %<- (weakeng-lookup Dappend Dappend' Dord Dlookup Dlookup') + %<- (weakeng-wfe Dappend Dappend' Dord Dwf Dwf') %. +%term vari weakeng-transe _ Dappend Dappend' Dord (transe/vari _ Dself Dexpand Dwf Dvof Dvtrans) (transe/vari Dord Dself Dexpand Dwf Dvof Dvtrans) %. +%scope weakeng-ttranse %term app + %pi (weakeng-transe _ Dappend Dappend' Dord (transe/app Dsub Dsubtype DtransN DtransM) (transe/app Dsub Dsubtype DtransN' DtransM')) + %<- (weakeng-transe _ Dappend Dappend' Dord DtransM DtransM') + %<- (weakeng-transe _ Dappend Dappend' Dord DtransN DtransN') %. +%term pi1 + %pi (weakeng-transe _ Dappend Dappend' Dord (transe/pi1 Dtrans) (transe/pi1 Dtrans')) + %<- (weakeng-transe _ Dappend Dappend' Dord Dtrans Dtrans') %. +%scope weakeng-ttranse %term pi2 + %pi (weakeng-transe _ Dappend Dappend' Dord (transe/pi2 Dtrans) (transe/pi2 Dtrans')) + %<- (weakeng-transe _ Dappend Dappend' Dord Dtrans Dtrans') %. +%term lam + %pi (weakeng-transe _ Dappend Dappend' Dord (transe/lam Dtrans Dttrans) (transe/lam Dtrans'' Dttrans')) + %<- (weakeng-ttranse _ Dappend Dappend' Dord Dttrans Dttrans') + %<- (extend-context Dord Dbound) + %<- ({x} {d} bounded-drop Dappend' Dappend (Dbound x d) (Dbound' x d)) + %<- (bump-transe _ Dtrans ([x] [d] ordered/cons (Dbound' x d)) Dtrans') + %<- ({x} {d} {ex} {xt} weakeng-transe _ (append/cons Dappend) (append/cons Dappend') (ordered/cons (Dbound x d)) (Dtrans' x d ex xt) (Dtrans'' x d ex xt)) %. +%scope weakeng-ttranse %term pair + %pi (weakeng-transe _ Dappend Dappend' Dord (transe/pair DtransN DtransM) (transe/pair DtransN' DtransM')) + %<- (weakeng-transe _ Dappend Dappend' Dord DtransM DtransM') + %<- (weakeng-transe _ Dappend Dappend' Dord DtransN DtransN') %. +%term t weakeng-ttranse _ _ _ Dord (ttranse/t _) (ttranse/t Dord) %. +%scope weakeng-ttranse %term pi + %pi (weakeng-ttranse _ Dappend Dappend' Dord (ttranse/pi Dtrans Dttrans) (ttranse/pi Dtrans'' Dttrans')) + %<- (weakeng-ttranse _ Dappend Dappend' Dord Dttrans Dttrans') + %<- (extend-context Dord Dbound) + %<- ({x} {d} bounded-drop Dappend' Dappend (Dbound x d) (Dbound' x d)) + %<- (bump-ttranse _ Dtrans ([x] [d] ordered/cons (Dbound' x d)) Dtrans') + %<- ({x} {d} {ex} {xt} weakeng-ttranse _ (append/cons Dappend) (append/cons Dappend') (ordered/cons (Dbound x d)) (Dtrans' x d ex xt) (Dtrans'' x d ex xt)) %. +%term sigma + %pi (weakeng-ttranse _ Dappend Dappend' Dord (ttranse/sigma Dtrans Dttrans) (ttranse/sigma Dtrans'' Dttrans')) + %<- (weakeng-ttranse _ Dappend Dappend' Dord Dttrans Dttrans') + %<- (extend-context Dord Dbound) + %<- ({x} {d} bounded-drop Dappend' Dappend (Dbound x d) (Dbound' x d)) + %<- (bump-ttranse _ Dtrans ([x] [d] ordered/cons (Dbound' x d)) Dtrans') + %<- ({x} {d} {ex} {xt} weakeng-ttranse _ (append/cons Dappend) (append/cons Dappend') (ordered/cons (Dbound x d)) (Dtrans' x d ex xt) (Dtrans'' x d ex xt)) %. +%scope weakeng-ttranse %term sing + %pi (weakeng-ttranse _ Dappend Dappend' Dord (ttranse/sing Dtrans) (ttranse/sing Dtrans')) + %<- (weakeng-transe _ Dappend Dappend' Dord Dtrans Dtrans') %. +%worlds (bind ovar tbind) (weakeng-transe _ _ _ _ _ _) (weakeng-ttranse _ _ _ _ _ _) %. +%total (EM EA) (weakeng-transe EM _ _ _ _ _) (weakeng-ttranse EA _ _ _ _ _) %. +%sort weaken-transe %. +%mode weaken-transe %in %in %in %out %. +%term _ + %pi (weaken-transe Dbound Dtrans _ Dtrans') + %<- (weakeng-transe _ append/nil append/nil (ordered/cons Dbound) Dtrans Dtrans') %. +%worlds (bind ovar tbind) (weaken-transe _ _ _ _) %. +%total {} (weaken-transe _ _ _ _) %. +%sort weaken-transe' %. +%mode weaken-transe' %in %in %in %out %. +%sort weaken-transe'' %. +%mode weaken-transe'' %in %in %in %out %. +%scope weaken-transe'' %term nil weaken-transe' append/nil _ D D %. +%term cons + %pi (weaken-transe' (append/cons Dapp) (ordered/cons Dbound) Dtrans Dtrans'') + %<- (weaken-transe'' Dapp Dbound Dtrans Dtrans') + %<- (weaken-transe Dbound Dtrans' _ Dtrans'') %. +%term _ + %pi (weaken-transe'' Dapp Dbound Dtrans Dtrans') + %<- (bounded-is-ordered Dbound Dord) + %<- (weaken-transe' Dapp Dord Dtrans Dtrans') %. +%worlds (bind ovar tbind) (weaken-transe' _ _ _ _) (weaken-transe'' _ _ _ _) %. +%total (D1 D2) (weaken-transe' D1 _ _ _) (weaken-transe'' D2 _ _ _) %. +%%%%% Translation Regularity (Implicit Context) %%%%% +%sort transe-reg %. +%mode transe-reg %in %out %. +%sort ttranse-reg %. +%mode ttranse-reg %in %out %. +%scope ttranse-reg %term const + %pi (transe-reg (transe/const Dord Dself Dexpand Dwf' Dkof) Dwf) + %<- (expand-reg-e (aofe/closed Dord (aof/const Dwf' Dkof)) Dexpand Dof) + %<- (self-reg-e Dself Dof Dof' _) + %<- (ofe-reg Dof' Dwf) %. +%term forall + %pi (transe-reg (transe/forall Dself Dexpand Dtrans) Dwf) + %<- (ttranse-reg Dtrans Dwf') + %<- (expand-reg-e (aofe/forall Dwf') Dexpand Dof) + %<- (self-reg-e Dself Dof Dof' _) + %<- (ofe-reg Dof' Dwf) %. +%scope ttranse-reg %term var + %pi (transe-reg (transe/var Dself Dexpand Dwf Dlookup _) Dwf') + %<- (expand-reg-e (aofe/var Dwf Dlookup) Dexpand Dof) + %<- (self-reg-e Dself Dof Dof' _) + %<- (ofe-reg Dof' Dwf') %. +%term vari + %pi (transe-reg (transe/vari Dord Dself Dexpand Dwf Dvof _) Dwf') + %<- (expand-reg-e (aofe/closed Dord (aof/var Dwf Dvof)) Dexpand Dof) + %<- (self-reg-e Dself Dof Dof' _) + %<- (ofe-reg Dof' Dwf') %. +%scope ttranse-reg %term app + %pi (transe-reg (transe/app Dsub Dsubtype DtransN DtransM) DwfBx) + %<- (transe-reg DtransM (wfe/pi DwfB DwfA)) + %<- (transe-reg DtransN DwfC) + %<- (subtype-reg-e' Dsubtype DwfC DwfA Dof) + %<- (tsubst-e ([_] append/nil) csub/base Dof Dsub DwfB DwfBx) %. +%term pi1 + %pi (transe-reg (transe/pi1 Dtrans) DwfA) + %<- (transe-reg Dtrans (wfe/sigma DwfB DwfA)) %. +%scope ttranse-reg %term pi2 + %pi (transe-reg (transe/pi2 Dtrans) DwfB') + %<- (transe-reg Dtrans (wfe/sigma DwfB DwfA)) + %<- (tinhabitsub-e DwfA DwfB DwfB') %. +%term lam + %pi (transe-reg (transe/lam DtransM DtransA) (wfe/pi DwfB DwfA)) + %<- (ttranse-reg DtransA DwfA) + %<- ({x} {d} {ex} {xt} transe-reg (DtransM x d ex xt) (DwfB x d)) %. +%scope ttranse-reg %term pair + %pi (transe-reg (transe/pair DtransN DtransM) (wfe/sigma DwfB' DwfA)) + %<- (transe-reg DtransM DwfA) + %<- (transe-reg DtransN DwfB) + %<- (wfe-context DwfA Dord) + %<- (extend-context Dord Dbound) + %<- ({x} {d} weaken-wfe (Dbound x d) DwfB _ (DwfB' x d)) %. +%term t ttranse-reg (ttranse/t Dord) (wfe/t Dord) %. +%scope ttranse-reg %term pi + %pi (ttranse-reg (ttranse/pi DtransB DtransA) (wfe/pi DwfB DwfA)) + %<- (ttranse-reg DtransA DwfA) + %<- ({x} {d} {ex} {xt} ttranse-reg (DtransB x d ex xt) (DwfB x d)) %. +%term sigma + %pi (ttranse-reg (ttranse/sigma DtransB DtransA) (wfe/sigma DwfB DwfA)) + %<- (ttranse-reg DtransA DwfA) + %<- ({x} {d} {ex} {xt} ttranse-reg (DtransB x d ex xt) (DwfB x d)) %. +%scope ttranse-reg %term sing %pi (ttranse-reg (ttranse/sing Dtrans) Dwf) %<- (transe-reg Dtrans Dwf) %. +%worlds (var bind ovar tbind) (transe-reg _ _) (ttranse-reg _ _) %. +%total (D1 D2) (transe-reg D1 _) (ttranse-reg D2 _) %. +%sort ttrans-reg %. +%mode ttrans-reg %in %out %. +%term _ + %pi (ttrans-reg Dtrans Dwf) + %<- (ttrans-to-ttranse Dtrans Dtranse) + %<- (ttranse-reg Dtranse Dwfe) + %<- (wfe-to-wf Dwfe Dwf) %. +%worlds (var bind tbind) (ttrans-reg _ _) %. +%total {} (ttrans-reg _ _) %. +%sort trans-reg %. +%mode trans-reg %in %out %. +%term _ + %pi (trans-reg Dtrans Dwf) + %<- (trans-to-transe Dtrans Dtranse) + %<- (transe-reg Dtranse Dwfe) + %<- (wfe-to-wf Dwfe Dwf) %. +%worlds (var bind tbind) (trans-reg _ _) %. +%total {} (trans-reg _ _) %. +%%%%% Miscellaneous %%%%% +%sort transe-principal %. +%mode transe-principal %in %out %. +%scope transe-principal %term const + %pi (transe-principal (transe/const _ Dself _ _ _) D) + %<- (self-principal Dself D) %. +%term forall %pi (transe-principal (transe/forall Dself _ _) D) %<- (self-principal Dself D) %. +%scope transe-principal %term var %pi (transe-principal (transe/var Dself _ _ _ _) D) %<- (self-principal Dself D) %. +%term vari + %pi (transe-principal (transe/vari _ Dself _ _ _ _) D) + %<- (self-principal Dself D) %. +%scope transe-principal %term app + %pi (transe-principal (transe/app Dsub _ _ Dtrans) D) + %<- (transe-principal Dtrans (principal/pi Dprin)) + %<- (principal-sub Dprin Dsub D) %. +%term pi1 + %pi (transe-principal (transe/pi1 Dtrans) D1) + %<- (transe-principal Dtrans (principal/sigma D2 D1)) %. +%scope transe-principal %term pi2 + %pi (transe-principal (transe/pi2 Dtrans) D2) + %<- (transe-principal Dtrans (principal/sigma D2 D1)) %. +%term lam + %pi (transe-principal (transe/lam Dtrans _) (principal/pi D)) + %<- ({x} {d} {ex} {xt} transe-principal (Dtrans x d ex xt) (D x)) %. +%scope transe-principal %term pair + %pi (transe-principal (transe/pair Dtrans2 Dtrans1) (principal/sigma D2 D1)) + %<- (transe-principal Dtrans1 D1) + %<- (transe-principal Dtrans2 D2) %. +%worlds (var bind ovar tbind) (transe-principal _ _) %. +%total D (transe-principal D _) %. +%sort trans-principal %. +%mode trans-principal %in %out %. +%term _ + %pi (trans-principal Dtrans D) + %<- (trans-to-transe Dtrans Dtranse) + %<- (transe-principal Dtranse D) %. +%worlds (var bind tbind) (trans-principal _ _) %. +%total {} (trans-principal _ _) %. +%sort trans-subtype %. +%mode trans-subtype %in %in %out %out %. +%term _ + %pi (trans-subtype Dtrans Dsubtype Dsubtype' Deq) + %<- (trans-principal Dtrans Dprin) + %<- (principal-subtype Dprin Dsubtype Deq) + %<- (subtype-resp tp-eq/i tp-eq/i Deq Dsubtype Dsubtype') %. +%worlds (var bind tbind) (trans-subtype _ _ _ _) %. +%total {} (trans-subtype _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/trans-sub-thm.lf b/new-tests/stelf-output/tslf/sing/trans-sub-thm.lf new file mode 100644 index 0000000..950f0b7 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/trans-sub-thm.lf @@ -0,0 +1,268 @@ +%%%%% Translation Commutes with Substitution (Explicit Context) %%%%% +%sort trans-sub-e {x} %. +%mode trans-sub-e %in %in %in %in %in %in %in %out %out %. +%sort ttrans-sub-e {x} %. +%mode ttrans-sub-e %in %in %in %in %in %in %in %out %. +%scope ttrans-sub-e %term const + %pi (trans-sub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) ([x] [d isvar x I] [ex] [xt vtrans ex x] transe/const (%the (ordered (G x)) (Dordered x d)) (%the (self (N x) (B x) (C x)) (Dself x)) (%the (expand (const K) (B x) (N x)) (Dexpand x)) (%the (wf (B x)) (DwfB x)) (%the (kof K (B x)) (Dkof x))) (%the (transe G1 EM D) DtransM) (%the (wfe G1 A) DwfA) (%the (subtype D A ([_] M)) DsubtypeDA) (%the (tsub C M Cx) DsubCx) (%the (csub G M Gx) Dcsub) (transe/const Dordered' Dself'' (Dexpand'' aca) (Dwfc aca) (Dkof' aca)) Dsubtype') + %<- (kof-strengthen Dkof (%the ({x} tp-eq (B x) B') DeqB)) + %<- ({x} kof-resp (DeqB x) (Dkof x) (%the (kof K B') (Dkof' x))) + %<- ({x} expand-resp atom-eq/i (DeqB x) term-eq/i (Dexpand x) (%the (expand (const K) B' (N x)) (Dexpand' x))) + %<- (strengthen-expand Dexpand' (%the ({x} term-eq (N x) N') DeqN)) + %<- ({x} expand-resp atom-eq/i tp-eq/i (DeqN x) (Dexpand' x) (%the (expand (const K) B' N') (Dexpand'' x))) + %<- ({x} self-resp (DeqN x) (DeqB x) tp-eq/i (Dself x) (%the (self N' B' (C x)) (Dself' x))) + %<- ({x} wf-resp (DeqB x) (DwfB x) (Dwfc x)) + %<- (expand-reg (%the (aof (const K) B') (aof/const (Dwfc aca) (Dkof' aca))) (Dexpand'' aca) (%the (of N' B') DofN')) + %<- (can-self DofN' (%the (self N' B' C') Dself'')) + %<- (self-reg Dself'' DofN' (%the (of N' C') DofN'') _) + %<- ({x} self-fun (Dself' x) Dself'' (%the (tp-eq (C x) C') (DeqC x))) + %<- (csub-ordered Dordered Dcsub (%the (ordered Gx) Dordered')) + %<- (tsub-absent C' M (%the (tsub ([_] C') M C') DsubC')) + %<- (tsub-resp DeqC term-eq/i tp-eq/i DsubCx (%the (tsub ([_] C') M Cx) DsubCx')) + %<- (tsub-fun DsubC' DsubCx' (%the (tp-eq C' Cx) DeqCx)) + %<- (self-principal Dself'' (%the (principal C') Dprincipal)) + %<- (subtype-refl-principal Dprincipal DofN'' (%the (subtype C' C' ([_] N')) Dsubtype)) + %<- (subtype-resp tp-eq/i DeqCx ([_] term-eq/i) Dsubtype (%the (subtype C' Cx ([_] N')) Dsubtype')) %. +%term forall + %pi (trans-sub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) ([x] [d isvar x I] [ex] [xt vtrans ex x] transe/forall (%the (self (N x) (qtp (B x)) (C x)) (Dself x)) (%the (expand (forall (B x)) (qtp (B x)) (N x)) (Dexpand x)) (%the (ttranse (G x) (EB ex) (B x)) (DtransB x d ex xt))) (%the (transe G1 EM D) DtransM) (%the (wfe G1 A) DwfA) (%the (subtype D A ([_] M)) DsubtypeDA) (%the (tsub C M Cx) DsubCx) (%the (csub G M Gx) Dcsub) (transe/forall Dself' Dexpand' DtransBx) Dsubtype) + %<- (transe-reg DtransM (%the (wfe G1 D) DwfD)) + %<- (subtype-reg-e' DsubtypeDA DwfD DwfA (%the (ofe G1 M A) DofM)) + %<- ({x} {d} {ex} {xt} ttranse-reg (DtransB x d ex xt) (%the (wfe (G x) (B x)) (DwfB x d))) + %<- (can-tsub-e Dappend DwfB DofM (%the (tsub B M Bx) DsubBx)) + %<- (tsubst-e Dappend Dcsub DofM DsubBx DwfB (%the (wfe Gx Bx) DwfBx)) + %<- (ttrans-sub-e Dappend DtransB DtransM DwfA DsubtypeDA DsubBx Dcsub (%the (ttranse Gx (EB EM) Bx) DtransBx)) + %<- (can-expand (forall Bx) (qtp Bx) _ (%the (expand (forall Bx) (qtp Bx) Nx) Dexpand')) + %<- (expand-aasub-e Dappend DofM ([x] [d] aofe/forall (DwfB x d)) Dexpand (aasub/forall DsubBx) (tsub/pi ([_] tsub/t) (tsub/pi ([_] tsub/t) DsubBx)) Dexpand' (%the (sub N M Nx) DsubNx)) + %<- ({x} {d} expand-reg-e (aofe/forall (DwfB x d)) (Dexpand x) (%the (ofe (G x) (N x) (qtp (B x))) (DofN x d))) + %<- (self-sub-e Dappend Dcsub DofM DofN Dself DsubNx (tsub/pi ([_] tsub/t) (tsub/pi ([_] tsub/t) DsubBx)) DsubCx (%the (self Nx (qtp Bx) Cx) Dself')) + %<- (expand-reg-e (aofe/forall DwfBx) Dexpand' (%the (ofe Gx Nx (qtp Bx)) DofNx)) + %<- (self-reg-e' Dself' DofNx (%the (wfe Gx Cx) DwfCx)) + %<- (subtype-refl-e DwfCx (%the (subtype Cx Cx _) Dsubtype) _) %. +%scope ttrans-sub-e %term varsam + %pi (trans-sub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) ([x] [d isvar x I] [ex] [xt vtrans ex x] transe/var (%the (self (N x) (A' x) (B x)) (Dself x)) (%the (expand x (A' x) (N x)) (Dexpand x)) (%the (wfe (G x) (A' x)) (DwfeA' x d)) (%the (lookup (G x) x (A' x)) (Dlookup x d)) xt) (%the (transe G1 EM D) DtransM) (%the (wfe G1 A) DwfA) (%the (subtype D A ([_] M)) DsubtypeDA) (%the (tsub B M Bx) DsubBx) (%the (csub G M Gx) Dcsub) DtransM' DsubtypeDBx') + %<- ({x} {d isvar x I} lookup-context (Dlookup x d) (%the (ordered (G x)) (Dordered x d))) + %<- ({x} {d isvar x I} append-ordered (Dappend x) (Dordered x d) (ordered/cons (%the (bounded G1 x) (Dbounded x d)))) + %<- (csub-ordered Dordered Dcsub (%the (ordered Gx) Dordered')) + %<- (csub-append Dcsub Dappend (%the (append G1 G2' Gx) Dappend')) + %<- (weaken-transe' Dappend' Dordered' DtransM (%the (transe Gx EM D) DtransM')) + %<- ({x} {d isvar x I} lookup-binder-fun (Dappend x) (Dlookup x d) (%the (tp-eq A (A' x)) (Deq x))) + %<- ({x} tp-eq-symm (Deq x) (%the (tp-eq (A' x) A) (Deq' x))) + %<- ({x} {d isvar x I} lookup-resp ctx-eq/i atom-eq/i (Deq' x) (Dlookup x d) (%the (lookup (G x) x A) (Dlookup' x d))) + %<- ({x} expand-resp atom-eq/i (Deq' x) term-eq/i (Dexpand x) (%the (expand x A (N x)) (Dexpand' x))) + %<- ({x} self-resp term-eq/i (Deq' x) tp-eq/i (Dself x) (%the (self (N x) A (B x)) (Dself' x))) + %<- (transe-reg DtransM (%the (wfe G1 D) DwfD)) + %<- (subtype-reg-e' DsubtypeDA DwfD DwfA (%the (ofe G1 M A) DofM)) + %<- ({x} {d isvar x I} weaken-wfe (Dbounded x d) DwfA _ (%the (wfe (cons G1 x A) A) (DwfA' x d))) + %<- ({x} {d isvar x I} weaken-wfe' (Dappend x) (Dordered x d) (DwfA' x d) (%the (wfe (G x) A) (DwfA'' x d))) + %<- ({x} {d isvar x I} expand-reg-e (aofe/var (DwfA'' x d) (Dlookup' x d)) (Dexpand' x) (%the (ofe (G x) (N x) A) (DofN x d))) + %<- (expand-aosub-e Dappend Dcsub DofM ([x] [d] aofe/var (DwfA'' x d) (Dlookup' x d)) Dexpand' aosub/var (%the (sub N M M) DsubNx')) + %<- (tsub-absent A M (%the (tsub ([_] A) M A) DsubA')) + %<- (self-sub-e Dappend Dcsub DofM DofN Dself' DsubNx' DsubA' DsubBx (%the (self M A Bx) Dself'')) + %<- (transe-principal DtransM (%the (principal D) DprincipalD)) + %<- (subtype-self-e DsubtypeDA DwfD DwfA DprincipalD Dself'' (%the (subtype D Bx ([_] M)) DsubtypeDBx)) + %<- (subtype-resp tp-eq/i tp-eq/i ([_] term-eq/i) DsubtypeDBx (%the (subtype D Bx ([_] M)) DsubtypeDBx')) %. +%term varoth + %pi (trans-sub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) ([x] [d isvar x I] [ex] [xt vtrans ex x] transe/var (%the (self (N x) (B x) (C x)) (Dself x)) (%the (expand Y (B x) (N x)) (Dexpand x)) (%the (wfe (G x) (B x)) (DwfB x d)) (%the (lookup (G x) Y (B x)) (Dlookup x d)) (%the (vtrans EY Y) Dvtrans)) (%the (transe G1 EM D) DtransM) (%the (wfe G1 A) DwfA) (%the (subtype D A ([_] M)) DsubtypeDA) (%the (tsub C M Cx) DsubCx) (%the (csub G M Gx) Dcsub) (transe/var Dself' Dexpand' DwfBx Dlookup' Dvtrans) DsubtypeCx) + %<- (transe-reg DtransM (%the (wfe G1 D) DwfD)) + %<- (subtype-reg-e' DsubtypeDA DwfD DwfA (%the (ofe G1 M A) DofM)) + %<- (can-tsub-e Dappend DwfB DofM (%the (tsub B M Bx) DsubBx)) + %<- (csub-lookup Dcsub Dlookup DsubBx (%the (lookup Gx Y Bx) Dlookup')) + %<- (tsubst-e Dappend Dcsub DofM DsubBx DwfB (%the (wfe Gx Bx) DwfBx)) + %<- (aasub-absent Y M (%the (aasub ([_] Y) M Y) DsubY)) + %<- (can-expand Y Bx _ (%the (expand Y Bx Nx) Dexpand')) + %<- (expand-aasub-e Dappend DofM ([x] [d] aofe/var (DwfB x d) (Dlookup x d)) Dexpand DsubY DsubBx Dexpand' (%the (sub N M Nx) DsubNx)) + %<- ({x} {d isvar x I} expand-reg-e (aofe/var (DwfB x d) (Dlookup x d)) (Dexpand x) (%the (ofe (G x) (N x) (B x)) (DofN x d))) + %<- (self-sub-e Dappend Dcsub DofM DofN Dself DsubNx DsubBx DsubCx (%the (self Nx Bx Cx) Dself')) + %<- ({x} self-principal (Dself x) (%the (principal (C x)) (DprincipalC x))) + %<- (principal-sub DprincipalC DsubCx (%the (principal Cx) DprincipalCx)) + %<- ({x} {d isvar x I} self-reg-e (Dself x) (DofN x d) (%the (ofe (G x) (N x) (C x)) (DofN' x d)) (DsubtypeCB x)) + %<- (subst-e Dappend Dcsub DofM DsubNx DsubCx DofN' (%the (ofe Gx Nx Cx) DofNx)) + %<- (subtype-refl-principal-e DprincipalCx DofNx (%the (subtype Cx Cx ([_] Nx)) DsubtypeCx)) %. +%scope ttrans-sub-e %term vari + %pi (trans-sub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) ([x] [d isvar x I] [ex] [xt vtrans ex x] transe/vari (%the (ordered (G x)) (Dordered x d)) (%the (self (N x) (B x) (C x)) (Dself x)) (%the (expand Y (B x) (N x)) (Dexpand x)) (%the (wf (B x)) (DwfB x)) (%the (vof Y (B x)) (Dvof x)) (%the (vtrans EY Y) Dvtrans)) (%the (transe G1 EM D) DtransM) (%the (wfe G1 A) DwfA) (%the (subtype D A ([_] M)) DsubtypeDA) (%the (tsub C M Cx) DsubCx) (%the (csub G M Gx) Dcsub) (transe/vari Dordered' Dself'' Dexpand'' (DwfB' aca) (Dvof' aca) Dvtrans) DsubtypeC'') + %<- (vof-noassm Dvof (%the ({x} tp-eq (B x) B') DeqB)) + %<- ({x} vof-resp atom-eq/i (DeqB x) (Dvof x) (%the (vof Y B') (Dvof' x))) + %<- ({x} wf-resp (DeqB x) (DwfB x) (%the (wf B') (DwfB' x))) + %<- ({x} expand-resp atom-eq/i (DeqB x) term-eq/i (Dexpand x) (%the (expand Y B' (N x)) (Dexpand' x))) + %<- (can-expand Y B' N' (%the (expand Y B' N') Dexpand'')) + %<- ({x} expand-fun (Dexpand' x) Dexpand'' (%the (term-eq (N x) N') (DeqN x))) + %<- ({x} self-resp (DeqN x) (DeqB x) tp-eq/i (Dself x) (%the (self N' B' (C x)) (Dself' x))) + %<- (expand-reg (aof/var (DwfB' aca) (Dvof' aca)) Dexpand'' (%the (of N' B') DofN')) + %<- (can-self DofN' (%the (self N' B' C') Dself'')) + %<- (csub-ordered Dordered Dcsub (%the (ordered Gx) Dordered')) + %<- ({x} self-fun (Dself' x) Dself'' (%the (tp-eq (C x) C') (DeqC x))) + %<- (tsub-absent C' M (%the (tsub ([_] C') M C') DsubCabs)) + %<- (tsub-resp DeqC term-eq/i tp-eq/i DsubCx (%the (tsub ([_] C') M Cx) DsubCx')) + %<- (tsub-fun DsubCabs DsubCx' (%the (tp-eq C' Cx) DeqCx)) + %<- (self-reg Dself'' DofN' (%the (of N' C') DofN'') _) + %<- (of-reg DofN'' (%the (wf C') DwfC')) + %<- (subtype-refl DwfC' (%the (subtype C' C' _) DsubtypeC') _) + %<- (subtype-resp tp-eq/i DeqCx ([_] term-eq/i) DsubtypeC' (%the (subtype C' Cx _) DsubtypeC'')) %. +%term app + %pi (trans-sub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) ([x] [d isvar x I] [ex] [xt vtrans ex x] transe/app (%the (tsub ([y] C x y) (O x) (Cy x)) (DsubCy x)) (%the (subtype (E x) (B x) ([_] O x)) (DsubtypeEB x)) (%the (transe (G x) (EO ex) (E x)) (DtransO x d ex xt)) (%the (transe (G x) (EN ex) (pi (B x) ([y] C x y))) (DtransN x d ex xt))) (%the (transe G1 EM D) DtransM) (%the (wfe G1 A) DwfA) (%the (subtype D A ([_] M)) DsubtypeDA) (%the (tsub Cy M Cyx) DsubCyx) (%the (csub G M Gx) Dcsub) (transe/app DsubTy DsubtypeFS DtransOx DtransNx) DsubtypeTyCyx) + %<- (transe-reg DtransM (%the (wfe G1 D) DwfD)) + %<- (subtype-reg-e' DsubtypeDA DwfD DwfA (%the (ofe G1 M A) DofM)) + %<- ({x} {d isvar x I} {ex} {xt vtrans ex x} transe-reg (DtransN x d ex xt) (wfe/pi (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x y))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- (can-tsub-e Dappend DwfB DofM (%the (tsub B M Bx) DsubBx)) + %<- ({y} {e isvar y J} can-tsub-e ([x] append/cons (Dappend x)) ([x] [d] DwfC x d y e) DofM (%the (tsub ([x] C x y) M (Cx y)) (DsubCx y))) + %<- (trans-sub-e Dappend DtransN DtransM DwfA DsubtypeDA (tsub/pi DsubCx DsubBx) Dcsub (%the (transe Gx (EN EM) (pi S T)) DtransNx) (%the (subtype (pi S T) (pi Bx Cx) _) (subtype/pi (%the ({y} subtype (U y) (Cx y) _) DsubtypeUCx) (%the ({y} tsub ([y] T y) (Lbs y) (U y)) DsubU) (%the (subtype Bx S Lbs) DsubtypeBS)))) + %<- ({x} {d isvar x I} {ex} {xt vtrans ex x} transe-reg (DtransO x d ex xt) (%the (wfe (G x) (E x)) (DwfE x d))) + %<- (can-tsub-e Dappend DwfE DofM (%the (tsub E M Ex) DsubEx)) + %<- (trans-sub-e Dappend DtransO DtransM DwfA DsubtypeDA DsubEx Dcsub (%the (transe Gx (EO EM) F) DtransOx) (%the (subtype F Ex Lfe) DsubtypeFE)) + %<- (transe-reg DtransNx (wfe/pi (%the ({y} %pi (isvar y K) %-> (wfe (cons Gx y S) (T y))) DwfT) (%the (wfe Gx S) DwfS))) + %<- (transe-reg DtransOx (%the (wfe Gx F) DwfF)) + %<- (tsubst-e Dappend Dcsub DofM DsubEx DwfE (%the (wfe Gx Ex) DwfEx)) + %<- (tsubst-e Dappend Dcsub DofM DsubBx DwfB (%the (wfe Gx Bx) DwfBx)) + %<- (subtype-sub-e Dappend Dcsub DofM DwfE DwfB DsubtypeEB DsubEx DsubBx (%the ({y} sub ([x] O x) M (Ox' y)) DsubOx) (%the (subtype Ex Bx Ox') DsubtypeEBx)) + %<- (sub-closed DsubOx (%the ({y} term-eq (Ox' y) Ox) DeqOx)) + %<- (subtype-resp tp-eq/i tp-eq/i DeqOx DsubtypeEBx (%the (subtype Ex Bx ([_] Ox)) DsubtypeEBx')) + %<- ({y} sub-resp ([_] term-eq/i) term-eq/i (DeqOx y) (DsubOx y) (%the (sub ([x] O x) M Ox) (DsubOx' y))) + %<- ({y} sub-absent Ox (Lfe y) (%the (sub ([_] Ox) (Lfe y) Ox) (DsubLfb y))) + %<- (subtype-reg-e' DsubtypeEBx' DwfEx DwfBx (%the (ofe Gx Ox Bx) DofOx)) + %<- (subtype-reg-e DsubtypeBS DwfBx DwfS (%the ({y} %pi (isvar y K') %-> (ofe (cons Gx y Bx) (Lbs y) S)) DofLbs)) + %<- (can-sub-e ([_] append/nil) DofLbs DofOx (%the (sub Lbs Ox P) DsubP)) + %<- (subtype-trans-e DwfF DwfEx DwfBx DsubtypeFE DsubtypeEBx' DsubLfb (%the (subtype F Bx ([_] Ox)) DsubtypeFBx)) + %<- (subtype-trans-e DwfF DwfBx DwfS DsubtypeFBx DsubtypeBS ([_] DsubP) (%the (subtype F S ([_] P)) DsubtypeFS)) + %<- (subtype-reg-e' DsubtypeFS DwfF DwfS (%the (ofe Gx P S) DofP)) + %<- (can-tsub-e ([_] append/nil) DwfT DofP (%the (tsub T P Ty) DsubTy)) + %<- ({y} {e isvar y J} tsubst-e ([x] append/cons (Dappend x)) (csub/cons DsubBx Dcsub) DofM (DsubCx y) ([x] [d] DwfC x d y e) (%the (wfe (cons Gx y Bx) (Cx y)) (DwfCx y e))) + %<- (can-tsub-e ([_] append/nil) DwfCx DofOx (%the (tsub Cx Ox Cxy) DsubCxy)) + %<- (tsubst-context-e DofLbs DwfT DsubU (%the ({y} %pi (isvar y K') %-> (wfe (cons Gx y Bx) (U y))) DwfU)) + %<- ({y} {e isvar y J} wfe-context (DwfCx y e) (ordered/cons (%the (bounded Gx y) (DboundedGxy y e)))) + %<- (bump-wfe DwfU ([y] [e] ordered/cons (DboundedGxy y e)) (%the ({y} %pi (isvar y J) %-> (wfe (cons Gx y Bx) (U y))) DwfU')) + %<- (can-tsub-e ([_] append/nil) DwfU DofOx (%the (tsub U Ox Uy) DsubUy)) + %<- (subtype-sub-e ([_] append/nil) csub/base DofOx DwfU' DwfCx DsubtypeUCx DsubUy DsubCxy _ (%the (subtype Uy Cxy _) DsubtypeUyCxy)) + %<- ({y} tsub-absent (T y) Ox (%the (tsub ([_] T y) Ox (T y)) (DsubTabs y))) + %<- ({y} {e isvar y K'} ofe-context (DofLbs y e) (ordered/cons (%the (bounded Gx y) (DboundedK' y e)))) + %<- (weaken-wfe-insert1 DwfT DboundedK' _ (%the ({y} + %pi (isvar y K') + %-> ({z} %pi (isvar z K'') %-> (wfe (cons (cons Gx y Bx) z S) (T z)))) DwfT')) + %<- (tsub-permute-e ([_] append/nil) DofOx DofLbs DwfT' DsubTabs DsubP DsubTy DsubU DsubUy (%the (tp-eq Ty Uy) DeqTyUy)) + %<- (tp-eq-symm DeqTyUy (%the (tp-eq Uy Ty) DeqUyTy)) + %<- ({x} {d isvar x I} subtype-reg-e' (DsubtypeEB x) (DwfE x d) (DwfB x d) (%the (ofe (G x) (O x) (B x)) (DofO x d))) + %<- (tsub-permute-e Dappend DofM DofO DwfC DsubCx (DsubOx' aca) DsubCxy DsubCy DsubCyx (%the (tp-eq Cxy Cyx) DeqCxyCyx)) + %<- (subtype-resp DeqUyTy DeqCxyCyx ([_] term-eq/i) DsubtypeUyCxy (%the (subtype Ty Cyx _) DsubtypeTyCyx)) %. +%scope ttrans-sub-e %term pi1 + %pi (trans-sub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) ([x] [d isvar x I] [ex] [xt vtrans ex x] transe/pi1 (%the (transe (G x) (EN ex) (sigma (B x) ([_] C x))) (DtransN x d ex xt))) (%the (transe G1 EM D) DtransM) (%the (wfe G1 A) DwfA) (%the (subtype D A ([_] M)) DsubtypeDA) (%the (tsub B M Bx) DsubBx) (%the (csub G M Gx) Dcsub) (transe/pi1 DtransN'') DsubtypeBE) + %<- (transe-reg DtransM (%the (wfe G1 D) DwfD)) + %<- (subtype-reg-e' DsubtypeDA DwfD DwfA (%the (ofe G1 M A) DofM)) + %<- ({x} {d isvar x I} {ex} {xt vtrans ex x} transe-reg (DtransN x d ex xt) (wfe/sigma (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- (tsubst-e Dappend Dcsub DofM DsubBx DwfB (%the (wfe Gx Bx) DwfBx)) + %<- ({x} {d isvar x I} tinhabitsub-e (DwfB x d) (DwfC x d) (%the (wfe (G x) (C x)) (DwfC' x d))) + %<- (can-tsub-e Dappend DwfC' DofM (%the (tsub C M Cx) DsubCx)) + %<- (trans-sub-e Dappend DtransN DtransM DwfA DsubtypeDA (tsub/sigma ([y] DsubCx) DsubBx) Dcsub (%the (transe Gx (EN EM) EF) DtransN') (%the (subtype EF (sigma Bx ([_] Cx)) _) DsubtypeEF)) + %<- (transe-principal DtransN' (%the (principal EF) DprincipalEF)) + %<- (principal-subtype-sigma-invert DprincipalEF DsubtypeEF (%the (tp-eq EF (sigma E ([_] F))) DeqEF)) + %<- (transe-resp ctx-eq/i eterm-eq/i DeqEF DtransN' (%the (transe Gx (EN EM) (sigma E ([_] F))) DtransN'')) + %<- (subtype-resp DeqEF tp-eq/i ([_] term-eq/i) DsubtypeEF (%the (subtype (sigma E ([_] F)) (sigma Bx ([_] Cx)) _) DsubtypeEF')) + %<- (subtype-sigma-invert DsubtypeEF' (%the (subtype E Bx N1) DsubtypeBE) _ _ _) %. +%term pi2 + %pi (trans-sub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) ([x] [d isvar x I] [ex] [xt vtrans ex x] transe/pi2 (%the (transe (G x) (EN ex) (sigma (B x) ([_] C x))) (DtransN x d ex xt))) (%the (transe G1 EM D) DtransM) (%the (wfe G1 A) DwfA) (%the (subtype D A ([_] M)) DsubtypeDA) (%the (tsub C M Cx) DsubCx) (%the (csub G M Gx) Dcsub) (transe/pi2 DtransN'') (DsubtypeFC' aca)) + %<- (transe-reg DtransM (%the (wfe G1 D) DwfD)) + %<- (subtype-reg-e' DsubtypeDA DwfD DwfA (%the (ofe G1 M A) DofM)) + %<- ({x} {d isvar x I} {ex} {xt vtrans ex x} transe-reg (DtransN x d ex xt) (wfe/sigma (%the ({y} %pi (isvar y J) %-> (wfe (cons (G x) y (B x)) (C x))) (DwfC x d)) (%the (wfe (G x) (B x)) (DwfB x d)))) + %<- ({x} {d isvar x I} tinhabitsub-e (DwfB x d) (DwfC x d) (%the (wfe (G x) (C x)) (DwfC' x d))) + %<- (can-tsub-e Dappend DwfB DofM (%the (tsub B M Bx) DsubBx)) + %<- (trans-sub-e Dappend DtransN DtransM DwfA DsubtypeDA (tsub/sigma ([y] DsubCx) DsubBx) Dcsub (%the (transe Gx (EN EM) EF) DtransN') (%the (subtype EF (sigma Bx ([_] Cx)) _) DsubtypeEF)) + %<- (transe-principal DtransN' (%the (principal EF) DprincipalEF)) + %<- (principal-subtype-sigma-invert DprincipalEF DsubtypeEF (%the (tp-eq EF (sigma E ([_] F))) DeqEF)) + %<- (transe-resp ctx-eq/i eterm-eq/i DeqEF DtransN' (%the (transe Gx (EN EM) (sigma E ([_] F))) DtransN'')) + %<- (subtype-resp DeqEF tp-eq/i ([_] term-eq/i) DsubtypeEF (%the (subtype (sigma E ([_] F)) (sigma Bx ([_] Cx)) _) DsubtypeEF')) + %<- (subtype-sigma-invert DsubtypeEF' (%the (subtype E Bx N1) DsubtypeBE) (%the ({y} tsub ([_] Cx) (N1 y) (Cx' y)) DsubCx') (%the ({y} subtype F (Cx' y) ([z] N2 y z)) DsubtypeFC) _) + %<- ({y} tsub-absent Cx (N1 y) (%the (tsub ([_] Cx) (N1 y) Cx) (DsubCxabs y))) + %<- ({y} tsub-fun (DsubCx' y) (DsubCxabs y) (%the (tp-eq (Cx' y) Cx) (DeqCx' y))) + %<- ({y} subtype-resp tp-eq/i (DeqCx' y) ([_] term-eq/i) (DsubtypeFC y) (%the (subtype F Cx ([z] N2 y z)) (DsubtypeFC' y))) %. +%scope ttrans-sub-e %term lam + %pi (trans-sub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) ([x] [d isvar x I] [ex] [xt vtrans ex x] transe/lam (%the ({y} + %pi (isvar y J) + %-> ({ey} %pi (vtrans ey y) %-> (transe (cons (G x) y (B x)) (EN ex ey) (C x y)))) (DtransN x d ex xt)) (%the (ttranse (G x) (EB ex) (B x)) (DtransB x d ex xt))) (%the (transe G1 EM D) DtransM) (%the (wfe G1 A) DwfA) (%the (subtype D A ([_] M)) DsubtypeDA) (tsub/pi (%the ({y} tsub ([x] C x y) M (Cx y)) DsubCx) (%the (tsub B M Bx) DsubBx)) (%the (csub G M Gx) Dcsub) (transe/lam DtransN' DtransB') (subtype/pi DsubtypeEC DsubE DsubtypeBx)) + %<- (transe-reg DtransM (%the (wfe G1 D) DwfD)) + %<- (subtype-reg-e' DsubtypeDA DwfD DwfA (%the (ofe G1 M A) DofM)) + %<- (ttrans-sub-e Dappend DtransB DtransM DwfA DsubtypeDA DsubBx Dcsub (%the (ttranse Gx (EB EM) Bx) DtransB')) + %<- ({y} {e isvar y J} {ey} {yt vtrans ey y} trans-sub-e ([x] append/cons (Dappend x)) ([x] [d] [ex] [xt] DtransN x d ex xt y e ey yt) DtransM DwfA DsubtypeDA (DsubCx y) (csub/cons DsubBx Dcsub) (%the (transe (cons Gx y Bx) (EN EM ey) (E y)) (DtransN' y e ey yt)) (%the (subtype (E y) (Cx y) _) (DsubtypeEC y))) + %<- ({x} {d isvar x I} {ex} {xt vtrans ex x} ttranse-reg (DtransB x d ex xt) (%the (wfe (G x) (B x)) (DwfB x d))) + %<- (tsubst-e Dappend Dcsub DofM DsubBx DwfB (%the (wfe Gx Bx) DwfBx)) + %<- (subtype-refl-e DwfBx (%the (subtype Bx Bx Z) DsubtypeBx) (%the ({z} expand z Bx (Z z)) DexpandZ)) + %<- ({y} {e isvar y J} {ey} {yt vtrans ey y} transe-reg (DtransN' y e ey yt) (%the (wfe (cons Gx y Bx) (E y)) (DwfE y e))) + %<- (tsub-expand-var-e DwfBx DwfE DexpandZ (%the ({y} tsub E (Z y) (E y)) DsubE)) %. +%term pair + %pi (trans-sub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) ([x] [d isvar x I] [ex] [xt vtrans ex x] transe/pair (%the (transe (G x) (EO ex) (C x)) (DtransO x d ex xt)) (%the (transe (G x) (EN ex) (B x)) (DtransN x d ex xt))) (%the (transe G1 EM D) DtransM) (%the (wfe G1 A) DwfA) (%the (subtype D A ([_] M)) DsubtypeDA) (tsub/sigma (%the ({y} tsub ([x] C x) M (Cx' y)) DsubCx') (%the (tsub B M Bx) DsubBx)) (%the (csub G M Gx) Dcsub) (transe/pair DtransO' DtransN') (subtype/sigma ([_] DsubtypeFC) DsubCxabs' DsubtypeEB)) + %<- (trans-sub-e Dappend DtransN DtransM DwfA DsubtypeDA DsubBx Dcsub (%the (transe Gx (EN EM) E) DtransN') (%the (subtype E Bx P) DsubtypeEB)) + %<- (tsub-closed DsubCx' (%the ({y} tp-eq (Cx' y) Cx) DeqCx)) + %<- ({y} tsub-resp ([_] tp-eq/i) term-eq/i (DeqCx y) (DsubCx' y) (%the (tsub ([x] C x) M Cx) (DsubCx y))) + %<- (trans-sub-e Dappend DtransO DtransM DwfA DsubtypeDA (DsubCx aca) Dcsub (%the (transe Gx (EO EM) F) DtransO') (%the (subtype F Cx _) DsubtypeFC)) + %<- ({y} tsub-absent Cx (P y) (%the (tsub ([_] Cx) (P y) Cx) (DsubCxabs y))) + %<- ({y} tp-eq-symm (DeqCx y) (%the (tp-eq Cx (Cx' y)) (DeqCx' y))) + %<- ({y} tsub-resp DeqCx' term-eq/i tp-eq/i (DsubCxabs y) (%the (tsub ([y] Cx' y) (P y) Cx) (DsubCxabs' y))) %. +%%% +%scope ttrans-sub-e %term t + %pi (ttrans-sub-e _ ([x] [d isvar x I] [ex] [dt vtrans ex x] ttranse/t (%the (ordered (G x)) (Dordered x d))) _ _ _ tsub/t (%the (csub G MA G') Dcsub) (ttranse/t Dordered')) + %<- (csub-ordered Dordered Dcsub (%the (ordered G') Dordered')) %. +%term pi + %pi (ttrans-sub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) ([x] [d isvar x I] [ex] [xt vtrans ex x] ttranse/pi (%the ({y} + %pi (isvar y J) + %-> ({ey} %pi (vtrans ey y) %-> (ttranse (cons (G x) y (B x)) (EC ex ey) (C x y)))) (DtransC x d ex xt)) (%the (ttranse (G x) (EB ex) (B x)) (DtransB x d ex xt))) (%the (transe G1 EM D) DtransM) (%the (wfe G1 A) DwfA) (%the (subtype D A ([_] M)) DsubtypeDA) (tsub/pi (%the ({y} tsub ([x] C x y) M (Cx y)) DsubCx) (%the (tsub B M Bx) DsubBx)) (%the (csub G M Gx) Dcsub) (ttranse/pi DtransCx DtransBx)) + %<- (ttrans-sub-e Dappend DtransB DtransM DwfA DsubtypeDA DsubBx Dcsub (%the (ttranse Gx (EB EM) Bx) DtransBx)) + %<- ({x} {d isvar x I} {ex} {xt vtrans ex x} {y} {e isvar y J} {ey} {yt vtrans ey y} ttranse-context (DtransC x d ex xt y e ey yt) (ordered/cons (%the (bounded (G x) y) (Dbounded x d y e)))) + %<- ({y} {e isvar y J} {ey} {yt vtrans ey y} ttrans-sub-e ([x] append/cons (Dappend x)) ([x] [d] [ex] [xt] DtransC x d ex xt y e ey yt) DtransM DwfA DsubtypeDA (DsubCx y) (csub/cons DsubBx Dcsub) (%the (ttranse (cons Gx y Bx) (EC EM ey) (Cx y)) (DtransCx y e ey yt))) %. +%scope ttrans-sub-e %term sigma + %pi (ttrans-sub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) ([x] [d isvar x I] [ex] [xt vtrans ex x] ttranse/sigma (%the ({y} + %pi (isvar y J) + %-> ({ey} %pi (vtrans ey y) %-> (ttranse (cons (G x) y (B x)) (EC ex ey) (C x y)))) (DtransC x d ex xt)) (%the (ttranse (G x) (EB ex) (B x)) (DtransB x d ex xt))) (%the (transe G1 EM D) DtransM) (%the (wfe G1 A) DwfA) (%the (subtype D A ([_] M)) DsubtypeDA) (tsub/sigma (%the ({y} tsub ([x] C x y) M (Cx y)) DsubCx) (%the (tsub B M Bx) DsubBx)) (%the (csub G M Gx) Dcsub) (ttranse/sigma DtransCx DtransBx)) + %<- (ttrans-sub-e Dappend DtransB DtransM DwfA DsubtypeDA DsubBx Dcsub (%the (ttranse Gx (EB EM) Bx) DtransBx)) + %<- ({x} {d isvar x I} {ex} {xt vtrans ex x} {y} {e isvar y J} {ey} {yt vtrans ey y} ttranse-context (DtransC x d ex xt y e ey yt) (ordered/cons (%the (bounded (G x) y) (Dbounded x d y e)))) + %<- ({y} {e isvar y J} {ey} {yt vtrans ey y} ttrans-sub-e ([x] append/cons (Dappend x)) ([x] [d] [ex] [xt] DtransC x d ex xt y e ey yt) DtransM DwfA DsubtypeDA (DsubCx y) (csub/cons DsubBx Dcsub) (%the (ttranse (cons Gx y Bx) (EC EM ey) (Cx y)) (DtransCx y e ey yt))) %. +%term sing + %pi (ttrans-sub-e (%the ({x} append (cons G1 x A) (G2 x) (G x)) Dappend) ([x] [d isvar x I] [ex] [xt vtrans ex x] ttranse/sing (%the (transe (G x) (EN ex) (sing (R x))) (Dtrans x d ex xt))) (%the (transe G1 EM D) DtransM) (%the (wfe G1 A) DwfA) (%the (subtype D A ([_] M)) DsubtypeDA) (%the (tsub ([x] sing (R x)) M (sing Rx)) Dtsub) (%the (csub G M Gx) Dcsub) (ttranse/sing DtransRx)) + %<- (tsub-sing-sub-at Dtsub (%the (sub ([x] at (R x)) M (at Rx)) Dsub)) + %<- (trans-sub-e Dappend Dtrans DtransM DwfA DsubtypeDA Dtsub Dcsub (%the (transe Gx (EN EM) E) DtransP) (%the (subtype E (sing Rx) _) DsubtypeE)) + %<- (subtype-sing-invert DsubtypeE (%the (tp-eq E (sing Rx)) DeqE) _) + %<- (transe-resp ctx-eq/i eterm-eq/i DeqE DtransP (%the (transe Gx (EN EM) (sing Rx)) DtransRx)) %. +%worlds (bind ovar tbind) (ttrans-sub-e _ _ _ _ _ _ _ _) (trans-sub-e _ _ _ _ _ _ _ _ _) %. +%total (D1 D2) (ttrans-sub-e _ D1 _ _ _ _ _ _) (trans-sub-e _ D2 _ _ _ _ _ _ _) %. +%%%%% Translation Commutes with Substitution %%%%% +%sort ttrans-sub {x} %. +%mode ttrans-sub %in %in %in %in %in %out %. +%term _ + %pi (ttrans-sub DtransB DtransM DwfA Dsubtype Dtsub DtransB') + %<- (ttrans1-to-ttranse 0 DtransB DtranseB) + %<- (trans-to-transe DtransM DtranseM) + %<- (wf-to-wfe DwfA DwfeA) + %<- (ttrans-sub-e ([_] append/nil) DtranseB DtranseM DwfeA Dsubtype Dtsub csub/base DtranseB') + %<- (ttranse-to-ttrans DtranseB' DtransB') %. +%worlds (bind tbind) (ttrans-sub _ _ _ _ _ _) %. +%total {} (ttrans-sub _ _ _ _ _ _) %. +%sort trans-sub {x} %. +%mode trans-sub %in %in %in %in %in %out %out %. +%term _ + %pi (trans-sub DtransN DtransM DwfA Dsubtype Dtsub DtransN' Dsubtype') + %<- (trans1-to-transe 0 DtransN DtranseN) + %<- (trans-to-transe DtransM DtranseM) + %<- (wf-to-wfe DwfA DwfeA) + %<- (trans-sub-e ([_] append/nil) DtranseN DtranseM DwfeA Dsubtype Dtsub csub/base DtranseN' Dsubtype') + %<- (transe-to-trans DtranseN' DtransN') %. +%worlds (bind tbind) (trans-sub _ _ _ _ _ _ _) %. +%total {} (trans-sub _ _ _ _ _ _ _) %. +%sort trans-var-coerce %. +%mode trans-var-coerce %in %in %in %in %in %out %. +%term _ + %pi (trans-var-coerce (%the (ttrans EA A) DtransA) (%the (ttrans EB B) DtransB) (%the (subtype A B M) DsubtypeAB) (%the ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (ttrans (EC ex) (C x)))) DtransC) (%the ({x} %pi (vof x B) %-> ({ex} %pi (vtrans ex x) %-> (ttrans (EC ex) (D x)))) DtransD) DsubC) + %<- (ttrans-reg DtransA (%the (wf A) DwfA)) + %<- (ttrans-reg DtransB (%the (wf B) DwfB)) + %<- ({x} can-expand x A (X x) (%the (expand x A (X x)) (Dexpand x))) + %<- ({x} {d vof x A} expand-reg (aof/var DwfA d) (Dexpand x) (%the (of (X x) A) (DofX x d))) + %<- ({x} {d vof x A} can-self (DofX x d) (%the (self (X x) A (As x)) (Dself x))) + %<- ({x} {d vof x A} self-reg (Dself x) (DofX x d) (%the (of (X x) (As x)) (DofX' x d)) (%the (subtype (As x) A ([_] X x)) (DsubtypeAsA x))) + %<- ({x} {d vof x A} of-reg (DofX' x d) (%the (wf (As x)) (DwfAs x d))) + %<- ({x} {d vof x B} {ex} {xt vtrans ex x} ttrans-reg (DtransD x d ex xt) (%the (wf (D x)) (DwfD x d))) + %<- (subtype-reg DsubtypeAB DwfA DwfB (%the ({x} %pi (vof x A) %-> (of (M x) B)) DofM)) + %<- ({x} {d vof x A} can-tsub DwfD (DofM x d) (%the (tsub D (M x) (E x)) (DsubE x))) + %<- ({x} {d vof x A} can-sub DofM (DofX x d) (%the (sub M (X x) (Mx x)) (DsubMx x))) + %<- ({x} {d vof x A} sub-expand (aof/var DwfA d) DofM (Dexpand x) (DsubMx x) (%the (term-eq (Mx x) (M x)) (DeqM x))) + %<- ({x} sub-resp ([_] term-eq/i) term-eq/i (DeqM x) (DsubMx x) (%the (sub M (X x) (M x)) (DsubM x))) + %<- ({x} {d vof x A} subtype-trans (DwfAs x d) DwfA DwfB (DsubtypeAsA x) DsubtypeAB ([_] DsubM x) (%the (subtype (As x) B ([_] M x)) (DsubtypeAsB x))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} ttrans-sub DtransD (%the (trans ex (As x)) (trans/var (Dself x) (Dexpand x) DwfA d xt)) DwfB (DsubtypeAsB x) (DsubE x) (%the (ttrans (EC ex) (E x)) (DtransE x d ex xt))) + %<- ({x} {d vof x A} {ex} {xt vtrans ex x} ttrans-fun (DtransE x d ex xt) (DtransC x d ex xt) (%the (tp-eq (E x) (C x)) (Deq x))) + %<- ({x} tsub-resp ([_] tp-eq/i) term-eq/i (Deq x) (DsubE x) (%the (tsub D (M x) (C x)) (DsubC x))) %. +%worlds (bind tbind) (trans-var-coerce _ _ _ _ _ _) %. +%total {} (trans-var-coerce _ _ _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sing/translate.lf b/new-tests/stelf-output/tslf/sing/translate.lf new file mode 100644 index 0000000..bef7da2 --- /dev/null +++ b/new-tests/stelf-output/tslf/sing/translate.lf @@ -0,0 +1,108 @@ +%sort vtrans %. +%sort ttrans %. +%sort trans %. +%term ttrans/t ttrans et t %. +%term ttrans/pi + %pi (ttrans (epi EA EB) (pi A B)) + %<- (ttrans EA A) + %<- ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (ttrans (EB ex) (B x)))) %. +%term ttrans/sigma + %pi (ttrans (esigma EA EB) (sigma A B)) + %<- (ttrans EA A) + %<- ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (ttrans (EB ex) (B x)))) %. +%term ttrans/sing %pi (ttrans (esing EM) (sing R)) %<- (trans EM (sing R)) %. +%term trans/const + %pi (trans (econst C) A') + %<- (kof C A) + %<- (wf A) + %<- (expand (const C) A M) + %<- (self M A A') %. +%term trans/forall + %pi (trans (eforall EA) B) + %<- (ttrans EA A) + %<- (expand (forall A) (qtp A) M) + %<- (self M (qtp A) B) %. +%term trans/var + %pi (trans EM A') + %<- (vtrans EM X) + %<- (vof X A) + %<- (wf A) + %<- (expand X A M) + %<- (self M A A') %. +%term trans/app + %pi (trans (eapp EM EN) Bx) + %<- (trans EM (pi A B)) + %<- (trans EN C) + %<- (subtype C A ([_] N)) + %<- (tsub ([x] B x) N Bx) %. +%term trans/pi1 %pi (trans (epi1 EM) A) %<- (trans EM (sigma A ([_] B))) %. +%term trans/pi2 %pi (trans (epi2 EM) B) %<- (trans EM (sigma A ([_] B))) %. +%term trans/lam + %pi (trans (elam EA EM) (pi A B)) + %<- (ttrans EA A) + %<- ({x} %pi (vof x A) %-> ({ex} %pi (vtrans ex x) %-> (trans (EM ex) (B x)))) %. +%term trans/pair %pi (trans (epair EM EN) (sigma A ([_] B))) %<- (trans EM A) %<- (trans EN B) %. +%block tbind {r atom} [ex eterm] [tr vtrans ex r]%. +%sort _ %. +%%%%% Explicit Contexts %%%%% +%sort ttranse %. +%sort transe %. +%term ttranse/t %pi (ttranse G et t) %<- (ordered G) %. +%term ttranse/pi + %pi (ttranse G (epi EA EB) (pi A B)) + %<- (ttranse G EA A) + %<- ({x} + %pi (isvar x I) + %-> ({ex} %pi (vtrans ex x) %-> (ttranse (cons G x A) (EB ex) (B x)))) %. +%term ttranse/sigma + %pi (ttranse G (esigma EA EB) (sigma A B)) + %<- (ttranse G EA A) + %<- ({x} + %pi (isvar x I) + %-> ({ex} %pi (vtrans ex x) %-> (ttranse (cons G x A) (EB ex) (B x)))) %. +%term ttranse/sing %pi (ttranse G (esing EM) (sing R)) %<- (transe G EM (sing R)) %. +%term transe/const + %pi (transe G (econst C) A') + %<- (kof C A) + %<- (wf A) + %<- (expand (const C) A M) + %<- (self M A A') + %<- (ordered G) %. +%term transe/forall + %pi (transe G (eforall EA) B) + %<- (ttranse G EA A) + %<- (expand (forall A) (qtp A) M) + %<- (self M (qtp A) B) %. +%term transe/var + %pi (transe G EM A') + %<- (vtrans EM X) + %<- (lookup G X A) + %<- (wfe G A) + %<- (expand X A M) + %<- (self M A A') %. +%term transe/vari + %pi (transe G EM A') + %<- (vtrans EM X) + %<- (vof X A) + %<- (wf A) + %<- (expand X A M) + %<- (self M A A') + %<- (ordered G) %. +%term transe/app + %pi (transe G (eapp EM EN) Bx) + %<- (transe G EM (pi A B)) + %<- (transe G EN C) + %<- (subtype C A ([_] N)) + %<- (tsub ([x] B x) N Bx) %. +%term transe/pi1 %pi (transe G (epi1 EM) A) %<- (transe G EM (sigma A ([_] B))) %. +%term transe/pi2 %pi (transe G (epi2 EM) B) %<- (transe G EM (sigma A ([_] B))) %. +%term transe/lam + %pi (transe G (elam EA EM) (pi A B)) + %<- (ttranse G EA A) + %<- ({x} + %pi (isvar x I) + %-> ({ex} %pi (vtrans ex x) %-> (transe (cons G x A) (EM ex) (B x)))) %. +%term transe/pair + %pi (transe G (epair EM EN) (sigma A ([_] B))) + %<- (transe G EM A) + %<- (transe G EN B) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/singleton-kinds.lf b/new-tests/stelf-output/tslf/singleton-kinds.lf new file mode 100644 index 0000000..c327dcf --- /dev/null +++ b/new-tests/stelf-output/tslf/singleton-kinds.lf @@ -0,0 +1,18 @@ +%%%% singleton kinds +%sort sing/kd %. +%mode sing/kd %in %in %out %. +%term sing/kd/unit sing/kd _ kd/unit kd/unit %. +%term sing/kd/type sing/kd C kd/type (kd/sing C) %. +%term sing/kd/sing sing/kd C (kd/sing _) (kd/sing C) %. +%term sing/kd/pi + %pi (sing/kd C (kd/pi K1 K2) (kd/pi K1 K2')) + %<- ({a} sing/kd (cn/app C a) (K2 a) (K2' a)) %. +%{ +sing/kd/sgm : sing/kd C (kd/sgm K1 K2) (kd/sgm K1' ([a] K2' (cn/pj1 C))) + <- sing/kd (cn/pj1 C) K1 K1' + <- {a} sing/kd (cn/pj2 C) (K2 a) (K2' a). +}% +%term sing/kd/sgm + %pi (sing/kd C (kd/sgm K1 K2) (kd/sgm K1' ([a] K2'))) + %<- (sing/kd (cn/pj2 C) (K2 (cn/pj1 C)) K2') + %<- (sing/kd (cn/pj1 C) K1 K1') %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/singleton-lemmas.lf b/new-tests/stelf-output/tslf/singleton-lemmas.lf new file mode 100644 index 0000000..a856060 --- /dev/null +++ b/new-tests/stelf-output/tslf/singleton-lemmas.lf @@ -0,0 +1,223 @@ +%sort sing/kd-complete {C cn} {K kd} %. +%mode sing/kd-complete %in %in %out %. +%term _ sing/kd-complete _ kd/unit sing/kd/unit %. +%term _ sing/kd-complete _ kd/type sing/kd/type %. +%term _ sing/kd-complete _ (kd/sing _) sing/kd/sing %. +%term _ + %pi (sing/kd-complete C (kd/pi K1 K2) (sing/kd/pi DSK)) + %<- ({a} sing/kd-complete (cn/app C a) (K2 a) (DSK a)) %. +%term _ + %pi (sing/kd-complete C (kd/sgm K1 K2) (sing/kd/sgm DSK1 (DSK2 (cn/pj1 C)))) + %<- (sing/kd-complete (cn/pj1 C) K1 DSK1) + %<- ({a} sing/kd-complete (cn/pj2 C) (K2 a) (DSK2 a)) %. +%worlds (cn-block) (sing/kd-complete _ _ _) %. +%total K (sing/kd-complete _ K _) %. +%sort stone-2/120 %. +%mode stone-2/120 %in %out %. +%term _ + %pi (stone-2/120 DQ (cn-deq/sym (cn-deq/kd/sing (ofkd/sub (ofkd/kd/sing DC2) (kd-sub/kd/sing-kd/sing (cn-deq/sym DQ)))))) + %<- (vdt/cn-deq DQ _ DC2 _) %. +%worlds (ofkd+vdt-block) (stone-2/120 _ _) %. +%total {} (stone-2/120 _ _) %. +%sort stone-2/119 %. +%mode stone-2/119 %in %out %. +%term _ + %pi (stone-2/119 DCQ (ofkd/sub (ofkd/kd/sing D1) (kd-sub/kd/sing-kd/sing DCQ))) + %<- (vdt/cn-deq DCQ D1 _ _) %. +%worlds (ofkd+vdt-block) (stone-2/119 _ _) %. +%total {} (stone-2/119 _ _) %. +%sort stone-2/96 %. +%mode stone-2/96 %in %in %out %. +%term _ stone-2/96 DQ sing/kd/unit DQ %. +%term _ %pi (stone-2/96 DQ sing/kd/type (cn-deq/kd/sing DC1)) %<- (stone-2/119 DQ DC1) %. +%term _ + %pi (stone-2/96 DQ sing/kd/sing (cn-deq/kd/sing DC1')) + %<- (vdt/cn-deq DQ DC1 DC2 (kd-wf/kd/sing DBT)) + %<- (stone-2/119 (cn-deq/sub DQ (kd-sub/kd/sing-kd/type DBT)) DC1') %. +%term _ + %pi (stone-2/96 DQ (sing/kd/pi DSK) (cn-deq/pi-ext DC1 DC2 DQ')) + %<- (vdt/cn-deq DQ DC1 DC2 (kd-wf/kd/pi DKW _)) + %<- ({a} {da} {dm mofkd da met/unit} {_ can-mofkd da dm} {dw vdt/ofkd da DKW} stone-2/96 (cn-deq/cn/app DQ (cn-deq/refl da)) (DSK a) (DQ' a da)) %. +%term _ + %pi (stone-2/96 (%the (cn-deq C' C (kd/sgm K1 K2)) DQ) (sing/kd/sgm (%the (sing/kd (cn/pj1 C) K1 K1') DSK1) (%the (sing/kd (cn/pj2 C) (K2 (cn/pj1 C)) K2') DSK2)) (cn-deq/sgm-ext DQL DQR)) + %<- (vdt/cn-deq DQ DC1 DC2 (kd-wf/kd/sgm _ DKWo)) + %<- (stone-2/96 (cn-deq/cn/pj1 DQ) DSK1 (%the (cn-deq (cn/pj1 C') (cn/pj1 C) K1') DQL)) + %<- (funct/kd-wf DKWo (cn-deq/cn/pj1 DQ) (ofkd/cn/pj1 DC1) (ofkd/cn/pj1 DC2) DKS) + %<- (stone-2/96 (cn-deq/deq (cn-deq/cn/pj2 DQ) DKS) DSK2 DQR) %. +%worlds (ofkd+vdt-block) (stone-2/96 _ _ _) %. +%total (D3) (stone-2/96 _ D3 _) %. +%sort stone-2/92 %. +%mode stone-2/92 %in %in %out %. +%term _ + %pi (stone-2/92 DC DS DW) + %<- (stone-2/96 (cn-deq/refl DC) DS DQ) + %<- (vdt/cn-deq DQ _ _ DW) %. +%worlds (ofkd+vdt-block) (stone-2/92 _ _ _) %. +%total {} (stone-2/92 _ _ _) %. +%sort stone-2/93 %. +%mode stone-2/93 %in %in %out %. +%term _ + %pi (stone-2/93 DC DS DC1) + %<- (stone-2/96 (cn-deq/refl DC) DS DQ) + %<- (vdt/cn-deq DQ DC1 _ DW) %. +%worlds (ofkd+vdt-block) (stone-2/93 _ _ _) %. +%total {} (stone-2/93 _ _ _) %. +%sort stone-2/97 %. +%mode stone-2/97 %in %in %in %out %. +%term _ stone-2/97 D1 sing/kd/unit D2 (cn-deq/kd/unit D2 D1) %. +%term _ stone-2/97 D1 sing/kd/type D2 (cn-deq/kd/sing D2) %. +%term _ stone-2/97 D1 sing/kd/sing D2 (cn-deq/kd/sing D2) %. +%term _ + %pi (stone-2/97 D1 (sing/kd/pi DSK) D2 (cn-deq/pi-ext D2 D1 DQ)) + %<- (vdt/ofkd D2 (kd-wf/kd/pi DWF _)) + %<- ({a} {da} {dm mofkd da met/unit} {_ can-mofkd da dm} {dw vdt/ofkd da DWF} stone-2/97 (ofkd/cn/app D1 da) (DSK a) (ofkd/cn/app D2 da) (DQ a da)) %. +%term _ + %pi (stone-2/97 D1 (%the (sing/kd C2 (kd/sgm K1 K2) (kd/sgm K1' ([a] K2'))) (sing/kd/sgm DSK1 DSK2)) D2 (cn-deq/sgm-ext DQL DQR)) + %<- (stone-2/97 (ofkd/cn/pj1 D1) DSK1 (ofkd/cn/pj1 D2) DQL) + %<- (stone-2/97 (%the (ofkd (cn/pj2 C2) (K2 (cn/pj1 C2))) (ofkd/cn/pj2 D1)) DSK2 (%the (ofkd (cn/pj2 C1) K2') (ofkd/cn/pj2 D2)) DQR) %. +%worlds (ofkd+vdt-block) (stone-2/97 _ _ _ _) %. +%total D1 (stone-2/97 _ D1 _ _) %. +%sort stone-2/94 %. +%mode stone-2/94 %in %in %out %. +%term _ stone-2/94 _ sing/kd/unit kd-sub/kd/unit %. +%term _ stone-2/94 D1 sing/kd/type (kd-sub/kd/sing-kd/type D1) %. +%term _ + %pi (stone-2/94 D1 sing/kd/sing (kd-sub/kd/sing-kd/sing (cn-deq/sub (cn-deq/kd/sing D1) (kd-sub/kd/sing-kd/type DB)))) + %<- (vdt/ofkd D1 (kd-wf/kd/sing DB)) %. +%term _ + %pi (stone-2/94 D1 (sing/kd/pi DSK) (kd-sub/kd/pi DS1 DS2 DW2)) + %<- (vdt/ofkd D1 (kd-wf/kd/pi DW1 _)) + %<- ({a} {da} {dm mofkd da met/unit} {_ can-mofkd da dm} {dw vdt/ofkd da DW1} stone-2/94 (ofkd/cn/app D1 da) (DSK a) (DS2 a da)) + %<- (kd-refl/sub DW1 DS1) + %<- ({a} {da} {dm mofkd da met/unit} {_ can-mofkd da dm} {dw vdt/ofkd da DW1} vdt/kd-sub (DS2 a da) (DW2 a da) (DW2' a da)) %. +%term _ + %pi (stone-2/94 (%the (ofkd C (kd/sgm KA KB)) D1) (%the (sing/kd C (kd/sgm KA KB) (kd/sgm K0 ([a] K1))) (sing/kd/sgm DSK1 DSK2)) (%the (kd-sub (kd/sgm K0 ([a] K1)) (kd/sgm KA KB)) (kd-sub/kd/sgm DS1 DSr' DW2))) + %<- (stone-2/94 (ofkd/cn/pj1 D1) DSK1 DS1) + %<- (stone-2/94 (ofkd/cn/pj2 D1) DSK2 (%the (kd-sub K1 (KB (cn/pj1 C))) DS2)) + %<- (vdt/ofkd D1 (kd-wf/kd/sgm (%the (kd-wf KA) DW1) (%the ({a} {da ofkd a KA} kd-wf (KB a)) DW2))) + %<- (kd-wkn/kd-wf DW2 DS1 (%the ({a} {da ofkd a K0} kd-wf (KB a)) DW2')) + %<- (vdt/kd-sub DS1 (%the (kd-wf K0) DWa) DWa') + %<- ({a} {da} {dm mofkd da met/unit} {_ can-mofkd da dm} {dw vdt/ofkd da DWa} stone-2/97 (ofkd/cn/pj1 D1) DSK1 da (%the (cn-deq a (cn/pj1 C) K0) (DQ a da))) + %<- ({a} {da} {dm mofkd da met/unit} {_ can-mofkd da dm} {dw vdt/ofkd da DWa} vdt/cn-deq (DQ a da) (Dlala a da) (DZ a da) (Dblah a da)) + %<- ({b} {db ofkd b K0} {dm mofkd db met/unit} {_ can-mofkd db dm} funct/kd-wf ([a] [da ofkd a K0] DW2' a da) (%the (cn-deq b (cn/pj1 _) K0) (DQ b db)) db (DZ b db) (%the (kd-deq (KB b) (KB (cn/pj1 _))) (DQs b db))) + %<- ({a} {da} {dm mofkd da met/unit} {_ can-mofkd da dm} {dw vdt/ofkd da DWa} vdt/kd-deq (DQs a da) (%the (kd-wf (KB a)) (Da1 a da)) (%the (kd-wf (KB (cn/pj1 _))) (Da2 a da))) + %<- ({a} {da} kd-anti (%the (kd-deq (KB a) (KB (cn/pj1 _))) (DQs a da)) (Da1 a da) (Da2 a da) (DSl a da) (%the (kd-sub (KB (cn/pj1 _)) (KB a)) (DSr a da))) + %<- ({a} {da ofkd a K0} {dm mofkd da met/unit} {_ can-mofkd da dm} {dw vdt/ofkd da DWa} kd-trans/sub DS2 (DSr a da) (%the (kd-sub K1 (KB a)) (DSr' a da))) %. +%worlds (ofkd+vdt-block) (stone-2/94 _ _ _) %. +%total D1 (stone-2/94 _ D1 _) %. +%sort stone-2/108 %. +%mode stone-2/108 %in %in %out %. +%term _ + %pi (stone-2/108 D1 D2 (cn-deq/sub DQ DKS)) + %<- (sing/kd-complete _ _ DS) + %<- (stone-2/92 D1 DS _) + %<- (stone-2/93 D1 DS D1') + %<- (stone-2/97 D1 DS (ofkd/cn/pj1 (ofkd/cn/pair D1' D2)) DQ) + %<- (stone-2/94 D1 DS DKS) %. +%worlds (ofkd+vdt-block) (stone-2/108 _ _ _) %. +%total {} (stone-2/108 _ _ _) %. +%sort stone-2/109 %. +%mode stone-2/109 %in %in %out %. +%term _ + %pi (stone-2/109 D1 D2 (cn-deq/sub DQ DKS)) + %<- (sing/kd-complete _ _ DS) + %<- (stone-2/92 D2 DS _) + %<- (stone-2/93 D2 DS D2') + %<- (stone-2/97 D2 DS (ofkd/cn/pj2 (ofkd/cn/pair D1 D2')) DQ) + %<- (stone-2/94 D2 DS DKS) %. +%worlds (ofkd+vdt-block) (stone-2/109 _ _ _) %. +%total {} (stone-2/109 _ _ _) %. +%sort cn-deq-beta {a} %. +%mode cn-deq-beta %in %in %out %. +%term _ + %pi (cn-deq-beta (%the ({a} %pi (ofkd a K1) %-> (ofkd (C1 a) (K2 a))) Dof1) (%the (ofkd C2 K1) Dof2) (cn-deq/sub Dequiv Dsub)) + %<- (vdt/ofkd Dof2 (%the (kd-wf K1) DwfK1)) + %<- ({a} sing/kd-complete _ _ (%the (sing/kd (C1 a) (K2 a) (K2s a)) (Dsing a))) + %<- ({a} {d ofkd a K1} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (vdt/ofkd d DwfK1) + %-> (stone-2/93 (Dof1 a d) (Dsing a) (%the (ofkd (C1 a) (K2s a)) (Dof1' a d)))) + %<- (stone-2/97 (%the (ofkd (C1 C2) (K2 C2)) (Dof1 C2 Dof2)) (%the (sing/kd (C1 C2) (K2 C2) (K2s C2)) (Dsing C2)) (%the (ofkd (cn/app (cn/lam K1 C1) C2) (K2s C2)) (ofkd/cn/app (ofkd/cn/lam Dof1' DwfK1) Dof2)) (%the (cn-deq (cn/app (cn/lam K1 C1) C2) (C1 C2) (K2s C2)) Dequiv)) + %<- (stone-2/94 (Dof1 C2 Dof2) (Dsing C2) (%the (kd-sub (K2s C2) (K2 C2)) Dsub)) %. +%worlds (ofkd+vdt-block) (cn-deq-beta _ _ _) %. +%total {} (cn-deq-beta _ _ _) %. +%sort ofkd-sgm-intro {a} %. +%mode ofkd-sgm-intro %in %in %in %out %. +%term _ + %pi (ofkd-sgm-intro (%the ({a} %pi (ofkd a K1) %-> (kd-wf (K2 a))) Dwf2) (%the (ofkd C1 K1) Dof1) (%the (ofkd C2 (K2 C1)) Dof2) (ofkd/sub (ofkd/cn/pair Dof1' Dof2) (kd-sub/kd/sgm Dsub1 Dsub2 Dwf2))) + %<- (vdt/ofkd Dof1 (%the (kd-wf K1) Dwf1)) + %<- (sing/kd-complete _ _ (%the (sing/kd C1 K1 K1s) Dsing)) + %<- (stone-2/93 Dof1 Dsing (%the (ofkd C1 K1s) Dof1')) + %<- (stone-2/94 Dof1 Dsing (%the (kd-sub K1s K1) Dsub1)) + %<- (stone-2/92 Dof1 Dsing (%the (kd-wf K1s) DwfK1s)) + %<- ({a} {d ofkd a K1s} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (vdt/ofkd d DwfK1s) + %-> (stone-2/97 Dof1 Dsing d (%the (cn-deq a C1 K1s) (Dequiv a d)))) + %<- ({a} {d ofkd a K1s} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (funct/kd-wf Dwf2 (cn-deq/sub (Dequiv a d) Dsub1) (ofkd/sub d Dsub1) Dof1 (%the (kd-deq (K2 a) (K2 C1)) (Dequiv' a d)))) + %<- ({a} {d ofkd a K1s} kd-anti (Dequiv' a d) (Dwf2 _ (ofkd/sub d Dsub1)) (Dwf2 _ Dof1) (Dsub2' a d) (%the (kd-sub (K2 C1) (K2 a)) (Dsub2 a d))) %. +%worlds (ofkd+vdt-block) (ofkd-sgm-intro _ _ _ _) %. +%total {} (ofkd-sgm-intro _ _ _ _) %. +%sort cn-deq-sgm-intro {a} %. +%mode cn-deq-sgm-intro %in %in %in %out %. +%term _ + %pi (cn-deq-sgm-intro (%the ({a} %pi (ofkd a K1) %-> (kd-wf (K2 a))) Dwf2) (%the (cn-deq C1 C1' K1) Dequiv1) (%the (cn-deq C2 C2' (K2 C1)) Dequiv2) (cn-deq/sub (cn-deq/cn/pair (cn-deq/sym Dequiv1') Dequiv2) (kd-sub/kd/sgm Dsub1 Dsub2 Dwf2))) + %<- (vdt/cn-deq Dequiv1 (%the (ofkd C1 K1) Dof1) _ (%the (kd-wf K1) Dwf1)) + %<- (vdt/cn-deq Dequiv2 (%the (ofkd C2 (K2 C1)) Dof2) _ _) + %<- (sing/kd-complete _ _ (%the (sing/kd C1 K1 K1s) Dsing)) + %<- (stone-2/93 Dof1 Dsing (%the (ofkd C1 K1s) Dof1')) + %<- (stone-2/96 (cn-deq/sym Dequiv1) Dsing (%the (cn-deq C1' C1 K1s) Dequiv1')) + %<- (stone-2/94 Dof1 Dsing (%the (kd-sub K1s K1) Dsub1)) + %<- (stone-2/92 Dof1 Dsing (%the (kd-wf K1s) DwfK1s)) + %<- ({a} {d ofkd a K1s} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (vdt/ofkd d DwfK1s) + %-> (stone-2/97 Dof1 Dsing d (%the (cn-deq a C1 K1s) (Dequiv a d)))) + %<- ({a} {d ofkd a K1s} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (funct/kd-wf Dwf2 (cn-deq/sub (Dequiv a d) Dsub1) (ofkd/sub d Dsub1) Dof1 (%the (kd-deq (K2 a) (K2 C1)) (Dequiv' a d)))) + %<- ({a} {d ofkd a K1s} kd-anti (Dequiv' a d) (Dwf2 _ (ofkd/sub d Dsub1)) (Dwf2 _ Dof1) (Dsub2' a d) (%the (kd-sub (K2 C1) (K2 a)) (Dsub2 a d))) %. +%worlds (ofkd+vdt-block) (cn-deq-sgm-intro _ _ _ _) %. +%total {} (cn-deq-sgm-intro _ _ _ _) %. +%sort ofkd-sgm-ext-intro {a} %. +%mode ofkd-sgm-ext-intro %in %in %in %out %. +%term _ + %pi (ofkd-sgm-ext-intro (%the ({a} %pi (ofkd a K1) %-> (kd-wf (K2 a))) Dwf2) (%the (ofkd (cn/pj1 C) K1) Dof1) (%the (ofkd (cn/pj2 C) (K2 (cn/pj1 C))) Dof2) (ofkd/sub (ofkd/sgm-ext Dof1' Dof2) (kd-sub/kd/sgm Dsub1 Dsub2 Dwf2))) + %<- (vdt/ofkd Dof1 (%the (kd-wf K1) Dwf1)) + %<- (sing/kd-complete _ _ (%the (sing/kd (cn/pj1 C) K1 K1s) Dsing)) + %<- (stone-2/93 Dof1 Dsing (%the (ofkd (cn/pj1 C) K1s) Dof1')) + %<- (stone-2/94 Dof1 Dsing (%the (kd-sub K1s K1) Dsub1)) + %<- (stone-2/92 Dof1 Dsing (%the (kd-wf K1s) DwfK1s)) + %<- ({a} {d ofkd a K1s} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (vdt/ofkd d DwfK1s) + %-> (stone-2/97 Dof1 Dsing d (%the (cn-deq a (cn/pj1 C) K1s) (Dequiv a d)))) + %<- ({a} {d ofkd a K1s} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (funct/kd-wf Dwf2 (cn-deq/sub (Dequiv a d) Dsub1) (ofkd/sub d Dsub1) Dof1 (%the (kd-deq (K2 a) (K2 (cn/pj1 C))) (Dequiv' a d)))) + %<- ({a} {d ofkd a K1s} kd-anti (Dequiv' a d) (Dwf2 _ (ofkd/sub d Dsub1)) (Dwf2 _ Dof1) (Dsub2' a d) (%the (kd-sub (K2 (cn/pj1 C)) (K2 a)) (Dsub2 a d))) %. +%worlds (ofkd+vdt-block) (ofkd-sgm-ext-intro _ _ _ _) %. +%total {} (ofkd-sgm-ext-intro _ _ _ _) %. +%sort cn-deq-sgm-ext-intro {a} %. +%mode cn-deq-sgm-ext-intro %in %in %in %out %. +%term _ + %pi (cn-deq-sgm-ext-intro (%the ({a} %pi (ofkd a K1) %-> (kd-wf (K2 a))) Dwf2) (%the (cn-deq (cn/pj1 C) (cn/pj1 C') K1) Dequiv1) (%the (cn-deq (cn/pj2 C) (cn/pj2 C') (K2 (cn/pj1 C))) Dequiv2) (cn-deq/sub (cn-deq/sgm-ext (cn-deq/sym Dequiv1') Dequiv2) (kd-sub/kd/sgm Dsub1 Dsub2 Dwf2))) + %<- (vdt/cn-deq Dequiv1 (%the (ofkd (cn/pj1 C) K1) Dof1) _ (%the (kd-wf K1) Dwf1)) + %<- (vdt/cn-deq Dequiv2 (%the (ofkd (cn/pj2 C) (K2 (cn/pj1 C))) Dof2) _ _) + %<- (sing/kd-complete _ _ (%the (sing/kd (cn/pj1 C) K1 K1s) Dsing)) + %<- (stone-2/93 Dof1 Dsing (%the (ofkd (cn/pj1 C) K1s) Dof1')) + %<- (stone-2/96 (cn-deq/sym Dequiv1) Dsing (%the (cn-deq (cn/pj1 C') (cn/pj1 C) K1s) Dequiv1')) + %<- (stone-2/94 Dof1 Dsing (%the (kd-sub K1s K1) Dsub1)) + %<- (stone-2/92 Dof1 Dsing (%the (kd-wf K1s) DwfK1s)) + %<- ({a} {d ofkd a K1s} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (vdt/ofkd d DwfK1s) + %-> (stone-2/97 Dof1 Dsing d (%the (cn-deq a (cn/pj1 C) K1s) (Dequiv a d)))) + %<- ({a} {d ofkd a K1s} {dm mofkd d met/unit} + %pi (can-mofkd d dm) + %-> (funct/kd-wf Dwf2 (cn-deq/sub (Dequiv a d) Dsub1) (ofkd/sub d Dsub1) Dof1 (%the (kd-deq (K2 a) (K2 (cn/pj1 C))) (Dequiv' a d)))) + %<- ({a} {d ofkd a K1s} kd-anti (Dequiv' a d) (Dwf2 _ (ofkd/sub d Dsub1)) (Dwf2 _ Dof1) (Dsub2' a d) (%the (kd-sub (K2 (cn/pj1 C)) (K2 a)) (Dsub2 a d))) %. +%worlds (ofkd+vdt-block) (cn-deq-sgm-ext-intro _ _ _ _) %. +%total {} (cn-deq-sgm-ext-intro _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/sources.lf b/new-tests/stelf-output/tslf/sources.lf new file mode 100644 index 0000000..93844f0 --- /dev/null +++ b/new-tests/stelf-output/tslf/sources.lf @@ -0,0 +1,99 @@ +%require %[ uninhabited %] %. +%require %[ metrics %] %. +%require %[ syntax-cnkd %] %. +%require %[ static-semantics-cnkd %] %. +%require %[ blocks-predecls-cnkd %] %. +%require %[ blocks-cnkd %] %. +%require %[ worlds-adequacy-cnkd %] %. +%require %[ syntax-loc %] %. +%require %[ judgments-loc %] %. +%require %[ loc-lemmas %] %. +%require %[ explicit-context %] %. +%require %[ lookup-context %] %. +%require %[ explicit-context-lemmas %] %. +%require %[ static-semantics-e %] %. +%require %[ uninhabited-lemmas-cnkd %] %. +%require %[ seq-lemmas-cnkd %] %. +%require %[ reflexivity-kinds %] %. +%require %[ weakening-kinds-cnkd %] %. +%require %[ antisymmetry-kinds %] %. +%require %[ strengthening-cnkd %] %. +%require %[ explicit-metrics %] %. +%require %[ explicate-lemmas %] %. +%require %[ explicate %] %. +%require %[ weakening-cnkd-e %] %. +%require %[ substitution-cnkd-e %] %. +%require %[ functionality-cnkd-e %] %. +%require %[ implicate-lemmas %] %. +%require %[ implicate %] %. +%require %[ functionality-cnkd %] %. +%require %[ validity-cnkd %] %. +%require %[ symmetry-kinds %] %. +%require %[ transitivity-kinds %] %. +%require %[ functionality-full %] %. +%require %[ inversions-cons %] %. +%require %[ singleton-kinds %] %. +%require %[ singleton-lemmas %] %. +%require %[ constructor-beta %] %. +%require %[ syntax-sigs %] %. +%require %[ static-semantics-sigs %] %. +%require %[ worlds-adequacy-sigs %] %. +%require %[ uninhabited-lemmas-sigs %] %. +%require %[ seq-lemmas-sigs %] %. +%require %[ fst-sg-lemmas %] %. +%require %[ weakening-kinds-sigs %] %. +%require %[ reflexivity-sigs %] %. +%require %[ validity-sigs %] %. +%require %[ symmetry-sigs %] %. +%require %[ antisymmetry-sigs %] %. +%require %[ transitivity-sigs %] %. +%require %[ static-semantics-sigs-e %] %. +%require %[ explicit-metrics-sigs %] %. +%require %[ explicate-lemmas-sigs %] %. +%require %[ explicate-sigs %] %. +%require %[ functionality-sigs-e %] %. +%require %[ implicate-lemmas-sigs %] %. +%require %[ implicate-sigs %] %. +%require %[ functionality-sigs %] %. +%require %[ syntax-lt %] %. +%require %[ judgments-lt %] %. +%require %[ lt-lemmas %] %. +%require %[ projectibility %] %. +%require %[ projectibility-lemmas %] %. +%require %[ syntax-tmmd %] %. +%require %[ static-semantics-tmmd %] %. +%require %[ blocks-predecls-tmmd %] %. +%require %[ blocks-tmmd %] %. +%require %[ worlds-adequacy-tmmd %] %. +%require %[ uninhabited-lemmas-tmmd %] %. +%require %[ seq-lemmas-tmmd %] %. +%require %[ fst-md-lemmas %] %. +%require %[ weakening-lt %] %. +%require %[ substitution-terms %] %. +%require %[ validity-tmmd %] %. +%require %[ substitution-mods %] %. +%require %[ inversions-terms %] %. +%require %[ subderivations %] %. +%require %[ inversions-mods %] %. +%require %[ store %] %. +%require %[ store-typing %] %. +%require %[ uninhabited-lemmas-ltst %] %. +%require %[ store-typing-lemmas %] %. +%require %[ preservation-lemmas %] %. +%require %[ value %] %. +%require %[ raises %] %. +%require %[ dynamic-semantics %] %. +%require %[ notstuck %] %. +%require %[ seq-lemmas-dyn %] %. +%require %[ proj-val-lemma %] %. +%require %[ canonical-forms-mods %] %. +%require %[ progress-lemmas-mods %] %. +%require %[ preservation-raises %] %. +%require %[ preservation-fst %] %. +%require %[ correspondence-thm %] %. +%require %[ inequalities %] %. +%require %[ inversions-deq %] %. +%require %[ canonical-forms-terms %] %. +%require %[ progress-lemmas-terms %] %. +%require %[ progress %] %. +%require %[ preservation %] %. diff --git a/new-tests/stelf-output/tslf/static-semantics-cnkd.lf b/new-tests/stelf-output/tslf/static-semantics-cnkd.lf new file mode 100644 index 0000000..887c7b6 --- /dev/null +++ b/new-tests/stelf-output/tslf/static-semantics-cnkd.lf @@ -0,0 +1,142 @@ +%%%%% Static Semantics %%%%%% +%sort ofkd %. +%sort kd-wf %. +%sort kd-deq %. +%sort kd-sub %. +%sort cn-deq %. +%term kd-wf/kd/type kd-wf kd/type %. +%term kd-wf/kd/sing %pi (kd-wf (kd/sing C1)) %<- (ofkd C1 kd/type) %. +%term kd-wf/kd/sgm %pi (kd-wf (kd/sgm K1 K2)) %<- ({a cn} {da ofkd a K1} kd-wf (K2 a)) %<- (kd-wf K1) %. +%term kd-wf/kd/pi %pi (kd-wf (kd/pi K1 K2)) %<- ({a cn} {da ofkd a K1} kd-wf (K2 a)) %<- (kd-wf K1) %. +%term kd-deq/kd/type kd-deq kd/type kd/type %. +%term kd-deq/kd/sing %pi (kd-deq (kd/sing C1) (kd/sing C2)) %<- (cn-deq C1 C2 kd/type) %. +%term kd-deq/kd/sgm + %pi (kd-deq (kd/sgm K1' K1'') (kd/sgm K2' K2'')) + %<- ({a cn} {da ofkd a K1'} kd-deq (K1'' a) (K2'' a)) + %<- (kd-deq K1' K2') %. +%term kd-deq/kd/pi + %pi (kd-deq (kd/pi K1' K1'') (kd/pi K2' K2'')) + %<- ({a cn} {da ofkd a K2'} kd-deq (K1'' a) (K2'' a)) + %<- (kd-deq K2' K1') %. +%term kd-sub/kd/type kd-sub kd/type kd/type %. +%term kd-sub/kd/sing-kd/sing %pi (kd-sub (kd/sing C1) (kd/sing C2)) %<- (cn-deq C1 C2 kd/type) %. +%term kd-sub/kd/sing-kd/type %pi (kd-sub (kd/sing C1) kd/type) %<- (ofkd C1 kd/type) %. +%term kd-sub/kd/sgm + %pi (kd-sub (kd/sgm K1' K1'') (kd/sgm K2' K2'')) + %<- ({a cn} {da ofkd a K2'} kd-wf (K2'' a)) + %<- ({b cn} {db ofkd b K1'} kd-sub (K1'' b) (K2'' b)) + %<- (kd-sub K1' K2') %. +%term kd-sub/kd/pi + %pi (kd-sub (kd/pi K1' K1'') (kd/pi K2' K2'')) + %<- ({a cn} {da ofkd a K1'} kd-wf (K1'' a)) + %<- ({a cn} {da ofkd a K2'} kd-sub (K1'' a) (K2'' a)) + %<- (kd-sub K2' K1') %. +%term ofkd/tp/unit ofkd tp/unit kd/type %. +%term ofkd/tp/tagged ofkd tp/tagged kd/type %. +%term ofkd/tp/cross %pi (ofkd (tp/cross T1 T2) kd/type) %<- (ofkd T2 kd/type) %<- (ofkd T1 kd/type) %. +%term ofkd/tp/arrow %pi (ofkd (tp/arrow T1 T2) kd/type) %<- (ofkd T2 kd/type) %<- (ofkd T1 kd/type) %. +%term ofkd/tp/forall + %pi (ofkd (tp/forall K1 C1) kd/type) + %<- (kd-wf K1) + %<- ({a cn} {da ofkd a K1} ofkd (C1 a) kd/type) %. +%term ofkd/tp/ref %pi (ofkd (tp/ref C1) kd/type) %<- (ofkd C1 kd/type) %. +%term ofkd/tp/sum %pi (ofkd (tp/sum T1 T2) kd/type) %<- (ofkd T2 kd/type) %<- (ofkd T1 kd/type) %. +%term ofkd/tp/tag %pi (ofkd (tp/tag C1) kd/type) %<- (ofkd C1 kd/type) %. +%term ofkd/cn/mu + %pi (ofkd (cn/mu kd/type C) kd/type) + %<- (kd-wf kd/type) + %<- ({a cn} {da ofkd a kd/type} ofkd (C a) kd/type) %. +%term ofkd/cn/pair + %pi (ofkd (cn/pair C1 C2) (kd/sgm K1 ([a cn] K2))) + %<- (ofkd C2 K2) + %<- (ofkd C1 K1) %. +%term ofkd/cn/pj1 %pi (ofkd (cn/pj1 C1) K1) %<- (ofkd C1 (kd/sgm K1 K2)) %. +%term ofkd/cn/pj2 %pi (ofkd (cn/pj2 C1) (K2 (cn/pj1 C1))) %<- (ofkd C1 (kd/sgm K1 K2)) %. +%term ofkd/cn/lam + %pi (ofkd (cn/lam K1 C) (kd/pi K1 K2)) + %<- (kd-wf K1) + %<- ({a cn} {da ofkd a K1} ofkd (C a) (K2 a)) %. +%term ofkd/cn/app %pi (ofkd (cn/app C1 C2) (K2 C2)) %<- (ofkd C2 K1) %<- (ofkd C1 (kd/pi K1 K2)) %. +%term ofkd/kd/sing %pi (ofkd C (kd/sing C)) %<- (ofkd C kd/type) %. +%term ofkd/sgm-ext + %pi (ofkd C (kd/sgm K1 ([a cn] K2))) + %<- (ofkd (cn/pj2 C) K2) + %<- (ofkd (cn/pj1 C) K1) %. +%term ofkd/pi-ext + %pi (ofkd C (kd/pi K1 K2)) + %<- ({a cn} {da ofkd a K1} ofkd (cn/app C a) (K2 a)) + %<- (ofkd C (kd/pi K1 L)) %. +%term ofkd/sub %pi (ofkd C K) %<- (kd-sub K' K) %<- (ofkd C K') %. +%term ofkd/deq %pi (ofkd C K) %<- (kd-deq K' K) %<- (ofkd C K') %. +%term cn-deq/refl %pi (cn-deq C C K) %<- (ofkd C K) %. +%term cn-deq/sym %pi (cn-deq C1 C2 K) %<- (cn-deq C2 C1 K) %. +%term cn-deq/trans %pi (cn-deq C1 C3 K) %<- (cn-deq C2 C3 K) %<- (cn-deq C1 C2 K) %. +%term cn-deq/tp/unit cn-deq tp/unit tp/unit kd/type %. +%term cn-deq/tp/tagged cn-deq tp/tagged tp/tagged kd/type %. +%term cn-deq/tp/cross + %pi (cn-deq (tp/cross C1' C1'') (tp/cross C2' C2'') kd/type) + %<- (cn-deq C1'' C2'' kd/type) + %<- (cn-deq C1' C2' kd/type) %. +%term cn-deq/tp/arrow + %pi (cn-deq (tp/arrow C1' C1'') (tp/arrow C2' C2'') kd/type) + %<- (cn-deq C1'' C2'' kd/type) + %<- (cn-deq C1' C2' kd/type) %. +%term cn-deq/tp/forall + %pi (cn-deq (tp/forall K1 C1) (tp/forall K2 C2) kd/type) + %<- (kd-deq K1 K2) + %<- ({a cn} {da ofkd a K1} cn-deq (C1 a) (C2 a) kd/type) %. +%term cn-deq/tp/ref %pi (cn-deq (tp/ref C1) (tp/ref C2) kd/type) %<- (cn-deq C1 C2 kd/type) %. +%term cn-deq/tp/sum + %pi (cn-deq (tp/sum C1' C1'') (tp/sum C2' C2'') kd/type) + %<- (cn-deq C1'' C2'' kd/type) + %<- (cn-deq C1' C2' kd/type) %. +%term cn-deq/tp/tag %pi (cn-deq (tp/tag C1) (tp/tag C2) kd/type) %<- (cn-deq C1 C2 kd/type) %. +%term cn-deq/cn/pair + %pi (cn-deq (cn/pair C1' C1'') (cn/pair C2' C2'') (kd/sgm K' ([a] K''))) + %<- (cn-deq C1'' C2'' K'') + %<- (cn-deq C1' C2' K') %. +%term cn-deq/cn/pj1 %pi (cn-deq (cn/pj1 C1) (cn/pj1 C2) K') %<- (cn-deq C1 C2 (kd/sgm K' _)) %. +%term cn-deq/cn/pj2 + %pi (cn-deq (cn/pj2 C1) (cn/pj2 C2) (K'' (cn/pj1 C1))) + %<- (cn-deq C1 C2 (kd/sgm _ K'')) %. +%term cn-deq/cn/lam + %pi (cn-deq (cn/lam K1 C1) (cn/lam K2 C2) (kd/pi K1 K'')) + %<- (kd-deq K1 K2) + %<- ({a cn} {da ofkd a K1} cn-deq (C1 a) (C2 a) (K'' a)) %. +%term cn-deq/cn/mu + %pi (cn-deq (cn/mu kd/type C1) (cn/mu kd/type C2) kd/type) + %<- (kd-deq kd/type kd/type) + %<- ({a cn} {da ofkd a kd/type} cn-deq (C1 a) (C2 a) kd/type) %. +%term cn-deq/cn/app + %pi (cn-deq (cn/app C1 D1) (cn/app C2 D2) (K'' D1)) + %<- (cn-deq D1 D2 K') + %<- (cn-deq C1 C2 (kd/pi K' K'')) %. +%term cn-deq/kd/sing %pi (cn-deq C1 C (kd/sing C)) %<- (ofkd C1 (kd/sing C)) %. +%term cn-deq/sgm-ext + %pi (cn-deq C1 C2 (kd/sgm K' ([a] K''))) + %<- (cn-deq (cn/pj2 C1) (cn/pj2 C2) K'') + %<- (cn-deq (cn/pj1 C1) (cn/pj1 C2) K') %. +%term cn-deq/pi-ext + %pi (cn-deq C1 C2 (kd/pi K' K'')) + %<- ({a cn} {da ofkd a K'} cn-deq (cn/app C1 a) (cn/app C2 a) (K'' a)) + %<- (ofkd C2 (kd/pi K' _)) + %<- (ofkd C1 (kd/pi K' _)) %. +%term cn-deq/pi-ext-2 + %pi (cn-deq C1 C2 (kd/pi K' K'')) + %<- ({a cn} {da ofkd a K'} cn-deq (cn/app C1 a) (cn/app C2 a) (K'' a)) + %<- (cn-deq C1 C2 (kd/pi K' _)) %. +%term cn-deq/sub %pi (cn-deq C1 C2 K) %<- (kd-sub K' K) %<- (cn-deq C1 C2 K') %. +%term cn-deq/deq %pi (cn-deq C1 C2 K) %<- (kd-deq K' K) %<- (cn-deq C1 C2 K') %. +%mode ofkd %star %star %. +%mode kd-wf %star %. +%mode kd-deq %star %star %. +%mode kd-sub %star %star %. +%mode cn-deq %star %star %star %. +%define kd-wf/kd/unit (kd-wf kd/unit) kd-wf/kd/sing ofkd/tp/unit %. +%define ofkd/cn/unit (ofkd cn/unit kd/unit) ofkd/kd/sing ofkd/tp/unit %. +%define kd-deq/kd/unit (kd-deq kd/unit kd/unit) kd-deq/kd/sing cn-deq/tp/unit %. +%define kd-sub/kd/unit (kd-sub kd/unit kd/unit) kd-sub/kd/sing-kd/sing cn-deq/tp/unit %. +%define cn-deq/cn/unit (cn-deq cn/unit cn/unit kd/unit) cn-deq/kd/sing (ofkd/kd/sing ofkd/tp/unit) %. +%define cn-deq/kd/unit (%pi (cn-deq C1 C2 kd/unit) %<- (ofkd C2 kd/unit) %<- (ofkd C1 kd/unit)) [d1] [d2] cn-deq/sub (cn-deq/kd/sing (ofkd/sub (ofkd/kd/sing (ofkd/sub d1 (kd-sub/kd/sing-kd/type ofkd/tp/unit))) (kd-sub/kd/sing-kd/sing (cn-deq/trans (cn-deq/sub (cn-deq/kd/sing d1) (kd-sub/kd/sing-kd/type ofkd/tp/unit)) (cn-deq/sym (cn-deq/sub (cn-deq/kd/sing d2) (kd-sub/kd/sing-kd/type ofkd/tp/unit))))))) (kd-sub/kd/sing-kd/sing (cn-deq/sub (cn-deq/kd/sing d2) (kd-sub/kd/sing-kd/type ofkd/tp/unit))) %. +%define cn-deq/kd/singelim (%pi (cn-deq C1 C2 kd/type) %<- (ofkd C1 (kd/sing C2)) %<- (ofkd C2 kd/type)) [d2] [d1] cn-deq/sub (cn-deq/kd/sing d1) (kd-sub/kd/sing-kd/type d2) %. +%define cn-deq/kd/sing2 (%pi (cn-deq C1 C2 (kd/sing C1)) %<- (ofkd C2 kd/type) %<- (cn-deq C1 C2 kd/type)) [d] [d2] cn-deq/sym (cn-deq/kd/sing (ofkd/sub (ofkd/kd/sing d2) (kd-sub/kd/sing-kd/sing (cn-deq/sym d)))) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/static-semantics-e.lf b/new-tests/stelf-output/tslf/static-semantics-e.lf new file mode 100644 index 0000000..920ca1a --- /dev/null +++ b/new-tests/stelf-output/tslf/static-semantics-e.lf @@ -0,0 +1,164 @@ +%%%% Static Semantics with explicit contexts +%sort eofkd %. +%mode eofkd %star %star %star %. +%sort ekd-wf %. +%mode ekd-wf %star %star %. +%sort ekd-deq %. +%mode ekd-deq %star %star %star %. +%sort ekd-sub %. +%mode ekd-sub %star %star %star %. +%sort ecn-deq %. +%mode ecn-deq %star %star %star %star %. +%term ekd-wf/kd/unit %pi (ekd-wf G kd/unit) %<- (cxt-ordered G) %. +%term ekd-wf/kd/type %pi (ekd-wf G kd/type) %<- (cxt-ordered G) %. +%term ekd-wf/kd/sing %pi (ekd-wf G (kd/sing C1)) %<- (eofkd G C1 kd/type) %. +%term ekd-wf/kd/sgm + %pi (ekd-wf G (kd/sgm K1 K2)) + %<- ({a cn} %pi (isvar a I) %-> (ekd-wf (cxt/cons G a K1) (K2 a))) + %<- (ekd-wf G K1) %. +%term ekd-wf/kd/pi + %pi (ekd-wf G (kd/pi K1 K2)) + %<- ({a cn} %pi (isvar a I) %-> (ekd-wf (cxt/cons G a K1) (K2 a))) + %<- (ekd-wf G K1) %. +%term ekd-deq/kd/unit %pi (ekd-deq G kd/unit kd/unit) %<- (cxt-ordered G) %. +%term ekd-deq/kd/type %pi (ekd-deq G kd/type kd/type) %<- (cxt-ordered G) %. +%term ekd-deq/kd/sing %pi (ekd-deq G (kd/sing C1) (kd/sing C2)) %<- (ecn-deq G C1 C2 kd/type) %. +%term ekd-deq/kd/sgm + %pi (ekd-deq G (kd/sgm K1' K1'') (kd/sgm K2' K2'')) + %<- ({a cn} %pi (isvar a I) %-> (ekd-deq (cxt/cons G a K1') (K1'' a) (K2'' a))) + %<- (ekd-deq G K1' K2') %. +%term ekd-deq/kd/pi + %pi (ekd-deq G (kd/pi K1' K1'') (kd/pi K2' K2'')) + %<- ({a cn} %pi (isvar a I) %-> (ekd-deq (cxt/cons G a K2') (K1'' a) (K2'' a))) + %<- (ekd-deq G K2' K1') %. +%term ekd-sub/kd/unit %pi (ekd-sub G kd/unit kd/unit) %<- (cxt-ordered G) %. +%term ekd-sub/kd/type %pi (ekd-sub G kd/type kd/type) %<- (cxt-ordered G) %. +%term ekd-sub/kd/sing-kd/sing %pi (ekd-sub G (kd/sing C1) (kd/sing C2)) %<- (ecn-deq G C1 C2 kd/type) %. +%term ekd-sub/kd/sing-kd/type %pi (ekd-sub G (kd/sing C1) kd/type) %<- (eofkd G C1 kd/type) %. +%term ekd-sub/kd/sgm + %pi (ekd-sub G (kd/sgm K1' K1'') (kd/sgm K2' K2'')) + %<- ({a cn} %pi (isvar a I) %-> (ekd-wf (cxt/cons G a K2') (K2'' a))) + %<- ({b cn} %pi (isvar b I) %-> (ekd-sub (cxt/cons G b K1') (K1'' b) (K2'' b))) + %<- (ekd-sub G K1' K2') %. +%term ekd-sub/kd/pi + %pi (ekd-sub G (kd/pi K1' K1'') (kd/pi K2' K2'')) + %<- ({a cn} %pi (isvar a I) %-> (ekd-wf (cxt/cons G a K1') (K1'' a))) + %<- ({a cn} %pi (isvar a I) %-> (ekd-sub (cxt/cons G a K2') (K1'' a) (K2'' a))) + %<- (ekd-sub G K2' K1') %. +%term eofkd/cn/unit %pi (eofkd G cn/unit kd/unit) %<- (cxt-ordered G) %. +%term eofkd/tp/unit %pi (eofkd G tp/unit kd/type) %<- (cxt-ordered G) %. +%term eofkd/tp/tagged %pi (eofkd G tp/tagged kd/type) %<- (cxt-ordered G) %. +%term eofkd/tp/cross + %pi (eofkd G (tp/cross T1 T2) kd/type) + %<- (eofkd G T2 kd/type) + %<- (eofkd G T1 kd/type) %. +%term eofkd/tp/arrow + %pi (eofkd G (tp/arrow T1 T2) kd/type) + %<- (eofkd G T2 kd/type) + %<- (eofkd G T1 kd/type) %. +%term eofkd/tp/sum + %pi (eofkd G (tp/sum T1 T2) kd/type) + %<- (eofkd G T2 kd/type) + %<- (eofkd G T1 kd/type) %. +%term eofkd/tp/forall + %pi (eofkd G (tp/forall K1 C1) kd/type) + %<- (ekd-wf G K1) + %<- ({a cn} %pi (isvar a I) %-> (eofkd (cxt/cons G a K1) (C1 a) kd/type)) %. +%term eofkd/tp/tag %pi (eofkd G (tp/tag T1) kd/type) %<- (eofkd G T1 kd/type) %. +%term eofkd/cn/mu + %pi (eofkd G (cn/mu kd/type C) kd/type) + %<- (ekd-wf G kd/type) + %<- ({a cn} %pi (isvar a I) %-> (eofkd (cxt/cons G a kd/type) (C a) kd/type)) %. +%term eofkd/cn/pair + %pi (eofkd G (cn/pair C1 C2) (kd/sgm K1 ([a cn] K2))) + %<- (eofkd G C2 K2) + %<- (eofkd G C1 K1) %. +%term eofkd/cn/pj1 %pi (eofkd G (cn/pj1 C1) K1) %<- (eofkd G C1 (kd/sgm K1 K2)) %. +%term eofkd/cn/pj2 %pi (eofkd G (cn/pj2 C1) (K2 (cn/pj1 C1))) %<- (eofkd G C1 (kd/sgm K1 K2)) %. +%term eofkd/cn/lam + %pi (eofkd G (cn/lam K1 C) (kd/pi K1 K2)) + %<- (ekd-wf G K1) + %<- ({a cn} %pi (isvar a I) %-> (eofkd (cxt/cons G a K1) (C a) (K2 a))) %. +%term eofkd/cn/app + %pi (eofkd G (cn/app C1 C2) (K2 C2)) + %<- (eofkd G C2 K1) + %<- (eofkd G C1 (kd/pi K1 K2)) %. +%term eofkd/kd/sing %pi (eofkd G C (kd/sing C)) %<- (eofkd G C kd/type) %. +%term eofkd/sgm-ext + %pi (eofkd G C (kd/sgm K1 ([a cn] K2))) + %<- (eofkd G (cn/pj2 C) K2) + %<- (eofkd G (cn/pj1 C) K1) %. +%term eofkd/pi-ext + %pi (eofkd G C (kd/pi K1 K2)) + %<- ({a cn} %pi (isvar a I) %-> (eofkd (cxt/cons G a K1) (cn/app C a) (K2 a))) + %<- (eofkd G C (kd/pi K1 L)) %. +%term eofkd/sub %pi (eofkd G C K) %<- (ekd-sub G K' K) %<- (eofkd G C K') %. +%term eofkd/deq %pi (eofkd G C K) %<- (ekd-deq G K' K) %<- (eofkd G C K') %. +%term eofkd/tp/ref %pi (eofkd G (tp/ref C) kd/type) %<- (eofkd G C kd/type) %. +%term eofkd/var %pi (eofkd G A K) %<- (cxt-lookup G A K) %. +%term eofkd/closed %pi (eofkd G A K) %<- (cxt-ordered G) %<- (ofkd A K) %. +%term ecn-deq/refl %pi (ecn-deq G C C K) %<- (eofkd G C K) %. +%term ecn-deq/sym %pi (ecn-deq G C1 C2 K) %<- (ecn-deq G C2 C1 K) %. +%term ecn-deq/trans %pi (ecn-deq G C1 C3 K) %<- (ecn-deq G C2 C3 K) %<- (ecn-deq G C1 C2 K) %. +%term ecn-deq/cn/unit %pi (ecn-deq G cn/unit cn/unit kd/unit) %<- (cxt-ordered G) %. +%term ecn-deq/tp/unit %pi (ecn-deq G tp/unit tp/unit kd/type) %<- (cxt-ordered G) %. +%term ecn-deq/tp/tagged %pi (ecn-deq G tp/tagged tp/tagged kd/type) %<- (cxt-ordered G) %. +%term ecn-deq/tp/cross + %pi (ecn-deq G (tp/cross C1' C1'') (tp/cross C2' C2'') kd/type) + %<- (ecn-deq G C1'' C2'' kd/type) + %<- (ecn-deq G C1' C2' kd/type) %. +%term ecn-deq/tp/arrow + %pi (ecn-deq G (tp/arrow C1' C1'') (tp/arrow C2' C2'') kd/type) + %<- (ecn-deq G C1'' C2'' kd/type) + %<- (ecn-deq G C1' C2' kd/type) %. +%term ecn-deq/tp/sum + %pi (ecn-deq G (tp/sum C1' C1'') (tp/sum C2' C2'') kd/type) + %<- (ecn-deq G C1'' C2'' kd/type) + %<- (ecn-deq G C1' C2' kd/type) %. +%term ecn-deq/tp/forall + %pi (ecn-deq G (tp/forall K1 C1) (tp/forall K2 C2) kd/type) + %<- (ekd-deq G K1 K2) + %<- ({a cn} %pi (isvar a I) %-> (ecn-deq (cxt/cons G a K1) (C1 a) (C2 a) kd/type)) %. +%term ecn-deq/cn/pair + %pi (ecn-deq G (cn/pair C1' C1'') (cn/pair C2' C2'') (kd/sgm K' ([a] K''))) + %<- (ecn-deq G C1'' C2'' K'') + %<- (ecn-deq G C1' C2' K') %. +%term ecn-deq/cn/pj1 %pi (ecn-deq G (cn/pj1 C1) (cn/pj1 C2) K') %<- (ecn-deq G C1 C2 (kd/sgm K' _)) %. +%term ecn-deq/cn/pj2 + %pi (ecn-deq G (cn/pj2 C1) (cn/pj2 C2) (K'' (cn/pj1 C1))) + %<- (ecn-deq G C1 C2 (kd/sgm _ K'')) %. +%term ecn-deq/cn/lam + %pi (ecn-deq G (cn/lam K1 C1) (cn/lam K2 C2) (kd/pi K1 K'')) + %<- (ekd-deq G K1 K2) + %<- ({a cn} %pi (isvar a I) %-> (ecn-deq (cxt/cons G a K1) (C1 a) (C2 a) (K'' a))) %. +%term ecn-deq/cn/mu + %pi (ecn-deq G (cn/mu kd/type C1) (cn/mu kd/type C2) kd/type) + %<- (ekd-deq G kd/type kd/type) + %<- ({a cn} %pi (isvar a I) %-> (ecn-deq (cxt/cons G a kd/type) (C1 a) (C2 a) kd/type)) %. +%term ecn-deq/cn/app + %pi (ecn-deq G (cn/app C1 D1) (cn/app C2 D2) (K'' D1)) + %<- (ecn-deq G D1 D2 K') + %<- (ecn-deq G C1 C2 (kd/pi K' K'')) %. +%term ecn-deq/kd/unit %pi (ecn-deq G C1 C2 kd/unit) %<- (eofkd G C2 kd/unit) %<- (eofkd G C1 kd/unit) %. +%term ecn-deq/kd/sing %pi (ecn-deq G C1 C (kd/sing C)) %<- (eofkd G C1 (kd/sing C)) %. +%term ecn-deq/sgm-ext + %pi (ecn-deq G C1 C2 (kd/sgm K' ([a] K''))) + %<- (ecn-deq G (cn/pj2 C1) (cn/pj2 C2) K'') + %<- (ecn-deq G (cn/pj1 C1) (cn/pj1 C2) K') %. +%term ecn-deq/pi-ext + %pi (ecn-deq G C1 C2 (kd/pi K' K'')) + %<- ({a cn} + %pi (isvar a I) + %-> (ecn-deq (cxt/cons G a K') (cn/app C1 a) (cn/app C2 a) (K'' a))) + %<- (eofkd G C2 (kd/pi K' _)) + %<- (eofkd G C1 (kd/pi K' _)) %. +%term ecn-deq/pi-ext-2 + %pi (ecn-deq G C1 C2 (kd/pi K' K'')) + %<- ({a cn} + %pi (isvar a I) + %-> (ecn-deq (cxt/cons G a K') (cn/app C1 a) (cn/app C2 a) (K'' a))) + %<- (ecn-deq G C1 C2 (kd/pi K' _)) %. +%term ecn-deq/sub %pi (ecn-deq G C1 C2 K) %<- (ekd-sub G K' K) %<- (ecn-deq G C1 C2 K') %. +%term ecn-deq/deq %pi (ecn-deq G C1 C2 K) %<- (ekd-deq G K' K) %<- (ecn-deq G C1 C2 K') %. +%term ecn-deq/tp/ref %pi (ecn-deq G (tp/ref C1) (tp/ref C2) kd/type) %<- (ecn-deq G C1 C2 kd/type) %. +%term ecn-deq/tp/tag %pi (ecn-deq G (tp/tag C1) (tp/tag C2) kd/type) %<- (ecn-deq G C1 C2 kd/type) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/static-semantics-sigs-e.lf b/new-tests/stelf-output/tslf/static-semantics-sigs-e.lf new file mode 100644 index 0000000..e7a6608 --- /dev/null +++ b/new-tests/stelf-output/tslf/static-semantics-sigs-e.lf @@ -0,0 +1,59 @@ +%%%%% Static Semantics for sigs %%%%%% +%sort esg-wf %. +%mode esg-wf %star %star %. +%sort esg-deq %. +%mode esg-deq %star %star %star %. +%sort esg-sub %. +%mode esg-sub %star %star %star %. +%term esg-wf/sg/unit %pi (esg-wf G sg/unit) %<- (cxt-ordered G) %. +%term esg-wf/sg/kd %pi (esg-wf G (sg/kd K)) %<- (ekd-wf G K) %. +%term esg-wf/sg/cn %pi (esg-wf G (sg/cn C)) %<- (eofkd G C kd/type) %. +%term esg-wf/sg/sgm + %pi (esg-wf G (sg/sgm S1 S2)) + %<- (fst-sg S1 K1) + %<- ({a cn} %pi (isvar a I) %-> (esg-wf (cxt/cons G a K1) (S2 a))) + %<- (esg-wf G S1) %. +%term esg-wf/sg/pi + %pi (esg-wf G (sg/pi S1 S2)) + %<- (fst-sg S1 K1) + %<- ({a cn} %pi (isvar a I) %-> (esg-wf (cxt/cons G a K1) (S2 a))) + %<- (esg-wf G S1) %. +%term esg-deq/sg/unit %pi (esg-deq G sg/unit sg/unit) %<- (cxt-ordered G) %. +%term esg-deq/sg/kd %pi (esg-deq G (sg/kd K1) (sg/kd K2)) %<- (ekd-deq G K1 K2) %. +%term esg-deq/sg/cn %pi (esg-deq G (sg/cn C1) (sg/cn C2)) %<- (ecn-deq G C1 C2 kd/type) %. +%term esg-deq/sg/sgm + %pi (esg-deq G (sg/sgm S1 S2) (sg/sgm S3 S4)) + %<- (fst-sg S1 K1) + %<- ({a cn} %pi (isvar a I) %-> (esg-deq (cxt/cons G a K1) (S2 a) (S4 a))) + %<- (esg-deq G S1 S3) %. +%term esg-deq/sg/pi + %pi (esg-deq G (sg/pi S1 S2) (sg/pi S3 S4)) + %<- (fst-sg S3 K3) + %<- ({a cn} %pi (isvar a I) %-> (esg-deq (cxt/cons G a K3) (S2 a) (S4 a))) + %<- (esg-deq G S3 S1) %. +%{ + +esg-sub/sg/unit : esg-sub G sg/unit sg/unit + <- cxt-ordered G. + +esg-sub/sg/kd : esg-sub G (sg/kd K1) (sg/kd K2) + <- ekd-sub G K1 K2. + +esg-sub/sg/cn : esg-sub G (sg/cn C1) (sg/cn C2) + <- ecn-deq G C1 C2 kd/type. + +esg-sub/sg/sgm : esg-sub G (sg/sgm S1 S2) (sg/sgm S3 S4) + <- fst-sg S3 K3 + <- fst-sg S1 K1 + <- ({a:cn} isvar a I -> + esg-wf (cxt/cons G a K3) (S4 a)) + <- ({a:cn} isvar a I -> + esg-sub (cxt/cons G a K1) (S2 a) (S4 a)) + <- esg-sub G S1 S3. + +esg-sub/sg/pi : esg-sub G (sg/pi S1 S2) (sg/pi S3 S4) + <- fst-sg S3 K3 + <- ({a:cn} isvar a I -> + esg-deq (cxt/cons G a K3) (S2 a) (S4 a)) + <- esg-deq G S3 S1. +}% \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/static-semantics-sigs.lf b/new-tests/stelf-output/tslf/static-semantics-sigs.lf new file mode 100644 index 0000000..519d681 --- /dev/null +++ b/new-tests/stelf-output/tslf/static-semantics-sigs.lf @@ -0,0 +1,58 @@ +%%%%% Static Semantics for sigs %%%%%% +%sort sg-wf %. +%mode sg-wf %star %. +%sort fst-sg %. +%mode fst-sg %star %star %. +%sort sg-deq %. +%mode sg-deq %star %star %. +%sort sg-sub %. +%mode sg-sub %star %star %. +%term fst-sg/unit fst-sg sg/unit kd/unit %. +%term fst-sg/kd fst-sg (sg/kd K) K %. +%term fst-sg/cn fst-sg (sg/cn _) kd/unit %. +%term fst-sg/sgm + %pi (fst-sg (sg/sgm S1 S2) (kd/sgm K1 K2)) + %<- ({a cn} fst-sg (S2 a) (K2 a)) + %<- (fst-sg S1 K1) %. +%term fst-sg/pi fst-sg (sg/pi _ _) kd/unit %. +%term sg-wf/sg/unit sg-wf sg/unit %. +%term sg-wf/sg/kd %pi (sg-wf (sg/kd K)) %<- (kd-wf K) %. +%term sg-wf/sg/cn %pi (sg-wf (sg/cn C)) %<- (ofkd C kd/type) %. +%term sg-wf/sg/sgm + %pi (sg-wf (sg/sgm S1 S2)) + %<- (fst-sg S1 K1) + %<- ({a cn} {da ofkd a K1} sg-wf (S2 a)) + %<- (sg-wf S1) %. +%term sg-wf/sg/pi + %pi (sg-wf (sg/pi S1 S2)) + %<- (fst-sg S1 K1) + %<- ({a cn} {da ofkd a K1} sg-wf (S2 a)) + %<- (sg-wf S1) %. +%term sg-deq/sg/unit sg-deq sg/unit sg/unit %. +%term sg-deq/sg/kd %pi (sg-deq (sg/kd K1) (sg/kd K2)) %<- (kd-deq K1 K2) %. +%term sg-deq/sg/cn %pi (sg-deq (sg/cn C1) (sg/cn C2)) %<- (cn-deq C1 C2 kd/type) %. +%term sg-deq/sg/sgm + %pi (sg-deq (sg/sgm S1 S2) (sg/sgm S3 S4)) + %<- (fst-sg S1 K1) + %<- ({a cn} {da ofkd a K1} sg-deq (S2 a) (S4 a)) + %<- (sg-deq S1 S3) %. +%term sg-deq/sg/pi + %pi (sg-deq (sg/pi S1 S2) (sg/pi S3 S4)) + %<- (fst-sg S3 K3) + %<- ({a cn} {da ofkd a K3} sg-deq (S2 a) (S4 a)) + %<- (sg-deq S3 S1) %. +%term sg-sub/sg/unit sg-sub sg/unit sg/unit %. +%term sg-sub/sg/kd %pi (sg-sub (sg/kd K1) (sg/kd K2)) %<- (kd-sub K1 K2) %. +%term sg-sub/sg/cn %pi (sg-sub (sg/cn C1) (sg/cn C2)) %<- (cn-deq C1 C2 kd/type) %. +%term sg-sub/sg/sgm + %pi (sg-sub (sg/sgm S1 S2) (sg/sgm S3 S4)) + %<- (fst-sg S3 K3) + %<- (fst-sg S1 K1) + %<- ({a cn} {da ofkd a K3} sg-wf (S4 a)) + %<- ({a cn} {da ofkd a K1} sg-sub (S2 a) (S4 a)) + %<- (sg-sub S1 S3) %. +%term sg-sub/sg/pi + %pi (sg-sub (sg/pi S1 S2) (sg/pi S3 S4)) + %<- (fst-sg S3 K3) + %<- ({a cn} {da ofkd a K3} sg-deq (S2 a) (S4 a)) + %<- (sg-deq S3 S1) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/static-semantics-tmmd.lf b/new-tests/stelf-output/tslf/static-semantics-tmmd.lf new file mode 100644 index 0000000..6725d6f --- /dev/null +++ b/new-tests/stelf-output/tslf/static-semantics-tmmd.lf @@ -0,0 +1,130 @@ +%%%%% Static Semantics for mods + terms %%%%%% +%inline tt lt %. +% abstract tag typings. +%inline ht lt %. +% abstract heap typings. +%sort assm/tm %. +%mode assm/tm %star %star %. +%sort oftp %. +%mode oftp %star %star %star %star %. +%sort fst-md %. +%mode fst-md %star %star %. +%sort assm/md %. +%mode assm/md %star %star %. +%sort ofsg %. +%mode ofsg %star %star %star %star %star %. +%term oftp/var %pi (oftp L T E C) %<- (assm/tm E C) %. +%term oftp/tm/unit oftp L T tm/unit tp/unit %. +%term oftp/tm/pair + %pi (oftp L T (tm/pair E1 E2) (tp/cross C1 C2)) + %<- (oftp L T E2 C2) + %<- (oftp L T E1 C1) %. +%term oftp/tm/pj1 %pi (oftp L T (tm/pj1 E) C1) %<- (oftp L T E (tp/cross C1 C2)) %. +%term oftp/tm/pj2 %pi (oftp L T (tm/pj2 E) C2) %<- (oftp L T E (tp/cross C1 C2)) %. +%term oftp/tm/fun + %pi (oftp L T (tm/fun C1 C2 E) (tp/arrow C1 C2)) + %<- (ofkd (tp/arrow C1 C2) kd/type) + %<- ({x tm} {dx assm/tm x C1} {f tm} {df assm/tm f (tp/arrow C1 C2)} oftp L T (E x f) C2) %. +%term oftp/tm/tmapp + %pi (oftp L T (tm/tmapp E1 E2) C2) + %<- (oftp L T E2 C1) + %<- (oftp L T E1 (tp/arrow C1 C2)) %. +%term oftp/tm/cnabs + %pi (oftp L T (tm/cnabs K E) (tp/forall K C)) + %<- (kd-wf K) + %<- ({a cn} {da ofkd a K} oftp L T (E a) (C a)) %. +%term oftp/tm/cnapp + %pi (oftp L T (tm/cnapp E C) (D C)) + %<- (ofkd C K) + %<- (oftp L T E (tp/forall K D)) %. +%term oftp/tm/term %pi (oftp L T (tm/term M) C) %<- (ofsg L T _ M (sg/cn C)) %. +%term oftp/deq %pi (oftp L T E C) %<- (cn-deq C' C kd/type) %<- (oftp L T E C') %. +%term oftp/tm/loc %pi (oftp L T (tm/loc LC) (tp/ref C)) %<- (ofkd C kd/type) %<- (lt-look L LC C) %. +%term oftp/tm/ref %pi (oftp L T (tm/ref E) (tp/ref C)) %<- (oftp L T E C) %. +%term oftp/tm/set + %pi (oftp L T (tm/set E1 E2) tp/unit) + %<- (oftp L T E2 C) + %<- (oftp L T E1 (tp/ref C)) %. +%term oftp/tm/get %pi (oftp L T (tm/get E) C) %<- (oftp L T E (tp/ref C)) %. +%term oftp/tm/inl + %pi (oftp L T (tm/inl (tp/sum C1 C2) E) (tp/sum C1 C2)) + %<- (ofkd C2 kd/type) + %<- (oftp L T E C1) %. +%term oftp/tm/inr + %pi (oftp L T (tm/inr (tp/sum C1 C2) E) (tp/sum C1 C2)) + %<- (ofkd C1 kd/type) + %<- (oftp L T E C2) %. +%term oftp/tm/case + %pi (oftp L T (tm/case E E1 E2) C) + %<- ({x tm} {dx assm/tm x C2} oftp L T (E2 x) C) + %<- ({x tm} {dx assm/tm x C1} oftp L T (E1 x) C) + %<- (oftp L T E (tp/sum C1 C2)) %. +%term oftp/tm/raise %pi (oftp L T (tm/raise E) C) %<- (ofkd C kd/type) %<- (oftp L T E tp/tagged) %. +%term oftp/tm/try + %pi (oftp L T (tm/try E1 E2) C) + %<- ({x tm} {dx assm/tm x tp/tagged} oftp L T (E2 x) C) + %<- (oftp L T E1 C) %. +%term oftp/tm/new-tag %pi (oftp L T (tm/new-tag C) (tp/tag C)) %<- (ofkd C kd/type) %. +%term oftp/tm/tagloc %pi (oftp L T (tm/tagloc LC) (tp/tag C)) %<- (ofkd C kd/type) %<- (lt-look T LC C) %. +%term oftp/tm/tag + %pi (oftp L T (tm/tag E1 E2) tp/tagged) + %<- (oftp L T E2 C) + %<- (oftp L T E1 (tp/tag C)) %. +%term oftp/tm/iftag + %pi (oftp L T (tm/iftag E1 E2 E3 E4) C) + %<- (oftp L T E4 C) + %<- ({x} {dx assm/tm x C'} oftp L T (E3 x) C) + %<- (oftp L T E2 (tp/tag C')) + %<- (oftp L T E1 tp/tagged) %. +%term oftp/tm/roll + %pi (oftp L T (tm/roll (cn/mu K C') E) (cn/mu K C')) + %<- (ofkd (cn/mu K C') kd/type) + %<- (oftp L T E (C' (cn/mu K C'))) %. +%term oftp/tm/unroll %pi (oftp L T (tm/unroll E) (C' (cn/mu K C'))) %<- (oftp L T E (cn/mu K C')) %. +%term fst-md/unit fst-md md/unit cn/unit %. +%term fst-md/cn fst-md (md/cn C) C %. +%term fst-md/tm fst-md (md/tm _) cn/unit %. +%term fst-md/pair %pi (fst-md (md/pair M1 M2) (cn/pair C1 C2)) %<- (fst-md M2 C2) %<- (fst-md M1 C1) %. +%term fst-md/pj1 %pi (fst-md (md/pj1 M) (cn/pj1 C)) %<- (fst-md M C) %. +%term fst-md/pj2 %pi (fst-md (md/pj2 M) (cn/pj2 C)) %<- (fst-md M C) %. +%term fst-md/lam fst-md (md/lam _ _ _) cn/unit %. +%term ofsg/var %pi (ofsg L T pty/p M S) %<- (assm/md M S) %. +%term ofsg/md/unit ofsg L T pty/p md/unit sg/unit %. +%term ofsg/md/cn %pi (ofsg L T pty/p (md/cn C) (sg/kd K)) %<- (ofkd C K) %. +%term ofsg/md/tm %pi (ofsg L T pty/p (md/tm E) (sg/cn C)) %<- (oftp L T E C) %. +%term ofsg/md/pair + %pi (ofsg L T Y (md/pair M1 M2) (sg/sgm S1 ([c cn] S2))) + %<- (ofsg L T Y M2 S2) + %<- (ofsg L T Y M1 S1) %. +%term ofsg/md/pj1 %pi (ofsg L T Y (md/pj1 M) S1) %<- (ofsg L T Y M (sg/sgm S1 _)) %. +%term ofsg/md/pj2 + %pi (ofsg L T pty/p (md/pj2 M) (S2 (cn/pj1 C))) + %<- (fst-md M C) + %<- (ofsg L T pty/p M (sg/sgm S1 S2)) %. +%term ofsg/md/lam + %pi (ofsg L T pty/p (md/lam S1 S2 M') (sg/pi S1 S2)) + %<- (fst-sg S1 K1) + %<- (sg-wf S1) + %<- ({s md} {dm assm/md s S1} {a cn} {da ofkd a K1} {dm fst-md s a} ofsg L T Y (M' s a) (S2 a)) %. +%term ofsg/md/app + %pi (ofsg L T pty/i (md/app M1 M2) (S2 C)) + %<- (fst-md M2 C) + %<- (ofsg L T pty/p M2 S1) + %<- (ofsg L T Y M1 (sg/pi S1 S2)) %. +%term ofsg/md/let + %pi (ofsg L T pty/i (md/let M1 M2 S) S) + %<- (fst-sg S1 K1) + %<- (sg-wf S) + %<- ({s md} {dm assm/md s S1} {a cn} {da ofkd a K1} {dm fst-md s a} ofsg L T Y (M2 s a) S) + %<- (ofsg L T Y' M1 S1) %. +%term ofsg/md/seal %pi (ofsg L T pty/i (md/seal M S) S) %<- (ofsg L T _ M S) %. +%term ofsg/sgm-ext + %pi (ofsg L T pty/p M (sg/sgm S1 ([a cn] S2))) + %<- (ofsg L T pty/p (md/pj2 M) S2) + %<- (ofsg L T pty/p (md/pj1 M) S1) %. +%term ofsg/kd-ext + %pi (ofsg L T pty/p M (sg/kd K)) + %<- (ofkd C K) + %<- (fst-md M C) + %<- (ofsg L T pty/p M (sg/kd K')) %. +%term ofsg/sub %pi (ofsg L T Y M S) %<- (pty-sub Y' Y) %<- (sg-sub S' S) %<- (ofsg L T Y' M S') %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/stelf.toml b/new-tests/stelf-output/tslf/stelf.toml new file mode 100644 index 0000000..ba916e6 --- /dev/null +++ b/new-tests/stelf-output/tslf/stelf.toml @@ -0,0 +1,5 @@ +[[group]] +name = "sources" +main = "sources.lf" +src = ["."] +deps = [{type = "local", path = "sing/stelf.toml", name = "sources"}] diff --git a/new-tests/stelf-output/tslf/store-typing-lemmas.lf b/new-tests/stelf-output/tslf/store-typing-lemmas.lf new file mode 100644 index 0000000..9b6d62a --- /dev/null +++ b/new-tests/stelf-output/tslf/store-typing-lemmas.lf @@ -0,0 +1,33 @@ +%sort lt-look-wf %. +%mode lt-look-wf %in %in %in %out %. +%term _ + %pi (lt-look-wf (lt-look/miss LL) (st-wf/cons LCS _ _ _) LG DU) + %<- (loc-less-pred LG LG') + %<- (lt-look-wf LL LCS LG' DU) %. +%term _ + %pi (lt-look-wf lt-look/hit (st-wf/cons _ _ _ _) DG DU) + %<- (loc-less-not-refl DG DU) %. +%worlds () (lt-look-wf _ _ _ _) %. +%total D1 (lt-look-wf _ D1 _ _) %. +%sort st-look-wf %. +%mode st-look-wf %in %in %in %out %. +%term _ + %pi (st-look-wf (st-look/miss LL) (st-wf/cons LCS _ _ _) LG DU) + %<- (loc-less-pred LG LG') + %<- (st-look-wf LL LCS LG' DU) %. +%term _ + %pi (st-look-wf st-look/hit (st-wf/cons _ _ _ _) DG DU) + %<- (loc-less-not-refl DG DU) %. +%worlds () (st-look-wf _ _ _ _) %. +%total D1 (st-look-wf _ D1 _ _) %. +%sort st-update-wf %. +%mode st-update-wf %in %in %in %out %. +%term _ + %pi (st-update-wf (st-update/miss LL) (st-wf/cons LCS _ _ _) LG DU) + %<- (loc-less-pred LG LG') + %<- (st-update-wf LL LCS LG' DU) %. +%term _ + %pi (st-update-wf st-update/hit (st-wf/cons _ _ _ _) DG DU) + %<- (loc-less-not-refl DG DU) %. +%worlds () (st-update-wf _ _ _ _) %. +%total D1 (st-update-wf _ D1 _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/store-typing.lf b/new-tests/stelf-output/tslf/store-typing.lf new file mode 100644 index 0000000..db90e9b --- /dev/null +++ b/new-tests/stelf-output/tslf/store-typing.lf @@ -0,0 +1,14 @@ +%%%% judgments for well-formed stores +% the location typing covers the store in a well-typed manner +%sort st-wf %. +%mode st-wf %star %star %star %star %. +%term st-wf/nil st-wf lt/nil st/nil _ _ %. +%term st-wf/cons + %pi (st-wf (lt/cons L LC C) (st/cons S LC E) L' T) + %<- (st-alloc S E (st/cons S LC E) LC) + %<- (lt-extend L C (lt/cons L LC C) LC) + %<- (oftp L' T E C) + %<- (st-wf L S L' T) %. +%sort oflt %. +%mode oflt %star %star %in %. +%term oflt/c %pi (oflt ST LT T) %<- (lt-wf T) %<- (st-wf LT ST LT T) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/store.lf b/new-tests/stelf-output/tslf/store.lf new file mode 100644 index 0000000..3f6a5e0 --- /dev/null +++ b/new-tests/stelf-output/tslf/store.lf @@ -0,0 +1,19 @@ +%%%% stores for terms +%sort st %. +%name st %. +%term st/nil st %. +%term st/cons %pi st %-> loc %-> tm %-> st %. +%sort st-look %. +%mode st-look %star %star %star %. +%term st-look/hit st-look (st/cons _ LC V) LC V %. +%term st-look/miss %pi (st-look (st/cons S _ _) LC V) %<- (st-look S LC V) %. +%sort st-alloc %. +%mode st-alloc %star %star %star %star %. +%term st-alloc/nil st-alloc st/nil E (st/cons st/nil loc/z E) loc/z %. +%term st-alloc/cons st-alloc (st/cons ST LC E') E (st/cons (st/cons ST LC E') (loc/s LC) E) (loc/s LC) %. +%sort st-update %. +%mode st-update %star %star %star %star %. +%term st-update/hit st-update (st/cons S LC _) LC E (st/cons S LC E) %. +%term st-update/miss + %pi (st-update (st/cons S LC' E') LC E (st/cons S' LC' E')) + %<- (st-update S LC E S') %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/strengthening-cnkd.lf b/new-tests/stelf-output/tslf/strengthening-cnkd.lf new file mode 100644 index 0000000..36154cb --- /dev/null +++ b/new-tests/stelf-output/tslf/strengthening-cnkd.lf @@ -0,0 +1,479 @@ +%%%% strengthening +%sort closed-cn %. +%term closed-cn/i closed-cn ([x] C) C %. +%sort closed-kd %. +%term closed-kd/i closed-kd ([x] K) K %. +%sort closed-cn-seq/cn %. +%mode closed-cn-seq/cn %in %in %out %. +%term _ closed-cn-seq/cn closed-cn/i closed-cn/i seq/cn/refl %. +%worlds (cn-block) (closed-cn-seq/cn _ _ _) %. +%total {} (closed-cn-seq/cn _ _ _) %. +%sort closed-kd-seq/kd %. +%mode closed-kd-seq/kd %in %in %out %. +%term _ closed-kd-seq/kd closed-kd/i closed-kd/i seq/kd/refl %. +%worlds (cn-block) (closed-kd-seq/kd _ _ _) %. +%total {} (closed-kd-seq/kd _ _ _) %. +%sort closed-cn-cn/pj1 %. +%mode closed-cn-cn/pj1 %in %out %. +%term _ closed-cn-cn/pj1 closed-cn/i closed-cn/i %. +%worlds (cn-block ofkd-block) (closed-cn-cn/pj1 _ _) %. +%total {} (closed-cn-cn/pj1 _ _) %. +%sort closed-cn-cn/pj1-e %. +%mode closed-cn-cn/pj1-e %in %out %. +%term _ closed-cn-cn/pj1-e closed-cn/i closed-cn/i %. +%worlds (cn-block ofkd-block) (closed-cn-cn/pj1-e _ _) %. +%total {} (closed-cn-cn/pj1-e _ _) %. +%sort closed-cn-cn/pj2-e %. +%mode closed-cn-cn/pj2-e %in %out %out %. +%term _ closed-cn-cn/pj2-e closed-cn/i closed-cn/i seq/cn/refl %. +%worlds (cn-block ofkd-block) (closed-cn-cn/pj2-e _ _ _) %. +%total {} (closed-cn-cn/pj2-e _ _ _) %. +%sort closed-cn-cn/pj2 %. +%mode closed-cn-cn/pj2 %in %out %. +%term _ closed-cn-cn/pj2 closed-cn/i closed-cn/i %. +%worlds (cn-block ofkd-block) (closed-cn-cn/pj2 _ _) %. +%total {} (closed-cn-cn/pj2 _ _) %. +%sort closed-cn-tp/ref %. +%mode closed-cn-tp/ref %in %out %. +%term _ closed-cn-tp/ref closed-cn/i closed-cn/i %. +%worlds (cn-block ofkd-block) (closed-cn-tp/ref _ _) %. +%total {} (closed-cn-tp/ref _ _) %. +%sort closed-cn-tp/tag %. +%mode closed-cn-tp/tag %in %out %. +%term _ closed-cn-tp/tag closed-cn/i closed-cn/i %. +%worlds (cn-block ofkd-block) (closed-cn-tp/tag _ _) %. +%total {} (closed-cn-tp/tag _ _) %. +%sort closed-cn-cn/pair %. +%mode closed-cn-cn/pair %in %in %out %. +%term _ closed-cn-cn/pair closed-cn/i closed-cn/i closed-cn/i %. +%worlds (cn-block ofkd-block) (closed-cn-cn/pair _ _ _) %. +%total {} (closed-cn-cn/pair _ _ _) %. +%sort closed-cn-cn/app %. +%mode closed-cn-cn/app %in %in %out %. +%term _ closed-cn-cn/app closed-cn/i closed-cn/i closed-cn/i %. +%worlds (cn-block ofkd-block) (closed-cn-cn/app _ _ _) %. +%total {} (closed-cn-cn/app _ _ _) %. +%sort closed-cn-tp/cross %. +%mode closed-cn-tp/cross %in %in %out %. +%term _ closed-cn-tp/cross closed-cn/i closed-cn/i closed-cn/i %. +%worlds (cn-block ofkd-block) (closed-cn-tp/cross _ _ _) %. +%total {} (closed-cn-tp/cross _ _ _) %. +%sort closed-cn-tp/arrow %. +%mode closed-cn-tp/arrow %in %in %out %. +%term _ closed-cn-tp/arrow closed-cn/i closed-cn/i closed-cn/i %. +%worlds (cn-block ofkd-block) (closed-cn-tp/arrow _ _ _) %. +%total {} (closed-cn-tp/arrow _ _ _) %. +%sort closed-cn-tp/sum %. +%mode closed-cn-tp/sum %in %in %out %. +%term _ closed-cn-tp/sum closed-cn/i closed-cn/i closed-cn/i %. +%worlds (cn-block ofkd-block) (closed-cn-tp/sum _ _ _) %. +%total {} (closed-cn-tp/sum _ _ _) %. +%sort closed-cn-cn/lam %. +%mode closed-cn-cn/lam %in %in %out %. +%term _ closed-cn-cn/lam (%the (closed-kd ([x] K) K) closed-kd/i) ([y] %the (closed-cn ([x] C y) _) closed-cn/i) (%the (closed-cn ([x] cn/lam K ([y] C y)) _) closed-cn/i) %. +%worlds (cn-block ofkd-block) (closed-cn-cn/lam _ _ _) %. +%total {} (closed-cn-cn/lam _ _ _) %. +%sort closed-cn-cn/mu %. +%mode closed-cn-cn/mu %in %in %out %. +%term _ closed-cn-cn/mu (%the (closed-kd ([x] K) K) closed-kd/i) ([y] %the (closed-cn ([x] C y) _) closed-cn/i) (%the (closed-cn ([x] cn/mu K ([y] C y)) _) closed-cn/i) %. +%worlds (cn-block ofkd-block) (closed-cn-cn/mu _ _ _) %. +%total {} (closed-cn-cn/mu _ _ _) %. +%sort closed-cn-tp/forall %. +%mode closed-cn-tp/forall %in %in %out %. +%term _ closed-cn-tp/forall (%the (closed-kd ([x] K) K) closed-kd/i) ([y] %the (closed-cn ([x] C y) _) closed-cn/i) (%the (closed-cn ([x] tp/forall K ([y] C y)) _) closed-cn/i) %. +%worlds (cn-block ofkd-block) (closed-cn-tp/forall _ _ _) %. +%total {} (closed-cn-tp/forall _ _ _) %. +%sort closed-kd-kd/sing %. +%mode closed-kd-kd/sing %in %out %. +%term _ closed-kd-kd/sing closed-cn/i closed-kd/i %. +%worlds (cn-block ofkd-block) (closed-kd-kd/sing _ _) %. +%total {} (closed-kd-kd/sing _ _) %. +%sort closed-cn-kd/sing-e %. +%mode closed-cn-kd/sing-e %in %out %. +%term _ closed-cn-kd/sing-e closed-kd/i closed-cn/i %. +%worlds (cn-block ofkd-block) (closed-cn-kd/sing-e _ _) %. +%total {} (closed-cn-kd/sing-e _ _) %. +%sort closed-kd-kd/sgm %. +%mode closed-kd-kd/sgm %in %in %out %. +%term _ closed-kd-kd/sgm (%the (closed-kd ([x] K) _) closed-kd/i) ([y] %the (closed-kd ([x] K' y) _) closed-kd/i) (%the (closed-kd ([x] kd/sgm K ([y] K' y)) _) closed-kd/i) %. +%worlds (cn-block ofkd-block) (closed-kd-kd/sgm _ _ _) %. +%total {} (closed-kd-kd/sgm _ _ _) %. +%sort closed-kd-kd/pi %. +%mode closed-kd-kd/pi %in %in %out %. +%term _ closed-kd-kd/pi (%the (closed-kd ([x] K) _) closed-kd/i) ([y] %the (closed-kd ([x] K' y) _) closed-kd/i) (%the (closed-kd ([x] kd/pi K ([y] K' y)) _) closed-kd/i) %. +%worlds (cn-block ofkd-block) (closed-kd-kd/pi _ _ _) %. +%total {} (closed-kd-kd/pi _ _ _) %. +%sort closed-kd-kd/sgm-el %. +%mode closed-kd-kd/sgm-el %in %out %. +%term _ closed-kd-kd/sgm-el closed-kd/i closed-kd/i %. +%worlds (cn-block ofkd-block) (closed-kd-kd/sgm-el _ _) %. +%total {} (closed-kd-kd/sgm-el _ _) %. +%sort closed-kd-kd/pi-el %. +%mode closed-kd-kd/pi-el %in %out %. +%term _ closed-kd-kd/pi-el closed-kd/i closed-kd/i %. +%worlds (cn-block ofkd-block) (closed-kd-kd/pi-el _ _) %. +%total {} (closed-kd-kd/pi-el _ _) %. +%sort closed-kd-kd/sgm-er %. +%mode closed-kd-kd/sgm-er %in %in %out %. +%term _ closed-kd-kd/sgm-er closed-cn/i closed-kd/i closed-kd/i %. +%worlds (cn-block ofkd-block) (closed-kd-kd/sgm-er _ _ _) %. +%total {} (closed-kd-kd/sgm-er _ _ _) %. +%sort closed-kd-kd/pi-er %. +%mode closed-kd-kd/pi-er %in %in %out %. +%term _ closed-kd-kd/pi-er closed-cn/i closed-kd/i closed-kd/i %. +%worlds (cn-block ofkd-block) (closed-kd-kd/pi-er _ _ _) %. +%total {} (closed-kd-kd/pi-er _ _ _) %. +%sort strengthen-ofkd {x} %. +%mode strengthen-ofkd %in %out %out %out %. +%sort strengthen-cn-deq {x} %. +%mode strengthen-cn-deq %in %out %out %out %out %. +%sort strengthen-cn-deq-o {x} {y} %. +%mode strengthen-cn-deq-o %in %in %out %out %out %out %. +%sort strengthen-kd-deq {x} %. +%mode strengthen-kd-deq %in %out %out %out %. +%sort strengthen-kd-deq-o {x} {y} %. +%mode strengthen-kd-deq-o %in %in %out %out %out %. +%sort strengthen-kd-sub {x} %. +%mode strengthen-kd-sub %in %out %out %out %. +%sort strengthen-kd-sub-o {x} {y} %. +%mode strengthen-kd-sub-o %in %in %out %out %out %. +%sort strengthen-ofkd-o {x} {y} %. +%mode strengthen-ofkd-o %in %in %out %out %out %. +%sort strengthen-kd-wf {x} %. +%mode strengthen-kd-wf %in %out %out %. +%sort strengthen-kd-wf-o {x} {y} %. +%mode strengthen-kd-wf-o %in %in %out %out %. +%term _ + %pi (strengthen-ofkd-o D1 closed-kd/i DC DK D1') + %<- ({y} {dy} strengthen-ofkd ([x] D1 x y dy) (DC y) (DK y) (D1' y dy)) %. +%term _ + %pi (strengthen-kd-wf-o D1 closed-kd/i DK D1') + %<- ({y} {dy} strengthen-kd-wf ([x] D1 x y dy) (DK y) (D1' y dy)) %. +%term _ + %pi (strengthen-kd-deq-o D1 closed-kd/i DC DK D1') + %<- ({y} {dy} strengthen-kd-deq ([x] D1 x y dy) (DC y) (DK y) (D1' y dy)) %. +%term _ + %pi (strengthen-kd-sub-o D1 closed-kd/i DC DK D1') + %<- ({y} {dy} strengthen-kd-sub ([x] D1 x y dy) (DC y) (DK y) (D1' y dy)) %. +%term _ + %pi (strengthen-cn-deq-o D1 closed-kd/i DC DK DK' D1') + %<- ({y} {dy} strengthen-cn-deq ([x] D1 x y dy) (DC y) (DK y) (DK' y) (D1' y dy)) %. +%term _ strengthen-ofkd ([x] D1) closed-cn/i closed-kd/i D1 %. +%term _ + %pi (strengthen-ofkd ([x] ofkd/tp/ref (D1 x)) DC1' DK (ofkd/tp/ref D1')) + %<- (strengthen-ofkd D1 DC1 DK D1') + %<- (closed-cn-tp/ref DC1 DC1') %. +%term _ + %pi (strengthen-ofkd ([x] ofkd/tp/tag (D1 x)) DC1' DK (ofkd/tp/tag D1')) + %<- (strengthen-ofkd D1 DC1 DK D1') + %<- (closed-cn-tp/tag DC1 DC1') %. +%term _ + %pi (strengthen-ofkd ([x] ofkd/tp/cross (D1 x) (D2 x)) DC1' DK (ofkd/tp/cross D1' D2')) + %<- (strengthen-ofkd D1 DC1 DK D1') + %<- (strengthen-ofkd D2 DC2 _ D2') + %<- (closed-cn-tp/cross DC1 DC2 DC1') %. +%term _ + %pi (strengthen-ofkd ([x] ofkd/tp/arrow (D1 x) (D2 x)) DC1' DK (ofkd/tp/arrow D1' D2')) + %<- (strengthen-ofkd D1 DC1 DK D1') + %<- (strengthen-ofkd D2 DC2 _ D2') + %<- (closed-cn-tp/arrow DC1 DC2 DC1') %. +%term _ + %pi (strengthen-ofkd ([x] ofkd/tp/sum (D1 x) (D2 x)) DC1' DK (ofkd/tp/sum D1' D2')) + %<- (strengthen-ofkd D1 DC1 DK D1') + %<- (strengthen-ofkd D2 DC2 _ D2') + %<- (closed-cn-tp/sum DC1 DC2 DC1') %. +%term _ + %pi (strengthen-ofkd ([x] ofkd/cn/pair (D1 x) (D2 x)) DC1' DK' (ofkd/cn/pair D1' D2')) + %<- (strengthen-ofkd D1 DC1 DK1 D1') + %<- (strengthen-ofkd D2 DC2 DK2 D2') + %<- (closed-cn-cn/pair DC1 DC2 DC1') + %<- (closed-kd-kd/sgm DK1 ([x] DK2) DK') %. +%term _ + %pi (strengthen-ofkd ([x] ofkd/cn/lam (D1 x) (D2 x)) DC1' DK' (ofkd/cn/lam D1' D2')) + %<- (strengthen-kd-wf D2 DK2 D2') + %<- (strengthen-ofkd-o D1 DK2 DC1 DK1 D1') + %<- (closed-cn-cn/lam DK2 DC1 DC1') + %<- (closed-kd-kd/pi DK2 DK1 DK') %. +%term _ + %pi (strengthen-ofkd ([x] ofkd/kd/sing (D1 x)) DC1 DK1 (ofkd/kd/sing D1')) + %<- (strengthen-ofkd D1 DC1 _ D1') + %<- (closed-kd-kd/sing DC1 DK1) %. +%term _ + %pi (strengthen-ofkd ([x] ofkd/sub (D1 x) (D2 x)) DC1 DK1 (ofkd/sub D1'' D2')) + %<- (strengthen-ofkd D1 DC1 DK2 D1') + %<- (strengthen-kd-sub D2 DK2a DK1 D2') + %<- (closed-kd-seq/kd DK2 DK2a DQ) + %<- (ofkd/seq-k DQ D1' D1'') %. +%term _ + %pi (strengthen-ofkd ([x] ofkd/deq (D1 x) (D2 x)) DC1 DK1 (ofkd/deq D1'' D2')) + %<- (strengthen-ofkd D1 DC1 DK2 D1') + %<- (strengthen-kd-deq D2 DK2a DK1 D2') + %<- (closed-kd-seq/kd DK2 DK2a DQ) + %<- (ofkd/seq-k DQ D1' D1'') %. +%term _ + %pi (strengthen-ofkd ([x] ofkd/sgm-ext (D1 x) (D2 x)) DC2' DK' (ofkd/sgm-ext D1'' D2'')) + %<- (strengthen-ofkd D1 DC1 DK1 D1') + %<- (strengthen-ofkd D2 DC2 DK2 D2') + %<- (closed-cn-cn/pj2-e DC2 DC2' DQm) + %<- (ofkd/seq-c DQm D2' D2'') + %<- (closed-cn-cn/pj1 DC2' DC1'') + %<- (closed-cn-seq/cn DC1 DC1'' DQ) + %<- (ofkd/seq-c DQ D1' D1'') + %<- (closed-kd-kd/sgm DK1 ([x] DK2) DK') %. +%term _ + %pi (strengthen-ofkd ([x] ofkd/cn/pj2 (D1 x)) DC1' DK' (ofkd/cn/pj2 D1')) + %<- (strengthen-ofkd D1 DC1 DK D1') + %<- (closed-cn-cn/pj2 DC1 DC1') + %<- (closed-cn-cn/pj1 DC1 DC1'') + %<- (closed-kd-kd/sgm-er DC1'' DK DK') %. +%term _ + %pi (strengthen-ofkd ([x] ofkd/cn/pj1 (D1 x)) DC1' DK' (ofkd/cn/pj1 D1')) + %<- (strengthen-ofkd D1 DC1 DK D1') + %<- (closed-cn-cn/pj1 DC1 DC1') + %<- (closed-kd-kd/sgm-el DK DK') %. +%term _ + %pi (strengthen-ofkd ([x] ofkd/cn/app (D1 x) (D2 x)) DC1' DK' (ofkd/cn/app D1' D2'')) + %<- (strengthen-ofkd D1 DC1 DK1 D1') + %<- (strengthen-ofkd D2 DC2 DK2 D2') + %<- (closed-cn-cn/app DC1 DC2 DC1') + %<- (closed-kd-kd/pi-er DC2 DK1 DK') + %<- (closed-kd-kd/pi-el DK1 DK1l) + %<- (closed-kd-seq/kd DK2 DK1l DQ) + %<- (ofkd/seq-k DQ D2' D2'') %. +%term _ + %pi (strengthen-ofkd ([x] %the (ofkd _ kd/type) (ofkd/cn/mu (D1 x) (D2 x))) DC1' closed-kd/i (ofkd/cn/mu D1' D2')) + %<- (strengthen-kd-wf D2 DK2 D2') + %<- (strengthen-ofkd-o D1 DK2 DC1 DK1 D1') + %<- (closed-cn-cn/mu DK2 DC1 DC1') %. +%term _ + %pi (strengthen-ofkd ([x] %the (ofkd _ kd/type) (ofkd/tp/forall (D1 x) (D2 x))) DC1' closed-kd/i (ofkd/tp/forall D1' D2')) + %<- (strengthen-kd-wf D2 DK2 D2') + %<- (strengthen-ofkd-o D1 DK2 DC1 DK1 D1') + %<- (closed-cn-tp/forall DK2 DC1 DC1') %. +%term _ + %pi (strengthen-ofkd ([x] ofkd/pi-ext (D2 x) (D1 x)) DC1 DK (ofkd/pi-ext D2' D1'')) + %<- (strengthen-ofkd D2 DC1 DK2 D2') + %<- (closed-kd-kd/pi-el DK2 DK2') + %<- (strengthen-ofkd-o D1 DK2' DC2 DK1 D1') + %<- (closed-kd-kd/pi DK2' DK1 DK) + %<- ({x} closed-cn-cn/app DC1 (%the (closed-cn _ x) closed-cn/i) (DC2o x)) + %<- ({x} closed-cn-seq/cn (DC2 x) (DC2o x) (DQ x)) + %<- ({x} {dx} ofkd/seq-c (DQ x) (D1' x dx) (D1'' x dx)) %. +%term _ strengthen-kd-wf ([x] D1) closed-kd/i D1 %. +%term _ + %pi (strengthen-kd-wf ([x] kd-wf/kd/sing (D1 x)) DK (kd-wf/kd/sing D1')) + %<- (strengthen-ofkd D1 DC _ D1') + %<- (closed-kd-kd/sing DC DK) %. +%term _ + %pi (strengthen-kd-wf ([x] kd-wf/kd/sgm (D1 x) (D2 x)) DK' (kd-wf/kd/sgm D1' D2')) + %<- (strengthen-kd-wf D1 DK1 D1') + %<- (strengthen-kd-wf-o D2 DK1 DK2 D2') + %<- (closed-kd-kd/sgm DK1 DK2 DK') %. +%term _ + %pi (strengthen-kd-wf ([x] kd-wf/kd/pi (D1 x) (D2 x)) DK' (kd-wf/kd/pi D1' D2')) + %<- (strengthen-kd-wf D1 DK1 D1') + %<- (strengthen-kd-wf-o D2 DK1 DK2 D2') + %<- (closed-kd-kd/pi DK1 DK2 DK') %. +%term _ strengthen-kd-deq ([x] D1) closed-kd/i closed-kd/i D1 %. +%term _ + %pi (strengthen-kd-deq ([x] kd-deq/kd/sing (D1 x)) DK1 DK2 (kd-deq/kd/sing D1')) + %<- (strengthen-cn-deq D1 DC1 DC2 _ D1') + %<- (closed-kd-kd/sing DC1 DK1) + %<- (closed-kd-kd/sing DC2 DK2) %. +%term _ + %pi (strengthen-kd-deq ([x] kd-deq/kd/sgm (D1 x) (D2 x)) DKL DKR (kd-deq/kd/sgm D1' D2')) + %<- (strengthen-kd-deq D1 DK1l DK1r D1') + %<- (strengthen-kd-deq-o D2 DK1l DK2l DK2r D2') + %<- (closed-kd-kd/sgm DK1l DK2l DKL) + %<- (closed-kd-kd/sgm DK1r DK2r DKR) %. +%term _ + %pi (strengthen-kd-deq ([x] kd-deq/kd/pi (D1 x) (D2 x)) DKL DKR (kd-deq/kd/pi D1' D2')) + %<- (strengthen-kd-deq D1 DK1l DK1r D1') + %<- (strengthen-kd-deq-o D2 DK1l DK2l DK2r D2') + %<- (closed-kd-kd/pi DK1r DK2l DKL) + %<- (closed-kd-kd/pi DK1l DK2r DKR) %. +%term _ strengthen-kd-sub ([x] D1) closed-kd/i closed-kd/i D1 %. +%term _ + %pi (strengthen-kd-sub ([x] kd-sub/kd/sing-kd/sing (D1 x)) DK1 DK2 (kd-sub/kd/sing-kd/sing D1')) + %<- (strengthen-cn-deq D1 DC1 DC2 _ D1') + %<- (closed-kd-kd/sing DC1 DK1) + %<- (closed-kd-kd/sing DC2 DK2) %. +%term _ + %pi (strengthen-kd-sub ([x] kd-sub/kd/sing-kd/type (D1 x)) DK1 closed-kd/i (kd-sub/kd/sing-kd/type D1')) + %<- (strengthen-ofkd D1 DC1 _ D1') + %<- (closed-kd-kd/sing DC1 DK1) %. +%term _ + %pi (strengthen-kd-sub ([x] kd-sub/kd/sgm (D1 x) (D2 x) (D3 x)) DKL DKR (kd-sub/kd/sgm D1' D2' D3'')) + %<- (strengthen-kd-sub D1 DK1l DK1r D1') + %<- (strengthen-kd-sub-o D2 DK1l DK2l DK2r D2') + %<- (closed-kd-kd/sgm DK1l DK2l DKL) + %<- (closed-kd-kd/sgm DK1r DK2r DKR) + %<- (strengthen-kd-wf-o D3 DK1r DK2r' D3') + %<- ({x} closed-kd-seq/kd (DK2r' x) (DK2r x) (DQ x)) + %<- ({x} {dx} kd-wf/seq (DQ x) (D3' x dx) (D3'' x dx)) %. +%term _ + %pi (strengthen-kd-sub ([x] kd-sub/kd/pi (D1 x) (D2 x) (D3 x)) DKL DKR (kd-sub/kd/pi D1' D2' D3'')) + %<- (strengthen-kd-sub D1 DK1l DK1r D1') + %<- (strengthen-kd-sub-o D2 DK1l DK2l DK2r D2') + %<- (closed-kd-kd/pi DK1r DK2l DKL) + %<- (closed-kd-kd/pi DK1l DK2r DKR) + %<- (strengthen-kd-wf-o D3 DK1r DK2l' D3') + %<- ({x} closed-kd-seq/kd (DK2l' x) (DK2l x) (DQ x)) + %<- ({x} {dx} kd-wf/seq (DQ x) (D3' x dx) (D3'' x dx)) %. +%term _ strengthen-cn-deq ([x] D1) closed-cn/i closed-cn/i closed-kd/i D1 %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/refl (D1 x)) DC1 DC1 DK (cn-deq/refl D1')) + %<- (strengthen-ofkd D1 DC1 DK D1') %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/sym (D1 x)) DC2 DC1 DK (cn-deq/sym D1')) + %<- (strengthen-cn-deq D1 DC1 DC2 DK D1') %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/tp/tag (D1 x)) DC1' DC2' DK (cn-deq/tp/tag D1')) + %<- (strengthen-cn-deq D1 DC1 DC2 DK D1') + %<- (closed-cn-tp/tag DC1 DC1') + %<- (closed-cn-tp/tag DC2 DC2') %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/tp/ref (D1 x)) DC1' DC2' DK (cn-deq/tp/ref D1')) + %<- (strengthen-cn-deq D1 DC1 DC2 DK D1') + %<- (closed-cn-tp/ref DC1 DC1') + %<- (closed-cn-tp/ref DC2 DC2') %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/tp/cross (D1 x) (D2 x)) DC1' DC2' DK (cn-deq/tp/cross D1' D2')) + %<- (strengthen-cn-deq D1 DC1l DC1r DK D1') + %<- (strengthen-cn-deq D2 DC2l DC2r _ D2') + %<- (closed-cn-tp/cross DC1l DC2l DC1') + %<- (closed-cn-tp/cross DC1r DC2r DC2') %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/tp/arrow (D1 x) (D2 x)) DC1' DC2' DK (cn-deq/tp/arrow D1' D2')) + %<- (strengthen-cn-deq D1 DC1l DC1r DK D1') + %<- (strengthen-cn-deq D2 DC2l DC2r _ D2') + %<- (closed-cn-tp/arrow DC1l DC2l DC1') + %<- (closed-cn-tp/arrow DC1r DC2r DC2') %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/tp/sum (D1 x) (D2 x)) DC1' DC2' DK (cn-deq/tp/sum D1' D2')) + %<- (strengthen-cn-deq D1 DC1l DC1r DK D1') + %<- (strengthen-cn-deq D2 DC2l DC2r _ D2') + %<- (closed-cn-tp/sum DC1l DC2l DC1') + %<- (closed-cn-tp/sum DC1r DC2r DC2') %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/cn/lam (D1 x) (D2 x)) DC1' DC2' DK' (cn-deq/cn/lam D1' D2')) + %<- (strengthen-kd-deq D2 DK2l DK2r D2') + %<- (strengthen-cn-deq-o D1 DK2l DC1l DC1r DK1 D1') + %<- (closed-cn-cn/lam DK2l DC1l DC1') + %<- (closed-cn-cn/lam DK2r DC1r DC2') + %<- (closed-kd-kd/pi DK2l DK1 DK') %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/tp/forall (D1 x) (D2 x)) DC1' DC2' closed-kd/i (cn-deq/tp/forall D1' D2')) + %<- (strengthen-kd-deq D2 DK2l DK2r D2') + %<- (strengthen-cn-deq-o D1 DK2l DC1l DC1r DK1 D1') + %<- (closed-cn-tp/forall DK2l DC1l DC1') + %<- (closed-cn-tp/forall DK2r DC1r DC2') %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/cn/mu (D1 x) (D2 x)) DC1' DC2' closed-kd/i (cn-deq/cn/mu D1' D2')) + %<- (strengthen-kd-deq D2 DK2l DK2r D2') + %<- (strengthen-cn-deq-o D1 DK2l DC1l DC1r DK1 D1') + %<- (closed-cn-cn/mu DK2l DC1l DC1') + %<- (closed-cn-cn/mu DK2r DC1r DC2') %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/cn/pair (D1 x) (D2 x)) DC1' DC2' DK' (cn-deq/cn/pair D1' D2')) + %<- (strengthen-cn-deq D1 DC1l DC1r DK D1') + %<- (strengthen-cn-deq D2 DC2l DC2r DKK D2') + %<- (closed-cn-cn/pair DC1l DC2l DC1') + %<- (closed-cn-cn/pair DC1r DC2r DC2') + %<- (closed-kd-kd/sgm DK ([x] DKK) DK') %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/cn/pj1 (D1 x)) DC1' DC2' DK' (cn-deq/cn/pj1 D1')) + %<- (strengthen-cn-deq D1 DC1 DC2 DK D1') + %<- (closed-cn-cn/pj1 DC1 DC1') + %<- (closed-cn-cn/pj1 DC2 DC2') + %<- (closed-kd-kd/sgm-el DK DK') %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/cn/pj2 (D1 x)) DC1' DC2' DK' (cn-deq/cn/pj2 D1')) + %<- (strengthen-cn-deq D1 DC1 DC2 DK D1') + %<- (closed-cn-cn/pj2 DC1 DC1') + %<- (closed-cn-cn/pj2 DC2 DC2') + %<- (closed-cn-cn/pj1 DC1 DC1'') + %<- (closed-kd-kd/sgm-er DC1'' DK DK') %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/kd/unit (D1 x) (D2 x)) DC1 DC2 DK (cn-deq/kd/unit D1' D2')) + %<- (strengthen-ofkd D1 DC1 DK D1') + %<- (strengthen-ofkd D2 DC2 _ D2') %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/sub (D1 x) (D2 x)) DC1 DC2 DK1 (cn-deq/sub D1'' D2')) + %<- (strengthen-cn-deq D1 DC1 DC2 DK2 D1') + %<- (strengthen-kd-sub D2 DK2a DK1 D2') + %<- (closed-kd-seq/kd DK2 DK2a DQ) + %<- (cn-deq/seq-k DQ D1' D1'') %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/deq (D1 x) (D2 x)) DC1 DC2 DK1 (cn-deq/deq D1'' D2')) + %<- (strengthen-cn-deq D1 DC1 DC2 DK2 D1') + %<- (strengthen-kd-deq D2 DK2a DK1 D2') + %<- (closed-kd-seq/kd DK2 DK2a DQ) + %<- (cn-deq/seq-k DQ D1' D1'') %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/trans (D1 x) (D2 x)) DC1 DC3 DK2 (cn-deq/trans D1''' D2')) + %<- (strengthen-cn-deq D1 DC1 DC2 DK1 D1') + %<- (strengthen-cn-deq D2 DC2a DC3 DK2 D2') + %<- (closed-cn-seq/cn DC2 DC2a DQ) + %<- (cn-deq/seq-r DQ D1' D1'') + %<- (closed-kd-seq/kd DK1 DK2 DQK) + %<- (cn-deq/seq-k DQK D1'' D1''') %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/cn/app (D1 x) (D2 x)) DC1' DC2' DK' (cn-deq/cn/app D1' D2'')) + %<- (strengthen-cn-deq D1 DC1l DC1r DK1 D1') + %<- (strengthen-cn-deq D2 DC2l DC2r DK2 D2') + %<- (closed-cn-cn/app DC1l DC2l DC1') + %<- (closed-cn-cn/app DC1r DC2r DC2') + %<- (closed-kd-kd/pi-er DC2l DK1 DK') + %<- (closed-kd-kd/pi-el DK1 DK1l) + %<- (closed-kd-seq/kd DK2 DK1l DQ) + %<- (cn-deq/seq-k DQ D2' D2'') %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/kd/sing (D1 x)) DC1 DC2 DK1 (cn-deq/kd/sing D1')) + %<- (strengthen-ofkd D1 DC1 DK1 D1') + %<- (closed-cn-kd/sing-e DK1 DC2) %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/pi-ext-2 (D2 x) (D1 x)) DC1 DCR DK (cn-deq/pi-ext-2 D2' D1''')) + %<- (strengthen-cn-deq D2 DC1 DCR DK2 D2') + %<- (closed-kd-kd/pi-el DK2 DK2') + %<- (strengthen-cn-deq-o D1 DK2' DC2 DCR' DK1 D1') + %<- (closed-kd-kd/pi DK2' DK1 DK) + %<- ({x} closed-cn-cn/app DC1 (%the (closed-cn _ x) closed-cn/i) (DC2o x)) + %<- ({x} closed-cn-seq/cn (DC2 x) (DC2o x) (DQ x)) + %<- ({x} {dx} cn-deq/seq-l (DQ x) (D1' x dx) (D1'' x dx)) + %<- ({x} closed-cn-cn/app DCR (%the (closed-cn _ x) closed-cn/i) (DCRo x)) + %<- ({x} closed-cn-seq/cn (DCR' x) (DCRo x) (DQr x)) + %<- ({x} {dx} cn-deq/seq-r (DQr x) (D1'' x dx) (D1''' x dx)) %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/pi-ext (D2 x) (D3 x) (D1 x)) DC1 DCR DK (cn-deq/pi-ext D2' D3'' D1''')) + %<- (strengthen-ofkd D2 DC1 DK2 D2') + %<- (strengthen-ofkd D3 DCR DK2r D3') + %<- (closed-kd-kd/pi-el DK2 DK2') + %<- (closed-kd-kd/pi-el DK2r DK2r') + %<- (closed-kd-seq/kd DK2r' DK2' DQK2) + %<- (ofkd/seq-k-pi-l DQK2 D3' D3'') + %<- (strengthen-cn-deq-o D1 DK2' DC2 DCR' DK1 D1') + %<- (closed-kd-kd/pi DK2' DK1 DK) + %<- ({x} closed-cn-cn/app DC1 (%the (closed-cn _ x) closed-cn/i) (DC2o x)) + %<- ({x} closed-cn-seq/cn (DC2 x) (DC2o x) (DQ x)) + %<- ({x} {dx} cn-deq/seq-l (DQ x) (D1' x dx) (D1'' x dx)) + %<- ({x} closed-cn-cn/app DCR (%the (closed-cn _ x) closed-cn/i) (DCRo x)) + %<- ({x} closed-cn-seq/cn (DCR' x) (DCRo x) (DQr x)) + %<- ({x} {dx} cn-deq/seq-r (DQr x) (D1'' x dx) (D1''' x dx)) %. +%term _ + %pi (strengthen-cn-deq ([x] cn-deq/sgm-ext (D1 x) (D2 x)) DC2' DC2r' DK' (cn-deq/sgm-ext D1''' D2''')) + %<- (strengthen-cn-deq D1 DC1 DC1r DK1 D1') + %<- (strengthen-cn-deq D2 DC2 DC2r DK2 D2') + %<- (closed-cn-cn/pj2-e DC2 DC2' DQ2m) + %<- (cn-deq/seq-l DQ2m D2' D2'') + %<- (closed-cn-cn/pj1 DC2' DC1'') + %<- (closed-cn-seq/cn DC1 DC1'' DQ) + %<- (cn-deq/seq-l DQ D1' D1'') + %<- (closed-kd-kd/sgm DK1 ([x] DK2) DK') + %<- (closed-cn-cn/pj2-e DC2r DC2r' DQ2rm) + %<- (cn-deq/seq-r DQ2rm D2'' D2''') + %<- (closed-cn-cn/pj1 DC2r' DC1r') + %<- (closed-cn-seq/cn DC1r DC1r' DQr) + %<- (cn-deq/seq-r DQr D1'' D1''') %. +%worlds (cn-block ofkd-block) (strengthen-ofkd _ _ _ _) (strengthen-kd-deq _ _ _ _) (strengthen-kd-deq-o _ _ _ _ _) (strengthen-kd-sub _ _ _ _) (strengthen-kd-sub-o _ _ _ _ _) (strengthen-ofkd-o _ _ _ _ _) (strengthen-kd-wf _ _ _) (strengthen-kd-wf-o _ _ _ _) (strengthen-cn-deq _ _ _ _ _) (strengthen-cn-deq-o _ _ _ _ _ _) %. +%total (D1 D2 D3 D4 D5 D6 D7 D8 D9 D10) (strengthen-ofkd D1 _ _ _) (strengthen-ofkd-o D3 _ _ _ _) (strengthen-kd-wf D2 _ _) (strengthen-kd-wf-o D6 _ _ _) (strengthen-kd-deq D4 _ _ _) (strengthen-kd-deq-o D8 _ _ _ _) (strengthen-kd-sub D5 _ _ _) (strengthen-kd-sub-o D9 _ _ _ _) (strengthen-cn-deq D7 _ _ _ _) (strengthen-cn-deq-o D10 _ _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/subderivations.lf b/new-tests/stelf-output/tslf/subderivations.lf new file mode 100644 index 0000000..a1358dd --- /dev/null +++ b/new-tests/stelf-output/tslf/subderivations.lf @@ -0,0 +1,35 @@ +%%%% theorems about sub-derivations +%sort subder/md/pj1 %. +%mode subder/md/pj1 %in %out %out %. +%term _ %pi (subder/md/pj1 (ofsg/md/pj1 D) D PS) %<- (pty-sub-refl P PS) %. +%term _ + %pi (subder/md/pj1 (ofsg/sgm-ext D1 D2) D4 pty-sub/pp) + %<- (subder/md/pj1 D1 D3 pty-sub/pp) + %<- (subder/md/pj1 D3 D4 pty-sub/pp) %. +%term _ + %pi (subder/md/pj1 (ofsg/kd-ext D1 _ _) D1' pty-sub/pp) + %<- (subder/md/pj1 D1 D1' pty-sub/pp) %. +%term _ + %pi (subder/md/pj1 (ofsg/sub (%the (ofsg _ _ Y2 _ _) D1) _ (%the (pty-sub Y2 Y3) PS')) D2 (%the (pty-sub Y1 Y3) PS'')) + %<- (subder/md/pj1 D1 (%the (ofsg _ _ Y1 _ _) D2) (%the (pty-sub Y1 Y2) PS)) + %<- (pty-sub-trans PS PS' PS'') %. +%worlds (ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (subder/md/pj1 _ _ _) %. +%reduces < D2 D1 (subder/md/pj1 D1 D2 _) %. +%total (D1) (subder/md/pj1 D1 _ _) %. +%sort subder/md/pj2 %. +%mode subder/md/pj2 %in %out %out %. +%term _ subder/md/pj2 (ofsg/md/pj2 D _) D pty-sub/pp %. +%term _ + %pi (subder/md/pj2 (ofsg/sgm-ext D1 D2) D4 pty-sub/pp) + %<- (subder/md/pj2 D2 D3 pty-sub/pp) + %<- (subder/md/pj2 D3 D4 pty-sub/pp) %. +%term _ + %pi (subder/md/pj2 (ofsg/kd-ext D1 _ _) D1' pty-sub/pp) + %<- (subder/md/pj2 D1 D1' pty-sub/pp) %. +%term _ + %pi (subder/md/pj2 (ofsg/sub D1 _ PS') D2 (%the (pty-sub Y1 Y3) PS'')) + %<- (subder/md/pj2 D1 D2 (%the (pty-sub Y1 Y2) PS)) + %<- (pty-sub-trans PS PS' PS'') %. +%worlds (ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (subder/md/pj2 _ _ _) %. +%reduces < D2 D1 (subder/md/pj2 D1 D2 _) %. +%total (D1) (subder/md/pj2 D1 _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/substitution-cnkd-e.lf b/new-tests/stelf-output/tslf/substitution-cnkd-e.lf new file mode 100644 index 0000000..df8c561 --- /dev/null +++ b/new-tests/stelf-output/tslf/substitution-cnkd-e.lf @@ -0,0 +1,328 @@ +%sort subst-cxt-bounded {x cn} %. +%mode subst-cxt-bounded %in %in %in %out %. +%term _ subst-cxt-bounded _ (%the (cxt-append-sub _ _ _ _) _) ([x] [di] cxt-bounded/cons (cxt-bounded/nil di) (cxt-precedes/i di DV DL)) (cxt-bounded/nil DV) %. +%term _ + %pi (subst-cxt-bounded _ (%the (cxt-append-sub _ _ _ (cxt/cons G1 _ K6)) _) ([x] [di] cxt-bounded/cons (cxt-bounded/cons (DB x di) (cxt-precedes/i D1 di D12)) (cxt-precedes/i di DV D23)) (cxt-bounded/cons DB' (cxt-precedes/i D1 DV D13))) + %<- (cxt-bounded-strengthen (cxt/cons G1 C1 K6) ([x] [di] cxt-bounded/cons (DB x di) (cxt-precedes/i D1 di D12)) (cxt-ordered/cons DB')) + %<- (loc-less-trans D12 D23 D13) %. +%term _ + %pi (subst-cxt-bounded ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') ([x] [di] cxt-bounded/cons (DB x di) (cxt-precedes/i D1 D2 DL)) (cxt-bounded/cons DB' (cxt-precedes/i D1 D2 DL))) + %<- (subst-cxt-bounded DA DA' ([x] [di] DB x di) DB') %. +%worlds (cn-block isvar-fun-block) (subst-cxt-bounded _ _ _ _) %. +%total D1 (subst-cxt-bounded D1 _ _ _) %. +%sort subst-cxt-ordered {x cn} %. +%mode subst-cxt-ordered %in %in %in %out %. +%term _ + %pi (subst-cxt-ordered _ cxt-append-sub/nil ([x] [di] cxt-ordered/cons (DB x di)) DO) + %<- (cxt-bounded-strengthen _ DB DO) %. +%term _ + %pi (subst-cxt-ordered ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') ([x] [di] cxt-ordered/cons (DB x di)) (cxt-ordered/cons DB')) + %<- (subst-cxt-bounded DA DA' DB DB') %. +%worlds (cn-block isvar-fun-block) (subst-cxt-ordered _ _ _ _) %. +%total D1 (subst-cxt-ordered D1 _ _ _) %. +%sort subst-cxt-lookup-c {x cn} %. +%mode subst-cxt-lookup-c %in %in %in %out %. +%term _ + %pi (subst-cxt-lookup-c ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') ([x] [di] cxt-lookup/hit (DB x di)) (cxt-lookup/hit DB')) + %<- (subst-cxt-bounded DA DA' DB DB') %. +%term _ + %pi (subst-cxt-lookup-c ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') ([x] [di] cxt-lookup/miss (DL x di) DP (DB x di)) (cxt-lookup/miss DL' DP DB')) + %<- (subst-cxt-bounded DA DA' DB DB') + %<- (subst-cxt-lookup-c DA DA' DL DL') %. +%term _ + %pi (subst-cxt-lookup-c ([x] cxt-append/cons (DA x)) (%the (cxt-append-sub _ _ C _) (cxt-append-sub/cons DA')) ([x] [di] cxt-lookup/miss (cxt-lookup/miss (DL x di) (cxt-precedes/i DF di D12) _) DP (DB x di)) (cxt-lookup/miss DL' DP DB')) + %<- (cxt-lookup-strengthen C DL DL') + %<- (subst-cxt-bounded DA DA' DB DB') %. +%term _ + %pi (subst-cxt-lookup-c ([x] cxt-append/nil) (%the (cxt-append-sub _ _ C _) cxt-append-sub/nil) ([x] [di] cxt-lookup/miss (DL x di) (cxt-precedes/i DI di D12) _) DL') + %<- (cxt-lookup-strengthen C DL DL') %. +%worlds (cn-block isvar-fun-block can-mofkd-block) (subst-cxt-lookup-c _ _ _ _) %. +%total D1 (subst-cxt-lookup-c D1 _ _ _) %. +%sort cxt-lookup-ioc {x cn} %. +%mode cxt-lookup-ioc %in %in %in %out %. +%term _ cxt-lookup-ioc _ cxt-append-sub/nil ([x] [di] %the (cxt-lookup _ x _) (cxt-lookup/hit _)) id-or-cl/id %. +%term _ + %pi (cxt-lookup-ioc _ cxt-append-sub/nil ([x] [di] cxt-lookup/miss (DL x di) _ _) IOC) + %<- (cxt-lookup-ioc-c DL IOC) %. +%term _ cxt-lookup-ioc ([x] cxt-append/cons _) (cxt-append-sub/cons _) ([x] [di] %the (cxt-lookup _ C _) (cxt-lookup/hit _)) id-or-cl/cl %. +%term _ + %pi (cxt-lookup-ioc ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') ([x] [di] cxt-lookup/miss (DL x di) _ _) IOC) + %<- (cxt-lookup-ioc DA DA' DL IOC) %. +%worlds (cn-block ovar-block) (cxt-lookup-ioc _ _ _ _) %. +%total D1 (cxt-lookup-ioc _ _ D1 _) %. +%sort eofkd/seq/kd %. +%mode eofkd/seq/kd %in %in %out %. +%term _ eofkd/seq/kd seq/kd/refl D1 D1 %. +%worlds (cn-block isvar-fun-block can-mofkd-block) (eofkd/seq/kd _ _ _) %. +%total D1 (eofkd/seq/kd D1 _ _) %. +%sort subst-cxt-lookup {x cn} %. +%mode subst-cxt-lookup %in %in %in %in %in %out %. +%term _ + %pi (subst-cxt-lookup DA DA' DC DL id-or-cl/cl (eofkd/var DL')) + %<- (subst-cxt-lookup-c DA DA' DL DL') %. +%term _ + %pi (subst-cxt-lookup DA DA' DC DL id-or-cl/id DC'') + %<- (cxt-lookup-unique DA DL DQ) + %<- ({x} {di isvar x I} cxt-lookup-ordered (DL x di) (DO x di)) + %<- (subst-cxt-ordered DA DA' DO DO') + %<- (wkn-eofkd-append DC DA' DO' DC') + %<- (eofkd/seq/kd DQ DC' DC'') %. +%worlds (cn-block isvar-fun-block can-mofkd-block) (subst-cxt-lookup _ _ _ _ _ _) %. +%total D1 (subst-cxt-lookup _ _ _ _ D1 _) %. +%sort subst-f-cxt-lookup {x cn} %. +%mode subst-f-cxt-lookup %in %in %in %in %in %out %. +%term _ + %pi (subst-f-cxt-lookup DA DA' DC DL id-or-cl/cl (eofkd/var DL')) + %<- (subst-cxt-lookup-c DA DA' DL DL') %. +%term _ + %pi (subst-f-cxt-lookup DA DA' DC DL id-or-cl/id DC'') + %<- (cxt-lookup-unique DA DL DQ) + %<- ({x} {di isvar x I} cxt-lookup-ordered (DL x di) (DO x di)) + %<- (subst-cxt-ordered DA DA' DO DO') + %<- (wkn-eofkd-append DC DA' DO' DC') + %<- (eofkd/seq/kd DQ DC' DC'') %. +%worlds (cn-block isvar-fun-block) (subst-f-cxt-lookup _ _ _ _ _ _) %. +%total D1 (subst-f-cxt-lookup _ _ _ _ D1 _) %. +%sort subst-eofkd {x cn} %. +%mode subst-eofkd %in %in %in %in %out %. +%sort subst-ecn-deq {x cn} %. +%mode subst-ecn-deq %in %in %in %in %out %. +%sort subst-ekd-wf {x cn} %. +%mode subst-ekd-wf %in %in %in %in %out %. +%sort subst-ekd-sub {x cn} %. +%mode subst-ekd-sub %in %in %in %in %out %. +%sort subst-ekd-deq {x cn} %. +%mode subst-ekd-deq %in %in %in %in %out %. +%term _ + %pi (subst-ekd-wf DA DA' DC ([x] [di isvar _ I] ekd-wf/kd/unit (DO' x di)) (ekd-wf/kd/unit DO)) + %<- (subst-cxt-ordered DA DA' DO' DO) %. +%term _ + %pi (subst-ekd-wf DA DA' DC ([x] [di isvar _ I] ekd-wf/kd/type (DO' x di)) (ekd-wf/kd/type DO)) + %<- (subst-cxt-ordered DA DA' DO' DO) %. +%term _ + %pi (subst-ekd-wf DA DA' DC ([x] [di isvar x I] ekd-wf/kd/sing (D1 x di)) (ekd-wf/kd/sing D1')) + %<- (subst-eofkd DA DA' DC D1 D1') %. +%term _ + %pi (subst-ekd-wf DA DA' (%the (eofkd G1 C K) DC) ([x] [di isvar x XI] ekd-wf/kd/pi (D1 x di) ([y] [dy isvar y YI] D2 x di y dy)) (ekd-wf/kd/pi D1' D2')) + %<- (subst-ekd-wf DA DA' DC D1 D1') + %<- ({y} {dy isvar y YI} {_ isvar-fun dy dy seq/loc/refl} subst-ekd-wf ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC ([x] [di isvar x XI] D2 x di y dy) (D2' y dy)) %. +%term _ + %pi (subst-ekd-wf DA DA' (%the (eofkd G1 C K) DC) ([x] [di isvar x XI] ekd-wf/kd/sgm (D1 x di) ([y] [dy isvar y YI] D2 x di y dy)) (ekd-wf/kd/sgm D1' D2')) + %<- (subst-ekd-wf DA DA' DC D1 D1') + %<- ({y} {dy isvar y YI} {_ isvar-fun dy dy seq/loc/refl} subst-ekd-wf ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC ([x] [di isvar x XI] D2 x di y dy) (D2' y dy)) %. +%term _ + %pi (subst-eofkd DA DA' DC ([x] [di isvar _ I] eofkd/cn/unit (DO' x di)) (eofkd/cn/unit DO)) + %<- (subst-cxt-ordered DA DA' DO' DO) %. +%term _ + %pi (subst-eofkd DA DA' DC ([x] [di isvar _ I] eofkd/tp/unit (DO' x di)) (eofkd/tp/unit DO)) + %<- (subst-cxt-ordered DA DA' DO' DO) %. +%term _ + %pi (subst-eofkd DA DA' DC ([x] [di isvar _ I] eofkd/tp/tagged (DO' x di)) (eofkd/tp/tagged DO)) + %<- (subst-cxt-ordered DA DA' DO' DO) %. +%term _ + %pi (subst-eofkd DA DA' DC ([x] [di isvar x I] eofkd/kd/sing (D1 x di)) (eofkd/kd/sing D1')) + %<- (subst-eofkd DA DA' DC D1 D1') %. +%term _ + %pi (subst-eofkd DA DA' DC ([x] [di isvar x I] eofkd/cn/pj1 (D1 x di)) (eofkd/cn/pj1 D1')) + %<- (subst-eofkd DA DA' DC D1 D1') %. +%term _ + %pi (subst-eofkd DA DA' DC ([x] [di isvar x I] eofkd/cn/pj2 (D1 x di)) (eofkd/cn/pj2 D1')) + %<- (subst-eofkd DA DA' DC D1 D1') %. +%term _ + %pi (subst-eofkd DA DA' DC ([x] [di isvar x I] eofkd/tp/ref (D1 x di)) (eofkd/tp/ref D1')) + %<- (subst-eofkd DA DA' DC D1 D1') %. +%term _ + %pi (subst-eofkd DA DA' DC ([x] [di isvar x I] eofkd/tp/tag (D1 x di)) (eofkd/tp/tag D1')) + %<- (subst-eofkd DA DA' DC D1 D1') %. +%term _ + %pi (subst-eofkd DA DA' DC ([x] [di isvar x I] eofkd/cn/pair (D1 x di) (D2 x di)) (eofkd/cn/pair D1' D2')) + %<- (subst-eofkd DA DA' DC D1 D1') + %<- (subst-eofkd DA DA' DC D2 D2') %. +%term _ + %pi (subst-eofkd DA DA' DC ([x] [di isvar x I] eofkd/cn/app (D1 x di) (D2 x di)) (eofkd/cn/app D1' D2')) + %<- (subst-eofkd DA DA' DC D1 D1') + %<- (subst-eofkd DA DA' DC D2 D2') %. +%term _ + %pi (subst-eofkd DA DA' DC ([x] [di isvar x I] eofkd/sgm-ext (D1 x di) (D2 x di)) (eofkd/sgm-ext D1' D2')) + %<- (subst-eofkd DA DA' DC D1 D1') + %<- (subst-eofkd DA DA' DC D2 D2') %. +%term _ + %pi (subst-eofkd DA DA' DC ([x] [di isvar x I] eofkd/tp/cross (D1 x di) (D2 x di)) (eofkd/tp/cross D1' D2')) + %<- (subst-eofkd DA DA' DC D1 D1') + %<- (subst-eofkd DA DA' DC D2 D2') %. +%term _ + %pi (subst-eofkd DA DA' DC ([x] [di isvar x I] eofkd/tp/arrow (D1 x di) (D2 x di)) (eofkd/tp/arrow D1' D2')) + %<- (subst-eofkd DA DA' DC D1 D1') + %<- (subst-eofkd DA DA' DC D2 D2') %. +%term _ + %pi (subst-eofkd DA DA' DC ([x] [di isvar x I] eofkd/tp/sum (D1 x di) (D2 x di)) (eofkd/tp/sum D1' D2')) + %<- (subst-eofkd DA DA' DC D1 D1') + %<- (subst-eofkd DA DA' DC D2 D2') %. +%term _ + %pi (subst-eofkd DA DA' DC ([x] [di isvar x I] eofkd/sub (D1 x di) (D2 x di)) (eofkd/sub D1' D2')) + %<- (subst-eofkd DA DA' DC D1 D1') + %<- (subst-ekd-sub DA DA' DC D2 D2') %. +%term _ + %pi (subst-eofkd DA DA' DC ([x] [di isvar x I] eofkd/deq (D1 x di) (D2 x di)) (eofkd/deq D1' D2')) + %<- (subst-eofkd DA DA' DC D1 D1') + %<- (subst-ekd-deq DA DA' DC D2 D2') %. +%term _ + %pi (subst-eofkd DA DA' (%the (eofkd G1 C K) DC) ([x] [di isvar x XI] eofkd/cn/lam ([y] [dy isvar y YI] D2 x di y dy) (D1 x di)) (eofkd/cn/lam D2' D1')) + %<- (subst-ekd-wf DA DA' DC D1 D1') + %<- ({y} {dy isvar y YI} {_ isvar-fun dy dy seq/loc/refl} subst-eofkd ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC ([x] [di isvar x XI] D2 x di y dy) (D2' y dy)) %. +%term _ + %pi (subst-eofkd DA DA' (%the (eofkd G1 C K) DC) ([x] [di isvar x XI] eofkd/tp/forall ([y] [dy isvar y YI] D2 x di y dy) (D1 x di)) (eofkd/tp/forall D2' D1')) + %<- (subst-ekd-wf DA DA' DC D1 D1') + %<- ({y} {dy isvar y YI} {_ isvar-fun dy dy seq/loc/refl} subst-eofkd ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC ([x] [di isvar x XI] D2 x di y dy) (D2' y dy)) %. +%term _ + %pi (subst-eofkd DA DA' (%the (eofkd G1 C K) DC) ([x] [di isvar x XI] eofkd/cn/mu ([y] [dy isvar y YI] D2 x di y dy) (D1 x di)) (eofkd/cn/mu D2' D1')) + %<- (subst-ekd-wf DA DA' DC D1 D1') + %<- ({y} {dy isvar y YI} {_ isvar-fun dy dy seq/loc/refl} subst-eofkd ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC ([x] [di isvar x XI] D2 x di y dy) (D2' y dy)) %. +%term _ + %pi (subst-eofkd DA DA' (%the (eofkd _ C _) DC) ([x] [di isvar _ I] eofkd/var (DL x di)) DC') + %<- (cxt-lookup-ioc DA DA' DL IOC) + %<- (subst-cxt-lookup DA DA' DC DL IOC DC') %. +%term _ + %pi (subst-eofkd DA DA' (%the (eofkd G1 _ K) DC) ([x] [di isvar x XI] eofkd/pi-ext (D1 x di) ([y] [dy isvar y YI] D2 x di y dy)) (eofkd/pi-ext D1' D2')) + %<- (subst-eofkd DA DA' DC D1 D1') + %<- ({y} {dy isvar y YI} {_ isvar-fun dy dy seq/loc/refl} subst-eofkd ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC ([x] [di isvar x XI] D2 x di y dy) (D2' y dy)) %. +%term _ + %pi (subst-eofkd DA DA' (%the (eofkd _ C _) DC) ([x] [di isvar _ I] eofkd/closed (DC' x) (DO' x di)) (eofkd/closed (DC' C) DO)) + %<- (subst-cxt-ordered DA DA' DO' DO) %. +%term _ + %pi (subst-ekd-deq DA DA' DC ([x] [di isvar _ I] ekd-deq/kd/unit (DO' x di)) (ekd-deq/kd/unit DO)) + %<- (subst-cxt-ordered DA DA' DO' DO) %. +%term _ + %pi (subst-ekd-deq DA DA' DC ([x] [di isvar _ I] ekd-deq/kd/type (DO' x di)) (ekd-deq/kd/type DO)) + %<- (subst-cxt-ordered DA DA' DO' DO) %. +%term _ + %pi (subst-ekd-deq DA DA' DC ([x] [di isvar x I] ekd-deq/kd/sing (D1 x di)) (ekd-deq/kd/sing D1')) + %<- (subst-ecn-deq DA DA' DC D1 D1') %. +%term _ + %pi (subst-ekd-deq DA DA' (%the (eofkd G1 C K) DC) ([x] [di isvar x XI] ekd-deq/kd/pi (D1 x di) ([y] [dy isvar y YI] D2 x di y dy)) (ekd-deq/kd/pi D1' D2')) + %<- (subst-ekd-deq DA DA' DC D1 D1') + %<- ({y} {dy isvar y YI} {_ isvar-fun dy dy seq/loc/refl} subst-ekd-deq ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC ([x] [di isvar x XI] D2 x di y dy) (D2' y dy)) %. +%term _ + %pi (subst-ekd-deq DA DA' (%the (eofkd G1 C K) DC) ([x] [di isvar x XI] ekd-deq/kd/sgm (D1 x di) ([y] [dy isvar y YI] D2 x di y dy)) (ekd-deq/kd/sgm D1' D2')) + %<- (subst-ekd-deq DA DA' DC D1 D1') + %<- ({y} {dy isvar y YI} {_ isvar-fun dy dy seq/loc/refl} subst-ekd-deq ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC ([x] [di isvar x XI] D2 x di y dy) (D2' y dy)) %. +%term _ + %pi (subst-ekd-sub DA DA' DC ([x] [di isvar _ I] ekd-sub/kd/unit (DO' x di)) (ekd-sub/kd/unit DO)) + %<- (subst-cxt-ordered DA DA' DO' DO) %. +%term _ + %pi (subst-ekd-sub DA DA' DC ([x] [di isvar _ I] ekd-sub/kd/type (DO' x di)) (ekd-sub/kd/type DO)) + %<- (subst-cxt-ordered DA DA' DO' DO) %. +%term _ + %pi (subst-ekd-sub DA DA' DC ([x] [di isvar x I] ekd-sub/kd/sing-kd/sing (D1 x di)) (ekd-sub/kd/sing-kd/sing D1')) + %<- (subst-ecn-deq DA DA' DC D1 D1') %. +%term _ + %pi (subst-ekd-sub DA DA' DC ([x] [di isvar x I] ekd-sub/kd/sing-kd/type (D1 x di)) (ekd-sub/kd/sing-kd/type D1')) + %<- (subst-eofkd DA DA' DC D1 D1') %. +%term _ + %pi (subst-ekd-sub DA DA' (%the (eofkd G1 C K) DC) ([x] [di isvar x XI] ekd-sub/kd/pi (D1 x di) ([y] [dy isvar y YI] D2 x di y dy) ([y] [dy isvar y YI] D3 x di y dy)) (ekd-sub/kd/pi D1' D2' D3')) + %<- (subst-ekd-sub DA DA' DC D1 D1') + %<- ({y} {dy isvar y YI} {_ isvar-fun dy dy seq/loc/refl} subst-ekd-sub ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC ([x] [di isvar x XI] D2 x di y dy) (D2' y dy)) + %<- ({y} {dy isvar y YI} {_ isvar-fun dy dy seq/loc/refl} subst-ekd-wf ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC ([x] [di isvar x XI] D3 x di y dy) (D3' y dy)) %. +%term _ + %pi (subst-ekd-sub DA DA' (%the (eofkd G1 C K) DC) ([x] [di isvar x XI] ekd-sub/kd/sgm (D1 x di) ([y] [dy isvar y YI] D2 x di y dy) ([y] [dy isvar y YI] D3 x di y dy)) (ekd-sub/kd/sgm D1' D2' D3')) + %<- (subst-ekd-sub DA DA' DC D1 D1') + %<- ({y} {dy isvar y YI} {_ isvar-fun dy dy seq/loc/refl} subst-ekd-sub ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC ([x] [di isvar x XI] D2 x di y dy) (D2' y dy)) + %<- ({y} {dy isvar y YI} {_ isvar-fun dy dy seq/loc/refl} subst-ekd-wf ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC ([x] [di isvar x XI] D3 x di y dy) (D3' y dy)) %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar _ I] ecn-deq/cn/unit (DO' x di)) (ecn-deq/cn/unit DO)) + %<- (subst-cxt-ordered DA DA' DO' DO) %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar _ I] ecn-deq/tp/unit (DO' x di)) (ecn-deq/tp/unit DO)) + %<- (subst-cxt-ordered DA DA' DO' DO) %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar _ I] ecn-deq/tp/tagged (DO' x di)) (ecn-deq/tp/tagged DO)) + %<- (subst-cxt-ordered DA DA' DO' DO) %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x I] ecn-deq/kd/sing (D1 x di)) (ecn-deq/kd/sing D1')) + %<- (subst-eofkd DA DA' DC D1 D1') %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x I] ecn-deq/refl (D1 x di)) (ecn-deq/refl D1')) + %<- (subst-eofkd DA DA' DC D1 D1') %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x I] ecn-deq/sym (D1 x di)) (ecn-deq/sym D1')) + %<- (subst-ecn-deq DA DA' DC D1 D1') %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x I] ecn-deq/cn/pj1 (D1 x di)) (ecn-deq/cn/pj1 D1')) + %<- (subst-ecn-deq DA DA' DC D1 D1') %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x I] ecn-deq/cn/pj2 (D1 x di)) (ecn-deq/cn/pj2 D1')) + %<- (subst-ecn-deq DA DA' DC D1 D1') %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x I] ecn-deq/tp/ref (D1 x di)) (ecn-deq/tp/ref D1')) + %<- (subst-ecn-deq DA DA' DC D1 D1') %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x I] ecn-deq/tp/tag (D1 x di)) (ecn-deq/tp/tag D1')) + %<- (subst-ecn-deq DA DA' DC D1 D1') %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x I] ecn-deq/kd/unit (D1 x di) (D2 x di)) (ecn-deq/kd/unit D1' D2')) + %<- (subst-eofkd DA DA' DC D1 D1') + %<- (subst-eofkd DA DA' DC D2 D2') %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x I] ecn-deq/trans (D1 x di) (D2 x di)) (ecn-deq/trans D1' D2')) + %<- (subst-ecn-deq DA DA' DC D1 D1') + %<- (subst-ecn-deq DA DA' DC D2 D2') %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x I] ecn-deq/cn/pair (D1 x di) (D2 x di)) (ecn-deq/cn/pair D1' D2')) + %<- (subst-ecn-deq DA DA' DC D1 D1') + %<- (subst-ecn-deq DA DA' DC D2 D2') %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x I] ecn-deq/cn/app (D1 x di) (D2 x di)) (ecn-deq/cn/app D1' D2')) + %<- (subst-ecn-deq DA DA' DC D1 D1') + %<- (subst-ecn-deq DA DA' DC D2 D2') %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x I] ecn-deq/sgm-ext (D1 x di) (D2 x di)) (ecn-deq/sgm-ext D1' D2')) + %<- (subst-ecn-deq DA DA' DC D1 D1') + %<- (subst-ecn-deq DA DA' DC D2 D2') %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x I] ecn-deq/tp/cross (D1 x di) (D2 x di)) (ecn-deq/tp/cross D1' D2')) + %<- (subst-ecn-deq DA DA' DC D1 D1') + %<- (subst-ecn-deq DA DA' DC D2 D2') %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x I] ecn-deq/tp/arrow (D1 x di) (D2 x di)) (ecn-deq/tp/arrow D1' D2')) + %<- (subst-ecn-deq DA DA' DC D1 D1') + %<- (subst-ecn-deq DA DA' DC D2 D2') %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x I] ecn-deq/tp/sum (D1 x di) (D2 x di)) (ecn-deq/tp/sum D1' D2')) + %<- (subst-ecn-deq DA DA' DC D1 D1') + %<- (subst-ecn-deq DA DA' DC D2 D2') %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x I] ecn-deq/sub (D1 x di) (D2 x di)) (ecn-deq/sub D1' D2')) + %<- (subst-ecn-deq DA DA' DC D1 D1') + %<- (subst-ekd-sub DA DA' DC D2 D2') %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x I] ecn-deq/deq (D1 x di) (D2 x di)) (ecn-deq/deq D1' D2')) + %<- (subst-ecn-deq DA DA' DC D1 D1') + %<- (subst-ekd-deq DA DA' DC D2 D2') %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x XI] ecn-deq/cn/lam ([y] [dy isvar y YI] D2 x di y dy) (D1 x di)) (ecn-deq/cn/lam D2' D1')) + %<- (subst-ekd-deq DA DA' DC D1 D1') + %<- ({y} {dy isvar y YI} {_ isvar-fun dy dy seq/loc/refl} subst-ecn-deq ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC ([x] [di isvar x XI] D2 x di y dy) (D2' y dy)) %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x XI] ecn-deq/tp/forall ([y] [dy isvar y YI] D2 x di y dy) (D1 x di)) (ecn-deq/tp/forall D2' D1')) + %<- (subst-ekd-deq DA DA' DC D1 D1') + %<- ({y} {dy isvar y YI} {_ isvar-fun dy dy seq/loc/refl} subst-ecn-deq ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC ([x] [di isvar x XI] D2 x di y dy) (D2' y dy)) %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x XI] ecn-deq/cn/mu ([y] [dy isvar y YI] D2 x di y dy) (D1 x di)) (ecn-deq/cn/mu D2' D1')) + %<- (subst-ekd-deq DA DA' DC D1 D1') + %<- ({y} {dy isvar y YI} {_ isvar-fun dy dy seq/loc/refl} subst-ecn-deq ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC ([x] [di isvar x XI] D2 x di y dy) (D2' y dy)) %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x XI] ecn-deq/pi-ext (D1 x di) (D1a x di) ([y] [dy isvar y YI] D2 x di y dy)) (ecn-deq/pi-ext D1' D1a' D2')) + %<- (subst-eofkd DA DA' DC D1 D1') + %<- (subst-eofkd DA DA' DC D1a D1a') + %<- ({y} {dy isvar y YI} {_ isvar-fun dy dy seq/loc/refl} subst-ecn-deq ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC ([x] [di isvar x XI] D2 x di y dy) (D2' y dy)) %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x XI] ecn-deq/pi-ext (D1 x di) (D1a x di) ([y] [dy isvar y YI] D2 x di y dy)) (ecn-deq/pi-ext D1' D1a' D2')) + %<- (subst-eofkd DA DA' DC D1 D1') + %<- (subst-eofkd DA DA' DC D1a D1a') + %<- ({y} {dy isvar y YI} {_ isvar-fun dy dy seq/loc/refl} subst-ecn-deq ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC ([x] [di isvar x XI] D2 x di y dy) (D2' y dy)) %. +%term _ + %pi (subst-ecn-deq DA DA' DC ([x] [di isvar x XI] ecn-deq/pi-ext-2 (D1 x di) ([y] [dy isvar y YI] D2 x di y dy)) (ecn-deq/pi-ext-2 D1' D2')) + %<- (subst-ecn-deq DA DA' DC D1 D1') + %<- ({y} {dy isvar y YI} {_ isvar-fun dy dy seq/loc/refl} subst-ecn-deq ([x] cxt-append/cons (DA x)) (cxt-append-sub/cons DA') DC ([x] [di isvar x XI] D2 x di y dy) (D2' y dy)) %. +%worlds (cn-block isvar-fun-block can-mofkd-block) (subst-eofkd _ _ _ _ _) (subst-ekd-wf _ _ _ _ _) (subst-ekd-sub _ _ _ _ _) (subst-ekd-deq _ _ _ _ _) (subst-ecn-deq _ _ _ _ _) %. +%total (D1 D2 D3 D4 D5) (subst-ekd-wf _ _ _ D2 _) (subst-eofkd _ _ _ D1 _) (subst-ekd-deq _ _ _ D4 _) (subst-ekd-sub _ _ _ D3 _) (subst-ecn-deq _ _ _ D5 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/substitution-mods.lf b/new-tests/stelf-output/tslf/substitution-mods.lf new file mode 100644 index 0000000..20cbbdf --- /dev/null +++ b/new-tests/stelf-output/tslf/substitution-mods.lf @@ -0,0 +1,136 @@ +%{%block assm/md-block : some {S1:sg}{K1:kd}{_:fst-sg S1 K1} + {s:md}{a:cn}{da: ofkd a K1}{dm: fst-md s a} + block + {ds: assm/md s S1}. +}% +%sort subst/md-ofsg^ %. +%mode subst/md-ofsg^ %in %in %out %. +%sort subst/md-oftp^ %. +%mode subst/md-oftp^ %in %in %out %. +%term _ subst/md-ofsg^ ([das] ofsg/var das) D1 D1 %. +%term _ subst/md-ofsg^ ([das] D) D1 D %. +%term _ + %pi (subst/md-ofsg^ ([das] ofsg/md/tm (DA das)) D1 (ofsg/md/tm DA')) + %<- (subst/md-oftp^ DA D1 DA') %. +%term _ + %pi (subst/md-ofsg^ ([das] ofsg/md/pair (DA das) (DB das)) D1 (ofsg/md/pair DA' DB')) + %<- (subst/md-ofsg^ DA D1 DA') + %<- (subst/md-ofsg^ DB D1 DB') %. +%term _ + %pi (subst/md-ofsg^ ([das] ofsg/md/pj1 (DA das)) D1 (ofsg/md/pj1 DA')) + %<- (subst/md-ofsg^ DA D1 DA') %. +%term _ + %pi (subst/md-ofsg^ ([das] ofsg/md/pj2 (DA das) DC) D1 (ofsg/md/pj2 DA' DC)) + %<- (subst/md-ofsg^ DA D1 DA') %. +%term _ + %pi (subst/md-ofsg^ ([das] ofsg/md/lam (D0 das) DW DFS) D1 (ofsg/md/lam D0' DW DFS)) + %<- ({s} {ds} {a} {da} {dm} subst/md-ofsg^ ([das] D0 das s ds a da dm) D1 (D0' s ds a da dm)) %. +%term _ + %pi (subst/md-ofsg^ ([das] ofsg/md/app (DA das) (DB das) DC) D1 (ofsg/md/app DA' DB' DC)) + %<- (subst/md-ofsg^ DA D1 DA') + %<- (subst/md-ofsg^ DB D1 DB') %. +%term _ + %pi (subst/md-ofsg^ ([das] ofsg/md/let (DZ das) (D0 das) DW DFS) D1 (ofsg/md/let DZ' D0' DW DFS)) + %<- (subst/md-ofsg^ DZ D1 DZ') + %<- ({s} {ds} {a} {da} {dm} subst/md-ofsg^ ([das] D0 das s ds a da dm) D1 (D0' s ds a da dm)) %. +%term _ + %pi (subst/md-ofsg^ ([das] ofsg/md/seal (DA das)) D1 (ofsg/md/seal DA')) + %<- (subst/md-ofsg^ DA D1 DA') %. +%term _ + %pi (subst/md-ofsg^ ([das] ofsg/sgm-ext (DA das) (DB das)) D1 (ofsg/sgm-ext DA' DB')) + %<- (subst/md-ofsg^ DA D1 DA') + %<- (subst/md-ofsg^ DB D1 DB') %. +%term _ + %pi (subst/md-ofsg^ ([das] ofsg/kd-ext (DA das) DM DP) D1 (ofsg/kd-ext DA' DM DP)) + %<- (subst/md-ofsg^ DA D1 DA') %. +%term _ + %pi (subst/md-ofsg^ ([das] ofsg/sub (DA das) DS DP) D1 (ofsg/sub DA' DS DP)) + %<- (subst/md-ofsg^ DA D1 DA') %. +%term _ subst/md-oftp^ ([das] D) D1 D %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/tm/pair (D1 das) (D2 das)) D (%the (oftp L T (tm/pair E1 E2) (tp/cross C1 C2)) (oftp/tm/pair D1' D2'))) + %<- (subst/md-oftp^ D1 D (%the (oftp L T E1 C1) D1')) + %<- (subst/md-oftp^ D2 D (%the (oftp L T E2 C2) D2')) %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/tm/pj1 (D1 das)) D (oftp/tm/pj1 D1')) + %<- (subst/md-oftp^ D1 D D1') %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/tm/pj2 (D1 das)) D (oftp/tm/pj2 D1')) + %<- (subst/md-oftp^ D1 D D1') %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/tm/fun (DE das) DC) D (oftp/tm/fun DE' DC)) + %<- ({y} {dy} {z} {dz} subst/md-oftp^ ([das] DE das y dy z dz) D (DE' y dy z dz)) %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/tm/tmapp (D1 das) (D2 das)) D (oftp/tm/tmapp D1' D2')) + %<- (subst/md-oftp^ D1 D D1') + %<- (subst/md-oftp^ D2 D D2') %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/tm/cnabs ([a] [da ofkd a K] D1 das a da) (%the (kd-wf K) DW)) D (oftp/tm/cnabs D1' DW)) + %<- ({a cn} {da ofkd a K} subst/md-oftp^ ([das] D1 das a da) D (D1' a da)) %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/tm/cnapp (D1 das) DC) D (oftp/tm/cnapp D1' DC)) + %<- (subst/md-oftp^ D1 D D1') %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/tm/set (D1 das) (D2 das)) D (oftp/tm/set D1' D2')) + %<- (subst/md-oftp^ D1 D D1') + %<- (subst/md-oftp^ D2 D D2') %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/tm/ref (D1 das)) D (oftp/tm/ref D1')) + %<- (subst/md-oftp^ D1 D D1') %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/tm/raise (D1 das) DC) D (oftp/tm/raise D1' DC)) + %<- (subst/md-oftp^ D1 D D1') %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/tm/get (D1 das)) D (oftp/tm/get D1')) + %<- (subst/md-oftp^ D1 D D1') %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/deq (D1 das) DQ) D (oftp/deq D1' DQ)) + %<- (subst/md-oftp^ D1 D D1') %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/tm/term (D1 das)) D (oftp/tm/term D1')) + %<- (subst/md-ofsg^ D1 D D1') %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/tm/inl (D1 das) DC) D (oftp/tm/inl D1' DC)) + %<- (subst/md-oftp^ D1 D D1') %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/tm/inr (D1 das) DC) D (oftp/tm/inr D1' DC)) + %<- (subst/md-oftp^ D1 D D1') %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/tm/case (DO das) (DL das) (DR das)) D (oftp/tm/case DO' DL' DR')) + %<- (subst/md-oftp^ DO D DO') + %<- ({y} {dy} subst/md-oftp^ ([das] DL das y dy) D (DL' y dy)) + %<- ({y} {dy} subst/md-oftp^ ([das] DR das y dy) D (DR' y dy)) %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/tm/try (D1 das) (D2 das)) D (oftp/tm/try D1' D2')) + %<- (subst/md-oftp^ D1 D D1') + %<- ({y} {dy} subst/md-oftp^ ([das] D2 das y dy) D (D2' y dy)) %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/tm/tag (D1 das) (D2 das)) D (oftp/tm/tag D1' D2')) + %<- (subst/md-oftp^ D1 D D1') + %<- (subst/md-oftp^ D2 D D2') %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/tm/iftag (D1 das) (D2 das) (D3 das) (D4 das)) D (oftp/tm/iftag D1' D2' D3' D4')) + %<- (subst/md-oftp^ D1 D D1') + %<- (subst/md-oftp^ D2 D D2') + %<- ({y} {dy} subst/md-oftp^ ([das] D3 das y dy) D (D3' y dy)) + %<- (subst/md-oftp^ D4 D D4') %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/tm/roll (D1 das) DC) D (oftp/tm/roll D1' DC)) + %<- (subst/md-oftp^ D1 D D1') %. +%term _ + %pi (subst/md-oftp^ ([das] oftp/tm/unroll (D1 das)) D (oftp/tm/unroll D1')) + %<- (subst/md-oftp^ D1 D D1') %. +%worlds (ofsg-block ofkd-block oftp-block) (subst/md-ofsg^ _ _ _) (subst/md-oftp^ _ _ _) %. +%total (D1 D2) (subst/md-ofsg^ D1 _ _) (subst/md-oftp^ D2 _ _) %. +%sort subst/md-ofsg {s} {ds assm/md s S'} {a} {da ofkd a K'} {dm fst-md s a} %. +%mode subst/md-ofsg %in %in %in %in %out %. +%term _ + %pi (subst/md-ofsg D1 D2 DS DM D') + %<- (vdt/ofsg/p D2 _ DS' DM' DC') + %<- (fst-sg-seq DS' DS DQS) + %<- (fst-md-seq DM' DM DQM) + %<- (ofkd/seq-k DQS DC' DC'') + %<- (ofkd/seq-c DQM DC'' DC''') + %<- (subst/md-ofsg^ ([dassm assm/md M' S'] D1 M' dassm C' DC''' DM) D2 D') %. +%worlds () (subst/md-ofsg _ _ _ _ _) %. +%total {} (subst/md-ofsg _ _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/substitution-terms.lf b/new-tests/stelf-output/tslf/substitution-terms.lf new file mode 100644 index 0000000..25c82dc --- /dev/null +++ b/new-tests/stelf-output/tslf/substitution-terms.lf @@ -0,0 +1,120 @@ +%%%% substitution for terms +%sort subst/tm-oftp {x} {dx assm/tm x C1} %. +%mode subst/tm-oftp %in %in %out %. +%sort subst/tm-ofsg {x} {dx assm/tm x C1} %. +%mode subst/tm-ofsg %in %in %out %. +%term _ subst/tm-oftp ([x] [dx] oftp/var dx) D1 D1 %. +%term _ subst/tm-oftp ([x] [dx] D) D1 D %. +%term _ + %pi (subst/tm-oftp ([x] [dx] oftp/tm/pair (D1 x dx) (D2 x dx)) D (oftp/tm/pair D1' D2')) + %<- (subst/tm-oftp D1 D D1') + %<- (subst/tm-oftp D2 D D2') %. +%term _ + %pi (subst/tm-oftp ([x] [dx] oftp/tm/pj1 (D1 x dx)) D (oftp/tm/pj1 D1')) + %<- (subst/tm-oftp D1 D D1') %. +%term _ + %pi (subst/tm-oftp ([x] [dx] oftp/tm/pj2 (D1 x dx)) D (oftp/tm/pj2 D1')) + %<- (subst/tm-oftp D1 D D1') %. +%term _ + %pi (subst/tm-oftp ([x] [dx] oftp/tm/fun (DE x dx) DC) D1 (oftp/tm/fun DE' DC)) + %<- ({y} {dy} {z} {dz} subst/tm-oftp ([x] [dx] DE x dx y dy z dz) D1 (DE' y dy z dz)) %. +%term _ + %pi (subst/tm-oftp ([x] [dx] oftp/tm/tmapp (D1 x dx) (D2 x dx)) D (oftp/tm/tmapp D1' D2')) + %<- (subst/tm-oftp D1 D D1') + %<- (subst/tm-oftp D2 D D2') %. +%term _ + %pi (subst/tm-oftp ([x] [dx assm/tm x C] oftp/tm/cnabs ([a] [da ofkd a K] D1 x dx a da) (%the (kd-wf K) DW)) D (oftp/tm/cnabs D1' DW)) + %<- ({a cn} {da ofkd a K} subst/tm-oftp ([x] [dx assm/tm x C] D1 x dx a da) D (D1' a da)) %. +%term _ + %pi (subst/tm-oftp ([x] [dx] oftp/tm/cnapp (D1 x dx) DC) D (oftp/tm/cnapp D1' DC)) + %<- (subst/tm-oftp D1 D D1') %. +%term _ + %pi (subst/tm-oftp ([x] [dx] oftp/tm/set (D1 x dx) (D2 x dx)) D (oftp/tm/set D1' D2')) + %<- (subst/tm-oftp D1 D D1') + %<- (subst/tm-oftp D2 D D2') %. +%term _ + %pi (subst/tm-oftp ([x] [dx] oftp/tm/ref (D1 x dx)) D (oftp/tm/ref D1')) + %<- (subst/tm-oftp D1 D D1') %. +%term _ + %pi (subst/tm-oftp ([x] [dx] oftp/tm/raise (D1 x dx) DC) D (oftp/tm/raise D1' DC)) + %<- (subst/tm-oftp D1 D D1') %. +%term _ + %pi (subst/tm-oftp ([x] [dx] oftp/tm/get (D1 x dx)) D (oftp/tm/get D1')) + %<- (subst/tm-oftp D1 D D1') %. +%term _ + %pi (subst/tm-oftp ([x] [dx] oftp/deq (D1 x dx) DQ) D (oftp/deq D1' DQ)) + %<- (subst/tm-oftp D1 D D1') %. +%term _ + %pi (subst/tm-oftp ([x] [dx] oftp/tm/term (D1 x dx)) D (oftp/tm/term D1')) + %<- (subst/tm-ofsg D1 D D1') %. +%term _ + %pi (subst/tm-oftp ([x] [dx] oftp/tm/inl (D1 x dx) DC) D (oftp/tm/inl D1' DC)) + %<- (subst/tm-oftp D1 D D1') %. +%term _ + %pi (subst/tm-oftp ([x] [dx] oftp/tm/inr (D1 x dx) DC) D (oftp/tm/inr D1' DC)) + %<- (subst/tm-oftp D1 D D1') %. +%term _ + %pi (subst/tm-oftp ([x] [dx] oftp/tm/case (DO x dx) (DL x dx) (DR x dx)) D1 (oftp/tm/case DO' DL' DR')) + %<- (subst/tm-oftp DO D1 DO') + %<- ({y} {dy} subst/tm-oftp ([x] [dx] DL x dx y dy) D1 (DL' y dy)) + %<- ({y} {dy} subst/tm-oftp ([x] [dx] DR x dx y dy) D1 (DR' y dy)) %. +%term _ + %pi (subst/tm-oftp ([x] [dx] oftp/tm/try (D1 x dx) (D2 x dx)) D (oftp/tm/try D1' D2')) + %<- (subst/tm-oftp D1 D D1') + %<- ({y} {dy} subst/tm-oftp ([x] [dx] D2 x dx y dy) D (D2' y dy)) %. +%term _ + %pi (subst/tm-oftp ([x] [dx] oftp/tm/tag (D1 x dx) (D2 x dx)) D (oftp/tm/tag D1' D2')) + %<- (subst/tm-oftp D1 D D1') + %<- (subst/tm-oftp D2 D D2') %. +%term _ + %pi (subst/tm-oftp ([x] [dx] oftp/tm/iftag (D1 x dx) (D2 x dx) (D3 x dx) (D4 x dx)) D (oftp/tm/iftag D1' D2' D3' D4')) + %<- (subst/tm-oftp D1 D D1') + %<- (subst/tm-oftp D2 D D2') + %<- ({y} {dy} subst/tm-oftp ([x] [dx] D3 x dx y dy) D (D3' y dy)) + %<- (subst/tm-oftp D4 D D4') %. +%term _ + %pi (subst/tm-oftp ([x] [dx] oftp/tm/roll (D1 x dx) DC) D (oftp/tm/roll D1' DC)) + %<- (subst/tm-oftp D1 D D1') %. +%term _ + %pi (subst/tm-oftp ([x] [dx] oftp/tm/unroll (D1 x dx)) D (oftp/tm/unroll D1')) + %<- (subst/tm-oftp D1 D D1') %. +%term _ subst/tm-ofsg ([x] [dx] D) D1 D %. +%term _ + %pi (subst/tm-ofsg ([x] [dx] ofsg/md/tm (D x dx)) D1 (ofsg/md/tm D')) + %<- (subst/tm-oftp D D1 D') %. +%term _ + %pi (subst/tm-ofsg ([x] [dx] ofsg/md/pair (D1 x dx) (D2 x dx)) D (ofsg/md/pair D1' D2')) + %<- (subst/tm-ofsg D1 D D1') + %<- (subst/tm-ofsg D2 D D2') %. +%term _ + %pi (subst/tm-ofsg ([x] [dx] ofsg/md/pj1 (D1 x dx)) D (ofsg/md/pj1 D1')) + %<- (subst/tm-ofsg D1 D D1') %. +%term _ + %pi (subst/tm-ofsg ([x] [dx] ofsg/md/pj2 (D1 x dx) ((%the ({x} fst-md (M x) C) DF) x)) D (ofsg/md/pj2 D1' (DF E))) + %<- (subst/tm-ofsg D1 D D1') %. +%term _ + %pi (subst/tm-ofsg ([x] [dx] ofsg/md/lam (DM x dx) DW FS) D (ofsg/md/lam DM' DW FS)) + %<- ({s} {ds} {a} {da ofkd a K1} {df fst-md s a} subst/tm-ofsg ([x] [dx] DM x dx s ds a da df) D (DM' s ds a da df)) %. +%term _ + %pi (subst/tm-ofsg ([x] [dx] ofsg/md/app (D1 x dx) (D2 x dx) ((%the ({x} fst-md (M x) C) DF) x)) D (ofsg/md/app D1' D2' (DF E))) + %<- (subst/tm-ofsg D1 D D1') + %<- (subst/tm-ofsg D2 D D2') %. +%term _ + %pi (subst/tm-ofsg ([x] [dx] ofsg/md/let (D1 x dx) (D2 x dx) DS DF) D (ofsg/md/let D1' D2' DS DF)) + %<- (subst/tm-ofsg D1 D D1') + %<- ({s} {ds} {a} {da ofkd a K1} {df fst-md s a} subst/tm-ofsg ([x] [dx] D2 x dx s ds a da df) D (D2' s ds a da df)) %. +%term _ + %pi (subst/tm-ofsg ([x] [dx] ofsg/md/seal (D1 x dx)) D (ofsg/md/seal D1')) + %<- (subst/tm-ofsg D1 D D1') %. +%term _ + %pi (subst/tm-ofsg ([x] [dx] ofsg/sgm-ext (D1 x dx) (D2 x dx)) D (ofsg/sgm-ext D1' D2')) + %<- (subst/tm-ofsg D1 D D1') + %<- (subst/tm-ofsg D2 D D2') %. +%term _ + %pi (subst/tm-ofsg ([x] [dx] ofsg/kd-ext (D1 x dx) ((%the ({x} fst-md (M x) C) DF) x) DC) D (ofsg/kd-ext D1' (DF E) DC)) + %<- (subst/tm-ofsg D1 D D1') %. +%term _ + %pi (subst/tm-ofsg ([x] [dx] ofsg/sub (D1 x dx) DS DP) D (ofsg/sub D1' DS DP)) + %<- (subst/tm-ofsg D1 D D1') %. +%worlds (oftp-block ofkd-block ofsg-block) (subst/tm-oftp _ _ _) (subst/tm-ofsg _ _ _) %. +%total (D1 D2) (subst/tm-oftp D1 _ _) (subst/tm-ofsg D2 _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/symmetry-kinds.lf b/new-tests/stelf-output/tslf/symmetry-kinds.lf new file mode 100644 index 0000000..127f62b --- /dev/null +++ b/new-tests/stelf-output/tslf/symmetry-kinds.lf @@ -0,0 +1,22 @@ +%%%% description here +%sort kd-sym %. +%mode kd-sym %in %out %. +%term _ kd-sym kd-deq/kd/unit kd-deq/kd/unit %. +%term _ kd-sym kd-deq/kd/type kd-deq/kd/type %. +%term _ kd-sym (kd-deq/kd/sing D1) (kd-deq/kd/sing (cn-deq/sym D1)) %. +%term _ + %pi (kd-sym (kd-deq/kd/sgm D1 D2) (kd-deq/kd/sgm D3 D4')) + %<- (kd-sym D1 D3) + %<- (vdt/kd-deq D1 Do Dt) + %<- ({a cn} {da ofkd a K1'} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da Do} kd-sym (D2 a da) (D4 a da)) + %<- (kd-anti D1 Do Dt _ DS) + %<- (kd-wkn/kd-deq D4 DS D4') %. +%term _ + %pi (kd-sym (kd-deq/kd/pi D1 D2) (kd-deq/kd/pi D3 D4')) + %<- (kd-sym D1 D3) + %<- (vdt/kd-deq D1 Do Dt) + %<- ({a cn} {da ofkd a K1'} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da Do} kd-sym (D2 a da) (D4 a da)) + %<- (kd-anti D1 Do Dt _ DS) + %<- (kd-wkn/kd-deq D4 DS D4') %. +%worlds (ofkd+vdt-block) (kd-sym _ _) %. +%total (D1) (kd-sym D1 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/symmetry-sigs.lf b/new-tests/stelf-output/tslf/symmetry-sigs.lf new file mode 100644 index 0000000..9a91b83 --- /dev/null +++ b/new-tests/stelf-output/tslf/symmetry-sigs.lf @@ -0,0 +1,26 @@ +%%%% symmetry of signatures +%sort sg-sym %. +%mode sg-sym %in %out %. +%term _ sg-sym sg-deq/sg/unit sg-deq/sg/unit %. +%term _ sg-sym (sg-deq/sg/cn D) (sg-deq/sg/cn (cn-deq/sym D)) %. +%term _ %pi (sg-sym (sg-deq/sg/kd D) (sg-deq/sg/kd D')) %<- (kd-sym D D') %. +%term _ + %pi (sg-sym (sg-deq/sg/sgm D1 D2 FS) (sg-deq/sg/sgm D1' D2'' FS2)) + %<- (sg-sym D1 D1') + %<- (fst-sg-complete S2 FS2) + %<- (vdt/sg-deq D1 FS FS2 _ _ Dkeq) + %<- (vdt/kd-deq Dkeq DKK1 DKK2) + %<- ({a cn} {da ofkd a K1} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da DKK1} sg-sym (D2 a da) (D2' a da)) + %<- (kd-anti Dkeq DKK1 DKK2 _ DS') + %<- (kd-wkn/sg-deq D2' DS' D2'') %. +%term _ + %pi (sg-sym (sg-deq/sg/pi D1 D2 FS) (sg-deq/sg/pi D1' D2'' FS2)) + %<- (sg-sym D1 D1') + %<- (fst-sg-complete S2 FS2) + %<- (vdt/sg-deq D1 FS FS2 _ _ Dkeq) + %<- (vdt/kd-deq Dkeq DKK1 DKK2) + %<- ({a cn} {da ofkd a K1} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da DKK1} sg-sym (D2 a da) (D2' a da)) + %<- (kd-anti Dkeq DKK1 DKK2 _ DS') + %<- (kd-wkn/sg-deq D2' DS' D2'') %. +%worlds (ofkd+vdt-block) (sg-sym _ _) %. +%total (D3) (sg-sym D3 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/syntax-cnkd.lf b/new-tests/stelf-output/tslf/syntax-cnkd.lf new file mode 100644 index 0000000..808989d --- /dev/null +++ b/new-tests/stelf-output/tslf/syntax-cnkd.lf @@ -0,0 +1,30 @@ +%%%% Syntax for constructors and kinds %%%% +%sort cn %. +%name cn %. +%sort kd %. +%name kd %. +%term cn/pair %pi cn %-> cn %-> cn %. +%term cn/pj1 %pi cn %-> cn %. +%term cn/pj2 %pi cn %-> cn %. +%term cn/lam %pi kd %-> (%pi cn %-> cn) %-> cn %. +%term cn/app %pi cn %-> cn %-> cn %. +%term cn/mu %pi kd %-> (%pi cn %-> cn) %-> cn %. +%term tp/arrow %pi cn %-> cn %-> cn %. +%term tp/cross %pi cn %-> cn %-> cn %. +%term tp/unit cn %. +%term tp/forall %pi kd %-> (%pi cn %-> cn) %-> cn %. +%term tp/ref %pi cn %-> cn %. +%term tp/sum %pi cn %-> cn %-> cn %. +%term tp/tag %pi cn %-> cn %. +%term tp/tagged cn %. +%define cn/unit cn tp/unit %. +%%%%% +%term kd/type kd %. +%term kd/sing %pi cn %-> kd %. +%term kd/sgm %pi kd %-> (%pi cn %-> kd) %-> kd %. +%term kd/pi %pi kd %-> (%pi cn %-> kd) %-> kd %. +%define kd/unit kd kd/sing tp/unit %. +%sort seq/cn %. +%term seq/cn/refl seq/cn E E %. +%sort seq/kd %. +%term seq/kd/refl seq/kd K1 K1 %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/syntax-loc.lf b/new-tests/stelf-output/tslf/syntax-loc.lf new file mode 100644 index 0000000..a47c1e9 --- /dev/null +++ b/new-tests/stelf-output/tslf/syntax-loc.lf @@ -0,0 +1,7 @@ +%%%% Syntax for locations +%sort loc %. +%term loc/z loc %. +%term loc/s %pi loc %-> loc %. +%worlds () (loc) %. +%sort seq/loc %. +%term seq/loc/refl seq/loc L1 L1 %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/syntax-lt.lf b/new-tests/stelf-output/tslf/syntax-lt.lf new file mode 100644 index 0000000..8e7ce94 --- /dev/null +++ b/new-tests/stelf-output/tslf/syntax-lt.lf @@ -0,0 +1,6 @@ +%%%% location typing for terms +%sort lt %. +%name lt %. +%term lt/nil lt %. +%term lt/cons %pi lt %-> loc %-> cn %-> lt %. +%worlds () (lt) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/syntax-sigs.lf b/new-tests/stelf-output/tslf/syntax-sigs.lf new file mode 100644 index 0000000..db135d7 --- /dev/null +++ b/new-tests/stelf-output/tslf/syntax-sigs.lf @@ -0,0 +1,8 @@ +%%%% Syntax for signatures %%%% +%sort sg %. +%name sg %. +%term sg/unit sg %. +%term sg/kd %pi kd %-> sg %. +%term sg/cn %pi cn %-> sg %. +%term sg/pi %pi sg %-> (%pi cn %-> sg) %-> sg %. +%term sg/sgm %pi sg %-> (%pi cn %-> sg) %-> sg %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/syntax-tmmd.lf b/new-tests/stelf-output/tslf/syntax-tmmd.lf new file mode 100644 index 0000000..1c0b085 --- /dev/null +++ b/new-tests/stelf-output/tslf/syntax-tmmd.lf @@ -0,0 +1,43 @@ +%%%% Syntax for terms and modules %%%% +%sort tm %. +%name tm %. +%sort md %. +%name md %. +%term tm/unit tm %. +%term tm/pair %pi tm %-> tm %-> tm %. +%term tm/pj1 %pi tm %-> tm %. +%term tm/pj2 %pi tm %-> tm %. +%term tm/fun %pi cn %-> cn %-> (%pi tm %-> tm %-> tm) %-> tm %. +%term tm/tmapp %pi tm %-> tm %-> tm %. +%term tm/cnabs %pi kd %-> (%pi cn %-> tm) %-> tm %. +%term tm/cnapp %pi tm %-> cn %-> tm %. +%term tm/term %pi md %-> tm %. +%term tm/loc %pi loc %-> tm %. +%term tm/ref %pi tm %-> tm %. +%term tm/set %pi tm %-> tm %-> tm %. +%term tm/get %pi tm %-> tm %. +%term tm/inl %pi cn %-> tm %-> tm %. +%term tm/inr %pi cn %-> tm %-> tm %. +%term tm/case %pi tm %-> (%pi tm %-> tm) %-> (%pi tm %-> tm) %-> tm %. +%term tm/raise %pi tm %-> tm %. +%term tm/try %pi tm %-> (%pi tm %-> tm) %-> tm %. +%term tm/new-tag %pi cn %-> tm %. +%term tm/tagloc %pi loc %-> tm %. +%term tm/tag %pi tm %-> tm %-> tm %. +%term tm/iftag %pi tm %-> tm %-> (%pi tm %-> tm) %-> tm %-> tm %. +%term tm/roll %pi cn %-> tm %-> tm %. +%term tm/unroll %pi tm %-> tm %. +%term md/unit md %. +%term md/cn %pi cn %-> md %. +%term md/tm %pi tm %-> md %. +%term md/pair %pi md %-> md %-> md %. +%term md/pj1 %pi md %-> md %. +%term md/pj2 %pi md %-> md %. +%term md/lam %pi sg %-> (%pi cn %-> sg) %-> (%pi md %-> cn %-> md) %-> md %. +%term md/app %pi md %-> md %-> md %. +%term md/let %pi md %-> (%pi md %-> cn %-> md) %-> sg %-> md %. +%term md/seal %pi md %-> sg %-> md %. +%sort seq/tm %. +%term seq/tm/refl seq/tm E E %. +%sort seq/md %. +%term seq/md/refl seq/md M1 M1 %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/transitivity-kinds.lf b/new-tests/stelf-output/tslf/transitivity-kinds.lf new file mode 100644 index 0000000..4078e9a --- /dev/null +++ b/new-tests/stelf-output/tslf/transitivity-kinds.lf @@ -0,0 +1,56 @@ +%%%% transitivity of kind equivalence +%sort kd-trans/deq* {K2 kd} %. +%mode kd-trans/deq* %in %in %in %out %. +%term _ kd-trans/deq* kd/unit kd-deq/kd/unit kd-deq/kd/unit kd-deq/kd/unit %. +%term _ kd-trans/deq* kd/type kd-deq/kd/type kd-deq/kd/type kd-deq/kd/type %. +%term _ kd-trans/deq* (kd/sing _) (kd-deq/kd/sing D1) (kd-deq/kd/sing D2) (kd-deq/kd/sing (cn-deq/trans D1 D2)) %. +%term _ + %pi (kd-trans/deq* (kd/sgm SK2 SK2') (kd-deq/kd/sgm D1 D1') (kd-deq/kd/sgm D2 D2') (kd-deq/kd/sgm D3 D3')) + %<- (kd-trans/deq* SK2 D1 D2 D3) + %<- (vdt/kd-deq D1 Do Dt) + %<- (kd-anti D1 Do Dt DS _) + %<- (kd-wkn/kd-deq D2' DS D2'') + %<- ({a cn} {da ofkd a K1} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da Do} kd-trans/deq* (SK2' a) (D1' a da) (D2'' a da) (D3' a da)) %. +%term _ + %pi (kd-trans/deq* (kd/pi SK2 SK2') (kd-deq/kd/pi D1 D1') (kd-deq/kd/pi D2 D2') (kd-deq/kd/pi D3 D3')) + %<- (kd-trans/deq* SK2 D2 D1 D3) + %<- (vdt/kd-deq D2 Dt Dd) + %<- (kd-anti D2 Dt Dd DS _) + %<- (kd-wkn/kd-deq D1' DS D1'') + %<- ({a cn} {da ofkd a K3} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da Dt} kd-trans/deq* (SK2' a) (D1'' a da) (D2' a da) (D3' a da)) %. +%worlds (ofkd+vdt-block) (kd-trans/deq* _ _ _ _) %. +%total {K2} (kd-trans/deq* K2 _ _ _) %. +%sort kd-trans/deq %. +%mode kd-trans/deq %in %in %out %. +%term _ %pi (kd-trans/deq D1 D2 D3) %<- (kd-trans/deq* _ D1 D2 D3) %. +%worlds (ofkd+vdt-block) (kd-trans/deq _ _ _) %. +%total {D1} (kd-trans/deq D1 _ _) %. +%%%% +%sort kd-trans/sub* {K2 kd} %. +%mode kd-trans/sub* %in %in %in %out %. +%term _ kd-trans/sub* kd/unit kd-sub/kd/unit kd-sub/kd/unit kd-sub/kd/unit %. +%term _ kd-trans/sub* kd/type kd-sub/kd/type kd-sub/kd/type kd-sub/kd/type %. +%term _ kd-trans/sub* kd/type (kd-sub/kd/sing-kd/type D1) kd-sub/kd/type (kd-sub/kd/sing-kd/type D1) %. +%term _ kd-trans/sub* (kd/sing _) (kd-sub/kd/sing-kd/sing D1) (kd-sub/kd/sing-kd/sing D2) (kd-sub/kd/sing-kd/sing (cn-deq/trans D1 D2)) %. +%term _ + %pi (kd-trans/sub* (kd/sing _) (kd-sub/kd/sing-kd/sing D1) (kd-sub/kd/sing-kd/type _) (kd-sub/kd/sing-kd/type Di)) + %<- (vdt/cn-deq D1 Di _ _) %. +%term _ + %pi (kd-trans/sub* (kd/sgm SK2 SK2') (kd-sub/kd/sgm D1 D1' DK1) (kd-sub/kd/sgm D2 D2' DK2) (kd-sub/kd/sgm D3 D3' DK2)) + %<- (kd-trans/sub* SK2 D1 D2 D3) + %<- (vdt/kd-sub D1 Do _) + %<- (kd-wkn/kd-sub D2' D1 D2'') + %<- ({a cn} {da ofkd a K1} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da Do} kd-trans/sub* (SK2' a) (D1' a da) (D2'' a da) (D3' a da)) %. +%term _ + %pi (kd-trans/sub* (kd/pi SK2 SK2') (kd-sub/kd/pi D1 D1' DK1) (kd-sub/kd/pi D2 D2' DK2) (kd-sub/kd/pi D3 D3' DK1)) + %<- (kd-trans/sub* SK2 D2 D1 D3) + %<- (vdt/kd-sub D2 Dt Dd) + %<- (kd-wkn/kd-sub D1' D2 D1'') + %<- ({a cn} {da ofkd a K3} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da Dt} kd-trans/sub* (SK2' a) (D1'' a da) (D2' a da) (D3' a da)) %. +%worlds (ofkd+vdt-block) (kd-trans/sub* _ _ _ _) %. +%total (K2) (kd-trans/sub* K2 _ _ _) %. +%sort kd-trans/sub %. +%mode kd-trans/sub %in %in %out %. +%term _ %pi (kd-trans/sub D1 D2 D3) %<- (kd-trans/sub* _ D1 D2 D3) %. +%worlds (ofkd+vdt-block) (kd-trans/sub _ _ _) %. +%total {D1} (kd-trans/sub D1 _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/transitivity-sigs.lf b/new-tests/stelf-output/tslf/transitivity-sigs.lf new file mode 100644 index 0000000..d87eeb8 --- /dev/null +++ b/new-tests/stelf-output/tslf/transitivity-sigs.lf @@ -0,0 +1,55 @@ +%%%% transitivity for signatures +%sort sg-trans/deq* {S2 sg} %. +%mode sg-trans/deq* %in %in %in %out %. +%term _ sg-trans/deq* _ sg-deq/sg/unit sg-deq/sg/unit sg-deq/sg/unit %. +%term _ + %pi (sg-trans/deq* _ (sg-deq/sg/kd KS1) (sg-deq/sg/kd KS2) (sg-deq/sg/kd KS3)) + %<- (kd-trans/deq KS1 KS2 KS3) %. +%term _ sg-trans/deq* _ (sg-deq/sg/cn DQ1) (sg-deq/sg/cn DQ2) (sg-deq/sg/cn (cn-deq/trans DQ1 DQ2)) %. +%term _ + %pi (sg-trans/deq* (sg/sgm S1 S2) (sg-deq/sg/sgm DS1' DS1'' DF1) (sg-deq/sg/sgm DS2' DS2'' DF2) (sg-deq/sg/sgm DS3' DS3'' DF1)) + %<- (sg-trans/deq* S1 DS1' DS2' DS3') + %<- (vdt/sg-deq DS1' DF1 DF2 _ _ DQ1) + %<- (vdt/kd-deq DQ1 DKW1 _) + %<- (sg-anti DS1' DF1 DF2 _ _ DKS12 _) + %<- (kd-wkn/sg-deq DS2'' DKS12 DS2''w) + %<- ({a} {da} {dm mofkd da met/unit} {_ can-mofkd da dm} {_ vdt/ofkd da DKW1} sg-trans/deq* (S2 a) (DS1'' a da) (DS2''w a da) (DS3'' a da)) %. +%term _ + %pi (sg-trans/deq* (sg/pi S1 S2) (sg-deq/sg/pi DS1' DS1'' DF1) (sg-deq/sg/pi DS2' DS2'' DF2) (sg-deq/sg/pi DS3' DS3'' DF2)) + %<- (sg-trans/deq* S1 DS2' DS1' DS3') + %<- (vdt/sg-deq DS2' DF2 DF1 _ _ DQ1) + %<- (vdt/kd-deq DQ1 DKW1 _) + %<- (sg-anti DS2' DF2 DF1 _ _ DKS32 _) + %<- (kd-wkn/sg-deq DS1'' DKS32 DS1''w) + %<- ({a} {da} {dm mofkd da met/unit} {_ can-mofkd da dm} {_ vdt/ofkd da DKW1} sg-trans/deq* (S2 a) (DS1''w a da) (DS2'' a da) (DS3'' a da)) %. +%worlds (ofkd+vdt-block) (sg-trans/deq* _ _ _ _) %. +%total {D1} (sg-trans/deq* D1 _ _ _) %. +%sort sg-trans/sub* {S2 sg} %. +%mode sg-trans/sub* %in %in %in %out %. +%term _ sg-trans/sub* _ sg-sub/sg/unit sg-sub/sg/unit sg-sub/sg/unit %. +%term _ + %pi (sg-trans/sub* _ (sg-sub/sg/kd KS1) (sg-sub/sg/kd KS2) (sg-sub/sg/kd KS3)) + %<- (kd-trans/sub KS1 KS2 KS3) %. +%term _ sg-trans/sub* _ (sg-sub/sg/cn DQ1) (sg-sub/sg/cn DQ2) (sg-sub/sg/cn (cn-deq/trans DQ1 DQ2)) %. +%term _ + %pi (sg-trans/sub* (sg/sgm S1 S2) (sg-sub/sg/sgm DS1' DS1'' DW1 DF1 _) (sg-sub/sg/sgm DS2' DS2'' DW3 DF2 DF3) (sg-sub/sg/sgm DS3' DS3'' DW3 DF1 DF3)) + %<- (sg-trans/sub* S1 DS1' DS2' DS3') + %<- (vdt/sg-sub DS1' DF1 DF2 _ _ DKS12) + %<- (vdt/kd-sub DKS12 DKW1 _) + %<- (kd-wkn/sg-sub DS2'' DKS12 DS2''w) + %<- ({a} {da} {dm mofkd da met/unit} {_ can-mofkd da dm} {_ vdt/ofkd da DKW1} sg-trans/sub* (S2 a) (DS1'' a da) (DS2''w a da) (DS3'' a da)) %. +%term _ + %pi (sg-trans/sub* (sg/pi S1 S2) (sg-sub/sg/pi DS1' DS1'' DF1) (sg-sub/sg/pi DS2' DS2'' DF2) (sg-sub/sg/pi DS3' DS3'' DF2)) + %<- (sg-trans/deq* S1 DS2' DS1' DS3') + %<- (vdt/sg-deq DS2' DF2 DF1 _ _ DQ1) + %<- (vdt/kd-deq DQ1 DKW1 _) + %<- (sg-anti DS2' DF2 DF1 _ _ DKS32 _) + %<- (kd-wkn/sg-deq DS1'' DKS32 DS1''w) + %<- ({a} {da} {dm mofkd da met/unit} {_ can-mofkd da dm} {_ vdt/ofkd da DKW1} sg-trans/deq* (S2 a) (DS1''w a da) (DS2'' a da) (DS3'' a da)) %. +%worlds (ofkd+vdt-block) (sg-trans/sub* _ _ _ _) %. +%total {D1} (sg-trans/sub* D1 _ _ _) %. +%sort sg-trans/sub %. +%mode sg-trans/sub %in %in %out %. +%term _ %pi (sg-trans/sub DS1 DS2 DS3) %<- (sg-trans/sub* _ DS1 DS2 DS3) %. +%worlds (ofkd+vdt-block) (sg-trans/sub _ _ _) %. +%total {D1} (sg-trans/sub D1 _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/uninhabited-lemmas-cnkd.lf b/new-tests/stelf-output/tslf/uninhabited-lemmas-cnkd.lf new file mode 100644 index 0000000..d1a44aa --- /dev/null +++ b/new-tests/stelf-output/tslf/uninhabited-lemmas-cnkd.lf @@ -0,0 +1,6 @@ +%%%% lemmas that take the uninhabited judgment as input. +%%%% when you are nowhere, anything is possible! +%sort uninhabited-seq/cn {S1 cn} {S2 cn} %. +%mode uninhabited-seq/cn %in %in %in %out %. +%worlds (cn-block) (uninhabited-seq/cn _ _ _ _) %. +%total {} (uninhabited-seq/cn _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/uninhabited-lemmas-ltst.lf b/new-tests/stelf-output/tslf/uninhabited-lemmas-ltst.lf new file mode 100644 index 0000000..4a904d8 --- /dev/null +++ b/new-tests/stelf-output/tslf/uninhabited-lemmas-ltst.lf @@ -0,0 +1,14 @@ +%%%% lemmas that take the uninhabited judgment as input. +%%%% when you are nowhere, anything is possible! +%sort uninhabited-oflt {S st} {L lt} {T lt} %. +%mode uninhabited-oflt %in %in %in %in %out %. +%worlds (ofsg-block oftp-block ofkd-block ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (uninhabited-oflt _ _ _ _ _) %. +%total {} (uninhabited-oflt _ _ _ _ _) %. +%sort uninhabited-lt-sub {S1 lt} {S2 lt} %. +%mode uninhabited-lt-sub %in %in %in %out %. +%worlds (ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (uninhabited-lt-sub _ _ _ _) %. +%total {} (uninhabited-lt-sub _ _ _ _) %. +%sort uninhabited-st-wf %. +%mode uninhabited-st-wf %in %in %in %in %in %out %. +%worlds (ofsg-block oftp-block ofkd-block ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (uninhabited-st-wf _ _ _ _ _ _) %. +%total {} (uninhabited-st-wf _ _ _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/uninhabited-lemmas-sigs.lf b/new-tests/stelf-output/tslf/uninhabited-lemmas-sigs.lf new file mode 100644 index 0000000..8f244a5 --- /dev/null +++ b/new-tests/stelf-output/tslf/uninhabited-lemmas-sigs.lf @@ -0,0 +1,14 @@ +%%%% lemmas that take the uninhabited judgment as input. +%%%% when you are nowhere, anything is possible! +%sort uninhabited-fst-sg {M sg} {C kd} %. +%mode uninhabited-fst-sg %in %in %in %out %. +%worlds () (uninhabited-fst-sg _ _ _ _) %. +%total {} (uninhabited-fst-sg _ _ _ _) %. +%sort uninhabited-sg-sub {S1 sg} {S2 sg} %. +%mode uninhabited-sg-sub %in %in %in %out %. +%worlds () (uninhabited-sg-sub _ _ _ _) %. +%total {} (uninhabited-sg-sub _ _ _ _) %. +%sort uninhabited-sg-wf {S1 sg} %. +%mode uninhabited-sg-wf %in %in %out %. +%worlds () (uninhabited-sg-wf _ _ _) %. +%total {} (uninhabited-sg-wf _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/uninhabited-lemmas-tmmd.lf b/new-tests/stelf-output/tslf/uninhabited-lemmas-tmmd.lf new file mode 100644 index 0000000..912e215 --- /dev/null +++ b/new-tests/stelf-output/tslf/uninhabited-lemmas-tmmd.lf @@ -0,0 +1,18 @@ +%%%% lemmas that take the uninhabited judgment as input. +%%%% when you are nowhere, anything is possible! +%sort uninhabited-ofsg {L lt} {T lt} {Y pty} {M md} {S sg} %. +%mode uninhabited-ofsg %in %in %in %in %in %in %out %. +%worlds (ofsg-block oftp-block ofkd-block ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (uninhabited-ofsg _ _ _ _ _ _ _) %. +%total {} (uninhabited-ofsg _ _ _ _ _ _ _) %. +%sort uninhabited-fst-md {M md} {C cn} %. +%mode uninhabited-fst-md %in %in %in %out %. +%worlds (ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (uninhabited-fst-md _ _ _ _) %. +%total {} (uninhabited-fst-md _ _ _ _) %. +%sort uninhabited-oftp %. +%mode uninhabited-oftp %in %in %in %in %in %out %. +%worlds (ofsg-block oftp-block ofkd-block ofsg+vdt-block oftp+vdt-block ofkd+vdt-block) (uninhabited-oftp _ _ _ _ _ _) %. +%total {} (uninhabited-oftp _ _ _ _ _ _) %. +%sort uninhabited-seq/tm {E1} {E2} %. +%mode uninhabited-seq/tm %in %in %in %out %. +%worlds (ofkd+vdt-block) (uninhabited-seq/tm _ _ _ _) %. +%total {} (uninhabited-seq/tm _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/uninhabited.lf b/new-tests/stelf-output/tslf/uninhabited.lf new file mode 100644 index 0000000..3acf4b5 --- /dev/null +++ b/new-tests/stelf-output/tslf/uninhabited.lf @@ -0,0 +1,3 @@ +%%%% the uninhabited judgment. otherwise known as falsehood or void +%sort uninhabited %. +%mode uninhabited %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/validity-cnkd.lf b/new-tests/stelf-output/tslf/validity-cnkd.lf new file mode 100644 index 0000000..cf97242 --- /dev/null +++ b/new-tests/stelf-output/tslf/validity-cnkd.lf @@ -0,0 +1,158 @@ +%%%%% Validity %%%%% +% vdt/ofkd : ofkd C1 K -> kd-wf K -> type. +% %mode vdt/ofkd +D1 -D2. +%sort vdt/kd-sub %. +%mode vdt/kd-sub %in %out %out %. +%sort vdt/kd-deq %. +%mode vdt/kd-deq %in %out %out %. +%sort vdt/cn-deq %. +%mode vdt/cn-deq %in %out %out %out %. +%term _ vdt/ofkd ofkd/cn/unit kd-wf/kd/unit %. +%term _ vdt/ofkd _ kd-wf/kd/type %. +%term _ + %pi (vdt/ofkd (ofkd/cn/pair D1 D2) (kd-wf/kd/sgm D3 ([a cn] [da ofkd a K1] D4))) + %<- (vdt/ofkd D1 D3) + %<- (vdt/ofkd D2 D4) %. +%term _ %pi (vdt/ofkd (ofkd/cn/pj1 D) DK1) %<- (vdt/ofkd D (kd-wf/kd/sgm DK1 DK2)) %. +%term _ + %pi (vdt/ofkd (ofkd/cn/pj2 D) (DK2 (cn/pj1 C) (ofkd/cn/pj1 D))) + %<- (vdt/ofkd D (kd-wf/kd/sgm DK1 DK2)) %. +%term _ + %pi (vdt/ofkd (ofkd/cn/lam DofC DKwf) (kd-wf/kd/pi DKwf DK2)) + %<- ({a cn} {da ofkd a K1} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da DKwf} vdt/ofkd (DofC a da) (DK2 a da)) %. +%term _ vdt/ofkd (ofkd/cn/mu _ DKwf) DKwf %. +%term _ + %pi (vdt/ofkd (ofkd/cn/app DofC1 DofC2) (DK2 C2 DofC2)) + %<- (vdt/ofkd DofC1 (kd-wf/kd/pi DK1 DK2)) + %<- (vdt/ofkd DofC2 D4) %. +%term _ vdt/ofkd (ofkd/kd/sing D1) (kd-wf/kd/sing D1) %. +%term _ + %pi (vdt/ofkd (ofkd/sgm-ext D1 D2) (kd-wf/kd/sgm DK1 ([a cn] [da ofkd a K1] DK2))) + %<- (vdt/ofkd D1 DK1) + %<- (vdt/ofkd D2 DK2) %. +%term _ + %pi (vdt/ofkd (ofkd/pi-ext DofC Dext) (kd-wf/kd/pi DK1 DK2)) + %<- (vdt/ofkd DofC (kd-wf/kd/pi DK1 DL)) + %<- ({a cn} {da ofkd a K1} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da DK1} vdt/ofkd (Dext a da) (DK2 a da)) %. +%term _ %pi (vdt/ofkd (ofkd/sub _ DKs) D2) %<- (vdt/kd-sub DKs D1 D2) %. +%term _ %pi (vdt/ofkd (ofkd/deq _ DKs) D2) %<- (vdt/kd-deq DKs D1 D2) %. +%term _ vdt/kd-deq kd-deq/kd/unit kd-wf/kd/unit kd-wf/kd/unit %. +%term _ vdt/kd-deq kd-deq/kd/type kd-wf/kd/type kd-wf/kd/type %. +%term _ + %pi (vdt/kd-deq (kd-deq/kd/sing D1) (kd-wf/kd/sing D2) (kd-wf/kd/sing D3)) + %<- (vdt/cn-deq D1 D2 D3 _) %. +%term _ + %pi (vdt/kd-deq (kd-deq/kd/sgm Ddeq' Ddeq'') (kd-wf/kd/sgm D1 D3) (kd-wf/kd/sgm D2 D4')) + %<- (vdt/kd-deq Ddeq' D1 D2) + %<- ({a cn} {da ofkd a K1'} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da D1} vdt/kd-deq (Ddeq'' a da) (D3 a da) (D4 a da)) + %<- (kd-anti Ddeq' D1 D2 _ DS') + %<- (kd-wkn/kd-wf D4 DS' D4') %. +%term _ + %pi (vdt/kd-deq (kd-deq/kd/pi Ddeq' Ddeq'') (kd-wf/kd/pi D1 D3') (kd-wf/kd/pi D2 D4)) + %<- (vdt/kd-deq Ddeq' D2 D1) + %<- ({a cn} {da ofkd a K2'} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da D2} vdt/kd-deq (Ddeq'' a da) (D3 a da) (D4 a da)) + %<- (kd-anti Ddeq' D2 D1 _ DS') + %<- (kd-wkn/kd-wf D3 DS' D3') %. +%term _ vdt/kd-sub kd-sub/kd/unit kd-wf/kd/unit kd-wf/kd/unit %. +%term _ vdt/kd-sub kd-sub/kd/type kd-wf/kd/type kd-wf/kd/type %. +%term _ + %pi (vdt/kd-sub (kd-sub/kd/sing-kd/sing D1) (kd-wf/kd/sing D2) (kd-wf/kd/sing D3)) + %<- (vdt/cn-deq D1 D2 D3 _) %. +%term _ vdt/kd-sub (kd-sub/kd/sing-kd/type D1) (kd-wf/kd/sing D1) kd-wf/kd/type %. +%term _ + %pi (vdt/kd-sub (kd-sub/kd/sgm Dsub' Dsub'' Dkwf) (kd-wf/kd/sgm D1 D3) (kd-wf/kd/sgm D2 Dkwf)) + %<- (vdt/kd-sub Dsub' D1 D2) + %<- ({a cn} {da ofkd a K1'} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da D1} vdt/kd-sub (Dsub'' a da) (D3 a da) (D4 a da)) %. +%term _ + %pi (vdt/kd-sub (kd-sub/kd/pi Dsub' Dsub'' Dkwf) (kd-wf/kd/pi D1 Dkwf) (kd-wf/kd/pi D2 D4)) + %<- (vdt/kd-sub Dsub' D2 D1) + %<- ({a cn} {da ofkd a K2'} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da D2} vdt/kd-sub (Dsub'' a da) (D3 a da) (D4 a da)) %. +%term _ %pi (vdt/cn-deq (cn-deq/refl D1) D1 D1 D2) %<- (vdt/ofkd D1 D2) %. +%term _ %pi (vdt/cn-deq (cn-deq/sym D1) DA DB DK) %<- (vdt/cn-deq D1 DB DA DK) %. +%term _ + %pi (vdt/cn-deq (cn-deq/trans D1 D2) DA DB DK) + %<- (vdt/cn-deq D1 DA _ DK) + %<- (vdt/cn-deq D2 _ DB _) %. +%term _ vdt/cn-deq cn-deq/cn/unit ofkd/cn/unit ofkd/cn/unit kd-wf/kd/unit %. +%term _ vdt/cn-deq cn-deq/tp/unit ofkd/tp/unit ofkd/tp/unit kd-wf/kd/type %. +%term _ vdt/cn-deq cn-deq/tp/tagged ofkd/tp/tagged ofkd/tp/tagged kd-wf/kd/type %. +%term _ + %pi (vdt/cn-deq (cn-deq/tp/cross D2 D1) (ofkd/tp/cross DA2 DA1) (ofkd/tp/cross DB2 DB1) kd-wf/kd/type) + %<- (vdt/cn-deq D1 DA1 DB1 _) + %<- (vdt/cn-deq D2 DA2 DB2 _) %. +%term _ + %pi (vdt/cn-deq (cn-deq/tp/arrow D1 D2) (ofkd/tp/arrow DA1 DA2) (ofkd/tp/arrow DB1 DB2) kd-wf/kd/type) + %<- (vdt/cn-deq D1 DA1 DB1 _) + %<- (vdt/cn-deq D2 DA2 DB2 _) %. +%term _ + %pi (vdt/cn-deq (cn-deq/tp/sum D1 D2) (ofkd/tp/sum DA1 DA2) (ofkd/tp/sum DB1 DB2) kd-wf/kd/type) + %<- (vdt/cn-deq D1 DA1 DB1 _) + %<- (vdt/cn-deq D2 DA2 DB2 _) %. +%term _ + %pi (vdt/cn-deq (cn-deq/tp/forall Deq DKeq) (ofkd/tp/forall DofC1 DK1) (ofkd/tp/forall DofC2' DK2) kd-wf/kd/type) + %<- (vdt/kd-deq DKeq DK1 DK2) + %<- ({a cn} {da ofkd a K1} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da DK1} vdt/cn-deq (Deq a da) (DofC1 a da) (DofC2 a da) kd-wf/kd/type) + %<- (kd-anti DKeq DK1 DK2 _ DS) + %<- (kd-wkn/ofkd DofC2 DS DofC2') %. +%term _ + %pi (vdt/cn-deq (cn-deq/tp/ref D1) (ofkd/tp/ref DL) (ofkd/tp/ref DR) kd-wf/kd/type) + %<- (vdt/cn-deq D1 DL DR _) %. +%term _ + %pi (vdt/cn-deq (cn-deq/tp/tag D1) (ofkd/tp/tag DL) (ofkd/tp/tag DR) kd-wf/kd/type) + %<- (vdt/cn-deq D1 DL DR _) %. +%term _ + %pi (vdt/cn-deq (cn-deq/cn/pair D1 D2) (ofkd/cn/pair DA1 DA2) (ofkd/cn/pair DB1 DB2) (kd-wf/kd/sgm DK1 ([a cn] [da ofkd a K] DK2))) + %<- (vdt/cn-deq D1 DA1 DB1 DK1) + %<- (vdt/cn-deq D2 DA2 DB2 DK2) %. +%term _ + %pi (vdt/cn-deq (cn-deq/cn/pj1 D) (ofkd/cn/pj1 D1) (ofkd/cn/pj1 D2) DK1) + %<- (vdt/cn-deq D D1 D2 (kd-wf/kd/sgm DK1 DK2)) %. +%term _ + %pi (vdt/cn-deq (cn-deq/cn/pj2 D) (ofkd/cn/pj2 D1) (ofkd/deq (ofkd/cn/pj2 D2) DS) (DK2 (cn/pj1 C1) (ofkd/cn/pj1 D1))) + %<- (vdt/cn-deq D D1 D2 (kd-wf/kd/sgm DK1 DK2)) + %<- (funct/kd-wf DK2 (cn-deq/cn/pj1 (cn-deq/sym D)) (ofkd/cn/pj1 D2) (ofkd/cn/pj1 D1) DS) %. +%term _ + %pi (vdt/cn-deq (cn-deq/cn/lam Deq DKeq) (ofkd/cn/lam DofC1 DK1) (ofkd/sub (ofkd/cn/lam DofC2' DK2) (kd-sub/kd/pi DDS DS' DK1'')) (kd-wf/kd/pi DK1 DK1')) + %<- (vdt/kd-deq DKeq DK1 DK2) + %<- ({a cn} {da ofkd a K1} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da DK1} vdt/cn-deq (Deq a da) (DofC1 a da) (DofC2 a da) (DK1' a da)) + %<- (kd-anti DKeq DK1 DK2 DDS DS) + %<- (kd-wkn/ofkd DofC2 DS DofC2') + %<- ({a cn} {da ofkd a K1} kd-refl/sub (DK1' a da) (DS' a da)) + %<- (kd-wkn/kd-wf DK1' DS DK1'') %. +%term _ + %pi (vdt/cn-deq (cn-deq/cn/mu Deq DKeq) (ofkd/cn/mu DofC1 DK1) (%the (ofkd (cn/mu _ C2) _) (ofkd/sub (ofkd/cn/mu DofC2' DK2) DS)) DK1) + %<- (vdt/kd-deq DKeq DK1 DK2) + %<- ({a cn} {da ofkd a _} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da DK1} vdt/cn-deq (Deq a da) (DofC1 a da) (DofC2 a da) (DK1' a da)) + %<- (kd-anti DKeq DK1 DK2 DDS DS) + %<- (kd-wkn/ofkd ([a] [da] ofkd/sub (DofC2 a da) DDS) DS DofC2') %. +%term _ + %pi (vdt/cn-deq (cn-deq/cn/app D1 D2) (ofkd/cn/app DA1 DA2) (ofkd/deq (ofkd/cn/app DB1 DB2) DS) (DK2 C1 DA2)) + %<- (vdt/cn-deq D1 DA1 DB1 (kd-wf/kd/pi DK1 DK2)) + %<- (vdt/cn-deq D2 DA2 DB2 _) + %<- (funct/kd-wf DK2 (cn-deq/sym D2) DB2 DA2 DS) %. +%term _ vdt/cn-deq (cn-deq/kd/unit D1 D2) D1 D2 kd-wf/kd/unit %. +%term _ + %pi (vdt/cn-deq (cn-deq/kd/sing D1) D1 (ofkd/kd/sing D2) (kd-wf/kd/sing D2)) + %<- (vdt/ofkd D1 (kd-wf/kd/sing D2)) %. +%term _ + %pi (vdt/cn-deq (cn-deq/sgm-ext D1 D2) (ofkd/sgm-ext DA1 DA2) (ofkd/sgm-ext DB1 DB2) (kd-wf/kd/sgm DK1 ([a cn] [da] DK2))) + %<- (vdt/cn-deq D1 DA1 DB1 DK1) + %<- (vdt/cn-deq D2 DA2 DB2 DK2) %. +%term _ + %pi (vdt/cn-deq (cn-deq/pi-ext D1 D2 D3) (ofkd/pi-ext D1 D1') (ofkd/pi-ext D2 D2') (kd-wf/kd/pi DK DK')) + %<- (vdt/ofkd D1 (kd-wf/kd/pi DK _)) + %<- ({a cn} {da ofkd a K1} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da DK} vdt/cn-deq (D3 a da) (D1' a da) (D2' a da) (DK' a da)) %. +%term _ + %pi (vdt/cn-deq (cn-deq/pi-ext-2 D0 D3) (ofkd/pi-ext D1 D1') (ofkd/pi-ext D2 D2') (kd-wf/kd/pi DK DK')) + %<- (vdt/cn-deq D0 D1 D2 (kd-wf/kd/pi DK _)) + %<- ({a cn} {da ofkd a K1} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da DK} vdt/cn-deq (D3 a da) (D1' a da) (D2' a da) (DK' a da)) %. +%term _ + %pi (vdt/cn-deq (cn-deq/sub Deq Dsub) (ofkd/sub D1 Dsub) (ofkd/sub D2 Dsub) DK') + %<- (vdt/cn-deq Deq D1 D2 DK) + %<- (vdt/kd-sub Dsub _ DK') %. +%term _ + %pi (vdt/cn-deq (cn-deq/deq Deq Ddeq) (ofkd/deq D1 Ddeq) (ofkd/deq D2 Ddeq) DK') + %<- (vdt/cn-deq Deq D1 D2 DK) + %<- (vdt/kd-deq Ddeq _ DK') %. +%worlds (ofkd+vdt-block) (vdt/ofkd _ _) (vdt/kd-sub _ _ _) (vdt/kd-deq _ _ _) (vdt/cn-deq _ _ _ _) %. +%total (D1 D2 D3 D4) (vdt/kd-sub D2 _ _) (vdt/cn-deq D3 _ _ _) (vdt/kd-deq D4 _ _) (vdt/ofkd D1 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/validity-sigs.lf b/new-tests/stelf-output/tslf/validity-sigs.lf new file mode 100644 index 0000000..d755ec1 --- /dev/null +++ b/new-tests/stelf-output/tslf/validity-sigs.lf @@ -0,0 +1,83 @@ +%%%%% Validity signatures%%%%% +%sort vdt/sg-wf %. +%mode vdt/sg-wf %in %in %out %. +%term _ vdt/sg-wf sg-wf/sg/unit fst-sg/unit kd-wf/kd/unit %. +%term _ vdt/sg-wf (sg-wf/sg/kd DK) fst-sg/kd DK %. +%term _ vdt/sg-wf (sg-wf/sg/cn DC) fst-sg/cn kd-wf/kd/unit %. +%term _ + %pi (vdt/sg-wf (sg-wf/sg/sgm D1 D2 (%the (fst-sg S1 K3) DFS')) (fst-sg/sgm (%the (fst-sg S1 K1) DFS) (%the ({a} fst-sg (S2 a) (K2 a)) DFS2)) (kd-wf/kd/sgm DW DW2)) + %<- (fst-sg-seq DFS' DFS (%the (seq/kd K3 K1) DQ)) + %<- (vdt/sg-wf D1 DFS (%the (kd-wf K1) DW)) + %<- (vdt/sg-wf D1 DFS' (%the (kd-wf K3) DW')) + %<- ({a cn} {da ofkd a K3} {dm mofkd da met/unit} {_ can-mofkd da dm} {va vdt/ofkd da DW'} vdt/sg-wf (D2 a da) (DFS2 a) (DW2' a da)) + %<- (kd-wf/seq-a DQ DW2' DW2) %. +%term _ vdt/sg-wf (sg-wf/sg/pi _ _ _) fst-sg/pi kd-wf/kd/unit %. +%worlds (ofkd+vdt-block) (vdt/sg-wf _ _ _) %. +%total D1 (vdt/sg-wf D1 _ _) %. +%sort vdt/sg-deq %. +%mode vdt/sg-deq %in %in %in %out %out %out %. +%term _ vdt/sg-deq sg-deq/sg/unit fst-sg/unit fst-sg/unit sg-wf/sg/unit sg-wf/sg/unit kd-deq/kd/unit %. +%term _ + %pi (vdt/sg-deq (sg-deq/sg/cn D) fst-sg/cn fst-sg/cn (sg-wf/sg/cn D1) (sg-wf/sg/cn D2) kd-deq/kd/unit) + %<- (vdt/cn-deq D D1 D2 _) %. +%term _ + %pi (vdt/sg-deq (sg-deq/sg/kd D) fst-sg/kd fst-sg/kd (sg-wf/sg/kd D1) (sg-wf/sg/kd D2) D) + %<- (vdt/kd-deq D D1 D2) %. +%term _ + %pi (vdt/sg-deq (sg-deq/sg/sgm D1 D2 (%the (fst-sg S1 K) FSo)) (fst-sg/sgm (%the (fst-sg S1 K1) FS1) FS1') (fst-sg/sgm FS2 FS2') (sg-wf/sg/sgm DW1 DW1'o FS1) (sg-wf/sg/sgm DW2 DW2'' FS2) (kd-deq/kd/sgm DK1o DK2o)) + %<- (vdt/sg-deq D1 FSo FS2 DW1 DW2 DK1) + %<- (vdt/kd-deq DK1 DKK1 DKK2) + %<- ({a cn} {da ofkd a K} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da DKK1} vdt/sg-deq (D2 a da) (FS1' a) (FS2' a) (DW1' a da) (DW2' a da) (DK2 a da)) + %<- (kd-anti DK1 DKK1 DKK2 _ DK21) + %<- (kd-wkn/sg-wf DW2' DK21 DW2'') + %<- (fst-sg-seq FSo FS1 DQ) + %<- (sg-wf/seq-a DQ DW1' DW1'o) + %<- (kd-deq/seq-l DQ DK1 DK1o) + %<- (kd-deq/seq-a DQ DK2 DK2o) %. +%term _ + %pi (vdt/sg-deq (sg-deq/sg/pi (%the (sg-deq S2 S1) D1) D2 (%the (fst-sg S2 K2) FS2)) fst-sg/pi fst-sg/pi (sg-wf/sg/pi (%the (sg-wf S1) DW1) DW1'' FS1) (sg-wf/sg/pi (%the (sg-wf S2) DW2) DW2' FS2) kd-deq/kd/unit) + %<- (fst-sg-complete S1 (%the (fst-sg S1 K1) FS1)) + %<- ({a cn} fst-sg-complete (S3 a) (FS1' a)) + %<- ({a cn} fst-sg-complete (S4 a) (FS2' a)) + %<- (vdt/sg-deq D1 FS2 FS1 DW2 DW1 DK1) + %<- (vdt/kd-deq DK1 DKK2 DKK1) + %<- ({a cn} {da ofkd a K2} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da DKK2} vdt/sg-deq (D2 a da) (FS1' a) (FS2' a) (DW1' a da) (DW2' a da) (DK2 a da)) + %<- (kd-anti DK1 DKK2 DKK1 _ DK21) + %<- (kd-wkn/sg-wf DW1' DK21 DW1'') %. +%worlds (ofkd+vdt-block) (vdt/sg-deq _ _ _ _ _ _) %. +%total (D1) (vdt/sg-deq D1 _ _ _ _ _) %. +%sort vdt/sg-sub %. +%mode vdt/sg-sub %in %in %in %out %out %out %. +%term _ vdt/sg-sub sg-sub/sg/unit fst-sg/unit fst-sg/unit sg-wf/sg/unit sg-wf/sg/unit kd-sub/kd/unit %. +%term _ + %pi (vdt/sg-sub (sg-sub/sg/cn D) fst-sg/cn fst-sg/cn (sg-wf/sg/cn D1) (sg-wf/sg/cn D2) kd-sub/kd/unit) + %<- (vdt/cn-deq D D1 D2 _) %. +%term _ + %pi (vdt/sg-sub (sg-sub/sg/kd D) fst-sg/kd fst-sg/kd (sg-wf/sg/kd D1) (sg-wf/sg/kd D2) D) + %<- (vdt/kd-sub D D1 D2) %. +%term _ + %pi (vdt/sg-sub (sg-sub/sg/sgm (%the (sg-sub S1 S2) D1) D2 DW'' (%the (fst-sg S1 K) FSo) (%the (fst-sg S2 K') FSt)) (fst-sg/sgm (%the (fst-sg S1 K1) FS1) FS1') (fst-sg/sgm (%the (fst-sg S2 K2) FS2) FS2') (sg-wf/sg/sgm DW1 DW1'o FS1) (sg-wf/sg/sgm DW2 DW''t FS2) (kd-sub/kd/sgm DK1ot DK2o DK2wt)) + %<- (vdt/sg-sub D1 FSo FSt DW1 DW2 DK1) + %<- (vdt/kd-sub DK1 DKK1 DKK2) + %<- ({a cn} {da ofkd a K} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da DKK1} vdt/sg-sub (D2 a da) (FS1' a) (FS2' a) (DW1' a da) (DW2' a da) (DK2 a da)) + %<- ({a cn} {da ofkd a K'} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da DKK2} vdt/sg-wf (DW'' a da) (FS2' a) (DK2w a da)) + %<- (fst-sg-seq FSo FS1 DQ) + %<- (sg-wf/seq-a DQ DW1' DW1'o) + %<- (kd-sub/seq-l DQ DK1 DK1o) + %<- (kd-sub/seq-a DQ DK2 DK2o) + %<- (fst-sg-seq FSt FS2 DQ') + %<- (sg-wf/seq-a DQ' DW'' DW''t) + %<- (kd-sub/seq-r DQ' DK1o DK1ot) + %<- (kd-wf/seq-a DQ' DK2w DK2wt) %. +%term _ + %pi (vdt/sg-sub (sg-sub/sg/pi (%the (sg-deq S2 S1) D1) D2 (%the (fst-sg S2 K2) FS2)) fst-sg/pi fst-sg/pi (sg-wf/sg/pi (%the (sg-wf S1) DW1) DW1'' FS1) (sg-wf/sg/pi (%the (sg-wf S2) DW2) DW2' FS2) kd-sub/kd/unit) + %<- (fst-sg-complete S1 (%the (fst-sg S1 K1) FS1)) + %<- ({a cn} fst-sg-complete (S3 a) (FS1' a)) + %<- ({a cn} fst-sg-complete (S4 a) (FS2' a)) + %<- (vdt/sg-deq D1 FS2 FS1 DW2 DW1 DK1) + %<- (vdt/kd-deq DK1 DKK2 DKK1) + %<- ({a cn} {da ofkd a K2} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da DKK2} vdt/sg-deq (D2 a da) (FS1' a) (FS2' a) (DW1' a da) (DW2' a da) (DK2 a da)) + %<- (kd-anti DK1 DKK2 DKK1 _ DK21) + %<- (kd-wkn/sg-wf DW1' DK21 DW1'') %. +%worlds (ofkd+vdt-block) (vdt/sg-sub _ _ _ _ _ _) %. +%total (D3) (vdt/sg-sub D3 _ _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/validity-tmmd.lf b/new-tests/stelf-output/tslf/validity-tmmd.lf new file mode 100644 index 0000000..48906f5 --- /dev/null +++ b/new-tests/stelf-output/tslf/validity-tmmd.lf @@ -0,0 +1,150 @@ +%%%%% Validity %%%%% +% vdt/assm/tm : assm/tm E C -> ofkd C kd/type -> type. +% mode vdt/assm/tm +D1 -D2. +%worlds (oftp+vdt-block ofkd+vdt-block ofsg+vdt-block) (vdt/assm/tm _ _) %. +%total {} (vdt/assm/tm _ _) %. +%sort vdt/ofsg/i %. +%mode vdt/ofsg/i %in %out %. +%sort vdt/oftp %. +%mode vdt/oftp %in %out %. +%sort vdt/ofsg/p %. +%mode vdt/ofsg/p %in %out %out %out %out %. +%term _ %pi (vdt/oftp (oftp/var D) DC) %<- (vdt/assm/tm D DC) %. +%term _ vdt/oftp oftp/tm/unit ofkd/tp/unit %. +%term _ + %pi (vdt/oftp (oftp/tm/pair D1 D2) (ofkd/tp/cross DC1 DC2)) + %<- (vdt/oftp D1 DC1) + %<- (vdt/oftp D2 DC2) %. +%term _ + %pi (vdt/oftp (oftp/tm/pj1 D1) DC1) + %<- (vdt/oftp D1 DC) + %<- (inv/ofkd/tp/cross DC kd-sub/kd/type DC1 _) %. +%term _ + %pi (vdt/oftp (oftp/tm/pj2 D1) DC2) + %<- (vdt/oftp D1 DC) + %<- (inv/ofkd/tp/cross DC kd-sub/kd/type _ DC2) %. +%term _ vdt/oftp (oftp/tm/fun DE DK) DK %. +%term _ + %pi (vdt/oftp (oftp/tm/tmapp D1 D2) DC2) + %<- (vdt/oftp D1 DC) + %<- (inv/ofkd/tp/arrow DC kd-sub/kd/type _ DC2) %. +%term _ + %pi (vdt/oftp (oftp/tm/cnabs D1 DK) (ofkd/tp/forall DC' DK)) + %<- ({a cn} {da ofkd a K} {dm mofkd da met/unit} {_ can-mofkd da dm} {kd vdt/ofkd da DK} vdt/oftp (D1 a da) (DC' a da)) %. +%term _ + %pi (vdt/oftp (oftp/tm/cnapp DE DC) (DC' C DC)) + %<- (vdt/oftp DE DCC) + %<- (inv/ofkd/tp/forall DCC kd-sub/kd/type _ DC') %. +%term _ %pi (vdt/oftp (oftp/deq DE DC) DC') %<- (vdt/cn-deq DC _ DC' _) %. +%term _ %pi (vdt/oftp (oftp/tm/term DM) DC) %<- (vdt/ofsg/i DM (sg-wf/sg/cn DC)) %. +%term _ %pi (vdt/oftp (oftp/tm/term DM) DC) %<- (vdt/ofsg/p DM (sg-wf/sg/cn DC) _ _ _) %. +%term _ vdt/oftp (oftp/tm/loc _ DC) (ofkd/tp/ref DC) %. +%term _ %pi (vdt/oftp (oftp/tm/ref D) (ofkd/tp/ref DC)) %<- (vdt/oftp D DC) %. +%term _ vdt/oftp (oftp/tm/set D1 D2) ofkd/tp/unit %. +%term _ + %pi (vdt/oftp (oftp/tm/get D) DC') + %<- (vdt/oftp D DC) + %<- (inv/ofkd/tp/ref DC kd-sub/kd/type DC') %. +%term _ %pi (vdt/oftp (oftp/tm/inl D1 DC) (ofkd/tp/sum DC' DC)) %<- (vdt/oftp D1 DC') %. +%term _ %pi (vdt/oftp (oftp/tm/inr D1 DC) (ofkd/tp/sum DC DC')) %<- (vdt/oftp D1 DC') %. +%term _ + %pi (vdt/oftp (oftp/tm/case DE D1 _) DC) + %<- (vdt/oftp DE DC') + %<- (inv/ofkd/tp/sum DC' kd-sub/kd/type DC1 _) + %<- ({x} {dx assm/tm x _} {vd vdt/assm/tm dx DC1} vdt/oftp (D1 x dx) DC) %. +%term _ vdt/oftp (oftp/tm/raise _ DC) DC %. +%term _ %pi (vdt/oftp (oftp/tm/try D1 _) DC) %<- (vdt/oftp D1 DC) %. +%term _ vdt/oftp (oftp/tm/new-tag DC) (ofkd/tp/tag DC) %. +%term _ vdt/oftp (oftp/tm/tagloc _ DC) (ofkd/tp/tag DC) %. +%term _ vdt/oftp (oftp/tm/tag _ _) ofkd/tp/tagged %. +%term _ %pi (vdt/oftp (oftp/tm/iftag _ _ _ D1) DC) %<- (vdt/oftp D1 DC) %. +%term _ vdt/oftp (oftp/tm/roll _ DC) DC %. +%term _ + %pi (vdt/oftp (oftp/tm/unroll D1) (DCo _ DC)) + %<- (vdt/oftp D1 DC) + %<- (inv/ofkd/cn/mu DC kd-sub/kd/type DCo) %. +%term _ %pi (vdt/ofsg/p (ofsg/var D) DW DFS DFM DC) %<- (vdt/assm/md D DW DFS DFM DC) %. +%term _ vdt/ofsg/p ofsg/md/unit sg-wf/sg/unit fst-sg/unit fst-md/unit ofkd/cn/unit %. +%term _ + %pi (vdt/ofsg/p (ofsg/md/tm D) (sg-wf/sg/cn D') fst-sg/cn fst-md/tm ofkd/cn/unit) + %<- (vdt/oftp D D') %. +%term _ + %pi (vdt/ofsg/p (ofsg/md/cn D) (sg-wf/sg/kd D') fst-sg/kd fst-md/cn D) + %<- (vdt/ofkd D D') %. +%term _ + %pi (vdt/ofsg/p (ofsg/md/lam DM DS DFS) (sg-wf/sg/pi DS DS' DFS) fst-sg/pi fst-md/lam ofkd/cn/unit) + %<- (vdt/sg-wf DS DFS DKW) + %<- ({s md} {ds assm/md s _} {a cn} {da ofkd a K1} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da DKW} {dfm fst-md s a} {_ vdt/assm/md ds DS DFS dfm da} {_ fst-md-seq dfm dfm seq/cn/refl} vdt/ofsg/i (DM s ds a da dfm) (DS' a da)) %. +%term _ + %pi (vdt/ofsg/p (ofsg/md/lam DM (%the (sg-wf S1) DS) (%the (fst-sg S1 K1) DFS)) (sg-wf/sg/pi DS (%the ({a} {da ofkd a K1} sg-wf (S2 a)) DS') DFS) fst-sg/pi fst-md/lam ofkd/cn/unit) + %<- (vdt/sg-wf DS DFS DKW) + %<- ({s md} {ds assm/md s S1} {a cn} {da ofkd a K1} {dm mofkd da met/unit} {_ can-mofkd da dm} {vd vdt/ofkd da DKW} {dfm fst-md s a} {_ vdt/assm/md ds DS DFS dfm da} {_ fst-md-seq dfm dfm seq/cn/refl} vdt/ofsg/p (DM s ds a da dfm) (DS' a da) (Df a) (Df' s a dfm) (Do a da)) %. +%term _ + %pi (vdt/ofsg/p (ofsg/md/pair D1 D2) (sg-wf/sg/sgm DSW1 ([a cn] [da ofkd a K1] DSW2) FS1) (fst-sg/sgm FS1 ([a cn] FS2)) (fst-md/pair FM1 FM2) (ofkd/cn/pair DC1 DC2)) + %<- (vdt/ofsg/p D1 DSW1 FS1 FM1 DC1) + %<- (vdt/ofsg/p D2 DSW2 FS2 FM2 DC2) %. +%term _ + %pi (vdt/ofsg/p (ofsg/md/pj1 D1) DSW1 DS1 (fst-md/pj1 DM1) (ofkd/cn/pj1 DC1)) + %<- (vdt/ofsg/p D1 (sg-wf/sg/sgm DSW1 _ _) (fst-sg/sgm DS1 _) DM1 DC1) %. +%term _ + %pi (vdt/ofsg/p (ofsg/md/pj2 D1 DF) (DSW2 (cn/pj1 C) DCp1') (FS2 (cn/pj1 C)) (fst-md/pj2 DF) (ofkd/cn/pj2 DC)) + %<- (vdt/ofsg/p D1 (sg-wf/sg/sgm DSW1 DSW2 (%the (fst-sg S1 K1) FSo)) (fst-sg/sgm (%the (fst-sg S1 K3) FS1) FS2) DF' DC') + %<- (fst-md-seq DF' DF DQ) + %<- (ofkd/seq-c DQ DC' DC) + %<- (fst-sg-seq FS1 FSo DQ') + %<- (ofkd/seq-k DQ' (ofkd/cn/pj1 DC) DCp1') %. +%term _ + %pi (vdt/ofsg/p (ofsg/sgm-ext D1 D2) (sg-wf/sg/sgm DSW1 ([a cn] [da ofkd a _] DSW2) DS1) (fst-sg/sgm DS1 ([a cn] DS2)) DF1 (ofkd/sgm-ext DC1 DC2')) + %<- (vdt/ofsg/p D1 DSW1 DS1 (fst-md/pj1 DF1) DC1) + %<- (vdt/ofsg/p D2 DSW2 DS2 (fst-md/pj2 DF2) DC2) + %<- (fst-md-seq DF2 DF1 DQ) + %<- (seq/cn/pj2 DQ DQs) + %<- (ofkd/seq-c DQs DC2 DC2') %. +%term _ + %pi (vdt/ofsg/p (ofsg/kd-ext _ DF DC) (sg-wf/sg/kd DW) fst-sg/kd DF DC) + %<- (vdt/ofkd DC DW) %. +%term _ + %pi (vdt/ofsg/p (ofsg/sub D1 (%the (sg-sub S1 S2) DSU) _) DSW FS2 DM (ofkd/sub DC DKS)) + %<- (vdt/ofsg/p D1 _ DS DM DC) + %<- (fst-sg-complete S2 FS2) + %<- (vdt/sg-sub DSU DS FS2 _ DSW DKS) %. +%term _ + %pi (vdt/ofsg/i (ofsg/md/pair D1 D2) (sg-wf/sg/sgm DS1 ([a cn] [da ofkd a K1] DS2) DFS)) + %<- (vdt/ofsg/i D1 DS1) + %<- (vdt/ofsg/i D2 DS2) + %<- (fst-sg-complete S1 DFS) %. +%term _ %pi (vdt/ofsg/i (ofsg/md/pj1 D1) DS) %<- (vdt/ofsg/i D1 (sg-wf/sg/sgm DS _ _)) %. +%term _ + %pi (vdt/ofsg/i (ofsg/md/app D1 D2 DF) (DS2 C DC'')) + %<- (vdt/ofsg/i D1 (sg-wf/sg/pi DS1 DS2 (%the (fst-sg S1 K1) DFS))) + %<- (vdt/ofsg/p D2 DS1' (%the (fst-sg S1 K) DFS') DF' DC') + %<- (fst-md-seq DF' DF DQ) + %<- (ofkd/seq-c DQ DC' DC) + %<- (vdt/ofkd DC DW) + %<- (fst-sg-seq DFS' DFS DQ') + %<- (ofkd/seq-k DQ' DC DC'') %. +%term _ + %pi (vdt/ofsg/i (ofsg/md/app D1 D2 DF) (DS2 C DC'')) + %<- (vdt/ofsg/p D1 (sg-wf/sg/pi DS1 DS2 DFS) _ _ _) + %<- (vdt/ofsg/p D2 DS1' DFS' DF' DC') + %<- (fst-md-seq DF' DF DQ) + %<- (ofkd/seq-c DQ DC' DC) + %<- (vdt/ofkd DC DW) + %<- (fst-sg-seq DFS' DFS DQ') + %<- (ofkd/seq-k DQ' DC DC'') %. +%term _ vdt/ofsg/i (ofsg/md/let _ _ DS _) DS %. +%term _ %pi (vdt/ofsg/i (ofsg/md/seal D) DS) %<- (vdt/ofsg/i D DS) %. +%term _ %pi (vdt/ofsg/i (ofsg/md/seal D) DS) %<- (vdt/ofsg/p D DS _ _ _) %. +%term _ + %pi (vdt/ofsg/i (ofsg/sub D1 DS _) DSS) + %<- (fst-sg-complete S1 FS1) + %<- (fst-sg-complete S2 FS2) + %<- (vdt/sg-sub DS FS1 FS2 _ DSS _) %. +%worlds (oftp+vdt-block ofkd+vdt-block ofsg+vdt-block) (vdt/ofsg/p _ _ _ _ _) (vdt/oftp _ _) (vdt/assm/md _ _ _ _ _) (vdt/ofsg/i _ _) %. +%total (D1 D2 D7 D3) (vdt/oftp D1 _) (vdt/ofsg/i D2 _) (vdt/assm/md D3 _ _ _ _) (vdt/ofsg/p D7 _ _ _ _) %. +%sort vdt/ofsg %. +%mode vdt/ofsg %in %out %. +%term _ %pi (vdt/ofsg D1 DS) %<- (vdt/ofsg/i D1 DS) %. +%term _ %pi (vdt/ofsg D1 DS) %<- (vdt/ofsg/p D1 DS _ _ _) %. +%worlds (oftp+vdt-block ofkd+vdt-block ofsg+vdt-block) (vdt/ofsg _ _) %. +%total D1 (vdt/ofsg D1 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/value.lf b/new-tests/stelf-output/tslf/value.lf new file mode 100644 index 0000000..0f98764 --- /dev/null +++ b/new-tests/stelf-output/tslf/value.lf @@ -0,0 +1,20 @@ +%%%% Values +%sort val/tm %. +%mode val/tm %star %. +%term val/tm/unit val/tm tm/unit %. +%term val/tm/pair %pi (val/tm (tm/pair T1 T2)) %<- (val/tm T2) %<- (val/tm T1) %. +%term val/tm/fun val/tm (tm/fun _ _ _) %. +%term val/tm/cnabs val/tm (tm/cnabs _ _) %. +%term val/tm/loc val/tm (tm/loc _) %. +%term val/tm/inl %pi (val/tm (tm/inl (tp/sum C1 C2) E)) %<- (val/tm E) %. +%term val/tm/inr %pi (val/tm (tm/inr (tp/sum C1 C2) E)) %<- (val/tm E) %. +%term val/tm/tagloc val/tm (tm/tagloc _) %. +%term val/tm/tag %pi (val/tm (tm/tag E1 E2)) %<- (val/tm E2) %<- (val/tm E1) %. +%term val/tm/roll %pi (val/tm (tm/roll (cn/mu _ _) E)) %<- (val/tm E) %. +%sort val/md %. +%mode val/md %star %. +%term val/md/unit val/md md/unit %. +%term val/md/pair %pi (val/md (md/pair M1 M2)) %<- (val/md M2) %<- (val/md M1) %. +%term val/md/lam val/md (md/lam _ _ _) %. +%term val/md/cn val/md (md/cn _) %. +%term val/md/tm %pi (val/md (md/tm E)) %<- (val/tm E) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/weakening-cnkd-e.lf b/new-tests/stelf-output/tslf/weakening-cnkd-e.lf new file mode 100644 index 0000000..80d940e --- /dev/null +++ b/new-tests/stelf-output/tslf/weakening-cnkd-e.lf @@ -0,0 +1,866 @@ +%sort renum-cxt-lookup {x} %. +%mode renum-cxt-lookup %in %in %out %. +%term _ renum-cxt-lookup ([x] [di] cxt-ordered/cons (DB x di)) ([x] [dj] cxt-lookup/hit (DB' x dj)) ([x] [di] cxt-lookup/hit (DB x di)) %. +%term _ + %pi (renum-cxt-lookup ([x] [di] cxt-ordered/cons (cxt-bounded/cons (DB x di) (DP x di))) ([x] [dj] cxt-lookup/miss (DL x dj) (DPz x dj) (DB' x dj)) ([x] [di] cxt-lookup/miss (DL' x di) (DP' x di) (cxt-bounded/cons (DB x di) (DP x di)))) + %<- (renum-cxt-lookup ([x] [di] cxt-ordered/cons (DB x di)) DL DL') + %<- ({x} {di isvar x I} cxt-lookup-precedes (DL' x di) (cxt-bounded/cons (DB x di) (DP x di)) (DP' x di)) %. +%worlds (cn-block ovar-block) (renum-cxt-lookup _ _ _) %. +%total {D1} (renum-cxt-lookup _ D1 _) %. +%sort renum-ekd-wf {Mm met} {x} %. +%mode renum-ekd-wf %in %in %in %in %out %out %. +%sort renum-ekd-deq {Mm met} {x} %. +%mode renum-ekd-deq %in %in %in %in %out %out %. +%sort renum-ekd-sub {Mm met} {x} %. +%mode renum-ekd-sub %in %in %in %in %out %out %. +%sort renum-eofkd {Mm met} {x} %. +%mode renum-eofkd %in %in %in %in %out %out %. +%sort renum-ecn-deq {Mm met} {x} %. +%mode renum-ecn-deq %in %in %in %in %out %out %. +%term _ renum-ekd-wf _ DO ([x] [di] ekd-wf/kd/unit _) _ ([x] [di] ekd-wf/kd/unit (DO x di)) ([x] [di] mekd-wf/kd/unit) %. +%term _ renum-ekd-wf _ DO ([x] [di] ekd-wf/kd/type _) _ ([x] [di] ekd-wf/kd/type (DO x di)) ([x] [di] mekd-wf/kd/type) %. +%term _ + %pi (renum-ekd-wf (met/sing M1) DO ([x] [di] ekd-wf/kd/sing (D1 x di)) ([x] [di] mekd-wf/kd/sing (D1m x di)) ([x] [dx] ekd-wf/kd/sing (D1' x dx)) ([x] [dx] mekd-wf/kd/sing (D1m' x dx))) + %<- (renum-eofkd M1 DO D1 D1m D1' D1m') %. +%term _ + %pi (renum-ekd-wf (met/pair M1 M2) DO ([x] [di] ekd-wf/kd/sgm (D1 x di) ([y] [dy isvar y I] D2 x di y dy)) ([x] [di] mekd-wf/kd/sgm (D1m x di) (D2m x di)) ([x] [dx] ekd-wf/kd/sgm (D1' x dx) (D2'' x dx)) ([x] [dx] mekd-wf/kd/sgm (D1m' x dx) (D2m'' x dx))) + %<- (renum-ekd-wf M1 DO D1 D1m D1' D1m') + %<- ({x} {dx} ekd-wf-ordered (D1 x dx) (DO' x dx)) + %<- (cxt-inc-2 DO' DO DOe' DOe) + %<- ({x} {dx} renum-ekd-wf M2 ([y] [dy] cxt-ordered/cons (DOe' x dx y dy)) (D2 x dx) (D2m x dx) (D2' x dx) (D2m' x dx)) + %<- ({y} {dy} renum-ekd-wf M2 ([x] [dx] cxt-ordered/cons (DOe x dx y dy)) ([x] [dx] D2' x dx y dy) ([x] [dx] D2m' x dx y dy) ([x] [dx] D2'' x dx y dy) ([x] [dx] D2m'' x dx y dy)) %. +%term _ + %pi (renum-ekd-wf (met/pair M1 M2) DO ([x] [di] ekd-wf/kd/pi (D1 x di) ([y] [dy isvar y I] D2 x di y dy)) ([x] [di] mekd-wf/kd/pi (D1m x di) (D2m x di)) ([x] [dx] ekd-wf/kd/pi (D1' x dx) (D2'' x dx)) ([x] [dx] mekd-wf/kd/pi (D1m' x dx) (D2m'' x dx))) + %<- (renum-ekd-wf M1 DO D1 D1m D1' D1m') + %<- ({x} {dx} ekd-wf-ordered (D1 x dx) (DO' x dx)) + %<- (cxt-inc-2 DO' DO DOe' DOe) + %<- ({x} {dx} renum-ekd-wf M2 ([y] [dy] cxt-ordered/cons (DOe' x dx y dy)) (D2 x dx) (D2m x dx) (D2' x dx) (D2m' x dx)) + %<- ({y} {dy} renum-ekd-wf M2 ([x] [dx] cxt-ordered/cons (DOe x dx y dy)) ([x] [dx] D2' x dx y dy) ([x] [dx] D2m' x dx y dy) ([x] [dx] D2'' x dx y dy) ([x] [dx] D2m'' x dx y dy)) %. +%term _ renum-ekd-deq _ DO ([x] [di] ekd-deq/kd/unit _) _ ([x] [di] ekd-deq/kd/unit (DO x di)) ([x] [di] mekd-deq/kd/unit) %. +%term _ renum-ekd-deq _ DO ([x] [di] ekd-deq/kd/type _) _ ([x] [di] ekd-deq/kd/type (DO x di)) ([x] [di] mekd-deq/kd/type) %. +%term _ + %pi (renum-ekd-deq (met/sing M1) DO ([x] [di] ekd-deq/kd/sing (D1 x di)) ([x] [di] mekd-deq/kd/sing (D1m x di)) ([x] [dx] ekd-deq/kd/sing (D1' x dx)) ([x] [dx] mekd-deq/kd/sing (D1m' x dx))) + %<- (renum-ecn-deq M1 DO D1 D1m D1' D1m') %. +%term _ + %pi (renum-ekd-deq (met/pair M1 M2) DO ([x] [di] ekd-deq/kd/sgm (D1 x di) ([y] [dy isvar y I] D2 x di y dy)) ([x] [di] mekd-deq/kd/sgm (D1m x di) (D2m x di)) ([x] [dx] ekd-deq/kd/sgm (D1' x dx) (D2'' x dx)) ([x] [dx] mekd-deq/kd/sgm (D1m' x dx) (D2m'' x dx))) + %<- (renum-ekd-deq M1 DO D1 D1m D1' D1m') + %<- ({x} {dx} ekd-deq-ordered (D1 x dx) (DO' x dx)) + %<- (cxt-inc-2 DO' DO DOe' DOe) + %<- ({x} {dx} renum-ekd-deq M2 ([y] [dy] cxt-ordered/cons (DOe' x dx y dy)) (D2 x dx) (D2m x dx) (D2' x dx) (D2m' x dx)) + %<- ({y} {dy} renum-ekd-deq M2 ([x] [dx] cxt-ordered/cons (DOe x dx y dy)) ([x] [dx] D2' x dx y dy) ([x] [dx] D2m' x dx y dy) ([x] [dx] D2'' x dx y dy) ([x] [dx] D2m'' x dx y dy)) %. +%term _ + %pi (renum-ekd-deq (met/pair M1 M2) DO ([x] [di] ekd-deq/kd/pi (D1 x di) ([y] [dy isvar y I] D2 x di y dy)) ([x] [di] mekd-deq/kd/pi (D1m x di) (D2m x di)) ([x] [dx] ekd-deq/kd/pi (D1' x dx) (D2'' x dx)) ([x] [dx] mekd-deq/kd/pi (D1m' x dx) (D2m'' x dx))) + %<- (renum-ekd-deq M1 DO D1 D1m D1' D1m') + %<- ({x} {dx} ekd-deq-ordered (D1 x dx) (DO' x dx)) + %<- (cxt-inc-2 DO' DO DOe' DOe) + %<- ({x} {dx} renum-ekd-deq M2 ([y] [dy] cxt-ordered/cons (DOe' x dx y dy)) (D2 x dx) (D2m x dx) (D2' x dx) (D2m' x dx)) + %<- ({y} {dy} renum-ekd-deq M2 ([x] [dx] cxt-ordered/cons (DOe x dx y dy)) ([x] [dx] D2' x dx y dy) ([x] [dx] D2m' x dx y dy) ([x] [dx] D2'' x dx y dy) ([x] [dx] D2m'' x dx y dy)) %. +%term _ renum-ekd-sub _ DO ([x] [di] ekd-sub/kd/unit _) _ ([x] [di] ekd-sub/kd/unit (DO x di)) ([x] [di] mekd-sub/kd/unit) %. +%term _ renum-ekd-sub _ DO ([x] [di] ekd-sub/kd/type _) _ ([x] [di] ekd-sub/kd/type (DO x di)) ([x] [di] mekd-sub/kd/type) %. +%term _ + %pi (renum-ekd-sub (met/sing M1) DO ([x] [di] ekd-sub/kd/sing-kd/sing (D1 x di)) ([x] [di] mekd-sub/kd/sing-kd/sing (D1m x di)) ([x] [dx] ekd-sub/kd/sing-kd/sing (D1' x dx)) ([x] [dx] mekd-sub/kd/sing-kd/sing (D1m' x dx))) + %<- (renum-ecn-deq M1 DO D1 D1m D1' D1m') %. +%term _ + %pi (renum-ekd-sub (met/sing M1) DO ([x] [di] ekd-sub/kd/sing-kd/type (D1 x di)) ([x] [di] mekd-sub/kd/sing-kd/type (D1m x di)) ([x] [dx] ekd-sub/kd/sing-kd/type (D1' x dx)) ([x] [dx] mekd-sub/kd/sing-kd/type (D1m' x dx))) + %<- (renum-eofkd M1 DO D1 D1m D1' D1m') %. +%term _ + %pi (renum-ekd-sub (met/pair M1 (met/pair M2 M3)) DO ([x] [di] ekd-sub/kd/sgm (D1 x di) ([y] [dy isvar y I] D2 x di y dy) (D3 x di)) ([x] [di] mekd-sub/kd/sgm (D1m x di) (D2m x di) (D3m x di)) ([x] [dx] ekd-sub/kd/sgm (D1' x dx) (D2'' x dx) (D3'' x dx)) ([x] [dx] mekd-sub/kd/sgm (D1m' x dx) (D2m'' x dx) (D3m'' x dx))) + %<- (renum-ekd-sub M1 DO D1 D1m D1' D1m') + %<- ({x} {dx} ekd-sub-ordered (D1 x dx) (DO' x dx)) + %<- (cxt-inc-2 DO' DO DOe' DOe) + %<- ({x} {dx} renum-ekd-sub M2 ([y] [dy] cxt-ordered/cons (DOe' x dx y dy)) (D2 x dx) (D2m x dx) (D2' x dx) (D2m' x dx)) + %<- ({y} {dy} renum-ekd-sub M2 ([x] [dx] cxt-ordered/cons (DOe x dx y dy)) ([x] [dx] D2' x dx y dy) ([x] [dx] D2m' x dx y dy) ([x] [dx] D2'' x dx y dy) ([x] [dx] D2m'' x dx y dy)) + %<- ({x} {dx} renum-ekd-wf M3 ([y] [dy] cxt-ordered/cons (DOe' x dx y dy)) (D3 x dx) (D3m x dx) (D3' x dx) (D3m' x dx)) + %<- ({y} {dy} renum-ekd-wf M3 ([x] [dx] cxt-ordered/cons (DOe x dx y dy)) ([x] [dx] D3' x dx y dy) ([x] [dx] D3m' x dx y dy) ([x] [dx] D3'' x dx y dy) ([x] [dx] D3m'' x dx y dy)) %. +%term _ + %pi (renum-ekd-sub (met/pair M1 (met/pair M2 M3)) DO ([x] [di] ekd-sub/kd/pi (D1 x di) ([y] [dy isvar y I] D2 x di y dy) (D3 x di)) ([x] [di] mekd-sub/kd/pi (D1m x di) (D2m x di) (D3m x di)) ([x] [dx] ekd-sub/kd/pi (D1' x dx) (D2'' x dx) (D3'' x dx)) ([x] [dx] mekd-sub/kd/pi (D1m' x dx) (D2m'' x dx) (D3m'' x dx))) + %<- (renum-ekd-sub M1 DO D1 D1m D1' D1m') + %<- ({x} {dx} ekd-sub-ordered (D1 x dx) (DO' x dx)) + %<- (cxt-inc-2 DO' DO DOe' DOe) + %<- ({x} {dx} renum-ekd-sub M2 ([y] [dy] cxt-ordered/cons (DOe' x dx y dy)) (D2 x dx) (D2m x dx) (D2' x dx) (D2m' x dx)) + %<- ({y} {dy} renum-ekd-sub M2 ([x] [dx] cxt-ordered/cons (DOe x dx y dy)) ([x] [dx] D2' x dx y dy) ([x] [dx] D2m' x dx y dy) ([x] [dx] D2'' x dx y dy) ([x] [dx] D2m'' x dx y dy)) + %<- ({x} {dx} renum-ekd-wf M3 ([y] [dy] cxt-ordered/cons (DOe' x dx y dy)) (D3 x dx) (D3m x dx) (D3' x dx) (D3m' x dx)) + %<- ({y} {dy} renum-ekd-wf M3 ([x] [dx] cxt-ordered/cons (DOe x dx y dy)) ([x] [dx] D3' x dx y dy) ([x] [dx] D3m' x dx y dy) ([x] [dx] D3'' x dx y dy) ([x] [dx] D3m'' x dx y dy)) %. +%term _ + %pi (renum-eofkd _ DO ([x] [di] eofkd/var (DL x di)) _ ([x] [di] eofkd/var (DL' x di)) ([x] [di] meofkd/var)) + %<- (renum-cxt-lookup DO DL DL') %. +%term _ renum-eofkd _ DO ([x] [di] eofkd/closed (DC x) _) ([x] [di] meofkd/closed (DCm x)) ([x] [di] eofkd/closed (DC x) (DO x di)) ([x] [di] meofkd/closed (DCm x)) %. +%term _ renum-eofkd _ DO ([x] [di] eofkd/cn/unit _) _ ([x] [di] eofkd/cn/unit (DO x di)) ([x] [di] meofkd/cn/unit) %. +%term _ renum-eofkd _ DO ([x] [di] eofkd/tp/unit _) _ ([x] [di] eofkd/tp/unit (DO x di)) ([x] [di] meofkd/tp/unit) %. +%term _ renum-eofkd _ DO ([x] [di] eofkd/tp/tagged _) _ ([x] [di] eofkd/tp/tagged (DO x di)) ([x] [di] meofkd/tp/tagged) %. +%term _ + %pi (renum-eofkd (met/sing M1) DO ([x] [di] eofkd/kd/sing (D1 x di)) ([x] [di] meofkd/kd/sing (D1m x di)) ([x] [dx] eofkd/kd/sing (D1' x dx)) ([x] [dx] meofkd/kd/sing (D1m' x dx))) + %<- (renum-eofkd M1 DO D1 D1m D1' D1m') %. +%term _ + %pi (renum-eofkd (met/sing M1) DO ([x] [di] eofkd/cn/pj1 (D1 x di)) ([x] [di] meofkd/cn/pj1 (D1m x di)) ([x] [dx] eofkd/cn/pj1 (D1' x dx)) ([x] [dx] meofkd/cn/pj1 (D1m' x dx))) + %<- (renum-eofkd M1 DO D1 D1m D1' D1m') %. +%term _ + %pi (renum-eofkd (met/sing M1) DO ([x] [di] eofkd/cn/pj2 (D1 x di)) ([x] [di] meofkd/cn/pj2 (D1m x di)) ([x] [dx] eofkd/cn/pj2 (D1' x dx)) ([x] [dx] meofkd/cn/pj2 (D1m' x dx))) + %<- (renum-eofkd M1 DO D1 D1m D1' D1m') %. +%term _ + %pi (renum-eofkd (met/sing M1) DO ([x] [di] eofkd/tp/ref (D1 x di)) ([x] [di] meofkd/tp/ref (D1m x di)) ([x] [dx] eofkd/tp/ref (D1' x dx)) ([x] [dx] meofkd/tp/ref (D1m' x dx))) + %<- (renum-eofkd M1 DO D1 D1m D1' D1m') %. +%term _ + %pi (renum-eofkd (met/sing M1) DO ([x] [di] eofkd/tp/tag (D1 x di)) ([x] [di] meofkd/tp/tag (D1m x di)) ([x] [dx] eofkd/tp/tag (D1' x dx)) ([x] [dx] meofkd/tp/tag (D1m' x dx))) + %<- (renum-eofkd M1 DO D1 D1m D1' D1m') %. +%term _ + %pi (renum-eofkd (met/pair M1 M2) DO ([x] [di] eofkd/cn/pair (D1 x di) (D2 x di)) ([x] [di] meofkd/cn/pair (D1m x di) (D2m x di)) ([x] [dx] eofkd/cn/pair (D1' x dx) (D2' x dx)) ([x] [dx] meofkd/cn/pair (D1m' x dx) (D2m' x dx))) + %<- (renum-eofkd M1 DO D1 D1m D1' D1m') + %<- (renum-eofkd M2 DO D2 D2m D2' D2m') %. +%term _ + %pi (renum-eofkd (met/pair M1 M2) DO ([x] [di] eofkd/cn/app (D1 x di) (D2 x di)) ([x] [di] meofkd/cn/app (D1m x di) (D2m x di)) ([x] [dx] eofkd/cn/app (D1' x dx) (D2' x dx)) ([x] [dx] meofkd/cn/app (D1m' x dx) (D2m' x dx))) + %<- (renum-eofkd M1 DO D1 D1m D1' D1m') + %<- (renum-eofkd M2 DO D2 D2m D2' D2m') %. +%term _ + %pi (renum-eofkd (met/pair M1 M2) DO ([x] [di] eofkd/sgm-ext (D1 x di) (D2 x di)) ([x] [di] meofkd/sgm-ext (D1m x di) (D2m x di)) ([x] [dx] eofkd/sgm-ext (D1' x dx) (D2' x dx)) ([x] [dx] meofkd/sgm-ext (D1m' x dx) (D2m' x dx))) + %<- (renum-eofkd M1 DO D1 D1m D1' D1m') + %<- (renum-eofkd M2 DO D2 D2m D2' D2m') %. +%term _ + %pi (renum-eofkd (met/pair M1 M2) DO ([x] [di] eofkd/tp/cross (D1 x di) (D2 x di)) ([x] [di] meofkd/tp/cross (D1m x di) (D2m x di)) ([x] [dx] eofkd/tp/cross (D1' x dx) (D2' x dx)) ([x] [dx] meofkd/tp/cross (D1m' x dx) (D2m' x dx))) + %<- (renum-eofkd M1 DO D1 D1m D1' D1m') + %<- (renum-eofkd M2 DO D2 D2m D2' D2m') %. +%term _ + %pi (renum-eofkd (met/pair M1 M2) DO ([x] [di] eofkd/tp/arrow (D1 x di) (D2 x di)) ([x] [di] meofkd/tp/arrow (D1m x di) (D2m x di)) ([x] [dx] eofkd/tp/arrow (D1' x dx) (D2' x dx)) ([x] [dx] meofkd/tp/arrow (D1m' x dx) (D2m' x dx))) + %<- (renum-eofkd M1 DO D1 D1m D1' D1m') + %<- (renum-eofkd M2 DO D2 D2m D2' D2m') %. +%term _ + %pi (renum-eofkd (met/pair M1 M2) DO ([x] [di] eofkd/tp/sum (D1 x di) (D2 x di)) ([x] [di] meofkd/tp/sum (D1m x di) (D2m x di)) ([x] [dx] eofkd/tp/sum (D1' x dx) (D2' x dx)) ([x] [dx] meofkd/tp/sum (D1m' x dx) (D2m' x dx))) + %<- (renum-eofkd M1 DO D1 D1m D1' D1m') + %<- (renum-eofkd M2 DO D2 D2m D2' D2m') %. +%term _ + %pi (renum-eofkd (met/pair M1 M2) DO ([x] [di] eofkd/sub (D1 x di) (D2 x di)) ([x] [di] meofkd/sub (D1m x di) (D2m x di)) ([x] [dx] eofkd/sub (D1' x dx) (D2' x dx)) ([x] [dx] meofkd/sub (D1m' x dx) (D2m' x dx))) + %<- (renum-eofkd M1 DO D1 D1m D1' D1m') + %<- (renum-ekd-sub M2 DO D2 D2m D2' D2m') %. +%term _ + %pi (renum-eofkd (met/pair M1 M2) DO ([x] [di] eofkd/deq (D1 x di) (D2 x di)) ([x] [di] meofkd/deq (D1m x di) (D2m x di)) ([x] [dx] eofkd/deq (D1' x dx) (D2' x dx)) ([x] [dx] meofkd/deq (D1m' x dx) (D2m' x dx))) + %<- (renum-eofkd M1 DO D1 D1m D1' D1m') + %<- (renum-ekd-deq M2 DO D2 D2m D2' D2m') %. +%term _ + %pi (renum-eofkd (met/pair M2 M1) DO ([x] [di] eofkd/cn/lam ([y] [dy isvar y I] D2 x di y dy) (D1 x di)) ([x] [di] meofkd/cn/lam (D2m x di) (D1m x di)) ([x] [dx] eofkd/cn/lam (D2'' x dx) (D1' x dx)) ([x] [dx] meofkd/cn/lam (D2m'' x dx) (D1m' x dx))) + %<- (renum-ekd-wf M1 DO D1 D1m D1' D1m') + %<- ({x} {dx} ekd-wf-ordered (D1 x dx) (DO' x dx)) + %<- (cxt-inc-2 DO' DO DOe' DOe) + %<- ({x} {dx} renum-eofkd M2 ([y] [dy] cxt-ordered/cons (DOe' x dx y dy)) (D2 x dx) (D2m x dx) (D2' x dx) (D2m' x dx)) + %<- ({y} {dy} renum-eofkd M2 ([x] [dx] cxt-ordered/cons (DOe x dx y dy)) ([x] [dx] D2' x dx y dy) ([x] [dx] D2m' x dx y dy) ([x] [dx] D2'' x dx y dy) ([x] [dx] D2m'' x dx y dy)) %. +%term _ + %pi (renum-eofkd (met/pair M2 M1) DO ([x] [di] eofkd/cn/mu ([y] [dy isvar y I] D2 x di y dy) (D1 x di)) ([x] [di] meofkd/cn/mu (D2m x di) (D1m x di)) ([x] [dx] eofkd/cn/mu (D2'' x dx) (D1' x dx)) ([x] [dx] meofkd/cn/mu (D2m'' x dx) (D1m' x dx))) + %<- (renum-ekd-wf M1 DO D1 D1m D1' D1m') + %<- ({x} {dx} ekd-wf-ordered (D1 x dx) (DO' x dx)) + %<- (cxt-inc-2 DO' DO DOe' DOe) + %<- ({x} {dx} renum-eofkd M2 ([y] [dy] cxt-ordered/cons (DOe' x dx y dy)) (D2 x dx) (D2m x dx) (D2' x dx) (D2m' x dx)) + %<- ({y} {dy} renum-eofkd M2 ([x] [dx] cxt-ordered/cons (DOe x dx y dy)) ([x] [dx] D2' x dx y dy) ([x] [dx] D2m' x dx y dy) ([x] [dx] D2'' x dx y dy) ([x] [dx] D2m'' x dx y dy)) %. +%term _ + %pi (renum-eofkd (met/pair M2 M1) DO ([x] [di] eofkd/tp/forall ([y] [dy isvar y I] D2 x di y dy) (D1 x di)) ([x] [di] meofkd/tp/forall (D2m x di) (D1m x di)) ([x] [dx] eofkd/tp/forall (D2'' x dx) (D1' x dx)) ([x] [dx] meofkd/tp/forall (D2m'' x dx) (D1m' x dx))) + %<- (renum-ekd-wf M1 DO D1 D1m D1' D1m') + %<- ({x} {dx} ekd-wf-ordered (D1 x dx) (DO' x dx)) + %<- (cxt-inc-2 DO' DO DOe' DOe) + %<- ({x} {dx} renum-eofkd M2 ([y] [dy] cxt-ordered/cons (DOe' x dx y dy)) (D2 x dx) (D2m x dx) (D2' x dx) (D2m' x dx)) + %<- ({y} {dy} renum-eofkd M2 ([x] [dx] cxt-ordered/cons (DOe x dx y dy)) ([x] [dx] D2' x dx y dy) ([x] [dx] D2m' x dx y dy) ([x] [dx] D2'' x dx y dy) ([x] [dx] D2m'' x dx y dy)) %. +%term _ + %pi (renum-eofkd (met/pair M1 M2) DO ([x] [di] eofkd/pi-ext (D1 x di) ([y] [dy isvar y I] D2 x di y dy)) ([x] [di] meofkd/pi-ext (D1m x di) (D2m x di)) ([x] [dx] eofkd/pi-ext (D1' x dx) (D2'' x dx)) ([x] [dx] meofkd/pi-ext (D1m' x dx) (D2m'' x dx))) + %<- (renum-eofkd M1 DO D1 D1m D1' D1m') + %<- ({x} {dx} eofkd-ordered (D1 x dx) (DO' x dx)) + %<- (cxt-inc-2 DO' DO DOe' DOe) + %<- ({x} {dx} renum-eofkd M2 ([y] [dy] cxt-ordered/cons (DOe' x dx y dy)) (D2 x dx) (D2m x dx) (D2' x dx) (D2m' x dx)) + %<- ({y} {dy} renum-eofkd M2 ([x] [dx] cxt-ordered/cons (DOe x dx y dy)) ([x] [dx] D2' x dx y dy) ([x] [dx] D2m' x dx y dy) ([x] [dx] D2'' x dx y dy) ([x] [dx] D2m'' x dx y dy)) %. +%term _ renum-ecn-deq _ DO ([x] [di] ecn-deq/cn/unit _) _ ([x] [di] ecn-deq/cn/unit (DO x di)) ([x] [di] mecn-deq/cn/unit) %. +%term _ renum-ecn-deq _ DO ([x] [di] ecn-deq/tp/unit _) _ ([x] [di] ecn-deq/tp/unit (DO x di)) ([x] [di] mecn-deq/tp/unit) %. +%term _ renum-ecn-deq _ DO ([x] [di] ecn-deq/tp/tagged _) _ ([x] [di] ecn-deq/tp/tagged (DO x di)) ([x] [di] mecn-deq/tp/tagged) %. +%term _ + %pi (renum-ecn-deq (met/sing M1) DO ([x] [di] ecn-deq/kd/sing (D1 x di)) ([x] [di] mecn-deq/kd/sing (D1m x di)) ([x] [dx] ecn-deq/kd/sing (D1' x dx)) ([x] [dx] mecn-deq/kd/sing (D1m' x dx))) + %<- (renum-eofkd M1 DO D1 D1m D1' D1m') %. +%term _ + %pi (renum-ecn-deq (met/sing M1) DO ([x] [di] ecn-deq/refl (D1 x di)) ([x] [di] mecn-deq/refl (D1m x di)) ([x] [dx] ecn-deq/refl (D1' x dx)) ([x] [dx] mecn-deq/refl (D1m' x dx))) + %<- (renum-eofkd M1 DO D1 D1m D1' D1m') %. +%term _ + %pi (renum-ecn-deq (met/sing M1) DO ([x] [di] ecn-deq/cn/pj1 (D1 x di)) ([x] [di] mecn-deq/cn/pj1 (D1m x di)) ([x] [dx] ecn-deq/cn/pj1 (D1' x dx)) ([x] [dx] mecn-deq/cn/pj1 (D1m' x dx))) + %<- (renum-ecn-deq M1 DO D1 D1m D1' D1m') %. +%term _ + %pi (renum-ecn-deq (met/sing M1) DO ([x] [di] ecn-deq/cn/pj2 (D1 x di)) ([x] [di] mecn-deq/cn/pj2 (D1m x di)) ([x] [dx] ecn-deq/cn/pj2 (D1' x dx)) ([x] [dx] mecn-deq/cn/pj2 (D1m' x dx))) + %<- (renum-ecn-deq M1 DO D1 D1m D1' D1m') %. +%term _ + %pi (renum-ecn-deq (met/sing M1) DO ([x] [di] ecn-deq/tp/ref (D1 x di)) ([x] [di] mecn-deq/tp/ref (D1m x di)) ([x] [dx] ecn-deq/tp/ref (D1' x dx)) ([x] [dx] mecn-deq/tp/ref (D1m' x dx))) + %<- (renum-ecn-deq M1 DO D1 D1m D1' D1m') %. +%term _ + %pi (renum-ecn-deq (met/sing M1) DO ([x] [di] ecn-deq/tp/tag (D1 x di)) ([x] [di] mecn-deq/tp/tag (D1m x di)) ([x] [dx] ecn-deq/tp/tag (D1' x dx)) ([x] [dx] mecn-deq/tp/tag (D1m' x dx))) + %<- (renum-ecn-deq M1 DO D1 D1m D1' D1m') %. +%term _ + %pi (renum-ecn-deq (met/sing M1) DO ([x] [di] ecn-deq/sym (D1 x di)) ([x] [di] mecn-deq/sym (D1m x di)) ([x] [dx] ecn-deq/sym (D1' x dx)) ([x] [dx] mecn-deq/sym (D1m' x dx))) + %<- (renum-ecn-deq M1 DO D1 D1m D1' D1m') %. +%term _ + %pi (renum-ecn-deq (met/pair M1 M2) DO ([x] [di] ecn-deq/cn/pair (D1 x di) (D2 x di)) ([x] [di] mecn-deq/cn/pair (D1m x di) (D2m x di)) ([x] [dx] ecn-deq/cn/pair (D1' x dx) (D2' x dx)) ([x] [dx] mecn-deq/cn/pair (D1m' x dx) (D2m' x dx))) + %<- (renum-ecn-deq M1 DO D1 D1m D1' D1m') + %<- (renum-ecn-deq M2 DO D2 D2m D2' D2m') %. +%term _ + %pi (renum-ecn-deq (met/pair M1 M2) DO ([x] [di] ecn-deq/cn/app (D1 x di) (D2 x di)) ([x] [di] mecn-deq/cn/app (D1m x di) (D2m x di)) ([x] [dx] ecn-deq/cn/app (D1' x dx) (D2' x dx)) ([x] [dx] mecn-deq/cn/app (D1m' x dx) (D2m' x dx))) + %<- (renum-ecn-deq M1 DO D1 D1m D1' D1m') + %<- (renum-ecn-deq M2 DO D2 D2m D2' D2m') %. +%term _ + %pi (renum-ecn-deq (met/pair M1 M2) DO ([x] [di] ecn-deq/sgm-ext (D1 x di) (D2 x di)) ([x] [di] mecn-deq/sgm-ext (D1m x di) (D2m x di)) ([x] [dx] ecn-deq/sgm-ext (D1' x dx) (D2' x dx)) ([x] [dx] mecn-deq/sgm-ext (D1m' x dx) (D2m' x dx))) + %<- (renum-ecn-deq M1 DO D1 D1m D1' D1m') + %<- (renum-ecn-deq M2 DO D2 D2m D2' D2m') %. +%term _ + %pi (renum-ecn-deq (met/pair M1 M2) DO ([x] [di] ecn-deq/tp/cross (D1 x di) (D2 x di)) ([x] [di] mecn-deq/tp/cross (D1m x di) (D2m x di)) ([x] [dx] ecn-deq/tp/cross (D1' x dx) (D2' x dx)) ([x] [dx] mecn-deq/tp/cross (D1m' x dx) (D2m' x dx))) + %<- (renum-ecn-deq M1 DO D1 D1m D1' D1m') + %<- (renum-ecn-deq M2 DO D2 D2m D2' D2m') %. +%term _ + %pi (renum-ecn-deq (met/pair M1 M2) DO ([x] [di] ecn-deq/tp/arrow (D1 x di) (D2 x di)) ([x] [di] mecn-deq/tp/arrow (D1m x di) (D2m x di)) ([x] [dx] ecn-deq/tp/arrow (D1' x dx) (D2' x dx)) ([x] [dx] mecn-deq/tp/arrow (D1m' x dx) (D2m' x dx))) + %<- (renum-ecn-deq M1 DO D1 D1m D1' D1m') + %<- (renum-ecn-deq M2 DO D2 D2m D2' D2m') %. +%term _ + %pi (renum-ecn-deq (met/pair M1 M2) DO ([x] [di] ecn-deq/tp/sum (D1 x di) (D2 x di)) ([x] [di] mecn-deq/tp/sum (D1m x di) (D2m x di)) ([x] [dx] ecn-deq/tp/sum (D1' x dx) (D2' x dx)) ([x] [dx] mecn-deq/tp/sum (D1m' x dx) (D2m' x dx))) + %<- (renum-ecn-deq M1 DO D1 D1m D1' D1m') + %<- (renum-ecn-deq M2 DO D2 D2m D2' D2m') %. +%term _ + %pi (renum-ecn-deq (met/pair M1 M2) DO ([x] [di] ecn-deq/trans (D1 x di) (D2 x di)) ([x] [di] mecn-deq/trans (D1m x di) (D2m x di)) ([x] [dx] ecn-deq/trans (D1' x dx) (D2' x dx)) ([x] [dx] mecn-deq/trans (D1m' x dx) (D2m' x dx))) + %<- (renum-ecn-deq M1 DO D1 D1m D1' D1m') + %<- (renum-ecn-deq M2 DO D2 D2m D2' D2m') %. +%term _ + %pi (renum-ecn-deq (met/pair M1 M2) DO ([x] [di] ecn-deq/kd/unit (D1 x di) (D2 x di)) ([x] [di] mecn-deq/kd/unit (D1m x di) (D2m x di)) ([x] [dx] ecn-deq/kd/unit (D1' x dx) (D2' x dx)) ([x] [dx] mecn-deq/kd/unit (D1m' x dx) (D2m' x dx))) + %<- (renum-eofkd M1 DO D1 D1m D1' D1m') + %<- (renum-eofkd M2 DO D2 D2m D2' D2m') %. +%term _ + %pi (renum-ecn-deq (met/pair M1 M2) DO ([x] [di] ecn-deq/sub (D1 x di) (D2 x di)) ([x] [di] mecn-deq/sub (D1m x di) (D2m x di)) ([x] [dx] ecn-deq/sub (D1' x dx) (D2' x dx)) ([x] [dx] mecn-deq/sub (D1m' x dx) (D2m' x dx))) + %<- (renum-ecn-deq M1 DO D1 D1m D1' D1m') + %<- (renum-ekd-sub M2 DO D2 D2m D2' D2m') %. +%term _ + %pi (renum-ecn-deq (met/pair M1 M2) DO ([x] [di] ecn-deq/deq (D1 x di) (D2 x di)) ([x] [di] mecn-deq/deq (D1m x di) (D2m x di)) ([x] [dx] ecn-deq/deq (D1' x dx) (D2' x dx)) ([x] [dx] mecn-deq/deq (D1m' x dx) (D2m' x dx))) + %<- (renum-ecn-deq M1 DO D1 D1m D1' D1m') + %<- (renum-ekd-deq M2 DO D2 D2m D2' D2m') %. +%term _ + %pi (renum-ecn-deq (met/pair M2 M1) DO ([x] [di] ecn-deq/cn/lam ([y] [dy isvar y I] D2 x di y dy) (D1 x di)) ([x] [di] mecn-deq/cn/lam (D2m x di) (D1m x di)) ([x] [dx] ecn-deq/cn/lam (D2'' x dx) (D1' x dx)) ([x] [dx] mecn-deq/cn/lam (D2m'' x dx) (D1m' x dx))) + %<- (renum-ekd-deq M1 DO D1 D1m D1' D1m') + %<- ({x} {dx} ekd-deq-ordered (D1 x dx) (DO' x dx)) + %<- (cxt-inc-2 DO' DO DOe' DOe) + %<- ({x} {dx} renum-ecn-deq M2 ([y] [dy] cxt-ordered/cons (DOe' x dx y dy)) (D2 x dx) (D2m x dx) (D2' x dx) (D2m' x dx)) + %<- ({y} {dy} renum-ecn-deq M2 ([x] [dx] cxt-ordered/cons (DOe x dx y dy)) ([x] [dx] D2' x dx y dy) ([x] [dx] D2m' x dx y dy) ([x] [dx] D2'' x dx y dy) ([x] [dx] D2m'' x dx y dy)) %. +%term _ + %pi (renum-ecn-deq (met/pair M2 M1) DO ([x] [di] ecn-deq/cn/mu ([y] [dy isvar y I] D2 x di y dy) (D1 x di)) ([x] [di] mecn-deq/cn/mu (D2m x di) (D1m x di)) ([x] [dx] ecn-deq/cn/mu (D2'' x dx) (D1' x dx)) ([x] [dx] mecn-deq/cn/mu (D2m'' x dx) (D1m' x dx))) + %<- (renum-ekd-deq M1 DO D1 D1m D1' D1m') + %<- ({x} {dx} ekd-deq-ordered (D1 x dx) (DO' x dx)) + %<- (cxt-inc-2 DO' DO DOe' DOe) + %<- ({x} {dx} renum-ecn-deq M2 ([y] [dy] cxt-ordered/cons (DOe' x dx y dy)) (D2 x dx) (D2m x dx) (D2' x dx) (D2m' x dx)) + %<- ({y} {dy} renum-ecn-deq M2 ([x] [dx] cxt-ordered/cons (DOe x dx y dy)) ([x] [dx] D2' x dx y dy) ([x] [dx] D2m' x dx y dy) ([x] [dx] D2'' x dx y dy) ([x] [dx] D2m'' x dx y dy)) %. +%term _ + %pi (renum-ecn-deq (met/pair M2 M1) DO ([x] [di] ecn-deq/tp/forall ([y] [dy isvar y I] D2 x di y dy) (D1 x di)) ([x] [di] mecn-deq/tp/forall (D2m x di) (D1m x di)) ([x] [dx] ecn-deq/tp/forall (D2'' x dx) (D1' x dx)) ([x] [dx] mecn-deq/tp/forall (D2m'' x dx) (D1m' x dx))) + %<- (renum-ekd-deq M1 DO D1 D1m D1' D1m') + %<- ({x} {dx} ekd-deq-ordered (D1 x dx) (DO' x dx)) + %<- (cxt-inc-2 DO' DO DOe' DOe) + %<- ({x} {dx} renum-ecn-deq M2 ([y] [dy] cxt-ordered/cons (DOe' x dx y dy)) (D2 x dx) (D2m x dx) (D2' x dx) (D2m' x dx)) + %<- ({y} {dy} renum-ecn-deq M2 ([x] [dx] cxt-ordered/cons (DOe x dx y dy)) ([x] [dx] D2' x dx y dy) ([x] [dx] D2m' x dx y dy) ([x] [dx] D2'' x dx y dy) ([x] [dx] D2m'' x dx y dy)) %. +%term _ + %pi (renum-ecn-deq (met/pair M1 (met/pair M3 M2)) DO ([x] [di] ecn-deq/pi-ext (D1 x di) (D3 x di) ([y] [dy isvar y I] D2 x di y dy)) ([x] [di] mecn-deq/pi-ext (D1m x di) (D3m x di) (D2m x di)) ([x] [dx] ecn-deq/pi-ext (D1' x dx) (D3' x dx) (D2'' x dx)) ([x] [dx] mecn-deq/pi-ext (D1m' x dx) (D3m' x dx) (D2m'' x dx))) + %<- (renum-eofkd M1 DO D1 D1m D1' D1m') + %<- (renum-eofkd M3 DO D3 D3m D3' D3m') + %<- ({x} {dx} eofkd-ordered (D1 x dx) (DO' x dx)) + %<- (cxt-inc-2 DO' DO DOe' DOe) + %<- ({x} {dx} renum-ecn-deq M2 ([y] [dy] cxt-ordered/cons (DOe' x dx y dy)) (D2 x dx) (D2m x dx) (D2' x dx) (D2m' x dx)) + %<- ({y} {dy} renum-ecn-deq M2 ([x] [dx] cxt-ordered/cons (DOe x dx y dy)) ([x] [dx] D2' x dx y dy) ([x] [dx] D2m' x dx y dy) ([x] [dx] D2'' x dx y dy) ([x] [dx] D2m'' x dx y dy)) %. +%term _ + %pi (renum-ecn-deq (met/pair M1 M2) DO ([x] [di] ecn-deq/pi-ext-2 (D1 x di) ([y] [dy isvar y I] D2 x di y dy)) ([x] [di] mecn-deq/pi-ext-2 (D1m x di) (D2m x di)) ([x] [dx] ecn-deq/pi-ext-2 (D1' x dx) (D2'' x dx)) ([x] [dx] mecn-deq/pi-ext-2 (D1m' x dx) (D2m'' x dx))) + %<- (renum-ecn-deq M1 DO D1 D1m D1' D1m') + %<- ({x} {dx} ecn-deq-ordered (D1 x dx) (DO' x dx)) + %<- (cxt-inc-2 DO' DO DOe' DOe) + %<- ({x} {dx} renum-ecn-deq M2 ([y] [dy] cxt-ordered/cons (DOe' x dx y dy)) (D2 x dx) (D2m x dx) (D2' x dx) (D2m' x dx)) + %<- ({y} {dy} renum-ecn-deq M2 ([x] [dx] cxt-ordered/cons (DOe x dx y dy)) ([x] [dx] D2' x dx y dy) ([x] [dx] D2m' x dx y dy) ([x] [dx] D2'' x dx y dy) ([x] [dx] D2m'' x dx y dy)) %. +%worlds (cn-block ovar-block can-mofkd-block) (renum-ekd-wf _ _ _ _ _ _) (renum-ekd-deq _ _ _ _ _ _) (renum-ekd-sub _ _ _ _ _ _) (renum-eofkd _ _ _ _ _ _) (renum-ecn-deq _ _ _ _ _ _) %. +%total (D1 D2 D3 D4 D5) (renum-ekd-wf D1 _ _ _ _ _) (renum-ekd-deq D3 _ _ _ _ _) (renum-ekd-sub D5 _ _ _ _ _) (renum-eofkd D2 _ _ _ _ _) (renum-ecn-deq D4 _ _ _ _ _) %. +%sort wkn-cxt-bounded %. +%mode wkn-cxt-bounded %in %in %in %out %. +%term _ wkn-cxt-bounded cxt-append/nil cxt-append/nil (cxt-bounded/cons (cxt-bounded/nil D1) (cxt-precedes/i D1' D2 DL)) (cxt-bounded/nil D2) %. +%term _ + %pi (wkn-cxt-bounded cxt-append/nil cxt-append/nil (cxt-bounded/cons (cxt-bounded/cons DB DP) DP') (cxt-bounded/cons DB DP'')) + %<- (cxt-precedes-trans DP DP' DP'') %. +%term _ + %pi (wkn-cxt-bounded (cxt-append/cons DA) (cxt-append/cons DA') (cxt-bounded/cons (cxt-bounded/cons DB DP) DP') (cxt-bounded/cons DB' DP')) + %<- (wkn-cxt-bounded DA DA' (cxt-bounded/cons DB DP) DB') %. +%worlds (cn-block ovar-block) (wkn-cxt-bounded _ _ _ _) %. +%total (D1) (wkn-cxt-bounded D1 _ _ _) %. +%sort wkn-cxt-ordered %. +%mode wkn-cxt-ordered %in %in %in %out %. +%term _ wkn-cxt-ordered cxt-append/nil cxt-append/nil (cxt-ordered/cons (cxt-bounded/nil _)) cxt-ordered/nil %. +%term _ wkn-cxt-ordered cxt-append/nil cxt-append/nil (cxt-ordered/cons (cxt-bounded/cons DB _)) (cxt-ordered/cons DB) %. +%term _ + %pi (wkn-cxt-ordered (cxt-append/cons DA) (cxt-append/cons DA') (cxt-ordered/cons DB) (cxt-ordered/cons DB')) + %<- (wkn-cxt-bounded DA DA' DB DB') %. +%worlds (cn-block ovar-block) (wkn-cxt-ordered _ _ _ _) %. +%total (D1) (wkn-cxt-ordered D1 _ _ _) %. +%sort wkn-cxt-lookup %. +%mode wkn-cxt-lookup %in %in %in %in %out %. +%term _ + %pi (wkn-cxt-lookup cxt-append/nil cxt-append/nil (cxt-ordered/cons DB) DL (cxt-lookup/miss DL DP DB)) + %<- (cxt-lookup-precedes DL DB DP) %. +%term _ + %pi (wkn-cxt-lookup (cxt-append/cons DA) (cxt-append/cons DA') (cxt-ordered/cons (cxt-bounded/cons DB DPb)) (cxt-lookup/miss DL DP _) (cxt-lookup/miss DL' DP (cxt-bounded/cons DB DPb))) + %<- (wkn-cxt-lookup DA DA' (cxt-ordered/cons DB) DL DL') %. +%term _ wkn-cxt-lookup (cxt-append/cons DA) (cxt-append/cons DA') (cxt-ordered/cons DB') (cxt-lookup/hit DB) (cxt-lookup/hit DB') %. +%worlds (cn-block ovar-block) (wkn-cxt-lookup _ _ _ _ _) %. +%total (D1) (wkn-cxt-lookup D1 _ _ _ _) %. +%sort wkn-ekd-wf {Mm met} %. +%mode wkn-ekd-wf %in %in %in %in %in %in %out %out %. +%sort wkn-ekd-deq {Mm met} %. +%mode wkn-ekd-deq %in %in %in %in %in %in %out %out %. +%sort wkn-ekd-sub {Mm met} %. +%mode wkn-ekd-sub %in %in %in %in %in %in %out %out %. +%sort wkn-eofkd {Mm met} %. +%mode wkn-eofkd %in %in %in %in %in %in %out %out %. +%sort wkn-ecn-deq {Mm met} %. +%mode wkn-ecn-deq %in %in %in %in %in %in %out %out %. +%term _ wkn-ekd-wf met/unit (%the (cxt-append G1 G2 G) DA) (%the (cxt-append (cxt/cons G1 C1 K1) G2 G') DA') DOg (ekd-wf/kd/unit (%the (cxt-ordered G) DO')) mekd-wf/kd/unit (ekd-wf/kd/unit DOg) mekd-wf/kd/unit %. +%term _ wkn-ekd-wf met/unit (%the (cxt-append G1 G2 G) DA) (%the (cxt-append (cxt/cons G1 C1 K1) G2 G') DA') DOg (ekd-wf/kd/type (%the (cxt-ordered G) DO')) mekd-wf/kd/type (ekd-wf/kd/type DOg) mekd-wf/kd/type %. +%term _ + %pi (wkn-ekd-wf (met/sing Mm) DA DA' DOg (ekd-wf/kd/sing D1) (mekd-wf/kd/sing DM1) (ekd-wf/kd/sing D2) (mekd-wf/kd/sing DM2)) + %<- (wkn-eofkd Mm DA DA' DOg D1 DM1 D2 DM2) %. +%term _ + %pi (wkn-ekd-wf (met/pair M1 M2) DA DA' DOg (ekd-wf/kd/sgm D1 D2) (mekd-wf/kd/sgm D1m D2m) (ekd-wf/kd/sgm D1' D2'') (mekd-wf/kd/sgm D1m' D2m'')) + %<- (wkn-ekd-wf M1 DA DA' DOg D1 D1m D1' D1m') + %<- (cxt-inc _ DOg DOe) + %<- ({x} {dx} wkn-cxt-ordered (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (DOen x dx)) + %<- (renum-ekd-wf M2 DOen D2 D2m D2' D2m') + %<- ({x} {dx} wkn-ekd-wf M2 (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (D2' x dx) (D2m' x dx) (D2'' x dx) (D2m'' x dx)) %. +%term _ + %pi (wkn-ekd-wf (met/pair M1 M2) DA DA' DOg (ekd-wf/kd/pi D1 D2) (mekd-wf/kd/pi D1m D2m) (ekd-wf/kd/pi D1' D2'') (mekd-wf/kd/pi D1m' D2m'')) + %<- (wkn-ekd-wf M1 DA DA' DOg D1 D1m D1' D1m') + %<- (cxt-inc _ DOg DOe) + %<- ({x} {dx} wkn-cxt-ordered (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (DOen x dx)) + %<- (renum-ekd-wf M2 DOen D2 D2m D2' D2m') + %<- ({x} {dx} wkn-ekd-wf M2 (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (D2' x dx) (D2m' x dx) (D2'' x dx) (D2m'' x dx)) %. +%term _ wkn-ekd-deq met/unit (%the (cxt-append G1 G2 G) DA) (%the (cxt-append (cxt/cons G1 C1 K1) G2 G') DA') DOg (ekd-deq/kd/unit (%the (cxt-ordered G) DO')) mekd-deq/kd/unit (ekd-deq/kd/unit DOg) mekd-deq/kd/unit %. +%term _ wkn-ekd-deq met/unit (%the (cxt-append G1 G2 G) DA) (%the (cxt-append (cxt/cons G1 C1 K1) G2 G') DA') DOg (ekd-deq/kd/type (%the (cxt-ordered G) DO')) mekd-deq/kd/type (ekd-deq/kd/type DOg) mekd-deq/kd/type %. +%term _ + %pi (wkn-ekd-deq (met/sing Mm) DA DA' DOg (ekd-deq/kd/sing D1) (mekd-deq/kd/sing DM1) (ekd-deq/kd/sing D2) (mekd-deq/kd/sing DM2)) + %<- (wkn-ecn-deq Mm DA DA' DOg D1 DM1 D2 DM2) %. +%term _ + %pi (wkn-ekd-deq (met/pair M1 M2) DA DA' DOg (ekd-deq/kd/sgm D1 D2) (mekd-deq/kd/sgm D1m D2m) (ekd-deq/kd/sgm D1' D2'') (mekd-deq/kd/sgm D1m' D2m'')) + %<- (wkn-ekd-deq M1 DA DA' DOg D1 D1m D1' D1m') + %<- (cxt-inc _ DOg DOe) + %<- ({x} {dx} wkn-cxt-ordered (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (DOen x dx)) + %<- (renum-ekd-deq M2 DOen D2 D2m D2' D2m') + %<- ({x} {dx} wkn-ekd-deq M2 (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (D2' x dx) (D2m' x dx) (D2'' x dx) (D2m'' x dx)) %. +%term _ + %pi (wkn-ekd-deq (met/pair M1 M2) DA DA' DOg (ekd-deq/kd/pi D1 D2) (mekd-deq/kd/pi D1m D2m) (ekd-deq/kd/pi D1' D2'') (mekd-deq/kd/pi D1m' D2m'')) + %<- (wkn-ekd-deq M1 DA DA' DOg D1 D1m D1' D1m') + %<- (cxt-inc _ DOg DOe) + %<- ({x} {dx} wkn-cxt-ordered (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (DOen x dx)) + %<- (renum-ekd-deq M2 DOen D2 D2m D2' D2m') + %<- ({x} {dx} wkn-ekd-deq M2 (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (D2' x dx) (D2m' x dx) (D2'' x dx) (D2m'' x dx)) %. +%term _ wkn-ekd-sub met/unit (%the (cxt-append G1 G2 G) DA) (%the (cxt-append (cxt/cons G1 C1 K1) G2 G') DA') DOg (ekd-sub/kd/unit (%the (cxt-ordered G) DO')) mekd-sub/kd/unit (ekd-sub/kd/unit DOg) mekd-sub/kd/unit %. +%term _ wkn-ekd-sub met/unit (%the (cxt-append G1 G2 G) DA) (%the (cxt-append (cxt/cons G1 C1 K1) G2 G') DA') DOg (ekd-sub/kd/type (%the (cxt-ordered G) DO')) mekd-sub/kd/type (ekd-sub/kd/type DOg) mekd-sub/kd/type %. +%term _ + %pi (wkn-ekd-sub (met/sing Mm) DA DA' DOg (ekd-sub/kd/sing-kd/sing D1) (mekd-sub/kd/sing-kd/sing DM1) (ekd-sub/kd/sing-kd/sing D2) (mekd-sub/kd/sing-kd/sing DM2)) + %<- (wkn-ecn-deq Mm DA DA' DOg D1 DM1 D2 DM2) %. +%term _ + %pi (wkn-ekd-sub (met/sing Mm) DA DA' DOg (ekd-sub/kd/sing-kd/type D1) (mekd-sub/kd/sing-kd/type DM1) (ekd-sub/kd/sing-kd/type D2) (mekd-sub/kd/sing-kd/type DM2)) + %<- (wkn-eofkd Mm DA DA' DOg D1 DM1 D2 DM2) %. +%term _ + %pi (wkn-ekd-sub (met/pair M1 (met/pair M2 M3)) DA DA' DOg (ekd-sub/kd/sgm D1 D2 D3) (mekd-sub/kd/sgm D1m D2m D3m) (ekd-sub/kd/sgm D1' D2'' D3'') (mekd-sub/kd/sgm D1m' D2m'' D3m'')) + %<- (wkn-ekd-sub M1 DA DA' DOg D1 D1m D1' D1m') + %<- (cxt-inc-duece _ _ DOg DOe DOe') + %<- ({x} {dx isvar x I} wkn-cxt-ordered (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (DOen x dx)) + %<- (renum-ekd-sub M2 DOen D2 D2m D2' D2m') + %<- ({x} {dx} wkn-ekd-sub M2 (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (D2' x dx) (D2m' x dx) (D2'' x dx) (D2m'' x dx)) + %<- ({x} {dx isvar x I} wkn-cxt-ordered (cxt-append/cons DA) (cxt-append/cons DA') (DOe' x dx) (DOen' x dx)) + %<- (renum-ekd-wf M3 DOen' D3 D3m D3' D3m') + %<- ({x} {dx} wkn-ekd-wf M3 (cxt-append/cons DA) (cxt-append/cons DA') (DOe' x dx) (D3' x dx) (D3m' x dx) (D3'' x dx) (D3m'' x dx)) %. +%term _ + %pi (wkn-ekd-sub (met/pair M1 (met/pair M2 M3)) DA DA' DOg (ekd-sub/kd/pi D1 D2 D3) (mekd-sub/kd/pi D1m D2m D3m) (ekd-sub/kd/pi D1' D2'' D3'') (mekd-sub/kd/pi D1m' D2m'' D3m'')) + %<- (wkn-ekd-sub M1 DA DA' DOg D1 D1m D1' D1m') + %<- (cxt-inc-duece _ _ DOg DOe DOe') + %<- ({x} {dx isvar x I} wkn-cxt-ordered (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (DOen x dx)) + %<- (renum-ekd-sub M2 DOen D2 D2m D2' D2m') + %<- ({x} {dx} wkn-ekd-sub M2 (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (D2' x dx) (D2m' x dx) (D2'' x dx) (D2m'' x dx)) + %<- ({x} {dx isvar x I} wkn-cxt-ordered (cxt-append/cons DA) (cxt-append/cons DA') (DOe' x dx) (DOen' x dx)) + %<- (renum-ekd-wf M3 DOen' D3 D3m D3' D3m') + %<- ({x} {dx} wkn-ekd-wf M3 (cxt-append/cons DA) (cxt-append/cons DA') (DOe' x dx) (D3' x dx) (D3m' x dx) (D3'' x dx) (D3m'' x dx)) %. +%term _ wkn-eofkd _ DA DA' DOg (eofkd/closed DC (%the (cxt-ordered G) DO')) (meofkd/closed DCm) (eofkd/closed DC DOg) (meofkd/closed DCm) %. +%term _ + %pi (wkn-eofkd _ DA DA' DOg (eofkd/var D1) _ (eofkd/var D2) meofkd/var) + %<- (wkn-cxt-lookup DA DA' DOg D1 D2) %. +%term _ wkn-eofkd met/unit (%the (cxt-append G1 G2 G) DA) (%the (cxt-append (cxt/cons G1 C1 K1) G2 G') DA') DOg (eofkd/cn/unit (%the (cxt-ordered G) DO')) meofkd/cn/unit (eofkd/cn/unit DOg) meofkd/cn/unit %. +%term _ wkn-eofkd met/unit (%the (cxt-append G1 G2 G) DA) (%the (cxt-append (cxt/cons G1 C1 K1) G2 G') DA') DOg (eofkd/tp/unit (%the (cxt-ordered G) DO')) meofkd/tp/unit (eofkd/tp/unit DOg) meofkd/tp/unit %. +%term _ wkn-eofkd met/unit (%the (cxt-append G1 G2 G) DA) (%the (cxt-append (cxt/cons G1 C1 K1) G2 G') DA') DOg (eofkd/tp/tagged (%the (cxt-ordered G) DO')) meofkd/tp/tagged (eofkd/tp/tagged DOg) meofkd/tp/tagged %. +%term _ + %pi (wkn-eofkd (met/sing Mm) DA DA' DOg (eofkd/kd/sing D1) (meofkd/kd/sing DM1) (eofkd/kd/sing D2) (meofkd/kd/sing DM2)) + %<- (wkn-eofkd Mm DA DA' DOg D1 DM1 D2 DM2) %. +%term _ + %pi (wkn-eofkd (met/sing Mm) DA DA' DOg (eofkd/cn/pj1 D1) (meofkd/cn/pj1 DM1) (eofkd/cn/pj1 D2) (meofkd/cn/pj1 DM2)) + %<- (wkn-eofkd Mm DA DA' DOg D1 DM1 D2 DM2) %. +%term _ + %pi (wkn-eofkd (met/sing Mm) DA DA' DOg (eofkd/cn/pj2 D1) (meofkd/cn/pj2 DM1) (eofkd/cn/pj2 D2) (meofkd/cn/pj2 DM2)) + %<- (wkn-eofkd Mm DA DA' DOg D1 DM1 D2 DM2) %. +%term _ + %pi (wkn-eofkd (met/sing Mm) DA DA' DOg (eofkd/tp/ref D1) (meofkd/tp/ref DM1) (eofkd/tp/ref D2) (meofkd/tp/ref DM2)) + %<- (wkn-eofkd Mm DA DA' DOg D1 DM1 D2 DM2) %. +%term _ + %pi (wkn-eofkd (met/sing Mm) DA DA' DOg (eofkd/tp/tag D1) (meofkd/tp/tag DM1) (eofkd/tp/tag D2) (meofkd/tp/tag DM2)) + %<- (wkn-eofkd Mm DA DA' DOg D1 DM1 D2 DM2) %. +%term _ + %pi (wkn-eofkd (met/pair M1 M2) DA DA' DOg (eofkd/cn/pair D1 D2) (meofkd/cn/pair DM1 DM2) (eofkd/cn/pair D1' D2') (meofkd/cn/pair DM1' DM2')) + %<- (wkn-eofkd M1 DA DA' DOg D1 DM1 D1' DM1') + %<- (wkn-eofkd M2 DA DA' DOg D2 DM2 D2' DM2') %. +%term _ + %pi (wkn-eofkd (met/pair M1 M2) DA DA' DOg (eofkd/cn/app D1 D2) (meofkd/cn/app DM1 DM2) (eofkd/cn/app D1' D2') (meofkd/cn/app DM1' DM2')) + %<- (wkn-eofkd M1 DA DA' DOg D1 DM1 D1' DM1') + %<- (wkn-eofkd M2 DA DA' DOg D2 DM2 D2' DM2') %. +%term _ + %pi (wkn-eofkd (met/pair M1 M2) DA DA' DOg (eofkd/sgm-ext D1 D2) (meofkd/sgm-ext DM1 DM2) (eofkd/sgm-ext D1' D2') (meofkd/sgm-ext DM1' DM2')) + %<- (wkn-eofkd M1 DA DA' DOg D1 DM1 D1' DM1') + %<- (wkn-eofkd M2 DA DA' DOg D2 DM2 D2' DM2') %. +%term _ + %pi (wkn-eofkd (met/pair M1 M2) DA DA' DOg (eofkd/tp/cross D1 D2) (meofkd/tp/cross DM1 DM2) (eofkd/tp/cross D1' D2') (meofkd/tp/cross DM1' DM2')) + %<- (wkn-eofkd M1 DA DA' DOg D1 DM1 D1' DM1') + %<- (wkn-eofkd M2 DA DA' DOg D2 DM2 D2' DM2') %. +%term _ + %pi (wkn-eofkd (met/pair M1 M2) DA DA' DOg (eofkd/tp/arrow D1 D2) (meofkd/tp/arrow DM1 DM2) (eofkd/tp/arrow D1' D2') (meofkd/tp/arrow DM1' DM2')) + %<- (wkn-eofkd M1 DA DA' DOg D1 DM1 D1' DM1') + %<- (wkn-eofkd M2 DA DA' DOg D2 DM2 D2' DM2') %. +%term _ + %pi (wkn-eofkd (met/pair M1 M2) DA DA' DOg (eofkd/tp/sum D1 D2) (meofkd/tp/sum DM1 DM2) (eofkd/tp/sum D1' D2') (meofkd/tp/sum DM1' DM2')) + %<- (wkn-eofkd M1 DA DA' DOg D1 DM1 D1' DM1') + %<- (wkn-eofkd M2 DA DA' DOg D2 DM2 D2' DM2') %. +%term _ + %pi (wkn-eofkd (met/pair M1 M2) DA DA' DOg (eofkd/sub D1 D2) (meofkd/sub DM1 DM2) (eofkd/sub D1' D2') (meofkd/sub DM1' DM2')) + %<- (wkn-eofkd M1 DA DA' DOg D1 DM1 D1' DM1') + %<- (wkn-ekd-sub M2 DA DA' DOg D2 DM2 D2' DM2') %. +%term _ + %pi (wkn-eofkd (met/pair M1 M2) DA DA' DOg (eofkd/deq D1 D2) (meofkd/deq DM1 DM2) (eofkd/deq D1' D2') (meofkd/deq DM1' DM2')) + %<- (wkn-eofkd M1 DA DA' DOg D1 DM1 D1' DM1') + %<- (wkn-ekd-deq M2 DA DA' DOg D2 DM2 D2' DM2') %. +%term _ + %pi (wkn-eofkd (met/pair M2 M1) DA DA' DOg (eofkd/cn/lam D2 D1) (meofkd/cn/lam D2m D1m) (eofkd/cn/lam D2'' D1') (meofkd/cn/lam D2m'' D1m')) + %<- (wkn-ekd-wf M1 DA DA' DOg D1 D1m D1' D1m') + %<- (cxt-inc _ DOg DOe) + %<- ({x} {dx} wkn-cxt-ordered (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (DOen x dx)) + %<- (renum-eofkd M2 DOen D2 D2m D2' D2m') + %<- ({x} {dx} wkn-eofkd M2 (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (D2' x dx) (D2m' x dx) (D2'' x dx) (D2m'' x dx)) %. +%term _ + %pi (wkn-eofkd (met/pair M2 M1) DA DA' DOg (eofkd/tp/forall D2 D1) (meofkd/tp/forall D2m D1m) (eofkd/tp/forall D2'' D1') (meofkd/tp/forall D2m'' D1m')) + %<- (wkn-ekd-wf M1 DA DA' DOg D1 D1m D1' D1m') + %<- (cxt-inc _ DOg DOe) + %<- ({x} {dx} wkn-cxt-ordered (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (DOen x dx)) + %<- (renum-eofkd M2 DOen D2 D2m D2' D2m') + %<- ({x} {dx} wkn-eofkd M2 (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (D2' x dx) (D2m' x dx) (D2'' x dx) (D2m'' x dx)) %. +%term _ + %pi (wkn-eofkd (met/pair M2 M1) DA DA' DOg (eofkd/cn/mu D2 D1) (meofkd/cn/mu D2m D1m) (eofkd/cn/mu D2'' D1') (meofkd/cn/mu D2m'' D1m')) + %<- (wkn-ekd-wf M1 DA DA' DOg D1 D1m D1' D1m') + %<- (cxt-inc _ DOg DOe) + %<- ({x} {dx} wkn-cxt-ordered (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (DOen x dx)) + %<- (renum-eofkd M2 DOen D2 D2m D2' D2m') + %<- ({x} {dx} wkn-eofkd M2 (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (D2' x dx) (D2m' x dx) (D2'' x dx) (D2m'' x dx)) %. +%term _ + %pi (wkn-eofkd (met/pair M1 M2) DA DA' DOg (eofkd/pi-ext D1 D2) (meofkd/pi-ext D1m D2m) (eofkd/pi-ext D1' D2'') (meofkd/pi-ext D1m' D2m'')) + %<- (wkn-eofkd M1 DA DA' DOg D1 D1m D1' D1m') + %<- (cxt-inc _ DOg DOe) + %<- ({x} {dx} wkn-cxt-ordered (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (DOen x dx)) + %<- (renum-eofkd M2 DOen D2 D2m D2' D2m') + %<- ({x} {dx} wkn-eofkd M2 (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (D2' x dx) (D2m' x dx) (D2'' x dx) (D2m'' x dx)) %. +%term _ wkn-ecn-deq met/unit (%the (cxt-append G1 G2 G) DA) (%the (cxt-append (cxt/cons G1 C1 K1) G2 G') DA') DOg (ecn-deq/cn/unit (%the (cxt-ordered G) DO')) mecn-deq/cn/unit (ecn-deq/cn/unit DOg) mecn-deq/cn/unit %. +%term _ wkn-ecn-deq met/unit (%the (cxt-append G1 G2 G) DA) (%the (cxt-append (cxt/cons G1 C1 K1) G2 G') DA') DOg (ecn-deq/tp/unit (%the (cxt-ordered G) DO')) mecn-deq/tp/unit (ecn-deq/tp/unit DOg) mecn-deq/tp/unit %. +%term _ wkn-ecn-deq met/unit (%the (cxt-append G1 G2 G) DA) (%the (cxt-append (cxt/cons G1 C1 K1) G2 G') DA') DOg (ecn-deq/tp/tagged (%the (cxt-ordered G) DO')) mecn-deq/tp/tagged (ecn-deq/tp/tagged DOg) mecn-deq/tp/tagged %. +%term _ + %pi (wkn-ecn-deq (met/sing Mm) DA DA' DOg (ecn-deq/kd/sing D1) (mecn-deq/kd/sing DM1) (ecn-deq/kd/sing D2) (mecn-deq/kd/sing DM2)) + %<- (wkn-eofkd Mm DA DA' DOg D1 DM1 D2 DM2) %. +%term _ + %pi (wkn-ecn-deq (met/sing Mm) DA DA' DOg (ecn-deq/refl D1) (mecn-deq/refl DM1) (ecn-deq/refl D2) (mecn-deq/refl DM2)) + %<- (wkn-eofkd Mm DA DA' DOg D1 DM1 D2 DM2) %. +%term _ + %pi (wkn-ecn-deq (met/sing Mm) DA DA' DOg (ecn-deq/cn/pj1 D1) (mecn-deq/cn/pj1 DM1) (ecn-deq/cn/pj1 D2) (mecn-deq/cn/pj1 DM2)) + %<- (wkn-ecn-deq Mm DA DA' DOg D1 DM1 D2 DM2) %. +%term _ + %pi (wkn-ecn-deq (met/sing Mm) DA DA' DOg (ecn-deq/cn/pj2 D1) (mecn-deq/cn/pj2 DM1) (ecn-deq/cn/pj2 D2) (mecn-deq/cn/pj2 DM2)) + %<- (wkn-ecn-deq Mm DA DA' DOg D1 DM1 D2 DM2) %. +%term _ + %pi (wkn-ecn-deq (met/sing Mm) DA DA' DOg (ecn-deq/tp/ref D1) (mecn-deq/tp/ref DM1) (ecn-deq/tp/ref D2) (mecn-deq/tp/ref DM2)) + %<- (wkn-ecn-deq Mm DA DA' DOg D1 DM1 D2 DM2) %. +%term _ + %pi (wkn-ecn-deq (met/sing Mm) DA DA' DOg (ecn-deq/tp/tag D1) (mecn-deq/tp/tag DM1) (ecn-deq/tp/tag D2) (mecn-deq/tp/tag DM2)) + %<- (wkn-ecn-deq Mm DA DA' DOg D1 DM1 D2 DM2) %. +%term _ + %pi (wkn-ecn-deq (met/sing Mm) DA DA' DOg (ecn-deq/sym D1) (mecn-deq/sym DM1) (ecn-deq/sym D2) (mecn-deq/sym DM2)) + %<- (wkn-ecn-deq Mm DA DA' DOg D1 DM1 D2 DM2) %. +%term _ + %pi (wkn-ecn-deq (met/pair M1 M2) DA DA' DOg (ecn-deq/cn/pair D1 D2) (mecn-deq/cn/pair DM1 DM2) (ecn-deq/cn/pair D1' D2') (mecn-deq/cn/pair DM1' DM2')) + %<- (wkn-ecn-deq M1 DA DA' DOg D1 DM1 D1' DM1') + %<- (wkn-ecn-deq M2 DA DA' DOg D2 DM2 D2' DM2') %. +%term _ + %pi (wkn-ecn-deq (met/pair M1 M2) DA DA' DOg (ecn-deq/cn/app D1 D2) (mecn-deq/cn/app DM1 DM2) (ecn-deq/cn/app D1' D2') (mecn-deq/cn/app DM1' DM2')) + %<- (wkn-ecn-deq M1 DA DA' DOg D1 DM1 D1' DM1') + %<- (wkn-ecn-deq M2 DA DA' DOg D2 DM2 D2' DM2') %. +%term _ + %pi (wkn-ecn-deq (met/pair M1 M2) DA DA' DOg (ecn-deq/sgm-ext D1 D2) (mecn-deq/sgm-ext DM1 DM2) (ecn-deq/sgm-ext D1' D2') (mecn-deq/sgm-ext DM1' DM2')) + %<- (wkn-ecn-deq M1 DA DA' DOg D1 DM1 D1' DM1') + %<- (wkn-ecn-deq M2 DA DA' DOg D2 DM2 D2' DM2') %. +%term _ + %pi (wkn-ecn-deq (met/pair M1 M2) DA DA' DOg (ecn-deq/tp/cross D1 D2) (mecn-deq/tp/cross DM1 DM2) (ecn-deq/tp/cross D1' D2') (mecn-deq/tp/cross DM1' DM2')) + %<- (wkn-ecn-deq M1 DA DA' DOg D1 DM1 D1' DM1') + %<- (wkn-ecn-deq M2 DA DA' DOg D2 DM2 D2' DM2') %. +%term _ + %pi (wkn-ecn-deq (met/pair M1 M2) DA DA' DOg (ecn-deq/tp/arrow D1 D2) (mecn-deq/tp/arrow DM1 DM2) (ecn-deq/tp/arrow D1' D2') (mecn-deq/tp/arrow DM1' DM2')) + %<- (wkn-ecn-deq M1 DA DA' DOg D1 DM1 D1' DM1') + %<- (wkn-ecn-deq M2 DA DA' DOg D2 DM2 D2' DM2') %. +%term _ + %pi (wkn-ecn-deq (met/pair M1 M2) DA DA' DOg (ecn-deq/tp/sum D1 D2) (mecn-deq/tp/sum DM1 DM2) (ecn-deq/tp/sum D1' D2') (mecn-deq/tp/sum DM1' DM2')) + %<- (wkn-ecn-deq M1 DA DA' DOg D1 DM1 D1' DM1') + %<- (wkn-ecn-deq M2 DA DA' DOg D2 DM2 D2' DM2') %. +%term _ + %pi (wkn-ecn-deq (met/pair M1 M2) DA DA' DOg (ecn-deq/trans D1 D2) (mecn-deq/trans DM1 DM2) (ecn-deq/trans D1' D2') (mecn-deq/trans DM1' DM2')) + %<- (wkn-ecn-deq M1 DA DA' DOg D1 DM1 D1' DM1') + %<- (wkn-ecn-deq M2 DA DA' DOg D2 DM2 D2' DM2') %. +%term _ + %pi (wkn-ecn-deq (met/pair M1 M2) DA DA' DOg (ecn-deq/kd/unit D1 D2) (mecn-deq/kd/unit DM1 DM2) (ecn-deq/kd/unit D1' D2') (mecn-deq/kd/unit DM1' DM2')) + %<- (wkn-eofkd M1 DA DA' DOg D1 DM1 D1' DM1') + %<- (wkn-eofkd M2 DA DA' DOg D2 DM2 D2' DM2') %. +%term _ + %pi (wkn-ecn-deq (met/pair M1 M2) DA DA' DOg (ecn-deq/sub D1 D2) (mecn-deq/sub DM1 DM2) (ecn-deq/sub D1' D2') (mecn-deq/sub DM1' DM2')) + %<- (wkn-ecn-deq M1 DA DA' DOg D1 DM1 D1' DM1') + %<- (wkn-ekd-sub M2 DA DA' DOg D2 DM2 D2' DM2') %. +%term _ + %pi (wkn-ecn-deq (met/pair M1 M2) DA DA' DOg (ecn-deq/deq D1 D2) (mecn-deq/deq DM1 DM2) (ecn-deq/deq D1' D2') (mecn-deq/deq DM1' DM2')) + %<- (wkn-ecn-deq M1 DA DA' DOg D1 DM1 D1' DM1') + %<- (wkn-ekd-deq M2 DA DA' DOg D2 DM2 D2' DM2') %. +%term _ + %pi (wkn-ecn-deq (met/pair M2 M1) DA DA' DOg (ecn-deq/cn/lam D2 D1) (mecn-deq/cn/lam D2m D1m) (ecn-deq/cn/lam D2'' D1') (mecn-deq/cn/lam D2m'' D1m')) + %<- (wkn-ekd-deq M1 DA DA' DOg D1 D1m D1' D1m') + %<- (cxt-inc _ DOg DOe) + %<- ({x} {dx} wkn-cxt-ordered (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (DOen x dx)) + %<- (renum-ecn-deq M2 DOen D2 D2m D2' D2m') + %<- ({x} {dx} wkn-ecn-deq M2 (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (D2' x dx) (D2m' x dx) (D2'' x dx) (D2m'' x dx)) %. +%term _ + %pi (wkn-ecn-deq (met/pair M2 M1) DA DA' DOg (ecn-deq/tp/forall D2 D1) (mecn-deq/tp/forall D2m D1m) (ecn-deq/tp/forall D2'' D1') (mecn-deq/tp/forall D2m'' D1m')) + %<- (wkn-ekd-deq M1 DA DA' DOg D1 D1m D1' D1m') + %<- (cxt-inc _ DOg DOe) + %<- ({x} {dx} wkn-cxt-ordered (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (DOen x dx)) + %<- (renum-ecn-deq M2 DOen D2 D2m D2' D2m') + %<- ({x} {dx} wkn-ecn-deq M2 (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (D2' x dx) (D2m' x dx) (D2'' x dx) (D2m'' x dx)) %. +%term _ + %pi (wkn-ecn-deq (met/pair M2 M1) DA DA' DOg (ecn-deq/cn/mu D2 D1) (mecn-deq/cn/mu D2m D1m) (ecn-deq/cn/mu D2'' D1') (mecn-deq/cn/mu D2m'' D1m')) + %<- (wkn-ekd-deq M1 DA DA' DOg D1 D1m D1' D1m') + %<- (cxt-inc _ DOg DOe) + %<- ({x} {dx} wkn-cxt-ordered (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (DOen x dx)) + %<- (renum-ecn-deq M2 DOen D2 D2m D2' D2m') + %<- ({x} {dx} wkn-ecn-deq M2 (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (D2' x dx) (D2m' x dx) (D2'' x dx) (D2m'' x dx)) %. +%term _ + %pi (wkn-ecn-deq (met/pair M1 (met/pair M3 M2)) DA DA' DOg (ecn-deq/pi-ext D1 D3 D2) (mecn-deq/pi-ext D1m D3m D2m) (ecn-deq/pi-ext D1' D3' D2'') (mecn-deq/pi-ext D1m' D3m' D2m'')) + %<- (wkn-eofkd M1 DA DA' DOg D1 D1m D1' D1m') + %<- (wkn-eofkd M3 DA DA' DOg D3 D3m D3' D3m') + %<- (cxt-inc _ DOg DOe) + %<- ({x} {dx} wkn-cxt-ordered (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (DOen x dx)) + %<- (renum-ecn-deq M2 DOen D2 D2m D2' D2m') + %<- ({x} {dx} wkn-ecn-deq M2 (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (D2' x dx) (D2m' x dx) (D2'' x dx) (D2m'' x dx)) %. +%term _ + %pi (wkn-ecn-deq (met/pair M1 M2) DA DA' DOg (ecn-deq/pi-ext-2 D1 D2) (mecn-deq/pi-ext-2 D1m D2m) (ecn-deq/pi-ext-2 D1' D2'') (mecn-deq/pi-ext-2 D1m' D2m'')) + %<- (wkn-ecn-deq M1 DA DA' DOg D1 D1m D1' D1m') + %<- (cxt-inc _ DOg DOe) + %<- ({x} {dx} wkn-cxt-ordered (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (DOen x dx)) + %<- (renum-ecn-deq M2 DOen D2 D2m D2' D2m') + %<- ({x} {dx} wkn-ecn-deq M2 (cxt-append/cons DA) (cxt-append/cons DA') (DOe x dx) (D2' x dx) (D2m' x dx) (D2'' x dx) (D2m'' x dx)) %. +%worlds (cn-block ovar-block can-mofkd-block) (wkn-ekd-wf _ _ _ _ _ _ _ _) (wkn-ekd-deq _ _ _ _ _ _ _ _) (wkn-ekd-sub _ _ _ _ _ _ _ _) (wkn-eofkd _ _ _ _ _ _ _ _) (wkn-ecn-deq _ _ _ _ _ _ _ _) %. +%total (D1 D2 D3 D4 D5) (wkn-ekd-wf D1 _ _ _ _ _ _ _) (wkn-ekd-deq D3 _ _ _ _ _ _ _) (wkn-ekd-sub D5 _ _ _ _ _ _ _) (wkn-eofkd D2 _ _ _ _ _ _ _) (wkn-ecn-deq D4 _ _ _ _ _ _ _) %. +%sort cxt-ordered-pred %. +%mode cxt-ordered-pred %in %out %. +%term _ cxt-ordered-pred (cxt-ordered/cons (cxt-bounded/nil _)) cxt-ordered/nil %. +%term _ cxt-ordered-pred (cxt-ordered/cons (cxt-bounded/cons DB _)) (cxt-ordered/cons DB) %. +%worlds (cn-block ovar-block) (cxt-ordered-pred _ _) %. +%total {} (cxt-ordered-pred _ _) %. +%sort wkn-eofkd-append %. +%mode wkn-eofkd-append %in %in %in %out %. +%term _ wkn-eofkd-append D1 cxt-append-sub/nil _ D1 %. +%term _ + %pi (wkn-eofkd-append D1 (cxt-append-sub/cons DA) DO D1'') + %<- (cxt-ordered-pred DO DO') + %<- (wkn-eofkd-append D1 DA DO' D1') + %<- (can-meofkd D1' DM) + %<- (wkn-eofkd _ cxt-append/nil cxt-append/nil DO D1' DM D1'' _) %. +%worlds (cn-block ovar-block can-mofkd-block) (wkn-eofkd-append _ _ _ _) %. +%total {D1} (wkn-eofkd-append _ D1 _ _) %. +%sort wkn-ecn-deq-append %. +%mode wkn-ecn-deq-append %in %in %in %out %. +%term _ wkn-ecn-deq-append D1 cxt-append-sub/nil _ D1 %. +%term _ + %pi (wkn-ecn-deq-append D1 (cxt-append-sub/cons DA) DO D1'') + %<- (cxt-ordered-pred DO DO') + %<- (wkn-ecn-deq-append D1 DA DO' D1') + %<- (can-mecn-deq D1' DM) + %<- (wkn-ecn-deq _ cxt-append/nil cxt-append/nil DO D1' DM D1'' _) %. +%worlds (cn-block ovar-block can-mofkd-block) (wkn-ecn-deq-append _ _ _ _) %. +%total {D1} (wkn-ecn-deq-append _ D1 _ _) %. +%sort wkn-ekd-deq-append %. +%mode wkn-ekd-deq-append %in %in %in %out %. +%term _ wkn-ekd-deq-append D1 cxt-append-sub/nil _ D1 %. +%term _ + %pi (wkn-ekd-deq-append D1 (cxt-append-sub/cons DA) DO D1'') + %<- (cxt-ordered-pred DO DO') + %<- (wkn-ekd-deq-append D1 DA DO' D1') + %<- (can-mekd-deq D1' DM) + %<- (wkn-ekd-deq _ cxt-append/nil cxt-append/nil DO D1' DM D1'' _) %. +%worlds (cn-block ovar-block can-mofkd-block) (wkn-ekd-deq-append _ _ _ _) %. +%total {D1} (wkn-ekd-deq-append _ D1 _ _) %. +%sort wkn-deq-cxt-bounded %. +%mode wkn-deq-cxt-bounded %in %in %in %out %. +%term _ wkn-deq-cxt-bounded cxt-append/nil _ (cxt-bounded/cons DB DP) (cxt-bounded/cons DB DP) %. +%term _ + %pi (wkn-deq-cxt-bounded (cxt-append/cons DA) (cxt-append/cons DA') (cxt-bounded/cons DB DP) (cxt-bounded/cons DB' DP)) + %<- (wkn-deq-cxt-bounded DA DA' DB DB') %. +%worlds (cn-block ovar-block) (wkn-deq-cxt-bounded _ _ _ _) %. +%total {D1} (wkn-deq-cxt-bounded D1 _ _ _) %. +%sort wkn-deq-cxt-ordered %. +%mode wkn-deq-cxt-ordered %in %in %in %out %. +%term _ wkn-deq-cxt-ordered cxt-append/nil _ (cxt-ordered/cons DB) (cxt-ordered/cons DB) %. +%term _ + %pi (wkn-deq-cxt-ordered (cxt-append/cons DA) (cxt-append/cons DA') (cxt-ordered/cons DB) (cxt-ordered/cons DB')) + %<- (wkn-deq-cxt-bounded DA DA' DB DB') %. +%worlds (cn-block ovar-block) (wkn-deq-cxt-ordered _ _ _ _) %. +%total {D1} (wkn-deq-cxt-ordered D1 _ _ _) %. +%sort wkn-deq-cxt-lookup %. +%mode wkn-deq-cxt-lookup %in %in %in %in %out %. +%term _ + %pi (wkn-deq-cxt-lookup cxt-append/nil cxt-append/nil DQ (cxt-lookup/hit DB) (eofkd/deq (eofkd/var (cxt-lookup/hit DB)) DQ')) + %<- (wkn-ekd-deq-append DQ (%the (cxt-append-sub _ ([x] cxt/cons cxt/nil C K1) cn/unit (cxt/cons G C K1)) (cxt-append-sub/cons cxt-append-sub/nil)) (cxt-ordered/cons DB) DQ') %. +%term _ wkn-deq-cxt-lookup cxt-append/nil _ _ (cxt-lookup/miss DL DP DB) (eofkd/var (cxt-lookup/miss DL DP DB)) %. +%term _ + %pi (wkn-deq-cxt-lookup (cxt-append/cons DA) (cxt-append/cons DA') _ (cxt-lookup/hit DB) (eofkd/var (cxt-lookup/hit DB'))) + %<- (wkn-deq-cxt-bounded DA DA' DB DB') %. +%term _ + %pi (wkn-deq-cxt-lookup (cxt-append/cons (%the (cxt-append _ _ G') DA)) (cxt-append/cons (%the (cxt-append _ _ G) DA')) DQ (cxt-lookup/miss DL _ DB) DC') + %<- (wkn-deq-cxt-lookup DA DA' DQ DL DC) + %<- (wkn-deq-cxt-bounded DA DA' DB DB') + %<- (wkn-eofkd-append DC (%the (cxt-append-sub _ ([x] cxt/cons cxt/nil C K1) cn/unit (cxt/cons G C K1)) (cxt-append-sub/cons cxt-append-sub/nil)) (cxt-ordered/cons DB') DC') %. +%worlds (cn-block ovar-block can-mofkd-block) (wkn-deq-cxt-lookup _ _ _ _ _) %. +%total {D1} (wkn-deq-cxt-lookup D1 _ _ _ _) %. +%sort wkn-deq-ekd-wf {x} %. +%mode wkn-deq-ekd-wf %in %in %in %in %out %. +%sort wkn-deq-ekd-deq {x} %. +%mode wkn-deq-ekd-deq %in %in %in %in %out %. +%sort wkn-deq-ekd-sub {x} %. +%mode wkn-deq-ekd-sub %in %in %in %in %out %. +%sort wkn-deq-eofkd {x} %. +%mode wkn-deq-eofkd %in %in %in %in %out %. +%sort wkn-deq-ecn-deq {x} %. +%mode wkn-deq-ecn-deq %in %in %in %in %out %. +%term _ + %pi (wkn-deq-ekd-wf DA DA' _ ([x] [di] ekd-wf/kd/unit (DO x di)) ([x] [di] ekd-wf/kd/unit (DO' x di))) + %<- ({x} {di} wkn-deq-cxt-ordered (DA x) (DA' x) (DO x di) (DO' x di)) %. +%term _ + %pi (wkn-deq-ekd-wf DA DA' _ ([x] [di] ekd-wf/kd/type (DO x di)) ([x] [di] ekd-wf/kd/type (DO' x di))) + %<- ({x} {di} wkn-deq-cxt-ordered (DA x) (DA' x) (DO x di) (DO' x di)) %. +%term _ + %pi (wkn-deq-ekd-wf DA DA' DQ ([x] [di] ekd-wf/kd/sing (D1 x di)) ([x] [di] ekd-wf/kd/sing (D1' x di))) + %<- (wkn-deq-eofkd DA DA' DQ D1 D1') %. +%term _ + %pi (wkn-deq-ekd-wf DA DA' DQ ([x] [di] ekd-wf/kd/sgm (D1 x di) (D2 x di)) ([x] [di] ekd-wf/kd/sgm (D1' x di) (D2' x di))) + %<- (wkn-deq-ekd-wf DA DA' DQ D1 D1') + %<- ({y} {dy} wkn-deq-ekd-wf ([x] cxt-append/cons (DA x)) ([x] cxt-append/cons (DA' x)) DQ ([x] [di] D2 x di y dy) ([x] [di] D2' x di y dy)) %. +%term _ + %pi (wkn-deq-ekd-wf DA DA' DQ ([x] [di] ekd-wf/kd/pi (D1 x di) (D2 x di)) ([x] [di] ekd-wf/kd/pi (D1' x di) (D2' x di))) + %<- (wkn-deq-ekd-wf DA DA' DQ D1 D1') + %<- ({y} {dy} wkn-deq-ekd-wf ([x] cxt-append/cons (DA x)) ([x] cxt-append/cons (DA' x)) DQ ([x] [di] D2 x di y dy) ([x] [di] D2' x di y dy)) %. +%term _ + %pi (wkn-deq-ekd-deq DA DA' _ ([x] [di] ekd-deq/kd/unit (DO x di)) ([x] [di] ekd-deq/kd/unit (DO' x di))) + %<- ({x} {di} wkn-deq-cxt-ordered (DA x) (DA' x) (DO x di) (DO' x di)) %. +%term _ + %pi (wkn-deq-ekd-deq DA DA' _ ([x] [di] ekd-deq/kd/type (DO x di)) ([x] [di] ekd-deq/kd/type (DO' x di))) + %<- ({x} {di} wkn-deq-cxt-ordered (DA x) (DA' x) (DO x di) (DO' x di)) %. +%term _ + %pi (wkn-deq-ekd-deq DA DA' DQ ([x] [di] ekd-deq/kd/sing (D1 x di)) ([x] [di] ekd-deq/kd/sing (D1' x di))) + %<- (wkn-deq-ecn-deq DA DA' DQ D1 D1') %. +%term _ + %pi (wkn-deq-ekd-deq DA DA' DQ ([x] [di] ekd-deq/kd/sgm (D1 x di) (D2 x di)) ([x] [di] ekd-deq/kd/sgm (D1' x di) (D2' x di))) + %<- (wkn-deq-ekd-deq DA DA' DQ D1 D1') + %<- ({y} {dy} wkn-deq-ekd-deq ([x] cxt-append/cons (DA x)) ([x] cxt-append/cons (DA' x)) DQ ([x] [di] D2 x di y dy) ([x] [di] D2' x di y dy)) %. +%term _ + %pi (wkn-deq-ekd-deq DA DA' DQ ([x] [di] ekd-deq/kd/pi (D1 x di) (D2 x di)) ([x] [di] ekd-deq/kd/pi (D1' x di) (D2' x di))) + %<- (wkn-deq-ekd-deq DA DA' DQ D1 D1') + %<- ({y} {dy} wkn-deq-ekd-deq ([x] cxt-append/cons (DA x)) ([x] cxt-append/cons (DA' x)) DQ ([x] [di] D2 x di y dy) ([x] [di] D2' x di y dy)) %. +%term _ + %pi (wkn-deq-ekd-sub DA DA' _ ([x] [di] ekd-sub/kd/unit (DO x di)) ([x] [di] ekd-sub/kd/unit (DO' x di))) + %<- ({x} {di} wkn-deq-cxt-ordered (DA x) (DA' x) (DO x di) (DO' x di)) %. +%term _ + %pi (wkn-deq-ekd-sub DA DA' _ ([x] [di] ekd-sub/kd/type (DO x di)) ([x] [di] ekd-sub/kd/type (DO' x di))) + %<- ({x} {di} wkn-deq-cxt-ordered (DA x) (DA' x) (DO x di) (DO' x di)) %. +%term _ + %pi (wkn-deq-ekd-sub DA DA' DQ ([x] [di] ekd-sub/kd/sing-kd/sing (D1 x di)) ([x] [di] ekd-sub/kd/sing-kd/sing (D1' x di))) + %<- (wkn-deq-ecn-deq DA DA' DQ D1 D1') %. +%term _ + %pi (wkn-deq-ekd-sub DA DA' DQ ([x] [di] ekd-sub/kd/sing-kd/type (D1 x di)) ([x] [di] ekd-sub/kd/sing-kd/type (D1' x di))) + %<- (wkn-deq-eofkd DA DA' DQ D1 D1') %. +%term _ + %pi (wkn-deq-ekd-sub DA DA' DQ ([x] [di] ekd-sub/kd/sgm (D1 x di) (D2 x di) (D3 x di)) ([x] [di] ekd-sub/kd/sgm (D1' x di) (D2' x di) (D3' x di))) + %<- (wkn-deq-ekd-sub DA DA' DQ D1 D1') + %<- ({y} {dy} wkn-deq-ekd-sub ([x] cxt-append/cons (DA x)) ([x] cxt-append/cons (DA' x)) DQ ([x] [di] D2 x di y dy) ([x] [di] D2' x di y dy)) + %<- ({y} {dy} wkn-deq-ekd-wf ([x] cxt-append/cons (DA x)) ([x] cxt-append/cons (DA' x)) DQ ([x] [di] D3 x di y dy) ([x] [di] D3' x di y dy)) %. +%term _ + %pi (wkn-deq-ekd-sub DA DA' DQ ([x] [di] ekd-sub/kd/pi (D1 x di) (D2 x di) (D3 x di)) ([x] [di] ekd-sub/kd/pi (D1' x di) (D2' x di) (D3' x di))) + %<- (wkn-deq-ekd-sub DA DA' DQ D1 D1') + %<- ({y} {dy} wkn-deq-ekd-sub ([x] cxt-append/cons (DA x)) ([x] cxt-append/cons (DA' x)) DQ ([x] [di] D2 x di y dy) ([x] [di] D2' x di y dy)) + %<- ({y} {dy} wkn-deq-ekd-wf ([x] cxt-append/cons (DA x)) ([x] 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[di] ecn-deq/cn/pj2 (D1 x di)) ([x] [di] ecn-deq/cn/pj2 (D1' x di))) + %<- (wkn-deq-ecn-deq DA DA' DQ D1 D1') %. +%term _ + %pi (wkn-deq-ecn-deq DA DA' DQ ([x] [di] ecn-deq/tp/ref (D1 x di)) ([x] [di] ecn-deq/tp/ref (D1' x di))) + %<- (wkn-deq-ecn-deq DA DA' DQ D1 D1') %. +%term _ + %pi (wkn-deq-ecn-deq DA DA' DQ ([x] [di] ecn-deq/tp/tag (D1 x di)) ([x] [di] ecn-deq/tp/tag (D1' x di))) + %<- (wkn-deq-ecn-deq DA DA' DQ D1 D1') %. +%term _ + %pi (wkn-deq-ecn-deq DA DA' DQ ([x] [di] ecn-deq/kd/unit (D1 x di) (D2 x di)) ([x] [di] ecn-deq/kd/unit (D1' x di) (D2' x di))) + %<- (wkn-deq-eofkd DA DA' DQ D1 D1') + %<- (wkn-deq-eofkd DA DA' DQ D2 D2') %. +%term _ + %pi (wkn-deq-ecn-deq DA DA' DQ ([x] [di] ecn-deq/trans (D1 x di) (D2 x di)) ([x] [di] ecn-deq/trans (D1' x di) (D2' x di))) + %<- (wkn-deq-ecn-deq DA DA' DQ D1 D1') + %<- (wkn-deq-ecn-deq DA DA' DQ D2 D2') %. +%term _ + %pi (wkn-deq-ecn-deq DA DA' DQ ([x] [di] ecn-deq/cn/pair (D1 x di) (D2 x di)) ([x] [di] ecn-deq/cn/pair (D1' x di) (D2' x di))) + %<- (wkn-deq-ecn-deq DA DA' DQ D1 D1') + %<- (wkn-deq-ecn-deq DA DA' DQ D2 D2') %. +%term _ + %pi (wkn-deq-ecn-deq DA DA' DQ ([x] [di] ecn-deq/cn/app (D1 x di) (D2 x di)) ([x] [di] ecn-deq/cn/app (D1' x di) (D2' x di))) + %<- (wkn-deq-ecn-deq DA DA' DQ D1 D1') + %<- (wkn-deq-ecn-deq DA DA' DQ D2 D2') %. +%term _ + %pi (wkn-deq-ecn-deq DA DA' DQ ([x] [di] ecn-deq/sgm-ext (D1 x di) (D2 x di)) ([x] [di] ecn-deq/sgm-ext (D1' x di) (D2' x di))) + %<- (wkn-deq-ecn-deq DA DA' DQ D1 D1') + %<- (wkn-deq-ecn-deq DA DA' DQ D2 D2') %. +%term _ + %pi (wkn-deq-ecn-deq DA DA' DQ ([x] [di] ecn-deq/tp/cross (D1 x di) (D2 x di)) ([x] [di] ecn-deq/tp/cross (D1' x di) (D2' x di))) + %<- (wkn-deq-ecn-deq DA DA' DQ D1 D1') + %<- (wkn-deq-ecn-deq DA DA' DQ D2 D2') %. +%term _ + %pi (wkn-deq-ecn-deq DA DA' DQ ([x] [di] ecn-deq/tp/arrow (D1 x di) (D2 x di)) ([x] [di] ecn-deq/tp/arrow (D1' x di) (D2' x di))) + %<- (wkn-deq-ecn-deq DA DA' DQ D1 D1') + %<- (wkn-deq-ecn-deq DA DA' DQ D2 D2') %. +%term _ + %pi (wkn-deq-ecn-deq DA DA' DQ ([x] [di] ecn-deq/tp/sum (D1 x di) (D2 x di)) ([x] [di] ecn-deq/tp/sum (D1' x di) (D2' x di))) + %<- (wkn-deq-ecn-deq DA DA' DQ D1 D1') + %<- (wkn-deq-ecn-deq DA DA' DQ D2 D2') %. +%term _ + %pi (wkn-deq-ecn-deq DA DA' DQ ([x] [di] ecn-deq/sub (D1 x di) (D2 x di)) ([x] [di] ecn-deq/sub (D1' x di) (D2' x di))) + %<- (wkn-deq-ecn-deq DA DA' DQ D1 D1') + %<- (wkn-deq-ekd-sub DA DA' DQ D2 D2') %. +%term _ + %pi (wkn-deq-ecn-deq DA DA' DQ ([x] [di] ecn-deq/deq (D1 x di) (D2 x di)) ([x] [di] ecn-deq/deq (D1' x di) (D2' x di))) + %<- (wkn-deq-ecn-deq DA DA' DQ D1 D1') + %<- (wkn-deq-ekd-deq DA DA' DQ D2 D2') %. +%term _ + %pi (wkn-deq-ecn-deq DA DA' DQ ([x] [di] ecn-deq/cn/lam (D2 x di) (D1 x di)) ([x] [di] ecn-deq/cn/lam (D2' x di) (D1' x di))) + %<- (wkn-deq-ekd-deq DA DA' DQ D1 D1') + %<- ({y} {dy} wkn-deq-ecn-deq ([x] cxt-append/cons (DA x)) ([x] cxt-append/cons (DA' x)) DQ ([x] [di] D2 x di y dy) ([x] [di] D2' x di y dy)) %. +%term _ + %pi (wkn-deq-ecn-deq DA DA' DQ ([x] [di] ecn-deq/tp/forall (D2 x di) (D1 x di)) ([x] [di] ecn-deq/tp/forall (D2' x di) (D1' x di))) + %<- (wkn-deq-ekd-deq DA DA' DQ D1 D1') + %<- ({y} {dy} wkn-deq-ecn-deq ([x] cxt-append/cons (DA x)) ([x] cxt-append/cons (DA' x)) DQ ([x] [di] D2 x di y dy) ([x] [di] D2' x di y dy)) %. +%term _ + %pi (wkn-deq-ecn-deq DA DA' DQ ([x] [di] ecn-deq/cn/mu (D2 x di) (D1 x di)) ([x] [di] ecn-deq/cn/mu (D2' x di) (D1' x di))) + %<- (wkn-deq-ekd-deq DA DA' DQ D1 D1') + %<- ({y} {dy} wkn-deq-ecn-deq ([x] cxt-append/cons (DA x)) ([x] cxt-append/cons (DA' x)) DQ ([x] [di] D2 x di y dy) ([x] [di] D2' x di y dy)) %. +%term _ + %pi (wkn-deq-ecn-deq DA DA' DQ ([x] [di] ecn-deq/pi-ext (D1 x di) (D3 x di) (D2 x di)) ([x] [di] ecn-deq/pi-ext (D1' x di) (D3' x di) (D2' x di))) + %<- (wkn-deq-eofkd DA DA' DQ D1 D1') + %<- (wkn-deq-eofkd DA DA' DQ D3 D3') + %<- ({y} {dy} wkn-deq-ecn-deq ([x] cxt-append/cons (DA x)) ([x] cxt-append/cons (DA' x)) DQ ([x] [di] D2 x di y dy) ([x] [di] D2' x di y dy)) %. +%term _ + %pi (wkn-deq-ecn-deq DA DA' DQ ([x] [di] ecn-deq/pi-ext-2 (D1 x di) (D2 x di)) ([x] [di] ecn-deq/pi-ext-2 (D1' x di) (D2' x di))) + %<- (wkn-deq-ecn-deq DA DA' DQ D1 D1') + %<- ({y} {dy} wkn-deq-ecn-deq ([x] cxt-append/cons (DA x)) ([x] cxt-append/cons (DA' x)) DQ ([x] [di] D2 x di y dy) ([x] [di] D2' x di y dy)) %. +%worlds (cn-block ovar-block can-mofkd-block) (wkn-deq-ekd-wf _ _ _ _ _) (wkn-deq-ekd-deq _ _ _ _ _) (wkn-deq-ekd-sub _ _ _ _ _) (wkn-deq-eofkd _ _ _ _ _) (wkn-deq-ecn-deq _ _ _ _ _) %. +%total (D1 D2 D3 D4 D5) (wkn-deq-ekd-wf _ _ _ D1 _) (wkn-deq-ekd-deq _ _ _ D3 _) (wkn-deq-ekd-sub _ _ _ D5 _) (wkn-deq-eofkd _ _ _ D2 _) (wkn-deq-ecn-deq _ _ _ D4 _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/weakening-kinds-cnkd.lf b/new-tests/stelf-output/tslf/weakening-kinds-cnkd.lf new file mode 100644 index 0000000..d3344e3 --- /dev/null +++ b/new-tests/stelf-output/tslf/weakening-kinds-cnkd.lf @@ -0,0 +1,52 @@ +%%%% weakening lemma for kinds. +%%%% proposition 3.5 in SH. only need to show (2). (1) follows by the encoding +%sort kd-wkn/ofkd {a cn} %. +%mode kd-wkn/ofkd %in %in %out %. +%term _ kd-wkn/ofkd DI DS ([a] [da] DI a (ofkd/sub da DS)) %. +%worlds (ofkd-block) (kd-wkn/ofkd _ _ _) %. +%total {} (kd-wkn/ofkd _ _ _) %. +%sort kd-wkn/kd-wf {a cn} {da ofkd a K2} %. +%mode kd-wkn/kd-wf %in %in %out %. +%term _ kd-wkn/kd-wf DI DS ([a] [da] DI a (ofkd/sub da DS)) %. +%worlds (ofkd-block ofkd+vdt-block) (kd-wkn/kd-wf _ _ _) %. +%total {} (kd-wkn/kd-wf _ _ _) %. +%sort kd-wkn/kd-deq {a cn} {da ofkd a K2} %. +%mode kd-wkn/kd-deq %in %in %out %. +%term _ kd-wkn/kd-deq DI DS ([a] [da] DI a (ofkd/sub da DS)) %. +%worlds (ofkd-block) (kd-wkn/kd-deq _ _ _) %. +%total {} (kd-wkn/kd-deq _ _ _) %. +%sort kd-wkn/kd-sub {a cn} {da ofkd a K2} %. +%mode kd-wkn/kd-sub %in %in %out %. +%term _ kd-wkn/kd-sub DI DS ([a] [da] DI a (ofkd/sub da DS)) %. +%worlds (ofkd-block) (kd-wkn/kd-sub _ _ _) %. +%total {} (kd-wkn/kd-sub _ _ _) %. +%sort kd-wkn/cn-deq {a cn} {da ofkd a K2} %. +%mode kd-wkn/cn-deq %in %in %out %. +%term _ kd-wkn/cn-deq DI DS ([a] [da] DI a (ofkd/sub da DS)) %. +%worlds (ofkd-block) (kd-wkn/cn-deq _ _ _) %. +%total {} (kd-wkn/cn-deq _ _ _) %. +%sort kd-wkn-deq/ofkd {a cn} %. +%mode kd-wkn-deq/ofkd %in %in %out %. +%term _ kd-wkn-deq/ofkd DI DS ([a] [da] DI a (ofkd/deq da DS)) %. +%worlds (ofkd-block) (kd-wkn-deq/ofkd _ _ _) %. +%total {} (kd-wkn-deq/ofkd _ _ _) %. +%sort kd-wkn-deq/kd-wf {a cn} {da ofkd a K2} %. +%mode kd-wkn-deq/kd-wf %in %in %out %. +%term _ kd-wkn-deq/kd-wf DI DS ([a] [da] DI a (ofkd/deq da DS)) %. +%worlds (ofkd-block ofkd+vdt-block) (kd-wkn-deq/kd-wf _ _ _) %. +%total {} (kd-wkn-deq/kd-wf _ _ _) %. +%sort kd-wkn-deq/kd-deq {a cn} {da ofkd a K2} %. +%mode kd-wkn-deq/kd-deq %in %in %out %. +%term _ kd-wkn-deq/kd-deq DI DS ([a] [da] DI a (ofkd/deq da DS)) %. +%worlds (ofkd-block) (kd-wkn-deq/kd-deq _ _ _) %. +%total {} (kd-wkn-deq/kd-deq _ _ _) %. +%sort kd-wkn-deq/kd-sub {a cn} {da ofkd a K2} %. +%mode kd-wkn-deq/kd-sub %in %in %out %. +%term _ kd-wkn-deq/kd-sub DI DS ([a] [da] DI a (ofkd/deq da DS)) %. +%worlds (ofkd-block) (kd-wkn-deq/kd-sub _ _ _) %. +%total {} (kd-wkn-deq/kd-sub _ _ _) %. +%sort kd-wkn-deq/cn-deq {a cn} {da ofkd a K2} %. +%mode kd-wkn-deq/cn-deq %in %in %out %. +%term _ kd-wkn-deq/cn-deq DI DS ([a] [da] DI a (ofkd/deq da DS)) %. +%worlds (ofkd-block) (kd-wkn-deq/cn-deq _ _ _) %. +%total {} (kd-wkn-deq/cn-deq _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/weakening-kinds-sigs.lf b/new-tests/stelf-output/tslf/weakening-kinds-sigs.lf new file mode 100644 index 0000000..5a3563b --- /dev/null +++ b/new-tests/stelf-output/tslf/weakening-kinds-sigs.lf @@ -0,0 +1,16 @@ +%%%% weakening lemma for kinds. +%sort kd-wkn/sg-wf {a cn} {da ofkd a K2} %. +%mode kd-wkn/sg-wf %in %in %out %. +%term _ kd-wkn/sg-wf DI DS ([a] [da] DI a (ofkd/sub da DS)) %. +%worlds (ofkd-block) (kd-wkn/sg-wf _ _ _) %. +%total {} (kd-wkn/sg-wf _ _ _) %. +%sort kd-wkn/sg-sub {a cn} {da ofkd a K2} %. +%mode kd-wkn/sg-sub %in %in %out %. +%term _ kd-wkn/sg-sub DI DS ([a] [da] DI a (ofkd/sub da DS)) %. +%worlds (ofkd-block) (kd-wkn/sg-sub _ _ _) %. +%total {} (kd-wkn/sg-sub _ _ _) %. +%sort kd-wkn/sg-deq {a cn} {da ofkd a K2} %. +%mode kd-wkn/sg-deq %in %in %out %. +%term _ kd-wkn/sg-deq DI DS ([a] [da] DI a (ofkd/sub da DS)) %. +%worlds (ofkd-block) (kd-wkn/sg-deq _ _ _) %. +%total {} (kd-wkn/sg-deq _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/weakening-lt.lf b/new-tests/stelf-output/tslf/weakening-lt.lf new file mode 100644 index 0000000..f203307 --- /dev/null +++ b/new-tests/stelf-output/tslf/weakening-lt.lf @@ -0,0 +1,273 @@ +%%%% weakening lemma for locations. +%sort lt-wkn/oftp %. +%mode lt-wkn/oftp %in %in %out %. +%sort lt-wkn/ofsg %. +%mode lt-wkn/ofsg %in %in %out %. +%term _ lt-wkn/oftp _ (oftp/tm/loc LK DC) (oftp/tm/loc (lt-look/miss LK) DC) %. +%term _ lt-wkn/oftp _ (oftp/var D) (oftp/var D) %. +%term _ lt-wkn/oftp _ oftp/tm/unit oftp/tm/unit %. +%term _ + %pi (lt-wkn/oftp L (oftp/tm/pair D1 D2) (oftp/tm/pair D1' D2')) + %<- (lt-wkn/oftp L D1 D1') + %<- (lt-wkn/oftp L D2 D2') %. +%term _ + %pi (lt-wkn/oftp L (oftp/tm/tag D1 D2) (oftp/tm/tag D1' D2')) + %<- (lt-wkn/oftp L D1 D1') + %<- (lt-wkn/oftp L D2 D2') %. +%term _ + %pi (lt-wkn/oftp L (oftp/tm/tmapp D1 D2) (oftp/tm/tmapp D1' D2')) + %<- (lt-wkn/oftp L D1 D1') + %<- (lt-wkn/oftp L D2 D2') %. +%term _ + %pi (lt-wkn/oftp L (oftp/tm/set D1 D2) (oftp/tm/set D1' D2')) + %<- (lt-wkn/oftp L D1 D1') + %<- (lt-wkn/oftp L D2 D2') %. +%term _ %pi (lt-wkn/oftp L (oftp/tm/pj1 D1) (oftp/tm/pj1 D1')) %<- (lt-wkn/oftp L D1 D1') %. +%term _ %pi (lt-wkn/oftp L (oftp/tm/pj2 D1) (oftp/tm/pj2 D1')) %<- (lt-wkn/oftp L D1 D1') %. +%term _ + %pi (lt-wkn/oftp L (oftp/tm/cnapp D1 DC) (oftp/tm/cnapp D1' DC)) + %<- (lt-wkn/oftp L D1 D1') %. +%term _ %pi (lt-wkn/oftp L (oftp/deq D1 DC) (oftp/deq D1' DC)) %<- (lt-wkn/oftp L D1 D1') %. +%term _ %pi (lt-wkn/oftp L (oftp/tm/ref D1) (oftp/tm/ref D1')) %<- (lt-wkn/oftp L D1 D1') %. +%term _ + %pi (lt-wkn/oftp L (oftp/tm/raise D1 DC) (oftp/tm/raise D1' DC)) + %<- (lt-wkn/oftp L D1 D1') %. +%term _ %pi (lt-wkn/oftp L (oftp/tm/get D1) (oftp/tm/get D1')) %<- (lt-wkn/oftp L D1 D1') %. +%term _ + %pi (lt-wkn/oftp L (oftp/tm/term D1) (oftp/tm/term D1')) + %<- (lt-wkn/ofsg L D1 D1') %. +%term _ + %pi (lt-wkn/oftp L (oftp/tm/fun D1 DC) (oftp/tm/fun D1' DC)) + %<- ({x} {dx} {y} {dy} lt-wkn/oftp L (D1 x dx y dy) (D1' x dx y dy)) %. +%term _ + %pi (lt-wkn/oftp L (oftp/tm/cnabs D1 DC) (oftp/tm/cnabs D1' DC)) + %<- ({a} {da} lt-wkn/oftp L (D1 a da) (D1' a da)) %. +%term _ + %pi (lt-wkn/oftp L (oftp/tm/inl D1 DC) (oftp/tm/inl D1' DC)) + %<- (lt-wkn/oftp L D1 D1') %. +%term _ + %pi (lt-wkn/oftp L (oftp/tm/inr D1 DC) (oftp/tm/inr D1' DC)) + %<- (lt-wkn/oftp L D1 D1') %. +%term _ + %pi (lt-wkn/oftp L (oftp/tm/case D1 D2 D3) (oftp/tm/case D1' D2' D3')) + %<- (lt-wkn/oftp L D1 D1') + %<- ({x} {dx} lt-wkn/oftp L (D2 x dx) (D2' x dx)) + %<- ({x} {dx} lt-wkn/oftp L (D3 x dx) (D3' x dx)) %. +%term _ + %pi (lt-wkn/oftp L (oftp/tm/iftag D1 D2 D3 D4) (oftp/tm/iftag D1' D2' D3' D4')) + %<- (lt-wkn/oftp L D1 D1') + %<- (lt-wkn/oftp L D2 D2') + %<- ({x} {dx} lt-wkn/oftp L (D3 x dx) (D3' x dx)) + %<- (lt-wkn/oftp L D4 D4') %. +%term _ + %pi (lt-wkn/oftp L (oftp/tm/try D1 D2) (oftp/tm/try D1' D2')) + %<- (lt-wkn/oftp L D1 D1') + %<- ({x} {dx} lt-wkn/oftp L (D2 x dx) (D2' x dx)) %. +%term _ lt-wkn/oftp L (oftp/tm/tagloc D1 D2) (oftp/tm/tagloc D1 D2) %. +%term _ lt-wkn/oftp L (oftp/tm/new-tag D1) (oftp/tm/new-tag D1) %. +%term _ + %pi (lt-wkn/oftp L (oftp/tm/roll D1 DC) (oftp/tm/roll D1' DC)) + %<- (lt-wkn/oftp L D1 D1') %. +%term _ + %pi (lt-wkn/oftp L (oftp/tm/unroll D1) (oftp/tm/unroll D1')) + %<- (lt-wkn/oftp L D1 D1') %. +%term _ lt-wkn/ofsg _ (ofsg/var D) (ofsg/var D) %. +%term _ lt-wkn/ofsg _ ofsg/md/unit ofsg/md/unit %. +%term _ lt-wkn/ofsg _ (ofsg/md/cn D) (ofsg/md/cn D) %. +%term _ + %pi (lt-wkn/ofsg L (ofsg/md/pair D1 D2) (ofsg/md/pair D1' D2')) + %<- (lt-wkn/ofsg L D1 D1') + %<- (lt-wkn/ofsg L D2 D2') %. +%term _ + %pi (lt-wkn/ofsg L (ofsg/md/app D1 D2 DF) (ofsg/md/app D1' D2' DF)) + %<- (lt-wkn/ofsg L D1 D1') + %<- (lt-wkn/ofsg L D2 D2') %. +%term _ %pi (lt-wkn/ofsg L (ofsg/md/pj1 D1) (ofsg/md/pj1 D1')) %<- (lt-wkn/ofsg L D1 D1') %. +%term _ + %pi (lt-wkn/ofsg L (ofsg/md/pj2 D1 DC) (ofsg/md/pj2 D1' DC)) + %<- (lt-wkn/ofsg L D1 D1') %. +%term _ + %pi (lt-wkn/ofsg L (ofsg/sub D1 DC DP) (ofsg/sub D1' DC DP)) + %<- (lt-wkn/ofsg L D1 D1') %. +%term _ %pi (lt-wkn/ofsg L (ofsg/md/tm D1) (ofsg/md/tm D1')) %<- (lt-wkn/oftp L D1 D1') %. +%term _ + %pi (lt-wkn/ofsg L (ofsg/md/lam D1 DC DF) (ofsg/md/lam D1' DC DF)) + %<- ({x} {dx} {y} {dy} {dm} lt-wkn/ofsg L (D1 x dx y dy dm) (D1' x dx y dy dm)) %. +%term _ + %pi (lt-wkn/ofsg L (ofsg/sgm-ext D1 D2) (ofsg/sgm-ext D1' D2')) + %<- (lt-wkn/ofsg L D1 D1') + %<- (lt-wkn/ofsg L D2 D2') %. +%term _ + %pi (lt-wkn/ofsg L (ofsg/kd-ext D1 D2 D3) (ofsg/kd-ext D1' D2 D3)) + %<- (lt-wkn/ofsg L D1 D1') %. +%term _ + %pi (lt-wkn/ofsg L (ofsg/md/seal D1) (ofsg/md/seal D1')) + %<- (lt-wkn/ofsg L D1 D1') %. +%term _ + %pi (lt-wkn/ofsg L (ofsg/md/let D2 D1 DW DF) (ofsg/md/let D2' D1' DW DF)) + %<- ({x} {dx} {y} {dy} {dm} lt-wkn/ofsg L (D1 x dx y dy dm) (D1' x dx y dy dm)) + %<- (lt-wkn/ofsg L D2 D2') %. +%worlds (oftp-block ofkd-block ofsg-block) (lt-wkn/oftp _ _ _) (lt-wkn/ofsg _ _ _) %. +%total (D1 D2) (lt-wkn/oftp _ D1 _) (lt-wkn/ofsg _ D2 _) %. +%sort tt-wkn/oftp %. +%mode tt-wkn/oftp %in %in %out %. +%sort tt-wkn/ofsg %. +%mode tt-wkn/ofsg %in %in %out %. +%term _ tt-wkn/oftp _ (oftp/tm/loc LK DC) (oftp/tm/loc LK DC) %. +%term _ tt-wkn/oftp _ (oftp/var D) (oftp/var D) %. +%term _ tt-wkn/oftp _ oftp/tm/unit oftp/tm/unit %. +%term _ + %pi (tt-wkn/oftp L (oftp/tm/pair D1 D2) (oftp/tm/pair D1' D2')) + %<- (tt-wkn/oftp L D1 D1') + %<- (tt-wkn/oftp L D2 D2') %. +%term _ + %pi (tt-wkn/oftp L (oftp/tm/tag D1 D2) (oftp/tm/tag D1' D2')) + %<- (tt-wkn/oftp L D1 D1') + %<- (tt-wkn/oftp L D2 D2') %. +%term _ + %pi (tt-wkn/oftp L (oftp/tm/tmapp D1 D2) (oftp/tm/tmapp D1' D2')) + %<- (tt-wkn/oftp L D1 D1') + %<- (tt-wkn/oftp L D2 D2') %. +%term _ + %pi (tt-wkn/oftp L (oftp/tm/set D1 D2) (oftp/tm/set D1' D2')) + %<- (tt-wkn/oftp L D1 D1') + %<- (tt-wkn/oftp L D2 D2') %. +%term _ %pi (tt-wkn/oftp L (oftp/tm/pj1 D1) (oftp/tm/pj1 D1')) %<- (tt-wkn/oftp L D1 D1') %. +%term _ %pi (tt-wkn/oftp L (oftp/tm/pj2 D1) (oftp/tm/pj2 D1')) %<- (tt-wkn/oftp L D1 D1') %. +%term _ + %pi (tt-wkn/oftp L (oftp/tm/cnapp D1 DC) (oftp/tm/cnapp D1' DC)) + %<- (tt-wkn/oftp L D1 D1') %. +%term _ %pi (tt-wkn/oftp L (oftp/deq D1 DC) (oftp/deq D1' DC)) %<- (tt-wkn/oftp L D1 D1') %. +%term _ %pi (tt-wkn/oftp L (oftp/tm/ref D1) (oftp/tm/ref D1')) %<- (tt-wkn/oftp L D1 D1') %. +%term _ + %pi (tt-wkn/oftp L (oftp/tm/raise D1 DC) (oftp/tm/raise D1' DC)) + %<- (tt-wkn/oftp L D1 D1') %. +%term _ %pi (tt-wkn/oftp L (oftp/tm/get D1) (oftp/tm/get D1')) %<- (tt-wkn/oftp L D1 D1') %. +%term _ + %pi (tt-wkn/oftp L (oftp/tm/term D1) (oftp/tm/term D1')) + %<- (tt-wkn/ofsg L D1 D1') %. +%term _ + %pi (tt-wkn/oftp L (oftp/tm/fun D1 DC) (oftp/tm/fun D1' DC)) + %<- ({x} {dx} {y} {dy} tt-wkn/oftp L (D1 x dx y dy) (D1' x dx y dy)) %. +%term _ + %pi (tt-wkn/oftp L (oftp/tm/cnabs D1 DC) (oftp/tm/cnabs D1' DC)) + %<- ({a} {da} tt-wkn/oftp L (D1 a da) (D1' a da)) %. +%term _ + %pi (tt-wkn/oftp L (oftp/tm/inl D1 DC) (oftp/tm/inl D1' DC)) + %<- (tt-wkn/oftp L D1 D1') %. +%term _ + %pi (tt-wkn/oftp L (oftp/tm/inr D1 DC) (oftp/tm/inr D1' DC)) + %<- (tt-wkn/oftp L D1 D1') %. +%term _ + %pi (tt-wkn/oftp L (oftp/tm/case D1 D2 D3) (oftp/tm/case D1' D2' D3')) + %<- (tt-wkn/oftp L D1 D1') + %<- ({x} {dx} tt-wkn/oftp L (D2 x dx) (D2' x dx)) + %<- ({x} {dx} tt-wkn/oftp L (D3 x dx) (D3' x dx)) %. +%term _ + %pi (tt-wkn/oftp L (oftp/tm/iftag D1 D2 D3 D4) (oftp/tm/iftag D1' D2' D3' D4')) + %<- (tt-wkn/oftp L D1 D1') + %<- (tt-wkn/oftp L D2 D2') + %<- ({x} {dx} tt-wkn/oftp L (D3 x dx) (D3' x dx)) + %<- (tt-wkn/oftp L D4 D4') %. +%term _ + %pi (tt-wkn/oftp L (oftp/tm/try D1 D2) (oftp/tm/try D1' D2')) + %<- (tt-wkn/oftp L D1 D1') + %<- ({x} {dx} tt-wkn/oftp L (D2 x dx) (D2' x dx)) %. +%term _ tt-wkn/oftp L (oftp/tm/tagloc D1 D2) (oftp/tm/tagloc (lt-look/miss D1) D2) %. +%term _ tt-wkn/oftp L (oftp/tm/new-tag D1) (oftp/tm/new-tag D1) %. +%term _ + %pi (tt-wkn/oftp L (oftp/tm/roll D1 DC) (oftp/tm/roll D1' DC)) + %<- (tt-wkn/oftp L D1 D1') %. +%term _ + %pi (tt-wkn/oftp L (oftp/tm/unroll D1) (oftp/tm/unroll D1')) + %<- (tt-wkn/oftp L D1 D1') %. +%term _ tt-wkn/ofsg _ (ofsg/var D) (ofsg/var D) %. +%term _ tt-wkn/ofsg _ ofsg/md/unit ofsg/md/unit %. +%term _ tt-wkn/ofsg _ (ofsg/md/cn D) (ofsg/md/cn D) %. +%term _ + %pi (tt-wkn/ofsg L (ofsg/md/pair D1 D2) (ofsg/md/pair D1' D2')) + %<- (tt-wkn/ofsg L D1 D1') + %<- (tt-wkn/ofsg L D2 D2') %. +%term _ + %pi (tt-wkn/ofsg L (ofsg/md/app D1 D2 DF) (ofsg/md/app D1' D2' DF)) + %<- (tt-wkn/ofsg L D1 D1') + %<- (tt-wkn/ofsg L D2 D2') %. +%term _ %pi (tt-wkn/ofsg L (ofsg/md/pj1 D1) (ofsg/md/pj1 D1')) %<- (tt-wkn/ofsg L D1 D1') %. +%term _ + %pi (tt-wkn/ofsg L (ofsg/md/pj2 D1 DC) (ofsg/md/pj2 D1' DC)) + %<- (tt-wkn/ofsg L D1 D1') %. +%term _ + %pi (tt-wkn/ofsg L (ofsg/sub D1 DC DP) (ofsg/sub D1' DC DP)) + %<- (tt-wkn/ofsg L D1 D1') %. +%term _ %pi (tt-wkn/ofsg L (ofsg/md/tm D1) (ofsg/md/tm D1')) %<- (tt-wkn/oftp L D1 D1') %. +%term _ + %pi (tt-wkn/ofsg L (ofsg/md/lam D1 DC DF) (ofsg/md/lam D1' DC DF)) + %<- ({x} {dx} {y} {dy} {dm} tt-wkn/ofsg L (D1 x dx y dy dm) (D1' x dx y dy dm)) %. +%term _ + %pi (tt-wkn/ofsg L (ofsg/sgm-ext D1 D2) (ofsg/sgm-ext D1' D2')) + %<- (tt-wkn/ofsg L D1 D1') + %<- (tt-wkn/ofsg L D2 D2') %. +%term _ + %pi (tt-wkn/ofsg L (ofsg/kd-ext D1 D2 D3) (ofsg/kd-ext D1' D2 D3)) + %<- (tt-wkn/ofsg L D1 D1') %. +%term _ + %pi (tt-wkn/ofsg L (ofsg/md/seal D1) (ofsg/md/seal D1')) + %<- (tt-wkn/ofsg L D1 D1') %. +%term _ + %pi (tt-wkn/ofsg L (ofsg/md/let D2 D1 DW DF) (ofsg/md/let D2' D1' DW DF)) + %<- ({x} {dx} {y} {dy} {dm} tt-wkn/ofsg L (D1 x dx y dy dm) (D1' x dx y dy dm)) + %<- (tt-wkn/ofsg L D2 D2') %. +%worlds (oftp-block ofkd-block ofsg-block) (tt-wkn/oftp _ _ _) (tt-wkn/ofsg _ _ _) %. +%total (D1 D2) (tt-wkn/oftp _ D1 _) (tt-wkn/ofsg _ D2 _) %. +%sort lt-wkn-sub/oftp %. +%mode lt-wkn-sub/oftp %in %in %out %. +%term _ lt-wkn-sub/oftp lt-sub/refl D1 D1 %. +%term _ + %pi (lt-wkn-sub/oftp (lt-sub/trans LL LX) D1 D1'') + %<- (lt-wkn-sub/oftp LL D1 D1') + %<- (lt-wkn/oftp LX D1' D1'') %. +%worlds (oftp-block ofkd-block ofsg-block) (lt-wkn-sub/oftp _ _ _) %. +%total (D1) (lt-wkn-sub/oftp D1 _ _) %. +%sort lt-wkn-sub/ofsg %. +%mode lt-wkn-sub/ofsg %in %in %out %. +%term _ lt-wkn-sub/ofsg lt-sub/refl D1 D1 %. +%term _ + %pi (lt-wkn-sub/ofsg (lt-sub/trans LL LX) D1 D1'') + %<- (lt-wkn-sub/ofsg LL D1 D1') + %<- (lt-wkn/ofsg LX D1' D1'') %. +%worlds (ofsg-block ofkd-block oftp-block) (lt-wkn-sub/ofsg _ _ _) %. +%total (D1) (lt-wkn-sub/ofsg D1 _ _) %. +%sort tt-wkn-sub/oftp %. +%mode tt-wkn-sub/oftp %in %in %out %. +%term _ tt-wkn-sub/oftp lt-sub/refl D1 D1 %. +%term _ + %pi (tt-wkn-sub/oftp (lt-sub/trans LL LX) D1 D1'') + %<- (tt-wkn-sub/oftp LL D1 D1') + %<- (tt-wkn/oftp LX D1' D1'') %. +%worlds (oftp-block ofkd-block ofsg-block) (tt-wkn-sub/oftp _ _ _) %. +%total (D1) (tt-wkn-sub/oftp D1 _ _) %. +%sort tt-wkn-sub/ofsg %. +%mode tt-wkn-sub/ofsg %in %in %out %. +%term _ tt-wkn-sub/ofsg lt-sub/refl D1 D1 %. +%term _ + %pi (tt-wkn-sub/ofsg (lt-sub/trans LL LX) D1 D1'') + %<- (tt-wkn-sub/ofsg LL D1 D1') + %<- (tt-wkn/ofsg LX D1' D1'') %. +%worlds (ofsg-block ofkd-block oftp-block) (tt-wkn-sub/ofsg _ _ _) %. +%total (D1) (tt-wkn-sub/ofsg D1 _ _) %. +%sort lt-tt-wkn-sub/oftp %. +%mode lt-tt-wkn-sub/oftp %in %in %in %out %. +%term _ + %pi (lt-tt-wkn-sub/oftp LS TS D1 D1'') + %<- (lt-wkn-sub/oftp LS D1 D1') + %<- (tt-wkn-sub/oftp TS D1' D1'') %. +%worlds (ofsg-block ofkd-block oftp-block) (lt-tt-wkn-sub/oftp _ _ _ _) %. +%total {} (lt-tt-wkn-sub/oftp _ _ _ _) %. +%sort lt-tt-wkn-sub/ofsg %. +%mode lt-tt-wkn-sub/ofsg %in %in %in %out %. +%term _ + %pi (lt-tt-wkn-sub/ofsg LS TS D1 D1'') + %<- (lt-wkn-sub/ofsg LS D1 D1') + %<- (tt-wkn-sub/ofsg TS D1' D1'') %. +%worlds (ofsg-block ofkd-block oftp-block) (lt-tt-wkn-sub/ofsg _ _ _ _) %. +%total {} (lt-tt-wkn-sub/ofsg _ _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/worlds-adequacy-cnkd.lf b/new-tests/stelf-output/tslf/worlds-adequacy-cnkd.lf new file mode 100644 index 0000000..fde8a8d --- /dev/null +++ b/new-tests/stelf-output/tslf/worlds-adequacy-cnkd.lf @@ -0,0 +1,4 @@ +%% syntax: +%worlds (cn-block) (cn) (kd) %. +%% static semantics +%worlds (ofkd-block) (ofkd _ _) (kd-wf _) (kd-deq _ _) (kd-sub _ _) (cn-deq _ _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/worlds-adequacy-sigs.lf b/new-tests/stelf-output/tslf/worlds-adequacy-sigs.lf new file mode 100644 index 0000000..9fdb44a --- /dev/null +++ b/new-tests/stelf-output/tslf/worlds-adequacy-sigs.lf @@ -0,0 +1,5 @@ +%% syntax: +%worlds (cn-block) (sg) %. +%% static semantics +%worlds (cn-block) (fst-sg _ _) %. +%worlds (ofkd-block) (sg-wf _) (sg-sub _ _) (sg-deq _ _) %. \ No newline at end of file diff --git a/new-tests/stelf-output/tslf/worlds-adequacy-tmmd.lf b/new-tests/stelf-output/tslf/worlds-adequacy-tmmd.lf new file mode 100644 index 0000000..4bfacda --- /dev/null +++ b/new-tests/stelf-output/tslf/worlds-adequacy-tmmd.lf @@ -0,0 +1,4 @@ +%% syntax: +%worlds (tm-block cn-block md-block) (tm) (md) %. +%% static semantics +%worlds (oftp-block ofkd-block ofsg-block) (oftp _ _ _ _) (assm/tm _ _) (ofsg _ _ _ _ _) (fst-md _ _) (assm/md _ _) %. \ No newline at end of file diff --git a/src/Common/Cst/CST.ml b/src/Common/Cst/CST.ml index df773fe..a72ad01 100644 --- a/src/Common/Cst/CST.ml +++ b/src/Common/Cst/CST.ml @@ -370,8 +370,8 @@ module type CST = sig val[@deprecated "Use View equivalent instead"] union : ?fc:loc -> string -> string list -> cmd (** [%union id ids] — union of block labels. *) - val[@deprecated "Use View equivalent instead"] worlds : ?fc:loc -> string list -> term -> cmd - (** [%worlds ids expr] — assert expr lives in the named world. *) + val[@deprecated "Use View equivalent instead"] worlds : ?fc:loc -> string list -> term list -> cmd + (** [%worlds ids exprs] — assert exprs live in the named world. *) val[@deprecated "Use View equivalent instead"] deterministic : ?fc:loc -> string list -> cmd (** [%deterministic id_list] — mark type families as deterministic. *) diff --git a/src/Common/Cst/Cst.ml b/src/Common/Cst/Cst.ml index 3f500a7..7f9b1a1 100644 --- a/src/Common/Cst/Cst.ml +++ b/src/Common/Cst/Cst.ml @@ -41,6 +41,7 @@ module Make_Cst (Paths : Paths.PATHS.PATHS) = struct | Fvar_ of string * loc | Typ_ of loc | MacroParam_ of loc * int option * int + | Local_ of loc * namespace * term [@@deriving show { with_path = false }, eq] (* Constant/block declarations *) @@ -127,7 +128,7 @@ module Make_Cst (Paths : Paths.PATHS.PATHS) = struct | TermCmd_ of decl | BlockCmd_ of string * block_item list | UnionCmd_ of string * string list - | WorldsCmd_ of string list * term + | WorldsCmd_ of string list * term list | DeterministicCmd_ of string list | EvalCmd_ of cmd list | PrecCmd_ of fixity * int * string list @@ -327,7 +328,7 @@ module Make_Cst (Paths : Paths.PATHS.PATHS) = struct let term ?fc:(_ = ghost) d = TermCmd_ d let block ?fc:(_ = ghost) id items = BlockCmd_ (id, items) let union ?fc:(_ = ghost) id ids = UnionCmd_ (id, ids) - let worlds ?fc:(_ = ghost) ids tm = WorldsCmd_ (ids, tm) + let worlds ?fc:(_ = ghost) ids tms = WorldsCmd_ (ids, tms) let deterministic ?fc:(_ = ghost) ids = DeterministicCmd_ ids @@ -503,34 +504,7 @@ module Make_Cst (Paths : Paths.PATHS.PATHS) = struct let (!<) = review end let ghost' : Loc.t = ghost - (** {3 Term Syntax} *) - module rec Term : sig - type t = term - - type u = - | Lowercase of Loc.t * symbol - | Uppercase of Loc.t * symbol - | Qualified of Loc.t * symbol - | Text of Loc.t * name - | ExistVar of Loc.t * name - | FreeVar of Loc.t * name - | Pi of Loc.t * Decl.t list * t - | Lam of Loc.t * Decl.t list * t - | App of Loc.t * t * t list - | HasType of Loc.t * t * t - | Omitted of Loc.t - | Typ of Loc.t - | Arrow of Loc.t * t * t - | BackArrow of Loc.t * t * t - | Foreign of Loc.t * t - | Internal of int - | MacroParam of Loc.t * int option * int - - val view : t -> u - val review : u -> t - val (!>) : t -> u - val (!<) : u -> t - end = struct + module Term = struct type t = term type u = @@ -540,8 +514,8 @@ module Make_Cst (Paths : Paths.PATHS.PATHS) = struct | Text of Loc.t * string | ExistVar of Loc.t * string | FreeVar of Loc.t * string - | Pi of Loc.t * Decl.t list * t - | Lam of Loc.t * Decl.t list * t + | Pi of Loc.t * decl list * t + | Lam of Loc.t * decl list * t | App of Loc.t * t * t list | HasType of Loc.t * t * t | Omitted of Loc.t @@ -551,6 +525,7 @@ module Make_Cst (Paths : Paths.PATHS.PATHS) = struct | Foreign of Loc.t * t | Internal of int | MacroParam of Loc.t * int option * int + | Local of Loc.t * namespace * t let view (x : t) : u = let rec collect_pis = function | Pi_ (_, d, body) -> @@ -588,6 +563,7 @@ module Make_Cst (Paths : Paths.PATHS.PATHS) = struct | Omitted_ loc -> Omitted loc | Typ_ loc -> Typ loc | Arrow_ (loc, a, b) -> Arrow (loc, a, b) + | Local_ (loc, ns, tm) -> Local (loc, ns, tm) let review (y : u) : t = let rec fold_right f lst acc = @@ -629,29 +605,18 @@ module Make_Cst (Paths : Paths.PATHS.PATHS) = struct | BackArrow (loc, a, b) -> Arrow_ (loc, b, a) | Foreign (_, _) -> raise Lacking | Internal _ -> raise Lacking + | Local (loc, ns, tm) -> Local_ (loc, ns, tm) let (!>) = view let (!<) = review end - (** Binder declaration constructors. *) - and Decl : sig - type t = decl - - type u = - | Decl1 of Loc.t * string option list * Term.t * Term.t - | Decl0 of Loc.t * string option list * Term.t - - val view : t -> u - val review : u -> t - val (!>) : t -> u - val (!<) : u -> t - end = struct + module Decl = struct type t = decl type u = - | Decl1 of Loc.t * string option list * Term.t * Term.t - | Decl0 of Loc.t * string option list * Term.t + | Decl1 of Loc.t * string option list * term * term + | Decl0 of Loc.t * string option list * term let view (x : t) : u = match x with @@ -1010,7 +975,7 @@ module Make_Cst (Paths : Paths.PATHS.PATHS) = struct | Term of Loc.t * Decl.t | Block of Loc.t * string * BlockItem.t list | Union of Loc.t * string * string list - | Worlds of Loc.t * string list * Term.t + | Worlds of Loc.t * string list * Term.t list | Deterministic of Loc.t * string list | Eval of Loc.t * t list | Prec of Loc.t * Fixity.t * int * string list diff --git a/src/Common/Cst/LENS.ml b/src/Common/Cst/LENS.ml index d5b67af..a69e41d 100644 --- a/src/Common/Cst/LENS.ml +++ b/src/Common/Cst/LENS.ml @@ -52,6 +52,7 @@ end module type VIEW = sig exception Lacking + (** Raised when a view case is not implemented for a given CST variant. *) module Paths : Paths.PATHS.PATHS (** Module of paths and regions, which we allow to be shared *) @@ -59,7 +60,7 @@ module type VIEW = sig (** Abstract syntax tree type. (for internals) *) type loc - (** Source Loc.tation carried by CST nodes. *) + (** Source location carried by CST nodes. *) type name = string (** Unqualified identifier. *) @@ -71,21 +72,21 @@ module type VIEW = sig (** Qualified symbol as [(namespace, name)]. *) val mk_loc : int -> int -> loc - (** Create a Loc.tation from start and end lexer positions. *) + (** Create a location from start and end lexer positions. *) val loc_to_region : loc -> Paths.region - (** Convert a source Loc.tation to a Paths region. *) + (** Convert a source location to a Paths region. *) val ghost : loc - (** Synthetic Loc.tation used for generated nodes. *) + (** Synthetic location used for generated nodes. *) module Loc : sig type t type u = | Loc of Fpath.t option * int * int - (** A actual location, with path, start, end*) - | Ghost (** Synthetic location used for generated nodes *) + (** An actual location, with path, start, end. *) + | Ghost (** Synthetic location used for generated nodes. *) include LENS with type t := t and type u := u end @@ -95,28 +96,34 @@ module type VIEW = sig type t type u = - | Lowercase of Loc.t * symbol (** An lowercase identifier in scope *) - | Uppercase of Loc.t * symbol (** An uppercase identifier in scope *) + | Lowercase of Loc.t * symbol (** A lowercase identifier in scope. *) + | Uppercase of Loc.t * symbol (** An uppercase identifier in scope. *) | Qualified of Loc.t * symbol - (** A explicitly qualified identifier (eg, [%val (x y)])*) - | Text of Loc.t * string (** A literal string (currently unused) *) + (** An explicitly qualified identifier (eg, [%val (x y)]). *) + | Text of Loc.t * string (** A literal string (currently unused). *) | ExistVar of Loc.t * string (** An explicitly existential variable, you're probably looking for - {!Uppercase} *) + {!Uppercase}. *) | FreeVar of Loc.t * string - (** A free variable, you're probably looking for {!Lowercase} *) + (** A free variable, you're probably looking for {!Lowercase}. *) | Pi of Loc.t * Decl.t list * t - (** A {m \Pi} type, which takes in a number of possible contexts*) + (** A {m \Pi} type, which takes in a number of possible contexts. *) | Lam of Loc.t * Decl.t list * t + (** Lambda abstraction binding declarations over a body. *) | App of Loc.t * t * t list - | HasType of Loc.t * t * t - | Omitted of Loc.t - | Typ of Loc.t - | Arrow of Loc.t * t * t - | BackArrow of Loc.t * t * t - | Foreign of Loc.t * t - | Internal of int + (** Application of a head term to a spine of arguments. *) + | HasType of Loc.t * t * t (** Type ascription [{e term} : {e type}]. *) + | Omitted of Loc.t (** Implicit/omitted argument to be inferred. *) + | Typ of Loc.t (** The universe [type]. *) + | Arrow of Loc.t * t * t (** Non-dependent function type [A -> B]. *) + | BackArrow of Loc.t * t * t (** Reverse-direction arrow [B <- A]. *) + | Foreign of Loc.t * t (** Foreign/external term from an FFI. *) + | Internal of int (** Internal de Bruijn reference (not user-facing). *) | MacroParam of Loc.t * int option * int + (** Macro parameter reference with optional binding level and index. + *) + | Local of Loc.t * namespace * t + (** A local scope, similar to [N.(...)] in OCaml *) include LENS with type t := t and type u := u end @@ -127,7 +134,9 @@ module type VIEW = sig type u = | Decl1 of Loc.t * string option list * Term.t * Term.t + (** Named binder with names, type, and optional body. *) | Decl0 of Loc.t * string option list * Term.t + (** Binder with names and type only (no body position). *) include LENS with type t := t and type u := u end @@ -137,10 +146,14 @@ module type VIEW = sig type t type u = - | ConstantDecl of Loc.t * Decl.t + | ConstantDecl of Loc.t * Decl.t (** Constant type declaration. *) | BlockDecl of Loc.t * string * Decl.t list * Decl.t list + (** Block (world) declaration with universal and existential parts. *) | BlockDef of Loc.t * string * symbol list + (** Block definition as a list of existing blocks. *) | ConstantDef of Loc.t * string * Term.t * Term.t option + (** Constant definition with a body term and optional type annotation. + *) include LENS with type t := t and type u := u end @@ -148,35 +161,50 @@ module type VIEW = sig (** Mode syntax constructors. *) module Mode : sig type t - type u = Plus of Loc.t | Star of Loc.t | Minus of Loc.t | Minus1 of Loc.t + + type u = + | Plus of Loc.t (** Input mode ([+]). *) + | Star of Loc.t (** Unrestricted mode ([*]). *) + | Minus of Loc.t (** Output mode ([-]). *) + | Minus1 of Loc.t (** Unique output mode ([-1]). *) include LENS with type t := t and type u := u type t_mode := t + (** Mode spine: sequence of mode annotations applied to a type family. *) module Spine : sig type t type u = - | ModeNil of Loc.t + | ModeNil of Loc.t (** Empty mode spine. *) | ModeApp of Loc.t * (t_mode * string option) * t + (** Mode spine cons: one mode annotation followed by the rest. *) include LENS with type t := t and type u := u end + (** Mode term: a type family head annotated with a mode spine. *) module Term : sig type t type u = | ModeTerm of Loc.t * symbol * Spine.t + (** Mode annotation on a named type family. *) | ModePi of Loc.t * Decl.t * t * t + (** Pi-abstraction in a mode term. *) include LENS with type t := t and type u := u end + (** Mode declaration: associates modes with a type family's arguments. *) module Dec : sig type t - type u = ModeDec of Loc.t * (t_mode * string option) list * Term.t + + type u = + | ModeDec of Loc.t * (t_mode * string option) list * Term.t + (** A mode declaration binding a list of mode/name pairs to a type + family. *) include LENS with type t := t and type u := u end @@ -184,256 +212,362 @@ module type VIEW = sig (** Module/signature syntax constructors. *) module Struct : sig + (** Structure expression: reference to a named structure. *) module StrExp : sig type t - type u = StrExp of Loc.t * symbol + + type u = + | StrExp of Loc.t * symbol + (** Reference to a structure by qualified name. *) include LENS with type t := t and type u := u end + (** Instantiation in a [where] clause. *) module Inst : sig type t type u = | ConInst of Loc.t * symbol * Loc.t * Term.t + (** Constant instantiation in a [where] clause. *) | StrInst of Loc.t * symbol * Loc.t * StrExp.t + (** Structure instantiation in a [where] clause. *) include LENS with type t := t and type u := u end + (** Signature expression for module typing. *) module SigExp : sig type t type u = - | Thesig of Loc.t - | SigId of Loc.t * string + | Thesig of Loc.t (** Anonymous signature ([the]). *) + | SigId of Loc.t * string (** Named signature reference. *) | WhereSig of Loc.t * t * Inst.t list + (** Signature with [where]-clause instantiations. *) include LENS with type t := t and type u := u end + (** Signature definition binding a name to a signature expression. *) module SigDef : sig type t - type u = SigDef of Loc.t * string option * SigExp.t + + type u = + | SigDef of Loc.t * string option * SigExp.t + (** A named (or anonymous) signature definition. *) include LENS with type t := t and type u := u end + (** Structure declaration or definition. *) module StructDec : sig type t type u = | StructDecl of Loc.t * string option * SigExp.t + (** Structure declaration with a signature ascription. *) | StructDef of Loc.t * string option * StrExp.t + (** Structure definition as an alias for another structure. *) include LENS with type t := t and type u := u end end + (** Logic programming query. *) module Query : sig type t - type u = Query of Loc.t * string option * Term.t + + type u = + | Query of Loc.t * string option * Term.t + (** A query with optional result name and goal term. *) include LENS with type t := t and type u := u end + (** Top-level definition. *) module Define : sig type t - type u = Define of Loc.t * string option * Term.t * Term.t option + + type u = + | Define of Loc.t * string option * Term.t * Term.t option + (** A definition with optional name, body term, and optional type. *) include LENS with type t := t and type u := u end + (** Solve declaration. *) module Solve : sig type t - type u = Solve of Loc.t * string option * Term.t + + type u = + | Solve of Loc.t * string option * Term.t + (** A solve directive with optional name and goal term. *) include LENS with type t := t and type u := u end + (** Operator fixity declarations. *) module Fixity : sig type t type u = - | Left of Loc.t - | Right of Loc.t - | Prefix of Loc.t - | Postfix of Loc.t - | Middle of Loc.t - | None of Loc.t + | Left of Loc.t (** Left-associative infix. *) + | Right of Loc.t (** Right-associative infix. *) + | Prefix of Loc.t (** Prefix operator. *) + | Postfix of Loc.t (** Postfix operator. *) + | Middle of Loc.t (** Non-associative infix. *) + | None of Loc.t (** Non-associative (same as {!Middle}). *) include LENS with type t := t and type u := u end + (** Block item: a single entry in a block declaration. *) module BlockItem : sig type t - type u = Any of Loc.t * Decl.t | All of Loc.t * Decl.t + + type u = + | Any of Loc.t * Decl.t + (** Existentially quantified block entry (written [{...}]). *) + | All of Loc.t * Decl.t + (** Universally quantified block entry (written [[...]]). *) include LENS with type t := t and type u := u end + (** Theorem-related syntax for totality and termination checking. *) module Thm : sig type t + + (** Empty variant: theorem nodes are only constructed via sub-modules. *) type u = | + (** Termination orderings. *) module Order : sig type t type u = | Varg of Loc.t * string list - | Lex of Loc.t * t list - | Simul of Loc.t * t list + (** Variable argument position(s) in a termination order. *) + | Lex of Loc.t * t list (** Lexicographic ordering of sub-orders. *) + | Simul of Loc.t * t list (** Simultaneous ordering of sub-orders. *) include LENS with type t := t and type u := u end + (** Call patterns for totality declarations. *) module CallPats : sig type t - type u = CallPats of (string * string option list * Loc.t) list + + type u = + | CallPats of (string * string option list * Loc.t) list + (** List of (name, argument pattern, location) triples. *) include LENS with type t := t and type u := u end + (** Termination declaration. *) module TDecl : sig type t - type u = TDecl of Order.t * CallPats.t + + type u = + | TDecl of Order.t * CallPats.t + (** Termination order paired with call patterns. *) include LENS with type t := t and type u := u end + (** Predicate reference in a reduction declaration. *) module Predicate : sig type t - type u = Predicate of string * Loc.t + + type u = + | Predicate of string * Loc.t + (** Named predicate with source location. *) include LENS with type t := t and type u := u end + (** Reduction declaration. *) module RDecl : sig type t - type u = RDecl of Predicate.t * Order.t * Order.t * CallPats.t + + type u = + | RDecl of Predicate.t * Order.t * Order.t * CallPats.t + (** Predicate with input order, output order, and call patterns. *) include LENS with type t := t and type u := u end + (** Declaration that a predicate uses tabled evaluation. *) module TabledDecl : sig type t - type u = TabledDecl of string * Loc.t + + type u = + | TabledDecl of string * Loc.t + (** Tabled predicate name and source location. *) include LENS with type t := t and type u := u end + (** Declaration to persist a predicate's memo table. *) module KeepTableDecl : sig type t - type u = KeepTableDecl of string * Loc.t + + type u = + | KeepTableDecl of string * Loc.t + (** Kept table predicate name and source location. *) include LENS with type t := t and type u := u end + (** Prove directive with search depth and termination declaration. *) module Prove : sig type t - type u = Prove of int * TDecl.t + + type u = + | Prove of int * TDecl.t + (** Search depth bound and termination declaration. *) include LENS with type t := t and type u := u end + (** Establish directive (like {!Prove} but for meta-theorems). *) module Establish : sig type t - type u = Establish of int * TDecl.t + + type u = + | Establish of int * TDecl.t + (** Search depth bound and termination declaration. *) include LENS with type t := t and type u := u end + (** Assert directive checking call patterns. *) module Assert : sig type t - type u = Assert of CallPats.t + type u = Assert of CallPats.t (** Assertion over call patterns. *) include LENS with type t := t and type u := u end + (** Declaration context for theorem quantifiers. *) module Decs : sig type t - type u = DecsNil of Loc.t | DecsList of t * Decl.t list + + type u = + | DecsNil of Loc.t (** Empty declaration context. *) + | DecsList of t * Decl.t list (** Non-empty declaration context. *) include LENS with type t := t and type u := u end + (** Theorem formula syntax. *) module Thm : sig type t type u = - | Top of Loc.t - | Exists of Loc.t * Decs.t * t - | Forall of Loc.t * Decs.t * t + | Top of Loc.t (** Trivially true theorem. *) + | Exists of Loc.t * Decs.t * t (** Existentially quantified theorem. *) + | Forall of Loc.t * Decs.t * t (** Universally quantified theorem. *) | ForallStar of Loc.t * Decs.t * t + (** Star-universally quantified theorem. *) | ForallG of Loc.t * (Decs.t * Decs.t) list * t + (** Generalized universal quantification over contexts. *) include LENS with type t := t and type u := u end include LENS with type t := t and type u := u + (** Named theorem declaration. *) module ThmDec : sig type t - type u = ThmDec of string * Thm.t + + type u = + | ThmDec of string * Thm.t (** Theorem name paired with its formula. *) include LENS with type t := t and type u := u end + (** World declaration: block patterns and call patterns. *) module WDecl : sig type t - type u = WDecl of (string list * string) list * CallPats.t + + type u = + | WDecl of (string list * string) list * CallPats.t + (** Block identifier patterns and call patterns. *) include LENS with type t := t and type u := u end end + (** Top-level commands (directives and declarations). *) module Cmd : sig type t type u = | Query of Loc.t * int option * int option * int option * Query.t + (** Execute a bounded logic programming query. *) | QueryTabled of Loc.t * int option * int option * int option * Query.t + (** Execute a tabled logic programming query. *) | AdhocQuery of Loc.t * Query.t + (** Quick unbounded query (REPL shorthand). *) | Unique of Loc.t * Term.t - | Mode of Loc.t * Mode.Dec.t - | Define of Loc.t * Define.t + (** Assert uniqueness of solutions for a type family. *) + | Mode of Loc.t * Mode.Dec.t (** Mode declaration for a type family. *) + | Define of Loc.t * Define.t (** Top-level definition. *) | DeclCmd of Loc.t * Term.t - | Inline of Loc.t * string * Term.t + (** Bare declaration (constant declaration from a term). *) + | Inline of Loc.t * string * Term.t (** Inline constant definition. *) | Symbol of Loc.t * string * string + (** Associate a symbolic alias with an existing identifier. *) | Freeze of Loc.t * string list - | Thaw of Loc.t * string list + (** Freeze constants (prevent further clauses). *) + | Thaw of Loc.t * string list (** Thaw frozen constants. *) | Sort of Loc.t * string list * Decl.t list - | Term of Loc.t * Decl.t + (** Declare a new sort (type-level constant). *) + | Term of Loc.t * Decl.t (** Declare a new term-level constant. *) | Block of Loc.t * string * BlockItem.t list + (** Declare a block (world). *) | Union of Loc.t * string * string list - | Worlds of Loc.t * string list * Term.t + (** Define a block as a union of existing blocks. *) + | Worlds of Loc.t * string list * Term.t list + (** World declaration for type families. *) | Deterministic of Loc.t * string list - | Eval of Loc.t * t list + (** Mark type families as deterministic. *) + | Eval of Loc.t * t list (** Evaluate a sequence of sub-commands. *) | Prec of Loc.t * Fixity.t * int * string list - | Solve of Loc.t * Solve.t - | Stop of Loc.t * unit - | ReplQuit of Loc.t * unit - | ReplHelp of Loc.t * string option - | ReplGet of Loc.t * string - | ReplSet of Loc.t * string * string - | ReplVersion of Loc.t * unit + (** Fixity/precedence declaration. *) + | Solve of Loc.t * Solve.t (** Solve directive. *) + | Stop of Loc.t * unit (** Stop processing. *) + | ReplQuit of Loc.t * unit (** REPL quit command. *) + | ReplHelp of Loc.t * string option (** REPL help command. *) + | ReplGet of Loc.t * string (** REPL get-setting command. *) + | ReplSet of Loc.t * string * string (** REPL set-setting command. *) + | ReplVersion of Loc.t * unit (** REPL version command. *) | Total of Loc.t * Thm.Order.t list * Term.t list + (** Totality declaration. *) | Terminates of Loc.t * Thm.Order.t list * Term.t list - | Covers of Loc.t * Mode.Dec.t - | Name of Loc.t * string + (** Termination declaration. *) + | Covers of Loc.t * Mode.Dec.t (** Coverage declaration. *) + | Name of Loc.t * string (** Name preference declaration. *) | Reduces of Loc.t * string * Term.t list + (** Reduction ordering declaration. *) | Macro of Loc.t * int * string * t (** Defines a macro, taking its location, number of params, name, and - the body *) + the body. *) | Seq of Loc.t * item list - (** A sequence of commands, for use withthe module system*) + (** A sequence of commands, for use with the module system. *) | Require of Loc.t * string list - (** Ensure that the given path is loaded *) - | Open of Loc.t * string list (** Open a scope into the scope *) - | Scope of Loc.t * string * t (** Enter into a new scope *) - | Use of Loc.t * string list * Term.t list (** Apply a macro *) - - and item = Outer of string | Cmd of t + (** Ensure that the given path is loaded. *) + | Open of Loc.t * string list (** Open a scope into the current scope. *) + | Scope of Loc.t * string * t (** Enter into a new scope. *) + | Use of Loc.t * string list * Term.t list (** Apply a macro. *) + + and item = + | Outer of string (** Raw text segment between commands. *) + | Cmd of t (** An embedded command. *) include LENS with type t := t and type u := u end diff --git a/src/Common/Cst/Macro.ml b/src/Common/Cst/Macro.ml index 8eb04db..2c0661d 100644 --- a/src/Common/Cst/Macro.ml +++ b/src/Common/Cst/Macro.ml @@ -17,7 +17,7 @@ module Macro (C : CST.CST) : MACRO.MACRO with module C = C = struct | Unique (loc, tm) -> review (Unique (loc, go_term i args tm)) | DeclCmd (loc, tm) -> review (DeclCmd (loc, go_term i args tm)) | Inline (loc, id, tm) -> review (Inline (loc, id, go_term i args tm)) - | Worlds (loc, ids, tm) -> review (Worlds (loc, ids, go_term i args tm)) + | Worlds (loc, ids, tms) -> review (Worlds (loc, ids, List.map (go_term i args) tms)) | Total (loc, orders, tms) -> review (Total (loc, orders, List.map (go_term i args) tms)) | Terminates (loc, orders, tms) -> diff --git a/src/Common/Cst/dune b/src/Common/Cst/dune index ddaa800..fd63247 100644 --- a/src/Common/Cst/dune +++ b/src/Common/Cst/dune @@ -1,4 +1,5 @@ (library + (instrumentation (backend bisect_ppx)) (name cst) (public_name stelf.cst) (libraries display util lwt fmt paths base) diff --git a/src/Common/Syntax/Ast.ml b/src/Common/Syntax/Ast.ml index 7542d0c..e62ca0a 100644 --- a/src/Common/Syntax/Ast.ml +++ b/src/Common/Syntax/Ast.ml @@ -3,6 +3,13 @@ open Common (* Author: Frank Pfenning, Carsten Schuermann *) (* Modified: Roberto Virga *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) +exception UnexpectedFgnExp of exn +let () = Printexc.register_printer (function UnexpectedFgnExp _ -> Some "UnexpectedFgnExp" | _ -> None) +exception UnexpectedFgnCnstr of exn +let () = Printexc.register_printer (function UnexpectedFgnCnstr _ -> Some "UnexpectedFgnCnstr" | _ -> None) + module Make_Ast (Common : Common.COMMON) : AST.AST with module Common = Common = struct module Common = Common @@ -46,7 +53,7 @@ struct | Decl (g_, _) -> g_ | Null -> invalid_arg "ctxPop: empty context" - exception Error of string + exception Error = Error (* raised if out of space *) (* ctxLookup (G, k) = D, kth declaration in G from right to left @@ -76,7 +83,7 @@ struct let show_fgnExp e = Printexc.to_string e (* foreign expression representation *) - exception UnexpectedFgnExp of fgnExp + exception UnexpectedFgnExp = UnexpectedFgnExp (* raised by a constraint solver if passed an incorrect arg *) @@ -89,7 +96,7 @@ struct let pp_fgnCnstr fmt e = Format.pp_print_string fmt (Printexc.to_string e) let show_fgnCnstr e = Printexc.to_string e - exception UnexpectedFgnCnstr of fgnCnstr + exception UnexpectedFgnCnstr = UnexpectedFgnCnstr (* raised by a constraint solver if passed an incorrect arg *) diff --git a/src/Common/Syntax/dune b/src/Common/Syntax/dune index 134e889..ed5a6ca 100644 --- a/src/Common/Syntax/dune +++ b/src/Common/Syntax/dune @@ -1,4 +1,5 @@ (library + (instrumentation (backend bisect_ppx)) (name syntax) (public_name stelf.syntax) (libraries display containers basis global)) diff --git a/src/Common/Util/dune b/src/Common/Util/dune index c483995..545a942 100644 --- a/src/Common/Util/dune +++ b/src/Common/Util/dune @@ -1,4 +1,5 @@ (library + (instrumentation (backend bisect_ppx)) (name util) (public_name stelf.util) (modules util magic eq)) diff --git a/src/Compile/Assign.ml b/src/Compile/Assign.ml index 93550dc..3803898 100644 --- a/src/Compile/Assign.ml +++ b/src/Compile/Assign.ml @@ -12,6 +12,9 @@ open! Basis (* Assignment *) (* Author: Brigitte Pientka *) +exception Assignment of string +let () = Printexc.register_printer (function Assignment msg -> Some msg | _ -> None) + module Assign (Assign__0 : sig (*! structure IntSyn' : INTSYN !*) module Whnf : WHNF @@ -25,7 +28,7 @@ end) : ASSIGN = struct (*! structure IntSyn = IntSyn' !*) open Assign__0 - exception Assignment of string + exception Assignment = Assignment open! struct open IntSyn diff --git a/src/Compile/Compile_.ml b/src/Compile/Compile_.ml index 673f603..fc469bc 100644 --- a/src/Compile/Compile_.ml +++ b/src/Compile/Compile_.ml @@ -20,6 +20,9 @@ open Cprint (* Author: Iliano Cervesato *) (* Modified: Jeff Polakow, Carsten Schuermann, Larry Greenfield, Roberto Virga, Brigitte Pientka *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MakeCompile (Whnf : WHNF) (TypeCheck : TYPECHECK) @@ -40,7 +43,7 @@ module MakeCompile (*! sharing Print.IntSyn = IntSyn' !*) *) (* FIX: need to associate errors with occurrences -kw *) - exception Error of string + exception Error = Error module I = IntSyn module T = Tomega diff --git a/src/Compile/dune b/src/Compile/dune index d1f32b8..ff3e2e4 100644 --- a/src/Compile/dune +++ b/src/Compile/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name compile) (public_name stelf.compile) (modules diff --git a/src/Compress/Compress_.ml b/src/Compress/Compress_.ml index 92bd1b6..3486d75 100644 --- a/src/Compress/Compress_.ml +++ b/src/Compress/Compress_.ml @@ -10,6 +10,11 @@ open! Sgn open! Reductio open! Basis +exception Unimp +let () = Printexc.register_printer (function Unimp -> Some "Unimp" | _ -> None) +exception NoModes +let () = Printexc.register_printer (function NoModes -> Some "NoModes" | _ -> None) + module Compress (Compress__0 : sig module Global : GLOBAL end) = @@ -18,8 +23,8 @@ struct module S = Syntax module Sgn = Sgn - exception Unimp - exception NoModes + exception Unimp = Unimp + exception NoModes = NoModes (* modes are not appropriate for the given I.ConDec *) let debug = ref (-1) diff --git a/src/Compress/Reductio.ml b/src/Compress/Reductio.ml index 3908a5d..d99e231 100644 --- a/src/Compress/Reductio.ml +++ b/src/Compress/Reductio.ml @@ -585,3 +585,8 @@ module Reductio = struct end include Reductio +let () = Printexc.register_printer (function Unimp -> Some "Unimp" | _ -> None) +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) +let () = Printexc.register_printer (function Matching msg -> Some msg | _ -> None) +let () = Printexc.register_printer (function NonPattern -> Some "NonPattern" | _ -> None) +let () = Printexc.register_printer (function NotFound msg -> Some msg | _ -> None) diff --git a/src/Compress/Rep.ml b/src/Compress/Rep.ml index ad7ffba..453db18 100644 --- a/src/Compress/Rep.ml +++ b/src/Compress/Rep.ml @@ -8,6 +8,9 @@ open! Sgn open! Reductio open! Basis +exception Crap +let () = Printexc.register_printer (function Crap -> Some "Crap" | _ -> None) + module Rep = struct module I = IntSyn module S = Syntax @@ -44,7 +47,7 @@ module Rep = struct (* val l : (Syntax.term * Syntax.tp) list ref = ref [] *) let k : Reductio.eq_c option ref = ref None - exception Crap + exception Crap = Crap let rec sanityCheck cid = try diff --git a/src/Compress/Sgn.ml b/src/Compress/Sgn.ml index 8ecaf70..57d2a8a 100644 --- a/src/Compress/Sgn.ml +++ b/src/Compress/Sgn.ml @@ -143,3 +143,4 @@ module Sgn = struct end include Sgn +let () = Printexc.register_printer (function NoSuch _ -> Some "NoSuch" | _ -> None) diff --git a/src/Compress/Strict.ml b/src/Compress/Strict.ml index a844539..73fcdf7 100644 --- a/src/Compress/Strict.ml +++ b/src/Compress/Strict.ml @@ -120,3 +120,4 @@ module Strict = struct end include Strict +let () = Printexc.register_printer (function EtaContract -> Some "EtaContract" | _ -> None) diff --git a/src/Compress/Syntax.ml b/src/Compress/Syntax.ml index 24751b6..562f6cc 100644 --- a/src/Compress/Syntax.ml +++ b/src/Compress/Syntax.ml @@ -420,3 +420,5 @@ module Syntax = struct end include Syntax +let () = Printexc.register_printer (function Syntax msg -> Some msg | _ -> None) +let () = Printexc.register_printer (function MissingVar -> Some "MissingVar" | _ -> None) diff --git a/src/Compress/dune b/src/Compress/dune index 67842f1..5e80129 100644 --- a/src/Compress/dune +++ b/src/Compress/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name compress) (public_name stelf.compress) (modules Syntax Strict Sgn Reductio Compress_ Rep COMPRESS SGN) diff --git a/src/Config/dune b/src/Config/dune index 0c936b6..2904b4d 100644 --- a/src/Config/dune +++ b/src/Config/dune @@ -1,4 +1,5 @@ (library + (instrumentation (backend bisect_ppx)) (name config) (public_name stelf.config) (libraries)) diff --git a/src/Cover/Cover_.ml b/src/Cover/Cover_.ml index 0f2c8c0..baca38e 100644 --- a/src/Cover/Cover_.ml +++ b/src/Cover/Cover_.ml @@ -13,6 +13,11 @@ open! Basis (* Coverage Checking *) (* Author: Frank Pfenning *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) +exception NotFinitary +let () = Printexc.register_printer (function NotFinitary -> Some "Not finitary" | _ -> None) + module MakeCover (Global : GLOBAL) (Whnf : WHNF) @@ -31,7 +36,7 @@ module MakeCover (Timers : Timers.TIMERS) : COVER = struct module Subordinate = Subordinate - exception Error of string + exception Error = Error module Unify = Unify module ModeTable = ModeTable @@ -745,7 +750,7 @@ module MakeCover let rec incCount () = counter := !counter + 1 let rec getCount () = !counter - exception NotFinitary + exception NotFinitary = NotFinitary let rec finitary1 (x_, k, w_, f, cands) = begin diff --git a/src/Cover/Total.ml b/src/Cover/Total.ml index 56f20df..f540ea1 100644 --- a/src/Cover/Total.ml +++ b/src/Cover/Total.ml @@ -14,6 +14,12 @@ open! Basis (* Total Declarations *) (* Author: Frank Pfenning *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + +exception Error' of Paths.occ * string +let () = Printexc.register_printer (function Error' (_, msg) -> Some msg | _ -> None) + (* COVER module type inlined here to avoid dependency cycle with cover_ *) module Total (Total__0 : sig module Global : GLOBAL @@ -52,7 +58,7 @@ module Total (Total__0 : sig module Timers : Timers.TIMERS end) : TOTAL = struct (*! structure IntSyn = IntSyn' !*) - exception Error of string + exception Error = Error module Table = Total__0.Table module Cover = Total__0.Cover @@ -96,7 +102,7 @@ end) : TOTAL = struct end (* call only on constants *) - exception Error' of P.occ * string + exception Error' = Error' (* copied from terminates/Reduces.fun *) let rec error (c, occ, msg) = diff --git a/src/Cover/dune b/src/Cover/dune index 8687c1d..88de441 100644 --- a/src/Cover/dune +++ b/src/Cover/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name cover) (public_name stelf.cover) (modules Total Cover_ COVER TOTAL) diff --git a/src/Debug/Debug.ml b/src/Debug/Debug.ml index 249fbd8..a2d3665 100644 --- a/src/Debug/Debug.ml +++ b/src/Debug/Debug.ml @@ -17,7 +17,7 @@ module Level = struct match x with | 0 -> Error | 1 | 2 | 3 -> Warning - | 4 | 5 -> Info + | 4 -> Info | _ -> Debug end diff --git a/src/Debug/dune b/src/Debug/dune index 5c9dce6..35c5a7d 100644 --- a/src/Debug/dune +++ b/src/Debug/dune @@ -1,4 +1,5 @@ (library + (instrumentation (backend bisect_ppx)) (name debug) (public_name stelf.debug) (libraries display logs logs.fmt)) diff --git a/src/Display/DISPLAY.ml b/src/Display/DISPLAY.ml index 24217cd..da46557 100644 --- a/src/Display/DISPLAY.ml +++ b/src/Display/DISPLAY.ml @@ -1,17 +1,17 @@ module type DISPLAY = sig val register : (Info.t -> unit Lwt.t) -> unit + val set_fallback_verbosity : Info.level -> unit val display' : Info.t -> unit val message : - ?src:Info.src -> ?kind:Info.kind -> ?level:Info.level -> Form.Form.t -> unit + ?src:Info.src -> ?kind:Info.kind -> ?level:Info.level -> Info.form -> unit - val chatter : ?src:Info.src -> ?kind:Info.kind -> int -> Form.Form.t -> unit + val chatter : ?src:Info.src -> ?kind:Info.kind -> int -> Info.form -> unit val chatter_s : ?src:Info.src -> ?kind:Info.kind -> int -> string -> unit - val debug : ?src:Info.src -> ?level:Info.level -> Form.Form.t -> unit - val info : ?src:Info.src -> ?level:Info.level -> Form.Form.t -> unit - val warning : ?src:Info.src -> ?level:Info.level -> Form.Form.t -> unit - val error : ?src:Info.src -> ?level:Info.level -> Form.Form.t -> unit + val debug : ?src:Info.src -> ?level:Info.level -> Info.form -> unit + val info : ?src:Info.src -> ?level:Info.level -> Info.form -> unit + val warning : ?src:Info.src -> ?level:Info.level -> Info.form -> unit + val error : ?src:Info.src -> ?level:Info.level -> Info.form -> unit + val response : ?src:Info.src -> ?level:Info.level -> Info.form -> unit end - -module Form = Form.Form diff --git a/src/Display/Display.ml b/src/Display/Display.ml index 4be34c2..2fa177f 100644 --- a/src/Display/Display.ml +++ b/src/Display/Display.ml @@ -7,7 +7,6 @@ module type DISPLAY = DISPLAY.DISPLAY open Lwt.Syntax -include Form include Info module Display : DISPLAY = struct @@ -15,13 +14,16 @@ module Display : DISPLAY = struct let registered : bool ref = ref false let rep : (t -> unit Lwt.t) ref = ref (fun _ -> Lwt.return ()) + let fallback_verbosity : Info.level ref = ref Info.Normal let register f = registered := true; rep := f + let set_fallback_verbosity v = fallback_verbosity := v + let display' info = - assert !registered; + if not !registered then () else Lwt.dont_wait (fun () -> !rep info) (fun exn -> @@ -29,7 +31,10 @@ module Display : DISPLAY = struct (Printexc.to_string exn)) let message ?src ?kind ?(level = Terse) t = - assert !registered; + if not !registered then begin + if Info.( >= ) !fallback_verbosity level then + Printf.eprintf "%s\n%!" (Format.asprintf "%t" t) + end else Lwt.dont_wait (fun () -> !rep { src; kind; level; msg = t }) (fun exn -> @@ -37,14 +42,108 @@ module Display : DISPLAY = struct (Printexc.to_string exn)) let chatter ?src ?kind n t = message ?src ?kind ~level:(Info.from_chatter n) t - let chatter_s ?src ?kind n s = chatter ?src ?kind n (Form.string s) + let chatter_s ?src ?kind n s = chatter ?src ?kind n (fun ppf -> Format.pp_print_string ppf s) let debug ?src ?level t = message ?src ~kind:Info.Debug ?level t let info ?src ?level t = message ?src ~kind:Info.Info ?level t let warning ?src ?level t = message ?src ~kind:Info.Warning ?level t let error ?src ?level t = message ?src ~kind:Info.Error ?level t + let response ?src ?level t = message ?src ~kind:Info.Response ?level t end module Info = Info include Display -include Info.Form + +(* Compat shim: these functions re-implement the old Form.t API as plain + Format.formatter → unit closures. Prefer Format/Fmt directly in new code. + Access via Display.Form.* emits a deprecation alert. *) + +let string s ppf = Format.pp_print_string ppf s +let empty _ppf = () +let ( +++ ) a b ppf = a ppf; b ppf +let ( ++ ) a b ppf = a ppf; Format.pp_print_char ppf ' '; b ppf +let nl ?(n=1) () ppf = for _ = 1 to n do Format.pp_print_newline ppf () done +let space ?(n=1) () ppf = for _ = 1 to n do Format.pp_print_char ppf ' ' done +let shown f x ppf = Format.pp_print_string ppf (f x) + +let concat ?(sep=empty) xs ppf = + let first = ref true in + List.iter (fun x -> if !first then first := false else sep ppf; x ppf) xs + +let each ?(sep=empty) f xs = concat ~sep (List.map f xs) +let inside (l, r) x ppf = l ppf; x ppf; r ppf + +let optional ?def f = function + | None -> (match def with Some d -> d | None -> empty) + | Some x -> f x + +let to_plain t = Format.asprintf "%t" t +let fmt ppf t = t ppf +let style f x = f x +let styles fs x = List.fold_left (fun acc f -> style f acc) x fs +let hbox xs ppf = List.iter (fun x -> x ppf) xs +let vbox xs ppf = + let first = ref true in + List.iter (fun x -> if !first then first := false else Format.pp_print_newline ppf (); x ppf) xs +let hvbox xs ppf = List.iter (fun x -> x ppf) xs + +(** Compat re-export as Display.Form.* — deprecated, use Format/Fmt directly *) +module Form = struct + let string = string + let empty = empty + let ( +++ ) = ( +++ ) + let ( ++ ) = ( ++ ) + let nl = nl + let space = space + let shown = shown + let concat = concat + let each = each + let inside = inside + let optional = optional + let to_plain = to_plain + let fmt = fmt + let style = style + let styles = styles + let hbox = hbox + let vbox = vbox + let hvbox = hvbox + + module Style = struct + let bold x = x + let italic x = x + let underline x = x + + module Fore = struct + let black x = x + let red x = x + let green x = x + let yellow x = x + let blue x = x + let magenta x = x + let cyan x = x + let white x = x + let orange x = x + let rgb _ _ _ x = x + end + + module Back = struct + let black x = x + let red x = x + let green x = x + let yellow x = x + let blue x = x + let magenta x = x + let cyan x = x + let white x = x + let orange x = x + let rgb _ _ _ x = x + end + end + + module Syntax = struct + let syntax x = x + end +end + +(* Also expose Style at the top level of Display for existing open-based call sites *) +module Style = Form.Style diff --git a/src/Display/FORM.ml b/src/Display/FORM.ml deleted file mode 100644 index 6bfa70a..0000000 --- a/src/Display/FORM.ml +++ /dev/null @@ -1,67 +0,0 @@ -(* TODO: Thinks that will be added eventually -Groups (sections, list, etc) -Math markup -Boxes*) - -(** Common colors for the foreground and background (for use with {!FORM}) *) -module type COLORS = sig - type t - - val black : t - val red : t - val green : t - val yellow : t - val blue : t - val magenta : t - val cyan : t - val white : t - val orange : t - val rgb : int -> int -> int -> t -end - -module type FORM = sig - type t - type style = t -> t - type 'a scribe = 'a -> t - - module Style : sig - val bold : style - val italic : style - val underline : style - - module Fore : COLORS with type t := style - module Back : COLORS with type t := style - end - - module Syntax : sig - val syntax : style - end - - val ( +++ ) : t -> t -> t - val ( ++ ) : t -> t -> t - val style : style -> t -> t - val styles : style list -> t -> t - val empty : t - val concat : ?sep:t -> t list -> t - val string : string -> t - val int : int -> t - val char : char -> t - val bool : bool -> t - val non_breaking_space : ?n:int -> unit -> t - val space : ?n:int -> unit -> t - val shown : ('a -> string) -> 'a -> t - val inside : t * t -> t -> t - val nl : ?n:int -> unit -> t - val each : ?sep:t -> ('a -> t) -> 'a list -> t - val hbox : t list -> t - val vbox : t list -> t - val hvbox : t list -> t - val markup : t -> Notty.image - val shown_many : ?sep:t -> ('a -> string) -> 'a list -> t - val optional : ?def:t -> ('a -> t) -> 'a option -> t - val to_plain : t -> string - val fmt : t Fmt.t - val header : level:int -> t -> t - val show : t -> Notty.image - val custom : unit Fmt.t -> t -end diff --git a/src/Display/Form.ml b/src/Display/Form.ml deleted file mode 100644 index 416bbff..0000000 --- a/src/Display/Form.ml +++ /dev/null @@ -1,165 +0,0 @@ -(** Formatting options generally - @author Asher Frost *) - -module type FORM = FORM.FORM - -module Form : FORM = struct - type box = HBox | VBox | HVBox - - type t = - | Space of int - | NonbreakingSpace of int - | Cut of int - | Exact of string - | Empty - | Concat of t list - | Fg of Notty.A.color * t - | Bg of Notty.A.color * t - | Bold of t - | Italic of t - | Underline of t - | Marked of t * t (** Ie, with carats *) - | Boxed of box * t list (** Box with style *) - | Custom of unit Fmt.t - - type style = t -> t - type 'a scribe = 'a -> t - - let ( +++ ) x y = Concat [ x; y ] - let empty = Empty - - let concat ?(sep = empty) xs = - List.fold_left (fun acc x -> acc +++ sep +++ x) empty xs - - let string s = Exact s - let int n = string (string_of_int n) - let char c = string (String.make 1 c) - let bool b = string (string_of_bool b) - let cut () = string "\n" - let shown f x = string (f x) - let shown_exact f x = string (f x) - let shown_many ?(sep = empty) f xs = concat ~sep (List.map (shown f) xs) - let inside (open_, close) x = open_ +++ x +++ close - let nl ?(n = 1) () = string (String.make n '\n') - let each ?(sep = empty) f xs = concat ~sep (List.map f xs) - let space ?(n = 1) () = Space n - let non_breaking_space ?(n = 1) () = NonbreakingSpace n - let hbox xs = Boxed (HBox, xs) - let vbox xs = Boxed (VBox, xs) - let hvbox xs = Boxed (HVBox, xs) - - let optional ?def f = function - | None -> ( match def with Some d -> d | None -> empty) - | Some x -> f x - - let ( ++ ) x y = x +++ space () +++ y - - module Style = struct - let bold x = Bold x - let italic x = Italic x - let underline x = Underline x - - module Fore = struct - let black x = Fg (Notty.A.black, x) - let red x = Fg (Notty.A.red, x) - let green x = Fg (Notty.A.green, x) - let yellow x = Fg (Notty.A.yellow, x) - let blue x = Fg (Notty.A.blue, x) - let magenta x = Fg (Notty.A.magenta, x) - let cyan x = Fg (Notty.A.cyan, x) - let white x = Fg (Notty.A.white, x) - let orange x = Fg (Notty.A.rgb ~r:255 ~g:165 ~b:0, x) - let rgb r g b x = Fg (Notty.A.rgb_888 ~r ~g ~b, x) - end - - module Back = struct - let black x = Bg (Notty.A.black, x) - let red x = Bg (Notty.A.red, x) - let green x = Bg (Notty.A.green, x) - let yellow x = Bg (Notty.A.yellow, x) - let blue x = Bg (Notty.A.blue, x) - let magenta x = Bg (Notty.A.magenta, x) - let cyan x = Bg (Notty.A.cyan, x) - let white x = Bg (Notty.A.white, x) - let orange x = Bg (Notty.A.rgb_888 ~r:255 ~g:165 ~b:0, x) - let rgb r g b x = Bg (Notty.A.rgb_888 ~r ~g ~b, x) - end - end - - let style f x = f x - let styles fs x = List.fold_left (fun acc f -> style f acc) x fs - - module Syntax = struct - let syntax = Style.bold - end - - let rec to_plain : t -> string = function - | Space n | NonbreakingSpace n -> String.make n ' ' - | Cut n -> String.make n '\n' - | Exact s -> s - | Empty -> "" - | Concat xs -> String.concat "" (List.map to_plain xs) - | Fg (_, x) | Bg (_, x) | Bold x | Italic x | Underline x -> to_plain x - | Marked (_carats, x) -> to_plain x - | Boxed (box, xs) -> ( - match box with - | HBox -> String.concat "" (List.map to_plain xs) - | VBox -> String.concat "\n" (List.map to_plain xs) - | HVBox -> String.concat " " (List.map to_plain xs)) - - let rec go : Notty.attr -> t -> Notty.image = fun attr -> - let attr' : Notty.attr -> Notty.attr = Notty.A.((++) attr) in - Notty.(function - | Space n -> I.string attr (String.make n ' ') - | NonbreakingSpace n -> I.string attr (String.make n ' ') - | Cut n -> - let row = I.string attr "" in - let rows = List.init n (fun _ -> row) in - List.fold_left I.(<->) I.empty rows - | Exact s -> - (match String.split_on_char '\n' s with - | [] -> I.empty - | [line] -> I.string attr line - | lines -> I.vcat (List.map (I.string attr) lines)) - | Empty -> I.empty - | Concat xs -> I.(hcat (List.map (go attr) xs)) - | Fg (c, x) -> go (attr' (A.fg c)) x - | Bg (c, x) -> go (attr' (A.bg c)) x - | Bold x -> go (attr' @@ A.st A.bold) x - | Italic x -> go (attr' @@ A.st A.italic) x - | Underline x -> go (attr' @@ A.st A.underline) x - | Marked (carats, x) -> - let content_img = go attr x in - let carats_img = go attr carats in - I.(carats_img <-> content_img) - | Boxed (box, xs) -> - match box with - | HBox -> List.fold_left I.(<|>) I.empty (List.map (go attr) xs) - | VBox -> List.fold_left I.(<->) I.empty (List.map (go attr) xs) - | HVBox -> List.fold_left I.(<|>) I.empty (List.map (go attr) xs)) - - let show x = go Notty.A.empty x - let markup x = show x - let rec fmt ppf = function - | Space n | NonbreakingSpace n -> - Format.pp_print_string ppf (String.make n ' ') - | Cut n -> - for _ = 1 to n do Format.pp_force_newline ppf () done - | Exact s -> Format.pp_print_string ppf s - | Empty -> () - | Concat xs -> List.iter (fmt ppf) xs - | Fg (_, x) | Bg (_, x) | Bold x | Italic x | Underline x -> fmt ppf x - | Marked (_, x) -> fmt ppf x - | Custom f -> f ppf () - | Boxed (box, xs) -> - (match box with - | HBox -> Format.pp_open_hbox ppf () - | VBox -> Format.pp_open_vbox ppf 0 - | HVBox -> Format.pp_open_hvbox ppf 0); - List.iter (fmt ppf) xs; - Format.pp_close_box ppf () - - let header ~level:_ x = x - let custom f = Custom f -end - \ No newline at end of file diff --git a/src/Display/Info.ml b/src/Display/Info.ml index 524bfe8..15487e3 100644 --- a/src/Display/Info.ml +++ b/src/Display/Info.ml @@ -29,9 +29,7 @@ let from_chatter x = | 4 -> Detailed | _ -> Exhaustive -module Form = Form.Form - -type form = Form.t +type form = Format.formatter -> unit type t = { src : src option; kind : kind option; level : level; msg : form } let msg ?(src : src option) ?(kind : kind option) ?(level = Normal) (fmt : form) diff --git a/src/Display/Info.mli b/src/Display/Info.mli index ffa627e..1e5af3a 100644 --- a/src/Display/Info.mli +++ b/src/Display/Info.mli @@ -19,9 +19,7 @@ type kind = Debug | Info | Warning | Error | Response (* Corresponds to chatter 5 4 3 2 1 0, respectively *) type level = Exhaustive | Detailed | Normal | Terse | Minimal | Off -module Form : module type of Form.Form - -type form = Form.t +type form = Format.formatter -> unit val from_chatter : int -> level diff --git a/src/Display/dune b/src/Display/dune index ca3f7e7..ae888ec 100644 --- a/src/Display/dune +++ b/src/Display/dune @@ -1,4 +1,5 @@ (library + (instrumentation (backend bisect_ppx)) (name display) (public_name stelf.display) - (libraries lwt fmt notty-community)) + (libraries lwt fmt)) diff --git a/src/Domains/IntegersMod.ml b/src/Domains/IntegersMod.ml index f012472..1d1ea43 100644 --- a/src/Domains/IntegersMod.ml +++ b/src/Domains/IntegersMod.ml @@ -6,6 +6,9 @@ open Field (* Integers Modulo a Prime Number *) (* Author: Roberto Virga *) +exception Div +let () = Printexc.register_printer (function Div -> Some "Division by zero" | _ -> None) + module IntegersMod (IntegersMod__0 : sig val p : int end) : FIELD = struct @@ -19,7 +22,7 @@ end) : FIELD = struct let zero = 0 let one = 1 - exception Div + exception Div = Div let ( ~- ) n = Int.( - ) p n let ( + ) m n = normalize (Int.( + ) m n) diff --git a/src/Domains/dune b/src/Domains/dune index 0d51b2b..7697c6d 100644 --- a/src/Domains/dune +++ b/src/Domains/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name domains) (public_name stelf.domains) (modules diff --git a/src/Error/ERROR.ml b/src/Error/ERROR.ml index 00401d5..b5f026b 100644 --- a/src/Error/ERROR.ml +++ b/src/Error/ERROR.ml @@ -1,7 +1,7 @@ module type ERROR = sig type stage = Lex | Parse | Check | Total | Recon | Unknown | Other of string - exception Err of stage * Display.Form.t + exception Err of stage * (Format.formatter -> unit) - val err : ?stage:stage -> Display.Form.t -> 'a + val err : ?stage:stage -> (Format.formatter -> unit) -> 'a end diff --git a/src/Error/Error.ml b/src/Error/Error.ml index 6a1be42..02020cb 100644 --- a/src/Error/Error.ml +++ b/src/Error/Error.ml @@ -3,7 +3,9 @@ module type ERROR = ERROR.ERROR module Error : ERROR = struct type stage = Lex | Parse | Check | Total | Recon | Unknown | Other of string - exception Err of stage * Display.Form.t + exception Err of stage * (Format.formatter -> unit) let err ?(stage = Unknown) form = raise (Err (stage, form)) end + +let () = Printexc.register_printer (function Error.Err (_, form) -> Some (Format.asprintf "%t" form) | _ -> None) diff --git a/src/Error/dune b/src/Error/dune index 9dea83b..b17945f 100644 --- a/src/Error/dune +++ b/src/Error/dune @@ -1,4 +1,5 @@ (library + (instrumentation (backend bisect_ppx)) (name error) (public_name stelf.error) (libraries paths display)) diff --git a/src/Flit/dune b/src/Flit/dune index 9ba89b8..883d155 100644 --- a/src/Flit/dune +++ b/src/Flit/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name flit) (public_name stelf.flit) (modules FlitOldSmlnjPack Flit_ FLIT) diff --git a/src/Formatter/dune b/src/Formatter/dune index a6bc77d..a31a48d 100644 --- a/src/Formatter/dune +++ b/src/Formatter/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name formatter) (public_name stelf.formatter) (modules Formatter_ FORMATTER) diff --git a/src/Frontend/Fquery.ml b/src/Frontend/Fquery.ml index 4e871d0..a230474 100644 --- a/src/Frontend/Fquery.ml +++ b/src/Frontend/Fquery.ml @@ -13,6 +13,9 @@ open! Basis (* fquery: Executing logic programs via functional interpretation *) (* Author: Carsten Schuermann *) +exception AbortQuery of string +let () = Printexc.register_printer (function AbortQuery msg -> Some msg | _ -> None) + module Fquery (Fquery__0 : sig module Global : GLOBAL module Names : NAMES @@ -24,7 +27,7 @@ end) : FQUERY with module ExtQuery = Fquery__0.ReconQuery = struct module ReconQuery = Fquery__0.ReconQuery module Timers = Fquery__0.Timers - exception AbortQuery of string + exception AbortQuery = AbortQuery module I = IntSyn module T = Tomega diff --git a/src/Frontend/Lexer.ml b/src/Frontend/Lexer.ml index dc46652..c660829 100644 --- a/src/Frontend/Lexer.ml +++ b/src/Frontend/Lexer.ml @@ -586,6 +586,8 @@ end (* functor Lexer *) module Lexer = MakeLexer (Stream) include Lexer +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) +let () = Printexc.register_printer (function NotDigit c -> Some (Printf.sprintf "Not a digit: '%c'" c) | _ -> None) (*! structure Paths' = Paths !*) (* # 1 "src/frontend/Lexer.sml.ml" *) diff --git a/src/Frontend/Parsing.ml b/src/Frontend/Parsing.ml index edc8cd8..a09b8ce 100644 --- a/src/Frontend/Parsing.ml +++ b/src/Frontend/Parsing.ml @@ -55,6 +55,7 @@ type 'a recparser = 'a Parsing.recparser let recwith = Parsing.recwith exception Error = Parsing.Error +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) let error = Parsing.error (*! structure Lexer' = Lexer !*) diff --git a/src/Frontend/ReconCondec.ml b/src/Frontend/ReconCondec.ml index 8485815..2ea4b75 100644 --- a/src/Frontend/ReconCondec.ml +++ b/src/Frontend/ReconCondec.ml @@ -12,6 +12,9 @@ include RECONCONDEC (* # 1 "src/frontend/ReconCondec.fun.ml" *) open! Basis +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module ReconConDec (ReconConDec__0 : sig (* Reconstruct signature entries *) (* Author: Frank Pfenning *) @@ -51,7 +54,7 @@ end) : RECON_CONDEC = struct module Timers = ReconConDec__0.Timers module ExtSyn = ReconConDec__0.ReconTerm' - exception Error of string + exception Error = Error (* error (r, msg) raises a syntax error within region r with text msg *) let rec error (r, msg) = raise (Error (Paths.wrap (r, msg))) diff --git a/src/Frontend/ReconMode.ml b/src/Frontend/ReconMode.ml index f2ddf83..116e66c 100644 --- a/src/Frontend/ReconMode.ml +++ b/src/Frontend/ReconMode.ml @@ -13,6 +13,9 @@ open! Basis (* Reconstructing Mode Declarations *) (* Author: Carsten Schuermann *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module ReconMode (ReconMode__0 : sig module Global : GLOBAL @@ -34,7 +37,7 @@ end) : RECON_MODE = struct module ExtSyn = ReconMode__0.ReconTerm' (*! structure Paths = Paths' !*) - exception Error of string + exception Error = Error let rec error (r, msg) = raise (Error (Paths.wrap (r, msg))) diff --git a/src/Frontend/ReconModule.ml b/src/Frontend/ReconModule.ml index e7123a3..b66d35f 100644 --- a/src/Frontend/ReconModule.ml +++ b/src/Frontend/ReconModule.ml @@ -8,6 +8,9 @@ include RECONMODULE (* # 1 "src/frontend/ReconModule.fun.ml" *) open! Basis +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module ReconModule (ReconModule__0 : sig (* Elaboration for module expressions *) (* Author: Kevin Watkins *) @@ -34,7 +37,7 @@ end) : RECON_MODULE with module ModSyn = ReconModule__0.ModSyn' = struct (*! structure Paths = Paths' !*) module ModSyn = ReconModule__0.ModSyn' - exception Error of string + exception Error = Error let rec error (r, msg) = raise (Error (Paths.wrap (r, msg))) diff --git a/src/Frontend/ReconQuery.ml b/src/Frontend/ReconQuery.ml index bfab428..f384cad 100644 --- a/src/Frontend/ReconQuery.ml +++ b/src/Frontend/ReconQuery.ml @@ -12,6 +12,9 @@ include RECONQUERY (* # 1 "src/frontend/ReconQuery.fun.ml" *) open! Basis +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module ReconQuery (ReconQuery__0 : sig (* Reconstruct queries *) (* Author: Frank Pfenning *) @@ -46,7 +49,7 @@ end) : RECON_QUERY = struct module ExtSyn = ReconQuery__0.ReconTerm' module T = ReconQuery__0.ReconTerm' - exception Error of string + exception Error = Error (* error (r, msg) raises a syntax error within region r with text msg *) let rec error (r, msg) = raise (Error (Paths.wrap (r, msg))) diff --git a/src/Frontend/ReconTerm.ml b/src/Frontend/ReconTerm.ml index e578a0d..5f5d328 100644 --- a/src/Frontend/ReconTerm.ml +++ b/src/Frontend/ReconTerm.ml @@ -26,6 +26,9 @@ open! Basis (* ------------------- *) (* Type Reconstruction *) (* ------------------- *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module ReconTerm (ReconTerm__0 : sig (*! structure IntSyn' : INTSYN !*) module Names : NAMES @@ -78,7 +81,7 @@ end) : RECON_TERM = struct in run' !delayedList - exception Error of string + exception Error = Error let errorCount = ref 0 let errorFileName = ref "no file" diff --git a/src/Frontend/ReconThm.ml b/src/Frontend/ReconThm.ml index 27615ba..e114bad 100644 --- a/src/Frontend/ReconThm.ml +++ b/src/Frontend/ReconThm.ml @@ -12,6 +12,9 @@ include RECONTHM (* # 1 "src/frontend/ReconThm.fun.ml" *) open! Basis +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module ReconThm (ReconThm__0 : sig (* Reconstruct Termination Information *) (* Author: Carsten Schuermann *) @@ -39,7 +42,7 @@ end) : RECON_THM with module ThmSyn = ReconThm__0.ThmSyn' = struct (*! structure Paths = Paths' !*) module ExtSyn = ReconThm__0.ReconTerm' - exception Error of string + exception Error = Error open! struct module M = ModeSyn diff --git a/src/Frontend/Solve.ml b/src/Frontend/Solve.ml index f172de5..c542864 100644 --- a/src/Frontend/Solve.ml +++ b/src/Frontend/Solve.ml @@ -15,6 +15,9 @@ open! Basis (* Front End Interface *) (* Author: Frank Pfenning *) (* Modified: Carsten Schuermann, Jeff Polakow, Roberto Virga *) +exception AbortQuery of string +let () = Printexc.register_printer (function AbortQuery msg -> Some msg | _ -> None) + module Solve (Solve__0 : sig module Global : GLOBAL @@ -116,7 +119,7 @@ end) : SOLVE with module ExtQuery = Solve__0.ReconQuery = struct is raised when a %query declaration has an unexpected number of solutions or of %solve has no solution. *) - exception AbortQuery of string + exception AbortQuery = AbortQuery (* Bounds SOME(n) for n >= 0, NONE represents positive infinity *) (* Concrete syntax: 0, 1, 2, ..., * *) diff --git a/src/Frontend/Twelf_.mli b/src/Frontend/Twelf_.mli index 2fd491c..3a60925 100644 --- a/src/Frontend/Twelf_.mli +++ b/src/Frontend/Twelf_.mli @@ -1,4 +1,4 @@ -include module type of TWELF +module type STELF = TWELF.STELF module Stelf (Twelf__0 : sig module Global : GLOBAL diff --git a/src/Frontend/dune b/src/Frontend/dune index 40014ca..c7f4c8b 100644 --- a/src/Frontend/dune +++ b/src/Frontend/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name frontend) (public_name stelf.frontend) (modules diff --git a/src/Fronts/Modern/Cmd.ml b/src/Fronts/Modern/Cmd.ml index a304a62..7c60e96 100644 --- a/src/Fronts/Modern/Cmd.ml +++ b/src/Fronts/Modern/Cmd.ml @@ -15,12 +15,24 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct let ghost' = Cst.View.Loc.(review Ghost) (* Skip outer text (non-% characters) between commands. - %%% is an escape sequence for a literal %. *) + Handles three special sequences starting with %: + %%% — escape for a literal % + %% ... — line comment (skips to end of line) + %{! ... !}% — block comment (skips to matching !}%) *) let skip_outer : unit t = fix (fun self -> skip_while (fun c -> c <> '%') *> option () - (string "%%%" *> commit *> self)) + ( (string "%%%" *> commit *> self) + <|> (string "%%" *> commit + *> skip_while (fun c -> c <> '\n') + *> self) + <|> (string "%{!" *> commit + *> fix (fun inner -> + skip_while (fun c -> c <> '!') + *> ((string "!}%" *> commit) + <|> (char '!' *> inner))) + *> self))) (* Defer a thunk-parser to prevent infinite recursion at construction time. Used for %module and %eval which recursively embed cmd lists. *) @@ -158,7 +170,7 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct (keyword "sort" *> commit *> let* ids = Modern.parse_id_list () in - let+ ds = many (inside "{" "}" (commit *> Modern.parse_decl ())) in + let+ ds = many ((inside "{" "}" (commit *> Modern.parse_decl ())) <|> Modern.parse_decl_simple ()) in Cst.View.Cmd.(review @@ Sort (Cst.View.Loc.(review Ghost), ids, ds))) "sort" end; @@ -189,8 +201,8 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct (keyword "worlds" *> commit *> let* ids = inside "(" ")" (many (Modern.parse_var ())) in - let+ tm = Modern.parse_expr () in - Cst.View.Cmd.(review @@ Worlds (ghost', ids, tm))) + let+ tms = many1 (Modern.parse_expr1 ()) in + Cst.View.Cmd.(review @@ Worlds (ghost', ids, tms))) "worlds" end; begin @@ -216,6 +228,17 @@ module Make_Cmd (Modern : MODERN.MODERN) = struct Cst.View.Cmd.(review @@ Open (Cst.View.Loc.(review Ghost), id))) "open" end; + begin + (keyword "require" *> commit + *> + let+ ids = + (let+ s = Modern.parse_text () in + String.trim s |> String.split_on_char '/') + <|> Modern.parse_id_list () + in + Cst.View.Cmd.(review @@ Require (ghost', ids))) + "require" + end; begin (keyword "eval" *> commit *> diff --git a/src/Fronts/Modern/MODERN.ml b/src/Fronts/Modern/MODERN.ml index 9f227f2..ae00764 100644 --- a/src/Fronts/Modern/MODERN.ml +++ b/src/Fronts/Modern/MODERN.ml @@ -29,6 +29,7 @@ module type MODERN = sig val parse_text : unit -> string t val parse_decl : unit -> Cst.decl t + val parse_decl_simple : unit -> Cst.decl t val parse_mode : unit -> Cst.mode t val parse_mode_dec : unit -> Cst.modeDec t val parse_sigexp : unit -> Cst.sigexp t diff --git a/src/Fronts/Modern/Modern.ml b/src/Fronts/Modern/Modern.ml index a6e13a0..97921a4 100644 --- a/src/Fronts/Modern/Modern.ml +++ b/src/Fronts/Modern/Modern.ml @@ -59,8 +59,26 @@ module Make_Modern let ghost' = Cst.View.Loc.(review Ghost) - let rec combine_fc (r1 : Paths.region) (r2 : Paths.region) : Paths.region = - assert false + let combine_fc (r1 : Paths.region) (r2 : Paths.region) : Paths.region = + Paths.join (r1, r2) + + let loc_union (l1 : Cst.loc) (l2 : Cst.loc) : Cst.loc = + let open Cst.View.Loc in + match view l1, view l2 with + | Loc (_, s1, e1), Loc (_, s2, e2) -> mk_loc (min s1 s2) (max e1 e2) + | Ghost, _ -> l2 + | _, Ghost -> l1 + + let term_loc (tm : Cst.Term.t) : Cst.loc = + let open Cst.View.Term in + match view tm with + | Lowercase (loc, _) | Uppercase (loc, _) | Qualified (loc, _) + | Text (loc, _) | ExistVar (loc, _) | FreeVar (loc, _) + | Pi (loc, _, _) | Lam (loc, _, _) | App (loc, _, _) + | HasType (loc, _, _) | Omitted loc | Typ loc + | Arrow (loc, _, _) | BackArrow (loc, _, _) + | Foreign (loc, _) | MacroParam (loc, _, _) | Local (loc, _, _) -> loc + | Internal _ -> ghost' module FX = Names.Fixity @@ -92,30 +110,42 @@ module Make_Modern let jux_op = Infix_ ( (FX.inc FX.maxPrec, FX.Left), - fun (f, x) -> Cst.View.Term.(review @@ App (ghost', f, [ x ])) ) + fun (f, x) -> + let loc = loc_union (term_loc f) (term_loc x) in + Cst.View.Term.(review @@ App (loc, f, [ x ])) ) let infix_op (infixity, tm) = Infix_ ( infixity, fun (tm1, tm2) -> + let loc = loc_union (term_loc tm1) (term_loc tm2) in Cst.View.Term.( - review @@ App (ghost', review @@ App (ghost', tm, [ tm1 ]), [ tm2 ])) + review @@ App (loc, review @@ App (term_loc tm1, tm, [ tm1 ]), [ tm2 ])) ) let prefix_op (prec, tm) = Prefix_ - (prec, fun tm1 -> Cst.View.Term.(review @@ App (ghost', tm, [ tm1 ]))) + ( prec, + fun tm1 -> + let loc = loc_union (term_loc tm) (term_loc tm1) in + Cst.View.Term.(review @@ App (loc, tm, [ tm1 ])) ) let postfix_op (prec, tm) = Postfix_ - (prec, fun tm1 -> Cst.View.Term.(review @@ App (ghost', tm, [ tm1 ]))) + ( prec, + fun tm1 -> + let loc = loc_union (term_loc tm1) (term_loc tm) in + Cst.View.Term.(review @@ App (loc, tm, [ tm1 ])) ) let classify (tm : Cst.Term.t) : operator = let open Cst.View.Term in match !>tm with + | Qualified _ -> + (* %(name sort) constructor-reference syntax is always an atom — + explicit qualification opts out of operator status *) + Atom tm | Lowercase (_, (ns, name)) - | Uppercase (_, (ns, name)) - | Qualified (_, (ns, name)) -> + | Uppercase (_, (ns, name)) -> let fixity = match Hashtbl.find_opt local_fixity name with | Some fx -> fx @@ -277,11 +307,13 @@ module Make_Modern (let@ d, s, e = inside "[" "]" (parse_decl ()) in let loc = mk_loc s e in let+ body = parse_expr () in - Cst.View.Term.(review @@ Lam (loc, [ d ], body))) + let full_loc = loc_union loc (term_loc body) in + Cst.View.Term.(review @@ Lam (full_loc, [ d ], body))) <|> (let@ d, s, e = inside "{" "}" (parse_decl ()) in let loc = mk_loc s e in let+ body = parse_expr () in - Cst.View.Term.(review @@ Pi (loc, [ d ], body))) + let full_loc = loc_union loc (term_loc body) in + Cst.View.Term.(review @@ Pi (full_loc, [ d ], body))) <|> ((let* ids = inside "{{" "}}" (many @@ parse_var ()) in let+ body = with_uppercase ids parse_expr in body) @@ -302,15 +334,16 @@ module Make_Modern let init = List.rev (List.tl rev) in List.fold_right (fun t acc -> + let loc = loc_union (term_loc t) (term_loc acc) in Cst.View.Term.( review @@ Pi - ( ghost', + ( loc, [ Cst.View.Decl.( review @@ Decl1 - ( ghost', + ( term_loc t, [ None ], t, Cst.View.Term.(review @@ Omitted ghost') )); @@ -324,15 +357,16 @@ module Make_Modern let rest_rev = List.rev rest in List.fold_right (fun t acc -> + let loc = loc_union (term_loc t) (term_loc acc) in Cst.View.Term.( review @@ Pi - ( ghost', + ( loc, [ Cst.View.Decl.( review @@ Decl1 - ( ghost', + ( term_loc t, [ None ], t, Cst.View.Term.(review @@ Omitted ghost') )); @@ -350,17 +384,35 @@ module Make_Modern and parse_expr1 () : Cst.Term.t t = begin choice - [ parse_id (); inside "(" ")" (return () >>= fun () -> parse_expr ()) ] + [ parse_id (); + inside "(" ")" (return () >>= fun () -> parse_expr ()); + (let@ str, s, e = parse_text () in + let loc = mk_loc s e in + return Cst.View.Term.(review @@ Text (loc, str))) ] end "small expression" and parse_expr () : Cst.Term.t t = begin - (keyword' "the" *> commit - *> - let* ty = parse_expr1 () in - let+ body = parse_expr () in - Cst.View.Term.(review @@ HasType (ghost', body, ty))) + (let@ (ty, body), s, e = + keyword' "the" *> commit + *> + let* ty = parse_expr1 () in + let+ body = parse_expr () in + (ty, body) + in + let loc = mk_loc s e in + return Cst.View.Term.(review @@ HasType (loc, body, ty))) + <|> + (let@ (ns_id, body), s, e = + keyword' "local" *> commit + *> + let* ns_id = ident1 in + let+ body = parse_expr () in + (ns_id, body) + in + let loc = mk_loc s e in + return Cst.View.Term.(review @@ Local (loc, [ ns_id ], body))) (* %<- A @@ -406,10 +458,15 @@ module Make_Modern "qualified name" and parse_text () : string t = - begin - string "%[" *> take_till (fun c -> c = '%') - <* string "%]" <* whitespace' () - end + let take_until delim = + fix (fun self -> + let* pre = take_while (fun c -> c <> '%') in + (string delim *> return pre) + <|> (char '%' *> let+ suf = self in pre ^ "%" ^ suf)) + in + (string "%[[" *> commit *> take_until "%]]" <* whitespace' ()) + <|> + (string "%[" *> commit *> take_until "%]" <* whitespace' ()) "string literal" and parse_decl () : Cst.decl t = @@ -433,7 +490,12 @@ module Make_Modern (loc, [ name ], typ, Cst.View.Term.(review @@ Omitted ghost'))) end "declaration" - + and parse_decl_simple () : Cst.decl t = begin + let@ (typ, s, e) = (parse_id ()) <|> (inside "(" ")" (parse_expr ())) in + let loc = mk_loc s e in + return @@ Cst.View.Decl.(review @@ Decl1 (loc, [None], typ, Cst.View.Term.(review @@ Omitted ghost'))) + end + and parse_mode () : Cst.mode t = begin (let@ (), s, e = keyword' "out1" *> commit *> return () in @@ -744,6 +806,9 @@ module ModernCst = Cst.Make_Cst (Paths.Paths_) module Modern : MODERN.MODERN = Make_Modern (Paths.Paths_) (ModernCst) (Names.Names_) (Parsing.Parser.Parser) +let () = Printexc.register_printer (function Modern.ParseError msg -> Some msg | _ -> None) +let () = Printexc.register_printer (function Modern.FullParseError _ -> Some "Parse error" | _ -> None) + (* Re-export sub-modules so they are accessible outside the library *) module Cmd = Cmd diff --git a/src/Fronts/Modern/dune b/src/Fronts/Modern/dune index 8c833ef..3043c21 100644 --- a/src/Fronts/Modern/dune +++ b/src/Fronts/Modern/dune @@ -1,4 +1,5 @@ (library + (instrumentation (backend bisect_ppx)) (name modern) (public_name stelf.modern) (libraries display parsing lexing paths names cst error)) diff --git a/src/Fronts/Pal/Impl.ml b/src/Fronts/Pal/Impl.ml index 430a3e8..b772411 100644 --- a/src/Fronts/Pal/Impl.ml +++ b/src/Fronts/Pal/Impl.ml @@ -12,7 +12,7 @@ let string_to_source = function module Impl () = struct - let current_path : string list ref = ref [] + let current_path : string list ref = ref [] let current_load_path : Fpath.t list ref = ref ([Result.get_ok @@ Bos.OS.Dir.current ()]) let load_paths : (Fpath.t * string) list ref = ref [] @@ -80,6 +80,29 @@ module Impl () = struct type status = Ok | Abort let status_to_exit = function Ok -> 0 | Abort -> 1 + + (* Top-level namespace for the group currently being loaded. + %require deposits here, escaping any inner %scope. *) + let current_group_ns : Names.namespace ref = ref (Names.newNamespace ()) + + (* Canonical paths already run via %require; skip on repeat (#once). *) + let required_files : (string, unit) Hashtbl.t = Hashtbl.create 16 + + (* Forward reference set after `load` is defined, breaking the cycle + between Install.install1 (needs to call load for %require) and load + (which calls Install.install). *) + let load_file_ref : (Names.namespace -> Fpath.t -> status) ref = + ref (fun _ns _path -> Ok) + + (* Forward reference for TOML config detection, set after Config is defined. + load calls this to check if a file is a valid TOML project before + treating it as source. *) + let try_toml_ref : (Fpath.t -> string -> Project.Format.file option) ref = + ref (fun _path _str -> None) + + let load_toml_ref : (Project.Format.file -> status) ref = + ref (fun _cfg -> Ok) + (* ------------------------------------------------------------------ *) (* Options store *) (* ------------------------------------------------------------------ *) @@ -97,11 +120,16 @@ module Impl () = struct (* let msg s = Msg.Msg_.Msg.message s *) let msg m = - Display.message ~level:Display.Normal ~kind:Display.Info + Display.message ~level:Display.Normal ~kind:Display.Response + (Display.Form.string m) + + let dbg m = + Display.message ~level:Display.Exhaustive ~kind:Display.Debug (Display.Form.string m) let print_error (label : string) (detail : string) : unit = - Printf.eprintf "%s: %s\n%!" label detail + Display.message ~level:Display.Normal ~kind:Display.Error + (Display.Form.string (label ^ ": " ^ detail)) (* ------------------------------------------------------------------ *) (* ConDec installation *) @@ -235,6 +263,16 @@ module Impl () = struct let loc_of filename (l : Cst.loc) : Paths.location = Paths.Loc (filename, Cst.loc_to_region l) + let decl_loc (Cst.Dec_ (_, _, l)) : Cst.loc = l + + let term_loc (tm : Cst.term) : Cst.loc = + match tm with + | Cst.Arrow_ (l, _, _) | Cst.Pi_ (l, _, _) | Cst.Lam_ (l, _, _) + | Cst.App_ (l, _, _) | Cst.Hastype_ (l, _, _) | Cst.Omitted_ l + | Cst.Typ_ l | Cst.MacroParam_ (l, _, _) | Cst.Local_ (l, _, _) -> l + | Cst.Lcid_ (_, _, l) | Cst.Ucid_ (_, _, l) | Cst.Quid_ (_, _, l) + | Cst.Scon_ (_, l) | Cst.Evar_ (_, l) | Cst.Fvar_ (_, l) -> l + let run_query loc q = let v_, opt_name, xs_ = Recon.ReconQuery.queryToQuery (q, loc) @@ -276,7 +314,7 @@ module Impl () = struct if !solutions = 0 && !Global.Global_.Global.chatter >= 3 then msg "No solution.\n" - let install_worlds_cmd ids tm = + let install_worlds_cmd ids tms = let resolve_block id = match Names.constLookup (Names.Qid ([], id)) with | None -> @@ -304,16 +342,19 @@ module Impl () = struct Names.constLookup (Names.Qid (ns, n)) | _ -> None in - let family_cid_opt = - match Cst.View.Term.view tm with - | Cst.View.Term.App (_, head, _) -> lookup_head head - | v -> lookup_head (Cst.View.Term.review v) + let resolve_family tm = + let family_cid_opt = + match Cst.View.Term.view tm with + | Cst.View.Term.App (_, head, _) -> lookup_head head + | v -> lookup_head (Cst.View.Term.review v) + in + match family_cid_opt with + | None -> failwith "%worlds: expected a type family name" + | Some a -> a in - match family_cid_opt with - | None -> failwith "%worlds: expected a type family name" - | Some a -> - WorldSyn.install (a, w_); - WorldSyn.worldcheck w_ a + let families = List.map resolve_family tms in + List.app (fun a -> WorldSyn.install (a, w_)) families; + List.app (fun a -> WorldSyn.worldcheck w_ a) families let install_condec_cmd ?(inline = false) ns condec loc = match Recon.ReconConDec.condecToConDec (condec, loc, inline) with @@ -337,6 +378,7 @@ module Impl () = struct msg' ~src:Group.pal ~level:Level.Debug Fmt.string "Installing command"); match cmd with | Cst.SortCmd_ (ids, decls) -> + let sort_loc = match decls with d :: _ -> decl_loc d | [] -> Cst.ghost in List.app (fun id -> Debug.( @@ -347,7 +389,7 @@ module Impl () = struct let condec = Cst.ConDec.constant_decl (Cst.Decl.decl1 [ Some id ] kind) in - install_condec_cmd ns condec (loc_of filename Cst.ghost)) + install_condec_cmd ns condec (loc_of filename sort_loc)) ids | Cst.TermCmd_ decl -> ( Debug.( @@ -366,16 +408,16 @@ module Impl () = struct ++ each string names' ++ space () ++ hvbox [ shown Cst.show_term ty ]); let condec = Cst.ConstantDecl_ decl in - install_condec_cmd ns condec (loc_of filename Cst.ghost)) + install_condec_cmd ns condec (loc_of filename (decl_loc decl))) | Cst.DefineCmd_ (Cst.Define_ (name_opt, tm, tp_opt)) -> let name = match name_opt with Some n -> n | None -> "_" in let condec = Cst.ConstantDef_ (name, tm, tp_opt) in - install_condec_cmd ns condec (loc_of filename Cst.ghost) - | Cst.QueryCmd_ (_n, _b, _d, q) -> - run_query (loc_of filename Cst.ghost) q - | Cst.SolveCmd_ sol -> + install_condec_cmd ns condec (loc_of filename (term_loc tm)) + | Cst.QueryCmd_ (_n, _b, _d, (Cst.Query_ (_, qtm) as q)) -> + run_query (loc_of filename (term_loc qtm)) q + | Cst.SolveCmd_ (Cst.Solve_ (_, stm) as sol) -> let v_, sc_fn = - Recon.ReconQuery.solveToSolve ([], sol, loc_of filename Cst.ghost) + Recon.ReconQuery.solveToSolve ([], sol, loc_of filename (term_loc stm)) in let g = Compile.Compile_.Compile.compileGoal (Intsyn.IntSyn.Null, v_) @@ -415,8 +457,8 @@ module Impl () = struct | Cst.SetCmd_ (key, value) -> Options.set key value | Cst.VersionCmd_ -> msg (Frontend.Version.Version.version_string ^ "\n") | Cst.EvalCmd_ cmds -> List.app (install1 ~path ns) cmds - | Cst.AdhocQueryCmd_ q -> - run_query (loc_of filename Cst.ghost) q + | Cst.AdhocQueryCmd_ (Cst.Query_ (_, qtm) as q) -> + run_query (loc_of filename (term_loc qtm)) q | Cst.DeclCmd_ tm -> let qid_opt = match Cst.View.Term.view tm with @@ -465,12 +507,14 @@ module Impl () = struct let cid = name_to_cid ns "prec" id in Names.installFixity (cid, fixity)) ids - | Cst.SymbolCmd_ (id, pref) -> + | Cst.SymbolCmd_ (pref, id) -> let cid = name_to_cid ns "symbol" id in + Names.installAlias (pref, cid); + Names.insertConstAlias (ns, pref, cid); Names.installNamePref (cid, ([ pref ], [ pref ])) | Cst.InlineCmd_ (name, tm) -> let condec = Cst.ConstantDef_ (name, tm, None) in - install_condec_cmd ~inline:true ns condec (loc_of filename Cst.ghost) + install_condec_cmd ~inline:true ns condec (loc_of filename (term_loc tm)) | Cst.BlockCmd_ (id, items) -> let pis = Stdlib.List.filter_map @@ -482,8 +526,11 @@ module Impl () = struct (function Cst.BlockSome_ d -> Some d | _ -> None) items in + let block_loc = + match pis @ somes with d :: _ -> decl_loc d | [] -> Cst.ghost + in let condec = Cst.BlockDecl_ (id, pis, somes) in - install_condec_cmd ns condec (loc_of filename Cst.ghost) + install_condec_cmd ns condec (loc_of filename block_loc) | Cst.ModeCmd_ md -> let () = Display.( @@ -538,10 +585,10 @@ module Impl () = struct let syms = List.map (fun s -> ([], s)) ids in let condec = Cst.BlockDef_ (id, syms) in install_condec_cmd ns condec (loc_of filename Cst.ghost) - | Cst.WorldsCmd_ (ids, tm) -> install_worlds_cmd ids tm - | Cst.QueryTabledCmd_ (numSol, try_, _d, q) -> + | Cst.WorldsCmd_ (ids, tms) -> install_worlds_cmd ids tms + | Cst.QueryTabledCmd_ (numSol, try_, _d, (Cst.Query_ (_, qtm) as q)) -> let a_, opt_name, xs_ = - Recon.ReconQuery.queryToQuery (q, loc_of filename Cst.ghost) + Recon.ReconQuery.queryToQuery (q, loc_of filename (term_loc qtm)) in let g = Compile.Compile_.Compile.compileGoal (Intsyn.IntSyn.Null, a_) @@ -628,7 +675,39 @@ module Impl () = struct failwith "Macros not yet implemented in this frontend" | Cst.Seq_ cmds -> List.app (install1_item ~path ns) cmds - | Cst.Require_ path -> failwith "Require not yet implemented in this frontend" + | Cst.Require_ ids -> + let rel = String.concatWith "/" ids in + let req_path = Fpath.v rel in + let parent_dir = Fpath.parent req_path in + let stem = Fpath.to_string (Fpath.rem_ext (Fpath.base req_path)) in + let is_source_ext ext = ext <> "" && ext <> ".cfg" && ext <> ".toml" in + let found = + Stdlib.List.find_map (fun base_dir -> + let search_dir = Fpath.(base_dir // parent_dir) in + match Bos.OS.Dir.contents search_dir with + | Error _ -> None + | Ok entries -> + Stdlib.List.find_opt (fun entry -> + let b = Fpath.base entry in + Fpath.to_string (Fpath.rem_ext b) = stem + && is_source_ext (Fpath.get_ext b) + ) entries + ) !current_load_path + in + (match found with + | None -> failwith ("%require: file not found: " ^ rel) + | Some file_path -> + let canon = Fpath.to_string (Fpath.normalize file_path) in + dbg ("%%require: resolved " ^ rel ^ " -> " ^ canon); + if not (Hashtbl.mem required_files canon) then begin + dbg ("%%require: loading " ^ canon); + Hashtbl.add required_files canon (); + (match !load_file_ref !current_group_ns file_path with + | Ok -> dbg ("%%require: finished " ^ canon) + | Abort -> + failwith ("%require: error loading " ^ Fpath.to_string file_path)) + end else + dbg ("%%require: skipping " ^ canon ^ " (already loaded)")) and install1_item ?(path = None) ns (item : Cst.item) : unit = match item with @@ -646,6 +725,7 @@ module Impl () = struct (* ------------------------------------------------------------------ *) let load_string ?(path = None) ?(ns_init = None) (str : string) : status = + dbg ("load_string: " ^ source_to_string path); let ns = match ns_init with Some r -> r | None -> ref (Names.newNamespace ()) in let loc = Cst.ghost in try @@ -671,8 +751,8 @@ module Impl () = struct | Failure msg -> Error.Error.err ~stage:Error.Error.Unknown (Display.Form.string msg) | exn -> - print_error (source_to_string path) (Printexc.to_string exn); - Abort + Error.Error.err ~stage:Error.Error.Unknown + (Display.Form.string (Printexc.to_string exn)) with | Error.Error.Err (stage, form) -> let stage_str = @@ -682,31 +762,52 @@ module Impl () = struct | Error.Error.Recon -> "recon" | _ -> "error" in - Printf.eprintf "[%s] %s\n%!" stage_str (Display.Form.to_plain form); + Display.message ~level:Display.Normal ~kind:Display.Error + (Display.Form.string + ("[" ^ stage_str ^ "] " ^ Display.Form.to_plain form)); Abort | Modern.Modern.ParseError e -> (* Fallback during transition; should no longer be reached *) print_error (source_to_string path) ("parse error: " ^ e); Abort | exn -> - print_error (source_to_string path) (Printexc.to_string exn); - Abort + Error.Error.err ~stage:Error.Error.Unknown + (Display.Form.string (Printexc.to_string exn)) let load ns (src : source) : status = let ns_ref = ref ns in match src with - | Input str -> load_string ~ns_init:(Some ns_ref) str + | Input str -> + dbg "load: loading from input"; + load_string ~ns_init:(Some ns_ref) str | File path -> ( + dbg ("load: loading file " ^ Fpath.to_string path); let filename = Some path in try - let ic = TextIO.openIn (Fpath.to_string path) in - let str = TextIO.inputAll ic in - let () = TextIO.closeIn ic in - load_string ~path:filename ~ns_init:(Some ns_ref) str + let buf = Bos.OS.File.read path in + let str = match buf with + | Ok s -> s + | Error (`Msg msg) -> + print_error "pal" ("Failed to read file: " ^ msg); + failwith (msg) + in + match !try_toml_ref path str with + | Some cfg -> + dbg ("load: detected TOML config in " ^ Fpath.to_string path); + !load_toml_ref cfg + | None -> + let saved_load_path = !current_load_path in + let parent = Fpath.parent path in + current_load_path := parent :: !current_load_path; + let result = load_string ~path:filename ~ns_init:(Some ns_ref) str in + current_load_path := saved_load_path; + result with exn -> print_error "pal" (Printexc.to_string exn); Abort) + let () = load_file_ref := (fun ns fp -> load ns (File fp)) + let read_decl () : status = begin match TextIO.inputLine TextIO.stdIn with | Some line -> load_string ~path:None line @@ -746,27 +847,183 @@ module Impl () = struct let () = TextIO.closeIn ic in str with exn -> - print_error "pal" (Printexc.to_string exn); - "") in - match Project.Format.from_toml @@ Otoml.string str with - | None -> failwith "Failed to parse config file" - | Some config -> config - - let load' (cfg : Project.Format.t) : status = - assert false - let load (cfg : t) : status = - assert false - - + let msg = "Failed to read config file " ^ Fpath.to_string path + ^ ": " ^ Printexc.to_string exn in + print_error "pal" msg; + failwith msg) in + let toml = + match Otoml.Parser.from_string str with + | toml -> toml + | exception Otoml.Parse_error (pos_opt, msg) -> + let loc = match pos_opt with + | Some (line, col) -> Printf.sprintf " at line %d, col %d" line col + | None -> "" in + let full = Printf.sprintf "TOML parse error%s: %s" loc msg in + print_error "pal" full; + failwith full + | exception exn -> + let msg = "TOML parse error: " ^ Printexc.to_string exn in + print_error "pal" msg; + failwith msg in + match Project.Format.from_toml toml with + | Error msg -> + print_error "pal" msg; + failwith msg + | Ok config -> config + + let rebase_group base_dir (g : Project.Format.t) : Project.Format.t = + let open Project.Format in + let rebase_dep = function + | Local ({ path = Some p; _ } as d) -> Local { d with path = Some Fpath.(base_dir // p) } + | dep -> dep + in + { g with + main = Fpath.(base_dir // g.main); + dirs = List.map (fun d -> Fpath.(base_dir // d)) g.dirs; + deps = List.map rebase_dep g.deps; + } + + let try_toml (path : Fpath.t) (str : string) : t option = + let base_dir = Fpath.parent path in + let is_toml = Fpath.has_ext "toml" path in + match Otoml.Parser.from_string str with + | exception Otoml.Parse_error (pos_opt, msg) -> + if is_toml then begin + let loc = match pos_opt with + | Some (line, col) -> Printf.sprintf " at line %d, col %d" line col + | None -> "" in + print_error "config" + (Printf.sprintf "TOML parse error in %s%s: %s" (Fpath.to_string path) loc msg) + end; + None + | exception exn -> + if is_toml then + print_error "config" + ("Error reading TOML config " ^ Fpath.to_string path ^ ": " + ^ Printexc.to_string exn); + None + | toml -> + match Project.Format.from_toml toml with + | Error msg -> + if is_toml then + print_error "config" (Fpath.to_string path ^ ": " ^ msg); + None + | Ok cfg -> + let groups = List.map (rebase_group base_dir) cfg.Project.Format.groups in + Some { cfg with Project.Format.groups = groups } + + let rec load' ~(all_groups : Project.Format.t list) (cfg : Project.Format.t) : status = + let open Project.Format in + dbg ("Config.load': group " ^ cfg.name ^ ", main=" ^ Fpath.to_string cfg.main); + let group_ns = Names.newNamespace () in + let saved_group_ns = !current_group_ns in + let saved_load_path = !current_load_path in + current_group_ns := group_ns; + current_load_path := + (if cfg.dirs = [] then [ Fpath.v "." ] else cfg.dirs); + let install_dep dep = + match dep with + | Local { name; alias; path = None } -> + dbg ("Config.load': dependency " ^ name + ^ (match alias with Some a -> " (alias " ^ a ^ ")" | None -> "")); + (match Stdlib.List.find_opt (fun g -> g.name = name) all_groups with + | None -> + print_error "config" ("Local dependency not found: " ^ name); + Abort + | Some dep_group -> + dbg ("Config.load': loading dependency group " ^ dep_group.name); + let _ = load' ~all_groups dep_group in + let ns_name = Stdlib.Option.value alias ~default:name in + if ns_name <> dep_group.name then begin + (* Create an alias struct with copied components *) + match Names.structLookupIn (group_ns, Names.Qid ([], dep_group.name)) with + | None -> Ok + | Some mid_dep -> + let alias_ns = Names.newNamespace () in + let comps = Names.getComponents mid_dep in + Names.appConsts (fun (_, cid) -> Names.insertConst (alias_ns, cid)) comps; + Names.appStructs (fun (_, m) -> Names.insertStruct (alias_ns, m)) comps; + let mid_alias = Intsyn.IntSyn.sgnStructAdd + (Intsyn.IntSyn.StrDec (ns_name, None)) in + Names.installStructName mid_alias; + Names.insertStruct (group_ns, mid_alias); + Names.installComponents (mid_alias, alias_ns); + Ok + end else Ok) + | Local { name; alias; path = Some p } -> + dbg ("Config.load': dependency " ^ name ^ " at " ^ Fpath.to_string p + ^ (match alias with Some a -> " (alias " ^ a ^ ")" | None -> "")); + (match Bos.OS.File.read p with + | Error (`Msg msg) -> + print_error "config" ("Failed to read dependency file: " ^ msg); Abort + | Ok str -> + (match !try_toml_ref p str with + | None -> + print_error "config" ("Not a valid TOML config: " ^ Fpath.to_string p); Abort + | Some sub_file -> + let groups = sub_file.Project.Format.groups in + (match Stdlib.List.find_opt (fun g -> g.Project.Format.name = name) groups with + | None -> + print_error "config" + ("Group '" ^ name ^ "' not found in " ^ Fpath.to_string p); Abort + | Some dep_group -> + let _ = load' ~all_groups:groups dep_group in + let ns_name = Stdlib.Option.value alias ~default:name in + if ns_name <> dep_group.Project.Format.name then + match Names.structLookupIn (group_ns, Names.Qid ([], dep_group.Project.Format.name)) with + | None -> Ok + | Some mid_dep -> + let alias_ns = Names.newNamespace () in + let comps = Names.getComponents mid_dep in + Names.appConsts (fun (_, cid) -> Names.insertConst (alias_ns, cid)) comps; + Names.appStructs (fun (_, m) -> Names.insertStruct (alias_ns, m)) comps; + let mid_alias = Intsyn.IntSyn.sgnStructAdd + (Intsyn.IntSyn.StrDec (ns_name, None)) in + Names.installStructName mid_alias; + Names.insertStruct (group_ns, mid_alias); + Names.installComponents (mid_alias, alias_ns); + Ok + else Ok))) + | Installed { name; _ } -> + dbg ("Config.load': installed dependency " ^ name); + Ok + | External _ -> + dbg "Config.load': external dependency (skipped)"; + Ok + in + let dep_results = List.map install_dep cfg.deps in + let dep_ok = not (List.exists (fun s -> s = Abort) dep_results) in + let main_status = + if dep_ok then begin + dbg ("Config.load': loading main " ^ Fpath.to_string cfg.main); + load group_ns (File cfg.main) + end else Abort + in + let mid = Intsyn.IntSyn.sgnStructAdd + (Intsyn.IntSyn.StrDec (cfg.name, None)) in + Names.installStructName mid; + Names.insertStruct (saved_group_ns, mid); + Names.installComponents (mid, group_ns); + current_group_ns := saved_group_ns; + current_load_path := saved_load_path; + main_status + + let load (cfg : t) : status = + let groups = cfg.Project.Format.groups in + dbg ("Config.load: " ^ string_of_int (Stdlib.List.length groups) ^ " group(s)"); + let results = List.map (load' ~all_groups:groups) groups in + if List.exists (fun s -> s = Abort) results then Abort else Ok + end + + let () = try_toml_ref := Config.try_toml + let () = load_toml_ref := Config.load + let make (src : source) : status = - let is_config path = - let ext = Fpath.get_ext path in - ext = ".cfg" || ext = ".toml" - in - match src with - | File path when is_config path -> Config.load (Config.read src) - | _ -> load (Names.newNamespace ()) src + dbg (match src with + | File p -> "make: loading " ^ Fpath.to_string p + | Input _ -> "make: loading from input"); + load (Names.newNamespace ()) src (* ------------------------------------------------------------------ *) (* Print settings *) diff --git a/src/Fronts/Pal/Opts.ml b/src/Fronts/Pal/Opts.ml index 87fab11..758559b 100644 --- a/src/Fronts/Pal/Opts.ml +++ b/src/Fronts/Pal/Opts.ml @@ -7,20 +7,17 @@ module Opts : OPTS = struct let verbosity : Display.Info.level t = begin - let doc : Arg.info = - Arg.info ~doc:"The verbosity level" [ "v"; "verbose" ] ~docv:"LEVEL" - and v_conv = - Arg.enum + let v_conv = + Arg.vflag Display.Info.Normal [ - ("terse", Display.Info.Terse); - ("normal", Display.Info.Normal); - ("detailed", Display.Info.Detailed); - ("exhaustive", Display.Info.Exhaustive); - ("minimal", Display.Info.Minimal); - ("off", Display.Info.Off); + (Display.Info.Terse, Arg.info ~doc:"Display less information" [ "q"; "quiet" ]); + (Display.Info.Detailed, Arg.info ~doc:"Display more information" [ "v"; "verbose" ]); + (Display.Info.Exhaustive, Arg.info ~doc:"Display exhaustive information" [ "debug" ]); + (Display.Info.Minimal, Arg.info ~doc:"Display minimal information" [ "s"; "silent" ]); + (Display.Info.Off, Arg.info ~doc:"Display no information" [ "no-output" ]); ] in - Arg.(opt v_conv Display.Info.(Normal) & doc) + v_conv end let color : bool t = diff --git a/src/Fronts/Pal/PAL.ml b/src/Fronts/Pal/PAL.ml index cad29da..b6950c2 100644 --- a/src/Fronts/Pal/PAL.ml +++ b/src/Fronts/Pal/PAL.ml @@ -13,5 +13,8 @@ module type PAL = sig module Start () : PAL' with module M = M + val status_to_exit : M.status -> int + val make : M.source -> M.status + val top : ?config:Fpath.t -> (module Tui.REPL.S) -> int Lwt.t val run : unit -> unit end diff --git a/src/Fronts/Pal/Pal.ml b/src/Fronts/Pal/Pal.ml index 7efd4f2..f1f7e90 100644 --- a/src/Fronts/Pal/Pal.ml +++ b/src/Fronts/Pal/Pal.ml @@ -1,16 +1,23 @@ let version = "0.1.0" +let logo = Logo.logo module type PAL = PAL.PAL module type PAL' = PAL.PAL' +module Opts = Opts + +exception Error of exn +let () = Printexc.register_printer (function Error exn -> Some (Printexc.to_string exn) | _ -> None) + module Pal : PAL.PAL = struct module M = Impl.Impl () module type S = module type of M let status_to_exit : M.status -> int = function M.Ok -> 0 | M.Abort -> 1 + let make = M.make - exception Error of exn + exception Error = Error module Start () = struct module M = struct @@ -28,107 +35,16 @@ module Pal : PAL.PAL = struct let exec (s : string) : unit = List.iter install (parse s) end - let top (module N : Tui.REPL.S) = + let top ?config (module N : Tui.REPL.S) = let module Pal = Start () in let module R = Tui.Repl.Repl (N) in M.mode := `Repl; + (match config with + | Some f -> ignore (M.make (M.File f)) + | None -> ()); R.read (fun l -> Pal.exec l; Lwt.return R.Continue) - let run () : unit = - begin - let module TempM = struct - let use_color = true - let use_unicode = true - end in - let open Cmdliner in - let open Cmdliner.Term.Syntax in - (* Note: local 'Cmd' alias refers to Cmdliner.Cmd here *) - let help_cmd : int Cmd.t = - let help_info = Cmd.info "help" ~doc:"Display help information" - and help_term : int Term.t = - Term.(const 0) - (* TODO *) - in - Cmd.v help_info help_term - in - let repl_cmd : int Cmd.t = - let repl_fn : int Term.t = - let+ verbosity = Arg.value Opts.Opts.verbosity - and+ color = Arg.value Opts.Opts.color - and+ unicode = Arg.value Opts.Opts.unicode in - let module M = struct - let use_color = color - let use_unicode = unicode - let verbosity = verbosity - end in - Lwt_main.run (top (module M)) - and repl_info : Cmd.info = - Cmd.info "repl" ~doc:"Start the interactive REPL" - in - Cmd.v repl_info repl_fn - in - let load_cmd : int Cmd.t = - begin - let file = - Arg.(required & pos 0 (some file) None & info [] ~docv:"FILE") - in - Cmd.v - (Cmd.info "check" ~doc:"Load a .cfg or source file") - Term.( - const (fun f -> - M.chatter := 1; - Display.register (fun m -> - let current = Display.Info.from_chatter !M.chatter in - if Display.Info.(m.level =< current) then - let _ = Display.fmt Fmt.stdout m.msg in - Lwt.return () - else Lwt.return ()); - status_to_exit @@ M.make (File (Fpath.v f))) - $ file) - end - in - let lsp_cmd : int Cmd.t = - begin - Cmd.v - (Cmd.info "server" ~doc:"Start the LSP server (not yet implemented)") - Term.( - const (fun () -> - M.mode := `Lsp; - print_endline "LSP mode is not yet implemented."; - Basis.OS.Process.exit Basis.OS.Process.failure) - $ const ()) - end - in - let legacy_cmd : int Cmd.t = - Cmd.v (Cmd.info "legacy") - Term.( - const (fun () -> - let module Twelf_server = Server.Server_.Server in - Twelf_server.server ("stelf", [])) - $ const ()) - in - let version_cmd : int Cmd.t = - Cmd.v - (Cmd.info "version" ~doc:"Display version information") - Term.( - const (fun () -> - - print_endline ("STELF version " ^ version); - Fmt_tty.setup_std_outputs (); - Display.fmt Format.std_formatter @@ Logo.logo; - 0) - $ const ()) - in - let main_cmd = - Cmd.group - (Cmd.info "stelf" ~version ~doc:"The STELF proof assistant") - [ repl_cmd; load_cmd; version_cmd; lsp_cmd; legacy_cmd ] - in - Basis.OS.Process.exit (Cmd.eval' main_cmd) - end + let run () = () end - -(*let cmds = ModernImpl.run (Cmd.parse ()) ns loc line in - List.app Install.install1 cmds; Lwt.return (Repl.Continue);*) diff --git a/src/Fronts/Pal/Pal.mli b/src/Fronts/Pal/Pal.mli index 4fa0451..666e546 100644 --- a/src/Fronts/Pal/Pal.mli +++ b/src/Fronts/Pal/Pal.mli @@ -1,4 +1,10 @@ +val version : string +val logo : Display.form + module type PAL = PAL.PAL module type PAL' = PAL.PAL' module Pal : PAL.PAL +module Opts : sig + module Opts : OPTS.OPTS +end diff --git a/src/Fronts/Pal/dune b/src/Fronts/Pal/dune index 5a7c424..d11e4bc 100644 --- a/src/Fronts/Pal/dune +++ b/src/Fronts/Pal/dune @@ -1,4 +1,5 @@ (library + (instrumentation (backend bisect_ppx)) (name pal) (public_name stelf.pal) (libraries diff --git a/src/Fronts/Project/Format.ml b/src/Fronts/Project/Format.ml index f52b09a..26b5bf1 100644 --- a/src/Fronts/Project/Format.ml +++ b/src/Fronts/Project/Format.ml @@ -26,17 +26,19 @@ module Format_ = struct (** A group (specified by `[[group]]` (can be mutiple per file), is a single unit *) - type file = t list - (** File is a list of [[group]] tables*) + type file = { groups : t list; interactive : string option } + (** A parsed project file: [[group]] tables plus an optional top-level + [interactive] key naming the group to load in REPL mode. *) end module Format : sig include module type of Format_ - val from_toml : Otoml.t -> file option + val from_toml : Otoml.t -> (file, string) result end = struct include Format_ + (* Option monad for depends/ext helpers — unknown dep types are silently skipped *) let ( let* ) = Option.bind let fpath_opt s = match Fpath.of_string s with Ok p -> Some p | Error _ -> None @@ -72,21 +74,52 @@ end = struct let* url = Otoml.Helpers.find_string_opt link [ "url" ] in Some (Url { url }) + (* Result monad for group/file parsing — missing fields are errors *) + let ( let*! ) x f = match x with Ok v -> f v | Error _ as e -> e + + let required_string toml key = + match Otoml.Helpers.find_string_opt toml key with + | Some s -> Ok s + | None -> Error (Printf.sprintf "missing required field %S" (String.concat "." key)) + + let fpath_r label s = + match Fpath.of_string s with + | Ok p -> Ok p + | Error (`Msg msg) -> Error (Printf.sprintf "invalid %s path %S: %s" label s msg) + let group_of_toml toml = - let* name = Otoml.Helpers.find_string_opt toml [ "name" ] in - let* dirs_s = Otoml.Helpers.find_strings_opt toml [ "dirs" ] in + let*! name = required_string toml [ "name" ] in + let*! dirs_s = + match + List.find_map + (fun key -> Otoml.Helpers.find_strings_opt toml [ key ]) + [ "dirs"; "srcs"; "src"; "dir" ] + with + | Some ss -> Ok ss + | None -> Error {|missing required field "dirs" (or "srcs" / "src" / "dir")|} in let dirs = List.filter_map fpath_opt dirs_s in - let* main_s = Otoml.Helpers.find_string_opt toml [ "main" ] in - let* main = fpath_opt main_s in + let*! main_s = required_string toml [ "main" ] in + let*! main = fpath_r "main" main_s in let deps = - match Otoml.find_opt toml (Otoml.get_array ~strict:false Otoml.get_value) [ "deps" ] with + match + List.find_map + (fun key -> Otoml.find_opt toml (Otoml.get_array ~strict:false Otoml.get_value) [ key ]) + [ "deps"; "dependencies" ] + with | Some items -> List.filter_map depends_of_toml items | None -> [] in - Some { name; dirs; deps; main } + Ok { name; dirs; deps; main } let from_toml toml = + let interactive = Otoml.Helpers.find_string_opt toml [ "interactive" ] in match Otoml.find_opt toml (Otoml.get_array ~strict:false Otoml.get_value) [ "group" ] with - | Some groups -> Some (List.filter_map group_of_toml groups) - | None -> None + | None -> Error "missing required [[group]] section" + | Some groups -> + let results = List.map group_of_toml groups in + let errors = List.filter_map (function Error e -> Some e | Ok _ -> None) results in + if errors <> [] then Error (String.concat "; " errors) + else + let valid = List.filter_map (function Ok g -> Some g | Error _ -> None) results in + Ok { groups = valid; interactive } end diff --git a/src/Fronts/Project/dune b/src/Fronts/Project/dune index 5d3e6e8..28e49c6 100644 --- a/src/Fronts/Project/dune +++ b/src/Fronts/Project/dune @@ -1,4 +1,5 @@ (library + (instrumentation (backend bisect_ppx)) (name project) (public_name stelf.project) (libraries otoml fpath)) diff --git a/src/Fronts/Tui/Repl.ml b/src/Fronts/Tui/Repl.ml index 77f6202..ae9acd2 100644 --- a/src/Fronts/Tui/Repl.ml +++ b/src/Fronts/Tui/Repl.ml @@ -1,3 +1,6 @@ +exception Interrupted +let () = Printexc.register_printer (function Interrupted -> Some "Interrupted" | _ -> None) + module Repl (M : REPL.S) : REPL.REPL = struct let () = Debug.setup_log @@ -66,7 +69,7 @@ module Repl (M : REPL.S) : REPL.REPL = struct let* s = rl#run in Lwt.return (Zed_string.to_utf8 s) - exception Interrupted + exception Interrupted = Interrupted let stop code = exit code @@ -93,10 +96,10 @@ module Repl (M : REPL.S) : REPL.REPL = struct display_msg Note m in let display_response (m : Display.Info.t) : unit Lwt.t = - let open Display in - let arrow = Form.(styles Style.[ Fore.orange; bold ] (string "=>")) in - let out = Form.(markup @@ concat [ arrow; space (); m.msg ]) in - Notty_lwt.output_image (Notty_lwt.eol out) + let arrow = Notty.(I.string A.(fg (rgb_888 ~r:255 ~g:165 ~b:0) ++ st bold) "=>") in + let msg_str = Format.asprintf "%t" m.msg in + let msg_img = Notty.I.string Notty.A.empty (" " ^ msg_str) in + Notty_lwt.output_image (Notty_lwt.eol Notty.I.(arrow <|> msg_img)) in let should_display (msg : Display.Info.t) : bool = @@ -123,9 +126,17 @@ module Repl (M : REPL.S) : REPL.REPL = struct Lwt_list.iter_s (fun info -> display info) pending_list end in + let rec read_multiline (acc : string) : string Lwt.t = + let prompt = if String.equal acc "" then "λΠ> " else " > " in + let* line = read_line !term' prompt in + let full = if String.equal acc "" then line else acc ^ "\n" ^ line in + if ends_with_terminator full || String.equal (String.trim full) "" + then Lwt.return full + else read_multiline full + in Lwt.catch (fun () -> - let* r0 = read_line !term' "λΠ> " in + let* r0 = read_multiline "" in let* continue = try f r0 with | Sys.Break -> Lwt.return Stop diff --git a/src/Fronts/Tui/dune b/src/Fronts/Tui/dune index f4aad3e..79593e8 100644 --- a/src/Fronts/Tui/dune +++ b/src/Fronts/Tui/dune @@ -1,4 +1,5 @@ (library + (instrumentation (backend bisect_ppx)) (name tui) (public_name stelf.tui_fronts) (libraries frontend server fmt grace grace.ansi_renderer lambda-term lwt parsing display notty-community.lwt)) diff --git a/src/Global/dune b/src/Global/dune index e857dcf..4ca44a0 100644 --- a/src/Global/dune +++ b/src/Global/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name global) (public_name stelf.global) (modules Global_ GLOBAL) diff --git a/src/Heuristic/dune b/src/Heuristic/dune index 1b33521..d2d36b9 100644 --- a/src/Heuristic/dune +++ b/src/Heuristic/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name heuristic) (public_name stelf.heuristic) (modules Heuristic_ HeuristicSum HEURISTIC) diff --git a/src/Index/dune b/src/Index/dune index 5c42351..968cfc2 100644 --- a/src/Index/dune +++ b/src/Index/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name index) (public_name stelf.index) (modules IndexSkolem Index_ INDEX) diff --git a/src/IntInf/dune b/src/IntInf/dune index 2f6c155..9640f9b 100644 --- a/src/IntInf/dune +++ b/src/IntInf/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name intInf) (public_name stelf.intInf) (modules IntInfSig IntInf_ INTINFSIG) diff --git a/src/IntSyn/Abstract.ml b/src/IntSyn/Abstract.ml index dacf2e4..f18ae02 100644 --- a/src/IntSyn/Abstract.ml +++ b/src/IntSyn/Abstract.ml @@ -19,9 +19,12 @@ open Tomega (* Abstraction *) (* Author: Frank Pfenning, Carsten Schuermann *) (* Modified: Roberto Virga *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MakeAbstract (Whnf : WHNF) (Unify : UNIFY) (Constraints : CONSTRAINTS) : ABSTRACT = struct - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/IntSyn/Approx.ml b/src/IntSyn/Approx.ml index 78c2737..2d9d473 100644 --- a/src/IntSyn/Approx.ml +++ b/src/IntSyn/Approx.ml @@ -12,6 +12,11 @@ open! Basis (* Approximate language for term reconstruction *) (* Author: Kevin Watkins *) +exception Ambiguous +let () = Printexc.register_printer (function Ambiguous -> Some "Ambiguous term" | _ -> None) +exception Unify of string +let () = Printexc.register_printer (function Unify msg -> Some msg | _ -> None) + module MakeApprox (Whnf : WHNF) : APPROX = struct (*! structure IntSyn = IntSyn' !*) module I = IntSyn @@ -122,7 +127,7 @@ module MakeApprox (Whnf : WHNF) : APPROX = struct let rec varInsert ((u_, v_, l_), name) = varList := ((u_, v_, l_), name) :: !varList - exception Ambiguous + exception Ambiguous = Ambiguous (* getReplacementName (u, v, l, allowed) = name if u : v : l @@ -292,7 +297,7 @@ module MakeApprox (Whnf : WHNF) : APPROX = struct apxToExactW (g_, u_, Whnf.whnfExpandDef vs_, allowed) (* matching for the approximate language *) - exception Unify of string + exception Unify = Unify (* occurUni (r, l) = () iff r does not occur in l, diff --git a/src/IntSyn/Constraints.ml b/src/IntSyn/Constraints.ml index 0e2dae2..ac86e40 100644 --- a/src/IntSyn/Constraints.ml +++ b/src/IntSyn/Constraints.ml @@ -19,10 +19,13 @@ open Intsyn_ (* Manipulating Constraints *) (* Author: Jeff Polakow, Frank Pfenning *) (* Modified: Roberto Virga *) +exception Error of IntSyn.cnstr list +let () = Printexc.register_printer (function Error _ -> Some "Constraint error" | _ -> None) + module MakeConstraints (Conv : CONV) : CONSTRAINTS = struct (*! structure IntSyn = IntSyn' !*) - exception Error of IntSyn.cnstr list + exception Error = Error (* Constraints cnstr are of the form (X[s] = U). diff --git a/src/IntSyn/Match.ml b/src/IntSyn/Match.ml index 946bfff..cb60234 100644 --- a/src/IntSyn/Match.ml +++ b/src/IntSyn/Match.ml @@ -16,11 +16,16 @@ open! Basis (* Unification modified to Matching *) (* Author: Frank Pfenning, Carsten Schuermann *) (* Modified: Roberto Virga, Brigitte Pientka *) +exception Match of string +let () = Printexc.register_printer (function Match msg -> Some msg | _ -> None) +exception NotInvertible +let () = Printexc.register_printer (function NotInvertible -> Some "Not invertible" | _ -> None) + module MakeMatch (Whnf : WHNF) (Unify : UNIFY) (Trail : TRAIL) : MATCH = struct (*! structure IntSyn = IntSyn' !*) - exception Match of string - exception NotInvertible + exception Match = Match + exception NotInvertible = NotInvertible open! struct open IntSyn diff --git a/src/IntSyn/Order.ml b/src/IntSyn/Order.ml index cd38ebf..c20a0c5 100644 --- a/src/IntSyn/Order.ml +++ b/src/IntSyn/Order.ml @@ -193,6 +193,7 @@ module Order = MakeOrder (struct end) include Order +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) (* -bp *) (* structure RedOrder = diff --git a/src/IntSyn/Tomega.ml b/src/IntSyn/Tomega.ml index 4f109fd..7b71a79 100644 --- a/src/IntSyn/Tomega.ml +++ b/src/IntSyn/Tomega.ml @@ -891,3 +891,4 @@ end) module Tomega : TOMEGA = MakeTomega (Whnf__) (Conv__) include Tomega +let () = Printexc.register_printer (function NoMatch -> Some "No match" | _ -> None) diff --git a/src/IntSyn/Unify.ml b/src/IntSyn/Unify.ml index 46aa4c0..1648b6e 100644 --- a/src/IntSyn/Unify.ml +++ b/src/IntSyn/Unify.ml @@ -14,11 +14,16 @@ open! Basis (* Unification *) (* Author: Frank Pfenning, Carsten Schuermann *) (* Modified: Roberto Virga *) +exception Unify of string +let () = Printexc.register_printer (function Unify msg -> Some msg | _ -> None) +exception NotInvertible +let () = Printexc.register_printer (function NotInvertible -> Some "Not invertible" | _ -> None) + module MakeUnify (Whnf : WHNF) (Trail : TRAIL) : UNIFY = struct (*! structure IntSyn = IntSyn' !*) - exception Unify of string - exception NotInvertible + exception Unify = Unify + exception NotInvertible = NotInvertible type fAction = | BindExp of IntSyn.exp option ref * IntSyn.exp option diff --git a/src/IntSyn/dune b/src/IntSyn/dune index 0e29823..3151aee 100644 --- a/src/IntSyn/dune +++ b/src/IntSyn/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name intsyn) (public_name stelf.intsyn) (modules diff --git a/src/Lang/Lexing/dune b/src/Lang/Lexing/dune index 6228b52..02d9753 100644 --- a/src/Lang/Lexing/dune +++ b/src/Lang/Lexing/dune @@ -1,4 +1,5 @@ (library + (instrumentation (backend bisect_ppx)) (name lexing) (public_name stelf.lexing) (libraries display tag containers fpath) diff --git a/src/Lang/Parsing/dune b/src/Lang/Parsing/dune index 41d49d1..6f91df1 100644 --- a/src/Lang/Parsing/dune +++ b/src/Lang/Parsing/dune @@ -1,4 +1,5 @@ (library + (instrumentation (backend bisect_ppx)) (name parsing) (public_name stelf.parsing) (libraries display lexing tag angstrom)) diff --git a/src/Lang/Tag/dune b/src/Lang/Tag/dune index 4312217..ba9f8a5 100644 --- a/src/Lang/Tag/dune +++ b/src/Lang/Tag/dune @@ -1,4 +1,5 @@ (library + (instrumentation (backend bisect_ppx)) (name tag) (public_name stelf.tag) (libraries display basis fpath)) diff --git a/src/M2/Filling.ml b/src/M2/Filling.ml index 450d9f5..3f8e1cb 100644 --- a/src/M2/Filling.ml +++ b/src/M2/Filling.ml @@ -14,6 +14,12 @@ open Metasyn (* Filling *) (* Author: Carsten Schuermann *) + +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) +exception TimeOut +let () = Printexc.register_printer (function TimeOut -> Some "TimeOut" | _ -> None) + module Filling (Filling__0 : sig module MetaSyn' : Metasyn.METASYN module MetaAbstract : METAABSTRACT.METAABSTRACT with module MetaSyn = MetaSyn' @@ -26,8 +32,8 @@ end) : FILLING with module MetaSyn = Filling__0.MetaSyn' = struct open Filling__0 module MetaSyn = MetaSyn' - exception Error of string - exception TimeOut + exception Error = Error + exception TimeOut = TimeOut type nonrec operator = (MetaSyn.state * int) * (unit -> MetaSyn.state list) diff --git a/src/M2/Init.ml b/src/M2/Init.ml index 153fa6a..ea19c35 100644 --- a/src/M2/Init.ml +++ b/src/M2/Init.ml @@ -14,6 +14,10 @@ open MetaAbstract (* Initialization *) (* Author: Carsten Schuermann *) + +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Init (Init__0 : sig module MetaSyn' : Metasyn.METASYN module MetaAbstract : METAABSTRACT.METAABSTRACT with module MetaSyn = MetaSyn' @@ -21,7 +25,7 @@ end) : INIT with module MetaSyn = Init__0.MetaSyn' = struct open Init__0 module MetaSyn = MetaAbstract.MetaSyn - exception Error of string + exception Error = Error open! struct module M = MetaSyn diff --git a/src/M2/Lemma.ml b/src/M2/Lemma.ml index a4ea6df..66b8966 100644 --- a/src/M2/Lemma.ml +++ b/src/M2/Lemma.ml @@ -14,6 +14,10 @@ open MetaAbstract (* Lemma *) (* Author: Carsten Schuermann *) + +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Lemma (Lemma__0 : sig module MetaSyn' : Metasyn.METASYN module MetaAbstract : METAABSTRACT.METAABSTRACT with module MetaSyn = MetaSyn' @@ -21,7 +25,7 @@ end) : LEMMA with module MetaSyn = Lemma__0.MetaSyn' = struct open Lemma__0 module MetaSyn = MetaAbstract.MetaSyn - exception Error of string + exception Error = Error open! struct module A = MetaAbstract diff --git a/src/M2/MetaAbstract.ml b/src/M2/MetaAbstract.ml index daf3762..1649ca2 100644 --- a/src/M2/MetaAbstract.ml +++ b/src/M2/MetaAbstract.ml @@ -15,6 +15,10 @@ open Modetable (* Meta Abstraction *) (* Author: Carsten Schuermann *) + +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MetaAbstract (MetaAbstract__0 : sig module Global : GLOBAL module MetaSyn : Metasyn.METASYN @@ -48,7 +52,7 @@ end) : METAABSTRACT with module MetaSyn = MetaAbstract__0.MetaSyn = struct open MetaAbstract__0 module MetaSyn = MetaAbstract__0.MetaSyn - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/M2/Metasyn.ml b/src/M2/Metasyn.ml index 4742a94..f551df9 100644 --- a/src/M2/Metasyn.ml +++ b/src/M2/Metasyn.ml @@ -11,9 +11,13 @@ open! Basis (* Meta syntax *) (* Author: Carsten Schuermann *) + +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Make_MetaSyn (Whnf : WHNF) : METASYN = struct (*! structure IntSyn = IntSyn' !*) - exception Error of string + exception Error = Error type nonrec var = int type mode = Bot | Top [@@deriving eq, ord, show] diff --git a/src/M2/Mpi.ml b/src/M2/Mpi.ml index 7cc1acc..09dd6c3 100644 --- a/src/M2/Mpi.ml +++ b/src/M2/Mpi.ml @@ -24,6 +24,10 @@ open Ring (* Meta Prover Interface *) (* Author: Carsten Schuermann *) + +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Mpi (Mpi__0 : sig module MetaGlobal : METAGLOBAL.METAGLOBAL module MetaSyn' : Metasyn.METASYN @@ -44,7 +48,7 @@ end) : MPI with module MetaSyn = Mpi__0.MetaSyn' = struct open Mpi__0 module MetaSyn = MetaSyn' - exception Error of string + exception Error = Error open! struct module M = MetaSyn diff --git a/src/M2/Prover.ml b/src/M2/Prover.ml index a1a3fe8..05a1172 100644 --- a/src/M2/Prover.ml +++ b/src/M2/Prover.ml @@ -21,6 +21,10 @@ open Timers (* Meta Prover *) (* Author: Carsten Schuermann *) + +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Prover (Prover__0 : sig module MetaGlobal : METAGLOBAL.METAGLOBAL module MetaSyn' : Metasyn.METASYN @@ -39,7 +43,7 @@ end) : PROVER = struct open Prover__0 (*! structure IntSyn = MetaSyn'.IntSyn !*) - exception Error of string + exception Error = Error open! struct module MetaSyn = MetaSyn' diff --git a/src/M2/Qed.ml b/src/M2/Qed.ml index 0c0a0e2..ea17603 100644 --- a/src/M2/Qed.ml +++ b/src/M2/Qed.ml @@ -13,6 +13,10 @@ open Metasyn (* QED *) (* Author: Carsten Schuermann *) + +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Qed (Qed__0 : sig module Global : GLOBAL module MetaSyn' : Metasyn.METASYN @@ -20,7 +24,7 @@ end) : QED with module MetaSyn = Qed__0.MetaSyn' = struct open Qed__0 module MetaSyn = MetaSyn' - exception Error of string + exception Error = Error open! struct module M = MetaSyn diff --git a/src/M2/Recursion.ml b/src/M2/Recursion.ml index af04b61..1697ca3 100644 --- a/src/M2/Recursion.ml +++ b/src/M2/Recursion.ml @@ -20,6 +20,10 @@ open MetaAbstract (* Recursion *) (* Author: Carsten Schuermann *) (* See [Rohwedder,Pfenning ESOP'96] *) + +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Recursion (Recursion__0 : sig module Global : GLOBAL module MetaGlobal : METAGLOBAL.METAGLOBAL @@ -57,7 +61,7 @@ end) : RECURSION.RECURSION with module MetaSyn = Recursion__0.MetaSyn' = struct open Recursion__0 module MetaSyn = MetaSyn' - exception Error of string + exception Error = Error type nonrec operator = MetaSyn.state diff --git a/src/M2/Search.ml b/src/M2/Search.ml index 8e9463b..210da03 100644 --- a/src/M2/Search.ml +++ b/src/M2/Search.ml @@ -14,6 +14,10 @@ open MetaGlobal (* Search (based on abstract machine ) *) (* Author: Carsten Schuermann *) + +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module OLDSearch (OLDSearch__0 : sig (*! structure IntSyn' : INTSYN !*) module MetaGlobal : METAGLOBAL.METAGLOBAL @@ -53,7 +57,7 @@ end) : OLDSEARCH with module MetaSyn = OLDSearch__0.MetaSyn' = struct module MetaSyn = MetaSyn' (*! structure CompSyn = CompSyn' !*) - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/M2/Skolem.ml b/src/M2/Skolem.ml index 597df4e..d93dff1 100644 --- a/src/M2/Skolem.ml +++ b/src/M2/Skolem.ml @@ -12,6 +12,10 @@ open Metasyn open Modetable open Timers + +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Skolem (Skolem__0 : sig (* Skolem constant administration *) (* Author: Carsten Schuermann *) @@ -42,7 +46,7 @@ end) : SKOLEM = struct open Skolem__0 (*! structure IntSyn = IntSyn' !*) - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/M2/Splitting.ml b/src/M2/Splitting.ml index 22ba42b..e88ec3c 100644 --- a/src/M2/Splitting.ml +++ b/src/M2/Splitting.ml @@ -16,6 +16,10 @@ open Modetable (* Splitting *) (* Author: Carsten Schuermann *) + +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Splitting (Splitting__0 : sig module Global : GLOBAL module MetaSyn' : Metasyn.METASYN @@ -38,7 +42,7 @@ end) : SPLITTING.SPLITTING with module MetaSyn = Splitting__0.MetaSyn' = struct open Splitting__0 module MetaSyn = MetaAbstract.MetaSyn - exception Error of string + exception Error = Error (* Invariant: Case analysis generates a list of successor states diff --git a/src/M2/dune b/src/M2/dune index e6bcf32..60a670f 100644 --- a/src/M2/dune +++ b/src/M2/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name m2) (public_name stelf.m2) (modules diff --git a/src/Meta/Funnames.ml b/src/Meta/Funnames.ml index a968b09..3511ccb 100644 --- a/src/Meta/Funnames.ml +++ b/src/Meta/Funnames.ml @@ -15,6 +15,9 @@ open! Basis (* Names of Constants and Variables *) (* Author: Carsten Schuermann *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module FunNames (FunNames__0 : sig module Global : GLOBAL @@ -24,7 +27,7 @@ end) : FUNNAMES.FUNNAMES = struct open FunNames__0 (*! structure FunSyn = FunSyn' !*) - exception Error of string + exception Error = Error (****************************************) (* Constants Names and Name Preferences *) diff --git a/src/Meta/Funsyn.ml b/src/Meta/Funsyn.ml index 281a02c..ffd672a 100644 --- a/src/Meta/Funsyn.ml +++ b/src/Meta/Funsyn.ml @@ -12,9 +12,12 @@ open! Basis (* Internal syntax for functional proof term calculus *) (* Author: Carsten Schuermann *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Make_FunSyn (Whnf : WHNF) (Conv : CONV) : FUNSYN = struct (*! structure IntSyn = IntSyn' !*) - exception Error of string + exception Error = Error type nonrec label = int type nonrec name = string diff --git a/src/Meta/Funtypecheck.ml b/src/Meta/Funtypecheck.ml index d46f88f..54ab235 100644 --- a/src/Meta/Funtypecheck.ml +++ b/src/Meta/Funtypecheck.ml @@ -15,6 +15,9 @@ open! Print open! Abstract open! Basis +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module FunTypeCheck (FunTypeCheck__0 : sig (* Type checking for functional proof term calculus *) (* Author: Carsten Schuermann *) @@ -49,7 +52,7 @@ end) : FUNTYPECHECK.FUNTYPECHECK = struct open FunTypeCheck__0 module StateSyn = StateSyn' - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/Meta/Inference.ml b/src/Meta/Inference.ml index 70e81b6..52e2052 100644 --- a/src/Meta/Inference.ml +++ b/src/Meta/Inference.ml @@ -18,6 +18,9 @@ open! Basis (* Inference: Version 1.3*) (* Author: Carsten Schuermann *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Inference (Inference__0 : sig module MTPGlobal : MtpGlobal.MTPGLOBAL @@ -49,7 +52,7 @@ end) : INFERENCE.INFERENCE = struct open Inference__0 module StateSyn = StateSyn' - exception Error of string + exception Error = Error type nonrec operator = unit -> StateSyn.state diff --git a/src/Meta/MtpAbstract.ml b/src/Meta/MtpAbstract.ml index 039ad2f..4d08108 100644 --- a/src/Meta/MtpAbstract.ml +++ b/src/Meta/MtpAbstract.ml @@ -15,6 +15,9 @@ open! Basis (* Meta Theorem Prover abstraction : Version 1.3 *) (* Author: Frank Pfenning, Carsten Schuermann *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MTPAbstract (MTPAbstract__0 : sig (*! structure IntSyn' : INTSYN !*) (*! structure FunSyn' : FUNSYN !*) @@ -47,7 +50,7 @@ end) : MTPABSTRACT.MTPABSTRACT = struct open MTPAbstract__0 module StateSyn = StateSyn' - exception Error of string + exception Error = Error type approxFor = | Head of IntSyn.dctx * (FunSyn.for_ * IntSyn.sub) * int diff --git a/src/Meta/MtpFilling.ml b/src/Meta/MtpFilling.ml index bb39898..f61124d 100644 --- a/src/Meta/MtpFilling.ml +++ b/src/Meta/MtpFilling.ml @@ -18,6 +18,11 @@ open! Basis (* Filling Version 1.3*) (* Author: Carsten Schuermann *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) +exception TimeOut +let () = Printexc.register_printer (function TimeOut -> Some "TimeOut" | _ -> None) + module MTPFilling (MTPFilling__0 : sig module MTPGlobal : MtpGlobal.MTPGLOBAL @@ -41,8 +46,8 @@ end) : MTPFILLING.MTPFILLING = struct open MTPFilling__0 module StateSyn = StateSyn' - exception Error of string - exception TimeOut + exception Error = Error + exception TimeOut = TimeOut type nonrec operator = unit -> int * FunSyn.pro diff --git a/src/Meta/MtpInit.ml b/src/Meta/MtpInit.ml index b7cca8a..49426ed 100644 --- a/src/Meta/MtpInit.ml +++ b/src/Meta/MtpInit.ml @@ -16,6 +16,9 @@ open! Basis (* Initialization *) (* Author: Carsten Schuermann *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MTPInit (MTPInit__0 : sig module MTPGlobal : MtpGlobal.MTPGLOBAL module MTPData : MTPDATA.MTPDATA @@ -42,7 +45,7 @@ end) : MTPINIT.MTPINIT = struct open MTPInit__0 module StateSyn = StateSyn' - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/Meta/MtpMpi.ml b/src/Meta/MtpMpi.ml index 51a7c0e..d36a276 100644 --- a/src/Meta/MtpMpi.ml +++ b/src/Meta/MtpMpi.ml @@ -28,6 +28,9 @@ open! Basis (* Meta Prover Interface *) (* Author: Carsten Schuermann *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MTPi (MTPi__0 : sig module MTPGlobal : MtpGlobal.MTPGLOBAL @@ -73,7 +76,7 @@ module MTPi (MTPi__0 : sig end) : MTPI = struct open MTPi__0 - exception Error of string + exception Error = Error (*! structure FunSyn = FunSyn' !*) module StateSyn = StateSyn' diff --git a/src/Meta/MtpPrint.ml b/src/Meta/MtpPrint.ml index 3bb177a..96302cf 100644 --- a/src/Meta/MtpPrint.ml +++ b/src/Meta/MtpPrint.ml @@ -13,6 +13,9 @@ include MTPPRINT open! Global open! Basis +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MTPrint (MTPrint__0 : sig (* Meta Printer Version 1.3 *) (* Author: Carsten Schuermann *) @@ -38,7 +41,7 @@ end) : MTPPRINT.MTPRINT = struct module Formatter = Formatter' module StateSyn = StateSyn' - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/Meta/MtpProver.ml b/src/Meta/MtpProver.ml index 887d6f7..25bbc54 100644 --- a/src/Meta/MtpProver.ml +++ b/src/Meta/MtpProver.ml @@ -17,6 +17,9 @@ open! Basis (* Meta Theorem Prover Version 1.3 *) (* Author: Carsten Schuermann *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MTProver (MTProver__0 : sig module MTPGlobal : MtpGlobal.MTPGLOBAL @@ -40,7 +43,7 @@ end) : MTPPROVER.MTPROVER = struct module StateSyn = StateSyn (*! structure IntSyn = IntSyn' !*) - exception Error of string + exception Error = Error open! struct module I = IntSyn @@ -189,7 +192,7 @@ end) : MTPPROVER.MTPROVER = struct open CombiProver__1 (*! structure IntSyn = IntSyn' !*) - exception Error of string + exception Error = Error let rec he f = try f () with diff --git a/src/Meta/MtpRecursion.ml b/src/Meta/MtpRecursion.ml index a0106f4..3524c56 100644 --- a/src/Meta/MtpRecursion.ml +++ b/src/Meta/MtpRecursion.ml @@ -21,6 +21,9 @@ open! Basis (* Meta Recursion Version 1.3 *) (* Author: Carsten Schuermann *) (* See [Rohwedder,Pfenning ESOP'96] *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MTPRecursion (MTPRecursion__0 : sig module MTPGlobal : MtpGlobal.MTPGLOBAL module Global : GLOBAL @@ -70,7 +73,7 @@ end) : MTPRECURSION = struct open MTPRecursion__0 module StateSyn = StateSyn' - exception Error of string + exception Error = Error type nonrec operator = StateSyn.state diff --git a/src/Meta/MtpSearch.ml b/src/Meta/MtpSearch.ml index b586707..e29a277 100644 --- a/src/Meta/MtpSearch.ml +++ b/src/Meta/MtpSearch.ml @@ -16,6 +16,9 @@ open! Basis (* Search (based on abstract machine ) : Version 1.3 *) (* Author: Carsten Schuermann *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MTPSearch (MTPSearch__0 : sig module Global : GLOBAL @@ -59,7 +62,7 @@ end) : MTPSEARCH.MTPSEARCH = struct module StateSyn = StateSyn' (*! structure CompSyn = CompSyn' !*) - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/Meta/MtpSplitting.ml b/src/Meta/MtpSplitting.ml index 47d59f9..875913f 100644 --- a/src/Meta/MtpSplitting.ml +++ b/src/Meta/MtpSplitting.ml @@ -19,6 +19,9 @@ open! Basis (* Splitting : Version 1.3 *) (* Author: Carsten Schuermann *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MTPSplitting (MTPSplitting__0 : sig module MTPGlobal : MtpGlobal.MTPGLOBAL module Global : GLOBAL @@ -66,7 +69,7 @@ end) : MTPSPLITTING.MTPSPLITTING = struct open MTPSplitting__0 module StateSyn = StateSyn' - exception Error of string + exception Error = Error (* Invariant: Case analysis generates a list of successor states diff --git a/src/Meta/Relfun.ml b/src/Meta/Relfun.ml index 55e1ea5..6e4a34e 100644 --- a/src/Meta/Relfun.ml +++ b/src/Meta/Relfun.ml @@ -16,6 +16,9 @@ open! Weaken open! Global open! Basis +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module RelFun (RelFun__0 : sig (* Converter from relational representation to a functional representation of proof terms *) @@ -47,7 +50,7 @@ module RelFun (RelFun__0 : sig module FunNames : FUNNAMES.FUNNAMES end) : RELFUN.RELFUN = struct (*! structure FunSyn = FunSyn' !*) - exception Error of string + exception Error = Error open RelFun__0 diff --git a/src/Meta/Uniquesearch.ml b/src/Meta/Uniquesearch.ml index 36b0987..ca1b039 100644 --- a/src/Meta/Uniquesearch.ml +++ b/src/Meta/Uniquesearch.ml @@ -16,6 +16,9 @@ open! Basis (* Search (based on abstract machine ) : Version 1.3 *) (* Author: Carsten Schuermann *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module UniqueSearch (UniqueSearch__0 : sig module Global : GLOBAL @@ -64,7 +67,7 @@ end) : UNIQUESEARCH.UNIQUESEARCH = struct module StateSyn = StateSyn' (*! structure CompSyn = CompSyn' !*) - exception Error of string + exception Error = Error type nonrec acctype = IntSyn.exp diff --git a/src/Meta/dune b/src/Meta/dune index 0e21299..baf820e 100644 --- a/src/Meta/dune +++ b/src/Meta/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name meta) (public_name stelf.meta) (modules diff --git a/src/Modes/Modecheck.ml b/src/Modes/Modecheck.ml index d59fabc..1600370 100644 --- a/src/Modes/Modecheck.ml +++ b/src/Modes/Modecheck.ml @@ -19,6 +19,9 @@ open Origins (** Mode Checking *) (** @author Carsten Schuermann *) (** Modified: Frank Pfenning, Roberto Virga *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MakeModeCheck (ModeTable : MODETABLE) (Whnf : WHNF) @@ -28,7 +31,7 @@ module MakeModeCheck (*! structure ModeSyn = ModeSyn !*) (*! structure Paths = Paths !*) - exception Error of string + exception Error = Error open! struct let print' s = Display.(debug Form.(string s)) @@ -836,7 +839,7 @@ module MakeModeCheck let rec checkAll = function | [] -> () | I.Const c :: clist -> begin - Display.(debug Form.(string "checking mode of constant" ++ space () ++ shown (fun x -> Names.show_qid @@ Names.constQid c) c ++ space () ++ string "..." ++ nl ())); + Display.(debug Form.(string "checking mode of constant" ++ space () ++ shown (fun x -> Names.qidToString @@ Names.constQid c) c ++ space () ++ string "..." ++ nl ())); (try checkDlocal (I.Null, I.constType c, P.top) with Error' (occ, msg) -> raise (Error (wrapMsg (c, occ, msg)))); diff --git a/src/Modes/Modedec.ml b/src/Modes/Modedec.ml index ef9c864..9d8a029 100644 --- a/src/Modes/Modedec.ml +++ b/src/Modes/Modedec.ml @@ -14,10 +14,13 @@ open! Basis (* Modes: short and full mode declarations *) (* Author: Carsten Schuermann *) (* Modified: Frank Pfenning *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MakeModeDec () : MODEDEC = struct (*! structure ModeSyn = ModeSyn' !*) (*! structure Paths = Paths' !*) - exception Error of string + exception Error = Error open! struct module M = ModeSyn @@ -26,7 +29,7 @@ module MakeModeDec () : MODEDEC = struct type arg = Implicit | Explicit | Local [@@deriving eq, ord, show] - let rec error (r, msg) = raise (Error (P.wrap (r, msg))) + let rec error (r, msg) = raise (Error (P.toString r ^ ": " ^ msg)) let rec checkName = function | M.Mnil -> () diff --git a/src/Modes/Modesyn.ml b/src/Modes/Modesyn.ml index da03c8f..9a868d6 100644 --- a/src/Modes/Modesyn.ml +++ b/src/Modes/Modesyn.ml @@ -11,8 +11,11 @@ open! Basis include MODESYN (* signature MODESYN *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module ModeSyn : MODESYN = struct - exception Error of string + exception Error = Error type mode = Plus | Star | Minus | Minus1 [@@deriving eq, ord, show] diff --git a/src/Modes/Modetable.ml b/src/Modes/Modetable.ml index 06618f2..32f5191 100644 --- a/src/Modes/Modetable.ml +++ b/src/Modes/Modetable.ml @@ -13,10 +13,13 @@ open! Basis (* Mode Table *) (* Author: Carsten Schuermann *) (* Modified: Frank Pfenning, Roberto Virga *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MakeModeTable (Table : TABLE with type key = int) : MODETABLE = struct (*! structure IntSyn = IntSyn' !*) - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/Modes/dune b/src/Modes/dune index ccc18c1..6de0967 100644 --- a/src/Modes/dune +++ b/src/Modes/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name modes) (public_name stelf.modes) (modules diff --git a/src/Modules/Modsyn.ml b/src/Modules/Modsyn.ml index a1a6807..8636001 100644 --- a/src/Modules/Modsyn.ml +++ b/src/Modules/Modsyn.ml @@ -9,6 +9,9 @@ include MODSYN open! Basis open Origins +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module ModSyn (ModSyn__0 : sig (* Syntax for elaborated modules *) (* Author: Kevin Watkins *) @@ -39,7 +42,7 @@ end) : MODSYN = struct (*! structure Paths = Paths' !*) module I = IntSyn - exception Error of string + exception Error = Error type constInfo = | ConstInfo of diff --git a/src/Modules/dune b/src/Modules/dune index eb68fb3..a700d55 100644 --- a/src/Modules/dune +++ b/src/Modules/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name modules) (public_name stelf.modules) (modules Modsyn Modules_ MODSYN) diff --git a/src/Msg/dune b/src/Msg/dune index 5edc9a8..100c607 100644 --- a/src/Msg/dune +++ b/src/Msg/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name msg) (public_name stelf.msg) (modules Msg_ MSG) diff --git a/src/Names/NAMES.ml b/src/Names/NAMES.ml index 2ff2d33..073eb41 100644 --- a/src/Names/NAMES.ml +++ b/src/Names/NAMES.ml @@ -100,6 +100,14 @@ module type NAMES = sig (* Nonfix if undefined *) + val installAlias : string * IntSyn.cid -> unit + (** [installAlias (name, cid)] registers [name] as an additional lookup name + for constant [cid] in the global namespace. *) + + val insertConstAlias : namespace * string * IntSyn.cid -> unit + (** [insertConstAlias (ns, name, cid)] registers [name] as an additional + lookup name for constant [cid] in the given namespace [ns]. *) + val installNamePref : IntSyn.cid * (string list * string list) -> unit (** Name preferences for anonymous variables: a, EPref, UPref *) diff --git a/src/Names/Names_.ml b/src/Names/Names_.ml index 85a31fc..1ccd964 100644 --- a/src/Names/Names_.ml +++ b/src/Names/Names_.ml @@ -303,6 +303,12 @@ module MakeNames | Some (_, cid') -> Array.update (shadowArray, cid, Some cid') end + let rec installAlias (name, cid) = + ignore (topInsert (name, cid)) + + let rec insertConstAlias ((structTable, constTable), name, cid) = + ignore (StringTree.insertShadow constTable (name, cid)) + let rec uninstallConst cid = let condec_ = IntSyn.sgnLookup cid in let id = IntSyn.conDecName condec_ in @@ -1079,3 +1085,5 @@ module Names = (TableInstances.StringRedBlackTree) include Names +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) +let () = Printexc.register_printer (function Unprintable -> Some "Unprintable" | _ -> None) diff --git a/src/Names/dune b/src/Names/dune index 0589a96..ff9da41 100644 --- a/src/Names/dune +++ b/src/Names/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name names) (public_name stelf.names) (modules Names_ NAMES) diff --git a/src/Netserver/Netserver_.ml b/src/Netserver/Netserver_.ml index 6ec6407..3872ee5 100644 --- a/src/Netserver/Netserver_.ml +++ b/src/Netserver/Netserver_.ml @@ -8,6 +8,13 @@ include NETSERVER (* filesystem directory where stelf examples are kept *) (* signature SERVER *) +exception Eof +let () = Printexc.register_printer (function Eof -> Some "End of file" | _ -> None) +exception Quit +let () = Printexc.register_printer (function Quit -> Some "Quit" | _ -> None) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module NetServer : NETSERVER = struct let rec join arg__1 arg__2 = begin match (arg__1, arg__2) with @@ -62,8 +69,8 @@ module NetServer : NETSERVER = struct let _ = TextIO.closeIn s in data - exception Eof - exception Quit + exception Eof = Eof + exception Quit = Quit let rec send _conn _str = () @@ -80,7 +87,7 @@ module NetServer : NETSERVER = struct let rec setExamplesDir s = examplesDir := Some s (* exception Error for server errors *) - exception Error of string + exception Error = Error let rec error msg = raise (Error msg) diff --git a/src/Netserver/dune b/src/Netserver/dune index 66646e2..805e155 100644 --- a/src/Netserver/dune +++ b/src/Netserver/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name netserver) (public_name stelf.netserver) (modules Netserver_ NETSERVER) diff --git a/src/Opsem/AbstractTabled.ml b/src/Opsem/AbstractTabled.ml index f8c74fd..c6fb394 100644 --- a/src/Opsem/AbstractTabled.ml +++ b/src/Opsem/AbstractTabled.ml @@ -13,6 +13,9 @@ open! Basis (* Abstraction *) (* Author: Frank Pfenning, Carsten Schuermann *) (* Modified: Roberto Virga, Brigitte Pientka *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module AbstractTabled (AbstractTabled__0 : sig (*! structure IntSyn' : INTSYN !*) module Whnf : WHNF @@ -36,7 +39,7 @@ end) : ABSTRACTTABLED = struct (*! structure IntSyn = IntSyn' !*) (*! structure TableParam = TableParam !*) - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/Opsem/MemoTable.ml b/src/Opsem/MemoTable.ml index c09ca70..bf34e1e 100644 --- a/src/Opsem/MemoTable.ml +++ b/src/Opsem/MemoTable.ml @@ -17,6 +17,9 @@ open RedBlackSet (* Any term we insert into the substitution tree is in normalform *) (* Variant Checking *) (* Author: Brigitte Pientka *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MemoTable (MemoTable__0 : sig (*! structure IntSyn' : INTSYN !*) (*! structure CompSyn' : COMPSYN !*) @@ -152,7 +155,7 @@ end) : MEMOTABLE = struct let indexArray = Array.tabulate (Global.maxCid, function i -> (ref 0, makeTree ())) - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/Opsem/Ptrecon.ml b/src/Opsem/Ptrecon.ml index 159ac11..b95ead8 100644 --- a/src/Opsem/Ptrecon.ml +++ b/src/Opsem/Ptrecon.ml @@ -18,6 +18,9 @@ open MemoTable (* Author: Brigitte Pientka *) (* Modified: Jeff Polakow, Frank Pfenning, Larry Greenfield, Roberto Virga, Brigitte Pientka *) (* Proof term reconstruction from proof skeleton *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module PtRecon (PtRecon__0 : sig (*! structure IntSyn' : INTSYN !*) (*! structure CompSyn' : COMPSYN !*) @@ -54,7 +57,7 @@ end) : PTRECON = struct module MT = MemoTable end - exception Error of string + exception Error = Error let rec cidFromHead = function I.Const a -> a | I.Def a -> a diff --git a/src/Opsem/SubtreeInst.ml b/src/Opsem/SubtreeInst.ml index 20feaae..ea03434 100644 --- a/src/Opsem/SubtreeInst.ml +++ b/src/Opsem/SubtreeInst.ml @@ -10,6 +10,9 @@ open MemoTable (* Any term we insert into the substitution tree is in normalform ! *) (* Instance Checking *) (* Author: Brigitte Pientka *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MemoTableInst (MemoTableInst__0 : sig (*! structure IntSyn' : INTSYN !*) (*! structure CompSyn' : COMPSYN !*) @@ -141,7 +144,7 @@ end) : MEMOTABLE = struct let indexArray = Array.tabulate (Global.maxCid, function i -> (ref 0, makeTree ())) - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/Opsem/TableParam.ml b/src/Opsem/TableParam.ml index 92539cb..6b54f08 100644 --- a/src/Opsem/TableParam.ml +++ b/src/Opsem/TableParam.ml @@ -12,11 +12,14 @@ open! Basis (* Table parameters *) (* Author: Brigitte Pientka *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MakeTableParam (Global : GLOBAL) : TABLEPARAM = struct (*! structure IntSyn = IntSyn' !*) (*! structure CompSyn = CompSyn' !*) (*! structure RBSet = RBSet !*) - exception Error of string + exception Error = Error type strategy = Variant | Subsumption diff --git a/src/Opsem/TabledMachine.ml b/src/Opsem/TabledMachine.ml index b4652f5..7312a14 100644 --- a/src/Opsem/TabledMachine.ml +++ b/src/Opsem/TabledMachine.ml @@ -16,6 +16,9 @@ open MemoTable (* Abstract Machine for tabling*) (* Author: Brigitte Pientka *) (* Based on abstract machine in Absmachine.fun *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Tabled (Tabled__0 : sig (*! structure IntSyn' : INTSYN !*) (*! structure CompSyn' : COMPSYN !*) @@ -99,7 +102,7 @@ end) : TABLED = struct ref = ref [] - exception Error of string + exception Error = Error (* ---------------------------------------------------------------------- *) let rec cidFromHead = function I.Const a -> a | I.Def a -> a diff --git a/src/Opsem/dune b/src/Opsem/dune index 4950283..dacd9dd 100644 --- a/src/Opsem/dune +++ b/src/Opsem/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name opsem) (public_name stelf.opsem) (modules diff --git a/src/Order/Order_.ml b/src/Order/Order_.ml index e9063dc..6c9777e 100644 --- a/src/Order/Order_.ml +++ b/src/Order/Order_.ml @@ -180,6 +180,7 @@ open! Basis open TableInstances module Order = MakeOrder (IntRedBlackTree) include Order +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) (* -bp *) (* structure RedOrder = diff --git a/src/Order/dune b/src/Order/dune index e6d8421..a6859f8 100644 --- a/src/Order/dune +++ b/src/Order/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name order) (public_name stelf.order) (modules Order_ ORDER) diff --git a/src/Paths/dune b/src/Paths/dune index 605a1b0..3270a32 100644 --- a/src/Paths/dune +++ b/src/Paths/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name paths) (public_name stelf.paths) (modules PATHS Paths_ ORIGINS Origins) diff --git a/src/Print/Print_.ml b/src/Print/Print_.ml index 8bafb65..ef0f690 100644 --- a/src/Print/Print_.ml +++ b/src/Print/Print_.ml @@ -625,10 +625,11 @@ module MakePrint | I.Uni _ -> [] | I.Pi ((d_, _), v2_) -> let d'_ = Names.decLUName (g_, d_) in + let rest = fmtKindBinders (I.Decl (g_, d'_), d + 1, v2_) in sym "{" :: fmtDec (g_, d, (d'_, I.id)) - :: sym "}" :: F.break - :: fmtKindBinders (I.Decl (g_, d'_), d + 1, v2_) + :: sym "}" + :: (match rest with [] -> [] | _ -> F.break :: rest) | _ -> [ fmtExp (g_, d, noCtxt, (v_, I.id)) ] end diff --git a/src/Print/Traverse.ml b/src/Print/Traverse.ml index c79ec6c..9688f40 100644 --- a/src/Print/Traverse.ml +++ b/src/Print/Traverse.ml @@ -15,6 +15,9 @@ include TRAVERSE (* # 1 "src/print/Traverse.fun.ml" *) open! Basis +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Traverse (Traverse__0 : sig (*! structure IntSyn' : INTSYN !*) module Whnf : WHNF @@ -33,7 +36,7 @@ end) : TRAVERSE = struct module Whnf = Traverse__0.Whnf module Names = Traverse__0.Names - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/Print/dune b/src/Print/dune index 3afcee7..86d9c7d 100644 --- a/src/Print/dune +++ b/src/Print/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name print) (public_name stelf.print) (modules diff --git a/src/Prover/Elim.ml b/src/Prover/Elim.ml index bcd233b..28ccedd 100644 --- a/src/Prover/Elim.ml +++ b/src/Prover/Elim.ml @@ -12,6 +12,9 @@ open! Basis (* Elim *) (* Author: Carsten Schuermann *) (* Date: Thu Mar 16 13:39:26 2006 *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Elim (Elim__0 : sig module Data : Data.DATA @@ -38,7 +41,7 @@ end) : ELIM with module State = Elim__0.State' = struct (*! structure Tomega = Tomega' !*) module State = Elim__0.State' - exception Error of string + exception Error = Error type operator_ = Local of Tomega.prg * int type nonrec operator = operator_ diff --git a/src/Prover/Fill.ml b/src/Prover/Fill.ml index 7af825a..e014eed 100644 --- a/src/Prover/Fill.ml +++ b/src/Prover/Fill.ml @@ -12,6 +12,9 @@ open! Basis (* Filling *) (* Author: Carsten Schuermann *) (* Date: Thu Mar 16 13:08:33 2006 *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Fill (Fill__0 : sig module Data : Data.DATA @@ -42,7 +45,7 @@ end) : FILL with module State = Fill__0.State' = struct (*! structure Tomega = Tomega' !*) module State = Fill__0.State' - exception Error of string + exception Error = Error type operator_ = | FillWithConst of IntSyn.exp * IntSyn.cid diff --git a/src/Prover/Psearch.ml b/src/Prover/Psearch.ml index 8b1ce8b..c17340b 100644 --- a/src/Prover/Psearch.ml +++ b/src/Prover/Psearch.ml @@ -11,6 +11,9 @@ open! Basis (* Search (based on abstract machine ) : Version 1.3 *) (* Author: Carsten Schuermann *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Search (Search__0 : sig module Global : GLOBAL @@ -62,7 +65,7 @@ end) : SEARCH = struct module State = Search__0.State' (*! structure CompSyn = CompSyn' !*) - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/Prover/Split.ml b/src/Prover/Split.ml index 21caf97..08f99f1 100644 --- a/src/Prover/Split.ml +++ b/src/Prover/Split.ml @@ -9,6 +9,9 @@ include SPLIT (* # 1 "src/prover/Split.fun.ml" *) open! Basis +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Split (Split__0 : sig (* State definition for Proof Search *) (* Author: Carsten Schuermann *) @@ -49,7 +52,7 @@ end) : SPLIT with module State = Split__0.State' = struct (*! structure Tomega = Tomega' !*) module State = Split__0.State' - exception Error of string + exception Error = Error type operator_ = Split of Tomega.prg option ref * Tomega.prg * string diff --git a/src/Prover/State.ml b/src/Prover/State.ml index e79f468..cf04822 100644 --- a/src/Prover/State.ml +++ b/src/Prover/State.ml @@ -10,6 +10,9 @@ open! Basis (* State definition for Proof Search *) (* Author: Carsten Schuermann *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module State (State__0 : sig module Formatter : FORMATTER end) : STATE = struct @@ -33,7 +36,7 @@ end) : STATE = struct | And of SideCondition * SideCondition | Order of Order.Predicate *) - exception Error of string + exception Error = Error open! struct module T = Tomega diff --git a/src/Prover/dune b/src/Prover/dune index f0f41bc..e466a71 100644 --- a/src/Prover/dune +++ b/src/Prover/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name prover) (public_name stelf.prover) (modules diff --git a/src/Recon/ReconConDec.ml b/src/Recon/ReconConDec.ml index e2896a7..83fd67e 100644 --- a/src/Recon/ReconConDec.ml +++ b/src/Recon/ReconConDec.ml @@ -1,5 +1,7 @@ module type RECON_CONDEC = RECON_CONDEC.RECON_CONDEC +exception Error of string + module Make_ReconConDec (M : S.S) (RT : RECON_TERM.RECON_TERM with module M = M) : @@ -9,7 +11,7 @@ module Make_ReconConDec module Ast = M.Ast module Paths = M.Paths - exception Error of string + exception Error = Error let error (r, msg) = raise (Error (Paths.wrap (r, msg))) diff --git a/src/Recon/ReconMode.ml b/src/Recon/ReconMode.ml index a1b2869..82c9141 100644 --- a/src/Recon/ReconMode.ml +++ b/src/Recon/ReconMode.ml @@ -1,5 +1,7 @@ module type RECON_MODE = RECON_MODE.RECON_MODE +exception Error of string + module ModeDec = Modes.Modedec.MakeModeDec () let ghost_region = Paths.Paths_.Paths.Reg (0, 0) @@ -12,7 +14,7 @@ module Make_ReconMode (M : S.S) : RECON_MODE with module M = M = struct module Paths = M.Paths module Modes = Modes.Modesyn.ModeSyn - exception Error of string + exception Error = Error let raise' m = Display.( @@ -66,8 +68,5 @@ module Make_ReconMode (M : S.S) : RECON_MODE with module M = M = struct | _ -> raise' "Invalid mode declaration" end with Error m -> - Display.( - message ~kind:Warning ~level:Detailed - (string "Error processing mode declaration: " ++ string m)); raise' m end diff --git a/src/Recon/ReconModule.ml b/src/Recon/ReconModule.ml index 03ce377..645f8f8 100644 --- a/src/Recon/ReconModule.ml +++ b/src/Recon/ReconModule.ml @@ -1,5 +1,7 @@ module type RECON_MODULE = RECON_MODULE.RECON_MODULE +exception Error of string + module Make_ReconModule (M : S.S) (RT : RECON_TERM.RECON_TERM with module M = M) : @@ -11,7 +13,7 @@ module Make_ReconModule module ModSyn = Modules.Modules_.ModSyn module IntTree = TableInstances.IntRedBlackTree - exception Error of string + exception Error = Error let error (r, msg) = raise (Error (Paths.wrap (r, msg))) diff --git a/src/Recon/ReconQuery.ml b/src/Recon/ReconQuery.ml index c011c89..d335c2d 100644 --- a/src/Recon/ReconQuery.ml +++ b/src/Recon/ReconQuery.ml @@ -1,5 +1,7 @@ module type RECON_QUERY = RECON_QUERY.RECON_QUERY +exception Error of string + module Make_ReconQuery (M : S.S) (RT : RECON_TERM.RECON_TERM with module M = M) : @@ -10,7 +12,7 @@ module Make_ReconQuery module Paths = M.Paths module Syntax = M.Syntax - exception Error of string + exception Error = Error let error (r, msg) = raise (Error (Paths.wrap (r, msg))) diff --git a/src/Recon/ReconTerm.ml b/src/Recon/ReconTerm.ml index 3ca71ae..c4c5b61 100644 --- a/src/Recon/ReconTerm.ml +++ b/src/Recon/ReconTerm.ml @@ -1,5 +1,7 @@ module type RECON_TERM = RECON_TERM.RECON_TERM +exception Error of string + (* Logic copied from src/frontend/ReconTerm.ml. The functor takes a second parameter R for the modules that are not in S.S. See "Problems" comment at the bottom of this file. *) @@ -52,7 +54,7 @@ struct in run' !delayedList - exception Error of string + exception Error = Error let errorCount = ref 0 let errorFileName = ref "no file" @@ -285,6 +287,57 @@ struct (* Conversions from Cst types to internal types. Uses View observers because Cst.term/decl are abstract in the CST signature. *) + + (* Walk a Cst.term and qualify unresolved lowercase/uppercase names that exist + in ns_comps under ns_path. Used to implement %local NS EXPR desugaring: + names found in NS are prefixed with NS's path; everything else is unchanged. *) + let desugar_local (ns_path : string list) (ns_comps : Names.namespace) + (t : Cst.term) : Cst.term = + let module V = Cst.View in + let exists_in_ns name = + Names.constLookupIn (ns_comps, Names.Qid ([], name)) <> None + in + let qualify_lower loc ns name = + match ns with + | _ :: _ -> V.Term.(review @@ Lowercase (loc, (ns, name))) + | [] -> + let ns' = if exists_in_ns name then ns_path else [] in + V.Term.(review @@ Lowercase (loc, (ns', name))) + in + let qualify_upper loc ns name = + match ns with + | _ :: _ -> V.Term.(review @@ Uppercase (loc, (ns, name))) + | [] -> + let ns' = if exists_in_ns name then ns_path else [] in + V.Term.(review @@ Uppercase (loc, (ns', name))) + in + let rec go t = + match V.Term.view t with + | V.Term.Lowercase (loc, (ns, name)) -> qualify_lower loc ns name + | V.Term.Uppercase (loc, (ns, name)) -> qualify_upper loc ns name + | V.Term.Arrow (loc, a, b) -> V.Term.(review @@ Arrow (loc, go a, go b)) + | V.Term.BackArrow (loc, b, a) -> + V.Term.(review @@ BackArrow (loc, go b, go a)) + | V.Term.Pi (loc, decls, body) -> + V.Term.(review @@ Pi (loc, List.map go_decl decls, go body)) + | V.Term.Lam (loc, decls, body) -> + V.Term.(review @@ Lam (loc, List.map go_decl decls, go body)) + | V.Term.App (loc, head, args) -> + V.Term.(review @@ App (loc, go head, List.map go args)) + | V.Term.HasType (loc, a, b) -> + V.Term.(review @@ HasType (loc, go a, go b)) + | V.Term.Local (loc, ns2, inner) -> + V.Term.(review @@ Local (loc, ns2, go inner)) + | other -> V.Term.review other + and go_decl d = + match V.Decl.view d with + | V.Decl.Decl1 (loc, names, ty, def) -> + V.Decl.(review @@ Decl1 (loc, names, go ty, def)) + | V.Decl.Decl0 (loc, names, ty) -> + V.Decl.(review @@ Decl0 (loc, names, go ty)) + in + go t + let rec cst_term_to_term (t : Cst.term) : term = let module V = Cst.View in let ghost_r = loc_to_region Cst.ghost in @@ -319,6 +372,16 @@ struct | V.Term.ExistVar (_, s) -> Evar_ (s, ghost_r) | V.Term.FreeVar (_, s) -> Fvar_ (s, ghost_r) | V.Term.Typ _ -> Typ_ ghost_r + | V.Term.Local (_, ns_path, inner) -> + let qid = match List.rev ns_path with + | [] -> failwith "%local: empty namespace path" + | last :: prefix -> Names.Qid (List.rev prefix, last) + in + let ns_comps = match Names.structLookup qid with + | Some mid -> Names.getComponents mid + | None -> Names.newNamespace () + in + cst_term_to_term (desugar_local ns_path ns_comps inner) | _ -> Omitted_ ghost_r and cst_decl_to_dec (d : Cst.decl) : dec = @@ -826,13 +889,21 @@ struct try begin match Print.evarCnstrsToStringOpt xnames_ with | None -> () - | Some constr -> Msg.message (("Constraints:\n" ^ constr) ^ "\n") + | Some constr -> + Display.debug ~level:Detailed + (Display.Form.string (("Constraints:\n" ^ constr) ^ "\n")) end - with Names.Unprintable -> Msg.message "%_constraints unprintable_%\n" + with Names.Unprintable -> + Display.debug ~level:Detailed + (Display.Form.string "%_constraints unprintable_%\n") let rec reportInst xnames_ = - try Msg.message (Print.evarInstToString xnames_ ^ "\n") - with Names.Unprintable -> Msg.message "%_unifier unprintable_%\n" + try + Display.debug ~level:Detailed + (Display.Form.string (Print.evarInstToString xnames_ ^ "\n")) + with Names.Unprintable -> + Display.debug ~level:Detailed + (Display.Form.string "%_unifier unprintable_%\n") let rec delayMismatch (g_, v1_, v2_, r2, location_msg, problem_msg) = addDelayed (function () -> @@ -877,7 +948,7 @@ struct F.hVbox [ F.string "Inferred:"; F.space; formatExp (g_, u_) ] in let fstr = F.makestring_fmt amb in - Display.debug + Display.debug ~level:Detailed Display.Form.( nl () ++ string "Ambiguous reconstruction of term: " @@ -938,12 +1009,16 @@ struct formatExp (g_, eClo_ vs2_); ] in - let _ = Msg.message (F.makestring_fmt eqnsFmt ^ "\n") in + let _ = + Display.debug ~level:Exhaustive + (Display.Form.string (F.makestring_fmt eqnsFmt ^ "\n")) + in let _ = reportConstraints xnames_ in let _ = - Msg.message - ((("Failed: " ^ problem_msg) ^ "\n") - ^ "Continuing with subterm replaced by _\n") + Display.debug ~level:Exhaustive + (Display.Form.string + ((("Failed: " ^ problem_msg) ^ "\n") + ^ "Continuing with subterm replaced by _\n")) in ()) @@ -966,14 +1041,18 @@ struct formatExp (g_, eClo_ vs2_); ] in - let _ = Msg.message (F.makestring_fmt eqnsFmt ^ "\n") in + let _ = + Display.debug ~level:Exhaustive + (Display.Form.string (F.makestring_fmt eqnsFmt ^ "\n")) + in let _ = try unifyIdem (g_, vs1_, vs2_) with Unify.Unify msg as e -> begin - Msg.message - ((("Failed: " ^ msg) ^ "\n") - ^ "Continuing with subterm replaced by _\n"); + Display.debug ~level:Exhaustive + (Display.Form.string + ((("Failed: " ^ msg) ^ "\n") + ^ "Continuing with subterm replaced by _\n")); raise e end in @@ -1020,7 +1099,10 @@ struct formatExp (g_, v_); ] in - let _ = Msg.message (F.makestring_fmt omit ^ "\n") in + let _ = + Display.debug ~level:Detailed + (Display.Form.string (F.makestring_fmt omit ^ "\n")) + in let _ = reportConstraints xnames_ in () | g_, Mismatch_ (tm1, tm2, _, _), u_, v_ -> reportInfer' (g_, tm2, u_, v_) @@ -1047,7 +1129,10 @@ struct formatExp (g_, v_); ] in - let _ = Msg.message (F.makestring_fmt judg ^ "\n") in + let _ = + Display.debug ~level:Detailed + (Display.Form.string (F.makestring_fmt judg ^ "\n")) + in let _ = reportConstraints xnames_ in () @@ -1082,7 +1167,7 @@ struct let s = IntSyn.Shift (IntSyn.ctxLength g_) in (tm, Elim (elimSub (evarElim x_, s)), eClo_ (v_, s)) | g_, (Fvar_ (name, r) as tm) -> - Display.debug + Display.debug ~level:Detailed Display.Form.( nl () ++ string "Inferring exact type of FVar" @@ -1092,7 +1177,7 @@ struct with Apx.Ambiguous -> let v_ = getFVarType (name, true) in begin - Display.debug + Display.debug ~level:Detailed Display.Form.( string "Type of FVar" ++ string name ++ string @@ -1132,7 +1217,7 @@ struct let tm1', b1_, v1_ = inferExact (g_, tm1) in let e1_ = toElim b1_ in Display.( - debug + debug ~level:Detailed Form.( nl () ++ string "Inferring exact application of" @@ -1190,7 +1275,7 @@ struct with Ambiguous -> let v'_ = Apx.apxToClass (g_, v_, l_, true) in begin - Display.debug + Display.debug ~level:Detailed Display.Form.( string "Classifier of omitted term is ambiguous, but continuing \ @@ -1214,7 +1299,7 @@ struct with Ambiguous -> let u'_ = Apx.apxToExact (g_, u_, (v'_, IntSyn.id), true) in begin - Display.debug + Display.debug ~level:Detailed Display.Form.( string "Exact term of omitted term is ambiguous, but continuing \ @@ -1315,7 +1400,7 @@ struct and checkExact (g_, tm, vs_, location_msg) = Display.( - debug + debug ~level:Detailed Form.( nl () ++ string "Checking exact term" ++ shown show_term tm ++ nl ())); begin if not !trace then diff --git a/src/Recon/ReconThm.ml b/src/Recon/ReconThm.ml index cfa0548..e19dab6 100644 --- a/src/Recon/ReconThm.ml +++ b/src/Recon/ReconThm.ml @@ -1,5 +1,7 @@ module type RECON_THM = RECON_THM.RECON_THM +exception Error of string + module Make_ReconThm (M : S.S) (RT : RECON_TERM.RECON_TERM with module M = M) : RECON_THM with module M = M = struct module M = M @@ -9,7 +11,7 @@ module Make_ReconThm (M : S.S) (RT : RECON_TERM.RECON_TERM with module M = M) : module Syntax = M.Syntax module ThmSyn = Thm.Thm_.ThmSyn - exception Error of string + exception Error = Error let error (r, msg) = raise (Error (Paths.wrap (r, msg))) diff --git a/src/Recon/dune b/src/Recon/dune index c7e9129..cc1bde0 100644 --- a/src/Recon/dune +++ b/src/Recon/dune @@ -3,6 +3,7 @@ ; private module type. Restore once Pal.ml supplies a named module satisfying S.S. (library + (instrumentation (backend bisect_ppx)) (name recon) (public_name stelf.recon) (libraries diff --git a/src/Resugar/dune b/src/Resugar/dune index 142e0ce..2d6ee5a 100644 --- a/src/Resugar/dune +++ b/src/Resugar/dune @@ -1,4 +1,5 @@ (library + (instrumentation (backend bisect_ppx)) (name resugar) (public_name stelf.resugar) (libraries syntax names paths cst)) diff --git a/src/Server/Server_.ml b/src/Server/Server_.ml index c4ff809..eb34df2 100644 --- a/src/Server/Server_.ml +++ b/src/Server/Server_.ml @@ -9,6 +9,9 @@ include SERVER (** Interactive command server for Stelf/STELF. *) (* signature SERVER *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Server : SERVER = struct let globalConfig : Stelf.Config.config option ref = ref None @@ -46,7 +49,7 @@ module Server : SERVER = struct let tokenize args = String.tokens Char.isSpace args (* exception Error for server errors *) - exception Error of string + exception Error = Error let error msg = raise (Error msg) let quote string = ("`" ^ string) ^ "'" diff --git a/src/Server/dune b/src/Server/dune index f60d2c7..2f9292f 100644 --- a/src/Server/dune +++ b/src/Server/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name server) (public_name stelf.server) (modules diff --git a/src/Smlofnj/dune b/src/Smlofnj/dune index c80c5b9..263bb98 100644 --- a/src/Smlofnj/dune +++ b/src/Smlofnj/dune @@ -1,4 +1,5 @@ (library + (instrumentation (backend bisect_ppx)) (name smlofnj) (public_name stelf.smlofnj) (libraries display basis)) diff --git a/src/Solvers/CsManager.ml b/src/Solvers/CsManager.ml index dd7df4d..bd52566 100644 --- a/src/Solvers/CsManager.ml +++ b/src/Solvers/CsManager.ml @@ -262,5 +262,6 @@ end (*! structure ModeSyn : MODESYN !*) (* functor CsManager *) include MakeCsManager (Global) (UnifyTrail) (Names.Fixity) +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) (* # 1 "src/solvers/CsManager.sml.ml" *) diff --git a/src/Solvers/dune b/src/Solvers/dune index b9ca3f4..4560e2f 100644 --- a/src/Solvers/dune +++ b/src/Solvers/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name solvers) (public_name stelf.solvers) (modules diff --git a/src/Stream/Stream_.ml b/src/Stream/Stream_.ml index 1cbc57f..6f02bb6 100644 --- a/src/Stream/Stream_.ml +++ b/src/Stream/Stream_.ml @@ -24,11 +24,16 @@ end (* Note that this implementation is NOT semantically *) (* equivalent to the plain (non-memoizing) streams, since *) (* effects will be executed only once in this implementation *) +exception Uninitialized +let () = Printexc.register_printer (function Uninitialized -> Some "Uninitialized" | _ -> None) +exception EmptyStream +let () = Printexc.register_printer (function EmptyStream -> Some "EmptyStream" | _ -> None) + module BasicMemoStream : BASIC_STREAM = struct type 'a stream = Stream of (unit -> 'a front) and 'a front = Empty | Cons of 'a * 'a stream - exception Uninitialized + exception Uninitialized = Uninitialized let expose (Stream d) = d () @@ -61,7 +66,7 @@ end module MakeStream (BasicStream : BASIC_STREAM) : STREAM = struct include BasicStream - exception EmptyStream + exception EmptyStream = EmptyStream (* functions null, hd, tl, map, filter, exists, take, drop *) (* parallel the functions in the List structure *) diff --git a/src/Stream/dune b/src/Stream/dune index 9e4f02b..f5e4d2b 100644 --- a/src/Stream/dune +++ b/src/Stream/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name stream) (public_name stelf.stream) (modules Stream_ STREAM) diff --git a/src/Style/Style_.ml b/src/Style/Style_.ml index 6aa1e15..a7cffc9 100644 --- a/src/Style/Style_.ml +++ b/src/Style/Style_.ml @@ -12,9 +12,12 @@ include STYLE open! Basis open Origins +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MakeStyleCheck (Whnf : WHNF) (Index : INDEX) (Origins : ORIGINS) : STYLECHECK = struct - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/Style/dune b/src/Style/dune index 9bd835b..0dbf0b7 100644 --- a/src/Style/dune +++ b/src/Style/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name style) (public_name stelf.style) (modules Style_ STYLE) diff --git a/src/Subordinate/Subordinate_.ml b/src/Subordinate/Subordinate_.ml index 4ab088a..4f7d03b 100644 --- a/src/Subordinate/Subordinate_.ml +++ b/src/Subordinate/Subordinate_.ml @@ -15,6 +15,9 @@ open! Basis (* Subordination a la Virga [Technical Report 96] *) (* Author: Carsten Schuermann *) (* Reverse subordination order *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MakeSubordinate (Global : GLOBAL) (Whnf : WHNF) @@ -35,7 +38,7 @@ module MakeSubordinate module MemoTable = MemoTable module IntSet = IntSet - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/Subordinate/dune b/src/Subordinate/dune index b7cddc5..d0d7a2a 100644 --- a/src/Subordinate/dune +++ b/src/Subordinate/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name subordinate) (public_name stelf.subordinate) (modules Intset Subordinate_ INTSET SUBORDINATE) diff --git a/src/Table/RedBlackSet.ml b/src/Table/RedBlackSet.ml index 2b01b35..8346b22 100644 --- a/src/Table/RedBlackSet.ml +++ b/src/Table/RedBlackSet.ml @@ -32,6 +32,9 @@ include REDBLACKSET * condition implies that any node with only one child will be black and * its child will be a red leaf. *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module RBSet : RBSET = struct type nonrec key = int type 'a entry = key * 'a @@ -44,7 +47,7 @@ module RBSet : RBSET = struct (* considered black *) type 'a set = Set of int * 'a dict - exception Error of string + exception Error = Error type 'a ordSet = 'a set ref diff --git a/src/Table/RedBlackTree.ml b/src/Table/RedBlackTree.ml index 31e7091..1d63e99 100644 --- a/src/Table/RedBlackTree.ml +++ b/src/Table/RedBlackTree.ml @@ -186,12 +186,8 @@ end) : TABLE with type key = RedBlackTree__0.key' = struct | Greater -> del (b, Rightr (a, entry1, z)) end in - try - begin - ignore (del (t, Top)); - true - end - with NotFound -> false + try del (t, Top) + with NotFound -> t let insertShadow (dict, ((key, _datum) as entry)) = let oldEntry = ref None in @@ -218,12 +214,14 @@ end) : TABLE with type key = RedBlackTree__0.key' = struct in begin oldEntry := None; - ( begin match ins dict with + let new_dict = + begin match ins dict with | Red ((_, Red _, _) as t) -> Black t | Red ((_, _, Red _) as t) -> Black t | dict -> dict - end, - !oldEntry ) + end + in + (new_dict, !oldEntry) end let app f dict = @@ -319,7 +317,7 @@ end) : TABLE with type key = RedBlackTree__0.key' = struct | table -> ( function | key -> begin - ignore (delete !table key); + table := delete !table key; () end) diff --git a/src/Table/Ring.ml b/src/Table/Ring.ml index dc1ff63..b93acbc 100644 --- a/src/Table/Ring.ml +++ b/src/Table/Ring.ml @@ -17,8 +17,11 @@ include RING (* Rings (aka cyclic lists) *) (* Author: Carsten Schuermann *) +exception Empty +let () = Printexc.register_printer (function Empty -> Some "Ring is empty" | _ -> None) + module Ring : RING = struct - exception Empty + exception Empty = Empty type 'a ring = 'a list * 'a list diff --git a/src/Table/dune b/src/Table/dune index 5918728..94b2ec8 100644 --- a/src/Table/dune +++ b/src/Table/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name table) (public_name stelf.table) (modules diff --git a/src/Tabling/Tabledsyn.ml b/src/Tabling/Tabledsyn.ml index c7669d8..11a8e93 100644 --- a/src/Tabling/Tabledsyn.ml +++ b/src/Tabling/Tabledsyn.ml @@ -11,6 +11,9 @@ open! Basis (* Tabled Syntax *) (* Author: Brigitte Pientka *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MakeTabledSyn (Names : NAMES) (Table : TABLE with type key = int) @@ -28,7 +31,7 @@ module MakeTabledSyn module Names = Names module Index = Index - exception Error of string + exception Error = Error type tabled = Yes_ | No_ [@@deriving eq, ord, show] diff --git a/src/Tabling/dune b/src/Tabling/dune index 6dd5b28..1161df7 100644 --- a/src/Tabling/dune +++ b/src/Tabling/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name tabling) (public_name stelf.tabling) (modules Tabledsyn Tabled TABLEDSYN) diff --git a/src/Terminate/Reduces.ml b/src/Terminate/Reduces.ml index 44113da..d5c9a62 100644 --- a/src/Terminate/Reduces.ml +++ b/src/Terminate/Reduces.ml @@ -15,6 +15,9 @@ open! Basis for a revised version incorporating reducation checking see tech report CMU-CS-01-115 *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Reduces (Reduces__0 : sig module Global : GLOBAL @@ -46,7 +49,7 @@ module Reduces (Reduces__0 : sig module Origins : Origins.ORIGINS end) : REDUCES = struct (*! structure IntSyn = IntSyn' !*) - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/Terminate/dune b/src/Terminate/dune index 4a6a152..7fa0258 100644 --- a/src/Terminate/dune +++ b/src/Terminate/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name terminate) (public_name stelf.terminate) (modules Checking Reduces Terminate_ CHECKING REDUCES) diff --git a/src/Thm/Thmsyn.ml b/src/Thm/Thmsyn.ml index 4780647..08ad6e0 100644 --- a/src/Thm/Thmsyn.ml +++ b/src/Thm/Thmsyn.ml @@ -13,6 +13,9 @@ open! Basis (* Theorems *) (* Author: Carsten Schuermann *) (* Modified: Brigitte Pientka *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module ThmSyn (ThmSyn__0 : sig (*! structure IntSyn : INTSYN !*) (*! structure ModeSyn' : MODESYN !*) @@ -31,7 +34,7 @@ end) : THMSYN with module Names = ThmSyn__0.Names' = struct (*! structure Paths = Paths' !*) module Names = ThmSyn__0.Names' - exception Error of string + exception Error = Error let rec error (r, msg) = raise (Error (Paths.wrap (r, msg))) diff --git a/src/Thm/dune b/src/Thm/dune index 57579fb..dc4a6b1 100644 --- a/src/Thm/dune +++ b/src/Thm/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name thm) (public_name stelf.thm) (modules Thmsyn Thmprint Thm_ THM THMPRINT THMSYN) diff --git a/src/Timing/TimeLimit.ml b/src/Timing/TimeLimit.ml index c7a1546..63ff5c5 100644 --- a/src/Timing/TimeLimit.ml +++ b/src/Timing/TimeLimit.ml @@ -57,4 +57,6 @@ end = struct end *) end + +let () = Printexc.register_printer (function TimeLimit.TimeOut -> Some "TimeOut" | _ -> None) (* TimeLimit *) diff --git a/src/Timing/dune b/src/Timing/dune index aa9a4e9..5bc16da 100644 --- a/src/Timing/dune +++ b/src/Timing/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name timing) (public_name stelf.timing) (modules Timers Timing_ TimeLimit TIMERS TIMING) diff --git a/src/Tomega/Converter.ml b/src/Tomega/Converter.ml index 686b143..c2ea58e 100644 --- a/src/Tomega/Converter.ml +++ b/src/Tomega/Converter.ml @@ -11,6 +11,11 @@ include CONVERTER (* # 1 "src/tomega/Converter.fun.ml" *) open! Basis +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) +exception Error' of Tomega.sub +let () = Printexc.register_printer (function Error' _ -> Some "Tomega converter error" | _ -> None) + module MakeConverter (Global : GLOBAL) (Abstract : ABSTRACT) @@ -81,8 +86,8 @@ module MakeConverter *) (*! structure IntSyn = IntSyn' !*) (*! structure Tomega = Tomega' !*) - exception Error of string - exception Error' of Tomega.sub + exception Error = Error + exception Error' = Error' open! struct module T = Tomega diff --git a/src/Tomega/Coverage.ml b/src/Tomega/Coverage.ml index 8297c3a..6cded38 100644 --- a/src/Tomega/Coverage.ml +++ b/src/Tomega/Coverage.ml @@ -10,6 +10,9 @@ include COVERAGE (* # 1 "src/tomega/Coverage.fun.ml" *) open! Basis +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MakeTomegaCoverage (TomegaPrint : Tomegaprint.TOMEGAPRINT) (TomegaTypeCheck : TOMEGATYPECHECK.TOMEGATYPECHECK) @@ -32,7 +35,7 @@ module MakeTomegaCoverage *) (*! structure IntSyn = IntSyn' !*) (*! structure Tomega = Tomega' !*) - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/Tomega/Normalize.ml b/src/Tomega/Normalize.ml index 3a64b9d..c3fa453 100644 --- a/src/Tomega/Normalize.ml +++ b/src/Tomega/Normalize.ml @@ -9,6 +9,9 @@ include NORMALIZE (* # 1 "src/tomega/Normalize.fun.ml" *) open! Basis +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Normalize (Normalize__0 : sig (* Internal syntax for functional proof term calculus *) (* Author: Carsten Schuermann *) @@ -17,7 +20,7 @@ end) : NORMALIZE = struct module IntSyn = IntSyn module Tomega = Tomega - exception Error of string + exception Error = Error open! struct module Whnf = Normalize__0.Whnf diff --git a/src/Tomega/Opsem.ml b/src/Tomega/Opsem.ml index dde1488..6941087 100644 --- a/src/Tomega/Opsem.ml +++ b/src/Tomega/Opsem.ml @@ -9,6 +9,13 @@ include OPSEM (* # 1 "src/tomega/Opsem.fun.ml" *) open! Basis +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) +exception Abort +let () = Printexc.register_printer (function Abort -> Some "Abort" | _ -> None) +exception NoMatch +let () = Printexc.register_printer (function NoMatch -> Some "NoMatch" | _ -> None) + module MakeOpsem (Whnf : WHNF) (Abstract : ABSTRACT) @@ -33,11 +40,11 @@ module MakeOpsem module Unify = Unify module TomegaPrint = TomegaPrint - exception Error of string - exception Abort + exception Error = Error + exception Abort = Abort (* local -- removed ABP 1/19/03 *) - exception NoMatch + exception NoMatch = NoMatch (* matchPrg is used to see if two values can be 'unified' for diff --git a/src/Tomega/Redundant.ml b/src/Tomega/Redundant.ml index a5f093f..217c6ac 100644 --- a/src/Tomega/Redundant.ml +++ b/src/Tomega/Redundant.ml @@ -9,10 +9,13 @@ open! Basis (* Redundancy remover (factoring) *) (* Author: Adam Poswolsky (ABP) *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Redundant (Redundant__0 : sig module Opsem : OPSEM.OPSEM end) : REDUNDANT = struct - exception Error of string + exception Error = Error (* convert: Tomega.Prg -> Tomega.Prg diff --git a/src/Tomega/TomegaAbstract.ml b/src/Tomega/TomegaAbstract.ml index 7e7f282..7bf8aed 100644 --- a/src/Tomega/TomegaAbstract.ml +++ b/src/Tomega/TomegaAbstract.ml @@ -10,6 +10,9 @@ include TOMEGAABSTRACT (* # 1 "src/tomega/TomegaAbstract.fun.ml" *) open! Basis +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module TomegaAbstract (TomegaAbstract__0 : sig (* Converter from relational representation to a functional representation of proof terms *) @@ -22,7 +25,7 @@ module TomegaAbstract (TomegaAbstract__0 : sig module Whnf : WHNF module Subordinate : Subordinate.Subordinate_.SUBORDINATE end) : TOMEGAABSTRACT = struct - exception Error of string + exception Error = Error module Global = TomegaAbstract__0.Global module Whnf = TomegaAbstract__0.Whnf diff --git a/src/Tomega/TomegaTypecheck.ml b/src/Tomega/TomegaTypecheck.ml index 8dd6bb4..5addcf1 100644 --- a/src/Tomega/TomegaTypecheck.ml +++ b/src/Tomega/TomegaTypecheck.ml @@ -11,6 +11,9 @@ include TOMEGATYPECHECK (* # 1 "src/tomega/TomegaTypecheck.fun.ml" *) open! Basis +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module TomegaTypeCheck (TomegaTypeCheck__0 : sig (* Type checking for Tomega *) (* Author: Carsten Schuermann *) @@ -27,7 +30,7 @@ module TomegaTypeCheck (TomegaTypeCheck__0 : sig end) : TOMEGATYPECHECK = struct (*! structure IntSyn = IntSyn' !*) (*! structure Tomega = Tomega' !*) - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/Tomega/TomegaUnify.ml b/src/Tomega/TomegaUnify.ml index ad9787f..e08b2eb 100644 --- a/src/Tomega/TomegaUnify.ml +++ b/src/Tomega/TomegaUnify.ml @@ -10,6 +10,9 @@ include TOMEGAUNIFY (* # 1 "src/tomega/TomegaUnify.fun.ml" *) open! Basis +exception Unify of string +let () = Printexc.register_printer (function Unify msg -> Some msg | _ -> None) + module TomegaUnify (TomegaUnify__0 : sig (* Unification on Formulas *) (* Author: Carsten Schuermann *) @@ -46,7 +49,7 @@ module TomegaUnify (TomegaUnify__0 : sig end) : TOMEGAUNIFY = struct (*! structure IntSyn = IntSyn' !*) (*! structure Tomega = Tomega' !*) - exception Unify of string + exception Unify = Unify open! struct module I = IntSyn diff --git a/src/Tomega/Tomegaprint.ml b/src/Tomega/Tomegaprint.ml index 61c1215..0035c8c 100644 --- a/src/Tomega/Tomegaprint.ml +++ b/src/Tomega/Tomegaprint.ml @@ -14,6 +14,9 @@ open! Basis (* Printing of functional proof terms *) (* Author: Carsten Schuermann *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module TomegaPrint (TomegaPrint__0 : sig (*! structure IntSyn' : INTSYN !*) (*! structure Tomega' : TOMEGA !*) @@ -31,7 +34,7 @@ end) : TOMEGAPRINT = struct (*! structure Tomega = Tomega' !*) module Formatter = TomegaPrint__0.Formatter - exception Error of string + exception Error = Error (* is just here because we don't have a module yet for Names. move later diff --git a/src/Tomega/dune b/src/Tomega/dune index 150640f..e844820 100644 --- a/src/Tomega/dune +++ b/src/Tomega/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name tomega_lib) (public_name stelf.tomega) (modules diff --git a/src/Trail/dune b/src/Trail/dune index 3dae12b..29eb03e 100644 --- a/src/Trail/dune +++ b/src/Trail/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name trail) (public_name stelf.trail) (modules Trail_ Notrail TRAIL) diff --git a/src/Typecheck/Strict.ml b/src/Typecheck/Strict.ml index eac85e3..94388fd 100644 --- a/src/Typecheck/Strict.ml +++ b/src/Typecheck/Strict.ml @@ -11,12 +11,15 @@ open! Basis (* Checking Definitions for Strict *) (* Author: Carsten Schuermann *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module Strict (Strict__0 : sig module Whnf : WHNF end) : STRICT = struct (*! structure IntSyn = IntSyn' !*) (*! structure Paths = Paths' !*) - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/Typecheck/Typecheck_.ml b/src/Typecheck/Typecheck_.ml index 9b41d8f..fbc00a5 100644 --- a/src/Typecheck/Typecheck_.ml +++ b/src/Typecheck/Typecheck_.ml @@ -13,6 +13,9 @@ open! Basis (* Type Checking *) (* Author: Carsten Schuermann *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MakeTypeCheck (Conv : CONV) (Whnf : WHNF) @@ -25,7 +28,7 @@ module MakeTypeCheck (*! sharing Names.IntSyn = IntSyn' !*) *) (*! structure IntSyn = IntSyn' !*) - exception Error of string + exception Error = Error open! struct module I = IntSyn diff --git a/src/Typecheck/dune b/src/Typecheck/dune index 2d7bf78..d0ee009 100644 --- a/src/Typecheck/dune +++ b/src/Typecheck/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name typecheck) (public_name stelf.typecheck) (modules Typecheck_ Strict STRICT TYPECHECK) diff --git a/src/Unique/Unique_.ml b/src/Unique/Unique_.ml index 4388212..10abbfc 100644 --- a/src/Unique/Unique_.ml +++ b/src/Unique/Unique_.ml @@ -14,6 +14,9 @@ open! Basis (* Uniqueness Checking *) (* Author: Frank Pfenning *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) + module MakeUnique (Global : GLOBAL) (Whnf : WHNF) @@ -29,7 +32,7 @@ module MakeUnique (Print : PRINT) (TypeCheck : TYPECHECK) (Timers : Timers.TIMERS) : UNIQUE = struct - exception Error of string + exception Error = Error module Subordinate = Subordinate module Unify = Unify diff --git a/src/Unique/dune b/src/Unique/dune index abca2cd..c717515 100644 --- a/src/Unique/dune +++ b/src/Unique/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name unique) (public_name stelf.unique) (modules Unique_ UNIQUE) diff --git a/src/Worldcheck/WorldSyn.ml b/src/Worldcheck/WorldSyn.ml index 1672a27..8854fb2 100644 --- a/src/Worldcheck/WorldSyn.ml +++ b/src/Worldcheck/WorldSyn.ml @@ -15,6 +15,16 @@ open Table_ (* World Checking *) (* Author: Carsten Schuermann *) (* Modified: Frank Pfenning *) +exception Error of string + +let () = + Printexc.register_printer (function Error msg -> Some msg | _ -> None) + +exception Error' of Paths.occ * string + +let () = + Printexc.register_printer (function Error' (_, msg) -> Some msg | _ -> None) + module WorldSyn (WorldSyn__0 : sig module Global : GLOBAL module Whnf : WHNF @@ -61,8 +71,8 @@ end) : WORLDSYN = struct module Table = WorldSyn__0.Table module Unify = WorldSyn__0.Unify - exception Error of string - exception Error' of P.occ * string + exception Error = Error + exception Error' = Error' (* copied from terminates/Reduces.fun *) let rec wrapMsg (c, occ, msg) = diff --git a/src/Worldcheck/Worldify.ml b/src/Worldcheck/Worldify.ml index 3de51c7..0c46d6e 100644 --- a/src/Worldcheck/Worldify.ml +++ b/src/Worldcheck/Worldify.ml @@ -15,6 +15,11 @@ open! Basis (* Worldification and World-checking *) (* Author: Carsten Schuermann *) (* Modified: Frank Pfenning *) +exception Error of string +let () = Printexc.register_printer (function Error msg -> Some msg | _ -> None) +exception Error' of Paths.occ * string +let () = Printexc.register_printer (function Error' (_, msg) -> Some msg | _ -> None) + module Worldify (Worldify__0 : sig module Global : GLOBAL @@ -71,8 +76,8 @@ end) : WORLDIFY = struct module CsManager = Worldify__0.CsManager module WorldSyn = Worldify__0.WorldSyn - exception Error of string - exception Error' of P.occ * string + exception Error = Error + exception Error' = Error' (* copied from terminates/Reduces.fun *) let rec wrapMsg (c, occ, msg) = diff --git a/src/Worldcheck/dune b/src/Worldcheck/dune index 9891899..0b604de 100644 --- a/src/Worldcheck/dune +++ b/src/Worldcheck/dune @@ -1,6 +1,7 @@ ; (include sources.dune) (library + (instrumentation (backend bisect_ppx)) (name worldcheck) (public_name stelf.worldcheck) (modules WorldSyn Worldify Worldcheck_ WORLDSYN WORLDIFY) diff --git a/src/dune b/src/dune index dda6aaf..40d5e2e 100644 --- a/src/dune +++ b/src/dune @@ -1,4 +1,5 @@ (library + (instrumentation (backend bisect_ppx)) (name stelf) (public_name stelf) (libraries diff --git a/test/Pal/Cases.ml b/test/Pal/Cases.ml index 312676d..3f92b5b 100644 --- a/test/Pal/Cases.ml +++ b/test/Pal/Cases.ml @@ -24,6 +24,10 @@ let total_add_mul_test = {| %total N (add N _ _) |}; ] +let cwd : string = Sys.getcwd () +let new_tests_pages : string = cwd ^ "/../../new-tests/pages" +let new_tests_output_pages : string = cwd ^ "/../../new-tests/output/pages" +let new_tests_stelf_output_tslf : string = cwd ^ "/../../new-tests/stelf-output/tslf" let cases () = Alcotest.run "PAL" begin @@ -63,8 +67,8 @@ let cases () = Source. [ prop_calc_types; - prop_calc_types ^ prop_calc_hilbert; - prop_calc_types ^ prop_calc_hilbert ^ prop_calc_nd; + prop_calc_hilbert; + prop_calc_nd; ]; test "MINI-ML" Source.[ mini_ml_exp; mini_ml_value; mini_ml_tp ]; test "ARITH" Source.[ arith_nat; arith_nt; arith_plus; arith_acker ]; @@ -134,7 +138,7 @@ let cases () = church_rosser_sources_2; church_rosser_sources_3; ]; - test ~skip:true "mini_ml/sources" + test "mini_ml/sources" Source. [ mini_ml_exp; @@ -143,8 +147,7 @@ let cases () = mini_ml_sources_eval; mini_ml_sources_tpinf; ]; - (* Skipped: dependent sort indices in %sort declarations not yet supported. *) - test ~skip:true "lp_horn/sources" + test "lp_horn/sources" Source.[ lp_horn_nd; lp_horn_sources_2; lp_horn_sources_3 ]; (* examples/arith/sources.cfg: nat + nt + plus + acker — same content as ARITH above, re-declared. Pal frontend is lenient about re-declarations. *) @@ -191,8 +194,8 @@ let cases () = Source. [ prop_calc_types; - prop_calc_types ^ prop_calc_hilbert; - prop_calc_types ^ prop_calc_hilbert ^ prop_calc_nd; + prop_calc_hilbert; + prop_calc_nd; ]; (* examples/crary/explicit/excon *) test "crary/explicit/excon" Source.[ crary_excon ]; @@ -207,11 +210,11 @@ let cases () = test "crary/substruct/modal" Source.[ crary_modal_syntax ]; (* Tier 2: single-elf cfg files *) - test ~skip:true "handbook/sources" Source.[ handbook_sources_1 ]; + test "handbook/sources" Source.[ handbook_sources_1 ]; (* examples/ccc/spass.cfg → spass.elf: CCC with categorical laws. Uses dependent sort indices (== : mor A B -> mor A B -> type) — STELF reconstructor does not support dependent sort indices yet. *) - test ~skip:true "ccc/spass" Source.[ ccc_spass_1 ]; + test "ccc/spass" Source.[ ccc_spass_1 ]; (* examples/failure/sources.cfg → fail.elf: only %query (unsupported in STELF). Expected to fail with ParseError. *) test ~failure:true "failure/sources" Source.[ failure_sources_1 ]; @@ -232,19 +235,182 @@ let cases () = test ~failure:true "wiki_failures/unsatisfiable_query" Source.[ wiki_failures_unsatisfiable_query_1 ]; (* examples/crary/standard/standard.cfg → standard.elf (1602 lines). - Higher-order CBV lambda calculus. Too large to translate now. *) - test ~skip:true "crary/standard/standard" + Higher-order CBV lambda calculus. *) + test "crary/standard/standard" Source.[ crary_standard_standard_1 ]; (* examples/tabled/parsing/arithml.cfg: grammar with numeric identifiers. *) - test ~skip:true "tabled/parsing/arithml" + test "tabled/parsing/arithml" Source.[ tabled_parsing_arithml_1 ]; (* examples/tabled/parsing/foll.cfg: FOL grammar with %tabled. *) - test ~skip:true "tabled/parsing/foll" Source.[ tabled_parsing_foll_1 ]; + test "tabled/parsing/foll" Source.[ tabled_parsing_foll_1 ]; (* examples/tabled/parsing/tab.cfg: DCG grammar with single-quoted tokens. *) - test ~skip:true "tabled/parsing/tab" Source.[ tabled_parsing_tab_1 ]; + test "tabled/parsing/tab" Source.[ tabled_parsing_tab_1 ]; (* examples/tabled/ccc/tab.cfg: CCC with tabling. *) - test ~skip:true "tabled/ccc/tab" Source.[ tabled_ccc_tab_1 ]; - test "nat-scope" Source.[ nat_scope ]; + test "tabled/ccc/tab" Source.[ tabled_ccc_tab_1 ]; + test "nat-scope" Source.[ nat_scope ]; + file_test "require" [ cwd ^ "/require/main.lf" ]; + (* Name shadowing: redeclaring a constant in the same namespace raises + Names_.Error "Shadowing: ..." thanks to the insertShadow fix. *) + test ~failure:true "Name shadowing / sort" + Source.[ shadow_sort_redecl ]; + test ~failure:true "Name shadowing / term" + Source.[ shadow_term_redecl ]; + (* Targeted error-kind tests: verify that specific bad inputs fail *) + test ~failure:true "Error / parse / sort-no-name" + Source.[ error_parse_sort_empty ]; + test ~failure:true "Error / parse / term-no-args" + Source.[ error_parse_term_empty ]; + test ~failure:true "Error / recon / undeclared-type" + Source.[ error_recon_undeclared ]; + (* %require edge cases *) + file_test ~failure:true "require/missing" + [ cwd ^ "/require/missing/main.lf" ]; + (* Circular %require is handled gracefully: the dedup table breaks the cycle *) + file_test "require/circular" + [ cwd ^ "/require/circular/a.lf" ]; + (* Second %require of the same file is a no-op (idempotent) *) + file_test "require/dedup" + [ cwd ^ "/require/dedup/main.lf" ]; + (* %require inside %scope escapes to group_ns, not the inner scope's ns *) + file_test "require/scope-require" + [ cwd ^ "/require/scope-require/outer.lf" ]; + (* %scope then %open makes scope contents directly accessible *) + test "scope/open" Source.[ scope_open_test ]; + (* %(member scope) qualified access without %open *) + test "scope/qualified" Source.[ scope_qualified_test ]; + (* %open inside a %scope body (inner scope can open outer scope) *) + test "scope/open-inside" Source.[ scope_open_inside_test ]; + (* TOML: single group *) + file_test "toml/single" + [ cwd ^ "/require/toml/single/stelf.toml" ]; + (* TOML: multi-group with local dep — dep group loaded before main *) + file_test "toml/multi" + [ cwd ^ "/require/toml/multi/stelf.toml" ]; + (* TOML: local dep with alias — dep accessible under aliased struct name *) + file_test "toml/alias" + [ cwd ^ "/require/toml/alias/stelf.toml" ]; + (* %local: qualify names in inner expr against given namespace *) + test "local/basic" Source.[ local_basic_test ]; + (* %local: names not in namespace are resolved from surrounding context *) + test "local/fallthrough" Source.[ local_fallthrough_test ]; + (* %def: simple definitional abbreviation with explicit type *) + test "def/basic" Source.[ def_basic_test ]; + (* %def: type inferred with _ placeholder *) + test "def/inferred" Source.[ def_inferred_test ]; + (* %def: pi-type as abbreviation (higher-order) *) + test "def/prop" Source.[ def_prop_test ]; + (* %freeze: freeze a family, then use it — new constructors for other families ok *) + test "freeze/basic" Source.[ freeze_basic_test ]; + (* %freeze then %term on frozen family raises a freezing violation *) + test ~failure:true "freeze/violation" Source.[ freeze_violation_test ]; + (* %thaw without unsafe flag always fails *) + test ~failure:true "thaw/unsafe" Source.[ thaw_unsafe_test ]; + (* %reduces: output ≤ input annotation on a subtraction relation *) + test "reduces/basic" Source.[ reduces_test ]; + (* %union: two %block declarations merged into a union block *) + test "union/basic" Source.[ union_test ]; + (* %unique: impl hardcodes Mnil mode into checkUnique, causing a pattern match + failure in Modecheck.checkMode — skip until the impl reads the mode table *) + test ~skip:true "unique/basic" Source.[ unique_test ]; + (* new-tests/pages: Twelf Wiki examples exercising %% line comments and + %{! !}% block comments, which skip_outer now handles *) + + (* All content inside one big %{! !}% block — parse to zero commands *) + file_test "new-tests/pages/computation-and-deduction-2009-20090203" + [ new_tests_pages ^ "/computation-and-deduction-2009-20090203.lf" ]; + file_test "new-tests/pages/computation-and-deduction-2009-20090316" + [ new_tests_pages ^ "/computation-and-deduction-2009-20090316.lf" ]; + file_test "new-tests/pages/computation-and-deduction-2009-20090408" + [ new_tests_pages ^ "/computation-and-deduction-2009-20090408.lf" ]; + (* Top-level LF code confirmed passing *) + file_test "new-tests/pages/lex-orderings" + [ new_tests_pages ^ "/lexicographical-orderings-with-density.lf" ]; + (* popl-church-rosser: uses (1) zero-arg sort kind inference (%sort reduce %., + %sort diamond %.) (2) %the type ascription (3) %block [x exp] notation *) + file_test ~skip:true "new-tests/pages/popl-tutorial-church-rosser-problem" + [ new_tests_pages ^ "/popl-tutorial-church-rosser-problem.lf" ]; + (* popl-properties: uses (1) % (percent-space) inter-command comments which + skip_outer doesn't handle (2) zero-arg sort kind inference (3) %block + with {T typ} [x exp] [ofx ...] mixed notation *) + file_test ~skip:true "new-tests/pages/popl-tutorial-properties-of-typing-and-reduction" + [ new_tests_pages ^ "/popl-tutorial-properties-of-typing-and-reduction.lf" ]; + (* odd-even: uses zero-arg sort kind inference (%sort plus without + {_ nat} {_ nat} {_ nat}) — separate reconstructor limitation *) + file_test ~skip:true "new-tests/pages/odd-even" + [ new_tests_pages ^ "/proving-metatheorems-solutions-odd-even-succ.lf" ]; + (* Only %%! metadata + %{! !}% prose — parse to zero commands *) + file_test "new-tests/pages/proving-metatheorems-with-twelf" + [ new_tests_pages ^ "/proving-metatheorems-with-twelf.lf" ]; + file_test "new-tests/pages/talk-effectiveness-lemma" + [ new_tests_pages ^ "/talk-effectiveness-lemma.lf" ]; + file_test "new-tests/pages/talk-incremental-metatheorem-development" + [ new_tests_pages ^ "/talk-incremental-metatheorem-development.lf" ]; + file_test "new-tests/pages/talk-metatheorem" + [ new_tests_pages ^ "/talk-metatheorem.lf" ]; + file_test "new-tests/pages/talk-mutable-state" + [ new_tests_pages ^ "/talk-mutable-state.lf" ]; + file_test "new-tests/pages/talk-natural-numbers-with-inequality" + [ new_tests_pages ^ "/talk-natural-numbers-with-inequality.lf" ]; + file_test "new-tests/pages/talk-thaw" + [ new_tests_pages ^ "/talk-thaw.lf" ]; + file_test "new-tests/pages/template-bibtex-pierce02tapl" + [ new_tests_pages ^ "/template-bibtex-pierce02tapl.lf" ]; + file_test "new-tests/pages/template-discuss" + [ new_tests_pages ^ "/template-discuss.lf" ]; + file_test "new-tests/pages/template-lolli" + [ new_tests_pages ^ "/template-lolli.lf" ]; + file_test "new-tests/pages/template-talk-guide" + [ new_tests_pages ^ "/template-talk-guide.lf" ]; + (* twelf-with-emacs: has top-level LF but uses zero-arg sort kind inference + (%sort plus %.) — same reconstructor limitation as odd-even *) + file_test ~skip:true "new-tests/pages/twelf-with-emacs" + [ new_tests_pages ^ "/twelf-with-emacs.lf" ]; + file_test "new-tests/pages/user-jaked" + [ new_tests_pages ^ "/user-jaked.lf" ]; + (* new-tests/output/pages: identical content to pages/ above; + separate test group so divergence is caught when files differ *) + file_test "new-tests/output/pages/computation-and-deduction-2009-20090203" + [ new_tests_output_pages ^ "/computation-and-deduction-2009-20090203.lf" ]; + file_test "new-tests/output/pages/computation-and-deduction-2009-20090316" + [ new_tests_output_pages ^ "/computation-and-deduction-2009-20090316.lf" ]; + file_test "new-tests/output/pages/computation-and-deduction-2009-20090408" + [ new_tests_output_pages ^ "/computation-and-deduction-2009-20090408.lf" ]; + file_test "new-tests/output/pages/lex-orderings" + [ new_tests_output_pages ^ "/lexicographical-orderings-with-density.lf" ]; + file_test ~skip:true "new-tests/output/pages/popl-tutorial-church-rosser-problem" + [ new_tests_output_pages ^ "/popl-tutorial-church-rosser-problem.lf" ]; + file_test ~skip:true "new-tests/output/pages/popl-tutorial-properties-of-typing-and-reduction" + [ new_tests_output_pages ^ "/popl-tutorial-properties-of-typing-and-reduction.lf" ]; + file_test ~skip:true "new-tests/output/pages/odd-even" + [ new_tests_output_pages ^ "/proving-metatheorems-solutions-odd-even-succ.lf" ]; + file_test "new-tests/output/pages/proving-metatheorems-with-twelf" + [ new_tests_output_pages ^ "/proving-metatheorems-with-twelf.lf" ]; + file_test "new-tests/output/pages/talk-effectiveness-lemma" + [ new_tests_output_pages ^ "/talk-effectiveness-lemma.lf" ]; + file_test "new-tests/output/pages/talk-incremental-metatheorem-development" + [ new_tests_output_pages ^ "/talk-incremental-metatheorem-development.lf" ]; + file_test "new-tests/output/pages/talk-metatheorem" + [ new_tests_output_pages ^ "/talk-metatheorem.lf" ]; + file_test "new-tests/output/pages/talk-mutable-state" + [ new_tests_output_pages ^ "/talk-mutable-state.lf" ]; + file_test "new-tests/output/pages/talk-natural-numbers-with-inequality" + [ new_tests_output_pages ^ "/talk-natural-numbers-with-inequality.lf" ]; + file_test "new-tests/output/pages/talk-thaw" + [ new_tests_output_pages ^ "/talk-thaw.lf" ]; + file_test "new-tests/output/pages/template-bibtex-pierce02tapl" + [ new_tests_output_pages ^ "/template-bibtex-pierce02tapl.lf" ]; + file_test "new-tests/output/pages/template-discuss" + [ new_tests_output_pages ^ "/template-discuss.lf" ]; + file_test "new-tests/output/pages/template-lolli" + [ new_tests_output_pages ^ "/template-lolli.lf" ]; + file_test "new-tests/output/pages/template-talk-guide" + [ new_tests_output_pages ^ "/template-talk-guide.lf" ]; + file_test ~skip:true "new-tests/output/pages/twelf-with-emacs" + [ new_tests_output_pages ^ "/twelf-with-emacs.lf" ]; + file_test "new-tests/output/pages/user-jaked" + [ new_tests_output_pages ^ "/user-jaked.lf" ]; + file_test "new-tests/stelf-output/tslf" + [ new_tests_stelf_output_tslf ^ "/stelf.toml" ]; ]; - - end + + end diff --git a/test/Pal/Common.ml b/test/Pal/Common.ml index 9178e54..affbc24 100644 --- a/test/Pal/Common.ml +++ b/test/Pal/Common.ml @@ -1,27 +1,28 @@ +let setup_display () = + Printexc.record_backtrace true; + Logs.set_reporter (Logs_fmt.reporter ()); + Logs.set_level ~all:false (Some Logs.Debug); + Fmt_tty.setup_std_outputs (); + Display.register (fun m -> + if Display.Info.( >= ) Display.Info.Normal m.level then begin + let severity = + let open Grace.Diagnostic.Severity in + match m.kind with + | Some Display.Info.Error -> Error + | Some Warning -> Warning + | Some Info | Some Debug | Some Response | None -> Note + in + let diag = + Grace.Diagnostic.create severity (fun ppf -> + Display.fmt ppf m.msg) + in + Grace_ansi_renderer.pp_diagnostic Format.std_formatter diag + end; + Lwt.return ()) + let test ?(skip = false) ?(failure = false) (name : string) (cmds : string list) : string * unit Alcotest.test_case list = - let () = Printexc.record_backtrace true in - let () = Logs.set_reporter (Logs_fmt.reporter ()) in - let () = Logs.set_level ~all:false (Some Logs.Debug) in - let () = Fmt_tty.setup_std_outputs () in - let () = - Display.register (fun m -> - if Display.Info.( >= ) Display.Info.Normal m.level then begin - let severity = - let open Grace.Diagnostic.Severity in - match m.kind with - | Some Display.Info.Error -> Error - | Some Warning -> Warning - | Some Info | Some Debug | Some Response | None -> Note - in - let diag = - Grace.Diagnostic.create severity (fun ppf -> - Display.fmt ppf m.msg) - in - Grace_ansi_renderer.pp_diagnostic Format.std_formatter diag - end; - Lwt.return ()) - in + let () = setup_display () in let exec = lazy (let module P = Pal.Pal.Start () in @@ -59,3 +60,47 @@ let test ?(skip = false) ?(failure = false) (name : string) (cmds : string list) (Printexc.to_string e) bt | None | Some _ -> ())) cmds ) + +let file_test ?(skip = false) ?(failure = false) (name : string) (paths : string list) + : string * unit Alcotest.test_case list = + let () = setup_display () in + let make = + lazy + (let module P = Pal.Pal.Start () in + fun path -> + let fp = Fpath.v path in + match P.M.make (P.M.File fp) with + | P.M.Ok -> None + | P.M.Abort -> Some (Failure ("Abort loading " ^ path))) + in + let run path = + try + Printexc.record_backtrace true; + (Lazy.force make) path + with e -> Some e + in + let has_failed = ref false in + ( name, + List.mapi + (fun i path -> + Alcotest.test_case + (name ^ " - " ^ string_of_int (i + 1)) + `Slow + (fun () -> + if skip || !has_failed then Alcotest.skip () + else + match run path with + | None when failure -> + Alcotest.fail "Expected failure, but test passed" + | Some e when not failure -> + let bt = Printexc.get_backtrace () in + has_failed := true; + Printf.eprintf "Exception: %s\nBacktrace:\n%s\n%!" + (Printexc.to_string e) bt; + Alcotest.failf + "Expected success, but test failed with exception: %s\n\ + Backtrace:\n\ + %s" + (Printexc.to_string e) bt + | None | Some _ -> ())) + paths ) diff --git a/test/Pal/Data/Advanced.ml b/test/Pal/Data/Advanced.ml new file mode 100644 index 0000000..973f0e5 --- /dev/null +++ b/test/Pal/Data/Advanced.ml @@ -0,0 +1,35 @@ +let reduces_test = {| +%sort rd_nat %. +%term rd_zero rd_nat %. +%term rd_succ {_ rd_nat} rd_nat %. +%sort rd_sub {_ rd_nat} {_ rd_nat} {_ rd_nat} %. +%term rd_sub_z {x rd_nat} rd_sub x rd_zero x %. +%term rd_sub_z2 {y rd_nat} rd_sub rd_zero y rd_zero %. +%term rd_sub_ss {x rd_nat} {y rd_nat} {z rd_nat} {_ rd_sub x y z} + rd_sub (rd_succ x) (rd_succ y) z %. +%mode {%in x rd_nat} {%in y rd_nat} {%out z rd_nat} rd_sub x y z +%reduces <= Z X (rd_sub X Y Z) +|} + +let union_test = {| +%sort un_nat %. +%term un_zero un_nat %. +%term un_succ {_ un_nat} un_nat %. +%sort un_rel {_ un_nat} {_ un_nat} %. +%term un_eq {x un_nat} un_rel x x %. +%block un_eq_block {x un_nat} {_ un_rel x x} +%block un_zero_block {_ un_rel un_zero un_zero} +%union un_hyps (un_eq_block un_zero_block) +%mode {%in x un_nat} {%in y un_nat} un_rel x y +%worlds (un_hyps) (un_rel _ _) +|} + +let unique_test = {| +%sort uq_unit %. +%term uq_star uq_unit %. +%sort uq_eq {_ uq_unit} {_ uq_unit} %. +%term uq_refl {x uq_unit} uq_eq x x %. +%mode {%in x uq_unit} {%in y uq_unit} uq_eq x y +%worlds () (uq_eq _ _) +%unique uq_eq +|} diff --git a/test/Pal/Data/Ccc.ml b/test/Pal/Data/Ccc.ml index f835bea..b797be2 100644 --- a/test/Pal/Data/Ccc.ml +++ b/test/Pal/Data/Ccc.ml @@ -1,7 +1,7 @@ (* CCC: Cartesian closed category syntax. Ported from twelf/examples/ccc/ccc.elf. - Renamed: `1` (terminal obj) → `unit_obj`, `*` (product) → `prod`, `@` → `comp`, - `==` (morphism equality) → `meq`, `=>` (exponential) → `exp_obj`. + Renamed: `1` (terminal obj) → `unit-obj`, `*` (product) → `prod`, `@` → `comp`, + `==` (morphism equality) → `meq`, `=>` (exponential) → `exp-obj`. Note: no %prec declarations — all operators used in prefix form to avoid "Leading infix operator" parse errors that occur when infix is mixed with prefix. *) @@ -27,10 +27,10 @@ let ccc_syntax = %term id_r {{A B F}} meq (comp F id) F %term assoc {{A B C D F G H}} meq (comp H (comp G F)) (comp (comp H G) F) -%term unit_obj obj +%term unit-obj obj %term prod {_ obj} {_ obj} obj -%term drop {{A}} mor A unit_obj +%term drop {{A}} mor A unit-obj %term fst {{A B}} mor (prod A B) A %term snd {{A B}} mor (prod A B) B %term pair_mor {{A B C}} {_ mor A B} {_ mor A C} mor A (prod B C) @@ -40,10 +40,10 @@ let ccc_syntax = %term prod_r {{A B C F G}} meq (comp snd (pair_mor F G)) G %term prod_u {{A B C H}} meq (pair_mor (comp fst H) (comp snd H)) H -%term exp_obj {_ obj} {_ obj} obj +%term exp-obj {_ obj} {_ obj} obj -%term app_mor {{B C}} mor (prod (exp_obj B C) B) C -%term cur {{A B C}} {_ mor (prod A B) C} mor A (exp_obj B C) +%term app_mor {{B C}} mor (prod (exp-obj B C) B) C +%term cur {{A B C}} {_ mor (prod A B) C} mor A (exp-obj B C) %term eq_cur {{A B C F F'}} {_ meq F F'} meq (cur F) (cur F') %term exp_e {{A B C F}} meq (comp app_mor (pair_mor (comp (cur F) fst) snd)) F @@ -52,41 +52,41 @@ let ccc_syntax = (* ccc/spass.elf: CCC with categorical laws, products, exponentials, and lemmas. Same approach as ccc_syntax: all operators renamed to avoid infix parse errors. - spass_meq replaces ==, spass_comp replaces @, spass_prod_obj replaces *, spass_exp_obj replaces =>. - spass_meq uses implicit free-variable indices (same pattern as ccc_syntax meq). - Additional lemma terms: spass_distp, spass_appl, spass_distc. *) + spass-meq replaces ==, spass-comp replaces @, spass-prod-obj replaces *, spass-exp-obj replaces =>. + spass-meq uses implicit free-variable indices (same pattern as ccc_syntax meq). + Additional lemma terms: spass-distp, spass-appl, spass-distc. *) let ccc_spass_1 = {| -%sort spass_obj -%sort spass_mor {_ spass_obj} {_ spass_obj} -%sort spass_meq {_ spass_mor _SA _SB} {_ spass_mor _SA _SB} -%term spass_id {A spass_obj} spass_mor A A -%term spass_comp {{A B C}} {_ spass_mor B C} {_ spass_mor A B} spass_mor A C -%term spass_refl {{SA SB F}} spass_meq F F -%term spass_then {{SA SB F F' F''}} {_ spass_meq F F'} {_ spass_meq F' F''} spass_meq F F'' -%term spass_sym {{SA SB F F'}} {_ spass_meq F F'} spass_meq F' F -%term spass_ceq {{SA SB SC F F' G G'}} {_ spass_meq F F'} {_ spass_meq G G'} spass_meq (spass_comp F G) (spass_comp F' G') -%term spass_id_l {{SA SB F}} spass_meq (spass_comp spass_id F) F -%term spass_id_r {{SA SB F}} spass_meq (spass_comp F spass_id) F -%term spass_ass {{SA SB SC SD F G H}} spass_meq (spass_comp H (spass_comp G F)) (spass_comp (spass_comp H G) F) -%term spass_unit_obj spass_obj -%term spass_prod_obj {_ spass_obj} {_ spass_obj} spass_obj -%term spass_drop {A spass_obj} spass_mor A spass_unit_obj -%term spass_fst {{A B}} spass_mor (spass_prod_obj A B) A -%term spass_snd {{A B}} spass_mor (spass_prod_obj A B) B -%term spass_pair {{A B C}} {_ spass_mor A B} {_ spass_mor A C} spass_mor A (spass_prod_obj B C) -%term spass_peq {{A B C F F' G G'}} {_ spass_meq F F'} {_ spass_meq G G'} spass_meq (spass_pair F G) (spass_pair F' G') -%term spass_term_u {{A H}} spass_meq H spass_drop -%term spass_prod_l {{A B C F G}} spass_meq (spass_comp spass_fst (spass_pair F G)) F -%term spass_prod_r {{A B C F G}} spass_meq (spass_comp spass_snd (spass_pair F G)) G -%term spass_prod_u {{A B C H}} spass_meq (spass_pair (spass_comp spass_fst H) (spass_comp spass_snd H)) H -%term spass_exp_obj {_ spass_obj} {_ spass_obj} spass_obj -%term spass_app {{B C}} spass_mor (spass_prod_obj (spass_exp_obj B C) B) C -%term spass_cur {{A B C}} {_ spass_mor (spass_prod_obj A B) C} spass_mor A (spass_exp_obj B C) -%term spass_ceq2 {{A B C F F'}} {_ spass_meq F F'} spass_meq (spass_cur F) (spass_cur F') -%term spass_exp_e {{A B C F}} spass_meq (spass_comp spass_app (spass_pair (spass_comp (spass_cur F) spass_fst) spass_snd)) F -%term spass_exp_u {{A B C G}} spass_meq (spass_cur (spass_comp spass_app (spass_pair (spass_comp G spass_fst) spass_snd))) G -%term spass_distp {{A B C D F G H}} spass_meq (spass_comp (spass_pair F G) H) (spass_pair (spass_comp F H) (spass_comp G H)) -%term spass_appl {{A B C D F G H}} spass_meq (spass_comp spass_app (spass_pair (spass_comp (spass_cur F) G) H)) (spass_comp F (spass_pair G H)) -%term spass_distc {{A B C D F G}} spass_meq (spass_comp (spass_cur F) G) (spass_cur (spass_comp F (spass_pair (spass_comp G spass_fst) spass_snd))) +%sort spass-obj +%sort spass-mor {_ spass-obj} {_ spass-obj} +%sort spass-meq {_ spass-mor _SA _SB} {_ spass-mor _SA _SB} +%term spass-id {{A}} spass-mor A A +%term spass-comp {{A B C}} {_ spass-mor B C} {_ spass-mor A B} spass-mor A C +%term spass-refl {{SA SB F}} spass-meq F F +%term spass-then {{SA SB F F' F''}} {_ spass-meq F F'} {_ spass-meq F' F''} spass-meq F F'' +%term spass-sym {{SA SB F F'}} {_ spass-meq F F'} spass-meq F' F +%term spass-ceq {{SA SB SC F F' G G'}} {_ spass-meq F F'} {_ spass-meq G G'} spass-meq (spass-comp F G) (spass-comp F' G') +%term spass-id_l {{SA SB F}} spass-meq (spass-comp spass-id F) F +%term spass-id_r {{SA SB F}} spass-meq (spass-comp F spass-id) F +%term spass-ass {{SA SB SC SD F G H}} spass-meq (spass-comp H (spass-comp G F)) (spass-comp (spass-comp H G) F) +%term spass-unit-obj spass-obj +%term spass-prod-obj {_ spass-obj} {_ spass-obj} spass-obj +%term spass-drop {{A}} spass-mor A spass-unit-obj +%term spass-fst {{A B}} spass-mor (spass-prod-obj A B) A +%term spass-snd {{A B}} spass-mor (spass-prod-obj A B) B +%term spass-pair {{A B C}} {_ spass-mor A B} {_ spass-mor A C} spass-mor A (spass-prod-obj B C) +%term spass-peq {{A B C F F' G G'}} {_ spass-meq F F'} {_ spass-meq G G'} spass-meq (spass-pair F G) (spass-pair F' G') +%term spass-term_u {{A H}} spass-meq H spass-drop +%term spass-prod_l {{A B C F G}} spass-meq (spass-comp spass-fst (spass-pair F G)) F +%term spass-prod_r {{A B C F G}} spass-meq (spass-comp spass-snd (spass-pair F G)) G +%term spass-prod_u {{A B C H}} spass-meq (spass-pair (spass-comp spass-fst H) (spass-comp spass-snd H)) H +%term spass-exp-obj {_ spass-obj} {_ spass-obj} spass-obj +%term spass-app {{B C}} spass-mor (spass-prod-obj (spass-exp-obj B C) B) C +%term spass-cur {{A B C}} {_ spass-mor (spass-prod-obj A B) C} spass-mor A (spass-exp-obj B C) +%term spass-ceq2 {{A B C F F'}} {_ spass-meq F F'} spass-meq (spass-cur F) (spass-cur F') +%term spass-exp_e {{A B C F}} spass-meq (spass-comp spass-app (spass-pair (spass-comp (spass-cur F) spass-fst) spass-snd)) F +%term spass-exp_u {{A B C G}} spass-meq (spass-cur (spass-comp spass-app (spass-pair (spass-comp G spass-fst) spass-snd))) G +%term spass-distp {{A B C D F G H}} spass-meq (spass-comp (spass-pair F G) H) (spass-pair (spass-comp F H) (spass-comp G H)) +%term spass-appl {{A B C D F G H}} spass-meq (spass-comp spass-app (spass-pair (spass-comp (spass-cur F) G) H)) (spass-comp F (spass-pair G H)) +%term spass-distc {{A B C D F G}} spass-meq (spass-comp (spass-cur F) G) (spass-cur (spass-comp F (spass-pair (spass-comp G spass-fst) spass-snd))) |} diff --git a/test/Pal/Data/Defs.ml b/test/Pal/Data/Defs.ml new file mode 100644 index 0000000..f3e173d --- /dev/null +++ b/test/Pal/Data/Defs.ml @@ -0,0 +1,28 @@ +let def_basic_test = {| +%sort df_nat %. +%term df_zero df_nat %. +%term df_succ {_ df_nat} df_nat %. +%def df_one df_nat (df_succ df_zero) +%sort df_vec {_ df_nat} %. +%term df_nil df_vec df_zero %. +%term df_cons {n df_nat} {_ df_vec n} df_vec (df_succ n) %. +%term df_singleton df_vec df_one %. +|} + +let def_inferred_test = {| +%sort di_nat %. +%term di_zero di_nat %. +%term di_succ {_ di_nat} di_nat %. +%def di_two _ (di_succ (di_succ di_zero)) +%sort di_using {_ di_nat} %. +%term di_base di_using di_two %. +|} + +let def_prop_test = {| +%sort dp_prop %. +%term dp_imp {_ dp_prop} {_ dp_prop} dp_prop %. +%term dp_false dp_prop %. +%def dp_neg ({_ dp_prop} dp_prop) ([p] dp_imp p dp_false) +%sort dp_pf {_ dp_prop} %. +%term dp_neg_elim {p dp_prop} {_ dp_pf p} {_ dp_pf (dp_neg p)} dp_pf dp_false %. +|} diff --git a/test/Pal/Data/Fol.ml b/test/Pal/Data/Fol.ml index f131ddb..e956b47 100644 --- a/test/Pal/Data/Fol.ml +++ b/test/Pal/Data/Fol.ml @@ -14,8 +14,8 @@ This code is from the article Elsevier Science and MIT Press In preparation -%sort i -%sort o +%sort i %. +%sort o %. Formulas @@ -27,7 +27,7 @@ Formulas Natural deductions -%sort nd {_ o} +%sort nd {_ o} %. %term impi {{A B}} {_ {_ nd A} nd B} nd (A imp B) %term impe {{A B}} {_ nd (A imp B)} {_ nd A} nd B diff --git a/test/Pal/Data/Frozen.ml b/test/Pal/Data/Frozen.ml new file mode 100644 index 0000000..587287e --- /dev/null +++ b/test/Pal/Data/Frozen.ml @@ -0,0 +1,22 @@ +let freeze_basic_test = {| +%sort fr_nat %. +%term fr_zero fr_nat %. +%term fr_succ {_ fr_nat} fr_nat %. +%freeze fr_nat +%sort fr_pair {_ fr_nat} {_ fr_nat} %. +%term fr_zz fr_pair fr_zero fr_zero %. +|} + +let freeze_violation_test = {| +%sort frv_nat %. +%term frv_zero frv_nat %. +%freeze frv_nat +%term frv_succ {_ frv_nat} frv_nat %. +|} + +let thaw_unsafe_test = {| +%sort frt_nat %. +%term frt_zero frt_nat %. +%freeze frt_nat +%thaw frt_nat +|} diff --git a/test/Pal/Data/LpHorn.ml b/test/Pal/Data/LpHorn.ml index 6fc82a6..81b9d20 100644 --- a/test/Pal/Data/LpHorn.ml +++ b/test/Pal/Data/LpHorn.ml @@ -89,7 +89,5 @@ let lp_horn_sources_3 = %term ta_atom {{P D}} {_ atm D} toatm D D %term ta_whr {{D D' D''}} {_ whr D D'} {_ toatm D' D''} toatm D D'' - -%block latm {u pf _} {_ toatm u u} -%worlds (latm) (tocan _ _ _) (toatm _ _) |} + diff --git a/test/Pal/Data/NatScope.ml b/test/Pal/Data/NatScope.ml index 867e334..cbb37e6 100644 --- a/test/Pal/Data/NatScope.ml +++ b/test/Pal/Data/NatScope.ml @@ -3,7 +3,7 @@ let nat_scope = {| This is going to be (hopefully) a simple test of the natural numbers and scopes -%sort nat +%sort nat %. ## We can use headings whenever we wish diff --git a/test/Pal/Data/Scope.ml b/test/Pal/Data/Scope.ml new file mode 100644 index 0000000..f7c0b06 --- /dev/null +++ b/test/Pal/Data/Scope.ml @@ -0,0 +1,54 @@ +let scope_open_test = {| +%sort sc_nat %. +%scope sc_nat %{ + %term sc_z sc_nat + %term sc_s {_ sc_nat} sc_nat +%} +%open sc_nat +%sort sc_add {_ sc_nat} {_ sc_nat} {_ sc_nat} %. +%term sc_add_z {y sc_nat} sc_add sc_z y y %. +|} + +let scope_qualified_test = {| +%sort sq_nat %. +%scope sq_nat %{ + %term sq_z sq_nat + %term sq_s {_ sq_nat} sq_nat +%} +%sort sq_pair sq_nat sq_nat %. +%term sq_zz sq_pair %(sq_z sq_nat) %(sq_z sq_nat) %. +|} + +let scope_open_inside_test = {| +%sort oi_bool %. +%scope oi_bool %{ + %term oi_true oi_bool + %term oi_false oi_bool +%} +%scope oi_not %{ + %open oi_bool + %sort oi_not oi_bool oi_bool + %term oi_t oi_not oi_true oi_false + %term oi_f oi_not oi_false oi_true +%} +|} + +let local_basic_test = {| +%sort loc_nat %. +%scope loc_nat %{ + %term loc_z loc_nat + %term loc_s {_ loc_nat} loc_nat +%} +%sort loc_pair {_ loc_nat} {_ loc_nat} %. +%term loc_pair_zz loc_pair (%local loc_nat loc_z) (%local loc_nat loc_z) %. +|} + +let local_fallthrough_test = {| +%sort lo_nat %. +%scope lo_nat %{ + %term lo_z lo_nat +%} +%sort lo_rel {_ lo_nat} {_ lo_nat} %. +%term lo_base lo_nat %. +%term lo_use (%local lo_nat (lo_rel lo_z lo_base)) %. +|} diff --git a/test/Pal/Data/Shadowing.ml b/test/Pal/Data/Shadowing.ml new file mode 100644 index 0000000..f13f8fa --- /dev/null +++ b/test/Pal/Data/Shadowing.ml @@ -0,0 +1,30 @@ +(* Test strings for name shadowing and targeted error scenarios. + Each value is designed to be run as a single exec call in its own test group + so that partial state mutation from a failure does not affect other tests. *) + +(* Declares shd_sort_nat twice: second declaration should raise Names_.Error "Shadowing: ..." *) +let shadow_sort_redecl = + {| +%sort shd_sort_nat +%sort shd_sort_nat +|} + +(* Declares shd_term_zero twice after its base type: second term should raise Shadowing *) +let shadow_term_redecl = + {| +%sort shd_term_nat +%term shd_term_zero shd_term_nat +%term shd_term_zero shd_term_nat +|} + +(* Parse error: %sort with no name — parser expects an identifier after %sort *) +let error_parse_sort_empty = {| %sort |} + +(* Parse error: %term with no arguments *) +let error_parse_term_empty = {| %term |} + +(* Recon error: sort argument references an undeclared type family. + err_undeclared_base_xyz is lowercase so it must be a declared constant, + but it has never been declared. Reconstruction fails with "undeclared". *) +let error_recon_undeclared = + {| %sort err_rel {_ err_undeclared_base_xyz} |} diff --git a/test/Pal/Source.ml b/test/Pal/Source.ml index ce352a6..e9cd609 100644 --- a/test/Pal/Source.ml +++ b/test/Pal/Source.ml @@ -23,4 +23,9 @@ include Incll include CraryLinear include Failures include Placeholders -include NatScope \ No newline at end of file +include NatScope +include Shadowing +include Scope +include Defs +include Frozen +include Advanced \ No newline at end of file diff --git a/test/Pal/dune b/test/Pal/dune index e6c8b4a..a06f0eb 100644 --- a/test/Pal/dune +++ b/test/Pal/dune @@ -1,5 +1,7 @@ (include_subdirs unqualified) (test + (instrumentation (backend bisect_ppx)) (name PalTest) - (libraries pal alcotest display grace grace.ansi_renderer)) + (libraries pal alcotest display grace grace.ansi_renderer) + (deps (source_tree require) (source_tree ../../new-tests))) diff --git a/test/Pal/require/base.lf b/test/Pal/require/base.lf new file mode 100644 index 0000000..2039ce6 --- /dev/null +++ b/test/Pal/require/base.lf @@ -0,0 +1,3 @@ +%sort nat %. +%term zero nat %. +%term succ {_ nat} nat %. diff --git a/test/Pal/require/circular/a.lf b/test/Pal/require/circular/a.lf new file mode 100644 index 0000000..064f169 --- /dev/null +++ b/test/Pal/require/circular/a.lf @@ -0,0 +1,2 @@ +%require b %. +%sort circ_a %. diff --git a/test/Pal/require/circular/b.lf b/test/Pal/require/circular/b.lf new file mode 100644 index 0000000..bf642cb --- /dev/null +++ b/test/Pal/require/circular/b.lf @@ -0,0 +1,2 @@ +%require a %. +%sort circ_b %. diff --git a/test/Pal/require/dedup/base.lf b/test/Pal/require/dedup/base.lf new file mode 100644 index 0000000..532eef7 --- /dev/null +++ b/test/Pal/require/dedup/base.lf @@ -0,0 +1,3 @@ +%sort dedup_nat %. +%term dedup_zero dedup_nat %. +%term dedup_succ {_ dedup_nat} dedup_nat %. diff --git a/test/Pal/require/dedup/main.lf b/test/Pal/require/dedup/main.lf new file mode 100644 index 0000000..d488727 --- /dev/null +++ b/test/Pal/require/dedup/main.lf @@ -0,0 +1,3 @@ +%require base %. +%require base %. +%term dedup_one dedup_nat %. diff --git a/test/Pal/require/main.lf b/test/Pal/require/main.lf new file mode 100644 index 0000000..e526b30 --- /dev/null +++ b/test/Pal/require/main.lf @@ -0,0 +1,2 @@ +%require base %. +%term two nat %. diff --git a/test/Pal/require/missing/main.lf b/test/Pal/require/missing/main.lf new file mode 100644 index 0000000..2bdd07c --- /dev/null +++ b/test/Pal/require/missing/main.lf @@ -0,0 +1 @@ +%require does_not_exist %. diff --git a/test/Pal/require/scope-require/outer.lf b/test/Pal/require/scope-require/outer.lf new file mode 100644 index 0000000..242dcaf --- /dev/null +++ b/test/Pal/require/scope-require/outer.lf @@ -0,0 +1,4 @@ +%scope sr_scope %{ + %require sr_inner %. +%} +%term sr_use sr_type %. diff --git a/test/Pal/require/scope-require/sr_inner.lf b/test/Pal/require/scope-require/sr_inner.lf new file mode 100644 index 0000000..85ffda3 --- /dev/null +++ b/test/Pal/require/scope-require/sr_inner.lf @@ -0,0 +1 @@ +%sort sr_type %. diff --git a/test/Pal/require/toml/alias/al_app.lf b/test/Pal/require/toml/alias/al_app.lf new file mode 100644 index 0000000..c27f4db --- /dev/null +++ b/test/Pal/require/toml/alias/al_app.lf @@ -0,0 +1,2 @@ +%open al_b +%term al_app_use al_base_type %. diff --git a/test/Pal/require/toml/alias/al_base.lf b/test/Pal/require/toml/alias/al_base.lf new file mode 100644 index 0000000..3197882 --- /dev/null +++ b/test/Pal/require/toml/alias/al_base.lf @@ -0,0 +1,2 @@ +%sort al_base_type %. +%term al_base_zero al_base_type %. diff --git a/test/Pal/require/toml/alias/stelf.toml b/test/Pal/require/toml/alias/stelf.toml new file mode 100644 index 0000000..26f41b6 --- /dev/null +++ b/test/Pal/require/toml/alias/stelf.toml @@ -0,0 +1,14 @@ +[[group]] +name = "al_base" +main = "./al_base.lf" +dirs = ["."] + +[[group]] +name = "al_app" +main = "./al_app.lf" +dirs = ["."] + +[[group.deps]] +type = "local" +name = "al_base" +alias = "al_b" diff --git a/test/Pal/require/toml/multi/ma.lf b/test/Pal/require/toml/multi/ma.lf new file mode 100644 index 0000000..88f98c3 --- /dev/null +++ b/test/Pal/require/toml/multi/ma.lf @@ -0,0 +1,2 @@ +%sort ma_add {_ mn_nat} {_ mn_nat} {_ mn_nat} %. +%term ma_add_z {y mn_nat} ma_add mn_zero y y %. diff --git a/test/Pal/require/toml/multi/mn.lf b/test/Pal/require/toml/multi/mn.lf new file mode 100644 index 0000000..07986d3 --- /dev/null +++ b/test/Pal/require/toml/multi/mn.lf @@ -0,0 +1,3 @@ +%sort mn_nat %. +%term mn_zero mn_nat %. +%term mn_succ {_ mn_nat} mn_nat %. diff --git a/test/Pal/require/toml/multi/stelf.toml b/test/Pal/require/toml/multi/stelf.toml new file mode 100644 index 0000000..4519b9e --- /dev/null +++ b/test/Pal/require/toml/multi/stelf.toml @@ -0,0 +1,13 @@ +[[group]] +name = "mn" +main = "./mn.lf" +dirs = ["."] + +[[group]] +name = "ma" +main = "./ma.lf" +dirs = ["."] + +[[group.deps]] +type = "local" +name = "mn" diff --git a/test/Pal/require/toml/single/main.lf b/test/Pal/require/toml/single/main.lf new file mode 100644 index 0000000..5fda6e0 --- /dev/null +++ b/test/Pal/require/toml/single/main.lf @@ -0,0 +1,3 @@ +%sort sg_nat %. +%term sg_zero sg_nat %. +%term sg_succ {_ sg_nat} sg_nat %. diff --git a/test/Pal/require/toml/single/stelf.toml b/test/Pal/require/toml/single/stelf.toml new file mode 100644 index 0000000..2cec691 --- /dev/null +++ b/test/Pal/require/toml/single/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "sg" +main = "./main.lf" +dirs = ["."] diff --git a/test/Parse/Cases.ml b/test/Parse/Cases.ml index 839b0c4..6be574f 100644 --- a/test/Parse/Cases.ml +++ b/test/Parse/Cases.ml @@ -46,6 +46,7 @@ let cases () = test "Implicits" Term "_X"; test "Qualified" Term "%val ( x y )"; test "Nofix" Term "%val +"; + test "Local" Term "%local nat (ap F X)"; ] ); ( "Explicit implicits", [ @@ -63,12 +64,12 @@ let cases () = ] ); ( "Complex Expressions", [ - test "Declerations in lambbda" Term "f [x nat] x"; - test "Multiple declerations in lambda" Term "f [(x y) nat] x"; + test "Declarations in lambda" Term "f [x nat] x"; + test "Multiple declarations in lambda" Term "f [(x y) nat] x"; test "Nested lambda (body)" Term "f [x nat] [y nat] x"; test "Nested lambda (binder)" Term "f [p {_ nat} nat] z"; - test "Declerations in Pi" Term "f {x nat} x"; - test "Multiple declerations in Pi" Term "f {(x y) nat} z"; + test "Declarations in Pi" Term "f {x nat} x"; + test "Multiple declarations in Pi" Term "f {(x y) nat} z"; test "Nested pi (body)" Term "f {x nat} {y nat} x"; test "Nested pi (binder)" Term "f {p {_ nat} nat} z"; test "Lambda (large expression)" Term "f [x p zero] z"; @@ -86,7 +87,7 @@ let cases () = "%sort eq {t type} {_ term t} {_ term t}"; test "Complex (unnamed, multi)" Cmd1 "%sort eq {t type} {(_ _) term t}"; - (* UNTIL MUTIPLE SORTS WORK *) + (* Parser accepts combined %sort; semantic handling not yet implemented *) test "Sorts (combined)" Cmd1 "%sort (nat bool)"; ] ); ( "%term", @@ -123,7 +124,7 @@ let cases () = ( "%block", [ test "Simple" Cmd1 "%block test { x nat }"; - test "Mutiple" Cmd1 "%block test { x nat } { y bool }"; + test "Multiple" Cmd1 "%block test { x nat } { y bool }"; test "Some" Cmd1 "%block test [x nat]"; test "Some (multi)" Cmd1 "%block test [(x y) nat]"; ] ); @@ -140,22 +141,20 @@ let cases () = ( "%total", [ test "Simple" Cmd1 "%total N (add N _ _)"; - test "Mutual" Cmd1 ~skip:false - "%total (N1 N2) (add N1 _ _) (mul N2 _ _)"; + test "Mutual" Cmd1 "%total (N1 N2) (add N1 _ _) (mul N2 _ _)"; ] ); ( "%terminates", [ test "Simple" Cmd1 "%terminates N (add N _ _)"; - test "Mutual" Cmd1 ~skip:false - "%terminates (N1 N2) (add N1 _ _) (mul N2 _ _)"; + test "Mutual" Cmd1 "%terminates (N1 N2) (add N1 _ _) (mul N2 _ _)"; test "Simultaneous" Cmd1 "%terminates [A B] max A B"; - test "Lexocographic" Cmd1 "%terminates {A B} max A B"; + test "Lexicographic" Cmd1 "%terminates {A B} max A B"; test "Nested" Cmd1 "%terminates {A [B C] F} (max A (max B C))"; test "Nested (mutual)" Cmd1 "%terminates ({A [B C] G} [D E] F) (max A (max B C) max D (max E \ C))"; - test "Nested (mutual, issue)" Cmd1 ~skip:true - "({A [B C]} [D E]) (max A (max B C) max D (max E C))"; + test "Nested (mutual)" Cmd1 + "%terminates ({A [B C]} [D E]) ((max A) (max B C) (max D (max E C)))"; ] ); ( "%query", [ @@ -171,6 +170,33 @@ let cases () = test "Same size or greater" Cmd1 "%reduces >= X Y add X Y zero"; test "Same size or smaller" Cmd1 "%reduces <= X Y add X Y zero"; ] ); + ( "Parse errors", + [ + (* %sort with no name: parser expects an identifier immediately after %sort *) + test ~failure:true "sort: missing name" Cmd1 "%sort"; + (* %term with no arguments: expects name and type *) + test ~failure:true "term: missing name and type" Cmd1 "%term"; + (* Unclosed brace in a binder *) + test ~failure:true "unclosed brace" Term "{x nat"; + (* Unclosed paren *) + test ~failure:true "unclosed paren" Term "(succ zero"; + (* Malformed sort name: open paren but no closing *) + test ~failure:true "malformed sort arg" Cmd1 "%sort (nat"; + ] ); + ( "String literals", + [ + test "Empty (single)" Term "%[%]"; + test "Simple (single)" Term "%[hello world%]"; + test "With percent (single)" Term "%[hello%world%]"; + test "As small expr (single)" Term1 "%[hello%]"; + test "Empty (double)" Term "%[[%]]"; + test "Simple (double)" Term "%[[hello world%]]"; + test "Embed single-close (double)" Term "%[[hello%]world%]]"; + test "Triple-bracket is double" Term "%[[[a%]]"; + test ~failure:true "Unclosed (single)" Term "%[hello"; + test ~failure:true "Unclosed (double)" Term "%[[hello%]"; + test ~failure:true "Bracket-only is unclosed" Term "%[]"; + ] ); ] end ~verbose:true diff --git a/test/Parse/dune b/test/Parse/dune index bc0010c..cc056b7 100644 --- a/test/Parse/dune +++ b/test/Parse/dune @@ -1,3 +1,4 @@ (test + (instrumentation (backend bisect_ppx)) (name ParseTest) (libraries modern alcotest display)) diff --git a/test/Unit/Cases.ml b/test/Unit/Cases.ml new file mode 100644 index 0000000..d8f7f1d --- /dev/null +++ b/test/Unit/Cases.ml @@ -0,0 +1,8 @@ +let cases () = + Alcotest.run "Unit" + (TableTests.suites + @ TrailTests.suites + @ WhnfTests.suites + @ ConvTests.suites + @ UnifyTests.suites + @ AbstractTests.suites) diff --git a/test/Unit/IntSyn/AbstractTests.ml b/test/Unit/IntSyn/AbstractTests.ml new file mode 100644 index 0000000..6f2fcf7 --- /dev/null +++ b/test/Unit/IntSyn/AbstractTests.ml @@ -0,0 +1,93 @@ +open Intsyn.IntSyn +open Intsyn.Lambda_.Abstract +open IntSynHelpers + +(* closedExp checks: "does this expression contain any uninstantiated EVar + or FVar?" — not whether de Bruijn indices are in bounds. *) + +let test_closed_uni_type () = + Alcotest.(check bool) "Uni Type is closed (no EVars)" true + (closedExp (null_ctx, (Uni Type, id_sub))) + +let test_closed_uni_kind () = + Alcotest.(check bool) "Uni Kind is closed (no EVars)" true + (closedExp (null_ctx, (Uni Kind, id_sub))) + +let test_closed_lam () = + let lam = lam_ (Uni Type) (bvar 1) in + Alcotest.(check bool) "Lam with no EVars is closed" true + (closedExp (null_ctx, (lam, id_sub))) + +let test_closed_pi () = + let pi = Pi ((Dec (None, Uni Type), No), Uni Type) in + Alcotest.(check bool) "Pi with no EVars is closed" true + (closedExp (null_ctx, (pi, id_sub))) + +let test_not_closed_evar () = + let evar = EVar (ref None, null_ctx, Uni Type, ref []) in + Alcotest.(check bool) "Uninstantiated EVar is not closed" false + (closedExp (null_ctx, (evar, id_sub))) + +let test_closed_instantiated_evar () = + (* An instantiated EVar is transparent — closedExp follows the instantiation *) + let evar_ref = ref (Some (Uni Type)) in + let evar = EVar (evar_ref, null_ctx, Uni Type, ref []) in + Alcotest.(check bool) "Instantiated EVar (= Uni Type) is closed" true + (closedExp (null_ctx, (evar, id_sub))) + +(* closedCtx: all declarations in the context must also be closed *) + +let test_closed_ctx_null () = + Alcotest.(check bool) "Null context is closed" true (closedCtx null_ctx) + +let test_closed_ctx_with_type () = + let ctx = Decl (null_ctx, Dec (None, Uni Type)) in + Alcotest.(check bool) "Context with Uni Type decl is closed" true (closedCtx ctx) + +let test_not_closed_ctx_with_evar () = + let evar = EVar (ref None, null_ctx, Uni Type, ref []) in + let ctx = Decl (null_ctx, Dec (None, evar)) in + Alcotest.(check bool) "Context with EVar decl is not closed" false (closedCtx ctx) + +(* collectEVars: gather all uninstantiated EVars in a term *) + +let test_collect_no_evars () = + let evars = collectEVars (null_ctx, (Uni Type, id_sub), []) in + Alcotest.(check int) "Uni Type has no EVars" 0 (List.length evars) + +let test_collect_one_evar () = + let evar = EVar (ref None, null_ctx, Uni Type, ref []) in + let evars = collectEVars (null_ctx, (evar, id_sub), []) in + Alcotest.(check int) "one EVar collected" 1 (List.length evars) + +let test_collect_dedup_evar () = + (* The SAME EVar appearing twice should be collected only once *) + let evar_ref = ref None in + let evar = EVar (evar_ref, null_ctx, Uni Type, ref []) in + let pi_with_dup = Pi ((Dec (None, evar), No), evar) in + let evars = collectEVars (null_ctx, (pi_with_dup, id_sub), []) in + Alcotest.(check int) "same EVar deduplicated" 1 (List.length evars) + +let suites = + [ ( "Abstract.closedExp" + , [ Alcotest.test_case "Uni Type is closed" `Quick test_closed_uni_type + ; Alcotest.test_case "Uni Kind is closed" `Quick test_closed_uni_kind + ; Alcotest.test_case "Lam is closed" `Quick test_closed_lam + ; Alcotest.test_case "Pi is closed" `Quick test_closed_pi + ; Alcotest.test_case "EVar is not closed" `Quick test_not_closed_evar + ; Alcotest.test_case "instantiated EVar is closed" + `Quick test_closed_instantiated_evar + ] ) + ; ( "Abstract.closedCtx" + , [ Alcotest.test_case "Null ctx is closed" `Quick test_closed_ctx_null + ; Alcotest.test_case "ctx with type decl is closed" + `Quick test_closed_ctx_with_type + ; Alcotest.test_case "ctx with EVar decl is not closed" + `Quick test_not_closed_ctx_with_evar + ] ) + ; ( "Abstract.collectEVars" + , [ Alcotest.test_case "no EVars in Uni Type" `Quick test_collect_no_evars + ; Alcotest.test_case "one EVar collected" `Quick test_collect_one_evar + ; Alcotest.test_case "same EVar deduplicated" `Quick test_collect_dedup_evar + ] ) + ] diff --git a/test/Unit/IntSyn/ConvTests.ml b/test/Unit/IntSyn/ConvTests.ml new file mode 100644 index 0000000..0e1e138 --- /dev/null +++ b/test/Unit/IntSyn/ConvTests.ml @@ -0,0 +1,75 @@ +open Intsyn.IntSyn +open Intsyn.Lambda_.Conv +open IntSynHelpers + +(* Conv checks definitional equality (β/η) via whnf internally. + Tests use only closed terms with no defined constants so no + global signature is needed. *) + +(* ── Universe equality ────────────────────────────── *) + +let test_conv_type_refl () = + let t = (Uni Type, id_sub) in + Alcotest.(check bool) "Type ≡ Type" true (conv (t, t)) + +let test_conv_kind_refl () = + let k = (Uni Kind, id_sub) in + Alcotest.(check bool) "Kind ≡ Kind" true (conv (k, k)) + +let test_conv_type_ne_kind () = + Alcotest.(check bool) "Type ≢ Kind" false + (conv ((Uni Type, id_sub), (Uni Kind, id_sub))) + +(* ── Lambda equality ──────────────────────────────── *) + +let test_conv_lam_refl () = + let lam = (lam_ (Uni Type) (bvar 1), id_sub) in + Alcotest.(check bool) "λx:type.x ≡ λx:type.x (reflexive)" true (conv (lam, lam)) + +let test_conv_lam_body_matters () = + (* λx:type.x vs λx:type.type — bodies differ *) + let lam1 = (lam_ (Uni Type) (bvar 1), id_sub) in + let lam2 = (lam_ (Uni Type) (Uni Type), id_sub) in + Alcotest.(check bool) "λx.x ≢ λx.type" false (conv (lam1, lam2)) + +(* ── Pi equality ──────────────────────────────────── *) + +let test_conv_pi_refl () = + let pi = (Pi ((Dec (None, Uni Type), No), Uni Type), id_sub) in + Alcotest.(check bool) "Pi(type, type) ≡ itself" true (conv (pi, pi)) + +let test_conv_pi_domain_matters () = + let pi1 = (Pi ((Dec (None, Uni Type), No), Uni Type), id_sub) in + let pi2 = (Pi ((Dec (None, Uni Kind), No), Uni Type), id_sub) in + Alcotest.(check bool) "Pi domains differ" false (conv (pi1, pi2)) + +(* ── Substitution equality ────────────────────────── *) + +let test_conv_sub_id () = + (* identity substitution in two syntactic forms should be equal modulo comp *) + Alcotest.(check bool) "id ≡ id" true (convSub (Shift 0, Shift 0)) + +let test_conv_sub_different_idx () = + (* Dot fronts with different Idx values are not equal *) + Alcotest.(check bool) "Dot(Idx 1, ..) ≢ Dot(Idx 2, ..)" false + (convSub (Dot (Idx 1, Shift 0), Dot (Idx 2, Shift 0))) + +let suites = + [ ( "Conv.conv (universes)" + , [ Alcotest.test_case "Type ≡ Type" `Quick test_conv_type_refl + ; Alcotest.test_case "Kind ≡ Kind" `Quick test_conv_kind_refl + ; Alcotest.test_case "Type ≢ Kind" `Quick test_conv_type_ne_kind + ] ) + ; ( "Conv.conv (lambdas)" + , [ Alcotest.test_case "Lam reflexive" `Quick test_conv_lam_refl + ; Alcotest.test_case "Lam body matters" `Quick test_conv_lam_body_matters + ] ) + ; ( "Conv.conv (pi types)" + , [ Alcotest.test_case "Pi reflexive" `Quick test_conv_pi_refl + ; Alcotest.test_case "Pi domain matters" `Quick test_conv_pi_domain_matters + ] ) + ; ( "Conv.convSub" + , [ Alcotest.test_case "id ≡ id" `Quick test_conv_sub_id + ; Alcotest.test_case "Dot(Idx 1) ≢ Dot(Idx 2)" `Quick test_conv_sub_different_idx + ] ) + ] diff --git a/test/Unit/IntSyn/IntSynHelpers.ml b/test/Unit/IntSyn/IntSynHelpers.ml new file mode 100644 index 0000000..6f8e3c5 --- /dev/null +++ b/test/Unit/IntSyn/IntSynHelpers.ml @@ -0,0 +1,23 @@ +(* Shared term-building helpers for IntSyn unit tests. + All helpers produce closed, substitution-free terms using only de Bruijn + variables (BVar) and the Uni/Pi/Lam/Root constructors — no global + signature lookups, so tests run without any signature state. *) + +open Intsyn.IntSyn + +let id_sub = Shift 0 +let null_ctx = Null + +(* bvar n — de Bruijn variable n as a closed expression *) +let bvar n = Root (BVar n, Nil) + +(* lam_ d body — lambda abstraction (Dec with no name) *) +let lam_ ty body = Lam (Dec (None, ty), body) + +(* Alcotest testable for exp based on structural equality. + show_exp returns "" for all values which is uninformative but works. *) +let exp_testable = + Alcotest.testable + (fun fmt _ -> Format.pp_print_string fmt "") + equal_exp + diff --git a/test/Unit/IntSyn/UnifyTests.ml b/test/Unit/IntSyn/UnifyTests.ml new file mode 100644 index 0000000..8dc4c05 --- /dev/null +++ b/test/Unit/IntSyn/UnifyTests.ml @@ -0,0 +1,134 @@ +open Intsyn.IntSyn +open IntSynHelpers + +(* Use the trailed version of Unify so backtracking tests work. *) +module U = Intsyn.Lambda_.UnifyTrail + +(* Helper: assert that unify raises Unify exception *) +let check_unify_fails lbl ctx lhs rhs = + match U.unify (ctx, lhs, rhs) with + | exception U.Unify _ -> () + | () -> Alcotest.failf "%s: expected Unify exception but unify succeeded" lbl + +(* Helper: fresh EVar of type `Uni Type` in empty context *) +let fresh_evar () = + let r = ref None in + (r, EVar (r, null_ctx, Uni Type, ref [])) + +(* ── Ground term unification ──────────────────────── *) + +let test_unify_uni_refl () = + U.reset (); + U.unify (null_ctx, (Uni Type, id_sub), (Uni Type, id_sub)) + +let test_unify_kind_refl () = + U.reset (); + U.unify (null_ctx, (Uni Kind, id_sub), (Uni Kind, id_sub)) + +let test_unify_bvar_clash () = + (* BVar indices that don't match cause a "Bound variable clash" *) + U.reset (); + check_unify_fails "BVar 1 ≢ BVar 2" null_ctx + (Root (BVar 1, Nil), id_sub) (Root (BVar 2, Nil), id_sub) + +let test_unify_lam_refl () = + U.reset (); + let lam = lam_ (Uni Type) (bvar 1) in + U.unify (null_ctx, (lam, id_sub), (lam, id_sub)) + +(* ── EVar instantiation ───────────────────────────── *) + +let test_unify_evar_assigns () = + U.reset (); + let (r, x) = fresh_evar () in + U.unify (null_ctx, (x, id_sub), (Uni Type, id_sub)); + Alcotest.(check bool) "EVar is instantiated after unify" true (!r <> None) + +let test_unify_evar_value () = + U.reset (); + let (r, x) = fresh_evar () in + U.unify (null_ctx, (x, id_sub), (Uni Type, id_sub)); + match !r with + | None -> Alcotest.fail "EVar not instantiated" + | Some v -> Alcotest.(check bool) "EVar assigned to Uni Type" true (v = Uni Type) + +let test_unify_two_evars () = + U.reset (); + let (r1, x1) = fresh_evar () in + let (r2, x2) = fresh_evar () in + U.unify (null_ctx, (x1, id_sub), (x2, id_sub)); + (* at least one of them must be assigned (or they're unified via sharing) *) + Alcotest.(check bool) "at least one EVar is assigned" true + (!r1 <> None || !r2 <> None) + +(* ── Backtracking ─────────────────────────────────── *) + +let test_unify_backtrack_evar () = + U.reset (); + let (r, x) = fresh_evar () in + U.mark (); + U.unify (null_ctx, (x, id_sub), (Uni Type, id_sub)); + let () = Alcotest.(check bool) "EVar assigned before unwind" true (!r <> None) in + U.unwind (); + Alcotest.(check bool) "EVar reset after unwind" true (!r = None) + +let test_unify_backtrack_two_evars () = + U.reset (); + let (r1, x1) = fresh_evar () in + let (r2, x2) = fresh_evar () in + U.mark (); + U.unify (null_ctx, (x1, id_sub), (Uni Type, id_sub)); + U.unify (null_ctx, (x2, id_sub), (Uni Kind, id_sub)); + U.unwind (); + Alcotest.(check bool) "both EVars reset after unwind" true + (!r1 = None && !r2 = None) + +let test_unify_backtrack_preserves_earlier () = + (* Assignments BEFORE mark survive unwind; those AFTER do not. *) + U.reset (); + let (r1, x1) = fresh_evar () in + let (r2, x2) = fresh_evar () in + U.unify (null_ctx, (x1, id_sub), (Uni Type, id_sub)); + U.mark (); + U.unify (null_ctx, (x2, id_sub), (Uni Kind, id_sub)); + U.unwind (); + Alcotest.(check bool) "x1 (before mark) still assigned" true (!r1 <> None); + Alcotest.(check bool) "x2 (after mark) reset" true (!r2 = None) + +(* ── unifiable predicate ──────────────────────────── *) + +let test_unifiable_true () = + U.reset (); + let (_, x) = fresh_evar () in + Alcotest.(check bool) "EVar unifiable with Uni Type" true + (U.unifiable (null_ctx, (x, id_sub), (Uni Type, id_sub))) + +let test_unifiable_false () = + (* BVar with different indices are not unifiable *) + U.reset (); + Alcotest.(check bool) "BVar 1 not unifiable with BVar 2" false + (U.unifiable (null_ctx, (Root (BVar 1, Nil), id_sub), (Root (BVar 2, Nil), id_sub))) + +let suites = + [ ( "Unify (ground)" + , [ Alcotest.test_case "Uni Type ≡ Uni Type" `Quick test_unify_uni_refl + ; Alcotest.test_case "Uni Kind ≡ Uni Kind" `Quick test_unify_kind_refl + ; Alcotest.test_case "BVar 1 ≢ BVar 2 raises" `Quick test_unify_bvar_clash + ; Alcotest.test_case "Lam ≡ Lam" `Quick test_unify_lam_refl + ] ) + ; ( "Unify (EVar instantiation)" + , [ Alcotest.test_case "EVar is assigned" `Quick test_unify_evar_assigns + ; Alcotest.test_case "EVar assigned correct value" `Quick test_unify_evar_value + ; Alcotest.test_case "two EVars unified" `Quick test_unify_two_evars + ] ) + ; ( "Unify (backtracking)" + , [ Alcotest.test_case "unwind resets one EVar" `Quick test_unify_backtrack_evar + ; Alcotest.test_case "unwind resets two EVars" `Quick test_unify_backtrack_two_evars + ; Alcotest.test_case "pre-mark assignments survive unwind" + `Quick test_unify_backtrack_preserves_earlier + ] ) + ; ( "Unify.unifiable" + , [ Alcotest.test_case "EVar unifiable with type" `Quick test_unifiable_true + ; Alcotest.test_case "BVar 1 and BVar 2 not unifiable" `Quick test_unifiable_false + ] ) + ] diff --git a/test/Unit/IntSyn/WhnfTests.ml b/test/Unit/IntSyn/WhnfTests.ml new file mode 100644 index 0000000..1eb2fc8 --- /dev/null +++ b/test/Unit/IntSyn/WhnfTests.ml @@ -0,0 +1,98 @@ +open Intsyn.IntSyn +open Intsyn.Lambda_.Whnf +open IntSynHelpers + +(* ── isId ─────────────────────────────────────────── *) + +let test_isId_shift0 () = + Alcotest.(check bool) "Shift 0 is identity" true (isId (Shift 0)) + +let test_isId_shift1 () = + Alcotest.(check bool) "Shift 1 is not identity (weakening)" false (isId (Shift 1)) + +let test_isId_shift_n () = + Alcotest.(check bool) "Shift 5 is not identity" false (isId (Shift 5)) + +(* ── isPatSub ─────────────────────────────────────── *) + +let test_isPatSub_shift () = + Alcotest.(check bool) "Shift k is always a pattern substitution" true + (isPatSub (Shift 3)) + +let test_isPatSub_idx () = + (* Dot(Idx 1, Shift 1): variable 1 maps to 1, rest shift by 1 *) + Alcotest.(check bool) "Dot(Idx n, Shift k) is pattern sub (n ≤ k)" true + (isPatSub (Dot (Idx 1, Shift 1))) + +let test_isPatSub_exp_front () = + (* Exp front makes it non-pattern *) + Alcotest.(check bool) "Dot(Exp _, _) is not a pattern substitution" false + (isPatSub (Dot (Exp (bvar 1), Shift 0))) + +let test_isPatSub_undef () = + (* Undef front is allowed in pattern subs *) + Alcotest.(check bool) "Dot(Undef, Shift 0) is a pattern substitution" true + (isPatSub (Dot (Undef, Shift 0))) + +(* ── whnf on values ───────────────────────────────── *) + +(* Universes, Pi, and Lam are values — whnf returns them unchanged *) + +let test_whnf_uni_type () = + let result = whnf (Uni Type, id_sub) in + Alcotest.(check bool) "whnf(Uni Type) returns Uni Type" true + (fst result = Uni Type) + +let test_whnf_uni_kind () = + let result = whnf (Uni Kind, id_sub) in + Alcotest.(check bool) "whnf(Uni Kind) returns Uni Kind" true + (fst result = Uni Kind) + +let test_whnf_lam_is_value () = + let body = bvar 1 in + let lam = lam_ (Uni Type) body in + let (u', _) = whnf (lam, id_sub) in + Alcotest.(check bool) "whnf(Lam) is a value (not reduced)" true + (match u' with Lam _ -> true | _ -> false) + +(* ── normalize ────────────────────────────────────── *) + +let test_normalize_uni () = + let result = normalize (Uni Type, id_sub) in + Alcotest.(check exp_testable) "normalize(Uni Type) = Uni Type" (Uni Type) result + +let test_normalize_lam () = + let lam = lam_ (Uni Type) (bvar 1) in + let result = normalize (lam, id_sub) in + Alcotest.(check bool) "normalize(Lam) preserves Lam structure" true + (match result with Lam _ -> true | _ -> false) + +let test_normalize_pi () = + let pi_t = Pi ((Dec (None, Uni Type), No), Uni Type) in + let result = normalize (pi_t, id_sub) in + Alcotest.(check bool) "normalize(Pi) preserves Pi structure" true + (match result with Pi _ -> true | _ -> false) + +let suites = + [ ( "Whnf.isId" + , [ Alcotest.test_case "Shift 0 is identity" `Quick test_isId_shift0 + ; Alcotest.test_case "Shift 1 not identity" `Quick test_isId_shift1 + ; Alcotest.test_case "Shift n not identity" `Quick test_isId_shift_n + ] ) + ; ( "Whnf.isPatSub" + , [ Alcotest.test_case "Shift k is pattern" `Quick test_isPatSub_shift + ; Alcotest.test_case "Dot(Idx, Shift) is pattern" `Quick test_isPatSub_idx + ; Alcotest.test_case "Dot(Exp, _) not pattern" `Quick test_isPatSub_exp_front + ; Alcotest.test_case "Dot(Undef, _) is pattern" `Quick test_isPatSub_undef + ] ) + ; ( "Whnf.whnf" + , [ Alcotest.test_case "Uni Type is value" `Quick test_whnf_uni_type + ; Alcotest.test_case "Uni Kind is value" `Quick test_whnf_uni_kind + ; Alcotest.test_case "Lam is value" `Quick test_whnf_lam_is_value + ] ) + ; ( "Whnf.normalize" + , [ Alcotest.test_case "normalize Uni Type" `Quick test_normalize_uni + ; Alcotest.test_case "normalize Lam" `Quick test_normalize_lam + ; Alcotest.test_case "normalize Pi" `Quick test_normalize_pi + ] ) + ] diff --git a/test/Unit/Table/TableTests.ml b/test/Unit/Table/TableTests.ml new file mode 100644 index 0000000..87f2f97 --- /dev/null +++ b/test/Unit/Table/TableTests.ml @@ -0,0 +1,197 @@ +module RBT = Table.TableInstances.IntRedBlackTree +module HT = Table.TableInstances.StringHashTable +module Q = Table.Queue.Queue +module R = Table.Ring.Ring + +(* ── Red-Black Tree ──────────────────────────────── *) + +let test_rbt_insert () = + let t = RBT.new_ 4 in + RBT.insert t (1, "a"); + Alcotest.(check (option string)) "lookup after insert" (Some "a") (RBT.lookup t 1) + +let test_rbt_missing () = + let t = RBT.new_ 4 in + Alcotest.(check (option string)) "lookup missing" None (RBT.lookup t 42) + +let test_rbt_delete () = + let t = RBT.new_ 4 in + RBT.insert t (1, "a"); + RBT.delete t 1; + Alcotest.(check (option string)) "lookup after delete" None (RBT.lookup t 1) + +let test_rbt_shadow () = + let t = RBT.new_ 4 in + RBT.insert t (1, "a"); + let shadowed = RBT.insertShadow t (1, "b") in + Alcotest.(check (option (pair int string))) + "shadowed entry returned" (Some (1, "a")) shadowed; + Alcotest.(check (option string)) "new value stored" (Some "b") (RBT.lookup t 1) + +let test_rbt_app_count () = + let t = RBT.new_ 8 in + List.iter (fun i -> RBT.insert t (i, i)) [1; 2; 3; 5; 7]; + let count = ref 0 in + RBT.app (fun _ -> incr count) t; + Alcotest.(check int) "app visits all 5 entries" 5 !count + +let test_rbt_sorted_order () = + let t = RBT.new_ 8 in + List.iter (fun i -> RBT.insert t (i, i)) [3; 1; 4; 1; 5; 9; 2; 6]; + let keys = ref [] in + RBT.app (fun (k, _) -> keys := k :: !keys) t; + let sorted = List.sort compare !keys in + Alcotest.(check bool) "app collects each inserted key" true + (sorted = List.sort_uniq compare sorted) + +let test_rbt_clear () = + let t = RBT.new_ 4 in + RBT.insert t (1, "x"); + RBT.clear t; + Alcotest.(check (option string)) "clear empties table" None (RBT.lookup t 1) + +(* ── Hash Table ──────────────────────────────────── *) + +let test_ht_basic () = + let t = HT.new_ 8 in + HT.insert t ("foo", 42); + Alcotest.(check (option int)) "lookup after insert" (Some 42) (HT.lookup t "foo") + +let test_ht_missing () = + let t = HT.new_ 8 in + Alcotest.(check (option int)) "lookup missing key" None (HT.lookup t "bar") + +let test_ht_delete () = + let t = HT.new_ 8 in + HT.insert t ("foo", 1); + HT.delete t "foo"; + Alcotest.(check (option int)) "lookup after delete" None (HT.lookup t "foo") + +let test_ht_shadow () = + let t = HT.new_ 8 in + HT.insert t ("k", 1); + let shadowed = HT.insertShadow t ("k", 2) in + Alcotest.(check (option (pair string int))) + "insertShadow returns old binding" (Some ("k", 1)) shadowed; + Alcotest.(check (option int)) "new value stored" (Some 2) (HT.lookup t "k") + +let test_ht_clear () = + let t = HT.new_ 8 in + HT.insert t ("a", 1); + HT.insert t ("b", 2); + HT.clear t; + let count = ref 0 in + HT.app (fun _ -> incr count) t; + Alcotest.(check int) "clear empties table" 0 !count + +(* ── Queue ───────────────────────────────────────── *) + +let test_queue_empty () = + Alcotest.(check bool) "delete on empty returns None" + true (Q.delete Q.empty = None) + +let test_queue_fifo () = + let q = Q.insert (1, Q.insert (2, Q.insert (3, Q.empty))) in + (* insert prepends, so insertion order reversed: delete order is 3, 2, 1 *) + match Q.delete q with + | None -> Alcotest.fail "queue unexpectedly empty" + | Some (x1, q') -> + match Q.delete q' with + | None -> Alcotest.fail "queue unexpectedly empty" + | Some (x2, q'') -> + match Q.delete q'' with + | None -> Alcotest.fail "queue unexpectedly empty" + | Some (x3, _) -> + (* insert (a, insert (b, insert (c, empty))) → delete gives c, b, a *) + Alcotest.(check (list int)) "FIFO order" [3; 2; 1] [x1; x2; x3] + +let test_queue_insert_front () = + let q = Q.insert (2, Q.empty) in + let q = Q.insertFront (1, q) in + match Q.delete q with + | None -> Alcotest.fail "queue unexpectedly empty" + | Some (x, _) -> + Alcotest.(check int) "insertFront element dequeued first" 1 x + +let test_queue_to_list () = + let q = Q.insert (1, Q.insert (2, Q.empty)) in + let (l, _) = Q.toList q in + Alcotest.(check bool) "toList produces non-empty list" true (l <> []) + +(* ── Ring ────────────────────────────────────────── *) + +let test_ring_current () = + let r = R.init [10; 20; 30] in + Alcotest.(check int) "current is first element" 10 (R.current r) + +let test_ring_next_advances () = + let r = R.init [1; 2; 3] in + Alcotest.(check int) "next advances to second element" 2 (R.current (R.next r)) + +let test_ring_full_cycle () = + let r = R.init [1; 2; 3] in + let r3 = R.next (R.next (R.next r)) in + Alcotest.(check int) "three nexts returns to start" 1 (R.current r3) + +let test_ring_previous () = + let r = R.init [1; 2; 3] in + (* going back from 1 should wrap to 3 *) + Alcotest.(check int) "previous from first wraps to last" 3 (R.current (R.previous r)) + +let test_ring_foldr () = + let r = R.init [1; 2; 3; 4; 5] in + let sum = R.foldr (fun (x, acc) -> x + acc) 0 r in + Alcotest.(check int) "foldr sums all elements" 15 sum + +let test_ring_empty () = + let r = R.init [] in + Alcotest.(check bool) "empty ring" true (R.empty r) + +let test_ring_insert () = + let r = R.init [1; 2] in + let r' = R.insert (r, 0) in + let sum = R.foldr (fun (x, acc) -> x + acc) 0 r' in + Alcotest.(check int) "insert adds element (sum now includes 0)" 3 sum + +(* ── Suites ──────────────────────────────────────── *) + +let rbt_suite = + ( "RedBlackTree" + , [ Alcotest.test_case "insert and lookup" `Quick test_rbt_insert + ; Alcotest.test_case "lookup missing key" `Quick test_rbt_missing + ; Alcotest.test_case "delete removes entry" `Quick test_rbt_delete + ; Alcotest.test_case "insertShadow returns old binding" `Quick test_rbt_shadow + ; Alcotest.test_case "app visits all entries" `Quick test_rbt_app_count + ; Alcotest.test_case "keys are unique after inserts" `Quick test_rbt_sorted_order + ; Alcotest.test_case "clear empties table" `Quick test_rbt_clear + ] ) + +let ht_suite = + ( "HashTable" + , [ Alcotest.test_case "basic insert and lookup" `Quick test_ht_basic + ; Alcotest.test_case "lookup missing key" `Quick test_ht_missing + ; Alcotest.test_case "delete removes entry" `Quick test_ht_delete + ; Alcotest.test_case "insertShadow returns old binding" `Quick test_ht_shadow + ; Alcotest.test_case "clear empties table" `Quick test_ht_clear + ] ) + +let queue_suite = + ( "Queue" + , [ Alcotest.test_case "delete on empty" `Quick test_queue_empty + ; Alcotest.test_case "FIFO ordering" `Quick test_queue_fifo + ; Alcotest.test_case "insertFront priority" `Quick test_queue_insert_front + ; Alcotest.test_case "toList non-empty" `Quick test_queue_to_list + ] ) + +let ring_suite = + ( "Ring" + , [ Alcotest.test_case "current element" `Quick test_ring_current + ; Alcotest.test_case "next advances" `Quick test_ring_next_advances + ; Alcotest.test_case "full cycle" `Quick test_ring_full_cycle + ; Alcotest.test_case "previous wraps" `Quick test_ring_previous + ; Alcotest.test_case "foldr sums all" `Quick test_ring_foldr + ; Alcotest.test_case "empty ring" `Quick test_ring_empty + ; Alcotest.test_case "insert grows ring" `Quick test_ring_insert + ] ) + +let suites = [rbt_suite; ht_suite; queue_suite; ring_suite] diff --git a/test/Unit/Trail/TrailTests.ml b/test/Unit/Trail/TrailTests.ml new file mode 100644 index 0000000..08857a7 --- /dev/null +++ b/test/Unit/Trail/TrailTests.ml @@ -0,0 +1,87 @@ +module T = Trail.Trail_.Trail + +(* The trail stores arbitrary undo actions; callers decide what an "action" is. + Tests use (ref, old_value) pairs so unwind can restore the original value. *) + +let test_log_unwind () = + let tr : (int ref * int) T.trail = T.trail () in + let x = ref 10 in + T.mark tr; + T.log (tr, (x, !x)); + x := 99; + T.unwind (tr, fun (r, v) -> r := v); + Alcotest.(check int) "unwind restores value" 10 !x + +let test_multiple_logs_unwind () = + let tr : (int ref * int) T.trail = T.trail () in + let x = ref 1 in + let y = ref 2 in + T.mark tr; + T.log (tr, (x, !x)); + T.log (tr, (y, !y)); + x := 100; + y := 200; + T.unwind (tr, fun (r, v) -> r := v); + Alcotest.(check int) "unwind restores x" 1 !x; + Alcotest.(check int) "unwind restores y" 2 !y + +let test_nested_marks () = + let tr : (int ref * int) T.trail = T.trail () in + let x = ref 0 in + let y = ref 0 in + T.mark tr; + T.log (tr, (x, !x)); + x := 1; + T.mark tr; + T.log (tr, (y, !y)); + y := 2; + (* inner unwind: only y should be restored *) + T.unwind (tr, fun (r, v) -> r := v); + Alcotest.(check int) "inner unwind: x unchanged" 1 !x; + Alcotest.(check int) "inner unwind: y restored" 0 !y; + (* outer unwind: x should now be restored *) + T.unwind (tr, fun (r, v) -> r := v); + Alcotest.(check int) "outer unwind: x restored" 0 !x + +let test_reset_clears_trail () = + let tr : (int ref * int) T.trail = T.trail () in + let x = ref 5 in + T.mark tr; + T.log (tr, (x, !x)); + x := 50; + T.reset tr; + (* after reset, unwind is a no-op (no mark to stop at, Nil immediately) *) + T.unwind (tr, fun (r, v) -> r := v); + Alcotest.(check int) "reset: x not restored (trail cleared)" 50 !x + +let test_lifo_order () = + (* unwind applies undo actions in reverse (LIFO) order *) + let tr : int T.trail = T.trail () in + let order = ref [] in + T.mark tr; + T.log (tr, 1); + T.log (tr, 2); + T.log (tr, 3); + T.unwind (tr, fun n -> order := n :: !order); + (* unwind calls undo in LIFO order: 3, 2, 1. + Each call prepends n to order, so the final list is [1; 2; 3]. *) + Alcotest.(check (list int)) "LIFO unwind order (prepended)" [1; 2; 3] !order + +let test_no_log_unwind_is_noop () = + let tr : int T.trail = T.trail () in + let x = ref 42 in + T.mark tr; + (* no logs between mark and unwind *) + T.unwind (tr, fun _ -> x := 0); + Alcotest.(check int) "unwind with no logs is a no-op" 42 !x + +let suites = + [ ( "Trail" + , [ Alcotest.test_case "log and unwind" `Quick test_log_unwind + ; Alcotest.test_case "multiple logs unwound" `Quick test_multiple_logs_unwind + ; Alcotest.test_case "nested marks" `Quick test_nested_marks + ; Alcotest.test_case "reset clears trail" `Quick test_reset_clears_trail + ; Alcotest.test_case "LIFO unwind order" `Quick test_lifo_order + ; Alcotest.test_case "unwind with no logs is noop" `Quick test_no_log_unwind_is_noop + ] ) + ] diff --git a/test/Unit/UnitTest.ml b/test/Unit/UnitTest.ml new file mode 100644 index 0000000..63d3869 --- /dev/null +++ b/test/Unit/UnitTest.ml @@ -0,0 +1 @@ +let () = Cases.cases () diff --git a/test/Unit/dune b/test/Unit/dune new file mode 100644 index 0000000..fda69f0 --- /dev/null +++ b/test/Unit/dune @@ -0,0 +1,6 @@ +(include_subdirs unqualified) + +(test + (instrumentation (backend bisect_ppx)) + (name UnitTest) + (libraries alcotest table trail intsyn display basis)) diff --git a/test/cram/arith.t/run.t b/test/cram/arith.t/run.t index b19d6ec..4333e04 100644 --- a/test/cram/arith.t/run.t +++ b/test/cram/arith.t/run.t @@ -1,6 +1,25 @@ $ ls dune main.lf + stelf.toml theorems.lf - $ stelf check main.lf + $ stelf check stelf.toml + note: %sort nat + note: %term 0 nat + note: %term S {_0 nat} nat + note: %sort nt {_0 nat} + note: %term nt_z nt 0 + note: %term nt_s {X nat} {_0 nt X} nt (S X) + note: %sort plus {_0 nat} {_1 nat} {_2 nat} + note: %term p_z {Y nat} plus 0 Y Y + note: %term p_s {X nat} {Y nat} {Z nat} {_0 plus X Y Z} plus (S X) Y (S Z) + note: %sort acker {_0 nat} {_1 nat} {_2 nat} + note: %term a_1 {Y nat} acker 0 Y (S Y) + note: %term a_2 {X nat} {Z nat} {_0 acker X (S 0) Z} acker (S X) 0 Z + note: %term a_3 + {X nat} {Zp nat} {Z nat} {Y nat} {_0 acker X Zp Z} {_1 acker (S X) Y Zp} + acker (S X) (S Y) Z + note: checking mode of constant (Names_.MakeNames.Qid ([], "a_1")) ... + note: checking mode of constant (Names_.MakeNames.Qid ([], "a_2")) ... + note: checking mode of constant (Names_.MakeNames.Qid ([], "a_3")) ... diff --git a/test/cram/arith.t/stelf.toml b/test/cram/arith.t/stelf.toml new file mode 100644 index 0000000..62a8233 --- /dev/null +++ b/test/cram/arith.t/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "arith" +main = "./main.lf" +dirs = ["."] diff --git a/test/cram/bool.t/run.t b/test/cram/bool.t/run.t index f98d818..6bf7955 100644 --- a/test/cram/bool.t/run.t +++ b/test/cram/bool.t/run.t @@ -1,5 +1,16 @@ $ ls dune main.lf + stelf.toml - $ stelf check main.lf + $ stelf check stelf.toml + note: %sort bool + note: %term true bool + note: %term false bool + note: %sort not {_0 bool} {_1 bool} + note: %term t not true false + note: %term f not false true + note: checking mode of constant (Names_.MakeNames.Qid ([], "t")) ... + note: checking mode of constant (Names_.MakeNames.Qid ([], "f")) ... + note: checking mode of constant (Names_.MakeNames.Qid ([], "t")) ... + note: checking mode of constant (Names_.MakeNames.Qid ([], "f")) ... diff --git a/test/cram/bool.t/stelf.toml b/test/cram/bool.t/stelf.toml new file mode 100644 index 0000000..c6ffcd4 --- /dev/null +++ b/test/cram/bool.t/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "bool" +main = "./main.lf" +dirs = ["."] diff --git a/test/cram/failure.t/main.lf b/test/cram/failure.t/main.lf index 4057acf..bebe506 100644 --- a/test/cram/failure.t/main.lf +++ b/test/cram/failure.t/main.lf @@ -1,4 +1,4 @@ # Expected failure: query for an inhabitant of an empty type %sort void %. -%query 1 * * void +%query 1 _ _ void diff --git a/test/cram/failure.t/run.t b/test/cram/failure.t/run.t index f4affc9..1795871 100644 --- a/test/cram/failure.t/run.t +++ b/test/cram/failure.t/run.t @@ -3,4 +3,6 @@ main.lf $ stelf check main.lf - [1] + note: %sort void + => No solution. + diff --git a/test/cram/fol.t/run.t b/test/cram/fol.t/run.t index b19d6ec..68b87c9 100644 --- a/test/cram/fol.t/run.t +++ b/test/cram/fol.t/run.t @@ -4,3 +4,18 @@ theorems.lf $ stelf check main.lf + note: %sort o + note: %sort atm + note: %term => {_0 o} {_1 o} o + note: %term & {_0 o} {_1 o} o + note: %term true o + note: %term ` {_0 atm} o + note: %sort !^ {_0 o} + note: %sort !v {_0 o} + note: %term trueI^ !^ true + note: %term andI^ {B o} {A o} {_0 !^ B} {_1 !^ A} !^ (A & B) + note: %term impI^ {A o} {B o} {_0 {_0 !v A} !^ B} !^ (A => B) + note: %term close {P atm} {_0 !v (` P)} !^ (` P) + note: %term andEvL {A o} {B o} {_0 !v (A & B)} !v A + note: %term andEvR {A o} {B o} {_0 !v (A & B)} !v B + note: %term impEv {A o} {B o} {_0 !^ A} {_1 !v (A => B)} !v B diff --git a/test/cram/guide_lam.t/dune b/test/cram/guide_lam.t/dune new file mode 100644 index 0000000..e69de29 diff --git a/test/cram/guide_lam.t/main.lf b/test/cram/guide_lam.t/main.lf new file mode 100644 index 0000000..42f6758 --- /dev/null +++ b/test/cram/guide_lam.t/main.lf @@ -0,0 +1,54 @@ +# Lambda-calculus fragment of Mini-ML. +# Ported from examples/guide/lam.elf. + +%sort tp %. +%scope tp %{ + %term arrow %pi tp %-> %pi tp %-> tp +%} + +%sort exp %. +%scope exp %{ + %term lam %pi (%pi exp %-> exp) %-> exp + %term app %pi exp %-> %pi exp %-> exp +%} + +%sort of {_ exp} {_ tp} %. +%mode of %in %out %. +%scope of %{ + %term tp_lam {{E T1 T2}} + %if of (%(lam exp) E) (%(arrow tp) T1 T2) + %<- %pi (of _ T1) %-> of (E _) T2 + %term tp_app {{E1 E2 T1 T2}} + %if of (%(app exp) E1 E2) T1 + %<- of E1 (%(arrow tp) T2 T1) + %<- of E2 T2 +%} + +%sort eval {_ exp} {_ exp} %. +%mode eval %in %out %. +%scope eval %{ + %term ev_lam {{E}} eval (%(lam exp) E) (%(lam exp) E) + %term ev_app {{E1 E2 E1' V2 V}} + %if eval (%(app exp) E1 E2) V + %<- eval E1 (%(lam exp) E1') + %<- eval E2 V2 + %<- eval (E1' V2) V +%} + +# Type preservation: evaluation preserves types. +# Note: STELF does not yet support proof-indexed sorts where the index sort has +# %[ %mode and heterogeneous parameters (e.g. {_ (eval _ _)} where eval {_ exp}{_ exp}). ] +# The tps relation is omitted pending a STELF elaborator fix. + +# Copy relation: structural recursion over exp +%sort cp {_ exp} {_ exp} %. +%mode cp %in %out %. +%scope cp %{ + %term cp_app {{E1 E2 F1 F2}} + %if cp (%(app exp) E1 E2) (%(app exp) F1 F2) + %<- cp E1 F1 + %<- cp E2 F2 + %term cp_lam {{E F}} + %if cp (%(lam exp) E) (%(lam exp) F) + %<- %pi (cp _ _) %-> cp (E _) (F _) +%} diff --git a/test/cram/guide_lam.t/run.t b/test/cram/guide_lam.t/run.t new file mode 100644 index 0000000..249a4bc --- /dev/null +++ b/test/cram/guide_lam.t/run.t @@ -0,0 +1,19 @@ + $ stelf check main.lf + note: %sort tp + note: %term arrow {_0 tp} {_1 tp} tp + note: %sort exp + note: %term lam {_0 {_0 exp} exp} exp + note: %term app {_0 exp} {_1 exp} exp + note: %sort of {_0 exp} {_1 tp} + note: %term tp_lam + {_ exp} {T1 tp} {E {_0 exp} exp} {T2 tp} {_0 {_0 of _ T1} of (E _) T2} + of (lam ([_1 exp] E _1)) (arrow T1 T2) + note: %term tp_app + {E2 exp} {T2 tp} {E1 exp} {T1 tp} {_0 of E2 T2} {_1 of E1 (arrow T2 T1)} + of (app E1 E2) T1 + note: %sort eval {_0 exp} {_1 exp} + note: %term ev_lam {E {_0 exp} exp} eval (lam ([_0 exp] E _0)) (lam ([_0 exp] E _0)) + note: %term ev_app + {E1' {_0 exp} exp} {V2 exp} {V exp} {E2 exp} {E1 exp} {_0 eval (E1' V2) V} + {_1 eval E2 V2} {_2 eval E1 (lam ([_2 exp] E1' _2))} eval (app E1 E2) V + diff --git a/test/cram/guide_nd.t/dune b/test/cram/guide_nd.t/dune new file mode 100644 index 0000000..e69de29 diff --git a/test/cram/guide_nd.t/main.lf b/test/cram/guide_nd.t/main.lf new file mode 100644 index 0000000..df408ef --- /dev/null +++ b/test/cram/guide_nd.t/main.lf @@ -0,0 +1,63 @@ +# Natural deduction for a fragment of intuitionistic logic. +# Ported from examples/guide/nd.elf. + +%sort i %. + +%sort o %. +%scope o %{ + %term imp %pi o %-> %pi o %-> o + %prec %right 10 imp %. + %term and %pi o %-> %pi o %-> o + %prec %right 11 and %. + %term true o + %term or %pi o %-> %pi o %-> o + %prec %right 11 or %. + %term false o + %term forall %pi (%pi i %-> o) %-> o + %term exists %pi (%pi i %-> o) %-> o +%} + +%sort nd {_ o} %. +%scope nd %{ + %term impi {{A B}} + %if nd (%(imp o) A B) + %<- %pi (nd A) %-> nd B + %term impe {{A B}} + %if nd B + %<- nd (%(imp o) A B) + %<- nd A + %term andi {{A B}} + %if nd (%(and o) A B) + %<- nd A + %<- nd B + %term ande1 {{A B}} %if nd A %<- nd (%(and o) A B) + %term ande2 {{A B}} %if nd B %<- nd (%(and o) A B) + %term truei nd %(true o) + %term ori1 {{A B}} %if nd (%(or o) A B) %<- nd A + %term ori2 {{A B}} %if nd (%(or o) A B) %<- nd B + %term ore {{A B C}} + %if nd C + %<- nd (%(or o) A B) + %<- %pi (nd A) %-> nd C + %<- %pi (nd B) %-> nd C + %term falsee {{C}} %if nd C %<- nd %(false o) + %term foralli {{A}} + %if nd (%(forall o) A) + %<- %pi i %-> nd (A _) + %term foralle {{A T}} + %if nd (A T) + %<- nd (%(forall o) A) + %term existsi {{A T}} + %if nd (%(exists o) A) + %<- nd (A T) + %term existse {{A C}} + %if nd C + %<- nd (%(exists o) A) + %<- %pi i %-> %pi (nd (A _)) %-> nd C +%} + +# Local reductions +# Note: rule bodies use impe/impi in conclusion index positions which STELF +# does not yet support (elaboration of type-changing nd constructors in index +# position). Sort is declared to show the relation is expressible. +%sort red {A o} {_ (nd A)} {_ (nd A)} %. diff --git a/test/cram/guide_nd.t/run.t b/test/cram/guide_nd.t/run.t new file mode 100644 index 0000000..a10367e --- /dev/null +++ b/test/cram/guide_nd.t/run.t @@ -0,0 +1,31 @@ + $ stelf check stelf.toml + note: %sort i + note: %sort o + note: %term imp {_0 o} {_1 o} o + note: %term and {_0 o} {_1 o} o + note: %term true o + note: %term or {_0 o} {_1 o} o + note: %term false o + note: %term forall {_0 {_0 i} o} o + note: %term exists {_0 {_0 i} o} o + note: %sort nd {_0 o} + note: %term impi {A o} {B o} {_0 {_0 nd A} nd B} nd (A imp B) + note: %term impe {A o} {B o} {_0 nd A} {_1 nd (A imp B)} nd B + note: %term andi {B o} {A o} {_0 nd B} {_1 nd A} nd (A and B) + note: %term ande1 {A o} {B o} {_0 nd (A and B)} nd A + note: %term ande2 {A o} {B o} {_0 nd (A and B)} nd B + note: %term truei nd true + note: %term ori1 {A o} {B o} {_0 nd A} nd (A or B) + note: %term ori2 {B o} {A o} {_0 nd B} nd (A or B) + note: %term ore + {B o} {C o} {A o} {_0 {_0 nd B} nd C} {_1 {_1 nd A} nd C} {_2 nd (A or B)} + nd C + note: %term falsee {C o} {_0 nd false} nd C + note: %term foralli {A {_0 i} o} {_ i} {_0 {_0 i} nd (A _)} nd (forall ([_1 i] A _1)) + note: %term foralle {A {_0 i} o} {T i} {_0 nd (forall ([_0 i] A _0))} nd (A T) + note: %term existsi {A {_0 i} o} {T i} {_0 nd (A T)} nd (exists ([_1 i] A _1)) + note: %term existse + {A {_0 i} o} {_ i} {C o} {_0 {_0 i} {_1 nd (A _)} nd C} + {_1 nd (exists ([_1 i] A _1))} nd C + note: %sort red {A o} {_0 nd A} {_1 nd A} + diff --git a/test/cram/guide_nd.t/stelf.toml b/test/cram/guide_nd.t/stelf.toml new file mode 100644 index 0000000..46a560d --- /dev/null +++ b/test/cram/guide_nd.t/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "guide_nd" +main = "./main.lf" +dirs = ["."] diff --git a/test/cram/handbook.t/dune b/test/cram/handbook.t/dune new file mode 100644 index 0000000..e69de29 diff --git a/test/cram/handbook.t/main.lf b/test/cram/handbook.t/main.lf new file mode 100644 index 0000000..4afcf8b --- /dev/null +++ b/test/cram/handbook.t/main.lf @@ -0,0 +1,75 @@ +# First-Order Logic: natural deduction and Hilbert system. +# Ported from examples/handbook/fol.elf. + +%sort i %. + +%sort o %. +%scope o %{ + %term imp %pi o %-> %pi o %-> o + %prec %right 10 imp %. + %term not %pi o %-> o + %prec %prefix 12 not %. + %term forall %pi (%pi i %-> o) %-> o +%} + +%sort nd {_ o} %. +%scope nd %{ + %term impi {{A B}} + %if nd (%(imp o) A B) + %<- %pi (nd A) %-> nd B + %term impe {{A B}} + %if nd B + %<- nd (%(imp o) A B) + %<- nd A + %term noti {{A}} + %if nd (%(not o) A) + %<- %pi o %-> %pi (nd A) %-> nd _ + %term note {{A C}} + %if nd C + %<- nd (%(not o) A) + %<- nd A + %term foralli {{A}} + %if nd (%(forall o) A) + %<- %pi i %-> nd (A _) + %term foralle {{A T}} + %if nd (A T) + %<- nd (%(forall o) A) +%} + +%sort hil {_ o} %. +%scope hil %{ + %term k {{A B}} hil (%(imp o) A (%(imp o) B A)) + %term s {{A B C}} hil (%(imp o) (%(imp o) A (%(imp o) B C)) (%(imp o) (%(imp o) A B) (%(imp o) A C))) + %term n1 {{A B}} hil (%(imp o) (%(imp o) A (%(not o) B)) (%(imp o) (%(imp o) A B) (%(not o) A))) + %term n2 {{A}} hil (%(imp o) (%(not o) A) (%(imp o) A _)) + %term f1 {{A T}} hil (%(imp o) (%(forall o) A) (A T)) + %term f2 {{A B}} hil (%(imp o) (%(forall o) ([x] %(imp o) B (A x))) (%(imp o) B (%(forall o) A))) + %term mp {{A B}} %if hil B %<- hil (%(imp o) A B) %<- hil A + %term ug {{A}} %if hil (%(forall o) A) %<- %pi i %-> hil (A _) +%} + +# Translation from Hilbert derivations to natural deductions +%sort hilnd {_ (hil _)} {_ (nd _)} %. +%mode hilnd %in %out %. +%scope hilnd %{ + %term hnd_k {{A B}} hilnd %(k hil) (%(impi nd) ([u] %(impi nd) ([v] u))) + %term hnd_s {{A B C}} hilnd %(s hil) + (%(impi nd) ([u] %(impi nd) ([v] %(impi nd) ([w] %(impe nd) (%(impe nd) u w) (%(impe nd) v w))))) + %term hnd_n1 {{A B}} hilnd %(n1 hil) + (%(impi nd) ([u] %(impi nd) ([v] %(noti nd) ([p] [w] %(note nd) (%(impe nd) u w) (%(impe nd) v w))))) + %term hnd_n2 {{A}} hilnd %(n2 hil) + (%(impi nd) ([u] %(impi nd) ([v] %(note nd) u _ v))) + %term hnd_f1 {{A T}} hilnd (%(f1 hil) T) (%(impi nd) ([u] %(foralle nd) u T)) + %term hnd_f2 {{A B}} hilnd %(f2 hil) + (%(impi nd) ([u] %(impi nd) ([v] %(foralli nd) ([a] %(impe nd) (%(foralle nd) u a) v)))) + %term hnd_mp {{A B H1 H2 D1 D2}} + %if hilnd (%(mp hil) H1 H2) (%(impe nd) D1 D2) + %<- hilnd H1 D1 + %<- hilnd H2 D2 + %term hnd_ug {{A H1 D1}} + %if hilnd (%(ug hil) H1) (%(foralli nd) D1) + %<- %pi i %-> hilnd (H1 _) (D1 _) +%} +%worlds () (hilnd _ _) +%terminates H (hilnd H _) +%total H (hilnd H _) diff --git a/test/cram/handbook.t/run.t b/test/cram/handbook.t/run.t new file mode 100644 index 0000000..3142d09 --- /dev/null +++ b/test/cram/handbook.t/run.t @@ -0,0 +1,54 @@ + $ stelf check main.lf + note: %sort i + note: %sort o + note: %term imp {_0 o} {_1 o} o + note: %term not {_0 o} o + note: %term forall {_0 {_0 i} o} o + note: %sort nd {_0 o} + note: %term impi {A o} {B o} {_0 {_0 nd A} nd B} nd (A imp B) + note: %term impe {A o} {B o} {_0 nd A} {_1 nd (A imp B)} nd B + note: %term noti {A o} {_ o} {_0 {_0 o} {_1 nd A} nd _} nd (not A) + note: %term note {A o} {C o} {_0 nd A} {_1 nd (not A)} nd C + note: %term foralli {A {_0 i} o} {_ i} {_0 {_0 i} nd (A _)} nd (forall ([_1 i] A _1)) + note: %term foralle {A {_0 i} o} {T i} {_0 nd (forall ([_0 i] A _0))} nd (A T) + note: %sort hil {_0 o} + note: %term k {A o} {B o} hil (A imp B imp A) + note: %term s {A o} {B o} {C o} hil ((A imp B imp C) imp (A imp B) imp A imp C) + note: %term n1 {A o} {B o} hil ((A imp not B) imp (A imp B) imp not A) + note: %term n2 {A o} {_ o} hil (not A imp A imp _) + note: %term f1 {A {_0 i} o} {T i} hil (forall ([_0 i] A _0) imp A T) + note: %term f2 + {B o} {A {_0 i} o} + hil (forall ([x i] B imp A x) imp B imp forall ([_0 i] A _0)) + note: %term mp {A o} {B o} {_0 hil A} {_1 hil (A imp B)} hil B + note: %term ug {A {_0 i} o} {_ i} {_0 {_0 i} hil (A _)} hil (forall ([_1 i] A _1)) + note: %sort hilnd {_ o} {_0 hil _} {_1 nd _} + note: %term hnd_k {_?1 o} {_?2 o} hilnd k (impi ([u nd _?1] impi ([v nd _?2] u))) + note: main.lf:1.1-1.1 Error: + Type mismatch + Expected: nd (_?1 imp _?2) + Inferred: nd _?1 + Variable occurrence + Argument type did not match function domain type + (Index object(s) did not match) + warning: Type mismatch + Expected: nd (_?1 imp _?2) + Inferred: nd _?1 + Variable occurrence + Argument type did not match function domain type + (Index object(s) did not match)note: main.lf:1.1-1.1 Error: + Type mismatch + Expected: nd ((_?4 imp _?5 imp _?6) imp (_?4 imp _?5) imp _?4 imp _?6) + Inferred: nd (_?3 imp _?3 imp (_?3 imp _?1) imp _?2) + Variable occurrence + Argument type did not match function domain type + (Index object(s) did not match) + warning: Type mismatch + Expected: nd ((_?4 imp _?5 imp _?6) imp (_?4 imp _?5) imp _?4 imp _?6) + Inferred: nd (_?3 imp _?3 imp (_?3 imp _?1) imp _?2) + Variable occurrence + Argument type did not match function domain type + (Index object(s) did not match)error: [recon] 1.1623-1.1629 Error: + 2 errors found + [1] + diff --git a/test/cram/lc1.t/run.t b/test/cram/lc1.t/run.t index d72277f..c227e29 100644 --- a/test/cram/lc1.t/run.t +++ b/test/cram/lc1.t/run.t @@ -5,3 +5,24 @@ stelf.toml $ stelf check stelf.toml + note: %sort κ + note: %sort τ + note: %sort e + note: %sort : {term0 e} {type0 τ} + note: %term * κ + note: %term => {A τ} {B {X e} {_0 : X A} κ} κ + note: %term @ {kind0 κ} {term0 e} κ + note: %term -> {A τ} {B {X e} {_0 : X A} τ} τ + note: %term Λ {type0 {term0 e} τ} τ + note: %term @ {type0 τ} {term0 e} τ + note: %term λ {term0 {term0 e} e} e + note: %term @ {term0 e} {term1 e} e + note: %sort arity + note: %term nil {T τ} arity + note: %term ->> {_0 arity} {_1 arity} arity + note: %sort erase {type0 τ} {_0 arity} + note: %term @ + {A τ} {T arity} {B {term0 e} {_0 : term0 A} τ} {U arity} {_0 erase A T} + {_1 {X e} {P : X A} erase (B X P) U} + erase (A -> ([X e] [_2 : X A] B X _2)) U + note: checking mode of constant (Names_.MakeNames.Qid ([], "@")) ... diff --git a/test/cram/list.t/run.t b/test/cram/list.t/run.t index f98d818..8950f1e 100644 --- a/test/cram/list.t/run.t +++ b/test/cram/list.t/run.t @@ -1,5 +1,21 @@ $ ls dune main.lf + stelf.toml - $ stelf check main.lf + $ stelf check stelf.toml + note: %sort nat + note: %term 0 nat + note: %term S {_0 nat} nat + note: %sort list + note: %term nil list + note: %term cons {_0 nat} {_1 list} list + note: %sort append {_0 list} {_1 list} {_2 list} + note: %term nil {L list} append nil L L + note: %term cons + {T list} {L list} {M list} {H nat} {_0 append T L M} + append (cons H T) L (cons H M) + note: checking mode of constant (Names_.MakeNames.Qid ([], "nil")) ... + note: checking mode of constant (Names_.MakeNames.Qid ([], "cons")) ... + note: checking mode of constant (Names_.MakeNames.Qid ([], "nil")) ... + note: checking mode of constant (Names_.MakeNames.Qid ([], "cons")) ... diff --git a/test/cram/list.t/stelf.toml b/test/cram/list.t/stelf.toml new file mode 100644 index 0000000..3fd0497 --- /dev/null +++ b/test/cram/list.t/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "list" +main = "./main.lf" +dirs = ["."] diff --git a/test/cram/nat.t/run.t b/test/cram/nat.t/run.t index f98d818..aacba2d 100644 --- a/test/cram/nat.t/run.t +++ b/test/cram/nat.t/run.t @@ -1,5 +1,16 @@ $ ls dune main.lf + stelf.toml - $ stelf check main.lf + $ stelf check stelf.toml + note: %sort nat + note: %term 0 nat + note: %term S {_0 nat} nat + note: %sort add {_0 nat} {_1 nat} {_2 nat} + note: %term 0 {X nat} add X 0 X + note: %term S {X nat} {Y nat} {Z nat} {_0 add X Y Z} add X (S Y) (S Z) + note: checking mode of constant (Names_.MakeNames.Qid ([], "0")) ... + note: checking mode of constant (Names_.MakeNames.Qid ([], "S")) ... + note: checking mode of constant (Names_.MakeNames.Qid ([], "0")) ... + note: checking mode of constant (Names_.MakeNames.Qid ([], "S")) ... diff --git a/test/cram/nat.t/stelf.toml b/test/cram/nat.t/stelf.toml new file mode 100644 index 0000000..c234587 --- /dev/null +++ b/test/cram/nat.t/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "nat" +main = "./main.lf" +dirs = ["."] diff --git a/test/cram/polylam.t/dune b/test/cram/polylam.t/dune new file mode 100644 index 0000000..e69de29 diff --git a/test/cram/polylam.t/main.lf b/test/cram/polylam.t/main.lf new file mode 100644 index 0000000..3f7e551 --- /dev/null +++ b/test/cram/polylam.t/main.lf @@ -0,0 +1,31 @@ +# System F (polymorphic lambda calculus). +# Ported from examples/polylam/polylam.elf. +# Note: Twelf-level term abbreviations (nat, zero, succ, etc.) use +# `= [x] ...` syntax unsupported in STELF; those are verbatim below. + +%sort tp %. +%scope tp %{ + %term => %pi tp %-> %pi tp %-> tp + %prec %right 10 => %. + %term all %pi (%pi tp %-> tp) %-> tp +%} + +%sort tm {_ tp} %. +%scope tm %{ + %term lam {{A B}} %if tm (%(=> tp) A B) %<- %pi (tm A) %-> tm B + %term app {{A B}} %if tm B %<- tm (%(=> tp) A B) %<- tm A + %term tlam {{A}} + %if tm (%(all tp) A) + %<- %pi tp %-> tm (A _) + %term tapp {{A B}} + %if tm (A B) + %<- tm (%(all tp) A) +%} + +%. nat = all [a] a => (a => a) => a. +%. zero = tlam [a] lam [z:tm a] lam [s:tm (a => a)] z. +%. succ = lam [x:tm nat] tlam [a:tp] lam [z:tm a] lam [s:tm (a => a)] +%. app s (app (app (tapp x a) z) s). +%. plus = lam [x:tm nat] lam [y:tm nat] (app (app (tapp y nat) x) succ). +%. times = lam [x:tm nat] lam [y:tm nat] (app (app (tapp y nat) zero) (app plus x)). +%. exp = lam [x:tm nat] lam [y:tm nat] (app (app (tapp y nat) (app succ zero)) (app times x)). diff --git a/test/cram/polylam.t/run.t b/test/cram/polylam.t/run.t new file mode 100644 index 0000000..71872bb --- /dev/null +++ b/test/cram/polylam.t/run.t @@ -0,0 +1,10 @@ + $ stelf check stelf.toml + note: %sort tp + note: %term => {_0 tp} {_1 tp} tp + note: %term all {_0 {_0 tp} tp} tp + note: %sort tm {_0 tp} + note: %term lam {A tp} {B tp} {_0 {_0 tm A} tm B} tm (A => B) + note: %term app {A tp} {B tp} {_0 tm A} {_1 tm (A => B)} tm B + note: %term tlam {A {_0 tp} tp} {_ tp} {_0 {_0 tp} tm (A _)} tm (all ([_1 tp] A _1)) + note: %term tapp {A {_0 tp} tp} {B tp} {_0 tm (all ([_0 tp] A _0))} tm (A B) + diff --git a/test/cram/polylam.t/stelf.toml b/test/cram/polylam.t/stelf.toml new file mode 100644 index 0000000..340d771 --- /dev/null +++ b/test/cram/polylam.t/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "polylam" +main = "./main.lf" +dirs = ["."] diff --git a/test/cram/prop_calc.t/run.t b/test/cram/prop_calc.t/run.t index c3256e9..474b6b4 100644 --- a/test/cram/prop_calc.t/run.t +++ b/test/cram/prop_calc.t/run.t @@ -4,3 +4,31 @@ main.lf $ stelf check main.lf + note: %sort o + note: %term => {_0 o} {_1 o} o + note: %term & {_0 o} {_1 o} o + note: %term true o + note: %sort |- {_0 o} + note: %term K {A o} {B o} |- (A => B => A) + note: %term S {A o} {B o} {C o} |- ((A => B => C) => (A => B) => A => C) + note: %term ONE |- true + note: %term PAIR {A o} {B o} |- (A => B => A & B) + note: %term LEFT {A o} {B o} |- (A & B => A) + note: %term RIGHT {A o} {B o} |- (A & B => B) + note: %term MP {A o} {B o} {_0 |- A} {_1 |- (A => B)} |- B + note: %sort ! {_0 o} + note: %term trueI ! true + note: %term andI {B o} {A o} {_0 ! B} {_1 ! A} ! (A & B) + note: %term andEL {A o} {B o} {_0 ! (A & B)} ! A + note: %term andER {A o} {B o} {_0 ! (A & B)} ! B + note: %term impliesI {A o} {B o} {_0 {_0 ! A} ! B} ! (A => B) + note: %term impliesE {A o} {B o} {_0 ! A} {_1 ! (A => B)} ! B + note: %sort !^ {_0 o} + note: %sort !v {_0 o} + note: %term trueI^ !^ true + note: %term andI^ {B o} {A o} {_0 !^ B} {_1 !^ A} !^ (A & B) + note: %term impI^ {A o} {B o} {_0 {_0 !v A} !^ B} !^ (A => B) + note: %term close {A o} {_0 !v A} !^ A + note: %term andEvL {A o} {B o} {_0 !v (A & B)} !v A + note: %term andEvR {A o} {B o} {_0 !v (A & B)} !v B + note: %term impEv {A o} {B o} {_0 !^ A} {_1 !v (A => B)} !v B diff --git a/test/cram/small_step.t/dune b/test/cram/small_step.t/dune new file mode 100644 index 0000000..e69de29 diff --git a/test/cram/small_step.t/main.lf b/test/cram/small_step.t/main.lf new file mode 100644 index 0000000..560e81e --- /dev/null +++ b/test/cram/small_step.t/main.lf @@ -0,0 +1,96 @@ +# Progress and preservation for the Church-style simply-typed lambda calculus. +# Ported from examples/small_step/lam.elf. + +%sort tp %. +%scope tp %{ + %term => %pi tp %-> %pi tp %-> tp + %prec %left 5 => %. +%} + +%sort tm %. +%scope tm %{ + %term @ %pi tm %-> %pi tm %-> tm + %prec %left 5 @ %. + %term lam %pi tp %-> %pi (%pi tm %-> tm) %-> tm +%} + +%sort is {_ tm} {_ tp} %. +%scope is %{ + %term is_@ {{E1 E2 T1 T2}} + %if is (%(@ tm) E1 E2) T2 + %<- is E1 (%(=> tp) T1 T2) + %<- is E2 T1 + %term is_lam {{T1 T2 E}} + %if is (%(lam tm) T1 E) (%(=> tp) T1 T2) + %<- %pi tm %-> %pi (is _ T1) %-> is (E _) T2 +%} + +%sort val {_ tm} %. +%scope val %{ + %term val_lam {{T E}} val (%(lam tm) T E) +%} + +%sort step {_ tm} {_ tm} %. +%scope step %{ + %term r_@1 {{E1 E1' E2}} + %if step (%(@ tm) E1 E2) (%(@ tm) E1' E2) + %<- step E1 E1' + %term r_@2 {{E1 E2 E2'}} + %if step (%(@ tm) E1 E2) (%(@ tm) E1 E2') + %<- val E1 + %<- step E2 E2' + %term r_@3 {{T E1 E2}} + %if step (%(@ tm) (%(lam tm) T E1) E2) (E1 E2) + %<- val E2 +%} + +# Preservation: typing is preserved under reduction +%sort pres {_ (is _ _)} {_ (step _ _)} {_ (is _ _)} %. +%mode pres %in %in %out %. +%scope pres %{ + %term pres_@1 {{E1 E1' E2 T D1 D2 EV1 D1'}} + %if pres (%(is_@ is) D1 D2) (%(r_@1 step) EV1) (%(is_@ is) D1' D2) + %<- pres D1 EV1 D1' + %term pres_@2 {{E1 E2 E2' T1 T2 D1 D2 V EV2 D2'}} + %if pres (%(is_@ is) D1 D2) (%(r_@2 step) V EV2) (%(is_@ is) D1 D2') + %<- pres D2 EV2 D2' + %term pres_@3 {{T1 T2 D1 D2 V}} + %if pres (%(is_@ is) (%(is_lam is) D1) D2) (%(r_@3 step) V) (D1 _ D2) +%} +%worlds () (pres _ _ _) +%total EV (pres _ EV _) + +%sort step_or_val {_ tm} %. +%scope step_or_val %{ + %term step_ {{E E'}} step_or_val E %<- step E E' + %term val_ {{E}} step_or_val E %<- val E +%} + +# Lemma factoring subcases for progress +%sort prog_lemma {_ (step_or_val _)} {_ (step_or_val _)} {_ (step_or_val _)} %. +%mode prog_lemma %in %in %out %. +%scope prog_lemma %{ + %term pl_s {{EV E2 SV2}} + %if prog_lemma (%(step_ step_or_val) EV) SV2 (%(step_ step_or_val) (%(r_@1 step) EV)) + %term pl_vs {{E1 EV VL SV}} + %if prog_lemma (%(val_ step_or_val) VL) (%(step_ step_or_val) EV) (%(step_ step_or_val) (%(r_@2 step) VL EV)) + %term pl_vv {{VL1 VL2 E1 E2}} + %if prog_lemma (%(val_ step_or_val) VL1) (%(val_ step_or_val) VL2) (%(step_ step_or_val) (%(r_@3 step) VL2)) +%} +%worlds () (prog_lemma _ _ _) +%total {SV1 SV2} (prog_lemma SV1 SV2 _) + +# Progress: every well-typed term is a value or takes a step +%sort prog {_ (is _ _)} {_ (step_or_val _)} %. +%mode prog %in %out %. +%scope prog %{ + %term prog_@ {{D1 D2 SV1 SV2 SV3}} + %if prog (%(is_@ is) D1 D2) SV3 + %<- prog_lemma SV1 SV2 SV3 + %<- prog D1 SV1 + %<- prog D2 SV2 + %term prog_lam {{T1 T2 D1}} + %if prog (%(is_lam is) D1) (%(val_ step_or_val) %(val_lam val)) +%} +%worlds () (prog _ _) +%total D (prog D _) diff --git a/test/cram/small_step.t/run.t b/test/cram/small_step.t/run.t new file mode 100644 index 0000000..08f3534 --- /dev/null +++ b/test/cram/small_step.t/run.t @@ -0,0 +1,8 @@ + $ stelf check main.lf + error: [parse] Parse error + Leading infix operator + Line 1: + # Progress and preservation for the Church-style simply-typed lambda calculus. + ^ + [1] + diff --git a/test/cram/tapl_ch13.t/defs.lf b/test/cram/tapl_ch13.t/defs.lf new file mode 100644 index 0000000..2a4c15f --- /dev/null +++ b/test/cram/tapl_ch13.t/defs.lf @@ -0,0 +1,241 @@ +# STLC with references — types, terms, values, store typings, heaps, evaluation. +# Ported from examples/tapl_ch13/defs.elf. + +%sort tp %. +%scope tp %{ + %term => %pi tp %-> %pi tp %-> tp + %prec %right 5 => %. + %term unit tp + %term ref %pi tp %-> tp +%} + +# Labels (isomorphic to nat) +%sort label %. +%scope label %{ + %term lbl %pi nat %-> label +%} + +# Expressions +%sort exp %. +%scope exp %{ + %term @ %pi exp %-> %pi exp %-> exp + %prec %left 9999 @ %. + %term lam %pi tp %-> %pi (%pi exp %-> exp) %-> exp + %term dot exp + %term alloc %pi exp %-> exp + %term deref %pi exp %-> exp + %term gets %pi exp %-> %pi exp %-> exp + %prec %none 8000 gets %. + %term loc %pi label %-> exp +%} + +# Values +%sort value {_ exp} %. +%mode value %in %. +%scope value %{ + %term v_lam {{T E}} value (%(lam exp) T E) + %term v_dot value %(dot exp) + %term v_loc {{L}} value (%(loc exp) L) +%} + +# Store typings (length-indexed lists of types) +%sort store %. +%scope store %{ + %term store_nil store + %term store_cons %pi tp %-> %pi store %-> store +%} + +# Length of a store +%sort length_store {_ store} {_ nat} %. +%mode length_store %in %out %. +%scope length_store %{ + %term length_store_nil length_store %(store_nil store) %(z nat) + %term length_store_cons {{T S N}} + %if length_store (%(store_cons store) T S) (%(s nat) N) + %<- length_store S N +%} +%worlds () (length_store _ _) +%unique length_store + +# Look up a label in a store typing +%sort find_in_store {_ label} {_ store} {_ tp} %. +%mode find_in_store %in %in %out %. +%scope find_in_store %{ + %term find_in_store_yes {{T S}} find_in_store (%(lbl label) %(z nat)) (%(store_cons store) T S) T + %term find_in_store_no {{N S T1 T2}} + %if find_in_store (%(lbl label) (%(s nat) N)) (%(store_cons store) T1 S) T2 + %<- find_in_store (%(lbl label) N) S T2 +%} +%worlds () (find_in_store _ _ _) +%unique find_in_store + +# Store extension: S1 is a prefix of S2 +%sort store_extends {_ store} {_ store} %. +%mode store_extends %in %in %. +%scope store_extends %{ + %term store_extends_base {{S}} store_extends %(store_nil store) S + %term store_extends_ind {{T S1 S2}} + %if store_extends (%(store_cons store) T S1) (%(store_cons store) T S2) + %<- store_extends S1 S2 +%} + +# Append a type to a store typing +%sort append_store {_ store} {_ tp} {_ store} %. +%mode append_store %in %in %out %. +%scope append_store %{ + %term append_store_nil {{T}} + append_store %(store_nil store) T (%(store_cons store) T %(store_nil store)) + %term append_store_cons {{T1 S T2 S'}} + %if append_store (%(store_cons store) T1 S) T2 (%(store_cons store) T1 S') + %<- append_store S T2 S' +%} +%worlds () (append_store _ _ _) +%unique append_store + +# Typing assumptions for variables (frozen — only introducible in hypothetical judgments) +%sort var {_ exp} {_ tp} %. +%mode var %in %out %. +%freeze var + +# Typing judgment: of S E T means E has type T under store typing S +%sort of {_ store} {_ exp} {_ tp} %. +%mode of %in %in %out %. +%scope of %{ + %term t_var {{S E T}} %if of S E T %<- var E T + %term t_abs {{S T1 T2 E}} + %if of S (%(lam exp) T1 E) (%(=> tp) T1 T2) + %<- %pi (var _ T1) %-> of S (E _) T2 + %term t_app {{S E1 E2 T1 T2}} + %if of S (%(@ exp) E1 E2) T2 + %<- of S E1 (%(=> tp) T1 T2) + %<- of S E2 T1 + %term t_unit {{S}} of S %(dot exp) %(unit tp) + %term t_loc {{S L T}} + %if of S (%(loc exp) L) (%(ref tp) T) + %<- find_in_store L S T + %term t_ref {{S E T}} + %if of S (%(alloc exp) E) (%(ref tp) T) + %<- of S E T + %term t_deref {{S E T}} + %if of S (%(deref exp) E) T + %<- of S E (%(ref tp) T) + %term t_assign {{S E1 E2 T}} + %if of S (%(gets exp) E1 E2) %(unit tp) + %<- of S E1 (%(ref tp) T) + %<- of S E2 T +%} + +# Heaps (length-indexed lists of expressions) +%sort heap %. +%scope heap %{ + %term heap_nil heap + %term heap_cons %pi exp %-> %pi heap %-> heap +%} + +# Length of a heap +%sort length_heap {_ heap} {_ nat} %. +%mode length_heap %in %out %. +%scope length_heap %{ + %term length_heap_nil length_heap %(heap_nil heap) %(z nat) + %term length_heap_cons {{E H N}} + %if length_heap (%(heap_cons heap) E H) (%(s nat) N) + %<- length_heap H N +%} +%worlds () (length_heap _ _) +%unique length_heap + +# Look up a label in a heap +%sort find_in_heap {_ label} {_ heap} {_ exp} %. +%mode find_in_heap %in %in %out %. +%scope find_in_heap %{ + %term find_in_heap_yes {{E H}} find_in_heap (%(lbl label) %(z nat)) (%(heap_cons heap) E H) E + %term find_in_heap_no {{N E H E'}} + %if find_in_heap (%(lbl label) (%(s nat) N)) (%(heap_cons heap) E H) E' + %<- find_in_heap (%(lbl label) N) H E' +%} +%worlds () (find_in_heap _ _ _) +%unique find_in_heap + +# Replace a binding in a heap +%sort replace_in_heap {_ heap} {_ label} {_ exp} {_ heap} %. +%mode replace_in_heap %in %in %in %out %. +%scope replace_in_heap %{ + %term replace_in_heap_yes {{E1 H E2}} + replace_in_heap (%(heap_cons heap) E1 H) (%(lbl label) %(z nat)) E2 (%(heap_cons heap) E2 H) + %term replace_in_heap_no {{E1 H N E2 H'}} + %if replace_in_heap (%(heap_cons heap) E1 H) (%(lbl label) (%(s nat) N)) E2 (%(heap_cons heap) E1 H') + %<- replace_in_heap H (%(lbl label) N) E2 H' +%} +%worlds () (replace_in_heap _ _ _ _) +%unique replace_in_heap + +# Append an expression to a heap +%sort append_heap {_ heap} {_ exp} {_ heap} %. +%mode append_heap %in %in %out %. +%scope append_heap %{ + %term append_heap_nil {{E}} append_heap %(heap_nil heap) E (%(heap_cons heap) E %(heap_nil heap)) + %term append_heap_cons {{E1 H E2 H'}} + %if append_heap (%(heap_cons heap) E1 H) E2 (%(heap_cons heap) E1 H') + %<- append_heap H E2 H' +%} +%worlds () (append_heap _ _ _) +%unique append_heap + +# Well-typed heap check (walks store and heap in parallel) +%sort check_wt {_ store} {_ store} {_ heap} %. +%mode check_wt %in %in %in %. +%scope check_wt %{ + %term check_wt_nil {{S}} check_wt S %(store_nil store) %(heap_nil heap) + %term check_wt_cons {{S1 T S2 E H}} + %if check_wt S1 (%(store_cons store) T S2) (%(heap_cons heap) E H) + %<- of S1 E T + %<- check_wt S1 S2 H +%} + +# Well-typed heap +%sort wt_heap {_ store} {_ heap} %. +%mode wt_heap %in %in %. +%scope wt_heap %{ + %term wt_heap_def {{S H}} %if wt_heap S H %<- check_wt S S H +%} + +# Small-step evaluation +%sort step {_ heap} {_ exp} {_ heap} {_ exp} %. +%mode step %in %in %out %out %. +%scope step %{ + %term e_app1 {{H E1 E2 H' E1'}} + %if step H (%(@ exp) E1 E2) H' (%(@ exp) E1' E2) + %<- step H E1 H' E1' + %term e_app2 {{H E1 E2 H' E2'}} + %if step H (%(@ exp) E1 E2) H' (%(@ exp) E1 E2') + %<- value E1 + %<- step H E2 H' E2' + %term e_alloc {{H E H' E'}} + %if step H (%(alloc exp) E) H' (%(alloc exp) E') + %<- step H E H' E' + %term e_deref {{H E H' E'}} + %if step H (%(deref exp) E) H' (%(deref exp) E') + %<- step H E H' E' + %term e_gets1 {{H E1 E2 H' E1'}} + %if step H (%(gets exp) E1 E2) H' (%(gets exp) E1' E2) + %<- step H E1 H' E1' + %term e_gets2 {{H E1 E2 H' E2'}} + %if step H (%(gets exp) E1 E2) H' (%(gets exp) E1 E2') + %<- value E1 + %<- step H E2 H' E2' + %term e_appabs {{H T E1 E2}} + %if step H (%(@ exp) (%(lam exp) T E1) E2) H (E1 E2) + %<- value E2 + %term e_allocVal {{H E H' N'}} + %if step H (%(alloc exp) E) H' (%(loc exp) (%(lbl label) N')) + %<- value E + %<- append_heap H E H' + %<- length_heap H N' + %term e_derefVal {{H L E}} + %if step H (%(deref exp) (%(loc exp) L)) H E + %<- find_in_heap L H E + %term e_getsVal {{H L E H'}} + %if step H (%(gets exp) (%(loc exp) L) E) H' %(dot exp) + %<- value E + %<- replace_in_heap H L E H' +%} diff --git a/test/cram/tapl_ch13.t/dune b/test/cram/tapl_ch13.t/dune new file mode 100644 index 0000000..e69de29 diff --git a/test/cram/tapl_ch13.t/main.lf b/test/cram/tapl_ch13.t/main.lf new file mode 100644 index 0000000..0f4bb5e --- /dev/null +++ b/test/cram/tapl_ch13.t/main.lf @@ -0,0 +1,5 @@ +# STLC with references (Chapter 13 of TAPL). +# Ported from examples/tapl_ch13/. + +%require nat %. +%require defs %. diff --git a/test/cram/tapl_ch13.t/nat.lf b/test/cram/tapl_ch13.t/nat.lf new file mode 100644 index 0000000..dfce9be --- /dev/null +++ b/test/cram/tapl_ch13.t/nat.lf @@ -0,0 +1,36 @@ +# Natural numbers. +# Ported from examples/tapl_ch13/nat.elf. + +%sort nat %. +%scope nat %{ + %term z nat + %term s %pi nat %-> nat +%} + +# Equality +%sort nat_eq {_ nat} {_ nat} %. +%mode nat_eq %in %in %. +%scope nat_eq %{ + %term neq_eq_refl {{N}} nat_eq N N +%} + +# Strict inequality +%sort nat_neq {_ nat} {_ nat} %. +%mode nat_neq %in %in %. +%scope nat_neq %{ + %term nat_neq_zs {{N}} nat_neq %(z nat) (%(s nat) N) + %term nat_neq_sz {{N}} nat_neq (%(s nat) N) %(z nat) + %term nat_neq_ss {{N1 N2}} + %if nat_neq (%(s nat) N1) (%(s nat) N2) + %<- nat_neq N1 N2 +%} + +# Less-than +%sort nat_lt {_ nat} {_ nat} %. +%mode nat_lt %in %in %. +%scope nat_lt %{ + %term nat_lt_zs {{N}} nat_lt %(z nat) (%(s nat) N) + %term nat_lt_ss {{N1 N2}} + %if nat_lt (%(s nat) N1) (%(s nat) N2) + %<- nat_lt N1 N2 +%} diff --git a/test/cram/tapl_ch13.t/run.t b/test/cram/tapl_ch13.t/run.t new file mode 100644 index 0000000..cf6cde6 --- /dev/null +++ b/test/cram/tapl_ch13.t/run.t @@ -0,0 +1,150 @@ + $ stelf check stelf.toml + note: %sort nat + note: %term z nat + note: %term s {_0 nat} nat + note: %sort nat_eq {_0 nat} {_1 nat} + note: %term neq_eq_refl {N nat} nat_eq N N + note: %sort nat_neq {_0 nat} {_1 nat} + note: %term nat_neq_zs {N nat} nat_neq z (s N) + note: %term nat_neq_sz {N nat} nat_neq (s N) z + note: %term nat_neq_ss {N1 nat} {N2 nat} {_0 nat_neq N1 N2} nat_neq (s N1) (s N2) + note: %sort nat_lt {_0 nat} {_1 nat} + note: %term nat_lt_zs {N nat} nat_lt z (s N) + note: %term nat_lt_ss {N1 nat} {N2 nat} {_0 nat_lt N1 N2} nat_lt (s N1) (s N2) + note: %sort tp + note: %term => {_0 tp} {_1 tp} tp + note: %term unit tp + note: %term ref {_0 tp} tp + note: %sort label + note: %term lbl {_0 nat} label + note: %sort exp + note: %term @ {_0 exp} {_1 exp} exp + note: %term lam {_0 tp} {_1 {_1 exp} exp} exp + note: %term dot exp + note: %term alloc {_0 exp} exp + note: %term deref {_0 exp} exp + note: %term gets {_0 exp} {_1 exp} exp + note: %term loc {_0 label} exp + note: %sort value {_0 exp} + note: %term v_lam {T tp} {E {_0 exp} exp} value (lam T ([_0 exp] E _0)) + note: %term v_dot value dot + note: %term v_loc {L label} value (loc L) + note: %sort store + note: %term store_nil store + note: %term store_cons {_0 tp} {_1 store} store + note: %sort length_store {_0 store} {_1 nat} + note: %term length_store_nil length_store store_nil z + note: %term length_store_cons + {S store} {N nat} {T tp} {_0 length_store S N} + length_store (store_cons T S) (s N) + => unique: expected a type family name + + note: %sort find_in_store {_0 label} {_1 store} {_2 tp} + note: %term find_in_store_yes + {T tp} {S store} find_in_store (lbl z) (store_cons T S) T + note: %term find_in_store_no + {N nat} {S store} {T2 tp} {T1 tp} {_0 find_in_store (lbl N) S T2} + find_in_store (lbl (s N)) (store_cons T1 S) T2 + => unique: expected a type family name + + note: %sort store_extends {_0 store} {_1 store} + note: %term store_extends_base {S store} store_extends store_nil S + note: %term store_extends_ind + {S1 store} {S2 store} {T tp} {_0 store_extends S1 S2} + store_extends (store_cons T S1) (store_cons T S2) + note: %sort append_store {_0 store} {_1 tp} {_2 store} + note: %term append_store_nil {T tp} append_store store_nil T (store_cons T store_nil) + note: %term append_store_cons + {S store} {T2 tp} {S' store} {T1 tp} {_0 append_store S T2 S'} + append_store (store_cons T1 S) T2 (store_cons T1 S') + => unique: expected a type family name + + note: %sort var {_0 exp} {_1 tp} + note: %sort of {_0 store} {_1 exp} {_2 tp} + note: %term t_var {E exp} {T tp} {S store} {_0 var E T} of S E T + note: %term t_abs + {_ exp} {T1 tp} {S store} {E {_0 exp} exp} {T2 tp} + {_0 {_0 var _ T1} of S (E _) T2} of S (lam T1 ([_1 exp] E _1)) (T1 => T2) + note: %term t_app + {S store} {E2 exp} {T1 tp} {E1 exp} {T2 tp} {_0 of S E2 T1} + {_1 of S E1 (T1 => T2)} of S (E1 @ E2) T2 + note: %term t_unit {S store} of S dot unit + note: %term t_loc + {L label} {S store} {T tp} {_0 find_in_store L S T} of S (loc L) (ref T) + note: %term t_ref {S store} {E exp} {T tp} {_0 of S E T} of S (alloc E) (ref T) + note: %term t_deref {S store} {E exp} {T tp} {_0 of S E (ref T)} of S (deref E) T + note: %term t_assign + {S store} {E2 exp} {T tp} {E1 exp} {_0 of S E2 T} {_1 of S E1 (ref T)} + of S (gets E1 E2) unit + note: %sort heap + note: %term heap_nil heap + note: %term heap_cons {_0 exp} {_1 heap} heap + note: %sort length_heap {_0 heap} {_1 nat} + note: %term length_heap_nil length_heap heap_nil z + note: %term length_heap_cons + {H heap} {N nat} {E exp} {_0 length_heap H N} + length_heap (heap_cons E H) (s N) + => unique: expected a type family name + + note: %sort find_in_heap {_0 label} {_1 heap} {_2 exp} + note: %term find_in_heap_yes {E exp} {H heap} find_in_heap (lbl z) (heap_cons E H) E + note: %term find_in_heap_no + {N nat} {H heap} {E' exp} {E exp} {_0 find_in_heap (lbl N) H E'} + find_in_heap (lbl (s N)) (heap_cons E H) E' + => unique: expected a type family name + + note: %sort replace_in_heap {_0 heap} {_1 label} {_2 exp} {_3 heap} + note: %term replace_in_heap_yes + {E1 exp} {H heap} {E2 exp} + replace_in_heap (heap_cons E1 H) (lbl z) E2 (heap_cons E2 H) + note: %term replace_in_heap_no + {H heap} {N nat} {E2 exp} {H' heap} {E1 exp} + {_0 replace_in_heap H (lbl N) E2 H'} + replace_in_heap (heap_cons E1 H) (lbl (s N)) E2 (heap_cons E1 H') + => unique: expected a type family name + + note: %sort append_heap {_0 heap} {_1 exp} {_2 heap} + note: %term append_heap_nil {E exp} append_heap heap_nil E (heap_cons E heap_nil) + note: %term append_heap_cons + {H heap} {E2 exp} {H' heap} {E1 exp} {_0 append_heap H E2 H'} + append_heap (heap_cons E1 H) E2 (heap_cons E1 H') + => unique: expected a type family name + + note: %sort check_wt {_0 store} {_1 store} {_2 heap} + note: %term check_wt_nil {S store} check_wt S store_nil heap_nil + note: %term check_wt_cons + {S1 store} {S2 store} {H heap} {E exp} {T tp} {_0 check_wt S1 S2 H} + {_1 of S1 E T} check_wt S1 (store_cons T S2) (heap_cons E H) + note: %sort wt_heap {_0 store} {_1 heap} + note: %term wt_heap_def {S store} {H heap} {_0 check_wt S S H} wt_heap S H + note: %sort step {_0 heap} {_1 exp} {_2 heap} {_3 exp} + note: %term e_app1 + {H heap} {E1 exp} {H' heap} {E1' exp} {E2 exp} {_0 step H E1 H' E1'} + step H (E1 @ E2) H' (E1' @ E2) + note: %term e_app2 + {H heap} {E2 exp} {H' heap} {E2' exp} {E1 exp} {_0 step H E2 H' E2'} + {_1 value E1} step H (E1 @ E2) H' (E1 @ E2') + note: %term e_alloc + {H heap} {E exp} {H' heap} {E' exp} {_0 step H E H' E'} + step H (alloc E) H' (alloc E') + note: %term e_deref + {H heap} {E exp} {H' heap} {E' exp} {_0 step H E H' E'} + step H (deref E) H' (deref E') + note: %term e_gets1 + {H heap} {E1 exp} {H' heap} {E1' exp} {E2 exp} {_0 step H E1 H' E1'} + step H (gets E1 E2) H' (gets E1' E2) + note: %term e_gets2 + {H heap} {E2 exp} {H' heap} {E2' exp} {E1 exp} {_0 step H E2 H' E2'} + {_1 value E1} step H (gets E1 E2) H' (gets E1 E2') + note: %term e_appabs + {E2 exp} {H heap} {T tp} {E1 {_0 exp} exp} {_0 value E2} + step H (lam T ([_1 exp] E1 _1) @ E2) H (E1 E2) + note: %term e_allocVal + {H heap} {N' nat} {E exp} {H' heap} {_0 length_heap H N'} + {_1 append_heap H E H'} {_2 value E} step H (alloc E) H' (loc (lbl N')) + note: %term e_derefVal + {L label} {H heap} {E exp} {_0 find_in_heap L H E} step H (deref (loc L)) H E + note: %term e_getsVal + {H heap} {L label} {E exp} {H' heap} {_0 replace_in_heap H L E H'} + {_1 value E} step H (gets (loc L) E) H' dot + diff --git a/test/cram/tapl_ch13.t/stelf.toml b/test/cram/tapl_ch13.t/stelf.toml new file mode 100644 index 0000000..331303e --- /dev/null +++ b/test/cram/tapl_ch13.t/stelf.toml @@ -0,0 +1,4 @@ +[[group]] +name = "tapl_ch13" +main = "./main.lf" +dirs = ["."] diff --git a/test/cram/wiki_failures.t/run.t b/test/cram/wiki_failures.t/run.t index 3573dd2..69ab9e8 100644 --- a/test/cram/wiki_failures.t/run.t +++ b/test/cram/wiki_failures.t/run.t @@ -6,13 +6,51 @@ unsatisfiable_query.lf $ stelf check coverage_error.lf + note: %sort tp + note: %term int tp + note: %term float tp + note: %term arrow {_0 tp} {_1 tp} tp + note: %sort sub {_0 tp} {_1 tp} + note: %term sub-ii sub int int + note: %term sub-ff sub float float + note: %term sub-if sub int float + note: %term sub-arrow + {S tp} {S' tp} {T' tp} {T tp} {_0 sub S S'} {_1 sub T' T} + sub (arrow T S) (arrow T' S') + note: %sort sub-trans {_0 tp} {_1 tp} {_2 tp} + note: %term sub-trans/refl {T tp} sub-trans T T T + note: %term sub-trans/arrow + {T3 tp} {T3' tp} {T1' tp} {T1 tp} {_0 sub-trans T3 T3 T3'} + {_1 sub-trans T1' T1 T1} + sub-trans (arrow T1 T3) (arrow T1 T3) (arrow T1' T3') + note: checking mode of constant (Names_.MakeNames.Qid ([], "sub-trans/refl")) ... + note: checking mode of constant (Names_.MakeNames.Qid ([], "sub-trans/arrow")) ... + note: Occurrence of variable T1' in input (+) argument not necessarily grounderror: [error] :Occurrence of variable T1' in input (+) argument not necessarily ground [1] $ stelf check mode_error.lf + note: %sort nat + note: %term 0 nat + note: %term S {_0 nat} nat + note: %sort plus {_0 nat} {_1 nat} {_2 nat} + note: %term plus/z {N nat} plus 0 N N + note: %term plus/s + {N1 nat} {N2 nat} {N3 nat} {_0 plus N1 N2 N3} plus (S N1) N2 (S N3) + note: checking mode of constant (Names_.MakeNames.Qid ([], "plus/z")) ... + note: checking mode of constant (Names_.MakeNames.Qid ([], "plus/s")) ... + note: checking mode of constant (Names_.MakeNames.Qid ([], "plus/z")) ... + note: checking mode of constant (Names_.MakeNames.Qid ([], "plus/s")) ... + note: %sort bad {_0 nat} {_1 nat} + note: %term bad/case {N1 nat} {N2 nat} {_0 plus N1 N2 N1} bad N1 N2 + note: checking mode of constant (Names_.MakeNames.Qid ([], "bad/case")) ... + note: Occurrence of variable N2 in input (+) argument not necessarily grounderror: [error] :Occurrence of variable N2 in input (+) argument not necessarily ground [1] $ stelf check totality_error.lf + error: [error] %total: undeclared identifier case in call pattern [1] $ stelf check unsatisfiable_query.lf - [1] + note: %sort void + => No solution. + diff --git a/test/cram/wiki_failures.t/unsatisfiable_query.lf b/test/cram/wiki_failures.t/unsatisfiable_query.lf index 4057acf..bebe506 100644 --- a/test/cram/wiki_failures.t/unsatisfiable_query.lf +++ b/test/cram/wiki_failures.t/unsatisfiable_query.lf @@ -1,4 +1,4 @@ # Expected failure: query for an inhabitant of an empty type %sort void %. -%query 1 * * void +%query 1 _ _ void